LMFDB - Embedded newform 5070.2.b.s.1351.2

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			1351.2
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.s.1351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.19806i q^{11} +1.00000 q^{12} -1.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.04892 q^{17} -1.00000i q^{18} -5.34481i q^{19} +1.00000i q^{20} +1.35690i q^{21} -3.19806 q^{22} +8.32304 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.35690i q^{28} +8.07069 q^{29} +1.00000 q^{30} -0.185981i q^{31} -1.00000i q^{32} +3.19806i q^{33} -2.04892i q^{34} -1.35690 q^{35} -1.00000 q^{36} +2.46681i q^{37} -5.34481 q^{38} +1.00000 q^{40} +12.2349i q^{41} +1.35690 q^{42} +2.91185 q^{43} +3.19806i q^{44} -1.00000i q^{45} -8.32304i q^{46} -13.0978i q^{47} -1.00000 q^{48} +5.15883 q^{49} +1.00000i q^{50} -2.04892 q^{51} +5.74094 q^{53} +1.00000i q^{54} -3.19806 q^{55} +1.35690 q^{56} +5.34481i q^{57} -8.07069i q^{58} +1.62565i q^{59} -1.00000i q^{60} +5.28382 q^{61} -0.185981 q^{62} -1.35690i q^{63} -1.00000 q^{64} +3.19806 q^{66} +1.55496i q^{67} -2.04892 q^{68} -8.32304 q^{69} +1.35690i q^{70} -2.32304i q^{71} +1.00000i q^{72} +5.11529i q^{73} +2.46681 q^{74} +1.00000 q^{75} +5.34481i q^{76} -4.33944 q^{77} -1.42327 q^{79} -1.00000i q^{80} +1.00000 q^{81} +12.2349 q^{82} +11.1196i q^{83} -1.35690i q^{84} -2.04892i q^{85} -2.91185i q^{86} -8.07069 q^{87} +3.19806 q^{88} -3.37867i q^{89} -1.00000 q^{90} -8.32304 q^{92} +0.185981i q^{93} -13.0978 q^{94} -5.34481 q^{95} +1.00000i q^{96} -4.74094i q^{97} -5.15883i q^{98} -3.19806i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{16} - 6 q^{17} - 28 q^{22} + 10 q^{23} - 6 q^{25} - 6 q^{27} + 24 q^{29} + 6 q^{30} - 6 q^{36} + 14 q^{38} + 6 q^{40} + 10 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 6 q^{53} - 28 q^{55} - 34 q^{61} + 28 q^{62} - 6 q^{64} + 28 q^{66} + 6 q^{68} - 10 q^{69} + 8 q^{74} + 6 q^{75} + 14 q^{77} - 14 q^{79} + 6 q^{81} + 26 q^{82} - 24 q^{87} + 28 q^{88} - 6 q^{90} - 10 q^{92} - 42 q^{94} + 14 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 28 * q^22 + 10 * q^23 - 6 * q^25 - 6 * q^27 + 24 * q^29 + 6 * q^30 - 6 * q^36 + 14 * q^38 + 6 * q^40 + 10 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 6 * q^53 - 28 * q^55 - 34 * q^61 + 28 * q^62 - 6 * q^64 + 28 * q^66 + 6 * q^68 - 10 * q^69 + 8 * q^74 + 6 * q^75 + 14 * q^77 - 14 * q^79 + 6 * q^81 + 26 * q^82 - 24 * q^87 + 28 * q^88 - 6 * q^90 - 10 * q^92 - 42 * q^94 + 14 * 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
$7$	− 1.35690i	− 0.512858i	−0.966563	−	0.256429i	$-0.917454\pi$
$7$	− 1.35690i	− 0.512858i	0.966563	−	0.256429i	$-0.0825461\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 3.19806i	− 0.964252i	−0.876102	−	0.482126i	$-0.839865\pi$
$11$	− 3.19806i	− 0.964252i	0.876102	−	0.482126i	$-0.160135\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−1.35690	−0.362646
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	2.04892	0.496935	0.248468	−	0.968640i	$-0.420073\pi$
$17$	2.04892	0.496935	0.248468	+	0.968640i	$0.420073\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 5.34481i	− 1.22618i	−0.790011	−	0.613092i	$-0.789925\pi$
$19$	− 5.34481i	− 1.22618i	0.790011	−	0.613092i	$-0.210075\pi$
$20$	1.00000i	0.223607i
$21$	1.35690i	0.296099i
$22$	−3.19806	−0.681829
$23$	8.32304	1.73547	0.867737	−	0.497023i	$-0.165574\pi$
$23$	8.32304	1.73547	0.867737	+	0.497023i	$0.165574\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.35690i	0.256429i
$29$	8.07069	1.49869	0.749345	−	0.662180i	$-0.230369\pi$
$29$	8.07069	1.49869	0.749345	+	0.662180i	$0.230369\pi$
$30$	1.00000	0.182574
$31$	− 0.185981i	− 0.0334031i	−0.999861	−	0.0167016i	$-0.994683\pi$
$31$	− 0.185981i	− 0.0334031i	0.999861	−	0.0167016i	$-0.00531652\pi$
$32$	− 1.00000i	− 0.176777i
$33$	3.19806i	0.556711i
$34$	− 2.04892i	− 0.351386i
$35$	−1.35690	−0.229357
$36$	−1.00000	−0.166667
$37$	2.46681i	0.405541i	0.979226	+	0.202771i	$0.0649946\pi$
$37$	2.46681i	0.405541i	−0.979226	+	0.202771i	$0.935005\pi$
$38$	−5.34481	−0.867043
$39$	0	0
$40$	1.00000	0.158114
$41$	12.2349i	1.91077i	0.295363	+	0.955385i	$0.404559\pi$
$41$	12.2349i	1.91077i	−0.295363	+	0.955385i	$0.595441\pi$
$42$	1.35690	0.209374
$43$	2.91185	0.444054	0.222027	−	0.975041i	$-0.428733\pi$
$43$	2.91185	0.444054	0.222027	+	0.975041i	$0.428733\pi$
$44$	3.19806i	0.482126i
$45$	− 1.00000i	− 0.149071i
$46$	− 8.32304i	− 1.22717i
$47$	− 13.0978i	− 1.91052i	−0.295775	−	0.955258i	$-0.595578\pi$
$47$	− 13.0978i	− 1.91052i	0.295775	−	0.955258i	$-0.404422\pi$
$48$	−1.00000	−0.144338
$49$	5.15883	0.736976
$50$	1.00000i	0.141421i
$51$	−2.04892	−0.286906
$52$	0	0
$53$	5.74094	0.788579	0.394289	−	0.918986i	$-0.370991\pi$
$53$	5.74094	0.788579	0.394289	+	0.918986i	$0.370991\pi$
$54$	1.00000i	0.136083i
$55$	−3.19806	−0.431227
$56$	1.35690	0.181323
$57$	5.34481i	0.707938i
$58$	− 8.07069i	− 1.05973i
$59$	1.62565i	0.211641i	0.994385	+	0.105821i	$0.0337469\pi$
$59$	1.62565i	0.211641i	−0.994385	+	0.105821i	$0.966253\pi$
$60$	− 1.00000i	− 0.129099i
$61$	5.28382	0.676523	0.338262	−	0.941052i	$-0.390161\pi$
$61$	5.28382	0.676523	0.338262	+	0.941052i	$0.390161\pi$
$62$	−0.185981	−0.0236196
$63$	− 1.35690i	− 0.170953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.19806	0.393654
$67$	1.55496i	0.189968i	0.995479	+	0.0949842i	$0.0302801\pi$
$67$	1.55496i	0.189968i	−0.995479	+	0.0949842i	$0.969720\pi$
$68$	−2.04892	−0.248468
$69$	−8.32304	−1.00198
$70$	1.35690i	0.162180i
$71$	− 2.32304i	− 0.275695i	−0.990453	−	0.137847i	$-0.955982\pi$
$71$	− 2.32304i	− 0.275695i	0.990453	−	0.137847i	$-0.0440183\pi$
$72$	1.00000i	0.117851i
$73$	5.11529i	0.598700i	0.954143	+	0.299350i	$0.0967698\pi$
$73$	5.11529i	0.598700i	−0.954143	+	0.299350i	$0.903230\pi$
