LMFDB - Embedded newform 5070.2.b.t.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.t.1351.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.29590i q^{11} +1.00000 q^{12} -1.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.04892 q^{17} -1.00000i q^{18} -3.24698i q^{19} +1.00000i q^{20} +1.35690i q^{21} -5.29590 q^{22} -1.33513 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.35690i q^{28} -5.13706 q^{29} +1.00000 q^{30} -2.80194i q^{31} -1.00000i q^{32} +5.29590i q^{33} -2.04892i q^{34} -1.35690 q^{35} -1.00000 q^{36} -2.24698i q^{37} -3.24698 q^{38} +1.00000 q^{40} -10.6310i q^{41} +1.35690 q^{42} -4.41789 q^{43} +5.29590i q^{44} -1.00000i q^{45} +1.33513i q^{46} +4.82371i q^{47} -1.00000 q^{48} +5.15883 q^{49} +1.00000i q^{50} -2.04892 q^{51} +14.2349 q^{53} +1.00000i q^{54} -5.29590 q^{55} +1.35690 q^{56} +3.24698i q^{57} +5.13706i q^{58} +13.8998i q^{59} -1.00000i q^{60} -3.62565 q^{61} -2.80194 q^{62} -1.35690i q^{63} -1.00000 q^{64} +5.29590 q^{66} -9.03684i q^{67} -2.04892 q^{68} +1.33513 q^{69} +1.35690i q^{70} -11.7506i q^{71} +1.00000i q^{72} -12.1032i q^{73} -2.24698 q^{74} +1.00000 q^{75} +3.24698i q^{76} -7.18598 q^{77} +0.259061 q^{79} -1.00000i q^{80} +1.00000 q^{81} -10.6310 q^{82} -3.02177i q^{83} -1.35690i q^{84} -2.04892i q^{85} +4.41789i q^{86} +5.13706 q^{87} +5.29590 q^{88} -5.47650i q^{89} -1.00000 q^{90} +1.33513 q^{92} +2.80194i q^{93} +4.82371 q^{94} -3.24698 q^{95} +1.00000i q^{96} +5.85086i q^{97} -5.15883i q^{98} -5.29590i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{22} - 6 q^{23} - 6 q^{25} - 6 q^{27} - 20 q^{29} + 6 q^{30} - 6 q^{36} - 10 q^{38} + 6 q^{40} - 38 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 38 q^{53} - 4 q^{55} + 2 q^{61} - 8 q^{62} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 6 q^{69} - 4 q^{74} + 6 q^{75} - 14 q^{77} + 30 q^{79} + 6 q^{81} - 34 q^{82} + 20 q^{87} + 4 q^{88} - 6 q^{90} + 6 q^{92} + 14 q^{94} - 10 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 - 4 * q^22 - 6 * q^23 - 6 * q^25 - 6 * q^27 - 20 * q^29 + 6 * q^30 - 6 * q^36 - 10 * q^38 + 6 * q^40 - 38 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 38 * q^53 - 4 * q^55 + 2 * q^61 - 8 * q^62 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 6 * q^69 - 4 * q^74 + 6 * q^75 - 14 * q^77 + 30 * q^79 + 6 * q^81 - 34 * q^82 + 20 * q^87 + 4 * q^88 - 6 * q^90 + 6 * q^92 + 14 * q^94 - 10 * 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
$11$	− 5.29590i	− 1.59677i	−0.602145	−	0.798387i	$-0.705687\pi$
$11$	− 5.29590i	− 1.59677i	0.602145	−	0.798387i	$-0.294313\pi$
$12$	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
$19$	− 3.24698i	− 0.744908i	−0.928051	−	0.372454i	$-0.878516\pi$
$19$	− 3.24698i	− 0.744908i	0.928051	−	0.372454i	$-0.121484\pi$
$20$	1.00000i	0.223607i
$21$	1.35690i	0.296099i
$22$	−5.29590	−1.12909
$23$	−1.33513	−0.278393	−0.139196	−	0.990265i	$-0.544452\pi$
$23$	−1.33513	−0.278393	−0.139196	+	0.990265i	$0.544452\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.35690i	0.256429i
$29$	−5.13706	−0.953929	−0.476964	−	0.878923i	$-0.658263\pi$
$29$	−5.13706	−0.953929	−0.476964	+	0.878923i	$0.658263\pi$
$30$	1.00000	0.182574
$31$	− 2.80194i	− 0.503243i	−0.967826	−	0.251621i	$-0.919036\pi$
$31$	− 2.80194i	− 0.503243i	0.967826	−	0.251621i	$-0.0809638\pi$
$32$	− 1.00000i	− 0.176777i
$33$	5.29590i	0.921897i
$34$	− 2.04892i	− 0.351386i
$35$	−1.35690	−0.229357
$36$	−1.00000	−0.166667
$37$	− 2.24698i	− 0.369401i	−0.982795	−	0.184701i	$-0.940868\pi$
$37$	− 2.24698i	− 0.369401i	0.982795	−	0.184701i	$-0.0591315\pi$
$38$	−3.24698	−0.526730
$39$	0	0
$40$	1.00000	0.158114
$41$	− 10.6310i	− 1.66029i	−0.557550	−	0.830143i	$-0.688258\pi$
$41$	− 10.6310i	− 1.66029i	0.557550	−	0.830143i	$-0.311742\pi$
$42$	1.35690	0.209374
$43$	−4.41789	−0.673723	−0.336861	−	0.941554i	$-0.609365\pi$
$43$	−4.41789	−0.673723	−0.336861	+	0.941554i	$0.609365\pi$
$44$	5.29590i	0.798387i
$45$	− 1.00000i	− 0.149071i
$46$	1.33513i	0.196854i
$47$	4.82371i	0.703610i	0.936073	+	0.351805i	$0.114432\pi$
$47$	4.82371i	0.703610i	−0.936073	+	0.351805i	$0.885568\pi$
$48$	−1.00000	−0.144338
$49$	5.15883	0.736976
$50$	1.00000i	0.141421i
$51$	−2.04892	−0.286906
$52$	0	0
$53$	14.2349	1.95531	0.977657	−	0.210207i	$-0.0674139\pi$
$53$	14.2349	1.95531	0.977657	+	0.210207i	$0.0674139\pi$
$54$	1.00000i	0.136083i
$55$	−5.29590	−0.714099
$56$	1.35690	0.181323
$57$	3.24698i	0.430073i
$58$	5.13706i	0.674529i
$59$	13.8998i	1.80960i	0.425841	+	0.904798i	$0.359978\pi$
$59$	13.8998i	1.80960i	−0.425841	+	0.904798i	$0.640022\pi$
$60$	− 1.00000i	− 0.129099i
$61$	−3.62565	−0.464216	−0.232108	−	0.972690i	$-0.574562\pi$
$61$	−3.62565	−0.464216	−0.232108	+	0.972690i	$0.574562\pi$
$62$	−2.80194	−0.355846
$63$	− 1.35690i	− 0.170953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	5.29590	0.651880
$67$	− 9.03684i	− 1.10403i	−0.833836	−	0.552013i	$-0.813860\pi$
$67$	− 9.03684i	− 1.10403i	0.833836	−	0.552013i	$-0.186140\pi$
$68$	−2.04892	−0.248468
$69$	1.33513	0.160730
$70$	1.35690i	0.162180i
$71$	− 11.7506i	− 1.39454i	−0.716807	−	0.697271i	$-0.754397\pi$
$71$	− 11.7506i	− 1.39454i	0.716807	−	0.697271i	$-0.245603\pi$
$72$	1.00000i	0.117851i
$73$	− 12.1032i	− 1.41657i	−0.705925	−	0.708287i	$-0.749468\pi$
$73$	− 12.1032i	− 1.41657i	0.705925	−	0.708287i	$-0.250532\pi$
$74$	−2.24698	−0.261206
$75$	1.00000	0.115470
$76$	3.24698i	0.372454i
$77$	−7.18598	−0.818919
$78$	0	0
$79$	0.259061	0.0291467	0.0145733	−	0.999894i	$-0.495361\pi$
$79$	0.259061	0.0291467	0.0145733	+	0.999894i	$0.495361\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−10.6310	−1.17400
$83$	− 3.02177i	− 0.331682i	−0.986152	−	0.165841i	$-0.946966\pi$
