LMFDB - Embedded newform 920.2.e.b.369.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} + \cdots + 8 $ x^14 - 2x^11 + 39x^10 - 10x^9 + 2x^8 - 26x^7 + 297x^6 - 116x^5 + 24x^4 - 20x^3 + 64x^2 - 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.5
Root			$0.416087 - 0.416087i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.b.369.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40334i q^{3} +(0.466981 - 2.18676i) q^{5} -1.57117i q^{7} +1.03063 q^{9} +O(q^{10})$	$q-1.40334i q^{3} +(0.466981 - 2.18676i) q^{5} -1.57117i q^{7} +1.03063 q^{9} +4.35401 q^{11} +0.964590i q^{13} +(-3.06878 - 0.655335i) q^{15} +0.300242i q^{17} +8.62443 q^{19} -2.20489 q^{21} -1.00000i q^{23} +(-4.56386 - 2.04235i) q^{25} -5.65635i q^{27} -4.76644 q^{29} -5.59148 q^{31} -6.11017i q^{33} +(-3.43577 - 0.733706i) q^{35} +4.38462i q^{37} +1.35365 q^{39} -6.62014 q^{41} -1.72988i q^{43} +(0.481284 - 2.25374i) q^{45} +0.687333i q^{47} +4.53143 q^{49} +0.421342 q^{51} +8.05208i q^{53} +(2.03324 - 9.52118i) q^{55} -12.1030i q^{57} -5.74620 q^{59} -13.6547 q^{61} -1.61929i q^{63} +(2.10933 + 0.450445i) q^{65} -6.49053i q^{67} -1.40334 q^{69} +9.89977 q^{71} +6.35994i q^{73} +(-2.86612 + 6.40466i) q^{75} -6.84088i q^{77} +6.95266 q^{79} -4.84592 q^{81} -0.185320i q^{83} +(0.656557 + 0.140207i) q^{85} +6.68895i q^{87} +1.64878 q^{89} +1.51553 q^{91} +7.84676i q^{93} +(4.02744 - 18.8596i) q^{95} +9.21689i q^{97} +4.48736 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) $ 14 * q + 2 * q^5 - 4 * q^9	$ 14 q + 2 q^{5} - 4 q^{9} - 14 q^{11} - 6 q^{15} + 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} - 20 q^{31} - 2 q^{35} + 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} - 14 q^{51} - 38 q^{55} + 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} - 28 q^{71} - 24 q^{75} + 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} - 14 q^{91} - 30 q^{95} + 122 q^{99}+O(q^{100}) $ 14 * q + 2 * q^5 - 4 * q^9 - 14 * q^11 - 6 * q^15 + 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 - 20 * q^31 - 2 * q^35 + 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 - 14 * q^51 - 38 * q^55 + 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 - 28 * q^71 - 24 * q^75 + 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 - 14 * q^91 - 30 * q^95 + 122 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.40334i		−	0.810221i	−0.914268	−	0.405110i	$-0.867233\pi$
$3$		−	1.40334i		−	0.810221i	0.914268	−	0.405110i	$-0.132767\pi$
$4$		0			0
$5$	0.466981	−	2.18676i	0.208840	−	0.977950i
$5$	0.466981	−	2.18676i	0.208840	−	0.977950i
$6$		0			0
$7$		−	1.57117i		−	0.593846i	−0.954901	−	0.296923i	$-0.904040\pi$
$7$		−	1.57117i		−	0.593846i	0.954901	−	0.296923i	$-0.0959605\pi$
$8$		0			0
$9$	1.03063			0.343543
$10$		0			0
$11$	4.35401			1.31278			0.656391	−	0.754421i	$-0.272082\pi$
$11$	4.35401			1.31278			0.656391	+	0.754421i	$0.272082\pi$
$12$		0			0
$13$			0.964590i			0.267529i	0.991013	+	0.133765i	$0.0427066\pi$
$13$			0.964590i			0.267529i	−0.991013	+	0.133765i	$0.957293\pi$
$14$		0			0
$15$	−3.06878	−	0.655335i	−0.792355	−	0.169207i
$16$		0			0
$17$			0.300242i			0.0728193i	0.999337	+	0.0364096i	$0.0115921\pi$
$17$			0.300242i			0.0728193i	−0.999337	+	0.0364096i	$0.988408\pi$
$18$		0			0
$19$	8.62443			1.97858			0.989290	−	0.145963i	$-0.0466280\pi$
$19$	8.62443			1.97858			0.989290	+	0.145963i	$0.0466280\pi$
$20$		0			0
$21$	−2.20489			−0.481146
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.56386	−	2.04235i	−0.912772	−	0.408471i
$26$		0			0
$27$		−	5.65635i		−	1.08857i
$28$		0			0
$29$	−4.76644			−0.885106			−0.442553	−	0.896742i	$-0.645927\pi$
$29$	−4.76644			−0.885106			−0.442553	+	0.896742i	$0.645927\pi$
$30$		0			0
$31$	−5.59148			−1.00426			−0.502129	−	0.864792i	$-0.667450\pi$
$31$	−5.59148			−1.00426			−0.502129	+	0.864792i	$0.667450\pi$
$32$		0			0
$33$		−	6.11017i		−	1.06364i
$34$		0			0
$35$	−3.43577	−	0.733706i	−0.580752	−	0.124019i
$36$		0			0
$37$			4.38462i			0.720827i	0.932793	+	0.360413i	$0.117364\pi$
$37$			4.38462i			0.720827i	−0.932793	+	0.360413i	$0.882636\pi$
$38$		0			0
$39$	1.35365			0.216758
$40$		0			0
$41$	−6.62014			−1.03389			−0.516946	−	0.856018i	$-0.672931\pi$
$41$	−6.62014			−1.03389			−0.516946	+	0.856018i	$0.672931\pi$
$42$		0			0
$43$		−	1.72988i		−	0.263804i	−0.991263	−	0.131902i	$-0.957892\pi$
$43$		−	1.72988i		−	0.263804i	0.991263	−	0.131902i	$-0.0421085\pi$
$44$		0			0
$45$	0.481284	−	2.25374i	0.0717455	−	0.335967i
$46$		0			0
$47$			0.687333i			0.100258i	0.998743	+	0.0501289i	$0.0159632\pi$
$47$			0.687333i			0.100258i	−0.998743	+	0.0501289i	$0.984037\pi$
$48$		0			0
$49$	4.53143			0.647347
$50$		0			0
$51$	0.421342			0.0589997
$52$		0			0
$53$			8.05208i			1.10604i	0.833168	+	0.553019i	$0.186524\pi$
$53$			8.05208i			1.10604i	−0.833168	+	0.553019i	$0.813476\pi$
$54$		0			0
$55$	2.03324	−	9.52118i	0.274162	−	1.28384i
$56$		0			0
$57$		−	12.1030i		−	1.60309i
$58$		0			0
$59$	−5.74620			−0.748091			−0.374046	−	0.927410i	$-0.622030\pi$
$59$	−5.74620			−0.748091			−0.374046	+	0.927410i	$0.622030\pi$
$60$		0			0
$61$	−13.6547			−1.74831			−0.874153	−	0.485651i	$-0.838583\pi$
$61$	−13.6547			−1.74831			−0.874153	+	0.485651i	$0.838583\pi$
$62$		0			0
$63$		−	1.61929i		−	0.204011i
$64$		0			0
$65$	2.10933	+	0.450445i	0.261630	+	0.0558708i
$66$		0			0
$67$		−	6.49053i		−	0.792945i	−0.918047	−	0.396472i	$-0.870234\pi$
$67$		−	6.49053i		−	0.792945i	0.918047	−	0.396472i	$-0.129766\pi$
$68$		0			0
$69$	−1.40334			−0.168943
$70$		0			0
$71$	9.89977			1.17489			0.587443	−	0.809265i	$-0.300134\pi$
