LMFDB - Embedded newform 2340.2.u.i.577.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2340,2,Mod(73,2340)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2340, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2340.73");

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.u (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.6849940730$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 780)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.6
Character	$\chi$	$=$	2340.577
Dual form			2340.2.u.i.73.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.17614 - 1.90176i) q^{5} -4.16665i q^{7} +O(q^{10})$	$q+(-1.17614 - 1.90176i) q^{5} -4.16665i q^{7} +(3.15147 + 3.15147i) q^{11} +(1.82867 - 3.10741i) q^{13} +(4.18610 - 4.18610i) q^{17} +(-4.37727 - 4.37727i) q^{19} +(1.31199 + 1.31199i) q^{23} +(-2.23337 + 4.47348i) q^{25} +3.93181i q^{29} +(3.84918 - 3.84918i) q^{31} +(-7.92397 + 4.90058i) q^{35} -8.44004i q^{37} +(-1.55464 + 1.55464i) q^{41} +(8.11673 + 8.11673i) q^{43} +1.02013i q^{47} -10.3610 q^{49} +(1.36494 - 1.36494i) q^{53} +(2.28676 - 9.69993i) q^{55} +(1.43740 - 1.43740i) q^{59} +0.293533 q^{61} +(-8.06031 + 0.177065i) q^{65} -11.8746 q^{67} +(-11.3963 + 11.3963i) q^{71} +6.46158 q^{73} +(13.1311 - 13.1311i) q^{77} -8.32607i q^{79} -15.1492i q^{83} +(-12.8844 - 3.03750i) q^{85} +(-6.44206 + 6.44206i) q^{89} +(-12.9475 - 7.61943i) q^{91} +(-3.17622 + 13.4728i) q^{95} -10.9521 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q+O(q^{10}) $ 28 * q	$ 28 q - 8 q^{11} - 4 q^{13} + 4 q^{17} - 16 q^{19} + 8 q^{23} + 12 q^{25} - 12 q^{41} + 16 q^{43} - 36 q^{49} - 36 q^{53} + 40 q^{55} - 16 q^{59} + 8 q^{61} + 40 q^{65} + 48 q^{67} - 8 q^{71} + 48 q^{73} + 48 q^{77} - 4 q^{85} + 36 q^{89} - 24 q^{91} - 72 q^{95} - 72 q^{97}+O(q^{100}) $ 28 * q - 8 * q^11 - 4 * q^13 + 4 * q^17 - 16 * q^19 + 8 * q^23 + 12 * q^25 - 12 * q^41 + 16 * q^43 - 36 * q^49 - 36 * q^53 + 40 * q^55 - 16 * q^59 + 8 * q^61 + 40 * q^65 + 48 * q^67 - 8 * q^71 + 48 * q^73 + 48 * q^77 - 4 * q^85 + 36 * q^89 - 24 * q^91 - 72 * q^95 - 72 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	−1.17614	−	1.90176i	−0.525987	−	0.850492i
$5$	−1.17614	−	1.90176i	−0.525987	−	0.850492i
$6$		0			0
$7$		−	4.16665i		−	1.57485i	−0.616413	−	0.787423i	$-0.711415\pi$
$7$		−	4.16665i		−	1.57485i	0.616413	−	0.787423i	$-0.288585\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.15147	+	3.15147i	0.950205	+	0.950205i	0.998818	−	0.0486129i	$-0.0154801\pi$
$11$	3.15147	+	3.15147i	0.950205	+	0.950205i	−0.0486129	+	0.998818i	$0.515480\pi$
$12$		0			0
$13$	1.82867	−	3.10741i	0.507182	−	0.861839i
$13$	1.82867	−	3.10741i	0.507182	−	0.861839i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.18610	−	4.18610i	1.01528	−	1.01528i	0.0153976	−	0.999881i	$-0.495099\pi$
$17$	4.18610	−	4.18610i	1.01528	−	1.01528i	0.999881	−	0.0153976i	$-0.00490140\pi$
$18$		0			0
$19$	−4.37727	−	4.37727i	−1.00422	−	1.00422i	−0.999991	−	0.00422428i	$-0.998655\pi$
$19$	−4.37727	−	4.37727i	−1.00422	−	1.00422i	−0.00422428	−	0.999991i	$-0.501345\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.31199	+	1.31199i	0.273569	+	0.273569i	0.830535	−	0.556966i	$-0.188035\pi$
$23$	1.31199	+	1.31199i	0.273569	+	0.273569i	−0.556966	+	0.830535i	$0.688035\pi$
$24$		0			0
$25$	−2.23337	+	4.47348i	−0.446675	+	0.894696i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.93181i			0.730119i	0.930984	+	0.365060i	$0.118951\pi$
$29$			3.93181i			0.730119i	−0.930984	+	0.365060i	$0.881049\pi$
$30$		0			0
$31$	3.84918	−	3.84918i	0.691334	−	0.691334i	−0.271192	−	0.962525i	$-0.587418\pi$
$31$	3.84918	−	3.84918i	0.691334	−	0.691334i	0.962525	+	0.271192i	$0.0874177\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.92397	+	4.90058i	−1.33939	+	0.828349i
$36$		0			0
$37$		−	8.44004i		−	1.38753i	−0.720200	−	0.693767i	$-0.755950\pi$
$37$		−	8.44004i		−	1.38753i	0.720200	−	0.693767i	$-0.244050\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.55464	+	1.55464i	−0.242794	+	0.242794i	−0.818005	−	0.575211i	$-0.804920\pi$
$41$	−1.55464	+	1.55464i	−0.242794	+	0.242794i	0.575211	+	0.818005i	$0.304920\pi$
$42$		0			0
$43$	8.11673	+	8.11673i	1.23779	+	1.23779i	0.960903	+	0.276887i	$0.0893026\pi$
$43$	8.11673	+	8.11673i	1.23779	+	1.23779i	0.276887	+	0.960903i	$0.410697\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.02013i			0.148801i	0.997228	+	0.0744003i	$0.0237043\pi$
$47$			1.02013i			0.148801i	−0.997228	+	0.0744003i	$0.976296\pi$
$48$		0			0
$49$	−10.3610			−1.48014
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.36494	−	1.36494i	0.187489	−	0.187489i	−0.607121	−	0.794610i	$-0.707676\pi$
$53$	1.36494	−	1.36494i	0.187489	−	0.187489i	0.794610	+	0.607121i	$0.207676\pi$
$54$		0			0
$55$	2.28676	−	9.69993i	0.308346	−	1.30794i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.43740	−	1.43740i	0.187134	−	0.187134i	−0.607322	−	0.794456i	$-0.707756\pi$
$59$	1.43740	−	1.43740i	0.187134	−	0.187134i	0.794456	+	0.607322i	$0.207756\pi$
$60$		0			0
$61$	0.293533			0.0375831			0.0187915	−	0.999823i	$-0.494018\pi$
$61$	0.293533			0.0375831			0.0187915	+	0.999823i	$0.494018\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.06031	+	0.177065i	−0.999759	+	0.0219623i
$66$		0			0
$67$	−11.8746			−1.45071			−0.725357	−	0.688373i	$-0.758325\pi$
$67$	−11.8746			−1.45071			−0.725357	+	0.688373i	$0.758325\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.3963	+	11.3963i	−1.35249	+	1.35249i	−0.469625	+	0.882866i	$0.655611\pi$
$71$	−11.3963	+	11.3963i	−1.35249	+	1.35249i	−0.882866	+	0.469625i	$0.844389\pi$
$72$		0			0
$73$	6.46158			0.756271			0.378135	−	0.925750i	$-0.376565\pi$
$73$	6.46158			0.756271			0.378135	+	0.925750i	$0.376565\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13.1311	−	13.1311i	1.49643	−	1.49643i
