LMFDB - Embedded newform 1275.2.g.e.526.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1275,2,Mod(526,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1275.526"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,16,0,0,0,0,-12,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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			526.9
Root			$1.74480i$ of defining polynomial
Character	$\chi$	$=$	1275.526
Dual form			1275.2.g.e.526.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.74480 q^{2} -1.00000i q^{3} +1.04434 q^{4} -1.74480i q^{6} +4.98357i q^{7} -1.66744 q^{8} -1.00000 q^{9} +2.23877i q^{11} -1.04434i q^{12} -2.92477 q^{13} +8.69535i q^{14} -4.99803 q^{16} +(-3.95816 + 1.15455i) q^{17} -1.74480 q^{18} -1.52050 q^{19} +4.98357 q^{21} +3.90621i q^{22} +1.30964i q^{23} +1.66744i q^{24} -5.10314 q^{26} +1.00000i q^{27} +5.20454i q^{28} +1.82004i q^{29} +1.39136i q^{31} -5.38571 q^{32} +2.23877 q^{33} +(-6.90621 + 2.01446i) q^{34} -1.04434 q^{36} +3.54628i q^{37} -2.65298 q^{38} +2.92477i q^{39} -8.48551i q^{41} +8.69535 q^{42} +9.13105 q^{43} +2.33803i q^{44} +2.28507i q^{46} -0.166519 q^{47} +4.99803i q^{48} -17.8360 q^{49} +(1.15455 + 3.95816i) q^{51} -3.05445 q^{52} +7.58067 q^{53} +1.74480i q^{54} -8.30981i q^{56} +1.52050i q^{57} +3.17561i q^{58} +13.7233 q^{59} -4.50092i q^{61} +2.42765i q^{62} -4.98357i q^{63} +0.599071 q^{64} +3.90621 q^{66} +1.03226 q^{67} +(-4.13366 + 1.20574i) q^{68} +1.30964 q^{69} -1.78218i q^{71} +1.66744 q^{72} +11.6402i q^{73} +6.18756i q^{74} -1.58792 q^{76} -11.1570 q^{77} +5.10314i q^{78} +13.9481i q^{79} +1.00000 q^{81} -14.8055i q^{82} -4.47514 q^{83} +5.20454 q^{84} +15.9319 q^{86} +1.82004 q^{87} -3.73301i q^{88} +14.6490 q^{89} -14.5758i q^{91} +1.36771i q^{92} +1.39136 q^{93} -0.290543 q^{94} +5.38571i q^{96} +14.7821i q^{97} -31.1203 q^{98} -2.23877i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{4} - 12 q^{9} - 4 q^{13} + 32 q^{16} + 10 q^{17} - 4 q^{19} - 36 q^{26} - 20 q^{32} - 12 q^{33} - 24 q^{34} - 16 q^{36} - 44 q^{38} + 28 q^{42} + 28 q^{43} - 16 q^{47} - 16 q^{49} + 6 q^{51}+ \cdots - 8 q^{98}+O(q^{100}) $ 12 * q + 16 * q^4 - 12 * q^9 - 4 * q^13 + 32 * q^16 + 10 * q^17 - 4 * q^19 - 36 * q^26 - 20 * q^32 - 12 * q^33 - 24 * q^34 - 16 * q^36 - 44 * q^38 + 28 * q^42 + 28 * q^43 - 16 * q^47 - 16 * q^49 + 6 * q^51 + 16 * q^52 - 12 * q^53 + 32 * q^59 + 56 * q^64 - 12 * q^66 - 20 * q^67 + 56 * q^68 - 16 * q^69 + 64 * q^76 - 72 * q^77 + 12 * q^81 - 44 * q^83 - 36 * q^84 + 20 * q^86 + 32 * q^87 + 4 * q^89 + 8 * q^93 + 40 * q^94 - 8 * q^98

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 $

$2$	1.74480			1.23376			0.616881	−	0.787056i	$-0.288396\pi$
$2$	1.74480			1.23376			0.616881	+	0.787056i	$0.288396\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	1.04434			0.522169
$5$		0			0
$5$		0			0
$6$		−	1.74480i		−	0.712313i
$7$			4.98357i			1.88361i	0.336157	+	0.941806i	$0.390873\pi$
$7$			4.98357i			1.88361i	−0.336157	+	0.941806i	$0.609127\pi$
$8$	−1.66744			−0.589529
$9$	−1.00000			−0.333333
$10$		0			0
$11$			2.23877i			0.675013i	0.941323	+	0.337507i	$0.109584\pi$
$11$			2.23877i			0.675013i	−0.941323	+	0.337507i	$0.890416\pi$
$12$		−	1.04434i		−	0.301475i
$13$	−2.92477			−0.811184			−0.405592	−	0.914054i	$-0.632935\pi$
$13$	−2.92477			−0.811184			−0.405592	+	0.914054i	$0.632935\pi$
$14$			8.69535i			2.32393i
$15$		0			0
$16$	−4.99803			−1.24951
$17$	−3.95816	+	1.15455i	−0.959994	+	0.280020i
$17$	−3.95816	+	1.15455i	−0.959994	+	0.280020i
$18$	−1.74480			−0.411254
$19$	−1.52050			−0.348827			−0.174413	−	0.984673i	$-0.555803\pi$
$19$	−1.52050			−0.348827			−0.174413	+	0.984673i	$0.555803\pi$
$20$		0			0
$21$	4.98357			1.08750
$22$			3.90621i			0.832806i
$23$			1.30964i			0.273080i	0.990635	+	0.136540i	$0.0435982\pi$
$23$			1.30964i			0.273080i	−0.990635	+	0.136540i	$0.956402\pi$
$24$			1.66744i			0.340365i
$25$		0			0
$26$	−5.10314			−1.00081
$27$			1.00000i			0.192450i
$28$			5.20454i			0.983565i
$29$			1.82004i			0.337972i	0.985618	+	0.168986i	$0.0540493\pi$
$29$			1.82004i			0.337972i	−0.985618	+	0.168986i	$0.945951\pi$
$30$		0			0
$31$			1.39136i			0.249896i	0.992163	+	0.124948i	$0.0398764\pi$
$31$			1.39136i			0.249896i	−0.992163	+	0.124948i	$0.960124\pi$
$32$	−5.38571			−0.952067
$33$	2.23877			0.389719
$34$	−6.90621	+	2.01446i	−1.18440	+	0.345478i
$35$		0			0
$36$	−1.04434			−0.174056
$37$			3.54628i			0.583005i	0.956570	+	0.291503i	$0.0941552\pi$
$37$			3.54628i			0.583005i	−0.956570	+	0.291503i	$0.905845\pi$
$38$	−2.65298			−0.430370
$39$			2.92477i			0.468337i
$40$		0			0
$41$		−	8.48551i		−	1.32521i	−0.748967	−	0.662607i	$-0.769450\pi$
$41$		−	8.48551i		−	1.32521i	0.748967	−	0.662607i	$-0.230550\pi$
$42$	8.69535			1.34172
$43$	9.13105			1.39247			0.696236	−	0.717813i	$-0.254857\pi$
$43$	9.13105			1.39247			0.696236	+	0.717813i	$0.254857\pi$
$44$			2.33803i			0.352471i
$45$		0			0
$46$			2.28507i			0.336915i
$47$	−0.166519			−0.0242893			−0.0121446	−	0.999926i	$-0.503866\pi$
$47$	−0.166519			−0.0242893			−0.0121446	+	0.999926i	$0.503866\pi$
$48$			4.99803i			0.721404i
$49$	−17.8360			−2.54800
$50$		0			0
$51$	1.15455	+	3.95816i	0.161669	+	0.554253i
$52$	−3.05445			−0.423576
$53$	7.58067			1.04129			0.520643	−	0.853775i	$-0.325692\pi$
$53$	7.58067			1.04129			0.520643	+	0.853775i	$0.325692\pi$
$54$			1.74480i			0.237438i
$55$		0			0
$56$		−	8.30981i		−	1.11044i
$57$			1.52050i			0.201395i
$58$			3.17561i			0.416978i
$59$	13.7233			1.78662			0.893308	−	0.449445i	$-0.148378\pi$
$59$	13.7233			1.78662			0.893308	+	0.449445i	$0.148378\pi$
$60$		0			0
$61$		−	4.50092i		−	0.576284i	−0.957588	−	0.288142i	$-0.906962\pi$
$61$		−	4.50092i		−	0.576284i	0.957588	−	0.288142i	$-0.0930375\pi$
$62$			2.42765i			0.308312i
$63$		−	4.98357i		−	0.627871i
$64$	0.599071			0.0748839
$65$		0			0
$66$	3.90621			0.480821
$67$	1.03226			0.126111			0.0630556	−	0.998010i	$-0.479915\pi$
$67$	1.03226			0.126111			0.0630556	+	0.998010i	$0.479915\pi$
$68$	−4.13366	+	1.20574i	−0.501280	+	0.146218i
$69$	1.30964			0.157663
$70$		0			0
