LMFDB - Embedded newform 1100.3.e.b.549.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,3,Mod(549,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1100.549");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.e (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:	$29.9728290796$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 85x^{14} + 2456x^{12} + 32605x^{10} + 215801x^{8} + 712960x^{6} + 1098976x^{4} + 633600x^{2} + 92416 $ x^16 + 85x^14 + 2456x^12 + 32605x^10 + 215801x^8 + 712960x^6 + 1098976x^4 + 633600*x^2 + 92416
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.10
Root			$-0.885333i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.b.549.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.712855i q^{3} +7.86633 q^{7} +8.49184 q^{9} +O(q^{10})$	$q+0.712855i q^{3} +7.86633 q^{7} +8.49184 q^{9} +(2.47679 + 10.7175i) q^{11} -1.76214 q^{13} +15.1046 q^{17} -3.61913i q^{19} +5.60755i q^{21} +10.7158i q^{23} +12.4691i q^{27} -26.5900i q^{29} -21.3957 q^{31} +(-7.64005 + 1.76559i) q^{33} -1.48215i q^{37} -1.25615i q^{39} +23.4235i q^{41} +60.3649 q^{43} +1.97588i q^{47} +12.8792 q^{49} +10.7674i q^{51} -16.5981i q^{53} +2.57991 q^{57} -78.3187 q^{59} +27.2323i q^{61} +66.7996 q^{63} -56.0870i q^{67} -7.63885 q^{69} -41.7570 q^{71} +106.496 q^{73} +(19.4832 + 84.3077i) q^{77} +89.4259i q^{79} +67.5379 q^{81} +132.432 q^{83} +18.9548 q^{87} +59.5747 q^{89} -13.8616 q^{91} -15.2520i q^{93} -46.9636i q^{97} +(21.0325 + 91.0116i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 56 q^{9}+O(q^{10}) $ 16 * q - 56 * q^9	$ 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) $ 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			0.712855i			0.237618i	0.992917	+	0.118809i	$0.0379077\pi$
$3$			0.712855i			0.237618i	−0.992917	+	0.118809i	$0.962092\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.86633			1.12376			0.561881	−	0.827218i	$-0.310078\pi$
$7$	7.86633			1.12376			0.561881	+	0.827218i	$0.310078\pi$
$8$		0			0
$9$	8.49184			0.943538
$10$		0			0
$11$	2.47679	+	10.7175i	0.225162	+	0.974321i
$11$	2.47679	+	10.7175i	0.225162	+	0.974321i
$12$		0			0
$13$	−1.76214			−0.135549			−0.0677746	−	0.997701i	$-0.521590\pi$
$13$	−1.76214			−0.135549			−0.0677746	+	0.997701i	$0.521590\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.1046			0.888505			0.444253	−	0.895902i	$-0.353469\pi$
$17$	15.1046			0.888505			0.444253	+	0.895902i	$0.353469\pi$
$18$		0			0
$19$		−	3.61913i		−	0.190480i	−0.995454	−	0.0952402i	$-0.969638\pi$
$19$		−	3.61913i		−	0.190480i	0.995454	−	0.0952402i	$-0.0303619\pi$
$20$		0			0
$21$			5.60755i			0.267026i
$22$		0			0
$23$			10.7158i			0.465906i	0.972488	+	0.232953i	$0.0748389\pi$
$23$			10.7158i			0.465906i	−0.972488	+	0.232953i	$0.925161\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			12.4691i			0.461820i
$28$		0			0
$29$		−	26.5900i		−	0.916898i	−0.888721	−	0.458449i	$-0.848405\pi$
$29$		−	26.5900i		−	0.916898i	0.888721	−	0.458449i	$-0.151595\pi$
$30$		0			0
$31$	−21.3957			−0.690183			−0.345091	−	0.938569i	$-0.612152\pi$
$31$	−21.3957			−0.690183			−0.345091	+	0.938569i	$0.612152\pi$
$32$		0			0
$33$	−7.64005	+	1.76559i	−0.231517	+	0.0535027i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	1.48215i		−	0.0400581i	−0.999799	−	0.0200291i	$-0.993624\pi$
$37$		−	1.48215i		−	0.0400581i	0.999799	−	0.0200291i	$-0.00637588\pi$
$38$		0			0
$39$		−	1.25615i		−	0.0322090i
$40$		0			0
$41$			23.4235i			0.571305i	0.958333	+	0.285652i	$0.0922103\pi$
$41$			23.4235i			0.571305i	−0.958333	+	0.285652i	$0.907790\pi$
$42$		0			0
$43$	60.3649			1.40384			0.701918	−	0.712258i	$-0.252327\pi$
$43$	60.3649			1.40384			0.701918	+	0.712258i	$0.252327\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.97588i			0.0420401i	0.999779	+	0.0210200i	$0.00669138\pi$
$47$			1.97588i			0.0420401i	−0.999779	+	0.0210200i	$0.993309\pi$
$48$		0			0
$49$	12.8792			0.262840
$50$		0			0
$51$			10.7674i			0.211125i
$52$		0			0
$53$		−	16.5981i		−	0.313171i	−0.987664	−	0.156586i	$-0.949951\pi$
$53$		−	16.5981i		−	0.313171i	0.987664	−	0.156586i	$-0.0500487\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.57991			0.0452616
$58$		0			0
$59$	−78.3187			−1.32744			−0.663718	−	0.747983i	$-0.731023\pi$
$59$	−78.3187			−1.32744			−0.663718	+	0.747983i	$0.731023\pi$
$60$		0			0
$61$			27.2323i			0.446431i	0.974769	+	0.223216i	$0.0716554\pi$
$61$			27.2323i			0.446431i	−0.974769	+	0.223216i	$0.928345\pi$
$62$		0			0
$63$	66.7996			1.06031
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	56.0870i		−	0.837119i	−0.908189	−	0.418560i	$-0.862535\pi$
$67$		−	56.0870i		−	0.837119i	0.908189	−	0.418560i	$-0.137465\pi$
$68$		0			0
$69$	−7.63885			−0.110708
$70$		0			0
$71$	−41.7570			−0.588127			−0.294063	−	0.955786i	$-0.595008\pi$
$71$	−41.7570			−0.588127			−0.294063	+	0.955786i	$0.595008\pi$
$72$		0			0
$73$	106.496			1.45886			0.729428	−	0.684058i	$-0.239786\pi$
$73$	106.496			1.45886			0.729428	+	0.684058i	$0.239786\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	19.4832	+	84.3077i	0.253029	+	1.09490i
$78$		0			0
$79$			89.4259i			1.13197i	0.824415	+	0.565986i	$0.191505\pi$
