LMFDB - Embedded newform 289.4.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(1,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("289.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.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:	$17.0515519917$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156 $ x^12 - 4x^11 - 58x^10 + 204x^9 + 1191x^8 - 3456x^7 - 10364x^6 + 21448x^5 + 38476x^4 - 32336x^3 - 57024x^2 - 15776*x + 1156
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.83828$ of defining polynomial
Character	$\chi$	$=$	289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.68604 q^{2} -4.33137 q^{3} -0.785167 q^{4} +2.08666 q^{5} -11.6343 q^{6} +24.9985 q^{7} -23.5973 q^{8} -8.23920 q^{9} +O(q^{10})$	\(q+2.68604 q^{2} -4.33137 q^{3} -0.785167 q^{4} +2.08666 q^{5} -11.6343 q^{6} +24.9985 q^{7} -23.5973 q^{8} -8.23920 q^{9} +5.60485 q^{10} -3.82158 q^{11} +3.40085 q^{12} +17.6726 q^{13} +67.1470 q^{14} -9.03809 q^{15} -57.1022 q^{16} -22.1309 q^{18} -160.915 q^{19} -1.63837 q^{20} -108.278 q^{21} -10.2649 q^{22} -99.9676 q^{23} +102.209 q^{24} -120.646 q^{25} +47.4693 q^{26} +152.634 q^{27} -19.6280 q^{28} -200.583 q^{29} -24.2767 q^{30} +76.5382 q^{31} +35.3998 q^{32} +16.5527 q^{33} +52.1632 q^{35} +6.46915 q^{36} -244.759 q^{37} -432.224 q^{38} -76.5465 q^{39} -49.2395 q^{40} -54.1132 q^{41} -290.839 q^{42} +142.087 q^{43} +3.00058 q^{44} -17.1924 q^{45} -268.517 q^{46} -468.451 q^{47} +247.331 q^{48} +281.924 q^{49} -324.060 q^{50} -13.8759 q^{52} -96.5673 q^{53} +409.982 q^{54} -7.97432 q^{55} -589.898 q^{56} +696.981 q^{57} -538.776 q^{58} +364.484 q^{59} +7.09641 q^{60} +707.872 q^{61} +205.585 q^{62} -205.967 q^{63} +551.903 q^{64} +36.8766 q^{65} +44.4612 q^{66} -304.454 q^{67} +432.997 q^{69} +140.113 q^{70} -470.003 q^{71} +194.423 q^{72} -142.056 q^{73} -657.432 q^{74} +522.562 q^{75} +126.345 q^{76} -95.5336 q^{77} -205.607 q^{78} +717.865 q^{79} -119.153 q^{80} -438.657 q^{81} -145.350 q^{82} -367.639 q^{83} +85.0162 q^{84} +381.653 q^{86} +868.802 q^{87} +90.1791 q^{88} +1042.28 q^{89} -46.1795 q^{90} +441.788 q^{91} +78.4913 q^{92} -331.516 q^{93} -1258.28 q^{94} -335.773 q^{95} -153.330 q^{96} +903.534 q^{97} +757.260 q^{98} +31.4867 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10}) $ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9	$ 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100}) $ 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * q^98

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 $

$2$	2.68604	0.949660	0.474830	−	0.880078i	$-0.342510\pi$
$2$	2.68604	0.949660	0.474830	+	0.880078i	$0.342510\pi$
$3$	−4.33137	−0.833573	−0.416787	−	0.909004i	$-0.636844\pi$
$3$	−4.33137	−0.833573	−0.416787	+	0.909004i	$0.636844\pi$
$4$	−0.785167	−0.0981459
$5$	2.08666	0.186636	0.0933181	−	0.995636i	$-0.470253\pi$
$5$	2.08666	0.186636	0.0933181	+	0.995636i	$0.470253\pi$
$6$	−11.6343	−0.791611
$7$	24.9985	1.34979	0.674895	−	0.737913i	$-0.264189\pi$
$7$	24.9985	1.34979	0.674895	+	0.737913i	$0.264189\pi$
$8$	−23.5973	−1.04287
$9$	−8.23920	−0.305156
$10$	5.60485	0.177241
$11$	−3.82158	−0.104750	−0.0523749	−	0.998627i	$-0.516679\pi$
$11$	−3.82158	−0.104750	−0.0523749	+	0.998627i	$0.516679\pi$
$12$	3.40085	0.0818118
$13$	17.6726	0.377038	0.188519	−	0.982070i	$-0.439631\pi$
$13$	17.6726	0.377038	0.188519	+	0.982070i	$0.439631\pi$
$14$	67.1470	1.28184
$15$	−9.03809	−0.155575
$16$	−57.1022	−0.892221
$17$	0	0
$17$	0	0
$18$	−22.1309	−0.289794
$19$	−160.915	−1.94297	−0.971483	−	0.237111i	$-0.923799\pi$
$19$	−160.915	−1.94297	−0.971483	+	0.237111i	$0.923799\pi$
$20$	−1.63837	−0.0183176
$21$	−108.278	−1.12515
$22$	−10.2649	−0.0994768
$23$	−99.9676	−0.906291	−0.453145	−	0.891437i	$-0.649698\pi$
$23$	−99.9676	−0.906291	−0.453145	+	0.891437i	$0.649698\pi$
$24$	102.209	0.869305
$25$	−120.646	−0.965167
$26$	47.4693	0.358058
$27$	152.634	1.08794
$28$	−19.6280	−0.132476
$29$	−200.583	−1.28439	−0.642197	−	0.766540i	$-0.721977\pi$
$29$	−200.583	−1.28439	−0.642197	+	0.766540i	$0.721977\pi$
$30$	−24.2767	−0.147743
$31$	76.5382	0.443441	0.221720	−	0.975110i	$-0.428833\pi$
$31$	76.5382	0.443441	0.221720	+	0.975110i	$0.428833\pi$
$32$	35.3998	0.195558
$33$	16.5527	0.0873167
$34$	0	0
$35$	52.1632	0.251920
$36$	6.46915	0.0299498
$37$	−244.759	−1.08752	−0.543758	−	0.839242i	$-0.682999\pi$
$37$	−244.759	−1.08752	−0.543758	+	0.839242i	$0.682999\pi$
$38$	−432.224	−1.84516
$39$	−76.5465	−0.314289
$40$	−49.2395	−0.194636
$41$	−54.1132	−0.206123	−0.103062	−	0.994675i	$-0.532864\pi$
$41$	−54.1132	−0.206123	−0.103062	+	0.994675i	$0.532864\pi$
$42$	−290.839	−1.06851
$43$	142.087	0.503909	0.251955	−	0.967739i	$-0.418927\pi$
$43$	142.087	0.503909	0.251955	+	0.967739i	$0.418927\pi$
$44$	3.00058	0.0102808
$45$	−17.1924	−0.0569531
$46$	−268.517	−0.860668
$47$	−468.451	−1.45384	−0.726921	−	0.686721i	$-0.759049\pi$
$47$	−468.451	−1.45384	−0.726921	+	0.686721i	$0.759049\pi$
$48$	247.331	0.743732
$49$	281.924	0.821935
$50$	−324.060	−0.916580
$51$	0	0
$52$	−13.8759	−0.0370047
$53$	−96.5673	−0.250274	−0.125137	−	0.992139i	$-0.539937\pi$
$53$	−96.5673	−0.250274	−0.125137	+	0.992139i	$0.539937\pi$
$54$	409.982	1.03318
$55$	−7.97432	−0.0195501
$56$	−589.898	−1.40765
$57$	696.981	1.61960
$58$	−538.776	−1.21974
$59$	364.484	0.804268	0.402134	−	0.915581i	$-0.368269\pi$
$59$	364.484	0.804268	0.402134	+	0.915581i	$0.368269\pi$
$60$	7.09641	0.0152690
$61$	707.872	1.48580	0.742899	−	0.669404i	$-0.233450\pi$
$61$	707.872	1.48580	0.742899	+	0.669404i	$0.233450\pi$
$62$	205.585	0.421118
$63$	−205.967	−0.411896
$64$	551.903	1.07794
$65$	36.8766	0.0703689
$66$	44.4612	0.0829212
