LMFDB - Embedded newform 1512.2.bl.d.1025.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(593,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("1512.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1512 = 2^{3} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1512.bl (of order $6$ , degree $2$ , 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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.81094542259068665856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{14} + 9x^{12} - 8x^{10} + 44x^{8} - 32x^{6} + 144x^{4} - 64x^{2} + 256$ x^16 - x^14 + 9x^12 - 8x^10 + 44x^8 - 32x^6 + 144x^4 - 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$2^{16}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.4
Root			$-1.12988 - 0.850516i$ of defining polynomial
Character	$\chi$	$=$	1512.1025
Dual form			1512.2.bl.d.593.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.310225 - 0.537326i) q^{5} +(-1.44109 + 2.21884i) q^{7} +(-1.91649 - 1.10649i) q^{11} -0.963711i q^{13} +(0.653486 - 1.13187i) q^{17} +(-1.57941 + 0.911873i) q^{19} +(5.27562 - 3.04588i) q^{23} +(2.30752 - 3.99674i) q^{25} -6.68969i q^{29} +(3.48866 + 2.01418i) q^{31} +(1.63930 + 0.0859913i) q^{35} +(0.165402 + 0.286485i) q^{37} +11.7190 q^{41} -2.18909 q^{43} +(-1.27394 - 2.20652i) q^{47} +(-2.84654 - 6.39509i) q^{49} +(0.985818 + 0.569162i) q^{53} +1.37304i q^{55} +(6.20629 - 10.7496i) q^{59} +(0.603098 - 0.348199i) q^{61} +(-0.517827 + 0.298967i) q^{65} +(-3.61028 + 6.25319i) q^{67} -7.40207i q^{71} +(-2.17396 - 1.25513i) q^{73} +(5.21696 - 2.65786i) q^{77} +(4.52429 + 7.83630i) q^{79} +8.85562 q^{83} -0.810911 q^{85} +(-4.20928 - 7.29069i) q^{89} +(2.13832 + 1.38879i) q^{91} +(0.979946 + 0.565772i) q^{95} -12.4027i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 4 q^{7} + 12 q^{19} - 12 q^{31} + 16 q^{37} - 16 q^{43} - 28 q^{49} - 60 q^{61} - 4 q^{67} + 12 q^{73} - 32 q^{79} - 32 q^{85} + 24 q^{91}+O(q^{100})$ 16 * q + 4 * q^7 + 12 * q^19 - 12 * q^31 + 16 * q^37 - 16 * q^43 - 28 * q^49 - 60 * q^61 - 4 * q^67 + 12 * q^73 - 32 * q^79 - 32 * q^85 + 24 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$ .

$n$	$757$	$785$	$1081$	$1135$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.310225	−	0.537326i	−0.138737	−	0.240299i	0.788282	−	0.615314i	$-0.210971\pi$
$5$	−0.310225	−	0.537326i	−0.138737	−	0.240299i	−0.927019	+	0.375015i	$0.877638\pi$
$6$		0			0
$7$	−1.44109	+	2.21884i	−0.544679	+	0.838644i
$7$	−1.44109	+	2.21884i	−0.544679	+	0.838644i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.91649	−	1.10649i	−0.577845	−	0.333619i	0.182432	−	0.983219i	$-0.441603\pi$
$11$	−1.91649	−	1.10649i	−0.577845	−	0.333619i	−0.760276	+	0.649600i	$0.774936\pi$
$12$		0			0
$13$		−	0.963711i		−	0.267285i	−0.991030	−	0.133643i	$-0.957333\pi$
$13$		−	0.963711i		−	0.267285i	0.991030	−	0.133643i	$-0.0426674\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.653486	−	1.13187i	0.158494	−	0.274519i	−0.775832	−	0.630939i	$-0.782670\pi$
$17$	0.653486	−	1.13187i	0.158494	−	0.274519i	0.934326	+	0.356421i	$0.116003\pi$
$18$		0			0
$19$	−1.57941	+	0.911873i	−0.362342	+	0.209198i	−0.670107	−	0.742264i	$-0.733752\pi$
$19$	−1.57941	+	0.911873i	−0.362342	+	0.209198i	0.307766	+	0.951462i	$0.400419\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.27562	−	3.04588i	1.10004	−	0.635110i	0.163810	−	0.986492i	$-0.447622\pi$
$23$	5.27562	−	3.04588i	1.10004	−	0.635110i	0.936232	+	0.351382i	$0.114288\pi$
$24$		0			0
$25$	2.30752	−	3.99674i	0.461504	−	0.799349i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.68969i		−	1.24224i	−0.783714	−	0.621122i	$-0.786677\pi$
$29$		−	6.68969i		−	1.24224i	0.783714	−	0.621122i	$-0.213323\pi$
$30$		0			0
$31$	3.48866	+	2.01418i	0.626582	+	0.361757i	0.779427	−	0.626493i	$-0.215510\pi$
$31$	3.48866	+	2.01418i	0.626582	+	0.361757i	−0.152845	+	0.988250i	$0.548844\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.63930	+	0.0859913i	0.277093	+	0.0145352i
$36$		0			0
$37$	0.165402	+	0.286485i	0.0271919	+	0.0470978i	0.879301	−	0.476266i	$-0.158010\pi$
