LMFDB - Embedded newform 325.6.a.i.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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.28147$ 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)

$f(q)$	$=$	\(q-7.28147 q^{2} +14.9935 q^{3} +21.0198 q^{4} -109.175 q^{6} -135.391 q^{7} +79.9521 q^{8} -18.1955 q^{9} +191.317 q^{11} +315.160 q^{12} -169.000 q^{13} +985.848 q^{14} -1254.80 q^{16} +874.338 q^{17} +132.490 q^{18} +1992.26 q^{19} -2029.99 q^{21} -1393.07 q^{22} +2091.74 q^{23} +1198.76 q^{24} +1230.57 q^{26} -3916.23 q^{27} -2845.90 q^{28} -46.3621 q^{29} -9450.42 q^{31} +6578.33 q^{32} +2868.52 q^{33} -6366.46 q^{34} -382.466 q^{36} +3374.82 q^{37} -14506.6 q^{38} -2533.90 q^{39} -10511.6 q^{41} +14781.3 q^{42} -4056.27 q^{43} +4021.45 q^{44} -15230.9 q^{46} +17035.2 q^{47} -18813.8 q^{48} +1523.82 q^{49} +13109.4 q^{51} -3552.34 q^{52} +23984.5 q^{53} +28515.9 q^{54} -10824.8 q^{56} +29870.9 q^{57} +337.584 q^{58} -10197.8 q^{59} -27440.0 q^{61} +68812.9 q^{62} +2463.52 q^{63} -7746.27 q^{64} -20887.0 q^{66} +317.851 q^{67} +18378.4 q^{68} +31362.5 q^{69} +39664.7 q^{71} -1454.77 q^{72} -4764.06 q^{73} -24573.7 q^{74} +41876.9 q^{76} -25902.7 q^{77} +18450.5 q^{78} -74545.9 q^{79} -54296.4 q^{81} +76539.5 q^{82} +14234.5 q^{83} -42669.9 q^{84} +29535.6 q^{86} -695.130 q^{87} +15296.2 q^{88} -132052. q^{89} +22881.1 q^{91} +43967.9 q^{92} -141695. q^{93} -124041. q^{94} +98632.1 q^{96} -19366.7 q^{97} -11095.6 q^{98} -3481.12 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * 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 $

$2$	−7.28147	−1.28719	−0.643597	−	0.765365i	$-0.722559\pi$
$2$	−7.28147	−1.28719	−0.643597	+	0.765365i	$0.722559\pi$
$3$	14.9935	0.961832	0.480916	−	0.876767i	$-0.340304\pi$
$3$	14.9935	0.961832	0.480916	+	0.876767i	$0.340304\pi$
$4$	21.0198	0.656868
$5$	0	0
$5$	0	0
$6$	−109.175	−1.23806
$7$	−135.391	−1.04435	−0.522175	−	0.852839i	$-0.674879\pi$
$7$	−135.391	−1.04435	−0.522175	+	0.852839i	$0.674879\pi$
$8$	79.9521	0.441677
$9$	−18.1955	−0.0748787
$10$	0	0
$11$	191.317	0.476731	0.238365	−	0.971176i	$-0.423388\pi$
$11$	191.317	0.476731	0.238365	+	0.971176i	$0.423388\pi$
$12$	315.160	0.631797
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	985.848	1.34428
$15$	0	0
$16$	−1254.80	−1.22539
$17$	874.338	0.733765	0.366883	−	0.930267i	$-0.380425\pi$
$17$	874.338	0.733765	0.366883	+	0.930267i	$0.380425\pi$
$18$	132.490	0.0963834
$19$	1992.26	1.26608	0.633042	−	0.774118i	$-0.281806\pi$
$19$	1992.26	1.26608	0.633042	+	0.774118i	$0.281806\pi$
$20$	0	0
$21$	−2029.99	−1.00449
$22$	−1393.07	−0.613645
$23$	2091.74	0.824495	0.412248	−	0.911072i	$-0.364744\pi$
$23$	2091.74	0.824495	0.412248	+	0.911072i	$0.364744\pi$
$24$	1198.76	0.424819
$25$	0	0
$26$	1230.57	0.357003
$27$	−3916.23	−1.03385
$28$	−2845.90	−0.686000
$29$	−46.3621	−0.0102369	−0.00511845	−	0.999987i	$-0.501629\pi$
$29$	−46.3621	−0.0102369	−0.00511845	+	0.999987i	$0.501629\pi$
$30$	0	0
$31$	−9450.42	−1.76623	−0.883114	−	0.469158i	$-0.844558\pi$
$31$	−9450.42	−1.76623	−0.883114	+	0.469158i	$0.844558\pi$
$32$	6578.33	1.13564
$33$	2868.52	0.458535
$34$	−6366.46	−0.944498
$35$	0	0
$36$	−382.466	−0.0491854
$37$	3374.82	0.405272	0.202636	−	0.979254i	$-0.435049\pi$
$37$	3374.82	0.405272	0.202636	+	0.979254i	$0.435049\pi$
$38$	−14506.6	−1.62969
$39$	−2533.90	−0.266764
$40$	0	0
$41$	−10511.6	−0.976578	−0.488289	−	0.872682i	$-0.662379\pi$
$41$	−10511.6	−0.976578	−0.488289	+	0.872682i	$0.662379\pi$
$42$	14781.3	1.29297
$43$	−4056.27	−0.334546	−0.167273	−	0.985911i	$-0.553496\pi$
$43$	−4056.27	−0.334546	−0.167273	+	0.985911i	$0.553496\pi$
$44$	4021.45	0.313149
$45$	0	0
$46$	−15230.9	−1.06129
$47$	17035.2	1.12487	0.562435	−	0.826842i	$-0.309865\pi$
$47$	17035.2	1.12487	0.562435	+	0.826842i	$0.309865\pi$
$48$	−18813.8	−1.17862
$49$	1523.82	0.0906658
$50$	0	0
$51$	13109.4	0.705759
$52$	−3552.34	−0.182182
$53$	23984.5	1.17285	0.586424	−	0.810004i	$-0.300535\pi$
$53$	23984.5	1.17285	0.586424	+	0.810004i	$0.300535\pi$
$54$	28515.9	1.33077
$55$	0	0
$56$	−10824.8	−0.461265
$57$	29870.9	1.21776
$58$	337.584	0.0131769
$59$	−10197.8	−0.381398	−0.190699	−	0.981649i	$-0.561075\pi$
$59$	−10197.8	−0.381398	−0.190699	+	0.981649i	$0.561075\pi$
$60$	0	0
$61$	−27440.0	−0.944191	−0.472095	−	0.881547i	$-0.656502\pi$
$61$	−27440.0	−0.944191	−0.472095	+	0.881547i	$0.656502\pi$
$62$	68812.9	2.27348
$63$	2463.52	0.0781995
$64$	−7746.27	−0.236397
$65$	0	0
$66$	−20887.0	−0.590223
$67$	317.851	0.00865040	0.00432520	−	0.999991i	$-0.498623\pi$
$67$	317.851	0.00865040	0.00432520	+	0.999991i	$0.498623\pi$
$68$	18378.4	0.481987
$69$	31362.5	0.793026
$70$	0	0
$71$	39664.7	0.933809	0.466905	−	0.884308i	$-0.345369\pi$
$71$	39664.7	0.933809	0.466905	+	0.884308i	$0.345369\pi$
$72$	−1454.77	−0.0330722
$73$	−4764.06	−0.104633	−0.0523167	−	0.998631i	$-0.516661\pi$
$73$	−4764.06	−0.104633	−0.0523167	+	0.998631i	$0.516661\pi$
$74$	−24573.7	−0.521664
$75$	0	0
$76$	41876.9	0.831650
$77$	−25902.7	−0.497873
$78$	18450.5	0.343377
$79$	−74545.9	−1.34387	−0.671933	−	0.740612i	$-0.734536\pi$
$79$	−74545.9	−1.34387	−0.671933	+	0.740612i	$0.734536\pi$
$80$	0	0
$81$	−54296.4	−0.919514
$82$	76539.5	1.25705
$83$	14234.5	0.226803	0.113401	−	0.993549i	$-0.463825\pi$
