LMFDB - Embedded newform 48.28.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,28,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 28, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 28 $
Character orbit:	$[\chi]$	$=$	48.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:	$221.690675922$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{30001}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7500 $ x^2 - x - 7500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$87.1040$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59432e6 q^{3} -1.07803e9 q^{5} +1.13904e10 q^{7} +2.54187e12 q^{9} +O(q^{10})$	$q+1.59432e6 q^{3} -1.07803e9 q^{5} +1.13904e10 q^{7} +2.54187e12 q^{9} +5.05749e13 q^{11} +1.21304e15 q^{13} -1.71872e15 q^{15} +7.00452e16 q^{17} -2.51027e16 q^{19} +1.81600e16 q^{21} +3.40370e18 q^{23} -6.28844e18 q^{25} +4.05256e18 q^{27} +5.74556e19 q^{29} -1.08167e20 q^{31} +8.06328e19 q^{33} -1.22791e19 q^{35} -1.90073e21 q^{37} +1.93398e21 q^{39} +6.66148e21 q^{41} -1.03265e21 q^{43} -2.74020e21 q^{45} -6.43133e22 q^{47} -6.55826e22 q^{49} +1.11675e23 q^{51} +2.13714e23 q^{53} -5.45211e22 q^{55} -4.00218e22 q^{57} -1.07026e24 q^{59} +8.16628e23 q^{61} +2.89528e22 q^{63} -1.30769e24 q^{65} -1.68630e24 q^{67} +5.42659e24 q^{69} +3.18547e24 q^{71} +2.09853e25 q^{73} -1.00258e25 q^{75} +5.76068e23 q^{77} +8.88148e22 q^{79} +6.46108e24 q^{81} +6.50402e25 q^{83} -7.55105e25 q^{85} +9.16028e25 q^{87} +3.38617e26 q^{89} +1.38170e25 q^{91} -1.72453e26 q^{93} +2.70614e25 q^{95} -4.16124e26 q^{97} +1.28555e26 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9}+O(q^{10}) $ 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9	$ 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9} - 75762335668248 q^{11} - 103021079177588 q^{13} - 28\!\cdots\!00 q^{15}+ \cdots - 19\!\cdots\!92 q^{99}+O(q^{100}) $ 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9 - 75762335668248 * q^11 - 103021079177588 * q^13 - 2825054421990300 * q^15 + 34691611579147908 * q^17 - 111620476558642216 * q^19 - 589365730506544992 * q^21 + 2892743075819387952 * q^23 - 13257495997920171250 * q^25 + 8105110306037952534 * q^27 + 29959552473322806972 * q^29 - 10367463257055494032 * q^31 - 120789634289608156104 * q^33 + 252142979584030920000 * q^35 - 3703660908819524869316 * q^37 - 164248876017649632924 * q^39 + 1481814928479478034772 * q^41 - 9778778623457380249240 * q^43 - 4504049241230841066900 * q^45 - 89494998518809740323520 * q^47 + 13908377820361409575890 * q^49 + 55309634247701830126284 * q^51 + 425671275824601813476748 * q^53 + 33146870181194328159600 * q^55 - 177959093048404133739768 * q^57 - 369418769643380130292824 * q^59 + 925133397428892745249420 * q^61 - 939639339558386331280416 * q^63 - 394447136075147294825400 * q^65 - 4151706674555213948132968 * q^67 + 4611966818869594057796496 * q^69 - 9760455855870867537668784 * q^71 + 28449956347640075157409588 * q^73 - 21136730791892081187813750 * q^75 + 48717589762471785388668288 * q^77 + 49553221898675501744231120 * q^79 + 12922163778453346597864482 * q^81 + 120473408203228735926347736 * q^83 - 50977978054175038499879400 * q^85 + 47765203577925437580019956 * q^87 + 205644052209697133056985556 * q^89 + 515309018103795393250147648 * q^91 - 16529085122378486411580336 * q^93 + 87097784513216888583651600 * q^95 - 1478339006400409594836427772 * q^97 - 192577692109510944264197592 * 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 $

$2$	0	0
$2$	0	0
$3$	1.59432e6	0.577350
$3$	1.59432e6	0.577350
$4$	0	0
$5$	−1.07803e9	−0.394943	−0.197471	−	0.980309i	$-0.563273\pi$
$5$	−1.07803e9	−0.394943	−0.197471	+	0.980309i	$0.563273\pi$
$6$	0	0
$7$	1.13904e10	0.0444340	0.0222170	−	0.999753i	$-0.492928\pi$
$7$	1.13904e10	0.0444340	0.0222170	+	0.999753i	$0.492928\pi$
$8$	0	0
$9$	2.54187e12	0.333333
$10$	0	0
$11$	5.05749e13	0.441707	0.220853	−	0.975307i	$-0.429116\pi$
$11$	5.05749e13	0.441707	0.220853	+	0.975307i	$0.429116\pi$
$12$	0	0
$13$	1.21304e15	1.11081	0.555405	−	0.831580i	$-0.312563\pi$
$13$	1.21304e15	1.11081	0.555405	+	0.831580i	$0.312563\pi$
$14$	0	0
$15$	−1.71872e15	−0.228020
$16$	0	0
$17$	7.00452e16	1.71521	0.857606	−	0.514307i	$-0.171951\pi$
$17$	7.00452e16	1.71521	0.857606	+	0.514307i	$0.171951\pi$
$18$	0	0
$19$	−2.51027e16	−0.136945	−0.0684726	−	0.997653i	$-0.521813\pi$
$19$	−2.51027e16	−0.136945	−0.0684726	+	0.997653i	$0.521813\pi$
$20$	0	0
$21$	1.81600e16	0.0256540
$22$	0	0
$23$	3.40370e18	1.40807	0.704037	−	0.710164i	$-0.251379\pi$
$23$	3.40370e18	1.40807	0.704037	+	0.710164i	$0.251379\pi$
$24$	0	0
$25$	−6.28844e18	−0.844020
$26$	0	0
$27$	4.05256e18	0.192450
$28$	0	0
$29$	5.74556e19	1.03982	0.519912	−	0.854220i	$-0.325965\pi$
$29$	5.74556e19	1.03982	0.519912	+	0.854220i	$0.325965\pi$
$30$	0	0
$31$	−1.08167e20	−0.795632	−0.397816	−	0.917465i	$-0.630232\pi$
$31$	−1.08167e20	−0.795632	−0.397816	+	0.917465i	$0.630232\pi$
$32$	0	0
$33$	8.06328e19	0.255020
$34$	0	0
$35$	−1.22791e19	−0.0175489
$36$	0	0
$37$	−1.90073e21	−1.28292	−0.641458	−	0.767159i	$-0.721670\pi$
$37$	−1.90073e21	−1.28292	−0.641458	+	0.767159i	$0.721670\pi$
$38$	0	0
$39$	1.93398e21	0.641326
$40$	0	0
$41$	6.66148e21	1.12458	0.562288	−	0.826942i	$-0.309921\pi$
$41$	6.66148e21	1.12458	0.562288	+	0.826942i	$0.309921\pi$
