LMFDB - Embedded newform 5070.2.b.z.1351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5070.1351");

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.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.z.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.04892i q^{11} -1.00000 q^{12} +3.69202 q^{14} -1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} +1.00000i q^{18} +0.911854i q^{19} +1.00000i q^{20} -3.69202i q^{21} -3.04892 q^{22} +0.356896 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +3.69202i q^{28} +10.5036 q^{29} +1.00000 q^{30} +2.06100i q^{31} +1.00000i q^{32} +3.04892i q^{33} +6.85086i q^{34} -3.69202 q^{35} -1.00000 q^{36} +0.899772i q^{37} -0.911854 q^{38} -1.00000 q^{40} +10.2959i q^{41} +3.69202 q^{42} -9.43296 q^{43} -3.04892i q^{44} -1.00000i q^{45} +0.356896i q^{46} -11.5918i q^{47} +1.00000 q^{48} -6.63102 q^{49} -1.00000i q^{50} +6.85086 q^{51} -3.40581 q^{53} +1.00000i q^{54} +3.04892 q^{55} -3.69202 q^{56} +0.911854i q^{57} +10.5036i q^{58} -8.54288i q^{59} +1.00000i q^{60} +1.55496 q^{61} -2.06100 q^{62} -3.69202i q^{63} -1.00000 q^{64} -3.04892 q^{66} +1.14914i q^{67} -6.85086 q^{68} +0.356896 q^{69} -3.69202i q^{70} -4.13706i q^{71} -1.00000i q^{72} -11.1685i q^{73} -0.899772 q^{74} -1.00000 q^{75} -0.911854i q^{76} +11.2567 q^{77} +9.62565 q^{79} -1.00000i q^{80} +1.00000 q^{81} -10.2959 q^{82} -9.86054i q^{83} +3.69202i q^{84} -6.85086i q^{85} -9.43296i q^{86} +10.5036 q^{87} +3.04892 q^{88} +6.41119i q^{89} +1.00000 q^{90} -0.356896 q^{92} +2.06100i q^{93} +11.5918 q^{94} +0.911854 q^{95} +1.00000i q^{96} +6.97823i q^{97} -6.63102i q^{98} +3.04892i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} + 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 6 q^{30} - 12 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} - 12 q^{56} + 10 q^{61} - 32 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} + 40 q^{74} - 6 q^{75} + 14 q^{77} + 34 q^{79} + 6 q^{81} - 34 q^{82} + 6 q^{90} + 6 q^{92} + 14 q^{94} - 2 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 + 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 6 * q^30 - 12 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 - 12 * q^56 + 10 * q^61 - 32 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 + 40 * q^74 - 6 * q^75 + 14 * q^77 + 34 * q^79 + 6 * q^81 - 34 * q^82 + 6 * q^90 + 6 * q^92 + 14 * q^94 - 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	1.00000i	0.408248i
$7$	− 3.69202i	− 1.39545i	−0.716364	−	0.697726i	$-0.754195\pi$
$7$	− 3.69202i	− 1.39545i	0.716364	−	0.697726i	$-0.245805\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.04892i	0.919283i	0.888105	+	0.459642i	$0.152022\pi$
$11$	3.04892i	0.919283i	−0.888105	+	0.459642i	$0.847978\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	3.69202	0.986734
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	6.85086	1.66158	0.830788	−	0.556589i	$-0.187890\pi$
$17$	6.85086	1.66158	0.830788	+	0.556589i	$0.187890\pi$
$18$	1.00000i	0.235702i
$19$	0.911854i	0.209194i	0.994515	+	0.104597i	$0.0333552\pi$
$19$	0.911854i	0.209194i	−0.994515	+	0.104597i	$0.966645\pi$
$20$	1.00000i	0.223607i
$21$	− 3.69202i	− 0.805665i
$22$	−3.04892	−0.650031
$23$	0.356896	0.0744179	0.0372090	−	0.999308i	$-0.488153\pi$
$23$	0.356896	0.0744179	0.0372090	+	0.999308i	$0.488153\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	3.69202i	0.697726i
$29$	10.5036	1.95048	0.975239	−	0.221153i	$-0.0709819\pi$
$29$	10.5036	1.95048	0.975239	+	0.221153i	$0.0709819\pi$
$30$	1.00000	0.182574
$31$	2.06100i	0.370166i	0.982723	+	0.185083i	$0.0592555\pi$
$31$	2.06100i	0.370166i	−0.982723	+	0.185083i	$0.940745\pi$
$32$	1.00000i	0.176777i
$33$	3.04892i	0.530748i
$34$	6.85086i	1.17491i
$35$	−3.69202	−0.624066
$36$	−1.00000	−0.166667
$37$	0.899772i	0.147922i	0.997261	+	0.0739608i	$0.0235640\pi$
$37$	0.899772i	0.147922i	−0.997261	+	0.0739608i	$0.976436\pi$
$38$	−0.911854	−0.147922
$39$	0	0
$40$	−1.00000	−0.158114
$41$	10.2959i	1.60795i	0.594664	+	0.803974i	$0.297285\pi$
$41$	10.2959i	1.60795i	−0.594664	+	0.803974i	$0.702715\pi$
$42$	3.69202	0.569691
$43$	−9.43296	−1.43851	−0.719256	−	0.694745i	$-0.755517\pi$
$43$	−9.43296	−1.43851	−0.719256	+	0.694745i	$0.755517\pi$
$44$	− 3.04892i	− 0.459642i
$45$	− 1.00000i	− 0.149071i
$46$	0.356896i	0.0526214i
$47$	− 11.5918i	− 1.69084i	−0.534105	−	0.845418i	$-0.679351\pi$
$47$	− 11.5918i	− 1.69084i	0.534105	−	0.845418i	$-0.320649\pi$
$48$	1.00000	0.144338
$49$	−6.63102	−0.947289
$50$	− 1.00000i	− 0.141421i
$51$	6.85086	0.959312
$52$	0	0
$53$	−3.40581	−0.467824	−0.233912	−	0.972258i	$-0.575153\pi$
$53$	−3.40581	−0.467824	−0.233912	+	0.972258i	$0.575153\pi$
$54$	1.00000i	0.136083i
$55$	3.04892	0.411116
$56$	−3.69202	−0.493367
$57$	0.911854i	0.120778i
$58$	10.5036i	1.37920i
$59$	− 8.54288i	− 1.11219i	−0.831119	−	0.556094i	$-0.812299\pi$
$59$	− 8.54288i	− 1.11219i	0.831119	−	0.556094i	$-0.187701\pi$
$60$	1.00000i	0.129099i
$61$	1.55496	0.199092	0.0995460	−	0.995033i	$-0.468261\pi$
$61$	1.55496	0.199092	0.0995460	+	0.995033i	$0.468261\pi$
$62$	−2.06100	−0.261747
$63$	− 3.69202i	− 0.465151i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.04892	−0.375296
$67$	1.14914i	0.140390i	0.997533	+	0.0701952i	$0.0223622\pi$
$67$	1.14914i	0.140390i	−0.997533	+	0.0701952i	$0.977638\pi$
$68$	−6.85086	−0.830788
$69$	0.356896	0.0429652
$70$	− 3.69202i	− 0.441281i
$71$	− 4.13706i	− 0.490979i	−0.969399	−	0.245490i	$-0.921051\pi$
$71$	− 4.13706i	− 0.490979i	0.969399	−	0.245490i	$-0.0789487\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 11.1685i	− 1.30718i	−0.756850	−	0.653588i	$-0.773263\pi$
$73$	− 11.1685i	− 1.30718i	0.756850	−	0.653588i	$-0.226737\pi$
$74$	−0.899772	−0.104596
$75$	−1.00000	−0.115470
