LMFDB - Embedded newform 1152.2.k.c.289.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(289,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("1152.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			289.4
Root			$0.500000 - 0.0297061i$ of defining polynomial
Character	$\chi$	$=$	1152.289
Dual form			1152.2.k.c.865.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.74912 - 1.74912i) q^{5} +2.55765i q^{7} +(0.473626 - 0.473626i) q^{11} +(-2.88784 - 2.88784i) q^{13} +6.44549 q^{17} +(4.55765 + 4.55765i) q^{19} -2.82843i q^{23} -1.11882i q^{25} +(-3.07931 - 3.07931i) q^{29} +6.55765 q^{31} +(4.47363 + 4.47363i) q^{35} +(2.72922 - 2.72922i) q^{37} -0.788632i q^{41} +(0.389604 - 0.389604i) q^{43} -2.82843 q^{47} +0.458440 q^{49} +(-2.57754 + 2.57754i) q^{53} -1.65685i q^{55} +(4.00000 - 4.00000i) q^{59} +(4.38607 + 4.38607i) q^{61} -10.1023 q^{65} +(2.11882 + 2.11882i) q^{67} +5.11529i q^{71} -14.7721i q^{73} +(1.21137 + 1.21137i) q^{77} -6.32000 q^{79} +(0.641669 + 0.641669i) q^{83} +(11.2739 - 11.2739i) q^{85} -6.31724i q^{89} +(7.38607 - 7.38607i) q^{91} +15.9437 q^{95} +12.6533 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{11} + 8 q^{19} - 16 q^{29} + 24 q^{31} + 24 q^{35} + 16 q^{37} + 8 q^{43} - 8 q^{49} + 16 q^{53} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{67} + 16 q^{77} - 24 q^{79} - 40 q^{83} + 16 q^{85}+ \cdots + 48 q^{95}+O(q^{100}) $ 8 * q - 8 * q^11 + 8 * q^19 - 16 * q^29 + 24 * q^31 + 24 * q^35 + 16 * q^37 + 8 * q^43 - 8 * q^49 + 16 * q^53 + 32 * q^59 - 16 * q^61 + 16 * q^65 + 16 * q^67 + 16 * q^77 - 24 * q^79 - 40 * q^83 + 16 * q^85 + 8 * q^91 + 48 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.74912	−	1.74912i	0.782229	−	0.782229i	−0.197977	−	0.980207i	$-0.563437\pi$
$5$	1.74912	−	1.74912i	0.782229	−	0.782229i	0.980207	+	0.197977i	$0.0634373\pi$
$6$		0			0
$7$			2.55765i			0.966700i	0.875427	+	0.483350i	$0.160580\pi$
$7$			2.55765i			0.966700i	−0.875427	+	0.483350i	$0.839420\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.473626	−	0.473626i	0.142804	−	0.142804i	−0.632091	−	0.774894i	$-0.717803\pi$
$11$	0.473626	−	0.473626i	0.142804	−	0.142804i	0.774894	+	0.632091i	$0.217803\pi$
$12$		0			0
$13$	−2.88784	−	2.88784i	−0.800943	−	0.800943i	0.182300	−	0.983243i	$-0.441646\pi$
$13$	−2.88784	−	2.88784i	−0.800943	−	0.800943i	−0.983243	+	0.182300i	$0.941646\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.44549			1.56326			0.781630	−	0.623742i	$-0.214389\pi$
$17$	6.44549			1.56326			0.781630	+	0.623742i	$0.214389\pi$
$18$		0			0
$19$	4.55765	+	4.55765i	1.04560	+	1.04560i	0.998910	+	0.0466864i	$0.0148661\pi$
$19$	4.55765	+	4.55765i	1.04560	+	1.04560i	0.0466864	+	0.998910i	$0.485134\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$		−	1.11882i		−	0.223765i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.07931	−	3.07931i	−0.571813	−	0.571813i	0.360821	−	0.932635i	$-0.382496\pi$
$29$	−3.07931	−	3.07931i	−0.571813	−	0.571813i	−0.932635	+	0.360821i	$0.882496\pi$
$30$		0			0
$31$	6.55765			1.17779			0.588894	−	0.808210i	$-0.299563\pi$
$31$	6.55765			1.17779			0.588894	+	0.808210i	$0.299563\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.47363	+	4.47363i	0.756181	+	0.756181i
$36$		0			0
$37$	2.72922	−	2.72922i	0.448681	−	0.448681i	−0.446235	−	0.894916i	$-0.647235\pi$
$37$	2.72922	−	2.72922i	0.448681	−	0.448681i	0.894916	+	0.446235i	$0.147235\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.788632i		−	0.123164i	−0.998102	−	0.0615818i	$-0.980385\pi$
$41$		−	0.788632i		−	0.123164i	0.998102	−	0.0615818i	$-0.0196145\pi$
$42$		0			0
$43$	0.389604	−	0.389604i	0.0594141	−	0.0594141i	−0.676775	−	0.736190i	$-0.736623\pi$
$43$	0.389604	−	0.389604i	0.0594141	−	0.0594141i	0.736190	+	0.676775i	$0.236623\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	0.458440			0.0654915
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.57754	+	2.57754i	−0.354053	+	0.354053i	−0.861615	−	0.507562i	$-0.830547\pi$
$53$	−2.57754	+	2.57754i	−0.354053	+	0.354053i	0.507562	+	0.861615i	$0.330547\pi$
$54$		0			0
$55$		−	1.65685i		−	0.223410i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000	−	4.00000i	0.520756	−	0.520756i	−0.397044	−	0.917800i	$-0.629964\pi$
$59$	4.00000	−	4.00000i	0.520756	−	0.520756i	0.917800	+	0.397044i	$0.129964\pi$
$60$		0			0
$61$	4.38607	+	4.38607i	0.561579	+	0.561579i	0.929756	−	0.368177i	$-0.120018\pi$
$61$	4.38607	+	4.38607i	0.561579	+	0.561579i	−0.368177	+	0.929756i	$0.620018\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.1023			−1.25304
$66$		0			0
$67$	2.11882	+	2.11882i	0.258856	+	0.258856i	0.824589	−	0.565733i	$-0.191407\pi$
$67$	2.11882	+	2.11882i	0.258856	+	0.258856i	−0.565733	+	0.824589i	$0.691407\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.11529i			0.607074i	0.952820	+	0.303537i	$0.0981676\pi$
$71$			5.11529i			0.607074i	−0.952820	+	0.303537i	$0.901832\pi$
$72$		0			0
$73$		−	14.7721i		−	1.72895i	−0.502676	−	0.864475i	$-0.667651\pi$
$73$		−	14.7721i		−	1.72895i	0.502676	−	0.864475i	$-0.332349\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.21137	+	1.21137i	0.138048	+	0.138048i
$78$		0			0
$79$	−6.32000			−0.711055			−0.355528	−	0.934666i	$-0.615699\pi$
$79$	−6.32000			−0.711055			−0.355528	+	0.934666i	$0.615699\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.641669	+	0.641669i	0.0704323	+	0.0704323i	0.741445	−	0.671013i	$-0.234141\pi$
$83$	0.641669	+	0.641669i	0.0704323	+	0.0704323i	−0.671013	+	0.741445i	$0.734141\pi$
$84$		0			0
$85$	11.2739	−	11.2739i	1.22283	−	1.22283i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.31724i		−	0.669626i	−0.942285	−	0.334813i	$-0.891327\pi$
