LMFDB - Embedded newform 8470.2.a.co.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.4
Root			$2.14896$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.67989 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.67989 q^{9} -1.00000 q^{10} +1.14896 q^{12} +3.79720 q^{13} +1.00000 q^{14} +1.14896 q^{15} +1.00000 q^{16} +6.47709 q^{17} +1.67989 q^{18} +2.67989 q^{19} +1.00000 q^{20} -1.14896 q^{21} +2.65626 q^{23} -1.14896 q^{24} +1.00000 q^{25} -3.79720 q^{26} -5.37701 q^{27} -1.00000 q^{28} +0.386922 q^{29} -1.14896 q^{30} -6.01867 q^{31} -1.00000 q^{32} -6.47709 q^{34} -1.00000 q^{35} -1.67989 q^{36} +2.38692 q^{37} -2.67989 q^{38} +4.36284 q^{39} -1.00000 q^{40} +9.47709 q^{41} +1.14896 q^{42} -4.66428 q^{43} -1.67989 q^{45} -2.65626 q^{46} -4.48205 q^{47} +1.14896 q^{48} +1.00000 q^{49} -1.00000 q^{50} +7.44193 q^{51} +3.79720 q^{52} -5.09819 q^{53} +5.37701 q^{54} +1.00000 q^{56} +3.07909 q^{57} -0.386922 q^{58} +4.74643 q^{59} +1.14896 q^{60} +11.6593 q^{61} +6.01867 q^{62} +1.67989 q^{63} +1.00000 q^{64} +3.79720 q^{65} +5.71316 q^{67} +6.47709 q^{68} +3.05194 q^{69} +1.00000 q^{70} +3.33119 q^{71} +1.67989 q^{72} +15.0127 q^{73} -2.38692 q^{74} +1.14896 q^{75} +2.67989 q^{76} -4.36284 q^{78} -6.03633 q^{79} +1.00000 q^{80} -1.13831 q^{81} -9.47709 q^{82} -11.1551 q^{83} -1.14896 q^{84} +6.47709 q^{85} +4.66428 q^{86} +0.444559 q^{87} -5.87277 q^{89} +1.67989 q^{90} -3.79720 q^{91} +2.65626 q^{92} -6.91522 q^{93} +4.48205 q^{94} +2.67989 q^{95} -1.14896 q^{96} -3.30450 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 - 4 * q^10 - 4 * q^12 + 14 * q^13 + 4 * q^14 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 6 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 + 4 * q^24 + 4 * q^25 - 14 * q^26 - 22 * q^27 - 4 * q^28 + 10 * q^29 + 4 * q^30 - 18 * q^31 - 4 * q^32 - 12 * q^34 - 4 * q^35 + 6 * q^36 + 18 * q^37 + 2 * q^38 - 30 * q^39 - 4 * q^40 + 24 * q^41 - 4 * q^42 + 10 * q^43 + 6 * q^45 - 8 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 14 * q^52 + 20 * q^53 + 22 * q^54 + 4 * q^56 + 30 * q^57 - 10 * q^58 - 14 * q^59 - 4 * q^60 + 14 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 14 * q^65 + 12 * q^68 - 4 * q^69 + 4 * q^70 - 14 * q^71 - 6 * q^72 + 30 * q^73 - 18 * q^74 - 4 * q^75 - 2 * q^76 + 30 * q^78 + 8 * q^79 + 4 * q^80 + 16 * q^81 - 24 * q^82 + 8 * q^83 + 4 * q^84 + 12 * q^85 - 10 * q^86 + 2 * q^87 - 4 * q^89 - 6 * q^90 - 14 * q^91 - 2 * q^93 + 8 * q^94 - 2 * q^95 + 4 * 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
$3$	1.14896	0.663353	0.331677	−	0.943393i	$-0.392386\pi$
$3$	1.14896	0.663353	0.331677	+	0.943393i	$0.392386\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.14896	−0.469061
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−1.67989	−0.559963
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.14896	0.331677
$13$	3.79720	1.05315	0.526577	−	0.850127i	$-0.323475\pi$
$13$	3.79720	1.05315	0.526577	+	0.850127i	$0.323475\pi$
$14$	1.00000	0.267261
$15$	1.14896	0.296660
$16$	1.00000	0.250000
$17$	6.47709	1.57093	0.785463	−	0.618909i	$-0.212425\pi$
$17$	6.47709	1.57093	0.785463	+	0.618909i	$0.212425\pi$
$18$	1.67989	0.395953
$19$	2.67989	0.614809	0.307404	−	0.951579i	$-0.400540\pi$
$19$	2.67989	0.614809	0.307404	+	0.951579i	$0.400540\pi$
$20$	1.00000	0.223607
$21$	−1.14896	−0.250724
$22$	0	0
$23$	2.65626	0.553869	0.276934	−	0.960889i	$-0.410682\pi$
$23$	2.65626	0.553869	0.276934	+	0.960889i	$0.410682\pi$
$24$	−1.14896	−0.234531
$25$	1.00000	0.200000
$26$	−3.79720	−0.744693
$27$	−5.37701	−1.03481
$28$	−1.00000	−0.188982
$29$	0.386922	0.0718497	0.0359248	−	0.999354i	$-0.488562\pi$
$29$	0.386922	0.0718497	0.0359248	+	0.999354i	$0.488562\pi$
$30$	−1.14896	−0.209771
$31$	−6.01867	−1.08099	−0.540493	−	0.841349i	$-0.681762\pi$
$31$	−6.01867	−1.08099	−0.540493	+	0.841349i	$0.681762\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.47709	−1.11081
$35$	−1.00000	−0.169031
$36$	−1.67989	−0.279981
$37$	2.38692	0.392408	0.196204	−	0.980563i	$-0.437139\pi$
$37$	2.38692	0.392408	0.196204	+	0.980563i	$0.437139\pi$
$38$	−2.67989	−0.434735
$39$	4.36284	0.698613
$40$	−1.00000	−0.158114
$41$	9.47709	1.48007	0.740037	−	0.672567i	$-0.234808\pi$
$41$	9.47709	1.48007	0.740037	+	0.672567i	$0.234808\pi$
$42$	1.14896	0.177289
$43$	−4.66428	−0.711296	−0.355648	−	0.934620i	$-0.615740\pi$
$43$	−4.66428	−0.711296	−0.355648	+	0.934620i	$0.615740\pi$
$44$	0	0
$45$	−1.67989	−0.250423
$46$	−2.65626	−0.391644
$47$	−4.48205	−0.653774	−0.326887	−	0.945063i	$-0.606000\pi$
$47$	−4.48205	−0.653774	−0.326887	+	0.945063i	$0.606000\pi$
$48$	1.14896	0.165838
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	7.44193	1.04208
$52$	3.79720	0.526577
$53$	−5.09819	−0.700290	−0.350145	−	0.936695i	$-0.613868\pi$
$53$	−5.09819	−0.700290	−0.350145	+	0.936695i	$0.613868\pi$
$54$	5.37701	0.731718
$55$	0	0
$56$	1.00000	0.133631
$57$	3.07909	0.407835
$58$	−0.386922	−0.0508054
$59$	4.74643	0.617933	0.308966	−	0.951073i	$-0.400017\pi$
$59$	4.74643	0.617933	0.308966	+	0.951073i	$0.400017\pi$
$60$	1.14896	0.148330
$61$	11.6593	1.49282	0.746412	−	0.665484i	$-0.231775\pi$
$61$	11.6593	1.49282	0.746412	+	0.665484i	$0.231775\pi$
$62$	6.01867	0.764372
$63$	1.67989	0.211646
$64$	1.00000	0.125000
$65$	3.79720	0.470985
$66$	0	0
$67$	5.71316	0.697974	0.348987	−	0.937128i	$-0.386526\pi$
$67$	5.71316	0.697974	0.348987	+	0.937128i	$0.386526\pi$
$68$	6.47709	0.785463
$69$	3.05194	0.367411
$70$	1.00000	0.119523
$71$	3.33119	0.395340	0.197670	−	0.980269i	$-0.436663\pi$
