LMFDB - Embedded newform 425.6.a.d.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,6,Mod(1,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.1631234205$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.95319$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.95319 q^{2} +9.94571 q^{3} -7.46587 q^{4} +49.2631 q^{6} -13.5050 q^{7} -195.482 q^{8} -144.083 q^{9} +742.882 q^{11} -74.2534 q^{12} +179.248 q^{13} -66.8928 q^{14} -729.353 q^{16} -289.000 q^{17} -713.670 q^{18} -1855.97 q^{19} -134.317 q^{21} +3679.64 q^{22} +2557.44 q^{23} -1944.21 q^{24} +887.850 q^{26} -3849.81 q^{27} +100.826 q^{28} -1108.67 q^{29} -8774.81 q^{31} +2642.80 q^{32} +7388.50 q^{33} -1431.47 q^{34} +1075.70 q^{36} +5237.19 q^{37} -9192.99 q^{38} +1782.75 q^{39} -16598.4 q^{41} -665.297 q^{42} -14484.7 q^{43} -5546.26 q^{44} +12667.5 q^{46} -16088.0 q^{47} -7253.94 q^{48} -16624.6 q^{49} -2874.31 q^{51} -1338.24 q^{52} -11804.4 q^{53} -19068.9 q^{54} +2639.98 q^{56} -18459.0 q^{57} -5491.46 q^{58} -29390.1 q^{59} +29333.5 q^{61} -43463.3 q^{62} +1945.84 q^{63} +36429.6 q^{64} +36596.7 q^{66} -4122.11 q^{67} +2157.64 q^{68} +25435.6 q^{69} -13649.5 q^{71} +28165.6 q^{72} +33582.1 q^{73} +25940.8 q^{74} +13856.4 q^{76} -10032.6 q^{77} +8830.30 q^{78} -86281.4 q^{79} -3277.05 q^{81} -82215.3 q^{82} +73620.8 q^{83} +1002.79 q^{84} -71745.7 q^{86} -11026.5 q^{87} -145220. q^{88} -100199. q^{89} -2420.74 q^{91} -19093.5 q^{92} -87271.8 q^{93} -79687.2 q^{94} +26284.5 q^{96} -4407.67 q^{97} -82344.9 q^{98} -107037. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) $ 5 * q + 7 * q^2 + 36 * q^3 + 43 * q^4 - 105 * q^6 + 204 * q^7 + 63 * q^8 + 531 * q^9 - 792 * q^11 - 785 * q^12 - 88 * q^13 + 860 * q^14 - 2365 * q^16 - 1445 * q^17 + 2052 * q^18 - 5160 * q^19 - 6428 * q^21 + 3058 * q^22 + 6140 * q^23 - 11751 * q^24 - 9643 * q^26 + 13380 * q^27 + 4568 * q^28 - 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 13072 * q^33 - 2023 * q^34 + 2048 * q^36 - 9834 * q^37 - 10625 * q^38 + 10108 * q^39 - 5302 * q^41 - 16222 * q^42 + 4916 * q^43 - 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 45557 * q^48 - 26483 * q^49 - 10404 * q^51 + 11525 * q^52 - 56492 * q^53 + 64635 * q^54 + 73228 * q^56 - 67554 * q^57 - 54213 * q^58 - 96200 * q^59 - 7284 * q^61 - 122335 * q^62 - 64108 * q^63 + 28483 * q^64 + 115232 * q^66 + 60308 * q^67 - 12427 * q^68 - 89076 * q^69 - 25304 * q^71 - 118722 * q^72 - 163328 * q^73 - 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 134895 * q^78 - 50940 * q^79 + 92445 * q^81 - 191390 * q^82 + 179060 * q^83 - 128054 * q^84 + 87086 * q^86 - 55764 * q^87 - 167634 * q^88 - 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 202558 * q^93 + 100515 * q^94 + 428113 * q^96 + 106076 * q^97 - 49329 * q^98 - 535112 * q^99

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 $

$2$	4.95319	0.875609	0.437805	−	0.899070i	$-0.355756\pi$
$2$	4.95319	0.875609	0.437805	+	0.899070i	$0.355756\pi$
$3$	9.94571	0.638018	0.319009	−	0.947752i	$-0.396650\pi$
$3$	9.94571	0.638018	0.319009	+	0.947752i	$0.396650\pi$
$4$	−7.46587	−0.233308
$5$	0	0
$5$	0	0
$6$	49.2631	0.558654
$7$	−13.5050	−0.104172	−0.0520858	−	0.998643i	$-0.516587\pi$
$7$	−13.5050	−0.104172	−0.0520858	+	0.998643i	$0.516587\pi$
$8$	−195.482	−1.07990
$9$	−144.083	−0.592933
$10$	0	0
$11$	742.882	1.85114	0.925568	−	0.378581i	$-0.123588\pi$
$11$	742.882	1.85114	0.925568	+	0.378581i	$0.123588\pi$
$12$	−74.2534	−0.148855
$13$	179.248	0.294168	0.147084	−	0.989124i	$-0.453011\pi$
$13$	179.248	0.294168	0.147084	+	0.989124i	$0.453011\pi$
$14$	−66.8928	−0.0912135
$15$	0	0
$16$	−729.353	−0.712259
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	−713.670	−0.519178
$19$	−1855.97	−1.17947	−0.589735	−	0.807596i	$-0.700768\pi$
$19$	−1855.97	−1.17947	−0.589735	+	0.807596i	$0.700768\pi$
$20$	0	0
$21$	−134.317	−0.0664633
$22$	3679.64	1.62087
$23$	2557.44	1.00806	0.504029	−	0.863687i	$-0.331850\pi$
$23$	2557.44	1.00806	0.504029	+	0.863687i	$0.331850\pi$
$24$	−1944.21	−0.688993
$25$	0	0
$26$	887.850	0.257577
$27$	−3849.81	−1.01632
$28$	100.826	0.0243041
$29$	−1108.67	−0.244798	−0.122399	−	0.992481i	$-0.539059\pi$
$29$	−1108.67	−0.244798	−0.122399	+	0.992481i	$0.539059\pi$
$30$	0	0
$31$	−8774.81	−1.63996	−0.819980	−	0.572391i	$-0.806016\pi$
$31$	−8774.81	−1.63996	−0.819980	+	0.572391i	$0.806016\pi$
$32$	2642.80	0.456236
$33$	7388.50	1.18106
$34$	−1431.47	−0.212366
$35$	0	0
$36$	1075.70	0.138336
$37$	5237.19	0.628919	0.314459	−	0.949271i	$-0.398177\pi$
$37$	5237.19	0.628919	0.314459	+	0.949271i	$0.398177\pi$
$38$	−9192.99	−1.03276
$39$	1782.75	0.187685
$40$	0	0
$41$	−16598.4	−1.54208	−0.771042	−	0.636785i	$-0.780264\pi$
$41$	−16598.4	−1.54208	−0.771042	+	0.636785i	$0.780264\pi$
$42$	−665.297	−0.0581959
$43$	−14484.7	−1.19465	−0.597324	−	0.802000i	$-0.703769\pi$
$43$	−14484.7	−1.19465	−0.597324	+	0.802000i	$0.703769\pi$
$44$	−5546.26	−0.431886
$45$	0	0
$46$	12667.5	0.882665
$47$	−16088.0	−1.06233	−0.531164	−	0.847269i	$-0.678245\pi$
$47$	−16088.0	−1.06233	−0.531164	+	0.847269i	$0.678245\pi$
$48$	−7253.94	−0.454434
$49$	−16624.6	−0.989148
$50$	0	0
$51$	−2874.31	−0.154742
$52$	−1338.24	−0.0686319
$53$	−11804.4	−0.577237	−0.288618	−	0.957444i	$-0.593196\pi$
$53$	−11804.4	−0.577237	−0.288618	+	0.957444i	$0.593196\pi$
$54$	−19068.9	−0.889899
$55$	0	0
$56$	2639.98	0.112494
$57$	−18459.0	−0.752524
$58$	−5491.46	−0.214347
$59$	−29390.1	−1.09919	−0.549593	−	0.835433i	$-0.685217\pi$
$59$	−29390.1	−1.09919	−0.549593	+	0.835433i	$0.685217\pi$
$60$	0	0
