LMFDB - Embedded newform 325.4.b.e.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,4,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	325.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:	$19.1756207519$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.4.b.e.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{2} -3.68466i q^{3} +1.43845 q^{4} -9.43845 q^{6} -18.1771i q^{7} -24.1771i q^{8} +13.4233 q^{9} +O(q^{10})$	\(q-2.56155i q^{2} -3.68466i q^{3} +1.43845 q^{4} -9.43845 q^{6} -18.1771i q^{7} -24.1771i q^{8} +13.4233 q^{9} +64.7386 q^{11} -5.30019i q^{12} -13.0000i q^{13} -46.5616 q^{14} -50.4233 q^{16} +25.5464i q^{17} -34.3845i q^{18} +107.970 q^{19} -66.9763 q^{21} -165.831i q^{22} +73.2614i q^{23} -89.0843 q^{24} -33.3002 q^{26} -148.946i q^{27} -26.1468i q^{28} -175.909 q^{29} -113.093 q^{31} -64.2547i q^{32} -238.540i q^{33} +65.4384 q^{34} +19.3087 q^{36} -114.808i q^{37} -276.570i q^{38} -47.9006 q^{39} -69.6458 q^{41} +171.563i q^{42} +438.302i q^{43} +93.1231 q^{44} +187.663 q^{46} +31.9479i q^{47} +185.793i q^{48} +12.5937 q^{49} +94.1298 q^{51} -18.6998i q^{52} +2.84658i q^{53} -381.533 q^{54} -439.469 q^{56} -397.831i q^{57} +450.600i q^{58} -71.6325 q^{59} -920.695 q^{61} +289.693i q^{62} -243.996i q^{63} -567.978 q^{64} -611.032 q^{66} +444.280i q^{67} +36.7471i q^{68} +269.943 q^{69} -541.719 q^{71} -324.536i q^{72} +764.004i q^{73} -294.086 q^{74} +155.309 q^{76} -1176.76i q^{77} +122.700i q^{78} +421.538 q^{79} -186.386 q^{81} +178.401i q^{82} +603.797i q^{83} -96.3419 q^{84} +1122.73 q^{86} +648.165i q^{87} -1565.19i q^{88} +1159.88 q^{89} -236.302 q^{91} +105.383i q^{92} +416.708i q^{93} +81.8362 q^{94} -236.757 q^{96} -583.269i q^{97} -32.2595i q^{98} +869.006 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{4} - 46 q^{6} - 70 q^{9}+O(q^{10}) $ 4 * q + 14 * q^4 - 46 * q^6 - 70 * q^9	$ 4 q + 14 q^{4} - 46 q^{6} - 70 q^{9} + 160 q^{11} - 178 q^{14} - 78 q^{16} + 168 q^{19} - 606 q^{21} - 546 q^{24} - 26 q^{26} + 88 q^{29} - 172 q^{31} + 270 q^{34} - 500 q^{36} + 130 q^{39} - 460 q^{41} + 356 q^{44} - 8 q^{46} - 766 q^{49} + 962 q^{51} - 182 q^{54} - 2030 q^{56} + 736 q^{59} - 2116 q^{61} - 1538 q^{64} - 1636 q^{66} - 1592 q^{69} - 262 q^{71} - 294 q^{74} + 44 q^{76} + 2016 q^{79} + 244 q^{81} - 2818 q^{84} + 2718 q^{86} + 1440 q^{89} + 234 q^{91} + 1622 q^{94} - 3726 q^{96} + 260 q^{99}+O(q^{100}) $ 4 * q + 14 * q^4 - 46 * q^6 - 70 * q^9 + 160 * q^11 - 178 * q^14 - 78 * q^16 + 168 * q^19 - 606 * q^21 - 546 * q^24 - 26 * q^26 + 88 * q^29 - 172 * q^31 + 270 * q^34 - 500 * q^36 + 130 * q^39 - 460 * q^41 + 356 * q^44 - 8 * q^46 - 766 * q^49 + 962 * q^51 - 182 * q^54 - 2030 * q^56 + 736 * q^59 - 2116 * q^61 - 1538 * q^64 - 1636 * q^66 - 1592 * q^69 - 262 * q^71 - 294 * q^74 + 44 * q^76 + 2016 * q^79 + 244 * q^81 - 2818 * q^84 + 2718 * q^86 + 1440 * q^89 + 234 * q^91 + 1622 * q^94 - 3726 * q^96 + 260 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.56155i	− 0.905646i	−0.891601	−	0.452823i	$-0.850417\pi$
$2$	− 2.56155i	− 0.905646i	0.891601	−	0.452823i	$-0.149583\pi$
$3$	− 3.68466i	− 0.709113i	−0.935035	−	0.354556i	$-0.884632\pi$
$3$	− 3.68466i	− 0.709113i	0.935035	−	0.354556i	$-0.115368\pi$
$4$	1.43845	0.179806
$5$	0	0
$5$	0	0
$6$	−9.43845	−0.642205
$7$	− 18.1771i	− 0.981470i	−0.871309	−	0.490735i	$-0.836728\pi$
$7$	− 18.1771i	− 0.981470i	0.871309	−	0.490735i	$-0.163272\pi$
$8$	− 24.1771i	− 1.06849i
$9$	13.4233	0.497159
$10$	0	0
$11$	64.7386	1.77449	0.887247	−	0.461295i	$-0.152615\pi$
$11$	64.7386	1.77449	0.887247	+	0.461295i	$0.152615\pi$
$12$	− 5.30019i	− 0.127503i
$13$	− 13.0000i	− 0.277350i
$13$	− 13.0000i	− 0.277350i
$14$	−46.5616	−0.888864
$15$	0	0
$16$	−50.4233	−0.787864
$17$	25.5464i	0.364465i	0.983255	+	0.182233i	$0.0583324\pi$
$17$	25.5464i	0.364465i	−0.983255	+	0.182233i	$0.941668\pi$
$18$	− 34.3845i	− 0.450250i
$19$	107.970	1.30368	0.651841	−	0.758356i	$-0.273997\pi$
$19$	107.970	1.30368	0.651841	+	0.758356i	$0.273997\pi$
$20$	0	0
$21$	−66.9763	−0.695973
$22$	− 165.831i	− 1.60706i
$23$	73.2614i	0.664176i	0.943248	+	0.332088i	$0.107753\pi$
$23$	73.2614i	0.664176i	−0.943248	+	0.332088i	$0.892247\pi$
$24$	−89.0843	−0.757677
$25$	0	0
$26$	−33.3002	−0.251181
$27$	− 148.946i	− 1.06165i
$28$	− 26.1468i	− 0.176474i
$29$	−175.909	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$29$	−175.909	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$30$	0	0
$31$	−113.093	−0.655228	−0.327614	−	0.944812i	$-0.606245\pi$
$31$	−113.093	−0.655228	−0.327614	+	0.944812i	$0.606245\pi$
$32$	− 64.2547i	− 0.354961i
$33$	− 238.540i	− 1.25832i
$34$	65.4384	0.330077
$35$	0	0
$36$	19.3087	0.0893921
$37$	− 114.808i	− 0.510116i	−0.966926	−	0.255058i	$-0.917905\pi$
$37$	− 114.808i	− 0.510116i	0.966926	−	0.255058i	$-0.0820945\pi$
$38$	− 276.570i	− 1.18067i
$39$	−47.9006	−0.196673
$40$	0	0
$41$	−69.6458	−0.265289	−0.132645	−	0.991164i	$-0.542347\pi$
$41$	−69.6458	−0.265289	−0.132645	+	0.991164i	$0.542347\pi$
$42$	171.563i	0.630305i
$43$	438.302i	1.55443i	0.629236	+	0.777214i	$0.283368\pi$
$43$	438.302i	1.55443i	−0.629236	+	0.777214i	$0.716632\pi$
$44$	93.1231	0.319064
$45$	0	0
$46$	187.663	0.601508
$47$	31.9479i	0.0991506i	0.998770	+	0.0495753i	$0.0157868\pi$
$47$	31.9479i	0.0991506i	−0.998770	+	0.0495753i	$0.984213\pi$
$48$	185.793i	0.558684i
$49$	12.5937	0.0367164
$50$	0	0
$51$	94.1298	0.258447
$52$	− 18.6998i	− 0.0498692i
$53$	2.84658i	0.00737752i	0.999993	+	0.00368876i	$0.00117417\pi$
$53$	2.84658i	0.00737752i	−0.999993	+	0.00368876i	$0.998826\pi$
$54$	−381.533	−0.961483
$55$	0	0
$56$	−439.469	−1.04869
$57$	− 397.831i	− 0.924457i
$58$	450.600i	1.02012i
$59$	−71.6325	−0.158064	−0.0790319	−	0.996872i	$-0.525183\pi$
$59$	−71.6325	−0.158064	−0.0790319	+	0.996872i	$0.525183\pi$