$76$	− 0.911854i	− 0.104597i
$77$	11.2567	1.28282
$78$	0	0
$79$	9.62565	1.08297	0.541485	−	0.840710i	$-0.317862\pi$
$79$	9.62565	1.08297	0.541485	+	0.840710i	$0.317862\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−10.2959	−1.13699
$83$	− 9.86054i	− 1.08234i	−0.840915	−	0.541168i	$-0.817982\pi$
$83$	− 9.86054i	− 1.08234i	0.840915	−	0.541168i	$-0.182018\pi$
$84$	3.69202i	0.402833i
$85$	− 6.85086i	− 0.743080i
$86$	− 9.43296i	− 1.01718i
$87$	10.5036	1.12611
$88$	3.04892	0.325016
$89$	6.41119i	0.679585i	0.940500	+	0.339792i	$0.110357\pi$
$89$	6.41119i	0.679585i	−0.940500	+	0.339792i	$0.889643\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−0.356896	−0.0372090
$93$	2.06100i	0.213716i
$94$	11.5918	1.19560
$95$	0.911854	0.0935542
$96$	1.00000i	0.102062i
$97$	6.97823i	0.708532i	0.935145	+	0.354266i	$0.115269\pi$
$97$	6.97823i	0.708532i	−0.935145	+	0.354266i	$0.884731\pi$
$98$	− 6.63102i	− 0.669834i
$99$	3.04892i	0.306428i
$100$	1.00000	0.100000
$101$	4.09783	0.407750	0.203875	−	0.978997i	$-0.434646\pi$
$101$	4.09783	0.407750	0.203875	+	0.978997i	$0.434646\pi$
$102$	6.85086i	0.678336i
$103$	16.9215	1.66733	0.833665	−	0.552271i	$-0.186239\pi$
$103$	16.9215	1.66733	0.833665	+	0.552271i	$0.186239\pi$
$104$	0	0
$105$	−3.69202	−0.360304
$106$	− 3.40581i	− 0.330802i
$107$	−12.7192	−1.22961	−0.614804	−	0.788680i	$-0.710765\pi$
$107$	−12.7192	−1.22961	−0.614804	+	0.788680i	$0.710765\pi$
$108$	−1.00000	−0.0962250
$109$	19.9922i	1.91491i	0.288585	+	0.957454i	$0.406815\pi$
$109$	19.9922i	1.91491i	−0.288585	+	0.957454i	$0.593185\pi$
$110$	3.04892i	0.290703i
$111$	0.899772i	0.0854026i
$112$	− 3.69202i	− 0.348863i
$113$	−4.49934	−0.423262	−0.211631	−	0.977350i	$-0.567877\pi$
$113$	−4.49934	−0.423262	−0.211631	+	0.977350i	$0.567877\pi$
$114$	−0.911854	−0.0854030
$115$	− 0.356896i	− 0.0332807i
$116$	−10.5036	−0.975239
$117$	0	0
$118$	8.54288	0.786436
$119$	− 25.2935i	− 2.31865i
$120$	−1.00000	−0.0912871
$121$	1.70410	0.154918
$122$	1.55496i	0.140779i
$123$	10.2959i	0.928350i
$124$	− 2.06100i	− 0.185083i
$125$	1.00000i	0.0894427i
$126$	3.69202	0.328911
$127$	7.08815	0.628971	0.314486	−	0.949262i	$-0.398168\pi$
$127$	7.08815	0.628971	0.314486	+	0.949262i	$0.398168\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−9.43296	−0.830526
$130$	0	0
$131$	19.7506	1.72562	0.862810	−	0.505528i	$-0.168702\pi$
$131$	19.7506	1.72562	0.862810	+	0.505528i	$0.168702\pi$
$132$	− 3.04892i	− 0.265374i
$133$	3.36658	0.291920
$134$	−1.14914	−0.0992710
$135$	− 1.00000i	− 0.0860663i
$136$	− 6.85086i	− 0.587456i
$137$	7.94331i	0.678643i	0.940670	+	0.339322i	$0.110197\pi$
$137$	7.94331i	0.678643i	−0.940670	+	0.339322i	$0.889803\pi$
$138$	0.356896i	0.0303810i
$139$	−1.71379	−0.145362	−0.0726810	−	0.997355i	$-0.523155\pi$
$139$	−1.71379	−0.145362	−0.0726810	+	0.997355i	$0.523155\pi$
$140$	3.69202	0.312033
$141$	− 11.5918i	− 0.976205i
$142$	4.13706	0.347175
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 10.5036i	− 0.872280i
$146$	11.1685	0.924313
$147$	−6.63102	−0.546918
$148$	− 0.899772i	− 0.0739608i
$149$	− 6.91185i	− 0.566241i	−0.959084	−	0.283121i	$-0.908630\pi$
$149$	− 6.91185i	− 0.566241i	0.959084	−	0.283121i	$-0.0913697\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 5.43296i	− 0.442128i	−0.975259	−	0.221064i	$-0.929047\pi$
$151$	− 5.43296i	− 0.442128i	0.975259	−	0.221064i	$-0.0709530\pi$
$152$	0.911854	0.0739611
$153$	6.85086	0.553859
$154$	11.2567i	0.907088i
$155$	2.06100	0.165543
$156$	0	0
$157$	−9.13706	−0.729217	−0.364609	−	0.931161i	$-0.618797\pi$
$157$	−9.13706	−0.729217	−0.364609	+	0.931161i	$0.618797\pi$
$158$	9.62565i	0.765775i
$159$	−3.40581	−0.270099
$160$	1.00000	0.0790569
$161$	− 1.31767i	− 0.103847i
$162$	1.00000i	0.0785674i
$163$	− 14.3773i	− 1.12612i	−0.826416	−	0.563060i	$-0.809624\pi$
$163$	− 14.3773i	− 1.12612i	0.826416	−	0.563060i	$-0.190376\pi$
$164$	− 10.2959i	− 0.803974i
$165$	3.04892	0.237358
$166$	9.86054	0.765327
$167$	3.00538i	0.232563i	0.993216	+	0.116282i	$0.0370975\pi$
$167$	3.00538i	0.232563i	−0.993216	+	0.116282i	$0.962903\pi$
$168$	−3.69202	−0.284846
$169$	0	0
$170$	6.85086	0.525437
$171$	0.911854i	0.0697312i
$172$	9.43296	0.719256
$173$	−6.89546	−0.524252	−0.262126	−	0.965034i	$-0.584424\pi$
$173$	−6.89546	−0.524252	−0.262126	+	0.965034i	$0.584424\pi$
$174$	10.5036i	0.796279i
$175$	3.69202i	0.279091i
$176$	3.04892i	0.229821i
$177$	− 8.54288i	− 0.642122i
$178$	−6.41119	−0.480539
$179$	25.6461	1.91688	0.958439	−	0.285296i	$-0.0920921\pi$
$179$	25.6461	1.91688	0.958439	+	0.285296i	$0.0920921\pi$
$180$	1.00000i	0.0745356i
$181$	25.8092	1.91839	0.959193	−	0.282753i	$-0.0912478\pi$
$181$	25.8092	1.91839	0.959193	+	0.282753i	$0.0912478\pi$
$182$	0	0
$183$	1.55496	0.114946
$184$	− 0.356896i	− 0.0263107i
$185$	0.899772	0.0661526
$186$	−2.06100	−0.151120
$187$	20.8877i	1.52746i
$188$	11.5918i	0.845418i
$189$	− 3.69202i	− 0.268555i
$190$	0.911854i	0.0661528i
$191$	−14.5700	−1.05425	−0.527125	−	0.849788i	$-0.676730\pi$
$191$	−14.5700	−1.05425	−0.527125	+	0.849788i	$0.676730\pi$
$192$	−1.00000	−0.0721688
$193$	− 8.32544i	− 0.599278i	−0.954053	−	0.299639i	$-0.903134\pi$
$193$	− 8.32544i	− 0.599278i	0.954053	−	0.299639i	$-0.0968662\pi$
$194$	−6.97823	−0.501008
$195$	0	0
$196$	6.63102	0.473644
$197$	− 26.8810i	− 1.91519i	−0.288116	−	0.957595i	$-0.593029\pi$
$197$	− 26.8810i	− 1.91519i	0.288116	−	0.957595i	$-0.406971\pi$
$198$	−3.04892	−0.216677
$199$	18.7047	1.32594	0.662970	−	0.748646i	$-0.269296\pi$
$199$	18.7047	1.32594	0.662970	+	0.748646i	$0.269296\pi$
$200$	1.00000i	0.0707107i
$201$	1.14914i	0.0810544i
$202$	4.09783i	0.288323i
$203$	− 38.7797i	− 2.72180i
$204$	−6.85086	−0.479656
$205$	10.2959	0.719097
$206$	16.9215i	1.17898i
$207$	0.356896	0.0248060
$208$	0	0
$209$	−2.78017	−0.192308
