LMFDB - Embedded newform 6171.2.a.br.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6171,2,Mod(1,6171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6171, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6171 = 3 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6171.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.2756830873$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 34 x^{18} + 31 x^{17} + 488 x^{16} - 395 x^{15} - 3853 x^{14} + 2660 x^{13} + \cdots + 400 $ x^20 - x^19 - 34x^18 + 31x^17 + 488x^16 - 395x^15 - 3853x^14 + 2660x^13 + 18296x^12 - 10136x^11 - 53706x^10 + 21732x^9 + 96366x^8 - 24556x^7 - 100129x^6 + 12495x^5 + 53335x^4 - 2210x^3 - 11300x^2 + 200x + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 561)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.18586$ of defining polynomial
Character	$\chi$	$=$	6171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18586 q^{2} +1.00000 q^{3} +2.77797 q^{4} -0.765047 q^{5} +2.18586 q^{6} -2.36983 q^{7} +1.70052 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.18586 q^{2} +1.00000 q^{3} +2.77797 q^{4} -0.765047 q^{5} +2.18586 q^{6} -2.36983 q^{7} +1.70052 q^{8} +1.00000 q^{9} -1.67228 q^{10} +2.77797 q^{12} +0.0983702 q^{13} -5.18011 q^{14} -0.765047 q^{15} -1.83883 q^{16} +1.00000 q^{17} +2.18586 q^{18} +6.26048 q^{19} -2.12528 q^{20} -2.36983 q^{21} +7.44527 q^{23} +1.70052 q^{24} -4.41470 q^{25} +0.215023 q^{26} +1.00000 q^{27} -6.58331 q^{28} +0.421939 q^{29} -1.67228 q^{30} +1.61072 q^{31} -7.42047 q^{32} +2.18586 q^{34} +1.81303 q^{35} +2.77797 q^{36} +3.90065 q^{37} +13.6845 q^{38} +0.0983702 q^{39} -1.30098 q^{40} +5.31409 q^{41} -5.18011 q^{42} +11.0176 q^{43} -0.765047 q^{45} +16.2743 q^{46} +4.58842 q^{47} -1.83883 q^{48} -1.38391 q^{49} -9.64991 q^{50} +1.00000 q^{51} +0.273269 q^{52} +3.24539 q^{53} +2.18586 q^{54} -4.02995 q^{56} +6.26048 q^{57} +0.922298 q^{58} +8.16386 q^{59} -2.12528 q^{60} +10.4400 q^{61} +3.52080 q^{62} -2.36983 q^{63} -12.5424 q^{64} -0.0752578 q^{65} +13.5364 q^{67} +2.77797 q^{68} +7.44527 q^{69} +3.96302 q^{70} -5.66896 q^{71} +1.70052 q^{72} -14.9470 q^{73} +8.52626 q^{74} -4.41470 q^{75} +17.3914 q^{76} +0.215023 q^{78} +5.57564 q^{79} +1.40679 q^{80} +1.00000 q^{81} +11.6158 q^{82} -9.54860 q^{83} -6.58331 q^{84} -0.765047 q^{85} +24.0829 q^{86} +0.421939 q^{87} +13.4266 q^{89} -1.67228 q^{90} -0.233120 q^{91} +20.6827 q^{92} +1.61072 q^{93} +10.0296 q^{94} -4.78956 q^{95} -7.42047 q^{96} +6.54636 q^{97} -3.02503 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{2} + 20 q^{3} + 29 q^{4} + 17 q^{5} + q^{6} + q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) $ 20 * q + q^2 + 20 * q^3 + 29 * q^4 + 17 * q^5 + q^6 + q^7 + 6 * q^8 + 20 * q^9	\( 20 q + q^{2} + 20 q^{3} + 29 q^{4} + 17 q^{5} + q^{6} + q^{7} + 6 q^{8} + 20 q^{9} + 6 q^{10} + 29 q^{12} + 8 q^{13} + 21 q^{14} + 17 q^{15} + 39 q^{16} + 20 q^{17} + q^{18} + 6 q^{19} + 32 q^{20} + q^{21} + 36 q^{23} + 6 q^{24} + 39 q^{25} + 17 q^{26} + 20 q^{27} - 7 q^{28} - 4 q^{29} + 6 q^{30} + 7 q^{31} - 7 q^{32} + q^{34} - 2 q^{35} + 29 q^{36} + 12 q^{37} + 24 q^{38} + 8 q^{39} - 15 q^{40} - 6 q^{41} + 21 q^{42} - 6 q^{43} + 17 q^{45} + 9 q^{46} + 52 q^{47} + 39 q^{48} + 45 q^{49} + 18 q^{50} + 20 q^{51} + 36 q^{52} + 60 q^{53} + q^{54} + 51 q^{56} + 6 q^{57} - 15 q^{58} + 42 q^{59} + 32 q^{60} - 18 q^{61} - 55 q^{62} + q^{63} + 40 q^{64} - 8 q^{65} - 15 q^{67} + 29 q^{68} + 36 q^{69} - 31 q^{70} + 50 q^{71} + 6 q^{72} - 17 q^{73} + 2 q^{74} + 39 q^{75} + 11 q^{76} + 17 q^{78} + 3 q^{79} + 37 q^{80} + 20 q^{81} - 5 q^{82} + 4 q^{83} - 7 q^{84} + 17 q^{85} + 32 q^{86} - 4 q^{87} + 16 q^{89} + 6 q^{90} + 13 q^{91} + 86 q^{92} + 7 q^{93} - 24 q^{94} - 64 q^{95} - 7 q^{96} + 15 q^{97} - 69 q^{98}+O(q^{100}) \) 20 * q + q^2 + 20 * q^3 + 29 * q^4 + 17 * q^5 + q^6 + q^7 + 6 * q^8 + 20 * q^9 + 6 * q^10 + 29 * q^12 + 8 * q^13 + 21 * q^14 + 17 * q^15 + 39 * q^16 + 20 * q^17 + q^18 + 6 * q^19 + 32 * q^20 + q^21 + 36 * q^23 + 6 * q^24 + 39 * q^25 + 17 * q^26 + 20 * q^27 - 7 * q^28 - 4 * q^29 + 6 * q^30 + 7 * q^31 - 7 * q^32 + q^34 - 2 * q^35 + 29 * q^36 + 12 * q^37 + 24 * q^38 + 8 * q^39 - 15 * q^40 - 6 * q^41 + 21 * q^42 - 6 * q^43 + 17 * q^45 + 9 * q^46 + 52 * q^47 + 39 * q^48 + 45 * q^49 + 18 * q^50 + 20 * q^51 + 36 * q^52 + 60 * q^53 + q^54 + 51 * q^56 + 6 * q^57 - 15 * q^58 + 42 * q^59 + 32 * q^60 - 18 * q^61 - 55 * q^62 + q^63 + 40 * q^64 - 8 * q^65 - 15 * q^67 + 29 * q^68 + 36 * q^69 - 31 * q^70 + 50 * q^71 + 6 * q^72 - 17 * q^73 + 2 * q^74 + 39 * q^75 + 11 * q^76 + 17 * q^78 + 3 * q^79 + 37 * q^80 + 20 * q^81 - 5 * q^82 + 4 * q^83 - 7 * q^84 + 17 * q^85 + 32 * q^86 - 4 * q^87 + 16 * q^89 + 6 * q^90 + 13 * q^91 + 86 * q^92 + 7 * q^93 - 24 * q^94 - 64 * q^95 - 7 * q^96 + 15 * q^97 - 69 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.18586	1.54563	0.772817	−	0.634629i	$-0.218847\pi$
$2$	2.18586	1.54563	0.772817	+	0.634629i	$0.218847\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.77797	1.38898
$5$	−0.765047	−0.342139	−0.171070	−	0.985259i	$-0.554722\pi$
$5$	−0.765047	−0.342139	−0.171070	+	0.985259i	$0.554722\pi$
$6$	2.18586	0.892372
$7$	−2.36983	−0.895711	−0.447856	−	0.894106i	$-0.647812\pi$
$7$	−2.36983	−0.895711	−0.447856	+	0.894106i	$0.647812\pi$
$8$	1.70052	0.601226
$9$	1.00000	0.333333
$10$	−1.67228	−0.528822
$11$	0	0
$11$	0	0
$12$	2.77797	0.801930
$13$	0.0983702	0.0272830	0.0136415	−	0.999907i	$-0.495658\pi$
$13$	0.0983702	0.0272830	0.0136415	+	0.999907i	$0.495658\pi$
$14$	−5.18011	−1.38444
$15$	−0.765047	−0.197534
$16$	−1.83883	−0.459708
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.18586	0.515211
$19$	6.26048	1.43625	0.718126	−	0.695913i	$-0.245000\pi$
$19$	6.26048	1.43625	0.718126	+	0.695913i	$0.245000\pi$
$20$	−2.12528	−0.475226
$21$	−2.36983	−0.517139
$22$	0	0
$23$	7.44527	1.55245	0.776223	−	0.630458i	$-0.217133\pi$
$23$	7.44527	1.55245	0.776223	+	0.630458i	$0.217133\pi$
$24$	1.70052	0.347118
$25$	−4.41470	−0.882941
$26$	0.215023	0.0421695