$67$	−304.454	−0.555150	−0.277575	−	0.960704i	$-0.589531\pi$
$67$	−304.454	−0.555150	−0.277575	+	0.960704i	$0.589531\pi$
$68$	0	0
$69$	432.997	0.755460
$70$	140.113	0.239238
$71$	−470.003	−0.785621	−0.392810	−	0.919619i	$-0.628497\pi$
$71$	−470.003	−0.785621	−0.392810	+	0.919619i	$0.628497\pi$
$72$	194.423	0.318236
$73$	−142.056	−0.227759	−0.113880	−	0.993495i	$-0.536328\pi$
$73$	−142.056	−0.227759	−0.113880	+	0.993495i	$0.536328\pi$
$74$	−657.432	−1.03277
$75$	522.562	0.804537
$76$	126.345	0.190694
$77$	−95.5336	−0.141390
$78$	−205.607	−0.298467
$79$	717.865	1.02236	0.511178	−	0.859475i	$-0.329209\pi$
$79$	717.865	1.02236	0.511178	+	0.859475i	$0.329209\pi$
$80$	−119.153	−0.166521
$81$	−438.657	−0.601725
$82$	−145.350	−0.195747
$83$	−367.639	−0.486188	−0.243094	−	0.970003i	$-0.578162\pi$
$83$	−367.639	−0.486188	−0.243094	+	0.970003i	$0.578162\pi$
$84$	85.0162	0.110429
$85$	0	0
$86$	381.653	0.478542
$87$	868.802	1.07064
$88$	90.1791	0.109240
$89$	1042.28	1.24137	0.620684	−	0.784061i	$-0.286855\pi$
$89$	1042.28	1.24137	0.620684	+	0.784061i	$0.286855\pi$
$90$	−46.1795	−0.0540861
$91$	441.788	0.508922
$92$	78.4913	0.0889488
$93$	−331.516	−0.369640
$94$	−1258.28	−1.38066
$95$	−335.773	−0.362628
$96$	−153.330	−0.163012
$97$	903.534	0.945773	0.472887	−	0.881123i	$-0.343212\pi$
$97$	903.534	0.945773	0.472887	+	0.881123i	$0.343212\pi$
$98$	757.260	0.780559
$99$	31.4867	0.0319650
$100$	94.7272	0.0947272
$101$	443.233	0.436667	0.218333	−	0.975874i	$-0.429938\pi$
$101$	443.233	0.436667	0.218333	+	0.975874i	$0.429938\pi$
$102$	0	0
$103$	−1396.36	−1.33580	−0.667901	−	0.744250i	$-0.732807\pi$
$103$	−1396.36	−1.33580	−0.667901	+	0.744250i	$0.732807\pi$
$104$	−417.026	−0.393200
$105$	−225.938	−0.209994
$106$	−259.384	−0.237676
$107$	1843.89	1.66594	0.832972	−	0.553315i	$-0.186637\pi$
$107$	1843.89	1.66594	0.832972	+	0.553315i	$0.186637\pi$
$108$	−119.843	−0.106777
$109$	372.960	0.327735	0.163867	−	0.986482i	$-0.447603\pi$
$109$	372.960	0.327735	0.163867	+	0.986482i	$0.447603\pi$
$110$	−21.4194	−0.0185660
$111$	1060.14	0.906524
$112$	−1427.47	−1.20431
$113$	−1779.13	−1.48112	−0.740560	−	0.671990i	$-0.765440\pi$
$113$	−1779.13	−1.48112	−0.740560	+	0.671990i	$0.765440\pi$
$114$	1872.12	1.53807
$115$	−208.598	−0.169147
$116$	157.492	0.126058
$117$	−145.608	−0.115055
$118$	979.020	0.763781
$119$	0	0
$120$	213.275	0.162244
$121$	−1316.40	−0.989027
$122$	1901.37	1.41100
$123$	234.384	0.171819
$124$	−60.0953	−0.0435219
$125$	−512.578	−0.366771
$126$	−553.238	−0.391161
$127$	1735.00	1.21225	0.606126	−	0.795368i	$-0.292723\pi$
$127$	1735.00	1.21225	0.606126	+	0.795368i	$0.292723\pi$
$128$	1199.24	0.828114
$129$	−615.433	−0.420045
$130$	99.0522	0.0668265
$131$	−565.413	−0.377102	−0.188551	−	0.982063i	$-0.560379\pi$
$131$	−565.413	−0.377102	−0.188551	+	0.982063i	$0.560379\pi$
$132$	−12.9966	−0.00856978
$133$	−4022.62	−2.62260
$134$	−817.778	−0.527203
$135$	318.495	0.203050
$136$	0	0
$137$	−1150.24	−0.717314	−0.358657	−	0.933469i	$-0.616765\pi$
$137$	−1150.24	−0.717314	−0.358657	+	0.933469i	$0.616765\pi$
$138$	1163.05	0.717430
$139$	−984.663	−0.600849	−0.300425	−	0.953806i	$-0.597128\pi$
$139$	−984.663	−0.600849	−0.300425	+	0.953806i	$0.597128\pi$
$140$	−40.9569	−0.0247249
$141$	2029.04	1.21188
$142$	−1262.45	−0.746073
$143$	−67.5371	−0.0394947
$144$	470.476	0.272266
$145$	−418.549	−0.239714
$146$	−381.569	−0.216294
$147$	−1221.12	−0.685143
$148$	192.176	0.106735
$149$	1772.18	0.974382	0.487191	−	0.873295i	$-0.338021\pi$
$149$	1772.18	0.974382	0.487191	+	0.873295i	$0.338021\pi$
$150$	1403.63	0.764037
$151$	−1892.09	−1.01971	−0.509855	−	0.860260i	$-0.670301\pi$
$151$	−1892.09	−1.01971	−0.509855	+	0.860260i	$0.670301\pi$
$152$	3797.16	2.02625
$153$	0	0
$154$	−256.607	−0.134273
$155$	159.709	0.0827621
$156$	60.1019	0.0308462
$157$	289.955	0.147395	0.0736973	−	0.997281i	$-0.476520\pi$
$157$	289.955	0.147395	0.0736973	+	0.997281i	$0.476520\pi$
$158$	1928.22	0.970891
$159$	418.269	0.208622
$160$	73.8673	0.0364982
$161$	−2499.04	−1.22330
$162$	−1178.25	−0.571434
$163$	−1691.93	−0.813022	−0.406511	−	0.913646i	$-0.633255\pi$
$163$	−1691.93	−0.813022	−0.406511	+	0.913646i	$0.633255\pi$
$164$	42.4879	0.0202302
$165$	34.5397	0.0162965
$166$	−987.495	−0.461714
$167$	−1255.82	−0.581907	−0.290953	−	0.956737i	$-0.593972\pi$
$167$	−1255.82	−0.581907	−0.290953	+	0.956737i	$0.593972\pi$
$168$	2555.07	1.17338
$169$	−1884.68	−0.857842
$170$	0	0
$171$	1325.81	0.592907
$172$	−111.562	−0.0494566
$173$	401.192	0.176313	0.0881564	−	0.996107i	$-0.471902\pi$
$173$	401.192	0.176313	0.0881564	+	0.996107i	$0.471902\pi$
$174$	2333.64	1.01674
$175$	−3015.96	−1.30277
$176$	218.220	0.0934601
$177$	−1578.72	−0.670416
$178$	2799.62	1.17888
$179$	−769.231	−0.321201	−0.160601	−	0.987019i	$-0.551343\pi$
$179$	−769.231	−0.321201	−0.160601	+	0.987019i	$0.551343\pi$
$180$	13.4989	0.00558971
$181$	2627.89	1.07917	0.539585	−	0.841931i	$-0.318581\pi$
$181$	2627.89	1.07917	0.539585	+	0.841931i	$0.318581\pi$
$182$	1186.66	0.483303
$183$	−3066.06	−1.23852
$184$	2358.97	0.945139
$185$	−510.727	−0.202970
$186$	−890.465	−0.351033
$187$	0	0
$188$	367.812	0.142689
$189$	3815.62	1.46850
$190$	−901.902	−0.344373
$191$	207.856	0.0787430	0.0393715	−	0.999225i	$-0.487464\pi$
$191$	207.856	0.0787430	0.0393715	+	0.999225i	$0.487464\pi$
$192$	−2390.50	−0.898538
$193$	4577.63	1.70728	0.853640	−	0.520864i	$-0.174390\pi$
$193$	4577.63	1.70728	0.853640	+	0.520864i	$0.174390\pi$
$194$	2426.93	0.898163
$195$	−159.726	−0.0586576
$196$	−221.357	−0.0806696
$197$	2640.99	0.955140	0.477570	−	0.878594i	$-0.341518\pi$
$197$	2640.99	0.955140	0.477570	+	0.878594i	$0.341518\pi$
$198$	84.5748	0.0303559
