LMFDB - Embedded newform 289.4.b.f.288.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(288,289)]

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([1]))

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.8
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25752 q^{2} +6.51757i q^{3} +2.61146 q^{4} -19.1073i q^{5} -21.2311i q^{6} +22.0995i q^{7} +17.5533 q^{8} -15.4787 q^{9} +O(q^{10})$	\(q-3.25752 q^{2} +6.51757i q^{3} +2.61146 q^{4} -19.1073i q^{5} -21.2311i q^{6} +22.0995i q^{7} +17.5533 q^{8} -15.4787 q^{9} +62.2426i q^{10} -8.25972i q^{11} +17.0204i q^{12} -0.398269 q^{13} -71.9898i q^{14} +124.533 q^{15} -78.0720 q^{16} +50.4221 q^{18} -103.371 q^{19} -49.8980i q^{20} -144.035 q^{21} +26.9062i q^{22} +138.311i q^{23} +114.405i q^{24} -240.090 q^{25} +1.29737 q^{26} +75.0910i q^{27} +57.7120i q^{28} -222.296i q^{29} -405.670 q^{30} -49.1520i q^{31} +113.895 q^{32} +53.8333 q^{33} +422.263 q^{35} -40.4219 q^{36} -61.3077i q^{37} +336.734 q^{38} -2.59575i q^{39} -335.397i q^{40} -387.391i q^{41} +469.198 q^{42} +20.3847 q^{43} -21.5699i q^{44} +295.756i q^{45} -450.553i q^{46} +44.1322 q^{47} -508.839i q^{48} -145.389 q^{49} +782.100 q^{50} -1.04006 q^{52} -59.6456 q^{53} -244.611i q^{54} -157.821 q^{55} +387.920i q^{56} -673.728i q^{57} +724.134i q^{58} -238.069 q^{59} +325.214 q^{60} -595.816i q^{61} +160.114i q^{62} -342.071i q^{63} +253.561 q^{64} +7.60987i q^{65} -175.363 q^{66} -408.815 q^{67} -901.454 q^{69} -1375.53 q^{70} -1037.38i q^{71} -271.702 q^{72} -22.3789i q^{73} +199.711i q^{74} -1564.80i q^{75} -269.949 q^{76} +182.536 q^{77} +8.45571i q^{78} -682.688i q^{79} +1491.75i q^{80} -907.335 q^{81} +1261.93i q^{82} +312.352 q^{83} -376.142 q^{84} -66.4037 q^{86} +1448.83 q^{87} -144.985i q^{88} -904.392 q^{89} -963.433i q^{90} -8.80157i q^{91} +361.195i q^{92} +320.351 q^{93} -143.762 q^{94} +1975.15i q^{95} +742.317i q^{96} -1006.72i q^{97} +473.610 q^{98} +127.849i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−3.25752	−1.15171	−0.575854	−	0.817552i	$-0.695330\pi$
$2$	−3.25752	−1.15171	−0.575854	+	0.817552i	$0.695330\pi$
$3$	6.51757i	1.25431i	0.778896	+	0.627153i	$0.215780\pi$
$3$	6.51757i	1.25431i	−0.778896	+	0.627153i	$0.784220\pi$
$4$	2.61146	0.326432
$5$	− 19.1073i	− 1.70901i	−0.519441	−	0.854506i	$-0.673860\pi$
$5$	− 19.1073i	− 1.70901i	0.519441	−	0.854506i	$-0.326140\pi$
$6$	− 21.2311i	− 1.44460i
$7$	22.0995i	1.19326i	0.802515	+	0.596631i	$0.203495\pi$
$7$	22.0995i	1.19326i	−0.802515	+	0.596631i	$0.796505\pi$
$8$	17.5533	0.775753
$9$	−15.4787	−0.573284
$10$	62.2426i	1.96828i
$11$	− 8.25972i	− 0.226400i	−0.993572	−	0.113200i	$-0.963890\pi$
$11$	− 8.25972i	− 0.226400i	0.993572	−	0.113200i	$-0.0361101\pi$
$12$	17.0204i	0.409446i
$13$	−0.398269	−0.00849693	−0.00424846	−	0.999991i	$-0.501352\pi$
$13$	−0.398269	−0.00849693	−0.00424846	+	0.999991i	$0.501352\pi$
$14$	− 71.9898i	− 1.37429i
$15$	124.533	2.14362
$16$	−78.0720	−1.21987
$17$	0	0
$17$	0	0
$18$	50.4221	0.660256
$19$	−103.371	−1.24816	−0.624078	−	0.781362i	$-0.714525\pi$
$19$	−103.371	−1.24816	−0.624078	+	0.781362i	$0.714525\pi$
$20$	− 49.8980i	− 0.557877i
$21$	−144.035	−1.49672
$22$	26.9062i	0.260747i
$23$	138.311i	1.25391i	0.779056	+	0.626955i	$0.215699\pi$
$23$	138.311i	1.25391i	−0.779056	+	0.626955i	$0.784301\pi$
$24$	114.405i	0.973032i
$25$	−240.090	−1.92072
$26$	1.29737	0.00978598
$27$	75.0910i	0.535233i
$28$	57.7120i	0.389520i
$29$	− 222.296i	− 1.42342i	−0.702471	−	0.711712i	$-0.747920\pi$
$29$	− 222.296i	− 1.42342i	0.702471	−	0.711712i	$-0.252080\pi$
$30$	−405.670	−2.46883
$31$	− 49.1520i	− 0.284773i	−0.989811	−	0.142386i	$-0.954522\pi$
$31$	− 49.1520i	− 0.284773i	0.989811	−	0.142386i	$-0.0454776\pi$
$32$	113.895	0.629186
$33$	53.8333	0.283975
$34$	0	0
$35$	422.263	2.03930
$36$	−40.4219	−0.187138
$37$	− 61.3077i	− 0.272403i	−0.990681	−	0.136202i	$-0.956510\pi$
$37$	− 61.3077i	− 0.272403i	0.990681	−	0.136202i	$-0.0434895\pi$
$38$	336.734	1.43751
$39$	− 2.59575i	− 0.0106577i
$40$	− 335.397i	− 1.32577i
$41$	− 387.391i	− 1.47562i	−0.675011	−	0.737808i	$-0.735861\pi$
$41$	− 387.391i	− 1.47562i	0.675011	−	0.737808i	$-0.264139\pi$
$42$	469.198	1.72378
$43$	20.3847	0.0722939	0.0361470	−	0.999346i	$-0.488492\pi$
$43$	20.3847	0.0722939	0.0361470	+	0.999346i	$0.488492\pi$
$44$	− 21.5699i	− 0.0739043i
$45$	295.756i	0.979749i
$46$	− 450.553i	− 1.44414i
$47$	44.1322	0.136965	0.0684824	−	0.997652i	$-0.478184\pi$
$47$	44.1322	0.136965	0.0684824	+	0.997652i	$0.478184\pi$
$48$	− 508.839i	− 1.53010i
$49$	−145.389	−0.423876
$50$	782.100	2.21211
$51$	0	0
$52$	−1.04006	−0.00277367
$53$	−59.6456	−0.154584	−0.0772920	−	0.997009i	$-0.524627\pi$
$53$	−59.6456	−0.154584	−0.0772920	+	0.997009i	$0.524627\pi$
$54$	− 244.611i	− 0.616432i
$55$	−157.821	−0.386920
$56$	387.920i	0.925678i
$57$	− 673.728i	− 1.56557i
$58$	724.134i	1.63937i
$59$	−238.069	−0.525321	−0.262660	−	0.964888i	$-0.584600\pi$
$59$	−238.069	−0.525321	−0.262660	+	0.964888i	$0.584600\pi$
$60$	325.214	0.699749
$61$	− 595.816i	− 1.25060i	−0.780386	−	0.625298i	$-0.784977\pi$
$61$	− 595.816i	− 1.25060i	0.780386	−	0.625298i	$-0.215023\pi$
$62$	160.114i	0.327975i
$63$	− 342.071i	− 0.684078i
$64$	253.561	0.495235
$65$	7.60987i	0.0145214i
$66$	−175.363	−0.327056
$67$	−408.815	−0.745442	−0.372721	−	0.927943i	$-0.621575\pi$
$67$	−408.815	−0.745442	−0.372721	+	0.927943i	$0.621575\pi$
$68$	0	0
$69$	−901.454	−1.57279
$70$	−1375.53	−2.34868
$71$	− 1037.38i	− 1.73400i	−0.498309	−	0.866999i	$-0.666046\pi$
$71$	− 1037.38i	− 1.73400i	0.498309	−	0.866999i	$-0.333954\pi$
$72$	−271.702	−0.444727
$73$	− 22.3789i	− 0.0358802i	−0.999839	−	0.0179401i	$-0.994289\pi$
