LMFDB - Embedded newform 8470.2.a.cq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.10522$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.91913 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.91913 q^{6} -1.00000 q^{7} -1.00000 q^{8} +0.683063 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.91913 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.91913 q^{6} -1.00000 q^{7} -1.00000 q^{8} +0.683063 q^{9} +1.00000 q^{10} +1.91913 q^{12} -0.869151 q^{13} +1.00000 q^{14} -1.91913 q^{15} +1.00000 q^{16} -4.15520 q^{17} -0.683063 q^{18} +7.56151 q^{19} -1.00000 q^{20} -1.91913 q^{21} -2.00000 q^{23} -1.91913 q^{24} +1.00000 q^{25} +0.869151 q^{26} -4.44651 q^{27} -1.00000 q^{28} -1.19588 q^{29} +1.91913 q^{30} +7.70820 q^{31} -1.00000 q^{32} +4.15520 q^{34} +1.00000 q^{35} +0.683063 q^{36} +6.81263 q^{37} -7.56151 q^{38} -1.66801 q^{39} +1.00000 q^{40} -9.86261 q^{41} +1.91913 q^{42} -6.62654 q^{43} -0.683063 q^{45} +2.00000 q^{46} +10.7529 q^{47} +1.91913 q^{48} +1.00000 q^{49} -1.00000 q^{50} -7.97437 q^{51} -0.869151 q^{52} -1.57656 q^{53} +4.44651 q^{54} +1.00000 q^{56} +14.5115 q^{57} +1.19588 q^{58} -7.43146 q^{59} -1.91913 q^{60} -7.73830 q^{61} -7.70820 q^{62} -0.683063 q^{63} +1.00000 q^{64} +0.869151 q^{65} +3.56151 q^{67} -4.15520 q^{68} -3.83826 q^{69} -1.00000 q^{70} -2.23528 q^{71} -0.683063 q^{72} +17.0389 q^{73} -6.81263 q^{74} +1.91913 q^{75} +7.56151 q^{76} +1.66801 q^{78} -3.13085 q^{79} -1.00000 q^{80} -10.5826 q^{81} +9.86261 q^{82} -8.22953 q^{83} -1.91913 q^{84} +4.15520 q^{85} +6.62654 q^{86} -2.29505 q^{87} -9.89429 q^{89} +0.683063 q^{90} +0.869151 q^{91} -2.00000 q^{92} +14.7931 q^{93} -10.7529 q^{94} -7.56151 q^{95} -1.91913 q^{96} +14.7984 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{12} - q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 6 q^{18} + 3 q^{19} - 4 q^{20} - 2 q^{21} - 8 q^{23} - 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} - 4 q^{28} - 15 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{32} + 2 q^{34} + 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} + 4 q^{40} - 11 q^{41} + 2 q^{42} - 7 q^{43} - 6 q^{45} + 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} - 4 q^{50} - 18 q^{51} - q^{52} + 10 q^{53} - 14 q^{54} + 4 q^{56} + 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} - 26 q^{61} - 4 q^{62} - 6 q^{63} + 4 q^{64} + q^{65} - 13 q^{67} - 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} - 6 q^{72} + 18 q^{73} - 2 q^{74} + 2 q^{75} + 3 q^{76} - q^{78} - 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} + 2 q^{83} - 2 q^{84} + 2 q^{85} + 7 q^{86} - 26 q^{87} - 7 q^{89} + 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} - 5 q^{94} - 3 q^{95} - 2 q^{96} + 10 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 + 4 * q^10 + 2 * q^12 - q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 6 * q^18 + 3 * q^19 - 4 * q^20 - 2 * q^21 - 8 * q^23 - 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 - 4 * q^28 - 15 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^32 + 2 * q^34 + 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 + q^39 + 4 * q^40 - 11 * q^41 + 2 * q^42 - 7 * q^43 - 6 * q^45 + 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 - 4 * q^50 - 18 * q^51 - q^52 + 10 * q^53 - 14 * q^54 + 4 * q^56 + 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 - 26 * q^61 - 4 * q^62 - 6 * q^63 + 4 * q^64 + q^65 - 13 * q^67 - 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 - 6 * q^72 + 18 * q^73 - 2 * q^74 + 2 * q^75 + 3 * q^76 - q^78 - 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 + 2 * q^83 - 2 * q^84 + 2 * q^85 + 7 * q^86 - 26 * q^87 - 7 * q^89 + 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 - 5 * q^94 - 3 * q^95 - 2 * q^96 + 10 * q^97 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.91913	1.10801	0.554005	−	0.832513i	$-0.313099\pi$
$3$	1.91913	1.10801	0.554005	+	0.832513i	$0.313099\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.91913	−0.783482
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0.683063	0.227688
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	1.91913	0.554005
$13$	−0.869151	−0.241059	−0.120530	−	0.992710i	$-0.538459\pi$
$13$	−0.869151	−0.241059	−0.120530	+	0.992710i	$0.538459\pi$
$14$	1.00000	0.267261
$15$	−1.91913	−0.495517
$16$	1.00000	0.250000
$17$	−4.15520	−1.00778	−0.503892	−	0.863767i	$-0.668099\pi$
$17$	−4.15520	−1.00778	−0.503892	+	0.863767i	$0.668099\pi$
$18$	−0.683063	−0.160999
$19$	7.56151	1.73473	0.867365	−	0.497672i	$-0.165812\pi$
$19$	7.56151	1.73473	0.867365	+	0.497672i	$0.165812\pi$
$20$	−1.00000	−0.223607
$21$	−1.91913	−0.418789
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−1.91913	−0.391741
$25$	1.00000	0.200000
$26$	0.869151	0.170454
$27$	−4.44651	−0.855730
$28$	−1.00000	−0.188982
$29$	−1.19588	−0.222069	−0.111034	−	0.993817i	$-0.535416\pi$
$29$	−1.19588	−0.222069	−0.111034	+	0.993817i	$0.535416\pi$
$30$	1.91913	0.350384
$31$	7.70820	1.38443	0.692217	−	0.721689i	$-0.256634\pi$
$31$	7.70820	1.38443	0.692217	+	0.721689i	$0.256634\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.15520	0.712611
$35$	1.00000	0.169031
$36$	0.683063	0.113844
$37$	6.81263	1.11999	0.559995	−	0.828496i	$-0.310803\pi$
$37$	6.81263	1.11999	0.559995	+	0.828496i	$0.310803\pi$
$38$	−7.56151	−1.22664
$39$	−1.66801	−0.267096
$40$	1.00000	0.158114
$41$	−9.86261	−1.54028	−0.770141	−	0.637874i	$-0.779814\pi$
$41$	−9.86261	−1.54028	−0.770141	+	0.637874i	$0.779814\pi$
$42$	1.91913	0.296128
$43$	−6.62654	−1.01054	−0.505269	−	0.862962i	$-0.668607\pi$
$43$	−6.62654	−1.01054	−0.505269	+	0.862962i	$0.668607\pi$
$44$	0	0
$45$	−0.683063	−0.101825
$46$	2.00000	0.294884
$47$	10.7529	1.56847	0.784233	−	0.620466i	$-0.213057\pi$
$47$	10.7529	1.56847	0.784233	+	0.620466i	$0.213057\pi$
$48$	1.91913	0.277003
