LMFDB - Embedded newform 880.3.i.g.769.7

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			769.7
Root			$1.41421 + 3.43731i$ of defining polynomial
Character	$\chi$	$=$	880.769
Dual form			880.3.i.g.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.43731i q^{3} +(-3.90754 - 3.11948i) q^{5} -2.82843 q^{7} -2.81507 q^{9} +O(q^{10})$	$q+3.43731i q^{3} +(-3.90754 - 3.11948i) q^{5} -2.82843 q^{7} -2.81507 q^{9} +(7.81507 + 7.74110i) q^{11} +3.98111 q^{13} +(10.7226 - 13.4314i) q^{15} +23.2571 q^{17} -33.1286i q^{19} -9.72217i q^{21} +2.16600i q^{23} +(5.53768 + 24.3790i) q^{25} +21.2595i q^{27} +11.5201i q^{29} -36.7055 q^{31} +(-26.6085 + 26.8628i) q^{33} +(11.0522 + 8.82322i) q^{35} +17.4456i q^{37} +13.6843i q^{39} +45.0151i q^{41} +63.4847 q^{43} +(11.0000 + 8.78157i) q^{45} +46.5919i q^{47} -41.0000 q^{49} +79.9416i q^{51} -11.2066i q^{53} +(-6.38947 - 54.6276i) q^{55} +113.873 q^{57} +44.7055 q^{59} +38.8887i q^{61} +7.96223 q^{63} +(-15.5563 - 12.4190i) q^{65} +91.5360i q^{67} -7.44522 q^{69} -54.5548 q^{71} +56.1521 q^{73} +(-83.7980 + 19.0347i) q^{75} +(-22.1044 - 21.8951i) q^{77} +101.917i q^{79} -98.4110 q^{81} -15.7113 q^{83} +(-90.8778 - 72.5499i) q^{85} -39.5980 q^{87} -28.7055 q^{89} -11.2603 q^{91} -126.168i q^{93} +(-103.344 + 129.451i) q^{95} +76.2564i q^{97} +(-22.0000 - 21.7918i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			3.43731i			1.14577i	0.819636	+	0.572884i	$0.194176\pi$
$3$			3.43731i			1.14577i	−0.819636	+	0.572884i	$0.805824\pi$
$4$		0			0
$5$	−3.90754	−	3.11948i	−0.781507	−	0.623896i
$5$	−3.90754	−	3.11948i	−0.781507	−	0.623896i
$6$		0			0
$7$	−2.82843			−0.404061			−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−2.82843			−0.404061			−0.202031	+	0.979379i	$0.564754\pi$
$8$		0			0
$9$	−2.81507			−0.312786
$10$		0			0
$11$	7.81507	+	7.74110i	0.710461	+	0.703736i
$11$	7.81507	+	7.74110i	0.710461	+	0.703736i
$12$		0			0
$13$	3.98111			0.306240			0.153120	−	0.988208i	$-0.451068\pi$
$13$	3.98111			0.306240			0.153120	+	0.988208i	$0.451068\pi$
$14$		0			0
$15$	10.7226	−	13.4314i	0.714841	−	0.895427i
$16$		0			0
$17$	23.2571			1.36806			0.684031	−	0.729453i	$-0.260225\pi$
$17$	23.2571			1.36806			0.684031	+	0.729453i	$0.260225\pi$
$18$		0			0
$19$		−	33.1286i		−	1.74361i	−0.489850	−	0.871807i	$-0.662949\pi$
$19$		−	33.1286i		−	1.74361i	0.489850	−	0.871807i	$-0.337051\pi$
$20$		0			0
$21$		−	9.72217i		−	0.462960i
$22$		0			0
$23$			2.16600i			0.0941741i	0.998891	+	0.0470870i	$0.0149938\pi$
$23$			2.16600i			0.0941741i	−0.998891	+	0.0470870i	$0.985006\pi$
$24$		0			0
$25$	5.53768	+	24.3790i	0.221507	+	0.975159i
$26$		0			0
$27$			21.2595i			0.787388i
$28$		0			0
$29$			11.5201i			0.397244i	0.980076	+	0.198622i	$0.0636465\pi$
$29$			11.5201i			0.397244i	−0.980076	+	0.198622i	$0.936353\pi$
$30$		0			0
$31$	−36.7055			−1.18405			−0.592024	−	0.805920i	$-0.701671\pi$
$31$	−36.7055			−1.18405			−0.592024	+	0.805920i	$0.701671\pi$
$32$		0			0
$33$	−26.6085	+	26.8628i	−0.806319	+	0.814024i
$34$		0			0
$35$	11.0522	+	8.82322i	0.315777	+	0.252092i
$36$		0			0
$37$			17.4456i			0.471502i	0.971813	+	0.235751i	$0.0757550\pi$
$37$			17.4456i			0.471502i	−0.971813	+	0.235751i	$0.924245\pi$
$38$		0			0
$39$			13.6843i			0.350880i
$40$		0			0
$41$			45.0151i			1.09793i	0.835846	+	0.548964i	$0.184978\pi$
$41$			45.0151i			1.09793i	−0.835846	+	0.548964i	$0.815022\pi$
$42$		0			0
$43$	63.4847			1.47639			0.738194	−	0.674589i	$-0.235679\pi$
$43$	63.4847			1.47639			0.738194	+	0.674589i	$0.235679\pi$
$44$		0			0
$45$	11.0000	+	8.78157i	0.244444	+	0.195146i
$46$		0			0
$47$			46.5919i			0.991318i	0.868517	+	0.495659i	$0.165073\pi$
$47$			46.5919i			0.991318i	−0.868517	+	0.495659i	$0.834927\pi$
$48$		0			0
$49$	−41.0000			−0.836735
$50$		0			0
$51$			79.9416i			1.56748i
$52$		0			0
$53$		−	11.2066i		−	0.211446i	−0.994396	−	0.105723i	$-0.966284\pi$
$53$		−	11.2066i		−	0.211446i	0.994396	−	0.105723i	$-0.0337156\pi$
$54$		0			0
$55$	−6.38947	−	54.6276i	−0.116172	−	0.993229i
$56$		0			0
$57$	113.873			1.99778
$58$		0			0
$59$	44.7055			0.757721			0.378860	−	0.925454i	$-0.376316\pi$
$59$	44.7055			0.757721			0.378860	+	0.925454i	$0.376316\pi$
$60$		0			0
$61$			38.8887i			0.637519i	0.947836	+	0.318760i	$0.103266\pi$
$61$			38.8887i			0.637519i	−0.947836	+	0.318760i	$0.896734\pi$
$62$		0			0
$63$	7.96223			0.126385
$64$		0			0
$65$	−15.5563	−	12.4190i	−0.239328	−	0.191062i
$66$		0			0
$67$			91.5360i			1.36621i	0.730321	+	0.683104i	$0.239371\pi$
$67$			91.5360i			1.36621i	−0.730321	+	0.683104i	$0.760629\pi$
$68$		0			0
$69$	−7.44522			−0.107902
$70$		0			0
$71$	−54.5548			−0.768377			−0.384189	−	0.923255i	$-0.625519\pi$
$71$	−54.5548			−0.768377			−0.384189	+	0.923255i	$0.625519\pi$
$72$		0			0
$73$	56.1521			0.769206			0.384603	−	0.923082i	$-0.374338\pi$
$73$	56.1521			0.769206			0.384603	+	0.923082i	$0.374338\pi$
$74$		0			0
$75$	−83.7980	+	19.0347i	−1.11731	+	0.253796i
$76$		0			0
$77$	−22.1044	−	21.8951i	−0.287070	−	0.284352i
$78$		0			0
$79$			101.917i			1.29008i	0.764148	+	0.645041i	$0.223160\pi$
$79$			101.917i			1.29008i	−0.764148	+	0.645041i	$0.776840\pi$
$80$		0			0
$81$	−98.4110			−1.21495
$82$		0			0
$83$	−15.7113			−0.189293			−0.0946463	−	0.995511i	$-0.530172\pi$
$83$	−15.7113			−0.189293			−0.0946463	+	0.995511i	$0.530172\pi$
$84$		0			0
$85$	−90.8778	−	72.5499i	−1.06915	−	0.853528i
$86$		0			0
$87$	−39.5980			−0.455149
$88$		0			0
$89$	−28.7055			−0.322534			−0.161267	−	0.986911i	$-0.551558\pi$