$71$	9.89977			1.17489			0.587443	+	0.809265i	$0.300134\pi$
$72$		0			0
$73$			6.35994i			0.744375i	0.928158	+	0.372187i	$0.121392\pi$
$73$			6.35994i			0.744375i	−0.928158	+	0.372187i	$0.878608\pi$
$74$		0			0
$75$	−2.86612	+	6.40466i	−0.330951	+	0.739546i
$76$		0			0
$77$		−	6.84088i		−	0.779591i
$78$		0			0
$79$	6.95266			0.782236			0.391118	−	0.920341i	$-0.372088\pi$
$79$	6.95266			0.782236			0.391118	+	0.920341i	$0.372088\pi$
$80$		0			0
$81$	−4.84592			−0.538436
$82$		0			0
$83$		−	0.185320i		−	0.0203415i	−0.999948	−	0.0101707i	$-0.996762\pi$
$83$		−	0.185320i		−	0.0203415i	0.999948	−	0.0101707i	$-0.00323750\pi$
$84$		0			0
$85$	0.656557	+	0.140207i	0.0712136	+	0.0152076i
$86$		0			0
$87$			6.68895i			0.717131i
$88$		0			0
$89$	1.64878			0.174770			0.0873852	−	0.996175i	$-0.472149\pi$
$89$	1.64878			0.174770			0.0873852	+	0.996175i	$0.472149\pi$
$90$		0			0
$91$	1.51553			0.158871
$92$		0			0
$93$			7.84676i			0.813671i
$94$		0			0
$95$	4.02744	−	18.8596i	0.413207	−	1.93495i
$96$		0			0
$97$			9.21689i			0.935834i	0.883773	+	0.467917i	$0.154995\pi$
$97$			9.21689i			0.935834i	−0.883773	+	0.467917i	$0.845005\pi$
$98$		0			0
$99$	4.48736			0.450997
$100$		0			0
$101$	−0.719780			−0.0716208			−0.0358104	−	0.999359i	$-0.511401\pi$
$101$	−0.719780			−0.0716208			−0.0358104	+	0.999359i	$0.511401\pi$
$102$		0			0
$103$			9.16137i			0.902696i	0.892348	+	0.451348i	$0.149057\pi$
$103$			9.16137i			0.902696i	−0.892348	+	0.451348i	$0.850943\pi$
$104$		0			0
$105$	−1.02964	+	4.82157i	−0.100483	+	0.470537i
$106$		0			0
$107$			12.7486i			1.23245i	0.787570	+	0.616226i	$0.211339\pi$
$107$			12.7486i			1.23245i	−0.787570	+	0.616226i	$0.788661\pi$
$108$		0			0
$109$	3.38019			0.323763			0.161882	−	0.986810i	$-0.448244\pi$
$109$	3.38019			0.323763			0.161882	+	0.986810i	$0.448244\pi$
$110$		0			0
$111$	6.15312			0.584029
$112$		0			0
$113$		−	15.8547i		−	1.49148i	−0.666237	−	0.745741i	$-0.732096\pi$
$113$		−	15.8547i		−	1.49148i	0.666237	−	0.745741i	$-0.267904\pi$
$114$		0			0
$115$	−2.18676	−	0.466981i	−0.203917	−	0.0435462i
$116$		0			0
$117$			0.994133i			0.0919076i
$118$		0			0
$119$	0.471730			0.0432434
$120$		0			0
$121$	7.95739			0.723399
$122$		0			0
$123$			9.29032i			0.837680i
$124$		0			0
$125$	−6.59737	+	9.02633i	−0.590087	+	0.807340i
$126$		0			0
$127$		−	15.2222i		−	1.35075i	−0.737473	−	0.675377i	$-0.763981\pi$
$127$		−	15.2222i		−	1.35075i	0.737473	−	0.675377i	$-0.236019\pi$
$128$		0			0
$129$	−2.42762			−0.213740
$130$		0			0
$131$	−12.8695			−1.12441			−0.562207	−	0.826997i	$-0.690047\pi$
$131$	−12.8695			−1.12441			−0.562207	+	0.826997i	$0.690047\pi$
$132$		0			0
$133$		−	13.5504i		−	1.17497i
$134$		0			0
$135$	−12.3691	−	2.64141i	−1.06456	−	0.227336i
$136$		0			0
$137$		−	16.7433i		−	1.43048i	−0.698881	−	0.715238i	$-0.746318\pi$
$137$		−	16.7433i		−	1.43048i	0.698881	−	0.715238i	$-0.253682\pi$
$138$		0			0
$139$	11.8293			1.00335			0.501676	−	0.865056i	$-0.332717\pi$
$139$	11.8293			1.00335			0.501676	+	0.865056i	$0.332717\pi$
$140$		0			0
$141$	0.964565			0.0812310
$142$		0			0
$143$			4.19983i			0.351208i
$144$		0			0
$145$	−2.22584	+	10.4231i	−0.184846	+	0.865589i
$146$		0			0
$147$		−	6.35915i		−	0.524494i
$148$		0			0
$149$	−1.46021			−0.119625			−0.0598125	−	0.998210i	$-0.519050\pi$
$149$	−1.46021			−0.119625			−0.0598125	+	0.998210i	$0.519050\pi$
$150$		0			0
$151$	17.7897			1.44771			0.723854	−	0.689953i	$-0.242369\pi$
$151$	17.7897			1.44771			0.723854	+	0.689953i	$0.242369\pi$
$152$		0			0
$153$			0.309437i			0.0250165i
$154$		0			0
$155$	−2.61111	+	12.2272i	−0.209730	+	0.982115i
$156$		0			0
$157$			20.0359i			1.59904i	0.600642	+	0.799518i	$0.294912\pi$
$157$			20.0359i			1.59904i	−0.600642	+	0.799518i	$0.705088\pi$
$158$		0			0
$159$	11.2998			0.896135
$160$		0			0
$161$	−1.57117			−0.123825
$162$		0			0
$163$			22.9076i			1.79426i	0.441766	+	0.897130i	$0.354352\pi$
$163$			22.9076i			1.79426i	−0.441766	+	0.897130i	$0.645648\pi$
$164$		0			0
$165$	−13.3615	−	2.85333i	−1.04019	−	0.222132i
$166$		0			0
$167$			0.826887i			0.0639864i	0.999488	+	0.0319932i	$0.0101855\pi$
$167$			0.826887i			0.0639864i	−0.999488	+	0.0319932i	$0.989815\pi$
$168$		0			0
$169$	12.0696			0.928428
$170$		0			0
$171$	8.88858			0.679727
$172$		0			0
$173$		−	8.03904i		−	0.611197i	−0.952160	−	0.305599i	$-0.901143\pi$
$173$		−	8.03904i		−	0.611197i	0.952160	−	0.305599i	$-0.0988565\pi$
$174$		0			0
$175$	−3.20888	+	7.17059i	−0.242569	+	0.542046i
$176$		0			0
$177$			8.06389i			0.606119i
$178$		0			0
$179$	21.6941			1.62149			0.810747	−	0.585397i	$-0.199061\pi$
$179$	21.6941			1.62149			0.810747	+	0.585397i	$0.199061\pi$
$180$		0			0
$181$	−14.9499			−1.11122			−0.555609	−	0.831444i	$-0.687515\pi$
$181$	−14.9499			−1.11122			−0.555609	+	0.831444i	$0.687515\pi$
$182$		0			0
$183$			19.1622i			1.41651i
$184$		0			0
$185$	9.58812	+	2.04753i	0.704932	+	0.150538i
$186$		0			0
$187$			1.30725i			0.0955959i
$188$		0			0
$189$	−8.88709			−0.646441
$190$		0			0
$191$	−7.86406			−0.569024			−0.284512	−	0.958673i	$-0.591831\pi$
$191$	−7.86406			−0.569024			−0.284512	+	0.958673i	$0.591831\pi$
$192$		0			0
$193$			15.1864i			1.09314i	0.837413	+	0.546571i	$0.184067\pi$
$193$			15.1864i			1.09314i	−0.837413	+	0.546571i	$0.815933\pi$
$194$		0			0
$195$	0.632129	−	2.96011i	0.0452677	−	0.211978i
$196$		0			0
$197$			12.2911i			0.875707i	0.899046	+	0.437853i	$0.144261\pi$
$197$			12.2911i			0.875707i	−0.899046	+	0.437853i	$0.855739\pi$
$198$		0			0
$199$	4.37123			0.309869			0.154934	−	0.987925i	$-0.450483\pi$