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−7.97437	−1.11664
$52$	−0.869151	−0.120530
$53$	−1.57656	−0.216558	−0.108279	−	0.994121i	$-0.534534\pi$
$53$	−1.57656	−0.216558	−0.108279	+	0.994121i	$0.534534\pi$
$54$	4.44651	0.605093
$55$	0	0
$56$	1.00000	0.133631
$57$	14.5115	1.92210
$58$	1.19588	0.157026
$59$	−7.43146	−0.967493	−0.483747	−	0.875208i	$-0.660724\pi$
$59$	−7.43146	−0.967493	−0.483747	+	0.875208i	$0.660724\pi$
$60$	−1.91913	−0.247759
$61$	−7.73830	−0.990788	−0.495394	−	0.868668i	$-0.664976\pi$
$61$	−7.73830	−0.990788	−0.495394	+	0.868668i	$0.664976\pi$
$62$	−7.70820	−0.978943
$63$	−0.683063	−0.0860578
$64$	1.00000	0.125000
$65$	0.869151	0.107805
$66$	0	0
$67$	3.56151	0.435108	0.217554	−	0.976048i	$-0.430192\pi$
$67$	3.56151	0.435108	0.217554	+	0.976048i	$0.430192\pi$
$68$	−4.15520	−0.503892
$69$	−3.83826	−0.462072
$70$	−1.00000	−0.119523
$71$	−2.23528	−0.265278	−0.132639	−	0.991164i	$-0.542345\pi$
$71$	−2.23528	−0.265278	−0.132639	+	0.991164i	$0.542345\pi$
$72$	−0.683063	−0.0804997
$73$	17.0389	1.99425	0.997127	−	0.0757515i	$-0.0241356\pi$
$73$	17.0389	1.99425	0.997127	+	0.0757515i	$0.0241356\pi$
$74$	−6.81263	−0.791952
$75$	1.91913	0.221602
$76$	7.56151	0.867365
$77$	0	0
$78$	1.66801	0.188865
$79$	−3.13085	−0.352248	−0.176124	−	0.984368i	$-0.556356\pi$
$79$	−3.13085	−0.352248	−0.176124	+	0.984368i	$0.556356\pi$
$80$	−1.00000	−0.111803
$81$	−10.5826	−1.17585
$82$	9.86261	1.08914
$83$	−8.22953	−0.903308	−0.451654	−	0.892193i	$-0.649166\pi$
$83$	−8.22953	−0.903308	−0.451654	+	0.892193i	$0.649166\pi$
$84$	−1.91913	−0.209394
$85$	4.15520	0.450695
$86$	6.62654	0.714559
$87$	−2.29505	−0.246055
$88$	0	0
$89$	−9.89429	−1.04879	−0.524396	−	0.851474i	$-0.675709\pi$
$89$	−9.89429	−1.04879	−0.524396	+	0.851474i	$0.675709\pi$
$90$	0.683063	0.0720011
$91$	0.869151	0.0911118
$92$	−2.00000	−0.208514
$93$	14.7931	1.53397
$94$	−10.7529	−1.10907
$95$	−7.56151	−0.775795
$96$	−1.91913	−0.195870
$97$	14.7984	1.50255	0.751274	−	0.659991i	$-0.229440\pi$
$97$	14.7984	1.50255	0.751274	+	0.659991i	$0.229440\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−17.6082	−1.75209	−0.876043	−	0.482233i	$-0.839826\pi$
$101$	−17.6082	−1.75209	−0.876043	+	0.482233i	$0.839826\pi$
$102$	7.97437	0.789580
$103$	−4.85282	−0.478163	−0.239081	−	0.971000i	$-0.576846\pi$
$103$	−4.85282	−0.478163	−0.239081	+	0.971000i	$0.576846\pi$
$104$	0.869151	0.0852272
$105$	1.91913	0.187288
$106$	1.57656	0.153129
$107$	−5.97309	−0.577440	−0.288720	−	0.957414i	$-0.593230\pi$
$107$	−5.97309	−0.577440	−0.288720	+	0.957414i	$0.593230\pi$
$108$	−4.44651	−0.427865
$109$	−19.3651	−1.85484	−0.927422	−	0.374016i	$-0.877981\pi$
$109$	−19.3651	−1.85484	−0.927422	+	0.374016i	$0.877981\pi$
$110$	0	0
$111$	13.0743	1.24096
$112$	−1.00000	−0.0944911
$113$	13.4932	1.26934	0.634668	−	0.772785i	$-0.281137\pi$
$113$	13.4932	1.26934	0.634668	+	0.772785i	$0.281137\pi$
$114$	−14.5115	−1.35913
$115$	2.00000	0.186501
$116$	−1.19588	−0.111034
$117$	−0.593685	−0.0548862
$118$	7.43146	0.684121
$119$	4.15520	0.380906
$120$	1.91913	0.175192
$121$	0	0
$122$	7.73830	0.700593
$123$	−18.9276	−1.70665
$124$	7.70820	0.692217
$125$	−1.00000	−0.0894427
$126$	0.683063	0.0608521
$127$	−2.68257	−0.238040	−0.119020	−	0.992892i	$-0.537975\pi$
$127$	−2.68257	−0.238040	−0.119020	+	0.992892i	$0.537975\pi$
$128$	−1.00000	−0.0883883
$129$	−12.7172	−1.11969
$130$	−0.869151	−0.0762296
$131$	1.19986	0.104832	0.0524159	−	0.998625i	$-0.483308\pi$
$131$	1.19986	0.104832	0.0524159	+	0.998625i	$0.483308\pi$
$132$	0	0
$133$	−7.56151	−0.655666
$134$	−3.56151	−0.307668
$135$	4.44651	0.382694
$136$	4.15520	0.356305
$137$	−4.65822	−0.397979	−0.198989	−	0.980002i	$-0.563766\pi$
$137$	−4.65822	−0.397979	−0.198989	+	0.980002i	$0.563766\pi$
$138$	3.83826	0.326735
$139$	−12.6208	−1.07048	−0.535241	−	0.844699i	$-0.679779\pi$
$139$	−12.6208	−1.07048	−0.535241	+	0.844699i	$0.679779\pi$
$140$	1.00000	0.0845154
$141$	20.6361	1.73788
$142$	2.23528	0.187580
$143$	0	0
$144$	0.683063	0.0569219
$145$	1.19588	0.0993123
$146$	−17.0389	−1.41015
$147$	1.91913	0.158287
$148$	6.81263	0.559995
$149$	16.1238	1.32091	0.660457	−	0.750864i	$-0.270362\pi$
$149$	16.1238	1.32091	0.660457	+	0.750864i	$0.270362\pi$
$150$	−1.91913	−0.156696
$151$	9.81017	0.798341	0.399170	−	0.916877i	$-0.369298\pi$
$151$	9.81017	0.798341	0.399170	+	0.916877i	$0.369298\pi$
$152$	−7.56151	−0.613320
$153$	−2.83826	−0.229460
$154$	0	0
$155$	−7.70820	−0.619138
$156$	−1.66801	−0.133548
$157$	−2.74234	−0.218863	−0.109431	−	0.993994i	$-0.534903\pi$
$157$	−2.74234	−0.218863	−0.109431	+	0.993994i	$0.534903\pi$
$158$	3.13085	0.249077
$159$	−3.02563	−0.239948
$160$	1.00000	0.0790569
$161$	2.00000	0.157622
$162$	10.5826	0.831449
$163$	−9.38020	−0.734714	−0.367357	−	0.930080i	$-0.619737\pi$
$163$	−9.38020	−0.734714	−0.367357	+	0.930080i	$0.619737\pi$
$164$	−9.86261	−0.770141
$165$	0	0
$166$	8.22953	0.638735
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	1.91913	0.148064
$169$	−12.2446	−0.941891
$170$	−4.15520	−0.318689
$171$	5.16499	0.394977
$172$	−6.62654	−0.505269
$173$	9.14787	0.695499	0.347750	−	0.937587i	$-0.386946\pi$
$173$	9.14787	0.695499	0.347750	+	0.937587i	$0.386946\pi$
$174$	2.29505	0.173987
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−14.2619	−1.07199
$178$	9.89429	0.741609
$179$	3.71474	0.277653	0.138826	−	0.990317i	$-0.455667\pi$
$179$	3.71474	0.277653	0.138826	+	0.990317i	$0.455667\pi$
$180$	−0.683063	−0.0509125
$181$	−21.4439	−1.59392	−0.796958	−	0.604035i	$-0.793559\pi$
$181$	−21.4439	−1.59392	−0.796958	+	0.604035i	$0.793559\pi$
$182$	−0.869151	−0.0644257
$183$	−14.8508	−1.09780
$184$	2.00000	0.147442
$185$	−6.81263	−0.500875
$186$	−14.7931	−1.08468
$187$	0	0
$188$	10.7529	0.784233
$189$	4.44651	0.323436
$190$	7.56151	0.548570
$191$	−21.8922	−1.58407	−0.792033	−	0.610479i	$-0.790977\pi$
