LMFDB - Embedded newform 5070.2.a.bl.1.3

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			1.3
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.91185 q^{11} -1.00000 q^{12} -4.24698 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.33513 q^{17} -1.00000 q^{18} +4.85086 q^{19} +1.00000 q^{20} -4.24698 q^{21} +1.91185 q^{22} -0.445042 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.24698 q^{28} -8.56465 q^{29} +1.00000 q^{30} -5.29590 q^{31} -1.00000 q^{32} +1.91185 q^{33} +3.33513 q^{34} +4.24698 q^{35} +1.00000 q^{36} -1.75302 q^{37} -4.85086 q^{38} -1.00000 q^{40} -3.24698 q^{41} +4.24698 q^{42} -1.97823 q^{43} -1.91185 q^{44} +1.00000 q^{45} +0.445042 q^{46} -10.3840 q^{47} -1.00000 q^{48} +11.0368 q^{49} -1.00000 q^{50} +3.33513 q^{51} -1.14914 q^{53} +1.00000 q^{54} -1.91185 q^{55} -4.24698 q^{56} -4.85086 q^{57} +8.56465 q^{58} -13.4601 q^{59} -1.00000 q^{60} +1.13169 q^{61} +5.29590 q^{62} +4.24698 q^{63} +1.00000 q^{64} -1.91185 q^{66} -13.5308 q^{67} -3.33513 q^{68} +0.445042 q^{69} -4.24698 q^{70} +2.14675 q^{71} -1.00000 q^{72} -5.15883 q^{73} +1.75302 q^{74} -1.00000 q^{75} +4.85086 q^{76} -8.11960 q^{77} -14.5526 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.24698 q^{82} +9.49157 q^{83} -4.24698 q^{84} -3.33513 q^{85} +1.97823 q^{86} +8.56465 q^{87} +1.91185 q^{88} +1.25667 q^{89} -1.00000 q^{90} -0.445042 q^{92} +5.29590 q^{93} +10.3840 q^{94} +4.85086 q^{95} +1.00000 q^{96} +5.01507 q^{97} -11.0368 q^{98} -1.91185 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 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 + 8 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{14} - 3 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} + q^{19} + 3 q^{20} - 8 q^{21} + 2 q^{22} - q^{23} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 8 q^{28} - 4 q^{29} + 3 q^{30} - 2 q^{31} - 3 q^{32} + 2 q^{33} + 9 q^{34} + 8 q^{35} + 3 q^{36} - 10 q^{37} - q^{38} - 3 q^{40} - 5 q^{41} + 8 q^{42} - 9 q^{43} - 2 q^{44} + 3 q^{45} + q^{46} - 21 q^{47} - 3 q^{48} + 5 q^{49} - 3 q^{50} + 9 q^{51} - 17 q^{53} + 3 q^{54} - 2 q^{55} - 8 q^{56} - q^{57} + 4 q^{58} - 15 q^{59} - 3 q^{60} + q^{61} + 2 q^{62} + 8 q^{63} + 3 q^{64} - 2 q^{66} - 3 q^{67} - 9 q^{68} + q^{69} - 8 q^{70} - 21 q^{71} - 3 q^{72} - 7 q^{73} + 10 q^{74} - 3 q^{75} + q^{76} - 3 q^{77} - 3 q^{79} + 3 q^{80} + 3 q^{81} + 5 q^{82} - 22 q^{83} - 8 q^{84} - 9 q^{85} + 9 q^{86} + 4 q^{87} + 2 q^{88} - 23 q^{89} - 3 q^{90} - q^{92} + 2 q^{93} + 21 q^{94} + q^{95} + 3 q^{96} - 10 q^{97} - 5 q^{98} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 8 * q^14 - 3 * q^15 + 3 * q^16 - 9 * q^17 - 3 * q^18 + q^19 + 3 * q^20 - 8 * q^21 + 2 * q^22 - q^23 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 8 * q^28 - 4 * q^29 + 3 * q^30 - 2 * q^31 - 3 * q^32 + 2 * q^33 + 9 * q^34 + 8 * q^35 + 3 * q^36 - 10 * q^37 - q^38 - 3 * q^40 - 5 * q^41 + 8 * q^42 - 9 * q^43 - 2 * q^44 + 3 * q^45 + q^46 - 21 * q^47 - 3 * q^48 + 5 * q^49 - 3 * q^50 + 9 * q^51 - 17 * q^53 + 3 * q^54 - 2 * q^55 - 8 * q^56 - q^57 + 4 * q^58 - 15 * q^59 - 3 * q^60 + q^61 + 2 * q^62 + 8 * q^63 + 3 * q^64 - 2 * q^66 - 3 * q^67 - 9 * q^68 + q^69 - 8 * q^70 - 21 * q^71 - 3 * q^72 - 7 * q^73 + 10 * q^74 - 3 * q^75 + q^76 - 3 * q^77 - 3 * q^79 + 3 * q^80 + 3 * q^81 + 5 * q^82 - 22 * q^83 - 8 * q^84 - 9 * q^85 + 9 * q^86 + 4 * q^87 + 2 * q^88 - 23 * q^89 - 3 * q^90 - q^92 + 2 * q^93 + 21 * q^94 + q^95 + 3 * q^96 - 10 * q^97 - 5 * q^98 - 2 * 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
$7$	4.24698	1.60521	0.802604	−	0.596513i	$-0.203447\pi$
$7$	4.24698	1.60521	0.802604	+	0.596513i	$0.203447\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.91185	−0.576446	−0.288223	−	0.957563i	$-0.593064\pi$
$11$	−1.91185	−0.576446	−0.288223	+	0.957563i	$0.593064\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−4.24698	−1.13505
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−3.33513	−0.808887	−0.404443	−	0.914563i	$-0.632535\pi$
$17$	−3.33513	−0.808887	−0.404443	+	0.914563i	$0.632535\pi$
$18$	−1.00000	−0.235702
$19$	4.85086	1.11286	0.556431	−	0.830894i	$-0.312170\pi$
$19$	4.85086	1.11286	0.556431	+	0.830894i	$0.312170\pi$
$20$	1.00000	0.223607
$21$	−4.24698	−0.926767
$22$	1.91185	0.407609
$23$	−0.445042	−0.0927976	−0.0463988	−	0.998923i	$-0.514775\pi$
$23$	−0.445042	−0.0927976	−0.0463988	+	0.998923i	$0.514775\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	4.24698	0.802604
$29$	−8.56465	−1.59041	−0.795207	−	0.606337i	$-0.792638\pi$
$29$	−8.56465	−1.59041	−0.795207	+	0.606337i	$0.792638\pi$
$30$	1.00000	0.182574
$31$	−5.29590	−0.951171	−0.475586	−	0.879669i	$-0.657764\pi$
$31$	−5.29590	−0.951171	−0.475586	+	0.879669i	$0.657764\pi$
$32$	−1.00000	−0.176777
$33$	1.91185	0.332811
$34$	3.33513	0.571969
$35$	4.24698	0.717871
$36$	1.00000	0.166667
$37$	−1.75302	−0.288195	−0.144097	−	0.989564i	$-0.546028\pi$
$37$	−1.75302	−0.288195	−0.144097	+	0.989564i	$0.546028\pi$
$38$	−4.85086	−0.786913
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−3.24698	−0.507093	−0.253547	−	0.967323i	$-0.581597\pi$
$41$	−3.24698	−0.507093	−0.253547	+	0.967323i	$0.581597\pi$
$42$	4.24698	0.655323
$43$	−1.97823	−0.301677	−0.150839	−	0.988558i	$-0.548197\pi$
$43$	−1.97823	−0.301677	−0.150839	+	0.988558i	$0.548197\pi$
$44$	−1.91185	−0.288223
$45$	1.00000	0.149071
$46$	0.445042	0.0656178
$47$	−10.3840	−1.51467	−0.757334	−	0.653028i	$-0.773499\pi$
$47$	−10.3840	−1.51467	−0.757334	+	0.653028i	$0.773499\pi$
$48$	−1.00000	−0.144338
$49$	11.0368	1.57669
$50$	−1.00000	−0.141421
$51$	3.33513	0.467011
$52$	0	0
$53$	−1.14914	−0.157847	−0.0789236	−	0.996881i	$-0.525148\pi$
$53$	−1.14914	−0.157847	−0.0789236	+	0.996881i	$0.525148\pi$
$54$	1.00000	0.136083
