LMFDB - Embedded newform 2640.2.f.b.1121.7

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			1121.7
Root			$2.08963i$ of defining polynomial
Character	$\chi$	$=$	2640.1121
Dual form			2640.2.f.b.1121.8

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72809 - 0.117016i) q^{3} -1.00000i q^{5} -0.723074i q^{7} +(2.97261 - 0.404430i) q^{9} +(2.32366 + 2.36656i) q^{11} +4.17926i q^{13} +(-0.117016 - 1.72809i) q^{15} -4.49908 q^{17} +7.94523i q^{19} +(-0.0846115 - 1.24954i) q^{21} +8.91237i q^{23} -1.00000 q^{25} +(5.08963 - 1.04674i) q^{27} -4.98812 q^{29} +5.85944 q^{31} +(4.29243 + 3.81772i) q^{33} -0.723074 q^{35} -5.54197 q^{37} +(0.489042 + 7.22215i) q^{39} +3.53193 q^{41} +10.4443i q^{43} +(-0.404430 - 2.97261i) q^{45} -4.44431i q^{47} +6.47716 q^{49} +(-7.77483 + 0.526466i) q^{51} -2.41329i q^{53} +(2.36656 - 2.32366i) q^{55} +(0.929721 + 13.7301i) q^{57} -9.71120i q^{59} -2.59256i q^{61} +(-0.292433 - 2.14942i) q^{63} +4.17926 q^{65} +9.24313 q^{67} +(1.04289 + 15.4014i) q^{69} -3.54197i q^{71} -4.56154i q^{73} +(-1.72809 + 0.117016i) q^{75} +(1.71120 - 1.68018i) q^{77} -11.1774i q^{79} +(8.67287 - 2.40443i) q^{81} -15.3804 q^{83} +4.49908i q^{85} +(-8.61994 + 0.583692i) q^{87} -2.10351i q^{89} +3.02192 q^{91} +(10.1257 - 0.685650i) q^{93} +7.94523 q^{95} +8.27274 q^{97} +(7.86446 + 6.09510i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 2 q^{3} + 2 q^{9} + 6 q^{11} - 4 q^{15} - 4 q^{17} + 8 q^{21} - 8 q^{25} + 22 q^{27} + 4 q^{29} + 4 q^{31} + 18 q^{33} - 12 q^{37} - 8 q^{39} + 16 q^{41} - 4 q^{49} - 28 q^{51} + 6 q^{55} - 14 q^{57}+ \cdots + 10 q^{99}+O(q^{100})$ 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 + 18 * 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 + 10 * 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
$3$	1.72809	−	0.117016i	0.997715	−	0.0675594i
$3$	1.72809	−	0.117016i	0.997715	−	0.0675594i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	0.723074i		−	0.273296i	−0.990620	−	0.136648i	$-0.956367\pi$
$7$		−	0.723074i		−	0.273296i	0.990620	−	0.136648i	$-0.0436330\pi$
$8$		0			0
$9$	2.97261	−	0.404430i	0.990871	−	0.134810i
$10$		0			0
$11$	2.32366	+	2.36656i	0.700611	+	0.713544i
$11$	2.32366	+	2.36656i	0.700611	+	0.713544i
$12$		0			0
$13$			4.17926i			1.15912i	0.814930	+	0.579559i	$0.196775\pi$
$13$			4.17926i			1.15912i	−0.814930	+	0.579559i	$0.803225\pi$
$14$		0			0
$15$	−0.117016	−	1.72809i	−0.0302135	−	0.446192i
$16$		0			0
$17$	−4.49908			−1.09119			−0.545594	−	0.838050i	$-0.683696\pi$
$17$	−4.49908			−1.09119			−0.545594	+	0.838050i	$0.683696\pi$
$18$		0			0
$19$			7.94523i			1.82276i	0.411565	+	0.911380i	$0.364982\pi$
$19$			7.94523i			1.82276i	−0.411565	+	0.911380i	$0.635018\pi$
$20$		0			0
$21$	−0.0846115	−	1.24954i	−0.0184637	−	0.272672i
$22$		0			0
$23$			8.91237i			1.85836i	0.369629	+	0.929179i	$0.379485\pi$
$23$			8.91237i			1.85836i	−0.369629	+	0.929179i	$0.620515\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.08963	−	1.04674i	0.979500	−	0.201445i
$28$		0			0
$29$	−4.98812			−0.926271			−0.463135	−	0.886287i	$-0.653276\pi$
$29$	−4.98812			−0.926271			−0.463135	+	0.886287i	$0.653276\pi$
$30$		0			0
$31$	5.85944			1.05239			0.526193	−	0.850365i	$-0.323619\pi$
$31$	5.85944			1.05239			0.526193	+	0.850365i	$0.323619\pi$
$32$		0			0
$33$	4.29243	+	3.81772i	0.747217	+	0.664580i
$34$		0			0
$35$	−0.723074			−0.122222
$36$		0			0
$37$	−5.54197			−0.911095			−0.455547	−	0.890212i	$-0.650556\pi$
$37$	−5.54197			−0.911095			−0.455547	+	0.890212i	$0.650556\pi$
$38$		0			0
$39$	0.489042	+	7.22215i	0.0783093	+	1.15647i
$40$		0			0
$41$	3.53193			0.551596			0.275798	−	0.961216i	$-0.411058\pi$
$41$	3.53193			0.551596			0.275798	+	0.961216i	$0.411058\pi$
$42$		0			0
$43$			10.4443i			1.59274i	0.604808	+	0.796371i	$0.293250\pi$
$43$			10.4443i			1.59274i	−0.604808	+	0.796371i	$0.706750\pi$
$44$		0			0
$45$	−0.404430	−	2.97261i	−0.0602889	−	0.443131i
$46$		0			0
$47$		−	4.44431i		−	0.648269i	−0.946011	−	0.324135i	$-0.894927\pi$
$47$		−	4.44431i		−	0.648269i	0.946011	−	0.324135i	$-0.105073\pi$
$48$		0			0
$49$	6.47716			0.925309
$50$		0			0
$51$	−7.77483	+	0.526466i	−1.08869	+	0.0737199i
$52$		0			0
$53$		−	2.41329i		−	0.331491i	−0.986169	−	0.165746i	$-0.946997\pi$
$53$		−	2.41329i		−	0.331491i	0.986169	−	0.165746i	$-0.0530031\pi$
$54$		0			0
$55$	2.36656	−	2.32366i	0.319106	−	0.313323i
$56$		0			0
$57$	0.929721	+	13.7301i	0.123145	+	1.81860i
$58$		0			0
$59$		−	9.71120i		−	1.26429i	−0.774850	−	0.632145i	$-0.782175\pi$
$59$		−	9.71120i		−	1.26429i	0.774850	−	0.632145i	$-0.217825\pi$
$60$		0			0
$61$		−	2.59256i		−	0.331943i	−0.986131	−	0.165971i	$-0.946924\pi$
$61$		−	2.59256i		−	0.331943i	0.986131	−	0.165971i	$-0.0530759\pi$
$62$		0			0
$63$	−0.292433	−	2.14942i	−0.0368431	−	0.270802i
$64$		0			0
$65$	4.17926			0.518374
$66$		0			0
$67$	9.24313			1.12923			0.564614	−	0.825355i	$-0.309025\pi$
$67$	9.24313			1.12923			0.564614	+	0.825355i	$0.309025\pi$
$68$		0			0
$69$	1.04289	+	15.4014i	0.125550	+	1.85411i
$70$		0			0
$71$		−	3.54197i		−	0.420355i	−0.977663	−	0.210177i	$-0.932596\pi$
$71$		−	3.54197i		−	0.420355i	0.977663	−	0.210177i	$-0.0674042\pi$
$72$		0			0
$73$		−	4.56154i		−	0.533888i	−0.963712	−	0.266944i	$-0.913986\pi$
$73$		−	4.56154i		−	0.533888i	0.963712	−	0.266944i	$-0.0860139\pi$
$74$		0			0
$75$	−1.72809	+	0.117016i	−0.199543	+	0.0135119i
$76$		0			0
$77$	1.71120	−	1.68018i	0.195009	−	0.191474i
$78$		0			0
$79$		−	11.1774i		−	1.25756i	−0.777584	−	0.628779i	$-0.783555\pi$
$79$		−	11.1774i		−	1.25756i	0.777584	−	0.628779i	$-0.216445\pi$
$80$		0			0
$81$	8.67287	−	2.40443i	0.963652	−	0.267159i