$191$	−21.8922	−1.58407	−0.792033	+	0.610479i	$0.790977\pi$
$192$	1.91913	0.138501
$193$	4.07880	0.293598	0.146799	−	0.989166i	$-0.453103\pi$
$193$	4.07880	0.293598	0.146799	+	0.989166i	$0.453103\pi$
$194$	−14.7984	−1.06246
$195$	1.66801	0.119449
$196$	1.00000	0.0714286
$197$	14.2275	1.01366	0.506832	−	0.862045i	$-0.330816\pi$
$197$	14.2275	1.01366	0.506832	+	0.862045i	$0.330816\pi$
$198$	0	0
$199$	−8.75085	−0.620331	−0.310166	−	0.950682i	$-0.600385\pi$
$199$	−8.75085	−0.620331	−0.310166	+	0.950682i	$0.600385\pi$
$200$	−1.00000	−0.0707107
$201$	6.83501	0.482105
$202$	17.6082	1.23891
$203$	1.19588	0.0839342
$204$	−7.97437	−0.558318
$205$	9.86261	0.688835
$206$	4.85282	0.338112
$207$	−1.36613	−0.0949523
$208$	−0.869151	−0.0602648
$209$	0	0
$210$	−1.91913	−0.132433
$211$	−19.7695	−1.36099	−0.680494	−	0.732754i	$-0.738235\pi$
$211$	−19.7695	−1.36099	−0.680494	+	0.732754i	$0.738235\pi$
$212$	−1.57656	−0.108279
$213$	−4.28979	−0.293931
$214$	5.97309	0.408312
$215$	6.62654	0.451926
$216$	4.44651	0.302546
$217$	−7.70820	−0.523267
$218$	19.3651	1.31157
$219$	32.6999	2.20965
$220$	0	0
$221$	3.61149	0.242935
$222$	−13.0743	−0.877492
$223$	10.9231	0.731465	0.365733	−	0.930720i	$-0.380818\pi$
$223$	10.9231	0.731465	0.365733	+	0.930720i	$0.380818\pi$
$224$	1.00000	0.0668153
$225$	0.683063	0.0455375
$226$	−13.4932	−0.897557
$227$	−19.5668	−1.29869	−0.649346	−	0.760493i	$-0.724957\pi$
$227$	−19.5668	−1.29869	−0.649346	+	0.760493i	$0.724957\pi$
$228$	14.5115	0.961050
$229$	−14.8613	−0.982064	−0.491032	−	0.871141i	$-0.663380\pi$
$229$	−14.8613	−0.982064	−0.491032	+	0.871141i	$0.663380\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	1.19588	0.0785132
$233$	4.57784	0.299905	0.149952	−	0.988693i	$-0.452088\pi$
$233$	4.57784	0.299905	0.149952	+	0.988693i	$0.452088\pi$
$234$	0.593685	0.0388104
$235$	−10.7529	−0.701439
$236$	−7.43146	−0.483747
$237$	−6.00851	−0.390295
$238$	−4.15520	−0.269342
$239$	−16.9479	−1.09627	−0.548136	−	0.836389i	$-0.684662\pi$
$239$	−16.9479	−1.09627	−0.548136	+	0.836389i	$0.684662\pi$
$240$	−1.91913	−0.123879
$241$	27.4289	1.76685	0.883425	−	0.468572i	$-0.155231\pi$
$241$	27.4289	1.76685	0.883425	+	0.468572i	$0.155231\pi$
$242$	0	0
$243$	−6.96990	−0.447119
$244$	−7.73830	−0.495394
$245$	−1.00000	−0.0638877
$246$	18.9276	1.20678
$247$	−6.57210	−0.418172
$248$	−7.70820	−0.489471
$249$	−15.7935	−1.00087
$250$	1.00000	0.0632456
$251$	5.21490	0.329162	0.164581	−	0.986364i	$-0.447373\pi$
$251$	5.21490	0.329162	0.164581	+	0.986364i	$0.447373\pi$
$252$	−0.683063	−0.0430289
$253$	0	0
$254$	2.68257	0.168320
$255$	7.97437	0.499374
$256$	1.00000	0.0625000
$257$	−1.77372	−0.110642	−0.0553209	−	0.998469i	$-0.517618\pi$
$257$	−1.77372	−0.110642	−0.0553209	+	0.998469i	$0.517618\pi$
$258$	12.7172	0.791738
$259$	−6.81263	−0.423316
$260$	0.869151	0.0539024
$261$	−0.816860	−0.0505623
$262$	−1.19986	−0.0741273
$263$	−14.0317	−0.865231	−0.432615	−	0.901579i	$-0.642409\pi$
$263$	−14.0317	−0.865231	−0.432615	+	0.901579i	$0.642409\pi$
$264$	0	0
$265$	1.57656	0.0968475
$266$	7.56151	0.463626
$267$	−18.9884	−1.16207
$268$	3.56151	0.217554
$269$	−14.0382	−0.855923	−0.427962	−	0.903797i	$-0.640768\pi$
$269$	−14.0382	−0.855923	−0.427962	+	0.903797i	$0.640768\pi$
$270$	−4.44651	−0.270606
$271$	21.5635	1.30989	0.654944	−	0.755677i	$-0.272692\pi$
$271$	21.5635	1.30989	0.654944	+	0.755677i	$0.272692\pi$
$272$	−4.15520	−0.251946
$273$	1.66801	0.100953
$274$	4.65822	0.281414
$275$	0	0
$276$	−3.83826	−0.231036
$277$	−10.4696	−0.629056	−0.314528	−	0.949248i	$-0.601846\pi$
$277$	−10.4696	−0.629056	−0.314528	+	0.949248i	$0.601846\pi$
$278$	12.6208	0.756945
$279$	5.26519	0.315218
$280$	−1.00000	−0.0597614
$281$	27.5391	1.64285	0.821423	−	0.570319i	$-0.193180\pi$
$281$	27.5391	1.64285	0.821423	+	0.570319i	$0.193180\pi$
$282$	−20.6361	−1.22886
$283$	−12.6529	−0.752137	−0.376068	−	0.926592i	$-0.622724\pi$
$283$	−12.6529	−0.752137	−0.376068	+	0.926592i	$0.622724\pi$
$284$	−2.23528	−0.132639
$285$	−14.5115	−0.859589
$286$	0	0
$287$	9.86261	0.582172
$288$	−0.683063	−0.0402499
$289$	0.265676	0.0156280
$290$	−1.19588	−0.0702244
$291$	28.4000	1.66484
$292$	17.0389	0.997127
$293$	−26.4209	−1.54352	−0.771762	−	0.635911i	$-0.780624\pi$
$293$	−26.4209	−1.54352	−0.771762	+	0.635911i	$0.780624\pi$
$294$	−1.91913	−0.111926
$295$	7.43146	0.432676
$296$	−6.81263	−0.395976
$297$	0	0
$298$	−16.1238	−0.934028
$299$	1.73830	0.100529
$300$	1.91913	0.110801
$301$	6.62654	0.381948
$302$	−9.81017	−0.564512
$303$	−33.7925	−1.94133
$304$	7.56151	0.433683
$305$	7.73830	0.443094
$306$	2.83826	0.162253
$307$	−24.8324	−1.41726	−0.708630	−	0.705580i	$-0.750687\pi$
$307$	−24.8324	−1.41726	−0.708630	+	0.705580i	$0.750687\pi$
$308$	0	0
$309$	−9.31320	−0.529809
$310$	7.70820	0.437797
$311$	5.74086	0.325535	0.162767	−	0.986664i	$-0.447958\pi$
$311$	5.74086	0.325535	0.162767	+	0.986664i	$0.447958\pi$
$312$	1.66801	0.0944327
$313$	−21.7694	−1.23048	−0.615239	−	0.788340i	$-0.710941\pi$
$313$	−21.7694	−1.23048	−0.615239	+	0.788340i	$0.710941\pi$
$314$	2.74234	0.154759
$315$	0.683063	0.0384862
$316$	−3.13085	−0.176124
$317$	6.86291	0.385460	0.192730	−	0.981252i	$-0.438266\pi$
$317$	6.86291	0.385460	0.192730	+	0.981252i	$0.438266\pi$
$318$	3.02563	0.169669
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−11.4631	−0.639810
$322$	−2.00000	−0.111456
$323$	−31.4196	−1.74823
$324$	−10.5826	−0.587923
$325$	−0.869151	−0.0482118
$326$	9.38020	0.519521
$327$	−37.1642	−2.05519
$328$	9.86261	0.544572
$329$	−10.7529	−0.592824
$330$	0	0
$331$	−20.6399	−1.13447	−0.567235	−	0.823556i	$-0.691987\pi$
$331$	−20.6399	−1.13447	−0.567235	+	0.823556i	$0.691987\pi$
$332$	−8.22953	−0.451654
$333$	4.65345	0.255008
$334$	12.0000	0.656611
$335$	−3.56151	−0.194586
$336$	−1.91913	−0.104697
$337$	0.0255665	0.00139269	0.000696347	−	1.00000i	$-0.499778\pi$
$337$	0.0255665	0.00139269	0.000696347		1.00000i	$0.499778\pi$