$89$		−	6.31724i		−	0.669626i	0.942285	−	0.334813i	$-0.108673\pi$
$90$		0			0
$91$	7.38607	−	7.38607i	0.774271	−	0.774271i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.9437			1.63579
$96$		0			0
$97$	12.6533			1.28475			0.642375	−	0.766390i	$-0.277949\pi$
$97$	12.6533			1.28475			0.642375	+	0.766390i	$0.277949\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.52480	−	7.52480i	0.748745	−	0.748745i	−0.225498	−	0.974244i	$-0.572401\pi$
$101$	7.52480	−	7.52480i	0.748745	−	0.748745i	0.974244	+	0.225498i	$0.0724010\pi$
$102$		0			0
$103$		−	3.33686i		−	0.328790i	−0.986395	−	0.164395i	$-0.947433\pi$
$103$		−	3.33686i		−	0.328790i	0.986395	−	0.164395i	$-0.0525672\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.0625	+	14.0625i	−1.35948	+	1.35948i	−0.484918	+	0.874560i	$0.661151\pi$
$107$	−14.0625	+	14.0625i	−1.35948	+	1.35948i	−0.874560	+	0.484918i	$0.838849\pi$
$108$		0			0
$109$	−2.76901	−	2.76901i	−0.265224	−	0.265224i	0.561949	−	0.827172i	$-0.310052\pi$
$109$	−2.76901	−	2.76901i	−0.265224	−	0.265224i	−0.827172	+	0.561949i	$0.810052\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.23765			−0.210500			−0.105250	−	0.994446i	$-0.533564\pi$
$113$	−2.23765			−0.210500			−0.105250	+	0.994446i	$0.533564\pi$
$114$		0			0
$115$	−4.94725	−	4.94725i	−0.461334	−	0.461334i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			16.4853i			1.51120i
$120$		0			0
$121$			10.5514i			0.959214i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.78863	+	6.78863i	0.607194	+	0.607194i
$126$		0			0
$127$	12.2145			1.08386			0.541931	−	0.840423i	$-0.317693\pi$
$127$	12.2145			1.08386			0.541931	+	0.840423i	$0.317693\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.77568	−	3.77568i	−0.329883	−	0.329883i	0.522659	−	0.852542i	$-0.324940\pi$
$131$	−3.77568	−	3.77568i	−0.329883	−	0.329883i	−0.852542	+	0.522659i	$0.824940\pi$
$132$		0			0
$133$	−11.6569	+	11.6569i	−1.01078	+	1.01078i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.10587i			0.436224i	0.975924	+	0.218112i	$0.0699898\pi$
$137$			5.10587i			0.436224i	−0.975924	+	0.218112i	$0.930010\pi$
$138$		0			0
$139$	−11.7757	+	11.7757i	−0.998800	+	0.998800i	−0.999999	−	0.00119925i	$-0.999618\pi$
$139$	−11.7757	+	11.7757i	−0.998800	+	0.998800i	0.00119925	+	0.999999i	$0.499618\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.73551			−0.228755
$144$		0			0
$145$	−10.7721			−0.894578
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.90774	+	7.90774i	−0.647827	+	0.647827i	−0.952467	−	0.304640i	$-0.901464\pi$
$149$	−7.90774	+	7.90774i	−0.647827	+	0.647827i	0.304640	+	0.952467i	$0.401464\pi$
$150$		0			0
$151$			14.6506i			1.19225i	0.802893	+	0.596123i	$0.203293\pi$
$151$			14.6506i			1.19225i	−0.802893	+	0.596123i	$0.796707\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.4701	−	11.4701i	0.921300	−	0.921300i
$156$		0			0
$157$	3.15196	+	3.15196i	0.251553	+	0.251553i	0.821607	−	0.570054i	$-0.193078\pi$
$157$	3.15196	+	3.15196i	0.251553	+	0.251553i	−0.570054	+	0.821607i	$0.693078\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.23412			0.570128
$162$		0			0
$163$	−5.50490	−	5.50490i	−0.431177	−	0.431177i	0.457852	−	0.889029i	$-0.348619\pi$
$163$	−5.50490	−	5.50490i	−0.431177	−	0.431177i	−0.889029	+	0.457852i	$0.848619\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	20.1814i		−	1.56168i	−0.624730	−	0.780841i	$-0.714791\pi$
$167$		−	20.1814i		−	1.56168i	0.624730	−	0.780841i	$-0.285209\pi$
$168$		0			0
$169$			3.67923i			0.283018i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.35322	+	4.35322i	0.330969	+	0.330969i	0.852955	−	0.521985i	$-0.174808\pi$
$173$	4.35322	+	4.35322i	0.330969	+	0.330969i	−0.521985	+	0.852955i	$0.674808\pi$
$174$		0			0
$175$	2.86156			0.216313
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.2833	−	13.2833i	−0.992843	−	0.992843i	0.00713130	−	0.999975i	$-0.497730\pi$
$179$	−13.2833	−	13.2833i	−0.992843	−	0.992843i	−0.999975	+	0.00713130i	$0.997730\pi$
$180$		0			0
$181$	−6.34628	+	6.34628i	−0.471715	+	0.471715i	−0.902469	−	0.430754i	$-0.858248\pi$
$181$	−6.34628	+	6.34628i	−0.471715	+	0.471715i	0.430754	+	0.902469i	$0.358248\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	9.54745i		−	0.701943i
$186$		0			0
$187$	3.05275	−	3.05275i	0.223239	−	0.223239i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.60058			−0.405243			−0.202622	−	0.979257i	$-0.564946\pi$
$191$	−5.60058			−0.405243			−0.202622	+	0.979257i	$0.564946\pi$
$192$		0			0
$193$	−19.4514			−1.40014			−0.700071	−	0.714074i	$-0.746848\pi$
$193$	−19.4514			−1.40014			−0.700071	+	0.714074i	$0.746848\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.23793	−	1.23793i	0.0881988	−	0.0881988i	−0.661631	−	0.749830i	$-0.730135\pi$
$197$	1.23793	−	1.23793i	0.0881988	−	0.0881988i	0.749830	+	0.661631i	$0.230135\pi$
$198$		0			0
$199$			0.993710i			0.0704422i	0.999380	+	0.0352211i	$0.0112135\pi$
$199$			0.993710i			0.0704422i	−0.999380	+	0.0352211i	$0.988786\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.87579	−	7.87579i	0.552772	−	0.552772i
$204$		0			0
$205$	−1.37941	−	1.37941i	−0.0963422	−	0.0963422i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.31724			0.298630
$210$		0			0
$211$	−4.22432	−	4.22432i	−0.290814	−	0.290814i	0.546588	−	0.837402i	$-0.315927\pi$
$211$	−4.22432	−	4.22432i	−0.290814	−	0.290814i	−0.837402	+	0.546588i	$0.815927\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.36293i		−	0.0929509i
$216$		0			0
$217$			16.7721i			1.13857i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.6135	−	18.6135i	−1.25208	−	1.25208i
$222$		0			0
$223$	−23.7659			−1.59148			−0.795740	−	0.605639i	$-0.792918\pi$