$60$	0	0
$61$	−920.695	−1.93251	−0.966253	−	0.257593i	$-0.917071\pi$
$61$	−920.695	−1.93251	−0.966253	+	0.257593i	$0.917071\pi$
$62$	289.693i	0.593404i
$63$	− 243.996i	− 0.487947i
$64$	−567.978	−1.10933
$65$	0	0
$66$	−611.032	−1.13959
$67$	444.280i	0.810112i	0.914292	+	0.405056i	$0.132748\pi$
$67$	444.280i	0.810112i	−0.914292	+	0.405056i	$0.867252\pi$
$68$	36.7471i	0.0655330i
$69$	269.943	0.470976
$70$	0	0
$71$	−541.719	−0.905496	−0.452748	−	0.891639i	$-0.649556\pi$
$71$	−541.719	−0.905496	−0.452748	+	0.891639i	$0.649556\pi$
$72$	− 324.536i	− 0.531207i
$73$	764.004i	1.22493i	0.790498	+	0.612465i	$0.209822\pi$
$73$	764.004i	1.22493i	−0.790498	+	0.612465i	$0.790178\pi$
$74$	−294.086	−0.461984
$75$	0	0
$76$	155.309	0.234410
$77$	− 1176.76i	− 1.74161i
$78$	122.700i	0.178116i
$79$	421.538	0.600338	0.300169	−	0.953886i	$-0.402957\pi$
$79$	421.538	0.600338	0.300169	+	0.953886i	$0.402957\pi$
$80$	0	0
$81$	−186.386	−0.255674
$82$	178.401i	0.240258i
$83$	603.797i	0.798498i	0.916842	+	0.399249i	$0.130729\pi$
$83$	603.797i	0.798498i	−0.916842	+	0.399249i	$0.869271\pi$
$84$	−96.3419	−0.125140
$85$	0	0
$86$	1122.73	1.40776
$87$	648.165i	0.798742i
$88$	− 1565.19i	− 1.89602i
$89$	1159.88	1.38143	0.690715	−	0.723127i	$-0.257296\pi$
$89$	1159.88	1.38143	0.690715	+	0.723127i	$0.257296\pi$
$90$	0	0
$91$	−236.302	−0.272211
$92$	105.383i	0.119423i
$93$	416.708i	0.464631i
$94$	81.8362	0.0897953
$95$	0	0
$96$	−236.757	−0.251707
$97$	− 583.269i	− 0.610536i	−0.952267	−	0.305268i	$-0.901254\pi$
$97$	− 583.269i	− 0.610536i	0.952267	−	0.305268i	$-0.0987460\pi$
$98$	− 32.2595i	− 0.0332521i
$99$	869.006	0.882206
$100$	0	0
$101$	921.740	0.908085	0.454043	−	0.890980i	$-0.349981\pi$
$101$	921.740	0.908085	0.454043	+	0.890980i	$0.349981\pi$
$102$	− 241.118i	− 0.234061i
$103$	− 930.712i	− 0.890347i	−0.895444	−	0.445174i	$-0.853142\pi$
$103$	− 930.712i	− 0.890347i	0.895444	−	0.445174i	$-0.146858\pi$
$104$	−314.302	−0.296345
$105$	0	0
$106$	7.29168	0.00668142
$107$	− 857.383i	− 0.774638i	−0.921946	−	0.387319i	$-0.873401\pi$
$107$	− 857.383i	− 0.774638i	0.921946	−	0.387319i	$-0.126599\pi$
$108$	− 214.251i	− 0.190892i
$109$	−671.853	−0.590384	−0.295192	−	0.955438i	$-0.595384\pi$
$109$	−671.853	−0.590384	−0.295192	+	0.955438i	$0.595384\pi$
$110$	0	0
$111$	−423.027	−0.361730
$112$	916.548i	0.773265i
$113$	641.474i	0.534024i	0.963693	+	0.267012i	$0.0860364\pi$
$113$	641.474i	0.534024i	−0.963693	+	0.267012i	$0.913964\pi$
$114$	−1019.07	−0.837231
$115$	0	0
$116$	−253.036	−0.202533
$117$	− 174.503i	− 0.137887i
$118$	183.491i	0.143150i
$119$	464.359	0.357712
$120$	0	0
$121$	2860.09	2.14883
$122$	2358.41i	1.75017i
$123$	256.621i	0.188120i
$124$	−162.678	−0.117814
$125$	0	0
$126$	−625.009	−0.441907
$127$	553.174i	0.386506i	0.981149	+	0.193253i	$0.0619038\pi$
$127$	553.174i	0.386506i	−0.981149	+	0.193253i	$0.938096\pi$
$128$	940.868i	0.649702i
$129$	1614.99	1.10227
$130$	0	0
$131$	2056.40	1.37152	0.685758	−	0.727830i	$-0.259471\pi$
$131$	2056.40	1.37152	0.685758	+	0.727830i	$0.259471\pi$
$132$	− 343.127i	− 0.226253i
$133$	− 1962.57i	− 1.27952i
$134$	1138.05	0.733674
$135$	0	0
$136$	617.637	0.389426
$137$	1808.57i	1.12786i	0.825824	+	0.563928i	$0.190710\pi$
$137$	1808.57i	1.12786i	−0.825824	+	0.563928i	$0.809290\pi$
$138$	− 691.474i	− 0.426537i
$139$	−1493.64	−0.911428	−0.455714	−	0.890126i	$-0.650616\pi$
$139$	−1493.64	−0.911428	−0.455714	+	0.890126i	$0.650616\pi$
$140$	0	0
$141$	117.717	0.0703090
$142$	1387.64i	0.820058i
$143$	− 841.602i	− 0.492156i
$144$	−676.847	−0.391694
$145$	0	0
$146$	1957.04	1.10935
$147$	− 46.4036i	− 0.0260361i
$148$	− 165.145i	− 0.0917218i
$149$	2759.02	1.51696	0.758482	−	0.651694i	$-0.225941\pi$
$149$	2759.02	1.51696	0.758482	+	0.651694i	$0.225941\pi$
$150$	0	0
$151$	−976.355	−0.526190	−0.263095	−	0.964770i	$-0.584743\pi$
$151$	−976.355	−0.526190	−0.263095	+	0.964770i	$0.584743\pi$
$152$	− 2610.39i	− 1.39297i
$153$	342.917i	0.181197i
$154$	−3014.33	−1.57728
$155$	0	0
$156$	−68.9024	−0.0353629
$157$	564.875i	0.287146i	0.989640	+	0.143573i	$0.0458592\pi$
$157$	564.875i	0.287146i	−0.989640	+	0.143573i	$0.954141\pi$
$158$	− 1079.79i	− 0.543694i
$159$	10.4887	0.00523149
$160$	0	0
$161$	1331.68	0.651869
$162$	477.438i	0.231550i
$163$	1508.53i	0.724892i	0.932005	+	0.362446i	$0.118058\pi$
$163$	1508.53i	0.724892i	−0.932005	+	0.362446i	$0.881942\pi$
$164$	−100.182	−0.0477005
$165$	0	0
$166$	1546.66	0.723157
$167$	− 592.521i	− 0.274555i	−0.990533	−	0.137277i	$-0.956165\pi$
$167$	− 592.521i	− 0.274555i	0.990533	−	0.137277i	$-0.0438351\pi$
$168$	1619.29i	0.743638i
$169$	−169.000	−0.0769231
$170$	0	0
$171$	1449.31	0.648137
$172$	630.474i	0.279495i
$173$	− 4495.57i	− 1.97568i	−0.155488	−	0.987838i	$-0.549695\pi$
$173$	− 4495.57i	− 1.97568i	0.155488	−	0.987838i	$-0.450305\pi$
$174$	1660.31	0.723377
$175$	0	0
$176$	−3264.34	−1.39806
$177$	263.941i	0.112085i
$178$	− 2971.10i	− 1.25109i
$179$	154.285	0.0644235	0.0322117	−	0.999481i	$-0.489745\pi$
$179$	154.285	0.0644235	0.0322117	+	0.999481i	$0.489745\pi$
$180$	0	0
$181$	1071.35	0.439959	0.219979	−	0.975505i	$-0.429401\pi$
$181$	1071.35	0.439959	0.219979	+	0.975505i	$0.429401\pi$
$182$	605.300i	0.246527i
$183$	3392.45i	1.37037i
$184$	1771.25	0.709663
$185$	0	0
$186$	1067.42	0.420791
$187$	1653.84i	0.646742i
$188$	45.9554i	0.0178279i
$189$	−2707.40	−1.04198
$190$	0	0
$191$	677.203	0.256548	0.128274	−	0.991739i	$-0.459056\pi$
$191$	677.203	0.256548	0.128274	+	0.991739i	$0.459056\pi$
$192$	2092.81i	0.786642i
$193$	1321.68i	0.492936i	0.969151	+	0.246468i	$0.0792700\pi$
$193$	1321.68i	0.492936i	−0.969151	+	0.246468i	$0.920730\pi$
$194$	−1494.07	−0.552929
$195$	0	0
$196$	18.1154	0.00660183
$197$	− 1267.37i	− 0.458356i	−0.973385	−	0.229178i	$-0.926396\pi$
$197$	− 1267.37i	− 0.458356i	0.973385	−	0.229178i	$-0.0736038\pi$
