LMFDB - Embedded newform 276.2.g.a.137.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [276,2,Mod(137,276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(276, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("276.137");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 276 = 2^{2} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	276.g (of order $2$, degree $1$, 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:	$2.20387109579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14453810176.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 15x^{4} - 30x^{2} + 64 $ x^8 + 9x^6 + 15x^4 - 30*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			137.2
Root			$0.331077 + 2.33494i$ of defining polynomial
Character	$\chi$	$=$	276.137
Dual form			276.2.g.a.137.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.780776 - 1.54609i) q^{3} +3.02045 q^{5} -2.62238i q^{7} +(-1.78078 + 2.41430i) q^{9} +O(q^{10})$	\(q+(-0.780776 - 1.54609i) q^{3} +3.02045 q^{5} -2.62238i q^{7} +(-1.78078 + 2.41430i) q^{9} +1.32431 q^{11} -1.56155 q^{13} +(-2.35829 - 4.66988i) q^{15} +0.371834 q^{17} -6.71737i q^{19} +(-4.05444 + 2.04750i) q^{21} +(4.71659 + 0.868210i) q^{23} +4.12311 q^{25} +(5.12311 + 0.868210i) q^{27} -4.82860i q^{29} -8.68466 q^{31} +(-1.03399 - 2.04750i) q^{33} -7.92077i q^{35} +9.33976i q^{37} +(1.21922 + 2.41430i) q^{39} +1.35576i q^{41} +6.71737i q^{43} +(-5.37874 + 7.29226i) q^{45} +11.0129i q^{47} +0.123106 q^{49} +(-0.290319 - 0.574888i) q^{51} +11.1293 q^{53} +4.00000 q^{55} +(-10.3857 + 5.24477i) q^{57} +6.18435i q^{59} +5.24477i q^{61} +(6.33122 + 4.66988i) q^{63} -4.71659 q^{65} +7.86715i q^{67} +(-2.34027 - 7.97014i) q^{69} -3.09218i q^{71} +6.68466 q^{73} +(-3.21922 - 6.37468i) q^{75} -3.47284i q^{77} -11.9621i q^{79} +(-2.65767 - 8.59865i) q^{81} -10.7575 q^{83} +1.12311 q^{85} +(-7.46543 + 3.77005i) q^{87} -7.73704 q^{89} +4.09499i q^{91} +(6.78078 + 13.4272i) q^{93} -20.2895i q^{95} -4.09499i q^{97} +(-2.35829 + 3.19727i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 6 q^{9}+O(q^{10}) $ 8 * q + 2 * q^3 - 6 * q^9	$ 8 q + 2 q^{3} - 6 q^{9} + 4 q^{13} + 8 q^{27} - 20 q^{31} + 18 q^{39} - 32 q^{49} + 32 q^{55} - 14 q^{69} + 4 q^{73} - 34 q^{75} - 46 q^{81} - 24 q^{85} - 2 q^{87} + 46 q^{93}+O(q^{100}) $ 8 * q + 2 * q^3 - 6 * q^9 + 4 * q^13 + 8 * q^27 - 20 * q^31 + 18 * q^39 - 32 * q^49 + 32 * q^55 - 14 * q^69 + 4 * q^73 - 34 * q^75 - 46 * q^81 - 24 * q^85 - 2 * q^87 + 46 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/276\mathbb{Z}\right)^\times$.

$n$	$97$	$139$	$185$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.780776	−	1.54609i	−0.450781	−	0.892634i
$3$	−0.780776	−	1.54609i	−0.450781	−	0.892634i
$4$		0			0
$5$	3.02045			1.35079			0.675393	−	0.737458i	$-0.263974\pi$
$5$	3.02045			1.35079			0.675393	+	0.737458i	$0.263974\pi$
$6$		0			0
$7$		−	2.62238i		−	0.991168i	−0.868560	−	0.495584i	$-0.834954\pi$
$7$		−	2.62238i		−	0.991168i	0.868560	−	0.495584i	$-0.165046\pi$
$8$		0			0
$9$	−1.78078	+	2.41430i	−0.593592	+	0.804766i
$10$		0			0
$11$	1.32431			0.399294			0.199647	−	0.979868i	$-0.436021\pi$
$11$	1.32431			0.399294			0.199647	+	0.979868i	$0.436021\pi$
$12$		0			0
$13$	−1.56155			−0.433097			−0.216548	−	0.976272i	$-0.569480\pi$
$13$	−1.56155			−0.433097			−0.216548	+	0.976272i	$0.569480\pi$
$14$		0			0
$15$	−2.35829	−	4.66988i	−0.608909	−	1.20576i
$16$		0			0
$17$	0.371834			0.0901830			0.0450915	−	0.998983i	$-0.485642\pi$
$17$	0.371834			0.0901830			0.0450915	+	0.998983i	$0.485642\pi$
$18$		0			0
$19$		−	6.71737i		−	1.54107i	−0.637397	−	0.770536i	$-0.719989\pi$
$19$		−	6.71737i		−	1.54107i	0.637397	−	0.770536i	$-0.280011\pi$
$20$		0			0
$21$	−4.05444	+	2.04750i	−0.884750	+	0.446800i
$22$		0			0
$23$	4.71659	+	0.868210i	0.983477	+	0.181034i
$23$	4.71659	+	0.868210i	0.983477	+	0.181034i
$24$		0			0
$25$	4.12311			0.824621
$26$		0			0
$27$	5.12311	+	0.868210i	0.985942	+	0.167087i
$28$		0			0
$29$		−	4.82860i		−	0.896648i	−0.893871	−	0.448324i	$-0.852021\pi$
$29$		−	4.82860i		−	0.896648i	0.893871	−	0.448324i	$-0.147979\pi$
$30$		0			0
$31$	−8.68466			−1.55981			−0.779905	−	0.625897i	$-0.784733\pi$
$31$	−8.68466			−1.55981			−0.779905	+	0.625897i	$0.784733\pi$
$32$		0			0
$33$	−1.03399	−	2.04750i	−0.179994	−	0.356423i
$34$		0			0
$35$		−	7.92077i		−	1.33885i
$36$		0			0
$37$			9.33976i			1.53545i	0.640782	+	0.767723i	$0.278610\pi$
$37$			9.33976i			1.53545i	−0.640782	+	0.767723i	$0.721390\pi$
$38$		0			0
$39$	1.21922	+	2.41430i	0.195232	+	0.386597i
$40$		0			0
$41$			1.35576i			0.211733i	0.994380	+	0.105867i	$0.0337617\pi$
$41$			1.35576i			0.211733i	−0.994380	+	0.105867i	$0.966238\pi$
$42$		0			0
$43$			6.71737i			1.02439i	0.858869	+	0.512195i	$0.171167\pi$
$43$			6.71737i			1.02439i	−0.858869	+	0.512195i	$0.828833\pi$
$44$		0			0
$45$	−5.37874	+	7.29226i	−0.801816	+	1.08707i
$46$		0			0
$47$			11.0129i			1.60640i	0.595707	+	0.803202i	$0.296872\pi$
$47$			11.0129i			1.60640i	−0.595707	+	0.803202i	$0.703128\pi$
$48$		0			0
$49$	0.123106			0.0175865
$50$		0			0
$51$	−0.290319	−	0.574888i	−0.0406528	−	0.0805005i
$52$		0			0
$53$	11.1293			1.52873			0.764365	−	0.644784i	$-0.223053\pi$
$53$	11.1293			1.52873			0.764365	+	0.644784i	$0.223053\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$	−10.3857	+	5.24477i	−1.37561	+	0.694686i
$58$		0			0
$59$			6.18435i			0.805134i	0.915391	+	0.402567i	$0.131882\pi$
$59$			6.18435i			0.805134i	−0.915391	+	0.402567i	$0.868118\pi$
$60$		0			0
$61$			5.24477i			0.671524i	0.941947	+	0.335762i	$0.108994\pi$
$61$			5.24477i			0.671524i	−0.941947	+	0.335762i	$0.891006\pi$
$62$		0			0
$63$	6.33122	+	4.66988i	0.797658	+	0.588349i
$64$		0			0
$65$	−4.71659			−0.585021
$66$		0			0