$89$	−28.7055			−0.322534			−0.161267	+	0.986911i	$0.551558\pi$
$90$		0			0
$91$	−11.2603			−0.123739
$92$		0			0
$93$		−	126.168i		−	1.35665i
$94$		0			0
$95$	−103.344	+	129.451i	−1.08783	+	1.36265i
$96$		0			0
$97$			76.2564i			0.786148i	0.919507	+	0.393074i	$0.128588\pi$
$97$			76.2564i			0.786148i	−0.919507	+	0.393074i	$0.871412\pi$
$98$		0			0
$99$	−22.0000	−	21.7918i	−0.222222	−	0.220119i
$100$		0			0
$101$		−	76.7122i		−	0.759526i	−0.925084	−	0.379763i	$-0.876005\pi$
$101$		−	76.7122i		−	0.759526i	0.925084	−	0.379763i	$-0.123995\pi$
$102$		0			0
$103$		−	113.314i		−	1.10013i	−0.835121	−	0.550066i	$-0.814603\pi$
$103$		−	113.314i		−	1.10013i	0.835121	−	0.550066i	$-0.185397\pi$
$104$		0			0
$105$	−30.3281	+	37.9897i	−0.288839	+	0.361807i
$106$		0			0
$107$	157.036			1.46763			0.733813	−	0.679352i	$-0.237739\pi$
$107$	157.036			1.46763			0.733813	+	0.679352i	$0.237739\pi$
$108$		0			0
$109$			171.403i			1.57251i	0.617904	+	0.786254i	$0.287982\pi$
$109$			171.403i			1.57251i	−0.617904	+	0.786254i	$0.712018\pi$
$110$		0			0
$111$	−59.9658			−0.540232
$112$		0			0
$113$			83.6491i			0.740258i	0.928980	+	0.370129i	$0.120686\pi$
$113$			83.6491i			0.740258i	−0.928980	+	0.370129i	$0.879314\pi$
$114$		0			0
$115$	6.75681	−	8.46374i	0.0587548	−	0.0735977i
$116$		0			0
$117$	−11.2071			−0.0957874
$118$		0			0
$119$	−65.7809			−0.552780
$120$		0			0
$121$	1.15073	+	120.995i	0.00951016	+	0.999955i
$122$		0			0
$123$	−154.731			−1.25797
$124$		0			0
$125$	54.4110	−	112.536i	0.435288	−	0.900291i
$126$		0			0
$127$	132.936			1.04674			0.523370	−	0.852105i	$-0.324674\pi$
$127$	132.936			1.04674			0.523370	+	0.852105i	$0.324674\pi$
$128$		0			0
$129$			218.216i			1.69160i
$130$		0			0
$131$		−	49.3436i		−	0.376668i	−0.982105	−	0.188334i	$-0.939691\pi$
$131$		−	49.3436i		−	0.376668i	0.982105	−	0.188334i	$-0.0603088\pi$
$132$		0			0
$133$			93.7020i			0.704526i
$134$		0			0
$135$	66.3186	−	83.0722i	0.491249	−	0.615350i
$136$		0			0
$137$		−	22.2957i		−	0.162742i	−0.996684	−	0.0813711i	$-0.974070\pi$
$137$		−	22.2957i		−	0.162742i	0.996684	−	0.0813711i	$-0.0259299\pi$
$138$		0			0
$139$		−	261.766i		−	1.88321i	−0.336723	−	0.941604i	$-0.609318\pi$
$139$		−	261.766i		−	1.88321i	0.336723	−	0.941604i	$-0.390682\pi$
$140$		0			0
$141$	−160.151			−1.13582
$142$		0			0
$143$	31.1127	+	30.8182i	0.217571	+	0.215512i
$144$		0			0
$145$	35.9366	−	45.0151i	0.247839	−	0.310449i
$146$		0			0
$147$		−	140.930i		−	0.958704i
$148$		0			0
$149$		−	266.827i		−	1.79079i	−0.445277	−	0.895393i	$-0.646895\pi$
$149$		−	266.827i		−	1.79079i	0.445277	−	0.895393i	$-0.353105\pi$
$150$		0			0
$151$			123.858i			0.820249i	0.912030	+	0.410125i	$0.134515\pi$
$151$			123.858i			0.820249i	−0.912030	+	0.410125i	$0.865485\pi$
$152$		0			0
$153$	−65.4703			−0.427910
$154$		0			0
$155$	143.428	+	114.502i	0.925343	+	0.738723i
$156$		0			0
$157$			224.955i			1.43284i	0.697671	+	0.716418i	$0.254220\pi$
$157$			224.955i			1.43284i	−0.697671	+	0.716418i	$0.745780\pi$
$158$		0			0
$159$	38.5206			0.242268
$160$		0			0
$161$		−	6.12638i		−	0.0380521i
$162$		0			0
$163$			62.8838i			0.385790i	0.981219	+	0.192895i	$0.0617877\pi$
$163$			62.8838i			0.385790i	−0.981219	+	0.192895i	$0.938212\pi$
$164$		0			0
$165$	187.772	−	21.9626i	1.13801	−	0.133106i
$166$		0			0
$167$	−104.652			−0.626658			−0.313329	−	0.949645i	$-0.601444\pi$
$167$	−104.652			−0.626658			−0.313329	+	0.949645i	$0.601444\pi$
$168$		0			0
$169$	−153.151			−0.906217
$170$		0			0
$171$			93.2596i			0.545378i
$172$		0			0
$173$	−109.059			−0.630400			−0.315200	−	0.949025i	$-0.602072\pi$
$173$	−109.059			−0.630400			−0.315200	+	0.949025i	$0.602072\pi$
$174$		0			0
$175$	−15.6629	−	68.9541i	−0.0895025	−	0.394024i
$176$		0			0
$177$			153.667i			0.868172i
$178$		0			0
$179$	98.1165			0.548137			0.274069	−	0.961710i	$-0.411630\pi$
$179$	98.1165			0.548137			0.274069	+	0.961710i	$0.411630\pi$
$180$		0			0
$181$	357.377			1.97446			0.987229	−	0.159309i	$-0.0509266\pi$
$181$	357.377			1.97446			0.987229	+	0.159309i	$0.0509266\pi$
$182$		0			0
$183$	−133.672			−0.730450
$184$		0			0
$185$	54.4212	−	68.1692i	0.294168	−	0.368482i
$186$		0			0
$187$	181.756	+	180.035i	0.971955	+	0.962755i
$188$		0			0
$189$		−	60.1309i		−	0.318153i
$190$		0			0
$191$	−164.706			−0.862333			−0.431166	−	0.902272i	$-0.641898\pi$
$191$	−164.706			−0.862333			−0.431166	+	0.902272i	$0.641898\pi$
$192$		0			0
$193$	312.919			1.62134			0.810671	−	0.585501i	$-0.199102\pi$
$193$	312.919			1.62134			0.810671	+	0.585501i	$0.199102\pi$
$194$		0			0
$195$	42.6879	−	53.4719i	0.218912	−	0.274215i
$196$		0			0
$197$	126.436			0.641809			0.320905	−	0.947112i	$-0.396013\pi$
$197$	126.436			0.641809			0.320905	+	0.947112i	$0.396013\pi$
$198$		0			0
$199$	197.411			0.992015			0.496008	−	0.868318i	$-0.334799\pi$
$199$	197.411			0.992015			0.496008	+	0.868318i	$0.334799\pi$
$200$		0			0
$201$	−314.637			−1.56536
$202$		0			0
$203$		−	32.5837i		−	0.160511i
$204$		0			0
$205$	140.424	−	175.898i	0.684993	−	0.858039i
$206$		0			0
$207$		−	6.09746i		−	0.0294563i
$208$		0			0
$209$	256.452	−	258.903i	1.22704	−	1.23877i
$210$		0			0
$211$			46.0802i			0.218390i	0.994020	+	0.109195i	$0.0348273\pi$
$211$			46.0802i			0.218390i	−0.994020	+	0.109195i	$0.965173\pi$
$212$		0			0
$213$		−	187.521i		−	0.880383i
$214$		0			0
$215$	−248.069	−	198.039i	−1.15381	−	0.921112i
$216$		0			0
$217$	103.819			0.478428
$218$		0			0
$219$			193.012i			0.881333i
$220$		0			0
$221$	92.5890			0.418955
$222$		0			0
$223$			392.748i			1.76120i	0.473860	+	0.880600i	$0.342860\pi$
