LMFDB - Embedded newform 6025.2.a.o.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.810170 q^{2} +2.88822 q^{3} -1.34362 q^{4} -2.33995 q^{6} +3.73380 q^{7} +2.70891 q^{8} +5.34179 q^{9} +O(q^{10})$	\(q-0.810170 q^{2} +2.88822 q^{3} -1.34362 q^{4} -2.33995 q^{6} +3.73380 q^{7} +2.70891 q^{8} +5.34179 q^{9} +0.363723 q^{11} -3.88067 q^{12} +1.90909 q^{13} -3.02501 q^{14} +0.492573 q^{16} +1.56783 q^{17} -4.32776 q^{18} -4.74495 q^{19} +10.7840 q^{21} -0.294678 q^{22} -0.210707 q^{23} +7.82390 q^{24} -1.54669 q^{26} +6.76358 q^{27} -5.01682 q^{28} +3.72590 q^{29} +2.95609 q^{31} -5.81688 q^{32} +1.05051 q^{33} -1.27021 q^{34} -7.17735 q^{36} -2.19189 q^{37} +3.84422 q^{38} +5.51386 q^{39} +11.8082 q^{41} -8.73688 q^{42} +5.79164 q^{43} -0.488707 q^{44} +0.170708 q^{46} -3.70475 q^{47} +1.42266 q^{48} +6.94123 q^{49} +4.52822 q^{51} -2.56510 q^{52} -2.34533 q^{53} -5.47966 q^{54} +10.1145 q^{56} -13.7044 q^{57} -3.01862 q^{58} -5.13708 q^{59} +3.62512 q^{61} -2.39493 q^{62} +19.9451 q^{63} +3.72752 q^{64} -0.851093 q^{66} -1.38724 q^{67} -2.10657 q^{68} -0.608567 q^{69} -0.880976 q^{71} +14.4704 q^{72} +6.39905 q^{73} +1.77581 q^{74} +6.37543 q^{76} +1.35807 q^{77} -4.46716 q^{78} -0.286499 q^{79} +3.50933 q^{81} -9.56662 q^{82} +6.19569 q^{83} -14.4896 q^{84} -4.69222 q^{86} +10.7612 q^{87} +0.985291 q^{88} -2.54350 q^{89} +7.12814 q^{91} +0.283111 q^{92} +8.53781 q^{93} +3.00148 q^{94} -16.8004 q^{96} -8.23089 q^{97} -5.62358 q^{98} +1.94293 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	$ 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * 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 $

$2$	−0.810170	−0.572877	−0.286439	−	0.958099i	$-0.592471\pi$
$2$	−0.810170	−0.572877	−0.286439	+	0.958099i	$0.592471\pi$
$3$	2.88822	1.66751	0.833756	−	0.552133i	$-0.186186\pi$
$3$	2.88822	1.66751	0.833756	+	0.552133i	$0.186186\pi$
$4$	−1.34362	−0.671812
$5$	0	0
$5$	0	0
$6$	−2.33995	−0.955279
$7$	3.73380	1.41124	0.705621	−	0.708589i	$-0.250668\pi$
$7$	3.73380	1.41124	0.705621	+	0.708589i	$0.250668\pi$
$8$	2.70891	0.957743
$9$	5.34179	1.78060
$10$	0	0
$11$	0.363723	0.109667	0.0548333	−	0.998496i	$-0.482537\pi$
$11$	0.363723	0.109667	0.0548333	+	0.998496i	$0.482537\pi$
$12$	−3.88067	−1.12025
$13$	1.90909	0.529486	0.264743	−	0.964319i	$-0.414713\pi$
$13$	1.90909	0.529486	0.264743	+	0.964319i	$0.414713\pi$
$14$	−3.02501	−0.808468
$15$	0	0
$16$	0.492573	0.123143
$17$	1.56783	0.380254	0.190127	−	0.981760i	$-0.439110\pi$
$17$	1.56783	0.380254	0.190127	+	0.981760i	$0.439110\pi$
$18$	−4.32776	−1.02006
$19$	−4.74495	−1.08857	−0.544283	−	0.838902i	$-0.683198\pi$
$19$	−4.74495	−1.08857	−0.544283	+	0.838902i	$0.683198\pi$
$20$	0	0
$21$	10.7840	2.35326
$22$	−0.294678	−0.0628255
$23$	−0.210707	−0.0439354	−0.0219677	−	0.999759i	$-0.506993\pi$
$23$	−0.210707	−0.0439354	−0.0219677	+	0.999759i	$0.506993\pi$
$24$	7.82390	1.59705
$25$	0	0
$26$	−1.54669	−0.303330
$27$	6.76358	1.30165
$28$	−5.01682	−0.948089
$29$	3.72590	0.691883	0.345941	−	0.938256i	$-0.387560\pi$
$29$	3.72590	0.691883	0.345941	+	0.938256i	$0.387560\pi$
$30$	0	0
$31$	2.95609	0.530929	0.265464	−	0.964121i	$-0.414475\pi$
$31$	2.95609	0.530929	0.265464	+	0.964121i	$0.414475\pi$
$32$	−5.81688	−1.02829
$33$	1.05051	0.182870
$34$	−1.27021	−0.217839
$35$	0	0
$36$	−7.17735	−1.19623
$37$	−2.19189	−0.360345	−0.180173	−	0.983635i	$-0.557666\pi$
$37$	−2.19189	−0.360345	−0.180173	+	0.983635i	$0.557666\pi$
$38$	3.84422	0.623614
$39$	5.51386	0.882924
$40$	0	0
$41$	11.8082	1.84412	0.922062	−	0.387042i	$-0.126503\pi$
$41$	11.8082	1.84412	0.922062	+	0.387042i	$0.126503\pi$
$42$	−8.73688	−1.34813
$43$	5.79164	0.883217	0.441609	−	0.897208i	$-0.354408\pi$
$43$	5.79164	0.883217	0.441609	+	0.897208i	$0.354408\pi$
$44$	−0.488707	−0.0736754
$45$	0	0
$46$	0.170708	0.0251696
$47$	−3.70475	−0.540394	−0.270197	−	0.962805i	$-0.587089\pi$
$47$	−3.70475	−0.540394	−0.270197	+	0.962805i	$0.587089\pi$
$48$	1.42266	0.205343
$49$	6.94123	0.991604
$50$	0	0
$51$	4.52822	0.634077
$52$	−2.56510	−0.355715
$53$	−2.34533	−0.322156	−0.161078	−	0.986942i	$-0.551497\pi$
$53$	−2.34533	−0.322156	−0.161078	+	0.986942i	$0.551497\pi$
$54$	−5.47966	−0.745687
$55$	0	0
$56$	10.1145	1.35161
$57$	−13.7044	−1.81520
$58$	−3.01862	−0.396364
$59$	−5.13708	−0.668791	−0.334395	−	0.942433i	$-0.608532\pi$
$59$	−5.13708	−0.668791	−0.334395	+	0.942433i	$0.608532\pi$
$60$	0	0
$61$	3.62512	0.464149	0.232074	−	0.972698i	$-0.425449\pi$
$61$	3.62512	0.464149	0.232074	+	0.972698i	$0.425449\pi$
$62$	−2.39493	−0.304157
$63$	19.9451	2.51285
$64$	3.72752	0.465940
$65$	0	0
$66$	−0.851093	−0.104762
$67$	−1.38724	−0.169478	−0.0847392	−	0.996403i	$-0.527006\pi$
$67$	−1.38724	−0.169478	−0.0847392	+	0.996403i	$0.527006\pi$
$68$	−2.10657	−0.255459
$69$	−0.608567	−0.0732628
$70$	0	0
$71$	−0.880976	−0.104553	−0.0522763	−	0.998633i	$-0.516648\pi$
$71$	−0.880976	−0.104553	−0.0522763	+	0.998633i	$0.516648\pi$
$72$	14.4704	1.70535
$73$	6.39905	0.748953	0.374476	−	0.927236i	$-0.377823\pi$
$73$	6.39905	0.748953	0.374476	+	0.927236i	$0.377823\pi$
$74$	1.77581	0.206433
$75$	0	0
$76$	6.37543	0.731312
$77$	1.35807	0.154766
$78$	−4.46716	−0.505807
$79$	−0.286499	−0.0322337	−0.0161168	−	0.999870i	$-0.505130\pi$
$79$	−0.286499	−0.0322337	−0.0161168	+	0.999870i	$0.505130\pi$
$80$	0	0
$81$	3.50933	0.389925
$82$	−9.56662	−1.05646