$83$	14234.5	0.226803	0.113401	+	0.993549i	$0.463825\pi$
$84$	−42669.9	−0.659817
$85$	0	0
$86$	29535.6	0.430625
$87$	−695.130	−0.00984618
$88$	15296.2	0.210561
$89$	−132052.	−1.76713	−0.883566	−	0.468307i	$-0.844864\pi$
$89$	−132052.	−1.76713	−0.883566	+	0.468307i	$0.844864\pi$
$90$	0	0
$91$	22881.1	0.289650
$92$	43967.9	0.541585
$93$	−141695.	−1.69882
$94$	−124041.	−1.44793
$95$	0	0
$96$	98632.1	1.09230
$97$	−19366.7	−0.208990	−0.104495	−	0.994525i	$-0.533323\pi$
$97$	−19366.7	−0.208990	−0.104495	+	0.994525i	$0.533323\pi$
$98$	−11095.6	−0.116704
$99$	−3481.12	−0.0356970
$100$	0	0
$101$	−90391.7	−0.881709	−0.440855	−	0.897579i	$-0.645325\pi$
$101$	−90391.7	−0.881709	−0.440855	+	0.897579i	$0.645325\pi$
$102$	−95455.5	−0.908449
$103$	−59762.2	−0.555051	−0.277526	−	0.960718i	$-0.589514\pi$
$103$	−59762.2	−0.555051	−0.277526	+	0.960718i	$0.589514\pi$
$104$	−13511.9	−0.122499
$105$	0	0
$106$	−174643.	−1.50968
$107$	−178407.	−1.50645	−0.753223	−	0.657766i	$-0.771502\pi$
$107$	−178407.	−1.50645	−0.753223	+	0.657766i	$0.771502\pi$
$108$	−82318.3	−0.679105
$109$	−37689.5	−0.303847	−0.151923	−	0.988392i	$-0.548547\pi$
$109$	−37689.5	−0.303847	−0.151923	+	0.988392i	$0.548547\pi$
$110$	0	0
$111$	50600.3	0.389804
$112$	169889.	1.27974
$113$	38616.5	0.284497	0.142248	−	0.989831i	$-0.454567\pi$
$113$	38616.5	0.284497	0.142248	+	0.989831i	$0.454567\pi$
$114$	−217504.	−1.56749
$115$	0	0
$116$	−974.522	−0.00672429
$117$	3075.04	0.0207676
$118$	74255.3	0.490933
$119$	−118378.	−0.766307
$120$	0	0
$121$	−124449.	−0.772728
$122$	199804.	1.21536
$123$	−157605.	−0.939305
$124$	−198646.	−1.16018
$125$	0	0
$126$	−17938.0	−0.100658
$127$	−301119.	−1.65664	−0.828320	−	0.560255i	$-0.810703\pi$
$127$	−301119.	−1.65664	−0.828320	+	0.560255i	$0.810703\pi$
$128$	−154102.	−0.831351
$129$	−60817.6	−0.321777
$130$	0	0
$131$	−100245.	−0.510371	−0.255186	−	0.966892i	$-0.582137\pi$
$131$	−100245.	−0.510371	−0.255186	+	0.966892i	$0.582137\pi$
$132$	60295.6	0.301197
$133$	−269735.	−1.32223
$134$	−2314.42	−0.0111347
$135$	0	0
$136$	69905.2	0.324087
$137$	−72376.9	−0.329457	−0.164728	−	0.986339i	$-0.552675\pi$
$137$	−72376.9	−0.329457	−0.164728	+	0.986339i	$0.552675\pi$
$138$	−228365.	−1.02078
$139$	296858.	1.30320	0.651601	−	0.758562i	$-0.274098\pi$
$139$	296858.	1.30320	0.651601	+	0.758562i	$0.274098\pi$
$140$	0	0
$141$	255417.	1.08194
$142$	−288817.	−1.20199
$143$	−32332.7	−0.132221
$144$	22831.8	0.0917558
$145$	0	0
$146$	34689.3	0.134683
$147$	22847.4	0.0872053
$148$	70938.0	0.266210
$149$	16259.4	0.0599981	0.0299991	−	0.999550i	$-0.490450\pi$
$149$	16259.4	0.0599981	0.0299991	+	0.999550i	$0.490450\pi$
$150$	0	0
$151$	383495.	1.36873	0.684364	−	0.729140i	$-0.260080\pi$
$151$	383495.	1.36873	0.684364	+	0.729140i	$0.260080\pi$
$152$	159285.	0.559200
$153$	−15909.0	−0.0549434
$154$	188610.	0.640859
$155$	0	0
$156$	−53262.0	−0.175229
$157$	−235502.	−0.762510	−0.381255	−	0.924470i	$-0.624508\pi$
$157$	−235502.	−0.762510	−0.381255	+	0.924470i	$0.624508\pi$
$158$	542804.	1.72982
$159$	359612.	1.12808
$160$	0	0
$161$	−283204.	−0.861061
$162$	395358.	1.18359
$163$	−205165.	−0.604832	−0.302416	−	0.953176i	$-0.597793\pi$
$163$	−205165.	−0.604832	−0.302416	+	0.953176i	$0.597793\pi$
$164$	−220951.	−0.641483
$165$	0	0
$166$	−103648.	−0.291939
$167$	−68965.6	−0.191356	−0.0956778	−	0.995412i	$-0.530502\pi$
$167$	−68965.6	−0.191356	−0.0956778	+	0.995412i	$0.530502\pi$
$168$	−162302.	−0.443660
$169$	28561.0	0.0769231
$170$	0	0
$171$	−36250.2	−0.0948027
$172$	−85261.9	−0.219753
$173$	−613056.	−1.55735	−0.778673	−	0.627430i	$-0.784107\pi$
$173$	−613056.	−1.55735	−0.778673	+	0.627430i	$0.784107\pi$
$174$	5061.56	0.0126739
$175$	0	0
$176$	−240066.	−0.584182
$177$	−152901.	−0.366841
$178$	961530.	2.27464
$179$	−637734.	−1.48767	−0.743836	−	0.668362i	$-0.766996\pi$
$179$	−637734.	−1.48767	−0.743836	+	0.668362i	$0.766996\pi$
$180$	0	0
$181$	145338.	0.329749	0.164875	−	0.986315i	$-0.447278\pi$
$181$	145338.	0.329749	0.164875	+	0.986315i	$0.447278\pi$
$182$	−166608.	−0.372836
$183$	−411421.	−0.908153
$184$	167239.	0.364161
$185$	0	0
$186$	1.03175e6	2.18671
$187$	167276.	0.349808
$188$	358076.	0.738891
$189$	530224.	1.07970
$190$	0	0
$191$	−967379.	−1.91873	−0.959364	−	0.282173i	$-0.908945\pi$
$191$	−967379.	−1.91873	−0.959364	+	0.282173i	$0.908945\pi$
$192$	−116143.	−0.227375
$193$	9104.15	0.0175932	0.00879662	−	0.999961i	$-0.497200\pi$
$193$	9104.15	0.0175932	0.00879662	+	0.999961i	$0.497200\pi$
$194$	141018.	0.269011
$195$	0	0
$196$	32030.4	0.0595555
$197$	567218.	1.04132	0.520660	−	0.853764i	$-0.325686\pi$
$197$	567218.	1.04132	0.520660	+	0.853764i	$0.325686\pi$
$198$	25347.7	0.0459489
$199$	−321297.	−0.575141	−0.287571	−	0.957759i	$-0.592848\pi$
$199$	−321297.	−0.575141	−0.287571	+	0.957759i	$0.592848\pi$
$200$	0	0
$201$	4765.69	0.00832023
$202$	658185.	1.13493
$203$	6277.03	0.0106909
$204$	275556.	0.463591
$205$	0	0
$206$	435156.	0.714459
$207$	−38060.3	−0.0617371
$208$	212061.	0.339863
$209$	381154.	0.603580
$210$	0	0
$211$	−165479.	−0.255880	−0.127940	−	0.991782i	$-0.540836\pi$
$211$	−165479.	−0.255880	−0.127940	+	0.991782i	$0.540836\pi$
$212$	504150.	0.770407
$213$	594712.	0.898168
$214$	1.29907e6	1.93909
$215$	0	0