$338$	12.2446	0.666017
$339$	25.8953	1.40644
$340$	4.15520	0.225347
$341$	0	0
$342$	−5.16499	−0.279291
$343$	−1.00000	−0.0539949
$344$	6.62654	0.357279
$345$	3.83826	0.206645
$346$	−9.14787	−0.491792
$347$	27.7393	1.48912	0.744562	−	0.667554i	$-0.232659\pi$
$347$	27.7393	1.48912	0.744562	+	0.667554i	$0.232659\pi$
$348$	−2.29505	−0.123027
$349$	7.76742	0.415780	0.207890	−	0.978152i	$-0.433340\pi$
$349$	7.76742	0.415780	0.207890	+	0.978152i	$0.433340\pi$
$350$	1.00000	0.0534522
$351$	3.86468	0.206282
$352$	0	0
$353$	−23.1303	−1.23110	−0.615550	−	0.788098i	$-0.711066\pi$
$353$	−23.1303	−1.23110	−0.615550	+	0.788098i	$0.711066\pi$
$354$	14.2619	0.758013
$355$	2.23528	0.118636
$356$	−9.89429	−0.524396
$357$	7.97437	0.422048
$358$	−3.71474	−0.196330
$359$	−12.6100	−0.665531	−0.332766	−	0.943010i	$-0.607982\pi$
$359$	−12.6100	−0.665531	−0.332766	+	0.943010i	$0.607982\pi$
$360$	0.683063	0.0360006
$361$	38.1765	2.00929
$362$	21.4439	1.12707
$363$	0	0
$364$	0.869151	0.0455559
$365$	−17.0389	−0.891857
$366$	14.8508	0.776264
$367$	3.51354	0.183405	0.0917027	−	0.995786i	$-0.470769\pi$
$367$	3.51354	0.183405	0.0917027	+	0.995786i	$0.470769\pi$
$368$	−2.00000	−0.104257
$369$	−6.73678	−0.350703
$370$	6.81263	0.354172
$371$	1.57656	0.0818511
$372$	14.7931	0.766984
$373$	−8.48915	−0.439552	−0.219776	−	0.975550i	$-0.570533\pi$
$373$	−8.48915	−0.439552	−0.219776	+	0.975550i	$0.570533\pi$
$374$	0	0
$375$	−1.91913	−0.0991035
$376$	−10.7529	−0.554536
$377$	1.03940	0.0535317
$378$	−4.44651	−0.228704
$379$	−19.0706	−0.979590	−0.489795	−	0.871838i	$-0.662928\pi$
$379$	−19.0706	−0.979590	−0.489795	+	0.871838i	$0.662928\pi$
$380$	−7.56151	−0.387897
$381$	−5.14821	−0.263751
$382$	21.8922	1.12010
$383$	1.33479	0.0682044	0.0341022	−	0.999418i	$-0.489143\pi$
$383$	1.33479	0.0682044	0.0341022	+	0.999418i	$0.489143\pi$
$384$	−1.91913	−0.0979352
$385$	0	0
$386$	−4.07880	−0.207605
$387$	−4.52634	−0.230087
$388$	14.7984	0.751274
$389$	−1.18211	−0.0599354	−0.0299677	−	0.999551i	$-0.509540\pi$
$389$	−1.18211	−0.0599354	−0.0299677	+	0.999551i	$0.509540\pi$
$390$	−1.66801	−0.0844632
$391$	8.31040	0.420275
$392$	−1.00000	−0.0505076
$393$	2.30268	0.116155
$394$	−14.2275	−0.716769
$395$	3.13085	0.157530
$396$	0	0
$397$	−15.9540	−0.800708	−0.400354	−	0.916361i	$-0.631113\pi$
$397$	−15.9540	−0.800708	−0.400354	+	0.916361i	$0.631113\pi$
$398$	8.75085	0.438641
$399$	−14.5115	−0.726485
$400$	1.00000	0.0500000
$401$	−11.8135	−0.589937	−0.294969	−	0.955507i	$-0.595309\pi$
$401$	−11.8135	−0.589937	−0.294969	+	0.955507i	$0.595309\pi$
$402$	−6.83501	−0.340899
$403$	−6.69959	−0.333730
$404$	−17.6082	−0.876043
$405$	10.5826	0.525854
$406$	−1.19588	−0.0593504
$407$	0	0
$408$	7.97437	0.394790
$409$	−17.2060	−0.850780	−0.425390	−	0.905010i	$-0.639863\pi$
$409$	−17.2060	−0.850780	−0.425390	+	0.905010i	$0.639863\pi$
$410$	−9.86261	−0.487080
$411$	−8.93974	−0.440965
$412$	−4.85282	−0.239081
$413$	7.43146	0.365678
$414$	1.36613	0.0671414
$415$	8.22953	0.403972
$416$	0.869151	0.0426136
$417$	−24.2210	−1.18611
$418$	0	0
$419$	−7.07910	−0.345837	−0.172918	−	0.984936i	$-0.555320\pi$
$419$	−7.07910	−0.345837	−0.172918	+	0.984936i	$0.555320\pi$
$420$	1.91913	0.0936440
$421$	−15.8752	−0.773710	−0.386855	−	0.922141i	$-0.626439\pi$
$421$	−15.8752	−0.773710	−0.386855	+	0.922141i	$0.626439\pi$
$422$	19.7695	0.962364
$423$	7.34488	0.357120
$424$	1.57656	0.0765647
$425$	−4.15520	−0.201557
$426$	4.28979	0.207841
$427$	7.73830	0.374483
$428$	−5.97309	−0.288720
$429$	0	0
$430$	−6.62654	−0.319560
$431$	−17.4697	−0.841485	−0.420742	−	0.907180i	$-0.638230\pi$
$431$	−17.4697	−0.841485	−0.420742	+	0.907180i	$0.638230\pi$
$432$	−4.44651	−0.213933
$433$	18.1892	0.874115	0.437057	−	0.899434i	$-0.356021\pi$
$433$	18.1892	0.874115	0.437057	+	0.899434i	$0.356021\pi$
$434$	7.70820	0.370006
$435$	2.29505	0.110039
$436$	−19.3651	−0.927422
$437$	−15.1230	−0.723433
$438$	−32.6999	−1.56246
$439$	−16.5826	−0.791445	−0.395722	−	0.918370i	$-0.629506\pi$
$439$	−16.5826	−0.791445	−0.395722	+	0.918370i	$0.629506\pi$
$440$	0	0
$441$	0.683063	0.0325268
$442$	−3.61149	−0.171781
$443$	−37.1961	−1.76724	−0.883619	−	0.468206i	$-0.844901\pi$
$443$	−37.1961	−1.76724	−0.883619	+	0.468206i	$0.844901\pi$
$444$	13.0743	0.620480
$445$	9.89429	0.469034
$446$	−10.9231	−0.517224
$447$	30.9437	1.46359
$448$	−1.00000	−0.0472456
$449$	10.5728	0.498962	0.249481	−	0.968380i	$-0.419740\pi$
$449$	10.5728	0.498962	0.249481	+	0.968380i	$0.419740\pi$
$450$	−0.683063	−0.0321999
$451$	0	0
$452$	13.4932	0.634668
$453$	18.8270	0.884570
$454$	19.5668	0.918314
$455$	−0.869151	−0.0407464
$456$	−14.5115	−0.679565
$457$	−1.57179	−0.0735254	−0.0367627	−	0.999324i	$-0.511705\pi$
$457$	−1.57179	−0.0735254	−0.0367627	+	0.999324i	$0.511705\pi$
$458$	14.8613	0.694424
$459$	18.4761	0.862391
$460$	2.00000	0.0932505
$461$	15.9056	0.740796	0.370398	−	0.928873i	$-0.379221\pi$
$461$	15.9056	0.740796	0.370398	+	0.928873i	$0.379221\pi$
$462$	0	0
$463$	22.1265	1.02831	0.514153	−	0.857698i	$-0.328106\pi$
$463$	22.1265	1.02831	0.514153	+	0.857698i	$0.328106\pi$
$464$	−1.19588	−0.0555172
$465$	−14.7931	−0.686011
$466$	−4.57784	−0.212065
$467$	3.31217	0.153269	0.0766344	−	0.997059i	$-0.475583\pi$
$467$	3.31217	0.153269	0.0766344	+	0.997059i	$0.475583\pi$
$468$	−0.593685	−0.0274431
$469$	−3.56151	−0.164455
$470$	10.7529	0.495992
$471$	−5.26292	−0.242502
$472$	7.43146	0.342061
$473$	0	0
$474$	6.00851	0.275980
$475$	7.56151	0.346946
$476$	4.15520	0.190453
$477$	−1.07689	−0.0493075
$478$	16.9479	0.775181
$479$	−26.3656	−1.20467	−0.602337	−	0.798242i	$-0.705764\pi$
$479$	−26.3656	−1.20467	−0.602337	+	0.798242i	$0.705764\pi$
$480$	1.91913	0.0875959
$481$	−5.92120	−0.269984
$482$	−27.4289	−1.24935
$483$	3.83826	0.174647
$484$	0	0
$485$	−14.7984	−0.671960
$486$	6.96990	0.316161
$487$	12.4706	0.565095	0.282547	−	0.959253i	$-0.408821\pi$