$67$			7.86715i			0.961125i	0.876960	+	0.480563i	$0.159568\pi$
$67$			7.86715i			0.961125i	−0.876960	+	0.480563i	$0.840432\pi$
$68$		0			0
$69$	−2.34027	−	7.97014i	−0.281736	−	0.959492i
$70$		0			0
$71$		−	3.09218i		−	0.366974i	−0.983022	−	0.183487i	$-0.941262\pi$
$71$		−	3.09218i		−	0.366974i	0.983022	−	0.183487i	$-0.0587385\pi$
$72$		0			0
$73$	6.68466			0.782380			0.391190	−	0.920310i	$-0.372064\pi$
$73$	6.68466			0.782380			0.391190	+	0.920310i	$0.372064\pi$
$74$		0			0
$75$	−3.21922	−	6.37468i	−0.371724	−	0.736085i
$76$		0			0
$77$		−	3.47284i		−	0.395767i
$78$		0			0
$79$		−	11.9621i		−	1.34585i	−0.739713	−	0.672923i	$-0.765039\pi$
$79$		−	11.9621i		−	1.34585i	0.739713	−	0.672923i	$-0.234961\pi$
$80$		0			0
$81$	−2.65767	−	8.59865i	−0.295297	−	0.955406i
$82$		0			0
$83$	−10.7575			−1.18079			−0.590394	−	0.807115i	$-0.701027\pi$
$83$	−10.7575			−1.18079			−0.590394	+	0.807115i	$0.701027\pi$
$84$		0			0
$85$	1.12311			0.121818
$86$		0			0
$87$	−7.46543	+	3.77005i	−0.800379	+	0.404192i
$88$		0			0
$89$	−7.73704			−0.820124			−0.410062	−	0.912058i	$-0.634493\pi$
$89$	−7.73704			−0.820124			−0.410062	+	0.912058i	$0.634493\pi$
$90$		0			0
$91$			4.09499i			0.429272i
$92$		0			0
$93$	6.78078	+	13.4272i	0.703134	+	1.39234i
$94$		0			0
$95$		−	20.2895i		−	2.08166i
$96$		0			0
$97$		−	4.09499i		−	0.415783i	−0.978152	−	0.207892i	$-0.933340\pi$
$97$		−	4.09499i		−	0.415783i	0.978152	−	0.207892i	$-0.0666601\pi$
$98$		0			0
$99$	−2.35829	+	3.19727i	−0.237018	+	0.321338i
$100$		0			0
$101$			15.8415i			1.57629i	0.615488	+	0.788146i	$0.288959\pi$
$101$			15.8415i			1.57629i	−0.615488	+	0.788146i	$0.711041\pi$
$102$		0			0
$103$		−	7.86715i		−	0.775173i	−0.921833	−	0.387587i	$-0.873309\pi$
$103$		−	7.86715i		−	0.775173i	0.921833	−	0.387587i	$-0.126691\pi$
$104$		0			0
$105$	−12.2462	+	6.18435i	−1.19511	+	0.603531i
$106$		0			0
$107$	6.04090			0.583995			0.291998	−	0.956419i	$-0.405680\pi$
$107$	6.04090			0.583995			0.291998	+	0.956419i	$0.405680\pi$
$108$		0			0
$109$		−	5.24477i		−	0.502358i	−0.967941	−	0.251179i	$-0.919182\pi$
$109$		−	5.24477i		−	0.502358i	0.967941	−	0.251179i	$-0.0808182\pi$
$110$		0			0
$111$	14.4401	−	7.29226i	1.37059	−	0.692151i
$112$		0			0
$113$	6.41273			0.603259			0.301629	−	0.953425i	$-0.402469\pi$
$113$	6.41273			0.603259			0.301629	+	0.953425i	$0.402469\pi$
$114$		0			0
$115$	14.2462	+	2.62238i	1.32847	+	0.244539i
$116$		0			0
$117$	2.78078	−	3.77005i	0.257083	−	0.348542i
$118$		0			0
$119$		−	0.975092i		−	0.0893865i
$120$		0			0
$121$	−9.24621			−0.840565
$122$		0			0
$123$	2.09612	−	1.05854i	0.189001	−	0.0954455i
$124$		0			0
$125$	−2.64861			−0.236899
$126$		0			0
$127$	4.68466			0.415696			0.207848	−	0.978161i	$-0.433354\pi$
$127$	4.68466			0.415696			0.207848	+	0.978161i	$0.433354\pi$
$128$		0			0
$129$	10.3857	−	5.24477i	0.914406	−	0.461776i
$130$		0			0
$131$			12.7494i			1.11392i	0.830540	+	0.556959i	$0.188032\pi$
$131$			12.7494i			1.11392i	−0.830540	+	0.556959i	$0.811968\pi$
$132$		0			0
$133$	−17.6155			−1.52746
$134$		0			0
$135$	15.4741	+	2.62238i	1.33180	+	0.225699i
$136$		0			0
$137$	0.952473			0.0813752			0.0406876	−	0.999172i	$-0.487045\pi$
$137$	0.952473			0.0813752			0.0406876	+	0.999172i	$0.487045\pi$
$138$		0			0
$139$	−14.9309			−1.26642			−0.633210	−	0.773980i	$-0.718263\pi$
$139$	−14.9309			−1.26642			−0.633210	+	0.773980i	$0.718263\pi$
$140$		0			0
$141$	17.0270	−	8.59865i	1.43393	−	0.724137i
$142$		0			0
$143$	−2.06798			−0.172933
$144$		0			0
$145$		−	14.5845i		−	1.21118i
$146$		0			0
$147$	−0.0961180	−	0.190332i	−0.00792768	−	0.0156983i
$148$		0			0
$149$	16.4265			1.34572			0.672858	−	0.739772i	$-0.265067\pi$
$149$	16.4265			1.34572			0.672858	+	0.739772i	$0.265067\pi$
$150$		0			0
$151$	10.4384			0.849469			0.424734	−	0.905318i	$-0.360367\pi$
$151$	10.4384			0.849469			0.424734	+	0.905318i	$0.360367\pi$
$152$		0			0
$153$	−0.662153	+	0.897718i	−0.0535319	+	0.0725762i
$154$		0			0
$155$	−26.2316			−2.10697
$156$		0			0
$157$			13.4347i			1.07221i	0.844151	+	0.536105i	$0.180105\pi$
$157$			13.4347i			1.07221i	−0.844151	+	0.536105i	$0.819895\pi$
$158$		0			0
$159$	−8.68951	−	17.2069i	−0.689123	−	1.36460i
$160$		0			0
$161$	2.27678	−	12.3687i	0.179435	−	0.974790i
$162$		0			0
$163$	−5.56155			−0.435614			−0.217807	−	0.975992i	$-0.569890\pi$
$163$	−5.56155			−0.435614			−0.217807	+	0.975992i	$0.569890\pi$
$164$		0			0
$165$	−3.12311	−	6.18435i	−0.243133	−	0.481451i
$166$		0			0
$167$		−	14.1051i		−	1.09149i	−0.837952	−	0.545744i	$-0.816247\pi$
$167$		−	14.1051i		−	1.09149i	0.837952	−	0.545744i	$-0.183753\pi$
$168$		0			0
$169$	−10.5616			−0.812427
$170$		0			0
$171$	16.2177	+	11.9621i	1.24020	+	0.914768i
$172$		0			0
$173$		−	24.7374i		−	1.88075i	−0.340139	−	0.940375i	$-0.610474\pi$
$173$		−	24.7374i		−	1.88075i	0.340139	−	0.940375i	$-0.389526\pi$
$174$		0			0
$175$		−	10.8124i		−	0.817338i
$176$		0			0
$177$	9.56155	−	4.82860i	0.718690	−	0.362940i
$178$		0			0
$179$			23.3817i			1.74763i	0.486261	+	0.873813i	$0.338360\pi$
$179$			23.3817i			1.74763i	−0.486261	+	0.873813i	$0.661640\pi$
$180$		0			0
$181$		−	17.5297i		−	1.30298i	−0.758659	−	0.651488i	$-0.774145\pi$
$181$		−	17.5297i		−	1.30298i	0.758659	−	0.651488i	$-0.225855\pi$
$182$		0			0
$183$	8.10887	−	4.09499i	0.599425	−	0.302710i
$184$		0			0
$185$			28.2102i			2.07406i
$186$		0			0
$187$	0.492423			0.0360095
$188$		0			0
$189$	2.27678	−	13.4347i	0.165611	−	0.977234i
$190$		0			0
$191$	−20.9343			−1.51476			−0.757378	−	0.652977i	$-0.773520\pi$
$191$	−20.9343			−1.51476			−0.757378	+	0.652977i	$0.773520\pi$
$192$		0			0
$193$	14.4384			1.03930			0.519651	−	0.854379i	$-0.326062\pi$
$193$	14.4384			1.03930			0.519651	+	0.854379i	$0.326062\pi$
$194$		0			0
$195$	3.68260	+	7.29226i	0.263717	+	0.522210i
