LMFDB - Embedded newform 2640.2.f.a.1121.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(1121,2640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2640.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2640.f (of order $2$ , degree $1$ , not 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:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2051727616.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4$ x^8 + 11x^6 + 37x^4 + 36*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 330)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.5
Root			$2.25619i$ of defining polynomial
Character	$\chi$	$=$	2640.1121
Dual form			2640.2.f.a.1121.6

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0828988 - 1.73007i) q^{3} -1.00000i q^{5} -4.34658i q^{7} +(-2.98626 + 0.286841i) q^{9} +(-1.20394 + 3.09039i) q^{11} +4.51238i q^{13} +(-1.73007 + 0.0828988i) q^{15} +2.72065 q^{17} +3.97251i q^{19} +(-7.51987 + 0.360326i) q^{21} +1.66840i q^{23} -1.00000 q^{25} +(0.743810 + 5.14264i) q^{27} -5.08606 q^{29} -9.74541 q^{31} +(5.44639 + 1.82671i) q^{33} -4.34658 q^{35} -5.60710 q^{37} +(7.80671 - 0.374071i) q^{39} +2.92026 q^{41} +3.25186i q^{43} +(0.286841 + 2.98626i) q^{45} +9.25186i q^{47} -11.8928 q^{49} +(-0.225539 - 4.70691i) q^{51} +3.05225i q^{53} +(3.09039 + 1.20394i) q^{55} +(6.87271 - 0.329316i) q^{57} +5.43264i q^{59} -11.5646i q^{61} +(1.24678 + 12.9800i) q^{63} +4.51238 q^{65} -12.3529 q^{67} +(2.88645 - 0.138309i) q^{69} -3.60710i q^{71} -1.36502i q^{73} +(0.0828988 + 1.73007i) q^{75} +(13.4326 + 5.23303i) q^{77} -1.07107i q^{79} +(8.83544 - 1.71316i) q^{81} +14.5887 q^{83} -2.72065i q^{85} +(0.421628 + 8.79922i) q^{87} +3.75791i q^{89} +19.6134 q^{91} +(0.807883 + 16.8602i) q^{93} +3.97251 q^{95} -12.7977 q^{97} +(2.70883 - 9.57404i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 2 q^{3} + 2 q^{9} - 6 q^{11} - 4 q^{15} + 4 q^{17} - 8 q^{21} - 8 q^{25} + 22 q^{27} - 4 q^{29} + 4 q^{31} - 2 q^{33} - 12 q^{37} + 8 q^{39} - 16 q^{41} - 4 q^{49} + 28 q^{51} + 6 q^{55} + 14 q^{57}+ \cdots + 6 q^{99}+O(q^{100})$ 8 * q - 2 * q^3 + 2 * q^9 - 6 * q^11 - 4 * q^15 + 4 * q^17 - 8 * q^21 - 8 * q^25 + 22 * q^27 - 4 * q^29 + 4 * q^31 - 2 * q^33 - 12 * q^37 + 8 * q^39 - 16 * q^41 - 4 * q^49 + 28 * q^51 + 6 * q^55 + 14 * q^57 - 14 * q^63 + 4 * q^65 + 12 * q^67 + 8 * q^69 + 2 * q^75 + 36 * q^77 + 2 * q^81 + 72 * q^83 + 22 * q^87 + 48 * q^91 + 8 * q^93 - 20 * q^95 - 8 * q^97 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$ .

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

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
$3$	−0.0828988	−	1.73007i	−0.0478616	−	0.998854i
$3$	−0.0828988	−	1.73007i	−0.0478616	−	0.998854i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	4.34658i		−	1.64285i	−0.570314	−	0.821427i	$-0.693178\pi$
$7$		−	4.34658i		−	1.64285i	0.570314	−	0.821427i	$-0.306822\pi$
$8$		0			0
$9$	−2.98626	+	0.286841i	−0.995419	+	0.0956136i
$10$		0			0
$11$	−1.20394	+	3.09039i	−0.363002	+	0.931788i
$11$	−1.20394	+	3.09039i	−0.363002	+	0.931788i
$12$		0			0
$13$			4.51238i			1.25151i	0.780020	+	0.625754i	$0.215209\pi$
$13$			4.51238i			1.25151i	−0.780020	+	0.625754i	$0.784791\pi$
$14$		0			0
$15$	−1.73007	+	0.0828988i	−0.446701	+	0.0214044i
$16$		0			0
$17$	2.72065			0.659855			0.329928	−	0.944006i	$-0.392976\pi$
$17$	2.72065			0.659855			0.329928	+	0.944006i	$0.392976\pi$
$18$		0			0
$19$			3.97251i			0.911357i	0.890145	+	0.455678i	$0.150603\pi$
$19$			3.97251i			0.911357i	−0.890145	+	0.455678i	$0.849397\pi$
$20$		0			0
$21$	−7.51987	+	0.360326i	−1.64097	+	0.0786297i
$22$		0			0
$23$			1.66840i			0.347886i	0.984756	+	0.173943i	$0.0556509\pi$
$23$			1.66840i			0.347886i	−0.984756	+	0.173943i	$0.944349\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	0.743810	+	5.14264i	0.143146	+	0.989702i
$28$		0			0
$29$	−5.08606			−0.944458			−0.472229	−	0.881476i	$-0.656550\pi$
$29$	−5.08606			−0.944458			−0.472229	+	0.881476i	$0.656550\pi$
$30$		0			0
$31$	−9.74541			−1.75033			−0.875164	−	0.483827i	$-0.839246\pi$
$31$	−9.74541			−1.75033			−0.875164	+	0.483827i	$0.839246\pi$
$32$		0			0
$33$	5.44639	+	1.82671i	0.948094	+	0.317989i
$34$		0			0
$35$	−4.34658			−0.734706
$36$		0			0
$37$	−5.60710			−0.921802			−0.460901	−	0.887452i	$-0.652474\pi$
$37$	−5.60710			−0.921802			−0.460901	+	0.887452i	$0.652474\pi$
$38$		0			0
$39$	7.80671	−	0.374071i	1.25007	−	0.0598993i
$40$		0			0
$41$	2.92026			0.456069			0.228034	−	0.973653i	$-0.426770\pi$
$41$	2.92026			0.456069			0.228034	+	0.973653i	$0.426770\pi$
$42$		0			0
$43$			3.25186i			0.495904i	0.968772	+	0.247952i	$0.0797575\pi$
$43$			3.25186i			0.495904i	−0.968772	+	0.247952i	$0.920243\pi$
$44$		0			0
$45$	0.286841	+	2.98626i	0.0427597	+	0.445165i
$46$		0			0
$47$			9.25186i			1.34952i	0.738036	+	0.674761i	$0.235753\pi$
$47$			9.25186i			1.34952i	−0.738036	+	0.674761i	$0.764247\pi$
$48$		0			0
$49$	−11.8928			−1.69897
$50$		0			0
$51$	−0.225539	−	4.70691i	−0.0315817	−	0.659099i
$52$		0			0
$53$			3.05225i			0.419258i	0.977781	+	0.209629i	$0.0672257\pi$
$53$			3.05225i			0.419258i	−0.977781	+	0.209629i	$0.932774\pi$
$54$		0			0
$55$	3.09039	+	1.20394i	0.416708	+	0.162339i
$56$		0			0
$57$	6.87271	−	0.329316i	0.910312	−	0.0436190i
$58$		0			0
$59$			5.43264i			0.707270i	0.935384	+	0.353635i	$0.115054\pi$
$59$			5.43264i			0.707270i	−0.935384	+	0.353635i	$0.884946\pi$
$60$		0			0
$61$		−	11.5646i		−	1.48070i	−0.672222	−	0.740349i	$-0.734660\pi$
$61$		−	11.5646i		−	1.48070i	0.672222	−	0.740349i	$-0.265340\pi$
$62$		0			0
$63$	1.24678	+	12.9800i	0.157079	+	1.63533i
$64$		0			0
$65$	4.51238			0.559692
$66$		0			0
$67$	−12.3529			−1.50915			−0.754574	−	0.656215i	$-0.772156\pi$
$67$	−12.3529			−1.50915			−0.754574	+	0.656215i	$0.772156\pi$