$223$			392.748i			1.76120i	−0.473860	+	0.880600i	$0.657140\pi$
$224$		0			0
$225$	−15.5890	−	68.6286i	−0.0692844	−	0.305016i
$226$		0			0
$227$	268.701			1.18370			0.591851	−	0.806047i	$-0.298397\pi$
$227$	268.701			1.18370			0.591851	+	0.806047i	$0.298397\pi$
$228$		0			0
$229$	−205.528			−0.897500			−0.448750	−	0.893657i	$-0.648131\pi$
$229$	−205.528			−0.897500			−0.448750	+	0.893657i	$0.648131\pi$
$230$		0			0
$231$	75.2603	−	75.9795i	0.325802	−	0.328915i
$232$		0			0
$233$	−303.698			−1.30342			−0.651712	−	0.758467i	$-0.725949\pi$
$233$	−303.698			−1.30342			−0.651712	+	0.758467i	$0.725949\pi$
$234$		0			0
$235$	145.343	−	182.060i	0.618479	−	0.774722i
$236$		0			0
$237$	−350.318			−1.47814
$238$		0			0
$239$		−	186.885i		−	0.781948i	−0.920402	−	0.390974i	$-0.872138\pi$
$239$		−	186.885i		−	0.781948i	0.920402	−	0.390974i	$-0.127862\pi$
$240$		0			0
$241$		−	20.1770i		−	0.0837222i	−0.999123	−	0.0418611i	$-0.986671\pi$
$241$		−	20.1770i		−	0.0837222i	0.999123	−	0.0418611i	$-0.0133287\pi$
$242$		0			0
$243$		−	146.933i		−	0.604664i
$244$		0			0
$245$	160.209	+	127.899i	0.653914	+	0.522036i
$246$		0			0
$247$		−	131.889i		−	0.533963i
$248$		0			0
$249$		−	54.0045i		−	0.216886i
$250$		0			0
$251$	−104.939			−0.418082			−0.209041	−	0.977907i	$-0.567034\pi$
$251$	−104.939			−0.418082			−0.209041	+	0.977907i	$0.567034\pi$
$252$		0			0
$253$	−16.7673	+	16.9275i	−0.0662737	+	0.0669070i
$254$		0			0
$255$	249.376	−	312.375i	0.977946	−	1.22500i
$256$		0			0
$257$		−	115.079i		−	0.447778i	−0.974615	−	0.223889i	$-0.928125\pi$
$257$		−	115.079i		−	0.447778i	0.974615	−	0.223889i	$-0.0718753\pi$
$258$		0			0
$259$		−	49.3436i		−	0.190516i
$260$		0			0
$261$		−	32.4298i		−	0.124252i
$262$		0			0
$263$	229.103			0.871113			0.435556	−	0.900162i	$-0.356552\pi$
$263$	229.103			0.871113			0.435556	+	0.900162i	$0.356552\pi$
$264$		0			0
$265$	−34.9588	+	43.7903i	−0.131920	+	0.165246i
$266$		0			0
$267$		−	98.6696i		−	0.369549i
$268$		0			0
$269$	500.836			1.86184			0.930922	−	0.365218i	$-0.119006\pi$
$269$	500.836			1.86184			0.930922	+	0.365218i	$0.119006\pi$
$270$		0			0
$271$			222.877i			0.822425i	0.911539	+	0.411213i	$0.134895\pi$
$271$			222.877i			0.822425i	−0.911539	+	0.411213i	$0.865105\pi$
$272$		0			0
$273$		−	38.7051i		−	0.141777i
$274$		0			0
$275$	−145.443	+	233.391i	−0.528882	+	0.848695i
$276$		0			0
$277$	162.470			0.586533			0.293267	−	0.956031i	$-0.405258\pi$
$277$	162.470			0.586533			0.293267	+	0.956031i	$0.405258\pi$
$278$		0			0
$279$	103.329			0.370354
$280$		0			0
$281$			352.861i			1.25573i	0.778321	+	0.627867i	$0.216072\pi$
$281$			352.861i			1.25573i	−0.778321	+	0.627867i	$0.783928\pi$
$282$		0			0
$283$	−281.390			−0.994311			−0.497155	−	0.867661i	$-0.665622\pi$
$283$	−281.390			−0.994311			−0.497155	+	0.867661i	$0.665622\pi$
$284$		0			0
$285$	−444.964	−	355.226i	−1.56128	−	1.24641i
$286$		0			0
$287$		−	127.322i		−	0.443630i
$288$		0			0
$289$	251.890			0.871593
$290$		0			0
$291$	−262.117			−0.900744
$292$		0			0
$293$	−242.422			−0.827378			−0.413689	−	0.910418i	$-0.635760\pi$
$293$	−242.422			−0.827378			−0.413689	+	0.910418i	$0.635760\pi$
$294$		0			0
$295$	−174.688	−	139.458i	−0.592164	−	0.472739i
$296$		0			0
$297$	−164.572	+	166.144i	−0.554114	+	0.559409i
$298$		0			0
$299$			8.62311i			0.0288398i
$300$		0			0
$301$	−179.562			−0.596551
$302$		0			0
$303$	263.683			0.870242
$304$		0			0
$305$	121.312	−	151.959i	0.397746	−	0.498226i
$306$		0			0
$307$	−487.226			−1.58705			−0.793527	−	0.608535i	$-0.791758\pi$
$307$	−487.226			−1.58705			−0.793527	+	0.608535i	$0.791758\pi$
$308$		0			0
$309$	389.493			1.26050
$310$		0			0
$311$	421.411			1.35502			0.677510	−	0.735514i	$-0.263059\pi$
$311$	421.411			1.35502			0.677510	+	0.735514i	$0.263059\pi$
$312$		0			0
$313$		−	502.247i		−	1.60462i	−0.596905	−	0.802312i	$-0.703603\pi$
$313$		−	502.247i		−	1.60462i	0.596905	−	0.802312i	$-0.296397\pi$
$314$		0			0
$315$	−31.1127	−	24.8380i	−0.0987705	−	0.0788508i
$316$		0			0
$317$		−	616.290i		−	1.94413i	−0.234706	−	0.972066i	$-0.575413\pi$
$317$		−	616.290i		−	1.94413i	0.234706	−	0.972066i	$-0.424587\pi$
$318$		0			0
$319$	−89.1780	+	90.0301i	−0.279555	+	0.282226i
$320$		0			0
$321$			539.781i			1.68156i
$322$		0			0
$323$		−	770.475i		−	2.38537i
$324$		0			0
$325$	22.0461	+	97.0555i	0.0678343	+	0.298632i
$326$		0			0
$327$	−589.165			−1.80173
$328$		0			0
$329$		−	131.782i		−	0.400553i
$330$		0			0
$331$	−197.883			−0.597835			−0.298918	−	0.954279i	$-0.596626\pi$
$331$	−197.883			−0.597835			−0.298918	+	0.954279i	$0.596626\pi$
$332$		0			0
$333$		−	49.1106i		−	0.147479i
$334$		0			0
$335$	285.545	−	357.680i	0.852372	−	1.06770i
$336$		0			0
$337$	−375.861			−1.11531			−0.557657	−	0.830071i	$-0.688300\pi$
$337$	−375.861			−1.11531			−0.557657	+	0.830071i	$0.688300\pi$
$338$		0			0
$339$	−287.528			−0.848164
$340$		0			0
$341$	−286.856	−	284.141i	−0.841221	−	0.833258i
$342$		0			0
$343$	254.558			0.742153
$344$		0			0
$345$	29.0925	+	23.2252i	0.0843260	+	0.0673195i
$346$		0			0
$347$	−310.624			−0.895169			−0.447584	−	0.894242i	$-0.647716\pi$
$347$	−310.624			−0.895169			−0.447584	+	0.894242i	$0.647716\pi$
$348$		0			0
$349$		−	474.256i		−	1.35890i	−0.733722	−	0.679450i	$-0.762219\pi$
$349$		−	474.256i		−	1.35890i	0.733722	−	0.679450i	$-0.237781\pi$
$350$		0			0
$351$			84.6364i			0.241129i
$352$		0			0
$353$			460.434i			1.30434i	0.758071	+	0.652172i	$0.226142\pi$
$353$			460.434i			1.30434i	−0.758071	+	0.652172i	$0.773858\pi$
