LMFDB - Embedded newform 325.6.a.h.1.9

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			1.9
Root			$9.23858$ 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)

$f(q)$	$=$	$q+8.23858 q^{2} +11.6375 q^{3} +35.8743 q^{4} +95.8765 q^{6} -195.071 q^{7} +31.9185 q^{8} -107.569 q^{9} +O(q^{10})$	\(q+8.23858 q^{2} +11.6375 q^{3} +35.8743 q^{4} +95.8765 q^{6} -195.071 q^{7} +31.9185 q^{8} -107.569 q^{9} +64.5579 q^{11} +417.487 q^{12} +169.000 q^{13} -1607.11 q^{14} -885.013 q^{16} +426.010 q^{17} -886.215 q^{18} -959.570 q^{19} -2270.14 q^{21} +531.866 q^{22} -499.768 q^{23} +371.452 q^{24} +1392.32 q^{26} -4079.74 q^{27} -6998.03 q^{28} +1288.17 q^{29} -6738.17 q^{31} -8312.65 q^{32} +751.292 q^{33} +3509.72 q^{34} -3858.95 q^{36} -6218.21 q^{37} -7905.50 q^{38} +1966.74 q^{39} -6498.90 q^{41} -18702.7 q^{42} -15640.4 q^{43} +2315.97 q^{44} -4117.38 q^{46} -6299.21 q^{47} -10299.3 q^{48} +21245.7 q^{49} +4957.69 q^{51} +6062.75 q^{52} +40397.3 q^{53} -33611.3 q^{54} -6226.38 q^{56} -11167.0 q^{57} +10612.7 q^{58} +25614.1 q^{59} +24197.9 q^{61} -55513.0 q^{62} +20983.6 q^{63} -40164.0 q^{64} +6189.58 q^{66} -39183.3 q^{67} +15282.8 q^{68} -5816.04 q^{69} -32681.2 q^{71} -3433.44 q^{72} +14507.0 q^{73} -51229.3 q^{74} -34423.9 q^{76} -12593.4 q^{77} +16203.1 q^{78} +79037.9 q^{79} -21338.7 q^{81} -53541.7 q^{82} +102366. q^{83} -81439.5 q^{84} -128855. q^{86} +14991.1 q^{87} +2060.59 q^{88} -48103.8 q^{89} -32967.0 q^{91} -17928.8 q^{92} -78415.4 q^{93} -51896.6 q^{94} -96738.4 q^{96} +73361.2 q^{97} +175035. q^{98} -6944.42 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21} + 4345 q^{22} + 1839 q^{23} - 8469 q^{24} - 845 q^{26} - 7649 q^{27} - 2836 q^{28} - 8737 q^{29} + 748 q^{31} - 423 q^{32} - 356 q^{33} - 17789 q^{34} + 512 q^{36} - 15486 q^{37} - 3425 q^{38} - 1859 q^{39} - 28676 q^{41} + 6876 q^{42} + 28665 q^{43} - 30599 q^{44} - 12056 q^{46} - 29452 q^{47} + 64759 q^{48} - 40907 q^{49} - 31006 q^{51} + 15379 q^{52} + 75977 q^{53} - 102761 q^{54} - 23002 q^{56} - 38038 q^{57} + 142384 q^{58} - 88142 q^{59} + 28165 q^{61} - 137308 q^{62} + 41492 q^{63} - 100845 q^{64} + 42577 q^{66} - 94754 q^{67} + 89267 q^{68} - 181747 q^{69} - 70562 q^{71} - 263778 q^{72} + 60602 q^{73} - 135676 q^{74} + 46373 q^{76} - 140292 q^{77} - 14027 q^{78} - 164073 q^{79} - 69935 q^{81} - 72887 q^{82} - 22458 q^{83} - 345656 q^{84} - 294920 q^{86} - 87031 q^{87} + 430607 q^{88} - 252698 q^{89} + 2028 q^{91} - 237824 q^{92} + 56556 q^{93} - 501606 q^{94} - 319181 q^{96} + 137986 q^{97} + 378699 q^{98} - 757776 q^{99}+O(q^{100}) \) 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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.23858	1.45639	0.728195	−	0.685370i	$-0.240360\pi$
$2$	8.23858	1.45639	0.728195	+	0.685370i	$0.240360\pi$
$3$	11.6375	0.746545	0.373273	−	0.927722i	$-0.378236\pi$
$3$	11.6375	0.746545	0.373273	+	0.927722i	$0.378236\pi$
$4$	35.8743	1.12107
$5$	0	0
$5$	0	0
$6$	95.8765	1.08726
$7$	−195.071	−1.50469	−0.752346	−	0.658768i	$-0.771078\pi$
$7$	−195.071	−1.50469	−0.752346	+	0.658768i	$0.771078\pi$
$8$	31.9185	0.176327
$9$	−107.569	−0.442670
$10$	0	0
$11$	64.5579	0.160867	0.0804337	−	0.996760i	$-0.474369\pi$
$11$	64.5579	0.160867	0.0804337	+	0.996760i	$0.474369\pi$
$12$	417.487	0.836930
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−1607.11	−2.19142
$15$	0	0
$16$	−885.013	−0.864271
$17$	426.010	0.357518	0.178759	−	0.983893i	$-0.442792\pi$
$17$	426.010	0.357518	0.178759	+	0.983893i	$0.442792\pi$
$18$	−886.215	−0.644700
$19$	−959.570	−0.609807	−0.304904	−	0.952383i	$-0.598624\pi$
$19$	−959.570	−0.609807	−0.304904	+	0.952383i	$0.598624\pi$
$20$	0	0
$21$	−2270.14	−1.12332
$22$	531.866	0.234286
$23$	−499.768	−0.196992	−0.0984960	−	0.995137i	$-0.531403\pi$
$23$	−499.768	−0.196992	−0.0984960	+	0.995137i	$0.531403\pi$
$24$	371.452	0.131636
$25$	0	0
$26$	1392.32	0.403930
$27$	−4079.74	−1.07702
$28$	−6998.03	−1.68687
$29$	1288.17	0.284432	0.142216	−	0.989836i	$-0.454577\pi$
$29$	1288.17	0.284432	0.142216	+	0.989836i	$0.454577\pi$
$30$	0	0
$31$	−6738.17	−1.25932	−0.629662	−	0.776869i	$-0.716807\pi$
$31$	−6738.17	−1.25932	−0.629662	+	0.776869i	$0.716807\pi$
$32$	−8312.65	−1.43504
$33$	751.292	0.120095
$34$	3509.72	0.520685
$35$	0	0
$36$	−3858.95	−0.496264
$37$	−6218.21	−0.746726	−0.373363	−	0.927685i	$-0.621795\pi$
$37$	−6218.21	−0.746726	−0.373363	+	0.927685i	$0.621795\pi$
$38$	−7905.50	−0.888117
$39$	1966.74	0.207054
$40$	0	0
$41$	−6498.90	−0.603782	−0.301891	−	0.953342i	$-0.597618\pi$
$41$	−6498.90	−0.603782	−0.301891	+	0.953342i	$0.597618\pi$
$42$	−18702.7	−1.63599
$43$	−15640.4	−1.28996	−0.644980	−	0.764200i	$-0.723134\pi$
$43$	−15640.4	−1.28996	−0.644980	+	0.764200i	$0.723134\pi$
$44$	2315.97	0.180344
$45$	0	0
$46$	−4117.38	−0.286897
$47$	−6299.21	−0.415951	−0.207975	−	0.978134i	$-0.566687\pi$
$47$	−6299.21	−0.415951	−0.207975	+	0.978134i	$0.566687\pi$
$48$	−10299.3	−0.645217
$49$	21245.7	1.26410
$50$	0	0
$51$	4957.69	0.266903
$52$	6062.75	0.310929
$53$	40397.3	1.97543	0.987717	−	0.156254i	$-0.0499418\pi$
$53$	40397.3	1.97543	0.987717	+	0.156254i	$0.0499418\pi$
$54$	−33611.3	−1.56856
$55$	0	0
$56$	−6226.38	−0.265317
$57$	−11167.0	−0.455249
$58$	10612.7	0.414244
$59$	25614.1	0.957965	0.478982	−	0.877824i	$-0.341006\pi$
$59$	25614.1	0.957965	0.478982	+	0.877824i	$0.341006\pi$
$60$	0	0
$61$	24197.9	0.832632	0.416316	−	0.909220i	$-0.363321\pi$
$61$	24197.9	0.832632	0.416316	+	0.909220i	$0.363321\pi$
$62$	−55513.0	−1.83407
$63$	20983.6	0.666082
