LMFDB - Embedded newform 5070.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, 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("5070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.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:	$40.4841538248$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	5070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.55496 q^{11} -1.00000 q^{12} -1.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.35690 q^{17} +1.00000 q^{18} -6.51573 q^{19} +1.00000 q^{20} +1.69202 q^{21} -4.55496 q^{22} +8.94869 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +9.07606 q^{29} -1.00000 q^{30} -10.6528 q^{31} +1.00000 q^{32} +4.55496 q^{33} +2.35690 q^{34} -1.69202 q^{35} +1.00000 q^{36} +6.18598 q^{37} -6.51573 q^{38} +1.00000 q^{40} +3.00969 q^{41} +1.69202 q^{42} -4.93900 q^{43} -4.55496 q^{44} +1.00000 q^{45} +8.94869 q^{46} -4.28621 q^{47} -1.00000 q^{48} -4.13706 q^{49} +1.00000 q^{50} -2.35690 q^{51} -3.40581 q^{53} -1.00000 q^{54} -4.55496 q^{55} -1.69202 q^{56} +6.51573 q^{57} +9.07606 q^{58} +4.32304 q^{59} -1.00000 q^{60} -14.3666 q^{61} -10.6528 q^{62} -1.69202 q^{63} +1.00000 q^{64} +4.55496 q^{66} -3.24698 q^{67} +2.35690 q^{68} -8.94869 q^{69} -1.69202 q^{70} -14.9487 q^{71} +1.00000 q^{72} -6.72886 q^{73} +6.18598 q^{74} -1.00000 q^{75} -6.51573 q^{76} +7.70709 q^{77} +5.67994 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.00969 q^{82} -7.71917 q^{83} +1.69202 q^{84} +2.35690 q^{85} -4.93900 q^{86} -9.07606 q^{87} -4.55496 q^{88} -9.12498 q^{89} +1.00000 q^{90} +8.94869 q^{92} +10.6528 q^{93} -4.28621 q^{94} -6.51573 q^{95} -1.00000 q^{96} -4.40581 q^{97} -4.13706 q^{98} -4.55496 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 14 q^{11} - 3 q^{12} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 7 q^{19} + 3 q^{20} - 14 q^{22} - 5 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 12 q^{29} - 3 q^{30} - 14 q^{31} + 3 q^{32} + 14 q^{33} + 3 q^{34} + 3 q^{36} + 4 q^{37} - 7 q^{38} + 3 q^{40} - 13 q^{41} - 5 q^{43} - 14 q^{44} + 3 q^{45} - 5 q^{46} - 21 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 3 q^{51} + 3 q^{53} - 3 q^{54} - 14 q^{55} + 7 q^{57} + 12 q^{58} - 7 q^{59} - 3 q^{60} - 17 q^{61} - 14 q^{62} + 3 q^{64} + 14 q^{66} - 5 q^{67} + 3 q^{68} + 5 q^{69} - 13 q^{71} + 3 q^{72} + 13 q^{73} + 4 q^{74} - 3 q^{75} - 7 q^{76} - 7 q^{77} - 7 q^{79} + 3 q^{80} + 3 q^{81} - 13 q^{82} - 12 q^{83} + 3 q^{85} - 5 q^{86} - 12 q^{87} - 14 q^{88} - 3 q^{89} + 3 q^{90} - 5 q^{92} + 14 q^{93} - 21 q^{94} - 7 q^{95} - 3 q^{96} - 7 q^{98} - 14 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 14 * q^11 - 3 * q^12 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 7 * q^19 + 3 * q^20 - 14 * q^22 - 5 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 12 * q^29 - 3 * q^30 - 14 * q^31 + 3 * q^32 + 14 * q^33 + 3 * q^34 + 3 * q^36 + 4 * q^37 - 7 * q^38 + 3 * q^40 - 13 * q^41 - 5 * q^43 - 14 * q^44 + 3 * q^45 - 5 * q^46 - 21 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 3 * q^51 + 3 * q^53 - 3 * q^54 - 14 * q^55 + 7 * q^57 + 12 * q^58 - 7 * q^59 - 3 * q^60 - 17 * q^61 - 14 * q^62 + 3 * q^64 + 14 * q^66 - 5 * q^67 + 3 * q^68 + 5 * q^69 - 13 * q^71 + 3 * q^72 + 13 * q^73 + 4 * q^74 - 3 * q^75 - 7 * q^76 - 7 * q^77 - 7 * q^79 + 3 * q^80 + 3 * q^81 - 13 * q^82 - 12 * q^83 + 3 * q^85 - 5 * q^86 - 12 * q^87 - 14 * q^88 - 3 * q^89 + 3 * q^90 - 5 * q^92 + 14 * q^93 - 21 * q^94 - 7 * q^95 - 3 * q^96 - 7 * q^98 - 14 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−1.69202	−0.639524	−0.319762	−	0.947498i	$-0.603603\pi$
$7$	−1.69202	−0.639524	−0.319762	+	0.947498i	$0.603603\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.55496	−1.37337	−0.686686	−	0.726954i	$-0.740935\pi$
$11$	−4.55496	−1.37337	−0.686686	+	0.726954i	$0.740935\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.69202	−0.452212
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	2.35690	0.571631	0.285816	−	0.958285i	$-0.407736\pi$
$17$	2.35690	0.571631	0.285816	+	0.958285i	$0.407736\pi$
$18$	1.00000	0.235702
$19$	−6.51573	−1.49481	−0.747405	−	0.664368i	$-0.768701\pi$
$19$	−6.51573	−1.49481	−0.747405	+	0.664368i	$0.768701\pi$
$20$	1.00000	0.223607
$21$	1.69202	0.369229
$22$	−4.55496	−0.971120
$23$	8.94869	1.86593	0.932965	−	0.359966i	$-0.117212\pi$
$23$	8.94869	1.86593	0.932965	+	0.359966i	$0.117212\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.69202	−0.319762
$29$	9.07606	1.68538	0.842691	−	0.538397i	$-0.180970\pi$
$29$	9.07606	1.68538	0.842691	+	0.538397i	$0.180970\pi$
$30$	−1.00000	−0.182574
$31$	−10.6528	−1.91330	−0.956649	−	0.291243i	$-0.905931\pi$
$31$	−10.6528	−1.91330	−0.956649	+	0.291243i	$0.905931\pi$
$32$	1.00000	0.176777
$33$	4.55496	0.792916
$34$	2.35690	0.404204
$35$	−1.69202	−0.286004
$36$	1.00000	0.166667
$37$	6.18598	1.01697	0.508484	−	0.861071i	$-0.330206\pi$
$37$	6.18598	1.01697	0.508484	+	0.861071i	$0.330206\pi$
$38$	−6.51573	−1.05699
$39$	0	0
$40$	1.00000	0.158114
$41$	3.00969	0.470034	0.235017	−	0.971991i	$-0.424485\pi$
$41$	3.00969	0.470034	0.235017	+	0.971991i	$0.424485\pi$
$42$	1.69202	0.261085
$43$	−4.93900	−0.753191	−0.376595	−	0.926378i	$-0.622905\pi$
$43$	−4.93900	−0.753191	−0.376595	+	0.926378i	$0.622905\pi$
$44$	−4.55496	−0.686686
$45$	1.00000	0.149071
$46$	8.94869	1.31941
$47$	−4.28621	−0.625208	−0.312604	−	0.949884i	$-0.601201\pi$
$47$	−4.28621	−0.625208	−0.312604	+	0.949884i	$0.601201\pi$
$48$	−1.00000	−0.144338
$49$	−4.13706	−0.591009
$50$	1.00000	0.141421
$51$	−2.35690	−0.330031
$52$	0	0
$53$	−3.40581	−0.467824	−0.233912	−	0.972258i	$-0.575153\pi$
$53$	−3.40581	−0.467824	−0.233912	+	0.972258i	$0.575153\pi$
$54$	−1.00000	−0.136083
$55$	−4.55496	−0.614190
$56$	−1.69202	−0.226106
