LMFDB - Embedded newform 325.6.a.j.1.3

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 $ x^11 - 5x^10 - 257x^9 + 1165x^8 + 22234x^7 - 90282x^6 - 751180x^5 + 2564400x^4 + 9571584x^3 - 23191072x^2 - 33154496x + 44115200
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.07859$ 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)

$f(q)$	$=$	\(q-6.07859 q^{2} -25.6203 q^{3} +4.94922 q^{4} +155.735 q^{6} -137.301 q^{7} +164.431 q^{8} +413.398 q^{9} +169.031 q^{11} -126.800 q^{12} -169.000 q^{13} +834.595 q^{14} -1157.88 q^{16} +487.128 q^{17} -2512.87 q^{18} +2385.49 q^{19} +3517.68 q^{21} -1027.47 q^{22} -4144.47 q^{23} -4212.75 q^{24} +1027.28 q^{26} -4365.63 q^{27} -679.532 q^{28} -4570.45 q^{29} -2957.68 q^{31} +1776.50 q^{32} -4330.61 q^{33} -2961.05 q^{34} +2046.00 q^{36} -5733.02 q^{37} -14500.4 q^{38} +4329.82 q^{39} -1705.07 q^{41} -21382.5 q^{42} -10922.2 q^{43} +836.570 q^{44} +25192.5 q^{46} +7912.12 q^{47} +29665.2 q^{48} +2044.52 q^{49} -12480.4 q^{51} -836.418 q^{52} -26431.1 q^{53} +26536.9 q^{54} -22576.4 q^{56} -61116.8 q^{57} +27781.9 q^{58} +11248.5 q^{59} -40298.6 q^{61} +17978.5 q^{62} -56759.8 q^{63} +26253.6 q^{64} +26324.0 q^{66} +64887.2 q^{67} +2410.91 q^{68} +106182. q^{69} -47076.1 q^{71} +67975.2 q^{72} -42771.8 q^{73} +34848.7 q^{74} +11806.3 q^{76} -23208.1 q^{77} -26319.2 q^{78} +90057.0 q^{79} +11392.9 q^{81} +10364.4 q^{82} +16427.0 q^{83} +17409.8 q^{84} +66391.5 q^{86} +117096. q^{87} +27793.8 q^{88} -35632.7 q^{89} +23203.8 q^{91} -20511.9 q^{92} +75776.6 q^{93} -48094.5 q^{94} -45514.4 q^{96} +121030. q^{97} -12427.8 q^{98} +69876.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) $ 11 * q - 5 * q^2 - 11 * q^3 + 187 * q^4 + 351 * q^6 - 208 * q^7 - 165 * q^8 + 1372 * q^9 + 1276 * q^11 - 1533 * q^12 - 1859 * q^13 + 578 * q^14 + 5707 * q^16 - 2218 * q^17 + 6776 * q^18 + 3520 * q^19 + 1706 * q^21 + 6785 * q^22 + 1215 * q^23 + 7723 * q^24 + 845 * q^26 - 161 * q^27 - 12746 * q^28 + 18425 * q^29 + 9068 * q^31 - 21165 * q^32 - 19748 * q^33 - 14455 * q^34 + 37684 * q^36 + 2280 * q^37 + 23783 * q^38 + 1859 * q^39 - 10010 * q^41 + 44184 * q^42 - 6763 * q^43 + 60651 * q^44 - 9858 * q^46 - 4110 * q^47 - 63289 * q^48 + 73313 * q^49 + 36606 * q^51 - 31603 * q^52 - 15633 * q^53 + 48013 * q^54 - 1262 * q^56 + 205974 * q^57 - 77652 * q^58 + 131642 * q^59 - 43365 * q^61 + 115940 * q^62 - 218466 * q^63 + 180071 * q^64 - 58607 * q^66 + 100904 * q^67 - 371277 * q^68 + 59345 * q^69 + 68420 * q^71 + 210708 * q^72 - 124436 * q^73 + 371396 * q^74 + 22517 * q^76 + 150978 * q^77 - 59319 * q^78 + 52239 * q^79 + 107507 * q^81 + 351137 * q^82 - 211538 * q^83 + 852716 * q^84 + 202246 * q^86 + 28273 * q^87 + 28107 * q^88 + 327712 * q^89 + 35152 * q^91 + 413252 * q^92 - 243338 * q^93 + 649974 * q^94 + 899663 * q^96 + 110796 * q^97 - 27143 * q^98 + 426698 * 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 $

$2$	−6.07859	−1.07455	−0.537276	−	0.843406i	$-0.680547\pi$
$2$	−6.07859	−1.07455	−0.537276	+	0.843406i	$0.680547\pi$
$3$	−25.6203	−1.64354	−0.821770	−	0.569819i	$-0.807013\pi$
$3$	−25.6203	−1.64354	−0.821770	+	0.569819i	$0.807013\pi$
$4$	4.94922	0.154663
$5$	0	0
$5$	0	0
$6$	155.735	1.76607
$7$	−137.301	−1.05908	−0.529539	−	0.848286i	$-0.677635\pi$
$7$	−137.301	−1.05908	−0.529539	+	0.848286i	$0.677635\pi$
$8$	164.431	0.908359
$9$	413.398	1.70122
$10$	0	0
$11$	169.031	0.421196	0.210598	−	0.977573i	$-0.432459\pi$
$11$	169.031	0.421196	0.210598	+	0.977573i	$0.432459\pi$
$12$	−126.800	−0.254195
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	834.595	1.13804
$15$	0	0
$16$	−1157.88	−1.13074
$17$	487.128	0.408810	0.204405	−	0.978886i	$-0.434474\pi$
$17$	487.128	0.408810	0.204405	+	0.978886i	$0.434474\pi$
$18$	−2512.87	−1.82806
$19$	2385.49	1.51598	0.757990	−	0.652267i	$-0.226182\pi$
$19$	2385.49	1.51598	0.757990	+	0.652267i	$0.226182\pi$
$20$	0	0
$21$	3517.68	1.74064
$22$	−1027.47	−0.452597
$23$	−4144.47	−1.63361	−0.816807	−	0.576911i	$-0.804258\pi$
$23$	−4144.47	−1.63361	−0.816807	+	0.576911i	$0.804258\pi$
$24$	−4212.75	−1.49292
$25$	0	0
$26$	1027.28	0.298027
$27$	−4365.63	−1.15249
$28$	−679.532	−0.163800
$29$	−4570.45	−1.00917	−0.504585	−	0.863362i	$-0.668354\pi$
$29$	−4570.45	−1.00917	−0.504585	+	0.863362i	$0.668354\pi$
$30$	0	0
$31$	−2957.68	−0.552774	−0.276387	−	0.961046i	$-0.589137\pi$
$31$	−2957.68	−0.552774	−0.276387	+	0.961046i	$0.589137\pi$
$32$	1776.50	0.306683
$33$	−4330.61	−0.692252
$34$	−2961.05	−0.439287
$35$	0	0
$36$	2046.00	0.263117
$37$	−5733.02	−0.688461	−0.344230	−	0.938885i	$-0.611860\pi$
$37$	−5733.02	−0.688461	−0.344230	+	0.938885i	$0.611860\pi$
$38$	−14500.4	−1.62900
$39$	4329.82	0.455836
$40$	0	0
$41$	−1705.07	−0.158410	−0.0792051	−	0.996858i	$-0.525238\pi$
$41$	−1705.07	−0.158410	−0.0792051	+	0.996858i	$0.525238\pi$
$42$	−21382.5	−1.87041
$43$	−10922.2	−0.900821	−0.450410	−	0.892822i	$-0.648722\pi$
$43$	−10922.2	−0.900821	−0.450410	+	0.892822i	$0.648722\pi$
$44$	836.570	0.0651434
$45$	0	0
$46$	25192.5	1.75540
$47$	7912.12	0.522454	0.261227	−	0.965277i	$-0.415873\pi$
$47$	7912.12	0.522454	0.261227	+	0.965277i	$0.415873\pi$
$48$	29665.2	1.85842
$49$	2044.52	0.121647
$50$	0	0
$51$	−12480.4	−0.671895
$52$	−836.418	−0.0428958
$53$	−26431.1	−1.29249	−0.646244	−	0.763131i	$-0.723661\pi$
$53$	−26431.1	−1.29249	−0.646244	+	0.763131i	$0.723661\pi$
$54$	26536.9	1.23841
$55$	0	0
$56$	−22576.4	−0.962023
$57$	−61116.8	−2.49157
$58$	27781.9	1.08441
$59$	11248.5	0.420692	0.210346	−	0.977627i	$-0.432541\pi$
$59$	11248.5	0.420692	0.210346	+	0.977627i	$0.432541\pi$
