LMFDB - Embedded newform 8007.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27923 q^{2} -1.00000 q^{3} +3.19488 q^{4} +1.61810 q^{5} +2.27923 q^{6} +2.49401 q^{7} -2.72340 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.27923 q^{2} -1.00000 q^{3} +3.19488 q^{4} +1.61810 q^{5} +2.27923 q^{6} +2.49401 q^{7} -2.72340 q^{8} +1.00000 q^{9} -3.68801 q^{10} +5.10414 q^{11} -3.19488 q^{12} +5.82809 q^{13} -5.68442 q^{14} -1.61810 q^{15} -0.182510 q^{16} -1.00000 q^{17} -2.27923 q^{18} +1.00423 q^{19} +5.16962 q^{20} -2.49401 q^{21} -11.6335 q^{22} +4.97512 q^{23} +2.72340 q^{24} -2.38176 q^{25} -13.2835 q^{26} -1.00000 q^{27} +7.96807 q^{28} -0.936385 q^{29} +3.68801 q^{30} +1.80187 q^{31} +5.86278 q^{32} -5.10414 q^{33} +2.27923 q^{34} +4.03555 q^{35} +3.19488 q^{36} +0.543475 q^{37} -2.28886 q^{38} -5.82809 q^{39} -4.40672 q^{40} -1.20995 q^{41} +5.68442 q^{42} -1.38590 q^{43} +16.3071 q^{44} +1.61810 q^{45} -11.3394 q^{46} -9.36940 q^{47} +0.182510 q^{48} -0.779896 q^{49} +5.42858 q^{50} +1.00000 q^{51} +18.6200 q^{52} -4.71778 q^{53} +2.27923 q^{54} +8.25898 q^{55} -6.79220 q^{56} -1.00423 q^{57} +2.13423 q^{58} +8.46224 q^{59} -5.16962 q^{60} +5.26173 q^{61} -4.10686 q^{62} +2.49401 q^{63} -12.9976 q^{64} +9.43041 q^{65} +11.6335 q^{66} +9.18723 q^{67} -3.19488 q^{68} -4.97512 q^{69} -9.19795 q^{70} -8.93610 q^{71} -2.72340 q^{72} +12.7624 q^{73} -1.23870 q^{74} +2.38176 q^{75} +3.20838 q^{76} +12.7298 q^{77} +13.2835 q^{78} +10.0211 q^{79} -0.295318 q^{80} +1.00000 q^{81} +2.75774 q^{82} -2.66450 q^{83} -7.96807 q^{84} -1.61810 q^{85} +3.15877 q^{86} +0.936385 q^{87} -13.9006 q^{88} -15.8097 q^{89} -3.68801 q^{90} +14.5353 q^{91} +15.8949 q^{92} -1.80187 q^{93} +21.3550 q^{94} +1.62494 q^{95} -5.86278 q^{96} -4.75805 q^{97} +1.77756 q^{98} +5.10414 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) $ 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9 + 8 * q^10 - 7 * q^11 - 61 * q^12 + 8 * q^13 - 8 * q^14 - q^15 + 71 * q^16 - 56 * q^17 + q^18 - 2 * q^19 - 4 * q^20 - 19 * q^21 + 47 * q^22 + 16 * q^23 + 85 * q^25 - 11 * q^26 - 56 * q^27 + 52 * q^28 + 17 * q^29 - 8 * q^30 + 23 * q^31 + 11 * q^32 + 7 * q^33 - q^34 - 41 * q^35 + 61 * q^36 + 58 * q^37 - 22 * q^38 - 8 * q^39 + 38 * q^40 - q^41 + 8 * q^42 + 27 * q^43 + 2 * q^44 + q^45 + 46 * q^46 + 5 * q^47 - 71 * q^48 + 59 * q^49 - 4 * q^50 + 56 * q^51 + 25 * q^52 + 15 * q^53 - q^54 + 9 * q^55 - 36 * q^56 + 2 * q^57 + 89 * q^58 - 61 * q^59 + 4 * q^60 + 47 * q^61 + 8 * q^62 + 19 * q^63 + 88 * q^64 + 39 * q^65 - 47 * q^66 + 20 * q^67 - 61 * q^68 - 16 * q^69 + 36 * q^70 - 2 * q^71 + 93 * q^73 + 48 * q^74 - 85 * q^75 + 38 * q^76 + 26 * q^77 + 11 * q^78 + 72 * q^79 + 42 * q^80 + 56 * q^81 + 33 * q^82 - 11 * q^83 - 52 * q^84 - q^85 - 4 * q^86 - 17 * q^87 + 130 * q^88 - 6 * q^89 + 8 * q^90 + 37 * q^91 + 132 * q^92 - 23 * q^93 - 32 * q^94 + 12 * q^95 - 11 * q^96 + 100 * q^97 + 42 * q^98 - 7 * q^99

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 $

$2$	−2.27923	−1.61166	−0.805829	−	0.592149i	$-0.798280\pi$
$2$	−2.27923	−1.61166	−0.805829	+	0.592149i	$0.798280\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.19488	1.59744
$5$	1.61810	0.723635	0.361817	−	0.932249i	$-0.382156\pi$
$5$	1.61810	0.723635	0.361817	+	0.932249i	$0.382156\pi$
$6$	2.27923	0.930491
$7$	2.49401	0.942649	0.471324	−	0.881960i	$-0.343776\pi$
$7$	2.49401	0.942649	0.471324	+	0.881960i	$0.343776\pi$
$8$	−2.72340	−0.962867
$9$	1.00000	0.333333
$10$	−3.68801	−1.16625
$11$	5.10414	1.53895	0.769477	−	0.638674i	$-0.220517\pi$
$11$	5.10414	1.53895	0.769477	+	0.638674i	$0.220517\pi$
$12$	−3.19488	−0.922282
$13$	5.82809	1.61642	0.808211	−	0.588893i	$-0.200436\pi$
$13$	5.82809	1.61642	0.808211	+	0.588893i	$0.200436\pi$
$14$	−5.68442	−1.51923
$15$	−1.61810	−0.417791
$16$	−0.182510	−0.0456274
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.27923	−0.537219
$19$	1.00423	0.230385	0.115193	−	0.993343i	$-0.463251\pi$
$19$	1.00423	0.230385	0.115193	+	0.993343i	$0.463251\pi$
$20$	5.16962	1.15596
$21$	−2.49401	−0.544238
$22$	−11.6335	−2.48027
$23$	4.97512	1.03738	0.518692	−	0.854961i	$-0.326419\pi$
$23$	4.97512	1.03738	0.518692	+	0.854961i	$0.326419\pi$
$24$	2.72340	0.555912
$25$	−2.38176	−0.476353
$26$	−13.2835	−2.60512
$27$	−1.00000	−0.192450
$28$	7.96807	1.50582
$29$	−0.936385	−0.173882	−0.0869411	−	0.996213i	$-0.527709\pi$
$29$	−0.936385	−0.173882	−0.0869411	+	0.996213i	$0.527709\pi$
$30$	3.68801	0.673335
$31$	1.80187	0.323625	0.161812	−	0.986822i	$-0.448266\pi$
$31$	1.80187	0.323625	0.161812	+	0.986822i	$0.448266\pi$
$32$	5.86278	1.03640
$33$	−5.10414	−0.888516
$34$	2.27923	0.390884
$35$	4.03555	0.682133
$36$	3.19488	0.532480
$37$	0.543475	0.0893467	0.0446734	−	0.999002i	$-0.485775\pi$
$37$	0.543475	0.0893467	0.0446734	+	0.999002i	$0.485775\pi$
$38$	−2.28886	−0.371302
$39$	−5.82809	−0.933241
$40$	−4.40672	−0.696764
$41$	−1.20995	−0.188962	−0.0944809	−	0.995527i	$-0.530119\pi$
$41$	−1.20995	−0.188962	−0.0944809	+	0.995527i	$0.530119\pi$
$42$	5.68442	0.877126
$43$	−1.38590	−0.211347	−0.105674	−	0.994401i	$-0.533700\pi$
$43$	−1.38590	−0.211347	−0.105674	+	0.994401i	$0.533700\pi$
$44$	16.3071	2.45839
$45$	1.61810	0.241212
$46$	−11.3394	−1.67191
$47$	−9.36940	−1.36667	−0.683334	−	0.730106i	$-0.739471\pi$
$47$	−9.36940	−1.36667	−0.683334	+	0.730106i	$0.739471\pi$
$48$	0.182510	0.0263430
$49$	−0.779896	−0.111414
$50$	5.42858	0.767717
$51$	1.00000	0.140028
$52$	18.6200	2.58214
$53$	−4.71778	−0.648037	−0.324018	−	0.946051i	$-0.605034\pi$
$53$	−4.71778	−0.648037	−0.324018	+	0.946051i	$0.605034\pi$
$54$	2.27923	0.310164
$55$	8.25898	1.11364
$56$	−6.79220	−0.907645
$57$	−1.00423	−0.133013
$58$	2.13423	0.280239
$59$	8.46224	1.10169	0.550845	−	0.834607i	$-0.314305\pi$
$59$	8.46224	1.10169	0.550845	+	0.834607i	$0.314305\pi$
$60$	−5.16962	−0.667395
$61$	5.26173	0.673696	0.336848	−	0.941559i	$-0.390639\pi$