$83$	6.19569	0.680065	0.340032	−	0.940414i	$-0.389562\pi$
$83$	6.19569	0.680065	0.340032	+	0.940414i	$0.389562\pi$
$84$	−14.4896	−1.58095
$85$	0	0
$86$	−4.69222	−0.505975
$87$	10.7612	1.15372
$88$	0.985291	0.105032
$89$	−2.54350	−0.269610	−0.134805	−	0.990872i	$-0.543041\pi$
$89$	−2.54350	−0.269610	−0.134805	+	0.990872i	$0.543041\pi$
$90$	0	0
$91$	7.12814	0.747233
$92$	0.283111	0.0295163
$93$	8.53781	0.885330
$94$	3.00148	0.309579
$95$	0	0
$96$	−16.8004	−1.71468
$97$	−8.23089	−0.835720	−0.417860	−	0.908511i	$-0.637220\pi$
$97$	−8.23089	−0.835720	−0.417860	+	0.908511i	$0.637220\pi$
$98$	−5.62358	−0.568067
$99$	1.94293	0.195272
$100$	0	0
$101$	15.3898	1.53135	0.765673	−	0.643230i	$-0.222406\pi$
$101$	15.3898	1.53135	0.765673	+	0.643230i	$0.222406\pi$
$102$	−3.66863	−0.363248
$103$	−9.05525	−0.892240	−0.446120	−	0.894973i	$-0.647195\pi$
$103$	−9.05525	−0.892240	−0.446120	+	0.894973i	$0.647195\pi$
$104$	5.17154	0.507111
$105$	0	0
$106$	1.90012	0.184556
$107$	−15.2953	−1.47865	−0.739327	−	0.673347i	$-0.764856\pi$
$107$	−15.2953	−1.47865	−0.739327	+	0.673347i	$0.764856\pi$
$108$	−9.08771	−0.874466
$109$	6.31702	0.605061	0.302530	−	0.953140i	$-0.402169\pi$
$109$	6.31702	0.605061	0.302530	+	0.953140i	$0.402169\pi$
$110$	0	0
$111$	−6.33066	−0.600880
$112$	1.83917	0.173785
$113$	−5.72064	−0.538153	−0.269076	−	0.963119i	$-0.586718\pi$
$113$	−5.72064	−0.538153	−0.269076	+	0.963119i	$0.586718\pi$
$114$	11.1029	1.03988
$115$	0	0
$116$	−5.00621	−0.464815
$117$	10.1979	0.942800
$118$	4.16191	0.383135
$119$	5.85394	0.536630
$120$	0	0
$121$	−10.8677	−0.987973
$122$	−2.93696	−0.265900
$123$	34.1045	3.07510
$124$	−3.97187	−0.356684
$125$	0	0
$126$	−16.1590	−1.43955
$127$	13.6961	1.21534	0.607668	−	0.794191i	$-0.292105\pi$
$127$	13.6961	1.21534	0.607668	+	0.794191i	$0.292105\pi$
$128$	8.61383	0.761362
$129$	16.7275	1.47277
$130$	0	0
$131$	15.8734	1.38686	0.693431	−	0.720523i	$-0.256098\pi$
$131$	15.8734	1.38686	0.693431	+	0.720523i	$0.256098\pi$
$132$	−1.41149	−0.122855
$133$	−17.7167	−1.53623
$134$	1.12390	0.0970903
$135$	0	0
$136$	4.24709	0.364185
$137$	1.09189	0.0932863	0.0466432	−	0.998912i	$-0.485148\pi$
$137$	1.09189	0.0932863	0.0466432	+	0.998912i	$0.485148\pi$
$138$	0.493043	0.0419706
$139$	10.9871	0.931910	0.465955	−	0.884808i	$-0.345711\pi$
$139$	10.9871	0.931910	0.465955	+	0.884808i	$0.345711\pi$
$140$	0	0
$141$	−10.7001	−0.901113
$142$	0.713740	0.0598958
$143$	0.694379	0.0580669
$144$	2.63122	0.219268
$145$	0	0
$146$	−5.18432	−0.429058
$147$	20.0478	1.65351
$148$	2.94508	0.242084
$149$	9.67482	0.792592	0.396296	−	0.918123i	$-0.370295\pi$
$149$	9.67482	0.792592	0.396296	+	0.918123i	$0.370295\pi$
$150$	0	0
$151$	−10.2525	−0.834339	−0.417170	−	0.908829i	$-0.636978\pi$
$151$	−10.2525	−0.834339	−0.417170	+	0.908829i	$0.636978\pi$
$152$	−12.8536	−1.04257
$153$	8.37499	0.677078
$154$	−1.10027	−0.0886620
$155$	0	0
$156$	−7.40855	−0.593159
$157$	−1.76906	−0.141186	−0.0705930	−	0.997505i	$-0.522489\pi$
$157$	−1.76906	−0.141186	−0.0705930	+	0.997505i	$0.522489\pi$
$158$	0.232113	0.0184659
$159$	−6.77382	−0.537199
$160$	0	0
$161$	−0.786736	−0.0620035
$162$	−2.84315	−0.223379
$163$	9.69938	0.759714	0.379857	−	0.925045i	$-0.375973\pi$
$163$	9.69938	0.759714	0.379857	+	0.925045i	$0.375973\pi$
$164$	−15.8657	−1.23890
$165$	0	0
$166$	−5.01956	−0.389594
$167$	−11.6371	−0.900505	−0.450252	−	0.892901i	$-0.648666\pi$
$167$	−11.6371	−0.900505	−0.450252	+	0.892901i	$0.648666\pi$
$168$	29.2128	2.25382
$169$	−9.35538	−0.719645
$170$	0	0
$171$	−25.3465	−1.93830
$172$	−7.78179	−0.593356
$173$	3.10327	0.235937	0.117969	−	0.993017i	$-0.462362\pi$
$173$	3.10327	0.235937	0.117969	+	0.993017i	$0.462362\pi$
$174$	−8.71841	−0.660941
$175$	0	0
$176$	0.179160	0.0135047
$177$	−14.8370	−1.11522
$178$	2.06067	0.154454
$179$	−4.47412	−0.334411	−0.167206	−	0.985922i	$-0.553474\pi$
$179$	−4.47412	−0.334411	−0.167206	+	0.985922i	$0.553474\pi$
$180$	0	0
$181$	−25.1406	−1.86868	−0.934342	−	0.356378i	$-0.884012\pi$
$181$	−25.1406	−1.86868	−0.934342	+	0.356378i	$0.884012\pi$
$182$	−5.77501	−0.428072
$183$	10.4701	0.773973
$184$	−0.570785	−0.0420788
$185$	0	0
$186$	−6.91708	−0.507185
$187$	0.570254	0.0417011
$188$	4.97779	0.363043
$189$	25.2538	1.83695
$190$	0	0
$191$	−20.6068	−1.49105	−0.745527	−	0.666475i	$-0.767802\pi$
$191$	−20.6068	−1.49105	−0.745527	+	0.666475i	$0.767802\pi$
$192$	10.7659	0.776960
$193$	0.420838	0.0302926	0.0151463	−	0.999885i	$-0.495179\pi$
$193$	0.420838	0.0302926	0.0151463	+	0.999885i	$0.495179\pi$
$194$	6.66842	0.478765
$195$	0	0
$196$	−9.32640	−0.666172
$197$	−17.1310	−1.22053	−0.610266	−	0.792197i	$-0.708937\pi$
$197$	−17.1310	−1.22053	−0.610266	+	0.792197i	$0.708937\pi$
$198$	−1.57411	−0.111867
$199$	4.57679	0.324440	0.162220	−	0.986755i	$-0.448135\pi$
$199$	4.57679	0.324440	0.162220	+	0.986755i	$0.448135\pi$
$200$	0	0
$201$	−4.00665	−0.282607
$202$	−12.4684	−0.877273
$203$	13.9118	0.976414
$204$	−6.08422	−0.425981
$205$	0	0
$206$	7.33630	0.511144
$207$	−1.12555	−0.0782312
$208$	0.940365	0.0652026
$209$	−1.72585	−0.119379
$210$	0	0
$211$	−10.9965	−0.757029	−0.378515	−	0.925595i	$-0.623565\pi$
$211$	−10.9965	−0.757029	−0.378515	+	0.925595i	$0.623565\pi$
$212$	3.15124	0.216428
$213$	−2.54445	−0.174343
$214$	12.3918	0.847087
$215$	0	0
$216$	18.3219	1.24665
$217$	11.0374	0.749269
$218$	−5.11786	−0.346625
$219$	18.4818	1.24889
$220$	0	0