$79$			89.4259i			1.13197i	−0.824415	+	0.565986i	$0.808495\pi$
$80$		0			0
$81$	67.5379			0.833801
$82$		0			0
$83$	132.432			1.59557			0.797785	−	0.602942i	$-0.206005\pi$
$83$	132.432			1.59557			0.797785	+	0.602942i	$0.206005\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	18.9548			0.217872
$88$		0			0
$89$	59.5747			0.669379			0.334689	−	0.942328i	$-0.391369\pi$
$89$	59.5747			0.669379			0.334689	+	0.942328i	$0.391369\pi$
$90$		0			0
$91$	−13.8616			−0.152325
$92$		0			0
$93$		−	15.2520i		−	0.164000i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	46.9636i		−	0.484161i	−0.970256	−	0.242081i	$-0.922170\pi$
$97$		−	46.9636i		−	0.484161i	0.970256	−	0.242081i	$-0.0778299\pi$
$98$		0			0
$99$	21.0325	+	91.0116i	0.212449	+	0.919309i
$100$		0			0
$101$			176.702i			1.74952i	0.484552	+	0.874762i	$0.338983\pi$
$101$			176.702i			1.74952i	−0.484552	+	0.874762i	$0.661017\pi$
$102$		0			0
$103$			154.350i			1.49854i	0.662265	+	0.749270i	$0.269595\pi$
$103$			154.350i			1.49854i	−0.662265	+	0.749270i	$0.730405\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	79.0080			0.738393			0.369196	−	0.929351i	$-0.379633\pi$
$107$	79.0080			0.738393			0.369196	+	0.929351i	$0.379633\pi$
$108$		0			0
$109$		−	154.724i		−	1.41949i	−0.704459	−	0.709745i	$-0.748810\pi$
$109$		−	154.724i		−	1.41949i	0.704459	−	0.709745i	$-0.251190\pi$
$110$		0			0
$111$	1.05656			0.00951855
$112$		0			0
$113$			123.672i			1.09444i	0.836987	+	0.547222i	$0.184315\pi$
$113$			123.672i			1.09444i	−0.836987	+	0.547222i	$0.815685\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−14.9638			−0.127896
$118$		0			0
$119$	118.818			0.998468
$120$		0			0
$121$	−108.731	+	53.0901i	−0.898604	+	0.438761i
$122$		0			0
$123$	−16.6976			−0.135752
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−20.3036			−0.159871			−0.0799356	−	0.996800i	$-0.525471\pi$
$127$	−20.3036			−0.159871			−0.0799356	+	0.996800i	$0.525471\pi$
$128$		0			0
$129$			43.0314i			0.333577i
$130$		0			0
$131$			103.194i			0.787738i	0.919167	+	0.393869i	$0.128864\pi$
$131$			103.194i			0.787738i	−0.919167	+	0.393869i	$0.871136\pi$
$132$		0			0
$133$		−	28.4693i		−	0.214055i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			174.377i			1.27282i	0.771350	+	0.636411i	$0.219582\pi$
$137$			174.377i			1.27282i	−0.771350	+	0.636411i	$0.780418\pi$
$138$		0			0
$139$		−	17.5314i		−	0.126125i	−0.998010	−	0.0630626i	$-0.979913\pi$
$139$		−	17.5314i		−	0.126125i	0.998010	−	0.0630626i	$-0.0200868\pi$
$140$		0			0
$141$	−1.40852			−0.00998950
$142$		0			0
$143$	−4.36444	−	18.8858i	−0.0305206	−	0.132068i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			9.18099i			0.0624557i
$148$		0			0
$149$		−	287.015i		−	1.92628i	−0.269003	−	0.963139i	$-0.586694\pi$
$149$		−	287.015i		−	1.92628i	0.269003	−	0.963139i	$-0.413306\pi$
$150$		0			0
$151$			137.003i			0.907306i	0.891178	+	0.453653i	$0.149879\pi$
$151$			137.003i			0.907306i	−0.891178	+	0.453653i	$0.850121\pi$
$152$		0			0
$153$	128.266			0.838338
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	224.331i		−	1.42886i	−0.699708	−	0.714429i	$-0.746687\pi$
$157$		−	224.331i		−	1.42886i	0.699708	−	0.714429i	$-0.253313\pi$
$158$		0			0
$159$	11.8320			0.0744153
$160$		0			0
$161$			84.2944i			0.523568i
$162$		0			0
$163$		−	253.133i		−	1.55297i	−0.630138	−	0.776483i	$-0.717002\pi$
$163$		−	253.133i		−	1.55297i	0.630138	−	0.776483i	$-0.282998\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	189.467			1.13453			0.567267	−	0.823534i	$-0.308001\pi$
$167$	189.467			1.13453			0.567267	+	0.823534i	$0.308001\pi$
$168$		0			0
$169$	−165.895			−0.981626
$170$		0			0
$171$		−	30.7331i		−	0.179725i
$172$		0			0
$173$	−115.886			−0.669863			−0.334932	−	0.942242i	$-0.608713\pi$
$173$	−115.886			−0.669863			−0.334932	+	0.942242i	$0.608713\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	55.8299i		−	0.315423i
$178$		0			0
$179$	−177.845			−0.993549			−0.496775	−	0.867880i	$-0.665482\pi$
$179$	−177.845			−0.993549			−0.496775	+	0.867880i	$0.665482\pi$
$180$		0			0
$181$	−220.643			−1.21902			−0.609511	−	0.792777i	$-0.708634\pi$
$181$	−220.643			−1.21902			−0.609511	+	0.792777i	$0.708634\pi$
$182$		0			0
$183$	−19.4127			−0.106080
$184$		0			0
$185$		0			0
$186$		0			0
$187$	37.4108	+	161.884i	0.200058	+	0.865690i
$188$		0			0
$189$			98.0864i			0.518976i
$190$		0			0
$191$	257.495			1.34814			0.674071	−	0.738667i	$-0.264544\pi$
$191$	257.495			1.34814			0.674071	+	0.738667i	$0.264544\pi$
$192$		0			0
$193$	165.563			0.857838			0.428919	−	0.903343i	$-0.358895\pi$
$193$	165.563			0.857838			0.428919	+	0.903343i	$0.358895\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−47.5240			−0.241238			−0.120619	−	0.992699i	$-0.538488\pi$
$197$	−47.5240			−0.241238			−0.120619	+	0.992699i	$0.538488\pi$
$198$		0			0
$199$	71.8992			0.361302			0.180651	−	0.983547i	$-0.442180\pi$
$199$	71.8992			0.361302			0.180651	+	0.983547i	$0.442180\pi$
$200$		0			0
$201$	39.9819			0.198915
$202$		0			0
$203$		−	209.166i		−	1.03038i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			90.9972i			0.439600i
