LMFDB - Embedded newform 4410.2.d.a.4409.10

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:	$16$
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} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 $ x^16 - 6x^15 + 21x^14 - 54x^13 + 113x^12 - 168x^11 + 186x^10 - 84x^9 - 6x^8 - 420x^7 + 4650x^6 - 21000x^5 + 70625x^4 - 168750x^3 + 328125x^2 - 468750*x + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4409.10
Root			$1.68760 + 1.46697i$ of defining polynomial
Character	$\chi$	$=$	4410.4409
Dual form			4410.2.d.a.4409.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +(0.426635 + 2.19499i) q^{5} -1.00000 q^{8} +(-0.426635 - 2.19499i) q^{10} +1.28310i q^{11} +6.14864 q^{13} +1.00000 q^{16} -6.52257i q^{17} +6.03756i q^{19} +(0.426635 + 2.19499i) q^{20} -1.28310i q^{22} -2.87917 q^{23} +(-4.63597 + 1.87292i) q^{25} -6.14864 q^{26} +1.35052i q^{29} +8.65515i q^{31} -1.00000 q^{32} +6.52257i q^{34} +9.53381i q^{37} -6.03756i q^{38} +(-0.426635 - 2.19499i) q^{40} -8.71759 q^{41} -5.35859i q^{43} +1.28310i q^{44} +2.87917 q^{46} -0.806172i q^{47} +(4.63597 - 1.87292i) q^{50} +6.14864 q^{52} +6.66826 q^{53} +(-2.81639 + 0.547415i) q^{55} -1.35052i q^{58} -1.59622 q^{59} -6.35361i q^{61} -8.65515i q^{62} +1.00000 q^{64} +(2.62322 + 13.4962i) q^{65} -5.39463i q^{67} -6.52257i q^{68} +15.6787i q^{71} +12.4167 q^{73} -9.53381i q^{74} +6.03756i q^{76} +11.1819 q^{79} +(0.426635 + 2.19499i) q^{80} +8.71759 q^{82} -3.74493i q^{83} +(14.3170 - 2.78275i) q^{85} +5.35859i q^{86} -1.28310i q^{88} +3.63941 q^{89} -2.87917 q^{92} +0.806172i q^{94} +(-13.2524 + 2.57583i) q^{95} -8.76818 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} + 16 q^{16} - 16 q^{23} + 12 q^{25} - 16 q^{32} + 16 q^{46} - 12 q^{50} + 32 q^{53} + 16 q^{64} - 40 q^{65} - 8 q^{79} + 64 q^{85} - 16 q^{92} - 24 q^{95}+O(q^{100}) $ 16 * q - 16 * q^2 + 16 * q^4 - 16 * q^8 + 16 * q^16 - 16 * q^23 + 12 * q^25 - 16 * q^32 + 16 * q^46 - 12 * q^50 + 32 * q^53 + 16 * q^64 - 40 * q^65 - 8 * q^79 + 64 * q^85 - 16 * q^92 - 24 * q^95

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
$5$	0.426635	+	2.19499i	0.190797	+	0.981630i
$5$	0.426635	+	2.19499i	0.190797	+	0.981630i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−0.426635	−	2.19499i	−0.134914	−	0.694117i
$11$			1.28310i			0.386869i	0.981113	+	0.193435i	$0.0619627\pi$
$11$			1.28310i			0.386869i	−0.981113	+	0.193435i	$0.938037\pi$
$12$		0			0
$13$	6.14864			1.70533			0.852663	−	0.522462i	$-0.174986\pi$
$13$	6.14864			1.70533			0.852663	+	0.522462i	$0.174986\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.52257i		−	1.58195i	−0.611846	−	0.790977i	$-0.709573\pi$
$17$		−	6.52257i		−	1.58195i	0.611846	−	0.790977i	$-0.290427\pi$
$18$		0			0
$19$			6.03756i			1.38511i	0.721365	+	0.692555i	$0.243515\pi$
$19$			6.03756i			1.38511i	−0.721365	+	0.692555i	$0.756485\pi$
$20$	0.426635	+	2.19499i	0.0953985	+	0.490815i
$21$		0			0
$22$		−	1.28310i		−	0.273558i
$23$	−2.87917			−0.600348			−0.300174	−	0.953884i	$-0.597045\pi$
$23$	−2.87917			−0.600348			−0.300174	+	0.953884i	$0.597045\pi$
$24$		0			0
$25$	−4.63597	+	1.87292i	−0.927193	+	0.374584i
$26$	−6.14864			−1.20585
$27$		0			0
$28$		0			0
$29$			1.35052i			0.250785i	0.992107	+	0.125393i	$0.0400191\pi$
$29$			1.35052i			0.250785i	−0.992107	+	0.125393i	$0.959981\pi$
$30$		0			0
$31$			8.65515i			1.55451i	0.629185	+	0.777255i	$0.283389\pi$
$31$			8.65515i			1.55451i	−0.629185	+	0.777255i	$0.716611\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			6.52257i			1.11861i
$35$		0			0
$36$		0			0
$37$			9.53381i			1.56735i	0.621172	+	0.783674i	$0.286657\pi$
$37$			9.53381i			1.56735i	−0.621172	+	0.783674i	$0.713343\pi$
$38$		−	6.03756i		−	0.979421i
$39$		0			0
$40$	−0.426635	−	2.19499i	−0.0674569	−	0.347058i
$41$	−8.71759			−1.36146			−0.680730	−	0.732535i	$-0.738337\pi$
$41$	−8.71759			−1.36146			−0.680730	+	0.732535i	$0.738337\pi$
$42$		0			0
$43$		−	5.35859i		−	0.817177i	−0.912719	−	0.408588i	$-0.866021\pi$
$43$		−	5.35859i		−	0.817177i	0.912719	−	0.408588i	$-0.133979\pi$
$44$			1.28310i			0.193435i
$45$		0			0
$46$	2.87917			0.424510
$47$		−	0.806172i		−	0.117592i	−0.998270	−	0.0587961i	$-0.981274\pi$
$47$		−	0.806172i		−	0.117592i	0.998270	−	0.0587961i	$-0.0187262\pi$
$48$		0			0
$49$		0			0
$50$	4.63597	−	1.87292i	0.655625	−	0.264871i
$51$		0			0
$52$	6.14864			0.852663
$53$	6.66826			0.915956			0.457978	−	0.888964i	$-0.348574\pi$
$53$	6.66826			0.915956			0.457978	+	0.888964i	$0.348574\pi$
$54$		0			0
$55$	−2.81639	+	0.547415i	−0.379762	+	0.0738134i
$56$		0			0
$57$		0			0
$58$		−	1.35052i		−	0.177332i
$59$	−1.59622			−0.207810			−0.103905	−	0.994587i	$-0.533134\pi$
$59$	−1.59622			−0.207810			−0.103905	+	0.994587i	$0.533134\pi$
$60$		0			0
$61$		−	6.35361i		−	0.813497i	−0.913540	−	0.406748i	$-0.866663\pi$
$61$		−	6.35361i		−	0.813497i	0.913540	−	0.406748i	$-0.133337\pi$
$62$		−	8.65515i		−	1.09921i
$63$		0			0
$64$	1.00000			0.125000
$65$	2.62322	+	13.4962i	0.325371	+	1.67400i
$66$		0			0
$67$		−	5.39463i		−	0.659059i	−0.944145	−	0.329529i	$-0.893110\pi$
$67$		−	5.39463i		−	0.659059i	0.944145	−	0.329529i	$-0.106890\pi$
$68$		−	6.52257i		−	0.790977i
$69$		0			0
$70$		0			0
$71$			15.6787i			1.86072i	0.366650	+	0.930359i	$0.380505\pi$
$71$			15.6787i			1.86072i	−0.366650	+	0.930359i	$0.619495\pi$
$72$		0			0
$73$	12.4167			1.45327			0.726635	−	0.687024i	$-0.241083\pi$
$73$	12.4167			1.45327			0.726635	+	0.687024i	$0.241083\pi$
$74$		−	9.53381i		−	1.10828i
$75$		0			0
$76$			6.03756i			0.692555i
$77$		0			0
$78$		0			0
$79$	11.1819			1.25806			0.629028	−	0.777382i	$-0.283453\pi$
$79$	11.1819			1.25806			0.629028	+	0.777382i	$0.283453\pi$
$80$	0.426635	+	2.19499i	0.0476992	+	0.245407i
$81$		0			0
$82$	8.71759			0.962697