$55$	−1.91185	−0.257794
$56$	−4.24698	−0.567527
$57$	−4.85086	−0.642511
$58$	8.56465	1.12459
$59$	−13.4601	−1.75236	−0.876178	−	0.481987i	$-0.839915\pi$
$59$	−13.4601	−1.75236	−0.876178	+	0.481987i	$0.839915\pi$
$60$	−1.00000	−0.129099
$61$	1.13169	0.144898	0.0724488	−	0.997372i	$-0.476919\pi$
$61$	1.13169	0.144898	0.0724488	+	0.997372i	$0.476919\pi$
$62$	5.29590	0.672580
$63$	4.24698	0.535069
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.91185	−0.235333
$67$	−13.5308	−1.65305	−0.826525	−	0.562900i	$-0.809686\pi$
$67$	−13.5308	−1.65305	−0.826525	+	0.562900i	$0.809686\pi$
$68$	−3.33513	−0.404443
$69$	0.445042	0.0535767
$70$	−4.24698	−0.507611
$71$	2.14675	0.254773	0.127386	−	0.991853i	$-0.459341\pi$
$71$	2.14675	0.254773	0.127386	+	0.991853i	$0.459341\pi$
$72$	−1.00000	−0.117851
$73$	−5.15883	−0.603796	−0.301898	−	0.953340i	$-0.597620\pi$
$73$	−5.15883	−0.603796	−0.301898	+	0.953340i	$0.597620\pi$
$74$	1.75302	0.203784
$75$	−1.00000	−0.115470
$76$	4.85086	0.556431
$77$	−8.11960	−0.925315
$78$	0	0
$79$	−14.5526	−1.63729	−0.818646	−	0.574299i	$-0.805275\pi$
$79$	−14.5526	−1.63729	−0.818646	+	0.574299i	$0.805275\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	3.24698	0.358569
$83$	9.49157	1.04183	0.520917	−	0.853607i	$-0.325590\pi$
$83$	9.49157	1.04183	0.520917	+	0.853607i	$0.325590\pi$
$84$	−4.24698	−0.463383
$85$	−3.33513	−0.361745
$86$	1.97823	0.213318
$87$	8.56465	0.918227
$88$	1.91185	0.203804
$89$	1.25667	0.133207	0.0666033	−	0.997780i	$-0.478784\pi$
$89$	1.25667	0.133207	0.0666033	+	0.997780i	$0.478784\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−0.445042	−0.0463988
$93$	5.29590	0.549159
$94$	10.3840	1.07103
$95$	4.85086	0.497687
$96$	1.00000	0.102062
$97$	5.01507	0.509203	0.254601	−	0.967046i	$-0.418056\pi$
$97$	5.01507	0.509203	0.254601	+	0.967046i	$0.418056\pi$
$98$	−11.0368	−1.11489
$99$	−1.91185	−0.192149
$100$	1.00000	0.100000
$101$	14.8116	1.47381	0.736906	−	0.675995i	$-0.236286\pi$
$101$	14.8116	1.47381	0.736906	+	0.675995i	$0.236286\pi$
$102$	−3.33513	−0.330227
$103$	−13.6896	−1.34888	−0.674440	−	0.738330i	$-0.735615\pi$
$103$	−13.6896	−1.34888	−0.674440	+	0.738330i	$0.735615\pi$
$104$	0	0
$105$	−4.24698	−0.414463
$106$	1.14914	0.111615
$107$	−12.8334	−1.24065	−0.620326	−	0.784344i	$-0.713000\pi$
$107$	−12.8334	−1.24065	−0.620326	+	0.784344i	$0.713000\pi$
$108$	−1.00000	−0.0962250
$109$	−10.8726	−1.04141	−0.520704	−	0.853737i	$-0.674331\pi$
$109$	−10.8726	−1.04141	−0.520704	+	0.853737i	$0.674331\pi$
$110$	1.91185	0.182288
$111$	1.75302	0.166389
$112$	4.24698	0.401302
$113$	−11.2295	−1.05638	−0.528192	−	0.849125i	$-0.677130\pi$
$113$	−11.2295	−1.05638	−0.528192	+	0.849125i	$0.677130\pi$
$114$	4.85086	0.454324
$115$	−0.445042	−0.0415004
$116$	−8.56465	−0.795207
$117$	0	0
$118$	13.4601	1.23910
$119$	−14.1642	−1.29843
$120$	1.00000	0.0912871
$121$	−7.34481	−0.667710
$122$	−1.13169	−0.102458
$123$	3.24698	0.292770
$124$	−5.29590	−0.475586
$125$	1.00000	0.0894427
$126$	−4.24698	−0.378351
$127$	20.8267	1.84807	0.924035	−	0.382308i	$-0.124871\pi$
$127$	20.8267	1.84807	0.924035	+	0.382308i	$0.124871\pi$
$128$	−1.00000	−0.0883883
$129$	1.97823	0.174173
$130$	0	0
$131$	17.5036	1.52930	0.764650	−	0.644445i	$-0.222912\pi$
$131$	17.5036	1.52930	0.764650	+	0.644445i	$0.222912\pi$
$132$	1.91185	0.166406
$133$	20.6015	1.78638
$134$	13.5308	1.16888
$135$	−1.00000	−0.0860663
$136$	3.33513	0.285985
$137$	0.929312	0.0793965	0.0396983	−	0.999212i	$-0.487360\pi$
$137$	0.929312	0.0793965	0.0396983	+	0.999212i	$0.487360\pi$
$138$	−0.445042	−0.0378845
$139$	3.12200	0.264804	0.132402	−	0.991196i	$-0.457731\pi$
$139$	3.12200	0.264804	0.132402	+	0.991196i	$0.457731\pi$
$140$	4.24698	0.358935
$141$	10.3840	0.874494
$142$	−2.14675	−0.180151
$143$	0	0
$144$	1.00000	0.0833333
$145$	−8.56465	−0.711255
$146$	5.15883	0.426948
$147$	−11.0368	−0.910303
$148$	−1.75302	−0.144097
$149$	−6.57673	−0.538787	−0.269393	−	0.963030i	$-0.586823\pi$
$149$	−6.57673	−0.538787	−0.269393	+	0.963030i	$0.586823\pi$
$150$	1.00000	0.0816497
$151$	−6.29590	−0.512353	−0.256176	−	0.966630i	$-0.582463\pi$
$151$	−6.29590	−0.512353	−0.256176	+	0.966630i	$0.582463\pi$
$152$	−4.85086	−0.393456
$153$	−3.33513	−0.269629
$154$	8.11960	0.654296
$155$	−5.29590	−0.425377
$156$	0	0
$157$	19.9148	1.58938	0.794689	−	0.607017i	$-0.207634\pi$
$157$	19.9148	1.58938	0.794689	+	0.607017i	$0.207634\pi$
$158$	14.5526	1.15774
$159$	1.14914	0.0911331
$160$	−1.00000	−0.0790569
$161$	−1.89008	−0.148959
$162$	−1.00000	−0.0785674
$163$	−13.1075	−1.02666	−0.513330	−	0.858191i	$-0.671588\pi$
$163$	−13.1075	−1.02666	−0.513330	+	0.858191i	$0.671588\pi$
$164$	−3.24698	−0.253547
$165$	1.91185	0.148838
$166$	−9.49157	−0.736688
$167$	−21.5351	−1.66644	−0.833218	−	0.552944i	$-0.813504\pi$
$167$	−21.5351	−1.66644	−0.833218	+	0.552944i	$0.813504\pi$
$168$	4.24698	0.327662
$169$	0	0
$170$	3.33513	0.255792
$171$	4.85086	0.370954
$172$	−1.97823	−0.150839
$173$	−13.9541	−1.06091	−0.530454	−	0.847714i	$-0.677979\pi$
$173$	−13.9541	−1.06091	−0.530454	+	0.847714i	$0.677979\pi$
$174$	−8.56465	−0.649284
$175$	4.24698	0.321041
$176$	−1.91185	−0.144111
$177$	13.4601	1.01172
$178$	−1.25667	−0.0941913
$179$	8.32975	0.622595	0.311297	−	0.950313i	$-0.399236\pi$
$179$	8.32975	0.622595	0.311297	+	0.950313i	$0.399236\pi$
$180$	1.00000	0.0745356
$181$	17.1371	1.27379	0.636894	−	0.770951i	$-0.280219\pi$
$181$	17.1371	1.27379	0.636894	+	0.770951i	$0.280219\pi$
$182$	0	0
$183$	−1.13169	−0.0836567
$184$	0.445042	0.0328089
$185$	−1.75302	−0.128885
$186$	−5.29590	−0.388314
$187$	6.37627	0.466279
$188$	−10.3840	−0.757334
$189$	−4.24698	−0.308922
$190$	−4.85086	−0.351918
$191$	23.1685	1.67642	0.838208	−	0.545351i	$-0.183604\pi$
$191$	23.1685	1.67642	0.838208	+	0.545351i	$0.183604\pi$