$73$	− 22.3789i	− 0.0358802i	0.999839	−	0.0179401i	$-0.00571082\pi$
$74$	199.711i	0.313729i
$75$	− 1564.80i	− 2.40917i
$76$	−269.949	−0.407438
$77$	182.536	0.270155
$78$	8.45571i	0.0122746i
$79$	− 682.688i	− 0.972259i	−0.873887	−	0.486129i	$-0.838408\pi$
$79$	− 682.688i	− 0.972259i	0.873887	−	0.486129i	$-0.161592\pi$
$80$	1491.75i	2.08478i
$81$	−907.335	−1.24463
$82$	1261.93i	1.69948i
$83$	312.352	0.413073	0.206536	−	0.978439i	$-0.433781\pi$
$83$	312.352	0.413073	0.206536	+	0.978439i	$0.433781\pi$
$84$	−376.142	−0.488577
$85$	0	0
$86$	−66.4037	−0.0832616
$87$	1448.83	1.78541
$88$	− 144.985i	− 0.175630i
$89$	−904.392	−1.07714	−0.538570	−	0.842581i	$-0.681035\pi$
$89$	−904.392	−1.07714	−0.538570	+	0.842581i	$0.681035\pi$
$90$	− 963.433i	− 1.12839i
$91$	− 8.80157i	− 0.0101391i
$92$	361.195i	0.409317i
$93$	320.351	0.357192
$94$	−143.762	−0.157744
$95$	1975.15i	2.13311i
$96$	742.317i	0.789192i
$97$	− 1006.72i	− 1.05378i	−0.849933	−	0.526891i	$-0.823358\pi$
$97$	− 1006.72i	− 1.05378i	0.849933	−	0.526891i	$-0.176642\pi$
$98$	473.610	0.488182
$99$	127.849i	0.129791i
$100$	−626.986	−0.626986
$101$	−1124.60	−1.10794	−0.553968	−	0.832538i	$-0.686887\pi$
$101$	−1124.60	−1.10794	−0.553968	+	0.832538i	$0.686887\pi$
$102$	0	0
$103$	−396.677	−0.379473	−0.189736	−	0.981835i	$-0.560763\pi$
$103$	−396.677	−0.379473	−0.189736	+	0.981835i	$0.560763\pi$
$104$	−6.99094	−0.00659152
$105$	2752.13i	2.55791i
$106$	194.297	0.178036
$107$	288.906i	0.261024i	0.991447	+	0.130512i	$0.0416621\pi$
$107$	288.906i	0.261024i	−0.991447	+	0.130512i	$0.958338\pi$
$108$	196.097i	0.174717i
$109$	1379.15i	1.21191i	0.795499	+	0.605955i	$0.207209\pi$
$109$	1379.15i	1.21191i	−0.795499	+	0.605955i	$0.792791\pi$
$110$	514.106	0.445619
$111$	399.577	0.341677
$112$	− 1725.35i	− 1.45563i
$113$	− 1364.42i	− 1.13587i	−0.823072	−	0.567937i	$-0.807742\pi$
$113$	− 1364.42i	− 1.13587i	0.823072	−	0.567937i	$-0.192258\pi$
$114$	2194.68i	1.80308i
$115$	2642.76	2.14295
$116$	− 580.517i	− 0.464652i
$117$	6.16468	0.00487115
$118$	775.515	0.605016
$119$	0	0
$120$	2185.97	1.66292
$121$	1262.78	0.948743
$122$	1940.88i	1.44032i
$123$	2524.84	1.85087
$124$	− 128.358i	− 0.0929591i
$125$	2199.07i	1.57353i
$126$	1114.31i	0.787859i
$127$	−1003.33	−0.701032	−0.350516	−	0.936557i	$-0.613994\pi$
$127$	−1003.33	−0.701032	−0.350516	+	0.936557i	$0.613994\pi$
$128$	−1737.14	−1.19955
$129$	132.859i	0.0906787i
$130$	− 24.7893i	− 0.0167244i
$131$	218.014i	0.145404i	0.997354	+	0.0727022i	$0.0231623\pi$
$131$	218.014i	0.145404i	−0.997354	+	0.0727022i	$0.976838\pi$
$132$	140.583	0.0926986
$133$	− 2284.45i	− 1.48938i
$134$	1331.72	0.858532
$135$	1434.79	0.914719
$136$	0	0
$137$	−1625.34	−1.01359	−0.506797	−	0.862066i	$-0.669171\pi$
$137$	−1625.34	−1.01359	−0.506797	+	0.862066i	$0.669171\pi$
$138$	2936.51	1.81139
$139$	− 1034.35i	− 0.631166i	−0.948898	−	0.315583i	$-0.897800\pi$
$139$	− 1034.35i	− 0.631166i	0.948898	−	0.315583i	$-0.102200\pi$
$140$	1102.72	0.665694
$141$	287.635i	0.171796i
$142$	3379.28i	1.99706i
$143$	3.28959i	0.00192370i
$144$	1208.45	0.699334
$145$	−4247.48	−2.43265
$146$	72.8999i	0.0413236i
$147$	− 947.585i	− 0.531670i
$148$	− 160.103i	− 0.0889213i
$149$	−650.200	−0.357493	−0.178747	−	0.983895i	$-0.557204\pi$
$149$	−650.200	−0.357493	−0.178747	+	0.983895i	$0.557204\pi$
$150$	5097.39i	2.77467i
$151$	−1023.67	−0.551689	−0.275845	−	0.961202i	$-0.588958\pi$
$151$	−1023.67	−0.551689	−0.275845	+	0.961202i	$0.588958\pi$
$152$	−1814.50	−0.968261
$153$	0	0
$154$	−594.615	−0.311139
$155$	−939.164	−0.486680
$156$	− 6.77869i	− 0.00347903i
$157$	671.612	0.341404	0.170702	−	0.985323i	$-0.445396\pi$
$157$	671.612	0.341404	0.170702	+	0.985323i	$0.445396\pi$
$158$	2223.87i	1.11976i
$159$	− 388.744i	− 0.193896i
$160$	− 2176.23i	− 1.07529i
$161$	−3056.62	−1.49624
$162$	2955.66	1.43345
$163$	− 1893.45i	− 0.909857i	−0.890528	−	0.454929i	$-0.849665\pi$
$163$	− 1893.45i	− 0.909857i	0.890528	−	0.454929i	$-0.150335\pi$
$164$	− 1011.65i	− 0.481689i
$165$	− 1028.61i	− 0.485316i
$166$	−1017.49	−0.475739
$167$	− 290.246i	− 0.134491i	−0.997736	−	0.0672453i	$-0.978579\pi$
$167$	− 290.246i	− 0.134491i	0.997736	−	0.0672453i	$-0.0214210\pi$
$168$	−2528.29	−1.16108
$169$	−2196.84	−0.999928
$170$	0	0
$171$	1600.05	0.715547
$172$	53.2339	0.0235991
$173$	3665.86i	1.61104i	0.592567	+	0.805521i	$0.298114\pi$
$173$	3665.86i	1.61104i	−0.592567	+	0.805521i	$0.701886\pi$
$174$	−4719.59	−2.05627
$175$	− 5305.89i	− 2.29193i
$176$	644.852i	0.276179i
$177$	− 1551.63i	− 0.658913i
$178$	2946.08	1.24055
$179$	3628.67	1.51519	0.757597	−	0.652723i	$-0.226373\pi$
$179$	3628.67	1.51519	0.757597	+	0.652723i	$0.226373\pi$
$180$	772.355i	0.319822i
$181$	3827.78i	1.57191i	0.618281	+	0.785957i	$0.287829\pi$
$181$	3827.78i	1.57191i	−0.618281	+	0.785957i	$0.712171\pi$
$182$	28.6713i	0.0116773i
$183$	3883.27	1.56863
$184$	2427.82i	0.972725i
$185$	−1171.43	−0.465541
$186$	−1043.55	−0.411382
$187$	0	0
$188$	115.249	0.0447097
$189$	−1659.48	−0.638673
$190$	− 6434.08i	− 2.45672i
$191$	−877.247	−0.332332	−0.166166	−	0.986098i	$-0.553139\pi$
$191$	−877.247	−0.332332	−0.166166	+	0.986098i	$0.553139\pi$
$192$	1652.60i	0.621177i
$193$	− 4769.70i	− 1.77891i	−0.457018	−	0.889457i	$-0.651083\pi$
$193$	− 4769.70i	− 1.77891i	0.457018	−	0.889457i	$-0.348917\pi$
$194$	3279.41i	1.21365i
$195$	−49.5978	−0.0182142
$196$	−379.679	−0.138367
$197$	1492.97i	0.539949i	0.962867	+	0.269974i	$0.0870152\pi$
$197$	1492.97i	0.539949i	−0.962867	+	0.269974i	$0.912985\pi$
$198$	− 416.473i	− 0.149482i
$199$	3091.28i	1.10118i	0.834775	+	0.550591i	$0.185598\pi$
$199$	3091.28i	1.10118i	−0.834775	+	0.550591i	$0.814402\pi$
$200$	−4214.38	−1.49001
$201$	− 2664.48i	− 0.935013i
