LMFDB - Embedded newform 1275.2.d.i.424.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1275,2,Mod(424,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1275.424"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-12,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.1809262577$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 268x^{2} + 36 $ x^12 + 20x^10 + 144x^8 + 452x^6 + 604x^4 + 268*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.1
Root			$-2.67247i$ of defining polynomial
Character	$\chi$	$=$	1275.424
Dual form			1275.2.d.i.424.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.67247i q^{2} -1.00000 q^{3} -5.14207 q^{4} +2.67247i q^{6} +3.95896 q^{7} +8.39707i q^{8} +1.00000 q^{9} -2.28650i q^{11} +5.14207 q^{12} -1.90324i q^{13} -10.5802i q^{14} +12.1567 q^{16} +(-2.31913 + 3.40905i) q^{17} -2.67247i q^{18} -6.58376 q^{19} -3.95896 q^{21} -6.11058 q^{22} -7.11415 q^{23} -8.39707i q^{24} -5.08635 q^{26} -1.00000 q^{27} -20.3573 q^{28} -2.23078i q^{29} -9.45279i q^{31} -15.6943i q^{32} +2.28650i q^{33} +(9.11058 + 6.19779i) q^{34} -5.14207 q^{36} -6.57214 q^{37} +17.5949i q^{38} +1.90324i q^{39} +1.52889i q^{41} +10.5802i q^{42} -4.26946i q^{43} +11.7573i q^{44} +19.0123i q^{46} +3.61168i q^{47} -12.1567 q^{48} +8.67337 q^{49} +(2.31913 - 3.40905i) q^{51} +9.78662i q^{52} +1.48664i q^{53} +2.67247i q^{54} +33.2437i q^{56} +6.58376 q^{57} -5.96168 q^{58} -10.7633 q^{59} -15.0087i q^{61} -25.2623 q^{62} +3.95896 q^{63} -17.6291 q^{64} +6.11058 q^{66} +4.91401i q^{67} +(11.9251 - 17.5296i) q^{68} +7.11415 q^{69} +9.15317i q^{71} +8.39707i q^{72} +7.94910 q^{73} +17.5638i q^{74} +33.8541 q^{76} -9.05214i q^{77} +5.08635 q^{78} -3.92218i q^{79} +1.00000 q^{81} +4.08592 q^{82} +3.85286i q^{83} +20.3573 q^{84} -11.4100 q^{86} +2.23078i q^{87} +19.1999 q^{88} +4.71863 q^{89} -7.53487i q^{91} +36.5815 q^{92} +9.45279i q^{93} +9.65208 q^{94} +15.6943i q^{96} -7.70758 q^{97} -23.1793i q^{98} -2.28650i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 16 q^{4} + 12 q^{9} + 16 q^{12} + 32 q^{16} - 6 q^{17} + 4 q^{19} + 12 q^{22} - 16 q^{23} - 36 q^{26} - 12 q^{27} - 36 q^{28} + 24 q^{34} - 16 q^{36} - 4 q^{37} - 32 q^{48} + 16 q^{49}+ \cdots - 16 q^{97}+O(q^{100}) $ 12 * q - 12 * q^3 - 16 * q^4 + 12 * q^9 + 16 * q^12 + 32 * q^16 - 6 * q^17 + 4 * q^19 + 12 * q^22 - 16 * q^23 - 36 * q^26 - 12 * q^27 - 36 * q^28 + 24 * q^34 - 16 * q^36 - 4 * q^37 - 32 * q^48 + 16 * q^49 + 6 * q^51 - 4 * q^57 + 4 * q^58 - 32 * q^59 - 56 * q^62 - 56 * q^64 - 12 * q^66 - 20 * q^68 + 16 * q^69 + 44 * q^73 + 64 * q^76 + 36 * q^78 + 12 * q^81 - 40 * q^82 + 36 * q^84 + 20 * q^86 - 8 * q^88 - 4 * q^89 - 40 * q^94 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times$.

$n$	$52$	$751$	$851$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		−	2.67247i		−	1.88972i	−0.327477	−	0.944859i	$-0.606198\pi$
$2$		−	2.67247i		−	1.88972i	0.327477	−	0.944859i	$-0.393802\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−5.14207			−2.57104
$5$		0			0
$5$		0			0
$6$			2.67247i			1.09103i
$7$	3.95896			1.49635			0.748173	−	0.663504i	$-0.230931\pi$
$7$	3.95896			1.49635			0.748173	+	0.663504i	$0.230931\pi$
$8$			8.39707i			2.96881i
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.28650i		−	0.689404i	−0.938712	−	0.344702i	$-0.887980\pi$
$11$		−	2.28650i		−	0.689404i	0.938712	−	0.344702i	$-0.112020\pi$
$12$	5.14207			1.48439
$13$		−	1.90324i		−	0.527865i	−0.964541	−	0.263933i	$-0.914980\pi$
$13$		−	1.90324i		−	0.527865i	0.964541	−	0.263933i	$-0.0850196\pi$
$14$		−	10.5802i		−	2.82767i
$15$		0			0
$16$	12.1567			3.03919
$17$	−2.31913	+	3.40905i	−0.562471	+	0.826817i
$17$	−2.31913	+	3.40905i	−0.562471	+	0.826817i
$18$		−	2.67247i		−	0.629906i
$19$	−6.58376			−1.51042			−0.755209	−	0.655484i	$-0.772464\pi$
$19$	−6.58376			−1.51042			−0.755209	+	0.655484i	$0.772464\pi$
$20$		0			0
$21$	−3.95896			−0.863916
$22$	−6.11058			−1.30278
$23$	−7.11415			−1.48340			−0.741702	−	0.670730i	$-0.765981\pi$
$23$	−7.11415			−1.48340			−0.741702	+	0.670730i	$0.765981\pi$
$24$		−	8.39707i		−	1.71405i
$25$		0			0
$26$	−5.08635			−0.997516
$27$	−1.00000			−0.192450
$28$	−20.3573			−3.84716
$29$		−	2.23078i		−	0.414245i	−0.978315	−	0.207123i	$-0.933590\pi$
$29$		−	2.23078i		−	0.414245i	0.978315	−	0.207123i	$-0.0664099\pi$
$30$		0			0
$31$		−	9.45279i		−	1.69777i	−0.528577	−	0.848886i	$-0.677274\pi$
$31$		−	9.45279i		−	1.69777i	0.528577	−	0.848886i	$-0.322726\pi$
$32$		−	15.6943i		−	2.77439i
$33$			2.28650i			0.398028i
$34$	9.11058	+	6.19779i	1.56245	+	1.06291i
$35$		0			0
$36$	−5.14207			−0.857012
$37$	−6.57214			−1.08045			−0.540226	−	0.841520i	$-0.681661\pi$
$37$	−6.57214			−1.08045			−0.540226	+	0.841520i	$0.681661\pi$
$38$			17.5949i			2.85426i
$39$			1.90324i			0.304763i
$40$		0			0
$41$			1.52889i			0.238773i	0.992848	+	0.119387i	$0.0380928\pi$
$41$			1.52889i			0.238773i	−0.992848	+	0.119387i	$0.961907\pi$
$42$			10.5802i			1.63256i
$43$		−	4.26946i		−	0.651087i	−0.945527	−	0.325544i	$-0.894453\pi$
$43$		−	4.26946i		−	0.651087i	0.945527	−	0.325544i	$-0.105547\pi$
$44$			11.7573i			1.77248i
$45$		0			0
$46$			19.0123i			2.80321i
$47$			3.61168i			0.526817i	0.964684	+	0.263409i	$0.0848467\pi$
$47$			3.61168i			0.526817i	−0.964684	+	0.263409i	$0.915153\pi$
$48$	−12.1567			−1.75468
$49$	8.67337			1.23905
$50$		0			0
$51$	2.31913	−	3.40905i	0.324743	−	0.477363i
$52$			9.78662i			1.35716i
$53$			1.48664i			0.204205i	0.994774	+	0.102103i	$0.0325570\pi$
$53$			1.48664i			0.204205i	−0.994774	+	0.102103i	$0.967443\pi$
$54$			2.67247i			0.363676i
$55$		0			0
$56$			33.2437i			4.44237i
$57$	6.58376			0.872040
$58$	−5.96168			−0.782807
$59$	−10.7633			−1.40126			−0.700631	−	0.713524i	$-0.747098\pi$
$59$	−10.7633			−1.40126			−0.700631	+	0.713524i	$0.747098\pi$
$60$		0			0
$61$		−	15.0087i		−	1.92167i	−0.277115	−	0.960837i	$-0.589378\pi$
$61$		−	15.0087i		−	1.92167i	0.277115	−	0.960837i	$-0.410622\pi$
$62$	−25.2623			−3.20831
$63$	3.95896			0.498782
$64$	−17.6291			−2.20363
$65$		0			0
$66$	6.11058			0.752160