$37$	0.165402	+	0.286485i	0.0271919	+	0.0470978i	−0.852109	+	0.523364i	$0.824677\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.7190			1.83020			0.915102	−	0.403221i	$-0.132110\pi$
$41$	11.7190			1.83020			0.915102	+	0.403221i	$0.132110\pi$
$42$		0			0
$43$	−2.18909			−0.333833			−0.166916	−	0.985971i	$-0.553381\pi$
$43$	−2.18909			−0.333833			−0.166916	+	0.985971i	$0.553381\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.27394	−	2.20652i	−0.185823	−	0.321854i	0.758031	−	0.652219i	$-0.226162\pi$
$47$	−1.27394	−	2.20652i	−0.185823	−	0.321854i	−0.943853	+	0.330364i	$0.892828\pi$
$48$		0			0
$49$	−2.84654	−	6.39509i	−0.406649	−	0.913585i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.985818	+	0.569162i	0.135413	+	0.0781804i	0.566176	−	0.824284i	$-0.308422\pi$
$53$	0.985818	+	0.569162i	0.135413	+	0.0781804i	−0.430763	+	0.902465i	$0.641756\pi$
$54$		0			0
$55$			1.37304i			0.185141i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.20629	−	10.7496i	0.807991	−	1.39948i	−0.106263	−	0.994338i	$-0.533889\pi$
$59$	6.20629	−	10.7496i	0.807991	−	1.39948i	0.914254	−	0.405143i	$-0.132778\pi$
$60$		0			0
$61$	0.603098	−	0.348199i	0.0772188	−	0.0445823i	−0.460893	−	0.887455i	$-0.652471\pi$
$61$	0.603098	−	0.348199i	0.0772188	−	0.0445823i	0.538112	+	0.842873i	$0.319138\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.517827	+	0.298967i	−0.0642285	+	0.0370823i
$66$		0			0
$67$	−3.61028	+	6.25319i	−0.441066	+	0.763949i	−0.997769	−	0.0667628i	$-0.978733\pi$
$67$	−3.61028	+	6.25319i	−0.441066	+	0.763949i	0.556703	+	0.830712i	$0.312066\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.40207i		−	0.878464i	−0.898374	−	0.439232i	$-0.855251\pi$
$71$		−	7.40207i		−	0.878464i	0.898374	−	0.439232i	$-0.144749\pi$
$72$		0			0
$73$	−2.17396	−	1.25513i	−0.254442	−	0.146902i	0.367354	−	0.930081i	$-0.380264\pi$
$73$	−2.17396	−	1.25513i	−0.254442	−	0.146902i	−0.621797	+	0.783179i	$0.713597\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	5.21696	−	2.65786i	0.594527	−	0.302891i
$78$		0			0
$79$	4.52429	+	7.83630i	0.509023	+	0.881653i	0.999945	+	0.0104498i	$0.00332634\pi$
$79$	4.52429	+	7.83630i	0.509023	+	0.881653i	−0.490923	+	0.871203i	$0.663340\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.85562			0.972031			0.486016	−	0.873950i	$-0.338450\pi$
$83$	8.85562			0.972031			0.486016	+	0.873950i	$0.338450\pi$
$84$		0			0
$85$	−0.810911			−0.0879556
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.20928	−	7.29069i	−0.446183	−	0.772812i	0.551951	−	0.833877i	$-0.313884\pi$
$89$	−4.20928	−	7.29069i	−0.446183	−	0.772812i	−0.998134	+	0.0610650i	$0.980550\pi$
$90$		0			0
$91$	2.13832	+	1.38879i	0.224157	+	0.145585i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.979946	+	0.565772i	0.100540	+	0.0580470i
$96$		0			0
$97$		−	12.4027i		−	1.25930i	−0.776877	−	0.629652i	$-0.783197\pi$
$97$		−	12.4027i		−	1.25930i	0.776877	−	0.629652i	$-0.216803\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.82480	+	3.16064i	−0.181574	+	0.314496i	−0.942417	−	0.334441i	$-0.891453\pi$
$101$	−1.82480	+	3.16064i	−0.181574	+	0.314496i	0.760843	+	0.648936i	$0.224786\pi$
$102$		0			0
$103$	3.12562	−	1.80458i	0.307976	−	0.177810i	−0.338044	−	0.941130i	$-0.609765\pi$
$103$	3.12562	−	1.80458i	0.307976	−	0.177810i	0.646020	+	0.763320i	$0.276432\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.1599	+	7.02050i	−1.17554	+	0.678697i	−0.954978	−	0.296676i	$-0.904122\pi$
$107$	−12.1599	+	7.02050i	−1.17554	+	0.678697i	−0.220560	+	0.975373i	$0.570788\pi$
$108$		0			0
$109$	−0.681140	+	1.17977i	−0.0652413	+	0.113001i	−0.896801	−	0.442434i	$-0.854115\pi$
$109$	−0.681140	+	1.17977i	−0.0652413	+	0.113001i	0.831560	+	0.555435i	$0.187448\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	16.1266i		−	1.51707i	−0.651635	−	0.758533i	$-0.725917\pi$
$113$		−	16.1266i		−	1.51707i	0.651635	−	0.758533i	$-0.274083\pi$
$114$		0			0
$115$	−3.27326	−	1.88982i	−0.305233	−	0.176226i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.56972	+	3.08111i	0.143896	+	0.282444i
$120$		0			0
$121$	−3.05137	−	5.28513i	−0.277397	−	0.480466i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.96566			−0.533585