$78$		0			0
$79$		−	8.32607i		−	0.936756i	−0.883528	−	0.468378i	$-0.844838\pi$
$79$		−	8.32607i		−	0.936756i	0.883528	−	0.468378i	$-0.155162\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	15.1492i		−	1.66285i	−0.555640	−	0.831423i	$-0.687527\pi$
$83$		−	15.1492i		−	1.66285i	0.555640	−	0.831423i	$-0.312473\pi$
$84$		0			0
$85$	−12.8844	−	3.03750i	−1.39751	−	0.329463i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.44206	+	6.44206i	−0.682857	+	0.682857i	−0.960643	−	0.277786i	$-0.910399\pi$
$89$	−6.44206	+	6.44206i	−0.682857	+	0.682857i	0.277786	+	0.960643i	$0.410399\pi$
$90$		0			0
$91$	−12.9475	−	7.61943i	−1.35726	−	0.798733i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.17622	+	13.4728i	−0.325873	+	1.38228i
$96$		0			0
$97$	−10.9521			−1.11201			−0.556007	−	0.831178i	$-0.687667\pi$
$97$	−10.9521			−1.11201			−0.556007	+	0.831178i	$0.687667\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	8.05908i		−	0.801908i	−0.916098	−	0.400954i	$-0.868679\pi$
$101$		−	8.05908i		−	0.801908i	0.916098	−	0.400954i	$-0.131321\pi$
$102$		0			0
$103$	4.15942	+	4.15942i	0.409840	+	0.409840i	0.881683	−	0.471843i	$-0.156411\pi$
$103$	4.15942	+	4.15942i	0.409840	+	0.409840i	−0.471843	+	0.881683i	$0.656411\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.88527	−	1.88527i	−0.182256	−	0.182256i	0.610082	−	0.792338i	$-0.291137\pi$
$107$	−1.88527	−	1.88527i	−0.182256	−	0.182256i	−0.792338	+	0.610082i	$0.791137\pi$
$108$		0			0
$109$	−3.25319	−	3.25319i	−0.311599	−	0.311599i	0.533930	−	0.845529i	$-0.320715\pi$
$109$	−3.25319	−	3.25319i	−0.311599	−	0.311599i	−0.845529	+	0.533930i	$0.820715\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.79437	+	7.79437i	−0.733233	+	0.733233i	−0.971259	−	0.238026i	$-0.923500\pi$
$113$	−7.79437	+	7.79437i	−0.733233	+	0.733233i	0.238026	+	0.971259i	$0.423500\pi$
$114$		0			0
$115$	0.952001	−	4.03818i	0.0887746	−	0.376562i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.4420	−	17.4420i	−1.59891	−	1.59891i
$120$		0			0
$121$			8.86356i			0.805778i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1343	−	1.01412i	0.995878	−	0.0907052i
$126$		0			0
$127$	−4.51457	+	4.51457i	−0.400604	+	0.400604i	−0.878446	−	0.477842i	$-0.841419\pi$
$127$	−4.51457	+	4.51457i	−0.400604	+	0.400604i	0.477842	+	0.878446i	$0.341419\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.47998			−0.566158			−0.283079	−	0.959097i	$-0.591356\pi$
$131$	−6.47998			−0.566158			−0.283079	+	0.959097i	$0.591356\pi$
$132$		0			0
$133$	−18.2386	+	18.2386i	−1.58148	+	1.58148i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	14.7302i		−	1.25848i	−0.777209	−	0.629242i	$-0.783365\pi$
$137$		−	14.7302i		−	1.25848i	0.777209	−	0.629242i	$-0.216635\pi$
$138$		0			0
$139$		−	7.19379i		−	0.610169i	−0.952325	−	0.305085i	$-0.901315\pi$
$139$		−	7.19379i		−	0.610169i	0.952325	−	0.305085i	$-0.0986848\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	15.5559	−	4.02990i	1.30085	−	0.336997i
$144$		0			0
$145$	7.47736	−	4.62438i	0.620961	−	0.384033i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.19801	−	7.19801i	−0.589684	−	0.589684i	0.347862	−	0.937546i	$-0.386908\pi$
$149$	−7.19801	−	7.19801i	−0.589684	−	0.589684i	−0.937546	+	0.347862i	$0.886908\pi$
$150$		0			0
$151$	6.77767	+	6.77767i	0.551559	+	0.551559i	0.926891	−	0.375332i	$-0.122471\pi$
$151$	6.77767	+	6.77767i	0.551559	+	0.551559i	−0.375332	+	0.926891i	$0.622471\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.8474	−	2.79303i	−0.951607	−	0.224341i
$156$		0			0
$157$	11.4226	+	11.4226i	0.911626	+	0.911626i	0.996400	−	0.0847741i	$-0.0270169\pi$
$157$	11.4226	+	11.4226i	0.911626	+	0.911626i	−0.0847741	+	0.996400i	$0.527017\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.46661	−	5.46661i	0.430829	−	0.430829i
$162$		0			0
$163$	−15.5052			−1.21446			−0.607230	−	0.794526i	$-0.707719\pi$
$163$	−15.5052			−1.21446			−0.607230	+	0.794526i	$0.707719\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			10.8047i			0.836092i	0.908426	+	0.418046i	$0.137285\pi$
$167$			10.8047i			0.836092i	−0.908426	+	0.418046i	$0.862715\pi$
$168$		0			0
$169$	−6.31194	−	11.3648i	−0.485534	−	0.874218i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.61397	−	7.61397i	−0.578879	−	0.578879i	0.355715	−	0.934594i	$-0.384237\pi$
$173$	−7.61397	−	7.61397i	−0.578879	−	0.578879i	−0.934594	+	0.355715i	$0.884237\pi$
$174$		0			0
$175$	18.6394	+	9.30569i	1.40901	+	0.703444i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.2476			0.990175			0.495087	−	0.868843i	$-0.335136\pi$
$179$	13.2476			0.990175			0.495087	+	0.868843i	$0.335136\pi$
$180$		0			0
$181$		−	13.5363i		−	1.00614i	−0.864245	−	0.503072i	$-0.832203\pi$
$181$		−	13.5363i		−	1.00614i	0.864245	−	0.503072i	$-0.167797\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.0509	+	9.92669i	−1.18009	+	0.729825i
$186$		0			0
$187$	26.3848			1.92945
$188$		0			0
$189$		0			0
$190$		0			0
$191$	23.5256			1.70226			0.851128	−	0.524959i	$-0.175919\pi$
$191$	23.5256			1.70226			0.851128	+	0.524959i	$0.175919\pi$
$192$		0			0
$193$	12.6069			0.907466			0.453733	−	0.891138i	$-0.350092\pi$
$193$	12.6069			0.907466			0.453733	+	0.891138i	$0.350092\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.4136			0.955681			0.477841	−	0.878447i	$-0.341420\pi$
$197$	13.4136			0.955681			0.477841	+	0.878447i	$0.341420\pi$
$198$		0			0
$199$	22.5835			1.60091			0.800453	−	0.599396i	$-0.204592\pi$
$199$	22.5835			1.60091			0.800453	+	0.599396i	$0.204592\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	16.3825			1.14983
$204$		0			0
$205$	4.78503	+	1.12807i	0.334201	+	0.0787879i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	27.5897i		−	1.90842i