$64$	−40164.0	−1.22571
$65$	0	0
$66$	6189.58	0.174905
$67$	−39183.3	−1.06639	−0.533193	−	0.845994i	$-0.679008\pi$
$67$	−39183.3	−1.06639	−0.533193	+	0.845994i	$0.679008\pi$
$68$	15282.8	0.400803
$69$	−5816.04	−0.147064
$70$	0	0
$71$	−32681.2	−0.769401	−0.384700	−	0.923041i	$-0.625695\pi$
$71$	−32681.2	−0.769401	−0.384700	+	0.923041i	$0.625695\pi$
$72$	−3433.44	−0.0780545
$73$	14507.0	0.318617	0.159309	−	0.987229i	$-0.449073\pi$
$73$	14507.0	0.318617	0.159309	+	0.987229i	$0.449073\pi$
$74$	−51229.3	−1.08752
$75$	0	0
$76$	−34423.9	−0.683637
$77$	−12593.4	−0.242056
$78$	16203.1	0.301552
$79$	79037.9	1.42484	0.712422	−	0.701751i	$-0.247598\pi$
$79$	79037.9	1.42484	0.712422	+	0.701751i	$0.247598\pi$
$80$	0	0
$81$	−21338.7	−0.361373
$82$	−53541.7	−0.879342
$83$	102366.	1.63102	0.815509	−	0.578744i	$-0.196457\pi$
$83$	102366.	1.63102	0.815509	+	0.578744i	$0.196457\pi$
$84$	−81439.5	−1.25932
$85$	0	0
$86$	−128855.	−1.87868
$87$	14991.1	0.212341
$88$	2060.59	0.0283652
$89$	−48103.8	−0.643731	−0.321866	−	0.946785i	$-0.604310\pi$
$89$	−48103.8	−0.643731	−0.321866	+	0.946785i	$0.604310\pi$
$90$	0	0
$91$	−32967.0	−0.417327
$92$	−17928.8	−0.220842
$93$	−78415.4	−0.940143
$94$	−51896.6	−0.605786
$95$	0	0
$96$	−96738.4	−1.07132
$97$	73361.2	0.791657	0.395829	−	0.918324i	$-0.370457\pi$
$97$	73361.2	0.791657	0.395829	+	0.918324i	$0.370457\pi$
$98$	175035.	1.84102
$99$	−6944.42	−0.0712111
$100$	0	0
$101$	22470.8	0.219187	0.109594	−	0.993976i	$-0.465045\pi$
$101$	22470.8	0.219187	0.109594	+	0.993976i	$0.465045\pi$
$102$	40844.3	0.388715
$103$	−116368.	−1.08078	−0.540392	−	0.841413i	$-0.681724\pi$
$103$	−116368.	−1.08078	−0.540392	+	0.841413i	$0.681724\pi$
$104$	5394.23	0.0489042
$105$	0	0
$106$	332816.	2.87700
$107$	−33242.7	−0.280697	−0.140348	−	0.990102i	$-0.544822\pi$
$107$	−33242.7	−0.280697	−0.140348	+	0.990102i	$0.544822\pi$
$108$	−146358.	−1.20741
$109$	−87314.5	−0.703915	−0.351957	−	0.936016i	$-0.614484\pi$
$109$	−87314.5	−0.703915	−0.351957	+	0.936016i	$0.614484\pi$
$110$	0	0
$111$	−72364.4	−0.557465
$112$	172640.	1.30046
$113$	211485.	1.55806	0.779029	−	0.626988i	$-0.215713\pi$
$113$	211485.	1.55806	0.779029	+	0.626988i	$0.215713\pi$
$114$	−92000.1	−0.663020
$115$	0	0
$116$	46212.2	0.318868
$117$	−18179.1	−0.122775
$118$	211024.	1.39517
$119$	−83102.3	−0.537954
$120$	0	0
$121$	−156883.	−0.974122
$122$	199356.	1.21264
$123$	−75630.9	−0.450751
$124$	−241727.	−1.41179
$125$	0	0
$126$	172875.	0.970075
$127$	−74724.8	−0.411107	−0.205554	−	0.978646i	$-0.565900\pi$
$127$	−74724.8	−0.411107	−0.205554	+	0.978646i	$0.565900\pi$
$128$	−64890.1	−0.350069
$129$	−182015.	−0.963013
$130$	0	0
$131$	27550.2	0.140264	0.0701320	−	0.997538i	$-0.477658\pi$
$131$	27550.2	0.140264	0.0701320	+	0.997538i	$0.477658\pi$
$132$	26952.1	0.134635
$133$	187184.	0.917572
$134$	−322815.	−1.55307
$135$	0	0
$136$	13597.6	0.0630399
$137$	266022.	1.21092	0.605460	−	0.795876i	$-0.292989\pi$
$137$	266022.	1.21092	0.605460	+	0.795876i	$0.292989\pi$
$138$	−47916.0	−0.214182
$139$	−263800.	−1.15808	−0.579039	−	0.815300i	$-0.696572\pi$
$139$	−263800.	−1.15808	−0.579039	+	0.815300i	$0.696572\pi$
$140$	0	0
$141$	−73307.0	−0.310526
$142$	−269247.	−1.12055
$143$	10910.3	0.0446166
$144$	95199.8	0.382587
$145$	0	0
$146$	119517.	0.464031
$147$	247247.	0.943708
$148$	−223074.	−0.837133
$149$	4746.74	0.0175158	0.00875790	−	0.999962i	$-0.497212\pi$
$149$	4746.74	0.0175158	0.00875790	+	0.999962i	$0.497212\pi$
$150$	0	0
$151$	−94651.9	−0.337821	−0.168911	−	0.985631i	$-0.554025\pi$
$151$	−94651.9	−0.337821	−0.168911	+	0.985631i	$0.554025\pi$
$152$	−30628.1	−0.107525
$153$	−45825.4	−0.158262
$154$	−103752.	−0.352528
$155$	0	0
$156$	70555.2	0.232123
$157$	−264887.	−0.857652	−0.428826	−	0.903387i	$-0.641073\pi$
$157$	−264887.	−0.857652	−0.428826	+	0.903387i	$0.641073\pi$
$158$	651160.	2.07513
$159$	470123.	1.47475
$160$	0	0
$161$	97490.2	0.296412
$162$	−175801.	−0.526300
$163$	121957.	0.359533	0.179766	−	0.983709i	$-0.442466\pi$
$163$	121957.	0.359533	0.179766	+	0.983709i	$0.442466\pi$
$164$	−233143.	−0.676883
$165$	0	0
$166$	843348.	2.37540
$167$	−465583.	−1.29183	−0.645916	−	0.763408i	$-0.723524\pi$
$167$	−465583.	−1.29183	−0.645916	+	0.763408i	$0.723524\pi$
$168$	−72459.5	−0.198071
$169$	28561.0	0.0769231
$170$	0	0
$171$	103220.	0.269943
$172$	−561087.	−1.44614
$173$	69307.4	0.176062	0.0880308	−	0.996118i	$-0.471943\pi$
$173$	69307.4	0.176062	0.0880308	+	0.996118i	$0.471943\pi$
$174$	123505.	0.309252
$175$	0	0
$176$	−57134.6	−0.139033
$177$	298084.	0.715164
$178$	−396307.	−0.937524
$179$	305640.	0.712981	0.356491	−	0.934299i	$-0.383973\pi$
$179$	305640.	0.712981	0.356491	+	0.934299i	$0.383973\pi$
$180$	0	0
$181$	−284602.	−0.645715	−0.322858	−	0.946448i	$-0.604644\pi$
$181$	−284602.	−0.645715	−0.322858	+	0.946448i	$0.604644\pi$
$182$	−271601.	−0.607790
$183$	281603.	0.621598
$184$	−15951.9	−0.0347350
$185$	0	0
$186$	−646032.	−1.36921
$187$	27502.3	0.0575129
$188$	−225980.	−0.466310
$189$	795839.	1.62058
$190$	0	0
$191$	−536766.	−1.06464	−0.532319	−	0.846544i	$-0.678679\pi$
$191$	−536766.	−1.06464	−0.532319	+	0.846544i	$0.678679\pi$
$192$	−467409.	−0.915048
$193$	−931130.	−1.79936	−0.899678	−	0.436555i	$-0.856199\pi$
$193$	−931130.	−1.79936	−0.899678	+	0.436555i	$0.856199\pi$
$194$	604393.	1.15296
$195$	0	0
$196$	762175.	1.41715
$197$	210297.	0.386071	0.193036	−	0.981192i	$-0.438167\pi$
$197$	210297.	0.386071	0.193036	+	0.981192i	$0.438167\pi$
$198$	−57212.2	−0.103711
$199$	853286.	1.52743	0.763716	−	0.645552i	$-0.223373\pi$