$60$	0	0
$61$	−40298.6	−1.38665	−0.693323	−	0.720627i	$-0.743854\pi$
$61$	−40298.6	−1.38665	−0.693323	+	0.720627i	$0.743854\pi$
$62$	17978.5	0.593984
$63$	−56759.8	−1.80173
$64$	26253.6	0.801195
$65$	0	0
$66$	26324.0	0.743861
$67$	64887.2	1.76592	0.882962	−	0.469444i	$-0.155546\pi$
$67$	64887.2	1.76592	0.882962	+	0.469444i	$0.155546\pi$
$68$	2410.91	0.0632278
$69$	106182.	2.68491
$70$	0	0
$71$	−47076.1	−1.10829	−0.554147	−	0.832419i	$-0.686955\pi$
$71$	−47076.1	−1.10829	−0.554147	+	0.832419i	$0.686955\pi$
$72$	67975.2	1.54532
$73$	−42771.8	−0.939400	−0.469700	−	0.882826i	$-0.655638\pi$
$73$	−42771.8	−0.939400	−0.469700	+	0.882826i	$0.655638\pi$
$74$	34848.7	0.739787
$75$	0	0
$76$	11806.3	0.234466
$77$	−23208.1	−0.446079
$78$	−26319.2	−0.489820
$79$	90057.0	1.62349	0.811745	−	0.584012i	$-0.198518\pi$
$79$	90057.0	1.62349	0.811745	+	0.584012i	$0.198518\pi$
$80$	0	0
$81$	11392.9	0.192941
$82$	10364.4	0.170220
$83$	16427.0	0.261735	0.130868	−	0.991400i	$-0.458224\pi$
$83$	16427.0	0.261735	0.130868	+	0.991400i	$0.458224\pi$
$84$	17409.8	0.269212
$85$	0	0
$86$	66391.5	0.967979
$87$	117096.	1.65861
$88$	27793.8	0.382597
$89$	−35632.7	−0.476841	−0.238421	−	0.971162i	$-0.576630\pi$
$89$	−35632.7	−0.476841	−0.238421	+	0.971162i	$0.576630\pi$
$90$	0	0
$91$	23203.8	0.293735
$92$	−20511.9	−0.252660
$93$	75776.6	0.908506
$94$	−48094.5	−0.561405
$95$	0	0
$96$	−45514.4	−0.504046
$97$	121030.	1.30606	0.653029	−	0.757333i	$-0.273498\pi$
$97$	121030.	1.30606	0.653029	+	0.757333i	$0.273498\pi$
$98$	−12427.8	−0.130716
$99$	69876.9	0.716549
$100$	0	0
$101$	−33043.6	−0.322318	−0.161159	−	0.986928i	$-0.551523\pi$
$101$	−33043.6	−0.322318	−0.161159	+	0.986928i	$0.551523\pi$
$102$	75862.9	0.721987
$103$	198815.	1.84653	0.923265	−	0.384164i	$-0.125510\pi$
$103$	198815.	1.84653	0.923265	+	0.384164i	$0.125510\pi$
$104$	−27788.8	−0.251933
$105$	0	0
$106$	160664.	1.38885
$107$	−183355.	−1.54822	−0.774112	−	0.633049i	$-0.781803\pi$
$107$	−183355.	−1.54822	−0.774112	+	0.633049i	$0.781803\pi$
$108$	−21606.5	−0.178248
$109$	−178328.	−1.43765	−0.718827	−	0.695189i	$-0.755321\pi$
$109$	−178328.	−1.43765	−0.718827	+	0.695189i	$0.755321\pi$
$110$	0	0
$111$	146881.	1.13151
$112$	158978.	1.19754
$113$	22426.7	0.165223	0.0826114	−	0.996582i	$-0.473674\pi$
$113$	22426.7	0.165223	0.0826114	+	0.996582i	$0.473674\pi$
$114$	371504.	2.67733
$115$	0	0
$116$	−22620.2	−0.156081
$117$	−69864.2	−0.471835
$118$	−68374.9	−0.452056
$119$	−66883.1	−0.432961
$120$	0	0
$121$	−132480.	−0.822594
$122$	244959.	1.49002
$123$	43684.4	0.260354
$124$	−14638.2	−0.0854937
$125$	0	0
$126$	345020.	1.93605
$127$	−90854.6	−0.499847	−0.249924	−	0.968266i	$-0.580406\pi$
$127$	−90854.6	−0.499847	−0.249924	+	0.968266i	$0.580406\pi$
$128$	−216433.	−1.16761
$129$	279829.	1.48054
$130$	0	0
$131$	−78399.9	−0.399151	−0.199575	−	0.979882i	$-0.563956\pi$
$131$	−78399.9	−0.399151	−0.199575	+	0.979882i	$0.563956\pi$
$132$	−21433.1	−0.107066
$133$	−327530.	−1.60554
$134$	−394423.	−1.89758
$135$	0	0
$136$	80098.8	0.371346
$137$	111261.	0.506456	0.253228	−	0.967407i	$-0.418508\pi$
$137$	111261.	0.506456	0.253228	+	0.967407i	$0.418508\pi$
$138$	−645439.	−2.88508
$139$	−226363.	−0.993728	−0.496864	−	0.867828i	$-0.665515\pi$
$139$	−226363.	−0.993728	−0.496864	+	0.867828i	$0.665515\pi$
$140$	0	0
$141$	−202711.	−0.858675
$142$	286156.	1.19092
$143$	−28566.2	−0.116819
$144$	−478665.	−1.92365
$145$	0	0
$146$	259992.	1.00943
$147$	−52381.1	−0.199931
$148$	−28374.0	−0.106479
$149$	443036.	1.63483	0.817416	−	0.576047i	$-0.195406\pi$
$149$	443036.	1.63483	0.817416	+	0.576047i	$0.195406\pi$
$150$	0	0
$151$	−1334.00	−0.00476118	−0.00238059	−	0.999997i	$-0.500758\pi$
$151$	−1334.00	−0.00476118	−0.00238059	+	0.999997i	$0.500758\pi$
$152$	392247.	1.37705
$153$	201378.	0.695477
$154$	141072.	0.479336
$155$	0	0
$156$	21429.2	0.0705010
$157$	−66009.4	−0.213726	−0.106863	−	0.994274i	$-0.534081\pi$
$157$	−66009.4	−0.213726	−0.106863	+	0.994274i	$0.534081\pi$
$158$	−547419.	−1.74453
$159$	677173.	2.12425
$160$	0	0
$161$	569039.	1.73012
$162$	−69253.0	−0.207325
$163$	−57419.5	−0.169274	−0.0846371	−	0.996412i	$-0.526973\pi$
$163$	−57419.5	−0.169274	−0.0846371	+	0.996412i	$0.526973\pi$
$164$	−8438.78	−0.0245002
$165$	0	0
$166$	−99852.7	−0.281248
$167$	−298875.	−0.829275	−0.414637	−	0.909987i	$-0.636092\pi$
$167$	−298875.	−0.829275	−0.414637	+	0.909987i	$0.636092\pi$
$168$	578414.	1.58112
$169$	28561.0	0.0769231
$170$	0	0
$171$	986155.	2.57902
$172$	−54056.3	−0.139324
$173$	−16389.5	−0.0416342	−0.0208171	−	0.999783i	$-0.506627\pi$
$173$	−16389.5	−0.0416342	−0.0208171	+	0.999783i	$0.506627\pi$
$174$	−711779.	−1.78226
$175$	0	0
$176$	−195717.	−0.476264
$177$	−288189.	−0.691424
$178$	216596.	0.512391
$179$	420541.	0.981015	0.490507	−	0.871437i	$-0.336811\pi$
$179$	420541.	0.981015	0.490507	+	0.871437i	$0.336811\pi$
$180$	0	0
$181$	−598369.	−1.35760	−0.678801	−	0.734322i	$-0.737500\pi$
$181$	−598369.	−1.35760	−0.678801	+	0.734322i	$0.737500\pi$
$182$	−141047.	−0.315634
$183$	1.03246e6	2.27901
$184$	−681477.	−1.48391
$185$	0	0
$186$	−460615.	−0.976237
$187$	82339.7	0.172189
$188$	39158.8	0.0808044
$189$	599405.	1.22058
$190$	0	0
$191$	207706.	0.411971	0.205985	−	0.978555i	$-0.433960\pi$
$191$	207706.	0.411971	0.205985	+	0.978555i	$0.433960\pi$
$192$	−672623.	−1.31680
$193$	−545285.	−1.05373	−0.526867	−	0.849948i	$-0.676633\pi$
$193$	−545285.	−1.05373	−0.526867	+	0.849948i	$0.676633\pi$
$194$	−735690.	−1.40343
$195$	0	0
$196$	10118.8	0.0188143
$197$	559003.	1.02624	0.513119	−	0.858317i	$-0.328490\pi$