$192$	−1.00000	−0.0721688
$193$	−6.38942	−0.459920	−0.229960	−	0.973200i	$-0.573860\pi$
$193$	−6.38942	−0.459920	−0.229960	+	0.973200i	$0.573860\pi$
$194$	−5.01507	−0.360061
$195$	0	0
$196$	11.0368	0.788345
$197$	14.8605	1.05877	0.529385	−	0.848382i	$-0.322423\pi$
$197$	14.8605	1.05877	0.529385	+	0.848382i	$0.322423\pi$
$198$	1.91185	0.135870
$199$	26.3860	1.87045	0.935226	−	0.354052i	$-0.115197\pi$
$199$	26.3860	1.87045	0.935226	+	0.354052i	$0.115197\pi$
$200$	−1.00000	−0.0707107
$201$	13.5308	0.954389
$202$	−14.8116	−1.04214
$203$	−36.3739	−2.55295
$204$	3.33513	0.233505
$205$	−3.24698	−0.226779
$206$	13.6896	0.953802
$207$	−0.445042	−0.0309325
$208$	0	0
$209$	−9.27413	−0.641505
$210$	4.24698	0.293069
$211$	17.4383	1.20050	0.600252	−	0.799811i	$-0.295067\pi$
$211$	17.4383	1.20050	0.600252	+	0.799811i	$0.295067\pi$
$212$	−1.14914	−0.0789236
$213$	−2.14675	−0.147093
$214$	12.8334	0.877273
$215$	−1.97823	−0.134914
$216$	1.00000	0.0680414
$217$	−22.4916	−1.52683
$218$	10.8726	0.736387
$219$	5.15883	0.348602
$220$	−1.91185	−0.128897
$221$	0	0
$222$	−1.75302	−0.117655
$223$	9.95838	0.666862	0.333431	−	0.942774i	$-0.391794\pi$
$223$	9.95838	0.666862	0.333431	+	0.942774i	$0.391794\pi$
$224$	−4.24698	−0.283763
$225$	1.00000	0.0666667
$226$	11.2295	0.746977
$227$	19.6112	1.30164	0.650820	−	0.759232i	$-0.274425\pi$
$227$	19.6112	1.30164	0.650820	+	0.759232i	$0.274425\pi$
$228$	−4.85086	−0.321256
$229$	8.72886	0.576819	0.288410	−	0.957507i	$-0.406874\pi$
$229$	8.72886	0.576819	0.288410	+	0.957507i	$0.406874\pi$
$230$	0.445042	0.0293452
$231$	8.11960	0.534231
$232$	8.56465	0.562297
$233$	0.975837	0.0639292	0.0319646	−	0.999489i	$-0.489824\pi$
$233$	0.975837	0.0639292	0.0319646	+	0.999489i	$0.489824\pi$
$234$	0	0
$235$	−10.3840	−0.677380
$236$	−13.4601	−0.876178
$237$	14.5526	0.945291
$238$	14.1642	0.918129
$239$	7.39373	0.478261	0.239130	−	0.970987i	$-0.423138\pi$
$239$	7.39373	0.478261	0.239130	+	0.970987i	$0.423138\pi$
$240$	−1.00000	−0.0645497
$241$	2.11529	0.136258	0.0681290	−	0.997677i	$-0.478297\pi$
$241$	2.11529	0.136258	0.0681290	+	0.997677i	$0.478297\pi$
$242$	7.34481	0.472143
$243$	−1.00000	−0.0641500
$244$	1.13169	0.0724488
$245$	11.0368	0.705118
$246$	−3.24698	−0.207020
$247$	0	0
$248$	5.29590	0.336290
$249$	−9.49157	−0.601504
$250$	−1.00000	−0.0632456
$251$	−1.62863	−0.102798	−0.0513991	−	0.998678i	$-0.516368\pi$
$251$	−1.62863	−0.102798	−0.0513991	+	0.998678i	$0.516368\pi$
$252$	4.24698	0.267535
$253$	0.850855	0.0534928
$254$	−20.8267	−1.30678
$255$	3.33513	0.208854
$256$	1.00000	0.0625000
$257$	−8.87263	−0.553459	−0.276730	−	0.960948i	$-0.589251\pi$
$257$	−8.87263	−0.553459	−0.276730	+	0.960948i	$0.589251\pi$
$258$	−1.97823	−0.123159
$259$	−7.44504	−0.462612
$260$	0	0
$261$	−8.56465	−0.530138
$262$	−17.5036	−1.08138
$263$	22.4185	1.38238	0.691192	−	0.722672i	$-0.257086\pi$
$263$	22.4185	1.38238	0.691192	+	0.722672i	$0.257086\pi$
$264$	−1.91185	−0.117666
$265$	−1.14914	−0.0705914
$266$	−20.6015	−1.26316
$267$	−1.25667	−0.0769068
$268$	−13.5308	−0.826525
$269$	26.9071	1.64055	0.820276	−	0.571967i	$-0.193820\pi$
$269$	26.9071	1.64055	0.820276	+	0.571967i	$0.193820\pi$
$270$	1.00000	0.0608581
$271$	−31.7265	−1.92725	−0.963623	−	0.267266i	$-0.913880\pi$
$271$	−31.7265	−1.92725	−0.963623	+	0.267266i	$0.913880\pi$
$272$	−3.33513	−0.202222
$273$	0	0
$274$	−0.929312	−0.0561418
$275$	−1.91185	−0.115289
$276$	0.445042	0.0267884
$277$	−30.6069	−1.83899	−0.919494	−	0.393104i	$-0.871401\pi$
$277$	−30.6069	−1.83899	−0.919494	+	0.393104i	$0.871401\pi$
$278$	−3.12200	−0.187245
$279$	−5.29590	−0.317057
$280$	−4.24698	−0.253806
$281$	−16.6649	−0.994143	−0.497072	−	0.867710i	$-0.665591\pi$
$281$	−16.6649	−0.994143	−0.497072	+	0.867710i	$0.665591\pi$
$282$	−10.3840	−0.618361
$283$	−4.32304	−0.256978	−0.128489	−	0.991711i	$-0.541013\pi$
$283$	−4.32304	−0.256978	−0.128489	+	0.991711i	$0.541013\pi$
$284$	2.14675	0.127386
$285$	−4.85086	−0.287340
$286$	0	0
$287$	−13.7899	−0.813989
$288$	−1.00000	−0.0589256
$289$	−5.87694	−0.345702
$290$	8.56465	0.502933
$291$	−5.01507	−0.293988
$292$	−5.15883	−0.301898
$293$	−5.62804	−0.328794	−0.164397	−	0.986394i	$-0.552568\pi$
$293$	−5.62804	−0.328794	−0.164397	+	0.986394i	$0.552568\pi$
$294$	11.0368	0.643681
$295$	−13.4601	−0.783678
$296$	1.75302	0.101892
$297$	1.91185	0.110937
$298$	6.57673	0.380980
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−8.40150	−0.484254
$302$	6.29590	0.362288
$303$	−14.8116	−0.850906
$304$	4.85086	0.278216
$305$	1.13169	0.0648002
$306$	3.33513	0.190656
$307$	−12.6635	−0.722747	−0.361373	−	0.932421i	$-0.617692\pi$
$307$	−12.6635	−0.722747	−0.361373	+	0.932421i	$0.617692\pi$
$308$	−8.11960	−0.462657
$309$	13.6896	0.778776
$310$	5.29590	0.300787
$311$	8.49635	0.481784	0.240892	−	0.970552i	$-0.422560\pi$
$311$	8.49635	0.481784	0.240892	+	0.970552i	$0.422560\pi$
$312$	0	0
$313$	−25.2610	−1.42784	−0.713918	−	0.700230i	$-0.753081\pi$
$313$	−25.2610	−1.42784	−0.713918	+	0.700230i	$0.753081\pi$
$314$	−19.9148	−1.12386
$315$	4.24698	0.239290
$316$	−14.5526	−0.818646
$317$	−4.96615	−0.278927	−0.139463	−	0.990227i	$-0.544538\pi$
$317$	−4.96615	−0.278927	−0.139463	+	0.990227i	$0.544538\pi$
$318$	−1.14914	−0.0644408
$319$	16.3744	0.916788
$320$	1.00000	0.0559017
$321$	12.8334	0.716290
$322$	1.89008	0.105330
$323$	−16.1782	−0.900180
$324$	1.00000	0.0555556
$325$	0	0
$326$	13.1075	0.725959
$327$	10.8726	0.601258
$328$	3.24698	0.179284
$329$	−44.1008	−2.43136
$330$	−1.91185	−0.105244
$331$	35.3749	1.94438	0.972191	−	0.234188i	$-0.0752430\pi$
$331$	35.3749	1.94438	0.972191	+	0.234188i	$0.0752430\pi$
$332$	9.49157	0.520917
$333$	−1.75302	−0.0960649
$334$	21.5351	1.17835
$335$	−13.5308	−0.739266
$336$	−4.24698	−0.231692