$210$		0			0
$211$	−12.0590			−0.830175			−0.415088	−	0.909781i	$-0.636249\pi$
$211$	−12.0590			−0.830175			−0.415088	+	0.909781i	$0.636249\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.88962	−	24.9825i	0.401669	−	1.70379i
$216$		0			0
$217$	−16.0382	−	16.0382i	−1.08874	−	1.08874i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.35292	−	20.6629i	−0.360076	−	1.38994i
$222$		0			0
$223$		−	9.25360i		−	0.619667i	−0.950791	−	0.309834i	$-0.899727\pi$
$223$		−	9.25360i		−	0.619667i	0.950791	−	0.309834i	$-0.100273\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.0754			1.06696			0.533481	−	0.845812i	$-0.320884\pi$
$227$	16.0754			1.06696			0.533481	+	0.845812i	$0.320884\pi$
$228$		0			0
$229$	−5.43244	+	5.43244i	−0.358986	+	0.358986i	−0.863439	−	0.504453i	$-0.831694\pi$
$229$	−5.43244	+	5.43244i	−0.358986	+	0.358986i	0.504453	+	0.863439i	$0.331694\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.12524	+	1.12524i	0.0737170	+	0.0737170i	0.743004	−	0.669287i	$-0.233400\pi$
$233$	1.12524	+	1.12524i	0.0737170	+	0.0737170i	−0.669287	+	0.743004i	$0.733400\pi$
$234$		0			0
$235$	1.94003	−	1.19981i	0.126554	−	0.0782672i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.4781	+	12.4781i	0.807144	+	0.807144i	0.984201	−	0.177057i	$-0.0566576\pi$
$239$	12.4781	+	12.4781i	0.807144	+	0.807144i	−0.177057	+	0.984201i	$0.556658\pi$
$240$		0			0
$241$	1.27702	+	1.27702i	0.0822603	+	0.0822603i	0.747040	−	0.664779i	$-0.231474\pi$
$241$	1.27702	+	1.27702i	0.0822603	+	0.0822603i	−0.664779	+	0.747040i	$0.731474\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.1860	+	19.7041i	0.778535	+	1.25885i
$246$		0			0
$247$	−21.6065	+	5.59738i	−1.37479	+	0.356153i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			19.6566i			1.24072i	0.784319	+	0.620358i	$0.213013\pi$
$251$			19.6566i			1.24072i	−0.784319	+	0.620358i	$0.786987\pi$
$252$		0			0
$253$			8.26940i			0.519893i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.8698	+	20.8698i	−1.30182	+	1.30182i	−0.374662	+	0.927162i	$0.622241\pi$
$257$	−20.8698	+	20.8698i	−1.30182	+	1.30182i	−0.927162	+	0.374662i	$0.877759\pi$
$258$		0			0
$259$	−35.1667			−2.18515
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.80349	+	3.80349i	−0.234533	+	0.234533i	−0.814582	−	0.580049i	$-0.803034\pi$
$263$	−3.80349	+	3.80349i	−0.234533	+	0.234533i	0.580049	+	0.814582i	$0.303034\pi$
$264$		0			0
$265$	−4.20115	−	0.990421i	−0.258074	−	0.0608411i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	3.31848i		−	0.202332i	−0.994870	−	0.101166i	$-0.967743\pi$
$269$		−	3.31848i		−	0.202332i	0.994870	−	0.101166i	$-0.0322572\pi$
$270$		0			0
$271$	5.56659	+	5.56659i	0.338146	+	0.338146i	0.855669	−	0.517523i	$-0.173146\pi$
$271$	5.56659	+	5.56659i	0.338146	+	0.338146i	−0.517523	+	0.855669i	$0.673146\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−21.1365	+	7.05964i	−1.27458	+	0.425712i
$276$		0			0
$277$	−3.00278	+	3.00278i	−0.180419	+	0.180419i	−0.791539	−	0.611119i	$-0.790720\pi$
$277$	−3.00278	+	3.00278i	−0.180419	+	0.180419i	0.611119	+	0.791539i	$0.290720\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.34615	−	2.34615i	−0.139960	−	0.139960i	0.633656	−	0.773615i	$-0.281554\pi$
$281$	−2.34615	−	2.34615i	−0.139960	−	0.139960i	−0.773615	+	0.633656i	$0.781554\pi$
$282$		0			0
$283$	16.7918	+	16.7918i	0.998167	+	0.998167i	0.999998	−	0.00183106i	$-0.000582846\pi$
$283$	16.7918	+	16.7918i	0.998167	+	0.998167i	−0.00183106	+	0.999998i	$0.500583\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.47764	+	6.47764i	0.382363	+	0.382363i
$288$		0			0
$289$		−	18.0469i		−	1.06158i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.6729			−0.623516			−0.311758	−	0.950162i	$-0.600918\pi$
$293$	−10.6729			−0.623516			−0.311758	+	0.950162i	$0.600918\pi$
$294$		0			0
$295$	−4.42419	−	1.04300i	−0.257586	−	0.0607260i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.47608	−	1.67769i	0.374522	−	0.0970233i
$300$		0			0
$301$	33.8196	−	33.8196i	1.94933	−	1.94933i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.345237	−	0.558229i	−0.0197682	−	0.0319641i
$306$		0			0
$307$			9.95047i			0.567903i	0.958839	+	0.283952i	$0.0916455\pi$
$307$			9.95047i			0.567903i	−0.958839	+	0.283952i	$0.908355\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	5.16651i		−	0.292966i	−0.989213	−	0.146483i	$-0.953205\pi$
$311$		−	5.16651i		−	0.292966i	0.989213	−	0.146483i	$-0.0467954\pi$
$312$		0			0
$313$	2.76718	−	2.76718i	0.156410	−	0.156410i	−0.624564	−	0.780974i	$-0.714723\pi$
$313$	2.76718	−	2.76718i	0.156410	−	0.156410i	0.780974	+	0.624564i	$0.214723\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−29.1994			−1.64000			−0.820000	−	0.572363i	$-0.806027\pi$
$317$	−29.1994			−1.64000			−0.820000	+	0.572363i	$0.806027\pi$
$318$		0			0
$319$	−12.3910	+	12.3910i	−0.693763	+	0.693763i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.6474			−2.03912
$324$		0			0
$325$	9.81682	+	15.1205i	0.544539	+	0.838735i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.25051			0.234338
$330$		0			0
$331$	−8.69091	+	8.69091i	−0.477696	+	0.477696i	−0.904394	−	0.426698i	$-0.859677\pi$
$331$	−8.69091	+	8.69091i	−0.477696	+	0.477696i	0.426698	+	0.904394i	$0.359677\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	13.9662	+	22.5826i	0.763057	+	1.23382i
$336$		0			0
$337$	−7.49283	+	7.49283i	−0.408161	+	0.408161i	−0.881097	−	0.472936i	$-0.843194\pi$
$337$	−7.49283	+	7.49283i	−0.408161	+	0.408161i	0.472936	+	0.881097i	$0.343194\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	24.2612			1.31382
$342$		0			0
$343$			14.0040i			0.756147i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.92612	+	9.92612i	0.532862	+	0.532862i	0.921423	−	0.388561i	$-0.127028\pi$