$67$			4.91401i			0.600342i	0.953885	+	0.300171i	$0.0970437\pi$
$67$			4.91401i			0.600342i	−0.953885	+	0.300171i	$0.902956\pi$
$68$	11.9251	−	17.5296i	1.44613	−	2.12578i
$69$	7.11415			0.856443
$70$		0			0
$71$			9.15317i			1.08628i	0.839642	+	0.543141i	$0.182765\pi$
$71$			9.15317i			1.08628i	−0.839642	+	0.543141i	$0.817235\pi$
$72$			8.39707i			0.989605i
$73$	7.94910			0.930371			0.465186	−	0.885213i	$-0.345988\pi$
$73$	7.94910			0.930371			0.465186	+	0.885213i	$0.345988\pi$
$74$			17.5638i			2.04175i
$75$		0			0
$76$	33.8541			3.88334
$77$		−	9.05214i		−	1.03159i
$78$	5.08635			0.575916
$79$		−	3.92218i		−	0.441280i	−0.975355	−	0.220640i	$-0.929185\pi$
$79$		−	3.92218i		−	0.441280i	0.975355	−	0.220640i	$-0.0708145\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	4.08592			0.451214
$83$			3.85286i			0.422906i	0.977388	+	0.211453i	$0.0678195\pi$
$83$			3.85286i			0.422906i	−0.977388	+	0.211453i	$0.932180\pi$
$84$	20.3573			2.22116
$85$		0			0
$86$	−11.4100			−1.23037
$87$			2.23078i			0.239165i
$88$	19.1999			2.04671
$89$	4.71863			0.500174			0.250087	−	0.968223i	$-0.419541\pi$
$89$	4.71863			0.500174			0.250087	+	0.968223i	$0.419541\pi$
$90$		0			0
$91$		−	7.53487i		−	0.789869i
$92$	36.5815			3.81388
$93$			9.45279i			0.980209i
$94$	9.65208			0.995536
$95$		0			0
$96$			15.6943i			1.60180i
$97$	−7.70758			−0.782586			−0.391293	−	0.920266i	$-0.627972\pi$
$97$	−7.70758			−0.782586			−0.391293	+	0.920266i	$0.627972\pi$
$98$		−	23.1793i		−	2.34146i
$99$		−	2.28650i		−	0.229801i
$100$		0			0
$101$	−16.4474			−1.63658			−0.818288	−	0.574809i	$-0.805076\pi$
$101$	−16.4474			−1.63658			−0.818288	+	0.574809i	$0.805076\pi$
$102$	−9.11058	−	6.19779i	−0.902082	−	0.613672i
$103$			14.1528i			1.39452i	0.716818	+	0.697260i	$0.245598\pi$
$103$			14.1528i			1.39452i	−0.716818	+	0.697260i	$0.754402\pi$
$104$	15.9817			1.56713
$105$		0			0
$106$	3.97298			0.385890
$107$	2.31765			0.224056			0.112028	−	0.993705i	$-0.464265\pi$
$107$	2.31765			0.224056			0.112028	+	0.993705i	$0.464265\pi$
$108$	5.14207			0.494796
$109$		−	1.14870i		−	0.110025i	−0.998486	−	0.0550127i	$-0.982480\pi$
$109$		−	1.14870i		−	0.110025i	0.998486	−	0.0550127i	$-0.0175199\pi$
$110$		0			0
$111$	6.57214			0.623800
$112$	48.1281			4.54768
$113$	−10.0421			−0.944677			−0.472339	−	0.881417i	$-0.656590\pi$
$113$	−10.0421			−0.944677			−0.472339	+	0.881417i	$0.656590\pi$
$114$		−	17.5949i		−	1.64791i
$115$		0			0
$116$			11.4708i			1.06504i
$117$		−	1.90324i		−	0.175955i
$118$			28.7645i			2.64799i
$119$	−9.18133	+	13.4963i	−0.841651	+	1.23720i
$120$		0			0
$121$	5.77194			0.524722
$122$	−40.1104			−3.63142
$123$		−	1.52889i		−	0.137856i
$124$			48.6069i			4.36503i
$125$		0			0
$126$		−	10.5802i		−	0.942558i
$127$		−	15.5700i		−	1.38161i	−0.723039	−	0.690807i	$-0.757255\pi$
$127$		−	15.5700i		−	1.38161i	0.723039	−	0.690807i	$-0.242745\pi$
$128$			15.7244i			1.38985i
$129$			4.26946i			0.375905i
$130$		0			0
$131$		−	22.0324i		−	1.92498i	−0.271312	−	0.962491i	$-0.587457\pi$
$131$		−	22.0324i		−	1.92498i	0.271312	−	0.962491i	$-0.412543\pi$
$132$		−	11.7573i		−	1.02334i
$133$	−26.0648			−2.26011
$134$	13.1325			1.13448
$135$		0			0
$136$	−28.6261	−	19.4739i	−2.45467	−	1.66987i
$137$			10.7934i			0.922140i	0.887364	+	0.461070i	$0.152534\pi$
$137$			10.7934i			0.922140i	−0.887364	+	0.461070i	$0.847466\pi$
$138$		−	19.0123i		−	1.61844i
$139$			0.219574i			0.0186240i	0.999957	+	0.00931200i	$0.00296415\pi$
$139$			0.219574i			0.0186240i	−0.999957	+	0.00931200i	$0.997036\pi$
$140$		0			0
$141$		−	3.61168i		−	0.304158i
$142$	24.4615			2.05277
$143$	−4.35176			−0.363912
$144$	12.1567			1.01306
$145$		0			0
$146$		−	21.2437i		−	1.75814i
$147$	−8.67337			−0.715367
$148$	33.7944			2.77788
$149$	−15.2763			−1.25148			−0.625742	−	0.780030i	$-0.715204\pi$
$149$	−15.2763			−1.25148			−0.625742	+	0.780030i	$0.715204\pi$
$150$		0			0
$151$	−23.1265			−1.88201			−0.941004	−	0.338396i	$-0.890116\pi$
$151$	−23.1265			−1.88201			−0.941004	+	0.338396i	$0.890116\pi$
$152$		−	55.2843i		−	4.48415i
$153$	−2.31913	+	3.40905i	−0.187490	+	0.275606i
$154$	−24.1915			−1.94941
$155$		0			0
$156$		−	9.78662i		−	0.783556i
$157$			9.09777i			0.726081i	0.931773	+	0.363041i	$0.118261\pi$
$157$			9.09777i			0.726081i	−0.931773	+	0.363041i	$0.881739\pi$
$158$	−10.4819			−0.833894
$159$		−	1.48664i		−	0.117898i
$160$		0			0
$161$	−28.1646			−2.21968
$162$		−	2.67247i		−	0.209969i
$163$	19.0597			1.49287			0.746434	−	0.665460i	$-0.231765\pi$
$163$	19.0597			1.49287			0.746434	+	0.665460i	$0.231765\pi$
$164$		−	7.86168i		−	0.613894i
$165$		0			0
$166$	10.2966			0.799173
$167$	13.4792			1.04305			0.521525	−	0.853236i	$-0.325363\pi$
$167$	13.4792			1.04305			0.521525	+	0.853236i	$0.325363\pi$
$168$		−	33.2437i		−	2.56481i
$169$	9.37766			0.721359
$170$		0			0
$171$	−6.58376			−0.503472
$172$			21.9539i			1.67397i
$173$	1.11240			0.0845740			0.0422870	−	0.999106i	$-0.486536\pi$
$173$	1.11240			0.0845740			0.0422870	+	0.999106i	$0.486536\pi$
$174$	5.96168			0.451954
$175$		0			0
$176$		−	27.7963i		−	2.09523i
$177$	10.7633			0.809019
$178$		−	12.6104i		−	0.945188i
$179$	4.06356			0.303725			0.151863	−	0.988402i	$-0.451473\pi$
$179$	4.06356			0.303725			0.151863	+	0.988402i	$0.451473\pi$
$180$		0			0
$181$			13.2839i			0.987388i	0.869636	+	0.493694i	$0.164354\pi$
$181$			13.2839i			0.987388i	−0.869636	+	0.493694i	$0.835646\pi$
$182$	−20.1367			−1.49263
$183$			15.0087i			1.10948i
$184$		−	59.7381i		−	4.40395i
$185$		0			0
$186$	25.2623			1.85232
$187$	7.79479	+	5.30267i	0.570011	+	0.387770i
$188$		−	18.5715i		−	1.35447i
$189$	−3.95896			−0.287972
$190$		0			0
$191$	−11.4755			−0.830335			−0.415168	−	0.909745i	$-0.636277\pi$
$191$	−11.4755			−0.830335			−0.415168	+	0.909745i	$0.636277\pi$
$192$	17.6291			1.27227
$193$	−8.50463			−0.612177			−0.306088	−	0.952003i	$-0.599020\pi$
$193$	−8.50463			−0.612177			−0.306088	+	0.952003i	$0.599020\pi$
$194$			20.5982i			1.47887i
$195$		0			0
$196$	−44.5991			−3.18565
$197$	16.8014			1.19705			0.598526	−	0.801104i	$-0.295753\pi$
$197$	16.8014			1.19705			0.598526	+	0.801104i	$0.295753\pi$