$68$		0			0
$69$	2.88645	−	0.138309i	0.347488	−	0.0166504i
$70$		0			0
$71$		−	3.60710i		−	0.428084i	−0.976824	−	0.214042i	$-0.931337\pi$
$71$		−	3.60710i		−	0.428084i	0.976824	−	0.214042i	$-0.0686630\pi$
$72$		0			0
$73$		−	1.36502i		−	0.159763i	−0.996804	−	0.0798816i	$-0.974546\pi$
$73$		−	1.36502i		−	0.159763i	0.996804	−	0.0798816i	$-0.0254542\pi$
$74$		0			0
$75$	0.0828988	+	1.73007i	0.00957233	+	0.199771i
$76$		0			0
$77$	13.4326	+	5.23303i	1.53079	+	0.596359i
$78$		0			0
$79$		−	1.07107i		−	0.120505i	−0.998183	−	0.0602526i	$-0.980809\pi$
$79$		−	1.07107i		−	0.120505i	0.998183	−	0.0602526i	$-0.0191906\pi$
$80$		0			0
$81$	8.83544	−	1.71316i	0.981716	−	0.190351i
$82$		0			0
$83$	14.5887			1.60131			0.800657	−	0.599123i	$-0.204484\pi$
$83$	14.5887			1.60131			0.800657	+	0.599123i	$0.204484\pi$
$84$		0			0
$85$		−	2.72065i		−	0.295096i
$86$		0			0
$87$	0.421628	+	8.79922i	0.0452033	+	0.943375i
$88$		0			0
$89$			3.75791i			0.398338i	0.979965	+	0.199169i	$0.0638243\pi$
$89$			3.75791i			0.398338i	−0.979965	+	0.199169i	$0.936176\pi$
$90$		0			0
$91$	19.6134			2.05605
$92$		0			0
$93$	0.807883	+	16.8602i	0.0837735	+	1.74832i
$94$		0			0
$95$	3.97251			0.407571
$96$		0			0
$97$	−12.7977			−1.29941			−0.649703	−	0.760188i	$-0.725107\pi$
$97$	−12.7977			−1.29941			−0.649703	+	0.760188i	$0.725107\pi$
$98$		0			0
$99$	2.70883	−	9.57404i	0.272247	−	0.962227i
$100$		0			0
$101$	6.73948			0.670603			0.335302	−	0.942111i	$-0.391162\pi$
$101$	6.73948			0.670603			0.335302	+	0.942111i	$0.391162\pi$
$102$		0			0
$103$	5.25186			0.517481			0.258740	−	0.965947i	$-0.416693\pi$
$103$	5.25186			0.517481			0.258740	+	0.965947i	$0.416693\pi$
$104$		0			0
$105$	0.360326	+	7.51987i	0.0351643	+	0.733864i
$106$		0			0
$107$	−6.25707			−0.604894			−0.302447	−	0.953166i	$-0.597804\pi$
$107$	−6.25707			−0.604894			−0.302447	+	0.953166i	$0.597804\pi$
$108$		0			0
$109$			17.9168i			1.71612i	0.513550	+	0.858060i	$0.328330\pi$
$109$			17.9168i			1.71612i	−0.513550	+	0.858060i	$0.671670\pi$
$110$		0			0
$111$	0.464822	+	9.70066i	0.0441190	+	0.920746i
$112$		0			0
$113$		−	8.51238i		−	0.800777i	−0.916345	−	0.400389i	$-0.868875\pi$
$113$		−	8.51238i		−	0.800777i	0.916345	−	0.400389i	$-0.131125\pi$
$114$		0			0
$115$	1.66840			0.155580
$116$		0			0
$117$	−1.29433	−	13.4751i	−0.119661	−	1.24578i
$118$		0			0
$119$		−	11.8255i		−	1.08405i
$120$		0			0
$121$	−8.10105	−	7.44131i	−0.736459	−	0.676482i
$122$		0			0
$123$	−0.242086	−	5.05225i	−0.0218282	−	0.455546i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	8.95024i		−	0.794205i	−0.917774	−	0.397103i	$-0.870016\pi$
$127$		−	8.95024i		−	0.794205i	0.917774	−	0.397103i	$-0.129984\pi$
$128$		0			0
$129$	5.62593	−	0.269575i	0.495335	−	0.0237348i
$130$		0			0
$131$	12.3379			1.07797			0.538985	−	0.842316i	$-0.318808\pi$
$131$	12.3379			1.07797			0.538985	+	0.842316i	$0.318808\pi$
$132$		0			0
$133$	17.2668			1.49723
$134$		0			0
$135$	5.14264	−	0.743810i	0.442608	−	0.0640170i
$136$		0			0
$137$			20.9416i			1.78916i	0.446908	+	0.894580i	$0.352525\pi$
$137$			20.9416i			1.78916i	−0.446908	+	0.894580i	$0.647475\pi$
$138$		0			0
$139$		−	15.5458i		−	1.31858i	−0.751890	−	0.659289i	$-0.770857\pi$
$139$		−	15.5458i		−	1.31858i	0.751890	−	0.659289i	$-0.229143\pi$
$140$		0			0
$141$	16.0063	−	0.766968i	1.34798	−	0.0645903i
$142$		0			0
$143$	−13.9450	−	5.43264i	−1.16614	−	0.454300i
$144$		0			0
$145$			5.08606i			0.422374i
$146$		0			0
$147$	0.985897	+	20.5753i	0.0813154	+	1.69702i
$148$		0			0
$149$	−23.2882			−1.90784			−0.953920	−	0.300061i	$-0.902993\pi$
$149$	−23.2882			−1.90784			−0.953920	+	0.300061i	$0.902993\pi$
$150$		0			0
$151$		−	10.8440i		−	0.882470i	−0.897392	−	0.441235i	$-0.854540\pi$
$151$		−	10.8440i		−	0.882470i	0.897392	−	0.441235i	$-0.145460\pi$
$152$		0			0
$153$	−8.12456	+	0.780394i	−0.656832	+	0.0630911i
$154$		0			0
$155$			9.74541i			0.782770i
$156$		0			0
$157$	11.0861			0.884764			0.442382	−	0.896827i	$-0.354134\pi$
$157$	11.0861			0.884764			0.442382	+	0.896827i	$0.354134\pi$
$158$		0			0
$159$	5.28059	−	0.253028i	0.418778	−	0.0200664i
$160$		0			0
$161$	7.25186			0.571527
$162$		0			0
$163$	−4.16580			−0.326290			−0.163145	−	0.986602i	$-0.552164\pi$
$163$	−4.16580			−0.326290			−0.163145	+	0.986602i	$0.552164\pi$
$164$		0			0
$165$	1.82671	−	5.44639i	0.142209	−	0.424001i
$166$		0			0
$167$	−8.78828			−0.680057			−0.340029	−	0.940415i	$-0.610437\pi$
$167$	−8.78828			−0.680057			−0.340029	+	0.940415i	$0.610437\pi$
$168$		0			0
$169$	−7.36157			−0.566275
$170$		0			0
$171$	−1.13948	−	11.8629i	−0.0871381	−	0.907181i
$172$		0			0
$173$	2.66319			0.202479			0.101239	−	0.994862i	$-0.467719\pi$
$173$	2.66319			0.202479			0.101239	+	0.994862i	$0.467719\pi$
$174$		0			0
$175$			4.34658i			0.328571i
$176$		0			0
$177$	9.39883	−	0.450359i	0.706459	−	0.0338511i
$178$		0			0
$179$			11.2605i			0.841651i	0.907142	+	0.420825i	$0.138260\pi$
$179$			11.2605i			0.841651i	−0.907142	+	0.420825i	$0.861740\pi$
$180$		0			0
$181$	−0.711272			−0.0528684			−0.0264342	−	0.999651i	$-0.508415\pi$
$181$	−0.711272			−0.0528684			−0.0264342	+	0.999651i	$0.508415\pi$
$182$		0			0
$183$	−20.0076	+	0.958694i	−1.47900	+	0.0708687i
$184$		0			0
$185$			5.60710i			0.412242i
$186$		0			0
$187$	−3.27551	+	8.40788i	−0.239529	+	0.614845i
$188$		0			0
$189$	22.3529	−	3.23303i	1.62593	−	0.235168i
$190$		0			0
$191$		−	19.5852i		−	1.41714i	−0.705642	−	0.708568i	$-0.749341\pi$
$191$		−	19.5852i		−	1.41714i	0.705642	−	0.708568i	$-0.250659\pi$
$192$		0			0
$193$		−	8.82899i		−	0.635524i	−0.948170	−	0.317762i	$-0.897069\pi$
$193$		−	8.82899i		−	0.635524i	0.948170	−	0.317762i	$-0.102931\pi$
$194$		0			0
$195$	−0.374071	−	7.80671i	−0.0267878	−	0.559050i
$196$		0			0