$57$	6.51573	0.863029
$58$	9.07606	1.19175
$59$	4.32304	0.562812	0.281406	−	0.959589i	$-0.409199\pi$
$59$	4.32304	0.562812	0.281406	+	0.959589i	$0.409199\pi$
$60$	−1.00000	−0.129099
$61$	−14.3666	−1.83945	−0.919726	−	0.392560i	$-0.871589\pi$
$61$	−14.3666	−1.83945	−0.919726	+	0.392560i	$0.871589\pi$
$62$	−10.6528	−1.35291
$63$	−1.69202	−0.213175
$64$	1.00000	0.125000
$65$	0	0
$66$	4.55496	0.560677
$67$	−3.24698	−0.396682	−0.198341	−	0.980133i	$-0.563555\pi$
$67$	−3.24698	−0.396682	−0.198341	+	0.980133i	$0.563555\pi$
$68$	2.35690	0.285816
$69$	−8.94869	−1.07730
$70$	−1.69202	−0.202235
$71$	−14.9487	−1.77408	−0.887042	−	0.461690i	$-0.847243\pi$
$71$	−14.9487	−1.77408	−0.887042	+	0.461690i	$0.847243\pi$
$72$	1.00000	0.117851
$73$	−6.72886	−0.787553	−0.393777	−	0.919206i	$-0.628832\pi$
$73$	−6.72886	−0.787553	−0.393777	+	0.919206i	$0.628832\pi$
$74$	6.18598	0.719106
$75$	−1.00000	−0.115470
$76$	−6.51573	−0.747405
$77$	7.70709	0.878304
$78$	0	0
$79$	5.67994	0.639043	0.319522	−	0.947579i	$-0.396478\pi$
$79$	5.67994	0.639043	0.319522	+	0.947579i	$0.396478\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	3.00969	0.332365
$83$	−7.71917	−0.847289	−0.423644	−	0.905829i	$-0.639249\pi$
$83$	−7.71917	−0.847289	−0.423644	+	0.905829i	$0.639249\pi$
$84$	1.69202	0.184615
$85$	2.35690	0.255641
$86$	−4.93900	−0.532586
$87$	−9.07606	−0.973056
$88$	−4.55496	−0.485560
$89$	−9.12498	−0.967246	−0.483623	−	0.875276i	$-0.660679\pi$
$89$	−9.12498	−0.967246	−0.483623	+	0.875276i	$0.660679\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	8.94869	0.932965
$93$	10.6528	1.10464
$94$	−4.28621	−0.442089
$95$	−6.51573	−0.668500
$96$	−1.00000	−0.102062
$97$	−4.40581	−0.447343	−0.223671	−	0.974665i	$-0.571804\pi$
$97$	−4.40581	−0.447343	−0.223671	+	0.974665i	$0.571804\pi$
$98$	−4.13706	−0.417907
$99$	−4.55496	−0.457791
$100$	1.00000	0.100000
$101$	2.31767	0.230617	0.115308	−	0.993330i	$-0.463214\pi$
$101$	2.31767	0.230617	0.115308	+	0.993330i	$0.463214\pi$
$102$	−2.35690	−0.233367
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	1.69202	0.165124
$106$	−3.40581	−0.330802
$107$	−3.06100	−0.295918	−0.147959	−	0.988994i	$-0.547270\pi$
$107$	−3.06100	−0.295918	−0.147959	+	0.988994i	$0.547270\pi$
$108$	−1.00000	−0.0962250
$109$	7.45473	0.714034	0.357017	−	0.934098i	$-0.383794\pi$
$109$	7.45473	0.714034	0.357017	+	0.934098i	$0.383794\pi$
$110$	−4.55496	−0.434298
$111$	−6.18598	−0.587147
$112$	−1.69202	−0.159881
$113$	−19.9812	−1.87967	−0.939837	−	0.341623i	$-0.889024\pi$
$113$	−19.9812	−1.87967	−0.939837	+	0.341623i	$0.889024\pi$
$114$	6.51573	0.610254
$115$	8.94869	0.834470
$116$	9.07606	0.842691
$117$	0	0
$118$	4.32304	0.397968
$119$	−3.98792	−0.365572
$120$	−1.00000	−0.0912871
$121$	9.74764	0.886149
$122$	−14.3666	−1.30069
$123$	−3.00969	−0.271374
$124$	−10.6528	−0.956649
$125$	1.00000	0.0894427
$126$	−1.69202	−0.150737
$127$	−3.14244	−0.278846	−0.139423	−	0.990233i	$-0.544525\pi$
$127$	−3.14244	−0.278846	−0.139423	+	0.990233i	$0.544525\pi$
$128$	1.00000	0.0883883
$129$	4.93900	0.434855
$130$	0	0
$131$	−3.77479	−0.329805	−0.164902	−	0.986310i	$-0.552731\pi$
$131$	−3.77479	−0.329805	−0.164902	+	0.986310i	$0.552731\pi$
$132$	4.55496	0.396458
$133$	11.0248	0.955967
$134$	−3.24698	−0.280496
$135$	−1.00000	−0.0860663
$136$	2.35690	0.202102
$137$	14.5894	1.24646	0.623228	−	0.782040i	$-0.285821\pi$
$137$	14.5894	1.24646	0.623228	+	0.782040i	$0.285821\pi$
$138$	−8.94869	−0.761763
$139$	−14.3599	−1.21799	−0.608995	−	0.793174i	$-0.708427\pi$
$139$	−14.3599	−1.21799	−0.608995	+	0.793174i	$0.708427\pi$
$140$	−1.69202	−0.143002
$141$	4.28621	0.360964
$142$	−14.9487	−1.25447
$143$	0	0
$144$	1.00000	0.0833333
$145$	9.07606	0.753726
$146$	−6.72886	−0.556884
$147$	4.13706	0.341219
$148$	6.18598	0.508484
$149$	0.252356	0.0206738	0.0103369	−	0.999947i	$-0.496710\pi$
$149$	0.252356	0.0206738	0.0103369	+	0.999947i	$0.496710\pi$
$150$	−1.00000	−0.0816497
$151$	11.0804	0.901708	0.450854	−	0.892598i	$-0.351119\pi$
$151$	11.0804	0.901708	0.450854	+	0.892598i	$0.351119\pi$
$152$	−6.51573	−0.528495
$153$	2.35690	0.190544
$154$	7.70709	0.621055
$155$	−10.6528	−0.855653
$156$	0	0
$157$	5.03923	0.402174	0.201087	−	0.979573i	$-0.435553\pi$
$157$	5.03923	0.402174	0.201087	+	0.979573i	$0.435553\pi$
$158$	5.67994	0.451872
$159$	3.40581	0.270099
$160$	1.00000	0.0790569
$161$	−15.1414	−1.19331
$162$	1.00000	0.0785674
$163$	20.9825	1.64348	0.821740	−	0.569863i	$-0.193004\pi$
$163$	20.9825	1.64348	0.821740	+	0.569863i	$0.193004\pi$
$164$	3.00969	0.235017
$165$	4.55496	0.354603
$166$	−7.71917	−0.599124
$167$	−22.2325	−1.72040	−0.860201	−	0.509954i	$-0.829662\pi$
$167$	−22.2325	−1.72040	−0.860201	+	0.509954i	$0.829662\pi$
$168$	1.69202	0.130542
$169$	0	0
$170$	2.35690	0.180766
$171$	−6.51573	−0.498270
$172$	−4.93900	−0.376595
$173$	−11.6286	−0.884108	−0.442054	−	0.896988i	$-0.645750\pi$
$173$	−11.6286	−0.884108	−0.442054	+	0.896988i	$0.645750\pi$
$174$	−9.07606	−0.688055
$175$	−1.69202	−0.127905
$176$	−4.55496	−0.343343
$177$	−4.32304	−0.324940
$178$	−9.12498	−0.683946
$179$	−15.7875	−1.18001	−0.590005	−	0.807399i	$-0.700874\pi$
$179$	−15.7875	−1.18001	−0.590005	+	0.807399i	$0.700874\pi$
$180$	1.00000	0.0745356
$181$	−1.91185	−0.142107	−0.0710535	−	0.997473i	$-0.522636\pi$
$181$	−1.91185	−0.142107	−0.0710535	+	0.997473i	$0.522636\pi$
$182$	0	0
$183$	14.3666	1.06201
$184$	8.94869	0.659706
$185$	6.18598	0.454802
$186$	10.6528	0.781101
$187$	−10.7356	−0.785062
$188$	−4.28621	−0.312604
$189$	1.69202	0.123076
$190$	−6.51573	−0.472701
$191$	24.3260	1.76017	0.880085	−	0.474817i	$-0.157486\pi$
$191$	24.3260	1.76017	0.880085	+	0.474817i	$0.157486\pi$
$192$	−1.00000	−0.0721688
$193$	−0.521106	−0.0375101	−0.0187550	−	0.999824i	$-0.505970\pi$
