LMFDB - Embedded newform 2034.2.c.h.451.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,2,Mod(451,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.451"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2034.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,10,0,10,0,0,0,10,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2415717711$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 25x^{8} + 190x^{6} + 482x^{4} + 368x^{2} + 32 $ x^10 + 25x^8 + 190x^6 + 482x^4 + 368x^2 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.8
Root			$1.63968i$ of defining polynomial
Character	$\chi$	$=$	2034.451
Dual form			2034.2.c.h.451.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.63968i q^{5} +3.55588 q^{7} +1.00000 q^{8} +1.63968i q^{10} +3.33279 q^{11} -1.82544 q^{13} +3.55588 q^{14} +1.00000 q^{16} -0.106462i q^{17} +4.06519i q^{19} +1.63968i q^{20} +3.33279 q^{22} -8.58345i q^{23} +2.31146 q^{25} -1.82544 q^{26} +3.55588 q^{28} -3.75172i q^{29} +4.64426 q^{31} +1.00000 q^{32} -0.106462i q^{34} +5.83049i q^{35} +4.24482i q^{37} +4.06519i q^{38} +1.63968i q^{40} -7.62573 q^{41} +11.8407i q^{43} +3.33279 q^{44} -8.58345i q^{46} +3.97272i q^{47} +5.64426 q^{49} +2.31146 q^{50} -1.82544 q^{52} -7.55807 q^{53} +5.46470i q^{55} +3.55588 q^{56} -3.75172i q^{58} +3.60391i q^{59} -1.82544 q^{61} +4.64426 q^{62} +1.00000 q^{64} -2.99312i q^{65} -10.3901i q^{67} -0.106462i q^{68} +5.83049i q^{70} -2.25853i q^{71} -14.1005i q^{73} +4.24482i q^{74} +4.06519i q^{76} +11.8510 q^{77} +4.19372i q^{79} +1.63968i q^{80} -7.62573 q^{82} +8.51821 q^{83} +0.174564 q^{85} +11.8407i q^{86} +3.33279 q^{88} -10.6622i q^{89} -6.49102 q^{91} -8.58345i q^{92} +3.97272i q^{94} -6.66559 q^{95} +17.1838 q^{97} +5.64426 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} + 10 q^{8} + 2 q^{11} + 2 q^{13} + 10 q^{16} + 2 q^{22} + 2 q^{26} - 8 q^{31} + 10 q^{32} - 8 q^{41} + 2 q^{44} + 2 q^{49} + 2 q^{52} + 18 q^{53} + 2 q^{61} - 8 q^{62} + 10 q^{64}+ \cdots + 2 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^8 + 2 * q^11 + 2 * q^13 + 10 * q^16 + 2 * q^22 + 2 * q^26 - 8 * q^31 + 10 * q^32 - 8 * q^41 + 2 * q^44 + 2 * q^49 + 2 * q^52 + 18 * q^53 + 2 * q^61 - 8 * q^62 + 10 * q^64 - 12 * q^77 - 8 * q^82 - 14 * q^83 + 22 * q^85 + 2 * q^88 + 18 * q^91 - 4 * q^95 + 10 * q^97 + 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times$.

$n$	$227$	$1585$
$\chi(n)$	$1$	$-1$

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
$5$			1.63968i			0.733286i	0.930362	+	0.366643i	$0.119493\pi$
$5$			1.63968i			0.733286i	−0.930362	+	0.366643i	$0.880507\pi$
$6$		0			0
$7$	3.55588			1.34399			0.671997	−	0.740553i	$-0.265437\pi$
$7$	3.55588			1.34399			0.671997	+	0.740553i	$0.265437\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$			1.63968i			0.518511i
$11$	3.33279			1.00488			0.502438	−	0.864613i	$-0.332437\pi$
$11$	3.33279			1.00488			0.502438	+	0.864613i	$0.332437\pi$
$12$		0			0
$13$	−1.82544			−0.506285			−0.253142	−	0.967429i	$-0.581464\pi$
$13$	−1.82544			−0.506285			−0.253142	+	0.967429i	$0.581464\pi$
$14$	3.55588			0.950348
$15$		0			0
$16$	1.00000			0.250000
$17$		−	0.106462i		−	0.0258209i	−0.999917	−	0.0129105i	$-0.995890\pi$
$17$		−	0.106462i		−	0.0258209i	0.999917	−	0.0129105i	$-0.00410964\pi$
$18$		0			0
$19$			4.06519i			0.932617i	0.884622	+	0.466309i	$0.154416\pi$
$19$			4.06519i			0.932617i	−0.884622	+	0.466309i	$0.845584\pi$
$20$			1.63968i			0.366643i
$21$		0			0
$22$	3.33279			0.710554
$23$		−	8.58345i		−	1.78977i	−0.446293	−	0.894887i	$-0.647256\pi$
$23$		−	8.58345i		−	1.78977i	0.446293	−	0.894887i	$-0.352744\pi$
$24$		0			0
$25$	2.31146			0.462292
$26$	−1.82544			−0.357997
$27$		0			0
$28$	3.55588			0.671997
$29$		−	3.75172i		−	0.696678i	−0.937369	−	0.348339i	$-0.886746\pi$
$29$		−	3.75172i		−	0.696678i	0.937369	−	0.348339i	$-0.113254\pi$
$30$		0			0
$31$	4.64426			0.834133			0.417066	−	0.908876i	$-0.363058\pi$
$31$	4.64426			0.834133			0.417066	+	0.908876i	$0.363058\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	0.106462i		−	0.0182582i
$35$			5.83049i			0.985532i
$36$		0			0
$37$			4.24482i			0.697845i	0.937152	+	0.348922i	$0.113452\pi$
$37$			4.24482i			0.697845i	−0.937152	+	0.348922i	$0.886548\pi$
$38$			4.06519i			0.659460i
$39$		0			0
$40$			1.63968i			0.259256i
$41$	−7.62573			−1.19094			−0.595469	−	0.803378i	$-0.703034\pi$
$41$	−7.62573			−1.19094			−0.595469	+	0.803378i	$0.703034\pi$
$42$		0			0
$43$			11.8407i			1.80570i	0.429960	+	0.902848i	$0.358528\pi$
$43$			11.8407i			1.80570i	−0.429960	+	0.902848i	$0.641472\pi$
$44$	3.33279			0.502438
$45$		0			0
$46$		−	8.58345i		−	1.26556i
$47$			3.97272i			0.579481i	0.957105	+	0.289740i	$0.0935690\pi$
$47$			3.97272i			0.579481i	−0.957105	+	0.289740i	$0.906431\pi$
$48$		0			0
$49$	5.64426			0.806322
$50$	2.31146			0.326890
$51$		0			0
$52$	−1.82544			−0.253142
$53$	−7.55807			−1.03818			−0.519090	−	0.854719i	$-0.673729\pi$
$53$	−7.55807			−1.03818			−0.519090	+	0.854719i	$0.673729\pi$
$54$		0			0
$55$			5.46470i			0.736861i
$56$	3.55588			0.475174
$57$		0			0
$58$		−	3.75172i		−	0.492626i
$59$			3.60391i			0.469189i	0.972093	+	0.234594i	$0.0753762\pi$
$59$			3.60391i			0.469189i	−0.972093	+	0.234594i	$0.924624\pi$
$60$		0			0
$61$	−1.82544			−0.233723			−0.116862	−	0.993148i	$-0.537283\pi$
$61$	−1.82544			−0.233723			−0.116862	+	0.993148i	$0.537283\pi$
$62$	4.64426			0.589821
$63$		0			0
$64$	1.00000			0.125000
$65$		−	2.99312i		−	0.371251i
$66$		0			0
$67$		−	10.3901i		−	1.26935i	−0.772778	−	0.634677i	$-0.781133\pi$
$67$		−	10.3901i		−	1.26935i	0.772778	−	0.634677i	$-0.218867\pi$