$199$	1650.96	0.588106	0.294053	−	0.955789i	$-0.404996\pi$
$199$	1650.96	0.588106	0.294053	+	0.955789i	$0.404996\pi$
$200$	2846.92	1.00654
$201$	1318.71	0.462758
$202$	1190.54	0.414685
$203$	−5014.28	−1.73366
$204$	0	0
$205$	−112.916	−0.0384701
$206$	−3750.69	−1.26856
$207$	823.653	0.276560
$208$	−1009.14	−0.336401
$209$	614.947	0.203525
$210$	−606.881	−0.199423
$211$	−1104.55	−0.360381	−0.180191	−	0.983632i	$-0.557672\pi$
$211$	−1104.55	−0.360381	−0.180191	+	0.983632i	$0.557672\pi$
$212$	75.8215	0.0245634
$213$	2035.76	0.654873
$214$	4952.78	1.58208
$215$	296.487	0.0940477
$216$	−3601.76	−1.13458
$217$	1913.34	0.598552
$218$	1001.79	0.311237
$219$	615.299	0.189854
$220$	6.26117	0.00191876
$221$	0	0
$222$	2847.59	0.860889
$223$	2164.79	0.650067	0.325033	−	0.945703i	$-0.394624\pi$
$223$	2164.79	0.650067	0.325033	+	0.945703i	$0.394624\pi$
$224$	884.942	0.263963
$225$	994.025	0.294526
$226$	−4778.83	−1.40656
$227$	−2605.70	−0.761879	−0.380940	−	0.924600i	$-0.624399\pi$
$227$	−2605.70	−0.761879	−0.380940	+	0.924600i	$0.624399\pi$
$228$	−547.247	−0.158958
$229$	−1128.94	−0.325774	−0.162887	−	0.986645i	$-0.552081\pi$
$229$	−1128.94	−0.325774	−0.162887	+	0.986645i	$0.552081\pi$
$230$	−560.304	−0.160632
$231$	413.792	0.117859
$232$	4733.24	1.33945
$233$	−4315.97	−1.21351	−0.606756	−	0.794888i	$-0.707530\pi$
$233$	−4315.97	−1.21351	−0.606756	+	0.794888i	$0.707530\pi$
$234$	−391.109	−0.109263
$235$	−977.496	−0.271340
$236$	−286.181	−0.0789356
$237$	−3109.34	−0.852209
$238$	0	0
$239$	788.197	0.213323	0.106662	−	0.994295i	$-0.465984\pi$
$239$	788.197	0.213323	0.106662	+	0.994295i	$0.465984\pi$
$240$	516.094	0.138807
$241$	−3622.86	−0.968335	−0.484167	−	0.874975i	$-0.660877\pi$
$241$	−3622.86	−0.968335	−0.484167	+	0.874975i	$0.660877\pi$
$242$	−3535.90	−0.939240
$243$	−2221.13	−0.586361
$244$	−555.798	−0.145825
$245$	588.278	0.153403
$246$	629.567	0.163170
$247$	−2843.78	−0.732571
$248$	−1806.10	−0.462449
$249$	1592.38	0.405274
$250$	−1376.81	−0.348308
$251$	−5974.99	−1.50254	−0.751271	−	0.659994i	$-0.770559\pi$
$251$	−5974.99	−1.50254	−0.751271	+	0.659994i	$0.770559\pi$
$252$	161.719	0.0404259
$253$	382.034	0.0949339
$254$	4660.28	1.15123
$255$	0	0
$256$	−1194.02	−0.291509
$257$	−3750.91	−0.910411	−0.455205	−	0.890386i	$-0.650434\pi$
$257$	−3750.91	−0.910411	−0.455205	+	0.890386i	$0.650434\pi$
$258$	−1653.08	−0.398900
$259$	−6118.59	−1.46792
$260$	−28.9543	−0.00690642
$261$	1652.65	0.391940
$262$	−1518.73	−0.358119
$263$	−7566.43	−1.77401	−0.887007	−	0.461755i	$-0.847220\pi$
$263$	−7566.43	−1.77401	−0.887007	+	0.461755i	$0.847220\pi$
$264$	−390.599	−0.0910596
$265$	−201.503	−0.0467103
$266$	−10804.9	−2.49057
$267$	−4514.52	−1.03477
$268$	239.048	0.0544857
$269$	6990.63	1.58449	0.792243	−	0.610206i	$-0.208913\pi$
$269$	6990.63	1.58449	0.792243	+	0.610206i	$0.208913\pi$
$270$	855.492	0.192828
$271$	−1356.64	−0.304097	−0.152049	−	0.988373i	$-0.548587\pi$
$271$	−1356.64	−0.304097	−0.152049	+	0.988373i	$0.548587\pi$
$272$	0	0
$273$	−1913.55	−0.424224
$274$	−3089.61	−0.681204
$275$	461.057	0.101101
$276$	−339.975	−0.0741453
$277$	7174.85	1.55630	0.778150	−	0.628078i	$-0.216158\pi$
$277$	7174.85	1.55630	0.778150	+	0.628078i	$0.216158\pi$
$278$	−2644.85	−0.570603
$279$	−630.613	−0.135318
$280$	−1230.91	−0.262718
$281$	6611.42	1.40357	0.701787	−	0.712387i	$-0.252386\pi$
$281$	6611.42	1.40357	0.701787	+	0.712387i	$0.252386\pi$
$282$	5450.08	1.15088
$283$	2310.88	0.485398	0.242699	−	0.970102i	$-0.421967\pi$
$283$	2310.88	0.485398	0.242699	+	0.970102i	$0.421967\pi$
$284$	369.031	0.0771055
$285$	1454.36	0.302277
$286$	−181.408	−0.0375065
$287$	−1352.75	−0.278223
$288$	−291.666	−0.0596757
$289$	0	0
$290$	−1124.24	−0.227647
$291$	−3913.55	−0.788371
$292$	111.538	0.0223537
$293$	−6445.85	−1.28522	−0.642612	−	0.766192i	$-0.722149\pi$
$293$	−6445.85	−1.28522	−0.642612	+	0.766192i	$0.722149\pi$
$294$	−3279.98	−0.650653
$295$	760.553	0.150105
$296$	5775.65	1.13413
$297$	−583.303	−0.113962
$298$	4760.16	0.925332
$299$	−1766.69	−0.341706
$300$	−410.299	−0.0789621
$301$	3551.96	0.680172
$302$	−5082.24	−0.968378
$303$	−1919.81	−0.363994
$304$	9188.57	1.73356
$305$	1477.08	0.277304
$306$	0	0
$307$	221.421	0.0411634	0.0205817	−	0.999788i	$-0.493448\pi$
$307$	221.421	0.0411634	0.0205817	+	0.999788i	$0.493448\pi$
$308$	75.0099	0.0138769
$309$	6048.17	1.11349
$310$	428.985	0.0785959
$311$	1293.87	0.235912	0.117956	−	0.993019i	$-0.462366\pi$
$311$	1293.87	0.235912	0.117956	+	0.993019i	$0.462366\pi$
$312$	1806.30	0.327761
$313$	−385.448	−0.0696064	−0.0348032	−	0.999394i	$-0.511080\pi$
$313$	−385.448	−0.0696064	−0.0348032	+	0.999394i	$0.511080\pi$
$314$	778.833	0.139975
$315$	−429.783	−0.0768747
$316$	−563.644	−0.100340
$317$	−649.047	−0.114997	−0.0574986	−	0.998346i	$-0.518312\pi$
$317$	−649.047	−0.114997	−0.0574986	+	0.998346i	$0.518312\pi$
$318$	1123.49	0.198120
$319$	766.545	0.134540
$320$	1151.63	0.201182
$321$	−7986.60	−1.38869
$322$	−6712.53	−1.16172
$323$	0	0
$324$	344.419	0.0590568
$325$	−2132.12	−0.363904
$326$	−4544.61	−0.772094
$327$	−1615.43	−0.273191
$328$	1276.93	0.214959
$329$	−11710.6	−1.96238
$330$	92.7753	0.0154761
$331$	1429.71	0.237415	0.118707	−	0.992929i	$-0.462125\pi$
$331$	1429.71	0.237415	0.118707	+	0.992929i	$0.462125\pi$
$332$	288.658	0.0477174
$333$	2016.61	0.331861
$334$	−3373.19	−0.552614
$335$	−635.292	−0.103611
$336$	6182.89	1.00388
$337$	−565.364	−0.0913867	−0.0456934	−	0.998956i	$-0.514550\pi$
$337$	−565.364	−0.0913867	−0.0456934	+	0.998956i	$0.514550\pi$
$338$	−5062.33	−0.814659
$339$	7706.08	1.23462
$340$	0	0
$341$	−292.497	−0.0464504
$342$	3561.18	0.563060
$343$	−1526.81	−0.240350
$344$	−3352.88	−0.525509
$345$	903.516	0.140996