$193$	−0.521106	−0.0375101	−0.0187550	+	0.999824i	$0.505970\pi$
$194$	−4.40581	−0.316319
$195$	0	0
$196$	−4.13706	−0.295505
$197$	−4.87502	−0.347331	−0.173665	−	0.984805i	$-0.555561\pi$
$197$	−4.87502	−0.347331	−0.173665	+	0.984805i	$0.555561\pi$
$198$	−4.55496	−0.323707
$199$	−18.4983	−1.31131	−0.655654	−	0.755062i	$-0.727607\pi$
$199$	−18.4983	−1.31131	−0.655654	+	0.755062i	$0.727607\pi$
$200$	1.00000	0.0707107
$201$	3.24698	0.229024
$202$	2.31767	0.163070
$203$	−15.3569	−1.07784
$204$	−2.35690	−0.165016
$205$	3.00969	0.210206
$206$	−9.00000	−0.627060
$207$	8.94869	0.621977
$208$	0	0
$209$	29.6789	2.05293
$210$	1.69202	0.116761
$211$	18.2500	1.25638	0.628190	−	0.778060i	$-0.283796\pi$
$211$	18.2500	1.25638	0.628190	+	0.778060i	$0.283796\pi$
$212$	−3.40581	−0.233912
$213$	14.9487	1.02427
$214$	−3.06100	−0.209246
$215$	−4.93900	−0.336837
$216$	−1.00000	−0.0680414
$217$	18.0248	1.22360
$218$	7.45473	0.504898
$219$	6.72886	0.454694
$220$	−4.55496	−0.307095
$221$	0	0
$222$	−6.18598	−0.415176
$223$	8.54048	0.571913	0.285957	−	0.958243i	$-0.407689\pi$
$223$	8.54048	0.571913	0.285957	+	0.958243i	$0.407689\pi$
$224$	−1.69202	−0.113053
$225$	1.00000	0.0666667
$226$	−19.9812	−1.32913
$227$	−9.61596	−0.638233	−0.319117	−	0.947715i	$-0.603386\pi$
$227$	−9.61596	−0.638233	−0.319117	+	0.947715i	$0.603386\pi$
$228$	6.51573	0.431515
$229$	−13.4058	−0.885881	−0.442941	−	0.896551i	$-0.646065\pi$
$229$	−13.4058	−0.885881	−0.442941	+	0.896551i	$0.646065\pi$
$230$	8.94869	0.590059
$231$	−7.70709	−0.507089
$232$	9.07606	0.595873
$233$	−0.153457	−0.0100533	−0.00502664	−	0.999987i	$-0.501600\pi$
$233$	−0.153457	−0.0100533	−0.00502664	+	0.999987i	$0.501600\pi$
$234$	0	0
$235$	−4.28621	−0.279601
$236$	4.32304	0.281406
$237$	−5.67994	−0.368952
$238$	−3.98792	−0.258498
$239$	14.5851	0.943431	0.471715	−	0.881751i	$-0.343635\pi$
$239$	14.5851	0.943431	0.471715	+	0.881751i	$0.343635\pi$
$240$	−1.00000	−0.0645497
$241$	3.40044	0.219041	0.109521	−	0.993985i	$-0.465068\pi$
$241$	3.40044	0.219041	0.109521	+	0.993985i	$0.465068\pi$
$242$	9.74764	0.626602
$243$	−1.00000	−0.0641500
$244$	−14.3666	−0.919726
$245$	−4.13706	−0.264307
$246$	−3.00969	−0.191891
$247$	0	0
$248$	−10.6528	−0.676453
$249$	7.71917	0.489182
$250$	1.00000	0.0632456
$251$	1.80864	0.114161	0.0570803	−	0.998370i	$-0.481821\pi$
$251$	1.80864	0.114161	0.0570803	+	0.998370i	$0.481821\pi$
$252$	−1.69202	−0.106587
$253$	−40.7609	−2.56262
$254$	−3.14244	−0.197174
$255$	−2.35690	−0.147595
$256$	1.00000	0.0625000
$257$	−22.2664	−1.38894	−0.694469	−	0.719523i	$-0.744360\pi$
$257$	−22.2664	−1.38894	−0.694469	+	0.719523i	$0.744360\pi$
$258$	4.93900	0.307489
$259$	−10.4668	−0.650376
$260$	0	0
$261$	9.07606	0.561794
$262$	−3.77479	−0.233207
$263$	8.08815	0.498736	0.249368	−	0.968409i	$-0.419777\pi$
$263$	8.08815	0.498736	0.249368	+	0.968409i	$0.419777\pi$
$264$	4.55496	0.280338
$265$	−3.40581	−0.209217
$266$	11.0248	0.675971
$267$	9.12498	0.558440
$268$	−3.24698	−0.198341
$269$	16.6843	1.01726	0.508628	−	0.860986i	$-0.330153\pi$
$269$	16.6843	1.01726	0.508628	+	0.860986i	$0.330153\pi$
$270$	−1.00000	−0.0608581
$271$	27.4077	1.66490	0.832451	−	0.554099i	$-0.186937\pi$
$271$	27.4077	1.66490	0.832451	+	0.554099i	$0.186937\pi$
$272$	2.35690	0.142908
$273$	0	0
$274$	14.5894	0.881378
$275$	−4.55496	−0.274674
$276$	−8.94869	−0.538648
$277$	−16.0411	−0.963819	−0.481910	−	0.876221i	$-0.660057\pi$
$277$	−16.0411	−0.963819	−0.481910	+	0.876221i	$0.660057\pi$
$278$	−14.3599	−0.861248
$279$	−10.6528	−0.637766
$280$	−1.69202	−0.101118
$281$	−10.0935	−0.602129	−0.301065	−	0.953604i	$-0.597342\pi$
$281$	−10.0935	−0.602129	−0.301065	+	0.953604i	$0.597342\pi$
$282$	4.28621	0.255240
$283$	12.1021	0.719398	0.359699	−	0.933068i	$-0.382879\pi$
$283$	12.1021	0.719398	0.359699	+	0.933068i	$0.382879\pi$
$284$	−14.9487	−0.887042
$285$	6.51573	0.385959
$286$	0	0
$287$	−5.09246	−0.300598
$288$	1.00000	0.0589256
$289$	−11.4450	−0.673238
$290$	9.07606	0.532965
$291$	4.40581	0.258273
$292$	−6.72886	−0.393777
$293$	4.01075	0.234311	0.117155	−	0.993114i	$-0.462622\pi$
$293$	4.01075	0.234311	0.117155	+	0.993114i	$0.462622\pi$
$294$	4.13706	0.241278
$295$	4.32304	0.251697
$296$	6.18598	0.359553
$297$	4.55496	0.264305
$298$	0.252356	0.0146186
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.35690	0.481683
$302$	11.0804	0.637604
$303$	−2.31767	−0.133147
$304$	−6.51573	−0.373703
$305$	−14.3666	−0.822628
$306$	2.35690	0.134735
$307$	−15.2784	−0.871987	−0.435993	−	0.899950i	$-0.643603\pi$
$307$	−15.2784	−0.871987	−0.435993	+	0.899950i	$0.643603\pi$
$308$	7.70709	0.439152
$309$	9.00000	0.511992
$310$	−10.6528	−0.605038
$311$	25.4330	1.44217	0.721085	−	0.692846i	$-0.243644\pi$
$311$	25.4330	1.44217	0.721085	+	0.692846i	$0.243644\pi$
$312$	0	0
$313$	−19.1226	−1.08087	−0.540436	−	0.841385i	$-0.681741\pi$
$313$	−19.1226	−1.08087	−0.540436	+	0.841385i	$0.681741\pi$
$314$	5.03923	0.284380
$315$	−1.69202	−0.0953346
$316$	5.67994	0.319522
$317$	−20.3230	−1.14146	−0.570728	−	0.821139i	$-0.693339\pi$
$317$	−20.3230	−1.14146	−0.570728	+	0.821139i	$0.693339\pi$
$318$	3.40581	0.190989
$319$	−41.3411	−2.31466
$320$	1.00000	0.0559017
$321$	3.06100	0.170848
$322$	−15.1414	−0.843796
$323$	−15.3569	−0.854481
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.9825	1.16212
$327$	−7.45473	−0.412248
$328$	3.00969	0.166182
$329$	7.25236	0.399835
$330$	4.55496	0.250742
$331$	−22.3588	−1.22895	−0.614476	−	0.788936i	$-0.710632\pi$
$331$	−22.3588	−1.22895	−0.614476	+	0.788936i	$0.710632\pi$
$332$	−7.71917	−0.423644
$333$	6.18598	0.338990
$334$	−22.2325	−1.21651
$335$	−3.24698	−0.177401
$336$	1.69202	0.0923073
$337$	−3.67025	−0.199931	−0.0999657	−	0.994991i	$-0.531873\pi$
$337$	−3.67025	−0.199931	−0.0999657	+	0.994991i	$0.531873\pi$