$74$	2.46681	0.286761
$75$	1.00000	0.115470
$76$	5.34481i	0.613092i
$77$	−4.33944	−0.494525
$78$	0	0
$79$	−1.42327	−0.160131	−0.0800653	−	0.996790i	$-0.525513\pi$
$79$	−1.42327	−0.160131	−0.0800653	+	0.996790i	$0.525513\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	12.2349	1.35112
$83$	11.1196i	1.22054i	0.792195	+	0.610268i	$0.208938\pi$
$83$	11.1196i	1.22054i	−0.792195	+	0.610268i	$0.791062\pi$
$84$	− 1.35690i	− 0.148049i
$85$	− 2.04892i	− 0.222236i
$86$	− 2.91185i	− 0.313993i
$87$	−8.07069	−0.865269
$88$	3.19806	0.340915
$89$	− 3.37867i	− 0.358138i	−0.983837	−	0.179069i	$-0.942691\pi$
$89$	− 3.37867i	− 0.358138i	0.983837	−	0.179069i	$-0.0573085\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−8.32304	−0.867737
$93$	0.185981i	0.0192853i
$94$	−13.0978	−1.35094
$95$	−5.34481	−0.548366
$96$	1.00000i	0.102062i
$97$	− 4.74094i	− 0.481369i	−0.970603	−	0.240685i	$-0.922628\pi$
$97$	− 4.74094i	− 0.481369i	0.970603	−	0.240685i	$-0.0773720\pi$
$98$	− 5.15883i	− 0.521121i
$99$	− 3.19806i	− 0.321417i
$100$	1.00000	0.100000
$101$	12.5918	1.25293	0.626465	−	0.779449i	$-0.284501\pi$
$101$	12.5918	1.25293	0.626465	+	0.779449i	$0.284501\pi$
$102$	2.04892i	0.202873i
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	0	0
$105$	1.35690	0.132419
$106$	− 5.74094i	− 0.557609i
$107$	−5.08815	−0.491890	−0.245945	−	0.969284i	$-0.579098\pi$
$107$	−5.08815	−0.491890	−0.245945	+	0.969284i	$0.579098\pi$
$108$	1.00000	0.0962250
$109$	6.43296i	0.616166i	0.951360	+	0.308083i	$0.0996874\pi$
$109$	6.43296i	0.616166i	−0.951360	+	0.308083i	$0.900313\pi$
$110$	3.19806i	0.304923i
$111$	− 2.46681i	− 0.234139i
$112$	− 1.35690i	− 0.128215i
$113$	−18.6571	−1.75511	−0.877556	−	0.479473i	$-0.840828\pi$
$113$	−18.6571	−1.75511	−0.877556	+	0.479473i	$0.840828\pi$
$114$	5.34481	0.500588
$115$	− 8.32304i	− 0.776128i
$116$	−8.07069	−0.749345
$117$	0	0
$118$	1.62565	0.149653
$119$	− 2.78017i	− 0.254858i
$120$	−1.00000	−0.0912871
$121$	0.772398	0.0702180
$122$	− 5.28382i	− 0.478374i
$123$	− 12.2349i	− 1.10318i
$124$	0.185981i	0.0167016i
$125$	1.00000i	0.0894427i
$126$	−1.35690	−0.120882
$127$	16.3763	1.45316	0.726580	−	0.687082i	$-0.241109\pi$
$127$	16.3763	1.45316	0.726580	+	0.687082i	$0.241109\pi$
$128$	1.00000i	0.0883883i
$129$	−2.91185	−0.256374
$130$	0	0
$131$	3.00969	0.262958	0.131479	−	0.991319i	$-0.458027\pi$
$131$	3.00969	0.262958	0.131479	+	0.991319i	$0.458027\pi$
$132$	− 3.19806i	− 0.278356i
$133$	−7.25236	−0.628859
$134$	1.55496	0.134328
$135$	1.00000i	0.0860663i
$136$	2.04892i	0.175693i
$137$	− 20.2892i	− 1.73342i	−0.498810	−	0.866711i	$-0.666229\pi$
$137$	− 20.2892i	− 1.73342i	0.498810	−	0.866711i	$-0.333771\pi$
$138$	8.32304i	0.708505i
$139$	−0.153457	−0.0130160	−0.00650802	−	0.999979i	$-0.502072\pi$
$139$	−0.153457	−0.0130160	−0.00650802	+	0.999979i	$0.502072\pi$
$140$	1.35690	0.114679
$141$	13.0978i	1.10304i
$142$	−2.32304	−0.194946
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 8.07069i	− 0.670234i
$146$	5.11529	0.423345
$147$	−5.15883	−0.425493
$148$	− 2.46681i	− 0.202771i
$149$	− 10.7724i	− 0.882509i	−0.897382	−	0.441255i	$-0.854534\pi$
$149$	− 10.7724i	− 0.882509i	0.897382	−	0.441255i	$-0.145466\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 17.3817i	− 1.41450i	−0.706964	−	0.707249i	$-0.749936\pi$
$151$	− 17.3817i	− 1.41450i	0.706964	−	0.707249i	$-0.250064\pi$
$152$	5.34481	0.433522
$153$	2.04892	0.165645
$154$	4.33944i	0.349682i
$155$	−0.185981	−0.0149383
$156$	0	0
$157$	15.5429	1.24046	0.620228	−	0.784421i	$-0.287040\pi$
$157$	15.5429	1.24046	0.620228	+	0.784421i	$0.287040\pi$
$158$	1.42327i	0.113229i
$159$	−5.74094	−0.455286
$160$	−1.00000	−0.0790569
$161$	− 11.2935i	− 0.890053i
$162$	− 1.00000i	− 0.0785674i
$163$	2.00239i	0.156840i	0.996920	+	0.0784198i	$0.0249874\pi$
$163$	2.00239i	0.156840i	−0.996920	+	0.0784198i	$0.975013\pi$
$164$	− 12.2349i	− 0.955385i
$165$	3.19806	0.248969
$166$	11.1196	0.863049
$167$	8.24027i	0.637652i	0.947813	+	0.318826i	$0.103288\pi$
$167$	8.24027i	0.637652i	−0.947813	+	0.318826i	$0.896712\pi$
$168$	−1.35690	−0.104687
$169$	0	0
$170$	−2.04892	−0.157145
$171$	− 5.34481i	− 0.408728i
$172$	−2.91185	−0.222027
$173$	−12.7463	−0.969084	−0.484542	−	0.874768i	$-0.661014\pi$
$173$	−12.7463	−0.969084	−0.484542	+	0.874768i	$0.661014\pi$
$174$	8.07069i	0.611837i
$175$	1.35690i	0.102572i
$176$	− 3.19806i	− 0.241063i
$177$	− 1.62565i	− 0.122191i
$178$	−3.37867	−0.253242
$179$	−16.0422	−1.19905	−0.599526	−	0.800356i	$-0.704644\pi$
$179$	−16.0422	−1.19905	−0.599526	+	0.800356i	$0.704644\pi$
$180$	1.00000i	0.0745356i
$181$	−3.85086	−0.286232	−0.143116	−	0.989706i	$-0.545712\pi$
$181$	−3.85086	−0.286232	−0.143116	+	0.989706i	$0.545712\pi$
$182$	0	0
$183$	−5.28382	−0.390591
$184$	8.32304i	0.613583i
$185$	2.46681	0.181364
$186$	0.185981	0.0136368
$187$	− 6.55257i	− 0.479171i
$188$	13.0978i	0.955258i
$189$	1.35690i	0.0986997i
$190$	5.34481i	0.387754i
$191$	11.8280	0.855845	0.427923	−	0.903815i	$-0.359246\pi$
$191$	11.8280	0.855845	0.427923	+	0.903815i	$0.359246\pi$
$192$	1.00000	0.0721688
$193$	− 0.872625i	− 0.0628129i	−0.999507	−	0.0314065i	$-0.990001\pi$
$193$	− 0.872625i	− 0.0628129i	0.999507	−	0.0314065i	$-0.00999864\pi$
$194$	−4.74094	−0.340380
$195$	0	0
$196$	−5.15883	−0.368488
$197$	10.6213i	0.756739i	0.925655	+	0.378369i	$0.123515\pi$
$197$	10.6213i	0.756739i	−0.925655	+	0.378369i	$0.876485\pi$
$198$	−3.19806	−0.227276
$199$	−12.3424	−0.874931	−0.437466	−	0.899235i	$-0.644124\pi$
$199$	−12.3424	−0.874931	−0.437466	+	0.899235i	$0.644124\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 1.55496i	− 0.109678i
$202$	− 12.5918i	− 0.885956i
$203$	− 10.9511i	− 0.768615i
$204$	2.04892	0.143453
$205$	12.2349	0.854522
$206$	− 9.00000i	− 0.627060i
