LMFDB - Embedded newform 25.38.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,38,Mod(1,25)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 38, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 38);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 38 $
Character orbit:	$[\chi]$	$=$	25.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$216.785095312$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 1079718948039 x^{10} + \cdots + 10\!\cdots\!06 $ x^12 - 5x^11 - 1079718948039x^10 + 10523476561398225x^9 + 415385046259603643543340x^8 - 8636185588829497996867930050x^7 - 68770856851085140062601719864245230x^6 + 2312306571920800052487868143409528250450x^5 + 4835884795296394010629330683048755920289543865x^4 - 201024142009016313906197472595808556435000680966025x^3 - 136393513979458617639702315997099766935880495831011313019x^2 + 5228013294981499078281972392176520164642954203262819983955245*x + 1076315358414104676142423545906669947059711093813538189890789906506
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{52}\cdot 3^{17}\cdot 5^{36}\cdot 7^{3}\cdot 19 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$160003.$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+165648. q^{2} +1.34135e9 q^{3} -1.10000e11 q^{4} +2.22192e14 q^{6} +1.00021e15 q^{7} -4.09878e16 q^{8} +1.34892e18 q^{9} +O(q^{10})$	\(q+165648. q^{2} +1.34135e9 q^{3} -1.10000e11 q^{4} +2.22192e14 q^{6} +1.00021e15 q^{7} -4.09878e16 q^{8} +1.34892e18 q^{9} +4.76870e18 q^{11} -1.47547e20 q^{12} +4.71691e20 q^{13} +1.65682e20 q^{14} +8.32866e21 q^{16} -3.06565e22 q^{17} +2.23447e23 q^{18} -4.89773e23 q^{19} +1.34162e24 q^{21} +7.89927e23 q^{22} +1.91621e25 q^{23} -5.49788e25 q^{24} +7.81349e25 q^{26} +1.20538e27 q^{27} -1.10022e26 q^{28} -9.36618e26 q^{29} +4.06162e27 q^{31} +7.01295e27 q^{32} +6.39647e27 q^{33} -5.07821e27 q^{34} -1.48381e29 q^{36} -1.44633e29 q^{37} -8.11302e28 q^{38} +6.32701e29 q^{39} +1.75478e29 q^{41} +2.22237e29 q^{42} +2.39963e30 q^{43} -5.24555e29 q^{44} +3.17417e30 q^{46} +9.06911e30 q^{47} +1.11716e31 q^{48} -1.75617e31 q^{49} -4.11210e31 q^{51} -5.18858e31 q^{52} -1.30823e32 q^{53} +1.99670e32 q^{54} -4.09962e31 q^{56} -6.56955e32 q^{57} -1.55149e32 q^{58} -2.48222e32 q^{59} +1.11589e33 q^{61} +6.72800e32 q^{62} +1.34920e33 q^{63} +1.70006e31 q^{64} +1.05957e33 q^{66} -5.23541e33 q^{67} +3.37221e33 q^{68} +2.57030e34 q^{69} -3.42169e33 q^{71} -5.52894e34 q^{72} +1.86541e34 q^{73} -2.39582e34 q^{74} +5.38749e34 q^{76} +4.76968e33 q^{77} +1.04806e35 q^{78} +1.28199e35 q^{79} +1.00944e36 q^{81} +2.90677e34 q^{82} +5.27592e35 q^{83} -1.47578e35 q^{84} +3.97495e35 q^{86} -1.25633e36 q^{87} -1.95458e35 q^{88} +1.17022e36 q^{89} +4.71788e35 q^{91} -2.10782e36 q^{92} +5.44803e36 q^{93} +1.50228e36 q^{94} +9.40678e36 q^{96} +6.60283e36 q^{97} -2.90907e36 q^{98} +6.43261e36 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 67745 q^{2} + 724208180 q^{3} + 510552903189 q^{4} + 15880293089869 q^{6} - 22\!\cdots\!00 q^{7}+ \cdots + 13\!\cdots\!56 q^{9}+O(q^{10}) $ 12 * q + 67745 * q^2 + 724208180 * q^3 + 510552903189 * q^4 + 15880293089869 * q^6 - 2280592823278800 * q^7 - 13603597852867065 * q^8 + 1359108258754272256 * q^9	$ 12 q + 67745 q^{2} + 724208180 q^{3} + 510552903189 q^{4} + 15880293089869 q^{6} - 22\!\cdots\!00 q^{7}+ \cdots - 99\!\cdots\!68 q^{99}+O(q^{100}) $ 12 * q + 67745 * q^2 + 724208180 * q^3 + 510552903189 * q^4 + 15880293089869 * q^6 - 2280592823278800 * q^7 - 13603597852867065 * q^8 + 1359108258754272256 * q^9 + 2554656505188935564 * q^11 + 38296868672496581965 * q^12 + 83357727454160617560 * q^13 + 541210680522824521098 * q^14 + 5889259433625089080017 * q^16 + 27158393512650748861460 * q^17 + 32304995024093396237630 * q^18 - 1307297534999360105100 * q^19 + 1927702939801406301291504 * q^21 + 8101495479590718049976265 * q^22 + 6561364838658797127044040 * q^23 - 29787767561390801566152825 * q^24 + 164742777145327712449309724 * q^26 + 803078304022385000040963860 * q^27 + 706360659695833214077901610 * q^28 - 2618773509396722286923297000 * q^29 + 3495844606583171918858800344 * q^31 + 6373892627582681144636589295 * q^32 + 1200396846201406692144438460 * q^33 + 3609863853608971653493888413 * q^34 - 91889542016499788236065200218 * q^36 - 218845170107098937703383787120 * q^37 - 54297488357007537741487211935 * q^38 + 125348774149234247086164192152 * q^39 - 51474601681994565845304037316 * q^41 + 279007056776373014503679203410 * q^42 + 1095093974210612352636183955800 * q^43 - 2189164920482983205730699871367 * q^44 + 10107503169848980559418679324254 * q^46 + 23907862328757523727528033674040 * q^47 + 6657319366828177502039371224865 * q^48 - 14951602913621438213797405279716 * q^49 + 22208888829315215979428699940124 * q^51 + 48235173815673852004466444424300 * q^52 + 16838816611239414826974664431080 * q^53 - 57545129090023056489658919639825 * q^54 + 187574880792755877609939226588350 * q^56 + 487409130824695345102950044427620 * q^57 + 537922378980272521289962414359360 * q^58 - 1336700779068696398361473055992200 * q^59 + 3485099106494564728317108451692864 * q^61 + 6971612944345799706023880876891690 * q^62 + 4069695817379180445347658914088720 * q^63 - 7333556814672452956681284356117151 * q^64 + 11216229648698841479202248762840093 * q^66 + 14753973559495590347225839865512260 * q^67 + 8785613970183933622822253658267505 * q^68 - 2522802561523836072474822533544408 * q^69 + 10576084284690246863128363185276904 * q^71 - 9674318002594923678160001962177470 * q^72 - 4825945256111753902227483456687060 * q^73 + 21873580722917055024973698667460818 * q^74 + 59441092708215293905864123640191425 * q^76 - 29028478500644775653137476920193600 * q^77 + 87450040981989737501234198133333100 * q^78 + 31047245994101750554934024487283200 * q^79 + 25973485616223818269003376033095972 * q^81 - 701481290104386131269144762106398035 * q^82 + 68699132245521973559209397648692020 * q^83 + 650071241509187672385081824537914938 * q^84 - 415357678037782103677821291329558836 * q^86 - 2554515878920505052325980738506323720 * q^87 - 1479450788257499500717123093674302805 * q^88 + 3353246041445492030040660620069119500 * q^89 - 2001623473367880972919402485939696816 * q^91 - 4351655844619574190087420297293008290 * q^92 + 2158939592433745585172010945825503160 * q^93 - 788004521663479731556746949918537092 * q^94 + 10002124457561358992178222672331390079 * q^96 + 14103135523445618268336692441237736840 * q^97 + 1821220898206765147319159644986861565 * q^98 - 9936699890312541409796157011630543968 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	165648.	0.446820	0.223410	−	0.974725i	$-0.428281\pi$