$82$		0			0
$83$	−15.3804			−1.68822			−0.844111	−	0.536168i	$-0.819871\pi$
$83$	−15.3804			−1.68822			−0.844111	+	0.536168i	$0.819871\pi$
$84$		0			0
$85$			4.49908i			0.487994i
$86$		0			0
$87$	−8.61994	+	0.583692i	−0.924155	+	0.0625783i
$88$		0			0
$89$		−	2.10351i		−	0.222972i	−0.993766	−	0.111486i	$-0.964439\pi$
$89$		−	2.10351i		−	0.222972i	0.993766	−	0.111486i	$-0.0355610\pi$
$90$		0			0
$91$	3.02192			0.316783
$92$		0			0
$93$	10.1257	−	0.685650i	1.04998	−	0.0710986i
$94$		0			0
$95$	7.94523			0.815163
$96$		0			0
$97$	8.27274			0.839969			0.419985	−	0.907531i	$-0.362035\pi$
$97$	8.27274			0.839969			0.419985	+	0.907531i	$0.362035\pi$
$98$		0			0
$99$	7.86446	+	6.09510i	0.790408	+	0.612581i
$100$		0			0
$101$	−1.73495			−0.172634			−0.0863171	−	0.996268i	$-0.527510\pi$
$101$	−1.73495			−0.172634			−0.0863171	+	0.996268i	$0.527510\pi$
$102$		0			0
$103$	−8.44431			−0.832042			−0.416021	−	0.909355i	$-0.636576\pi$
$103$	−8.44431			−0.832042			−0.416021	+	0.909355i	$0.636576\pi$
$104$		0			0
$105$	−1.24954	+	0.0846115i	−0.121943	+	0.00825724i
$106$		0			0
$107$	14.2928			1.38174			0.690869	−	0.722980i	$-0.257228\pi$
$107$	14.2928			1.38174			0.690869	+	0.722980i	$0.257228\pi$
$108$		0			0
$109$		−	14.4958i		−	1.38845i	−0.719759	−	0.694224i	$-0.755748\pi$
$109$		−	14.4958i		−	1.38845i	0.719759	−	0.694224i	$-0.244252\pi$
$110$		0			0
$111$	−9.57705	+	0.648501i	−0.909013	+	0.0615530i
$112$		0			0
$113$			0.179261i			0.0168635i	0.999964	+	0.00843174i	$0.00268394\pi$
$113$			0.179261i			0.0168635i	−0.999964	+	0.00843174i	$0.997316\pi$
$114$		0			0
$115$	8.91237			0.831083
$116$		0			0
$117$	1.69022	+	12.4233i	0.156261	+	1.14854i
$118$		0			0
$119$			3.25317i			0.298218i
$120$		0			0
$121$	−0.201178	+	10.9982i	−0.0182889	+	0.999833i
$122$		0			0
$123$	6.10351	−	0.413294i	0.550335	−	0.0372655i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			6.84666i			0.607543i	0.952745	+	0.303772i	$0.0982460\pi$
$127$			6.84666i			0.607543i	−0.952745	+	0.303772i	$0.901754\pi$
$128$		0			0
$129$	1.22215	+	18.0487i	0.107605	+	1.58910i
$130$		0			0
$131$	11.4324			0.998856			0.499428	−	0.866355i	$-0.333544\pi$
$131$	11.4324			0.998856			0.499428	+	0.866355i	$0.333544\pi$
$132$		0			0
$133$	5.74499			0.498154
$134$		0			0
$135$	−1.04674	−	5.08963i	−0.0900888	−	0.438046i
$136$		0			0
$137$			0.137308i			0.0117310i	0.999983	+	0.00586552i	$0.00186706\pi$
$137$			0.137308i			0.0117310i	−0.999983	+	0.00586552i	$0.998133\pi$
$138$		0			0
$139$			8.17157i			0.693104i	0.938031	+	0.346552i	$0.112648\pi$
$139$			8.17157i			0.693104i	−0.938031	+	0.346552i	$0.887352\pi$
$140$		0			0
$141$	−0.520057	−	7.68018i	−0.0437967	−	0.646788i
$142$		0			0
$143$	−9.89046	+	9.71120i	−0.827082	+	0.812091i
$144$		0			0
$145$			4.98812i			0.414241i
$146$		0			0
$147$	11.1931	−	0.757934i	0.923195	−	0.0625133i
$148$		0			0
$149$	−2.58577			−0.211834			−0.105917	−	0.994375i	$-0.533778\pi$
$149$	−2.58577			−0.211834			−0.105917	+	0.994375i	$0.533778\pi$
$150$		0			0
$151$		−	5.09164i		−	0.414351i	−0.978304	−	0.207176i	$-0.933573\pi$
$151$		−	5.09164i		−	0.414351i	0.978304	−	0.207176i	$-0.0664272\pi$
$152$		0			0
$153$	−13.3740	+	1.81956i	−1.08123	+	0.147103i
$154$		0			0
$155$		−	5.85944i		−	0.470642i
$156$		0			0
$157$	1.01188			0.0807567			0.0403783	−	0.999184i	$-0.487144\pi$
$157$	1.01188			0.0807567			0.0403783	+	0.999184i	$0.487144\pi$
$158$		0			0
$159$	−0.282395	−	4.17040i	−0.0223954	−	0.330734i
$160$		0			0
$161$	6.44431			0.507883
$162$		0			0
$163$	−0.543813			−0.0425947			−0.0212974	−	0.999773i	$-0.506780\pi$
$163$	−0.543813			−0.0425947			−0.0212974	+	0.999773i	$0.506780\pi$
$164$		0			0
$165$	3.81772	−	4.29243i	0.297209	−	0.334165i
$166$		0			0
$167$	1.34942			0.104422			0.0522108	−	0.998636i	$-0.483373\pi$
$167$	1.34942			0.104422			0.0522108	+	0.998636i	$0.483373\pi$
$168$		0			0
$169$	−4.46623			−0.343556
$170$		0			0
$171$	3.21329	+	23.6181i	0.245726	+	1.80612i
$172$		0			0
$173$	11.8247			0.899019			0.449509	−	0.893276i	$-0.351599\pi$
$173$	11.8247			0.899019			0.449509	+	0.893276i	$0.351599\pi$
$174$		0			0
$175$			0.723074i			0.0546593i
$176$		0			0
$177$	−1.13637	−	16.7819i	−0.0854147	−	1.26140i
$178$		0			0
$179$			16.2650i			1.21571i	0.794049	+	0.607853i	$0.207969\pi$
$179$			16.2650i			1.21571i	−0.794049	+	0.607853i	$0.792031\pi$
$180$		0			0
$181$	−16.1212			−1.19828			−0.599141	−	0.800643i	$-0.704491\pi$
$181$	−16.1212			−1.19828			−0.599141	+	0.800643i	$0.704491\pi$
$182$		0			0
$183$	−0.303371	−	4.48018i	−0.0224258	−	0.331184i
$184$		0			0
$185$			5.54197i			0.407454i
$186$		0			0
$187$	−10.4543	−	10.6473i	−0.764498	−	0.778610i
$188$		0			0
$189$	−0.756869	−	3.68018i	−0.0550541	−	0.267694i
$190$		0			0
$191$		−	23.4082i		−	1.69376i	−0.531785	−	0.846879i	$-0.678479\pi$
$191$		−	23.4082i		−	1.69376i	0.531785	−	0.846879i	$-0.321521\pi$
$192$		0			0
$193$		−	9.28094i		−	0.668056i	−0.942563	−	0.334028i	$-0.891592\pi$
$193$		−	9.28094i		−	0.668056i	0.942563	−	0.334028i	$-0.108408\pi$
$194$		0			0
$195$	7.22215	−	0.489042i	0.517189	−	0.0350210i
$196$		0			0
$197$	10.8467			0.772793			0.386396	−	0.922333i	$-0.373720\pi$
$197$	10.8467			0.772793			0.386396	+	0.922333i	$0.373720\pi$
$198$		0			0
$199$	−10.3275			−0.732098			−0.366049	−	0.930596i	$-0.619290\pi$
$199$	−10.3275			−0.732098			−0.366049	+	0.930596i	$0.619290\pi$
$200$		0			0
$201$	15.9730	−	1.08160i	1.12665	−	0.0762900i
$202$		0			0
$203$			3.60678i			0.253147i
$204$		0			0
$205$		−	3.53193i		−	0.246681i
$206$		0			0
$207$	3.60443	+	26.4931i	0.250525	+	1.84139i
$208$		0			0
