LMFDB - Embedded newform 2340.2.fo.b.1241.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,2,Mod(1241,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 6, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.1241"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2340.fo (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$10$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			1241.6
Character	$\chi$	$=$	2340.1241
Dual form			2340.2.fo.b.1961.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{5} +(0.891313 - 3.32643i) q^{7} +(0.508619 + 1.89819i) q^{11} +(-2.17134 + 2.87842i) q^{13} +(-2.85207 - 4.93992i) q^{17} +(5.92027 + 1.58633i) q^{19} +(-2.03246 + 3.52032i) q^{23} +1.00000i q^{25} +(3.49080 + 2.01541i) q^{29} +(6.96300 - 6.96300i) q^{31} +(2.98239 - 1.72188i) q^{35} +(6.75423 - 1.80979i) q^{37} +(8.34356 - 2.23565i) q^{41} +(1.37368 - 0.793097i) q^{43} +(-4.85319 + 4.85319i) q^{47} +(-4.20849 - 2.42977i) q^{49} +7.98860i q^{53} +(-0.982576 + 1.70187i) q^{55} +(6.45232 + 1.72889i) q^{59} +(-4.91891 - 8.51980i) q^{61} +(-3.57072 + 0.499981i) q^{65} +(-2.21585 - 8.26968i) q^{67} +(3.02496 - 11.2893i) q^{71} +(-8.23540 - 8.23540i) q^{73} +6.76753 q^{77} -7.09045 q^{79} +(10.5452 + 10.5452i) q^{83} +(1.47634 - 5.50977i) q^{85} +(-0.499670 - 1.86479i) q^{89} +(7.63950 + 9.78837i) q^{91} +(3.06456 + 5.30797i) q^{95} +(0.729538 + 0.195479i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 8 q^{7} + 12 q^{13} + 28 q^{19} + 36 q^{31} - 40 q^{37} - 72 q^{43} - 36 q^{49} - 32 q^{61} + 16 q^{67} + 28 q^{73} + 48 q^{79} - 20 q^{85} + 132 q^{91} + 48 q^{97}+O(q^{100}) $ 40 * q + 8 * q^7 + 12 * q^13 + 28 * q^19 + 36 * q^31 - 40 * q^37 - 72 * q^43 - 36 * q^49 - 32 * q^61 + 16 * q^67 + 28 * q^73 + 48 * q^79 - 20 * q^85 + 132 * q^91 + 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$1$	$e\left(\frac{5}{12}\right)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	0.891313	−	3.32643i	0.336885	−	1.25727i	−0.564927	−	0.825141i	$-0.691096\pi$
$7$	0.891313	−	3.32643i	0.336885	−	1.25727i	0.901811	−	0.432130i	$-0.142238\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.508619	+	1.89819i	0.153354	+	0.572326i	0.999241	+	0.0389631i	$0.0124055\pi$
$11$	0.508619	+	1.89819i	0.153354	+	0.572326i	−0.845886	+	0.533363i	$0.820928\pi$
$12$		0			0
$13$	−2.17134	+	2.87842i	−0.602221	+	0.798330i
$13$	−2.17134	+	2.87842i	−0.602221	+	0.798330i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.85207	−	4.93992i	−0.691727	−	1.19811i	−0.971271	−	0.237974i	$-0.923517\pi$
$17$	−2.85207	−	4.93992i	−0.691727	−	1.19811i	0.279544	−	0.960133i	$-0.409817\pi$
$18$		0			0
$19$	5.92027	+	1.58633i	1.35820	+	0.363930i	0.863156	−	0.504937i	$-0.168484\pi$
$19$	5.92027	+	1.58633i	1.35820	+	0.363930i	0.495047	+	0.868866i	$0.335151\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.03246	+	3.52032i	−0.423796	+	0.734037i	−0.996307	−	0.0858604i	$-0.972636\pi$
$23$	−2.03246	+	3.52032i	−0.423796	+	0.734037i	0.572511	+	0.819897i	$0.305969\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.49080	+	2.01541i	0.648225	+	0.374253i	0.787776	−	0.615962i	$-0.211232\pi$
$29$	3.49080	+	2.01541i	0.648225	+	0.374253i	−0.139551	+	0.990215i	$0.544566\pi$
$30$		0			0
$31$	6.96300	−	6.96300i	1.25059	−	1.25059i	0.295137	−	0.955455i	$-0.404635\pi$
$31$	6.96300	−	6.96300i	1.25059	−	1.25059i	0.955455	−	0.295137i	$-0.0953654\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.98239	−	1.72188i	0.504116	−	0.291052i
$36$		0			0
$37$	6.75423	−	1.80979i	1.11039	−	0.297528i	0.343401	−	0.939189i	$-0.388421\pi$
$37$	6.75423	−	1.80979i	1.11039	−	0.297528i	0.766988	+	0.641661i	$0.221754\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.34356	−	2.23565i	1.30304	−	0.349150i	0.460444	−	0.887689i	$-0.347690\pi$
$41$	8.34356	−	2.23565i	1.30304	−	0.349150i	0.842601	+	0.538539i	$0.181023\pi$
$42$		0			0
$43$	1.37368	−	0.793097i	0.209485	−	0.120946i	−0.391587	−	0.920141i	$-0.628074\pi$
$43$	1.37368	−	0.793097i	0.209485	−	0.120946i	0.601072	+	0.799195i	$0.294740\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.85319	+	4.85319i	−0.707910	+	0.707910i	−0.966095	−	0.258185i	$-0.916876\pi$
$47$	−4.85319	+	4.85319i	−0.707910	+	0.707910i	0.258185	+	0.966095i	$0.416876\pi$
$48$		0			0
$49$	−4.20849	−	2.42977i	−0.601213	−	0.347110i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.98860i			1.09732i	0.836046	+	0.548659i	$0.184861\pi$
$53$			7.98860i			1.09732i	−0.836046	+	0.548659i	$0.815139\pi$
$54$		0			0
$55$	−0.982576	+	1.70187i	−0.132491	+	0.229480i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.45232	+	1.72889i	0.840020	+	0.225083i	0.653080	−	0.757289i	$-0.273476\pi$
$59$	6.45232	+	1.72889i	0.840020	+	0.225083i	0.186940	+	0.982371i	$0.440143\pi$
$60$		0			0
$61$	−4.91891	−	8.51980i	−0.629802	−	1.09085i	−0.987591	−	0.157046i	$-0.949803\pi$
$61$	−4.91891	−	8.51980i	−0.629802	−	1.09085i	0.357790	−	0.933802i	$-0.383531\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.57072	+	0.499981i	−0.442893	+	0.0620151i
$66$		0			0
$67$	−2.21585	−	8.26968i	−0.270710	−	1.01030i	−0.958662	−	0.284546i	$-0.908157\pi$
$67$	−2.21585	−	8.26968i	−0.270710	−	1.01030i	0.687953	−	0.725756i	$-0.258510\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.02496	−	11.2893i	0.358997	−	1.33979i	−0.516383	−	0.856358i	$-0.672722\pi$
$71$	3.02496	−	11.2893i	0.358997	−	1.33979i	0.875379	−	0.483436i	$-0.160612\pi$
$72$		0			0
$73$	−8.23540	−	8.23540i	−0.963880	−	0.963880i	0.0354896	−	0.999370i	$-0.488701\pi$
$73$	−8.23540	−	8.23540i	−0.963880	−	0.963880i	−0.999370	+	0.0354896i	$0.988701\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.76753			0.771232
$78$		0			0
$79$	−7.09045			−0.797738			−0.398869	−	0.917008i	$-0.630597\pi$
$79$	−7.09045			−0.797738			−0.398869	+	0.917008i	$0.630597\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.5452	+	10.5452i	1.15749	+	1.15749i	0.985014	+	0.172473i	$0.0551757\pi$