$198$	−6.11058			−0.434260
$199$		−	8.91705i		−	0.632113i	−0.948740	−	0.316056i	$-0.897641\pi$
$199$		−	8.91705i		−	0.632113i	0.948740	−	0.316056i	$-0.102359\pi$
$200$		0			0
$201$		−	4.91401i		−	0.346608i
$202$			43.9550i			3.09267i
$203$		−	8.83157i		−	0.619854i
$204$	−11.9251	+	17.5296i	−0.834925	+	1.22732i
$205$		0			0
$206$	37.8230			2.63525
$207$	−7.11415			−0.494468
$208$		−	23.1373i		−	1.60428i
$209$			15.0537i			1.04129i
$210$		0			0
$211$		−	12.3592i		−	0.850845i	−0.904995	−	0.425423i	$-0.860125\pi$
$211$		−	12.3592i		−	0.850845i	0.904995	−	0.425423i	$-0.139875\pi$
$212$		−	7.64439i		−	0.525019i
$213$		−	9.15317i		−	0.627165i
$214$		−	6.19384i		−	0.423402i
$215$		0			0
$216$		−	8.39707i		−	0.571348i
$217$		−	37.4232i		−	2.54045i
$218$	−3.06986			−0.207917
$219$	−7.94910			−0.537150
$220$		0			0
$221$	6.48826	+	4.41387i	0.436448	+	0.296909i
$222$		−	17.5638i		−	1.17881i
$223$		−	9.31429i		−	0.623731i	−0.950126	−	0.311866i	$-0.899046\pi$
$223$		−	9.31429i		−	0.623731i	0.950126	−	0.311866i	$-0.100954\pi$
$224$		−	62.1332i		−	4.15145i
$225$		0			0
$226$			26.8371i			1.78517i
$227$	0.990817			0.0657628			0.0328814	−	0.999459i	$-0.489532\pi$
$227$	0.990817			0.0657628			0.0328814	+	0.999459i	$0.489532\pi$
$228$	−33.8541			−2.24205
$229$	12.8425			0.848653			0.424327	−	0.905509i	$-0.360511\pi$
$229$	12.8425			0.848653			0.424327	+	0.905509i	$0.360511\pi$
$230$		0			0
$231$			9.05214i			0.595587i
$232$	18.7320			1.22982
$233$	−11.2347			−0.736007			−0.368003	−	0.929824i	$-0.619958\pi$
$233$	−11.2347			−0.736007			−0.368003	+	0.929824i	$0.619958\pi$
$234$	−5.08635			−0.332505
$235$		0			0
$236$	55.3456			3.60269
$237$			3.92218i			0.254773i
$238$	36.0684	+	24.5368i	2.33797	+	1.59048i
$239$	24.7670			1.60205			0.801023	−	0.598633i	$-0.204289\pi$
$239$	24.7670			1.60205			0.801023	+	0.598633i	$0.204289\pi$
$240$		0			0
$241$			14.1406i			0.910877i	0.890267	+	0.455439i	$0.150518\pi$
$241$			14.1406i			0.910877i	−0.890267	+	0.455439i	$0.849482\pi$
$242$		−	15.4253i		−	0.991576i
$243$	−1.00000			−0.0641500
$244$			77.1760i			4.94069i
$245$		0			0
$246$	−4.08592			−0.260508
$247$			12.5305i			0.797296i
$248$	79.3758			5.04037
$249$		−	3.85286i		−	0.244165i
$250$		0			0
$251$	0.534118			0.0337132			0.0168566	−	0.999858i	$-0.494634\pi$
$251$	0.534118			0.0337132			0.0168566	+	0.999858i	$0.494634\pi$
$252$	−20.3573			−1.28239
$253$			16.2665i			1.02266i
$254$	−41.6103			−2.61086
$255$		0			0
$256$	6.76479			0.422800
$257$		−	10.3708i		−	0.646915i	−0.946243	−	0.323457i	$-0.895155\pi$
$257$		−	10.3708i		−	0.646915i	0.946243	−	0.323457i	$-0.104845\pi$
$258$	11.4100			0.710355
$259$	−26.0188			−1.61673
$260$		0			0
$261$		−	2.23078i		−	0.138082i
$262$	−58.8809			−3.63768
$263$		−	10.1290i		−	0.624583i	−0.949986	−	0.312291i	$-0.898904\pi$
$263$		−	10.1290i		−	0.624583i	0.949986	−	0.312291i	$-0.101096\pi$
$264$	−19.1999			−1.18167
$265$		0			0
$266$			69.6574i			4.27097i
$267$	−4.71863			−0.288776
$268$		−	25.2682i		−	1.54350i
$269$		−	25.4852i		−	1.55386i	−0.629587	−	0.776930i	$-0.716776\pi$
$269$		−	25.4852i		−	1.55386i	0.629587	−	0.776930i	$-0.283224\pi$
$270$		0			0
$271$	−6.58885			−0.400244			−0.200122	−	0.979771i	$-0.564134\pi$
$271$	−6.58885			−0.400244			−0.200122	+	0.979771i	$0.564134\pi$
$272$	−28.1930	+	41.4430i	−1.70945	+	2.51285i
$273$			7.53487i			0.456031i
$274$	28.8449			1.74258
$275$		0			0
$276$	−36.5815			−2.20195
$277$	−2.46779			−0.148275			−0.0741375	−	0.997248i	$-0.523620\pi$
$277$	−2.46779			−0.148275			−0.0741375	+	0.997248i	$0.523620\pi$
$278$	0.586803			0.0351941
$279$		−	9.45279i		−	0.565924i
$280$		0			0
$281$	26.2937			1.56855			0.784274	−	0.620414i	$-0.213035\pi$
$281$	26.2937			1.56855			0.784274	+	0.620414i	$0.213035\pi$
$282$	−9.65208			−0.574773
$283$	−18.9028			−1.12365			−0.561826	−	0.827255i	$-0.689901\pi$
$283$	−18.9028			−1.12365			−0.561826	+	0.827255i	$0.689901\pi$
$284$		−	47.0663i		−	2.79287i
$285$		0			0
$286$			11.6299i			0.687692i
$287$			6.05283i			0.357287i
$288$		−	15.6943i		−	0.924798i
$289$	−6.24330	−	15.8121i	−0.367253	−	0.930121i
$290$		0			0
$291$	7.70758			0.451826
$292$	−40.8748			−2.39202
$293$			5.45486i			0.318676i	0.987224	+	0.159338i	$0.0509360\pi$
$293$			5.45486i			0.318676i	−0.987224	+	0.159338i	$0.949064\pi$
$294$			23.1793i			1.35184i
$295$		0			0
$296$		−	55.1867i		−	3.20766i
$297$			2.28650i			0.132676i
$298$			40.8254i			2.36495i
$299$			13.5400i			0.783037i
$300$		0			0
$301$		−	16.9026i		−	0.974252i
$302$			61.8047i			3.55646i
$303$	16.4474			0.944877
$304$	−80.0371			−4.59044
$305$		0			0
$306$	9.11058	+	6.19779i	0.520817	+	0.354304i
$307$		−	10.7040i		−	0.610910i	−0.952207	−	0.305455i	$-0.901191\pi$
$307$		−	10.7040i		−	0.610910i	0.952207	−	0.305455i	$-0.0988085\pi$
$308$			46.5468i			2.65225i
$309$		−	14.1528i		−	0.805127i
$310$		0			0
$311$		−	20.1434i		−	1.14223i	−0.820871	−	0.571113i	$-0.806512\pi$
$311$		−	20.1434i		−	1.14223i	0.820871	−	0.571113i	$-0.193488\pi$
$312$	−15.9817			−0.904785
$313$	15.0079			0.848296			0.424148	−	0.905593i	$-0.360574\pi$
$313$	15.0079			0.848296			0.424148	+	0.905593i	$0.360574\pi$
$314$	24.3135			1.37209
$315$		0			0
$316$			20.1681i			1.13455i
$317$	5.58449			0.313656			0.156828	−	0.987626i	$-0.449873\pi$
$317$	5.58449			0.313656			0.156828	+	0.987626i	$0.449873\pi$
$318$	−3.97298			−0.222794
$319$	−5.10067			−0.285582
$320$		0			0
$321$	−2.31765			−0.129359
$322$			75.2690i			4.19458i
$323$	15.2686	−	22.4444i	0.849566	−	1.24884i
$324$	−5.14207			−0.285671
$325$		0			0
$326$		−	50.9363i		−	2.82110i
$327$			1.14870i			0.0635232i
$328$	−12.8382			−0.708873
$329$			14.2985i			0.788301i
$330$		0			0
$331$	34.7988			1.91271			0.956357	−	0.292200i	$-0.0943874\pi$
$331$	34.7988			1.91271			0.956357	+	0.292200i	$0.0943874\pi$
$332$		−	19.8117i		−	1.08731i
$333$	−6.57214			−0.360151
$334$		−	36.0226i		−	1.97107i
$335$		0			0
$336$	−48.1281			−2.62560
$337$	1.89214			0.103071			0.0515357	−	0.998671i	$-0.483588\pi$
$337$	1.89214			0.103071			0.0515357	+	0.998671i	$0.483588\pi$