$209$	−18.8028	+	18.4620i	−1.30062	+	1.27705i
$210$		0			0
$211$		−	10.8156i		−	0.744580i	−0.928116	−	0.372290i	$-0.878573\pi$
$211$		−	10.8156i		−	0.744580i	0.928116	−	0.372290i	$-0.121427\pi$
$212$		0			0
$213$	−0.414469	−	6.12086i	−0.0283989	−	0.419394i
$214$		0			0
$215$	10.4443			0.712296
$216$		0			0
$217$		−	4.23681i		−	0.287614i
$218$		0			0
$219$	−0.533775	−	7.88277i	−0.0360691	−	0.532668i
$220$		0			0
$221$		−	18.8028i		−	1.26482i
$222$		0			0
$223$	11.3366			0.759155			0.379578	−	0.925160i	$-0.376069\pi$
$223$	11.3366			0.759155			0.379578	+	0.925160i	$0.376069\pi$
$224$		0			0
$225$	−2.97261	+	0.404430i	−0.198174	+	0.0269620i
$226$		0			0
$227$	−20.5502			−1.36396			−0.681981	−	0.731370i	$-0.738882\pi$
$227$	−20.5502			−1.36396			−0.681981	+	0.731370i	$0.738882\pi$
$228$		0			0
$229$	−2.15150			−0.142175			−0.0710874	−	0.997470i	$-0.522647\pi$
$229$	−2.15150			−0.142175			−0.0710874	+	0.997470i	$0.522647\pi$
$230$		0			0
$231$	2.76050	−	3.10375i	0.181627	−	0.204212i
$232$		0			0
$233$	−16.6860			−1.09314			−0.546569	−	0.837414i	$-0.684066\pi$
$233$	−16.6860			−1.09314			−0.546569	+	0.837414i	$0.684066\pi$
$234$		0			0
$235$	−4.44431			−0.289915
$236$		0			0
$237$	−1.30794	−	19.3156i	−0.0849599	−	1.25468i
$238$		0			0
$239$	−21.5436			−1.39354			−0.696771	−	0.717294i	$-0.745381\pi$
$239$	−21.5436			−1.39354			−0.696771	+	0.717294i	$0.745381\pi$
$240$		0			0
$241$			5.44615i			0.350817i	0.984496	+	0.175409i	$0.0561247\pi$
$241$			5.44615i			0.350817i	−0.984496	+	0.175409i	$0.943875\pi$
$242$		0			0
$243$	14.7062	−	5.16995i	0.943402	−	0.331652i
$244$		0			0
$245$		−	6.47716i		−	0.413811i
$246$		0			0
$247$	−33.2052			−2.11280
$248$		0			0
$249$	−26.5788	+	1.79976i	−1.68437	+	0.114055i
$250$		0			0
$251$			6.19934i			0.391299i	0.980674	+	0.195649i	$0.0626814\pi$
$251$			6.19934i			0.391299i	−0.980674	+	0.195649i	$0.937319\pi$
$252$		0			0
$253$	−21.0916	+	20.7094i	−1.32602	+	1.30199i
$254$		0			0
$255$	0.526466	+	7.77483i	0.0329686	+	0.486879i
$256$		0			0
$257$			12.3508i			0.770424i	0.922828	+	0.385212i	$0.125872\pi$
$257$			12.3508i			0.770424i	−0.922828	+	0.385212i	$0.874128\pi$
$258$		0			0
$259$			4.00726i			0.248999i
$260$		0			0
$261$	−14.8278	+	2.01735i	−0.917815	+	0.124871i
$262$		0			0
$263$	11.3676			0.700958			0.350479	−	0.936571i	$-0.386019\pi$
$263$	11.3676			0.700958			0.350479	+	0.936571i	$0.386019\pi$
$264$		0			0
$265$	−2.41329			−0.148247
$266$		0			0
$267$	−0.246145	−	3.63507i	−0.0150639	−	0.222463i
$268$		0			0
$269$			11.3567i			0.692429i	0.938155	+	0.346215i	$0.112533\pi$
$269$			11.3567i			0.692429i	−0.938155	+	0.346215i	$0.887467\pi$
$270$		0			0
$271$			4.07340i			0.247441i	0.992317	+	0.123721i	$0.0394827\pi$
$271$			4.07340i			0.247441i	−0.992317	+	0.123721i	$0.960517\pi$
$272$		0			0
$273$	5.22215	−	0.353614i	0.316059	−	0.0214017i
$274$		0			0
$275$	−2.32366	−	2.36656i	−0.140122	−	0.142709i
$276$		0			0
$277$		−	7.71120i		−	0.463321i	−0.972797	−	0.231660i	$-0.925584\pi$
$277$		−	7.71120i		−	0.463321i	0.972797	−	0.231660i	$-0.0744158\pi$
$278$		0			0
$279$	17.4179	−	2.36974i	1.04278	−	0.141872i
$280$		0			0
$281$	24.3786			1.45431			0.727153	−	0.686476i	$-0.240843\pi$
$281$	24.3786			1.45431			0.727153	+	0.686476i	$0.240843\pi$
$282$		0			0
$283$		−	19.5674i		−	1.16316i	−0.813489	−	0.581580i	$-0.802435\pi$
$283$		−	19.5674i		−	1.16316i	0.813489	−	0.581580i	$-0.197565\pi$
$284$		0			0
$285$	13.7301	−	0.929721i	0.813301	−	0.0550719i
$286$		0			0
$287$		−	2.55385i		−	0.150749i
$288$		0			0
$289$	3.24172			0.190690
$290$		0			0
$291$	14.2961	−	0.968045i	0.838050	−	0.0567478i
$292$		0			0
$293$	13.8704			0.810316			0.405158	−	0.914247i	$-0.367216\pi$
$293$	13.8704			0.810316			0.405158	+	0.914247i	$0.367216\pi$
$294$		0			0
$295$	−9.71120			−0.565408
$296$		0			0
$297$	14.3038	+	9.61263i	0.829988	+	0.557781i
$298$		0			0
$299$	−37.2471			−2.15406
$300$		0			0
$301$	7.55201			0.435291
$302$		0			0
$303$	−2.99816	+	0.203018i	−0.172240	+	0.0116631i
$304$		0			0
$305$	−2.59256			−0.148449
$306$		0			0
$307$			19.4844i			1.11203i	0.831171	+	0.556017i	$0.187671\pi$
$307$			19.4844i			1.11203i	−0.831171	+	0.556017i	$0.812329\pi$
$308$		0			0
$309$	−14.5926	+	0.988122i	−0.830141	+	0.0562123i
$310$		0			0
$311$			7.37040i			0.417937i	0.977922	+	0.208969i	$0.0670106\pi$
$311$			7.37040i			0.417937i	−0.977922	+	0.208969i	$0.932989\pi$
$312$		0			0
$313$	8.80283			0.497565			0.248783	−	0.968559i	$-0.419969\pi$
$313$	8.80283			0.497565			0.248783	+	0.968559i	$0.419969\pi$
$314$		0			0
$315$	−2.14942	+	0.292433i	−0.121106	+	0.0164767i
$316$		0			0
$317$			21.7737i			1.22293i	0.791271	+	0.611465i	$0.209420\pi$
$317$			21.7737i			1.22293i	−0.791271	+	0.611465i	$0.790580\pi$
$318$		0			0
$319$	−11.5907	−	11.8047i	−0.648955	−	0.660935i
$320$		0			0
$321$	24.6993	−	1.67249i	1.37858	−	0.0933494i
$322$		0			0
$323$		−	35.7462i		−	1.98897i
$324$		0			0
$325$		−	4.17926i		−	0.231824i
$326$		0			0
$327$	−1.69625	−	25.0502i	−0.0938028	−	1.38528i
$328$		0			0
$329$	−3.21357			−0.177170
$330$		0			0
$331$	−2.44431			−0.134351			−0.0671757	−	0.997741i	$-0.521399\pi$
$331$	−2.44431			−0.134351			−0.0671757	+	0.997741i	$0.521399\pi$
$332$		0			0
$333$	−16.4741	+	2.24134i	−0.902778	+	0.122825i
$334$		0			0
$335$		−	9.24313i		−	0.505006i
$336$		0			0
$337$		−	34.8786i		−	1.89996i	−0.312315	−	0.949979i	$-0.601104\pi$
$337$		−	34.8786i		−	1.89996i	0.312315	−	0.949979i	$-0.398896\pi$
$338$		0			0
$339$	0.0209765	+	0.309780i	0.00113929	+	0.0168250i
$340$		0			0
$341$	13.6154	+	13.8667i	0.737314	+	0.750924i
$342$		0			0
$343$		−	9.74499i		−	0.526180i
$344$		0			0