$197$	−12.9502			−0.922666			−0.461333	−	0.887227i	$-0.652629\pi$
$197$	−12.9502			−0.922666			−0.461333	+	0.887227i	$0.652629\pi$
$198$		0			0
$199$	−1.17485			−0.0832830			−0.0416415	−	0.999133i	$-0.513259\pi$
$199$	−1.17485			−0.0832830			−0.0416415	+	0.999133i	$0.513259\pi$
$200$		0			0
$201$	1.02404	+	21.3713i	0.0722303	+	1.50742i
$202$		0			0
$203$			22.1070i			1.55161i
$204$		0			0
$205$		−	2.92026i		−	0.203960i
$206$		0			0
$207$	−0.478566	−	4.98228i	−0.0332627	−	0.346293i
$208$		0			0
$209$	−12.2766	−	4.78267i	−0.849191	−	0.330824i
$210$		0			0
$211$			21.1498i			1.45602i	0.685569	+	0.728008i	$0.259554\pi$
$211$			21.1498i			1.45602i	−0.685569	+	0.728008i	$0.740446\pi$
$212$		0			0
$213$	−6.24053	+	0.299024i	−0.427594	+	0.0204888i
$214$		0			0
$215$	3.25186			0.221775
$216$		0			0
$217$			42.3592i			2.87553i
$218$		0			0
$219$	−2.36157	+	0.113158i	−0.159580	+	0.00764652i
$220$		0			0
$221$			12.2766i			0.825815i
$222$		0			0
$223$	−22.6382			−1.51597			−0.757983	−	0.652275i	$-0.773815\pi$
$223$	−22.6382			−1.51597			−0.757983	+	0.652275i	$0.773815\pi$
$224$		0			0
$225$	2.98626	−	0.286841i	0.199084	−	0.0191227i
$226$		0			0
$227$	23.5758			1.56478			0.782390	−	0.622789i	$-0.214001\pi$
$227$	23.5758			1.56478			0.782390	+	0.622789i	$0.214001\pi$
$228$		0			0
$229$	3.50893			0.231877			0.115938	−	0.993256i	$-0.463012\pi$
$229$	3.50893			0.231877			0.115938	+	0.993256i	$0.463012\pi$
$230$		0			0
$231$	7.93994	−	23.6732i	0.522410	−	1.55758i
$232$		0			0
$233$	−9.84991			−0.645289			−0.322644	−	0.946520i	$-0.604572\pi$
$233$	−9.84991			−0.645289			−0.322644	+	0.946520i	$0.604572\pi$
$234$		0			0
$235$	9.25186			0.603525
$236$		0			0
$237$	−1.85303	+	0.0887907i	−0.120367	+	0.00576758i
$238$		0			0
$239$	−24.1540			−1.56239			−0.781197	−	0.624285i	$-0.785390\pi$
$239$	−24.1540			−1.56239			−0.781197	+	0.624285i	$0.785390\pi$
$240$		0			0
$241$			4.69316i			0.302313i	0.988510	+	0.151157i	$0.0482998\pi$
$241$			4.69316i			0.302313i	−0.988510	+	0.151157i	$0.951700\pi$
$242$		0			0
$243$	−3.69633	−	15.1439i	−0.237119	−	0.971481i
$244$		0			0
$245$			11.8928i			0.759801i
$246$		0			0
$247$	−17.9255			−1.14057
$248$		0			0
$249$	−1.20938	−	25.2394i	−0.0766415	−	1.59948i
$250$		0			0
$251$			10.5424i			0.665427i	0.943028	+	0.332714i	$0.107964\pi$
$251$			10.5424i			0.665427i	−0.943028	+	0.332714i	$0.892036\pi$
$252$		0			0
$253$	−5.15603	−	2.00866i	−0.324157	−	0.126284i
$254$		0			0
$255$	−4.70691	+	0.225539i	−0.294758	+	0.0141238i
$256$		0			0
$257$			11.0334i			0.688246i	0.938925	+	0.344123i	$0.111824\pi$
$257$			11.0334i			0.688246i	−0.938925	+	0.344123i	$0.888176\pi$
$258$		0			0
$259$			24.3717i			1.51439i
$260$		0			0
$261$	15.1883	−	1.45889i	0.940131	−	0.0903030i
$262$		0			0
$263$	30.8378			1.90154			0.950770	−	0.309897i	$-0.100295\pi$
$263$	30.8378			1.90154			0.950770	+	0.309897i	$0.100295\pi$
$264$		0			0
$265$	3.05225			0.187498
$266$		0			0
$267$	6.50144	−	0.311526i	0.397882	−	0.0190651i
$268$		0			0
$269$		−	9.58345i		−	0.584313i	−0.956370	−	0.292157i	$-0.905627\pi$
$269$		−	9.58345i		−	0.584313i	0.956370	−	0.292157i	$-0.0943729\pi$
$270$		0			0
$271$			21.3400i			1.29631i	0.761507	+	0.648157i	$0.224460\pi$
$271$			21.3400i			1.29631i	−0.761507	+	0.648157i	$0.775540\pi$
$272$		0			0
$273$	−1.62593	−	33.9325i	−0.0984057	−	2.05369i
$274$		0			0
$275$	1.20394	−	3.09039i	0.0726004	−	0.186358i
$276$		0			0
$277$		−	7.43264i		−	0.446584i	−0.974752	−	0.223292i	$-0.928320\pi$
$277$		−	7.43264i		−	0.446584i	0.974752	−	0.223292i	$-0.0716804\pi$
$278$		0			0
$279$	29.1023	−	2.79538i	1.74231	−	0.167355i
$280$		0			0
$281$	−20.0300			−1.19489			−0.597444	−	0.801911i	$-0.703817\pi$
$281$	−20.0300			−1.19489			−0.597444	+	0.801911i	$0.703817\pi$
$282$		0			0
$283$		−	5.98189i		−	0.355587i	−0.984068	−	0.177793i	$-0.943104\pi$
$283$		−	5.98189i		−	0.355587i	0.984068	−	0.177793i	$-0.0568958\pi$
$284$		0			0
$285$	−0.329316	−	6.87271i	−0.0195070	−	0.407104i
$286$		0			0
$287$		−	12.6932i		−	0.749254i
$288$		0			0
$289$	−9.59805			−0.564591
$290$		0			0
$291$	1.06091	+	22.1408i	0.0621917	+	1.29792i
$292$		0			0
$293$	22.9998			1.34366			0.671830	−	0.740705i	$-0.265509\pi$
$293$	22.9998			1.34366			0.671830	+	0.740705i	$0.265509\pi$
$294$		0			0
$295$	5.43264			0.316301
$296$		0			0
$297$	−16.7883	−	3.89277i	−0.974155	−	0.225882i
$298$		0			0
$299$	−7.52848			−0.435383
$300$		0			0
$301$	14.1345			0.814697
$302$		0			0
$303$	−0.558695	−	11.6597i	−0.0320962	−	0.669835i
$304$		0			0
$305$	−11.5646			−0.662189
$306$		0			0
$307$			27.2645i			1.55607i	0.628222	+	0.778034i	$0.283783\pi$
$307$			27.2645i			1.55607i	−0.628222	+	0.778034i	$0.716217\pi$
$308$		0			0
$309$	−0.435373	−	9.08606i	−0.0247675	−	0.516888i
$310$		0			0
$311$			0.0613019i			0.00347611i	0.999998	+	0.00173806i	$0.000553241\pi$
$311$			0.0613019i			0.00347611i	−0.999998	+	0.00173806i	$0.999447\pi$
$312$		0			0
$313$	−22.2766			−1.25915			−0.629574	−	0.776940i	$-0.716771\pi$
$313$	−22.2766			−1.25915			−0.629574	+	0.776940i	$0.716771\pi$
$314$		0			0
$315$	12.9800	−	1.24678i	0.731340	−	0.0702479i
$316$		0			0
$317$			2.48169i			0.139385i	0.997569	+	0.0696927i	$0.0222019\pi$
$317$			2.48169i			0.139385i	−0.997569	+	0.0696927i	$0.977798\pi$
$318$		0			0
$319$	6.12332	−	15.7179i	0.342840	−	0.880035i
$320$		0			0
$321$	0.518704	+	10.8251i	0.0289512	+	0.604201i
$322$		0			0
$323$			10.8078i			0.601363i
$324$		0			0
$325$		−	4.51238i		−	0.250302i
$326$		0			0
$327$	30.9973	−	1.48528i	1.71415	−	0.0821363i
$328$		0			0
$329$	40.2140			2.21707
$330$		0			0
$331$	11.2519			0.618458			0.309229	−	0.950988i	$-0.399929\pi$
$331$	11.2519			0.618458			0.309229	+	0.950988i	$0.399929\pi$
$332$		0			0
$333$	16.7442	−	1.60835i	0.917579	−	0.0881368i
$334$		0			0