$221$	2.99312	0.201339
$222$	5.12891	0.344230
$223$	15.9121	1.06555	0.532775	−	0.846257i	$-0.321149\pi$
$223$	15.9121	1.06555	0.532775	+	0.846257i	$0.321149\pi$
$224$	−21.7190	−1.45116
$225$	0	0
$226$	4.63470	0.308295
$227$	3.82315	0.253751	0.126876	−	0.991919i	$-0.459505\pi$
$227$	3.82315	0.253751	0.126876	+	0.991919i	$0.459505\pi$
$228$	18.4136	1.21947
$229$	−2.04619	−0.135216	−0.0676080	−	0.997712i	$-0.521537\pi$
$229$	−2.04619	−0.135216	−0.0676080	+	0.997712i	$0.521537\pi$
$230$	0	0
$231$	3.92239	0.258074
$232$	10.0931	0.662645
$233$	7.43671	0.487195	0.243597	−	0.969876i	$-0.421672\pi$
$233$	7.43671	0.487195	0.243597	+	0.969876i	$0.421672\pi$
$234$	−8.26207	−0.540108
$235$	0	0
$236$	6.90230	0.449302
$237$	−0.827472	−0.0537501
$238$	−4.74269	−0.307423
$239$	−24.2338	−1.56755	−0.783777	−	0.621042i	$-0.786709\pi$
$239$	−24.2338	−1.56755	−0.783777	+	0.621042i	$0.786709\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	8.80469	0.565987
$243$	−10.1551	−0.651448
$244$	−4.87079	−0.311821
$245$	0	0
$246$	−27.6305	−1.76165
$247$	−9.05853	−0.576380
$248$	8.00775	0.508493
$249$	17.8945	1.13402
$250$	0	0
$251$	29.7129	1.87546	0.937730	−	0.347366i	$-0.112924\pi$
$251$	29.7129	1.87546	0.937730	+	0.347366i	$0.112924\pi$
$252$	−26.7988	−1.68816
$253$	−0.0766390	−0.00481825
$254$	−11.0962	−0.696238
$255$	0	0
$256$	−14.4337	−0.902107
$257$	−9.33011	−0.581996	−0.290998	−	0.956724i	$-0.593987\pi$
$257$	−9.33011	−0.581996	−0.290998	+	0.956724i	$0.593987\pi$
$258$	−13.5521	−0.843719
$259$	−8.18408	−0.508534
$260$	0	0
$261$	19.9030	1.23196
$262$	−12.8601	−0.794502
$263$	10.0899	0.622173	0.311086	−	0.950382i	$-0.399307\pi$
$263$	10.0899	0.622173	0.311086	+	0.950382i	$0.399307\pi$
$264$	2.84573	0.175143
$265$	0	0
$266$	14.3535	0.880071
$267$	−7.34617	−0.449578
$268$	1.86393	0.113858
$269$	−8.13602	−0.496062	−0.248031	−	0.968752i	$-0.579784\pi$
$269$	−8.13602	−0.496062	−0.248031	+	0.968752i	$0.579784\pi$
$270$	0	0
$271$	−6.40501	−0.389077	−0.194538	−	0.980895i	$-0.562321\pi$
$271$	−6.40501	−0.389077	−0.194538	+	0.980895i	$0.562321\pi$
$272$	0.772268	0.0468256
$273$	20.5876	1.24602
$274$	−0.884616	−0.0534416
$275$	0	0
$276$	0.817685	0.0492188
$277$	19.3496	1.16261	0.581304	−	0.813687i	$-0.302543\pi$
$277$	19.3496	1.16261	0.581304	+	0.813687i	$0.302543\pi$
$278$	−8.90139	−0.533870
$279$	15.7908	0.945369
$280$	0	0
$281$	−5.41886	−0.323262	−0.161631	−	0.986851i	$-0.551675\pi$
$281$	−5.41886	−0.323262	−0.161631	+	0.986851i	$0.551675\pi$
$282$	8.66892	0.516227
$283$	18.4017	1.09387	0.546935	−	0.837175i	$-0.315794\pi$
$283$	18.4017	1.09387	0.546935	+	0.837175i	$0.315794\pi$
$284$	1.18370	0.0702397
$285$	0	0
$286$	−0.562566	−0.0332652
$287$	44.0892	2.60251
$288$	−31.0725	−1.83097
$289$	−14.5419	−0.855407
$290$	0	0
$291$	−23.7726	−1.39357
$292$	−8.59792	−0.503155
$293$	11.9473	0.697971	0.348985	−	0.937128i	$-0.386526\pi$
$293$	11.9473	0.697971	0.348985	+	0.937128i	$0.386526\pi$
$294$	−16.2421	−0.947259
$295$	0	0
$296$	−5.93763	−0.345118
$297$	2.46007	0.142748
$298$	−7.83825	−0.454058
$299$	−0.402258	−0.0232632
$300$	0	0
$301$	21.6248	1.24643
$302$	8.30630	0.477974
$303$	44.4491	2.55354
$304$	−2.33723	−0.134049
$305$	0	0
$306$	−6.78517	−0.387882
$307$	28.9265	1.65092	0.825462	−	0.564458i	$-0.190915\pi$
$307$	28.9265	1.65092	0.825462	+	0.564458i	$0.190915\pi$
$308$	−1.82473	−0.103974
$309$	−26.1535	−1.48782
$310$	0	0
$311$	10.9958	0.623512	0.311756	−	0.950162i	$-0.399083\pi$
$311$	10.9958	0.623512	0.311756	+	0.950162i	$0.399083\pi$
$312$	14.9365	0.845614
$313$	−29.3670	−1.65992	−0.829960	−	0.557822i	$-0.811637\pi$
$313$	−29.3670	−1.65992	−0.829960	+	0.557822i	$0.811637\pi$
$314$	1.43324	0.0808823
$315$	0	0
$316$	0.384947	0.0216550
$317$	9.21630	0.517639	0.258820	−	0.965926i	$-0.416667\pi$
$317$	9.21630	0.517639	0.258820	+	0.965926i	$0.416667\pi$
$318$	5.48795	0.307749
$319$	1.35520	0.0758764
$320$	0	0
$321$	−44.1762	−2.46567
$322$	0.637391	0.0355204
$323$	−7.43925	−0.413931
$324$	−4.71521	−0.261956
$325$	0	0
$326$	−7.85815	−0.435223
$327$	18.2449	1.00895
$328$	31.9872	1.76620
$329$	−13.8328	−0.762626
$330$	0	0
$331$	31.4522	1.72877	0.864386	−	0.502829i	$-0.167708\pi$
$331$	31.4522	1.72877	0.864386	+	0.502829i	$0.167708\pi$
$332$	−8.32467	−0.456876
$333$	−11.7086	−0.641629
$334$	9.42802	0.515878
$335$	0	0
$336$	5.31191	0.289788
$337$	−8.48911	−0.462431	−0.231216	−	0.972903i	$-0.574270\pi$
$337$	−8.48911	−0.462431	−0.231216	+	0.972903i	$0.574270\pi$
$338$	7.57945	0.412268
$339$	−16.5224	−0.897376
$340$	0	0
$341$	1.07520	0.0582252
$342$	20.5350	1.11041
$343$	−0.219435	−0.0118484
$344$	15.6890	0.845895
$345$	0	0
$346$	−2.51418	−0.135163
$347$	9.35352	0.502123	0.251062	−	0.967971i	$-0.419220\pi$
$347$	9.35352	0.502123	0.251062	+	0.967971i	$0.419220\pi$
$348$	−14.4590	−0.775084
$349$	6.74485	0.361044	0.180522	−	0.983571i	$-0.442221\pi$
$349$	6.74485	0.361044	0.180522	+	0.983571i	$0.442221\pi$
$350$	0	0
$351$	12.9123	0.689206
$352$	−2.11573	−0.112769
$353$	27.5391	1.46576	0.732878	−	0.680360i	$-0.238177\pi$
$353$	27.5391	1.46576	0.732878	+	0.680360i	$0.238177\pi$
$354$	12.0205	0.638882
$355$	0	0
$356$	3.41750	0.181127
$357$	16.9074	0.894837
$358$	3.62480	0.191577
$359$	6.29764	0.332377	0.166188	−	0.986094i	$-0.446854\pi$
$359$	6.29764	0.332377	0.166188	+	0.986094i	$0.446854\pi$
$360$	0	0
$361$	3.51454	0.184976
$362$	20.3681	1.07053