$199$	4.37123			0.309869			0.154934	+	0.987925i	$0.450483\pi$
$200$		0			0
$201$	−9.10845			−0.642460
$202$		0			0
$203$			7.48888i			0.525617i
$204$		0			0
$205$	−3.09148	+	14.4767i	−0.215918	+	1.01109i
$206$		0			0
$207$		−	1.03063i		−	0.0716336i
$208$		0			0
$209$	37.5508			2.59745
$210$		0			0
$211$	−8.93960			−0.615427			−0.307714	−	0.951479i	$-0.599564\pi$
$211$	−8.93960			−0.615427			−0.307714	+	0.951479i	$0.599564\pi$
$212$		0			0
$213$		−	13.8928i		−	0.951917i
$214$		0			0
$215$	−3.78284	−	0.807822i	−0.257987	−	0.0550930i
$216$		0			0
$217$			8.78516i			0.596375i
$218$		0			0
$219$	8.92518			0.603108
$220$		0			0
$221$	−0.289610			−0.0194813
$222$		0			0
$223$			11.9586i			0.800806i	0.916339	+	0.400403i	$0.131130\pi$
$223$			11.9586i			0.800806i	−0.916339	+	0.400403i	$0.868870\pi$
$224$		0			0
$225$	−4.70364	−	2.10491i	−0.313576	−	0.140327i
$226$		0			0
$227$			1.85445i			0.123084i	0.998104	+	0.0615419i	$0.0196018\pi$
$227$			1.85445i			0.123084i	−0.998104	+	0.0615419i	$0.980398\pi$
$228$		0			0
$229$	16.5190			1.09161			0.545804	−	0.837913i	$-0.316224\pi$
$229$	16.5190			1.09161			0.545804	+	0.837913i	$0.316224\pi$
$230$		0			0
$231$	−9.60011			−0.631641
$232$		0			0
$233$		−	24.5906i		−	1.61099i	−0.592606	−	0.805493i	$-0.701901\pi$
$233$		−	24.5906i		−	1.61099i	0.592606	−	0.805493i	$-0.298099\pi$
$234$		0			0
$235$	1.50303	+	0.320972i	0.0980472	+	0.0209379i
$236$		0			0
$237$		−	9.75697i		−	0.633783i
$238$		0			0
$239$	−27.1169			−1.75405			−0.877025	−	0.480446i	$-0.840475\pi$
$239$	−27.1169			−1.75405			−0.877025	+	0.480446i	$0.840475\pi$
$240$		0			0
$241$	−12.8855			−0.830030			−0.415015	−	0.909815i	$-0.636224\pi$
$241$	−12.8855			−0.830030			−0.415015	+	0.909815i	$0.636224\pi$
$242$		0			0
$243$		−	10.1686i		−	0.652314i
$244$		0			0
$245$	2.11609	−	9.90915i	0.135192	−	0.633073i
$246$		0			0
$247$			8.31904i			0.529328i
$248$		0			0
$249$	−0.260067			−0.0164811
$250$		0			0
$251$	−12.4610			−0.786533			−0.393267	−	0.919424i	$-0.628655\pi$
$251$	−12.4610			−0.786533			−0.393267	+	0.919424i	$0.628655\pi$
$252$		0			0
$253$		−	4.35401i		−	0.273734i
$254$		0			0
$255$	0.196759	−	0.921374i	0.0123215	−	0.0576987i
$256$		0			0
$257$			13.5073i			0.842564i	0.906930	+	0.421282i	$0.138420\pi$
$257$			13.5073i			0.842564i	−0.906930	+	0.421282i	$0.861580\pi$
$258$		0			0
$259$	6.88898			0.428060
$260$		0			0
$261$	−4.91243			−0.304072
$262$		0			0
$263$		−	27.2579i		−	1.68080i	−0.541969	−	0.840399i	$-0.682321\pi$
$263$		−	27.2579i		−	1.68080i	0.541969	−	0.840399i	$-0.317679\pi$
$264$		0			0
$265$	17.6080	+	3.76017i	1.08165	+	0.230985i
$266$		0			0
$267$		−	2.31380i		−	0.141603i
$268$		0			0
$269$	17.8409			1.08778			0.543889	−	0.839157i	$-0.316951\pi$
$269$	17.8409			1.08778			0.543889	+	0.839157i	$0.316951\pi$
$270$		0			0
$271$	2.31906			0.140873			0.0704364	−	0.997516i	$-0.477561\pi$
$271$	2.31906			0.140873			0.0704364	+	0.997516i	$0.477561\pi$
$272$		0			0
$273$		−	2.12681i		−	0.128721i
$274$		0			0
$275$	−19.8711	−	8.89242i	−1.19827	−	0.536233i
$276$		0			0
$277$			32.0762i			1.92727i	0.267215	+	0.963637i	$0.413897\pi$
$277$			32.0762i			1.92727i	−0.267215	+	0.963637i	$0.586103\pi$
$278$		0			0
$279$	−5.76273			−0.345006
$280$		0			0
$281$	12.9448			0.772221			0.386110	−	0.922453i	$-0.373818\pi$
$281$	12.9448			0.772221			0.386110	+	0.922453i	$0.373818\pi$
$282$		0			0
$283$			22.6070i			1.34385i	0.740621	+	0.671923i	$0.234532\pi$
$283$			22.6070i			1.34385i	−0.740621	+	0.671923i	$0.765468\pi$
$284$		0			0
$285$	−26.4665	−	5.65189i	−1.56774	−	0.334789i
$286$		0			0
$287$			10.4014i			0.613972i
$288$		0			0
$289$	16.9099			0.994697
$290$		0			0
$291$	12.9345			0.758232
$292$		0			0
$293$		−	5.61612i		−	0.328097i	−0.986452	−	0.164049i	$-0.947545\pi$
$293$		−	5.61612i		−	0.328097i	0.986452	−	0.164049i	$-0.0524554\pi$
$294$		0			0
$295$	−2.68336	+	12.5656i	−0.156232	+	0.731596i
$296$		0			0
$297$		−	24.6278i		−	1.42905i
$298$		0			0
$299$	0.964590			0.0557837
$300$		0			0
$301$	−2.71794			−0.156659
$302$		0			0
$303$			1.01010i			0.0580286i
$304$		0			0
$305$	−6.37649	+	29.8596i	−0.365117	+	1.70976i
$306$		0			0
$307$			2.42467i			0.138383i	0.997603	+	0.0691915i	$0.0220419\pi$
$307$			2.42467i			0.138383i	−0.997603	+	0.0691915i	$0.977958\pi$
$308$		0			0
$309$	12.8565			0.731383
$310$		0			0
$311$	3.57886			0.202938			0.101469	−	0.994839i	$-0.467646\pi$
$311$	3.57886			0.202938			0.101469	+	0.994839i	$0.467646\pi$
$312$		0			0
$313$			5.51000i			0.311443i	0.987801	+	0.155722i	$0.0497703\pi$
$313$			5.51000i			0.311443i	−0.987801	+	0.155722i	$0.950230\pi$
$314$		0			0
$315$	−3.54100	−	0.756178i	−0.199513	−	0.0426058i
$316$		0			0
$317$			5.88368i			0.330461i	0.986255	+	0.165230i	$0.0528367\pi$
$317$			5.88368i			0.330461i	−0.986255	+	0.165230i	$0.947163\pi$
$318$		0			0
$319$	−20.7531			−1.16195
$320$		0			0
$321$	17.8906			0.998558
$322$		0			0
$323$			2.58941i			0.144079i
$324$		0			0
$325$	1.97003	−	4.40225i	0.109278	−	0.244193i
$326$		0			0
$327$		−	4.74356i		−	0.262320i
$328$		0			0
$329$	1.07992			0.0595377
$330$		0			0
$331$	−18.6138			−1.02311			−0.511553	−	0.859252i	$-0.670930\pi$
$331$	−18.6138			−1.02311			−0.511553	+	0.859252i	$0.670930\pi$
$332$		0			0
$333$			4.51891i			0.247635i
$334$		0			0
$335$	−14.1933	−	3.03096i	−0.775460	−	0.165599i
$336$		0			0
$337$			29.9571i			1.63187i	0.578146	+	0.815933i	$0.303776\pi$
$337$			29.9571i			1.63187i	−0.578146	+	0.815933i	$0.696224\pi$
$338$		0			0
$339$	−22.2495			−1.20843
$340$		0			0
$341$	−24.3453			−1.31837
$342$		0			0
$343$		−	18.1178i		−	0.978270i