$346$	1077.62	0.167437
$347$	3788.72	0.586135	0.293068	−	0.956092i	$-0.405324\pi$
$347$	3788.72	0.586135	0.293068	+	0.956092i	$0.405324\pi$
$348$	−682.155	−0.105079
$349$	−6387.30	−0.979669	−0.489834	−	0.871816i	$-0.662943\pi$
$349$	−6387.30	−0.979669	−0.489834	+	0.871816i	$0.662943\pi$
$350$	−8101.01	−1.23719
$351$	2697.44	0.410196
$352$	−135.283	−0.0204847
$353$	6291.34	0.948595	0.474298	−	0.880365i	$-0.342702\pi$
$353$	6291.34	0.948595	0.474298	+	0.880365i	$0.342702\pi$
$354$	−4240.50	−0.636667
$355$	−980.735	−0.146625
$356$	−818.367	−0.121835
$357$	0	0
$358$	−2066.19	−0.305032
$359$	−3369.51	−0.495364	−0.247682	−	0.968841i	$-0.579669\pi$
$359$	−3369.51	−0.495364	−0.247682	+	0.968841i	$0.579669\pi$
$360$	405.694	0.0593944
$361$	19034.5	2.77511
$362$	7058.63	1.02484
$363$	5701.80	0.824427
$364$	−346.877	−0.0499486
$365$	−296.423	−0.0425081
$366$	−8235.56	−1.17617
$367$	997.517	0.141880	0.0709400	−	0.997481i	$-0.477400\pi$
$367$	997.517	0.141880	0.0709400	+	0.997481i	$0.477400\pi$
$368$	5708.37	0.808612
$369$	445.849	0.0628997
$370$	−1371.84	−0.192752
$371$	−2414.04	−0.337818
$372$	260.295	0.0362787
$373$	3180.59	0.441514	0.220757	−	0.975329i	$-0.429147\pi$
$373$	3180.59	0.441514	0.220757	+	0.975329i	$0.429147\pi$
$374$	0	0
$375$	2220.17	0.305731
$376$	11054.2	1.51616
$377$	−3544.83	−0.484265
$378$	10248.9	1.39457
$379$	−10833.9	−1.46834	−0.734170	−	0.678965i	$-0.762429\pi$
$379$	−10833.9	−1.46834	−0.734170	+	0.678965i	$0.762429\pi$
$380$	263.638	0.0355904
$381$	−7514.92	−1.01050
$382$	558.310	0.0747791
$383$	−7567.93	−1.00967	−0.504834	−	0.863216i	$-0.668446\pi$
$383$	−7567.93	−1.00967	−0.504834	+	0.863216i	$0.668446\pi$
$384$	−5194.34	−0.690294
$385$	−199.346	−0.0263886
$386$	12295.7	1.62134
$387$	−1170.69	−0.153771
$388$	−709.426	−0.0928238
$389$	9599.36	1.25117	0.625587	−	0.780154i	$-0.284859\pi$
$389$	9599.36	1.25117	0.625587	+	0.780154i	$0.284859\pi$
$390$	−429.032	−0.0557048
$391$	0	0
$392$	−6652.65	−0.857168
$393$	2449.02	0.314342
$394$	7093.81	0.907058
$395$	1497.94	0.190809
$396$	−24.7224	−0.00313724
$397$	5272.03	0.666487	0.333244	−	0.942841i	$-0.391857\pi$
$397$	5272.03	0.666487	0.333244	+	0.942841i	$0.391857\pi$
$398$	4434.54	0.558501
$399$	17423.5	2.18613
$400$	6889.14	0.861143
$401$	−3600.34	−0.448360	−0.224180	−	0.974548i	$-0.571970\pi$
$401$	−3600.34	−0.448360	−0.224180	+	0.974548i	$0.571970\pi$
$402$	3542.10	0.439463
$403$	1352.63	0.167194
$404$	−348.012	−0.0428571
$405$	−915.327	−0.112304
$406$	−13468.6	−1.64639
$407$	935.364	0.113917
$408$	0	0
$409$	9516.13	1.15047	0.575235	−	0.817988i	$-0.304911\pi$
$409$	9516.13	1.15047	0.575235	+	0.817988i	$0.304911\pi$
$410$	−303.296	−0.0365335
$411$	4982.14	0.597933
$412$	1096.38	0.131104
$413$	9111.55	1.08559
$414$	2212.37	0.262638
$415$	−767.137	−0.0907404
$416$	625.606	0.0737329
$417$	4264.95	0.500852
$418$	1651.78	0.193280
$419$	−6310.52	−0.735774	−0.367887	−	0.929871i	$-0.619919\pi$
$419$	−6310.52	−0.735774	−0.367887	+	0.929871i	$0.619919\pi$
$420$	177.400	0.0206100
$421$	−1544.81	−0.178835	−0.0894177	−	0.995994i	$-0.528501\pi$
$421$	−1544.81	−0.178835	−0.0894177	+	0.995994i	$0.528501\pi$
$422$	−2966.88	−0.342240
$423$	3859.66	0.443648
$424$	2278.73	0.261002
$425$	0	0
$426$	5468.14	0.621906
$427$	17695.7	2.00552
$428$	−1447.77	−0.163506
$429$	292.528	0.0329217
$430$	796.378	0.0893134
$431$	897.334	0.100286	0.0501428	−	0.998742i	$-0.484032\pi$
$431$	897.334	0.100286	0.0501428	+	0.998742i	$0.484032\pi$
$432$	−8715.74	−0.970686
$433$	4896.47	0.543440	0.271720	−	0.962376i	$-0.412408\pi$
$433$	4896.47	0.543440	0.271720	+	0.962376i	$0.412408\pi$
$434$	5139.31	0.568421
$435$	1812.89	0.199819
$436$	−292.836	−0.0321659
$437$	16086.2	1.76089
$438$	1652.72	0.180297
$439$	−1404.01	−0.152642	−0.0763210	−	0.997083i	$-0.524317\pi$
$439$	−1404.01	−0.152642	−0.0763210	+	0.997083i	$0.524317\pi$
$440$	188.173	0.0203881
$441$	−2322.83	−0.250818
$442$	0	0
$443$	−6453.57	−0.692141	−0.346070	−	0.938209i	$-0.612484\pi$
$443$	−6453.57	−0.692141	−0.346070	+	0.938209i	$0.612484\pi$
$444$	−832.388	−0.0889716
$445$	2174.89	0.231684
$446$	5814.71	0.617342
$447$	−7675.99	−0.812219
$448$	13796.7	1.45499
$449$	−4409.11	−0.463427	−0.231714	−	0.972784i	$-0.574433\pi$
$449$	−4409.11	−0.463427	−0.231714	+	0.972784i	$0.574433\pi$
$450$	2670.00	0.279700
$451$	206.798	0.0215914
$452$	1396.92	0.145366
$453$	8195.36	0.850003
$454$	−6999.03	−0.723526
$455$	921.859	0.0949833
$456$	−16446.9	−1.68903
$457$	12571.6	1.28681	0.643406	−	0.765525i	$-0.277521\pi$
$457$	12571.6	1.28681	0.643406	+	0.765525i	$0.277521\pi$
$458$	−3032.37	−0.309374
$459$	0	0
$460$	163.784	0.0166011
$461$	−5115.00	−0.516767	−0.258383	−	0.966042i	$-0.583190\pi$
$461$	−5115.00	−0.516767	−0.258383	+	0.966042i	$0.583190\pi$
$462$	1111.46	0.111926
$463$	5280.96	0.530080	0.265040	−	0.964237i	$-0.414615\pi$
$463$	5280.96	0.530080	0.265040	+	0.964237i	$0.414615\pi$
$464$	11453.8	1.14596
$465$	−691.759	−0.0689883
$466$	−11592.9	−1.15242
$467$	−4430.45	−0.439008	−0.219504	−	0.975612i	$-0.570444\pi$
$467$	−4430.45	−0.439008	−0.219504	+	0.975612i	$0.570444\pi$
$468$	114.327	0.0112922
$469$	−7610.90	−0.749336
$470$	−2625.60	−0.257680
$471$	−1255.91	−0.122864
$472$	−8600.86	−0.838743
$473$	−542.997	−0.0527844
$474$	−8351.83	−0.809308
$475$	19413.7	1.87529
$476$	0	0
$477$	795.637	0.0763726
$478$	2117.13	0.202584
$479$	8697.09	0.829604	0.414802	−	0.909912i	$-0.363851\pi$
$479$	8697.09	0.829604	0.414802	+	0.909912i	$0.363851\pi$
$480$	−319.947	−0.0304240
$481$	−4325.52	−0.410034
$482$	−9731.15	−0.919589
$483$	10824.3	1.01971
$484$	1033.59	0.0970690
$485$	1885.37	0.176516
$486$	−5966.06	−0.556844
$487$	−13031.1	−1.21252	−0.606258	−	0.795268i	$-0.707330\pi$
