LMFDB - Embedded newform 4410.2.d.c.4409.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(4409,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4410.4409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4409.3
Character	$\chi$	$=$	4410.4409
Dual form			4410.2.d.c.4409.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(-2.00118 - 0.997634i) q^{5} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +(-2.00118 - 0.997634i) q^{5} -1.00000 q^{8} +(2.00118 + 0.997634i) q^{10} +0.311751i q^{11} +1.62784 q^{13} +1.00000 q^{16} +1.60916i q^{17} -4.54781i q^{19} +(-2.00118 - 0.997634i) q^{20} -0.311751i q^{22} -4.42207 q^{23} +(3.00945 + 3.99289i) q^{25} -1.62784 q^{26} +1.79249i q^{29} +0.415259i q^{31} -1.00000 q^{32} -1.60916i q^{34} -2.16685i q^{37} +4.54781i q^{38} +(2.00118 + 0.997634i) q^{40} -3.39718 q^{41} -0.812195i q^{43} +0.311751i q^{44} +4.42207 q^{46} -0.316426i q^{47} +(-3.00945 - 3.99289i) q^{50} +1.62784 q^{52} -11.5449 q^{53} +(0.311013 - 0.623869i) q^{55} -1.79249i q^{58} +10.0691 q^{59} -9.94358i q^{61} -0.415259i q^{62} +1.00000 q^{64} +(-3.25760 - 1.62399i) q^{65} +14.8833i q^{67} +1.60916i q^{68} +2.38340i q^{71} +6.13913 q^{73} +2.16685i q^{74} -4.54781i q^{76} +9.32022 q^{79} +(-2.00118 - 0.997634i) q^{80} +3.39718 q^{82} -9.49304i q^{83} +(1.60535 - 3.22021i) q^{85} +0.812195i q^{86} -0.311751i q^{88} -10.0245 q^{89} -4.42207 q^{92} +0.316426i q^{94} +(-4.53705 + 9.10099i) q^{95} -12.7011 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8}+O(q^{10}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{16} + 32 q^{23} - 16 q^{25} - 24 q^{32} - 32 q^{46} + 16 q^{50} - 32 q^{53} + 24 q^{64} - 32 q^{85} + 32 q^{92} + 32 q^{95}+O(q^{100}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^16 + 32 * q^23 - 16 * q^25 - 24 * q^32 - 32 * q^46 + 16 * q^50 - 32 * q^53 + 24 * q^64 - 32 * q^85 + 32 * q^92 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$.

$n$	$1081$	$2647$	$3431$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−2.00118	−	0.997634i	−0.894955	−	0.446156i
$5$	−2.00118	−	0.997634i	−0.894955	−	0.446156i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$		0			0
$10$	2.00118	+	0.997634i	0.632829	+	0.315480i
$11$			0.311751i			0.0939963i	0.998895	+	0.0469982i	$0.0149655\pi$
$11$			0.311751i			0.0939963i	−0.998895	+	0.0469982i	$0.985035\pi$
$12$		0			0
$13$	1.62784			0.451481			0.225741	−	0.974187i	$-0.427520\pi$
$13$	1.62784			0.451481			0.225741	+	0.974187i	$0.427520\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			1.60916i			0.390278i	0.980776	+	0.195139i	$0.0625157\pi$
$17$			1.60916i			0.390278i	−0.980776	+	0.195139i	$0.937484\pi$
$18$		0			0
$19$		−	4.54781i		−	1.04334i	−0.853148	−	0.521670i	$-0.825309\pi$
$19$		−	4.54781i		−	1.04334i	0.853148	−	0.521670i	$-0.174691\pi$
$20$	−2.00118	−	0.997634i	−0.447478	−	0.223078i
$21$		0			0
$22$		−	0.311751i		−	0.0664654i
$23$	−4.42207			−0.922065			−0.461033	−	0.887383i	$-0.652521\pi$
$23$	−4.42207			−0.922065			−0.461033	+	0.887383i	$0.652521\pi$
$24$		0			0
$25$	3.00945	+	3.99289i	0.601890	+	0.798579i
$26$	−1.62784			−0.319246
$27$		0			0
$28$		0			0
$29$			1.79249i			0.332858i	0.986053	+	0.166429i	$0.0532236\pi$
$29$			1.79249i			0.332858i	−0.986053	+	0.166429i	$0.946776\pi$
$30$		0			0
$31$			0.415259i			0.0745827i	0.999304	+	0.0372913i	$0.0118730\pi$
$31$			0.415259i			0.0745827i	−0.999304	+	0.0372913i	$0.988127\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		−	1.60916i		−	0.275968i
$35$		0			0
$36$		0			0
$37$		−	2.16685i		−	0.356227i	−0.984010	−	0.178114i	$-0.943001\pi$
$37$		−	2.16685i		−	0.356227i	0.984010	−	0.178114i	$-0.0569995\pi$
$38$			4.54781i			0.737752i
$39$		0			0
$40$	2.00118	+	0.997634i	0.316415	+	0.157740i
$41$	−3.39718			−0.530550			−0.265275	−	0.964173i	$-0.585463\pi$
$41$	−3.39718			−0.530550			−0.265275	+	0.964173i	$0.585463\pi$
$42$		0			0
$43$		−	0.812195i		−	0.123859i	−0.998081	−	0.0619293i	$-0.980275\pi$
$43$		−	0.812195i		−	0.123859i	0.998081	−	0.0619293i	$-0.0197253\pi$
$44$			0.311751i			0.0469982i
$45$		0			0
$46$	4.42207			0.651998
$47$		−	0.316426i		−	0.0461555i	−0.999734	−	0.0230777i	$-0.992653\pi$
$47$		−	0.316426i		−	0.0461555i	0.999734	−	0.0230777i	$-0.00734652\pi$
$48$		0			0
$49$		0			0
$50$	−3.00945	−	3.99289i	−0.425601	−	0.564681i
$51$		0			0
$52$	1.62784			0.225741
$53$	−11.5449			−1.58581			−0.792907	−	0.609342i	$-0.791434\pi$
$53$	−11.5449			−1.58581			−0.792907	+	0.609342i	$0.791434\pi$
$54$		0			0
$55$	0.311013	−	0.623869i	0.0419370	−	0.0841225i
$56$		0			0
$57$		0			0
$58$		−	1.79249i		−	0.235366i
$59$	10.0691			1.31088			0.655441	−	0.755246i	$-0.272483\pi$
$59$	10.0691			1.31088			0.655441	+	0.755246i	$0.272483\pi$
$60$		0			0
$61$		−	9.94358i		−	1.27314i	−0.771217	−	0.636572i	$-0.780352\pi$
$61$		−	9.94358i		−	1.27314i	0.771217	−	0.636572i	$-0.219648\pi$
$62$		−	0.415259i		−	0.0527379i
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.25760	−	1.62399i	−0.404056	−	0.201431i
$66$		0			0
$67$			14.8833i			1.81829i	0.416482	+	0.909144i	$0.363263\pi$
$67$			14.8833i			1.81829i	−0.416482	+	0.909144i	$0.636737\pi$
$68$			1.60916i			0.195139i
$69$		0			0
$70$		0			0
$71$			2.38340i			0.282858i	0.989948	+	0.141429i	$0.0451697\pi$
$71$			2.38340i			0.282858i	−0.989948	+	0.141429i	$0.954830\pi$
$72$		0			0
$73$	6.13913			0.718531			0.359265	−	0.933235i	$-0.383027\pi$
$73$	6.13913			0.718531			0.359265	+	0.933235i	$0.383027\pi$
$74$			2.16685i			0.251891i
$75$		0			0
$76$		−	4.54781i		−	0.521670i
$77$		0			0
$78$		0			0
$79$	9.32022			1.04861			0.524303	−	0.851532i	$-0.324326\pi$
$79$	9.32022			1.04861			0.524303	+	0.851532i	$0.324326\pi$