$61$	5.26173	0.673696	0.336848	+	0.941559i	$0.390639\pi$
$62$	−4.10686	−0.521572
$63$	2.49401	0.314216
$64$	−12.9976	−1.62470
$65$	9.43041	1.16970
$66$	11.6335	1.43198
$67$	9.18723	1.12240	0.561199	−	0.827681i	$-0.310340\pi$
$67$	9.18723	1.12240	0.561199	+	0.827681i	$0.310340\pi$
$68$	−3.19488	−0.387436
$69$	−4.97512	−0.598934
$70$	−9.19795	−1.09937
$71$	−8.93610	−1.06052	−0.530260	−	0.847835i	$-0.677906\pi$
$71$	−8.93610	−1.06052	−0.530260	+	0.847835i	$0.677906\pi$
$72$	−2.72340	−0.320956
$73$	12.7624	1.49373	0.746865	−	0.664976i	$-0.231558\pi$
$73$	12.7624	1.49373	0.746865	+	0.664976i	$0.231558\pi$
$74$	−1.23870	−0.143996
$75$	2.38176	0.275022
$76$	3.20838	0.368027
$77$	12.7298	1.45069
$78$	13.2835	1.50407
$79$	10.0211	1.12746	0.563730	−	0.825959i	$-0.309366\pi$
$79$	10.0211	1.12746	0.563730	+	0.825959i	$0.309366\pi$
$80$	−0.295318	−0.0330176
$81$	1.00000	0.111111
$82$	2.75774	0.304542
$83$	−2.66450	−0.292467	−0.146234	−	0.989250i	$-0.546715\pi$
$83$	−2.66450	−0.292467	−0.146234	+	0.989250i	$0.546715\pi$
$84$	−7.96807	−0.869388
$85$	−1.61810	−0.175507
$86$	3.15877	0.340619
$87$	0.936385	0.100391
$88$	−13.9006	−1.48181
$89$	−15.8097	−1.67583	−0.837914	−	0.545802i	$-0.816225\pi$
$89$	−15.8097	−1.67583	−0.837914	+	0.545802i	$0.816225\pi$
$90$	−3.68801	−0.388750
$91$	14.5353	1.52372
$92$	15.8949	1.65716
$93$	−1.80187	−0.186845
$94$	21.3550	2.20260
$95$	1.62494	0.166715
$96$	−5.86278	−0.598367
$97$	−4.75805	−0.483107	−0.241553	−	0.970388i	$-0.577657\pi$
$97$	−4.75805	−0.483107	−0.241553	+	0.970388i	$0.577657\pi$
$98$	1.77756	0.179561
$99$	5.10414	0.512985
$100$	−7.60944	−0.760944
$101$	19.0941	1.89993	0.949967	−	0.312349i	$-0.101116\pi$
$101$	19.0941	1.89993	0.949967	+	0.312349i	$0.101116\pi$
$102$	−2.27923	−0.225677
$103$	−9.42015	−0.928195	−0.464098	−	0.885784i	$-0.653621\pi$
$103$	−9.42015	−0.928195	−0.464098	+	0.885784i	$0.653621\pi$
$104$	−15.8722	−1.55640
$105$	−4.03555	−0.393830
$106$	10.7529	1.04441
$107$	1.79764	0.173784	0.0868922	−	0.996218i	$-0.472306\pi$
$107$	1.79764	0.173784	0.0868922	+	0.996218i	$0.472306\pi$
$108$	−3.19488	−0.307427
$109$	17.2251	1.64986	0.824931	−	0.565233i	$-0.191214\pi$
$109$	17.2251	1.64986	0.824931	+	0.565233i	$0.191214\pi$
$110$	−18.8241	−1.79481
$111$	−0.543475	−0.0515844
$112$	−0.455182	−0.0430106
$113$	−8.47701	−0.797450	−0.398725	−	0.917071i	$-0.630547\pi$
$113$	−8.47701	−0.797450	−0.398725	+	0.917071i	$0.630547\pi$
$114$	2.28886	0.214371
$115$	8.05023	0.750688
$116$	−2.99163	−0.277766
$117$	5.82809	0.538807
$118$	−19.2874	−1.77555
$119$	−2.49401	−0.228626
$120$	4.40672	0.402277
$121$	15.0522	1.36838
$122$	−11.9927	−1.08577
$123$	1.20995	0.109097
$124$	5.75674	0.516971
$125$	−11.9444	−1.06834
$126$	−5.68442	−0.506409
$127$	6.10070	0.541350	0.270675	−	0.962671i	$-0.412753\pi$
$127$	6.10070	0.541350	0.270675	+	0.962671i	$0.412753\pi$
$128$	17.8989	1.58205
$129$	1.38590	0.122021
$130$	−21.4941	−1.88515
$131$	−5.53395	−0.483503	−0.241752	−	0.970338i	$-0.577722\pi$
$131$	−5.53395	−0.483503	−0.241752	+	0.970338i	$0.577722\pi$
$132$	−16.3071	−1.41935
$133$	2.50455	0.217172
$134$	−20.9398	−1.80892
$135$	−1.61810	−0.139264
$136$	2.72340	0.233530
$137$	9.06120	0.774150	0.387075	−	0.922048i	$-0.373485\pi$
$137$	9.06120	0.774150	0.387075	+	0.922048i	$0.373485\pi$
$138$	11.3394	0.965277
$139$	4.38099	0.371591	0.185795	−	0.982588i	$-0.440514\pi$
$139$	4.38099	0.371591	0.185795	+	0.982588i	$0.440514\pi$
$140$	12.8931	1.08967
$141$	9.36940	0.789046
$142$	20.3674	1.70920
$143$	29.7474	2.48760
$144$	−0.182510	−0.0152091
$145$	−1.51516	−0.125827
$146$	−29.0885	−2.40738
$147$	0.779896	0.0643247
$148$	1.73634	0.142726
$149$	12.3340	1.01044	0.505220	−	0.862991i	$-0.331411\pi$
$149$	12.3340	1.01044	0.505220	+	0.862991i	$0.331411\pi$
$150$	−5.42858	−0.443242
$151$	9.12639	0.742695	0.371348	−	0.928494i	$-0.378896\pi$
$151$	9.12639	0.742695	0.371348	+	0.928494i	$0.378896\pi$
$152$	−2.73491	−0.221830
$153$	−1.00000	−0.0808452
$154$	−29.0141	−2.33802
$155$	2.91559	0.234186
$156$	−18.6200	−1.49080
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	−22.8403	−1.81708
$159$	4.71778	0.374144
$160$	9.48654	0.749977
$161$	12.4080	0.977889
$162$	−2.27923	−0.179073
$163$	20.6916	1.62069	0.810344	−	0.585955i	$-0.199280\pi$
$163$	20.6916	1.62069	0.810344	+	0.585955i	$0.199280\pi$
$164$	−3.86563	−0.301855
$165$	−8.25898	−0.642961
$166$	6.07300	0.471357
$167$	5.06653	0.392060	0.196030	−	0.980598i	$-0.437195\pi$
$167$	5.06653	0.392060	0.196030	+	0.980598i	$0.437195\pi$
$168$	6.79220	0.524029
$169$	20.9666	1.61282
$170$	3.68801	0.282857
$171$	1.00423	0.0767951
$172$	−4.42777	−0.337614
$173$	−17.5347	−1.33314	−0.666569	−	0.745443i	$-0.732238\pi$
$173$	−17.5347	−1.33314	−0.666569	+	0.745443i	$0.732238\pi$
$174$	−2.13423	−0.161796
$175$	−5.94015	−0.449033
$176$	−0.931554	−0.0702185
$177$	−8.46224	−0.636061
$178$	36.0340	2.70086
$179$	−0.598950	−0.0447676	−0.0223838	−	0.999749i	$-0.507126\pi$
$179$	−0.598950	−0.0447676	−0.0223838	+	0.999749i	$0.507126\pi$
$180$	5.16962	0.385321
$181$	−13.7944	−1.02533	−0.512664	−	0.858590i	$-0.671341\pi$
$181$	−13.7944	−1.02533	−0.512664	+	0.858590i	$0.671341\pi$
$182$	−33.1293	−2.45571
$183$	−5.26173	−0.388958
$184$	−13.5492	−0.998864
$185$	0.879395	0.0646544
$186$	4.10686	0.301130
$187$	−5.10414	−0.373251
$188$	−29.9341	−2.18317
$189$	−2.49401	−0.181413
$190$	−3.70360	−0.268687
$191$	−5.60521	−0.405579	−0.202790	−	0.979222i	$-0.565001\pi$
$191$	−5.60521	−0.405579	−0.202790	+	0.979222i	$0.565001\pi$
$192$	12.9976	0.938020
$193$	2.13388	0.153600	0.0768002	−	0.997047i	$-0.475530\pi$
$193$	2.13388	0.153600	0.0768002	+	0.997047i	$0.475530\pi$
$194$	10.8447	0.778603
$195$	−9.43041	−0.675326
$196$	−2.49167	−0.177977
$197$	6.77843	0.482943	0.241472	−	0.970408i	$-0.422370\pi$
$197$	6.77843	0.482943	0.241472	+	0.970408i	$0.422370\pi$
$198$	−11.6335	−0.826756
$199$	1.46778	0.104048	0.0520241	−	0.998646i	$-0.483433\pi$