$83$	− 3.02177i	− 0.331682i	0.986152	−	0.165841i	$-0.0530339\pi$
$84$	− 1.35690i	− 0.148049i
$85$	− 2.04892i	− 0.222236i
$86$	4.41789i	0.476394i
$87$	5.13706	0.550751
$88$	5.29590	0.564545
$89$	− 5.47650i	− 0.580508i	−0.956950	−	0.290254i	$-0.906260\pi$
$89$	− 5.47650i	− 0.580508i	0.956950	−	0.290254i	$-0.0937398\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	1.33513	0.139196
$93$	2.80194i	0.290547i
$94$	4.82371	0.497527
$95$	−3.24698	−0.333133
$96$	1.00000i	0.102062i
$97$	5.85086i	0.594064i	0.954867	+	0.297032i	$0.0959969\pi$
$97$	5.85086i	0.594064i	−0.954867	+	0.297032i	$0.904003\pi$
$98$	− 5.15883i	− 0.521121i
$99$	− 5.29590i	− 0.532258i
$100$	1.00000	0.100000
$101$	8.39612	0.835446	0.417723	−	0.908575i	$-0.362828\pi$
$101$	8.39612	0.835446	0.417723	+	0.908575i	$0.362828\pi$
$102$	2.04892i	0.202873i
$103$	−13.6353	−1.34353	−0.671765	−	0.740765i	$-0.734463\pi$
$103$	−13.6353	−1.34353	−0.671765	+	0.740765i	$0.734463\pi$
$104$	0	0
$105$	1.35690	0.132419
$106$	− 14.2349i	− 1.38262i
$107$	3.92394	0.379341	0.189671	−	0.981848i	$-0.439258\pi$
$107$	3.92394	0.379341	0.189671	+	0.981848i	$0.439258\pi$
$108$	1.00000	0.0962250
$109$	0.554958i	0.0531553i	0.999647	+	0.0265777i	$0.00846093\pi$
$109$	0.554958i	0.0531553i	−0.999647	+	0.0265777i	$0.991539\pi$
$110$	5.29590i	0.504944i
$111$	2.24698i	0.213274i
$112$	− 1.35690i	− 0.128215i
$113$	4.39373	0.413328	0.206664	−	0.978412i	$-0.433739\pi$
$113$	4.39373	0.413328	0.206664	+	0.978412i	$0.433739\pi$
$114$	3.24698	0.304108
$115$	1.33513i	0.124501i
$116$	5.13706	0.476964
$117$	0	0
$118$	13.8998	1.27958
$119$	− 2.78017i	− 0.254858i
$120$	−1.00000	−0.0912871
$121$	−17.0465	−1.54968
$122$	3.62565i	0.328251i
$123$	10.6310i	0.958567i
$124$	2.80194i	0.251621i
$125$	1.00000i	0.0894427i
$126$	−1.35690	−0.120882
$127$	2.23490	0.198315	0.0991576	−	0.995072i	$-0.468385\pi$
$127$	2.23490	0.198315	0.0991576	+	0.995072i	$0.468385\pi$
$128$	1.00000i	0.0883883i
$129$	4.41789	0.388974
$130$	0	0
$131$	−8.51573	−0.744023	−0.372011	−	0.928228i	$-0.621332\pi$
$131$	−8.51573	−0.744023	−0.372011	+	0.928228i	$0.621332\pi$
$132$	− 5.29590i	− 0.460949i
$133$	−4.40581	−0.382032
$134$	−9.03684	−0.780664
$135$	1.00000i	0.0860663i
$136$	2.04892i	0.175693i
$137$	13.1685i	1.12506i	0.826776	+	0.562531i	$0.190172\pi$
$137$	13.1685i	1.12506i	−0.826776	+	0.562531i	$0.809828\pi$
$138$	− 1.33513i	− 0.113653i
$139$	7.17629	0.608685	0.304343	−	0.952563i	$-0.401563\pi$
$139$	7.17629	0.608685	0.304343	+	0.952563i	$0.401563\pi$
$140$	1.35690	0.114679
$141$	− 4.82371i	− 0.406229i
$142$	−11.7506	−0.986091
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.13706i	0.426610i
$146$	−12.1032	−1.00167
$147$	−5.15883	−0.425493
$148$	2.24698i	0.184701i
$149$	9.66248i	0.791581i	0.918341	+	0.395791i	$0.129529\pi$
$149$	9.66248i	0.791581i	−0.918341	+	0.395791i	$0.870471\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	7.35152i	0.598258i	0.954213	+	0.299129i	$0.0966961\pi$
$151$	7.35152i	0.598258i	−0.954213	+	0.299129i	$0.903304\pi$
$152$	3.24698	0.263365
$153$	2.04892	0.165645
$154$	7.18598i	0.579063i
$155$	−2.80194	−0.225057
$156$	0	0
$157$	2.85325	0.227714	0.113857	−	0.993497i	$-0.463679\pi$
$157$	2.85325	0.227714	0.113857	+	0.993497i	$0.463679\pi$
$158$	− 0.259061i	− 0.0206098i
$159$	−14.2349	−1.12890
$160$	−1.00000	−0.0790569
$161$	1.81163i	0.142776i
$162$	− 1.00000i	− 0.0785674i
$163$	15.1075i	1.18331i	0.806190	+	0.591656i	$0.201526\pi$
$163$	15.1075i	1.18331i	−0.806190	+	0.591656i	$0.798474\pi$
$164$	10.6310i	0.830143i
$165$	5.29590	0.412285
$166$	−3.02177	−0.234535
$167$	16.0881i	1.24494i	0.782644	+	0.622469i	$0.213870\pi$
$167$	16.0881i	1.24494i	−0.782644	+	0.622469i	$0.786130\pi$
$168$	−1.35690	−0.104687
$169$	0	0
$170$	−2.04892	−0.157145
$171$	− 3.24698i	− 0.248303i
$172$	4.41789	0.336861
$173$	15.4795	1.17688	0.588442	−	0.808540i	$-0.299742\pi$
$173$	15.4795	1.17688	0.588442	+	0.808540i	$0.299742\pi$
$174$	− 5.13706i	− 0.389440i
$175$	1.35690i	0.102572i
$176$	− 5.29590i	− 0.399193i
$177$	− 13.8998i	− 1.04477i
$178$	−5.47650	−0.410481
$179$	−19.0737	−1.42563	−0.712817	−	0.701351i	$-0.752581\pi$
$179$	−19.0737	−1.42563	−0.712817	+	0.701351i	$0.752581\pi$
$180$	1.00000i	0.0745356i
$181$	−2.16852	−0.161185	−0.0805925	−	0.996747i	$-0.525681\pi$
$181$	−2.16852	−0.161185	−0.0805925	+	0.996747i	$0.525681\pi$
$182$	0	0
$183$	3.62565	0.268015
$184$	− 1.33513i	− 0.0984268i
$185$	−2.24698	−0.165201
$186$	2.80194	0.205448
$187$	− 10.8509i	− 0.793493i
$188$	− 4.82371i	− 0.351805i
$189$	1.35690i	0.0986997i
$190$	3.24698i	0.235561i
$191$	−11.7409	−0.849545	−0.424772	−	0.905300i	$-0.639646\pi$
$191$	−11.7409	−0.849545	−0.424772	+	0.905300i	$0.639646\pi$
$192$	1.00000	0.0721688
$193$	− 24.6722i	− 1.77594i	−0.459900	−	0.887971i	$-0.652115\pi$
$193$	− 24.6722i	− 1.77594i	0.459900	−	0.887971i	$-0.347885\pi$
$194$	5.85086	0.420067
$195$	0	0
$196$	−5.15883	−0.368488
$197$	− 26.2010i	− 1.86675i	−0.358906	−	0.933374i	$-0.616850\pi$
$197$	− 26.2010i	− 1.86675i	0.358906	−	0.933374i	$-0.383150\pi$
$198$	−5.29590	−0.376363
$199$	−19.6722	−1.39452	−0.697262	−	0.716817i	$-0.745598\pi$
$199$	−19.6722	−1.39452	−0.697262	+	0.716817i	$0.745598\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	9.03684i	0.637409i
$202$	− 8.39612i	− 0.590749i
$203$	6.97046i	0.489230i
$204$	2.04892	0.143453
$205$	−10.6310	−0.742503
$206$	13.6353i	0.950019i
$207$	−1.33513	−0.0927976
$208$	0	0
$209$	−17.1957	−1.18945
$210$	− 1.35690i	− 0.0936347i
$211$	−16.2935	−1.12169	−0.560846	−	0.827920i	$-0.689524\pi$
$211$	−16.2935	−1.12169	−0.560846	+	0.827920i	$0.689524\pi$
$212$	−14.2349	−0.977657
$213$	11.7506i	0.805140i