$126$		0			0
$127$	8.16641			0.724652			0.362326	−	0.932051i	$-0.381983\pi$
$127$	8.16641			0.724652			0.362326	+	0.932051i	$0.381983\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.50391	−	4.33690i	−0.218768	−	0.378916i	0.735664	−	0.677347i	$-0.236870\pi$
$131$	−2.50391	−	4.33690i	−0.218768	−	0.378916i	−0.954431	+	0.298430i	$0.903537\pi$
$132$		0			0
$133$	0.252762	−	4.81855i	0.0219172	−	0.417822i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.08103	−	2.35619i	−0.348666	−	0.201303i	0.315431	−	0.948948i	$-0.397851\pi$
$137$	−4.08103	−	2.35619i	−0.348666	−	0.201303i	−0.664098	+	0.747646i	$0.731184\pi$
$138$		0			0
$139$			10.3716i			0.879709i	0.898069	+	0.439854i	$0.144970\pi$
$139$			10.3716i			0.879709i	−0.898069	+	0.439854i	$0.855030\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.06633	+	1.84695i	−0.0891714	+	0.154449i
$144$		0			0
$145$	−3.59454	+	2.07531i	−0.298511	+	0.172345i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.01781	−	4.62908i	0.656844	−	0.379229i	−0.134229	−	0.990950i	$-0.542856\pi$
$149$	8.01781	−	4.62908i	0.656844	−	0.379229i	0.791074	+	0.611721i	$0.209522\pi$
$150$		0			0
$151$	−8.62359	+	14.9365i	−0.701778	+	1.21552i	0.266064	+	0.963955i	$0.414277\pi$
$151$	−8.62359	+	14.9365i	−0.701778	+	1.21552i	−0.967842	+	0.251560i	$0.919056\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.49940i		−	0.200756i
$156$		0			0
$157$	3.59454	+	2.07531i	0.286876	+	0.165628i	0.636532	−	0.771250i	$-0.280368\pi$
$157$	3.59454	+	2.07531i	0.286876	+	0.165628i	−0.349656	+	0.936878i	$0.613702\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.844287	+	16.0951i	−0.0665391	+	1.26848i
$162$		0			0
$163$	−6.38217	−	11.0542i	−0.499890	−	0.865835i	0.500110	−	0.865962i	$-0.333293\pi$
$163$	−6.38217	−	11.0542i	−0.499890	−	0.865835i	−1.00000		0.000126653i	$0.999960\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20.5086			1.58700			0.793501	−	0.608569i	$-0.208256\pi$
$167$	20.5086			1.58700			0.793501	+	0.608569i	$0.208256\pi$
$168$		0			0
$169$	12.0713			0.928559
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.81088	+	6.60064i	0.289736	+	0.501837i	0.973747	−	0.227635i	$-0.0730992\pi$
$173$	3.81088	+	6.60064i	0.289736	+	0.501837i	−0.684011	+	0.729472i	$0.739766\pi$
$174$		0			0
$175$	5.54282	+	10.8797i	0.418997	+	0.822427i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.78757	−	3.34146i	−0.432583	−	0.249752i	0.267863	−	0.963457i	$-0.413682\pi$
$179$	−5.78757	−	3.34146i	−0.432583	−	0.249752i	−0.700447	+	0.713705i	$0.747016\pi$
$180$		0			0
$181$		−	5.28785i		−	0.393043i	−0.980500	−	0.196521i	$-0.937036\pi$
$181$		−	5.28785i		−	0.393043i	0.980500	−	0.196521i	$-0.0629645\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.102624	−	0.177749i	0.00754504	−	0.0130684i
$186$		0			0
$187$	−2.50480	+	1.44615i	−0.183169	+	0.105753i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.45434	+	3.72641i	−0.467020	+	0.269634i	−0.714991	−	0.699133i	$-0.753569\pi$
$191$	−6.45434	+	3.72641i	−0.467020	+	0.269634i	0.247972	+	0.968767i	$0.420236\pi$
$192$		0			0
$193$	9.74522	−	16.8792i	0.701476	−	1.21499i	−0.266472	−	0.963843i	$-0.585858\pi$
$193$	9.74522	−	16.8792i	0.701476	−	1.21499i	0.967948	−	0.251150i	$-0.0808086\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.62061i		−	0.186711i	−0.995633	−	0.0933555i	$-0.970241\pi$
$197$		−	2.62061i		−	0.186711i	0.995633	−	0.0933555i	$-0.0297593\pi$
$198$		0			0
$199$	7.10031	+	4.09937i	0.503328	+	0.290596i	0.730087	−	0.683355i	$-0.239480\pi$
$199$	7.10031	+	4.09937i	0.503328	+	0.290596i	−0.226759	+	0.973951i	$0.572813\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.8434	+	9.64042i	1.04180	+	0.676625i
$204$		0			0
$205$	−3.63554	−	6.29694i	−0.253917	−	0.439797i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.03591			0.279170
$210$		0			0
$211$	−19.6095			−1.34997			−0.674986	−	0.737831i	$-0.735850\pi$
$211$	−19.6095			−1.34997			−0.674986	+	0.737831i	$0.735850\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.679111	+	1.17625i	0.0463150	+	0.0802199i
$216$		0			0