$83$		−	3.74493i		−	0.411059i	−0.978651	−	0.205530i	$-0.934108\pi$
$83$		−	3.74493i		−	0.411059i	0.978651	−	0.205530i	$-0.0658917\pi$
$84$		0			0
$85$	14.3170	−	2.78275i	1.55289	−	0.301832i
$86$			5.35859i			0.577831i
$87$		0			0
$88$		−	1.28310i		−	0.136779i
$89$	3.63941			0.385777			0.192889	−	0.981221i	$-0.438214\pi$
$89$	3.63941			0.385777			0.192889	+	0.981221i	$0.438214\pi$
$90$		0			0
$91$		0			0
$92$	−2.87917			−0.300174
$93$		0			0
$94$			0.806172i			0.0831503i
$95$	−13.2524	+	2.57583i	−1.35967	+	0.264275i
$96$		0			0
$97$	−8.76818			−0.890274			−0.445137	−	0.895463i	$-0.646845\pi$
$97$	−8.76818			−0.890274			−0.445137	+	0.895463i	$0.646845\pi$
$98$		0			0
$99$		0			0
$100$	−4.63597	+	1.87292i	−0.463597	+	0.187292i
$101$	1.70654			0.169807			0.0849035	−	0.996389i	$-0.472942\pi$
$101$	1.70654			0.169807			0.0849035	+	0.996389i	$0.472942\pi$
$102$		0			0
$103$	−10.3644			−1.02123			−0.510617	−	0.859808i	$-0.670583\pi$
$103$	−10.3644			−1.02123			−0.510617	+	0.859808i	$0.670583\pi$
$104$	−6.14864			−0.602923
$105$		0			0
$106$	−6.66826			−0.647679
$107$	11.5136			1.11306			0.556531	−	0.830827i	$-0.312132\pi$
$107$	11.5136			1.11306			0.556531	+	0.830827i	$0.312132\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$	2.81639	−	0.547415i	0.268532	−	0.0521940i
$111$		0			0
$112$		0			0
$113$	−13.3875			−1.25939			−0.629695	−	0.776843i	$-0.716820\pi$
$113$	−13.3875			−1.25939			−0.629695	+	0.776843i	$0.716820\pi$
$114$		0			0
$115$	−1.22835	−	6.31975i	−0.114545	−	0.589319i
$116$			1.35052i			0.125393i
$117$		0			0
$118$	1.59622			0.146944
$119$		0			0
$120$		0			0
$121$	9.35366			0.850332
$122$			6.35361i			0.575229i
$123$		0			0
$124$			8.65515i			0.777255i
$125$	−6.08890	−	9.37685i	−0.544608	−	0.838691i
$126$		0			0
$127$			5.65313i			0.501634i	0.968035	+	0.250817i	$0.0806992\pi$
$127$			5.65313i			0.501634i	−0.968035	+	0.250817i	$0.919301\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	−2.62322	−	13.4962i	−0.230072	−	1.18369i
$131$	−12.4673			−1.08928			−0.544638	−	0.838671i	$-0.683333\pi$
$131$	−12.4673			−1.08928			−0.544638	+	0.838671i	$0.683333\pi$
$132$		0			0
$133$		0			0
$134$			5.39463i			0.466025i
$135$		0			0
$136$			6.52257i			0.559305i
$137$	0.0509686			0.00435454			0.00217727	−	0.999998i	$-0.499307\pi$
$137$	0.0509686			0.00435454			0.00217727	+	0.999998i	$0.499307\pi$
$138$		0			0
$139$			1.05069i			0.0891184i	0.999007	+	0.0445592i	$0.0141883\pi$
$139$			1.05069i			0.0891184i	−0.999007	+	0.0445592i	$0.985812\pi$
$140$		0			0
$141$		0			0
$142$		−	15.6787i		−	1.31573i
$143$			7.88931i			0.659738i
$144$		0			0
$145$	−2.96438	+	0.576179i	−0.246178	+	0.0478491i
$146$	−12.4167			−1.02762
$147$		0			0
$148$			9.53381i			0.783674i
$149$			8.22305i			0.673659i	0.941566	+	0.336829i	$0.109355\pi$
$149$			8.22305i			0.673659i	−0.941566	+	0.336829i	$0.890645\pi$
$150$		0			0
$151$	−4.26715			−0.347256			−0.173628	−	0.984811i	$-0.555549\pi$
$151$	−4.26715			−0.347256			−0.173628	+	0.984811i	$0.555549\pi$
$152$		−	6.03756i		−	0.489710i
$153$		0			0
$154$		0			0
$155$	−18.9980	+	3.69259i	−1.52595	+	0.296596i
$156$		0			0
$157$	−4.83203			−0.385638			−0.192819	−	0.981234i	$-0.561763\pi$
$157$	−4.83203			−0.385638			−0.192819	+	0.981234i	$0.561763\pi$
$158$	−11.1819			−0.889581
$159$		0			0
$160$	−0.426635	−	2.19499i	−0.0337285	−	0.173529i
$161$		0			0
$162$		0			0
$163$			10.7532i			0.842257i	0.907001	+	0.421128i	$0.138366\pi$
$163$			10.7532i			0.842257i	−0.907001	+	0.421128i	$0.861634\pi$
$164$	−8.71759			−0.680730
$165$		0			0
$166$			3.74493i			0.290663i
$167$			18.9267i			1.46459i	0.680988	+	0.732295i	$0.261551\pi$
$167$			18.9267i			1.46459i	−0.680988	+	0.732295i	$0.738449\pi$
$168$		0			0
$169$	24.8057			1.90813
$170$	−14.3170	+	2.78275i	−1.09806	+	0.213427i
$171$		0			0
$172$		−	5.35859i		−	0.408588i
$173$			11.8435i			0.900447i	0.892916	+	0.450223i	$0.148656\pi$
$173$			11.8435i			0.900447i	−0.892916	+	0.450223i	$0.851344\pi$
$174$		0			0
$175$		0			0
$176$			1.28310i			0.0967173i
$177$		0			0
$178$	−3.63941			−0.272786
$179$		−	3.78561i		−	0.282949i	−0.989942	−	0.141475i	$-0.954816\pi$
$179$		−	3.78561i		−	0.282949i	0.989942	−	0.141475i	$-0.0451844\pi$
$180$		0			0
$181$			19.0033i			1.41251i	0.707960	+	0.706253i	$0.249616\pi$
$181$			19.0033i			1.41251i	−0.707960	+	0.706253i	$0.750384\pi$
$182$		0			0
$183$		0			0
$184$	2.87917			0.212255
$185$	−20.9266	+	4.06745i	−1.53856	+	0.299045i
$186$		0			0
$187$	8.36910			0.612009
$188$		−	0.806172i		−	0.0587961i
$189$		0			0
$190$	13.2524	−	2.57583i	0.961428	−	0.186871i
$191$			2.82843i			0.204658i	0.994751	+	0.102329i	$0.0326294\pi$
$191$			2.82843i			0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$		0			0
$193$		−	3.53892i		−	0.254737i	−0.991855	−	0.127368i	$-0.959347\pi$
$193$		−	3.53892i		−	0.254737i	0.991855	−	0.127368i	$-0.0406530\pi$
$194$	8.76818			0.629519
$195$		0			0
$196$		0			0
$197$	2.36728			0.168662			0.0843308	−	0.996438i	$-0.473125\pi$
$197$	2.36728			0.168662			0.0843308	+	0.996438i	$0.473125\pi$
$198$		0			0
$199$			3.46410i			0.245564i	0.992434	+	0.122782i	$0.0391815\pi$
$199$			3.46410i			0.245564i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	4.63597	−	1.87292i	0.327812	−	0.132435i
$201$		0			0
$202$	−1.70654			−0.120072
$203$		0			0
$204$		0			0
$205$	−3.71923	−	19.1350i	−0.259762	−	1.33645i
$206$	10.3644			0.722122
$207$		0			0
$208$	6.14864			0.426331
$209$	−7.74679			−0.535856
$210$		0			0
$211$	−14.3620			−0.988721			−0.494361	−	0.869257i	$-0.664598\pi$
$211$	−14.3620			−0.988721			−0.494361	+	0.869257i	$0.664598\pi$
$212$	6.66826			0.457978
$213$		0			0
$214$	−11.5136			−0.787053
$215$	11.7620	−	2.28616i	0.802165	−	0.155915i
$216$		0			0
$217$		0			0
$218$	−2.00000			−0.135457
$219$		0			0