$223$	−23.7659			−1.59148			−0.795740	+	0.605639i	$0.792918\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.641669	−	0.641669i	−0.0425891	−	0.0425891i	0.685492	−	0.728081i	$-0.259587\pi$
$227$	−0.641669	−	0.641669i	−0.0425891	−	0.0425891i	−0.728081	+	0.685492i	$0.759587\pi$
$228$		0			0
$229$	−5.34275	+	5.34275i	−0.353059	+	0.353059i	−0.861246	−	0.508188i	$-0.830316\pi$
$229$	−5.34275	+	5.34275i	−0.353059	+	0.353059i	0.508188	+	0.861246i	$0.330316\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			23.2271i			1.52166i	0.648954	+	0.760828i	$0.275207\pi$
$233$			23.2271i			1.52166i	−0.648954	+	0.760828i	$0.724793\pi$
$234$		0			0
$235$	−4.94725	+	4.94725i	−0.322723	+	0.322723i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.9213			−1.74140			−0.870698	−	0.491817i	$-0.836333\pi$
$239$	−26.9213			−1.74140			−0.870698	+	0.491817i	$0.836333\pi$
$240$		0			0
$241$	−10.3494			−0.666664			−0.333332	−	0.942809i	$-0.608173\pi$
$241$	−10.3494			−0.666664			−0.333332	+	0.942809i	$0.608173\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.801866	−	0.801866i	0.0512293	−	0.0512293i
$246$		0			0
$247$		−	26.3235i		−	1.67492i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.75696	+	9.75696i	−0.615854	+	0.615854i	−0.944465	−	0.328611i	$-0.893419\pi$
$251$	−9.75696	+	9.75696i	−0.615854	+	0.615854i	0.328611	+	0.944465i	$0.393419\pi$
$252$		0			0
$253$	−1.33962	−	1.33962i	−0.0842209	−	0.0842209i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.9965			−1.06021			−0.530105	−	0.847932i	$-0.677848\pi$
$257$	−16.9965			−1.06021			−0.530105	+	0.847932i	$0.677848\pi$
$258$		0			0
$259$	6.98038	+	6.98038i	0.433740	+	0.433740i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	29.9929i		−	1.84944i	−0.380643	−	0.924722i	$-0.624297\pi$
$263$		−	29.9929i		−	1.84944i	0.380643	−	0.924722i	$-0.375703\pi$
$264$		0			0
$265$			9.01686i			0.553901i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	20.6003	+	20.6003i	1.25602	+	1.25602i	0.952976	+	0.303046i	$0.0980037\pi$
$269$	20.6003	+	20.6003i	1.25602	+	1.25602i	0.303046	+	0.952976i	$0.401996\pi$
$270$		0			0
$271$	−26.6506			−1.61891			−0.809453	−	0.587184i	$-0.800236\pi$
$271$	−26.6506			−1.61891			−0.809453	+	0.587184i	$0.800236\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.529904	−	0.529904i	−0.0319544	−	0.0319544i
$276$		0			0
$277$	−12.1220	+	12.1220i	−0.728338	+	0.728338i	−0.970289	−	0.241951i	$-0.922213\pi$
$277$	−12.1220	+	12.1220i	−0.728338	+	0.728338i	0.241951	+	0.970289i	$0.422213\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.76588i			0.164999i	0.996591	+	0.0824993i	$0.0262902\pi$
$281$			2.76588i			0.164999i	−0.996591	+	0.0824993i	$0.973710\pi$
$282$		0			0
$283$	−4.48528	+	4.48528i	−0.266622	+	0.266622i	−0.827738	−	0.561115i	$-0.810372\pi$
$283$	−4.48528	+	4.48528i	−0.266622	+	0.266622i	0.561115	+	0.827738i	$0.310372\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.01704			0.119062
$288$		0			0
$289$	24.5443			1.44378
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.20793	+	8.20793i	−0.479512	+	0.479512i	−0.904976	−	0.425463i	$-0.860111\pi$
$293$	−8.20793	+	8.20793i	−0.479512	+	0.479512i	0.425463	+	0.904976i	$0.360111\pi$
$294$		0			0
$295$		−	13.9929i		−	0.814700i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.16804	+	8.16804i	−0.472370	+	0.472370i
$300$		0			0
$301$	0.996470	+	0.996470i	0.0574356	+	0.0574356i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.3435			0.878567
$306$		0			0
$307$	−10.4549	−	10.4549i	−0.596693	−	0.596693i	0.342738	−	0.939431i	$-0.388646\pi$
$307$	−10.4549	−	10.4549i	−0.596693	−	0.596693i	−0.939431	+	0.342738i	$0.888646\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.0761i			0.854885i	0.904043	+	0.427442i	$0.140585\pi$
$311$			15.0761i			0.854885i	−0.904043	+	0.427442i	$0.859415\pi$
$312$		0			0
$313$		−	23.0027i		−	1.30019i	−0.759852	−	0.650096i	$-0.774729\pi$
$313$		−	23.0027i		−	1.30019i	0.759852	−	0.650096i	$-0.225271\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.75892	−	6.75892i	−0.379618	−	0.379618i	0.491346	−	0.870964i	$-0.336505\pi$
$317$	−6.75892	−	6.75892i	−0.379618	−	0.379618i	−0.870964	+	0.491346i	$0.836505\pi$
$318$		0			0
$319$	−2.91688			−0.163314
$320$		0			0
$321$		0			0
$322$		0			0
$323$	29.3763	+	29.3763i	1.63454	+	1.63454i
$324$		0			0
$325$	−3.23099	+	3.23099i	−0.179223	+	0.179223i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	7.23412i		−	0.398830i
$330$		0			0
$331$	19.6631	−	19.6631i	1.08078	−	1.08078i	0.0843464	−	0.996436i	$-0.473120\pi$
$331$	19.6631	−	19.6631i	1.08078	−	1.08078i	0.996436	−	0.0843464i	$-0.0268802\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.41215			0.404969
$336$		0			0
$337$	3.00980			0.163954			0.0819771	−	0.996634i	$-0.473877\pi$
$337$	3.00980			0.163954			0.0819771	+	0.996634i	$0.473877\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.10587	−	3.10587i	0.168192	−	0.168192i
$342$		0			0
$343$			19.0761i			1.03001i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.27521	−	6.27521i	0.336871	−	0.336871i	−0.518317	−	0.855188i	$-0.673441\pi$
$347$	6.27521	−	6.27521i	0.336871	−	0.336871i	0.855188	+	0.518317i	$0.173441\pi$
$348$		0			0
$349$	4.74255	+	4.74255i	0.253863	+	0.253863i	0.822552	−	0.568690i	$-0.192549\pi$
$349$	4.74255	+	4.74255i	0.253863	+	0.253863i	−0.568690	+	0.822552i	$0.692549\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8.75882			0.466185			0.233093	−	0.972455i	$-0.425116\pi$
$353$	8.75882			0.466185			0.233093	+	0.972455i	$0.425116\pi$
$354$		0			0
$355$	8.94725	+	8.94725i	0.474871	+	0.474871i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			32.7917i			1.73068i	0.501184	+	0.865341i	$0.332898\pi$
