LMFDB - Embedded newform 325.6.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.56155 q^{2} +1.63068 q^{3} -11.1922 q^{4} +7.43845 q^{6} -126.309 q^{7} -197.024 q^{8} -240.341 q^{9} -14.8296 q^{11} -18.2510 q^{12} +169.000 q^{13} -576.164 q^{14} -540.582 q^{16} +1051.12 q^{17} -1096.33 q^{18} -213.723 q^{19} -205.969 q^{21} -67.6458 q^{22} +4231.16 q^{23} -321.283 q^{24} +770.902 q^{26} -788.176 q^{27} +1413.68 q^{28} -504.955 q^{29} +4783.58 q^{31} +3838.86 q^{32} -24.1823 q^{33} +4794.75 q^{34} +2689.95 q^{36} +4635.74 q^{37} -974.911 q^{38} +275.585 q^{39} +7944.15 q^{41} -939.541 q^{42} +8516.41 q^{43} +165.976 q^{44} +19300.7 q^{46} -24921.2 q^{47} -881.518 q^{48} -853.113 q^{49} +1714.05 q^{51} -1891.49 q^{52} +7808.46 q^{53} -3595.31 q^{54} +24885.8 q^{56} -348.515 q^{57} -2303.38 q^{58} -37337.5 q^{59} -18172.2 q^{61} +21820.5 q^{62} +30357.1 q^{63} +34809.8 q^{64} -110.309 q^{66} +34559.9 q^{67} -11764.4 q^{68} +6899.68 q^{69} +41255.7 q^{71} +47352.8 q^{72} +1056.42 q^{73} +21146.2 q^{74} +2392.04 q^{76} +1873.10 q^{77} +1257.10 q^{78} -47719.3 q^{79} +57117.6 q^{81} +36237.7 q^{82} +74799.0 q^{83} +2305.26 q^{84} +38848.0 q^{86} -823.421 q^{87} +2921.78 q^{88} +9799.26 q^{89} -21346.2 q^{91} -47356.1 q^{92} +7800.50 q^{93} -113679. q^{94} +6259.97 q^{96} +138432. q^{97} -3891.52 q^{98} +3564.15 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} + 28 q^{3} - 43 q^{4} + 19 q^{6} + 36 q^{7} - 225 q^{8} + 212 q^{9} - 376 q^{11} - 857 q^{12} + 338 q^{13} - 505 q^{14} + 465 q^{16} + 2630 q^{17} - 898 q^{18} - 312 q^{19} + 4074 q^{21} - 226 q^{22}+ \cdots - 159808 q^{99}+O(q^{100}) $ 2 * q + 5 * q^2 + 28 * q^3 - 43 * q^4 + 19 * q^6 + 36 * q^7 - 225 * q^8 + 212 * q^9 - 376 * q^11 - 857 * q^12 + 338 * q^13 - 505 * q^14 + 465 * q^16 + 2630 * q^17 - 898 * q^18 - 312 * q^19 + 4074 * q^21 - 226 * q^22 + 2624 * q^23 - 1059 * q^24 + 845 * q^26 + 4732 * q^27 - 3749 * q^28 - 812 * q^29 + 7720 * q^31 + 5175 * q^32 - 9548 * q^33 + 5487 * q^34 - 11698 * q^36 + 16858 * q^37 - 1018 * q^38 + 4732 * q^39 + 7840 * q^41 + 937 * q^42 - 2420 * q^43 + 11654 * q^44 + 18596 * q^46 - 9972 * q^47 + 25635 * q^48 + 8684 * q^49 + 43348 * q^51 - 7267 * q^52 + 43720 * q^53 - 1175 * q^54 + 20345 * q^56 - 2940 * q^57 - 2438 * q^58 - 38936 * q^59 + 1984 * q^61 + 23108 * q^62 + 103776 * q^63 + 3217 * q^64 - 4286 * q^66 + 69928 * q^67 - 61985 * q^68 - 35480 * q^69 + 67396 * q^71 + 34698 * q^72 - 74412 * q^73 + 26505 * q^74 + 5518 * q^76 - 56748 * q^77 + 3211 * q^78 - 55296 * q^79 + 92762 * q^81 + 36192 * q^82 + 75712 * q^83 - 133831 * q^84 + 34053 * q^86 - 8920 * q^87 + 13026 * q^88 + 116508 * q^89 + 6084 * q^91 + 3764 * q^92 + 85232 * q^93 - 107125 * q^94 + 41493 * q^96 + 34756 * q^97 + 290 * q^98 - 159808 * 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 $

$2$	4.56155	0.806376	0.403188	−	0.915117i	$-0.367902\pi$
$2$	4.56155	0.806376	0.403188	+	0.915117i	$0.367902\pi$
$3$	1.63068	0.104608	0.0523042	−	0.998631i	$-0.483343\pi$
$3$	1.63068	0.104608	0.0523042	+	0.998631i	$0.483343\pi$
$4$	−11.1922	−0.349757
$5$	0	0
$5$	0	0
$6$	7.43845	0.0843537
$7$	−126.309	−0.974290	−0.487145	−	0.873321i	$-0.661962\pi$
$7$	−126.309	−0.974290	−0.487145	+	0.873321i	$0.661962\pi$
$8$	−197.024	−1.08841
$9$	−240.341	−0.989057
$10$	0	0
$11$	−14.8296	−0.0369527	−0.0184764	−	0.999829i	$-0.505882\pi$
$11$	−14.8296	−0.0369527	−0.0184764	+	0.999829i	$0.505882\pi$
$12$	−18.2510	−0.0365875
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−576.164	−0.785644
$15$	0	0
$16$	−540.582	−0.527912
$17$	1051.12	0.882126	0.441063	−	0.897476i	$-0.354602\pi$
$17$	1051.12	0.882126	0.441063	+	0.897476i	$0.354602\pi$
$18$	−1096.33	−0.797552
$19$	−213.723	−0.135821	−0.0679107	−	0.997691i	$-0.521633\pi$
$19$	−213.723	−0.135821	−0.0679107	+	0.997691i	$0.521633\pi$
$20$	0	0
$21$	−205.969	−0.101919
$22$	−67.6458	−0.0297978
$23$	4231.16	1.66778	0.833892	−	0.551928i	$-0.186108\pi$
$23$	4231.16	1.66778	0.833892	+	0.551928i	$0.186108\pi$
$24$	−321.283	−0.113857
$25$	0	0
$26$	770.902	0.223649
$27$	−788.176	−0.208072
$28$	1413.68	0.340765
$29$	−504.955	−0.111495	−0.0557477	−	0.998445i	$-0.517754\pi$
$29$	−504.955	−0.111495	−0.0557477	+	0.998445i	$0.517754\pi$
$30$	0	0
$31$	4783.58	0.894022	0.447011	−	0.894528i	$-0.352488\pi$
$31$	4783.58	0.894022	0.447011	+	0.894528i	$0.352488\pi$
$32$	3838.86	0.662716
$33$	−24.1823	−0.00386557
$34$	4794.75	0.711325
$35$	0	0
$36$	2689.95	0.345930
$37$	4635.74	0.556692	0.278346	−	0.960481i	$-0.410214\pi$
$37$	4635.74	0.556692	0.278346	+	0.960481i	$0.410214\pi$
$38$	−974.911	−0.109523
$39$	275.585	0.0290131
$40$	0	0
$41$	7944.15	0.738054	0.369027	−	0.929419i	$-0.379691\pi$
$41$	7944.15	0.738054	0.369027	+	0.929419i	$0.379691\pi$
$42$	−939.541	−0.0821850
$43$	8516.41	0.702401	0.351201	−	0.936300i	$-0.385774\pi$
$43$	8516.41	0.702401	0.351201	+	0.936300i	$0.385774\pi$
$44$	165.976	0.0129245
$45$	0	0
$46$	19300.7	1.34486
$47$	−24921.2	−1.64560	−0.822801	−	0.568330i	$-0.807590\pi$
$47$	−24921.2	−1.64560	−0.822801	+	0.568330i	$0.807590\pi$
$48$	−881.518	−0.0552241
$49$	−853.113	−0.0507594
$50$	0	0
$51$	1714.05	0.0922777
$52$	−1891.49	−0.0970052
$53$	7808.46	0.381835	0.190917	−	0.981606i	$-0.438854\pi$
$53$	7808.46	0.381835	0.190917	+	0.981606i	$0.438854\pi$
$54$	−3595.31	−0.167784
$55$	0	0
$56$	24885.8	1.06043
$57$	−348.515	−0.0142081
$58$	−2303.38	−0.0899073
$59$	−37337.5	−1.39642	−0.698209	−	0.715894i	$-0.746020\pi$
$59$	−37337.5	−1.39642	−0.698209	+	0.715894i	$0.746020\pi$
$60$	0	0
$61$	−18172.2	−0.625292	−0.312646	−	0.949870i	$-0.601215\pi$
$61$	−18172.2	−0.625292	−0.312646	+	0.949870i	$0.601215\pi$