$196$		0			0
$197$			13.7245i			0.977827i	0.872332	+	0.488914i	$0.162607\pi$
$197$			13.7245i			0.977827i	−0.872332	+	0.488914i	$0.837393\pi$
$198$		0			0
$199$		−	2.62238i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$199$		−	2.62238i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$200$		0			0
$201$	12.1633	−	6.14249i	0.857933	−	0.433257i
$202$		0			0
$203$	−12.6624			−0.888728
$204$		0			0
$205$			4.09499i			0.286007i
$206$		0			0
$207$	−10.4953	+	9.84116i	−0.729474	+	0.684008i
$208$		0			0
$209$		−	8.89586i		−	0.615340i
$210$		0			0
$211$	8.49242			0.584642			0.292321	−	0.956320i	$-0.405572\pi$
$211$	8.49242			0.584642			0.292321	+	0.956320i	$0.405572\pi$
$212$		0			0
$213$	−4.78078	+	2.41430i	−0.327573	+	0.165425i
$214$		0			0
$215$			20.2895i			1.38373i
$216$		0			0
$217$			22.7745i			1.54603i
$218$		0			0
$219$	−5.21922	−	10.3351i	−0.352682	−	0.698379i
$220$		0			0
$221$	−0.580639			−0.0390580
$222$		0			0
$223$	−14.2462			−0.953997			−0.476998	−	0.878904i	$-0.658275\pi$
$223$	−14.2462			−0.953997			−0.476998	+	0.878904i	$0.658275\pi$
$224$		0			0
$225$	−7.34233	+	9.95441i	−0.489489	+	0.663627i
$226$		0			0
$227$	−22.8393			−1.51590			−0.757948	−	0.652315i	$-0.773798\pi$
$227$	−22.8393			−1.51590			−0.757948	+	0.652315i	$0.773798\pi$
$228$		0			0
$229$		−	23.9243i		−	1.58096i	−0.612487	−	0.790480i	$-0.709831\pi$
$229$		−	23.9243i		−	1.58096i	0.612487	−	0.790480i	$-0.290169\pi$
$230$		0			0
$231$	−5.36932	+	2.71151i	−0.353275	+	0.178404i
$232$		0			0
$233$		−	7.54011i		−	0.493969i	−0.969019	−	0.246984i	$-0.920560\pi$
$233$		−	7.54011i		−	0.493969i	0.969019	−	0.246984i	$-0.0794397\pi$
$234$		0			0
$235$			33.2640i			2.16991i
$236$		0			0
$237$	−18.4945	+	9.33976i	−1.20135	+	0.606682i
$238$		0			0
$239$		−	6.56502i		−	0.424656i	−0.977198	−	0.212328i	$-0.931896\pi$
$239$		−	6.56502i		−	0.424656i	0.977198	−	0.212328i	$-0.0681045\pi$
$240$		0			0
$241$			4.09499i			0.263781i	0.991264	+	0.131891i	$0.0421048\pi$
$241$			4.09499i			0.263781i	−0.991264	+	0.131891i	$0.957895\pi$
$242$		0			0
$243$	−11.2192	+	10.8226i	−0.719714	+	0.694271i
$244$		0			0
$245$	0.371834			0.0237556
$246$		0			0
$247$			10.4895i			0.667433i
$248$		0			0
$249$	8.39919	+	16.6320i	0.532277	+	1.05401i
$250$		0			0
$251$	18.7033			1.18054			0.590272	−	0.807205i	$-0.299021\pi$
$251$	18.7033			1.18054			0.590272	+	0.807205i	$0.299021\pi$
$252$		0			0
$253$	6.24621	+	1.14978i	0.392696	+	0.0722858i
$254$		0			0
$255$	−0.876894	−	1.73642i	−0.0549133	−	0.108739i
$256$		0			0
$257$		−	20.6701i		−	1.28937i	−0.764449	−	0.644684i	$-0.776989\pi$
$257$		−	20.6701i		−	1.28937i	0.764449	−	0.644684i	$-0.223011\pi$
$258$		0			0
$259$	24.4924			1.52189
$260$		0			0
$261$	11.6577	+	8.59865i	0.721592	+	0.532243i
$262$		0			0
$263$	−4.13595			−0.255034			−0.127517	−	0.991836i	$-0.540701\pi$
$263$	−4.13595			−0.255034			−0.127517	+	0.991836i	$0.540701\pi$
$264$		0			0
$265$	33.6155			2.06499
$266$		0			0
$267$	6.04090	+	11.9621i	0.369697	+	0.732071i
$268$		0			0
$269$		−	20.6701i		−	1.26028i	−0.776481	−	0.630140i	$-0.782997\pi$
$269$		−	20.6701i		−	1.26028i	0.776481	−	0.630140i	$-0.217003\pi$
$270$		0			0
$271$	−2.24621			−0.136448			−0.0682238	−	0.997670i	$-0.521733\pi$
$271$	−2.24621			−0.136448			−0.0682238	+	0.997670i	$0.521733\pi$
$272$		0			0
$273$	6.33122	−	3.19727i	0.383183	−	0.193508i
$274$		0			0
$275$	5.46026			0.329266
$276$		0			0
$277$	−7.56155			−0.454330			−0.227165	−	0.973856i	$-0.572946\pi$
$277$	−7.56155			−0.454330			−0.227165	+	0.973856i	$0.572946\pi$
$278$		0			0
$279$	15.4654	−	20.9674i	0.925891	−	1.25528i
$280$		0			0
$281$	−26.4404			−1.57730			−0.788650	−	0.614843i	$-0.789220\pi$
$281$	−26.4404			−1.57730			−0.788650	+	0.614843i	$0.789220\pi$
$282$		0			0
$283$			26.5467i			1.57804i	0.614370	+	0.789018i	$0.289410\pi$
$283$			26.5467i			1.57804i	−0.614370	+	0.789018i	$0.710590\pi$
$284$		0			0
$285$	−31.3693	+	15.8415i	−1.85816	+	0.938372i
$286$		0			0
$287$	3.55531			0.209863
$288$		0			0
$289$	−16.8617			−0.991867
$290$		0			0
$291$	−6.33122	+	3.19727i	−0.371142	+	0.187427i
$292$		0			0
$293$	15.6829			0.916204			0.458102	−	0.888900i	$-0.348530\pi$
$293$	15.6829			0.916204			0.458102	+	0.888900i	$0.348530\pi$
$294$		0			0
$295$			18.6795i			1.08756i
$296$		0			0
$297$	6.78456	+	1.14978i	0.393680	+	0.0667168i
$298$		0			0
$299$	−7.36520	−	1.35576i	−0.425941	−	0.0784054i
$300$		0			0
$301$	17.6155			1.01534
$302$		0			0
$303$	24.4924	−	12.3687i	1.40705	−	0.710563i
$304$		0			0
$305$			15.8415i			0.907084i
$306$		0			0
$307$	−22.2462			−1.26966			−0.634829	−	0.772653i	$-0.718930\pi$
$307$	−22.2462			−1.26966			−0.634829	+	0.772653i	$0.718930\pi$
$308$		0			0
$309$	−12.1633	+	6.14249i	−0.691946	+	0.349434i
$310$		0			0
$311$		−	1.35576i		−	0.0768779i	−0.999261	−	0.0384389i	$-0.987761\pi$
$311$		−	1.35576i		−	0.0768779i	0.999261	−	0.0384389i	$-0.0122385\pi$
$312$		0			0
$313$		−	18.6795i		−	1.05583i	−0.849298	−	0.527914i	$-0.822974\pi$
$313$		−	18.6795i		−	1.05583i	0.849298	−	0.527914i	$-0.177026\pi$
$314$		0			0
$315$	19.1231	+	14.1051i	1.07746	+	0.794734i
$316$		0			0
$317$			24.7374i			1.38939i	0.719304	+	0.694696i	$0.244461\pi$
$317$			24.7374i			1.38939i	−0.719304	+	0.694696i	$0.755539\pi$
$318$		0			0
$319$		−	6.39454i		−	0.358026i
$320$		0			0
$321$	−4.71659	−	9.33976i	−0.263254	−	0.521294i
$322$		0			0
$323$		−	2.49775i		−	0.138978i
$324$		0			0
$325$	−6.43845			−0.357141
$326$		0			0
$327$	−8.10887	+	4.09499i	−0.448422	+	0.226453i
$328$		0			0
$329$	28.8802			1.59222
$330$		0			0
$331$	−7.80776			−0.429154			−0.214577	−	0.976707i	$-0.568837\pi$
$331$	−7.80776			−0.429154			−0.214577	+	0.976707i	$0.568837\pi$
$332$		0			0
$333$	−22.5490	−	16.6320i	−1.23568	−	0.911429i
$334$		0			0