$71$	3.33119	0.395340	0.197670	+	0.980269i	$0.436663\pi$
$72$	1.67989	0.197977
$73$	15.0127	1.75710	0.878552	−	0.477646i	$-0.158510\pi$
$73$	15.0127	1.75710	0.878552	+	0.477646i	$0.158510\pi$
$74$	−2.38692	−0.277474
$75$	1.14896	0.132671
$76$	2.67989	0.307404
$77$	0	0
$78$	−4.36284	−0.493994
$79$	−6.03633	−0.679141	−0.339570	−	0.940581i	$-0.610282\pi$
$79$	−6.03633	−0.679141	−0.339570	+	0.940581i	$0.610282\pi$
$80$	1.00000	0.111803
$81$	−1.13831	−0.126479
$82$	−9.47709	−1.04657
$83$	−11.1551	−1.22443	−0.612215	−	0.790691i	$-0.709721\pi$
$83$	−11.1551	−1.22443	−0.612215	+	0.790691i	$0.709721\pi$
$84$	−1.14896	−0.125362
$85$	6.47709	0.702539
$86$	4.66428	0.502962
$87$	0.444559	0.0476617
$88$	0	0
$89$	−5.87277	−0.622513	−0.311256	−	0.950326i	$-0.600750\pi$
$89$	−5.87277	−0.622513	−0.311256	+	0.950326i	$0.600750\pi$
$90$	1.67989	0.177076
$91$	−3.79720	−0.398055
$92$	2.65626	0.276934
$93$	−6.91522	−0.717075
$94$	4.48205	0.462288
$95$	2.67989	0.274951
$96$	−1.14896	−0.117265
$97$	−3.30450	−0.335522	−0.167761	−	0.985828i	$-0.553654\pi$
$97$	−3.30450	−0.335522	−0.167761	+	0.985828i	$0.553654\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−5.56420	−0.553658	−0.276829	−	0.960919i	$-0.589284\pi$
$101$	−5.56420	−0.553658	−0.276829	+	0.960919i	$0.589284\pi$
$102$	−7.44193	−0.736861
$103$	−7.95988	−0.784310	−0.392155	−	0.919899i	$-0.628270\pi$
$103$	−7.95988	−0.784310	−0.392155	+	0.919899i	$0.628270\pi$
$104$	−3.79720	−0.372346
$105$	−1.14896	−0.112127
$106$	5.09819	0.495180
$107$	−17.4244	−1.68448	−0.842241	−	0.539100i	$-0.818764\pi$
$107$	−17.4244	−1.68448	−0.842241	+	0.539100i	$0.818764\pi$
$108$	−5.37701	−0.517403
$109$	4.92398	0.471631	0.235816	−	0.971798i	$-0.424224\pi$
$109$	4.92398	0.471631	0.235816	+	0.971798i	$0.424224\pi$
$110$	0	0
$111$	2.74248	0.260305
$112$	−1.00000	−0.0944911
$113$	−11.3824	−1.07077	−0.535383	−	0.844609i	$-0.679833\pi$
$113$	−11.3824	−1.07077	−0.535383	+	0.844609i	$0.679833\pi$
$114$	−3.07909	−0.288383
$115$	2.65626	0.247698
$116$	0.386922	0.0359248
$117$	−6.37888	−0.589728
$118$	−4.74643	−0.436944
$119$	−6.47709	−0.593754
$120$	−1.14896	−0.104885
$121$	0	0
$122$	−11.6593	−1.05559
$123$	10.8888	0.981811
$124$	−6.01867	−0.540493
$125$	1.00000	0.0894427
$126$	−1.67989	−0.149656
$127$	−1.43724	−0.127534	−0.0637671	−	0.997965i	$-0.520311\pi$
$127$	−1.43724	−0.127534	−0.0637671	+	0.997965i	$0.520311\pi$
$128$	−1.00000	−0.0883883
$129$	−5.35908	−0.471841
$130$	−3.79720	−0.333037
$131$	19.5165	1.70516	0.852582	−	0.522594i	$-0.175036\pi$
$131$	19.5165	1.70516	0.852582	+	0.522594i	$0.175036\pi$
$132$	0	0
$133$	−2.67989	−0.232376
$134$	−5.71316	−0.493542
$135$	−5.37701	−0.462779
$136$	−6.47709	−0.555406
$137$	18.9809	1.62165	0.810823	−	0.585292i	$-0.199020\pi$
$137$	18.9809	1.62165	0.810823	+	0.585292i	$0.199020\pi$
$138$	−3.05194	−0.259799
$139$	6.58593	0.558611	0.279306	−	0.960202i	$-0.409896\pi$
$139$	6.58593	0.558611	0.279306	+	0.960202i	$0.409896\pi$
$140$	−1.00000	−0.0845154
$141$	−5.14970	−0.433683
$142$	−3.33119	−0.279548
$143$	0	0
$144$	−1.67989	−0.139991
$145$	0.386922	0.0321321
$146$	−15.0127	−1.24246
$147$	1.14896	0.0947647
$148$	2.38692	0.196204
$149$	10.1138	0.828554	0.414277	−	0.910151i	$-0.364034\pi$
$149$	10.1138	0.828554	0.414277	+	0.910151i	$0.364034\pi$
$150$	−1.14896	−0.0938123
$151$	−11.8335	−0.963000	−0.481500	−	0.876446i	$-0.659908\pi$
$151$	−11.8335	−0.963000	−0.481500	+	0.876446i	$0.659908\pi$
$152$	−2.67989	−0.217368
$153$	−10.8808	−0.879660
$154$	0	0
$155$	−6.01867	−0.483431
$156$	4.36284	0.349307
$157$	1.69054	0.134920	0.0674599	−	0.997722i	$-0.478511\pi$
$157$	1.69054	0.134920	0.0674599	+	0.997722i	$0.478511\pi$
$158$	6.03633	0.480225
$159$	−5.85762	−0.464540
$160$	−1.00000	−0.0790569
$161$	−2.65626	−0.209343
$162$	1.13831	0.0894341
$163$	−8.06159	−0.631432	−0.315716	−	0.948854i	$-0.602245\pi$
$163$	−8.06159	−0.631432	−0.315716	+	0.948854i	$0.602245\pi$
$164$	9.47709	0.740037
$165$	0	0
$166$	11.1551	0.865803
$167$	16.2178	1.25497	0.627487	−	0.778627i	$-0.284084\pi$
$167$	16.2178	1.25497	0.627487	+	0.778627i	$0.284084\pi$
$168$	1.14896	0.0886443
$169$	1.41876	0.109135
$170$	−6.47709	−0.496770
$171$	−4.50191	−0.344270
$172$	−4.66428	−0.355648
$173$	12.1600	0.924511	0.462255	−	0.886747i	$-0.347040\pi$
$173$	12.1600	0.924511	0.462255	+	0.886747i	$0.347040\pi$
$174$	−0.444559	−0.0337019
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	5.45347	0.409908
$178$	5.87277	0.440183
$179$	−23.3033	−1.74177	−0.870886	−	0.491486i	$-0.836454\pi$
$179$	−23.3033	−1.74177	−0.870886	+	0.491486i	$0.836454\pi$
$180$	−1.67989	−0.125211
$181$	22.1288	1.64482	0.822411	−	0.568893i	$-0.192628\pi$
$181$	22.1288	1.64482	0.822411	+	0.568893i	$0.192628\pi$
$182$	3.79720	0.281467
$183$	13.3961	0.990269
$184$	−2.65626	−0.195822
$185$	2.38692	0.175490
$186$	6.91522	0.507048
$187$	0	0
$188$	−4.48205	−0.326887
$189$	5.37701	0.391120
$190$	−2.67989	−0.194420
$191$	20.2864	1.46787	0.733936	−	0.679219i	$-0.237681\pi$
$191$	20.2864	1.46787	0.733936	+	0.679219i	$0.237681\pi$
$192$	1.14896	0.0829191
$193$	−3.84508	−0.276775	−0.138387	−	0.990378i	$-0.544192\pi$
$193$	−3.84508	−0.276775	−0.138387	+	0.990378i	$0.544192\pi$
$194$	3.30450	0.237250
$195$	4.36284	0.312429
$196$	1.00000	0.0714286
$197$	11.8159	0.841846	0.420923	−	0.907096i	$-0.361706\pi$
$197$	11.8159	0.841846	0.420923	+	0.907096i	$0.361706\pi$
$198$	0	0
$199$	12.5496	0.889617	0.444809	−	0.895626i	$-0.353272\pi$
$199$	12.5496	0.889617	0.444809	+	0.895626i	$0.353272\pi$