$216$	−313111.	−0.456629
$217$	1.27951e6	1.84456
$218$	274435.	0.391109
$219$	−71429.8	−0.100640
$220$	0	0
$221$	−147763.	−0.203510
$222$	−368445.	−0.501753
$223$	1.14125e6	1.53680	0.768402	−	0.639968i	$-0.221052\pi$
$223$	1.14125e6	1.53680	0.768402	+	0.639968i	$0.221052\pi$
$224$	−890649.	−1.18601
$225$	0	0
$226$	−281185.	−0.366202
$227$	−573565.	−0.738785	−0.369393	−	0.929273i	$-0.620434\pi$
$227$	−573565.	−0.738785	−0.369393	+	0.929273i	$0.620434\pi$
$228$	627880.	0.799908
$229$	1.11647e6	1.40688	0.703440	−	0.710755i	$-0.251647\pi$
$229$	1.11647e6	1.40688	0.703440	+	0.710755i	$0.251647\pi$
$230$	0	0
$231$	−388372.	−0.478871
$232$	−3706.75	−0.00452140
$233$	1.08552e6	1.30993	0.654967	−	0.755658i	$-0.272683\pi$
$233$	1.08552e6	1.30993	0.654967	+	0.755658i	$0.272683\pi$
$234$	−22390.8	−0.0267319
$235$	0	0
$236$	−214356.	−0.250528
$237$	−1.11770e6	−1.29257
$238$	861964.	0.986386
$239$	497543.	0.563425	0.281712	−	0.959499i	$-0.409098\pi$
$239$	497543.	0.563425	0.281712	+	0.959499i	$0.409098\pi$
$240$	0	0
$241$	−1.68881e6	−1.87300	−0.936499	−	0.350669i	$-0.885954\pi$
$241$	−1.68881e6	−1.87300	−0.936499	+	0.350669i	$0.885954\pi$
$242$	906169.	0.994651
$243$	137552.	0.149434
$244$	−576783.	−0.620209
$245$	0	0
$246$	1.14759e6	1.20907
$247$	−336692.	−0.351148
$248$	−755581.	−0.780103
$249$	213425.	0.218146
$250$	0	0
$251$	−358692.	−0.359366	−0.179683	−	0.983725i	$-0.557507\pi$
$251$	−358692.	−0.359366	−0.179683	+	0.983725i	$0.557507\pi$
$252$	51782.6	0.0513668
$253$	400187.	0.393062
$254$	2.19259e6	2.13242
$255$	0	0
$256$	1.36997e6	1.30651
$257$	−715783.	−0.676003	−0.338001	−	0.941146i	$-0.609751\pi$
$257$	−715783.	−0.676003	−0.338001	+	0.941146i	$0.609751\pi$
$258$	442841.	0.414189
$259$	−456922.	−0.423246
$260$	0	0
$261$	843.583	0.000766525 0
$262$	729933.	0.656946
$263$	−167988.	−0.149757	−0.0748787	−	0.997193i	$-0.523857\pi$
$263$	−167988.	−0.149757	−0.0748787	+	0.997193i	$0.523857\pi$
$264$	229344.	0.202524
$265$	0	0
$266$	1.96407e6	1.70197
$267$	−1.97991e6	−1.69968
$268$	6681.15	0.00568217
$269$	1.93165e6	1.62760	0.813800	−	0.581145i	$-0.197395\pi$
$269$	1.93165e6	1.62760	0.813800	+	0.581145i	$0.197395\pi$
$270$	0	0
$271$	2.04198e6	1.68900	0.844499	−	0.535558i	$-0.179899\pi$
$271$	2.04198e6	1.68900	0.844499	+	0.535558i	$0.179899\pi$
$272$	−1.09712e6	−0.899150
$273$	343068.	0.278595
$274$	527010.	0.424075
$275$	0	0
$276$	659232.	0.520914
$277$	−1.05919e6	−0.829416	−0.414708	−	0.909955i	$-0.636116\pi$
$277$	−1.05919e6	−0.829416	−0.414708	+	0.909955i	$0.636116\pi$
$278$	−2.16156e6	−1.67747
$279$	171955.	0.132253
$280$	0	0
$281$	−935578.	−0.706829	−0.353414	−	0.935467i	$-0.614979\pi$
$281$	−935578.	−0.706829	−0.353414	+	0.935467i	$0.614979\pi$
$282$	−1.85981e6	−1.39266
$283$	827252.	0.614005	0.307002	−	0.951709i	$-0.400674\pi$
$283$	827252.	0.614005	0.307002	+	0.951709i	$0.400674\pi$
$284$	833743.	0.613390
$285$	0	0
$286$	235429.	0.170194
$287$	1.42317e6	1.01989
$288$	−119696.	−0.0850353
$289$	−655390.	−0.461589
$290$	0	0
$291$	−290374.	−0.201013
$292$	−100139.	−0.0687303
$293$	−1.77525e6	−1.20806	−0.604032	−	0.796960i	$-0.706440\pi$
$293$	−1.77525e6	−1.20806	−0.604032	+	0.796960i	$0.706440\pi$
$294$	−166362.	−0.112250
$295$	0	0
$296$	269824.	0.178999
$297$	−749243.	−0.492869
$298$	−118392.	−0.0772292
$299$	−353504.	−0.228674
$300$	0	0
$301$	549184.	0.349383
$302$	−2.79241e6	−1.76182
$303$	−1.35529e6	−0.848056
$304$	−2.49989e6	−1.55145
$305$	0	0
$306$	115841.	0.0707228
$307$	293202.	0.177550	0.0887750	−	0.996052i	$-0.471705\pi$
$307$	293202.	0.177550	0.0887750	+	0.996052i	$0.471705\pi$
$308$	−544470.	−0.327037
$309$	−896043.	−0.533866
$310$	0	0
$311$	−2.34227e6	−1.37321	−0.686605	−	0.727031i	$-0.740900\pi$
$311$	−2.34227e6	−1.37321	−0.686605	+	0.727031i	$0.740900\pi$
$312$	−202590.	−0.117824
$313$	−1.20634e6	−0.695997	−0.347999	−	0.937495i	$-0.613139\pi$
$313$	−1.20634e6	−0.695997	−0.347999	+	0.937495i	$0.613139\pi$
$314$	1.71480e6	0.981499
$315$	0	0
$316$	−1.56694e6	−0.882743
$317$	2.21969e6	1.24063	0.620316	−	0.784352i	$-0.287004\pi$
$317$	2.21969e6	1.24063	0.620316	+	0.784352i	$0.287004\pi$
$318$	−2.61850e6	−1.45206
$319$	−8869.88	−0.00488024
$320$	0	0
$321$	−2.67495e6	−1.44895
$322$	2.06214e6	1.10835
$323$	1.74191e6	0.929008
$324$	−1.14130e6	−0.604000
$325$	0	0
$326$	1.49390e6	0.778536
$327$	−565097.	−0.292249
$328$	−840420.	−0.431332
$329$	−2.30642e6	−1.17476
$330$	0	0
$331$	1.37355e6	0.689087	0.344544	−	0.938770i	$-0.388034\pi$
$331$	1.37355e6	0.689087	0.344544	+	0.938770i	$0.388034\pi$
$332$	299207.	0.148980
$333$	−61406.7	−0.0303462
$334$	502171.	0.246312
$335$	0	0
$336$	2.54723e6	1.23089
$337$	−428681.	−0.205617	−0.102809	−	0.994701i	$-0.532783\pi$
$337$	−428681.	−0.205617	−0.102809	+	0.994701i	$0.532783\pi$
$338$	−207966.	−0.0990149
$339$	578996.	0.273638
$340$	0	0
$341$	−1.80803e6	−0.842015
$342$	263955.	0.122029
$343$	2.06921e6	0.949663
$344$	−324307.	−0.147761
$345$	0	0
$346$	4.46395e6	2.00461
$347$	−3.34049e6	−1.48932	−0.744658	−	0.667446i	$-0.767388\pi$
$347$	−3.34049e6	−1.48932	−0.744658	+	0.667446i	$0.767388\pi$
$348$	−14611.5	−0.00646764
$349$	−1.65843e6	−0.728840	−0.364420	−	0.931235i	$-0.618733\pi$
$349$	−1.65843e6	−0.728840	−0.364420	+	0.931235i	$0.618733\pi$