$208$		0			0
$209$	38.7881	−	8.96381i	0.185589	−	0.0428890i
$210$		0			0
$211$		−	196.874i		−	0.933053i	−0.884507	−	0.466527i	$-0.845505\pi$
$211$		−	196.874i		−	0.933053i	0.884507	−	0.466527i	$-0.154495\pi$
$212$		0			0
$213$		−	29.7667i		−	0.139750i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−168.305			−0.775601
$218$		0			0
$219$			75.9165i			0.346651i
$220$		0			0
$221$	−26.6164			−0.120436
$222$		0			0
$223$			344.833i			1.54634i	0.634200	+	0.773169i	$0.281330\pi$
$223$			344.833i			1.54634i	−0.634200	+	0.773169i	$0.718670\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	191.765			0.844782			0.422391	−	0.906414i	$-0.361191\pi$
$227$	191.765			0.844782			0.422391	+	0.906414i	$0.361191\pi$
$228$		0			0
$229$	−0.909566			−0.00397190			−0.00198595	−	0.999998i	$-0.500632\pi$
$229$	−0.909566			−0.00397190			−0.00198595	+	0.999998i	$0.500632\pi$
$230$		0			0
$231$	−60.0991	+	13.8887i	−0.260169	+	0.0601243i
$232$		0			0
$233$	−74.3310			−0.319017			−0.159508	−	0.987197i	$-0.550991\pi$
$233$	−74.3310			−0.319017			−0.159508	+	0.987197i	$0.550991\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−63.7477			−0.268977
$238$		0			0
$239$			246.430i			1.03109i	0.856863	+	0.515544i	$0.172410\pi$
$239$			246.430i			1.03109i	−0.856863	+	0.515544i	$0.827590\pi$
$240$		0			0
$241$		−	337.399i		−	1.39999i	−0.714145	−	0.699997i	$-0.753184\pi$
$241$		−	337.399i		−	1.39999i	0.714145	−	0.699997i	$-0.246816\pi$
$242$		0			0
$243$			160.367i			0.659946i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			6.37741i			0.0258195i
$248$		0			0
$249$			94.4050i			0.379137i
$250$		0			0
$251$	75.4459			0.300581			0.150291	−	0.988642i	$-0.451979\pi$
$251$	75.4459			0.300581			0.150291	+	0.988642i	$0.451979\pi$
$252$		0			0
$253$	−114.847	+	26.5409i	−0.453943	+	0.104905i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	330.263i		−	1.28507i	−0.766256	−	0.642535i	$-0.777883\pi$
$257$		−	330.263i		−	1.28507i	0.766256	−	0.642535i	$-0.222117\pi$
$258$		0			0
$259$		−	11.6591i		−	0.0450158i
$260$		0			0
$261$		−	225.798i		−	0.865128i
$262$		0			0
$263$	34.4497			0.130987			0.0654937	−	0.997853i	$-0.479138\pi$
$263$	34.4497			0.130987			0.0654937	+	0.997853i	$0.479138\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			42.4681i			0.159057i
$268$		0			0
$269$	402.928			1.49787			0.748936	−	0.662642i	$-0.230565\pi$
$269$	402.928			1.49787			0.748936	+	0.662642i	$0.230565\pi$
$270$		0			0
$271$		−	143.258i		−	0.528627i	−0.964437	−	0.264313i	$-0.914855\pi$
$271$		−	143.258i		−	0.528627i	0.964437	−	0.264313i	$-0.0851454\pi$
$272$		0			0
$273$		−	9.88129i		−	0.0361952i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−522.435			−1.88605			−0.943024	−	0.332725i	$-0.892032\pi$
$277$	−522.435			−1.88605			−0.943024	+	0.332725i	$0.892032\pi$
$278$		0			0
$279$	−181.689			−0.651213
$280$		0			0
$281$		−	291.412i		−	1.03705i	−0.855061	−	0.518527i	$-0.826481\pi$
$281$		−	291.412i		−	1.03705i	0.855061	−	0.518527i	$-0.173519\pi$
$282$		0			0
$283$	−215.587			−0.761792			−0.380896	−	0.924618i	$-0.624384\pi$
$283$	−215.587			−0.761792			−0.380896	+	0.924618i	$0.624384\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			184.257i			0.642010i
$288$		0			0
$289$	−60.8514			−0.210558
$290$		0			0
$291$	33.4783			0.115046
$292$		0			0
$293$	−354.843			−1.21107			−0.605535	−	0.795819i	$-0.707041\pi$
$293$	−354.843			−1.21107			−0.605535	+	0.795819i	$0.707041\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−133.638	+	30.8834i	−0.449961	+	0.103985i
$298$		0			0
$299$		−	18.8828i		−	0.0631532i
$300$		0			0
$301$	474.851			1.57758
$302$		0			0
$303$	−125.963			−0.415719
$304$		0			0
$305$		0			0
$306$		0			0
$307$	32.9549			0.107345			0.0536725	−	0.998559i	$-0.482907\pi$
$307$	32.9549			0.107345			0.0536725	+	0.998559i	$0.482907\pi$
$308$		0			0
$309$	−110.029			−0.356080
$310$		0			0
$311$	−291.436			−0.937094			−0.468547	−	0.883439i	$-0.655222\pi$
$311$	−291.436			−0.937094			−0.468547	+	0.883439i	$0.655222\pi$
$312$		0			0
$313$		−	267.480i		−	0.854569i	−0.904117	−	0.427284i	$-0.859470\pi$
$313$		−	267.480i		−	0.854569i	0.904117	−	0.427284i	$-0.140530\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	241.219i		−	0.760945i	−0.924792	−	0.380472i	$-0.875761\pi$
$317$		−	241.219i		−	0.760945i	0.924792	−	0.380472i	$-0.124239\pi$
$318$		0			0
$319$	284.980	−	65.8579i	0.893353	−	0.206451i
$320$		0			0
$321$			56.3212i			0.175456i
$322$		0			0
$323$		−	54.6655i		−	0.169243i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	110.296			0.337297
$328$		0			0
$329$			15.5430i			0.0472431i
$330$		0			0
$331$	85.9292			0.259605			0.129802	−	0.991540i	$-0.458566\pi$
$331$	85.9292			0.259605			0.129802	+	0.991540i	$0.458566\pi$
$332$		0			0
$333$		−	12.5862i		−	0.0377964i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−517.174			−1.53464			−0.767321	−	0.641263i	$-0.778411\pi$
$337$	−517.174			−1.53464			−0.767321	+	0.641263i	$0.778411\pi$
$338$		0			0
$339$	−88.1604			−0.260060
$340$		0			0
$341$	−52.9925	−	229.309i	−0.155403	−	0.672460i
$342$		0			0
$343$	−284.138			−0.828392
$344$		0			0
$345$		0			0