$83$	10.5452	+	10.5452i	1.15749	+	1.15749i	0.172473	+	0.985014i	$0.444824\pi$
$84$		0			0
$85$	1.47634	−	5.50977i	0.160131	−	0.597618i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.499670	−	1.86479i	−0.0529649	−	0.197668i	0.934374	−	0.356295i	$-0.115960\pi$
$89$	−0.499670	−	1.86479i	−0.0529649	−	0.197668i	−0.987339	+	0.158627i	$0.949293\pi$
$90$		0			0
$91$	7.63950	+	9.78837i	0.800837	+	1.02610i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.06456	+	5.30797i	0.314417	+	0.544586i
$96$		0			0
$97$	0.729538	+	0.195479i	0.0740734	+	0.0198479i	0.295665	−	0.955292i	$-0.404459\pi$
$97$	0.729538	+	0.195479i	0.0740734	+	0.0198479i	−0.221592	+	0.975139i	$0.571125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.425498	−	0.736984i	0.0423386	−	0.0733326i	−0.844080	−	0.536218i	$-0.819853\pi$
$101$	0.425498	−	0.736984i	0.0423386	−	0.0733326i	0.886418	+	0.462885i	$0.153186\pi$
$102$		0			0
$103$			7.59154i			0.748017i	0.927425	+	0.374008i	$0.122017\pi$
$103$			7.59154i			0.748017i	−0.927425	+	0.374008i	$0.877983\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.774703	+	0.447275i	0.0748933	+	0.0432397i	0.536979	−	0.843596i	$-0.319565\pi$
$107$	0.774703	+	0.447275i	0.0748933	+	0.0432397i	−0.462086	+	0.886835i	$0.652899\pi$
$108$		0			0
$109$	12.7047	−	12.7047i	1.21689	−	1.21689i	0.248175	−	0.968715i	$-0.420169\pi$
$109$	12.7047	−	12.7047i	1.21689	−	1.21689i	0.968715	−	0.248175i	$-0.0798308\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.9437	−	8.62774i	1.40578	−	0.811629i	0.410805	−	0.911723i	$-0.365248\pi$
$113$	14.9437	−	8.62774i	1.40578	−	0.811629i	0.994978	+	0.100094i	$0.0319143\pi$
$114$		0			0
$115$	−3.92640	+	1.05208i	−0.366139	+	0.0981066i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−18.9744	+	5.08417i	−1.73938	+	0.466065i
$120$		0			0
$121$	6.18184	−	3.56909i	0.561985	−	0.324462i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$	1.73957	+	1.00434i	0.154362	+	0.0891207i	0.575191	−	0.818019i	$-0.304928\pi$
$127$	1.73957	+	1.00434i	0.154362	+	0.0891207i	−0.420830	+	0.907140i	$0.638261\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.38899i			0.208727i	0.994539	+	0.104364i	$0.0332806\pi$
$131$			2.38899i			0.208727i	−0.994539	+	0.104364i	$0.966719\pi$
$132$		0			0
$133$	10.5536	−	18.2794i	0.915116	−	1.58503i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−17.5322	−	4.69775i	−1.49788	−	0.401356i	−0.585491	−	0.810679i	$-0.699098\pi$
$137$	−17.5322	−	4.69775i	−1.49788	−	0.401356i	−0.912389	+	0.409323i	$0.865765\pi$
$138$		0			0
$139$	−4.27973	−	7.41271i	−0.363002	−	0.628738i	0.625451	−	0.780263i	$-0.284915\pi$
$139$	−4.27973	−	7.41271i	−0.363002	−	0.628738i	−0.988453	+	0.151525i	$0.951581\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.56817	−	2.65760i	−0.549258	−	0.222239i
$144$		0			0
$145$	1.04326	+	3.89348i	0.0866377	+	0.323336i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.913623	−	3.40969i	0.0748469	−	0.279332i	−0.918352	−	0.395765i	$-0.870479\pi$
$149$	0.913623	−	3.40969i	0.0748469	−	0.279332i	0.993199	+	0.116433i	$0.0371460\pi$
$150$		0			0
$151$	12.3902	+	12.3902i	1.00830	+	1.00830i	0.999965	+	0.00833776i	$0.00265402\pi$
$151$	12.3902	+	12.3902i	1.00830	+	1.00830i	0.00833776	+	0.999965i	$0.497346\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.84717			0.790944
$156$		0			0
$157$	5.67466			0.452887			0.226443	−	0.974024i	$-0.427290\pi$
$157$	5.67466			0.452887			0.226443	+	0.974024i	$0.427290\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.89852	+	9.89852i	0.780112	+	0.780112i
$162$		0			0
$163$	−0.529406	+	1.97577i	−0.0414663	+	0.154754i	−0.983555	−	0.180610i	$-0.942193\pi$
$163$	−0.529406	+	1.97577i	−0.0414663	+	0.154754i	0.942088	+	0.335364i	$0.108859\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.65709	+	17.3805i	0.360377	+	1.34494i	0.873582	+	0.486678i	$0.161791\pi$
$167$	4.65709	+	17.3805i	0.360377	+	1.34494i	−0.513205	+	0.858266i	$0.671542\pi$
$168$		0			0
$169$	−3.57058	−	12.5000i	−0.274660	−	0.961541i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.5102	+	18.2042i	0.799078	+	1.38404i	0.920217	+	0.391408i	$0.128012\pi$
$173$	10.5102	+	18.2042i	0.799078	+	1.38404i	−0.121140	+	0.992635i	$0.538655\pi$
$174$		0			0
$175$	3.32643	+	0.891313i	0.251454	+	0.0673769i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.49390	+	7.78366i	−0.335890	+	0.581778i	−0.983655	−	0.180062i	$-0.942370\pi$
$179$	−4.49390	+	7.78366i	−0.335890	+	0.581778i	0.647766	+	0.761840i	$0.275704\pi$
$180$		0			0
$181$			7.26581i			0.540063i	0.962851	+	0.270032i	$0.0870342\pi$
$181$			7.26581i			0.540063i	−0.962851	+	0.270032i	$0.912966\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.05568	+	3.49625i	0.445222	+	0.257049i
$186$		0			0
$187$	7.92631	−	7.92631i	0.579629	−	0.579629i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.12248	−	4.68952i	0.587722	−	0.339322i	−0.176474	−	0.984305i	$-0.556469\pi$
$191$	8.12248	−	4.68952i	0.587722	−	0.339322i	0.764196	+	0.644984i	$0.223136\pi$
$192$		0			0
$193$	17.2472	−	4.62137i	1.24148	−	0.332654i	0.422441	−	0.906391i	$-0.361174\pi$
$193$	17.2472	−	4.62137i	1.24148	−	0.332654i	0.819040	+	0.573737i	$0.194507\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.13761	+	1.91252i	−0.508534	+	0.136261i	−0.503958	−	0.863728i	$-0.668123\pi$
$197$	−7.13761	+	1.91252i	−0.508534	+	0.136261i	−0.00457606	+	0.999990i	$0.501457\pi$
$198$		0			0
$199$	9.69092	−	5.59506i	0.686971	−	0.396623i	−0.115505	−	0.993307i	$-0.536849\pi$
$199$	9.69092	−	5.59506i	0.686971	−	0.396623i	0.802476	+	0.596684i	$0.203515\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.81552	−	9.81552i	0.688915	−	0.688915i
$204$		0			0
$205$	7.48063	+	4.31894i	0.522470	+	0.301648i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			12.0446i			0.833146i
$210$		0			0
$211$	−11.4167	+	19.7744i	−0.785960	+	1.36132i	0.142464	+	0.989800i	$0.454498\pi$