$199$	1.46778	0.104048	0.0520241	+	0.998646i	$0.483433\pi$
$200$	6.48649	0.458664
$201$	−9.18723	−0.648017
$202$	−43.5198	−3.06204
$203$	−2.33536	−0.163910
$204$	3.19488	0.223686
$205$	−1.95781	−0.136739
$206$	21.4707	1.49593
$207$	4.97512	0.345795
$208$	−1.06368	−0.0737532
$209$	5.12571	0.354553
$210$	9.19795	0.634719
$211$	0.900992	0.0620268	0.0310134	−	0.999519i	$-0.490127\pi$
$211$	0.900992	0.0620268	0.0310134	+	0.999519i	$0.490127\pi$
$212$	−15.0727	−1.03520
$213$	8.93610	0.612292
$214$	−4.09723	−0.280081
$215$	−2.24251	−0.152938
$216$	2.72340	0.185304
$217$	4.49388	0.305064
$218$	−39.2599	−2.65901
$219$	−12.7624	−0.862405
$220$	26.3864	1.77897
$221$	−5.82809	−0.392040
$222$	1.23870	0.0831363
$223$	−1.14873	−0.0769244	−0.0384622	−	0.999260i	$-0.512246\pi$
$223$	−1.14873	−0.0769244	−0.0384622	+	0.999260i	$0.512246\pi$
$224$	14.6219	0.976964
$225$	−2.38176	−0.158784
$226$	19.3210	1.28522
$227$	−9.94668	−0.660184	−0.330092	−	0.943949i	$-0.607080\pi$
$227$	−9.94668	−0.660184	−0.330092	+	0.943949i	$0.607080\pi$
$228$	−3.20838	−0.212480
$229$	−4.13983	−0.273568	−0.136784	−	0.990601i	$-0.543677\pi$
$229$	−4.13983	−0.273568	−0.136784	+	0.990601i	$0.543677\pi$
$230$	−18.3483	−1.20985
$231$	−12.7298	−0.837558
$232$	2.55015	0.167425
$233$	−26.4889	−1.73535	−0.867673	−	0.497136i	$-0.834385\pi$
$233$	−26.4889	−1.73535	−0.867673	+	0.497136i	$0.834385\pi$
$234$	−13.2835	−0.868373
$235$	−15.1606	−0.988968
$236$	27.0358	1.75988
$237$	−10.0211	−0.650939
$238$	5.68442	0.368467
$239$	15.0055	0.970627	0.485313	−	0.874340i	$-0.338706\pi$
$239$	15.0055	0.970627	0.485313	+	0.874340i	$0.338706\pi$
$240$	0.295318	0.0190627
$241$	−18.8343	−1.21323	−0.606613	−	0.794997i	$-0.707472\pi$
$241$	−18.8343	−1.21323	−0.606613	+	0.794997i	$0.707472\pi$
$242$	−34.3074	−2.20536
$243$	−1.00000	−0.0641500
$244$	16.8106	1.07619
$245$	−1.26195	−0.0806228
$246$	−2.75774	−0.175827
$247$	5.85272	0.372400
$248$	−4.90720	−0.311608
$249$	2.66450	0.168856
$250$	27.2240	1.72180
$251$	−11.3018	−0.713366	−0.356683	−	0.934225i	$-0.616092\pi$
$251$	−11.3018	−0.713366	−0.356683	+	0.934225i	$0.616092\pi$
$252$	7.96807	0.501941
$253$	25.3937	1.59649
$254$	−13.9049	−0.872470
$255$	1.61810	0.101329
$256$	−14.8005	−0.925031
$257$	25.3547	1.58158	0.790790	−	0.612087i	$-0.209670\pi$
$257$	25.3547	1.58158	0.790790	+	0.612087i	$0.209670\pi$
$258$	−3.15877	−0.196657
$259$	1.35543	0.0842226
$260$	30.1290	1.86852
$261$	−0.936385	−0.0579608
$262$	12.6131	0.779241
$263$	16.2080	0.999430	0.499715	−	0.866190i	$-0.333438\pi$
$263$	16.2080	0.999430	0.499715	+	0.866190i	$0.333438\pi$
$264$	13.9006	0.855523
$265$	−7.63382	−0.468942
$266$	−5.70845	−0.350007
$267$	15.8097	0.967540
$268$	29.3521	1.79296
$269$	−25.5722	−1.55917	−0.779583	−	0.626299i	$-0.784569\pi$
$269$	−25.5722	−1.55917	−0.779583	+	0.626299i	$0.784569\pi$
$270$	3.68801	0.224445
$271$	−24.8244	−1.50798	−0.753989	−	0.656887i	$-0.771873\pi$
$271$	−24.8244	−1.50798	−0.753989	+	0.656887i	$0.771873\pi$
$272$	0.182510	0.0110663
$273$	−14.5353	−0.879719
$274$	−20.6525	−1.24767
$275$	−12.1568	−0.733085
$276$	−15.8949	−0.956761
$277$	19.4328	1.16760	0.583801	−	0.811897i	$-0.301565\pi$
$277$	19.4328	1.16760	0.583801	+	0.811897i	$0.301565\pi$
$278$	−9.98528	−0.598877
$279$	1.80187	0.107875
$280$	−10.9904	−0.656804
$281$	−8.67292	−0.517383	−0.258692	−	0.965960i	$-0.583291\pi$
$281$	−8.67292	−0.517383	−0.258692	+	0.965960i	$0.583291\pi$
$282$	−21.3550	−1.27167
$283$	8.27525	0.491913	0.245956	−	0.969281i	$-0.420898\pi$
$283$	8.27525	0.491913	0.245956	+	0.969281i	$0.420898\pi$
$284$	−28.5498	−1.69412
$285$	−1.62494	−0.0962529
$286$	−67.8010	−4.00916
$287$	−3.01762	−0.178125
$288$	5.86278	0.345468
$289$	1.00000	0.0588235
$290$	3.45340	0.202790
$291$	4.75805	0.278922
$292$	40.7744	2.38614
$293$	27.3902	1.60015	0.800075	−	0.599900i	$-0.204793\pi$
$293$	27.3902	1.60015	0.800075	+	0.599900i	$0.204793\pi$
$294$	−1.77756	−0.103669
$295$	13.6927	0.797222
$296$	−1.48010	−0.0860290
$297$	−5.10414	−0.296172
$298$	−28.1120	−1.62848
$299$	28.9955	1.67685
$300$	7.60944	0.439331
$301$	−3.45644	−0.199226
$302$	−20.8011	−1.19697
$303$	−19.0941	−1.09693
$304$	−0.183281	−0.0105119
$305$	8.51399	0.487510
$306$	2.27923	0.130295
$307$	20.5395	1.17225	0.586126	−	0.810220i	$-0.300652\pi$
$307$	20.5395	1.17225	0.586126	+	0.810220i	$0.300652\pi$
$308$	40.6701	2.31739
$309$	9.42015	0.535894
$310$	−6.64530	−0.377428
$311$	4.38429	0.248610	0.124305	−	0.992244i	$-0.460330\pi$
$311$	4.38429	0.248610	0.124305	+	0.992244i	$0.460330\pi$
$312$	15.8722	0.898587
$313$	−10.8710	−0.614462	−0.307231	−	0.951635i	$-0.599403\pi$
$313$	−10.8710	−0.614462	−0.307231	+	0.951635i	$0.599403\pi$
$314$	−2.27923	−0.128624
$315$	4.03555	0.227378
$316$	32.0161	1.80105
$317$	10.2512	0.575766	0.287883	−	0.957666i	$-0.407049\pi$
$317$	10.2512	0.575766	0.287883	+	0.957666i	$0.407049\pi$
$318$	−10.7529	−0.602992
$319$	−4.77943	−0.267597
$320$	−21.0314	−1.17569
$321$	−1.79764	−0.100334
$322$	−28.2807	−1.57602
$323$	−1.00423	−0.0558766
$324$	3.19488	0.177493
$325$	−13.8811	−0.769987
$326$	−47.1608	−2.61199
$327$	−17.2251	−0.952548
$328$	3.29517	0.181945
$329$	−23.3674	−1.28829
$330$	18.8241	1.03623
$331$	25.0808	1.37856	0.689282	−	0.724494i	$-0.257926\pi$
$331$	25.0808	1.37856	0.689282	+	0.724494i	$0.257926\pi$
$332$	−8.51276	−0.467198
$333$	0.543475	0.0297822
$334$	−11.5478	−0.631866
$335$	14.8658	0.812207
$336$	0.455182	0.0248322
$337$	25.5703	1.39290	0.696452	−	0.717603i	$-0.254761\pi$
$337$	25.5703	1.39290	0.696452	+	0.717603i	$0.254761\pi$
$338$	−47.7878	−2.59931
$339$	8.47701	0.460408
$340$	−5.16962	−0.280362
$341$	9.19697	0.498044
$342$	−2.28886	−0.123767
$343$	−19.4032	−1.04767
$344$	3.77435	0.203499
$345$	−8.05023	−0.433410
$346$	39.9656	2.14856
$347$	−18.0086	−0.966754	−0.483377	−	0.875412i	$-0.660590\pi$
$347$	−18.0086	−0.966754	−0.483377	+	0.875412i	$0.660590\pi$
$348$	2.99163	0.160368
$349$	10.1268	0.542074	0.271037	−	0.962569i	$-0.412633\pi$