$197$	559003.	1.02624	0.513119	+	0.858317i	$0.328490\pi$
$198$	−424753.	−0.769969
$199$	−800588.	−1.43310	−0.716549	−	0.697536i	$-0.754280\pi$
$199$	−800588.	−1.43310	−0.716549	+	0.697536i	$0.754280\pi$
$200$	0	0
$201$	−1.66243e6	−2.90237
$202$	200858.	0.346347
$203$	627527.	1.06879
$204$	−61768.0	−0.103917
$205$	0	0
$206$	−1.20851e6	−1.98419
$207$	−1.71331e6	−2.77914
$208$	195682.	0.313612
$209$	403221.	0.638524
$210$	0	0
$211$	−971904.	−1.50285	−0.751427	−	0.659816i	$-0.770634\pi$
$211$	−971904.	−1.50285	−0.751427	+	0.659816i	$0.770634\pi$
$212$	−130814.	−0.199900
$213$	1.20610e6	1.82153
$214$	1.11454e6	1.66365
$215$	0	0
$216$	−717843.	−1.04688
$217$	406092.	0.585431
$218$	1.08398e6	1.54483
$219$	1.09582e6	1.54394
$220$	0	0
$221$	−82324.7	−0.113383
$222$	−892831.	−1.21587
$223$	401286.	0.540371	0.270186	−	0.962808i	$-0.412915\pi$
$223$	401286.	0.540371	0.270186	+	0.962808i	$0.412915\pi$
$224$	−243915.	−0.324802
$225$	0	0
$226$	−136323.	−0.177541
$227$	1.30062e6	1.67527	0.837637	−	0.546227i	$-0.183936\pi$
$227$	1.30062e6	1.67527	0.837637	+	0.546227i	$0.183936\pi$
$228$	−302481.	−0.385354
$229$	−864830.	−1.08979	−0.544894	−	0.838505i	$-0.683430\pi$
$229$	−864830.	−1.08979	−0.544894	+	0.838505i	$0.683430\pi$
$230$	0	0
$231$	594596.	0.733149
$232$	−751521.	−0.916688
$233$	−1.19230e6	−1.43878	−0.719390	−	0.694606i	$-0.755579\pi$
$233$	−1.19230e6	−1.43878	−0.719390	+	0.694606i	$0.755579\pi$
$234$	424676.	0.507011
$235$	0	0
$236$	55671.3	0.0650655
$237$	−2.30728e6	−2.66827
$238$	406555.	0.465240
$239$	−1.23057e6	−1.39352	−0.696759	−	0.717305i	$-0.745375\pi$
$239$	−1.23057e6	−1.39352	−0.696759	+	0.717305i	$0.745375\pi$
$240$	0	0
$241$	1.64306e6	1.82226	0.911129	−	0.412122i	$-0.135212\pi$
$241$	1.64306e6	1.82226	0.911129	+	0.412122i	$0.135212\pi$
$242$	805289.	0.883921
$243$	768958.	0.835385
$244$	−199447.	−0.214463
$245$	0	0
$246$	−265539.	−0.279764
$247$	−403148.	−0.420457
$248$	−486333.	−0.502117
$249$	−420863.	−0.430172
$250$	0	0
$251$	−447259.	−0.448100	−0.224050	−	0.974578i	$-0.571928\pi$
$251$	−447259.	−0.448100	−0.224050	+	0.974578i	$0.571928\pi$
$252$	−280917.	−0.278661
$253$	−700543.	−0.688071
$254$	552267.	0.537112
$255$	0	0
$256$	475490.	0.453463
$257$	−465085.	−0.439237	−0.219619	−	0.975586i	$-0.570481\pi$
$257$	−465085.	−0.439237	−0.219619	+	0.975586i	$0.570481\pi$
$258$	−1.70097e6	−1.59091
$259$	787148.	0.729134
$260$	0	0
$261$	−1.88941e6	−1.71682
$262$	476560.	0.428909
$263$	1.88192e6	1.67769	0.838846	−	0.544369i	$-0.183231\pi$
$263$	1.88192e6	1.67769	0.838846	+	0.544369i	$0.183231\pi$
$264$	−712085.	−0.628813
$265$	0	0
$266$	1.99092e6	1.72524
$267$	912919.	0.783707
$268$	321141.	0.273123
$269$	39508.5	0.0332897	0.0166448	−	0.999861i	$-0.494702\pi$
$269$	39508.5	0.0332897	0.0166448	+	0.999861i	$0.494702\pi$
$270$	0	0
$271$	2.01156e6	1.66383	0.831916	−	0.554902i	$-0.187244\pi$
$271$	2.01156e6	1.66383	0.831916	+	0.554902i	$0.187244\pi$
$272$	−564036.	−0.462258
$273$	−594488.	−0.482766
$274$	−676310.	−0.544214
$275$	0	0
$276$	525520.	0.415256
$277$	750989.	0.588077	0.294039	−	0.955794i	$-0.405001\pi$
$277$	750989.	0.588077	0.294039	+	0.955794i	$0.405001\pi$
$278$	1.37596e6	1.06781
$279$	−1.22270e6	−0.940392
$280$	0	0
$281$	−116949.	−0.0883550	−0.0441775	−	0.999024i	$-0.514067\pi$
$281$	−116949.	−0.0883550	−0.0441775	+	0.999024i	$0.514067\pi$
$282$	1.23219e6	0.922691
$283$	−1.08953e6	−0.808671	−0.404335	−	0.914611i	$-0.632497\pi$
$283$	−1.08953e6	−0.808671	−0.404335	+	0.914611i	$0.632497\pi$
$284$	−232990.	−0.171412
$285$	0	0
$286$	173642.	0.125528
$287$	234108.	0.167769
$288$	734400.	0.521737
$289$	−1.18256e6	−0.832875
$290$	0	0
$291$	−3.10081e6	−2.14656
$292$	−211687.	−0.145290
$293$	2.35419e6	1.60204	0.801020	−	0.598638i	$-0.204291\pi$
$293$	2.35419e6	1.60204	0.801020	+	0.598638i	$0.204291\pi$
$294$	318403.	0.214837
$295$	0	0
$296$	−942683.	−0.625369
$297$	−737926.	−0.485424
$298$	−2.69303e6	−1.75671
$299$	700415.	0.453083
$300$	0	0
$301$	1.49963e6	0.954040
$302$	8108.85	0.00511613
$303$	846585.	0.529742
$304$	−2.76211e6	−1.71418
$305$	0	0
$306$	−1.22409e6	−0.747327
$307$	1.40614e6	0.851495	0.425748	−	0.904842i	$-0.360011\pi$
$307$	1.40614e6	0.851495	0.425748	+	0.904842i	$0.360011\pi$
$308$	−114862.	−0.0689920
$309$	−5.09369e6	−3.03485
$310$	0	0
$311$	1.46661e6	0.859832	0.429916	−	0.902869i	$-0.358543\pi$
$311$	1.46661e6	0.859832	0.429916	+	0.902869i	$0.358543\pi$
$312$	711955.	0.414063
$313$	−310576.	−0.179187	−0.0895935	−	0.995978i	$-0.528557\pi$
$313$	−310576.	−0.179187	−0.0895935	+	0.995978i	$0.528557\pi$
$314$	401244.	0.229659
$315$	0	0
$316$	445712.	0.251094
$317$	−2.21756e6	−1.23944	−0.619721	−	0.784822i	$-0.712754\pi$
$317$	−2.21756e6	−1.23944	−0.619721	+	0.784822i	$0.712754\pi$
$318$	−4.11625e6	−2.28262
$319$	−772546.	−0.425058
$320$	0	0
$321$	4.69761e6	2.54457
$322$	−3.45895e6	−1.85911
$323$	1.16204e6	0.619747
$324$	56386.2	0.0298408
$325$	0	0
$326$	349030.	0.181894
$327$	4.56882e6	2.36284
$328$	−280366.	−0.143893
$329$	−1.08634e6	−0.553320
$330$	0	0
$331$	439684.	0.220582	0.110291	−	0.993899i	$-0.464822\pi$
$331$	439684.	0.220582	0.110291	+	0.993899i	$0.464822\pi$
$332$	81300.7	0.0404808
$333$	−2.37002e6	−1.17123
$334$	1.81674e6	0.891099
$335$	0	0
$336$	−4.07306e6	−1.96821
$337$	−65125.6	−0.0312376	−0.0156188	−	0.999878i	$-0.504972\pi$
$337$	−65125.6	−0.0312376	−0.0156188	+	0.999878i	$0.504972\pi$
$338$	−173611.	−0.0826579
$339$	−574579.	−0.271550
$340$	0	0
$341$	−499939.	−0.232826
$342$	−5.99443e6	−2.77129
$343$	2.02690e6	0.930245