$198$	− 2226.00i	− 0.798966i
$199$	−2396.24	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$199$	−2396.24	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$200$	0	0
$201$	1637.02	0.574460
$202$	− 2361.09i	− 0.822403i
$203$	3197.51i	1.10552i
$204$	135.401	0.0464703
$205$	0	0
$206$	−2384.07	−0.806339
$207$	983.409i	0.330201i
$208$	655.503i	0.218514i
$209$	6989.81	2.31337
$210$	0	0
$211$	−91.5539	−0.0298712	−0.0149356	−	0.999888i	$-0.504754\pi$
$211$	−91.5539	−0.0298712	−0.0149356	+	0.999888i	$0.504754\pi$
$212$	4.09466i	0.00132652i
$213$	1996.05i	0.642099i
$214$	−2196.23	−0.701548
$215$	0	0
$216$	−3601.08	−1.13436
$217$	2055.70i	0.643087i
$218$	1720.99i	0.534679i
$219$	2815.09	0.868613
$220$	0	0
$221$	332.103	0.101085
$222$	1083.61i	0.327599i
$223$	1235.42i	0.370985i	0.982646	+	0.185493i	$0.0593880\pi$
$223$	1235.42i	0.370985i	−0.982646	+	0.185493i	$0.940612\pi$
$224$	−1167.96	−0.348383
$225$	0	0
$226$	1643.17	0.483637
$227$	− 3301.66i	− 0.965370i	−0.875794	−	0.482685i	$-0.839662\pi$
$227$	− 3301.66i	− 0.965370i	0.875794	−	0.482685i	$-0.160338\pi$
$228$	− 572.260i	− 0.166223i
$229$	−211.283	−0.0609694	−0.0304847	−	0.999535i	$-0.509705\pi$
$229$	−211.283	−0.0609694	−0.0304847	+	0.999535i	$0.509705\pi$
$230$	0	0
$231$	−4335.96	−1.23500
$232$	4252.97i	1.20354i
$233$	− 256.724i	− 0.0721827i	−0.999348	−	0.0360913i	$-0.988509\pi$
$233$	− 256.724i	− 0.0721827i	0.999348	−	0.0360913i	$-0.0114907\pi$
$234$	−446.998	−0.124877
$235$	0	0
$236$	−103.040	−0.0284208
$237$	− 1553.22i	− 0.425708i
$238$	− 1189.48i	− 0.323960i
$239$	3549.62	0.960694	0.480347	−	0.877078i	$-0.340511\pi$
$239$	3549.62	0.960694	0.480347	+	0.877078i	$0.340511\pi$
$240$	0	0
$241$	−5030.10	−1.34447	−0.672235	−	0.740338i	$-0.734665\pi$
$241$	−5030.10	−1.34447	−0.672235	+	0.740338i	$0.734665\pi$
$242$	− 7326.27i	− 1.94608i
$243$	− 3334.77i	− 0.880353i
$244$	−1324.37	−0.347476
$245$	0	0
$246$	657.349	0.170370
$247$	− 1403.61i	− 0.361576i
$248$	2734.25i	0.700102i
$249$	2224.79	0.566226
$250$	0	0
$251$	−718.784	−0.180754	−0.0903770	−	0.995908i	$-0.528807\pi$
$251$	−718.784	−0.180754	−0.0903770	+	0.995908i	$0.528807\pi$
$252$	− 350.976i	− 0.0877357i
$253$	4742.84i	1.17858i
$254$	1416.98	0.350038
$255$	0	0
$256$	−2133.74	−0.520933
$257$	− 1280.79i	− 0.310871i	−0.987846	−	0.155435i	$-0.950322\pi$
$257$	− 1280.79i	− 0.310871i	0.987846	−	0.155435i	$-0.0496780\pi$
$258$	− 4136.89i	− 0.998262i
$259$	−2086.87	−0.500663
$260$	0	0
$261$	−2361.28	−0.559998
$262$	− 5267.58i	− 1.24211i
$263$	5225.55i	1.22517i	0.790403	+	0.612587i	$0.209871\pi$
$263$	5225.55i	1.22517i	−0.790403	+	0.612587i	$0.790129\pi$
$264$	−5767.19	−1.34449
$265$	0	0
$266$	−5027.24	−1.15880
$267$	− 4273.77i	− 0.979590i
$268$	639.074i	0.145663i
$269$	−6443.80	−1.46054	−0.730270	−	0.683158i	$-0.760606\pi$
$269$	−6443.80	−1.46054	−0.730270	+	0.683158i	$0.760606\pi$
$270$	0	0
$271$	−3929.93	−0.880909	−0.440455	−	0.897775i	$-0.645183\pi$
$271$	−3929.93	−0.880909	−0.440455	+	0.897775i	$0.645183\pi$
$272$	− 1288.13i	− 0.287149i
$273$	870.692i	0.193028i
$274$	4632.74	1.02144
$275$	0	0
$276$	388.299	0.0846842
$277$	5884.40i	1.27639i	0.769876	+	0.638194i	$0.220318\pi$
$277$	5884.40i	1.27639i	−0.769876	+	0.638194i	$0.779682\pi$
$278$	3826.03i	0.825431i
$279$	−1518.08	−0.325752
$280$	0	0
$281$	3529.79	0.749358	0.374679	−	0.927155i	$-0.377753\pi$
$281$	3529.79	0.749358	0.374679	+	0.927155i	$0.377753\pi$
$282$	− 301.538i	− 0.0636750i
$283$	− 2611.00i	− 0.548438i	−0.961667	−	0.274219i	$-0.911581\pi$
$283$	− 2611.00i	− 0.548438i	0.961667	−	0.274219i	$-0.0884193\pi$
$284$	−779.234	−0.162813
$285$	0	0
$286$	−2155.81	−0.445719
$287$	1265.96i	0.260373i
$288$	− 862.510i	− 0.176472i
$289$	4260.38	0.867165
$290$	0	0
$291$	−2149.15	−0.432939
$292$	1098.98i	0.220250i
$293$	− 5491.03i	− 1.09484i	−0.836857	−	0.547422i	$-0.815609\pi$
$293$	− 5491.03i	− 1.09484i	0.836857	−	0.547422i	$-0.184391\pi$
$294$	−118.865	−0.0235795
$295$	0	0
$296$	−2775.72	−0.545052
$297$	− 9642.56i	− 1.88390i
$298$	− 7067.37i	− 1.37383i
$299$	952.398	0.184209
$300$	0	0
$301$	7967.05	1.52563
$302$	2500.99i	0.476542i
$303$	− 3396.30i	− 0.643935i
$304$	−5444.19	−1.02712
$305$	0	0
$306$	878.399	0.164100
$307$	7307.59i	1.35852i	0.733897	+	0.679261i	$0.237700\pi$
$307$	7307.59i	1.35852i	−0.733897	+	0.679261i	$0.762300\pi$
$308$	− 1692.71i	− 0.313152i
$309$	−3429.36	−0.631357
$310$	0	0
$311$	7904.92	1.44131	0.720654	−	0.693295i	$-0.243842\pi$
$311$	7904.92	1.44131	0.720654	+	0.693295i	$0.243842\pi$
$312$	1158.10i	0.210142i
$313$	10002.4i	1.80629i	0.429336	+	0.903145i	$0.358748\pi$
$313$	10002.4i	1.80629i	−0.429336	+	0.903145i	$0.641252\pi$
$314$	1446.96	0.260053
$315$	0	0
$316$	606.360	0.107944
$317$	6230.81i	1.10397i	0.833856	+	0.551983i	$0.186129\pi$
$317$	6230.81i	1.10397i	−0.833856	+	0.551983i	$0.813871\pi$
$318$	− 26.8673i	− 0.00473788i
$319$	−11388.1	−1.99878
$320$	0	0
$321$	−3159.16	−0.549306
$322$	− 3411.16i	− 0.590362i
$323$	2758.24i	0.475147i
$324$	−268.107	−0.0459717
$325$	0	0
$326$	3864.19	0.656495
$327$	2475.55i	0.418649i
$328$	1683.83i	0.283458i
$329$	580.719	0.0973134
$330$	0	0
$331$	−4634.51	−0.769594	−0.384797	−	0.923001i	$-0.625729\pi$
$331$	−4634.51	−0.769594	−0.384797	+	0.923001i	$0.625729\pi$
$332$	868.531i	0.143575i
$333$	− 1541.10i	− 0.253609i
$334$	−1517.77	−0.248649
$335$	0	0
$336$	3377.17	0.548332
$337$	− 3029.82i	− 0.489747i	−0.969555	−	0.244874i	$-0.921254\pi$
$337$	− 3029.82i	− 0.489747i	0.969555	−	0.244874i	$-0.0787464\pi$
$338$	432.902i	0.0696651i
$339$	2363.61	0.378684
$340$	0	0
$341$	−7321.47	−1.16270
$342$	− 3712.48i	− 0.586982i
$343$	− 6463.66i	− 1.01751i
$344$	10596.9	1.66089
$345$	0	0
$346$	−11515.6	−1.78926
$347$	− 2841.60i	− 0.439611i	−0.975544	−	0.219805i	$-0.929458\pi$