$487$	−13031.1	−1.21252	−0.606258	+	0.795268i	$0.707330\pi$
$488$	−16703.9	−1.54949
$489$	7328.40	0.677713
$490$	1580.14	0.145681
$491$	−11760.3	−1.08093	−0.540464	−	0.841367i	$-0.681751\pi$
$491$	−11760.3	−1.08093	−0.540464	+	0.841367i	$0.681751\pi$
$492$	−184.031	−0.0168633
$493$	0	0
$494$	−7638.51	−0.695694
$495$	65.7020	0.00596583
$496$	−4370.50	−0.395647
$497$	−11749.4	−1.06042
$498$	4277.21	0.384872
$499$	−10724.6	−0.962118	−0.481059	−	0.876688i	$-0.659748\pi$
$499$	−10724.6	−0.962118	−0.481059	+	0.876688i	$0.659748\pi$
$500$	402.460	0.0359971
$501$	5439.43	0.485062
$502$	−16049.1	−1.42690
$503$	−3904.96	−0.346150	−0.173075	−	0.984909i	$-0.555370\pi$
$503$	−3904.96	−0.346150	−0.173075	+	0.984909i	$0.555370\pi$
$504$	4860.29	0.429552
$505$	924.875	0.0814978
$506$	1026.16	0.0901549
$507$	8163.25	0.715075
$508$	−1362.26	−0.118978
$509$	15132.3	1.31774	0.658870	−	0.752257i	$-0.271035\pi$
$509$	15132.3	1.31774	0.658870	+	0.752257i	$0.271035\pi$
$510$	0	0
$511$	−3551.19	−0.307427
$512$	−12801.1	−1.10495
$513$	−24561.1	−2.11383
$514$	−10075.1	−0.864581
$515$	−2913.73	−0.249309
$516$	483.218	0.0412257
$517$	1790.22	0.152290
$518$	−16434.8	−1.39402
$519$	−1737.71	−0.146970
$520$	−870.190	−0.0733853
$521$	−16534.9	−1.39041	−0.695207	−	0.718810i	$-0.744687\pi$
$521$	−16534.9	−1.39041	−0.695207	+	0.718810i	$0.744687\pi$
$522$	4439.08	0.372210
$523$	−8724.36	−0.729426	−0.364713	−	0.931120i	$-0.618833\pi$
$523$	−8724.36	−0.729426	−0.364713	+	0.931120i	$0.618833\pi$
$524$	443.944	0.0370111
$525$	13063.3	1.08596
$526$	−20323.8	−1.68471
$527$	0	0
$528$	−945.194	−0.0779058
$529$	−2173.47	−0.178637
$530$	−541.245	−0.0443589
$531$	−3003.06	−0.245427
$532$	3158.43	0.257397
$533$	−956.319	−0.0777163
$534$	−12126.2	−0.982681
$535$	3847.57	0.310925
$536$	7184.32	0.578946
$537$	3331.83	0.267745
$538$	18777.2	1.50472
$539$	−1077.39	−0.0860976
$540$	−250.072	−0.0199285
$541$	8246.12	0.655320	0.327660	−	0.944796i	$-0.393740\pi$
$541$	8246.12	0.655320	0.327660	+	0.944796i	$0.393740\pi$
$542$	−3644.01	−0.288789
$543$	−11382.4	−0.899567
$544$	0	0
$545$	778.240	0.0611672
$546$	−5139.87	−0.402868
$547$	23842.2	1.86366	0.931829	−	0.362898i	$-0.118213\pi$
$547$	23842.2	1.86366	0.931829	+	0.362898i	$0.118213\pi$
$548$	903.134	0.0704014
$549$	−5832.30	−0.453399
$550$	1238.42	0.0960117
$551$	32276.8	2.49553
$552$	−10217.6	−0.787843
$553$	17945.5	1.37997
$554$	19272.0	1.47796
$555$	2212.15	0.169190
$556$	773.126	0.0589709
$557$	19341.5	1.47132	0.735661	−	0.677350i	$-0.236872\pi$
$557$	19341.5	1.47132	0.735661	+	0.677350i	$0.236872\pi$
$558$	−1693.86	−0.128506
$559$	2511.05	0.189993
$560$	−2978.63	−0.224768
$561$	0	0
$562$	17758.6	1.33292
$563$	−14210.3	−1.06376	−0.531878	−	0.846821i	$-0.678513\pi$
$563$	−14210.3	−1.06376	−0.531878	+	0.846821i	$0.678513\pi$
$564$	−1593.13	−0.118942
$565$	−3712.44	−0.276431
$566$	6207.12	0.460963
$567$	−10965.8	−0.812202
$568$	11090.8	0.819297
$569$	−9576.15	−0.705542	−0.352771	−	0.935710i	$-0.614761\pi$
$569$	−9576.15	−0.705542	−0.352771	+	0.935710i	$0.614761\pi$
$570$	3906.48	0.287060
$571$	14785.0	1.08359	0.541796	−	0.840510i	$-0.317745\pi$
$571$	14785.0	1.08359	0.541796	+	0.840510i	$0.317745\pi$
$572$	53.0279	0.00387624
$573$	−900.301	−0.0656381
$574$	−3633.54	−0.264218
$575$	12060.7	0.874722
$576$	−4547.24	−0.328938
$577$	−11438.5	−0.825284	−0.412642	−	0.910893i	$-0.635394\pi$
$577$	−11438.5	−0.825284	−0.412642	+	0.910893i	$0.635394\pi$
$578$	0	0
$579$	−19827.4	−1.42314
$580$	328.631	0.0235270
$581$	−9190.42	−0.656253
$582$	−10512.0	−0.748685
$583$	369.039	0.0262162
$584$	3352.15	0.237522
$585$	−303.834	−0.0214735
$586$	−17313.8	−1.22053
$587$	−7892.09	−0.554925	−0.277463	−	0.960736i	$-0.589493\pi$
$587$	−7892.09	−0.554925	−0.277463	+	0.960736i	$0.589493\pi$
$588$	958.782	0.0672440
$589$	−12316.1	−0.861590
$590$	2042.88	0.142549
$591$	−11439.1	−0.796179
$592$	13976.2	0.970304
$593$	7829.44	0.542186	0.271093	−	0.962553i	$-0.412615\pi$
$593$	7829.44	0.542186	0.271093	+	0.962553i	$0.412615\pi$
$594$	−1566.78	−0.108225
$595$	0	0
$596$	−1391.46	−0.0956317
$597$	−7150.91	−0.490230
$598$	−4745.40	−0.324504
$599$	−18939.7	−1.29191	−0.645956	−	0.763375i	$-0.723541\pi$
$599$	−18939.7	−1.29191	−0.645956	+	0.763375i	$0.723541\pi$
$600$	−12331.1	−0.839024
$601$	−17777.4	−1.20658	−0.603290	−	0.797522i	$-0.706144\pi$
$601$	−17777.4	−1.20658	−0.603290	+	0.797522i	$0.706144\pi$
$602$	9540.73	0.645932
$603$	2508.46	0.169407
$604$	1485.61	0.100080
$605$	−2746.86	−0.184588
$606$	−5156.69	−0.345670
$607$	−17713.3	−1.18445	−0.592223	−	0.805774i	$-0.701750\pi$
$607$	−17713.3	−1.18445	−0.592223	+	0.805774i	$0.701750\pi$
$608$	−5696.35	−0.379963
$609$	21718.7	1.44513
$610$	3967.51	0.263344
$611$	−8278.74	−0.548154
$612$	0	0
$613$	25427.3	1.67537	0.837683	−	0.546156i	$-0.183910\pi$
$613$	25427.3	1.67537	0.837683	+	0.546156i	$0.183910\pi$
$614$	594.746	0.0390912
$615$	489.080	0.0320676
$616$	2254.34	0.147451
$617$	−10028.4	−0.654343	−0.327172	−	0.944965i	$-0.606096\pi$
$617$	−10028.4	−0.654343	−0.327172	+	0.944965i	$0.606096\pi$
$618$	16245.6	1.05744
$619$	−4460.94	−0.289661	−0.144831	−	0.989456i	$-0.546264\pi$
$619$	−4460.94	−0.289661	−0.144831	+	0.989456i	$0.546264\pi$
$620$	−125.398	−0.00812276
$621$	−15258.5	−0.985993
$622$	3475.39	0.224036
$623$	26055.5	1.67559
$624$	4370.97	0.280415
$625$	14011.2	0.896714
$626$	−1035.33	−0.0661024
$627$	−2663.57	−0.169653
$628$	−227.664	−0.0144662
$629$	0	0
$630$	−1154.42	−0.0730049
$631$	−1098.49	−0.0693032	−0.0346516	−	0.999399i	$-0.511032\pi$
$631$	−1098.49	−0.0693032	−0.0346516	+	0.999399i	$0.511032\pi$
$632$	−16939.7	−1.06618
$633$	4784.23	0.300404