$338$	0	0
$339$	19.9812	1.08523
$340$	2.35690	0.127821
$341$	48.5230	2.62767
$342$	−6.51573	−0.352330
$343$	18.8442	1.01749
$344$	−4.93900	−0.266293
$345$	−8.94869	−0.481781
$346$	−11.6286	−0.625159
$347$	−21.7832	−1.16938	−0.584690	−	0.811257i	$-0.698784\pi$
$347$	−21.7832	−1.16938	−0.584690	+	0.811257i	$0.698784\pi$
$348$	−9.07606	−0.486528
$349$	−1.74333	−0.0933184	−0.0466592	−	0.998911i	$-0.514857\pi$
$349$	−1.74333	−0.0933184	−0.0466592	+	0.998911i	$0.514857\pi$
$350$	−1.69202	−0.0904424
$351$	0	0
$352$	−4.55496	−0.242780
$353$	6.09544	0.324428	0.162214	−	0.986756i	$-0.448137\pi$
$353$	6.09544	0.324428	0.162214	+	0.986756i	$0.448137\pi$
$354$	−4.32304	−0.229767
$355$	−14.9487	−0.793394
$356$	−9.12498	−0.483623
$357$	3.98792	0.211063
$358$	−15.7875	−0.834393
$359$	25.9627	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$359$	25.9627	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$360$	1.00000	0.0527046
$361$	23.4547	1.23446
$362$	−1.91185	−0.100485
$363$	−9.74764	−0.511619
$364$	0	0
$365$	−6.72886	−0.352204
$366$	14.3666	0.750953
$367$	27.8799	1.45532	0.727660	−	0.685938i	$-0.240608\pi$
$367$	27.8799	1.45532	0.727660	+	0.685938i	$0.240608\pi$
$368$	8.94869	0.466483
$369$	3.00969	0.156678
$370$	6.18598	0.321594
$371$	5.76271	0.299185
$372$	10.6528	0.552322
$373$	−25.1400	−1.30170	−0.650851	−	0.759205i	$-0.725588\pi$
$373$	−25.1400	−1.30170	−0.650851	+	0.759205i	$0.725588\pi$
$374$	−10.7356	−0.555123
$375$	−1.00000	−0.0516398
$376$	−4.28621	−0.221044
$377$	0	0
$378$	1.69202	0.0870282
$379$	−15.7463	−0.808834	−0.404417	−	0.914575i	$-0.632526\pi$
$379$	−15.7463	−0.808834	−0.404417	+	0.914575i	$0.632526\pi$
$380$	−6.51573	−0.334250
$381$	3.14244	0.160992
$382$	24.3260	1.24463
$383$	11.8183	0.603889	0.301944	−	0.953326i	$-0.402364\pi$
$383$	11.8183	0.603889	0.301944	+	0.953326i	$0.402364\pi$
$384$	−1.00000	−0.0510310
$385$	7.70709	0.392790
$386$	−0.521106	−0.0265236
$387$	−4.93900	−0.251064
$388$	−4.40581	−0.223671
$389$	−6.14675	−0.311653	−0.155826	−	0.987784i	$-0.549804\pi$
$389$	−6.14675	−0.311653	−0.155826	+	0.987784i	$0.549804\pi$
$390$	0	0
$391$	21.0911	1.06662
$392$	−4.13706	−0.208953
$393$	3.77479	0.190413
$394$	−4.87502	−0.245600
$395$	5.67994	0.285789
$396$	−4.55496	−0.228895
$397$	−5.92825	−0.297530	−0.148765	−	0.988873i	$-0.547530\pi$
$397$	−5.92825	−0.297530	−0.148765	+	0.988873i	$0.547530\pi$
$398$	−18.4983	−0.927235
$399$	−11.0248	−0.551928
$400$	1.00000	0.0500000
$401$	9.49934	0.474374	0.237187	−	0.971464i	$-0.423775\pi$
$401$	9.49934	0.474374	0.237187	+	0.971464i	$0.423775\pi$
$402$	3.24698	0.161945
$403$	0	0
$404$	2.31767	0.115308
$405$	1.00000	0.0496904
$406$	−15.3569	−0.762150
$407$	−28.1769	−1.39668
$408$	−2.35690	−0.116684
$409$	−9.00000	−0.445021	−0.222511	−	0.974930i	$-0.571425\pi$
$409$	−9.00000	−0.445021	−0.222511	+	0.974930i	$0.571425\pi$
$410$	3.00969	0.148638
$411$	−14.5894	−0.719642
$412$	−9.00000	−0.443398
$413$	−7.31468	−0.359932
$414$	8.94869	0.439804
$415$	−7.71917	−0.378919
$416$	0	0
$417$	14.3599	0.703206
$418$	29.6789	1.45164
$419$	30.6450	1.49711	0.748554	−	0.663074i	$-0.230749\pi$
$419$	30.6450	1.49711	0.748554	+	0.663074i	$0.230749\pi$
$420$	1.69202	0.0825622
$421$	4.03252	0.196533	0.0982666	−	0.995160i	$-0.468670\pi$
$421$	4.03252	0.196533	0.0982666	+	0.995160i	$0.468670\pi$
$422$	18.2500	0.888394
$423$	−4.28621	−0.208403
$424$	−3.40581	−0.165401
$425$	2.35690	0.114326
$426$	14.9487	0.724266
$427$	24.3086	1.17637
$428$	−3.06100	−0.147959
$429$	0	0
$430$	−4.93900	−0.238180
$431$	−27.6179	−1.33031	−0.665153	−	0.746707i	$-0.731634\pi$
$431$	−27.6179	−1.33031	−0.665153	+	0.746707i	$0.731634\pi$
$432$	−1.00000	−0.0481125
$433$	10.6756	0.513038	0.256519	−	0.966539i	$-0.417424\pi$
$433$	10.6756	0.513038	0.256519	+	0.966539i	$0.417424\pi$
$434$	18.0248	0.865216
$435$	−9.07606	−0.435164
$436$	7.45473	0.357017
$437$	−58.3072	−2.78921
$438$	6.72886	0.321517
$439$	25.9299	1.23757	0.618783	−	0.785562i	$-0.287626\pi$
$439$	25.9299	1.23757	0.618783	+	0.785562i	$0.287626\pi$
$440$	−4.55496	−0.217149
$441$	−4.13706	−0.197003
$442$	0	0
$443$	−35.3749	−1.68071	−0.840357	−	0.542033i	$-0.817655\pi$
$443$	−35.3749	−1.68071	−0.840357	+	0.542033i	$0.817655\pi$
$444$	−6.18598	−0.293574
$445$	−9.12498	−0.432566
$446$	8.54048	0.404404
$447$	−0.252356	−0.0119360
$448$	−1.69202	−0.0799405
$449$	−6.24698	−0.294813	−0.147407	−	0.989076i	$-0.547093\pi$
$449$	−6.24698	−0.294813	−0.147407	+	0.989076i	$0.547093\pi$
$450$	1.00000	0.0471405
$451$	−13.7090	−0.645532
$452$	−19.9812	−0.939837
$453$	−11.0804	−0.520601
$454$	−9.61596	−0.451299
$455$	0	0
$456$	6.51573	0.305127
$457$	−29.6383	−1.38642	−0.693211	−	0.720735i	$-0.743805\pi$
$457$	−29.6383	−1.38642	−0.693211	+	0.720735i	$0.743805\pi$
$458$	−13.4058	−0.626413
$459$	−2.35690	−0.110010
$460$	8.94869	0.417235
$461$	34.2911	1.59710	0.798548	−	0.601931i	$-0.205602\pi$
$461$	34.2911	1.59710	0.798548	+	0.601931i	$0.205602\pi$
$462$	−7.70709	−0.358566
$463$	24.1769	1.12360	0.561798	−	0.827275i	$-0.310110\pi$
$463$	24.1769	1.12360	0.561798	+	0.827275i	$0.310110\pi$
$464$	9.07606	0.421346
$465$	10.6528	0.494011
$466$	−0.153457	−0.00710875
$467$	−0.521106	−0.0241139	−0.0120570	−	0.999927i	$-0.503838\pi$
$467$	−0.521106	−0.0241139	−0.0120570	+	0.999927i	$0.503838\pi$
$468$	0	0
$469$	5.49396	0.253687
$470$	−4.28621	−0.197708
$471$	−5.03923	−0.232195
$472$	4.32304	0.198984
$473$	22.4969	1.03441
$474$	−5.67994	−0.260888
$475$	−6.51573	−0.298962
$476$	−3.98792	−0.182786
$477$	−3.40581	−0.155941
$478$	14.5851	0.667106
$479$	10.4168	0.475957	0.237979	−	0.971270i	$-0.423515\pi$
$479$	10.4168	0.475957	0.237979	+	0.971270i	$0.423515\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	3.40044	0.154886
$483$	15.1414	0.688956