$200$	−1.00000	−0.0707107
$201$	6.56420	0.463003
$202$	5.56420	0.391496
$203$	−0.386922	−0.0271566
$204$	7.44193	0.521039
$205$	9.47709	0.661909
$206$	7.95988	0.554591
$207$	−4.46222	−0.310146
$208$	3.79720	0.263289
$209$	0	0
$210$	1.14896	0.0792859
$211$	3.99973	0.275353	0.137676	−	0.990477i	$-0.456037\pi$
$211$	3.99973	0.275353	0.137676	+	0.990477i	$0.456037\pi$
$212$	−5.09819	−0.350145
$213$	3.82741	0.262250
$214$	17.4244	1.19111
$215$	−4.66428	−0.318101
$216$	5.37701	0.365859
$217$	6.01867	0.408574
$218$	−4.92398	−0.333494
$219$	17.2490	1.16558
$220$	0	0
$221$	24.5948	1.65443
$222$	−2.74248	−0.184063
$223$	17.8782	1.19721	0.598605	−	0.801044i	$-0.295722\pi$
$223$	17.8782	1.19721	0.598605	+	0.801044i	$0.295722\pi$
$224$	1.00000	0.0668153
$225$	−1.67989	−0.111993
$226$	11.3824	0.757146
$227$	−5.76204	−0.382440	−0.191220	−	0.981547i	$-0.561244\pi$
$227$	−5.76204	−0.382440	−0.191220	+	0.981547i	$0.561244\pi$
$228$	3.07909	0.203918
$229$	−8.57617	−0.566729	−0.283365	−	0.959012i	$-0.591451\pi$
$229$	−8.57617	−0.566729	−0.283365	+	0.959012i	$0.591451\pi$
$230$	−2.65626	−0.175149
$231$	0	0
$232$	−0.386922	−0.0254027
$233$	14.4011	0.943445	0.471723	−	0.881747i	$-0.343632\pi$
$233$	14.4011	0.943445	0.471723	+	0.881747i	$0.343632\pi$
$234$	6.37888	0.417000
$235$	−4.48205	−0.292377
$236$	4.74643	0.308966
$237$	−6.93551	−0.450510
$238$	6.47709	0.419848
$239$	−24.0096	−1.55305	−0.776527	−	0.630084i	$-0.783020\pi$
$239$	−24.0096	−1.55305	−0.776527	+	0.630084i	$0.783020\pi$
$240$	1.14896	0.0741651
$241$	−16.7641	−1.07987	−0.539935	−	0.841707i	$-0.681551\pi$
$241$	−16.7641	−1.07987	−0.539935	+	0.841707i	$0.681551\pi$
$242$	0	0
$243$	14.8232	0.950906
$244$	11.6593	0.746412
$245$	1.00000	0.0638877
$246$	−10.8888	−0.694245
$247$	10.1761	0.647489
$248$	6.01867	0.382186
$249$	−12.8168	−0.812229
$250$	−1.00000	−0.0632456
$251$	−2.42488	−0.153057	−0.0765286	−	0.997067i	$-0.524384\pi$
$251$	−2.42488	−0.153057	−0.0765286	+	0.997067i	$0.524384\pi$
$252$	1.67989	0.105823
$253$	0	0
$254$	1.43724	0.0901803
$255$	7.44193	0.466032
$256$	1.00000	0.0625000
$257$	5.18367	0.323348	0.161674	−	0.986844i	$-0.448311\pi$
$257$	5.18367	0.323348	0.161674	+	0.986844i	$0.448311\pi$
$258$	5.35908	0.333642
$259$	−2.38692	−0.148316
$260$	3.79720	0.235493
$261$	−0.649986	−0.0402331
$262$	−19.5165	−1.20573
$263$	20.3770	1.25650	0.628250	−	0.778011i	$-0.283771\pi$
$263$	20.3770	1.25650	0.628250	+	0.778011i	$0.283771\pi$
$264$	0	0
$265$	−5.09819	−0.313179
$266$	2.67989	0.164314
$267$	−6.74759	−0.412946
$268$	5.71316	0.348987
$269$	−11.4504	−0.698143	−0.349072	−	0.937096i	$-0.613503\pi$
$269$	−11.4504	−0.698143	−0.349072	+	0.937096i	$0.613503\pi$
$270$	5.37701	0.327234
$271$	14.8199	0.900247	0.450124	−	0.892966i	$-0.351380\pi$
$271$	14.8199	0.900247	0.450124	+	0.892966i	$0.351380\pi$
$272$	6.47709	0.392731
$273$	−4.36284	−0.264051
$274$	−18.9809	−1.14668
$275$	0	0
$276$	3.05194	0.183705
$277$	24.5158	1.47301	0.736504	−	0.676433i	$-0.236475\pi$
$277$	24.5158	1.47301	0.736504	+	0.676433i	$0.236475\pi$
$278$	−6.58593	−0.394998
$279$	10.1107	0.605311
$280$	1.00000	0.0597614
$281$	−28.5929	−1.70571	−0.852856	−	0.522146i	$-0.825132\pi$
$281$	−28.5929	−1.70571	−0.852856	+	0.522146i	$0.825132\pi$
$282$	5.14970	0.306660
$283$	−5.19494	−0.308807	−0.154404	−	0.988008i	$-0.549346\pi$
$283$	−5.19494	−0.308807	−0.154404	+	0.988008i	$0.549346\pi$
$284$	3.33119	0.197670
$285$	3.07909	0.182389
$286$	0	0
$287$	−9.47709	−0.559415
$288$	1.67989	0.0989884
$289$	24.9527	1.46781
$290$	−0.386922	−0.0227209
$291$	−3.79675	−0.222569
$292$	15.0127	0.878552
$293$	18.2205	1.06445	0.532225	−	0.846603i	$-0.321356\pi$
$293$	18.2205	1.06445	0.532225	+	0.846603i	$0.321356\pi$
$294$	−1.14896	−0.0670088
$295$	4.74643	0.276348
$296$	−2.38692	−0.138737
$297$	0	0
$298$	−10.1138	−0.585876
$299$	10.0864	0.583310
$300$	1.14896	0.0663353
$301$	4.66428	0.268845
$302$	11.8335	0.680944
$303$	−6.39305	−0.367271
$304$	2.67989	0.153702
$305$	11.6593	0.667611
$306$	10.8808	0.622013
$307$	−10.9710	−0.626146	−0.313073	−	0.949729i	$-0.601358\pi$
$307$	−10.9710	−0.626146	−0.313073	+	0.949729i	$0.601358\pi$
$308$	0	0
$309$	−9.14559	−0.520275
$310$	6.01867	0.341837
$311$	20.8687	1.18335	0.591677	−	0.806175i	$-0.298466\pi$
$311$	20.8687	1.18335	0.591677	+	0.806175i	$0.298466\pi$
$312$	−4.36284	−0.246997
$313$	8.64805	0.488817	0.244408	−	0.969672i	$-0.421406\pi$
$313$	8.64805	0.488817	0.244408	+	0.969672i	$0.421406\pi$
$314$	−1.69054	−0.0954026
$315$	1.67989	0.0946510
$316$	−6.03633	−0.339570
$317$	3.53662	0.198636	0.0993182	−	0.995056i	$-0.468334\pi$
$317$	3.53662	0.198636	0.0993182	+	0.995056i	$0.468334\pi$
$318$	5.85762	0.328479
$319$	0	0
$320$	1.00000	0.0559017
$321$	−20.0200	−1.11741
$322$	2.65626	0.148028
$323$	17.3579	0.965818
$324$	−1.13831	−0.0632395
$325$	3.79720	0.210631
$326$	8.06159	0.446490
$327$	5.65746	0.312858
$328$	−9.47709	−0.523285
$329$	4.48205	0.247103
$330$	0	0
$331$	10.1966	0.560458	0.280229	−	0.959933i	$-0.409590\pi$
$331$	10.1966	0.560458	0.280229	+	0.959933i	$0.409590\pi$
$332$	−11.1551	−0.612215
$333$	−4.00976	−0.219734
$334$	−16.2178	−0.887400
$335$	5.71316	0.312143
$336$	−1.14896	−0.0626810
$337$	−14.1072	−0.768469	−0.384234	−	0.923236i	$-0.625535\pi$
$337$	−14.1072	−0.768469	−0.384234	+	0.923236i	$0.625535\pi$
$338$	−1.41876	−0.0771702
$339$	−13.0779	−0.710296
$340$	6.47709	0.351270
$341$	0	0
$342$	4.50191	0.243436
$343$	−1.00000	−0.0539949
$344$	4.66428	0.251481
$345$	3.05194	0.164311