$349$	10.1268	0.542074	0.271037	+	0.962569i	$0.412633\pi$
$350$	13.5390	0.723688
$351$	−5.82809	−0.311080
$352$	29.9244	1.59498
$353$	−18.3651	−0.977477	−0.488738	−	0.872430i	$-0.662543\pi$
$353$	−18.3651	−0.977477	−0.488738	+	0.872430i	$0.662543\pi$
$354$	19.2874	1.02511
$355$	−14.4595	−0.767429
$356$	−50.5102	−2.67703
$357$	2.49401	0.131997
$358$	1.36514	0.0721501
$359$	−14.9137	−0.787116	−0.393558	−	0.919300i	$-0.628756\pi$
$359$	−14.9137	−0.787116	−0.393558	+	0.919300i	$0.628756\pi$
$360$	−4.40672	−0.232255
$361$	−17.9915	−0.946923
$362$	31.4405	1.65248
$363$	−15.0522	−0.790035
$364$	46.4386	2.43405
$365$	20.6508	1.08091
$366$	11.9927	0.626868
$367$	−9.20204	−0.480342	−0.240171	−	0.970731i	$-0.577204\pi$
$367$	−9.20204	−0.480342	−0.240171	+	0.970731i	$0.577204\pi$
$368$	−0.908008	−0.0473332
$369$	−1.20995	−0.0629873
$370$	−2.00434	−0.104201
$371$	−11.7662	−0.610871
$372$	−5.75674	−0.298473
$373$	−8.45097	−0.437575	−0.218787	−	0.975773i	$-0.570210\pi$
$373$	−8.45097	−0.437575	−0.218787	+	0.975773i	$0.570210\pi$
$374$	11.6335	0.601553
$375$	11.9444	0.616806
$376$	25.5166	1.31592
$377$	−5.45733	−0.281067
$378$	5.68442	0.292375
$379$	−16.1905	−0.831650	−0.415825	−	0.909445i	$-0.636507\pi$
$379$	−16.1905	−0.831650	−0.415825	+	0.909445i	$0.636507\pi$
$380$	5.19147	0.266317
$381$	−6.10070	−0.312548
$382$	12.7756	0.653654
$383$	−13.5881	−0.694319	−0.347160	−	0.937806i	$-0.612854\pi$
$383$	−13.5881	−0.694319	−0.347160	+	0.937806i	$0.612854\pi$
$384$	−17.8989	−0.913400
$385$	20.5980	1.04977
$386$	−4.86361	−0.247551
$387$	−1.38590	−0.0704491
$388$	−15.2014	−0.771734
$389$	−32.6807	−1.65698	−0.828489	−	0.560006i	$-0.810799\pi$
$389$	−32.6807	−1.65698	−0.828489	+	0.560006i	$0.810799\pi$
$390$	21.4941	1.08839
$391$	−4.97512	−0.251603
$392$	2.12397	0.107277
$393$	5.53395	0.279151
$394$	−15.4496	−0.778339
$395$	16.2151	0.815869
$396$	16.3071	0.819462
$397$	−4.24523	−0.213062	−0.106531	−	0.994309i	$-0.533974\pi$
$397$	−4.24523	−0.213062	−0.106531	+	0.994309i	$0.533974\pi$
$398$	−3.34541	−0.167690
$399$	−2.50455	−0.125385
$400$	0.434695	0.0217347
$401$	11.7828	0.588404	0.294202	−	0.955743i	$-0.404946\pi$
$401$	11.7828	0.588404	0.294202	+	0.955743i	$0.404946\pi$
$402$	20.9398	1.04438
$403$	10.5014	0.523114
$404$	61.0034	3.03503
$405$	1.61810	0.0804039
$406$	5.32281	0.264167
$407$	2.77397	0.137501
$408$	−2.72340	−0.134828
$409$	−22.9821	−1.13639	−0.568197	−	0.822893i	$-0.692359\pi$
$409$	−22.9821	−1.13639	−0.568197	+	0.822893i	$0.692359\pi$
$410$	4.46229	0.220377
$411$	−9.06120	−0.446956
$412$	−30.0962	−1.48274
$413$	21.1050	1.03851
$414$	−11.3394	−0.557303
$415$	−4.31142	−0.211639
$416$	34.1688	1.67526
$417$	−4.38099	−0.214538
$418$	−11.6827	−0.571417
$419$	12.7876	0.624717	0.312359	−	0.949964i	$-0.398881\pi$
$419$	12.7876	0.624717	0.312359	+	0.949964i	$0.398881\pi$
$420$	−12.8931	−0.629119
$421$	−1.59470	−0.0777207	−0.0388604	−	0.999245i	$-0.512373\pi$
$421$	−1.59470	−0.0777207	−0.0388604	+	0.999245i	$0.512373\pi$
$422$	−2.05357	−0.0999660
$423$	−9.36940	−0.455556
$424$	12.8484	0.623973
$425$	2.38176	0.115532
$426$	−20.3674	−0.986804
$427$	13.1228	0.635058
$428$	5.74324	0.277610
$429$	−29.7474	−1.43622
$430$	5.11120	0.246484
$431$	−37.8394	−1.82266	−0.911330	−	0.411676i	$-0.864944\pi$
$431$	−37.8394	−1.82266	−0.911330	+	0.411676i	$0.864944\pi$
$432$	0.182510	0.00878100
$433$	−30.7272	−1.47666	−0.738329	−	0.674441i	$-0.764385\pi$
$433$	−30.7272	−1.47666	−0.738329	+	0.674441i	$0.764385\pi$
$434$	−10.2426	−0.491659
$435$	1.51516	0.0726464
$436$	55.0320	2.63555
$437$	4.99615	0.238998
$438$	29.0885	1.38990
$439$	0.587884	0.0280582	0.0140291	−	0.999902i	$-0.495534\pi$
$439$	0.587884	0.0280582	0.0140291	+	0.999902i	$0.495534\pi$
$440$	−22.4925	−1.07229
$441$	−0.779896	−0.0371379
$442$	13.2835	0.631834
$443$	−12.3669	−0.587568	−0.293784	−	0.955872i	$-0.594915\pi$
$443$	−12.3669	−0.587568	−0.293784	+	0.955872i	$0.594915\pi$
$444$	−1.73634	−0.0824029
$445$	−25.5817	−1.21269
$446$	2.61821	0.123976
$447$	−12.3340	−0.583378
$448$	−32.4162	−1.53152
$449$	10.5592	0.498318	0.249159	−	0.968463i	$-0.419846\pi$
$449$	10.5592	0.498318	0.249159	+	0.968463i	$0.419846\pi$
$450$	5.42858	0.255906
$451$	−6.17573	−0.290804
$452$	−27.0830	−1.27388
$453$	−9.12639	−0.428795
$454$	22.6707	1.06399
$455$	23.5196	1.10262
$456$	2.73491	0.128074
$457$	17.1973	0.804455	0.402228	−	0.915540i	$-0.368236\pi$
$457$	17.1973	0.804455	0.402228	+	0.915540i	$0.368236\pi$
$458$	9.43561	0.440897
$459$	1.00000	0.0466760
$460$	25.7195	1.19918
$461$	−20.5032	−0.954931	−0.477466	−	0.878650i	$-0.658445\pi$
$461$	−20.5032	−0.954931	−0.477466	+	0.878650i	$0.658445\pi$
$462$	29.0141	1.34986
$463$	−21.4465	−0.996703	−0.498351	−	0.866975i	$-0.666061\pi$
$463$	−21.4465	−0.996703	−0.498351	+	0.866975i	$0.666061\pi$
$464$	0.170899	0.00793380
$465$	−2.91559	−0.135207
$466$	60.3742	2.79678
$467$	14.9581	0.692179	0.346090	−	0.938201i	$-0.387509\pi$
$467$	14.9581	0.692179	0.346090	+	0.938201i	$0.387509\pi$
$468$	18.6200	0.860712
$469$	22.9131	1.05803
$470$	34.5544	1.59388
$471$	−1.00000	−0.0460776
$472$	−23.0461	−1.06078
$473$	−7.07380	−0.325254
$474$	22.8403	1.04909
$475$	−2.39183	−0.109745
$476$	−7.96807	−0.365216
$477$	−4.71778	−0.216012
$478$	−34.2010	−1.56432
$479$	−34.7336	−1.58702	−0.793508	−	0.608559i	$-0.791748\pi$
$479$	−34.7336	−1.58702	−0.793508	+	0.608559i	$0.791748\pi$
$480$	−9.48654	−0.433000
$481$	3.16742	0.144422
$482$	42.9277	1.95530
$483$	−12.4080	−0.564585
$484$	48.0899	2.18591
$485$	−7.69899	−0.349593
$486$	2.27923	0.103388
$487$	17.4959	0.792814	0.396407	−	0.918075i	$-0.370257\pi$
$487$	17.4959	0.792814	0.396407	+	0.918075i	$0.370257\pi$
$488$	−14.3298	−0.648679
$489$	−20.6916	−0.935704
$490$	2.87626	0.129936
$491$	16.9167	0.763441	0.381720	−	0.924278i	$-0.375332\pi$
$491$	16.9167	0.763441	0.381720	+	0.924278i	$0.375332\pi$
$492$	3.86563	0.174276
$493$	0.936385	0.0421726
$494$	−13.3397	−0.600181