$207$	8.32304	0.578492
$208$	0	0
$209$	−17.0930	−1.18235
$210$	− 1.35690i	− 0.0936347i
$211$	6.75733	0.465194	0.232597	−	0.972573i	$-0.425278\pi$
$211$	6.75733	0.465194	0.232597	+	0.972573i	$0.425278\pi$
$212$	−5.74094	−0.394289
$213$	2.32304i	0.159172i
$214$	5.08815i	0.347819i
$215$	− 2.91185i	− 0.198587i
$216$	− 1.00000i	− 0.0680414i
$217$	−0.252356	−0.0171311
$218$	6.43296	0.435695
$219$	− 5.11529i	− 0.345659i
$220$	3.19806	0.215613
$221$	0	0
$222$	−2.46681	−0.165562
$223$	21.5972i	1.44625i	0.690715	+	0.723127i	$0.257296\pi$
$223$	21.5972i	1.44625i	−0.690715	+	0.723127i	$0.742704\pi$
$224$	−1.35690	−0.0906614
$225$	−1.00000	−0.0666667
$226$	18.6571i	1.24105i
$227$	10.2862i	0.682720i	0.939933	+	0.341360i	$0.110887\pi$
$227$	10.2862i	0.682720i	−0.939933	+	0.341360i	$0.889113\pi$
$228$	− 5.34481i	− 0.353969i
$229$	− 4.25906i	− 0.281447i	−0.990049	−	0.140723i	$-0.955057\pi$
$229$	− 4.25906i	− 0.281447i	0.990049	−	0.140723i	$-0.0449428\pi$
$230$	−8.32304	−0.548805
$231$	4.33944	0.285514
$232$	8.07069i	0.529867i
$233$	−19.5133	−1.27836	−0.639181	−	0.769057i	$-0.720726\pi$
$233$	−19.5133	−1.27836	−0.639181	+	0.769057i	$0.720726\pi$
$234$	0	0
$235$	−13.0978	−0.854409
$236$	− 1.62565i	− 0.105821i
$237$	1.42327	0.0924514
$238$	−2.78017	−0.180211
$239$	− 7.16315i	− 0.463345i	−0.972794	−	0.231673i	$-0.925580\pi$
$239$	− 7.16315i	− 0.463345i	0.972794	−	0.231673i	$-0.0744198\pi$
$240$	1.00000i	0.0645497i
$241$	− 17.9584i	− 1.15680i	−0.815753	−	0.578400i	$-0.803677\pi$
$241$	− 17.9584i	− 1.15680i	0.815753	−	0.578400i	$-0.196323\pi$
$242$	− 0.772398i	− 0.0496516i
$243$	−1.00000	−0.0641500
$244$	−5.28382	−0.338262
$245$	− 5.15883i	− 0.329586i
$246$	−12.2349	−0.780069
$247$	0	0
$248$	0.185981	0.0118098
$249$	− 11.1196i	− 0.704676i
$250$	1.00000	0.0632456
$251$	6.68425	0.421906	0.210953	−	0.977496i	$-0.432343\pi$
$251$	6.68425	0.421906	0.210953	+	0.977496i	$0.432343\pi$
$252$	1.35690i	0.0854764i
$253$	− 26.6176i	− 1.67343i
$254$	− 16.3763i	− 1.02754i
$255$	2.04892i	0.128308i
$256$	1.00000	0.0625000
$257$	−9.91484	−0.618471	−0.309235	−	0.950986i	$-0.600073\pi$
$257$	−9.91484	−0.618471	−0.309235	+	0.950986i	$0.600073\pi$
$258$	2.91185i	0.181284i
$259$	3.34721	0.207985
$260$	0	0
$261$	8.07069	0.499563
$262$	− 3.00969i	− 0.185939i
$263$	13.8509	0.854080	0.427040	−	0.904233i	$-0.359556\pi$
$263$	13.8509	0.854080	0.427040	+	0.904233i	$0.359556\pi$
$264$	−3.19806	−0.196827
$265$	− 5.74094i	− 0.352663i
$266$	7.25236i	0.444671i
$267$	3.37867i	0.206771i
$268$	− 1.55496i	− 0.0949842i
$269$	−17.8756	−1.08990	−0.544948	−	0.838470i	$-0.683450\pi$
$269$	−17.8756	−1.08990	−0.544948	+	0.838470i	$0.683450\pi$
$270$	1.00000	0.0608581
$271$	− 31.2083i	− 1.89577i	−0.318610	−	0.947886i	$-0.603216\pi$
$271$	− 31.2083i	− 1.89577i	0.318610	−	0.947886i	$-0.396784\pi$
$272$	2.04892	0.124234
$273$	0	0
$274$	−20.2892	−1.22571
$275$	3.19806i	0.192850i
$276$	8.32304	0.500988
$277$	−22.9245	−1.37740	−0.688701	−	0.725046i	$-0.741819\pi$
$277$	−22.9245	−1.37740	−0.688701	+	0.725046i	$0.741819\pi$
$278$	0.153457i	0.00920373i
$279$	− 0.185981i	− 0.0111344i
$280$	− 1.35690i	− 0.0810900i
$281$	− 28.0683i	− 1.67441i	−0.546886	−	0.837207i	$-0.684187\pi$
$281$	− 28.0683i	− 1.67441i	0.546886	−	0.837207i	$-0.315813\pi$
$282$	13.0978	0.779965
$283$	24.8364	1.47637	0.738185	−	0.674599i	$-0.235683\pi$
$283$	24.8364	1.47637	0.738185	+	0.674599i	$0.235683\pi$
$284$	2.32304i	0.137847i
$285$	5.34481	0.316599
$286$	0	0
$287$	16.6015	0.979955
$288$	− 1.00000i	− 0.0589256i
$289$	−12.8019	−0.753055
$290$	−8.07069	−0.473927
$291$	4.74094i	0.277919i
$292$	− 5.11529i	− 0.299350i
$293$	28.4349i	1.66118i	0.556882	+	0.830592i	$0.311998\pi$
$293$	28.4349i	1.66118i	−0.556882	+	0.830592i	$0.688002\pi$
$294$	5.15883i	0.300869i
$295$	1.62565	0.0946488
$296$	−2.46681	−0.143381
$297$	3.19806i	0.185570i
$298$	−10.7724	−0.624028
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	− 3.95108i	− 0.227737i
$302$	−17.3817	−1.00020
$303$	−12.5918	−0.723380
$304$	− 5.34481i	− 0.306546i
$305$	− 5.28382i	− 0.302550i
$306$	− 2.04892i	− 0.117129i
$307$	10.1347i	0.578416i	0.957266	+	0.289208i	$0.0933920\pi$
$307$	10.1347i	0.578416i	−0.957266	+	0.289208i	$0.906608\pi$
$308$	4.33944	0.247262
$309$	−9.00000	−0.511992
$310$	0.185981i	0.0105630i
$311$	−20.0218	−1.13533	−0.567665	−	0.823259i	$-0.692153\pi$
$311$	−20.0218	−1.13533	−0.567665	+	0.823259i	$0.692153\pi$
$312$	0	0
$313$	8.63640	0.488158	0.244079	−	0.969755i	$-0.421514\pi$
$313$	8.63640	0.488158	0.244079	+	0.969755i	$0.421514\pi$
$314$	− 15.5429i	− 0.877135i
$315$	−1.35690	−0.0764524
$316$	1.42327	0.0800653
$317$	17.6256i	0.989955i	0.868906	+	0.494977i	$0.164824\pi$
$317$	17.6256i	0.989955i	−0.868906	+	0.494977i	$0.835176\pi$
$318$	5.74094i	0.321936i
$319$	− 25.8106i	− 1.44511i
$320$	1.00000i	0.0559017i
$321$	5.08815	0.283993
$322$	−11.2935	−0.629362
$323$	− 10.9511i	− 0.609335i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	2.00239	0.110902
$327$	− 6.43296i	− 0.355744i
$328$	−12.2349	−0.675559
$329$	−17.7724	−0.979824
$330$	− 3.19806i	− 0.176048i
$331$	− 31.5163i	− 1.73229i	−0.499790	−	0.866147i	$-0.666589\pi$
$331$	− 31.5163i	− 1.73229i	0.499790	−	0.866147i	$-0.333411\pi$
$332$	− 11.1196i	− 0.610268i
$333$	2.46681i	0.135180i
$334$	8.24027	0.450888
$335$	1.55496	0.0849564
$336$	1.35690i	0.0740247i
$337$	11.8116	0.643420	0.321710	−	0.946838i	$-0.395742\pi$
$337$	11.8116	0.643420	0.321710	+	0.946838i	$0.395742\pi$
$338$	0	0
$339$	18.6571	1.01331
$340$	2.04892i	0.111118i
$341$	−0.594778	−0.0322090
$342$	−5.34481	−0.289014
$343$	− 16.4983i	− 0.890823i
$344$	2.91185i	0.156997i
$345$	8.32304i	0.448098i
$346$	12.7463i	0.685246i
$347$	−17.4101	−0.934624	−0.467312	−	0.884092i	$-0.654778\pi$