$346$		0			0
$347$	202.703			0.584158			0.292079	−	0.956394i	$-0.405653\pi$
$347$	202.703			0.584158			0.292079	+	0.956394i	$0.405653\pi$
$348$		0			0
$349$		−	411.006i		−	1.17767i	−0.808254	−	0.588834i	$-0.799587\pi$
$349$		−	411.006i		−	1.17767i	0.808254	−	0.588834i	$-0.200413\pi$
$350$		0			0
$351$		−	21.9724i		−	0.0625993i
$352$		0			0
$353$			119.131i			0.337483i	0.985660	+	0.168741i	$0.0539703\pi$
$353$			119.131i			0.337483i	−0.985660	+	0.168741i	$0.946030\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			84.6998i			0.237254i
$358$		0			0
$359$			154.881i			0.431423i	0.976457	+	0.215711i	$0.0692071\pi$
$359$			154.881i			0.431423i	−0.976457	+	0.215711i	$0.930793\pi$
$360$		0			0
$361$	347.902			0.963717
$362$		0			0
$363$	−37.8455	−	77.5095i	−0.104258	−	0.213525i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	401.386i		−	1.09370i	−0.837232	−	0.546848i	$-0.815828\pi$
$367$		−	401.386i		−	1.09370i	0.837232	−	0.546848i	$-0.184172\pi$
$368$		0			0
$369$			198.908i			0.539047i
$370$		0			0
$371$		−	130.566i		−	0.351930i
$372$		0			0
$373$	−710.613			−1.90513			−0.952565	−	0.304336i	$-0.901566\pi$
$373$	−710.613			−1.90513			−0.952565	+	0.304336i	$0.901566\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			46.8554i			0.124285i
$378$		0			0
$379$	−206.360			−0.544484			−0.272242	−	0.962229i	$-0.587765\pi$
$379$	−206.360			−0.544484			−0.272242	+	0.962229i	$0.587765\pi$
$380$		0			0
$381$		−	14.4735i		−	0.0379883i
$382$		0			0
$383$		−	660.006i		−	1.72325i	−0.507543	−	0.861627i	$-0.669446\pi$
$383$		−	660.006i		−	1.72325i	0.507543	−	0.861627i	$-0.330554\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	512.609			1.32457
$388$		0			0
$389$	−3.37671			−0.00868049			−0.00434024	−	0.999991i	$-0.501382\pi$
$389$	−3.37671			−0.00868049			−0.00434024	+	0.999991i	$0.501382\pi$
$390$		0			0
$391$			161.858i			0.413960i
$392$		0			0
$393$	−73.5621			−0.187181
$394$		0			0
$395$		0			0
$396$		0			0
$397$			525.780i			1.32438i	0.749335	+	0.662191i	$0.230373\pi$
$397$			525.780i			1.32438i	−0.749335	+	0.662191i	$0.769627\pi$
$398$		0			0
$399$	20.2945			0.0508633
$400$		0			0
$401$	−440.318			−1.09805			−0.549025	−	0.835806i	$-0.685001\pi$
$401$	−440.318			−1.09805			−0.549025	+	0.835806i	$0.685001\pi$
$402$		0			0
$403$	37.7022			0.0935537
$404$		0			0
$405$		0			0
$406$		0			0
$407$	15.8850	−	3.67097i	0.0390295	−	0.00901959i
$408$		0			0
$409$		−	275.710i		−	0.674107i	−0.941485	−	0.337054i	$-0.890570\pi$
$409$		−	275.710i		−	0.674107i	0.941485	−	0.337054i	$-0.109430\pi$
$410$		0			0
$411$	−124.305			−0.302446
$412$		0			0
$413$	−616.081			−1.49172
$414$		0			0
$415$		0			0
$416$		0			0
$417$	12.4973			0.0299696
$418$		0			0
$419$	−393.548			−0.939254			−0.469627	−	0.882865i	$-0.655612\pi$
$419$	−393.548			−0.939254			−0.469627	+	0.882865i	$0.655612\pi$
$420$		0			0
$421$	−125.449			−0.297978			−0.148989	−	0.988839i	$-0.547602\pi$
$421$	−125.449			−0.297978			−0.148989	+	0.988839i	$0.547602\pi$
$422$		0			0
$423$			16.7789i			0.0396664i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			214.218i			0.501683i
$428$		0			0
$429$	13.4628	−	3.11122i	0.0313819	−	0.00725225i
$430$		0			0
$431$		−	401.945i		−	0.932586i	−0.884630	−	0.466293i	$-0.845589\pi$
$431$		−	401.945i		−	0.932586i	0.884630	−	0.466293i	$-0.154411\pi$
$432$		0			0
$433$		−	708.096i		−	1.63533i	−0.575698	−	0.817663i	$-0.695269\pi$
$433$		−	708.096i		−	1.63533i	0.575698	−	0.817663i	$-0.304731\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	38.7820			0.0887461
$438$		0			0
$439$		−	1.96949i		−	0.00448630i	−0.999997	−	0.00224315i	$-0.999286\pi$
$439$		−	1.96949i		−	0.00448630i	0.999997	−	0.00224315i	$-0.000714018\pi$
$440$		0			0
$441$	109.368			0.248000
$442$		0			0
$443$			571.522i			1.29012i	0.764133	+	0.645059i	$0.223167\pi$
$443$			571.522i			1.29012i	−0.764133	+	0.645059i	$0.776833\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	204.600			0.457719
$448$		0			0
$449$	−303.444			−0.675822			−0.337911	−	0.941178i	$-0.609720\pi$
$449$	−303.444			−0.675822			−0.337911	+	0.941178i	$0.609720\pi$
$450$		0			0
$451$	−251.042	+	58.0150i	−0.556634	+	0.128636i
$452$		0			0
$453$	−97.6634			−0.215593
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−643.024			−1.40706			−0.703528	−	0.710668i	$-0.748393\pi$
$457$	−643.024			−1.40706			−0.703528	+	0.710668i	$0.748393\pi$
$458$		0			0
$459$			188.341i			0.410330i
$460$		0			0
$461$			294.839i			0.639564i	0.947491	+	0.319782i	$0.103610\pi$
$461$			294.839i			0.639564i	−0.947491	+	0.319782i	$0.896390\pi$
$462$		0			0
$463$			253.803i			0.548172i	0.961705	+	0.274086i	$0.0883752\pi$
$463$			253.803i			0.548172i	−0.961705	+	0.274086i	$0.911625\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	476.878i		−	1.02115i	−0.859833	−	0.510576i	$-0.829432\pi$
$467$		−	476.878i		−	1.02115i	0.859833	−	0.510576i	$-0.170568\pi$
$468$		0			0
$469$		−	441.199i		−	0.940723i
$470$		0			0
$471$	159.915			0.339523
$472$		0			0
$473$	149.511	+	646.963i	0.316091	+	1.36779i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	140.948i		−	0.295489i
$478$		0			0