$345$	15.4014	−	1.04289i	0.829184	−	0.0561475i
$346$		0			0
$347$	24.1176			1.29470			0.647349	−	0.762193i	$-0.275878\pi$
$347$	24.1176			1.29470			0.647349	+	0.762193i	$0.275878\pi$
$348$		0			0
$349$		−	1.86269i		−	0.0997076i	−0.998757	−	0.0498538i	$-0.984124\pi$
$349$		−	1.86269i		−	0.0997076i	0.998757	−	0.0498538i	$-0.0158755\pi$
$350$		0			0
$351$	4.37459	+	21.2709i	0.233498	+	1.13536i
$352$		0			0
$353$			1.56338i			0.0832103i	0.999134	+	0.0416052i	$0.0132472\pi$
$353$			1.56338i			0.0832103i	−0.999134	+	0.0416052i	$0.986753\pi$
$354$		0			0
$355$	−3.54197			−0.187988
$356$		0			0
$357$	0.380674	+	5.62178i	0.0201474	+	0.297536i
$358$		0			0
$359$	27.8010			1.46728			0.733640	−	0.679538i	$-0.237820\pi$
$359$	27.8010			1.46728			0.733640	+	0.679538i	$0.237820\pi$
$360$		0			0
$361$	−44.1267			−2.32246
$362$		0			0
$363$	0.939310	+	19.0294i	0.0493010	+	0.998784i
$364$		0			0
$365$	−4.56154			−0.238762
$366$		0			0
$367$	12.1212			0.632724			0.316362	−	0.948639i	$-0.397539\pi$
$367$	12.1212			0.632724			0.316362	+	0.948639i	$0.397539\pi$
$368$		0			0
$369$	10.4991	−	1.42842i	0.546560	−	0.0743606i
$370$		0			0
$371$	−1.74499			−0.0905954
$372$		0			0
$373$		−	5.43846i		−	0.281593i	−0.990039	−	0.140796i	$-0.955034\pi$
$373$		−	5.43846i		−	0.281593i	0.990039	−	0.140796i	$-0.0449663\pi$
$374$		0			0
$375$	0.117016	+	1.72809i	0.00604270	+	0.0892384i
$376$		0			0
$377$		−	20.8467i		−	1.07366i
$378$		0			0
$379$	27.0164			1.38774			0.693869	−	0.720102i	$-0.255905\pi$
$379$	27.0164			1.38774			0.693869	+	0.720102i	$0.255905\pi$
$380$		0			0
$381$	0.801171	+	11.8317i	0.0410453	+	0.606155i
$382$		0			0
$383$			27.8401i			1.42256i	0.702907	+	0.711282i	$0.251885\pi$
$383$			27.8401i			1.42256i	−0.702907	+	0.711282i	$0.748115\pi$
$384$		0			0
$385$	−1.68018	−	1.71120i	−0.0856300	−	0.0872106i
$386$		0			0
$387$	4.22399	+	31.0469i	0.214718	+	1.57820i
$388$		0			0
$389$		−	10.6396i		−	0.539451i	−0.962937	−	0.269726i	$-0.913067\pi$
$389$		−	10.6396i		−	0.539451i	0.962937	−	0.269726i	$-0.0869330\pi$
$390$		0			0
$391$		−	40.0975i		−	2.02782i
$392$		0			0
$393$	19.7563	−	1.33778i	0.996574	−	0.0674821i
$394$		0			0
$395$	−11.1774			−0.562397
$396$		0			0
$397$	15.0478			0.755228			0.377614	−	0.925963i	$-0.376745\pi$
$397$	15.0478			0.755228			0.377614	+	0.925963i	$0.376745\pi$
$398$		0			0
$399$	9.92788	−	0.672258i	0.497016	−	0.0336550i
$400$		0			0
$401$		−	35.9082i		−	1.79317i	−0.442873	−	0.896585i	$-0.646041\pi$
$401$		−	35.9082i		−	1.79317i	0.442873	−	0.896585i	$-0.353959\pi$
$402$		0			0
$403$			24.4881i			1.21984i
$404$		0			0
$405$	−2.40443	−	8.67287i	−0.119477	−	0.430958i
$406$		0			0
$407$	−12.8777	−	13.1154i	−0.638323	−	0.650106i
$408$		0			0
$409$			7.53193i			0.372430i	0.982509	+	0.186215i	$0.0596221\pi$
$409$			7.53193i			0.372430i	−0.982509	+	0.186215i	$0.940378\pi$
$410$		0			0
$411$	0.0160673	+	0.237282i	0.000792542	+	0.0117042i
$412$		0			0
$413$	−7.02192			−0.345526
$414$		0			0
$415$			15.3804i			0.754996i
$416$		0			0
$417$	0.956207	+	14.1212i	0.0468257	+	0.691520i
$418$		0			0
$419$			20.5816i			1.00548i	0.864438	+	0.502739i	$0.167674\pi$
$419$			20.5816i			1.00548i	−0.864438	+	0.502739i	$0.832326\pi$
$420$		0			0
$421$	22.3987			1.09164			0.545822	−	0.837901i	$-0.316217\pi$
$421$	22.3987			1.09164			0.545822	+	0.837901i	$0.316217\pi$
$422$		0			0
$423$	−1.79741	−	13.2112i	−0.0873932	−	0.642351i
$424$		0			0
$425$	4.49908			0.218237
$426$		0			0
$427$	−1.87461			−0.0907187
$428$		0			0
$429$	−15.9553	+	17.9392i	−0.770328	+	0.866113i
$430$		0			0
$431$	27.4224			1.32089			0.660445	−	0.750874i	$-0.270368\pi$
$431$	27.4224			1.32089			0.660445	+	0.750874i	$0.270368\pi$
$432$		0			0
$433$	−19.2947			−0.927242			−0.463621	−	0.886034i	$-0.653450\pi$
$433$	−19.2947			−0.927242			−0.463621	+	0.886034i	$0.653450\pi$
$434$		0			0
$435$	0.583692	+	8.61994i	0.0279859	+	0.413295i
$436$		0			0
$437$	−70.8109			−3.38734
$438$		0			0
$439$		−	30.3033i		−	1.44630i	−0.690692	−	0.723149i	$-0.742694\pi$
$439$		−	30.3033i		−	1.44630i	0.690692	−	0.723149i	$-0.257306\pi$
$440$		0			0
$441$	19.2541	−	2.61956i	0.916862	−	0.124741i
$442$		0			0
$443$			33.6874i			1.60054i	0.599641	+	0.800269i	$0.295310\pi$
$443$			33.6874i			1.60054i	−0.599641	+	0.800269i	$0.704690\pi$
$444$		0			0
$445$	−2.10351			−0.0997161
$446$		0			0
$447$	−4.46845	+	0.302577i	−0.211350	+	0.0143114i
$448$		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
$451$	8.20703	+	8.35852i	0.386454	+	0.393587i
$452$		0			0
$453$	−0.595804	−	8.79882i	−0.0279933	−	0.413405i
$454$		0			0
$455$		−	3.02192i		−	0.141670i
$456$		0			0
$457$			13.9424i			0.652200i	0.945335	+	0.326100i	$0.105735\pi$
$457$			13.9424i			0.652200i	−0.945335	+	0.326100i	$0.894265\pi$
$458$		0			0
$459$	−22.8987	+	4.70936i	−1.06882	+	0.219814i
$460$		0			0
$461$	8.62960			0.401921			0.200960	−	0.979599i	$-0.435594\pi$
$461$	8.62960			0.401921			0.200960	+	0.979599i	$0.435594\pi$
$462$		0			0
$463$	−22.2070			−1.03205			−0.516024	−	0.856574i	$-0.672588\pi$
$463$	−22.2070			−1.03205			−0.516024	+	0.856574i	$0.672588\pi$
$464$		0			0
$465$	−0.685650	−	10.1257i	−0.0317963	−	0.469566i
$466$		0			0
$467$			5.06062i			0.234178i	0.993121	+	0.117089i	$0.0373562\pi$
$467$			5.06062i			0.234178i	−0.993121	+	0.117089i	$0.962644\pi$
$468$		0			0
$469$		−	6.68347i		−	0.308614i
$470$		0			0
$471$	1.74862	−	0.118406i	0.0805722	−	0.00545587i
$472$		0			0
$473$	−24.7170	+	24.2691i	−1.13649	+	1.11589i
$474$		0			0
$475$		−	7.94523i		−	0.364552i
$476$		0			0
$477$	−0.976009	−	7.17379i	−0.0446884	−	0.328465i
$478$		0			0
$479$	−9.71521			−0.443899			−0.221950	−	0.975058i	$-0.571242\pi$