$210$	− 3.69202i	− 0.254774i
$211$	2.93362	0.201959	0.100980	−	0.994889i	$-0.467802\pi$
$211$	2.93362	0.201959	0.100980	+	0.994889i	$0.467802\pi$
$212$	3.40581	0.233912
$213$	− 4.13706i	− 0.283467i
$214$	− 12.7192i	− 0.869464i
$215$	9.43296i	0.643323i
$216$	− 1.00000i	− 0.0680414i
$217$	7.60925	0.516550
$218$	−19.9922	−1.35404
$219$	− 11.1685i	− 0.754699i
$220$	−3.04892	−0.205558
$221$	0	0
$222$	−0.899772	−0.0603888
$223$	− 18.1444i	− 1.21504i	−0.794306	−	0.607518i	$-0.792165\pi$
$223$	− 18.1444i	− 1.21504i	0.794306	−	0.607518i	$-0.207835\pi$
$224$	3.69202	0.246684
$225$	−1.00000	−0.0666667
$226$	− 4.49934i	− 0.299291i
$227$	12.3056i	0.816750i	0.912814	+	0.408375i	$0.133904\pi$
$227$	12.3056i	0.816750i	−0.912814	+	0.408375i	$0.866096\pi$
$228$	− 0.911854i	− 0.0603890i
$229$	− 4.64848i	− 0.307180i	−0.988135	−	0.153590i	$-0.950916\pi$
$229$	− 4.64848i	− 0.307180i	0.988135	−	0.153590i	$-0.0490835\pi$
$230$	0.356896	0.0235330
$231$	11.2567	0.740634
$232$	− 10.5036i	− 0.689598i
$233$	−0.658170	−0.0431181	−0.0215591	−	0.999768i	$-0.506863\pi$
$233$	−0.658170	−0.0431181	−0.0215591	+	0.999768i	$0.506863\pi$
$234$	0	0
$235$	−11.5918	−0.756165
$236$	8.54288i	0.556094i
$237$	9.62565	0.625253
$238$	25.2935	1.63953
$239$	− 18.7748i	− 1.21444i	−0.794534	−	0.607220i	$-0.792285\pi$
$239$	− 18.7748i	− 1.21444i	0.794534	−	0.607220i	$-0.207715\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 18.0271i	− 1.16123i	−0.814178	−	0.580615i	$-0.802812\pi$
$241$	− 18.0271i	− 1.16123i	0.814178	−	0.580615i	$-0.197188\pi$
$242$	1.70410i	0.109544i
$243$	1.00000	0.0641500
$244$	−1.55496	−0.0995460
$245$	6.63102i	0.423640i
$246$	−10.2959	−0.656442
$247$	0	0
$248$	2.06100	0.130874
$249$	− 9.86054i	− 0.624887i
$250$	−1.00000	−0.0632456
$251$	2.18060	0.137638	0.0688192	−	0.997629i	$-0.478077\pi$
$251$	2.18060	0.137638	0.0688192	+	0.997629i	$0.478077\pi$
$252$	3.69202i	0.232575i
$253$	1.08815i	0.0684112i
$254$	7.08815i	0.444750i
$255$	− 6.85086i	− 0.429017i
$256$	1.00000	0.0625000
$257$	9.58748	0.598051	0.299025	−	0.954245i	$-0.403338\pi$
$257$	9.58748	0.598051	0.299025	+	0.954245i	$0.403338\pi$
$258$	− 9.43296i	− 0.587270i
$259$	3.32198	0.206418
$260$	0	0
$261$	10.5036	0.650159
$262$	19.7506i	1.22020i
$263$	−1.41789	−0.0874311	−0.0437156	−	0.999044i	$-0.513920\pi$
$263$	−1.41789	−0.0874311	−0.0437156	+	0.999044i	$0.513920\pi$
$264$	3.04892	0.187648
$265$	3.40581i	0.209217i
$266$	3.36658i	0.206419i
$267$	6.41119i	0.392358i
$268$	− 1.14914i	− 0.0701952i
$269$	−4.14675	−0.252832	−0.126416	−	0.991977i	$-0.540347\pi$
$269$	−4.14675	−0.252832	−0.126416	+	0.991977i	$0.540347\pi$
$270$	1.00000	0.0608581
$271$	− 6.40044i	− 0.388799i	−0.980922	−	0.194399i	$-0.937724\pi$
$271$	− 6.40044i	− 0.388799i	0.980922	−	0.194399i	$-0.0622758\pi$
$272$	6.85086	0.415394
$273$	0	0
$274$	−7.94331	−0.479873
$275$	− 3.04892i	− 0.183857i
$276$	−0.356896	−0.0214826
$277$	−10.3394	−0.621237	−0.310618	−	0.950535i	$-0.600536\pi$
$277$	−10.3394	−0.621237	−0.310618	+	0.950535i	$0.600536\pi$
$278$	− 1.71379i	− 0.102786i
$279$	2.06100i	0.123389i
$280$	3.69202i	0.220640i
$281$	− 5.08144i	− 0.303133i	−0.988447	−	0.151567i	$-0.951568\pi$
$281$	− 5.08144i	− 0.303133i	0.988447	−	0.151567i	$-0.0484318\pi$
$282$	11.5918	0.690281
$283$	−24.5284	−1.45806	−0.729031	−	0.684481i	$-0.760029\pi$
$283$	−24.5284	−1.45806	−0.729031	+	0.684481i	$0.760029\pi$
$284$	4.13706i	0.245490i
$285$	0.911854	0.0540136
$286$	0	0
$287$	38.0127	2.24382
$288$	1.00000i	0.0589256i
$289$	29.9342	1.76084
$290$	10.5036	0.616795
$291$	6.97823i	0.409071i
$292$	11.1685i	0.653588i
$293$	− 15.8538i	− 0.926191i	−0.886308	−	0.463096i	$-0.846739\pi$
$293$	− 15.8538i	− 0.926191i	0.886308	−	0.463096i	$-0.153261\pi$
$294$	− 6.63102i	− 0.386729i
$295$	−8.54288	−0.497386
$296$	0.899772	0.0522982
$297$	3.04892i	0.176916i
$298$	6.91185	0.400393
$299$	0	0
$300$	1.00000	0.0577350
$301$	34.8267i	2.00738i
$302$	5.43296	0.312632
$303$	4.09783	0.235414
$304$	0.911854i	0.0522984i
$305$	− 1.55496i	− 0.0890366i
$306$	6.85086i	0.391637i
$307$	14.2121i	0.811125i	0.914067	+	0.405563i	$0.132924\pi$
$307$	14.2121i	0.811125i	−0.914067	+	0.405563i	$0.867076\pi$
$308$	−11.2567	−0.641408
$309$	16.9215	0.962633
$310$	2.06100i	0.117057i
$311$	−15.0127	−0.851291	−0.425645	−	0.904890i	$-0.639953\pi$
$311$	−15.0127	−0.851291	−0.425645	+	0.904890i	$0.639953\pi$
$312$	0	0
$313$	−13.4993	−0.763028	−0.381514	−	0.924363i	$-0.624597\pi$
$313$	−13.4993	−0.763028	−0.381514	+	0.924363i	$0.624597\pi$
$314$	− 9.13706i	− 0.515634i
$315$	−3.69202	−0.208022
$316$	−9.62565	−0.541485
$317$	33.9463i	1.90661i	0.302003	+	0.953307i	$0.402345\pi$
$317$	33.9463i	1.90661i	−0.302003	+	0.953307i	$0.597655\pi$
$318$	− 3.40581i	− 0.190989i
$319$	32.0248i	1.79304i
$320$	1.00000i	0.0559017i
$321$	−12.7192	−0.709915
$322$	1.31767	0.0734307
$323$	6.24698i	0.347591i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	14.3773	0.796287
$327$	19.9922i	1.10557i
$328$	10.2959	0.568496
$329$	−42.7972	−2.35948
$330$	3.04892i	0.167837i
$331$	− 6.84415i	− 0.376189i	−0.982151	−	0.188094i	$-0.939769\pi$
$331$	− 6.84415i	− 0.376189i	0.982151	−	0.188094i	$-0.0602310\pi$
$332$	9.86054i	0.541168i
$333$	0.899772i	0.0493072i
$334$	−3.00538	−0.164447
$335$	1.14914	0.0627845
$336$	− 3.69202i	− 0.201416i
$337$	−27.7332	−1.51072	−0.755361	−	0.655309i	$-0.772538\pi$
$337$	−27.7332	−1.51072	−0.755361	+	0.655309i	$0.772538\pi$
$338$	0	0
$339$	−4.49934	−0.244370
$340$	6.85086i	0.371540i
$341$	−6.28382	−0.340288
$342$	−0.911854	−0.0493074
$343$	− 1.36227i	− 0.0735558i
$344$	9.43296i	0.508591i
$345$	− 0.356896i	− 0.0192146i
$346$	− 6.89546i	− 0.370702i
$347$	−10.4862	−0.562928	−0.281464	−	0.959572i	$-0.590820\pi$
$347$	−10.4862	−0.562928	−0.281464	+	0.959572i	$0.590820\pi$