$80$	−2.00118	−	0.997634i	−0.223739	−	0.111539i
$81$		0			0
$82$	3.39718			0.375155
$83$		−	9.49304i		−	1.04200i	−0.853558	−	0.520998i	$-0.825560\pi$
$83$		−	9.49304i		−	1.04200i	0.853558	−	0.520998i	$-0.174440\pi$
$84$		0			0
$85$	1.60535	−	3.22021i	0.174125	−	0.349281i
$86$			0.812195i			0.0875812i
$87$		0			0
$88$		−	0.311751i		−	0.0332327i
$89$	−10.0245			−1.06259			−0.531297	−	0.847186i	$-0.678295\pi$
$89$	−10.0245			−1.06259			−0.531297	+	0.847186i	$0.678295\pi$
$90$		0			0
$91$		0			0
$92$	−4.42207			−0.461033
$93$		0			0
$94$			0.316426i			0.0326368i
$95$	−4.53705	+	9.10099i	−0.465492	+	0.933742i
$96$		0			0
$97$	−12.7011			−1.28960			−0.644802	−	0.764350i	$-0.723060\pi$
$97$	−12.7011			−1.28960			−0.644802	+	0.764350i	$0.723060\pi$
$98$		0			0
$99$		0			0
$100$	3.00945	+	3.99289i	0.300945	+	0.399289i
$101$	14.0918			1.40219			0.701094	−	0.713069i	$-0.252696\pi$
$101$	14.0918			1.40219			0.701094	+	0.713069i	$0.252696\pi$
$102$		0			0
$103$	−9.46782			−0.932892			−0.466446	−	0.884550i	$-0.654466\pi$
$103$	−9.46782			−0.932892			−0.466446	+	0.884550i	$0.654466\pi$
$104$	−1.62784			−0.159623
$105$		0			0
$106$	11.5449			1.12134
$107$	−15.1045			−1.46021			−0.730103	−	0.683337i	$-0.760528\pi$
$107$	−15.1045			−1.46021			−0.730103	+	0.683337i	$0.760528\pi$
$108$		0			0
$109$	−2.15856			−0.206752			−0.103376	−	0.994642i	$-0.532965\pi$
$109$	−2.15856			−0.206752			−0.103376	+	0.994642i	$0.532965\pi$
$110$	−0.311013	+	0.623869i	−0.0296539	+	0.0594836i
$111$		0			0
$112$		0			0
$113$	−0.600087			−0.0564514			−0.0282257	−	0.999602i	$-0.508986\pi$
$113$	−0.600087			−0.0564514			−0.0282257	+	0.999602i	$0.508986\pi$
$114$		0			0
$115$	8.84936	+	4.41161i	0.825207	+	0.411385i
$116$			1.79249i			0.166429i
$117$		0			0
$118$	−10.0691			−0.926934
$119$		0			0
$120$		0			0
$121$	10.9028			0.991165
$122$			9.94358i			0.900249i
$123$		0			0
$124$			0.415259i			0.0372913i
$125$	−2.03901	−	10.9928i	−0.182374	−	0.983229i
$126$		0			0
$127$			4.39770i			0.390233i	0.980780	+	0.195117i	$0.0625085\pi$
$127$			4.39770i			0.390233i	−0.980780	+	0.195117i	$0.937492\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	3.25760	+	1.62399i	0.285711	+	0.142433i
$131$	10.4665			0.914463			0.457231	−	0.889348i	$-0.348841\pi$
$131$	10.4665			0.914463			0.457231	+	0.889348i	$0.348841\pi$
$132$		0			0
$133$		0			0
$134$		−	14.8833i		−	1.28572i
$135$		0			0
$136$		−	1.60916i		−	0.137984i
$137$	−14.0199			−1.19780			−0.598898	−	0.800825i	$-0.704395\pi$
$137$	−14.0199			−1.19780			−0.598898	+	0.800825i	$0.704395\pi$
$138$		0			0
$139$			7.00761i			0.594378i	0.954819	+	0.297189i	$0.0960491\pi$
$139$			7.00761i			0.594378i	−0.954819	+	0.297189i	$0.903951\pi$
$140$		0			0
$141$		0			0
$142$		−	2.38340i		−	0.200011i
$143$			0.507480i			0.0424376i
$144$		0			0
$145$	1.78825	−	3.58710i	0.148506	−	0.297893i
$146$	−6.13913			−0.508078
$147$		0			0
$148$		−	2.16685i		−	0.178114i
$149$			4.79900i			0.393150i	0.980489	+	0.196575i	$0.0629819\pi$
$149$			4.79900i			0.393150i	−0.980489	+	0.196575i	$0.937018\pi$
$150$		0			0
$151$	−9.72228			−0.791188			−0.395594	−	0.918426i	$-0.629461\pi$
$151$	−9.72228			−0.791188			−0.395594	+	0.918426i	$0.629461\pi$
$152$			4.54781i			0.368876i
$153$		0			0
$154$		0			0
$155$	0.414276	−	0.831008i	0.0332755	−	0.0667482i
$156$		0			0
$157$	3.49721			0.279107			0.139554	−	0.990215i	$-0.455433\pi$
$157$	3.49721			0.279107			0.139554	+	0.990215i	$0.455433\pi$
$158$	−9.32022			−0.741477
$159$		0			0
$160$	2.00118	+	0.997634i	0.158207	+	0.0788699i
$161$		0			0
$162$		0			0
$163$			23.3218i			1.82670i	0.407172	+	0.913351i	$0.366515\pi$
$163$			23.3218i			1.82670i	−0.407172	+	0.913351i	$0.633485\pi$
$164$	−3.39718			−0.265275
$165$		0			0
$166$			9.49304i			0.736803i
$167$		−	4.94867i		−	0.382940i	−0.981499	−	0.191470i	$-0.938675\pi$
$167$		−	4.94867i		−	0.382940i	0.981499	−	0.191470i	$-0.0613254\pi$
$168$		0			0
$169$	−10.3501			−0.796164
$170$	−1.60535	+	3.22021i	−0.123125	+	0.246979i
$171$		0			0
$172$		−	0.812195i		−	0.0619293i
$173$			22.8768i			1.73929i	0.493676	+	0.869646i	$0.335653\pi$
$173$			22.8768i			1.73929i	−0.493676	+	0.869646i	$0.664347\pi$
$174$		0			0
$175$		0			0
$176$			0.311751i			0.0234991i
$177$		0			0
$178$	10.0245			0.751367
$179$			21.9568i			1.64113i	0.571552	+	0.820566i	$0.306342\pi$
$179$			21.9568i			1.64113i	−0.571552	+	0.820566i	$0.693658\pi$
$180$		0			0
$181$			19.6496i			1.46054i	0.683157	+	0.730272i	$0.260606\pi$
$181$			19.6496i			1.46054i	−0.683157	+	0.730272i	$0.739394\pi$
$182$		0			0
$183$		0			0
$184$	4.42207			0.325999
$185$	−2.16172	+	4.33625i	−0.158933	+	0.318808i
$186$		0			0
$187$	−0.501655			−0.0366847
$188$		−	0.316426i		−	0.0230777i
$189$		0			0
$190$	4.53705	−	9.10099i	0.329152	−	0.660255i
$191$		−	9.21677i		−	0.666902i	−0.942767	−	0.333451i	$-0.891787\pi$
$191$		−	9.21677i		−	0.666902i	0.942767	−	0.333451i	$-0.108213\pi$
$192$		0			0
$193$			11.1432i			0.802102i	0.916056	+	0.401051i	$0.131355\pi$
$193$			11.1432i			0.802102i	−0.916056	+	0.401051i	$0.868645\pi$
$194$	12.7011			0.911887
$195$		0			0
$196$		0			0
$197$	22.0051			1.56780			0.783901	−	0.620886i	$-0.213227\pi$
$197$	22.0051			1.56780			0.783901	+	0.620886i	$0.213227\pi$
$198$		0			0
$199$			6.47821i			0.459228i	0.973282	+	0.229614i	$0.0737464\pi$
$199$			6.47821i			0.459228i	−0.973282	+	0.229614i	$0.926254\pi$
$200$	−3.00945	−	3.99289i	−0.212800	−	0.282340i
$201$		0			0
$202$	−14.0918			−0.991496
$203$		0			0
$204$		0			0
$205$	6.79837	+	3.38914i	0.474818	+	0.236708i
$206$	9.46782			0.659654
$207$		0			0
$208$	1.62784			0.112870
$209$	1.41778			0.0980700
$210$		0			0
$211$	−2.26234			−0.155746			−0.0778730	−	0.996963i	$-0.524813\pi$