$61$	29333.5	1.00934	0.504671	−	0.863311i	$-0.331614\pi$
$61$	29333.5	1.00934	0.504671	+	0.863311i	$0.331614\pi$
$62$	−43463.3	−1.43597
$63$	1945.84	0.0617667
$64$	36429.6	1.11174
$65$	0	0
$66$	36596.7	1.03415
$67$	−4122.11	−0.112184	−0.0560922	−	0.998426i	$-0.517864\pi$
$67$	−4122.11	−0.112184	−0.0560922	+	0.998426i	$0.517864\pi$
$68$	2157.64	0.0565856
$69$	25435.6	0.643159
$70$	0	0
$71$	−13649.5	−0.321343	−0.160672	−	0.987008i	$-0.551366\pi$
$71$	−13649.5	−0.321343	−0.160672	+	0.987008i	$0.551366\pi$
$72$	28165.6	0.640306
$73$	33582.1	0.737566	0.368783	−	0.929516i	$-0.379775\pi$
$73$	33582.1	0.737566	0.368783	+	0.929516i	$0.379775\pi$
$74$	25940.8	0.550687
$75$	0	0
$76$	13856.4	0.275180
$77$	−10032.6	−0.192836
$78$	8830.30	0.164338
$79$	−86281.4	−1.55543	−0.777713	−	0.628619i	$-0.783621\pi$
$79$	−86281.4	−1.55543	−0.777713	+	0.628619i	$0.783621\pi$
$80$	0	0
$81$	−3277.05	−0.0554971
$82$	−82215.3	−1.35026
$83$	73620.8	1.17302	0.586510	−	0.809942i	$-0.300501\pi$
$83$	73620.8	1.17302	0.586510	+	0.809942i	$0.300501\pi$
$84$	1002.79	0.0155064
$85$	0	0
$86$	−71745.7	−1.04604
$87$	−11026.5	−0.156185
$88$	−145220.	−1.99903
$89$	−100199.	−1.34087	−0.670435	−	0.741968i	$-0.733893\pi$
$89$	−100199.	−1.34087	−0.670435	+	0.741968i	$0.733893\pi$
$90$	0	0
$91$	−2420.74	−0.0306440
$92$	−19093.5	−0.235188
$93$	−87271.8	−1.04632
$94$	−79687.2	−0.930184
$95$	0	0
$96$	26284.5	0.291087
$97$	−4407.67	−0.0475642	−0.0237821	−	0.999717i	$-0.507571\pi$
$97$	−4407.67	−0.0475642	−0.0237821	+	0.999717i	$0.507571\pi$
$98$	−82344.9	−0.866107
$99$	−107037.	−1.09760
$100$	0	0
$101$	42746.3	0.416960	0.208480	−	0.978027i	$-0.433148\pi$
$101$	42746.3	0.416960	0.208480	+	0.978027i	$0.433148\pi$
$102$	−14237.0	−0.135494
$103$	192307.	1.78609	0.893043	−	0.449972i	$-0.148566\pi$
$103$	192307.	1.78609	0.893043	+	0.449972i	$0.148566\pi$
$104$	−35039.8	−0.317671
$105$	0	0
$106$	−58469.4	−0.505434
$107$	114157.	0.963927	0.481963	−	0.876191i	$-0.339924\pi$
$107$	114157.	0.963927	0.481963	+	0.876191i	$0.339924\pi$
$108$	28742.2	0.237116
$109$	−98113.3	−0.790973	−0.395487	−	0.918472i	$-0.629424\pi$
$109$	−98113.3	−0.790973	−0.395487	+	0.918472i	$0.629424\pi$
$110$	0	0
$111$	52087.6	0.401261
$112$	9849.90	0.0741971
$113$	−133620.	−0.984412	−0.492206	−	0.870479i	$-0.663809\pi$
$113$	−133620.	−0.984412	−0.492206	+	0.870479i	$0.663809\pi$
$114$	−91430.8	−0.658917
$115$	0	0
$116$	8277.19	0.0571134
$117$	−25826.5	−0.174422
$118$	−145575.	−0.962457
$119$	3902.94	0.0252653
$120$	0	0
$121$	390823.	2.42670
$122$	145294.	0.883790
$123$	−165083.	−0.983877
$124$	65511.6	0.382617
$125$	0	0
$126$	9638.10	0.0540835
$127$	−60761.4	−0.334286	−0.167143	−	0.985933i	$-0.553454\pi$
$127$	−60761.4	−0.334286	−0.167143	+	0.985933i	$0.553454\pi$
$128$	95873.3	0.517217
$129$	−144061.	−0.762206
$130$	0	0
$131$	−283562.	−1.44368	−0.721838	−	0.692062i	$-0.756703\pi$
$131$	−283562.	−1.44368	−0.721838	+	0.692062i	$0.756703\pi$
$132$	−55161.5	−0.275551
$133$	25064.9	0.122867
$134$	−20417.6	−0.0982296
$135$	0	0
$136$	56494.3	0.261913
$137$	116549.	0.530528	0.265264	−	0.964176i	$-0.414541\pi$
$137$	116549.	0.530528	0.265264	+	0.964176i	$0.414541\pi$
$138$	125987.	0.563156
$139$	212209.	0.931595	0.465798	−	0.884891i	$-0.345767\pi$
$139$	212209.	0.931595	0.465798	+	0.884891i	$0.345767\pi$
$140$	0	0
$141$	−160007.	−0.677784
$142$	−67608.4	−0.281371
$143$	133160.	0.544546
$144$	105087.	0.422322
$145$	0	0
$146$	166339.	0.645820
$147$	−165344.	−0.631094
$148$	−39100.2	−0.146732
$149$	8825.81	0.0325678	0.0162839	−	0.999867i	$-0.494816\pi$
$149$	8825.81	0.0325678	0.0162839	+	0.999867i	$0.494816\pi$
$150$	0	0
$151$	353306.	1.26098	0.630491	−	0.776196i	$-0.282853\pi$
$151$	353306.	1.26098	0.630491	+	0.776196i	$0.282853\pi$
$152$	362809.	1.27371
$153$	41639.9	0.143807
$154$	−49693.5	−0.168849
$155$	0	0
$156$	−13309.8	−0.0437884
$157$	−55282.7	−0.178995	−0.0894974	−	0.995987i	$-0.528526\pi$
$157$	−55282.7	−0.178995	−0.0894974	+	0.995987i	$0.528526\pi$
$158$	−427369.	−1.36195
$159$	−117403.	−0.368287
$160$	0	0
$161$	−34538.2	−0.105011
$162$	−16231.9	−0.0485938
$163$	−306844.	−0.904585	−0.452292	−	0.891870i	$-0.649394\pi$
$163$	−306844.	−0.904585	−0.452292	+	0.891870i	$0.649394\pi$
$164$	123922.	0.359781
$165$	0	0
$166$	364658.	1.02711
$167$	589063.	1.63444	0.817222	−	0.576322i	$-0.195513\pi$
$167$	589063.	1.63444	0.817222	+	0.576322i	$0.195513\pi$
$168$	26256.5	0.0717735
$169$	−339163.	−0.913465
$170$	0	0
$171$	267413.	0.699347
$172$	108141.	0.278721
$173$	−647356.	−1.64448	−0.822238	−	0.569143i	$-0.807275\pi$
$173$	−647356.	−1.64448	−0.822238	+	0.569143i	$0.807275\pi$
$174$	−54616.5	−0.136757
$175$	0	0
$176$	−541824.	−1.31849
$177$	−292306.	−0.701300
$178$	−496303.	−1.17408
$179$	385367.	0.898963	0.449482	−	0.893290i	$-0.351609\pi$
$179$	385367.	0.898963	0.449482	+	0.893290i	$0.351609\pi$
$180$	0	0
$181$	94431.6	0.214250	0.107125	−	0.994246i	$-0.465836\pi$
$181$	94431.6	0.214250	0.107125	+	0.994246i	$0.465836\pi$
$182$	−11990.4	−0.0268321
$183$	291742.	0.643979
$184$	−499933.	−1.08860
$185$	0	0
$186$	−432274.	−0.916171
$187$	−214693.	−0.448966
$188$	120111.	0.247850
$189$	51991.7	0.105872
$190$	0	0
$191$	−284007.	−0.563308	−0.281654	−	0.959516i	$-0.590883\pi$
$191$	−284007.	−0.563308	−0.281654	+	0.959516i	$0.590883\pi$
$192$	362318.	0.709312
$193$	−847831.	−1.63838	−0.819192	−	0.573519i	$-0.805578\pi$
$193$	−847831.	−1.63838	−0.819192	+	0.573519i	$0.805578\pi$
$194$	−21832.1	−0.0416476
$195$	0	0
$196$	124117.	0.230777
$197$	−812264.	−1.49119	−0.745593	−	0.666402i	$-0.767833\pi$