$220$	−2.81639	+	0.547415i	−0.189881	+	0.0369067i
$221$		−	40.1049i		−	2.69775i
$222$		0			0
$223$	−13.5975			−0.910557			−0.455279	−	0.890349i	$-0.650460\pi$
$223$	−13.5975			−0.910557			−0.455279	+	0.890349i	$0.650460\pi$
$224$		0			0
$225$		0			0
$226$	13.3875			0.890523
$227$		−	12.3559i		−	0.820092i	−0.912065	−	0.410046i	$-0.865513\pi$
$227$		−	12.3559i		−	0.820092i	0.912065	−	0.410046i	$-0.134487\pi$
$228$		0			0
$229$		−	2.41341i		−	0.159483i	−0.996816	−	0.0797414i	$-0.974591\pi$
$229$		−	2.41341i		−	0.159483i	0.996816	−	0.0797414i	$-0.0254094\pi$
$230$	1.22835	+	6.31975i	0.0809952	+	0.416712i
$231$		0			0
$232$		−	1.35052i		−	0.0886660i
$233$	12.8756			0.843509			0.421754	−	0.906710i	$-0.361414\pi$
$233$	12.8756			0.843509			0.421754	+	0.906710i	$0.361414\pi$
$234$		0			0
$235$	1.76954	−	0.343941i	0.115432	−	0.0224362i
$236$	−1.59622			−0.103905
$237$		0			0
$238$		0			0
$239$			27.2546i			1.76296i	0.472226	+	0.881478i	$0.343451\pi$
$239$			27.2546i			1.76296i	−0.472226	+	0.881478i	$0.656549\pi$
$240$		0			0
$241$		−	2.61759i		−	0.168614i	−0.996440	−	0.0843070i	$-0.973132\pi$
$241$		−	2.61759i		−	0.168614i	0.996440	−	0.0843070i	$-0.0268677\pi$
$242$	−9.35366			−0.601276
$243$		0			0
$244$		−	6.35361i		−	0.406748i
$245$		0			0
$246$		0			0
$247$			37.1227i			2.36206i
$248$		−	8.65515i		−	0.549603i
$249$		0			0
$250$	6.08890	+	9.37685i	0.385096	+	0.593044i
$251$	−26.7173			−1.68638			−0.843190	−	0.537615i	$-0.819325\pi$
$251$	−26.7173			−1.68638			−0.843190	+	0.537615i	$0.819325\pi$
$252$		0			0
$253$		−	3.69426i		−	0.232256i
$254$		−	5.65313i		−	0.354709i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	10.5613i		−	0.658793i	−0.944192	−	0.329397i	$-0.893155\pi$
$257$		−	10.5613i		−	0.658793i	0.944192	−	0.329397i	$-0.106845\pi$
$258$		0			0
$259$		0			0
$260$	2.62322	+	13.4962i	0.162685	+	0.836999i
$261$		0			0
$262$	12.4673			0.770234
$263$	8.33174			0.513757			0.256879	−	0.966444i	$-0.417306\pi$
$263$	8.33174			0.513757			0.256879	+	0.966444i	$0.417306\pi$
$264$		0			0
$265$	2.84491	+	14.6368i	0.174762	+	0.899130i
$266$		0			0
$267$		0			0
$268$		−	5.39463i		−	0.329529i
$269$	−14.2834			−0.870876			−0.435438	−	0.900219i	$-0.643406\pi$
$269$	−14.2834			−0.870876			−0.435438	+	0.900219i	$0.643406\pi$
$270$		0			0
$271$		−	7.60446i		−	0.461938i	−0.972961	−	0.230969i	$-0.925810\pi$
$271$		−	7.60446i		−	0.461938i	0.972961	−	0.230969i	$-0.0741896\pi$
$272$		−	6.52257i		−	0.395489i
$273$		0			0
$274$	−0.0509686			−0.00307912
$275$	−2.40314	−	5.94841i	−0.144915	−	0.358702i
$276$		0			0
$277$			12.0400i			0.723416i	0.932291	+	0.361708i	$0.117806\pi$
$277$			12.0400i			0.723416i	−0.932291	+	0.361708i	$0.882194\pi$
$278$		−	1.05069i		−	0.0630162i
$279$		0			0
$280$		0			0
$281$		−	12.7879i		−	0.762861i	−0.924397	−	0.381431i	$-0.875431\pi$
$281$		−	12.7879i		−	0.762861i	0.924397	−	0.381431i	$-0.124569\pi$
$282$		0			0
$283$	−15.5521			−0.924479			−0.462239	−	0.886755i	$-0.652954\pi$
$283$	−15.5521			−0.924479			−0.462239	+	0.886755i	$0.652954\pi$
$284$			15.6787i			0.930359i
$285$		0			0
$286$		−	7.88931i		−	0.466505i
$287$		0			0
$288$		0			0
$289$	−25.5439			−1.50258
$290$	2.96438	−	0.576179i	0.174074	−	0.0338344i
$291$		0			0
$292$	12.4167			0.726635
$293$		−	11.5536i		−	0.674967i	−0.941331	−	0.337483i	$-0.890424\pi$
$293$		−	11.5536i		−	0.674967i	0.941331	−	0.337483i	$-0.109576\pi$
$294$		0			0
$295$	−0.681003	−	3.50369i	−0.0396495	−	0.203993i
$296$		−	9.53381i		−	0.554141i
$297$		0			0
$298$		−	8.22305i		−	0.476349i
$299$	−17.7030			−1.02379
$300$		0			0
$301$		0			0
$302$	4.26715			0.245547
$303$		0			0
$304$			6.03756i			0.346278i
$305$	13.9461	−	2.71067i	0.798553	−	0.155213i
$306$		0			0
$307$	−2.57617			−0.147030			−0.0735150	−	0.997294i	$-0.523422\pi$
$307$	−2.57617			−0.147030			−0.0735150	+	0.997294i	$0.523422\pi$
$308$		0			0
$309$		0			0
$310$	18.9980	−	3.69259i	1.07901	−	0.209725i
$311$	24.7140			1.40140			0.700702	−	0.713454i	$-0.252870\pi$
$311$	24.7140			1.40140			0.700702	+	0.713454i	$0.252870\pi$
$312$		0			0
$313$	−9.00173			−0.508808			−0.254404	−	0.967098i	$-0.581879\pi$
$313$	−9.00173			−0.508808			−0.254404	+	0.967098i	$0.581879\pi$
$314$	4.83203			0.272687
$315$		0			0
$316$	11.1819			0.629028
$317$	−23.9059			−1.34269			−0.671344	−	0.741146i	$-0.734283\pi$
$317$	−23.9059			−1.34269			−0.671344	+	0.741146i	$0.734283\pi$
$318$		0			0
$319$	−1.73285			−0.0970211
$320$	0.426635	+	2.19499i	0.0238496	+	0.122704i
$321$		0			0
$322$		0			0
$323$	39.3804			2.19118
$324$		0			0
$325$	−28.5049	+	11.5159i	−1.58117	+	0.638787i
$326$		−	10.7532i		−	0.595565i
$327$		0			0
$328$	8.71759			0.481349
$329$		0			0
$330$		0			0
$331$	4.36728			0.240047			0.120024	−	0.992771i	$-0.461703\pi$
$331$	4.36728			0.240047			0.120024	+	0.992771i	$0.461703\pi$
$332$		−	3.74493i		−	0.205530i
$333$		0			0
$334$		−	18.9267i		−	1.03562i
$335$	11.8412	−	2.30154i	0.646951	−	0.125746i
$336$		0			0
$337$		−	19.5159i		−	1.06310i	−0.847028	−	0.531549i	$-0.821610\pi$
$337$		−	19.5159i		−	1.06310i	0.847028	−	0.531549i	$-0.178390\pi$
$338$	−24.8057			−1.34925
$339$		0			0
$340$	14.3170	−	2.78275i	0.776447	−	0.150916i
$341$	−11.1054			−0.601392
$342$		0			0
$343$		0			0
$344$			5.35859i			0.288916i
$345$		0			0
$346$		−	11.8435i		−	0.636712i
$347$	26.9147			1.44486			0.722429	−	0.691446i	$-0.243026\pi$
$347$	26.9147			1.44486			0.722429	+	0.691446i	$0.243026\pi$
$348$		0			0
$349$			10.3821i			0.555741i	0.960619	+	0.277870i	$0.0896286\pi$
$349$			10.3821i			0.555741i	−0.960619	+	0.277870i	$0.910371\pi$
$350$		0			0
$351$		0			0
$352$		−	1.28310i		−	0.0683894i
$353$		−	0.168951i		−	0.00899236i	−0.999990	−	0.00449618i	$-0.998569\pi$
$353$		−	0.168951i		−	0.00899236i	0.999990	−	0.00449618i	$-0.00143118\pi$
$354$		0			0
$355$	−34.4146	+	6.68908i	−1.82654	+	0.355019i