$68$		−	0.106462i		−	0.0129105i
$69$		0			0
$70$			5.83049i			0.696876i
$71$		−	2.25853i		−	0.268038i	−0.990979	−	0.134019i	$-0.957212\pi$
$71$		−	2.25853i		−	0.268038i	0.990979	−	0.134019i	$-0.0427883\pi$
$72$		0			0
$73$		−	14.1005i		−	1.65034i	−0.564887	−	0.825168i	$-0.691080\pi$
$73$		−	14.1005i		−	1.65034i	0.564887	−	0.825168i	$-0.308920\pi$
$74$			4.24482i			0.493451i
$75$		0			0
$76$			4.06519i			0.466309i
$77$	11.8510			1.35055
$78$		0			0
$79$			4.19372i			0.471830i	0.971774	+	0.235915i	$0.0758087\pi$
$79$			4.19372i			0.471830i	−0.971774	+	0.235915i	$0.924191\pi$
$80$			1.63968i			0.183321i
$81$		0			0
$82$	−7.62573			−0.842121
$83$	8.51821			0.934995			0.467497	−	0.883994i	$-0.345156\pi$
$83$	8.51821			0.934995			0.467497	+	0.883994i	$0.345156\pi$
$84$		0			0
$85$	0.174564			0.0189341
$86$			11.8407i			1.27682i
$87$		0			0
$88$	3.33279			0.355277
$89$		−	10.6622i		−	1.13019i	−0.825025	−	0.565096i	$-0.808839\pi$
$89$		−	10.6622i		−	1.13019i	0.825025	−	0.565096i	$-0.191161\pi$
$90$		0			0
$91$	−6.49102			−0.680444
$92$		−	8.58345i		−	0.894887i
$93$		0			0
$94$			3.97272i			0.409755i
$95$	−6.66559			−0.683875
$96$		0			0
$97$	17.1838			1.74475			0.872375	−	0.488837i	$-0.162579\pi$
$97$	17.1838			1.74475			0.872375	+	0.488837i	$0.162579\pi$
$98$	5.64426			0.570156
$99$		0			0
$100$	2.31146			0.231146
$101$			17.7126i			1.76247i	0.472681	+	0.881234i	$0.343286\pi$
$101$			17.7126i			1.76247i	−0.472681	+	0.881234i	$0.656714\pi$
$102$		0			0
$103$			10.9757i			1.08147i	0.841194	+	0.540733i	$0.181853\pi$
$103$			10.9757i			1.08147i	−0.841194	+	0.540733i	$0.818147\pi$
$104$	−1.82544			−0.178999
$105$		0			0
$106$	−7.55807			−0.734105
$107$			2.15328i			0.208165i	0.994569	+	0.104083i	$0.0331906\pi$
$107$			2.15328i			0.208165i	−0.994569	+	0.104083i	$0.966809\pi$
$108$		0			0
$109$	2.66339			0.255107			0.127553	−	0.991832i	$-0.459288\pi$
$109$	2.66339			0.255107			0.127553	+	0.991832i	$0.459288\pi$
$110$			5.46470i			0.521039i
$111$		0			0
$112$	3.55588			0.335999
$113$	−9.37470	+	5.01149i	−0.881897	+	0.471442i
$113$	−9.37470	+	5.01149i	−0.881897	+	0.471442i
$114$		0			0
$115$	14.0741			1.31242
$116$		−	3.75172i		−	0.348339i
$117$		0			0
$118$			3.60391i			0.331766i
$119$		−	0.378567i		−	0.0347032i
$120$		0			0
$121$	0.107518			0.00977433
$122$	−1.82544			−0.165267
$123$		0			0
$124$	4.64426			0.417066
$125$			11.9884i			1.07228i
$126$		0			0
$127$	−2.29293			−0.203465			−0.101732	−	0.994812i	$-0.532439\pi$
$127$	−2.29293			−0.203465			−0.101732	+	0.994812i	$0.532439\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	2.99312i		−	0.262514i
$131$	−6.89248			−0.602199			−0.301099	−	0.953593i	$-0.597354\pi$
$131$	−6.89248			−0.602199			−0.301099	+	0.953593i	$0.597354\pi$
$132$		0			0
$133$			14.4553i			1.25343i
$134$		−	10.3901i		−	0.897569i
$135$		0			0
$136$		−	0.106462i		−	0.00912908i
$137$		−	9.80333i		−	0.837555i	−0.908089	−	0.418778i	$-0.862459\pi$
$137$		−	9.80333i		−	0.837555i	0.908089	−	0.418778i	$-0.137541\pi$
$138$		0			0
$139$	18.7397			1.58948			0.794739	−	0.606951i	$-0.207608\pi$
$139$	18.7397			1.58948			0.794739	+	0.606951i	$0.207608\pi$
$140$			5.83049i			0.492766i
$141$		0			0
$142$		−	2.25853i		−	0.189531i
$143$	−6.08380			−0.508753
$144$		0			0
$145$	6.15161			0.510864
$146$		−	14.1005i		−	1.16696i
$147$		0			0
$148$			4.24482i			0.348922i
$149$	−10.9585			−0.897757			−0.448879	−	0.893593i	$-0.648176\pi$
$149$	−10.9585			−0.897757			−0.448879	+	0.893593i	$0.648176\pi$
$150$		0			0
$151$		−	10.9757i		−	0.893188i	−0.894737	−	0.446594i	$-0.852637\pi$
$151$		−	10.9757i		−	0.893188i	0.894737	−	0.446594i	$-0.147363\pi$
$152$			4.06519i			0.329730i
$153$		0			0
$154$	11.8510			0.954981
$155$			7.61508i			0.611658i
$156$		0			0
$157$	−6.58587			−0.525609			−0.262805	−	0.964849i	$-0.584647\pi$
$157$	−6.58587			−0.525609			−0.262805	+	0.964849i	$0.584647\pi$
$158$			4.19372i			0.333634i
$159$		0			0
$160$			1.63968i			0.129628i
$161$		−	30.5217i		−	2.40545i
$162$		0			0
$163$	0.351323			0.0275177			0.0137589	−	0.999905i	$-0.495620\pi$
$163$	0.351323			0.0275177			0.0137589	+	0.999905i	$0.495620\pi$
$164$	−7.62573			−0.595469
$165$		0			0
$166$	8.51821			0.661141
$167$			3.23643i			0.250442i	0.992129	+	0.125221i	$0.0399641\pi$
$167$			3.23643i			0.250442i	−0.992129	+	0.125221i	$0.960036\pi$
$168$		0			0
$169$	−9.66778			−0.743676
$170$	0.174564			0.0133884
$171$		0			0
$172$			11.8407i			0.902848i
$173$	10.9585			0.833161			0.416580	−	0.909099i	$-0.363228\pi$
$173$	10.9585			0.833161			0.416580	+	0.909099i	$0.363228\pi$
$174$		0			0
$175$	8.21927			0.621318
$176$	3.33279			0.251219
$177$		0			0
$178$		−	10.6622i		−	0.799167i
$179$		−	23.5698i		−	1.76169i	−0.473402	−	0.880847i	$-0.656974\pi$
$179$		−	23.5698i		−	1.76169i	0.473402	−	0.880847i	$-0.343026\pi$
$180$		0			0
$181$		−	0.865062i		−	0.0642996i	−0.999483	−	0.0321498i	$-0.989765\pi$
$181$		−	0.865062i		−	0.0642996i	0.999483	−	0.0321498i	$-0.0102354\pi$
$182$	−6.49102			−0.481147
$183$		0			0
$184$		−	8.58345i		−	0.632781i
$185$	−6.96014			−0.511720
$186$		0			0
$187$		−	0.354817i		−	0.0259468i
$188$			3.97272i			0.289740i
$189$		0			0
$190$	−6.66559			−0.483573
$191$		−	1.73227i		−	0.125343i	−0.998034	−	0.0626713i	$-0.980038\pi$
$191$		−	1.73227i		−	0.125343i	0.998034	−	0.0626713i	$-0.0199620\pi$
$192$		0			0
$193$			9.83806i			0.708159i	0.935215	+	0.354079i	$0.115206\pi$
$193$			9.83806i			0.708159i	−0.935215	+	0.354079i	$0.884794\pi$
$194$	17.1838			1.23372
$195$		0			0
$196$	5.64426			0.403161
$197$		−	15.9984i		−	1.13984i	−0.821701	−	0.569919i	$-0.806975\pi$
$197$		−	15.9984i		−	1.13984i	0.821701	−	0.569919i	$-0.193025\pi$