$354$		0			0
$355$	213.175	+	170.183i	0.600492	+	0.479388i
$356$		0			0
$357$		−	226.109i		−	0.633359i
$358$		0			0
$359$		−	335.649i		−	0.934955i	−0.884005	−	0.467477i	$-0.845163\pi$
$359$		−	335.649i		−	0.934955i	0.884005	−	0.467477i	$-0.154837\pi$
$360$		0			0
$361$	−736.507			−2.04019
$362$		0			0
$363$	−415.895	+	3.95541i	−1.14572	+	0.0108964i
$364$		0			0
$365$	−219.416	−	175.165i	−0.601140	−	0.479905i
$366$		0			0
$367$			175.255i			0.477533i	0.971077	+	0.238767i	$0.0767431\pi$
$367$			175.255i			0.477533i	−0.971077	+	0.238767i	$0.923257\pi$
$368$		0			0
$369$		−	126.721i		−	0.343416i
$370$		0			0
$371$			31.6971i			0.0854369i
$372$		0			0
$373$	236.338			0.633615			0.316808	−	0.948490i	$-0.397389\pi$
$373$	236.338			0.633615			0.316808	+	0.948490i	$0.397389\pi$
$374$		0			0
$375$	386.822	+	187.027i	1.03153	+	0.498740i
$376$		0			0
$377$			45.8627i			0.121652i
$378$		0			0
$379$	416.267			1.09833			0.549165	−	0.835714i	$-0.314946\pi$
$379$	416.267			1.09833			0.549165	+	0.835714i	$0.314946\pi$
$380$		0			0
$381$			456.942i			1.19932i
$382$		0			0
$383$			166.451i			0.434599i	0.976105	+	0.217299i	$0.0697248\pi$
$383$			166.451i			0.434599i	−0.976105	+	0.217299i	$0.930275\pi$
$384$		0			0
$385$	18.0722	+	154.510i	0.0469407	+	0.401325i
$386$		0			0
$387$	−178.714			−0.461793
$388$		0			0
$389$	−232.939			−0.598814			−0.299407	−	0.954126i	$-0.596789\pi$
$389$	−232.939			−0.598814			−0.299407	+	0.954126i	$0.596789\pi$
$390$		0			0
$391$			50.3749i			0.128836i
$392$		0			0
$393$	169.609			0.431575
$394$		0			0
$395$	317.927	−	398.243i	0.804878	−	1.00821i
$396$		0			0
$397$		−	705.825i		−	1.77790i	−0.458006	−	0.888949i	$-0.651436\pi$
$397$		−	705.825i		−	1.77790i	0.458006	−	0.888949i	$-0.348564\pi$
$398$		0			0
$399$	−322.082			−0.807224
$400$		0			0
$401$	−92.2331			−0.230008			−0.115004	−	0.993365i	$-0.536688\pi$
$401$	−92.2331			−0.230008			−0.115004	+	0.993365i	$0.536688\pi$
$402$		0			0
$403$	−146.129			−0.362603
$404$		0			0
$405$	384.545	+	306.991i	0.949493	+	0.758003i
$406$		0			0
$407$	−135.048	+	136.338i	−0.331813	+	0.334984i
$408$		0			0
$409$			111.673i			0.273038i	0.990637	+	0.136519i	$0.0435915\pi$
$409$			111.673i			0.273038i	−0.990637	+	0.136519i	$0.956409\pi$
$410$		0			0
$411$	76.6371			0.186465
$412$		0			0
$413$	−126.446			−0.306165
$414$		0			0
$415$	61.3925	+	49.0111i	0.147934	+	0.118099i
$416$		0			0
$417$	899.769			2.15772
$418$		0			0
$419$	70.1507			0.167424			0.0837121	−	0.996490i	$-0.473322\pi$
$419$	70.1507			0.167424			0.0837121	+	0.996490i	$0.473322\pi$
$420$		0			0
$421$	299.644			0.711744			0.355872	−	0.934535i	$-0.384184\pi$
$421$	299.644			0.711744			0.355872	+	0.934535i	$0.384184\pi$
$422$		0			0
$423$		−	131.160i		−	0.310070i
$424$		0			0
$425$	128.790	+	566.983i	0.303036	+	1.33408i
$426$		0			0
$427$		−	109.994i		−	0.257597i
$428$		0			0
$429$	−105.932	+	106.944i	−0.246927	+	0.249286i
$430$		0			0
$431$			39.5875i			0.0918504i	0.998945	+	0.0459252i	$0.0146236\pi$
$431$			39.5875i			0.0918504i	−0.998945	+	0.0459252i	$0.985376\pi$
$432$		0			0
$433$			387.168i			0.894153i	0.894496	+	0.447076i	$0.147535\pi$
$433$			387.168i			0.894153i	−0.894496	+	0.447076i	$0.852465\pi$
$434$		0			0
$435$	154.731	+	123.525i	0.355702	+	0.283966i
$436$		0			0
$437$	71.7568			0.164203
$438$		0			0
$439$		−	84.9689i		−	0.193551i	−0.995306	−	0.0967755i	$-0.969147\pi$
$439$		−	84.9689i		−	0.193551i	0.995306	−	0.0967755i	$-0.0308529\pi$
$440$		0			0
$441$	115.418			0.261719
$442$		0			0
$443$			113.196i			0.255521i	0.991805	+	0.127761i	$0.0407790\pi$
$443$			113.196i			0.255521i	−0.991805	+	0.127761i	$0.959221\pi$
$444$		0			0
$445$	112.168	+	89.5463i	0.252063	+	0.201228i
$446$		0			0
$447$	917.166			2.05183
$448$		0			0
$449$	−112.390			−0.250312			−0.125156	−	0.992137i	$-0.539943\pi$
$449$	−112.390			−0.250312			−0.125156	+	0.992137i	$0.539943\pi$
$450$		0			0
$451$	−348.466	+	351.796i	−0.772652	+	0.780035i
$452$		0			0
$453$	−425.737			−0.939816
$454$		0			0
$455$	44.0000	+	35.1263i	0.0967033	+	0.0772006i
$456$		0			0
$457$	−409.705			−0.896511			−0.448255	−	0.893906i	$-0.647954\pi$
$457$	−409.705			−0.896511			−0.448255	+	0.893906i	$0.647954\pi$
$458$		0			0
$459$			494.433i			1.07720i
$460$		0			0
$461$			80.6404i			0.174925i	0.996168	+	0.0874625i	$0.0278758\pi$
$461$			80.6404i			0.174925i	−0.996168	+	0.0874625i	$0.972124\pi$
$462$		0			0
$463$			112.913i			0.243873i	0.992538	+	0.121936i	$0.0389104\pi$
$463$			112.913i			0.243873i	−0.992538	+	0.121936i	$0.961090\pi$
$464$		0			0
$465$	−393.579	+	493.006i	−0.846406	+	1.06023i
$466$		0			0
$467$		−	65.8748i		−	0.141060i	−0.997510	−	0.0705298i	$-0.977531\pi$
$467$		−	65.8748i		−	0.141060i	0.997510	−	0.0705298i	$-0.0224690\pi$
$468$		0			0
$469$		−	258.903i		−	0.552032i
$470$		0			0
$471$	−773.240			−1.64170
$472$		0			0
$473$	496.137	+	491.441i	1.04892	+	1.03899i
$474$		0			0
$475$	807.642	−	183.456i	1.70030	−	0.386223i
$476$		0			0
$477$			31.5475i			0.0661372i
$478$		0			0
$479$			148.397i			0.309806i	0.987930	+	0.154903i	$0.0495065\pi$
$479$			148.397i			0.309806i	−0.987930	+	0.154903i	$0.950494\pi$
$480$		0			0
$481$			69.4529i			0.144393i
$482$		0			0
$483$	21.0583			0.0435989
$484$		0			0
$485$	237.880	−	297.975i	0.490475	−	0.614381i
$486$		0			0
$487$			104.062i			0.213679i	0.994276	+	0.106840i	$0.0340731\pi$
$487$			104.062i			0.213679i	−0.994276	+	0.106840i	$0.965927\pi$
$488$		0			0
$489$	−216.151			−0.442026
$490$		0			0
$491$		−	330.255i		−	0.672617i	−0.941752	−	0.336309i	$-0.890821\pi$