$363$	−31.3883	−1.64746
$364$	−9.57755	−0.502000
$365$	0	0
$366$	−8.48258	−0.443391
$367$	9.13283	0.476730	0.238365	−	0.971176i	$-0.423389\pi$
$367$	9.13283	0.476730	0.238365	+	0.971176i	$0.423389\pi$
$368$	−0.103788	−0.00541035
$369$	63.0767	3.28364
$370$	0	0
$371$	−8.75698	−0.454640
$372$	−11.4716	−0.594775
$373$	−25.1732	−1.30342	−0.651709	−	0.758469i	$-0.725948\pi$
$373$	−25.1732	−1.30342	−0.651709	+	0.758469i	$0.725948\pi$
$374$	−0.462003	−0.0238896
$375$	0	0
$376$	−10.0358	−0.517558
$377$	7.11307	0.366342
$378$	−20.4599	−1.05234
$379$	−33.6035	−1.72610	−0.863048	−	0.505122i	$-0.831447\pi$
$379$	−33.6035	−1.72610	−0.863048	+	0.505122i	$0.831447\pi$
$380$	0	0
$381$	39.5574	2.02659
$382$	16.6950	0.854190
$383$	18.3265	0.936443	0.468221	−	0.883611i	$-0.344895\pi$
$383$	18.3265	0.936443	0.468221	+	0.883611i	$0.344895\pi$
$384$	24.8786	1.26958
$385$	0	0
$386$	−0.340950	−0.0173539
$387$	30.9377	1.57265
$388$	11.0592	0.561447
$389$	30.4975	1.54629	0.773143	−	0.634232i	$-0.218684\pi$
$389$	30.4975	1.54629	0.773143	+	0.634232i	$0.218684\pi$
$390$	0	0
$391$	−0.330352	−0.0167066
$392$	18.8031	0.949702
$393$	45.8457	2.31261
$394$	13.8790	0.699214
$395$	0	0
$396$	−2.61057	−0.131186
$397$	6.27221	0.314793	0.157397	−	0.987535i	$-0.449690\pi$
$397$	6.27221	0.314793	0.157397	+	0.987535i	$0.449690\pi$
$398$	−3.70798	−0.185864
$399$	−51.1696	−2.56168
$400$	0	0
$401$	11.1113	0.554870	0.277435	−	0.960744i	$-0.410516\pi$
$401$	11.1113	0.554870	0.277435	+	0.960744i	$0.410516\pi$
$402$	3.24607	0.161899
$403$	5.64343	0.281119
$404$	−20.6781	−1.02878
$405$	0	0
$406$	−11.2709	−0.559365
$407$	−0.797242	−0.0395178
$408$	12.2665	0.607283
$409$	5.94001	0.293714	0.146857	−	0.989158i	$-0.453084\pi$
$409$	5.94001	0.293714	0.146857	+	0.989158i	$0.453084\pi$
$410$	0	0
$411$	3.15361	0.155556
$412$	12.1668	0.599418
$413$	−19.1808	−0.943826
$414$	0.911888	0.0448169
$415$	0	0
$416$	−11.1049	−0.544464
$417$	31.7330	1.55397
$418$	1.39823	0.0683897
$419$	17.6829	0.863865	0.431933	−	0.901906i	$-0.357832\pi$
$419$	17.6829	0.863865	0.431933	+	0.901906i	$0.357832\pi$
$420$	0	0
$421$	−35.0390	−1.70770	−0.853849	−	0.520521i	$-0.825738\pi$
$421$	−35.0390	−1.70770	−0.853849	+	0.520521i	$0.825738\pi$
$422$	8.90903	0.433685
$423$	−19.7900	−0.962222
$424$	−6.35328	−0.308542
$425$	0	0
$426$	2.06144	0.0998769
$427$	13.5354	0.655026
$428$	20.5511	0.993377
$429$	2.00552	0.0968273
$430$	0	0
$431$	3.76649	0.181426	0.0907128	−	0.995877i	$-0.471085\pi$
$431$	3.76649	0.181426	0.0907128	+	0.995877i	$0.471085\pi$
$432$	3.33156	0.160290
$433$	−18.9073	−0.908629	−0.454315	−	0.890841i	$-0.650116\pi$
$433$	−18.9073	−0.908629	−0.454315	+	0.890841i	$0.650116\pi$
$434$	−8.94219	−0.429239
$435$	0	0
$436$	−8.48770	−0.406487
$437$	0.999793	0.0478266
$438$	−14.9734	−0.715459
$439$	21.9399	1.04714	0.523568	−	0.851984i	$-0.324601\pi$
$439$	21.9399	1.04714	0.523568	+	0.851984i	$0.324601\pi$
$440$	0	0
$441$	37.0786	1.76565
$442$	−2.42494	−0.115342
$443$	−1.03097	−0.0489829	−0.0244915	−	0.999700i	$-0.507797\pi$
$443$	−1.03097	−0.0489829	−0.0244915	+	0.999700i	$0.507797\pi$
$444$	8.50603	0.403678
$445$	0	0
$446$	−12.8915	−0.610430
$447$	27.9430	1.32166
$448$	13.9178	0.657554
$449$	−15.7218	−0.741960	−0.370980	−	0.928641i	$-0.620978\pi$
$449$	−15.7218	−0.741960	−0.370980	+	0.928641i	$0.620978\pi$
$450$	0	0
$451$	4.29490	0.202239
$452$	7.68639	0.361537
$453$	−29.6115	−1.39127
$454$	−3.09740	−0.145368
$455$	0	0
$456$	−37.1240	−1.73849
$457$	9.35883	0.437787	0.218894	−	0.975749i	$-0.429755\pi$
$457$	9.35883	0.437787	0.218894	+	0.975749i	$0.429755\pi$
$458$	1.65776	0.0774621
$459$	10.6041	0.494958
$460$	0	0
$461$	29.5562	1.37657	0.688286	−	0.725440i	$-0.258364\pi$
$461$	29.5562	1.37657	0.688286	+	0.725440i	$0.258364\pi$
$462$	−3.17781	−0.147845
$463$	−10.4336	−0.484890	−0.242445	−	0.970165i	$-0.577949\pi$
$463$	−10.4336	−0.484890	−0.242445	+	0.970165i	$0.577949\pi$
$464$	1.83528	0.0852006
$465$	0	0
$466$	−6.02500	−0.279103
$467$	0.425612	0.0196950	0.00984748	−	0.999952i	$-0.496865\pi$
$467$	0.425612	0.0196950	0.00984748	+	0.999952i	$0.496865\pi$
$468$	−13.7022	−0.633384
$469$	−5.17967	−0.239175
$470$	0	0
$471$	−5.10942	−0.235429
$472$	−13.9159	−0.640529
$473$	2.10655	0.0968595
$474$	0.670393	0.0307922
$475$	0	0
$476$	−7.86549	−0.360514
$477$	−12.5283	−0.573629
$478$	19.6335	0.898016
$479$	33.5508	1.53298	0.766488	−	0.642258i	$-0.222002\pi$
$479$	33.5508	1.53298	0.766488	+	0.642258i	$0.222002\pi$
$480$	0	0
$481$	−4.18452	−0.190798
$482$	0.810170	0.0369023
$483$	−2.27226	−0.103392
$484$	14.6021	0.663732
$485$	0	0
$486$	8.22733	0.373199
$487$	−9.18309	−0.416126	−0.208063	−	0.978115i	$-0.566716\pi$
$487$	−9.18309	−0.416126	−0.208063	+	0.978115i	$0.566716\pi$
$488$	9.82010	0.444535
$489$	28.0139	1.26683
$490$	0	0
$491$	19.2635	0.869349	0.434675	−	0.900588i	$-0.356863\pi$
$491$	19.2635	0.869349	0.434675	+	0.900588i	$0.356863\pi$
$492$	−45.8236	−2.06589
$493$	5.84156	0.263091
$494$	7.33895	0.330195
$495$	0	0
$496$	1.45609	0.0653802
$497$	−3.28938	−0.147549
$498$	−14.4976	−0.649652
$499$	−12.8283	−0.574273	−0.287137	−	0.957890i	$-0.592703\pi$
$499$	−12.8283	−0.574273	−0.287137	+	0.957890i	$0.592703\pi$
$500$	0	0
$501$	−33.6104	−1.50160
$502$	−24.0725	−1.07441
$503$	0.634256	0.0282801	0.0141400	−	0.999900i	$-0.495499\pi$
$503$	0.634256	0.0282801	0.0141400	+	0.999900i	$0.495499\pi$
$504$	54.0295	2.40666