$359$			32.7917i			1.73068i	−0.501184	+	0.865341i	$0.667102\pi$
$360$		0			0
$361$			22.5443i			1.18654i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−25.8382	−	25.8382i	−1.35243	−	1.35243i
$366$		0			0
$367$	20.6435			1.07758			0.538791	−	0.842439i	$-0.318881\pi$
$367$	20.6435			1.07758			0.538791	+	0.842439i	$0.318881\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.59245	−	6.59245i	−0.342263	−	0.342263i
$372$		0			0
$373$	16.6167	−	16.6167i	0.860378	−	0.860378i	−0.131004	−	0.991382i	$-0.541820\pi$
$373$	16.6167	−	16.6167i	0.860378	−	0.860378i	0.991382	+	0.131004i	$0.0418200\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			17.7851i			0.915979i
$378$		0			0
$379$	−7.77844	+	7.77844i	−0.399552	+	0.399552i	−0.878075	−	0.478523i	$-0.841172\pi$
$379$	−7.77844	+	7.77844i	−0.399552	+	0.399552i	0.478523	+	0.878075i	$0.341172\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.2037			0.879070			0.439535	−	0.898225i	$-0.355143\pi$
$383$	17.2037			0.879070			0.439535	+	0.898225i	$0.355143\pi$
$384$		0			0
$385$	4.23765			0.215971
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.8515	+	23.8515i	−1.20932	+	1.20932i	−0.238069	+	0.971248i	$0.576514\pi$
$389$	−23.8515	+	23.8515i	−1.20932	+	1.20932i	−0.971248	+	0.238069i	$0.923486\pi$
$390$		0			0
$391$		−	18.2306i		−	0.921961i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.0544	+	11.0544i	−0.556208	+	0.556208i
$396$		0			0
$397$	10.2673	+	10.2673i	0.515299	+	0.515299i	0.916145	−	0.400847i	$-0.131284\pi$
$397$	10.2673	+	10.2673i	0.515299	+	0.515299i	−0.400847	+	0.916145i	$0.631284\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−32.2274			−1.60936			−0.804681	−	0.593708i	$-0.797663\pi$
$401$	−32.2274			−1.60936			−0.804681	+	0.593708i	$0.797663\pi$
$402$		0			0
$403$	−18.9374	−	18.9374i	−0.943341	−	0.943341i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.58526i		−	0.128146i
$408$		0			0
$409$			11.5702i			0.572110i	0.958213	+	0.286055i	$0.0923440\pi$
$409$			11.5702i			0.572110i	−0.958213	+	0.286055i	$0.907656\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10.2306	+	10.2306i	0.503414	+	0.503414i
$414$		0			0
$415$	2.24471			0.110188
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.74717	+	6.74717i	0.329621	+	0.329621i	0.852442	−	0.522822i	$-0.175121\pi$
$419$	6.74717	+	6.74717i	0.329621	+	0.329621i	−0.522822	+	0.852442i	$0.675121\pi$
$420$		0			0
$421$	17.2239	−	17.2239i	0.839443	−	0.839443i	−0.149343	−	0.988785i	$-0.547716\pi$
$421$	17.2239	−	17.2239i	0.839443	−	0.839443i	0.988785	+	0.149343i	$0.0477158\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	7.21137i		−	0.349803i
$426$		0			0
$427$	−11.2180	+	11.2180i	−0.542879	+	0.542879i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−40.7088			−1.96087			−0.980437	−	0.196832i	$-0.936935\pi$
$431$	−40.7088			−1.96087			−0.980437	+	0.196832i	$0.936935\pi$
$432$		0			0
$433$	7.31371			0.351474			0.175737	−	0.984437i	$-0.443769\pi$
$433$	7.31371			0.351474			0.175737	+	0.984437i	$0.443769\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.8910	−	12.8910i	0.616659	−	0.616659i
$438$		0			0
$439$			17.7122i			0.845356i	0.906280	+	0.422678i	$0.138910\pi$
$439$			17.7122i			0.845356i	−0.906280	+	0.422678i	$0.861090\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.6944	−	15.6944i	0.745664	−	0.745664i	−0.227997	−	0.973662i	$-0.573218\pi$
$443$	15.6944	−	15.6944i	0.745664	−	0.745664i	0.973662	+	0.227997i	$0.0732178\pi$
$444$		0			0
$445$	−11.0496	−	11.0496i	−0.523801	−	0.523801i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	28.3400			1.33745			0.668723	−	0.743511i	$-0.266841\pi$
$449$	28.3400			1.33745			0.668723	+	0.743511i	$0.266841\pi$
$450$		0			0
$451$	−0.373517	−	0.373517i	−0.0175882	−	0.0175882i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	25.8382i		−	1.21131i
$456$		0			0
$457$		−	17.3396i		−	0.811113i	−0.914070	−	0.405557i	$-0.867078\pi$
$457$		−	17.3396i		−	0.811113i	0.914070	−	0.405557i	$-0.132922\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.69284	−	1.69284i	−0.0788434	−	0.0788434i	0.666585	−	0.745429i	$-0.267755\pi$
$461$	−1.69284	−	1.69284i	−0.0788434	−	0.0788434i	−0.745429	+	0.666585i	$0.767755\pi$
$462$		0			0
$463$	2.70238			0.125590			0.0627951	−	0.998026i	$-0.479999\pi$
$463$	2.70238			0.125590			0.0627951	+	0.998026i	$0.479999\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.1136	−	17.1136i	−0.791924	−	0.791924i	0.189883	−	0.981807i	$-0.439189\pi$
$467$	−17.1136	−	17.1136i	−0.791924	−	0.791924i	−0.981807	+	0.189883i	$0.939189\pi$
$468$		0			0
$469$	−5.41921	+	5.41921i	−0.250236	+	0.250236i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	0.369053i		−	0.0169691i
$474$		0			0
$475$	5.09921	−	5.09921i	0.233968	−	0.233968i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	22.2251			1.01549			0.507745	−	0.861508i	$-0.330479\pi$
$479$	22.2251			1.01549			0.507745	+	0.861508i	$0.330479\pi$
$480$		0			0
$481$	−15.7631			−0.718735
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.1322	−	22.1322i	1.00497	−	1.00497i
$486$		0			0
$487$		−	13.9839i		−	0.633672i	−0.948480	−	0.316836i	$-0.897380\pi$
$487$		−	13.9839i		−	0.633672i	0.948480	−	0.316836i	$-0.102620\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.23412	−	7.23412i	0.326471	−	0.326471i	−0.524772	−	0.851243i	$-0.675849\pi$
$491$	7.23412	−	7.23412i	0.326471	−	0.326471i	0.851243	+	0.524772i	$0.175849\pi$
$492$		0			0
$493$	−19.8476	−	19.8476i	−0.893893	−	0.893893i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.0831			−0.586858
$498$		0			0