$348$	−10.5036	−0.563055
$349$	8.13467i	0.435439i	0.976011	+	0.217719i	$0.0698618\pi$
$349$	8.13467i	0.435439i	−0.976011	+	0.217719i	$0.930138\pi$
$350$	−3.69202	−0.197347
$351$	0	0
$352$	−3.04892	−0.162508
$353$	− 25.8649i	− 1.37665i	−0.725404	−	0.688324i	$-0.758347\pi$
$353$	− 25.8649i	− 1.37665i	0.725404	−	0.688324i	$-0.241653\pi$
$354$	8.54288	0.454049
$355$	−4.13706	−0.219573
$356$	− 6.41119i	− 0.339792i
$357$	− 25.2935i	− 1.33867i
$358$	25.6461i	1.35544i
$359$	30.2959i	1.59896i	0.600695	+	0.799478i	$0.294891\pi$
$359$	30.2959i	1.59896i	−0.600695	+	0.799478i	$0.705109\pi$
$360$	−1.00000	−0.0527046
$361$	18.1685	0.956238
$362$	25.8092i	1.35650i
$363$	1.70410	0.0894422
$364$	0	0
$365$	−11.1685	−0.584587
$366$	1.55496i	0.0812790i
$367$	−3.41013	−0.178007	−0.0890035	−	0.996031i	$-0.528368\pi$
$367$	−3.41013	−0.178007	−0.0890035	+	0.996031i	$0.528368\pi$
$368$	0.356896	0.0186045
$369$	10.2959i	0.535983i
$370$	0.899772i	0.0467769i
$371$	12.5743i	0.652827i
$372$	− 2.06100i	− 0.106858i
$373$	3.35988	0.173968	0.0869840	−	0.996210i	$-0.472277\pi$
$373$	3.35988	0.173968	0.0869840	+	0.996210i	$0.472277\pi$
$374$	−20.8877	−1.08008
$375$	1.00000i	0.0516398i
$376$	−11.5918	−0.597801
$377$	0	0
$378$	3.69202	0.189897
$379$	− 31.4916i	− 1.61761i	−0.588075	−	0.808807i	$-0.700114\pi$
$379$	− 31.4916i	− 1.61761i	0.588075	−	0.808807i	$-0.299886\pi$
$380$	−0.911854	−0.0467771
$381$	7.08815	0.363137
$382$	− 14.5700i	− 0.745467i
$383$	21.8485i	1.11640i	0.829705	+	0.558202i	$0.188509\pi$
$383$	21.8485i	1.11640i	−0.829705	+	0.558202i	$0.811491\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	− 11.2567i	− 0.573693i
$386$	8.32544	0.423754
$387$	−9.43296	−0.479504
$388$	− 6.97823i	− 0.354266i
$389$	11.3787	0.576921	0.288461	−	0.957492i	$-0.406857\pi$
$389$	11.3787	0.576921	0.288461	+	0.957492i	$0.406857\pi$
$390$	0	0
$391$	2.44504	0.123651
$392$	6.63102i	0.334917i
$393$	19.7506	0.996287
$394$	26.8810	1.35424
$395$	− 9.62565i	− 0.484319i
$396$	− 3.04892i	− 0.153214i
$397$	− 22.0476i	− 1.10654i	−0.833003	−	0.553268i	$-0.813380\pi$
$397$	− 22.0476i	− 1.10654i	0.833003	−	0.553268i	$-0.186620\pi$
$398$	18.7047i	0.937582i
$399$	3.36658	0.168540
$400$	−1.00000	−0.0500000
$401$	− 10.2459i	− 0.511657i	−0.966722	−	0.255828i	$-0.917652\pi$
$401$	− 10.2459i	− 0.511657i	0.966722	−	0.255828i	$-0.0823482\pi$
$402$	−1.14914	−0.0573141
$403$	0	0
$404$	−4.09783	−0.203875
$405$	− 1.00000i	− 0.0496904i
$406$	38.7797	1.92460
$407$	−2.74333	−0.135982
$408$	− 6.85086i	− 0.339168i
$409$	22.8189i	1.12832i	0.825664	+	0.564162i	$0.190800\pi$
$409$	22.8189i	1.12832i	−0.825664	+	0.564162i	$0.809200\pi$
$410$	10.2959i	0.508478i
$411$	7.94331i	0.391815i
$412$	−16.9215	−0.833665
$413$	−31.5405	−1.55201
$414$	0.356896i	0.0175405i
$415$	−9.86054	−0.484035
$416$	0	0
$417$	−1.71379	−0.0839247
$418$	− 2.78017i	− 0.135982i
$419$	−28.5773	−1.39609	−0.698047	−	0.716052i	$-0.745947\pi$
$419$	−28.5773	−1.39609	−0.698047	+	0.716052i	$0.745947\pi$
$420$	3.69202	0.180152
$421$	32.3937i	1.57877i	0.613896	+	0.789387i	$0.289601\pi$
$421$	32.3937i	1.57877i	−0.613896	+	0.789387i	$0.710399\pi$
$422$	2.93362i	0.142807i
$423$	− 11.5918i	− 0.563612i
$424$	3.40581i	0.165401i
$425$	−6.85086	−0.332315
$426$	4.13706	0.200441
$427$	− 5.74094i	− 0.277824i
$428$	12.7192	0.614804
$429$	0	0
$430$	−9.43296	−0.454898
$431$	− 14.6455i	− 0.705449i	−0.935727	−	0.352724i	$-0.885255\pi$
$431$	− 14.6455i	− 0.705449i	0.935727	−	0.352724i	$-0.114745\pi$
$432$	1.00000	0.0481125
$433$	35.8485	1.72277	0.861384	−	0.507955i	$-0.169598\pi$
$433$	35.8485	1.72277	0.861384	+	0.507955i	$0.169598\pi$
$434$	7.60925i	0.365256i
$435$	− 10.5036i	− 0.503611i
$436$	− 19.9922i	− 0.957454i
$437$	0.325437i	0.0155678i
$438$	11.1685	0.533653
$439$	−2.55927	−0.122147	−0.0610736	−	0.998133i	$-0.519452\pi$
$439$	−2.55927	−0.122147	−0.0610736	+	0.998133i	$0.519452\pi$
$440$	− 3.04892i	− 0.145351i
$441$	−6.63102	−0.315763
$442$	0	0
$443$	23.2857	1.10634	0.553169	−	0.833069i	$-0.313418\pi$
$443$	23.2857	1.10634	0.553169	+	0.833069i	$0.313418\pi$
$444$	− 0.899772i	− 0.0427013i
$445$	6.41119	0.303920
$446$	18.1444	0.859160
$447$	− 6.91185i	− 0.326919i
$448$	3.69202i	0.174432i
$449$	27.3327i	1.28991i	0.764220	+	0.644956i	$0.223124\pi$
$449$	27.3327i	1.28991i	−0.764220	+	0.644956i	$0.776876\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−31.3913	−1.47816
$452$	4.49934	0.211631
$453$	− 5.43296i	− 0.255263i
$454$	−12.3056	−0.577530
$455$	0	0
$456$	0.911854	0.0427015
$457$	− 7.74094i	− 0.362106i	−0.983473	−	0.181053i	$-0.942049\pi$
$457$	− 7.74094i	− 0.362106i	0.983473	−	0.181053i	$-0.0579505\pi$
$458$	4.64848	0.217209
$459$	6.85086	0.319771
$460$	0.356896i	0.0166404i
$461$	17.1511i	0.798805i	0.916776	+	0.399402i	$0.130782\pi$
$461$	17.1511i	0.798805i	−0.916776	+	0.399402i	$0.869218\pi$
$462$	11.2567i	0.523708i
$463$	34.2747i	1.59288i	0.604717	+	0.796441i	$0.293286\pi$
$463$	34.2747i	1.59288i	−0.604717	+	0.796441i	$0.706714\pi$
$464$	10.5036	0.487620
$465$	2.06100	0.0955765
$466$	− 0.658170i	− 0.0304891i
$467$	−20.9474	−0.969328	−0.484664	−	0.874700i	$-0.661058\pi$
$467$	−20.9474	−0.969328	−0.484664	+	0.874700i	$0.661058\pi$
$468$	0	0
$469$	4.24267	0.195908
$470$	− 11.5918i	− 0.534690i
$471$	−9.13706	−0.421014
$472$	−8.54288	−0.393218
$473$	− 28.7603i	− 1.32240i
$474$	9.62565i	0.442121i
$475$	− 0.911854i	− 0.0418387i
$476$	25.2935i	1.15933i
$477$	−3.40581	−0.155941
$478$	18.7748	0.858739
$479$	− 15.8310i	− 0.723337i	−0.932307	−	0.361669i	$-0.882207\pi$
$479$	− 15.8310i	− 0.723337i	0.932307	−	0.361669i	$-0.117793\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	18.0271	0.821114
$483$	− 1.31767i	− 0.0599559i
$484$	−1.70410	−0.0774592
$485$	6.97823	0.316865
$486$	1.00000i	0.0453609i