$347$	9.92612	+	9.92612i	0.532862	+	0.532862i	−0.388561	+	0.921423i	$0.627028\pi$
$348$		0			0
$349$	7.52765	−	7.52765i	0.402946	−	0.402946i	−0.476324	−	0.879270i	$-0.658031\pi$
$349$	7.52765	−	7.52765i	0.402946	−	0.402946i	0.879270	+	0.476324i	$0.158031\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	7.33990i		−	0.390664i	−0.980737	−	0.195332i	$-0.937422\pi$
$353$		−	7.33990i		−	0.390664i	0.980737	−	0.195332i	$-0.0625784\pi$
$354$		0			0
$355$	35.0767	+	8.26933i	1.86168	+	0.438890i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.1816	+	24.1816i	−1.27626	+	1.27626i	−0.333509	+	0.942747i	$0.608233\pi$
$359$	−24.1816	+	24.1816i	−1.27626	+	1.27626i	−0.942747	+	0.333509i	$0.891767\pi$
$360$		0			0
$361$			19.3210i			1.01690i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.59974	−	12.2884i	−0.397789	−	0.643203i
$366$		0			0
$367$	17.7690	+	17.7690i	0.927537	+	0.927537i	0.997546	−	0.0700097i	$-0.0223030\pi$
$367$	17.7690	+	17.7690i	0.927537	+	0.927537i	−0.0700097	+	0.997546i	$0.522303\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.68722	−	5.68722i	−0.295266	−	0.295266i
$372$		0			0
$373$	−3.42778	+	3.42778i	−0.177484	+	0.177484i	−0.790258	−	0.612774i	$-0.790054\pi$
$373$	−3.42778	+	3.42778i	−0.177484	+	0.177484i	0.612774	+	0.790258i	$0.290054\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.2177	+	7.18999i	0.629245	+	0.370303i
$378$		0			0
$379$	6.32325	+	6.32325i	0.324804	+	0.324804i	0.850606	−	0.525803i	$-0.176235\pi$
$379$	6.32325	+	6.32325i	0.324804	+	0.324804i	−0.525803	+	0.850606i	$0.676235\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			24.6950i			1.26186i	0.775841	+	0.630928i	$0.217326\pi$
$383$			24.6950i			1.26186i	−0.775841	+	0.630928i	$0.782674\pi$
$384$		0			0
$385$	−40.4162	−	9.52812i	−2.05980	−	0.485598i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.4062			1.38955			0.694775	−	0.719228i	$-0.255504\pi$
$389$	27.4062			1.38955			0.694775	+	0.719228i	$0.255504\pi$
$390$		0			0
$391$	10.9843			0.555498
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.8342	+	9.79265i	−0.796704	+	0.492722i
$396$		0			0
$397$		−	39.1875i		−	1.96676i	−0.181554	−	0.983381i	$-0.558113\pi$
$397$		−	39.1875i		−	1.96676i	0.181554	−	0.983381i	$-0.441887\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−21.9890	−	21.9890i	−1.09808	−	1.09808i	−0.994636	−	0.103441i	$-0.967015\pi$
$401$	−21.9890	−	21.9890i	−1.09808	−	1.09808i	−0.103441	−	0.994636i	$-0.532985\pi$
$402$		0			0
$403$	−4.92209	−	18.9999i	−0.245187	−	0.946450i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	26.5985	−	26.5985i	1.31844	−	1.31844i
$408$		0			0
$409$	−2.71373	−	2.71373i	−0.134185	−	0.134185i	0.636824	−	0.771009i	$-0.280248\pi$
$409$	−2.71373	−	2.71373i	−0.134185	−	0.134185i	−0.771009	+	0.636824i	$0.780248\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.98916	−	5.98916i	−0.294707	−	0.294707i
$414$		0			0
$415$	−28.8102	+	17.8177i	−1.41424	+	0.874636i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			34.1751i			1.66956i	0.550583	+	0.834780i	$0.314405\pi$
$419$			34.1751i			1.66956i	−0.550583	+	0.834780i	$0.685595\pi$
$420$		0			0
$421$	15.5333	−	15.5333i	0.757049	−	0.757049i	−0.218736	−	0.975784i	$-0.570193\pi$
$421$	15.5333	−	15.5333i	0.757049	−	0.757049i	0.975784	+	0.218736i	$0.0701932\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	9.37732	+	28.0756i	0.454867	+	1.36187i
$426$		0			0
$427$		−	1.22305i		−	0.0591875i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.4717	−	26.4717i	1.27510	−	1.27510i	0.331722	−	0.943377i	$-0.392371\pi$
$431$	26.4717	−	26.4717i	1.27510	−	1.27510i	0.943377	−	0.331722i	$-0.107629\pi$
$432$		0			0
$433$	−26.0961	−	26.0961i	−1.25410	−	1.25410i	−0.953866	−	0.300232i	$-0.902936\pi$
$433$	−26.0961	−	26.0961i	−1.25410	−	1.25410i	−0.300232	−	0.953866i	$-0.597064\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	11.4859i		−	0.549444i
$438$		0			0
$439$	−12.5133			−0.597226			−0.298613	−	0.954374i	$-0.596524\pi$
$439$	−12.5133			−0.597226			−0.298613	+	0.954374i	$0.596524\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.48719	−	1.48719i	0.0706585	−	0.0706585i	−0.670894	−	0.741553i	$-0.734090\pi$
$443$	1.48719	−	1.48719i	0.0706585	−	0.0706585i	0.741553	+	0.670894i	$0.234090\pi$
$444$		0			0
$445$	19.8280	+	4.67446i	0.939939	+	0.221591i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11.5904	+	11.5904i	−0.546986	+	0.546986i	−0.925568	−	0.378582i	$-0.876412\pi$
$449$	−11.5904	+	11.5904i	−0.546986	+	0.546986i	0.378582	+	0.925568i	$0.376412\pi$
$450$		0			0
$451$	−9.79880			−0.461408
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.737770	+	33.5845i	0.0345872	+	1.57447i
$456$		0			0
$457$	34.6269			1.61978			0.809888	−	0.586584i	$-0.199528\pi$
$457$	34.6269			1.61978			0.809888	+	0.586584i	$0.199528\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.8332	−	18.8332i	0.877149	−	0.877149i	−0.116090	−	0.993239i	$-0.537036\pi$
$461$	18.8332	−	18.8332i	0.877149	−	0.877149i	0.993239	+	0.116090i	$0.0370361\pi$
$462$		0			0
$463$	13.5047			0.627615			0.313808	−	0.949487i	$-0.398395\pi$
$463$	13.5047			0.627615			0.313808	+	0.949487i	$0.398395\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.20739	−	7.20739i	0.333518	−	0.333518i	−0.520403	−	0.853921i	$-0.674218\pi$
$467$	7.20739	−	7.20739i	0.333518	−	0.333518i	0.853921	+	0.520403i	$0.174218\pi$
$468$		0			0
$469$			49.4773i			2.28465i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			51.1593i			2.35231i
$474$		0			0
$475$	29.3577	−	9.80556i	1.34703	−	0.449910i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.3993	+	12.3993i	−0.566537	+	0.566537i	−0.931157	−	0.364620i	$-0.881199\pi$