$42$	0	0
$43$	−1.03265e21	−0.0916490	−0.0458245	−	0.998950i	$-0.514592\pi$
$43$	−1.03265e21	−0.0916490	−0.0458245	+	0.998950i	$0.514592\pi$
$44$	0	0
$45$	−2.74020e21	−0.131648
$46$	0	0
$47$	−6.43133e22	−1.71783	−0.858914	−	0.512120i	$-0.828860\pi$
$47$	−6.43133e22	−1.71783	−0.858914	+	0.512120i	$0.828860\pi$
$48$	0	0
$49$	−6.55826e22	−0.998026
$50$	0	0
$51$	1.11675e23	0.990278
$52$	0	0
$53$	2.13714e23	1.12748	0.563740	−	0.825953i	$-0.309362\pi$
$53$	2.13714e23	1.12748	0.563740	+	0.825953i	$0.309362\pi$
$54$	0	0
$55$	−5.45211e22	−0.174449
$56$	0	0
$57$	−4.00218e22	−0.0790653
$58$	0	0
$59$	−1.07026e24	−1.32736	−0.663679	−	0.748017i	$-0.731006\pi$
$59$	−1.07026e24	−1.32736	−0.663679	+	0.748017i	$0.731006\pi$
$60$	0	0
$61$	8.16628e23	0.645761	0.322880	−	0.946440i	$-0.395349\pi$
$61$	8.16628e23	0.645761	0.322880	+	0.946440i	$0.395349\pi$
$62$	0	0
$63$	2.89528e22	0.0148113
$64$	0	0
$65$	−1.30769e24	−0.438706
$66$	0	0
$67$	−1.68630e24	−0.375773	−0.187886	−	0.982191i	$-0.560164\pi$
$67$	−1.68630e24	−0.375773	−0.187886	+	0.982191i	$0.560164\pi$
$68$	0	0
$69$	5.42659e24	0.812951
$70$	0	0
$71$	3.18547e24	0.324478	0.162239	−	0.986751i	$-0.448128\pi$
$71$	3.18547e24	0.324478	0.162239	+	0.986751i	$0.448128\pi$
$72$	0	0
$73$	2.09853e25	1.46912	0.734559	−	0.678545i	$-0.237389\pi$
$73$	2.09853e25	1.46912	0.734559	+	0.678545i	$0.237389\pi$
$74$	0	0
$75$	−1.00258e25	−0.487295
$76$	0	0
$77$	5.76068e23	0.0196268
$78$	0	0
$79$	8.88148e22	0.00214052	0.00107026	−	0.999999i	$-0.499659\pi$
$79$	8.88148e22	0.00214052	0.00107026	+	0.999999i	$0.499659\pi$
$80$	0	0
$81$	6.46108e24	0.111111
$82$	0	0
$83$	6.50402e25	0.804689	0.402344	−	0.915488i	$-0.368195\pi$
$83$	6.50402e25	0.804689	0.402344	+	0.915488i	$0.368195\pi$
$84$	0	0
$85$	−7.55105e25	−0.677411
$86$	0	0
$87$	9.16028e25	0.600342
$88$	0	0
$89$	3.38617e26	1.63284	0.816420	−	0.577459i	$-0.195956\pi$
$89$	3.38617e26	1.63284	0.816420	+	0.577459i	$0.195956\pi$
$90$	0	0
$91$	1.38170e25	0.0493577
$92$	0	0
$93$	−1.72453e26	−0.459358
$94$	0	0
$95$	2.70614e25	0.0540855
$96$	0	0
$97$	−4.16124e26	−0.627775	−0.313888	−	0.949460i	$-0.601632\pi$
$97$	−4.16124e26	−0.627775	−0.313888	+	0.949460i	$0.601632\pi$
$98$	0	0
$99$	1.28555e26	0.147236
$100$	0	0
$101$	−9.90798e26	−0.866256	−0.433128	−	0.901332i	$-0.642590\pi$
$101$	−9.90798e26	−0.866256	−0.433128	+	0.901332i	$0.642590\pi$
$102$	0	0
$103$	2.01221e27	1.35012	0.675059	−	0.737764i	$-0.264118\pi$
$103$	2.01221e27	1.35012	0.675059	+	0.737764i	$0.264118\pi$
$104$	0	0
$105$	−1.95769e25	−0.0101318
$106$	0	0
$107$	−2.23219e25	−0.00895470	−0.00447735	−	0.999990i	$-0.501425\pi$
$107$	−2.23219e25	−0.00895470	−0.00447735	+	0.999990i	$0.501425\pi$
$108$	0	0
$109$	5.40319e27	1.68808	0.844039	−	0.536281i	$-0.180171\pi$
$109$	5.40319e27	1.68808	0.844039	+	0.536281i	$0.180171\pi$
$110$	0	0
$111$	−3.03038e27	−0.740691
$112$	0	0
$113$	−3.71413e27	−0.713342	−0.356671	−	0.934230i	$-0.616088\pi$
$113$	−3.71413e27	−0.713342	−0.356671	+	0.934230i	$0.616088\pi$
$114$	0	0
$115$	−3.66927e27	−0.556108
$116$	0	0
$117$	3.08338e27	0.370270
$118$	0	0
$119$	7.97842e26	0.0762137
$120$	0	0
$121$	−1.05522e28	−0.804895
$122$	0	0
$123$	1.06205e28	0.649274
$124$	0	0
$125$	1.48110e28	0.728282
$126$	0	0
$127$	−2.13925e28	−0.849007	−0.424503	−	0.905426i	$-0.639551\pi$
$127$	−2.13925e28	−0.849007	−0.424503	+	0.905426i	$0.639551\pi$
$128$	0	0
$129$	−1.64637e27	−0.0529136
$130$	0	0
$131$	−5.14435e28	−1.34328	−0.671642	−	0.740876i	$-0.734411\pi$
$131$	−5.14435e28	−1.34328	−0.671642	+	0.740876i	$0.734411\pi$
$132$	0	0
$133$	−2.85930e26	−0.00608502
$134$	0	0
$135$	−4.36876e27	−0.0760068
$136$	0	0
$137$	3.71968e28	0.530612	0.265306	−	0.964164i	$-0.414527\pi$
$137$	3.71968e28	0.530612	0.265306	+	0.964164i	$0.414527\pi$
$138$	0	0
$139$	1.00529e29	1.17921	0.589605	−	0.807692i	$-0.299283\pi$
$139$	1.00529e29	1.17921	0.589605	+	0.807692i	$0.299283\pi$
$140$	0	0
$141$	−1.02536e29	−0.991788
$142$	0	0
$143$	6.13494e28	0.490652
$144$	0	0
$145$	−6.19387e28	−0.410671
$146$	0	0
$147$	−1.04560e29	−0.576210
$148$	0	0
$149$	−2.06193e29	−0.946801	−0.473400	−	0.880847i	$-0.656974\pi$
$149$	−2.06193e29	−0.946801	−0.473400	+	0.880847i	$0.656974\pi$
$150$	0	0
$151$	−8.73857e28	−0.335160	−0.167580	−	0.985859i	$-0.553595\pi$
$151$	−8.73857e28	−0.335160	−0.167580	+	0.985859i	$0.553595\pi$
$152$	0	0
$153$	1.78045e29	0.571737
$154$	0	0
$155$	1.16607e29	0.314229
$156$	0	0
$157$	3.66647e29	0.831004	0.415502	−	0.909592i	$-0.363606\pi$
$157$	3.66647e29	0.831004	0.415502	+	0.909592i	$0.363606\pi$
$158$	0	0
$159$	3.40729e29	0.650950
$160$	0	0
$161$	3.87694e28	0.0625663
$162$	0	0
$163$	9.33235e29	1.27485	0.637424	−	0.770513i	$-0.280000\pi$
$163$	9.33235e29	1.27485	0.637424	+	0.770513i	$0.280000\pi$
$164$	0	0
$165$	−8.69242e28	−0.100718
$166$	0	0
$167$	−3.46815e29	−0.341527	−0.170764	−	0.985312i	$-0.554623\pi$
$167$	−3.46815e29	−0.341527	−0.170764	+	0.985312i	$0.554623\pi$
$168$	0	0
$169$	2.78930e29	0.233897
$170$	0	0
$171$	−6.38077e28	−0.0456484
$172$	0	0
$173$	2.65803e30	1.62532	0.812659	−	0.582740i	$-0.198019\pi$
$173$	2.65803e30	1.62532	0.812659	+	0.582740i	$0.198019\pi$
$174$	0	0
$175$	−7.16278e28	−0.0375032
$176$	0	0
$177$	−1.70634e30	−0.766351
$178$	0	0