$487$	33.7060i	1.52737i	0.645592	+	0.763683i	$0.276611\pi$
$487$	33.7060i	1.52737i	−0.645592	+	0.763683i	$0.723389\pi$
$488$	− 1.55496i	− 0.0703896i
$489$	− 14.3773i	− 0.650166i
$490$	−6.63102	−0.299559
$491$	−3.98925	−0.180032	−0.0900161	−	0.995940i	$-0.528692\pi$
$491$	−3.98925	−0.180032	−0.0900161	+	0.995940i	$0.528692\pi$
$492$	− 10.2959i	− 0.464175i
$493$	71.9590	3.24087
$494$	0	0
$495$	3.04892	0.137039
$496$	2.06100i	0.0925416i
$497$	−15.2741	−0.685138
$498$	9.86054	0.441862
$499$	20.4698i	0.916354i	0.888861	+	0.458177i	$0.151497\pi$
$499$	20.4698i	0.916354i	−0.888861	+	0.458177i	$0.848503\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	3.00538i	0.134270i
$502$	2.18060i	0.0973251i
$503$	22.5351	1.00479	0.502395	−	0.864638i	$-0.332452\pi$
$503$	22.5351	1.00479	0.502395	+	0.864638i	$0.332452\pi$
$504$	−3.69202	−0.164456
$505$	− 4.09783i	− 0.182351i
$506$	−1.08815	−0.0483740
$507$	0	0
$508$	−7.08815	−0.314486
$509$	− 10.9718i	− 0.486316i	−0.969987	−	0.243158i	$-0.921817\pi$
$509$	− 10.9718i	− 0.486316i	0.969987	−	0.243158i	$-0.0781833\pi$
$510$	6.85086	0.303361
$511$	−41.2344	−1.82410
$512$	1.00000i	0.0441942i
$513$	0.911854i	0.0402593i
$514$	9.58748i	0.422886i
$515$	− 16.9215i	− 0.745652i
$516$	9.43296	0.415263
$517$	35.3424	1.55436
$518$	3.32198i	0.145959i
$519$	−6.89546	−0.302677
$520$	0	0
$521$	15.9691	0.699620	0.349810	−	0.936821i	$-0.386246\pi$
$521$	15.9691	0.699620	0.349810	+	0.936821i	$0.386246\pi$
$522$	10.5036i	0.459732i
$523$	−24.7681	−1.08303	−0.541516	−	0.840690i	$-0.682150\pi$
$523$	−24.7681	−1.08303	−0.541516	+	0.840690i	$0.682150\pi$
$524$	−19.7506	−0.862810
$525$	3.69202i	0.161133i
$526$	− 1.41789i	− 0.0618232i
$527$	14.1196i	0.615060i
$528$	3.04892i	0.132687i
$529$	−22.8726	−0.994462
$530$	−3.40581	−0.147939
$531$	− 8.54288i	− 0.370729i
$532$	−3.36658	−0.145960
$533$	0	0
$534$	−6.41119	−0.277439
$535$	12.7192i	0.549898i
$536$	1.14914	0.0496355
$537$	25.6461	1.10671
$538$	− 4.14675i	− 0.178779i
$539$	− 20.2174i	− 0.870827i
$540$	1.00000i	0.0430331i
$541$	− 0.460107i	− 0.0197816i	−0.999951	−	0.00989078i	$-0.996852\pi$
$541$	− 0.460107i	− 0.0197816i	0.999951	−	0.00989078i	$-0.00314838\pi$
$542$	6.40044	0.274922
$543$	25.8092	1.10758
$544$	6.85086i	0.293728i
$545$	19.9922	0.856373
$546$	0	0
$547$	−8.70410	−0.372161	−0.186080	−	0.982535i	$-0.559578\pi$
$547$	−8.70410	−0.372161	−0.186080	+	0.982535i	$0.559578\pi$
$548$	− 7.94331i	− 0.339322i
$549$	1.55496	0.0663640
$550$	3.04892	0.130006
$551$	9.57779i	0.408028i
$552$	− 0.356896i	− 0.0151905i
$553$	− 35.5381i	− 1.51123i
$554$	− 10.3394i	− 0.439281i
$555$	0.899772	0.0381932
$556$	1.71379	0.0726810
$557$	36.7415i	1.55679i	0.627776	+	0.778394i	$0.283966\pi$
$557$	36.7415i	1.55679i	−0.627776	+	0.778394i	$0.716034\pi$
$558$	−2.06100	−0.0872490
$559$	0	0
$560$	−3.69202	−0.156016
$561$	20.8877i	0.881879i
$562$	5.08144	0.214348
$563$	27.7713	1.17042	0.585211	−	0.810881i	$-0.301012\pi$
$563$	27.7713	1.17042	0.585211	+	0.810881i	$0.301012\pi$
$564$	11.5918i	0.488103i
$565$	4.49934i	0.189288i
$566$	− 24.5284i	− 1.03101i
$567$	− 3.69202i	− 0.155050i
$568$	−4.13706	−0.173587
$569$	−38.4379	−1.61140	−0.805700	−	0.592324i	$-0.798210\pi$
$569$	−38.4379	−1.61140	−0.805700	+	0.592324i	$0.798210\pi$
$570$	0.911854i	0.0381934i
$571$	−10.8086	−0.452328	−0.226164	−	0.974089i	$-0.572618\pi$
$571$	−10.8086	−0.452328	−0.226164	+	0.974089i	$0.572618\pi$
$572$	0	0
$573$	−14.5700	−0.608671
$574$	38.0127i	1.58662i
$575$	−0.356896	−0.0148836
$576$	−1.00000	−0.0416667
$577$	35.5991i	1.48201i	0.671500	+	0.741005i	$0.265650\pi$
$577$	35.5991i	1.48201i	−0.671500	+	0.741005i	$0.734350\pi$
$578$	29.9342i	1.24510i
$579$	− 8.32544i	− 0.345993i
$580$	10.5036i	0.436140i
$581$	−36.4053	−1.51035
$582$	−6.97823	−0.289257
$583$	− 10.3840i	− 0.430063i
$584$	−11.1685	−0.462157
$585$	0	0
$586$	15.8538	0.654916
$587$	11.4741i	0.473587i	0.971560	+	0.236794i	$0.0760965\pi$
$587$	11.4741i	0.473587i	−0.971560	+	0.236794i	$0.923903\pi$
$588$	6.63102	0.273459
$589$	−1.87933	−0.0774364
$590$	− 8.54288i	− 0.351705i
$591$	− 26.8810i	− 1.10574i
$592$	0.899772i	0.0369804i
$593$	− 15.5147i	− 0.637111i	−0.947904	−	0.318555i	$-0.896802\pi$
$593$	− 15.5147i	− 0.637111i	0.947904	−	0.318555i	$-0.103198\pi$
$594$	−3.04892	−0.125099
$595$	−25.2935	−1.03693
$596$	6.91185i	0.283121i
$597$	18.7047	0.765532
$598$	0	0
$599$	−14.2034	−0.580337	−0.290168	−	0.956976i	$-0.593711\pi$
$599$	−14.2034	−0.580337	−0.290168	+	0.956976i	$0.593711\pi$
$600$	1.00000i	0.0408248i
$601$	−39.5579	−1.61360	−0.806801	−	0.590823i	$-0.798803\pi$
$601$	−39.5579	−1.61360	−0.806801	+	0.590823i	$0.798803\pi$
$602$	−34.8267	−1.41943
$603$	1.14914i	0.0467968i
$604$	5.43296i	0.221064i
$605$	− 1.70410i	− 0.0692816i
$606$	4.09783i	0.166463i
$607$	4.77346	0.193749	0.0968744	−	0.995297i	$-0.469115\pi$
$607$	4.77346	0.193749	0.0968744	+	0.995297i	$0.469115\pi$
$608$	−0.911854	−0.0369806
$609$	− 38.7797i	− 1.57143i
$610$	1.55496	0.0629584
$611$	0	0
$612$	−6.85086	−0.276929
$613$	5.43429i	0.219489i	0.993960	+	0.109744i	$0.0350032\pi$
$613$	5.43429i	0.219489i	−0.993960	+	0.109744i	$0.964997\pi$
$614$	−14.2121	−0.573552
$615$	10.2959	0.415171
$616$	− 11.2567i	− 0.453544i
$617$	− 20.6692i	− 0.832110i	−0.909339	−	0.416055i	$-0.863412\pi$
$617$	− 20.6692i	− 0.832110i	0.909339	−	0.416055i	$-0.136588\pi$
$618$	16.9215i	0.680684i
$619$	− 24.4077i	− 0.981030i	−0.871433	−	0.490515i	$-0.836809\pi$
$619$	− 24.4077i	− 0.981030i	0.871433	−	0.490515i	$-0.163191\pi$
$620$	−2.06100	−0.0827717
$621$	0.356896	0.0143217
$622$	− 15.0127i	− 0.601953i
$623$	23.6703	0.948329
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 13.4993i	− 0.539542i
$627$	−2.78017	−0.111029
$628$	9.13706	0.364609
$629$	6.16421i	0.245783i
$630$	− 3.69202i	− 0.147094i