$479$	−12.3993	+	12.3993i	−0.566537	+	0.566537i	0.364620	+	0.931157i	$0.381199\pi$
$480$		0			0
$481$	−26.2266	−	15.4340i	−1.19583	−	0.703731i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.8812	+	20.8282i	0.584905	+	0.945760i
$486$		0			0
$487$	32.1455			1.45665			0.728326	−	0.685230i	$-0.240298\pi$
$487$	32.1455			1.45665			0.728326	+	0.685230i	$0.240298\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			3.73137i			0.168395i	0.996449	+	0.0841973i	$0.0268326\pi$
$491$			3.73137i			0.168395i	−0.996449	+	0.0841973i	$0.973167\pi$
$492$		0			0
$493$	16.4590	+	16.4590i	0.741275	+	0.741275i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	47.4844	+	47.4844i	2.12997	+	2.12997i
$498$		0			0
$499$	5.85645	+	5.85645i	0.262171	+	0.262171i	0.825936	−	0.563765i	$-0.190647\pi$
$499$	5.85645	+	5.85645i	0.262171	+	0.262171i	−0.563765	+	0.825936i	$0.690647\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.0951	−	21.0951i	0.940586	−	0.940586i	−0.0577458	−	0.998331i	$-0.518391\pi$
$503$	21.0951	−	21.0951i	0.940586	−	0.940586i	0.998331	+	0.0577458i	$0.0183913\pi$
$504$		0			0
$505$	−15.3264	+	9.47863i	−0.682017	+	0.421793i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−23.2392	−	23.2392i	−1.03006	−	1.03006i	−0.999534	−	0.0305246i	$-0.990282\pi$
$509$	−23.2392	−	23.2392i	−1.03006	−	1.03006i	−0.0305246	−	0.999534i	$-0.509718\pi$
$510$		0			0
$511$		−	26.9231i		−	1.19101i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.01814	−	12.8023i	0.132995	−	0.564137i
$516$		0			0
$517$	−3.21490	+	3.21490i	−0.141391	+	0.141391i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.5024			−0.942038			−0.471019	−	0.882123i	$-0.656114\pi$
$521$	−21.5024			−0.942038			−0.471019	+	0.882123i	$0.656114\pi$
$522$		0			0
$523$	14.0736	−	14.0736i	0.615395	−	0.615395i	−0.328952	−	0.944347i	$-0.606695\pi$
$523$	14.0736	−	14.0736i	0.615395	−	0.615395i	0.944347	+	0.328952i	$0.106695\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	32.2262i		−	1.40379i
$528$		0			0
$529$		−	19.5574i		−	0.850320i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.98797	+	7.67381i	0.0861087	+	0.332390i
$534$		0			0
$535$	−1.36798	+	5.80269i	−0.0591431	+	0.250872i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−32.6523	−	32.6523i	−1.40644	−	1.40644i
$540$		0			0
$541$	11.4516	+	11.4516i	0.492344	+	0.492344i	0.909044	−	0.416700i	$-0.136813\pi$
$541$	11.4516	+	11.4516i	0.492344	+	0.492344i	−0.416700	+	0.909044i	$0.636813\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.36057	+	10.0130i	−0.101116	+	0.428910i
$546$		0			0
$547$	16.7429	+	16.7429i	0.715874	+	0.715874i	0.967757	−	0.251884i	$-0.0810501\pi$
$547$	16.7429	+	16.7429i	0.715874	+	0.715874i	−0.251884	+	0.967757i	$0.581050\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.2106	−	17.2106i	0.733197	−	0.733197i
$552$		0			0
$553$	−34.6918			−1.47525
$554$		0			0
$555$		0			0
$556$		0			0
$557$			38.6119i			1.63604i	0.575189	+	0.818020i	$0.304928\pi$
$557$			38.6119i			1.63604i	−0.575189	+	0.818020i	$0.695072\pi$
$558$		0			0
$559$	40.0648	−	10.3792i	1.69456	−	0.438991i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.62289	−	9.62289i	−0.405556	−	0.405556i	0.474629	−	0.880186i	$-0.342582\pi$
$563$	−9.62289	−	9.62289i	−0.405556	−	0.405556i	−0.880186	+	0.474629i	$0.842582\pi$
$564$		0			0
$565$	23.9903	+	5.65572i	1.00928	+	0.237938i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−23.7899			−0.997326			−0.498663	−	0.866796i	$-0.666175\pi$
$569$	−23.7899			−0.997326			−0.498663	+	0.866796i	$0.666175\pi$
$570$		0			0
$571$		−	11.2205i		−	0.469562i	−0.972048	−	0.234781i	$-0.924563\pi$
$571$		−	11.2205i		−	0.469562i	0.972048	−	0.234781i	$-0.0754373\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.79933	+	2.93900i	−0.366957	+	0.122565i
$576$		0			0
$577$	40.0865			1.66882			0.834411	−	0.551143i	$-0.185808\pi$
$577$	40.0865			1.66882			0.834411	+	0.551143i	$0.185808\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−63.1216			−2.61873
$582$		0			0
$583$	8.60314			0.356305
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.7208			−0.483770			−0.241885	−	0.970305i	$-0.577766\pi$
$587$	−11.7208			−0.483770			−0.241885	+	0.970305i	$0.577766\pi$
$588$		0			0
$589$	−33.6979			−1.38850
$590$		0			0
$591$		0			0
$592$		0			0
$593$	15.4834			0.635826			0.317913	−	0.948120i	$-0.397018\pi$
$593$	15.4834			0.635826			0.317913	+	0.948120i	$0.397018\pi$
$594$		0			0
$595$	−12.6562	+	53.6849i	−0.518854	+	2.20086i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	13.5127i		−	0.552115i	−0.961141	−	0.276058i	$-0.910972\pi$
$599$		−	13.5127i		−	0.552115i	0.961141	−	0.276058i	$-0.0890280\pi$
$600$		0			0
$601$	27.4625			1.12022			0.560108	−	0.828419i	$-0.310759\pi$
$601$	27.4625			1.12022			0.560108	+	0.828419i	$0.310759\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	16.8564	−	10.4248i	0.685309	−	0.423829i
$606$		0			0
$607$	9.34983	+	9.34983i	0.379498	+	0.379498i	0.870921	−	0.491423i	$-0.163523\pi$
$607$	9.34983	+	9.34983i	0.379498	+	0.379498i	−0.491423	+	0.870921i	$0.663523\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.16994	+	1.86547i	0.128242	+	0.0754689i
$612$		0			0
$613$		−	16.7293i		−	0.675690i	−0.941202	−	0.337845i	$-0.890302\pi$
$613$		−	16.7293i		−	0.675690i	0.941202	−	0.337845i	$-0.109698\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	33.3846			1.34401			0.672007	−	0.740544i	$-0.265432\pi$
$617$	33.3846			1.34401			0.672007	+	0.740544i	$0.265432\pi$
$618$		0			0
$619$	12.9465	−	12.9465i	0.520365	−	0.520365i	−0.397316	−	0.917682i	$-0.630058\pi$
$619$	12.9465	−	12.9465i	0.520365	−	0.520365i	0.917682	+	0.397316i	$0.130058\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	26.8418	+	26.8418i	1.07540	+	1.07540i
$624$		0			0