$350$	0	0
$351$	661843.	0.286739
$352$	1.25855e6	0.541394
$353$	−2.62106e6	−1.11954	−0.559771	−	0.828648i	$-0.689111\pi$
$353$	−2.62106e6	−1.11954	−0.559771	+	0.828648i	$0.689111\pi$
$354$	1.11335e6	0.472195
$355$	0	0
$356$	−2.77570e6	−1.16077
$357$	−1.77490e6	−0.737059
$358$	4.64364e6	1.91492
$359$	−818855.	−0.335329	−0.167664	−	0.985844i	$-0.553623\pi$
$359$	−818855.	−0.335329	−0.167664	+	0.985844i	$0.553623\pi$
$360$	0	0
$361$	1.49300e6	0.602966
$362$	−1.05828e6	−0.424451
$363$	−1.86592e6	−0.743235
$364$	480957.	0.190262
$365$	0	0
$366$	2.99575e6	1.16897
$367$	2.81469e6	1.09085	0.545425	−	0.838160i	$-0.316368\pi$
$367$	2.81469e6	1.09085	0.545425	+	0.838160i	$0.316368\pi$
$368$	−2.62472e6	−1.01033
$369$	191263.	0.0731249
$370$	0	0
$371$	−3.24730e6	−1.22486
$372$	−2.97839e6	−1.11590
$373$	4.64910e6	1.73020	0.865101	−	0.501598i	$-0.167254\pi$
$373$	4.64910e6	1.73020	0.865101	+	0.501598i	$0.167254\pi$
$374$	−1.21802e6	−0.450271
$375$	0	0
$376$	1.36200e6	0.496829
$377$	7835.20	0.00283920
$378$	−3.86081e6	−1.38979
$379$	−2.33478e6	−0.834924	−0.417462	−	0.908694i	$-0.637080\pi$
$379$	−2.33478e6	−0.834924	−0.417462	+	0.908694i	$0.637080\pi$
$380$	0	0
$381$	−4.51482e6	−1.59341
$382$	7.04394e6	2.46977
$383$	3.05707e6	1.06490	0.532449	−	0.846462i	$-0.321272\pi$
$383$	3.05707e6	1.06490	0.532449	+	0.846462i	$0.321272\pi$
$384$	−2.31053e6	−0.799621
$385$	0	0
$386$	−66291.6	−0.0226459
$387$	73806.0	0.0250504
$388$	−407083.	−0.137279
$389$	1.25272e6	0.419741	0.209871	−	0.977729i	$-0.432696\pi$
$389$	1.25272e6	0.419741	0.209871	+	0.977729i	$0.432696\pi$
$390$	0	0
$391$	1.82889e6	0.604986
$392$	121833.	0.0400450
$393$	−1.50303e6	−0.490891
$394$	−4.13018e6	−1.34038
$395$	0	0
$396$	−73172.4	−0.0234482
$397$	−5.44444e6	−1.73371	−0.866856	−	0.498559i	$-0.833863\pi$
$397$	−5.44444e6	−1.73371	−0.866856	+	0.498559i	$0.833863\pi$
$398$	2.33952e6	0.740318
$399$	−4.04427e6	−1.27177
$400$	0	0
$401$	−3.75387e6	−1.16578	−0.582892	−	0.812549i	$-0.698079\pi$
$401$	−3.75387e6	−1.16578	−0.582892	+	0.812549i	$0.698079\pi$
$402$	−34701.2	−0.0107098
$403$	1.59712e6	0.489864
$404$	−1.90001e6	−0.579167
$405$	0	0
$406$	−45706.0	−0.0137613
$407$	645663.	0.193206
$408$	1.04812e6	0.311718
$409$	4.09010e6	1.20900	0.604500	−	0.796605i	$-0.293373\pi$
$409$	4.09010e6	1.20900	0.604500	+	0.796605i	$0.293373\pi$
$410$	0	0
$411$	−1.08518e6	−0.316882
$412$	−1.25619e6	−0.364596
$413$	1.38070e6	0.398313
$414$	277135.	0.0794677
$415$	0	0
$416$	−1.11174e6	−0.314970
$417$	4.45094e6	1.25346
$418$	−2.77536e6	−0.776925
$419$	3.67162e6	1.02170	0.510849	−	0.859671i	$-0.329331\pi$
$419$	3.67162e6	1.02170	0.510849	+	0.859671i	$0.329331\pi$
$420$	0	0
$421$	1.59907e6	0.439705	0.219852	−	0.975533i	$-0.429442\pi$
$421$	1.59907e6	0.439705	0.219852	+	0.975533i	$0.429442\pi$
$422$	1.20493e6	0.329367
$423$	−309964.	−0.0842288
$424$	1.91761e6	0.518020
$425$	0	0
$426$	−4.33037e6	−1.15612
$427$	3.71514e6	0.986065
$428$	−3.75009e6	−0.989536
$429$	−484779.	−0.127175
$430$	0	0
$431$	6.53931e6	1.69566	0.847830	−	0.530268i	$-0.177909\pi$
$431$	6.53931e6	1.69566	0.847830	+	0.530268i	$0.177909\pi$
$432$	4.91409e6	1.26688
$433$	1.60206e6	0.410639	0.205319	−	0.978695i	$-0.434177\pi$
$433$	1.60206e6	0.410639	0.205319	+	0.978695i	$0.434177\pi$
$434$	−9.31668e6	−2.37431
$435$	0	0
$436$	−792226.	−0.199587
$437$	4.16729e6	1.04388
$438$	520114.	0.129543
$439$	−4.84652e6	−1.20024	−0.600121	−	0.799909i	$-0.704881\pi$
$439$	−4.84652e6	−1.20024	−0.600121	+	0.799909i	$0.704881\pi$
$440$	0	0
$441$	−27726.7	−0.00678894
$442$	1.07593e6	0.261957
$443$	580247.	0.140476	0.0702382	−	0.997530i	$-0.477624\pi$
$443$	580247.	0.140476	0.0702382	+	0.997530i	$0.477624\pi$
$444$	1.06361e6	0.256050
$445$	0	0
$446$	−8.30997e6	−1.97816
$447$	243784.	0.0577081
$448$	1.04878e6	0.246881
$449$	2.04347e6	0.478356	0.239178	−	0.970976i	$-0.423122\pi$
$449$	2.04347e6	0.478356	0.239178	+	0.970976i	$0.423122\pi$
$450$	0	0
$451$	−2.01104e6	−0.465565
$452$	811711.	0.186877
$453$	5.74993e6	1.31649
$454$	4.17640e6	0.950960
$455$	0	0
$456$	2.38824e6	0.537857
$457$	1.52332e6	0.341194	0.170597	−	0.985341i	$-0.445430\pi$
$457$	1.52332e6	0.341194	0.170597	+	0.985341i	$0.445430\pi$
$458$	−8.12951e6	−1.81093
$459$	−3.42411e6	−0.758605
$460$	0	0
$461$	7.87768e6	1.72642	0.863209	−	0.504847i	$-0.168451\pi$
$461$	7.87768e6	1.72642	0.863209	+	0.504847i	$0.168451\pi$
$462$	2.82792e6	0.616399
$463$	136322.	0.0295539	0.0147770	−	0.999891i	$-0.495296\pi$
$463$	136322.	0.0295539	0.0147770	+	0.999891i	$0.495296\pi$
$464$	58175.3	0.0125442
$465$	0	0
$466$	−7.90420e6	−1.68614
$467$	5.56784e6	1.18139	0.590697	−	0.806894i	$-0.298853\pi$
$467$	5.56784e6	1.18139	0.590697	+	0.806894i	$0.298853\pi$
$468$	64636.8	0.0136416
$469$	−43034.2	−0.00903404
$470$	0	0
$471$	−3.53100e6	−0.733407
$472$	−815339.	−0.168455
$473$	−776035.	−0.159488
$474$	8.13852e6	1.66379
$475$	0	0
$476$	−2.48828e6	−0.503363
$477$	−436411.	−0.0878213
$478$	−3.62284e6	−0.725237
$479$	3.92083e6	0.780799	0.390399	−	0.920646i	$-0.372337\pi$
$479$	3.92083e6	0.780799	0.390399	+	0.920646i	$0.372337\pi$
$480$	0	0
$481$	−570345.	−0.112402
$482$	1.22970e7	2.41091
$483$	−4.24621e6	−0.828197
$484$	−2.61588e6	−0.507581
$485$	0	0
$486$	−1.00158e6	−0.192351