$71$		−	1.78218i		−	0.211506i	−0.994392	−	0.105753i	$-0.966275\pi$
$71$		−	1.78218i		−	0.211506i	0.994392	−	0.105753i	$-0.0337253\pi$
$72$	1.66744			0.196510
$73$			11.6402i			1.36238i	0.732107	+	0.681189i	$0.238537\pi$
$73$			11.6402i			1.36238i	−0.732107	+	0.681189i	$0.761463\pi$
$74$			6.18756i			0.719290i
$75$		0			0
$76$	−1.58792			−0.182147
$77$	−11.1570			−1.27146
$78$			5.10314i			0.577817i
$79$			13.9481i			1.56928i	0.619949	+	0.784642i	$0.287153\pi$
$79$			13.9481i			1.56928i	−0.619949	+	0.784642i	$0.712847\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		−	14.8055i		−	1.63500i
$83$	−4.47514			−0.491211			−0.245605	−	0.969370i	$-0.578987\pi$
$83$	−4.47514			−0.491211			−0.245605	+	0.969370i	$0.578987\pi$
$84$	5.20454			0.567861
$85$		0			0
$86$	15.9319			1.71798
$87$	1.82004			0.195128
$88$		−	3.73301i		−	0.397940i
$89$	14.6490			1.55280			0.776398	−	0.630243i	$-0.217045\pi$
$89$	14.6490			1.55280			0.776398	+	0.630243i	$0.217045\pi$
$90$		0			0
$91$		−	14.5758i		−	1.52796i
$92$			1.36771i			0.142594i
$93$	1.39136			0.144278
$94$	−0.290543			−0.0299672
$95$		0			0
$96$			5.38571i			0.549676i
$97$			14.7821i			1.50089i	0.660932	+	0.750445i	$0.270161\pi$
$97$			14.7821i			1.50089i	−0.660932	+	0.750445i	$0.729839\pi$
$98$	−31.1203			−3.14362
$99$		−	2.23877i		−	0.225004i
$100$		0			0
$101$	−9.91167			−0.986248			−0.493124	−	0.869959i	$-0.664145\pi$
$101$	−9.91167			−0.986248			−0.493124	+	0.869959i	$0.664145\pi$
$102$	2.01446	+	6.90621i	0.199462	+	0.683816i
$103$	−11.0014			−1.08400			−0.541999	−	0.840379i	$-0.682332\pi$
$103$	−11.0014			−1.08400			−0.541999	+	0.840379i	$0.682332\pi$
$104$	4.87687			0.478217
$105$		0			0
$106$	13.2268			1.28470
$107$		−	3.68292i		−	0.356041i	−0.984027	−	0.178021i	$-0.943031\pi$
$107$		−	3.68292i		−	0.356041i	0.984027	−	0.178021i	$-0.0569694\pi$
$108$			1.04434i			0.100492i
$109$		−	17.1471i		−	1.64239i	−0.570644	−	0.821197i	$-0.693307\pi$
$109$		−	17.1471i		−	1.64239i	0.570644	−	0.821197i	$-0.306693\pi$
$110$		0			0
$111$	3.54628			0.336598
$112$		−	24.9081i		−	2.35359i
$113$			16.1338i			1.51774i	0.651245	+	0.758868i	$0.274247\pi$
$113$			16.1338i			1.51774i	−0.651245	+	0.758868i	$0.725753\pi$
$114$			2.65298i			0.248474i
$115$		0			0
$116$			1.90074i			0.176479i
$117$	2.92477			0.270395
$118$	23.9444			2.20426
$119$	−5.75379	−	19.7258i	−0.527449	−	1.80826i
$120$		0			0
$121$	5.98793			0.544357
$122$		−	7.85322i		−	0.710997i
$123$	−8.48551			−0.765113
$124$			1.45305i			0.130488i
$125$		0			0
$126$		−	8.69535i		−	0.774643i
$127$	11.6766			1.03613			0.518065	−	0.855341i	$-0.326652\pi$
$127$	11.6766			1.03613			0.518065	+	0.855341i	$0.326652\pi$
$128$	11.8167			1.04446
$129$		−	9.13105i		−	0.803944i
$130$		0			0
$131$			8.14138i			0.711315i	0.934616	+	0.355658i	$0.115743\pi$
$131$			8.14138i			0.711315i	−0.934616	+	0.355658i	$0.884257\pi$
$132$	2.33803			0.203499
$133$		−	7.57753i		−	0.657055i
$134$	1.80110			0.155591
$135$		0			0
$136$	6.59999	−	1.92515i	0.565945	−	0.165080i
$137$	−3.47912			−0.297241			−0.148621	−	0.988894i	$-0.547483\pi$
$137$	−3.47912			−0.297241			−0.148621	+	0.988894i	$0.547483\pi$
$138$	2.28507			0.194518
$139$		−	9.56541i		−	0.811328i	−0.914022	−	0.405664i	$-0.867040\pi$
$139$		−	9.56541i		−	0.811328i	0.914022	−	0.405664i	$-0.132960\pi$
$140$		0			0
$141$			0.166519i			0.0140234i
$142$		−	3.10956i		−	0.260948i
$143$		−	6.54787i		−	0.547560i
$144$	4.99803			0.416503
$145$		0			0
$146$			20.3098i			1.68085i
$147$			17.8360i			1.47109i
$148$			3.70352i			0.304427i
$149$	−14.6107			−1.19696			−0.598478	−	0.801139i	$-0.704228\pi$
$149$	−14.6107			−1.19696			−0.598478	+	0.801139i	$0.704228\pi$
$150$		0			0
$151$	−13.6611			−1.11172			−0.555860	−	0.831276i	$-0.687611\pi$
$151$	−13.6611			−1.11172			−0.555860	+	0.831276i	$0.687611\pi$
$152$	2.53535			0.205644
$153$	3.95816	−	1.15455i	0.319998	−	0.0933399i
$154$	−19.4669			−1.56868
$155$		0			0
$156$			3.05445i			0.244552i
$157$	−21.1926			−1.69135			−0.845675	−	0.533698i	$-0.820802\pi$
$157$	−21.1926			−1.69135			−0.845675	+	0.533698i	$0.820802\pi$
$158$			24.3367i			1.93612i
$159$		−	7.58067i		−	0.601187i
$160$		0			0
$161$	−6.52670			−0.514376
$162$	1.74480			0.137085
$163$			4.86075i			0.380724i	0.981714	+	0.190362i	$0.0609661\pi$
$163$			4.86075i			0.380724i	−0.981714	+	0.190362i	$0.939034\pi$
$164$		−	8.86175i		−	0.691986i
$165$		0			0
$166$	−7.80824			−0.606037
$167$			21.1447i			1.63622i	0.575060	+	0.818111i	$0.304979\pi$
$167$			21.1447i			1.63622i	−0.575060	+	0.818111i	$0.695021\pi$
$168$	−8.30981			−0.641116
$169$	−4.44574			−0.341980
$170$		0			0
$171$	1.52050			0.116276
$172$	9.53591			0.727106
$173$		−	14.8666i		−	1.13029i	−0.824993	−	0.565143i	$-0.808821\pi$
$173$		−	14.8666i		−	1.13029i	0.824993	−	0.565143i	$-0.191179\pi$
$174$	3.17561			0.240742
$175$		0			0
$176$		−	11.1894i		−	0.843435i
$177$		−	13.7233i		−	1.03150i
$178$	25.5597			1.91578
$179$	24.1387			1.80421			0.902104	−	0.431519i	$-0.142022\pi$
$179$	24.1387			1.80421			0.902104	+	0.431519i	$0.142022\pi$
$180$		0			0
$181$		−	20.1402i		−	1.49701i	−0.663128	−	0.748506i	$-0.730771\pi$
$181$		−	20.1402i		−	1.49701i	0.663128	−	0.748506i	$-0.269229\pi$
$182$		−	25.4319i		−	1.88514i
$183$	−4.50092			−0.332718
$184$		−	2.18375i		−	0.160988i
$185$		0			0
$186$	2.42765			0.178004
$187$	−2.58477	−	8.86139i	−0.189017	−	0.648009i
$188$	−0.173902			−0.0126831
$189$	−4.98357			−0.362501
$190$		0			0
$191$	−16.0837			−1.16377			−0.581886	−	0.813270i	$-0.697685\pi$
$191$	−16.0837			−1.16377			−0.581886	+	0.813270i	$0.697685\pi$
$192$		−	0.599071i		−	0.0432342i
$193$			17.6995i			1.27404i	0.770849	+	0.637018i	$0.219832\pi$
$193$			17.6995i			1.27404i	−0.770849	+	0.637018i	$0.780168\pi$
$194$			25.7918i			1.85174i
$195$		0			0
$196$	−18.6268			−1.33049
$197$		−	3.66995i		−	0.261473i	−0.991417	−	0.130737i	$-0.958266\pi$
$197$		−	3.66995i		−	0.261473i	0.991417	−	0.130737i	$-0.0417342\pi$
$198$		−	3.90621i		−	0.277602i
$199$		−	18.6289i		−	1.32057i	−0.751015	−	0.660286i	$-0.770435\pi$
$199$		−	18.6289i		−	1.32057i	0.751015	−	0.660286i	$-0.229565\pi$