$344$		0			0
$345$	−0.655335	+	3.06878i	−0.0352820	+	0.165217i
$346$		0			0
$347$		−	26.7363i		−	1.43528i	−0.696415	−	0.717640i	$-0.745222\pi$
$347$		−	26.7363i		−	1.43528i	0.696415	−	0.717640i	$-0.254778\pi$
$348$		0			0
$349$	−28.7392			−1.53837			−0.769186	−	0.639025i	$-0.779338\pi$
$349$	−28.7392			−1.53837			−0.769186	+	0.639025i	$0.779338\pi$
$350$		0			0
$351$	5.45606			0.291223
$352$		0			0
$353$			0.662874i			0.0352812i	0.999844	+	0.0176406i	$0.00561547\pi$
$353$			0.662874i			0.0352812i	−0.999844	+	0.0176406i	$0.994385\pi$
$354$		0			0
$355$	4.62300	−	21.6484i	0.245364	−	1.14898i
$356$		0			0
$357$		−	0.661999i		−	0.0350367i
$358$		0			0
$359$	16.0576			0.847485			0.423743	−	0.905783i	$-0.360716\pi$
$359$	16.0576			0.847485			0.423743	+	0.905783i	$0.360716\pi$
$360$		0			0
$361$	55.3808			2.91478
$362$		0			0
$363$		−	11.1669i		−	0.586113i
$364$		0			0
$365$	13.9077	+	2.96997i	0.727961	+	0.155455i
$366$		0			0
$367$		−	27.5855i		−	1.43995i	−0.693999	−	0.719976i	$-0.744153\pi$
$367$		−	27.5855i		−	1.43995i	0.693999	−	0.719976i	$-0.255847\pi$
$368$		0			0
$369$	−6.82290			−0.355186
$370$		0			0
$371$	12.6512			0.656817
$372$		0			0
$373$		−	33.1014i		−	1.71393i	−0.515376	−	0.856964i	$-0.672348\pi$
$373$		−	33.1014i		−	1.71393i	0.515376	−	0.856964i	$-0.327652\pi$
$374$		0			0
$375$	12.6670	+	9.25838i	0.654123	+	0.478101i
$376$		0			0
$377$		−	4.59766i		−	0.236791i
$378$		0			0
$379$	−2.17121			−0.111527			−0.0557637	−	0.998444i	$-0.517759\pi$
$379$	−2.17121			−0.111527			−0.0557637	+	0.998444i	$0.517759\pi$
$380$		0			0
$381$	−21.3620			−1.09441
$382$		0			0
$383$		−	5.52673i		−	0.282403i	−0.989981	−	0.141201i	$-0.954903\pi$
$383$		−	5.52673i		−	0.282403i	0.989981	−	0.141201i	$-0.0450965\pi$
$384$		0			0
$385$	−14.9594	−	3.19456i	−0.762401	−	0.162810i
$386$		0			0
$387$		−	1.78286i		−	0.0906281i
$388$		0			0
$389$	−17.1036			−0.867185			−0.433593	−	0.901109i	$-0.642754\pi$
$389$	−17.1036			−0.867185			−0.433593	+	0.901109i	$0.642754\pi$
$390$		0			0
$391$	0.300242			0.0151839
$392$		0			0
$393$			18.0603i			0.911023i
$394$		0			0
$395$	3.24676	−	15.2038i	0.163362	−	0.764987i
$396$		0			0
$397$		−	12.7419i		−	0.639497i	−0.947503	−	0.319748i	$-0.896402\pi$
$397$		−	12.7419i		−	0.639497i	0.947503	−	0.319748i	$-0.103598\pi$
$398$		0			0
$399$	−19.0159			−0.951987
$400$		0			0
$401$	14.1997			0.709098			0.354549	−	0.935038i	$-0.384634\pi$
$401$	14.1997			0.709098			0.354549	+	0.935038i	$0.384634\pi$
$402$		0			0
$403$		−	5.39348i		−	0.268668i
$404$		0			0
$405$	−2.26295	+	10.5969i	−0.112447	+	0.526563i
$406$		0			0
$407$			19.0907i			0.946289i
$408$		0			0
$409$	0.0235143			0.00116271			0.000581354	−	1.00000i	$-0.499815\pi$
$409$	0.0235143			0.00116271			0.000581354		1.00000i	$0.499815\pi$
$410$		0			0
$411$	−23.4966			−1.15900
$412$		0			0
$413$			9.02825i			0.444251i
$414$		0			0
$415$	−0.405250	−	0.0865407i	−0.0198929	−	0.00424812i
$416$		0			0
$417$		−	16.6006i		−	0.812936i
$418$		0			0
$419$	20.2763			0.990562			0.495281	−	0.868733i	$-0.335065\pi$
$419$	20.2763			0.990562			0.495281	+	0.868733i	$0.335065\pi$
$420$		0			0
$421$	12.5320			0.610771			0.305386	−	0.952229i	$-0.401215\pi$
$421$	12.5320			0.610771			0.305386	+	0.952229i	$0.401215\pi$
$422$		0			0
$423$			0.708385i			0.0344429i
$424$		0			0
$425$	0.613199	−	1.37026i	0.0297445	−	0.0664673i
$426$		0			0
$427$			21.4539i			1.03822i
$428$		0			0
$429$	5.89380			0.284556
$430$		0			0
$431$	−22.8635			−1.10130			−0.550648	−	0.834737i	$-0.685619\pi$
$431$	−22.8635			−1.10130			−0.550648	+	0.834737i	$0.685619\pi$
$432$		0			0
$433$		−	23.7997i		−	1.14374i	−0.820343	−	0.571871i	$-0.806218\pi$
$433$		−	23.7997i		−	1.14374i	0.820343	−	0.571871i	$-0.193782\pi$
$434$		0			0
$435$	14.6271	+	3.12361i	0.701318	+	0.149766i
$436$		0			0
$437$		−	8.62443i		−	0.412562i
$438$		0			0
$439$	−4.66686			−0.222737			−0.111368	−	0.993779i	$-0.535523\pi$
$439$	−4.66686			−0.222737			−0.111368	+	0.993779i	$0.535523\pi$
$440$		0			0
$441$	4.67022			0.222391
$442$		0			0
$443$		−	13.6616i		−	0.649082i	−0.945872	−	0.324541i	$-0.894790\pi$
$443$		−	13.6616i		−	0.649082i	0.945872	−	0.324541i	$-0.105210\pi$
$444$		0			0
$445$	0.769949	−	3.60549i	0.0364991	−	0.170917i
$446$		0			0
$447$			2.04917i			0.0969226i
$448$		0			0
$449$	25.8921			1.22192			0.610961	−	0.791661i	$-0.290783\pi$
$449$	25.8921			1.22192			0.610961	+	0.791661i	$0.290783\pi$
$450$		0			0
$451$	−28.8241			−1.35727
$452$		0			0
$453$		−	24.9651i		−	1.17296i
$454$		0			0
$455$	0.707725	−	3.31411i	0.0331787	−	0.155368i
$456$		0			0
$457$		−	12.6856i		−	0.593405i	−0.954970	−	0.296703i	$-0.904113\pi$
$457$		−	12.6856i		−	0.593405i	0.954970	−	0.296703i	$-0.0958870\pi$
$458$		0			0
$459$	1.69827			0.0792686
$460$		0			0
$461$	26.5769			1.23781			0.618904	−	0.785466i	$-0.287577\pi$
$461$	26.5769			1.23781			0.618904	+	0.785466i	$0.287577\pi$
$462$		0			0
$463$			14.6461i			0.680664i	0.940305	+	0.340332i	$0.110539\pi$
$463$			14.6461i			0.680664i	−0.940305	+	0.340332i	$0.889461\pi$
$464$		0			0
$465$	17.1590	+	3.66429i	0.795730	+	0.169927i
$466$		0			0
$467$		−	12.7601i		−	0.590467i	−0.955425	−	0.295233i	$-0.904603\pi$
$467$		−	12.7601i		−	0.590467i	0.955425	−	0.295233i	$-0.0953974\pi$
$468$		0			0
$469$	−10.1977			−0.470887
$470$		0			0
$471$	28.1172			1.29557
$472$		0			0
$473$		−	7.53192i		−	0.346318i
$474$		0			0
$475$	−39.3607	−	17.6141i	−1.80599	−	0.808192i
$476$		0			0
$477$			8.29870i			0.379971i
$478$		0			0
$479$	−10.2315			−0.467488			−0.233744	−	0.972298i	$-0.575098\pi$