$198$		0			0
$199$			22.4105i			1.58864i	0.607501	+	0.794319i	$0.292172\pi$
$199$			22.4105i			1.58864i	−0.607501	+	0.794319i	$0.707828\pi$
$200$	2.31146			0.163445
$201$		0			0
$202$			17.7126i			1.24625i
$203$		−	13.3407i		−	0.936331i
$204$		0			0
$205$		−	12.5037i		−	0.873298i
$206$			10.9757i			0.764712i
$207$		0			0
$208$	−1.82544			−0.126571
$209$			13.5484i			0.937164i
$210$		0			0
$211$	−15.1566			−1.04342			−0.521712	−	0.853122i	$-0.674706\pi$
$211$	−15.1566			−1.04342			−0.521712	+	0.853122i	$0.674706\pi$
$212$	−7.55807			−0.519090
$213$		0			0
$214$			2.15328i			0.147195i
$215$	−19.4150			−1.32409
$216$		0			0
$217$	16.5144			1.12107
$218$	2.66339			0.180388
$219$		0			0
$220$			5.46470i			0.368430i
$221$			0.194340i			0.0130727i
$222$		0			0
$223$		−	15.6285i		−	1.04656i	−0.852160	−	0.523282i	$-0.824708\pi$
$223$		−	15.6285i		−	1.04656i	0.852160	−	0.523282i	$-0.175292\pi$
$224$	3.55588			0.237587
$225$		0			0
$226$	−9.37470	+	5.01149i	−0.623595	+	0.333360i
$227$	−7.55807			−0.501647			−0.250823	−	0.968033i	$-0.580701\pi$
$227$	−7.55807			−0.501647			−0.250823	+	0.968033i	$0.580701\pi$
$228$		0			0
$229$			6.60232i			0.436294i	0.975916	+	0.218147i	$0.0700012\pi$
$229$			6.60232i			0.436294i	−0.975916	+	0.218147i	$0.929999\pi$
$230$	14.0741			0.928018
$231$		0			0
$232$		−	3.75172i		−	0.246313i
$233$	−7.04394			−0.461464			−0.230732	−	0.973017i	$-0.574112\pi$
$233$	−7.04394			−0.461464			−0.230732	+	0.973017i	$0.574112\pi$
$234$		0			0
$235$	−6.51397			−0.424925
$236$			3.60391i			0.234594i
$237$		0			0
$238$		−	0.378567i		−	0.0245389i
$239$	−12.7494			−0.824690			−0.412345	−	0.911028i	$-0.635290\pi$
$239$	−12.7494			−0.824690			−0.412345	+	0.911028i	$0.635290\pi$
$240$		0			0
$241$	−21.7079			−1.39833			−0.699165	−	0.714961i	$-0.746445\pi$
$241$	−21.7079			−1.39833			−0.699165	+	0.714961i	$0.746445\pi$
$242$	0.107518			0.00691150
$243$		0			0
$244$	−1.82544			−0.116862
$245$			9.25475i			0.591264i
$246$		0			0
$247$		−	7.42074i		−	0.472170i
$248$	4.64426			0.294911
$249$		0			0
$250$			11.9884i			0.758215i
$251$	−17.0423			−1.07570			−0.537851	−	0.843040i	$-0.680764\pi$
$251$	−17.0423			−1.07570			−0.537851	+	0.843040i	$0.680764\pi$
$252$		0			0
$253$		−	28.6069i		−	1.79850i
$254$	−2.29293			−0.143871
$255$		0			0
$256$	1.00000			0.0625000
$257$	14.2913			0.891468			0.445734	−	0.895166i	$-0.352943\pi$
$257$	14.2913			0.891468			0.445734	+	0.895166i	$0.352943\pi$
$258$		0			0
$259$			15.0941i			0.937900i
$260$		−	2.99312i		−	0.185626i
$261$		0			0
$262$	−6.89248			−0.425819
$263$		−	29.0808i		−	1.79320i	−0.442841	−	0.896600i	$-0.646029\pi$
$263$		−	29.0808i		−	1.79320i	0.442841	−	0.896600i	$-0.353971\pi$
$264$		0			0
$265$		−	12.3928i		−	0.761283i
$266$			14.4553i			0.886311i
$267$		0			0
$268$		−	10.3901i		−	0.634677i
$269$		−	0.504192i		−	0.0307411i	−0.999882	−	0.0153706i	$-0.995107\pi$
$269$		−	0.504192i		−	0.0307411i	0.999882	−	0.0153706i	$-0.00489279\pi$
$270$		0			0
$271$		−	31.8893i		−	1.93714i	−0.248751	−	0.968568i	$-0.580020\pi$
$271$		−	31.8893i		−	1.93714i	0.248751	−	0.968568i	$-0.419980\pi$
$272$		−	0.106462i		−	0.00645523i
$273$		0			0
$274$		−	9.80333i		−	0.592241i
$275$	7.70362			0.464546
$276$		0			0
$277$	−5.73248			−0.344431			−0.172216	−	0.985059i	$-0.555093\pi$
$277$	−5.73248			−0.344431			−0.172216	+	0.985059i	$0.555093\pi$
$278$	18.7397			1.12393
$279$		0			0
$280$			5.83049i			0.348438i
$281$		−	5.73815i		−	0.342309i	−0.985244	−	0.171155i	$-0.945250\pi$
$281$		−	5.73815i		−	0.342309i	0.985244	−	0.171155i	$-0.0547498\pi$
$282$		0			0
$283$	−20.6056			−1.22487			−0.612437	−	0.790519i	$-0.709811\pi$
$283$	−20.6056			−1.22487			−0.612437	+	0.790519i	$0.709811\pi$
$284$		−	2.25853i		−	0.134019i
$285$		0			0
$286$	−6.08380			−0.359743
$287$	−27.1161			−1.60062
$288$		0			0
$289$	16.9887			0.999333
$290$	6.15161			0.361235
$291$		0			0
$292$		−	14.1005i		−	0.825168i
$293$		−	2.37440i		−	0.138714i	−0.997592	−	0.0693570i	$-0.977905\pi$
$293$		−	2.37440i		−	0.138714i	0.997592	−	0.0693570i	$-0.0220947\pi$
$294$		0			0
$295$	−5.90924			−0.344049
$296$			4.24482i			0.246725i
$297$		0			0
$298$	−10.9585			−0.634810
$299$			15.6685i			0.906135i
$300$		0			0
$301$			42.1042i			2.42685i
$302$		−	10.9757i		−	0.631579i
$303$		0			0
$304$			4.06519i			0.233154i
$305$		−	2.99312i		−	0.171386i
$306$		0			0
$307$	9.60278			0.548059			0.274030	−	0.961721i	$-0.411643\pi$
$307$	9.60278			0.548059			0.274030	+	0.961721i	$0.411643\pi$
$308$	11.8510			0.675274
$309$		0			0
$310$			7.61508i			0.432507i
$311$	−15.1161			−0.857158			−0.428579	−	0.903504i	$-0.640986\pi$
$311$	−15.1161			−0.857158			−0.428579	+	0.903504i	$0.640986\pi$
$312$		0			0
$313$	3.79867			0.214714			0.107357	−	0.994221i	$-0.465761\pi$
$313$	3.79867			0.214714			0.107357	+	0.994221i	$0.465761\pi$
$314$	−6.58587			−0.371662
$315$		0			0
$316$			4.19372i			0.235915i
$317$	−12.8788			−0.723345			−0.361673	−	0.932305i	$-0.617794\pi$
$317$	−12.8788			−0.723345			−0.361673	+	0.932305i	$0.617794\pi$
$318$		0			0
$319$		−	12.5037i		−	0.700074i
$320$			1.63968i			0.0916607i
$321$		0			0
$322$		−	30.5217i		−	1.70091i
$323$	0.432790			0.0240811
$324$		0			0
$325$	−4.21942			−0.234052
$326$	0.351323			0.0194580
$327$		0			0
$328$	−7.62573			−0.421060
$329$			14.1265i			0.778819i
$330$		0			0
$331$	−27.5350			−1.51346			−0.756729	−	0.653728i	$-0.773204\pi$
$331$	−27.5350			−1.51346			−0.756729	+	0.653728i	$0.773204\pi$
$332$	8.51821			0.467497
$333$		0			0
$334$			3.23643i			0.177090i
$335$	17.0364			0.930799
$336$		0			0
$337$	10.7898			0.587758			0.293879	−	0.955843i	$-0.405054\pi$