$356$	3.63941			0.192889
$357$		0			0
$358$			3.78561i			0.200075i
$359$			37.6661i			1.98794i	0.109655	+	0.993970i	$0.465025\pi$
$359$			37.6661i			1.98794i	−0.109655	+	0.993970i	$0.534975\pi$
$360$		0			0
$361$	−17.4521			−0.918531
$362$		−	19.0033i		−	0.998792i
$363$		0			0
$364$		0			0
$365$	5.29741	+	27.2546i	0.277279	+	1.42657i
$366$		0			0
$367$	15.1557			0.791124			0.395562	−	0.918439i	$-0.370550\pi$
$367$	15.1557			0.791124			0.395562	+	0.918439i	$0.370550\pi$
$368$	−2.87917			−0.150087
$369$		0			0
$370$	20.9266	−	4.06745i	1.08792	−	0.211457i
$371$		0			0
$372$		0			0
$373$		−	12.0400i		−	0.623410i	−0.950179	−	0.311705i	$-0.899100\pi$
$373$		−	12.0400i		−	0.623410i	0.950179	−	0.311705i	$-0.100900\pi$
$374$	−8.36910			−0.432756
$375$		0			0
$376$			0.806172i			0.0415751i
$377$			8.30386i			0.427671i
$378$		0			0
$379$	18.0918			0.929312			0.464656	−	0.885491i	$-0.346178\pi$
$379$	18.0918			0.929312			0.464656	+	0.885491i	$0.346178\pi$
$380$	−13.2524	+	2.57583i	−0.679833	+	0.132137i
$381$		0			0
$382$		−	2.82843i		−	0.144715i
$383$			29.1538i			1.48969i	0.667237	+	0.744846i	$0.267477\pi$
$383$			29.1538i			1.48969i	−0.667237	+	0.744846i	$0.732523\pi$
$384$		0			0
$385$		0			0
$386$			3.53892i			0.180126i
$387$		0			0
$388$	−8.76818			−0.445137
$389$			19.6641i			0.997012i	0.866886	+	0.498506i	$0.166118\pi$
$389$			19.6641i			0.997012i	−0.866886	+	0.498506i	$0.833882\pi$
$390$		0			0
$391$			18.7796i			0.949723i
$392$		0			0
$393$		0			0
$394$	−2.36728			−0.119262
$395$	4.77057	+	24.5441i	0.240033	+	1.23495i
$396$		0			0
$397$	−5.68454			−0.285299			−0.142649	−	0.989773i	$-0.545562\pi$
$397$	−5.68454			−0.285299			−0.142649	+	0.989773i	$0.545562\pi$
$398$		−	3.46410i		−	0.173640i
$399$		0			0
$400$	−4.63597	+	1.87292i	−0.231798	+	0.0936460i
$401$			10.2166i			0.510195i	0.966915	+	0.255097i	$0.0821075\pi$
$401$			10.2166i			0.510195i	−0.966915	+	0.255097i	$0.917892\pi$
$402$		0			0
$403$			53.2174i			2.65095i
$404$	1.70654			0.0849035
$405$		0			0
$406$		0			0
$407$	−12.2328			−0.606359
$408$		0			0
$409$		−	13.9683i		−	0.690687i	−0.938476	−	0.345344i	$-0.887762\pi$
$409$		−	13.9683i		−	0.690687i	0.938476	−	0.345344i	$-0.112238\pi$
$410$	3.71923	+	19.1350i	0.183680	+	0.945012i
$411$		0			0
$412$	−10.3644			−0.510617
$413$		0			0
$414$		0			0
$415$	8.22008	−	1.59772i	0.403508	−	0.0784288i
$416$	−6.14864			−0.301462
$417$		0			0
$418$	7.74679			0.378908
$419$	−11.3181			−0.552924			−0.276462	−	0.961025i	$-0.589162\pi$
$419$	−11.3181			−0.552924			−0.276462	+	0.961025i	$0.589162\pi$
$420$		0			0
$421$	−13.9099			−0.677928			−0.338964	−	0.940799i	$-0.610077\pi$
$421$	−13.9099			−0.677928			−0.338964	+	0.940799i	$0.610077\pi$
$422$	14.3620			0.699132
$423$		0			0
$424$	−6.66826			−0.323839
$425$	12.2162	+	30.2384i	0.592575	+	1.46678i
$426$		0			0
$427$		0			0
$428$	11.5136			0.556531
$429$		0			0
$430$	−11.7620	+	2.28616i	−0.567216	+	0.110248i
$431$			26.0901i			1.25672i	0.777924	+	0.628359i	$0.216273\pi$
$431$			26.0901i			1.25672i	−0.777924	+	0.628359i	$0.783727\pi$
$432$		0			0
$433$	8.42614			0.404935			0.202467	−	0.979289i	$-0.435104\pi$
$433$	8.42614			0.404935			0.202467	+	0.979289i	$0.435104\pi$
$434$		0			0
$435$		0			0
$436$	2.00000			0.0957826
$437$		−	17.3831i		−	0.831548i
$438$		0			0
$439$		−	27.6585i		−	1.32007i	−0.751236	−	0.660033i	$-0.770542\pi$
$439$		−	27.6585i		−	1.32007i	0.751236	−	0.660033i	$-0.229458\pi$
$440$	2.81639	−	0.547415i	0.134266	−	0.0260970i
$441$		0			0
$442$			40.1049i			1.90759i
$443$	−2.35722			−0.111995			−0.0559975	−	0.998431i	$-0.517834\pi$
$443$	−2.35722			−0.111995			−0.0559975	+	0.998431i	$0.517834\pi$
$444$		0			0
$445$	1.55270	+	7.98848i	0.0736051	+	0.378690i
$446$	13.5975			0.643861
$447$		0			0
$448$		0			0
$449$		−	12.8021i		−	0.604169i	−0.953281	−	0.302084i	$-0.902318\pi$
$449$		−	12.8021i		−	0.604169i	0.953281	−	0.302084i	$-0.0976824\pi$
$450$		0			0
$451$		−	11.1855i		−	0.526707i
$452$	−13.3875			−0.629695
$453$		0			0
$454$			12.3559i			0.579893i
$455$		0			0
$456$		0			0
$457$			34.5242i			1.61498i	0.589884	+	0.807488i	$0.299173\pi$
$457$			34.5242i			1.61498i	−0.589884	+	0.807488i	$0.700827\pi$
$458$			2.41341i			0.112771i
$459$		0			0
$460$	−1.22835	−	6.31975i	−0.0572723	−	0.294660i
$461$	−20.2692			−0.944032			−0.472016	−	0.881590i	$-0.656474\pi$
$461$	−20.2692			−0.944032			−0.472016	+	0.881590i	$0.656474\pi$
$462$		0			0
$463$			13.1246i			0.609952i	0.952360	+	0.304976i	$0.0986485\pi$
$463$			13.1246i			0.609952i	−0.952360	+	0.304976i	$0.901352\pi$
$464$			1.35052i			0.0626963i
$465$		0			0
$466$	−12.8756			−0.596451
$467$		−	1.53167i		−	0.0708773i	−0.999372	−	0.0354387i	$-0.988717\pi$
$467$		−	1.53167i		−	0.0708773i	0.999372	−	0.0354387i	$-0.0112828\pi$
$468$		0			0
$469$		0			0
$470$	−1.76954	+	0.343941i	−0.0816228	+	0.0158648i
$471$		0			0
$472$	1.59622			0.0734720
$473$	6.87560			0.316140
$474$		0			0
$475$	−11.3079	−	27.9899i	−0.518840	−	1.28426i
$476$		0			0
$477$		0			0
$478$		−	27.2546i		−	1.24660i
$479$	29.3170			1.33953			0.669764	−	0.742574i	$-0.266395\pi$
$479$	29.3170			1.33953			0.669764	+	0.742574i	$0.266395\pi$
$480$		0			0
$481$			58.6199i			2.67284i
$482$			2.61759i			0.119228i
$483$		0			0
$484$	9.35366			0.425166
$485$	−3.74081	−	19.2461i	−0.169862	−	0.873919i
$486$		0			0
$487$			8.19540i			0.371369i	0.982609	+	0.185685i	$0.0594502\pi$
$487$			8.19540i			0.371369i	−0.982609	+	0.185685i	$0.940550\pi$
$488$			6.35361i			0.287615i
$489$		0			0
$490$		0			0
$491$		−	40.4383i		−	1.82495i	−0.409129	−	0.912477i	$-0.634167\pi$
$491$		−	40.4383i		−	1.82495i	0.409129	−	0.912477i	$-0.365833\pi$
$492$		0			0
$493$	8.80886			0.396731
$494$		−	37.1227i		−	1.67023i
$495$		0			0
$496$			8.65515i			0.388628i
$497$		0			0
