LMFDB - Embedded newform 880.3.i.g.769.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,3,Mod(769,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("880.769");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	880.i (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:	$23.9782632637$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.130897030168576.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 169x^{4} - 112x^{2} + 1936 $ x^8 + 18x^6 + 169x^4 - 112*x^2 + 1936
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.6
Root			$-1.41421 + 1.08854i$ of defining polynomial
Character	$\chi$	$=$	880.769
Dual form			880.3.i.g.769.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08854i q^{3} +(1.40754 + 4.79780i) q^{5} +2.82843 q^{7} +7.81507 q^{9} +O(q^{10})$	$q+1.08854i q^{3} +(1.40754 + 4.79780i) q^{5} +2.82843 q^{7} +7.81507 q^{9} +(-2.81507 - 10.6337i) q^{11} +11.0522 q^{13} +(-5.22261 + 1.53216i) q^{15} +21.8428 q^{17} -5.87305i q^{19} +3.07887i q^{21} -22.4568i q^{23} +(-21.0377 + 13.5061i) q^{25} +18.3039i q^{27} -36.3770i q^{29} +37.7055 q^{31} +(11.5752 - 3.06433i) q^{33} +(3.98111 + 13.5702i) q^{35} -52.3321i q^{37} +12.0308i q^{39} +51.2020i q^{41} +26.7151 q^{43} +(11.0000 + 37.4951i) q^{45} +49.4691i q^{47} -41.0000 q^{49} +23.7769i q^{51} +42.7365i q^{53} +(47.0560 - 28.4735i) q^{55} +6.39307 q^{57} -29.7055 q^{59} -12.3155i q^{61} +22.1044 q^{63} +(15.5563 + 53.0261i) q^{65} +5.84532i q^{67} +24.4452 q^{69} -86.4452 q^{71} +49.0810 q^{73} +(-14.7020 - 22.9004i) q^{75} +(-7.96223 - 30.0766i) q^{77} +147.733i q^{79} +50.4110 q^{81} +105.911 q^{83} +(30.7446 + 104.797i) q^{85} +39.5980 q^{87} +45.7055 q^{89} +31.2603 q^{91} +41.0441i q^{93} +(28.1777 - 8.26653i) q^{95} +35.7206i q^{97} +(-22.0000 - 83.1031i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{5} + 20 q^{9}+O(q^{10}) $ 8 * q - 10 * q^5 + 20 * q^9	$ 8 q - 10 q^{5} + 20 q^{9} + 20 q^{11} + 22 q^{15} - 62 q^{25} + 4 q^{31} + 88 q^{45} - 328 q^{49} - 138 q^{55} + 60 q^{59} + 68 q^{69} - 564 q^{71} - 394 q^{75} - 192 q^{81} + 68 q^{89} + 80 q^{91} - 176 q^{99}+O(q^{100}) $ 8 * q - 10 * q^5 + 20 * q^9 + 20 * q^11 + 22 * q^15 - 62 * q^25 + 4 * q^31 + 88 * q^45 - 328 * q^49 - 138 * q^55 + 60 * q^59 + 68 * q^69 - 564 * q^71 - 394 * q^75 - 192 * q^81 + 68 * q^89 + 80 * q^91 - 176 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$			1.08854i			0.362848i	0.983405	+	0.181424i	$0.0580706\pi$
$3$			1.08854i			0.362848i	−0.983405	+	0.181424i	$0.941929\pi$
$4$		0			0
$5$	1.40754	+	4.79780i	0.281507	+	0.959559i
$5$	1.40754	+	4.79780i	0.281507	+	0.959559i
$6$		0			0
$7$	2.82843			0.404061			0.202031	−	0.979379i	$-0.435246\pi$
$7$	2.82843			0.404061			0.202031	+	0.979379i	$0.435246\pi$
$8$		0			0
$9$	7.81507			0.868341
$10$		0			0
$11$	−2.81507	−	10.6337i	−0.255916	−	0.966699i
$11$	−2.81507	−	10.6337i	−0.255916	−	0.966699i
$12$		0			0
$13$	11.0522			0.850168			0.425084	−	0.905154i	$-0.360245\pi$
$13$	11.0522			0.850168			0.425084	+	0.905154i	$0.360245\pi$
$14$		0			0
$15$	−5.22261	+	1.53216i	−0.348174	+	0.102144i
$16$		0			0
$17$	21.8428			1.28487			0.642436	−	0.766339i	$-0.277924\pi$
$17$	21.8428			1.28487			0.642436	+	0.766339i	$0.277924\pi$
$18$		0			0
$19$		−	5.87305i		−	0.309108i	−0.987984	−	0.154554i	$-0.950606\pi$
$19$		−	5.87305i		−	0.309108i	0.987984	−	0.154554i	$-0.0493940\pi$
$20$		0			0
$21$			3.07887i			0.146613i
$22$		0			0
$23$		−	22.4568i		−	0.976383i	−0.872736	−	0.488192i	$-0.837657\pi$
$23$		−	22.4568i		−	0.976383i	0.872736	−	0.488192i	$-0.162343\pi$
$24$		0			0
$25$	−21.0377	+	13.5061i	−0.841507	+	0.540246i
$26$		0			0
$27$			18.3039i			0.677924i
$28$		0			0
$29$		−	36.3770i		−	1.25438i	−0.778866	−	0.627190i	$-0.784205\pi$
$29$		−	36.3770i		−	1.25438i	0.778866	−	0.627190i	$-0.215795\pi$
$30$		0			0
$31$	37.7055			1.21631			0.608153	−	0.793820i	$-0.291911\pi$
$31$	37.7055			1.21631			0.608153	+	0.793820i	$0.291911\pi$
$32$		0			0
$33$	11.5752	−	3.06433i	0.350765	−	0.0928585i
$34$		0			0
$35$	3.98111	+	13.5702i	0.113746	+	0.387720i
$36$		0			0
$37$		−	52.3321i		−	1.41438i	−0.707023	−	0.707191i	$-0.749962\pi$
$37$		−	52.3321i		−	1.41438i	0.707023	−	0.707191i	$-0.250038\pi$
$38$		0			0
$39$			12.0308i			0.308482i
$40$		0			0
$41$			51.2020i			1.24883i	0.781093	+	0.624415i	$0.214662\pi$
$41$			51.2020i			1.24883i	−0.781093	+	0.624415i	$0.785338\pi$
$42$		0			0
$43$	26.7151			0.621282			0.310641	−	0.950527i	$-0.399456\pi$
$43$	26.7151			0.621282			0.310641	+	0.950527i	$0.399456\pi$
$44$		0			0
$45$	11.0000	+	37.4951i	0.244444	+	0.833225i
$46$		0			0
$47$			49.4691i			1.05253i	0.850319	+	0.526267i	$0.176409\pi$
$47$			49.4691i			1.05253i	−0.850319	+	0.526267i	$0.823591\pi$
$48$		0			0
$49$	−41.0000			−0.836735
$50$		0			0
$51$			23.7769i			0.466213i
$52$		0			0
$53$			42.7365i			0.806350i	0.915123	+	0.403175i	$0.132093\pi$
$53$			42.7365i			0.806350i	−0.915123	+	0.403175i	$0.867907\pi$
$54$		0			0
$55$	47.0560	−	28.4735i	0.855563	−	0.517699i
$56$		0			0
$57$	6.39307			0.112159
$58$		0			0
$59$	−29.7055			−0.503483			−0.251742	−	0.967794i	$-0.581003\pi$
$59$	−29.7055			−0.503483			−0.251742	+	0.967794i	$0.581003\pi$
$60$		0			0
$61$		−	12.3155i		−	0.201893i	−0.994892	−	0.100946i	$-0.967813\pi$
$61$		−	12.3155i		−	0.201893i	0.994892	−	0.100946i	$-0.0321871\pi$
$62$		0			0
$63$	22.1044			0.350863
$64$		0			0
$65$	15.5563	+	53.0261i	0.239328	+	0.815786i
$66$		0			0
$67$			5.84532i			0.0872436i	0.999048	+	0.0436218i	$0.0138897\pi$
$67$			5.84532i			0.0872436i	−0.999048	+	0.0436218i	$0.986110\pi$
$68$		0			0
$69$	24.4452			0.354279
$70$		0			0
$71$	−86.4452			−1.21754			−0.608769	−	0.793347i	$-0.708336\pi$
$71$	−86.4452			−1.21754			−0.608769	+	0.793347i	$0.708336\pi$