$179$	−4.30907e30	−1.66291	−0.831454	−	0.555594i	$-0.812491\pi$
$179$	−4.30907e30	−1.66291	−0.831454	+	0.555594i	$0.812491\pi$
$180$	0	0
$181$	−5.56401e29	−0.184811	−0.0924053	−	0.995721i	$-0.529456\pi$
$181$	−5.56401e29	−0.184811	−0.0924053	+	0.995721i	$0.529456\pi$
$182$	0	0
$183$	1.30197e30	0.372830
$184$	0	0
$185$	2.04904e30	0.506678
$186$	0	0
$187$	3.54253e30	0.757621
$188$	0	0
$189$	4.61602e28	0.00855132
$190$	0	0
$191$	4.19469e30	0.674139	0.337070	−	0.941480i	$-0.390564\pi$
$191$	4.19469e30	0.674139	0.337070	+	0.941480i	$0.390564\pi$
$192$	0	0
$193$	4.85628e28	0.00678078	0.00339039	−	0.999994i	$-0.498921\pi$
$193$	4.85628e28	0.00678078	0.00339039	+	0.999994i	$0.498921\pi$
$194$	0	0
$195$	−2.08488e30	−0.253287
$196$	0	0
$197$	−2.40097e29	−0.0254151	−0.0127075	−	0.999919i	$-0.504045\pi$
$197$	−2.40097e29	−0.0254151	−0.0127075	+	0.999919i	$0.504045\pi$
$198$	0	0
$199$	−7.40632e30	−0.684047	−0.342024	−	0.939691i	$-0.611112\pi$
$199$	−7.40632e30	−0.684047	−0.342024	+	0.939691i	$0.611112\pi$
$200$	0	0
$201$	−2.68851e30	−0.216953
$202$	0	0
$203$	6.54442e29	0.0462035
$204$	0	0
$205$	−7.18125e30	−0.444143
$206$	0	0
$207$	8.65174e30	0.469358
$208$	0	0
$209$	−1.26957e30	−0.0604896
$210$	0	0
$211$	3.11843e31	1.30654	0.653271	−	0.757124i	$-0.273396\pi$
$211$	3.11843e31	1.30654	0.653271	+	0.757124i	$0.273396\pi$
$212$	0	0
$213$	5.07866e30	0.187337
$214$	0	0
$215$	1.11322e30	0.0361961
$216$	0	0
$217$	−1.23207e30	−0.0353531
$218$	0	0
$219$	3.34574e31	0.848196
$220$	0	0
$221$	8.49675e31	1.90527
$222$	0	0
$223$	2.26940e31	0.450605	0.225302	−	0.974289i	$-0.427663\pi$
$223$	2.26940e31	0.450605	0.225302	+	0.974289i	$0.427663\pi$
$224$	0	0
$225$	−1.59844e31	−0.281340
$226$	0	0
$227$	−8.41908e31	−1.31497	−0.657484	−	0.753468i	$-0.728379\pi$
$227$	−8.41908e31	−1.31497	−0.657484	+	0.753468i	$0.728379\pi$
$228$	0	0
$229$	4.09888e31	0.568703	0.284352	−	0.958720i	$-0.408222\pi$
$229$	4.09888e31	0.568703	0.284352	+	0.958720i	$0.408222\pi$
$230$	0	0
$231$	9.18439e29	0.0113315
$232$	0	0
$233$	6.99046e31	0.767717	0.383858	−	0.923392i	$-0.374595\pi$
$233$	6.99046e31	0.767717	0.383858	+	0.923392i	$0.374595\pi$
$234$	0	0
$235$	6.93314e31	0.678444
$236$	0	0
$237$	1.41599e29	0.00123583
$238$	0	0
$239$	1.89074e32	1.47320	0.736598	−	0.676330i	$-0.236431\pi$
$239$	1.89074e32	1.47320	0.736598	+	0.676330i	$0.236431\pi$
$240$	0	0
$241$	1.01767e32	0.708563	0.354281	−	0.935139i	$-0.384725\pi$
$241$	1.01767e32	0.708563	0.354281	+	0.935139i	$0.384725\pi$
$242$	0	0
$243$	1.03011e31	0.0641500
$244$	0	0
$245$	7.06998e31	0.394163
$246$	0	0
$247$	−3.04506e31	−0.152120
$248$	0	0
$249$	1.03695e32	0.464587
$250$	0	0
$251$	2.25120e32	0.905356	0.452678	−	0.891674i	$-0.350469\pi$
$251$	2.25120e32	0.905356	0.452678	+	0.891674i	$0.350469\pi$
$252$	0	0
$253$	1.72142e32	0.621955
$254$	0	0
$255$	−1.20388e32	−0.391103
$256$	0	0
$257$	1.28310e32	0.375113	0.187557	−	0.982254i	$-0.439943\pi$
$257$	1.28310e32	0.375113	0.187557	+	0.982254i	$0.439943\pi$
$258$	0	0
$259$	−2.16501e31	−0.0570050
$260$	0	0
$261$	1.46045e32	0.346608
$262$	0	0
$263$	6.17345e32	1.32168	0.660838	−	0.750528i	$-0.270201\pi$
$263$	6.17345e32	1.32168	0.660838	+	0.750528i	$0.270201\pi$
$264$	0	0
$265$	−2.30389e32	−0.445290
$266$	0	0
$267$	5.39865e32	0.942720
$268$	0	0
$269$	−1.70730e31	−0.0269559	−0.0134779	−	0.999909i	$-0.504290\pi$
$269$	−1.70730e31	−0.0269559	−0.0134779	+	0.999909i	$0.504290\pi$
$270$	0	0
$271$	1.06709e33	1.52446	0.762229	−	0.647307i	$-0.224105\pi$
$271$	1.06709e33	1.52446	0.762229	+	0.647307i	$0.224105\pi$
$272$	0	0
$273$	2.20287e31	0.0284967
$274$	0	0
$275$	−3.18037e32	−0.372809
$276$	0	0
$277$	5.90772e32	0.627976	0.313988	−	0.949427i	$-0.398335\pi$
$277$	5.90772e32	0.627976	0.313988	+	0.949427i	$0.398335\pi$
$278$	0	0
$279$	−2.74946e32	−0.265211
$280$	0	0
$281$	3.49531e32	0.306161	0.153081	−	0.988214i	$-0.451081\pi$
$281$	3.49531e32	0.306161	0.153081	+	0.988214i	$0.451081\pi$
$282$	0	0
$283$	1.04098e32	0.0828565	0.0414282	−	0.999141i	$-0.486809\pi$
$283$	1.04098e32	0.0828565	0.0414282	+	0.999141i	$0.486809\pi$
$284$	0	0
$285$	4.31446e31	0.0312263
$286$	0	0
$287$	7.58768e31	0.0499693
$288$	0	0
$289$	3.23862e33	1.94195
$290$	0	0
$291$	−6.63436e32	−0.362446
$292$	0	0
$293$	−4.57358e32	−0.227794	−0.113897	−	0.993493i	$-0.536333\pi$
$293$	−4.57358e32	−0.227794	−0.113897	+	0.993493i	$0.536333\pi$
$294$	0	0
$295$	1.15377e33	0.524231
$296$	0	0
$297$	2.04958e32	0.0850065
$298$	0	0
$299$	4.12882e33	1.56410
$300$	0	0
$301$	−1.17623e31	−0.00407233
$302$	0	0
$303$	−1.57965e33	−0.500133
$304$	0	0
$305$	−8.80347e32	−0.255039
$306$	0	0
$307$	−2.50586e33	−0.664642	−0.332321	−	0.943166i	$-0.607832\pi$
$307$	−2.50586e33	−0.664642	−0.332321	+	0.943166i	$0.607832\pi$
$308$	0	0
$309$	3.20812e33	0.779491
$310$	0	0
$311$	4.09104e33	0.911104	0.455552	−	0.890209i	$-0.349442\pi$
$311$	4.09104e33	0.911104	0.455552	+	0.890209i	$0.349442\pi$
$312$	0	0
$313$	5.57324e33	1.13831	0.569154	−	0.822231i	$-0.307271\pi$
$313$	5.57324e33	1.13831	0.569154	+	0.822231i	$0.307271\pi$
$314$	0	0
$315$	−3.12119e31	−0.00584962
$316$	0	0
$317$	6.22580e33	1.07126	0.535631	−	0.844452i	$-0.320074\pi$
$317$	6.22580e33	1.07126	0.535631	+	0.844452i	$0.320074\pi$
$318$	0	0
$319$	2.90581e33	0.459297
$320$	0	0
$321$	−3.55884e31	−0.00517000