$347$	− 2841.60i	− 0.439611i	0.975544	−	0.219805i	$-0.0705423\pi$
$348$	932.351i	0.143619i
$349$	−7565.68	−1.16040	−0.580202	−	0.814472i	$-0.697027\pi$
$349$	−7565.68	−1.16040	−0.580202	+	0.814472i	$0.697027\pi$
$350$	0	0
$351$	−1936.30	−0.294450
$352$	− 4159.76i	− 0.629875i
$353$	− 2339.44i	− 0.352736i	−0.984324	−	0.176368i	$-0.943565\pi$
$353$	− 2339.44i	− 0.352736i	0.984324	−	0.176368i	$-0.0564348\pi$
$354$	676.100	0.101509
$355$	0	0
$356$	1668.43	0.248389
$357$	− 1711.00i	− 0.253658i
$358$	− 395.209i	− 0.0583449i
$359$	2531.68	0.372192	0.186096	−	0.982532i	$-0.440417\pi$
$359$	2531.68	0.372192	0.186096	+	0.982532i	$0.440417\pi$
$360$	0	0
$361$	4798.45	0.699585
$362$	− 2744.31i	− 0.398447i
$363$	− 10538.5i	− 1.52376i
$364$	−339.908	−0.0489451
$365$	0	0
$366$	8689.93	1.24107
$367$	− 6577.81i	− 0.935583i	−0.883839	−	0.467792i	$-0.845050\pi$
$367$	− 6577.81i	− 0.935583i	0.883839	−	0.467792i	$-0.154950\pi$
$368$	− 3694.08i	− 0.523280i
$369$	−934.876	−0.131891
$370$	0	0
$371$	51.7426	0.00724081
$372$	599.413i	0.0835433i
$373$	2902.72i	0.402942i	0.979495	+	0.201471i	$0.0645721\pi$
$373$	2902.72i	0.402942i	−0.979495	+	0.201471i	$0.935428\pi$
$374$	4236.40	0.585719
$375$	0	0
$376$	772.407	0.105941
$377$	2286.82i	0.312406i
$378$	6935.16i	0.943667i
$379$	1865.73	0.252866	0.126433	−	0.991975i	$-0.459647\pi$
$379$	1865.73	0.252866	0.126433	+	0.991975i	$0.459647\pi$
$380$	0	0
$381$	2038.26	0.274076
$382$	− 1734.69i	− 0.232342i
$383$	10836.0i	1.44567i	0.691019	+	0.722837i	$0.257162\pi$
$383$	10836.0i	1.44567i	−0.691019	+	0.722837i	$0.742838\pi$
$384$	3466.78	0.460712
$385$	0	0
$386$	3385.55	0.446425
$387$	5883.46i	0.772798i
$388$	− 839.001i	− 0.109778i
$389$	9520.34	1.24088	0.620438	−	0.784256i	$-0.286955\pi$
$389$	9520.34	1.24088	0.620438	+	0.784256i	$0.286955\pi$
$390$	0	0
$391$	−1871.56	−0.242069
$392$	− 304.480i	− 0.0392310i
$393$	− 7577.13i	− 0.972559i
$394$	−3246.43	−0.415108
$395$	0	0
$396$	1250.02	0.158626
$397$	10108.8i	1.27796i	0.769225	+	0.638978i	$0.220642\pi$
$397$	10108.8i	1.27796i	−0.769225	+	0.638978i	$0.779358\pi$
$398$	6138.10i	0.773053i
$399$	−7231.41	−0.907327
$400$	0	0
$401$	2084.38	0.259573	0.129787	−	0.991542i	$-0.458571\pi$
$401$	2084.38	0.259573	0.129787	+	0.991542i	$0.458571\pi$
$402$	− 4193.32i	− 0.520258i
$403$	1470.21i	0.181728i
$404$	1325.88	0.163279
$405$	0	0
$406$	8190.60	1.00121
$407$	− 7432.50i	− 0.905197i
$408$	− 2275.78i	− 0.276147i
$409$	9716.53	1.17470	0.587349	−	0.809334i	$-0.300172\pi$
$409$	9716.53	1.17470	0.587349	+	0.809334i	$0.300172\pi$
$410$	0	0
$411$	6663.95	0.799777
$412$	− 1338.78i	− 0.160090i
$413$	1302.07i	0.155135i
$414$	2519.05	0.299045
$415$	0	0
$416$	−835.311	−0.0984483
$417$	5503.54i	0.646305i
$418$	− 17904.8i	− 2.09510i
$419$	−13381.9	−1.56026	−0.780129	−	0.625619i	$-0.784847\pi$
$419$	−13381.9	−1.56026	−0.780129	+	0.625619i	$0.784847\pi$
$420$	0	0
$421$	−9463.37	−1.09553	−0.547763	−	0.836633i	$-0.684521\pi$
$421$	−9463.37	−1.09553	−0.547763	+	0.836633i	$0.684521\pi$
$422$	234.520i	0.0270527i
$423$	428.846i	0.0492936i
$424$	68.8221	0.00788278
$425$	0	0
$426$	5112.98	0.581514
$427$	16735.5i	1.89670i
$428$	− 1233.30i	− 0.139285i
$429$	−3101.02	−0.348994
$430$	0	0
$431$	4852.28	0.542288	0.271144	−	0.962539i	$-0.412598\pi$
$431$	4852.28	0.542288	0.271144	+	0.962539i	$0.412598\pi$
$432$	7510.35i	0.836439i
$433$	− 8208.00i	− 0.910973i	−0.890243	−	0.455486i	$-0.849465\pi$
$433$	− 8208.00i	− 0.910973i	0.890243	−	0.455486i	$-0.150535\pi$
$434$	5265.78	0.582409
$435$	0	0
$436$	−966.425	−0.106155
$437$	7910.01i	0.865874i
$438$	− 7211.01i	− 0.786656i
$439$	2993.80	0.325481	0.162741	−	0.986669i	$-0.447967\pi$
$439$	2993.80	0.325481	0.162741	+	0.986669i	$0.447967\pi$
$440$	0	0
$441$	169.049	0.0182539
$442$	− 850.700i	− 0.0915468i
$443$	9743.67i	1.04500i	0.852639	+	0.522501i	$0.175001\pi$
$443$	9743.67i	1.04500i	−0.852639	+	0.522501i	$0.824999\pi$
$444$	−608.503	−0.0650411
$445$	0	0
$446$	3164.59	0.335981
$447$	− 10166.0i	− 1.07570i
$448$	10324.2i	1.08878i
$449$	561.459	0.0590131	0.0295065	−	0.999565i	$-0.490606\pi$
$449$	561.459	0.0590131	0.0295065	+	0.999565i	$0.490606\pi$
$450$	0	0
$451$	−4508.78	−0.470754
$452$	922.726i	0.0960207i
$453$	3597.54i	0.373128i
$454$	−8457.38	−0.874283
$455$	0	0
$456$	−9618.40	−0.987770
$457$	− 13758.4i	− 1.40830i	−0.710054	−	0.704148i	$-0.751329\pi$
$457$	− 13758.4i	− 1.40830i	0.710054	−	0.704148i	$-0.248671\pi$
$458$	541.213i	0.0552166i
$459$	3805.03	0.386936
$460$	0	0
$461$	12009.2	1.21329	0.606644	−	0.794974i	$-0.292515\pi$
$461$	12009.2	1.21329	0.606644	+	0.794974i	$0.292515\pi$
$462$	11106.8i	1.11847i
$463$	13635.7i	1.36870i	0.729156	+	0.684348i	$0.239913\pi$
$463$	13635.7i	1.36870i	−0.729156	+	0.684348i	$0.760087\pi$
$464$	8869.91	0.887447
$465$	0	0
$466$	−657.613	−0.0653719
$467$	− 8821.95i	− 0.874157i	−0.899423	−	0.437079i	$-0.856013\pi$
$467$	− 8821.95i	− 0.874157i	0.899423	−	0.437079i	$-0.143987\pi$
$468$	− 251.013i	− 0.0247929i
$469$	8075.72	0.795100
$470$	0	0
$471$	2081.37	0.203619
$472$	1731.87i	0.168889i
$473$	28375.1i	2.75832i
$474$	−3978.66	−0.385540
$475$	0	0
$476$	667.956	0.0643187
$477$	38.2105i	0.00366780i
$478$	− 9092.54i	− 0.870049i
$479$	14620.0	1.39459	0.697293	−	0.716786i	$-0.254388\pi$
$479$	14620.0	1.39459	0.697293	+	0.716786i	$0.254388\pi$
$480$	0	0
$481$	−1492.50	−0.141481
$482$	12884.9i	1.21761i
$483$	− 4906.78i	− 0.462249i
$484$	4114.09	0.386372
$485$	0	0
$486$	−8542.20	−0.797288
$487$	9798.86i	0.911763i	0.890040	+	0.455882i	$0.150676\pi$
$487$	9798.86i	0.911763i	−0.890040	+	0.455882i	$0.849324\pi$
$488$	22259.7i	2.06486i
$489$	5558.43	0.514030
$490$	0	0
$491$	−10836.1	−0.995977	−0.497989	−	0.867184i	$-0.665928\pi$
$491$	−10836.1	−0.995977	−0.497989	+	0.867184i	$0.665928\pi$
$492$	369.136i	0.0338251i
$493$	− 4493.84i	− 0.410532i
$494$	−3595.41	−0.327460