$211$	−2.26234			−0.155746			−0.0778730	+	0.996963i	$0.524813\pi$
$212$	−11.5449			−0.792907
$213$		0			0
$214$	15.1045			1.03252
$215$	−0.810274	+	1.62535i	−0.0552602	+	0.110848i
$216$		0			0
$217$		0			0
$218$	2.15856			0.146196
$219$		0			0
$220$	0.311013	−	0.623869i	0.0209685	−	0.0420613i
$221$			2.61945i			0.176203i
$222$		0			0
$223$	−14.8667			−0.995549			−0.497775	−	0.867306i	$-0.665849\pi$
$223$	−14.8667			−0.995549			−0.497775	+	0.867306i	$0.665849\pi$
$224$		0			0
$225$		0			0
$226$	0.600087			0.0399172
$227$			24.5838i			1.63168i	0.578277	+	0.815840i	$0.303725\pi$
$227$			24.5838i			1.63168i	−0.578277	+	0.815840i	$0.696275\pi$
$228$		0			0
$229$			5.88697i			0.389022i	0.980900	+	0.194511i	$0.0623120\pi$
$229$			5.88697i			0.389022i	−0.980900	+	0.194511i	$0.937688\pi$
$230$	−8.84936	−	4.41161i	−0.583510	−	0.290893i
$231$		0			0
$232$		−	1.79249i		−	0.117683i
$233$	−2.00586			−0.131408			−0.0657041	−	0.997839i	$-0.520929\pi$
$233$	−2.00586			−0.131408			−0.0657041	+	0.997839i	$0.520929\pi$
$234$		0			0
$235$	−0.315677	+	0.633226i	−0.0205925	+	0.0413071i
$236$	10.0691			0.655441
$237$		0			0
$238$		0			0
$239$		−	3.89695i		−	0.252073i	−0.992026	−	0.126036i	$-0.959774\pi$
$239$		−	3.89695i		−	0.252073i	0.992026	−	0.126036i	$-0.0402256\pi$
$240$		0			0
$241$		−	17.8215i		−	1.14799i	−0.818860	−	0.573993i	$-0.805394\pi$
$241$		−	17.8215i		−	1.14799i	0.818860	−	0.573993i	$-0.194606\pi$
$242$	−10.9028			−0.700859
$243$		0			0
$244$		−	9.94358i		−	0.636572i
$245$		0			0
$246$		0			0
$247$		−	7.40310i		−	0.471048i
$248$		−	0.415259i		−	0.0263690i
$249$		0			0
$250$	2.03901	+	10.9928i	0.128958	+	0.695248i
$251$	−4.07378			−0.257134			−0.128567	−	0.991701i	$-0.541038\pi$
$251$	−4.07378			−0.257134			−0.128567	+	0.991701i	$0.541038\pi$
$252$		0			0
$253$		−	1.37858i		−	0.0866707i
$254$		−	4.39770i		−	0.275936i
$255$		0			0
$256$	1.00000			0.0625000
$257$			13.4552i			0.839314i	0.907683	+	0.419657i	$0.137850\pi$
$257$			13.4552i			0.839314i	−0.907683	+	0.419657i	$0.862150\pi$
$258$		0			0
$259$		0			0
$260$	−3.25760	−	1.62399i	−0.202028	−	0.100716i
$261$		0			0
$262$	−10.4665			−0.646623
$263$	−10.2357			−0.631159			−0.315579	−	0.948899i	$-0.602199\pi$
$263$	−10.2357			−0.631159			−0.315579	+	0.948899i	$0.602199\pi$
$264$		0			0
$265$	23.1034	+	11.5176i	1.41923	+	0.707520i
$266$		0			0
$267$		0			0
$268$			14.8833i			0.909144i
$269$	−14.1286			−0.861436			−0.430718	−	0.902486i	$-0.641740\pi$
$269$	−14.1286			−0.861436			−0.430718	+	0.902486i	$0.641740\pi$
$270$		0			0
$271$		−	20.2909i		−	1.23258i	−0.787517	−	0.616292i	$-0.788634\pi$
$271$		−	20.2909i		−	1.23258i	0.787517	−	0.616292i	$-0.211366\pi$
$272$			1.60916i			0.0975694i
$273$		0			0
$274$	14.0199			0.846970
$275$	−1.24479	+	0.938198i	−0.0750635	+	0.0565755i
$276$		0			0
$277$			9.75883i			0.586351i	0.956059	+	0.293176i	$0.0947121\pi$
$277$			9.75883i			0.586351i	−0.956059	+	0.293176i	$0.905288\pi$
$278$		−	7.00761i		−	0.420289i
$279$		0			0
$280$		0			0
$281$			18.0765i			1.07835i	0.842193	+	0.539177i	$0.181264\pi$
$281$			18.0765i			1.07835i	−0.842193	+	0.539177i	$0.818736\pi$
$282$		0			0
$283$	20.5468			1.22138			0.610691	−	0.791869i	$-0.290892\pi$
$283$	20.5468			1.22138			0.610691	+	0.791869i	$0.290892\pi$
$284$			2.38340i			0.141429i
$285$		0			0
$286$		−	0.507480i		−	0.0300079i
$287$		0			0
$288$		0			0
$289$	14.4106			0.847683
$290$	−1.78825	+	3.58710i	−0.105010	+	0.210642i
$291$		0			0
$292$	6.13913			0.359265
$293$		−	10.3254i		−	0.603217i	−0.953432	−	0.301608i	$-0.902477\pi$
$293$		−	10.3254i		−	0.603217i	0.953432	−	0.301608i	$-0.0975235\pi$
$294$		0			0
$295$	−20.1501	−	10.0453i	−1.17318	−	0.584858i
$296$			2.16685i			0.125945i
$297$		0			0
$298$		−	4.79900i		−	0.277999i
$299$	−7.19842			−0.416295
$300$		0			0
$301$		0			0
$302$	9.72228			0.559454
$303$		0			0
$304$		−	4.54781i		−	0.260835i
$305$	−9.92006	+	19.8989i	−0.568021	+	1.13941i
$306$		0			0
$307$	−26.7069			−1.52424			−0.762122	−	0.647433i	$-0.775842\pi$
$307$	−26.7069			−1.52424			−0.762122	+	0.647433i	$0.775842\pi$
$308$		0			0
$309$		0			0
$310$	−0.414276	+	0.831008i	−0.0235293	+	0.0471981i
$311$	23.4808			1.33148			0.665738	−	0.746186i	$-0.268117\pi$
$311$	23.4808			1.33148			0.665738	+	0.746186i	$0.268117\pi$
$312$		0			0
$313$	−3.08356			−0.174293			−0.0871465	−	0.996196i	$-0.527775\pi$
$313$	−3.08356			−0.174293			−0.0871465	+	0.996196i	$0.527775\pi$
$314$	−3.49721			−0.197359
$315$		0			0
$316$	9.32022			0.524303
$317$	−21.9010			−1.23008			−0.615041	−	0.788495i	$-0.710861\pi$
$317$	−21.9010			−1.23008			−0.615041	+	0.788495i	$0.710861\pi$
$318$		0			0
$319$	−0.558811			−0.0312874
$320$	−2.00118	−	0.997634i	−0.111869	−	0.0557695i
$321$		0			0
$322$		0			0
$323$	7.31813			0.407192
$324$		0			0
$325$	4.89890	+	6.49979i	0.271742	+	0.360544i
$326$		−	23.3218i		−	1.29167i
$327$		0			0
$328$	3.39718			0.187578
$329$		0			0
$330$		0			0
$331$	−24.7179			−1.35862			−0.679308	−	0.733853i	$-0.737720\pi$
$331$	−24.7179			−1.35862			−0.679308	+	0.733853i	$0.737720\pi$
$332$		−	9.49304i		−	0.520998i
$333$		0			0
$334$			4.94867i			0.270779i
$335$	14.8481	−	29.7842i	0.811239	−	1.62729i
$336$		0			0
$337$			33.6111i			1.83091i	0.402417	+	0.915457i	$0.368170\pi$
$337$			33.6111i			1.83091i	−0.402417	+	0.915457i	$0.631830\pi$
$338$	10.3501			0.562973
$339$		0			0
$340$	1.60535	−	3.22021i	0.0870623	−	0.174640i
$341$	−0.129457			−0.00701050
$342$		0			0
$343$		0			0
$344$			0.812195i			0.0437906i
$345$		0			0
$346$		−	22.8768i		−	1.22986i
$347$	14.6233			0.785017			0.392509	−	0.919748i	$-0.371607\pi$
$347$	14.6233			0.785017			0.392509	+	0.919748i	$0.371607\pi$