$199$	853286.	1.52743	0.763716	+	0.645552i	$0.223373\pi$
$200$	0	0
$201$	−455995.	−0.796105
$202$	185128.	0.319222
$203$	−251285.	−0.427983
$204$	177854.	0.299218
$205$	0	0
$206$	−958704.	−1.57404
$207$	53759.4	0.0872025
$208$	−149567.	−0.239706
$209$	−61947.8	−0.0980981
$210$	0	0
$211$	−913530.	−1.41259	−0.706296	−	0.707917i	$-0.749635\pi$
$211$	−913530.	−1.41259	−0.706296	+	0.707917i	$0.749635\pi$
$212$	1.44922e6	2.21460
$213$	−380328.	−0.574393
$214$	−273873.	−0.408804
$215$	0	0
$216$	−130219.	−0.189907
$217$	1.31442e6	1.89490
$218$	−719348.	−1.02517
$219$	168825.	0.237862
$220$	0	0
$221$	71995.7	0.0991576
$222$	−596180.	−0.811886
$223$	412081.	0.554908	0.277454	−	0.960739i	$-0.410509\pi$
$223$	412081.	0.554908	0.277454	+	0.960739i	$0.410509\pi$
$224$	1.62156e6	2.15930
$225$	0	0
$226$	1.74234e6	2.26914
$227$	−864103.	−1.11302	−0.556508	−	0.830843i	$-0.687859\pi$
$227$	−864103.	−1.11302	−0.556508	+	0.830843i	$0.687859\pi$
$228$	−400607.	−0.510366
$229$	−236005.	−0.297395	−0.148697	−	0.988883i	$-0.547508\pi$
$229$	−236005.	−0.297395	−0.148697	+	0.988883i	$0.547508\pi$
$230$	0	0
$231$	−146555.	−0.180706
$232$	41116.5	0.0501529
$233$	1.24095e6	1.49749	0.748744	−	0.662859i	$-0.230657\pi$
$233$	1.24095e6	1.49749	0.748744	+	0.662859i	$0.230657\pi$
$234$	−149770.	−0.178808
$235$	0	0
$236$	918888.	1.07395
$237$	919803.	1.06371
$238$	−684645.	−0.783471
$239$	−868458.	−0.983454	−0.491727	−	0.870749i	$-0.663634\pi$
$239$	−868458.	−0.983454	−0.491727	+	0.870749i	$0.663634\pi$
$240$	0	0
$241$	−836767.	−0.928029	−0.464015	−	0.885827i	$-0.653592\pi$
$241$	−836767.	−0.928029	−0.464015	+	0.885827i	$0.653592\pi$
$242$	−1.29250e6	−1.41870
$243$	743048.	0.807237
$244$	868082.	0.933440
$245$	0	0
$246$	−623092.	−0.656469
$247$	−162167.	−0.169130
$248$	−215072.	−0.222052
$249$	1.19128e6	1.21763
$250$	0	0
$251$	−1.94946e6	−1.95313	−0.976565	−	0.215224i	$-0.930952\pi$
$251$	−1.94946e6	−1.95313	−0.976565	+	0.215224i	$0.930952\pi$
$252$	752770.	0.746725
$253$	−32264.0	−0.0316896
$254$	−615626.	−0.598733
$255$	0	0
$256$	750647.	0.715873
$257$	−358481.	−0.338558	−0.169279	−	0.985568i	$-0.554144\pi$
$257$	−358481.	−0.338558	−0.169279	+	0.985568i	$0.554144\pi$
$258$	−1.49954e6	−1.40252
$259$	1.21299e6	1.12359
$260$	0	0
$261$	−138567.	−0.125909
$262$	226975.	0.204279
$263$	2.02422e6	1.80454	0.902272	−	0.431167i	$-0.141898\pi$
$263$	2.02422e6	1.80454	0.902272	+	0.431167i	$0.141898\pi$
$264$	23980.2	0.0211759
$265$	0	0
$266$	1.54213e6	1.33634
$267$	−559808.	−0.480575
$268$	−1.40567e6	−1.19549
$269$	−1.72406e6	−1.45268	−0.726341	−	0.687334i	$-0.758781\pi$
$269$	−1.72406e6	−1.45268	−0.726341	+	0.687334i	$0.758781\pi$
$270$	0	0
$271$	−618168.	−0.511309	−0.255654	−	0.966768i	$-0.582291\pi$
$271$	−618168.	−0.511309	−0.255654	+	0.966768i	$0.582291\pi$
$272$	−377025.	−0.308992
$273$	−383653.	−0.311553
$274$	2.19164e6	1.76357
$275$	0	0
$276$	−208646.	−0.164869
$277$	535823.	0.419587	0.209793	−	0.977746i	$-0.432721\pi$
$277$	535823.	0.419587	0.209793	+	0.977746i	$0.432721\pi$
$278$	−2.17334e6	−1.68661
$279$	724817.	0.557465
$280$	0	0
$281$	1.19483e6	0.902695	0.451347	−	0.892348i	$-0.350944\pi$
$281$	1.19483e6	0.902695	0.451347	+	0.892348i	$0.350944\pi$
$282$	−603946.	−0.452247
$283$	−2.59984e6	−1.92966	−0.964830	−	0.262875i	$-0.915329\pi$
$283$	−2.59984e6	−1.92966	−0.964830	+	0.262875i	$0.915329\pi$
$284$	−1.17242e6	−0.862553
$285$	0	0
$286$	89885.3	0.0649791
$287$	1.26775e6	0.908506
$288$	894182.	0.635250
$289$	−1.23837e6	−0.872181
$290$	0	0
$291$	853741.	0.591008
$292$	520427.	0.357193
$293$	2.12976e6	1.44931	0.724654	−	0.689113i	$-0.242000\pi$
$293$	2.12976e6	1.44931	0.724654	+	0.689113i	$0.242000\pi$
$294$	2.03696e6	1.37441
$295$	0	0
$296$	−198476.	−0.131668
$297$	−263380.	−0.173257
$298$	39106.4	0.0255098
$299$	−84460.8	−0.0546358
$300$	0	0
$301$	3.05098e6	1.94099
$302$	−779798.	−0.492000
$303$	261504.	0.163633
$304$	849232.	0.527039
$305$	0	0
$306$	−377536.	−0.230492
$307$	1.52136e6	0.921270	0.460635	−	0.887590i	$-0.347622\pi$
$307$	1.52136e6	0.921270	0.460635	+	0.887590i	$0.347622\pi$
$308$	−451778.	−0.271362
$309$	−1.35423e6	−0.806854
$310$	0	0
$311$	1.78856e6	1.04858	0.524290	−	0.851540i	$-0.324331\pi$
$311$	1.78856e6	1.04858	0.524290	+	0.851540i	$0.324331\pi$
$312$	62775.3	0.0365092
$313$	451359.	0.260412	0.130206	−	0.991487i	$-0.458436\pi$
$313$	451359.	0.260412	0.130206	+	0.991487i	$0.458436\pi$
$314$	−2.18229e6	−1.24908
$315$	0	0
$316$	2.83543e6	1.59735
$317$	1.39653e6	0.780554	0.390277	−	0.920697i	$-0.372379\pi$
$317$	1.39653e6	0.780554	0.390277	+	0.920697i	$0.372379\pi$
$318$	3.87315e6	2.14781
$319$	83161.6	0.0457558
$320$	0	0
$321$	−386862.	−0.209553
$322$	803182.	0.431692
$323$	−408786.	−0.218017
$324$	−765512.	−0.405125
$325$	0	0
$326$	1.00475e6	0.523620
$327$	−1.01612e6	−0.525504
$328$	−207435.	−0.106463
$329$	1.22879e6	0.625878
$330$	0	0
$331$	1.37630e6	0.690467	0.345233	−	0.938517i	$-0.387800\pi$
$331$	1.37630e6	0.690467	0.345233	+	0.938517i	$0.387800\pi$
$332$	3.67229e6	1.82849
$333$	668885.	0.330553
$334$	−3.83575e6	−1.88141
$335$	0	0
$336$	2.00910e6	0.970854
$337$	1.93389e6	0.927590	0.463795	−	0.885943i	$-0.346487\pi$
$337$	1.93389e6	0.927590	0.463795	+	0.885943i	$0.346487\pi$
$338$	235302.	0.112030
$339$	2.46115e6	1.16316
$340$	0	0
$341$	−435002.	−0.202584
$342$	850385.	0.393143
$343$	−865866.	−0.397388
$344$	−499218.	−0.227454
$345$	0	0
$346$	570995.	0.256414
$347$	4.24482e6	1.89250	0.946250	−	0.323435i	$-0.104838\pi$