$479$			788.853i			1.64687i	0.567407	+	0.823437i	$0.307947\pi$
$479$			788.853i			1.64687i	−0.567407	+	0.823437i	$0.692053\pi$
$480$		0			0
$481$			2.61176i			0.00542985i
$482$		0			0
$483$	−60.0897			−0.124409
$484$		0			0
$485$		0			0
$486$		0			0
$487$			199.251i			0.409139i	0.978852	+	0.204569i	$0.0655794\pi$
$487$			199.251i			0.409139i	−0.978852	+	0.204569i	$0.934421\pi$
$488$		0			0
$489$	180.447			0.369013
$490$		0			0
$491$		−	683.145i		−	1.39133i	−0.718365	−	0.695667i	$-0.755109\pi$
$491$		−	683.145i		−	1.39133i	0.718365	−	0.695667i	$-0.244891\pi$
$492$		0			0
$493$		−	401.632i		−	0.814669i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−328.474			−0.660914
$498$		0			0
$499$	769.021			1.54112			0.770562	−	0.637365i	$-0.219976\pi$
$499$	769.021			1.54112			0.770562	+	0.637365i	$0.219976\pi$
$500$		0			0
$501$			135.063i			0.269586i
$502$		0			0
$503$	521.400			1.03658			0.518290	−	0.855205i	$-0.326569\pi$
$503$	521.400			1.03658			0.518290	+	0.855205i	$0.326569\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	118.259i		−	0.233252i
$508$		0			0
$509$	19.3080			0.0379332			0.0189666	−	0.999820i	$-0.493962\pi$
$509$	19.3080			0.0379332			0.0189666	+	0.999820i	$0.493962\pi$
$510$		0			0
$511$	837.737			1.63941
$512$		0			0
$513$	45.1274			0.0879677
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−21.1766	+	4.89385i	−0.0409606	+	0.00946585i
$518$		0			0
$519$		−	82.6101i		−	0.159172i
$520$		0			0
$521$	363.483			0.697664			0.348832	−	0.937185i	$-0.386578\pi$
$521$	363.483			0.697664			0.348832	+	0.937185i	$0.386578\pi$
$522$		0			0
$523$	−513.531			−0.981894			−0.490947	−	0.871189i	$-0.663349\pi$
$523$	−513.531			−0.981894			−0.490947	+	0.871189i	$0.663349\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−323.173			−0.613231
$528$		0			0
$529$	414.171			0.782931
$530$		0			0
$531$	−665.070			−1.25249
$532$		0			0
$533$		−	41.2755i		−	0.0774399i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	126.778i		−	0.236085i
$538$		0			0
$539$	31.8990	+	138.033i	0.0591818	+	0.256091i
$540$		0			0
$541$		−	25.8111i		−	0.0477100i	−0.999715	−	0.0238550i	$-0.992406\pi$
$541$		−	25.8111i		−	0.0477100i	0.999715	−	0.0238550i	$-0.00759401\pi$
$542$		0			0
$543$		−	157.287i		−	0.289662i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−631.814			−1.15505			−0.577527	−	0.816372i	$-0.695982\pi$
$547$	−631.814			−1.15505			−0.577527	+	0.816372i	$0.695982\pi$
$548$		0			0
$549$			231.252i			0.421225i
$550$		0			0
$551$	−96.2328			−0.174651
$552$		0			0
$553$			703.453i			1.27207i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	75.6524			0.135821			0.0679106	−	0.997691i	$-0.478367\pi$
$557$	75.6524			0.135821			0.0679106	+	0.997691i	$0.478367\pi$
$558$		0			0
$559$	−106.371			−0.190289
$560$		0			0
$561$	−115.400	+	26.6685i	−0.205704	+	0.0475374i
$562$		0			0
$563$	954.016			1.69452			0.847261	−	0.531177i	$-0.178250\pi$
$563$	954.016			1.69452			0.847261	+	0.531177i	$0.178250\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	531.275			0.936993
$568$		0			0
$569$		−	360.243i		−	0.633115i	−0.948573	−	0.316558i	$-0.897473\pi$
$569$		−	360.243i		−	0.633115i	0.948573	−	0.316558i	$-0.102527\pi$
$570$		0			0
$571$		−	86.4148i		−	0.151339i	−0.997133	−	0.0756697i	$-0.975891\pi$
$571$		−	86.4148i		−	0.151339i	0.997133	−	0.0756697i	$-0.0241095\pi$
$572$		0			0
$573$			183.557i			0.320343i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	570.772i		−	0.989207i	−0.869119	−	0.494603i	$-0.835313\pi$
$577$		−	570.772i		−	0.989207i	0.869119	−	0.494603i	$-0.164687\pi$
$578$		0			0
$579$			118.022i			0.203838i
$580$		0			0
$581$	1041.76			1.79304
$582$		0			0
$583$	177.891	−	41.1099i	0.305130	−	0.0705144i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			302.867i			0.515957i	0.966151	+	0.257979i	$0.0830564\pi$
$587$			302.867i			0.515957i	−0.966151	+	0.257979i	$0.916944\pi$
$588$		0			0
$589$			77.4337i			0.131466i
$590$		0			0
$591$		−	33.8777i		−	0.0573227i
$592$		0			0
$593$	427.786			0.721393			0.360697	−	0.932683i	$-0.382539\pi$
$593$	427.786			0.721393			0.360697	+	0.932683i	$0.382539\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			51.2537i			0.0858520i
$598$		0			0
$599$	−723.483			−1.20782			−0.603909	−	0.797053i	$-0.706391\pi$
$599$	−723.483			−1.20782			−0.603909	+	0.797053i	$0.706391\pi$
$600$		0			0
$601$			818.046i			1.36114i	0.732683	+	0.680570i	$0.238268\pi$
$601$			818.046i			1.36114i	−0.732683	+	0.680570i	$0.761732\pi$
$602$		0			0
$603$		−	476.282i		−	0.789853i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	136.664			0.225147			0.112573	−	0.993643i	$-0.464091\pi$
$607$	136.664			0.225147			0.112573	+	0.993643i	$0.464091\pi$
$608$		0			0
$609$	149.105			0.244836
$610$		0			0
$611$		−	3.48178i		−	0.00569850i
$612$		0			0
$613$	147.320			0.240326			0.120163	−	0.992754i	$-0.461658\pi$
$613$	147.320			0.240326			0.120163	+	0.992754i	$0.461658\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			653.924i			1.05984i	0.848046	+	0.529922i	$0.177779\pi$
$617$			653.924i			1.05984i	−0.848046	+	0.529922i	$0.822221\pi$
$618$		0			0
$619$	287.719			0.464813			0.232407	−	0.972619i	$-0.425340\pi$