$487$	12.4706	0.565095	0.282547	+	0.959253i	$0.408821\pi$
$488$	7.73830	0.350296
$489$	−18.0018	−0.814071
$490$	1.00000	0.0451754
$491$	36.6954	1.65604	0.828021	−	0.560698i	$-0.189467\pi$
$491$	36.6954	1.65604	0.828021	+	0.560698i	$0.189467\pi$
$492$	−18.9276	−0.853324
$493$	4.96911	0.223797
$494$	6.57210	0.295693
$495$	0	0
$496$	7.70820	0.346109
$497$	2.23528	0.100266
$498$	15.7935	0.707725
$499$	22.0831	0.988577	0.494289	−	0.869298i	$-0.335429\pi$
$499$	22.0831	0.988577	0.494289	+	0.869298i	$0.335429\pi$
$500$	−1.00000	−0.0447214
$501$	−23.0296	−1.02889
$502$	−5.21490	−0.232753
$503$	21.6517	0.965401	0.482700	−	0.875786i	$-0.339656\pi$
$503$	21.6517	0.965401	0.482700	+	0.875786i	$0.339656\pi$
$504$	0.683063	0.0304260
$505$	17.6082	0.783557
$506$	0	0
$507$	−23.4989	−1.04362
$508$	−2.68257	−0.119020
$509$	10.8021	0.478795	0.239398	−	0.970922i	$-0.423050\pi$
$509$	10.8021	0.478795	0.239398	+	0.970922i	$0.423050\pi$
$510$	−7.97437	−0.353111
$511$	−17.0389	−0.753757
$512$	−1.00000	−0.0441942
$513$	−33.6223	−1.48446
$514$	1.77372	0.0782355
$515$	4.85282	0.213841
$516$	−12.7172	−0.559844
$517$	0	0
$518$	6.81263	0.299330
$519$	17.5560	0.770621
$520$	−0.869151	−0.0381148
$521$	25.9337	1.13618	0.568088	−	0.822968i	$-0.307683\pi$
$521$	25.9337	1.13618	0.568088	+	0.822968i	$0.307683\pi$
$522$	0.816860	0.0357530
$523$	23.7789	1.03978	0.519889	−	0.854234i	$-0.325973\pi$
$523$	23.7789	1.03978	0.519889	+	0.854234i	$0.325973\pi$
$524$	1.19986	0.0524159
$525$	−1.91913	−0.0837577
$526$	14.0317	0.611810
$527$	−32.0291	−1.39521
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−1.57656	−0.0684815
$531$	−5.07615	−0.220286
$532$	−7.56151	−0.327833
$533$	8.57210	0.371299
$534$	18.9884	0.821710
$535$	5.97309	0.258239
$536$	−3.56151	−0.153834
$537$	7.12908	0.307642
$538$	14.0382	0.605229
$539$	0	0
$540$	4.44651	0.191347
$541$	−33.1087	−1.42345	−0.711727	−	0.702456i	$-0.752087\pi$
$541$	−33.1087	−1.42345	−0.711727	+	0.702456i	$0.752087\pi$
$542$	−21.5635	−0.926231
$543$	−41.1537	−1.76608
$544$	4.15520	0.178153
$545$	19.3651	0.829512
$546$	−1.66801	−0.0713844
$547$	26.2035	1.12038	0.560190	−	0.828364i	$-0.310728\pi$
$547$	26.2035	1.12038	0.560190	+	0.828364i	$0.310728\pi$
$548$	−4.65822	−0.198989
$549$	−5.28575	−0.225590
$550$	0	0
$551$	−9.04265	−0.385230
$552$	3.83826	0.163367
$553$	3.13085	0.133137
$554$	10.4696	0.444810
$555$	−13.0743	−0.554974
$556$	−12.6208	−0.535241
$557$	−1.03615	−0.0439030	−0.0219515	−	0.999759i	$-0.506988\pi$
$557$	−1.03615	−0.0439030	−0.0219515	+	0.999759i	$0.506988\pi$
$558$	−5.26519	−0.222893
$559$	5.75946	0.243599
$560$	1.00000	0.0422577
$561$	0	0
$562$	−27.5391	−1.16167
$563$	16.5668	0.698206	0.349103	−	0.937084i	$-0.386486\pi$
$563$	16.5668	0.698206	0.349103	+	0.937084i	$0.386486\pi$
$564$	20.6361	0.868939
$565$	−13.4932	−0.567665
$566$	12.6529	0.531841
$567$	10.5826	0.444428
$568$	2.23528	0.0937901
$569$	−28.6394	−1.20063	−0.600314	−	0.799765i	$-0.704958\pi$
$569$	−28.6394	−1.20063	−0.600314	+	0.799765i	$0.704958\pi$
$570$	14.5115	0.607821
$571$	16.0059	0.669828	0.334914	−	0.942249i	$-0.391293\pi$
$571$	16.0059	0.669828	0.334914	+	0.942249i	$0.391293\pi$
$572$	0	0
$573$	−42.0140	−1.75516
$574$	−9.86261	−0.411657
$575$	−2.00000	−0.0834058
$576$	0.683063	0.0284609
$577$	35.1456	1.46313	0.731565	−	0.681771i	$-0.238790\pi$
$577$	35.1456	1.46313	0.731565	+	0.681771i	$0.238790\pi$
$578$	−0.265676	−0.0110507
$579$	7.82774	0.325310
$580$	1.19588	0.0496561
$581$	8.22953	0.341418
$582$	−28.4000	−1.17722
$583$	0	0
$584$	−17.0389	−0.705075
$585$	0.593685	0.0245458
$586$	26.4209	1.09144
$587$	30.2584	1.24890	0.624449	−	0.781066i	$-0.285324\pi$
$587$	30.2584	1.24890	0.624449	+	0.781066i	$0.285324\pi$
$588$	1.91913	0.0791436
$589$	58.2857	2.40162
$590$	−7.43146	−0.305948
$591$	27.3043	1.12315
$592$	6.81263	0.279997
$593$	16.7760	0.688907	0.344454	−	0.938803i	$-0.388064\pi$
$593$	16.7760	0.688907	0.344454	+	0.938803i	$0.388064\pi$
$594$	0	0
$595$	−4.15520	−0.170347
$596$	16.1238	0.660457
$597$	−16.7940	−0.687334
$598$	−1.73830	−0.0710844
$599$	−22.6333	−0.924772	−0.462386	−	0.886679i	$-0.653007\pi$
$599$	−22.6333	−0.924772	−0.462386	+	0.886679i	$0.653007\pi$
$600$	−1.91913	−0.0783482
$601$	−31.7235	−1.29403	−0.647014	−	0.762478i	$-0.723983\pi$
$601$	−31.7235	−1.29403	−0.647014	+	0.762478i	$0.723983\pi$
$602$	−6.62654	−0.270078
$603$	2.43274	0.0990687
$604$	9.81017	0.399170
$605$	0	0
$606$	33.7925	1.37273
$607$	15.2933	0.620735	0.310367	−	0.950617i	$-0.399548\pi$
$607$	15.2933	0.620735	0.310367	+	0.950617i	$0.399548\pi$
$608$	−7.56151	−0.306660
$609$	2.29505	0.0930000
$610$	−7.73830	−0.313315
$611$	−9.34586	−0.378093
$612$	−2.83826	−0.114730
$613$	9.73988	0.393390	0.196695	−	0.980465i	$-0.436979\pi$
$613$	9.73988	0.393390	0.196695	+	0.980465i	$0.436979\pi$
$614$	24.8324	1.00215
$615$	18.9276	0.763236
$616$	0	0
$617$	30.2611	1.21827	0.609133	−	0.793068i	$-0.291518\pi$
$617$	30.2611	1.21827	0.609133	+	0.793068i	$0.291518\pi$
$618$	9.31320	0.374632
$619$	−27.0250	−1.08623	−0.543114	−	0.839659i	$-0.682755\pi$
$619$	−27.0250	−1.08623	−0.543114	+	0.839659i	$0.682755\pi$
$620$	−7.70820	−0.309569
$621$	8.89301	0.356864
$622$	−5.74086	−0.230188
$623$	9.89429	0.396406
$624$	−1.66801	−0.0667740
$625$	1.00000	0.0400000
$626$	21.7694	0.870080
$627$	0	0
$628$	−2.74234	−0.109431
$629$	−28.3078	−1.12871
$630$	−0.683063	−0.0272139
$631$	10.8554	0.432146	0.216073	−	0.976377i	$-0.430675\pi$
$631$	10.8554	0.432146	0.216073	+	0.976377i	$0.430675\pi$
$632$	3.13085	0.124538
$633$	−37.9402	−1.50799
$634$	−6.86291	−0.272561
$635$	2.68257	0.106455
$636$	−3.02563	−0.119974
$637$	−0.869151	−0.0344370
$638$	0	0
$639$	−1.52683	−0.0604006
$640$	1.00000	0.0395285
$641$	−16.0629	−0.634445	−0.317223	−	0.948351i	$-0.602750\pi$
$641$	−16.0629	−0.634445	−0.317223	+	0.948351i	$0.602750\pi$
$642$	11.4631	0.452414