$2$	165648.	0.446820	0.223410	+	0.974725i	$0.428281\pi$
$3$	1.34135e9	1.99893	0.999464	−	0.0327273i	$-0.0104193\pi$
$3$	1.34135e9	1.99893	0.999464	+	0.0327273i	$0.0104193\pi$
$4$	−1.10000e11	−0.800352
$5$	0	0
$5$	0	0
$6$	2.22192e14	0.893161
$7$	1.00021e15	0.232154	0.116077	−	0.993240i	$-0.462968\pi$
$7$	1.00021e15	0.232154	0.116077	+	0.993240i	$0.462968\pi$
$8$	−4.09878e16	−0.804433
$9$	1.34892e18	2.99572
$10$	0	0
$11$	4.76870e18	0.258604	0.129302	−	0.991605i	$-0.458726\pi$
$11$	4.76870e18	0.258604	0.129302	+	0.991605i	$0.458726\pi$
$12$	−1.47547e20	−1.59985
$13$	4.71691e20	1.16334	0.581669	−	0.813426i	$-0.302400\pi$
$13$	4.71691e20	1.16334	0.581669	+	0.813426i	$0.302400\pi$
$14$	1.65682e20	0.103731
$15$	0	0
$16$	8.32866e21	0.440916
$17$	−3.06565e22	−0.528711	−0.264355	−	0.964425i	$-0.585159\pi$
$17$	−3.06565e22	−0.528711	−0.264355	+	0.964425i	$0.585159\pi$
$18$	2.23447e23	1.33854
$19$	−4.89773e23	−1.07908	−0.539540	−	0.841960i	$-0.681402\pi$
$19$	−4.89773e23	−1.07908	−0.539540	+	0.841960i	$0.681402\pi$
$20$	0	0
$21$	1.34162e24	0.464059
$22$	7.89927e23	0.115549
$23$	1.91621e25	1.23162	0.615811	−	0.787894i	$-0.288828\pi$
$23$	1.91621e25	1.23162	0.615811	+	0.787894i	$0.288828\pi$
$24$	−5.49788e25	−1.60800
$25$	0	0
$26$	7.81349e25	0.519802
$27$	1.20538e27	3.98929
$28$	−1.10022e26	−0.185805
$29$	−9.36618e26	−0.826417	−0.413209	−	0.910636i	$-0.635592\pi$
$29$	−9.36618e26	−0.826417	−0.413209	+	0.910636i	$0.635592\pi$
$30$	0	0
$31$	4.06162e27	1.04354	0.521768	−	0.853087i	$-0.325273\pi$
$31$	4.06162e27	1.04354	0.521768	+	0.853087i	$0.325273\pi$
$32$	7.01295e27	1.00144
$33$	6.39647e27	0.516931
$34$	−5.07821e27	−0.236238
$35$	0	0
$36$	−1.48381e29	−2.39763
$37$	−1.44633e29	−1.40778	−0.703889	−	0.710310i	$-0.748555\pi$
$37$	−1.44633e29	−1.40778	−0.703889	+	0.710310i	$0.748555\pi$
$38$	−8.11302e28	−0.482154
$39$	6.32701e29	2.32543
$40$	0	0
$41$	1.75478e29	0.255695	0.127847	−	0.991794i	$-0.459193\pi$
$41$	1.75478e29	0.255695	0.127847	+	0.991794i	$0.459193\pi$
$42$	2.22237e29	0.207350
$43$	2.39963e30	1.44870	0.724349	−	0.689433i	$-0.242140\pi$
$43$	2.39963e30	1.44870	0.724349	+	0.689433i	$0.242140\pi$
$44$	−5.24555e29	−0.206974
$45$	0	0
$46$	3.17417e30	0.550313
$47$	9.06911e30	1.05622	0.528110	−	0.849176i	$-0.322901\pi$
$47$	9.06911e30	1.05622	0.528110	+	0.849176i	$0.322901\pi$
$48$	1.11716e31	0.881359
$49$	−1.75617e31	−0.946105
$50$	0	0
$51$	−4.11210e31	−1.05686
$52$	−5.18858e31	−0.931080
$53$	−1.30823e32	−1.65037	−0.825183	−	0.564865i	$-0.808928\pi$
$53$	−1.30823e32	−1.65037	−0.825183	+	0.564865i	$0.808928\pi$
$54$	1.99670e32	1.78249
$55$	0	0
$56$	−4.09962e31	−0.186752
$57$	−6.56955e32	−2.15700
$58$	−1.55149e32	−0.369259
$59$	−2.48222e32	−0.430605	−0.215302	−	0.976547i	$-0.569074\pi$
$59$	−2.48222e32	−0.430605	−0.215302	+	0.976547i	$0.569074\pi$
$60$	0	0
$61$	1.11589e33	1.04477	0.522383	−	0.852711i	$-0.325043\pi$
$61$	1.11589e33	1.04477	0.522383	+	0.852711i	$0.325043\pi$
$62$	6.72800e32	0.466272
$63$	1.34920e33	0.695466
$64$	1.70006e31	0.00654839
$65$	0	0
$66$	1.05957e33	0.230975
$67$	−5.23541e33	−0.864106	−0.432053	−	0.901848i	$-0.642211\pi$
$67$	−5.23541e33	−0.864106	−0.432053	+	0.901848i	$0.642211\pi$
$68$	3.37221e33	0.423155
$69$	2.57030e34	2.46193
$70$	0	0
$71$	−3.42169e33	−0.193179	−0.0965897	−	0.995324i	$-0.530793\pi$
$71$	−3.42169e33	−0.193179	−0.0965897	+	0.995324i	$0.530793\pi$
$72$	−5.52894e34	−2.40985
$73$	1.86541e34	0.629943	0.314972	−	0.949101i	$-0.398005\pi$
$73$	1.86541e34	0.629943	0.314972	+	0.949101i	$0.398005\pi$
$74$	−2.39582e34	−0.629023
$75$	0	0
$76$	5.38749e34	0.863644
$77$	4.76968e33	0.0600359
$78$	1.04806e35	1.03905
$79$	1.28199e35	1.00412	0.502058	−	0.864834i	$-0.332576\pi$
$79$	1.28199e35	1.00412	0.502058	+	0.864834i	$0.332576\pi$
$80$	0	0
$81$	1.00944e36	4.97860
$82$	2.90677e34	0.114249
$83$	5.27592e35	1.65712	0.828559	−	0.559902i	$-0.189161\pi$
$83$	5.27592e35	1.65712	0.828559	+	0.559902i	$0.189161\pi$
$84$	−1.47578e35	−0.371410
$85$	0	0
$86$	3.97495e35	0.647307
$87$	−1.25633e36	−1.65195
$88$	−1.95458e35	−0.208030
$89$	1.17022e36	1.01054	0.505269	−	0.862962i	$-0.331393\pi$
$89$	1.17022e36	1.01054	0.505269	+	0.862962i	$0.331393\pi$
$90$	0	0
$91$	4.71788e35	0.270073
$92$	−2.10782e36	−0.985732
$93$	5.44803e36	2.08595
$94$	1.50228e36	0.471940
$95$	0	0
$96$	9.40678e36	2.00181
$97$	6.60283e36	1.15999	0.579993	−	0.814621i	$-0.303055\pi$
$97$	6.60283e36	1.15999	0.579993	+	0.814621i	$0.303055\pi$
$98$	−2.90907e36	−0.422738
$99$	6.43261e36	0.774704
$100$	0	0
$101$	−6.69284e36	−0.556756	−0.278378	−	0.960472i	$-0.589797\pi$
$101$	−6.69284e36	−0.556756	−0.278378	+	0.960472i	$0.589797\pi$
$102$	−6.81163e36	−0.472224
$103$	7.34537e36	0.425133	0.212567	−	0.977147i	$-0.431818\pi$
$103$	7.34537e36	0.425133	0.212567	+	0.977147i	$0.431818\pi$
$104$	−1.93336e37	−0.935827
$105$	0	0
$106$	−2.16706e37	−0.737416
$107$	−1.06607e37	−0.304919	−0.152460	−	0.988310i	$-0.548719\pi$
$107$	−1.06607e37	−0.304919	−0.152460	+	0.988310i	$0.548719\pi$
$108$	−1.32592e38	−3.19284
$109$	6.50897e37	1.32167	0.660833	−	0.750533i	$-0.270203\pi$
$109$	6.50897e37	1.32167	0.660833	+	0.750533i	$0.270203\pi$
$110$	0	0
$111$	−1.94002e38	−2.81405
$112$	8.33038e36	0.102360
$113$	−3.26477e37	−0.340330	−0.170165	−	0.985416i	$-0.554430\pi$
$113$	−3.26477e37	−0.340330	−0.170165	+	0.985416i	$0.554430\pi$
$114$	−1.08824e38	−0.963792
$115$	0	0
$116$	1.03028e38	0.661425
$117$	6.36275e38	3.48503
$118$	−4.11176e37	−0.192403
$119$	−3.06628e37	−0.122742
$120$	0	0
$121$	−3.17299e38	−0.933124
$122$	1.84845e38	0.466822
$123$	2.35377e38	0.511115
$124$	−4.46776e38	−0.835196
$125$	0	0
$126$	2.23493e38	0.310748
$127$	−4.50990e38	−0.541749	−0.270875	−	0.962615i	$-0.587313\pi$
$127$	−4.50990e38	−0.541749	−0.270875	+	0.962615i	$0.587313\pi$
$128$	−9.61036e38	−0.998517
$129$	3.21873e39	2.89584
$130$	0	0
$131$	1.79467e39	1.21469	0.607344	−	0.794439i	$-0.292235\pi$
$131$	1.79467e39	1.21469	0.607344	+	0.794439i	$0.292235\pi$
$132$	−7.03609e38	−0.413727
$133$	−4.89874e38	−0.250512
$134$	−8.67238e38	−0.386100
$135$	0	0
$136$	1.25654e39	0.425312
$137$	−1.37700e39	−0.407010	−0.203505	−	0.979074i	$-0.565233\pi$
$137$	−1.37700e39	−0.407010	−0.203505	+	0.979074i	$0.565233\pi$
$138$	4.25765e39	1.10004
$139$	−1.64229e39	−0.371259	−0.185630	−	0.982620i	$-0.559432\pi$
$139$	−1.64229e39	−0.371259	−0.185630	+	0.982620i	$0.559432\pi$
$140$	0	0
$141$	1.21648e40	2.11131
$142$	−5.66797e38	−0.0863163
$143$	2.24935e39	0.300844
$144$	1.12347e40	1.32086
$145$	0	0
$146$	3.09002e39	0.281471