$347$	4.24482e6	1.89250	0.946250	+	0.323435i	$0.104838\pi$
$348$	537794.	0.238050
$349$	−3.06057e6	−1.34505	−0.672525	−	0.740074i	$-0.734790\pi$
$349$	−3.06057e6	−1.34505	−0.672525	+	0.740074i	$0.734790\pi$
$350$	0	0
$351$	−689476.	−0.298711
$352$	−536647.	−0.230851
$353$	−4.53104e6	−1.93536	−0.967679	−	0.252186i	$-0.918851\pi$
$353$	−4.53104e6	−1.93536	−0.967679	+	0.252186i	$0.918851\pi$
$354$	2.45579e6	1.04156
$355$	0	0
$356$	−1.72569e6	−0.721668
$357$	−967102.	−0.401607
$358$	2.51804e6	1.03838
$359$	−2.68308e6	−1.09875	−0.549373	−	0.835577i	$-0.685133\pi$
$359$	−2.68308e6	−1.09875	−0.549373	+	0.835577i	$0.685133\pi$
$360$	0	0
$361$	−1.55532e6	−0.628135
$362$	−2.34472e6	−0.940413
$363$	−1.82573e6	−0.727226
$364$	−1.18267e6	−0.467853
$365$	0	0
$366$	2.32001e6	0.905288
$367$	659030.	0.255411	0.127706	−	0.991812i	$-0.459239\pi$
$367$	659030.	0.255411	0.127706	+	0.991812i	$0.459239\pi$
$368$	442301.	0.170254
$369$	699079.	0.267276
$370$	0	0
$371$	−7.88034e6	−2.97242
$372$	−2.81309e6	−1.05397
$373$	−1.73787e6	−0.646764	−0.323382	−	0.946268i	$-0.604820\pi$
$373$	−1.73787e6	−0.646764	−0.323382	+	0.946268i	$0.604820\pi$
$374$	226580.	0.0837612
$375$	0	0
$376$	−201062.	−0.0733432
$377$	217701.	0.0788872
$378$	6.55659e6	2.36020
$379$	−4.61878e6	−1.65169	−0.825847	−	0.563894i	$-0.809303\pi$
$379$	−4.61878e6	−1.65169	−0.825847	+	0.563894i	$0.809303\pi$
$380$	0	0
$381$	−869609.	−0.306910
$382$	−4.42219e6	−1.55053
$383$	656507.	0.228688	0.114344	−	0.993441i	$-0.463523\pi$
$383$	656507.	0.228688	0.114344	+	0.993441i	$0.463523\pi$
$384$	−755158.	−0.261342
$385$	0	0
$386$	−7.67119e6	−2.62056
$387$	1.68242e6	0.571026
$388$	2.63178e6	0.887504
$389$	−1.50525e6	−0.504352	−0.252176	−	0.967681i	$-0.581146\pi$
$389$	−1.50525e6	−0.504352	−0.252176	+	0.967681i	$0.581146\pi$
$390$	0	0
$391$	−212906.	−0.0704282
$392$	678132.	0.222894
$393$	320615.	0.104714
$394$	1.73255e6	0.562270
$395$	0	0
$396$	−249126.	−0.0798327
$397$	389426.	0.124008	0.0620038	−	0.998076i	$-0.480251\pi$
$397$	389426.	0.124008	0.0620038	+	0.998076i	$0.480251\pi$
$398$	7.02987e6	2.22454
$399$	2.17836e6	0.685009
$400$	0	0
$401$	3.45560e6	1.07316	0.536578	−	0.843851i	$-0.319717\pi$
$401$	3.45560e6	1.07316	0.536578	+	0.843851i	$0.319717\pi$
$402$	−3.75676e6	−1.15944
$403$	−1.13875e6	−0.349274
$404$	806125.	0.245725
$405$	0	0
$406$	−2.07023e6	−0.623309
$407$	−401435.	−0.120124
$408$	158242.	0.0470622
$409$	−5.22273e6	−1.54379	−0.771896	−	0.635748i	$-0.780692\pi$
$409$	−5.22273e6	−1.54379	−0.771896	+	0.635748i	$0.780692\pi$
$410$	0	0
$411$	3.09582e6	0.904007
$412$	−4.17460e6	−1.21164
$413$	−4.99658e6	−1.44144
$414$	442902.	0.127001
$415$	0	0
$416$	−1.40484e6	−0.398009
$417$	−3.06997e6	−0.864558
$418$	−510362.	−0.142869
$419$	−2.19501e6	−0.610805	−0.305402	−	0.952223i	$-0.598791\pi$
$419$	−2.19501e6	−0.610805	−0.305402	+	0.952223i	$0.598791\pi$
$420$	0	0
$421$	−4.06919e6	−1.11893	−0.559465	−	0.828854i	$-0.688993\pi$
$421$	−4.06919e6	−1.11893	−0.559465	+	0.828854i	$0.688993\pi$
$422$	−7.52619e6	−2.05728
$423$	677599.	0.184129
$424$	1.28942e6	0.348322
$425$	0	0
$426$	−3.13336e6	−0.836540
$427$	−4.72031e6	−1.25285
$428$	−1.19256e6	−0.314681
$429$	126968.	0.0333083
$430$	0	0
$431$	−3.69314e6	−0.957641	−0.478821	−	0.877913i	$-0.658936\pi$
$431$	−3.69314e6	−0.957641	−0.478821	+	0.877913i	$0.658936\pi$
$432$	3.61062e6	0.930836
$433$	−7.32688e6	−1.87802	−0.939008	−	0.343896i	$-0.888253\pi$
$433$	−7.32688e6	−1.87802	−0.939008	+	0.343896i	$0.888253\pi$
$434$	1.08290e7	2.75971
$435$	0	0
$436$	−3.13234e6	−0.789139
$437$	479562.	0.120127
$438$	1.39088e6	0.346420
$439$	−4.33121e6	−1.07263	−0.536313	−	0.844019i	$-0.680183\pi$
$439$	−4.33121e6	−1.07263	−0.536313	+	0.844019i	$0.680183\pi$
$440$	0	0
$441$	−2.28538e6	−0.559579
$442$	593143.	0.144412
$443$	6.00225e6	1.45313	0.726566	−	0.687097i	$-0.241115\pi$
$443$	6.00225e6	1.45313	0.726566	+	0.687097i	$0.241115\pi$
$444$	−2.59602e6	−0.624958
$445$	0	0
$446$	3.39497e6	0.808162
$447$	55240.2	0.0130763
$448$	7.83484e6	1.84432
$449$	1.63461e6	0.382647	0.191324	−	0.981527i	$-0.438722\pi$
$449$	1.63461e6	0.382647	0.191324	+	0.981527i	$0.438722\pi$
$450$	0	0
$451$	−419556.	−0.0971288
$452$	7.58687e6	1.74669
$453$	−1.10151e6	−0.252199
$454$	−7.11899e6	−1.62098
$455$	0	0
$456$	−356434.	−0.0802725
$457$	−8.14010e6	−1.82322	−0.911610	−	0.411056i	$-0.865160\pi$
$457$	−8.14010e6	−1.82322	−0.911610	+	0.411056i	$0.865160\pi$
$458$	−1.94435e6	−0.433123
$459$	−1.73801e6	−0.385053
$460$	0	0
$461$	1.30948e6	0.286977	0.143489	−	0.989652i	$-0.454168\pi$
$461$	1.30948e6	0.286977	0.143489	+	0.989652i	$0.454168\pi$
$462$	−1.20741e6	−0.263178
$463$	−1.54820e6	−0.335641	−0.167820	−	0.985818i	$-0.553673\pi$
$463$	−1.54820e6	−0.335641	−0.167820	+	0.985818i	$0.553673\pi$
$464$	−1.14005e6	−0.245826
$465$	0	0
$466$	1.02236e7	2.18093
$467$	6.32397e6	1.34183	0.670915	−	0.741534i	$-0.265902\pi$
$467$	6.32397e6	1.34183	0.670915	+	0.741534i	$0.265902\pi$
$468$	−652163.	−0.137639
$469$	7.64353e6	1.60458
$470$	0	0
$471$	−3.08262e6	−0.640276
$472$	817566.	0.168915
$473$	−1.00971e6	−0.207512
$474$	7.57787e6	1.54918
$475$	0	0
$476$	−2.98123e6	−0.603085
$477$	−4.34549e6	−0.874465
$478$	−7.15486e6	−1.43229
$479$	−803117.	−0.159934	−0.0799669	−	0.996798i	$-0.525481\pi$
$479$	−803117.	−0.159934	−0.0799669	+	0.996798i	$0.525481\pi$
$480$	0	0
$481$	−1.05088e6	−0.207104
$482$	−6.89377e6	−1.35157
$483$	1.13454e6	0.221285
$484$	−5.62807e6	−1.09206
$485$	0	0
$486$	6.12166e6	1.17565
$487$	−6.13178e6	−1.17156	−0.585779	−	0.810471i	$-0.699211\pi$