$335$			23.7623i			1.29827i
$336$		0			0
$337$			10.4895i			0.571401i	0.958319	+	0.285701i	$0.0922263\pi$
$337$			10.4895i			0.571401i	−0.958319	+	0.285701i	$0.907774\pi$
$338$		0			0
$339$	−5.00691	−	9.91465i	−0.271938	−	0.538490i
$340$		0			0
$341$	−11.5012			−0.622822
$342$		0			0
$343$		−	18.6795i		−	1.00860i
$344$		0			0
$345$	−7.06867	−	24.0734i	−0.380564	−	1.29607i
$346$		0			0
$347$		−	13.1300i		−	0.704857i	−0.935839	−	0.352429i	$-0.885356\pi$
$347$		−	13.1300i		−	0.704857i	0.935839	−	0.352429i	$-0.114644\pi$
$348$		0			0
$349$	−23.8078			−1.27440			−0.637200	−	0.770698i	$-0.719908\pi$
$349$	−23.8078			−1.27440			−0.637200	+	0.770698i	$0.719908\pi$
$350$		0			0
$351$	−8.00000	−	1.35576i	−0.427008	−	0.0723649i
$352$		0			0
$353$			14.4858i			0.771001i	0.922708	+	0.385500i	$0.125971\pi$
$353$			14.4858i			0.771001i	−0.922708	+	0.385500i	$0.874029\pi$
$354$		0			0
$355$		−	9.33976i		−	0.495703i
$356$		0			0
$357$	−1.50758	+	0.761329i	−0.0797895	+	0.0402938i
$358$		0			0
$359$	17.3790			0.917230			0.458615	−	0.888635i	$-0.348346\pi$
$359$	17.3790			0.917230			0.458615	+	0.888635i	$0.348346\pi$
$360$		0			0
$361$	−26.1231			−1.37490
$362$		0			0
$363$	7.21922	+	14.2955i	0.378911	+	0.750317i
$364$		0			0
$365$	20.1907			1.05683
$366$		0			0
$367$		−	5.56760i		−	0.290626i	−0.989386	−	0.145313i	$-0.953581\pi$
$367$		−	5.56760i		−	0.290626i	0.989386	−	0.145313i	$-0.0464189\pi$
$368$		0			0
$369$	−3.27320	−	2.41430i	−0.170396	−	0.125683i
$370$		0			0
$371$		−	29.1853i		−	1.51523i
$372$		0			0
$373$		−	30.3188i		−	1.56985i	−0.619591	−	0.784925i	$-0.712702\pi$
$373$		−	30.3188i		−	1.56985i	0.619591	−	0.784925i	$-0.287298\pi$
$374$		0			0
$375$	2.06798	+	4.09499i	0.106790	+	0.211464i
$376$		0			0
$377$			7.54011i			0.388335i
$378$		0			0
$379$		−	9.66259i		−	0.496334i	−0.968717	−	0.248167i	$-0.920172\pi$
$379$		−	9.66259i		−	0.496334i	0.968717	−	0.248167i	$-0.0798281\pi$
$380$		0			0
$381$	−3.65767	−	7.24289i	−0.187388	−	0.371065i
$382$		0			0
$383$	14.7304			0.752689			0.376344	−	0.926480i	$-0.377181\pi$
$383$	14.7304			0.752689			0.376344	+	0.926480i	$0.377181\pi$
$384$		0			0
$385$		−	10.4895i		−	0.534596i
$386$		0			0
$387$	−16.2177	−	11.9621i	−0.824394	−	0.608070i
$388$		0			0
$389$	−4.34475			−0.220288			−0.110144	−	0.993916i	$-0.535131\pi$
$389$	−4.34475			−0.220288			−0.110144	+	0.993916i	$0.535131\pi$
$390$		0			0
$391$	1.75379	+	0.322830i	0.0886929	+	0.0163262i
$392$		0			0
$393$	19.7116	−	9.95441i	0.994321	−	0.502133i
$394$		0			0
$395$		−	36.1310i		−	1.81795i
$396$		0			0
$397$	28.9309			1.45200			0.725999	−	0.687695i	$-0.241378\pi$
$397$	28.9309			1.45200			0.725999	+	0.687695i	$0.241378\pi$
$398$		0			0
$399$	13.7538	+	27.2352i	0.688551	+	1.36346i
$400$		0			0
$401$	−9.06134			−0.452502			−0.226251	−	0.974069i	$-0.572647\pi$
$401$	−9.06134			−0.452502			−0.226251	+	0.974069i	$0.572647\pi$
$402$		0			0
$403$	13.5616			0.675549
$404$		0			0
$405$	−8.02736	−	25.9718i	−0.398883	−	1.29055i
$406$		0			0
$407$			12.3687i			0.613094i
$408$		0			0
$409$	20.6847			1.02279			0.511395	−	0.859346i	$-0.329129\pi$
$409$	20.6847			1.02279			0.511395	+	0.859346i	$0.329129\pi$
$410$		0			0
$411$	−0.743668	−	1.47261i	−0.0366825	−	0.0726383i
$412$		0			0
$413$	16.2177			0.798023
$414$		0			0
$415$	−32.4924			−1.59499
$416$		0			0
$417$	11.6577	+	23.0844i	0.570879	+	1.13045i
$418$		0			0
$419$	36.9890			1.80703			0.903517	−	0.428553i	$-0.140977\pi$
$419$	36.9890			1.80703			0.903517	+	0.428553i	$0.140977\pi$
$420$		0			0
$421$			9.33976i			0.455192i	0.973756	+	0.227596i	$0.0730865\pi$
$421$			9.33976i			0.455192i	−0.973756	+	0.227596i	$0.926913\pi$
$422$		0			0
$423$	−26.5885	−	19.6116i	−1.29278	−	0.953549i
$424$		0			0
$425$	1.53311			0.0743668
$426$		0			0
$427$	13.7538			0.665592
$428$		0			0
$429$	1.61463	+	3.19727i	0.0779549	+	0.154366i
$430$		0			0
$431$	13.5691			0.653602			0.326801	−	0.945093i	$-0.394029\pi$
$431$	13.5691			0.653602			0.326801	+	0.945093i	$0.394029\pi$
$432$		0			0
$433$		−	22.7745i		−	1.09447i	−0.836978	−	0.547236i	$-0.815680\pi$
$433$		−	22.7745i		−	1.09447i	0.836978	−	0.547236i	$-0.184320\pi$
$434$		0			0
$435$	−22.5490	+	11.3873i	−1.08114	+	0.545977i
$436$		0			0
$437$	5.83209	−	31.6831i	0.278987	−	1.51561i
$438$		0			0
$439$	−2.93087			−0.139883			−0.0699414	−	0.997551i	$-0.522281\pi$
$439$	−2.93087			−0.139883			−0.0699414	+	0.997551i	$0.522281\pi$
$440$		0			0
$441$	−0.219224	+	0.297214i	−0.0104392	+	0.0141530i
$442$		0			0
$443$			3.85350i			0.183086i	0.995801	+	0.0915428i	$0.0291798\pi$
$443$			3.85350i			0.183086i	−0.995801	+	0.0915428i	$0.970820\pi$
$444$		0			0
$445$	−23.3693			−1.10781
$446$		0			0
$447$	−12.8255	−	25.3969i	−0.606623	−	1.20123i
$448$		0			0
$449$		−	8.89586i		−	0.419822i	−0.977721	−	0.209911i	$-0.932683\pi$
$449$		−	8.89586i		−	0.419822i	0.977721	−	0.209911i	$-0.0673174\pi$
$450$		0			0
$451$			1.79544i			0.0845438i
$452$		0			0
$453$	−8.15009	−	16.1388i	−0.382925	−	0.758265i
$454$		0			0
$455$			12.3687i			0.579854i
$456$		0			0
$457$			33.2640i			1.55603i	0.628248	+	0.778013i	$0.283772\pi$
$457$			33.2640i			1.55603i	−0.628248	+	0.778013i	$0.716228\pi$
$458$		0			0
$459$	1.90495	+	0.322830i	0.0889152	+	0.0150684i
$460$		0			0
$461$			1.35576i			0.0631438i	0.999501	+	0.0315719i	$0.0100513\pi$
$461$			1.35576i			0.0631438i	−0.999501	+	0.0315719i	$0.989949\pi$
$462$		0			0
$463$	−7.50758			−0.348907			−0.174453	−	0.984665i	$-0.555816\pi$
$463$	−7.50758			−0.348907			−0.174453	+	0.984665i	$0.555816\pi$
$464$		0			0
$465$	20.4810	+	40.5563i	0.949783	+	1.88075i
$466$		0			0
$467$	10.1768			0.470928			0.235464	−	0.971883i	$-0.424339\pi$
$467$	10.1768			0.470928			0.235464	+	0.971883i	$0.424339\pi$
$468$		0			0
$469$	20.6307			0.952636
$470$		0			0