$322$	0	0
$323$	−1.75832e33	−0.234890
$324$	0	0
$325$	−7.62812e33	−0.937545
$326$	0	0
$327$	8.61443e33	0.974613
$328$	0	0
$329$	−7.32553e32	−0.0763299
$330$	0	0
$331$	6.69049e33	0.642363	0.321181	−	0.947018i	$-0.395920\pi$
$331$	6.69049e33	0.642363	0.321181	+	0.947018i	$0.395920\pi$
$332$	0	0
$333$	−4.83141e33	−0.427638
$334$	0	0
$335$	1.81788e33	0.148409
$336$	0	0
$337$	−8.45330e33	−0.636829	−0.318414	−	0.947952i	$-0.603150\pi$
$337$	−8.45330e33	−0.636829	−0.318414	+	0.947952i	$0.603150\pi$
$338$	0	0
$339$	−5.92152e33	−0.411848
$340$	0	0
$341$	−5.47054e33	−0.351436
$342$	0	0
$343$	−1.49550e33	−0.0887802
$344$	0	0
$345$	−5.85001e33	−0.321069
$346$	0	0
$347$	−3.54455e34	−1.79934	−0.899671	−	0.436568i	$-0.856194\pi$
$347$	−3.54455e34	−1.79934	−0.899671	+	0.436568i	$0.856194\pi$
$348$	0	0
$349$	1.62742e34	0.764462	0.382231	−	0.924067i	$-0.375156\pi$
$349$	1.62742e34	0.764462	0.382231	+	0.924067i	$0.375156\pi$
$350$	0	0
$351$	4.91591e33	0.213775
$352$	0	0
$353$	−6.68547e33	−0.269261	−0.134630	−	0.990896i	$-0.542985\pi$
$353$	−6.68547e33	−0.269261	−0.134630	+	0.990896i	$0.542985\pi$
$354$	0	0
$355$	−3.43402e33	−0.128150
$356$	0	0
$357$	1.27202e33	0.0440020
$358$	0	0
$359$	7.80470e33	0.250369	0.125185	−	0.992133i	$-0.460048\pi$
$359$	7.80470e33	0.250369	0.125185	+	0.992133i	$0.460048\pi$
$360$	0	0
$361$	−3.29705e34	−0.981246
$362$	0	0
$363$	−1.68236e34	−0.464706
$364$	0	0
$365$	−2.26227e34	−0.580218
$366$	0	0
$367$	−6.61615e34	−1.57621	−0.788104	−	0.615542i	$-0.788937\pi$
$367$	−6.61615e34	−1.57621	−0.788104	+	0.615542i	$0.788937\pi$
$368$	0	0
$369$	1.69326e34	0.374858
$370$	0	0
$371$	2.43428e33	0.0500984
$372$	0	0
$373$	−6.79473e34	−1.30048	−0.650239	−	0.759729i	$-0.725331\pi$
$373$	−6.79473e34	−1.30048	−0.650239	+	0.759729i	$0.725331\pi$
$374$	0	0
$375$	2.36136e34	0.420474
$376$	0	0
$377$	6.96959e34	1.15505
$378$	0	0
$379$	−7.98595e34	−1.23224	−0.616122	−	0.787650i	$-0.711297\pi$
$379$	−7.98595e34	−1.23224	−0.616122	+	0.787650i	$0.711297\pi$
$380$	0	0
$381$	−3.41065e34	−0.490174
$382$	0	0
$383$	−1.19033e35	−1.59399	−0.796994	−	0.603987i	$-0.793578\pi$
$383$	−1.19033e35	−1.59399	−0.796994	+	0.603987i	$0.793578\pi$
$384$	0	0
$385$	−6.21016e32	−0.00775146
$386$	0	0
$387$	−2.62485e33	−0.0305497
$388$	0	0
$389$	1.48935e34	0.161688	0.0808441	−	0.996727i	$-0.474238\pi$
$389$	1.48935e34	0.161688	0.0808441	+	0.996727i	$0.474238\pi$
$390$	0	0
$391$	2.38413e35	2.41514
$392$	0	0
$393$	−8.20175e34	−0.775545
$394$	0	0
$395$	−9.57447e31	−0.000845384 0
$396$	0	0
$397$	4.07374e34	0.335986	0.167993	−	0.985788i	$-0.446271\pi$
$397$	4.07374e34	0.335986	0.167993	+	0.985788i	$0.446271\pi$
$398$	0	0
$399$	−4.55864e32	−0.00351319
$400$	0	0
$401$	−1.64933e35	−1.18812	−0.594058	−	0.804422i	$-0.702475\pi$
$401$	−1.64933e35	−1.18812	−0.594058	+	0.804422i	$0.702475\pi$
$402$	0	0
$403$	−1.31211e35	−0.883796
$404$	0	0
$405$	−6.96521e33	−0.0438825
$406$	0	0
$407$	−9.61295e34	−0.566672
$408$	0	0
$409$	−2.13655e35	−1.17882	−0.589411	−	0.807833i	$-0.700640\pi$
$409$	−2.13655e35	−1.17882	−0.589411	+	0.807833i	$0.700640\pi$
$410$	0	0
$411$	5.93036e34	0.306349
$412$	0	0
$413$	−1.21907e34	−0.0589798
$414$	0	0
$415$	−7.01151e34	−0.317806
$416$	0	0
$417$	1.60276e35	0.680817
$418$	0	0
$419$	−6.25908e34	−0.249242	−0.124621	−	0.992204i	$-0.539772\pi$
$419$	−6.25908e34	−0.249242	−0.124621	+	0.992204i	$0.539772\pi$
$420$	0	0
$421$	1.62873e35	0.608191	0.304096	−	0.952641i	$-0.401646\pi$
$421$	1.62873e35	0.608191	0.304096	+	0.952641i	$0.401646\pi$
$422$	0	0
$423$	−1.63476e35	−0.572609
$424$	0	0
$425$	−4.40475e35	−1.44767
$426$	0	0
$427$	9.30171e33	0.0286937
$428$	0	0
$429$	9.78107e34	0.283278
$430$	0	0
$431$	6.21025e35	1.68914	0.844571	−	0.535444i	$-0.179856\pi$
$431$	6.21025e35	1.68914	0.844571	+	0.535444i	$0.179856\pi$
$432$	0	0
$433$	1.04792e35	0.267757	0.133878	−	0.990998i	$-0.457257\pi$
$433$	1.04792e35	0.267757	0.133878	+	0.990998i	$0.457257\pi$
$434$	0	0
$435$	−9.87502e34	−0.237101
$436$	0	0
$437$	−8.54420e34	−0.192829
$438$	0	0
$439$	−6.93788e35	−1.47216	−0.736081	−	0.676893i	$-0.763326\pi$
$439$	−6.93788e35	−1.47216	−0.736081	+	0.676893i	$0.763326\pi$
$440$	0	0
$441$	−1.66702e35	−0.332675
$442$	0	0
$443$	−5.15844e35	−0.968430	−0.484215	−	0.874949i	$-0.660895\pi$
$443$	−5.15844e35	−0.968430	−0.484215	+	0.874949i	$0.660895\pi$
$444$	0	0
$445$	−3.65038e35	−0.644878
$446$	0	0
$447$	−3.28738e35	−0.546636
$448$	0	0
$449$	9.16926e35	1.43552	0.717759	−	0.696292i	$-0.245168\pi$
$449$	9.16926e35	1.43552	0.717759	+	0.696292i	$0.245168\pi$
$450$	0	0
$451$	3.36904e35	0.496732
$452$	0	0
$453$	−1.39321e35	−0.193504
$454$	0	0
$455$	−1.48951e34	−0.0194934
$456$	0	0
$457$	1.55254e36	1.91502	0.957510	−	0.288400i	$-0.0931232\pi$
$457$	1.55254e36	1.91502	0.957510	+	0.288400i	$0.0931232\pi$
$458$	0	0
$459$	2.83862e35	0.330093
$460$	0	0
$461$	−5.06840e35	−0.555787	−0.277893	−	0.960612i	$-0.589636\pi$
$461$	−5.06840e35	−0.555787	−0.277893	+	0.960612i	$0.589636\pi$
$462$	0	0
$463$	1.45476e36	1.50469	0.752347	−	0.658767i	$-0.228922\pi$
$463$	1.45476e36	1.50469	0.752347	+	0.658767i	$0.228922\pi$
$464$	0	0
$465$	1.85909e35	0.181420
$466$	0	0
$467$	6.89399e35	0.634881	0.317441	−	0.948278i	$-0.397177\pi$
$467$	6.89399e35	0.634881	0.317441	+	0.948278i	$0.397177\pi$