$62$	21820.5	0.720918
$63$	30357.1	0.963628
$64$	34809.8	1.06231
$65$	0	0
$66$	−110.309	−0.00311710
$67$	34559.9	0.940559	0.470279	−	0.882518i	$-0.344153\pi$
$67$	34559.9	0.940559	0.470279	+	0.882518i	$0.344153\pi$
$68$	−11764.4	−0.308530
$69$	6899.68	0.174464
$70$	0	0
$71$	41255.7	0.971265	0.485632	−	0.874163i	$-0.338589\pi$
$71$	41255.7	0.971265	0.485632	+	0.874163i	$0.338589\pi$
$72$	47352.8	1.07650
$73$	1056.42	0.0232022	0.0116011	−	0.999933i	$-0.496307\pi$
$73$	1056.42	0.0232022	0.0116011	+	0.999933i	$0.496307\pi$
$74$	21146.2	0.448903
$75$	0	0
$76$	2392.04	0.0475045
$77$	1873.10	0.0360027
$78$	1257.10	0.0233955
$79$	−47719.3	−0.860253	−0.430126	−	0.902769i	$-0.641531\pi$
$79$	−47719.3	−0.860253	−0.430126	+	0.902769i	$0.641531\pi$
$80$	0	0
$81$	57117.6	0.967291
$82$	36237.7	0.595149
$83$	74799.0	1.19179	0.595896	−	0.803061i	$-0.296797\pi$
$83$	74799.0	1.19179	0.595896	+	0.803061i	$0.296797\pi$
$84$	2305.26	0.0356469
$85$	0	0
$86$	38848.0	0.566400
$87$	−823.421	−0.0116634
$88$	2921.78	0.0402198
$89$	9799.26	0.131135	0.0655675	−	0.997848i	$-0.479114\pi$
$89$	9799.26	0.131135	0.0655675	+	0.997848i	$0.479114\pi$
$90$	0	0
$91$	−21346.2	−0.270219
$92$	−47356.1	−0.583320
$93$	7800.50	0.0935222
$94$	−113679.	−1.32697
$95$	0	0
$96$	6259.97	0.0693257
$97$	138432.	1.49385	0.746927	−	0.664906i	$-0.231528\pi$
$97$	138432.	1.49385	0.746927	+	0.664906i	$0.231528\pi$
$98$	−3891.52	−0.0409312
$99$	3564.15	0.0365484
$100$	0	0
$101$	139151.	1.35733	0.678663	−	0.734450i	$-0.262560\pi$
$101$	139151.	1.35733	0.678663	+	0.734450i	$0.262560\pi$
$102$	7818.71	0.0744106
$103$	−98512.2	−0.914950	−0.457475	−	0.889223i	$-0.651246\pi$
$103$	−98512.2	−0.914950	−0.457475	+	0.889223i	$0.651246\pi$
$104$	−33297.0	−0.301871
$105$	0	0
$106$	35618.7	0.307902
$107$	26848.4	0.226704	0.113352	−	0.993555i	$-0.463841\pi$
$107$	26848.4	0.226704	0.113352	+	0.993555i	$0.463841\pi$
$108$	8821.45	0.0727747
$109$	−63220.9	−0.509676	−0.254838	−	0.966984i	$-0.582022\pi$
$109$	−63220.9	−0.509676	−0.254838	+	0.966984i	$0.582022\pi$
$110$	0	0
$111$	7559.43	0.0582346
$112$	68280.2	0.514340
$113$	−114434.	−0.843058	−0.421529	−	0.906815i	$-0.638506\pi$
$113$	−114434.	−0.843058	−0.421529	+	0.906815i	$0.638506\pi$
$114$	−1589.77	−0.0114570
$115$	0	0
$116$	5651.57	0.0389964
$117$	−40617.6	−0.274315
$118$	−170317.	−1.12604
$119$	−132766.	−0.859446
$120$	0	0
$121$	−160831.	−0.998634
$122$	−82893.4	−0.504221
$123$	12954.4	0.0772066
$124$	−53538.9	−0.312691
$125$	0	0
$126$	138476.	0.777047
$127$	248871.	1.36919	0.684596	−	0.728922i	$-0.259978\pi$
$127$	248871.	1.36919	0.684596	+	0.728922i	$0.259978\pi$
$128$	35943.2	0.193906
$129$	13887.6	0.0734770
$130$	0	0
$131$	102963.	0.524205	0.262102	−	0.965040i	$-0.415584\pi$
$131$	102963.	0.524205	0.262102	+	0.965040i	$0.415584\pi$
$132$	270.654	0.00135201
$133$	26995.1	0.132329
$134$	157647.	0.758444
$135$	0	0
$136$	−207096.	−0.960117
$137$	36037.4	0.164041	0.0820204	−	0.996631i	$-0.473863\pi$
$137$	36037.4	0.164041	0.0820204	+	0.996631i	$0.473863\pi$
$138$	31473.3	0.140684
$139$	152655.	0.670151	0.335076	−	0.942191i	$-0.391238\pi$
$139$	152655.	0.670151	0.335076	+	0.942191i	$0.391238\pi$
$140$	0	0
$141$	−40638.6	−0.172144
$142$	188190.	0.783205
$143$	−2506.20	−0.0102488
$144$	129924.	0.522136
$145$	0	0
$146$	4818.92	0.0187097
$147$	−1391.16	−0.00530986
$148$	−51884.3	−0.194707
$149$	72547.1	0.267704	0.133852	−	0.991001i	$-0.457265\pi$
$149$	72547.1	0.267704	0.133852	+	0.991001i	$0.457265\pi$
$150$	0	0
$151$	489021.	1.74536	0.872681	−	0.488291i	$-0.162379\pi$
$151$	489021.	1.74536	0.872681	+	0.488291i	$0.162379\pi$
$152$	42108.6	0.147830
$153$	−252627.	−0.872473
$154$	8544.26	0.0290317
$155$	0	0
$156$	−3084.42	−0.0101476
$157$	−89467.9	−0.289680	−0.144840	−	0.989455i	$-0.546267\pi$
$157$	−89467.9	−0.289680	−0.144840	+	0.989455i	$0.546267\pi$
$158$	−217674.	−0.693687
$159$	12733.1	0.0399431
$160$	0	0
$161$	−534432.	−1.62490
$162$	260545.	0.780000
$163$	225668.	0.665275	0.332637	−	0.943055i	$-0.392061\pi$
$163$	225668.	0.665275	0.332637	+	0.943055i	$0.392061\pi$
$164$	−88912.8	−0.258140
$165$	0	0
$166$	341200.	0.961033
$167$	−209528.	−0.581367	−0.290683	−	0.956819i	$-0.593883\pi$
$167$	−209528.	−0.581367	−0.290683	+	0.956819i	$0.593883\pi$
$168$	40580.9	0.110930
$169$	28561.0	0.0769231
$170$	0	0
$171$	51366.5	0.134335
$172$	−95317.6	−0.245670
$173$	465184.	1.18171	0.590853	−	0.806779i	$-0.298791\pi$
$173$	465184.	1.18171	0.590853	+	0.806779i	$0.298791\pi$
$174$	−3756.08	−0.00940506
$175$	0	0
$176$	8016.60	0.0195078
$177$	−60885.7	−0.146077
$178$	44699.9	0.105744
$179$	−472573.	−1.10239	−0.551197	−	0.834375i	$-0.685829\pi$
$179$	−472573.	−1.10239	−0.551197	+	0.834375i	$0.685829\pi$
$180$	0	0
$181$	−74099.4	−0.168120	−0.0840598	−	0.996461i	$-0.526789\pi$
$181$	−74099.4	−0.168120	−0.0840598	+	0.996461i	$0.526789\pi$
$182$	−97371.7	−0.217898
$183$	−29633.1	−0.0654108
$184$	−833638.	−1.81524
$185$	0	0
$186$	35582.4	0.0754141
$187$	−15587.7	−0.0325970
$188$	278924.	0.575561
$189$	99553.5	0.202722
$190$	0	0
$191$	−224128.	−0.444543	−0.222271	−	0.974985i	$-0.571347\pi$
$191$	−224128.	−0.444543	−0.222271	+	0.974985i	$0.571347\pi$
$192$	56763.8	0.111127
$193$	662265.	1.27979	0.639895	−	0.768463i	$-0.278978\pi$
$193$	662265.	1.27979	0.639895	+	0.768463i	$0.278978\pi$
$194$	631467.	1.20461
$195$	0	0
$196$	9548.24	0.0177535
$197$	−666464.	−1.22352	−0.611760	−	0.791043i	$-0.709538\pi$
$197$	−666464.	−1.22352	−0.611760	+	0.791043i	$0.709538\pi$
$198$	16258.1	0.0294717
$199$	645720.	1.15588	0.577938	−	0.816081i	$-0.303858\pi$
$199$	645720.	1.15588	0.577938	+	0.816081i	$0.303858\pi$
$200$	0	0