$27$	1.00000	0.192450
$28$	−6.58331	−1.24413
$29$	0.421939	0.0783521	0.0391761	−	0.999232i	$-0.487527\pi$
$29$	0.421939	0.0783521	0.0391761	+	0.999232i	$0.487527\pi$
$30$	−1.67228	−0.305316
$31$	1.61072	0.289294	0.144647	−	0.989483i	$-0.453795\pi$
$31$	1.61072	0.289294	0.144647	+	0.989483i	$0.453795\pi$
$32$	−7.42047	−1.31177
$33$	0	0
$34$	2.18586	0.374871
$35$	1.81303	0.306458
$36$	2.77797	0.462995
$37$	3.90065	0.641263	0.320632	−	0.947204i	$-0.396105\pi$
$37$	3.90065	0.641263	0.320632	+	0.947204i	$0.396105\pi$
$38$	13.6845	2.21992
$39$	0.0983702	0.0157518
$40$	−1.30098	−0.205703
$41$	5.31409	0.829922	0.414961	−	0.909839i	$-0.363795\pi$
$41$	5.31409	0.829922	0.414961	+	0.909839i	$0.363795\pi$
$42$	−5.18011	−0.799308
$43$	11.0176	1.68017	0.840085	−	0.542455i	$-0.182505\pi$
$43$	11.0176	1.68017	0.840085	+	0.542455i	$0.182505\pi$
$44$	0	0
$45$	−0.765047	−0.114046
$46$	16.2743	2.39951
$47$	4.58842	0.669290	0.334645	−	0.942344i	$-0.391384\pi$
$47$	4.58842	0.669290	0.334645	+	0.942344i	$0.391384\pi$
$48$	−1.83883	−0.265413
$49$	−1.38391	−0.197701
$50$	−9.64991	−1.36470
$51$	1.00000	0.140028
$52$	0.273269	0.0378956
$53$	3.24539	0.445788	0.222894	−	0.974843i	$-0.428450\pi$
$53$	3.24539	0.445788	0.222894	+	0.974843i	$0.428450\pi$
$54$	2.18586	0.297457
$55$	0	0
$56$	−4.02995	−0.538525
$57$	6.26048	0.829221
$58$	0.922298	0.121104
$59$	8.16386	1.06284	0.531422	−	0.847107i	$-0.321658\pi$
$59$	8.16386	1.06284	0.531422	+	0.847107i	$0.321658\pi$
$60$	−2.12528	−0.274372
$61$	10.4400	1.33670	0.668350	−	0.743847i	$-0.267001\pi$
$61$	10.4400	1.33670	0.668350	+	0.743847i	$0.267001\pi$
$62$	3.52080	0.447143
$63$	−2.36983	−0.298570
$64$	−12.5424	−1.56780
$65$	−0.0752578	−0.00933458
$66$	0	0
$67$	13.5364	1.65373	0.826867	−	0.562398i	$-0.190121\pi$
$67$	13.5364	1.65373	0.826867	+	0.562398i	$0.190121\pi$
$68$	2.77797	0.336878
$69$	7.44527	0.896306
$70$	3.96302	0.473672
$71$	−5.66896	−0.672781	−0.336391	−	0.941723i	$-0.609206\pi$
$71$	−5.66896	−0.672781	−0.336391	+	0.941723i	$0.609206\pi$
$72$	1.70052	0.200409
$73$	−14.9470	−1.74941	−0.874704	−	0.484657i	$-0.838944\pi$
$73$	−14.9470	−1.74941	−0.874704	+	0.484657i	$0.838944\pi$
$74$	8.52626	0.991158
$75$	−4.41470	−0.509766
$76$	17.3914	1.99493
$77$	0	0
$78$	0.215023	0.0243466
$79$	5.57564	0.627308	0.313654	−	0.949537i	$-0.398447\pi$
$79$	5.57564	0.627308	0.313654	+	0.949537i	$0.398447\pi$
$80$	1.40679	0.157284
$81$	1.00000	0.111111
$82$	11.6158	1.28275
$83$	−9.54860	−1.04810	−0.524048	−	0.851689i	$-0.675579\pi$
$83$	−9.54860	−1.04810	−0.524048	+	0.851689i	$0.675579\pi$
$84$	−6.58331	−0.718298
$85$	−0.765047	−0.0829810
$86$	24.0829	2.59693
$87$	0.421939	0.0452366
$88$	0	0
$89$	13.4266	1.42322	0.711608	−	0.702577i	$-0.247967\pi$
$89$	13.4266	1.42322	0.711608	+	0.702577i	$0.247967\pi$
$90$	−1.67228	−0.176274
$91$	−0.233120	−0.0244377
$92$	20.6827	2.15632
$93$	1.61072	0.167024
$94$	10.0296	1.03448
$95$	−4.78956	−0.491399
$96$	−7.42047	−0.757349
$97$	6.54636	0.664682	0.332341	−	0.943159i	$-0.392162\pi$
$97$	6.54636	0.664682	0.332341	+	0.943159i	$0.392162\pi$
$98$	−3.02503	−0.305574
$99$	0	0
$100$	−12.2639	−1.22639
$101$	−11.6557	−1.15978	−0.579890	−	0.814694i	$-0.696905\pi$
$101$	−11.6557	−1.15978	−0.579890	+	0.814694i	$0.696905\pi$
$102$	2.18586	0.216432
$103$	−0.989913	−0.0975390	−0.0487695	−	0.998810i	$-0.515530\pi$
$103$	−0.989913	−0.0975390	−0.0487695	+	0.998810i	$0.515530\pi$
$104$	0.167281	0.0164032
$105$	1.81303	0.176934
$106$	7.09395	0.689026
$107$	−15.6889	−1.51670	−0.758352	−	0.651846i	$-0.773995\pi$
$107$	−15.6889	−1.51670	−0.758352	+	0.651846i	$0.773995\pi$
$108$	2.77797	0.267310
$109$	−0.591991	−0.0567024	−0.0283512	−	0.999598i	$-0.509026\pi$
$109$	−0.591991	−0.0567024	−0.0283512	+	0.999598i	$0.509026\pi$
$110$	0	0
$111$	3.90065	0.370233
$112$	4.35772	0.411766
$113$	−0.994406	−0.0935458	−0.0467729	−	0.998906i	$-0.514894\pi$
$113$	−0.994406	−0.0935458	−0.0467729	+	0.998906i	$0.514894\pi$
$114$	13.6845	1.28167
$115$	−5.69599	−0.531153
$116$	1.17213	0.108830
$117$	0.0983702	0.00909432
$118$	17.8450	1.64277
$119$	−2.36983	−0.217242
$120$	−1.30098	−0.118763
$121$	0	0
$122$	22.8202	2.06605
$123$	5.31409	0.479156
$124$	4.47453	0.401825
$125$	7.20269	0.644228
$126$	−5.18011	−0.461480
$127$	−11.3066	−1.00330	−0.501649	−	0.865071i	$-0.667273\pi$
$127$	−11.3066	−1.00330	−0.501649	+	0.865071i	$0.667273\pi$
$128$	−12.5750	−1.11148
$129$	11.0176	0.970047
$130$	−0.164503	−0.0144278
$131$	4.75497	0.415444	0.207722	−	0.978188i	$-0.433395\pi$
$131$	4.75497	0.415444	0.207722	+	0.978188i	$0.433395\pi$
$132$	0	0
$133$	−14.8363	−1.28647
$134$	29.5886	2.55607
$135$	−0.765047	−0.0658448
$136$	1.70052	0.145819
$137$	−9.35753	−0.799468	−0.399734	−	0.916631i	$-0.630897\pi$
$137$	−9.35753	−0.799468	−0.399734	+	0.916631i	$0.630897\pi$
$138$	16.2743	1.38536
$139$	−15.3556	−1.30245	−0.651224	−	0.758886i	$-0.725744\pi$
$139$	−15.3556	−1.30245	−0.651224	+	0.758886i	$0.725744\pi$
$140$	5.03654	0.425665
$141$	4.58842	0.386415
$142$	−12.3915	−1.03987
$143$	0	0
$144$	−1.83883	−0.153236
$145$	−0.322803	−0.0268074
$146$	−32.6719	−2.70394
$147$	−1.38391	−0.114143
$148$	10.8359	0.890704
$149$	6.98910	0.572569	0.286285	−	0.958145i	$-0.407580\pi$
$149$	6.98910	0.572569	0.286285	+	0.958145i	$0.407580\pi$
$150$	−9.64991	−0.787911
$151$	15.4391	1.25641	0.628206	−	0.778047i	$-0.283789\pi$
$151$	15.4391	1.25641	0.628206	+	0.778047i	$0.283789\pi$
$152$	10.6461	0.863512
$153$	1.00000	0.0808452
$154$	0	0
$155$	−1.23228	−0.0989789
$156$	0.273269	0.0218790
$157$	−17.9409	−1.43184	−0.715918	−	0.698184i	$-0.753992\pi$
$157$	−17.9409	−1.43184	−0.715918	+	0.698184i	$0.753992\pi$
$158$	12.1875	0.969589
$159$	3.24539	0.257376
$160$	5.67701	0.448807
$161$	−17.6440	−1.39054
$162$	2.18586	0.171737
$163$	13.3671	1.04700	0.523498	−	0.852027i	$-0.324627\pi$
$163$	13.3671	1.04700	0.523498	+	0.852027i	$0.324627\pi$
$164$	14.7624	1.15275
$165$	0	0
$166$	−20.8719	−1.61997
$167$	−5.88665	−0.455523	−0.227761	−	0.973717i	$-0.573141\pi$
$167$	−5.88665	−0.455523	−0.227761	+	0.973717i	$0.573141\pi$
$168$	−4.02995	−0.310917