$487$	8.77882e6	1.67731	0.838656	−	0.544661i	$-0.183342\pi$
$487$	8.77882e6	1.67731	0.838656	+	0.544661i	$0.183342\pi$
$488$	−2.19389e6	−0.417027
$489$	−3.07614e6	−0.581747
$490$	0	0
$491$	−9.13377e6	−1.70980	−0.854902	−	0.518789i	$-0.826383\pi$
$491$	−9.13377e6	−1.70980	−0.854902	+	0.518789i	$0.826383\pi$
$492$	−3.31282e6	−0.616999
$493$	−40536.2	−0.00751148
$494$	2.45161e6	0.451996
$495$	0	0
$496$	1.18584e7	2.16432
$497$	−5.37026e6	−0.975223
$498$	−1.55405e6	−0.280796
$499$	−9.41419e6	−1.69251	−0.846256	−	0.532777i	$-0.821148\pi$
$499$	−9.41419e6	−1.69251	−0.846256	+	0.532777i	$0.821148\pi$
$500$	0	0
$501$	−1.03403e6	−0.184052
$502$	2.61180e6	0.462574
$503$	5.68086e6	1.00114	0.500569	−	0.865697i	$-0.333124\pi$
$503$	5.68086e6	1.00114	0.500569	+	0.865697i	$0.333124\pi$
$504$	196963.	0.0345389
$505$	0	0
$506$	−2.91395e6	−0.505947
$507$	428229.	0.0739871
$508$	−6.32945e6	−1.08819
$509$	2.85772e6	0.488906	0.244453	−	0.969661i	$-0.421392\pi$
$509$	2.85772e6	0.488906	0.244453	+	0.969661i	$0.421392\pi$
$510$	0	0
$511$	645012.	0.109274
$512$	−5.04413e6	−0.850377
$513$	−7.80215e6	−1.30894
$514$	5.21195e6	0.870147
$515$	0	0
$516$	−1.27837e6	−0.211365
$517$	3.25913e6	0.536260
$518$	3.32706e6	0.544799
$519$	−9.19185e6	−1.49791
$520$	0	0
$521$	2.64216e6	0.426446	0.213223	−	0.977004i	$-0.431604\pi$
$521$	2.64216e6	0.426446	0.213223	+	0.977004i	$0.431604\pi$
$522$	−6142.52	−0.000986667 0
$523$	1.02124e7	1.63258	0.816288	−	0.577645i	$-0.196028\pi$
$523$	1.02124e7	1.63258	0.816288	+	0.577645i	$0.196028\pi$
$524$	−2.10714e6	−0.335247
$525$	0	0
$526$	1.22320e6	0.192767
$527$	−8.26286e6	−1.29600
$528$	−3.59942e6	−0.561885
$529$	−2.06096e6	−0.320207
$530$	0	0
$531$	185555.	0.0285586
$532$	−5.66977e6	−0.868533
$533$	1.77645e6	0.270854
$534$	1.44167e7	2.18782
$535$	0	0
$536$	25412.8	0.00382068
$537$	−9.56186e6	−1.43089
$538$	−1.40652e7	−2.09504
$539$	291533.	0.0432232
$540$	0	0
$541$	−2.02275e6	−0.297132	−0.148566	−	0.988902i	$-0.547466\pi$
$541$	−2.02275e6	−0.297132	−0.148566	+	0.988902i	$0.547466\pi$
$542$	−1.48686e7	−2.17407
$543$	2.17913e6	0.317163
$544$	5.75169e6	0.833293
$545$	0	0
$546$	−2.49804e6	−0.358606
$547$	2.12905e6	0.304241	0.152120	−	0.988362i	$-0.451390\pi$
$547$	2.12905e6	0.304241	0.152120	+	0.988362i	$0.451390\pi$
$548$	−1.52135e6	−0.216410
$549$	499285.	0.0706998
$550$	0	0
$551$	−92365.4	−0.0129608
$552$	2.50750e6	0.350262
$553$	1.00929e7	1.40347
$554$	7.71242e6	1.06762
$555$	0	0
$556$	6.23989e6	0.856032
$557$	−8.58830e6	−1.17292	−0.586461	−	0.809977i	$-0.699479\pi$
$557$	−8.58830e6	−1.17292	−0.586461	+	0.809977i	$0.699479\pi$
$558$	−1.25209e6	−0.170235
$559$	685510.	0.0927863
$560$	0	0
$561$	2.50805e6	0.336457
$562$	6.81238e6	0.909826
$563$	1.02986e7	1.36933	0.684664	−	0.728859i	$-0.259949\pi$
$563$	1.02986e7	1.36933	0.684664	+	0.728859i	$0.259949\pi$
$564$	5.36880e6	0.710689
$565$	0	0
$566$	−6.02361e6	−0.790343
$567$	7.35126e6	0.960294
$568$	3.17127e6	0.412442
$569$	4.16791e6	0.539681	0.269841	−	0.962905i	$-0.413029\pi$
$569$	4.16791e6	0.539681	0.269841	+	0.962905i	$0.413029\pi$
$570$	0	0
$571$	7.69686e6	0.987923	0.493962	−	0.869484i	$-0.335548\pi$
$571$	7.69686e6	0.987923	0.493962	+	0.869484i	$0.335548\pi$
$572$	−679625.	−0.0868519
$573$	−1.45044e7	−1.84549
$574$	−1.03628e7	−1.31280
$575$	0	0
$576$	140947.	0.0177011
$577$	−1.42934e7	−1.78729	−0.893645	−	0.448775i	$-0.851860\pi$
$577$	−1.42934e7	−1.78729	−0.893645	+	0.448775i	$0.851860\pi$
$578$	4.77220e6	0.594154
$579$	136503.	0.0169217
$580$	0	0
$581$	−1.92723e6	−0.236861
$582$	2.11435e6	0.258743
$583$	4.58866e6	0.559133
$584$	−380897.	−0.0462141
$585$	0	0
$586$	1.29264e7	1.55501
$587$	−1.52860e7	−1.83104	−0.915522	−	0.402268i	$-0.868222\pi$
$587$	−1.52860e7	−1.83104	−0.915522	+	0.402268i	$0.868222\pi$
$588$	480247.	0.0572824
$589$	−1.88277e7	−2.23619
$590$	0	0
$591$	8.50457e6	1.00157
$592$	−4.23473e6	−0.496617
$593$	−8.42604e6	−0.983981	−0.491991	−	0.870601i	$-0.663731\pi$
$593$	−8.42604e6	−0.983981	−0.491991	+	0.870601i	$0.663731\pi$
$594$	5.45559e6	0.634418
$595$	0	0
$596$	341768.	0.0394109
$597$	−4.81737e6	−0.553189
$598$	2.57403e6	0.294348
$599$	−1.40447e6	−0.159936	−0.0799680	−	0.996797i	$-0.525482\pi$
$599$	−1.40447e6	−0.159936	−0.0799680	+	0.996797i	$0.525482\pi$
$600$	0	0
$601$	3.45723e6	0.390429	0.195214	−	0.980761i	$-0.437460\pi$
$601$	3.45723e6	0.390429	0.195214	+	0.980761i	$0.437460\pi$
$602$	−3.99887e6	−0.449723
$603$	−5783.46	−0.000647731 0
$604$	8.06098e6	0.899075
$605$	0	0
$606$	9.86848e6	1.09161
$607$	−1.56253e7	−1.72130	−0.860652	−	0.509194i	$-0.829944\pi$
$607$	−1.56253e7	−1.72130	−0.860652	+	0.509194i	$0.829944\pi$
$608$	1.31058e7	1.43782
$609$	94114.5	0.0102828
$610$	0	0
$611$	−2.87895e6	−0.311983
$612$	−334405.	−0.0360906
$613$	−3.97422e6	−0.427170	−0.213585	−	0.976924i	$-0.568514\pi$
$613$	−3.97422e6	−0.427170	−0.213585	+	0.976924i	$0.568514\pi$
$614$	−2.13494e6	−0.228541
$615$	0	0
$616$	−2.07098e6	−0.219899
$617$	−5.17583e6	−0.547353	−0.273676	−	0.961822i	$-0.588240\pi$
$617$	−5.17583e6	−0.547353	−0.273676	+	0.961822i	$0.588240\pi$
$618$	6.52451e6	0.687189
$619$	2.06480e6	0.216597	0.108298	−	0.994118i	$-0.465460\pi$
$619$	2.06480e6	0.216597	0.108298	+	0.994118i	$0.465460\pi$
$620$	0	0
$621$	−8.19174e6	−0.852407