$631$	32.0538i	1.27604i	0.770019	+	0.638021i	$0.220247\pi$
$631$	32.0538i	1.27604i	−0.770019	+	0.638021i	$0.779753\pi$
$632$	− 9.62565i	− 0.382888i
$633$	2.93362	0.116601
$634$	−33.9463	−1.34818
$635$	− 7.08815i	− 0.281284i
$636$	3.40581	0.135049
$637$	0	0
$638$	−32.0248	−1.26787
$639$	− 4.13706i	− 0.163660i
$640$	−1.00000	−0.0395285
$641$	−18.5327	−0.731998	−0.365999	−	0.930615i	$-0.619273\pi$
$641$	−18.5327	−0.731998	−0.365999	+	0.930615i	$0.619273\pi$
$642$	− 12.7192i	− 0.501986i
$643$	− 22.3454i	− 0.881217i	−0.897699	−	0.440608i	$-0.854763\pi$
$643$	− 22.3454i	− 0.881217i	0.897699	−	0.440608i	$-0.145237\pi$
$644$	1.31767i	0.0519234i
$645$	9.43296i	0.371422i
$646$	−6.24698	−0.245784
$647$	−7.86964	−0.309388	−0.154694	−	0.987962i	$-0.549439\pi$
$647$	−7.86964	−0.309388	−0.154694	+	0.987962i	$0.549439\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	26.0465	1.02242
$650$	0	0
$651$	7.60925	0.298230
$652$	14.3773i	0.563060i
$653$	−47.2271	−1.84814	−0.924070	−	0.382223i	$-0.875159\pi$
$653$	−47.2271	−1.84814	−0.924070	+	0.382223i	$0.875159\pi$
$654$	−19.9922	−0.781758
$655$	− 19.7506i	− 0.771721i
$656$	10.2959i	0.401987i
$657$	− 11.1685i	− 0.435726i
$658$	− 42.7972i	− 1.66841i
$659$	−46.6276	−1.81635	−0.908176	−	0.418588i	$-0.862525\pi$
$659$	−46.6276	−1.81635	−0.908176	+	0.418588i	$0.862525\pi$
$660$	−3.04892	−0.118679
$661$	39.9415i	1.55354i	0.629781	+	0.776772i	$0.283144\pi$
$661$	39.9415i	1.55354i	−0.629781	+	0.776772i	$0.716856\pi$
$662$	6.84415	0.266005
$663$	0	0
$664$	−9.86054	−0.382663
$665$	− 3.36658i	− 0.130551i
$666$	−0.899772	−0.0348655
$667$	3.74871	0.145151
$668$	− 3.00538i	− 0.116282i
$669$	− 18.1444i	− 0.701501i
$670$	1.14914i	0.0443953i
$671$	4.74094i	0.183022i
$672$	3.69202	0.142423
$673$	35.3612	1.36307	0.681537	−	0.731783i	$-0.261312\pi$
$673$	35.3612	1.36307	0.681537	+	0.731783i	$0.261312\pi$
$674$	− 27.7332i	− 1.06824i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	19.1715	0.736821	0.368410	−	0.929663i	$-0.379902\pi$
$677$	19.1715	0.736821	0.368410	+	0.929663i	$0.379902\pi$
$678$	− 4.49934i	− 0.172796i
$679$	25.7638	0.988723
$680$	−6.85086	−0.262718
$681$	12.3056i	0.471551i
$682$	− 6.28382i	− 0.240620i
$683$	1.13946i	0.0436001i	0.999762	+	0.0218000i	$0.00693972\pi$
$683$	1.13946i	0.0436001i	−0.999762	+	0.0218000i	$0.993060\pi$
$684$	− 0.911854i	− 0.0348656i
$685$	7.94331	0.303498
$686$	1.36227	0.0520118
$687$	− 4.64848i	− 0.177351i
$688$	−9.43296	−0.359628
$689$	0	0
$690$	0.356896	0.0135868
$691$	11.9071i	0.452966i	0.974015	+	0.226483i	$0.0727228\pi$
$691$	11.9071i	0.452966i	−0.974015	+	0.226483i	$0.927277\pi$
$692$	6.89546	0.262126
$693$	11.2567	0.427605
$694$	− 10.4862i	− 0.398050i
$695$	1.71379i	0.0650078i
$696$	− 10.5036i	− 0.398140i
$697$	70.5357i	2.67173i
$698$	−8.13467	−0.307902
$699$	−0.658170	−0.0248943
$700$	− 3.69202i	− 0.139545i
$701$	−38.0575	−1.43741	−0.718707	−	0.695313i	$-0.755266\pi$
$701$	−38.0575	−1.43741	−0.718707	+	0.695313i	$0.755266\pi$
$702$	0	0
$703$	−0.820461	−0.0309443
$704$	− 3.04892i	− 0.114910i
$705$	−11.5918	−0.436572
$706$	25.8649	0.973437
$707$	− 15.1293i	− 0.568996i
$708$	8.54288i	0.321061i
$709$	26.1215i	0.981014i	0.871437	+	0.490507i	$0.163188\pi$
$709$	26.1215i	0.981014i	−0.871437	+	0.490507i	$0.836812\pi$
$710$	− 4.13706i	− 0.155261i
$711$	9.62565	0.360990
$712$	6.41119	0.240270
$713$	0.735562i	0.0275470i
$714$	25.2935	0.946586
$715$	0	0
$716$	−25.6461	−0.958439
$717$	− 18.7748i	− 0.701157i
$718$	−30.2959	−1.13063
$719$	−21.3739	−0.797111	−0.398556	−	0.917144i	$-0.630488\pi$
$719$	−21.3739	−0.797111	−0.398556	+	0.917144i	$0.630488\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 62.4747i	− 2.32668i
$722$	18.1685i	0.676162i
$723$	− 18.0271i	− 0.670437i
$724$	−25.8092	−0.959193
$725$	−10.5036	−0.390096
$726$	1.70410i	0.0632452i
$727$	17.9124	0.664336	0.332168	−	0.943220i	$-0.392220\pi$
$727$	17.9124	0.664336	0.332168	+	0.943220i	$0.392220\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 11.1685i	− 0.413366i
$731$	−64.6238	−2.39020
$732$	−1.55496	−0.0574729
$733$	− 27.5386i	− 1.01716i	−0.861015	−	0.508580i	$-0.830171\pi$
$733$	− 27.5386i	− 1.01716i	0.861015	−	0.508580i	$-0.169829\pi$
$734$	− 3.41013i	− 0.125870i
$735$	6.63102i	0.244589i
$736$	0.356896i	0.0131554i
$737$	−3.50365	−0.129059
$738$	−10.2959	−0.378997
$739$	− 7.99090i	− 0.293950i	−0.989140	−	0.146975i	$-0.953046\pi$
$739$	− 7.99090i	− 0.293950i	0.989140	−	0.146975i	$-0.0469537\pi$
$740$	−0.899772	−0.0330763
$741$	0	0
$742$	−12.5743	−0.461618
$743$	− 19.8398i	− 0.727853i	−0.931428	−	0.363927i	$-0.881436\pi$
$743$	− 19.8398i	− 0.727853i	0.931428	−	0.363927i	$-0.118564\pi$
$744$	2.06100	0.0755599
$745$	−6.91185	−0.253231
$746$	3.35988i	0.123014i
$747$	− 9.86054i	− 0.360778i
$748$	− 20.8877i	− 0.763730i
$749$	46.9594i	1.71586i
$750$	−1.00000	−0.0365148
$751$	25.7525	0.939724	0.469862	−	0.882740i	$-0.344304\pi$
$751$	25.7525	0.939724	0.469862	+	0.882740i	$0.344304\pi$
$752$	− 11.5918i	− 0.422709i
$753$	2.18060	0.0794656
$754$	0	0
$755$	−5.43296	−0.197726
$756$	3.69202i	0.134278i
$757$	−51.5096	−1.87215	−0.936074	−	0.351802i	$-0.885569\pi$
$757$	−51.5096	−1.87215	−0.936074	+	0.351802i	$0.885569\pi$
$758$	31.4916	1.14383
$759$	1.08815i	0.0394972i
$760$	− 0.911854i	− 0.0330764i
$761$	− 31.4765i	− 1.14102i	−0.821290	−	0.570511i	$-0.806745\pi$
$761$	− 31.4765i	− 1.14102i	0.821290	−	0.570511i	$-0.193255\pi$
$762$	7.08815i	0.256776i
$763$	73.8117	2.67216
$764$	14.5700	0.527125
$765$	− 6.85086i	− 0.247693i
$766$	−21.8485	−0.789417
$767$	0	0
$768$	1.00000	0.0360844
$769$	47.1584i	1.70057i	0.526319	+	0.850287i	$0.323572\pi$
$769$	47.1584i	1.70057i	−0.526319	+	0.850287i	$0.676428\pi$
$770$	11.2567	0.405662
$771$	9.58748	0.345285
$772$	8.32544i	0.299639i