$344$	−1.79594e6	−0.818269
$345$	0	0
$346$	99625.0	0.0447382
$347$	3.63163e6	1.61912	0.809558	−	0.587039i	$-0.199707\pi$
$347$	3.63163e6	1.61912	0.809558	+	0.587039i	$0.199707\pi$
$348$	579534.	0.256526
$349$	755061.	0.331832	0.165916	−	0.986140i	$-0.446942\pi$
$349$	755061.	0.331832	0.165916	+	0.986140i	$0.446942\pi$
$350$	0	0
$351$	737791.	0.319643
$352$	300283.	0.129174
$353$	3.06050e6	1.30724	0.653620	−	0.756823i	$-0.273249\pi$
$353$	3.06050e6	1.30724	0.653620	+	0.756823i	$0.273249\pi$
$354$	1.75178e6	0.742972
$355$	0	0
$356$	−176354.	−0.0737497
$357$	1.71356e6	0.711590
$358$	−2.55629e6	−1.05415
$359$	3.68034e6	1.50713	0.753567	−	0.657371i	$-0.228332\pi$
$359$	3.68034e6	1.50713	0.753567	+	0.657371i	$0.228332\pi$
$360$	0	0
$361$	3.21446e6	1.29819
$362$	3.63724e6	1.45882
$363$	3.39416e6	1.35197
$364$	114841.	0.0454300
$365$	0	0
$366$	−6.27591e6	−2.44891
$367$	2.22414e6	0.861978	0.430989	−	0.902357i	$-0.358165\pi$
$367$	2.22414e6	0.861978	0.430989	+	0.902357i	$0.358165\pi$
$368$	4.79880e6	1.84720
$369$	−704873.	−0.269491
$370$	0	0
$371$	3.62902e6	1.36885
$372$	375035.	0.140512
$373$	−3.73922e6	−1.39158	−0.695791	−	0.718244i	$-0.744946\pi$
$373$	−3.73922e6	−1.39158	−0.695791	+	0.718244i	$0.744946\pi$
$374$	−500509.	−0.185026
$375$	0	0
$376$	1.30099e6	0.474576
$377$	772406.	0.279893
$378$	−3.64353e6	−1.31158
$379$	656559.	0.234788	0.117394	−	0.993085i	$-0.462546\pi$
$379$	656559.	0.234788	0.117394	+	0.993085i	$0.462546\pi$
$380$	0	0
$381$	2.32772e6	0.821519
$382$	−1.26256e6	−0.442684
$383$	−3.33584e6	−1.16200	−0.581002	−	0.813902i	$-0.697339\pi$
$383$	−3.33584e6	−1.16200	−0.581002	+	0.813902i	$0.697339\pi$
$384$	5.54506e6	1.91901
$385$	0	0
$386$	3.31456e6	1.13229
$387$	−4.51521e6	−1.53250
$388$	599003.	0.201999
$389$	4.44083e6	1.48796	0.743978	−	0.668204i	$-0.232937\pi$
$389$	4.44083e6	1.48796	0.743978	+	0.668204i	$0.232937\pi$
$390$	0	0
$391$	−2.01889e6	−0.667837
$392$	336181.	0.110499
$393$	2.00863e6	0.656021
$394$	−3.39795e6	−1.10275
$395$	0	0
$396$	345836.	0.110824
$397$	−2.72643e6	−0.868196	−0.434098	−	0.900866i	$-0.642933\pi$
$397$	−2.72643e6	−0.868196	−0.434098	+	0.900866i	$0.642933\pi$
$398$	4.86644e6	1.53994
$399$	8.39139e6	2.63877
$400$	0	0
$401$	5.27026e6	1.63671	0.818353	−	0.574715i	$-0.194887\pi$
$401$	5.27026e6	1.63671	0.818353	+	0.574715i	$0.194887\pi$
$402$	1.01052e7	3.11875
$403$	499848.	0.153312
$404$	−163540.	−0.0498506
$405$	0	0
$406$	−3.81447e6	−1.14847
$407$	−969056.	−0.289977
$408$	−2.05215e6	−0.610322
$409$	4.58065e6	1.35400	0.677000	−	0.735983i	$-0.263280\pi$
$409$	4.58065e6	1.35400	0.677000	+	0.735983i	$0.263280\pi$
$410$	0	0
$411$	−2.85054e6	−0.832381
$412$	983979.	0.285590
$413$	−1.54443e6	−0.445546
$414$	1.04145e7	2.98634
$415$	0	0
$416$	−300228.	−0.0850586
$417$	5.79947e6	1.63323
$418$	−2.45101e6	−0.686128
$419$	3.94543e6	1.09789	0.548945	−	0.835858i	$-0.315030\pi$
$419$	3.94543e6	1.09789	0.548945	+	0.835858i	$0.315030\pi$
$420$	0	0
$421$	4.17743e6	1.14869	0.574347	−	0.818612i	$-0.305256\pi$
$421$	4.17743e6	1.14869	0.574347	+	0.818612i	$0.305256\pi$
$422$	5.90780e6	1.61490
$423$	3.27085e6	0.888812
$424$	−4.34609e6	−1.17404
$425$	0	0
$426$	−7.33140e6	−1.95732
$427$	5.53304e6	1.46857
$428$	−907465.	−0.239453
$429$	731873.	0.191996
$430$	0	0
$431$	−7.66563e6	−1.98772	−0.993859	−	0.110652i	$-0.964706\pi$
$431$	−7.66563e6	−1.98772	−0.993859	+	0.110652i	$0.964706\pi$
$432$	5.05488e6	1.30317
$433$	5.49209e6	1.40773	0.703863	−	0.710336i	$-0.251457\pi$
$433$	5.49209e6	1.40773	0.703863	+	0.710336i	$0.251457\pi$
$434$	−2.46847e6	−0.629076
$435$	0	0
$436$	−882586.	−0.222352
$437$	−9.88658e6	−2.47652
$438$	−6.66107e6	−1.65905
$439$	−2.56331e6	−0.634805	−0.317403	−	0.948291i	$-0.602811\pi$
$439$	−2.56331e6	−0.634805	−0.317403	+	0.948291i	$0.602811\pi$
$440$	0	0
$441$	845199.	0.206949
$442$	500418.	0.121836
$443$	−1.41896e6	−0.343526	−0.171763	−	0.985138i	$-0.554946\pi$
$443$	−1.41896e6	−0.343526	−0.171763	+	0.985138i	$0.554946\pi$
$444$	726948.	0.175003
$445$	0	0
$446$	−2.43925e6	−0.580657
$447$	−1.13507e7	−2.68691
$448$	−3.60464e6	−0.848528
$449$	3.56619e6	0.834813	0.417406	−	0.908720i	$-0.362939\pi$
$449$	3.56619e6	0.834813	0.417406	+	0.908720i	$0.362939\pi$
$450$	0	0
$451$	−288210.	−0.0667217
$452$	110995.	0.0255539
$453$	34177.5	0.00782519
$454$	−7.90593e6	−1.80017
$455$	0	0
$456$	−1.00495e7	−2.26324
$457$	−3.71538e6	−0.832172	−0.416086	−	0.909325i	$-0.636598\pi$
$457$	−3.71538e6	−0.832172	−0.416086	+	0.909325i	$0.636598\pi$
$458$	5.25694e6	1.17103
$459$	−2.12662e6	−0.471149
$460$	0	0
$461$	3.74184e6	0.820035	0.410018	−	0.912078i	$-0.365523\pi$
$461$	3.74184e6	0.820035	0.410018	+	0.912078i	$0.365523\pi$
$462$	−3.61431e6	−0.787807
$463$	−3.53774e6	−0.766962	−0.383481	−	0.923549i	$-0.625275\pi$
$463$	−3.53774e6	−0.766962	−0.383481	+	0.923549i	$0.625275\pi$
$464$	5.29203e6	1.14111
$465$	0	0
$466$	7.24748e6	1.54605
$467$	−1.97789e6	−0.419671	−0.209836	−	0.977737i	$-0.567293\pi$
$467$	−1.97789e6	−0.419671	−0.209836	+	0.977737i	$0.567293\pi$
$468$	−345773.	−0.0729754
$469$	−8.90907e6	−1.87025
$470$	0	0
$471$	1.69118e6	0.351267
$472$	1.84960e6	0.382139
$473$	−1.84619e6	−0.379422
$474$	1.40250e7	2.86720
$475$	0	0
$476$	−331019.	−0.0669632
$477$	−1.09266e7	−2.19881
$478$	7.48014e6	1.49741
$479$	−2.88196e6	−0.573918	−0.286959	−	0.957943i	$-0.592644\pi$
$479$	−2.88196e6	−0.573918	−0.286959	+	0.957943i	$0.592644\pi$
$480$	0	0
$481$	968880.	0.190945
$482$	−9.98746e6	−1.95811
$483$	−1.45789e7	−2.84353
$484$	−655671.	−0.127225
$485$	0	0
$486$	−4.67418e6	−0.897665