$491$		−	330.255i		−	0.672617i	0.941752	−	0.336309i	$-0.109179\pi$
$492$		0			0
$493$			267.923i			0.543454i
$494$		0			0
$495$	17.9868	+	153.781i	0.0363370	+	0.310668i
$496$		0			0
$497$	154.304			0.310471
$498$		0			0
$499$	214.589			0.430038			0.215019	−	0.976610i	$-0.431019\pi$
$499$	214.589			0.430038			0.215019	+	0.976610i	$0.431019\pi$
$500$		0			0
$501$		−	359.720i		−	0.718005i
$502$		0			0
$503$	848.122			1.68613			0.843063	−	0.537815i	$-0.180750\pi$
$503$	848.122			1.68613			0.843063	+	0.537815i	$0.180750\pi$
$504$		0			0
$505$	−239.302	+	299.756i	−0.473866	+	0.593575i
$506$		0			0
$507$		−	526.426i		−	1.03832i
$508$		0			0
$509$	130.472			0.256331			0.128165	−	0.991753i	$-0.459091\pi$
$509$	130.472			0.256331			0.128165	+	0.991753i	$0.459091\pi$
$510$		0			0
$511$	−158.822			−0.310806
$512$		0			0
$513$	704.298			1.37290
$514$		0			0
$515$	−353.479	+	442.777i	−0.686368	+	0.859761i
$516$		0			0
$517$	−360.673	+	364.119i	−0.697626	+	0.704293i
$518$		0			0
$519$		−	374.870i		−	0.722293i
$520$		0			0
$521$	168.939			0.324258			0.162129	−	0.986770i	$-0.448164\pi$
$521$	168.939			0.324258			0.162129	+	0.986770i	$0.448164\pi$
$522$		0			0
$523$	18.1331			0.0346713			0.0173357	−	0.999850i	$-0.494482\pi$
$523$	18.1331			0.0346713			0.0173357	+	0.999850i	$0.494482\pi$
$524$		0			0
$525$	237.016	−	53.8383i	0.451460	−	0.102549i
$526$		0			0
$527$	−853.662			−1.61985
$528$		0			0
$529$	524.308			0.991131
$530$		0			0
$531$	−125.849			−0.237004
$532$		0			0
$533$			179.210i			0.336229i
$534$		0			0
$535$	−613.624	−	489.871i	−1.14696	−	0.915646i
$536$		0			0
$537$			337.257i			0.628038i
$538$		0			0
$539$	−320.418	−	317.385i	−0.594468	−	0.588841i
$540$		0			0
$541$		−	98.2869i		−	0.181676i	−0.995866	−	0.0908382i	$-0.971045\pi$
$541$		−	98.2869i		−	0.181676i	0.995866	−	0.0908382i	$-0.0289546\pi$
$542$		0			0
$543$			1228.41i			2.26227i
$544$		0			0
$545$	534.689	−	669.765i	0.981081	−	1.22893i
$546$		0			0
$547$	207.231			0.378850			0.189425	−	0.981895i	$-0.439338\pi$
$547$	207.231			0.378850			0.189425	+	0.981895i	$0.439338\pi$
$548$		0			0
$549$		−	109.474i		−	0.199407i
$550$		0			0
$551$	381.644			0.692639
$552$		0			0
$553$		−	288.264i		−	0.521272i
$554$		0			0
$555$	234.319	+	187.062i	0.422196	+	0.337049i
$556$		0			0
$557$	−959.370			−1.72239			−0.861194	−	0.508277i	$-0.830283\pi$
$557$	−959.370			−1.72239			−0.861194	+	0.508277i	$0.830283\pi$
$558$		0			0
$559$	252.740			0.452128
$560$		0			0
$561$	−618.836	+	624.749i	−1.10309	+	1.11364i
$562$		0			0
$563$	−51.7446			−0.0919088			−0.0459544	−	0.998944i	$-0.514633\pi$
$563$	−51.7446			−0.0919088			−0.0459544	+	0.998944i	$0.514633\pi$
$564$		0			0
$565$	260.942	−	326.862i	0.461844	−	0.578517i
$566$		0			0
$567$	278.348			0.490914
$568$		0			0
$569$		−	362.183i		−	0.636526i	−0.948003	−	0.318263i	$-0.896901\pi$
$569$		−	362.183i		−	0.636526i	0.948003	−	0.318263i	$-0.103099\pi$
$570$		0			0
$571$			698.164i			1.22270i	0.791358	+	0.611352i	$0.209374\pi$
$571$			698.164i			1.22270i	−0.791358	+	0.611352i	$0.790626\pi$
$572$		0			0
$573$		−	566.143i		−	0.988034i
$574$		0			0
$575$	−52.8049	+	11.9946i	−0.0918347	+	0.0208602i
$576$		0			0
$577$		−	19.2829i		−	0.0334192i	−0.999860	−	0.0167096i	$-0.994681\pi$
$577$		−	19.2829i		−	0.0334192i	0.999860	−	0.0167096i	$-0.00531908\pi$
$578$		0			0
$579$			1075.60i			1.85768i
$580$		0			0
$581$	44.4383			0.0764858
$582$		0			0
$583$	86.7516	−	87.5806i	0.148802	−	0.150224i
$584$		0			0
$585$	43.7923	+	34.9604i	0.0748586	+	0.0597614i
$586$		0			0
$587$			137.281i			0.233869i	0.993140	+	0.116934i	$0.0373068\pi$
$587$			137.281i			0.233869i	−0.993140	+	0.116934i	$0.962693\pi$
$588$		0			0
$589$			1216.00i			2.06452i
$590$		0			0
$591$			434.601i			0.735365i
$592$		0			0
$593$	−307.049			−0.517789			−0.258895	−	0.965906i	$-0.583358\pi$
$593$	−307.049			−0.517789			−0.258895	+	0.965906i	$0.583358\pi$
$594$		0			0
$595$	257.041	+	205.202i	0.432002	+	0.344878i
$596$		0			0
$597$			678.562i			1.13662i
$598$		0			0
$599$	810.343			1.35283			0.676413	−	0.736522i	$-0.263533\pi$
$599$	810.343			1.35283			0.676413	+	0.736522i	$0.263533\pi$
$600$		0			0
$601$		−	382.095i		−	0.635766i	−0.948130	−	0.317883i	$-0.897028\pi$
$601$		−	382.095i		−	0.635766i	0.948130	−	0.317883i	$-0.102972\pi$
$602$		0			0
$603$		−	257.680i		−	0.427331i
$604$		0			0
$605$	372.944	−	476.380i	0.616436	−	0.787405i
$606$		0			0
$607$	−302.312			−0.498043			−0.249022	−	0.968498i	$-0.580109\pi$
$607$	−302.312			−0.498043			−0.249022	+	0.968498i	$0.580109\pi$
$608$		0			0
$609$	112.000			0.183908
$610$		0			0
$611$			185.488i			0.303581i
$612$		0			0
$613$	520.344			0.848848			0.424424	−	0.905464i	$-0.360477\pi$
$613$	520.344			0.848848			0.424424	+	0.905464i	$0.360477\pi$
$614$		0			0
$615$	604.615	+	482.679i	0.983114	+	0.784844i
$616$		0			0
$617$		−	1040.89i		−	1.68702i	−0.537113	−	0.843511i	$-0.680485\pi$
$617$		−	1040.89i		−	1.68702i	0.537113	−	0.843511i	$-0.319515\pi$
$618$		0			0
$619$	−1076.13			−1.73850			−0.869249	−	0.494374i	$-0.835397\pi$
$619$	−1076.13			−1.73850			−0.869249	+	0.494374i	$0.835397\pi$
$620$		0			0
$621$	−46.0481			−0.0741516
$622$		0			0
$623$	81.1914			0.130323
$624$		0			0
$625$	−563.668	+	270.006i	−0.901869	+	0.432010i
$626$		0			0
$627$	889.928	+	881.505i	1.41934	+	1.40591i
$628$		0			0
$629$			405.733i			0.645044i
$630$		0			0
$631$	−705.843			−1.11861			−0.559305	−	0.828962i	$-0.688932\pi$
$631$	−705.843			−1.11861			−0.559305	+	0.828962i	$0.688932\pi$