$773$	3.71917i	0.133769i	0.997761	+	0.0668846i	$0.0213059\pi$
$773$	3.71917i	0.133769i	−0.997761	+	0.0668846i	$0.978694\pi$
$774$	− 9.43296i	− 0.339061i
$775$	− 2.06100i	− 0.0740333i
$776$	6.97823	0.250504
$777$	3.32198	0.119175
$778$	11.3787i	0.407945i
$779$	−9.38835	−0.336373
$780$	0	0
$781$	12.6136	0.451349
$782$	2.44504i	0.0874345i
$783$	10.5036	0.375370
$784$	−6.63102	−0.236822
$785$	9.13706i	0.326116i
$786$	19.7506i	0.704482i
$787$	47.2282i	1.68350i	0.539865	+	0.841752i	$0.318475\pi$
$787$	47.2282i	1.68350i	−0.539865	+	0.841752i	$0.681525\pi$
$788$	26.8810i	0.957595i
$789$	−1.41789	−0.0504784
$790$	9.62565	0.342465
$791$	16.6116i	0.590642i
$792$	3.04892	0.108339
$793$	0	0
$794$	22.0476	0.782440
$795$	3.40581i	0.120792i
$796$	−18.7047	−0.662970
$797$	−34.4359	−1.21978	−0.609892	−	0.792485i	$-0.708787\pi$
$797$	−34.4359	−1.21978	−0.609892	+	0.792485i	$0.708787\pi$
$798$	3.36658i	0.119176i
$799$	− 79.4137i	− 2.80945i
$800$	− 1.00000i	− 0.0353553i
$801$	6.41119i	0.226528i
$802$	10.2459	0.361796
$803$	34.0519	1.20167
$804$	− 1.14914i	− 0.0405272i
$805$	−1.31767	−0.0464417
$806$	0	0
$807$	−4.14675	−0.145973
$808$	− 4.09783i	− 0.144161i
$809$	−32.3846	−1.13858	−0.569292	−	0.822136i	$-0.692782\pi$
$809$	−32.3846	−1.13858	−0.569292	+	0.822136i	$0.692782\pi$
$810$	1.00000	0.0351364
$811$	37.8297i	1.32838i	0.747564	+	0.664190i	$0.231223\pi$
$811$	37.8297i	1.32838i	−0.747564	+	0.664190i	$0.768777\pi$
$812$	38.7797i	1.36090i
$813$	− 6.40044i	− 0.224473i
$814$	− 2.74333i	− 0.0961537i
$815$	−14.3773	−0.503616
$816$	6.85086	0.239828
$817$	− 8.60148i	− 0.300928i
$818$	−22.8189	−0.797845
$819$	0	0
$820$	−10.2959	−0.359548
$821$	43.2820i	1.51055i	0.655406	+	0.755276i	$0.272497\pi$
$821$	43.2820i	1.51055i	−0.655406	+	0.755276i	$0.727503\pi$
$822$	−7.94331	−0.277055
$823$	26.7399	0.932093	0.466047	−	0.884760i	$-0.345678\pi$
$823$	26.7399	0.932093	0.466047	+	0.884760i	$0.345678\pi$
$824$	− 16.9215i	− 0.589490i
$825$	− 3.04892i	− 0.106150i
$826$	− 31.5405i	− 1.09743i
$827$	− 21.4437i	− 0.745671i	−0.927897	−	0.372835i	$-0.878386\pi$
$827$	− 21.4437i	− 0.745671i	0.927897	−	0.372835i	$-0.121614\pi$
$828$	−0.356896	−0.0124030
$829$	−22.4547	−0.779885	−0.389943	−	0.920839i	$-0.627505\pi$
$829$	−22.4547	−0.779885	−0.389943	+	0.920839i	$0.627505\pi$
$830$	− 9.86054i	− 0.342264i
$831$	−10.3394	−0.358671
$832$	0	0
$833$	−45.4282	−1.57399
$834$	− 1.71379i	− 0.0593438i
$835$	3.00538	0.104005
$836$	2.78017	0.0961541
$837$	2.06100i	0.0712385i
$838$	− 28.5773i	− 0.987187i
$839$	− 24.4034i	− 0.842500i	−0.906945	−	0.421250i	$-0.861592\pi$
$839$	− 24.4034i	− 0.842500i	0.906945	−	0.421250i	$-0.138408\pi$
$840$	3.69202i	0.127387i
$841$	81.3266	2.80437
$842$	−32.3937	−1.11636
$843$	− 5.08144i	− 0.175014i
$844$	−2.93362	−0.100980
$845$	0	0
$846$	11.5918	0.398534
$847$	− 6.29159i	− 0.216181i
$848$	−3.40581	−0.116956
$849$	−24.5284	−0.841813
$850$	− 6.85086i	− 0.234982i
$851$	0.321125i	0.0110080i
$852$	4.13706i	0.141733i
$853$	7.96184i	0.272608i	0.990667	+	0.136304i	$0.0435224\pi$
$853$	7.96184i	0.272608i	−0.990667	+	0.136304i	$0.956478\pi$
$854$	5.74094	0.196451
$855$	0.911854	0.0311847
$856$	12.7192i	0.434732i
$857$	8.12737	0.277626	0.138813	−	0.990319i	$-0.455671\pi$
$857$	8.12737	0.277626	0.138813	+	0.990319i	$0.455671\pi$
$858$	0	0
$859$	17.1666	0.585717	0.292858	−	0.956156i	$-0.405394\pi$
$859$	17.1666	0.585717	0.292858	+	0.956156i	$0.405394\pi$
$860$	− 9.43296i	− 0.321661i
$861$	38.0127	1.29547
$862$	14.6455	0.498828
$863$	− 46.2857i	− 1.57558i	−0.615941	−	0.787792i	$-0.711224\pi$
$863$	− 46.2857i	− 1.57558i	0.615941	−	0.787792i	$-0.288776\pi$
$864$	1.00000i	0.0340207i
$865$	6.89546i	0.234453i
$866$	35.8485i	1.21818i
$867$	29.9342	1.01662
$868$	−7.60925	−0.258275
$869$	29.3478i	0.995556i
$870$	10.5036	0.356107
$871$	0	0
$872$	19.9922	0.677022
$873$	6.97823i	0.236177i
$874$	−0.325437	−0.0110081
$875$	3.69202	0.124813
$876$	11.1685i	0.377349i
$877$	2.72156i	0.0919006i	0.998944	+	0.0459503i	$0.0146316\pi$
$877$	2.72156i	0.0919006i	−0.998944	+	0.0459503i	$0.985368\pi$
$878$	− 2.55927i	− 0.0863712i
$879$	− 15.8538i	− 0.534737i
$880$	3.04892	0.102779
$881$	16.7241	0.563448	0.281724	−	0.959495i	$-0.409094\pi$
$881$	16.7241	0.563448	0.281724	+	0.959495i	$0.409094\pi$
$882$	− 6.63102i	− 0.223278i
$883$	2.88364	0.0970423	0.0485211	−	0.998822i	$-0.484549\pi$
$883$	2.88364	0.0970423	0.0485211	+	0.998822i	$0.484549\pi$
$884$	0	0
$885$	−8.54288	−0.287166
$886$	23.2857i	0.782300i
$887$	55.8866	1.87649	0.938245	−	0.345973i	$-0.112451\pi$
$887$	55.8866	1.87649	0.938245	+	0.345973i	$0.112451\pi$
$888$	0.899772	0.0301944
$889$	− 26.1696i	− 0.877700i
$890$	6.41119i	0.214904i
$891$	3.04892i	0.102143i
$892$	18.1444i	0.607518i
$893$	10.5700	0.353712
$894$	6.91185	0.231167
$895$	− 25.6461i	− 0.857254i
$896$	−3.69202	−0.123342
$897$	0	0
$898$	−27.3327	−0.912105
$899$	21.6480i	0.722001i
$900$	1.00000	0.0333333
$901$	−23.3327	−0.777326
$902$	− 31.3913i	− 1.04522i
$903$	34.8267i	1.15896i
$904$	4.49934i	0.149646i
$905$	− 25.8092i	− 0.857928i
$906$	5.43296	0.180498
$907$	−49.5381	−1.64489	−0.822443	−	0.568848i	$-0.807389\pi$
$907$	−49.5381	−1.64489	−0.822443	+	0.568848i	$0.807389\pi$
$908$	− 12.3056i	− 0.408375i
$909$	4.09783	0.135917
$910$	0	0
$911$	37.0603	1.22786	0.613931	−	0.789360i	$-0.289587\pi$
$911$	37.0603	1.22786	0.613931	+	0.789360i	$0.289587\pi$
$912$	0.911854i	0.0301945i
$913$	30.0640	0.994973
$914$	7.74094	0.256047
$915$	− 1.55496i	− 0.0514053i
$916$	4.64848i	0.153590i
$917$	− 72.9197i	− 2.40802i
$918$	6.85086i	0.226112i
$919$	53.6228	1.76885	0.884426	−	0.466680i	$-0.154550\pi$
$919$	53.6228	1.76885	0.884426	+	0.466680i	$0.154550\pi$
$920$	−0.356896	−0.0117665
$921$	14.2121i	0.468303i
$922$	−17.1511	−0.564840
$923$	0	0