$498$		0			0
$499$	17.4490			0.781125			0.390562	−	0.920576i	$-0.372281\pi$
$499$	17.4490			0.781125			0.390562	+	0.920576i	$0.372281\pi$
$500$	−6.08890	−	9.37685i	−0.272304	−	0.419345i
$501$		0			0
$502$	26.7173			1.19245
$503$		−	1.91949i		−	0.0855860i	−0.999084	−	0.0427930i	$-0.986374\pi$
$503$		−	1.91949i		−	0.0855860i	0.999084	−	0.0427930i	$-0.0136256\pi$
$504$		0			0
$505$	0.728069	+	3.74584i	0.0323987	+	0.166688i
$506$			3.69426i			0.164230i
$507$		0			0
$508$			5.65313i			0.250817i
$509$	−11.0910			−0.491599			−0.245800	−	0.969321i	$-0.579051\pi$
$509$	−11.0910			−0.491599			−0.245800	+	0.969321i	$0.579051\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			10.5613i			0.465837i
$515$	−4.42181	−	22.7498i	−0.194848	−	1.00247i
$516$		0			0
$517$	1.03440			0.0454928
$518$		0			0
$519$		0			0
$520$	−2.62322	−	13.4962i	−0.115036	−	0.591847i
$521$	−39.7412			−1.74109			−0.870546	−	0.492088i	$-0.836234\pi$
$521$	−39.7412			−1.74109			−0.870546	+	0.492088i	$0.836234\pi$
$522$		0			0
$523$	−31.2808			−1.36781			−0.683907	−	0.729570i	$-0.739720\pi$
$523$	−31.2808			−1.36781			−0.683907	+	0.729570i	$0.739720\pi$
$524$	−12.4673			−0.544638
$525$		0			0
$526$	−8.33174			−0.363281
$527$	56.4538			2.45917
$528$		0			0
$529$	−14.7104			−0.639582
$530$	−2.84491	−	14.6368i	−0.123575	−	0.635781i
$531$		0			0
$532$		0			0
$533$	−53.6013			−2.32173
$534$		0			0
$535$	4.91210	+	25.2722i	0.212369	+	1.09261i
$536$			5.39463i			0.233012i
$537$		0			0
$538$	14.2834			0.615802
$539$		0			0
$540$		0			0
$541$	−24.7974			−1.06612			−0.533061	−	0.846077i	$-0.678959\pi$
$541$	−24.7974			−1.06612			−0.533061	+	0.846077i	$0.678959\pi$
$542$			7.60446i			0.326640i
$543$		0			0
$544$			6.52257i			0.279653i
$545$	0.853270	+	4.38998i	0.0365501	+	0.188046i
$546$		0			0
$547$			40.5380i			1.73328i	0.498934	+	0.866640i	$0.333725\pi$
$547$			40.5380i			1.73328i	−0.498934	+	0.866640i	$0.666275\pi$
$548$	0.0509686			0.00217727
$549$		0			0
$550$	2.40314	+	5.94841i	0.102470	+	0.253641i
$551$	−8.15384			−0.347365
$552$		0			0
$553$		0			0
$554$		−	12.0400i		−	0.511532i
$555$		0			0
$556$			1.05069i			0.0445592i
$557$	23.7464			1.00617			0.503084	−	0.864238i	$-0.332199\pi$
$557$	23.7464			1.00617			0.503084	+	0.864238i	$0.332199\pi$
$558$		0			0
$559$		−	32.9480i		−	1.39355i
$560$		0			0
$561$		0			0
$562$			12.7879i			0.539424i
$563$			8.42867i			0.355226i	0.984100	+	0.177613i	$0.0568375\pi$
$563$			8.42867i			0.355226i	−0.984100	+	0.177613i	$0.943162\pi$
$564$		0			0
$565$	−5.71157	−	29.3854i	−0.240288	−	1.23625i
$566$	15.5521			0.653705
$567$		0			0
$568$		−	15.6787i		−	0.657863i
$569$		−	23.4565i		−	0.983348i	−0.870779	−	0.491674i	$-0.836385\pi$
$569$		−	23.4565i		−	0.983348i	0.870779	−	0.491674i	$-0.163615\pi$
$570$		0			0
$571$	15.3589			0.642751			0.321376	−	0.946952i	$-0.395855\pi$
$571$	15.3589			0.642751			0.321376	+	0.946952i	$0.395855\pi$
$572$			7.88931i			0.329869i
$573$		0			0
$574$		0			0
$575$	13.3477	−	5.39245i	0.556639	−	0.224881i
$576$		0			0
$577$	14.2408			0.592853			0.296426	−	0.955056i	$-0.404205\pi$
$577$	14.2408			0.592853			0.296426	+	0.955056i	$0.404205\pi$
$578$	25.5439			1.06248
$579$		0			0
$580$	−2.96438	+	0.576179i	−0.123089	+	0.0239245i
$581$		0			0
$582$		0			0
$583$			8.55604i			0.354355i
$584$	−12.4167			−0.513808
$585$		0			0
$586$			11.5536i			0.477274i
$587$			12.5249i			0.516958i	0.966017	+	0.258479i	$0.0832212\pi$
$587$			12.5249i			0.516958i	−0.966017	+	0.258479i	$0.916779\pi$
$588$		0			0
$589$	−52.2560			−2.15317
$590$	0.681003	+	3.50369i	0.0280365	+	0.144245i
$591$		0			0
$592$			9.53381i			0.391837i
$593$		−	14.7150i		−	0.604273i	−0.953265	−	0.302136i	$-0.902300\pi$
$593$		−	14.7150i		−	0.604273i	0.953265	−	0.302136i	$-0.0976998\pi$
$594$		0			0
$595$		0			0
$596$			8.22305i			0.336829i
$597$		0			0
$598$	17.7030			0.723928
$599$			27.6324i			1.12903i	0.825423	+	0.564515i	$0.190937\pi$
$599$			27.6324i			1.12903i	−0.825423	+	0.564515i	$0.809063\pi$
$600$		0			0
$601$			8.82137i			0.359831i	0.983682	+	0.179916i	$0.0575825\pi$
$601$			8.82137i			0.359831i	−0.983682	+	0.179916i	$0.942418\pi$
$602$		0			0
$603$		0			0
$604$	−4.26715			−0.173628
$605$	3.99060	+	20.5312i	0.162241	+	0.834711i
$606$		0			0
$607$	26.5354			1.07704			0.538519	−	0.842613i	$-0.318984\pi$
$607$	26.5354			1.07704			0.538519	+	0.842613i	$0.318984\pi$
$608$		−	6.03756i		−	0.244855i
$609$		0			0
$610$	−13.9461	+	2.71067i	−0.564662	+	0.109752i
$611$		−	4.95686i		−	0.200533i
$612$		0			0
$613$		−	0.913683i		−	0.0369033i	−0.999830	−	0.0184517i	$-0.994126\pi$
$613$		−	0.913683i		−	0.0369033i	0.999830	−	0.0184517i	$-0.00587368\pi$
$614$	2.57617			0.103966
$615$		0			0
$616$		0			0
$617$	35.0558			1.41129			0.705646	−	0.708565i	$-0.250657\pi$
$617$	35.0558			1.41129			0.705646	+	0.708565i	$0.250657\pi$
$618$		0			0
$619$		−	19.0967i		−	0.767561i	−0.923424	−	0.383781i	$-0.874622\pi$
$619$		−	19.0967i		−	0.767561i	0.923424	−	0.383781i	$-0.125378\pi$
$620$	−18.9980	+	3.69259i	−0.762977	+	0.148298i
$621$		0			0
$622$	−24.7140			−0.990942
$623$		0			0
$624$		0			0
$625$	17.9843	−	17.3656i	0.719374	−	0.694623i
$626$	9.00173			0.359782
$627$		0			0
$628$	−4.83203			−0.192819
$629$	62.1849			2.47947
$630$		0			0
$631$	21.5350			0.857296			0.428648	−	0.903472i	$-0.358990\pi$
$631$	21.5350			0.857296			0.428648	+	0.903472i	$0.358990\pi$
$632$	−11.1819			−0.444790
$633$		0			0
$634$	23.9059			0.949423
$635$	−12.4086	+	2.41182i	−0.492419	+	0.0957102i
$636$		0			0
$637$		0			0
$638$	1.73285			0.0686043
$639$		0			0
$640$	−0.426635	−	2.19499i	−0.0168642	−	0.0867646i
$641$			33.1131i			1.30789i	0.756543	+	0.653943i	$0.226887\pi$
$641$			33.1131i			1.30789i	−0.756543	+	0.653943i	$0.773113\pi$
$642$		0			0
$643$	−3.31191			−0.130609			−0.0653044	−	0.997865i	$-0.520802\pi$