$335$			12.3529i			0.674911i
$336$		0			0
$337$			0.968917i			0.0527803i	0.999652	+	0.0263901i	$0.00840121\pi$
$337$			0.968917i			0.0527803i	−0.999652	+	0.0263901i	$0.991599\pi$
$338$		0			0
$339$	−14.7270	+	0.705666i	−0.799860	+	0.0383265i
$340$		0			0
$341$	11.7329	−	30.1171i	0.635373	−	1.63093i
$342$		0			0
$343$			21.2668i			1.14830i
$344$		0			0
$345$	−0.138309	−	2.88645i	−0.00744629	−	0.155401i
$346$		0			0
$347$	−1.59388			−0.0855641			−0.0427821	−	0.999084i	$-0.513622\pi$
$347$	−1.59388			−0.0855641			−0.0427821	+	0.999084i	$0.513622\pi$
$348$		0			0
$349$		−	18.9416i		−	1.01392i	−0.861970	−	0.506960i	$-0.830769\pi$
$349$		−	18.9416i		−	1.01392i	0.861970	−	0.506960i	$-0.169231\pi$
$350$		0			0
$351$	−23.2055	+	3.35635i	−1.23862	+	0.179149i
$352$		0			0
$353$		−	0.806322i		−	0.0429162i	−0.999770	−	0.0214581i	$-0.993169\pi$
$353$		−	0.806322i		−	0.0429162i	0.999770	−	0.0214581i	$-0.00683085\pi$
$354$		0			0
$355$	−3.60710			−0.191445
$356$		0			0
$357$	−20.4590	+	0.980323i	−1.08280	+	0.0518842i
$358$		0			0
$359$	6.83531			0.360754			0.180377	−	0.983598i	$-0.442268\pi$
$359$	6.83531			0.360754			0.180377	+	0.983598i	$0.442268\pi$
$360$		0			0
$361$	3.21916			0.169429
$362$		0			0
$363$	−12.2024	+	14.6322i	−0.640459	+	0.767992i
$364$		0			0
$365$	−1.36502			−0.0714482
$366$		0			0
$367$	−3.28873			−0.171670			−0.0858351	−	0.996309i	$-0.527356\pi$
$367$	−3.28873			−0.171670			−0.0858351	+	0.996309i	$0.527356\pi$
$368$		0			0
$369$	−8.72065	+	0.837650i	−0.453979	+	0.0436063i
$370$		0			0
$371$	13.2668			0.688780
$372$		0			0
$373$			11.3650i			0.588458i	0.955735	+	0.294229i	$0.0950629\pi$
$373$			11.3650i			0.588458i	−0.955735	+	0.294229i	$0.904937\pi$
$374$		0			0
$375$	1.73007	−	0.0828988i	0.0893402	−	0.00428087i
$376$		0			0
$377$		−	22.9502i		−	1.18200i
$378$		0			0
$379$	−26.1848			−1.34502			−0.672511	−	0.740087i	$-0.734784\pi$
$379$	−26.1848			−1.34502			−0.672511	+	0.740087i	$0.734784\pi$
$380$		0			0
$381$	−15.4845	+	0.741964i	−0.793295	+	0.0380120i
$382$		0			0
$383$		−	18.7796i		−	0.959590i	−0.877381	−	0.479795i	$-0.840711\pi$
$383$		−	18.7796i		−	0.959590i	0.877381	−	0.479795i	$-0.159289\pi$
$384$		0			0
$385$	5.23303	−	13.4326i	0.266700	−	0.684591i
$386$		0			0
$387$	−0.932765	−	9.71088i	−0.0474151	−	0.493632i
$388$		0			0
$389$		−	24.4661i		−	1.24048i	−0.784413	−	0.620239i	$-0.787035\pi$
$389$		−	24.4661i		−	1.24048i	0.784413	−	0.620239i	$-0.212965\pi$
$390$		0			0
$391$			4.53915i			0.229555i
$392$		0			0
$393$	−1.02280	−	21.3454i	−0.0515934	−	1.07673i
$394$		0			0
$395$	−1.07107			−0.0538916
$396$		0			0
$397$	−34.0708			−1.70997			−0.854983	−	0.518656i	$-0.826433\pi$
$397$	−34.0708			−1.70997			−0.854983	+	0.518656i	$0.826433\pi$
$398$		0			0
$399$	−1.43140	−	29.8728i	−0.0716597	−	1.49551i
$400$		0			0
$401$		−	2.52416i		−	0.126051i	−0.998012	−	0.0630254i	$-0.979925\pi$
$401$		−	2.52416i		−	0.126051i	0.998012	−	0.0630254i	$-0.0200749\pi$
$402$		0			0
$403$		−	43.9750i		−	2.19055i
$404$		0			0
$405$	−1.71316	−	8.83544i	−0.0851276	−	0.439037i
$406$		0			0
$407$	6.75063	−	17.3281i	0.334616	−	0.858924i
$408$		0			0
$409$		−	1.07974i		−	0.0533895i	−0.999644	−	0.0266948i	$-0.991502\pi$
$409$		−	1.07974i		−	0.0533895i	0.999644	−	0.0266948i	$-0.00849822\pi$
$410$		0			0
$411$	36.2303	−	1.73603i	1.78711	−	0.0856321i
$412$		0			0
$413$	23.6134			1.16194
$414$		0			0
$415$		−	14.5887i		−	0.716130i
$416$		0			0
$417$	−26.8953	+	1.28873i	−1.31707	+	0.0631093i
$418$		0			0
$419$			27.6897i			1.35273i	0.736566	+	0.676366i	$0.236446\pi$
$419$			27.6897i			1.35273i	−0.736566	+	0.676366i	$0.763554\pi$
$420$		0			0
$421$	31.0847			1.51498			0.757488	−	0.652849i	$-0.226426\pi$
$421$	31.0847			1.51498			0.757488	+	0.652849i	$0.226426\pi$
$422$		0			0
$423$	−2.65381	−	27.6284i	−0.129033	−	1.34334i
$424$		0			0
$425$	−2.72065			−0.131971
$426$		0			0
$427$	−50.2666			−2.43257
$428$		0			0
$429$	−8.24280	+	24.5762i	−0.397966	+	1.18655i
$430$		0			0
$431$	2.86529			0.138016			0.0690080	−	0.997616i	$-0.478017\pi$
$431$	2.86529			0.138016			0.0690080	+	0.997616i	$0.478017\pi$
$432$		0			0
$433$	−14.8158			−0.712000			−0.356000	−	0.934486i	$-0.615860\pi$
$433$	−14.8158			−0.712000			−0.356000	+	0.934486i	$0.615860\pi$
$434$		0			0
$435$	8.79922	−	0.421628i	0.421890	−	0.0202155i
$436$		0			0
$437$	−6.62776			−0.317049
$438$		0			0
$439$		−	11.3108i		−	0.539836i	−0.962883	−	0.269918i	$-0.913003\pi$
$439$		−	11.3108i		−	0.539836i	0.962883	−	0.269918i	$-0.0869966\pi$
$440$		0			0
$441$	35.5149	−	3.41133i	1.69118	−	0.162444i
$442$		0			0
$443$		−	1.60476i		−	0.0762447i	−0.999273	−	0.0381223i	$-0.987862\pi$
$443$		−	1.60476i		−	0.0762447i	0.999273	−	0.0381223i	$-0.0121377\pi$
$444$		0			0
$445$	3.75791			0.178142
$446$		0			0
$447$	1.93056	+	40.2900i	0.0913123	+	1.90565i
$448$		0			0
$449$			4.00000i			0.188772i	0.995536	+	0.0943858i	$0.0300887\pi$
$449$			4.00000i			0.188772i	−0.995536	+	0.0943858i	$0.969911\pi$
$450$		0			0
$451$	−3.51583	+	9.02476i	−0.165554	+	0.424959i
$452$		0			0
$453$	−18.7608	+	0.898952i	−0.881459	+	0.0422365i
$454$		0			0
$455$		−	19.6134i		−	0.919492i
$456$		0			0
$457$			32.8716i			1.53767i	0.639448	+	0.768835i	$0.279163\pi$
$457$			32.8716i			1.53767i	−0.639448	+	0.768835i	$0.720837\pi$
$458$		0			0
$459$	2.02365	+	13.9913i	0.0944559	+	0.653060i
$460$		0			0
$461$	−15.9387			−0.742339			−0.371170	−	0.928565i	$-0.621043\pi$
$461$	−15.9387			−0.742339			−0.371170	+	0.928565i	$0.621043\pi$
$462$		0			0
$463$	−10.4842			−0.487241			−0.243620	−	0.969871i	$-0.578335\pi$
$463$	−10.4842			−0.487241			−0.243620	+	0.969871i	$0.578335\pi$
$464$		0			0
$465$	16.8602	−	0.807883i	0.781873	−	0.0374647i
$466$		0			0
$467$		−	2.64436i		−	0.122367i	−0.998127	−	0.0611833i	$-0.980513\pi$