$202$	3663.40	1.27602
$203$	4912.63	1.69852
$204$	0	0
$205$	−7402.00	−2.52185
$206$	1292.18	0.437042
$207$	− 2140.88i	− 0.718846i
$208$	31.0937	0.0103652
$209$	853.816i	0.282582i
$210$	− 8965.12i	− 2.94596i
$211$	− 2776.38i	− 0.905848i	−0.891549	−	0.452924i	$-0.850381\pi$
$211$	− 2776.38i	− 0.905848i	0.891549	−	0.452924i	$-0.149619\pi$
$212$	−155.762	−0.0504612
$213$	6761.17	2.17497
$214$	− 941.118i	− 0.300624i
$215$	− 389.498i	− 0.123551i
$216$	1318.10i	0.415209i
$217$	1086.24	0.339809
$218$	− 4492.60i	− 1.39577i
$219$	145.856	0.0450048
$220$	−412.144	−0.126303
$221$	0	0
$222$	−1301.63	−0.393513
$223$	−4641.22	−1.39372	−0.696859	−	0.717209i	$-0.745419\pi$
$223$	−4641.22	−1.39372	−0.696859	+	0.717209i	$0.745419\pi$
$224$	2517.02i	0.750784i
$225$	3716.28	1.10112
$226$	4444.63i	1.30820i
$227$	1712.30i	0.500658i	0.968161	+	0.250329i	$0.0805388\pi$
$227$	1712.30i	0.500658i	−0.968161	+	0.250329i	$0.919461\pi$
$228$	− 1759.41i	− 0.511052i
$229$	6028.18	1.73954	0.869768	−	0.493461i	$-0.164269\pi$
$229$	6028.18	1.73954	0.869768	+	0.493461i	$0.164269\pi$
$230$	−8608.86	−2.46805
$231$	1189.69i	0.338857i
$232$	− 3902.02i	− 1.10423i
$233$	− 2663.60i	− 0.748919i	−0.927243	−	0.374459i	$-0.877828\pi$
$233$	− 2663.60i	− 0.748919i	0.927243	−	0.374459i	$-0.122172\pi$
$234$	−20.0816	−0.00561015
$235$	− 843.249i	− 0.234075i
$236$	−621.707	−0.171482
$237$	4449.47	1.21951
$238$	0	0
$239$	6020.73	1.62949	0.814747	−	0.579817i	$-0.196876\pi$
$239$	6020.73	1.62949	0.814747	+	0.579817i	$0.196876\pi$
$240$	−9722.56	−2.61495
$241$	− 6080.97i	− 1.62535i	−0.582716	−	0.812676i	$-0.698010\pi$
$241$	− 6080.97i	− 1.62535i	0.582716	−	0.812676i	$-0.301990\pi$
$242$	−4113.53	−1.09268
$243$	− 3886.16i	− 1.02591i
$244$	− 1555.95i	− 0.408235i
$245$	2778.01i	0.724409i
$246$	−8224.74	−2.13167
$247$	41.1695	0.0106055
$248$	− 862.780i	− 0.220914i
$249$	2035.77i	0.518120i
$250$	− 7163.52i	− 1.81224i
$251$	−3296.82	−0.829058	−0.414529	−	0.910036i	$-0.636054\pi$
$251$	−3296.82	−0.829058	−0.414529	+	0.910036i	$0.636054\pi$
$252$	− 893.305i	− 0.223305i
$253$	1142.41	0.283885
$254$	3268.37	0.807384
$255$	0	0
$256$	3630.28	0.886300
$257$	1243.37	0.301786	0.150893	−	0.988550i	$-0.451785\pi$
$257$	1243.37	0.301786	0.150893	+	0.988550i	$0.451785\pi$
$258$	− 432.790i	− 0.104435i
$259$	1354.87	0.325049
$260$	19.8729i	0.00474024i
$261$	3440.84i	0.816026i
$262$	− 710.186i	− 0.167463i
$263$	−4523.81	−1.06065	−0.530324	−	0.847795i	$-0.677930\pi$
$263$	−4523.81	−1.06065	−0.530324	+	0.847795i	$0.677930\pi$
$264$	944.951	0.220294
$265$	1139.67i	0.264186i
$266$	7441.66i	1.71533i
$267$	− 5894.44i	− 1.35106i
$268$	−1067.60	−0.243337
$269$	2155.56i	0.488576i	0.969703	+	0.244288i	$0.0785542\pi$
$269$	2155.56i	0.488576i	−0.969703	+	0.244288i	$0.921446\pi$
$270$	−4673.86	−1.05349
$271$	−5433.12	−1.21785	−0.608927	−	0.793226i	$-0.708400\pi$
$271$	−5433.12	−1.21785	−0.608927	+	0.793226i	$0.708400\pi$
$272$	0	0
$273$	57.3648	0.0127175
$274$	5294.59	1.16736
$275$	1983.08i	0.434851i
$276$	−2354.11	−0.513408
$277$	3349.75i	0.726595i	0.931673	+	0.363297i	$0.118349\pi$
$277$	3349.75i	0.726595i	−0.931673	+	0.363297i	$0.881651\pi$
$278$	3369.40i	0.726919i
$279$	760.807i	0.163256i
$280$	7412.11	1.58199
$281$	2668.18	0.566441	0.283221	−	0.959055i	$-0.408597\pi$
$281$	2668.18	0.566441	0.283221	+	0.959055i	$0.408597\pi$
$282$	− 936.977i	− 0.197859i
$283$	5452.65i	1.14532i	0.819792	+	0.572662i	$0.194089\pi$
$283$	5452.65i	1.14532i	−0.819792	+	0.572662i	$0.805911\pi$
$284$	− 2709.07i	− 0.566033i
$285$	−12873.1	−2.67558
$286$	− 10.7159i	− 0.00221555i
$287$	8561.15	1.76080
$288$	−1762.94	−0.360702
$289$	0	0
$290$	13836.3	2.80170
$291$	6561.36	1.32177
$292$	− 58.4417i	− 0.0117125i
$293$	6341.24	1.26437	0.632183	−	0.774819i	$-0.282159\pi$
$293$	6341.24	1.26437	0.632183	+	0.774819i	$0.282159\pi$
$294$	3086.78i	0.612329i
$295$	4548.86i	0.897780i
$296$	− 1076.15i	− 0.211318i
$297$	620.231	0.121177
$298$	2118.04	0.411728
$299$	− 55.0852i	− 0.0106544i
$300$	− 4086.42i	− 0.786433i
$301$	450.493i	0.0862657i
$302$	3334.63	0.635385
$303$	− 7329.63i	− 1.38969i
$304$	8070.38	1.52259
$305$	−11384.5	−2.13729
$306$	0	0
$307$	820.602	0.152555	0.0762773	−	0.997087i	$-0.475697\pi$
$307$	820.602	0.152555	0.0762773	+	0.997087i	$0.475697\pi$
$308$	476.685	0.0881872
$309$	− 2585.37i	− 0.475975i
$310$	3059.35	0.560514
$311$	− 7492.46i	− 1.36610i	−0.730369	−	0.683052i	$-0.760652\pi$
$311$	− 7492.46i	− 1.36610i	0.730369	−	0.683052i	$-0.239348\pi$
$312$	− 45.5639i	− 0.00826779i
$313$	1217.76i	0.219910i	0.993937	+	0.109955i	$0.0350707\pi$
$313$	1217.76i	0.219910i	−0.993937	+	0.109955i	$0.964929\pi$
$314$	−2187.79	−0.393198
$315$	−6536.07	−1.16910
$316$	− 1782.81i	− 0.317377i
$317$	− 5682.18i	− 1.00676i	−0.864065	−	0.503380i	$-0.832090\pi$
$317$	− 5682.18i	− 1.00676i	0.864065	−	0.503380i	$-0.167910\pi$
$318$	1266.34i	0.223311i
$319$	−1836.10	−0.322263
$320$	− 4844.87i	− 0.846363i
$321$	−1882.96	−0.327404
$322$	9957.00	1.72324
$323$	0	0
$324$	−2369.47	−0.406287
$325$	95.6206	0.0163202
$326$	6167.97i	1.04789i
$327$	−8988.68	−1.52011
$328$	− 6799.98i	− 1.14471i
$329$	975.301i	0.163435i
$330$	3350.72i	0.558943i
$331$	−7387.94	−1.22682	−0.613411	−	0.789764i	$-0.710203\pi$
$331$	−7387.94	−1.22682	−0.613411	+	0.789764i	$0.710203\pi$
$332$	815.693	0.134840
$333$	948.962i	0.156165i
$334$	945.483i	0.154894i
$335$	7811.36i	1.27397i
$336$	11245.1	1.82581
$337$	5104.29i	0.825069i	0.910942	+	0.412534i	$0.135356\pi$
$337$	5104.29i	0.825069i	−0.910942	+	0.412534i	$0.864644\pi$
$338$	7156.26	1.15163
$339$	8892.69	1.42473
$340$	0	0
$341$	−405.982	−0.0644725
$342$	−5212.19	−0.824102
$343$	4367.10i	0.687467i