$505$	0	0
$506$	0.0620906	0.00276026
$507$	−27.0204	−1.20002
$508$	−18.4025	−0.816478
$509$	−24.5805	−1.08951	−0.544756	−	0.838595i	$-0.683378\pi$
$509$	−24.5805	−1.08951	−0.544756	+	0.838595i	$0.683378\pi$
$510$	0	0
$511$	23.8928	1.05695
$512$	−5.53390	−0.244566
$513$	−32.0929	−1.41693
$514$	7.55898	0.333412
$515$	0	0
$516$	−22.4755	−0.989428
$517$	−1.34750	−0.0592631
$518$	6.63050	0.291327
$519$	8.96290	0.393428
$520$	0	0
$521$	11.1620	0.489017	0.244509	−	0.969647i	$-0.421373\pi$
$521$	11.1620	0.489017	0.244509	+	0.969647i	$0.421373\pi$
$522$	−16.1248	−0.705763
$523$	−24.1026	−1.05393	−0.526967	−	0.849886i	$-0.676671\pi$
$523$	−24.1026	−1.05393	−0.526967	+	0.849886i	$0.676671\pi$
$524$	−21.3278	−0.931711
$525$	0	0
$526$	−8.17458	−0.356429
$527$	4.63463	0.201887
$528$	0.517453	0.0225192
$529$	−22.9556	−0.998070
$530$	0	0
$531$	−27.4412	−1.19085
$532$	23.8045	1.03206
$533$	22.5428	0.976438
$534$	5.95165	0.257553
$535$	0	0
$536$	−3.75790	−0.162317
$537$	−12.9222	−0.557635
$538$	6.59157	0.284183
$539$	2.52469	0.108746
$540$	0	0
$541$	−29.4447	−1.26593	−0.632963	−	0.774182i	$-0.718162\pi$
$541$	−29.4447	−1.26593	−0.632963	+	0.774182i	$0.718162\pi$
$542$	5.18915	0.222893
$543$	−72.6114	−3.11605
$544$	−9.11985	−0.391010
$545$	0	0
$546$	−16.6795	−0.713816
$547$	3.78570	0.161865	0.0809324	−	0.996720i	$-0.474210\pi$
$547$	3.78570	0.161865	0.0809324	+	0.996720i	$0.474210\pi$
$548$	−1.46709	−0.0626709
$549$	19.3646	0.826461
$550$	0	0
$551$	−17.6792	−0.753160
$552$	−1.64855	−0.0701669
$553$	−1.06973	−0.0454895
$554$	−15.6765	−0.666031
$555$	0	0
$556$	−14.7625	−0.626068
$557$	−33.1324	−1.40387	−0.701933	−	0.712243i	$-0.747679\pi$
$557$	−33.1324	−1.40387	−0.701933	+	0.712243i	$0.747679\pi$
$558$	−12.7932	−0.541580
$559$	11.0568	0.467651
$560$	0	0
$561$	1.64702	0.0695371
$562$	4.39020	0.185190
$563$	−35.1874	−1.48297	−0.741486	−	0.670968i	$-0.765879\pi$
$563$	−35.1874	−1.48297	−0.741486	+	0.670968i	$0.765879\pi$
$564$	14.3769	0.605378
$565$	0	0
$566$	−14.9086	−0.626653
$567$	13.1031	0.550279
$568$	−2.38648	−0.100134
$569$	4.18986	0.175648	0.0878239	−	0.996136i	$-0.472009\pi$
$569$	4.18986	0.175648	0.0878239	+	0.996136i	$0.472009\pi$
$570$	0	0
$571$	23.7814	0.995221	0.497610	−	0.867401i	$-0.334211\pi$
$571$	23.7814	0.995221	0.497610	+	0.867401i	$0.334211\pi$
$572$	−0.932985	−0.0390101
$573$	−59.5168	−2.48635
$574$	−35.7198	−1.49092
$575$	0	0
$576$	19.9116	0.829650
$577$	30.9793	1.28968	0.644842	−	0.764316i	$-0.276923\pi$
$577$	30.9793	1.28968	0.644842	+	0.764316i	$0.276923\pi$
$578$	11.7814	0.490043
$579$	1.21547	0.0505132
$580$	0	0
$581$	23.1334	0.959736
$582$	19.2598	0.798346
$583$	−0.853051	−0.0353298
$584$	17.3344	0.717304
$585$	0	0
$586$	−9.67938	−0.399851
$587$	−11.5211	−0.475527	−0.237763	−	0.971323i	$-0.576414\pi$
$587$	−11.5211	−0.475527	−0.237763	+	0.971323i	$0.576414\pi$
$588$	−26.9367	−1.11085
$589$	−14.0265	−0.577951
$590$	0	0
$591$	−49.4779	−2.03525
$592$	−1.07967	−0.0443740
$593$	−16.2886	−0.668894	−0.334447	−	0.942415i	$-0.608550\pi$
$593$	−16.2886	−0.668894	−0.334447	+	0.942415i	$0.608550\pi$
$594$	−1.99308	−0.0817770
$595$	0	0
$596$	−12.9993	−0.532473
$597$	13.2188	0.541008
$598$	0.325898	0.0133269
$599$	18.8786	0.771358	0.385679	−	0.922633i	$-0.373967\pi$
$599$	18.8786	0.771358	0.385679	+	0.922633i	$0.373967\pi$
$600$	0	0
$601$	7.39444	0.301626	0.150813	−	0.988562i	$-0.451811\pi$
$601$	7.39444	0.301626	0.150813	+	0.988562i	$0.451811\pi$
$602$	−17.5198	−0.714053
$603$	−7.41034	−0.301773
$604$	13.7756	0.560519
$605$	0	0
$606$	−36.0114	−1.46286
$607$	14.8548	0.602939	0.301469	−	0.953476i	$-0.402523\pi$
$607$	14.8548	0.602939	0.301469	+	0.953476i	$0.402523\pi$
$608$	27.6008	1.11936
$609$	40.1801	1.62818
$610$	0	0
$611$	−7.07270	−0.286131
$612$	−11.2528	−0.454869
$613$	−12.4420	−0.502529	−0.251264	−	0.967919i	$-0.580846\pi$
$613$	−12.4420	−0.502529	−0.251264	+	0.967919i	$0.580846\pi$
$614$	−23.4354	−0.945776
$615$	0	0
$616$	3.67888	0.148226
$617$	21.2628	0.856008	0.428004	−	0.903777i	$-0.359217\pi$
$617$	21.2628	0.856008	0.428004	+	0.903777i	$0.359217\pi$
$618$	21.1888	0.852339
$619$	−29.0488	−1.16757	−0.583785	−	0.811909i	$-0.698429\pi$
$619$	−29.0488	−1.16757	−0.583785	+	0.811909i	$0.698429\pi$
$620$	0	0
$621$	−1.42513	−0.0571886
$622$	−8.90844	−0.357196
$623$	−9.49690	−0.380485
$624$	2.71598	0.108726
$625$	0	0
$626$	23.7923	0.950930
$627$	−4.98462	−0.199067
$628$	2.37695	0.0948505
$629$	−3.43651	−0.137022
$630$	0	0
$631$	−43.0424	−1.71349	−0.856746	−	0.515738i	$-0.827518\pi$
$631$	−43.0424	−1.71349	−0.856746	+	0.515738i	$0.827518\pi$
$632$	−0.776100	−0.0308716
$633$	−31.7602	−1.26236
$634$	−7.46678	−0.296544
$635$	0	0
$636$	9.10146	0.360897
$637$	13.2514	0.525040
$638$	−1.09794	−0.0434679
$639$	−4.70598	−0.186166
$640$	0	0
$641$	1.23954	0.0489589	0.0244795	−	0.999700i	$-0.492207\pi$
$641$	1.23954	0.0489589	0.0244795	+	0.999700i	$0.492207\pi$
$642$	35.7902	1.41253
$643$	0.818429	0.0322757	0.0161378	−	0.999870i	$-0.494863\pi$
$643$	0.818429	0.0322757	0.0161378	+	0.999870i	$0.494863\pi$
$644$	1.05708	0.0416547
$645$	0	0
$646$	6.02706	0.237132
$647$	−47.0773	−1.85080	−0.925400	−	0.378991i	$-0.876271\pi$
$647$	−47.0773	−1.85080	−0.925400	+	0.378991i	$0.876271\pi$
$648$	9.50643	0.373448
$649$	−1.86847	−0.0733440
$650$	0	0
$651$	31.8784	1.24941
$652$	−13.0323	−0.510385