$467$		−	2.64436i		−	0.122367i	0.998127	−	0.0611833i	$-0.0194874\pi$
$468$		0			0
$469$			53.6929i			2.47931i
$470$		0			0
$471$	−0.919021	−	19.1796i	−0.0423463	−	0.883750i
$472$		0			0
$473$	−10.0495	−	3.91505i	−0.462077	−	0.180014i
$474$		0			0
$475$		−	3.97251i		−	0.182271i
$476$		0			0
$477$	−0.875509	−	9.11479i	−0.0400868	−	0.417338i
$478$		0			0
$479$	−28.6082			−1.30714			−0.653571	−	0.756865i	$-0.726730\pi$
$479$	−28.6082			−1.30714			−0.653571	+	0.756865i	$0.726730\pi$
$480$		0			0
$481$		−	25.3014i		−	1.15364i
$482$		0			0
$483$	−0.601170	−	12.5462i	−0.0273542	−	0.570872i
$484$		0			0
$485$			12.7977i			0.581112i
$486$		0			0
$487$	25.5584			1.15816			0.579082	−	0.815269i	$-0.303411\pi$
$487$	25.5584			1.15816			0.579082	+	0.815269i	$0.303411\pi$
$488$		0			0
$489$	0.345340	+	7.20710i	0.0156168	+	0.325917i
$490$		0			0
$491$	−30.0732			−1.35718			−0.678592	−	0.734516i	$-0.737409\pi$
$491$	−30.0732			−1.35718			−0.678592	+	0.734516i	$0.737409\pi$
$492$		0			0
$493$	−13.8374			−0.623205
$494$		0			0
$495$	−9.57404	−	2.70883i	−0.430321	−	0.121753i
$496$		0			0
$497$	−15.6786			−0.703280
$498$		0			0
$499$	−13.6503			−0.611071			−0.305536	−	0.952181i	$-0.598835\pi$
$499$	−13.6503			−0.611071			−0.305536	+	0.952181i	$0.598835\pi$
$500$		0			0
$501$	0.728538	+	15.2043i	0.0325487	+	0.679278i
$502$		0			0
$503$	−22.0495			−0.983139			−0.491570	−	0.870838i	$-0.663577\pi$
$503$	−22.0495			−0.983139			−0.491570	+	0.870838i	$0.663577\pi$
$504$		0			0
$505$		−	6.73948i		−	0.299903i
$506$		0			0
$507$	0.610265	+	12.7360i	0.0271028	+	0.565626i
$508$		0			0
$509$		−	17.7660i		−	0.787464i	−0.919225	−	0.393732i	$-0.871184\pi$
$509$		−	17.7660i		−	0.787464i	0.919225	−	0.393732i	$-0.128816\pi$
$510$		0			0
$511$	−5.93316			−0.262467
$512$		0			0
$513$	−20.4292	+	2.95479i	−0.901971	+	0.130457i
$514$		0			0
$515$		−	5.25186i		−	0.231425i
$516$		0			0
$517$	−28.5919	−	11.1387i	−1.25747	−	0.489879i
$518$		0			0
$519$	−0.220775	−	4.60749i	−0.00969096	−	0.202247i
$520$		0			0
$521$			21.0074i			0.920352i	0.887828	+	0.460176i	$0.152214\pi$
$521$			21.0074i			0.920352i	−0.887828	+	0.460176i	$0.847786\pi$
$522$		0			0
$523$			41.7779i			1.82682i	0.407042	+	0.913409i	$0.366560\pi$
$523$			41.7779i			1.82682i	−0.407042	+	0.913409i	$0.633440\pi$
$524$		0			0
$525$	7.51987	−	0.360326i	0.328194	−	0.0157259i
$526$		0			0
$527$	−26.5139			−1.15496
$528$		0			0
$529$	20.2164			0.878975
$530$		0			0
$531$	−1.55830	−	16.2233i	−0.0676246	−	0.704030i
$532$		0			0
$533$			13.1773i			0.570774i
$534$		0			0
$535$			6.25707i			0.270517i
$536$		0			0
$537$	19.4814	−	0.933483i	0.840686	−	0.0402828i
$538$		0			0
$539$	14.3182	−	36.7533i	0.616729	−	1.58308i
$540$		0			0
$541$			6.18327i			0.265840i	0.991127	+	0.132920i	$0.0424353\pi$
$541$			6.18327i			0.265840i	−0.991127	+	0.132920i	$0.957565\pi$
$542$		0			0
$543$	0.0589636	+	1.23055i	0.00253037	+	0.0528078i
$544$		0			0
$545$	17.9168			0.767472
$546$		0			0
$547$		−	21.2999i		−	0.910720i	−0.890307	−	0.455360i	$-0.849511\pi$
$547$		−	21.2999i		−	0.910720i	0.890307	−	0.455360i	$-0.150489\pi$
$548$		0			0
$549$	3.31721	+	34.5349i	0.141575	+	1.47392i
$550$		0			0
$551$		−	20.2044i		−	0.860738i
$552$		0			0
$553$	−4.65551			−0.197972
$554$		0			0
$555$	9.70066	−	0.464822i	0.411770	−	0.0197306i
$556$		0			0
$557$	5.87740			0.249033			0.124517	−	0.992218i	$-0.460262\pi$
$557$	5.87740			0.249033			0.124517	+	0.992218i	$0.460262\pi$
$558$		0			0
$559$	−14.6736			−0.620628
$560$		0			0
$561$	14.8177	+	4.96984i	0.625605	+	0.209827i
$562$		0			0
$563$	−21.9442			−0.924839			−0.462420	−	0.886661i	$-0.653019\pi$
$563$	−21.9442			−0.924839			−0.462420	+	0.886661i	$0.653019\pi$
$564$		0			0
$565$	−8.51238			−0.358118
$566$		0			0
$567$	−7.44639	−	38.4040i	−0.312719	−	1.61282i
$568$		0			0
$569$	−15.7284			−0.659367			−0.329683	−	0.944092i	$-0.606942\pi$
$569$	−15.7284			−0.659367			−0.329683	+	0.944092i	$0.606942\pi$
$570$		0			0
$571$		−	39.4633i		−	1.65149i	−0.564044	−	0.825745i	$-0.690755\pi$
$571$		−	39.4633i		−	1.65149i	0.564044	−	0.825745i	$-0.309245\pi$
$572$		0			0
$573$	−33.8837	+	1.62359i	−1.41551	+	0.0678265i
$574$		0			0
$575$		−	1.66840i		−	0.0695773i
$576$		0			0
$577$	−12.1721			−0.506732			−0.253366	−	0.967370i	$-0.581538\pi$
$577$	−12.1721			−0.506732			−0.253366	+	0.967370i	$0.581538\pi$
$578$		0			0
$579$	−15.2747	+	0.731912i	−0.634796	+	0.0304172i
$580$		0			0
$581$		−	63.4108i		−	2.63072i
$582$		0			0
$583$	−9.43264	−	3.67473i	−0.390660	−	0.152192i
$584$		0			0
$585$	−13.4751	+	1.29433i	−0.557128	+	0.0535141i
$586$		0			0
$587$			3.76175i			0.155264i	0.996982	+	0.0776321i	$0.0247359\pi$
$587$			3.76175i			0.155264i	−0.996982	+	0.0776321i	$0.975264\pi$
$588$		0			0
$589$		−	38.7138i		−	1.59517i
$590$		0			0
$591$	1.07356	+	22.4048i	0.0441603	+	0.921609i
$592$		0			0
$593$	19.5389			0.802367			0.401183	−	0.915998i	$-0.368599\pi$
$593$	19.5389			0.802367			0.401183	+	0.915998i	$0.368599\pi$
$594$		0			0
$595$	−11.8255			−0.484800
$596$		0			0
$597$	0.0973938	+	2.03257i	0.00398606	+	0.0831876i
$598$		0			0
$599$			33.4776i			1.36786i	0.729549	+	0.683929i	$0.239730\pi$
$599$			33.4776i			1.36786i	−0.729549	+	0.683929i	$0.760270\pi$
$600$		0			0
$601$			22.9203i			0.934937i	0.884010	+	0.467469i	$0.154834\pi$
$601$			22.9203i			0.934937i	−0.884010	+	0.467469i	$0.845166\pi$
$602$		0			0
$603$	36.8889	−	3.54332i	1.50223	−	0.144295i
$604$		0			0
$605$	−7.44131	+	8.10105i	−0.302532	+	0.329354i
$606$		0			0
$607$		−	7.21187i		−	0.292721i	−0.989231	−	0.146360i	$-0.953244\pi$
$607$		−	7.21187i		−	0.292721i	0.989231	−	0.146360i	$-0.0467559\pi$
$608$		0			0
$609$	38.2465	−	1.83264i	1.54983	−	0.0742624i
$610$		0			0
$611$	−41.7479			−1.68894
$612$		0			0