$147$	−2.35563e40	−1.89120
$148$	1.59095e40	1.12672
$149$	2.39548e40	1.49778	0.748888	−	0.662696i	$-0.230588\pi$
$149$	2.39548e40	1.49778	0.748888	+	0.662696i	$0.230588\pi$
$150$	0	0
$151$	−1.31983e40	−0.644827	−0.322414	−	0.946599i	$-0.604494\pi$
$151$	−1.31983e40	−0.644827	−0.322414	+	0.946599i	$0.604494\pi$
$152$	2.00747e40	0.868047
$153$	−4.13533e40	−1.58387
$154$	7.90090e38	0.0268252
$155$	0	0
$156$	−6.95968e40	−1.86116
$157$	1.33645e40	0.317549	0.158774	−	0.987315i	$-0.449246\pi$
$157$	1.33645e40	0.317549	0.158774	+	0.987315i	$0.449246\pi$
$158$	2.12360e40	0.448659
$159$	−1.75479e41	−3.29896
$160$	0	0
$161$	1.91660e40	0.285926
$162$	1.67212e41	2.22453
$163$	−3.64303e40	−0.432506	−0.216253	−	0.976337i	$-0.569384\pi$
$163$	−3.64303e40	−0.432506	−0.216253	+	0.976337i	$0.569384\pi$
$164$	−1.93025e40	−0.204646
$165$	0	0
$166$	8.73947e40	0.740433
$167$	3.67786e40	0.278830	0.139415	−	0.990234i	$-0.455478\pi$
$167$	3.67786e40	0.278830	0.139415	+	0.990234i	$0.455478\pi$
$168$	−5.49901e40	−0.373304
$169$	5.80917e40	0.353354
$170$	0	0
$171$	−6.60667e41	−3.23262
$172$	−2.63958e41	−1.15947
$173$	−1.43523e40	−0.0566329	−0.0283165	−	0.999599i	$-0.509015\pi$
$173$	−1.43523e40	−0.0566329	−0.0283165	+	0.999599i	$0.509015\pi$
$174$	−2.08109e41	−0.738123
$175$	0	0
$176$	3.97169e40	0.114023
$177$	−3.32952e41	−0.860748
$178$	1.93845e41	0.451528
$179$	8.11355e41	1.70384	0.851922	−	0.523668i	$-0.175437\pi$
$179$	8.11355e41	1.70384	0.851922	+	0.523668i	$0.175437\pi$
$180$	0	0
$181$	2.77760e41	0.474916	0.237458	−	0.971398i	$-0.423686\pi$
$181$	2.77760e41	0.474916	0.237458	+	0.971398i	$0.423686\pi$
$182$	7.81509e40	0.120674
$183$	1.49679e42	2.08841
$184$	−7.85411e41	−0.990758
$185$	0	0
$186$	9.02457e41	0.932045
$187$	−1.46192e41	−0.136727
$188$	−9.97598e41	−0.845348
$189$	1.20563e42	0.926129
$190$	0	0
$191$	2.20534e42	1.39431	0.697154	−	0.716921i	$-0.254449\pi$
$191$	2.20534e42	1.39431	0.697154	+	0.716921i	$0.254449\pi$
$192$	2.28037e40	0.0130898
$193$	2.68658e42	1.40085	0.700423	−	0.713728i	$-0.252995\pi$
$193$	2.68658e42	1.40085	0.700423	+	0.713728i	$0.252995\pi$
$194$	1.09375e42	0.518305
$195$	0	0
$196$	1.93178e42	0.757217
$197$	−4.28438e42	−1.52849	−0.764245	−	0.644925i	$-0.776888\pi$
$197$	−4.28438e42	−1.52849	−0.764245	+	0.644925i	$0.776888\pi$
$198$	1.06555e42	0.346153
$199$	4.80396e41	0.142173	0.0710865	−	0.997470i	$-0.477353\pi$
$199$	4.80396e41	0.142173	0.0710865	+	0.997470i	$0.477353\pi$
$200$	0	0
$201$	−7.02249e42	−1.72729
$202$	−1.10866e42	−0.248770
$203$	−9.36810e41	−0.191856
$204$	4.52329e42	0.845856
$205$	0	0
$206$	1.21675e42	0.189958
$207$	2.58482e43	3.68959
$208$	3.92856e42	0.512934
$209$	−2.33558e42	−0.279054
$210$	0	0
$211$	−1.52865e42	−0.153138	−0.0765690	−	0.997064i	$-0.524397\pi$
$211$	−1.52865e42	−0.153138	−0.0765690	+	0.997064i	$0.524397\pi$
$212$	1.43905e43	1.32087
$213$	−4.58966e42	−0.386152
$214$	−1.76592e42	−0.136244
$215$	0	0
$216$	−4.94060e43	−3.20912
$217$	4.06245e42	0.242261
$218$	1.07820e43	0.590547
$219$	2.50216e43	1.25921
$220$	0	0
$221$	−1.44604e43	−0.615069
$222$	−3.21362e43	−1.25737
$223$	−2.93237e43	−1.05579	−0.527897	−	0.849308i	$-0.677019\pi$
$223$	−2.93237e43	−1.05579	−0.527897	+	0.849308i	$0.677019\pi$
$224$	7.01439e42	0.232489
$225$	0	0
$226$	−5.40803e42	−0.152066
$227$	−3.59899e43	−0.932614	−0.466307	−	0.884623i	$-0.654416\pi$
$227$	−3.59899e43	−0.932614	−0.466307	+	0.884623i	$0.654416\pi$
$228$	7.22648e43	1.72636
$229$	1.38698e43	0.305572	0.152786	−	0.988259i	$-0.451175\pi$
$229$	1.38698e43	0.305572	0.152786	+	0.988259i	$0.451175\pi$
$230$	0	0
$231$	6.39779e42	0.120007
$232$	3.83899e43	0.664797
$233$	6.48781e43	1.03756	0.518781	−	0.854907i	$-0.326386\pi$
$233$	6.48781e43	1.03756	0.518781	+	0.854907i	$0.326386\pi$
$234$	1.05398e44	1.55718
$235$	0	0
$236$	2.73044e43	0.344635
$237$	1.71959e44	2.00716
$238$	−5.07925e42	−0.0548436
$239$	−5.70438e43	−0.569964	−0.284982	−	0.958533i	$-0.591988\pi$
$239$	−5.70438e43	−0.569964	−0.284982	+	0.958533i	$0.591988\pi$
$240$	0	0
$241$	1.87323e43	0.160426	0.0802132	−	0.996778i	$-0.474440\pi$
$241$	1.87323e43	0.160426	0.0802132	+	0.996778i	$0.474440\pi$
$242$	−5.25601e43	−0.416938
$243$	8.11240e44	5.96257
$244$	−1.22747e44	−0.836180
$245$	0	0
$246$	3.89898e43	0.228376
$247$	−2.31022e44	−1.25533
$248$	−1.66477e44	−0.839455
$249$	7.07683e44	3.31246
$250$	0	0
$251$	−1.46757e44	−0.592429	−0.296214	−	0.955121i	$-0.595724\pi$
$251$	−1.46757e44	−0.592429	−0.296214	+	0.955121i	$0.595724\pi$
$252$	−1.48411e44	−0.556618
$253$	9.13782e43	0.318503
$254$	−7.47058e43	−0.242064
$255$	0	0
$256$	−1.61531e44	−0.452705
$257$	4.85248e44	1.26532	0.632661	−	0.774429i	$-0.281963\pi$
$257$	4.85248e44	1.26532	0.632661	+	0.774429i	$0.281963\pi$
$258$	5.33178e44	1.29392
$259$	−1.44662e44	−0.326821
$260$	0	0
$261$	−1.26342e45	−2.47571
$262$	2.97284e44	0.542746
$263$	−3.81575e44	−0.649230	−0.324615	−	0.945846i	$-0.605235\pi$
$263$	−3.81575e44	−0.649230	−0.324615	+	0.945846i	$0.605235\pi$
$264$	−2.62177e44	−0.415836
$265$	0	0
$266$	−8.11469e43	−0.111934
$267$	1.56967e45	2.01999
$268$	5.75893e44	0.691589
$269$	5.92938e44	0.664649	0.332324	−	0.943165i	$-0.392167\pi$
$269$	5.92938e44	0.664649	0.332324	+	0.943165i	$0.392167\pi$
$270$	0	0
$271$	−1.35956e45	−1.32882	−0.664409	−	0.747369i	$-0.731317\pi$
$271$	−1.35956e45	−1.32882	−0.664409	+	0.747369i	$0.731317\pi$
$272$	−2.55328e44	−0.233117
$273$	6.32831e44	0.539857
$274$	−2.28098e44	−0.181860
$275$	0	0
$276$	−2.82731e45	−1.97041
$277$	−4.41862e44	−0.288013	−0.144006	−	0.989577i	$-0.545999\pi$
$277$	−4.41862e44	−0.288013	−0.144006	+	0.989577i	$0.545999\pi$
$278$	−2.72043e44	−0.165886
$279$	5.47880e45	3.12614
$280$	0	0
$281$	9.24406e43	0.0462163	0.0231082	−	0.999733i	$-0.492644\pi$
$281$	9.24406e43	0.0462163	0.0231082	+	0.999733i	$0.492644\pi$
$282$	2.01508e45	0.943374
$283$	−1.09688e45	−0.480961	−0.240481	−	0.970654i	$-0.577305\pi$
$283$	−1.09688e45	−0.480961	−0.240481	+	0.970654i	$0.577305\pi$
$284$	3.76384e44	0.154612
$285$	0	0
$286$	3.72602e44	0.134423
$287$	1.75514e44	0.0593605
$288$	9.45992e45	3.00004
$289$	−2.42227e45	−0.720465
$290$	0	0
$291$	8.85667e45	2.31873
$292$	−2.05194e45	−0.504176
$293$	−4.00593e44	−0.0923959	−0.0461979	−	0.998932i	$-0.514710\pi$
$293$	−4.00593e44	−0.0923959	−0.0461979	+	0.998932i	$0.514710\pi$
$294$	−3.90206e45	−0.845023
$295$	0	0
$296$	5.92817e45	1.13246
$297$	5.74812e45	1.03165
$298$	3.96808e45	0.669236
$299$	9.03858e45	1.43279
$300$	0	0
$301$	2.40012e45	0.336321
$302$	−2.18627e45	−0.288121
$303$	−8.97741e45	−1.11292
$304$	−4.07916e45	−0.475783
$305$	0	0
$306$	−6.85011e45	−0.707703