$211$	−11.4167	+	19.7744i	−0.785960	+	1.36132i	−0.928424	+	0.371523i	$0.878836\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.53215	+	0.410537i	0.104491	+	0.0279984i
$216$		0			0
$217$	−16.9557	−	29.3681i	−1.15103	−	1.99364i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	20.4120	+	2.51680i	1.37306	+	0.169298i
$222$		0			0
$223$	1.37050	+	5.11477i	0.0917754	+	0.342511i	0.996511	−	0.0834634i	$-0.0265982\pi$
$223$	1.37050	+	5.11477i	0.0917754	+	0.342511i	−0.904735	+	0.425974i	$0.859932\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.57212	+	5.86722i	−0.104345	+	0.389421i	−0.998270	−	0.0587955i	$-0.981274\pi$
$227$	−1.57212	+	5.86722i	−0.104345	+	0.389421i	0.893925	+	0.448217i	$0.147941\pi$
$228$		0			0
$229$	18.3920	+	18.3920i	1.21538	+	1.21538i	0.969232	+	0.246149i	$0.0791652\pi$
$229$	18.3920	+	18.3920i	1.21538	+	1.21538i	0.246149	+	0.969232i	$0.420835\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.1923			−1.25733			−0.628665	−	0.777676i	$-0.716398\pi$
$233$	−19.1923			−1.25733			−0.628665	+	0.777676i	$0.716398\pi$
$234$		0			0
$235$	−6.86345			−0.447722
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.9770	−	18.9770i	−1.22752	−	1.22752i	−0.964899	−	0.262621i	$-0.915413\pi$
$239$	−18.9770	−	18.9770i	−1.22752	−	1.22752i	−0.262621	−	0.964899i	$-0.584587\pi$
$240$		0			0
$241$	−2.99434	+	11.1750i	−0.192882	+	0.719847i	0.799922	+	0.600103i	$0.204874\pi$
$241$	−2.99434	+	11.1750i	−0.192882	+	0.719847i	−0.992805	+	0.119744i	$0.961793\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.25774	−	4.69396i	−0.0803543	−	0.299886i
$246$		0			0
$247$	−17.4210	+	13.5966i	−1.10847	+	0.865128i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.581224	−	1.00671i	−0.0366865	−	0.0635429i	0.847099	−	0.531435i	$-0.178347\pi$
$251$	−0.581224	−	1.00671i	−0.0366865	−	0.0635429i	−0.883786	+	0.467892i	$0.845014\pi$
$252$		0			0
$253$	−7.71598	−	2.06749i	−0.485100	−	0.129982i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.14605	+	5.44912i	−0.196245	+	0.339907i	−0.947308	−	0.320324i	$-0.896208\pi$
$257$	−3.14605	+	5.44912i	−0.196245	+	0.339907i	0.751063	+	0.660231i	$0.229542\pi$
$258$		0			0
$259$		−	24.0805i		−	1.49629i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20.5988	−	11.8927i	−1.27017	−	0.733336i	−0.295154	−	0.955450i	$-0.595371\pi$
$263$	−20.5988	−	11.8927i	−1.27017	−	0.733336i	−0.975021	+	0.222114i	$0.928704\pi$
$264$		0			0
$265$	−5.64879	+	5.64879i	−0.347002	+	0.347002i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.8032	−	14.8975i	1.57325	−	0.908316i	0.577482	−	0.816403i	$-0.304035\pi$
$269$	25.8032	−	14.8975i	1.57325	−	0.908316i	0.995767	−	0.0919125i	$-0.0292980\pi$
$270$		0			0
$271$	−19.0477	+	5.10382i	−1.15707	+	0.310035i	−0.785794	−	0.618489i	$-0.787745\pi$
$271$	−19.0477	+	5.10382i	−1.15707	+	0.310035i	−0.371273	+	0.928524i	$0.621079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.89819	+	0.508619i	−0.114465	+	0.0306709i
$276$		0			0
$277$	12.9355	−	7.46830i	0.777217	−	0.448727i	−0.0582260	−	0.998303i	$-0.518544\pi$
$277$	12.9355	−	7.46830i	0.777217	−	0.448727i	0.835443	+	0.549577i	$0.185211\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0170	−	10.0170i	0.597565	−	0.597565i	−0.342099	−	0.939664i	$-0.611138\pi$
$281$	10.0170	−	10.0170i	0.597565	−	0.597565i	0.939664	+	0.342099i	$0.111138\pi$
$282$		0			0
$283$	−5.88093	−	3.39535i	−0.349585	−	0.201833i	0.314918	−	0.949119i	$-0.398023\pi$
$283$	−5.88093	−	3.39535i	−0.349585	−	0.201833i	−0.664502	+	0.747286i	$0.731356\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	29.7469i		−	1.75590i
$288$		0			0
$289$	−7.76855	+	13.4555i	−0.456974	+	0.791502i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−27.7119	−	7.42539i	−1.61895	−	0.433796i	−0.668256	−	0.743931i	$-0.732959\pi$
$293$	−27.7119	−	7.42539i	−1.61895	−	0.433796i	−0.950693	+	0.310135i	$0.899626\pi$
$294$		0			0
$295$	3.33997	+	5.78499i	0.194460	+	0.336815i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.71980	−	13.4941i	−0.330784	−	0.780381i
$300$		0			0
$301$	−1.41380	−	5.27636i	−0.0814898	−	0.304124i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.54621	−	9.50260i	0.145796	−	0.544117i
$306$		0			0
$307$	−4.69587	−	4.69587i	−0.268007	−	0.268007i	0.560290	−	0.828297i	$-0.310690\pi$
$307$	−4.69587	−	4.69587i	−0.268007	−	0.268007i	−0.828297	+	0.560290i	$0.810690\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.58130			−0.543306			−0.271653	−	0.962395i	$-0.587570\pi$
$311$	−9.58130			−0.543306			−0.271653	+	0.962395i	$0.587570\pi$
$312$		0			0
$313$	1.53943			0.0870135			0.0435068	−	0.999053i	$-0.486147\pi$
$313$	1.53943			0.0870135			0.0435068	+	0.999053i	$0.486147\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.75192	+	6.75192i	0.379225	+	0.379225i	0.870823	−	0.491597i	$-0.163587\pi$
$317$	6.75192	+	6.75192i	0.379225	+	0.379225i	−0.491597	+	0.870823i	$0.663587\pi$
$318$		0			0
$319$	−2.05016	+	7.65129i	−0.114787	+	0.428390i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9.04864	−	33.7700i	−0.503480	−	1.87901i
$324$		0			0
$325$	−2.87842	−	2.17134i	−0.159666	−	0.120444i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.8181	+	20.4695i	0.651551	+	1.12852i
$330$		0			0
$331$	−12.8935	−	3.45481i	−0.708692	−	0.189893i	−0.113571	−	0.993530i	$-0.536229\pi$
$331$	−12.8935	−	3.45481i	−0.708692	−	0.189893i	−0.595121	+	0.803636i	$0.702896\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.28070	−	7.41439i	0.233880	−	0.405091i
$336$		0			0
$337$		−	22.8636i		−	1.24546i	−0.782436	−	0.622731i	$-0.786023\pi$
$337$		−	22.8636i		−	1.24546i	0.782436	−	0.622731i	$-0.213977\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.7586	+	9.67560i	0.907531	+	0.523963i
$342$		0			0
$343$	5.21225	−	5.21225i	0.281435	−	0.281435i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.8104	+	11.4375i	−1.06348	+	0.613998i	−0.926392	−	0.376562i	$-0.877106\pi$