$613$		−	10.9911i		−	0.443926i	−0.975055	−	0.221963i	$-0.928754\pi$
$613$		−	10.9911i		−	0.443926i	0.975055	−	0.221963i	$-0.0712464\pi$
$614$		0			0
$615$	−5.05225	+	0.242086i	−0.203726	+	0.00976186i
$616$		0			0
$617$			9.52449i			0.383442i	0.981450	+	0.191721i	$0.0614068\pi$
$617$			9.52449i			0.383442i	−0.981450	+	0.191721i	$0.938593\pi$
$618$		0			0
$619$	−20.5329			−0.825287			−0.412644	−	0.910893i	$-0.635395\pi$
$619$	−20.5329			−0.825287			−0.412644	+	0.910893i	$0.635395\pi$
$620$		0			0
$621$	−8.58001	+	1.24098i	−0.344304	+	0.0497987i
$622$		0			0
$623$	16.3341			0.654411
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−7.25662	+	21.6358i	−0.289802	+	0.864052i
$628$		0			0
$629$	−15.2550			−0.608256
$630$		0			0
$631$	29.6731			1.18127			0.590634	−	0.806940i	$-0.298878\pi$
$631$	29.6731			1.18127			0.590634	+	0.806940i	$0.298878\pi$
$632$		0			0
$633$	36.5906	−	1.75330i	1.45435	−	0.0696873i
$634$		0			0
$635$	−8.95024			−0.355179
$636$		0			0
$637$		−	53.6647i		−	2.12627i
$638$		0			0
$639$	1.03466	+	10.7717i	0.0409307	+	0.426123i
$640$		0			0
$641$		−	19.0947i		−	0.754196i	−0.926173	−	0.377098i	$-0.876922\pi$
$641$		−	19.0947i		−	0.754196i	0.926173	−	0.377098i	$-0.123078\pi$
$642$		0			0
$643$	−22.0663			−0.870209			−0.435104	−	0.900380i	$-0.643289\pi$
$643$	−22.0663			−0.870209			−0.435104	+	0.900380i	$0.643289\pi$
$644$		0			0
$645$	−0.269575	−	5.62593i	−0.0106145	−	0.221521i
$646$		0			0
$647$		−	18.5832i		−	0.730581i	−0.930894	−	0.365291i	$-0.880970\pi$
$647$		−	18.5832i		−	0.730581i	0.930894	−	0.365291i	$-0.119030\pi$
$648$		0			0
$649$	−16.7890	−	6.54059i	−0.659026	−	0.256740i
$650$		0			0
$651$	73.2843	−	3.51153i	2.87224	−	0.137628i
$652$		0			0
$653$			31.2063i			1.22120i	0.791941	+	0.610598i	$0.209071\pi$
$653$			31.2063i			1.22120i	−0.791941	+	0.610598i	$0.790929\pi$
$654$		0			0
$655$		−	12.3379i		−	0.482082i
$656$		0			0
$657$	0.391542	+	4.07629i	0.0152755	+	0.159031i
$658$		0			0
$659$	26.6138			1.03672			0.518362	−	0.855161i	$-0.326542\pi$
$659$	26.6138			1.03672			0.518362	+	0.855161i	$0.326542\pi$
$660$		0			0
$661$	−4.07452			−0.158481			−0.0792403	−	0.996856i	$-0.525249\pi$
$661$	−4.07452			−0.158481			−0.0792403	+	0.996856i	$0.525249\pi$
$662$		0			0
$663$	21.2394	−	1.01772i	0.824868	−	0.0395248i
$664$		0			0
$665$		−	17.2668i		−	0.669580i
$666$		0			0
$667$		−	8.48561i		−	0.328564i
$668$		0			0
$669$	1.87668	+	39.1656i	0.0725566	+	1.51423i
$670$		0			0
$671$	35.7392	+	13.9231i	1.37970	+	0.537497i
$672$		0			0
$673$		−	0.828988i		−	0.0319551i	−0.999872	−	0.0159776i	$-0.994914\pi$
$673$		−	0.828988i		−	0.0319551i	0.999872	−	0.0159776i	$-0.00508603\pi$
$674$		0			0
$675$	−0.743810	−	5.14264i	−0.0286293	−	0.197940i
$676$		0			0
$677$	6.51415			0.250359			0.125179	−	0.992134i	$-0.460049\pi$
$677$	6.51415			0.250359			0.125179	+	0.992134i	$0.460049\pi$
$678$		0			0
$679$			55.6261i			2.13473i
$680$		0			0
$681$	−1.95440	−	40.7876i	−0.0748929	−	1.56299i
$682$		0			0
$683$		−	8.99407i		−	0.344148i	−0.985084	−	0.172074i	$-0.944953\pi$
$683$		−	8.99407i		−	0.344148i	0.985084	−	0.172074i	$-0.0550469\pi$
$684$		0			0
$685$	20.9416			0.800136
$686$		0			0
$687$	−0.290886	−	6.07068i	−0.0110980	−	0.231611i
$688$		0			0
$689$	−13.7729			−0.524706
$690$		0			0
$691$	39.7848			1.51348			0.756742	−	0.653714i	$-0.226790\pi$
$691$	39.7848			1.51348			0.756742	+	0.653714i	$0.226790\pi$
$692$		0			0
$693$	−41.6144	−	11.7741i	−1.58080	−	0.447263i
$694$		0			0
$695$	−15.5458			−0.589686
$696$		0			0
$697$	7.94502			0.300939
$698$		0			0
$699$	0.816545	+	17.0410i	0.0308846	+	0.644549i
$700$		0			0
$701$	−40.4528			−1.52788			−0.763941	−	0.645286i	$-0.776738\pi$
$701$	−40.4528			−1.52788			−0.763941	+	0.645286i	$0.776738\pi$
$702$		0			0
$703$		−	22.2743i		−	0.840090i
$704$		0			0
$705$	−0.766968	−	16.0063i	−0.0288857	−	0.602833i
$706$		0			0
$707$		−	29.2937i		−	1.10170i
$708$		0			0
$709$	−31.3126			−1.17597			−0.587984	−	0.808872i	$-0.700078\pi$
$709$	−31.3126			−1.17597			−0.587984	+	0.808872i	$0.700078\pi$
$710$		0			0
$711$	0.307228	+	3.19850i	0.0115219	+	0.119953i
$712$		0			0
$713$		−	16.2593i		−	0.608915i
$714$		0			0
$715$	−5.43264	+	13.9450i	−0.203169	+	0.521514i
$716$		0			0
$717$	2.00234	+	41.7880i	0.0747787	+	1.56060i
$718$		0			0
$719$		−	49.2324i		−	1.83606i	−0.396513	−	0.918029i	$-0.629780\pi$
$719$		−	49.2324i		−	1.83606i	0.396513	−	0.918029i	$-0.370220\pi$
$720$		0			0
$721$		−	22.8276i		−	0.850145i
$722$		0			0
$723$	8.11948	−	0.389058i	0.301967	−	0.0144692i
$724$		0			0
$725$	5.08606			0.188892
$726$		0			0
$727$	−22.6682			−0.840716			−0.420358	−	0.907358i	$-0.638095\pi$
$727$	−22.6682			−0.840716			−0.420358	+	0.907358i	$0.638095\pi$
$728$		0			0
$729$	−25.8935	+	7.65030i	−0.959018	+	0.283344i
$730$		0			0
$731$			8.84718i			0.327225i
$732$		0			0
$733$		−	0.210758i		−	0.00778452i	−0.999992	−	0.00389226i	$-0.998761\pi$
$733$		−	0.210758i		−	0.00778452i	0.999992	−	0.00389226i	$-0.00123895\pi$
$734$		0			0
$735$	20.5753	−	0.985897i	0.758931	−	0.0363653i
$736$		0			0
$737$	14.8722	−	38.1753i	0.547824	−	1.40621i
$738$		0			0
$739$			14.8053i			0.544623i	0.962209	+	0.272312i	$0.0877881\pi$
$739$			14.8053i			0.544623i	−0.962209	+	0.272312i	$0.912212\pi$
$740$		0			0
$741$	1.48600	+	31.0123i	0.0545896	+	1.13926i
$742$		0			0
$743$	19.2717			0.707009			0.353504	−	0.935433i	$-0.384990\pi$
$743$	19.2717			0.707009			0.353504	+	0.935433i	$0.384990\pi$
$744$		0			0
$745$			23.2882i			0.853212i
$746$		0			0
$747$	−43.5655	+	4.18462i	−1.59398	+	0.153107i
$748$		0			0
$749$			27.1969i			0.993752i
$750$		0			0
$751$	24.6107			0.898057			0.449029	−	0.893517i	$-0.351770\pi$
$751$	24.6107			0.898057			0.449029	+	0.893517i	$0.351770\pi$
$752$		0			0
$753$	18.2390	−	0.873948i	0.664665	−	0.0318484i