$337$	10.7898			0.587758			0.293879	+	0.955843i	$0.405054\pi$
$338$	−9.66778			−0.525858
$339$		0			0
$340$	0.174564			0.00946706
$341$	15.4783			0.838200
$342$		0			0
$343$	−4.82086			−0.260302
$344$			11.8407i			0.638410i
$345$		0			0
$346$	10.9585			0.589134
$347$	−8.66967			−0.465412			−0.232706	−	0.972547i	$-0.574758\pi$
$347$	−8.66967			−0.465412			−0.232706	+	0.972547i	$0.574758\pi$
$348$		0			0
$349$		−	17.8399i		−	0.954948i	−0.878646	−	0.477474i	$-0.841552\pi$
$349$		−	17.8399i		−	0.954948i	0.878646	−	0.477474i	$-0.158448\pi$
$350$	8.21927			0.439338
$351$		0			0
$352$	3.33279			0.177639
$353$	−8.20751			−0.436842			−0.218421	−	0.975855i	$-0.570091\pi$
$353$	−8.20751			−0.436842			−0.218421	+	0.975855i	$0.570091\pi$
$354$		0			0
$355$	3.70326			0.196548
$356$		−	10.6622i		−	0.565096i
$357$		0			0
$358$		−	23.5698i		−	1.24571i
$359$		−	8.17173i		−	0.431287i	−0.976472	−	0.215644i	$-0.930815\pi$
$359$		−	8.17173i		−	0.431287i	0.976472	−	0.215644i	$-0.0691849\pi$
$360$		0			0
$361$	2.47427			0.130225
$362$		−	0.865062i		−	0.0454667i
$363$		0			0
$364$	−6.49102			−0.340222
$365$	23.1202			1.21017
$366$		0			0
$367$	−18.3749			−0.959159			−0.479580	−	0.877498i	$-0.659211\pi$
$367$	−18.3749			−0.959159			−0.479580	+	0.877498i	$0.659211\pi$
$368$		−	8.58345i		−	0.447443i
$369$		0			0
$370$	−6.96014			−0.361840
$371$	−26.8756			−1.39531
$372$		0			0
$373$		−	12.3241i		−	0.638117i	−0.947735	−	0.319058i	$-0.896633\pi$
$373$		−	12.3241i		−	0.638117i	0.947735	−	0.319058i	$-0.103367\pi$
$374$		−	0.354817i		−	0.0183472i
$375$		0			0
$376$			3.97272i			0.204877i
$377$			6.84853i			0.352717i
$378$		0			0
$379$			31.9069i			1.63895i	0.573118	+	0.819473i	$0.305734\pi$
$379$			31.9069i			1.63895i	−0.573118	+	0.819473i	$0.694266\pi$
$380$	−6.66559			−0.341937
$381$		0			0
$382$		−	1.73227i		−	0.0886306i
$383$	−28.0008			−1.43078			−0.715388	−	0.698727i	$-0.753750\pi$
$383$	−28.0008			−1.43078			−0.715388	+	0.698727i	$0.753750\pi$
$384$		0			0
$385$			19.4318i			0.990337i
$386$			9.83806i			0.500744i
$387$		0			0
$388$	17.1838			0.872375
$389$	10.2988			0.522172			0.261086	−	0.965316i	$-0.415919\pi$
$389$	10.2988			0.522172			0.261086	+	0.965316i	$0.415919\pi$
$390$		0			0
$391$	−0.913815			−0.0462136
$392$	5.64426			0.285078
$393$		0			0
$394$		−	15.9984i		−	0.805987i
$395$	−6.87634			−0.345986
$396$		0			0
$397$		−	19.0594i		−	0.956564i	−0.878206	−	0.478282i	$-0.841260\pi$
$397$		−	19.0594i		−	0.956564i	0.878206	−	0.478282i	$-0.158740\pi$
$398$			22.4105i			1.12334i
$399$		0			0
$400$	2.31146			0.115573
$401$	−8.36675			−0.417816			−0.208908	−	0.977935i	$-0.566991\pi$
$401$	−8.36675			−0.417816			−0.208908	+	0.977935i	$0.566991\pi$
$402$		0			0
$403$	−8.47779			−0.422309
$404$			17.7126i			0.881234i
$405$		0			0
$406$		−	13.3407i		−	0.662086i
$407$			14.1471i			0.701247i
$408$		0			0
$409$		−	0.359278i		−	0.0177652i	−0.999961	−	0.00888259i	$-0.997173\pi$
$409$		−	0.359278i		−	0.0177652i	0.999961	−	0.00888259i	$-0.00282745\pi$
$410$		−	12.5037i		−	0.617515i
$411$		0			0
$412$			10.9757i			0.540733i
$413$			12.8150i			0.630587i
$414$		0			0
$415$			13.9671i			0.685618i
$416$	−1.82544			−0.0894994
$417$		0			0
$418$			13.5484i			0.662675i
$419$		−	13.9889i		−	0.683400i	−0.939809	−	0.341700i	$-0.888997\pi$
$419$		−	13.9889i		−	0.683400i	0.939809	−	0.341700i	$-0.111003\pi$
$420$		0			0
$421$	−37.7292			−1.83881			−0.919405	−	0.393311i	$-0.871330\pi$
$421$	−37.7292			−1.83881			−0.919405	+	0.393311i	$0.871330\pi$
$422$	−15.1566			−0.737812
$423$		0			0
$424$	−7.55807			−0.367052
$425$		−	0.246084i		−	0.0119368i
$426$		0			0
$427$	−6.49102			−0.314123
$428$			2.15328i			0.104083i
$429$		0			0
$430$	−19.4150			−0.936273
$431$		−	17.6158i		−	0.848521i	−0.905540	−	0.424261i	$-0.860534\pi$
$431$		−	17.6158i		−	0.848521i	0.905540	−	0.424261i	$-0.139466\pi$
$432$		0			0
$433$			18.6072i			0.894205i	0.894483	+	0.447103i	$0.147544\pi$
$433$			18.6072i			0.894205i	−0.894483	+	0.447103i	$0.852456\pi$
$434$	16.5144			0.792716
$435$		0			0
$436$	2.66339			0.127553
$437$	34.8933			1.66917
$438$		0			0
$439$	−16.5225			−0.788576			−0.394288	−	0.918987i	$-0.629009\pi$
$439$	−16.5225			−0.788576			−0.394288	+	0.918987i	$0.629009\pi$
$440$			5.46470i			0.260520i
$441$		0			0
$442$			0.194340i			0.00924383i
$443$	26.1482			1.24234			0.621170	−	0.783676i	$-0.286658\pi$
$443$	26.1482			1.24234			0.621170	+	0.783676i	$0.286658\pi$
$444$		0			0
$445$	17.4826			0.828754
$446$		−	15.6285i		−	0.740032i
$447$		0			0
$448$	3.55588			0.167999
$449$			36.2457i			1.71054i	0.518182	+	0.855271i	$0.326609\pi$
$449$			36.2457i			1.71054i	−0.518182	+	0.855271i	$0.673391\pi$
$450$		0			0
$451$	−25.4150			−1.19674
$452$	−9.37470	+	5.01149i	−0.440949	+	0.235721i
$453$		0			0
$454$	−7.55807			−0.354718
$455$		−	10.6432i		−	0.498960i
$456$		0			0
$457$		−	2.51954i		−	0.117859i	−0.998262	−	0.0589296i	$-0.981231\pi$
$457$		−	2.51954i		−	0.117859i	0.998262	−	0.0589296i	$-0.0187687\pi$
$458$			6.60232i			0.308506i
$459$		0			0
$460$	14.0741			0.656208
$461$	42.2245			1.96659			0.983296	−	0.182016i	$-0.0582622\pi$
$461$	42.2245			1.96659			0.983296	+	0.182016i	$0.0582622\pi$
$462$		0			0
$463$	27.6497			1.28499			0.642495	−	0.766290i	$-0.277899\pi$
$463$	27.6497			1.28499			0.642495	+	0.766290i	$0.277899\pi$
$464$		−	3.75172i		−	0.174169i
$465$		0			0
$466$	−7.04394			−0.326304
$467$	22.6581			1.04849			0.524245	−	0.851567i	$-0.324348\pi$
$467$	22.6581			1.04849			0.524245	+	0.851567i	$0.324348\pi$
$468$		0			0
$469$		−	36.9459i		−	1.70601i
$470$	−6.51397			−0.300467
$471$		0			0
$472$			3.60391i			0.165883i