$479$	−9.71521			−0.443899			−0.221950	+	0.975058i	$0.571242\pi$
$480$		0			0
$481$		−	23.1614i		−	1.05607i
$482$		0			0
$483$	11.1364	−	0.754089i	0.506722	−	0.0343122i
$484$		0			0
$485$		−	8.27274i		−	0.375646i
$486$		0			0
$487$	−14.8685			−0.673758			−0.336879	−	0.941548i	$-0.609371\pi$
$487$	−14.8685			−0.673758			−0.336879	+	0.941548i	$0.609371\pi$
$488$		0			0
$489$	−0.939760	+	0.0636350i	−0.0424974	+	0.00287767i
$490$		0			0
$491$	16.1816			0.730266			0.365133	−	0.930955i	$-0.381024\pi$
$491$	16.1816			0.730266			0.365133	+	0.930955i	$0.381024\pi$
$492$		0			0
$493$	22.4420			1.01074
$494$		0			0
$495$	6.09510	−	7.86446i	0.273954	−	0.353481i
$496$		0			0
$497$	−2.56111			−0.114881
$498$		0			0
$499$	4.65502			0.208387			0.104194	−	0.994557i	$-0.466774\pi$
$499$	4.65502			0.208387			0.104194	+	0.994557i	$0.466774\pi$
$500$		0			0
$501$	2.33193	−	0.157905i	0.104183	−	0.00705466i
$502$		0			0
$503$	−12.7170			−0.567025			−0.283513	−	0.958969i	$-0.591500\pi$
$503$	−12.7170			−0.567025			−0.283513	+	0.958969i	$0.591500\pi$
$504$		0			0
$505$			1.73495i			0.0772044i
$506$		0			0
$507$	−7.71806	+	0.522621i	−0.342771	+	0.0232104i
$508$		0			0
$509$		−	20.1413i		−	0.892748i	−0.894846	−	0.446374i	$-0.852715\pi$
$509$		−	20.1413i		−	0.892748i	0.894846	−	0.446374i	$-0.147285\pi$
$510$		0			0
$511$	−3.29833			−0.145910
$512$		0			0
$513$	8.31657	+	40.4383i	0.367186	+	1.78539i
$514$		0			0
$515$			8.44431i			0.372101i
$516$		0			0
$517$	10.5177	−	10.3271i	0.462568	−	0.454184i
$518$		0			0
$519$	20.4343	−	1.38369i	0.896965	−	0.0607372i
$520$		0			0
$521$		−	33.7772i		−	1.47981i	−0.672713	−	0.739904i	$-0.734871\pi$
$521$		−	33.7772i		−	1.47981i	0.672713	−	0.739904i	$-0.265129\pi$
$522$		0			0
$523$		−	22.9525i		−	1.00364i	−0.864971	−	0.501822i	$-0.832663\pi$
$523$		−	22.9525i		−	1.00364i	0.864971	−	0.501822i	$-0.167337\pi$
$524$		0			0
$525$	0.0846115	+	1.24954i	0.00369275	+	0.0545344i
$526$		0			0
$527$	−26.3621			−1.14835
$528$		0			0
$529$	−56.4304			−2.45350
$530$		0			0
$531$	−3.92750	−	28.8676i	−0.170439	−	1.25275i
$532$		0			0
$533$			14.7609i			0.639365i
$534$		0			0
$535$		−	14.2928i		−	0.617932i
$536$		0			0
$537$	1.90328	+	28.1075i	0.0821324	+	1.21293i
$538$		0			0
$539$	15.0507	+	15.3286i	0.648282	+	0.660248i
$540$		0			0
$541$			2.83352i			0.121822i	0.998143	+	0.0609112i	$0.0194006\pi$
$541$			2.83352i			0.121822i	−0.998143	+	0.0609112i	$0.980599\pi$
$542$		0			0
$543$	−27.8590	+	1.88645i	−1.19554	+	0.0809552i
$544$		0			0
$545$	−14.4958			−0.620933
$546$		0			0
$547$			37.5017i			1.60346i	0.597689	+	0.801728i	$0.296086\pi$
$547$			37.5017i			1.60346i	−0.597689	+	0.801728i	$0.703914\pi$
$548$		0			0
$549$	−1.04851	−	7.70667i	−0.0447492	−	0.328913i
$550$		0			0
$551$		−	39.6318i		−	1.68837i
$552$		0			0
$553$	−8.08211			−0.343686
$554$		0			0
$555$	0.648501	+	9.57705i	0.0275273	+	0.406523i
$556$		0			0
$557$	8.74080			0.370360			0.185180	−	0.982705i	$-0.440713\pi$
$557$	8.74080			0.370360			0.185180	+	0.982705i	$0.440713\pi$
$558$		0			0
$559$	−43.6495			−1.84618
$560$		0			0
$561$	−19.3120	−	17.1762i	−0.815353	−	0.725182i
$562$		0			0
$563$	9.96087			0.419800			0.209900	−	0.977723i	$-0.432686\pi$
$563$	9.96087			0.419800			0.209900	+	0.977723i	$0.432686\pi$
$564$		0			0
$565$	0.179261			0.00754158
$566$		0			0
$567$	−1.73858	−	6.27113i	−0.0730136	−	0.263363i
$568$		0			0
$569$	31.6696			1.32766			0.663829	−	0.747885i	$-0.268930\pi$
$569$	31.6696			1.32766			0.663829	+	0.747885i	$0.268930\pi$
$570$		0			0
$571$		−	3.66411i		−	0.153338i	−0.997057	−	0.0766692i	$-0.975571\pi$
$571$		−	3.66411i		−	0.153338i	0.997057	−	0.0766692i	$-0.0244285\pi$
$572$		0			0
$573$	−2.73914	−	40.4516i	−0.114429	−	1.68989i
$574$		0			0
$575$		−	8.91237i		−	0.371672i
$576$		0			0
$577$	7.97624			0.332055			0.166028	−	0.986121i	$-0.446906\pi$
$577$	7.97624			0.332055			0.166028	+	0.986121i	$0.446906\pi$
$578$		0			0
$579$	−1.08602	−	16.0383i	−0.0451335	−	0.666530i
$580$		0			0
$581$			11.1212i			0.461385i
$582$		0			0
$583$	5.71120	−	5.60768i	0.236534	−	0.232246i
$584$		0			0
$585$	12.4233	−	1.69022i	0.513642	−	0.0698820i
$586$		0			0
$587$		−	11.0569i		−	0.456369i	−0.973618	−	0.228184i	$-0.926721\pi$
$587$		−	11.0569i		−	0.456369i	0.973618	−	0.228184i	$-0.0732789\pi$
$588$		0			0
$589$			46.5546i			1.91825i
$590$		0			0
$591$	18.7441	−	1.26924i	0.771027	−	0.0522094i
$592$		0			0
$593$	−18.2271			−0.748497			−0.374249	−	0.927328i	$-0.622099\pi$
$593$	−18.2271			−0.748497			−0.374249	+	0.927328i	$0.622099\pi$
$594$		0			0
$595$	3.25317			0.133367
$596$		0			0
$597$	−17.8469	+	1.20849i	−0.730425	+	0.0494601i
$598$		0			0
$599$			24.8567i			1.01562i	0.861470	+	0.507809i	$0.169544\pi$
$599$			24.8567i			1.01562i	−0.861470	+	0.507809i	$0.830456\pi$
$600$		0			0
$601$		−	16.4681i		−	0.671746i	−0.941907	−	0.335873i	$-0.890969\pi$
$601$		−	16.4681i		−	0.671746i	0.941907	−	0.335873i	$-0.109031\pi$
$602$		0			0
$603$	27.4763	−	3.73820i	1.11892	−	0.152231i
$604$		0			0
$605$	10.9982	+	0.201178i	0.447139	+	0.00817905i
$606$		0			0
$607$		−	28.1455i		−	1.14239i	−0.820815	−	0.571195i	$-0.806480\pi$
$607$		−	28.1455i		−	1.14239i	0.820815	−	0.571195i	$-0.193520\pi$
$608$		0			0
$609$	0.422052	+	6.23286i	0.0171024	+	0.252568i
$610$		0			0
$611$	18.5739			0.751421
$612$		0			0
$613$		−	44.5797i		−	1.80056i	−0.435313	−	0.900279i	$-0.643362\pi$
$613$		−	44.5797i		−	1.80056i	0.435313	−	0.900279i	$-0.356638\pi$
$614$		0			0
$615$	−0.413294	−	6.10351i	−0.0166656	−	0.246117i
$616$		0			0
$617$			16.5023i			0.664359i	0.943216	+	0.332179i	$0.107784\pi$