$169$	−12.9903	−0.999256
$170$	−1.67228	−0.128258
$171$	6.26048	0.478751
$172$	30.6066	2.33373
$173$	−21.3512	−1.62330	−0.811651	−	0.584142i	$-0.801431\pi$
$173$	−21.3512	−1.62330	−0.811651	+	0.584142i	$0.801431\pi$
$174$	0.922298	0.0699192
$175$	10.4621	0.790860
$176$	0	0
$177$	8.16386	0.613633
$178$	29.3486	2.19977
$179$	5.99638	0.448190	0.224095	−	0.974567i	$-0.428057\pi$
$179$	5.99638	0.448190	0.224095	+	0.974567i	$0.428057\pi$
$180$	−2.12528	−0.158409
$181$	8.15705	0.606308	0.303154	−	0.952942i	$-0.401960\pi$
$181$	8.15705	0.606308	0.303154	+	0.952942i	$0.401960\pi$
$182$	−0.509568	−0.0377717
$183$	10.4400	0.771744
$184$	12.6609	0.933372
$185$	−2.98418	−0.219401
$186$	3.52080	0.258158
$187$	0	0
$188$	12.7465	0.929633
$189$	−2.36983	−0.172380
$190$	−10.4693	−0.759522
$191$	25.7685	1.86454	0.932270	−	0.361763i	$-0.117825\pi$
$191$	25.7685	1.86454	0.932270	+	0.361763i	$0.117825\pi$
$192$	−12.5424	−0.905171
$193$	−18.6223	−1.34046	−0.670230	−	0.742154i	$-0.733804\pi$
$193$	−18.6223	−1.34046	−0.670230	+	0.742154i	$0.733804\pi$
$194$	14.3094	1.02736
$195$	−0.0752578	−0.00538932
$196$	−3.84446	−0.274604
$197$	3.26588	0.232684	0.116342	−	0.993209i	$-0.462883\pi$
$197$	3.26588	0.232684	0.116342	+	0.993209i	$0.462883\pi$
$198$	0	0
$199$	3.79001	0.268667	0.134333	−	0.990936i	$-0.457111\pi$
$199$	3.79001	0.268667	0.134333	+	0.990936i	$0.457111\pi$
$200$	−7.50731	−0.530847
$201$	13.5364	0.954783
$202$	−25.4776	−1.79260
$203$	−0.999924	−0.0701809
$204$	2.77797	0.194497
$205$	−4.06553	−0.283949
$206$	−2.16381	−0.150760
$207$	7.44527	0.517482
$208$	−0.180886	−0.0125422
$209$	0	0
$210$	3.96302	0.273475
$211$	5.74982	0.395834	0.197917	−	0.980219i	$-0.436582\pi$
$211$	5.74982	0.395834	0.197917	+	0.980219i	$0.436582\pi$
$212$	9.01558	0.619193
$213$	−5.66896	−0.388430
$214$	−34.2937	−2.34427
$215$	−8.42899	−0.574852
$216$	1.70052	0.115706
$217$	−3.81713	−0.259124
$218$	−1.29401	−0.0876412
$219$	−14.9470	−1.01002
$220$	0	0
$221$	0.0983702	0.00661709
$222$	8.52626	0.572245
$223$	10.5389	0.705736	0.352868	−	0.935673i	$-0.385206\pi$
$223$	10.5389	0.705736	0.352868	+	0.935673i	$0.385206\pi$
$224$	17.5853	1.17496
$225$	−4.41470	−0.294314
$226$	−2.17363	−0.144588
$227$	10.0242	0.665330	0.332665	−	0.943045i	$-0.392052\pi$
$227$	10.0242	0.665330	0.332665	+	0.943045i	$0.392052\pi$
$228$	17.3914	1.15177
$229$	14.0891	0.931033	0.465516	−	0.885039i	$-0.345869\pi$
$229$	14.0891	0.931033	0.465516	+	0.885039i	$0.345869\pi$
$230$	−12.4506	−0.820969
$231$	0	0
$232$	0.717518	0.0471073
$233$	22.6565	1.48428	0.742138	−	0.670247i	$-0.233812\pi$
$233$	22.6565	1.48428	0.742138	+	0.670247i	$0.233812\pi$
$234$	0.215023	0.0140565
$235$	−3.51036	−0.228991
$236$	22.6789	1.47627
$237$	5.57564	0.362177
$238$	−5.18011	−0.335776
$239$	−2.23124	−0.144327	−0.0721636	−	0.997393i	$-0.522990\pi$
$239$	−2.23124	−0.144327	−0.0721636	+	0.997393i	$0.522990\pi$
$240$	1.40679	0.0908082
$241$	−2.26466	−0.145880	−0.0729399	−	0.997336i	$-0.523238\pi$
$241$	−2.26466	−0.145880	−0.0729399	+	0.997336i	$0.523238\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	29.0019	1.85665
$245$	1.05876	0.0676415
$246$	11.6158	0.740599
$247$	0.615844	0.0391852
$248$	2.73907	0.173931
$249$	−9.54860	−0.605118
$250$	15.7440	0.995741
$251$	−5.08123	−0.320724	−0.160362	−	0.987058i	$-0.551266\pi$
$251$	−5.08123	−0.320724	−0.160362	+	0.987058i	$0.551266\pi$
$252$	−6.58331	−0.414709
$253$	0	0
$254$	−24.7146	−1.55073
$255$	−0.765047	−0.0479091
$256$	−2.40226	−0.150141
$257$	−23.0800	−1.43969	−0.719844	−	0.694136i	$-0.755787\pi$
$257$	−23.0800	−1.43969	−0.719844	+	0.694136i	$0.755787\pi$
$258$	24.0829	1.49934
$259$	−9.24388	−0.574387
$260$	−0.209064	−0.0129656
$261$	0.421939	0.0261174
$262$	10.3937	0.642124
$263$	8.90815	0.549300	0.274650	−	0.961544i	$-0.411438\pi$
$263$	8.90815	0.549300	0.274650	+	0.961544i	$0.411438\pi$
$264$	0	0
$265$	−2.48288	−0.152522
$266$	−32.4299	−1.98841
$267$	13.4266	0.821694
$268$	37.6036	2.29701
$269$	−28.4451	−1.73433	−0.867166	−	0.498020i	$-0.834061\pi$
$269$	−28.4451	−1.73433	−0.867166	+	0.498020i	$0.834061\pi$
$270$	−1.67228	−0.101772
$271$	22.0114	1.33710	0.668548	−	0.743669i	$-0.266916\pi$
$271$	22.0114	1.33710	0.668548	+	0.743669i	$0.266916\pi$
$272$	−1.83883	−0.111496
$273$	−0.233120	−0.0141091
$274$	−20.4542	−1.23568
$275$	0	0
$276$	20.6827	1.24495
$277$	−8.85549	−0.532075	−0.266037	−	0.963963i	$-0.585714\pi$
$277$	−8.85549	−0.532075	−0.266037	+	0.963963i	$0.585714\pi$
$278$	−33.5652	−2.01311
$279$	1.61072	0.0964313
$280$	3.08310	0.184251
$281$	−0.758849	−0.0452691	−0.0226346	−	0.999744i	$-0.507205\pi$
$281$	−0.758849	−0.0452691	−0.0226346	+	0.999744i	$0.507205\pi$
$282$	10.0296	0.597256
$283$	−9.48059	−0.563562	−0.281781	−	0.959479i	$-0.590925\pi$
$283$	−9.48059	−0.563562	−0.281781	+	0.959479i	$0.590925\pi$
$284$	−15.7482	−0.934482
$285$	−4.78956	−0.283709
$286$	0	0
$287$	−12.5935	−0.743370
$288$	−7.42047	−0.437256
$289$	1.00000	0.0588235
$290$	−0.705602	−0.0414344
$291$	6.54636	0.383754
$292$	−41.5221	−2.42990
$293$	2.70411	0.157976	0.0789879	−	0.996876i	$-0.474831\pi$
$293$	2.70411	0.157976	0.0789879	+	0.996876i	$0.474831\pi$
$294$	−3.02503	−0.176423
$295$	−6.24574	−0.363641
$296$	6.63315	0.385544
$297$	0	0
$298$	15.2772	0.884983
$299$	0.732393	0.0423554
$300$	−12.2639	−0.708057
$301$	−26.1099	−1.50495
$302$	33.7476	1.94195
$303$	−11.6557	−0.669600
$304$	−11.5120	−0.660257
$305$	−7.98706	−0.457338
$306$	2.18586	0.124957
$307$	−34.9112	−1.99249	−0.996244	−	0.0865854i	$-0.972404\pi$
$307$	−34.9112	−1.99249	−0.996244	+	0.0865854i	$0.972404\pi$
$308$	0	0
$309$	−0.989913	−0.0563142
$310$	−2.69358	−0.152985
$311$	−2.13121	−0.120850	−0.0604250	−	0.998173i	$-0.519246\pi$
$311$	−2.13121	−0.120850	−0.0604250	+	0.998173i	$0.519246\pi$
$312$	0.167281	0.00947041
$313$	6.54819	0.370125	0.185063	−	0.982727i	$-0.440751\pi$
$313$	6.54819	0.370125	0.185063	+	0.982727i	$0.440751\pi$
$314$	−39.2161	−2.21309
$315$	1.81303	0.102153
$316$	15.4889	0.871321
$317$	8.01428	0.450127	0.225063	−	0.974344i	$-0.427741\pi$