$217$	−9.49661	+	4.83819i	−0.644672	+	0.328438i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.09080	−	0.629771i	−0.0733749	−	0.0423630i
$222$		0			0
$223$			23.9924i			1.60665i	0.595539	+	0.803326i	$0.296938\pi$
$223$			23.9924i			1.60665i	−0.595539	+	0.803326i	$0.703062\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.8243	+	22.2123i	−0.851177	+	1.47428i	0.0289703	+	0.999580i	$0.490777\pi$
$227$	−12.8243	+	22.2123i	−0.851177	+	1.47428i	−0.880147	+	0.474701i	$0.842556\pi$
$228$		0			0
$229$	8.65507	−	4.99701i	0.571943	−	0.330212i	−0.185982	−	0.982553i	$-0.559547\pi$
$229$	8.65507	−	4.99701i	0.571943	−	0.330212i	0.757925	+	0.652342i	$0.226213\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−22.4800	+	12.9788i	−1.47271	+	0.850271i	−0.999529	−	0.0306937i	$-0.990228\pi$
$233$	−22.4800	+	12.9788i	−1.47271	+	0.850271i	−0.473183	+	0.880964i	$0.656895\pi$
$234$		0			0
$235$	−0.790414	+	1.36904i	−0.0515609	+	0.0893062i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	18.1554i		−	1.17438i	−0.809450	−	0.587189i	$-0.800235\pi$
$239$		−	18.1554i		−	1.17438i	0.809450	−	0.587189i	$-0.199765\pi$
$240$		0			0
$241$	3.44245	+	1.98750i	0.221748	+	0.128026i	0.606759	−	0.794886i	$-0.292469\pi$
$241$	3.44245	+	1.98750i	0.221748	+	0.128026i	−0.385011	+	0.922912i	$0.625802\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.55318	+	3.51344i	−0.163117	+	0.224465i
$246$		0			0
$247$	0.878782	+	1.52210i	0.0559156	+	0.0968486i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.403459			−0.0254661			−0.0127331	−	0.999919i	$-0.504053\pi$
$251$	−0.403459			−0.0254661			−0.0127331	+	0.999919i	$0.504053\pi$
$252$		0			0
$253$	−13.4809			−0.847538
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.22515	+	12.5143i	0.450692	+	0.780622i	0.998429	−	0.0560288i	$-0.0178439\pi$
$257$	7.22515	+	12.5143i	0.450692	+	0.780622i	−0.547737	+	0.836651i	$0.684511\pi$
$258$		0			0
$259$	−0.874023	−	0.0458477i	−0.0543091	−	0.00284884i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.3400	−	9.43390i	−1.00757	−	0.581719i	−0.0970889	−	0.995276i	$-0.530953\pi$
$263$	−16.3400	−	9.43390i	−1.00757	−	0.581719i	−0.910478	+	0.413556i	$0.864286\pi$
$264$		0			0
$265$		−	0.706274i		−	0.0433861i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.65868	+	8.06908i	−0.284045	+	0.491980i	−0.972377	−	0.233415i	$-0.925010\pi$
$269$	−4.65868	+	8.06908i	−0.284045	+	0.491980i	0.688332	+	0.725396i	$0.258343\pi$
$270$		0			0
$271$	23.0776	−	13.3239i	1.40187	−	0.809368i	0.407282	−	0.913302i	$-0.366477\pi$
$271$	23.0776	−	13.3239i	1.40187	−	0.809368i	0.994584	+	0.103934i	$0.0331432\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.84470	+	5.10649i	−0.533355	+	0.307933i
$276$		0			0
$277$	9.63457	−	16.6876i	0.578885	−	1.00266i	−0.416722	−	0.909034i	$-0.636821\pi$
$277$	9.63457	−	16.6876i	0.578885	−	1.00266i	0.995608	−	0.0936250i	$-0.0298455\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1.20740i			0.0720276i	0.999351	+	0.0360138i	$0.0114660\pi$
$281$			1.20740i			0.0720276i	−0.999351	+	0.0360138i	$0.988534\pi$
$282$		0			0
$283$	−13.2610	−	7.65622i	−0.788282	−	0.455115i	0.0510754	−	0.998695i	$-0.483735\pi$
$283$	−13.2610	−	7.65622i	−0.788282	−	0.455115i	−0.839357	+	0.543580i	$0.817068\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.8881	+	26.0027i	−0.996875	+	1.53489i
$288$		0			0
$289$	7.64591	+	13.2431i	0.449760	+	0.779006i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.9303			1.10592			0.552960	−	0.833208i	$-0.313498\pi$
$293$	18.9303			1.10592			0.552960	+	0.833208i	$0.313498\pi$
$294$		0			0
$295$	−7.70139			−0.448392
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.93535	−	5.08417i	−0.169755	−	0.294025i
$300$		0			0
$301$	3.15467	−	4.85725i	0.181832	−	0.279967i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.374192	−	0.216040i	−0.0214262	−	0.0123704i
$306$		0			0
$307$		−	19.4828i		−	1.11194i	−0.831202	−	0.555971i	$-0.812346\pi$
$307$		−	19.4828i		−	1.11194i	0.831202	−	0.555971i	$-0.187654\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.06853	+	15.7072i	−0.514229	+	0.890671i	0.485634	+	0.874162i	$0.338589\pi$