$348$		0			0
$349$			23.1804i			1.24082i	0.784277	+	0.620410i	$0.213034\pi$
$349$			23.1804i			1.24082i	−0.784277	+	0.620410i	$0.786966\pi$
$350$		0			0
$351$		0			0
$352$		−	0.311751i		−	0.0166164i
$353$			11.4999i			0.612078i	0.952019	+	0.306039i	$0.0990038\pi$
$353$			11.4999i			0.612078i	−0.952019	+	0.306039i	$0.900996\pi$
$354$		0			0
$355$	2.37776	−	4.76962i	0.126199	−	0.253145i
$356$	−10.0245			−0.531297
$357$		0			0
$358$		−	21.9568i		−	1.16046i
$359$			31.8652i			1.68178i	0.541206	+	0.840890i	$0.317968\pi$
$359$			31.8652i			1.68178i	−0.541206	+	0.840890i	$0.682032\pi$
$360$		0			0
$361$	−1.68257			−0.0885564
$362$		−	19.6496i		−	1.03276i
$363$		0			0
$364$		0			0
$365$	−12.2855	−	6.12461i	−0.643053	−	0.320577i
$366$		0			0
$367$	−13.8580			−0.723381			−0.361690	−	0.932298i	$-0.617800\pi$
$367$	−13.8580			−0.723381			−0.361690	+	0.932298i	$0.617800\pi$
$368$	−4.42207			−0.230516
$369$		0			0
$370$	2.16172	−	4.33625i	0.112382	−	0.225431i
$371$		0			0
$372$		0			0
$373$			7.50062i			0.388368i	0.980965	+	0.194184i	$0.0622058\pi$
$373$			7.50062i			0.388368i	−0.980965	+	0.194184i	$0.937794\pi$
$374$	0.501655			0.0259400
$375$		0			0
$376$			0.316426i			0.0163184i
$377$			2.91789i			0.150279i
$378$		0			0
$379$	−21.9559			−1.12780			−0.563899	−	0.825844i	$-0.690699\pi$
$379$	−21.9559			−1.12780			−0.563899	+	0.825844i	$0.690699\pi$
$380$	−4.53705	+	9.10099i	−0.232746	+	0.466871i
$381$		0			0
$382$			9.21677i			0.471571i
$383$		−	2.73722i		−	0.139865i	−0.997552	−	0.0699327i	$-0.977722\pi$
$383$		−	2.73722i		−	0.139865i	0.997552	−	0.0699327i	$-0.0222784\pi$
$384$		0			0
$385$		0			0
$386$		−	11.1432i		−	0.567171i
$387$		0			0
$388$	−12.7011			−0.644802
$389$		−	23.4497i		−	1.18895i	−0.804114	−	0.594474i	$-0.797360\pi$
$389$		−	23.4497i		−	1.18895i	0.804114	−	0.594474i	$-0.202640\pi$
$390$		0			0
$391$		−	7.11580i		−	0.359861i
$392$		0			0
$393$		0			0
$394$	−22.0051			−1.10860
$395$	−18.6514	−	9.29817i	−0.938456	−	0.467842i
$396$		0			0
$397$	23.8664			1.19782			0.598909	−	0.800817i	$-0.295601\pi$
$397$	23.8664			1.19782			0.598909	+	0.800817i	$0.295601\pi$
$398$		−	6.47821i		−	0.324723i
$399$		0			0
$400$	3.00945	+	3.99289i	0.150473	+	0.199645i
$401$			11.1789i			0.558247i	0.960255	+	0.279124i	$0.0900439\pi$
$401$			11.1789i			0.558247i	−0.960255	+	0.279124i	$0.909956\pi$
$402$		0			0
$403$			0.675975i			0.0336727i
$404$	14.0918			0.701094
$405$		0			0
$406$		0			0
$407$	0.675515			0.0334841
$408$		0			0
$409$		−	3.49480i		−	0.172807i	−0.996260	−	0.0864033i	$-0.972463\pi$
$409$		−	3.49480i		−	0.172807i	0.996260	−	0.0864033i	$-0.0275374\pi$
$410$	−6.79837	−	3.38914i	−0.335747	−	0.167378i
$411$		0			0
$412$	−9.46782			−0.466446
$413$		0			0
$414$		0			0
$415$	−9.47059	+	18.9973i	−0.464893	+	0.932540i
$416$	−1.62784			−0.0798114
$417$		0			0
$418$	−1.41778			−0.0693460
$419$	−19.4084			−0.948160			−0.474080	−	0.880482i	$-0.657219\pi$
$419$	−19.4084			−0.948160			−0.474080	+	0.880482i	$0.657219\pi$
$420$		0			0
$421$	−21.6055			−1.05299			−0.526493	−	0.850179i	$-0.676493\pi$
$421$	−21.6055			−1.05299			−0.526493	+	0.850179i	$0.676493\pi$
$422$	2.26234			0.110129
$423$		0			0
$424$	11.5449			0.560670
$425$	−6.42519	+	4.84267i	−0.311667	+	0.234904i
$426$		0			0
$427$		0			0
$428$	−15.1045			−0.730103
$429$		0			0
$430$	0.810274	−	1.62535i	0.0390749	−	0.0783813i
$431$			15.4999i			0.746606i	0.927709	+	0.373303i	$0.121775\pi$
$431$			15.4999i			0.746606i	−0.927709	+	0.373303i	$0.878225\pi$
$432$		0			0
$433$	17.2674			0.829819			0.414910	−	0.909863i	$-0.363813\pi$
$433$	17.2674			0.829819			0.414910	+	0.909863i	$0.363813\pi$
$434$		0			0
$435$		0			0
$436$	−2.15856			−0.103376
$437$			20.1107i			0.962027i
$438$		0			0
$439$			6.71063i			0.320281i	0.987094	+	0.160141i	$0.0511948\pi$
$439$			6.71063i			0.320281i	−0.987094	+	0.160141i	$0.948805\pi$
$440$	−0.311013	+	0.623869i	−0.0148270	+	0.0297418i
$441$		0			0
$442$		−	2.61945i		−	0.124594i
$443$	10.9536			0.520423			0.260211	−	0.965552i	$-0.416208\pi$
$443$	10.9536			0.520423			0.260211	+	0.965552i	$0.416208\pi$
$444$		0			0
$445$	20.0608	+	10.0008i	0.950974	+	0.474082i
$446$	14.8667			0.703960
$447$		0			0
$448$		0			0
$449$			17.1293i			0.808384i	0.914674	+	0.404192i	$0.132447\pi$
$449$			17.1293i			0.808384i	−0.914674	+	0.404192i	$0.867553\pi$
$450$		0			0
$451$		−	1.05907i		−	0.0498697i
$452$	−0.600087			−0.0282257
$453$		0			0
$454$		−	24.5838i		−	1.15377i
$455$		0			0
$456$		0			0
$457$		−	32.8719i		−	1.53768i	−0.639440	−	0.768841i	$-0.720834\pi$
$457$		−	32.8719i		−	1.53768i	0.639440	−	0.768841i	$-0.279166\pi$
$458$		−	5.88697i		−	0.275080i
$459$		0			0
$460$	8.84936	+	4.41161i	0.412604	+	0.205692i
$461$	30.3170			1.41200			0.706001	−	0.708211i	$-0.250498\pi$
$461$	30.3170			1.41200			0.706001	+	0.708211i	$0.250498\pi$
$462$		0			0
$463$			22.1702i			1.03034i	0.857089	+	0.515169i	$0.172271\pi$
$463$			22.1702i			1.03034i	−0.857089	+	0.515169i	$0.827729\pi$
$464$			1.79249i			0.0832144i
$465$		0			0
$466$	2.00586			0.0929196
$467$		−	19.6615i		−	0.909825i	−0.890536	−	0.454913i	$-0.849670\pi$
$467$		−	19.6615i		−	0.909825i	0.890536	−	0.454913i	$-0.150330\pi$
$468$		0			0
$469$		0			0
$470$	0.315677	−	0.633226i	0.0145611	−	0.0292085i
$471$		0			0
$472$	−10.0691			−0.463467
$473$	0.253202			0.0116423
$474$		0			0
$475$	18.1589	−	13.6864i	0.833189	−	0.627976i
$476$		0			0
$477$		0			0
$478$			3.89695i			0.178242i
$479$	−0.840055			−0.0383831			−0.0191915	−	0.999816i	$-0.506109\pi$
$479$	−0.840055			−0.0383831			−0.0191915	+	0.999816i	$0.506109\pi$
$480$		0			0
$481$		−	3.52728i		−	0.160830i
$482$			17.8215i			0.811749i
$483$		0			0