$499$	−2.59078	−	2.59078i	−0.115979	−	0.115979i	0.646735	−	0.762715i	$-0.276134\pi$
$499$	−2.59078	−	2.59078i	−0.115979	−	0.115979i	−0.762715	+	0.646735i	$0.776134\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			39.6443i			1.76765i	0.467817	+	0.883825i	$0.345041\pi$
$503$			39.6443i			1.76765i	−0.467817	+	0.883825i	$0.654959\pi$
$504$		0			0
$505$		−	26.3235i		−	1.17138i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	20.2875	+	20.2875i	0.899229	+	0.899229i	0.995368	−	0.0961393i	$-0.0306494\pi$
$509$	20.2875	+	20.2875i	0.899229	+	0.899229i	−0.0961393	+	0.995368i	$0.530649\pi$
$510$		0			0
$511$	37.7819			1.67137
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.83655	−	5.83655i	−0.257189	−	0.257189i
$516$		0			0
$517$	−1.33962	+	1.33962i	−0.0589162	+	0.0589162i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	23.1784i		−	1.01546i	−0.861515	−	0.507732i	$-0.830484\pi$
$521$		−	23.1784i		−	1.01546i	0.861515	−	0.507732i	$-0.169516\pi$
$522$		0			0
$523$	5.78550	−	5.78550i	0.252982	−	0.252982i	−0.569210	−	0.822192i	$-0.692751\pi$
$523$	5.78550	−	5.78550i	0.252982	−	0.252982i	0.822192	+	0.569210i	$0.192751\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	42.2672			1.84119
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.27744	+	2.27744i	−0.0986470	+	0.0986470i
$534$		0			0
$535$			49.1941i			2.12685i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.217129	−	0.217129i	0.00935241	−	0.00935241i
$540$		0			0
$541$	−4.55175	−	4.55175i	−0.195695	−	0.195695i	0.602457	−	0.798152i	$-0.294189\pi$
$541$	−4.55175	−	4.55175i	−0.195695	−	0.195695i	−0.798152	+	0.602457i	$0.794189\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.68667			−0.414931
$546$		0			0
$547$	27.7355	+	27.7355i	1.18588	+	1.18588i	0.978195	+	0.207689i	$0.0665942\pi$
$547$	27.7355	+	27.7355i	1.18588	+	1.18588i	0.207689	+	0.978195i	$0.433406\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	28.0688i		−	1.19577i
$552$		0			0
$553$		−	16.1643i		−	0.687377i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.17538	−	1.17538i	−0.0498026	−	0.0498026i	0.681767	−	0.731569i	$-0.261212\pi$
$557$	−1.17538	−	1.17538i	−0.0498026	−	0.0498026i	−0.731569	+	0.681767i	$0.761212\pi$
$558$		0			0
$559$	−2.25023			−0.0951745
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.7346	−	28.7346i	−1.21102	−	1.21102i	−0.970692	−	0.240326i	$-0.922746\pi$
$563$	−28.7346	−	28.7346i	−1.21102	−	1.21102i	−0.240326	−	0.970692i	$-0.577254\pi$
$564$		0			0
$565$	−3.91391	+	3.91391i	−0.164659	+	0.164659i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	27.0004i		−	1.13191i	−0.824435	−	0.565957i	$-0.808507\pi$
$569$		−	27.0004i		−	1.13191i	0.824435	−	0.565957i	$-0.191493\pi$
$570$		0			0
$571$	14.8284	−	14.8284i	0.620550	−	0.620550i	−0.325122	−	0.945672i	$-0.605405\pi$
$571$	14.8284	−	14.8284i	0.620550	−	0.620550i	0.945672	+	0.325122i	$0.105405\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.16451			−0.131969
$576$		0			0
$577$	−37.6372			−1.56686			−0.783429	−	0.621481i	$-0.786531\pi$
$577$	−37.6372			−1.56686			−0.783429	+	0.621481i	$0.786531\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.64116	+	1.64116i	−0.0680869	+	0.0680869i
$582$		0			0
$583$			2.44158i			0.101120i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−31.2574	+	31.2574i	−1.29013	+	1.29013i	−0.355429	+	0.934703i	$0.615665\pi$
$587$	−31.2574	+	31.2574i	−1.29013	+	1.29013i	−0.934703	+	0.355429i	$0.884335\pi$
$588$		0			0
$589$	29.8874	+	29.8874i	1.23149	+	1.23149i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.59611			0.147675			0.0738373	−	0.997270i	$-0.476475\pi$
$593$	3.59611			0.147675			0.0738373	+	0.997270i	$0.476475\pi$
$594$		0			0
$595$	28.8347	+	28.8347i	1.18211	+	1.18211i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	22.0296i		−	0.900104i	−0.893002	−	0.450052i	$-0.851405\pi$
$599$		−	22.0296i		−	0.900104i	0.893002	−	0.450052i	$-0.148595\pi$
$600$		0			0
$601$		−	10.7721i		−	0.439405i	−0.975567	−	0.219703i	$-0.929491\pi$
$601$		−	10.7721i		−	0.439405i	0.975567	−	0.219703i	$-0.0705087\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	18.4556	+	18.4556i	0.750325	+	0.750325i
$606$		0			0
$607$	5.47453			0.222204			0.111102	−	0.993809i	$-0.464562\pi$
$607$	5.47453			0.222204			0.111102	+	0.993809i	$0.464562\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.16804	+	8.16804i	0.330444	+	0.330444i
$612$		0			0
$613$	10.5049	−	10.5049i	0.424289	−	0.424289i	−0.462389	−	0.886677i	$-0.653007\pi$
$613$	10.5049	−	10.5049i	0.424289	−	0.424289i	0.886677	+	0.462389i	$0.153007\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22.2235i		−	0.894686i	−0.894363	−	0.447343i	$-0.852370\pi$
$617$		−	22.2235i		−	0.894686i	0.894363	−	0.447343i	$-0.147630\pi$
$618$		0			0
$619$	−11.6398	+	11.6398i	−0.467843	+	0.467843i	−0.901215	−	0.433372i	$-0.857324\pi$
$619$	−11.6398	+	11.6398i	−0.467843	+	0.467843i	0.433372	+	0.901215i	$0.357324\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.1573			0.647327
$624$		0			0
$625$	29.3424			1.17369
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.5912	−	17.5912i	0.701405	−	0.701405i
$630$		0			0
$631$		−	4.06977i		−	0.162015i	−0.996713	−	0.0810075i	$-0.974186\pi$
$631$		−	4.06977i		−	0.162015i	0.996713	−	0.0810075i	$-0.0258138\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	21.3646	−	21.3646i	0.847828	−	0.847828i
$636$		0			0
$637$	−1.32390	−	1.32390i	−0.0524549	−	0.0524549i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.41958			0.332553			0.166277	−	0.986079i	$-0.446826\pi$
$641$	8.41958			0.332553			0.166277	+	0.986079i	$0.446826\pi$
$642$		0			0
$643$	7.37275	+	7.37275i	0.290753	+	0.290753i	0.837378	−	0.546625i	$-0.184088\pi$