$201$	56356.3	0.0983903
$202$	634746.	1.09451
$203$	63780.1	0.108629
$204$	−19184.0	−0.0322748
$205$	0	0
$206$	−449369.	−0.737794
$207$	−1.01692e6	−1.64953
$208$	−91358.4	−0.146417
$209$	3169.43	0.00501897
$210$	0	0
$211$	−868021.	−1.34222	−0.671110	−	0.741357i	$-0.734182\pi$
$211$	−868021.	−1.34222	−0.671110	+	0.741357i	$0.734182\pi$
$212$	−87394.1	−0.133550
$213$	67274.9	0.101602
$214$	122470.	0.182808
$215$	0	0
$216$	155289.	0.226468
$217$	−604207.	−0.871037
$218$	−288385.	−0.410991
$219$	1722.69	0.00242715
$220$	0	0
$221$	177639.	0.244658
$222$	34482.7	0.0469590
$223$	58342.9	0.0785644	0.0392822	−	0.999228i	$-0.487493\pi$
$223$	58342.9	0.0785644	0.0392822	+	0.999228i	$0.487493\pi$
$224$	−484882.	−0.645678
$225$	0	0
$226$	−521995.	−0.679822
$227$	−111768.	−0.143964	−0.0719820	−	0.997406i	$-0.522932\pi$
$227$	−111768.	−0.143964	−0.0719820	+	0.997406i	$0.522932\pi$
$228$	3900.67	0.00496937
$229$	1.14984e6	1.44893	0.724467	−	0.689309i	$-0.242086\pi$
$229$	1.14984e6	1.44893	0.724467	+	0.689309i	$0.242086\pi$
$230$	0	0
$231$	3054.44	0.00376618
$232$	99488.0	0.121353
$233$	−630470.	−0.760807	−0.380404	−	0.924821i	$-0.624215\pi$
$233$	−630470.	−0.760807	−0.380404	+	0.924821i	$0.624215\pi$
$234$	−185279.	−0.221201
$235$	0	0
$236$	417891.	0.488408
$237$	−77815.0	−0.0899897
$238$	−605618.	−0.693037
$239$	−165628.	−0.187559	−0.0937797	−	0.995593i	$-0.529895\pi$
$239$	−165628.	−0.187559	−0.0937797	+	0.995593i	$0.529895\pi$
$240$	0	0
$241$	690968.	0.766329	0.383165	−	0.923680i	$-0.374834\pi$
$241$	690968.	0.766329	0.383165	+	0.923680i	$0.374834\pi$
$242$	−733639.	−0.805275
$243$	284667.	0.309259
$244$	203387.	0.218700
$245$	0	0
$246$	59092.1	0.0622575
$247$	−36119.3	−0.0376701
$248$	−942478.	−0.973065
$249$	121974.	0.124671
$250$	0	0
$251$	−386887.	−0.387615	−0.193807	−	0.981040i	$-0.562084\pi$
$251$	−386887.	−0.387615	−0.193807	+	0.981040i	$0.562084\pi$
$252$	−339764.	−0.337036
$253$	−62746.2	−0.0616292
$254$	1.13524e6	1.10408
$255$	0	0
$256$	−949957.	−0.905950
$257$	−258260.	−0.243907	−0.121953	−	0.992536i	$-0.538916\pi$
$257$	−258260.	−0.243907	−0.121953	+	0.992536i	$0.538916\pi$
$258$	63348.8	0.0592501
$259$	−585535.	−0.542379
$260$	0	0
$261$	121361.	0.110275
$262$	469669.	0.422706
$263$	1.19053e6	1.06133	0.530665	−	0.847582i	$-0.321942\pi$
$263$	1.19053e6	1.06133	0.530665	+	0.847582i	$0.321942\pi$
$264$	4764.49	0.00420733
$265$	0	0
$266$	123140.	0.106707
$267$	15979.5	0.0137178
$268$	−386803.	−0.328967
$269$	850968.	0.717022	0.358511	−	0.933525i	$-0.383285\pi$
$269$	850968.	0.717022	0.358511	+	0.933525i	$0.383285\pi$
$270$	0	0
$271$	−40926.1	−0.0338514	−0.0169257	−	0.999857i	$-0.505388\pi$
$271$	−40926.1	−0.0338514	−0.0169257	+	0.999857i	$0.505388\pi$
$272$	−568218.	−0.465685
$273$	−34808.8	−0.0282672
$274$	164387.	0.132279
$275$	0	0
$276$	−77222.8	−0.0610201
$277$	−1.00054e6	−0.783490	−0.391745	−	0.920074i	$-0.628128\pi$
$277$	−1.00054e6	−0.783490	−0.391745	+	0.920074i	$0.628128\pi$
$278$	696342.	0.540394
$279$	−1.14969e6	−0.884239
$280$	0	0
$281$	−1.73596e6	−1.31151	−0.655757	−	0.754972i	$-0.727650\pi$
$281$	−1.73596e6	−1.31151	−0.655757	+	0.754972i	$0.727650\pi$
$282$	−185375.	−0.138813
$283$	1.27363e6	0.945319	0.472660	−	0.881245i	$-0.343294\pi$
$283$	1.27363e6	0.945319	0.472660	+	0.881245i	$0.343294\pi$
$284$	−461743.	−0.339707
$285$	0	0
$286$	−11432.1	−0.00826443
$287$	−1.00342e6	−0.719078
$288$	−922636.	−0.655464
$289$	−315001.	−0.221854
$290$	0	0
$291$	225739.	0.156270
$292$	−11823.7	−0.00811515
$293$	2043.70	0.00139075	0.000695374	−	1.00000i	$-0.499779\pi$
$293$	2043.70	0.00139075	0.000695374		1.00000i	$0.499779\pi$
$294$	−6345.84	−0.00428174
$295$	0	0
$296$	−913351.	−0.605910
$297$	11688.3	0.00768883
$298$	330927.	0.215870
$299$	715066.	0.462560
$300$	0	0
$301$	−1.07570e6	−0.684342
$302$	2.23070e6	1.40742
$303$	226912.	0.141988
$304$	115535.	0.0717018
$305$	0	0
$306$	−1.15237e6	−0.703541
$307$	401308.	0.243014	0.121507	−	0.992591i	$-0.461227\pi$
$307$	401308.	0.243014	0.121507	+	0.992591i	$0.461227\pi$
$308$	−20964.2	−0.0125922
$309$	−160642.	−0.0957114
$310$	0	0
$311$	−1.92628e6	−1.12933	−0.564663	−	0.825322i	$-0.690994\pi$
$311$	−1.92628e6	−1.12933	−0.564663	+	0.825322i	$0.690994\pi$
$312$	−54296.9	−0.0315783
$313$	−1.64519e6	−0.949196	−0.474598	−	0.880203i	$-0.657407\pi$
$313$	−1.64519e6	−0.949196	−0.474598	+	0.880203i	$0.657407\pi$
$314$	−408113.	−0.233591
$315$	0	0
$316$	534085.	0.300880
$317$	1.99476e6	1.11492	0.557459	−	0.830205i	$-0.311777\pi$
$317$	1.99476e6	1.11492	0.557459	+	0.830205i	$0.311777\pi$
$318$	58082.8	0.0322092
$319$	7488.26	0.00412006
$320$	0	0
$321$	43781.2	0.0237151
$322$	−2.43784e6	−1.31028
$323$	−224649.	−0.119812
$324$	−639273.	−0.338317
$325$	0	0
$326$	1.02940e6	0.536462
$327$	−103093.	−0.0533164
$328$	−1.56519e6	−0.803306
$329$	3.14777e6	1.60329
$330$	0	0
$331$	−675924.	−0.339100	−0.169550	−	0.985522i	$-0.554231\pi$
$331$	−675924.	−0.339100	−0.169550	+	0.985522i	$0.554231\pi$
$332$	−837168.	−0.416838
$333$	−1.11416e6	−0.550600
$334$	−955772.	−0.468800
$335$	0	0
$336$	111343.	0.0538042
$337$	2.13552e6	1.02430	0.512152	−	0.858895i	$-0.328848\pi$
$337$	2.13552e6	1.02430	0.512152	+	0.858895i	$0.328848\pi$
$338$	130283.	0.0620289
$339$	−186605.	−0.0881909
$340$	0	0
$341$	−70938.3	−0.0330366
$342$	234311.	0.108325
$343$	2.23063e6	1.02374
$344$	−1.67793e6	−0.764502
$345$	0	0
$346$	2.12196e6	0.952899
$347$	2.57257e6	1.14695	0.573473	−	0.819225i	$-0.305596\pi$
$347$	2.57257e6	1.14695	0.573473	+	0.819225i	$0.305596\pi$
$348$	9215.92	0.00407935
$349$	2.02363e6	0.889339	0.444670	−	0.895695i	$-0.353321\pi$