$307$	−1.34219e46	−1.30543	−0.652714	−	0.757604i	$-0.726370\pi$
$307$	−1.34219e46	−1.30543	−0.652714	+	0.757604i	$0.726370\pi$
$308$	−5.24663e44	−0.0480498
$309$	9.85268e45	0.849811
$310$	0	0
$311$	2.20245e45	0.168592	0.0842962	−	0.996441i	$-0.473136\pi$
$311$	2.20245e45	0.168592	0.0842962	+	0.996441i	$0.473136\pi$
$312$	−2.59330e46	−1.87065
$313$	4.52346e45	0.307539	0.153770	−	0.988107i	$-0.450859\pi$
$313$	4.52346e45	0.307539	0.153770	+	0.988107i	$0.450859\pi$
$314$	2.21381e45	0.141887
$315$	0	0
$316$	−1.41019e46	−0.803647
$317$	1.24472e46	0.669077	0.334538	−	0.942382i	$-0.391420\pi$
$317$	1.24472e46	0.669077	0.334538	+	0.942382i	$0.391420\pi$
$318$	−2.90678e46	−1.47404
$319$	−4.46645e45	−0.213715
$320$	0	0
$321$	−1.42996e46	−0.609511
$322$	3.17482e45	0.127757
$323$	1.50148e46	0.570521
$324$	−1.11038e47	−3.98463
$325$	0	0
$326$	−6.03462e45	−0.193252
$327$	8.73077e46	2.64192
$328$	−7.19246e45	−0.205689
$329$	9.07098e45	0.245205
$330$	0	0
$331$	1.06900e46	0.258321	0.129161	−	0.991624i	$-0.458772\pi$
$331$	1.06900e46	0.258321	0.129161	+	0.991624i	$0.458772\pi$
$332$	−5.80349e46	−1.32628
$333$	−1.95098e47	−4.21730
$334$	6.09231e45	0.124587
$335$	0	0
$336$	1.11739e46	0.204611
$337$	3.78201e45	0.0655495	0.0327748	−	0.999463i	$-0.489566\pi$
$337$	3.78201e45	0.0655495	0.0327748	+	0.999463i	$0.489566\pi$
$338$	9.62279e45	0.157885
$339$	−4.37918e46	−0.680296
$340$	0	0
$341$	1.93686e46	0.269863
$342$	−1.09438e47	−1.44440
$343$	−3.61312e46	−0.451795
$344$	−9.83556e46	−1.16538
$345$	0	0
$346$	−2.37744e45	−0.0253047
$347$	−6.79140e46	−0.685274	−0.342637	−	0.939468i	$-0.611320\pi$
$347$	−6.79140e46	−0.685274	−0.342637	+	0.939468i	$0.611320\pi$
$348$	1.38195e47	1.32214
$349$	1.49122e47	1.35291	0.676457	−	0.736482i	$-0.263514\pi$
$349$	1.49122e47	1.35291	0.676457	+	0.736482i	$0.263514\pi$
$350$	0	0
$351$	5.68569e47	4.64089
$352$	3.34426e46	0.258977
$353$	−1.06866e47	−0.785253	−0.392626	−	0.919698i	$-0.628433\pi$
$353$	−1.06866e47	−0.785253	−0.392626	+	0.919698i	$0.628433\pi$
$354$	−5.51529e46	−0.384599
$355$	0	0
$356$	−1.28724e47	−0.808786
$357$	−4.11295e46	−0.245353
$358$	1.34400e47	0.761311
$359$	7.00844e46	0.377028	0.188514	−	0.982070i	$-0.439633\pi$
$359$	7.00844e46	0.377028	0.188514	+	0.982070i	$0.439633\pi$
$360$	0	0
$361$	3.38705e46	0.164414
$362$	4.60105e46	0.212202
$363$	−4.25607e47	−1.86525
$364$	−5.18965e46	−0.216154
$365$	0	0
$366$	2.47941e47	0.933143
$367$	−4.91071e46	−0.175720	−0.0878601	−	0.996133i	$-0.528003\pi$
$367$	−4.91071e46	−0.175720	−0.0878601	+	0.996133i	$0.528003\pi$
$368$	1.59595e47	0.543042
$369$	2.36706e47	0.765989
$370$	0	0
$371$	−1.30850e47	−0.383139
$372$	−5.99281e47	−1.66950
$373$	−5.42348e47	−1.43769	−0.718844	−	0.695171i	$-0.755329\pi$
$373$	−5.42348e47	−1.43769	−0.718844	+	0.695171i	$0.755329\pi$
$374$	−2.42164e46	−0.0610922
$375$	0	0
$376$	−3.71723e47	−0.849658
$377$	−4.41794e47	−0.961402
$378$	1.99711e47	0.413813
$379$	1.27990e47	0.252552	0.126276	−	0.991995i	$-0.459698\pi$
$379$	1.27990e47	0.252552	0.126276	+	0.991995i	$0.459698\pi$
$380$	0	0
$381$	−6.04933e47	−1.08292
$382$	3.65312e47	0.623005
$383$	−4.38481e47	−0.712482	−0.356241	−	0.934394i	$-0.615942\pi$
$383$	−4.38481e47	−0.712482	−0.356241	+	0.934394i	$0.615942\pi$
$384$	−1.28908e48	−1.99596
$385$	0	0
$386$	4.45028e47	0.625925
$387$	3.23692e48	4.33989
$388$	−7.26309e47	−0.928398
$389$	6.97639e47	0.850282	0.425141	−	0.905127i	$-0.360225\pi$
$389$	6.97639e47	0.850282	0.425141	+	0.905127i	$0.360225\pi$
$390$	0	0
$391$	−5.87443e47	−0.651172
$392$	7.19815e47	0.761078
$393$	2.40727e48	2.42807
$394$	−7.09701e47	−0.682960
$395$	0	0
$396$	−7.07584e47	−0.620036
$397$	−1.26101e48	−1.05462	−0.527308	−	0.849674i	$-0.676799\pi$
$397$	−1.26101e48	−1.05462	−0.527308	+	0.849674i	$0.676799\pi$
$398$	7.95768e46	0.0635257
$399$	−6.57090e47	−0.500756
$400$	0	0
$401$	−2.19814e48	−1.52716	−0.763581	−	0.645712i	$-0.776561\pi$
$401$	−2.19814e48	−1.52716	−0.763581	+	0.645712i	$0.776561\pi$
$402$	−1.16326e48	−0.771786
$403$	1.91583e48	1.21398
$404$	7.36210e47	0.445601
$405$	0	0
$406$	−1.55181e47	−0.0857249
$407$	−6.89710e47	−0.364057
$408$	1.68546e48	0.850169
$409$	3.31580e48	1.59848	0.799239	−	0.601014i	$-0.205236\pi$
$409$	3.31580e48	1.59848	0.799239	+	0.601014i	$0.205236\pi$
$410$	0	0
$411$	−1.84704e48	−0.813583
$412$	−8.07988e47	−0.340256
$413$	−2.48273e47	−0.0999664
$414$	4.28171e48	1.64858
$415$	0	0
$416$	3.30795e48	1.16502
$417$	−2.20288e48	−0.742120
$418$	−3.86886e47	−0.124687
$419$	−4.59569e48	−1.41707	−0.708536	−	0.705675i	$-0.750644\pi$
$419$	−4.59569e48	−1.41707	−0.708536	+	0.705675i	$0.750644\pi$
$420$	0	0
$421$	5.96931e48	1.68541	0.842704	−	0.538377i	$-0.180962\pi$
$421$	5.96931e48	1.68541	0.842704	+	0.538377i	$0.180962\pi$
$422$	−2.53219e47	−0.0684251
$423$	1.22335e49	3.16413
$424$	5.36214e48	1.32761
$425$	0	0
$426$	−7.60270e47	−0.172540
$427$	1.11612e48	0.242546
$428$	1.17267e48	0.244043
$429$	3.01716e48	0.601365
$430$	0	0
$431$	3.74409e48	0.684725	0.342363	−	0.939568i	$-0.388773\pi$
$431$	3.74409e48	0.684725	0.342363	+	0.939568i	$0.388773\pi$
$432$	1.00392e49	1.75894
$433$	−4.16543e47	−0.0699252	−0.0349626	−	0.999389i	$-0.511131\pi$
$433$	−4.16543e47	−0.0699252	−0.0349626	+	0.999389i	$0.511131\pi$
$434$	6.72938e47	0.108247
$435$	0	0
$436$	−7.15984e48	−1.05780
$437$	−9.38508e48	−1.32902
$438$	4.14478e48	0.562640
$439$	−1.06778e49	−1.38960	−0.694798	−	0.719205i	$-0.744506\pi$
$439$	−1.06778e49	−1.38960	−0.694798	+	0.719205i	$0.744506\pi$
$440$	0	0
$441$	−2.36894e49	−2.83426
$442$	−2.39535e48	−0.274825
$443$	−9.44868e48	−1.03969	−0.519843	−	0.854262i	$-0.674010\pi$
$443$	−9.44868e48	−1.03969	−0.519843	+	0.854262i	$0.674010\pi$
$444$	2.13402e49	2.25223
$445$	0	0
$446$	−4.85743e48	−0.471749
$447$	3.21317e49	2.99395
$448$	1.70041e46	0.00152023
$449$	3.88062e48	0.332923	0.166461	−	0.986048i	$-0.446766\pi$
$449$	3.88062e48	0.332923	0.166461	+	0.986048i	$0.446766\pi$
$450$	0	0
$451$	8.36803e47	0.0661237
$452$	3.59123e48	0.272384
$453$	−1.77034e49	−1.28896
$454$	−5.96166e48	−0.416710
$455$	0	0
$456$	2.69271e49	1.73516
$457$	−4.03847e48	−0.249901	−0.124950	−	0.992163i	$-0.539877\pi$
$457$	−4.03847e48	−0.249901	−0.124950	+	0.992163i	$0.539877\pi$
$458$	2.29751e48	0.136536
$459$	−3.69529e49	−2.10918
$460$	0	0
$461$	−2.12842e49	−1.12096	−0.560481	−	0.828167i	$-0.689384\pi$
$461$	−2.12842e49	−1.12096	−0.560481	+	0.828167i	$0.689384\pi$
$462$	1.05978e48	0.0536217
$463$	2.04790e49	0.995543	0.497771	−	0.867308i	$-0.334152\pi$
$463$	2.04790e49	0.995543	0.497771	+	0.867308i	$0.334152\pi$
$464$	−7.80077e48	−0.364380
$465$	0	0