$338$		−	25.0615i		−	1.36316i
$339$	10.0421			0.545410
$340$		0			0
$341$	−21.6138			−1.17045
$342$			17.5949i			0.951421i
$343$	6.62479			0.357705
$344$	35.8510			1.93296
$345$		0			0
$346$		−	2.97284i		−	0.159821i
$347$	−23.6057			−1.26722			−0.633610	−	0.773652i	$-0.718428\pi$
$347$	−23.6057			−1.26722			−0.633610	+	0.773652i	$0.718428\pi$
$348$		−	11.4708i		−	0.614901i
$349$	12.6402			0.676616			0.338308	−	0.941035i	$-0.390145\pi$
$349$	12.6402			0.676616			0.338308	+	0.941035i	$0.390145\pi$
$350$		0			0
$351$			1.90324i			0.101588i
$352$	−35.8850			−1.91268
$353$		−	1.54340i		−	0.0821470i	−0.999156	−	0.0410735i	$-0.986922\pi$
$353$		−	1.54340i		−	0.0821470i	0.999156	−	0.0410735i	$-0.0130778\pi$
$354$		−	28.7645i		−	1.52882i
$355$		0			0
$356$	−24.2635			−1.28597
$357$	9.18133	−	13.4963i	0.485928	−	0.714300i
$358$		−	10.8597i		−	0.573955i
$359$	−22.2202			−1.17274			−0.586370	−	0.810044i	$-0.699443\pi$
$359$	−22.2202			−1.17274			−0.586370	+	0.810044i	$0.699443\pi$
$360$		0			0
$361$	24.3458			1.28136
$362$	35.5009			1.86588
$363$	−5.77194			−0.302948
$364$			38.7448i			2.03078i
$365$		0			0
$366$	40.1104			2.09660
$367$	−10.4553			−0.545763			−0.272881	−	0.962048i	$-0.587977\pi$
$367$	−10.4553			−0.545763			−0.272881	+	0.962048i	$0.587977\pi$
$368$	−86.4849			−4.50834
$369$			1.52889i			0.0795910i
$370$		0			0
$371$			5.88553i			0.305562i
$372$		−	48.6069i		−	2.52015i
$373$			25.9610i			1.34421i	0.740455	+	0.672106i	$0.234610\pi$
$373$			25.9610i			1.34421i	−0.740455	+	0.672106i	$0.765390\pi$
$374$	14.1712	−	20.8313i	0.732776	−	1.07716i
$375$		0			0
$376$	−30.3275			−1.56402
$377$	−4.24572			−0.218666
$378$			10.5802i			0.544186i
$379$			4.86700i			0.250001i	0.992157	+	0.125000i	$0.0398932\pi$
$379$			4.86700i			0.250001i	−0.992157	+	0.125000i	$0.960107\pi$
$380$		0			0
$381$			15.5700i			0.797675i
$382$			30.6678i			1.56910i
$383$		−	4.65321i		−	0.237768i	−0.992908	−	0.118884i	$-0.962068\pi$
$383$		−	4.65321i		−	0.237768i	0.992908	−	0.118884i	$-0.0379317\pi$
$384$		−	15.7244i		−	0.802433i
$385$		0			0
$386$			22.7283i			1.15684i
$387$		−	4.26946i		−	0.217029i
$388$	39.6329			2.01206
$389$	−7.85767			−0.398400			−0.199200	−	0.979959i	$-0.563834\pi$
$389$	−7.85767			−0.398400			−0.199200	+	0.979959i	$0.563834\pi$
$390$		0			0
$391$	16.4986	−	24.2525i	0.834371	−	1.22650i
$392$			72.8309i			3.67852i
$393$			22.0324i			1.11139i
$394$		−	44.9012i		−	2.26209i
$395$		0			0
$396$			11.7573i			0.590827i
$397$	8.84619			0.443978			0.221989	−	0.975049i	$-0.428745\pi$
$397$	8.84619			0.443978			0.221989	+	0.975049i	$0.428745\pi$
$398$	−23.8305			−1.19451
$399$	26.0648			1.30487
$400$		0			0
$401$			35.1967i			1.75764i	0.477154	+	0.878820i	$0.341669\pi$
$401$			35.1967i			1.75764i	−0.477154	+	0.878820i	$0.658331\pi$
$402$	−13.1325			−0.654991
$403$	−17.9910			−0.896194
$404$	84.5736			4.20769
$405$		0			0
$406$	−23.6021			−1.17135
$407$			15.0272i			0.744869i
$408$	28.6261	+	19.4739i	1.41720	+	0.964101i
$409$	−5.72269			−0.282969			−0.141484	−	0.989940i	$-0.545188\pi$
$409$	−5.72269			−0.282969			−0.141484	+	0.989940i	$0.545188\pi$
$410$		0			0
$411$		−	10.7934i		−	0.532398i
$412$		−	72.7749i		−	3.58536i
$413$	−42.6115			−2.09677
$414$			19.0123i			0.934405i
$415$		0			0
$416$	−29.8702			−1.46450
$417$		−	0.219574i		−	0.0107526i
$418$	40.2306			1.96774
$419$		−	15.4486i		−	0.754714i	−0.926068	−	0.377357i	$-0.876833\pi$
$419$		−	15.4486i		−	0.754714i	0.926068	−	0.377357i	$-0.123167\pi$
$420$		0			0
$421$	10.0582			0.490205			0.245103	−	0.969497i	$-0.421178\pi$
$421$	10.0582			0.490205			0.245103	+	0.969497i	$0.421178\pi$
$422$	−33.0296			−1.60786
$423$			3.61168i			0.175606i
$424$	−12.4834			−0.606247
$425$		0			0
$426$	−24.4615			−1.18517
$427$		−	59.4190i		−	2.87549i
$428$	−11.9175			−0.576055
$429$	4.35176			0.210105
$430$		0			0
$431$			30.4185i			1.46521i	0.680655	+	0.732604i	$0.261695\pi$
$431$			30.4185i			1.46521i	−0.680655	+	0.732604i	$0.738305\pi$
$432$	−12.1567			−0.584892
$433$		−	36.3734i		−	1.74799i	−0.485933	−	0.873996i	$-0.661520\pi$
$433$		−	36.3734i		−	1.74799i	0.485933	−	0.873996i	$-0.338480\pi$
$434$	−100.012			−4.80074
$435$		0			0
$436$			5.90669i			0.282879i
$437$	46.8378			2.24056
$438$			21.2437i			1.01506i
$439$		−	8.74032i		−	0.417153i	−0.978006	−	0.208576i	$-0.933117\pi$
$439$		−	8.74032i		−	0.417153i	0.978006	−	0.208576i	$-0.0668830\pi$
$440$		0			0
$441$	8.67337			0.413017
$442$	11.7959	−	17.3397i	0.561074	−	0.824763i
$443$		−	23.8933i		−	1.13520i	−0.823303	−	0.567602i	$-0.807871\pi$
$443$		−	23.8933i		−	1.13520i	0.823303	−	0.567602i	$-0.192129\pi$
$444$	−33.7944			−1.60381
$445$		0			0
$446$	−24.8921			−1.17868
$447$	15.2763			0.722545
$448$	−69.7928			−3.29740
$449$			13.3605i			0.630523i	0.949005	+	0.315262i	$0.102092\pi$
$449$			13.3605i			0.630523i	−0.949005	+	0.315262i	$0.897908\pi$
$450$		0			0
$451$	3.49581			0.164611
$452$	51.6370			2.42880
$453$	23.1265			1.08658
$454$		−	2.64792i		−	0.124273i
$455$		0			0
$456$			55.2843i			2.58892i
$457$			2.28323i			0.106805i	0.998573	+	0.0534026i	$0.0170067\pi$
$457$			2.28323i			0.106805i	−0.998573	+	0.0534026i	$0.982993\pi$
$458$		−	34.3210i		−	1.60372i
$459$	2.31913	−	3.40905i	0.108248	−	0.159121i
$460$		0			0
$461$	−7.74444			−0.360695			−0.180347	−	0.983603i	$-0.557722\pi$
$461$	−7.74444			−0.360695			−0.180347	+	0.983603i	$0.557722\pi$
$462$	24.1915			1.12549
$463$		−	1.05455i		−	0.0490092i	−0.999700	−	0.0245046i	$-0.992199\pi$
$463$		−	1.05455i		−	0.0490092i	0.999700	−	0.0245046i	$-0.00780084\pi$
$464$		−	27.1190i		−	1.25897i
$465$		0			0
$466$			30.0242i			1.39085i
$467$		−	20.8216i		−	0.963507i	−0.876307	−	0.481754i	$-0.840000\pi$
$467$		−	20.8216i		−	0.963507i	0.876307	−	0.481754i	$-0.160000\pi$
$468$			9.78662i			0.452387i
$469$			19.4544i			0.898319i
$470$		0			0
$471$		−	9.09777i		−	0.419203i
$472$		−	90.3802i		−	4.16008i
$473$	−9.76211			−0.448862
$474$	10.4819			0.481449
$475$		0			0
$476$	47.2111	−	69.3990i	2.16392	−	3.18090i
$477$			1.48664i			0.0680684i
$478$		−	66.1890i		−	3.02742i