$311$	−9.06853	+	15.7072i	−0.514229	+	0.890671i	−0.999864	+	0.0165093i	$0.994745\pi$
$312$		0			0
$313$	−16.2088	+	9.35817i	−0.916177	+	0.528955i	−0.882413	−	0.470475i	$-0.844083\pi$
$313$	−16.2088	+	9.35817i	−0.916177	+	0.528955i	−0.0337634	+	0.999430i	$0.510749\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−29.4904	+	17.0263i	−1.65634	+	0.956290i	−0.681962	+	0.731388i	$0.738873\pi$
$317$	−29.4904	+	17.0263i	−1.65634	+	0.956290i	−0.974381	+	0.224903i	$0.927794\pi$
$318$		0			0
$319$	−7.40207	+	12.8208i	−0.414436	+	0.717824i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.38358i			0.132626i
$324$		0			0
$325$	−3.85170	−	2.22378i	−0.213654	−	0.123353i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.73178	+	0.353122i	0.371135	+	0.0194683i
$330$		0			0
$331$	−8.76511	−	15.1816i	−0.481774	−	0.834457i	0.518007	−	0.855376i	$-0.326674\pi$
$331$	−8.76511	−	15.1816i	−0.481774	−	0.834457i	−0.999781	+	0.0209192i	$0.993341\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.48000			0.244769
$336$		0			0
$337$	9.53700			0.519513			0.259757	−	0.965674i	$-0.416358\pi$
$337$	9.53700			0.519513			0.259757	+	0.965674i	$0.416358\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.45733	−	7.72032i	−0.241378	−	0.418079i
$342$		0			0
$343$	18.2918	+	2.89985i	0.987666	+	0.156577i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.58768	−	2.64870i	−0.246280	−	0.142190i	0.371780	−	0.928321i	$-0.378748\pi$
$347$	−4.58768	−	2.64870i	−0.246280	−	0.142190i	−0.618060	+	0.786131i	$0.712081\pi$
$348$		0			0
$349$		−	20.7945i		−	1.11310i	−0.830813	−	0.556552i	$-0.812124\pi$
$349$		−	20.7945i		−	1.11310i	0.830813	−	0.556552i	$-0.187876\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.79161	+	16.9596i	−0.521155	+	0.902666i	0.478543	+	0.878064i	$0.341165\pi$
$353$	−9.79161	+	16.9596i	−0.521155	+	0.902666i	−0.999697	+	0.0246021i	$0.992168\pi$
$354$		0			0
$355$	−3.97732	+	2.29631i	−0.211094	+	0.121875i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−19.1760	+	11.0713i	−1.01207	+	0.584319i	−0.911797	−	0.410641i	$-0.865305\pi$
$359$	−19.1760	+	11.0713i	−1.01207	+	0.584319i	−0.100273	+	0.994960i	$0.531972\pi$
$360$		0			0
$361$	−7.83697	+	13.5740i	−0.412472	+	0.714423i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.55750i			0.0815231i
$366$		0			0
$367$	18.1684	+	10.4895i	0.948382	+	0.547549i	0.892578	−	0.450893i	$-0.148894\pi$
$367$	18.1684	+	10.4895i	0.948382	+	0.547549i	0.0558039	+	0.998442i	$0.482228\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.68353	+	1.36716i	−0.139322	+	0.0709797i
$372$		0			0
$373$	13.6507	+	23.6437i	0.706805	+	1.22422i	0.966036	+	0.258407i	$0.0831975\pi$
$373$	13.6507	+	23.6437i	0.706805	+	1.22422i	−0.259231	+	0.965815i	$0.583469\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.44693			−0.332034
$378$		0			0
$379$	1.96174			0.100768			0.0503840	−	0.998730i	$-0.483955\pi$
$379$	1.96174			0.100768			0.0503840	+	0.998730i	$0.483955\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.10367	+	10.5719i	0.311883	+	0.540197i	0.978770	−	0.204962i	$-0.0657070\pi$
$383$	6.10367	+	10.5719i	0.311883	+	0.540197i	−0.666887	+	0.745159i	$0.732374\pi$
$384$		0			0
$385$	−3.04657	−	1.97867i	−0.155267	−	0.100842i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.8845	−	13.7897i	−1.21099	−	0.699167i	−0.248017	−	0.968756i	$-0.579779\pi$
$389$	−23.8845	−	13.7897i	−1.21099	−	0.699167i	−0.962976	+	0.269589i	$0.913112\pi$
$390$		0			0
$391$		−	7.96175i		−	0.402643i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.80710	−	4.86204i	0.141240	−	0.244636i
$396$		0			0
$397$	−28.7563	+	16.6025i	−1.44324	+	0.833255i	−0.998064	−	0.0621887i	$-0.980192\pi$
$397$	−28.7563	+	16.6025i	−1.44324	+	0.833255i	−0.445175	+	0.895443i	$0.646859\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.1061	+	8.72153i	−0.754364	+	0.435532i	−0.827269	−	0.561807i	$-0.810107\pi$
$401$	−15.1061	+	8.72153i	−0.754364	+	0.435532i	0.0729044	+	0.997339i	$0.476773\pi$
$402$		0			0
$403$	1.94109	−	3.36206i	0.0966924	−	0.167476i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.732061i		−	0.0362869i
$408$		0			0
$409$	−14.3281	−	8.27231i	−0.708477	−	0.409039i	0.102020	−	0.994782i	$-0.467469\pi$