$466$	1.07470e49	0.463603
$467$	1.46601e49	0.607818	0.303909	−	0.952701i	$-0.401708\pi$
$467$	1.46601e49	0.607818	0.303909	+	0.952701i	$0.401708\pi$
$468$	−6.99900e49	−2.78925
$469$	−5.23649e48	−0.200605
$470$	0	0
$471$	1.79264e49	0.634757
$472$	1.01741e49	0.346392
$473$	1.14431e49	0.374639
$474$	2.84848e49	0.896837
$475$	0	0
$476$	3.37290e48	0.0982369
$477$	−1.76470e50	−4.94403
$478$	−9.44921e48	−0.254671
$479$	1.49122e48	0.0386666	0.0193333	−	0.999813i	$-0.493846\pi$
$479$	1.49122e48	0.0386666	0.0193333	+	0.999813i	$0.493846\pi$
$480$	0	0
$481$	−6.82219e49	−1.63772
$482$	3.10297e48	0.0716816
$483$	2.57082e49	0.571545
$484$	3.49028e49	0.746828
$485$	0	0
$486$	1.34381e50	2.66419
$487$	2.77121e49	0.528912	0.264456	−	0.964398i	$-0.414808\pi$
$487$	2.77121e49	0.528912	0.264456	+	0.964398i	$0.414808\pi$
$488$	−4.57378e49	−0.840444
$489$	−4.88656e49	−0.864549
$490$	0	0
$491$	1.12985e50	1.85360	0.926798	−	0.375561i	$-0.122550\pi$
$491$	1.12985e50	1.85360	0.926798	+	0.375561i	$0.122550\pi$
$492$	−2.58913e49	−0.409072
$493$	2.87135e49	0.436936
$494$	−3.82684e49	−0.560908
$495$	0	0
$496$	3.38278e49	0.460112
$497$	−3.42239e48	−0.0448473
$498$	1.17226e50	1.48007
$499$	−1.50252e50	−1.82793	−0.913965	−	0.405793i	$-0.866995\pi$
$499$	−1.50252e50	−1.82793	−0.913965	+	0.405793i	$0.866995\pi$
$500$	0	0
$501$	4.93327e49	0.557362
$502$	−2.43101e49	−0.264709
$503$	−1.24457e50	−1.30621	−0.653104	−	0.757268i	$-0.726534\pi$
$503$	−1.24457e50	−1.30621	−0.653104	+	0.757268i	$0.726534\pi$
$504$	−5.53007e49	−0.559456
$505$	0	0
$506$	1.51367e49	0.142313
$507$	7.79210e49	0.706329
$508$	4.96087e49	0.433590
$509$	8.75400e48	0.0737781	0.0368890	−	0.999319i	$-0.488255\pi$
$509$	8.75400e48	0.0737781	0.0368890	+	0.999319i	$0.488255\pi$
$510$	0	0
$511$	1.86579e49	0.146244
$512$	1.05327e50	0.796239
$513$	−5.90365e50	−4.30477
$514$	8.03806e49	0.565371
$515$	0	0
$516$	−3.54059e50	−2.31770
$517$	4.32479e49	0.273143
$518$	−2.39631e49	−0.146030
$519$	−1.92514e49	−0.113205
$520$	0	0
$521$	−5.31489e49	−0.291069	−0.145535	−	0.989353i	$-0.546490\pi$
$521$	−5.31489e49	−0.291069	−0.145535	+	0.989353i	$0.546490\pi$
$522$	−2.09284e50	−1.10620
$523$	4.61130e49	0.235257	0.117628	−	0.993058i	$-0.462471\pi$
$523$	4.61130e49	0.235257	0.117628	+	0.993058i	$0.462471\pi$
$524$	−1.97413e50	−0.972178
$525$	0	0
$526$	−6.32073e49	−0.290089
$527$	−1.24515e50	−0.551729
$528$	5.32741e49	0.227923
$529$	1.25121e50	0.516894
$530$	0	0
$531$	−3.34833e50	−1.28997
$532$	5.38859e49	0.200498
$533$	8.27715e49	0.297459
$534$	2.60013e50	0.902572
$535$	0	0
$536$	2.14588e50	0.695115
$537$	1.08831e51	3.40586
$538$	9.82192e49	0.296978
$539$	−8.37465e49	−0.244667
$540$	0	0
$541$	1.06064e50	0.289348	0.144674	−	0.989479i	$-0.453787\pi$
$541$	1.06064e50	0.289348	0.144674	+	0.989479i	$0.453787\pi$
$542$	−2.25208e50	−0.593742
$543$	3.72572e50	0.949322
$544$	−2.14993e50	−0.529474
$545$	0	0
$546$	1.04827e50	0.241219
$547$	−9.19571e49	−0.204559	−0.102280	−	0.994756i	$-0.532614\pi$
$547$	−9.19571e49	−0.204559	−0.102280	+	0.994756i	$0.532614\pi$
$548$	1.51470e50	0.325751
$549$	1.50525e51	3.12982
$550$	0	0
$551$	4.58730e50	0.891770
$552$	−1.05351e51	−1.98045
$553$	1.28226e50	0.233109
$554$	−7.31938e49	−0.128690
$555$	0	0
$556$	1.80652e50	0.297138
$557$	−4.80322e50	−0.764207	−0.382104	−	0.924119i	$-0.624800\pi$
$557$	−4.80322e50	−0.764207	−0.382104	+	0.924119i	$0.624800\pi$
$558$	9.07555e50	1.39682
$559$	1.13188e51	1.68532
$560$	0	0
$561$	−1.96094e50	−0.273307
$562$	1.53126e49	0.0206504
$563$	1.84536e50	0.240811	0.120405	−	0.992725i	$-0.461581\pi$
$563$	1.84536e50	0.240811	0.120405	+	0.992725i	$0.461581\pi$
$564$	−1.33812e51	−1.68979
$565$	0	0
$566$	−1.81696e50	−0.214903
$567$	1.00965e51	1.15580
$568$	1.40247e50	0.155400
$569$	−1.49495e51	−1.60343	−0.801717	−	0.597704i	$-0.796080\pi$
$569$	−1.49495e51	−1.60343	−0.801717	+	0.597704i	$0.796080\pi$
$570$	0	0
$571$	−4.42408e50	−0.444687	−0.222344	−	0.974968i	$-0.571371\pi$
$571$	−4.42408e50	−0.444687	−0.222344	+	0.974968i	$0.571371\pi$
$572$	−2.47428e50	−0.240781
$573$	2.95813e51	2.78712
$574$	2.90736e49	0.0265234
$575$	0	0
$576$	2.29325e49	0.0196171
$577$	2.16164e51	1.79074	0.895368	−	0.445327i	$-0.146913\pi$
$577$	2.16164e51	1.79074	0.895368	+	0.445327i	$0.146913\pi$
$578$	−4.01245e50	−0.321918
$579$	3.60364e51	2.80019
$580$	0	0
$581$	5.27700e50	0.384706
$582$	1.46709e51	1.03605
$583$	−6.23855e50	−0.426791
$584$	−7.64590e50	−0.506747
$585$	0	0
$586$	−6.63576e49	−0.0412843
$587$	−1.69000e51	−1.01878	−0.509391	−	0.860535i	$-0.670129\pi$
$587$	−1.69000e51	−1.01878	−0.509391	+	0.860535i	$0.670129\pi$
$588$	2.59118e51	1.51362
$589$	−1.98927e51	−1.12606
$590$	0	0
$591$	−5.74683e51	−3.05534
$592$	−1.20460e51	−0.620712
$593$	−2.92325e51	−1.46001	−0.730003	−	0.683444i	$-0.760481\pi$
$593$	−2.92325e51	−1.46001	−0.730003	+	0.683444i	$0.760481\pi$
$594$	9.52166e50	0.460960
$595$	0	0
$596$	−2.63502e51	−1.19875
$597$	6.44376e50	0.284194
$598$	1.49723e51	0.640200
$599$	−2.96346e50	−0.122858	−0.0614290	−	0.998111i	$-0.519566\pi$
$599$	−2.96346e50	−0.122858	−0.0614290	+	0.998111i	$0.519566\pi$
$600$	0	0
$601$	−2.64345e51	−1.03037	−0.515185	−	0.857079i	$-0.672277\pi$
$601$	−2.64345e51	−1.03037	−0.515185	+	0.857079i	$0.672277\pi$
$602$	3.97577e50	0.150275
$603$	−7.06217e51	−2.58862
$604$	1.45180e51	0.516089
$605$	0	0
$606$	−1.48709e51	−0.497273
$607$	−4.44383e51	−1.44134	−0.720670	−	0.693278i	$-0.756166\pi$
$607$	−4.44383e51	−1.44134	−0.720670	+	0.693278i	$0.756166\pi$
$608$	−3.43476e51	−1.08064
$609$	−1.25659e51	−0.383506
$610$	0	0
$611$	4.27782e51	1.22874
$612$	4.54885e51	1.26765
$613$	4.41079e51	1.19261	0.596304	−	0.802759i	$-0.296635\pi$
$613$	4.41079e51	1.19261	0.596304	+	0.802759i	$0.296635\pi$
$614$	−2.22331e51	−0.583291
$615$	0	0
$616$	−1.95499e50	−0.0482948
$617$	4.23551e51	1.01538	0.507692	−	0.861539i	$-0.330499\pi$
$617$	4.23551e51	1.01538	0.507692	+	0.861539i	$0.330499\pi$
$618$	1.63208e51	0.379712
$619$	−5.96008e51	−1.34578	−0.672892	−	0.739741i	$-0.734948\pi$
$619$	−5.96008e51	−1.34578	−0.672892	+	0.739741i	$0.734948\pi$
$620$	0	0
$621$	2.30977e52	4.91330
$622$	3.64832e50	0.0753304
$623$	1.17046e51	0.234600
$624$	5.26955e51	1.02532
$625$	0	0
$626$	7.49304e50	0.137415
$627$	−3.13282e51	−0.557810
$628$	−1.47009e51	−0.254151
$629$	4.43394e51	0.744308
$630$	0	0
$631$	−8.10206e51	−1.28249	−0.641243	−	0.767338i	$-0.721581\pi$
$631$	−8.10206e51	−1.28249	−0.641243	+	0.767338i	$0.721581\pi$
$632$	−5.25460e51	−0.807744
$633$	−2.05045e51	−0.306112
$634$	2.06186e51	0.298957
$635$	0	0
$636$	1.93026e52	2.64033
$637$	−8.28370e51	−1.10064