$200$		0			0
$201$		−	1.03226i		−	0.0728103i
$202$	−17.2939			−1.21680
$203$	−9.07028			−0.636609
$204$	1.20574	+	4.13366i	0.0844189	+	0.289414i
$205$		0			0
$206$	−19.1952			−1.33739
$207$		−	1.30964i		−	0.0910265i
$208$	14.6181			1.01358
$209$		−	3.40405i		−	0.235463i
$210$		0			0
$211$			5.66739i			0.390159i	0.980787	+	0.195079i	$0.0624965\pi$
$211$			5.66739i			0.390159i	−0.980787	+	0.195079i	$0.937504\pi$
$212$	7.91679			0.543728
$213$	−1.78218			−0.122113
$214$		−	6.42597i		−	0.439270i
$215$		0			0
$216$		−	1.66744i		−	0.113455i
$217$	−6.93395			−0.470707
$218$		−	29.9183i		−	2.02632i
$219$	11.6402			0.786570
$220$		0			0
$221$	11.5767	−	3.37679i	0.778732	−	0.227148i
$222$	6.18756			0.415282
$223$	−3.65155			−0.244526			−0.122263	−	0.992498i	$-0.539015\pi$
$223$	−3.65155			−0.244526			−0.122263	+	0.992498i	$0.539015\pi$
$224$		−	26.8400i		−	1.79333i
$225$		0			0
$226$			28.1502i			1.87253i
$227$			13.7100i			0.909967i	0.890500	+	0.454983i	$0.150355\pi$
$227$			13.7100i			0.909967i	−0.890500	+	0.454983i	$0.849645\pi$
$228$			1.58792i			0.105163i
$229$	−8.65352			−0.571841			−0.285920	−	0.958253i	$-0.592299\pi$
$229$	−8.65352			−0.571841			−0.285920	+	0.958253i	$0.592299\pi$
$230$		0			0
$231$			11.1570i			0.734080i
$232$		−	3.03480i		−	0.199245i
$233$		−	22.0694i		−	1.44581i	−0.690947	−	0.722906i	$-0.742806\pi$
$233$		−	22.0694i		−	1.44581i	0.690947	−	0.722906i	$-0.257194\pi$
$234$	5.10314			0.333603
$235$		0			0
$236$	14.3317			0.932916
$237$	13.9481			0.906027
$238$	−10.0392	−	34.4176i	−0.650746	−	2.23096i
$239$	7.58610			0.490704			0.245352	−	0.969434i	$-0.421096\pi$
$239$	7.58610			0.490704			0.245352	+	0.969434i	$0.421096\pi$
$240$		0			0
$241$			19.9526i			1.28526i	0.766178	+	0.642628i	$0.222156\pi$
$241$			19.9526i			1.28526i	−0.766178	+	0.642628i	$0.777844\pi$
$242$	10.4478			0.671607
$243$		−	1.00000i		−	0.0641500i
$244$		−	4.70049i		−	0.300918i
$245$		0			0
$246$	−14.8055			−0.943967
$247$	4.44711			0.282963
$248$		−	2.32001i		−	0.147321i
$249$			4.47514i			0.283601i
$250$		0			0
$251$	10.6807			0.674162			0.337081	−	0.941476i	$-0.390560\pi$
$251$	10.6807			0.674162			0.337081	+	0.941476i	$0.390560\pi$
$252$		−	5.20454i		−	0.327855i
$253$	−2.93199			−0.184332
$254$	20.3734			1.27834
$255$		0			0
$256$	19.4196			1.21373
$257$	5.59173			0.348803			0.174401	−	0.984675i	$-0.444201\pi$
$257$	5.59173			0.348803			0.174401	+	0.984675i	$0.444201\pi$
$258$		−	15.9319i		−	0.991876i
$259$	−17.6731			−1.09816
$260$		0			0
$261$		−	1.82004i		−	0.112657i
$262$			14.2051i			0.877594i
$263$	13.8769			0.855686			0.427843	−	0.903853i	$-0.359274\pi$
$263$	13.8769			0.855686			0.427843	+	0.903853i	$0.359274\pi$
$264$	−3.73301			−0.229751
$265$		0			0
$266$		−	13.2213i		−	0.810649i
$267$		−	14.6490i		−	0.896507i
$268$	1.07803			0.0658514
$269$			30.2465i			1.84416i	0.386997	+	0.922081i	$0.373512\pi$
$269$			30.2465i			1.84416i	−0.386997	+	0.922081i	$0.626488\pi$
$270$		0			0
$271$	−17.6456			−1.07190			−0.535948	−	0.844251i	$-0.680046\pi$
$271$	−17.6456			−1.07190			−0.535948	+	0.844251i	$0.680046\pi$
$272$	19.7830	−	5.77048i	1.19952	−	0.349887i
$273$	−14.5758			−0.882166
$274$	−6.07038			−0.366725
$275$		0			0
$276$	1.36771			0.0823266
$277$			16.5619i			0.995111i	0.867432	+	0.497556i	$0.165769\pi$
$277$			16.5619i			0.995111i	−0.867432	+	0.497556i	$0.834231\pi$
$278$		−	16.6898i		−	1.00099i
$279$		−	1.39136i		−	0.0832987i
$280$		0			0
$281$	8.40360			0.501317			0.250658	−	0.968076i	$-0.419353\pi$
$281$	8.40360			0.501317			0.250658	+	0.968076i	$0.419353\pi$
$282$			0.290543i			0.0173016i
$283$		−	16.2345i		−	0.965038i	−0.875885	−	0.482519i	$-0.839722\pi$
$283$		−	16.2345i		−	0.965038i	0.875885	−	0.482519i	$-0.160278\pi$
$284$		−	1.86120i		−	0.110442i
$285$		0			0
$286$		−	11.4247i		−	0.675559i
$287$	42.2881			2.49619
$288$	5.38571			0.317356
$289$	14.3340	−	9.13979i	0.843178	−	0.537635i
$290$		0			0
$291$	14.7821			0.866540
$292$			12.1563i			0.711393i
$293$	21.3036			1.24457			0.622284	−	0.782791i	$-0.286205\pi$
$293$	21.3036			1.24457			0.622284	+	0.782791i	$0.286205\pi$
$294$			31.1203i			1.81497i
$295$		0			0
$296$		−	5.91321i		−	0.343699i
$297$	−2.23877			−0.129906
$298$	−25.4928			−1.47676
$299$		−	3.83040i		−	0.221518i
$300$		0			0
$301$			45.5052i			2.62288i
$302$	−23.8358			−1.37160
$303$			9.91167i			0.569411i
$304$	7.59952			0.435862
$305$		0			0
$306$	6.90621	−	2.01446i	0.394802	−	0.115159i
$307$	−23.9979			−1.36963			−0.684817	−	0.728715i	$-0.740118\pi$
$307$	−23.9979			−1.36963			−0.684817	+	0.728715i	$0.740118\pi$
$308$	−11.6517			−0.663919
$309$			11.0014i			0.625846i
$310$		0			0
$311$			1.60781i			0.0911708i	0.998960	+	0.0455854i	$0.0145153\pi$
$311$			1.60781i			0.0911708i	−0.998960	+	0.0455854i	$0.985485\pi$
$312$		−	4.87687i		−	0.276099i
$313$			10.5247i			0.594893i	0.954738	+	0.297447i	$0.0961350\pi$
$313$			10.5247i			0.594893i	−0.954738	+	0.297447i	$0.903865\pi$
$314$	−36.9768			−2.08672
$315$		0			0
$316$			14.5665i			0.819432i
$317$			21.3111i			1.19695i	0.801142	+	0.598475i	$0.204226\pi$
$317$			21.3111i			1.19695i	−0.801142	+	0.598475i	$0.795774\pi$
$318$		−	13.2268i		−	0.741721i
$319$	−4.07464			−0.228136
$320$		0			0
$321$	−3.68292			−0.205560
$322$	−11.3878			−0.634618
$323$	6.01838	−	1.75550i	0.334872	−	0.0976784i
$324$	1.04434			0.0580188
$325$		0			0
$326$			8.48106i			0.469722i
$327$	−17.1471			−0.948237
$328$			14.1491i			0.781253i
$329$		−	0.829858i		−	0.0457515i
$330$		0			0
$331$	−0.393709			−0.0216402			−0.0108201	−	0.999941i	$-0.503444\pi$
$331$	−0.393709			−0.0216402			−0.0108201	+	0.999941i	$0.503444\pi$
$332$	−4.67357			−0.256495
$333$		−	3.54628i		−	0.194335i
$334$			36.8933i			2.01871i
$335$		0			0
$336$	−24.9081			−1.35885
$337$		−	3.90176i		−	0.212542i	−0.994337	−	0.106271i	$-0.966109\pi$
$337$		−	3.90176i		−	0.212542i	0.994337	−	0.106271i	$-0.0338911\pi$
$338$	−7.75695			−0.421922
$339$	16.1338			0.876265
$340$		0			0
$341$	−3.11494			−0.168683
$342$	2.65298			0.143457
$343$		−	54.0018i		−	2.91582i
$344$	−15.2255			−0.820903
$345$		0			0