$653$	−9.23889	−0.361546	−0.180773	−	0.983525i	$-0.557860\pi$
$653$	−9.23889	−0.361546	−0.180773	+	0.983525i	$0.557860\pi$
$654$	−14.7815	−0.578002
$655$	0	0
$656$	5.81638	0.227091
$657$	34.1824	1.33358
$658$	11.2069	0.436891
$659$	−22.1226	−0.861775	−0.430888	−	0.902406i	$-0.641799\pi$
$659$	−22.1226	−0.861775	−0.430888	+	0.902406i	$0.641799\pi$
$660$	0	0
$661$	12.0734	0.469601	0.234801	−	0.972044i	$-0.424556\pi$
$661$	12.0734	0.469601	0.234801	+	0.972044i	$0.424556\pi$
$662$	−25.4817	−0.990373
$663$	8.64477	0.335735
$664$	16.7835	0.651327
$665$	0	0
$666$	9.48598	0.367574
$667$	−0.785073	−0.0303982
$668$	15.6359	0.604970
$669$	45.9575	1.77682
$670$	0	0
$671$	1.31854	0.0509016
$672$	−62.7292	−2.41983
$673$	−46.8488	−1.80589	−0.902944	−	0.429759i	$-0.858598\pi$
$673$	−46.8488	−1.80589	−0.902944	+	0.429759i	$0.858598\pi$
$674$	6.87762	0.264916
$675$	0	0
$676$	12.5701	0.483466
$677$	−49.5645	−1.90492	−0.952459	−	0.304666i	$-0.901455\pi$
$677$	−49.5645	−1.90492	−0.952459	+	0.304666i	$0.901455\pi$
$678$	13.3860	0.514086
$679$	−30.7325	−1.17940
$680$	0	0
$681$	11.0421	0.423134
$682$	−0.871092	−0.0333559
$683$	−21.2955	−0.814851	−0.407426	−	0.913238i	$-0.633573\pi$
$683$	−21.2955	−0.814851	−0.407426	+	0.913238i	$0.633573\pi$
$684$	34.0562	1.30217
$685$	0	0
$686$	0.177780	0.00678767
$687$	−5.90984	−0.225474
$688$	2.85281	0.108762
$689$	−4.47744	−0.170577
$690$	0	0
$691$	49.2164	1.87228	0.936140	−	0.351629i	$-0.114372\pi$
$691$	49.2164	1.87228	0.936140	+	0.351629i	$0.114372\pi$
$692$	−4.16962	−0.158505
$693$	7.25451	0.275576
$694$	−7.57795	−0.287655
$695$	0	0
$696$	29.1511	1.10497
$697$	18.5131	0.701235
$698$	−5.46448	−0.206834
$699$	21.4788	0.812403
$700$	0	0
$701$	42.0057	1.58653	0.793267	−	0.608874i	$-0.208379\pi$
$701$	42.0057	1.58653	0.793267	+	0.608874i	$0.208379\pi$
$702$	−10.4611	−0.394831
$703$	10.4004	0.392259
$704$	1.35578	0.0510980
$705$	0	0
$706$	−22.3113	−0.839698
$707$	57.4625	2.16110
$708$	19.9353	0.749216
$709$	−3.83105	−0.143878	−0.0719391	−	0.997409i	$-0.522919\pi$
$709$	−3.83105	−0.143878	−0.0719391	+	0.997409i	$0.522919\pi$
$710$	0	0
$711$	−1.53042	−0.0573952
$712$	−6.89009	−0.258217
$713$	−0.622867	−0.0233266
$714$	−13.6979	−0.512631
$715$	0	0
$716$	6.01153	0.224662
$717$	−69.9924	−2.61391
$718$	−5.10216	−0.190411
$719$	16.0121	0.597151	0.298575	−	0.954386i	$-0.403489\pi$
$719$	16.0121	0.597151	0.298575	+	0.954386i	$0.403489\pi$
$720$	0	0
$721$	−33.8105	−1.25917
$722$	−2.84738	−0.105968
$723$	−2.88822	−0.107414
$724$	33.7795	1.25540
$725$	0	0
$726$	25.4298	0.943790
$727$	38.8351	1.44031	0.720157	−	0.693811i	$-0.244070\pi$
$727$	38.8351	1.44031	0.720157	+	0.693811i	$0.244070\pi$
$728$	19.3095	0.715657
$729$	−39.8580	−1.47622
$730$	0	0
$731$	9.08028	0.335846
$732$	−14.0679	−0.519964
$733$	5.45600	0.201522	0.100761	−	0.994911i	$-0.467872\pi$
$733$	5.45600	0.201522	0.100761	+	0.994911i	$0.467872\pi$
$734$	−7.39915	−0.273108
$735$	0	0
$736$	1.22566	0.0451783
$737$	−0.504571	−0.0185861
$738$	−51.1028	−1.88112
$739$	−37.1163	−1.36535	−0.682673	−	0.730724i	$-0.739183\pi$
$739$	−37.1163	−1.36535	−0.682673	+	0.730724i	$0.739183\pi$
$740$	0	0
$741$	−26.1630	−0.961121
$742$	7.09465	0.260453
$743$	19.5501	0.717223	0.358611	−	0.933487i	$-0.383250\pi$
$743$	19.5501	0.717223	0.358611	+	0.933487i	$0.383250\pi$
$744$	23.1281	0.847918
$745$	0	0
$746$	20.3946	0.746698
$747$	33.0960	1.21092
$748$	−0.766207	−0.0280153
$749$	−57.1096	−2.08674
$750$	0	0
$751$	3.61687	0.131981	0.0659907	−	0.997820i	$-0.478979\pi$
$751$	3.61687	0.131981	0.0659907	+	0.997820i	$0.478979\pi$
$752$	−1.82486	−0.0665458
$753$	85.8171	3.12735
$754$	−5.76280	−0.209869
$755$	0	0
$756$	−33.9317	−1.23408
$757$	−44.2825	−1.60947	−0.804737	−	0.593631i	$-0.797694\pi$
$757$	−44.2825	−1.60947	−0.804737	+	0.593631i	$0.797694\pi$
$758$	27.2246	0.988840
$759$	−0.221350	−0.00803449
$760$	0	0
$761$	18.3597	0.665539	0.332770	−	0.943008i	$-0.392017\pi$
$761$	18.3597	0.665539	0.332770	+	0.943008i	$0.392017\pi$
$762$	−32.0483	−1.16099
$763$	23.5865	0.853887
$764$	27.6878	1.00171
$765$	0	0
$766$	−14.8476	−0.536466
$767$	−9.80714	−0.354115
$768$	−41.6877	−1.50427
$769$	25.7436	0.928339	0.464170	−	0.885746i	$-0.346353\pi$
$769$	25.7436	0.928339	0.464170	+	0.885746i	$0.346353\pi$
$770$	0	0
$771$	−26.9474	−0.970486
$772$	−0.565447	−0.0203509
$773$	6.64313	0.238937	0.119468	−	0.992838i	$-0.461881\pi$
$773$	6.64313	0.238937	0.119468	+	0.992838i	$0.461881\pi$
$774$	−25.0648	−0.900936
$775$	0	0
$776$	−22.2967	−0.800405
$777$	−23.6374	−0.847987
$778$	−24.7082	−0.885831
$779$	−56.0291	−2.00745
$780$	0	0
$781$	−0.320431	−0.0114659
$782$	0.267641	0.00957083
$783$	25.2004	0.900591
$784$	3.41906	0.122109
$785$	0	0
$786$	−37.1428	−1.32484
$787$	34.7919	1.24020	0.620098	−	0.784524i	$-0.287093\pi$
$787$	34.7919	1.24020	0.620098	+	0.784524i	$0.287093\pi$
$788$	23.0176	0.819968
$789$	29.1419	1.03748
$790$	0	0
$791$	−21.3597	−0.759464
$792$	5.26322	0.187020
$793$	6.92067	0.245760
$794$	−5.08156	−0.180338
$795$	0	0
$796$	−6.14949	−0.217963
$797$	−6.42823	−0.227700	−0.113850	−	0.993498i	$-0.536318\pi$
$797$	−6.42823	−0.227700	−0.113850	+	0.993498i	$0.536318\pi$
$798$	41.4561	1.46753
$799$	−5.80840	−0.205487
$800$	0	0
$801$	−13.5868	−0.480067
$802$	−9.00202	−0.317872
$803$	2.32748	0.0821351
$804$	5.38343	0.189859
$805$	0	0
$806$	−4.57214	−0.161047
$807$	−23.4986	−0.827189
$808$	41.6896	1.46663