$643$	−3.31191			−0.130609			−0.0653044	+	0.997865i	$0.520802\pi$
$644$		0			0
$645$		0			0
$646$	−39.3804			−1.54940
$647$			23.3338i			0.917348i	0.888605	+	0.458674i	$0.151675\pi$
$647$			23.3338i			0.917348i	−0.888605	+	0.458674i	$0.848325\pi$
$648$		0			0
$649$		−	2.04811i		−	0.0803953i
$650$	28.5049	−	11.5159i	1.11805	−	0.451691i
$651$		0			0
$652$			10.7532i			0.421128i
$653$	17.4604			0.683277			0.341639	−	0.939831i	$-0.389018\pi$
$653$	17.4604			0.683277			0.341639	+	0.939831i	$0.389018\pi$
$654$		0			0
$655$	−5.31900	−	27.3657i	−0.207830	−	1.06926i
$656$	−8.71759			−0.340365
$657$		0			0
$658$		0			0
$659$			1.29864i			0.0505877i	0.999680	+	0.0252939i	$0.00805215\pi$
$659$			1.29864i			0.0505877i	−0.999680	+	0.0252939i	$0.991948\pi$
$660$		0			0
$661$		−	5.07917i		−	0.197557i	−0.995109	−	0.0987785i	$-0.968506\pi$
$661$		−	5.07917i		−	0.197557i	0.995109	−	0.0987785i	$-0.0314935\pi$
$662$	−4.36728			−0.169739
$663$		0			0
$664$			3.74493i			0.145331i
$665$		0			0
$666$		0			0
$667$		−	3.88838i		−	0.150559i
$668$			18.9267i			0.732295i
$669$		0			0
$670$	−11.8412	+	2.30154i	−0.457464	+	0.0889161i
$671$	8.15232			0.314717
$672$		0			0
$673$		−	18.4124i		−	0.709747i	−0.934914	−	0.354873i	$-0.884524\pi$
$673$		−	18.4124i		−	0.709747i	0.934914	−	0.354873i	$-0.115476\pi$
$674$			19.5159i			0.751723i
$675$		0			0
$676$	24.8057			0.954067
$677$		−	41.2795i		−	1.58650i	−0.608896	−	0.793250i	$-0.708388\pi$
$677$		−	41.2795i		−	1.58650i	0.608896	−	0.793250i	$-0.291612\pi$
$678$		0			0
$679$		0			0
$680$	−14.3170	+	2.78275i	−0.549031	+	0.106714i
$681$		0			0
$682$	11.1054			0.425249
$683$	−22.9793			−0.879278			−0.439639	−	0.898174i	$-0.644894\pi$
$683$	−22.9793			−0.879278			−0.439639	+	0.898174i	$0.644894\pi$
$684$		0			0
$685$	0.0217450	+	0.111876i	0.000830833	+	0.00427454i
$686$		0			0
$687$		0			0
$688$		−	5.35859i		−	0.204294i
$689$	41.0007			1.56200
$690$		0			0
$691$			12.7072i			0.483406i	0.970350	+	0.241703i	$0.0777059\pi$
$691$			12.7072i			0.483406i	−0.970350	+	0.241703i	$0.922294\pi$
$692$			11.8435i			0.450223i
$693$		0			0
$694$	−26.9147			−1.02167
$695$	−2.30625	+	0.448261i	−0.0874812	+	0.0170035i
$696$		0			0
$697$			56.8611i			2.15377i
$698$		−	10.3821i		−	0.392968i
$699$		0			0
$700$		0			0
$701$		−	26.5441i		−	1.00256i	−0.865286	−	0.501279i	$-0.832863\pi$
$701$		−	26.5441i		−	1.00256i	0.865286	−	0.501279i	$-0.167137\pi$
$702$		0			0
$703$	−57.5609			−2.17095
$704$			1.28310i			0.0483586i
$705$		0			0
$706$			0.168951i			0.00635856i
$707$		0			0
$708$		0			0
$709$	42.3532			1.59061			0.795303	−	0.606212i	$-0.207312\pi$
$709$	42.3532			1.59061			0.795303	+	0.606212i	$0.207312\pi$
$710$	34.4146	−	6.68908i	1.29156	−	0.251037i
$711$		0			0
$712$	−3.63941			−0.136393
$713$		−	24.9196i		−	0.933248i
$714$		0			0
$715$	−17.3170	+	3.36586i	−0.647618	+	0.125876i
$716$		−	3.78561i		−	0.141475i
$717$		0			0
$718$		−	37.6661i		−	1.40569i
$719$	23.9014			0.891373			0.445686	−	0.895189i	$-0.352960\pi$
$719$	23.9014			0.891373			0.445686	+	0.895189i	$0.352960\pi$
$720$		0			0
$721$		0			0
$722$	17.4521			0.649499
$723$		0			0
$724$			19.0033i			0.706253i
$725$	−2.52942	−	6.26097i	−0.0939401	−	0.232526i
$726$		0			0
$727$	−22.4529			−0.832731			−0.416365	−	0.909197i	$-0.636696\pi$
$727$	−22.4529			−0.832731			−0.416365	+	0.909197i	$0.636696\pi$
$728$		0			0
$729$		0			0
$730$	−5.29741	−	27.2546i	−0.196066	−	1.00874i
$731$	−34.9517			−1.29274
$732$		0			0
$733$	−27.5450			−1.01740			−0.508700	−	0.860944i	$-0.669874\pi$
$733$	−27.5450			−1.01740			−0.508700	+	0.860944i	$0.669874\pi$
$734$	−15.1557			−0.559409
$735$		0			0
$736$	2.87917			0.106128
$737$	6.92184			0.254969
$738$		0			0
$739$	31.0913			1.14371			0.571856	−	0.820354i	$-0.306224\pi$
$739$	31.0913			1.14371			0.571856	+	0.820354i	$0.306224\pi$
$740$	−20.9266	+	4.06745i	−0.769278	+	0.149523i
$741$		0			0
$742$		0			0
$743$	−1.12610			−0.0413127			−0.0206564	−	0.999787i	$-0.506576\pi$
$743$	−1.12610			−0.0413127			−0.0206564	+	0.999787i	$0.506576\pi$
$744$		0			0
$745$	−18.0495	+	3.50824i	−0.661283	+	0.128532i
$746$			12.0400i			0.440817i
$747$		0			0
$748$	8.36910			0.306005
$749$		0			0
$750$		0			0
$751$	−9.54216			−0.348198			−0.174099	−	0.984728i	$-0.555701\pi$
$751$	−9.54216			−0.348198			−0.174099	+	0.984728i	$0.555701\pi$
$752$		−	0.806172i		−	0.0293981i
$753$		0			0
$754$		−	8.30386i		−	0.302409i
$755$	−1.82051	−	9.36635i	−0.0662553	−	0.340876i
$756$		0			0
$757$		−	33.1687i		−	1.20554i	−0.797917	−	0.602768i	$-0.794065\pi$
$757$		−	33.1687i		−	1.20554i	0.797917	−	0.602768i	$-0.205935\pi$
$758$	−18.0918			−0.657123
$759$		0			0
$760$	13.2524	−	2.57583i	0.480714	−	0.0934353i
$761$	−4.34245			−0.157414			−0.0787068	−	0.996898i	$-0.525079\pi$
$761$	−4.34245			−0.157414			−0.0787068	+	0.996898i	$0.525079\pi$
$762$		0			0
$763$		0			0
$764$			2.82843i			0.102329i
$765$		0			0
$766$		−	29.1538i		−	1.05337i
$767$	−9.81458			−0.354384
$768$		0			0
$769$		−	46.1795i		−	1.66528i	−0.553818	−	0.832638i	$-0.686830\pi$
$769$		−	46.1795i		−	1.66528i	0.553818	−	0.832638i	$-0.313170\pi$
$770$		0			0
$771$		0			0
$772$		−	3.53892i		−	0.127368i
$773$			17.9745i			0.646497i	0.946314	+	0.323248i	$0.104775\pi$
$773$			17.9745i			0.646497i	−0.946314	+	0.323248i	$0.895225\pi$
$774$		0			0
$775$	−16.2104	−	40.1250i	−0.582295	−	1.44133i
$776$	8.76818			0.314759
$777$		0			0
$778$		−	19.6641i		−	0.704994i
$779$		−	52.6330i		−	1.88577i
$780$		0			0
$781$	−20.1173			−0.719854
$782$		−	18.7796i		−	0.671556i
$783$		0			0
$784$		0			0
$785$	−2.06151	−	10.6063i	−0.0735785	−	0.378554i
$786$		0			0
$787$	13.8417			0.493405			0.246702	−	0.969091i	$-0.420653\pi$
$787$	13.8417			0.493405			0.246702	+	0.969091i	$0.420653\pi$
$788$	2.36728			0.0843308