$487$	−6.13178e6	−1.17156	−0.585779	+	0.810471i	$0.699211\pi$
$488$	772362.	0.146815
$489$	1.41928e6	0.268407
$490$	0	0
$491$	−236451.	−0.0442626	−0.0221313	−	0.999755i	$-0.507045\pi$
$491$	−236451.	−0.0442626	−0.0221313	+	0.999755i	$0.507045\pi$
$492$	−2.71320e6	−0.505324
$493$	548774.	0.101689
$494$	−1.33603e6	−0.246319
$495$	0	0
$496$	5.96337e6	1.08840
$497$	6.37517e6	1.15771
$498$	9.81445e6	1.77334
$499$	−1.81339e6	−0.326018	−0.163009	−	0.986625i	$-0.552120\pi$
$499$	−1.81339e6	−0.326018	−0.163009	+	0.986625i	$0.552120\pi$
$500$	0	0
$501$	−5.41822e6	−0.964412
$502$	−1.60608e7	−2.84452
$503$	−4.69515e6	−0.827427	−0.413713	−	0.910407i	$-0.635768\pi$
$503$	−4.69515e6	−0.827427	−0.413713	+	0.910407i	$0.635768\pi$
$504$	669765.	0.117448
$505$	0	0
$506$	−265809.	−0.0461524
$507$	332378.	0.0574266
$508$	−2.68070e6	−0.460881
$509$	7.10737e6	1.21595	0.607973	−	0.793958i	$-0.291983\pi$
$509$	7.10737e6	1.21595	0.607973	+	0.793958i	$0.291983\pi$
$510$	0	0
$511$	−2.82989e6	−0.479421
$512$	8.26075e6	1.39266
$513$	3.91480e6	0.656774
$514$	−2.95338e6	−0.493073
$515$	0	0
$516$	−6.52965e6	−1.07961
$517$	−406664.	−0.0669128
$518$	9.99335e6	1.63639
$519$	806564.	0.131438
$520$	0	0
$521$	9.92332e6	1.60163	0.800816	−	0.598911i	$-0.204400\pi$
$521$	9.92332e6	1.60163	0.800816	+	0.598911i	$0.204400\pi$
$522$	−1.14160e6	−0.183373
$523$	−1.66096e6	−0.265525	−0.132763	−	0.991148i	$-0.542385\pi$
$523$	−1.66096e6	−0.265525	−0.132763	+	0.991148i	$0.542385\pi$
$524$	988343.	0.157246
$525$	0	0
$526$	1.66767e7	2.62812
$527$	−2.87053e6	−0.450231
$528$	−664904.	−0.103794
$529$	−6.18658e6	−0.961194
$530$	0	0
$531$	−2.75528e6	−0.424062
$532$	6.71510e6	1.02866
$533$	−1.09831e6	−0.167459
$534$	−4.61202e6	−0.699904
$535$	0	0
$536$	−1.25067e6	−0.188032
$537$	3.55689e6	0.532273
$538$	−1.42038e7	−2.11567
$539$	1.37158e6	0.203352
$540$	0	0
$541$	8.08498e6	1.18764	0.593822	−	0.804597i	$-0.297619\pi$
$541$	8.08498e6	1.18764	0.593822	+	0.804597i	$0.297619\pi$
$542$	−5.09283e6	−0.744665
$543$	−3.31205e6	−0.482056
$544$	−3.54127e6	−0.513053
$545$	0	0
$546$	−3.16076e6	−0.453743
$547$	−2.35716e6	−0.336838	−0.168419	−	0.985715i	$-0.553866\pi$
$547$	−2.35716e6	−0.336838	−0.168419	+	0.985715i	$0.553866\pi$
$548$	9.54333e6	1.35753
$549$	−2.60294e6	−0.368581
$550$	0	0
$551$	−1.23609e6	−0.173449
$552$	−185640.	−0.0259312
$553$	−1.54180e7	−2.14395
$554$	4.41442e6	0.611082
$555$	0	0
$556$	−9.46363e6	−1.29829
$557$	3.10870e6	0.424562	0.212281	−	0.977209i	$-0.431911\pi$
$557$	3.10870e6	0.424562	0.212281	+	0.977209i	$0.431911\pi$
$558$	5.97146e6	0.811886
$559$	−2.64322e6	−0.357770
$560$	0	0
$561$	320058.	0.0429360
$562$	9.84372e6	1.31468
$563$	−174928.	−0.0232589	−0.0116295	−	0.999932i	$-0.503702\pi$
$563$	−174928.	−0.0232589	−0.0116295	+	0.999932i	$0.503702\pi$
$564$	−2.62984e6	−0.348122
$565$	0	0
$566$	−2.14190e7	−2.81034
$567$	4.16257e6	0.543756
$568$	−1.04314e6	−0.135666
$569$	−7.95974e6	−1.03067	−0.515333	−	0.856990i	$-0.672332\pi$
$569$	−7.95974e6	−1.03067	−0.515333	+	0.856990i	$0.672332\pi$
$570$	0	0
$571$	3.22926e6	0.414488	0.207244	−	0.978289i	$-0.433551\pi$
$571$	3.22926e6	0.414488	0.207244	+	0.978289i	$0.433551\pi$
$572$	391399.	0.0500183
$573$	−6.24661e6	−0.794800
$574$	1.04444e7	1.32314
$575$	0	0
$576$	4.32040e6	0.542585
$577$	−1.13083e7	−1.41403	−0.707014	−	0.707200i	$-0.749958\pi$
$577$	−1.13083e7	−1.41403	−0.707014	+	0.707200i	$0.749958\pi$
$578$	−1.02024e7	−1.27024
$579$	−1.08360e7	−1.34330
$580$	0	0
$581$	−1.99686e7	−2.45418
$582$	7.03361e6	0.860738
$583$	2.60796e6	0.317783
$584$	463041.	0.0561807
$585$	0	0
$586$	1.75462e7	2.11076
$587$	9.47132e6	1.13453	0.567264	−	0.823536i	$-0.308002\pi$
$587$	9.47132e6	1.13453	0.567264	+	0.823536i	$0.308002\pi$
$588$	8.86980e6	1.05796
$589$	6.46574e6	0.767945
$590$	0	0
$591$	2.44733e6	0.288220
$592$	5.50320e6	0.645373
$593$	−5.00001e6	−0.583894	−0.291947	−	0.956434i	$-0.594303\pi$
$593$	−5.00001e6	−0.583894	−0.291947	+	0.956434i	$0.594303\pi$
$594$	−2.16988e6	−0.252330
$595$	0	0
$596$	170286.	0.0196365
$597$	9.93011e6	1.14030
$598$	−695837.	−0.0795710
$599$	9.49229e6	1.08095	0.540473	−	0.841361i	$-0.318245\pi$
$599$	9.49229e6	1.08095	0.540473	+	0.841361i	$0.318245\pi$
$600$	0	0
$601$	7.36100e6	0.831286	0.415643	−	0.909528i	$-0.363557\pi$
$601$	7.36100e6	0.831286	0.415643	+	0.909528i	$0.363557\pi$
$602$	2.51358e7	2.82684
$603$	4.21490e6	0.472057
$604$	−3.39557e6	−0.378722
$605$	0	0
$606$	2.15442e6	0.238314
$607$	1.14445e7	1.26074	0.630370	−	0.776295i	$-0.282903\pi$
$607$	1.14445e7	1.26074	0.630370	+	0.776295i	$0.282903\pi$
$608$	7.97657e6	0.875099
$609$	−2.92432e6	−0.319508
$610$	0	0
$611$	−1.06457e6	−0.115364
$612$	−1.64395e6	−0.177423
$613$	1.01622e7	1.09229	0.546144	−	0.837691i	$-0.316095\pi$
$613$	1.01622e7	1.09229	0.546144	+	0.837691i	$0.316095\pi$
$614$	1.25339e7	1.34173
$615$	0	0
$616$	−401962.	−0.0426809
$617$	−1.19269e7	−1.26129	−0.630644	−	0.776072i	$-0.717209\pi$
$617$	−1.19269e7	−1.26129	−0.630644	+	0.776072i	$0.717209\pi$
$618$	−1.11569e7	−1.17509
$619$	6.13819e6	0.643893	0.321946	−	0.946758i	$-0.395663\pi$
$619$	6.13819e6	0.643893	0.321946	+	0.946758i	$0.395663\pi$
$620$	0	0
$621$	2.03892e6	0.212164
$622$	1.47352e7	1.52714
$623$	9.38366e6	0.968617
$624$	−1.74059e6	−0.178951
$625$	0	0
$626$	3.71856e6	0.379262
$627$	−720917.	−0.0732347
$628$	−9.50262e6	−0.961489
$629$	−2.64902e6	−0.266968
$630$	0	0
$631$	9.80829e6	0.980663	0.490332	−	0.871536i	$-0.336876\pi$
$631$	9.80829e6	0.980663	0.490332	+	0.871536i	$0.336876\pi$
$632$	2.52277e6	0.251238