$346$		−	25.9393i		−	1.39451i
$347$			3.96587i			0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$347$			3.96587i			0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$348$	1.90074			0.101890
$349$	32.5572			1.74275			0.871374	−	0.490620i	$-0.163230\pi$
$349$	32.5572			1.74275			0.871374	+	0.490620i	$0.163230\pi$
$350$		0			0
$351$		−	2.92477i		−	0.156112i
$352$		−	12.0573i		−	0.642658i
$353$	18.6270			0.991416			0.495708	−	0.868489i	$-0.334909\pi$
$353$	18.6270			0.991416			0.495708	+	0.868489i	$0.334909\pi$
$354$		−	23.9444i		−	1.27263i
$355$		0			0
$356$	15.2986			0.810822
$357$	−19.7258	+	5.75379i	−1.04400	+	0.304523i
$358$	42.1172			2.22596
$359$	−16.2989			−0.860221			−0.430111	−	0.902776i	$-0.641525\pi$
$359$	−16.2989			−0.860221			−0.430111	+	0.902776i	$0.641525\pi$
$360$		0			0
$361$	−16.6881			−0.878320
$362$		−	35.1408i		−	1.84696i
$363$		−	5.98793i		−	0.314285i
$364$		−	15.2221i		−	0.797852i
$365$		0			0
$366$	−7.85322			−0.410495
$367$		−	25.8123i		−	1.34739i	−0.739009	−	0.673696i	$-0.764706\pi$
$367$		−	25.8123i		−	1.34739i	0.739009	−	0.673696i	$-0.235294\pi$
$368$		−	6.54564i		−	0.341215i
$369$			8.48551i			0.441738i
$370$		0			0
$371$			37.7788i			1.96138i
$372$	1.45305			0.0753373
$373$	−2.29788			−0.118980			−0.0594900	−	0.998229i	$-0.518947\pi$
$373$	−2.29788			−0.118980			−0.0594900	+	0.998229i	$0.518947\pi$
$374$	−4.50991	−	15.4614i	−0.233202	−	0.799489i
$375$		0			0
$376$	0.277660			0.0143192
$377$		−	5.32318i		−	0.274158i
$378$	−8.69535			−0.447241
$379$			25.0126i			1.28481i	0.766365	+	0.642405i	$0.222063\pi$
$379$			25.0126i			1.28481i	−0.766365	+	0.642405i	$0.777937\pi$
$380$		0			0
$381$		−	11.6766i		−	0.598210i
$382$	−28.0628			−1.43582
$383$	8.85151			0.452291			0.226145	−	0.974094i	$-0.427388\pi$
$383$	8.85151			0.452291			0.226145	+	0.974094i	$0.427388\pi$
$384$		−	11.8167i		−	0.603017i
$385$		0			0
$386$			30.8821i			1.57186i
$387$	−9.13105			−0.464157
$388$			15.4375i			0.783719i
$389$	8.05651			0.408481			0.204241	−	0.978921i	$-0.434528\pi$
$389$	8.05651			0.408481			0.204241	+	0.978921i	$0.434528\pi$
$390$		0			0
$391$	−1.51205	−	5.18378i	−0.0764677	−	0.262155i
$392$	29.7404			1.50212
$393$	8.14138			0.410678
$394$		−	6.40334i		−	0.322596i
$395$		0			0
$396$		−	2.33803i		−	0.117490i
$397$			30.6558i			1.53857i	0.638904	+	0.769286i	$0.279388\pi$
$397$			30.6558i			1.53857i	−0.638904	+	0.769286i	$0.720612\pi$
$398$		−	32.5039i		−	1.62927i
$399$	−7.57753			−0.379351
$400$		0			0
$401$			23.1614i			1.15663i	0.815815	+	0.578313i	$0.196289\pi$
$401$			23.1614i			1.15663i	−0.815815	+	0.578313i	$0.803711\pi$
$402$		−	1.80110i		−	0.0898306i
$403$		−	4.06941i		−	0.202712i
$404$	−10.3511			−0.514989
$405$		0			0
$406$	−15.8259			−0.785424
$407$	−7.93929			−0.393536
$408$	−1.92515	−	6.59999i	−0.0953089	−	0.326748i
$409$	−10.9627			−0.542072			−0.271036	−	0.962569i	$-0.587366\pi$
$409$	−10.9627			−0.542072			−0.271036	+	0.962569i	$0.587366\pi$
$410$		0			0
$411$			3.47912i			0.171612i
$412$	−11.4892			−0.566030
$413$			68.3908i			3.36529i
$414$		−	2.28507i		−	0.112305i
$415$		0			0
$416$	15.7519			0.772302
$417$	−9.56541			−0.468420
$418$		−	5.93939i		−	0.290505i
$419$			14.7089i			0.718576i	0.933227	+	0.359288i	$0.116980\pi$
$419$			14.7089i			0.718576i	−0.933227	+	0.359288i	$0.883020\pi$
$420$		0			0
$421$	1.85001			0.0901638			0.0450819	−	0.998983i	$-0.485645\pi$
$421$	1.85001			0.0901638			0.0450819	+	0.998983i	$0.485645\pi$
$422$			9.88848i			0.481363i
$423$	0.166519			0.00809642
$424$	−12.6403			−0.613868
$425$		0			0
$426$	−3.10956			−0.150659
$427$	22.4307			1.08550
$428$		−	3.84622i		−	0.185914i
$429$	−6.54787			−0.316134
$430$		0			0
$431$		−	7.41116i		−	0.356983i	−0.983941	−	0.178491i	$-0.942878\pi$
$431$		−	7.41116i		−	0.356983i	0.983941	−	0.178491i	$-0.0571217\pi$
$432$		−	4.99803i		−	0.240468i
$433$	−5.01269			−0.240894			−0.120447	−	0.992720i	$-0.538433\pi$
$433$	−5.01269			−0.240894			−0.120447	+	0.992720i	$0.538433\pi$
$434$	−12.0984			−0.580741
$435$		0			0
$436$		−	17.9074i		−	0.857608i
$437$		−	1.99132i		−	0.0952575i
$438$	20.3098			0.970440
$439$		−	21.2169i		−	1.01263i	−0.862349	−	0.506315i	$-0.831007\pi$
$439$		−	21.2169i		−	1.01263i	0.862349	−	0.506315i	$-0.168993\pi$
$440$		0			0
$441$	17.8360			0.849332
$442$	20.1990	−	5.89184i	0.960770	−	0.280246i
$443$	22.8512			1.08569			0.542846	−	0.839832i	$-0.317347\pi$
$443$	22.8512			1.08569			0.542846	+	0.839832i	$0.317347\pi$
$444$	3.70352			0.175761
$445$		0			0
$446$	−6.37124			−0.301687
$447$			14.6107i			0.691063i
$448$			2.98551i			0.141052i
$449$			20.1999i			0.953294i	0.879095	+	0.476647i	$0.158148\pi$
$449$			20.1999i			0.953294i	−0.879095	+	0.476647i	$0.841852\pi$
$450$		0			0
$451$	18.9971			0.894537
$452$			16.8491i			0.792515i
$453$			13.6611i			0.641852i
$454$			23.9213i			1.12268i
$455$		0			0
$456$		−	2.53535i		−	0.118728i
$457$	27.4808			1.28550			0.642749	−	0.766077i	$-0.277794\pi$
$457$	27.4808			1.28550			0.642749	+	0.766077i	$0.277794\pi$
$458$	−15.0987			−0.705515
$459$	−1.15455	−	3.95816i	−0.0538898	−	0.184751i
$460$		0			0
$461$	27.5990			1.28541			0.642706	−	0.766113i	$-0.277812\pi$
$461$	27.5990			1.28541			0.642706	+	0.766113i	$0.277812\pi$
$462$			19.4669i			0.905680i
$463$	−20.5387			−0.954516			−0.477258	−	0.878763i	$-0.658369\pi$
$463$	−20.5387			−0.954516			−0.477258	+	0.878763i	$0.658369\pi$
$464$		−	9.09661i		−	0.422299i
$465$		0			0
$466$		−	38.5067i		−	1.78379i
$467$	−12.6544			−0.585576			−0.292788	−	0.956177i	$-0.594583\pi$
$467$	−12.6544			−0.585576			−0.292788	+	0.956177i	$0.594583\pi$
$468$	3.05445			0.141192
$469$			5.14436i			0.237545i
$470$		0			0
$471$			21.1926i			0.976501i
$472$	−22.8827			−1.05326
$473$			20.4423i			0.939937i
$474$	24.3367			1.11782
$475$		0			0
$476$	−6.00890	−	20.6004i	−0.275418	−	0.944217i
$477$	−7.58067			−0.347095
$478$	13.2363			0.605412
$479$		−	25.6921i		−	1.17390i	−0.809622	−	0.586952i	$-0.800328\pi$
$479$		−	25.6921i		−	1.17390i	0.809622	−	0.586952i	$-0.199672\pi$
$480$		0			0
$481$		−	10.3720i		−	0.472924i