$495$	0	0
$496$	5702.51	0.516230
$497$	9846.86i	0.888717i
$498$	− 5698.91i	− 0.512800i
$499$	−2589.96	−0.232349	−0.116175	−	0.993229i	$-0.537063\pi$
$499$	−2589.96	−0.232349	−0.116175	+	0.993229i	$0.537063\pi$
$500$	0	0
$501$	−2183.24	−0.194690
$502$	1841.20i	0.163699i
$503$	− 17067.5i	− 1.51292i	−0.654038	−	0.756462i	$-0.726926\pi$
$503$	− 17067.5i	− 1.51292i	0.654038	−	0.756462i	$-0.273074\pi$
$504$	−5899.12	−0.521364
$505$	0	0
$506$	12149.0	1.06737
$507$	622.707i	0.0545471i
$508$	795.712i	0.0694961i
$509$	1012.89	0.0882038	0.0441019	−	0.999027i	$-0.485957\pi$
$509$	1012.89	0.0882038	0.0441019	+	0.999027i	$0.485957\pi$
$510$	0	0
$511$	13887.4	1.20223
$512$	12992.6i	1.12148i
$513$	− 16081.7i	− 1.38406i
$514$	−3280.82	−0.281539
$515$	0	0
$516$	2323.08	0.198194
$517$	2068.26i	0.175942i
$518$	5345.63i	0.453424i
$519$	−16564.6	−1.40098
$520$	0	0
$521$	−14367.7	−1.20818	−0.604089	−	0.796917i	$-0.706463\pi$
$521$	−14367.7	−1.20818	−0.604089	+	0.796917i	$0.706463\pi$
$522$	6048.54i	0.507160i
$523$	− 16219.9i	− 1.35611i	−0.735010	−	0.678057i	$-0.762822\pi$
$523$	− 16219.9i	− 1.35611i	0.735010	−	0.678057i	$-0.237178\pi$
$524$	2958.02	0.246607
$525$	0	0
$526$	13385.5	1.10957
$527$	− 2889.11i	− 0.238808i
$528$	12028.0i	0.991382i
$529$	6799.77	0.558870
$530$	0	0
$531$	−961.545	−0.0785828
$532$	− 2823.06i	− 0.230066i
$533$	905.396i	0.0735780i
$534$	−10947.5	−0.887161
$535$	0	0
$536$	10741.4	0.865593
$537$	− 568.488i	− 0.0456835i
$538$	16506.1i	1.32273i
$539$	815.301	0.0651530
$540$	0	0
$541$	17592.2	1.39806	0.699029	−	0.715094i	$-0.253616\pi$
$541$	17592.2	1.39806	0.699029	+	0.715094i	$0.253616\pi$
$542$	10066.7i	0.797792i
$543$	− 3947.55i	− 0.311980i
$544$	1641.48	0.129371
$545$	0	0
$546$	2230.32	0.174815
$547$	− 10504.6i	− 0.821103i	−0.911837	−	0.410552i	$-0.865336\pi$
$547$	− 10504.6i	− 0.821103i	0.911837	−	0.410552i	$-0.134664\pi$
$548$	2601.53i	0.202795i
$549$	−12358.8	−0.960763
$550$	0	0
$551$	−18992.8	−1.46846
$552$	− 6526.44i	− 0.503231i
$553$	− 7662.33i	− 0.589214i
$554$	15073.2	1.15596
$555$	0	0
$556$	−2148.52	−0.163880
$557$	507.558i	0.0386102i	0.999814	+	0.0193051i	$0.00614539\pi$
$557$	507.558i	0.0386102i	−0.999814	+	0.0193051i	$0.993855\pi$
$558$	3888.64i	0.295016i
$559$	5697.93	0.431121
$560$	0	0
$561$	6093.83	0.458613
$562$	− 9041.75i	− 0.678653i
$563$	− 3443.14i	− 0.257746i	−0.991661	−	0.128873i	$-0.958864\pi$
$563$	− 3443.14i	− 0.257746i	0.991661	−	0.128873i	$-0.0411360\pi$
$564$	169.330	0.0126420
$565$	0	0
$566$	−6688.21	−0.496690
$567$	3387.96i	0.250936i
$568$	13097.2i	0.967509i
$569$	−23972.2	−1.76620	−0.883098	−	0.469189i	$-0.844546\pi$
$569$	−23972.2	−1.76620	−0.883098	+	0.469189i	$0.844546\pi$
$570$	0	0
$571$	−7458.32	−0.546622	−0.273311	−	0.961926i	$-0.588119\pi$
$571$	−7458.32	−0.546622	−0.273311	+	0.961926i	$0.588119\pi$
$572$	− 1210.60i	− 0.0884926i
$573$	− 2495.26i	− 0.181922i
$574$	3242.82	0.235806
$575$	0	0
$576$	−7624.14	−0.551515
$577$	− 5669.57i	− 0.409059i	−0.978860	−	0.204530i	$-0.934434\pi$
$577$	− 5669.57i	− 0.409059i	0.978860	−	0.204530i	$-0.0655665\pi$
$578$	− 10913.2i	− 0.785344i
$579$	4869.94	0.349547
$580$	0	0
$581$	10975.3	0.783702
$582$	5505.15i	0.392089i
$583$	184.284i	0.0130914i
$584$	18471.4	1.30882
$585$	0	0
$586$	−14065.6	−0.991541
$587$	− 1017.39i	− 0.0715371i	−0.999360	−	0.0357685i	$-0.988612\pi$
$587$	− 1017.39i	− 0.0715371i	0.999360	−	0.0357685i	$-0.0113879\pi$
$588$	− 66.7491i	− 0.00468144i
$589$	−12210.6	−0.854208
$590$	0	0
$591$	−4669.81	−0.325026
$592$	5788.99i	0.401902i
$593$	− 10198.2i	− 0.706221i	−0.935582	−	0.353111i	$-0.885124\pi$
$593$	− 10198.2i	− 0.706221i	0.935582	−	0.353111i	$-0.114876\pi$
$594$	−24699.9	−1.70615
$595$	0	0
$596$	3968.70	0.272759
$597$	8829.33i	0.605294i
$598$	− 2439.62i	− 0.166828i
$599$	−12516.3	−0.853763	−0.426881	−	0.904308i	$-0.640388\pi$
$599$	−12516.3	−0.853763	−0.426881	+	0.904308i	$0.640388\pi$
$600$	0	0
$601$	9627.46	0.653431	0.326716	−	0.945123i	$-0.394058\pi$
$601$	9627.46	0.653431	0.326716	+	0.945123i	$0.394058\pi$
$602$	− 20408.0i	− 1.38168i
$603$	5963.70i	0.402754i
$604$	−1404.44	−0.0946120
$605$	0	0
$606$	−8699.80	−0.583177
$607$	− 6667.20i	− 0.445821i	−0.974839	−	0.222910i	$-0.928444\pi$
$607$	− 6667.20i	− 0.445821i	0.974839	−	0.222910i	$-0.0715557\pi$
$608$	− 6937.56i	− 0.462755i
$609$	11781.7	0.783942
$610$	0	0
$611$	415.323	0.0274994
$612$	493.268i	0.0325803i
$613$	− 23085.4i	− 1.52106i	−0.649302	−	0.760530i	$-0.724939\pi$
$613$	− 23085.4i	− 1.52106i	0.649302	−	0.760530i	$-0.275061\pi$
$614$	18718.8	1.23034
$615$	0	0
$616$	−28450.6	−1.86089
$617$	− 3049.24i	− 0.198959i	−0.995040	−	0.0994796i	$-0.968282\pi$
$617$	− 3049.24i	− 0.198959i	0.995040	−	0.0994796i	$-0.0317178\pi$
$618$	8784.48i	0.571786i
$619$	−7296.58	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$619$	−7296.58	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$620$	0	0
$621$	10912.0	0.705126
$622$	− 20248.9i	− 1.30531i
$623$	− 21083.3i	− 1.35583i
$624$	2415.30	0.154951
$625$	0	0
$626$	25621.7	1.63586
$627$	− 25755.1i	− 1.64044i
$628$	812.543i	0.0516305i
$629$	2932.92	0.185920
$630$	0	0
$631$	−23829.5	−1.50339	−0.751694	−	0.659512i	$-0.770763\pi$
$631$	−23829.5	−1.50339	−0.751694	+	0.659512i	$0.770763\pi$
$632$	− 10191.6i	− 0.641453i
$633$	337.345i	0.0211821i
$634$	15960.5	0.999801
$635$	0	0
$636$	15.0874	0.000940653 0
$637$	− 163.718i	− 0.0101833i
$638$	29171.3i	1.81019i
$639$	−7271.65	−0.450175
$640$	0	0
$641$	13405.3	0.826016	0.413008	−	0.910727i	$-0.364478\pi$
$641$	13405.3	0.826016	0.413008	+	0.910727i	$0.364478\pi$
$642$	8092.36i	0.497477i
$643$	5251.51i	0.322083i	0.986948	+	0.161042i	$0.0514853\pi$
$643$	5251.51i	0.322083i	−0.986948	+	0.161042i	$0.948515\pi$
$644$	1915.55	0.117210
$645$	0	0
$646$	7065.37	0.430315