$317$	8.01428	0.450127	0.225063	+	0.974344i	$0.427741\pi$
$318$	7.09395	0.397809
$319$	0	0
$320$	9.59554	0.536407
$321$	−15.6889	−0.875669
$322$	−38.5673	−2.14927
$323$	6.26048	0.348342
$324$	2.77797	0.154332
$325$	−0.434275	−0.0240892
$326$	29.2187	1.61827
$327$	−0.591991	−0.0327372
$328$	9.03674	0.498971
$329$	−10.8738	−0.599491
$330$	0	0
$331$	−4.33296	−0.238161	−0.119081	−	0.992885i	$-0.537995\pi$
$331$	−4.33296	−0.238161	−0.119081	+	0.992885i	$0.537995\pi$
$332$	−26.5257	−1.45579
$333$	3.90065	0.213754
$334$	−12.8674	−0.704072
$335$	−10.3560	−0.565807
$336$	4.35772	0.237733
$337$	−19.5339	−1.06408	−0.532040	−	0.846719i	$-0.678575\pi$
$337$	−19.5339	−1.06408	−0.532040	+	0.846719i	$0.678575\pi$
$338$	−28.3950	−1.54448
$339$	−0.994406	−0.0540087
$340$	−2.12528	−0.115259
$341$	0	0
$342$	13.6845	0.739973
$343$	19.8684	1.07279
$344$	18.7357	1.01016
$345$	−5.69599	−0.306662
$346$	−46.6707	−2.50903
$347$	−15.6200	−0.838526	−0.419263	−	0.907865i	$-0.637711\pi$
$347$	−15.6200	−0.838526	−0.419263	+	0.907865i	$0.637711\pi$
$348$	1.17213	0.0628329
$349$	21.6478	1.15878	0.579391	−	0.815050i	$-0.303290\pi$
$349$	21.6478	1.15878	0.579391	+	0.815050i	$0.303290\pi$
$350$	22.8686	1.22238
$351$	0.0983702	0.00525061
$352$	0	0
$353$	20.1659	1.07332	0.536661	−	0.843798i	$-0.319686\pi$
$353$	20.1659	1.07332	0.536661	+	0.843798i	$0.319686\pi$
$354$	17.8450	0.948452
$355$	4.33702	0.230185
$356$	37.2986	1.97682
$357$	−2.36983	−0.125425
$358$	13.1072	0.692738
$359$	−11.2895	−0.595839	−0.297920	−	0.954591i	$-0.596293\pi$
$359$	−11.2895	−0.595839	−0.297920	+	0.954591i	$0.596293\pi$
$360$	−1.30098	−0.0685677
$361$	20.1936	1.06282
$362$	17.8301	0.937131
$363$	0	0
$364$	−0.647601	−0.0339435
$365$	11.4351	0.598542
$366$	22.8202	1.19283
$367$	−14.6579	−0.765135	−0.382568	−	0.923927i	$-0.624960\pi$
$367$	−14.6579	−0.765135	−0.382568	+	0.923927i	$0.624960\pi$
$368$	−13.6906	−0.713673
$369$	5.31409	0.276641
$370$	−6.52299	−0.339114
$371$	−7.69102	−0.399298
$372$	4.47453	0.231994
$373$	−13.3810	−0.692839	−0.346420	−	0.938080i	$-0.612603\pi$
$373$	−13.3810	−0.692839	−0.346420	+	0.938080i	$0.612603\pi$
$374$	0	0
$375$	7.20269	0.371945
$376$	7.80272	0.402395
$377$	0.0415062	0.00213768
$378$	−5.18011	−0.266436
$379$	−5.36481	−0.275572	−0.137786	−	0.990462i	$-0.543999\pi$
$379$	−5.36481	−0.275572	−0.137786	+	0.990462i	$0.543999\pi$
$380$	−13.3052	−0.682545
$381$	−11.3066	−0.579254
$382$	56.3261	2.88190
$383$	29.6078	1.51289	0.756445	−	0.654057i	$-0.226934\pi$
$383$	29.6078	1.51289	0.756445	+	0.654057i	$0.226934\pi$
$384$	−12.5750	−0.641714
$385$	0	0
$386$	−40.7056	−2.07186
$387$	11.0176	0.560057
$388$	18.1856	0.923233
$389$	−7.20721	−0.365420	−0.182710	−	0.983167i	$-0.558487\pi$
$389$	−7.20721	−0.365420	−0.182710	+	0.983167i	$0.558487\pi$
$390$	−0.164503	−0.00832992
$391$	7.44527	0.376524
$392$	−2.35337	−0.118863
$393$	4.75497	0.239857
$394$	7.13875	0.359645
$395$	−4.26562	−0.214627
$396$	0	0
$397$	−26.1985	−1.31486	−0.657432	−	0.753514i	$-0.728357\pi$
$397$	−26.1985	−1.31486	−0.657432	+	0.753514i	$0.728357\pi$
$398$	8.28443	0.415261
$399$	−14.8363	−0.742742
$400$	8.11790	0.405895
$401$	−1.66094	−0.0829436	−0.0414718	−	0.999140i	$-0.513205\pi$
$401$	−1.66094	−0.0829436	−0.0414718	+	0.999140i	$0.513205\pi$
$402$	29.5886	1.47575
$403$	0.158447	0.00789280
$404$	−32.3790	−1.61092
$405$	−0.765047	−0.0380155
$406$	−2.18569	−0.108474
$407$	0	0
$408$	1.70052	0.0841885
$409$	−8.52450	−0.421509	−0.210755	−	0.977539i	$-0.567592\pi$
$409$	−8.52450	−0.421509	−0.210755	+	0.977539i	$0.567592\pi$
$410$	−8.88666	−0.438881
$411$	−9.35753	−0.461573
$412$	−2.74994	−0.135480
$413$	−19.3470	−0.952001
$414$	16.2743	0.799838
$415$	7.30513	0.358595
$416$	−0.729953	−0.0357889
$417$	−15.3556	−0.751968
$418$	0	0
$419$	−27.9695	−1.36640	−0.683200	−	0.730232i	$-0.739412\pi$
$419$	−27.9695	−1.36640	−0.683200	+	0.730232i	$0.739412\pi$
$420$	5.03654	0.245758
$421$	37.4430	1.82486	0.912431	−	0.409231i	$-0.134203\pi$
$421$	37.4430	1.82486	0.912431	+	0.409231i	$0.134203\pi$
$422$	12.5683	0.611814
$423$	4.58842	0.223097
$424$	5.51886	0.268020
$425$	−4.41470	−0.214145
$426$	−12.3915	−0.600371
$427$	−24.7409	−1.19730
$428$	−43.5833	−2.10668
$429$	0	0
$430$	−18.4246	−0.888511
$431$	−27.2341	−1.31182	−0.655911	−	0.754838i	$-0.727715\pi$
$431$	−27.2341	−1.31182	−0.655911	+	0.754838i	$0.727715\pi$
$432$	−1.83883	−0.0884709
$433$	−9.27257	−0.445611	−0.222806	−	0.974863i	$-0.571522\pi$
$433$	−9.27257	−0.445611	−0.222806	+	0.974863i	$0.571522\pi$
$434$	−8.34370	−0.400511
$435$	−0.322803	−0.0154772
$436$	−1.64453	−0.0787587
$437$	46.6110	2.22971
$438$	−32.6719	−1.56112
$439$	22.0443	1.05212	0.526059	−	0.850448i	$-0.323669\pi$
$439$	22.0443	1.05212	0.526059	+	0.850448i	$0.323669\pi$
$440$	0	0
$441$	−1.38391	−0.0659005
$442$	0.215023	0.0102276
$443$	14.0371	0.666924	0.333462	−	0.942764i	$-0.391783\pi$
$443$	14.0371	0.666924	0.333462	+	0.942764i	$0.391783\pi$
$444$	10.8359	0.514248
$445$	−10.2720	−0.486938
$446$	23.0365	1.09081
$447$	6.98910	0.330573
$448$	29.7234	1.40430
$449$	2.11684	0.0999000	0.0499500	−	0.998752i	$-0.484094\pi$
$449$	2.11684	0.0999000	0.0499500	+	0.998752i	$0.484094\pi$
$450$	−9.64991	−0.454901
$451$	0	0
$452$	−2.76243	−0.129934
$453$	15.4391	0.725390
$454$	21.9115	1.02836
$455$	0.178348	0.00836109
$456$	10.6461	0.498549
$457$	32.0021	1.49700	0.748498	−	0.663137i	$-0.230775\pi$
$457$	32.0021	1.49700	0.748498	+	0.663137i	$0.230775\pi$
$458$	30.7967	1.43904
$459$	1.00000	0.0466760
$460$	−15.8233	−0.737763
$461$	13.1106	0.610621	0.305311	−	0.952253i	$-0.401240\pi$
$461$	13.1106	0.610621	0.305311	+	0.952253i	$0.401240\pi$
$462$	0	0
$463$	−19.6290	−0.912235	−0.456117	−	0.889920i	$-0.650760\pi$
$463$	−19.6290	−0.912235	−0.456117	+	0.889920i	$0.650760\pi$
$464$	−0.775876	−0.0360191
$465$	−1.23228	−0.0571455
$466$	49.5238	2.29415
$467$	31.7617	1.46976	0.734878	−	0.678199i	$-0.237239\pi$
$467$	31.7617	1.46976	0.734878	+	0.678199i	$0.237239\pi$
$468$	0.273269	0.0126319
$469$	−32.0789	−1.48127
$470$	−7.67314	−0.353935
$471$	−17.9409	−0.826671