$214$	− 3.92394i	− 0.268235i
$215$	4.41789i	0.301298i
$216$	− 1.00000i	− 0.0680414i
$217$	−3.80194	−0.258092
$218$	0.554958	0.0375865
$219$	12.1032i	0.817859i
$220$	5.29590	0.357049
$221$	0	0
$222$	2.24698	0.150807
$223$	2.28083i	0.152736i	0.997080	+	0.0763679i	$0.0243323\pi$
$223$	2.28083i	0.152736i	−0.997080	+	0.0763679i	$0.975668\pi$
$224$	−1.35690	−0.0906614
$225$	−1.00000	−0.0666667
$226$	− 4.39373i	− 0.292267i
$227$	3.37196i	0.223805i	0.993719	+	0.111902i	$0.0356944\pi$
$227$	3.37196i	0.223805i	−0.993719	+	0.111902i	$0.964306\pi$
$228$	− 3.24698i	− 0.215036i
$229$	18.3763i	1.21434i	0.794573	+	0.607169i	$0.207695\pi$
$229$	18.3763i	1.21434i	−0.794573	+	0.607169i	$0.792305\pi$
$230$	1.33513	0.0880356
$231$	7.18598	0.472803
$232$	− 5.13706i	− 0.337265i
$233$	−10.0858	−0.660740	−0.330370	−	0.943852i	$-0.607173\pi$
$233$	−10.0858	−0.660740	−0.330370	+	0.943852i	$0.607173\pi$
$234$	0	0
$235$	4.82371	0.314664
$236$	− 13.8998i	− 0.904798i
$237$	−0.259061	−0.0168278
$238$	−2.78017	−0.180211
$239$	20.0834i	1.29908i	0.760325	+	0.649542i	$0.225039\pi$
$239$	20.0834i	1.29908i	−0.760325	+	0.649542i	$0.774961\pi$
$240$	1.00000i	0.0645497i
$241$	2.99462i	0.192901i	0.995338	+	0.0964503i	$0.0307489\pi$
$241$	2.99462i	0.192901i	−0.995338	+	0.0964503i	$0.969251\pi$
$242$	17.0465i	1.09579i
$243$	−1.00000	−0.0641500
$244$	3.62565	0.232108
$245$	− 5.15883i	− 0.329586i
$246$	10.6310	0.677809
$247$	0	0
$248$	2.80194	0.177923
$249$	3.02177i	0.191497i
$250$	1.00000	0.0632456
$251$	7.20237	0.454610	0.227305	−	0.973824i	$-0.427009\pi$
$251$	7.20237	0.454610	0.227305	+	0.973824i	$0.427009\pi$
$252$	1.35690i	0.0854764i
$253$	7.07069i	0.444530i
$254$	− 2.23490i	− 0.140230i
$255$	2.04892i	0.128308i
$256$	1.00000	0.0625000
$257$	18.8291	1.17453	0.587263	−	0.809396i	$-0.300205\pi$
$257$	18.8291	1.17453	0.587263	+	0.809396i	$0.300205\pi$
$258$	− 4.41789i	− 0.275046i
$259$	−3.04892	−0.189451
$260$	0	0
$261$	−5.13706	−0.317976
$262$	8.51573i	0.526104i
$263$	−17.9245	−1.10527	−0.552637	−	0.833422i	$-0.686378\pi$
$263$	−17.9245	−1.10527	−0.552637	+	0.833422i	$0.686378\pi$
$264$	−5.29590	−0.325940
$265$	− 14.2349i	− 0.874443i
$266$	4.40581i	0.270138i
$267$	5.47650i	0.335156i
$268$	9.03684i	0.552013i
$269$	−22.4045	−1.36603	−0.683013	−	0.730406i	$-0.739331\pi$
$269$	−22.4045	−1.36603	−0.683013	+	0.730406i	$0.739331\pi$
$270$	1.00000	0.0608581
$271$	− 23.2325i	− 1.41127i	−0.708573	−	0.705637i	$-0.750661\pi$
$271$	− 23.2325i	− 1.41127i	0.708573	−	0.705637i	$-0.249339\pi$
$272$	2.04892	0.124234
$273$	0	0
$274$	13.1685	0.795540
$275$	5.29590i	0.319355i
$276$	−1.33513	−0.0803651
$277$	9.78448	0.587892	0.293946	−	0.955822i	$-0.405031\pi$
$277$	9.78448	0.587892	0.293946	+	0.955822i	$0.405031\pi$
$278$	− 7.17629i	− 0.430405i
$279$	− 2.80194i	− 0.167748i
$280$	− 1.35690i	− 0.0810900i
$281$	21.6286i	1.29026i	0.764075	+	0.645128i	$0.223196\pi$
$281$	21.6286i	1.29026i	−0.764075	+	0.645128i	$0.776804\pi$
$282$	−4.82371	−0.287248
$283$	−3.38942	−0.201480	−0.100740	−	0.994913i	$-0.532121\pi$
$283$	−3.38942	−0.201480	−0.100740	+	0.994913i	$0.532121\pi$
$284$	11.7506i	0.697271i
$285$	3.24698	0.192334
$286$	0	0
$287$	−14.4252	−0.851492
$288$	− 1.00000i	− 0.0589256i
$289$	−12.8019	−0.753055
$290$	5.13706	0.301659
$291$	− 5.85086i	− 0.342983i
$292$	12.1032i	0.708287i
$293$	34.1280i	1.99378i	0.0788243	+	0.996889i	$0.474883\pi$
$293$	34.1280i	1.99378i	−0.0788243	+	0.996889i	$0.525117\pi$
$294$	5.15883i	0.300869i
$295$	13.8998	0.809276
$296$	2.24698	0.130603
$297$	5.29590i	0.307299i
$298$	9.66248	0.559733
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	5.99462i	0.345524i
$302$	7.35152	0.423032
$303$	−8.39612	−0.482345
$304$	− 3.24698i	− 0.186227i
$305$	3.62565i	0.207604i
$306$	− 2.04892i	− 0.117129i
$307$	− 15.6601i	− 0.893768i	−0.894592	−	0.446884i	$-0.852534\pi$
$307$	− 15.6601i	− 0.893768i	0.894592	−	0.446884i	$-0.147466\pi$
$308$	7.18598	0.409459
$309$	13.6353	0.775687
$310$	2.80194i	0.159139i
$311$	−25.7972	−1.46282	−0.731411	−	0.681937i	$-0.761138\pi$
$311$	−25.7972	−1.46282	−0.731411	+	0.681937i	$0.761138\pi$
$312$	0	0
$313$	−15.6243	−0.883139	−0.441569	−	0.897227i	$-0.645578\pi$
$313$	−15.6243	−0.883139	−0.441569	+	0.897227i	$0.645578\pi$
$314$	− 2.85325i	− 0.161018i
$315$	−1.35690	−0.0764524
$316$	−0.259061	−0.0145733
$317$	9.25965i	0.520074i	0.965599	+	0.260037i	$0.0837347\pi$
$317$	9.25965i	0.520074i	−0.965599	+	0.260037i	$0.916265\pi$
$318$	14.2349i	0.798253i
$319$	27.2054i	1.52321i
$320$	1.00000i	0.0559017i
$321$	−3.92394	−0.219013
$322$	1.81163	0.100958
$323$	− 6.65279i	− 0.370171i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	15.1075	0.836728
$327$	− 0.554958i	− 0.0306893i
$328$	10.6310	0.587000
$329$	6.54527	0.360852
$330$	− 5.29590i	− 0.291530i
$331$	− 21.8582i	− 1.20143i	−0.799462	−	0.600716i	$-0.794882\pi$
$331$	− 21.8582i	− 1.20143i	0.799462	−	0.600716i	$-0.205118\pi$
$332$	3.02177i	0.165841i
$333$	− 2.24698i	− 0.123134i
$334$	16.0881	0.880304
$335$	−9.03684	−0.493735
$336$	1.35690i	0.0740247i
$337$	−14.5013	−0.789934	−0.394967	−	0.918695i	$-0.629244\pi$
$337$	−14.5013	−0.789934	−0.394967	+	0.918695i	$0.629244\pi$
$338$	0	0
$339$	−4.39373	−0.238635
$340$	2.04892i	0.111118i
$341$	−14.8388	−0.803565
$342$	−3.24698	−0.175577
$343$	− 16.4983i	− 0.890823i
$344$	− 4.41789i	− 0.238197i
$345$	− 1.33513i	− 0.0718807i
$346$	− 15.4795i	− 0.832182i
$347$	7.84117	0.420936	0.210468	−	0.977601i	$-0.432501\pi$
$347$	7.84117	0.420936	0.210468	+	0.977601i	$0.432501\pi$
$348$	−5.13706	−0.275375
$349$	20.4252i	1.09334i	0.837350	+	0.546668i	$0.184104\pi$
$349$	20.4252i	1.09334i	−0.837350	+	0.546668i	$0.815896\pi$
$350$	1.35690	0.0725291
$351$	0	0