$337$	−4.70171	−0.256118	−0.128059	−	0.991767i	$-0.540875\pi$
$337$	−4.70171	−0.256118	−0.128059	+	0.991767i	$0.540875\pi$
$338$	0	0
$339$	11.2295	0.609904
$340$	−3.33513	−0.180873
$341$	10.1250	0.548299
$342$	−4.85086	−0.262304
$343$	17.1444	0.925708
$344$	1.97823	0.106659
$345$	0.445042	0.0239602
$346$	13.9541	0.750175
$347$	14.9028	0.800022	0.400011	−	0.916510i	$-0.369006\pi$
$347$	14.9028	0.800022	0.400011	+	0.916510i	$0.369006\pi$
$348$	8.56465	0.459113
$349$	−35.2669	−1.88780	−0.943898	−	0.330237i	$-0.892871\pi$
$349$	−35.2669	−1.88780	−0.943898	+	0.330237i	$0.892871\pi$
$350$	−4.24698	−0.227011
$351$	0	0
$352$	1.91185	0.101902
$353$	−10.5574	−0.561911	−0.280956	−	0.959721i	$-0.590651\pi$
$353$	−10.5574	−0.561911	−0.280956	+	0.959721i	$0.590651\pi$
$354$	−13.4601	−0.715397
$355$	2.14675	0.113938
$356$	1.25667	0.0666033
$357$	14.1642	0.749650
$358$	−8.32975	−0.440241
$359$	−28.7724	−1.51855	−0.759275	−	0.650770i	$-0.774446\pi$
$359$	−28.7724	−1.51855	−0.759275	+	0.650770i	$0.774446\pi$
$360$	−1.00000	−0.0527046
$361$	4.53079	0.238463
$362$	−17.1371	−0.900704
$363$	7.34481	0.385503
$364$	0	0
$365$	−5.15883	−0.270026
$366$	1.13169	0.0591542
$367$	−24.1511	−1.26068	−0.630338	−	0.776321i	$-0.717084\pi$
$367$	−24.1511	−1.26068	−0.630338	+	0.776321i	$0.717084\pi$
$368$	−0.445042	−0.0231994
$369$	−3.24698	−0.169031
$370$	1.75302	0.0911352
$371$	−4.88040	−0.253377
$372$	5.29590	0.274579
$373$	−21.0664	−1.09078	−0.545388	−	0.838184i	$-0.683618\pi$
$373$	−21.0664	−1.09078	−0.545388	+	0.838184i	$0.683618\pi$
$374$	−6.37627	−0.329709
$375$	−1.00000	−0.0516398
$376$	10.3840	0.535516
$377$	0	0
$378$	4.24698	0.218441
$379$	8.42327	0.432674	0.216337	−	0.976319i	$-0.430589\pi$
$379$	8.42327	0.432674	0.216337	+	0.976319i	$0.430589\pi$
$380$	4.85086	0.248844
$381$	−20.8267	−1.06698
$382$	−23.1685	−1.18540
$383$	−10.6377	−0.543562	−0.271781	−	0.962359i	$-0.587613\pi$
$383$	−10.6377	−0.543562	−0.271781	+	0.962359i	$0.587613\pi$
$384$	1.00000	0.0510310
$385$	−8.11960	−0.413813
$386$	6.38942	0.325213
$387$	−1.97823	−0.100559
$388$	5.01507	0.254601
$389$	−25.7429	−1.30521	−0.652607	−	0.757696i	$-0.726325\pi$
$389$	−25.7429	−1.30521	−0.652607	+	0.757696i	$0.726325\pi$
$390$	0	0
$391$	1.48427	0.0750628
$392$	−11.0368	−0.557444
$393$	−17.5036	−0.882942
$394$	−14.8605	−0.748663
$395$	−14.5526	−0.732219
$396$	−1.91185	−0.0960743
$397$	−11.4373	−0.574020	−0.287010	−	0.957928i	$-0.592661\pi$
$397$	−11.4373	−0.574020	−0.287010	+	0.957928i	$0.592661\pi$
$398$	−26.3860	−1.32261
$399$	−20.6015	−1.03136
$400$	1.00000	0.0500000
$401$	−22.1511	−1.10617	−0.553086	−	0.833124i	$-0.686550\pi$
$401$	−22.1511	−1.10617	−0.553086	+	0.833124i	$0.686550\pi$
$402$	−13.5308	−0.674855
$403$	0	0
$404$	14.8116	0.736906
$405$	1.00000	0.0496904
$406$	36.3739	1.80521
$407$	3.35152	0.166129
$408$	−3.33513	−0.165113
$409$	22.0965	1.09260	0.546301	−	0.837589i	$-0.316035\pi$
$409$	22.0965	1.09260	0.546301	+	0.837589i	$0.316035\pi$
$410$	3.24698	0.160357
$411$	−0.929312	−0.0458396
$412$	−13.6896	−0.674440
$413$	−57.1648	−2.81290
$414$	0.445042	0.0218726
$415$	9.49157	0.465923
$416$	0	0
$417$	−3.12200	−0.152885
$418$	9.27413	0.453612
$419$	9.03444	0.441361	0.220681	−	0.975346i	$-0.429172\pi$
$419$	9.03444	0.441361	0.220681	+	0.975346i	$0.429172\pi$
$420$	−4.24698	−0.207231
$421$	−7.17928	−0.349896	−0.174948	−	0.984578i	$-0.555976\pi$
$421$	−7.17928	−0.349896	−0.174948	+	0.984578i	$0.555976\pi$
$422$	−17.4383	−0.848885
$423$	−10.3840	−0.504889
$424$	1.14914	0.0558074
$425$	−3.33513	−0.161777
$426$	2.14675	0.104010
$427$	4.80625	0.232591
$428$	−12.8334	−0.620326
$429$	0	0
$430$	1.97823	0.0953987
$431$	−8.36360	−0.402860	−0.201430	−	0.979503i	$-0.564559\pi$
$431$	−8.36360	−0.402860	−0.201430	+	0.979503i	$0.564559\pi$
$432$	−1.00000	−0.0481125
$433$	−12.2851	−0.590386	−0.295193	−	0.955438i	$-0.595384\pi$
$433$	−12.2851	−0.590386	−0.295193	+	0.955438i	$0.595384\pi$
$434$	22.4916	1.07963
$435$	8.56465	0.410643
$436$	−10.8726	−0.520704
$437$	−2.15883	−0.103271
$438$	−5.15883	−0.246499
$439$	20.4849	0.977689	0.488845	−	0.872371i	$-0.337419\pi$
$439$	20.4849	0.977689	0.488845	+	0.872371i	$0.337419\pi$
$440$	1.91185	0.0911441
$441$	11.0368	0.525564
$442$	0	0
$443$	−0.928247	−0.0441024	−0.0220512	−	0.999757i	$-0.507020\pi$
$443$	−0.928247	−0.0441024	−0.0220512	+	0.999757i	$0.507020\pi$
$444$	1.75302	0.0831947
$445$	1.25667	0.0595718
$446$	−9.95838	−0.471543
$447$	6.57673	0.311069
$448$	4.24698	0.200651
$449$	22.9355	1.08240	0.541198	−	0.840895i	$-0.317971\pi$
$449$	22.9355	1.08240	0.541198	+	0.840895i	$0.317971\pi$
$450$	−1.00000	−0.0471405
$451$	6.20775	0.292312
$452$	−11.2295	−0.528192
$453$	6.29590	0.295807
$454$	−19.6112	−0.920398
$455$	0	0
$456$	4.85086	0.227162
$457$	−10.3773	−0.485431	−0.242716	−	0.970097i	$-0.578038\pi$
$457$	−10.3773	−0.485431	−0.242716	+	0.970097i	$0.578038\pi$
$458$	−8.72886	−0.407873
$459$	3.33513	0.155670
$460$	−0.445042	−0.0207502
$461$	−7.69740	−0.358504	−0.179252	−	0.983803i	$-0.557368\pi$
$461$	−7.69740	−0.358504	−0.179252	+	0.983803i	$0.557368\pi$
$462$	−8.11960	−0.377758
$463$	39.0640	1.81546	0.907729	−	0.419558i	$-0.137815\pi$
$463$	39.0640	1.81546	0.907729	+	0.419558i	$0.137815\pi$
$464$	−8.56465	−0.397604
$465$	5.29590	0.245591
$466$	−0.975837	−0.0452048
$467$	0.916166	0.0423951	0.0211976	−	0.999775i	$-0.493252\pi$
$467$	0.916166	0.0423951	0.0211976	+	0.999775i	$0.493252\pi$
$468$	0	0
$469$	−57.4650	−2.65349
$470$	10.3840	0.478980
$471$	−19.9148	−0.917627
$472$	13.4601	0.619552
$473$	3.78209	0.173901
$474$	−14.5526	−0.668421
$475$	4.85086	0.222572
$476$	−14.1642	−0.649216
$477$	−1.14914	−0.0526157
$478$	−7.39373	−0.338181
$479$	23.1715	1.05873	0.529367	−	0.848393i	$-0.322430\pi$