$619$	287.719			0.464813			0.232407	+	0.972619i	$0.425340\pi$
$620$		0			0
$621$	−133.617			−0.215165
$622$		0			0
$623$	468.635			0.752222
$624$		0			0
$625$		0			0
$626$		0			0
$627$	6.38990	+	27.6503i	0.0101912	+	0.0440994i
$628$		0			0
$629$		−	22.3873i		−	0.0355919i
$630$		0			0
$631$	129.366			0.205017			0.102508	−	0.994732i	$-0.467313\pi$
$631$	129.366			0.205017			0.102508	+	0.994732i	$0.467313\pi$
$632$		0			0
$633$	140.343			0.221710
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−22.6949			−0.0356278
$638$		0			0
$639$	−354.594			−0.554920
$640$		0			0
$641$	−1020.20			−1.59158			−0.795789	−	0.605574i	$-0.792944\pi$
$641$	−1020.20			−1.59158			−0.795789	+	0.605574i	$0.792944\pi$
$642$		0			0
$643$			694.490i			1.08008i	0.841640	+	0.540039i	$0.181590\pi$
$643$			694.490i			1.08008i	−0.841640	+	0.540039i	$0.818410\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	437.000i		−	0.675425i	−0.941249	−	0.337713i	$-0.890347\pi$
$647$		−	437.000i		−	0.675425i	0.941249	−	0.337713i	$-0.109653\pi$
$648$		0			0
$649$	−193.979	−	839.384i	−0.298889	−	1.29335i
$650$		0			0
$651$		−	119.977i		−	0.184297i
$652$		0			0
$653$		−	1174.79i		−	1.79906i	−0.436858	−	0.899531i	$-0.643909\pi$
$653$		−	1174.79i		−	1.79906i	0.436858	−	0.899531i	$-0.356091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	904.351			1.37649
$658$		0			0
$659$			1151.93i			1.74800i	0.485927	+	0.873999i	$0.338482\pi$
$659$			1151.93i			1.74800i	−0.485927	+	0.873999i	$0.661518\pi$
$660$		0			0
$661$	−1290.69			−1.95263			−0.976316	−	0.216351i	$-0.930584\pi$
$661$	−1290.69			−1.95263			−0.976316	+	0.216351i	$0.930584\pi$
$662$		0			0
$663$		−	18.9736i		−	0.0286178i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	284.935			0.427189
$668$		0			0
$669$	−245.816			−0.367438
$670$		0			0
$671$	−291.863	+	67.4487i	−0.434968	+	0.100520i
$672$		0			0
$673$	473.976			0.704274			0.352137	−	0.935949i	$-0.385455\pi$
$673$	473.976			0.704274			0.352137	+	0.935949i	$0.385455\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	677.090			1.00013			0.500066	−	0.865987i	$-0.333309\pi$
$677$	677.090			1.00013			0.500066	+	0.865987i	$0.333309\pi$
$678$		0			0
$679$		−	369.432i		−	0.544082i
$680$		0			0
$681$			136.701i			0.200736i
$682$		0			0
$683$			715.412i			1.04746i	0.851886	+	0.523728i	$0.175459\pi$
$683$			715.412i			1.04746i	−0.851886	+	0.523728i	$0.824541\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	0.648388i		−	0.000943797i
$688$		0			0
$689$			29.2481i			0.0424501i
$690$		0			0
$691$	698.129			1.01032			0.505158	−	0.863027i	$-0.331434\pi$
$691$	698.129			1.01032			0.505158	+	0.863027i	$0.331434\pi$
$692$		0			0
$693$	165.448	+	715.927i	0.238742	+	1.03308i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			353.802i			0.507607i
$698$		0			0
$699$		−	52.9872i		−	0.0758043i
$700$		0			0
$701$		−	286.516i		−	0.408724i	−0.978895	−	0.204362i	$-0.934488\pi$
$701$		−	286.516i		−	0.408724i	0.978895	−	0.204362i	$-0.0655120\pi$
$702$		0			0
$703$	−5.36410			−0.00763029
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1390.00i			1.96605i
$708$		0			0
$709$	−447.411			−0.631045			−0.315523	−	0.948918i	$-0.602180\pi$
$709$	−447.411			−0.631045			−0.315523	+	0.948918i	$0.602180\pi$
$710$		0			0
$711$			759.390i			1.06806i
$712$		0			0
$713$		−	229.273i		−	0.321561i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−175.669			−0.245005
$718$		0			0
$719$	−277.725			−0.386266			−0.193133	−	0.981173i	$-0.561865\pi$
$719$	−277.725			−0.386266			−0.193133	+	0.981173i	$0.561865\pi$
$720$		0			0
$721$			1214.17i			1.68400i
$722$		0			0
$723$	240.516			0.332664
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	53.3868i		−	0.0734344i	−0.999326	−	0.0367172i	$-0.988310\pi$
$727$		−	53.3868i		−	0.0734344i	0.999326	−	0.0367172i	$-0.0116901\pi$
$728$		0			0
$729$	493.522			0.676985
$730$		0			0
$731$	911.788			1.24732
$732$		0			0
$733$	−310.801			−0.424012			−0.212006	−	0.977268i	$-0.568000\pi$
$733$	−310.801			−0.424012			−0.212006	+	0.977268i	$0.568000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	601.114	−	138.916i	0.815623	−	0.188488i
$738$		0			0
$739$		−	1066.66i		−	1.44339i	−0.692213	−	0.721693i	$-0.743364\pi$
$739$		−	1066.66i		−	1.44339i	0.692213	−	0.721693i	$-0.256636\pi$
$740$		0			0
$741$	−4.54617			−0.00613518
$742$		0			0
$743$	827.964			1.11435			0.557176	−	0.830394i	$-0.311885\pi$
$743$	827.964			1.11435			0.557176	+	0.830394i	$0.311885\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1124.59			1.50548
$748$		0			0
$749$	621.503			0.829777
$750$		0			0
$751$	−1351.87			−1.80009			−0.900044	−	0.435799i	$-0.856466\pi$
$751$	−1351.87			−1.80009			−0.900044	+	0.435799i	$0.856466\pi$
$752$		0			0
$753$			53.7820i			0.0714236i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1248.88i		−	1.64978i	−0.565295	−	0.824889i	$-0.691238\pi$
$757$		−	1248.88i		−	1.64978i	0.565295	−	0.824889i	$-0.308762\pi$
$758$		0			0
$759$	−18.9198	−	81.8696i	−0.0249273	−	0.107865i
$760$		0			0
$761$			1341.51i			1.76283i	0.472347	+	0.881413i	$0.343407\pi$