$352$	−5.29590	−0.282272
$353$	− 13.2905i	− 0.707383i	−0.935362	−	0.353692i	$-0.884926\pi$
$353$	− 13.2905i	− 0.707383i	0.935362	−	0.353692i	$-0.115074\pi$
$354$	−13.8998	−0.738765
$355$	−11.7506	−0.623659
$356$	5.47650i	0.290254i
$357$	2.78017i	0.147142i
$358$	19.0737i	1.00807i
$359$	− 12.8267i	− 0.676967i	−0.940972	−	0.338483i	$-0.890086\pi$
$359$	− 12.8267i	− 0.676967i	0.940972	−	0.338483i	$-0.109914\pi$
$360$	1.00000	0.0527046
$361$	8.45712	0.445112
$362$	2.16852i	0.113975i
$363$	17.0465	0.894710
$364$	0	0
$365$	−12.1032	−0.633511
$366$	− 3.62565i	− 0.189516i
$367$	−17.9554	−0.937264	−0.468632	−	0.883393i	$-0.655253\pi$
$367$	−17.9554	−0.937264	−0.468632	+	0.883393i	$0.655253\pi$
$368$	−1.33513	−0.0695982
$369$	− 10.6310i	− 0.553429i
$370$	2.24698i	0.116815i
$371$	− 19.3153i	− 1.00280i
$372$	− 2.80194i	− 0.145274i
$373$	27.8732	1.44322	0.721610	−	0.692300i	$-0.243402\pi$
$373$	27.8732	1.44322	0.721610	+	0.692300i	$0.243402\pi$
$374$	−10.8509	−0.561084
$375$	− 1.00000i	− 0.0516398i
$376$	−4.82371	−0.248764
$377$	0	0
$378$	1.35690	0.0697912
$379$	13.8702i	0.712466i	0.934397	+	0.356233i	$0.115939\pi$
$379$	13.8702i	0.712466i	−0.934397	+	0.356233i	$0.884061\pi$
$380$	3.24698	0.166567
$381$	−2.23490	−0.114497
$382$	11.7409i	0.600719i
$383$	− 26.8984i	− 1.37445i	−0.726446	−	0.687223i	$-0.758829\pi$
$383$	− 26.8984i	− 1.37445i	0.726446	−	0.687223i	$-0.241171\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	7.18598i	0.366231i
$386$	−24.6722	−1.25578
$387$	−4.41789	−0.224574
$388$	− 5.85086i	− 0.297032i
$389$	0.187309	0.00949693	0.00474846	−	0.999989i	$-0.498489\pi$
$389$	0.187309	0.00949693	0.00474846	+	0.999989i	$0.498489\pi$
$390$	0	0
$391$	−2.73556	−0.138343
$392$	5.15883i	0.260560i
$393$	8.51573	0.429562
$394$	−26.2010	−1.31999
$395$	− 0.259061i	− 0.0130348i
$396$	5.29590i	0.266129i
$397$	8.58211i	0.430724i	0.976534	+	0.215362i	$0.0690931\pi$
$397$	8.58211i	0.430724i	−0.976534	+	0.215362i	$0.930907\pi$
$398$	19.6722i	0.986077i
$399$	4.40581	0.220567
$400$	−1.00000	−0.0500000
$401$	23.9748i	1.19724i	0.801032	+	0.598621i	$0.204285\pi$
$401$	23.9748i	1.19724i	−0.801032	+	0.598621i	$0.795715\pi$
$402$	9.03684	0.450716
$403$	0	0
$404$	−8.39612	−0.417723
$405$	− 1.00000i	− 0.0496904i
$406$	6.97046	0.345938
$407$	−11.8998	−0.589850
$408$	− 2.04892i	− 0.101437i
$409$	8.37926i	0.414328i	0.978306	+	0.207164i	$0.0664233\pi$
$409$	8.37926i	0.414328i	−0.978306	+	0.207164i	$0.933577\pi$
$410$	10.6310i	0.525029i
$411$	− 13.1685i	− 0.649555i
$412$	13.6353	0.671765
$413$	18.8605	0.928067
$414$	1.33513i	0.0656178i
$415$	−3.02177	−0.148333
$416$	0	0
$417$	−7.17629	−0.351425
$418$	17.1957i	0.841068i
$419$	37.0887	1.81190	0.905952	−	0.423381i	$-0.139157\pi$
$419$	37.0887	1.81190	0.905952	+	0.423381i	$0.139157\pi$
$420$	−1.35690	−0.0662097
$421$	− 4.90515i	− 0.239062i	−0.992830	−	0.119531i	$-0.961861\pi$
$421$	− 4.90515i	− 0.239062i	0.992830	−	0.119531i	$-0.0381391\pi$
$422$	16.2935i	0.793155i
$423$	4.82371i	0.234537i
$424$	14.2349i	0.691308i
$425$	−2.04892	−0.0993871
$426$	11.7506	0.569320
$427$	4.91962i	0.238077i
$428$	−3.92394	−0.189671
$429$	0	0
$430$	4.41789	0.213050
$431$	6.06531i	0.292156i	0.989273	+	0.146078i	$0.0466650\pi$
$431$	6.06531i	0.292156i	−0.989273	+	0.146078i	$0.953335\pi$
$432$	−1.00000	−0.0481125
$433$	14.8877	0.715457	0.357728	−	0.933826i	$-0.383551\pi$
$433$	14.8877	0.715457	0.357728	+	0.933826i	$0.383551\pi$
$434$	3.80194i	0.182499i
$435$	− 5.13706i	− 0.246303i
$436$	− 0.554958i	− 0.0265777i
$437$	4.33513i	0.207377i
$438$	12.1032	0.578314
$439$	21.1336	1.00865	0.504326	−	0.863513i	$-0.331741\pi$
$439$	21.1336	1.00865	0.504326	+	0.863513i	$0.331741\pi$
$440$	− 5.29590i	− 0.252472i
$441$	5.15883	0.245659
$442$	0	0
$443$	28.0683	1.33356	0.666782	−	0.745252i	$-0.267671\pi$
$443$	28.0683	1.33356	0.666782	+	0.745252i	$0.267671\pi$
$444$	− 2.24698i	− 0.106637i
$445$	−5.47650	−0.259611
$446$	2.28083	0.108000
$447$	− 9.66248i	− 0.457020i
$448$	1.35690i	0.0641073i
$449$	19.2446i	0.908208i	0.890949	+	0.454104i	$0.150041\pi$
$449$	19.2446i	0.908208i	−0.890949	+	0.454104i	$0.849959\pi$
$450$	1.00000i	0.0471405i
$451$	−56.3008	−2.65110
$452$	−4.39373	−0.206664
$453$	− 7.35152i	− 0.345404i
$454$	3.37196	0.158254
$455$	0	0
$456$	−3.24698	−0.152054
$457$	37.8485i	1.77048i	0.465138	+	0.885238i	$0.346005\pi$
$457$	37.8485i	1.77048i	−0.465138	+	0.885238i	$0.653995\pi$
$458$	18.3763	0.858667
$459$	−2.04892	−0.0956353
$460$	− 1.33513i	− 0.0622506i
$461$	11.2228i	0.522699i	0.965244	+	0.261349i	$0.0841675\pi$
$461$	11.2228i	0.522699i	−0.965244	+	0.261349i	$0.915833\pi$
$462$	− 7.18598i	− 0.334322i
$463$	25.0726i	1.16522i	0.812751	+	0.582611i	$0.197969\pi$
$463$	25.0726i	1.16522i	−0.812751	+	0.582611i	$0.802031\pi$
$464$	−5.13706	−0.238482
$465$	2.80194	0.129937
$466$	10.0858i	0.467213i
$467$	−36.9657	−1.71057	−0.855284	−	0.518160i	$-0.826617\pi$
$467$	−36.9657	−1.71057	−0.855284	+	0.518160i	$0.826617\pi$
$468$	0	0
$469$	−12.2620	−0.566209
$470$	− 4.82371i	− 0.222501i
$471$	−2.85325	−0.131471
$472$	−13.8998	−0.639789
$473$	23.3967i	1.07578i
$474$	0.259061i	0.0118991i
$475$	3.24698i	0.148982i
$476$	2.78017i	0.127429i
$477$	14.2349	0.651771
$478$	20.0834	0.918592
$479$	31.4590i	1.43740i	0.695320	+	0.718700i	$0.255263\pi$
$479$	31.4590i	1.43740i	−0.695320	+	0.718700i	$0.744737\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	2.99462	0.136401
$483$	− 1.81163i	− 0.0824319i
$484$	17.0465	0.774842
$485$	5.85086	0.265674
$486$	1.00000i	0.0453609i
$487$	− 18.8713i	− 0.855140i	−0.903982	−	0.427570i	$-0.859370\pi$
$487$	− 18.8713i	− 0.855140i	0.903982	−	0.427570i	$-0.140630\pi$
$488$	− 3.62565i	− 0.164125i
$489$	− 15.1075i	− 0.683186i