$484$	10.9028			0.495582
$485$	25.4172	+	12.6711i	1.15414	+	0.575364i
$486$		0			0
$487$			16.6252i			0.753359i	0.926344	+	0.376680i	$0.122934\pi$
$487$			16.6252i			0.753359i	−0.926344	+	0.376680i	$0.877066\pi$
$488$			9.94358i			0.450125i
$489$		0			0
$490$		0			0
$491$			40.4292i			1.82455i	0.409583	+	0.912273i	$0.365674\pi$
$491$			40.4292i			1.82455i	−0.409583	+	0.912273i	$0.634326\pi$
$492$		0			0
$493$	−2.88440			−0.129907
$494$			7.40310i			0.333081i
$495$		0			0
$496$			0.415259i			0.0186457i
$497$		0			0
$498$		0			0
$499$	−31.9074			−1.42837			−0.714187	−	0.699955i	$-0.753203\pi$
$499$	−31.9074			−1.42837			−0.714187	+	0.699955i	$0.753203\pi$
$500$	−2.03901	−	10.9928i	−0.0911872	−	0.491615i
$501$		0			0
$502$	4.07378			0.181822
$503$		−	5.66488i		−	0.252584i	−0.991993	−	0.126292i	$-0.959692\pi$
$503$		−	5.66488i		−	0.252584i	0.991993	−	0.126292i	$-0.0403077\pi$
$504$		0			0
$505$	−28.2003	−	14.0585i	−1.25489	−	0.625594i
$506$			1.37858i			0.0612855i
$507$		0			0
$508$			4.39770i			0.195117i
$509$	−29.8683			−1.32389			−0.661944	−	0.749553i	$-0.730268\pi$
$509$	−29.8683			−1.32389			−0.661944	+	0.749553i	$0.730268\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	13.4552i		−	0.593485i
$515$	18.9468	+	9.44542i	0.834896	+	0.416215i
$516$		0			0
$517$	0.0986459			0.00433844
$518$		0			0
$519$		0			0
$520$	3.25760	+	1.62399i	0.142855	+	0.0712166i
$521$	4.48791			0.196619			0.0983094	−	0.995156i	$-0.468657\pi$
$521$	4.48791			0.196619			0.0983094	+	0.995156i	$0.468657\pi$
$522$		0			0
$523$	11.0076			0.481327			0.240664	−	0.970609i	$-0.422635\pi$
$523$	11.0076			0.481327			0.240664	+	0.970609i	$0.422635\pi$
$524$	10.4665			0.457231
$525$		0			0
$526$	10.2357			0.446297
$527$	−0.668216			−0.0291079
$528$		0			0
$529$	−3.44531			−0.149796
$530$	−23.1034	−	11.5176i	−1.00355	−	0.500292i
$531$		0			0
$532$		0			0
$533$	−5.53006			−0.239533
$534$		0			0
$535$	30.2268	+	15.0688i	1.30682	+	0.651479i
$536$		−	14.8833i		−	0.642862i
$537$		0			0
$538$	14.1286			0.609127
$539$		0			0
$540$		0			0
$541$	−3.88022			−0.166824			−0.0834119	−	0.996515i	$-0.526582\pi$
$541$	−3.88022			−0.166824			−0.0834119	+	0.996515i	$0.526582\pi$
$542$			20.2909i			0.871569i
$543$		0			0
$544$		−	1.60916i		−	0.0689920i
$545$	4.31966	+	2.15345i	0.185034	+	0.0922437i
$546$		0			0
$547$			31.8176i			1.36042i	0.733017	+	0.680210i	$0.238112\pi$
$547$			31.8176i			1.36042i	−0.733017	+	0.680210i	$0.761888\pi$
$548$	−14.0199			−0.598898
$549$		0			0
$550$	1.24479	−	0.938198i	0.0530779	−	0.0400049i
$551$	8.15192			0.347283
$552$		0			0
$553$		0			0
$554$		−	9.75883i		−	0.414613i
$555$		0			0
$556$			7.00761i			0.297189i
$557$	13.9185			0.589745			0.294873	−	0.955537i	$-0.404723\pi$
$557$	13.9185			0.589745			0.294873	+	0.955537i	$0.404723\pi$
$558$		0			0
$559$		−	1.32212i		−	0.0559199i
$560$		0			0
$561$		0			0
$562$		−	18.0765i		−	0.762511i
$563$		−	9.50915i		−	0.400763i	−0.979718	−	0.200382i	$-0.935782\pi$
$563$		−	9.50915i		−	0.400763i	0.979718	−	0.200382i	$-0.0642182\pi$
$564$		0			0
$565$	1.20088	+	0.598667i	0.0505215	+	0.0251861i
$566$	−20.5468			−0.863647
$567$		0			0
$568$		−	2.38340i		−	0.100005i
$569$		−	18.7818i		−	0.787376i	−0.919244	−	0.393688i	$-0.871199\pi$
$569$		−	18.7818i		−	0.787376i	0.919244	−	0.393688i	$-0.128801\pi$
$570$		0			0
$571$	−16.1599			−0.676270			−0.338135	−	0.941098i	$-0.609796\pi$
$571$	−16.1599			−0.676270			−0.338135	+	0.941098i	$0.609796\pi$
$572$			0.507480i			0.0212188i
$573$		0			0
$574$		0			0
$575$	−13.3080	−	17.6569i	−0.554982	−	0.736342i
$576$		0			0
$577$	−41.0353			−1.70832			−0.854161	−	0.520008i	$-0.825929\pi$
$577$	−41.0353			−1.70832			−0.854161	+	0.520008i	$0.825929\pi$
$578$	−14.4106			−0.599403
$579$		0			0
$580$	1.78825	−	3.58710i	0.0742532	−	0.148946i
$581$		0			0
$582$		0			0
$583$		−	3.59913i		−	0.149061i
$584$	−6.13913			−0.254039
$585$		0			0
$586$			10.3254i			0.426539i
$587$			44.4357i			1.83406i	0.398820	+	0.917029i	$0.369420\pi$
$587$			44.4357i			1.83406i	−0.398820	+	0.917029i	$0.630580\pi$
$588$		0			0
$589$	1.88852			0.0778150
$590$	20.1501	+	10.0453i	0.829565	+	0.413557i
$591$		0			0
$592$		−	2.16685i		−	0.0890568i
$593$		−	11.0566i		−	0.454041i	−0.973890	−	0.227020i	$-0.927102\pi$
$593$		−	11.0566i		−	0.454041i	0.973890	−	0.227020i	$-0.0728984\pi$
$594$		0			0
$595$		0			0
$596$			4.79900i			0.196575i
$597$		0			0
$598$	7.19842			0.294365
$599$			18.2205i			0.744469i	0.928139	+	0.372234i	$0.121408\pi$
$599$			18.2205i			0.744469i	−0.928139	+	0.372234i	$0.878592\pi$
$600$		0			0
$601$		−	1.30846i		−	0.0533730i	−0.999644	−	0.0266865i	$-0.991504\pi$
$601$		−	1.30846i		−	0.0533730i	0.999644	−	0.0266865i	$-0.00849559\pi$
$602$		0			0
$603$		0			0
$604$	−9.72228			−0.395594
$605$	−21.8185	−	10.8770i	−0.887048	−	0.442214i
$606$		0			0
$607$	−30.3509			−1.23191			−0.615953	−	0.787783i	$-0.711229\pi$
$607$	−30.3509			−1.23191			−0.615953	+	0.787783i	$0.711229\pi$
$608$			4.54781i			0.184438i
$609$		0			0
$610$	9.92006	−	19.8989i	0.401651	−	0.805683i
$611$		−	0.515091i		−	0.0208383i
$612$		0			0
$613$		−	32.1506i		−	1.29855i	−0.760554	−	0.649275i	$-0.775073\pi$
$613$		−	32.1506i		−	1.29855i	0.760554	−	0.649275i	$-0.224927\pi$
$614$	26.7069			1.07780
$615$		0			0
$616$		0			0
$617$	21.8787			0.880803			0.440401	−	0.897801i	$-0.354836\pi$
$617$	21.8787			0.880803			0.440401	+	0.897801i	$0.354836\pi$
$618$		0			0
$619$			8.50875i			0.341996i	0.985271	+	0.170998i	$0.0546991\pi$
$619$			8.50875i			0.341996i	−0.985271	+	0.170998i	$0.945301\pi$
$620$	0.414276	−	0.831008i	0.0166377	−	0.0333741i
$621$		0			0
$622$	−23.4808			−0.941496
$623$		0			0
$624$		0			0
$625$	−6.88641	+	24.0328i	−0.275456	+	0.961314i