$479$	−10.2315			−0.467488			−0.233744	+	0.972298i	$0.575098\pi$
$480$		0			0
$481$	−4.22936			−0.192842
$482$		0			0
$483$			2.20489i			0.100326i
$484$		0			0
$485$	20.1552	+	4.30411i	0.915198	+	0.195440i
$486$		0			0
$487$			21.8299i			0.989207i	0.869119	+	0.494603i	$0.164687\pi$
$487$			21.8299i			0.989207i	−0.869119	+	0.494603i	$0.835313\pi$
$488$		0			0
$489$	32.1472			1.45375
$490$		0			0
$491$	−38.1315			−1.72085			−0.860426	−	0.509575i	$-0.829802\pi$
$491$	−38.1315			−1.72085			−0.860426	+	0.509575i	$0.829802\pi$
$492$		0			0
$493$		−	1.43108i		−	0.0644527i
$494$		0			0
$495$	2.09551	−	9.81279i	0.0941863	−	0.441052i
$496$		0			0
$497$		−	15.5542i		−	0.697702i
$498$		0			0
$499$	−6.36523			−0.284947			−0.142474	−	0.989799i	$-0.545506\pi$
$499$	−6.36523			−0.284947			−0.142474	+	0.989799i	$0.545506\pi$
$500$		0			0
$501$	1.16041			0.0518431
$502$		0			0
$503$		−	21.3160i		−	0.950433i	−0.879869	−	0.475217i	$-0.842370\pi$
$503$		−	21.3160i		−	0.950433i	0.879869	−	0.475217i	$-0.157630\pi$
$504$		0			0
$505$	−0.336123	+	1.57399i	−0.0149573	+	0.0700415i
$506$		0			0
$507$		−	16.9377i		−	0.752232i
$508$		0			0
$509$	25.5122			1.13081			0.565405	−	0.824814i	$-0.308720\pi$
$509$	25.5122			1.13081			0.565405	+	0.824814i	$0.308720\pi$
$510$		0			0
$511$	9.99254			0.442044
$512$		0			0
$513$		−	48.7828i		−	2.15381i
$514$		0			0
$515$	20.0337	+	4.27818i	0.882792	+	0.188519i
$516$		0			0
$517$			2.99266i			0.131617i
$518$		0			0
$519$	−11.2815			−0.495205
$520$		0			0
$521$	−40.7247			−1.78418			−0.892091	−	0.451856i	$-0.850762\pi$
$521$	−40.7247			−1.78418			−0.892091	+	0.451856i	$0.850762\pi$
$522$		0			0
$523$			23.3437i			1.02075i	0.859953	+	0.510374i	$0.170493\pi$
$523$			23.3437i			1.02075i	−0.859953	+	0.510374i	$0.829507\pi$
$524$		0			0
$525$	10.0628	+	4.50316i	0.439177	+	0.196534i
$526$		0			0
$527$		−	1.67879i		−	0.0731294i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−5.92219			−0.257001
$532$		0			0
$533$		−	6.38571i		−	0.276596i
$534$		0			0
$535$	27.8781	+	5.95334i	1.20528	+	0.257385i
$536$		0			0
$537$		−	30.4443i		−	1.31377i
$538$		0			0
$539$	19.7299			0.849826
$540$		0			0
$541$	1.23303			0.0530123			0.0265061	−	0.999649i	$-0.491562\pi$
$541$	1.23303			0.0530123			0.0265061	+	0.999649i	$0.491562\pi$
$542$		0			0
$543$			20.9798i			0.900331i
$544$		0			0
$545$	1.57848	−	7.39166i	0.0676148	−	0.316624i
$546$		0			0
$547$			1.56470i			0.0669018i	0.999440	+	0.0334509i	$0.0106497\pi$
$547$			1.56470i			0.0669018i	−0.999440	+	0.0334509i	$0.989350\pi$
$548$		0			0
$549$	−14.0729			−0.600618
$550$		0			0
$551$	−41.1078			−1.75125
$552$		0			0
$553$		−	10.9238i		−	0.464528i
$554$		0			0
$555$	2.87339	−	13.4554i	0.121969	−	0.571151i
$556$		0			0
$557$		−	22.4857i		−	0.952749i	−0.879242	−	0.476375i	$-0.841951\pi$
$557$		−	22.4857i		−	0.952749i	0.879242	−	0.476375i	$-0.158049\pi$
$558$		0			0
$559$	1.66863			0.0705753
$560$		0			0
$561$	1.83453			0.0774537
$562$		0			0
$563$			7.68774i			0.324000i	0.986791	+	0.162000i	$0.0517944\pi$
$563$			7.68774i			0.324000i	−0.986791	+	0.162000i	$0.948206\pi$
$564$		0			0
$565$	−34.6704	−	7.40382i	−1.45859	−	0.311481i
$566$		0			0
$567$			7.61376i			0.319748i
$568$		0			0
$569$	−28.1828			−1.18149			−0.590743	−	0.806860i	$-0.701165\pi$
$569$	−28.1828			−1.18149			−0.590743	+	0.806860i	$0.701165\pi$
$570$		0			0
$571$	31.1444			1.30335			0.651676	−	0.758497i	$-0.274066\pi$
$571$	31.1444			1.30335			0.651676	+	0.758497i	$0.274066\pi$
$572$		0			0
$573$			11.0360i			0.461035i
$574$		0			0
$575$	−2.04235	+	4.56386i	−0.0851720	+	0.190326i
$576$		0			0
$577$			1.20837i			0.0503051i	0.999684	+	0.0251525i	$0.00800714\pi$
$577$			1.20837i			0.0503051i	−0.999684	+	0.0251525i	$0.991993\pi$
$578$		0			0
$579$	21.3118			0.885687
$580$		0			0
$581$	−0.291168			−0.0120797
$582$		0			0
$583$			35.0588i			1.45199i
$584$		0			0
$585$	2.17393	+	0.464241i	0.0898810	+	0.0191940i
$586$		0			0
$587$			13.4162i			0.553746i	0.960906	+	0.276873i	$0.0892982\pi$
$587$			13.4162i			0.553746i	−0.960906	+	0.276873i	$0.910702\pi$
$588$		0			0
$589$	−48.2233			−1.98701
$590$		0			0
$591$	17.2487			0.709516
$592$		0			0
$593$		−	5.37465i		−	0.220711i	−0.993892	−	0.110355i	$-0.964801\pi$
$593$		−	5.37465i		−	0.220711i	0.993892	−	0.110355i	$-0.0351989\pi$
$594$		0			0
$595$	0.220289	−	1.03156i	0.00903097	−	0.0422899i
$596$		0			0
$597$		−	6.13434i		−	0.251062i
$598$		0			0
$599$	18.7001			0.764064			0.382032	−	0.924149i	$-0.375224\pi$
$599$	18.7001			0.764064			0.382032	+	0.924149i	$0.375224\pi$
$600$		0			0
$601$	18.4418			0.752255			0.376128	−	0.926568i	$-0.377255\pi$
$601$	18.4418			0.752255			0.376128	+	0.926568i	$0.377255\pi$
$602$		0			0
$603$		−	6.68932i		−	0.272410i
$604$		0			0
$605$	3.71595	−	17.4009i	0.151075	−	0.707448i
$606$		0			0
$607$			15.6501i			0.635218i	0.948222	+	0.317609i	$0.102880\pi$
$607$			15.6501i			0.635218i	−0.948222	+	0.317609i	$0.897120\pi$
$608$		0			0
$609$	10.5095			0.425865
$610$		0			0
$611$	−0.662995			−0.0268219
$612$		0			0
$613$			2.18552i			0.0882725i	0.999026	+	0.0441363i	$0.0140536\pi$
$613$			2.18552i			0.0882725i	−0.999026	+	0.0441363i	$0.985946\pi$
$614$		0			0
$615$	20.3157	+	4.33840i	0.819209	+	0.174941i
$616$		0			0
$617$		−	4.07539i		−	0.164069i	−0.996629	−	0.0820346i	$-0.973858\pi$
$617$		−	4.07539i		−	0.164069i	0.996629	−	0.0820346i	$-0.0261418\pi$
$618$		0			0
$619$	0.0289543			0.00116377			0.000581885	−	1.00000i	$-0.499815\pi$
$619$	0.0289543			0.00116377			0.000581885		1.00000i	$0.499815\pi$
$620$		0			0
$621$	−5.65635			−0.226982
$622$		0			0
$623$		−	2.59051i		−	0.103787i