$625$	−15.0241	−	19.9819i	−0.600963	−	0.799277i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−35.3309	−	35.3309i	−1.40873	−	1.40873i
$630$		0			0
$631$	29.0903	+	29.0903i	1.15807	+	1.15807i	0.984891	+	0.173174i	$0.0554024\pi$
$631$	29.0903	+	29.0903i	1.15807	+	1.15807i	0.173174	+	0.984891i	$0.444598\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13.8954	+	3.27585i	0.551423	+	0.129998i
$636$		0			0
$637$	−18.9468	+	32.1958i	−0.750700	+	1.27564i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			27.5037i			1.08633i	0.839626	+	0.543165i	$0.182774\pi$
$641$			27.5037i			1.08633i	−0.839626	+	0.543165i	$0.817226\pi$
$642$		0			0
$643$		−	3.97198i		−	0.156640i	−0.996928	−	0.0783198i	$-0.975044\pi$
$643$		−	3.97198i		−	0.156640i	0.996928	−	0.0783198i	$-0.0249555\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.2989	−	24.2989i	0.955290	−	0.955290i	−0.0437527	−	0.999042i	$-0.513931\pi$
$647$	24.2989	−	24.2989i	0.955290	−	0.955290i	0.999042	+	0.0437527i	$0.0139314\pi$
$648$		0			0
$649$	9.05987			0.355631
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.39282	−	6.39282i	0.250170	−	0.250170i	−0.570870	−	0.821040i	$-0.693394\pi$
$653$	6.39282	−	6.39282i	0.250170	−	0.250170i	0.821040	+	0.570870i	$0.193394\pi$
$654$		0			0
$655$	7.62138	+	12.3234i	0.297792	+	0.481513i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	11.4763i		−	0.447052i	−0.974698	−	0.223526i	$-0.928243\pi$
$659$		−	11.4763i		−	0.447052i	0.974698	−	0.223526i	$-0.0717567\pi$
$660$		0			0
$661$	−19.3374	−	19.3374i	−0.752139	−	0.752139i	0.222739	−	0.974878i	$-0.428500\pi$
$661$	−19.3374	−	19.3374i	−0.752139	−	0.752139i	−0.974878	+	0.222739i	$0.928500\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	56.1365	+	13.2342i	2.17688	+	0.513200i
$666$		0			0
$667$	−5.15850	+	5.15850i	−0.199738	+	0.199738i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.925062	+	0.925062i	0.0357116	+	0.0357116i
$672$		0			0
$673$	6.06351	+	6.06351i	0.233731	+	0.233731i	0.814248	−	0.580517i	$-0.197150\pi$
$673$	6.06351	+	6.06351i	0.233731	+	0.233731i	−0.580517	+	0.814248i	$0.697150\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.6902	+	10.6902i	0.410859	+	0.410859i	0.882038	−	0.471179i	$-0.156171\pi$
$677$	10.6902	+	10.6902i	0.410859	+	0.410859i	−0.471179	+	0.882038i	$0.656171\pi$
$678$		0			0
$679$			45.6334i			1.75125i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.96421			0.0751586			0.0375793	−	0.999294i	$-0.488035\pi$
$683$	1.96421			0.0751586			0.0375793	+	0.999294i	$0.488035\pi$
$684$		0			0
$685$	−28.0133	+	17.3248i	−1.07033	+	0.661947i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.74540	−	6.73744i	−0.0664943	−	0.256676i
$690$		0			0
$691$	−29.0749	+	29.0749i	−1.10606	+	1.10606i	−0.112400	+	0.993663i	$0.535854\pi$
$691$	−29.0749	+	29.0749i	−1.10606	+	1.10606i	−0.993663	+	0.112400i	$0.964146\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.6809	+	8.46093i	−0.518944	+	0.320941i
$696$		0			0
$697$			13.0158i			0.493007i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.8458i		−	1.20280i	−0.798949	−	0.601399i	$-0.794610\pi$
$701$		−	31.8458i		−	1.20280i	0.798949	−	0.601399i	$-0.205390\pi$
$702$		0			0
$703$	−36.9443	+	36.9443i	−1.39338	+	1.39338i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−33.5794			−1.26288
$708$		0			0
$709$	−20.4104	+	20.4104i	−0.766528	+	0.766528i	−0.977493	−	0.210966i	$-0.932339\pi$
$709$	−20.4104	+	20.4104i	−0.766528	+	0.766528i	0.210966	+	0.977493i	$0.432339\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.1002			0.378255
$714$		0			0
$715$	−25.9599	−	24.8438i	−0.970844	−	0.929107i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−31.3675			−1.16981			−0.584905	−	0.811102i	$-0.698868\pi$
$719$	−31.3675			−1.16981			−0.584905	+	0.811102i	$0.698868\pi$
$720$		0			0
$721$	17.3309	−	17.3309i	0.645435	−	0.645435i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−17.5889	−	8.78121i	−0.653235	−	0.326126i
$726$		0			0
$727$	13.3838	−	13.3838i	0.496379	−	0.496379i	−0.413930	−	0.910309i	$-0.635844\pi$
$727$	13.3838	−	13.3838i	0.496379	−	0.496379i	0.910309	+	0.413930i	$0.135844\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	67.9549			2.51340
$732$		0			0
$733$			23.1954i			0.856740i	0.903603	+	0.428370i	$0.140912\pi$
$733$			23.1954i			0.856740i	−0.903603	+	0.428370i	$0.859088\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−37.4225	−	37.4225i	−1.37847	−	1.37847i
$738$		0			0
$739$	−3.46527	+	3.46527i	−0.127472	+	0.127472i	−0.767965	−	0.640492i	$-0.778730\pi$
$739$	−3.46527	+	3.46527i	−0.127472	+	0.127472i	0.640492	+	0.767965i	$0.278730\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.29558i		−	0.0475301i	−0.999718	−	0.0237650i	$-0.992435\pi$
$743$		−	1.29558i		−	0.0475301i	0.999718	−	0.0237650i	$-0.00756536\pi$
$744$		0			0
$745$	−5.22299	+	22.1548i	−0.191356	+	0.811688i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.85528	+	7.85528i	−0.287026	+	0.287026i
$750$		0			0
$751$		−	1.38749i		−	0.0506301i	−0.999680	−	0.0253150i	$-0.991941\pi$
$751$		−	1.38749i		−	0.0506301i	0.999680	−	0.0253150i	$-0.00805889\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.91798	−	20.8610i	0.178984	−	0.759210i
$756$		0			0
$757$	−9.11127	−	9.11127i	−0.331155	−	0.331155i	0.521870	−	0.853025i	$-0.325235\pi$
$757$	−9.11127	−	9.11127i	−0.331155	−	0.331155i	−0.853025	+	0.521870i	$0.825235\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.8539	+	19.8539i	0.719703	+	0.719703i	0.968544	−	0.248841i	$-0.0800496\pi$
$761$	19.8539	+	19.8539i	0.719703	+	0.719703i	−0.248841	+	0.968544i	$0.580050\pi$
$762$		0			0
$763$	−13.5549	+	13.5549i	−0.490721	+	0.490721i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.83806	−	7.09513i	−0.0663685	−	0.256190i