$487$	−2.88578e6	−0.551366	−0.275683	−	0.961249i	$-0.588904\pi$
$487$	−2.88578e6	−0.551366	−0.275683	+	0.961249i	$0.588904\pi$
$488$	−6.62632e6	−1.25957
$489$	1.47110e6	0.278209
$490$	0	0
$491$	5.27189e6	0.986876	0.493438	−	0.869781i	$-0.335740\pi$
$491$	5.27189e6	0.986876	0.493438	+	0.869781i	$0.335740\pi$
$492$	216204.	0.0402671
$493$	−2.22640e6	−0.412558
$494$	2.45057e6	0.451803
$495$	0	0
$496$	3.42464e6	0.625045
$497$	6.46359e6	1.17377
$498$	2.55825e6	0.462243
$499$	−308093.	−0.0553898	−0.0276949	−	0.999616i	$-0.508817\pi$
$499$	−308093.	−0.0553898	−0.0276949	+	0.999616i	$0.508817\pi$
$500$	0	0
$501$	7.65725e6	1.36295
$502$	2.71870e6	0.481507
$503$	4.27447e6	0.753291	0.376645	−	0.926358i	$-0.377078\pi$
$503$	4.27447e6	0.753291	0.376645	+	0.926358i	$0.377078\pi$
$504$	−9.33305e6	−1.63662
$505$	0	0
$506$	4.25831e6	0.739368
$507$	−731740.	−0.126426
$508$	−449659.	−0.0773080
$509$	−405253.	−0.0693317	−0.0346658	−	0.999399i	$-0.511037\pi$
$509$	−405253.	−0.0693317	−0.0346658	+	0.999399i	$0.511037\pi$
$510$	0	0
$511$	5.87260e6	0.994898
$512$	4.03553e6	0.680340
$513$	−1.04142e7	−1.74715
$514$	2.82706e6	0.471984
$515$	0	0
$516$	1.38494e6	0.228984
$517$	1.33739e6	0.220056
$518$	−4.78475e6	−0.783492
$519$	419903.	0.0684275
$520$	0	0
$521$	−7.00100e6	−1.12997	−0.564983	−	0.825102i	$-0.691117\pi$
$521$	−7.00100e6	−1.12997	−0.564983	+	0.825102i	$0.691117\pi$
$522$	1.14850e7	1.84482
$523$	−7.78764e6	−1.24495	−0.622475	−	0.782640i	$-0.713873\pi$
$523$	−7.78764e6	−1.24495	−0.622475	+	0.782640i	$0.713873\pi$
$524$	−388018.	−0.0617339
$525$	0	0
$526$	−1.14394e7	−1.80277
$527$	−1.44077e6	−0.225979
$528$	5.01433e6	0.782759
$529$	1.07403e7	1.66869
$530$	0	0
$531$	4.65010e6	0.715692
$532$	−1.62102e6	−0.248318
$533$	288157.	0.0439351
$534$	−5.54926e6	−0.842135
$535$	0	0
$536$	1.06694e7	1.60409
$537$	−1.07744e7	−1.61234
$538$	−240156.	−0.0357715
$539$	345586.	0.0512371
$540$	0	0
$541$	3.36678e6	0.494562	0.247281	−	0.968944i	$-0.420463\pi$
$541$	3.36678e6	0.494562	0.247281	+	0.968944i	$0.420463\pi$
$542$	−1.22274e7	−1.78787
$543$	1.53304e7	2.23128
$544$	865383.	0.125375
$545$	0	0
$546$	3.61365e6	0.518757
$547$	1.32125e7	1.88806	0.944030	−	0.329860i	$-0.107001\pi$
$547$	1.32125e7	1.88806	0.944030	+	0.329860i	$0.107001\pi$
$548$	550656.	0.0783301
$549$	−1.66594e7	−2.35900
$550$	0	0
$551$	−1.09028e7	−1.52988
$552$	1.74596e7	2.43886
$553$	−1.23649e7	−1.71940
$554$	−4.56495e6	−0.631920
$555$	0	0
$556$	−1.12032e6	−0.153693
$557$	7.47598e6	1.02101	0.510505	−	0.859875i	$-0.329459\pi$
$557$	7.47598e6	1.02101	0.510505	+	0.859875i	$0.329459\pi$
$558$	7.43228e6	1.01050
$559$	1.84585e6	0.249843
$560$	0	0
$561$	−2.10956e6	−0.282999
$562$	710886.	0.0949421
$563$	1.32316e7	1.75931	0.879653	−	0.475617i	$-0.157775\pi$
$563$	1.32316e7	1.75931	0.879653	+	0.475617i	$0.157775\pi$
$564$	−1.00326e6	−0.132805
$565$	0	0
$566$	6.62278e6	0.868959
$567$	−1.56426e6	−0.204339
$568$	−7.74075e6	−1.00673
$569$	−9.80441e6	−1.26952	−0.634762	−	0.772708i	$-0.718902\pi$
$569$	−9.80441e6	−1.26952	−0.634762	+	0.772708i	$0.718902\pi$
$570$	0	0
$571$	2.98442e6	0.383063	0.191532	−	0.981486i	$-0.438655\pi$
$571$	2.98442e6	0.383063	0.191532	+	0.981486i	$0.438655\pi$
$572$	−141380.	−0.0180675
$573$	−5.32149e6	−0.677091
$574$	−1.42305e6	−0.180276
$575$	0	0
$576$	1.08532e7	1.36301
$577$	−1.47218e7	−1.84086	−0.920429	−	0.390911i	$-0.872160\pi$
$577$	−1.47218e7	−1.84086	−0.920429	+	0.390911i	$0.872160\pi$
$578$	7.18831e6	0.894968
$579$	1.39704e7	1.73185
$580$	0	0
$581$	−2.25544e6	−0.277198
$582$	1.88486e7	2.30659
$583$	−4.46768e6	−0.544390
$584$	−7.03299e6	−0.853312
$585$	0	0
$586$	−1.43102e7	−1.72148
$587$	−9.66987e6	−1.15831	−0.579156	−	0.815217i	$-0.696618\pi$
$587$	−9.66987e6	−1.15831	−0.579156	+	0.815217i	$0.696618\pi$
$588$	−259245.	−0.0309220
$589$	−7.05552e6	−0.837993
$590$	0	0
$591$	−1.43218e7	−1.68666
$592$	6.63815e6	0.778472
$593$	−1.52788e7	−1.78424	−0.892121	−	0.451797i	$-0.850783\pi$
$593$	−1.52788e7	−1.78424	−0.892121	+	0.451797i	$0.850783\pi$
$594$	4.48554e6	0.521614
$595$	0	0
$596$	2.19268e6	0.252848
$597$	2.05113e7	2.35536
$598$	−4.25754e6	−0.486861
$599$	7.95473e6	0.905854	0.452927	−	0.891548i	$-0.350380\pi$
$599$	7.95473e6	0.905854	0.452927	+	0.891548i	$0.350380\pi$
$600$	0	0
$601$	5.48231e6	0.619123	0.309562	−	0.950879i	$-0.399818\pi$
$601$	5.48231e6	0.619123	0.309562	+	0.950879i	$0.399818\pi$
$602$	−9.11560e6	−1.02517
$603$	2.68242e7	3.00423
$604$	−6602.27	−0.000736378 0
$605$	0	0
$606$	−5.14604e6	−0.569235
$607$	−1.49346e6	−0.164521	−0.0822605	−	0.996611i	$-0.526214\pi$
$607$	−1.49346e6	−0.164521	−0.0822605	+	0.996611i	$0.526214\pi$
$608$	4.23782e6	0.464926
$609$	−1.60774e7	−1.75660
$610$	0	0
$611$	−1.33715e6	−0.144903
$612$	996663.	0.107565
$613$	−9.31063e6	−1.00076	−0.500378	−	0.865807i	$-0.666805\pi$
$613$	−9.31063e6	−1.00076	−0.500378	+	0.865807i	$0.666805\pi$
$614$	−8.54733e6	−0.914976
$615$	0	0
$616$	−3.81611e6	−0.405200
$617$	245403.	0.0259517	0.0129759	−	0.999916i	$-0.495870\pi$
$617$	245403.	0.0259517	0.0129759	+	0.999916i	$0.495870\pi$
$618$	3.09625e7	3.26110
$619$	2.82756e6	0.296610	0.148305	−	0.988942i	$-0.452618\pi$
$619$	2.82756e6	0.296610	0.148305	+	0.988942i	$0.452618\pi$
$620$	0	0
$621$	1.80932e7	1.88272
$622$	−8.91491e6	−0.923934
$623$	4.89240e6	0.505012
$624$	−5.01342e6	−0.515433
$625$	0	0
$626$	1.88786e6	0.192546
$627$	−1.03306e7	−1.04944
$628$	−326695.	−0.0330555
$629$	−2.79272e6	−0.281449
$630$	0	0
$631$	−1.53391e7	−1.53366	−0.766828	−	0.641853i	$-0.778166\pi$
$631$	−1.53391e7	−1.53366	−0.766828	+	0.641853i	$0.778166\pi$