$624$		0			0
$625$	16.6576	+	18.6420i	0.666304	+	0.745681i
$626$		0			0
$627$		−	52.6967i		−	2.10450i
$628$		0			0
$629$	−1.31644			−0.0524901
$630$		0			0
$631$	33.3679			1.32836			0.664178	−	0.747575i	$-0.268782\pi$
$631$	33.3679			1.32836			0.664178	+	0.747575i	$0.268782\pi$
$632$		0			0
$633$			12.5453i			0.498632i
$634$		0			0
$635$	−33.2874	−	7.10849i	−1.32097	−	0.282092i
$636$		0			0
$637$			4.37097i			0.173184i
$638$		0			0
$639$	10.2030			0.403624
$640$		0			0
$641$	1.56295			0.0617327			0.0308663	−	0.999524i	$-0.490173\pi$
$641$	1.56295			0.0617327			0.0308663	+	0.999524i	$0.490173\pi$
$642$		0			0
$643$		−	33.3037i		−	1.31337i	−0.754166	−	0.656684i	$-0.771958\pi$
$643$		−	33.3037i		−	1.31337i	0.754166	−	0.656684i	$-0.228042\pi$
$644$		0			0
$645$	−1.13365	+	5.30862i	−0.0446375	+	0.209027i
$646$		0			0
$647$		−	1.54230i		−	0.0606343i	−0.999540	−	0.0303171i	$-0.990348\pi$
$647$		−	1.54230i		−	0.0606343i	0.999540	−	0.0303171i	$-0.00965172\pi$
$648$		0			0
$649$	−25.0190			−0.982081
$650$		0			0
$651$	12.3286			0.483195
$652$		0			0
$653$		−	13.1195i		−	0.513407i	−0.966490	−	0.256703i	$-0.917364\pi$
$653$		−	13.1195i		−	0.513407i	0.966490	−	0.256703i	$-0.0826364\pi$
$654$		0			0
$655$	−6.00981	+	28.1425i	−0.234823	+	1.09962i
$656$		0			0
$657$			6.55473i			0.255724i
$658$		0			0
$659$	−7.38016			−0.287490			−0.143745	−	0.989615i	$-0.545915\pi$
$659$	−7.38016			−0.287490			−0.143745	+	0.989615i	$0.545915\pi$
$660$		0			0
$661$	−25.0266			−0.973421			−0.486710	−	0.873563i	$-0.661803\pi$
$661$	−25.0266			−0.973421			−0.486710	+	0.873563i	$0.661803\pi$
$662$		0			0
$663$			0.406422i			0.0157841i
$664$		0			0
$665$	−29.6316	−	6.32780i	−1.14906	−	0.245381i
$666$		0			0
$667$			4.76644i			0.184557i
$668$		0			0
$669$	16.7820			0.648830
$670$		0			0
$671$	−59.4527			−2.29515
$672$		0			0
$673$			35.2500i			1.35879i	0.733773	+	0.679395i	$0.237757\pi$
$673$			35.2500i			1.35879i	−0.733773	+	0.679395i	$0.762243\pi$
$674$		0			0
$675$	−11.5523	+	25.8148i	−0.444647	+	0.993612i
$676$		0			0
$677$			6.04850i			0.232463i	0.993222	+	0.116231i	$0.0370814\pi$
$677$			6.04850i			0.232463i	−0.993222	+	0.116231i	$0.962919\pi$
$678$		0			0
$679$	14.4813			0.555741
$680$		0			0
$681$	2.60242			0.0997251
$682$		0			0
$683$			32.9347i			1.26021i	0.776510	+	0.630105i	$0.216988\pi$
$683$			32.9347i			1.26021i	−0.776510	+	0.630105i	$0.783012\pi$
$684$		0			0
$685$	−36.6136	−	7.81880i	−1.39893	−	0.298741i
$686$		0			0
$687$		−	23.1819i		−	0.884443i
$688$		0			0
$689$	−7.76696			−0.295897
$690$		0			0
$691$	11.7152			0.445668			0.222834	−	0.974856i	$-0.428469\pi$
$691$	11.7152			0.445668			0.222834	+	0.974856i	$0.428469\pi$
$692$		0			0
$693$		−	7.05040i		−	0.267823i
$694$		0			0
$695$	5.52408	−	25.8680i	0.209540	−	0.981228i
$696$		0			0
$697$		−	1.98764i		−	0.0752872i
$698$		0			0
$699$	−34.5091			−1.30525
$700$		0			0
$701$	43.8861			1.65756			0.828779	−	0.559577i	$-0.189036\pi$
$701$	43.8861			1.65756			0.828779	+	0.559577i	$0.189036\pi$
$702$		0			0
$703$			37.8148i			1.42621i
$704$		0			0
$705$	0.450433	−	2.10927i	0.0169643	−	0.0794398i
$706$		0			0
$707$			1.13090i			0.0425317i
$708$		0			0
$709$	−30.2394			−1.13567			−0.567833	−	0.823144i	$-0.692218\pi$
$709$	−30.2394			−1.13567			−0.567833	+	0.823144i	$0.692218\pi$
$710$		0			0
$711$	7.16561			0.268731
$712$		0			0
$713$			5.59148i			0.209402i
$714$		0			0
$715$	9.18403	+	1.96124i	0.343463	+	0.0733463i
$716$		0			0
$717$			38.0544i			1.42117i
$718$		0			0
$719$	−27.5338			−1.02684			−0.513419	−	0.858138i	$-0.671621\pi$
$719$	−27.5338			−1.02684			−0.513419	+	0.858138i	$0.671621\pi$
$720$		0			0
$721$	14.3941			0.536063
$722$		0			0
$723$			18.0828i			0.672507i
$724$		0			0
$725$	21.7534	+	9.73475i	0.807899	+	0.361540i
$726$		0			0
$727$		−	33.7995i		−	1.25356i	−0.779198	−	0.626778i	$-0.784373\pi$
$727$		−	33.7995i		−	1.25356i	0.779198	−	0.626778i	$-0.215627\pi$
$728$		0			0
$729$	−28.8078			−1.06695
$730$		0			0
$731$	0.519382			0.0192100
$732$		0			0
$733$			48.3133i			1.78449i	0.451550	+	0.892246i	$0.350871\pi$
$733$			48.3133i			1.78449i	−0.451550	+	0.892246i	$0.649129\pi$
$734$		0			0
$735$	−13.9059	−	2.96960i	−0.512928	−	0.109535i
$736$		0			0
$737$		−	28.2598i		−	1.04096i
$738$		0			0
$739$	22.7004			0.835048			0.417524	−	0.908666i	$-0.362898\pi$
$739$	22.7004			0.835048			0.417524	+	0.908666i	$0.362898\pi$
$740$		0			0
$741$	11.6745			0.428872
$742$		0			0
$743$			20.9953i			0.770242i	0.922866	+	0.385121i	$0.125840\pi$
$743$			20.9953i			0.770242i	−0.922866	+	0.385121i	$0.874160\pi$
$744$		0			0
$745$	−0.681890	+	3.19313i	−0.0249825	+	0.116987i
$746$		0			0
$747$		−	0.190995i		−	0.00698816i
$748$		0			0
$749$	20.0302			0.731887
$750$		0			0
$751$	−21.3909			−0.780565			−0.390283	−	0.920695i	$-0.627623\pi$
$751$	−21.3909			−0.780565			−0.390283	+	0.920695i	$0.627623\pi$
$752$		0			0
$753$			17.4871i			0.637266i
$754$		0			0
$755$	8.30747	−	38.9019i	0.302340	−	1.41579i
$756$		0			0
$757$			54.7364i			1.98943i	0.102676	+	0.994715i	$0.467260\pi$
$757$			54.7364i			1.98943i	−0.102676	+	0.994715i	$0.532740\pi$
$758$		0			0
$759$	−6.11017			−0.221785
$760$		0			0
$761$	7.87321			0.285404			0.142702	−	0.989766i	$-0.454421\pi$
$761$	7.87321			0.285404			0.142702	+	0.989766i	$0.454421\pi$
$762$		0			0
$763$		−	5.31084i		−	0.192265i
$764$		0			0
$765$	0.676666	+	0.144501i	0.0244649	+	0.00522446i
$766$		0			0
$767$		−	5.54272i		−	0.200136i
$768$		0			0
$769$	−32.7134			−1.17968			−0.589838	−	0.807522i	$-0.700808\pi$
$769$	−32.7134			−1.17968			−0.589838	+	0.807522i	$0.700808\pi$