$197$	−812264.	−1.49119	−0.745593	+	0.666402i	$0.767833\pi$
$198$	−530173.	−0.961069
$199$	413483.	0.740160	0.370080	−	0.929000i	$-0.379330\pi$
$199$	413483.	0.740160	0.370080	+	0.929000i	$0.379330\pi$
$200$	0	0
$201$	−40997.3	−0.0715756
$202$	211731.	0.365094
$203$	14972.6	0.0255010
$204$	21459.2	0.0361026
$205$	0	0
$206$	952534.	1.56391
$207$	−368483.	−0.597711
$208$	−130735.	−0.209524
$209$	−1.37877e6	−2.18336
$210$	0	0
$211$	−1.08516e6	−1.67798	−0.838990	−	0.544146i	$-0.816854\pi$
$211$	−1.08516e6	−1.67798	−0.838990	+	0.544146i	$0.816854\pi$
$212$	88130.0	0.134674
$213$	−135754.	−0.205023
$214$	565443.	0.844023
$215$	0	0
$216$	752570.	1.09752
$217$	118504.	0.170837
$218$	−485974.	−0.692584
$219$	333998.	0.470580
$220$	0	0
$221$	−51802.7	−0.0713463
$222$	258000.	0.351348
$223$	1.31798e6	1.77479	0.887395	−	0.461010i	$-0.152513\pi$
$223$	1.31798e6	1.77479	0.887395	+	0.461010i	$0.152513\pi$
$224$	−35691.0	−0.0475268
$225$	0	0
$226$	−661848.	−0.861960
$227$	813956.	1.04842	0.524211	−	0.851588i	$-0.324360\pi$
$227$	813956.	1.04842	0.524211	+	0.851588i	$0.324360\pi$
$228$	137812.	0.175570
$229$	1.19764e6	1.50917	0.754586	−	0.656201i	$-0.227838\pi$
$229$	1.19764e6	1.50917	0.754586	+	0.656201i	$0.227838\pi$
$230$	0	0
$231$	−99781.5	−0.123033
$232$	216725.	0.264356
$233$	1.07508e6	1.29733	0.648663	−	0.761076i	$-0.275328\pi$
$233$	1.07508e6	1.29733	0.648663	+	0.761076i	$0.275328\pi$
$234$	−127924.	−0.152726
$235$	0	0
$236$	219423.	0.256449
$237$	−858131.	−0.992390
$238$	19332.0	0.0221225
$239$	−1.50327e6	−1.70232	−0.851162	−	0.524903i	$-0.824101\pi$
$239$	−1.50327e6	−1.70232	−0.851162	+	0.524903i	$0.824101\pi$
$240$	0	0
$241$	−259714.	−0.288040	−0.144020	−	0.989575i	$-0.546003\pi$
$241$	−259714.	−0.288040	−0.144020	+	0.989575i	$0.546003\pi$
$242$	1.93582e6	2.12485
$243$	902912.	0.980912
$244$	−219000.	−0.235488
$245$	0	0
$246$	−817690.	−0.861491
$247$	−332679.	−0.346963
$248$	1.71532e6	1.77099
$249$	732212.	0.748408
$250$	0	0
$251$	417745.	0.418530	0.209265	−	0.977859i	$-0.432893\pi$
$251$	417745.	0.418530	0.209265	+	0.977859i	$0.432893\pi$
$252$	−14527.4	−0.0144107
$253$	1.89988e6	1.86605
$254$	−300963.	−0.292704
$255$	0	0
$256$	−690868.	−0.658863
$257$	−1.23266e6	−1.16415	−0.582076	−	0.813134i	$-0.697760\pi$
$257$	−1.23266e6	−1.16415	−0.582076	+	0.813134i	$0.697760\pi$
$258$	−713563.	−0.667395
$259$	−70728.2	−0.0655154
$260$	0	0
$261$	159740.	0.145149
$262$	−1.40454e6	−1.26410
$263$	−210447.	−0.187609	−0.0938044	−	0.995591i	$-0.529903\pi$
$263$	−210447.	−0.187609	−0.0938044	+	0.995591i	$0.529903\pi$
$264$	−1.44432e6	−1.27542
$265$	0	0
$266$	124151.	0.107584
$267$	−996547.	−0.855499
$268$	30775.1	0.0261735
$269$	1.19459e6	1.00656	0.503280	−	0.864123i	$-0.332126\pi$
$269$	1.19459e6	1.00656	0.503280	+	0.864123i	$0.332126\pi$
$270$	0	0
$271$	−486392.	−0.402312	−0.201156	−	0.979559i	$-0.564470\pi$
$271$	−486392.	−0.402312	−0.201156	+	0.979559i	$0.564470\pi$
$272$	210783.	0.172748
$273$	−24076.0	−0.0195514
$274$	577292.	0.464535
$275$	0	0
$276$	−189898.	−0.150054
$277$	681163.	0.533399	0.266699	−	0.963780i	$-0.414067\pi$
$277$	681163.	0.533399	0.266699	+	0.963780i	$0.414067\pi$
$278$	1.05111e6	0.815713
$279$	1.26430e6	0.972387
$280$	0	0
$281$	1.22238e6	0.923509	0.461754	−	0.887008i	$-0.347220\pi$
$281$	1.22238e6	0.923509	0.461754	+	0.887008i	$0.347220\pi$
$282$	−792546.	−0.593474
$283$	−44528.8	−0.0330503	−0.0165251	−	0.999863i	$-0.505260\pi$
$283$	−44528.8	−0.0330503	−0.0165251	+	0.999863i	$0.505260\pi$
$284$	101905.	0.0749721
$285$	0	0
$286$	659568.	0.476809
$287$	224162.	0.160641
$288$	−380782.	−0.270517
$289$	83521.0	0.0588235
$290$	0	0
$291$	−43837.4	−0.0303468
$292$	−250720.	−0.172080
$293$	−151289.	−0.102953	−0.0514763	−	0.998674i	$-0.516393\pi$
$293$	−151289.	−0.102953	−0.0514763	+	0.998674i	$0.516393\pi$
$294$	−818979.	−0.552592
$295$	0	0
$296$	−1.02378e6	−0.679167
$297$	−2.85996e6	−1.88135
$298$	43715.9	0.0285167
$299$	458416.	0.296539
$300$	0	0
$301$	195616.	0.124448
$302$	1.74999e6	1.10413
$303$	425142.	0.266028
$304$	1.35366e6	0.840089
$305$	0	0
$306$	206251.	0.125919
$307$	−20159.6	−0.0122077	−0.00610387	−	0.999981i	$-0.501943\pi$
$307$	−20159.6	−0.0122077	−0.00610387	+	0.999981i	$0.501943\pi$
$308$	74902.2	0.0449902
$309$	1.91263e6	1.13955
$310$	0	0
$311$	1.75609e6	1.02954	0.514772	−	0.857327i	$-0.327877\pi$
$311$	1.75609e6	1.02954	0.514772	+	0.857327i	$0.327877\pi$
$312$	−348496.	−0.202680
$313$	−683041.	−0.394081	−0.197041	−	0.980395i	$-0.563133\pi$
$313$	−683041.	−0.394081	−0.197041	+	0.980395i	$0.563133\pi$
$314$	−273826.	−0.156729
$315$	0	0
$316$	644166.	0.362894
$317$	−87223.8	−0.0487514	−0.0243757	−	0.999703i	$-0.507760\pi$
$317$	−87223.8	−0.0487514	−0.0243757	+	0.999703i	$0.507760\pi$
$318$	−581520.	−0.322476
$319$	−823612.	−0.453154
$320$	0	0
$321$	1.13538e6	0.615002
$322$	−171074.	−0.0919486
$323$	536376.	0.286064
$324$	24466.0	0.0129479
$325$	0	0
$326$	−1.51986e6	−0.792063
$327$	−975807.	−0.504655
$328$	3.24470e6	1.66529
$329$	217269.	0.110664
$330$	0	0
$331$	2.57320e6	1.29093	0.645467	−	0.763788i	$-0.276663\pi$
$331$	2.57320e6	1.29093	0.645467	+	0.763788i	$0.276663\pi$
$332$	−549643.	−0.273675
$333$	−754589.	−0.372907
$334$	2.91774e6	1.43114
$335$	0	0
$336$	97964.3	0.0473391
$337$	1.70829e6	0.819384	0.409692	−	0.912224i	$-0.365636\pi$
$337$	1.70829e6	0.819384	0.409692	+	0.912224i	$0.365636\pi$
$338$	−1.67994e6	−0.799838
$339$	−1.32895e6	−0.628072
$340$	0	0
$341$	−6.51865e6	−3.03579
$342$	1.32455e6	0.612355
$343$	451493.	0.207213
$344$	2.83151e6	1.29009