$761$			1341.51i			1.76283i	−0.472347	+	0.881413i	$0.656593\pi$
$762$		0			0
$763$		−	1217.11i		−	1.59517i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	138.009			0.179933
$768$		0			0
$769$		−	1023.05i		−	1.33036i	−0.746684	−	0.665179i	$-0.768355\pi$
$769$		−	1023.05i		−	1.33036i	0.746684	−	0.665179i	$-0.231645\pi$
$770$		0			0
$771$	235.430			0.305356
$772$		0			0
$773$			379.249i			0.490619i	0.969445	+	0.245310i	$0.0788896\pi$
$773$			379.249i			0.490619i	−0.969445	+	0.245310i	$0.921110\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	8.31124			0.0106966
$778$		0			0
$779$	84.7726			0.108822
$780$		0			0
$781$	−103.423	−	447.532i	−0.132424	−	0.573024i
$782$		0			0
$783$	331.555			0.423442
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−425.740			−0.540965			−0.270483	−	0.962725i	$-0.587183\pi$
$787$	−425.740			−0.540965			−0.270483	+	0.962725i	$0.587183\pi$
$788$		0			0
$789$			24.5576i			0.0311250i
$790$		0			0
$791$			972.847i			1.22990i
$792$		0			0
$793$		−	47.9871i		−	0.0605134i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			158.433i			0.198787i	0.995048	+	0.0993936i	$0.0316903\pi$
$797$			158.433i			0.198787i	−0.995048	+	0.0993936i	$0.968310\pi$
$798$		0			0
$799$			29.8449i			0.0373528i
$800$		0			0
$801$	505.899			0.631584
$802$		0			0
$803$	263.769	+	1141.38i	0.328480	+	1.42139i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			287.229i			0.355922i
$808$		0			0
$809$			340.085i			0.420377i	0.977661	+	0.210188i	$0.0674078\pi$
$809$			340.085i			0.420377i	−0.977661	+	0.210188i	$0.932592\pi$
$810$		0			0
$811$			1399.02i			1.72505i	0.506014	+	0.862525i	$0.331119\pi$
$811$			1399.02i			1.72505i	−0.506014	+	0.862525i	$0.668881\pi$
$812$		0			0
$813$	102.122			0.125611
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	218.468i		−	0.267403i
$818$		0			0
$819$	−117.710			−0.143724
$820$		0			0
$821$		−	401.984i		−	0.489627i	−0.969570	−	0.244814i	$-0.921273\pi$
$821$		−	401.984i		−	0.489627i	0.969570	−	0.244814i	$-0.0787268\pi$
$822$		0			0
$823$		−	1288.52i		−	1.56564i	−0.622248	−	0.782820i	$-0.713781\pi$
$823$		−	1288.52i		−	1.56564i	0.622248	−	0.782820i	$-0.286219\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.53374			−0.00790053			−0.00395027	−	0.999992i	$-0.501257\pi$
$827$	−6.53374			−0.00790053			−0.00395027	+	0.999992i	$0.501257\pi$
$828$		0			0
$829$	202.157			0.243856			0.121928	−	0.992539i	$-0.461092\pi$
$829$	202.157			0.243856			0.121928	+	0.992539i	$0.461092\pi$
$830$		0			0
$831$		−	372.421i		−	0.448160i
$832$		0			0
$833$	194.535			0.233535
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	266.786i		−	0.318740i
$838$		0			0
$839$	836.983			0.997596			0.498798	−	0.866718i	$-0.333775\pi$
$839$	836.983			0.997596			0.498798	+	0.866718i	$0.333775\pi$
$840$		0			0
$841$	133.969			0.159298
$842$		0			0
$843$	207.735			0.246423
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−855.315	+	417.624i	−1.00982	+	0.493063i
$848$		0			0
$849$		−	153.682i		−	0.181016i
$850$		0			0
$851$	15.8825			0.0186633
$852$		0			0
$853$	429.341			0.503331			0.251665	−	0.967814i	$-0.419022\pi$
$853$	429.341			0.503331			0.251665	+	0.967814i	$0.419022\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−34.2196			−0.0399295			−0.0199647	−	0.999801i	$-0.506355\pi$
$857$	−34.2196			−0.0399295			−0.0199647	+	0.999801i	$0.506355\pi$
$858$		0			0
$859$	725.382			0.844450			0.422225	−	0.906491i	$-0.361249\pi$
$859$	725.382			0.844450			0.422225	+	0.906491i	$0.361249\pi$
$860$		0			0
$861$	−131.348			−0.152553
$862$		0			0
$863$			396.795i			0.459786i	0.973216	+	0.229893i	$0.0738376\pi$
$863$			396.795i			0.459786i	−0.973216	+	0.229893i	$0.926162\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	43.3782i		−	0.0500325i
$868$		0			0
$869$	−958.425	+	221.489i	−1.10291	+	0.254878i
$870$		0			0
$871$			98.8331i			0.113471i
$872$		0			0
$873$		−	398.808i		−	0.456824i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−779.062			−0.888327			−0.444163	−	0.895946i	$-0.646499\pi$
$877$	−779.062			−0.888327			−0.444163	+	0.895946i	$0.646499\pi$
$878$		0			0
$879$		−	252.952i		−	0.287772i
$880$		0			0
$881$	−344.584			−0.391128			−0.195564	−	0.980691i	$-0.562654\pi$
$881$	−344.584			−0.391128			−0.195564	+	0.980691i	$0.562654\pi$
$882$		0			0
$883$			1360.03i			1.54024i	0.637902	+	0.770118i	$0.279803\pi$
$883$			1360.03i			1.54024i	−0.637902	+	0.770118i	$0.720197\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	653.214			0.736431			0.368216	−	0.929740i	$-0.379969\pi$
$887$	653.214			0.736431			0.368216	+	0.929740i	$0.379969\pi$
$888$		0			0
$889$	−159.715			−0.179657
$890$		0			0
$891$	167.277	+	723.839i	0.187741	+	0.812390i
$892$		0			0
$893$	7.15098			0.00800782
$894$		0			0
$895$		0			0
$896$		0			0
$897$	13.4607			0.0150064
$898$		0			0
$899$			568.912i			0.632828i
$900$		0			0
$901$		−	250.707i		−	0.278254i
$902$		0			0
$903$			338.500i			0.374861i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	959.647i		−	1.05805i	−0.848608	−	0.529023i	$-0.822559\pi$
$907$		−	959.647i		−	1.05805i	0.848608	−	0.529023i	$-0.177441\pi$
$908$		0			0
$909$			1500.52i			1.65074i
$910$		0			0