$632$		0			0
$633$	−158.392			−0.250224
$634$		0			0
$635$	−519.453	−	414.691i	−0.818036	−	0.653057i
$636$		0			0
$637$	−163.226			−0.256241
$638$		0			0
$639$	153.576			0.240338
$640$		0			0
$641$	−530.802			−0.828084			−0.414042	−	0.910258i	$-0.635883\pi$
$641$	−530.802			−0.828084			−0.414042	+	0.910258i	$0.635883\pi$
$642$		0			0
$643$			415.065i			0.645513i	0.946482	+	0.322757i	$0.104610\pi$
$643$			415.065i			0.645513i	−0.946482	+	0.322757i	$0.895390\pi$
$644$		0			0
$645$	680.721	−	852.688i	1.05538	−	1.32200i
$646$		0			0
$647$		−	890.968i		−	1.37708i	−0.725201	−	0.688538i	$-0.758253\pi$
$647$		−	890.968i		−	1.37708i	0.725201	−	0.688538i	$-0.241747\pi$
$648$		0			0
$649$	349.377	+	346.070i	0.538331	+	0.533236i
$650$		0			0
$651$			356.857i			0.548168i
$652$		0			0
$653$		−	368.900i		−	0.564931i	−0.959277	−	0.282465i	$-0.908848\pi$
$653$		−	368.900i		−	0.564931i	0.959277	−	0.282465i	$-0.0911522\pi$
$654$		0			0
$655$	−153.926	+	192.812i	−0.235002	+	0.294369i
$656$		0			0
$657$	−158.072			−0.240597
$658$		0			0
$659$		−	272.221i		−	0.413082i	−0.978438	−	0.206541i	$-0.933779\pi$
$659$		−	272.221i		−	0.413082i	0.978438	−	0.206541i	$-0.0662206\pi$
$660$		0			0
$661$	−784.336			−1.18659			−0.593295	−	0.804985i	$-0.702173\pi$
$661$	−784.336			−1.18659			−0.593295	+	0.804985i	$0.702173\pi$
$662$		0			0
$663$			318.257i			0.480025i
$664$		0			0
$665$	292.301	−	366.144i	0.439551	−	0.550592i
$666$		0			0
$667$	−24.9525			−0.0374100
$668$		0			0
$669$	−1349.99			−2.01793
$670$		0			0
$671$	−301.041	+	303.918i	−0.448646	+	0.452933i
$672$		0			0
$673$	−165.395			−0.245758			−0.122879	−	0.992422i	$-0.539213\pi$
$673$	−165.395			−0.245758			−0.122879	+	0.992422i	$0.539213\pi$
$674$		0			0
$675$	−518.284	+	117.728i	−0.767829	+	0.174412i
$676$		0			0
$677$	469.955			0.694173			0.347086	−	0.937833i	$-0.387171\pi$
$677$	469.955			0.694173			0.347086	+	0.937833i	$0.387171\pi$
$678$		0			0
$679$		−	215.686i		−	0.317652i
$680$		0			0
$681$			923.606i			1.35625i
$682$		0			0
$683$		−	592.512i		−	0.867514i	−0.901030	−	0.433757i	$-0.857188\pi$
$683$		−	592.512i		−	0.867514i	0.901030	−	0.433757i	$-0.142812\pi$
$684$		0			0
$685$	−69.5510	+	87.1212i	−0.101534	+	0.127184i
$686$		0			0
$687$		−	706.461i		−	1.02833i
$688$		0			0
$689$		−	44.6148i		−	0.0647530i
$690$		0			0
$691$	−586.418			−0.848651			−0.424326	−	0.905510i	$-0.639489\pi$
$691$	−586.418			−0.848651			−0.424326	+	0.905510i	$0.639489\pi$
$692$		0			0
$693$	62.2254	+	61.6364i	0.0897913	+	0.0889414i
$694$		0			0
$695$	−816.574	+	1022.86i	−1.17493	+	1.47174i
$696$		0			0
$697$			1046.92i			1.50203i
$698$		0			0
$699$		−	1043.90i		−	1.49342i
$700$		0			0
$701$			1185.11i			1.69060i	0.534296	+	0.845298i	$0.320577\pi$
$701$			1185.11i			1.69060i	−0.534296	+	0.845298i	$0.679423\pi$
$702$		0			0
$703$	577.949			0.822117
$704$		0			0
$705$	625.795	+	499.587i	0.887652	+	0.708634i
$706$		0			0
$707$			216.975i			0.306895i
$708$		0			0
$709$	230.583			0.325222			0.162611	−	0.986690i	$-0.448008\pi$
$709$	230.583			0.325222			0.162611	+	0.986690i	$0.448008\pi$
$710$		0			0
$711$		−	286.902i		−	0.403520i
$712$		0			0
$713$		−	79.5043i		−	0.111507i
$714$		0			0
$715$	−25.4372	−	217.479i	−0.0355765	−	0.304166i
$716$		0			0
$717$	642.383			0.895931
$718$		0			0
$719$	−647.117			−0.900024			−0.450012	−	0.893023i	$-0.648580\pi$
$719$	−647.117			−0.900024			−0.450012	+	0.893023i	$0.648580\pi$
$720$		0			0
$721$			320.499i			0.444520i
$722$		0			0
$723$	69.3547			0.0959263
$724$		0			0
$725$	−280.847	+	63.7944i	−0.387375	+	0.0879923i
$726$		0			0
$727$		−	1381.75i		−	1.90062i	−0.311304	−	0.950310i	$-0.600766\pi$
$727$		−	1381.75i		−	1.90062i	0.311304	−	0.950310i	$-0.399234\pi$
$728$		0			0
$729$	−380.644			−0.522146
$730$		0			0
$731$	1476.47			2.01979
$732$		0			0
$733$	354.823			0.484069			0.242034	−	0.970268i	$-0.422185\pi$
$733$	354.823			0.484069			0.242034	+	0.970268i	$0.422185\pi$
$734$		0			0
$735$	−439.627	+	550.687i	−0.598132	+	0.749234i
$736$		0			0
$737$	−708.589	+	715.360i	−0.961451	+	0.970638i
$738$		0			0
$739$		−	198.704i		−	0.268882i	−0.990922	−	0.134441i	$-0.957076\pi$
$739$		−	198.704i		−	0.268882i	0.990922	−	0.134441i	$-0.0429239\pi$
$740$		0			0
$741$	453.343			0.611798
$742$		0			0
$743$	−1120.12			−1.50756			−0.753779	−	0.657127i	$-0.771771\pi$
$743$	−1120.12			−1.50756			−0.753779	+	0.657127i	$0.771771\pi$
$744$		0			0
$745$	−832.362	+	1042.64i	−1.11726	+	1.39951i
$746$		0			0
$747$	44.2284			0.0592081
$748$		0			0
$749$	−444.165			−0.593010
$750$		0			0
$751$	−869.405			−1.15766			−0.578831	−	0.815447i	$-0.696491\pi$
$751$	−869.405			−1.15766			−0.578831	+	0.815447i	$0.696491\pi$
$752$		0			0
$753$		−	360.706i		−	0.479025i
$754$		0			0
$755$	386.371	−	483.978i	0.511750	−	0.641031i
$756$		0			0
$757$			874.961i			1.15583i	0.816098	+	0.577913i	$0.196133\pi$
$757$			874.961i			1.15583i	−0.816098	+	0.577913i	$0.803867\pi$
$758$		0			0
$759$	−58.1849	−	57.6342i	−0.0766600	−	0.0759344i
$760$		0			0
$761$		−	645.727i		−	0.848524i	−0.905539	−	0.424262i	$-0.860534\pi$
$761$		−	645.727i		−	0.848524i	0.905539	−	0.424262i	$-0.139466\pi$
$762$		0			0
$763$		−	484.802i		−	0.635389i
$764$		0			0
$765$	255.828	+	204.233i	0.334415	+	0.266972i
$766$		0			0
$767$	177.978			0.232044
$768$		0			0
$769$		−	670.097i		−	0.871388i	−0.900095	−	0.435694i	$-0.856503\pi$
$769$		−	670.097i		−	0.871388i	0.900095	−	0.435694i	$-0.143497\pi$
$770$		0			0
$771$	395.562			0.513050
$772$		0			0
$773$			612.123i			0.791880i	0.918276	+	0.395940i	$0.129581\pi$