$647$	− 21611.4i	− 1.31319i	−0.754244	−	0.656595i	$-0.771996\pi$
$647$	− 21611.4i	− 1.31319i	0.754244	−	0.656595i	$-0.228004\pi$
$648$	4506.28i	0.273184i
$649$	−4637.39	−0.280483
$650$	0	0
$651$	7574.54	0.456021
$652$	2169.94i	0.130340i
$653$	− 21595.8i	− 1.29420i	−0.762406	−	0.647099i	$-0.775982\pi$
$653$	− 21595.8i	− 1.29420i	0.762406	−	0.647099i	$-0.224018\pi$
$654$	6341.25	0.379148
$655$	0	0
$656$	3511.77	0.209012
$657$	10255.4i	0.608985i
$658$	− 1487.54i	− 0.0881314i
$659$	16642.6	0.983768	0.491884	−	0.870661i	$-0.336308\pi$
$659$	16642.6	0.983768	0.491884	+	0.870661i	$0.336308\pi$
$660$	0	0
$661$	26981.1	1.58766	0.793831	−	0.608139i	$-0.208084\pi$
$661$	26981.1	1.58766	0.793831	+	0.608139i	$0.208084\pi$
$662$	11871.5i	0.696980i
$663$	− 1223.69i	− 0.0716803i
$664$	14598.1	0.853185
$665$	0	0
$666$	−3947.60	−0.229680
$667$	− 12887.3i	− 0.748126i
$668$	− 852.310i	− 0.0493665i
$669$	4552.09	0.263070
$670$	0	0
$671$	−59604.5	−3.42922
$672$	4303.55i	0.247043i
$673$	11149.2i	0.638591i	0.947655	+	0.319296i	$0.103446\pi$
$673$	11149.2i	0.638591i	−0.947655	+	0.319296i	$0.896554\pi$
$674$	−7761.04	−0.443537
$675$	0	0
$676$	−243.098	−0.0138312
$677$	− 3314.33i	− 0.188154i	−0.995565	−	0.0940769i	$-0.970010\pi$
$677$	− 3314.33i	− 0.188154i	0.995565	−	0.0940769i	$-0.0299900\pi$
$678$	− 6054.51i	− 0.342953i
$679$	−10602.1	−0.599223
$680$	0	0
$681$	−12165.5	−0.684556
$682$	18754.3i	1.05299i
$683$	24505.2i	1.37287i	0.727193	+	0.686433i	$0.240824\pi$
$683$	24505.2i	1.37287i	−0.727193	+	0.686433i	$0.759176\pi$
$684$	2084.75	0.116539
$685$	0	0
$686$	−16557.0	−0.921500
$687$	778.506i	0.0432342i
$688$	− 22100.6i	− 1.22468i
$689$	37.0056	0.00204616
$690$	0	0
$691$	−21752.8	−1.19756	−0.598782	−	0.800912i	$-0.704348\pi$
$691$	−21752.8	−1.19756	−0.598782	+	0.800912i	$0.704348\pi$
$692$	− 6466.64i	− 0.355238i
$693$	− 15796.0i	− 0.865858i
$694$	−7278.90	−0.398132
$695$	0	0
$696$	15670.7	0.853445
$697$	− 1779.20i	− 0.0966887i
$698$	19379.9i	1.05092i
$699$	−945.941	−0.0511857
$700$	0	0
$701$	34250.9	1.84542	0.922709	−	0.385496i	$-0.125970\pi$
$701$	34250.9	1.84542	0.922709	+	0.385496i	$0.125970\pi$
$702$	4959.93i	0.266667i
$703$	− 12395.8i	− 0.665028i
$704$	−36770.1	−1.96850
$705$	0	0
$706$	−5992.59	−0.319454
$707$	− 16754.6i	− 0.891259i
$708$	379.666i	0.0201536i
$709$	5527.11	0.292771	0.146386	−	0.989228i	$-0.453236\pi$
$709$	5527.11	0.292771	0.146386	+	0.989228i	$0.453236\pi$
$710$	0	0
$711$	5658.43	0.298464
$712$	− 28042.6i	− 1.47604i
$713$	− 8285.33i	− 0.435187i
$714$	−4382.83	−0.229724
$715$	0	0
$716$	221.931	0.0115837
$717$	− 13079.1i	− 0.681241i
$718$	− 6485.02i	− 0.337074i
$719$	3777.78	0.195949	0.0979745	−	0.995189i	$-0.468764\pi$
$719$	3777.78	0.195949	0.0979745	+	0.995189i	$0.468764\pi$
$720$	0	0
$721$	−16917.6	−0.873849
$722$	− 12291.5i	− 0.633576i
$723$	18534.2i	0.953380i
$724$	1541.08	0.0791072
$725$	0	0
$726$	−26994.8	−1.37999
$727$	− 19076.8i	− 0.973204i	−0.873624	−	0.486602i	$-0.838236\pi$
$727$	− 19076.8i	− 0.973204i	0.873624	−	0.486602i	$-0.161764\pi$
$728$	5713.09i	0.290853i
$729$	−17319.9	−0.879944
$730$	0	0
$731$	−11197.0	−0.566535
$732$	4879.86i	0.246400i
$733$	7997.30i	0.402984i	0.979490	+	0.201492i	$0.0645790\pi$
$733$	7997.30i	0.402984i	−0.979490	+	0.201492i	$0.935421\pi$
$734$	−16849.4	−0.847307
$735$	0	0
$736$	4707.39	0.235756
$737$	28762.1i	1.43754i
$738$	2394.74i	0.119446i
$739$	−28983.6	−1.44273	−0.721367	−	0.692553i	$-0.756486\pi$
$739$	−28983.6	−1.44273	−0.721367	+	0.692553i	$0.756486\pi$
$740$	0	0
$741$	−5171.81	−0.256398
$742$	− 132.541i	− 0.00655761i
$743$	− 19145.4i	− 0.945324i	−0.881244	−	0.472662i	$-0.843293\pi$
$743$	− 19145.4i	− 0.945324i	0.881244	−	0.472662i	$-0.156707\pi$
$744$	10074.8	0.496451
$745$	0	0
$746$	7435.47	0.364922
$747$	8104.95i	0.396981i
$748$	2378.96i	0.116288i
$749$	−15584.7	−0.760284
$750$	0	0
$751$	−25516.9	−1.23985	−0.619923	−	0.784663i	$-0.712836\pi$
$751$	−25516.9	−1.23985	−0.619923	+	0.784663i	$0.712836\pi$
$752$	− 1610.92i	− 0.0781172i
$753$	2648.47i	0.128175i
$754$	5857.80	0.282929
$755$	0	0
$756$	−3894.46	−0.187355
$757$	17230.6i	0.827289i	0.910438	+	0.413645i	$0.135744\pi$
$757$	17230.6i	0.827289i	−0.910438	+	0.413645i	$0.864256\pi$
$758$	− 4779.17i	− 0.229007i
$759$	17475.7	0.835744
$760$	0	0
$761$	−2343.06	−0.111611	−0.0558053	−	0.998442i	$-0.517773\pi$
$761$	−2343.06	−0.111611	−0.0558053	+	0.998442i	$0.517773\pi$
$762$	− 5221.11i	− 0.248216i
$763$	12212.3i	0.579444i
$764$	974.121	0.0461289
$765$	0	0
$766$	27756.9	1.30927
$767$	931.223i	0.0438390i
$768$	7862.11i	0.369400i
$769$	7100.18	0.332950	0.166475	−	0.986046i	$-0.446761\pi$
$769$	7100.18	0.332950	0.166475	+	0.986046i	$0.446761\pi$
$770$	0	0
$771$	−4719.29	−0.220442
$772$	1901.17i	0.0886328i
$773$	12270.4i	0.570940i	0.958388	+	0.285470i	$0.0921498\pi$
$773$	12270.4i	0.570940i	−0.958388	+	0.285470i	$0.907850\pi$
$774$	15070.8	0.699881
$775$	0	0
$776$	−14101.7	−0.652349
$777$	7689.40i	0.355027i
$778$	− 24386.9i	− 1.12379i
$779$	−7519.64	−0.345852
$780$	0	0
$781$	−35070.1	−1.60680
$782$	4794.11i	0.219229i
$783$	26201.0i	1.19584i
$784$	−635.017	−0.0289275
$785$	0	0
$786$	−19409.2	−0.880794
$787$	− 3425.04i	− 0.155133i	−0.996987	−	0.0775663i	$-0.975285\pi$
$787$	− 3425.04i	− 0.155133i	0.996987	−	0.0775663i	$-0.0247150\pi$
$788$	− 1823.04i	− 0.0824151i
$789$	19254.3	0.868787
$790$	0	0
$791$	11660.1	0.524129
$792$	− 21010.0i	− 0.942624i
$793$	11969.0i	0.535981i
$794$	25894.3	1.15737
$795$	0	0
$796$	−3446.87	−0.153481
$797$	11781.1i	0.523600i	0.965122	+	0.261800i	$0.0843160\pi$
$797$	11781.1i	0.523600i	−0.965122	+	0.261800i	$0.915684\pi$
$798$	18523.6i	0.821717i
$799$	−816.154	−0.0361370
$800$	0	0
$801$	15569.4	0.686790
$802$	− 5339.24i	− 0.235081i
$803$	49460.6i	2.17363i
$804$	2354.77	0.103291
$805$	0	0
$806$	3766.01	0.164581