$626$	3.08356			0.123244
$627$		0			0
$628$	3.49721			0.139554
$629$	3.48679			0.139028
$630$		0			0
$631$	11.1644			0.444446			0.222223	−	0.974996i	$-0.428669\pi$
$631$	11.1644			0.444446			0.222223	+	0.974996i	$0.428669\pi$
$632$	−9.32022			−0.370738
$633$		0			0
$634$	21.9010			0.869800
$635$	4.38730	−	8.80060i	0.174105	−	0.349241i
$636$		0			0
$637$		0			0
$638$	0.558811			0.0221235
$639$		0			0
$640$	2.00118	+	0.997634i	0.0791036	+	0.0394350i
$641$		−	39.4806i		−	1.55939i	−0.626159	−	0.779696i	$-0.715374\pi$
$641$		−	39.4806i		−	1.55939i	0.626159	−	0.779696i	$-0.284626\pi$
$642$		0			0
$643$	48.6505			1.91859			0.959295	−	0.282406i	$-0.0911326\pi$
$643$	48.6505			1.91859			0.959295	+	0.282406i	$0.0911326\pi$
$644$		0			0
$645$		0			0
$646$	−7.31813			−0.287928
$647$		−	8.81882i		−	0.346703i	−0.984860	−	0.173352i	$-0.944540\pi$
$647$		−	8.81882i		−	0.346703i	0.984860	−	0.173352i	$-0.0554597\pi$
$648$		0			0
$649$			3.13904i			0.123218i
$650$	−4.89890	−	6.49979i	−0.192151	−	0.254943i
$651$		0			0
$652$			23.3218i			0.913351i
$653$	16.1157			0.630654			0.315327	−	0.948983i	$-0.397886\pi$
$653$	16.1157			0.630654			0.315327	+	0.948983i	$0.397886\pi$
$654$		0			0
$655$	−20.9454	−	10.4417i	−0.818403	−	0.407993i
$656$	−3.39718			−0.132637
$657$		0			0
$658$		0			0
$659$			14.8431i			0.578204i	0.957298	+	0.289102i	$0.0933567\pi$
$659$			14.8431i			0.578204i	−0.957298	+	0.289102i	$0.906643\pi$
$660$		0			0
$661$		−	31.2191i		−	1.21428i	−0.794594	−	0.607141i	$-0.792316\pi$
$661$		−	31.2191i		−	1.21428i	0.794594	−	0.607141i	$-0.207684\pi$
$662$	24.7179			0.960687
$663$		0			0
$664$			9.49304i			0.368401i
$665$		0			0
$666$		0			0
$667$		−	7.92653i		−	0.306916i
$668$		−	4.94867i		−	0.191470i
$669$		0			0
$670$	−14.8481	+	29.7842i	−0.573633	+	1.15067i
$671$	3.09992			0.119671
$672$		0			0
$673$		−	14.8577i		−	0.572722i	−0.958122	−	0.286361i	$-0.907554\pi$
$673$		−	14.8577i		−	0.572722i	0.958122	−	0.286361i	$-0.0924457\pi$
$674$		−	33.6111i		−	1.29465i
$675$		0			0
$676$	−10.3501			−0.398082
$677$		−	4.93030i		−	0.189487i	−0.995502	−	0.0947434i	$-0.969797\pi$
$677$		−	4.93030i		−	0.189487i	0.995502	−	0.0947434i	$-0.0302031\pi$
$678$		0			0
$679$		0			0
$680$	−1.60535	+	3.22021i	−0.0615623	+	0.123489i
$681$		0			0
$682$	0.129457			0.00495717
$683$	25.9148			0.991603			0.495801	−	0.868436i	$-0.334874\pi$
$683$	25.9148			0.991603			0.495801	+	0.868436i	$0.334874\pi$
$684$		0			0
$685$	28.0563	+	13.9867i	1.07197	+	0.534404i
$686$		0			0
$687$		0			0
$688$		−	0.812195i		−	0.0309646i
$689$	−18.7933			−0.715966
$690$		0			0
$691$		−	7.05734i		−	0.268474i	−0.990949	−	0.134237i	$-0.957142\pi$
$691$		−	7.05734i		−	0.268474i	0.990949	−	0.134237i	$-0.0428583\pi$
$692$			22.8768i			0.869646i
$693$		0			0
$694$	−14.6233			−0.555091
$695$	6.99104	−	14.0235i	0.265185	−	0.531942i
$696$		0			0
$697$		−	5.46659i		−	0.207062i
$698$		−	23.1804i		−	0.877393i
$699$		0			0
$700$		0			0
$701$			11.2951i			0.426610i	0.976986	+	0.213305i	$0.0684229\pi$
$701$			11.2951i			0.426610i	−0.976986	+	0.213305i	$0.931577\pi$
$702$		0			0
$703$	−9.85440			−0.371666
$704$			0.311751i			0.0117495i
$705$		0			0
$706$		−	11.4999i		−	0.432805i
$707$		0			0
$708$		0			0
$709$	17.5418			0.658797			0.329399	−	0.944191i	$-0.393154\pi$
$709$	17.5418			0.658797			0.329399	+	0.944191i	$0.393154\pi$
$710$	−2.37776	+	4.76962i	−0.0892359	+	0.179001i
$711$		0			0
$712$	10.0245			0.375684
$713$		−	1.83630i		−	0.0687701i
$714$		0			0
$715$	0.506279	−	1.01556i	0.0189338	−	0.0379798i
$716$			21.9568i			0.820566i
$717$		0			0
$718$		−	31.8652i		−	1.18920i
$719$	−14.4525			−0.538986			−0.269493	−	0.963002i	$-0.586856\pi$
$719$	−14.4525			−0.538986			−0.269493	+	0.963002i	$0.586856\pi$
$720$		0			0
$721$		0			0
$722$	1.68257			0.0626188
$723$		0			0
$724$			19.6496i			0.730272i
$725$	−7.15724	+	5.39442i	−0.265813	+	0.200344i
$726$		0			0
$727$	−47.6063			−1.76562			−0.882809	−	0.469731i	$-0.844351\pi$
$727$	−47.6063			−1.76562			−0.882809	+	0.469731i	$0.844351\pi$
$728$		0			0
$729$		0			0
$730$	12.2855	+	6.12461i	0.454707	+	0.226682i
$731$	1.30695			0.0483392
$732$		0			0
$733$	−44.2175			−1.63321			−0.816605	−	0.577196i	$-0.804147\pi$
$733$	−44.2175			−1.63321			−0.816605	+	0.577196i	$0.804147\pi$
$734$	13.8580			0.511507
$735$		0			0
$736$	4.42207			0.163000
$737$	−4.63988			−0.170912
$738$		0			0
$739$	−49.8496			−1.83375			−0.916873	−	0.399179i	$-0.869295\pi$
$739$	−49.8496			−1.83375			−0.916873	+	0.399179i	$0.869295\pi$
$740$	−2.16172	+	4.33625i	−0.0794664	+	0.159404i
$741$		0			0
$742$		0			0
$743$	−8.28900			−0.304094			−0.152047	−	0.988373i	$-0.548587\pi$
$743$	−8.28900			−0.304094			−0.152047	+	0.988373i	$0.548587\pi$
$744$		0			0
$745$	4.78765	−	9.60367i	0.175406	−	0.351851i
$746$		−	7.50062i		−	0.274617i
$747$		0			0
$748$	−0.501655			−0.0183423
$749$		0			0
$750$		0			0
$751$	−11.1961			−0.408551			−0.204275	−	0.978913i	$-0.565484\pi$
$751$	−11.1961			−0.408551			−0.204275	+	0.978913i	$0.565484\pi$
$752$		−	0.316426i		−	0.0115389i
$753$		0			0
$754$		−	2.91789i		−	0.106263i
$755$	19.4560	+	9.69928i	0.708078	+	0.352993i
$756$		0			0
$757$			11.1038i			0.403576i	0.979429	+	0.201788i	$0.0646752\pi$
$757$			11.1038i			0.403576i	−0.979429	+	0.201788i	$0.935325\pi$
$758$	21.9559			0.797473
$759$		0			0
$760$	4.53705	−	9.10099i	0.164576	−	0.330128i
$761$	25.0190			0.906937			0.453469	−	0.891272i	$-0.350186\pi$
$761$	25.0190			0.906937			0.453469	+	0.891272i	$0.350186\pi$
$762$		0			0
$763$		0			0
$764$		−	9.21677i		−	0.333451i
$765$		0			0
$766$			2.73722i			0.0988997i
$767$	16.3908			0.591839
$768$		0			0