$789$		0			0
$790$	−4.77057	−	24.5441i	−0.169729	−	0.873239i
$791$		0			0
$792$		0			0
$793$		−	39.0661i		−	1.38728i
$794$	5.68454			0.201737
$795$		0			0
$796$			3.46410i			0.122782i
$797$		−	7.46138i		−	0.264296i	−0.991230	−	0.132148i	$-0.957813\pi$
$797$		−	7.46138i		−	0.264296i	0.991230	−	0.132148i	$-0.0421874\pi$
$798$		0			0
$799$	−5.25831			−0.186026
$800$	4.63597	−	1.87292i	0.163906	−	0.0662177i
$801$		0			0
$802$		−	10.2166i		−	0.360762i
$803$			15.9319i			0.562225i
$804$		0			0
$805$		0			0
$806$		−	53.2174i		−	1.87450i
$807$		0			0
$808$	−1.70654			−0.0600359
$809$			1.48668i			0.0522689i	0.999658	+	0.0261344i	$0.00831980\pi$
$809$			1.48668i			0.0522689i	−0.999658	+	0.0261344i	$0.991680\pi$
$810$		0			0
$811$		−	31.5494i		−	1.10785i	−0.832567	−	0.553925i	$-0.813129\pi$
$811$		−	31.5494i		−	1.10785i	0.832567	−	0.553925i	$-0.186871\pi$
$812$		0			0
$813$		0			0
$814$	12.2328			0.428760
$815$	−23.6032	+	4.58770i	−0.826784	+	0.160700i
$816$		0			0
$817$	32.3528			1.13188
$818$			13.9683i			0.488390i
$819$		0			0
$820$	−3.71923	−	19.1350i	−0.129881	−	0.668224i
$821$		−	31.8039i		−	1.10996i	−0.831862	−	0.554982i	$-0.812725\pi$
$821$		−	31.8039i		−	1.10996i	0.831862	−	0.554982i	$-0.187275\pi$
$822$		0			0
$823$		−	36.6371i		−	1.27709i	−0.769585	−	0.638545i	$-0.779537\pi$
$823$		−	36.6371i		−	1.27709i	0.769585	−	0.638545i	$-0.220463\pi$
$824$	10.3644			0.361061
$825$		0			0
$826$		0			0
$827$	21.9864			0.764541			0.382271	−	0.924050i	$-0.375142\pi$
$827$	21.9864			0.764541			0.382271	+	0.924050i	$0.375142\pi$
$828$		0			0
$829$			24.3062i			0.844191i	0.906551	+	0.422095i	$0.138705\pi$
$829$			24.3062i			0.844191i	−0.906551	+	0.422095i	$0.861295\pi$
$830$	−8.22008	+	1.59772i	−0.285323	+	0.0554576i
$831$		0			0
$832$	6.14864			0.213166
$833$		0			0
$834$		0			0
$835$	−41.5439	+	8.07478i	−1.43768	+	0.279439i
$836$	−7.74679			−0.267928
$837$		0			0
$838$	11.3181			0.390977
$839$	27.3914			0.945655			0.472827	−	0.881155i	$-0.343233\pi$
$839$	27.3914			0.945655			0.472827	+	0.881155i	$0.343233\pi$
$840$		0			0
$841$	27.1761			0.937107
$842$	13.9099			0.479368
$843$		0			0
$844$	−14.3620			−0.494361
$845$	10.5830	+	54.4484i	0.364066	+	1.87308i
$846$		0			0
$847$		0			0
$848$	6.66826			0.228989
$849$		0			0
$850$	−12.2162	−	30.2384i	−0.419013	−	1.03717i
$851$		−	27.4494i		−	0.940954i
$852$		0			0
$853$	22.6073			0.774060			0.387030	−	0.922067i	$-0.373501\pi$
$853$	22.6073			0.774060			0.387030	+	0.922067i	$0.373501\pi$
$854$		0			0
$855$		0			0
$856$	−11.5136			−0.393527
$857$		−	47.8578i		−	1.63479i	−0.576076	−	0.817396i	$-0.695417\pi$
$857$		−	47.8578i		−	1.63479i	0.576076	−	0.817396i	$-0.304583\pi$
$858$		0			0
$859$			48.0090i			1.63804i	0.573762	+	0.819022i	$0.305484\pi$
$859$			48.0090i			1.63804i	−0.573762	+	0.819022i	$0.694516\pi$
$860$	11.7620	−	2.28616i	0.401082	−	0.0779574i
$861$		0			0
$862$		−	26.0901i		−	0.888634i
$863$	−47.5569			−1.61886			−0.809428	−	0.587218i	$-0.800223\pi$
$863$	−47.5569			−1.61886			−0.809428	+	0.587218i	$0.800223\pi$
$864$		0			0
$865$	−25.9964	+	5.05286i	−0.883905	+	0.171802i
$866$	−8.42614			−0.286332
$867$		0			0
$868$		0			0
$869$			14.3474i			0.486703i
$870$		0			0
$871$		−	33.1696i		−	1.12391i
$872$	−2.00000			−0.0677285
$873$		0			0
$874$			17.3831i			0.587993i
$875$		0			0
$876$		0			0
$877$			37.0004i			1.24941i	0.780859	+	0.624707i	$0.214782\pi$
$877$			37.0004i			1.24941i	−0.780859	+	0.624707i	$0.785218\pi$
$878$			27.6585i			0.933428i
$879$		0			0
$880$	−2.81639	+	0.547415i	−0.0949405	+	0.0184534i
$881$	−5.97213			−0.201206			−0.100603	−	0.994927i	$-0.532077\pi$
$881$	−5.97213			−0.201206			−0.100603	+	0.994927i	$0.532077\pi$
$882$		0			0
$883$			11.0589i			0.372163i	0.982534	+	0.186081i	$0.0595788\pi$
$883$			11.0589i			0.372163i	−0.982534	+	0.186081i	$0.940421\pi$
$884$		−	40.1049i		−	1.34887i
$885$		0			0
$886$	2.35722			0.0791925
$887$			5.95070i			0.199805i	0.994997	+	0.0999026i	$0.0318531\pi$
$887$			5.95070i			0.199805i	−0.994997	+	0.0999026i	$0.968147\pi$
$888$		0			0
$889$		0			0
$890$	−1.55270	−	7.98848i	−0.0520467	−	0.267774i
$891$		0			0
$892$	−13.5975			−0.455279
$893$	4.86731			0.162878
$894$		0			0
$895$	8.30937	−	1.61507i	0.277752	−	0.0539859i
$896$		0			0
$897$		0			0
$898$			12.8021i			0.427212i
$899$	−11.6890			−0.389849
$900$		0			0
$901$		−	43.4942i		−	1.44900i
$902$			11.1855i			0.372438i
$903$		0			0
$904$	13.3875			0.445261
$905$	−41.7121	+	8.10748i	−1.38656	+	0.269502i
$906$		0			0
$907$		−	55.3144i		−	1.83669i	−0.395786	−	0.918343i	$-0.629528\pi$
$907$		−	55.3144i		−	1.83669i	0.395786	−	0.918343i	$-0.370472\pi$
$908$		−	12.3559i		−	0.410046i
$909$		0			0
$910$		0			0
$911$		−	35.2080i		−	1.16649i	−0.812295	−	0.583246i	$-0.801782\pi$
$911$		−	35.2080i		−	1.16649i	0.812295	−	0.583246i	$-0.198218\pi$
$912$		0			0
$913$	4.80512			0.159026
$914$		−	34.5242i		−	1.14196i
$915$		0			0
$916$		−	2.41341i		−	0.0797414i
$917$		0			0
$918$		0			0
$919$	−50.8875			−1.67862			−0.839311	−	0.543651i	$-0.817041\pi$
$919$	−50.8875			−1.67862			−0.839311	+	0.543651i	$0.817041\pi$
$920$	1.22835	+	6.31975i	0.0404976	+	0.208356i
$921$		0			0
$922$	20.2692			0.667531
$923$			96.4026i			3.17313i
$924$		0			0
$925$	−17.8560	−	44.1984i	−0.587103	−	1.45323i
$926$		−	13.1246i		−	0.431301i
$927$		0			0
$928$		−	1.35052i		−	0.0443330i
$929$	59.6810			1.95807			0.979036	−	0.203689i	$-0.0652932\pi$
$929$	59.6810			1.95807			0.979036	+	0.203689i	$0.0652932\pi$
$930$		0			0
$931$		0			0
$932$	12.8756			0.421754
$933$		0			0
$934$			1.53167i			0.0501179i
$935$	3.57055	+	18.3701i	0.116770	+	0.600766i
$936$		0			0
$937$	24.2579			0.792471			0.396235	−	0.918149i	$-0.370316\pi$
$937$	24.2579			0.792471			0.396235	+	0.918149i	$0.370316\pi$
$938$		0			0
$939$		0			0