$495$	8.25898	0.371214
$496$	−0.328858	−0.0147662
$497$	−22.2868	−0.999698
$498$	−6.07300	−0.272138
$499$	4.82377	0.215942	0.107971	−	0.994154i	$-0.465565\pi$
$499$	4.82377	0.215942	0.107971	+	0.994154i	$0.465565\pi$
$500$	−38.1609	−1.70661
$501$	−5.06653	−0.226356
$502$	25.7595	1.14970
$503$	−29.8361	−1.33033	−0.665163	−	0.746698i	$-0.731638\pi$
$503$	−29.8361	−1.33033	−0.665163	+	0.746698i	$0.731638\pi$
$504$	−6.79220	−0.302548
$505$	30.8961	1.37486
$506$	−57.8780	−2.57299
$507$	−20.9666	−0.931161
$508$	19.4910	0.864773
$509$	4.41774	0.195813	0.0979065	−	0.995196i	$-0.468785\pi$
$509$	4.41774	0.195813	0.0979065	+	0.995196i	$0.468785\pi$
$510$	−3.68801	−0.163308
$511$	31.8297	1.40806
$512$	−2.06411	−0.0912215
$513$	−1.00423	−0.0443377
$514$	−57.7890	−2.54897
$515$	−15.2427	−0.671674
$516$	4.42777	0.194922
$517$	−47.8227	−2.10324
$518$	−3.08934	−0.135738
$519$	17.5347	0.769688
$520$	−25.6828	−1.12626
$521$	−14.8642	−0.651213	−0.325606	−	0.945505i	$-0.605568\pi$
$521$	−14.8642	−0.651213	−0.325606	+	0.945505i	$0.605568\pi$
$522$	2.13423	0.0934129
$523$	5.43930	0.237844	0.118922	−	0.992904i	$-0.462056\pi$
$523$	5.43930	0.237844	0.118922	+	0.992904i	$0.462056\pi$
$524$	−17.6803	−0.772367
$525$	5.94015	0.259249
$526$	−36.9418	−1.61074
$527$	−1.80187	−0.0784905
$528$	0.931554	0.0405407
$529$	1.75185	0.0761672
$530$	17.3992	0.755774
$531$	8.46224	0.367230
$532$	8.00175	0.346920
$533$	−7.05167	−0.305442
$534$	−36.0340	−1.55934
$535$	2.90875	0.125756
$536$	−25.0205	−1.08072
$537$	0.598950	0.0258466
$538$	58.2849	2.51284
$539$	−3.98069	−0.171461
$540$	−5.16962	−0.222465
$541$	14.8253	0.637390	0.318695	−	0.947857i	$-0.396755\pi$
$541$	14.8253	0.637390	0.318695	+	0.947857i	$0.396755\pi$
$542$	56.5806	2.43034
$543$	13.7944	0.591973
$544$	−5.86278	−0.251365
$545$	27.8718	1.19390
$546$	33.1293	1.41781
$547$	−6.96863	−0.297957	−0.148979	−	0.988840i	$-0.547599\pi$
$547$	−6.96863	−0.297957	−0.148979	+	0.988840i	$0.547599\pi$
$548$	28.9494	1.23666
$549$	5.26173	0.224565
$550$	27.7082	1.18148
$551$	−0.940342	−0.0400599
$552$	13.5492	0.576694
$553$	24.9927	1.06280
$554$	−44.2917	−1.88178
$555$	−0.879395	−0.0373282
$556$	13.9967	0.593594
$557$	19.0763	0.808289	0.404145	−	0.914695i	$-0.367569\pi$
$557$	19.0763	0.808289	0.404145	+	0.914695i	$0.367569\pi$
$558$	−4.10686	−0.173857
$559$	−8.07713	−0.341626
$560$	−0.736528	−0.0311240
$561$	5.10414	0.215497
$562$	19.7676	0.833844
$563$	17.9811	0.757815	0.378907	−	0.925435i	$-0.376300\pi$
$563$	17.9811	0.757815	0.378907	+	0.925435i	$0.376300\pi$
$564$	29.9341	1.26045
$565$	−13.7166	−0.577062
$566$	−18.8612	−0.792795
$567$	2.49401	0.104739
$568$	24.3366	1.02114
$569$	−8.58128	−0.359746	−0.179873	−	0.983690i	$-0.557569\pi$
$569$	−8.58128	−0.359746	−0.179873	+	0.983690i	$0.557569\pi$
$570$	3.70360	0.155127
$571$	3.96890	0.166093	0.0830467	−	0.996546i	$-0.473535\pi$
$571$	3.96890	0.166093	0.0830467	+	0.996546i	$0.473535\pi$
$572$	95.0392	3.97379
$573$	5.60521	0.234161
$574$	6.87785	0.287076
$575$	−11.8496	−0.494161
$576$	−12.9976	−0.541566
$577$	−16.1420	−0.671999	−0.335999	−	0.941862i	$-0.609074\pi$
$577$	−16.1420	−0.671999	−0.335999	+	0.941862i	$0.609074\pi$
$578$	−2.27923	−0.0948034
$579$	−2.13388	−0.0886812
$580$	−4.84075	−0.201001
$581$	−6.64530	−0.275694
$582$	−10.8447	−0.449526
$583$	−24.0802	−0.997299
$584$	−34.7572	−1.43826
$585$	9.43041	0.389900
$586$	−62.4284	−2.57889
$587$	35.1512	1.45085	0.725423	−	0.688303i	$-0.241644\pi$
$587$	35.1512	1.45085	0.725423	+	0.688303i	$0.241644\pi$
$588$	2.49167	0.102755
$589$	1.80948	0.0745584
$590$	−31.2088	−1.28485
$591$	−6.77843	−0.278827
$592$	−0.0991895	−0.00407666
$593$	−9.23874	−0.379389	−0.189695	−	0.981843i	$-0.560750\pi$
$593$	−9.23874	−0.379389	−0.189695	+	0.981843i	$0.560750\pi$
$594$	11.6335	0.477328
$595$	−4.03555	−0.165442
$596$	39.4056	1.61412
$597$	−1.46778	−0.0600723
$598$	−66.0873	−2.70251
$599$	−20.9640	−0.856565	−0.428283	−	0.903645i	$-0.640881\pi$
$599$	−20.9640	−0.856565	−0.428283	+	0.903645i	$0.640881\pi$
$600$	−6.48649	−0.264810
$601$	36.3925	1.48448	0.742241	−	0.670133i	$-0.233763\pi$
$601$	36.3925	1.48448	0.742241	+	0.670133i	$0.233763\pi$
$602$	7.87802	0.321084
$603$	9.18723	0.374133
$604$	29.1577	1.18641
$605$	24.3559	0.990208
$606$	43.5198	1.76787
$607$	39.9090	1.61986	0.809929	−	0.586528i	$-0.199505\pi$
$607$	39.9090	1.61986	0.809929	+	0.586528i	$0.199505\pi$
$608$	5.88756	0.238772
$609$	2.33536	0.0946334
$610$	−19.4053	−0.785699
$611$	−54.6057	−2.20911
$612$	−3.19488	−0.129145
$613$	25.5420	1.03163	0.515816	−	0.856699i	$-0.327489\pi$
$613$	25.5420	1.03163	0.515816	+	0.856699i	$0.327489\pi$
$614$	−46.8142	−1.88927
$615$	1.95781	0.0789465
$616$	−34.6683	−1.39682
$617$	9.11376	0.366906	0.183453	−	0.983028i	$-0.441272\pi$
$617$	9.11376	0.366906	0.183453	+	0.983028i	$0.441272\pi$
$618$	−21.4707	−0.863677
$619$	−34.2286	−1.37576	−0.687882	−	0.725822i	$-0.741459\pi$
$619$	−34.2286	−1.37576	−0.687882	+	0.725822i	$0.741459\pi$
$620$	9.31497	0.374098
$621$	−4.97512	−0.199645
$622$	−9.99280	−0.400675
$623$	−39.4297	−1.57972
$624$	1.06368	0.0425814
$625$	−7.41839	−0.296735
$626$	24.7774	0.990303
$627$	−5.12571	−0.204701
$628$	3.19488	0.127490
$629$	−0.543475	−0.0216698
$630$	−9.19795	−0.366455
$631$	3.36447	0.133937	0.0669687	−	0.997755i	$-0.478667\pi$
$631$	3.36447	0.133937	0.0669687	+	0.997755i	$0.478667\pi$
$632$	−27.2914	−1.08559
$633$	−0.900992	−0.0358112
$634$	−23.3649	−0.927938
$635$	9.87152	0.391740
$636$	15.0727	0.597673
$637$	−4.54530	−0.180092
$638$	10.8934	0.431274
$639$	−8.93610	−0.353507
$640$	28.9622	1.14483
$641$	−25.8452	−1.02082	−0.510412	−	0.859930i	$-0.670507\pi$
$641$	−25.8452	−1.02082	−0.510412	+	0.859930i	$0.670507\pi$
$642$	4.09723	0.161705
$643$	15.8609	0.625493	0.312747	−	0.949837i	$-0.398751\pi$
$643$	15.8609	0.625493	0.312747	+	0.949837i	$0.398751\pi$
$644$	39.6421	1.56212
$645$	2.24251	0.0882989
$646$	2.28886	0.0900540