$768$		0			0
$769$	22.6182	+	22.6182i	0.815634	+	0.815634i	0.985472	−	0.169838i	$-0.0543243\pi$
$769$	22.6182	+	22.6182i	0.815634	+	0.815634i	−0.169838	+	0.985472i	$0.554324\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	20.5557i		−	0.739336i	−0.929164	−	0.369668i	$-0.879471\pi$
$773$		−	20.5557i		−	0.739336i	0.929164	−	0.369668i	$-0.120529\pi$
$774$		0			0
$775$	8.62258	+	25.8159i	0.309732	+	0.927335i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.6102			0.487634
$780$		0			0
$781$	−71.8302			−2.57029
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.28845	−	35.1578i	0.295827	−	1.25483i
$786$		0			0
$787$		−	13.3191i		−	0.474774i	−0.971415	−	0.237387i	$-0.923709\pi$
$787$		−	13.3191i		−	0.474774i	0.971415	−	0.237387i	$-0.0762909\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	32.4764	+	32.4764i	1.15473	+	1.15473i
$792$		0			0
$793$	0.536775	−	0.912126i	0.0190614	−	0.0323906i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−20.3474	+	20.3474i	−0.720743	+	0.720743i	−0.968757	−	0.248014i	$-0.920222\pi$
$797$	−20.3474	+	20.3474i	−0.720743	+	0.720743i	0.248014	+	0.968757i	$0.420222\pi$
$798$		0			0
$799$	4.27035	+	4.27035i	0.151074	+	0.151074i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	20.3635	+	20.3635i	0.718612	+	0.718612i
$804$		0			0
$805$	−16.8257	−	3.96666i	−0.593027	−	0.139806i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	42.5767i		−	1.49692i	−0.663182	−	0.748458i	$-0.730795\pi$
$809$		−	42.5767i		−	1.49692i	0.663182	−	0.748458i	$-0.269205\pi$
$810$		0			0
$811$	22.7142	−	22.7142i	0.797603	−	0.797603i	−0.185114	−	0.982717i	$-0.559265\pi$
$811$	22.7142	−	22.7142i	0.797603	−	0.797603i	0.982717	+	0.185114i	$0.0592655\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.2363	+	29.4871i	0.638791	+	1.03289i
$816$		0			0
$817$		−	71.0583i		−	2.48601i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.60013	+	7.60013i	−0.265246	+	0.265246i	−0.827181	−	0.561935i	$-0.810057\pi$
$821$	−7.60013	+	7.60013i	−0.265246	+	0.265246i	0.561935	+	0.827181i	$0.310057\pi$
$822$		0			0
$823$	25.4074	+	25.4074i	0.885648	+	0.885648i	0.994101	−	0.108454i	$-0.0345900\pi$
$823$	25.4074	+	25.4074i	0.885648	+	0.885648i	−0.108454	+	0.994101i	$0.534590\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37.7770i		−	1.31363i	−0.754050	−	0.656817i	$-0.771902\pi$
$827$		−	37.7770i		−	1.31363i	0.754050	−	0.656817i	$-0.228098\pi$
$828$		0			0
$829$	−32.0493			−1.11312			−0.556559	−	0.830808i	$-0.687878\pi$
$829$	−32.0493			−1.11312			−0.556559	+	0.830808i	$0.687878\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−43.3721	+	43.3721i	−1.50276	+	1.50276i
$834$		0			0
$835$	20.5479	−	12.7079i	0.711090	−	0.439774i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.17521	−	9.17521i	0.316763	−	0.316763i	−0.530759	−	0.847523i	$-0.678093\pi$
$839$	9.17521	−	9.17521i	0.316763	−	0.316763i	0.847523	+	0.530759i	$0.178093\pi$
$840$		0			0
$841$	13.5408			0.466926
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.1894	+	25.3705i	−0.488131	+	0.872770i
$846$		0			0
$847$	36.9314			1.26898
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.0732	−	11.0732i	0.379586	−	0.379586i
$852$		0			0
$853$	25.1821			0.862220			0.431110	−	0.902299i	$-0.358122\pi$
$853$	25.1821			0.862220			0.431110	+	0.902299i	$0.358122\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.17990	−	7.17990i	0.245261	−	0.245261i	−0.573762	−	0.819022i	$-0.694516\pi$
$857$	7.17990	−	7.17990i	0.245261	−	0.245261i	0.819022	+	0.573762i	$0.194516\pi$
$858$		0			0
$859$		−	44.4341i		−	1.51607i	−0.652213	−	0.758036i	$-0.726159\pi$
$859$		−	44.4341i		−	1.51607i	0.652213	−	0.758036i	$-0.273841\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			21.1546i			0.720111i	0.932931	+	0.360055i	$0.117242\pi$
$863$			21.1546i			0.720111i	−0.932931	+	0.360055i	$0.882758\pi$
$864$		0			0
$865$	−5.52482	+	23.4351i	−0.187849	+	0.796816i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26.2394	−	26.2394i	0.890111	−	0.890111i
$870$		0			0
$871$	−21.7147	+	36.8992i	−0.735775	+	1.25028i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.22546	−	46.3925i	−0.142847	−	1.56835i
$876$		0			0
$877$	38.6352			1.30462			0.652309	−	0.757953i	$-0.273800\pi$
$877$	38.6352			1.30462			0.652309	+	0.757953i	$0.273800\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			17.4550i			0.588072i	0.955794	+	0.294036i	$0.0949986\pi$
$881$			17.4550i			0.588072i	−0.955794	+	0.294036i	$0.905001\pi$
$882$		0			0
$883$	−22.8569	−	22.8569i	−0.769196	−	0.769196i	0.208769	−	0.977965i	$-0.433054\pi$
$883$	−22.8569	−	22.8569i	−0.769196	−	0.769196i	−0.977965	+	0.208769i	$0.933054\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.6394	−	12.6394i	−0.424390	−	0.424390i	0.462322	−	0.886712i	$-0.347016\pi$
$887$	−12.6394	−	12.6394i	−0.424390	−	0.424390i	−0.886712	+	0.462322i	$0.847016\pi$
$888$		0			0
$889$	18.8107	+	18.8107i	0.630889	+	0.630889i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.46537	−	4.46537i	0.149428	−	0.149428i
$894$		0			0
$895$	−15.5811	−	25.1938i	−0.520819	−	0.842136i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	15.1343	+	15.1343i	0.504756	+	0.504756i
$900$		0			0
$901$		−	11.4275i		−	0.380707i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−25.7427	+	15.9206i	−0.855718	+	0.529219i
$906$		0			0
$907$	−16.4274	+	16.4274i	−0.545462	+	0.545462i	−0.925125	−	0.379663i	$-0.876040\pi$
$907$	−16.4274	+	16.4274i	−0.545462	+	0.545462i	0.379663	+	0.925125i	$0.376040\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.40393			0.0796456			0.0398228	−	0.999207i	$-0.487321\pi$
$911$	2.40393			0.0796456			0.0398228	+	0.999207i	$0.487321\pi$
$912$		0			0