$409$	−14.3281	−	8.27231i	−0.708477	−	0.409039i	−0.810497	+	0.585743i	$0.800803\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	14.9079	+	29.2619i	0.733571	+	1.43988i
$414$		0			0
$415$	−2.74724	−	4.75836i	−0.134857	−	0.233578i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	22.7781			1.11278			0.556392	−	0.830920i	$-0.312185\pi$
$419$	22.7781			1.11278			0.556392	+	0.830920i	$0.312185\pi$
$420$		0			0
$421$	15.6863			0.764504			0.382252	−	0.924058i	$-0.375149\pi$
$421$	15.6863			0.764504			0.382252	+	0.924058i	$0.375149\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.01586	−	5.22363i	−0.146291	−	0.253383i
$426$		0			0
$427$	−0.0965172	+	1.83996i	−0.00467079	+	0.0890421i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.9241	+	12.6579i	1.05604	+	0.609707i	0.924335	−	0.381581i	$-0.124620\pi$
$431$	21.9241	+	12.6579i	1.05604	+	0.609707i	0.131709	+	0.991288i	$0.457954\pi$
$432$		0			0
$433$			37.9831i			1.82535i	0.408686	+	0.912675i	$0.365987\pi$
$433$			37.9831i			1.82535i	−0.408686	+	0.912675i	$0.634013\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.55491	+	9.62139i	−0.265727	+	0.460253i
$438$		0			0
$439$	7.42813	−	4.28864i	0.354526	−	0.204685i	−0.312151	−	0.950032i	$-0.601050\pi$
$439$	7.42813	−	4.28864i	0.354526	−	0.204685i	0.666677	+	0.745347i	$0.267716\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.94380	+	5.16371i	−0.424933	+	0.245335i	−0.697186	−	0.716891i	$-0.745565\pi$
$443$	−8.94380	+	5.16371i	−0.424933	+	0.245335i	0.272253	+	0.962226i	$0.412231\pi$
$444$		0			0
$445$	−2.61165	+	4.52351i	−0.123804	+	0.214435i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	20.5336i		−	0.969041i	−0.874780	−	0.484520i	$-0.838994\pi$
$449$		−	20.5336i		−	0.969041i	0.874780	−	0.484520i	$-0.161006\pi$
$450$		0			0
$451$	−22.4594	−	12.9670i	−1.05757	−	0.610591i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.0828707	−	1.57981i	0.00388504	−	0.0740629i
$456$		0			0
$457$	12.4978	+	21.6469i	0.584623	+	1.01260i	0.994922	+	0.100646i	$0.0320910\pi$
$457$	12.4978	+	21.6469i	0.584623	+	1.01260i	−0.410299	+	0.911951i	$0.634576\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	42.6072			1.98442			0.992208	−	0.124596i	$-0.0397634\pi$
$461$	42.6072			1.98442			0.992208	+	0.124596i	$0.0397634\pi$
$462$		0			0
$463$	30.1942			1.40325			0.701623	−	0.712549i	$-0.252459\pi$
$463$	30.1942			1.40325			0.701623	+	0.712549i	$0.252459\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.90935	−	10.2353i	−0.273452	−	0.473633i	0.696291	−	0.717759i	$-0.254832\pi$
$467$	−5.90935	−	10.2353i	−0.273452	−	0.473633i	−0.969743	+	0.244126i	$0.921499\pi$
$468$		0			0
$469$	−8.67213	−	17.0220i	−0.400442	−	0.786005i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.19538	+	2.42220i	0.192904	+	0.111373i
$474$		0			0
$475$			8.41666i			0.386183i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.12346	+	7.14204i	−0.188406	+	0.326328i	−0.944719	−	0.327882i	$-0.893665\pi$
$479$	−4.12346	+	7.14204i	−0.188406	+	0.326328i	0.756313	+	0.654210i	$0.226999\pi$
$480$		0			0
$481$	0.276088	−	0.159400i	0.0125885	−	0.00726800i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.66430	+	3.84763i	−0.302610	+	0.174712i
$486$		0			0
$487$	12.5643	−	21.7620i	0.569344	−	0.986132i	−0.427287	−	0.904116i	$-0.640531\pi$
$487$	12.5643	−	21.7620i	0.569344	−	0.986132i	0.996631	−	0.0820163i	$-0.0261359\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.80970i		−	0.352447i	−0.984350	−	0.176223i	$-0.943612\pi$
$491$		−	7.80970i		−	0.352447i	0.984350	−	0.176223i	$-0.0563881\pi$
$492$		0			0
$493$	−7.57187	−	4.37162i	−0.341020	−	0.196888i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.4240	+	10.6670i	0.736718	+	0.478481i
$498$		0			0
$499$	−14.6425	−	25.3616i	−0.655489	−	1.13534i	−0.981771	−	0.190068i	$-0.939129\pi$
$499$	−14.6425	−	25.3616i	−0.655489	−	1.13534i	0.326282	−	0.945273i	$-0.394204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.32335			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$503$	−5.32335			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$504$		0			0