$479$	23.1715	1.05873	0.529367	+	0.848393i	$0.322430\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−2.11529	−0.0963490
$483$	1.89008	0.0860018
$484$	−7.34481	−0.333855
$485$	5.01507	0.227722
$486$	1.00000	0.0453609
$487$	11.9812	0.542921	0.271460	−	0.962450i	$-0.412493\pi$
$487$	11.9812	0.542921	0.271460	+	0.962450i	$0.412493\pi$
$488$	−1.13169	−0.0512290
$489$	13.1075	0.592743
$490$	−11.0368	−0.498593
$491$	−24.8745	−1.12257	−0.561286	−	0.827622i	$-0.689693\pi$
$491$	−24.8745	−1.12257	−0.561286	+	0.827622i	$0.689693\pi$
$492$	3.24698	0.146385
$493$	28.5642	1.28647
$494$	0	0
$495$	−1.91185	−0.0859314
$496$	−5.29590	−0.237793
$497$	9.11721	0.408963
$498$	9.49157	0.425327
$499$	42.0737	1.88348	0.941738	−	0.336347i	$-0.109192\pi$
$499$	42.0737	1.88348	0.941738	+	0.336347i	$0.109192\pi$
$500$	1.00000	0.0447214
$501$	21.5351	0.962118
$502$	1.62863	0.0726893
$503$	−20.0151	−0.892428	−0.446214	−	0.894926i	$-0.647228\pi$
$503$	−20.0151	−0.892428	−0.446214	+	0.894926i	$0.647228\pi$
$504$	−4.24698	−0.189176
$505$	14.8116	0.659109
$506$	−0.850855	−0.0378251
$507$	0	0
$508$	20.8267	0.924035
$509$	−25.6907	−1.13872	−0.569360	−	0.822088i	$-0.692809\pi$
$509$	−25.6907	−1.13872	−0.569360	+	0.822088i	$0.692809\pi$
$510$	−3.33513	−0.147682
$511$	−21.9095	−0.969217
$512$	−1.00000	−0.0441942
$513$	−4.85086	−0.214170
$514$	8.87263	0.391355
$515$	−13.6896	−0.603237
$516$	1.97823	0.0870867
$517$	19.8528	0.873124
$518$	7.44504	0.327116
$519$	13.9541	0.612516
$520$	0	0
$521$	−24.7342	−1.08363	−0.541813	−	0.840499i	$-0.682262\pi$
$521$	−24.7342	−1.08363	−0.541813	+	0.840499i	$0.682262\pi$
$522$	8.56465	0.374864
$523$	−18.7439	−0.819615	−0.409807	−	0.912172i	$-0.634404\pi$
$523$	−18.7439	−0.819615	−0.409807	+	0.912172i	$0.634404\pi$
$524$	17.5036	0.764650
$525$	−4.24698	−0.185353
$526$	−22.4185	−0.977492
$527$	17.6625	0.769390
$528$	1.91185	0.0832028
$529$	−22.8019	−0.991389
$530$	1.14914	0.0499157
$531$	−13.4601	−0.584119
$532$	20.6015	0.893188
$533$	0	0
$534$	1.25667	0.0543814
$535$	−12.8334	−0.554836
$536$	13.5308	0.584441
$537$	−8.32975	−0.359455
$538$	−26.9071	−1.16005
$539$	−21.1008	−0.908877
$540$	−1.00000	−0.0430331
$541$	43.7434	1.88068	0.940339	−	0.340239i	$-0.110508\pi$
$541$	43.7434	1.88068	0.940339	+	0.340239i	$0.110508\pi$
$542$	31.7265	1.36277
$543$	−17.1371	−0.735422
$544$	3.33513	0.142992
$545$	−10.8726	−0.465732
$546$	0	0
$547$	17.9681	0.768259	0.384130	−	0.923279i	$-0.374502\pi$
$547$	17.9681	0.768259	0.384130	+	0.923279i	$0.374502\pi$
$548$	0.929312	0.0396983
$549$	1.13169	0.0482992
$550$	1.91185	0.0815217
$551$	−41.5459	−1.76991
$552$	−0.445042	−0.0189422
$553$	−61.8044	−2.62819
$554$	30.6069	1.30036
$555$	1.75302	0.0744116
$556$	3.12200	0.132402
$557$	8.26981	0.350403	0.175202	−	0.984533i	$-0.443942\pi$
$557$	8.26981	0.350403	0.175202	+	0.984533i	$0.443942\pi$
$558$	5.29590	0.224193
$559$	0	0
$560$	4.24698	0.179468
$561$	−6.37627	−0.269206
$562$	16.6649	0.702965
$563$	19.8984	0.838619	0.419310	−	0.907843i	$-0.362272\pi$
$563$	19.8984	0.838619	0.419310	+	0.907843i	$0.362272\pi$
$564$	10.3840	0.437247
$565$	−11.2295	−0.472430
$566$	4.32304	0.181711
$567$	4.24698	0.178356
$568$	−2.14675	−0.0900757
$569$	14.6528	0.614277	0.307139	−	0.951665i	$-0.400629\pi$
$569$	14.6528	0.614277	0.307139	+	0.951665i	$0.400629\pi$
$570$	4.85086	0.203180
$571$	−9.47517	−0.396524	−0.198262	−	0.980149i	$-0.563530\pi$
$571$	−9.47517	−0.396524	−0.198262	+	0.980149i	$0.563530\pi$
$572$	0	0
$573$	−23.1685	−0.967879
$574$	13.7899	0.575577
$575$	−0.445042	−0.0185595
$576$	1.00000	0.0416667
$577$	8.79225	0.366026	0.183013	−	0.983110i	$-0.441415\pi$
$577$	8.79225	0.366026	0.183013	+	0.983110i	$0.441415\pi$
$578$	5.87694	0.244448
$579$	6.38942	0.265535
$580$	−8.56465	−0.355628
$581$	40.3105	1.67236
$582$	5.01507	0.207881
$583$	2.19700	0.0909903
$584$	5.15883	0.213474
$585$	0	0
$586$	5.62804	0.232492
$587$	2.38942	0.0986219	0.0493110	−	0.998783i	$-0.484297\pi$
$587$	2.38942	0.0986219	0.0493110	+	0.998783i	$0.484297\pi$
$588$	−11.0368	−0.455151
$589$	−25.6896	−1.05852
$590$	13.4601	0.554144
$591$	−14.8605	−0.611281
$592$	−1.75302	−0.0720487
$593$	−21.4034	−0.878933	−0.439467	−	0.898259i	$-0.644833\pi$
$593$	−21.4034	−0.878933	−0.439467	+	0.898259i	$0.644833\pi$
$594$	−1.91185	−0.0784443
$595$	−14.1642	−0.580676
$596$	−6.57673	−0.269393
$597$	−26.3860	−1.07991
$598$	0	0
$599$	12.4765	0.509776	0.254888	−	0.966971i	$-0.417961\pi$
$599$	12.4765	0.509776	0.254888	+	0.966971i	$0.417961\pi$
$600$	1.00000	0.0408248
$601$	−12.9487	−0.528188	−0.264094	−	0.964497i	$-0.585073\pi$
$601$	−12.9487	−0.528188	−0.264094	+	0.964497i	$0.585073\pi$
$602$	8.40150	0.342420
$603$	−13.5308	−0.551017
$604$	−6.29590	−0.256176
$605$	−7.34481	−0.298609
$606$	14.8116	0.601681
$607$	−11.5230	−0.467705	−0.233853	−	0.972272i	$-0.575133\pi$
$607$	−11.5230	−0.467705	−0.233853	+	0.972272i	$0.575133\pi$
$608$	−4.85086	−0.196728
$609$	36.3739	1.47394
$610$	−1.13169	−0.0458206
$611$	0	0
$612$	−3.33513	−0.134814
$613$	44.9842	1.81689	0.908447	−	0.417999i	$-0.137269\pi$
$613$	44.9842	1.81689	0.908447	+	0.417999i	$0.137269\pi$
$614$	12.6635	0.511059
$615$	3.24698	0.130931
$616$	8.11960	0.327148
$617$	−24.9318	−1.00372	−0.501859	−	0.864950i	$-0.667350\pi$
$617$	−24.9318	−1.00372	−0.501859	+	0.864950i	$0.667350\pi$
$618$	−13.6896	−0.550678
$619$	11.2929	0.453900	0.226950	−	0.973906i	$-0.427125\pi$
$619$	11.2929	0.453900	0.226950	+	0.973906i	$0.427125\pi$
$620$	−5.29590	−0.212688
$621$	0.445042	0.0178589
$622$	−8.49635	−0.340673
$623$	5.33704	0.213824
$624$	0	0
$625$	1.00000	0.0400000
$626$	25.2610	1.00963
$627$	9.27413	0.370373
$628$	19.9148	0.794689
$629$	5.84654	0.233117
$630$	−4.24698	−0.169204
$631$	20.6165	0.820732	0.410366	−	0.911921i	$-0.365401\pi$
$631$	20.6165	0.820732	0.410366	+	0.911921i	$0.365401\pi$