$643$	7.37275	+	7.37275i	0.290753	+	0.290753i	−0.546625	+	0.837378i	$0.684088\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			11.6132i			0.456560i	0.973595	+	0.228280i	$0.0733102\pi$
$647$			11.6132i			0.456560i	−0.973595	+	0.228280i	$0.926690\pi$
$648$		0			0
$649$		−	3.78901i		−	0.148731i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.93049	−	1.93049i	−0.0755458	−	0.0755458i	0.668324	−	0.743870i	$-0.267012\pi$
$653$	−1.93049	−	1.93049i	−0.0755458	−	0.0755458i	−0.743870	+	0.668324i	$0.767012\pi$
$654$		0			0
$655$	−13.2082			−0.516088
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.3102	+	22.3102i	0.869081	+	0.869081i	0.992371	−	0.123290i	$-0.0393444\pi$
$659$	22.3102	+	22.3102i	0.869081	+	0.869081i	−0.123290	+	0.992371i	$0.539344\pi$
$660$		0			0
$661$	−10.7033	+	10.7033i	−0.416311	+	0.416311i	−0.883930	−	0.467619i	$-0.845112\pi$
$661$	−10.7033	+	10.7033i	−0.416311	+	0.416311i	0.467619	+	0.883930i	$0.345112\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			40.7784i			1.58132i
$666$		0			0
$667$	−8.70960	+	8.70960i	−0.337237	+	0.337237i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.15472			0.160391
$672$		0			0
$673$	−20.6345			−0.795401			−0.397700	−	0.917515i	$-0.630192\pi$
$673$	−20.6345			−0.795401			−0.397700	+	0.917515i	$0.630192\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	26.8246	−	26.8246i	1.03095	−	1.03095i	0.0314484	−	0.999505i	$-0.489988\pi$
$677$	26.8246	−	26.8246i	1.03095	−	1.03095i	0.999505	−	0.0314484i	$-0.0100120\pi$
$678$		0			0
$679$			32.3627i			1.24197i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.9026	−	12.9026i	0.493705	−	0.493705i	−0.415766	−	0.909472i	$-0.636487\pi$
$683$	12.9026	−	12.9026i	0.493705	−	0.493705i	0.909472	+	0.415766i	$0.136487\pi$
$684$		0			0
$685$	8.93077	+	8.93077i	0.341227	+	0.341227i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.8871			0.567152
$690$		0			0
$691$	21.3923	+	21.3923i	0.813803	+	0.813803i	0.985202	−	0.171399i	$-0.0548286\pi$
$691$	21.3923	+	21.3923i	0.813803	+	0.813803i	−0.171399	+	0.985202i	$0.554829\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			41.1941i			1.56258i
$696$		0			0
$697$		−	5.08312i		−	0.192537i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.2040	+	14.2040i	0.536479	+	0.536479i	0.922493	−	0.386014i	$-0.126148\pi$
$701$	14.2040	+	14.2040i	0.536479	+	0.536479i	−0.386014	+	0.922493i	$0.626148\pi$
$702$		0			0
$703$	24.8776			0.938278
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.2458	+	19.2458i	0.723812	+	0.723812i
$708$		0			0
$709$	29.5474	−	29.5474i	1.10968	−	1.10968i	0.116485	−	0.993192i	$-0.462837\pi$
$709$	29.5474	−	29.5474i	1.10968	−	1.10968i	0.993192	−	0.116485i	$-0.0371626\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	18.5478i		−	0.694622i
$714$		0			0
$715$	−4.78473	+	4.78473i	−0.178939	+	0.178939i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−28.3683			−1.05796			−0.528979	−	0.848635i	$-0.677425\pi$
$719$	−28.3683			−1.05796			−0.528979	+	0.848635i	$0.677425\pi$
$720$		0			0
$721$	8.53450			0.317841
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.44521	+	3.44521i	−0.127952	+	0.127952i
$726$		0			0
$727$		−	20.4843i		−	0.759722i	−0.925044	−	0.379861i	$-0.875972\pi$
$727$		−	20.4843i		−	0.759722i	0.925044	−	0.379861i	$-0.124028\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.51119	−	2.51119i	0.0928797	−	0.0928797i
$732$		0			0
$733$	−33.9961	−	33.9961i	−1.25567	−	1.25567i	−0.953138	−	0.302536i	$-0.902167\pi$
$733$	−33.9961	−	33.9961i	−1.25567	−	1.25567i	−0.302536	−	0.953138i	$-0.597833\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.00706			0.0739310
$738$		0			0
$739$	−15.1645	−	15.1645i	−0.557836	−	0.557836i	0.370855	−	0.928691i	$-0.379065\pi$
$739$	−15.1645	−	15.1645i	−0.557836	−	0.557836i	−0.928691	+	0.370855i	$0.879065\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			2.17431i			0.0797677i	0.999204	+	0.0398839i	$0.0126988\pi$
$743$			2.17431i			0.0797677i	−0.999204	+	0.0398839i	$0.987301\pi$
$744$		0			0
$745$			27.6631i			1.01350i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−35.9670	−	35.9670i	−1.31421	−	1.31421i
$750$		0			0
$751$	29.8980			1.09099			0.545497	−	0.838113i	$-0.316341\pi$
$751$	29.8980			1.09099			0.545497	+	0.838113i	$0.316341\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25.6256	+	25.6256i	0.932610	+	0.932610i
$756$		0			0
$757$	15.3294	−	15.3294i	0.557157	−	0.557157i	−0.371340	−	0.928497i	$-0.621101\pi$
$757$	15.3294	−	15.3294i	0.557157	−	0.557157i	0.928497	+	0.371340i	$0.121101\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			4.29449i			0.155675i	0.996966	+	0.0778375i	$0.0248015\pi$
$761$			4.29449i			0.155675i	−0.996966	+	0.0778375i	$0.975198\pi$
$762$		0			0
$763$	7.08216	−	7.08216i	0.256392	−	0.256392i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23.1027			−0.834191
$768$		0			0
$769$	33.8819			1.22181			0.610907	−	0.791703i	$-0.290805\pi$
$769$	33.8819			1.22181			0.610907	+	0.791703i	$0.290805\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−35.0230	+	35.0230i	−1.25969	+	1.25969i	−0.308450	+	0.951240i	$0.599810\pi$
$773$	−35.0230	+	35.0230i	−1.25969	+	1.25969i	−0.951240	+	0.308450i	$0.900190\pi$
$774$		0			0
$775$		−	7.33686i		−	0.263548i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.59431	−	3.59431i	0.128779	−	0.128779i
$780$		0			0
$781$	2.42274	+	2.42274i	0.0866923	+	0.0866923i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.0263			0.393545
$786$		0			0
$787$	−24.1090	−	24.1090i	−0.859393	−	0.859393i	0.131873	−	0.991267i	$-0.457901\pi$