$633$	−1.06312e7	−1.05456
$634$	1.15055e7	1.13679
$635$	0	0
$636$	1.68653e7	1.65330
$637$	3.59053e6	0.350598
$638$	685134.	0.0666383
$639$	3.51548e6	0.340591
$640$	0	0
$641$	1.57763e6	0.151657	0.0758283	−	0.997121i	$-0.475840\pi$
$641$	1.57763e6	0.151657	0.0758283	+	0.997121i	$0.475840\pi$
$642$	−3.18719e6	−0.305190
$643$	1.87484e7	1.78829	0.894144	−	0.447779i	$-0.147785\pi$
$643$	1.87484e7	1.78829	0.894144	+	0.447779i	$0.147785\pi$
$644$	3.49739e6	0.332299
$645$	0	0
$646$	−3.36782e6	−0.317518
$647$	1.32643e7	1.24573	0.622863	−	0.782331i	$-0.285969\pi$
$647$	1.32643e7	1.24573	0.622863	+	0.782331i	$0.285969\pi$
$648$	−681101.	−0.0637198
$649$	1.65360e6	0.154105
$650$	0	0
$651$	1.52966e7	1.41463
$652$	4.37513e6	0.403062
$653$	7.19177e6	0.660013	0.330007	−	0.943979i	$-0.392949\pi$
$653$	7.19177e6	0.660013	0.330007	+	0.943979i	$0.392949\pi$
$654$	−8.37141e6	−0.765339
$655$	0	0
$656$	5.75161e6	0.521831
$657$	−1.56050e6	−0.141042
$658$	1.01235e7	0.911522
$659$	−2.13493e6	−0.191501	−0.0957503	−	0.995405i	$-0.530525\pi$
$659$	−2.13493e6	−0.191501	−0.0957503	+	0.995405i	$0.530525\pi$
$660$	0	0
$661$	−3.54104e6	−0.315230	−0.157615	−	0.987501i	$-0.550380\pi$
$661$	−3.54104e6	−0.315230	−0.157615	+	0.987501i	$0.550380\pi$
$662$	1.13388e7	1.00559
$663$	837850.	0.0740256
$664$	3.26736e6	0.287592
$665$	0	0
$666$	5.51067e6	0.481414
$667$	−643786.	−0.0560308
$668$	−1.67025e7	−1.44824
$669$	4.79559e6	0.414264
$670$	0	0
$671$	1.56217e6	0.133943
$672$	1.88709e7	1.61201
$673$	−1.59930e7	−1.36110	−0.680551	−	0.732700i	$-0.738260\pi$
$673$	−1.59930e7	−1.36110	−0.680551	+	0.732700i	$0.738260\pi$
$674$	1.59325e7	1.35093
$675$	0	0
$676$	1.02461e6	0.0862362
$677$	−6.51782e6	−0.546551	−0.273275	−	0.961936i	$-0.588107\pi$
$677$	−6.51782e6	−0.546551	−0.273275	+	0.961936i	$0.588107\pi$
$678$	2.02764e7	1.69401
$679$	−1.43107e7	−1.19120
$680$	0	0
$681$	−1.00560e7	−0.830916
$682$	−3.58380e6	−0.295041
$683$	−2.30581e7	−1.89135	−0.945673	−	0.325118i	$-0.894596\pi$
$683$	−2.30581e7	−1.89135	−0.945673	+	0.325118i	$0.894596\pi$
$684$	3.70293e6	0.302626
$685$	0	0
$686$	−7.13351e6	−0.578752
$687$	−2.74651e6	−0.222019
$688$	1.38419e7	1.11487
$689$	6.82714e6	0.547887
$690$	0	0
$691$	1.77699e7	1.41576	0.707880	−	0.706333i	$-0.249652\pi$
$691$	1.77699e7	1.41576	0.707880	+	0.706333i	$0.249652\pi$
$692$	2.48635e6	0.197377
$693$	1.35466e6	0.107151
$694$	3.49713e7	2.75622
$695$	0	0
$696$	478493.	0.0374414
$697$	−2.76860e6	−0.215863
$698$	−2.52148e7	−1.95892
$699$	1.44415e7	1.11794
$700$	0	0
$701$	−7.07397e6	−0.543711	−0.271856	−	0.962338i	$-0.587637\pi$
$701$	−7.07397e6	−0.543711	−0.271856	+	0.962338i	$0.587637\pi$
$702$	−5.68031e6	−0.435040
$703$	5.96681e6	0.455359
$704$	−2.59291e6	−0.197177
$705$	0	0
$706$	−3.73294e7	−2.81863
$707$	−4.38341e6	−0.329810
$708$	1.06936e7	0.801750
$709$	−8.17421e6	−0.610703	−0.305352	−	0.952240i	$-0.598774\pi$
$709$	−8.17421e6	−0.610703	−0.305352	+	0.952240i	$0.598774\pi$
$710$	0	0
$711$	−8.50201e6	−0.630736
$712$	−1.53540e6	−0.113507
$713$	3.36752e6	0.248077
$714$	−7.96755e6	−0.584897
$715$	0	0
$716$	1.09646e7	0.799303
$717$	−1.01067e7	−0.734193
$718$	−2.21048e7	−1.60020
$719$	−5.47393e6	−0.394891	−0.197446	−	0.980314i	$-0.563265\pi$
$719$	−5.47393e6	−0.394891	−0.197446	+	0.980314i	$0.563265\pi$
$720$	0	0
$721$	2.26999e7	1.62625
$722$	−1.28137e7	−0.914810
$723$	−9.73786e6	−0.692816
$724$	−1.02099e7	−0.723893
$725$	0	0
$726$	−1.50414e7	−1.05912
$727$	−1.55536e7	−1.09143	−0.545714	−	0.837972i	$-0.683741\pi$
$727$	−1.55536e7	−1.09143	−0.545714	+	0.837972i	$0.683741\pi$
$728$	−1.05226e6	−0.0735858
$729$	1.38325e7	0.964012
$730$	0	0
$731$	−6.66296e6	−0.461183
$732$	1.01023e7	0.696855
$733$	1.49050e7	1.02464	0.512320	−	0.858795i	$-0.328786\pi$
$733$	1.49050e7	1.02464	0.512320	+	0.858795i	$0.328786\pi$
$734$	5.42948e6	0.371979
$735$	0	0
$736$	4.15439e6	0.282692
$737$	−2.52959e6	−0.171547
$738$	5.75942e6	0.389258
$739$	1.31574e6	0.0886256	0.0443128	−	0.999018i	$-0.485890\pi$
$739$	1.31574e6	0.0886256	0.0443128	+	0.999018i	$0.485890\pi$
$740$	0	0
$741$	−1.88722e6	−0.126263
$742$	−6.49229e7	−4.32900
$743$	1.76450e7	1.17260	0.586300	−	0.810094i	$-0.300584\pi$
$743$	1.76450e7	1.17260	0.586300	+	0.810094i	$0.300584\pi$
$744$	−2.50290e6	−0.165772
$745$	0	0
$746$	−1.43176e7	−0.941941
$747$	−1.10113e7	−0.722003
$748$	986626.	0.0644761
$749$	6.48469e6	0.422362
$750$	0	0
$751$	−1.08240e7	−0.700308	−0.350154	−	0.936692i	$-0.613871\pi$
$751$	−1.08240e7	−0.700308	−0.350154	+	0.936692i	$0.613871\pi$
$752$	5.57489e6	0.359494
$753$	−2.26869e7	−1.45810
$754$	1.79355e6	0.114891
$755$	0	0
$756$	2.85502e7	1.81679
$757$	−2.74661e7	−1.74204	−0.871018	−	0.491251i	$-0.836540\pi$
$757$	−2.74661e7	−1.74204	−0.871018	+	0.491251i	$0.836540\pi$
$758$	−3.80522e7	−2.40551
$759$	−375472.	−0.0236577
$760$	0	0
$761$	3.08096e6	0.192852	0.0964261	−	0.995340i	$-0.469259\pi$
$761$	3.08096e6	0.192852	0.0964261	+	0.995340i	$0.469259\pi$
$762$	−7.16435e6	−0.446981
$763$	1.70325e7	1.05918
$764$	−1.92561e7	−1.19353
$765$	0	0
$766$	5.40869e6	0.333058
$767$	4.32879e6	0.265692
$768$	8.73565e6	0.534432
$769$	3.97374e6	0.242317	0.121159	−	0.992633i	$-0.461339\pi$
$769$	3.97374e6	0.242317	0.121159	+	0.992633i	$0.461339\pi$
$770$	0	0
$771$	−4.17182e6	−0.252749
$772$	−3.34036e7	−2.01721
$773$	1.29080e6	0.0776978	0.0388489	−	0.999245i	$-0.487631\pi$
$773$	1.29080e6	0.0776978	0.0388489	+	0.999245i	$0.487631\pi$
$774$	1.38607e7	0.831637
$775$	0	0
$776$	2.34158e6	0.139590
$777$	1.41162e7	0.838813
$778$	−1.24011e7	−0.734533