$484$	9.74764	0.443075
$485$	−4.40581	−0.200058
$486$	−1.00000	−0.0453609
$487$	30.2349	1.37007	0.685037	−	0.728508i	$-0.259786\pi$
$487$	30.2349	1.37007	0.685037	+	0.728508i	$0.259786\pi$
$488$	−14.3666	−0.650345
$489$	−20.9825	−0.948863
$490$	−4.13706	−0.186893
$491$	−31.5883	−1.42556	−0.712781	−	0.701387i	$-0.752565\pi$
$491$	−31.5883	−1.42556	−0.712781	+	0.701387i	$0.752565\pi$
$492$	−3.00969	−0.135687
$493$	21.3913	0.963417
$494$	0	0
$495$	−4.55496	−0.204730
$496$	−10.6528	−0.478325
$497$	25.2935	1.13457
$498$	7.71917	0.345904
$499$	−27.2862	−1.22150	−0.610749	−	0.791824i	$-0.709132\pi$
$499$	−27.2862	−1.22150	−0.610749	+	0.791824i	$0.709132\pi$
$500$	1.00000	0.0447214
$501$	22.2325	0.993275
$502$	1.80864	0.0807237
$503$	29.6752	1.32315	0.661575	−	0.749879i	$-0.269888\pi$
$503$	29.6752	1.32315	0.661575	+	0.749879i	$0.269888\pi$
$504$	−1.69202	−0.0753686
$505$	2.31767	0.103135
$506$	−40.7609	−1.81204
$507$	0	0
$508$	−3.14244	−0.139423
$509$	−44.1661	−1.95763	−0.978815	−	0.204748i	$-0.934362\pi$
$509$	−44.1661	−1.95763	−0.978815	+	0.204748i	$0.934362\pi$
$510$	−2.35690	−0.104365
$511$	11.3854	0.503659
$512$	1.00000	0.0441942
$513$	6.51573	0.287676
$514$	−22.2664	−0.982127
$515$	−9.00000	−0.396587
$516$	4.93900	0.217427
$517$	19.5235	0.858643
$518$	−10.4668	−0.459885
$519$	11.6286	0.510440
$520$	0	0
$521$	20.1153	0.881267	0.440633	−	0.897687i	$-0.354754\pi$
$521$	20.1153	0.881267	0.440633	+	0.897687i	$0.354754\pi$
$522$	9.07606	0.397249
$523$	6.46117	0.282527	0.141264	−	0.989972i	$-0.454883\pi$
$523$	6.46117	0.282527	0.141264	+	0.989972i	$0.454883\pi$
$524$	−3.77479	−0.164902
$525$	1.69202	0.0738459
$526$	8.08815	0.352660
$527$	−25.1075	−1.09370
$528$	4.55496	0.198229
$529$	57.0790	2.48170
$530$	−3.40581	−0.147939
$531$	4.32304	0.187604
$532$	11.0248	0.477984
$533$	0	0
$534$	9.12498	0.394877
$535$	−3.06100	−0.132339
$536$	−3.24698	−0.140248
$537$	15.7875	0.681279
$538$	16.6843	0.719309
$539$	18.8442	0.811675
$540$	−1.00000	−0.0430331
$541$	34.0097	1.46219	0.731095	−	0.682275i	$-0.239009\pi$
$541$	34.0097	1.46219	0.731095	+	0.682275i	$0.239009\pi$
$542$	27.4077	1.17726
$543$	1.91185	0.0820455
$544$	2.35690	0.101051
$545$	7.45473	0.319326
$546$	0	0
$547$	14.6907	0.628129	0.314064	−	0.949402i	$-0.398309\pi$
$547$	14.6907	0.628129	0.314064	+	0.949402i	$0.398309\pi$
$548$	14.5894	0.623228
$549$	−14.3666	−0.613151
$550$	−4.55496	−0.194224
$551$	−59.1372	−2.51933
$552$	−8.94869	−0.380882
$553$	−9.61058	−0.408683
$554$	−16.0411	−0.681523
$555$	−6.18598	−0.262580
$556$	−14.3599	−0.608995
$557$	7.00969	0.297010	0.148505	−	0.988912i	$-0.452554\pi$
$557$	7.00969	0.297010	0.148505	+	0.988912i	$0.452554\pi$
$558$	−10.6528	−0.450969
$559$	0	0
$560$	−1.69202	−0.0715010
$561$	10.7356	0.453256
$562$	−10.0935	−0.425770
$563$	−30.2131	−1.27333	−0.636666	−	0.771140i	$-0.719687\pi$
$563$	−30.2131	−1.27333	−0.636666	+	0.771140i	$0.719687\pi$
$564$	4.28621	0.180482
$565$	−19.9812	−0.840616
$566$	12.1021	0.508691
$567$	−1.69202	−0.0710582
$568$	−14.9487	−0.627233
$569$	14.8436	0.622274	0.311137	−	0.950365i	$-0.399290\pi$
$569$	14.8436	0.622274	0.311137	+	0.950365i	$0.399290\pi$
$570$	6.51573	0.272914
$571$	0.965557	0.0404073	0.0202037	−	0.999796i	$-0.493569\pi$
$571$	0.965557	0.0404073	0.0202037	+	0.999796i	$0.493569\pi$
$572$	0	0
$573$	−24.3260	−1.01623
$574$	−5.09246	−0.212555
$575$	8.94869	0.373186
$576$	1.00000	0.0416667
$577$	27.8431	1.15912	0.579561	−	0.814929i	$-0.303224\pi$
$577$	27.8431	1.15912	0.579561	+	0.814929i	$0.303224\pi$
$578$	−11.4450	−0.476051
$579$	0.521106	0.0216564
$580$	9.07606	0.376863
$581$	13.0610	0.541862
$582$	4.40581	0.182627
$583$	15.5133	0.642497
$584$	−6.72886	−0.278442
$585$	0	0
$586$	4.01075	0.165683
$587$	−1.47889	−0.0610405	−0.0305202	−	0.999534i	$-0.509716\pi$
$587$	−1.47889	−0.0610405	−0.0305202	+	0.999534i	$0.509716\pi$
$588$	4.13706	0.170610
$589$	69.4107	2.86002
$590$	4.32304	0.177977
$591$	4.87502	0.200531
$592$	6.18598	0.254242
$593$	−16.8552	−0.692159	−0.346079	−	0.938205i	$-0.612487\pi$
$593$	−16.8552	−0.692159	−0.346079	+	0.938205i	$0.612487\pi$
$594$	4.55496	0.186892
$595$	−3.98792	−0.163489
$596$	0.252356	0.0103369
$597$	18.4983	0.757084
$598$	0	0
$599$	−24.4862	−1.00048	−0.500239	−	0.865887i	$-0.666755\pi$
$599$	−24.4862	−1.00048	−0.500239	+	0.865887i	$0.666755\pi$
$600$	−1.00000	−0.0408248
$601$	−9.97344	−0.406825	−0.203413	−	0.979093i	$-0.565203\pi$
$601$	−9.97344	−0.406825	−0.203413	+	0.979093i	$0.565203\pi$
$602$	8.35690	0.340602
$603$	−3.24698	−0.132227
$604$	11.0804	0.450854
$605$	9.74764	0.396298
$606$	−2.31767	−0.0941488
$607$	−35.4476	−1.43877	−0.719386	−	0.694611i	$-0.755577\pi$
$607$	−35.4476	−1.43877	−0.719386	+	0.694611i	$0.755577\pi$
$608$	−6.51573	−0.264248
$609$	15.3569	0.622293
$610$	−14.3666	−0.581686
$611$	0	0
$612$	2.35690	0.0952719
$613$	16.1782	0.653432	0.326716	−	0.945123i	$-0.394058\pi$
$613$	16.1782	0.653432	0.326716	+	0.945123i	$0.394058\pi$
$614$	−15.2784	−0.616588
$615$	−3.00969	−0.121362
$616$	7.70709	0.310527
$617$	−17.6920	−0.712254	−0.356127	−	0.934438i	$-0.615903\pi$
$617$	−17.6920	−0.712254	−0.356127	+	0.934438i	$0.615903\pi$
$618$	9.00000	0.362033
$619$	1.23968	0.0498271	0.0249136	−	0.999690i	$-0.492069\pi$
$619$	1.23968	0.0498271	0.0249136	+	0.999690i	$0.492069\pi$
$620$	−10.6528	−0.427826
$621$	−8.94869	−0.359099
$622$	25.4330	1.01977
$623$	15.4397	0.618577
$624$	0	0
$625$	1.00000	0.0400000
$626$	−19.1226	−0.764292
$627$	−29.6789	−1.18526
$628$	5.03923	0.201087
$629$	14.5797	0.581331
$630$	−1.69202	−0.0674117
$631$	33.7845	1.34494	0.672469	−	0.740125i	$-0.265234\pi$
$631$	33.7845	1.34494	0.672469	+	0.740125i	$0.265234\pi$
$632$	5.67994	0.225936
$633$	−18.2500	−0.725371
$634$	−20.3230	−0.807131
$635$	−3.14244	−0.124704