$72$		0			0
$73$	49.0810			0.672343			0.336171	−	0.941801i	$-0.390868\pi$
$73$	49.0810			0.672343			0.336171	+	0.941801i	$0.390868\pi$
$74$		0			0
$75$	−14.7020	−	22.9004i	−0.196027	−	0.305339i
$76$		0			0
$77$	−7.96223	−	30.0766i	−0.103406	−	0.390605i
$78$		0			0
$79$			147.733i			1.87004i	0.354599	+	0.935019i	$0.384617\pi$
$79$			147.733i			1.87004i	−0.354599	+	0.935019i	$0.615383\pi$
$80$		0			0
$81$	50.4110			0.622358
$82$		0			0
$83$	105.911			1.27604			0.638019	−	0.770021i	$-0.279754\pi$
$83$	105.911			1.27604			0.638019	+	0.770021i	$0.279754\pi$
$84$		0			0
$85$	30.7446	+	104.797i	0.361701	+	1.23291i
$86$		0			0
$87$	39.5980			0.455149
$88$		0			0
$89$	45.7055			0.513545			0.256773	−	0.966472i	$-0.417341\pi$
$89$	45.7055			0.513545			0.256773	+	0.966472i	$0.417341\pi$
$90$		0			0
$91$	31.2603			0.343520
$92$		0			0
$93$			41.0441i			0.441334i
$94$		0			0
$95$	28.1777	−	8.26653i	0.296607	−	0.0870161i
$96$		0			0
$97$			35.7206i			0.368254i	0.982902	+	0.184127i	$0.0589458\pi$
$97$			35.7206i			0.368254i	−0.982902	+	0.184127i	$0.941054\pi$
$98$		0			0
$99$	−22.0000	−	83.1031i	−0.222222	−	0.839425i
$100$		0			0
$101$		−	172.079i		−	1.70375i	−0.523742	−	0.851877i	$-0.675464\pi$
$101$		−	172.079i		−	1.70375i	0.523742	−	0.851877i	$-0.324536\pi$
$102$		0			0
$103$			91.4004i			0.887383i	0.896180	+	0.443691i	$0.146331\pi$
$103$			91.4004i			0.887383i	−0.896180	+	0.443691i	$0.853669\pi$
$104$		0			0
$105$	−14.7718	+	4.33362i	−0.140684	+	0.0412725i
$106$		0			0
$107$	83.4968			0.780344			0.390172	−	0.920742i	$-0.372416\pi$
$107$	83.4968			0.780344			0.390172	+	0.920742i	$0.372416\pi$
$108$		0			0
$109$			11.1767i			0.102539i	0.998685	+	0.0512694i	$0.0163267\pi$
$109$			11.1767i			0.102539i	−0.998685	+	0.0512694i	$0.983673\pi$
$110$		0			0
$111$	56.9658			0.513205
$112$		0			0
$113$		−	77.6520i		−	0.687186i	−0.939119	−	0.343593i	$-0.888356\pi$
$113$		−	77.6520i		−	0.687186i	0.939119	−	0.343593i	$-0.111644\pi$
$114$		0			0
$115$	107.743	−	31.6088i	0.936897	−	0.274859i
$116$		0			0
$117$	86.3736			0.738236
$118$		0			0
$119$	61.7809			0.519167
$120$		0			0
$121$	−105.151	+	59.8692i	−0.869014	+	0.494787i
$122$		0			0
$123$	−55.7356			−0.453135
$124$		0			0
$125$	−94.4110	−	81.9241i	−0.755288	−	0.655393i
$126$		0			0
$127$	−132.936			−1.04674			−0.523370	−	0.852105i	$-0.675326\pi$
$127$	−132.936			−1.04674			−0.523370	+	0.852105i	$0.675326\pi$
$128$		0			0
$129$			29.0806i			0.225431i
$130$		0			0
$131$		−	148.018i		−	1.12991i	−0.825123	−	0.564953i	$-0.808894\pi$
$131$		−	148.018i		−	1.12991i	0.825123	−	0.564953i	$-0.191106\pi$
$132$		0			0
$133$		−	16.6115i		−	0.124898i
$134$		0			0
$135$	−87.8186	+	25.7635i	−0.650508	+	0.190840i
$136$		0			0
$137$			212.795i			1.55325i	0.629962	+	0.776626i	$0.283070\pi$
$137$			212.795i			1.55325i	−0.629962	+	0.776626i	$0.716930\pi$
$138$		0			0
$139$			246.541i			1.77368i	0.462078	+	0.886839i	$0.347104\pi$
$139$			246.541i			1.77368i	−0.462078	+	0.886839i	$0.652896\pi$
$140$		0			0
$141$	−53.8493			−0.381910
$142$		0			0
$143$	−31.1127	−	117.525i	−0.217571	−	0.821856i
$144$		0			0
$145$	174.530	−	51.2020i	1.20365	−	0.353117i
$146$		0			0
$147$		−	44.6303i		−	0.303607i
$148$		0			0
$149$		−	13.6862i		−	0.0918539i	−0.998945	−	0.0459270i	$-0.985376\pi$
$149$		−	13.6862i		−	0.0918539i	0.998945	−	0.0459270i	$-0.0146241\pi$
$150$		0			0
$151$		−	170.139i		−	1.12675i	−0.826202	−	0.563374i	$-0.809503\pi$
$151$		−	170.139i		−	1.12675i	0.826202	−	0.563374i	$-0.190497\pi$
$152$		0			0
$153$	170.703			1.11571
$154$		0			0
$155$	53.0719	+	180.903i	0.342399	+	1.16712i
$156$		0			0
$157$			36.5258i			0.232649i	0.993211	+	0.116324i	$0.0371112\pi$
$157$			36.5258i			0.232649i	−0.993211	+	0.116324i	$0.962889\pi$
$158$		0			0
$159$	−46.5206			−0.292582
$160$		0			0
$161$		−	63.5175i		−	0.394518i
$162$		0			0
$163$			100.914i			0.619104i	0.950882	+	0.309552i	$0.100179\pi$
$163$			100.914i			0.619104i	−0.950882	+	0.309552i	$0.899821\pi$
$164$		0			0
$165$	30.9946	+	51.2225i	0.187846	+	0.310439i
$166$		0			0
$167$	104.652			0.626658			0.313329	−	0.949645i	$-0.398556\pi$
$167$	104.652			0.626658			0.313329	+	0.949645i	$0.398556\pi$
$168$		0			0
$169$	−46.8493			−0.277215
$170$		0			0
$171$		−	45.8983i		−	0.268411i
$172$		0			0
$173$	−206.640			−1.19445			−0.597225	−	0.802073i	$-0.703730\pi$
$173$	−206.640			−1.19445			−0.597225	+	0.802073i	$0.703730\pi$
$174$		0			0
$175$	−59.5036	+	38.2011i	−0.340020	+	0.218292i
$176$		0			0
$177$		−	32.3357i		−	0.182688i
$178$		0			0
$179$	−125.117			−0.698975			−0.349488	−	0.936941i	$-0.613644\pi$
$179$	−125.117			−0.698975			−0.349488	+	0.936941i	$0.613644\pi$
$180$		0			0
$181$	91.6232			0.506205			0.253103	−	0.967439i	$-0.418549\pi$
$181$	91.6232			0.506205			0.253103	+	0.967439i	$0.418549\pi$
$182$		0			0
$183$	13.4059			0.0732564
$184$		0			0
$185$	251.079	−	73.6594i	1.35718	−	0.398159i
$186$		0			0
$187$	−61.4892	−	232.270i	−0.328819	−	1.24209i
$188$		0			0
$189$			51.7714i			0.273923i
$190$		0			0
$191$	−90.2945			−0.472746			−0.236373	−	0.971662i	$-0.575959\pi$
$191$	−90.2945			−0.472746			−0.236373	+	0.971662i	$0.575959\pi$
$192$		0			0
$193$	153.113			0.793332			0.396666	−	0.917963i	$-0.370167\pi$
$193$	153.113			0.793332			0.396666	+	0.917963i	$0.370167\pi$
$194$		0			0
$195$	−57.7212	+	16.9338i	−0.296006	+	0.0868398i
$196$		0			0
$197$	−321.869			−1.63385			−0.816927	−	0.576741i	$-0.804324\pi$
$197$	−321.869			−1.63385			−0.816927	+	0.576741i	$0.804324\pi$
$198$		0			0
$199$	48.5890			0.244166			0.122083	−	0.992520i	$-0.461043\pi$
$199$	48.5890			0.244166			0.122083	+	0.992520i	$0.461043\pi$
$200$		0			0
$201$	−6.36289			−0.0316561
$202$		0			0