$347$	−19.8104	+	11.4375i	−1.06348	+	0.613998i	−0.137084	+	0.990559i	$0.543773\pi$
$348$		0			0
$349$	−28.2139	+	7.55988i	−1.51025	+	0.404671i	−0.916519	−	0.399991i	$-0.869013\pi$
$349$	−28.2139	+	7.55988i	−1.51025	+	0.404671i	−0.593734	+	0.804662i	$0.702347\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.66047	+	0.444923i	−0.0883781	+	0.0236808i	−0.302737	−	0.953074i	$-0.597900\pi$
$353$	−1.66047	+	0.444923i	−0.0883781	+	0.0236808i	0.214359	+	0.976755i	$0.431234\pi$
$354$		0			0
$355$	10.1217	−	5.84378i	0.537205	−	0.310155i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.7340	+	24.7340i	−1.30541	+	1.30541i	−0.380719	+	0.924691i	$0.624323\pi$
$359$	−24.7340	+	24.7340i	−1.30541	+	1.30541i	−0.924691	+	0.380719i	$0.875677\pi$
$360$		0			0
$361$	16.0787	+	9.28303i	0.846246	+	0.488581i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	11.6466i		−	0.609612i
$366$		0			0
$367$	−14.0841	+	24.3943i	−0.735182	+	1.27337i	0.219462	+	0.975621i	$0.429570\pi$
$367$	−14.0841	+	24.3943i	−0.735182	+	1.27337i	−0.954643	+	0.297751i	$0.903763\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	26.5735	+	7.12034i	1.37963	+	0.369670i
$372$		0			0
$373$	−9.71397	−	16.8251i	−0.502970	−	0.871170i	−0.999994	−	0.00343336i	$-0.998907\pi$
$373$	−9.71397	−	16.8251i	−0.502970	−	0.871170i	0.497024	−	0.867737i	$-0.334426\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.3809	+	5.67184i	−0.689152	+	0.292115i
$378$		0			0
$379$	4.26288	+	15.9093i	0.218969	+	0.817205i	0.984731	+	0.174081i	$0.0556955\pi$
$379$	4.26288	+	15.9093i	0.218969	+	0.817205i	−0.765762	+	0.643124i	$0.777638\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.65734	+	24.8455i	−0.340174	+	1.26955i	0.557975	+	0.829857i	$0.311578\pi$
$383$	−6.65734	+	24.8455i	−0.340174	+	1.26955i	−0.898150	+	0.439690i	$0.855088\pi$
$384$		0			0
$385$	4.78537	+	4.78537i	0.243885	+	0.243885i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.3836			0.729275			0.364638	−	0.931149i	$-0.381193\pi$
$389$	14.3836			0.729275			0.364638	+	0.931149i	$0.381193\pi$
$390$		0			0
$391$	23.1868			1.17261
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.01371	−	5.01371i	−0.252267	−	0.252267i
$396$		0			0
$397$	4.49229	−	16.7655i	0.225462	−	0.841434i	−0.756758	−	0.653696i	$-0.773218\pi$
$397$	4.49229	−	16.7655i	0.225462	−	0.841434i	0.982219	−	0.187738i	$-0.0601157\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.978911	+	3.65335i	0.0488845	+	0.182439i	0.986051	−	0.166443i	$-0.0532281\pi$
$401$	0.978911	+	3.65335i	0.0488845	+	0.182439i	−0.937167	+	0.348882i	$0.886561\pi$
$402$		0			0
$403$	4.92340	+	35.1615i	0.245252	+	1.75152i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.87066	+	11.9003i	0.340566	+	0.589878i
$408$		0			0
$409$	7.18094	+	1.92413i	0.355075	+	0.0951420i	0.431947	−	0.901899i	$-0.357827\pi$
$409$	7.18094	+	1.92413i	0.355075	+	0.0951420i	−0.0768724	+	0.997041i	$0.524493\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.5021	−	19.9222i	0.565980	−	0.980306i
$414$		0			0
$415$			14.9132i			0.732059i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.7311	−	13.7012i	−1.15934	−	0.669346i	−0.208195	−	0.978087i	$-0.566759\pi$
$419$	−23.7311	−	13.7012i	−1.15934	−	0.669346i	−0.951146	+	0.308742i	$0.900092\pi$
$420$		0			0
$421$	−12.9016	+	12.9016i	−0.628787	+	0.628787i	−0.947763	−	0.318976i	$-0.896661\pi$
$421$	−12.9016	+	12.9016i	−0.628787	+	0.628787i	0.318976	+	0.947763i	$0.396661\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.93992	−	2.85207i	0.239621	−	0.138345i
$426$		0			0
$427$	−32.7248	+	8.76857i	−1.58366	+	0.424341i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	31.2983	−	8.38637i	1.50759	−	0.403957i	0.591955	−	0.805971i	$-0.298356\pi$
$431$	31.2983	−	8.38637i	1.50759	−	0.403957i	0.915634	+	0.402014i	$0.131690\pi$
$432$		0			0
$433$	−23.3516	+	13.4820i	−1.12221	+	0.647906i	−0.941963	−	0.335717i	$-0.891022\pi$
$433$	−23.3516	+	13.4820i	−1.12221	+	0.647906i	−0.180242	+	0.983622i	$0.557688\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−17.6171	+	17.6171i	−0.842739	+	0.842739i
$438$		0			0
$439$	−22.8690	−	13.2034i	−1.09148	−	0.630164i	−0.157508	−	0.987518i	$-0.550346\pi$
$439$	−22.8690	−	13.2034i	−1.09148	−	0.630164i	−0.933969	+	0.357353i	$0.883679\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	9.10315i		−	0.432504i	−0.976338	−	0.216252i	$-0.930617\pi$
$443$		−	9.10315i		−	0.432504i	0.976338	−	0.216252i	$-0.0693833\pi$
$444$		0			0
$445$	0.965289	−	1.67193i	0.0457591	−	0.0792570i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.70398	−	0.724529i	−0.127609	−	0.0341926i	0.194449	−	0.980913i	$-0.437708\pi$
$449$	−2.70398	−	0.724529i	−0.127609	−	0.0341926i	−0.322058	+	0.946720i	$0.604375\pi$
$450$		0			0
$451$	8.48738	+	14.7006i	0.399655	+	0.692223i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.51948	+	12.3234i	−0.0712341	+	0.577728i
$456$		0			0
$457$	0.549464	+	2.05063i	0.0257028	+	0.0959242i	0.977586	−	0.210538i	$-0.0675215\pi$
$457$	0.549464	+	2.05063i	0.0257028	+	0.0959242i	−0.951883	+	0.306462i	$0.900855\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.71401	+	28.7891i	−0.359277	+	1.34084i	0.515738	+	0.856746i	$0.327518\pi$
$461$	−7.71401	+	28.7891i	−0.359277	+	1.34084i	−0.875016	+	0.484095i	$0.839149\pi$
$462$		0			0
$463$	25.3632	+	25.3632i	1.17873	+	1.17873i	0.980068	+	0.198662i	$0.0636596\pi$
$463$	25.3632	+	25.3632i	1.17873	+	1.17873i	0.198662	+	0.980068i	$0.436340\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.9718			−1.15556			−0.577778	−	0.816194i	$-0.696080\pi$
$467$	−24.9718			−1.15556			−0.577778	+	0.816194i	$0.696080\pi$
$468$		0			0
$469$	−29.4835			−1.36142
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.20413	+	2.20413i	0.101346	+	0.101346i
$474$		0			0
$475$	−1.58633	+	5.92027i	−0.0727859	+	0.271641i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.59444	−	9.68258i	−0.118543	−	0.442409i	0.880985	−	0.473145i	$-0.156881\pi$