$632$	1.48081e7	1.47471
$633$	2.49004e7	2.47000
$634$	1.34796e7	1.33185
$635$	0	0
$636$	3.35148e6	0.328544
$637$	−345523.	−0.0337388
$638$	4.69599e6	0.456747
$639$	−1.94612e7	−1.88546
$640$	0	0
$641$	1.02617e7	0.986444	0.493222	−	0.869904i	$-0.335819\pi$
$641$	1.02617e7	0.986444	0.493222	+	0.869904i	$0.335819\pi$
$642$	−2.85548e7	−2.73427
$643$	5.26807e6	0.502486	0.251243	−	0.967924i	$-0.419161\pi$
$643$	5.26807e6	0.502486	0.251243	+	0.967924i	$0.419161\pi$
$644$	2.81630e6	0.267586
$645$	0	0
$646$	−7.06356e6	−0.665951
$647$	−2.03948e7	−1.91540	−0.957698	−	0.287775i	$-0.907085\pi$
$647$	−2.03948e7	−1.91540	−0.957698	+	0.287775i	$0.907085\pi$
$648$	1.87335e6	0.175259
$649$	1.90134e6	0.177194
$650$	0	0
$651$	−1.04042e7	−0.962179
$652$	−284182.	−0.0261805
$653$	8.69527e6	0.797994	0.398997	−	0.916952i	$-0.369358\pi$
$653$	8.69527e6	0.797994	0.398997	+	0.916952i	$0.369358\pi$
$654$	−2.77720e7	−2.53900
$655$	0	0
$656$	1.97427e6	0.179121
$657$	−1.76818e7	−1.59813
$658$	6.60342e6	0.594571
$659$	1.85493e7	1.66385	0.831926	−	0.554886i	$-0.187238\pi$
$659$	1.85493e7	1.66385	0.831926	+	0.554886i	$0.187238\pi$
$660$	0	0
$661$	1.00321e6	0.0893078	0.0446539	−	0.999003i	$-0.485782\pi$
$661$	1.00321e6	0.0893078	0.0446539	+	0.999003i	$0.485782\pi$
$662$	−2.67266e6	−0.237027
$663$	2.10918e6	0.186350
$664$	2.70109e6	0.237750
$665$	0	0
$666$	1.44063e7	1.25854
$667$	1.89421e7	1.64859
$668$	−1.47920e6	−0.128258
$669$	−1.02811e7	−0.888122
$670$	0	0
$671$	−6.81171e6	−0.584049
$672$	6.24916e6	0.533824
$673$	653810.	0.0556434	0.0278217	−	0.999613i	$-0.491143\pi$
$673$	653810.	0.0556434	0.0278217	+	0.999613i	$0.491143\pi$
$674$	395872.	0.0335664
$675$	0	0
$676$	141355.	0.0118972
$677$	−831237.	−0.0697033	−0.0348516	−	0.999392i	$-0.511096\pi$
$677$	−831237.	−0.0697033	−0.0348516	+	0.999392i	$0.511096\pi$
$678$	3.49263e6	0.291795
$679$	−1.66175e7	−1.38322
$680$	0	0
$681$	−3.33222e7	−2.75338
$682$	3.03892e6	0.250184
$683$	4.86860e6	0.399349	0.199675	−	0.979862i	$-0.436012\pi$
$683$	4.86860e6	0.399349	0.199675	+	0.979862i	$0.436012\pi$
$684$	4.88070e6	0.398879
$685$	0	0
$686$	−1.23207e7	−0.999597
$687$	2.21572e7	1.79111
$688$	1.26466e7	1.01860
$689$	4.46686e6	0.358471
$690$	0	0
$691$	−7.17359e6	−0.571533	−0.285767	−	0.958299i	$-0.592248\pi$
$691$	−7.17359e6	−0.571533	−0.285767	+	0.958299i	$0.592248\pi$
$692$	−81115.2	−0.00643928
$693$	−9.59416e6	−0.758881
$694$	−2.20752e7	−1.73983
$695$	0	0
$696$	1.92542e7	1.50661
$697$	−830589.	−0.0647596
$698$	−4.58970e6	−0.356571
$699$	3.05470e7	2.36469
$700$	0	0
$701$	−9.45026e6	−0.726355	−0.363177	−	0.931720i	$-0.618308\pi$
$701$	−9.45026e6	−0.726355	−0.363177	+	0.931720i	$0.618308\pi$
$702$	−4.48473e6	−0.343474
$703$	−1.36761e7	−1.04369
$704$	4.43766e6	0.337460
$705$	0	0
$706$	−1.86035e7	−1.40470
$707$	4.53691e6	0.341359
$708$	−1.42631e6	−0.106938
$709$	−2.98225e6	−0.222807	−0.111404	−	0.993775i	$-0.535535\pi$
$709$	−2.98225e6	−0.222807	−0.111404	+	0.993775i	$0.535535\pi$
$710$	0	0
$711$	3.72293e7	2.76192
$712$	−5.85910e6	−0.433143
$713$	1.22580e7	0.903019
$714$	−1.04160e7	−0.764640
$715$	0	0
$716$	2.08135e6	0.151727
$717$	3.15276e7	2.29030
$718$	−2.23713e7	−1.61949
$719$	1.65959e7	1.19723	0.598617	−	0.801036i	$-0.295717\pi$
$719$	1.65959e7	1.19723	0.598617	+	0.801036i	$0.295717\pi$
$720$	0	0
$721$	−2.72975e7	−1.95562
$722$	−1.95394e7	−1.39498
$723$	−4.20955e7	−2.99495
$724$	−2.96146e6	−0.209971
$725$	0	0
$726$	−2.06317e7	−1.45276
$727$	−47580.6	−0.00333882	−0.00166941	−	0.999999i	$-0.500531\pi$
$727$	−47580.6	−0.00333882	−0.00166941	+	0.999999i	$0.500531\pi$
$728$	3.81542e6	0.266817
$729$	−2.24694e7	−1.56593
$730$	0	0
$731$	−5.32051e6	−0.368264
$732$	5.10988e6	0.352479
$733$	−6.02222e6	−0.413996	−0.206998	−	0.978341i	$-0.566369\pi$
$733$	−6.02222e6	−0.413996	−0.206998	+	0.978341i	$0.566369\pi$
$734$	−1.35196e7	−0.926241
$735$	0	0
$736$	−7.36265e6	−0.501002
$737$	1.09679e7	0.743800
$738$	4.28463e6	0.289583
$739$	8.15662e6	0.549413	0.274707	−	0.961528i	$-0.411419\pi$
$739$	8.15662e6	0.549413	0.274707	+	0.961528i	$0.411419\pi$
$740$	0	0
$741$	1.03287e7	0.691038
$742$	−2.20593e7	−1.47090
$743$	−473118.	−0.0314411	−0.0157205	−	0.999876i	$-0.505004\pi$
$743$	−473118.	−0.0314411	−0.0157205	+	0.999876i	$0.505004\pi$
$744$	1.24600e7	0.825249
$745$	0	0
$746$	2.27292e7	1.49533
$747$	6.79087e6	0.445270
$748$	407517.	0.0266313
$749$	2.51748e7	1.63969
$750$	0	0
$751$	5.49506e6	0.355527	0.177764	−	0.984073i	$-0.443114\pi$
$751$	5.49506e6	0.355527	0.177764	+	0.984073i	$0.443114\pi$
$752$	−9.16129e6	−0.590761
$753$	1.14589e7	0.736470
$754$	−4.69514e6	−0.300760
$755$	0	0
$756$	2.96659e6	0.188778
$757$	−6.82041e6	−0.432585	−0.216292	−	0.976329i	$-0.569396\pi$
$757$	−6.82041e6	−0.432585	−0.216292	+	0.976329i	$0.569396\pi$
$758$	−3.99095e6	−0.252292
$759$	1.79481e7	1.13087
$760$	0	0
$761$	1.14435e7	0.716304	0.358152	−	0.933663i	$-0.383407\pi$
$761$	1.14435e7	0.716304	0.358152	+	0.933663i	$0.383407\pi$
$762$	−1.41492e7	−0.882766
$763$	2.44846e7	1.52259
$764$	1.02798e6	0.0637167
$765$	0	0
$766$	2.02772e7	1.24863
$767$	−1.90100e6	−0.116679
$768$	−1.21822e7	−0.745284
$769$	−5.05628e6	−0.308330	−0.154165	−	0.988045i	$-0.549269\pi$
$769$	−5.05628e6	−0.308330	−0.154165	+	0.988045i	$0.549269\pi$
$770$	0	0
$771$	1.19156e7	0.721904
$772$	−2.69874e6	−0.162974
$773$	−3.54919e6	−0.213639	−0.106820	−	0.994278i	$-0.534067\pi$
$773$	−3.54919e6	−0.213639	−0.106820	+	0.994278i	$0.534067\pi$
$774$	2.74461e7	1.64675
$775$	0	0
$776$	1.99010e7	1.18637
$777$	−2.01669e7	−1.19836
$778$	−2.69940e7	−1.59889