$203$		−	102.890i		−	0.506846i
$204$		0			0
$205$	−245.657	+	72.0687i	−1.19833	+	0.351554i
$206$		0			0
$207$		−	175.502i		−	0.847834i
$208$		0			0
$209$	−62.4522	+	16.5331i	−0.298814	+	0.0791056i
$210$		0			0
$211$		−	145.508i		−	0.689612i	−0.938674	−	0.344806i	$-0.887945\pi$
$211$		−	145.508i		−	0.689612i	0.938674	−	0.344806i	$-0.112055\pi$
$212$		0			0
$213$		−	94.0994i		−	0.441781i
$214$		0			0
$215$	37.6025	+	128.174i	0.174895	+	0.596156i
$216$		0			0
$217$	106.647			0.491462
$218$		0			0
$219$			53.4268i			0.243958i
$220$		0			0
$221$	241.411			1.09236
$222$		0			0
$223$			78.0918i			0.350187i	0.984552	+	0.175094i	$0.0560228\pi$
$223$			78.0918i			0.350187i	−0.984552	+	0.175094i	$0.943977\pi$
$224$		0			0
$225$	−164.411	+	105.551i	−0.730716	+	0.469118i
$226$		0			0
$227$	−268.701			−1.18370			−0.591851	−	0.806047i	$-0.701603\pi$
$227$	−268.701			−1.18370			−0.591851	+	0.806047i	$0.701603\pi$
$228$		0			0
$229$	166.528			0.727195			0.363597	−	0.931556i	$-0.381548\pi$
$229$	166.528			0.727195			0.363597	+	0.931556i	$0.381548\pi$
$230$		0			0
$231$	32.7397	−	8.66723i	0.141730	−	0.0375205i
$232$		0			0
$233$	−42.0682			−0.180550			−0.0902750	−	0.995917i	$-0.528775\pi$
$233$	−42.0682			−0.180550			−0.0902750	+	0.995917i	$0.528775\pi$
$234$		0			0
$235$	−237.343	+	69.6296i	−1.00997	+	0.296296i
$236$		0			0
$237$	−160.814			−0.678539
$238$		0			0
$239$			10.0906i			0.0422203i	0.999777	+	0.0211101i	$0.00672007\pi$
$239$			10.0906i			0.0422203i	−0.999777	+	0.0211101i	$0.993280\pi$
$240$		0			0
$241$		−	157.254i		−	0.652507i	−0.945282	−	0.326254i	$-0.894214\pi$
$241$		−	157.254i		−	0.652507i	0.945282	−	0.326254i	$-0.105786\pi$
$242$		0			0
$243$			219.610i			0.903745i
$244$		0			0
$245$	−57.7090	−	196.710i	−0.235547	−	0.802896i
$246$		0			0
$247$		−	64.9100i		−	0.262794i
$248$		0			0
$249$			115.289i			0.463007i
$250$		0			0
$251$	415.939			1.65713			0.828563	−	0.559896i	$-0.189159\pi$
$251$	415.939			1.65713			0.828563	+	0.559896i	$0.189159\pi$
$252$		0			0
$253$	−238.799	+	63.2176i	−0.943869	+	0.249872i
$254$		0			0
$255$	−114.077	+	33.4668i	−0.447359	+	0.131242i
$256$		0			0
$257$		−	129.015i		−	0.502003i	−0.967987	−	0.251002i	$-0.919240\pi$
$257$		−	129.015i		−	0.502003i	0.967987	−	0.251002i	$-0.0807599\pi$
$258$		0			0
$259$		−	148.018i		−	0.571497i
$260$		0			0
$261$		−	284.289i		−	1.08923i
$262$		0			0
$263$	−229.103			−0.871113			−0.435556	−	0.900162i	$-0.643448\pi$
$263$	−229.103			−0.871113			−0.435556	+	0.900162i	$0.643448\pi$
$264$		0			0
$265$	−205.041	+	60.1532i	−0.773740	+	0.226993i
$266$		0			0
$267$			49.7524i			0.186339i
$268$		0			0
$269$	−370.836			−1.37857			−0.689286	−	0.724489i	$-0.742076\pi$
$269$	−370.836			−1.37857			−0.689286	+	0.724489i	$0.742076\pi$
$270$		0			0
$271$		−	234.226i		−	0.864302i	−0.901801	−	0.432151i	$-0.857755\pi$
$271$		−	234.226i		−	0.864302i	0.901801	−	0.432151i	$-0.142245\pi$
$272$		0			0
$273$			34.0282i			0.124645i
$274$		0			0
$275$	202.843	+	185.687i	0.737610	+	0.675227i
$276$		0			0
$277$	−478.169			−1.72624			−0.863121	−	0.504997i	$-0.831493\pi$
$277$	−478.169			−1.72624			−0.863121	+	0.504997i	$0.831493\pi$
$278$		0			0
$279$	294.671			1.05617
$280$		0			0
$281$		−	340.847i		−	1.21298i	−0.795091	−	0.606490i	$-0.792577\pi$
$281$		−	340.847i		−	1.21298i	0.795091	−	0.606490i	$-0.207423\pi$
$282$		0			0
$283$	−289.875			−1.02429			−0.512147	−	0.858898i	$-0.671150\pi$
$283$	−289.875			−1.02429			−0.512147	+	0.858898i	$0.671150\pi$
$284$		0			0
$285$	8.99848	+	30.6726i	0.0315736	+	0.107623i
$286$		0			0
$287$			144.821i			0.504603i
$288$		0			0
$289$	188.110			0.650898
$290$		0			0
$291$	−38.8835			−0.133620
$292$		0			0
$293$	−373.944			−1.27626			−0.638129	−	0.769930i	$-0.720291\pi$
$293$	−373.944			−1.27626			−0.638129	+	0.769930i	$0.720291\pi$
$294$		0			0
$295$	−41.8116	−	142.521i	−0.141734	−	0.483122i
$296$		0			0
$297$	194.638	−	51.5269i	0.655348	−	0.173491i
$298$		0			0
$299$		−	248.197i		−	0.830090i
$300$		0			0
$301$	75.5617			0.251036
$302$		0			0
$303$	187.316			0.618204
$304$		0			0
$305$	59.0871	−	17.3345i	0.193728	−	0.0568343i
$306$		0			0
$307$	366.959			1.19531			0.597654	−	0.801754i	$-0.296100\pi$
$307$	366.959			1.19531			0.597654	+	0.801754i	$0.296100\pi$
$308$		0			0
$309$	−99.4934			−0.321985
$310$		0			0
$311$	272.589			0.876492			0.438246	−	0.898855i	$-0.355600\pi$
$311$	272.589			0.876492			0.438246	+	0.898855i	$0.355600\pi$
$312$		0			0
$313$			83.9447i			0.268194i	0.990968	+	0.134097i	$0.0428134\pi$
$313$			83.9447i			0.268194i	−0.990968	+	0.134097i	$0.957187\pi$
$314$		0			0
$315$	31.1127	+	106.052i	0.0987705	+	0.336674i
$316$		0			0
$317$		−	276.170i		−	0.871197i	−0.900141	−	0.435599i	$-0.856537\pi$
$317$		−	276.170i		−	0.871197i	0.900141	−	0.435599i	$-0.143463\pi$
$318$		0			0
$319$	−386.822	+	102.404i	−1.21261	+	0.321016i
$320$		0			0
$321$			90.8899i			0.283146i
$322$		0			0
$323$		−	128.284i		−	0.397164i
$324$		0			0
$325$	−232.512	+	149.272i	−0.715422	+	0.459300i
$326$		0			0
$327$	−12.1664			−0.0372060
$328$		0			0
$329$			139.920i			0.425288i
$330$		0			0
$331$	−421.117			−1.27226			−0.636128	−	0.771584i	$-0.719465\pi$
$331$	−421.117			−1.27226			−0.636128	+	0.771584i	$0.719465\pi$
$332$		0			0
$333$		−	408.979i		−	1.22817i
$334$		0			0
$335$	−28.0447	+	8.22750i	−0.0837154	+	0.0245597i
$336$		0			0
$337$	601.361			1.78445			0.892226	−	0.451589i	$-0.149143\pi$
$337$	601.361			1.78445			0.892226	+	0.451589i	$0.149143\pi$
$338$		0			0
$339$	84.5276			0.249344
$340$		0			0
$341$	−106.144	−	400.949i	−0.311272	−	1.17580i
$342$		0			0
$343$	−254.558			−0.742153
$344$		0			0
$345$	34.4075	+	117.283i	0.0997320	+	0.339951i
$346$		0			0