$647$	33.0176	1.29805	0.649027	−	0.760765i	$-0.275176\pi$
$647$	33.0176	1.29805	0.649027	+	0.760765i	$0.275176\pi$
$648$	−2.72340	−0.106985
$649$	43.1924	1.69545
$650$	31.6383	1.24095
$651$	−4.49388	−0.176129
$652$	66.1070	2.58895
$653$	48.9313	1.91483	0.957414	−	0.288718i	$-0.0932290\pi$
$653$	48.9313	1.91483	0.957414	+	0.288718i	$0.0932290\pi$
$654$	39.2599	1.53518
$655$	−8.95446	−0.349880
$656$	0.220827	0.00862184
$657$	12.7624	0.497910
$658$	53.2596	2.07628
$659$	13.8291	0.538706	0.269353	−	0.963042i	$-0.413190\pi$
$659$	13.8291	0.538706	0.269353	+	0.963042i	$0.413190\pi$
$660$	−26.3864	−1.02709
$661$	24.1397	0.938927	0.469463	−	0.882952i	$-0.344447\pi$
$661$	24.1397	0.938927	0.469463	+	0.882952i	$0.344447\pi$
$662$	−57.1648	−2.22177
$663$	5.82809	0.226344
$664$	7.25650	0.281607
$665$	4.05261	0.157154
$666$	−1.23870	−0.0479988
$667$	−4.65863	−0.180383
$668$	16.1869	0.626291
$669$	1.14873	0.0444123
$670$	−33.8826	−1.30900
$671$	26.8566	1.03679
$672$	−14.6219	−0.564050
$673$	−21.9066	−0.844438	−0.422219	−	0.906494i	$-0.638749\pi$
$673$	−21.9066	−0.844438	−0.422219	+	0.906494i	$0.638749\pi$
$674$	−58.2806	−2.24488
$675$	2.38176	0.0916741
$676$	66.9859	2.57638
$677$	41.7899	1.60612	0.803059	−	0.595899i	$-0.203204\pi$
$677$	41.7899	1.60612	0.803059	+	0.595899i	$0.203204\pi$
$678$	−19.3210	−0.742019
$679$	−11.8666	−0.455400
$680$	4.40672	0.168990
$681$	9.94668	0.381158
$682$	−20.9620	−0.802676
$683$	−43.4663	−1.66319	−0.831596	−	0.555381i	$-0.812572\pi$
$683$	−43.4663	−1.66319	−0.831596	+	0.555381i	$0.812572\pi$
$684$	3.20838	0.122676
$685$	14.6619	0.560202
$686$	44.2242	1.68849
$687$	4.13983	0.157944
$688$	0.252940	0.00964323
$689$	−27.4956	−1.04750
$690$	18.3483	0.698508
$691$	−16.0063	−0.608907	−0.304453	−	0.952527i	$-0.598474\pi$
$691$	−16.0063	−0.608907	−0.304453	+	0.952527i	$0.598474\pi$
$692$	−56.0212	−2.12961
$693$	12.7298	0.483564
$694$	41.0458	1.55808
$695$	7.08887	0.268896
$696$	−2.55015	−0.0966632
$697$	1.20995	0.0458300
$698$	−23.0812	−0.873638
$699$	26.4889	1.00190
$700$	−18.9781	−0.717303
$701$	−8.95820	−0.338346	−0.169173	−	0.985586i	$-0.554110\pi$
$701$	−8.95820	−0.338346	−0.169173	+	0.985586i	$0.554110\pi$
$702$	13.2835	0.501355
$703$	0.545772	0.0205842
$704$	−66.3415	−2.50034
$705$	15.1606	0.570981
$706$	41.8583	1.57536
$707$	47.6210	1.79097
$708$	−27.0358	−1.01607
$709$	29.7301	1.11654	0.558270	−	0.829660i	$-0.311465\pi$
$709$	29.7301	1.11654	0.558270	+	0.829660i	$0.311465\pi$
$710$	32.9564	1.23683
$711$	10.0211	0.375820
$712$	43.0562	1.61360
$713$	8.96451	0.335723
$714$	−5.68442	−0.212734
$715$	48.1341	1.80011
$716$	−1.91357	−0.0715135
$717$	−15.0055	−0.560392
$718$	33.9918	1.26856
$719$	−21.3516	−0.796282	−0.398141	−	0.917324i	$-0.630345\pi$
$719$	−21.3516	−0.796282	−0.398141	+	0.917324i	$0.630345\pi$
$720$	−0.295318	−0.0110059
$721$	−23.4940	−0.874962
$722$	41.0068	1.52611
$723$	18.8343	0.700456
$724$	−44.0713	−1.63790
$725$	2.23025	0.0828293
$726$	34.3074	1.27327
$727$	31.0489	1.15154	0.575771	−	0.817611i	$-0.304702\pi$
$727$	31.0489	1.15154	0.575771	+	0.817611i	$0.304702\pi$
$728$	−39.5855	−1.46714
$729$	1.00000	0.0370370
$730$	−47.0680	−1.74206
$731$	1.38590	0.0512592
$732$	−16.8106	−0.621337
$733$	52.9117	1.95434	0.977169	−	0.212464i	$-0.0681489\pi$
$733$	52.9117	1.95434	0.977169	+	0.212464i	$0.0681489\pi$
$734$	20.9735	0.774147
$735$	1.26195	0.0465476
$736$	29.1680	1.07515
$737$	46.8929	1.72732
$738$	2.75774	0.101514
$739$	7.58755	0.279113	0.139556	−	0.990214i	$-0.455432\pi$
$739$	7.58755	0.279113	0.139556	+	0.990214i	$0.455432\pi$
$740$	2.80956	0.103281
$741$	−5.85272	−0.215005
$742$	26.8179	0.984515
$743$	−28.3469	−1.03995	−0.519974	−	0.854182i	$-0.674058\pi$
$743$	−28.3469	−1.03995	−0.519974	+	0.854182i	$0.674058\pi$
$744$	4.90720	0.179907
$745$	19.9576	0.731190
$746$	19.2617	0.705221
$747$	−2.66450	−0.0974890
$748$	−16.3071	−0.596246
$749$	4.48334	0.163818
$750$	−27.2240	−0.994081
$751$	−32.2517	−1.17688	−0.588441	−	0.808540i	$-0.700258\pi$
$751$	−32.2517	−1.17688	−0.588441	+	0.808540i	$0.700258\pi$
$752$	1.71001	0.0623575
$753$	11.3018	0.411862
$754$	12.4385	0.452984
$755$	14.7674	0.537440
$756$	−7.96807	−0.289796
$757$	−1.06876	−0.0388448	−0.0194224	−	0.999811i	$-0.506183\pi$
$757$	−1.06876	−0.0388448	−0.0194224	+	0.999811i	$0.506183\pi$
$758$	36.9018	1.34034
$759$	−25.3937	−0.921733
$760$	−4.42535	−0.160524
$761$	20.3947	0.739308	0.369654	−	0.929170i	$-0.379476\pi$
$761$	20.3947	0.739308	0.369654	+	0.929170i	$0.379476\pi$
$762$	13.9049	0.503721
$763$	42.9596	1.55524
$764$	−17.9080	−0.647888
$765$	−1.61810	−0.0585024
$766$	30.9704	1.11900
$767$	49.3187	1.78080
$768$	14.8005	0.534067
$769$	−28.1214	−1.01408	−0.507042	−	0.861921i	$-0.669261\pi$
$769$	−28.1214	−1.01408	−0.507042	+	0.861921i	$0.669261\pi$
$770$	−46.9476	−1.69187
$771$	−25.3547	−0.913126
$772$	6.81750	0.245367
$773$	−0.341934	−0.0122985	−0.00614926	−	0.999981i	$-0.501957\pi$
$773$	−0.341934	−0.0122985	−0.00614926	+	0.999981i	$0.501957\pi$
$774$	3.15877	0.113540
$775$	−4.29162	−0.154160
$776$	12.9581	0.465168
$777$	−1.35543	−0.0486259
$778$	74.4868	2.67048
$779$	−1.21506	−0.0435340
$780$	−30.1290	−1.07879
$781$	−45.6111	−1.63209
$782$	11.3394	0.405497
$783$	0.936385	0.0334637
$784$	0.142339	0.00508352
$785$	1.61810	0.0577523
$786$	−12.6131	−0.449895
$787$	25.0269	0.892112	0.446056	−	0.895005i	$-0.352828\pi$
$787$	25.0269	0.892112	0.446056	+	0.895005i	$0.352828\pi$
$788$	21.6563	0.771472
$789$	−16.2080	−0.577021
$790$	−36.9578	−1.31490
$791$	−21.1418	−0.751715
$792$	−13.9006	−0.493936
$793$	30.6659	1.08898
$794$	9.67585	0.343383
$795$	7.63382	0.270744
$796$	4.68938	0.166211
$797$	−9.55924	−0.338605	−0.169303	−	0.985564i	$-0.554152\pi$
$797$	−9.55924	−0.338605	−0.169303	+	0.985564i	$0.554152\pi$
$798$	5.70845	0.202077
$799$	9.36940	0.331466
$800$	−13.9638	−0.493693
$801$	−15.8097	−0.558609
$802$	−26.8556	−0.948306
$803$	65.1412	2.29878