$345$	0	0
$346$	−3.20648e6	−1.43992
$347$	−1.00945e6	−0.450051	−0.225025	−	0.974353i	$-0.572247\pi$
$347$	−1.00945e6	−0.450051	−0.225025	+	0.974353i	$0.572247\pi$
$348$	82322.6	0.0364394
$349$	1.26376e6	0.555394	0.277697	−	0.960669i	$-0.410429\pi$
$349$	1.26376e6	0.555394	0.277697	+	0.960669i	$0.410429\pi$
$350$	0	0
$351$	−690072.	−0.298969
$352$	1.96329e6	0.844555
$353$	−1.36753e6	−0.584117	−0.292059	−	0.956400i	$-0.594340\pi$
$353$	−1.36753e6	−0.584117	−0.292059	+	0.956400i	$0.594340\pi$
$354$	−1.44785e6	−0.614065
$355$	0	0
$356$	748070.	0.312836
$357$	38817.5	0.0161197
$358$	1.90880e6	0.787141
$359$	−509814.	−0.208774	−0.104387	−	0.994537i	$-0.533288\pi$
$359$	−509814.	−0.208774	−0.104387	+	0.994537i	$0.533288\pi$
$360$	0	0
$361$	968531.	0.391152
$362$	467738.	0.187599
$363$	3.88702e6	1.54828
$364$	18072.9	0.00714949
$365$	0	0
$366$	1.44506e6	0.563874
$367$	−1.26468e6	−0.490136	−0.245068	−	0.969506i	$-0.578810\pi$
$367$	−1.26468e6	−0.490136	−0.245068	+	0.969506i	$0.578810\pi$
$368$	−1.86528e6	−0.717998
$369$	2.39155e6	0.914352
$370$	0	0
$371$	159418.	0.0601316
$372$	651559.	0.244116
$373$	2.53071e6	0.941827	0.470913	−	0.882179i	$-0.343924\pi$
$373$	2.53071e6	0.941827	0.470913	+	0.882179i	$0.343924\pi$
$374$	−1.06342e6	−0.393119
$375$	0	0
$376$	3.14492e6	1.14720
$377$	−198727.	−0.0720118
$378$	257525.	0.0927021
$379$	1.23704e6	0.442370	0.221185	−	0.975232i	$-0.429007\pi$
$379$	1.23704e6	0.442370	0.221185	+	0.975232i	$0.429007\pi$
$380$	0	0
$381$	−604315.	−0.213280
$382$	−1.40674e6	−0.493237
$383$	118138.	0.0411522	0.0205761	−	0.999788i	$-0.493450\pi$
$383$	118138.	0.0411522	0.0205761	+	0.999788i	$0.493450\pi$
$384$	953528.	0.329994
$385$	0	0
$386$	−4.19947e6	−1.43458
$387$	2.08700e6	0.708346
$388$	32907.1	0.0110971
$389$	752781.	0.252229	0.126114	−	0.992016i	$-0.459749\pi$
$389$	752781.	0.252229	0.126114	+	0.992016i	$0.459749\pi$
$390$	0	0
$391$	−739100.	−0.244490
$392$	3.24981e6	1.06818
$393$	−2.82023e6	−0.921091
$394$	−4.02330e6	−1.30570
$395$	0	0
$396$	799121.	0.256079
$397$	3.41832e6	1.08852	0.544261	−	0.838916i	$-0.316810\pi$
$397$	3.41832e6	1.08852	0.544261	+	0.838916i	$0.316810\pi$
$398$	2.04806e6	0.648091
$399$	249288.	0.0783915
$400$	0	0
$401$	−349725.	−0.108609	−0.0543044	−	0.998524i	$-0.517294\pi$
$401$	−349725.	−0.108609	−0.0543044	+	0.998524i	$0.517294\pi$
$402$	−203068.	−0.0626723
$403$	−1.57287e6	−0.482425
$404$	−319138.	−0.0972803
$405$	0	0
$406$	74162.1	0.0223289
$407$	3.89062e6	1.16421
$408$	561876.	0.167105
$409$	1.04602e6	0.309193	0.154597	−	0.987978i	$-0.450592\pi$
$409$	1.04602e6	0.309193	0.154597	+	0.987978i	$0.450592\pi$
$410$	0	0
$411$	1.15917e6	0.338487
$412$	−1.43574e6	−0.416709
$413$	396913.	0.114504
$414$	−1.82517e6	−0.523361
$415$	0	0
$416$	473717.	0.134210
$417$	2.11057e6	0.594374
$418$	−6.82931e6	−1.91177
$419$	−1.09315e6	−0.304189	−0.152095	−	0.988366i	$-0.548602\pi$
$419$	−1.09315e6	−0.304189	−0.152095	+	0.988366i	$0.548602\pi$
$420$	0	0
$421$	−3.13270e6	−0.861416	−0.430708	−	0.902491i	$-0.641736\pi$
$421$	−3.13270e6	−0.861416	−0.430708	+	0.902491i	$0.641736\pi$
$422$	−5.37500e6	−1.46926
$423$	2.31801e6	0.629889
$424$	2.30755e6	0.623356
$425$	0	0
$426$	−672414.	−0.179520
$427$	−396148.	−0.105145
$428$	−852283.	−0.224892
$429$	1.32437e6	0.347430
$430$	0	0
$431$	−5.19507e6	−1.34710	−0.673548	−	0.739144i	$-0.735230\pi$
$431$	−5.19507e6	−1.34710	−0.673548	+	0.739144i	$0.735230\pi$
$432$	2.80787e6	0.723883
$433$	−3.87822e6	−0.994061	−0.497030	−	0.867733i	$-0.665576\pi$
$433$	−3.87822e6	−0.994061	−0.497030	+	0.867733i	$0.665576\pi$
$434$	586972.	0.149587
$435$	0	0
$436$	732501.	0.184541
$437$	−4.74653e6	−1.18898
$438$	1.65436e6	0.412045
$439$	1.95323e6	0.483717	0.241858	−	0.970312i	$-0.422243\pi$
$439$	1.95323e6	0.483717	0.241858	+	0.970312i	$0.422243\pi$
$440$	0	0
$441$	2.39532e6	0.586499
$442$	−256589.	−0.0624715
$443$	−386945.	−0.0936786	−0.0468393	−	0.998902i	$-0.514915\pi$
$443$	−386945.	−0.0936786	−0.0468393	+	0.998902i	$0.514915\pi$
$444$	−388880.	−0.0936176
$445$	0	0
$446$	6.52822e6	1.55402
$447$	87779.0	0.0207789
$448$	−491981.	−0.115812
$449$	3.34050e6	0.781980	0.390990	−	0.920395i	$-0.372133\pi$
$449$	3.34050e6	0.781980	0.390990	+	0.920395i	$0.372133\pi$
$450$	0	0
$451$	−1.23307e7	−2.85461
$452$	997593.	0.229672
$453$	3.51388e6	0.804529
$454$	4.03168e6	0.918009
$455$	0	0
$456$	3.60840e6	0.812647
$457$	4.35918e6	0.976370	0.488185	−	0.872740i	$-0.337659\pi$
$457$	4.35918e6	0.976370	0.488185	+	0.872740i	$0.337659\pi$
$458$	5.93216e6	1.32145
$459$	1.11260e6	0.246494
$460$	0	0
$461$	−3.93227e6	−0.861769	−0.430884	−	0.902407i	$-0.641798\pi$
$461$	−3.93227e6	−0.861769	−0.430884	+	0.902407i	$0.641798\pi$
$462$	−494237.	−0.107728
$463$	4.47600e6	0.970370	0.485185	−	0.874412i	$-0.338752\pi$
$463$	4.47600e6	0.970370	0.485185	+	0.874412i	$0.338752\pi$
$464$	808612.	0.174359
$465$	0	0
$466$	5.32506e6	1.13595
$467$	5.93018e6	1.25828	0.629138	−	0.777294i	$-0.283408\pi$
$467$	5.93018e6	1.25828	0.629138	+	0.777294i	$0.283408\pi$
$468$	192818.	0.0406941
$469$	55669.0	0.0116864
$470$	0	0
$471$	−549826.	−0.114202
$472$	5.74524e6	1.18701
$473$	−1.07605e7	−2.21145
$474$	−4.25049e6	−0.868946
$475$	0	0
$476$	−29138.8	−0.00589461
$477$	1.70081e6	0.342263
$478$	−7.44599e6	−1.49057
$479$	2.00176e6	0.398632	0.199316	−	0.979935i	$-0.436128\pi$
$479$	2.00176e6	0.398632	0.199316	+	0.979935i	$0.436128\pi$
$480$	0	0
$481$	938757.	0.185008
$482$	−1.28641e6	−0.252211
$483$	−343507.	−0.0669989
$484$	−2.91783e6	−0.566171
$485$	0	0
$486$	4.47230e6	0.858895