$347$	−531.241			−1.53095			−0.765477	−	0.643463i	$-0.777497\pi$
$347$	−531.241			−1.53095			−0.765477	+	0.643463i	$0.777497\pi$
$348$		0			0
$349$		−	242.556i		−	0.695002i	−0.937680	−	0.347501i	$-0.887030\pi$
$349$		−	242.556i		−	0.695002i	0.937680	−	0.347501i	$-0.112970\pi$
$350$		0			0
$351$			202.298i			0.576349i
$352$		0			0
$353$			643.382i			1.82261i	0.411731	+	0.911305i	$0.364924\pi$
$353$			643.382i			1.82261i	−0.411731	+	0.911305i	$0.635076\pi$
$354$		0			0
$355$	−121.675	−	414.746i	−0.342746	−	1.16830i
$356$		0			0
$357$			67.2512i			0.188379i
$358$		0			0
$359$		−	597.374i		−	1.66399i	−0.554780	−	0.831997i	$-0.687197\pi$
$359$		−	597.374i		−	1.66399i	0.554780	−	0.831997i	$-0.312803\pi$
$360$		0			0
$361$	326.507			0.904452
$362$		0			0
$363$	−65.1703	−	114.461i	−0.179532	−	0.315320i
$364$		0			0
$365$	69.0833	+	235.481i	0.189269	+	0.645152i
$366$		0			0
$367$			680.355i			1.85383i	0.375274	+	0.926914i	$0.377549\pi$
$367$			680.355i			1.85383i	−0.375274	+	0.926914i	$0.622451\pi$
$368$		0			0
$369$			400.147i			1.08441i
$370$		0			0
$371$			120.877i			0.325815i
$372$		0			0
$373$	79.3608			0.212763			0.106382	−	0.994325i	$-0.466073\pi$
$373$	79.3608			0.212763			0.106382	+	0.994325i	$0.466073\pi$
$374$		0			0
$375$	89.1780	−	102.771i	0.237808	−	0.274055i
$376$		0			0
$377$		−	402.046i		−	1.06643i
$378$		0			0
$379$	86.7327			0.228846			0.114423	−	0.993432i	$-0.463498\pi$
$379$	86.7327			0.228846			0.114423	+	0.993432i	$0.463498\pi$
$380$		0			0
$381$		−	144.707i		−	0.379808i
$382$		0			0
$383$		−	734.141i		−	1.91682i	−0.285400	−	0.958409i	$-0.592126\pi$
$383$		−	734.141i		−	1.91682i	0.285400	−	0.958409i	$-0.407874\pi$
$384$		0			0
$385$	133.094	−	80.5351i	0.345700	−	0.209182i
$386$		0			0
$387$	208.781			0.539485
$388$		0			0
$389$	287.939			0.740202			0.370101	−	0.928992i	$-0.379323\pi$
$389$	287.939			0.740202			0.370101	+	0.928992i	$0.379323\pi$
$390$		0			0
$391$		−	490.521i		−	1.25453i
$392$		0			0
$393$	161.124			0.409984
$394$		0			0
$395$	−708.792	+	207.939i	−1.79441	+	0.526429i
$396$		0			0
$397$			193.045i			0.486260i	0.969994	+	0.243130i	$0.0781741\pi$
$397$			193.045i			0.486260i	−0.969994	+	0.243130i	$0.921826\pi$
$398$		0			0
$399$	18.0823			0.0453191
$400$		0			0
$401$	354.233			0.883374			0.441687	−	0.897169i	$-0.354380\pi$
$401$	354.233			0.883374			0.441687	+	0.897169i	$0.354380\pi$
$402$		0			0
$403$	416.728			1.03406
$404$		0			0
$405$	70.9553	+	241.862i	0.175198	+	0.597190i
$406$		0			0
$407$	−556.484	+	147.319i	−1.36728	+	0.361963i
$408$		0			0
$409$			586.482i			1.43394i	0.697103	+	0.716971i	$0.254472\pi$
$409$			586.482i			1.43394i	−0.697103	+	0.716971i	$0.745528\pi$
$410$		0			0
$411$	−231.637			−0.563594
$412$		0			0
$413$	−84.0199			−0.203438
$414$		0			0
$415$	149.074	+	508.140i	0.359214	+	1.22443i
$416$		0			0
$417$	−268.371			−0.643575
$418$		0			0
$419$	−36.1507			−0.0862786			−0.0431393	−	0.999069i	$-0.513736\pi$
$419$	−36.1507			−0.0862786			−0.0431393	+	0.999069i	$0.513736\pi$
$420$		0			0
$421$	−295.644			−0.702242			−0.351121	−	0.936330i	$-0.614200\pi$
$421$	−295.644			−0.702242			−0.351121	+	0.936330i	$0.614200\pi$
$422$		0			0
$423$			386.605i			0.913959i
$424$		0			0
$425$	−459.523	+	295.013i	−1.08123	+	0.694147i
$426$		0			0
$427$		−	34.8334i		−	0.0815770i
$428$		0			0
$429$	127.932	−	33.8675i	0.298209	−	0.0789453i
$430$		0			0
$431$		−	290.732i		−	0.674551i	−0.941406	−	0.337276i	$-0.890495\pi$
$431$		−	290.732i		−	0.674551i	0.941406	−	0.337276i	$-0.109505\pi$
$432$		0			0
$433$		−	212.960i		−	0.491823i	−0.969292	−	0.245912i	$-0.920913\pi$
$433$		−	212.960i		−	0.491823i	0.969292	−	0.245912i	$-0.0790873\pi$
$434$		0			0
$435$	55.7356	+	189.983i	0.128128	+	0.436743i
$436$		0			0
$437$	−131.890			−0.301808
$438$		0			0
$439$			157.824i			0.359507i	0.983712	+	0.179754i	$0.0575300\pi$
$439$			157.824i			0.359507i	−0.983712	+	0.179754i	$0.942470\pi$
$440$		0			0
$441$	−320.418			−0.726571
$442$		0			0
$443$		−	218.723i		−	0.493731i	−0.969050	−	0.246865i	$-0.920599\pi$
$443$		−	218.723i		−	0.493731i	0.969050	−	0.246865i	$-0.0794006\pi$
$444$		0			0
$445$	64.3322	+	219.286i	0.144567	+	0.492777i
$446$		0			0
$447$	14.8981			0.0333290
$448$		0			0
$449$	−824.610			−1.83655			−0.918274	−	0.395946i	$-0.870417\pi$
$449$	−824.610			−1.83655			−0.918274	+	0.395946i	$0.870417\pi$
$450$		0			0
$451$	544.466	−	144.137i	1.20724	−	0.319595i
$452$		0			0
$453$	185.204			0.408838
$454$		0			0
$455$	44.0000	+	149.980i	0.0967033	+	0.329627i
$456$		0			0
$457$	304.472			0.666242			0.333121	−	0.942884i	$-0.391898\pi$
$457$	304.472			0.666242			0.333121	+	0.942884i	$0.391898\pi$
$458$		0			0
$459$			399.810i			0.871046i
$460$		0			0
$461$		−	254.639i		−	0.552363i	−0.961106	−	0.276181i	$-0.910931\pi$
$461$		−	254.639i		−	0.552363i	0.961106	−	0.276181i	$-0.0890690\pi$
$462$		0			0
$463$			151.472i			0.327153i	0.986531	+	0.163576i	$0.0523030\pi$
$463$			151.472i			0.327153i	−0.986531	+	0.163576i	$0.947697\pi$
$464$		0			0
$465$	−196.921	+	57.7711i	−0.423486	+	0.124239i
$466$		0			0
$467$			719.707i			1.54113i	0.637363	+	0.770564i	$0.280025\pi$
$467$			719.707i			1.54113i	−0.637363	+	0.770564i	$0.719975\pi$
$468$		0			0
$469$			16.5331i			0.0352517i
$470$		0			0
$471$	−39.7600			−0.0844161
$472$		0			0
$473$	−75.2050	−	284.080i	−0.158996	−	0.600592i
$474$		0			0
$475$	79.3223	+	123.555i	0.166994	+	0.260117i
$476$		0			0
$477$			333.989i			0.700187i
$478$		0			0
$479$			525.759i			1.09762i	0.835948	+	0.548809i	$0.184918\pi$
$479$			525.759i			1.09762i	−0.835948	+	0.548809i	$0.815082\pi$
$480$		0			0
$481$		−	578.384i		−	1.20246i
$482$		0			0
$483$	69.1415			0.143150
$484$		0			0
$485$	−171.380	+	50.2781i	−0.353361	+	0.103666i
$486$		0			0