$809$	−0.268221	−0.00943014	−0.00471507	−	0.999989i	$-0.501501\pi$
$809$	−0.268221	−0.00943014	−0.00471507	+	0.999989i	$0.501501\pi$
$810$	0	0
$811$	−53.7359	−1.88692	−0.943461	−	0.331482i	$-0.892451\pi$
$811$	−53.7359	−1.88692	−0.943461	+	0.331482i	$0.892451\pi$
$812$	−18.6922	−0.655966
$813$	−18.4991	−0.648790
$814$	0.645902	0.0226389
$815$	0	0
$816$	2.23048	0.0780823
$817$	−27.4810	−0.961440
$818$	−4.81242	−0.168262
$819$	38.0770	1.33052
$820$	0	0
$821$	−17.8673	−0.623572	−0.311786	−	0.950152i	$-0.600927\pi$
$821$	−17.8673	−0.623572	−0.311786	+	0.950152i	$0.600927\pi$
$822$	−2.55496	−0.0891145
$823$	22.3833	0.780233	0.390117	−	0.920765i	$-0.372435\pi$
$823$	22.3833	0.780233	0.390117	+	0.920765i	$0.372435\pi$
$824$	−24.5298	−0.854537
$825$	0	0
$826$	15.5397	0.540696
$827$	31.0041	1.07812	0.539059	−	0.842268i	$-0.318780\pi$
$827$	31.0041	1.07812	0.539059	+	0.842268i	$0.318780\pi$
$828$	1.51232	0.0525567
$829$	−28.6243	−0.994162	−0.497081	−	0.867704i	$-0.665595\pi$
$829$	−28.6243	−0.994162	−0.497081	+	0.867704i	$0.665595\pi$
$830$	0	0
$831$	55.8859	1.93866
$832$	7.11616	0.246708
$833$	10.8826	0.377061
$834$	−25.7091	−0.890234
$835$	0	0
$836$	2.31889	0.0802005
$837$	19.9937	0.691084
$838$	−14.3261	−0.494888
$839$	−42.5973	−1.47062	−0.735311	−	0.677730i	$-0.762964\pi$
$839$	−42.5973	−1.47062	−0.735311	+	0.677730i	$0.762964\pi$
$840$	0	0
$841$	−15.1177	−0.521298
$842$	28.3876	0.978300
$843$	−15.6508	−0.539044
$844$	14.7751	0.508581
$845$	0	0
$846$	16.0333	0.551235
$847$	−40.5778	−1.39427
$848$	−1.15525	−0.0396713
$849$	53.1482	1.82404
$850$	0	0
$851$	0.461847	0.0158319
$852$	3.41878	0.117125
$853$	48.2378	1.65163	0.825815	−	0.563942i	$-0.190716\pi$
$853$	48.2378	1.65163	0.825815	+	0.563942i	$0.190716\pi$
$854$	−10.9660	−0.375249
$855$	0	0
$856$	−41.4336	−1.41617
$857$	−5.24655	−0.179219	−0.0896094	−	0.995977i	$-0.528562\pi$
$857$	−5.24655	−0.179219	−0.0896094	+	0.995977i	$0.528562\pi$
$858$	−1.62481	−0.0554701
$859$	14.5107	0.495098	0.247549	−	0.968875i	$-0.420375\pi$
$859$	14.5107	0.495098	0.247549	+	0.968875i	$0.420375\pi$
$860$	0	0
$861$	127.339	4.33971
$862$	−3.05150	−0.103935
$863$	−4.24273	−0.144424	−0.0722122	−	0.997389i	$-0.523006\pi$
$863$	−4.24273	−0.144424	−0.0722122	+	0.997389i	$0.523006\pi$
$864$	−39.3429	−1.33847
$865$	0	0
$866$	15.3182	0.520533
$867$	−42.0002	−1.42640
$868$	−14.8301	−0.503368
$869$	−0.104206	−0.00353496
$870$	0	0
$871$	−2.64836	−0.0897364
$872$	17.1122	0.579492
$873$	−43.9677	−1.48808
$874$	−0.810003	−0.0273988
$875$	0	0
$876$	−24.8326	−0.839017
$877$	43.9611	1.48446	0.742231	−	0.670144i	$-0.233768\pi$
$877$	43.9611	1.48446	0.742231	+	0.670144i	$0.233768\pi$
$878$	−17.7751	−0.599880
$879$	34.5065	1.16387
$880$	0	0
$881$	10.7746	0.363005	0.181503	−	0.983390i	$-0.441904\pi$
$881$	10.7746	0.363005	0.181503	+	0.983390i	$0.441904\pi$
$882$	−30.0400	−1.01150
$883$	−0.839677	−0.0282574	−0.0141287	−	0.999900i	$-0.504497\pi$
$883$	−0.839677	−0.0282574	−0.0141287	+	0.999900i	$0.504497\pi$
$884$	−4.02162	−0.135262
$885$	0	0
$886$	0.835262	0.0280612
$887$	23.0506	0.773965	0.386982	−	0.922087i	$-0.373517\pi$
$887$	23.0506	0.773965	0.386982	+	0.922087i	$0.373517\pi$
$888$	−17.1492	−0.575488
$889$	51.1386	1.71513
$890$	0	0
$891$	1.27642	0.0427618
$892$	−21.3798	−0.715850
$893$	17.5789	0.588254
$894$	−22.6386	−0.757147
$895$	0	0
$896$	32.1623	1.07447
$897$	−1.16181	−0.0387916
$898$	12.7374	0.425052
$899$	11.0141	0.367340
$900$	0	0
$901$	−3.67707	−0.122501
$902$	−3.47960	−0.115858
$903$	62.4571	2.07844
$904$	−15.4967	−0.515412
$905$	0	0
$906$	23.9904	0.797027
$907$	1.71072	0.0568036	0.0284018	−	0.999597i	$-0.490958\pi$
$907$	1.71072	0.0568036	0.0284018	+	0.999597i	$0.490958\pi$
$908$	−5.13688	−0.170473
$909$	82.2092	2.72671
$910$	0	0
$911$	−27.0421	−0.895944	−0.447972	−	0.894048i	$-0.647854\pi$
$911$	−27.0421	−0.895944	−0.447972	+	0.894048i	$0.647854\pi$
$912$	−6.75043	−0.223529
$913$	2.25351	0.0745804
$914$	−7.58224	−0.250798
$915$	0	0
$916$	2.74931	0.0908397
$917$	59.2679	1.95720
$918$	−8.59115	−0.283550
$919$	−31.9673	−1.05450	−0.527251	−	0.849709i	$-0.676777\pi$
$919$	−31.9673	−1.05450	−0.527251	+	0.849709i	$0.676777\pi$
$920$	0	0
$921$	83.5460	2.75293
$922$	−23.9456	−0.788606
$923$	−1.68186	−0.0553591
$924$	−5.27022	−0.173377
$925$	0	0
$926$	8.45299	0.277782
$927$	−48.3712	−1.58872
$928$	−21.6731	−0.711455
$929$	−13.0426	−0.427913	−0.213956	−	0.976843i	$-0.568635\pi$
$929$	−13.0426	−0.427913	−0.213956	+	0.976843i	$0.568635\pi$
$930$	0	0
$931$	−32.9358	−1.07943
$932$	−9.99214	−0.327303
$933$	31.7581	1.03971
$934$	−0.344818	−0.0112828
$935$	0	0
$936$	27.6253	0.902960
$937$	3.71766	0.121451	0.0607254	−	0.998155i	$-0.480659\pi$
$937$	3.71766	0.121451	0.0607254	+	0.998155i	$0.480659\pi$
$938$	4.19642	0.137018
$939$	−84.8182	−2.76794
$940$	0	0
$941$	−3.90979	−0.127455	−0.0637277	−	0.997967i	$-0.520299\pi$
$941$	−3.90979	−0.127455	−0.0637277	+	0.997967i	$0.520299\pi$
$942$	4.13950	0.134872
$943$	−2.48806	−0.0810224
$944$	−2.53039	−0.0823570
$945$	0	0
$946$	−1.70667	−0.0554886
$947$	−7.99856	−0.259918	−0.129959	−	0.991519i	$-0.541485\pi$
$947$	−7.99856	−0.259918	−0.129959	+	0.991519i	$0.541485\pi$
$948$	1.11181	0.0361099
$949$	12.2164	0.396560
$950$	0	0
$951$	26.6187	0.863169
$952$	15.8578	0.513953
$953$	6.18501	0.200352	0.100176	−	0.994970i	$-0.468059\pi$
$953$	6.18501	0.200352	0.100176	+	0.994970i	$0.468059\pi$