$479$	−2.59444	−	9.68258i	−0.118543	−	0.442409i	−0.999528	+	0.0307366i	$0.990215\pi$
$480$		0			0
$481$	−9.45639	+	23.3712i	−0.431174	+	1.06563i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.377637	+	0.654086i	0.0171476	+	0.0297005i
$486$		0			0
$487$	25.9869	+	6.96317i	1.17758	+	0.315531i	0.793966	−	0.607962i	$-0.208013\pi$
$487$	25.9869	+	6.96317i	1.17758	+	0.315531i	0.383614	+	0.923494i	$0.374679\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.94371	+	6.83070i	−0.177977	+	0.308265i	−0.941188	−	0.337885i	$-0.890289\pi$
$491$	−3.94371	+	6.83070i	−0.177977	+	0.308265i	0.763210	+	0.646150i	$0.223622\pi$
$492$		0			0
$493$		−	22.9924i		−	1.03552i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−34.8569	−	20.1246i	−1.56354	−	0.902712i
$498$		0			0
$499$	−2.97945	+	2.97945i	−0.133379	+	0.133379i	−0.770644	−	0.637266i	$-0.780065\pi$
$499$	−2.97945	+	2.97945i	−0.133379	+	0.133379i	0.637266	+	0.770644i	$0.280065\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.32824	+	0.766857i	−0.0592231	+	0.0341925i	−0.529319	−	0.848423i	$-0.677553\pi$
$503$	−1.32824	+	0.766857i	−0.0592231	+	0.0341925i	0.470096	+	0.882615i	$0.344219\pi$
$504$		0			0
$505$	0.821999	−	0.220254i	0.0365785	−	0.00980117i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.21825	+	2.47002i	−0.408592	+	0.109482i	−0.457260	−	0.889333i	$-0.651169\pi$
$509$	−9.21825	+	2.47002i	−0.408592	+	0.109482i	0.0486681	+	0.998815i	$0.484502\pi$
$510$		0			0
$511$	−34.7348	+	20.0541i	−1.53658	+	0.887142i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.36803	+	5.36803i	−0.236544	+	0.236544i
$516$		0			0
$517$	−11.6807	−	6.74386i	−0.513717	−	0.296595i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	31.1280i		−	1.36374i	−0.731472	−	0.681872i	$-0.761166\pi$
$521$		−	31.1280i		−	1.36374i	0.731472	−	0.681872i	$-0.238834\pi$
$522$		0			0
$523$	5.69837	−	9.86987i	0.249172	−	0.431579i	−0.714124	−	0.700019i	$-0.753175\pi$
$523$	5.69837	−	9.86987i	0.249172	−	0.431579i	0.963296	+	0.268440i	$0.0865081\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−54.2556	−	14.5378i	−2.36341	−	0.633274i
$528$		0			0
$529$	3.23825	+	5.60881i	0.140793	+	0.243861i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11.6815	+	28.8706i	−0.505984	+	1.25052i
$534$		0			0
$535$	0.231527	+	0.864069i	0.0100098	+	0.0373569i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.47166	−	9.22435i	0.106462	−	0.397321i
$540$		0			0
$541$	4.63639	+	4.63639i	0.199334	+	0.199334i	0.799714	−	0.600381i	$-0.204984\pi$
$541$	4.63639	+	4.63639i	0.199334	+	0.199334i	−0.600381	+	0.799714i	$0.704984\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	17.9672			0.769629
$546$		0			0
$547$	27.7823			1.18788			0.593942	−	0.804508i	$-0.297571\pi$
$547$	27.7823			1.18788			0.593942	+	0.804508i	$0.297571\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.4694	+	17.4694i	0.744220	+	0.744220i
$552$		0			0
$553$	−6.31981	+	23.5859i	−0.268746	+	1.00297i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.0561	−	37.5300i	−0.426092	−	1.59020i	−0.761527	−	0.648133i	$-0.775550\pi$
$557$	−10.0561	−	37.5300i	−0.426092	−	1.59020i	0.335435	−	0.942063i	$-0.391117\pi$
$558$		0			0
$559$	−0.699868	+	5.67612i	−0.0296013	+	0.240074i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.75877	−	9.97448i	−0.242703	−	0.420374i	0.718780	−	0.695237i	$-0.244701\pi$
$563$	−5.75877	−	9.97448i	−0.242703	−	0.420374i	−0.961483	+	0.274863i	$0.911367\pi$
$564$		0			0
$565$	16.6675	+	4.46605i	0.701207	+	0.187888i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.02988	+	13.9082i	−0.336630	+	0.583061i	−0.983797	−	0.179288i	$-0.942621\pi$
$569$	−8.02988	+	13.9082i	−0.336630	+	0.583061i	0.647166	+	0.762349i	$0.275954\pi$
$570$		0			0
$571$			15.5249i			0.649696i	0.945766	+	0.324848i	$0.105313\pi$
$571$			15.5249i			0.649696i	−0.945766	+	0.324848i	$0.894687\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.52032	−	2.03246i	−0.146807	−	0.0847593i
$576$		0			0
$577$	−9.79856	+	9.79856i	−0.407919	+	0.407919i	−0.881012	−	0.473093i	$-0.843137\pi$
$577$	−9.79856	+	9.79856i	−0.407919	+	0.407919i	0.473093	+	0.881012i	$0.343137\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	44.4769	−	25.6788i	1.84521	−	1.06533i
$582$		0			0
$583$	−15.1639	+	4.06315i	−0.628024	+	0.168279i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.866418	−	0.232156i	0.0357609	−	0.00958211i	−0.240894	−	0.970551i	$-0.577441\pi$
$587$	0.866418	−	0.232156i	0.0357609	−	0.00958211i	0.276655	+	0.960969i	$0.410774\pi$
$588$		0			0
$589$	52.2685	−	30.1772i	2.15369	−	1.24343i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	31.4118	−	31.4118i	1.28993	−	1.28993i	0.355101	−	0.934828i	$-0.384447\pi$
$593$	31.4118	−	31.4118i	1.28993	−	1.28993i	0.934828	−	0.355101i	$-0.115553\pi$
$594$		0			0
$595$	−17.0120	−	9.82185i	−0.697422	−	0.402657i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			45.3601i			1.85336i	0.375849	+	0.926681i	$0.377351\pi$
$599$			45.3601i			1.85336i	−0.375849	+	0.926681i	$0.622649\pi$
$600$		0			0
$601$	3.29047	−	5.69926i	0.134221	−	0.232478i	−0.791079	−	0.611715i	$-0.790480\pi$
$601$	3.29047	−	5.69926i	0.134221	−	0.232478i	0.925300	+	0.379237i	$0.123813\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.89495	+	1.84750i	0.280319	+	0.0751114i
$606$		0			0
$607$	−0.263386	−	0.456198i	−0.0106905	−	0.0185165i	0.860631	−	0.509230i	$-0.170070\pi$
$607$	−0.263386	−	0.456198i	−0.0106905	−	0.0185165i	−0.871321	+	0.490713i	$0.836736\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.43160	−	24.5074i	−0.138827	−	0.991464i
$612$		0			0
$613$	−2.31081	−	8.62406i	−0.0933327	−	0.348323i	0.903429	−	0.428738i	$-0.141042\pi$
$613$	−2.31081	−	8.62406i	−0.0933327	−	0.348323i	−0.996761	+	0.0804159i	$0.974375\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.36090	+	12.5430i	−0.135305	+	0.504964i	0.864692	+	0.502303i	$0.167514\pi$