$634$	−1743.37	−0.109208
$635$	3620.34	0.226250
$636$	−328.411	−0.0204754
$637$	4982.32	0.309901
$638$	2058.97	0.127767
$639$	3872.45	0.239737
$640$	2502.39	0.154556
$641$	−1063.17	−0.0655115	−0.0327557	−	0.999463i	$-0.510428\pi$
$641$	−1063.17	−0.0655115	−0.0327557	+	0.999463i	$0.510428\pi$
$642$	−21452.4	−1.31878
$643$	−6571.26	−0.403025	−0.201513	−	0.979486i	$-0.564586\pi$
$643$	−6571.26	−0.403025	−0.201513	+	0.979486i	$0.564586\pi$
$644$	1962.16	0.120062
$645$	−1284.20	−0.0783957
$646$	0	0
$647$	−14783.1	−0.898273	−0.449137	−	0.893463i	$-0.648268\pi$
$647$	−14783.1	−0.898273	−0.449137	+	0.893463i	$0.648268\pi$
$648$	10351.1	0.627518
$649$	−1392.90	−0.0842469
$650$	−5726.98	−0.345585
$651$	−8287.38	−0.498937
$652$	1328.45	0.0797948
$653$	−25912.0	−1.55285	−0.776427	−	0.630208i	$-0.782970\pi$
$653$	−25912.0	−1.55285	−0.776427	+	0.630208i	$0.782970\pi$
$654$	−4339.12	−0.259439
$655$	−1179.82	−0.0703809
$656$	3089.98	0.183908
$657$	1170.43	0.0695020
$658$	−31455.1	−1.86360
$659$	4302.30	0.254315	0.127158	−	0.991883i	$-0.459415\pi$
$659$	4302.30	0.254315	0.127158	+	0.991883i	$0.459415\pi$
$660$	−27.1195	−0.00159943
$661$	−7467.66	−0.439423	−0.219711	−	0.975565i	$-0.570512\pi$
$661$	−7467.66	−0.439423	−0.219711	+	0.975565i	$0.570512\pi$
$662$	3840.28	0.225463
$663$	0	0
$664$	8675.31	0.507029
$665$	−8393.82	−0.489471
$666$	5416.72	0.315155
$667$	20051.9	1.16403
$668$	986.031	0.0571118
$669$	−9376.50	−0.541878
$670$	−1706.42	−0.0983953
$671$	−2705.19	−0.155637
$672$	−3833.01	−0.220032
$673$	−4236.79	−0.242669	−0.121335	−	0.992612i	$-0.538717\pi$
$673$	−4236.79	−0.242669	−0.121335	+	0.992612i	$0.538717\pi$
$674$	−1518.59	−0.0867863
$675$	−18414.7	−1.05005
$676$	1479.79	0.0841938
$677$	−23130.5	−1.31311	−0.656555	−	0.754278i	$-0.727987\pi$
$677$	−23130.5	−1.31311	−0.656555	+	0.754278i	$0.727987\pi$
$678$	20698.9	1.17247
$679$	22587.0	1.27660
$680$	0	0
$681$	11286.3	0.635082
$682$	−785.659	−0.0441121
$683$	11474.5	0.642839	0.321419	−	0.946937i	$-0.395840\pi$
$683$	11474.5	0.642839	0.321419	+	0.946937i	$0.395840\pi$
$684$	−1040.98	−0.0581914
$685$	−2400.16	−0.133877
$686$	−4101.08	−0.228251
$687$	4889.84	0.271556
$688$	−8113.49	−0.449599
$689$	−1706.59	−0.0943629
$690$	2426.88	0.133898
$691$	−15221.6	−0.837997	−0.418999	−	0.907987i	$-0.637619\pi$
$691$	−15221.6	−0.837997	−0.418999	+	0.907987i	$0.637619\pi$
$692$	−315.003	−0.0173044
$693$	787.120	0.0431461
$694$	10176.7	0.556629
$695$	−2054.65	−0.112140
$696$	−20501.4	−1.11653
$697$	0	0
$698$	−17156.6	−0.930352
$699$	18694.1	1.01155
$700$	2368.04	0.127862
$701$	−23996.1	−1.29289	−0.646447	−	0.762959i	$-0.723746\pi$
$701$	−23996.1	−1.29289	−0.646447	+	0.762959i	$0.723746\pi$
$702$	7245.44	0.389546
$703$	39385.2	2.11300
$704$	−2109.14	−0.112914
$705$	4233.90	0.226181
$706$	16898.8	0.900843
$707$	11080.2	0.589409
$708$	1239.56	0.0657986
$709$	22346.8	1.18371	0.591856	−	0.806044i	$-0.298395\pi$
$709$	22346.8	1.18371	0.591856	+	0.806044i	$0.298395\pi$
$710$	−2634.30	−0.139244
$711$	−5914.63	−0.311978
$712$	−24595.1	−1.29458
$713$	−7651.34	−0.401886
$714$	0	0
$715$	−140.927	−0.00737113
$716$	603.975	0.0315246
$717$	−3413.98	−0.177820
$718$	−9050.64	−0.470427
$719$	−32468.2	−1.68409	−0.842045	−	0.539408i	$-0.818648\pi$
$719$	−32468.2	−1.68409	−0.842045	+	0.539408i	$0.818648\pi$
$720$	981.722	0.0508148
$721$	−34906.9	−1.80305
$722$	51127.5	2.63541
$723$	15691.9	0.807178
$724$	−2063.34	−0.105916
$725$	24199.6	1.23965
$726$	15315.3	0.782925
$727$	−2899.33	−0.147909	−0.0739547	−	0.997262i	$-0.523562\pi$
$727$	−2899.33	−0.147909	−0.0739547	+	0.997262i	$0.523562\pi$
$728$	−10425.0	−0.530737
$729$	21464.3	1.09050
$730$	−796.204	−0.0403683
$731$	0	0
$732$	2407.37	0.121556
$733$	16399.9	0.826390	0.413195	−	0.910643i	$-0.364413\pi$
$733$	16399.9	0.826390	0.413195	+	0.910643i	$0.364413\pi$
$734$	2679.37	0.134738
$735$	−2548.05	−0.127873
$736$	−3538.84	−0.177233
$737$	1163.50	0.0581519
$738$	1197.57	0.0597333
$739$	6470.65	0.322093	0.161047	−	0.986947i	$-0.448513\pi$
$739$	6470.65	0.322093	0.161047	+	0.986947i	$0.448513\pi$
$740$	401.006	0.0199207
$741$	12317.5	0.610652
$742$	−6484.21	−0.320812
$743$	−7520.20	−0.371318	−0.185659	−	0.982614i	$-0.559442\pi$
$743$	−7520.20	−0.371318	−0.185659	+	0.982614i	$0.559442\pi$
$744$	7822.89	0.385485
$745$	3697.94	0.181855
$746$	8543.20	0.419288
$747$	3029.05	0.148363
$748$	0	0
$749$	46094.6	2.24868
$750$	5963.47	0.290340
$751$	10921.9	0.530685	0.265342	−	0.964154i	$-0.414515\pi$
$751$	10921.9	0.530685	0.265342	+	0.964154i	$0.414515\pi$
$752$	26749.6	1.29715
$753$	25879.9	1.25248
$754$	−9521.56	−0.459887
$755$	−3948.14	−0.190315
$756$	−2995.90	−0.144127
$757$	17112.9	0.821636	0.410818	−	0.911717i	$-0.365243\pi$
$757$	17112.9	0.821636	0.410818	+	0.911717i	$0.365243\pi$
$758$	−29100.4	−1.39442
$759$	−1654.73	−0.0791343
$760$	7923.36	0.378172
$761$	−35934.3	−1.71172	−0.855860	−	0.517207i	$-0.826972\pi$
$761$	−35934.3	−1.71172	−0.855860	+	0.517207i	$0.826972\pi$
$762$	−20185.4	−0.959633
$763$	9323.44	0.442374
$764$	−163.202	−0.00772831
$765$	0	0
$766$	−20327.8	−0.958842
$767$	6441.37	0.303239
$768$	5171.75	0.242994
$769$	29085.3	1.36390	0.681952	−	0.731397i	$-0.261131\pi$
$769$	29085.3	1.36390	0.681952	+	0.731397i	$0.261131\pi$
$770$	−535.451	−0.0250602
$771$	16246.6	0.758894
$772$	−3594.21	−0.167563
$773$	−16502.8	−0.767873	−0.383936	−	0.923360i	$-0.625432\pi$
$773$	−16502.8	−0.767873	−0.383936	+	0.923360i	$0.625432\pi$
$774$	−3144.51	−0.146030
$775$	−9234.02	−0.427994
$776$	−21321.0	−0.986314
$777$	26501.9	1.22362
$778$	25784.3	1.18819
$779$	8707.60	0.400491
$780$	125.412	0.00575701
$781$	1796.15	0.0822937
$782$	0	0
$783$	−30615.9	−1.39735