$807$	23743.2i	1.03569i
$808$	− 22285.0i	− 0.970276i
$809$	−18910.1	−0.821810	−0.410905	−	0.911678i	$-0.634787\pi$
$809$	−18910.1	−0.821810	−0.410905	+	0.911678i	$0.634787\pi$
$810$	0	0
$811$	12803.3	0.554359	0.277180	−	0.960818i	$-0.410600\pi$
$811$	12803.3	0.554359	0.277180	+	0.960818i	$0.410600\pi$
$812$	4599.45i	0.198780i
$813$	14480.5i	0.624664i
$814$	−19038.7	−0.819788
$815$	0	0
$816$	−4746.33	−0.203621
$817$	47323.3i	2.02648i
$818$	− 24889.4i	− 1.06386i
$819$	−3171.95	−0.135332
$820$	0	0
$821$	19335.1	0.821923	0.410962	−	0.911653i	$-0.365193\pi$
$821$	19335.1	0.821923	0.410962	+	0.911653i	$0.365193\pi$
$822$	− 17070.1i	− 0.724315i
$823$	− 2125.90i	− 0.0900417i	−0.998986	−	0.0450209i	$-0.985665\pi$
$823$	− 2125.90i	− 0.0900417i	0.998986	−	0.0450209i	$-0.0143354\pi$
$824$	−22501.9	−0.951324
$825$	0	0
$826$	3335.32	0.140497
$827$	6989.24i	0.293881i	0.989145	+	0.146941i	$0.0469426\pi$
$827$	6989.24i	0.293881i	−0.989145	+	0.146941i	$0.953057\pi$
$828$	1414.58i	0.0593721i
$829$	32649.7	1.36788	0.683938	−	0.729540i	$-0.260266\pi$
$829$	32649.7	1.36788	0.683938	+	0.729540i	$0.260266\pi$
$830$	0	0
$831$	21682.0	0.905103
$832$	7383.72i	0.307673i
$833$	321.724i	0.0133819i
$834$	14097.6	0.585324
$835$	0	0
$836$	10054.5	0.415958
$837$	16844.7i	0.695626i
$838$	34278.4i	1.41304i
$839$	4038.23	0.166168	0.0830841	−	0.996543i	$-0.473523\pi$
$839$	4038.23	0.166168	0.0830841	+	0.996543i	$0.473523\pi$
$840$	0	0
$841$	6555.00	0.268769
$842$	24240.9i	0.992159i
$843$	− 13006.1i	− 0.531380i
$844$	−131.695	−0.00537102
$845$	0	0
$846$	1098.51	0.0446426
$847$	− 51988.1i	− 2.10901i
$848$	− 143.534i	− 0.00581248i
$849$	−9620.64	−0.388904
$850$	0	0
$851$	8410.97	0.338807
$852$	2871.21i	0.115453i
$853$	8114.12i	0.325700i	0.986651	+	0.162850i	$0.0520687\pi$
$853$	8114.12i	0.325700i	−0.986651	+	0.162850i	$0.947931\pi$
$854$	42869.0	1.71774
$855$	0	0
$856$	−20729.0	−0.827690
$857$	22298.1i	0.888786i	0.895832	+	0.444393i	$0.146581\pi$
$857$	22298.1i	0.888786i	−0.895832	+	0.444393i	$0.853419\pi$
$858$	7943.42i	0.316065i
$859$	−33550.5	−1.33263	−0.666315	−	0.745670i	$-0.732130\pi$
$859$	−33550.5	−1.33263	−0.666315	+	0.745670i	$0.732130\pi$
$860$	0	0
$861$	4664.62	0.184634
$862$	− 12429.4i	− 0.491121i
$863$	− 14120.5i	− 0.556972i	−0.960440	−	0.278486i	$-0.910167\pi$
$863$	− 14120.5i	− 0.556972i	0.960440	−	0.278486i	$-0.0898326\pi$
$864$	−9570.49	−0.376846
$865$	0	0
$866$	−21025.2	−0.825018
$867$	− 15698.1i	− 0.614918i
$868$	2957.01i	0.115631i
$869$	27289.8	1.06530
$870$	0	0
$871$	5775.64	0.224685
$872$	16243.4i	0.630817i
$873$	− 7829.39i	− 0.303533i
$874$	20261.9	0.784175
$875$	0	0
$876$	4049.36	0.156182
$877$	1941.69i	0.0747619i	0.999301	+	0.0373809i	$0.0119015\pi$
$877$	1941.69i	0.0747619i	−0.999301	+	0.0373809i	$0.988099\pi$
$878$	− 7668.78i	− 0.294771i
$879$	−20232.6	−0.776368
$880$	0	0
$881$	−790.231	−0.0302197	−0.0151099	−	0.999886i	$-0.504810\pi$
$881$	−790.231	−0.0302197	−0.0151099	+	0.999886i	$0.504810\pi$
$882$	− 433.029i	− 0.0165316i
$883$	− 36638.6i	− 1.39636i	−0.715922	−	0.698180i	$-0.753993\pi$
$883$	− 36638.6i	− 1.39636i	0.715922	−	0.698180i	$-0.246007\pi$
$884$	477.713	0.0181756
$885$	0	0
$886$	24958.9	0.946401
$887$	40686.3i	1.54015i	0.637954	+	0.770075i	$0.279781\pi$
$887$	40686.3i	1.54015i	−0.637954	+	0.770075i	$0.720219\pi$
$888$	10227.6i	0.386503i
$889$	10055.1	0.379344
$890$	0	0
$891$	−12066.4	−0.453692
$892$	1777.08i	0.0667053i
$893$	3449.40i	0.129261i
$894$	−26040.9	−0.974202
$895$	0	0
$896$	17102.2	0.637663
$897$	− 3509.26i	− 0.130625i
$898$	− 1438.21i	− 0.0534449i
$899$	19894.0	0.738046
$900$	0	0
$901$	−72.7200	−0.00268885
$902$	11549.5i	0.426336i
$903$	− 29355.9i	− 1.08184i
$904$	15509.0	0.570598
$905$	0	0
$906$	9215.28	0.337922
$907$	10464.4i	0.383093i	0.981484	+	0.191547i	$0.0613503\pi$
$907$	10464.4i	0.383093i	−0.981484	+	0.191547i	$0.938650\pi$
$908$	− 4749.26i	− 0.173579i
$909$	12372.8	0.451463
$910$	0	0
$911$	−35611.5	−1.29513	−0.647563	−	0.762011i	$-0.724212\pi$
$911$	−35611.5	−1.29513	−0.647563	+	0.762011i	$0.724212\pi$
$912$	20060.0i	0.728346i
$913$	39089.0i	1.41693i
$914$	−35242.9	−1.27542
$915$	0	0
$916$	−303.920	−0.0109627
$917$	− 37379.4i	− 1.34610i
$918$	− 9746.80i	− 0.350427i
$919$	−1077.25	−0.0386674	−0.0193337	−	0.999813i	$-0.506154\pi$
$919$	−1077.25	−0.0386674	−0.0193337	+	0.999813i	$0.506154\pi$
$920$	0	0
$921$	26926.0	0.963346
$922$	− 30762.3i	− 1.09881i
$923$	7042.34i	0.251139i
$924$	−6237.04	−0.222060
$925$	0	0
$926$	34928.6	1.23955
$927$	− 12493.2i	− 0.442644i
$928$	11303.0i	0.399826i
$929$	−55733.8	−1.96832	−0.984159	−	0.177290i	$-0.943267\pi$
$929$	−55733.8	−1.96832	−0.984159	+	0.177290i	$0.943267\pi$
$930$	0	0
$931$	1359.74	0.0478665
$932$	− 369.284i	− 0.0129789i
$933$	− 29126.9i	− 1.02205i
$934$	−22597.9	−0.791677
$935$	0	0
$936$	−4218.97	−0.147330
$937$	3198.60i	0.111519i	0.998444	+	0.0557596i	$0.0177581\pi$
$937$	3198.60i	0.111519i	−0.998444	+	0.0557596i	$0.982242\pi$
$938$	− 20686.4i	− 0.720079i
$939$	36855.4	1.28086
$940$	0	0
$941$	8823.35	0.305667	0.152834	−	0.988252i	$-0.451160\pi$
$941$	8823.35	0.305667	0.152834	+	0.988252i	$0.451160\pi$
$942$	− 5331.54i	− 0.184407i
$943$	− 5102.35i	− 0.176199i
$944$	3611.95	0.124533
$945$	0	0
$946$	72684.3	2.49806
$947$	− 28290.4i	− 0.970766i	−0.874301	−	0.485383i	$-0.838680\pi$
$947$	− 28290.4i	− 0.970766i	0.874301	−	0.485383i	$-0.161320\pi$
$948$	− 2234.23i	− 0.0765447i
$949$	9932.05	0.339734
$950$	0	0
$951$	22958.4	0.782836
$952$	− 11226.8i	− 0.382210i
$953$	− 12399.0i	− 0.421452i	−0.977545	−	0.210726i	$-0.932417\pi$
$953$	− 12399.0i	− 0.421452i	0.977545	−	0.210726i	$-0.0675828\pi$
$954$	97.8783	0.00332173
$955$	0	0
$956$	5105.94	0.172739
$957$	41961.3i	1.41736i
$958$	− 37450.0i	− 1.26300i
$959$	32874.5	1.10696
$960$	0	0
$961$	−17001.0	−0.570676