$940$	1.76954	−	0.343941i	0.0577160	−	0.0112181i
$941$	30.1079			0.981488			0.490744	−	0.871304i	$-0.336725\pi$
$941$	30.1079			0.981488			0.490744	+	0.871304i	$0.336725\pi$
$942$		0			0
$943$	25.0994			0.817350
$944$	−1.59622			−0.0519525
$945$		0			0
$946$	−6.87560			−0.223545
$947$	46.1625			1.50008			0.750039	−	0.661393i	$-0.230035\pi$
$947$	46.1625			1.50008			0.750039	+	0.661393i	$0.230035\pi$
$948$		0			0
$949$	76.3460			2.47830
$950$	11.3079	+	27.9899i	0.366875	+	0.908112i
$951$		0			0
$952$		0			0
$953$	−23.7121			−0.768110			−0.384055	−	0.923310i	$-0.625473\pi$
$953$	−23.7121			−0.768110			−0.384055	+	0.923310i	$0.625473\pi$
$954$		0			0
$955$	−6.20837	+	1.20671i	−0.200898	+	0.0390481i
$956$			27.2546i			0.881478i
$957$		0			0
$958$	−29.3170			−0.947190
$959$		0			0
$960$		0			0
$961$	−43.9116			−1.41650
$962$		−	58.6199i		−	1.88998i
$963$		0			0
$964$		−	2.61759i		−	0.0843070i
$965$	7.76789	−	1.50983i	0.250057	−	0.0486030i
$966$		0			0
$967$		−	0.0922780i		−	0.00296746i	−0.999999	−	0.00148373i	$-0.999528\pi$
$967$		−	0.0922780i		−	0.00296746i	0.999999	−	0.00148373i	$-0.000472286\pi$
$968$	−9.35366			−0.300638
$969$		0			0
$970$	3.74081	+	19.2461i	0.120110	+	0.617954i
$971$	39.7005			1.27405			0.637024	−	0.770844i	$-0.280165\pi$
$971$	39.7005			1.27405			0.637024	+	0.770844i	$0.280165\pi$
$972$		0			0
$973$		0			0
$974$		−	8.19540i		−	0.262598i
$975$		0			0
$976$		−	6.35361i		−	0.203374i
$977$	−36.8070			−1.17756			−0.588779	−	0.808294i	$-0.700391\pi$
$977$	−36.8070			−1.17756			−0.588779	+	0.808294i	$0.700391\pi$
$978$		0			0
$979$			4.66973i			0.149245i
$980$		0			0
$981$		0			0
$982$			40.4383i			1.29044i
$983$		−	62.5731i		−	1.99577i	−0.0650009	−	0.997885i	$-0.520705\pi$
$983$		−	62.5731i		−	1.99577i	0.0650009	−	0.997885i	$-0.479295\pi$
$984$		0			0
$985$	1.00996	+	5.19615i	0.0321801	+	0.165563i
$986$	−8.80886			−0.280531
$987$		0			0
$988$			37.1227i			1.18103i
$989$			15.4283i			0.490591i
$990$		0			0
$991$	−34.8039			−1.10558			−0.552791	−	0.833320i	$-0.686437\pi$
$991$	−34.8039			−1.10558			−0.552791	+	0.833320i	$0.686437\pi$
$992$		−	8.65515i		−	0.274801i
$993$		0			0
$994$		0			0
$995$	−7.60367	+	1.47791i	−0.241053	+	0.0468528i
$996$		0			0
$997$	48.8631			1.54751			0.773755	−	0.633485i	$-0.218376\pi$
$997$	48.8631			1.54751			0.773755	+	0.633485i	$0.218376\pi$
$998$	−17.4490			−0.552339
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.d.a.4409.10		16
3.2	odd	2		4410.2.d.b.4409.7		16
5.4	even	2		4410.2.d.b.4409.9		16
7.2	even	3		630.2.bo.b.269.6	yes	16
7.3	odd	6		630.2.bo.b.89.8	yes	16
7.6	odd	2	inner	4410.2.d.a.4409.7		16
15.14	odd	2	inner	4410.2.d.a.4409.8		16
21.2	odd	6		630.2.bo.a.269.3	yes	16
21.17	even	6		630.2.bo.a.89.1	✓	16
21.20	even	2		4410.2.d.b.4409.10		16
35.2	odd	12		3150.2.bf.f.1151.8		32
35.3	even	12		3150.2.bf.f.1601.7		32
35.9	even	6		630.2.bo.a.269.1	yes	16
35.17	even	12		3150.2.bf.f.1601.10		32
35.23	odd	12		3150.2.bf.f.1151.9		32
35.24	odd	6		630.2.bo.a.89.3	yes	16
35.34	odd	2		4410.2.d.b.4409.8		16
105.2	even	12		3150.2.bf.f.1151.10		32
105.17	odd	12		3150.2.bf.f.1601.8		32
105.23	even	12		3150.2.bf.f.1151.7		32
105.38	odd	12		3150.2.bf.f.1601.9		32
105.44	odd	6		630.2.bo.b.269.8	yes	16
105.59	even	6		630.2.bo.b.89.6	yes	16
105.104	even	2	inner	4410.2.d.a.4409.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.bo.a.89.1	✓	16	21.17	even	6
630.2.bo.a.89.3	yes	16	35.24	odd	6
630.2.bo.a.269.1	yes	16	35.9	even	6
630.2.bo.a.269.3	yes	16	21.2	odd	6
630.2.bo.b.89.6	yes	16	105.59	even	6
630.2.bo.b.89.8	yes	16	7.3	odd	6
630.2.bo.b.269.6	yes	16	7.2	even	3
630.2.bo.b.269.8	yes	16	105.44	odd	6
3150.2.bf.f.1151.7		32	105.23	even	12
3150.2.bf.f.1151.8		32	35.2	odd	12
3150.2.bf.f.1151.9		32	35.23	odd	12
3150.2.bf.f.1151.10		32	105.2	even	12
3150.2.bf.f.1601.7		32	35.3	even	12
3150.2.bf.f.1601.8		32	105.17	odd	12
3150.2.bf.f.1601.9		32	105.38	odd	12
3150.2.bf.f.1601.10		32	35.17	even	12
4410.2.d.a.4409.7		16	7.6	odd	2	inner
4410.2.d.a.4409.8		16	15.14	odd	2	inner
4410.2.d.a.4409.9		16	105.104	even	2	inner
4410.2.d.a.4409.10		16	1.1	even	1	trivial
4410.2.d.b.4409.7		16	3.2	odd	2
4410.2.d.b.4409.8		16	35.34	odd	2
4410.2.d.b.4409.9		16	5.4	even	2
4410.2.d.b.4409.10		16	21.20	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(0.426635 + 2.19499i) q^{5} -1.00000 q^{8} +(-0.426635 - 2.19499i) q^{10} +1.28310i q^{11} +6.14864 q^{13} +1.00000 q^{16} -6.52257i q^{17} +6.03756i q^{19} +(0.426635 + 2.19499i) q^{20} -1.28310i q^{22} -2.87917 q^{23} +(-4.63597 + 1.87292i) q^{25} -6.14864 q^{26} +1.35052i q^{29} +8.65515i q^{31} -1.00000 q^{32} +6.52257i q^{34} +9.53381i q^{37} -6.03756i q^{38} +(-0.426635 - 2.19499i) q^{40} -8.71759 q^{41} -5.35859i q^{43} +1.28310i q^{44} +2.87917 q^{46} -0.806172i q^{47} +(4.63597 - 1.87292i) q^{50} +6.14864 q^{52} +6.66826 q^{53} +(-2.81639 + 0.547415i) q^{55} -1.35052i q^{58} -1.59622 q^{59} -6.35361i q^{61} -8.65515i q^{62} +1.00000 q^{64} +(2.62322 + 13.4962i) q^{65} -5.39463i q^{67} -6.52257i q^{68} +15.6787i q^{71} +12.4167 q^{73} -9.53381i q^{74} +6.03756i q^{76} +11.1819 q^{79} +(0.426635 + 2.19499i) q^{80} +8.71759 q^{82} -3.74493i q^{83} +(14.3170 - 2.78275i) q^{85} +5.35859i q^{86} -1.28310i q^{88} +3.63941 q^{89} -2.87917 q^{92} +0.806172i q^{94} +(-13.2524 + 2.57583i) q^{95} -8.76818 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} + 16 q^{16} - 16 q^{23} + 12 q^{25} - 16 q^{32} + 16 q^{46} - 12 q^{50} + 32 q^{53} + 16 q^{64} - 40 q^{65} - 8 q^{79} + 64 q^{85} - 16 q^{92} - 24 q^{95}+O(q^{100}) \) 16 * q - 16 * q^2 + 16 * q^4 - 16 * q^8 + 16 * q^16 - 16 * q^23 + 12 * q^25 - 16 * q^32 + 16 * q^46 - 12 * q^50 + 32 * q^53 + 16 * q^64 - 40 * q^65 - 8 * q^79 + 64 * q^85 - 16 * q^92 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more