$471$	20.7713	−	10.4895i	0.957091	−	0.483332i
$472$		0			0
$473$			8.89586i			0.409032i
$474$		0			0
$475$		−	27.6964i		−	1.27080i
$476$		0			0
$477$	−19.8188	+	26.8695i	−0.907442	+	1.23027i
$478$		0			0
$479$	24.7442			1.13059			0.565296	−	0.824888i	$-0.308762\pi$
$479$	24.7442			1.13059			0.565296	+	0.824888i	$0.308762\pi$
$480$		0			0
$481$		−	14.5845i		−	0.664997i
$482$		0			0
$483$	−20.9008	+	6.13709i	−0.951018	+	0.279247i
$484$		0			0
$485$		−	12.3687i		−	0.561634i
$486$		0			0
$487$	14.0540			0.636846			0.318423	−	0.947949i	$-0.396847\pi$
$487$	14.0540			0.636846			0.318423	+	0.947949i	$0.396847\pi$
$488$		0			0
$489$	4.34233	+	8.59865i	0.196367	+	0.388844i
$490$		0			0
$491$			23.3817i			1.05520i	0.849493	+	0.527600i	$0.176908\pi$
$491$			23.3817i			1.05520i	−0.849493	+	0.527600i	$0.823092\pi$
$492$		0			0
$493$		−	1.79544i		−	0.0808624i
$494$		0			0
$495$	−7.12311	+	9.65719i	−0.320160	+	0.434059i
$496$		0			0
$497$	−8.10887			−0.363733
$498$		0			0
$499$	−10.4384			−0.467289			−0.233645	−	0.972322i	$-0.575065\pi$
$499$	−10.4384			−0.467289			−0.233645	+	0.972322i	$0.575065\pi$
$500$		0			0
$501$	−21.8078	+	11.0129i	−0.974299	+	0.492022i
$502$		0			0
$503$	23.0023			1.02562			0.512811	−	0.858501i	$-0.328604\pi$
$503$	23.0023			1.02562			0.512811	+	0.858501i	$0.328604\pi$
$504$		0			0
$505$			47.8486i			2.12923i
$506$		0			0
$507$	8.24621	+	16.3291i	0.366227	+	0.725200i
$508$		0			0
$509$			17.1973i			0.762257i	0.924522	+	0.381128i	$0.124464\pi$
$509$			17.1973i			0.762257i	−0.924522	+	0.381128i	$0.875536\pi$
$510$		0			0
$511$		−	17.5297i		−	0.775470i
$512$		0			0
$513$	5.83209	−	34.4138i	0.257493	−	1.51941i
$514$		0			0
$515$		−	23.7623i		−	1.04709i
$516$		0			0
$517$			14.5845i			0.641427i
$518$		0			0
$519$	−38.2462	+	19.3144i	−1.67882	+	0.847807i
$520$		0			0
$521$	−36.0366			−1.57879			−0.789395	−	0.613885i	$-0.789606\pi$
$521$	−36.0366			−1.57879			−0.789395	+	0.613885i	$0.789606\pi$
$522$		0			0
$523$		−	2.62238i		−	0.114669i	−0.998355	−	0.0573344i	$-0.981740\pi$
$523$		−	2.62238i		−	0.114669i	0.998355	−	0.0573344i	$-0.0182601\pi$
$524$		0			0
$525$	−16.7169	+	8.44204i	−0.729584	+	0.368441i
$526$		0			0
$527$	−3.22925			−0.140668
$528$		0			0
$529$	21.4924	+	8.18998i	0.934453	+	0.356086i
$530$		0			0
$531$	−14.9309	−	11.0129i	−0.647945	−	0.477921i
$532$		0			0
$533$		−	2.11708i		−	0.0917011i
$534$		0			0
$535$	18.2462			0.788853
$536$		0			0
$537$	36.1501	−	18.2558i	1.55999	−	0.787798i
$538$		0			0
$539$	0.163030			0.00702218
$540$		0			0
$541$	33.1771			1.42639			0.713197	−	0.700964i	$-0.247246\pi$
$541$	33.1771			1.42639			0.713197	+	0.700964i	$0.247246\pi$
$542$		0			0
$543$	−27.1025	+	13.6868i	−1.16308	+	0.587357i
$544$		0			0
$545$		−	15.8415i		−	0.678577i
$546$		0			0
$547$	−7.80776			−0.333836			−0.166918	−	0.985971i	$-0.553381\pi$
$547$	−7.80776			−0.333836			−0.166918	+	0.985971i	$0.553381\pi$
$548$		0			0
$549$	−12.6624	−	9.33976i	−0.540419	−	0.398611i
$550$		0			0
$551$	−32.4355			−1.38180
$552$		0			0
$553$	−31.3693			−1.33396
$554$		0			0
$555$	43.6155	−	22.0259i	1.85138	−	0.934947i
$556$		0			0
$557$	17.1702			0.727525			0.363763	−	0.931492i	$-0.381492\pi$
$557$	17.1702			0.727525			0.363763	+	0.931492i	$0.381492\pi$
$558$		0			0
$559$		−	10.4895i		−	0.443660i
$560$		0			0
$561$	−0.384472	−	0.761329i	−0.0162324	−	0.0321433i
$562$		0			0
$563$	−38.4764			−1.62159			−0.810793	−	0.585333i	$-0.800964\pi$
$563$	−38.4764			−1.62159			−0.810793	+	0.585333i	$0.800964\pi$
$564$		0			0
$565$	19.3693			0.814873
$566$		0			0
$567$	−22.5490	+	6.96943i	−0.946967	+	0.292689i
$568$		0			0
$569$	−5.66906			−0.237659			−0.118830	−	0.992915i	$-0.537914\pi$
$569$	−5.66906			−0.237659			−0.118830	+	0.992915i	$0.537914\pi$
$570$		0			0
$571$		−	24.2471i		−	1.01471i	−0.861737	−	0.507355i	$-0.830623\pi$
$571$		−	24.2471i		−	1.01471i	0.861737	−	0.507355i	$-0.169377\pi$
$572$		0			0
$573$	16.3450	+	32.3663i	0.682823	+	1.35212i
$574$		0			0
$575$	19.4470	+	3.57972i	0.810996	+	0.149285i
$576$		0			0
$577$	−19.3153			−0.804108			−0.402054	−	0.915616i	$-0.631704\pi$
$577$	−19.3153			−0.804108			−0.402054	+	0.915616i	$0.631704\pi$
$578$		0			0
$579$	−11.2732	−	22.3231i	−0.468498	−	0.927717i
$580$		0			0
$581$			28.2102i			1.17036i
$582$		0			0
$583$	14.7386			0.610412
$584$		0			0
$585$	8.39919	−	11.3873i	0.347264	−	0.470805i
$586$		0			0
$587$			25.1181i			1.03673i	0.855158	+	0.518367i	$0.173460\pi$
$587$			25.1181i			1.03673i	−0.855158	+	0.518367i	$0.826540\pi$
$588$		0			0
$589$			58.3381i			2.40378i
$590$		0			0
$591$	21.2192	−	10.7157i	0.872842	−	0.440786i
$592$		0			0
$593$			3.47284i			0.142612i	0.997454	+	0.0713062i	$0.0227168\pi$
$593$			3.47284i			0.142612i	−0.997454	+	0.0713062i	$0.977283\pi$
$594$		0			0
$595$		−	2.94521i		−	0.120742i
$596$		0			0
$597$	−4.05444	+	2.04750i	−0.165937	+	0.0837984i
$598$		0			0
$599$		−	21.0508i		−	0.860113i	−0.902802	−	0.430056i	$-0.858494\pi$
$599$		−	21.0508i		−	0.860113i	0.902802	−	0.430056i	$-0.141506\pi$
$600$		0			0
$601$	−25.5616			−1.04268			−0.521339	−	0.853350i	$-0.674567\pi$
$601$	−25.5616			−1.04268			−0.521339	+	0.853350i	$0.674567\pi$
$602$		0			0
$603$	−18.9936	−	14.0096i	−0.773481	−	0.570516i
$604$		0			0
$605$	−27.9277			−1.13542
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$	9.88653	+	19.5772i	0.400622	+	0.793309i
$610$		0			0
$611$		−	17.1973i		−	0.695728i
$612$		0			0
$613$			7.04020i			0.284351i	0.989841	+	0.142176i	$0.0454098\pi$
$613$			7.04020i			0.284351i	−0.989841	+	0.142176i	$0.954590\pi$
$614$		0			0
$615$	6.33122	−	3.19727i	0.255299	−	0.128926i
$616$		0			0
$617$	−3.60109			−0.144974			−0.0724871	−	0.997369i	$-0.523094\pi$
$617$	−3.60109			−0.144974			−0.0724871	+	0.997369i	$0.523094\pi$
$618$		0			0