$490$	−5.15883	−0.233052
$491$	16.3961	0.739947	0.369973	−	0.929042i	$-0.379367\pi$
$491$	16.3961	0.739947	0.369973	+	0.929042i	$0.379367\pi$
$492$	− 10.6310i	− 0.479284i
$493$	−10.5254	−0.474041
$494$	0	0
$495$	−5.29590	−0.238033
$496$	− 2.80194i	− 0.125811i
$497$	−15.9444	−0.715203
$498$	3.02177	0.135409
$499$	− 40.2452i	− 1.80162i	−0.434212	−	0.900811i	$-0.642973\pi$
$499$	− 40.2452i	− 1.80162i	0.434212	−	0.900811i	$-0.357027\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 16.0881i	− 0.718765i
$502$	− 7.20237i	− 0.321458i
$503$	18.4004	0.820435	0.410217	−	0.911988i	$-0.365453\pi$
$503$	18.4004	0.820435	0.410217	+	0.911988i	$0.365453\pi$
$504$	1.35690	0.0604409
$505$	− 8.39612i	− 0.373623i
$506$	7.07069	0.314330
$507$	0	0
$508$	−2.23490	−0.0991576
$509$	35.1377i	1.55745i	0.627366	+	0.778725i	$0.284133\pi$
$509$	35.1377i	1.55745i	−0.627366	+	0.778725i	$0.715867\pi$
$510$	2.04892	0.0907276
$511$	−16.4228	−0.726502
$512$	− 1.00000i	− 0.0441942i
$513$	3.24698i	0.143358i
$514$	− 18.8291i	− 0.830515i
$515$	13.6353i	0.600845i
$516$	−4.41789	−0.194487
$517$	25.5459	1.12351
$518$	3.04892i	0.133962i
$519$	−15.4795	−0.679474
$520$	0	0
$521$	−35.2610	−1.54481	−0.772406	−	0.635129i	$-0.780947\pi$
$521$	−35.2610	−1.54481	−0.772406	+	0.635129i	$0.780947\pi$
$522$	5.13706i	0.224843i
$523$	14.3043	0.625482	0.312741	−	0.949839i	$-0.398753\pi$
$523$	14.3043	0.625482	0.312741	+	0.949839i	$0.398753\pi$
$524$	8.51573	0.372011
$525$	− 1.35690i	− 0.0592198i
$526$	17.9245i	0.781546i
$527$	− 5.74094i	− 0.250079i
$528$	5.29590i	0.230474i
$529$	−21.2174	−0.922497
$530$	−14.2349	−0.618324
$531$	13.8998i	0.603199i
$532$	4.40581	0.191016
$533$	0	0
$534$	5.47650	0.236991
$535$	− 3.92394i	− 0.169647i
$536$	9.03684	0.390332
$537$	19.0737	0.823090
$538$	22.4045i	0.965926i
$539$	− 27.3207i	− 1.17678i
$540$	− 1.00000i	− 0.0430331i
$541$	− 44.2422i	− 1.90212i	−0.309006	−	0.951060i	$-0.599996\pi$
$541$	− 44.2422i	− 1.90212i	0.309006	−	0.951060i	$-0.400004\pi$
$542$	−23.2325	−0.997922
$543$	2.16852	0.0930602
$544$	− 2.04892i	− 0.0878466i
$545$	0.554958	0.0237718
$546$	0	0
$547$	6.34481	0.271285	0.135642	−	0.990758i	$-0.456690\pi$
$547$	6.34481	0.271285	0.135642	+	0.990758i	$0.456690\pi$
$548$	− 13.1685i	− 0.562531i
$549$	−3.62565	−0.154739
$550$	5.29590	0.225818
$551$	16.6799i	0.710589i
$552$	1.33513i	0.0568267i
$553$	− 0.351519i	− 0.0149481i
$554$	− 9.78448i	− 0.415703i
$555$	2.24698	0.0953790
$556$	−7.17629	−0.304343
$557$	31.3884i	1.32997i	0.746858	+	0.664984i	$0.231562\pi$
$557$	31.3884i	1.32997i	−0.746858	+	0.664984i	$0.768438\pi$
$558$	−2.80194	−0.118615
$559$	0	0
$560$	−1.35690	−0.0573393
$561$	10.8509i	0.458123i
$562$	21.6286	0.912349
$563$	44.4989	1.87540	0.937702	−	0.347441i	$-0.112949\pi$
$563$	44.4989	1.87540	0.937702	+	0.347441i	$0.112949\pi$
$564$	4.82371i	0.203115i
$565$	− 4.39373i	− 0.184846i
$566$	3.38942i	0.142468i
$567$	− 1.35690i	− 0.0569843i
$568$	11.7506	0.493045
$569$	−30.5394	−1.28028	−0.640140	−	0.768259i	$-0.721123\pi$
$569$	−30.5394	−1.28028	−0.640140	+	0.768259i	$0.721123\pi$
$570$	− 3.24698i	− 0.136001i
$571$	32.7614	1.37102	0.685511	−	0.728063i	$-0.259579\pi$
$571$	32.7614	1.37102	0.685511	+	0.728063i	$0.259579\pi$
$572$	0	0
$573$	11.7409	0.490485
$574$	14.4252i	0.602096i
$575$	1.33513	0.0556786
$576$	−1.00000	−0.0416667
$577$	− 36.5870i	− 1.52314i	−0.648084	−	0.761569i	$-0.724430\pi$
$577$	− 36.5870i	− 1.52314i	0.648084	−	0.761569i	$-0.275570\pi$
$578$	12.8019i	0.532490i
$579$	24.6722i	1.02534i
$580$	− 5.13706i	− 0.213305i
$581$	−4.10023	−0.170106
$582$	−5.85086	−0.242526
$583$	− 75.3866i	− 3.12219i
$584$	12.1032	0.500834
$585$	0	0
$586$	34.1280	1.40981
$587$	− 22.4873i	− 0.928148i	−0.885796	−	0.464074i	$-0.846387\pi$
$587$	− 22.4873i	− 0.928148i	0.885796	−	0.464074i	$-0.153613\pi$
$588$	5.15883	0.212747
$589$	−9.09783	−0.374870
$590$	− 13.8998i	− 0.572245i
$591$	26.2010i	1.07777i
$592$	− 2.24698i	− 0.0923503i
$593$	39.5749i	1.62515i	0.582858	+	0.812574i	$0.301934\pi$
$593$	39.5749i	1.62515i	−0.582858	+	0.812574i	$0.698066\pi$
$594$	5.29590	0.217293
$595$	−2.78017	−0.113976
$596$	− 9.66248i	− 0.395791i
$597$	19.6722	0.805128
$598$	0	0
$599$	−31.3424	−1.28062	−0.640308	−	0.768118i	$-0.721193\pi$
$599$	−31.3424	−1.28062	−0.640308	+	0.768118i	$0.721193\pi$
$600$	1.00000i	0.0408248i
$601$	−32.9487	−1.34401	−0.672003	−	0.740549i	$-0.734566\pi$
$601$	−32.9487	−1.34401	−0.672003	+	0.740549i	$0.734566\pi$
$602$	5.99462	0.244323
$603$	− 9.03684i	− 0.368008i
$604$	− 7.35152i	− 0.299129i
$605$	17.0465i	0.693040i
$606$	8.39612i	0.341069i
$607$	−40.1739	−1.63061	−0.815304	−	0.579033i	$-0.803430\pi$
$607$	−40.1739	−1.63061	−0.815304	+	0.579033i	$0.803430\pi$
$608$	−3.24698	−0.131682
$609$	− 6.97046i	− 0.282457i
$610$	3.62565	0.146798
$611$	0	0
$612$	−2.04892	−0.0828226
$613$	− 11.3588i	− 0.458778i	−0.973335	−	0.229389i	$-0.926327\pi$
$613$	− 11.3588i	− 0.458778i	0.973335	−	0.229389i	$-0.0736728\pi$
$614$	−15.6601	−0.631990
$615$	10.6310	0.428684
$616$	− 7.18598i	− 0.289531i
$617$	− 20.9172i	− 0.842096i	−0.907038	−	0.421048i	$-0.861662\pi$
$617$	− 20.9172i	− 0.842096i	0.907038	−	0.421048i	$-0.138338\pi$
$618$	− 13.6353i	− 0.548494i
$619$	− 14.0562i	− 0.564967i	−0.959272	−	0.282483i	$-0.908842\pi$
$619$	− 14.0562i	− 0.564967i	0.959272	−	0.282483i	$-0.0911582\pi$
$620$	2.80194	0.112529
$621$	1.33513	0.0535767
$622$	25.7972i	1.03437i
$623$	−7.43104	−0.297718
$624$	0	0
$625$	1.00000	0.0400000
$626$	15.6243i	0.624473i
$627$	17.1957	0.686729
$628$	−2.85325	−0.113857
$629$	− 4.60388i	− 0.183569i
$630$	1.35690i	0.0540600i
$631$	− 40.6601i	− 1.61865i	−0.587359	−	0.809326i	$-0.699832\pi$
$631$	− 40.6601i	− 1.61865i	0.587359	−	0.809326i	$-0.300168\pi$