$622$	1.70552e7	1.76759
$623$	1.78787e7	1.84550
$624$	3.17954e6	0.326891
$625$	0	0
$626$	8.78390e6	0.895883
$627$	5.71483e6	0.580543
$628$	−4.95020e6	−0.500869
$629$	2.95074e6	0.297374
$630$	0	0
$631$	−1.72030e7	−1.72001	−0.860003	−	0.510289i	$-0.829539\pi$
$631$	−1.72030e7	−1.72001	−0.860003	+	0.510289i	$0.829539\pi$
$632$	−5.96010e6	−0.593555
$633$	−2.48110e6	−0.246113
$634$	−1.61626e7	−1.59693
$635$	0	0
$636$	7.55896e6	0.741002
$637$	−257526.	−0.0251462
$638$	64585.8	0.00628182
$639$	−721720.	−0.0699224
$640$	0	0
$641$	−7.98432e6	−0.767526	−0.383763	−	0.923432i	$-0.625372\pi$
$641$	−7.98432e6	−0.767526	−0.383763	+	0.923432i	$0.625372\pi$
$642$	1.94775e7	1.86508
$643$	−5.66314e6	−0.540169	−0.270085	−	0.962837i	$-0.587052\pi$
$643$	−5.66314e6	−0.540169	−0.270085	+	0.962837i	$0.587052\pi$
$644$	−5.95288e6	−0.565604
$645$	0	0
$646$	−1.26837e7	−1.19581
$647$	−3.74193e6	−0.351427	−0.175713	−	0.984441i	$-0.556223\pi$
$647$	−3.74193e6	−0.351427	−0.175713	+	0.984441i	$0.556223\pi$
$648$	−4.34111e6	−0.406128
$649$	−1.95103e6	−0.181824
$650$	0	0
$651$	1.91842e7	1.77416
$652$	−4.31253e6	−0.397295
$653$	1.41967e7	1.30288	0.651438	−	0.758702i	$-0.274166\pi$
$653$	1.41967e7	1.30288	0.651438	+	0.758702i	$0.274166\pi$
$654$	4.11474e6	0.376182
$655$	0	0
$656$	1.31899e7	1.19669
$657$	86684.6	0.00783481
$658$	1.67941e7	1.51214
$659$	−7.43083e6	−0.666536	−0.333268	−	0.942832i	$-0.608151\pi$
$659$	−7.43083e6	−0.666536	−0.333268	+	0.942832i	$0.608151\pi$
$660$	0	0
$661$	1.44503e7	1.28639	0.643194	−	0.765703i	$-0.277609\pi$
$661$	1.44503e7	1.28639	0.643194	+	0.765703i	$0.277609\pi$
$662$	−1.00015e7	−0.886989
$663$	−2.21548e6	−0.195742
$664$	1.13808e6	0.100174
$665$	0	0
$666$	447131.	0.0390615
$667$	−96977.5	−0.00844027
$668$	−1.44964e6	−0.125695
$669$	1.71113e7	1.47815
$670$	0	0
$671$	−5.24975e6	−0.450125
$672$	−1.33539e7	−1.14074
$673$	−8.38558e6	−0.713667	−0.356834	−	0.934168i	$-0.616144\pi$
$673$	−8.38558e6	−0.713667	−0.356834	+	0.934168i	$0.616144\pi$
$674$	3.12143e6	0.264669
$675$	0	0
$676$	600346.	0.0505283
$677$	1.53211e7	1.28475	0.642376	−	0.766390i	$-0.277949\pi$
$677$	1.53211e7	1.28475	0.642376	+	0.766390i	$0.277949\pi$
$678$	−4.21594e6	−0.352225
$679$	2.62208e6	0.218259
$680$	0	0
$681$	−8.59974e6	−0.710587
$682$	1.31651e7	1.08384
$683$	1.08140e7	0.887021	0.443510	−	0.896269i	$-0.353733\pi$
$683$	1.08140e7	0.887021	0.443510	+	0.896269i	$0.353733\pi$
$684$	−761972.	−0.0622729
$685$	0	0
$686$	−1.50669e7	−1.22240
$687$	1.67397e7	1.35318
$688$	5.08981e6	0.409950
$689$	−4.05339e6	−0.325290
$690$	0	0
$691$	2.25890e7	1.79971	0.899853	−	0.436193i	$-0.143673\pi$
$691$	2.25890e7	1.79971	0.899853	+	0.436193i	$0.143673\pi$
$692$	−1.28863e7	−1.02297
$693$	471314.	0.0372801
$694$	2.43237e7	1.91704
$695$	0	0
$696$	−55577.1	−0.00434883
$697$	−9.19065e6	−0.716579
$698$	1.20758e7	0.938159
$699$	1.62758e7	1.25994
$700$	0	0
$701$	1.06616e7	0.819461	0.409731	−	0.912207i	$-0.365623\pi$
$701$	1.06616e7	0.819461	0.409731	+	0.912207i	$0.365623\pi$
$702$	−4.81919e6	−0.369089
$703$	6.72353e6	0.513108
$704$	−1.48200e6	−0.112698
$705$	0	0
$706$	1.90852e7	1.44107
$707$	1.22383e7	0.920812
$708$	−3.21395e6	−0.240966
$709$	−1.17029e7	−0.874333	−0.437166	−	0.899381i	$-0.644018\pi$
$709$	−1.17029e7	−0.874333	−0.437166	+	0.899381i	$0.644018\pi$
$710$	0	0
$711$	1.35640e6	0.100627
$712$	−1.05578e7	−0.780501
$713$	−1.97678e7	−1.45625
$714$	1.29238e7	0.948738
$715$	0	0
$716$	−1.34050e7	−0.977204
$717$	7.45990e6	0.541920
$718$	5.96247e6	0.431633
$719$	2.34149e6	0.168915	0.0844577	−	0.996427i	$-0.473084\pi$
$719$	2.34149e6	0.168915	0.0844577	+	0.996427i	$0.473084\pi$
$720$	0	0
$721$	8.09128e6	0.579668
$722$	−1.08713e7	−0.776135
$723$	−2.53211e7	−1.80151
$724$	3.05498e6	0.216602
$725$	0	0
$726$	1.35866e7	0.956687
$727$	−6.10487e6	−0.428391	−0.214195	−	0.976791i	$-0.568713\pi$
$727$	−6.10487e6	−0.428391	−0.214195	+	0.976791i	$0.568713\pi$
$728$	1.82940e6	0.127932
$729$	1.52564e7	1.06325
$730$	0	0
$731$	−3.54655e6	−0.245478
$732$	−8.64799e6	−0.596537
$733$	1.79993e7	1.23736	0.618681	−	0.785642i	$-0.287667\pi$
$733$	1.79993e7	1.23736	0.618681	+	0.785642i	$0.287667\pi$
$734$	−2.04951e7	−1.40414
$735$	0	0
$736$	1.37602e7	0.936330
$737$	60810.4	0.00412391
$738$	−1.39268e6	−0.0941260
$739$	2.02603e7	1.36469	0.682346	−	0.731030i	$-0.260960\pi$
$739$	2.02603e7	1.36469	0.682346	+	0.731030i	$0.260960\pi$
$740$	0	0
$741$	−5.04819e6	−0.337746
$742$	2.36451e7	1.57664
$743$	−2.77445e7	−1.84376	−0.921881	−	0.387472i	$-0.873348\pi$
$743$	−2.77445e7	−1.84376	−0.921881	+	0.387472i	$0.873348\pi$
$744$	−1.13288e7	−0.750328
$745$	0	0
$746$	−3.38523e7	−2.22711
$747$	−259005.	−0.0169827
$748$	3.51611e6	0.229778
$749$	2.41548e7	1.57326
$750$	0	0
$751$	−2.16593e7	−1.40134	−0.700672	−	0.713483i	$-0.747117\pi$
$751$	−2.16593e7	−1.40134	−0.700672	+	0.713483i	$0.747117\pi$
$752$	−2.13758e7	−1.37841
$753$	−5.37804e6	−0.345650
$754$	−57051.7	−0.00365461
$755$	0	0
$756$	1.11452e7	0.709223
$757$	−2.34452e7	−1.48701	−0.743505	−	0.668731i	$-0.766838\pi$
$757$	−2.34452e7	−1.48701	−0.743505	+	0.668731i	$0.766838\pi$
$758$	1.70006e7	1.07471
$759$	6.00019e6	0.378060
$760$	0	0
$761$	−2.00518e7	−1.25514	−0.627569	−	0.778561i	$-0.715950\pi$
$761$	−2.00518e7	−1.25514	−0.627569	+	0.778561i	$0.715950\pi$