$349$	2.02363e6	0.889339	0.444670	+	0.895695i	$0.353321\pi$
$350$	0	0
$351$	−133202.	−0.0577088
$352$	−56928.7	−0.0244892
$353$	−2.04810e6	−0.874810	−0.437405	−	0.899265i	$-0.644102\pi$
$353$	−2.04810e6	−0.874810	−0.437405	+	0.899265i	$0.644102\pi$
$354$	−277733.	−0.117793
$355$	0	0
$356$	−109676.	−0.0458654
$357$	−216499.	−0.0899053
$358$	−2.15567e6	−0.888944
$359$	−1.59901e6	−0.654808	−0.327404	−	0.944885i	$-0.606174\pi$
$359$	−1.59901e6	−0.654808	−0.327404	+	0.944885i	$0.606174\pi$
$360$	0	0
$361$	−2.43042e6	−0.981553
$362$	−338008.	−0.135568
$363$	−262265.	−0.104466
$364$	238911.	0.0945112
$365$	0	0
$366$	−135173.	−0.0527457
$367$	3.86389e6	1.49747	0.748737	−	0.662867i	$-0.230661\pi$
$367$	3.86389e6	1.49747	0.748737	+	0.662867i	$0.230661\pi$
$368$	−2.28729e6	−0.880444
$369$	−1.90930e6	−0.729977
$370$	0	0
$371$	−986276.	−0.372018
$372$	−87305.0	−0.0327101
$373$	1.56702e6	0.583179	0.291589	−	0.956544i	$-0.405816\pi$
$373$	1.56702e6	0.583179	0.291589	+	0.956544i	$0.405816\pi$
$374$	−71104.0	−0.0262854
$375$	0	0
$376$	4.91007e6	1.79109
$377$	−85337.3	−0.0309233
$378$	454118.	0.163471
$379$	3.19239e6	1.14161	0.570805	−	0.821086i	$-0.306631\pi$
$379$	3.19239e6	1.14161	0.570805	+	0.821086i	$0.306631\pi$
$380$	0	0
$381$	405829.	0.143229
$382$	−1.02237e6	−0.358469
$383$	−400432.	−0.139486	−0.0697432	−	0.997565i	$-0.522218\pi$
$383$	−400432.	−0.139486	−0.0697432	+	0.997565i	$0.522218\pi$
$384$	58611.9	0.0202842
$385$	0	0
$386$	3.02096e6	1.03199
$387$	−2.04684e6	−0.694715
$388$	−1.54937e6	−0.522487
$389$	413440.	0.138528	0.0692642	−	0.997598i	$-0.477935\pi$
$389$	413440.	0.138528	0.0692642	+	0.997598i	$0.477935\pi$
$390$	0	0
$391$	4.44746e6	1.47120
$392$	168083.	0.0552471
$393$	167899.	0.0548362
$394$	−3.04011e6	−0.986618
$395$	0	0
$396$	−39890.8	−0.0127831
$397$	102926.	0.0327753	0.0163877	−	0.999866i	$-0.494783\pi$
$397$	102926.	0.0327753	0.0163877	+	0.999866i	$0.494783\pi$
$398$	2.94548e6	0.932071
$399$	44020.5	0.0138428
$400$	0	0
$401$	2.23365e6	0.693671	0.346836	−	0.937926i	$-0.387256\pi$
$401$	2.23365e6	0.693671	0.346836	+	0.937926i	$0.387256\pi$
$402$	257072.	0.0793396
$403$	808424.	0.247957
$404$	−1.55741e6	−0.474734
$405$	0	0
$406$	290937.	0.0875958
$407$	−68746.0	−0.0205713
$408$	−337708.	−0.100436
$409$	−4.46150e6	−1.31878	−0.659390	−	0.751801i	$-0.729185\pi$
$409$	−4.46150e6	−1.31878	−0.659390	+	0.751801i	$0.729185\pi$
$410$	0	0
$411$	58765.6	0.0171600
$412$	1.10257e6	0.320010
$413$	4.71606e6	1.36052
$414$	−4.63874e6	−1.33014
$415$	0	0
$416$	648768.	0.183804
$417$	248931.	0.0701034
$418$	14457.5	0.00404718
$419$	4.22792e6	1.17650	0.588250	−	0.808679i	$-0.299817\pi$
$419$	4.22792e6	1.17650	0.588250	+	0.808679i	$0.299817\pi$
$420$	0	0
$421$	4.11791e6	1.13233	0.566163	−	0.824293i	$-0.308427\pi$
$421$	4.11791e6	1.13233	0.566163	+	0.824293i	$0.308427\pi$
$422$	−3.95952e6	−1.08233
$423$	5.98959e6	1.62759
$424$	−1.53845e6	−0.415594
$425$	0	0
$426$	306878.	0.0819298
$427$	2.29531e6	0.609216
$428$	−300493.	−0.0792912
$429$	−4086.81	−0.00107212
$430$	0	0
$431$	1.15324e6	0.299038	0.149519	−	0.988759i	$-0.452227\pi$
$431$	1.15324e6	0.299038	0.149519	+	0.988759i	$0.452227\pi$
$432$	426074.	0.109844
$433$	33734.3	0.00864673	0.00432337	−	0.999991i	$-0.498624\pi$
$433$	33734.3	0.00864673	0.00432337	+	0.999991i	$0.498624\pi$
$434$	−2.75612e6	−0.702383
$435$	0	0
$436$	707583.	0.178263
$437$	−904298.	−0.226521
$438$	7858.13	0.00195719
$439$	7.48363e6	1.85332	0.926661	−	0.375898i	$-0.122666\pi$
$439$	7.48363e6	1.85332	0.926661	+	0.375898i	$0.122666\pi$
$440$	0	0
$441$	205038.	0.0502039
$442$	810312.	0.197286
$443$	3.28028e6	0.794148	0.397074	−	0.917786i	$-0.370026\pi$
$443$	3.28028e6	0.794148	0.397074	+	0.917786i	$0.370026\pi$
$444$	−84606.9	−0.0203680
$445$	0	0
$446$	266134.	0.0633525
$447$	118301.	0.0280040
$448$	−4.39678e6	−1.03500
$449$	7.95356e6	1.86185	0.930927	−	0.365205i	$-0.119001\pi$
$449$	7.95356e6	1.86185	0.930927	+	0.365205i	$0.119001\pi$
$450$	0	0
$451$	−117808.	−0.0272731
$452$	1.28077e6	0.294866
$453$	797439.	0.182579
$454$	−509837.	−0.116089
$455$	0	0
$456$	68665.8	0.0154642
$457$	3.35187e6	0.750753	0.375377	−	0.926872i	$-0.377513\pi$
$457$	3.35187e6	0.750753	0.375377	+	0.926872i	$0.377513\pi$
$458$	5.24506e6	1.16839
$459$	−828468.	−0.183546
$460$	0	0
$461$	4.68627e6	1.02701	0.513505	−	0.858086i	$-0.328347\pi$
$461$	4.68627e6	1.02701	0.513505	+	0.858086i	$0.328347\pi$
$462$	13933.0	0.00303696
$463$	−6.64697e6	−1.44102	−0.720512	−	0.693442i	$-0.756093\pi$
$463$	−6.64697e6	−1.44102	−0.720512	+	0.693442i	$0.756093\pi$
$464$	272969.	0.0588599
$465$	0	0
$466$	−2.87592e6	−0.613497
$467$	3.14141e6	0.666549	0.333275	−	0.942830i	$-0.391846\pi$
$467$	3.14141e6	0.666549	0.333275	+	0.942830i	$0.391846\pi$
$468$	454602.	0.0959437
$469$	−4.36522e6	−0.916377
$470$	0	0
$471$	−145894.	−0.0303029
$472$	7.35638e6	1.51988
$473$	−126295.	−0.0259556
$474$	−354957.	−0.0725655
$475$	0	0
$476$	1.48595e6	0.300598
$477$	−1.87669e6	−0.377656
$478$	−755521.	−0.151243
$479$	−6.68286e6	−1.33083	−0.665416	−	0.746473i	$-0.731746\pi$
$479$	−6.68286e6	−1.33083	−0.665416	+	0.746473i	$0.731746\pi$
$480$	0	0
$481$	783441.	0.154399
$482$	3.15189e6	0.617950
$483$	−871489.	−0.169979
$484$	1.80006e6	0.349280
$485$	0	0
$486$	1.29853e6	0.249379
$487$	4.06478e6	0.776631	0.388316	−	0.921526i	$-0.373057\pi$
$487$	4.06478e6	0.776631	0.388316	+	0.921526i	$0.373057\pi$
$488$	3.58035e6	0.680575
$489$	367993.	0.0695933
$490$	0	0
$491$	−2.10434e6	−0.393923	−0.196962	−	0.980411i	$-0.563107\pi$
$491$	−2.10434e6	−0.393923	−0.196962	+	0.980411i	$0.563107\pi$
$492$	−144989.	−0.0270036
$493$	−530768.	−0.0983530
$494$	−164760.	−0.0303763
$495$	0	0