$346$	−12.1600	−0.653728
$347$	4.16926	0.223817	0.111909	−	0.993718i	$-0.464304\pi$
$347$	4.16926	0.223817	0.111909	+	0.993718i	$0.464304\pi$
$348$	0.444559	0.0238308
$349$	8.03734	0.430229	0.215114	−	0.976589i	$-0.430988\pi$
$349$	8.03734	0.430229	0.215114	+	0.976589i	$0.430988\pi$
$350$	1.00000	0.0534522
$351$	−20.4176	−1.08981
$352$	0	0
$353$	15.7957	0.840723	0.420361	−	0.907357i	$-0.361903\pi$
$353$	15.7957	0.840723	0.420361	+	0.907357i	$0.361903\pi$
$354$	−5.45347	−0.289848
$355$	3.33119	0.176801
$356$	−5.87277	−0.311256
$357$	−7.44193	−0.393869
$358$	23.3033	1.23162
$359$	6.64022	0.350458	0.175229	−	0.984528i	$-0.443933\pi$
$359$	6.64022	0.350458	0.175229	+	0.984528i	$0.443933\pi$
$360$	1.67989	0.0885379
$361$	−11.8182	−0.622010
$362$	−22.1288	−1.16307
$363$	0	0
$364$	−3.79720	−0.199028
$365$	15.0127	0.785801
$366$	−13.3961	−0.700226
$367$	16.6306	0.868108	0.434054	−	0.900887i	$-0.357083\pi$
$367$	16.6306	0.868108	0.434054	+	0.900887i	$0.357083\pi$
$368$	2.65626	0.138467
$369$	−15.9205	−0.828786
$370$	−2.38692	−0.124090
$371$	5.09819	0.264685
$372$	−6.91522	−0.358537
$373$	−7.30477	−0.378227	−0.189113	−	0.981955i	$-0.560561\pi$
$373$	−7.30477	−0.378227	−0.189113	+	0.981955i	$0.560561\pi$
$374$	0	0
$375$	1.14896	0.0593321
$376$	4.48205	0.231144
$377$	1.46922	0.0756688
$378$	−5.37701	−0.276564
$379$	−13.2798	−0.682138	−0.341069	−	0.940038i	$-0.610789\pi$
$379$	−13.2798	−0.682138	−0.341069	+	0.940038i	$0.610789\pi$
$380$	2.67989	0.137475
$381$	−1.65133	−0.0846002
$382$	−20.2864	−1.03794
$383$	17.1065	0.874100	0.437050	−	0.899437i	$-0.356023\pi$
$383$	17.1065	0.874100	0.437050	+	0.899437i	$0.356023\pi$
$384$	−1.14896	−0.0586327
$385$	0	0
$386$	3.84508	0.195709
$387$	7.83547	0.398299
$388$	−3.30450	−0.167761
$389$	8.54697	0.433348	0.216674	−	0.976244i	$-0.430479\pi$
$389$	8.54697	0.433348	0.216674	+	0.976244i	$0.430479\pi$
$390$	−4.36284	−0.220921
$391$	17.2049	0.870087
$392$	−1.00000	−0.0505076
$393$	22.4237	1.13113
$394$	−11.8159	−0.595275
$395$	−6.03633	−0.303721
$396$	0	0
$397$	17.0472	0.855572	0.427786	−	0.903880i	$-0.359294\pi$
$397$	17.0472	0.855572	0.427786	+	0.903880i	$0.359294\pi$
$398$	−12.5496	−0.629054
$399$	−3.07909	−0.154147
$400$	1.00000	0.0500000
$401$	−2.71053	−0.135357	−0.0676787	−	0.997707i	$-0.521559\pi$
$401$	−2.71053	−0.135357	−0.0676787	+	0.997707i	$0.521559\pi$
$402$	−6.56420	−0.327392
$403$	−22.8541	−1.13844
$404$	−5.56420	−0.276829
$405$	−1.13831	−0.0565631
$406$	0.386922	0.0192026
$407$	0	0
$408$	−7.44193	−0.368430
$409$	29.3149	1.44953	0.724763	−	0.688998i	$-0.241949\pi$
$409$	29.3149	1.44953	0.724763	+	0.688998i	$0.241949\pi$
$410$	−9.47709	−0.468040
$411$	21.8083	1.07572
$412$	−7.95988	−0.392155
$413$	−4.74643	−0.233557
$414$	4.46222	0.219306
$415$	−11.1551	−0.547582
$416$	−3.79720	−0.186173
$417$	7.56698	0.370557
$418$	0	0
$419$	38.3776	1.87487	0.937434	−	0.348162i	$-0.113194\pi$
$419$	38.3776	1.87487	0.937434	+	0.348162i	$0.113194\pi$
$420$	−1.14896	−0.0560636
$421$	−37.1605	−1.81109	−0.905545	−	0.424249i	$-0.860538\pi$
$421$	−37.1605	−1.81109	−0.905545	+	0.424249i	$0.860538\pi$
$422$	−3.99973	−0.194704
$423$	7.52934	0.366089
$424$	5.09819	0.247590
$425$	6.47709	0.314185
$426$	−3.82741	−0.185439
$427$	−11.6593	−0.564234
$428$	−17.4244	−0.842241
$429$	0	0
$430$	4.66428	0.224932
$431$	−30.7450	−1.48093	−0.740467	−	0.672093i	$-0.765395\pi$
$431$	−30.7450	−1.48093	−0.740467	+	0.672093i	$0.765395\pi$
$432$	−5.37701	−0.258702
$433$	−8.55355	−0.411057	−0.205529	−	0.978651i	$-0.565891\pi$
$433$	−8.55355	−0.411057	−0.205529	+	0.978651i	$0.565891\pi$
$434$	−6.01867	−0.288905
$435$	0.444559	0.0213150
$436$	4.92398	0.235816
$437$	7.11849	0.340523
$438$	−17.2490	−0.824190
$439$	−24.5405	−1.17126	−0.585628	−	0.810580i	$-0.699152\pi$
$439$	−24.5405	−1.17126	−0.585628	+	0.810580i	$0.699152\pi$
$440$	0	0
$441$	−1.67989	−0.0799947
$442$	−24.5948	−1.16986
$443$	26.9683	1.28130	0.640652	−	0.767831i	$-0.278664\pi$
$443$	26.9683	1.28130	0.640652	+	0.767831i	$0.278664\pi$
$444$	2.74248	0.130152
$445$	−5.87277	−0.278396
$446$	−17.8782	−0.846555
$447$	11.6204	0.549624
$448$	−1.00000	−0.0472456
$449$	−35.1803	−1.66026	−0.830131	−	0.557569i	$-0.811734\pi$
$449$	−35.1803	−1.66026	−0.830131	+	0.557569i	$0.811734\pi$
$450$	1.67989	0.0791907
$451$	0	0
$452$	−11.3824	−0.535383
$453$	−13.5963	−0.638809
$454$	5.76204	0.270426
$455$	−3.79720	−0.178016
$456$	−3.07909	−0.144191
$457$	20.4364	0.955974	0.477987	−	0.878367i	$-0.341367\pi$
$457$	20.4364	0.955974	0.477987	+	0.878367i	$0.341367\pi$
$458$	8.57617	0.400738
$459$	−34.8274	−1.62560
$460$	2.65626	0.123849
$461$	5.59541	0.260604	0.130302	−	0.991474i	$-0.458405\pi$
$461$	5.59541	0.260604	0.130302	+	0.991474i	$0.458405\pi$
$462$	0	0
$463$	−29.5079	−1.37135	−0.685674	−	0.727909i	$-0.740492\pi$
$463$	−29.5079	−1.37135	−0.685674	+	0.727909i	$0.740492\pi$
$464$	0.386922	0.0179624
$465$	−6.91522	−0.320686
$466$	−14.4011	−0.667117
$467$	31.3408	1.45028	0.725140	−	0.688601i	$-0.241775\pi$
$467$	31.3408	1.45028	0.725140	+	0.688601i	$0.241775\pi$
$468$	−6.37888	−0.294864
$469$	−5.71316	−0.263809
$470$	4.48205	0.206741
$471$	1.94236	0.0894994
$472$	−4.74643	−0.218472
$473$	0	0
$474$	6.93551	0.318559
$475$	2.67989	0.122962
$476$	−6.47709	−0.296877
$477$	8.56439	0.392136
$478$	24.0096	1.09817
$479$	6.75139	0.308479	0.154239	−	0.988034i	$-0.450707\pi$
$479$	6.75139	0.308479	0.154239	+	0.988034i	$0.450707\pi$
$480$	−1.14896	−0.0524427
$481$	9.06363	0.413266
$482$	16.7641	0.763584
$483$	−3.05194	−0.138868
$484$	0	0