$617$			16.5023i			0.664359i	−0.943216	+	0.332179i	$0.892216\pi$
$618$		0			0
$619$	−9.34130			−0.375459			−0.187729	−	0.982221i	$-0.560113\pi$
$619$	−9.34130			−0.375459			−0.187729	+	0.982221i	$0.560113\pi$
$620$		0			0
$621$	9.32892	+	45.3607i	0.374357	+	1.82026i
$622$		0			0
$623$	−1.52100			−0.0609374
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−30.3327	+	34.1044i	−1.21137	+	1.36200i
$628$		0			0
$629$	24.9338			0.994175
$630$		0			0
$631$	−47.1686			−1.87775			−0.938876	−	0.344254i	$-0.888132\pi$
$631$	−47.1686			−1.87775			−0.938876	+	0.344254i	$0.888132\pi$
$632$		0			0
$633$	−1.26561	−	18.6905i	−0.0503034	−	0.742879i
$634$		0			0
$635$	6.84666			0.271702
$636$		0			0
$637$			27.0698i			1.07254i
$638$		0			0
$639$	−1.43248	−	10.5289i	−0.0566681	−	0.416518i
$640$		0			0
$641$		−	27.7212i		−	1.09492i	−0.836831	−	0.547461i	$-0.815594\pi$
$641$		−	27.7212i		−	1.09492i	0.836831	−	0.547461i	$-0.184406\pi$
$642$		0			0
$643$	−14.2371			−0.561458			−0.280729	−	0.959787i	$-0.590576\pi$
$643$	−14.2371			−0.561458			−0.280729	+	0.959787i	$0.590576\pi$
$644$		0			0
$645$	18.0487	−	1.22215i	0.710669	−	0.0481223i
$646$		0			0
$647$			39.2271i			1.54218i	0.636729	+	0.771088i	$0.280287\pi$
$647$			39.2271i			1.54218i	−0.636729	+	0.771088i	$0.719713\pi$
$648$		0			0
$649$	22.9821	−	22.5655i	0.902126	−	0.885775i
$650$		0			0
$651$	−0.495776	−	7.32161i	−0.0194310	−	0.286956i
$652$		0			0
$653$		−	19.9569i		−	0.780975i	−0.920608	−	0.390487i	$-0.872306\pi$
$653$		−	19.9569i		−	0.780975i	0.920608	−	0.390487i	$-0.127694\pi$
$654$		0			0
$655$		−	11.4324i		−	0.446702i
$656$		0			0
$657$	−1.84482	−	13.5597i	−0.0719735	−	0.529014i
$658$		0			0
$659$	16.3839			0.638228			0.319114	−	0.947716i	$-0.396615\pi$
$659$	16.3839			0.638228			0.319114	+	0.947716i	$0.396615\pi$
$660$		0			0
$661$	11.2052			0.435831			0.217916	−	0.975968i	$-0.430074\pi$
$661$	11.2052			0.435831			0.217916	+	0.975968i	$0.430074\pi$
$662$		0			0
$663$	−2.20024	−	32.4931i	−0.0854502	−	1.26193i
$664$		0			0
$665$		−	5.74499i		−	0.222781i
$666$		0			0
$667$		−	44.4560i		−	1.72134i
$668$		0			0
$669$	19.5907	−	1.32657i	0.757421	−	0.0512881i
$670$		0			0
$671$	6.13543	−	6.02423i	0.236856	−	0.232563i
$672$		0			0
$673$		−	17.2809i		−	0.666131i	−0.942904	−	0.333066i	$-0.891917\pi$
$673$		−	17.2809i		−	0.666131i	0.942904	−	0.333066i	$-0.108083\pi$
$674$		0			0
$675$	−5.08963	+	1.04674i	−0.195900	+	0.0402889i
$676$		0			0
$677$	−22.5856			−0.868036			−0.434018	−	0.900904i	$-0.642905\pi$
$677$	−22.5856			−0.868036			−0.434018	+	0.900904i	$0.642905\pi$
$678$		0			0
$679$		−	5.98180i		−	0.229561i
$680$		0			0
$681$	−35.5126	+	2.40471i	−1.36085	+	0.0921485i
$682$		0			0
$683$		−	19.5944i		−	0.749759i	−0.927074	−	0.374879i	$-0.877684\pi$
$683$		−	19.5944i		−	0.749759i	0.927074	−	0.374879i	$-0.122316\pi$
$684$		0			0
$685$	0.137308			0.00524628
$686$		0			0
$687$	−3.71799	+	0.251760i	−0.141850	+	0.00960525i
$688$		0			0
$689$	10.0858			0.384238
$690$		0			0
$691$	14.8970			0.566708			0.283354	−	0.959015i	$-0.408553\pi$
$691$	14.8970			0.566708			0.283354	+	0.959015i	$0.408553\pi$
$692$		0			0
$693$	4.40721	−	5.68659i	0.167416	−	0.216016i
$694$		0			0
$695$	8.17157			0.309965
$696$		0			0
$697$	−15.8905			−0.601894
$698$		0			0
$699$	−28.8350	+	1.95254i	−1.09064	+	0.0738518i
$700$		0			0
$701$	49.2152			1.85883			0.929417	−	0.369032i	$-0.120311\pi$
$701$	49.2152			1.85883			0.929417	+	0.369032i	$0.120311\pi$
$702$		0			0
$703$		−	44.0322i		−	1.66071i
$704$		0			0
$705$	−7.68018	+	0.520057i	−0.289252	+	0.0195865i
$706$		0			0
$707$			1.25450i			0.0471803i
$708$		0			0
$709$	−14.4616			−0.543116			−0.271558	−	0.962422i	$-0.587539\pi$
$709$	−14.4616			−0.543116			−0.271558	+	0.962422i	$0.587539\pi$
$710$		0			0
$711$	−4.52049	−	33.2262i	−0.169532	−	1.24608i
$712$		0			0
$713$			52.2215i			1.95571i
$714$		0			0
$715$	9.71120	+	9.89046i	0.363178	+	0.369882i
$716$		0			0
$717$	−37.2294	+	2.52096i	−1.39036	+	0.0941468i
$718$		0			0
$719$		−	11.3751i		−	0.424220i	−0.977246	−	0.212110i	$-0.931967\pi$
$719$		−	11.3751i		−	0.424220i	0.977246	−	0.212110i	$-0.0680335\pi$
$720$		0			0
$721$			6.10586i			0.227394i
$722$		0			0
$723$	0.637288	+	9.41145i	0.0237010	+	0.350016i
$724$		0			0
$725$	4.98812			0.185254
$726$		0			0
$727$	6.95801			0.258058			0.129029	−	0.991641i	$-0.458814\pi$
$727$	6.95801			0.258058			0.129029	+	0.991641i	$0.458814\pi$
$728$		0			0
$729$	24.8087	−	10.6550i	0.918840	−	0.394630i
$730$		0			0
$731$		−	46.9898i		−	1.73798i
$732$		0			0
$733$			3.11171i			0.114934i	0.998347	+	0.0574669i	$0.0183024\pi$
$733$			3.11171i			0.114934i	−0.998347	+	0.0574669i	$0.981698\pi$
$734$		0			0
$735$	−0.757934	−	11.1931i	−0.0279568	−	0.412865i
$736$		0			0
$737$	21.4779	+	21.8744i	0.791149	+	0.805754i
$738$		0			0
$739$			24.1796i			0.889461i	0.895665	+	0.444730i	$0.146700\pi$
$739$			24.1796i			0.889461i	−0.895665	+	0.444730i	$0.853300\pi$
$740$		0			0
$741$	−57.3817	+	3.88555i	−2.10797	+	0.142739i
$742$		0			0
$743$	−35.4078			−1.29899			−0.649493	−	0.760368i	$-0.725019\pi$
$743$	−35.4078			−1.29899			−0.649493	+	0.760368i	$0.725019\pi$
$744$		0			0
$745$			2.58577i			0.0947351i
$746$		0			0
$747$	−45.7201	+	6.22031i	−1.67281	+	0.227589i
$748$		0			0
$749$		−	10.3348i		−	0.377624i
$750$		0			0
$751$	−21.2818			−0.776585			−0.388293	−	0.921536i	$-0.626935\pi$
$751$	−21.2818			−0.776585			−0.388293	+	0.921536i	$0.626935\pi$
$752$		0			0
$753$	0.725424	+	10.7130i	0.0264359	+	0.390405i
$754$		0			0
$755$	−5.09164			−0.185304
$756$		0			0
$757$	−34.8762			−1.26760			−0.633799	−	0.773498i	$-0.718505\pi$