$762$	3.28745e7	2.05103
$763$	5.10284e6	0.317322
$764$	−2.03341e7	−1.26035
$765$	0	0
$766$	−2.22599e7	−1.37073
$767$	1.72344e6	0.105781
$768$	2.05407e7	1.25664
$769$	2.69938e7	1.64607	0.823034	−	0.567992i	$-0.192280\pi$
$769$	2.69938e7	1.64607	0.823034	+	0.567992i	$0.192280\pi$
$770$	0	0
$771$	−1.07321e7	−0.650201
$772$	191367.	0.0115564
$773$	−1.55609e7	−0.936668	−0.468334	−	0.883552i	$-0.655146\pi$
$773$	−1.55609e7	−0.936668	−0.468334	+	0.883552i	$0.655146\pi$
$774$	−537416.	−0.0322447
$775$	0	0
$776$	−1.54841e6	−0.0923061
$777$	−6.85085e6	−0.407091
$778$	−9.12167e6	−0.540288
$779$	−2.09418e7	−1.23643
$780$	0	0
$781$	7.58855e6	0.445175
$782$	−1.33170e7	−0.778734
$783$	181565.	0.0105834
$784$	−1.91209e6	−0.111101
$785$	0	0
$786$	1.09442e7	0.631872
$787$	2.60132e7	1.49712	0.748560	−	0.663067i	$-0.230745\pi$
$787$	2.60132e7	1.49712	0.748560	+	0.663067i	$0.230745\pi$
$788$	1.19228e7	0.684010
$789$	−2.51872e6	−0.144041
$790$	0	0
$791$	−5.22834e6	−0.297114
$792$	−278323.	−0.0157665
$793$	4.63736e6	0.261871
$794$	3.96435e7	2.23162
$795$	0	0
$796$	−6.75360e6	−0.377792
$797$	1.36342e7	0.760296	0.380148	−	0.924926i	$-0.375873\pi$
$797$	1.36342e7	0.760296	0.380148	+	0.924926i	$0.375873\pi$
$798$	2.94482e7	1.63701
$799$	1.48945e7	0.825390
$800$	0	0
$801$	2.40275e6	0.132320
$802$	2.73337e7	1.50059
$803$	−911448.	−0.0498819
$804$	100174.	0.00546530
$805$	0	0
$806$	−1.16294e7	−0.630550
$807$	2.89621e7	1.56548
$808$	−7.22701e6	−0.389431
$809$	−7.65343e6	−0.411135	−0.205568	−	0.978643i	$-0.565904\pi$
$809$	−7.65343e6	−0.411135	−0.205568	+	0.978643i	$0.565904\pi$
$810$	0	0
$811$	1.95683e7	1.04472	0.522360	−	0.852725i	$-0.325052\pi$
$811$	1.95683e7	1.04472	0.522360	+	0.852725i	$0.325052\pi$
$812$	131942.	0.00702251
$813$	3.06164e7	1.62453
$814$	−4.70137e6	−0.248693
$815$	0	0
$816$	−1.64497e7	−0.864832
$817$	−8.08115e6	−0.423563
$818$	−2.97820e7	−1.55622
$819$	−416334.	−0.0216886
$820$	0	0
$821$	−2.71975e7	−1.40822	−0.704112	−	0.710089i	$-0.748654\pi$
$821$	−2.71975e7	−1.40822	−0.704112	+	0.710089i	$0.748654\pi$
$822$	7.90172e6	0.407889
$823$	−3.39792e7	−1.74870	−0.874348	−	0.485300i	$-0.838710\pi$
$823$	−3.39792e7	−1.74870	−0.874348	+	0.485300i	$0.838710\pi$
$824$	−4.77811e6	−0.245153
$825$	0	0
$826$	−1.00535e7	−0.512706
$827$	2.56121e7	1.30221	0.651105	−	0.758988i	$-0.274306\pi$
$827$	2.56121e7	1.30221	0.651105	+	0.758988i	$0.274306\pi$
$828$	−800020.	−0.0405532
$829$	1.33798e7	0.676180	0.338090	−	0.941114i	$-0.390219\pi$
$829$	1.33798e7	0.676180	0.338090	+	0.941114i	$0.390219\pi$
$830$	0	0
$831$	−1.58809e7	−0.797759
$832$	1.30912e6	0.0655648
$833$	1.33233e6	0.0665274
$834$	−3.24094e7	−1.61345
$835$	0	0
$836$	8.01178e6	0.396473
$837$	3.70100e7	1.82602
$838$	−2.67348e7	−1.31512
$839$	−1.59865e7	−0.784056	−0.392028	−	0.919953i	$-0.628226\pi$
$839$	−1.59865e7	−0.784056	−0.392028	+	0.919953i	$0.628226\pi$
$840$	0	0
$841$	−2.05090e7	−0.999895
$842$	−1.16435e7	−0.565985
$843$	−1.40276e7	−0.679851
$844$	−3.47832e6	−0.168079
$845$	0	0
$846$	2.25699e6	0.108419
$847$	1.68493e7	0.806998
$848$	−3.00958e7	−1.43720
$849$	1.24034e7	0.590570
$850$	0	0
$851$	7.05925e6	0.334145
$852$	1.25007e7	0.589978
$853$	−2.04688e7	−0.963209	−0.481605	−	0.876389i	$-0.659946\pi$
$853$	−2.04688e7	−0.963209	−0.481605	+	0.876389i	$0.659946\pi$
$854$	−2.70517e7	−1.26926
$855$	0	0
$856$	−1.42640e7	−0.665362
$857$	−1.31659e7	−0.612349	−0.306175	−	0.951975i	$-0.599049\pi$
$857$	−1.31659e7	−0.612349	−0.306175	+	0.951975i	$0.599049\pi$
$858$	3.52990e6	0.163698
$859$	−3.58429e7	−1.65737	−0.828686	−	0.559713i	$-0.810911\pi$
$859$	−3.58429e7	−1.65737	−0.828686	+	0.559713i	$0.810911\pi$
$860$	0	0
$861$	2.13383e7	0.980962
$862$	−4.76158e7	−2.18264
$863$	−9.29285e6	−0.424739	−0.212369	−	0.977189i	$-0.568118\pi$
$863$	−9.29285e6	−0.424739	−0.212369	+	0.977189i	$0.568118\pi$
$864$	−2.57623e7	−1.17409
$865$	0	0
$866$	−1.16654e7	−0.528572
$867$	−9.82658e6	−0.443971
$868$	2.68949e7	1.21163
$869$	−1.42619e7	−0.640662
$870$	0	0
$871$	−53716.8	−0.00239919
$872$	−3.01336e6	−0.134202
$873$	352387.	0.0156489
$874$	−3.03440e7	−1.34368
$875$	0	0
$876$	−1.50144e6	−0.0661070
$877$	3.01018e7	1.32158	0.660791	−	0.750570i	$-0.270221\pi$
$877$	3.01018e7	1.32158	0.660791	+	0.750570i	$0.270221\pi$
$878$	3.52898e7	1.54494
$879$	−2.66171e7	−1.16195
$880$	0	0
$881$	4.75023e6	0.206193	0.103097	−	0.994671i	$-0.467125\pi$
$881$	4.75023e6	0.206193	0.103097	+	0.994671i	$0.467125\pi$
$882$	201891.	0.00873868
$883$	−2.31803e7	−1.00050	−0.500250	−	0.865881i	$-0.666758\pi$
$883$	−2.31803e7	−1.00050	−0.500250	+	0.865881i	$0.666758\pi$
$884$	−3.10595e6	−0.133679
$885$	0	0
$886$	−4.22505e6	−0.180820
$887$	−3.60486e7	−1.53844	−0.769219	−	0.638986i	$-0.779354\pi$
$887$	−3.60486e7	−1.53844	−0.769219	+	0.638986i	$0.779354\pi$
$888$	4.04560e6	0.172167
$889$	4.07689e7	1.73011
$890$	0	0
$891$	−1.03879e7	−0.438361
$892$	2.39888e7	1.00948
$893$	3.39385e7	1.42418
$894$	−1.77511e6	−0.0742816
$895$	0	0
$896$	2.08641e7	0.868221
$897$	−5.30026e6	−0.219946
$898$	−1.48794e7	−0.615737
$899$	438142.	0.0180807
$900$	0	0
$901$	2.09706e7	0.860595
$902$	1.46433e7	0.599272
$903$	8.23418e6	0.336048
$904$	3.08747e6	0.125656
$905$	0	0
$906$	−4.18679e7	−1.69457