$482$			34.8133i			1.58570i
$483$			6.52670i			0.296975i
$484$	6.25342			0.284247
$485$		0			0
$486$		−	1.74480i		−	0.0791459i
$487$		−	7.98736i		−	0.361942i	−0.983488	−	0.180971i	$-0.942076\pi$
$487$		−	7.98736i		−	0.361942i	0.983488	−	0.180971i	$-0.0579240\pi$
$488$			7.50502i			0.339736i
$489$	4.86075			0.219811
$490$		0			0
$491$	23.4535			1.05844			0.529221	−	0.848484i	$-0.322484\pi$
$491$	23.4535			1.05844			0.529221	+	0.848484i	$0.322484\pi$
$492$	−8.86175			−0.399519
$493$	−2.10133	−	7.20399i	−0.0946389	−	0.324452i
$494$	7.75934			0.349109
$495$		0			0
$496$		−	6.95408i		−	0.312247i
$497$	8.88163			0.398396
$498$			7.80824i			0.349896i
$499$		−	2.55509i		−	0.114381i	−0.998363	−	0.0571907i	$-0.981786\pi$
$499$		−	2.55509i		−	0.114381i	0.998363	−	0.0571907i	$-0.0182143\pi$
$500$		0			0
$501$	21.1447			0.944674
$502$	18.6358			0.831756
$503$		−	22.5894i		−	1.00721i	−0.863934	−	0.503605i	$-0.832007\pi$
$503$		−	22.5894i		−	1.00721i	0.863934	−	0.503605i	$-0.167993\pi$
$504$			8.30981i			0.370148i
$505$		0			0
$506$	−5.11574			−0.227422
$507$			4.44574i			0.197442i
$508$	12.1943			0.541036
$509$	12.4294			0.550921			0.275461	−	0.961312i	$-0.411170\pi$
$509$	12.4294			0.550921			0.275461	+	0.961312i	$0.411170\pi$
$510$		0			0
$511$	−58.0096			−2.56619
$512$	10.2501			0.452994
$513$		−	1.52050i		−	0.0671318i
$514$	9.75647			0.430340
$515$		0			0
$516$		−	9.53591i		−	0.419795i
$517$		−	0.372797i		−	0.0163956i
$518$	−30.8362			−1.35486
$519$	−14.8666			−0.652571
$520$		0			0
$521$		−	15.6804i		−	0.686969i	−0.939158	−	0.343485i	$-0.888393\pi$
$521$		−	15.6804i		−	0.686969i	0.939158	−	0.343485i	$-0.111607\pi$
$522$		−	3.17561i		−	0.138993i
$523$	9.60165			0.419851			0.209925	−	0.977717i	$-0.432678\pi$
$523$	9.60165			0.419851			0.209925	+	0.977717i	$0.432678\pi$
$524$			8.50236i			0.371427i
$525$		0			0
$526$	24.2125			1.05571
$527$	−1.60640	−	5.50723i	−0.0699758	−	0.239899i
$528$	−11.1894			−0.486957
$529$	21.2848			0.925428
$530$		0			0
$531$	−13.7233			−0.595539
$532$		−	7.91350i		−	0.343094i
$533$			24.8181i			1.07499i
$534$		−	25.5597i		−	1.10608i
$535$		0			0
$536$	−1.72124			−0.0743462
$537$		−	24.1387i		−	1.04166i
$538$			52.7742i			2.27526i
$539$		−	39.9306i		−	1.71993i
$540$		0			0
$541$			32.9544i			1.41682i	0.705801	+	0.708410i	$0.250587\pi$
$541$			32.9544i			1.41682i	−0.705801	+	0.708410i	$0.749413\pi$
$542$	−30.7882			−1.32247
$543$	−20.1402			−0.864300
$544$	21.3175	−	6.21807i	0.913979	−	0.266598i
$545$		0			0
$546$	−25.4319			−1.08838
$547$		−	18.9841i		−	0.811701i	−0.913940	−	0.405850i	$-0.866975\pi$
$547$		−	18.9841i		−	0.811701i	0.913940	−	0.405850i	$-0.133025\pi$
$548$	−3.63338			−0.155210
$549$			4.50092i			0.192095i
$550$		0			0
$551$		−	2.76737i		−	0.117894i
$552$	−2.18375			−0.0929467
$553$	−69.5113			−2.95592
$554$			28.8973i			1.22773i
$555$		0			0
$556$		−	9.98953i		−	0.423651i
$557$	44.5929			1.88946			0.944731	−	0.327847i	$-0.106323\pi$
$557$	44.5929			1.88946			0.944731	+	0.327847i	$0.106323\pi$
$558$		−	2.42765i		−	0.102771i
$559$	−26.7062			−1.12955
$560$		0			0
$561$	−8.86139	+	2.58477i	−0.374128	+	0.109129i
$562$	14.6626			0.618506
$563$	−4.26474			−0.179737			−0.0898686	−	0.995954i	$-0.528645\pi$
$563$	−4.26474			−0.179737			−0.0898686	+	0.995954i	$0.528645\pi$
$564$			0.173902i			0.00732260i
$565$		0			0
$566$		−	28.3259i		−	1.19063i
$567$			4.98357i			0.209290i
$568$			2.97168i			0.124689i
$569$	5.81934			0.243959			0.121980	−	0.992533i	$-0.461076\pi$
$569$	5.81934			0.243959			0.121980	+	0.992533i	$0.461076\pi$
$570$		0			0
$571$			10.0595i			0.420976i	0.977596	+	0.210488i	$0.0675053\pi$
$571$			10.0595i			0.420976i	−0.977596	+	0.210488i	$0.932495\pi$
$572$		−	6.83819i		−	0.285919i
$573$			16.0837i			0.671905i
$574$	73.7845			3.07971
$575$		0			0
$576$	−0.599071			−0.0249613
$577$	−15.7300			−0.654847			−0.327423	−	0.944878i	$-0.606180\pi$
$577$	−15.7300			−0.654847			−0.327423	+	0.944878i	$0.606180\pi$
$578$	25.0101	−	15.9471i	1.04028	−	0.663313i
$579$	17.6995			0.735566
$580$		0			0
$581$		−	22.3022i		−	0.925250i
$582$	25.7918			1.06910
$583$			16.9714i			0.702882i
$584$		−	19.4093i		−	0.803162i
$585$		0			0
$586$	37.1706			1.53550
$587$	−30.6657			−1.26571			−0.632854	−	0.774271i	$-0.718117\pi$
$587$	−30.6657			−1.26571			−0.632854	+	0.774271i	$0.718117\pi$
$588$			18.6268i			0.768156i
$589$		−	2.11557i		−	0.0871705i
$590$		0			0
$591$	−3.66995			−0.150962
$592$		−	17.7244i		−	0.728470i
$593$	−9.22665			−0.378893			−0.189447	−	0.981891i	$-0.560669\pi$
$593$	−9.22665			−0.378893			−0.189447	+	0.981891i	$0.560669\pi$
$594$	−3.90621			−0.160274
$595$		0			0
$596$	−15.2585			−0.625014
$597$	−18.6289			−0.762432
$598$		−	6.68330i		−	0.273300i
$599$	12.2310			0.499745			0.249873	−	0.968279i	$-0.419611\pi$
$599$	12.2310			0.499745			0.249873	+	0.968279i	$0.419611\pi$
$600$		0			0
$601$		−	36.4552i		−	1.48704i	−0.668715	−	0.743519i	$-0.733155\pi$
$601$		−	36.4552i		−	1.48704i	0.668715	−	0.743519i	$-0.266845\pi$
$602$			79.3977i			3.23601i
$603$	−1.03226			−0.0420371
$604$	−14.2668			−0.580507
$605$		0			0
$606$			17.2939i			0.702517i
$607$		−	35.4434i		−	1.43860i	−0.694697	−	0.719302i	$-0.744462\pi$
$607$		−	35.4434i		−	1.43860i	0.694697	−	0.719302i	$-0.255538\pi$
$608$	8.18897			0.332107
$609$			9.07028i			0.367546i
$610$		0			0
$611$	0.487029			0.0197031
$612$	4.13366	−	1.20574i	0.167093	−	0.0487393i
$613$	−31.2013			−1.26021			−0.630104	−	0.776511i	$-0.716988\pi$
$613$	−31.2013			−1.26021			−0.630104	+	0.776511i	$0.716988\pi$
$614$	−41.8716			−1.68980
$615$		0			0
$616$	18.6037			0.749565
$617$		−	19.2867i		−	0.776452i	−0.921564	−	0.388226i	$-0.873088\pi$
$617$		−	19.2867i		−	0.776452i	0.921564	−	0.388226i	$-0.126912\pi$
$618$			19.1952i			0.772145i
$619$		−	5.51660i		−	0.221731i	−0.993835	−	0.110865i	$-0.964638\pi$
$619$		−	5.51660i		−	0.221731i	0.993835	−	0.110865i	$-0.0353622\pi$
$620$		0			0
$621$	−1.30964			−0.0525542
$622$			2.80532i			0.112483i
$623$			73.0045i			2.92486i
$624$		−	14.6181i		−	0.585192i
$625$		0			0
$626$			18.3636i			0.733957i
$627$	−3.40405			−0.135945