$344$	357.819	0.0560823
$345$	17224.4i	2.68791i
$346$	− 11941.6i	− 1.85545i
$347$	8986.15i	1.39021i	0.718910	+	0.695103i	$0.244641\pi$
$347$	8986.15i	1.39021i	−0.718910	+	0.695103i	$0.755359\pi$
$348$	3783.55	0.582816
$349$	−1924.65	−0.295198	−0.147599	−	0.989047i	$-0.547155\pi$
$349$	−1924.65	−0.295198	−0.147599	+	0.989047i	$0.547155\pi$
$350$	17284.0i	2.63963i
$351$	− 29.9065i	− 0.00454783i
$352$	− 940.739i	− 0.142448i
$353$	4574.33	0.689709	0.344854	−	0.938656i	$-0.387928\pi$
$353$	4574.33	0.689709	0.344854	+	0.938656i	$0.387928\pi$
$354$	5054.47i	0.758876i
$355$	−19821.5	−2.96343
$356$	−2361.78	−0.351613
$357$	0	0
$358$	−11820.5	−1.74506
$359$	−9358.89	−1.37589	−0.687943	−	0.725765i	$-0.741486\pi$
$359$	−9358.89	−1.37589	−0.687943	+	0.725765i	$0.741486\pi$
$360$	5191.50i	0.760044i
$361$	3826.58	0.557892
$362$	− 12469.1i	− 1.81039i
$363$	8230.23i	1.19001i
$364$	− 22.9849i	− 0.00330972i
$365$	−427.602	−0.0613198
$366$	−12649.8	−1.80661
$367$	− 1596.51i	− 0.227077i	−0.993534	−	0.113539i	$-0.963781\pi$
$367$	− 1596.51i	− 0.227077i	0.993534	−	0.113539i	$-0.0362186\pi$
$368$	− 10798.2i	− 1.52961i
$369$	5996.29i	0.845947i
$370$	3815.95	0.536167
$371$	− 1318.14i	− 0.184459i
$372$	836.585	0.116599
$373$	−10884.6	−1.51094	−0.755471	−	0.655182i	$-0.772592\pi$
$373$	−10884.6	−1.51094	−0.755471	+	0.655182i	$0.772592\pi$
$374$	0	0
$375$	−14332.6	−1.97368
$376$	774.666	0.106251
$377$	88.5336i	0.0120947i
$378$	5405.79	0.735565
$379$	1561.47i	0.211629i	0.994386	+	0.105814i	$0.0337449\pi$
$379$	1561.47i	0.211629i	−0.994386	+	0.105814i	$0.966255\pi$
$380$	5158.01i	0.696317i
$381$	− 6539.26i	− 0.879308i
$382$	2857.65	0.382749
$383$	4409.48	0.588287	0.294143	−	0.955761i	$-0.404966\pi$
$383$	4409.48	0.588287	0.294143	+	0.955761i	$0.404966\pi$
$384$	− 11321.9i	− 1.50461i
$385$	− 3487.78i	− 0.461697i
$386$	15537.4i	2.04879i
$387$	−315.528	−0.0414450
$388$	− 2629.01i	− 0.343989i
$389$	8017.21	1.04496	0.522479	−	0.852652i	$-0.325007\pi$
$389$	8017.21	1.04496	0.522479	+	0.852652i	$0.325007\pi$
$390$	161.566	0.0209775
$391$	0	0
$392$	−2552.06	−0.328823
$393$	−1420.92	−0.182382
$394$	− 4863.39i	− 0.621864i
$395$	−13044.4	−1.66160
$396$	333.874i	0.0423681i
$397$	− 9437.12i	− 1.19304i	−0.802599	−	0.596519i	$-0.796550\pi$
$397$	− 9437.12i	− 1.19304i	0.802599	−	0.596519i	$-0.203450\pi$
$398$	− 10069.9i	− 1.26824i
$399$	14889.1	1.86814
$400$	18744.3	2.34304
$401$	4509.28i	0.561553i	0.959773	+	0.280777i	$0.0905920\pi$
$401$	4509.28i	0.561553i	−0.959773	+	0.280777i	$0.909408\pi$
$402$	8679.59i	1.07686i
$403$	19.5757i	0.00241969i
$404$	−2936.84	−0.361666
$405$	17336.8i	2.12709i
$406$	−16003.0	−1.95620
$407$	−506.384	−0.0616721
$408$	0	0
$409$	12446.9	1.50480	0.752398	−	0.658709i	$-0.228897\pi$
$409$	12446.9	1.50480	0.752398	+	0.658709i	$0.228897\pi$
$410$	24112.2	2.90443
$411$	− 10593.3i	− 1.27136i
$412$	−1035.91	−0.123872
$413$	− 5261.21i	− 0.626846i
$414$	6973.95i	0.827901i
$415$	− 5968.21i	− 0.705946i
$416$	−45.3608	−0.00534615
$417$	6741.41	0.791675
$418$	− 2781.33i	− 0.325452i
$419$	1648.22i	0.192173i	0.995373	+	0.0960867i	$0.0306326\pi$
$419$	1648.22i	0.192173i	−0.995373	+	0.0960867i	$0.969367\pi$
$420$	7187.07i	0.834984i
$421$	14395.5	1.66649	0.833247	−	0.552901i	$-0.186479\pi$
$421$	14395.5	1.66649	0.833247	+	0.552901i	$0.186479\pi$
$422$	9044.13i	1.04327i
$423$	−683.108	−0.0785197
$424$	−1046.98	−0.119919
$425$	0	0
$426$	−22024.7	−2.50493
$427$	13167.3	1.49229
$428$	754.466i	0.0852068i
$429$	−21.4401	−0.00241291
$430$	1268.80i	0.142295i
$431$	− 5351.92i	− 0.598127i	−0.954233	−	0.299064i	$-0.903326\pi$
$431$	− 5351.92i	− 0.598127i	0.954233	−	0.299064i	$-0.0966743\pi$
$432$	− 5862.50i	− 0.652916i
$433$	−15578.0	−1.72894	−0.864472	−	0.502680i	$-0.832347\pi$
$433$	−15578.0	−1.72894	−0.864472	+	0.502680i	$0.832347\pi$
$434$	−3538.44	−0.391361
$435$	− 27683.2i	− 3.05129i
$436$	3601.58i	0.395607i
$437$	− 14297.4i	− 1.56507i
$438$	−475.130	−0.0518324
$439$	16584.8i	1.80307i	0.432707	+	0.901535i	$0.357558\pi$
$439$	16584.8i	1.80307i	−0.432707	+	0.901535i	$0.642442\pi$
$440$	−2770.28	−0.300155
$441$	2250.44	0.243001
$442$	0	0
$443$	5163.31	0.553761	0.276881	−	0.960904i	$-0.410699\pi$
$443$	5163.31	0.553761	0.276881	+	0.960904i	$0.410699\pi$
$444$	1043.48	0.111535
$445$	17280.5i	1.84084i
$446$	15118.9	1.60516
$447$	− 4237.72i	− 0.448406i
$448$	5603.57i	0.590946i
$449$	10771.9i	1.13220i	0.824337	+	0.566099i	$0.191548\pi$
$449$	10771.9i	1.13220i	−0.824337	+	0.566099i	$0.808452\pi$
$450$	−12105.9	−1.26817
$451$	−3199.74	−0.334079
$452$	− 3563.13i	− 0.370786i
$453$	− 6671.84i	− 0.691987i
$454$	− 5577.86i	− 0.576612i
$455$	−168.175	−0.0173278
$456$	− 11826.1i	− 1.21450i
$457$	−5024.59	−0.514312	−0.257156	−	0.966370i	$-0.582785\pi$
$457$	−5024.59	−0.514312	−0.257156	+	0.966370i	$0.582785\pi$
$458$	−19636.9	−2.00344
$459$	0	0
$460$	6901.47	0.699527
$461$	2366.96	0.239133	0.119567	−	0.992826i	$-0.461849\pi$
$461$	2366.96	0.239133	0.119567	+	0.992826i	$0.461849\pi$
$462$	− 3875.44i	− 0.390264i
$463$	−17712.9	−1.77795	−0.888974	−	0.457958i	$-0.848581\pi$
$463$	−17712.9	−1.77795	−0.888974	+	0.457958i	$0.848581\pi$
$464$	17355.1i	1.73640i
$465$	− 6121.06i	− 0.610446i
$466$	8676.73i	0.862536i
$467$	19592.7	1.94141	0.970707	−	0.240267i	$-0.0772349\pi$
$467$	19592.7	1.94141	0.970707	+	0.240267i	$0.0772349\pi$
$468$	16.0988	0.00159010
$469$	− 9034.61i	− 0.889509i
$470$	2746.90i	0.269586i
$471$	4377.27i	0.428225i
$472$	−4178.89	−0.407519
$473$	− 168.372i	− 0.0163673i
$474$	−14494.2	−1.40452
$475$	24818.4	2.39736
$476$	0	0
$477$	923.234	0.0886205
$478$	−19612.7	−1.87670
$479$	4060.41i	0.387317i	0.981069	+	0.193659i	$0.0620354\pi$