$632$	14.5526	0.578870
$633$	−17.4383	−0.693112
$634$	4.96615	0.197231
$635$	20.8267	0.826482
$636$	1.14914	0.0455666
$637$	0	0
$638$	−16.3744	−0.648267
$639$	2.14675	0.0849242
$640$	−1.00000	−0.0395285
$641$	47.1487	1.86226	0.931130	−	0.364687i	$-0.118824\pi$
$641$	47.1487	1.86226	0.931130	+	0.364687i	$0.118824\pi$
$642$	−12.8334	−0.506494
$643$	8.41981	0.332045	0.166023	−	0.986122i	$-0.446908\pi$
$643$	8.41981	0.332045	0.166023	+	0.986122i	$0.446908\pi$
$644$	−1.89008	−0.0744797
$645$	1.97823	0.0778927
$646$	16.1782	0.636523
$647$	−43.3836	−1.70558	−0.852792	−	0.522251i	$-0.825093\pi$
$647$	−43.3836	−1.70558	−0.852792	+	0.522251i	$0.825093\pi$
$648$	−1.00000	−0.0392837
$649$	25.7338	1.01014
$650$	0	0
$651$	22.4916	0.881514
$652$	−13.1075	−0.513330
$653$	−21.2862	−0.832994	−0.416497	−	0.909137i	$-0.636742\pi$
$653$	−21.2862	−0.832994	−0.416497	+	0.909137i	$0.636742\pi$
$654$	−10.8726	−0.425153
$655$	17.5036	0.683924
$656$	−3.24698	−0.126773
$657$	−5.15883	−0.201265
$658$	44.1008	1.71923
$659$	24.9075	0.970260	0.485130	−	0.874442i	$-0.338772\pi$
$659$	24.9075	0.970260	0.485130	+	0.874442i	$0.338772\pi$
$660$	1.91185	0.0744188
$661$	17.6209	0.685372	0.342686	−	0.939450i	$-0.388663\pi$
$661$	17.6209	0.685372	0.342686	+	0.939450i	$0.388663\pi$
$662$	−35.3749	−1.37489
$663$	0	0
$664$	−9.49157	−0.368344
$665$	20.6015	0.798891
$666$	1.75302	0.0679282
$667$	3.81163	0.147587
$668$	−21.5351	−0.833218
$669$	−9.95838	−0.385013
$670$	13.5308	0.522740
$671$	−2.16362	−0.0835256
$672$	4.24698	0.163831
$673$	−28.8888	−1.11358	−0.556790	−	0.830653i	$-0.687967\pi$
$673$	−28.8888	−1.11358	−0.556790	+	0.830653i	$0.687967\pi$
$674$	4.70171	0.181103
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	24.3927	0.937486	0.468743	−	0.883335i	$-0.344707\pi$
$677$	24.3927	0.937486	0.468743	+	0.883335i	$0.344707\pi$
$678$	−11.2295	−0.431267
$679$	21.2989	0.817376
$680$	3.33513	0.127896
$681$	−19.6112	−0.751502
$682$	−10.1250	−0.387706
$683$	−47.6558	−1.82350	−0.911749	−	0.410748i	$-0.865268\pi$
$683$	−47.6558	−1.82350	−0.911749	+	0.410748i	$0.865268\pi$
$684$	4.85086	0.185477
$685$	0.929312	0.0355072
$686$	−17.1444	−0.654575
$687$	−8.72886	−0.333027
$688$	−1.97823	−0.0754193
$689$	0	0
$690$	−0.445042	−0.0169425
$691$	−50.2180	−1.91038	−0.955192	−	0.295987i	$-0.904351\pi$
$691$	−50.2180	−1.91038	−0.955192	+	0.295987i	$0.904351\pi$
$692$	−13.9541	−0.530454
$693$	−8.11960	−0.308438
$694$	−14.9028	−0.565701
$695$	3.12200	0.118424
$696$	−8.56465	−0.324642
$697$	10.8291	0.410181
$698$	35.2669	1.33487
$699$	−0.975837	−0.0369095
$700$	4.24698	0.160521
$701$	−12.8140	−0.483979	−0.241989	−	0.970279i	$-0.577800\pi$
$701$	−12.8140	−0.483979	−0.241989	+	0.970279i	$0.577800\pi$
$702$	0	0
$703$	−8.50365	−0.320721
$704$	−1.91185	−0.0720557
$705$	10.3840	0.391086
$706$	10.5574	0.397331
$707$	62.9047	2.36577
$708$	13.4601	0.505862
$709$	34.5515	1.29761	0.648804	−	0.760955i	$-0.275269\pi$
$709$	34.5515	1.29761	0.648804	+	0.760955i	$0.275269\pi$
$710$	−2.14675	−0.0805662
$711$	−14.5526	−0.545764
$712$	−1.25667	−0.0470956
$713$	2.35690	0.0882664
$714$	−14.1642	−0.530082
$715$	0	0
$716$	8.32975	0.311297
$717$	−7.39373	−0.276124
$718$	28.7724	1.07378
$719$	20.8528	0.777677	0.388839	−	0.921306i	$-0.372876\pi$
$719$	20.8528	0.777677	0.388839	+	0.921306i	$0.372876\pi$
$720$	1.00000	0.0372678
$721$	−58.1396	−2.16523
$722$	−4.53079	−0.168619
$723$	−2.11529	−0.0786686
$724$	17.1371	0.636894
$725$	−8.56465	−0.318083
$726$	−7.34481	−0.272592
$727$	32.7192	1.21349	0.606743	−	0.794898i	$-0.292476\pi$
$727$	32.7192	1.21349	0.606743	+	0.794898i	$0.292476\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	5.15883	0.190937
$731$	6.59764	0.244023
$732$	−1.13169	−0.0418283
$733$	−52.2737	−1.93077	−0.965386	−	0.260826i	$-0.916005\pi$
$733$	−52.2737	−1.93077	−0.965386	+	0.260826i	$0.916005\pi$
$734$	24.1511	0.891432
$735$	−11.0368	−0.407100
$736$	0.445042	0.0164045
$737$	25.8689	0.952893
$738$	3.24698	0.119523
$739$	13.4571	0.495028	0.247514	−	0.968884i	$-0.420386\pi$
$739$	13.4571	0.495028	0.247514	+	0.968884i	$0.420386\pi$
$740$	−1.75302	−0.0644423
$741$	0	0
$742$	4.88040	0.179165
$743$	−22.9989	−0.843749	−0.421875	−	0.906654i	$-0.638628\pi$
$743$	−22.9989	−0.843749	−0.421875	+	0.906654i	$0.638628\pi$
$744$	−5.29590	−0.194157
$745$	−6.57673	−0.240953
$746$	21.0664	0.771295
$747$	9.49157	0.347278
$748$	6.37627	0.233140
$749$	−54.5032	−1.99150
$750$	1.00000	0.0365148
$751$	34.1159	1.24491	0.622453	−	0.782657i	$-0.286136\pi$
$751$	34.1159	1.24491	0.622453	+	0.782657i	$0.286136\pi$
$752$	−10.3840	−0.378667
$753$	1.62863	0.0593506
$754$	0	0
$755$	−6.29590	−0.229131
$756$	−4.24698	−0.154461
$757$	−38.9004	−1.41386	−0.706929	−	0.707285i	$-0.749920\pi$
$757$	−38.9004	−1.41386	−0.706929	+	0.707285i	$0.749920\pi$
$758$	−8.42327	−0.305947
$759$	−0.850855	−0.0308841
$760$	−4.85086	−0.175959
$761$	36.8528	1.33591	0.667956	−	0.744201i	$-0.267169\pi$
$761$	36.8528	1.33591	0.667956	+	0.744201i	$0.267169\pi$
$762$	20.8267	0.754471
$763$	−46.1758	−1.67168
$764$	23.1685	0.838208
$765$	−3.33513	−0.120582
$766$	10.6377	0.384357
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	23.4198	0.844540	0.422270	−	0.906470i	$-0.361233\pi$
$769$	23.4198	0.844540	0.422270	+	0.906470i	$0.361233\pi$
$770$	8.11960	0.292610
$771$	8.87263	0.319540
$772$	−6.38942	−0.229960
$773$	−0.275192	−0.00989795	−0.00494898	−	0.999988i	$-0.501575\pi$
$773$	−0.275192	−0.00989795	−0.00494898	+	0.999988i	$0.501575\pi$
$774$	1.97823	0.0711060
$775$	−5.29590	−0.190234
$776$	−5.01507	−0.180030
$777$	7.44504	0.267089
$778$	25.7429	0.922926
$779$	−15.7506	−0.564325
$780$	0	0
$781$	−4.10428	−0.146863
$782$	−1.48427	−0.0530774
$783$	8.56465	0.306076
$784$	11.0368	0.394173
$785$	19.9148	0.710791