$496$	−2.58592e6	−0.471966
$497$	−5.21095e6	−0.946293
$498$	556389.	0.100532
$499$	5.96715e6	1.07279	0.536396	−	0.843966i	$-0.319785\pi$
$499$	5.96715e6	1.07279	0.536396	+	0.843966i	$0.319785\pi$
$500$	0	0
$501$	−341673.	−0.0608158
$502$	−1.76481e6	−0.312563
$503$	1.00144e7	1.76483	0.882417	−	0.470467i	$-0.155915\pi$
$503$	1.00144e7	1.76483	0.882417	+	0.470467i	$0.155915\pi$
$504$	−5.98108e6	−1.04882
$505$	0	0
$506$	−286220.	−0.0496963
$507$	46573.9	0.00804680
$508$	−2.78542e6	−0.478885
$509$	−8.47321e6	−1.44962	−0.724809	−	0.688950i	$-0.758072\pi$
$509$	−8.47321e6	−1.44962	−0.724809	+	0.688950i	$0.758072\pi$
$510$	0	0
$511$	−133435.	−0.0226057
$512$	−5.48346e6	−0.924442
$513$	168452.	0.0282606
$514$	−1.17807e6	−0.196681
$515$	0	0
$516$	−155433.	−0.0256991
$517$	369571.	0.0608095
$518$	−2.67095e6	−0.437362
$519$	758567.	0.123616
$520$	0	0
$521$	−197614.	−0.0318951	−0.0159476	−	0.999873i	$-0.505076\pi$
$521$	−197614.	−0.0318951	−0.0159476	+	0.999873i	$0.505076\pi$
$522$	553596.	0.0889235
$523$	−8.27263e6	−1.32248	−0.661240	−	0.750174i	$-0.729970\pi$
$523$	−8.27263e6	−1.32248	−0.661240	+	0.750174i	$0.729970\pi$
$524$	−1.15238e6	−0.183345
$525$	0	0
$526$	5.43066e6	0.855831
$527$	5.02812e6	0.788640
$528$	13072.5	0.00204068
$529$	1.14664e7	1.78150
$530$	0	0
$531$	8.97374e6	1.38114
$532$	−302136.	−0.0462832
$533$	1.34256e6	0.204699
$534$	72891.3	0.0110617
$535$	0	0
$536$	−6.80913e6	−1.02372
$537$	−770618.	−0.115320
$538$	3.88174e6	0.578190
$539$	12651.3	0.00187570
$540$	0	0
$541$	363216.	0.0533546	0.0266773	−	0.999644i	$-0.491507\pi$
$541$	363216.	0.0533546	0.0266773	+	0.999644i	$0.491507\pi$
$542$	−186686.	−0.0272970
$543$	−120833.	−0.0175867
$544$	4.03511e6	0.584599
$545$	0	0
$546$	−158782.	−0.0227940
$547$	620452.	0.0886624	0.0443312	−	0.999017i	$-0.485884\pi$
$547$	620452.	0.0886624	0.0443312	+	0.999017i	$0.485884\pi$
$548$	−403339.	−0.0573745
$549$	4.36752e6	0.618449
$550$	0	0
$551$	107921.	0.0151435
$552$	−1.35940e6	−0.189889
$553$	6.02736e6	0.838136
$554$	−4.56400e6	−0.631788
$555$	0	0
$556$	−1.70855e6	−0.234390
$557$	−3.89737e6	−0.532272	−0.266136	−	0.963935i	$-0.585747\pi$
$557$	−3.89737e6	−0.532272	−0.266136	+	0.963935i	$0.585747\pi$
$558$	−5.24437e6	−0.713029
$559$	1.43927e6	0.194811
$560$	0	0
$561$	−25418.5	−0.00340992
$562$	−7.91865e6	−1.05757
$563$	−510725.	−0.0679073	−0.0339536	−	0.999423i	$-0.510810\pi$
$563$	−510725.	−0.0679073	−0.0339536	+	0.999423i	$0.510810\pi$
$564$	454837.	0.0602085
$565$	0	0
$566$	5.80975e6	0.762283
$567$	−7.21445e6	−0.942422
$568$	−8.12834e6	−1.05714
$569$	9.75625e6	1.26329	0.631644	−	0.775259i	$-0.282381\pi$
$569$	9.75625e6	1.26329	0.631644	+	0.775259i	$0.282381\pi$
$570$	0	0
$571$	−1.41952e7	−1.82201	−0.911006	−	0.412393i	$-0.864693\pi$
$571$	−1.41952e7	−1.82201	−0.911006	+	0.412393i	$0.864693\pi$
$572$	28049.9	0.00358461
$573$	−365482.	−0.0465029
$574$	−4.57713e6	−0.579847
$575$	0	0
$576$	−8.36622e6	−1.05069
$577$	1.16423e6	0.145579	0.0727896	−	0.997347i	$-0.476810\pi$
$577$	1.16423e6	0.145579	0.0727896	+	0.997347i	$0.476810\pi$
$578$	−1.43689e6	−0.178898
$579$	1.07994e6	0.133877
$580$	0	0
$581$	−9.44777e6	−1.16115
$582$	1.02972e6	0.126012
$583$	−115796.	−0.0141098
$584$	−208140.	−0.0252536
$585$	0	0
$586$	9322.45	0.00112147
$587$	−6.58038e6	−0.788234	−0.394117	−	0.919060i	$-0.628950\pi$
$587$	−6.58038e6	−0.788234	−0.394117	+	0.919060i	$0.628950\pi$
$588$	15570.2	0.00185716
$589$	−1.02236e6	−0.121427
$590$	0	0
$591$	−1.08679e6	−0.127991
$592$	−2.50600e6	−0.293885
$593$	1.91423e6	0.223541	0.111771	−	0.993734i	$-0.464348\pi$
$593$	1.91423e6	0.223541	0.111771	+	0.993734i	$0.464348\pi$
$594$	53316.8	0.00620009
$595$	0	0
$596$	−811964.	−0.0936313
$597$	1.05296e6	0.120914
$598$	3.26181e6	0.372997
$599$	−2.33678e6	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$599$	−2.33678e6	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$600$	0	0
$601$	1.04273e7	1.17757	0.588786	−	0.808289i	$-0.299606\pi$
$601$	1.04273e7	1.17757	0.588786	+	0.808289i	$0.299606\pi$
$602$	−4.90684e6	−0.551837
$603$	−8.30617e6	−0.930266
$604$	−5.47324e6	−0.610453
$605$	0	0
$606$	1.03507e6	0.114495
$607$	120274.	0.0132495	0.00662474	−	0.999978i	$-0.497891\pi$
$607$	120274.	0.0132495	0.00662474	+	0.999978i	$0.497891\pi$
$608$	−820455.	−0.0900111
$609$	104005.	0.0113635
$610$	0	0
$611$	−4.21169e6	−0.456408
$612$	2.82747e6	0.305154
$613$	−1.34576e7	−1.44649	−0.723245	−	0.690592i	$-0.757350\pi$
$613$	−1.34576e7	−1.44649	−0.723245	+	0.690592i	$0.757350\pi$
$614$	1.83059e6	0.195961
$615$	0	0
$616$	−369046.	−0.0391858
$617$	6.84879e6	0.724270	0.362135	−	0.932126i	$-0.382048\pi$
$617$	6.84879e6	0.724270	0.362135	+	0.932126i	$0.382048\pi$
$618$	−732778.	−0.0771794
$619$	−5.40663e6	−0.567153	−0.283577	−	0.958950i	$-0.591521\pi$
$619$	−5.40663e6	−0.567153	−0.283577	+	0.958950i	$0.591521\pi$
$620$	0	0
$621$	−3.33490e6	−0.347019
$622$	−8.78684e6	−0.910661
$623$	−1.23773e6	−0.127763
$624$	−148977.	−0.0153164
$625$	0	0
$626$	−7.50463e6	−0.765409
$627$	5168.33	0.000525027 0
$628$	1.00135e6	0.101318
$629$	4.87273e6	0.491072
$630$	0	0
$631$	−9.62552e6	−0.962390	−0.481195	−	0.876614i	$-0.659797\pi$
$631$	−9.62552e6	−0.962390	−0.481195	+	0.876614i	$0.659797\pi$
$632$	9.40183e6	0.936310
$633$	−1.41547e6	−0.140408
$634$	9.09920e6	0.899043
$635$	0	0
$636$	−142512.	−0.0139704
$637$	−144176.	−0.0140781
$638$	34158.1	0.00332232
$639$	−9.91542e6	−0.960636
$640$	0	0
$641$	−1.58752e7	−1.52607	−0.763037	−	0.646355i	$-0.776293\pi$
$641$	−1.58752e7	−1.52607	−0.763037	+	0.646355i	$0.776293\pi$
$642$	199710.	0.0191233
$643$	−1.57235e7	−1.49976	−0.749880	−	0.661574i	$-0.769889\pi$