$505$	2.26439			0.100764
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.7275	−	25.5087i	−0.652784	−	1.13066i	−0.982444	−	0.186556i	$-0.940268\pi$
$509$	−14.7275	−	25.5087i	−0.652784	−	1.13066i	0.329660	−	0.944100i	$-0.393066\pi$
$510$		0			0
$511$	5.91780	−	3.01491i	0.261788	−	0.133372i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.93929	−	1.11965i	−0.0854553	−	0.0493376i
$516$		0			0
$517$			5.63838i			0.247976i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.43563	−	16.3430i	0.413382	−	0.715999i	−0.581875	−	0.813278i	$-0.697681\pi$
$521$	9.43563	−	16.3430i	0.413382	−	0.715999i	0.995257	+	0.0972791i	$0.0310139\pi$
$522$		0			0
$523$	22.1439	−	12.7848i	0.968285	−	0.559039i	0.0695719	−	0.997577i	$-0.477837\pi$
$523$	22.1439	−	12.7848i	0.968285	−	0.559039i	0.898713	+	0.438537i	$0.144503\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.55958	−	2.63247i	0.198618	−	0.114672i
$528$		0			0
$529$	7.05476	−	12.2192i	0.306729	−	0.531270i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	11.2938i		−	0.489187i
$534$		0			0
$535$	7.54459	+	4.35587i	0.326181	+	0.188321i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.62071	+	15.4058i	−0.0698091	+	0.663575i
$540$		0			0
$541$	16.3471	+	28.3141i	0.702819	+	1.21732i	0.967473	+	0.252974i	$0.0814088\pi$
$541$	16.3471	+	28.3141i	0.702819	+	1.21732i	−0.264654	+	0.964343i	$0.585258\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.845227			0.0362055
$546$		0			0
$547$	−41.0972			−1.75719			−0.878594	−	0.477570i	$-0.841518\pi$
$547$	−41.0972			−1.75719			−0.878594	+	0.477570i	$0.841518\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.10015	+	10.5658i	0.259875	+	0.450117i
$552$		0			0
$553$	−23.9074	−	1.25409i	−1.01665	−	0.0533292i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.412224	−	0.237997i	−0.0174665	−	0.0100843i	0.491241	−	0.871024i	$-0.336543\pi$
$557$	−0.412224	−	0.237997i	−0.0174665	−	0.0100843i	−0.508708	+	0.860939i	$0.669877\pi$
$558$		0			0
$559$			2.10965i			0.0892286i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−7.43275	+	12.8739i	−0.313253	+	0.542570i	−0.979065	−	0.203550i	$-0.934752\pi$
$563$	−7.43275	+	12.8739i	−0.313253	+	0.542570i	0.665812	+	0.746120i	$0.268085\pi$
$564$		0			0
$565$	−8.66525	+	5.00288i	−0.364550	+	0.210473i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.0050	−	6.93112i	0.503278	−	0.290567i	−0.226788	−	0.973944i	$-0.572823\pi$
$569$	12.0050	−	6.93112i	0.503278	−	0.290567i	0.730066	+	0.683377i	$0.239489\pi$
$570$		0			0
$571$	−1.96776	+	3.40826i	−0.0823482	+	0.142631i	−0.904258	−	0.426986i	$-0.859575\pi$
$571$	−1.96776	+	3.40826i	−0.0823482	+	0.142631i	0.821910	+	0.569617i	$0.192909\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	28.1137i		−	1.17242i
$576$		0			0
$577$	26.6457	+	15.3839i	1.10928	+	0.640440i	0.938642	−	0.344894i	$-0.112085\pi$
$577$	26.6457	+	15.3839i	1.10928	+	0.640440i	0.170634	+	0.985335i	$0.445419\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.7617	+	19.6493i	−0.529445	+	0.815188i
$582$		0			0
$583$	−1.25954	−	2.18159i	−0.0521649	−	0.0903523i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	26.7214			1.10291			0.551455	−	0.834205i	$-0.314073\pi$
$587$	26.7214			1.10291			0.551455	+	0.834205i	$0.314073\pi$
$588$		0			0
$589$	−7.34670			−0.302716
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.21759	+	5.57303i	0.132131	+	0.228857i	0.924498	−	0.381188i	$-0.124485\pi$
$593$	3.21759	+	5.57303i	0.132131	+	0.228857i	−0.792367	+	0.610045i	$0.791151\pi$
$594$		0			0
$595$	1.16859	−	1.79929i	0.0479076	−	0.0737635i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−39.0065	−	22.5204i	−1.59376	−	0.920159i	−0.992653	−	0.120992i	$-0.961392\pi$
$599$	−39.0065	−	22.5204i	−1.59376	−	0.920159i	−0.601109	−	0.799167i	$-0.705274\pi$
$600$		0			0
$601$			6.29317i			0.256704i	0.991729	+	0.128352i	$0.0409687\pi$
$601$			6.29317i			0.256704i	−0.991729	+	0.128352i	$0.959031\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.89322	+	3.27916i	−0.0769704	+	0.133317i
$606$		0			0
$607$	−2.21834	+	1.28076i	−0.0900396	+	0.0519844i	−0.544344	−	0.838862i	$-0.683221\pi$