$779$	−4.06743e6	−0.240147
$780$	0	0
$781$	−7.95731e6	−0.466809
$782$	1.22720e7	0.717626
$783$	1.99529e7	1.16306
$784$	−2.36731e6	−0.137551
$785$	0	0
$786$	−1.22096e7	−0.704929
$787$	−2.14355e7	−1.23367	−0.616833	−	0.787094i	$-0.711585\pi$
$787$	−2.14355e7	−1.23367	−0.616833	+	0.787094i	$0.711585\pi$
$788$	2.76663e6	0.158721
$789$	−4.82153e7	−2.75735
$790$	0	0
$791$	−3.07921e6	−0.174984
$792$	1.14899e7	0.650883
$793$	6.81047e6	0.384586
$794$	1.65728e7	0.932923
$795$	0	0
$796$	−3.96228e6	−0.221648
$797$	−1.94471e7	−1.08445	−0.542223	−	0.840235i	$-0.682417\pi$
$797$	−1.94471e7	−1.08445	−0.542223	+	0.840235i	$0.682417\pi$
$798$	−5.10078e7	−2.83550
$799$	3.85422e6	0.213584
$800$	0	0
$801$	−1.47305e7	−0.811214
$802$	−3.20357e7	−1.75873
$803$	−7.22975e6	−0.395671
$804$	−8.22772e6	−0.448889
$805$	0	0
$806$	−3.03837e6	−0.164742
$807$	−1.01222e6	−0.0547129
$808$	−5.43338e6	−0.292780
$809$	1.45681e7	0.782585	0.391292	−	0.920266i	$-0.372028\pi$
$809$	1.45681e7	0.782585	0.391292	+	0.920266i	$0.372028\pi$
$810$	0	0
$811$	1.86252e7	0.994371	0.497185	−	0.867644i	$-0.334367\pi$
$811$	1.86252e7	0.994371	0.497185	+	0.867644i	$0.334367\pi$
$812$	3.10577e6	0.165302
$813$	−5.15366e7	−2.73457
$814$	5.89049e6	0.311595
$815$	0	0
$816$	1.44508e7	0.759740
$817$	−2.60548e7	−1.36563
$818$	−2.78439e7	−1.45494
$819$	9.59241e6	0.499710
$820$	0	0
$821$	1.61122e7	0.834252	0.417126	−	0.908849i	$-0.363037\pi$
$821$	1.61122e7	0.834252	0.417126	+	0.908849i	$0.363037\pi$
$822$	1.73272e7	0.894438
$823$	1.15862e7	0.596269	0.298134	−	0.954524i	$-0.403636\pi$
$823$	1.15862e7	0.596269	0.298134	+	0.954524i	$0.403636\pi$
$824$	3.26913e7	1.67731
$825$	0	0
$826$	9.38794e6	0.478762
$827$	−8.40345e6	−0.427262	−0.213631	−	0.976914i	$-0.568529\pi$
$827$	−8.40345e6	−0.427262	−0.213631	+	0.976914i	$0.568529\pi$
$828$	−8.47957e6	−0.429831
$829$	−4.77193e6	−0.241161	−0.120581	−	0.992704i	$-0.538476\pi$
$829$	−4.77193e6	−0.241161	−0.120581	+	0.992704i	$0.538476\pi$
$830$	0	0
$831$	−1.92405e7	−0.966528
$832$	−4.43685e6	−0.222212
$833$	995943.	0.0497304
$834$	−3.52526e7	−1.75499
$835$	0	0
$836$	1.99563e6	0.0987561
$837$	1.29121e7	0.637067
$838$	−2.39826e7	−1.17974
$839$	−2.32624e7	−1.14091	−0.570453	−	0.821330i	$-0.693232\pi$
$839$	−2.32624e7	−1.14091	−0.570453	+	0.821330i	$0.693232\pi$
$840$	0	0
$841$	377863.	0.0184223
$842$	−2.53929e7	−1.23433
$843$	2.99627e6	0.145215
$844$	−4.81016e6	−0.232436
$845$	0	0
$846$	−1.98822e7	−0.955075
$847$	1.81896e7	0.871192
$848$	3.06041e7	1.46147
$849$	2.79140e7	1.32908
$850$	0	0
$851$	2.37603e7	1.12468
$852$	5.96927e6	0.281723
$853$	6.88858e6	0.324158	0.162079	−	0.986778i	$-0.448180\pi$
$853$	6.88858e6	0.324158	0.162079	+	0.986778i	$0.448180\pi$
$854$	−3.36330e7	−1.57805
$855$	0	0
$856$	−3.01492e7	−1.40634
$857$	−782343.	−0.0363869	−0.0181935	−	0.999834i	$-0.505791\pi$
$857$	−782343.	−0.0363869	−0.0181935	+	0.999834i	$0.505791\pi$
$858$	−4.44876e6	−0.206310
$859$	1.23216e7	0.569751	0.284875	−	0.958565i	$-0.408048\pi$
$859$	1.23216e7	0.569751	0.284875	+	0.958565i	$0.408048\pi$
$860$	0	0
$861$	−5.99790e6	−0.275735
$862$	4.65962e7	2.13591
$863$	7.42942e6	0.339569	0.169785	−	0.985481i	$-0.445693\pi$
$863$	7.42942e6	0.339569	0.169785	+	0.985481i	$0.445693\pi$
$864$	−7.75554e6	−0.353450
$865$	0	0
$866$	−3.33842e7	−1.51268
$867$	3.02976e7	1.36886
$868$	2.00984e6	0.0905445
$869$	1.52224e7	0.683807
$870$	0	0
$871$	−1.09659e7	−0.489779
$872$	−2.93226e7	−1.30591
$873$	5.00334e7	2.22190
$874$	6.00965e7	2.66116
$875$	0	0
$876$	5.42348e6	0.238791
$877$	7.95778e6	0.349376	0.174688	−	0.984624i	$-0.444108\pi$
$877$	7.95778e6	0.349376	0.174688	+	0.984624i	$0.444108\pi$
$878$	1.55813e7	0.682132
$879$	−6.03151e7	−2.63302
$880$	0	0
$881$	3.11775e7	1.35333	0.676663	−	0.736293i	$-0.263426\pi$
$881$	3.11775e7	1.35333	0.676663	+	0.736293i	$0.263426\pi$
$882$	−5.13761e6	−0.222377
$883$	2.20671e7	0.952452	0.476226	−	0.879323i	$-0.342004\pi$
$883$	2.20671e7	0.952452	0.476226	+	0.879323i	$0.342004\pi$
$884$	−407443.	−0.0175362
$885$	0	0
$886$	8.62524e6	0.369137
$887$	1.08566e6	0.0463323	0.0231662	−	0.999732i	$-0.492625\pi$
$887$	1.08566e6	0.0463323	0.0231662	+	0.999732i	$0.492625\pi$
$888$	2.41518e7	1.02782
$889$	1.24744e7	0.529378
$890$	0	0
$891$	1.92576e6	0.0812657
$892$	1.98605e6	0.0835755
$893$	1.88743e7	0.792030
$894$	6.89962e7	2.88723
$895$	0	0
$896$	2.97164e7	1.23659
$897$	−1.79448e7	−0.744660
$898$	−2.16774e7	−0.897050
$899$	1.35179e7	0.557842
$900$	0	0
$901$	−1.28754e7	−0.528381
$902$	1.75191e6	0.0716960
$903$	−3.84208e7	−1.56800
$904$	3.68764e6	0.150082
$905$	0	0
$906$	−207751.	−0.00840857
$907$	3.82701e7	1.54469	0.772345	−	0.635203i	$-0.219084\pi$
$907$	3.82701e7	1.54469	0.772345	+	0.635203i	$0.219084\pi$
$908$	6.43706e6	0.259103
$909$	−1.36601e7	−0.548334
$910$	0	0
$911$	−4.01079e7	−1.60116	−0.800579	−	0.599227i	$-0.795475\pi$
$911$	−4.01079e7	−1.60116	−0.800579	+	0.599227i	$0.795475\pi$
$912$	7.07660e7	2.81733
$913$	2.77666e6	0.110242
$914$	2.25843e7	0.894212
$915$	0	0
$916$	−4.28023e6	−0.168550
$917$	1.07644e7	0.422732
$918$	1.29269e7	0.506275
$919$	1.38759e6	0.0541967	0.0270984	−	0.999633i	$-0.491373\pi$
$919$	1.38759e6	0.0541967	0.0270984	+	0.999633i	$0.491373\pi$
$920$	0	0
$921$	−3.60256e7	−1.39947
$922$	−2.27451e7	−0.881171
$923$	7.95587e6	0.307385
$924$	2.94279e6	0.113391
$925$	0	0
$926$	2.15045e7	0.824141
$927$	8.21897e7	3.14136
$928$	−8.11940e6	−0.309495
$929$	−7.82980e6	−0.297654	−0.148827	−	0.988863i	$-0.547550\pi$
$929$	−7.82980e6	−0.297654	−0.148827	+	0.988863i	$0.547550\pi$
$930$	0	0
$931$	4.87717e6	0.184414