$911$	834.675			0.916218			0.458109	−	0.888896i	$-0.348527\pi$
$911$	834.675			0.916218			0.458109	+	0.888896i	$0.348527\pi$
$912$		0			0
$913$	328.007	+	1419.35i	0.359262	+	1.55460i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			811.755i			0.885229i
$918$		0			0
$919$		−	1364.98i		−	1.48529i	−0.669688	−	0.742643i	$-0.733572\pi$
$919$		−	1364.98i		−	1.48529i	0.669688	−	0.742643i	$-0.266428\pi$
$920$		0			0
$921$			23.4921i			0.0255071i
$922$		0			0
$923$	73.5816			0.0797201
$924$		0			0
$925$		0			0
$926$		0			0
$927$			1310.71i			1.41393i
$928$		0			0
$929$	382.643			0.411887			0.205944	−	0.978564i	$-0.433974\pi$
$929$	382.643			0.411887			0.205944	+	0.978564i	$0.433974\pi$
$930$		0			0
$931$		−	46.6114i		−	0.0500660i
$932$		0			0
$933$		−	207.752i		−	0.222671i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	614.872			0.656213			0.328107	−	0.944641i	$-0.393589\pi$
$937$	614.872			0.656213			0.328107	+	0.944641i	$0.393589\pi$
$938$		0			0
$939$	190.674			0.203061
$940$		0			0
$941$		−	846.329i		−	0.899394i	−0.893181	−	0.449697i	$-0.851532\pi$
$941$		−	846.329i		−	0.899394i	0.893181	−	0.449697i	$-0.148468\pi$
$942$		0			0
$943$	−251.003			−0.266175
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1445.78i		−	1.52669i	−0.645989	−	0.763347i	$-0.723555\pi$
$947$		−	1445.78i		−	1.52669i	0.645989	−	0.763347i	$-0.276445\pi$
$948$		0			0
$949$	−187.662			−0.197747
$950$		0			0
$951$	171.954			0.180814
$952$		0			0
$953$	1329.83			1.39541			0.697707	−	0.716383i	$-0.254204\pi$
$953$	1329.83			1.39541			0.697707	+	0.716383i	$0.254204\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	46.9471	+	203.149i	0.0490566	+	0.212277i
$958$		0			0
$959$			1371.70i			1.43035i
$960$		0			0
$961$	−503.225			−0.523648
$962$		0			0
$963$	670.923			0.696701
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−691.917			−0.715529			−0.357765	−	0.933812i	$-0.616461\pi$
$967$	−691.917			−0.715529			−0.357765	+	0.933812i	$0.616461\pi$
$968$		0			0
$969$	38.9685			0.0402152
$970$		0			0
$971$	−1102.69			−1.13562			−0.567811	−	0.823159i	$-0.692210\pi$
$971$	−1102.69			−1.13562			−0.567811	+	0.823159i	$0.692210\pi$
$972$		0			0
$973$		−	137.908i		−	0.141735i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1064.89i		−	1.08996i	−0.838448	−	0.544982i	$-0.816537\pi$
$977$		−	1064.89i		−	1.08996i	0.838448	−	0.544982i	$-0.183463\pi$
$978$		0			0
$979$	147.554	+	638.494i	0.150719	+	0.652190i
$980$		0			0
$981$		−	1313.89i		−	1.33934i
$982$		0			0
$983$		−	309.648i		−	0.315003i	−0.987519	−	0.157502i	$-0.949656\pi$
$983$		−	309.648i		−	0.315003i	0.987519	−	0.157502i	$-0.0503439\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−11.0799			−0.0112258
$988$		0			0
$989$			646.862i			0.654056i
$990$		0			0
$991$	−1151.55			−1.16200			−0.581002	−	0.813902i	$-0.697339\pi$
$991$	−1151.55			−1.16200			−0.581002	+	0.813902i	$0.697339\pi$
$992$		0			0
$993$			61.2550i			0.0616868i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1373.10			−1.37723			−0.688614	−	0.725128i	$-0.741781\pi$
$997$	−1373.10			−1.37723			−0.688614	+	0.725128i	$0.741781\pi$
$998$		0			0
$999$	18.4812			0.0184997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.b.549.10		16
5.2	odd	4		220.3.f.a.21.6	yes	8
5.3	odd	4		1100.3.f.f.901.3		8
5.4	even	2	inner	1100.3.e.b.549.7		16
11.10	odd	2	inner	1100.3.e.b.549.9		16
15.2	even	4		1980.3.b.a.901.7		8
20.7	even	4		880.3.j.b.241.3		8
55.32	even	4		220.3.f.a.21.5	✓	8
55.43	even	4		1100.3.f.f.901.4		8
55.54	odd	2	inner	1100.3.e.b.549.8		16
165.32	odd	4		1980.3.b.a.901.6		8
220.87	odd	4		880.3.j.b.241.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.f.a.21.5	✓	8	55.32	even	4
220.3.f.a.21.6	yes	8	5.2	odd	4
880.3.j.b.241.3		8	20.7	even	4
880.3.j.b.241.4		8	220.87	odd	4
1100.3.e.b.549.7		16	5.4	even	2	inner
1100.3.e.b.549.8		16	55.54	odd	2	inner
1100.3.e.b.549.9		16	11.10	odd	2	inner
1100.3.e.b.549.10		16	1.1	even	1	trivial
1100.3.f.f.901.3		8	5.3	odd	4
1100.3.f.f.901.4		8	55.43	even	4
1980.3.b.a.901.6		8	165.32	odd	4
1980.3.b.a.901.7		8	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.712855i q^{3} +7.86633 q^{7} +8.49184 q^{9} +O(q^{10})\)	\(q+0.712855i q^{3} +7.86633 q^{7} +8.49184 q^{9} +(2.47679 + 10.7175i) q^{11} -1.76214 q^{13} +15.1046 q^{17} -3.61913i q^{19} +5.60755i q^{21} +10.7158i q^{23} +12.4691i q^{27} -26.5900i q^{29} -21.3957 q^{31} +(-7.64005 + 1.76559i) q^{33} -1.48215i q^{37} -1.25615i q^{39} +23.4235i q^{41} +60.3649 q^{43} +1.97588i q^{47} +12.8792 q^{49} +10.7674i q^{51} -16.5981i q^{53} +2.57991 q^{57} -78.3187 q^{59} +27.2323i q^{61} +66.7996 q^{63} -56.0870i q^{67} -7.63885 q^{69} -41.7570 q^{71} +106.496 q^{73} +(19.4832 + 84.3077i) q^{77} +89.4259i q^{79} +67.5379 q^{81} +132.432 q^{83} +18.9548 q^{87} +59.5747 q^{89} -13.8616 q^{91} -15.2520i q^{93} -46.9636i q^{97} +(21.0325 + 91.0116i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 56 q^{9}+O(q^{10}) \) 16 * q - 56 * q^9	\( 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) \) 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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