$617$	−3.36090	+	12.5430i	−0.135305	+	0.504964i	−0.999996	+	0.00266121i	$0.999153\pi$
$618$		0			0
$619$	−16.0926	−	16.0926i	−0.646815	−	0.646815i	0.305407	−	0.952222i	$-0.401208\pi$
$619$	−16.0926	−	16.0926i	−0.646815	−	0.646815i	−0.952222	+	0.305407i	$0.901208\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.64846			−0.266365
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−28.2037	−	28.2037i	−1.12456	−	1.12456i
$630$		0			0
$631$	−5.71929	+	21.3447i	−0.227681	+	0.849718i	0.753631	+	0.657297i	$0.228300\pi$
$631$	−5.71929	+	21.3447i	−0.227681	+	0.849718i	−0.981313	+	0.192420i	$0.938366\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.519885	+	1.94024i	0.0206310	+	0.0769959i
$636$		0			0
$637$	16.1320	−	6.83794i	0.639171	−	0.270929i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.855851	+	1.48238i	0.0338041	+	0.0585504i	0.882432	−	0.470439i	$-0.155904\pi$
$641$	0.855851	+	1.48238i	0.0338041	+	0.0585504i	−0.848628	+	0.528989i	$0.822571\pi$
$642$		0			0
$643$	−6.10902	−	1.63691i	−0.240916	−	0.0645533i	0.136340	−	0.990662i	$-0.456466\pi$
$643$	−6.10902	−	1.63691i	−0.240916	−	0.0645533i	−0.377257	+	0.926109i	$0.623133\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.72991	−	13.3886i	0.303894	−	0.526360i	−0.673120	−	0.739533i	$-0.735046\pi$
$647$	7.72991	−	13.3886i	0.303894	−	0.526360i	0.977014	+	0.213173i	$0.0683798\pi$
$648$		0			0
$649$			13.1271i			0.515283i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−21.4822	−	12.4027i	−0.840663	−	0.485357i	0.0168265	−	0.999858i	$-0.494644\pi$
$653$	−21.4822	−	12.4027i	−0.840663	−	0.485357i	−0.857490	+	0.514501i	$0.827977\pi$
$654$		0			0
$655$	−1.68927	+	1.68927i	−0.0660054	+	0.0660054i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−37.0981	+	21.4186i	−1.44514	+	0.834349i	−0.998186	−	0.0602102i	$-0.980823\pi$
$659$	−37.0981	+	21.4186i	−1.44514	+	0.834349i	−0.446949	+	0.894559i	$0.647490\pi$
$660$		0			0
$661$	−38.7038	+	10.3707i	−1.50541	+	0.403372i	−0.914906	−	0.403666i	$-0.867736\pi$
$661$	−38.7038	+	10.3707i	−1.50541	+	0.403372i	−0.590499	+	0.807038i	$0.701069\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.3880	−	5.46296i	0.790615	−	0.211845i
$666$		0			0
$667$	−14.1898	+	8.19248i	−0.549431	+	0.317214i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.6704	−	13.6704i	0.527739	−	0.527739i
$672$		0			0
$673$	−11.5984	−	6.69635i	−0.447086	−	0.258125i	0.259513	−	0.965740i	$-0.416438\pi$
$673$	−11.5984	−	6.69635i	−0.447086	−	0.258125i	−0.706599	+	0.707615i	$0.749771\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			34.8532i			1.33952i	0.742579	+	0.669759i	$0.233602\pi$
$677$			34.8532i			1.33952i	−0.742579	+	0.669759i	$0.766398\pi$
$678$		0			0
$679$	1.30049	−	2.25252i	0.0499084	−	0.0864439i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.0409	+	5.90583i	0.843371	+	0.225980i	0.654538	−	0.756029i	$-0.272863\pi$
$683$	22.0409	+	5.90583i	0.843371	+	0.225980i	0.188832	+	0.982009i	$0.439530\pi$
$684$		0			0
$685$	−9.07536	−	15.7190i	−0.346752	−	0.600591i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22.9945	−	17.3459i	−0.876022	−	0.660828i
$690$		0			0
$691$	5.98058	+	22.3198i	0.227512	+	0.849086i	0.981383	+	0.192063i	$0.0615178\pi$
$691$	5.98058	+	22.3198i	0.227512	+	0.849086i	−0.753871	+	0.657023i	$0.771815\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.21535	−	8.26780i	0.0840331	−	0.313616i
$696$		0			0
$697$	−34.8403	−	34.8403i	−1.31967	−	1.31967i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.6723			1.12071			0.560354	−	0.828253i	$-0.310665\pi$
$701$	29.6723			1.12071			0.560354	+	0.828253i	$0.310665\pi$
$702$		0			0
$703$	42.8578			1.61641
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.07227	−	2.07227i	−0.0779358	−	0.0779358i
$708$		0			0
$709$	−7.40747	+	27.6450i	−0.278193	+	1.03823i	0.675478	+	0.737380i	$0.263937\pi$
$709$	−7.40747	+	27.6450i	−0.278193	+	1.03823i	−0.953671	+	0.300851i	$0.902729\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.3600	+	38.6640i	0.387984	+	1.44798i
$714$		0			0
$715$	−2.76520	−	6.52361i	−0.103412	−	0.243969i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.7191	−	35.8866i	−0.772693	−	1.33834i	−0.936082	−	0.351782i	$-0.885576\pi$
$719$	−20.7191	−	35.8866i	−0.772693	−	1.33834i	0.163389	−	0.986562i	$-0.447757\pi$
$720$		0			0
$721$	25.2527	+	6.76644i	0.940460	+	0.251995i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.01541	+	3.49080i	−0.0748506	+	0.129645i
$726$		0			0
$727$		−	7.55994i		−	0.280383i	−0.990124	−	0.140191i	$-0.955228\pi$
$727$		−	7.55994i		−	0.280383i	0.990124	−	0.140191i	$-0.0447718\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.83568	−	4.52393i	−0.289813	−	0.167324i
$732$		0			0
$733$	−26.5047	+	26.5047i	−0.978973	+	0.978973i	−0.999783	−	0.0208106i	$-0.993375\pi$
$733$	−26.5047	+	26.5047i	−0.978973	+	0.978973i	0.0208106	+	0.999783i	$0.493375\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.5704	−	8.41223i	0.536708	−	0.309868i
$738$		0			0
$739$	−46.3532	+	12.4203i	−1.70513	+	0.456888i	−0.974222	−	0.225590i	$-0.927569\pi$
$739$	−46.3532	+	12.4203i	−1.70513	+	0.456888i	−0.730906	+	0.682478i	$0.760902\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.36364	+	1.43718i	−0.196773	+	0.0527251i	−0.355859	−	0.934540i	$-0.615812\pi$
$743$	−5.36364	+	1.43718i	−0.196773	+	0.0527251i	0.159086	+	0.987265i	$0.449145\pi$
$744$		0			0
$745$	3.05704	−	1.76498i	0.112001	−	0.0646640i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.17833	−	2.17833i	0.0795944	−	0.0795944i
$750$		0			0
$751$	20.6110	+	11.8998i	0.752106	+	0.434229i	0.826454	−	0.563004i	$-0.190354\pi$
$751$	20.6110	+	11.8998i	0.752106	+	0.434229i	−0.0743482	+	0.997232i	$0.523688\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			17.5224i			0.637707i
$756$		0			0
$757$	−4.35532	+	7.54364i	−0.158297	+	0.274178i	−0.934255	−	0.356607i	$-0.883934\pi$