$787$	−24.1090	−	24.1090i	−0.859393	−	0.859393i	−0.991267	+	0.131873i	$0.957901\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	5.72312i		−	0.203491i
$792$		0			0
$793$		−	25.3326i		−	0.899585i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.7722	−	28.7722i	−1.01917	−	1.01917i	−0.999813	−	0.0193524i	$-0.993840\pi$
$797$	−28.7722	−	28.7722i	−1.01917	−	1.01917i	−0.0193524	−	0.999813i	$-0.506160\pi$
$798$		0			0
$799$	−18.2306			−0.644952
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.99647	−	6.99647i	−0.246900	−	0.246900i
$804$		0			0
$805$	12.6533	−	12.6533i	0.445971	−	0.445971i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			10.9926i			0.386478i	0.981152	+	0.193239i	$0.0618993\pi$
$809$			10.9926i			0.386478i	−0.981152	+	0.193239i	$0.938101\pi$
$810$		0			0
$811$	15.0259	−	15.0259i	0.527630	−	0.527630i	−0.392235	−	0.919865i	$-0.628298\pi$
$811$	15.0259	−	15.0259i	0.527630	−	0.527630i	0.919865	+	0.392235i	$0.128298\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.2574			−0.674558
$816$		0			0
$817$	3.55136			0.124246
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.2536	+	21.2536i	−0.741756	+	0.741756i	−0.972916	−	0.231159i	$-0.925748\pi$
$821$	−21.2536	+	21.2536i	−0.741756	+	0.741756i	0.231159	+	0.972916i	$0.425748\pi$
$822$		0			0
$823$		−	55.0851i		−	1.92015i	−0.279751	−	0.960073i	$-0.590252\pi$
$823$		−	55.0851i		−	1.92015i	0.279751	−	0.960073i	$-0.409748\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.4290	+	24.4290i	−0.849480	+	0.849480i	−0.990068	−	0.140588i	$-0.955101\pi$
$827$	−24.4290	+	24.4290i	−0.849480	+	0.849480i	0.140588	+	0.990068i	$0.455101\pi$
$828$		0			0
$829$	21.9235	+	21.9235i	0.761436	+	0.761436i	0.976582	−	0.215146i	$-0.0690227\pi$
$829$	21.9235	+	21.9235i	0.761436	+	0.761436i	−0.215146	+	0.976582i	$0.569023\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.95487			0.102380
$834$		0			0
$835$	−35.2996	−	35.2996i	−1.22159	−	1.22159i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			5.14195i			0.177520i	0.996053	+	0.0887599i	$0.0282904\pi$
$839$			5.14195i			0.177520i	−0.996053	+	0.0887599i	$0.971710\pi$
$840$		0			0
$841$		−	10.0357i		−	0.346059i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.43541	+	6.43541i	0.221385	+	0.221385i
$846$		0			0
$847$	−26.9867			−0.927272
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.71940	−	7.71940i	−0.264618	−	0.264618i
$852$		0			0
$853$	−13.0857	+	13.0857i	−0.448046	+	0.448046i	−0.894704	−	0.446659i	$-0.852614\pi$
$853$	−13.0857	+	13.0857i	−0.448046	+	0.448046i	0.446659	+	0.894704i	$0.352614\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			22.8878i			0.781833i	0.920426	+	0.390916i	$0.127842\pi$
$857$			22.8878i			0.781833i	−0.920426	+	0.390916i	$0.872158\pi$
$858$		0			0
$859$	25.1225	−	25.1225i	0.857169	−	0.857169i	−0.133834	−	0.991004i	$-0.542729\pi$
$859$	25.1225	−	25.1225i	0.857169	−	0.857169i	0.991004	+	0.133834i	$0.0427290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	43.9296			1.49538			0.747691	−	0.664047i	$-0.231163\pi$
$863$	43.9296			1.49538			0.747691	+	0.664047i	$0.231163\pi$
$864$		0			0
$865$	15.2286			0.517788
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.99331	+	2.99331i	−0.101541	+	0.101541i
$870$		0			0
$871$		−	12.2376i		−	0.414657i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.3629	+	17.3629i	−0.586974	+	0.586974i
$876$		0			0
$877$	−15.2575	−	15.2575i	−0.515208	−	0.515208i	0.400910	−	0.916117i	$-0.368694\pi$
$877$	−15.2575	−	15.2575i	−0.515208	−	0.515208i	−0.916117	+	0.400910i	$0.868694\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−21.6686			−0.730035			−0.365018	−	0.931001i	$-0.618937\pi$
$881$	−21.6686			−0.730035			−0.365018	+	0.931001i	$0.618937\pi$
$882$		0			0
$883$	0.0590385	+	0.0590385i	0.00198680	+	0.00198680i	0.708099	−	0.706113i	$-0.249553\pi$
$883$	0.0590385	+	0.0590385i	0.00198680	+	0.00198680i	−0.706113	+	0.708099i	$0.749553\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	30.8043i		−	1.03431i	−0.855892	−	0.517154i	$-0.826991\pi$
$887$		−	30.8043i		−	1.03431i	0.855892	−	0.517154i	$-0.173009\pi$
$888$		0			0
$889$			31.2404i			1.04777i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.8910	−	12.8910i	−0.431380	−	0.431380i
$894$		0			0
$895$	−46.4682			−1.55326
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.1930	−	20.1930i	−0.673475	−	0.673475i
$900$		0			0
$901$	−16.6135	+	16.6135i	−0.553477	+	0.553477i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			22.2008i			0.737979i
$906$		0			0
$907$	−35.0170	+	35.0170i	−1.16272	+	1.16272i	−0.178844	+	0.983878i	$0.557236\pi$
$907$	−35.0170	+	35.0170i	−1.16272	+	1.16272i	−0.983878	+	0.178844i	$0.942764\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.0829331			−0.00274770			−0.00137385	−	0.999999i	$-0.500437\pi$
$911$	−0.0829331			−0.00274770			−0.00137385	+	0.999999i	$0.500437\pi$
$912$		0			0
$913$	0.607822			0.0201160
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.65685	−	9.65685i	0.318897	−	0.318897i
$918$		0			0
$919$			20.1161i			0.663568i	0.943355	+	0.331784i	$0.107650\pi$
$919$			20.1161i			0.663568i	−0.943355	+	0.331784i	$0.892350\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	14.7721	−	14.7721i	0.486231	−	0.486231i
$924$		0			0
$925$	−3.05352	−	3.05352i	−0.100399	−	0.100399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−8.55098			−0.280549			−0.140274	−	0.990113i	$-0.544798\pi$
$929$	−8.55098			−0.280549			−0.140274	+	0.990113i	$0.544798\pi$
$930$		0			0
$931$	2.08941	+	2.08941i	0.0684776	+	0.0684776i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	10.6792i		−	0.349248i
$936$		0			0
$937$			33.5780i			1.09695i	0.836168	+	0.548473i	$0.184791\pi$