$754$		0			0
$755$	−10.8440			−0.394653
$756$		0			0
$757$	21.6166			0.785670			0.392835	−	0.919609i	$-0.371494\pi$
$757$	21.6166			0.785670			0.392835	+	0.919609i	$0.371494\pi$
$758$		0			0
$759$	−3.04769	+	9.08678i	−0.110624	+	0.329829i
$760$		0			0
$761$	34.6751			1.25697			0.628485	−	0.777822i	$-0.283675\pi$
$761$	34.6751			1.25697			0.628485	+	0.777822i	$0.283675\pi$
$762$		0			0
$763$	77.8769			2.81933
$764$		0			0
$765$	0.780394	+	8.12456i	0.0282152	+	0.293744i
$766$		0			0
$767$	−24.5141			−0.885155
$768$		0			0
$769$		−	16.8772i		−	0.608606i	−0.952575	−	0.304303i	$-0.901577\pi$
$769$		−	16.8772i		−	0.608606i	0.952575	−	0.304303i	$-0.0984235\pi$
$770$		0			0
$771$	19.0885	−	0.914657i	0.687457	−	0.0329406i
$772$		0			0
$773$			8.07701i			0.290510i	0.989394	+	0.145255i	$0.0464002\pi$
$773$			8.07701i			0.290510i	−0.989394	+	0.145255i	$0.953600\pi$
$774$		0			0
$775$	9.74541			0.350066
$776$		0			0
$777$	42.1647	−	2.02039i	1.51265	−	0.0724810i
$778$		0			0
$779$			11.6008i			0.415641i
$780$		0			0
$781$	11.1474	+	4.34274i	0.398884	+	0.155396i
$782$		0			0
$783$	−3.78306	−	26.1558i	−0.135196	−	0.934731i
$784$		0			0
$785$		−	11.0861i		−	0.395678i
$786$		0			0
$787$			15.2992i			0.545356i	0.962105	+	0.272678i	$0.0879094\pi$
$787$			15.2992i			0.545356i	−0.962105	+	0.272678i	$0.912091\pi$
$788$		0			0
$789$	−2.55642	−	53.3514i	−0.0910108	−	1.89936i
$790$		0			0
$791$	−36.9998			−1.31556
$792$		0			0
$793$	52.1840			1.85311
$794$		0			0
$795$	−0.253028	−	5.28059i	−0.00897397	−	0.187283i
$796$		0			0
$797$			4.37200i			0.154864i	0.996998	+	0.0774321i	$0.0246721\pi$
$797$			4.37200i			0.154864i	−0.996998	+	0.0774321i	$0.975328\pi$
$798$		0			0
$799$			25.1711i			0.890489i
$800$		0			0
$801$	−1.07792	−	11.2221i	−0.0380865	−	0.396513i
$802$		0			0
$803$	4.21844	+	1.64340i	0.148865	+	0.0579944i
$804$		0			0
$805$		−	7.25186i		−	0.255594i
$806$		0			0
$807$	−16.5800	+	0.794457i	−0.583644	+	0.0279662i
$808$		0			0
$809$	16.6459			0.585237			0.292619	−	0.956229i	$-0.405473\pi$
$809$	16.6459			0.585237			0.292619	+	0.956229i	$0.405473\pi$
$810$		0			0
$811$		−	35.1018i		−	1.23259i	−0.787515	−	0.616295i	$-0.788633\pi$
$811$		−	35.1018i		−	1.23259i	0.787515	−	0.616295i	$-0.211367\pi$
$812$		0			0
$813$	36.9196	−	1.76906i	1.29483	−	0.0620437i
$814$		0			0
$815$			4.16580i			0.145922i
$816$		0			0
$817$	−12.9180			−0.451945
$818$		0			0
$819$	−58.5707	+	5.62593i	−2.04663	+	0.196586i
$820$		0			0
$821$	−4.78900			−0.167137			−0.0835686	−	0.996502i	$-0.526632\pi$
$821$	−4.78900			−0.167137			−0.0835686	+	0.996502i	$0.526632\pi$
$822$		0			0
$823$	−19.7053			−0.686883			−0.343441	−	0.939174i	$-0.611593\pi$
$823$	−19.7053			−0.686883			−0.343441	+	0.939174i	$0.611593\pi$
$824$		0			0
$825$	−5.44639	−	1.82671i	−0.189619	−	0.0635978i
$826$		0			0
$827$	49.7179			1.72886			0.864431	−	0.502752i	$-0.167679\pi$
$827$	49.7179			1.72886			0.864431	+	0.502752i	$0.167679\pi$
$828$		0			0
$829$	43.5766			1.51348			0.756738	−	0.653718i	$-0.226792\pi$
$829$	43.5766			1.51348			0.756738	+	0.653718i	$0.226792\pi$
$830$		0			0
$831$	−12.8590	+	0.616157i	−0.446072	+	0.0213743i
$832$		0			0
$833$	−32.3561			−1.12107
$834$		0			0
$835$			8.78828i			0.304131i
$836$		0			0
$837$	−7.24874	−	50.1171i	−0.250553	−	1.73230i
$838$		0			0
$839$		−	5.89727i		−	0.203596i	−0.994805	−	0.101798i	$-0.967540\pi$
$839$		−	5.89727i		−	0.203596i	0.994805	−	0.101798i	$-0.0324596\pi$
$840$		0			0
$841$	−3.13198			−0.107999
$842$		0			0
$843$	1.66046	+	34.6532i	0.0571893	+	1.19352i
$844$		0			0
$845$			7.36157i			0.253246i
$846$		0			0
$847$	−32.3442	+	35.2119i	−1.11136	+	1.20989i
$848$		0			0
$849$	−10.3491	+	0.495892i	−0.355179	+	0.0170190i
$850$		0			0
$851$		−	9.35492i		−	0.320682i
$852$		0			0
$853$			25.2919i			0.865979i	0.901399	+	0.432990i	$0.142541\pi$
$853$			25.2919i			0.865979i	−0.901399	+	0.432990i	$0.857459\pi$
$854$		0			0
$855$	−11.8629	+	1.13948i	−0.405704	+	0.0389693i
$856$		0			0
$857$	25.0718			0.856436			0.428218	−	0.903675i	$-0.359142\pi$
$857$	25.0718			0.856436			0.428218	+	0.903675i	$0.359142\pi$
$858$		0			0
$859$	−12.2263			−0.417157			−0.208578	−	0.978006i	$-0.566884\pi$
$859$	−12.2263			−0.417157			−0.208578	+	0.978006i	$0.566884\pi$
$860$		0			0
$861$	−21.9600	+	1.05225i	−0.748395	+	0.0358605i
$862$		0			0
$863$		−	16.5713i		−	0.564095i	−0.959400	−	0.282048i	$-0.908986\pi$
$863$		−	16.5713i		−	0.564095i	0.959400	−	0.282048i	$-0.0910136\pi$
$864$		0			0
$865$		−	2.66319i		−	0.0905512i
$866$		0			0
$867$	0.795667	+	16.6053i	0.0270223	+	0.563944i
$868$		0			0
$869$	3.31004	+	1.28951i	0.112285	+	0.0437437i
$870$		0			0
$871$		−	55.7410i		−	1.88871i
$872$		0			0
$873$	38.2171	−	3.67089i	1.29345	−	0.124241i
$874$		0			0
$875$	4.34658			0.146941
$876$		0			0
$877$			41.5248i			1.40219i	0.713067	+	0.701096i	$0.247306\pi$
$877$			41.5248i			1.40219i	−0.713067	+	0.701096i	$0.752694\pi$
$878$		0			0
$879$	−1.90665	−	39.7911i	−0.0643098	−	1.34212i
$880$		0			0
$881$			18.8053i			0.633568i	0.948498	+	0.316784i	$0.102603\pi$
$881$			18.8053i			0.633568i	−0.948498	+	0.316784i	$0.897397\pi$
$882$		0			0
$883$	12.9252			0.434966			0.217483	−	0.976064i	$-0.430215\pi$
$883$	12.9252			0.434966			0.217483	+	0.976064i	$0.430215\pi$
$884$		0			0
$885$	−0.450359	−	9.39883i	−0.0151387	−	0.315938i
$886$		0			0
$887$	−43.1169			−1.44772			−0.723862	−	0.689945i	$-0.757635\pi$
$887$	−43.1169			−1.44772			−0.723862	+	0.689945i	$0.757635\pi$
$888$		0			0
$889$	−38.9029			−1.30476
$890$		0			0
$891$	−5.34303	+	29.3675i	−0.178998	+	0.983849i
$892$		0			0
$893$	−36.7531			−1.22990
$894$		0			0
$895$	11.2605			0.376398
$896$		0			0
$897$	0.624101	+	13.0248i	0.0208381	+	0.434884i
$898$		0			0
$899$	49.5658			1.65311
$900$		0			0