$907$	−3.74734e7	−1.51253	−0.756266	−	0.654264i	$-0.772979\pi$
$907$	−3.74734e7	−1.51253	−0.756266	+	0.654264i	$0.772979\pi$
$908$	−1.20562e7	−0.485284
$909$	1.64473e6	0.0660212
$910$	0	0
$911$	2.58798e7	1.03315	0.516577	−	0.856241i	$-0.327206\pi$
$911$	2.58798e7	1.03315	0.516577	+	0.856241i	$0.327206\pi$
$912$	−3.74821e7	−1.49223
$913$	2.72332e6	0.108124
$914$	−1.10920e7	−0.439182
$915$	0	0
$916$	2.34679e7	0.924134
$917$	1.35724e7	0.533006
$918$	2.49325e7	0.976472
$919$	1.10504e6	0.0431606	0.0215803	−	0.999767i	$-0.493130\pi$
$919$	1.10504e6	0.0431606	0.0215803	+	0.999767i	$0.493130\pi$
$920$	0	0
$921$	4.39611e6	0.170773
$922$	−5.73611e7	−2.22224
$923$	−6.70333e6	−0.258992
$924$	−8.16350e6	−0.314555
$925$	0	0
$926$	−992628.	−0.0380416
$927$	1.08740e6	0.0415615
$928$	−304985.	−0.0116254
$929$	−2.92574e6	−0.111224	−0.0556118	−	0.998452i	$-0.517711\pi$
$929$	−2.92574e6	−0.111224	−0.0556118	+	0.998452i	$0.517711\pi$
$930$	0	0
$931$	3.03585e6	0.114790
$932$	2.28175e7	0.860453
$933$	−3.51189e7	−1.32080
$934$	−4.05421e7	−1.52068
$935$	0	0
$936$	245856.	0.00917258
$937$	2.45270e7	0.912632	0.456316	−	0.889818i	$-0.349169\pi$
$937$	2.45270e7	0.912632	0.456316	+	0.889818i	$0.349169\pi$
$938$	313352.	0.0116286
$939$	−1.80872e7	−0.669432
$940$	0	0
$941$	919876.	0.0338653	0.0169327	−	0.999857i	$-0.494610\pi$
$941$	919876.	0.0338653	0.0169327	+	0.999857i	$0.494610\pi$
$942$	2.57108e7	0.944037
$943$	−2.19874e7	−0.805184
$944$	1.27963e7	0.467362
$945$	0	0
$946$	5.65068e6	0.205292
$947$	9.61521e6	0.348405	0.174202	−	0.984710i	$-0.444265\pi$
$947$	9.61521e6	0.348405	0.174202	+	0.984710i	$0.444265\pi$
$948$	−2.34939e7	−0.849051
$949$	805126.	0.0290201
$950$	0	0
$951$	3.32808e7	1.19328
$952$	−9.46455e6	−0.338460
$953$	2.43193e7	0.867399	0.433699	−	0.901058i	$-0.357208\pi$
$953$	2.43193e7	0.867399	0.433699	+	0.901058i	$0.357208\pi$
$954$	3.17772e6	0.113043
$955$	0	0
$956$	1.04582e7	0.370096
$957$	−132990.	−0.00469397
$958$	−2.85494e7	−1.00504
$959$	9.79921e6	0.344068
$960$	0	0
$961$	6.06813e7	2.11956
$962$	4.15295e6	0.144683
$963$	3.24622e6	0.112801
$964$	−3.54984e7	−1.23031
$965$	0	0
$966$	3.09186e7	1.06605
$967$	−2.12248e7	−0.729922	−0.364961	−	0.931023i	$-0.618918\pi$
$967$	−2.12248e7	−0.729922	−0.364961	+	0.931023i	$0.618918\pi$
$968$	−9.94993e6	−0.341296
$969$	2.61173e7	0.893549
$970$	0	0
$971$	−3.64590e7	−1.24096	−0.620479	−	0.784223i	$-0.713062\pi$
$971$	−3.64590e7	−1.24096	−0.620479	+	0.784223i	$0.713062\pi$
$972$	2.89131e6	0.0981587
$973$	−4.01920e7	−1.36100
$974$	−6.39227e7	−2.15903
$975$	0	0
$976$	3.44318e7	1.15700
$977$	−1.86618e7	−0.625487	−0.312743	−	0.949838i	$-0.601248\pi$
$977$	−1.86618e7	−0.625487	−0.312743	+	0.949838i	$0.601248\pi$
$978$	2.23988e7	0.748821
$979$	−2.52638e7	−0.842445
$980$	0	0
$981$	685781.	0.0227516
$982$	6.65073e7	2.20085
$983$	4.07152e6	0.134392	0.0671959	−	0.997740i	$-0.478595\pi$
$983$	4.07152e6	0.134392	0.0671959	+	0.997740i	$0.478595\pi$
$984$	−1.26008e7	−0.414869
$985$	0	0
$986$	295163.	0.00966873
$987$	−3.45812e7	−1.12992
$988$	−7.07720e6	−0.230658
$989$	−8.48466e6	−0.275832
$990$	0	0
$991$	3.55164e7	1.14880	0.574401	−	0.818574i	$-0.305235\pi$
$991$	3.55164e7	1.14880	0.574401	+	0.818574i	$0.305235\pi$
$992$	−6.21680e7	−2.00580
$993$	2.05943e7	0.662786
$994$	3.91033e7	1.25530
$995$	0	0
$996$	4.48616e6	0.143293
$997$	2.88979e7	0.920721	0.460360	−	0.887732i	$-0.347720\pi$
$997$	2.88979e7	0.920721	0.460360	+	0.887732i	$0.347720\pi$
$998$	6.85491e7	2.17859
$999$	−1.32166e7	−0.418992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.i.1.2	yes	9
5.2	odd	4		325.6.b.h.274.5		18
5.3	odd	4		325.6.b.h.274.14		18
5.4	even	2		325.6.a.h.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.8	✓	9	5.4	even	2
325.6.a.i.1.2	yes	9	1.1	even	1	trivial
325.6.b.h.274.5		18	5.2	odd	4
325.6.b.h.274.14		18	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-7.28147 q^{2} +14.9935 q^{3} +21.0198 q^{4} -109.175 q^{6} -135.391 q^{7} +79.9521 q^{8} -18.1955 q^{9} +191.317 q^{11} +315.160 q^{12} -169.000 q^{13} +985.848 q^{14} -1254.80 q^{16} +874.338 q^{17} +132.490 q^{18} +1992.26 q^{19} -2029.99 q^{21} -1393.07 q^{22} +2091.74 q^{23} +1198.76 q^{24} +1230.57 q^{26} -3916.23 q^{27} -2845.90 q^{28} -46.3621 q^{29} -9450.42 q^{31} +6578.33 q^{32} +2868.52 q^{33} -6366.46 q^{34} -382.466 q^{36} +3374.82 q^{37} -14506.6 q^{38} -2533.90 q^{39} -10511.6 q^{41} +14781.3 q^{42} -4056.27 q^{43} +4021.45 q^{44} -15230.9 q^{46} +17035.2 q^{47} -18813.8 q^{48} +1523.82 q^{49} +13109.4 q^{51} -3552.34 q^{52} +23984.5 q^{53} +28515.9 q^{54} -10824.8 q^{56} +29870.9 q^{57} +337.584 q^{58} -10197.8 q^{59} -27440.0 q^{61} +68812.9 q^{62} +2463.52 q^{63} -7746.27 q^{64} -20887.0 q^{66} +317.851 q^{67} +18378.4 q^{68} +31362.5 q^{69} +39664.7 q^{71} -1454.77 q^{72} -4764.06 q^{73} -24573.7 q^{74} +41876.9 q^{76} -25902.7 q^{77} +18450.5 q^{78} -74545.9 q^{79} -54296.4 q^{81} +76539.5 q^{82} +14234.5 q^{83} -42669.9 q^{84} +29535.6 q^{86} -695.130 q^{87} +15296.2 q^{88} -132052. q^{89} +22881.1 q^{91} +43967.9 q^{92} -141695. q^{93} -124041. q^{94} +98632.1 q^{96} -19366.7 q^{97} -11095.6 q^{98} -3481.12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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