$804$	−29.3521	−1.03517
$805$	20.0774	0.707635
$806$	−23.9352	−0.843081
$807$	25.5722	0.900185
$808$	−52.0009	−1.82938
$809$	14.6893	0.516450	0.258225	−	0.966085i	$-0.416862\pi$
$809$	14.6893	0.516450	0.258225	+	0.966085i	$0.416862\pi$
$810$	−3.68801	−0.129583
$811$	16.7378	0.587745	0.293872	−	0.955845i	$-0.405056\pi$
$811$	16.7378	0.587745	0.293872	+	0.955845i	$0.405056\pi$
$812$	−7.46118	−0.261836
$813$	24.8244	0.870631
$814$	−6.32251	−0.221604
$815$	33.4809	1.17279
$816$	−0.182510	−0.00638912
$817$	−1.39175	−0.0486913
$818$	52.3815	1.83148
$819$	14.5353	0.507906
$820$	−6.25496	−0.218433
$821$	42.3118	1.47669	0.738346	−	0.674422i	$-0.235607\pi$
$821$	42.3118	1.47669	0.738346	+	0.674422i	$0.235607\pi$
$822$	20.6525	0.720340
$823$	23.2146	0.809209	0.404604	−	0.914492i	$-0.367409\pi$
$823$	23.2146	0.809209	0.404604	+	0.914492i	$0.367409\pi$
$824$	25.6548	0.893729
$825$	12.1568	0.423247
$826$	−48.1030	−1.67372
$827$	20.7179	0.720431	0.360215	−	0.932869i	$-0.382703\pi$
$827$	20.7179	0.720431	0.360215	+	0.932869i	$0.382703\pi$
$828$	15.8949	0.552386
$829$	−31.5342	−1.09523	−0.547615	−	0.836731i	$-0.684464\pi$
$829$	−31.5342	−1.09523	−0.547615	+	0.836731i	$0.684464\pi$
$830$	9.82671	0.341090
$831$	−19.4328	−0.674116
$832$	−75.7511	−2.62620
$833$	0.779896	0.0270218
$834$	9.98528	0.345762
$835$	8.19813	0.283708
$836$	16.3760	0.566376
$837$	−1.80187	−0.0622816
$838$	−29.1460	−1.00683
$839$	−7.35591	−0.253954	−0.126977	−	0.991906i	$-0.540527\pi$
$839$	−7.35591	−0.253954	−0.126977	+	0.991906i	$0.540527\pi$
$840$	10.9904	0.379206
$841$	−28.1232	−0.969765
$842$	3.63468	0.125259
$843$	8.67292	0.298711
$844$	2.87856	0.0990841
$845$	33.9261	1.16709
$846$	21.3550	0.734200
$847$	37.5404	1.28990
$848$	0.861040	0.0295683
$849$	−8.27525	−0.284006
$850$	−5.42858	−0.186199
$851$	2.70385	0.0926869
$852$	28.5498	0.978099
$853$	41.1815	1.41003	0.705014	−	0.709194i	$-0.250941\pi$
$853$	41.1815	1.41003	0.705014	+	0.709194i	$0.250941\pi$
$854$	−29.9099	−1.02350
$855$	1.62494	0.0555716
$856$	−4.89569	−0.167331
$857$	−0.232283	−0.00793465	−0.00396733	−	0.999992i	$-0.501263\pi$
$857$	−0.232283	−0.00793465	−0.00396733	+	0.999992i	$0.501263\pi$
$858$	67.8010	2.31469
$859$	48.8651	1.66726	0.833628	−	0.552326i	$-0.186260\pi$
$859$	48.8651	1.66726	0.833628	+	0.552326i	$0.186260\pi$
$860$	−7.16456	−0.244309
$861$	3.01762	0.102840
$862$	86.2446	2.93750
$863$	2.20015	0.0748941	0.0374470	−	0.999299i	$-0.488077\pi$
$863$	2.20015	0.0748941	0.0374470	+	0.999299i	$0.488077\pi$
$864$	−5.86278	−0.199456
$865$	−28.3728	−0.964705
$866$	70.0344	2.37987
$867$	−1.00000	−0.0339618
$868$	14.3574	0.487322
$869$	51.1489	1.73511
$870$	−3.45340	−0.117081
$871$	53.5440	1.81427
$872$	−46.9107	−1.58860
$873$	−4.75805	−0.161036
$874$	−11.3874	−0.385183
$875$	−29.7895	−1.00707
$876$	−40.7744	−1.37764
$877$	−3.47701	−0.117410	−0.0587051	−	0.998275i	$-0.518697\pi$
$877$	−3.47701	−0.117410	−0.0587051	+	0.998275i	$0.518697\pi$
$878$	−1.33992	−0.0452202
$879$	−27.3902	−0.923847
$880$	−1.50734	−0.0508126
$881$	47.9892	1.61680	0.808398	−	0.588636i	$-0.200335\pi$
$881$	47.9892	1.61680	0.808398	+	0.588636i	$0.200335\pi$
$882$	1.77756	0.0598536
$883$	−52.1806	−1.75602	−0.878008	−	0.478646i	$-0.841128\pi$
$883$	−52.1806	−1.75602	−0.878008	+	0.478646i	$0.841128\pi$
$884$	−18.6200	−0.626260
$885$	−13.6927	−0.460276
$886$	28.1869	0.946959
$887$	−25.1655	−0.844974	−0.422487	−	0.906369i	$-0.638843\pi$
$887$	−25.1655	−0.844974	−0.422487	+	0.906369i	$0.638843\pi$
$888$	1.48010	0.0496689
$889$	15.2152	0.510303
$890$	58.3065	1.95444
$891$	5.10414	0.170995
$892$	−3.67004	−0.122882
$893$	−9.40900	−0.314860
$894$	28.1120	0.940205
$895$	−0.969159	−0.0323954
$896$	44.6401	1.49132
$897$	−28.9955	−0.968131
$898$	−24.0667	−0.803117
$899$	−1.68724	−0.0562726
$900$	−7.60944	−0.253648
$901$	4.71778	0.157172
$902$	14.0759	0.468676
$903$	3.45644	0.115023
$904$	23.0863	0.767838
$905$	−22.3206	−0.741962
$906$	20.8011	0.691071
$907$	−31.3340	−1.04043	−0.520214	−	0.854036i	$-0.674148\pi$
$907$	−31.3340	−1.04043	−0.520214	+	0.854036i	$0.674148\pi$
$908$	−31.7784	−1.05460
$909$	19.0941	0.633312
$910$	−53.6065	−1.77704
$911$	−36.8801	−1.22189	−0.610946	−	0.791672i	$-0.709211\pi$
$911$	−36.8801	−1.22189	−0.610946	+	0.791672i	$0.709211\pi$
$912$	0.183281	0.00606904
$913$	−13.6000	−0.450093
$914$	−39.1965	−1.29651
$915$	−8.51399	−0.281464
$916$	−13.2263	−0.437008
$917$	−13.8017	−0.455774
$918$	−2.27923	−0.0752257
$919$	−40.7522	−1.34429	−0.672146	−	0.740419i	$-0.734627\pi$
$919$	−40.7522	−1.34429	−0.672146	+	0.740419i	$0.734627\pi$
$920$	−21.9240	−0.722812
$921$	−20.5395	−0.676800
$922$	46.7316	1.53902
$923$	−52.0804	−1.71425
$924$	−40.6701	−1.33795
$925$	−1.29443	−0.0425606
$926$	48.8814	1.60634
$927$	−9.42015	−0.309398
$928$	−5.48982	−0.180212
$929$	27.0371	0.887057	0.443529	−	0.896260i	$-0.353726\pi$
$929$	27.0371	0.887057	0.443529	+	0.896260i	$0.353726\pi$
$930$	6.64530	0.217908
$931$	−0.783192	−0.0256681
$932$	−84.6288	−2.77211
$933$	−4.38429	−0.143535
$934$	−34.0930	−1.11556
$935$	−8.25898	−0.270098
$936$	−15.8722	−0.518800
$937$	−40.2422	−1.31466	−0.657328	−	0.753605i	$-0.728313\pi$
$937$	−40.2422	−1.31466	−0.657328	+	0.753605i	$0.728313\pi$
$938$	−52.2241	−1.70518
$939$	10.8710	0.354760
$940$	−48.4362	−1.57982
$941$	−42.4863	−1.38501	−0.692507	−	0.721411i	$-0.743494\pi$
$941$	−42.4863	−1.38501	−0.692507	+	0.721411i	$0.743494\pi$
$942$	2.27923	0.0742612
$943$	−6.01963	−0.196026
$944$	−1.54444	−0.0502673
$945$	−4.03555	−0.131277
$946$	16.1228	0.524198
$947$	−33.2316	−1.07988	−0.539940	−	0.841703i	$-0.681553\pi$
$947$	−33.2316	−1.07988	−0.539940	+	0.841703i	$0.681553\pi$
$948$	−32.0161	−1.03984
$949$	74.3806	2.41450
$950$	5.45152	0.176871
$951$	−10.2512	−0.332419
$952$	6.79220	0.220136
$953$	38.2217	1.23812	0.619062	−	0.785342i	$-0.287513\pi$
$953$	38.2217	1.23812	0.619062	+	0.785342i	$0.287513\pi$
$954$	10.7529	0.348138
$955$	−9.06978	−0.293491