$643$	−1.57235e7	−1.49976	−0.749880	+	0.661574i	$0.769889\pi$
$644$	5.98149e6	0.568322
$645$	0	0
$646$	−1.02475e6	−0.0966132
$647$	1.58173e7	1.48549	0.742747	−	0.669572i	$-0.233523\pi$
$647$	1.58173e7	1.48549	0.742747	+	0.669572i	$0.233523\pi$
$648$	−1.12535e7	−1.05281
$649$	553699.	0.0516015
$650$	0	0
$651$	−985270.	−0.0911178
$652$	−2.52573e6	−0.232685
$653$	−5.31229e6	−0.487527	−0.243763	−	0.969835i	$-0.578382\pi$
$653$	−5.31229e6	−0.487527	−0.243763	+	0.969835i	$0.578382\pi$
$654$	−470265.	−0.0429931
$655$	0	0
$656$	−4.29447e6	−0.389628
$657$	−253901.	−0.0229483
$658$	1.43587e7	1.29286
$659$	9.90554e6	0.888514	0.444257	−	0.895899i	$-0.353468\pi$
$659$	9.90554e6	0.888514	0.444257	+	0.895899i	$0.353468\pi$
$660$	0	0
$661$	−1.29988e7	−1.15717	−0.578587	−	0.815621i	$-0.696396\pi$
$661$	−1.29988e7	−1.15717	−0.578587	+	0.815621i	$0.696396\pi$
$662$	−3.08326e6	−0.273442
$663$	289674.	0.0255932
$664$	−1.47372e7	−1.29716
$665$	0	0
$666$	−5.08229e6	−0.443991
$667$	−2.13654e6	−0.185950
$668$	2.34508e6	0.203337
$669$	95138.8	0.00821850
$670$	0	0
$671$	269486.	0.0231062
$672$	−790688.	−0.0675433
$673$	1.32503e7	1.12769	0.563844	−	0.825881i	$-0.309322\pi$
$673$	1.32503e7	1.12769	0.563844	+	0.825881i	$0.309322\pi$
$674$	9.74129e6	0.825975
$675$	0	0
$676$	−319661.	−0.0269044
$677$	−2.23310e7	−1.87257	−0.936284	−	0.351245i	$-0.885758\pi$
$677$	−2.23310e7	−1.87257	−0.936284	+	0.351245i	$0.885758\pi$
$678$	−851208.	−0.0711151
$679$	−1.74852e7	−1.45545
$680$	0	0
$681$	−182259.	−0.0150598
$682$	−323589.	−0.0266399
$683$	1.40049e7	1.14876	0.574380	−	0.818588i	$-0.305243\pi$
$683$	1.40049e7	1.14876	0.574380	+	0.818588i	$0.305243\pi$
$684$	−574906.	−0.0469847
$685$	0	0
$686$	1.01751e7	0.825523
$687$	1.87502e6	0.151571
$688$	−4.60382e6	−0.370806
$689$	1.31963e6	0.105902
$690$	0	0
$691$	−5.24817e6	−0.418132	−0.209066	−	0.977902i	$-0.567042\pi$
$691$	−5.24817e6	−0.418132	−0.209066	+	0.977902i	$0.567042\pi$
$692$	−5.20645e6	−0.413310
$693$	−450183.	−0.0356087
$694$	1.17349e7	0.924870
$695$	0	0
$696$	162233.	0.0126945
$697$	8.35027e6	0.651056
$698$	9.23090e6	0.717142
$699$	−1.02810e6	−0.0795868
$700$	0	0
$701$	−2.13994e7	−1.64477	−0.822386	−	0.568930i	$-0.807358\pi$
$701$	−2.13994e7	−1.64477	−0.822386	+	0.568930i	$0.807358\pi$
$702$	−607607.	−0.0465350
$703$	−990767.	−0.0756107
$704$	−516214.	−0.0392553
$705$	0	0
$706$	−9.34250e6	−0.705426
$707$	−1.75760e7	−1.32243
$708$	681447.	0.0510915
$709$	4.35333e6	0.325242	0.162621	−	0.986689i	$-0.448005\pi$
$709$	4.35333e6	0.325242	0.162621	+	0.986689i	$0.448005\pi$
$710$	0	0
$711$	1.14689e7	0.850839
$712$	−1.93069e6	−0.142729
$713$	2.02401e7	1.49104
$714$	−987571.	−0.0724975
$715$	0	0
$716$	5.28915e6	0.385570
$717$	−270087.	−0.0196203
$718$	−7.29395e6	−0.528021
$719$	2.21389e7	1.59710	0.798552	−	0.601926i	$-0.205600\pi$
$719$	2.21389e7	1.59710	0.798552	+	0.601926i	$0.205600\pi$
$720$	0	0
$721$	1.24430e7	0.891426
$722$	−1.10865e7	−0.791501
$723$	1.12675e6	0.0801645
$724$	829338.	0.0588010
$725$	0	0
$726$	−1.19633e6	−0.0842385
$727$	4.83218e6	0.339084	0.169542	−	0.985523i	$-0.445771\pi$
$727$	4.83218e6	0.339084	0.169542	+	0.985523i	$0.445771\pi$
$728$	4.20570e6	0.294110
$729$	−1.34154e7	−0.934940
$730$	0	0
$731$	8.95178e6	0.619606
$732$	331661.	0.0228779
$733$	2.47827e7	1.70368	0.851840	−	0.523803i	$-0.175487\pi$
$733$	2.47827e7	1.70368	0.851840	+	0.523803i	$0.175487\pi$
$734$	1.76253e7	1.20753
$735$	0	0
$736$	1.62428e7	1.10527
$737$	−512509.	−0.0347562
$738$	−8.70939e6	−0.588636
$739$	7.09289e6	0.477762	0.238881	−	0.971049i	$-0.423219\pi$
$739$	7.09289e6	0.477762	0.238881	+	0.971049i	$0.423219\pi$
$740$	0	0
$741$	−58899.1	−0.00394061
$742$	−4.49895e6	−0.299986
$743$	−1.95117e7	−1.29665	−0.648327	−	0.761362i	$-0.724531\pi$
$743$	−1.95117e7	−1.29665	−0.648327	+	0.761362i	$0.724531\pi$
$744$	−1.53688e6	−0.101791
$745$	0	0
$746$	7.14803e6	0.470262
$747$	−1.79773e7	−1.17875
$748$	174461.	0.0114010
$749$	−3.39118e6	−0.220875
$750$	0	0
$751$	1.66103e7	1.07468	0.537339	−	0.843366i	$-0.319430\pi$
$751$	1.66103e7	1.07468	0.537339	+	0.843366i	$0.319430\pi$
$752$	1.34720e7	0.868733
$753$	−630891.	−0.0405477
$754$	−389271.	−0.0249358
$755$	0	0
$756$	−1.11423e6	−0.0709037
$757$	−1.22902e6	−0.0779508	−0.0389754	−	0.999240i	$-0.512409\pi$
$757$	−1.22902e6	−0.0779508	−0.0389754	+	0.999240i	$0.512409\pi$
$758$	1.45622e7	0.920567
$759$	−102319.	−0.00644693
$760$	0	0
$761$	1.37482e7	0.860569	0.430284	−	0.902693i	$-0.358413\pi$
$761$	1.37482e7	0.860569	0.430284	+	0.902693i	$0.358413\pi$
$762$	1.85121e6	0.115496
$763$	7.98535e6	0.496572
$764$	2.50850e6	0.155482
$765$	0	0
$766$	−1.82659e6	−0.112478
$767$	−6.31004e6	−0.387297
$768$	−1.54908e6	−0.0947699
$769$	−1.01549e7	−0.619240	−0.309620	−	0.950860i	$-0.600202\pi$
$769$	−1.01549e7	−0.619240	−0.309620	+	0.950860i	$0.600202\pi$
$770$	0	0
$771$	−421140.	−0.0255147
$772$	−7.41222e6	−0.447616
$773$	−1.41511e7	−0.851807	−0.425903	−	0.904769i	$-0.640044\pi$
$773$	−1.41511e7	−0.851807	−0.425903	+	0.904769i	$0.640044\pi$
$774$	−9.33677e6	−0.560202
$775$	0	0
$776$	−2.72745e7	−1.62593
$777$	−954821.	−0.0567374
$778$	1.88593e6	0.111706
$779$	−1.69785e6	−0.100243
$780$	0	0
$781$	−611803.	−0.0358909
$782$	2.02873e7	1.18634
$783$	397993.	0.0231991
$784$	461178.	0.0267965
$785$	0	0
$786$	765882.	0.0442186
$787$	1.03095e6	0.0593338	0.0296669	−	0.999560i	$-0.490555\pi$
$787$	1.03095e6	0.0593338	0.0296669	+	0.999560i	$0.490555\pi$
$788$	7.45923e6	0.427935
$789$	1.94137e6	0.111024
$790$	0	0
$791$	1.44540e7	0.821383
$792$	−702222.	−0.0397797
$793$	−3.07110e6	−0.173425
$794$	469500.	0.0264292
$795$	0	0
$796$	−7.22705e6	−0.404276