$479$	4060.41i	0.387317i	−0.981069	+	0.193659i	$0.937965\pi$
$480$	14183.7	1.34874
$481$	24.4170i	0.00231459i
$482$	19808.9i	1.87193i
$483$	− 19921.7i	− 1.87675i
$484$	3297.69	0.309701
$485$	−19235.7	−1.80093
$486$	12659.2i	1.18155i
$487$	− 7256.96i	− 0.675244i	−0.941282	−	0.337622i	$-0.890377\pi$
$487$	− 7256.96i	− 0.675244i	0.941282	−	0.337622i	$-0.109623\pi$
$488$	− 10458.5i	− 0.970155i
$489$	12340.7	1.14124
$490$	− 9049.42i	− 0.834308i
$491$	769.855	0.0707598	0.0353799	−	0.999374i	$-0.488736\pi$
$491$	769.855	0.0707598	0.0353799	+	0.999374i	$0.488736\pi$
$492$	6593.53	0.604185
$493$	0	0
$494$	−134.111	−0.0122144
$495$	2442.86	0.221815
$496$	3837.39i	0.347387i
$497$	22925.5	2.06912
$498$	− 6631.58i	− 0.596723i
$499$	1432.66i	0.128526i	0.997933	+	0.0642630i	$0.0204697\pi$
$499$	1432.66i	0.128526i	−0.997933	+	0.0642630i	$0.979530\pi$
$500$	5742.78i	0.513650i
$501$	1891.70	0.168692
$502$	10739.5	0.954833
$503$	16206.3i	1.43659i	0.695739	+	0.718295i	$0.255077\pi$
$503$	16206.3i	1.43659i	−0.695739	+	0.718295i	$0.744923\pi$
$504$	− 6004.48i	− 0.530676i
$505$	21488.0i	1.89348i
$506$	−3721.44	−0.326953
$507$	− 14318.1i	− 1.25422i
$508$	−2620.15	−0.228839
$509$	15009.4	1.30703	0.653517	−	0.756912i	$-0.273293\pi$
$509$	15009.4	1.30703	0.653517	+	0.756912i	$0.273293\pi$
$510$	0	0
$511$	494.564	0.0428145
$512$	2071.37	0.178794
$513$	− 7762.24i	− 0.668053i
$514$	−4050.29	−0.347570
$515$	7579.44i	0.648524i
$516$	346.955i	0.0296005i
$517$	− 364.520i	− 0.0310088i
$518$	−4413.53	−0.374361
$519$	−23892.5	−2.02074
$520$	133.578i	0.0112650i
$521$	19626.5i	1.65039i	0.564846	+	0.825196i	$0.308936\pi$
$521$	19626.5i	1.65039i	−0.564846	+	0.825196i	$0.691064\pi$
$522$	− 11208.6i	− 0.939824i
$523$	−9267.41	−0.774829	−0.387414	−	0.921906i	$-0.626632\pi$
$523$	−9267.41	−0.774829	−0.387414	+	0.921906i	$0.626632\pi$
$524$	569.335i	0.0474647i
$525$	34581.5	2.87478
$526$	14736.4	1.22156
$527$	0	0
$528$	−4202.87	−0.346414
$529$	−6963.04	−0.572289
$530$	− 3712.50i	− 0.304265i
$531$	3684.99	0.301158
$532$	− 5965.76i	− 0.486181i
$533$	154.286i	0.0125382i
$534$	19201.3i	1.55603i
$535$	5520.22	0.446094
$536$	−7176.04	−0.578280
$537$	23650.1i	1.90052i
$538$	− 7021.79i	− 0.562697i
$539$	1200.88i	0.0959655i
$540$	3746.90	0.298594
$541$	907.733i	0.0721377i	0.999349	+	0.0360689i	$0.0114836\pi$
$541$	907.733i	0.0721377i	−0.999349	+	0.0360689i	$0.988516\pi$
$542$	17698.5	1.40261
$543$	−24947.8	−1.97166
$544$	0	0
$545$	26351.8	2.07117
$546$	−186.867	−0.0146468
$547$	11197.7i	0.875281i	0.899150	+	0.437640i	$0.144186\pi$
$547$	11197.7i	0.875281i	−0.899150	+	0.437640i	$0.855814\pi$
$548$	−4244.51	−0.330870
$549$	9222.44i	0.716947i
$550$	− 6459.92i	− 0.500822i
$551$	22979.0i	1.77665i
$552$	−15823.5	−1.22009
$553$	15087.1	1.16016
$554$	− 10911.9i	− 0.836825i
$555$	− 7634.85i	− 0.583931i
$556$	− 2701.15i	− 0.206033i
$557$	−3207.25	−0.243978	−0.121989	−	0.992531i	$-0.538927\pi$
$557$	−3207.25	−0.243978	−0.121989	+	0.992531i	$0.538927\pi$
$558$	− 2478.35i	− 0.188023i
$559$	−8.11861	−0.000614276 0
$560$	−32966.9	−2.48769
$561$	0	0
$562$	−8691.64	−0.652375
$563$	6522.79	0.488282	0.244141	−	0.969740i	$-0.421494\pi$
$563$	6522.79	0.488282	0.244141	+	0.969740i	$0.421494\pi$
$564$	751.146i	0.0560797i
$565$	−26070.4	−1.94122
$566$	− 17762.2i	− 1.31908i
$567$	− 20051.7i	− 1.48517i
$568$	− 18209.4i	− 1.34516i
$569$	3040.50	0.224015	0.112007	−	0.993707i	$-0.464272\pi$
$569$	3040.50	0.224015	0.112007	+	0.993707i	$0.464272\pi$
$570$	41934.6	3.08148
$571$	17273.2i	1.26596i	0.774169	+	0.632979i	$0.218168\pi$
$571$	17273.2i	1.26596i	−0.774169	+	0.632979i	$0.781832\pi$
$572$	8.59064i	0 0.000627959i
$573$	− 5717.51i	− 0.416846i
$574$	−27888.2	−2.02793
$575$	− 33207.2i	− 2.40841i
$576$	−3924.78	−0.283910
$577$	−5804.29	−0.418779	−0.209390	−	0.977832i	$-0.567148\pi$
$577$	−5804.29	−0.418779	−0.209390	+	0.977832i	$0.567148\pi$
$578$	0	0
$579$	31086.8	2.23130
$580$	−11092.1	−0.794096
$581$	6902.83i	0.492904i
$582$	−21373.8	−1.52229
$583$	492.656i	0.0349978i
$584$	− 392.824i	− 0.0278342i
$585$	− 117.791i	− 0.00832486i
$586$	−20656.8	−1.45618
$587$	−12862.4	−0.904407	−0.452204	−	0.891915i	$-0.649362\pi$
$587$	−12862.4	−0.904407	−0.452204	+	0.891915i	$0.649362\pi$
$588$	− 2474.58i	− 0.173554i
$589$	5080.89i	0.355441i
$590$	− 14818.0i	− 1.03398i
$591$	−9730.55	−0.677261
$592$	4786.41i	0.332298i
$593$	−1319.33	−0.0913631	−0.0456816	−	0.998956i	$-0.514546\pi$
$593$	−1319.33	−0.0913631	−0.0456816	+	0.998956i	$0.514546\pi$
$594$	−2020.42	−0.139560
$595$	0	0
$596$	−1697.97	−0.116697
$597$	−20147.6	−1.38122
$598$	179.441i	0.0122707i
$599$	−16269.3	−1.10976	−0.554881	−	0.831930i	$-0.687236\pi$
$599$	−16269.3	−1.10976	−0.554881	+	0.831930i	$0.687236\pi$
$600$	− 27467.5i	− 1.86893i
$601$	− 7427.38i	− 0.504108i	−0.967713	−	0.252054i	$-0.918894\pi$
$601$	− 7427.38i	− 0.504108i	0.967713	−	0.252054i	$-0.0811061\pi$
$602$	− 1467.49i	− 0.0993529i
$603$	6327.90	0.427350
$604$	−2673.27	−0.180089
$605$	− 24128.3i	− 1.62141i
$606$	23876.4i	1.60052i
$607$	7212.10i	0.482257i	0.970493	+	0.241129i	$0.0775176\pi$
$607$	7212.10i	0.482257i	−0.970493	+	0.241129i	$0.922482\pi$
$608$	−11773.4	−0.785322
$609$	32018.4i	2.13046i
$610$	37085.1	2.46153
$611$	−17.5765	−0.00116378
$612$	0	0
$613$	−15446.5	−1.01775	−0.508873	−	0.860842i	$-0.669938\pi$
$613$	−15446.5	−1.01775	−0.508873	+	0.860842i	$0.669938\pi$
$614$	−2673.13	−0.175698
$615$	− 48243.1i	− 3.16317i
$616$	3204.11	0.209573
$617$	8746.54i	0.570701i	0.958423	+	0.285350i	$0.0921100\pi$
$617$	8746.54i	0.570701i	−0.958423	+	0.285350i	$0.907890\pi$
$618$	8421.89i	0.548185i
$619$	4740.09i	0.307787i	0.988087	+	0.153894i	$0.0491813\pi$