$468$	0	0
$469$	−1.92076e34	−0.0166971
$470$	0	0
$471$	5.84554e35	0.479780
$472$	0	0
$473$	−5.22261e34	−0.0404820
$474$	0	0
$475$	1.57857e35	0.115585
$476$	0	0
$477$	5.43232e35	0.375826
$478$	0	0
$479$	6.34595e35	0.414923	0.207461	−	0.978243i	$-0.433480\pi$
$479$	6.34595e35	0.414923	0.207461	+	0.978243i	$0.433480\pi$
$480$	0	0
$481$	−2.30566e36	−1.42507
$482$	0	0
$483$	6.18110e34	0.0361226
$484$	0	0
$485$	4.48592e35	0.247935
$486$	0	0
$487$	−2.48287e36	−1.29812	−0.649058	−	0.760739i	$-0.724837\pi$
$487$	−2.48287e36	−1.29812	−0.649058	+	0.760739i	$0.724837\pi$
$488$	0	0
$489$	1.48788e36	0.736034
$490$	0	0
$491$	2.37415e36	1.11150	0.555748	−	0.831351i	$-0.312432\pi$
$491$	2.37415e36	1.11150	0.555748	+	0.831351i	$0.312432\pi$
$492$	0	0
$493$	4.02449e36	1.78352
$494$	0	0
$495$	−1.38585e35	−0.0581496
$496$	0	0
$497$	3.62837e34	0.0144178
$498$	0	0
$499$	5.00732e36	1.88472	0.942362	−	0.334595i	$-0.108600\pi$
$499$	5.00732e36	1.88472	0.942362	+	0.334595i	$0.108600\pi$
$500$	0	0
$501$	−5.52935e35	−0.197181
$502$	0	0
$503$	1.61560e36	0.545966	0.272983	−	0.962019i	$-0.411990\pi$
$503$	1.61560e36	0.545966	0.272983	+	0.962019i	$0.411990\pi$
$504$	0	0
$505$	1.06811e36	0.342122
$506$	0	0
$507$	4.44705e35	0.135041
$508$	0	0
$509$	1.80638e36	0.520139	0.260069	−	0.965590i	$-0.416255\pi$
$509$	1.80638e36	0.520139	0.260069	+	0.965590i	$0.416255\pi$
$510$	0	0
$511$	2.39031e35	0.0652787
$512$	0	0
$513$	−1.01730e35	−0.0263551
$514$	0	0
$515$	−2.16922e36	−0.533219
$516$	0	0
$517$	−3.25264e36	−0.758776
$518$	0	0
$519$	4.23776e36	0.938377
$520$	0	0
$521$	4.59028e36	0.965006	0.482503	−	0.875894i	$-0.339728\pi$
$521$	4.59028e36	0.965006	0.482503	+	0.875894i	$0.339728\pi$
$522$	0	0
$523$	−5.82915e35	−0.116368	−0.0581838	−	0.998306i	$-0.518531\pi$
$523$	−5.82915e35	−0.116368	−0.0581838	+	0.998306i	$0.518531\pi$
$524$	0	0
$525$	−1.14198e35	−0.0216525
$526$	0	0
$527$	−7.57658e36	−1.36468
$528$	0	0
$529$	5.74195e36	0.982670
$530$	0	0
$531$	−2.72046e36	−0.442453
$532$	0	0
$533$	8.08063e36	1.24919
$534$	0	0
$535$	2.40636e34	0.00353659
$536$	0	0
$537$	−6.87006e36	−0.960080
$538$	0	0
$539$	−3.31684e36	−0.440835
$540$	0	0
$541$	−6.32438e36	−0.799567	−0.399783	−	0.916610i	$-0.630915\pi$
$541$	−6.32438e36	−0.799567	−0.399783	+	0.916610i	$0.630915\pi$
$542$	0	0
$543$	−8.87082e35	−0.106700
$544$	0	0
$545$	−5.82478e36	−0.666695
$546$	0	0
$547$	9.05384e36	0.986289	0.493144	−	0.869948i	$-0.335847\pi$
$547$	9.05384e36	0.986289	0.493144	+	0.869948i	$0.335847\pi$
$548$	0	0
$549$	2.07576e36	0.215254
$550$	0	0
$551$	−1.44229e36	−0.142399
$552$	0	0
$553$	1.01164e33	9.51118e−5 0
$554$	0	0
$555$	3.26683e36	0.292531
$556$	0	0
$557$	−1.74285e37	−1.48668	−0.743338	−	0.668917i	$-0.766758\pi$
$557$	−1.74285e37	−1.48668	−0.743338	+	0.668917i	$0.766758\pi$
$558$	0	0
$559$	−1.25264e36	−0.101805
$560$	0	0
$561$	5.64794e36	0.437413
$562$	0	0
$563$	7.81650e36	0.576964	0.288482	−	0.957485i	$-0.406849\pi$
$563$	7.81650e36	0.576964	0.288482	+	0.957485i	$0.406849\pi$
$564$	0	0
$565$	4.00393e36	0.281729
$566$	0	0
$567$	7.35942e34	0.00493711
$568$	0	0
$569$	−1.42845e37	−0.913797	−0.456899	−	0.889519i	$-0.651040\pi$
$569$	−1.42845e37	−0.913797	−0.456899	+	0.889519i	$0.651040\pi$
$570$	0	0
$571$	−2.26975e37	−1.38481	−0.692406	−	0.721508i	$-0.743449\pi$
$571$	−2.26975e37	−1.38481	−0.692406	+	0.721508i	$0.743449\pi$
$572$	0	0
$573$	6.68769e36	0.389214
$574$	0	0
$575$	−2.14040e37	−1.18844
$576$	0	0
$577$	−1.49672e37	−0.792987	−0.396494	−	0.918038i	$-0.629773\pi$
$577$	−1.49672e37	−0.792987	−0.396494	+	0.918038i	$0.629773\pi$
$578$	0	0
$579$	7.74247e34	0.00391489
$580$	0	0
$581$	7.40834e35	0.0357555
$582$	0	0
$583$	1.08086e37	0.498015
$584$	0	0
$585$	−3.32397e36	−0.146235
$586$	0	0
$587$	7.65028e36	0.321413	0.160706	−	0.987002i	$-0.448623\pi$
$587$	7.65028e36	0.321413	0.160706	+	0.987002i	$0.448623\pi$
$588$	0	0
$589$	2.71529e36	0.108958
$590$	0	0
$591$	−3.82791e35	−0.0146734
$592$	0	0
$593$	−1.64866e37	−0.603799	−0.301900	−	0.953340i	$-0.597621\pi$
$593$	−1.64866e37	−0.603799	−0.301900	+	0.953340i	$0.597621\pi$
$594$	0	0
$595$	−8.60094e35	−0.0301000
$596$	0	0
$597$	−1.18081e37	−0.394935
$598$	0	0
$599$	5.74500e36	0.183665	0.0918327	−	0.995774i	$-0.470727\pi$
$599$	5.74500e36	0.183665	0.0918327	+	0.995774i	$0.470727\pi$
$600$	0	0
$601$	−5.09841e36	−0.155822	−0.0779112	−	0.996960i	$-0.524825\pi$
$601$	−5.09841e36	−0.155822	−0.0779112	+	0.996960i	$0.524825\pi$
$602$	0	0
$603$	−4.28635e36	−0.125258
$604$	0	0
$605$	1.13755e37	0.317888
$606$	0	0
$607$	−1.21769e37	−0.325453	−0.162727	−	0.986671i	$-0.552029\pi$
$607$	−1.21769e37	−0.325453	−0.162727	+	0.986671i	$0.552029\pi$
$608$	0	0
$609$	1.04339e36	0.0266756
$610$	0	0
$611$	−7.80145e37	−1.90818
$612$	0	0
$613$	2.55443e37	0.597829	0.298915	−	0.954280i	$-0.403375\pi$
$613$	2.55443e37	0.597829	0.298915	+	0.954280i	$0.403375\pi$
$614$	0	0
$615$	−1.14492e37	−0.256426
$616$	0	0
$617$	1.30077e37	0.278838	0.139419	−	0.990234i	$-0.455477\pi$
$617$	1.30077e37	0.278838	0.139419	+	0.990234i	$0.455477\pi$
$618$	0	0
$619$	2.04833e37	0.420316	0.210158	−	0.977667i	$-0.432602\pi$
$619$	2.04833e37	0.420316	0.210158	+	0.977667i	$0.432602\pi$
$620$	0	0
$621$	1.37937e37	0.270984
$622$	0	0
$623$	3.85698e36	0.0725535
$624$	0	0
$625$	3.08859e37	0.556390