$472$	13.8828	0.639010
$473$	0	0
$474$	12.1875	0.559792
$475$	−27.6382	−1.26813
$476$	−6.58331	−0.301745
$477$	3.24539	0.148596
$478$	−4.87718	−0.223077
$479$	−10.5924	−0.483978	−0.241989	−	0.970279i	$-0.577800\pi$
$479$	−10.5924	−0.483978	−0.241989	+	0.970279i	$0.577800\pi$
$480$	5.67701	0.259119
$481$	0.383708	0.0174956
$482$	−4.95023	−0.225477
$483$	−17.6440	−0.802831
$484$	0	0
$485$	−5.00827	−0.227414
$486$	2.18586	0.0991524
$487$	−15.4115	−0.698363	−0.349181	−	0.937055i	$-0.613540\pi$
$487$	−15.4115	−0.698363	−0.349181	+	0.937055i	$0.613540\pi$
$488$	17.7534	0.803658
$489$	13.3671	0.604483
$490$	2.31429	0.104549
$491$	−5.42674	−0.244905	−0.122453	−	0.992474i	$-0.539076\pi$
$491$	−5.42674	−0.244905	−0.122453	+	0.992474i	$0.539076\pi$
$492$	14.7624	0.665539
$493$	0.421939	0.0190032
$494$	1.34615	0.0605660
$495$	0	0
$496$	−2.96185	−0.132991
$497$	13.4345	0.602618
$498$	−20.8719	−0.935291
$499$	−17.6637	−0.790734	−0.395367	−	0.918523i	$-0.629383\pi$
$499$	−17.6637	−0.790734	−0.395367	+	0.918523i	$0.629383\pi$
$500$	20.0088	0.894822
$501$	−5.88665	−0.262996
$502$	−11.1068	−0.495722
$503$	14.6635	0.653813	0.326906	−	0.945057i	$-0.393994\pi$
$503$	14.6635	0.653813	0.326906	+	0.945057i	$0.393994\pi$
$504$	−4.02995	−0.179508
$505$	8.91712	0.396807
$506$	0	0
$507$	−12.9903	−0.576921
$508$	−31.4093	−1.39356
$509$	16.7932	0.744346	0.372173	−	0.928163i	$-0.378613\pi$
$509$	16.7932	0.744346	0.372173	+	0.928163i	$0.378613\pi$
$510$	−1.67228	−0.0740499
$511$	35.4217	1.56696
$512$	19.8990	0.879419
$513$	6.26048	0.276407
$514$	−50.4495	−2.22523
$515$	0.757330	0.0333719
$516$	30.6066	1.34738
$517$	0	0
$518$	−20.2058	−0.887791
$519$	−21.3512	−0.937214
$520$	−0.127978	−0.00561219
$521$	13.2990	0.582642	0.291321	−	0.956625i	$-0.405905\pi$
$521$	13.2990	0.582642	0.291321	+	0.956625i	$0.405905\pi$
$522$	0.922298	0.0403679
$523$	−10.5033	−0.459279	−0.229640	−	0.973276i	$-0.573755\pi$
$523$	−10.5033	−0.459279	−0.229640	+	0.973276i	$0.573755\pi$
$524$	13.2092	0.577045
$525$	10.4621	0.456603
$526$	19.4719	0.849016
$527$	1.61072	0.0701641
$528$	0	0
$529$	32.4321	1.41009
$530$	−5.42721	−0.235743
$531$	8.16386	0.354281
$532$	−41.2147	−1.78688
$533$	0.522748	0.0226427
$534$	29.3486	1.27004
$535$	12.0027	0.518924
$536$	23.0190	0.994268
$537$	5.99638	0.258763
$538$	−62.1770	−2.68064
$539$	0	0
$540$	−2.12528	−0.0914573
$541$	−35.5800	−1.52970	−0.764851	−	0.644207i	$-0.777188\pi$
$541$	−35.5800	−1.52970	−0.764851	+	0.644207i	$0.777188\pi$
$542$	48.1137	2.06666
$543$	8.15705	0.350052
$544$	−7.42047	−0.318150
$545$	0.452901	0.0194001
$546$	−0.509568	−0.0218075
$547$	−40.8265	−1.74562	−0.872808	−	0.488063i	$-0.837704\pi$
$547$	−40.8265	−1.74562	−0.872808	+	0.488063i	$0.837704\pi$
$548$	−25.9949	−1.11045
$549$	10.4400	0.445566
$550$	0	0
$551$	2.64154	0.112533
$552$	12.6609	0.538882
$553$	−13.2133	−0.561887
$554$	−19.3568	−0.822393
$555$	−2.98418	−0.126671
$556$	−42.6574	−1.80908
$557$	19.1238	0.810301	0.405150	−	0.914250i	$-0.367219\pi$
$557$	19.1238	0.810301	0.405150	+	0.914250i	$0.367219\pi$
$558$	3.52080	0.149048
$559$	1.08380	0.0458400
$560$	−3.33386	−0.140881
$561$	0	0
$562$	−1.65873	−0.0699695
$563$	25.4795	1.07383	0.536916	−	0.843635i	$-0.319589\pi$
$563$	25.4795	1.07383	0.536916	+	0.843635i	$0.319589\pi$
$564$	12.7465	0.536724
$565$	0.760767	0.0320057
$566$	−20.7232	−0.871061
$567$	−2.36983	−0.0995235
$568$	−9.64020	−0.404494
$569$	−39.0873	−1.63862	−0.819312	−	0.573347i	$-0.805645\pi$
$569$	−39.0873	−1.63862	−0.819312	+	0.573347i	$0.805645\pi$
$570$	−10.4693	−0.438510
$571$	−29.0925	−1.21748	−0.608741	−	0.793369i	$-0.708325\pi$
$571$	−29.0925	−1.21748	−0.608741	+	0.793369i	$0.708325\pi$
$572$	0	0
$573$	25.7685	1.07649
$574$	−27.5276	−1.14898
$575$	−32.8687	−1.37072
$576$	−12.5424	−0.522601
$577$	6.62410	0.275765	0.137882	−	0.990449i	$-0.455970\pi$
$577$	6.62410	0.275765	0.137882	+	0.990449i	$0.455970\pi$
$578$	2.18586	0.0909196
$579$	−18.6223	−0.773914
$580$	−0.896737	−0.0372350
$581$	22.6286	0.938791
$582$	14.3094	0.593144
$583$	0	0
$584$	−25.4177	−1.05179
$585$	−0.0752578	−0.00311153
$586$	5.91080	0.244173
$587$	7.51050	0.309992	0.154996	−	0.987915i	$-0.450464\pi$
$587$	7.51050	0.309992	0.154996	+	0.987915i	$0.450464\pi$
$588$	−3.84446	−0.158543
$589$	10.0839	0.415499
$590$	−13.6523	−0.562056
$591$	3.26588	0.134340
$592$	−7.17265	−0.294794
$593$	14.6291	0.600744	0.300372	−	0.953822i	$-0.402889\pi$
$593$	14.6291	0.600744	0.300372	+	0.953822i	$0.402889\pi$
$594$	0	0
$595$	1.81303	0.0743270
$596$	19.4155	0.795290
$597$	3.79001	0.155115
$598$	1.60091	0.0654659
$599$	16.8627	0.688991	0.344495	−	0.938788i	$-0.388050\pi$
$599$	16.8627	0.688991	0.344495	+	0.938788i	$0.388050\pi$
$600$	−7.50731	−0.306485
$601$	36.5266	1.48995	0.744975	−	0.667093i	$-0.232462\pi$
$601$	36.5266	1.48995	0.744975	+	0.667093i	$0.232462\pi$
$602$	−57.0724	−2.32610
$603$	13.5364	0.551244
$604$	42.8892	1.74514
$605$	0	0
$606$	−25.4776	−1.03496
$607$	−13.5832	−0.551326	−0.275663	−	0.961254i	$-0.588897\pi$
$607$	−13.5832	−0.551326	−0.275663	+	0.961254i	$0.588897\pi$
$608$	−46.4557	−1.88403
$609$	−0.999924	−0.0405189
$610$	−17.4586	−0.706876
$611$	0.451364	0.0182602
$612$	2.77797	0.112293
$613$	−31.2783	−1.26332	−0.631659	−	0.775247i	$-0.717626\pi$
$613$	−31.2783	−1.26332	−0.631659	+	0.775247i	$0.717626\pi$
$614$	−76.3109	−3.07966
$615$	−4.06553	−0.163938
$616$	0	0
$617$	−13.5176	−0.544200	−0.272100	−	0.962269i	$-0.587718\pi$
$617$	−13.5176	−0.544200	−0.272100	+	0.962269i	$0.587718\pi$
$618$	−2.16381	−0.0870411
$619$	−1.72050	−0.0691529	−0.0345765	−	0.999402i	$-0.511008\pi$
$619$	−1.72050	−0.0691529	−0.0345765	+	0.999402i	$0.511008\pi$
$620$	−3.42323	−0.137480
$621$	7.44527	0.298769
$622$	−4.65853	−0.186790
$623$	−31.8187	−1.27479
$624$	−0.180886	−0.00724125
$625$	16.5631	0.662525
$626$	14.3134	0.572078
$627$	0	0
$628$	−49.8391	−1.98880
$629$	3.90065	0.155529
$630$	3.96302	0.157891
$631$	−47.6902	−1.89852	−0.949259	−	0.314496i	$-0.898165\pi$
$631$	−47.6902	−1.89852	−0.949259	+	0.314496i	$0.898165\pi$
$632$	9.48150	0.377154