$485$	−3.30450	−0.150050
$486$	−14.8232	−0.672392
$487$	−25.3561	−1.14900	−0.574498	−	0.818506i	$-0.694803\pi$
$487$	−25.3561	−1.14900	−0.574498	+	0.818506i	$0.694803\pi$
$488$	−11.6593	−0.527793
$489$	−9.26245	−0.418863
$490$	−1.00000	−0.0451754
$491$	19.7873	0.892987	0.446493	−	0.894787i	$-0.352673\pi$
$491$	19.7873	0.892987	0.446493	+	0.894787i	$0.352673\pi$
$492$	10.8888	0.490905
$493$	2.50613	0.112870
$494$	−10.1761	−0.457844
$495$	0	0
$496$	−6.01867	−0.270246
$497$	−3.33119	−0.149424
$498$	12.8168	0.574333
$499$	−26.7727	−1.19851	−0.599255	−	0.800559i	$-0.704536\pi$
$499$	−26.7727	−1.19851	−0.599255	+	0.800559i	$0.704536\pi$
$500$	1.00000	0.0447214
$501$	18.6337	0.832490
$502$	2.42488	0.108228
$503$	−0.355087	−0.0158326	−0.00791628	−	0.999969i	$-0.502520\pi$
$503$	−0.355087	−0.0158326	−0.00791628	+	0.999969i	$0.502520\pi$
$504$	−1.67989	−0.0748282
$505$	−5.56420	−0.247604
$506$	0	0
$507$	1.63010	0.0723951
$508$	−1.43724	−0.0637671
$509$	−23.9743	−1.06264	−0.531321	−	0.847171i	$-0.678304\pi$
$509$	−23.9743	−1.06264	−0.531321	+	0.847171i	$0.678304\pi$
$510$	−7.44193	−0.329534
$511$	−15.0127	−0.664123
$512$	−1.00000	−0.0441942
$513$	−14.4098	−0.636208
$514$	−5.18367	−0.228642
$515$	−7.95988	−0.350754
$516$	−5.35908	−0.235920
$517$	0	0
$518$	2.38692	0.104875
$519$	13.9714	0.613277
$520$	−3.79720	−0.166518
$521$	−7.25606	−0.317894	−0.158947	−	0.987287i	$-0.550810\pi$
$521$	−7.25606	−0.317894	−0.158947	+	0.987287i	$0.550810\pi$
$522$	0.649986	0.0284491
$523$	−17.1723	−0.750893	−0.375447	−	0.926844i	$-0.622511\pi$
$523$	−17.1723	−0.750893	−0.375447	+	0.926844i	$0.622511\pi$
$524$	19.5165	0.852582
$525$	−1.14896	−0.0501448
$526$	−20.3770	−0.888480
$527$	−38.9835	−1.69815
$528$	0	0
$529$	−15.9443	−0.693229
$530$	5.09819	0.221451
$531$	−7.97348	−0.346019
$532$	−2.67989	−0.116188
$533$	35.9865	1.55875
$534$	6.74759	0.291997
$535$	−17.4244	−0.753324
$536$	−5.71316	−0.246771
$537$	−26.7746	−1.15541
$538$	11.4504	0.493662
$539$	0	0
$540$	−5.37701	−0.231390
$541$	−23.7882	−1.02273	−0.511366	−	0.859363i	$-0.670860\pi$
$541$	−23.7882	−1.02273	−0.511366	+	0.859363i	$0.670860\pi$
$542$	−14.8199	−0.636571
$543$	25.4252	1.09110
$544$	−6.47709	−0.277703
$545$	4.92398	0.210920
$546$	4.36284	0.186712
$547$	−39.2442	−1.67796	−0.838981	−	0.544161i	$-0.816848\pi$
$547$	−39.2442	−1.67796	−0.838981	+	0.544161i	$0.816848\pi$
$548$	18.9809	0.810823
$549$	−19.5864	−0.835926
$550$	0	0
$551$	1.03691	0.0441738
$552$	−3.05194	−0.129899
$553$	6.03633	0.256691
$554$	−24.5158	−1.04157
$555$	2.74248	0.116412
$556$	6.58593	0.279306
$557$	43.0812	1.82541	0.912705	−	0.408618i	$-0.133989\pi$
$557$	43.0812	1.82541	0.912705	+	0.408618i	$0.133989\pi$
$558$	−10.1107	−0.428020
$559$	−17.7112	−0.749105
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	28.5929	1.20612
$563$	40.8534	1.72176	0.860882	−	0.508804i	$-0.169912\pi$
$563$	40.8534	1.72176	0.860882	+	0.508804i	$0.169912\pi$
$564$	−5.14970	−0.216841
$565$	−11.3824	−0.478861
$566$	5.19494	0.218360
$567$	1.13831	0.0478045
$568$	−3.33119	−0.139774
$569$	−4.46455	−0.187164	−0.0935818	−	0.995612i	$-0.529832\pi$
$569$	−4.46455	−0.187164	−0.0935818	+	0.995612i	$0.529832\pi$
$570$	−3.07909	−0.128969
$571$	31.4272	1.31519	0.657593	−	0.753373i	$-0.271575\pi$
$571$	31.4272	1.31519	0.657593	+	0.753373i	$0.271575\pi$
$572$	0	0
$573$	23.3083	0.973717
$574$	9.47709	0.395566
$575$	2.65626	0.110774
$576$	−1.67989	−0.0699953
$577$	8.24246	0.343138	0.171569	−	0.985172i	$-0.445116\pi$
$577$	8.24246	0.343138	0.171569	+	0.985172i	$0.445116\pi$
$578$	−24.9527	−1.03790
$579$	−4.41784	−0.183599
$580$	0.386922	0.0160661
$581$	11.1551	0.462791
$582$	3.79675	0.157380
$583$	0	0
$584$	−15.0127	−0.621230
$585$	−6.37888	−0.263734
$586$	−18.2205	−0.752680
$587$	1.10956	0.0457966	0.0228983	−	0.999738i	$-0.492711\pi$
$587$	1.10956	0.0457966	0.0228983	+	0.999738i	$0.492711\pi$
$588$	1.14896	0.0473824
$589$	−16.1294	−0.664599
$590$	−4.74643	−0.195408
$591$	13.5760	0.558441
$592$	2.38692	0.0981019
$593$	46.0484	1.89098	0.945491	−	0.325648i	$-0.105582\pi$
$593$	46.0484	1.89098	0.945491	+	0.325648i	$0.105582\pi$
$594$	0	0
$595$	−6.47709	−0.265535
$596$	10.1138	0.414277
$597$	14.4190	0.590130
$598$	−10.0864	−0.412462
$599$	−29.5604	−1.20781	−0.603903	−	0.797058i	$-0.706389\pi$
$599$	−29.5604	−1.20781	−0.603903	+	0.797058i	$0.706389\pi$
$600$	−1.14896	−0.0469061
$601$	5.68051	0.231713	0.115856	−	0.993266i	$-0.463039\pi$
$601$	5.68051	0.231713	0.115856	+	0.993266i	$0.463039\pi$
$602$	−4.66428	−0.190102
$603$	−9.59747	−0.390839
$604$	−11.8335	−0.481500
$605$	0	0
$606$	6.39305	0.259700
$607$	24.3166	0.986980	0.493490	−	0.869751i	$-0.335721\pi$
$607$	24.3166	0.986980	0.493490	+	0.869751i	$0.335721\pi$
$608$	−2.67989	−0.108684
$609$	−0.444559	−0.0180144
$610$	−11.6593	−0.472072
$611$	−17.0193	−0.688525
$612$	−10.8808	−0.439830
$613$	30.4381	1.22938	0.614692	−	0.788767i	$-0.289280\pi$
$613$	30.4381	1.22938	0.614692	+	0.788767i	$0.289280\pi$
$614$	10.9710	0.442752
$615$	10.8888	0.439079
$616$	0	0
$617$	33.4297	1.34583	0.672914	−	0.739721i	$-0.265043\pi$
$617$	33.4297	1.34583	0.672914	+	0.739721i	$0.265043\pi$
$618$	9.14559	0.367890
$619$	−4.33525	−0.174248	−0.0871241	−	0.996197i	$-0.527768\pi$
$619$	−4.33525	−0.174248	−0.0871241	+	0.996197i	$0.527768\pi$
$620$	−6.01867	−0.241716
$621$	−14.2827	−0.573147
$622$	−20.8687	−0.836758
$623$	5.87277	0.235288
$624$	4.36284	0.174653
$625$	1.00000	0.0400000
$626$	−8.64805	−0.345646
$627$	0	0
$628$	1.69054	0.0674599
$629$	15.4603	0.616443