$347$	−17.4101	−0.934624	−0.467312	+	0.884092i	$0.654778\pi$
$348$	8.07069	0.432634
$349$	− 18.6799i	− 0.999914i	−0.866050	−	0.499957i	$-0.833349\pi$
$349$	− 18.6799i	− 0.999914i	0.866050	−	0.499957i	$-0.166651\pi$
$350$	1.35690	0.0725291
$351$	0	0
$352$	−3.19806	−0.170457
$353$	25.3991i	1.35186i	0.736967	+	0.675929i	$0.236257\pi$
$353$	25.3991i	1.35186i	−0.736967	+	0.675929i	$0.763743\pi$
$354$	−1.62565	−0.0864021
$355$	−2.32304	−0.123294
$356$	3.37867i	0.179069i
$357$	2.78017i	0.147142i
$358$	16.0422i	0.847857i
$359$	− 34.0103i	− 1.79499i	−0.441021	−	0.897497i	$-0.645383\pi$
$359$	− 34.0103i	− 1.79499i	0.441021	−	0.897497i	$-0.354617\pi$
$360$	1.00000	0.0527046
$361$	−9.56704	−0.503528
$362$	3.85086i	0.202396i
$363$	−0.772398	−0.0405404
$364$	0	0
$365$	5.11529	0.267747
$366$	5.28382i	0.276189i
$367$	−25.6437	−1.33859	−0.669295	−	0.742997i	$-0.733404\pi$
$367$	−25.6437	−1.33859	−0.669295	+	0.742997i	$0.733404\pi$
$368$	8.32304	0.433869
$369$	12.2349i	0.636923i
$370$	− 2.46681i	− 0.128243i
$371$	− 7.78986i	− 0.404429i
$372$	− 0.185981i	− 0.00964265i
$373$	−16.3612	−0.847151	−0.423576	−	0.905861i	$-0.639225\pi$
$373$	−16.3612	−0.847151	−0.423576	+	0.905861i	$0.639225\pi$
$374$	−6.55257	−0.338825
$375$	− 1.00000i	− 0.0516398i
$376$	13.0978	0.675469
$377$	0	0
$378$	1.35690	0.0697912
$379$	22.8823i	1.17539i	0.809084	+	0.587693i	$0.199964\pi$
$379$	22.8823i	1.17539i	−0.809084	+	0.587693i	$0.800036\pi$
$380$	5.34481	0.274183
$381$	−16.3763	−0.838982
$382$	− 11.8280i	− 0.605174i
$383$	11.9191i	0.609040i	0.952506	+	0.304520i	$0.0984961\pi$
$383$	11.9191i	0.609040i	−0.952506	+	0.304520i	$0.901504\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	4.33944i	0.221158i
$386$	−0.872625	−0.0444155
$387$	2.91185	0.148018
$388$	4.74094i	0.240685i
$389$	−8.07606	−0.409473	−0.204736	−	0.978817i	$-0.565634\pi$
$389$	−8.07606	−0.409473	−0.204736	+	0.978817i	$0.565634\pi$
$390$	0	0
$391$	17.0532	0.862419
$392$	5.15883i	0.260560i
$393$	−3.00969	−0.151819
$394$	10.6213	0.535095
$395$	1.42327i	0.0716126i
$396$	3.19806i	0.160709i
$397$	20.5230i	1.03002i	0.857184	+	0.515011i	$0.172212\pi$
$397$	20.5230i	1.03002i	−0.857184	+	0.515011i	$0.827788\pi$
$398$	12.3424i	0.618670i
$399$	7.25236	0.363072
$400$	−1.00000	−0.0500000
$401$	18.3274i	0.915224i	0.889152	+	0.457612i	$0.151295\pi$
$401$	18.3274i	0.915224i	−0.889152	+	0.457612i	$0.848705\pi$
$402$	−1.55496	−0.0775543
$403$	0	0
$404$	−12.5918	−0.626465
$405$	− 1.00000i	− 0.0496904i
$406$	−10.9511	−0.543493
$407$	7.88902	0.391044
$408$	− 2.04892i	− 0.101437i
$409$	9.00000i	0.445021i	0.974930	+	0.222511i	$0.0714252\pi$
$409$	9.00000i	0.445021i	−0.974930	+	0.222511i	$0.928575\pi$
$410$	− 12.2349i	− 0.604239i
$411$	20.2892i	1.00079i
$412$	−9.00000	−0.443398
$413$	2.20583	0.108542
$414$	− 8.32304i	− 0.409055i
$415$	11.1196	0.545840
$416$	0	0
$417$	0.153457	0.00751481
$418$	17.0930i	0.836048i
$419$	−14.4185	−0.704389	−0.352195	−	0.935927i	$-0.614564\pi$
$419$	−14.4185	−0.704389	−0.352195	+	0.935927i	$0.614564\pi$
$420$	−1.35690	−0.0662097
$421$	− 19.9801i	− 0.973773i	−0.873465	−	0.486886i	$-0.838133\pi$
$421$	− 19.9801i	− 0.973773i	0.873465	−	0.486886i	$-0.161867\pi$
$422$	− 6.75733i	− 0.328942i
$423$	− 13.0978i	− 0.636839i
$424$	5.74094i	0.278805i
$425$	−2.04892	−0.0993871
$426$	2.32304	0.112552
$427$	− 7.16959i	− 0.346961i
$428$	5.08815	0.245945
$429$	0	0
$430$	−2.91185	−0.140422
$431$	− 21.1812i	− 1.02026i	−0.860097	−	0.510131i	$-0.829597\pi$
$431$	− 21.1812i	− 1.02026i	0.860097	−	0.510131i	$-0.170403\pi$
$432$	−1.00000	−0.0481125
$433$	−31.0291	−1.49116	−0.745581	−	0.666415i	$-0.767828\pi$
$433$	−31.0291	−1.49116	−0.745581	+	0.666415i	$0.767828\pi$
$434$	0.252356i	0.0121135i
$435$	8.07069i	0.386960i
$436$	− 6.43296i	− 0.308083i
$437$	− 44.4851i	− 2.12801i
$438$	−5.11529	−0.244418
$439$	−7.33406	−0.350036	−0.175018	−	0.984565i	$-0.555998\pi$
$439$	−7.33406	−0.350036	−0.175018	+	0.984565i	$0.555998\pi$
$440$	− 3.19806i	− 0.152462i
$441$	5.15883	0.245659
$442$	0	0
$443$	−18.1360	−0.861667	−0.430834	−	0.902431i	$-0.641780\pi$
$443$	−18.1360	−0.861667	−0.430834	+	0.902431i	$0.641780\pi$
$444$	2.46681i	0.117070i
$445$	−3.37867	−0.160164
$446$	21.5972	1.02266
$447$	10.7724i	0.509517i
$448$	1.35690i	0.0641073i
$449$	− 4.55496i	− 0.214962i	−0.994207	−	0.107481i	$-0.965722\pi$
$449$	− 4.55496i	− 0.214962i	0.994207	−	0.107481i	$-0.0342784\pi$
$450$	1.00000i	0.0471405i
$451$	39.1280	1.84246
$452$	18.6571	0.877556
$453$	17.3817i	0.816661i
$454$	10.2862	0.482756
$455$	0	0
$456$	−5.34481	−0.250294
$457$	− 9.98121i	− 0.466901i	−0.972369	−	0.233451i	$-0.924998\pi$
$457$	− 9.98121i	− 0.466901i	0.972369	−	0.233451i	$-0.0750018\pi$
$458$	−4.25906	−0.199013
$459$	−2.04892	−0.0956353
$460$	8.32304i	0.388064i
$461$	16.1672i	0.752981i	0.926420	+	0.376491i	$0.122869\pi$
$461$	16.1672i	0.752981i	−0.926420	+	0.376491i	$0.877131\pi$
$462$	− 4.33944i	− 0.201889i
$463$	3.88902i	0.180738i	0.995908	+	0.0903690i	$0.0288046\pi$
$463$	3.88902i	0.180738i	−0.995908	+	0.0903690i	$0.971195\pi$
$464$	8.07069	0.374672
$465$	0.185981	0.00862465
$466$	19.5133i	0.903938i
$467$	0.872625	0.0403803	0.0201901	−	0.999796i	$-0.493573\pi$
$467$	0.872625	0.0403803	0.0201901	+	0.999796i	$0.493573\pi$
$468$	0	0
$469$	2.10992	0.0974269
$470$	13.0978i	0.604158i
$471$	−15.5429	−0.716178
$472$	−1.62565	−0.0748264
$473$	− 9.31229i	− 0.428180i
$474$	− 1.42327i	− 0.0653730i
$475$	5.34481i	0.245237i
$476$	2.78017i	0.127429i
$477$	5.74094	0.262860
$478$	−7.16315	−0.327635
$479$	− 31.6305i	− 1.44524i	−0.691247	−	0.722618i	$-0.742938\pi$
$479$	− 31.6305i	− 1.44524i	0.691247	−	0.722618i	$-0.257062\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	−17.9584	−0.817982
$483$	11.2935i	0.513872i
$484$	−0.772398	−0.0351090