$479$		−	33.5045i		−	1.53086i	−0.643518	−	0.765431i	$-0.722526\pi$
$479$		−	33.5045i		−	1.53086i	0.643518	−	0.765431i	$-0.277474\pi$
$480$		0			0
$481$			12.5084i			0.570333i
$482$	37.7903			1.72130
$483$	28.1646			1.28154
$484$	−29.6797			−1.34908
$485$		0			0
$486$			2.67247i			0.121225i
$487$	1.41364			0.0640580			0.0320290	−	0.999487i	$-0.489803\pi$
$487$	1.41364			0.0640580			0.0320290	+	0.999487i	$0.489803\pi$
$488$	126.030			5.70509
$489$	−19.0597			−0.861907
$490$		0			0
$491$	−28.0849			−1.26745			−0.633727	−	0.773557i	$-0.718476\pi$
$491$	−28.0849			−1.26745			−0.633727	+	0.773557i	$0.718476\pi$
$492$			7.86168i			0.354432i
$493$	7.60485	+	5.17346i	0.342505	+	0.233001i
$494$	33.4873			1.50667
$495$		0			0
$496$		−	114.915i		−	5.15984i
$497$			36.2371i			1.62545i
$498$	−10.2966			−0.461403
$499$		−	43.1265i		−	1.93061i	−0.261134	−	0.965303i	$-0.584096\pi$
$499$		−	43.1265i		−	1.93061i	0.261134	−	0.965303i	$-0.415904\pi$
$500$		0			0
$501$	−13.4792			−0.602205
$502$		−	1.42741i		−	0.0637085i
$503$	−4.47717			−0.199627			−0.0998137	−	0.995006i	$-0.531825\pi$
$503$	−4.47717			−0.199627			−0.0998137	+	0.995006i	$0.531825\pi$
$504$			33.2437i			1.48079i
$505$		0			0
$506$	43.4716			1.93255
$507$	−9.37766			−0.416477
$508$			80.0620i			3.55218i
$509$	−24.3389			−1.07880			−0.539400	−	0.842049i	$-0.681349\pi$
$509$	−24.3389			−1.07880			−0.539400	+	0.842049i	$0.681349\pi$
$510$		0			0
$511$	31.4702			1.39216
$512$			13.3701i			0.590882i
$513$	6.58376			0.290680
$514$	−27.7157			−1.22249
$515$		0			0
$516$		−	21.9539i		−	0.966466i
$517$	8.25808			0.363190
$518$			69.5344i			3.05517i
$519$	−1.11240			−0.0488288
$520$		0			0
$521$			1.01295i			0.0443781i	0.999754	+	0.0221891i	$0.00706358\pi$
$521$			1.01295i			0.0443781i	−0.999754	+	0.0221891i	$0.992936\pi$
$522$	−5.96168			−0.260936
$523$		−	2.18441i		−	0.0955174i	−0.998859	−	0.0477587i	$-0.984792\pi$
$523$		−	2.18441i		−	0.0955174i	0.998859	−	0.0477587i	$-0.0152078\pi$
$524$			113.292i			4.94920i
$525$		0			0
$526$	−27.0695			−1.18029
$527$	32.2251	+	21.9222i	1.40375	+	0.954947i
$528$			27.7963i			1.20968i
$529$	27.6111			1.20048
$530$		0			0
$531$	−10.7633			−0.467087
$532$	134.027			5.81082
$533$	2.90986			0.126040
$534$			12.6104i			0.545705i
$535$		0			0
$536$	−41.2633			−1.78230
$537$	−4.06356			−0.175356
$538$	−68.1083			−2.93636
$539$		−	19.8316i		−	0.854208i
$540$		0			0
$541$			29.4672i			1.26689i	0.773787	+	0.633446i	$0.218360\pi$
$541$			29.4672i			1.26689i	−0.773787	+	0.633446i	$0.781640\pi$
$542$			17.6085i			0.756349i
$543$		−	13.2839i		−	0.570068i
$544$	53.5028	+	36.3972i	2.29392	+	1.56052i
$545$		0			0
$546$	20.1367			0.861770
$547$	−1.36308			−0.0582810			−0.0291405	−	0.999575i	$-0.509277\pi$
$547$	−1.36308			−0.0582810			−0.0291405	+	0.999575i	$0.509277\pi$
$548$		−	55.5003i		−	2.37085i
$549$		−	15.0087i		−	0.640558i
$550$		0			0
$551$			14.6869i			0.625683i
$552$			59.7381i			2.54262i
$553$		−	15.5278i		−	0.660307i
$554$			6.59507i			0.280198i
$555$		0			0
$556$		−	1.12906i		−	0.0478830i
$557$			12.4667i			0.528230i	0.964491	+	0.264115i	$0.0850798\pi$
$557$			12.4667i			0.528230i	−0.964491	+	0.264115i	$0.914920\pi$
$558$	−25.2623			−1.06944
$559$	−8.12583			−0.343686
$560$		0			0
$561$	−7.79479	−	5.30267i	−0.329096	−	0.223879i
$562$		−	70.2689i		−	2.96412i
$563$		−	21.6084i		−	0.910686i	−0.890316	−	0.455343i	$-0.849517\pi$
$563$		−	21.6084i		−	0.910686i	0.890316	−	0.455343i	$-0.150483\pi$
$564$			18.5715i			0.782001i
$565$		0			0
$566$			50.5170i			2.12339i
$567$	3.95896			0.166261
$568$	−76.8599			−3.22497
$569$	−15.4303			−0.646871			−0.323435	−	0.946250i	$-0.604838\pi$
$569$	−15.4303			−0.646871			−0.323435	+	0.946250i	$0.604838\pi$
$570$		0			0
$571$		−	20.3659i		−	0.852287i	−0.904656	−	0.426144i	$-0.859872\pi$
$571$		−	20.3659i		−	0.852287i	0.904656	−	0.426144i	$-0.140128\pi$
$572$	22.3771			0.935631
$573$	11.4755			0.479394
$574$	16.1760			0.675172
$575$		0			0
$576$	−17.6291			−0.734545
$577$			2.85180i			0.118722i	0.998237	+	0.0593611i	$0.0189063\pi$
$577$			2.85180i			0.118722i	−0.998237	+	0.0593611i	$0.981094\pi$
$578$	−42.2572	+	16.6850i	−1.75767	+	0.694005i
$579$	8.50463			0.353440
$580$		0			0
$581$			15.2533i			0.632814i
$582$		−	20.5982i		−	0.853824i
$583$	3.39919			0.140780
$584$			66.7491i			2.76210i
$585$		0			0
$586$	14.5779			0.602209
$587$		−	38.8326i		−	1.60279i	−0.598134	−	0.801396i	$-0.704091\pi$
$587$		−	38.8326i		−	1.60279i	0.598134	−	0.801396i	$-0.295909\pi$
$588$	44.5991			1.83923
$589$			62.2349i			2.56434i
$590$		0			0
$591$	−16.8014			−0.691118
$592$	−79.8958			−3.28370
$593$		−	23.1318i		−	0.949907i	−0.880011	−	0.474954i	$-0.842465\pi$
$593$		−	23.1318i		−	0.949907i	0.880011	−	0.474954i	$-0.157535\pi$
$594$	6.11058			0.250720
$595$		0			0
$596$	78.5519			3.21761
$597$			8.91705i			0.364950i
$598$	36.1851			1.47972
$599$	−0.296194			−0.0121022			−0.00605108	−	0.999982i	$-0.501926\pi$
$599$	−0.296194			−0.0121022			−0.00605108	+	0.999982i	$0.501926\pi$
$600$		0			0
$601$		−	20.3712i		−	0.830958i	−0.909603	−	0.415479i	$-0.863614\pi$
$601$		−	20.3712i		−	0.830958i	0.909603	−	0.415479i	$-0.136386\pi$
$602$	−45.1717			−1.84106
$603$			4.91401i			0.200114i
$604$	118.918			4.83871
$605$		0			0
$606$		−	43.9550i		−	1.78555i
$607$	16.7639			0.680427			0.340214	−	0.940348i	$-0.389501\pi$
$607$	16.7639			0.680427			0.340214	+	0.940348i	$0.389501\pi$
$608$			103.328i			4.19049i
$609$			8.83157i			0.357873i
$610$		0			0
$611$	6.87390			0.278088
$612$	11.9251	−	17.5296i	0.482044	−	0.708592i
$613$		−	19.1793i		−	0.774645i	−0.921944	−	0.387322i	$-0.873400\pi$
$613$		−	19.1793i		−	0.774645i	0.921944	−	0.387322i	$-0.126600\pi$
$614$	−28.6061			−1.15445
$615$		0			0
$616$	76.0115			3.06259
$617$	19.0949			0.768732			0.384366	−	0.923181i	$-0.374420\pi$
$617$	19.0949			0.768732			0.384366	+	0.923181i	$0.374420\pi$
$618$	−37.8230			−1.52146
$619$			25.4024i			1.02101i	0.859875	+	0.510504i	$0.170541\pi$
$619$			25.4024i			1.02101i	−0.859875	+	0.510504i	$0.829459\pi$
$620$		0			0
$621$	7.11415			0.285481
$622$	−53.8325			−2.15849