$797$	1.43337e7	0.799303	0.399651	−	0.916667i	$-0.369131\pi$
$797$	1.43337e7	0.799303	0.399651	+	0.916667i	$0.369131\pi$
$798$	200802.	0.0111625
$799$	−2.61952e7	−1.45163
$800$	0	0
$801$	−2.35516e6	−0.129700
$802$	1.01889e7	0.559360
$803$	−15666.3	−0.000857386 0
$804$	−630753.	−0.0344127
$805$	0	0
$806$	3.68767e6	0.199947
$807$	1.38766e6	0.0750065
$808$	−2.74161e7	−1.47733
$809$	1.55020e7	0.832751	0.416376	−	0.909193i	$-0.363300\pi$
$809$	1.55020e7	0.832751	0.416376	+	0.909193i	$0.363300\pi$
$810$	0	0
$811$	2.45861e7	1.31261	0.656307	−	0.754494i	$-0.272118\pi$
$811$	2.45861e7	1.31261	0.656307	+	0.754494i	$0.272118\pi$
$812$	−713842.	−0.0379938
$813$	−66737.5	−0.00354114
$814$	−313589.	−0.0165882
$815$	0	0
$816$	−926583.	−0.0487146
$817$	−1.82016e6	−0.0954011
$818$	−2.03513e7	−1.06343
$819$	5.13036e6	0.267262
$820$	0	0
$821$	−3.33396e6	−0.172624	−0.0863122	−	0.996268i	$-0.527508\pi$
$821$	−3.33396e6	−0.172624	−0.0863122	+	0.996268i	$0.527508\pi$
$822$	268062.	0.0138375
$823$	−3.08787e6	−0.158913	−0.0794564	−	0.996838i	$-0.525318\pi$
$823$	−3.08787e6	−0.158913	−0.0794564	+	0.996838i	$0.525318\pi$
$824$	1.94092e7	0.995842
$825$	0	0
$826$	2.15125e7	1.09709
$827$	−6.89555e6	−0.350595	−0.175297	−	0.984516i	$-0.556089\pi$
$827$	−6.89555e6	−0.350595	−0.175297	+	0.984516i	$0.556089\pi$
$828$	1.13816e7	0.576936
$829$	−2.00007e7	−1.01078	−0.505391	−	0.862890i	$-0.668652\pi$
$829$	−2.00007e7	−1.01078	−0.505391	+	0.862890i	$0.668652\pi$
$830$	0	0
$831$	−1.63156e6	−0.0819596
$832$	5.88286e6	0.294632
$833$	−896725.	−0.0447762
$834$	1.13551e6	0.0565297
$835$	0	0
$836$	−35473.0	−0.00175542
$837$	−3.77030e6	−0.186021
$838$	1.92859e7	0.948701
$839$	5.23988e6	0.256990	0.128495	−	0.991710i	$-0.458985\pi$
$839$	5.23988e6	0.256990	0.128495	+	0.991710i	$0.458985\pi$
$840$	0	0
$841$	−2.02562e7	−0.987569
$842$	1.87841e7	0.913081
$843$	−2.83079e6	−0.137195
$844$	9.71509e6	0.469452
$845$	0	0
$846$	2.73218e7	1.31245
$847$	2.03144e7	0.972959
$848$	−4.22111e6	−0.201575
$849$	2.07689e6	0.0988883
$850$	0	0
$851$	1.96146e7	0.928442
$852$	−752957.	−0.0355362
$853$	1.32853e7	0.625170	0.312585	−	0.949890i	$-0.398805\pi$
$853$	1.32853e7	0.625170	0.312585	+	0.949890i	$0.398805\pi$
$854$	1.04702e7	0.491257
$855$	0	0
$856$	−5.28976e6	−0.246747
$857$	8.34473e6	0.388115	0.194057	−	0.980990i	$-0.437835\pi$
$857$	8.34473e6	0.388115	0.194057	+	0.980990i	$0.437835\pi$
$858$	−18642.2	−0.000864528 0
$859$	−4.07521e7	−1.88437	−0.942187	−	0.335088i	$-0.891234\pi$
$859$	−4.07521e7	−1.88437	−0.942187	+	0.335088i	$0.891234\pi$
$860$	0	0
$861$	−1.63625e6	−0.0752216
$862$	5.26056e6	0.241137
$863$	1.73853e7	0.794614	0.397307	−	0.917686i	$-0.369945\pi$
$863$	1.73853e7	0.794614	0.397307	+	0.917686i	$0.369945\pi$
$864$	−3.02570e6	−0.137893
$865$	0	0
$866$	153881.	0.00697252
$867$	−513667.	−0.0232078
$868$	6.76243e6	0.304652
$869$	707656.	0.0317887
$870$	0	0
$871$	5.84063e6	0.260864
$872$	1.24560e7	0.554738
$873$	−3.32710e7	−1.47751
$874$	−4.12500e6	−0.182661
$875$	0	0
$876$	−19280.7	−0.000848912 0
$877$	−2.58279e6	−0.113394	−0.0566971	−	0.998391i	$-0.518057\pi$
$877$	−2.58279e6	−0.113394	−0.0566971	+	0.998391i	$0.518057\pi$
$878$	3.41370e7	1.49447
$879$	3332.63	0.000145484 0
$880$	0	0
$881$	−1.66814e7	−0.724090	−0.362045	−	0.932161i	$-0.617921\pi$
$881$	−1.66814e7	−0.724090	−0.362045	+	0.932161i	$0.617921\pi$
$882$	935291.	0.0404833
$883$	−2.36384e7	−1.02027	−0.510137	−	0.860093i	$-0.670405\pi$
$883$	−2.36384e7	−1.02027	−0.510137	+	0.860093i	$0.670405\pi$
$884$	−1.98818e6	−0.0855708
$885$	0	0
$886$	1.49632e7	0.640382
$887$	5.25660e6	0.224334	0.112167	−	0.993689i	$-0.464221\pi$
$887$	5.25660e6	0.224334	0.112167	+	0.993689i	$0.464221\pi$
$888$	−1.48939e6	−0.0633833
$889$	−3.14345e7	−1.33399
$890$	0	0
$891$	−847029.	−0.0357441
$892$	−652988.	−0.0274785
$893$	5.32625e6	0.223508
$894$	539638.	0.0225818
$895$	0	0
$896$	−4.53994e6	−0.188921
$897$	1.16605e6	0.0483876
$898$	3.62806e7	1.50135
$899$	−2.41549e6	−0.0996795
$900$	0	0
$901$	8.20763e6	0.336826
$902$	−537389.	−0.0219924
$903$	−1.75412e6	−0.0715879
$904$	2.25461e7	0.917595
$905$	0	0
$906$	3.63756e6	0.147228
$907$	−3.22789e7	−1.30287	−0.651435	−	0.758705i	$-0.725833\pi$
$907$	−3.22789e7	−1.30287	−0.651435	+	0.758705i	$0.725833\pi$
$908$	1.25094e6	0.0503525
$909$	−3.34438e7	−1.34247
$910$	0	0
$911$	4.20975e7	1.68058	0.840292	−	0.542134i	$-0.182383\pi$
$911$	4.20975e7	1.68058	0.840292	+	0.542134i	$0.182383\pi$
$912$	188401.	0.00750061
$913$	−1.10924e6	−0.0440400
$914$	1.52897e7	0.605389
$915$	0	0
$916$	−1.28693e7	−0.506775
$917$	−1.30051e7	−0.510728
$918$	−3.77910e6	−0.148007
$919$	−2.19460e7	−0.857168	−0.428584	−	0.903502i	$-0.640987\pi$
$919$	−2.19460e7	−0.857168	−0.428584	+	0.903502i	$0.640987\pi$
$920$	0	0
$921$	654407.	0.0254213
$922$	2.13767e7	0.828157
$923$	6.97221e6	0.269380
$924$	−34186.0	−0.00131725
$925$	0	0
$926$	−3.03205e7	−1.16201
$927$	2.36765e7	0.904937
$928$	−1.93845e6	−0.0738899
$929$	−1.35921e7	−0.516710	−0.258355	−	0.966050i	$-0.583180\pi$
$929$	−1.35921e7	−0.516710	−0.258355	+	0.966050i	$0.583180\pi$
$930$	0	0
$931$	182330.	0.00689421
$932$	7.05637e6	0.266098
$933$	−3.14116e6	−0.118137
$934$	1.43297e7	0.537489
$935$	0	0
$936$	8.00263e6	0.298568
$937$	−3.01018e7	−1.12007	−0.560033	−	0.828470i	$-0.689212\pi$
$937$	−3.01018e7	−1.12007	−0.560033	+	0.828470i	$0.689212\pi$
$938$	−1.99122e7	−0.738944
$939$	−2.68279e6	−0.0992938
$940$	0	0
$941$	−1.06275e7	−0.391252	−0.195626	−	0.980679i	$-0.562674\pi$
$941$	−1.06275e7	−0.391252	−0.195626	+	0.980679i	$0.562674\pi$
$942$	−665503.	−0.0244356
$943$	3.36130e7	1.23091
$944$	2.01840e7	0.737187
$945$	0	0
$946$	−576099.	−0.0209300