$485$	−4.74094	−0.215275
$486$	1.00000i	0.0453609i
$487$	− 21.7748i	− 0.986710i	−0.869828	−	0.493355i	$-0.835770\pi$
$487$	− 21.7748i	− 0.986710i	0.869828	−	0.493355i	$-0.164230\pi$
$488$	5.28382i	0.239187i
$489$	− 2.00239i	− 0.0905513i
$490$	−5.15883	−0.233052
$491$	−42.9590	−1.93871	−0.969356	−	0.245662i	$-0.920995\pi$
$491$	−42.9590	−1.93871	−0.969356	+	0.245662i	$0.920995\pi$
$492$	12.2349i	0.551592i
$493$	16.5362	0.744752
$494$	0	0
$495$	−3.19806	−0.143742
$496$	− 0.185981i	− 0.00835078i
$497$	−3.15213	−0.141392
$498$	−11.1196	−0.498281
$499$	36.0978i	1.61596i	0.589209	+	0.807981i	$0.299439\pi$
$499$	36.0978i	1.61596i	−0.589209	+	0.807981i	$0.700561\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 8.24027i	− 0.368148i
$502$	− 6.68425i	− 0.298333i
$503$	−21.4534	−0.956560	−0.478280	−	0.878207i	$-0.658740\pi$
$503$	−21.4534	−0.956560	−0.478280	+	0.878207i	$0.658740\pi$
$504$	1.35690	0.0604409
$505$	− 12.5918i	− 0.560327i
$506$	−26.6176	−1.18330
$507$	0	0
$508$	−16.3763	−0.726580
$509$	− 0.545860i	− 0.0241948i	−0.999927	−	0.0120974i	$-0.996149\pi$
$509$	− 0.545860i	− 0.0241948i	0.999927	−	0.0120974i	$-0.00385082\pi$
$510$	2.04892	0.0907276
$511$	6.94092	0.307048
$512$	− 1.00000i	− 0.0441942i
$513$	5.34481i	0.235979i
$514$	9.91484i	0.437325i
$515$	− 9.00000i	− 0.396587i
$516$	2.91185	0.128187
$517$	−41.8877	−1.84222
$518$	− 3.34721i	− 0.147068i
$519$	12.7463	0.559501
$520$	0	0
$521$	29.6136	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$521$	29.6136	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$522$	− 8.07069i	− 0.353244i
$523$	44.4543	1.94385	0.971924	−	0.235293i	$-0.0756050\pi$
$523$	44.4543	1.94385	0.971924	+	0.235293i	$0.0756050\pi$
$524$	−3.00969	−0.131479
$525$	− 1.35690i	− 0.0592198i
$526$	− 13.8509i	− 0.603926i
$527$	− 0.381059i	− 0.0165992i
$528$	3.19806i	0.139178i
$529$	46.2731	2.01187
$530$	−5.74094	−0.249370
$531$	1.62565i	0.0705470i
$532$	7.25236	0.314430
$533$	0	0
$534$	3.37867	0.146209
$535$	5.08815i	0.219980i
$536$	−1.55496	−0.0671640
$537$	16.0422	0.692273
$538$	17.8756i	0.770672i
$539$	− 16.4983i	− 0.710631i
$540$	− 1.00000i	− 0.0430331i
$541$	18.7651i	0.806775i	0.915029	+	0.403387i	$0.132167\pi$
$541$	18.7651i	0.806775i	−0.915029	+	0.403387i	$0.867833\pi$
$542$	−31.2083	−1.34051
$543$	3.85086	0.165256
$544$	− 2.04892i	− 0.0878466i
$545$	6.43296	0.275558
$546$	0	0
$547$	32.0116	1.36872	0.684359	−	0.729145i	$-0.260082\pi$
$547$	32.0116	1.36872	0.684359	+	0.729145i	$0.260082\pi$
$548$	20.2892i	0.866711i
$549$	5.28382	0.225508
$550$	3.19806	0.136366
$551$	− 43.1363i	− 1.83767i
$552$	− 8.32304i	− 0.354252i
$553$	1.93123i	0.0821243i
$554$	22.9245i	0.973970i
$555$	−2.46681	−0.104710
$556$	0.153457	0.00650802
$557$	− 8.23490i	− 0.348924i	−0.984664	−	0.174462i	$-0.944181\pi$
$557$	− 8.23490i	− 0.348924i	0.984664	−	0.174462i	$-0.0558186\pi$
$558$	−0.185981	−0.00787319
$559$	0	0
$560$	−1.35690	−0.0573393
$561$	6.55257i	0.276650i
$562$	−28.0683	−1.18399
$563$	30.2295	1.27402	0.637011	−	0.770855i	$-0.280170\pi$
$563$	30.2295	1.27402	0.637011	+	0.770855i	$0.280170\pi$
$564$	− 13.0978i	− 0.551518i
$565$	18.6571i	0.784910i
$566$	− 24.8364i	− 1.04395i
$567$	− 1.35690i	− 0.0569843i
$568$	2.32304	0.0974728
$569$	−44.3110	−1.85761	−0.928806	−	0.370566i	$-0.879164\pi$
$569$	−44.3110	−1.85761	−0.928806	+	0.370566i	$0.879164\pi$
$570$	− 5.34481i	− 0.223870i
$571$	−34.4873	−1.44325	−0.721623	−	0.692286i	$-0.756604\pi$
$571$	−34.4873	−1.44325	−0.721623	+	0.692286i	$0.756604\pi$
$572$	0	0
$573$	−11.8280	−0.494123
$574$	− 16.6015i	− 0.692932i
$575$	−8.32304	−0.347095
$576$	−1.00000	−0.0416667
$577$	14.1715i	0.589968i	0.955502	+	0.294984i	$0.0953142\pi$
$577$	14.1715i	0.589968i	−0.955502	+	0.294984i	$0.904686\pi$
$578$	12.8019i	0.532490i
$579$	0.872625i	0.0362651i
$580$	8.07069i	0.335117i
$581$	15.0881	0.625962
$582$	4.74094	0.196518
$583$	− 18.3599i	− 0.760389i
$584$	−5.11529	−0.211672
$585$	0	0
$586$	28.4349	1.17463
$587$	1.12737i	0.0465317i	0.999729	+	0.0232659i	$0.00740642\pi$
$587$	1.12737i	0.0465317i	−0.999729	+	0.0232659i	$0.992594\pi$
$588$	5.15883	0.212747
$589$	−0.994032	−0.0409584
$590$	− 1.62565i	− 0.0669268i
$591$	− 10.6213i	− 0.436903i
$592$	2.46681i	0.101385i
$593$	18.3913i	0.755242i	0.925960	+	0.377621i	$0.123258\pi$
$593$	18.3913i	0.755242i	−0.925960	+	0.377621i	$0.876742\pi$
$594$	3.19806	0.131218
$595$	−2.78017	−0.113976
$596$	10.7724i	0.441255i
$597$	12.3424	0.505142
$598$	0	0
$599$	13.1226	0.536174	0.268087	−	0.963395i	$-0.413608\pi$
$599$	13.1226	0.536174	0.268087	+	0.963395i	$0.413608\pi$
$600$	1.00000i	0.0408248i
$601$	25.5754	1.04324	0.521621	−	0.853177i	$-0.325327\pi$
$601$	25.5754	1.04324	0.521621	+	0.853177i	$0.325327\pi$
$602$	−3.95108	−0.161034
$603$	1.55496i	0.0633228i
$604$	17.3817i	0.707249i
$605$	− 0.772398i	− 0.0314024i
$606$	12.5918i	0.511507i
$607$	44.4782	1.80531	0.902656	−	0.430362i	$-0.141614\pi$
$607$	44.4782	1.80531	0.902656	+	0.430362i	$0.141614\pi$
$608$	−5.34481	−0.216761
$609$	10.9511i	0.443760i
$610$	−5.28382	−0.213935
$611$	0	0
$612$	−2.04892	−0.0828226
$613$	21.7657i	0.879108i	0.898216	+	0.439554i	$0.144863\pi$
$613$	21.7657i	0.879108i	−0.898216	+	0.439554i	$0.855137\pi$
$614$	10.1347	0.409002
$615$	−12.2349	−0.493359
$616$	− 4.33944i	− 0.174841i
$617$	17.3569i	0.698762i	0.936981	+	0.349381i	$0.113608\pi$
$617$	17.3569i	0.698762i	−0.936981	+	0.349381i	$0.886392\pi$
$618$	9.00000i	0.362033i
$619$	36.8049i	1.47931i	0.672984	+	0.739657i	$0.265012\pi$
$619$	36.8049i	1.47931i	−0.672984	+	0.739657i	$0.734988\pi$
$620$	0.185981	0.00746916
$621$	−8.32304	−0.333992
$622$	20.0218i	0.802800i
$623$	−4.58450	−0.183674
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 8.63640i	− 0.345180i
$627$	17.0930	0.682631