$913$	47.7424	−	47.7424i	1.58004	−	1.58004i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			26.9998i			0.891612i
$918$		0			0
$919$			7.76783i			0.256237i	0.991759	+	0.128118i	$0.0408938\pi$
$919$			7.76783i			0.256237i	−0.991759	+	0.128118i	$0.959106\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14.5729	+	56.2530i	0.479671	+	1.85159i
$924$		0			0
$925$	37.7563	+	18.8498i	1.24142	+	0.619776i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.0133	−	17.0133i	−0.558189	−	0.558189i	0.370602	−	0.928792i	$-0.379151\pi$
$929$	−17.0133	−	17.0133i	−0.558189	−	0.558189i	−0.928792	+	0.370602i	$0.879151\pi$
$930$		0			0
$931$	45.3528	+	45.3528i	1.48638	+	1.48638i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.0323	−	50.1775i	−1.01486	−	1.64098i
$936$		0			0
$937$	6.62978	+	6.62978i	0.216586	+	0.216586i	0.807058	−	0.590472i	$-0.201059\pi$
$937$	6.62978	+	6.62978i	0.216586	+	0.216586i	−0.590472	+	0.807058i	$0.701059\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.5888	−	23.5888i	0.768973	−	0.768973i	−0.208953	−	0.977926i	$-0.567006\pi$
$941$	23.5888	−	23.5888i	0.768973	−	0.768973i	0.977926	+	0.208953i	$0.0670055\pi$
$942$		0			0
$943$	−4.07934			−0.132842
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	6.31026i		−	0.205056i	−0.994730	−	0.102528i	$-0.967307\pi$
$947$		−	6.31026i		−	0.205056i	0.994730	−	0.102528i	$-0.0326931\pi$
$948$		0			0
$949$	11.8161	−	20.0787i	0.383567	−	0.651784i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4.37109	+	4.37109i	0.141594	+	0.141594i	0.774350	−	0.632757i	$-0.218077\pi$
$953$	4.37109	+	4.37109i	0.141594	+	0.141594i	−0.632757	+	0.774350i	$0.718077\pi$
$954$		0			0
$955$	−27.6695	−	44.7401i	−0.895365	−	1.44776i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−61.3755			−1.98192
$960$		0			0
$961$			1.36757i			0.0441152i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.8275	−	23.9753i	−0.477315	−	0.771793i
$966$		0			0
$967$	−39.7361			−1.27783			−0.638914	−	0.769278i	$-0.720616\pi$
$967$	−39.7361			−1.27783			−0.638914	+	0.769278i	$0.720616\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−31.5420			−1.01223			−0.506115	−	0.862466i	$-0.668919\pi$
$971$	−31.5420			−1.01223			−0.506115	+	0.862466i	$0.668919\pi$
$972$		0			0
$973$	−29.9740			−0.960923
$974$		0			0
$975$		0			0
$976$		0			0
$977$	39.9665			1.27864			0.639322	−	0.768940i	$-0.279215\pi$
$977$	39.9665			1.27864			0.639322	+	0.768940i	$0.279215\pi$
$978$		0			0
$979$	−40.6040			−1.29771
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23.4700			0.748577			0.374289	−	0.927312i	$-0.377887\pi$
$983$	23.4700			0.748577			0.374289	+	0.927312i	$0.377887\pi$
$984$		0			0
$985$	−15.7763	−	25.5095i	−0.502676	−	0.812800i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			21.2981i			0.677241i
$990$		0			0
$991$	33.6588			1.06921			0.534604	−	0.845103i	$-0.320461\pi$
$991$	33.6588			1.06921			0.534604	+	0.845103i	$0.320461\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−26.5615	−	42.9485i	−0.842056	−	1.36156i
$996$		0			0
$997$	−16.7331	−	16.7331i	−0.529943	−	0.529943i	0.390613	−	0.920555i	$-0.372263\pi$
$997$	−16.7331	−	16.7331i	−0.529943	−	0.529943i	−0.920555	+	0.390613i	$0.872263\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2340.2.u.i.577.6		28
3.2	odd	2		780.2.r.a.577.5	yes	28
5.3	odd	4		2340.2.bp.i.1513.13		28
13.8	odd	4		2340.2.bp.i.1477.13		28
15.2	even	4		3900.2.bm.b.2293.14		28
15.8	even	4		780.2.bm.a.733.2	yes	28
15.14	odd	2		3900.2.r.b.1357.14		28
39.8	even	4		780.2.bm.a.697.2	yes	28
65.8	even	4	inner	2340.2.u.i.73.6		28
195.8	odd	4		780.2.r.a.73.5	✓	28
195.47	odd	4		3900.2.r.b.3193.14		28
195.164	even	4		3900.2.bm.b.2257.14		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
780.2.r.a.73.5	✓	28	195.8	odd	4
780.2.r.a.577.5	yes	28	3.2	odd	2
780.2.bm.a.697.2	yes	28	39.8	even	4
780.2.bm.a.733.2	yes	28	15.8	even	4
2340.2.u.i.73.6		28	65.8	even	4	inner
2340.2.u.i.577.6		28	1.1	even	1	trivial
2340.2.bp.i.1477.13		28	13.8	odd	4
2340.2.bp.i.1513.13		28	5.3	odd	4
3900.2.r.b.1357.14		28	15.14	odd	2
3900.2.r.b.3193.14		28	195.47	odd	4
3900.2.bm.b.2257.14		28	195.164	even	4
3900.2.bm.b.2293.14		28	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.17614 - 1.90176i) q^{5} -4.16665i q^{7} +O(q^{10})\)	\(q+(-1.17614 - 1.90176i) q^{5} -4.16665i q^{7} +(3.15147 + 3.15147i) q^{11} +(1.82867 - 3.10741i) q^{13} +(4.18610 - 4.18610i) q^{17} +(-4.37727 - 4.37727i) q^{19} +(1.31199 + 1.31199i) q^{23} +(-2.23337 + 4.47348i) q^{25} +3.93181i q^{29} +(3.84918 - 3.84918i) q^{31} +(-7.92397 + 4.90058i) q^{35} -8.44004i q^{37} +(-1.55464 + 1.55464i) q^{41} +(8.11673 + 8.11673i) q^{43} +1.02013i q^{47} -10.3610 q^{49} +(1.36494 - 1.36494i) q^{53} +(2.28676 - 9.69993i) q^{55} +(1.43740 - 1.43740i) q^{59} +0.293533 q^{61} +(-8.06031 + 0.177065i) q^{65} -11.8746 q^{67} +(-11.3963 + 11.3963i) q^{71} +6.46158 q^{73} +(13.1311 - 13.1311i) q^{77} -8.32607i q^{79} -15.1492i q^{83} +(-12.8844 - 3.03750i) q^{85} +(-6.44206 + 6.44206i) q^{89} +(-12.9475 - 7.61943i) q^{91} +(-3.17622 + 13.4728i) q^{95} -10.9521 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q+O(q^{10}) \) 28 * q	\( 28 q - 8 q^{11} - 4 q^{13} + 4 q^{17} - 16 q^{19} + 8 q^{23} + 12 q^{25} - 12 q^{41} + 16 q^{43} - 36 q^{49} - 36 q^{53} + 40 q^{55} - 16 q^{59} + 8 q^{61} + 40 q^{65} + 48 q^{67} - 8 q^{71} + 48 q^{73} + 48 q^{77} - 4 q^{85} + 36 q^{89} - 24 q^{91} - 72 q^{95} - 72 q^{97}+O(q^{100}) \) 28 * q - 8 * q^11 - 4 * q^13 + 4 * q^17 - 16 * q^19 + 8 * q^23 + 12 * q^25 - 12 * q^41 + 16 * q^43 - 36 * q^49 - 36 * q^53 + 40 * q^55 - 16 * q^59 + 8 * q^61 + 40 * q^65 + 48 * q^67 - 8 * q^71 + 48 * q^73 + 48 * q^77 - 4 * q^85 + 36 * q^89 - 24 * q^91 - 72 * q^95 - 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more