$786$	17.5036	0.624334
$787$	−27.2543	−0.971510	−0.485755	−	0.874095i	$-0.661455\pi$
$787$	−27.2543	−0.971510	−0.485755	+	0.874095i	$0.661455\pi$
$788$	14.8605	0.529385
$789$	−22.4185	−0.798119
$790$	14.5526	0.517757
$791$	−47.6915	−1.69572
$792$	1.91185	0.0679348
$793$	0	0
$794$	11.4373	0.405894
$795$	1.14914	0.0407560
$796$	26.3860	0.935226
$797$	14.2819	0.505891	0.252945	−	0.967481i	$-0.418601\pi$
$797$	14.2819	0.505891	0.252945	+	0.967481i	$0.418601\pi$
$798$	20.6015	0.729285
$799$	34.6321	1.22520
$800$	−1.00000	−0.0353553
$801$	1.25667	0.0444022
$802$	22.1511	0.782181
$803$	9.86294	0.348055
$804$	13.5308	0.477194
$805$	−1.89008	−0.0666167
$806$	0	0
$807$	−26.9071	−0.947174
$808$	−14.8116	−0.521071
$809$	−8.82610	−0.310309	−0.155155	−	0.987890i	$-0.549588\pi$
$809$	−8.82610	−0.310309	−0.155155	+	0.987890i	$0.549588\pi$
$810$	−1.00000	−0.0351364
$811$	−6.37196	−0.223750	−0.111875	−	0.993722i	$-0.535686\pi$
$811$	−6.37196	−0.223750	−0.111875	+	0.993722i	$0.535686\pi$
$812$	−36.3739	−1.27647
$813$	31.7265	1.11270
$814$	−3.35152	−0.117471
$815$	−13.1075	−0.459137
$816$	3.33513	0.116753
$817$	−9.59611	−0.335725
$818$	−22.0965	−0.772586
$819$	0	0
$820$	−3.24698	−0.113389
$821$	−13.7476	−0.479796	−0.239898	−	0.970798i	$-0.577114\pi$
$821$	−13.7476	−0.479796	−0.239898	+	0.970798i	$0.577114\pi$
$822$	0.929312	0.0324135
$823$	47.8587	1.66825	0.834125	−	0.551575i	$-0.185973\pi$
$823$	47.8587	1.66825	0.834125	+	0.551575i	$0.185973\pi$
$824$	13.6896	0.476901
$825$	1.91185	0.0665622
$826$	57.1648	1.98902
$827$	29.1317	1.01301	0.506504	−	0.862238i	$-0.330938\pi$
$827$	29.1317	1.01301	0.506504	+	0.862238i	$0.330938\pi$
$828$	−0.445042	−0.0154663
$829$	−25.7748	−0.895195	−0.447598	−	0.894235i	$-0.647720\pi$
$829$	−25.7748	−0.895195	−0.447598	+	0.894235i	$0.647720\pi$
$830$	−9.49157	−0.329457
$831$	30.6069	1.06174
$832$	0	0
$833$	−36.8092	−1.27536
$834$	3.12200	0.108106
$835$	−21.5351	−0.745253
$836$	−9.27413	−0.320752
$837$	5.29590	0.183053
$838$	−9.03444	−0.312090
$839$	2.91292	0.100565	0.0502826	−	0.998735i	$-0.483988\pi$
$839$	2.91292	0.100565	0.0502826	+	0.998735i	$0.483988\pi$
$840$	4.24698	0.146535
$841$	44.3532	1.52942
$842$	7.17928	0.247414
$843$	16.6649	0.573969
$844$	17.4383	0.600252
$845$	0	0
$846$	10.3840	0.357011
$847$	−31.1933	−1.07181
$848$	−1.14914	−0.0394618
$849$	4.32304	0.148366
$850$	3.33513	0.114394
$851$	0.780167	0.0267438
$852$	−2.14675	−0.0735465
$853$	−12.9661	−0.443952	−0.221976	−	0.975052i	$-0.571251\pi$
$853$	−12.9661	−0.443952	−0.221976	+	0.975052i	$0.571251\pi$
$854$	−4.80625	−0.164466
$855$	4.85086	0.165896
$856$	12.8334	0.438636
$857$	−9.85623	−0.336682	−0.168341	−	0.985729i	$-0.553841\pi$
$857$	−9.85623	−0.336682	−0.168341	+	0.985729i	$0.553841\pi$
$858$	0	0
$859$	30.6746	1.04660	0.523301	−	0.852148i	$-0.324700\pi$
$859$	30.6746	1.04660	0.523301	+	0.852148i	$0.324700\pi$
$860$	−1.97823	−0.0674571
$861$	13.7899	0.469957
$862$	8.36360	0.284865
$863$	−34.1588	−1.16278	−0.581390	−	0.813625i	$-0.697491\pi$
$863$	−34.1588	−1.16278	−0.581390	+	0.813625i	$0.697491\pi$
$864$	1.00000	0.0340207
$865$	−13.9541	−0.474452
$866$	12.2851	0.417466
$867$	5.87694	0.199591
$868$	−22.4916	−0.763414
$869$	27.8224	0.943810
$870$	−8.56465	−0.290369
$871$	0	0
$872$	10.8726	0.368194
$873$	5.01507	0.169734
$874$	2.15883	0.0730236
$875$	4.24698	0.143574
$876$	5.15883	0.174301
$877$	−20.1588	−0.680715	−0.340358	−	0.940296i	$-0.610548\pi$
$877$	−20.1588	−0.680715	−0.340358	+	0.940296i	$0.610548\pi$
$878$	−20.4849	−0.691331
$879$	5.62804	0.189829
$880$	−1.91185	−0.0644486
$881$	−2.79550	−0.0941827	−0.0470913	−	0.998891i	$-0.514995\pi$
$881$	−2.79550	−0.0941827	−0.0470913	+	0.998891i	$0.514995\pi$
$882$	−11.0368	−0.371630
$883$	−16.8605	−0.567402	−0.283701	−	0.958913i	$-0.591562\pi$
$883$	−16.8605	−0.567402	−0.283701	+	0.958913i	$0.591562\pi$
$884$	0	0
$885$	13.4601	0.452457
$886$	0.928247	0.0311851
$887$	−8.71379	−0.292580	−0.146290	−	0.989242i	$-0.546733\pi$
$887$	−8.71379	−0.292580	−0.146290	+	0.989242i	$0.546733\pi$
$888$	−1.75302	−0.0588275
$889$	88.4505	2.96654
$890$	−1.25667	−0.0421236
$891$	−1.91185	−0.0640495
$892$	9.95838	0.333431
$893$	−50.3715	−1.68562
$894$	−6.57673	−0.219959
$895$	8.32975	0.278433
$896$	−4.24698	−0.141882
$897$	0	0
$898$	−22.9355	−0.765369
$899$	45.3575	1.51276
$900$	1.00000	0.0333333
$901$	3.83254	0.127681
$902$	−6.20775	−0.206695
$903$	8.40150	0.279584
$904$	11.2295	0.373488
$905$	17.1371	0.569655
$906$	−6.29590	−0.209167
$907$	14.2295	0.472483	0.236242	−	0.971694i	$-0.424084\pi$
$907$	14.2295	0.472483	0.236242	+	0.971694i	$0.424084\pi$
$908$	19.6112	0.650820
$909$	14.8116	0.491271
$910$	0	0
$911$	−19.3672	−0.641663	−0.320832	−	0.947136i	$-0.603962\pi$
$911$	−19.3672	−0.641663	−0.320832	+	0.947136i	$0.603962\pi$
$912$	−4.85086	−0.160628
$913$	−18.1465	−0.600561
$914$	10.3773	0.343252
$915$	−1.13169	−0.0374124
$916$	8.72886	0.288410
$917$	74.3376	2.45484
$918$	−3.33513	−0.110076
$919$	−44.1420	−1.45611	−0.728055	−	0.685519i	$-0.759575\pi$
$919$	−44.1420	−1.45611	−0.728055	+	0.685519i	$0.759575\pi$
$920$	0.445042	0.0146726
$921$	12.6635	0.417278
$922$	7.69740	0.253500
$923$	0	0
$924$	8.11960	0.267115
$925$	−1.75302	−0.0576390
$926$	−39.0640	−1.28372
$927$	−13.6896	−0.449626
$928$	8.56465	0.281148
$929$	−14.8616	−0.487594	−0.243797	−	0.969826i	$-0.578393\pi$
$929$	−14.8616	−0.487594	−0.243797	+	0.969826i	$0.578393\pi$
$930$	−5.29590	−0.173659
$931$	53.5381	1.75464
$932$	0.975837	0.0319646
$933$	−8.49635	−0.278158
$934$	−0.916166	−0.0299779
$935$	6.37627	0.208526
$936$	0	0
$937$	54.3196	1.77454	0.887272	−	0.461247i	$-0.152598\pi$
$937$	54.3196	1.77454	0.887272	+	0.461247i	$0.152598\pi$
$938$	57.4650	1.87630
$939$	25.2610	0.824361
$940$	−10.3840	−0.338690