$757$	−4.35532	+	7.54364i	−0.158297	+	0.274178i	0.775958	+	0.630785i	$0.217267\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−47.7018	−	12.7817i	−1.72919	−	0.463335i	−0.749194	−	0.662351i	$-0.769559\pi$
$761$	−47.7018	−	12.7817i	−1.72919	−	0.463335i	−0.979996	+	0.199016i	$0.936226\pi$
$762$		0			0
$763$	−30.9374	−	53.5851i	−1.12001	−	1.93991i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.9866	+	14.8185i	−0.685568	+	0.535063i
$768$		0			0
$769$	−9.13661	−	34.0983i	−0.329475	−	1.22962i	−0.909736	−	0.415186i	$-0.863716\pi$
$769$	−9.13661	−	34.0983i	−0.329475	−	1.22962i	0.580262	−	0.814430i	$-0.302950\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2.45540	+	9.16368i	−0.0883146	+	0.329594i	−0.995921	−	0.0902271i	$-0.971241\pi$
$773$	−2.45540	+	9.16368i	−0.0883146	+	0.329594i	0.907607	+	0.419822i	$0.137907\pi$
$774$		0			0
$775$	6.96300	+	6.96300i	0.250118	+	0.250118i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	52.9426			1.89687
$780$		0			0
$781$	22.9678			0.821853
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.01259	+	4.01259i	0.143215	+	0.143215i
$786$		0			0
$787$	−6.34138	+	23.6664i	−0.226046	+	0.843615i	0.755937	+	0.654644i	$0.227182\pi$
$787$	−6.34138	+	23.6664i	−0.226046	+	0.843615i	−0.981983	+	0.188970i	$0.939485\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.3800	−	57.3991i	−0.546851	−	2.04088i
$792$		0			0
$793$	35.2042	+	4.34069i	1.25014	+	0.154142i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−10.2034	−	17.6728i	−0.361424	−	0.626004i	0.626772	−	0.779203i	$-0.284376\pi$
$797$	−10.2034	−	17.6728i	−0.361424	−	0.626004i	−0.988195	+	0.153199i	$0.951042\pi$
$798$		0			0
$799$	37.8160	+	10.1328i	1.33783	+	0.358471i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	11.4437	−	19.8210i	0.403839	−	0.699470i
$804$		0			0
$805$			13.9986i			0.493386i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7.77258	+	4.48750i	0.273269	+	0.157772i	0.630372	−	0.776293i	$-0.282902\pi$
$809$	7.77258	+	4.48750i	0.273269	+	0.157772i	−0.357103	+	0.934065i	$0.616236\pi$
$810$		0			0
$811$	22.7278	−	22.7278i	0.798082	−	0.798082i	−0.184711	−	0.982793i	$-0.559135\pi$
$811$	22.7278	−	22.7278i	0.798082	−	0.798082i	0.982793	+	0.184711i	$0.0591350\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.77143	+	1.02273i	−0.0620504	+	0.0358248i
$816$		0			0
$817$	9.39070	−	2.51623i	0.328539	−	0.0880318i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.96055	−	2.40097i	0.312725	−	0.0837945i	−0.0990428	−	0.995083i	$-0.531578\pi$
$821$	8.96055	−	2.40097i	0.312725	−	0.0837945i	0.411768	+	0.911289i	$0.364911\pi$
$822$		0			0
$823$	−12.4137	+	7.16707i	−0.432715	+	0.249828i	−0.700503	−	0.713650i	$-0.747041\pi$
$823$	−12.4137	+	7.16707i	−0.432715	+	0.249828i	0.267787	+	0.963478i	$0.413707\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.98500	−	1.98500i	0.0690252	−	0.0690252i	−0.671751	−	0.740777i	$-0.734458\pi$
$827$	1.98500	−	1.98500i	0.0690252	−	0.0690252i	0.740777	+	0.671751i	$0.234458\pi$
$828$		0			0
$829$	−11.7890	−	6.80640i	−0.409450	−	0.236396i	0.281103	−	0.959678i	$-0.409300\pi$
$829$	−11.7890	−	6.80640i	−0.409450	−	0.236396i	−0.690553	+	0.723281i	$0.742633\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			27.7195i			0.960423i
$834$		0			0
$835$	−8.99681	+	15.5829i	−0.311347	+	0.539270i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.95822	−	2.13240i	−0.274748	−	0.0736186i	0.118814	−	0.992917i	$-0.462091\pi$
$839$	−7.95822	−	2.13240i	−0.274748	−	0.0736186i	−0.393562	+	0.919298i	$0.628757\pi$
$840$		0			0
$841$	−6.37621	−	11.0439i	−0.219869	−	0.380825i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.31408	−	11.3636i	0.217211	−	0.390921i
$846$		0			0
$847$	−6.36235	−	23.7446i	−0.218613	−	0.815874i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.35664	+	27.4554i	−0.252182	+	0.941158i
$852$		0			0
$853$	−20.0299	−	20.0299i	−0.685810	−	0.685810i	0.275493	−	0.961303i	$-0.411159\pi$
$853$	−20.0299	−	20.0299i	−0.685810	−	0.685810i	−0.961303	+	0.275493i	$0.911159\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.9955			0.580556			0.290278	−	0.956942i	$-0.406252\pi$
$857$	16.9955			0.580556			0.290278	+	0.956942i	$0.406252\pi$
$858$		0			0
$859$	34.6252			1.18140			0.590698	−	0.806893i	$-0.298852\pi$
$859$	34.6252			1.18140			0.590698	+	0.806893i	$0.298852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.07272	−	3.07272i	−0.104597	−	0.104597i	0.652872	−	0.757468i	$-0.273564\pi$
$863$	−3.07272	−	3.07272i	−0.104597	−	0.104597i	−0.757468	+	0.652872i	$0.773564\pi$
$864$		0			0
$865$	−5.44049	+	20.3042i	−0.184982	+	0.690363i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.60634	−	13.4590i	−0.122337	−	0.456567i
$870$		0			0
$871$	28.6150	+	11.5781i	0.969581	+	0.392309i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.72188	+	2.98239i	0.0582103	+	0.100823i
$876$		0			0
$877$	−50.5323	−	13.5401i	−1.70635	−	0.457216i	−0.731828	−	0.681490i	$-0.761332\pi$
$877$	−50.5323	−	13.5401i	−1.70635	−	0.457216i	−0.974526	+	0.224274i	$0.927999\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5.53663	−	9.58972i	0.186534	−	0.323086i	−0.757558	−	0.652767i	$-0.773608\pi$
$881$	5.53663	−	9.58972i	0.186534	−	0.323086i	0.944092	+	0.329681i	$0.106941\pi$
$882$		0			0
$883$		−	16.6757i		−	0.561182i	−0.959827	−	0.280591i	$-0.909469\pi$
$883$		−	16.6757i		−	0.561182i	0.959827	−	0.280591i	$-0.0905305\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.8908	+	17.2575i	1.00364	+	0.579449i	0.909322	−	0.416093i	$-0.136601\pi$
$887$	29.8908	+	17.2575i	1.00364	+	0.579449i	0.0943136	+	0.995543i	$0.469934\pi$
$888$		0			0
$889$	4.89136	−	4.89136i	0.164051	−	0.164051i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−36.4310	+	21.0334i	−1.21912	+	0.703857i
$894$		0			0
$895$	−8.68154	+	2.32621i	−0.290192	+	0.0777567i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	38.3398	−	10.2731i	1.27870	−	0.342628i
$900$		0			0