$487$	−7.29169e6	−1.39317	−0.696587	−	0.717472i	$-0.745299\pi$
$487$	−7.29169e6	−1.39317	−0.696587	+	0.717472i	$0.745299\pi$
$488$	−5.73417e6	−1.08999
$489$	−3.05179e6	−0.577141
$490$	0	0
$491$	−814260.	−0.152426	−0.0762130	−	0.997092i	$-0.524283\pi$
$491$	−814260.	−0.152426	−0.0762130	+	0.997092i	$0.524283\pi$
$492$	1.23249e6	0.229547
$493$	320406.	0.0593722
$494$	−1.64782e6	−0.303804
$495$	0	0
$496$	6.39993e6	1.16808
$497$	184336.	0.0334748
$498$	3.62679e6	0.655313
$499$	−1.65977e6	−0.298399	−0.149199	−	0.988807i	$-0.547670\pi$
$499$	−1.65977e6	−0.298399	−0.149199	+	0.988807i	$0.547670\pi$
$500$	0	0
$501$	5.85865e6	1.04281
$502$	2.06917e6	0.366469
$503$	−1.07132e7	−1.88799	−0.943997	−	0.329955i	$-0.892966\pi$
$503$	−1.07132e7	−1.88799	−0.943997	+	0.329955i	$0.892966\pi$
$504$	−380376.	−0.0667017
$505$	0	0
$506$	9.41045e6	1.63393
$507$	−3.37322e6	−0.582807
$508$	453636.	0.0779917
$509$	−2.37549e6	−0.406405	−0.203202	−	0.979137i	$-0.565135\pi$
$509$	−2.37549e6	−0.406405	−0.203202	+	0.979137i	$0.565135\pi$
$510$	0	0
$511$	−453526.	−0.0768334
$512$	−6.48995e6	−1.09412
$513$	7.14515e6	1.19872
$514$	−6.10560e6	−1.01934
$515$	0	0
$516$	1.07554e6	0.177829
$517$	−1.19515e7	−1.96651
$518$	−350331.	−0.0573659
$519$	−6.43841e6	−1.04921
$520$	0	0
$521$	−1.78521e6	−0.288135	−0.144067	−	0.989568i	$-0.546018\pi$
$521$	−1.78521e6	−0.288135	−0.144067	+	0.989568i	$0.546018\pi$
$522$	791225.	0.127094
$523$	3.77324e6	0.603199	0.301599	−	0.953435i	$-0.402479\pi$
$523$	3.77324e6	0.603199	0.301599	+	0.953435i	$0.402479\pi$
$524$	2.11704e6	0.336822
$525$	0	0
$526$	−1.04238e6	−0.164272
$527$	2.53592e6	0.397749
$528$	−5.38882e6	−0.841219
$529$	104148.	0.0161812
$530$	0	0
$531$	4.23461e6	0.651744
$532$	−187131.	−0.0286660
$533$	−2.97524e6	−0.453632
$534$	−4.93609e6	−0.749083
$535$	0	0
$536$	805798.	0.121147
$537$	3.83275e6	0.573555
$538$	5.91706e6	0.881354
$539$	−1.23501e7	−1.83105
$540$	0	0
$541$	1.60332e6	0.235519	0.117760	−	0.993042i	$-0.462429\pi$
$541$	1.60332e6	0.235519	0.117760	+	0.993042i	$0.462429\pi$
$542$	−2.40919e6	−0.352268
$543$	939189.	0.136695
$544$	−763769.	−0.110653
$545$	0	0
$546$	−119253.	−0.0171194
$547$	−1.05509e7	−1.50772	−0.753861	−	0.657034i	$-0.771811\pi$
$547$	−1.05509e7	−1.50772	−0.753861	+	0.657034i	$0.771811\pi$
$548$	−870142.	−0.123777
$549$	−4.22645e6	−0.598473
$550$	0	0
$551$	2.05766e6	0.288732
$552$	−4.97220e6	−0.694545
$553$	1.16523e6	0.162031
$554$	3.37393e6	0.467049
$555$	0	0
$556$	−1.58433e6	−0.217349
$557$	−5.14612e6	−0.702817	−0.351409	−	0.936222i	$-0.614297\pi$
$557$	−5.14612e6	−0.702817	−0.351409	+	0.936222i	$0.614297\pi$
$558$	6.26232e6	0.851431
$559$	−2.59636e6	−0.351427
$560$	0	0
$561$	−2.13528e6	−0.286449
$562$	6.05469e6	0.808633
$563$	−1.22483e7	−1.62856	−0.814282	−	0.580470i	$-0.802869\pi$
$563$	−1.22483e7	−1.62856	−0.814282	+	0.580470i	$0.802869\pi$
$564$	1.19459e6	0.158133
$565$	0	0
$566$	−220560.	−0.0289391
$567$	44256.5	0.00578122
$568$	2.66822e6	0.347018
$569$	1.16148e6	0.150395	0.0751973	−	0.997169i	$-0.476041\pi$
$569$	1.16148e6	0.150395	0.0751973	+	0.997169i	$0.476041\pi$
$570$	0	0
$571$	7.48367e6	0.960560	0.480280	−	0.877115i	$-0.340535\pi$
$571$	7.48367e6	0.960560	0.480280	+	0.877115i	$0.340535\pi$
$572$	−994156.	−0.127047
$573$	−2.82465e6	−0.359400
$574$	1.11032e6	0.140659
$575$	0	0
$576$	−5.24888e6	−0.659189
$577$	−5.14705e6	−0.643604	−0.321802	−	0.946807i	$-0.604288\pi$
$577$	−5.14705e6	−0.643604	−0.321802	+	0.946807i	$0.604288\pi$
$578$	413696.	0.0515064
$579$	−8.43228e6	−1.04532
$580$	0	0
$581$	−994248.	−0.122195
$582$	−217135.	−0.0265719
$583$	−8.76927e6	−1.06854
$584$	−6.56470e6	−0.796495
$585$	0	0
$586$	−749362.	−0.0901463
$587$	1.18473e7	1.41913	0.709566	−	0.704639i	$-0.248891\pi$
$587$	1.18473e7	1.41913	0.709566	+	0.704639i	$0.248891\pi$
$588$	1.23443e6	0.147240
$589$	1.62858e7	1.93429
$590$	0	0
$591$	−8.07855e6	−0.951403
$592$	−3.81976e6	−0.447953
$593$	1.17061e7	1.36703	0.683513	−	0.729938i	$-0.260451\pi$
$593$	1.17061e7	1.36703	0.683513	+	0.729938i	$0.260451\pi$
$594$	−1.41659e7	−1.64732
$595$	0	0
$596$	−65892.3	−0.00759835
$597$	4.11239e6	0.472235
$598$	2.27062e6	0.259652
$599$	1.16813e7	1.33022	0.665110	−	0.746746i	$-0.268385\pi$
$599$	1.16813e7	1.33022	0.665110	+	0.746746i	$0.268385\pi$
$600$	0	0
$601$	5.65882e6	0.639057	0.319529	−	0.947577i	$-0.396475\pi$
$601$	5.65882e6	0.639057	0.319529	+	0.947577i	$0.396475\pi$
$602$	968925.	0.108968
$603$	593924.	0.0665178
$604$	−2.63774e6	−0.294198
$605$	0	0
$606$	2.10581e6	0.232937
$607$	−3.58036e6	−0.394416	−0.197208	−	0.980362i	$-0.563187\pi$
$607$	−3.58036e6	−0.394416	−0.197208	+	0.980362i	$0.563187\pi$
$608$	−4.90496e6	−0.538117
$609$	148913.	0.0162701
$610$	0	0
$611$	−2.88375e6	−0.312503
$612$	−310878.	−0.0335515
$613$	−4.02237e6	−0.432345	−0.216173	−	0.976355i	$-0.569357\pi$
$613$	−4.02237e6	−0.432345	−0.216173	+	0.976355i	$0.569357\pi$
$614$	−99854.2	−0.0106892
$615$	0	0
$616$	1.96120e6	0.208243
$617$	−1.12013e7	−1.18456	−0.592280	−	0.805732i	$-0.701772\pi$
$617$	−1.12013e7	−1.18456	−0.592280	+	0.805732i	$0.701772\pi$
$618$	9.47363e6	0.997804
$619$	−2.08785e6	−0.219015	−0.109507	−	0.993986i	$-0.534927\pi$
$619$	−2.08785e6	−0.219015	−0.109507	+	0.993986i	$0.534927\pi$
$620$	0	0
$621$	−9.84566e6	−1.02451
$622$	8.69824e6	0.901478
$623$	1.35318e6	0.139681
$624$	−1.30025e6	−0.133680
$625$	0	0
$626$	−3.38323e6	−0.345061
$627$	−1.37128e7	−1.39302
$628$	412733.	0.0417610
$629$	−1.51355e6	−0.152535
$630$	0	0
$631$	−1.62010e7	−1.61982	−0.809911	−	0.586553i	$-0.800484\pi$