$769$			28.9449i			1.04378i	0.853013	+	0.521890i	$0.174773\pi$
$769$			28.9449i			1.04378i	−0.853013	+	0.521890i	$0.825227\pi$
$770$		0			0
$771$		0			0
$772$			11.1432i			0.401051i
$773$		−	22.2586i		−	0.800587i	−0.916387	−	0.400294i	$-0.868908\pi$
$773$		−	22.2586i		−	0.800587i	0.916387	−	0.400294i	$-0.131092\pi$
$774$		0			0
$775$	−1.65808	+	1.24970i	−0.0595601	+	0.0448906i
$776$	12.7011			0.455944
$777$		0			0
$778$			23.4497i			0.840714i
$779$			15.4497i			0.553543i
$780$		0			0
$781$	−0.743027			−0.0265876
$782$			7.11580i			0.254460i
$783$		0			0
$784$		0			0
$785$	−6.99854	−	3.48893i	−0.249789	−	0.124525i
$786$		0			0
$787$	−28.2207			−1.00596			−0.502980	−	0.864298i	$-0.667763\pi$
$787$	−28.2207			−1.00596			−0.502980	+	0.864298i	$0.667763\pi$
$788$	22.0051			0.783901
$789$		0			0
$790$	18.6514	+	9.29817i	0.663588	+	0.330814i
$791$		0			0
$792$		0			0
$793$		−	16.1865i		−	0.574801i
$794$	−23.8664			−0.846985
$795$		0			0
$796$			6.47821i			0.229614i
$797$			44.6012i			1.57985i	0.613201	+	0.789927i	$0.289882\pi$
$797$			44.6012i			1.57985i	−0.613201	+	0.789927i	$0.710118\pi$
$798$		0			0
$799$	0.509179			0.0180134
$800$	−3.00945	−	3.99289i	−0.106400	−	0.141170i
$801$		0			0
$802$		−	11.1789i		−	0.394740i
$803$			1.91388i			0.0675392i
$804$		0			0
$805$		0			0
$806$		−	0.675975i		−	0.0238102i
$807$		0			0
$808$	−14.0918			−0.495748
$809$		−	30.0452i		−	1.05633i	−0.849141	−	0.528166i	$-0.822880\pi$
$809$		−	30.0452i		−	1.05633i	0.849141	−	0.528166i	$-0.177120\pi$
$810$		0			0
$811$			8.16474i			0.286703i	0.989672	+	0.143351i	$0.0457879\pi$
$811$			8.16474i			0.286703i	−0.989672	+	0.143351i	$0.954212\pi$
$812$		0			0
$813$		0			0
$814$	−0.675515			−0.0236768
$815$	23.2666	−	46.6711i	0.814994	−	1.63482i
$816$		0			0
$817$	−3.69371			−0.129226
$818$			3.49480i			0.122193i
$819$		0			0
$820$	6.79837	+	3.38914i	0.237409	+	0.118354i
$821$		−	51.7286i		−	1.80534i	−0.430333	−	0.902670i	$-0.641604\pi$
$821$		−	51.7286i		−	1.80534i	0.430333	−	0.902670i	$-0.358396\pi$
$822$		0			0
$823$			8.26998i			0.288274i	0.989558	+	0.144137i	$0.0460405\pi$
$823$			8.26998i			0.288274i	−0.989558	+	0.144137i	$0.953959\pi$
$824$	9.46782			0.329827
$825$		0			0
$826$		0			0
$827$	30.3367			1.05491			0.527455	−	0.849583i	$-0.323146\pi$
$827$	30.3367			1.05491			0.527455	+	0.849583i	$0.323146\pi$
$828$		0			0
$829$			10.7227i			0.372413i	0.982511	+	0.186207i	$0.0596194\pi$
$829$			10.7227i			0.372413i	−0.982511	+	0.186207i	$0.940381\pi$
$830$	9.47059	−	18.9973i	0.328729	−	0.659406i
$831$		0			0
$832$	1.62784			0.0564352
$833$		0			0
$834$		0			0
$835$	−4.93696	+	9.90318i	−0.170851	+	0.342714i
$836$	1.41778			0.0490350
$837$		0			0
$838$	19.4084			0.670450
$839$	4.75405			0.164128			0.0820640	−	0.996627i	$-0.473849\pi$
$839$	4.75405			0.164128			0.0820640	+	0.996627i	$0.473849\pi$
$840$		0			0
$841$	25.7870			0.889206
$842$	21.6055			0.744574
$843$		0			0
$844$	−2.26234			−0.0778730
$845$	20.7125	+	10.3257i	0.712532	+	0.355213i
$846$		0			0
$847$		0			0
$848$	−11.5449			−0.396454
$849$		0			0
$850$	6.42519	−	4.84267i	0.220382	−	0.166102i
$851$			9.58194i			0.328465i
$852$		0			0
$853$	23.7651			0.813702			0.406851	−	0.913495i	$-0.366627\pi$
$853$	23.7651			0.813702			0.406851	+	0.913495i	$0.366627\pi$
$854$		0			0
$855$		0			0
$856$	15.1045			0.516261
$857$		−	5.31367i		−	0.181511i	−0.995873	−	0.0907557i	$-0.971072\pi$
$857$		−	5.31367i		−	0.181511i	0.995873	−	0.0907557i	$-0.0289282\pi$
$858$		0			0
$859$			40.7115i			1.38906i	0.719464	+	0.694530i	$0.244388\pi$
$859$			40.7115i			1.38906i	−0.719464	+	0.694530i	$0.755612\pi$
$860$	−0.810274	+	1.62535i	−0.0276301	+	0.0554239i
$861$		0			0
$862$		−	15.4999i		−	0.527930i
$863$	40.9315			1.39332			0.696662	−	0.717400i	$-0.254668\pi$
$863$	40.9315			1.39332			0.696662	+	0.717400i	$0.254668\pi$
$864$		0			0
$865$	22.8227	−	45.7806i	0.775995	−	1.55659i
$866$	−17.2674			−0.586771
$867$		0			0
$868$		0			0
$869$			2.90558i			0.0985651i
$870$		0			0
$871$			24.2277i			0.820923i
$872$	2.15856			0.0730979
$873$		0			0
$874$		−	20.1107i		−	0.680255i
$875$		0			0
$876$		0			0
$877$		−	51.7755i		−	1.74833i	−0.485627	−	0.874166i	$-0.661408\pi$
$877$		−	51.7755i		−	1.74833i	0.485627	−	0.874166i	$-0.338592\pi$
$878$		−	6.71063i		−	0.226473i
$879$		0			0
$880$	0.311013	−	0.623869i	0.0104842	−	0.0210306i
$881$	35.1323			1.18364			0.591818	−	0.806071i	$-0.298410\pi$
$881$	35.1323			1.18364			0.591818	+	0.806071i	$0.298410\pi$
$882$		0			0
$883$		−	53.0391i		−	1.78491i	−0.451139	−	0.892454i	$-0.648982\pi$
$883$		−	53.0391i		−	1.78491i	0.451139	−	0.892454i	$-0.351018\pi$
$884$			2.61945i			0.0881015i
$885$		0			0
$886$	−10.9536			−0.367994
$887$			22.5760i			0.758029i	0.925391	+	0.379015i	$0.123737\pi$
$887$			22.5760i			0.758029i	−0.925391	+	0.379015i	$0.876263\pi$
$888$		0			0
$889$		0			0
$890$	−20.0608	−	10.0008i	−0.672440	−	0.335227i
$891$		0			0
$892$	−14.8667			−0.497775
$893$	−1.43904			−0.0481558
$894$		0			0
$895$	21.9049	−	43.9396i	0.732200	−	1.46874i
$896$		0			0
$897$		0			0
$898$		−	17.1293i		−	0.571614i
$899$	−0.744348			−0.0248254
$900$		0			0
$901$		−	18.5776i		−	0.618908i
$902$			1.05907i			0.0352632i
$903$		0			0
$904$	0.600087			0.0199586
$905$	19.6031	−	39.3224i	0.651630	−	1.30712i
$906$		0			0
$907$		−	15.6810i		−	0.520679i	−0.965517	−	0.260339i	$-0.916166\pi$
$907$		−	15.6810i		−	0.520679i	0.965517	−	0.260339i	$-0.0838344\pi$
$908$			24.5838i			0.815840i
$909$		0			0
$910$		0			0
$911$		−	13.4833i		−	0.446721i	−0.974736	−	0.223361i	$-0.928297\pi$
$911$		−	13.4833i		−	0.446721i	0.974736	−	0.223361i	$-0.0717027\pi$
$912$		0			0
$913$	2.95946			0.0979438
$914$			32.8719i			1.08730i
$915$		0			0