$636$	3.40581	0.135049
$637$	0	0
$638$	−41.3411	−1.63671
$639$	−14.9487	−0.591361
$640$	1.00000	0.0395285
$641$	12.0785	0.477070	0.238535	−	0.971134i	$-0.423333\pi$
$641$	12.0785	0.477070	0.238535	+	0.971134i	$0.423333\pi$
$642$	3.06100	0.120808
$643$	−41.1377	−1.62231	−0.811155	−	0.584831i	$-0.801161\pi$
$643$	−41.1377	−1.62231	−0.811155	+	0.584831i	$0.801161\pi$
$644$	−15.1414	−0.596654
$645$	4.93900	0.194473
$646$	−15.3569	−0.604209
$647$	23.2054	0.912297	0.456148	−	0.889904i	$-0.349229\pi$
$647$	23.2054	0.912297	0.456148	+	0.889904i	$0.349229\pi$
$648$	1.00000	0.0392837
$649$	−19.6913	−0.772951
$650$	0	0
$651$	−18.0248	−0.706446
$652$	20.9825	0.821740
$653$	7.77538	0.304274	0.152137	−	0.988359i	$-0.451384\pi$
$653$	7.77538	0.304274	0.152137	+	0.988359i	$0.451384\pi$
$654$	−7.45473	−0.291503
$655$	−3.77479	−0.147493
$656$	3.00969	0.117509
$657$	−6.72886	−0.262518
$658$	7.25236	0.282726
$659$	−0.349600	−0.0136185	−0.00680924	−	0.999977i	$-0.502167\pi$
$659$	−0.349600	−0.0136185	−0.00680924	+	0.999977i	$0.502167\pi$
$660$	4.55496	0.177302
$661$	19.7855	0.769568	0.384784	−	0.923007i	$-0.374276\pi$
$661$	19.7855	0.769568	0.384784	+	0.923007i	$0.374276\pi$
$662$	−22.3588	−0.869000
$663$	0	0
$664$	−7.71917	−0.299562
$665$	11.0248	0.427522
$666$	6.18598	0.239702
$667$	81.2189	3.14481
$668$	−22.2325	−0.860201
$669$	−8.54048	−0.330194
$670$	−3.24698	−0.125442
$671$	65.4392	2.52625
$672$	1.69202	0.0652711
$673$	−16.6146	−0.640447	−0.320223	−	0.947342i	$-0.603758\pi$
$673$	−16.6146	−0.640447	−0.320223	+	0.947342i	$0.603758\pi$
$674$	−3.67025	−0.141373
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−5.85517	−0.225032	−0.112516	−	0.993650i	$-0.535891\pi$
$677$	−5.85517	−0.225032	−0.112516	+	0.993650i	$0.535891\pi$
$678$	19.9812	0.767374
$679$	7.45473	0.286086
$680$	2.35690	0.0903828
$681$	9.61596	0.368484
$682$	48.5230	1.85804
$683$	−2.19029	−0.0838092	−0.0419046	−	0.999122i	$-0.513343\pi$
$683$	−2.19029	−0.0838092	−0.0419046	+	0.999122i	$0.513343\pi$
$684$	−6.51573	−0.249135
$685$	14.5894	0.557432
$686$	18.8442	0.719473
$687$	13.4058	0.511464
$688$	−4.93900	−0.188298
$689$	0	0
$690$	−8.94869	−0.340671
$691$	11.8576	0.451083	0.225541	−	0.974234i	$-0.427585\pi$
$691$	11.8576	0.451083	0.225541	+	0.974234i	$0.427585\pi$
$692$	−11.6286	−0.442054
$693$	7.70709	0.292768
$694$	−21.7832	−0.826877
$695$	−14.3599	−0.544701
$696$	−9.07606	−0.344027
$697$	7.09352	0.268686
$698$	−1.74333	−0.0659861
$699$	0.153457	0.00580427
$700$	−1.69202	−0.0639524
$701$	22.7627	0.859736	0.429868	−	0.902892i	$-0.358560\pi$
$701$	22.7627	0.859736	0.429868	+	0.902892i	$0.358560\pi$
$702$	0	0
$703$	−40.3062	−1.52018
$704$	−4.55496	−0.171671
$705$	4.28621	0.161428
$706$	6.09544	0.229405
$707$	−3.92154	−0.147485
$708$	−4.32304	−0.162470
$709$	18.5854	0.697988	0.348994	−	0.937125i	$-0.386523\pi$
$709$	18.5854	0.697988	0.348994	+	0.937125i	$0.386523\pi$
$710$	−14.9487	−0.561014
$711$	5.67994	0.213014
$712$	−9.12498	−0.341973
$713$	−95.3285	−3.57008
$714$	3.98792	0.149244
$715$	0	0
$716$	−15.7875	−0.590005
$717$	−14.5851	−0.544690
$718$	25.9627	0.968919
$719$	−36.1575	−1.34845	−0.674224	−	0.738527i	$-0.735522\pi$
$719$	−36.1575	−1.34845	−0.674224	+	0.738527i	$0.735522\pi$
$720$	1.00000	0.0372678
$721$	15.2282	0.567128
$722$	23.4547	0.872895
$723$	−3.40044	−0.126464
$724$	−1.91185	−0.0710535
$725$	9.07606	0.337077
$726$	−9.74764	−0.361769
$727$	19.0164	0.705279	0.352639	−	0.935759i	$-0.385284\pi$
$727$	19.0164	0.705279	0.352639	+	0.935759i	$0.385284\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.72886	−0.249046
$731$	−11.6407	−0.430547
$732$	14.3666	0.531004
$733$	8.89248	0.328451	0.164226	−	0.986423i	$-0.447488\pi$
$733$	8.89248	0.328451	0.164226	+	0.986423i	$0.447488\pi$
$734$	27.8799	1.02907
$735$	4.13706	0.152598
$736$	8.94869	0.329853
$737$	14.7899	0.544791
$738$	3.00969	0.110788
$739$	−30.8595	−1.13518	−0.567592	−	0.823310i	$-0.692125\pi$
$739$	−30.8595	−1.13518	−0.567592	+	0.823310i	$0.692125\pi$
$740$	6.18598	0.227401
$741$	0	0
$742$	5.76271	0.211556
$743$	18.6300	0.683467	0.341733	−	0.939797i	$-0.388986\pi$
$743$	18.6300	0.683467	0.341733	+	0.939797i	$0.388986\pi$
$744$	10.6528	0.390550
$745$	0.252356	0.00924562
$746$	−25.1400	−0.920443
$747$	−7.71917	−0.282430
$748$	−10.7356	−0.392531
$749$	5.17928	0.189247
$750$	−1.00000	−0.0365148
$751$	12.5147	0.456667	0.228333	−	0.973583i	$-0.426672\pi$
$751$	12.5147	0.456667	0.228333	+	0.973583i	$0.426672\pi$
$752$	−4.28621	−0.156302
$753$	−1.80864	−0.0659106
$754$	0	0
$755$	11.0804	0.403256
$756$	1.69202	0.0615382
$757$	−4.75494	−0.172821	−0.0864106	−	0.996260i	$-0.527540\pi$
$757$	−4.75494	−0.172821	−0.0864106	+	0.996260i	$0.527540\pi$
$758$	−15.7463	−0.571932
$759$	40.7609	1.47953
$760$	−6.51573	−0.236350
$761$	−21.7748	−0.789336	−0.394668	−	0.918824i	$-0.629140\pi$
$761$	−21.7748	−0.789336	−0.394668	+	0.918824i	$0.629140\pi$
$762$	3.14244	0.113839
$763$	−12.6136	−0.456642
$764$	24.3260	0.880085
$765$	2.35690	0.0852137
$766$	11.8183	0.427014
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	11.4045	0.411256	0.205628	−	0.978630i	$-0.434076\pi$
$769$	11.4045	0.411256	0.205628	+	0.978630i	$0.434076\pi$
$770$	7.70709	0.277744
$771$	22.2664	0.801903
$772$	−0.521106	−0.0187550
$773$	49.6118	1.78441	0.892206	−	0.451630i	$-0.149157\pi$
$773$	49.6118	1.78441	0.892206	+	0.451630i	$0.149157\pi$
$774$	−4.93900	−0.177529
$775$	−10.6528	−0.382660
$776$	−4.40581	−0.158159
$777$	10.4668	0.375495
$778$	−6.14675	−0.220372
$779$	−19.6103	−0.702613
$780$	0	0
$781$	68.0907	2.43648
$782$	21.0911	0.754217
$783$	−9.07606	−0.324352
$784$	−4.13706	−0.147752
$785$	5.03923	0.179858
$786$	3.77479	0.134642
$787$	26.5532	0.946518	0.473259	−	0.880923i	$-0.343078\pi$