$628$	−15.5429	−0.620228
$629$	5.05429i	0.201528i
$630$	1.35690i	0.0540600i
$631$	11.7554i	0.467976i	0.972240	+	0.233988i	$0.0751776\pi$
$631$	11.7554i	0.467976i	−0.972240	+	0.233988i	$0.924822\pi$
$632$	− 1.42327i	− 0.0566147i
$633$	−6.75733	−0.268580
$634$	17.6256	0.700004
$635$	− 16.3763i	− 0.649873i
$636$	5.74094	0.227643
$637$	0	0
$638$	−25.8106	−1.02185
$639$	− 2.32304i	− 0.0918982i
$640$	1.00000	0.0395285
$641$	−33.0858	−1.30681	−0.653404	−	0.757009i	$-0.726660\pi$
$641$	−33.0858	−1.30681	−0.653404	+	0.757009i	$0.726660\pi$
$642$	− 5.08815i	− 0.200813i
$643$	10.3461i	0.408012i	0.978970	+	0.204006i	$0.0653962\pi$
$643$	10.3461i	0.408012i	−0.978970	+	0.204006i	$0.934604\pi$
$644$	11.2935i	0.445026i
$645$	2.91185i	0.114654i
$646$	−10.9511	−0.430865
$647$	11.0030	0.432572	0.216286	−	0.976330i	$-0.430606\pi$
$647$	11.0030	0.432572	0.216286	+	0.976330i	$0.430606\pi$
$648$	1.00000i	0.0392837i
$649$	5.19892	0.204075
$650$	0	0
$651$	0.252356	0.00989063
$652$	− 2.00239i	− 0.0784198i
$653$	−30.8224	−1.20617	−0.603086	−	0.797676i	$-0.706063\pi$
$653$	−30.8224	−1.20617	−0.603086	+	0.797676i	$0.706063\pi$
$654$	−6.43296	−0.251549
$655$	− 3.00969i	− 0.117598i
$656$	12.2349i	0.477693i
$657$	5.11529i	0.199567i
$658$	17.7724i	0.692840i
$659$	−33.2010	−1.29333	−0.646665	−	0.762774i	$-0.723837\pi$
$659$	−33.2010	−1.29333	−0.646665	+	0.762774i	$0.723837\pi$
$660$	−3.19806	−0.124484
$661$	37.4252i	1.45567i	0.685752	+	0.727836i	$0.259474\pi$
$661$	37.4252i	1.45567i	−0.685752	+	0.727836i	$0.740526\pi$
$662$	−31.5163	−1.22492
$663$	0	0
$664$	−11.1196	−0.431524
$665$	7.25236i	0.281234i
$666$	2.46681	0.0955870
$667$	67.1727	2.60094
$668$	− 8.24027i	− 0.318826i
$669$	− 21.5972i	− 0.834995i
$670$	− 1.55496i	− 0.0600733i
$671$	− 16.8980i	− 0.652339i
$672$	1.35690	0.0523434
$673$	34.9409	1.34687	0.673437	−	0.739245i	$-0.264817\pi$
$673$	34.9409	1.34687	0.673437	+	0.739245i	$0.264817\pi$
$674$	− 11.8116i	− 0.454967i
$675$	1.00000	0.0384900
$676$	0	0
$677$	29.3913	1.12960	0.564800	−	0.825228i	$-0.308953\pi$
$677$	29.3913	1.12960	0.564800	+	0.825228i	$0.308953\pi$
$678$	− 18.6571i	− 0.716522i
$679$	−6.43296	−0.246874
$680$	2.04892	0.0785724
$681$	− 10.2862i	− 0.394168i
$682$	0.594778i	0.0227752i
$683$	28.9855i	1.10910i	0.832150	+	0.554550i	$0.187110\pi$
$683$	28.9855i	1.10910i	−0.832150	+	0.554550i	$0.812890\pi$
$684$	5.34481i	0.204364i
$685$	−20.2892	−0.775210
$686$	−16.4983	−0.629907
$687$	4.25906i	0.162493i
$688$	2.91185	0.111013
$689$	0	0
$690$	8.32304	0.316853
$691$	1.37627i	0.0523559i	0.999657	+	0.0261780i	$0.00833365\pi$
$691$	1.37627i	0.0523559i	−0.999657	+	0.0261780i	$0.991666\pi$
$692$	12.7463	0.484542
$693$	−4.33944	−0.164842
$694$	17.4101i	0.660879i
$695$	0.153457i	0.00582095i
$696$	− 8.07069i	− 0.305919i
$697$	25.0683i	0.949529i
$698$	−18.6799	−0.707046
$699$	19.5133	0.738062
$700$	− 1.35690i	− 0.0512858i
$701$	−9.21014	−0.347862	−0.173931	−	0.984758i	$-0.555647\pi$
$701$	−9.21014	−0.347862	−0.173931	+	0.984758i	$0.555647\pi$
$702$	0	0
$703$	13.1847	0.497269
$704$	3.19806i	0.120532i
$705$	13.0978	0.493293
$706$	25.3991	0.955908
$707$	− 17.0858i	− 0.642576i
$708$	1.62565i	0.0610955i
$709$	− 46.1613i	− 1.73363i	−0.498634	−	0.866813i	$-0.666165\pi$
$709$	− 46.1613i	− 1.73363i	0.498634	−	0.866813i	$-0.333835\pi$
$710$	2.32304i	0.0871823i
$711$	−1.42327	−0.0533769
$712$	3.37867	0.126621
$713$	− 1.54793i	− 0.0579703i
$714$	2.78017	0.104045
$715$	0	0
$716$	16.0422	0.599526
$717$	7.16315i	0.267513i
$718$	−34.0103	−1.26925
$719$	46.3588	1.72889	0.864446	−	0.502726i	$-0.167669\pi$
$719$	46.3588	1.72889	0.864446	+	0.502726i	$0.167669\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 12.2121i	− 0.454801i
$722$	9.56704i	0.356048i
$723$	17.9584i	0.667879i
$724$	3.85086	0.143116
$725$	−8.07069	−0.299738
$726$	0.772398i	0.0286664i
$727$	1.67217	0.0620174	0.0310087	−	0.999519i	$-0.490128\pi$
$727$	1.67217	0.0620174	0.0310087	+	0.999519i	$0.490128\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 5.11529i	− 0.189325i
$731$	5.96615	0.220666
$732$	5.28382	0.195295
$733$	− 33.6189i	− 1.24174i	−0.783912	−	0.620872i	$-0.786779\pi$
$733$	− 33.6189i	− 1.24174i	0.783912	−	0.620872i	$-0.213221\pi$
$734$	25.6437i	0.946526i
$735$	5.15883i	0.190286i
$736$	− 8.32304i	− 0.306791i
$737$	4.97285	0.183177
$738$	12.2349	0.450373
$739$	31.8437i	1.17139i	0.810532	+	0.585694i	$0.199178\pi$
$739$	31.8437i	1.17139i	−0.810532	+	0.585694i	$0.800822\pi$
$740$	−2.46681	−0.0906818
$741$	0	0
$742$	−7.78986	−0.285975
$743$	23.4010i	0.858500i	0.903186	+	0.429250i	$0.141222\pi$
$743$	23.4010i	0.858500i	−0.903186	+	0.429250i	$0.858778\pi$
$744$	−0.185981	−0.00681838
$745$	−10.7724	−0.394670
$746$	16.3612i	0.599026i
$747$	11.1196i	0.406845i
$748$	6.55257i	0.239586i
$749$	6.90408i	0.252270i
$750$	−1.00000	−0.0365148
$751$	39.0146	1.42366	0.711831	−	0.702350i	$-0.247866\pi$
$751$	39.0146	1.42366	0.711831	+	0.702350i	$0.247866\pi$
$752$	− 13.0978i	− 0.477629i
$753$	−6.68425	−0.243588
$754$	0	0
$755$	−17.3817	−0.632583
$756$	− 1.35690i	− 0.0493498i
$757$	43.0224	1.56367	0.781837	−	0.623483i	$-0.214283\pi$
$757$	43.0224	1.56367	0.781837	+	0.623483i	$0.214283\pi$
$758$	22.8823	0.831123
$759$	26.6176i	0.966158i
$760$	− 5.34481i	− 0.193877i
$761$	− 14.9903i	− 0.543398i	−0.962382	−	0.271699i	$-0.912414\pi$
$761$	− 14.9903i	− 0.543398i	0.962382	−	0.271699i	$-0.0875856\pi$
$762$	16.3763i	0.593250i
$763$	8.72886	0.316006
$764$	−11.8280	−0.427923
$765$	− 2.04892i	− 0.0740788i
$766$	11.9191	0.430656
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 19.9138i	− 0.718109i	−0.933317	−	0.359055i	$-0.883099\pi$
$769$	− 19.9138i	− 0.718109i	0.933317	−	0.359055i	$-0.116901\pi$
$770$	4.33944	0.156382
$771$	9.91484	0.357074
$772$	0.872625i	0.0314065i