$784$	−16098.5	−0.733348
$785$	605.037	0.0275092
$786$	6578.17	0.298518
$787$	27374.1	1.23987	0.619937	−	0.784651i	$-0.287158\pi$
$787$	27374.1	1.23987	0.619937	+	0.784651i	$0.287158\pi$
$788$	−2073.62	−0.0937431
$789$	32773.0	1.47877
$790$	4023.53	0.181203
$791$	−44475.6	−1.99920
$792$	−743.003	−0.0333352
$793$	12509.9	0.560202
$794$	14160.9	0.632936
$795$	872.784	0.0389364
$796$	−1296.28	−0.0577202
$797$	−31776.1	−1.41226	−0.706128	−	0.708084i	$-0.749560\pi$
$797$	−31776.1	−1.41226	−0.706128	+	0.708084i	$0.749560\pi$
$798$	46800.2	2.07608
$799$	0	0
$800$	−4270.84	−0.188746
$801$	−8587.58	−0.378810
$802$	−9670.68	−0.425790
$803$	542.879	0.0238578
$804$	−1035.41	−0.0454178
$805$	−5214.63	−0.228313
$806$	3633.22	0.158777
$807$	−30279.0	−1.32078
$808$	−10459.1	−0.455385
$809$	−34367.2	−1.49355	−0.746777	−	0.665074i	$-0.768400\pi$
$809$	−34367.2	−1.49355	−0.746777	+	0.665074i	$0.768400\pi$
$810$	−2458.61	−0.106650
$811$	−1316.99	−0.0570232	−0.0285116	−	0.999593i	$-0.509077\pi$
$811$	−1316.99	−0.0570232	−0.0285116	+	0.999593i	$0.509077\pi$
$812$	3937.05	0.170152
$813$	5876.14	0.253487
$814$	2512.43	0.108182
$815$	−3530.49	−0.151739
$816$	0	0
$817$	−22863.9	−0.979078
$818$	25560.7	1.09256
$819$	−3639.98	−0.155300
$820$	88.6577	0.00377568
$821$	33481.1	1.42326	0.711632	−	0.702552i	$-0.247956\pi$
$821$	33481.1	1.42326	0.711632	+	0.702552i	$0.247956\pi$
$822$	13382.2	0.567833
$823$	11746.8	0.497529	0.248765	−	0.968564i	$-0.419975\pi$
$823$	11746.8	0.497529	0.248765	+	0.968564i	$0.419975\pi$
$824$	32950.5	1.39306
$825$	−1997.01	−0.0842752
$826$	24474.0	1.03094
$827$	−4367.84	−0.183658	−0.0918288	−	0.995775i	$-0.529271\pi$
$827$	−4367.84	−0.183658	−0.0918288	+	0.995775i	$0.529271\pi$
$828$	−646.706	−0.0271432
$829$	19994.7	0.837689	0.418844	−	0.908058i	$-0.362435\pi$
$829$	19994.7	0.837689	0.418844	+	0.908058i	$0.362435\pi$
$830$	−2060.56	−0.0861725
$831$	−31077.0	−1.29729
$832$	9753.55	0.406422
$833$	0	0
$834$	11455.8	0.475639
$835$	−2620.47	−0.108605
$836$	−482.837	−0.0199752
$837$	11682.3	0.482438
$838$	−16950.3	−0.698735
$839$	−25672.1	−1.05638	−0.528189	−	0.849127i	$-0.677129\pi$
$839$	−25672.1	−1.05638	−0.528189	+	0.849127i	$0.677129\pi$
$840$	5331.55	0.218995
$841$	15844.7	0.649667
$842$	−4149.44	−0.169833
$843$	−28636.5	−1.16998
$844$	867.258	0.0353700
$845$	−3932.68	−0.160104
$846$	10367.2	0.421315
$847$	−32907.9	−1.33498
$848$	5514.20	0.223300
$849$	−10009.3	−0.404614
$850$	0	0
$851$	24467.9	0.985605
$852$	−1598.41	−0.0642731
$853$	−15504.6	−0.622355	−0.311178	−	0.950352i	$-0.600723\pi$
$853$	−15504.6	−0.622355	−0.311178	+	0.950352i	$0.600723\pi$
$854$	47531.5	1.90456
$855$	2766.50	0.110658
$856$	−43511.0	−1.73736
$857$	−35915.1	−1.43155	−0.715774	−	0.698332i	$-0.753926\pi$
$857$	−35915.1	−1.43155	−0.715774	+	0.698332i	$0.753926\pi$
$858$	785.744	0.0312644
$859$	−36575.0	−1.45276	−0.726382	−	0.687292i	$-0.758799\pi$
$859$	−36575.0	−1.45276	−0.726382	+	0.687292i	$0.758799\pi$
$860$	−232.792	−0.00923040
$861$	5859.25	0.231920
$862$	2410.28	0.0952372
$863$	11402.6	0.449768	0.224884	−	0.974386i	$-0.427800\pi$
$863$	11402.6	0.449768	0.224884	+	0.974386i	$0.427800\pi$
$864$	5403.22	0.212756
$865$	837.151	0.0329063
$866$	13152.1	0.516083
$867$	0	0
$868$	−1502.29	−0.0587455
$869$	−2743.38	−0.107092
$870$	4869.51	0.189761
$871$	−5380.50	−0.209312
$872$	−8800.87	−0.341783
$873$	−7444.40	−0.288608
$874$	43208.4	1.67225
$875$	−12813.7	−0.495065
$876$	−483.113	−0.0186334
$877$	−23424.6	−0.901932	−0.450966	−	0.892541i	$-0.648920\pi$
$877$	−23424.6	−0.901932	−0.450966	+	0.892541i	$0.648920\pi$
$878$	−3771.24	−0.144958
$879$	27919.4	1.07133
$880$	455.351	0.0174430
$881$	2452.52	0.0937882	0.0468941	−	0.998900i	$-0.485068\pi$
$881$	2452.52	0.0937882	0.0468941	+	0.998900i	$0.485068\pi$
$882$	−6239.21	−0.238192
$883$	3832.99	0.146082	0.0730409	−	0.997329i	$-0.476730\pi$
$883$	3832.99	0.146082	0.0730409	+	0.997329i	$0.476730\pi$
$884$	0	0
$885$	−3294.24	−0.125124
$886$	−17334.6	−0.657298
$887$	13334.4	0.504762	0.252381	−	0.967628i	$-0.418786\pi$
$887$	13334.4	0.504762	0.252381	+	0.967628i	$0.418786\pi$
$888$	−25016.5	−0.945382
$889$	43372.3	1.63629
$890$	5841.84	0.220021
$891$	1676.36	0.0630306
$892$	−1699.72	−0.0638014
$893$	75380.6	2.82477
$894$	−20618.1	−0.771332
$895$	−1605.12	−0.0599478
$896$	29979.1	1.11778
$897$	7652.18	0.284837
$898$	−11843.1	−0.440098
$899$	−15352.3	−0.569553
$900$	−780.476	−0.0289065
$901$	0	0
$902$	555.468	0.0205045
$903$	−15384.9	−0.566973
$904$	41982.8	1.54461
$905$	5483.51	0.201412
$906$	22013.1	0.807214
$907$	−14432.7	−0.528368	−0.264184	−	0.964472i	$-0.585103\pi$
$907$	−14432.7	−0.528368	−0.264184	+	0.964472i	$0.585103\pi$
$908$	2045.91	0.0747753
$909$	−3651.89	−0.133251
$910$	2476.15	0.0902018
$911$	50465.5	1.83534	0.917670	−	0.397343i	$-0.130068\pi$
$911$	50465.5	1.83534	0.917670	+	0.397343i	$0.130068\pi$
$912$	−39799.1	−1.44505
$913$	1404.96	0.0509282
$914$	33767.8	1.22203
$915$	−6397.81	−0.231153
$916$	886.404	0.0319734
$917$	−14134.5	−0.509009
$918$	0	0
$919$	−33955.2	−1.21880	−0.609401	−	0.792862i	$-0.708590\pi$
$919$	−33955.2	−1.21880	−0.609401	+	0.792862i	$0.708590\pi$
$920$	4922.36	0.176397
$921$	−959.057	−0.0343127
$922$	−13739.1	−0.490753
$923$	−8306.16	−0.296209
$924$	−324.896	−0.0115674
$925$	29529.1	1.04963
$926$	14184.9	0.503396
$927$	11504.9	0.407628
$928$	−7100.62	−0.251174
$929$	14807.7	0.522954	0.261477	−	0.965210i	$-0.415790\pi$
$929$	14807.7	0.522954	0.261477	+	0.965210i	$0.415790\pi$
$930$	−1858.09	−0.0655154
$931$	−45365.6	−1.59699
$932$	3388.76	0.119101
$933$	−5604.23	−0.196650
$934$	−11900.4	−0.416908
$935$	0	0
$936$	3435.96	0.119987