$643$	44.2034	1.74321	0.871605	−	0.490208i	$-0.163079\pi$
$643$	44.2034	1.74321	0.871605	+	0.490208i	$0.163079\pi$
$644$	2.00000	0.0788110
$645$	12.7172	0.500739
$646$	31.4196	1.23619
$647$	−44.6177	−1.75410	−0.877051	−	0.480397i	$-0.840492\pi$
$647$	−44.6177	−1.75410	−0.877051	+	0.480397i	$0.840492\pi$
$648$	10.5826	0.415724
$649$	0	0
$650$	0.869151	0.0340909
$651$	−14.7931	−0.579785
$652$	−9.38020	−0.367357
$653$	6.77997	0.265321	0.132660	−	0.991162i	$-0.457648\pi$
$653$	6.77997	0.265321	0.132660	+	0.991162i	$0.457648\pi$
$654$	37.1642	1.45324
$655$	−1.19986	−0.0468822
$656$	−9.86261	−0.385070
$657$	11.6386	0.454067
$658$	10.7529	0.419190
$659$	−2.83294	−0.110356	−0.0551778	−	0.998477i	$-0.517573\pi$
$659$	−2.83294	−0.110356	−0.0551778	+	0.998477i	$0.517573\pi$
$660$	0	0
$661$	36.4484	1.41768	0.708839	−	0.705370i	$-0.249219\pi$
$661$	36.4484	1.41768	0.708839	+	0.705370i	$0.249219\pi$
$662$	20.6399	0.802192
$663$	6.93093	0.269175
$664$	8.22953	0.319368
$665$	7.56151	0.293223
$666$	−4.65345	−0.180318
$667$	2.39176	0.0926092
$668$	−12.0000	−0.464294
$669$	20.9629	0.810472
$670$	3.56151	0.137593
$671$	0	0
$672$	1.91913	0.0740321
$673$	−5.13572	−0.197968	−0.0989838	−	0.995089i	$-0.531559\pi$
$673$	−5.13572	−0.197968	−0.0989838	+	0.995089i	$0.531559\pi$
$674$	−0.0255665	−0.000984783 0
$675$	−4.44651	−0.171146
$676$	−12.2446	−0.470945
$677$	28.3506	1.08960	0.544801	−	0.838565i	$-0.316605\pi$
$677$	28.3506	1.08960	0.544801	+	0.838565i	$0.316605\pi$
$678$	−25.8953	−0.994502
$679$	−14.7984	−0.567909
$680$	−4.15520	−0.159345
$681$	−37.5512	−1.43896
$682$	0	0
$683$	25.0913	0.960094	0.480047	−	0.877243i	$-0.340620\pi$
$683$	25.0913	0.960094	0.480047	+	0.877243i	$0.340620\pi$
$684$	5.16499	0.197488
$685$	4.65822	0.177982
$686$	1.00000	0.0381802
$687$	−28.5208	−1.08814
$688$	−6.62654	−0.252635
$689$	1.37027	0.0522032
$690$	−3.83826	−0.146120
$691$	−49.3947	−1.87906	−0.939531	−	0.342465i	$-0.888738\pi$
$691$	−49.3947	−1.87906	−0.939531	+	0.342465i	$0.888738\pi$
$692$	9.14787	0.347750
$693$	0	0
$694$	−27.7393	−1.05297
$695$	12.6208	0.478734
$696$	2.29505	0.0869935
$697$	40.9811	1.55227
$698$	−7.76742	−0.294001
$699$	8.78548	0.332297
$700$	−1.00000	−0.0377964
$701$	51.4479	1.94316	0.971580	−	0.236712i	$-0.0760698\pi$
$701$	51.4479	1.94316	0.971580	+	0.236712i	$0.0760698\pi$
$702$	−3.86468	−0.145863
$703$	51.5138	1.94288
$704$	0	0
$705$	−20.6361	−0.777202
$706$	23.1303	0.870519
$707$	17.6082	0.662226
$708$	−14.2619	−0.535996
$709$	−39.4201	−1.48045	−0.740226	−	0.672358i	$-0.765282\pi$
$709$	−39.4201	−1.48045	−0.740226	+	0.672358i	$0.765282\pi$
$710$	−2.23528	−0.0838884
$711$	−2.13857	−0.0802025
$712$	9.89429	0.370804
$713$	−15.4164	−0.577349
$714$	−7.97437	−0.298433
$715$	0	0
$716$	3.71474	0.138826
$717$	−32.5253	−1.21468
$718$	12.6100	0.470602
$719$	24.7392	0.922618	0.461309	−	0.887240i	$-0.347380\pi$
$719$	24.7392	0.922618	0.461309	+	0.887240i	$0.347380\pi$
$720$	−0.683063	−0.0254562
$721$	4.85282	0.180728
$722$	−38.1765	−1.42078
$723$	52.6396	1.95769
$724$	−21.4439	−0.796958
$725$	−1.19588	−0.0444138
$726$	0	0
$727$	−51.2679	−1.90142	−0.950710	−	0.310081i	$-0.899644\pi$
$727$	−51.2679	−1.90142	−0.950710	+	0.310081i	$0.899644\pi$
$728$	−0.869151	−0.0322129
$729$	18.3717	0.680433
$730$	17.0389	0.630638
$731$	27.5346	1.01840
$732$	−14.8508	−0.548902
$733$	−0.281620	−0.0104019	−0.00520094	−	0.999986i	$-0.501656\pi$
$733$	−0.281620	−0.0104019	−0.00520094	+	0.999986i	$0.501656\pi$
$734$	−3.51354	−0.129687
$735$	−1.91913	−0.0707882
$736$	2.00000	0.0737210
$737$	0	0
$738$	6.73678	0.247984
$739$	1.44847	0.0532830	0.0266415	−	0.999645i	$-0.491519\pi$
$739$	1.44847	0.0532830	0.0266415	+	0.999645i	$0.491519\pi$
$740$	−6.81263	−0.250437
$741$	−12.6127	−0.463340
$742$	−1.57656	−0.0578774
$743$	−10.9338	−0.401121	−0.200560	−	0.979681i	$-0.564276\pi$
$743$	−10.9338	−0.401121	−0.200560	+	0.979681i	$0.564276\pi$
$744$	−14.7931	−0.542340
$745$	−16.1238	−0.590731
$746$	8.48915	0.310810
$747$	−5.62128	−0.205672
$748$	0	0
$749$	5.97309	0.218252
$750$	1.91913	0.0700767
$751$	−39.4330	−1.43893	−0.719465	−	0.694528i	$-0.755613\pi$
$751$	−39.4330	−1.43893	−0.719465	+	0.694528i	$0.755613\pi$
$752$	10.7529	0.392116
$753$	10.0081	0.364715
$754$	−1.03940	−0.0378527
$755$	−9.81017	−0.357029
$756$	4.44651	0.161718
$757$	−22.1713	−0.805829	−0.402914	−	0.915238i	$-0.632003\pi$
$757$	−22.1713	−0.805829	−0.402914	+	0.915238i	$0.632003\pi$
$758$	19.0706	0.692675
$759$	0	0
$760$	7.56151	0.274285
$761$	−29.0999	−1.05487	−0.527435	−	0.849595i	$-0.676846\pi$
$761$	−29.0999	−1.05487	−0.527435	+	0.849595i	$0.676846\pi$
$762$	5.14821	0.186500
$763$	19.3651	0.701065
$764$	−21.8922	−0.792033
$765$	2.83826	0.102618
$766$	−1.33479	−0.0482278
$767$	6.45906	0.233223
$768$	1.91913	0.0692507
$769$	36.2631	1.30768	0.653840	−	0.756633i	$-0.273157\pi$
$769$	36.2631	1.30768	0.653840	+	0.756633i	$0.273157\pi$
$770$	0	0
$771$	−3.40400	−0.122592
$772$	4.07880	0.146799
$773$	−4.25546	−0.153058	−0.0765291	−	0.997067i	$-0.524384\pi$
$773$	−4.25546	−0.153058	−0.0765291	+	0.997067i	$0.524384\pi$
$774$	4.52634	0.162696
$775$	7.70820	0.276887
$776$	−14.7984	−0.531231
$777$	−13.0743	−0.469039
$778$	1.18211	0.0423807
$779$	−74.5763	−2.67197
$780$	1.66801	0.0597245
$781$	0	0
$782$	−8.31040	−0.297179
$783$	5.31748	0.190031
$784$	1.00000	0.0357143
$785$	2.74234	0.0978784
$786$	−2.30268	−0.0821339
$787$	−8.32019	−0.296583	−0.148291	−	0.988944i	$-0.547377\pi$
$787$	−8.32019	−0.296583	−0.148291	+	0.988944i	$0.547377\pi$
$788$	14.2275	0.506832
$789$	−26.9286	−0.958685
$790$	−3.13085	−0.111391
$791$	−13.4932	−0.479764
$792$	0	0
$793$	6.72575	0.238838
$794$	15.9540	0.566186
$795$	3.02563	0.107308
$796$	−8.75085	−0.310166
$797$	2.63097	0.0931938	0.0465969	−	0.998914i	$-0.485162\pi$
$797$	2.63097	0.0931938	0.0465969	+	0.998914i	$0.485162\pi$
$798$	14.5115	0.513703