$473$			39.4627i			1.81450i
$474$		0			0
$475$			9.39652i			0.431142i
$476$		−	0.378567i		−	0.0173516i
$477$		0			0
$478$	−12.7494			−0.583144
$479$			5.77817i			0.264011i	0.991249	+	0.132006i	$0.0421417\pi$
$479$			5.77817i			0.264011i	−0.991249	+	0.132006i	$0.957858\pi$
$480$		0			0
$481$		−	7.74866i		−	0.353308i
$482$	−21.7079			−0.988768
$483$		0			0
$484$	0.107518			0.00488717
$485$			28.1759i			1.27940i
$486$		0			0
$487$			28.1323i			1.27479i	0.770536	+	0.637397i	$0.219989\pi$
$487$			28.1323i			1.27479i	−0.770536	+	0.637397i	$0.780011\pi$
$488$	−1.82544			−0.0826336
$489$		0			0
$490$			9.25475i			0.418087i
$491$		−	23.7012i		−	1.06962i	−0.844972	−	0.534810i	$-0.820383\pi$
$491$		−	23.7012i		−	1.06962i	0.844972	−	0.534810i	$-0.179617\pi$
$492$		0			0
$493$	−0.399418			−0.0179889
$494$		−	7.42074i		−	0.333875i
$495$		0			0
$496$	4.64426			0.208533
$497$		−	8.03105i		−	0.360242i
$498$		0			0
$499$		−	27.0101i		−	1.20914i	−0.796552	−	0.604570i	$-0.793345\pi$
$499$		−	27.0101i		−	1.20914i	0.796552	−	0.604570i	$-0.206655\pi$
$500$			11.9884i			0.536139i
$501$		0			0
$502$	−17.0423			−0.760636
$503$	−33.3353			−1.48635			−0.743173	−	0.669100i	$-0.766680\pi$
$503$	−33.3353			−1.48635			−0.743173	+	0.669100i	$0.766680\pi$
$504$		0			0
$505$	−29.0429			−1.29239
$506$		−	28.6069i		−	1.27173i
$507$		0			0
$508$	−2.29293			−0.101732
$509$	23.8569			1.05744			0.528719	−	0.848797i	$-0.322672\pi$
$509$	23.8569			1.05744			0.528719	+	0.848797i	$0.322672\pi$
$510$		0			0
$511$		−	50.1396i		−	2.21804i
$512$	1.00000			0.0441942
$513$		0			0
$514$	14.2913			0.630363
$515$	−17.9966			−0.793023
$516$		0			0
$517$			13.2403i			0.582306i
$518$			15.0941i			0.663195i
$519$		0			0
$520$		−	2.99312i		−	0.131257i
$521$	21.4107			0.938019			0.469010	−	0.883193i	$-0.344611\pi$
$521$	21.4107			0.938019			0.469010	+	0.883193i	$0.344611\pi$
$522$		0			0
$523$		−	19.2346i		−	0.841069i	−0.907276	−	0.420535i	$-0.861843\pi$
$523$		−	19.2346i		−	0.841069i	0.907276	−	0.420535i	$-0.138157\pi$
$524$	−6.89248			−0.301099
$525$		0			0
$526$		−	29.0808i		−	1.26798i
$527$		−	0.494439i		−	0.0215381i
$528$		0			0
$529$	−50.6757			−2.20329
$530$		−	12.3928i		−	0.538308i
$531$		0			0
$532$			14.4553i			0.626717i
$533$	13.9203			0.602954
$534$		0			0
$535$	−3.53068			−0.152645
$536$		−	10.3901i		−	0.448784i
$537$		0			0
$538$		−	0.504192i		−	0.0217373i
$539$	18.8111			0.810253
$540$		0			0
$541$		−	14.4601i		−	0.621690i	−0.950461	−	0.310845i	$-0.899388\pi$
$541$		−	14.4601i		−	0.621690i	0.950461	−	0.310845i	$-0.100612\pi$
$542$		−	31.8893i		−	1.36976i
$543$		0			0
$544$		−	0.106462i		−	0.00456454i
$545$			4.36710i			0.187066i
$546$		0			0
$547$	18.6581			0.797761			0.398881	−	0.917003i	$-0.369399\pi$
$547$	18.6581			0.797761			0.398881	+	0.917003i	$0.369399\pi$
$548$		−	9.80333i		−	0.418778i
$549$		0			0
$550$	7.70362			0.328484
$551$	15.2515			0.649734
$552$		0			0
$553$			14.9123i			0.634137i
$554$	−5.73248			−0.243550
$555$		0			0
$556$	18.7397			0.794739
$557$	0.00590518			0.000250210			0.000125105	−	1.00000i	$-0.499960\pi$
$557$	0.00590518			0.000250210			0.000125105		1.00000i	$0.499960\pi$
$558$		0			0
$559$		−	21.6145i		−	0.914196i
$560$			5.83049i			0.246383i
$561$		0			0
$562$		−	5.73815i		−	0.242049i
$563$	−25.5444			−1.07657			−0.538284	−	0.842764i	$-0.680927\pi$
$563$	−25.5444			−1.07657			−0.538284	+	0.842764i	$0.680927\pi$
$564$		0			0
$565$	−8.21723	−	15.3715i	−0.345701	−	0.646682i
$566$	−20.6056			−0.866117
$567$		0			0
$568$		−	2.25853i		−	0.0947657i
$569$	40.8257			1.71150			0.855751	−	0.517389i	$-0.173096\pi$
$569$	40.8257			1.71150			0.855751	+	0.517389i	$0.173096\pi$
$570$		0			0
$571$		−	0.441540i		−	0.0184779i	−0.999957	−	0.00923894i	$-0.997059\pi$
$571$		−	0.441540i		−	0.0184779i	0.999957	−	0.00923894i	$-0.00294089\pi$
$572$	−6.08380			−0.254377
$573$		0			0
$574$	−27.1161			−1.13181
$575$		−	19.8403i		−	0.827399i
$576$		0			0
$577$			20.8824i			0.869347i	0.900588	+	0.434674i	$0.143136\pi$
$577$			20.8824i			0.869347i	−0.900588	+	0.434674i	$0.856864\pi$
$578$	16.9887			0.706635
$579$		0			0
$580$	6.15161			0.255432
$581$	30.2897			1.25663
$582$		0			0
$583$	−25.1895			−1.04324
$584$		−	14.1005i		−	0.583482i
$585$		0			0
$586$		−	2.37440i		−	0.0980855i
$587$	13.7772			0.568645			0.284323	−	0.958729i	$-0.408231\pi$
$587$	13.7772			0.568645			0.284323	+	0.958729i	$0.408231\pi$
$588$		0			0
$589$			18.8798i			0.777927i
$590$	−5.90924			−0.243280
$591$		0			0
$592$			4.24482i			0.174461i
$593$	−15.0407			−0.617648			−0.308824	−	0.951119i	$-0.599935\pi$
$593$	−15.0407			−0.617648			−0.308824	+	0.951119i	$0.599935\pi$
$594$		0			0
$595$	0.620728			0.0254474
$596$	−10.9585			−0.448879
$597$		0			0
$598$			15.6685i			0.640734i
$599$			7.30137i			0.298326i	0.988813	+	0.149163i	$0.0476579\pi$
$599$			7.30137i			0.298326i	−0.988813	+	0.149163i	$0.952342\pi$
$600$		0			0
$601$	−12.2229			−0.498584			−0.249292	−	0.968428i	$-0.580198\pi$
$601$	−12.2229			−0.498584			−0.249292	+	0.968428i	$0.580198\pi$
$602$			42.1042i			1.71604i
$603$		0			0
$604$		−	10.9757i		−	0.446594i
$605$			0.176294i			0.00716738i
$606$		0			0
$607$		−	0.0687047i		−	0.00278864i	−0.999999	−	0.00139432i	$-0.999556\pi$
$607$		−	0.0687047i		−	0.00278864i	0.999999	−	0.00139432i	$-0.000443826\pi$
$608$			4.06519i			0.164865i
$609$		0			0
$610$		−	2.99312i		−	0.121188i
$611$		−	7.25195i		−	0.293382i
$612$		0			0
$613$		−	12.2774i		−	0.495881i	−0.968775	−	0.247941i	$-0.920246\pi$
$613$		−	12.2774i		−	0.495881i	0.968775	−	0.247941i	$-0.0797538\pi$
$614$	9.60278			0.387537
$615$		0			0
$616$	11.8510			0.477491