$937$			33.5780i			1.09695i	−0.836168	+	0.548473i	$0.815209\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.48463	−	8.48463i	−0.276591	−	0.276591i	0.555156	−	0.831747i	$-0.312659\pi$
$941$	−8.48463	−	8.48463i	−0.276591	−	0.276591i	−0.831747	+	0.555156i	$0.812659\pi$
$942$		0			0
$943$	−2.23059			−0.0726380
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.8571	+	17.8571i	0.580277	+	0.580277i	0.934979	−	0.354702i	$-0.115418\pi$
$947$	17.8571	+	17.8571i	0.580277	+	0.580277i	−0.354702	+	0.934979i	$0.615418\pi$
$948$		0			0
$949$	−42.6596	+	42.6596i	−1.38479	+	1.38479i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			3.86469i			0.125190i	0.998039	+	0.0625948i	$0.0199376\pi$
$953$			3.86469i			0.125190i	−0.998039	+	0.0625948i	$0.980062\pi$
$954$		0			0
$955$	−9.79607	+	9.79607i	−0.316993	+	0.316993i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.0590			−0.421698
$960$		0			0
$961$	12.0027			0.387185
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−34.0228	+	34.0228i	−1.09523	+	1.09523i
$966$		0			0
$967$			37.8714i			1.21786i	0.793224	+	0.608930i	$0.208401\pi$
$967$			37.8714i			1.21786i	−0.793224	+	0.608930i	$0.791599\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.64873	−	2.64873i	0.0850017	−	0.0850017i	−0.663327	−	0.748329i	$-0.730856\pi$
$971$	2.64873	−	2.64873i	0.0850017	−	0.0850017i	0.748329	+	0.663327i	$0.230856\pi$
$972$		0			0
$973$	−30.1180	−	30.1180i	−0.965540	−	0.965540i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−17.6530			−0.564768			−0.282384	−	0.959301i	$-0.591125\pi$
$977$	−17.6530			−0.564768			−0.282384	+	0.959301i	$0.591125\pi$
$978$		0			0
$979$	−2.99201	−	2.99201i	−0.0956250	−	0.0956250i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	22.3557i		−	0.713035i	−0.934289	−	0.356518i	$-0.883964\pi$
$983$		−	22.3557i		−	0.713035i	0.934289	−	0.356518i	$-0.116036\pi$
$984$		0			0
$985$		−	4.33057i		−	0.137983i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.10197	−	1.10197i	−0.0350405	−	0.0350405i
$990$		0			0
$991$	17.8769			0.567878			0.283939	−	0.958842i	$-0.408359\pi$
$991$	17.8769			0.567878			0.283939	+	0.958842i	$0.408359\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.73812	+	1.73812i	0.0551020	+	0.0551020i
$996$		0			0
$997$	−4.28610	+	4.28610i	−0.135742	+	0.135742i	−0.771713	−	0.635971i	$-0.780600\pi$
$997$	−4.28610	+	4.28610i	−0.135742	+	0.135742i	0.635971	+	0.771713i	$0.280600\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.k.c.289.4		8
3.2	odd	2		384.2.j.b.289.3		8
4.3	odd	2		1152.2.k.f.289.4		8
8.3	odd	2		576.2.k.b.145.1		8
8.5	even	2		144.2.k.b.109.3		8
12.11	even	2		384.2.j.a.289.1		8
16.3	odd	4		576.2.k.b.433.1		8
16.5	even	4	inner	1152.2.k.c.865.4		8
16.11	odd	4		1152.2.k.f.865.4		8
16.13	even	4		144.2.k.b.37.3		8
24.5	odd	2		48.2.j.a.13.2	✓	8
24.11	even	2		192.2.j.a.145.4		8
32.5	even	8		9216.2.a.bo.1.3		4
32.11	odd	8		9216.2.a.x.1.2		4
32.21	even	8		9216.2.a.y.1.2		4
32.27	odd	8		9216.2.a.bn.1.3		4
48.5	odd	4		384.2.j.b.97.3		8
48.11	even	4		384.2.j.a.97.1		8
48.29	odd	4		48.2.j.a.37.2	yes	8
48.35	even	4		192.2.j.a.49.4		8
96.5	odd	8		3072.2.a.i.1.2		4
96.11	even	8		3072.2.a.n.1.3		4
96.29	odd	8		3072.2.d.f.1537.6		8
96.35	even	8		3072.2.d.i.1537.2		8
96.53	odd	8		3072.2.a.t.1.3		4
96.59	even	8		3072.2.a.o.1.2		4
96.77	odd	8		3072.2.d.f.1537.3		8
96.83	even	8		3072.2.d.i.1537.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.2	✓	8	24.5	odd	2
48.2.j.a.37.2	yes	8	48.29	odd	4
144.2.k.b.37.3		8	16.13	even	4
144.2.k.b.109.3		8	8.5	even	2
192.2.j.a.49.4		8	48.35	even	4
192.2.j.a.145.4		8	24.11	even	2
384.2.j.a.97.1		8	48.11	even	4
384.2.j.a.289.1		8	12.11	even	2
384.2.j.b.97.3		8	48.5	odd	4
384.2.j.b.289.3		8	3.2	odd	2
576.2.k.b.145.1		8	8.3	odd	2
576.2.k.b.433.1		8	16.3	odd	4
1152.2.k.c.289.4		8	1.1	even	1	trivial
1152.2.k.c.865.4		8	16.5	even	4	inner
1152.2.k.f.289.4		8	4.3	odd	2
1152.2.k.f.865.4		8	16.11	odd	4
3072.2.a.i.1.2		4	96.5	odd	8
3072.2.a.n.1.3		4	96.11	even	8
3072.2.a.o.1.2		4	96.59	even	8
3072.2.a.t.1.3		4	96.53	odd	8
3072.2.d.f.1537.3		8	96.77	odd	8
3072.2.d.f.1537.6		8	96.29	odd	8
3072.2.d.i.1537.2		8	96.35	even	8
3072.2.d.i.1537.7		8	96.83	even	8
9216.2.a.x.1.2		4	32.11	odd	8
9216.2.a.y.1.2		4	32.21	even	8
9216.2.a.bn.1.3		4	32.27	odd	8
9216.2.a.bo.1.3		4	32.5	even	8

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.74912 - 1.74912i) q^{5} +2.55765i q^{7} +(0.473626 - 0.473626i) q^{11} +(-2.88784 - 2.88784i) q^{13} +6.44549 q^{17} +(4.55765 + 4.55765i) q^{19} -2.82843i q^{23} -1.11882i q^{25} +(-3.07931 - 3.07931i) q^{29} +6.55765 q^{31} +(4.47363 + 4.47363i) q^{35} +(2.72922 - 2.72922i) q^{37} -0.788632i q^{41} +(0.389604 - 0.389604i) q^{43} -2.82843 q^{47} +0.458440 q^{49} +(-2.57754 + 2.57754i) q^{53} -1.65685i q^{55} +(4.00000 - 4.00000i) q^{59} +(4.38607 + 4.38607i) q^{61} -10.1023 q^{65} +(2.11882 + 2.11882i) q^{67} +5.11529i q^{71} -14.7721i q^{73} +(1.21137 + 1.21137i) q^{77} -6.32000 q^{79} +(0.641669 + 0.641669i) q^{83} +(11.2739 - 11.2739i) q^{85} -6.31724i q^{89} +(7.38607 - 7.38607i) q^{91} +15.9437 q^{95} +12.6533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{11} + 8 q^{19} - 16 q^{29} + 24 q^{31} + 24 q^{35} + 16 q^{37} + 8 q^{43} - 8 q^{49} + 16 q^{53} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{67} + 16 q^{77} - 24 q^{79} - 40 q^{83} + 16 q^{85}+ \cdots + 48 q^{95}+O(q^{100}) \) 8 * q - 8 * q^11 + 8 * q^19 - 16 * q^29 + 24 * q^31 + 24 * q^35 + 16 * q^37 + 8 * q^43 - 8 * q^49 + 16 * q^53 + 32 * q^59 - 16 * q^61 + 16 * q^65 + 16 * q^67 + 16 * q^77 - 24 * q^79 - 40 * q^83 + 16 * q^85 + 8 * q^91 + 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more