$916$			5.88697i			0.194511i
$917$		0			0
$918$		0			0
$919$	40.9427			1.35058			0.675288	−	0.737554i	$-0.264019\pi$
$919$	40.9427			1.35058			0.675288	+	0.737554i	$0.264019\pi$
$920$	−8.84936	−	4.41161i	−0.291755	−	0.145446i
$921$		0			0
$922$	−30.3170			−0.998436
$923$			3.87980i			0.127705i
$924$		0			0
$925$	8.65199	−	6.52102i	0.284476	−	0.214410i
$926$		−	22.1702i		−	0.728558i
$927$		0			0
$928$		−	1.79249i		−	0.0588415i
$929$	−43.1865			−1.41690			−0.708451	−	0.705760i	$-0.750606\pi$
$929$	−43.1865			−1.41690			−0.708451	+	0.705760i	$0.750606\pi$
$930$		0			0
$931$		0			0
$932$	−2.00586			−0.0657041
$933$		0			0
$934$			19.6615i			0.643344i
$935$	1.00390	+	0.500468i	0.0328311	+	0.0163671i
$936$		0			0
$937$	−40.6318			−1.32738			−0.663691	−	0.748007i	$-0.731011\pi$
$937$	−40.6318			−1.32738			−0.663691	+	0.748007i	$0.731011\pi$
$938$		0			0
$939$		0			0
$940$	−0.315677	+	0.633226i	−0.0102963	+	0.0206535i
$941$	14.9160			0.486249			0.243124	−	0.969995i	$-0.421828\pi$
$941$	14.9160			0.486249			0.243124	+	0.969995i	$0.421828\pi$
$942$		0			0
$943$	15.0226			0.489202
$944$	10.0691			0.327721
$945$		0			0
$946$	−0.253202			−0.00823231
$947$	−10.6868			−0.347274			−0.173637	−	0.984810i	$-0.555552\pi$
$947$	−10.6868			−0.347274			−0.173637	+	0.984810i	$0.555552\pi$
$948$		0			0
$949$	9.99352			0.324403
$950$	−18.1589	+	13.6864i	−0.589153	+	0.444046i
$951$		0			0
$952$		0			0
$953$	−11.0869			−0.359139			−0.179569	−	0.983745i	$-0.557470\pi$
$953$	−11.0869			−0.359139			−0.179569	+	0.983745i	$0.557470\pi$
$954$		0			0
$955$	−9.19497	+	18.4444i	−0.297542	+	0.596848i
$956$		−	3.89695i		−	0.126036i
$957$		0			0
$958$	0.840055			0.0271409
$959$		0			0
$960$		0			0
$961$	30.8276			0.994437
$962$			3.52728i			0.113724i
$963$		0			0
$964$		−	17.8215i		−	0.573993i
$965$	11.1168	−	22.2995i	0.357862	−	0.717845i
$966$		0			0
$967$			50.3280i			1.61844i	0.587507	+	0.809219i	$0.300109\pi$
$967$			50.3280i			1.61844i	−0.587507	+	0.809219i	$0.699891\pi$
$968$	−10.9028			−0.350430
$969$		0			0
$970$	−25.4172	−	12.6711i	−0.816099	−	0.406844i
$971$	31.2861			1.00402			0.502010	−	0.864862i	$-0.332594\pi$
$971$	31.2861			1.00402			0.502010	+	0.864862i	$0.332594\pi$
$972$		0			0
$973$		0			0
$974$		−	16.6252i		−	0.532706i
$975$		0			0
$976$		−	9.94358i		−	0.318286i
$977$	41.0683			1.31389			0.656946	−	0.753937i	$-0.271848\pi$
$977$	41.0683			1.31389			0.656946	+	0.753937i	$0.271848\pi$
$978$		0			0
$979$		−	3.12514i		−	0.0998799i
$980$		0			0
$981$		0			0
$982$		−	40.4292i		−	1.29015i
$983$		−	53.4630i		−	1.70520i	−0.522561	−	0.852602i	$-0.675023\pi$
$983$		−	53.4630i		−	1.70520i	0.522561	−	0.852602i	$-0.324977\pi$
$984$		0			0
$985$	−44.0363	−	21.9531i	−1.40311	−	0.699484i
$986$	2.88440			0.0918580
$987$		0			0
$988$		−	7.40310i		−	0.235524i
$989$			3.59158i			0.114206i
$990$		0			0
$991$	−58.0202			−1.84307			−0.921536	−	0.388292i	$-0.873065\pi$
$991$	−58.0202			−1.84307			−0.921536	+	0.388292i	$0.873065\pi$
$992$		−	0.415259i		−	0.0131845i
$993$		0			0
$994$		0			0
$995$	6.46288	−	12.9641i	0.204887	−	0.410989i
$996$		0			0
$997$	−59.2128			−1.87529			−0.937644	−	0.347598i	$-0.886997\pi$
$997$	−59.2128			−1.87529			−0.937644	+	0.347598i	$0.886997\pi$
$998$	31.9074			1.01001
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.d.c.4409.3	✓	24
3.2	odd	2		4410.2.d.d.4409.22	yes	24
5.4	even	2		4410.2.d.d.4409.4	yes	24
7.6	odd	2	inner	4410.2.d.c.4409.22	yes	24
15.14	odd	2	inner	4410.2.d.c.4409.21	yes	24
21.20	even	2		4410.2.d.d.4409.3	yes	24
35.34	odd	2		4410.2.d.d.4409.21	yes	24
105.104	even	2	inner	4410.2.d.c.4409.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.d.c.4409.3	✓	24	1.1	even	1	trivial
4410.2.d.c.4409.4	yes	24	105.104	even	2	inner
4410.2.d.c.4409.21	yes	24	15.14	odd	2	inner
4410.2.d.c.4409.22	yes	24	7.6	odd	2	inner
4410.2.d.d.4409.3	yes	24	21.20	even	2
4410.2.d.d.4409.4	yes	24	5.4	even	2
4410.2.d.d.4409.21	yes	24	35.34	odd	2
4410.2.d.d.4409.22	yes	24	3.2	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-2.00118 - 0.997634i) q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-2.00118 - 0.997634i) q^{5} -1.00000 q^{8} +(2.00118 + 0.997634i) q^{10} +0.311751i q^{11} +1.62784 q^{13} +1.00000 q^{16} +1.60916i q^{17} -4.54781i q^{19} +(-2.00118 - 0.997634i) q^{20} -0.311751i q^{22} -4.42207 q^{23} +(3.00945 + 3.99289i) q^{25} -1.62784 q^{26} +1.79249i q^{29} +0.415259i q^{31} -1.00000 q^{32} -1.60916i q^{34} -2.16685i q^{37} +4.54781i q^{38} +(2.00118 + 0.997634i) q^{40} -3.39718 q^{41} -0.812195i q^{43} +0.311751i q^{44} +4.42207 q^{46} -0.316426i q^{47} +(-3.00945 - 3.99289i) q^{50} +1.62784 q^{52} -11.5449 q^{53} +(0.311013 - 0.623869i) q^{55} -1.79249i q^{58} +10.0691 q^{59} -9.94358i q^{61} -0.415259i q^{62} +1.00000 q^{64} +(-3.25760 - 1.62399i) q^{65} +14.8833i q^{67} +1.60916i q^{68} +2.38340i q^{71} +6.13913 q^{73} +2.16685i q^{74} -4.54781i q^{76} +9.32022 q^{79} +(-2.00118 - 0.997634i) q^{80} +3.39718 q^{82} -9.49304i q^{83} +(1.60535 - 3.22021i) q^{85} +0.812195i q^{86} -0.311751i q^{88} -10.0245 q^{89} -4.42207 q^{92} +0.316426i q^{94} +(-4.53705 + 9.10099i) q^{95} -12.7011 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8}+O(q^{10}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{16} + 32 q^{23} - 16 q^{25} - 24 q^{32} - 32 q^{46} + 16 q^{50} - 32 q^{53} + 24 q^{64} - 32 q^{85} + 32 q^{92} + 32 q^{95}+O(q^{100}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^16 + 32 * q^23 - 16 * q^25 - 24 * q^32 - 32 * q^46 + 16 * q^50 - 32 * q^53 + 24 * q^64 - 32 * q^85 + 32 * q^92 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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