$487$		−	638.185i		−	1.31044i	−0.755437	−	0.655221i	$-0.772576\pi$
$487$		−	638.185i		−	1.31044i	0.755437	−	0.655221i	$-0.227424\pi$
$488$		0			0
$489$	−109.849			−0.224641
$490$		0			0
$491$		−	697.269i		−	1.42010i	−0.704152	−	0.710050i	$-0.748673\pi$
$491$		−	697.269i		−	1.42010i	0.704152	−	0.710050i	$-0.251327\pi$
$492$		0			0
$493$		−	794.578i		−	1.61172i
$494$		0			0
$495$	367.746	−	222.522i	0.742921	−	0.449540i
$496$		0			0
$497$	−244.504			−0.491960
$498$		0			0
$499$	363.411			0.728279			0.364139	−	0.931344i	$-0.381363\pi$
$499$	363.411			0.728279			0.364139	+	0.931344i	$0.381363\pi$
$500$		0			0
$501$			113.918i			0.227381i
$502$		0			0
$503$	−336.989			−0.669959			−0.334980	−	0.942225i	$-0.608729\pi$
$503$	−336.989			−0.669959			−0.334980	+	0.942225i	$0.608729\pi$
$504$		0			0
$505$	825.601	−	242.208i	1.63485	−	0.479619i
$506$		0			0
$507$		−	50.9975i		−	0.100587i
$508$		0			0
$509$	502.528			0.987284			0.493642	−	0.869665i	$-0.335665\pi$
$509$	502.528			0.987284			0.493642	+	0.869665i	$0.335665\pi$
$510$		0			0
$511$	138.822			0.271667
$512$		0			0
$513$	107.500			0.209552
$514$		0			0
$515$	−438.521	+	128.649i	−0.851496	+	0.249805i
$516$		0			0
$517$	526.039	−	139.259i	1.01748	−	0.269360i
$518$		0			0
$519$		−	224.937i		−	0.433404i
$520$		0			0
$521$	−351.939			−0.675506			−0.337753	−	0.941235i	$-0.609667\pi$
$521$	−351.939			−0.675506			−0.337753	+	0.941235i	$0.609667\pi$
$522$		0			0
$523$	402.799			0.770171			0.385085	−	0.922881i	$-0.374172\pi$
$523$	402.799			0.770171			0.385085	+	0.922881i	$0.374172\pi$
$524$		0			0
$525$	−41.5836	−	64.7722i	−0.0792069	−	0.123376i
$526$		0			0
$527$	823.595			1.56280
$528$		0			0
$529$	24.6916			0.0466759
$530$		0			0
$531$	−232.151			−0.437195
$532$		0			0
$533$			565.894i			1.06171i
$534$		0			0
$535$	117.525	+	400.601i	0.219673	+	0.748786i
$536$		0			0
$537$		−	136.195i		−	0.253622i
$538$		0			0
$539$	115.418	+	435.981i	0.214134	+	0.808871i
$540$		0			0
$541$			227.499i			0.420515i	0.977646	+	0.210258i	$0.0674303\pi$
$541$			227.499i			0.420515i	−0.977646	+	0.210258i	$0.932570\pi$
$542$		0			0
$543$			99.7358i			0.183676i
$544$		0			0
$545$	−53.6237	+	15.7317i	−0.0983921	+	0.0288654i
$546$		0			0
$547$	724.833			1.32511			0.662553	−	0.749015i	$-0.269473\pi$
$547$	724.833			1.32511			0.662553	+	0.749015i	$0.269473\pi$
$548$		0			0
$549$		−	96.2463i		−	0.175312i
$550$		0			0
$551$	−213.644			−0.387739
$552$		0			0
$553$			417.852i			0.755609i
$554$		0			0
$555$	80.1814	+	273.310i	0.144471	+	0.492451i
$556$		0			0
$557$	583.537			1.04764			0.523822	−	0.851828i	$-0.324506\pi$
$557$	583.537			1.04764			0.523822	+	0.851828i	$0.324506\pi$
$558$		0			0
$559$	295.260			0.528194
$560$		0			0
$561$	252.836	−	66.9337i	0.450688	−	0.119311i
$562$		0			0
$563$	262.211			0.465739			0.232869	−	0.972508i	$-0.425189\pi$
$563$	262.211			0.465739			0.232869	+	0.972508i	$0.425189\pi$
$564$		0			0
$565$	372.558	−	109.298i	0.659395	−	0.193448i
$566$		0			0
$567$	142.584			0.251471
$568$		0			0
$569$			867.460i			1.52453i	0.647262	+	0.762267i	$0.275914\pi$
$569$			867.460i			1.52453i	−0.647262	+	0.762267i	$0.724086\pi$
$570$		0			0
$571$		−	630.208i		−	1.10369i	−0.833946	−	0.551846i	$-0.813924\pi$
$571$		−	630.208i		−	1.10369i	0.833946	−	0.551846i	$-0.186076\pi$
$572$		0			0
$573$		−	98.2895i		−	0.171535i
$574$		0			0
$575$	303.305	+	472.439i	0.527487	+	0.821634i
$576$		0			0
$577$			769.176i			1.33306i	0.745478	+	0.666530i	$0.232221\pi$
$577$			769.176i			1.33306i	−0.745478	+	0.666530i	$0.767779\pi$
$578$		0			0
$579$			166.670i			0.287859i
$580$		0			0
$581$	299.562			0.515597
$582$		0			0
$583$	454.447	−	120.306i	0.779498	−	0.206358i
$584$		0			0
$585$	121.574	+	414.403i	0.207819	+	0.708381i
$586$		0			0
$587$		−	523.523i		−	0.891861i	−0.895068	−	0.445931i	$-0.852873\pi$
$587$		−	523.523i		−	0.891861i	0.895068	−	0.445931i	$-0.147127\pi$
$588$		0			0
$589$		−	221.446i		−	0.375970i
$590$		0			0
$591$		−	350.369i		−	0.592840i
$592$		0			0
$593$	−8.65008			−0.0145870			−0.00729349	−	0.999973i	$-0.502322\pi$
$593$	−8.65008			−0.0145870			−0.00729349	+	0.999973i	$0.502322\pi$
$594$		0			0
$595$	86.9588	+	296.412i	0.146149	+	0.498171i
$596$		0			0
$597$			52.8912i			0.0885950i
$598$		0			0
$599$	−1124.34			−1.87703			−0.938517	−	0.345233i	$-0.887800\pi$
$599$	−1124.34			−1.87703			−0.938517	+	0.345233i	$0.887800\pi$
$600$		0			0
$601$		−	533.572i		−	0.887807i	−0.896075	−	0.443903i	$-0.853593\pi$
$601$		−	533.572i		−	0.887807i	0.896075	−	0.443903i	$-0.146407\pi$
$602$		0			0
$603$			45.6816i			0.0757572i
$604$		0			0
$605$	−435.244	−	420.224i	−0.719411	−	0.694584i
$606$		0			0
$607$	933.711			1.53824			0.769119	−	0.639106i	$-0.220695\pi$
$607$	933.711			1.53824			0.769119	+	0.639106i	$0.220695\pi$
$608$		0			0
$609$	112.000			0.183908
$610$		0			0
$611$			546.742i			0.894831i
$612$		0			0
$613$	216.288			0.352835			0.176417	−	0.984315i	$-0.443549\pi$
$613$	216.288			0.352835			0.176417	+	0.984315i	$0.443549\pi$
$614$		0			0
$615$	−78.4499	−	267.408i	−0.127561	−	0.434810i
$616$		0			0
$617$		−	5.63643i		−	0.00913522i	−0.999990	−	0.00456761i	$-0.998546\pi$
$617$		−	5.63643i		−	0.00913522i	0.999990	−	0.00456761i	$-0.00145392\pi$
$618$		0			0
$619$	−278.870			−0.450516			−0.225258	−	0.974299i	$-0.572323\pi$
$619$	−278.870			−0.450516			−0.225258	+	0.974299i	$0.572323\pi$
$620$		0			0
$621$	411.048			0.661913
$622$		0			0
$623$	129.275			0.207504
$624$		0			0
$625$	260.168	−	568.276i	0.416269	−	0.909241i
$626$		0			0
$627$	−17.9970	−	67.9819i	−0.0287033	−	0.108424i
$628$		0			0
$629$		−	1143.08i		−	1.81730i
$630$		0			0
$631$	452.843			0.717659			0.358830	−	0.933403i	$-0.383176\pi$