$630$	−1.67989	−0.0669284
$631$	−15.4234	−0.613998	−0.306999	−	0.951710i	$-0.599325\pi$
$631$	−15.4234	−0.613998	−0.306999	+	0.951710i	$0.599325\pi$
$632$	6.03633	0.240113
$633$	4.59554	0.182656
$634$	−3.53662	−0.140457
$635$	−1.43724	−0.0570350
$636$	−5.85762	−0.232270
$637$	3.79720	0.150451
$638$	0	0
$639$	−5.59603	−0.221376
$640$	−1.00000	−0.0395285
$641$	35.5278	1.40327	0.701633	−	0.712539i	$-0.252455\pi$
$641$	35.5278	1.40327	0.701633	+	0.712539i	$0.252455\pi$
$642$	20.0200	0.790126
$643$	49.2065	1.94051	0.970257	−	0.242079i	$-0.0778292\pi$
$643$	49.2065	1.94051	0.970257	+	0.242079i	$0.0778292\pi$
$644$	−2.65626	−0.104671
$645$	−5.35908	−0.211013
$646$	−17.3579	−0.682937
$647$	11.9820	0.471063	0.235531	−	0.971867i	$-0.424317\pi$
$647$	11.9820	0.471063	0.235531	+	0.971867i	$0.424317\pi$
$648$	1.13831	0.0447171
$649$	0	0
$650$	−3.79720	−0.148939
$651$	6.91522	0.271029
$652$	−8.06159	−0.315716
$653$	−50.4638	−1.97480	−0.987401	−	0.158241i	$-0.949418\pi$
$653$	−50.4638	−1.97480	−0.987401	+	0.158241i	$0.949418\pi$
$654$	−5.65746	−0.221224
$655$	19.5165	0.762572
$656$	9.47709	0.370018
$657$	−25.2197	−0.983913
$658$	−4.48205	−0.174728
$659$	29.2967	1.14124	0.570618	−	0.821216i	$-0.306704\pi$
$659$	29.2967	1.14124	0.570618	+	0.821216i	$0.306704\pi$
$660$	0	0
$661$	35.0445	1.36307	0.681537	−	0.731784i	$-0.261312\pi$
$661$	35.0445	1.36307	0.681537	+	0.731784i	$0.261312\pi$
$662$	−10.1966	−0.396304
$663$	28.2585	1.09747
$664$	11.1551	0.432901
$665$	−2.67989	−0.103922
$666$	4.00976	0.155375
$667$	1.02777	0.0397953
$668$	16.2178	0.627487
$669$	20.5413	0.794173
$670$	−5.71316	−0.220719
$671$	0	0
$672$	1.14896	0.0443221
$673$	37.4744	1.44453	0.722265	−	0.691616i	$-0.243101\pi$
$673$	37.4744	1.44453	0.722265	+	0.691616i	$0.243101\pi$
$674$	14.1072	0.543390
$675$	−5.37701	−0.206961
$676$	1.41876	0.0545676
$677$	8.42279	0.323714	0.161857	−	0.986814i	$-0.448252\pi$
$677$	8.42279	0.323714	0.161857	+	0.986814i	$0.448252\pi$
$678$	13.0779	0.502255
$679$	3.30450	0.126815
$680$	−6.47709	−0.248385
$681$	−6.62036	−0.253693
$682$	0	0
$683$	14.9986	0.573904	0.286952	−	0.957945i	$-0.407358\pi$
$683$	14.9986	0.573904	0.286952	+	0.957945i	$0.407358\pi$
$684$	−4.50191	−0.172135
$685$	18.9809	0.725222
$686$	1.00000	0.0381802
$687$	−9.85369	−0.375942
$688$	−4.66428	−0.177824
$689$	−19.3589	−0.737514
$690$	−3.05194	−0.116185
$691$	43.8937	1.66980	0.834898	−	0.550404i	$-0.185526\pi$
$691$	43.8937	1.66980	0.834898	+	0.550404i	$0.185526\pi$
$692$	12.1600	0.462255
$693$	0	0
$694$	−4.16926	−0.158263
$695$	6.58593	0.249819
$696$	−0.444559	−0.0168510
$697$	61.3840	2.32508
$698$	−8.03734	−0.304218
$699$	16.5463	0.625837
$700$	−1.00000	−0.0377964
$701$	−43.3836	−1.63857	−0.819287	−	0.573383i	$-0.805631\pi$
$701$	−43.3836	−1.63857	−0.819287	+	0.573383i	$0.805631\pi$
$702$	20.4176	0.770613
$703$	6.39668	0.241256
$704$	0	0
$705$	−5.14970	−0.193949
$706$	−15.7957	−0.594481
$707$	5.56420	0.209263
$708$	5.45347	0.204954
$709$	6.61088	0.248277	0.124138	−	0.992265i	$-0.460383\pi$
$709$	6.61088	0.248277	0.124138	+	0.992265i	$0.460383\pi$
$710$	−3.33119	−0.125017
$711$	10.1404	0.380294
$712$	5.87277	0.220091
$713$	−15.9872	−0.598724
$714$	7.44193	0.278507
$715$	0	0
$716$	−23.3033	−0.870886
$717$	−27.5861	−1.03022
$718$	−6.64022	−0.247811
$719$	−27.5111	−1.02599	−0.512995	−	0.858392i	$-0.671464\pi$
$719$	−27.5111	−1.02599	−0.512995	+	0.858392i	$0.671464\pi$
$720$	−1.67989	−0.0626057
$721$	7.95988	0.296441
$722$	11.8182	0.439828
$723$	−19.2613	−0.716335
$724$	22.1288	0.822411
$725$	0.386922	0.0143699
$726$	0	0
$727$	−13.3895	−0.496590	−0.248295	−	0.968685i	$-0.579870\pi$
$727$	−13.3895	−0.496590	−0.248295	+	0.968685i	$0.579870\pi$
$728$	3.79720	0.140734
$729$	20.4462	0.757265
$730$	−15.0127	−0.555645
$731$	−30.2110	−1.11739
$732$	13.3961	0.495135
$733$	−45.9046	−1.69552	−0.847762	−	0.530376i	$-0.822051\pi$
$733$	−45.9046	−1.69552	−0.847762	+	0.530376i	$0.822051\pi$
$734$	−16.6306	−0.613845
$735$	1.14896	0.0423801
$736$	−2.65626	−0.0979111
$737$	0	0
$738$	15.9205	0.586040
$739$	−15.5601	−0.572388	−0.286194	−	0.958172i	$-0.592390\pi$
$739$	−15.5601	−0.572388	−0.286194	+	0.958172i	$0.592390\pi$
$740$	2.38692	0.0877450
$741$	11.6919	0.429514
$742$	−5.09819	−0.187160
$743$	31.9688	1.17282	0.586411	−	0.810014i	$-0.300541\pi$
$743$	31.9688	1.17282	0.586411	+	0.810014i	$0.300541\pi$
$744$	6.91522	0.253524
$745$	10.1138	0.370541
$746$	7.30477	0.267447
$747$	18.7393	0.685635
$748$	0	0
$749$	17.4244	0.636675
$750$	−1.14896	−0.0419541
$751$	31.7056	1.15695	0.578476	−	0.815699i	$-0.303647\pi$
$751$	31.7056	1.15695	0.578476	+	0.815699i	$0.303647\pi$
$752$	−4.48205	−0.163443
$753$	−2.78610	−0.101531
$754$	−1.46922	−0.0535059
$755$	−11.8335	−0.430667
$756$	5.37701	0.195560
$757$	−22.3306	−0.811620	−0.405810	−	0.913957i	$-0.633011\pi$
$757$	−22.3306	−0.811620	−0.405810	+	0.913957i	$0.633011\pi$
$758$	13.2798	0.482344
$759$	0	0
$760$	−2.67989	−0.0972098
$761$	17.3657	0.629508	0.314754	−	0.949173i	$-0.398078\pi$
$761$	17.3657	0.629508	0.314754	+	0.949173i	$0.398078\pi$
$762$	1.65133	0.0598214
$763$	−4.92398	−0.178260
$764$	20.2864	0.733936
$765$	−10.8808	−0.393396
$766$	−17.1065	−0.618082
$767$	18.0232	0.650779
$768$	1.14896	0.0414596
$769$	−36.3673	−1.31144	−0.655720	−	0.755004i	$-0.727635\pi$
$769$	−36.3673	−1.31144	−0.655720	+	0.755004i	$0.727635\pi$
$770$	0	0
$771$	5.95584	0.214494
$772$	−3.84508	−0.138387
$773$	−53.0722	−1.90887	−0.954437	−	0.298413i	$-0.903543\pi$
$773$	−53.0722	−1.90887	−0.954437	+	0.298413i	$0.903543\pi$