$638$	−7.39860e50	−0.0954920
$639$	−4.61559e51	−0.578710
$640$	0	0
$641$	−1.08597e52	−1.28512	−0.642560	−	0.766235i	$-0.722128\pi$
$641$	−1.08597e52	−1.28512	−0.642560	+	0.766235i	$0.722128\pi$
$642$	−2.36871e51	−0.272342
$643$	1.13677e52	1.26990	0.634950	−	0.772553i	$-0.281021\pi$
$643$	1.13677e52	1.26990	0.634950	+	0.772553i	$0.281021\pi$
$644$	−2.10825e51	−0.228841
$645$	0	0
$646$	2.48717e51	0.254920
$647$	−1.26880e52	−1.26376	−0.631881	−	0.775066i	$-0.717717\pi$
$647$	−1.26880e52	−1.26376	−0.631881	+	0.775066i	$0.717717\pi$
$648$	−4.13746e52	−4.00495
$649$	−1.18370e51	−0.111356
$650$	0	0
$651$	5.44915e51	0.484262
$652$	4.00732e51	0.346157
$653$	8.64561e51	0.725942	0.362971	−	0.931801i	$-0.381762\pi$
$653$	8.64561e51	0.725942	0.362971	+	0.931801i	$0.381762\pi$
$654$	1.44624e52	1.18046
$655$	0	0
$656$	1.46150e51	0.112740
$657$	2.51629e52	1.88713
$658$	1.50259e51	0.109563
$659$	−1.70229e52	−1.20685	−0.603426	−	0.797419i	$-0.706198\pi$
$659$	−1.70229e52	−1.20685	−0.603426	+	0.797419i	$0.706198\pi$
$660$	0	0
$661$	1.05729e52	0.708710	0.354355	−	0.935111i	$-0.384700\pi$
$661$	1.05729e52	0.708710	0.354355	+	0.935111i	$0.384700\pi$
$662$	1.77078e51	0.115423
$663$	−1.93964e52	−1.22948
$664$	−2.16248e52	−1.33304
$665$	0	0
$666$	−3.23177e52	−1.88437
$667$	−1.79475e52	−1.01783
$668$	−4.04563e51	−0.223162
$669$	−3.93332e52	−2.11046
$670$	0	0
$671$	5.32134e51	0.270181
$672$	9.40872e51	0.464728
$673$	−1.79873e52	−0.864345	−0.432173	−	0.901791i	$-0.642253\pi$
$673$	−1.79873e52	−0.864345	−0.432173	+	0.901791i	$0.642253\pi$
$674$	6.26484e50	0.0292888
$675$	0	0
$676$	−6.39006e51	−0.282808
$677$	−2.52276e52	−1.08639	−0.543194	−	0.839607i	$-0.682785\pi$
$677$	−2.52276e52	−1.08639	−0.543194	+	0.839607i	$0.682785\pi$
$678$	−7.25404e51	−0.303970
$679$	6.60419e51	0.269295
$680$	0	0
$681$	−4.82748e52	−1.86423
$682$	3.20838e51	0.120580
$683$	1.24840e52	0.456635	0.228318	−	0.973587i	$-0.426678\pi$
$683$	1.24840e52	0.456635	0.228318	+	0.973587i	$0.426678\pi$
$684$	7.26730e52	2.58723
$685$	0	0
$686$	−5.98508e51	−0.201871
$687$	1.86042e52	0.610817
$688$	1.99857e52	0.638754
$689$	−6.17080e52	−1.91993
$690$	0	0
$691$	−2.42733e52	−0.715789	−0.357895	−	0.933762i	$-0.616505\pi$
$691$	−2.42733e52	−0.715789	−0.357895	+	0.933762i	$0.616505\pi$
$692$	1.57875e51	0.0453263
$693$	6.43393e51	0.179850
$694$	−1.12499e52	−0.306194
$695$	0	0
$696$	5.14941e52	1.32888
$697$	−5.37955e51	−0.135189
$698$	2.47018e52	0.604509
$699$	8.70239e52	2.07401
$700$	0	0
$701$	3.38548e52	0.765310	0.382655	−	0.923891i	$-0.375010\pi$
$701$	3.38548e52	0.765310	0.382655	+	0.923891i	$0.375010\pi$
$702$	9.41826e52	2.07364
$703$	7.08372e52	1.51911
$704$	8.10708e49	0.00169344
$705$	0	0
$706$	−1.77023e52	−0.350866
$707$	−6.69422e51	−0.129253
$708$	3.66246e52	0.688901
$709$	−2.82821e52	−0.518270	−0.259135	−	0.965841i	$-0.583437\pi$
$709$	−2.82821e52	−0.518270	−0.259135	+	0.965841i	$0.583437\pi$
$710$	0	0
$711$	1.72931e53	3.00805
$712$	−4.79647e52	−0.812909
$713$	7.78290e52	1.28524
$714$	−6.81303e51	−0.109628
$715$	0	0
$716$	−8.92487e52	−1.36368
$717$	−7.65154e52	−1.13932
$718$	1.16094e52	0.168464
$719$	−3.47193e52	−0.491006	−0.245503	−	0.969396i	$-0.578953\pi$
$719$	−3.47193e52	−0.491006	−0.245503	+	0.969396i	$0.578953\pi$
$720$	0	0
$721$	7.34688e51	0.0986962
$722$	5.61059e51	0.0734633
$723$	2.51265e52	0.320681
$724$	−3.05535e52	−0.380100
$725$	0	0
$726$	−7.05012e52	−0.833430
$727$	5.27093e52	0.607436	0.303718	−	0.952762i	$-0.401772\pi$
$727$	5.27093e52	0.607436	0.303718	+	0.952762i	$0.401772\pi$
$728$	−1.93376e52	−0.217256
$729$	6.33619e53	6.94015
$730$	0	0
$731$	−7.35644e52	−0.765942
$732$	−1.64646e53	−1.67147
$733$	−1.22783e53	−1.21538	−0.607691	−	0.794173i	$-0.707904\pi$
$733$	−1.22783e53	−1.21538	−0.607691	+	0.794173i	$0.707904\pi$
$734$	−8.13450e51	−0.0785152
$735$	0	0
$736$	1.34383e53	1.23340
$737$	−2.49661e52	−0.223461
$738$	3.92100e52	0.342259
$739$	−1.99727e53	−1.70025	−0.850127	−	0.526578i	$-0.823475\pi$
$739$	−1.99727e53	−1.70025	−0.850127	+	0.526578i	$0.823475\pi$
$740$	0	0
$741$	−3.09880e53	−2.50932
$742$	−2.16751e52	−0.171194
$743$	1.96635e53	1.51485	0.757424	−	0.652924i	$-0.226458\pi$
$743$	1.96635e53	1.51485	0.757424	+	0.652924i	$0.226458\pi$
$744$	−2.23303e53	−1.67801
$745$	0	0
$746$	−8.98391e52	−0.642387
$747$	7.11681e53	4.96425
$748$	1.60810e52	0.109430
$749$	−1.06629e52	−0.0707881
$750$	0	0
$751$	−1.50150e53	−0.948825	−0.474413	−	0.880303i	$-0.657340\pi$
$751$	−1.50150e53	−0.948825	−0.474413	+	0.880303i	$0.657340\pi$
$752$	7.55336e52	0.465704
$753$	−1.96852e53	−1.18422
$754$	−7.31825e52	−0.429573
$755$	0	0
$756$	−1.32619e53	−0.741229
$757$	4.30019e52	0.234538	0.117269	−	0.993100i	$-0.462586\pi$
$757$	4.30019e52	0.234538	0.117269	+	0.993100i	$0.462586\pi$
$758$	2.12013e52	0.112845
$759$	1.22570e53	0.636664
$760$	0	0
$761$	1.11294e53	0.550627	0.275313	−	0.961355i	$-0.411218\pi$
$761$	1.11294e53	0.550627	0.275313	+	0.961355i	$0.411218\pi$
$762$	−1.00206e53	−0.483869
$763$	6.51030e52	0.306830
$764$	−2.42587e53	−1.11594
$765$	0	0
$766$	−7.26337e52	−0.318351
$767$	−1.17084e53	−0.500938
$768$	−2.16668e53	−0.904926
$769$	−2.56548e53	−1.04600	−0.522999	−	0.852333i	$-0.675187\pi$
$769$	−2.56548e53	−1.04600	−0.522999	+	0.852333i	$0.675187\pi$
$770$	0	0
$771$	6.50885e53	2.52929
$772$	−2.95523e53	−1.12117
$773$	2.02530e53	0.750187	0.375093	−	0.926987i	$-0.377611\pi$
$773$	2.02530e53	0.750187	0.375093	+	0.926987i	$0.377611\pi$
$774$	5.36190e53	1.93915
$775$	0	0
$776$	−2.70635e53	−0.933131
$777$	−1.94042e53	−0.653292
$778$	1.15563e53	0.379923
$779$	−8.59446e52	−0.275915
$780$	0	0
$781$	−1.63170e52	−0.0499570
$782$	−9.73090e52	−0.290957
$783$	−1.12898e54	−3.29682
$784$	−1.46266e53	−0.417153
$785$	0	0
$786$	3.98760e53	1.08491
$787$	−1.20989e53	−0.321525	−0.160762	−	0.986993i	$-0.551395\pi$
$787$	−1.20989e53	−0.321525	−0.160762	+	0.986993i	$0.551395\pi$
$788$	4.71280e53	1.22333
$789$	−5.11824e53	−1.29776
$790$	0	0
$791$	−3.26544e52	−0.0790090
$792$	−2.63658e53	−0.623197
$793$	5.26355e53	1.21541
$794$	−2.08884e53	−0.471223
$795$	0	0
$796$	−5.28433e52	−0.113789
$797$	−3.10004e53	−0.652211	−0.326105	−	0.945333i	$-0.605736\pi$
$797$	−3.10004e53	−0.652211	−0.326105	+	0.945333i	$0.605736\pi$
$798$	−1.08846e53	−0.223748
$799$	−2.78028e53	−0.558435
$800$	0	0
$801$	1.57854e54	3.02728
$802$	−3.64119e53	−0.682366
$803$	8.89558e52	0.162906
$804$	7.72471e53	1.38244
$805$	0	0
$806$	3.17354e53	0.542432
$807$	7.95335e53	1.32859
$808$	2.74325e53	0.447873
$809$	2.26227e52	0.0360991	0.0180496	−	0.999837i	$-0.494254\pi$
$809$	2.26227e52	0.0360991	0.0180496	+	0.999837i	$0.494254\pi$