$628$	−22.1322			−0.883171
$629$	−4.09436	−	14.0367i	−0.163253	−	0.559681i
$630$		0			0
$631$	−2.66900			−0.106251			−0.0531257	−	0.998588i	$-0.516918\pi$
$631$	−2.66900			−0.106251			−0.0531257	+	0.998588i	$0.516918\pi$
$632$		−	23.2576i		−	0.925139i
$633$	5.66739			0.225258
$634$			37.1837i			1.47675i
$635$		0			0
$636$		−	7.91679i		−	0.313921i
$637$	52.1660			2.06689
$638$	−7.10944			−0.281465
$639$			1.78218i			0.0705021i
$640$		0			0
$641$		−	22.4010i		−	0.884787i	−0.896821	−	0.442393i	$-0.854129\pi$
$641$		−	22.4010i		−	0.884787i	0.896821	−	0.442393i	$-0.145871\pi$
$642$	−6.42597			−0.253613
$643$		−	9.24620i		−	0.364635i	−0.983240	−	0.182317i	$-0.941640\pi$
$643$		−	9.24620i		−	0.364635i	0.983240	−	0.182317i	$-0.0583598\pi$
$644$	−6.81609			−0.268592
$645$		0			0
$646$	10.5009	−	3.06300i	0.413152	−	0.120512i
$647$	34.6465			1.36209			0.681047	−	0.732239i	$-0.261525\pi$
$647$	34.6465			1.36209			0.681047	+	0.732239i	$0.261525\pi$
$648$	−1.66744			−0.0655033
$649$			30.7232i			1.20599i
$650$		0			0
$651$			6.93395i			0.271763i
$652$			5.07627i			0.198802i
$653$		−	0.981384i		−	0.0384045i	−0.999816	−	0.0192023i	$-0.993887\pi$
$653$		−	0.981384i		−	0.0384045i	0.999816	−	0.0192023i	$-0.00611265\pi$
$654$	−29.9183			−1.16990
$655$		0			0
$656$			42.4109i			1.65587i
$657$		−	11.6402i		−	0.454126i
$658$		−	1.44794i		−	0.0564465i
$659$	−18.3247			−0.713828			−0.356914	−	0.934137i	$-0.616171\pi$
$659$	−18.3247			−0.713828			−0.356914	+	0.934137i	$0.616171\pi$
$660$		0			0
$661$	−22.4177			−0.871946			−0.435973	−	0.899960i	$-0.643596\pi$
$661$	−22.4177			−0.871946			−0.435973	+	0.899960i	$0.643596\pi$
$662$	−0.686944			−0.0266989
$663$	−3.37679	−	11.5767i	−0.131144	−	0.449601i
$664$	7.46203			0.289583
$665$		0			0
$666$		−	6.18756i		−	0.239763i
$667$	−2.38360			−0.0922934
$668$			22.0822i			0.854386i
$669$			3.65155i			0.141177i
$670$		0			0
$671$	10.0765			0.388999
$672$	−26.8400			−1.03538
$673$			50.8202i			1.95898i	0.201504	+	0.979488i	$0.435417\pi$
$673$			50.8202i			1.95898i	−0.201504	+	0.979488i	$0.564583\pi$
$674$		−	6.80780i		−	0.262227i
$675$		0			0
$676$	−4.64286			−0.178572
$677$		−	5.48996i		−	0.210996i	−0.994419	−	0.105498i	$-0.966356\pi$
$677$		−	5.48996i		−	0.210996i	0.994419	−	0.105498i	$-0.0336437\pi$
$678$	28.1502			1.08110
$679$	−73.6674			−2.82710
$680$		0			0
$681$	13.7100			0.525370
$682$	−5.43495			−0.208115
$683$		−	23.8590i		−	0.912940i	−0.889739	−	0.456470i	$-0.849114\pi$
$683$		−	23.8590i		−	0.912940i	0.889739	−	0.456470i	$-0.150886\pi$
$684$	1.58792			0.0607156
$685$		0			0
$686$		−	94.2225i		−	3.59743i
$687$			8.65352i			0.330152i
$688$	−45.6373			−1.73991
$689$	−22.1717			−0.844675
$690$		0			0
$691$			11.9837i			0.455882i	0.973675	+	0.227941i	$0.0731993\pi$
$691$			11.9837i			0.455882i	−0.973675	+	0.227941i	$0.926801\pi$
$692$		−	15.5258i		−	0.590201i
$693$	11.1570			0.423821
$694$			6.91966i			0.262667i
$695$		0			0
$696$	−3.03480			−0.115034
$697$	9.79696	+	33.5870i	0.371086	+	1.27220i
$698$	56.8059			2.15014
$699$	−22.0694			−0.834740
$700$		0			0
$701$	−21.9970			−0.830816			−0.415408	−	0.909635i	$-0.636361\pi$
$701$	−21.9970			−0.830816			−0.415408	+	0.909635i	$0.636361\pi$
$702$		−	5.10314i		−	0.192606i
$703$		−	5.39213i		−	0.203368i
$704$			1.34118i			0.0505476i
$705$		0			0
$706$	32.5005			1.22317
$707$		−	49.3955i		−	1.85771i
$708$		−	14.3317i		−	0.538619i
$709$		−	13.6855i		−	0.513969i	−0.966416	−	0.256984i	$-0.917271\pi$
$709$		−	13.6855i		−	0.513969i	0.966416	−	0.256984i	$-0.0827289\pi$
$710$		0			0
$711$		−	13.9481i		−	0.523095i
$712$	−24.4264			−0.915419
$713$	−1.82219			−0.0682415
$714$	−34.4176	+	10.0392i	−1.28804	+	0.375709i
$715$		0			0
$716$	25.2089			0.942102
$717$		−	7.58610i		−	0.283308i
$718$	−28.4383			−1.06131
$719$			22.9414i			0.855568i	0.903881	+	0.427784i	$0.140706\pi$
$719$			22.9414i			0.855568i	−0.903881	+	0.427784i	$0.859294\pi$
$720$		0			0
$721$		−	54.8261i		−	2.04183i
$722$	−29.1174			−1.08364
$723$	19.9526			0.742043
$724$		−	21.0332i		−	0.781694i
$725$		0			0
$726$		−	10.4478i		−	0.387752i
$727$	30.6381			1.13630			0.568152	−	0.822923i	$-0.307658\pi$
$727$	30.6381			1.13630			0.568152	+	0.822923i	$0.307658\pi$
$728$			24.3042i			0.900775i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−36.1421	+	10.5423i	−1.33677	+	0.389920i
$732$	−4.70049			−0.173735
$733$	−30.3471			−1.12090			−0.560449	−	0.828189i	$-0.689371\pi$
$733$	−30.3471			−1.12090			−0.560449	+	0.828189i	$0.689371\pi$
$734$		−	45.0374i		−	1.66236i
$735$		0			0
$736$		−	7.05336i		−	0.259990i
$737$			2.31100i			0.0851267i
$738$			14.8055i			0.545000i
$739$	17.7291			0.652174			0.326087	−	0.945340i	$-0.394270\pi$
$739$	17.7291			0.652174			0.326087	+	0.945340i	$0.394270\pi$
$740$		0			0
$741$		−	4.44711i		−	0.163369i
$742$			65.9166i			2.41988i
$743$		−	11.0856i		−	0.406693i	−0.979107	−	0.203346i	$-0.934818\pi$
$743$		−	11.0856i		−	0.406693i	0.979107	−	0.203346i	$-0.0651818\pi$
$744$	−2.32001			−0.0850559
$745$		0			0
$746$	−4.00936			−0.146793
$747$	4.47514			0.163737
$748$	−2.69938	−	9.25429i	−0.0986989	−	0.338371i
$749$	18.3541			0.670644
$750$		0			0
$751$			10.3719i			0.378474i	0.981931	+	0.189237i	$0.0606015\pi$
$751$			10.3719i			0.378474i	−0.981931	+	0.189237i	$0.939399\pi$
$752$	0.832267			0.0303496
$753$		−	10.6807i		−	0.389228i
$754$		−	9.28791i		−	0.338246i
$755$		0			0
$756$	−5.20454			−0.189287
$757$	5.08248			0.184726			0.0923629	−	0.995725i	$-0.470558\pi$
$757$	5.08248			0.184726			0.0923629	+	0.995725i	$0.470558\pi$
$758$			43.6421i			1.58515i
$759$			2.93199i			0.106424i
$760$		0			0
$761$	25.5981			0.927931			0.463965	−	0.885853i	$-0.346426\pi$
$761$	25.5981			0.927931			0.463965	+	0.885853i	$0.346426\pi$
$762$		−	20.3734i		−	0.738049i
$763$	85.4538			3.09363
$764$	−16.7968			−0.607687
$765$		0			0
$766$	15.4441			0.558019
$767$	−40.1373			−1.44927
$768$		−	19.4196i		−	0.700745i
$769$	15.0057			0.541121			0.270560	−	0.962703i	$-0.412791\pi$
$769$	15.0057			0.541121			0.270560	+	0.962703i	$0.412791\pi$
$770$		0			0
$771$		−	5.59173i		−	0.201381i