$932$	−5.90094e6	−0.222526
$933$	−3.75749e7	−1.41317
$934$	1.20228e7	0.450959
$935$	0	0
$936$	−1.14878e7	−0.428595
$937$	5.19939e7	1.93465	0.967327	−	0.253532i	$-0.0815925\pi$
$937$	5.19939e7	1.93465	0.967327	+	0.253532i	$0.0815925\pi$
$938$	5.41546e7	2.00968
$939$	7.95703e6	0.294501
$940$	0	0
$941$	2.54802e7	0.938055	0.469028	−	0.883184i	$-0.344605\pi$
$941$	2.54802e7	0.938055	0.469028	+	0.883184i	$0.344605\pi$
$942$	−1.02800e7	−0.377455
$943$	7.06662e6	0.258781
$944$	−1.30244e7	−0.475694
$945$	0	0
$946$	1.12222e7	0.407709
$947$	6.45145e6	0.233767	0.116883	−	0.993146i	$-0.462710\pi$
$947$	6.45145e6	0.233767	0.116883	+	0.993146i	$0.462710\pi$
$948$	−1.14193e7	−0.412683
$949$	7.22843e6	0.260543
$950$	0	0
$951$	5.68143e7	2.03707
$952$	−1.09976e7	−0.393284
$953$	−2.82313e7	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$953$	−2.82313e7	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$954$	6.64181e7	2.36274
$955$	0	0
$956$	−6.09038e6	−0.215526
$957$	1.97928e7	0.698599
$958$	1.75183e7	0.616705
$959$	−1.52762e7	−0.536377
$960$	0	0
$961$	−1.98813e7	−0.694441
$962$	−5.88942e6	−0.205180
$963$	−7.57986e7	−2.63388
$964$	8.13185e6	0.281836
$965$	0	0
$966$	8.86193e7	3.05552
$967$	5.33839e7	1.83588	0.917939	−	0.396722i	$-0.129852\pi$
$967$	5.33839e7	1.83588	0.917939	+	0.396722i	$0.129852\pi$
$968$	−2.17837e7	−0.747211
$969$	−2.97717e7	−1.01858
$970$	0	0
$971$	3.29212e7	1.12054	0.560270	−	0.828310i	$-0.310697\pi$
$971$	3.29212e7	1.12054	0.560270	+	0.828310i	$0.310697\pi$
$972$	3.80574e6	0.129203
$973$	3.10798e7	1.05244
$974$	1.75414e7	0.592472
$975$	0	0
$976$	4.66610e7	1.56794
$977$	4.57380e7	1.53300	0.766498	−	0.642247i	$-0.221997\pi$
$977$	4.57380e7	1.53300	0.766498	+	0.642247i	$0.221997\pi$
$978$	−8.94223e6	−0.298950
$979$	−6.02302e6	−0.200843
$980$	0	0
$981$	−7.37205e7	−2.44577
$982$	−3.20457e7	−1.06045
$983$	5.78671e7	1.91006	0.955031	−	0.296505i	$-0.0958211\pi$
$983$	5.78671e7	1.91006	0.955031	+	0.296505i	$0.0958211\pi$
$984$	7.18305e6	0.236495
$985$	0	0
$986$	1.35333e7	0.443315
$987$	2.78323e7	0.909404
$988$	−1.99527e6	−0.0650292
$989$	4.52667e7	1.47159
$990$	0	0
$991$	−4.51653e7	−1.46090	−0.730451	−	0.682965i	$-0.760690\pi$
$991$	−4.51653e7	−1.46090	−0.730451	+	0.682965i	$0.760690\pi$
$992$	−5.25432e6	−0.169526
$993$	−1.12648e7	−0.362536
$994$	−3.92895e7	−1.26128
$995$	0	0
$996$	−2.08294e6	−0.0665318
$997$	1.89237e7	0.602933	0.301467	−	0.953477i	$-0.402524\pi$
$997$	1.89237e7	0.602933	0.301467	+	0.953477i	$0.402524\pi$
$998$	1.87277e6	0.0595193
$999$	2.50282e7	0.793445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.j.1.3	✓	11
5.2	odd	4		325.6.b.i.274.5		22
5.3	odd	4		325.6.b.i.274.18		22
5.4	even	2		325.6.a.k.1.9	yes	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.3	✓	11	1.1	even	1	trivial
325.6.a.k.1.9	yes	11	5.4	even	2
325.6.b.i.274.5		22	5.2	odd	4
325.6.b.i.274.18		22	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)

\(f(q)\)	\(=\)	\(q-6.07859 q^{2} -25.6203 q^{3} +4.94922 q^{4} +155.735 q^{6} -137.301 q^{7} +164.431 q^{8} +413.398 q^{9} +169.031 q^{11} -126.800 q^{12} -169.000 q^{13} +834.595 q^{14} -1157.88 q^{16} +487.128 q^{17} -2512.87 q^{18} +2385.49 q^{19} +3517.68 q^{21} -1027.47 q^{22} -4144.47 q^{23} -4212.75 q^{24} +1027.28 q^{26} -4365.63 q^{27} -679.532 q^{28} -4570.45 q^{29} -2957.68 q^{31} +1776.50 q^{32} -4330.61 q^{33} -2961.05 q^{34} +2046.00 q^{36} -5733.02 q^{37} -14500.4 q^{38} +4329.82 q^{39} -1705.07 q^{41} -21382.5 q^{42} -10922.2 q^{43} +836.570 q^{44} +25192.5 q^{46} +7912.12 q^{47} +29665.2 q^{48} +2044.52 q^{49} -12480.4 q^{51} -836.418 q^{52} -26431.1 q^{53} +26536.9 q^{54} -22576.4 q^{56} -61116.8 q^{57} +27781.9 q^{58} +11248.5 q^{59} -40298.6 q^{61} +17978.5 q^{62} -56759.8 q^{63} +26253.6 q^{64} +26324.0 q^{66} +64887.2 q^{67} +2410.91 q^{68} +106182. q^{69} -47076.1 q^{71} +67975.2 q^{72} -42771.8 q^{73} +34848.7 q^{74} +11806.3 q^{76} -23208.1 q^{77} -26319.2 q^{78} +90057.0 q^{79} +11392.9 q^{81} +10364.4 q^{82} +16427.0 q^{83} +17409.8 q^{84} +66391.5 q^{86} +117096. q^{87} +27793.8 q^{88} -35632.7 q^{89} +23203.8 q^{91} -20511.9 q^{92} +75776.6 q^{93} -48094.5 q^{94} -45514.4 q^{96} +121030. q^{97} -12427.8 q^{98} +69876.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 11 * q - 5 * q^2 - 11 * q^3 + 187 * q^4 + 351 * q^6 - 208 * q^7 - 165 * q^8 + 1372 * q^9 + 1276 * q^11 - 1533 * q^12 - 1859 * q^13 + 578 * q^14 + 5707 * q^16 - 2218 * q^17 + 6776 * q^18 + 3520 * q^19 + 1706 * q^21 + 6785 * q^22 + 1215 * q^23 + 7723 * q^24 + 845 * q^26 - 161 * q^27 - 12746 * q^28 + 18425 * q^29 + 9068 * q^31 - 21165 * q^32 - 19748 * q^33 - 14455 * q^34 + 37684 * q^36 + 2280 * q^37 + 23783 * q^38 + 1859 * q^39 - 10010 * q^41 + 44184 * q^42 - 6763 * q^43 + 60651 * q^44 - 9858 * q^46 - 4110 * q^47 - 63289 * q^48 + 73313 * q^49 + 36606 * q^51 - 31603 * q^52 - 15633 * q^53 + 48013 * q^54 - 1262 * q^56 + 205974 * q^57 - 77652 * q^58 + 131642 * q^59 - 43365 * q^61 + 115940 * q^62 - 218466 * q^63 + 180071 * q^64 - 58607 * q^66 + 100904 * q^67 - 371277 * q^68 + 59345 * q^69 + 68420 * q^71 + 210708 * q^72 - 124436 * q^73 + 371396 * q^74 + 22517 * q^76 + 150978 * q^77 - 59319 * q^78 + 52239 * q^79 + 107507 * q^81 + 351137 * q^82 - 211538 * q^83 + 852716 * q^84 + 202246 * q^86 + 28273 * q^87 + 28107 * q^88 + 327712 * q^89 + 35152 * q^91 + 413252 * q^92 - 243338 * q^93 + 649974 * q^94 + 899663 * q^96 + 110796 * q^97 - 27143 * q^98 + 426698 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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