$941$	14.0121	0.456781	0.228390	−	0.973570i	$-0.426654\pi$
$941$	14.0121	0.456781	0.228390	+	0.973570i	$0.426654\pi$
$942$	19.9148	0.648860
$943$	1.44504	0.0470570
$944$	−13.4601	−0.438089
$945$	−4.24698	−0.138154
$946$	−3.78209	−0.122966
$947$	−56.1831	−1.82571	−0.912853	−	0.408289i	$-0.866126\pi$
$947$	−56.1831	−1.82571	−0.912853	+	0.408289i	$0.866126\pi$
$948$	14.5526	0.472645
$949$	0	0
$950$	−4.85086	−0.157383
$951$	4.96615	0.161038
$952$	14.1642	0.459065
$953$	42.9090	1.38996	0.694979	−	0.719030i	$-0.255414\pi$
$953$	42.9090	1.38996	0.694979	+	0.719030i	$0.255414\pi$
$954$	1.14914	0.0372049
$955$	23.1685	0.749716
$956$	7.39373	0.239130
$957$	−16.3744	−0.529308
$958$	−23.1715	−0.748637
$959$	3.94677	0.127448
$960$	−1.00000	−0.0322749
$961$	−2.95348	−0.0952734
$962$	0	0
$963$	−12.8334	−0.413550
$964$	2.11529	0.0681290
$965$	−6.38942	−0.205683
$966$	−1.89008	−0.0608124
$967$	−2.89977	−0.0932504	−0.0466252	−	0.998912i	$-0.514847\pi$
$967$	−2.89977	−0.0932504	−0.0466252	+	0.998912i	$0.514847\pi$
$968$	7.34481	0.236071
$969$	16.1782	0.519719
$970$	−5.01507	−0.161024
$971$	32.8146	1.05307	0.526535	−	0.850153i	$-0.323491\pi$
$971$	32.8146	1.05307	0.526535	+	0.850153i	$0.323491\pi$
$972$	−1.00000	−0.0320750
$973$	13.2591	0.425066
$974$	−11.9812	−0.383903
$975$	0	0
$976$	1.13169	0.0362244
$977$	13.1806	0.421685	0.210842	−	0.977520i	$-0.432379\pi$
$977$	13.1806	0.421685	0.210842	+	0.977520i	$0.432379\pi$
$978$	−13.1075	−0.419132
$979$	−2.40257	−0.0767864
$980$	11.0368	0.352559
$981$	−10.8726	−0.347136
$982$	24.8745	0.793779
$983$	−31.3726	−1.00063	−0.500315	−	0.865844i	$-0.666782\pi$
$983$	−31.3726	−1.00063	−0.500315	+	0.865844i	$0.666782\pi$
$984$	−3.24698	−0.103510
$985$	14.8605	0.473496
$986$	−28.5642	−0.909669
$987$	44.1008	1.40374
$988$	0	0
$989$	0.880395	0.0279949
$990$	1.91185	0.0607627
$991$	−44.1842	−1.40356	−0.701778	−	0.712395i	$-0.747610\pi$
$991$	−44.1842	−1.40356	−0.701778	+	0.712395i	$0.747610\pi$
$992$	5.29590	0.168145
$993$	−35.3749	−1.12259
$994$	−9.11721	−0.289180
$995$	26.3860	0.836491
$996$	−9.49157	−0.300752
$997$	50.8254	1.60966	0.804828	−	0.593509i	$-0.202258\pi$
$997$	50.8254	1.60966	0.804828	+	0.593509i	$0.202258\pi$
$998$	−42.0737	−1.33182
$999$	1.75302	0.0554631

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.24698 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.91185 q^{11} -1.00000 q^{12} -4.24698 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.33513 q^{17} -1.00000 q^{18} +4.85086 q^{19} +1.00000 q^{20} -4.24698 q^{21} +1.91185 q^{22} -0.445042 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.24698 q^{28} -8.56465 q^{29} +1.00000 q^{30} -5.29590 q^{31} -1.00000 q^{32} +1.91185 q^{33} +3.33513 q^{34} +4.24698 q^{35} +1.00000 q^{36} -1.75302 q^{37} -4.85086 q^{38} -1.00000 q^{40} -3.24698 q^{41} +4.24698 q^{42} -1.97823 q^{43} -1.91185 q^{44} +1.00000 q^{45} +0.445042 q^{46} -10.3840 q^{47} -1.00000 q^{48} +11.0368 q^{49} -1.00000 q^{50} +3.33513 q^{51} -1.14914 q^{53} +1.00000 q^{54} -1.91185 q^{55} -4.24698 q^{56} -4.85086 q^{57} +8.56465 q^{58} -13.4601 q^{59} -1.00000 q^{60} +1.13169 q^{61} +5.29590 q^{62} +4.24698 q^{63} +1.00000 q^{64} -1.91185 q^{66} -13.5308 q^{67} -3.33513 q^{68} +0.445042 q^{69} -4.24698 q^{70} +2.14675 q^{71} -1.00000 q^{72} -5.15883 q^{73} +1.75302 q^{74} -1.00000 q^{75} +4.85086 q^{76} -8.11960 q^{77} -14.5526 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.24698 q^{82} +9.49157 q^{83} -4.24698 q^{84} -3.33513 q^{85} +1.97823 q^{86} +8.56465 q^{87} +1.91185 q^{88} +1.25667 q^{89} -1.00000 q^{90} -0.445042 q^{92} +5.29590 q^{93} +10.3840 q^{94} +4.85086 q^{95} +1.00000 q^{96} +5.01507 q^{97} -11.0368 q^{98} -1.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 8 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 8 q^{14} - 3 q^{15} + 3 q^{16} - 9 q^{17} - 3 q^{18} + q^{19} + 3 q^{20} - 8 q^{21} + 2 q^{22} - q^{23} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 8 q^{28} - 4 q^{29} + 3 q^{30} - 2 q^{31} - 3 q^{32} + 2 q^{33} + 9 q^{34} + 8 q^{35} + 3 q^{36} - 10 q^{37} - q^{38} - 3 q^{40} - 5 q^{41} + 8 q^{42} - 9 q^{43} - 2 q^{44} + 3 q^{45} + q^{46} - 21 q^{47} - 3 q^{48} + 5 q^{49} - 3 q^{50} + 9 q^{51} - 17 q^{53} + 3 q^{54} - 2 q^{55} - 8 q^{56} - q^{57} + 4 q^{58} - 15 q^{59} - 3 q^{60} + q^{61} + 2 q^{62} + 8 q^{63} + 3 q^{64} - 2 q^{66} - 3 q^{67} - 9 q^{68} + q^{69} - 8 q^{70} - 21 q^{71} - 3 q^{72} - 7 q^{73} + 10 q^{74} - 3 q^{75} + q^{76} - 3 q^{77} - 3 q^{79} + 3 q^{80} + 3 q^{81} + 5 q^{82} - 22 q^{83} - 8 q^{84} - 9 q^{85} + 9 q^{86} + 4 q^{87} + 2 q^{88} - 23 q^{89} - 3 q^{90} - q^{92} + 2 q^{93} + 21 q^{94} + q^{95} + 3 q^{96} - 10 q^{97} - 5 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 8 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 8 * q^14 - 3 * q^15 + 3 * q^16 - 9 * q^17 - 3 * q^18 + q^19 + 3 * q^20 - 8 * q^21 + 2 * q^22 - q^23 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 8 * q^28 - 4 * q^29 + 3 * q^30 - 2 * q^31 - 3 * q^32 + 2 * q^33 + 9 * q^34 + 8 * q^35 + 3 * q^36 - 10 * q^37 - q^38 - 3 * q^40 - 5 * q^41 + 8 * q^42 - 9 * q^43 - 2 * q^44 + 3 * q^45 + q^46 - 21 * q^47 - 3 * q^48 + 5 * q^49 - 3 * q^50 + 9 * q^51 - 17 * q^53 + 3 * q^54 - 2 * q^55 - 8 * q^56 - q^57 + 4 * q^58 - 15 * q^59 - 3 * q^60 + q^61 + 2 * q^62 + 8 * q^63 + 3 * q^64 - 2 * q^66 - 3 * q^67 - 9 * q^68 + q^69 - 8 * q^70 - 21 * q^71 - 3 * q^72 - 7 * q^73 + 10 * q^74 - 3 * q^75 + q^76 - 3 * q^77 - 3 * q^79 + 3 * q^80 + 3 * q^81 + 5 * q^82 - 22 * q^83 - 8 * q^84 - 9 * q^85 + 9 * q^86 + 4 * q^87 + 2 * q^88 - 23 * q^89 - 3 * q^90 - q^92 + 2 * q^93 + 21 * q^94 + q^95 + 3 * q^96 - 10 * q^97 - 5 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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