$799$	−44.6803	−1.58067
$800$	−1.00000	−0.0353553
$801$	−6.75842	−0.238797
$802$	11.8135	0.417149
$803$	0	0
$804$	6.83501	0.241052
$805$	−2.00000	−0.0704907
$806$	6.69959	0.235983
$807$	−26.9411	−0.948372
$808$	17.6082	0.619456
$809$	−9.79837	−0.344492	−0.172246	−	0.985054i	$-0.555102\pi$
$809$	−9.79837	−0.344492	−0.172246	+	0.985054i	$0.555102\pi$
$810$	−10.5826	−0.371835
$811$	27.5065	0.965883	0.482941	−	0.875653i	$-0.339568\pi$
$811$	27.5065	0.965883	0.482941	+	0.875653i	$0.339568\pi$
$812$	1.19588	0.0419671
$813$	41.3831	1.45137
$814$	0	0
$815$	9.38020	0.328574
$816$	−7.97437	−0.279159
$817$	−50.1067	−1.75301
$818$	17.2060	0.601593
$819$	0.593685	0.0207450
$820$	9.86261	0.344417
$821$	44.1139	1.53958	0.769792	−	0.638295i	$-0.220360\pi$
$821$	44.1139	1.53958	0.769792	+	0.638295i	$0.220360\pi$
$822$	8.93974	0.311809
$823$	26.5906	0.926889	0.463444	−	0.886126i	$-0.346613\pi$
$823$	26.5906	0.926889	0.463444	+	0.886126i	$0.346613\pi$
$824$	4.85282	0.169056
$825$	0	0
$826$	−7.43146	−0.258573
$827$	−36.8252	−1.28054	−0.640270	−	0.768150i	$-0.721177\pi$
$827$	−36.8252	−1.28054	−0.640270	+	0.768150i	$0.721177\pi$
$828$	−1.36613	−0.0474761
$829$	−44.3460	−1.54020	−0.770100	−	0.637923i	$-0.779794\pi$
$829$	−44.3460	−1.54020	−0.770100	+	0.637923i	$0.779794\pi$
$830$	−8.22953	−0.285651
$831$	−20.0925	−0.697000
$832$	−0.869151	−0.0301324
$833$	−4.15520	−0.143969
$834$	24.2210	0.838703
$835$	12.0000	0.415277
$836$	0	0
$837$	−34.2746	−1.18470
$838$	7.07910	0.244543
$839$	18.4856	0.638194	0.319097	−	0.947722i	$-0.396620\pi$
$839$	18.4856	0.638194	0.319097	+	0.947722i	$0.396620\pi$
$840$	−1.91913	−0.0662163
$841$	−27.5699	−0.950685
$842$	15.8752	0.547096
$843$	52.8512	1.82029
$844$	−19.7695	−0.680494
$845$	12.2446	0.421226
$846$	−7.34488	−0.252522
$847$	0	0
$848$	−1.57656	−0.0541394
$849$	−24.2826	−0.833376
$850$	4.15520	0.142522
$851$	−13.6253	−0.467068
$852$	−4.28979	−0.146966
$853$	−20.2660	−0.693894	−0.346947	−	0.937885i	$-0.612782\pi$
$853$	−20.2660	−0.693894	−0.346947	+	0.937885i	$0.612782\pi$
$854$	−7.73830	−0.264799
$855$	−5.16499	−0.176639
$856$	5.97309	0.204156
$857$	−34.4745	−1.17763	−0.588813	−	0.808269i	$-0.700404\pi$
$857$	−34.4745	−1.17763	−0.588813	+	0.808269i	$0.700404\pi$
$858$	0	0
$859$	7.96220	0.271667	0.135833	−	0.990732i	$-0.456629\pi$
$859$	7.96220	0.271667	0.135833	+	0.990732i	$0.456629\pi$
$860$	6.62654	0.225963
$861$	18.9276	0.645052
$862$	17.4697	0.595020
$863$	−29.8949	−1.01763	−0.508817	−	0.860874i	$-0.669917\pi$
$863$	−29.8949	−1.01763	−0.508817	+	0.860874i	$0.669917\pi$
$864$	4.44651	0.151273
$865$	−9.14787	−0.311037
$866$	−18.1892	−0.618093
$867$	0.509868	0.0173160
$868$	−7.70820	−0.261633
$869$	0	0
$870$	−2.29505	−0.0778094
$871$	−3.09549	−0.104887
$872$	19.3651	0.655787
$873$	10.1082	0.342111
$874$	15.1230	0.511544
$875$	1.00000	0.0338062
$876$	32.6999	1.10483
$877$	2.51288	0.0848539	0.0424270	−	0.999100i	$-0.486491\pi$
$877$	2.51288	0.0848539	0.0424270	+	0.999100i	$0.486491\pi$
$878$	16.5826	0.559636
$879$	−50.7051	−1.71024
$880$	0	0
$881$	31.5748	1.06378	0.531890	−	0.846813i	$-0.321482\pi$
$881$	31.5748	1.06378	0.531890	+	0.846813i	$0.321482\pi$
$882$	−0.683063	−0.0229999
$883$	21.6431	0.728349	0.364174	−	0.931331i	$-0.381351\pi$
$883$	21.6431	0.728349	0.364174	+	0.931331i	$0.381351\pi$
$884$	3.61149	0.121468
$885$	14.2619	0.479410
$886$	37.1961	1.24963
$887$	−34.3585	−1.15365	−0.576823	−	0.816869i	$-0.695708\pi$
$887$	−34.3585	−1.15365	−0.576823	+	0.816869i	$0.695708\pi$
$888$	−13.0743	−0.438746
$889$	2.68257	0.0899706
$890$	−9.89429	−0.331657
$891$	0	0
$892$	10.9231	0.365733
$893$	81.3079	2.72087
$894$	−30.9437	−1.03491
$895$	−3.71474	−0.124170
$896$	1.00000	0.0334077
$897$	3.33603	0.111387
$898$	−10.5728	−0.352820
$899$	−9.21807	−0.307440
$900$	0.683063	0.0227688
$901$	6.55093	0.218243
$902$	0	0
$903$	12.7172	0.423202
$904$	−13.4932	−0.448778
$905$	21.4439	0.712821
$906$	−18.8270	−0.625485
$907$	15.8784	0.527234	0.263617	−	0.964627i	$-0.415084\pi$
$907$	15.8784	0.527234	0.263617	+	0.964627i	$0.415084\pi$
$908$	−19.5668	−0.649346
$909$	−12.0275	−0.398928
$910$	0.869151	0.0288121
$911$	19.4184	0.643361	0.321680	−	0.946848i	$-0.395752\pi$
$911$	19.4184	0.643361	0.321680	+	0.946848i	$0.395752\pi$
$912$	14.5115	0.480525
$913$	0	0
$914$	1.57179	0.0519903
$915$	14.8508	0.490953
$916$	−14.8613	−0.491032
$917$	−1.19986	−0.0396227
$918$	−18.4761	−0.609803
$919$	−32.2285	−1.06312	−0.531560	−	0.847021i	$-0.678394\pi$
$919$	−32.2285	−1.06312	−0.531560	+	0.847021i	$0.678394\pi$
$920$	−2.00000	−0.0659380
$921$	−47.6566	−1.57034
$922$	−15.9056	−0.523822
$923$	1.94279	0.0639478
$924$	0	0
$925$	6.81263	0.223998
$926$	−22.1265	−0.727122
$927$	−3.31478	−0.108872
$928$	1.19588	0.0392566
$929$	30.0808	0.986918	0.493459	−	0.869769i	$-0.335732\pi$
$929$	30.0808	0.986918	0.493459	+	0.869769i	$0.335732\pi$
$930$	14.7931	0.485083
$931$	7.56151	0.247819
$932$	4.57784	0.149952
$933$	11.0175	0.360696
$934$	−3.31217	−0.108377
$935$	0	0
$936$	0.593685	0.0194052
$937$	5.16706	0.168801	0.0844003	−	0.996432i	$-0.473103\pi$
$937$	5.16706	0.168801	0.0844003	+	0.996432i	$0.473103\pi$
$938$	3.56151	0.116288
$939$	−41.7783	−1.36338
$940$	−10.7529	−0.350720
$941$	20.8799	0.680666	0.340333	−	0.940305i	$-0.389460\pi$
$941$	20.8799	0.680666	0.340333	+	0.940305i	$0.389460\pi$
$942$	5.26292	0.171475
$943$	19.7252	0.642342
$944$	−7.43146	−0.241873
$945$	−4.44651	−0.144645
$946$	0	0
$947$	22.5179	0.731733	0.365867	−	0.930667i	$-0.380773\pi$
$947$	22.5179	0.731733	0.365867	+	0.930667i	$0.380773\pi$
$948$	−6.00851	−0.195147
$949$	−14.8094	−0.480733
$950$	−7.56151	−0.245328
$951$	13.1708	0.427093
$952$	−4.15520	−0.134671
$953$	3.73512	0.120992	0.0604961	−	0.998168i	$-0.480732\pi$
$953$	3.73512	0.120992	0.0604961	+	0.998168i	$0.480732\pi$
$954$	1.07689	0.0348656
$955$	21.8922	0.708416
$956$	−16.9479	−0.548136
$957$	0	0