$779$	6.23615e6	0.368191
$780$	0	0
$781$	−2.10983e6	−0.123771
$782$	−1.75405e6	−0.102571
$783$	−5.25540e6	−0.306339
$784$	−1.88027e7	−1.09252
$785$	0	0
$786$	2.64142e6	0.152504
$787$	2.52309e7	1.45210	0.726050	−	0.687642i	$-0.241354\pi$
$787$	2.52309e7	1.45210	0.726050	+	0.687642i	$0.241354\pi$
$788$	7.54425e6	0.432813
$789$	2.35568e7	1.34717
$790$	0	0
$791$	−4.12546e7	−2.34440
$792$	−221656.	−0.0125564
$793$	4.08944e6	0.230931
$794$	3.20832e6	0.180603
$795$	0	0
$796$	3.06110e7	1.71236
$797$	2.89490e7	1.61431	0.807155	−	0.590339i	$-0.201006\pi$
$797$	2.89490e7	1.61431	0.807155	+	0.590339i	$0.201006\pi$
$798$	1.79466e7	0.997641
$799$	−2.68353e6	−0.148710
$800$	0	0
$801$	5.17447e6	0.284960
$802$	2.84693e7	1.56293
$803$	936540.	0.0512551
$804$	−1.63585e7	−0.892490
$805$	0	0
$806$	−9.38169e6	−0.508679
$807$	−2.00637e7	−1.08449
$808$	717236.	0.0386486
$809$	−1.89823e7	−1.01971	−0.509855	−	0.860260i	$-0.670301\pi$
$809$	−1.89823e7	−1.01971	−0.509855	+	0.860260i	$0.670301\pi$
$810$	0	0
$811$	−2.83661e7	−1.51442	−0.757212	−	0.653169i	$-0.773439\pi$
$811$	−2.83661e7	−1.51442	−0.757212	+	0.653169i	$0.773439\pi$
$812$	−9.01466e6	−0.479799
$813$	−7.19392e6	−0.381715
$814$	−3.30725e6	−0.174947
$815$	0	0
$816$	−4.38762e6	−0.230677
$817$	1.50080e7	0.786627
$818$	−4.30279e7	−2.24836
$819$	3.54622e6	0.184738
$820$	0	0
$821$	3.27149e7	1.69390	0.846951	−	0.531671i	$-0.178436\pi$
$821$	3.27149e7	1.69390	0.846951	+	0.531671i	$0.178436\pi$
$822$	2.55052e7	1.31659
$823$	−1.40256e7	−0.721809	−0.360904	−	0.932603i	$-0.617532\pi$
$823$	−1.40256e7	−0.721809	−0.360904	+	0.932603i	$0.617532\pi$
$824$	−3.71428e6	−0.190571
$825$	0	0
$826$	−4.11647e7	−2.09930
$827$	−2.33094e7	−1.18513	−0.592566	−	0.805522i	$-0.701885\pi$
$827$	−2.33094e7	−1.18513	−0.592566	+	0.805522i	$0.701885\pi$
$828$	1.92858e6	0.0977602
$829$	2.47785e7	1.25224	0.626122	−	0.779725i	$-0.284641\pi$
$829$	2.47785e7	1.25224	0.626122	+	0.779725i	$0.284641\pi$
$830$	0	0
$831$	6.23563e6	0.313241
$832$	−6.78772e6	−0.339951
$833$	9.05089e6	0.451938
$834$	−2.52922e7	−1.25913
$835$	0	0
$836$	−2.22233e6	−0.109975
$837$	2.74900e7	1.35632
$838$	−1.80838e7	−0.889569
$839$	−2.58285e7	−1.26676	−0.633380	−	0.773841i	$-0.718333\pi$
$839$	−2.58285e7	−1.26676	−0.633380	+	0.773841i	$0.718333\pi$
$840$	0	0
$841$	−1.88518e7	−0.919098
$842$	−3.35244e7	−1.62960
$843$	1.39048e7	0.673903
$844$	−3.27722e7	−1.58361
$845$	0	0
$846$	5.58245e6	0.268163
$847$	3.06034e7	1.46575
$848$	−3.57521e7	−1.70731
$849$	−3.02556e7	−1.44058
$850$	0	0
$851$	3.10766e6	0.147099
$852$	−1.36440e7	−0.643935
$853$	1.49617e7	0.704060	0.352030	−	0.935989i	$-0.385492\pi$
$853$	1.49617e7	0.704060	0.352030	+	0.935989i	$0.385492\pi$
$854$	−3.88887e7	−1.82465
$855$	0	0
$856$	−1.06106e6	−0.0494943
$857$	7.36028e6	0.342328	0.171164	−	0.985243i	$-0.445247\pi$
$857$	7.36028e6	0.342328	0.171164	+	0.985243i	$0.445247\pi$
$858$	1.04604e6	0.0485099
$859$	2.71276e7	1.25438	0.627190	−	0.778866i	$-0.284205\pi$
$859$	2.71276e7	1.25438	0.627190	+	0.778866i	$0.284205\pi$
$860$	0	0
$861$	1.47534e7	0.678241
$862$	−3.04263e7	−1.39470
$863$	3.14458e7	1.43726	0.718630	−	0.695393i	$-0.244770\pi$
$863$	3.14458e7	1.43726	0.718630	+	0.695393i	$0.244770\pi$
$864$	3.39135e7	1.54557
$865$	0	0
$866$	−6.03631e7	−2.73512
$867$	−1.44115e7	−0.651123
$868$	4.71539e7	2.12431
$869$	5.10252e6	0.229211
$870$	0	0
$871$	−6.62198e6	−0.295762
$872$	−2.78695e6	−0.124119
$873$	−7.89138e6	−0.350443
$874$	3.95091e6	0.174952
$875$	0	0
$876$	6.05646e6	0.266661
$877$	3.94061e7	1.73007	0.865037	−	0.501708i	$-0.167295\pi$
$877$	3.94061e7	1.73007	0.865037	+	0.501708i	$0.167295\pi$
$878$	−3.56831e7	−1.56216
$879$	2.47850e7	1.08197
$880$	0	0
$881$	−2.67060e7	−1.15923	−0.579614	−	0.814891i	$-0.696797\pi$
$881$	−2.67060e7	−1.15923	−0.579614	+	0.814891i	$0.696797\pi$
$882$	−1.88283e7	−0.814965
$883$	4.33871e7	1.87266	0.936330	−	0.351120i	$-0.114199\pi$
$883$	4.33871e7	1.87266	0.936330	+	0.351120i	$0.114199\pi$
$884$	2.58279e6	0.111163
$885$	0	0
$886$	4.94501e7	2.11633
$887$	−3.25024e7	−1.38710	−0.693549	−	0.720410i	$-0.743954\pi$
$887$	−3.25024e7	−1.38710	−0.693549	+	0.720410i	$0.743954\pi$
$888$	−2.30976e6	−0.0982959
$889$	1.45766e7	0.618590
$890$	0	0
$891$	−1.37758e6	−0.0581332
$892$	1.47831e7	0.622091
$893$	6.04453e6	0.253650
$894$	455101.	0.0190443
$895$	0	0
$896$	1.26582e7	0.526746
$897$	−982911.	−0.0407881
$898$	1.34669e7	0.557283
$899$	−8.67991e6	−0.358192
$900$	0	0
$901$	1.72097e7	0.706253
$902$	−3.45654e6	−0.141457
$903$	3.55058e7	1.44904
$904$	6.75029e6	0.274727
$905$	0	0
$906$	−9.07489e6	−0.367300
$907$	−3.43395e7	−1.38604	−0.693020	−	0.720918i	$-0.743720\pi$
$907$	−3.43395e7	−1.38604	−0.693020	+	0.720918i	$0.743720\pi$
$908$	−3.09991e7	−1.24777
$909$	−2.41716e6	−0.0970277
$910$	0	0
$911$	3.18788e7	1.27264	0.636320	−	0.771425i	$-0.280456\pi$
$911$	3.18788e7	1.27264	0.636320	+	0.771425i	$0.280456\pi$
$912$	9.88293e6	0.393458
$913$	6.60851e6	0.262378
$914$	−6.70629e7	−2.65532
$915$	0	0
$916$	−8.46652e6	−0.333401
$917$	−5.37425e6	−0.211054
$918$	−1.43188e7	−0.560788
$919$	−2.12110e7	−0.828460	−0.414230	−	0.910172i	$-0.635949\pi$
$919$	−2.12110e7	−0.828460	−0.414230	+	0.910172i	$0.635949\pi$
$920$	0	0
$921$	1.77049e7	0.687770
$922$	1.07883e7	0.417951
$923$	−5.52313e6	−0.213393
$924$	−5.25757e6	−0.202584
$925$	0	0
$926$	−1.27550e7	−0.488823
$927$	1.25175e7	0.478431
$928$	−1.07081e7	−0.408172
$929$	1.66040e6	0.0631210	0.0315605	−	0.999502i	$-0.489952\pi$
$929$	1.66040e6	0.0631210	0.0315605	+	0.999502i	$0.489952\pi$