$619$			6.71737i			0.269994i	0.990846	+	0.134997i	$0.0431025\pi$
$619$			6.71737i			0.269994i	−0.990846	+	0.134997i	$0.956898\pi$
$620$		0			0
$621$	23.4098	+	8.54292i	0.939403	+	0.342816i
$622$		0			0
$623$			20.2895i			0.812881i
$624$		0			0
$625$	−28.6155			−1.14462
$626$		0			0
$627$	−13.7538	+	6.94568i	−0.549273	+	0.277384i
$628$		0			0
$629$			3.47284i			0.138471i
$630$		0			0
$631$			11.9621i			0.476205i	0.971240	+	0.238103i	$0.0765255\pi$
$631$			11.9621i			0.476205i	−0.971240	+	0.238103i	$0.923475\pi$
$632$		0			0
$633$	−6.63068	−	13.1300i	−0.263546	−	0.521872i
$634$		0			0
$635$	14.1498			0.561516
$636$		0			0
$637$	−0.192236			−0.00761667
$638$		0			0
$639$	7.46543	+	5.50647i	0.295328	+	0.217833i
$640$		0			0
$641$	−20.5625			−0.812170			−0.406085	−	0.913835i	$-0.633106\pi$
$641$	−20.5625			−0.812170			−0.406085	+	0.913835i	$0.633106\pi$
$642$		0			0
$643$			10.8124i			0.426398i	0.977009	+	0.213199i	$0.0683883\pi$
$643$			10.8124i			0.426398i	−0.977009	+	0.213199i	$0.931612\pi$
$644$		0			0
$645$	31.3693	−	15.8415i	1.23517	−	0.623760i
$646$		0			0
$647$		−	20.6701i		−	0.812627i	−0.913734	−	0.406314i	$-0.866814\pi$
$647$		−	20.6701i		−	0.812627i	0.913734	−	0.406314i	$-0.133186\pi$
$648$		0			0
$649$			8.18998i			0.321485i
$650$		0			0
$651$	35.2114	−	17.7818i	1.38004	−	0.696923i
$652$		0			0
$653$			23.3817i			0.914995i	0.889211	+	0.457497i	$0.151254\pi$
$653$			23.3817i			0.914995i	−0.889211	+	0.457497i	$0.848746\pi$
$654$		0			0
$655$			38.5088i			1.50466i
$656$		0			0
$657$	−11.9039	+	16.1388i	−0.464415	+	0.629633i
$658$		0			0
$659$	17.5420			0.683341			0.341671	−	0.939820i	$-0.389007\pi$
$659$	17.5420			0.683341			0.341671	+	0.939820i	$0.389007\pi$
$660$		0			0
$661$			26.2238i			1.01999i	0.860178	+	0.509994i	$0.170353\pi$
$661$			26.2238i			1.01999i	−0.860178	+	0.509994i	$0.829647\pi$
$662$		0			0
$663$	0.453349	+	0.897718i	0.0176066	+	0.0348645i
$664$		0			0
$665$	−53.2068			−2.06327
$666$		0			0
$667$	4.19224	−	22.7745i	0.162324	−	0.881832i
$668$		0			0
$669$	11.1231	+	22.0259i	0.430044	+	0.851570i
$670$		0			0
$671$			6.94568i			0.268135i
$672$		0			0
$673$	−34.5464			−1.33167			−0.665833	−	0.746101i	$-0.731924\pi$
$673$	−34.5464			−1.33167			−0.665833	+	0.746101i	$0.731924\pi$
$674$		0			0
$675$	21.1231	+	3.57972i	0.813029	+	0.137784i
$676$		0			0
$677$	50.0233			1.92255			0.961276	−	0.275588i	$-0.0888724\pi$
$677$	50.0233			1.92255			0.961276	+	0.275588i	$0.0888724\pi$
$678$		0			0
$679$	−10.7386			−0.412111
$680$		0			0
$681$	17.8324	+	35.3115i	0.683338	+	1.35314i
$682$		0			0
$683$			25.1181i			0.961116i	0.876963	+	0.480558i	$0.159566\pi$
$683$			25.1181i			0.961116i	−0.876963	+	0.480558i	$0.840434\pi$
$684$		0			0
$685$	2.87689			0.109920
$686$		0			0
$687$	−36.9890	+	18.6795i	−1.41122	+	0.712668i
$688$		0			0
$689$	−17.3790			−0.662088
$690$		0			0
$691$	−16.4924			−0.627401			−0.313701	−	0.949522i	$-0.601569\pi$
$691$	−16.4924			−0.627401			−0.313701	+	0.949522i	$0.601569\pi$
$692$		0			0
$693$	8.38447	+	6.18435i	0.318500	+	0.234924i
$694$		0			0
$695$	−45.0979			−1.71066
$696$		0			0
$697$			0.504116i			0.0190948i
$698$		0			0
$699$	−11.6577	+	5.88714i	−0.440934	+	0.222672i
$700$		0			0
$701$	−10.2226			−0.386103			−0.193052	−	0.981189i	$-0.561838\pi$
$701$	−10.2226			−0.386103			−0.193052	+	0.981189i	$0.561838\pi$
$702$		0			0
$703$	62.7386			2.36623
$704$		0			0
$705$	51.4291	−	25.9718i	1.93693	−	0.978154i
$706$		0			0
$707$	41.5426			1.56237
$708$		0			0
$709$		−	30.3188i		−	1.13865i	−0.822113	−	0.569324i	$-0.807205\pi$
$709$		−	30.3188i		−	1.13865i	0.822113	−	0.569324i	$-0.192795\pi$
$710$		0			0
$711$	28.8802	+	21.3019i	1.08309	+	0.798884i
$712$		0			0
$713$	−40.9620	−	7.54011i	−1.53404	−	0.282379i
$714$		0			0
$715$	−6.24621			−0.233595
$716$		0			0
$717$	−10.1501	+	5.12581i	−0.379062	+	0.191427i
$718$		0			0
$719$		−	37.8674i		−	1.41222i	−0.708104	−	0.706109i	$-0.750449\pi$
$719$		−	37.8674i		−	1.41222i	0.708104	−	0.706109i	$-0.249551\pi$
$720$		0			0
$721$	−20.6307			−0.768327
$722$		0			0
$723$	6.33122	−	3.19727i	0.235460	−	0.118908i
$724$		0			0
$725$		−	19.9088i		−	0.739395i
$726$		0			0
$727$			0.322830i			0.0119731i	0.999982	+	0.00598655i	$0.00190559\pi$
$727$			0.322830i			0.0119731i	−0.999982	+	0.00598655i	$0.998094\pi$
$728$		0			0
$729$	25.4924	+	8.89586i	0.944164	+	0.329476i
$730$		0			0
$731$			2.49775i			0.0923826i
$732$		0			0
$733$		−	3.44933i		−	0.127404i	−0.997969	−	0.0637020i	$-0.979709\pi$
$733$		−	3.44933i		−	0.127404i	0.997969	−	0.0637020i	$-0.0202907\pi$
$734$		0			0
$735$	−0.290319	−	0.574888i	−0.0107086	−	0.0212051i
$736$		0			0
$737$			10.4185i			0.383771i
$738$		0			0
$739$	7.80776			0.287213			0.143607	−	0.989635i	$-0.454130\pi$
$739$	7.80776			0.287213			0.143607	+	0.989635i	$0.454130\pi$
$740$		0			0
$741$	16.2177	−	8.18998i	0.595774	−	0.300866i
$742$		0			0
$743$	−0.580639			−0.0213016			−0.0106508	−	0.999943i	$-0.503390\pi$
$743$	−0.580639			−0.0213016			−0.0106508	+	0.999943i	$0.503390\pi$
$744$		0			0
$745$	49.6155			1.81777
$746$		0			0
$747$	19.1567	−	25.9718i	0.700906	−	0.950258i
$748$		0			0
$749$		−	15.8415i		−	0.578837i
$750$		0			0
$751$		−	6.71737i		−	0.245120i	−0.992461	−	0.122560i	$-0.960890\pi$
$751$		−	6.71737i		−	0.245120i	0.992461	−	0.122560i	$-0.0391105\pi$
$752$		0			0
$753$	−14.6031	−	28.9170i	−0.532167	−	1.05379i
$754$		0			0
$755$	31.5288			1.14745
$756$		0			0
$757$		−	11.6393i		−	0.423038i	−0.977374	−	0.211519i	$-0.932159\pi$
$757$		−	11.6393i		−	0.423038i	0.977374	−	0.211519i	$-0.0678410\pi$
$758$		0			0
$759$	−3.09924	−	10.5549i	−0.112495	−	0.383119i
$760$		0			0
$761$		−	22.6203i		−	0.819986i	−0.912089	−	0.409993i	$-0.865531\pi$
$761$		−	22.6203i		−	0.819986i	0.912089	−	0.409993i	$-0.134469\pi$