$956$	47.9408	1.55052
$957$	4.77943	0.154497
$958$	79.1657	2.55773
$959$	22.5988	0.729752
$960$	21.0314	0.678784
$961$	−27.7533	−0.895267
$962$	−7.21928	−0.232759
$963$	1.79764	0.0579281
$964$	−60.1734	−1.93805
$965$	3.45283	0.111151
$966$	28.2807	0.909917
$967$	21.3802	0.687542	0.343771	−	0.939054i	$-0.388296\pi$
$967$	21.3802	0.687542	0.343771	+	0.939054i	$0.388296\pi$
$968$	−40.9931	−1.31757
$969$	1.00423	0.0322604
$970$	17.5477	0.563424
$971$	−50.0941	−1.60760	−0.803798	−	0.594902i	$-0.797191\pi$
$971$	−50.0941	−1.60760	−0.803798	+	0.594902i	$0.797191\pi$
$972$	−3.19488	−0.102476
$973$	10.9263	0.350280
$974$	−39.8771	−1.27774
$975$	13.8811	0.444552
$976$	−0.960317	−0.0307390
$977$	27.9289	0.893526	0.446763	−	0.894652i	$-0.352577\pi$
$977$	27.9289	0.893526	0.446763	+	0.894652i	$0.352577\pi$
$978$	47.1608	1.50803
$979$	−80.6950	−2.57902
$980$	−4.03177	−0.128790
$981$	17.2251	0.549954
$982$	−38.5570	−1.23040
$983$	1.04480	0.0333241	0.0166620	−	0.999861i	$-0.494696\pi$
$983$	1.04480	0.0333241	0.0166620	+	0.999861i	$0.494696\pi$
$984$	−3.29517	−0.105046
$985$	10.9682	0.349475
$986$	−2.13423	−0.0679678
$987$	23.3674	0.743793
$988$	18.6987	0.594886
$989$	−6.89500	−0.219248
$990$	−18.8241	−0.598269
$991$	−22.9169	−0.727980	−0.363990	−	0.931403i	$-0.618586\pi$
$991$	−22.9169	−0.727980	−0.363990	+	0.931403i	$0.618586\pi$
$992$	10.5639	0.335406
$993$	−25.0808	−0.795914
$994$	50.7966	1.61117
$995$	2.37501	0.0752930
$996$	8.51276	0.269737
$997$	−12.5467	−0.397357	−0.198678	−	0.980065i	$-0.563665\pi$
$997$	−12.5467	−0.397357	−0.198678	+	0.980065i	$0.563665\pi$
$998$	−10.9945	−0.348024
$999$	−0.543475	−0.0171948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.g.1.8	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.g.1.8	✓	56	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27923 q^{2} -1.00000 q^{3} +3.19488 q^{4} +1.61810 q^{5} +2.27923 q^{6} +2.49401 q^{7} -2.72340 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.27923 q^{2} -1.00000 q^{3} +3.19488 q^{4} +1.61810 q^{5} +2.27923 q^{6} +2.49401 q^{7} -2.72340 q^{8} +1.00000 q^{9} -3.68801 q^{10} +5.10414 q^{11} -3.19488 q^{12} +5.82809 q^{13} -5.68442 q^{14} -1.61810 q^{15} -0.182510 q^{16} -1.00000 q^{17} -2.27923 q^{18} +1.00423 q^{19} +5.16962 q^{20} -2.49401 q^{21} -11.6335 q^{22} +4.97512 q^{23} +2.72340 q^{24} -2.38176 q^{25} -13.2835 q^{26} -1.00000 q^{27} +7.96807 q^{28} -0.936385 q^{29} +3.68801 q^{30} +1.80187 q^{31} +5.86278 q^{32} -5.10414 q^{33} +2.27923 q^{34} +4.03555 q^{35} +3.19488 q^{36} +0.543475 q^{37} -2.28886 q^{38} -5.82809 q^{39} -4.40672 q^{40} -1.20995 q^{41} +5.68442 q^{42} -1.38590 q^{43} +16.3071 q^{44} +1.61810 q^{45} -11.3394 q^{46} -9.36940 q^{47} +0.182510 q^{48} -0.779896 q^{49} +5.42858 q^{50} +1.00000 q^{51} +18.6200 q^{52} -4.71778 q^{53} +2.27923 q^{54} +8.25898 q^{55} -6.79220 q^{56} -1.00423 q^{57} +2.13423 q^{58} +8.46224 q^{59} -5.16962 q^{60} +5.26173 q^{61} -4.10686 q^{62} +2.49401 q^{63} -12.9976 q^{64} +9.43041 q^{65} +11.6335 q^{66} +9.18723 q^{67} -3.19488 q^{68} -4.97512 q^{69} -9.19795 q^{70} -8.93610 q^{71} -2.72340 q^{72} +12.7624 q^{73} -1.23870 q^{74} +2.38176 q^{75} +3.20838 q^{76} +12.7298 q^{77} +13.2835 q^{78} +10.0211 q^{79} -0.295318 q^{80} +1.00000 q^{81} +2.75774 q^{82} -2.66450 q^{83} -7.96807 q^{84} -1.61810 q^{85} +3.15877 q^{86} +0.936385 q^{87} -13.9006 q^{88} -15.8097 q^{89} -3.68801 q^{90} +14.5353 q^{91} +15.8949 q^{92} -1.80187 q^{93} +21.3550 q^{94} +1.62494 q^{95} -5.86278 q^{96} -4.75805 q^{97} +1.77756 q^{98} +5.10414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9}+O(q^{10}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9	\( 56 q + q^{2} - 56 q^{3} + 61 q^{4} + q^{5} - q^{6} + 19 q^{7} + 56 q^{9} + 8 q^{10} - 7 q^{11} - 61 q^{12} + 8 q^{13} - 8 q^{14} - q^{15} + 71 q^{16} - 56 q^{17} + q^{18} - 2 q^{19} - 4 q^{20} - 19 q^{21} + 47 q^{22} + 16 q^{23} + 85 q^{25} - 11 q^{26} - 56 q^{27} + 52 q^{28} + 17 q^{29} - 8 q^{30} + 23 q^{31} + 11 q^{32} + 7 q^{33} - q^{34} - 41 q^{35} + 61 q^{36} + 58 q^{37} - 22 q^{38} - 8 q^{39} + 38 q^{40} - q^{41} + 8 q^{42} + 27 q^{43} + 2 q^{44} + q^{45} + 46 q^{46} + 5 q^{47} - 71 q^{48} + 59 q^{49} - 4 q^{50} + 56 q^{51} + 25 q^{52} + 15 q^{53} - q^{54} + 9 q^{55} - 36 q^{56} + 2 q^{57} + 89 q^{58} - 61 q^{59} + 4 q^{60} + 47 q^{61} + 8 q^{62} + 19 q^{63} + 88 q^{64} + 39 q^{65} - 47 q^{66} + 20 q^{67} - 61 q^{68} - 16 q^{69} + 36 q^{70} - 2 q^{71} + 93 q^{73} + 48 q^{74} - 85 q^{75} + 38 q^{76} + 26 q^{77} + 11 q^{78} + 72 q^{79} + 42 q^{80} + 56 q^{81} + 33 q^{82} - 11 q^{83} - 52 q^{84} - q^{85} - 4 q^{86} - 17 q^{87} + 130 q^{88} - 6 q^{89} + 8 q^{90} + 37 q^{91} + 132 q^{92} - 23 q^{93} - 32 q^{94} + 12 q^{95} - 11 q^{96} + 100 q^{97} + 42 q^{98} - 7 q^{99}+O(q^{100}) \) 56 * q + q^2 - 56 * q^3 + 61 * q^4 + q^5 - q^6 + 19 * q^7 + 56 * q^9 + 8 * q^10 - 7 * q^11 - 61 * q^12 + 8 * q^13 - 8 * q^14 - q^15 + 71 * q^16 - 56 * q^17 + q^18 - 2 * q^19 - 4 * q^20 - 19 * q^21 + 47 * q^22 + 16 * q^23 + 85 * q^25 - 11 * q^26 - 56 * q^27 + 52 * q^28 + 17 * q^29 - 8 * q^30 + 23 * q^31 + 11 * q^32 + 7 * q^33 - q^34 - 41 * q^35 + 61 * q^36 + 58 * q^37 - 22 * q^38 - 8 * q^39 + 38 * q^40 - q^41 + 8 * q^42 + 27 * q^43 + 2 * q^44 + q^45 + 46 * q^46 + 5 * q^47 - 71 * q^48 + 59 * q^49 - 4 * q^50 + 56 * q^51 + 25 * q^52 + 15 * q^53 - q^54 + 9 * q^55 - 36 * q^56 + 2 * q^57 + 89 * q^58 - 61 * q^59 + 4 * q^60 + 47 * q^61 + 8 * q^62 + 19 * q^63 + 88 * q^64 + 39 * q^65 - 47 * q^66 + 20 * q^67 - 61 * q^68 - 16 * q^69 + 36 * q^70 - 2 * q^71 + 93 * q^73 + 48 * q^74 - 85 * q^75 + 38 * q^76 + 26 * q^77 + 11 * q^78 + 72 * q^79 + 42 * q^80 + 56 * q^81 + 33 * q^82 - 11 * q^83 - 52 * q^84 - q^85 - 4 * q^86 - 17 * q^87 + 130 * q^88 - 6 * q^89 + 8 * q^90 + 37 * q^91 + 132 * q^92 - 23 * q^93 - 32 * q^94 + 12 * q^95 - 11 * q^96 + 100 * q^97 + 42 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more