$623$	18.6809			0.748434
$624$			23.1373i			0.926232i
$625$		0			0
$626$		−	40.1081i		−	1.60304i
$627$		−	15.0537i		−	0.601188i
$628$		−	46.7814i		−	1.86678i
$629$	15.2416	−	22.4048i	0.607723	−	0.893337i
$630$		0			0
$631$	12.9340			0.514895			0.257447	−	0.966292i	$-0.417119\pi$
$631$	12.9340			0.514895			0.257447	+	0.966292i	$0.417119\pi$
$632$	32.9348			1.31008
$633$			12.3592i			0.491236i
$634$		−	14.9243i		−	0.592722i
$635$		0			0
$636$			7.64439i			0.303120i
$637$		−	16.5075i		−	0.654052i
$638$			13.6314i			0.539670i
$639$			9.15317i			0.362094i
$640$		0			0
$641$			7.97770i			0.315100i	0.987511	+	0.157550i	$0.0503596\pi$
$641$			7.97770i			0.315100i	−0.987511	+	0.157550i	$0.949640\pi$
$642$			6.19384i			0.244451i
$643$	−14.5100			−0.572219			−0.286109	−	0.958197i	$-0.592362\pi$
$643$	−14.5100			−0.572219			−0.286109	+	0.958197i	$0.592362\pi$
$644$	144.825			5.70689
$645$		0			0
$646$	−59.9818	−	40.8047i	−2.35995	−	1.60544i
$647$			19.9224i			0.783229i	0.920129	+	0.391614i	$0.128083\pi$
$647$			19.9224i			0.783229i	−0.920129	+	0.391614i	$0.871917\pi$
$648$			8.39707i			0.329868i
$649$			24.6102i			0.966036i
$650$		0			0
$651$			37.4232i			1.46673i
$652$	−98.0061			−3.83821
$653$	23.7364			0.928877			0.464438	−	0.885605i	$-0.346256\pi$
$653$	23.7364			0.928877			0.464438	+	0.885605i	$0.346256\pi$
$654$	3.06986			0.120041
$655$		0			0
$656$			18.5864i			0.725676i
$657$	7.94910			0.310124
$658$	38.2122			1.48967
$659$	8.04637			0.313442			0.156721	−	0.987643i	$-0.449908\pi$
$659$	8.04637			0.313442			0.156721	+	0.987643i	$0.449908\pi$
$660$		0			0
$661$	7.39195			0.287514			0.143757	−	0.989613i	$-0.454082\pi$
$661$	7.39195			0.287514			0.143757	+	0.989613i	$0.454082\pi$
$662$		−	92.9985i		−	3.61449i
$663$	−6.48826	−	4.41387i	−0.251983	−	0.171420i
$664$	−32.3527			−1.25553
$665$		0			0
$666$			17.5638i			0.680584i
$667$			15.8701i			0.614493i
$668$	−69.3108			−2.68172
$669$			9.31429i			0.360111i
$670$		0			0
$671$	−34.3174			−1.32481
$672$			62.1332i			2.39684i
$673$	14.6788			0.565824			0.282912	−	0.959146i	$-0.408700\pi$
$673$	14.6788			0.565824			0.282912	+	0.959146i	$0.408700\pi$
$674$		−	5.05668i		−	0.194776i
$675$		0			0
$676$	−48.2206			−1.85464
$677$	−7.30435			−0.280729			−0.140364	−	0.990100i	$-0.544827\pi$
$677$	−7.30435			−0.280729			−0.140364	+	0.990100i	$0.544827\pi$
$678$		−	26.8371i		−	1.03067i
$679$	−30.5140			−1.17102
$680$		0			0
$681$	−0.990817			−0.0379682
$682$			57.7620i			2.21182i
$683$	44.3563			1.69725			0.848623	−	0.528998i	$-0.177432\pi$
$683$	44.3563			1.69725			0.848623	+	0.528998i	$0.177432\pi$
$684$	33.8541			1.29445
$685$		0			0
$686$		−	17.7045i		−	0.675962i
$687$	−12.8425			−0.489970
$688$		−	51.9028i		−	1.97878i
$689$	2.82943			0.107793
$690$		0			0
$691$			7.45483i			0.283595i	0.989896	+	0.141798i	$0.0452882\pi$
$691$			7.45483i			0.283595i	−0.989896	+	0.141798i	$0.954712\pi$
$692$	−5.72002			−0.217443
$693$		−	9.05214i		−	0.343862i
$694$			63.0854i			2.39469i
$695$		0			0
$696$	−18.7320			−0.710035
$697$	−5.21208	−	3.54570i	−0.197422	−	0.134303i
$698$		−	33.7806i		−	1.27861i
$699$	11.2347			0.424934
$700$		0			0
$701$	38.2321			1.44401			0.722003	−	0.691890i	$-0.243222\pi$
$701$	38.2321			1.44401			0.722003	+	0.691890i	$0.243222\pi$
$702$	5.08635			0.191972
$703$	43.2693			1.63193
$704$			40.3088i			1.51919i
$705$		0			0
$706$	−4.12469			−0.155235
$707$	−65.1145			−2.44888
$708$	−55.3456			−2.08002
$709$		−	10.5468i		−	0.396093i	−0.980193	−	0.198046i	$-0.936540\pi$
$709$		−	10.5468i		−	0.396093i	0.980193	−	0.198046i	$-0.0634597\pi$
$710$		0			0
$711$		−	3.92218i		−	0.147093i
$712$			39.6227i			1.48492i
$713$			67.2486i			2.51848i
$714$	−36.0684	−	24.5368i	−1.34983	−	0.918266i
$715$		0			0
$716$	−20.8951			−0.780888
$717$	−24.7670			−0.924942
$718$			59.3828i			2.21615i
$719$		−	0.920080i		−	0.0343132i	−0.999853	−	0.0171566i	$-0.994539\pi$
$719$		−	0.920080i		−	0.0343132i	0.999853	−	0.0171566i	$-0.00546138\pi$
$720$		0			0
$721$			56.0305i			2.08669i
$722$		−	65.0634i		−	2.42141i
$723$		−	14.1406i		−	0.525895i
$724$		−	68.3070i		−	2.53861i
$725$		0			0
$726$			15.4253i			0.572487i
$727$			12.0729i			0.447758i	0.974617	+	0.223879i	$0.0718720\pi$
$727$			12.0729i			0.447758i	−0.974617	+	0.223879i	$0.928128\pi$
$728$	63.2708			2.34497
$729$	1.00000			0.0370370
$730$		0			0
$731$	14.5548	+	9.90143i	0.538330	+	0.366218i
$732$		−	77.1760i		−	2.85251i
$733$			34.9973i			1.29265i	0.763061	+	0.646327i	$0.223696\pi$
$733$			34.9973i			1.29265i	−0.763061	+	0.646327i	$0.776304\pi$
$734$			27.9415i			1.03134i
$735$		0			0
$736$			111.652i			4.11554i
$737$	11.2359			0.413878
$738$	4.08592			0.150405
$739$	−13.8600			−0.509849			−0.254925	−	0.966961i	$-0.582051\pi$
$739$	−13.8600			−0.509849			−0.254925	+	0.966961i	$0.582051\pi$
$740$		0			0
$741$		−	12.5305i		−	0.460319i
$742$	15.7289			0.577425
$743$	22.7473			0.834517			0.417258	−	0.908788i	$-0.362991\pi$
$743$	22.7473			0.834517			0.417258	+	0.908788i	$0.362991\pi$
$744$	−79.3758			−2.91006
$745$		0			0
$746$	69.3800			2.54018
$747$			3.85286i			0.140969i
$748$	−40.0813	−	27.2667i	−1.46552	−	0.996970i
$749$	9.17548			0.335265
$750$		0			0
$751$		−	3.48346i		−	0.127113i	−0.997978	−	0.0635566i	$-0.979756\pi$
$751$		−	3.48346i		−	0.127113i	0.997978	−	0.0635566i	$-0.0202443\pi$
$752$			43.9062i			1.60110i
$753$	−0.534118			−0.0194643
$754$			11.3465i			0.413216i
$755$		0			0
$756$	20.3573			0.740386
$757$		−	9.47036i		−	0.344206i	−0.985079	−	0.172103i	$-0.944944\pi$
$757$		−	9.47036i		−	0.344206i	0.985079	−	0.172103i	$-0.0550562\pi$
$758$	13.0069			0.472431
$759$		−	16.2665i		−	0.590436i
$760$		0			0
$761$	22.0781			0.800330			0.400165	−	0.916443i	$-0.368953\pi$
$761$	22.0781			0.800330			0.400165	+	0.916443i	$0.368953\pi$
$762$	41.6103			1.50738
$763$		−	4.54766i		−	0.164636i
$764$	59.0076			2.13482
$765$		0			0
$766$	−12.4356			−0.449315
$767$			20.4852i			0.739677i
$768$	−6.76479			−0.244104
$769$	19.4958			0.703035			0.351517	−	0.936181i	$-0.385666\pi$
$769$	19.4958			0.703035			0.351517	+	0.936181i	$0.385666\pi$