$757$	−34.8762			−1.26760			−0.633799	+	0.773498i	$0.718505\pi$
$758$		0			0
$759$	−34.0250	+	38.2558i	−1.23503	+	1.38860i
$760$		0			0
$761$	1.01354			0.0367407			0.0183704	−	0.999831i	$-0.494152\pi$
$761$	1.01354			0.0367407			0.0183704	+	0.999831i	$0.494152\pi$
$762$		0			0
$763$	−10.4816			−0.379458
$764$		0			0
$765$	1.81956	+	13.3740i	0.0657865	+	0.483539i
$766$		0			0
$767$	40.5856			1.46546
$768$		0			0
$769$		−	34.6112i		−	1.24811i	−0.781380	−	0.624056i	$-0.785484\pi$
$769$		−	34.6112i		−	1.24811i	0.781380	−	0.624056i	$-0.214516\pi$
$770$		0			0
$771$	1.44525	+	21.3434i	0.0520494	+	0.768664i
$772$		0			0
$773$		−	14.7718i		−	0.531305i	−0.964069	−	0.265653i	$-0.914413\pi$
$773$		−	14.7718i		−	0.531305i	0.964069	−	0.265653i	$-0.0855874\pi$
$774$		0			0
$775$	−5.85944			−0.210477
$776$		0			0
$777$	0.468915	+	6.92492i	0.0168222	+	0.248430i
$778$		0			0
$779$			28.0620i			1.00543i
$780$		0			0
$781$	8.38228	−	8.23035i	0.299942	−	0.294505i
$782$		0			0
$783$	−25.3877	+	5.22125i	−0.907282	+	0.186592i
$784$		0			0
$785$		−	1.01188i		−	0.0361155i
$786$		0			0
$787$		−	43.3530i		−	1.54537i	−0.634791	−	0.772684i	$-0.718914\pi$
$787$		−	43.3530i		−	1.54537i	0.634791	−	0.772684i	$-0.281086\pi$
$788$		0			0
$789$	19.6443	−	1.33020i	0.699356	−	0.0473563i
$790$		0			0
$791$	0.129619			0.00460873
$792$		0			0
$793$	10.8350			0.384761
$794$		0			0
$795$	−4.17040	+	0.282395i	−0.147909	+	0.0100155i
$796$		0			0
$797$			44.9405i			1.59187i	0.605380	+	0.795937i	$0.293021\pi$
$797$			44.9405i			1.59187i	−0.605380	+	0.795937i	$0.706979\pi$
$798$		0			0
$799$			19.9953i			0.707383i
$800$		0			0
$801$	−0.850724	−	6.25293i	−0.0300589	−	0.220937i
$802$		0			0
$803$	10.7951	−	10.5995i	0.380952	−	0.374048i
$804$		0			0
$805$		−	6.44431i		−	0.227132i
$806$		0			0
$807$	1.32892	+	19.6254i	0.0467801	+	0.690847i
$808$		0			0
$809$	35.2435			1.23909			0.619547	−	0.784960i	$-0.287316\pi$
$809$	35.2435			1.23909			0.619547	+	0.784960i	$0.287316\pi$
$810$		0			0
$811$		−	5.13034i		−	0.180151i	−0.995935	−	0.0900753i	$-0.971289\pi$
$811$		−	5.13034i		−	0.180151i	0.995935	−	0.0900753i	$-0.0287108\pi$
$812$		0			0
$813$	0.476654	+	7.03921i	0.0167170	+	0.246876i
$814$		0			0
$815$			0.543813i			0.0190489i
$816$		0			0
$817$	−82.9824			−2.90319
$818$		0			0
$819$	8.98299	−	1.22215i	0.313891	−	0.0427055i
$820$		0			0
$821$	−34.9821			−1.22088			−0.610442	−	0.792061i	$-0.709008\pi$
$821$	−34.9821			−1.22088			−0.610442	+	0.792061i	$0.709008\pi$
$822$		0			0
$823$	−25.2354			−0.879652			−0.439826	−	0.898083i	$-0.644960\pi$
$823$	−25.2354			−0.879652			−0.439826	+	0.898083i	$0.644960\pi$
$824$		0			0
$825$	−4.29243	−	3.81772i	−0.149443	−	0.132916i
$826$		0			0
$827$	−22.1953			−0.771807			−0.385904	−	0.922539i	$-0.626110\pi$
$827$	−22.1953			−0.771807			−0.385904	+	0.922539i	$0.626110\pi$
$828$		0			0
$829$	28.6988			0.996752			0.498376	−	0.866961i	$-0.333930\pi$
$829$	28.6988			0.996752			0.498376	+	0.866961i	$0.333930\pi$
$830$		0			0
$831$	−0.902336	−	13.3257i	−0.0313017	−	0.462262i
$832$		0			0
$833$	−29.1413			−1.00969
$834$		0			0
$835$		−	1.34942i		−	0.0466988i
$836$		0			0
$837$	29.8224	−	6.13330i	1.03081	−	0.211998i
$838$		0			0
$839$		−	41.5915i		−	1.43590i	−0.696096	−	0.717949i	$-0.745081\pi$
$839$		−	41.5915i		−	1.43590i	0.696096	−	0.717949i	$-0.254919\pi$
$840$		0			0
$841$	−4.11864			−0.142022
$842$		0			0
$843$	42.1285	−	2.85269i	1.45098	−	0.0982520i
$844$		0			0
$845$			4.46623i			0.153643i
$846$		0			0
$847$	7.95249	+	0.145467i	0.273251	+	0.00499829i
$848$		0			0
$849$	−2.28970	−	33.8143i	−0.0785824	−	1.16050i
$850$		0			0
$851$		−	49.3921i		−	1.69314i
$852$		0			0
$853$			30.0194i			1.02784i	0.857837	+	0.513922i	$0.171808\pi$
$853$			30.0194i			1.02784i	−0.857837	+	0.513922i	$0.828192\pi$
$854$		0			0
$855$	23.6181	−	3.21329i	0.807722	−	0.109892i
$856$		0			0
$857$	19.5089			0.666413			0.333206	−	0.942854i	$-0.391869\pi$
$857$	19.5089			0.666413			0.333206	+	0.942854i	$0.391869\pi$
$858$		0			0
$859$	−27.7655			−0.947348			−0.473674	−	0.880700i	$-0.657072\pi$
$859$	−27.7655			−0.947348			−0.473674	+	0.880700i	$0.657072\pi$
$860$		0			0
$861$	−0.298842	−	4.41329i	−0.0101845	−	0.150405i
$862$		0			0
$863$			20.0383i			0.682111i	0.940043	+	0.341055i	$0.110784\pi$
$863$			20.0383i			0.682111i	−0.940043	+	0.341055i	$0.889216\pi$
$864$		0			0
$865$		−	11.8247i		−	0.402053i
$866$		0			0
$867$	5.60200	−	0.379334i	0.190254	−	0.0128829i
$868$		0			0
$869$	26.4520	−	25.9726i	0.897323	−	0.881059i
$870$		0			0
$871$			38.6295i			1.30891i
$872$		0			0
$873$	24.5917	−	3.34574i	0.832301	−	0.113236i
$874$		0			0
$875$	0.723074			0.0244444
$876$		0			0
$877$			37.0898i			1.25243i	0.779650	+	0.626216i	$0.215397\pi$
$877$			37.0898i			1.25243i	−0.779650	+	0.626216i	$0.784603\pi$
$878$		0			0
$879$	23.9693	−	1.62306i	0.808465	−	0.0547445i
$880$		0			0
$881$		−	20.1796i		−	0.679868i	−0.940449	−	0.339934i	$-0.889595\pi$
$881$		−	20.1796i		−	0.679868i	0.940449	−	0.339934i	$-0.110405\pi$
$882$		0			0
$883$	54.6110			1.83781			0.918904	−	0.394481i	$-0.129076\pi$
$883$	54.6110			1.83781			0.918904	+	0.394481i	$0.129076\pi$
$884$		0			0
$885$	−16.7819	+	1.13637i	−0.564116	+	0.0381986i
$886$		0			0
$887$	−37.7371			−1.26709			−0.633544	−	0.773707i	$-0.718400\pi$
$887$	−37.7371			−1.26709			−0.633544	+	0.773707i	$0.718400\pi$
$888$		0			0
$889$	4.95065			0.166039
$890$		0			0
$891$	25.8431	+	14.9378i	0.865775	+	0.500434i
$892$		0			0
$893$	35.3111			1.18164
$894$		0			0
$895$	16.2650			0.543681
$896$		0			0
$897$	−64.3665	+	4.35852i	−2.14914	+	0.145527i
$898$		0			0
$899$	−29.2276			−0.974795
$900$		0			0
$901$			10.8576i			0.361719i