$774$	−7.83547	−0.281640
$775$	−6.01867	−0.216197
$776$	3.30450	0.118625
$777$	−2.74248	−0.0983860
$778$	−8.54697	−0.306424
$779$	25.3975	0.909962
$780$	4.36284	0.156215
$781$	0	0
$782$	−17.2049	−0.615244
$783$	−2.08048	−0.0743505
$784$	1.00000	0.0357143
$785$	1.69054	0.0603379
$786$	−22.4237	−0.799826
$787$	−38.3366	−1.36655	−0.683276	−	0.730160i	$-0.739445\pi$
$787$	−38.3366	−1.36655	−0.683276	+	0.730160i	$0.739445\pi$
$788$	11.8159	0.420923
$789$	23.4124	0.833503
$790$	6.03633	0.214763
$791$	11.3824	0.404711
$792$	0	0
$793$	44.2728	1.57217
$794$	−17.0472	−0.604981
$795$	−5.85762	−0.207748
$796$	12.5496	0.444809
$797$	−30.5021	−1.08044	−0.540219	−	0.841524i	$-0.681659\pi$
$797$	−30.5021	−1.08044	−0.540219	+	0.841524i	$0.681659\pi$
$798$	3.07909	0.108999
$799$	−29.0306	−1.02703
$800$	−1.00000	−0.0353553
$801$	9.86560	0.348584
$802$	2.71053	0.0957121
$803$	0	0
$804$	6.56420	0.231501
$805$	−2.65626	−0.0936209
$806$	22.8541	0.805002
$807$	−13.1561	−0.463116
$808$	5.56420	0.195748
$809$	−33.8990	−1.19183	−0.595913	−	0.803049i	$-0.703210\pi$
$809$	−33.8990	−1.19183	−0.595913	+	0.803049i	$0.703210\pi$
$810$	1.13831	0.0399962
$811$	42.9037	1.50655	0.753276	−	0.657705i	$-0.228473\pi$
$811$	42.9037	1.50655	0.753276	+	0.657705i	$0.228473\pi$
$812$	−0.386922	−0.0135783
$813$	17.0275	0.597182
$814$	0	0
$815$	−8.06159	−0.282385
$816$	7.44193	0.260520
$817$	−12.4998	−0.437311
$818$	−29.3149	−1.02497
$819$	6.37888	0.222896
$820$	9.47709	0.330954
$821$	50.2053	1.75218	0.876088	−	0.482151i	$-0.160144\pi$
$821$	50.2053	1.75218	0.876088	+	0.482151i	$0.160144\pi$
$822$	−21.8083	−0.760651
$823$	−19.9384	−0.695010	−0.347505	−	0.937678i	$-0.612971\pi$
$823$	−19.9384	−0.695010	−0.347505	+	0.937678i	$0.612971\pi$
$824$	7.95988	0.277296
$825$	0	0
$826$	4.74643	0.165149
$827$	−34.0394	−1.18366	−0.591832	−	0.806061i	$-0.701595\pi$
$827$	−34.0394	−1.18366	−0.591832	+	0.806061i	$0.701595\pi$
$828$	−4.46222	−0.155073
$829$	−12.9629	−0.450222	−0.225111	−	0.974333i	$-0.572274\pi$
$829$	−12.9629	−0.450222	−0.225111	+	0.974333i	$0.572274\pi$
$830$	11.1551	0.387199
$831$	28.1676	0.977125
$832$	3.79720	0.131644
$833$	6.47709	0.224418
$834$	−7.56698	−0.262023
$835$	16.2178	0.561241
$836$	0	0
$837$	32.3624	1.11861
$838$	−38.3776	−1.32573
$839$	−14.4663	−0.499430	−0.249715	−	0.968319i	$-0.580337\pi$
$839$	−14.4663	−0.499430	−0.249715	+	0.968319i	$0.580337\pi$
$840$	1.14896	0.0396429
$841$	−28.8503	−0.994838
$842$	37.1605	1.28063
$843$	−32.8522	−1.13149
$844$	3.99973	0.137676
$845$	1.41876	0.0488067
$846$	−7.52934	−0.258864
$847$	0	0
$848$	−5.09819	−0.175073
$849$	−5.96879	−0.204848
$850$	−6.47709	−0.222162
$851$	6.34029	0.217342
$852$	3.82741	0.131125
$853$	−51.1805	−1.75239	−0.876193	−	0.481960i	$-0.839925\pi$
$853$	−51.1805	−1.75239	−0.876193	+	0.481960i	$0.839925\pi$
$854$	11.6593	0.398974
$855$	−4.50191	−0.153962
$856$	17.4244	0.595555
$857$	−50.8960	−1.73857	−0.869287	−	0.494308i	$-0.835422\pi$
$857$	−50.8960	−1.73857	−0.869287	+	0.494308i	$0.835422\pi$
$858$	0	0
$859$	−20.8508	−0.711422	−0.355711	−	0.934596i	$-0.615761\pi$
$859$	−20.8508	−0.711422	−0.355711	+	0.934596i	$0.615761\pi$
$860$	−4.66428	−0.159051
$861$	−10.8888	−0.371090
$862$	30.7450	1.04718
$863$	54.0762	1.84077	0.920387	−	0.391009i	$-0.127874\pi$
$863$	54.0762	1.84077	0.920387	+	0.391009i	$0.127874\pi$
$864$	5.37701	0.182930
$865$	12.1600	0.413454
$866$	8.55355	0.290661
$867$	28.6697	0.973674
$868$	6.01867	0.204287
$869$	0	0
$870$	−0.444559	−0.0150719
$871$	21.6940	0.735074
$872$	−4.92398	−0.166747
$873$	5.55120	0.187880
$874$	−7.11849	−0.240786
$875$	−1.00000	−0.0338062
$876$	17.2490	0.582790
$877$	2.58531	0.0872997	0.0436499	−	0.999047i	$-0.486101\pi$
$877$	2.58531	0.0872997	0.0436499	+	0.999047i	$0.486101\pi$
$878$	24.5405	0.828203
$879$	20.9346	0.706107
$880$	0	0
$881$	42.8631	1.44410	0.722048	−	0.691843i	$-0.243201\pi$
$881$	42.8631	1.44410	0.722048	+	0.691843i	$0.243201\pi$
$882$	1.67989	0.0565648
$883$	−14.4297	−0.485596	−0.242798	−	0.970077i	$-0.578065\pi$
$883$	−14.4297	−0.485596	−0.242798	+	0.970077i	$0.578065\pi$
$884$	24.5948	0.827214
$885$	5.45347	0.183316
$886$	−26.9683	−0.906019
$887$	−8.49359	−0.285187	−0.142593	−	0.989781i	$-0.545544\pi$
$887$	−8.49359	−0.285187	−0.142593	+	0.989781i	$0.545544\pi$
$888$	−2.74248	−0.0920316
$889$	1.43724	0.0482034
$890$	5.87277	0.196856
$891$	0	0
$892$	17.8782	0.598605
$893$	−12.0114	−0.401946
$894$	−11.6204	−0.388643
$895$	−23.3033	−0.778944
$896$	1.00000	0.0334077
$897$	11.5888	0.386940
$898$	35.1803	1.17398
$899$	−2.32876	−0.0776684
$900$	−1.67989	−0.0559963
$901$	−33.0214	−1.10010
$902$	0	0
$903$	5.35908	0.178339
$904$	11.3824	0.378573
$905$	22.1288	0.735587
$906$	13.5963	0.451706
$907$	−47.4123	−1.57430	−0.787150	−	0.616762i	$-0.788444\pi$
$907$	−47.4123	−1.57430	−0.787150	+	0.616762i	$0.788444\pi$
$908$	−5.76204	−0.191220
$909$	9.34723	0.310028
$910$	3.79720	0.125876
$911$	14.7684	0.489300	0.244650	−	0.969611i	$-0.421327\pi$
$911$	14.7684	0.489300	0.244650	+	0.969611i	$0.421327\pi$
$912$	3.07909	0.101959
$913$	0	0
$914$	−20.4364	−0.675976
$915$	13.3961	0.442862
$916$	−8.57617	−0.283365
$917$	−19.5165	−0.644491
$918$	34.8274	1.14948
$919$	29.6648	0.978552	0.489276	−	0.872129i	$-0.337261\pi$
$919$	29.6648	0.978552	0.489276	+	0.872129i	$0.337261\pi$
$920$	−2.65626	−0.0875744
$921$	−12.6052	−0.415356
$922$	−5.59541	−0.184275
$923$	12.6492	0.416354
$924$	0	0
$925$	2.38692	0.0784815
$926$	29.5079	0.969690
$927$	13.3717	0.439184