$626$	0	0
$627$	−2.02410e36	−0.0349237
$628$	0	0
$629$	−1.33137e38	−2.20047
$630$	0	0
$631$	−7.16988e37	−1.13531	−0.567656	−	0.823266i	$-0.692150\pi$
$631$	−7.16988e37	−1.13531	−0.567656	+	0.823266i	$0.692150\pi$
$632$	0	0
$633$	4.97179e37	0.754332
$634$	0	0
$635$	2.30617e37	0.335309
$636$	0	0
$637$	−7.95543e37	−1.10862
$638$	0	0
$639$	8.09703e36	0.108159
$640$	0	0
$641$	−1.00079e38	−1.28161	−0.640807	−	0.767702i	$-0.721400\pi$
$641$	−1.00079e38	−1.28161	−0.640807	+	0.767702i	$0.721400\pi$
$642$	0	0
$643$	−7.58172e37	−0.930936	−0.465468	−	0.885065i	$-0.654114\pi$
$643$	−7.58172e37	−0.930936	−0.465468	+	0.885065i	$0.654114\pi$
$644$	0	0
$645$	1.77483e36	0.0208978
$646$	0	0
$647$	7.13094e37	0.805265	0.402632	−	0.915362i	$-0.368095\pi$
$647$	7.13094e37	0.805265	0.402632	+	0.915362i	$0.368095\pi$
$648$	0	0
$649$	−5.41285e37	−0.586303
$650$	0	0
$651$	−1.96431e36	−0.0204111
$652$	0	0
$653$	2.47590e37	0.246835	0.123417	−	0.992355i	$-0.460615\pi$
$653$	2.47590e37	0.246835	0.123417	+	0.992355i	$0.460615\pi$
$654$	0	0
$655$	5.54574e37	0.530520
$656$	0	0
$657$	5.33418e37	0.489706
$658$	0	0
$659$	−1.51836e38	−1.33789	−0.668946	−	0.743311i	$-0.733254\pi$
$659$	−1.51836e38	−1.33789	−0.668946	+	0.743311i	$0.733254\pi$
$660$	0	0
$661$	1.92148e38	1.62523	0.812617	−	0.582798i	$-0.198042\pi$
$661$	1.92148e38	1.62523	0.812617	+	0.582798i	$0.198042\pi$
$662$	0	0
$663$	1.35466e38	1.10001
$664$	0	0
$665$	3.08239e35	0.00240323
$666$	0	0
$667$	1.95562e38	1.46415
$668$	0	0
$669$	3.61816e37	0.260157
$670$	0	0
$671$	4.13009e37	0.285237
$672$	0	0
$673$	−9.37859e37	−0.622206	−0.311103	−	0.950376i	$-0.600698\pi$
$673$	−9.37859e37	−0.622206	−0.311103	+	0.950376i	$0.600698\pi$
$674$	0	0
$675$	−2.54843e37	−0.162432
$676$	0	0
$677$	−2.93781e37	−0.179919	−0.0899595	−	0.995945i	$-0.528674\pi$
$677$	−2.93781e37	−0.179919	−0.0899595	+	0.995945i	$0.528674\pi$
$678$	0	0
$679$	−4.73981e36	−0.0278945
$680$	0	0
$681$	−1.34227e38	−0.759197
$682$	0	0
$683$	−1.68599e38	−0.916591	−0.458295	−	0.888800i	$-0.651540\pi$
$683$	−1.68599e38	−0.916591	−0.458295	+	0.888800i	$0.651540\pi$
$684$	0	0
$685$	−4.00991e37	−0.209562
$686$	0	0
$687$	6.53493e37	0.328341
$688$	0	0
$689$	2.59243e38	1.25241
$690$	0	0
$691$	3.72820e38	1.73199	0.865996	−	0.500051i	$-0.166686\pi$
$691$	3.72820e38	1.73199	0.865996	+	0.500051i	$0.166686\pi$
$692$	0	0
$693$	1.46429e36	0.00654226
$694$	0	0
$695$	−1.08373e38	−0.465720
$696$	0	0
$697$	4.66604e38	1.92888
$698$	0	0
$699$	1.11451e38	0.443241
$700$	0	0
$701$	3.38534e38	1.29541	0.647707	−	0.761890i	$-0.275728\pi$
$701$	3.38534e38	1.29541	0.647707	+	0.761890i	$0.275728\pi$
$702$	0	0
$703$	4.77135e37	0.175689
$704$	0	0
$705$	1.10537e38	0.391700
$706$	0	0
$707$	−1.12856e37	−0.0384912
$708$	0	0
$709$	2.34891e38	0.771155	0.385578	−	0.922675i	$-0.374002\pi$
$709$	2.34891e38	0.771155	0.385578	+	0.922675i	$0.374002\pi$
$710$	0	0
$711$	2.25755e35	0.000713507 0
$712$	0	0
$713$	−3.68168e38	−1.12031
$714$	0	0
$715$	−6.61362e37	−0.193779
$716$	0	0
$717$	3.01445e38	0.850551
$718$	0	0
$719$	1.57531e38	0.428082	0.214041	−	0.976825i	$-0.431337\pi$
$719$	1.57531e38	0.428082	0.214041	+	0.976825i	$0.431337\pi$
$720$	0	0
$721$	2.29199e37	0.0599911
$722$	0	0
$723$	1.62250e38	0.409089
$724$	0	0
$725$	−3.61306e38	−0.877632
$726$	0	0
$727$	2.38862e38	0.559027	0.279513	−	0.960142i	$-0.409827\pi$
$727$	2.38862e38	0.559027	0.279513	+	0.960142i	$0.409827\pi$
$728$	0	0
$729$	1.64232e37	0.0370370
$730$	0	0
$731$	−7.23320e37	−0.157198
$732$	0	0
$733$	−1.39015e38	−0.291177	−0.145589	−	0.989345i	$-0.546508\pi$
$733$	−1.39015e38	−0.291177	−0.145589	+	0.989345i	$0.546508\pi$
$734$	0	0
$735$	1.12718e38	0.227570
$736$	0	0
$737$	−8.52846e37	−0.165981
$738$	0	0
$739$	−3.01753e37	−0.0566176	−0.0283088	−	0.999599i	$-0.509012\pi$
$739$	−3.01753e37	−0.0566176	−0.0283088	+	0.999599i	$0.509012\pi$
$740$	0	0
$741$	−4.85480e37	−0.0878265
$742$	0	0
$743$	−8.48049e38	−1.47935	−0.739677	−	0.672962i	$-0.765022\pi$
$743$	−8.48049e38	−1.47935	−0.739677	+	0.672962i	$0.765022\pi$
$744$	0	0
$745$	2.22281e38	0.373932
$746$	0	0
$747$	1.65324e38	0.268230
$748$	0	0
$749$	−2.54256e35	−0.000397893 0
$750$	0	0
$751$	1.00379e39	1.51532	0.757658	−	0.652651i	$-0.226343\pi$
$751$	1.00379e39	1.51532	0.757658	+	0.652651i	$0.226343\pi$
$752$	0	0
$753$	3.58914e38	0.522707
$754$	0	0
$755$	9.42041e37	0.132369
$756$	0	0
$757$	1.01436e39	1.37530	0.687652	−	0.726040i	$-0.258641\pi$
$757$	1.01436e39	1.37530	0.687652	+	0.726040i	$0.258641\pi$
$758$	0	0
$759$	2.74450e38	0.359086
$760$	0	0
$761$	−1.18512e39	−1.49648	−0.748240	−	0.663428i	$-0.769101\pi$
$761$	−1.18512e39	−1.49648	−0.748240	+	0.663428i	$0.769101\pi$
$762$	0	0
$763$	6.15445e37	0.0750080
$764$	0	0
$765$	−1.91938e38	−0.225804
$766$	0	0
$767$	−1.29827e39	−1.47444
$768$	0	0
$769$	−1.38650e39	−1.52024	−0.760122	−	0.649780i	$-0.774861\pi$
$769$	−1.38650e39	−1.52024	−0.760122	+	0.649780i	$0.774861\pi$
$770$	0	0
$771$	2.04567e38	0.216572
$772$	0	0
$773$	−3.84501e38	−0.393074	−0.196537	−	0.980496i	$-0.562970\pi$
$773$	−3.84501e38	−0.393074	−0.196537	+	0.980496i	$0.562970\pi$
$774$	0	0
$775$	6.80202e38	0.671530
$776$	0	0
$777$	−3.45172e37	−0.0329118
$778$	0	0
$779$	−1.67221e38	−0.154005
$780$	0	0
$781$	1.61105e38	0.143324
$782$	0	0
$783$	2.32842e38	0.200114