$762$		0			0
$763$	−13.7538			−0.497921
$764$		0			0
$765$	−2.00000	+	2.71151i	−0.0723102	+	0.0980349i
$766$		0			0
$767$		−	9.65719i		−	0.348701i
$768$		0			0
$769$		−	25.0741i		−	0.904194i	−0.891969	−	0.452097i	$-0.850676\pi$
$769$		−	25.0741i		−	0.904194i	0.891969	−	0.452097i	$-0.149324\pi$
$770$		0			0
$771$	−31.9579	+	16.1388i	−1.15093	+	0.581223i
$772$		0			0
$773$	−25.4421			−0.915089			−0.457545	−	0.889187i	$-0.651271\pi$
$773$	−25.4421			−0.915089			−0.457545	+	0.889187i	$0.651271\pi$
$774$		0			0
$775$	−35.8078			−1.28625
$776$		0			0
$777$	−19.1231	−	37.8674i	−0.686038	−	1.35849i
$778$		0			0
$779$	9.10712			0.326296
$780$		0			0
$781$		−	4.09499i		−	0.146530i
$782$		0			0
$783$	4.19224	−	24.7374i	0.149818	−	0.884043i
$784$		0			0
$785$			40.5790i			1.44832i
$786$		0			0
$787$		−	41.1312i		−	1.46617i	−0.680138	−	0.733084i	$-0.738080\pi$
$787$		−	41.1312i		−	1.46617i	0.680138	−	0.733084i	$-0.261920\pi$
$788$		0			0
$789$	3.22925	+	6.39454i	0.114964	+	0.227652i
$790$		0			0
$791$		−	16.8166i		−	0.597931i
$792$		0			0
$793$		−	8.18998i		−	0.290835i
$794$		0			0
$795$	−26.2462	−	51.9726i	−0.930857	−	1.84328i
$796$		0			0
$797$	−25.1161			−0.889656			−0.444828	−	0.895616i	$-0.646735\pi$
$797$	−25.1161			−0.889656			−0.444828	+	0.895616i	$0.646735\pi$
$798$		0			0
$799$			4.09499i			0.144870i
$800$		0			0
$801$	13.7779	−	18.6795i	0.486819	−	0.660008i
$802$		0			0
$803$	8.85254			0.312399
$804$		0			0
$805$	6.87689	−	37.3590i	0.242379	−	1.31673i
$806$		0			0
$807$	−31.9579	+	16.1388i	−1.12497	+	0.568111i
$808$		0			0
$809$			44.0518i			1.54878i	0.632709	+	0.774389i	$0.281943\pi$
$809$			44.0518i			1.54878i	−0.632709	+	0.774389i	$0.718057\pi$
$810$		0			0
$811$	27.3153			0.959171			0.479586	−	0.877495i	$-0.340787\pi$
$811$	27.3153			0.959171			0.479586	+	0.877495i	$0.340787\pi$
$812$		0			0
$813$	1.75379	+	3.47284i	0.0615081	+	0.121798i
$814$		0			0
$815$	−16.7984			−0.588422
$816$		0			0
$817$	45.1231			1.57866
$818$		0			0
$819$	−9.88653	−	7.29226i	−0.345463	−	0.254812i
$820$		0			0
$821$		−	33.6333i		−	1.17381i	−0.809656	−	0.586905i	$-0.800346\pi$
$821$		−	33.6333i		−	1.17381i	0.809656	−	0.586905i	$-0.199654\pi$
$822$		0			0
$823$	−18.4384			−0.642724			−0.321362	−	0.946956i	$-0.604141\pi$
$823$	−18.4384			−0.642724			−0.321362	+	0.946956i	$0.604141\pi$
$824$		0			0
$825$	−4.26324	−	8.44204i	−0.148427	−	0.293914i
$826$		0			0
$827$	−3.39228			−0.117961			−0.0589806	−	0.998259i	$-0.518785\pi$
$827$	−3.39228			−0.117961			−0.0589806	+	0.998259i	$0.518785\pi$
$828$		0			0
$829$	9.86174			0.342512			0.171256	−	0.985227i	$-0.445217\pi$
$829$	9.86174			0.342512			0.171256	+	0.985227i	$0.445217\pi$
$830$		0			0
$831$	5.90388	+	11.6908i	0.204803	+	0.405550i
$832$		0			0
$833$	0.0457749			0.00158601
$834$		0			0
$835$		−	42.6038i		−	1.47437i
$836$		0			0
$837$	−44.4924	−	7.54011i	−1.53788	−	0.260624i
$838$		0			0
$839$	−7.36520			−0.254275			−0.127138	−	0.991885i	$-0.540579\pi$
$839$	−7.36520			−0.254275			−0.127138	+	0.991885i	$0.540579\pi$
$840$		0			0
$841$	5.68466			0.196023
$842$		0			0
$843$	20.6440	+	40.8791i	0.711018	+	1.40795i
$844$		0			0
$845$	−31.9006			−1.09741
$846$		0			0
$847$			24.2471i			0.833141i
$848$		0			0
$849$	41.0435	−	20.7270i	1.40861	−	0.711349i
$850$		0			0
$851$	−8.10887	+	44.0518i	−0.277969	+	1.51008i
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$	48.9848	+	36.1310i	1.67525	+	1.23565i
$856$		0			0
$857$			7.54011i			0.257565i	0.991673	+	0.128783i	$0.0411069\pi$
$857$			7.54011i			0.257565i	−0.991673	+	0.128783i	$0.958893\pi$
$858$		0			0
$859$	−38.0540			−1.29838			−0.649192	−	0.760624i	$-0.724893\pi$
$859$	−38.0540			−1.29838			−0.649192	+	0.760624i	$0.724893\pi$
$860$		0			0
$861$	−2.77590	−	5.49682i	−0.0946025	−	0.187331i
$862$		0			0
$863$			28.8047i			0.980523i	0.871576	+	0.490261i	$0.163099\pi$
$863$			28.8047i			0.980523i	−0.871576	+	0.490261i	$0.836901\pi$
$864$		0			0
$865$		−	74.7181i		−	2.54049i
$866$		0			0
$867$	13.1652	+	26.0697i	0.447115	+	0.885375i
$868$		0			0
$869$		−	15.8415i		−	0.537388i
$870$		0			0
$871$		−	12.2850i		−	0.416260i
$872$		0			0
$873$	9.88653	+	7.29226i	0.334608	+	0.246806i
$874$		0			0
$875$			6.94568i			0.234807i
$876$		0			0
$877$	−54.9848			−1.85671			−0.928353	−	0.371699i	$-0.878775\pi$
$877$	−54.9848			−1.85671			−0.928353	+	0.371699i	$0.878775\pi$
$878$		0			0
$879$	−12.2448	−	24.2471i	−0.413008	−	0.817835i
$880$		0			0
$881$	−32.0636			−1.08025			−0.540126	−	0.841584i	$-0.681623\pi$
$881$	−32.0636			−1.08025			−0.540126	+	0.841584i	$0.681623\pi$
$882$		0			0
$883$	40.4924			1.36268			0.681339	−	0.731968i	$-0.261398\pi$
$883$	40.4924			1.36268			0.681339	+	0.731968i	$0.261398\pi$
$884$		0			0
$885$	28.8802	−	14.5845i	0.970796	−	0.490253i
$886$		0			0
$887$			5.58992i			0.187691i	0.995587	+	0.0938457i	$0.0299160\pi$
$887$			5.58992i			0.187691i	−0.995587	+	0.0938457i	$0.970084\pi$
$888$		0			0
$889$		−	12.2850i		−	0.412025i
$890$		0			0
$891$	−3.51957	−	11.3873i	−0.117910	−	0.381487i
$892$		0			0
$893$	73.9781			2.47558
$894$		0			0
$895$			70.6231i			2.36067i
$896$		0			0
$897$	3.65446	+	12.4458i	0.122019	+	0.415553i
$898$		0			0
$899$			41.9347i			1.39860i
$900$		0			0
$901$	4.13826			0.137865
$902$		0			0
$903$	−13.7538	−	27.2352i	−0.457697	−	0.906329i
$904$		0			0
$905$		−	52.9477i		−	1.76004i
$906$		0			0
$907$			28.3421i			0.941084i	0.882378	+	0.470542i	$0.155942\pi$
$907$			28.3421i			0.941084i	−0.882378	+	0.470542i	$0.844058\pi$
$908$		0			0
$909$	−38.2462	−	28.2102i	−1.26855	−	0.935675i
$910$		0			0
$911$	10.9205			0.361813			0.180906	−	0.983500i	$-0.442097\pi$
$911$	10.9205			0.361813			0.180906	+	0.983500i	$0.442097\pi$
$912$		0			0
$913$	−14.2462			−0.471481
$914$		0			0