$773$			612.123i			0.791880i	−0.918276	+	0.395940i	$0.870419\pi$
$774$		0			0
$775$	−203.263	−	894.842i	−0.262275	−	1.15464i
$776$		0			0
$777$	169.609			0.218287
$778$		0			0
$779$	1491.29			1.91436
$780$		0			0
$781$	−426.350	−	422.314i	−0.545902	−	0.540735i
$782$		0			0
$783$	−244.911			−0.312785
$784$		0			0
$785$	701.744	−	879.021i	0.893941	−	1.11977i
$786$		0			0
$787$	1391.41			1.76800			0.883998	−	0.467491i	$-0.154842\pi$
$787$	1391.41			1.76800			0.883998	+	0.467491i	$0.154842\pi$
$788$		0			0
$789$			787.496i			0.998093i
$790$		0			0
$791$		−	236.595i		−	0.299109i
$792$		0			0
$793$			154.820i			0.195234i
$794$		0			0
$795$	−150.521	−	120.164i	−0.189334	−	0.151150i
$796$		0			0
$797$			621.140i			0.779348i	0.920953	+	0.389674i	$0.127412\pi$
$797$			621.140i			0.779348i	−0.920953	+	0.389674i	$0.872588\pi$
$798$		0			0
$799$			1083.59i			1.35618i
$800$		0			0
$801$	80.8081			0.100884
$802$		0			0
$803$	438.833	+	434.679i	0.546491	+	0.541319i
$804$		0			0
$805$	−19.1111	+	23.9391i	−0.0237405	+	0.0297380i
$806$		0			0
$807$			1721.53i			2.13324i
$808$		0			0
$809$		−	492.303i		−	0.608532i	−0.952587	−	0.304266i	$-0.901589\pi$
$809$		−	492.303i		−	0.608532i	0.952587	−	0.304266i	$-0.0984112\pi$
$810$		0			0
$811$		−	581.566i		−	0.717098i	−0.933511	−	0.358549i	$-0.883272\pi$
$811$		−	581.566i		−	0.717098i	0.933511	−	0.358549i	$-0.116728\pi$
$812$		0			0
$813$	−766.097			−0.942309
$814$		0			0
$815$	196.165	−	245.721i	0.240693	−	0.301498i
$816$		0			0
$817$		−	2103.16i		−	2.57425i
$818$		0			0
$819$	31.6985			0.0387040
$820$		0			0
$821$			623.019i			0.758854i	0.925222	+	0.379427i	$0.123879\pi$
$821$			623.019i			0.758854i	−0.925222	+	0.379427i	$0.876121\pi$
$822$		0			0
$823$			1073.90i			1.30486i	0.757849	+	0.652430i	$0.226250\pi$
$823$			1073.90i			1.30486i	−0.757849	+	0.652430i	$0.773750\pi$
$824$		0			0
$825$	−802.237	−	499.931i	−0.972408	−	0.605977i
$826$		0			0
$827$	−1446.64			−1.74927			−0.874634	−	0.484784i	$-0.838898\pi$
$827$	−1446.64			−1.74927			−0.874634	+	0.484784i	$0.838898\pi$
$828$		0			0
$829$	−784.117			−0.945858			−0.472929	−	0.881100i	$-0.656803\pi$
$829$	−784.117			−0.945858			−0.472929	+	0.881100i	$0.656803\pi$
$830$		0			0
$831$			558.458i			0.672032i
$832$		0			0
$833$	−953.539			−1.14470
$834$		0			0
$835$	408.931	+	326.459i	0.489737	+	0.390969i
$836$		0			0
$837$		−	780.340i		−	0.932306i
$838$		0			0
$839$	−306.117			−0.364859			−0.182429	−	0.983219i	$-0.558396\pi$
$839$	−306.117			−0.364859			−0.182429	+	0.983219i	$0.558396\pi$
$840$		0			0
$841$	708.288			0.842198
$842$		0			0
$843$	−1212.89			−1.43878
$844$		0			0
$845$	598.442	+	477.751i	0.708215	+	0.565385i
$846$		0			0
$847$	−3.25475	−	342.224i	−0.00384268	−	0.404043i
$848$		0			0
$849$		−	967.224i		−	1.13925i
$850$		0			0
$851$	−37.7872			−0.0444033
$852$		0			0
$853$	−113.883			−0.133509			−0.0667545	−	0.997769i	$-0.521264\pi$
$853$	−113.883			−0.133509			−0.0667545	+	0.997769i	$0.521264\pi$
$854$		0			0
$855$	290.921	−	364.415i	0.340259	−	0.426217i
$856$		0			0
$857$	581.910			0.679008			0.339504	−	0.940605i	$-0.389741\pi$
$857$	581.910			0.679008			0.339504	+	0.940605i	$0.389741\pi$
$858$		0			0
$859$	118.281			0.137696			0.0688482	−	0.997627i	$-0.478068\pi$
$859$	118.281			0.137696			0.0688482	+	0.997627i	$0.478068\pi$
$860$		0			0
$861$	437.644			0.508297
$862$		0			0
$863$		−	1030.51i		−	1.19410i	−0.802203	−	0.597051i	$-0.796339\pi$
$863$		−	1030.51i		−	1.19410i	0.802203	−	0.597051i	$-0.203661\pi$
$864$		0			0
$865$	426.153	+	340.208i	0.492662	+	0.393304i
$866$		0			0
$867$			865.825i			0.998644i
$868$		0			0
$869$	−788.946	+	796.485i	−0.907878	+	0.916554i
$870$		0			0
$871$			364.415i			0.418387i
$872$		0			0
$873$		−	214.667i		−	0.245896i
$874$		0			0
$875$	−153.898	+	318.301i	−0.175883	+	0.363773i
$876$		0			0
$877$	378.593			0.431691			0.215845	−	0.976428i	$-0.430749\pi$
$877$	378.593			0.431691			0.215845	+	0.976428i	$0.430749\pi$
$878$		0			0
$879$		−	833.277i		−	0.947983i
$880$		0			0
$881$	810.076			0.919496			0.459748	−	0.888049i	$-0.347940\pi$
$881$	810.076			0.919496			0.459748	+	0.888049i	$0.347940\pi$
$882$		0			0
$883$		−	188.555i		−	0.213540i	−0.994284	−	0.106770i	$-0.965949\pi$
$883$		−	188.555i		−	0.213540i	0.994284	−	0.106770i	$-0.0340508\pi$
$884$		0			0
$885$	479.360	−	600.458i	0.541649	−	0.678483i
$886$		0			0
$887$	234.430			0.264295			0.132148	−	0.991230i	$-0.457813\pi$
$887$	234.430			0.264295			0.132148	+	0.991230i	$0.457813\pi$
$888$		0			0
$889$	−376.000			−0.422947
$890$		0			0
$891$	−769.089	−	761.810i	−0.863175	−	0.855005i
$892$		0			0
$893$	1543.53			1.72847
$894$		0			0
$895$	−383.394	−	306.073i	−0.428373	−	0.341981i
$896$		0			0
$897$	−29.6403			−0.0330438
$898$		0			0
$899$		−	422.850i		−	0.470356i
$900$		0			0
$901$		−	260.633i		−	0.289271i
$902$		0			0
$903$		−	617.209i		−	0.683509i
$904$		0			0
$905$	−1396.46	−	1114.83i	−1.54305	−	1.23186i
$906$		0			0
$907$			855.676i			0.943413i	0.881756	+	0.471707i	$0.156362\pi$
$907$			855.676i			0.943413i	−0.881756	+	0.471707i	$0.843638\pi$
$908$		0			0
$909$			215.950i			0.237569i
$910$		0			0
$911$	16.2609			0.0178495			0.00892477	−	0.999960i	$-0.497159\pi$
$911$	16.2609			0.0178495			0.00892477	+	0.999960i	$0.497159\pi$
$912$		0			0
$913$	−122.785	−	121.623i	−0.134485	−	0.133212i
$914$		0			0
$915$	522.329	+	416.988i	0.570852	+	0.455725i
$916$		0			0
$917$			139.565i			0.152197i
$918$		0			0
$919$			166.342i			0.181003i	0.995896	+	0.0905017i	$0.0288470\pi$