$631$	−1.62010e7	−1.61982	−0.809911	+	0.586553i	$0.800484\pi$
$632$	1.68665e7	1.67970
$633$	−1.07927e7	−1.07058
$634$	−432036.	−0.0426872
$635$	0	0
$636$	876516.	0.0859245
$637$	−2.97993e6	−0.290976
$638$	−4.07951e6	−0.396786
$639$	1.96665e6	0.190535
$640$	0	0
$641$	−8.40847e6	−0.808299	−0.404149	−	0.914693i	$-0.632432\pi$
$641$	−8.40847e6	−0.808299	−0.404149	+	0.914693i	$0.632432\pi$
$642$	5.62373e6	0.538502
$643$	1.66230e7	1.58556	0.792780	−	0.609508i	$-0.208633\pi$
$643$	1.66230e7	1.58556	0.792780	+	0.609508i	$0.208633\pi$
$644$	257857.	0.0244999
$645$	0	0
$646$	2.65677e6	0.250480
$647$	−5.98066e6	−0.561680	−0.280840	−	0.959755i	$-0.590613\pi$
$647$	−5.98066e6	−0.561680	−0.280840	+	0.959755i	$0.590613\pi$
$648$	640605.	0.0599312
$649$	−2.18334e7	−2.03474
$650$	0	0
$651$	1.17860e6	0.108997
$652$	2.29086e6	0.211047
$653$	−1.16116e7	−1.06564	−0.532819	−	0.846229i	$-0.678867\pi$
$653$	−1.16116e7	−1.06564	−0.532819	+	0.846229i	$0.678867\pi$
$654$	−4.83336e6	−0.441881
$655$	0	0
$656$	1.21061e7	1.09836
$657$	−4.83860e6	−0.437327
$658$	1.07617e6	0.0968986
$659$	−2.21305e7	−1.98508	−0.992541	−	0.121910i	$-0.961098\pi$
$659$	−2.21305e7	−1.98508	−0.992541	+	0.121910i	$0.961098\pi$
$660$	0	0
$661$	−9.38416e6	−0.835395	−0.417697	−	0.908586i	$-0.637163\pi$
$661$	−9.38416e6	−0.835395	−0.417697	+	0.908586i	$0.637163\pi$
$662$	1.27456e7	1.13035
$663$	−515215.	−0.0455202
$664$	−1.43916e7	−1.26674
$665$	0	0
$666$	−3.73763e6	−0.326521
$667$	−2.83536e6	−0.246770
$668$	−4.39786e6	−0.381330
$669$	1.31083e7	1.13235
$670$	0	0
$671$	2.17913e7	1.86843
$672$	−354972.	−0.0303229
$673$	−1.69192e6	−0.143993	−0.0719965	−	0.997405i	$-0.522937\pi$
$673$	−1.69192e6	−0.143993	−0.0719965	+	0.997405i	$0.522937\pi$
$674$	8.46150e6	0.717460
$675$	0	0
$676$	2.53215e6	0.213119
$677$	1.60015e7	1.34180	0.670902	−	0.741546i	$-0.265907\pi$
$677$	1.60015e7	1.34180	0.670902	+	0.741546i	$0.265907\pi$
$678$	−6.58255e6	−0.549946
$679$	59525.5	0.00495483
$680$	0	0
$681$	8.09538e6	0.668912
$682$	−3.22881e7	−2.65817
$683$	−1.85149e7	−1.51870	−0.759348	−	0.650685i	$-0.774482\pi$
$683$	−1.85149e7	−1.51870	−0.759348	+	0.650685i	$0.774482\pi$
$684$	−1.99647e6	−0.163164
$685$	0	0
$686$	2.23633e6	0.181437
$687$	1.19114e7	0.962879
$688$	1.05645e7	0.850898
$689$	−2.11591e6	−0.169805
$690$	0	0
$691$	−2.22261e7	−1.77079	−0.885397	−	0.464835i	$-0.846114\pi$
$691$	−2.22261e7	−1.77079	−0.885397	+	0.464835i	$0.846114\pi$
$692$	4.83307e6	0.383670
$693$	1.44553e6	0.114339
$694$	−5.00001e6	−0.394069
$695$	0	0
$696$	2.15549e6	0.168664
$697$	4.79695e6	0.374010
$698$	6.25965e6	0.486308
$699$	1.06924e7	0.827717
$700$	0	0
$701$	−1.66904e7	−1.28284	−0.641419	−	0.767190i	$-0.721654\pi$
$701$	−1.66904e7	−1.28284	−0.641419	+	0.767190i	$0.721654\pi$
$702$	−3.41806e6	−0.261780
$703$	−9.72008e6	−0.741791
$704$	2.70629e7	2.05799
$705$	0	0
$706$	−6.77364e6	−0.511458
$707$	−577288.	−0.0434354
$708$	2.18232e6	0.163619
$709$	9.38491e6	0.701156	0.350578	−	0.936534i	$-0.385985\pi$
$709$	9.38491e6	0.701156	0.350578	+	0.936534i	$0.385985\pi$
$710$	0	0
$711$	1.24317e7	0.922264
$712$	1.95870e7	1.44800
$713$	−2.24410e7	−1.65318
$714$	192271.	0.0141146
$715$	0	0
$716$	−2.87710e6	−0.209736
$717$	−1.49511e7	−1.08611
$718$	−2.52521e6	−0.182804
$719$	−2.29753e6	−0.165745	−0.0828724	−	0.996560i	$-0.526409\pi$
$719$	−2.29753e6	−0.165745	−0.0828724	+	0.996560i	$0.526409\pi$
$720$	0	0
$721$	−2.59710e6	−0.186059
$722$	4.79732e6	0.342496
$723$	−2.58304e6	−0.183775
$724$	−705014.	−0.0499863
$725$	0	0
$726$	1.92531e7	1.35569
$727$	−2.21473e6	−0.155412	−0.0777059	−	0.996976i	$-0.524760\pi$
$727$	−2.21473e6	−0.155412	−0.0777059	+	0.996976i	$0.524760\pi$
$728$	473212.	0.0330923
$729$	9.77643e6	0.681336
$730$	0	0
$731$	4.18609e6	0.289744
$732$	−2.17811e6	−0.150246
$733$	6.33827e6	0.435724	0.217862	−	0.975980i	$-0.430092\pi$
$733$	6.33827e6	0.435724	0.217862	+	0.975980i	$0.430092\pi$
$734$	−6.26422e6	−0.429168
$735$	0	0
$736$	6.75880e6	0.459912
$737$	−3.06224e6	−0.207668
$738$	1.18458e7	0.800615
$739$	−7.68253e6	−0.517480	−0.258740	−	0.965947i	$-0.583307\pi$
$739$	−7.68253e6	−0.517480	−0.258740	+	0.965947i	$0.583307\pi$
$740$	0	0
$741$	−3.30873e6	−0.221369
$742$	789629.	0.0526518
$743$	2.47626e7	1.64560	0.822801	−	0.568330i	$-0.192410\pi$
$743$	2.47626e7	1.64560	0.822801	+	0.568330i	$0.192410\pi$
$744$	1.70601e7	1.12992
$745$	0	0
$746$	1.25351e7	0.824672
$747$	−1.06075e7	−0.695522
$748$	1.60287e6	0.104748
$749$	−1.54169e6	−0.100414
$750$	0	0
$751$	−4.97740e6	−0.322035	−0.161017	−	0.986952i	$-0.551478\pi$
$751$	−4.97740e6	−0.322035	−0.161017	+	0.986952i	$0.551478\pi$
$752$	1.17339e7	0.756652
$753$	4.15477e6	0.267030
$754$	−984333.	−0.0630542
$755$	0	0
$756$	−388163.	−0.0247007
$757$	−2.57073e7	−1.63049	−0.815244	−	0.579118i	$-0.803397\pi$
$757$	−2.57073e7	−1.63049	−0.815244	+	0.579118i	$0.803397\pi$
$758$	6.12731e6	0.387344
$759$	1.88956e7	1.19058
$760$	0	0
$761$	2.72075e7	1.70305	0.851524	−	0.524316i	$-0.175679\pi$
$761$	2.72075e7	1.70305	0.851524	+	0.524316i	$0.175679\pi$
$762$	−2.99329e6	−0.186750
$763$	1.32502e6	0.0823969
$764$	2.12036e6	0.131424
$765$	0	0
$766$	585161.	0.0360333
$767$	−5.26812e6	−0.323346
$768$	−6.87118e6	−0.420367
$769$	1.70602e7	1.04032	0.520161	−	0.854068i	$-0.325872\pi$
$769$	1.70602e7	1.04032	0.520161	+	0.854068i	$0.325872\pi$
$770$	0	0
$771$	−1.22597e7	−0.742750
$772$	6.32979e6	0.382249
$773$	−2.42134e7	−1.45749	−0.728746	−	0.684784i	$-0.759897\pi$
$773$	−2.42134e7	−1.45749	−0.728746	+	0.684784i	$0.759897\pi$
$774$	1.03373e7	0.620234
$775$	0	0
$776$	861621.	0.0513644