$784$	0	0
$785$	−3.95255e38	−0.328199
$786$	0	0
$787$	−8.31177e38	−0.666860	−0.333430	−	0.942775i	$-0.608206\pi$
$787$	−8.31177e38	−0.666860	−0.333430	+	0.942775i	$0.608206\pi$
$788$	0	0
$789$	9.84247e38	0.763070
$790$	0	0
$791$	−4.23054e37	−0.0316966
$792$	0	0
$793$	9.90602e38	0.717317
$794$	0	0
$795$	−3.67315e38	−0.257088
$796$	0	0
$797$	−5.86889e38	−0.397071	−0.198536	−	0.980094i	$-0.563619\pi$
$797$	−5.86889e38	−0.397071	−0.198536	+	0.980094i	$0.563619\pi$
$798$	0	0
$799$	−4.50484e39	−2.94644
$800$	0	0
$801$	8.60719e38	0.544280
$802$	0	0
$803$	1.06133e39	0.648919
$804$	0	0
$805$	−4.17945e37	−0.0247101
$806$	0	0
$807$	−2.72198e37	−0.0155630
$808$	0	0
$809$	−2.02827e39	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$809$	−2.02827e39	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$810$	0	0
$811$	1.61063e39	0.861421	0.430710	−	0.902490i	$-0.358263\pi$
$811$	1.61063e39	0.861421	0.430710	+	0.902490i	$0.358263\pi$
$812$	0	0
$813$	1.70129e39	0.880147
$814$	0	0
$815$	−1.00605e39	−0.503492
$816$	0	0
$817$	2.59222e37	0.0125509
$818$	0	0
$819$	3.51209e37	0.0164526
$820$	0	0
$821$	−2.30031e39	−1.04268	−0.521342	−	0.853348i	$-0.674569\pi$
$821$	−2.30031e39	−1.04268	−0.521342	+	0.853348i	$0.674569\pi$
$822$	0	0
$823$	2.08776e39	0.915764	0.457882	−	0.889013i	$-0.348608\pi$
$823$	2.08776e39	0.915764	0.457882	+	0.889013i	$0.348608\pi$
$824$	0	0
$825$	−5.07054e38	−0.215242
$826$	0	0
$827$	−3.43290e39	−1.41038	−0.705192	−	0.709017i	$-0.749139\pi$
$827$	−3.43290e39	−1.41038	−0.705192	+	0.709017i	$0.749139\pi$
$828$	0	0
$829$	8.03456e38	0.319504	0.159752	−	0.987157i	$-0.448931\pi$
$829$	8.03456e38	0.319504	0.159752	+	0.987157i	$0.448931\pi$
$830$	0	0
$831$	9.41881e38	0.362562
$832$	0	0
$833$	−4.59375e39	−1.71183
$834$	0	0
$835$	3.73875e38	0.134884
$836$	0	0
$837$	−4.38353e38	−0.153119
$838$	0	0
$839$	−2.54669e39	−0.861369	−0.430685	−	0.902502i	$-0.641728\pi$
$839$	−2.54669e39	−0.861369	−0.430685	+	0.902502i	$0.641728\pi$
$840$	0	0
$841$	2.48015e38	0.0812329
$842$	0	0
$843$	5.57266e38	0.176762
$844$	0	0
$845$	−3.00694e38	−0.0923760
$846$	0	0
$847$	−1.20193e38	−0.0357647
$848$	0	0
$849$	1.65966e38	0.0478372
$850$	0	0
$851$	−6.46952e39	−1.80644
$852$	0	0
$853$	1.61849e39	0.437823	0.218912	−	0.975745i	$-0.429749\pi$
$853$	1.61849e39	0.437823	0.218912	+	0.975745i	$0.429749\pi$
$854$	0	0
$855$	6.87864e37	0.0180285
$856$	0	0
$857$	1.79678e38	0.0456304	0.0228152	−	0.999740i	$-0.492737\pi$
$857$	1.79678e38	0.0456304	0.0228152	+	0.999740i	$0.492737\pi$
$858$	0	0
$859$	6.83644e39	1.68237	0.841187	−	0.540744i	$-0.181857\pi$
$859$	6.83644e39	1.68237	0.841187	+	0.540744i	$0.181857\pi$
$860$	0	0
$861$	1.20972e38	0.0288498
$862$	0	0
$863$	8.93182e37	0.0206440	0.0103220	−	0.999947i	$-0.496714\pi$
$863$	8.93182e37	0.0206440	0.0103220	+	0.999947i	$0.496714\pi$
$864$	0	0
$865$	−2.86543e39	−0.641907
$866$	0	0
$867$	5.16340e39	1.12119
$868$	0	0
$869$	4.49180e36	0.000945483 0
$870$	0	0
$871$	−2.04555e39	−0.417412
$872$	0	0
$873$	−1.05773e39	−0.209258
$874$	0	0
$875$	1.68703e38	0.0323605
$876$	0	0
$877$	−4.74244e39	−0.882078	−0.441039	−	0.897488i	$-0.645390\pi$
$877$	−4.74244e39	−0.882078	−0.441039	+	0.897488i	$0.645390\pi$
$878$	0	0
$879$	−7.29176e38	−0.131517
$880$	0	0
$881$	−4.04555e39	−0.707624	−0.353812	−	0.935317i	$-0.615115\pi$
$881$	−4.04555e39	−0.707624	−0.353812	+	0.935317i	$0.615115\pi$
$882$	0	0
$883$	−7.15911e39	−1.21448	−0.607239	−	0.794519i	$-0.707723\pi$
$883$	−7.15911e39	−1.21448	−0.607239	+	0.794519i	$0.707723\pi$
$884$	0	0
$885$	1.83948e39	0.302665
$886$	0	0
$887$	3.43934e39	0.548916	0.274458	−	0.961599i	$-0.411502\pi$
$887$	3.43934e39	0.548916	0.274458	+	0.961599i	$0.411502\pi$
$888$	0	0
$889$	−2.43669e38	−0.0377247
$890$	0	0
$891$	3.26769e38	0.0490785
$892$	0	0
$893$	1.61444e39	0.235248
$894$	0	0
$895$	4.64529e39	0.656753
$896$	0	0
$897$	6.58267e39	0.903034
$898$	0	0
$899$	−6.21481e39	−0.827317
$900$	0	0
$901$	1.49696e40	1.93387
$902$	0	0
$903$	−1.87528e37	−0.00235116
$904$	0	0
$905$	5.99814e38	0.0729896
$906$	0	0
$907$	−2.71200e38	−0.0320325	−0.0160163	−	0.999872i	$-0.505098\pi$
$907$	−2.71200e38	−0.0320325	−0.0160163	+	0.999872i	$0.505098\pi$
$908$	0	0
$909$	−2.51848e39	−0.288752
$910$	0	0
$911$	−1.18591e40	−1.31994	−0.659970	−	0.751292i	$-0.729431\pi$
$911$	−1.18591e40	−1.31994	−0.659970	+	0.751292i	$0.729431\pi$
$912$	0	0
$913$	3.28941e39	0.355436
$914$	0	0
$915$	−1.40356e39	−0.147247
$916$	0	0
$917$	−5.85961e38	−0.0596874
$918$	0	0
$919$	1.70613e40	1.68753	0.843766	−	0.536711i	$-0.180333\pi$
$919$	1.70613e40	1.68753	0.843766	+	0.536711i	$0.180333\pi$
$920$	0	0
$921$	−3.99514e39	−0.383731
$922$	0	0
$923$	3.86409e39	0.360433
$924$	0	0
$925$	1.19526e40	1.08281
$926$	0	0
$927$	5.11478e39	0.450039
$928$	0	0
$929$	−8.89239e39	−0.759987	−0.379994	−	0.924989i	$-0.624074\pi$
$929$	−8.89239e39	−0.759987	−0.379994	+	0.924989i	$0.624074\pi$
$930$	0	0
$931$	1.64630e39	0.136675
$932$	0	0
$933$	6.52243e39	0.526026
$934$	0	0
$935$	−3.81894e39	−0.299217
$936$	0	0
$937$	−1.80674e40	−1.37534	−0.687670	−	0.726023i	$-0.741367\pi$
$937$	−1.80674e40	−1.37534	−0.687670	+	0.726023i	$0.741367\pi$
$938$	0	0
$939$	8.88555e39	0.657202
$940$	0	0
$941$	2.62451e40	1.88620	0.943101	−	0.332507i	$-0.107895\pi$