$810$	0	0
$811$	2.39772e53	0.365522	0.182761	−	0.983157i	$-0.441497\pi$
$811$	2.39772e53	0.365522	0.182761	+	0.983157i	$0.441497\pi$
$812$	1.03049e53	0.153552
$813$	−1.82363e54	−2.65621
$814$	−1.14249e53	−0.162668
$815$	0	0
$816$	−3.42483e53	−0.465984
$817$	−1.17528e54	−1.56326
$818$	5.49256e53	0.714231
$819$	6.36406e53	0.809062
$820$	0	0
$821$	−1.19153e54	−1.44796	−0.723980	−	0.689821i	$-0.757689\pi$
$821$	−1.19153e54	−1.44796	−0.723980	+	0.689821i	$0.757689\pi$
$822$	−3.05959e53	−0.363525
$823$	7.91118e51	0.00919061	0.00459531	−	0.999989i	$-0.498537\pi$
$823$	7.91118e51	0.00919061	0.00459531	+	0.999989i	$0.498537\pi$
$824$	−3.01071e53	−0.341991
$825$	0	0
$826$	−4.11261e52	−0.0446670
$827$	−1.25980e54	−1.33798	−0.668988	−	0.743274i	$-0.733272\pi$
$827$	−1.25980e54	−1.33798	−0.668988	+	0.743274i	$0.733272\pi$
$828$	−2.84329e54	−2.95297
$829$	−9.90567e53	−1.00606	−0.503031	−	0.864268i	$-0.667782\pi$
$829$	−9.90567e53	−1.00606	−0.503031	+	0.864268i	$0.667782\pi$
$830$	0	0
$831$	−5.92690e53	−0.575717
$832$	8.01903e51	0.00761799
$833$	5.38381e53	0.500216
$834$	−3.64904e53	−0.331594
$835$	0	0
$836$	2.56913e53	0.223342
$837$	4.89581e54	4.16297
$838$	−7.61269e53	−0.633175
$839$	−1.15575e54	−0.940300	−0.470150	−	0.882586i	$-0.655800\pi$
$839$	−1.15575e54	−0.940300	−0.470150	+	0.882586i	$0.655800\pi$
$840$	0	0
$841$	−4.07223e53	−0.317035
$842$	9.88806e53	0.753074
$843$	1.23995e53	0.0923832
$844$	1.68151e53	0.122564
$845$	0	0
$846$	2.02646e54	1.41380
$847$	−3.17364e53	−0.216628
$848$	−1.08958e54	−0.727673
$849$	−1.47129e54	−0.961407
$850$	0	0
$851$	−2.77146e54	−1.73385
$852$	5.04861e53	0.309057
$853$	1.32284e54	0.792410	0.396205	−	0.918162i	$-0.370327\pi$
$853$	1.32284e54	0.792410	0.396205	+	0.918162i	$0.370327\pi$
$854$	1.84883e53	0.108374
$855$	0	0
$856$	4.36957e53	0.245287
$857$	1.64071e54	0.901337	0.450668	−	0.892691i	$-0.351186\pi$
$857$	1.64071e54	0.901337	0.450668	+	0.892691i	$0.351186\pi$
$858$	4.99788e53	0.268702
$859$	−4.41837e53	−0.232482	−0.116241	−	0.993221i	$-0.537084\pi$
$859$	−4.41837e53	−0.232482	−0.116241	+	0.993221i	$0.537084\pi$
$860$	0	0
$861$	2.35425e53	0.118657
$862$	6.20202e53	0.305949
$863$	8.34757e53	0.403051	0.201526	−	0.979483i	$-0.435410\pi$
$863$	8.34757e53	0.403051	0.201526	+	0.979483i	$0.435410\pi$
$864$	8.45330e54	3.99505
$865$	0	0
$866$	−6.89997e52	−0.0312440
$867$	−3.24910e54	−1.44016
$868$	−4.46868e53	−0.193894
$869$	6.11344e53	0.259669
$870$	0	0
$871$	−2.46950e54	−1.00525
$872$	−2.66788e54	−1.06319
$873$	8.90671e54	3.47499
$874$	−1.55462e54	−0.593832
$875$	0	0
$876$	−2.75236e54	−1.00781
$877$	−3.82715e54	−1.37209	−0.686045	−	0.727559i	$-0.740655\pi$
$877$	−3.82715e54	−1.37209	−0.686045	+	0.727559i	$0.740655\pi$
$878$	−1.76876e54	−0.620899
$879$	−5.37334e53	−0.184693
$880$	0	0
$881$	5.27673e54	1.73904	0.869522	−	0.493894i	$-0.164427\pi$
$881$	5.27673e54	1.73904	0.869522	+	0.493894i	$0.164427\pi$
$882$	−3.92411e54	−1.26640
$883$	1.54562e54	0.488459	0.244230	−	0.969717i	$-0.421465\pi$
$883$	1.54562e54	0.488459	0.244230	+	0.969717i	$0.421465\pi$
$884$	1.59064e54	0.492272
$885$	0	0
$886$	−1.56516e54	−0.464552
$887$	−2.62619e54	−0.763377	−0.381689	−	0.924291i	$-0.624657\pi$
$887$	−2.62619e54	−0.763377	−0.381689	+	0.924291i	$0.624657\pi$
$888$	7.95172e54	2.26371
$889$	−4.51083e53	−0.125769
$890$	0	0
$891$	4.81371e54	1.28749
$892$	3.22560e54	0.845007
$893$	−4.44181e54	−1.13975
$894$	5.32256e54	1.33776
$895$	0	0
$896$	−9.61234e53	−0.231809
$897$	1.21239e55	2.86405
$898$	6.42818e53	0.148756
$899$	−3.80418e54	−0.862396
$900$	0	0
$901$	4.01058e54	0.872567
$902$	1.38615e53	0.0295454
$903$	3.21939e54	0.672281
$904$	1.33816e54	0.273773
$905$	0	0
$906$	−2.93254e54	−0.575934
$907$	4.80527e54	0.924662	0.462331	−	0.886707i	$-0.347013\pi$
$907$	4.80527e54	0.924662	0.462331	+	0.886707i	$0.347013\pi$
$908$	3.95887e54	0.746420
$909$	−9.02812e54	−1.66788
$910$	0	0
$911$	−7.64588e54	−1.35624	−0.678121	−	0.734950i	$-0.737206\pi$
$911$	−7.64588e54	−1.35624	−0.678121	+	0.734950i	$0.737206\pi$
$912$	−5.47156e54	−0.951057
$913$	2.51593e54	0.428537
$914$	−6.68966e53	−0.111660
$915$	0	0
$916$	−1.52567e54	−0.244565
$917$	1.79504e54	0.281994
$918$	−6.12119e54	−0.942424
$919$	1.15737e55	1.74637	0.873184	−	0.487391i	$-0.162051\pi$
$919$	1.15737e55	1.74637	0.873184	+	0.487391i	$0.162051\pi$
$920$	0	0
$921$	−1.80034e55	−2.60946
$922$	−3.52570e54	−0.500868
$923$	−1.61398e54	−0.224733
$924$	−7.03754e53	−0.0960482
$925$	0	0
$926$	3.39231e54	0.444828
$927$	9.90834e54	1.27358
$928$	−6.56845e54	−0.827609
$929$	−7.83147e54	−0.967281	−0.483640	−	0.875267i	$-0.660686\pi$
$929$	−7.83147e54	−0.967281	−0.483640	+	0.875267i	$0.660686\pi$
$930$	0	0
$931$	8.60126e54	1.02092
$932$	−7.13656e54	−0.830414
$933$	2.95424e54	0.337004
$934$	2.42842e54	0.271585
$935$	0	0
$936$	−2.60795e55	−2.80347
$937$	4.21239e54	0.443962	0.221981	−	0.975051i	$-0.428748\pi$
$937$	4.21239e54	0.443962	0.221981	+	0.975051i	$0.428748\pi$
$938$	−8.67416e53	−0.0896344
$939$	6.06752e54	0.614749
$940$	0	0
$941$	1.67422e55	1.63082	0.815408	−	0.578886i	$-0.196513\pi$
$941$	1.67422e55	1.63082	0.815408	+	0.578886i	$0.196513\pi$
$942$	2.96949e54	0.283622
$943$	3.36253e54	0.314919
$944$	−2.06736e54	−0.189860
$945$	0	0
$946$	1.89553e54	0.167396
$947$	1.77451e55	1.53675	0.768375	−	0.640000i	$-0.221066\pi$
$947$	1.77451e55	1.53675	0.768375	+	0.640000i	$0.221066\pi$
$948$	−1.89155e55	−1.60643
$949$	8.79897e54	0.732836
$950$	0	0
$951$	1.66960e55	1.33744
$952$	1.25680e54	0.0987378
$953$	−1.35728e55	−1.04581	−0.522903	−	0.852392i	$-0.675151\pi$
$953$	−1.35728e55	−1.04581	−0.522903	+	0.852392i	$0.675151\pi$
$954$	−2.92320e55	−2.20909
$955$	0	0
$956$	6.27479e54	0.456172
$957$	−5.99105e54	−0.427201
$958$	2.47019e53	0.0172770
$959$	−1.37729e54	−0.0944888
$960$	0	0
$961$	1.34776e54	0.0889675
$962$	−1.13009e55	−0.731766
$963$	−1.43804e55	−0.913451
$964$	−2.06054e54	−0.128398
$965$	0	0
$966$	4.25853e54	0.255378
$967$	−7.42660e54	−0.436918	−0.218459	−	0.975846i	$-0.570103\pi$
$967$	−7.42660e54	−0.436918	−0.218459	+	0.975846i	$0.570103\pi$
$968$	1.30054e55	0.750635
$969$	2.01400e55	1.14043
$970$	0	0
$971$	−3.22246e54	−0.175643	−0.0878217	−	0.996136i	$-0.527991\pi$
$971$	−3.22246e54	−0.175643	−0.0878217	+	0.996136i	$0.527991\pi$
$972$	−8.92360e55	−4.77215
$973$	−1.64263e54	−0.0861892
$974$	4.59046e54	0.236328
$975$	0	0
$976$	9.29386e54	0.460654
$977$	−2.30246e54	−0.111981	−0.0559903	−	0.998431i	$-0.517832\pi$
$977$	−2.30246e54	−0.111981	−0.0559903	+	0.998431i	$0.517832\pi$
$978$	−8.09451e54	−0.386297
$979$	5.58043e54	0.261329
$980$	0	0
$981$	8.78009e55	3.95934