$633$	5.74982	0.228535
$634$	17.5181	0.695731
$635$	8.65007	0.343268
$636$	9.01558	0.357491
$637$	−0.136135	−0.00539388
$638$	0	0
$639$	−5.66896	−0.224260
$640$	9.62045	0.380282
$641$	−26.6106	−1.05106	−0.525529	−	0.850776i	$-0.676132\pi$
$641$	−26.6106	−1.05106	−0.525529	+	0.850776i	$0.676132\pi$
$642$	−34.2937	−1.35346
$643$	3.29286	0.129858	0.0649289	−	0.997890i	$-0.479318\pi$
$643$	3.29286	0.129858	0.0649289	+	0.997890i	$0.479318\pi$
$644$	−49.0145	−1.93144
$645$	−8.42899	−0.331891
$646$	13.6845	0.538410
$647$	45.5027	1.78889	0.894447	−	0.447173i	$-0.147569\pi$
$647$	45.5027	1.78889	0.894447	+	0.447173i	$0.147569\pi$
$648$	1.70052	0.0668029
$649$	0	0
$650$	−0.949263	−0.0372331
$651$	−3.81713	−0.149605
$652$	37.1335	1.45426
$653$	24.2808	0.950180	0.475090	−	0.879937i	$-0.342416\pi$
$653$	24.2808	0.950180	0.475090	+	0.879937i	$0.342416\pi$
$654$	−1.29401	−0.0505997
$655$	−3.63778	−0.142140
$656$	−9.77173	−0.381522
$657$	−14.9470	−0.583136
$658$	−23.7685	−0.926593
$659$	−34.8860	−1.35896	−0.679482	−	0.733692i	$-0.737796\pi$
$659$	−34.8860	−1.35896	−0.679482	+	0.733692i	$0.737796\pi$
$660$	0	0
$661$	−6.60717	−0.256989	−0.128494	−	0.991710i	$-0.541014\pi$
$661$	−6.60717	−0.256989	−0.128494	+	0.991710i	$0.541014\pi$
$662$	−9.47124	−0.368110
$663$	0.0983702	0.00382038
$664$	−16.2376	−0.630142
$665$	11.3504	0.440151
$666$	8.52626	0.330386
$667$	3.14145	0.121638
$668$	−16.3529	−0.632714
$669$	10.5389	0.407457
$670$	−22.6367	−0.874531
$671$	0	0
$672$	17.5853	0.678366
$673$	10.4816	0.404036	0.202018	−	0.979382i	$-0.435250\pi$
$673$	10.4816	0.404036	0.202018	+	0.979382i	$0.435250\pi$
$674$	−42.6984	−1.64468
$675$	−4.41470	−0.169922
$676$	−36.0867	−1.38795
$677$	−0.419441	−0.0161204	−0.00806021	−	0.999968i	$-0.502566\pi$
$677$	−0.419441	−0.0161204	−0.00806021	+	0.999968i	$0.502566\pi$
$678$	−2.17363	−0.0834777
$679$	−15.5138	−0.595363
$680$	−1.30098	−0.0498903
$681$	10.0242	0.384129
$682$	0	0
$683$	−21.5652	−0.825169	−0.412585	−	0.910919i	$-0.635374\pi$
$683$	−21.5652	−0.825169	−0.412585	+	0.910919i	$0.635374\pi$
$684$	17.3914	0.664977
$685$	7.15895	0.273529
$686$	43.4295	1.65815
$687$	14.0891	0.537532
$688$	−20.2595	−0.772388
$689$	0.319249	0.0121624
$690$	−12.4506	−0.473986
$691$	−31.4700	−1.19717	−0.598587	−	0.801057i	$-0.704271\pi$
$691$	−31.4700	−1.19717	−0.598587	+	0.801057i	$0.704271\pi$
$692$	−59.3130	−2.25474
$693$	0	0
$694$	−34.1431	−1.29605
$695$	11.7478	0.445619
$696$	0.717518	0.0271974
$697$	5.31409	0.201286
$698$	47.3190	1.79105
$699$	22.6565	0.856947
$700$	29.0633	1.09849
$701$	−44.3576	−1.67536	−0.837682	−	0.546158i	$-0.816090\pi$
$701$	−44.3576	−1.67536	−0.837682	+	0.546158i	$0.816090\pi$
$702$	0.215023	0.00811552
$703$	24.4199	0.921016
$704$	0	0
$705$	−3.51036	−0.132208
$706$	44.0797	1.65896
$707$	27.6219	1.03883
$708$	22.6789	0.852327
$709$	47.6187	1.78836	0.894179	−	0.447709i	$-0.147760\pi$
$709$	47.6187	1.78836	0.894179	+	0.447709i	$0.147760\pi$
$710$	9.48010	0.355782
$711$	5.57564	0.209103
$712$	22.8322	0.855674
$713$	11.9923	0.449114
$714$	−5.18011	−0.193861
$715$	0	0
$716$	16.6577	0.622529
$717$	−2.23124	−0.0833273
$718$	−24.6773	−0.920949
$719$	18.8404	0.702627	0.351314	−	0.936258i	$-0.385735\pi$
$719$	18.8404	0.702627	0.351314	+	0.936258i	$0.385735\pi$
$720$	1.40679	0.0524281
$721$	2.34592	0.0873668
$722$	44.1403	1.64273
$723$	−2.26466	−0.0842238
$724$	22.6600	0.842152
$725$	−1.86274	−0.0691803
$726$	0	0
$727$	44.4482	1.64849	0.824246	−	0.566232i	$-0.191600\pi$
$727$	44.4482	1.64849	0.824246	+	0.566232i	$0.191600\pi$
$728$	−0.396427	−0.0146926
$729$	1.00000	0.0370370
$730$	24.9955	0.925126
$731$	11.0176	0.407501
$732$	29.0019	1.07194
$733$	−32.7188	−1.20849	−0.604247	−	0.796797i	$-0.706526\pi$
$733$	−32.7188	−1.20849	−0.604247	+	0.796797i	$0.706526\pi$
$734$	−32.0400	−1.18262
$735$	1.05876	0.0390528
$736$	−55.2475	−2.03645
$737$	0	0
$738$	11.6158	0.427585
$739$	−18.6930	−0.687634	−0.343817	−	0.939037i	$-0.611720\pi$
$739$	−18.6930	−0.687634	−0.343817	+	0.939037i	$0.611720\pi$
$740$	−8.28996	−0.304745
$741$	0.615844	0.0226236
$742$	−16.8115	−0.617168
$743$	52.6699	1.93227	0.966136	−	0.258033i	$-0.0830743\pi$
$743$	52.6699	1.93227	0.966136	+	0.258033i	$0.0830743\pi$
$744$	2.73907	0.100419
$745$	−5.34699	−0.195899
$746$	−29.2488	−1.07088
$747$	−9.54860	−0.349365
$748$	0	0
$749$	37.1800	1.35853
$750$	15.7440	0.574891
$751$	−6.02719	−0.219935	−0.109968	−	0.993935i	$-0.535075\pi$
$751$	−6.02719	−0.219935	−0.109968	+	0.993935i	$0.535075\pi$
$752$	−8.43734	−0.307678
$753$	−5.08123	−0.185170
$754$	0.0907266	0.00330407
$755$	−11.8116	−0.429868
$756$	−6.58331	−0.239433
$757$	21.6449	0.786697	0.393349	−	0.919389i	$-0.371317\pi$
$757$	21.6449	0.786697	0.393349	+	0.919389i	$0.371317\pi$
$758$	−11.7267	−0.425933
$759$	0	0
$760$	−8.14476	−0.295442
$761$	−17.0039	−0.616390	−0.308195	−	0.951323i	$-0.599725\pi$
$761$	−17.0039	−0.616390	−0.308195	+	0.951323i	$0.599725\pi$
$762$	−24.7146	−0.895314
$763$	1.40292	0.0507890
$764$	71.5839	2.58982
$765$	−0.765047	−0.0276603
$766$	64.7185	2.33837
$767$	0.803080	0.0289975
$768$	−2.40226	−0.0866840
$769$	20.3195	0.732738	0.366369	−	0.930470i	$-0.380601\pi$
$769$	20.3195	0.732738	0.366369	+	0.930470i	$0.380601\pi$
$770$	0	0
$771$	−23.0800	−0.831204
$772$	−51.7320	−1.86188
$773$	−33.4342	−1.20254	−0.601272	−	0.799044i	$-0.705339\pi$
$773$	−33.4342	−1.20254	−0.601272	+	0.799044i	$0.705339\pi$
$774$	24.0829	0.865642
$775$	−7.11085	−0.255429
$776$	11.1322	0.399624
$777$	−9.24388	−0.331622
$778$	−15.7539	−0.564805
$779$	33.2688	1.19198
$780$	−0.209064	−0.00748568
$781$	0	0
$782$	16.2743	0.581968
$783$	0.421939	0.0150789
$784$	2.54478	0.0908850
$785$	13.7256	0.489888
$786$	10.3937	0.370731
$787$	−5.24106	−0.186823	−0.0934117	−	0.995628i	$-0.529777\pi$
$787$	−5.24106	−0.186823	−0.0934117	+	0.995628i	$0.529777\pi$
$788$	9.07251	0.323195
$789$	8.90815	0.317138
$790$	−9.32404	−0.331735
$791$	2.35657	0.0837901
$792$	0	0
$793$	1.02698	0.0364691
$794$	−57.2661	−2.03230
$795$	−2.48288	−0.0880585
$796$	10.5285	0.373174