$773$	25.5566i	0.919208i	0.888124	+	0.459604i	$0.152009\pi$
$773$	25.5566i	0.919208i	−0.888124	+	0.459604i	$0.847991\pi$
$774$	− 2.91185i	− 0.104664i
$775$	0.185981i	0.00668062i
$776$	4.74094	0.170190
$777$	−3.34721	−0.120080
$778$	8.07606i	0.289541i
$779$	65.3933	2.34296
$780$	0	0
$781$	−7.42924	−0.265839
$782$	− 17.0532i	− 0.609822i
$783$	−8.07069	−0.288423
$784$	5.15883	0.184244
$785$	− 15.5429i	− 0.554749i
$786$	3.00969i	0.107352i
$787$	− 28.6297i	− 1.02054i	−0.860015	−	0.510269i	$-0.829546\pi$
$787$	− 28.6297i	− 1.02054i	0.860015	−	0.510269i	$-0.170454\pi$
$788$	− 10.6213i	− 0.378369i
$789$	−13.8509	−0.493103
$790$	1.42327	0.0506377
$791$	25.3157i	0.900124i
$792$	3.19806	0.113638
$793$	0	0
$794$	20.5230	0.728335
$795$	5.74094i	0.203610i
$796$	12.3424	0.437466
$797$	−1.00431	−0.0355746	−0.0177873	−	0.999842i	$-0.505662\pi$
$797$	−1.00431	−0.0355746	−0.0177873	+	0.999842i	$0.505662\pi$
$798$	− 7.25236i	− 0.256731i
$799$	− 26.8364i	− 0.949403i
$800$	1.00000i	0.0353553i
$801$	− 3.37867i	− 0.119379i
$802$	18.3274	0.647161
$803$	16.3590	0.577297
$804$	1.55496i	0.0548391i
$805$	−11.2935	−0.398044
$806$	0	0
$807$	17.8756	0.629251
$808$	12.5918i	0.442978i
$809$	49.8743	1.75349	0.876743	−	0.480959i	$-0.159711\pi$
$809$	49.8743	1.75349	0.876743	+	0.480959i	$0.159711\pi$
$810$	−1.00000	−0.0351364
$811$	− 7.95646i	− 0.279389i	−0.990195	−	0.139695i	$-0.955388\pi$
$811$	− 7.95646i	− 0.279389i	0.990195	−	0.139695i	$-0.0446121\pi$
$812$	10.9511i	0.384308i
$813$	31.2083i	1.09452i
$814$	− 7.88902i	− 0.276510i
$815$	2.00239	0.0701408
$816$	−2.04892	−0.0717265
$817$	− 15.5633i	− 0.544492i
$818$	9.00000	0.314678
$819$	0	0
$820$	−12.2349	−0.427261
$821$	− 52.9071i	− 1.84647i	−0.384236	−	0.923235i	$-0.625535\pi$
$821$	− 52.9071i	− 1.84647i	0.384236	−	0.923235i	$-0.374465\pi$
$822$	20.2892	0.707667
$823$	−21.8006	−0.759921	−0.379961	−	0.925003i	$-0.624063\pi$
$823$	−21.8006	−0.759921	−0.379961	+	0.925003i	$0.624063\pi$
$824$	9.00000i	0.313530i
$825$	− 3.19806i	− 0.111342i
$826$	− 2.20583i	− 0.0767507i
$827$	− 27.3139i	− 0.949799i	−0.880040	−	0.474899i	$-0.842484\pi$
$827$	− 27.3139i	− 0.949799i	0.880040	−	0.474899i	$-0.157516\pi$
$828$	−8.32304	−0.289246
$829$	−40.8471	−1.41868	−0.709340	−	0.704867i	$-0.751007\pi$
$829$	−40.8471	−1.41868	−0.709340	+	0.704867i	$0.751007\pi$
$830$	− 11.1196i	− 0.385967i
$831$	22.9245	0.795243
$832$	0	0
$833$	10.5700	0.366230
$834$	− 0.153457i	− 0.00531377i
$835$	8.24027	0.285166
$836$	17.0930	0.591175
$837$	0.185981i	0.00642843i
$838$	14.4185i	0.498078i
$839$	43.0823i	1.48737i	0.668532	+	0.743683i	$0.266923\pi$
$839$	43.0823i	1.48737i	−0.668532	+	0.743683i	$0.733077\pi$
$840$	1.35690i	0.0468174i
$841$	36.1360	1.24607
$842$	−19.9801	−0.688561
$843$	28.0683i	0.966723i
$844$	−6.75733	−0.232597
$845$	0	0
$846$	−13.0978	−0.450313
$847$	− 1.04806i	− 0.0360119i
$848$	5.74094	0.197145
$849$	−24.8364	−0.852382
$850$	2.04892i	0.0702773i
$851$	20.5314i	0.703807i
$852$	− 2.32304i	− 0.0795862i
$853$	− 24.6813i	− 0.845071i	−0.906346	−	0.422535i	$-0.861140\pi$
$853$	− 24.6813i	− 0.845071i	0.906346	−	0.422535i	$-0.138860\pi$
$854$	−7.16959	−0.245338
$855$	−5.34481	−0.182789
$856$	− 5.08815i	− 0.173909i
$857$	−43.1075	−1.47252	−0.736262	−	0.676696i	$-0.763411\pi$
$857$	−43.1075	−1.47252	−0.736262	+	0.676696i	$0.763411\pi$
$858$	0	0
$859$	−42.1323	−1.43753	−0.718767	−	0.695251i	$-0.755293\pi$
$859$	−42.1323	−1.43753	−0.718767	+	0.695251i	$0.755293\pi$
$860$	2.91185i	0.0992934i
$861$	−16.6015	−0.565777
$862$	−21.1812	−0.721434
$863$	− 47.9342i	− 1.63170i	−0.578264	−	0.815850i	$-0.696270\pi$
$863$	− 47.9342i	− 1.63170i	0.578264	−	0.815850i	$-0.303730\pi$
$864$	1.00000i	0.0340207i
$865$	12.7463i	0.433388i
$866$	31.0291i	1.05441i
$867$	12.8019	0.434777
$868$	0.252356	0.00856553
$869$	4.55171i	0.154406i
$870$	8.07069	0.273622
$871$	0	0
$872$	−6.43296	−0.217848
$873$	− 4.74094i	− 0.160456i
$874$	−44.4851	−1.50473
$875$	1.35690	0.0458715
$876$	5.11529i	0.172830i
$877$	38.6674i	1.30570i	0.757485	+	0.652852i	$0.226428\pi$
$877$	38.6674i	1.30570i	−0.757485	+	0.652852i	$0.773572\pi$
$878$	7.33406i	0.247513i
$879$	− 28.4349i	− 0.959085i
$880$	−3.19806	−0.107807
$881$	−25.4980	−0.859050	−0.429525	−	0.903055i	$-0.641319\pi$
$881$	−25.4980	−0.859050	−0.429525	+	0.903055i	$0.641319\pi$
$882$	− 5.15883i	− 0.173707i
$883$	31.8377	1.07142	0.535712	−	0.844401i	$-0.320043\pi$
$883$	31.8377	1.07142	0.535712	+	0.844401i	$0.320043\pi$
$884$	0	0
$885$	−1.62565	−0.0546455
$886$	18.1360i	0.609291i
$887$	−7.83446	−0.263055	−0.131528	−	0.991312i	$-0.541988\pi$
$887$	−7.83446	−0.263055	−0.131528	+	0.991312i	$0.541988\pi$
$888$	2.46681	0.0827808
$889$	− 22.2209i	− 0.745265i
$890$	3.37867i	0.113253i
$891$	− 3.19806i	− 0.107139i
$892$	− 21.5972i	− 0.723127i
$893$	−70.0055	−2.34264
$894$	10.7724	0.360283
$895$	16.0422i	0.536232i
$896$	1.35690	0.0453307
$897$	0	0
$898$	−4.55496	−0.152001
$899$	− 1.50099i	− 0.0500609i
$900$	1.00000	0.0333333
$901$	11.7627	0.391873
$902$	− 39.1280i	− 1.30282i
$903$	3.95108i	0.131484i
$904$	− 18.6571i	− 0.620526i
$905$	3.85086i	0.128007i
$906$	17.3817	0.577467
$907$	−33.1089	−1.09936	−0.549681	−	0.835375i	$-0.685251\pi$
$907$	−33.1089	−1.09936	−0.549681	+	0.835375i	$0.685251\pi$
$908$	− 10.2862i	− 0.341360i
$909$	12.5918	0.417643
$910$	0	0
$911$	−14.0556	−0.465684	−0.232842	−	0.972515i	$-0.574802\pi$
$911$	−14.0556	−0.465684	−0.232842	+	0.972515i	$0.574802\pi$
$912$	5.34481i	0.176984i
$913$	35.5612	1.17690
$914$	−9.98121	−0.330149
$915$	5.28382i	0.174678i
$916$	4.25906i	0.140723i
$917$	− 4.08383i	− 0.134860i
$918$	2.04892i	0.0676243i
$919$	−27.1830	−0.896684	−0.448342	−	0.893862i	$-0.647985\pi$
$919$	−27.1830	−0.896684	−0.448342	+	0.893862i	$0.647985\pi$
$920$	8.32304	0.274403