$772$			18.4843i			0.665263i
$773$	−39.0295			−1.40379			−0.701897	−	0.712278i	$-0.747664\pi$
$773$	−39.0295			−1.40379			−0.701897	+	0.712278i	$0.747664\pi$
$774$	−15.9319			−0.572660
$775$		0			0
$776$		−	24.6482i		−	0.884819i
$777$			17.6731i			0.634020i
$778$	14.0570			0.503969
$779$			12.9022i			0.462271i
$780$		0			0
$781$	3.98989			0.142770
$782$	−2.63823	−	9.04467i	−0.0943429	−	0.323437i
$783$	−1.82004			−0.0650428
$784$	89.1448			3.18374
$785$		0			0
$786$	14.2051			0.506679
$787$		−	21.7701i		−	0.776020i	−0.921655	−	0.388010i	$-0.873163\pi$
$787$		−	21.7701i		−	0.776020i	0.921655	−	0.388010i	$-0.126837\pi$
$788$		−	3.83267i		−	0.136533i
$789$		−	13.8769i		−	0.494031i
$790$		0			0
$791$	−80.4037			−2.85883
$792$			3.73301i			0.132647i
$793$			13.1641i			0.467472i
$794$			53.4884i			1.89823i
$795$		0			0
$796$		−	19.4549i		−	0.689562i
$797$	−40.0932			−1.42017			−0.710087	−	0.704114i	$-0.751344\pi$
$797$	−40.0932			−1.42017			−0.710087	+	0.704114i	$0.751344\pi$
$798$	−13.2213			−0.468029
$799$	0.659108	−	0.192254i	0.0233175	−	0.00680147i
$800$		0			0
$801$	−14.6490			−0.517599
$802$			40.4122i			1.42700i
$803$	−26.0596			−0.919624
$804$		−	1.07803i		−	0.0380193i
$805$		0			0
$806$		−	7.10032i		−	0.250098i
$807$	30.2465			1.06473
$808$	16.5271			0.581422
$809$			33.2747i			1.16988i	0.811078	+	0.584938i	$0.198881\pi$
$809$			33.2747i			1.16988i	−0.811078	+	0.584938i	$0.801119\pi$
$810$		0			0
$811$			1.18314i			0.0415456i	0.999784	+	0.0207728i	$0.00661266\pi$
$811$			1.18314i			0.0415456i	−0.999784	+	0.0207728i	$0.993387\pi$
$812$	−9.47245			−0.332418
$813$			17.6456i			0.618860i
$814$	−13.8525			−0.485530
$815$		0			0
$816$	−5.77048	−	19.7830i	−0.202007	−	0.692544i
$817$	−13.8838			−0.485732
$818$	−19.1278			−0.668788
$819$			14.5758i			0.509319i
$820$		0			0
$821$			50.4742i			1.76156i	0.473525	+	0.880780i	$0.342981\pi$
$821$			50.4742i			1.76156i	−0.473525	+	0.880780i	$0.657019\pi$
$822$			6.07038i			0.211729i
$823$		−	19.7606i		−	0.688810i	−0.938821	−	0.344405i	$-0.888081\pi$
$823$		−	19.7606i		−	0.688810i	0.938821	−	0.344405i	$-0.111919\pi$
$824$	18.3441			0.639048
$825$		0			0
$826$			119.329i			4.15197i
$827$		−	50.1878i		−	1.74520i	−0.488434	−	0.872601i	$-0.662432\pi$
$827$		−	50.1878i		−	1.74520i	0.488434	−	0.872601i	$-0.337568\pi$
$828$		−	1.36771i		−	0.0475313i
$829$	43.5230			1.51162			0.755809	−	0.654793i	$-0.227244\pi$
$829$	43.5230			1.51162			0.755809	+	0.654793i	$0.227244\pi$
$830$		0			0
$831$	16.5619			0.574528
$832$	−1.75214			−0.0607446
$833$	70.5976	−	20.5925i	2.44606	−	0.713489i
$834$	−16.6898			−0.577919
$835$		0			0
$836$		−	3.55498i		−	0.122952i
$837$	−1.39136			−0.0480925
$838$			25.6641i			0.886552i
$839$			36.9007i			1.27395i	0.770884	+	0.636976i	$0.219815\pi$
$839$			36.9007i			1.27395i	−0.770884	+	0.636976i	$0.780185\pi$
$840$		0			0
$841$	25.6875			0.885775
$842$	3.22790			0.111241
$843$		−	8.40360i		−	0.289435i
$844$			5.91867i			0.203729i
$845$		0			0
$846$	0.290543			0.00998906
$847$			29.8412i			1.02536i
$848$	−37.8885			−1.30110
$849$	−16.2345			−0.557165
$850$		0			0
$851$	−4.64437			−0.159207
$852$	−1.86120			−0.0637638
$853$		−	29.8013i		−	1.02038i	−0.860062	−	0.510189i	$-0.829575\pi$
$853$		−	29.8013i		−	1.02038i	0.860062	−	0.510189i	$-0.170425\pi$
$854$	39.1371			1.33924
$855$		0			0
$856$			6.14105i			0.209897i
$857$			5.32427i			0.181874i	0.995857	+	0.0909368i	$0.0289861\pi$
$857$			5.32427i			0.181874i	−0.995857	+	0.0909368i	$0.971014\pi$
$858$	−11.4247			−0.390034
$859$	24.3062			0.829316			0.414658	−	0.909977i	$-0.363901\pi$
$859$	24.3062			0.829316			0.414658	+	0.909977i	$0.363901\pi$
$860$		0			0
$861$		−	42.2881i		−	1.44118i
$862$		−	12.9310i		−	0.440432i
$863$	25.1361			0.855642			0.427821	−	0.903863i	$-0.359281\pi$
$863$	25.1361			0.855642			0.427821	+	0.903863i	$0.359281\pi$
$864$		−	5.38571i		−	0.183225i
$865$		0			0
$866$	−8.74615			−0.297206
$867$	−9.13979	−	14.3340i	−0.310404	−	0.486809i
$868$	−7.24140			−0.245789
$869$	−31.2265			−1.05929
$870$		0			0
$871$	−3.01913			−0.102299
$872$			28.5918i			0.968240i
$873$		−	14.7821i		−	0.500297i
$874$		−	3.47445i		−	0.117525i
$875$		0			0
$876$	12.1563			0.410723
$877$			25.5005i			0.861090i	0.902569	+	0.430545i	$0.141679\pi$
$877$			25.5005i			0.861090i	−0.902569	+	0.430545i	$0.858321\pi$
$878$		−	37.0194i		−	1.24934i
$879$		−	21.3036i		−	0.718552i
$880$		0			0
$881$		−	46.7352i		−	1.57455i	−0.616603	−	0.787274i	$-0.711492\pi$
$881$		−	46.7352i		−	1.57455i	0.616603	−	0.787274i	$-0.288508\pi$
$882$	31.1203			1.04787
$883$	−11.2968			−0.380166			−0.190083	−	0.981768i	$-0.560876\pi$
$883$	−11.2968			−0.380166			−0.190083	+	0.981768i	$0.560876\pi$
$884$	12.0900	−	3.52651i	0.406630	−	0.118610i
$885$		0			0
$886$	39.8708			1.33949
$887$		−	1.07297i		−	0.0360266i	−0.999838	−	0.0180133i	$-0.994266\pi$
$887$		−	1.07297i		−	0.0360266i	0.999838	−	0.0180133i	$-0.00573413\pi$
$888$	−5.91321			−0.198434
$889$			58.1911i			1.95167i
$890$		0			0
$891$			2.23877i			0.0750015i
$892$	−3.81346			−0.127684
$893$	0.253192			0.00847275
$894$			25.4928i			0.852607i
$895$		0			0
$896$			58.8892i			1.96735i
$897$	−3.83040			−0.127893
$898$			35.2449i			1.17614i
$899$	−2.53233			−0.0844580
$900$		0			0
$901$	−30.0055	+	8.75228i	−0.999628	+	0.291581i
$902$	33.1462			1.10365
$903$	45.5052			1.51432
$904$		−	26.9021i		−	0.894750i
$905$		0			0
$906$			23.8358i			0.791893i
$907$			46.6720i			1.54972i	0.632133	+	0.774860i	$0.282180\pi$
$907$			46.6720i			1.54972i	−0.632133	+	0.774860i	$0.717820\pi$
$908$			14.3179i			0.475157i
$909$	9.91167			0.328749
$910$		0			0
$911$			8.14382i			0.269817i	0.990858	+	0.134908i	$0.0430740\pi$
$911$			8.14382i			0.269817i	−0.990858	+	0.134908i	$0.956926\pi$
$912$		−	7.59952i		−	0.251645i
$913$		−	10.0188i		−	0.331574i
$914$	47.9486			1.58600
$915$		0			0
$916$	−9.03721			−0.298598
$917$	−40.5731			−1.33984
$918$	−2.01446	−	6.90621i	−0.0664872	−	0.227939i
$919$	3.96126			0.130670			0.0653350	−	0.997863i	$-0.479188\pi$
$919$	3.96126			0.130670			0.0653350	+	0.997863i	$0.479188\pi$