$937$	−5141.33	−0.179253	−0.0896265	−	0.995975i	$-0.528567\pi$
$937$	−5141.33	−0.179253	−0.0896265	+	0.995975i	$0.528567\pi$
$938$	−20443.2	−0.711614
$939$	1669.52	0.0580220
$940$	767.498	0.0266309
$941$	38958.4	1.34964	0.674819	−	0.737984i	$-0.264222\pi$
$941$	38958.4	1.34964	0.674819	+	0.737984i	$0.264222\pi$
$942$	−3373.42	−0.116679
$943$	5409.57	0.186808
$944$	−20812.8	−0.717585
$945$	7961.89	0.274074
$946$	−1458.51	−0.0501273
$947$	−14191.6	−0.486974	−0.243487	−	0.969904i	$-0.578291\pi$
$947$	−14191.6	−0.486974	−0.243487	+	0.969904i	$0.578291\pi$
$948$	2441.35	0.0836408
$949$	−2510.50	−0.0858739
$950$	52146.0	1.78088
$951$	2811.27	0.0958586
$952$	0	0
$953$	−32646.8	−1.10969	−0.554844	−	0.831955i	$-0.687222\pi$
$953$	−32646.8	−1.10969	−0.554844	+	0.831955i	$0.687222\pi$
$954$	2137.12	0.0725280
$955$	433.724	0.0146963
$956$	−618.867	−0.0209368
$957$	−3320.19	−0.112149
$958$	23360.8	0.787842
$959$	−28754.3	−0.968223
$960$	−4988.15	−0.167700
$961$	−23932.9	−0.803360
$962$	−11618.5	−0.389393
$963$	−15192.2	−0.508372
$964$	2844.55	0.0950381
$965$	9551.94	0.318640
$966$	29074.5	0.968381
$967$	−16183.8	−0.538196	−0.269098	−	0.963113i	$-0.586726\pi$
$967$	−16183.8	−0.538196	−0.269098	+	0.963113i	$0.586726\pi$
$968$	31063.4	1.03142
$969$	0	0
$970$	5064.18	0.167630
$971$	28807.7	0.952095	0.476048	−	0.879420i	$-0.342069\pi$
$971$	28807.7	0.952095	0.476048	+	0.879420i	$0.342069\pi$
$972$	1743.96	0.0575490
$973$	−24615.1	−0.811021
$974$	−35002.1	−1.15148
$975$	9235.02	0.303341
$976$	−40421.0	−1.32566
$977$	3155.66	0.103335	0.0516675	−	0.998664i	$-0.483546\pi$
$977$	3155.66	0.103335	0.0516675	+	0.998664i	$0.483546\pi$
$978$	19684.4	0.643597
$979$	−3983.16	−0.130033
$980$	−461.897	−0.0150559
$981$	−3072.89	−0.100010
$982$	−31588.8	−1.02651
$983$	−14818.7	−0.480816	−0.240408	−	0.970672i	$-0.577281\pi$
$983$	−14818.7	−0.480816	−0.240408	+	0.970672i	$0.577281\pi$
$984$	−5530.85	−0.179184
$985$	5510.83	0.178264
$986$	0	0
$987$	50722.8	1.63579
$988$	2232.84	0.0718989
$989$	−14204.1	−0.456688
$990$	176.478	0.00566551
$991$	888.414	0.0284777	0.0142389	−	0.999899i	$-0.495467\pi$
$991$	888.414	0.0284777	0.0142389	+	0.999899i	$0.495467\pi$
$992$	2709.44	0.0867185
$993$	−6192.63	−0.197902
$994$	−31559.3	−1.00704
$995$	3444.98	0.109762
$996$	−1250.29	−0.0397760
$997$	−5116.97	−0.162544	−0.0812718	−	0.996692i	$-0.525898\pi$
$997$	−5116.97	−0.162544	−0.0812718	+	0.996692i	$0.525898\pi$
$998$	−28806.6	−0.913685
$999$	−37358.5	−1.18315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.a.g.1.9		12
17.4	even	4		289.4.b.e.288.4		12
17.10	odd	16		17.4.d.a.15.1	yes	12
17.12	odd	16		17.4.d.a.8.1	✓	12
17.13	even	4		289.4.b.e.288.3		12
17.16	even	2	inner	289.4.a.g.1.10		12
51.29	even	16		153.4.l.a.127.3		12
51.44	even	16		153.4.l.a.100.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.d.a.8.1	✓	12	17.12	odd	16
17.4.d.a.15.1	yes	12	17.10	odd	16
153.4.l.a.100.3		12	51.44	even	16
153.4.l.a.127.3		12	51.29	even	16
289.4.a.g.1.9		12	1.1	even	1	trivial
289.4.a.g.1.10		12	17.16	even	2	inner
289.4.b.e.288.3		12	17.13	even	4
289.4.b.e.288.4		12	17.4	even	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:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68604 q^{2} -4.33137 q^{3} -0.785167 q^{4} +2.08666 q^{5} -11.6343 q^{6} +24.9985 q^{7} -23.5973 q^{8} -8.23920 q^{9} +O(q^{10})\)	\(q+2.68604 q^{2} -4.33137 q^{3} -0.785167 q^{4} +2.08666 q^{5} -11.6343 q^{6} +24.9985 q^{7} -23.5973 q^{8} -8.23920 q^{9} +5.60485 q^{10} -3.82158 q^{11} +3.40085 q^{12} +17.6726 q^{13} +67.1470 q^{14} -9.03809 q^{15} -57.1022 q^{16} -22.1309 q^{18} -160.915 q^{19} -1.63837 q^{20} -108.278 q^{21} -10.2649 q^{22} -99.9676 q^{23} +102.209 q^{24} -120.646 q^{25} +47.4693 q^{26} +152.634 q^{27} -19.6280 q^{28} -200.583 q^{29} -24.2767 q^{30} +76.5382 q^{31} +35.3998 q^{32} +16.5527 q^{33} +52.1632 q^{35} +6.46915 q^{36} -244.759 q^{37} -432.224 q^{38} -76.5465 q^{39} -49.2395 q^{40} -54.1132 q^{41} -290.839 q^{42} +142.087 q^{43} +3.00058 q^{44} -17.1924 q^{45} -268.517 q^{46} -468.451 q^{47} +247.331 q^{48} +281.924 q^{49} -324.060 q^{50} -13.8759 q^{52} -96.5673 q^{53} +409.982 q^{54} -7.97432 q^{55} -589.898 q^{56} +696.981 q^{57} -538.776 q^{58} +364.484 q^{59} +7.09641 q^{60} +707.872 q^{61} +205.585 q^{62} -205.967 q^{63} +551.903 q^{64} +36.8766 q^{65} +44.4612 q^{66} -304.454 q^{67} +432.997 q^{69} +140.113 q^{70} -470.003 q^{71} +194.423 q^{72} -142.056 q^{73} -657.432 q^{74} +522.562 q^{75} +126.345 q^{76} -95.5336 q^{77} -205.607 q^{78} +717.865 q^{79} -119.153 q^{80} -438.657 q^{81} -145.350 q^{82} -367.639 q^{83} +85.0162 q^{84} +381.653 q^{86} +868.802 q^{87} +90.1791 q^{88} +1042.28 q^{89} -46.1795 q^{90} +441.788 q^{91} +78.4913 q^{92} -331.516 q^{93} -1258.28 q^{94} -335.773 q^{95} -153.330 q^{96} +903.534 q^{97} +757.260 q^{98} +31.4867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10}) \) 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9	\( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100}) \) 12 * q - 8 * q^2 + 16 * q^4 - 96 * q^8 - 36 * q^9 - 8 * q^13 - 192 * q^15 - 184 * q^16 - 352 * q^19 - 256 * q^21 - 492 * q^25 - 784 * q^26 + 744 * q^30 + 24 * q^32 - 1400 * q^33 - 632 * q^35 - 856 * q^36 - 624 * q^38 - 1664 * q^42 - 1200 * q^43 - 1512 * q^47 - 1052 * q^49 - 2856 * q^50 + 792 * q^52 - 2504 * q^53 - 1424 * q^55 - 3408 * q^59 - 2808 * q^60 + 272 * q^64 + 272 * q^66 - 1080 * q^67 - 344 * q^69 + 2600 * q^70 + 248 * q^72 + 896 * q^76 + 848 * q^77 - 2404 * q^81 - 2960 * q^83 + 4768 * q^84 - 1200 * q^86 - 160 * q^87 - 2144 * q^89 + 3800 * q^93 + 5984 * q^94 + 3464 * q^98

Introduction

L-functions

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