$632$	0.259061i	0.0103049i
$633$	16.2935	0.647609
$634$	9.25965	0.367748
$635$	− 2.23490i	− 0.0886892i
$636$	14.2349	0.564450
$637$	0	0
$638$	27.2054	1.07707
$639$	− 11.7506i	− 0.464848i
$640$	1.00000	0.0395285
$641$	1.92633	0.0760854	0.0380427	−	0.999276i	$-0.487888\pi$
$641$	1.92633	0.0760854	0.0380427	+	0.999276i	$0.487888\pi$
$642$	3.92394i	0.154865i
$643$	− 3.37973i	− 0.133284i	−0.997777	−	0.0666418i	$-0.978772\pi$
$643$	− 3.37973i	− 0.133284i	0.997777	−	0.0666418i	$-0.0212285\pi$
$644$	− 1.81163i	− 0.0713881i
$645$	− 4.41789i	− 0.173954i
$646$	−6.65279	−0.261751
$647$	10.8181	0.425302	0.212651	−	0.977128i	$-0.431790\pi$
$647$	10.8181	0.425302	0.212651	+	0.977128i	$0.431790\pi$
$648$	1.00000i	0.0392837i
$649$	73.6118	2.88951
$650$	0	0
$651$	3.80194	0.149010
$652$	− 15.1075i	− 0.591656i
$653$	33.7646	1.32131	0.660656	−	0.750689i	$-0.270278\pi$
$653$	33.7646	1.32131	0.660656	+	0.750689i	$0.270278\pi$
$654$	−0.554958	−0.0217006
$655$	8.51573i	0.332737i
$656$	− 10.6310i	− 0.415072i
$657$	− 12.1032i	− 0.472191i
$658$	− 6.54527i	− 0.255161i
$659$	−16.8592	−0.656742	−0.328371	−	0.944549i	$-0.606500\pi$
$659$	−16.8592	−0.656742	−0.328371	+	0.944549i	$0.606500\pi$
$660$	−5.29590	−0.206143
$661$	− 4.71140i	− 0.183252i	−0.995793	−	0.0916261i	$-0.970794\pi$
$661$	− 4.71140i	− 0.183252i	0.995793	−	0.0916261i	$-0.0292065\pi$
$662$	−21.8582	−0.849541
$663$	0	0
$664$	3.02177	0.117267
$665$	4.40581i	0.170850i
$666$	−2.24698	−0.0870687
$667$	6.85862	0.265567
$668$	− 16.0881i	− 0.622469i
$669$	− 2.28083i	− 0.0881820i
$670$	9.03684i	0.349123i
$671$	19.2010i	0.741248i
$672$	1.35690	0.0523434
$673$	−3.12797	−0.120574	−0.0602871	−	0.998181i	$-0.519202\pi$
$673$	−3.12797	−0.120574	−0.0602871	+	0.998181i	$0.519202\pi$
$674$	14.5013i	0.558567i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−1.50737	−0.0579329	−0.0289664	−	0.999580i	$-0.509222\pi$
$677$	−1.50737	−0.0579329	−0.0289664	+	0.999580i	$0.509222\pi$
$678$	4.39373i	0.168740i
$679$	7.93900	0.304671
$680$	2.04892	0.0785724
$681$	− 3.37196i	− 0.129214i
$682$	14.8388i	0.568206i
$683$	− 31.0726i	− 1.18896i	−0.804110	−	0.594480i	$-0.797358\pi$
$683$	− 31.0726i	− 1.18896i	0.804110	−	0.594480i	$-0.202642\pi$
$684$	3.24698i	0.124151i
$685$	13.1685	0.503143
$686$	−16.4983	−0.629907
$687$	− 18.3763i	− 0.701099i
$688$	−4.41789	−0.168431
$689$	0	0
$690$	−1.33513	−0.0508274
$691$	− 26.7469i	− 1.01750i	−0.860914	−	0.508750i	$-0.830108\pi$
$691$	− 26.7469i	− 1.01750i	0.860914	−	0.508750i	$-0.169892\pi$
$692$	−15.4795	−0.588442
$693$	−7.18598	−0.272973
$694$	− 7.84117i	− 0.297647i
$695$	− 7.17629i	− 0.272212i
$696$	5.13706i	0.194720i
$697$	− 21.7821i	− 0.825055i
$698$	20.4252	0.773105
$699$	10.0858	0.381478
$700$	− 1.35690i	− 0.0512858i
$701$	−11.2054	−0.423221	−0.211610	−	0.977354i	$-0.567871\pi$
$701$	−11.2054	−0.423221	−0.211610	+	0.977354i	$0.567871\pi$
$702$	0	0
$703$	−7.29590	−0.275170
$704$	5.29590i	0.199597i
$705$	−4.82371	−0.181671
$706$	−13.2905	−0.500195
$707$	− 11.3927i	− 0.428465i
$708$	13.8998i	0.522385i
$709$	− 26.9933i	− 1.01375i	−0.862018	−	0.506877i	$-0.830800\pi$
$709$	− 26.9933i	− 1.01375i	0.862018	−	0.506877i	$-0.169200\pi$
$710$	11.7506i	0.440993i
$711$	0.259061	0.00971555
$712$	5.47650	0.205241
$713$	3.74094i	0.140099i
$714$	2.78017	0.104045
$715$	0	0
$716$	19.0737	0.712817
$717$	− 20.0834i	− 0.750027i
$718$	−12.8267	−0.478688
$719$	15.4601	0.576565	0.288282	−	0.957545i	$-0.406916\pi$
$719$	15.4601	0.576565	0.288282	+	0.957545i	$0.406916\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	18.5017i	0.689040i
$722$	− 8.45712i	− 0.314742i
$723$	− 2.99462i	− 0.111371i
$724$	2.16852	0.0805925
$725$	5.13706	0.190786
$726$	− 17.0465i	− 0.632656i
$727$	10.9148	0.404809	0.202404	−	0.979302i	$-0.435124\pi$
$727$	10.9148	0.404809	0.202404	+	0.979302i	$0.435124\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.1032i	0.447960i
$731$	−9.05190	−0.334797
$732$	−3.62565	−0.134008
$733$	− 13.5539i	− 0.500624i	−0.968165	−	0.250312i	$-0.919467\pi$
$733$	− 13.5539i	− 0.500624i	0.968165	−	0.250312i	$-0.0805333\pi$
$734$	17.9554i	0.662746i
$735$	5.15883i	0.190286i
$736$	1.33513i	0.0492134i
$737$	−47.8582	−1.76288
$738$	−10.6310	−0.391333
$739$	− 15.4679i	− 0.568995i	−0.958677	−	0.284498i	$-0.908173\pi$
$739$	− 15.4679i	− 0.568995i	0.958677	−	0.284498i	$-0.0918268\pi$
$740$	2.24698	0.0826006
$741$	0	0
$742$	−19.3153	−0.709086
$743$	40.2040i	1.47494i	0.675378	+	0.737471i	$0.263980\pi$
$743$	40.2040i	1.47494i	−0.675378	+	0.737471i	$0.736020\pi$
$744$	−2.80194	−0.102724
$745$	9.66248	0.354006
$746$	− 27.8732i	− 1.02051i
$747$	− 3.02177i	− 0.110561i
$748$	10.8509i	0.396747i
$749$	− 5.32437i	− 0.194548i
$750$	−1.00000	−0.0365148
$751$	19.8465	0.724211	0.362105	−	0.932137i	$-0.382058\pi$
$751$	19.8465	0.724211	0.362105	+	0.932137i	$0.382058\pi$
$752$	4.82371i	0.175903i
$753$	−7.20237	−0.262469
$754$	0	0
$755$	7.35152	0.267549
$756$	− 1.35690i	− 0.0493498i
$757$	39.8883	1.44976	0.724882	−	0.688873i	$-0.241894\pi$
$757$	39.8883	1.44976	0.724882	+	0.688873i	$0.241894\pi$
$758$	13.8702	0.503790
$759$	− 7.07069i	− 0.256650i
$760$	− 3.24698i	− 0.117780i
$761$	− 31.8756i	− 1.15549i	−0.816217	−	0.577745i	$-0.803933\pi$
$761$	− 31.8756i	− 1.15549i	0.816217	−	0.577745i	$-0.196067\pi$
$762$	2.23490i	0.0809618i
$763$	0.753020	0.0272612
$764$	11.7409	0.424772
$765$	− 2.04892i	− 0.0740788i
$766$	−26.8984	−0.971880
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 43.0672i	− 1.55304i	−0.630090	−	0.776522i	$-0.716982\pi$
$769$	− 43.0672i	− 1.55304i	0.630090	−	0.776522i	$-0.283018\pi$
$770$	7.18598	0.258965
$771$	−18.8291	−0.678113
$772$	24.6722i	0.887971i