$928$	−0.386922	−0.0127013
$929$	−14.6996	−0.482277	−0.241139	−	0.970491i	$-0.577521\pi$
$929$	−14.6996	−0.482277	−0.241139	+	0.970491i	$0.577521\pi$
$930$	6.91522	0.226759
$931$	2.67989	0.0878298
$932$	14.4011	0.471723
$933$	23.9773	0.784982
$934$	−31.3408	−1.02550
$935$	0	0
$936$	6.37888	0.208500
$937$	54.2518	1.77233	0.886164	−	0.463371i	$-0.153360\pi$
$937$	54.2518	1.77233	0.886164	+	0.463371i	$0.153360\pi$
$938$	5.71316	0.186541
$939$	9.93628	0.324258
$940$	−4.48205	−0.146188
$941$	31.8477	1.03820	0.519102	−	0.854712i	$-0.326266\pi$
$941$	31.8477	1.03820	0.519102	+	0.854712i	$0.326266\pi$
$942$	−1.94236	−0.0632856
$943$	25.1736	0.819767
$944$	4.74643	0.154483
$945$	5.37701	0.174914
$946$	0	0
$947$	−44.7035	−1.45267	−0.726335	−	0.687341i	$-0.758778\pi$
$947$	−44.7035	−1.45267	−0.726335	+	0.687341i	$0.758778\pi$
$948$	−6.93551	−0.225255
$949$	57.0063	1.85050
$950$	−2.67989	−0.0869471
$951$	4.06344	0.131766
$952$	6.47709	0.209924
$953$	−45.4259	−1.47149	−0.735745	−	0.677259i	$-0.763168\pi$
$953$	−45.4259	−1.47149	−0.735745	+	0.677259i	$0.763168\pi$
$954$	−8.56439	−0.277282
$955$	20.2864	0.656452
$956$	−24.0096	−0.776527
$957$	0	0
$958$	−6.75139	−0.218127
$959$	−18.9809	−0.612924
$960$	1.14896	0.0370826
$961$	5.22439	0.168529
$962$	−9.06363	−0.292223
$963$	29.2711	0.943248
$964$	−16.7641	−0.539935
$965$	−3.84508	−0.123777
$966$	3.05194	0.0981946
$967$	−34.6596	−1.11458	−0.557289	−	0.830319i	$-0.688158\pi$
$967$	−34.6596	−1.11458	−0.557289	+	0.830319i	$0.688158\pi$
$968$	0	0
$969$	19.9435	0.640679
$970$	3.30450	0.106101
$971$	4.29180	0.137730	0.0688651	−	0.997626i	$-0.478062\pi$
$971$	4.29180	0.137730	0.0688651	+	0.997626i	$0.478062\pi$
$972$	14.8232	0.475453
$973$	−6.58593	−0.211135
$974$	25.3561	0.812463
$975$	4.36284	0.139723
$976$	11.6593	0.373206
$977$	−37.8036	−1.20944	−0.604722	−	0.796437i	$-0.706716\pi$
$977$	−37.8036	−1.20944	−0.604722	+	0.796437i	$0.706716\pi$
$978$	9.26245	0.296181
$979$	0	0
$980$	1.00000	0.0319438
$981$	−8.27173	−0.264096
$982$	−19.7873	−0.631437
$983$	−14.4139	−0.459731	−0.229866	−	0.973222i	$-0.573829\pi$
$983$	−14.4139	−0.459731	−0.229866	+	0.973222i	$0.573829\pi$
$984$	−10.8888	−0.347123
$985$	11.8159	0.376485
$986$	−2.50613	−0.0798115
$987$	5.14970	0.163917
$988$	10.1761	0.323744
$989$	−12.3896	−0.393965
$990$	0	0
$991$	−24.3859	−0.774644	−0.387322	−	0.921944i	$-0.626600\pi$
$991$	−24.3859	−0.774644	−0.387322	+	0.921944i	$0.626600\pi$
$992$	6.01867	0.191093
$993$	11.7155	0.371782
$994$	3.33119	0.105659
$995$	12.5496	0.397849
$996$	−12.8168	−0.406115
$997$	15.5791	0.493395	0.246697	−	0.969093i	$-0.420655\pi$
$997$	15.5791	0.493395	0.246697	+	0.969093i	$0.420655\pi$
$998$	26.7727	0.847474
$999$	−12.8345	−0.406066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.co.1.4		4
11.3	even	5		770.2.n.f.141.1	yes	8
11.4	even	5		770.2.n.f.71.1	✓	8
11.10	odd	2		8470.2.a.cs.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.f.71.1	✓	8	11.4	even	5
770.2.n.f.141.1	yes	8	11.3	even	5
8470.2.a.co.1.4		4	1.1	even	1	trivial
8470.2.a.cs.1.4		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.67989 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.67989 q^{9} -1.00000 q^{10} +1.14896 q^{12} +3.79720 q^{13} +1.00000 q^{14} +1.14896 q^{15} +1.00000 q^{16} +6.47709 q^{17} +1.67989 q^{18} +2.67989 q^{19} +1.00000 q^{20} -1.14896 q^{21} +2.65626 q^{23} -1.14896 q^{24} +1.00000 q^{25} -3.79720 q^{26} -5.37701 q^{27} -1.00000 q^{28} +0.386922 q^{29} -1.14896 q^{30} -6.01867 q^{31} -1.00000 q^{32} -6.47709 q^{34} -1.00000 q^{35} -1.67989 q^{36} +2.38692 q^{37} -2.67989 q^{38} +4.36284 q^{39} -1.00000 q^{40} +9.47709 q^{41} +1.14896 q^{42} -4.66428 q^{43} -1.67989 q^{45} -2.65626 q^{46} -4.48205 q^{47} +1.14896 q^{48} +1.00000 q^{49} -1.00000 q^{50} +7.44193 q^{51} +3.79720 q^{52} -5.09819 q^{53} +5.37701 q^{54} +1.00000 q^{56} +3.07909 q^{57} -0.386922 q^{58} +4.74643 q^{59} +1.14896 q^{60} +11.6593 q^{61} +6.01867 q^{62} +1.67989 q^{63} +1.00000 q^{64} +3.79720 q^{65} +5.71316 q^{67} +6.47709 q^{68} +3.05194 q^{69} +1.00000 q^{70} +3.33119 q^{71} +1.67989 q^{72} +15.0127 q^{73} -2.38692 q^{74} +1.14896 q^{75} +2.67989 q^{76} -4.36284 q^{78} -6.03633 q^{79} +1.00000 q^{80} -1.13831 q^{81} -9.47709 q^{82} -11.1551 q^{83} -1.14896 q^{84} +6.47709 q^{85} +4.66428 q^{86} +0.444559 q^{87} -5.87277 q^{89} +1.67989 q^{90} -3.79720 q^{91} +2.65626 q^{92} -6.91522 q^{93} +4.48205 q^{94} +2.67989 q^{95} -1.14896 q^{96} -3.30450 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 - 4 * q^10 - 4 * q^12 + 14 * q^13 + 4 * q^14 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 6 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 + 4 * q^24 + 4 * q^25 - 14 * q^26 - 22 * q^27 - 4 * q^28 + 10 * q^29 + 4 * q^30 - 18 * q^31 - 4 * q^32 - 12 * q^34 - 4 * q^35 + 6 * q^36 + 18 * q^37 + 2 * q^38 - 30 * q^39 - 4 * q^40 + 24 * q^41 - 4 * q^42 + 10 * q^43 + 6 * q^45 - 8 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 14 * q^52 + 20 * q^53 + 22 * q^54 + 4 * q^56 + 30 * q^57 - 10 * q^58 - 14 * q^59 - 4 * q^60 + 14 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 14 * q^65 + 12 * q^68 - 4 * q^69 + 4 * q^70 - 14 * q^71 - 6 * q^72 + 30 * q^73 - 18 * q^74 - 4 * q^75 - 2 * q^76 + 30 * q^78 + 8 * q^79 + 4 * q^80 + 16 * q^81 - 24 * q^82 + 8 * q^83 + 4 * q^84 + 12 * q^85 - 10 * q^86 + 2 * q^87 - 4 * q^89 - 6 * q^90 - 14 * q^91 - 2 * q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 10 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more