$777$	−703443.	−0.0418000
$778$	3.72867e6	0.220854
$779$	3.08062e7	1.81884
$780$	0	0
$781$	−1.01399e7	−0.594850
$782$	−3.66090e6	−0.214078
$783$	4.26818e6	0.248793
$784$	1.21252e7	0.704530
$785$	0	0
$786$	−1.39691e7	−0.806516
$787$	801866.	0.0461493	0.0230746	−	0.999734i	$-0.492654\pi$
$787$	801866.	0.0461493	0.0230746	+	0.999734i	$0.492654\pi$
$788$	6.06426e6	0.347906
$789$	−2.09304e6	−0.119698
$790$	0	0
$791$	1.80454e6	0.102548
$792$	2.09237e7	1.18529
$793$	5.25796e6	0.296917
$794$	1.69316e7	0.953119
$795$	0	0
$796$	−3.08701e6	−0.172685
$797$	1.85153e7	1.03249	0.516245	−	0.856441i	$-0.327330\pi$
$797$	1.85153e7	1.03249	0.516245	+	0.856441i	$0.327330\pi$
$798$	1.23477e6	0.0686403
$799$	4.64944e6	0.257652
$800$	0	0
$801$	1.44369e7	0.795047
$802$	−1.73225e6	−0.0950990
$803$	2.49476e7	1.36534
$804$	306080.	0.0166992
$805$	0	0
$806$	−7.79072e6	−0.422415
$807$	1.18811e7	0.642204
$808$	−8.35613e6	−0.450274
$809$	−1.25719e7	−0.675349	−0.337674	−	0.941263i	$-0.609640\pi$
$809$	−1.25719e7	−0.675349	−0.337674	+	0.941263i	$0.609640\pi$
$810$	0	0
$811$	6.53924e6	0.349120	0.174560	−	0.984647i	$-0.444150\pi$
$811$	6.53924e6	0.349120	0.174560	+	0.984647i	$0.444150\pi$
$812$	−111783.	−0.00594959
$813$	−4.83751e6	−0.256682
$814$	1.92710e7	1.01940
$815$	0	0
$816$	2.09639e6	0.110216
$817$	2.68833e7	1.40905
$818$	5.18112e6	0.270732
$819$	348787.	0.0181698
$820$	0	0
$821$	−3.60810e7	−1.86819	−0.934095	−	0.357024i	$-0.883792\pi$
$821$	−3.60810e7	−1.86819	−0.934095	+	0.357024i	$0.883792\pi$
$822$	5.74158e6	0.296382
$823$	1.53622e7	0.790595	0.395297	−	0.918553i	$-0.370642\pi$
$823$	1.53622e7	0.790595	0.395297	+	0.918553i	$0.370642\pi$
$824$	−3.75926e7	−1.92879
$825$	0	0
$826$	1.96599e6	0.100261
$827$	1.40297e7	0.713319	0.356659	−	0.934234i	$-0.383916\pi$
$827$	1.40297e7	0.713319	0.356659	+	0.934234i	$0.383916\pi$
$828$	2.75104e6	0.139451
$829$	−1.95476e6	−0.0987886	−0.0493943	−	0.998779i	$-0.515729\pi$
$829$	−1.95476e6	−0.0987886	−0.0493943	+	0.998779i	$0.515729\pi$
$830$	0	0
$831$	6.77466e6	0.340318
$832$	6.52993e6	0.327040
$833$	4.80451e6	0.239904
$834$	1.04541e7	0.520440
$835$	0	0
$836$	1.02937e7	0.509397
$837$	3.37814e7	1.66672
$838$	−5.41457e6	−0.266351
$839$	3.06449e7	1.50298	0.751490	−	0.659744i	$-0.229335\pi$
$839$	3.06449e7	1.50298	0.751490	+	0.659744i	$0.229335\pi$
$840$	0	0
$841$	−1.92820e7	−0.940074
$842$	−1.55169e7	−0.754264
$843$	1.21575e7	0.589215
$844$	8.10165e6	0.391487
$845$	0	0
$846$	1.14815e7	0.551537
$847$	−5.27806e6	−0.252794
$848$	8.60957e6	0.411142
$849$	−442871.	−0.0210867
$850$	0	0
$851$	1.33938e7	0.633986
$852$	1.01352e6	0.0478335
$853$	−1.39171e7	−0.654902	−0.327451	−	0.944868i	$-0.606190\pi$
$853$	−1.39171e7	−0.654902	−0.327451	+	0.944868i	$0.606190\pi$
$854$	−1.96220e6	−0.0920657
$855$	0	0
$856$	−2.23157e7	−1.04094
$857$	4.54122e6	0.211213	0.105607	−	0.994408i	$-0.466322\pi$
$857$	4.54122e6	0.211213	0.105607	+	0.994408i	$0.466322\pi$
$858$	6.55988e6	0.304213
$859$	1.93388e7	0.894226	0.447113	−	0.894478i	$-0.352452\pi$
$859$	1.93388e7	0.894226	0.447113	+	0.894478i	$0.352452\pi$
$860$	0	0
$861$	2.22945e6	0.102492
$862$	−2.57322e7	−1.17953
$863$	2.61691e7	1.19608	0.598042	−	0.801465i	$-0.295946\pi$
$863$	2.61691e7	1.19608	0.598042	+	0.801465i	$0.295946\pi$
$864$	−1.01743e7	−0.463682
$865$	0	0
$866$	−1.92096e7	−0.870409
$867$	830676.	0.0375305
$868$	−884733.	−0.0398578
$869$	−6.40970e7	−2.87931
$870$	0	0
$871$	−738879.	−0.0330011
$872$	1.91794e7	0.854169
$873$	635069.	0.0282024
$874$	−2.35105e7	−1.04108
$875$	0	0
$876$	−2.49359e6	−0.109790
$877$	−8.15287e6	−0.357941	−0.178970	−	0.983854i	$-0.557277\pi$
$877$	−8.15287e6	−0.357941	−0.178970	+	0.983854i	$0.557277\pi$
$878$	9.67471e6	0.423547
$879$	−1.50467e6	−0.0656856
$880$	0	0
$881$	−2.60347e7	−1.13009	−0.565046	−	0.825060i	$-0.691141\pi$
$881$	−2.60347e7	−1.13009	−0.565046	+	0.825060i	$0.691141\pi$
$882$	1.18645e7	0.513544
$883$	−9.01584e6	−0.389139	−0.194569	−	0.980889i	$-0.562331\pi$
$883$	−9.01584e6	−0.389139	−0.194569	+	0.980889i	$0.562331\pi$
$884$	386752.	0.0166457
$885$	0	0
$886$	−1.91662e6	−0.0820258
$887$	1.25216e7	0.534380	0.267190	−	0.963644i	$-0.413905\pi$
$887$	1.25216e7	0.534380	0.267190	+	0.963644i	$0.413905\pi$
$888$	−1.01822e7	−0.433321
$889$	820581.	0.0348231
$890$	0	0
$891$	−2.43446e6	−0.102733
$892$	−9.83987e6	−0.414073
$893$	2.98589e7	1.25298
$894$	434786.	0.0181942
$895$	0	0
$896$	−1.29477e6	−0.0538793
$897$	4.55927e6	0.189197
$898$	1.65462e7	0.684709
$899$	9.72838e6	0.401459
$900$	0	0
$901$	3.41147e6	0.140000
$902$	−6.10763e7	−2.49952
$903$	1.94554e6	0.0794002
$904$	2.61204e7	1.06306
$905$	0	0
$906$	1.74050e7	0.704453
$907$	5.20915e6	0.210256	0.105128	−	0.994459i	$-0.466475\pi$
$907$	5.20915e6	0.210256	0.105128	+	0.994459i	$0.466475\pi$
$908$	−6.07689e6	−0.244606
$909$	−6.15900e6	−0.247230
$910$	0	0
$911$	1.33715e7	0.533808	0.266904	−	0.963723i	$-0.413999\pi$
$911$	1.33715e7	0.533808	0.266904	+	0.963723i	$0.413999\pi$
$912$	1.34631e7	0.535992
$913$	5.46916e7	2.17142
$914$	2.15919e7	0.854919
$915$	0	0
$916$	−8.94145e6	−0.352103
$917$	3.82950e6	0.150390
$918$	5.51091e6	0.215832
$919$	3.60916e7	1.40967	0.704835	−	0.709371i	$-0.251021\pi$
$919$	3.60916e7	1.40967	0.704835	+	0.709371i	$0.251021\pi$
$920$	0	0
$921$	−200501.	−0.00778875
$922$	−1.94773e7	−0.754573
$923$	−2.44664e6	−0.0945290
$924$	744956.	0.0287045
$925$	0	0
$926$	2.21705e7	0.849665
$927$	−2.77081e7	−1.05903
$928$	−2.93000e6	−0.111686
$929$	2.41737e7	0.918977	0.459489	−	0.888184i	$-0.348033\pi$
$929$	2.41737e7	0.918977	0.459489	+	0.888184i	$0.348033\pi$