$619$	4740.09i	0.307787i	−0.988087	+	0.153894i	$0.950819\pi$
$620$	−2452.59	−0.158868
$621$	−10385.9	−0.671133
$622$	24406.9i	1.57335i
$623$	− 19986.7i	− 1.28531i
$624$	202.655i	0.0130011i
$625$	12007.1	0.768453
$626$	− 3966.89i	− 0.253273i
$627$	−5564.80	−0.354445
$628$	1753.89	0.111445
$629$	0	0
$630$	21291.4	1.34646
$631$	17055.1	1.07600	0.537998	−	0.842946i	$-0.319181\pi$
$631$	17055.1	1.07600	0.537998	+	0.842946i	$0.319181\pi$
$632$	− 11983.4i	− 0.754233i
$633$	18095.2	1.13621
$634$	18509.8i	1.15949i
$635$	19170.9i	1.19807i
$636$	− 1015.19i	− 0.0632938i
$637$	57.9042	0.00360164
$638$	5981.14	0.371153
$639$	16057.2i	0.994074i
$640$	33192.1i	2.05005i
$641$	− 16050.5i	− 0.989012i	−0.869174	−	0.494506i	$-0.835349\pi$
$641$	− 16050.5i	− 0.989012i	0.869174	−	0.494506i	$-0.164651\pi$
$642$	6133.80	0.377074
$643$	− 14821.5i	− 0.909027i	−0.890740	−	0.454514i	$-0.849813\pi$
$643$	− 14821.5i	− 0.909027i	0.890740	−	0.454514i	$-0.150187\pi$
$644$	−7982.23	−0.488422
$645$	2538.58	0.154971
$646$	0	0
$647$	−30527.0	−1.85493	−0.927465	−	0.373910i	$-0.878017\pi$
$647$	−30527.0	−1.85493	−0.927465	+	0.373910i	$0.878017\pi$
$648$	−15926.7	−0.965526
$649$	1966.38i	0.118933i
$650$	−311.486	−0.0187962
$651$	7079.62i	0.426224i
$652$	− 4944.68i	− 0.297007i
$653$	− 20173.6i	− 1.20897i	−0.796618	−	0.604483i	$-0.793380\pi$
$653$	− 20173.6i	− 1.20897i	0.796618	−	0.604483i	$-0.206620\pi$
$654$	29280.8	1.75072
$655$	4165.67	0.248498
$656$	30244.3i	1.80007i
$657$	346.396i	0.0205696i
$658$	− 3177.07i	− 0.188229i
$659$	−21503.0	−1.27107	−0.635537	−	0.772071i	$-0.719221\pi$
$659$	−21503.0	−1.27107	−0.635537	+	0.772071i	$0.719221\pi$
$660$	− 2686.17i	− 0.158423i
$661$	1772.12	0.104278	0.0521389	−	0.998640i	$-0.483396\pi$
$661$	1772.12	0.104278	0.0521389	+	0.998640i	$0.483396\pi$
$662$	24066.4	1.41294
$663$	0	0
$664$	5482.80	0.320443
$665$	−43649.8	−2.54536
$666$	− 3091.26i	− 0.179856i
$667$	30746.0	1.78485
$668$	− 757.966i	− 0.0439021i
$669$	− 30249.4i	− 1.74815i
$670$	− 25445.7i	− 1.46724i
$671$	−4921.27	−0.283135
$672$	−16404.9	−0.941713
$673$	− 33449.4i	− 1.91587i	−0.286983	−	0.957936i	$-0.592652\pi$
$673$	− 33449.4i	− 1.91587i	0.286983	−	0.957936i	$-0.407348\pi$
$674$	− 16627.3i	− 0.950239i
$675$	− 18028.6i	− 1.02803i
$676$	−5736.96	−0.326409
$677$	− 805.765i	− 0.0457431i	−0.999738	−	0.0228716i	$-0.992719\pi$
$677$	− 805.765i	− 0.0457431i	0.999738	−	0.0228716i	$-0.00728088\pi$
$678$	−28968.2	−1.64088
$679$	22248.0	1.25744
$680$	0	0
$681$	−11160.0	−0.627978
$682$	1322.49	0.0742536
$683$	993.802i	0.0556761i	0.999612	+	0.0278380i	$0.00886227\pi$
$683$	993.802i	0.0556761i	−0.999612	+	0.0278380i	$0.991138\pi$
$684$	4178.46	0.233578
$685$	31056.0i	1.73224i
$686$	− 14225.9i	− 0.791762i
$687$	39289.1i	2.18191i
$688$	−1591.47	−0.0881895
$689$	23.7550	0.00131349
$690$	− 56108.8i	− 3.09569i
$691$	− 33450.1i	− 1.84154i	−0.390111	−	0.920768i	$-0.627563\pi$
$691$	− 33450.1i	− 1.84154i	0.390111	−	0.920768i	$-0.372437\pi$
$692$	9573.25i	0.525896i
$693$	−2825.41	−0.154875
$694$	− 29272.6i	− 1.60111i
$695$	−19763.6	−1.07867
$696$	25431.7	1.38504
$697$	0	0
$698$	6269.59	0.339982
$699$	17360.2	0.939373
$700$	− 13856.1i	− 0.748159i
$701$	−18290.5	−0.985481	−0.492740	−	0.870176i	$-0.664005\pi$
$701$	−18290.5	−0.985481	−0.492740	+	0.870176i	$0.664005\pi$
$702$	97.4210i	0.00523778i
$703$	6337.44i	0.340002i
$704$	− 2094.34i	− 0.112121i
$705$	5495.93	0.293601
$706$	−14901.0	−0.794343
$707$	− 24853.1i	− 1.32206i
$708$	− 4052.02i	− 0.215091i
$709$	6780.00i	0.359137i	0.983745	+	0.179569i	$0.0574702\pi$
$709$	6780.00i	0.359137i	−0.983745	+	0.179569i	$0.942530\pi$
$710$	64569.0	3.41300
$711$	10567.1i	0.557380i
$712$	−15875.1	−0.835595
$713$	6798.28	0.357079
$714$	0	0
$715$	62.8554	0.00328763
$716$	9476.13	0.494608
$717$	39240.5i	2.04388i
$718$	30486.8	1.58462
$719$	− 24591.0i	− 1.27551i	−0.770240	−	0.637754i	$-0.779864\pi$
$719$	− 24591.0i	− 1.27551i	0.770240	−	0.637754i	$-0.220136\pi$
$720$	− 23090.3i	− 1.19517i
$721$	− 8766.37i	− 0.452811i
$722$	−12465.2	−0.642528
$723$	39633.1	2.03869
$724$	9996.08i	0.513124i
$725$	53371.1i	2.73400i
$726$	− 26810.2i	− 1.37055i
$727$	−9819.79	−0.500957	−0.250479	−	0.968122i	$-0.580588\pi$
$727$	−9819.79	−0.500957	−0.250479	+	0.968122i	$0.580588\pi$
$728$	− 154.497i	− 0.00786542i
$729$	830.242	0.0421806
$730$	1392.92	0.0706225
$731$	0	0
$732$	10141.0	0.512052
$733$	10751.8	0.541783	0.270892	−	0.962610i	$-0.412682\pi$
$733$	10751.8	0.541783	0.270892	+	0.962610i	$0.412682\pi$
$734$	5200.68i	0.261527i
$735$	−18105.8	−0.908631
$736$	15753.0i	0.788942i
$737$	3376.69i	0.168768i
$738$	− 19533.1i	− 0.974284i
$739$	1809.70	0.0900823	0.0450412	−	0.998985i	$-0.485658\pi$
$739$	1809.70	0.0900823	0.0450412	+	0.998985i	$0.485658\pi$
$740$	−3059.13	−0.151968
$741$	268.325i	0.0133025i
$742$	4293.87i	0.212443i
$743$	− 895.790i	− 0.0442306i	−0.999755	−	0.0221153i	$-0.992960\pi$
$743$	− 895.790i	− 0.0442306i	0.999755	−	0.0221153i	$-0.00704010\pi$
$744$	5623.22	0.277093
$745$	12423.6i	0.610960i
$746$	35456.7	1.74016
$747$	−4834.79	−0.236808
$748$	0	0
$749$	−6384.69	−0.311470
$750$	46688.7	2.27311
$751$	− 11683.1i	− 0.567672i	−0.958873	−	0.283836i	$-0.908393\pi$
$751$	− 11683.1i	− 0.567672i	0.958873	−	0.283836i	$-0.0916071\pi$
$752$	−3445.49	−0.167080
$753$	− 21487.2i	− 1.03989i
$754$	− 288.400i	− 0.0139296i
$755$	19559.6i	0.942844i
$756$	−4333.66	−0.208484
$757$	12621.2	0.605976	0.302988	−	0.952994i	$-0.402016\pi$
$757$	12621.2	0.605976	0.302988	+	0.952994i	$0.402016\pi$
$758$	− 5086.53i	− 0.243735i
$759$	7445.75i	0.356079i
$760$	34670.3i	1.65477i
$761$	−5759.46	−0.274350	−0.137175	−	0.990547i	$-0.543802\pi$
$761$	−5759.46	−0.274350	−0.137175	+	0.990547i	$0.543802\pi$