$617$	22.7419			0.915553			0.457777	−	0.889067i	$-0.348646\pi$
$617$	22.7419			0.915553			0.457777	+	0.889067i	$0.348646\pi$
$618$		0			0
$619$		−	4.42000i		−	0.177655i	−0.996047	−	0.0888275i	$-0.971688\pi$
$619$		−	4.42000i		−	0.177655i	0.996047	−	0.0888275i	$-0.0283120\pi$
$620$			7.61508i			0.305829i
$621$		0			0
$622$	−15.1161			−0.606102
$623$		−	37.9135i		−	1.51897i
$624$		0			0
$625$	−8.09984			−0.323994
$626$	3.79867			0.151826
$627$		0			0
$628$	−6.58587			−0.262805
$629$	0.451914			0.0180190
$630$		0			0
$631$		−	32.2485i		−	1.28379i	−0.766791	−	0.641897i	$-0.778148\pi$
$631$		−	32.2485i		−	1.28379i	0.766791	−	0.641897i	$-0.221852\pi$
$632$			4.19372i			0.166817i
$633$		0			0
$634$	−12.8788			−0.511482
$635$		−	3.75967i		−	0.149198i
$636$		0			0
$637$	−10.3032			−0.408229
$638$		−	12.5037i		−	0.495027i
$639$		0			0
$640$			1.63968i			0.0648139i
$641$			24.9010i			0.983530i	0.870728	+	0.491765i	$0.163648\pi$
$641$			24.9010i			0.983530i	−0.870728	+	0.491765i	$0.836352\pi$
$642$		0			0
$643$			38.6536i			1.52435i	0.647371	+	0.762175i	$0.275869\pi$
$643$			38.6536i			1.52435i	−0.647371	+	0.762175i	$0.724131\pi$
$644$		−	30.5217i		−	1.20272i
$645$		0			0
$646$	0.432790			0.0170279
$647$	26.9535			1.05965			0.529825	−	0.848107i	$-0.322258\pi$
$647$	26.9535			1.05965			0.529825	+	0.848107i	$0.322258\pi$
$648$		0			0
$649$			12.0111i			0.471476i
$650$	−4.21942			−0.165499
$651$		0			0
$652$	0.351323			0.0137589
$653$	−12.7298			−0.498156			−0.249078	−	0.968483i	$-0.580127\pi$
$653$	−12.7298			−0.498156			−0.249078	+	0.968483i	$0.580127\pi$
$654$		0			0
$655$		−	11.3014i		−	0.441584i
$656$	−7.62573			−0.297735
$657$		0			0
$658$			14.1265i			0.550708i
$659$			48.7716i			1.89987i	0.312442	+	0.949937i	$0.398853\pi$
$659$			48.7716i			1.89987i	−0.312442	+	0.949937i	$0.601147\pi$
$660$		0			0
$661$		−	6.80827i		−	0.264811i	−0.991196	−	0.132406i	$-0.957730\pi$
$661$		−	6.80827i		−	0.264811i	0.991196	−	0.132406i	$-0.0422701\pi$
$662$	−27.5350			−1.07018
$663$		0			0
$664$	8.51821			0.330571
$665$	−23.7020			−0.919124
$666$		0			0
$667$	−32.2027			−1.24690
$668$			3.23643i			0.125221i
$669$		0			0
$670$	17.0364			0.658174
$671$	−6.08380			−0.234863
$672$		0			0
$673$		−	29.2699i		−	1.12827i	−0.825682	−	0.564135i	$-0.809210\pi$
$673$		−	29.2699i		−	1.12827i	0.825682	−	0.564135i	$-0.190790\pi$
$674$	10.7898			0.415608
$675$		0			0
$676$	−9.66778			−0.371838
$677$	−20.2041			−0.776506			−0.388253	−	0.921553i	$-0.626921\pi$
$677$	−20.2041			−0.776506			−0.388253	+	0.921553i	$0.626921\pi$
$678$		0			0
$679$	61.1035			2.34494
$680$	0.174564			0.00669422
$681$		0			0
$682$	15.4783			0.592697
$683$		−	9.36715i		−	0.358424i	−0.983810	−	0.179212i	$-0.942645\pi$
$683$		−	9.36715i		−	0.358424i	0.983810	−	0.179212i	$-0.0573548\pi$
$684$		0			0
$685$	16.0743			0.614167
$686$	−4.82086			−0.184061
$687$		0			0
$688$			11.8407i			0.451424i
$689$	13.7968			0.525615
$690$		0			0
$691$	33.9704			1.29229			0.646147	−	0.763213i	$-0.276379\pi$
$691$	33.9704			1.29229			0.646147	+	0.763213i	$0.276379\pi$
$692$	10.9585			0.416580
$693$		0			0
$694$	−8.66967			−0.329096
$695$			30.7270i			1.16554i
$696$		0			0
$697$			0.811854i			0.0307511i
$698$		−	17.8399i		−	0.675250i
$699$		0			0
$700$	8.21927			0.310659
$701$			27.3944i			1.03467i	0.855782	+	0.517336i	$0.173076\pi$
$701$			27.3944i			1.03467i	−0.855782	+	0.517336i	$0.826924\pi$
$702$		0			0
$703$	−17.2560			−0.650822
$704$	3.33279			0.125609
$705$		0			0
$706$	−8.20751			−0.308894
$707$			62.9837i			2.36875i
$708$		0			0
$709$	−22.2760			−0.836592			−0.418296	−	0.908311i	$-0.637373\pi$
$709$	−22.2760			−0.836592			−0.418296	+	0.908311i	$0.637373\pi$
$710$	3.70326			0.138981
$711$		0			0
$712$		−	10.6622i		−	0.399583i
$713$		−	39.8637i		−	1.49291i
$714$		0			0
$715$		−	9.97547i		−	0.373061i
$716$		−	23.5698i		−	0.880847i
$717$		0			0
$718$		−	8.17173i		−	0.304966i
$719$	−23.2555			−0.867285			−0.433643	−	0.901085i	$-0.642772\pi$
$719$	−23.2555			−0.867285			−0.433643	+	0.901085i	$0.642772\pi$
$720$		0			0
$721$			39.0281i			1.45348i
$722$	2.47427			0.0920827
$723$		0			0
$724$		−	0.865062i		−	0.0321498i
$725$		−	8.67197i		−	0.322069i
$726$		0			0
$727$	−10.9416			−0.405801			−0.202900	−	0.979199i	$-0.565037\pi$
$727$	−10.9416			−0.405801			−0.202900	+	0.979199i	$0.565037\pi$
$728$	−6.49102			−0.240573
$729$		0			0
$730$	23.1202			0.855718
$731$	1.26059			0.0466248
$732$		0			0
$733$			38.7822i			1.43245i	0.697868	+	0.716226i	$0.254132\pi$
$733$			38.7822i			1.43245i	−0.697868	+	0.716226i	$0.745868\pi$
$734$	−18.3749			−0.678228
$735$		0			0
$736$		−	8.58345i		−	0.316390i
$737$		−	34.6281i		−	1.27554i
$738$		0			0
$739$	24.2954			0.893721			0.446860	−	0.894604i	$-0.352542\pi$
$739$	24.2954			0.893721			0.446860	+	0.894604i	$0.352542\pi$
$740$	−6.96014			−0.255860
$741$		0			0
$742$	−26.8756			−0.986633
$743$		−	16.4399i		−	0.603123i	−0.953447	−	0.301561i	$-0.902492\pi$
$743$		−	16.4399i		−	0.603123i	0.953447	−	0.301561i	$-0.0975078\pi$
$744$		0			0
$745$		−	17.9684i		−	0.658312i
$746$		−	12.3241i		−	0.451217i
$747$		0			0
$748$		−	0.354817i		−	0.0129734i
$749$			7.65679i			0.279773i
$750$		0			0
$751$		−	12.5724i		−	0.458774i	−0.973335	−	0.229387i	$-0.926328\pi$
$751$		−	12.5724i		−	0.458774i	0.973335	−	0.229387i	$-0.0736722\pi$
$752$			3.97272i			0.144870i
$753$		0			0
$754$			6.84853i			0.249409i
$755$	17.9966			0.654962
$756$		0			0
$757$			50.5476i			1.83718i	0.395208	+	0.918592i	$0.370672\pi$
$757$			50.5476i			1.83718i	−0.395208	+	0.918592i	$0.629328\pi$
$758$			31.9069i			1.15891i
$759$		0			0
$760$	−6.66559			−0.241786