$930$	0	0
$931$	3.08548e7	1.16667
$932$	−8.02637e6	−0.302677
$933$	1.74655e7	0.656868
$934$	2.93733e7	1.10176
$935$	0	0
$936$	5.04863e6	0.188358
$937$	3.26851e7	1.21619	0.608094	−	0.793865i	$-0.291934\pi$
$937$	3.26851e7	1.21619	0.608094	+	0.793865i	$0.291934\pi$
$938$	275739.	0.0102327
$939$	−6.79333e6	−0.251431
$940$	0	0
$941$	−3.02020e7	−1.11189	−0.555944	−	0.831220i	$-0.687643\pi$
$941$	−3.02020e7	−1.11189	−0.555944	+	0.831220i	$0.687643\pi$
$942$	−2.72340e6	−0.0999962
$943$	−4.24495e7	−1.55451
$944$	2.14358e7	0.782905
$945$	0	0
$946$	−5.32986e7	−1.93637
$947$	−8.76265e6	−0.317512	−0.158756	−	0.987318i	$-0.550748\pi$
$947$	−8.76265e6	−0.317512	−0.158756	+	0.987318i	$0.550748\pi$
$948$	6.40669e6	0.231533
$949$	6.01953e6	0.216969
$950$	0	0
$951$	−867503.	−0.0311043
$952$	−762955.	−0.0272839
$953$	−267287.	−0.00953334	−0.00476667	−	0.999989i	$-0.501517\pi$
$953$	−267287.	−0.00953334	−0.00476667	+	0.999989i	$0.501517\pi$
$954$	8.42444e6	0.299688
$955$	0	0
$956$	1.12232e7	0.397166
$957$	−8.19141e6	−0.289120
$958$	9.91509e6	0.349046
$959$	−1.57400e6	−0.0552659
$960$	0	0
$961$	4.83681e7	1.68947
$962$	4.64984e6	0.161995
$963$	−1.64481e7	−0.571544
$964$	1.93899e6	0.0672022
$965$	0	0
$966$	−1.70146e6	−0.0586648
$967$	−4.05337e7	−1.39396	−0.696980	−	0.717090i	$-0.745474\pi$
$967$	−4.05337e7	−1.39396	−0.696980	+	0.717090i	$0.745474\pi$
$968$	−7.63990e7	−2.62059
$969$	5.33464e6	0.182514
$970$	0	0
$971$	−4.41436e7	−1.50252	−0.751259	−	0.660007i	$-0.770553\pi$
$971$	−4.41436e7	−1.50252	−0.751259	+	0.660007i	$0.770553\pi$
$972$	−6.74102e6	−0.228855
$973$	−2.86588e6	−0.0970457
$974$	−3.61171e7	−1.21988
$975$	0	0
$976$	−2.13944e7	−0.718913
$977$	7.20419e6	0.241462	0.120731	−	0.992685i	$-0.461476\pi$
$977$	7.20419e6	0.241462	0.120731	+	0.992685i	$0.461476\pi$
$978$	−1.51161e7	−0.505350
$979$	−7.44358e7	−2.48213
$980$	0	0
$981$	1.41364e7	0.468994
$982$	−4.03319e6	−0.133466
$983$	4.44557e7	1.46738	0.733692	−	0.679482i	$-0.237796\pi$
$983$	4.44557e7	1.46738	0.733692	+	0.679482i	$0.237796\pi$
$984$	3.22709e7	1.06248
$985$	0	0
$986$	1.58703e6	0.0519868
$987$	2.16089e6	0.0706058
$988$	2.48374e6	0.0809494
$989$	−3.70438e7	−1.20427
$990$	0	0
$991$	4.66195e7	1.50794	0.753970	−	0.656909i	$-0.228137\pi$
$991$	4.66195e7	1.50794	0.753970	+	0.656909i	$0.228137\pi$
$992$	−2.31901e7	−0.748209
$993$	2.55923e7	0.823639
$994$	913050.	0.0293109
$995$	0	0
$996$	−5.46660e6	−0.174610
$997$	−6.01277e7	−1.91574	−0.957870	−	0.287202i	$-0.907275\pi$
$997$	−6.01277e7	−1.91574	−0.957870	+	0.287202i	$0.907275\pi$
$998$	−8.22117e6	−0.261281
$999$	−2.01622e7	−0.639182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.d.1.4		5
5.4	even	2		85.6.a.a.1.2	✓	5
15.14	odd	2		765.6.a.g.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.2	✓	5	5.4	even	2
425.6.a.d.1.4		5	1.1	even	1	trivial
765.6.a.g.1.4		5	15.14	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+4.95319 q^{2} +9.94571 q^{3} -7.46587 q^{4} +49.2631 q^{6} -13.5050 q^{7} -195.482 q^{8} -144.083 q^{9} +742.882 q^{11} -74.2534 q^{12} +179.248 q^{13} -66.8928 q^{14} -729.353 q^{16} -289.000 q^{17} -713.670 q^{18} -1855.97 q^{19} -134.317 q^{21} +3679.64 q^{22} +2557.44 q^{23} -1944.21 q^{24} +887.850 q^{26} -3849.81 q^{27} +100.826 q^{28} -1108.67 q^{29} -8774.81 q^{31} +2642.80 q^{32} +7388.50 q^{33} -1431.47 q^{34} +1075.70 q^{36} +5237.19 q^{37} -9192.99 q^{38} +1782.75 q^{39} -16598.4 q^{41} -665.297 q^{42} -14484.7 q^{43} -5546.26 q^{44} +12667.5 q^{46} -16088.0 q^{47} -7253.94 q^{48} -16624.6 q^{49} -2874.31 q^{51} -1338.24 q^{52} -11804.4 q^{53} -19068.9 q^{54} +2639.98 q^{56} -18459.0 q^{57} -5491.46 q^{58} -29390.1 q^{59} +29333.5 q^{61} -43463.3 q^{62} +1945.84 q^{63} +36429.6 q^{64} +36596.7 q^{66} -4122.11 q^{67} +2157.64 q^{68} +25435.6 q^{69} -13649.5 q^{71} +28165.6 q^{72} +33582.1 q^{73} +25940.8 q^{74} +13856.4 q^{76} -10032.6 q^{77} +8830.30 q^{78} -86281.4 q^{79} -3277.05 q^{81} -82215.3 q^{82} +73620.8 q^{83} +1002.79 q^{84} -71745.7 q^{86} -11026.5 q^{87} -145220. q^{88} -100199. q^{89} -2420.74 q^{91} -19093.5 q^{92} -87271.8 q^{93} -79687.2 q^{94} +26284.5 q^{96} -4407.67 q^{97} -82344.9 q^{98} -107037. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) \) 5 * q + 7 * q^2 + 36 * q^3 + 43 * q^4 - 105 * q^6 + 204 * q^7 + 63 * q^8 + 531 * q^9 - 792 * q^11 - 785 * q^12 - 88 * q^13 + 860 * q^14 - 2365 * q^16 - 1445 * q^17 + 2052 * q^18 - 5160 * q^19 - 6428 * q^21 + 3058 * q^22 + 6140 * q^23 - 11751 * q^24 - 9643 * q^26 + 13380 * q^27 + 4568 * q^28 - 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 13072 * q^33 - 2023 * q^34 + 2048 * q^36 - 9834 * q^37 - 10625 * q^38 + 10108 * q^39 - 5302 * q^41 - 16222 * q^42 + 4916 * q^43 - 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 45557 * q^48 - 26483 * q^49 - 10404 * q^51 + 11525 * q^52 - 56492 * q^53 + 64635 * q^54 + 73228 * q^56 - 67554 * q^57 - 54213 * q^58 - 96200 * q^59 - 7284 * q^61 - 122335 * q^62 - 64108 * q^63 + 28483 * q^64 + 115232 * q^66 + 60308 * q^67 - 12427 * q^68 - 89076 * q^69 - 25304 * q^71 - 118722 * q^72 - 163328 * q^73 - 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 134895 * q^78 - 50940 * q^79 + 92445 * q^81 - 191390 * q^82 + 179060 * q^83 - 128054 * q^84 + 87086 * q^86 - 55764 * q^87 - 167634 * q^88 - 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 202558 * q^93 + 100515 * q^94 + 428113 * q^96 + 106076 * q^97 - 49329 * q^98 - 535112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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