$797$	−17.4954	−0.619718	−0.309859	−	0.950783i	$-0.600282\pi$
$797$	−17.4954	−0.619718	−0.309859	+	0.950783i	$0.600282\pi$
$798$	−32.4299	−1.14801
$799$	4.58842	0.162327
$800$	32.7592	1.15821
$801$	13.4266	0.474405
$802$	−3.63058	−0.128200
$803$	0	0
$804$	37.6036	1.32618
$805$	13.4985	0.475760
$806$	0.346342	0.0121994
$807$	−28.4451	−1.00132
$808$	−19.8207	−0.697290
$809$	4.87159	0.171276	0.0856380	−	0.996326i	$-0.472707\pi$
$809$	4.87159	0.171276	0.0856380	+	0.996326i	$0.472707\pi$
$810$	−1.67228	−0.0587580
$811$	5.30607	0.186321	0.0931607	−	0.995651i	$-0.470303\pi$
$811$	5.30607	0.186321	0.0931607	+	0.995651i	$0.470303\pi$
$812$	−2.77775	−0.0974801
$813$	22.0114	0.771973
$814$	0	0
$815$	−10.2265	−0.358219
$816$	−1.83883	−0.0643720
$817$	68.9755	2.41315
$818$	−18.6333	−0.651499
$819$	−0.233120	−0.00814589
$820$	−11.2939	−0.394400
$821$	29.0535	1.01397	0.506987	−	0.861954i	$-0.330759\pi$
$821$	29.0535	1.01397	0.506987	+	0.861954i	$0.330759\pi$
$822$	−20.4542	−0.713423
$823$	17.6746	0.616097	0.308049	−	0.951371i	$-0.400324\pi$
$823$	17.6746	0.616097	0.308049	+	0.951371i	$0.400324\pi$
$824$	−1.68337	−0.0586430
$825$	0	0
$826$	−42.2897	−1.47145
$827$	−14.5303	−0.505268	−0.252634	−	0.967562i	$-0.581297\pi$
$827$	−14.5303	−0.505268	−0.252634	+	0.967562i	$0.581297\pi$
$828$	20.6827	0.718774
$829$	−47.0647	−1.63462	−0.817312	−	0.576196i	$-0.804537\pi$
$829$	−47.0647	−1.63462	−0.817312	+	0.576196i	$0.804537\pi$
$830$	15.9680	0.554256
$831$	−8.85549	−0.307194
$832$	−1.23380	−0.0427743
$833$	−1.38391	−0.0479496
$834$	−33.5652	−1.16227
$835$	4.50357	0.155852
$836$	0	0
$837$	1.61072	0.0556747
$838$	−61.1373	−2.11195
$839$	−34.3797	−1.18692	−0.593459	−	0.804864i	$-0.702238\pi$
$839$	−34.3797	−1.18692	−0.593459	+	0.804864i	$0.702238\pi$
$840$	3.08310	0.106377
$841$	−28.8220	−0.993861
$842$	81.8451	2.82057
$843$	−0.758849	−0.0261361
$844$	15.9728	0.549807
$845$	9.93821	0.341885
$846$	10.0296	0.344826
$847$	0	0
$848$	−5.96773	−0.204933
$849$	−9.48059	−0.325373
$850$	−9.64991	−0.330989
$851$	29.0414	0.995527
$852$	−15.7482	−0.539523
$853$	8.49250	0.290778	0.145389	−	0.989375i	$-0.453557\pi$
$853$	8.49250	0.290778	0.145389	+	0.989375i	$0.453557\pi$
$854$	−54.0801	−1.85058
$855$	−4.78956	−0.163800
$856$	−26.6794	−0.911882
$857$	−55.2529	−1.88740	−0.943702	−	0.330798i	$-0.892682\pi$
$857$	−55.2529	−1.88740	−0.943702	+	0.330798i	$0.892682\pi$
$858$	0	0
$859$	54.7225	1.86711	0.933553	−	0.358439i	$-0.116691\pi$
$859$	54.7225	1.86711	0.933553	+	0.358439i	$0.116691\pi$
$860$	−23.4155	−0.798461
$861$	−12.5935	−0.429185
$862$	−59.5299	−2.02760
$863$	23.4823	0.799345	0.399673	−	0.916658i	$-0.369124\pi$
$863$	23.4823	0.799345	0.399673	+	0.916658i	$0.369124\pi$
$864$	−7.42047	−0.252450
$865$	16.3347	0.555396
$866$	−20.2685	−0.688752
$867$	1.00000	0.0339618
$868$	−10.6039	−0.359919
$869$	0	0
$870$	−0.705602	−0.0239221
$871$	1.33158	0.0451188
$872$	−1.00669	−0.0340910
$873$	6.54636	0.221561
$874$	101.885	3.44631
$875$	−17.0691	−0.577042
$876$	−41.5221	−1.40290
$877$	5.69964	0.192463	0.0962316	−	0.995359i	$-0.469321\pi$
$877$	5.69964	0.192463	0.0962316	+	0.995359i	$0.469321\pi$
$878$	48.1857	1.62619
$879$	2.70411	0.0912074
$880$	0	0
$881$	38.2432	1.28845	0.644223	−	0.764838i	$-0.277181\pi$
$881$	38.2432	1.28845	0.644223	+	0.764838i	$0.277181\pi$
$882$	−3.02503	−0.101858
$883$	32.8233	1.10459	0.552296	−	0.833648i	$-0.313752\pi$
$883$	32.8233	1.10459	0.552296	+	0.833648i	$0.313752\pi$
$884$	0.273269	0.00919103
$885$	−6.24574	−0.209948
$886$	30.6831	1.03082
$887$	−57.2241	−1.92140	−0.960698	−	0.277594i	$-0.910463\pi$
$887$	−57.2241	−1.92140	−0.960698	+	0.277594i	$0.910463\pi$
$888$	6.63315	0.222594
$889$	26.7947	0.898665
$890$	−22.4531	−0.752628
$891$	0	0
$892$	29.2767	0.980256
$893$	28.7257	0.961269
$894$	15.2772	0.510945
$895$	−4.58751	−0.153344
$896$	29.8006	0.995567
$897$	0.732393	0.0244539
$898$	4.62711	0.154409
$899$	0.679626	0.0226668
$900$	−12.2639	−0.408797
$901$	3.24539	0.108120
$902$	0	0
$903$	−26.1099	−0.868882
$904$	−1.69101	−0.0562422
$905$	−6.24052	−0.207442
$906$	33.7476	1.12119
$907$	−1.40201	−0.0465530	−0.0232765	−	0.999729i	$-0.507410\pi$
$907$	−1.40201	−0.0465530	−0.0232765	+	0.999729i	$0.507410\pi$
$908$	27.8469	0.924133
$909$	−11.6557	−0.386594
$910$	0.389843	0.0129232
$911$	12.6876	0.420360	0.210180	−	0.977663i	$-0.432595\pi$
$911$	12.6876	0.420360	0.210180	+	0.977663i	$0.432595\pi$
$912$	−11.5120	−0.381200
$913$	0	0
$914$	69.9521	2.31381
$915$	−7.98706	−0.264044
$916$	39.1390	1.29319
$917$	−11.2685	−0.372118
$918$	2.18586	0.0721440
$919$	38.4593	1.26866	0.634328	−	0.773064i	$-0.281277\pi$
$919$	38.4593	1.26866	0.634328	+	0.773064i	$0.281277\pi$
$920$	−9.68616	−0.319343
$921$	−34.9112	−1.15036
$922$	28.6579	0.943797
$923$	−0.557656	−0.0183555
$924$	0	0
$925$	−17.2202	−0.566197
$926$	−42.9061	−1.40998
$927$	−0.989913	−0.0325130
$928$	−3.13099	−0.102780
$929$	5.03366	0.165149	0.0825746	−	0.996585i	$-0.473686\pi$
$929$	5.03366	0.165149	0.0825746	+	0.996585i	$0.473686\pi$
$930$	−2.69358	−0.0883260
$931$	−8.66394	−0.283949
$932$	62.9390	2.06163
$933$	−2.13121	−0.0697728
$934$	69.4265	2.27170
$935$	0	0
$936$	0.167281	0.00546774
$937$	9.72891	0.317830	0.158915	−	0.987292i	$-0.449200\pi$
$937$	9.72891	0.317830	0.158915	+	0.987292i	$0.449200\pi$
$938$	−70.1199	−2.28950
$939$	6.54819	0.213692
$940$	−9.75166	−0.318064
$941$	−46.9846	−1.53165	−0.765827	−	0.643046i	$-0.777670\pi$
$941$	−46.9846	−1.53165	−0.765827	+	0.643046i	$0.777670\pi$
$942$	−39.2161	−1.27773
$943$	39.5649	1.28841
$944$	−15.0120	−0.488598
$945$	1.81303	0.0589779
$946$	0	0
$947$	12.0711	0.392258	0.196129	−	0.980578i	$-0.437163\pi$
$947$	12.0711	0.392258	0.196129	+	0.980578i	$0.437163\pi$
$948$	15.4889	0.503057
$949$	−1.47033	−0.0477291
$950$	−60.4130	−1.96006
$951$	8.01428	0.259881
$952$	−4.02995	−0.130611
$953$	−48.7867	−1.58036	−0.790178	−	0.612878i	$-0.790012\pi$
$953$	−48.7867	−1.58036	−0.790178	+	0.612878i	$0.790012\pi$
$954$	7.09395	0.229675
$955$	−19.7141	−0.637933
$956$	−6.19832	−0.200468
$957$	0	0
$958$	−23.1534	−0.748053