$958$	26.3656	0.851834
$959$	4.65822	0.150422
$960$	−1.91913	−0.0619397
$961$	28.4164	0.916658
$962$	5.92120	0.190907
$963$	−4.07999	−0.131476
$964$	27.4289	0.883425
$965$	−4.07880	−0.131301
$966$	−3.83826	−0.123494
$967$	17.0337	0.547767	0.273884	−	0.961763i	$-0.411692\pi$
$967$	17.0337	0.547767	0.273884	+	0.961763i	$0.411692\pi$
$968$	0	0
$969$	−60.2983	−1.93706
$970$	14.7984	0.475147
$971$	31.4811	1.01028	0.505139	−	0.863038i	$-0.331441\pi$
$971$	31.4811	1.01028	0.505139	+	0.863038i	$0.331441\pi$
$972$	−6.96990	−0.223560
$973$	12.6208	0.404604
$974$	−12.4706	−0.399582
$975$	−1.66801	−0.0534192
$976$	−7.73830	−0.247697
$977$	26.7352	0.855335	0.427668	−	0.903936i	$-0.359335\pi$
$977$	26.7352	0.855335	0.427668	+	0.903936i	$0.359335\pi$
$978$	18.0018	0.575635
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−13.2276	−0.422325
$982$	−36.6954	−1.17100
$983$	22.6689	0.723027	0.361513	−	0.932367i	$-0.382260\pi$
$983$	22.6689	0.723027	0.361513	+	0.932367i	$0.382260\pi$
$984$	18.9276	0.603391
$985$	−14.2275	−0.453324
$986$	−4.96911	−0.158249
$987$	−20.6361	−0.656856
$988$	−6.57210	−0.209086
$989$	13.2531	0.421424
$990$	0	0
$991$	5.56237	0.176695	0.0883473	−	0.996090i	$-0.471841\pi$
$991$	5.56237	0.176695	0.0883473	+	0.996090i	$0.471841\pi$
$992$	−7.70820	−0.244736
$993$	−39.6106	−1.25701
$994$	−2.23528	−0.0708987
$995$	8.75085	0.277421
$996$	−15.7935	−0.500437
$997$	40.0101	1.26713	0.633566	−	0.773689i	$-0.281590\pi$
$997$	40.0101	1.26713	0.633566	+	0.773689i	$0.281590\pi$
$998$	−22.0831	−0.699030
$999$	−30.2924	−0.958409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cq.1.3		4
11.3	even	5		770.2.n.e.141.1	yes	8
11.4	even	5		770.2.n.e.71.1	✓	8
11.10	odd	2		8470.2.a.ct.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.e.71.1	✓	8	11.4	even	5
770.2.n.e.141.1	yes	8	11.3	even	5
8470.2.a.cq.1.3		4	1.1	even	1	trivial
8470.2.a.ct.1.3		4	11.10	odd	2

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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.91913 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.91913 q^{6} -1.00000 q^{7} -1.00000 q^{8} +0.683063 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.91913 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.91913 q^{6} -1.00000 q^{7} -1.00000 q^{8} +0.683063 q^{9} +1.00000 q^{10} +1.91913 q^{12} -0.869151 q^{13} +1.00000 q^{14} -1.91913 q^{15} +1.00000 q^{16} -4.15520 q^{17} -0.683063 q^{18} +7.56151 q^{19} -1.00000 q^{20} -1.91913 q^{21} -2.00000 q^{23} -1.91913 q^{24} +1.00000 q^{25} +0.869151 q^{26} -4.44651 q^{27} -1.00000 q^{28} -1.19588 q^{29} +1.91913 q^{30} +7.70820 q^{31} -1.00000 q^{32} +4.15520 q^{34} +1.00000 q^{35} +0.683063 q^{36} +6.81263 q^{37} -7.56151 q^{38} -1.66801 q^{39} +1.00000 q^{40} -9.86261 q^{41} +1.91913 q^{42} -6.62654 q^{43} -0.683063 q^{45} +2.00000 q^{46} +10.7529 q^{47} +1.91913 q^{48} +1.00000 q^{49} -1.00000 q^{50} -7.97437 q^{51} -0.869151 q^{52} -1.57656 q^{53} +4.44651 q^{54} +1.00000 q^{56} +14.5115 q^{57} +1.19588 q^{58} -7.43146 q^{59} -1.91913 q^{60} -7.73830 q^{61} -7.70820 q^{62} -0.683063 q^{63} +1.00000 q^{64} +0.869151 q^{65} +3.56151 q^{67} -4.15520 q^{68} -3.83826 q^{69} -1.00000 q^{70} -2.23528 q^{71} -0.683063 q^{72} +17.0389 q^{73} -6.81263 q^{74} +1.91913 q^{75} +7.56151 q^{76} +1.66801 q^{78} -3.13085 q^{79} -1.00000 q^{80} -10.5826 q^{81} +9.86261 q^{82} -8.22953 q^{83} -1.91913 q^{84} +4.15520 q^{85} +6.62654 q^{86} -2.29505 q^{87} -9.89429 q^{89} +0.683063 q^{90} +0.869151 q^{91} -2.00000 q^{92} +14.7931 q^{93} -10.7529 q^{94} -7.56151 q^{95} -1.91913 q^{96} +14.7984 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + 4 q^{10} + 2 q^{12} - q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 6 q^{18} + 3 q^{19} - 4 q^{20} - 2 q^{21} - 8 q^{23} - 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} - 4 q^{28} - 15 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{32} + 2 q^{34} + 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} + q^{39} + 4 q^{40} - 11 q^{41} + 2 q^{42} - 7 q^{43} - 6 q^{45} + 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} - 4 q^{50} - 18 q^{51} - q^{52} + 10 q^{53} - 14 q^{54} + 4 q^{56} + 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} - 26 q^{61} - 4 q^{62} - 6 q^{63} + 4 q^{64} + q^{65} - 13 q^{67} - 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} - 6 q^{72} + 18 q^{73} - 2 q^{74} + 2 q^{75} + 3 q^{76} - q^{78} - 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} + 2 q^{83} - 2 q^{84} + 2 q^{85} + 7 q^{86} - 26 q^{87} - 7 q^{89} + 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} - 5 q^{94} - 3 q^{95} - 2 q^{96} + 10 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 + 4 * q^10 + 2 * q^12 - q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 6 * q^18 + 3 * q^19 - 4 * q^20 - 2 * q^21 - 8 * q^23 - 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 - 4 * q^28 - 15 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^32 + 2 * q^34 + 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 + q^39 + 4 * q^40 - 11 * q^41 + 2 * q^42 - 7 * q^43 - 6 * q^45 + 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 - 4 * q^50 - 18 * q^51 - q^52 + 10 * q^53 - 14 * q^54 + 4 * q^56 + 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 - 26 * q^61 - 4 * q^62 - 6 * q^63 + 4 * q^64 + q^65 - 13 * q^67 - 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 - 6 * q^72 + 18 * q^73 - 2 * q^74 + 2 * q^75 + 3 * q^76 - q^78 - 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 + 2 * q^83 - 2 * q^84 + 2 * q^85 + 7 * q^86 - 26 * q^87 - 7 * q^89 + 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 - 5 * q^94 - 3 * q^95 - 2 * q^96 + 10 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

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Database

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