$762$	21301.8i	1.01271i
$763$	−30478.5	−1.44613
$764$	−2290.89	−0.108484
$765$	0	0
$766$	−14364.0	−0.677535
$767$	94.8156	0.00446361
$768$	23660.6i	1.11169i
$769$	3245.56	0.152195	0.0760975	−	0.997100i	$-0.475754\pi$
$769$	3245.56	0.152195	0.0760975	+	0.997100i	$0.475754\pi$
$770$	11361.5i	0.531741i
$771$	8103.72i	0.378532i
$772$	− 12455.9i	− 0.580695i
$773$	2500.27	0.116337	0.0581684	−	0.998307i	$-0.481474\pi$
$773$	2500.27	0.116337	0.0581684	+	0.998307i	$0.481474\pi$
$774$	1027.84	0.0477325
$775$	11800.9i	0.546970i
$776$	− 17671.2i	− 0.817475i
$777$	8830.47i	0.407711i
$778$	−26116.2	−1.20349
$779$	40045.0i	1.84180i
$780$	−129.523	−0.00594571
$781$	−8568.44	−0.392577
$782$	0	0
$783$	16692.4	0.761863
$784$	11350.8	0.517075
$785$	− 12832.7i	− 0.583464i
$786$	4628.68	0.210050
$787$	15270.8i	0.691671i	0.938295	+	0.345835i	$0.112404\pi$
$787$	15270.8i	0.691671i	−0.938295	+	0.345835i	$0.887596\pi$
$788$	3898.84i	0.176257i
$789$	− 29484.2i	− 1.33038i
$790$	42492.3	1.91368
$791$	30153.0	1.35540
$792$	2244.18i	0.100686i
$793$	237.295i	0.0106262i
$794$	30741.7i	1.37403i
$795$	−7427.86	−0.331370
$796$	8072.76i	0.359462i
$797$	−6885.20	−0.306006	−0.153003	−	0.988226i	$-0.548894\pi$
$797$	−6885.20	−0.306006	−0.153003	+	0.988226i	$0.548894\pi$
$798$	−48501.5	−2.15155
$799$	0	0
$800$	−27345.1	−1.20849
$801$	13998.8	0.617507
$802$	− 14689.1i	− 0.646746i
$803$	−184.844	−0.00812328
$804$	− 6958.17i	− 0.305219i
$805$	58403.8i	2.55710i
$806$	− 63.7684i	− 0.00278678i
$807$	−14049.0	−0.612824
$808$	−19740.4	−0.859485
$809$	13606.4i	0.591318i	0.955294	+	0.295659i	$0.0955392\pi$
$809$	13606.4i	0.591318i	−0.955294	+	0.295659i	$0.904461\pi$
$810$	− 56474.9i	− 2.44978i
$811$	− 12718.7i	− 0.550694i	−0.961345	−	0.275347i	$-0.911207\pi$
$811$	− 12718.7i	− 0.550694i	0.961345	−	0.275347i	$-0.0887927\pi$
$812$	12829.1	0.554452
$813$	− 35410.7i	− 1.52756i
$814$	1649.56	0.0710283
$815$	−36178.9	−1.55496
$816$	0	0
$817$	−2107.19	−0.0902341
$818$	−40546.2	−1.73309
$819$	136.237i	0.00581257i
$820$	−19330.0	−0.823212
$821$	− 29119.6i	− 1.23786i	−0.785447	−	0.618929i	$-0.787567\pi$
$821$	− 29119.6i	− 1.23786i	0.785447	−	0.618929i	$-0.212433\pi$
$822$	34507.8i	1.46423i
$823$	− 6284.88i	− 0.266193i	−0.991103	−	0.133097i	$-0.957508\pi$
$823$	− 6284.88i	− 0.266193i	0.991103	−	0.133097i	$-0.0424921\pi$
$824$	−6962.98	−0.294377
$825$	−12924.8	−0.545437
$826$	17138.5i	0.721944i
$827$	5383.60i	0.226368i	0.993574	+	0.113184i	$0.0361049\pi$
$827$	5383.60i	0.226368i	−0.993574	+	0.113184i	$0.963895\pi$
$828$	− 5590.81i	− 0.234655i
$829$	25382.6	1.06342	0.531709	−	0.846927i	$-0.321550\pi$
$829$	25382.6	1.06342	0.531709	+	0.846927i	$0.321550\pi$
$830$	19441.6i	0.813045i
$831$	−21832.2	−0.911372
$832$	−100.985	−0.00420798
$833$	0	0
$834$	−21960.3	−0.911779
$835$	−5545.83	−0.229846
$836$	2229.71i	0.0922440i
$837$	3690.88	0.152420
$838$	− 5369.10i	− 0.221328i
$839$	2499.44i	0.102849i	0.998677	+	0.0514246i	$0.0163762\pi$
$839$	2499.44i	0.102849i	−0.998677	+	0.0514246i	$0.983624\pi$
$840$	48308.9i	1.98431i
$841$	−25026.4	−1.02614
$842$	−46893.7	−1.91932
$843$	17390.0i	0.710491i
$844$	− 7250.41i	− 0.295698i
$845$	41975.8i	1.70889i
$846$	2225.24	0.0904318
$847$	27906.8i	1.13210i
$848$	4656.65	0.188573
$849$	−35538.0	−1.43659
$850$	0	0
$851$	8479.55	0.341569
$852$	17656.5	0.709979
$853$	38293.9i	1.53711i	0.639781	+	0.768557i	$0.279025\pi$
$853$	38293.9i	1.53711i	−0.639781	+	0.768557i	$0.720975\pi$
$854$	−42892.6	−1.71868
$855$	− 30572.6i	− 1.22288i
$856$	5071.25i	0.202490i
$857$	426.303i	0.0169921i	0.999964	+	0.00849606i	$0.00270441\pi$
$857$	426.303i	0.0169921i	−0.999964	+	0.00849606i	$0.997296\pi$
$858$	69.8418	0.00277897
$859$	−30976.1	−1.23037	−0.615186	−	0.788382i	$-0.710919\pi$
$859$	−30976.1	−1.23037	−0.615186	+	0.788382i	$0.710919\pi$
$860$	− 1017.16i	− 0.0403311i
$861$	55797.9i	2.20858i
$862$	17434.0i	0.688868i
$863$	38201.4	1.50682	0.753412	−	0.657549i	$-0.228407\pi$
$863$	38201.4	1.50682	0.753412	+	0.657549i	$0.228407\pi$
$864$	8552.48i	0.336761i
$865$	70044.8	2.75329
$866$	50745.8	1.99124
$867$	0	0
$868$	2836.66	0.110925
$869$	−5638.81	−0.220119
$870$	90178.8i	3.51419i
$871$	162.818	0.00633397
$872$	24208.6i	0.940144i
$873$	15582.7i	0.604116i
$874$	46574.1i	1.80251i
$875$	−48598.4	−1.87763
$876$	380.898	0.0146910
$877$	13461.8i	0.518327i	0.965833	+	0.259163i	$0.0834468\pi$
$877$	13461.8i	0.518327i	−0.965833	+	0.259163i	$0.916553\pi$
$878$	− 54025.2i	− 2.07661i
$879$	41329.5i	1.58590i
$880$	12321.4	0.471994
$881$	13867.4i	0.530310i	0.964206	+	0.265155i	$0.0854231\pi$
$881$	13867.4i	0.530310i	−0.964206	+	0.265155i	$0.914577\pi$
$882$	−7330.85	−0.279867
$883$	−330.794	−0.0126072	−0.00630358	−	0.999980i	$-0.502007\pi$
$883$	−330.794	−0.0126072	−0.00630358	+	0.999980i	$0.502007\pi$
$884$	0	0
$885$	−29647.5	−1.12609
$886$	−16819.6	−0.637772
$887$	− 42906.5i	− 1.62419i	−0.583523	−	0.812097i	$-0.698326\pi$
$887$	− 42906.5i	− 1.62419i	0.583523	−	0.812097i	$-0.301674\pi$
$888$	7013.90	0.265057
$889$	− 22173.1i	− 0.836515i
$890$	− 56291.7i	− 2.12012i
$891$	7494.33i	0.281784i
$892$	−12120.4	−0.454954
$893$	−4561.99	−0.170953
$894$	13804.5i	0.516433i
$895$	− 69334.3i	− 2.58948i
$896$	− 38389.9i	− 1.43138i
$897$	359.021	0.0133639
$898$	− 35089.7i	− 1.30396i
$899$	−10926.3	−0.405353
$900$	9704.91	0.359441
$901$	0	0
$902$	10423.2	0.384762
$903$	−2936.12	−0.108204
$904$	− 23950.1i	− 0.881158i
$905$	73138.6	2.68642
$906$	21733.7i	0.796968i
$907$	6908.93i	0.252930i	0.991971	+	0.126465i	$0.0403631\pi$
$907$	6908.93i	0.252930i	−0.991971	+	0.126465i	$0.959637\pi$
$908$	4471.60i	0.163431i
$909$	17407.2	0.635162
$910$	547.833	0.0199566