$982$	1.87159e55	0.828223
$983$	−1.48713e55	−0.645816	−0.322908	−	0.946430i	$-0.604660\pi$
$983$	−1.48713e55	−0.645816	−0.322908	+	0.946430i	$0.604660\pi$
$984$	−9.64757e54	−0.411158
$985$	0	0
$986$	4.75634e54	0.195231
$987$	1.21673e55	0.490148
$988$	2.54123e55	1.00471
$989$	4.59819e55	1.78425
$990$	0	0
$991$	2.38601e55	0.891887	0.445944	−	0.895061i	$-0.352868\pi$
$991$	2.38601e55	0.891887	0.445944	+	0.895061i	$0.352868\pi$
$992$	2.84839e55	1.04504
$993$	1.43390e55	0.516366
$994$	−5.66913e53	−0.0200387
$995$	0	0
$996$	−7.78448e55	−2.65113
$997$	−2.88180e54	−0.0963392	−0.0481696	−	0.998839i	$-0.515339\pi$
$997$	−2.88180e54	−0.0963392	−0.0481696	+	0.998839i	$0.515339\pi$
$998$	−2.48890e55	−0.816755
$999$	−1.74338e56	−5.61604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.38.a.e.1.7	yes	12
5.2	odd	4		25.38.b.d.24.14		24
5.3	odd	4		25.38.b.d.24.11		24
5.4	even	2		25.38.a.d.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.38.a.d.1.6	✓	12	5.4	even	2
25.38.a.e.1.7	yes	12	1.1	even	1	trivial
25.38.b.d.24.11		24	5.3	odd	4
25.38.b.d.24.14		24	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 38 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q+165648. q^{2} +1.34135e9 q^{3} -1.10000e11 q^{4} +2.22192e14 q^{6} +1.00021e15 q^{7} -4.09878e16 q^{8} +1.34892e18 q^{9} +O(q^{10})\)	\(q+165648. q^{2} +1.34135e9 q^{3} -1.10000e11 q^{4} +2.22192e14 q^{6} +1.00021e15 q^{7} -4.09878e16 q^{8} +1.34892e18 q^{9} +4.76870e18 q^{11} -1.47547e20 q^{12} +4.71691e20 q^{13} +1.65682e20 q^{14} +8.32866e21 q^{16} -3.06565e22 q^{17} +2.23447e23 q^{18} -4.89773e23 q^{19} +1.34162e24 q^{21} +7.89927e23 q^{22} +1.91621e25 q^{23} -5.49788e25 q^{24} +7.81349e25 q^{26} +1.20538e27 q^{27} -1.10022e26 q^{28} -9.36618e26 q^{29} +4.06162e27 q^{31} +7.01295e27 q^{32} +6.39647e27 q^{33} -5.07821e27 q^{34} -1.48381e29 q^{36} -1.44633e29 q^{37} -8.11302e28 q^{38} +6.32701e29 q^{39} +1.75478e29 q^{41} +2.22237e29 q^{42} +2.39963e30 q^{43} -5.24555e29 q^{44} +3.17417e30 q^{46} +9.06911e30 q^{47} +1.11716e31 q^{48} -1.75617e31 q^{49} -4.11210e31 q^{51} -5.18858e31 q^{52} -1.30823e32 q^{53} +1.99670e32 q^{54} -4.09962e31 q^{56} -6.56955e32 q^{57} -1.55149e32 q^{58} -2.48222e32 q^{59} +1.11589e33 q^{61} +6.72800e32 q^{62} +1.34920e33 q^{63} +1.70006e31 q^{64} +1.05957e33 q^{66} -5.23541e33 q^{67} +3.37221e33 q^{68} +2.57030e34 q^{69} -3.42169e33 q^{71} -5.52894e34 q^{72} +1.86541e34 q^{73} -2.39582e34 q^{74} +5.38749e34 q^{76} +4.76968e33 q^{77} +1.04806e35 q^{78} +1.28199e35 q^{79} +1.00944e36 q^{81} +2.90677e34 q^{82} +5.27592e35 q^{83} -1.47578e35 q^{84} +3.97495e35 q^{86} -1.25633e36 q^{87} -1.95458e35 q^{88} +1.17022e36 q^{89} +4.71788e35 q^{91} -2.10782e36 q^{92} +5.44803e36 q^{93} +1.50228e36 q^{94} +9.40678e36 q^{96} +6.60283e36 q^{97} -2.90907e36 q^{98} +6.43261e36 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 67745 q^{2} + 724208180 q^{3} + 510552903189 q^{4} + 15880293089869 q^{6} - 22\!\cdots\!00 q^{7}+ \cdots + 13\!\cdots\!56 q^{9}+O(q^{10}) \) 12 * q + 67745 * q^2 + 724208180 * q^3 + 510552903189 * q^4 + 15880293089869 * q^6 - 2280592823278800 * q^7 - 13603597852867065 * q^8 + 1359108258754272256 * q^9	\( 12 q + 67745 q^{2} + 724208180 q^{3} + 510552903189 q^{4} + 15880293089869 q^{6} - 22\!\cdots\!00 q^{7}+ \cdots - 99\!\cdots\!68 q^{99}+O(q^{100}) \) 12 * q + 67745 * q^2 + 724208180 * q^3 + 510552903189 * q^4 + 15880293089869 * q^6 - 2280592823278800 * q^7 - 13603597852867065 * q^8 + 1359108258754272256 * q^9 + 2554656505188935564 * q^11 + 38296868672496581965 * q^12 + 83357727454160617560 * q^13 + 541210680522824521098 * q^14 + 5889259433625089080017 * q^16 + 27158393512650748861460 * q^17 + 32304995024093396237630 * q^18 - 1307297534999360105100 * q^19 + 1927702939801406301291504 * q^21 + 8101495479590718049976265 * q^22 + 6561364838658797127044040 * q^23 - 29787767561390801566152825 * q^24 + 164742777145327712449309724 * q^26 + 803078304022385000040963860 * q^27 + 706360659695833214077901610 * q^28 - 2618773509396722286923297000 * q^29 + 3495844606583171918858800344 * q^31 + 6373892627582681144636589295 * q^32 + 1200396846201406692144438460 * q^33 + 3609863853608971653493888413 * q^34 - 91889542016499788236065200218 * q^36 - 218845170107098937703383787120 * q^37 - 54297488357007537741487211935 * q^38 + 125348774149234247086164192152 * q^39 - 51474601681994565845304037316 * q^41 + 279007056776373014503679203410 * q^42 + 1095093974210612352636183955800 * q^43 - 2189164920482983205730699871367 * q^44 + 10107503169848980559418679324254 * q^46 + 23907862328757523727528033674040 * q^47 + 6657319366828177502039371224865 * q^48 - 14951602913621438213797405279716 * q^49 + 22208888829315215979428699940124 * q^51 + 48235173815673852004466444424300 * q^52 + 16838816611239414826974664431080 * q^53 - 57545129090023056489658919639825 * q^54 + 187574880792755877609939226588350 * q^56 + 487409130824695345102950044427620 * q^57 + 537922378980272521289962414359360 * q^58 - 1336700779068696398361473055992200 * q^59 + 3485099106494564728317108451692864 * q^61 + 6971612944345799706023880876891690 * q^62 + 4069695817379180445347658914088720 * q^63 - 7333556814672452956681284356117151 * q^64 + 11216229648698841479202248762840093 * q^66 + 14753973559495590347225839865512260 * q^67 + 8785613970183933622822253658267505 * q^68 - 2522802561523836072474822533544408 * q^69 + 10576084284690246863128363185276904 * q^71 - 9674318002594923678160001962177470 * q^72 - 4825945256111753902227483456687060 * q^73 + 21873580722917055024973698667460818 * q^74 + 59441092708215293905864123640191425 * q^76 - 29028478500644775653137476920193600 * q^77 + 87450040981989737501234198133333100 * q^78 + 31047245994101750554934024487283200 * q^79 + 25973485616223818269003376033095972 * q^81 - 701481290104386131269144762106398035 * q^82 + 68699132245521973559209397648692020 * q^83 + 650071241509187672385081824537914938 * q^84 - 415357678037782103677821291329558836 * q^86 - 2554515878920505052325980738506323720 * q^87 - 1479450788257499500717123093674302805 * q^88 + 3353246041445492030040660620069119500 * q^89 - 2001623473367880972919402485939696816 * q^91 - 4351655844619574190087420297293008290 * q^92 + 2158939592433745585172010945825503160 * q^93 - 788004521663479731556746949918537092 * q^94 + 10002124457561358992178222672331390079 * q^96 + 14103135523445618268336692441237736840 * q^97 + 1821220898206765147319159644986861565 * q^98 - 9936699890312541409796157011630543968 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more