$915$	24.4924	−	12.3687i	0.809695	−	0.408897i
$916$		0			0
$917$	33.4337			1.10408
$918$		0			0
$919$			10.1667i			0.335369i	0.985841	+	0.167684i	$0.0536289\pi$
$919$			10.1667i			0.335369i	−0.985841	+	0.167684i	$0.946371\pi$
$920$		0			0
$921$	17.3693	+	34.3946i	0.572338	+	1.13334i
$922$		0			0
$923$			4.82860i			0.158935i
$924$		0			0
$925$			38.5088i			1.26616i
$926$		0			0
$927$	18.9936	+	14.0096i	0.623833	+	0.460137i
$928$		0			0
$929$			8.30144i			0.272361i	0.990684	+	0.136181i	$0.0434828\pi$
$929$			8.30144i			0.272361i	−0.990684	+	0.136181i	$0.956517\pi$
$930$		0			0
$931$		−	0.826946i		−	0.0271021i
$932$		0			0
$933$	−2.09612	+	1.05854i	−0.0686238	+	0.0346551i
$934$		0			0
$935$	1.48734			0.0486411
$936$		0			0
$937$		−	43.7536i		−	1.42937i	−0.699448	−	0.714683i	$-0.746571\pi$
$937$		−	43.7536i		−	1.42937i	0.699448	−	0.714683i	$-0.253429\pi$
$938$		0			0
$939$	−28.8802	+	14.5845i	−0.942469	+	0.475948i
$940$		0			0
$941$	−42.6581			−1.39061			−0.695307	−	0.718713i	$-0.744732\pi$
$941$	−42.6581			−1.39061			−0.695307	+	0.718713i	$0.744732\pi$
$942$		0			0
$943$	−1.17708	+	6.39454i	−0.0383310	+	0.208235i
$944$		0			0
$945$	6.87689	−	40.5790i	0.223705	−	1.32003i
$946$		0			0
$947$		−	20.6701i		−	0.671689i	−0.941917	−	0.335845i	$-0.890978\pi$
$947$		−	20.6701i		−	0.671689i	0.941917	−	0.335845i	$-0.109022\pi$
$948$		0			0
$949$	−10.4384			−0.338846
$950$		0			0
$951$	38.2462	−	19.3144i	1.24022	−	0.626312i
$952$		0			0
$953$	−11.8730			−0.384604			−0.192302	−	0.981336i	$-0.561595\pi$
$953$	−11.8730			−0.384604			−0.192302	+	0.981336i	$0.561595\pi$
$954$		0			0
$955$	−63.2311			−2.04611
$956$		0			0
$957$	−9.88653	+	4.99271i	−0.319586	+	0.161391i
$958$		0			0
$959$		−	2.49775i		−	0.0806565i
$960$		0			0
$961$	44.4233			1.43301
$962$		0			0
$963$	−10.7575	+	14.5845i	−0.346655	+	0.469980i
$964$		0			0
$965$	43.6106			1.40387
$966$		0			0
$967$	54.1619			1.74173			0.870865	−	0.491522i	$-0.163559\pi$
$967$	54.1619			1.74173			0.870865	+	0.491522i	$0.163559\pi$
$968$		0			0
$969$	−3.86174	+	1.95018i	−0.124057	+	0.0626489i
$970$		0			0
$971$	30.2045			0.969308			0.484654	−	0.874706i	$-0.338946\pi$
$971$	30.2045			0.969308			0.484654	+	0.874706i	$0.338946\pi$
$972$		0			0
$973$			39.1545i			1.25523i
$974$		0			0
$975$	5.02699	+	9.95441i	0.160992	+	0.318796i
$976$		0			0
$977$	6.41273			0.205161			0.102581	−	0.994725i	$-0.467290\pi$
$977$	6.41273			0.205161			0.102581	+	0.994725i	$0.467290\pi$
$978$		0			0
$979$	−10.2462			−0.327470
$980$		0			0
$981$	12.6624	+	9.33976i	0.404280	+	0.298195i
$982$		0			0
$983$	−29.4608			−0.939654			−0.469827	−	0.882759i	$-0.655684\pi$
$983$	−29.4608			−0.939654			−0.469827	+	0.882759i	$0.655684\pi$
$984$		0			0
$985$			41.4540i			1.32083i
$986$		0			0
$987$	−22.5490	−	44.6513i	−0.717741	−	1.42127i
$988$		0			0
$989$	−5.83209	+	31.6831i	−0.185450	+	1.00746i
$990$		0			0
$991$	20.4924			0.650963			0.325482	−	0.945548i	$-0.394474\pi$
$991$	20.4924			0.650963			0.325482	+	0.945548i	$0.394474\pi$
$992$		0			0
$993$	6.09612	+	12.0715i	0.193454	+	0.383077i
$994$		0			0
$995$		−	7.92077i		−	0.251105i
$996$		0			0
$997$	−31.1231			−0.985679			−0.492839	−	0.870120i	$-0.664041\pi$
$997$	−31.1231			−0.985679			−0.492839	+	0.870120i	$0.664041\pi$
$998$		0			0
$999$	−8.10887	+	47.8486i	−0.256553	+	1.51386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	276.2.g.a.137.2	yes	8
3.2	odd	2	inner	276.2.g.a.137.3	yes	8
4.3	odd	2		1104.2.m.b.689.8		8
12.11	even	2		1104.2.m.b.689.5		8
23.22	odd	2	inner	276.2.g.a.137.1	✓	8
69.68	even	2	inner	276.2.g.a.137.4	yes	8
92.91	even	2		1104.2.m.b.689.7		8
276.275	odd	2		1104.2.m.b.689.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.g.a.137.1	✓	8	23.22	odd	2	inner
276.2.g.a.137.2	yes	8	1.1	even	1	trivial
276.2.g.a.137.3	yes	8	3.2	odd	2	inner
276.2.g.a.137.4	yes	8	69.68	even	2	inner
1104.2.m.b.689.5		8	12.11	even	2
1104.2.m.b.689.6		8	276.275	odd	2
1104.2.m.b.689.7		8	92.91	even	2
1104.2.m.b.689.8		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	276.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.780776 - 1.54609i) q^{3} +3.02045 q^{5} -2.62238i q^{7} +(-1.78078 + 2.41430i) q^{9} +O(q^{10})\)	\(q+(-0.780776 - 1.54609i) q^{3} +3.02045 q^{5} -2.62238i q^{7} +(-1.78078 + 2.41430i) q^{9} +1.32431 q^{11} -1.56155 q^{13} +(-2.35829 - 4.66988i) q^{15} +0.371834 q^{17} -6.71737i q^{19} +(-4.05444 + 2.04750i) q^{21} +(4.71659 + 0.868210i) q^{23} +4.12311 q^{25} +(5.12311 + 0.868210i) q^{27} -4.82860i q^{29} -8.68466 q^{31} +(-1.03399 - 2.04750i) q^{33} -7.92077i q^{35} +9.33976i q^{37} +(1.21922 + 2.41430i) q^{39} +1.35576i q^{41} +6.71737i q^{43} +(-5.37874 + 7.29226i) q^{45} +11.0129i q^{47} +0.123106 q^{49} +(-0.290319 - 0.574888i) q^{51} +11.1293 q^{53} +4.00000 q^{55} +(-10.3857 + 5.24477i) q^{57} +6.18435i q^{59} +5.24477i q^{61} +(6.33122 + 4.66988i) q^{63} -4.71659 q^{65} +7.86715i q^{67} +(-2.34027 - 7.97014i) q^{69} -3.09218i q^{71} +6.68466 q^{73} +(-3.21922 - 6.37468i) q^{75} -3.47284i q^{77} -11.9621i q^{79} +(-2.65767 - 8.59865i) q^{81} -10.7575 q^{83} +1.12311 q^{85} +(-7.46543 + 3.77005i) q^{87} -7.73704 q^{89} +4.09499i q^{91} +(6.78078 + 13.4272i) q^{93} -20.2895i q^{95} -4.09499i q^{97} +(-2.35829 + 3.19727i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 6 q^{9}+O(q^{10}) \) 8 * q + 2 * q^3 - 6 * q^9	\( 8 q + 2 q^{3} - 6 q^{9} + 4 q^{13} + 8 q^{27} - 20 q^{31} + 18 q^{39} - 32 q^{49} + 32 q^{55} - 14 q^{69} + 4 q^{73} - 34 q^{75} - 46 q^{81} - 24 q^{85} - 2 q^{87} + 46 q^{93}+O(q^{100}) \) 8 * q + 2 * q^3 - 6 * q^9 + 4 * q^13 + 8 * q^27 - 20 * q^31 + 18 * q^39 - 32 * q^49 + 32 * q^55 - 14 * q^69 + 4 * q^73 - 34 * q^75 - 46 * q^81 - 24 * q^85 - 2 * q^87 + 46 * q^93

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