$962$	3823.12i	0.128131i
$963$	− 11508.9i	− 0.385118i
$964$	−7235.53	−0.241744
$965$	0	0
$966$	−12569.0	−0.418634
$967$	26667.1i	0.886820i	0.896319	+	0.443410i	$0.146231\pi$
$967$	26667.1i	0.886820i	−0.896319	+	0.443410i	$0.853769\pi$
$968$	− 69148.6i	− 2.29599i
$969$	10163.2	0.336933
$970$	0	0
$971$	49420.7	1.63335	0.816676	−	0.577096i	$-0.195814\pi$
$971$	49420.7	1.63335	0.816676	+	0.577096i	$0.195814\pi$
$972$	− 4796.89i	− 0.158293i
$973$	27149.9i	0.894539i
$974$	25100.3	0.825735
$975$	0	0
$976$	46424.5	1.52255
$977$	− 778.759i	− 0.0255012i	−0.999919	−	0.0127506i	$-0.995941\pi$
$977$	− 778.759i	− 0.0255012i	0.999919	−	0.0127506i	$-0.00405876\pi$
$978$	− 14238.2i	− 0.465529i
$979$	75089.2	2.45134
$980$	0	0
$981$	−9018.48	−0.293515
$982$	27757.2i	0.902003i
$983$	5997.90i	0.194612i	0.995255	+	0.0973059i	$0.0310225\pi$
$983$	5997.90i	0.194612i	−0.995255	+	0.0973059i	$0.968977\pi$
$984$	6204.35	0.201004
$985$	0	0
$986$	−11511.2	−0.371797
$987$	− 2139.75i	− 0.0690062i
$988$	− 2019.01i	− 0.0650135i
$989$	−32110.6	−1.03241
$990$	0	0
$991$	8974.94	0.287688	0.143844	−	0.989600i	$-0.454054\pi$
$991$	8974.94	0.287688	0.143844	+	0.989600i	$0.454054\pi$
$992$	7266.75i	0.232580i
$993$	17076.6i	0.545729i
$994$	25223.3	0.804863
$995$	0	0
$996$	3200.24	0.101811
$997$	− 28530.2i	− 0.906280i	−0.891439	−	0.453140i	$-0.850304\pi$
$997$	− 28530.2i	− 0.906280i	0.891439	−	0.453140i	$-0.149696\pi$
$998$	6634.31i	0.210426i
$999$	−17100.2	−0.541567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.4.b.e.274.1		4
5.2	odd	4		13.4.a.b.1.2	✓	2
5.3	odd	4		325.4.a.f.1.1		2
5.4	even	2	inner	325.4.b.e.274.4		4
15.2	even	4		117.4.a.d.1.1		2
20.7	even	4		208.4.a.h.1.2		2
35.27	even	4		637.4.a.b.1.2		2
40.27	even	4		832.4.a.z.1.1		2
40.37	odd	4		832.4.a.s.1.2		2
55.32	even	4		1573.4.a.b.1.1		2
60.47	odd	4		1872.4.a.bb.1.1		2
65.2	even	12		169.4.e.f.147.4		8
65.7	even	12		169.4.e.f.23.4		8
65.12	odd	4		169.4.a.g.1.1		2
65.17	odd	12		169.4.c.j.146.2		4
65.22	odd	12		169.4.c.g.146.1		4
65.32	even	12		169.4.e.f.23.1		8
65.37	even	12		169.4.e.f.147.1		8
65.42	odd	12		169.4.c.g.22.1		4
65.47	even	4		169.4.b.f.168.4		4
65.57	even	4		169.4.b.f.168.1		4
65.62	odd	12		169.4.c.j.22.2		4
195.77	even	4		1521.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	5.2	odd	4
117.4.a.d.1.1		2	15.2	even	4
169.4.a.g.1.1		2	65.12	odd	4
169.4.b.f.168.1		4	65.57	even	4
169.4.b.f.168.4		4	65.47	even	4
169.4.c.g.22.1		4	65.42	odd	12
169.4.c.g.146.1		4	65.22	odd	12
169.4.c.j.22.2		4	65.62	odd	12
169.4.c.j.146.2		4	65.17	odd	12
169.4.e.f.23.1		8	65.32	even	12
169.4.e.f.23.4		8	65.7	even	12
169.4.e.f.147.1		8	65.37	even	12
169.4.e.f.147.4		8	65.2	even	12
208.4.a.h.1.2		2	20.7	even	4
325.4.a.f.1.1		2	5.3	odd	4
325.4.b.e.274.1		4	1.1	even	1	trivial
325.4.b.e.274.4		4	5.4	even	2	inner
637.4.a.b.1.2		2	35.27	even	4
832.4.a.s.1.2		2	40.37	odd	4
832.4.a.z.1.1		2	40.27	even	4
1521.4.a.r.1.2		2	195.77	even	4
1573.4.a.b.1.1		2	55.32	even	4
1872.4.a.bb.1.1		2	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56155i q^{2} -3.68466i q^{3} +1.43845 q^{4} -9.43845 q^{6} -18.1771i q^{7} -24.1771i q^{8} +13.4233 q^{9} +O(q^{10})\)	\(q-2.56155i q^{2} -3.68466i q^{3} +1.43845 q^{4} -9.43845 q^{6} -18.1771i q^{7} -24.1771i q^{8} +13.4233 q^{9} +64.7386 q^{11} -5.30019i q^{12} -13.0000i q^{13} -46.5616 q^{14} -50.4233 q^{16} +25.5464i q^{17} -34.3845i q^{18} +107.970 q^{19} -66.9763 q^{21} -165.831i q^{22} +73.2614i q^{23} -89.0843 q^{24} -33.3002 q^{26} -148.946i q^{27} -26.1468i q^{28} -175.909 q^{29} -113.093 q^{31} -64.2547i q^{32} -238.540i q^{33} +65.4384 q^{34} +19.3087 q^{36} -114.808i q^{37} -276.570i q^{38} -47.9006 q^{39} -69.6458 q^{41} +171.563i q^{42} +438.302i q^{43} +93.1231 q^{44} +187.663 q^{46} +31.9479i q^{47} +185.793i q^{48} +12.5937 q^{49} +94.1298 q^{51} -18.6998i q^{52} +2.84658i q^{53} -381.533 q^{54} -439.469 q^{56} -397.831i q^{57} +450.600i q^{58} -71.6325 q^{59} -920.695 q^{61} +289.693i q^{62} -243.996i q^{63} -567.978 q^{64} -611.032 q^{66} +444.280i q^{67} +36.7471i q^{68} +269.943 q^{69} -541.719 q^{71} -324.536i q^{72} +764.004i q^{73} -294.086 q^{74} +155.309 q^{76} -1176.76i q^{77} +122.700i q^{78} +421.538 q^{79} -186.386 q^{81} +178.401i q^{82} +603.797i q^{83} -96.3419 q^{84} +1122.73 q^{86} +648.165i q^{87} -1565.19i q^{88} +1159.88 q^{89} -236.302 q^{91} +105.383i q^{92} +416.708i q^{93} +81.8362 q^{94} -236.757 q^{96} -583.269i q^{97} -32.2595i q^{98} +869.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{4} - 46 q^{6} - 70 q^{9}+O(q^{10}) \) 4 * q + 14 * q^4 - 46 * q^6 - 70 * q^9	\( 4 q + 14 q^{4} - 46 q^{6} - 70 q^{9} + 160 q^{11} - 178 q^{14} - 78 q^{16} + 168 q^{19} - 606 q^{21} - 546 q^{24} - 26 q^{26} + 88 q^{29} - 172 q^{31} + 270 q^{34} - 500 q^{36} + 130 q^{39} - 460 q^{41} + 356 q^{44} - 8 q^{46} - 766 q^{49} + 962 q^{51} - 182 q^{54} - 2030 q^{56} + 736 q^{59} - 2116 q^{61} - 1538 q^{64} - 1636 q^{66} - 1592 q^{69} - 262 q^{71} - 294 q^{74} + 44 q^{76} + 2016 q^{79} + 244 q^{81} - 2818 q^{84} + 2718 q^{86} + 1440 q^{89} + 234 q^{91} + 1622 q^{94} - 3726 q^{96} + 260 q^{99}+O(q^{100}) \) 4 * q + 14 * q^4 - 46 * q^6 - 70 * q^9 + 160 * q^11 - 178 * q^14 - 78 * q^16 + 168 * q^19 - 606 * q^21 - 546 * q^24 - 26 * q^26 + 88 * q^29 - 172 * q^31 + 270 * q^34 - 500 * q^36 + 130 * q^39 - 460 * q^41 + 356 * q^44 - 8 * q^46 - 766 * q^49 + 962 * q^51 - 182 * q^54 - 2030 * q^56 + 736 * q^59 - 2116 * q^61 - 1538 * q^64 - 1636 * q^66 - 1592 * q^69 - 262 * q^71 - 294 * q^74 + 44 * q^76 + 2016 * q^79 + 244 * q^81 - 2818 * q^84 + 2718 * q^86 + 1440 * q^89 + 234 * q^91 + 1622 * q^94 - 3726 * q^96 + 260 * q^99

Introduction

L-functions

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Database

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