$787$	26.5532	0.946518	0.473259	+	0.880923i	$0.343078\pi$
$788$	−4.87502	−0.173665
$789$	−8.08815	−0.287946
$790$	5.67994	0.202083
$791$	33.8086	1.20210
$792$	−4.55496	−0.161853
$793$	0	0
$794$	−5.92825	−0.210386
$795$	3.40581	0.120792
$796$	−18.4983	−0.655654
$797$	9.44803	0.334666	0.167333	−	0.985900i	$-0.446484\pi$
$797$	9.44803	0.334666	0.167333	+	0.985900i	$0.446484\pi$
$798$	−11.0248	−0.390272
$799$	−10.1021	−0.357388
$800$	1.00000	0.0353553
$801$	−9.12498	−0.322415
$802$	9.49934	0.335433
$803$	30.6497	1.08160
$804$	3.24698	0.114512
$805$	−15.1414	−0.533663
$806$	0	0
$807$	−16.6843	−0.587313
$808$	2.31767	0.0815352
$809$	−27.3376	−0.961140	−0.480570	−	0.876956i	$-0.659570\pi$
$809$	−27.3376	−0.961140	−0.480570	+	0.876956i	$0.659570\pi$
$810$	1.00000	0.0351364
$811$	−2.86592	−0.100636	−0.0503180	−	0.998733i	$-0.516023\pi$
$811$	−2.86592	−0.100636	−0.0503180	+	0.998733i	$0.516023\pi$
$812$	−15.3569	−0.538921
$813$	−27.4077	−0.961231
$814$	−28.1769	−0.987599
$815$	20.9825	0.734986
$816$	−2.35690	−0.0825079
$817$	32.1812	1.12588
$818$	−9.00000	−0.314678
$819$	0	0
$820$	3.00969	0.105103
$821$	16.9739	0.592394	0.296197	−	0.955127i	$-0.404282\pi$
$821$	16.9739	0.592394	0.296197	+	0.955127i	$0.404282\pi$
$822$	−14.5894	−0.508864
$823$	−22.2083	−0.774134	−0.387067	−	0.922052i	$-0.626512\pi$
$823$	−22.2083	−0.774134	−0.387067	+	0.922052i	$0.626512\pi$
$824$	−9.00000	−0.313530
$825$	4.55496	0.158583
$826$	−7.31468	−0.254510
$827$	36.3618	1.26442	0.632212	−	0.774796i	$-0.282147\pi$
$827$	36.3618	1.26442	0.632212	+	0.774796i	$0.282147\pi$
$828$	8.94869	0.310988
$829$	−26.5478	−0.922042	−0.461021	−	0.887389i	$-0.652517\pi$
$829$	−26.5478	−0.922042	−0.461021	+	0.887389i	$0.652517\pi$
$830$	−7.71917	−0.267936
$831$	16.0411	0.556461
$832$	0	0
$833$	−9.75063	−0.337839
$834$	14.3599	0.497242
$835$	−22.2325	−0.769388
$836$	29.6789	1.02647
$837$	10.6528	0.368214
$838$	30.6450	1.05861
$839$	26.7668	0.924091	0.462046	−	0.886856i	$-0.347116\pi$
$839$	26.7668	0.924091	0.462046	+	0.886856i	$0.347116\pi$
$840$	1.69202	0.0583803
$841$	53.3749	1.84052
$842$	4.03252	0.138970
$843$	10.0935	0.347639
$844$	18.2500	0.628190
$845$	0	0
$846$	−4.28621	−0.147363
$847$	−16.4932	−0.566714
$848$	−3.40581	−0.116956
$849$	−12.1021	−0.415345
$850$	2.35690	0.0808409
$851$	55.3564	1.89759
$852$	14.9487	0.512134
$853$	−50.3967	−1.72555	−0.862775	−	0.505587i	$-0.831276\pi$
$853$	−50.3967	−1.72555	−0.862775	+	0.505587i	$0.831276\pi$
$854$	24.3086	0.831822
$855$	−6.51573	−0.222833
$856$	−3.06100	−0.104623
$857$	27.5114	0.939772	0.469886	−	0.882727i	$-0.344295\pi$
$857$	27.5114	0.939772	0.469886	+	0.882727i	$0.344295\pi$
$858$	0	0
$859$	2.26098	0.0771436	0.0385718	−	0.999256i	$-0.487719\pi$
$859$	2.26098	0.0771436	0.0385718	+	0.999256i	$0.487719\pi$
$860$	−4.93900	−0.168419
$861$	5.09246	0.173550
$862$	−27.6179	−0.940669
$863$	2.18406	0.0743463	0.0371732	−	0.999309i	$-0.488165\pi$
$863$	2.18406	0.0743463	0.0371732	+	0.999309i	$0.488165\pi$
$864$	−1.00000	−0.0340207
$865$	−11.6286	−0.395385
$866$	10.6756	0.362773
$867$	11.4450	0.388694
$868$	18.0248	0.611800
$869$	−25.8719	−0.877644
$870$	−9.07606	−0.307707
$871$	0	0
$872$	7.45473	0.252449
$873$	−4.40581	−0.149114
$874$	−58.3072	−1.97227
$875$	−1.69202	−0.0572008
$876$	6.72886	0.227347
$877$	19.9815	0.674727	0.337363	−	0.941375i	$-0.390465\pi$
$877$	19.9815	0.674727	0.337363	+	0.941375i	$0.390465\pi$
$878$	25.9299	0.875092
$879$	−4.01075	−0.135279
$880$	−4.55496	−0.153548
$881$	−38.4801	−1.29643	−0.648213	−	0.761459i	$-0.724484\pi$
$881$	−38.4801	−1.29643	−0.648213	+	0.761459i	$0.724484\pi$
$882$	−4.13706	−0.139302
$883$	−37.5512	−1.26370	−0.631850	−	0.775091i	$-0.717704\pi$
$883$	−37.5512	−1.26370	−0.631850	+	0.775091i	$0.717704\pi$
$884$	0	0
$885$	−4.32304	−0.145318
$886$	−35.3749	−1.18844
$887$	18.5676	0.623440	0.311720	−	0.950174i	$-0.399095\pi$
$887$	18.5676	0.623440	0.311720	+	0.950174i	$0.399095\pi$
$888$	−6.18598	−0.207588
$889$	5.31708	0.178329
$890$	−9.12498	−0.305870
$891$	−4.55496	−0.152597
$892$	8.54048	0.285957
$893$	27.9278	0.934567
$894$	−0.252356	−0.00844006
$895$	−15.7875	−0.527717
$896$	−1.69202	−0.0565265
$897$	0	0
$898$	−6.24698	−0.208464
$899$	−96.6854	−3.22464
$900$	1.00000	0.0333333
$901$	−8.02715	−0.267423
$902$	−13.7090	−0.456460
$903$	−8.35690	−0.278100
$904$	−19.9812	−0.664565
$905$	−1.91185	−0.0635522
$906$	−11.0804	−0.368121
$907$	60.1648	1.99774	0.998870	−	0.0475319i	$-0.0151356\pi$
$907$	60.1648	1.99774	0.998870	+	0.0475319i	$0.0151356\pi$
$908$	−9.61596	−0.319117
$909$	2.31767	0.0768722
$910$	0	0
$911$	−37.0737	−1.22831	−0.614153	−	0.789187i	$-0.710502\pi$
$911$	−37.0737	−1.22831	−0.614153	+	0.789187i	$0.710502\pi$
$912$	6.51573	0.215757
$913$	35.1605	1.16364
$914$	−29.6383	−0.980348
$915$	14.3666	0.474945
$916$	−13.4058	−0.442941
$917$	6.38703	0.210918
$918$	−2.35690	−0.0777892
$919$	−50.5526	−1.66758	−0.833788	−	0.552085i	$-0.813832\pi$
$919$	−50.5526	−1.66758	−0.833788	+	0.552085i	$0.813832\pi$
$920$	8.94869	0.295030
$921$	15.2784	0.503442
$922$	34.2911	1.12932
$923$	0	0
$924$	−7.70709	−0.253545
$925$	6.18598	0.203394
$926$	24.1769	0.794502
$927$	−9.00000	−0.295599
$928$	9.07606	0.297936
$929$	−12.7269	−0.417557	−0.208779	−	0.977963i	$-0.566949\pi$
$929$	−12.7269	−0.417557	−0.208779	+	0.977963i	$0.566949\pi$
$930$	10.6528	0.349319
$931$	26.9560	0.883447
$932$	−0.153457	−0.00502664
$933$	−25.4330	−0.832638
$934$	−0.521106	−0.0170511
$935$	−10.7356	−0.351090
$936$	0	0
$937$	−35.3279	−1.15411	−0.577057	−	0.816704i	$-0.695799\pi$
$937$	−35.3279	−1.15411	−0.577057	+	0.816704i	$0.695799\pi$
$938$	5.49396	0.179384
$939$	19.1226	0.624042
$940$	−4.28621	−0.139801
$941$	42.0267	1.37003	0.685015	−	0.728529i	$-0.259796\pi$
$941$	42.0267	1.37003	0.685015	+	0.728529i	$0.259796\pi$