$773$	33.1282i	1.19154i	0.803155	+	0.595770i	$0.203153\pi$
$773$	33.1282i	1.19154i	−0.803155	+	0.595770i	$0.796847\pi$
$774$	4.41789i	0.158798i
$775$	2.80194i	0.100649i
$776$	−5.85086	−0.210033
$777$	3.04892	0.109379
$778$	− 0.187309i	− 0.00671534i
$779$	−34.5187	−1.23676
$780$	0	0
$781$	−62.2301	−2.22677
$782$	2.73556i	0.0978235i
$783$	5.13706	0.183584
$784$	5.15883	0.184244
$785$	− 2.85325i	− 0.101837i
$786$	− 8.51573i	− 0.303746i
$787$	− 35.9138i	− 1.28019i	−0.768297	−	0.640094i	$-0.778895\pi$
$787$	− 35.9138i	− 1.28019i	0.768297	−	0.640094i	$-0.221105\pi$
$788$	26.2010i	0.933374i
$789$	17.9245	0.638130
$790$	−0.259061	−0.00921698
$791$	− 5.96184i	− 0.211978i
$792$	5.29590	0.188182
$793$	0	0
$794$	8.58211	0.304568
$795$	14.2349i	0.504860i
$796$	19.6722	0.697262
$797$	29.7918	1.05528	0.527639	−	0.849468i	$-0.323077\pi$
$797$	29.7918	1.05528	0.527639	+	0.849468i	$0.323077\pi$
$798$	− 4.40581i	− 0.155964i
$799$	9.88338i	0.349649i
$800$	1.00000i	0.0353553i
$801$	− 5.47650i	− 0.193503i
$802$	23.9748	0.846579
$803$	−64.0974	−2.26195
$804$	− 9.03684i	− 0.318705i
$805$	1.81163	0.0638514
$806$	0	0
$807$	22.4045	0.788675
$808$	8.39612i	0.295375i
$809$	1.52648	0.0536683	0.0268341	−	0.999640i	$-0.491457\pi$
$809$	1.52648	0.0536683	0.0268341	+	0.999640i	$0.491457\pi$
$810$	−1.00000	−0.0351364
$811$	51.6039i	1.81206i	0.423217	+	0.906029i	$0.360901\pi$
$811$	51.6039i	1.81206i	−0.423217	+	0.906029i	$0.639099\pi$
$812$	− 6.97046i	− 0.244615i
$813$	23.2325i	0.814800i
$814$	11.8998i	0.417087i
$815$	15.1075	0.529193
$816$	−2.04892	−0.0717265
$817$	14.3448i	0.501862i
$818$	8.37926	0.292974
$819$	0	0
$820$	10.6310	0.371251
$821$	− 17.4795i	− 0.610038i	−0.952346	−	0.305019i	$-0.901337\pi$
$821$	− 17.4795i	− 0.610038i	0.952346	−	0.305019i	$-0.0986629\pi$
$822$	−13.1685	−0.459305
$823$	25.5566	0.890848	0.445424	−	0.895320i	$-0.353053\pi$
$823$	25.5566	0.890848	0.445424	+	0.895320i	$0.353053\pi$
$824$	− 13.6353i	− 0.475009i
$825$	− 5.29590i	− 0.184379i
$826$	− 18.8605i	− 0.656242i
$827$	− 9.16182i	− 0.318588i	−0.987231	−	0.159294i	$-0.949078\pi$
$827$	− 9.16182i	− 0.318588i	0.987231	−	0.159294i	$-0.0509217\pi$
$828$	1.33513	0.0463988
$829$	−48.6493	−1.68966	−0.844831	−	0.535034i	$-0.820299\pi$
$829$	−48.6493	−1.68966	−0.844831	+	0.535034i	$0.820299\pi$
$830$	3.02177i	0.104887i
$831$	−9.78448	−0.339420
$832$	0	0
$833$	10.5700	0.366230
$834$	7.17629i	0.248495i
$835$	16.0881	0.556753
$836$	17.1957	0.594725
$837$	2.80194i	0.0968491i
$838$	− 37.0887i	− 1.28121i
$839$	− 8.71246i	− 0.300788i	−0.988626	−	0.150394i	$-0.951946\pi$
$839$	− 8.71246i	− 0.300788i	0.988626	−	0.150394i	$-0.0480542\pi$
$840$	1.35690i	0.0468174i
$841$	−2.61058	−0.0900200
$842$	−4.90515	−0.169043
$843$	− 21.6286i	− 0.744930i
$844$	16.2935	0.560846
$845$	0	0
$846$	4.82371	0.165842
$847$	23.1304i	0.794769i
$848$	14.2349	0.488828
$849$	3.38942	0.116325
$850$	2.04892i	0.0702773i
$851$	3.00000i	0.102839i
$852$	− 11.7506i	− 0.402570i
$853$	− 5.41060i	− 0.185255i	−0.995701	−	0.0926277i	$-0.970473\pi$
$853$	− 5.41060i	− 0.185255i	0.995701	−	0.0926277i	$-0.0295266\pi$
$854$	4.91962	0.168346
$855$	−3.24698	−0.111044
$856$	3.92394i	0.134117i
$857$	−0.165275	−0.00564570	−0.00282285	−	0.999996i	$-0.500899\pi$
$857$	−0.165275	−0.00564570	−0.00282285	+	0.999996i	$0.500899\pi$
$858$	0	0
$859$	47.0853	1.60653	0.803264	−	0.595622i	$-0.203095\pi$
$859$	47.0853	1.60653	0.803264	+	0.595622i	$0.203095\pi$
$860$	− 4.41789i	− 0.150649i
$861$	14.4252	0.491609
$862$	6.06531	0.206585
$863$	31.2553i	1.06394i	0.846762	+	0.531972i	$0.178549\pi$
$863$	31.2553i	1.06394i	−0.846762	+	0.531972i	$0.821451\pi$
$864$	1.00000i	0.0340207i
$865$	− 15.4795i	− 0.526318i
$866$	− 14.8877i	− 0.505904i
$867$	12.8019	0.434777
$868$	3.80194	0.129046
$869$	− 1.37196i	− 0.0465406i
$870$	−5.13706	−0.174163
$871$	0	0
$872$	−0.554958	−0.0187933
$873$	5.85086i	0.198021i
$874$	4.33513	0.146638
$875$	1.35690	0.0458715
$876$	− 12.1032i	− 0.408930i
$877$	− 16.9898i	− 0.573706i	−0.957975	−	0.286853i	$-0.907391\pi$
$877$	− 16.9898i	− 0.573706i	0.957975	−	0.286853i	$-0.0926091\pi$
$878$	− 21.1336i	− 0.713225i
$879$	− 34.1280i	− 1.15111i
$880$	−5.29590	−0.178525
$881$	3.29159	0.110896	0.0554482	−	0.998462i	$-0.482341\pi$
$881$	3.29159	0.110896	0.0554482	+	0.998462i	$0.482341\pi$
$882$	− 5.15883i	− 0.173707i
$883$	−17.3157	−0.582721	−0.291361	−	0.956613i	$-0.594108\pi$
$883$	−17.3157	−0.582721	−0.291361	+	0.956613i	$0.594108\pi$
$884$	0	0
$885$	−13.8998	−0.467236
$886$	− 28.0683i	− 0.942973i
$887$	15.6775	0.526401	0.263200	−	0.964741i	$-0.415222\pi$
$887$	15.6775	0.526401	0.263200	+	0.964741i	$0.415222\pi$
$888$	−2.24698	−0.0754037
$889$	− 3.03252i	− 0.101708i
$890$	5.47650i	0.183573i
$891$	− 5.29590i	− 0.177419i
$892$	− 2.28083i	− 0.0763679i
$893$	15.6625	0.524125
$894$	−9.66248	−0.323162
$895$	19.0737i	0.637563i
$896$	1.35690	0.0453307
$897$	0	0
$898$	19.2446	0.642200
$899$	14.3937i	0.480058i
$900$	1.00000	0.0333333
$901$	29.1661	0.971665
$902$	56.3008i	1.87461i
$903$	− 5.99462i	− 0.199489i
$904$	4.39373i	0.146133i
$905$	2.16852i	0.0720841i
$906$	−7.35152	−0.244238
$907$	25.8877	0.859587	0.429793	−	0.902927i	$-0.358586\pi$
$907$	25.8877	0.859587	0.429793	+	0.902927i	$0.358586\pi$
$908$	− 3.37196i	− 0.111902i
$909$	8.39612	0.278482
$910$	0	0
$911$	−37.5220	−1.24316	−0.621579	−	0.783351i	$-0.713509\pi$
$911$	−37.5220	−1.24316	−0.621579	+	0.783351i	$0.713509\pi$
$912$	3.24698i	0.107518i
$913$	−16.0030	−0.529621
$914$	37.8485	1.25192
$915$	− 3.62565i	− 0.119860i
$916$	− 18.3763i	− 0.607169i
$917$	11.5550i	0.381578i
$918$	2.04892i	0.0676243i
$919$	−5.56358	−0.183526	−0.0917628	−	0.995781i	$-0.529250\pi$