$770$		0			0
$771$	18.9554			0.682663
$772$		0			0
$773$			2.88974i			0.103937i	0.998649	+	0.0519684i	$0.0165495\pi$
$773$			2.88974i			0.103937i	−0.998649	+	0.0519684i	$0.983450\pi$
$774$		0			0
$775$	25.5187	+	11.4198i	0.916659	+	0.410210i
$776$		0			0
$777$		−	9.66760i		−	0.346823i
$778$		0			0
$779$	−57.0949			−2.04564
$780$		0			0
$781$	43.1037			1.54237
$782$		0			0
$783$			26.9607i			0.963496i
$784$		0			0
$785$	43.8137	+	9.35637i	1.56378	+	0.333943i
$786$		0			0
$787$			46.4472i			1.65566i	0.560976	+	0.827832i	$0.310426\pi$
$787$			46.4472i			1.65566i	−0.560976	+	0.827832i	$0.689574\pi$
$788$		0			0
$789$	−38.2523			−1.36182
$790$		0			0
$791$	−24.9104			−0.885710
$792$		0			0
$793$		−	13.1712i		−	0.467723i
$794$		0			0
$795$	5.27681	−	24.7101i	0.187149	−	0.876375i
$796$		0			0
$797$			45.6713i			1.61776i	0.587974	+	0.808880i	$0.299926\pi$
$797$			45.6713i			1.61776i	−0.587974	+	0.808880i	$0.700074\pi$
$798$		0			0
$799$	−0.206366			−0.00730070
$800$		0			0
$801$	1.69928			0.0600411
$802$		0			0
$803$			27.6912i			0.977202i
$804$		0			0
$805$	−0.733706	+	3.43577i	−0.0258597	+	0.121095i
$806$		0			0
$807$		−	25.0369i		−	0.881340i
$808$		0			0
$809$	−37.7080			−1.32574			−0.662872	−	0.748733i	$-0.730663\pi$
$809$	−37.7080			−1.32574			−0.662872	+	0.748733i	$0.730663\pi$
$810$		0			0
$811$	9.02541			0.316925			0.158462	−	0.987365i	$-0.449346\pi$
$811$	9.02541			0.316925			0.158462	+	0.987365i	$0.449346\pi$
$812$		0			0
$813$		−	3.25444i		−	0.114138i
$814$		0			0
$815$	50.0934	+	10.6974i	1.75470	+	0.374714i
$816$		0			0
$817$		−	14.9192i		−	0.521958i
$818$		0			0
$819$	1.56195			0.0545790
$820$		0			0
$821$	−34.2481			−1.19527			−0.597634	−	0.801769i	$-0.703892\pi$
$821$	−34.2481			−1.19527			−0.597634	+	0.801769i	$0.703892\pi$
$822$		0			0
$823$		−	10.1842i		−	0.354999i	−0.984121	−	0.177499i	$-0.943199\pi$
$823$		−	10.1842i		−	0.354999i	0.984121	−	0.177499i	$-0.0568008\pi$
$824$		0			0
$825$	−12.4791	+	27.8859i	−0.434467	+	0.970864i
$826$		0			0
$827$			34.0710i			1.18477i	0.805656	+	0.592383i	$0.201813\pi$
$827$			34.0710i			1.18477i	−0.805656	+	0.592383i	$0.798187\pi$
$828$		0			0
$829$	24.5131			0.851376			0.425688	−	0.904870i	$-0.360032\pi$
$829$	24.5131			0.851376			0.425688	+	0.904870i	$0.360032\pi$
$830$		0			0
$831$	45.0140			1.56152
$832$		0			0
$833$			1.36052i			0.0471393i
$834$		0			0
$835$	1.80820	+	0.386140i	0.0625755	+	0.0133629i
$836$		0			0
$837$			31.6274i			1.09320i
$838$		0			0
$839$	−19.2362			−0.664107			−0.332053	−	0.943261i	$-0.607741\pi$
$839$	−19.2362			−0.664107			−0.332053	+	0.943261i	$0.607741\pi$
$840$		0			0
$841$	−6.28105			−0.216588
$842$		0			0
$843$		−	18.1660i		−	0.625669i
$844$		0			0
$845$	5.63626	−	26.3933i	0.193893	−	0.907956i
$846$		0			0
$847$		−	12.5024i		−	0.429588i
$848$		0			0
$849$	31.7254			1.08881
$850$		0			0
$851$	4.38462			0.150303
$852$		0			0
$853$		−	34.8964i		−	1.19483i	−0.801932	−	0.597415i	$-0.796195\pi$
$853$		−	34.8964i		−	1.19483i	0.801932	−	0.597415i	$-0.203805\pi$
$854$		0			0
$855$	4.15080	−	19.4372i	0.141954	−	0.664738i
$856$		0			0
$857$			41.9040i			1.43141i	0.698401	+	0.715706i	$0.253895\pi$
$857$			41.9040i			1.43141i	−0.698401	+	0.715706i	$0.746105\pi$
$858$		0			0
$859$	19.1286			0.652658			0.326329	−	0.945256i	$-0.394188\pi$
$859$	19.1286			0.652658			0.326329	+	0.945256i	$0.394188\pi$
$860$		0			0
$861$	14.5967			0.497453
$862$		0			0
$863$			0.509621i			0.0173477i	0.999962	+	0.00867385i	$0.00276101\pi$
$863$			0.509621i			0.0173477i	−0.999962	+	0.00867385i	$0.997239\pi$
$864$		0			0
$865$	−17.5795	−	3.75408i	−0.597720	−	0.127643i
$866$		0			0
$867$		−	23.7303i		−	0.805924i
$868$		0			0
$869$	30.2719			1.02691
$870$		0			0
$871$	6.26070			0.212136
$872$		0			0
$873$			9.49919i			0.321499i
$874$		0			0
$875$	14.1819	+	10.3656i	0.479435	+	0.350421i
$876$		0			0
$877$			15.6710i			0.529171i	0.964362	+	0.264586i	$0.0852352\pi$
$877$			15.6710i			0.529171i	−0.964362	+	0.264586i	$0.914765\pi$
$878$		0			0
$879$	−7.88135			−0.265831
$880$		0			0
$881$	−2.00846			−0.0676667			−0.0338333	−	0.999427i	$-0.510772\pi$
$881$	−2.00846			−0.0676667			−0.0338333	+	0.999427i	$0.510772\pi$
$882$		0			0
$883$		−	16.7848i		−	0.564854i	−0.959289	−	0.282427i	$-0.908861\pi$
$883$		−	16.7848i		−	0.564854i	0.959289	−	0.282427i	$-0.0911394\pi$
$884$		0			0
$885$	17.6338	+	3.76568i	0.592754	+	0.126582i
$886$		0			0
$887$		−	51.7294i		−	1.73690i	−0.495774	−	0.868452i	$-0.665116\pi$
$887$		−	51.7294i		−	1.73690i	0.495774	−	0.868452i	$-0.334884\pi$
$888$		0			0
$889$	−23.9167			−0.802140
$890$		0			0
$891$	−21.0992			−0.706849
$892$		0			0
$893$			5.92786i			0.198368i
$894$		0			0
$895$	10.1307	−	47.4399i	0.338633	−	1.58574i
$896$		0			0
$897$		−	1.35365i		−	0.0451971i
$898$		0			0
$899$	26.6514			0.888875
$900$		0			0
$901$	−2.41757			−0.0805409
$902$		0			0
$903$			3.81420i			0.126929i
$904$		0			0
$905$	−6.98132	+	32.6919i	−0.232067	+	1.08672i
$906$		0			0
$907$			29.0849i			0.965747i	0.875690	+	0.482874i	$0.160407\pi$
$907$			29.0849i			0.965747i	−0.875690	+	0.482874i	$0.839593\pi$
$908$		0			0
$909$	−0.741825			−0.0246048
$910$		0			0
$911$	26.3517			0.873070			0.436535	−	0.899687i	$-0.356205\pi$
$911$	26.3517			0.873070			0.436535	+	0.899687i	$0.356205\pi$
$912$		0			0
$913$		−	0.806883i		−	0.0267039i
$914$		0			0
$915$	41.9033	+	8.94840i	1.38528	+	0.295825i
$916$		0			0
$917$			20.2202i			0.667728i
$918$		0			0
$919$	46.0148			1.51789			0.758945	−	0.651155i	$-0.225715\pi$
$919$	46.0148			1.51789			0.758945	+	0.651155i	$0.225715\pi$