$901$	39.4630	−	22.7840i	1.31470	−	0.759045i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.13770	+	5.13770i	−0.170783	+	0.170783i
$906$		0			0
$907$	1.12688	+	0.650606i	0.0374175	+	0.0216030i	0.518592	−	0.855022i	$-0.326456\pi$
$907$	1.12688	+	0.650606i	0.0374175	+	0.0216030i	−0.481175	+	0.876625i	$0.659790\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			38.6264i			1.27975i	0.768479	+	0.639875i	$0.221014\pi$
$911$			38.6264i			1.27975i	−0.768479	+	0.639875i	$0.778986\pi$
$912$		0			0
$913$	−14.6533	+	25.3803i	−0.484955	+	0.839966i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.94681	+	2.12934i	0.262427	+	0.0703170i
$918$		0			0
$919$	−5.16469	−	8.94550i	−0.170367	−	0.295085i	0.768181	−	0.640233i	$-0.221162\pi$
$919$	−5.16469	−	8.94550i	−0.170367	−	0.295085i	−0.938548	+	0.345148i	$0.887829\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	25.9271	+	33.2200i	0.853402	+	1.09345i
$924$		0			0
$925$	1.80979	+	6.75423i	0.0595056	+	0.222078i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.6165	−	43.3535i	0.381126	−	1.42238i	−0.463058	−	0.886328i	$-0.653248\pi$
$929$	11.6165	−	43.3535i	0.381126	−	1.42238i	0.844184	−	0.536054i	$-0.180086\pi$
$930$		0			0
$931$	−21.0610	−	21.0610i	−0.690246	−	0.690246i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.2095			0.366589
$936$		0			0
$937$	17.3699			0.567451			0.283726	−	0.958905i	$-0.408429\pi$
$937$	17.3699			0.567451			0.283726	+	0.958905i	$0.408429\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.2216	−	22.2216i	−0.724403	−	0.724403i	0.245096	−	0.969499i	$-0.421180\pi$
$941$	−22.2216	−	22.2216i	−0.724403	−	0.724403i	−0.969499	+	0.245096i	$0.921180\pi$
$942$		0			0
$943$	−9.08772	+	33.9158i	−0.295937	+	1.10445i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.51192	−	28.0349i	−0.244105	−	0.911011i	−0.973831	−	0.227272i	$-0.927019\pi$
$947$	−7.51192	−	28.0349i	−0.244105	−	0.911011i	0.729727	−	0.683739i	$-0.239647\pi$
$948$		0			0
$949$	41.5868	−	5.82309i	1.34996	−	0.189026i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	22.9684	+	39.7824i	0.744020	+	1.28868i	0.950651	+	0.310261i	$0.100416\pi$
$953$	22.9684	+	39.7824i	0.744020	+	1.28868i	−0.206632	+	0.978419i	$0.566250\pi$
$954$		0			0
$955$	9.05945	+	2.42747i	0.293157	+	0.0785512i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−31.2534	+	54.1325i	−1.00923	+	1.74803i
$960$		0			0
$961$		−	65.9668i		−	2.12796i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.4634	+	8.92780i	0.497785	+	0.287396i
$966$		0			0
$967$	39.6468	−	39.6468i	1.27496	−	1.27496i	0.331502	−	0.943455i	$-0.392445\pi$
$967$	39.6468	−	39.6468i	1.27496	−	1.27496i	0.943455	−	0.331502i	$-0.107555\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	40.4780	−	23.3700i	1.29900	−	0.749978i	0.318768	−	0.947833i	$-0.396731\pi$
$971$	40.4780	−	23.3700i	1.29900	−	0.749978i	0.980231	+	0.197855i	$0.0633975\pi$
$972$		0			0
$973$	−28.4724	+	7.62916i	−0.912783	+	0.244580i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.90189	−	2.65320i	0.316789	−	0.0848835i	−0.0969208	−	0.995292i	$-0.530899\pi$
$977$	9.90189	−	2.65320i	0.316789	−	0.0848835i	0.413710	+	0.910409i	$0.364233\pi$
$978$		0			0
$979$	3.28560	−	1.89694i	0.105008	−	0.0606265i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.59079	−	4.59079i	0.146423	−	0.146423i	−0.630095	−	0.776518i	$-0.716984\pi$
$983$	4.59079	−	4.59079i	0.146423	−	0.146423i	0.776518	+	0.630095i	$0.216984\pi$
$984$		0			0
$985$	−6.39941	−	3.69470i	−0.203902	−	0.117723i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			6.44774i			0.205026i
$990$		0			0
$991$	−14.0237	+	24.2898i	−0.445478	+	0.771590i	−0.998085	−	0.0618513i	$-0.980300\pi$
$991$	−14.0237	+	24.2898i	−0.445478	+	0.771590i	0.552607	+	0.833442i	$0.313633\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.8088	+	2.89621i	0.342663	+	0.0918161i
$996$		0			0
$997$	14.8959	+	25.8005i	0.471759	+	0.817111i	0.999478	−	0.0323083i	$-0.0102858\pi$
$997$	14.8959	+	25.8005i	0.471759	+	0.817111i	−0.527719	+	0.849419i	$0.676953\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.fo.b.1241.6	yes	40
3.2	odd	2	inner	2340.2.fo.b.1241.4	✓	40
13.11	odd	12	inner	2340.2.fo.b.1961.4	yes	40
39.11	even	12	inner	2340.2.fo.b.1961.6	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2340.2.fo.b.1241.4	✓	40	3.2	odd	2	inner
2340.2.fo.b.1241.6	yes	40	1.1	even	1	trivial
2340.2.fo.b.1961.4	yes	40	13.11	odd	12	inner
2340.2.fo.b.1961.6	yes	40	39.11	even	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2340.fo (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{5} +(0.891313 - 3.32643i) q^{7} +(0.508619 + 1.89819i) q^{11} +(-2.17134 + 2.87842i) q^{13} +(-2.85207 - 4.93992i) q^{17} +(5.92027 + 1.58633i) q^{19} +(-2.03246 + 3.52032i) q^{23} +1.00000i q^{25} +(3.49080 + 2.01541i) q^{29} +(6.96300 - 6.96300i) q^{31} +(2.98239 - 1.72188i) q^{35} +(6.75423 - 1.80979i) q^{37} +(8.34356 - 2.23565i) q^{41} +(1.37368 - 0.793097i) q^{43} +(-4.85319 + 4.85319i) q^{47} +(-4.20849 - 2.42977i) q^{49} +7.98860i q^{53} +(-0.982576 + 1.70187i) q^{55} +(6.45232 + 1.72889i) q^{59} +(-4.91891 - 8.51980i) q^{61} +(-3.57072 + 0.499981i) q^{65} +(-2.21585 - 8.26968i) q^{67} +(3.02496 - 11.2893i) q^{71} +(-8.23540 - 8.23540i) q^{73} +6.76753 q^{77} -7.09045 q^{79} +(10.5452 + 10.5452i) q^{83} +(1.47634 - 5.50977i) q^{85} +(-0.499670 - 1.86479i) q^{89} +(7.63950 + 9.78837i) q^{91} +(3.06456 + 5.30797i) q^{95} +(0.729538 + 0.195479i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 8 q^{7} + 12 q^{13} + 28 q^{19} + 36 q^{31} - 40 q^{37} - 72 q^{43} - 36 q^{49} - 32 q^{61} + 16 q^{67} + 28 q^{73} + 48 q^{79} - 20 q^{85} + 132 q^{91} + 48 q^{97}+O(q^{100}) \) 40 * q + 8 * q^7 + 12 * q^13 + 28 * q^19 + 36 * q^31 - 40 * q^37 - 72 * q^43 - 36 * q^49 - 32 * q^61 + 16 * q^67 + 28 * q^73 + 48 * q^79 - 20 * q^85 + 132 * q^91 + 48 * q^97

Introduction

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