$770$		0			0
$771$			10.3708i			0.373496i
$772$	43.7314			1.57393
$773$		−	0.306046i		−	0.0110077i	−0.999985	−	0.00550386i	$-0.998248\pi$
$773$		−	0.306046i		−	0.0110077i	0.999985	−	0.00550386i	$-0.00175194\pi$
$774$	−11.4100			−0.410124
$775$		0			0
$776$		−	64.7211i		−	2.32335i
$777$	26.0188			0.933420
$778$			20.9993i			0.752863i
$779$		−	10.0659i		−	0.360647i
$780$		0			0
$781$	20.9287			0.748887
$782$	−64.8140	−	44.0920i	−2.31775	−	1.57673i
$783$			2.23078i			0.0797215i
$784$	105.440			3.76571
$785$		0			0
$786$	58.8809			2.10021
$787$	−8.72388			−0.310973			−0.155486	−	0.987838i	$-0.549694\pi$
$787$	−8.72388			−0.310973			−0.155486	+	0.987838i	$0.549694\pi$
$788$	−86.3941			−3.07766
$789$			10.1290i			0.360603i
$790$		0			0
$791$	−39.7561			−1.41356
$792$	19.1999			0.682238
$793$	−28.5653			−1.01438
$794$		−	23.6411i		−	0.838993i
$795$		0			0
$796$			45.8521i			1.62518i
$797$		−	19.1824i		−	0.679474i	−0.940520	−	0.339737i	$-0.889662\pi$
$797$		−	19.1824i		−	0.679474i	0.940520	−	0.339737i	$-0.110338\pi$
$798$		−	69.6574i		−	2.46584i
$799$	−12.3124	−	8.37593i	−0.435581	−	0.296319i
$800$		0			0
$801$	4.71863			0.166725
$802$	94.0620			3.32144
$803$		−	18.1756i		−	0.641402i
$804$			25.2682i			0.891140i
$805$		0			0
$806$			48.0802i			1.69355i
$807$			25.4852i			0.897122i
$808$		−	138.110i		−	4.85869i
$809$			23.4388i			0.824063i	0.911169	+	0.412032i	$0.135181\pi$
$809$			23.4388i			0.824063i	−0.911169	+	0.412032i	$0.864819\pi$
$810$		0			0
$811$		−	9.67154i		−	0.339614i	−0.985477	−	0.169807i	$-0.945686\pi$
$811$		−	9.67154i		−	0.339614i	0.985477	−	0.169807i	$-0.0543144\pi$
$812$			45.4125i			1.59367i
$813$	6.58885			0.231081
$814$	40.1596			1.40759
$815$		0			0
$816$	28.1930	−	41.4430i	0.986954	−	1.45080i
$817$			28.1091i			0.983413i
$818$			15.2937i			0.534732i
$819$		−	7.53487i		−	0.263290i
$820$		0			0
$821$			13.4718i			0.470170i	0.971975	+	0.235085i	$0.0755368\pi$
$821$			13.4718i			0.470170i	−0.971975	+	0.235085i	$0.924463\pi$
$822$	−28.8449			−1.00608
$823$	34.4453			1.20069			0.600344	−	0.799742i	$-0.295030\pi$
$823$	34.4453			1.20069			0.600344	+	0.799742i	$0.295030\pi$
$824$	−118.842			−4.14007
$825$		0			0
$826$			113.878i			3.96231i
$827$	6.29458			0.218884			0.109442	−	0.993993i	$-0.465094\pi$
$827$	6.29458			0.218884			0.109442	+	0.993993i	$0.465094\pi$
$828$	36.5815			1.27129
$829$	−11.9001			−0.413307			−0.206654	−	0.978414i	$-0.566257\pi$
$829$	−11.9001			−0.413307			−0.206654	+	0.978414i	$0.566257\pi$
$830$		0			0
$831$	2.46779			0.0856066
$832$			33.5524i			1.16322i
$833$	−20.1146	+	29.5680i	−0.696931	+	1.02447i
$834$	−0.586803			−0.0203193
$835$		0			0
$836$		−	77.4073i		−	2.67719i
$837$			9.45279i			0.326736i
$838$	−41.2859			−1.42620
$839$		−	14.0689i		−	0.485712i	−0.970062	−	0.242856i	$-0.921916\pi$
$839$		−	14.0689i		−	0.485712i	0.970062	−	0.242856i	$-0.0780843\pi$
$840$		0			0
$841$	24.0236			0.828401
$842$		−	26.8801i		−	0.926350i
$843$	−26.2937			−0.905602
$844$			63.5521i			2.18755i
$845$		0			0
$846$	9.65208			0.331845
$847$	22.8509			0.785166
$848$			18.0727i			0.620617i
$849$	18.9028			0.648741
$850$		0			0
$851$	46.7552			1.60275
$852$			47.0663i			1.61246i
$853$	−42.2959			−1.44818			−0.724092	−	0.689703i	$-0.757741\pi$
$853$	−42.2959			−1.44818			−0.724092	+	0.689703i	$0.757741\pi$
$854$	−158.795			−5.43386
$855$		0			0
$856$			19.4615i			0.665179i
$857$	−10.2126			−0.348855			−0.174427	−	0.984670i	$-0.555807\pi$
$857$	−10.2126			−0.348855			−0.174427	+	0.984670i	$0.555807\pi$
$858$		−	11.6299i		−	0.397039i
$859$	45.7519			1.56104			0.780518	−	0.625134i	$-0.214956\pi$
$859$	45.7519			1.56104			0.780518	+	0.625134i	$0.214956\pi$
$860$		0			0
$861$		−	6.05283i		−	0.206280i
$862$	81.2924			2.76883
$863$			15.5333i			0.528759i	0.964419	+	0.264380i	$0.0851672\pi$
$863$			15.5333i			0.528759i	−0.964419	+	0.264380i	$0.914833\pi$
$864$			15.6943i			0.533932i
$865$		0			0
$866$	−97.2065			−3.30321
$867$	6.24330	+	15.8121i	0.212034	+	0.537006i
$868$			192.433i			6.53160i
$869$	−8.96804			−0.304220
$870$		0			0
$871$	9.35256			0.316900
$872$	9.64572			0.326645
$873$	−7.70758			−0.260862
$874$		−	125.172i		−	4.23402i
$875$		0			0
$876$	40.8748			1.38103
$877$	−50.5232			−1.70605			−0.853024	−	0.521871i	$-0.825234\pi$
$877$	−50.5232			−1.70605			−0.853024	+	0.521871i	$0.825234\pi$
$878$	−23.3582			−0.788301
$879$		−	5.45486i		−	0.183988i
$880$		0			0
$881$		−	28.5095i		−	0.960508i	−0.877129	−	0.480254i	$-0.840544\pi$
$881$		−	28.5095i		−	0.960508i	0.877129	−	0.480254i	$-0.159456\pi$
$882$		−	23.1793i		−	0.780487i
$883$		−	9.96680i		−	0.335410i	−0.985837	−	0.167705i	$-0.946364\pi$
$883$		−	9.96680i		−	0.335410i	0.985837	−	0.167705i	$-0.0536355\pi$
$884$	−33.3631	−	22.6964i	−1.12212	−	0.763363i
$885$		0			0
$886$	−63.8540			−2.14522
$887$	−27.7267			−0.930971			−0.465486	−	0.885055i	$-0.654120\pi$
$887$	−27.7267			−0.930971			−0.465486	+	0.885055i	$0.654120\pi$
$888$			55.1867i			1.85194i
$889$		−	61.6410i		−	2.06737i
$890$		0			0
$891$		−	2.28650i		−	0.0766005i
$892$			47.8947i			1.60363i
$893$		−	23.7784i		−	0.795714i
$894$		−	40.8254i		−	1.36541i
$895$		0			0
$896$			62.2523i			2.07970i
$897$		−	13.5400i		−	0.452086i
$898$	35.7056			1.19151
$899$	−21.0871			−0.703294
$900$		0			0
$901$	−5.06802	−	3.44770i	−0.168840	−	0.114859i
$902$		−	9.34243i		−	0.311069i
$903$			16.9026i			0.562485i
$904$		−	84.3239i		−	2.80457i
$905$		0			0
$906$		−	61.8047i		−	2.05333i
$907$	−7.37527			−0.244892			−0.122446	−	0.992475i	$-0.539074\pi$
$907$	−7.37527			−0.244892			−0.122446	+	0.992475i	$0.539074\pi$
$908$	−5.09485			−0.169079
$909$	−16.4474			−0.545525
$910$		0			0
$911$			39.4870i			1.30826i	0.756382	+	0.654131i	$0.226965\pi$
$911$			39.4870i			1.30826i	−0.756382	+	0.654131i	$0.773035\pi$
$912$	80.0371			2.65029
$913$	8.80954			0.291553
$914$	6.10186			0.201832
$915$		0			0
$916$	−66.0368			−2.18192
$917$		−	87.2255i		−	2.88044i
$918$	−9.11058	−	6.19779i	−0.300694	−	0.204557i
$919$	−8.60852			−0.283969			−0.141985	−	0.989869i	$-0.545348\pi$