$954$	10.1500	0.328619
$955$	0	0
$956$	32.5611	1.05310
$957$	3.91410	0.126525
$958$	−27.1819	−0.878207
$959$	4.07689	0.131650
$960$	0	0
$961$	−22.2616	−0.718115
$962$	3.39017	0.109304
$963$	−81.7043	−2.63288
$964$	1.34362	0.0432752
$965$	0	0
$966$	1.84092	0.0592307
$967$	3.11816	0.100273	0.0501366	−	0.998742i	$-0.484034\pi$
$967$	3.11816	0.100273	0.0501366	+	0.998742i	$0.484034\pi$
$968$	−29.4396	−0.946224
$969$	−21.4862	−0.690235
$970$	0	0
$971$	−56.5229	−1.81391	−0.906953	−	0.421231i	$-0.861598\pi$
$971$	−56.5229	−1.81391	−0.906953	+	0.421231i	$0.861598\pi$
$972$	13.6446	0.437650
$973$	41.0234	1.31515
$974$	7.43987	0.238389
$975$	0	0
$976$	1.78563	0.0571567
$977$	4.97426	0.159141	0.0795703	−	0.996829i	$-0.474645\pi$
$977$	4.97426	0.159141	0.0795703	+	0.996829i	$0.474645\pi$
$978$	−22.6960	−0.725739
$979$	−0.925129	−0.0295673
$980$	0	0
$981$	33.7442	1.07737
$982$	−15.6067	−0.498030
$983$	−22.5018	−0.717696	−0.358848	−	0.933396i	$-0.616830\pi$
$983$	−22.5018	−0.717696	−0.358848	+	0.933396i	$0.616830\pi$
$984$	92.3859	2.94515
$985$	0	0
$986$	−4.73266	−0.150719
$987$	−39.9521	−1.27169
$988$	12.1713	0.387219
$989$	−1.22034	−0.0388045
$990$	0	0
$991$	4.13964	0.131500	0.0657500	−	0.997836i	$-0.479056\pi$
$991$	4.13964	0.131500	0.0657500	+	0.997836i	$0.479056\pi$
$992$	−17.1952	−0.545948
$993$	90.8408	2.88275
$994$	2.66496	0.0845274
$995$	0	0
$996$	−24.0434	−0.761846
$997$	2.34256	0.0741896	0.0370948	−	0.999312i	$-0.488190\pi$
$997$	2.34256	0.0741896	0.0370948	+	0.999312i	$0.488190\pi$
$998$	10.3931	0.328988
$999$	−14.8251	−0.469044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.o.1.13	yes	40
5.4	even	2		6025.2.a.l.1.28	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.28	✓	40	5.4	even	2
6025.2.a.o.1.13	yes	40	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.810170 q^{2} +2.88822 q^{3} -1.34362 q^{4} -2.33995 q^{6} +3.73380 q^{7} +2.70891 q^{8} +5.34179 q^{9} +O(q^{10})\)	\(q-0.810170 q^{2} +2.88822 q^{3} -1.34362 q^{4} -2.33995 q^{6} +3.73380 q^{7} +2.70891 q^{8} +5.34179 q^{9} +0.363723 q^{11} -3.88067 q^{12} +1.90909 q^{13} -3.02501 q^{14} +0.492573 q^{16} +1.56783 q^{17} -4.32776 q^{18} -4.74495 q^{19} +10.7840 q^{21} -0.294678 q^{22} -0.210707 q^{23} +7.82390 q^{24} -1.54669 q^{26} +6.76358 q^{27} -5.01682 q^{28} +3.72590 q^{29} +2.95609 q^{31} -5.81688 q^{32} +1.05051 q^{33} -1.27021 q^{34} -7.17735 q^{36} -2.19189 q^{37} +3.84422 q^{38} +5.51386 q^{39} +11.8082 q^{41} -8.73688 q^{42} +5.79164 q^{43} -0.488707 q^{44} +0.170708 q^{46} -3.70475 q^{47} +1.42266 q^{48} +6.94123 q^{49} +4.52822 q^{51} -2.56510 q^{52} -2.34533 q^{53} -5.47966 q^{54} +10.1145 q^{56} -13.7044 q^{57} -3.01862 q^{58} -5.13708 q^{59} +3.62512 q^{61} -2.39493 q^{62} +19.9451 q^{63} +3.72752 q^{64} -0.851093 q^{66} -1.38724 q^{67} -2.10657 q^{68} -0.608567 q^{69} -0.880976 q^{71} +14.4704 q^{72} +6.39905 q^{73} +1.77581 q^{74} +6.37543 q^{76} +1.35807 q^{77} -4.46716 q^{78} -0.286499 q^{79} +3.50933 q^{81} -9.56662 q^{82} +6.19569 q^{83} -14.4896 q^{84} -4.69222 q^{86} +10.7612 q^{87} +0.985291 q^{88} -2.54350 q^{89} +7.12814 q^{91} +0.283111 q^{92} +8.53781 q^{93} +3.00148 q^{94} -16.8004 q^{96} -8.23089 q^{97} -5.62358 q^{98} +1.94293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9	\( 40 q + 11 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 16 q^{7} + 33 q^{8} + 38 q^{9} + q^{11} + 14 q^{12} + 9 q^{13} - q^{14} + 43 q^{16} + 12 q^{17} + 42 q^{18} + 2 q^{21} + 5 q^{22} + 77 q^{23} - 2 q^{24} + 2 q^{26} + 38 q^{27} + 42 q^{28} + 2 q^{29} + q^{31} + 72 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 28 q^{37} + 23 q^{38} - 2 q^{39} - 2 q^{41} + 37 q^{42} + 31 q^{43} + 3 q^{44} + 14 q^{46} + 96 q^{47} + 13 q^{48} + 40 q^{49} - 10 q^{51} + 42 q^{52} + 54 q^{53} + 4 q^{54} - 15 q^{56} + 37 q^{57} + 27 q^{58} + q^{59} + 5 q^{61} + 39 q^{62} + 70 q^{63} + 65 q^{64} - 52 q^{66} + 34 q^{67} + 52 q^{68} + 21 q^{69} - 9 q^{71} + 70 q^{72} + 25 q^{73} + 22 q^{74} - 47 q^{76} + 54 q^{77} + 58 q^{78} + 13 q^{79} + 12 q^{81} - 5 q^{82} + 63 q^{83} + 95 q^{84} - 18 q^{86} + 47 q^{87} + 13 q^{88} + 19 q^{89} - 31 q^{91} + 137 q^{92} + 52 q^{93} + 120 q^{94} - 49 q^{96} + 36 q^{97} + 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q + 11 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 16 * q^7 + 33 * q^8 + 38 * q^9 + q^11 + 14 * q^12 + 9 * q^13 - q^14 + 43 * q^16 + 12 * q^17 + 42 * q^18 + 2 * q^21 + 5 * q^22 + 77 * q^23 - 2 * q^24 + 2 * q^26 + 38 * q^27 + 42 * q^28 + 2 * q^29 + q^31 + 72 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 28 * q^37 + 23 * q^38 - 2 * q^39 - 2 * q^41 + 37 * q^42 + 31 * q^43 + 3 * q^44 + 14 * q^46 + 96 * q^47 + 13 * q^48 + 40 * q^49 - 10 * q^51 + 42 * q^52 + 54 * q^53 + 4 * q^54 - 15 * q^56 + 37 * q^57 + 27 * q^58 + q^59 + 5 * q^61 + 39 * q^62 + 70 * q^63 + 65 * q^64 - 52 * q^66 + 34 * q^67 + 52 * q^68 + 21 * q^69 - 9 * q^71 + 70 * q^72 + 25 * q^73 + 22 * q^74 - 47 * q^76 + 54 * q^77 + 58 * q^78 + 13 * q^79 + 12 * q^81 - 5 * q^82 + 63 * q^83 + 95 * q^84 - 18 * q^86 + 47 * q^87 + 13 * q^88 + 19 * q^89 - 31 * q^91 + 137 * q^92 + 52 * q^93 + 120 * q^94 - 49 * q^96 + 36 * q^97 + 64 * q^98 + 16 * q^99

Introduction

L-functions

Modular forms

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Database

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