$947$	2.41709e7	0.875826	0.437913	−	0.899017i	$-0.355718\pi$
$947$	2.41709e7	0.875826	0.437913	+	0.899017i	$0.355718\pi$
$948$	870924.	0.0314745
$949$	178535.	0.00643514
$950$	0	0
$951$	3.25282e6	0.116630
$952$	2.61580e7	0.935432
$953$	−4.27043e7	−1.52314	−0.761569	−	0.648084i	$-0.775570\pi$
$953$	−4.27043e7	−1.52314	−0.761569	+	0.648084i	$0.775570\pi$
$954$	−8.56063e6	−0.304533
$955$	0	0
$956$	1.85375e6	0.0656003
$957$	12211.0	0.000430993 0
$958$	−3.04842e7	−1.07315
$959$	−4.55184e6	−0.159823
$960$	0	0
$961$	−5.74655e6	−0.200724
$962$	3.57371e6	0.124503
$963$	−6.45276e6	−0.224223
$964$	−7.73348e6	−0.268029
$965$	0	0
$966$	−3.97535e6	−0.137067
$967$	4.42692e7	1.52242	0.761211	−	0.648504i	$-0.224605\pi$
$967$	4.42692e7	1.52242	0.761211	+	0.648504i	$0.224605\pi$
$968$	3.16875e7	1.08693
$969$	−366332.	−0.0125333
$970$	0	0
$971$	3.88962e7	1.32391	0.661956	−	0.749542i	$-0.269726\pi$
$971$	3.88962e7	1.32391	0.661956	+	0.749542i	$0.269726\pi$
$972$	−3.18606e6	−0.108166
$973$	−1.92816e7	−0.652922
$974$	1.85417e7	0.626257
$975$	0	0
$976$	9.82357e6	0.330099
$977$	2.71611e7	0.910354	0.455177	−	0.890401i	$-0.349576\pi$
$977$	2.71611e7	0.910354	0.455177	+	0.890401i	$0.349576\pi$
$978$	1.67862e6	0.0561184
$979$	−145319.	−0.00484580
$980$	0	0
$981$	1.51946e7	0.504099
$982$	−9.59904e6	−0.317650
$983$	1.98048e7	0.653714	0.326857	−	0.945074i	$-0.394011\pi$
$983$	1.98048e7	0.653714	0.326857	+	0.945074i	$0.394011\pi$
$984$	−2.55232e6	−0.0840326
$985$	0	0
$986$	−2.42113e6	−0.0793096
$987$	5.13301e6	0.167718
$988$	404255.	0.0131754
$989$	3.60343e7	1.17145
$990$	0	0
$991$	−1.44104e7	−0.466115	−0.233057	−	0.972463i	$-0.574873\pi$
$991$	−1.44104e7	−0.466115	−0.233057	+	0.972463i	$0.574873\pi$
$992$	1.83635e7	0.592483
$993$	−1.10222e6	−0.0354727
$994$	−2.37700e7	−0.763068
$995$	0	0
$996$	−1.36516e6	−0.0436048
$997$	−3.16635e7	−1.00884	−0.504418	−	0.863459i	$-0.668293\pi$
$997$	−3.16635e7	−1.00884	−0.504418	+	0.863459i	$0.668293\pi$
$998$	2.72195e7	0.865074
$999$	−3.65378e6	−0.115832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.b.1.2		2
5.2	odd	4		325.6.b.b.274.4		4
5.3	odd	4		325.6.b.b.274.1		4
5.4	even	2		13.6.a.a.1.1	✓	2
15.14	odd	2		117.6.a.c.1.2		2
20.19	odd	2		208.6.a.h.1.1		2
35.34	odd	2		637.6.a.a.1.1		2
40.19	odd	2		832.6.a.i.1.2		2
40.29	even	2		832.6.a.p.1.1		2
65.34	odd	4		169.6.b.a.168.1		4
65.44	odd	4		169.6.b.a.168.4		4
65.64	even	2		169.6.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	5.4	even	2
117.6.a.c.1.2		2	15.14	odd	2
169.6.a.a.1.2		2	65.64	even	2
169.6.b.a.168.1		4	65.34	odd	4
169.6.b.a.168.4		4	65.44	odd	4
208.6.a.h.1.1		2	20.19	odd	2
325.6.a.b.1.2		2	1.1	even	1	trivial
325.6.b.b.274.1		4	5.3	odd	4
325.6.b.b.274.4		4	5.2	odd	4
637.6.a.a.1.1		2	35.34	odd	2
832.6.a.i.1.2		2	40.19	odd	2
832.6.a.p.1.1		2	40.29	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q+4.56155 q^{2} +1.63068 q^{3} -11.1922 q^{4} +7.43845 q^{6} -126.309 q^{7} -197.024 q^{8} -240.341 q^{9} -14.8296 q^{11} -18.2510 q^{12} +169.000 q^{13} -576.164 q^{14} -540.582 q^{16} +1051.12 q^{17} -1096.33 q^{18} -213.723 q^{19} -205.969 q^{21} -67.6458 q^{22} +4231.16 q^{23} -321.283 q^{24} +770.902 q^{26} -788.176 q^{27} +1413.68 q^{28} -504.955 q^{29} +4783.58 q^{31} +3838.86 q^{32} -24.1823 q^{33} +4794.75 q^{34} +2689.95 q^{36} +4635.74 q^{37} -974.911 q^{38} +275.585 q^{39} +7944.15 q^{41} -939.541 q^{42} +8516.41 q^{43} +165.976 q^{44} +19300.7 q^{46} -24921.2 q^{47} -881.518 q^{48} -853.113 q^{49} +1714.05 q^{51} -1891.49 q^{52} +7808.46 q^{53} -3595.31 q^{54} +24885.8 q^{56} -348.515 q^{57} -2303.38 q^{58} -37337.5 q^{59} -18172.2 q^{61} +21820.5 q^{62} +30357.1 q^{63} +34809.8 q^{64} -110.309 q^{66} +34559.9 q^{67} -11764.4 q^{68} +6899.68 q^{69} +41255.7 q^{71} +47352.8 q^{72} +1056.42 q^{73} +21146.2 q^{74} +2392.04 q^{76} +1873.10 q^{77} +1257.10 q^{78} -47719.3 q^{79} +57117.6 q^{81} +36237.7 q^{82} +74799.0 q^{83} +2305.26 q^{84} +38848.0 q^{86} -823.421 q^{87} +2921.78 q^{88} +9799.26 q^{89} -21346.2 q^{91} -47356.1 q^{92} +7800.50 q^{93} -113679. q^{94} +6259.97 q^{96} +138432. q^{97} -3891.52 q^{98} +3564.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 28 q^{3} - 43 q^{4} + 19 q^{6} + 36 q^{7} - 225 q^{8} + 212 q^{9} - 376 q^{11} - 857 q^{12} + 338 q^{13} - 505 q^{14} + 465 q^{16} + 2630 q^{17} - 898 q^{18} - 312 q^{19} + 4074 q^{21} - 226 q^{22}+ \cdots - 159808 q^{99}+O(q^{100}) \) 2 * q + 5 * q^2 + 28 * q^3 - 43 * q^4 + 19 * q^6 + 36 * q^7 - 225 * q^8 + 212 * q^9 - 376 * q^11 - 857 * q^12 + 338 * q^13 - 505 * q^14 + 465 * q^16 + 2630 * q^17 - 898 * q^18 - 312 * q^19 + 4074 * q^21 - 226 * q^22 + 2624 * q^23 - 1059 * q^24 + 845 * q^26 + 4732 * q^27 - 3749 * q^28 - 812 * q^29 + 7720 * q^31 + 5175 * q^32 - 9548 * q^33 + 5487 * q^34 - 11698 * q^36 + 16858 * q^37 - 1018 * q^38 + 4732 * q^39 + 7840 * q^41 + 937 * q^42 - 2420 * q^43 + 11654 * q^44 + 18596 * q^46 - 9972 * q^47 + 25635 * q^48 + 8684 * q^49 + 43348 * q^51 - 7267 * q^52 + 43720 * q^53 - 1175 * q^54 + 20345 * q^56 - 2940 * q^57 - 2438 * q^58 - 38936 * q^59 + 1984 * q^61 + 23108 * q^62 + 103776 * q^63 + 3217 * q^64 - 4286 * q^66 + 69928 * q^67 - 61985 * q^68 - 35480 * q^69 + 67396 * q^71 + 34698 * q^72 - 74412 * q^73 + 26505 * q^74 + 5518 * q^76 - 56748 * q^77 + 3211 * q^78 - 55296 * q^79 + 92762 * q^81 + 36192 * q^82 + 75712 * q^83 - 133831 * q^84 + 34053 * q^86 - 8920 * q^87 + 13026 * q^88 + 116508 * q^89 + 6084 * q^91 + 3764 * q^92 + 85232 * q^93 - 107125 * q^94 + 41493 * q^96 + 34756 * q^97 + 290 * q^98 - 159808 * q^99

Introduction

L-functions

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