$902$		0			0
$903$	13.0506	−	0.883708i	0.434296	−	0.0294080i
$904$		0			0
$905$			16.1212i			0.535888i
$906$		0			0
$907$	−0.352164			−0.0116934			−0.00584671	−	0.999983i	$-0.501861\pi$
$907$	−0.352164			−0.0116934			−0.00584671	+	0.999983i	$0.501861\pi$
$908$		0			0
$909$	−5.15734	+	0.701667i	−0.171058	+	0.0232728i
$910$		0			0
$911$			42.7252i			1.41555i	0.706438	+	0.707775i	$0.250301\pi$
$911$			42.7252i			1.41555i	−0.706438	+	0.707775i	$0.749699\pi$
$912$		0			0
$913$	−35.7390	−	36.3987i	−1.18279	−	1.20462i
$914$		0			0
$915$	−4.48018	+	0.303371i	−0.148110	+	0.0100291i
$916$		0			0
$917$		−	8.26650i		−	0.272984i
$918$		0			0
$919$			7.92660i			0.261474i	0.991417	+	0.130737i	$0.0417344\pi$
$919$			7.92660i			0.261474i	−0.991417	+	0.130737i	$0.958266\pi$
$920$		0			0
$921$	2.28000	+	33.6709i	0.0751284	+	1.10949i
$922$		0			0
$923$	14.8028			0.487241
$924$		0			0
$925$	5.54197			0.182219
$926$		0			0
$927$	−25.1017	+	3.41513i	−0.824447	+	0.112168i
$928$		0			0
$929$		−	12.1273i		−	0.397883i	−0.980011	−	0.198941i	$-0.936250\pi$
$929$		−	12.1273i		−	0.397883i	0.980011	−	0.198941i	$-0.0637503\pi$
$930$		0			0
$931$			51.4625i			1.68662i
$932$		0			0
$933$	0.862457	+	12.7367i	0.0282356	+	0.416982i
$934$		0			0
$935$	−10.6473	+	10.4543i	−0.348205	+	0.341894i
$936$		0			0
$937$		−	24.1709i		−	0.789628i	−0.918761	−	0.394814i	$-0.870809\pi$
$937$		−	24.1709i		−	0.789628i	0.918761	−	0.394814i	$-0.129191\pi$
$938$		0			0
$939$	15.2121	−	1.03007i	0.496429	−	0.0336152i
$940$		0			0
$941$	−7.35032			−0.239614			−0.119807	−	0.992797i	$-0.538228\pi$
$941$	−7.35032			−0.239614			−0.119807	+	0.992797i	$0.538228\pi$
$942$		0			0
$943$			31.4779i			1.02506i
$944$		0			0
$945$	−3.68018	+	0.756869i	−0.119716	+	0.0246210i
$946$		0			0
$947$			40.7166i			1.32311i	0.749896	+	0.661556i	$0.230104\pi$
$947$			40.7166i			1.32311i	−0.749896	+	0.661556i	$0.769896\pi$
$948$		0			0
$949$	19.0639			0.618839
$950$		0			0
$951$	2.54787	+	37.6269i	0.0826204	+	1.22014i
$952$		0			0
$953$	−44.4669			−1.44043			−0.720213	−	0.693753i	$-0.755956\pi$
$953$	−44.4669			−1.44043			−0.720213	+	0.693753i	$0.755956\pi$
$954$		0			0
$955$	−23.4082			−0.757472
$956$		0			0
$957$	−21.4112	−	19.0433i	−0.692125	−	0.615582i
$958$		0			0
$959$	0.0992841			0.00320605
$960$		0			0
$961$	3.33307			0.107518
$962$		0			0
$963$	42.4870	−	5.78045i	1.36913	−	0.186272i
$964$		0			0
$965$	−9.28094			−0.298764
$966$		0			0
$967$		−	21.7348i		−	0.698944i	−0.936947	−	0.349472i	$-0.886361\pi$
$967$		−	21.7348i		−	0.698944i	0.936947	−	0.349472i	$-0.113639\pi$
$968$		0			0
$969$	−4.18289	−	61.7728i	−0.134374	−	1.98443i
$970$		0			0
$971$		−	45.2511i		−	1.45218i	−0.687600	−	0.726089i	$-0.741336\pi$
$971$		−	45.2511i		−	1.45218i	0.687600	−	0.726089i	$-0.258664\pi$
$972$		0			0
$973$	5.90865			0.189423
$974$		0			0
$975$	−0.489042	−	7.22215i	−0.0156619	−	0.231294i
$976$		0			0
$977$		−	7.03610i		−	0.225105i	−0.993646	−	0.112552i	$-0.964097\pi$
$977$		−	7.03610i		−	0.225105i	0.993646	−	0.112552i	$-0.0359026\pi$
$978$		0			0
$979$	4.97808	−	4.88786i	0.159100	−	0.156217i
$980$		0			0
$981$	−5.86255	−	43.0905i	−0.187177	−	1.37577i
$982$		0			0
$983$			28.6360i			0.913345i	0.889635	+	0.456673i	$0.150959\pi$
$983$			28.6360i			0.913345i	−0.889635	+	0.456673i	$0.849041\pi$
$984$		0			0
$985$		−	10.8467i		−	0.345603i
$986$		0			0
$987$	−5.55334	+	0.376040i	−0.176765	+	0.0119695i
$988$		0			0
$989$	−93.0836			−2.95989
$990$		0			0
$991$	57.4187			1.82397			0.911983	−	0.410229i	$-0.134551\pi$
$991$	57.4187			1.82397			0.911983	+	0.410229i	$0.134551\pi$
$992$		0			0
$993$	−4.22399	+	0.286024i	−0.134044	+	0.00907670i
$994$		0			0
$995$			10.3275i			0.327404i
$996$		0			0
$997$		−	7.41470i		−	0.234826i	−0.993083	−	0.117413i	$-0.962540\pi$
$997$		−	7.41470i		−	0.234826i	0.993083	−	0.117413i	$-0.0374601\pi$
$998$		0			0
$999$	−28.2066	+	5.80099i	−0.892417	+	0.183535i

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.b.1121.7		8
3.2	odd	2		2640.2.f.a.1121.8		8
4.3	odd	2		330.2.d.a.131.2	yes	8
11.10	odd	2		2640.2.f.a.1121.7		8
12.11	even	2		330.2.d.b.131.1	yes	8
20.3	even	4		1650.2.f.c.1649.7		8
20.7	even	4		1650.2.f.e.1649.2		8
20.19	odd	2		1650.2.d.f.1451.7		8
33.32	even	2	inner	2640.2.f.b.1121.8		8
44.43	even	2		330.2.d.b.131.2	yes	8
60.23	odd	4		1650.2.f.f.1649.2		8
60.47	odd	4		1650.2.f.d.1649.7		8
60.59	even	2		1650.2.d.c.1451.8		8
132.131	odd	2		330.2.d.a.131.1	✓	8
220.43	odd	4		1650.2.f.d.1649.3		8
220.87	odd	4		1650.2.f.f.1649.6		8
220.219	even	2		1650.2.d.c.1451.7		8
660.263	even	4		1650.2.f.e.1649.6		8
660.527	even	4		1650.2.f.c.1649.3		8
660.659	odd	2		1650.2.d.f.1451.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
330.2.d.a.131.1	✓	8	132.131	odd	2
330.2.d.a.131.2	yes	8	4.3	odd	2
330.2.d.b.131.1	yes	8	12.11	even	2
330.2.d.b.131.2	yes	8	44.43	even	2
1650.2.d.c.1451.7		8	220.219	even	2
1650.2.d.c.1451.8		8	60.59	even	2
1650.2.d.f.1451.7		8	20.19	odd	2
1650.2.d.f.1451.8		8	660.659	odd	2
1650.2.f.c.1649.3		8	660.527	even	4
1650.2.f.c.1649.7		8	20.3	even	4
1650.2.f.d.1649.3		8	220.43	odd	4
1650.2.f.d.1649.7		8	60.47	odd	4
1650.2.f.e.1649.2		8	20.7	even	4
1650.2.f.e.1649.6		8	660.263	even	4
1650.2.f.f.1649.2		8	60.23	odd	4
1650.2.f.f.1649.6		8	220.87	odd	4
2640.2.f.a.1121.7		8	11.10	odd	2
2640.2.f.a.1121.8		8	3.2	odd	2
2640.2.f.b.1121.7		8	1.1	even	1	trivial
2640.2.f.b.1121.8		8	33.32	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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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.b.1121.7

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