$631$	452.843			0.717659			0.358830	+	0.933403i	$0.383176\pi$
$632$		0			0
$633$	158.392			0.250224
$634$		0			0
$635$	−187.112	−	637.800i	−0.294665	−	1.00441i
$636$		0			0
$637$	−453.139			−0.711365
$638$		0			0
$639$	−675.576			−1.05724
$640$		0			0
$641$	457.802			0.714199			0.357100	−	0.934066i	$-0.383766\pi$
$641$	457.802			0.714199			0.357100	+	0.934066i	$0.383766\pi$
$642$		0			0
$643$			1242.30i			1.93203i	0.258481	+	0.966016i	$0.416778\pi$
$643$			1242.30i			1.93203i	−0.258481	+	0.966016i	$0.583222\pi$
$644$		0			0
$645$	−139.523	+	40.9320i	−0.216314	+	0.0634604i
$646$		0			0
$647$		−	1045.87i		−	1.61649i	−0.588848	−	0.808243i	$-0.700419\pi$
$647$		−	1045.87i		−	1.61649i	0.588848	−	0.808243i	$-0.299581\pi$
$648$		0			0
$649$	83.6232	+	315.879i	0.128849	+	0.486717i
$650$		0			0
$651$			116.090i			0.178326i
$652$		0			0
$653$			1051.88i			1.61085i	0.592699	+	0.805424i	$0.298062\pi$
$653$			1051.88i			1.61085i	−0.592699	+	0.805424i	$0.701938\pi$
$654$		0			0
$655$	710.158	−	208.340i	1.08421	−	0.318077i
$656$		0			0
$657$	383.572			0.583823
$658$		0			0
$659$			86.2083i			0.130817i	0.997859	+	0.0654084i	$0.0208350\pi$
$659$			86.2083i			0.130817i	−0.997859	+	0.0654084i	$0.979165\pi$
$660$		0			0
$661$	−688.664			−1.04185			−0.520926	−	0.853602i	$-0.674413\pi$
$661$	−688.664			−1.04185			−0.520926	+	0.853602i	$0.674413\pi$
$662$		0			0
$663$			262.786i			0.396360i
$664$		0			0
$665$	79.6985	−	23.3813i	0.119847	−	0.0351598i
$666$		0			0
$667$	−816.912			−1.22476
$668$		0			0
$669$	−85.0063			−0.127065
$670$		0			0
$671$	−130.959	+	34.6689i	−0.195170	+	0.0516676i
$672$		0			0
$673$	811.827			1.20628			0.603140	−	0.797635i	$-0.293916\pi$
$673$	811.827			1.20628			0.603140	+	0.797635i	$0.293916\pi$
$674$		0			0
$675$	−247.216	−	385.072i	−0.366245	−	0.570478i
$676$		0			0
$677$	236.610			0.349498			0.174749	−	0.984613i	$-0.444089\pi$
$677$	236.610			0.349498			0.174749	+	0.984613i	$0.444089\pi$
$678$		0			0
$679$			101.033i			0.148797i
$680$		0			0
$681$		−	292.492i		−	0.429504i
$682$		0			0
$683$			101.645i			0.148821i	0.997228	+	0.0744105i	$0.0237075\pi$
$683$			101.645i			0.148821i	−0.997228	+	0.0744105i	$0.976292\pi$
$684$		0			0
$685$	−1020.95	+	299.517i	−1.49044	+	0.437252i
$686$		0			0
$687$			181.272i			0.263861i
$688$		0			0
$689$			472.332i			0.685533i
$690$		0			0
$691$	−150.582			−0.217919			−0.108959	−	0.994046i	$-0.534752\pi$
$691$	−150.582			−0.217919			−0.108959	+	0.994046i	$0.534752\pi$
$692$		0			0
$693$	−62.2254	−	235.051i	−0.0897913	−	0.339179i
$694$		0			0
$695$	−1182.85	+	347.016i	−1.70195	+	0.499303i
$696$		0			0
$697$			1118.40i			1.60459i
$698$		0			0
$699$		−	45.7930i		−	0.0655122i
$700$		0			0
$701$			704.744i			1.00534i	0.864478	+	0.502671i	$0.167649\pi$
$701$			704.744i			1.00534i	−0.864478	+	0.502671i	$0.832351\pi$
$702$		0			0
$703$	−307.349			−0.437197
$704$		0			0
$705$	−75.7948	−	258.358i	−0.107510	−	0.366465i
$706$		0			0
$707$		−	486.713i		−	0.688421i
$708$		0			0
$709$	−885.583			−1.24906			−0.624529	−	0.781001i	$-0.714709\pi$
$709$	−885.583			−1.24906			−0.624529	+	0.781001i	$0.714709\pi$
$710$		0			0
$711$			1154.54i			1.62383i
$712$		0			0
$713$		−	846.746i		−	1.18758i
$714$		0			0
$715$	520.071	−	314.694i	0.727372	−	0.440131i
$716$		0			0
$717$	−10.9841			−0.0153195
$718$		0			0
$719$	1128.12			1.56901			0.784504	−	0.620123i	$-0.212917\pi$
$719$	1128.12			1.56901			0.784504	+	0.620123i	$0.212917\pi$
$720$		0			0
$721$			258.519i			0.358557i
$722$		0			0
$723$	171.178			0.236761
$724$		0			0
$725$	491.313	+	765.288i	0.677674	+	1.05557i
$726$		0			0
$727$			626.987i			0.862430i	0.902249	+	0.431215i	$0.141915\pi$
$727$			626.987i			0.862430i	−0.902249	+	0.431215i	$0.858085\pi$
$728$		0			0
$729$	214.644			0.294436
$730$		0			0
$731$	583.534			0.798268
$732$		0			0
$733$	141.276			0.192737			0.0963685	−	0.995346i	$-0.469277\pi$
$733$	141.276			0.192737			0.0963685	+	0.995346i	$0.469277\pi$
$734$		0			0
$735$	214.127	−	62.8188i	0.291329	−	0.0854677i
$736$		0			0
$737$	62.1573	−	16.4550i	0.0843383	−	0.0223270i
$738$		0			0
$739$			848.418i			1.14806i	0.818834	+	0.574031i	$0.194621\pi$
$739$			848.418i			1.14806i	−0.818834	+	0.574031i	$0.805379\pi$
$740$		0			0
$741$	70.6574			0.0953541
$742$		0			0
$743$	−1315.28			−1.77023			−0.885113	−	0.465376i	$-0.845919\pi$
$743$	−1315.28			−1.77023			−0.885113	+	0.465376i	$0.845919\pi$
$744$		0			0
$745$	65.6637	−	19.2639i	0.0881393	−	0.0258575i
$746$		0			0
$747$	827.703			1.10804
$748$		0			0
$749$	236.165			0.315307
$750$		0			0
$751$	544.405			0.724906			0.362453	−	0.932002i	$-0.381939\pi$
$751$	544.405			0.724906			0.362453	+	0.932002i	$0.381939\pi$
$752$		0			0
$753$			452.767i			0.601285i
$754$		0			0
$755$	816.292	−	239.477i	1.08118	−	0.317188i
$756$		0			0
$757$		−	1065.19i		−	1.40712i	−0.710634	−	0.703562i	$-0.751592\pi$
$757$		−	1065.19i		−	1.40712i	0.710634	−	0.703562i	$-0.248408\pi$
$758$		0			0
$759$	−68.8151	−	259.943i	−0.0906654	−	0.342481i
$760$		0			0
$761$		−	1137.39i		−	1.49460i	−0.664488	−	0.747299i	$-0.731350\pi$
$761$		−	1137.39i		−	1.49460i	0.664488	−	0.747299i	$-0.268650\pi$
$762$		0			0
$763$			31.6126i			0.0414319i
$764$		0			0
$765$	240.271	+	819.000i	0.314080	+	1.07059i
$766$		0			0
$767$	−328.311			−0.428045
$768$		0			0
$769$			375.854i			0.488756i	0.969680	+	0.244378i	$0.0785838\pi$
$769$			375.854i			0.488756i	−0.969680	+	0.244378i	$0.921416\pi$
$770$		0			0
$771$	140.438			0.182151
$772$		0			0
$773$		−	176.434i		−	0.228245i	−0.993467	−	0.114123i	$-0.963594\pi$
$773$		−	176.434i		−	0.228245i	0.993467	−	0.114123i	$-0.0364057\pi$
$774$		0			0
$775$	−793.237	+	509.256i	−1.02353	+	0.657105i
$776$		0			0
$777$	161.124			0.207366
$778$		0			0
$779$	300.712			0.386023