$920$		0			0
$921$			23.9979i			0.790758i
$922$	48.1548			1.58589
$923$			5.21247i			0.171571i
$924$			11.6517i			0.383314i
$925$		0			0
$926$	−35.8360			−1.17765
$927$	11.0014			0.361332
$928$		−	9.80218i		−	0.321772i
$929$			32.9616i			1.08143i	0.841205	+	0.540717i	$0.181847\pi$
$929$			32.9616i			1.08143i	−0.841205	+	0.540717i	$0.818153\pi$
$930$		0			0
$931$	27.1196			0.888809
$932$		−	23.0479i		−	0.754959i
$933$	1.60781			0.0526375
$934$	−22.0795			−0.722462
$935$		0			0
$936$	−4.87687			−0.159406
$937$	49.6837			1.62310			0.811548	−	0.584285i	$-0.198625\pi$
$937$	49.6837			1.62310			0.811548	+	0.584285i	$0.198625\pi$
$938$			8.97590i			0.293074i
$939$	10.5247			0.343462
$940$		0			0
$941$			28.8461i			0.940356i	0.882572	+	0.470178i	$0.155810\pi$
$941$			28.8461i			0.940356i	−0.882572	+	0.470178i	$0.844190\pi$
$942$			36.9768i			1.20477i
$943$	11.1130			0.361889
$944$	−68.5893			−2.23239
$945$		0			0
$946$			35.6678i			1.15966i
$947$			49.1958i			1.59865i	0.600900	+	0.799324i	$0.294809\pi$
$947$			49.1958i			1.59865i	−0.600900	+	0.799324i	$0.705191\pi$
$948$	14.5665			0.473099
$949$		−	34.0448i		−	1.10514i
$950$		0			0
$951$	21.3111			0.691059
$952$	9.59410	+	32.8915i	0.310946	+	1.06602i
$953$	−44.1137			−1.42898			−0.714492	−	0.699644i	$-0.753342\pi$
$953$	−44.1137			−1.42898			−0.714492	+	0.699644i	$0.753342\pi$
$954$	−13.2268			−0.428233
$955$		0			0
$956$	7.92246			0.256231
$957$			4.07464i			0.131714i
$958$		−	44.8277i		−	1.44832i
$959$		−	17.3384i		−	0.559887i
$960$		0			0
$961$	29.0641			0.937552
$962$		−	18.0972i		−	0.583476i
$963$			3.68292i			0.118680i
$964$			20.8372i			0.671122i
$965$		0			0
$966$			11.3878i			0.366397i
$967$	−49.7270			−1.59911			−0.799557	−	0.600591i	$-0.794932\pi$
$967$	−49.7270			−1.59911			−0.799557	+	0.600591i	$0.794932\pi$
$968$	−9.98451			−0.320914
$969$	−1.75550	−	6.01838i	−0.0563947	−	0.193338i
$970$		0			0
$971$	37.0067			1.18760			0.593801	−	0.804612i	$-0.297627\pi$
$971$	37.0067			1.18760			0.593801	+	0.804612i	$0.297627\pi$
$972$		−	1.04434i		−	0.0334972i
$973$	47.6699			1.52823
$974$		−	13.9364i		−	0.446550i
$975$		0			0
$976$			22.4958i			0.720072i
$977$	−28.1018			−0.899056			−0.449528	−	0.893266i	$-0.648408\pi$
$977$	−28.1018			−0.899056			−0.449528	+	0.893266i	$0.648408\pi$
$978$	8.48106			0.271194
$979$			32.7958i			1.04816i
$980$		0			0
$981$			17.1471i			0.547465i
$982$	40.9218			1.30587
$983$		−	57.8272i		−	1.84440i	−0.386714	−	0.922200i	$-0.626390\pi$
$983$		−	57.8272i		−	1.84440i	0.386714	−	0.922200i	$-0.373610\pi$
$984$	14.1491			0.451056
$985$		0			0
$986$	−3.66640	−	12.5696i	−0.116762	−	0.400296i
$987$	−0.829858			−0.0264147
$988$	4.64429			0.147755
$989$			11.9584i			0.380256i
$990$		0			0
$991$		−	24.8426i		−	0.789151i	−0.918863	−	0.394576i	$-0.870892\pi$
$991$		−	24.8426i		−	0.789151i	0.918863	−	0.394576i	$-0.129108\pi$
$992$		−	7.49347i		−	0.237918i
$993$			0.393709i			0.0124940i
$994$	15.4967			0.491526
$995$		0			0
$996$			4.67357i			0.148088i
$997$			15.3101i			0.484877i	0.970167	+	0.242438i	$0.0779472\pi$
$997$			15.3101i			0.484877i	−0.970167	+	0.242438i	$0.922053\pi$
$998$		−	4.45813i		−	0.141119i
$999$	−3.54628			−0.112199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.g.e.526.9	✓	12
5.2	odd	4		1275.2.d.i.424.10		12
5.3	odd	4		1275.2.d.j.424.3		12
5.4	even	2		1275.2.g.f.526.4	yes	12
17.16	even	2	inner	1275.2.g.e.526.10	yes	12
85.33	odd	4		1275.2.d.i.424.3		12
85.67	odd	4		1275.2.d.j.424.10		12
85.84	even	2		1275.2.g.f.526.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1275.2.d.i.424.3		12	85.33	odd	4
1275.2.d.i.424.10		12	5.2	odd	4
1275.2.d.j.424.3		12	5.3	odd	4
1275.2.d.j.424.10		12	85.67	odd	4
1275.2.g.e.526.9	✓	12	1.1	even	1	trivial
1275.2.g.e.526.10	yes	12	17.16	even	2	inner
1275.2.g.f.526.3	yes	12	85.84	even	2
1275.2.g.f.526.4	yes	12	5.4	even	2

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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.74480 q^{2} -1.00000i q^{3} +1.04434 q^{4} -1.74480i q^{6} +4.98357i q^{7} -1.66744 q^{8} -1.00000 q^{9} +2.23877i q^{11} -1.04434i q^{12} -2.92477 q^{13} +8.69535i q^{14} -4.99803 q^{16} +(-3.95816 + 1.15455i) q^{17} -1.74480 q^{18} -1.52050 q^{19} +4.98357 q^{21} +3.90621i q^{22} +1.30964i q^{23} +1.66744i q^{24} -5.10314 q^{26} +1.00000i q^{27} +5.20454i q^{28} +1.82004i q^{29} +1.39136i q^{31} -5.38571 q^{32} +2.23877 q^{33} +(-6.90621 + 2.01446i) q^{34} -1.04434 q^{36} +3.54628i q^{37} -2.65298 q^{38} +2.92477i q^{39} -8.48551i q^{41} +8.69535 q^{42} +9.13105 q^{43} +2.33803i q^{44} +2.28507i q^{46} -0.166519 q^{47} +4.99803i q^{48} -17.8360 q^{49} +(1.15455 + 3.95816i) q^{51} -3.05445 q^{52} +7.58067 q^{53} +1.74480i q^{54} -8.30981i q^{56} +1.52050i q^{57} +3.17561i q^{58} +13.7233 q^{59} -4.50092i q^{61} +2.42765i q^{62} -4.98357i q^{63} +0.599071 q^{64} +3.90621 q^{66} +1.03226 q^{67} +(-4.13366 + 1.20574i) q^{68} +1.30964 q^{69} -1.78218i q^{71} +1.66744 q^{72} +11.6402i q^{73} +6.18756i q^{74} -1.58792 q^{76} -11.1570 q^{77} +5.10314i q^{78} +13.9481i q^{79} +1.00000 q^{81} -14.8055i q^{82} -4.47514 q^{83} +5.20454 q^{84} +15.9319 q^{86} +1.82004 q^{87} -3.73301i q^{88} +14.6490 q^{89} -14.5758i q^{91} +1.36771i q^{92} +1.39136 q^{93} -0.290543 q^{94} +5.38571i q^{96} +14.7821i q^{97} -31.1203 q^{98} -2.23877i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{4} - 12 q^{9} - 4 q^{13} + 32 q^{16} + 10 q^{17} - 4 q^{19} - 36 q^{26} - 20 q^{32} - 12 q^{33} - 24 q^{34} - 16 q^{36} - 44 q^{38} + 28 q^{42} + 28 q^{43} - 16 q^{47} - 16 q^{49} + 6 q^{51}+ \cdots - 8 q^{98}+O(q^{100}) \) 12 * q + 16 * q^4 - 12 * q^9 - 4 * q^13 + 32 * q^16 + 10 * q^17 - 4 * q^19 - 36 * q^26 - 20 * q^32 - 12 * q^33 - 24 * q^34 - 16 * q^36 - 44 * q^38 + 28 * q^42 + 28 * q^43 - 16 * q^47 - 16 * q^49 + 6 * q^51 + 16 * q^52 - 12 * q^53 + 32 * q^59 + 56 * q^64 - 12 * q^66 - 20 * q^67 + 56 * q^68 - 16 * q^69 + 64 * q^76 - 72 * q^77 + 12 * q^81 - 44 * q^83 - 36 * q^84 + 20 * q^86 + 32 * q^87 + 4 * q^89 + 8 * q^93 + 40 * q^94 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more