$930$	0	0
$931$	−2.03867e7	−0.770857
$932$	4.45181e7	1.67879
$933$	2.08143e7	0.782813
$934$	5.21005e7	1.95423
$935$	0	0
$936$	−580251.	−0.0216484
$937$	−6.93039e6	−0.257875	−0.128937	−	0.991653i	$-0.541157\pi$
$937$	−6.93039e6	−0.257875	−0.128937	+	0.991653i	$0.541157\pi$
$938$	6.29719e7	2.33690
$939$	5.25269e6	0.194410
$940$	0	0
$941$	−1.29119e7	−0.475352	−0.237676	−	0.971344i	$-0.576386\pi$
$941$	−1.29119e7	−0.475352	−0.237676	+	0.971344i	$0.576386\pi$
$942$	−2.53964e7	−0.932492
$943$	3.24794e6	0.118940
$944$	−2.26688e7	−0.827941
$945$	0	0
$946$	−8.31858e6	−0.302219
$947$	−2.15320e7	−0.780207	−0.390104	−	0.920771i	$-0.627561\pi$
$947$	−2.15320e7	−0.780207	−0.390104	+	0.920771i	$0.627561\pi$
$948$	3.29973e7	1.19250
$949$	2.45168e6	0.0883686
$950$	0	0
$951$	1.62521e7	0.582719
$952$	−2.65250e6	−0.0948557
$953$	3.37555e7	1.20396	0.601980	−	0.798511i	$-0.294379\pi$
$953$	3.37555e7	1.20396	0.601980	+	0.798511i	$0.294379\pi$
$954$	−3.58007e7	−1.27356
$955$	0	0
$956$	−3.11553e7	−1.10252
$957$	967793.	0.0341588
$958$	−6.61655e6	−0.232926
$959$	−5.18931e7	−1.82206
$960$	0	0
$961$	1.67737e7	0.585897
$962$	−8.65774e6	−0.301625
$963$	3.57588e6	0.124256
$964$	−3.00184e7	−1.04039
$965$	0	0
$966$	9.34702e6	0.322278
$967$	−2.32723e6	−0.0800336	−0.0400168	−	0.999199i	$-0.512741\pi$
$967$	−2.32723e6	−0.0800336	−0.0400168	+	0.999199i	$0.512741\pi$
$968$	−5.00748e6	−0.171764
$969$	−4.75725e6	−0.162760
$970$	0	0
$971$	2.32618e7	0.791762	0.395881	−	0.918302i	$-0.370439\pi$
$971$	2.32618e7	0.791762	0.395881	+	0.918302i	$0.370439\pi$
$972$	2.66563e7	0.904970
$973$	5.14597e7	1.74255
$974$	−5.05172e7	−1.70625
$975$	0	0
$976$	−2.14155e7	−0.719619
$977$	1.85849e7	0.622908	0.311454	−	0.950261i	$-0.399184\pi$
$977$	1.85849e7	0.622908	0.311454	+	0.950261i	$0.399184\pi$
$978$	1.16928e7	0.390906
$979$	−3.10548e6	−0.103555
$980$	0	0
$981$	9.39232e6	0.311602
$982$	−1.94802e6	−0.0644636
$983$	1.42083e6	0.0468986	0.0234493	−	0.999725i	$-0.492535\pi$
$983$	1.42083e6	0.0468986	0.0234493	+	0.999725i	$0.492535\pi$
$984$	−2.41403e6	−0.0794794
$985$	0	0
$986$	4.52112e6	0.148100
$987$	1.43001e7	0.467246
$988$	−5.81763e6	−0.189607
$989$	7.81656e6	0.254112
$990$	0	0
$991$	−2.86425e6	−0.0926462	−0.0463231	−	0.998927i	$-0.514750\pi$
$991$	−2.86425e6	−0.0926462	−0.0463231	+	0.998927i	$0.514750\pi$
$992$	5.60120e7	1.80718
$993$	1.60167e7	0.515465
$994$	5.25223e7	1.68608
$995$	0	0
$996$	4.27363e7	1.36505
$997$	4.95795e7	1.57966	0.789831	−	0.613325i	$-0.210168\pi$
$997$	4.95795e7	1.57966	0.789831	+	0.613325i	$0.210168\pi$
$998$	−1.49398e7	−0.474809
$999$	2.53687e7	0.804238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.h.1.9	✓	9
5.2	odd	4		325.6.b.h.274.17		18
5.3	odd	4		325.6.b.h.274.2		18
5.4	even	2		325.6.a.i.1.1	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.9	✓	9	1.1	even	1	trivial
325.6.a.i.1.1	yes	9	5.4	even	2
325.6.b.h.274.2		18	5.3	odd	4
325.6.b.h.274.17		18	5.2	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)

\(f(q)\)	\(=\)	\(q+8.23858 q^{2} +11.6375 q^{3} +35.8743 q^{4} +95.8765 q^{6} -195.071 q^{7} +31.9185 q^{8} -107.569 q^{9} +O(q^{10})\)	\(q+8.23858 q^{2} +11.6375 q^{3} +35.8743 q^{4} +95.8765 q^{6} -195.071 q^{7} +31.9185 q^{8} -107.569 q^{9} +64.5579 q^{11} +417.487 q^{12} +169.000 q^{13} -1607.11 q^{14} -885.013 q^{16} +426.010 q^{17} -886.215 q^{18} -959.570 q^{19} -2270.14 q^{21} +531.866 q^{22} -499.768 q^{23} +371.452 q^{24} +1392.32 q^{26} -4079.74 q^{27} -6998.03 q^{28} +1288.17 q^{29} -6738.17 q^{31} -8312.65 q^{32} +751.292 q^{33} +3509.72 q^{34} -3858.95 q^{36} -6218.21 q^{37} -7905.50 q^{38} +1966.74 q^{39} -6498.90 q^{41} -18702.7 q^{42} -15640.4 q^{43} +2315.97 q^{44} -4117.38 q^{46} -6299.21 q^{47} -10299.3 q^{48} +21245.7 q^{49} +4957.69 q^{51} +6062.75 q^{52} +40397.3 q^{53} -33611.3 q^{54} -6226.38 q^{56} -11167.0 q^{57} +10612.7 q^{58} +25614.1 q^{59} +24197.9 q^{61} -55513.0 q^{62} +20983.6 q^{63} -40164.0 q^{64} +6189.58 q^{66} -39183.3 q^{67} +15282.8 q^{68} -5816.04 q^{69} -32681.2 q^{71} -3433.44 q^{72} +14507.0 q^{73} -51229.3 q^{74} -34423.9 q^{76} -12593.4 q^{77} +16203.1 q^{78} +79037.9 q^{79} -21338.7 q^{81} -53541.7 q^{82} +102366. q^{83} -81439.5 q^{84} -128855. q^{86} +14991.1 q^{87} +2060.59 q^{88} -48103.8 q^{89} -32967.0 q^{91} -17928.8 q^{92} -78415.4 q^{93} -51896.6 q^{94} -96738.4 q^{96} +73361.2 q^{97} +175035. q^{98} -6944.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21} + 4345 q^{22} + 1839 q^{23} - 8469 q^{24} - 845 q^{26} - 7649 q^{27} - 2836 q^{28} - 8737 q^{29} + 748 q^{31} - 423 q^{32} - 356 q^{33} - 17789 q^{34} + 512 q^{36} - 15486 q^{37} - 3425 q^{38} - 1859 q^{39} - 28676 q^{41} + 6876 q^{42} + 28665 q^{43} - 30599 q^{44} - 12056 q^{46} - 29452 q^{47} + 64759 q^{48} - 40907 q^{49} - 31006 q^{51} + 15379 q^{52} + 75977 q^{53} - 102761 q^{54} - 23002 q^{56} - 38038 q^{57} + 142384 q^{58} - 88142 q^{59} + 28165 q^{61} - 137308 q^{62} + 41492 q^{63} - 100845 q^{64} + 42577 q^{66} - 94754 q^{67} + 89267 q^{68} - 181747 q^{69} - 70562 q^{71} - 263778 q^{72} + 60602 q^{73} - 135676 q^{74} + 46373 q^{76} - 140292 q^{77} - 14027 q^{78} - 164073 q^{79} - 69935 q^{81} - 72887 q^{82} - 22458 q^{83} - 345656 q^{84} - 294920 q^{86} - 87031 q^{87} + 430607 q^{88} - 252698 q^{89} + 2028 q^{91} - 237824 q^{92} + 56556 q^{93} - 501606 q^{94} - 319181 q^{96} + 137986 q^{97} + 378699 q^{98} - 757776 q^{99}+O(q^{100}) \) 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

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