$919$	−5.56358	−0.183526	−0.0917628	+	0.995781i	$0.529250\pi$
$920$	−1.33513	−0.0440178
$921$	15.6601i	0.516017i
$922$	11.2228	0.369604
$923$	0	0
$924$	−7.18598	−0.236401
$925$	2.24698i	0.0738802i
$926$	25.0726	0.823937
$927$	−13.6353	−0.447843
$928$	5.13706i	0.168632i
$929$	− 40.2602i	− 1.32090i	−0.750872	−	0.660448i	$-0.770366\pi$
$929$	− 40.2602i	− 1.32090i	0.750872	−	0.660448i	$-0.229634\pi$
$930$	− 2.80194i	− 0.0918792i
$931$	− 16.7506i	− 0.548980i
$932$	10.0858	0.330370
$933$	25.7972	0.844561
$934$	36.9657i	1.20955i
$935$	−10.8509	−0.354861
$936$	0	0
$937$	−30.5356	−0.997554	−0.498777	−	0.866730i	$-0.666217\pi$
$937$	−30.5356	−0.997554	−0.498777	+	0.866730i	$0.666217\pi$
$938$	12.2620i	0.400370i
$939$	15.6243	0.509880
$940$	−4.82371	−0.157332
$941$	41.6461i	1.35762i	0.734312	+	0.678812i	$0.237505\pi$
$941$	41.6461i	1.35762i	−0.734312	+	0.678812i	$0.762495\pi$
$942$	2.85325i	0.0929638i
$943$	14.1938i	0.462212i
$944$	13.8998i	0.452399i
$945$	1.35690	0.0441398
$946$	23.3967	0.760693
$947$	− 21.7724i	− 0.707508i	−0.935339	−	0.353754i	$-0.884905\pi$
$947$	− 21.7724i	− 0.707508i	0.935339	−	0.353754i	$-0.115095\pi$
$948$	0.259061	0.00841392
$949$	0	0
$950$	3.24698	0.105346
$951$	− 9.25965i	− 0.300265i
$952$	2.78017	0.0901057
$953$	41.9560	1.35909	0.679544	−	0.733635i	$-0.262178\pi$
$953$	41.9560	1.35909	0.679544	+	0.733635i	$0.262178\pi$
$954$	− 14.2349i	− 0.460872i
$955$	11.7409i	0.379928i
$956$	− 20.0834i	− 0.649542i
$957$	− 27.2054i	− 0.879424i
$958$	31.4590	1.01640
$959$	17.8683	0.576998
$960$	− 1.00000i	− 0.0322749i
$961$	23.1491	0.746747
$962$	0	0
$963$	3.92394	0.126447
$964$	− 2.99462i	− 0.0964503i
$965$	−24.6722	−0.794225
$966$	−1.81163	−0.0582881
$967$	16.7041i	0.537168i	0.963256	+	0.268584i	$0.0865556\pi$
$967$	16.7041i	0.537168i	−0.963256	+	0.268584i	$0.913444\pi$
$968$	− 17.0465i	− 0.547896i
$969$	6.65279i	0.213718i
$970$	− 5.85086i	− 0.187860i
$971$	−20.7275	−0.665178	−0.332589	−	0.943072i	$-0.607922\pi$
$971$	−20.7275	−0.665178	−0.332589	+	0.943072i	$0.607922\pi$
$972$	1.00000	0.0320750
$973$	− 9.73748i	− 0.312169i
$974$	−18.8713	−0.604675
$975$	0	0
$976$	−3.62565	−0.116054
$977$	− 33.4547i	− 1.07031i	−0.844753	−	0.535156i	$-0.820253\pi$
$977$	− 33.4547i	− 1.07031i	0.844753	−	0.535156i	$-0.179747\pi$
$978$	−15.1075	−0.483085
$979$	−29.0030	−0.926939
$980$	5.15883i	0.164793i
$981$	0.554958i	0.0177184i
$982$	− 16.3961i	− 0.523221i
$983$	− 31.8605i	− 1.01619i	−0.861300	−	0.508097i	$-0.830349\pi$
$983$	− 31.8605i	− 1.01619i	0.861300	−	0.508097i	$-0.169651\pi$
$984$	−10.6310	−0.338905
$985$	−26.2010	−0.834835
$986$	10.5254i	0.335198i
$987$	−6.54527	−0.208338
$988$	0	0
$989$	5.89844	0.187560
$990$	5.29590i	0.168315i
$991$	−21.0925	−0.670024	−0.335012	−	0.942214i	$-0.608740\pi$
$991$	−21.0925	−0.670024	−0.335012	+	0.942214i	$0.608740\pi$
$992$	−2.80194	−0.0889616
$993$	21.8582i	0.693647i
$994$	15.9444i	0.505725i
$995$	19.6722i	0.623650i
$996$	− 3.02177i	− 0.0957485i
$997$	17.7832	0.563198	0.281599	−	0.959532i	$-0.409135\pi$
$997$	17.7832	0.563198	0.281599	+	0.959532i	$0.409135\pi$
$998$	−40.2452	−1.27394
$999$	2.24698i	0.0710913i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bj.1.2	✓	3	13.5	odd	4
5070.2.a.bu.1.2	yes	3	13.8	odd	4
5070.2.b.t.1351.2		6	1.1	even	1	trivial
5070.2.b.t.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} -1.35690i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.29590i q^{11} +1.00000 q^{12} -1.35690 q^{14} +1.00000i q^{15} +1.00000 q^{16} +2.04892 q^{17} -1.00000i q^{18} -3.24698i q^{19} +1.00000i q^{20} +1.35690i q^{21} -5.29590 q^{22} -1.33513 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.35690i q^{28} -5.13706 q^{29} +1.00000 q^{30} -2.80194i q^{31} -1.00000i q^{32} +5.29590i q^{33} -2.04892i q^{34} -1.35690 q^{35} -1.00000 q^{36} -2.24698i q^{37} -3.24698 q^{38} +1.00000 q^{40} -10.6310i q^{41} +1.35690 q^{42} -4.41789 q^{43} +5.29590i q^{44} -1.00000i q^{45} +1.33513i q^{46} +4.82371i q^{47} -1.00000 q^{48} +5.15883 q^{49} +1.00000i q^{50} -2.04892 q^{51} +14.2349 q^{53} +1.00000i q^{54} -5.29590 q^{55} +1.35690 q^{56} +3.24698i q^{57} +5.13706i q^{58} +13.8998i q^{59} -1.00000i q^{60} -3.62565 q^{61} -2.80194 q^{62} -1.35690i q^{63} -1.00000 q^{64} +5.29590 q^{66} -9.03684i q^{67} -2.04892 q^{68} +1.33513 q^{69} +1.35690i q^{70} -11.7506i q^{71} +1.00000i q^{72} -12.1032i q^{73} -2.24698 q^{74} +1.00000 q^{75} +3.24698i q^{76} -7.18598 q^{77} +0.259061 q^{79} -1.00000i q^{80} +1.00000 q^{81} -10.6310 q^{82} -3.02177i q^{83} -1.35690i q^{84} -2.04892i q^{85} +4.41789i q^{86} +5.13706 q^{87} +5.29590 q^{88} -5.47650i q^{89} -1.00000 q^{90} +1.33513 q^{92} +2.80194i q^{93} +4.82371 q^{94} -3.24698 q^{95} +1.00000i q^{96} +5.85086i q^{97} -5.15883i q^{98} -5.29590i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 4 q^{22} - 6 q^{23} - 6 q^{25} - 6 q^{27} - 20 q^{29} + 6 q^{30} - 6 q^{36} - 10 q^{38} + 6 q^{40} - 38 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 38 q^{53} - 4 q^{55} + 2 q^{61} - 8 q^{62} - 6 q^{64} + 4 q^{66} + 6 q^{68} + 6 q^{69} - 4 q^{74} + 6 q^{75} - 14 q^{77} + 30 q^{79} + 6 q^{81} - 34 q^{82} + 20 q^{87} + 4 q^{88} - 6 q^{90} + 6 q^{92} + 14 q^{94} - 10 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 4 * q^22 - 6 * q^23 - 6 * q^25 - 6 * q^27 - 20 * q^29 + 6 * q^30 - 6 * q^36 - 10 * q^38 + 6 * q^40 - 38 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 38 * q^53 - 4 * q^55 + 2 * q^61 - 8 * q^62 - 6 * q^64 + 4 * q^66 + 6 * q^68 + 6 * q^69 - 4 * q^74 + 6 * q^75 - 14 * q^77 + 30 * q^79 + 6 * q^81 - 34 * q^82 + 20 * q^87 + 4 * q^88 - 6 * q^90 + 6 * q^92 + 14 * q^94 - 10 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more