$761$	39.0373			1.41510			0.707550	−	0.706663i	$-0.249800\pi$
$761$	39.0373			1.41510			0.707550	+	0.706663i	$0.249800\pi$
$762$		0			0
$763$	9.47070			0.342862
$764$		−	1.73227i		−	0.0626713i
$765$		0			0
$766$	−28.0008			−1.01171
$767$		−	6.57870i		−	0.237543i
$768$		0			0
$769$	25.2016			0.908792			0.454396	−	0.890800i	$-0.349855\pi$
$769$	25.2016			0.908792			0.454396	+	0.890800i	$0.349855\pi$
$770$			19.4318i			0.700274i
$771$		0			0
$772$			9.83806i			0.354079i
$773$	32.7476			1.17785			0.588924	−	0.808188i	$-0.299552\pi$
$773$	32.7476			1.17785			0.588924	+	0.808188i	$0.299552\pi$
$774$		0			0
$775$	10.7350			0.385613
$776$	17.1838			0.616862
$777$		0			0
$778$	10.2988			0.369231
$779$		−	31.0000i		−	1.11069i
$780$		0			0
$781$		−	7.52721i		−	0.269345i
$782$	−0.913815			−0.0326780
$783$		0			0
$784$	5.64426			0.201581
$785$		−	10.7987i		−	0.385422i
$786$		0			0
$787$	37.4913			1.33642			0.668210	−	0.743973i	$-0.267061\pi$
$787$	37.4913			1.33642			0.668210	+	0.743973i	$0.267061\pi$
$788$		−	15.9984i		−	0.569919i
$789$		0			0
$790$	−6.87634			−0.244649
$791$	−33.3353	+	17.8203i	−1.18527	+	0.633615i
$792$		0			0
$793$	3.33222			0.118330
$794$		−	19.0594i		−	0.676393i
$795$		0			0
$796$			22.4105i			0.794319i
$797$		−	0.593408i		−	0.0210196i	−0.999945	−	0.0105098i	$-0.996655\pi$
$797$		−	0.593408i		−	0.0210196i	0.999945	−	0.0105098i	$-0.00334543\pi$
$798$		0			0
$799$	0.422945			0.0149627
$800$	2.31146			0.0817225
$801$		0			0
$802$	−8.36675			−0.295440
$803$		−	46.9940i		−	1.65838i
$804$		0			0
$805$	50.0457			1.76388
$806$	−8.47779			−0.298617
$807$		0			0
$808$			17.7126i			0.623126i
$809$	24.3760			0.857013			0.428507	−	0.903539i	$-0.359040\pi$
$809$	24.3760			0.857013			0.428507	+	0.903539i	$0.359040\pi$
$810$		0			0
$811$		−	13.8055i		−	0.484776i	−0.970179	−	0.242388i	$-0.922069\pi$
$811$		−	13.8055i		−	0.484776i	0.970179	−	0.242388i	$-0.0779307\pi$
$812$		−	13.3407i		−	0.468166i
$813$		0			0
$814$			14.1471i			0.495857i
$815$			0.576055i			0.0201783i
$816$		0			0
$817$	−48.1348			−1.68402
$818$		−	0.359278i		−	0.0125619i
$819$		0			0
$820$		−	12.5037i		−	0.436649i
$821$	−7.86099			−0.274350			−0.137175	−	0.990547i	$-0.543802\pi$
$821$	−7.86099			−0.274350			−0.137175	+	0.990547i	$0.543802\pi$
$822$		0			0
$823$	53.7088			1.87217			0.936085	−	0.351774i	$-0.114422\pi$
$823$	53.7088			1.87217			0.936085	+	0.351774i	$0.114422\pi$
$824$			10.9757i			0.382356i
$825$		0			0
$826$			12.8150i			0.445892i
$827$	−49.1008			−1.70740			−0.853702	−	0.520762i	$-0.825648\pi$
$827$	−49.1008			−1.70740			−0.853702	+	0.520762i	$0.825648\pi$
$828$		0			0
$829$		−	25.5132i		−	0.886111i	−0.896494	−	0.443056i	$-0.853894\pi$
$829$		−	25.5132i		−	0.886111i	0.896494	−	0.443056i	$-0.146106\pi$
$830$			13.9671i			0.484805i
$831$		0			0
$832$	−1.82544			−0.0632856
$833$		−	0.600901i		−	0.0208200i
$834$		0			0
$835$	−5.30670			−0.183646
$836$			13.5484i			0.468582i
$837$		0			0
$838$		−	13.9889i		−	0.483237i
$839$			28.7881i			0.993876i	0.867786	+	0.496938i	$0.165542\pi$
$839$			28.7881i			0.993876i	−0.867786	+	0.496938i	$0.834458\pi$
$840$		0			0
$841$	14.9246			0.514640
$842$	−37.7292			−1.30024
$843$		0			0
$844$	−15.1566			−0.521712
$845$		−	15.8520i		−	0.545327i
$846$		0			0
$847$	0.382319			0.0131367
$848$	−7.55807			−0.259545
$849$		0			0
$850$		−	0.246084i		−	0.00844060i
$851$	36.4353			1.24898
$852$		0			0
$853$	34.6182			1.18530			0.592652	−	0.805459i	$-0.298081\pi$
$853$	34.6182			1.18530			0.592652	+	0.805459i	$0.298081\pi$
$854$	−6.49102			−0.222118
$855$		0			0
$856$			2.15328i			0.0735975i
$857$			2.79479i			0.0954683i	0.998860	+	0.0477342i	$0.0152000\pi$
$857$			2.79479i			0.0954683i	−0.998860	+	0.0477342i	$0.984800\pi$
$858$		0			0
$859$			17.1593i			0.585468i	0.956194	+	0.292734i	$0.0945651\pi$
$859$			17.1593i			0.585468i	−0.956194	+	0.292734i	$0.905435\pi$
$860$	−19.4150			−0.662045
$861$		0			0
$862$		−	17.6158i		−	0.599995i
$863$	−52.0008			−1.77013			−0.885065	−	0.465468i	$-0.845886\pi$
$863$	−52.0008			−1.77013			−0.885065	+	0.465468i	$0.845886\pi$
$864$		0			0
$865$			17.9684i			0.610945i
$866$			18.6072i			0.632299i
$867$		0			0
$868$	16.5144			0.560535
$869$			13.9768i			0.474130i
$870$		0			0
$871$			18.9665i			0.642655i
$872$	2.66339			0.0901939
$873$		0			0
$874$	34.8933			1.18028
$875$			42.6294i			1.44114i
$876$		0			0
$877$			3.64166i			0.122970i	0.998108	+	0.0614851i	$0.0195837\pi$
$877$			3.64166i			0.122970i	−0.998108	+	0.0614851i	$0.980416\pi$
$878$	−16.5225			−0.557607
$879$		0			0
$880$			5.46470i			0.184215i
$881$			31.1027i			1.04788i	0.851757	+	0.523938i	$0.175538\pi$
$881$			31.1027i			1.04788i	−0.851757	+	0.523938i	$0.824462\pi$
$882$		0			0
$883$		−	5.41359i		−	0.182182i	−0.995843	−	0.0910910i	$-0.970965\pi$
$883$		−	5.41359i		−	0.182182i	0.995843	−	0.0910910i	$-0.0290354\pi$
$884$			0.194340i			0.00653637i
$885$		0			0
$886$	26.1482			0.878467
$887$		−	12.4427i		−	0.417785i	−0.977939	−	0.208892i	$-0.933014\pi$
$887$		−	12.4427i		−	0.417785i	0.977939	−	0.208892i	$-0.0669858\pi$
$888$		0			0
$889$	−8.15339			−0.273456
$890$	17.4826			0.586018
$891$		0			0
$892$		−	15.6285i		−	0.523282i
$893$	−16.1498			−0.540434
$894$		0			0
$895$	38.6469			1.29182
$896$	3.55588			0.118793
$897$		0			0
$898$			36.2457i			1.20954i
$899$		−	17.4240i		−	0.581122i
$900$		0			0
$901$			0.804651i			0.0268068i
$902$	−25.4150			−0.846226
$903$		0			0
$904$	−9.37470	+	5.01149i	−0.311798	+	0.166680i
$905$	1.41842			0.0471499
$906$		0			0
$907$			3.88070i			0.128857i	0.997922	+	0.0644283i	$0.0205224\pi$
$907$			3.88070i			0.128857i	−0.997922	+	0.0644283i	$0.979478\pi$