$780$		0			0
$781$	243.350	+	919.232i	0.311587	+	1.17699i
$782$		0			0
$783$	665.843			0.850374
$784$		0			0
$785$	−175.244	+	51.4115i	−0.223240	+	0.0654923i
$786$		0			0
$787$	81.8506			0.104003			0.0520017	−	0.998647i	$-0.483440\pi$
$787$	81.8506			0.104003			0.0520017	+	0.998647i	$0.483440\pi$
$788$		0			0
$789$		−	249.388i		−	0.316081i
$790$		0			0
$791$		−	219.633i		−	0.277665i
$792$		0			0
$793$		−	136.113i		−	0.171643i
$794$		0			0
$795$	−65.4794	−	223.196i	−0.0823640	−	0.280750i
$796$		0			0
$797$			115.706i			0.145177i	0.997362	+	0.0725886i	$0.0231260\pi$
$797$			115.706i			0.145177i	−0.997362	+	0.0725886i	$0.976874\pi$
$798$		0			0
$799$			1080.55i			1.35237i
$800$		0			0
$801$	357.192			0.445932
$802$		0			0
$803$	−138.167	−	521.912i	−0.172063	−	0.649953i
$804$		0			0
$805$	304.744	−	89.4031i	0.378564	−	0.111060i
$806$		0			0
$807$		−	403.671i		−	0.500212i
$808$		0			0
$809$		−	793.230i		−	0.980507i	−0.871580	−	0.490254i	$-0.836904\pi$
$809$		−	793.230i		−	0.980507i	0.871580	−	0.490254i	$-0.163096\pi$
$810$		0			0
$811$		−	290.394i		−	0.358069i	−0.983843	−	0.179035i	$-0.942703\pi$
$811$		−	290.394i		−	0.358069i	0.983843	−	0.179035i	$-0.0572974\pi$
$812$		0			0
$813$	254.965			0.313610
$814$		0			0
$815$	−484.165	+	142.040i	−0.594067	+	0.174282i
$816$		0			0
$817$		−	156.899i		−	0.192043i
$818$		0			0
$819$	244.301			0.298292
$820$		0			0
$821$			850.021i			1.03535i	0.855578	+	0.517674i	$0.173202\pi$
$821$			850.021i			1.03535i	−0.855578	+	0.517674i	$0.826798\pi$
$822$		0			0
$823$		−	446.765i		−	0.542850i	−0.962460	−	0.271425i	$-0.912505\pi$
$823$		−	446.765i		−	0.542850i	0.962460	−	0.271425i	$-0.0874949\pi$
$824$		0			0
$825$	−202.129	+	220.803i	−0.245005	+	0.267640i
$826$		0			0
$827$	−1078.95			−1.30465			−0.652327	−	0.757938i	$-0.726207\pi$
$827$	−1078.95			−1.30465			−0.652327	+	0.757938i	$0.726207\pi$
$828$		0			0
$829$	−560.883			−0.676578			−0.338289	−	0.941042i	$-0.609848\pi$
$829$	−560.883			−0.676578			−0.338289	+	0.941042i	$0.609848\pi$
$830$		0			0
$831$		−	520.508i		−	0.626363i
$832$		0			0
$833$	−895.556			−1.07510
$834$		0			0
$835$	147.301	+	502.098i	0.176409	+	0.601315i
$836$		0			0
$837$			690.159i			0.824563i
$838$		0			0
$839$	−82.8835			−0.0987884			−0.0493942	−	0.998779i	$-0.515729\pi$
$839$	−82.8835			−0.0987884			−0.0493942	+	0.998779i	$0.515729\pi$
$840$		0			0
$841$	−482.288			−0.573470
$842$		0			0
$843$	371.027			0.440127
$844$		0			0
$845$	−65.9421	−	224.773i	−0.0780379	−	0.266004i
$846$		0			0
$847$	−297.411	+	169.336i	−0.351135	+	0.199924i
$848$		0			0
$849$		−	315.542i		−	0.371663i
$850$		0			0
$851$	−1175.21			−1.38098
$852$		0			0
$853$	−412.282			−0.483332			−0.241666	−	0.970359i	$-0.577694\pi$
$853$	−412.282			−0.483332			−0.241666	+	0.970359i	$0.577694\pi$
$854$		0			0
$855$	220.211	−	64.6035i	0.257556	−	0.0755597i
$856$		0			0
$857$	−1108.08			−1.29297			−0.646485	−	0.762927i	$-0.723762\pi$
$857$	−1108.08			−1.29297			−0.646485	+	0.762927i	$0.723762\pi$
$858$		0			0
$859$	−785.281			−0.914181			−0.457090	−	0.889420i	$-0.651108\pi$
$859$	−785.281			−0.914181			−0.457090	+	0.889420i	$0.651108\pi$
$860$		0			0
$861$	−157.644			−0.183094
$862$		0			0
$863$			749.791i			0.868819i	0.900716	+	0.434409i	$0.143043\pi$
$863$			749.791i			0.868819i	−0.900716	+	0.434409i	$0.856957\pi$
$864$		0			0
$865$	−290.853	−	991.416i	−0.336247	−	1.14615i
$866$		0			0
$867$			204.765i			0.236177i
$868$		0			0
$869$	1570.95	−	415.879i	1.80776	−	0.478572i
$870$		0			0
$871$			64.6035i			0.0741717i
$872$		0			0
$873$			279.159i			0.319770i
$874$		0			0
$875$	−267.035	−	231.716i	−0.305183	−	0.264819i
$876$		0			0
$877$	−273.360			−0.311699			−0.155849	−	0.987781i	$-0.549811\pi$
$877$	−273.360			−0.311699			−0.155849	+	0.987781i	$0.549811\pi$
$878$		0			0
$879$		−	407.054i		−	0.463087i
$880$		0			0
$881$	−795.076			−0.902470			−0.451235	−	0.892405i	$-0.649016\pi$
$881$	−795.076			−0.902470			−0.451235	+	0.892405i	$0.649016\pi$
$882$		0			0
$883$		−	1552.42i		−	1.75812i	−0.476711	−	0.879060i	$-0.658171\pi$
$883$		−	1552.42i		−	1.75812i	0.476711	−	0.879060i	$-0.341829\pi$
$884$		0			0
$885$	155.140	−	45.5137i	0.175300	−	0.0514280i
$886$		0			0
$887$	−865.828			−0.976131			−0.488066	−	0.872807i	$-0.662297\pi$
$887$	−865.828			−0.976131			−0.488066	+	0.872807i	$0.662297\pi$
$888$		0			0
$889$	−376.000			−0.422947
$890$		0			0
$891$	−141.911	−	536.055i	−0.159271	−	0.601633i
$892$		0			0
$893$	290.534			0.325347
$894$		0			0
$895$	−176.106	−	600.284i	−0.196767	−	0.670708i
$896$		0			0
$897$	270.173			0.301196
$898$		0			0
$899$		−	1371.61i		−	1.52571i
$900$		0			0
$901$			933.487i			1.03606i
$902$		0			0
$903$			82.2523i			0.0910878i
$904$		0			0
$905$	128.963	+	439.589i	0.142501	+	0.485734i
$906$		0			0
$907$			1393.40i			1.53628i	0.640284	+	0.768139i	$0.278817\pi$
$907$			1393.40i			1.53628i	−0.640284	+	0.768139i	$0.721183\pi$
$908$		0			0
$909$		−	1344.81i		−	1.47944i
$910$		0			0
$911$	−1578.26			−1.73245			−0.866224	−	0.499655i	$-0.833460\pi$
$911$	−1578.26			−1.73245			−0.866224	+	0.499655i	$0.833460\pi$
$912$		0			0
$913$	−298.147	−	1126.23i	−0.326558	−	1.23354i
$914$		0			0
$915$	18.8693	+	64.3189i	0.0206222	+	0.0702938i
$916$		0			0
$917$		−	418.657i		−	0.456551i
$918$		0			0
$919$		−	249.051i		−	0.271002i	−0.990777	−	0.135501i	$-0.956736\pi$
$919$		−	249.051i		−	0.271002i	0.990777	−	0.135501i	$-0.0432644\pi$
$920$		0			0
$921$			399.451i			0.433715i
$922$		0			0
$923$	−955.408			−1.03511
$924$		0			0
$925$	706.805	+	1100.95i	0.764114	+	1.19021i
$926$		0			0
$927$			714.301i			0.770551i
$928$		0			0
$929$	604.288			0.650471			0.325235	−	0.945633i	$-0.394556\pi$
$929$	604.288			0.650471			0.325235	+	0.945633i	$0.394556\pi$