$911$	− 33032.6i	− 1.20134i	−0.799497	−	0.600669i	$-0.794901\pi$
$911$	− 33032.6i	− 1.20134i	0.799497	−	0.600669i	$-0.205099\pi$
$912$	52599.2i	1.90980i
$913$	− 2579.94i	− 0.0935196i
$914$	16367.7	0.592337
$915$	− 74198.9i	− 2.68081i
$916$	15742.4	0.567841
$917$	−4818.01	−0.173506
$918$	0	0
$919$	8167.05	0.293151	0.146576	−	0.989199i	$-0.453175\pi$
$919$	8167.05	0.293151	0.146576	+	0.989199i	$0.453175\pi$
$920$	46389.2	1.66240
$921$	5348.33i	0.191350i
$922$	−7710.44	−0.275412
$923$	413.155i	0.0147337i
$924$	3106.83i	0.110614i
$925$	14719.4i	0.523211i
$926$	57700.3	2.04768
$927$	6140.03	0.217546
$928$	− 25318.3i	− 0.895599i
$929$	− 1864.03i	− 0.0658308i	−0.999458	−	0.0329154i	$-0.989521\pi$
$929$	− 1864.03i	− 0.0658308i	0.999458	−	0.0329154i	$-0.0104792\pi$
$930$	19939.5i	0.703056i
$931$	15029.1	0.529063
$932$	− 6955.87i	− 0.244471i
$933$	48832.6	1.71351
$934$	−63823.6	−2.23594
$935$	0	0
$936$	108.210	0.00377881
$937$	44071.0	1.53654	0.768270	−	0.640126i	$-0.221118\pi$
$937$	44071.0	1.53654	0.768270	+	0.640126i	$0.221118\pi$
$938$	29430.5i	1.02445i
$939$	−7936.84	−0.275835
$940$	− 2202.11i	− 0.0764095i
$941$	33030.1i	1.14426i	0.820162	+	0.572131i	$0.193883\pi$
$941$	33030.1i	1.14426i	−0.820162	+	0.572131i	$0.806117\pi$
$942$	− 14259.1i	− 0.493191i
$943$	53580.5	1.85029
$944$	18586.5	0.640825
$945$	31708.2i	1.09150i
$946$	548.476i	0.0188504i
$947$	9340.37i	0.320508i	0.987076	+	0.160254i	$0.0512314\pi$
$947$	9340.37i	0.320508i	−0.987076	+	0.160254i	$0.948769\pi$
$948$	11619.6	0.398088
$949$	8.91285i	0 0.000304872i
$950$	−80846.5	−2.76106
$951$	37034.0	1.26278
$952$	0	0
$953$	21211.2	0.720984	0.360492	−	0.932762i	$-0.382609\pi$
$953$	21211.2	0.720984	0.360492	+	0.932762i	$0.382609\pi$
$954$	−3007.46	−0.102065
$955$	16761.9i	0.567959i
$956$	15722.9	0.531919
$957$	− 11966.9i	− 0.404217i
$958$	− 13226.9i	− 0.446077i
$959$	− 35919.3i	− 1.20948i
$960$	31576.7	1.06160
$961$	27375.1	0.918904
$962$	− 79.5389i	− 0.00266574i
$963$	− 4471.88i	− 0.149641i
$964$	− 15880.2i	− 0.530567i
$965$	−91136.3	−3.04019
$966$	64895.4i	2.16147i
$967$	−21688.1	−0.721242	−0.360621	−	0.932712i	$-0.617435\pi$
$967$	−21688.1	−0.721242	−0.360621	+	0.932712i	$0.617435\pi$
$968$	22165.9	0.735991
$969$	0	0
$970$	62660.8	2.07414
$971$	22478.6	0.742918	0.371459	−	0.928449i	$-0.378858\pi$
$971$	22478.6	0.742918	0.371459	+	0.928449i	$0.378858\pi$
$972$	− 10148.5i	− 0.334892i
$973$	22858.5	0.753146
$974$	23639.7i	0.777685i
$975$	623.214i	0.0204706i
$976$	46516.5i	1.52557i
$977$	−21731.7	−0.711626	−0.355813	−	0.934557i	$-0.615796\pi$
$977$	−21731.7	−0.711626	−0.355813	+	0.934557i	$0.615796\pi$
$978$	−40200.2	−1.31438
$979$	7470.03i	0.243864i
$980$	7254.65i	0.236471i
$981$	− 21347.3i	− 0.694769i
$982$	−2507.82	−0.0814947
$983$	− 34128.8i	− 1.10737i	−0.832728	−	0.553683i	$-0.813222\pi$
$983$	− 34128.8i	− 1.10737i	0.832728	−	0.553683i	$-0.186778\pi$
$984$	44319.3	1.43582
$985$	28526.7	0.922779
$986$	0	0
$987$	−6356.59	−0.204998
$988$	107.513	0.00346197
$989$	2819.44i	0.0906501i
$990$	−7957.68	−0.255466
$991$	− 39750.8i	− 1.27419i	−0.770784	−	0.637097i	$-0.780135\pi$
$991$	− 39750.8i	− 1.27419i	0.770784	−	0.637097i	$-0.219865\pi$
$992$	− 5598.16i	− 0.179175i
$993$	− 48151.4i	− 1.53881i
$994$	−74680.5	−2.38302
$995$	59066.2	1.88193
$996$	5316.34i	0.169131i
$997$	54112.0i	1.71890i	0.511219	+	0.859450i	$0.329194\pi$
$997$	54112.0i	1.71890i	−0.511219	+	0.859450i	$0.670806\pi$
$998$	− 4666.91i	− 0.148024i
$999$	4603.66	0.145799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.8		24
17.4	even	4		289.4.a.h.1.9	✓	12
17.13	even	4		289.4.a.i.1.9	yes	12
17.16	even	2	inner	289.4.b.f.288.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.9	✓	12	17.4	even	4
289.4.a.i.1.9	yes	12	17.13	even	4
289.4.b.f.288.7		24	17.16	even	2	inner
289.4.b.f.288.8		24	1.1	even	1	trivial

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.25752 q^{2} +6.51757i q^{3} +2.61146 q^{4} -19.1073i q^{5} -21.2311i q^{6} +22.0995i q^{7} +17.5533 q^{8} -15.4787 q^{9} +O(q^{10})\)	\(q-3.25752 q^{2} +6.51757i q^{3} +2.61146 q^{4} -19.1073i q^{5} -21.2311i q^{6} +22.0995i q^{7} +17.5533 q^{8} -15.4787 q^{9} +62.2426i q^{10} -8.25972i q^{11} +17.0204i q^{12} -0.398269 q^{13} -71.9898i q^{14} +124.533 q^{15} -78.0720 q^{16} +50.4221 q^{18} -103.371 q^{19} -49.8980i q^{20} -144.035 q^{21} +26.9062i q^{22} +138.311i q^{23} +114.405i q^{24} -240.090 q^{25} +1.29737 q^{26} +75.0910i q^{27} +57.7120i q^{28} -222.296i q^{29} -405.670 q^{30} -49.1520i q^{31} +113.895 q^{32} +53.8333 q^{33} +422.263 q^{35} -40.4219 q^{36} -61.3077i q^{37} +336.734 q^{38} -2.59575i q^{39} -335.397i q^{40} -387.391i q^{41} +469.198 q^{42} +20.3847 q^{43} -21.5699i q^{44} +295.756i q^{45} -450.553i q^{46} +44.1322 q^{47} -508.839i q^{48} -145.389 q^{49} +782.100 q^{50} -1.04006 q^{52} -59.6456 q^{53} -244.611i q^{54} -157.821 q^{55} +387.920i q^{56} -673.728i q^{57} +724.134i q^{58} -238.069 q^{59} +325.214 q^{60} -595.816i q^{61} +160.114i q^{62} -342.071i q^{63} +253.561 q^{64} +7.60987i q^{65} -175.363 q^{66} -408.815 q^{67} -901.454 q^{69} -1375.53 q^{70} -1037.38i q^{71} -271.702 q^{72} -22.3789i q^{73} +199.711i q^{74} -1564.80i q^{75} -269.949 q^{76} +182.536 q^{77} +8.45571i q^{78} -682.688i q^{79} +1491.75i q^{80} -907.335 q^{81} +1261.93i q^{82} +312.352 q^{83} -376.142 q^{84} -66.4037 q^{86} +1448.83 q^{87} -144.985i q^{88} -904.392 q^{89} -963.433i q^{90} -8.80157i q^{91} +361.195i q^{92} +320.351 q^{93} -143.762 q^{94} +1975.15i q^{95} +742.317i q^{96} -1006.72i q^{97} +473.610 q^{98} +127.849i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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