$959$	22.1757	0.716092
$960$	9.59554	0.309695
$961$	−28.4056	−0.916309
$962$	0.838730	0.0270417
$963$	−15.6889	−0.505568
$964$	−6.29116	−0.202625
$965$	14.2469	0.458624
$966$	−38.5673	−1.24088
$967$	−19.9892	−0.642811	−0.321405	−	0.946942i	$-0.604155\pi$
$967$	−19.9892	−0.642811	−0.321405	+	0.946942i	$0.604155\pi$
$968$	0	0
$969$	6.26048	0.201116
$970$	−10.9474	−0.351499
$971$	43.3131	1.38998	0.694992	−	0.719018i	$-0.255408\pi$
$971$	43.3131	1.38998	0.694992	+	0.719018i	$0.255408\pi$
$972$	2.77797	0.0891033
$973$	36.3902	1.16662
$974$	−33.6874	−1.07941
$975$	−0.434275	−0.0139079
$976$	−19.1973	−0.614492
$977$	5.66490	0.181236	0.0906182	−	0.995886i	$-0.471116\pi$
$977$	5.66490	0.181236	0.0906182	+	0.995886i	$0.471116\pi$
$978$	29.2187	0.934310
$979$	0	0
$980$	2.94119	0.0939529
$981$	−0.591991	−0.0189008
$982$	−11.8621	−0.378534
$983$	−25.4592	−0.812021	−0.406010	−	0.913868i	$-0.633080\pi$
$983$	−25.4592	−0.812021	−0.406010	+	0.913868i	$0.633080\pi$
$984$	9.03674	0.288081
$985$	−2.49855	−0.0796105
$986$	0.922298	0.0293720
$987$	−10.8738	−0.346116
$988$	1.71079	0.0544276
$989$	82.0291	2.60837
$990$	0	0
$991$	−25.2217	−0.801195	−0.400597	−	0.916254i	$-0.631197\pi$
$991$	−25.2217	−0.801195	−0.400597	+	0.916254i	$0.631197\pi$
$992$	−11.9523	−0.379486
$993$	−4.33296	−0.137502
$994$	29.3658	0.931426
$995$	−2.89954	−0.0919216
$996$	−26.5257	−0.840499
$997$	43.4021	1.37456	0.687279	−	0.726394i	$-0.258805\pi$
$997$	43.4021	1.37456	0.687279	+	0.726394i	$0.258805\pi$
$998$	−38.6102	−1.22219
$999$	3.90065	0.123411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6171.2.a.br.1.16		20
11.7	odd	10		561.2.m.f.511.3	yes	40
11.8	odd	10		561.2.m.f.460.3	✓	40
11.10	odd	2		6171.2.a.bq.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
561.2.m.f.460.3	✓	40	11.8	odd	10
561.2.m.f.511.3	yes	40	11.7	odd	10
6171.2.a.bq.1.5		20	11.10	odd	2
6171.2.a.br.1.16		20	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:	\( N \)	\(=\)	\( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6171.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18586 q^{2} +1.00000 q^{3} +2.77797 q^{4} -0.765047 q^{5} +2.18586 q^{6} -2.36983 q^{7} +1.70052 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.18586 q^{2} +1.00000 q^{3} +2.77797 q^{4} -0.765047 q^{5} +2.18586 q^{6} -2.36983 q^{7} +1.70052 q^{8} +1.00000 q^{9} -1.67228 q^{10} +2.77797 q^{12} +0.0983702 q^{13} -5.18011 q^{14} -0.765047 q^{15} -1.83883 q^{16} +1.00000 q^{17} +2.18586 q^{18} +6.26048 q^{19} -2.12528 q^{20} -2.36983 q^{21} +7.44527 q^{23} +1.70052 q^{24} -4.41470 q^{25} +0.215023 q^{26} +1.00000 q^{27} -6.58331 q^{28} +0.421939 q^{29} -1.67228 q^{30} +1.61072 q^{31} -7.42047 q^{32} +2.18586 q^{34} +1.81303 q^{35} +2.77797 q^{36} +3.90065 q^{37} +13.6845 q^{38} +0.0983702 q^{39} -1.30098 q^{40} +5.31409 q^{41} -5.18011 q^{42} +11.0176 q^{43} -0.765047 q^{45} +16.2743 q^{46} +4.58842 q^{47} -1.83883 q^{48} -1.38391 q^{49} -9.64991 q^{50} +1.00000 q^{51} +0.273269 q^{52} +3.24539 q^{53} +2.18586 q^{54} -4.02995 q^{56} +6.26048 q^{57} +0.922298 q^{58} +8.16386 q^{59} -2.12528 q^{60} +10.4400 q^{61} +3.52080 q^{62} -2.36983 q^{63} -12.5424 q^{64} -0.0752578 q^{65} +13.5364 q^{67} +2.77797 q^{68} +7.44527 q^{69} +3.96302 q^{70} -5.66896 q^{71} +1.70052 q^{72} -14.9470 q^{73} +8.52626 q^{74} -4.41470 q^{75} +17.3914 q^{76} +0.215023 q^{78} +5.57564 q^{79} +1.40679 q^{80} +1.00000 q^{81} +11.6158 q^{82} -9.54860 q^{83} -6.58331 q^{84} -0.765047 q^{85} +24.0829 q^{86} +0.421939 q^{87} +13.4266 q^{89} -1.67228 q^{90} -0.233120 q^{91} +20.6827 q^{92} +1.61072 q^{93} +10.0296 q^{94} -4.78956 q^{95} -7.42047 q^{96} +6.54636 q^{97} -3.02503 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{2} + 20 q^{3} + 29 q^{4} + 17 q^{5} + q^{6} + q^{7} + 6 q^{8} + 20 q^{9}+O(q^{10}) \) 20 * q + q^2 + 20 * q^3 + 29 * q^4 + 17 * q^5 + q^6 + q^7 + 6 * q^8 + 20 * q^9	\( 20 q + q^{2} + 20 q^{3} + 29 q^{4} + 17 q^{5} + q^{6} + q^{7} + 6 q^{8} + 20 q^{9} + 6 q^{10} + 29 q^{12} + 8 q^{13} + 21 q^{14} + 17 q^{15} + 39 q^{16} + 20 q^{17} + q^{18} + 6 q^{19} + 32 q^{20} + q^{21} + 36 q^{23} + 6 q^{24} + 39 q^{25} + 17 q^{26} + 20 q^{27} - 7 q^{28} - 4 q^{29} + 6 q^{30} + 7 q^{31} - 7 q^{32} + q^{34} - 2 q^{35} + 29 q^{36} + 12 q^{37} + 24 q^{38} + 8 q^{39} - 15 q^{40} - 6 q^{41} + 21 q^{42} - 6 q^{43} + 17 q^{45} + 9 q^{46} + 52 q^{47} + 39 q^{48} + 45 q^{49} + 18 q^{50} + 20 q^{51} + 36 q^{52} + 60 q^{53} + q^{54} + 51 q^{56} + 6 q^{57} - 15 q^{58} + 42 q^{59} + 32 q^{60} - 18 q^{61} - 55 q^{62} + q^{63} + 40 q^{64} - 8 q^{65} - 15 q^{67} + 29 q^{68} + 36 q^{69} - 31 q^{70} + 50 q^{71} + 6 q^{72} - 17 q^{73} + 2 q^{74} + 39 q^{75} + 11 q^{76} + 17 q^{78} + 3 q^{79} + 37 q^{80} + 20 q^{81} - 5 q^{82} + 4 q^{83} - 7 q^{84} + 17 q^{85} + 32 q^{86} - 4 q^{87} + 16 q^{89} + 6 q^{90} + 13 q^{91} + 86 q^{92} + 7 q^{93} - 24 q^{94} - 64 q^{95} - 7 q^{96} + 15 q^{97} - 69 q^{98}+O(q^{100}) \) 20 * q + q^2 + 20 * q^3 + 29 * q^4 + 17 * q^5 + q^6 + q^7 + 6 * q^8 + 20 * q^9 + 6 * q^10 + 29 * q^12 + 8 * q^13 + 21 * q^14 + 17 * q^15 + 39 * q^16 + 20 * q^17 + q^18 + 6 * q^19 + 32 * q^20 + q^21 + 36 * q^23 + 6 * q^24 + 39 * q^25 + 17 * q^26 + 20 * q^27 - 7 * q^28 - 4 * q^29 + 6 * q^30 + 7 * q^31 - 7 * q^32 + q^34 - 2 * q^35 + 29 * q^36 + 12 * q^37 + 24 * q^38 + 8 * q^39 - 15 * q^40 - 6 * q^41 + 21 * q^42 - 6 * q^43 + 17 * q^45 + 9 * q^46 + 52 * q^47 + 39 * q^48 + 45 * q^49 + 18 * q^50 + 20 * q^51 + 36 * q^52 + 60 * q^53 + q^54 + 51 * q^56 + 6 * q^57 - 15 * q^58 + 42 * q^59 + 32 * q^60 - 18 * q^61 - 55 * q^62 + q^63 + 40 * q^64 - 8 * q^65 - 15 * q^67 + 29 * q^68 + 36 * q^69 - 31 * q^70 + 50 * q^71 + 6 * q^72 - 17 * q^73 + 2 * q^74 + 39 * q^75 + 11 * q^76 + 17 * q^78 + 3 * q^79 + 37 * q^80 + 20 * q^81 - 5 * q^82 + 4 * q^83 - 7 * q^84 + 17 * q^85 + 32 * q^86 - 4 * q^87 + 16 * q^89 + 6 * q^90 + 13 * q^91 + 86 * q^92 + 7 * q^93 - 24 * q^94 - 64 * q^95 - 7 * q^96 + 15 * q^97 - 69 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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