$930$		0			0
$931$			240.795i			0.258641i
$932$		0			0
$933$			296.725i			0.318033i
$934$		0			0
$935$	1027.84	−	621.941i	1.09929	−	0.665177i
$936$		0			0
$937$	−1248.76			−1.33272			−0.666361	−	0.745629i	$-0.732149\pi$
$937$	−1248.76			−1.33272			−0.666361	+	0.745629i	$0.732149\pi$
$938$		0			0
$939$	−91.3775			−0.0973136
$940$		0			0
$941$		−	615.457i		−	0.654046i	−0.945016	−	0.327023i	$-0.893955\pi$
$941$		−	615.457i		−	0.654046i	0.945016	−	0.327023i	$-0.106045\pi$
$942$		0			0
$943$	1149.83			1.21934
$944$		0			0
$945$	−248.388	+	72.8701i	−0.262845	+	0.0771112i
$946$		0			0
$947$		−	262.503i		−	0.277194i	−0.990349	−	0.138597i	$-0.955741\pi$
$947$		−	262.503i		−	0.277194i	0.990349	−	0.138597i	$-0.0442593\pi$
$948$		0			0
$949$	542.452			0.571604
$950$		0			0
$951$	300.623			0.316112
$952$		0			0
$953$	−528.907			−0.554992			−0.277496	−	0.960727i	$-0.589504\pi$
$953$	−528.907			−0.554992			−0.277496	+	0.960727i	$0.589504\pi$
$954$		0			0
$955$	−127.093	−	433.214i	−0.133081	−	0.453628i
$956$		0			0
$957$	−111.471	−	421.073i	−0.116480	−	0.439992i
$958$		0			0
$959$			601.876i			0.627608i
$960$		0			0
$961$	460.706			0.479402
$962$		0			0
$963$	652.534			0.677605
$964$		0			0
$965$	215.512	+	734.605i	0.223329	+	0.761249i
$966$		0			0
$967$	−735.954			−0.761069			−0.380534	−	0.924767i	$-0.624260\pi$
$967$	−735.954			−0.761069			−0.380534	+	0.924767i	$0.624260\pi$
$968$		0			0
$969$	139.643			0.144110
$970$		0			0
$971$	1451.82			1.49518			0.747588	−	0.664163i	$-0.231212\pi$
$971$	1451.82			1.49518			0.747588	+	0.664163i	$0.231212\pi$
$972$		0			0
$973$			697.324i			0.716674i
$974$		0			0
$975$	−162.489	−	253.100i	−0.166656	−	0.259590i
$976$		0			0
$977$		−	1340.06i		−	1.37161i	−0.727785	−	0.685806i	$-0.759450\pi$
$977$		−	1340.06i		−	1.37161i	0.727785	−	0.685806i	$-0.240550\pi$
$978$		0			0
$979$	−128.664	−	486.018i	−0.131424	−	0.496444i
$980$		0			0
$981$			87.3470i			0.0890387i
$982$		0			0
$983$		−	662.938i		−	0.674403i	−0.941432	−	0.337202i	$-0.890520\pi$
$983$		−	662.938i		−	0.674403i	0.941432	−	0.337202i	$-0.109480\pi$
$984$		0			0
$985$	−453.043	−	1544.26i	−0.459942	−	1.56778i
$986$		0			0
$987$	−152.309			−0.154315
$988$		0			0
$989$		−	599.936i		−	0.606609i
$990$		0			0
$991$	650.836			0.656747			0.328373	−	0.944548i	$-0.393500\pi$
$991$	650.836			0.656747			0.328373	+	0.944548i	$0.393500\pi$
$992$		0			0
$993$		−	458.404i		−	0.461635i
$994$		0			0
$995$	68.3908	+	233.120i	0.0687344	+	0.234291i
$996$		0			0
$997$	189.166			0.189735			0.0948676	−	0.995490i	$-0.469757\pi$
$997$	189.166			0.189735			0.0948676	+	0.995490i	$0.469757\pi$
$998$		0			0
$999$	957.884			0.958843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.3.i.g.769.6		8
4.3	odd	2		110.3.c.b.109.2	✓	8
5.4	even	2	inner	880.3.i.g.769.3		8
11.10	odd	2	inner	880.3.i.g.769.5		8
12.11	even	2		990.3.h.b.109.5		8
20.3	even	4		550.3.d.d.351.7		8
20.7	even	4		550.3.d.d.351.2		8
20.19	odd	2		110.3.c.b.109.7	yes	8
44.43	even	2		110.3.c.b.109.6	yes	8
55.54	odd	2	inner	880.3.i.g.769.4		8
60.59	even	2		990.3.h.b.109.2		8
132.131	odd	2		990.3.h.b.109.1		8
220.43	odd	4		550.3.d.d.351.3		8
220.87	odd	4		550.3.d.d.351.6		8
220.219	even	2		110.3.c.b.109.3	yes	8
660.659	odd	2		990.3.h.b.109.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.3.c.b.109.2	✓	8	4.3	odd	2
110.3.c.b.109.3	yes	8	220.219	even	2
110.3.c.b.109.6	yes	8	44.43	even	2
110.3.c.b.109.7	yes	8	20.19	odd	2
550.3.d.d.351.2		8	20.7	even	4
550.3.d.d.351.3		8	220.43	odd	4
550.3.d.d.351.6		8	220.87	odd	4
550.3.d.d.351.7		8	20.3	even	4
880.3.i.g.769.3		8	5.4	even	2	inner
880.3.i.g.769.4		8	55.54	odd	2	inner
880.3.i.g.769.5		8	11.10	odd	2	inner
880.3.i.g.769.6		8	1.1	even	1	trivial
990.3.h.b.109.1		8	132.131	odd	2
990.3.h.b.109.2		8	60.59	even	2
990.3.h.b.109.5		8	12.11	even	2
990.3.h.b.109.6		8	660.659	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	880.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.08854i q^{3} +(1.40754 + 4.79780i) q^{5} +2.82843 q^{7} +7.81507 q^{9} +O(q^{10})\)	\(q+1.08854i q^{3} +(1.40754 + 4.79780i) q^{5} +2.82843 q^{7} +7.81507 q^{9} +(-2.81507 - 10.6337i) q^{11} +11.0522 q^{13} +(-5.22261 + 1.53216i) q^{15} +21.8428 q^{17} -5.87305i q^{19} +3.07887i q^{21} -22.4568i q^{23} +(-21.0377 + 13.5061i) q^{25} +18.3039i q^{27} -36.3770i q^{29} +37.7055 q^{31} +(11.5752 - 3.06433i) q^{33} +(3.98111 + 13.5702i) q^{35} -52.3321i q^{37} +12.0308i q^{39} +51.2020i q^{41} +26.7151 q^{43} +(11.0000 + 37.4951i) q^{45} +49.4691i q^{47} -41.0000 q^{49} +23.7769i q^{51} +42.7365i q^{53} +(47.0560 - 28.4735i) q^{55} +6.39307 q^{57} -29.7055 q^{59} -12.3155i q^{61} +22.1044 q^{63} +(15.5563 + 53.0261i) q^{65} +5.84532i q^{67} +24.4452 q^{69} -86.4452 q^{71} +49.0810 q^{73} +(-14.7020 - 22.9004i) q^{75} +(-7.96223 - 30.0766i) q^{77} +147.733i q^{79} +50.4110 q^{81} +105.911 q^{83} +(30.7446 + 104.797i) q^{85} +39.5980 q^{87} +45.7055 q^{89} +31.2603 q^{91} +41.0441i q^{93} +(28.1777 - 8.26653i) q^{95} +35.7206i q^{97} +(-22.0000 - 83.1031i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{5} + 20 q^{9}+O(q^{10}) \) 8 * q - 10 * q^5 + 20 * q^9	\( 8 q - 10 q^{5} + 20 q^{9} + 20 q^{11} + 22 q^{15} - 62 q^{25} + 4 q^{31} + 88 q^{45} - 328 q^{49} - 138 q^{55} + 60 q^{59} + 68 q^{69} - 564 q^{71} - 394 q^{75} - 192 q^{81} + 68 q^{89} + 80 q^{91} - 176 q^{99}+O(q^{100}) \) 8 * q - 10 * q^5 + 20 * q^9 + 20 * q^11 + 22 * q^15 - 62 * q^25 + 4 * q^31 + 88 * q^45 - 328 * q^49 - 138 * q^55 + 60 * q^59 + 68 * q^69 - 564 * q^71 - 394 * q^75 - 192 * q^81 + 68 * q^89 + 80 * q^91 - 176 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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