$941$	2.62451e40	1.88620	0.943101	+	0.332507i	$0.107895\pi$
$942$	0	0
$943$	2.26737e40	1.58348
$944$	0	0
$945$	−4.97619e37	−0.00337728
$946$	0	0
$947$	6.71744e39	0.443077	0.221539	−	0.975152i	$-0.428892\pi$
$947$	6.71744e39	0.443077	0.221539	+	0.975152i	$0.428892\pi$
$948$	0	0
$949$	2.54560e40	1.63191
$950$	0	0
$951$	9.92594e39	0.618493
$952$	0	0
$953$	1.95322e39	0.118303	0.0591516	−	0.998249i	$-0.481160\pi$
$953$	1.95322e39	0.118303	0.0591516	+	0.998249i	$0.481160\pi$
$954$	0	0
$955$	−4.52199e39	−0.266246
$956$	0	0
$957$	4.63281e39	0.265175
$958$	0	0
$959$	4.23686e38	0.0235772
$960$	0	0
$961$	−6.78259e39	−0.366969
$962$	0	0
$963$	−5.67394e37	−0.00298490
$964$	0	0
$965$	−5.23519e37	−0.00267802
$966$	0	0
$967$	−1.48699e40	−0.739690	−0.369845	−	0.929093i	$-0.620589\pi$
$967$	−1.48699e40	−0.739690	−0.369845	+	0.929093i	$0.620589\pi$
$968$	0	0
$969$	−2.80333e39	−0.135614
$970$	0	0
$971$	1.17711e40	0.553804	0.276902	−	0.960898i	$-0.410692\pi$
$971$	1.17711e40	0.553804	0.276902	+	0.960898i	$0.410692\pi$
$972$	0	0
$973$	1.14506e39	0.0523970
$974$	0	0
$975$	−1.21617e40	−0.541292
$976$	0	0
$977$	−3.52154e40	−1.52460	−0.762300	−	0.647224i	$-0.775930\pi$
$977$	−3.52154e40	−1.52460	−0.762300	+	0.647224i	$0.775930\pi$
$978$	0	0
$979$	1.71255e40	0.721236
$980$	0	0
$981$	1.37342e40	0.562693
$982$	0	0
$983$	3.23914e40	1.29109	0.645547	−	0.763721i	$-0.276629\pi$
$983$	3.23914e40	1.29109	0.645547	+	0.763721i	$0.276629\pi$
$984$	0	0
$985$	2.58830e38	0.0100375
$986$	0	0
$987$	−1.16793e39	−0.0440691
$988$	0	0
$989$	−3.51482e39	−0.129049
$990$	0	0
$991$	−5.97175e38	−0.0213357	−0.0106679	−	0.999943i	$-0.503396\pi$
$991$	−5.97175e38	−0.0213357	−0.0106679	+	0.999943i	$0.503396\pi$
$992$	0	0
$993$	1.06668e40	0.370868
$994$	0	0
$995$	7.98421e39	0.270160
$996$	0	0
$997$	−1.51823e40	−0.499980	−0.249990	−	0.968248i	$-0.580427\pi$
$997$	−1.51823e40	−0.499980	−0.249990	+	0.968248i	$0.580427\pi$
$998$	0	0
$999$	−7.70283e39	−0.246897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.28.a.e.1.1		2
4.3	odd	2		3.28.a.b.1.1	✓	2
12.11	even	2		9.28.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.28.a.b.1.1	✓	2	4.3	odd	2
9.28.a.b.1.2		2	12.11	even	2
48.28.a.e.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59432e6 q^{3} -1.07803e9 q^{5} +1.13904e10 q^{7} +2.54187e12 q^{9} +O(q^{10})\)	\(q+1.59432e6 q^{3} -1.07803e9 q^{5} +1.13904e10 q^{7} +2.54187e12 q^{9} +5.05749e13 q^{11} +1.21304e15 q^{13} -1.71872e15 q^{15} +7.00452e16 q^{17} -2.51027e16 q^{19} +1.81600e16 q^{21} +3.40370e18 q^{23} -6.28844e18 q^{25} +4.05256e18 q^{27} +5.74556e19 q^{29} -1.08167e20 q^{31} +8.06328e19 q^{33} -1.22791e19 q^{35} -1.90073e21 q^{37} +1.93398e21 q^{39} +6.66148e21 q^{41} -1.03265e21 q^{43} -2.74020e21 q^{45} -6.43133e22 q^{47} -6.55826e22 q^{49} +1.11675e23 q^{51} +2.13714e23 q^{53} -5.45211e22 q^{55} -4.00218e22 q^{57} -1.07026e24 q^{59} +8.16628e23 q^{61} +2.89528e22 q^{63} -1.30769e24 q^{65} -1.68630e24 q^{67} +5.42659e24 q^{69} +3.18547e24 q^{71} +2.09853e25 q^{73} -1.00258e25 q^{75} +5.76068e23 q^{77} +8.88148e22 q^{79} +6.46108e24 q^{81} +6.50402e25 q^{83} -7.55105e25 q^{85} +9.16028e25 q^{87} +3.38617e26 q^{89} +1.38170e25 q^{91} -1.72453e26 q^{93} +2.70614e25 q^{95} -4.16124e26 q^{97} +1.28555e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9}+O(q^{10}) \) 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9	\( 2 q + 3188646 q^{3} - 1771946100 q^{5} - 369665199904 q^{7} + 5083731656658 q^{9} - 75762335668248 q^{11} - 103021079177588 q^{13} - 28\!\cdots\!00 q^{15}+ \cdots - 19\!\cdots\!92 q^{99}+O(q^{100}) \) 2 * q + 3188646 * q^3 - 1771946100 * q^5 - 369665199904 * q^7 + 5083731656658 * q^9 - 75762335668248 * q^11 - 103021079177588 * q^13 - 2825054421990300 * q^15 + 34691611579147908 * q^17 - 111620476558642216 * q^19 - 589365730506544992 * q^21 + 2892743075819387952 * q^23 - 13257495997920171250 * q^25 + 8105110306037952534 * q^27 + 29959552473322806972 * q^29 - 10367463257055494032 * q^31 - 120789634289608156104 * q^33 + 252142979584030920000 * q^35 - 3703660908819524869316 * q^37 - 164248876017649632924 * q^39 + 1481814928479478034772 * q^41 - 9778778623457380249240 * q^43 - 4504049241230841066900 * q^45 - 89494998518809740323520 * q^47 + 13908377820361409575890 * q^49 + 55309634247701830126284 * q^51 + 425671275824601813476748 * q^53 + 33146870181194328159600 * q^55 - 177959093048404133739768 * q^57 - 369418769643380130292824 * q^59 + 925133397428892745249420 * q^61 - 939639339558386331280416 * q^63 - 394447136075147294825400 * q^65 - 4151706674555213948132968 * q^67 + 4611966818869594057796496 * q^69 - 9760455855870867537668784 * q^71 + 28449956347640075157409588 * q^73 - 21136730791892081187813750 * q^75 + 48717589762471785388668288 * q^77 + 49553221898675501744231120 * q^79 + 12922163778453346597864482 * q^81 + 120473408203228735926347736 * q^83 - 50977978054175038499879400 * q^85 + 47765203577925437580019956 * q^87 + 205644052209697133056985556 * q^89 + 515309018103795393250147648 * q^91 - 16529085122378486411580336 * q^93 + 87097784513216888583651600 * q^95 - 1478339006400409594836427772 * q^97 - 192577692109510944264197592 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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