$920$		0			0
$921$	3.40264			0.112121
$922$		0			0
$923$			9.54922i			0.314316i
$924$		0			0
$925$	8.95494	−	20.0108i	0.294436	−	0.657950i
$926$		0			0
$927$			9.44196i			0.310115i
$928$		0			0
$929$	−26.1770			−0.858838			−0.429419	−	0.903105i	$-0.641282\pi$
$929$	−26.1770			−0.858838			−0.429419	+	0.903105i	$0.641282\pi$
$930$		0			0
$931$	39.0810			1.28083
$932$		0			0
$933$		−	5.02237i		−	0.164425i
$934$		0			0
$935$	2.85865	+	0.610463i	0.0934880	+	0.0199643i
$936$		0			0
$937$		−	44.4429i		−	1.45189i	−0.687754	−	0.725944i	$-0.741403\pi$
$937$		−	44.4429i		−	1.45189i	0.687754	−	0.725944i	$-0.258597\pi$
$938$		0			0
$939$	7.73242			0.252338
$940$		0			0
$941$	7.07346			0.230588			0.115294	−	0.993331i	$-0.463219\pi$
$941$	7.07346			0.230588			0.115294	+	0.993331i	$0.463219\pi$
$942$		0			0
$943$			6.62014i			0.215581i
$944$		0			0
$945$	−4.15010	+	19.4340i	−0.135003	+	0.632186i
$946$		0			0
$947$		−	14.4533i		−	0.469670i	−0.972035	−	0.234835i	$-0.924545\pi$
$947$		−	14.4533i		−	0.469670i	0.972035	−	0.234835i	$-0.0754549\pi$
$948$		0			0
$949$	−6.13473			−0.199142
$950$		0			0
$951$	8.25683			0.267746
$952$		0			0
$953$			32.3773i			1.04880i	0.851471	+	0.524402i	$0.175711\pi$
$953$			32.3773i			1.04880i	−0.851471	+	0.524402i	$0.824289\pi$
$954$		0			0
$955$	−3.67237	+	17.1968i	−0.118835	+	0.556476i
$956$		0			0
$957$			29.1237i			0.941437i
$958$		0			0
$959$	−26.3065			−0.849482
$960$		0			0
$961$	0.264608			0.00853575
$962$		0			0
$963$			13.1390i			0.423400i
$964$		0			0
$965$	33.2091	+	7.09177i	1.06904	+	0.228292i
$966$		0			0
$967$		−	25.8993i		−	0.832865i	−0.909167	−	0.416433i	$-0.863280\pi$
$967$		−	25.8993i		−	0.832865i	0.909167	−	0.416433i	$-0.136720\pi$
$968$		0			0
$969$	3.63383			0.116736
$970$		0			0
$971$	−25.2153			−0.809197			−0.404599	−	0.914494i	$-0.632589\pi$
$971$	−25.2153			−0.809197			−0.404599	+	0.914494i	$0.632589\pi$
$972$		0			0
$973$		−	18.5859i		−	0.595837i
$974$		0			0
$975$	−6.17787	−	2.76463i	−0.197850	−	0.0885391i
$976$		0			0
$977$		−	5.64235i		−	0.180515i	−0.995918	−	0.0902573i	$-0.971231\pi$
$977$		−	5.64235i		−	0.180515i	0.995918	−	0.0902573i	$-0.0287690\pi$
$978$		0			0
$979$	7.17880			0.229436
$980$		0			0
$981$	3.48371			0.111226
$982$		0			0
$983$			37.0322i			1.18114i	0.806986	+	0.590571i	$0.201097\pi$
$983$			37.0322i			1.18114i	−0.806986	+	0.590571i	$0.798903\pi$
$984$		0			0
$985$	26.8778	+	5.73972i	0.856397	+	0.182883i
$986$		0			0
$987$		−	1.51549i		−	0.0482387i
$988$		0			0
$989$	−1.72988			−0.0550070
$990$		0			0
$991$	7.14822			0.227071			0.113535	−	0.993534i	$-0.463782\pi$
$991$	7.14822			0.227071			0.113535	+	0.993534i	$0.463782\pi$
$992$		0			0
$993$			26.1215i			0.828942i
$994$		0			0
$995$	2.04128	−	9.55885i	0.0647130	−	0.303036i
$996$		0			0
$997$			4.96521i			0.157250i	0.996904	+	0.0786249i	$0.0250529\pi$
$997$			4.96521i			0.157250i	−0.996904	+	0.0786249i	$0.974947\pi$
$998$		0			0
$999$	24.8010			0.784667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.b.369.5	✓	14
4.3	odd	2		1840.2.e.g.369.10		14
5.2	odd	4		4600.2.a.bi.1.2		7
5.3	odd	4		4600.2.a.bh.1.6		7
5.4	even	2	inner	920.2.e.b.369.10	yes	14
20.3	even	4		9200.2.a.dc.1.2		7
20.7	even	4		9200.2.a.cz.1.6		7
20.19	odd	2		1840.2.e.g.369.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.5	✓	14	1.1	even	1	trivial
920.2.e.b.369.10	yes	14	5.4	even	2	inner
1840.2.e.g.369.5		14	20.19	odd	2
1840.2.e.g.369.10		14	4.3	odd	2
4600.2.a.bh.1.6		7	5.3	odd	4
4600.2.a.bi.1.2		7	5.2	odd	4
9200.2.a.cz.1.6		7	20.7	even	4
9200.2.a.dc.1.2		7	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.40334i q^{3} +(0.466981 - 2.18676i) q^{5} -1.57117i q^{7} +1.03063 q^{9} +O(q^{10})\)	\(q-1.40334i q^{3} +(0.466981 - 2.18676i) q^{5} -1.57117i q^{7} +1.03063 q^{9} +4.35401 q^{11} +0.964590i q^{13} +(-3.06878 - 0.655335i) q^{15} +0.300242i q^{17} +8.62443 q^{19} -2.20489 q^{21} -1.00000i q^{23} +(-4.56386 - 2.04235i) q^{25} -5.65635i q^{27} -4.76644 q^{29} -5.59148 q^{31} -6.11017i q^{33} +(-3.43577 - 0.733706i) q^{35} +4.38462i q^{37} +1.35365 q^{39} -6.62014 q^{41} -1.72988i q^{43} +(0.481284 - 2.25374i) q^{45} +0.687333i q^{47} +4.53143 q^{49} +0.421342 q^{51} +8.05208i q^{53} +(2.03324 - 9.52118i) q^{55} -12.1030i q^{57} -5.74620 q^{59} -13.6547 q^{61} -1.61929i q^{63} +(2.10933 + 0.450445i) q^{65} -6.49053i q^{67} -1.40334 q^{69} +9.89977 q^{71} +6.35994i q^{73} +(-2.86612 + 6.40466i) q^{75} -6.84088i q^{77} +6.95266 q^{79} -4.84592 q^{81} -0.185320i q^{83} +(0.656557 + 0.140207i) q^{85} +6.68895i q^{87} +1.64878 q^{89} +1.51553 q^{91} +7.84676i q^{93} +(4.02744 - 18.8596i) q^{95} +9.21689i q^{97} +4.48736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{5} - 4 q^{9}+O(q^{10}) \) 14 * q + 2 * q^5 - 4 * q^9	\( 14 q + 2 q^{5} - 4 q^{9} - 14 q^{11} - 6 q^{15} + 14 q^{19} - 12 q^{21} - 14 q^{25} + 22 q^{29} - 20 q^{31} - 2 q^{35} + 48 q^{39} - 32 q^{41} - 26 q^{45} + 34 q^{49} - 14 q^{51} - 38 q^{55} + 22 q^{59} + 10 q^{61} - 38 q^{65} + 6 q^{69} - 28 q^{71} - 24 q^{75} + 64 q^{79} - 10 q^{81} - 50 q^{85} + 48 q^{89} - 14 q^{91} - 30 q^{95} + 122 q^{99}+O(q^{100}) \) 14 * q + 2 * q^5 - 4 * q^9 - 14 * q^11 - 6 * q^15 + 14 * q^19 - 12 * q^21 - 14 * q^25 + 22 * q^29 - 20 * q^31 - 2 * q^35 + 48 * q^39 - 32 * q^41 - 26 * q^45 + 34 * q^49 - 14 * q^51 - 38 * q^55 + 22 * q^59 + 10 * q^61 - 38 * q^65 + 6 * q^69 - 28 * q^71 - 24 * q^75 + 64 * q^79 - 10 * q^81 - 50 * q^85 + 48 * q^89 - 14 * q^91 - 30 * q^95 + 122 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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