$924$	−11.2567	−0.370317
$925$	− 0.899772i	− 0.0295843i
$926$	−34.2747	−1.12634
$927$	16.9215	0.555776
$928$	10.5036i	0.344799i
$929$	− 10.6726i	− 0.350158i	−0.984554	−	0.175079i	$-0.943982\pi$
$929$	− 10.6726i	− 0.350158i	0.984554	−	0.175079i	$-0.0560181\pi$
$930$	2.06100i	0.0675828i
$931$	− 6.04652i	− 0.198167i
$932$	0.658170	0.0215591
$933$	−15.0127	−0.491493
$934$	− 20.9474i	− 0.685419i
$935$	20.8877	0.683101
$936$	0	0
$937$	17.0538	0.557124	0.278562	−	0.960418i	$-0.410142\pi$
$937$	17.0538	0.557124	0.278562	+	0.960418i	$0.410142\pi$
$938$	4.24267i	0.138528i
$939$	−13.4993	−0.440534
$940$	11.5918	0.378083
$941$	− 46.7706i	− 1.52468i	−0.647178	−	0.762339i	$-0.724051\pi$
$941$	− 46.7706i	− 1.52468i	0.647178	−	0.762339i	$-0.275949\pi$
$942$	− 9.13706i	− 0.297702i
$943$	3.67456i	0.119660i
$944$	− 8.54288i	− 0.278047i
$945$	−3.69202	−0.120101
$946$	28.7603	0.935079
$947$	30.0398i	0.976163i	0.872798	+	0.488081i	$0.162303\pi$
$947$	30.0398i	0.976163i	−0.872798	+	0.488081i	$0.837697\pi$
$948$	−9.62565	−0.312626
$949$	0	0
$950$	0.911854	0.0295845
$951$	33.9463i	1.10078i
$952$	−25.2935	−0.819767
$953$	29.4252	0.953175	0.476588	−	0.879127i	$-0.341873\pi$
$953$	29.4252	0.953175	0.476588	+	0.879127i	$0.341873\pi$
$954$	− 3.40581i	− 0.110267i
$955$	14.5700i	0.471475i
$956$	18.7748i	0.607220i
$957$	32.0248i	1.03521i
$958$	15.8310	0.511477
$959$	29.3269	0.947014
$960$	1.00000i	0.0322749i
$961$	26.7523	0.862977
$962$	0	0
$963$	−12.7192	−0.409869
$964$	18.0271i	0.580615i
$965$	−8.32544	−0.268005
$966$	1.31767	0.0423952
$967$	12.7023i	0.408478i	0.978921	+	0.204239i	$0.0654720\pi$
$967$	12.7023i	0.408478i	−0.978921	+	0.204239i	$0.934528\pi$
$968$	− 1.70410i	− 0.0547719i
$969$	6.24698i	0.200682i
$970$	6.97823i	0.224057i
$971$	22.9105	0.735234	0.367617	−	0.929977i	$-0.380174\pi$
$971$	22.9105	0.735234	0.367617	+	0.929977i	$0.380174\pi$
$972$	−1.00000	−0.0320750
$973$	6.32736i	0.202846i
$974$	−33.7060	−1.08001
$975$	0	0
$976$	1.55496	0.0497730
$977$	− 25.1390i	− 0.804267i	−0.915581	−	0.402134i	$-0.868269\pi$
$977$	− 25.1390i	− 0.804267i	0.915581	−	0.402134i	$-0.131731\pi$
$978$	14.3773	0.459737
$979$	−19.5472	−0.624731
$980$	− 6.63102i	− 0.211820i
$981$	19.9922i	0.638303i
$982$	− 3.98925i	− 0.127302i
$983$	50.3895i	1.60718i	0.595186	+	0.803588i	$0.297079\pi$
$983$	50.3895i	1.60718i	−0.595186	+	0.803588i	$0.702921\pi$
$984$	10.2959	0.328221
$985$	−26.8810	−0.856499
$986$	71.9590i	2.29164i
$987$	−42.7972	−1.36225
$988$	0	0
$989$	−3.36658	−0.107051
$990$	3.04892i	0.0969010i
$991$	23.3375	0.741341	0.370670	−	0.928764i	$-0.379128\pi$
$991$	23.3375	0.741341	0.370670	+	0.928764i	$0.379128\pi$
$992$	−2.06100	−0.0654368
$993$	− 6.84415i	− 0.217193i
$994$	− 15.2741i	− 0.484466i
$995$	− 18.7047i	− 0.592979i
$996$	9.86054i	0.312443i
$997$	22.8127	0.722485	0.361243	−	0.932472i	$-0.382353\pi$
$997$	22.8127	0.722485	0.361243	+	0.932472i	$0.382353\pi$
$998$	−20.4698	−0.647960
$999$	0.899772i	0.0284675i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.z.1351.4		6
13.5	odd	4		5070.2.a.bw.1.3	yes	3
13.8	odd	4		5070.2.a.bp.1.1	✓	3
13.12	even	2	inner	5070.2.b.z.1351.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bp.1.1	✓	3	13.8	odd	4
5070.2.a.bw.1.3	yes	3	13.5	odd	4
5070.2.b.z.1351.3		6	13.12	even	2	inner
5070.2.b.z.1351.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -3.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.04892i q^{11} -1.00000 q^{12} +3.69202 q^{14} -1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} +1.00000i q^{18} +0.911854i q^{19} +1.00000i q^{20} -3.69202i q^{21} -3.04892 q^{22} +0.356896 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +3.69202i q^{28} +10.5036 q^{29} +1.00000 q^{30} +2.06100i q^{31} +1.00000i q^{32} +3.04892i q^{33} +6.85086i q^{34} -3.69202 q^{35} -1.00000 q^{36} +0.899772i q^{37} -0.911854 q^{38} -1.00000 q^{40} +10.2959i q^{41} +3.69202 q^{42} -9.43296 q^{43} -3.04892i q^{44} -1.00000i q^{45} +0.356896i q^{46} -11.5918i q^{47} +1.00000 q^{48} -6.63102 q^{49} -1.00000i q^{50} +6.85086 q^{51} -3.40581 q^{53} +1.00000i q^{54} +3.04892 q^{55} -3.69202 q^{56} +0.911854i q^{57} +10.5036i q^{58} -8.54288i q^{59} +1.00000i q^{60} +1.55496 q^{61} -2.06100 q^{62} -3.69202i q^{63} -1.00000 q^{64} -3.04892 q^{66} +1.14914i q^{67} -6.85086 q^{68} +0.356896 q^{69} -3.69202i q^{70} -4.13706i q^{71} -1.00000i q^{72} -11.1685i q^{73} -0.899772 q^{74} -1.00000 q^{75} -0.911854i q^{76} +11.2567 q^{77} +9.62565 q^{79} -1.00000i q^{80} +1.00000 q^{81} -10.2959 q^{82} -9.86054i q^{83} +3.69202i q^{84} -6.85086i q^{85} -9.43296i q^{86} +10.5036 q^{87} +3.04892 q^{88} +6.41119i q^{89} +1.00000 q^{90} -0.356896 q^{92} +2.06100i q^{93} +11.5918 q^{94} +0.911854 q^{95} +1.00000i q^{96} +6.97823i q^{97} -6.63102i q^{98} +3.04892i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{4} + 6 q^{9} + 6 q^{10} - 6 q^{12} + 12 q^{14} + 6 q^{16} + 14 q^{17} - 6 q^{23} - 6 q^{25} + 6 q^{27} + 6 q^{30} - 12 q^{35} - 6 q^{36} + 2 q^{38} - 6 q^{40} + 12 q^{42} - 18 q^{43} + 6 q^{48} - 10 q^{49} + 14 q^{51} + 6 q^{53} - 12 q^{56} + 10 q^{61} - 32 q^{62} - 6 q^{64} - 14 q^{68} - 6 q^{69} + 40 q^{74} - 6 q^{75} + 14 q^{77} + 34 q^{79} + 6 q^{81} - 34 q^{82} + 6 q^{90} + 6 q^{92} + 14 q^{94} - 2 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 6 * q^9 + 6 * q^10 - 6 * q^12 + 12 * q^14 + 6 * q^16 + 14 * q^17 - 6 * q^23 - 6 * q^25 + 6 * q^27 + 6 * q^30 - 12 * q^35 - 6 * q^36 + 2 * q^38 - 6 * q^40 + 12 * q^42 - 18 * q^43 + 6 * q^48 - 10 * q^49 + 14 * q^51 + 6 * q^53 - 12 * q^56 + 10 * q^61 - 32 * q^62 - 6 * q^64 - 14 * q^68 - 6 * q^69 + 40 * q^74 - 6 * q^75 + 14 * q^77 + 34 * q^79 + 6 * q^81 - 34 * q^82 + 6 * q^90 + 6 * q^92 + 14 * q^94 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more