$908$	−7.55807			−0.250823
$909$		0			0
$910$		−	10.6432i		−	0.352818i
$911$	−10.3826			−0.343992			−0.171996	−	0.985098i	$-0.555022\pi$
$911$	−10.3826			−0.343992			−0.171996	+	0.985098i	$0.555022\pi$
$912$		0			0
$913$	28.3894			0.939553
$914$		−	2.51954i		−	0.0833390i
$915$		0			0
$916$			6.60232i			0.218147i
$917$	−24.5088			−0.809352
$918$		0			0
$919$	−2.73753			−0.0903029			−0.0451514	−	0.998980i	$-0.514377\pi$
$919$	−2.73753			−0.0903029			−0.0451514	+	0.998980i	$0.514377\pi$
$920$	14.0741			0.464009
$921$		0			0
$922$	42.2245			1.39059
$923$			4.12280i			0.135704i
$924$		0			0
$925$			9.81175i			0.322608i
$926$	27.6497			0.908625
$927$		0			0
$928$		−	3.75172i		−	0.123156i
$929$	14.3668			0.471358			0.235679	−	0.971831i	$-0.424269\pi$
$929$	14.3668			0.471358			0.235679	+	0.971831i	$0.424269\pi$
$930$		0			0
$931$			22.9449i			0.751990i
$932$	−7.04394			−0.230732
$933$		0			0
$934$	22.6581			0.741395
$935$	0.581786			0.0190264
$936$		0			0
$937$			4.22723i			0.138098i	0.997613	+	0.0690488i	$0.0219964\pi$
$937$			4.22723i			0.138098i	−0.997613	+	0.0690488i	$0.978004\pi$
$938$		−	36.9459i		−	1.20633i
$939$		0			0
$940$	−6.51397			−0.212462
$941$			13.4190i			0.437447i	0.975787	+	0.218724i	$0.0701893\pi$
$941$			13.4190i			0.437447i	−0.975787	+	0.218724i	$0.929811\pi$
$942$		0			0
$943$			65.4551i			2.13151i
$944$			3.60391i			0.117297i
$945$		0			0
$946$			39.4627i			1.28304i
$947$			24.4827i			0.795582i	0.917476	+	0.397791i	$0.130223\pi$
$947$			24.4827i			0.795582i	−0.917476	+	0.397791i	$0.869777\pi$
$948$		0			0
$949$			25.7395i			0.835540i
$950$			9.39652i			0.304863i
$951$		0			0
$952$		−	0.378567i		−	0.0122694i
$953$	−52.4473			−1.69894			−0.849468	−	0.527641i	$-0.823077\pi$
$953$	−52.4473			−1.69894			−0.849468	+	0.527641i	$0.823077\pi$
$954$		0			0
$955$	2.84036			0.0919119
$956$	−12.7494			−0.412345
$957$		0			0
$958$			5.77817i			0.186684i
$959$		−	34.8594i		−	1.12567i
$960$		0			0
$961$	−9.43089			−0.304222
$962$		−	7.74866i		−	0.249827i
$963$		0			0
$964$	−21.7079			−0.699165
$965$	−16.1312			−0.519283
$966$		0			0
$967$	8.66339			0.278596			0.139298	−	0.990251i	$-0.455515\pi$
$967$	8.66339			0.278596			0.139298	+	0.990251i	$0.455515\pi$
$968$	0.107518			0.00345575
$969$		0			0
$970$			28.1759i			0.904673i
$971$		−	6.07931i		−	0.195094i	−0.995231	−	0.0975472i	$-0.968900\pi$
$971$		−	6.07931i		−	0.195094i	0.995231	−	0.0975472i	$-0.0310997\pi$
$972$		0			0
$973$	66.6360			2.13625
$974$			28.1323i			0.901415i
$975$		0			0
$976$	−1.82544			−0.0584308
$977$			18.4104i			0.589002i	0.955651	+	0.294501i	$0.0951534\pi$
$977$			18.4104i			0.589002i	−0.955651	+	0.294501i	$0.904847\pi$
$978$		0			0
$979$		−	35.5350i		−	1.13570i
$980$			9.25475i			0.295632i
$981$		0			0
$982$		−	23.7012i		−	0.756336i
$983$			56.2812i			1.79509i	0.440921	+	0.897546i	$0.354652\pi$
$983$			56.2812i			1.79509i	−0.440921	+	0.897546i	$0.645348\pi$
$984$		0			0
$985$	26.2322			0.835827
$986$	−0.399418			−0.0127201
$987$		0			0
$988$		−	7.42074i		−	0.236085i
$989$	101.634			3.23179
$990$		0			0
$991$	−7.11295			−0.225950			−0.112975	−	0.993598i	$-0.536038\pi$
$991$	−7.11295			−0.225950			−0.112975	+	0.993598i	$0.536038\pi$
$992$	4.64426			0.147455
$993$		0			0
$994$		−	8.03105i		−	0.254729i
$995$	−36.7459			−1.16492
$996$		0			0
$997$		−	34.1051i		−	1.08012i	−0.841627	−	0.540060i	$-0.818402\pi$
$997$		−	34.1051i		−	1.08012i	0.841627	−	0.540060i	$-0.181598\pi$
$998$		−	27.0101i		−	0.854991i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.2.c.h.451.8	yes	10
3.2	odd	2		2034.2.c.g.451.3	✓	10
113.112	even	2	inner	2034.2.c.h.451.3	yes	10
339.338	odd	2		2034.2.c.g.451.8	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2034.2.c.g.451.3	✓	10	3.2	odd	2
2034.2.c.g.451.8	yes	10	339.338	odd	2
2034.2.c.h.451.3	yes	10	113.112	even	2	inner
2034.2.c.h.451.8	yes	10	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2034.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.63968i q^{5} +3.55588 q^{7} +1.00000 q^{8} +1.63968i q^{10} +3.33279 q^{11} -1.82544 q^{13} +3.55588 q^{14} +1.00000 q^{16} -0.106462i q^{17} +4.06519i q^{19} +1.63968i q^{20} +3.33279 q^{22} -8.58345i q^{23} +2.31146 q^{25} -1.82544 q^{26} +3.55588 q^{28} -3.75172i q^{29} +4.64426 q^{31} +1.00000 q^{32} -0.106462i q^{34} +5.83049i q^{35} +4.24482i q^{37} +4.06519i q^{38} +1.63968i q^{40} -7.62573 q^{41} +11.8407i q^{43} +3.33279 q^{44} -8.58345i q^{46} +3.97272i q^{47} +5.64426 q^{49} +2.31146 q^{50} -1.82544 q^{52} -7.55807 q^{53} +5.46470i q^{55} +3.55588 q^{56} -3.75172i q^{58} +3.60391i q^{59} -1.82544 q^{61} +4.64426 q^{62} +1.00000 q^{64} -2.99312i q^{65} -10.3901i q^{67} -0.106462i q^{68} +5.83049i q^{70} -2.25853i q^{71} -14.1005i q^{73} +4.24482i q^{74} +4.06519i q^{76} +11.8510 q^{77} +4.19372i q^{79} +1.63968i q^{80} -7.62573 q^{82} +8.51821 q^{83} +0.174564 q^{85} +11.8407i q^{86} +3.33279 q^{88} -10.6622i q^{89} -6.49102 q^{91} -8.58345i q^{92} +3.97272i q^{94} -6.66559 q^{95} +17.1838 q^{97} +5.64426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{8} + 2 q^{11} + 2 q^{13} + 10 q^{16} + 2 q^{22} + 2 q^{26} - 8 q^{31} + 10 q^{32} - 8 q^{41} + 2 q^{44} + 2 q^{49} + 2 q^{52} + 18 q^{53} + 2 q^{61} - 8 q^{62} + 10 q^{64}+ \cdots + 2 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^8 + 2 * q^11 + 2 * q^13 + 10 * q^16 + 2 * q^22 + 2 * q^26 - 8 * q^31 + 10 * q^32 - 8 * q^41 + 2 * q^44 + 2 * q^49 + 2 * q^52 + 18 * q^53 + 2 * q^61 - 8 * q^62 + 10 * q^64 - 12 * q^77 - 8 * q^82 - 14 * q^83 + 22 * q^85 + 2 * q^88 + 18 * q^91 - 4 * q^95 + 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more