$901$			8.30411i			0.276650i
$902$		0			0
$903$	−1.17173	−	24.4536i	−0.0389927	−	0.813764i
$904$		0			0
$905$			0.711272i			0.0236435i
$906$		0			0
$907$	16.4347			0.545706			0.272853	−	0.962056i	$-0.412033\pi$
$907$	16.4347			0.545706			0.272853	+	0.962056i	$0.412033\pi$
$908$		0			0
$909$	−20.1258	+	1.93316i	−0.667531	+	0.0641188i
$910$		0			0
$911$		−	48.2030i		−	1.59704i	−0.601971	−	0.798518i	$-0.705618\pi$
$911$		−	48.2030i		−	1.59704i	0.601971	−	0.798518i	$-0.294382\pi$
$912$		0			0
$913$	−17.5639	+	45.0847i	−0.581281	+	1.49209i
$914$		0			0
$915$	0.958694	+	20.0076i	0.0316934	+	0.661430i
$916$		0			0
$917$		−	53.6278i		−	1.77095i
$918$		0			0
$919$		−	33.3400i		−	1.09979i	−0.835235	−	0.549893i	$-0.814669\pi$
$919$		−	33.3400i		−	1.09979i	0.835235	−	0.549893i	$-0.185331\pi$
$920$		0			0
$921$	47.1694	−	2.26019i	1.55428	−	0.0744759i
$922$		0			0
$923$	16.2766			0.535751
$924$		0			0
$925$	5.60710			0.184360
$926$		0			0
$927$	−15.6834	+	1.50645i	−0.515110	+	0.0494782i
$928$		0			0
$929$		−	26.4142i		−	0.866622i	−0.901244	−	0.433311i	$-0.857345\pi$
$929$		−	26.4142i		−	0.866622i	0.901244	−	0.433311i	$-0.142655\pi$
$930$		0			0
$931$		−	47.2442i		−	1.54837i
$932$		0			0
$933$	0.106056	−	0.00508185i	0.00347213	−	0.000166372i
$934$		0			0
$935$	8.40788	+	3.27551i	0.274967	+	0.107121i
$936$		0			0
$937$		−	36.8009i		−	1.20223i	−0.799162	−	0.601116i	$-0.794723\pi$
$937$		−	36.8009i		−	1.20223i	0.799162	−	0.601116i	$-0.205277\pi$
$938$		0			0
$939$	1.84670	+	38.5400i	0.0602649	+	1.25771i
$940$		0			0
$941$	−12.9934			−0.423574			−0.211787	−	0.977316i	$-0.567928\pi$
$941$	−12.9934			−0.423574			−0.211787	+	0.977316i	$0.567928\pi$
$942$		0			0
$943$			4.87218i			0.158660i
$944$		0			0
$945$	−3.23303	−	22.3529i	−0.105171	−	0.727140i
$946$		0			0
$947$		−	6.36307i		−	0.206772i	−0.994641	−	0.103386i	$-0.967032\pi$
$947$		−	6.36307i		−	0.206772i	0.994641	−	0.103386i	$-0.0329677\pi$
$948$		0			0
$949$	6.15947			0.199945
$950$		0			0
$951$	4.29348	−	0.205729i	0.139226	−	0.00667122i
$952$		0			0
$953$	−29.7400			−0.963372			−0.481686	−	0.876344i	$-0.659975\pi$
$953$	−29.7400			−0.963372			−0.481686	+	0.876344i	$0.659975\pi$
$954$		0			0
$955$	−19.5852			−0.633763
$956$		0			0
$957$	−27.7007	−	9.29075i	−0.895435	−	0.300327i
$958$		0			0
$959$	91.0243			2.93933
$960$		0			0
$961$	63.9730			2.06365
$962$		0			0
$963$	18.6852	−	1.79478i	0.602123	−	0.0578361i
$964$		0			0
$965$	−8.82899			−0.284215
$966$		0			0
$967$			36.5418i			1.17510i	0.809186	+	0.587552i	$0.199908\pi$
$967$			36.5418i			1.17510i	−0.809186	+	0.587552i	$0.800092\pi$
$968$		0			0
$969$	18.6982	−	0.895955i	0.600674	−	0.0287822i
$970$		0			0
$971$			22.7040i			0.728607i	0.931280	+	0.364304i	$0.118693\pi$
$971$			22.7040i			0.728607i	−0.931280	+	0.364304i	$0.881307\pi$
$972$		0			0
$973$	−67.5711			−2.16623
$974$		0			0
$975$	−7.80671	+	0.374071i	−0.250015	+	0.0119799i
$976$		0			0
$977$			2.83708i			0.0907662i	0.998970	+	0.0453831i	$0.0144509\pi$
$977$			2.83708i			0.0907662i	−0.998970	+	0.0453831i	$0.985549\pi$
$978$		0			0
$979$	−11.6134	−	4.52431i	−0.371167	−	0.144598i
$980$		0			0
$981$	−5.13927	−	53.5042i	−0.164084	−	1.70826i
$982$		0			0
$983$			35.3487i			1.12745i	0.825963	+	0.563724i	$0.190632\pi$
$983$			35.3487i			1.12745i	−0.825963	+	0.563724i	$0.809368\pi$
$984$		0			0
$985$			12.9502i			0.412629i
$986$		0			0
$987$	−3.33369	−	69.5728i	−0.106112	−	2.21453i
$988$		0			0
$989$	−5.42542			−0.172518
$990$		0			0
$991$	20.0173			0.635871			0.317936	−	0.948112i	$-0.397010\pi$
$991$	20.0173			0.635871			0.317936	+	0.948112i	$0.397010\pi$
$992$		0			0
$993$	−0.932765	−	19.4665i	−0.0296004	−	0.617749i
$994$		0			0
$995$			1.17485i			0.0372453i
$996$		0			0
$997$		−	6.80711i		−	0.215583i	−0.994174	−	0.107792i	$-0.965622\pi$
$997$		−	6.80711i		−	0.215583i	0.994174	−	0.107792i	$-0.0343779\pi$
$998$		0			0
$999$	−4.17062	−	28.8353i	−0.131953	−	0.912309i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.f.a.1121.5		8
3.2	odd	2		2640.2.f.b.1121.6		8
4.3	odd	2		330.2.d.b.131.4	yes	8
11.10	odd	2		2640.2.f.b.1121.5		8
12.11	even	2		330.2.d.a.131.3	✓	8
20.3	even	4		1650.2.f.d.1649.1		8
20.7	even	4		1650.2.f.f.1649.8		8
20.19	odd	2		1650.2.d.c.1451.5		8
33.32	even	2	inner	2640.2.f.a.1121.6		8
44.43	even	2		330.2.d.a.131.4	yes	8
60.23	odd	4		1650.2.f.e.1649.8		8
60.47	odd	4		1650.2.f.c.1649.1		8
60.59	even	2		1650.2.d.f.1451.6		8
132.131	odd	2		330.2.d.b.131.3	yes	8
220.43	odd	4		1650.2.f.c.1649.5		8
220.87	odd	4		1650.2.f.e.1649.4		8
220.219	even	2		1650.2.d.f.1451.5		8
660.263	even	4		1650.2.f.f.1649.4		8
660.527	even	4		1650.2.f.d.1649.5		8
660.659	odd	2		1650.2.d.c.1451.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
330.2.d.a.131.3	✓	8	12.11	even	2
330.2.d.a.131.4	yes	8	44.43	even	2
330.2.d.b.131.3	yes	8	132.131	odd	2
330.2.d.b.131.4	yes	8	4.3	odd	2
1650.2.d.c.1451.5		8	20.19	odd	2
1650.2.d.c.1451.6		8	660.659	odd	2
1650.2.d.f.1451.5		8	220.219	even	2
1650.2.d.f.1451.6		8	60.59	even	2
1650.2.f.c.1649.1		8	60.47	odd	4
1650.2.f.c.1649.5		8	220.43	odd	4
1650.2.f.d.1649.1		8	20.3	even	4
1650.2.f.d.1649.5		8	660.527	even	4
1650.2.f.e.1649.4		8	220.87	odd	4
1650.2.f.e.1649.8		8	60.23	odd	4
1650.2.f.f.1649.4		8	660.263	even	4
1650.2.f.f.1649.8		8	20.7	even	4
2640.2.f.a.1121.5		8	1.1	even	1	trivial
2640.2.f.a.1121.6		8	33.32	even	2	inner
2640.2.f.b.1121.5		8	11.10	odd	2
2640.2.f.b.1121.6		8	3.2	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 $\F_{q}$

Introduction

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Modular forms

Varieties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2640.2.f.a.1121.5

Level	$2640$
Weight	$2$
Character	2640.1121
Analytic conductor	$21.081$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$