$919$			166.342i			0.181003i	−0.995896	+	0.0905017i	$0.971153\pi$
$920$		0			0
$921$		−	1674.74i		−	1.81840i
$922$		0			0
$923$	−217.189			−0.235307
$924$		0			0
$925$	−425.305	+	96.6081i	−0.459789	+	0.104441i
$926$		0			0
$927$			318.986i			0.344106i
$928$		0			0
$929$	965.712			1.03952			0.519759	−	0.854313i	$-0.326022\pi$
$929$	965.712			1.03952			0.519759	+	0.854313i	$0.326022\pi$
$930$		0			0
$931$			1358.27i			1.45894i
$932$		0			0
$933$			1448.52i			1.55254i
$934$		0			0
$935$	−148.600	−	1270.48i	−0.158931	−	1.35880i
$936$		0			0
$937$	842.861			0.899532			0.449766	−	0.893146i	$-0.351507\pi$
$937$	842.861			0.899532			0.449766	+	0.893146i	$0.351507\pi$
$938$		0			0
$939$	1726.38			1.83853
$940$		0			0
$941$		−	1467.05i		−	1.55903i	−0.626382	−	0.779517i	$-0.715465\pi$
$941$		−	1467.05i		−	1.55903i	0.626382	−	0.779517i	$-0.284535\pi$
$942$		0			0
$943$	−97.5028			−0.103396
$944$		0			0
$945$	−187.577	+	234.964i	−0.198494	+	0.248639i
$946$		0			0
$947$		−	463.518i		−	0.489460i	−0.969591	−	0.244730i	$-0.921301\pi$
$947$		−	463.518i		−	0.489460i	0.969591	−	0.244730i	$-0.0786993\pi$
$948$		0			0
$949$	223.548			0.235561
$950$		0			0
$951$	2118.38			2.22753
$952$		0			0
$953$	−1260.06			−1.32220			−0.661099	−	0.750298i	$-0.729910\pi$
$953$	−1260.06			−1.32220			−0.661099	+	0.750298i	$0.729910\pi$
$954$		0			0
$955$	643.593	+	513.796i	0.673919	+	0.538006i
$956$		0			0
$957$	−309.461	−	306.532i	−0.323366	−	0.320305i
$958$		0			0
$959$			63.0617i			0.0657578i
$960$		0			0
$961$	386.294			0.401971
$962$		0			0
$963$	−442.068			−0.459053
$964$		0			0
$965$	−1222.74	−	976.145i	−1.26709	−	1.01155i
$966$		0			0
$967$	−857.576			−0.886842			−0.443421	−	0.896314i	$-0.646235\pi$
$967$	−857.576			−0.886842			−0.443421	+	0.896314i	$0.646235\pi$
$968$		0			0
$969$	2648.36			2.73308
$970$		0			0
$971$	−110.816			−0.114125			−0.0570627	−	0.998371i	$-0.518173\pi$
$971$	−110.816			−0.114125			−0.0570627	+	0.998371i	$0.518173\pi$
$972$		0			0
$973$			740.386i			0.760931i
$974$		0			0
$975$	−333.609	+	75.7794i	−0.342163	+	0.0777224i
$976$		0			0
$977$		−	687.248i		−	0.703427i	−0.936108	−	0.351713i	$-0.885599\pi$
$977$		−	687.248i		−	0.703427i	0.936108	−	0.351713i	$-0.114401\pi$
$978$		0			0
$979$	−224.336	−	222.212i	−0.229148	−	0.226979i
$980$		0			0
$981$		−	482.513i		−	0.491858i
$982$		0			0
$983$		−	120.258i		−	0.122338i	−0.998127	−	0.0611688i	$-0.980517\pi$
$983$		−	120.258i		−	0.122338i	0.998127	−	0.0611688i	$-0.0194828\pi$
$984$		0			0
$985$	−494.055	−	394.416i	−0.501579	−	0.400422i
$986$		0			0
$987$	452.975			0.458941
$988$		0			0
$989$			137.508i			0.139037i
$990$		0			0
$991$	−220.836			−0.222842			−0.111421	−	0.993773i	$-0.535540\pi$
$991$	−220.836			−0.222842			−0.111421	+	0.993773i	$0.535540\pi$
$992$		0			0
$993$		−	680.186i		−	0.684981i
$994$		0			0
$995$	−771.391	−	615.820i	−0.775267	−	0.618914i
$996$		0			0
$997$	968.398			0.971312			0.485656	−	0.874150i	$-0.338581\pi$
$997$	968.398			0.971312			0.485656	+	0.874150i	$0.338581\pi$
$998$		0			0
$999$	−370.884			−0.371255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.3.i.g.769.7		8
4.3	odd	2		110.3.c.b.109.5	yes	8
5.4	even	2	inner	880.3.i.g.769.2		8
11.10	odd	2	inner	880.3.i.g.769.8		8
12.11	even	2		990.3.h.b.109.4		8
20.3	even	4		550.3.d.d.351.4		8
20.7	even	4		550.3.d.d.351.5		8
20.19	odd	2		110.3.c.b.109.4	yes	8
44.43	even	2		110.3.c.b.109.1	✓	8
55.54	odd	2	inner	880.3.i.g.769.1		8
60.59	even	2		990.3.h.b.109.7		8
132.131	odd	2		990.3.h.b.109.8		8
220.43	odd	4		550.3.d.d.351.8		8
220.87	odd	4		550.3.d.d.351.1		8
220.219	even	2		110.3.c.b.109.8	yes	8
660.659	odd	2		990.3.h.b.109.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.3.c.b.109.1	✓	8	44.43	even	2
110.3.c.b.109.4	yes	8	20.19	odd	2
110.3.c.b.109.5	yes	8	4.3	odd	2
110.3.c.b.109.8	yes	8	220.219	even	2
550.3.d.d.351.1		8	220.87	odd	4
550.3.d.d.351.4		8	20.3	even	4
550.3.d.d.351.5		8	20.7	even	4
550.3.d.d.351.8		8	220.43	odd	4
880.3.i.g.769.1		8	55.54	odd	2	inner
880.3.i.g.769.2		8	5.4	even	2	inner
880.3.i.g.769.7		8	1.1	even	1	trivial
880.3.i.g.769.8		8	11.10	odd	2	inner
990.3.h.b.109.3		8	660.659	odd	2
990.3.h.b.109.4		8	12.11	even	2
990.3.h.b.109.7		8	60.59	even	2
990.3.h.b.109.8		8	132.131	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)

\(f(q)\)	\(=\)	\(q+3.43731i q^{3} +(-3.90754 - 3.11948i) q^{5} -2.82843 q^{7} -2.81507 q^{9} +O(q^{10})\)	\(q+3.43731i q^{3} +(-3.90754 - 3.11948i) q^{5} -2.82843 q^{7} -2.81507 q^{9} +(7.81507 + 7.74110i) q^{11} +3.98111 q^{13} +(10.7226 - 13.4314i) q^{15} +23.2571 q^{17} -33.1286i q^{19} -9.72217i q^{21} +2.16600i q^{23} +(5.53768 + 24.3790i) q^{25} +21.2595i q^{27} +11.5201i q^{29} -36.7055 q^{31} +(-26.6085 + 26.8628i) q^{33} +(11.0522 + 8.82322i) q^{35} +17.4456i q^{37} +13.6843i q^{39} +45.0151i q^{41} +63.4847 q^{43} +(11.0000 + 8.78157i) q^{45} +46.5919i q^{47} -41.0000 q^{49} +79.9416i q^{51} -11.2066i q^{53} +(-6.38947 - 54.6276i) q^{55} +113.873 q^{57} +44.7055 q^{59} +38.8887i q^{61} +7.96223 q^{63} +(-15.5563 - 12.4190i) q^{65} +91.5360i q^{67} -7.44522 q^{69} -54.5548 q^{71} +56.1521 q^{73} +(-83.7980 + 19.0347i) q^{75} +(-22.1044 - 21.8951i) q^{77} +101.917i q^{79} -98.4110 q^{81} -15.7113 q^{83} +(-90.8778 - 72.5499i) q^{85} -39.5980 q^{87} -28.7055 q^{89} -11.2603 q^{91} -126.168i q^{93} +(-103.344 + 129.451i) q^{95} +76.2564i q^{97} +(-22.0000 - 21.7918i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{5} + 20 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more