$921$	− 10.1347i	− 0.333949i
$922$	16.1672	0.532438
$923$	0	0
$924$	−4.33944	−0.142757
$925$	− 2.46681i	− 0.0811083i
$926$	3.88902	0.127801
$927$	9.00000	0.295599
$928$	− 8.07069i	− 0.264933i
$929$	50.3521i	1.65200i	0.563671	+	0.826000i	$0.309389\pi$
$929$	50.3521i	1.65200i	−0.563671	+	0.826000i	$0.690611\pi$
$930$	− 0.185981i	− 0.00609855i
$931$	− 27.5730i	− 0.903669i
$932$	19.5133	0.639181
$933$	20.0218	0.655483
$934$	− 0.872625i	− 0.0285532i
$935$	−6.55257	−0.214292
$936$	0	0
$937$	26.6394	0.870271	0.435135	−	0.900365i	$-0.356701\pi$
$937$	26.6394	0.870271	0.435135	+	0.900365i	$0.356701\pi$
$938$	− 2.10992i	− 0.0688912i
$939$	−8.63640	−0.281838
$940$	13.0978	0.427204
$941$	25.7198i	0.838440i	0.907885	+	0.419220i	$0.137696\pi$
$941$	25.7198i	0.838440i	−0.907885	+	0.419220i	$0.862304\pi$
$942$	15.5429i	0.506414i
$943$	101.832i	3.31609i
$944$	1.62565i	0.0529103i
$945$	1.35690	0.0441398
$946$	−9.31229	−0.302769
$947$	14.4862i	0.470738i	0.971906	+	0.235369i	$0.0756298\pi$
$947$	14.4862i	0.470738i	−0.971906	+	0.235369i	$0.924370\pi$
$948$	−1.42327	−0.0462257
$949$	0	0
$950$	5.34481	0.173409
$951$	− 17.6256i	− 0.571551i
$952$	2.78017	0.0901057
$953$	29.3690	0.951354	0.475677	−	0.879620i	$-0.342203\pi$
$953$	29.3690	0.951354	0.475677	+	0.879620i	$0.342203\pi$
$954$	− 5.74094i	− 0.185870i
$955$	− 11.8280i	− 0.382746i
$956$	7.16315i	0.231673i
$957$	25.8106i	0.834337i
$958$	−31.6305	−1.02194
$959$	−27.5303	−0.889000
$960$	− 1.00000i	− 0.0322749i
$961$	30.9654	0.998884
$962$	0	0
$963$	−5.08815	−0.163963
$964$	17.9584i	0.578400i
$965$	−0.872625	−0.0280908
$966$	11.2935	0.363363
$967$	21.9360i	0.705415i	0.935734	+	0.352707i	$0.114739\pi$
$967$	21.9360i	0.705415i	−0.935734	+	0.352707i	$0.885261\pi$
$968$	0.772398i	0.0248258i
$969$	10.9511i	0.351799i
$970$	4.74094i	0.152222i
$971$	34.9754	1.12241	0.561206	−	0.827676i	$-0.310337\pi$
$971$	34.9754	1.12241	0.561206	+	0.827676i	$0.310337\pi$
$972$	1.00000	0.0320750
$973$	0.208225i	0.00667538i
$974$	−21.7748	−0.697709
$975$	0	0
$976$	5.28382	0.169131
$977$	7.72289i	0.247077i	0.992340	+	0.123539i	$0.0394243\pi$
$977$	7.72289i	0.247077i	−0.992340	+	0.123539i	$0.960576\pi$
$978$	−2.00239	−0.0640295
$979$	−10.8052	−0.345335
$980$	5.15883i	0.164793i
$981$	6.43296i	0.205389i
$982$	42.9590i	1.37088i
$983$	29.0489i	0.926517i	0.886223	+	0.463258i	$0.153320\pi$
$983$	29.0489i	0.926517i	−0.886223	+	0.463258i	$0.846680\pi$
$984$	12.2349	0.390034
$985$	10.6213	0.338424
$986$	− 16.5362i	− 0.526619i
$987$	17.7724	0.565702
$988$	0	0
$989$	24.2355	0.770644
$990$	3.19806i	0.101641i
$991$	37.7192	1.19819	0.599094	−	0.800678i	$-0.295527\pi$
$991$	37.7192	1.19819	0.599094	+	0.800678i	$0.295527\pi$
$992$	−0.185981	−0.00590489
$993$	31.5163i	1.00014i
$994$	3.15213i	0.0999795i
$995$	12.3424i	0.391281i
$996$	11.1196i	0.352338i
$997$	−7.18060	−0.227412	−0.113706	−	0.993514i	$-0.536272\pi$
$997$	−7.18060	−0.227412	−0.113706	+	0.993514i	$0.536272\pi$
$998$	36.0978	1.14266
$999$	− 2.46681i	− 0.0780465i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.s.1351.2		6
13.5	odd	4		5070.2.a.bk.1.2	✓	3
13.8	odd	4		5070.2.a.bt.1.2	yes	3
13.12	even	2	inner	5070.2.b.s.1351.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bk.1.2	✓	3	13.5	odd	4
5070.2.a.bt.1.2	yes	3	13.8	odd	4
5070.2.b.s.1351.2		6	1.1	even	1	trivial
5070.2.b.s.1351.5		6	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.19806i q^{11} +1.00000 q^{12} -1.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.04892 q^{17} -1.00000i q^{18} -5.34481i q^{19} +1.00000i q^{20} +1.35690i q^{21} -3.19806 q^{22} +8.32304 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.35690i q^{28} +8.07069 q^{29} +1.00000 q^{30} -0.185981i q^{31} -1.00000i q^{32} +3.19806i q^{33} -2.04892i q^{34} -1.35690 q^{35} -1.00000 q^{36} +2.46681i q^{37} -5.34481 q^{38} +1.00000 q^{40} +12.2349i q^{41} +1.35690 q^{42} +2.91185 q^{43} +3.19806i q^{44} -1.00000i q^{45} -8.32304i q^{46} -13.0978i q^{47} -1.00000 q^{48} +5.15883 q^{49} +1.00000i q^{50} -2.04892 q^{51} +5.74094 q^{53} +1.00000i q^{54} -3.19806 q^{55} +1.35690 q^{56} +5.34481i q^{57} -8.07069i q^{58} +1.62565i q^{59} -1.00000i q^{60} +5.28382 q^{61} -0.185981 q^{62} -1.35690i q^{63} -1.00000 q^{64} +3.19806 q^{66} +1.55496i q^{67} -2.04892 q^{68} -8.32304 q^{69} +1.35690i q^{70} -2.32304i q^{71} +1.00000i q^{72} +5.11529i q^{73} +2.46681 q^{74} +1.00000 q^{75} +5.34481i q^{76} -4.33944 q^{77} -1.42327 q^{79} -1.00000i q^{80} +1.00000 q^{81} +12.2349 q^{82} +11.1196i q^{83} -1.35690i q^{84} -2.04892i q^{85} -2.91185i q^{86} -8.07069 q^{87} +3.19806 q^{88} -3.37867i q^{89} -1.00000 q^{90} -8.32304 q^{92} +0.185981i q^{93} -13.0978 q^{94} -5.34481 q^{95} +1.00000i q^{96} -4.74094i q^{97} -5.15883i q^{98} -3.19806i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 28 q^{22} + 10 q^{23} - 6 q^{25} - 6 q^{27} + 24 q^{29} + 6 q^{30} - 6 q^{36} + 14 q^{38} + 6 q^{40} + 10 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 6 q^{53} - 28 q^{55} - 34 q^{61} + 28 q^{62} - 6 q^{64} + 28 q^{66} + 6 q^{68} - 10 q^{69} + 8 q^{74} + 6 q^{75} + 14 q^{77} - 14 q^{79} + 6 q^{81} + 26 q^{82} - 24 q^{87} + 28 q^{88} - 6 q^{90} - 10 q^{92} - 42 q^{94} + 14 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 28 * q^22 + 10 * q^23 - 6 * q^25 - 6 * q^27 + 24 * q^29 + 6 * q^30 - 6 * q^36 + 14 * q^38 + 6 * q^40 + 10 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 6 * q^53 - 28 * q^55 - 34 * q^61 + 28 * q^62 - 6 * q^64 + 28 * q^66 + 6 * q^68 - 10 * q^69 + 8 * q^74 + 6 * q^75 + 14 * q^77 - 14 * q^79 + 6 * q^81 + 26 * q^82 - 24 * q^87 + 28 * q^88 - 6 * q^90 - 10 * q^92 - 42 * q^94 + 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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