$919$	−8.60852			−0.283969			−0.141985	+	0.989869i	$0.545348\pi$
$920$		0			0
$921$			10.7040i			0.352709i
$922$			20.6967i			0.681611i
$923$	17.4207			0.573410
$924$		−	46.5468i		−	1.53128i
$925$		0			0
$926$	−2.81825			−0.0926136
$927$			14.1528i			0.464840i
$928$	−35.0106			−1.14928
$929$			42.2799i			1.38716i	0.720380	+	0.693580i	$0.243968\pi$
$929$			42.2799i			1.38716i	−0.720380	+	0.693580i	$0.756032\pi$
$930$		0			0
$931$	−57.1033			−1.87149
$932$	57.7694			1.89230
$933$			20.1434i			0.659465i
$934$	−55.6449			−1.82076
$935$		0			0
$936$	15.9817			0.522378
$937$			4.68521i			0.153059i	0.997067	+	0.0765295i	$0.0243839\pi$
$937$			4.68521i			0.153059i	−0.997067	+	0.0765295i	$0.975616\pi$
$938$	51.9911			1.69757
$939$	−15.0079			−0.489764
$940$		0			0
$941$		−	14.8271i		−	0.483348i	−0.970357	−	0.241674i	$-0.922304\pi$
$941$		−	14.8271i		−	0.483348i	0.970357	−	0.241674i	$-0.0776965\pi$
$942$	−24.3135			−0.792176
$943$		−	10.8768i		−	0.354197i
$944$	−130.847			−4.25870
$945$		0			0
$946$			26.0889i			0.848223i
$947$	−25.5008			−0.828666			−0.414333	−	0.910125i	$-0.635985\pi$
$947$	−25.5008			−0.828666			−0.414333	+	0.910125i	$0.635985\pi$
$948$		−	20.1681i		−	0.655030i
$949$		−	15.1291i		−	0.491110i
$950$		0			0
$951$	−5.58449			−0.181089
$952$	−113.330	−	77.0963i	−3.67303	−	2.49871i
$953$			8.10795i			0.262642i	0.991340	+	0.131321i	$0.0419219\pi$
$953$			8.10795i			0.262642i	−0.991340	+	0.131321i	$0.958078\pi$
$954$	3.97298			0.128630
$955$		0			0
$956$	−127.354			−4.11892
$957$	5.10067			0.164881
$958$	−89.5397			−2.89290
$959$			42.7305i			1.37984i
$960$		0			0
$961$	−58.3552			−1.88243
$962$	33.4282			1.07777
$963$	2.31765			0.0746852
$964$		−	72.7120i		−	2.34190i
$965$		0			0
$966$		−	75.2690i		−	2.42174i
$967$		−	11.5698i		−	0.372060i	−0.982544	−	0.186030i	$-0.940438\pi$
$967$		−	11.5698i		−	0.372060i	0.982544	−	0.186030i	$-0.0595621\pi$
$968$			48.4674i			1.55780i
$969$	−15.2686	+	22.4444i	−0.490497	+	0.721017i
$970$		0			0
$971$	−51.3311			−1.64729			−0.823647	−	0.567103i	$-0.808064\pi$
$971$	−51.3311			−1.64729			−0.823647	+	0.567103i	$0.808064\pi$
$972$	5.14207			0.164932
$973$			0.869284i			0.0278680i
$974$		−	3.77789i		−	0.121051i
$975$		0			0
$976$		−	182.458i		−	5.84032i
$977$			21.4460i			0.686117i	0.939314	+	0.343059i	$0.111463\pi$
$977$			21.4460i			0.686117i	−0.939314	+	0.343059i	$0.888537\pi$
$978$			50.9363i			1.62876i
$979$		−	10.7891i		−	0.344822i
$980$		0			0
$981$		−	1.14870i		−	0.0366752i
$982$			75.0559i			2.39513i
$983$	29.9981			0.956791			0.478396	−	0.878144i	$-0.341219\pi$
$983$	29.9981			0.956791			0.478396	+	0.878144i	$0.341219\pi$
$984$	12.8382			0.409268
$985$		0			0
$986$	13.8259	−	20.3237i	0.440306	−	0.647238i
$987$		−	14.2985i		−	0.455126i
$988$		−	64.4327i		−	2.04988i
$989$			30.3736i			0.965825i
$990$		0			0
$991$		−	48.6469i		−	1.54532i	−0.634820	−	0.772660i	$-0.718926\pi$
$991$		−	48.6469i		−	1.54532i	0.634820	−	0.772660i	$-0.281074\pi$
$992$	−148.355			−4.71028
$993$	−34.7988			−1.10431
$994$	96.8423			3.07165
$995$		0			0
$996$			19.8117i			0.627756i
$997$	5.63747			0.178541			0.0892703	−	0.996007i	$-0.471547\pi$
$997$	5.63747			0.178541			0.0892703	+	0.996007i	$0.471547\pi$
$998$	−115.254			−3.64830
$999$	6.57214			0.207933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.d.i.424.1		12
5.2	odd	4		1275.2.g.f.526.12	yes	12
5.3	odd	4		1275.2.g.e.526.1	✓	12
5.4	even	2		1275.2.d.j.424.12		12
17.16	even	2		1275.2.d.j.424.1		12
85.33	odd	4		1275.2.g.e.526.2	yes	12
85.67	odd	4		1275.2.g.f.526.11	yes	12
85.84	even	2	inner	1275.2.d.i.424.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1275.2.d.i.424.1		12	1.1	even	1	trivial
1275.2.d.i.424.12		12	85.84	even	2	inner
1275.2.d.j.424.1		12	17.16	even	2
1275.2.d.j.424.12		12	5.4	even	2
1275.2.g.e.526.1	✓	12	5.3	odd	4
1275.2.g.e.526.2	yes	12	85.33	odd	4
1275.2.g.f.526.11	yes	12	85.67	odd	4
1275.2.g.f.526.12	yes	12	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.67247i q^{2} -1.00000 q^{3} -5.14207 q^{4} +2.67247i q^{6} +3.95896 q^{7} +8.39707i q^{8} +1.00000 q^{9} -2.28650i q^{11} +5.14207 q^{12} -1.90324i q^{13} -10.5802i q^{14} +12.1567 q^{16} +(-2.31913 + 3.40905i) q^{17} -2.67247i q^{18} -6.58376 q^{19} -3.95896 q^{21} -6.11058 q^{22} -7.11415 q^{23} -8.39707i q^{24} -5.08635 q^{26} -1.00000 q^{27} -20.3573 q^{28} -2.23078i q^{29} -9.45279i q^{31} -15.6943i q^{32} +2.28650i q^{33} +(9.11058 + 6.19779i) q^{34} -5.14207 q^{36} -6.57214 q^{37} +17.5949i q^{38} +1.90324i q^{39} +1.52889i q^{41} +10.5802i q^{42} -4.26946i q^{43} +11.7573i q^{44} +19.0123i q^{46} +3.61168i q^{47} -12.1567 q^{48} +8.67337 q^{49} +(2.31913 - 3.40905i) q^{51} +9.78662i q^{52} +1.48664i q^{53} +2.67247i q^{54} +33.2437i q^{56} +6.58376 q^{57} -5.96168 q^{58} -10.7633 q^{59} -15.0087i q^{61} -25.2623 q^{62} +3.95896 q^{63} -17.6291 q^{64} +6.11058 q^{66} +4.91401i q^{67} +(11.9251 - 17.5296i) q^{68} +7.11415 q^{69} +9.15317i q^{71} +8.39707i q^{72} +7.94910 q^{73} +17.5638i q^{74} +33.8541 q^{76} -9.05214i q^{77} +5.08635 q^{78} -3.92218i q^{79} +1.00000 q^{81} +4.08592 q^{82} +3.85286i q^{83} +20.3573 q^{84} -11.4100 q^{86} +2.23078i q^{87} +19.1999 q^{88} +4.71863 q^{89} -7.53487i q^{91} +36.5815 q^{92} +9.45279i q^{93} +9.65208 q^{94} +15.6943i q^{96} -7.70758 q^{97} -23.1793i q^{98} -2.28650i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 16 q^{4} + 12 q^{9} + 16 q^{12} + 32 q^{16} - 6 q^{17} + 4 q^{19} + 12 q^{22} - 16 q^{23} - 36 q^{26} - 12 q^{27} - 36 q^{28} + 24 q^{34} - 16 q^{36} - 4 q^{37} - 32 q^{48} + 16 q^{49}+ \cdots - 16 q^{97}+O(q^{100}) \) 12 * q - 12 * q^3 - 16 * q^4 + 12 * q^9 + 16 * q^12 + 32 * q^16 - 6 * q^17 + 4 * q^19 + 12 * q^22 - 16 * q^23 - 36 * q^26 - 12 * q^27 - 36 * q^28 + 24 * q^34 - 16 * q^36 - 4 * q^37 - 32 * q^48 + 16 * q^49 + 6 * q^51 - 4 * q^57 + 4 * q^58 - 32 * q^59 - 56 * q^62 - 56 * q^64 - 12 * q^66 - 20 * q^68 + 16 * q^69 + 44 * q^73 + 64 * q^76 + 36 * q^78 + 12 * q^81 - 40 * q^82 + 36 * q^84 + 20 * q^86 - 8 * q^88 - 4 * q^89 - 40 * q^94 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more