$942$	−5.03923	−0.164187
$943$	26.9328	0.877052
$944$	4.32304	0.140703
$945$	1.69202	0.0550415
$946$	22.4969	0.731439
$947$	4.63640	0.150663	0.0753314	−	0.997159i	$-0.475999\pi$
$947$	4.63640	0.150663	0.0753314	+	0.997159i	$0.475999\pi$
$948$	−5.67994	−0.184476
$949$	0	0
$950$	−6.51573	−0.211398
$951$	20.3230	0.659020
$952$	−3.98792	−0.129249
$953$	−41.8998	−1.35727	−0.678633	−	0.734477i	$-0.737427\pi$
$953$	−41.8998	−1.35727	−0.678633	+	0.734477i	$0.737427\pi$
$954$	−3.40581	−0.110267
$955$	24.3260	0.787172
$956$	14.5851	0.471715
$957$	41.3411	1.33637
$958$	10.4168	0.336552
$959$	−24.6856	−0.797139
$960$	−1.00000	−0.0322749
$961$	82.4820	2.66071
$962$	0	0
$963$	−3.06100	−0.0986393
$964$	3.40044	0.109521
$965$	−0.521106	−0.0167750
$966$	15.1414	0.487166
$967$	−48.3545	−1.55498	−0.777488	−	0.628898i	$-0.783506\pi$
$967$	−48.3545	−1.55498	−0.777488	+	0.628898i	$0.783506\pi$
$968$	9.74764	0.313301
$969$	15.3569	0.493335
$970$	−4.40581	−0.141462
$971$	−18.9326	−0.607575	−0.303787	−	0.952740i	$-0.598251\pi$
$971$	−18.9326	−0.607575	−0.303787	+	0.952740i	$0.598251\pi$
$972$	−1.00000	−0.0320750
$973$	24.2972	0.778933
$974$	30.2349	0.968789
$975$	0	0
$976$	−14.3666	−0.459863
$977$	−54.7972	−1.75312	−0.876558	−	0.481296i	$-0.840166\pi$
$977$	−54.7972	−1.75312	−0.876558	+	0.481296i	$0.840166\pi$
$978$	−20.9825	−0.670948
$979$	41.5639	1.32839
$980$	−4.13706	−0.132154
$981$	7.45473	0.238011
$982$	−31.5883	−1.00802
$983$	24.6431	0.785993	0.392996	−	0.919540i	$-0.371438\pi$
$983$	24.6431	0.785993	0.392996	+	0.919540i	$0.371438\pi$
$984$	−3.00969	−0.0959454
$985$	−4.87502	−0.155331
$986$	21.3913	0.681239
$987$	−7.25236	−0.230845
$988$	0	0
$989$	−44.1976	−1.40540
$990$	−4.55496	−0.144766
$991$	23.1612	0.735741	0.367870	−	0.929877i	$-0.380087\pi$
$991$	23.1612	0.735741	0.367870	+	0.929877i	$0.380087\pi$
$992$	−10.6528	−0.338227
$993$	22.3588	0.709536
$994$	25.2935	0.802261
$995$	−18.4983	−0.586435
$996$	7.71917	0.244591
$997$	−11.5700	−0.366426	−0.183213	−	0.983073i	$-0.558650\pi$
$997$	−11.5700	−0.366426	−0.183213	+	0.983073i	$0.558650\pi$
$998$	−27.2862	−0.863730
$999$	−6.18598	−0.195716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.a.bt.1.1	yes	3
13.5	odd	4		5070.2.b.s.1351.1		6
13.8	odd	4		5070.2.b.s.1351.6		6
13.12	even	2		5070.2.a.bk.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bk.1.3	✓	3	13.12	even	2
5070.2.a.bt.1.1	yes	3	1.1	even	1	trivial
5070.2.b.s.1351.1		6	13.5	odd	4
5070.2.b.s.1351.6		6	13.8	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.55496 q^{11} -1.00000 q^{12} -1.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.35690 q^{17} +1.00000 q^{18} -6.51573 q^{19} +1.00000 q^{20} +1.69202 q^{21} -4.55496 q^{22} +8.94869 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +9.07606 q^{29} -1.00000 q^{30} -10.6528 q^{31} +1.00000 q^{32} +4.55496 q^{33} +2.35690 q^{34} -1.69202 q^{35} +1.00000 q^{36} +6.18598 q^{37} -6.51573 q^{38} +1.00000 q^{40} +3.00969 q^{41} +1.69202 q^{42} -4.93900 q^{43} -4.55496 q^{44} +1.00000 q^{45} +8.94869 q^{46} -4.28621 q^{47} -1.00000 q^{48} -4.13706 q^{49} +1.00000 q^{50} -2.35690 q^{51} -3.40581 q^{53} -1.00000 q^{54} -4.55496 q^{55} -1.69202 q^{56} +6.51573 q^{57} +9.07606 q^{58} +4.32304 q^{59} -1.00000 q^{60} -14.3666 q^{61} -10.6528 q^{62} -1.69202 q^{63} +1.00000 q^{64} +4.55496 q^{66} -3.24698 q^{67} +2.35690 q^{68} -8.94869 q^{69} -1.69202 q^{70} -14.9487 q^{71} +1.00000 q^{72} -6.72886 q^{73} +6.18598 q^{74} -1.00000 q^{75} -6.51573 q^{76} +7.70709 q^{77} +5.67994 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.00969 q^{82} -7.71917 q^{83} +1.69202 q^{84} +2.35690 q^{85} -4.93900 q^{86} -9.07606 q^{87} -4.55496 q^{88} -9.12498 q^{89} +1.00000 q^{90} +8.94869 q^{92} +10.6528 q^{93} -4.28621 q^{94} -6.51573 q^{95} -1.00000 q^{96} -4.40581 q^{97} -4.13706 q^{98} -4.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{8} + 3 q^{9} + 3 q^{10} - 14 q^{11} - 3 q^{12} - 3 q^{15} + 3 q^{16} + 3 q^{17} + 3 q^{18} - 7 q^{19} + 3 q^{20} - 14 q^{22} - 5 q^{23} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 12 q^{29} - 3 q^{30} - 14 q^{31} + 3 q^{32} + 14 q^{33} + 3 q^{34} + 3 q^{36} + 4 q^{37} - 7 q^{38} + 3 q^{40} - 13 q^{41} - 5 q^{43} - 14 q^{44} + 3 q^{45} - 5 q^{46} - 21 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 3 q^{51} + 3 q^{53} - 3 q^{54} - 14 q^{55} + 7 q^{57} + 12 q^{58} - 7 q^{59} - 3 q^{60} - 17 q^{61} - 14 q^{62} + 3 q^{64} + 14 q^{66} - 5 q^{67} + 3 q^{68} + 5 q^{69} - 13 q^{71} + 3 q^{72} + 13 q^{73} + 4 q^{74} - 3 q^{75} - 7 q^{76} - 7 q^{77} - 7 q^{79} + 3 q^{80} + 3 q^{81} - 13 q^{82} - 12 q^{83} + 3 q^{85} - 5 q^{86} - 12 q^{87} - 14 q^{88} - 3 q^{89} + 3 q^{90} - 5 q^{92} + 14 q^{93} - 21 q^{94} - 7 q^{95} - 3 q^{96} - 7 q^{98} - 14 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 - 3 * q^6 + 3 * q^8 + 3 * q^9 + 3 * q^10 - 14 * q^11 - 3 * q^12 - 3 * q^15 + 3 * q^16 + 3 * q^17 + 3 * q^18 - 7 * q^19 + 3 * q^20 - 14 * q^22 - 5 * q^23 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 12 * q^29 - 3 * q^30 - 14 * q^31 + 3 * q^32 + 14 * q^33 + 3 * q^34 + 3 * q^36 + 4 * q^37 - 7 * q^38 + 3 * q^40 - 13 * q^41 - 5 * q^43 - 14 * q^44 + 3 * q^45 - 5 * q^46 - 21 * q^47 - 3 * q^48 - 7 * q^49 + 3 * q^50 - 3 * q^51 + 3 * q^53 - 3 * q^54 - 14 * q^55 + 7 * q^57 + 12 * q^58 - 7 * q^59 - 3 * q^60 - 17 * q^61 - 14 * q^62 + 3 * q^64 + 14 * q^66 - 5 * q^67 + 3 * q^68 + 5 * q^69 - 13 * q^71 + 3 * q^72 + 13 * q^73 + 4 * q^74 - 3 * q^75 - 7 * q^76 - 7 * q^77 - 7 * q^79 + 3 * q^80 + 3 * q^81 - 13 * q^82 - 12 * q^83 + 3 * q^85 - 5 * q^86 - 12 * q^87 - 14 * q^88 - 3 * q^89 + 3 * q^90 - 5 * q^92 + 14 * q^93 - 21 * q^94 - 7 * q^95 - 3 * q^96 - 7 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more