$607$	−2.21834	+	1.28076i	−0.0900396	+	0.0519844i	0.454304	+	0.890847i	$0.349888\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.12645	+	1.22771i	−0.0860269	+	0.0496677i
$612$		0			0
$613$	5.95779	−	10.3192i	0.240633	−	0.416788i	−0.720262	−	0.693702i	$-0.755978\pi$
$613$	5.95779	−	10.3192i	0.240633	−	0.416788i	0.960895	+	0.276914i	$0.0893117\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	17.0171i		−	0.685083i	−0.939503	−	0.342541i	$-0.888712\pi$
$617$		−	17.0171i		−	0.685083i	0.939503	−	0.342541i	$-0.111288\pi$
$618$		0			0
$619$	−26.8253	−	15.4876i	−1.07820	−	0.622498i	−0.147789	−	0.989019i	$-0.547216\pi$
$619$	−26.8253	−	15.4876i	−1.07820	−	0.622498i	−0.930410	+	0.366521i	$0.880549\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	22.2428	+	1.16677i	0.891141	+	0.0467457i
$624$		0			0
$625$	−9.68691	−	16.7782i	−0.387476	−	0.671129i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.432351			0.0172390
$630$		0			0
$631$	−5.21976			−0.207795			−0.103898	−	0.994588i	$-0.533131\pi$
$631$	−5.21976			−0.207795			−0.103898	+	0.994588i	$0.533131\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.53343	−	4.38802i	−0.100536	−	0.174133i
$636$		0			0
$637$	−6.16302	+	2.74324i	−0.244188	+	0.108691i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.0187	−	11.5578i	−0.790693	−	0.456507i	0.0495138	−	0.998773i	$-0.484233\pi$
$641$	−20.0187	−	11.5578i	−0.790693	−	0.456507i	−0.840206	+	0.542267i	$0.817566\pi$
$642$		0			0
$643$		−	23.2939i		−	0.918623i	−0.888275	−	0.459312i	$-0.848096\pi$
$643$		−	23.2939i		−	0.918623i	0.888275	−	0.459312i	$-0.151904\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.2570	−	35.0862i	0.796386	−	1.37938i	−0.125570	−	0.992085i	$-0.540076\pi$
$647$	20.2570	−	35.0862i	0.796386	−	1.37938i	0.921956	−	0.387296i	$-0.126591\pi$
$648$		0			0
$649$	−23.7886	+	13.7344i	−0.933786	+	0.539121i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.3425	+	5.97123i	−0.404732	+	0.233672i	−0.688524	−	0.725214i	$-0.741741\pi$
$653$	−10.3425	+	5.97123i	−0.404732	+	0.233672i	0.283792	+	0.958886i	$0.408408\pi$
$654$		0			0
$655$	−1.55355	+	2.69083i	−0.0607023	+	0.105139i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	43.4636i		−	1.69310i	−0.532307	−	0.846552i	$-0.678675\pi$
$659$		−	43.4636i		−	1.69310i	0.532307	−	0.846552i	$-0.321325\pi$
$660$		0			0
$661$	43.1033	+	24.8857i	1.67653	+	0.967942i	0.963849	+	0.266450i	$0.0858506\pi$
$661$	43.1033	+	24.8857i	1.67653	+	0.967942i	0.712677	+	0.701493i	$0.247483\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.66755	+	1.35902i	−0.103443	+	0.0527006i
$666$		0			0
$667$	−20.3760	−	35.2923i	−0.788962	−	1.36652i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.54111			−0.0594939
$672$		0			0
$673$	32.8822			1.26751			0.633757	−	0.773532i	$-0.281512\pi$
$673$	32.8822			1.26751			0.633757	+	0.773532i	$0.281512\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.7744	+	22.1260i	0.490961	+	0.850370i	0.999946	−	0.0104056i	$-0.00331226\pi$
$677$	12.7744	+	22.1260i	0.490961	+	0.850370i	−0.508984	+	0.860776i	$0.669979\pi$
$678$		0			0
$679$	27.5197	+	17.8734i	1.05611	+	0.685917i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	37.4260	+	21.6079i	1.43207	+	0.826803i	0.997278	−	0.0737300i	$-0.0234903\pi$
$683$	37.4260	+	21.6079i	1.43207	+	0.826803i	0.434787	+	0.900533i	$0.356824\pi$
$684$		0			0
$685$			2.92379i			0.111712i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.548508	−	0.950043i	0.0208965	−	0.0361938i
$690$		0			0
$691$	18.2836	−	10.5561i	0.695542	−	0.401571i	−0.110143	−	0.993916i	$-0.535131\pi$
$691$	18.2836	−	10.5561i	0.695542	−	0.401571i	0.805685	+	0.592344i	$0.201797\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.57293	−	3.21754i	0.211393	−	0.122048i
$696$		0			0
$697$	7.65822	−	13.2644i	0.290076	−	0.502426i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.7546i		−	1.27489i	−0.770495	−	0.637446i	$-0.779991\pi$
$701$		−	33.7546i		−	1.27489i	0.770495	−	0.637446i	$-0.220009\pi$
$702$		0			0
$703$	−0.522475	−	0.301651i	−0.0197055	−	0.0113770i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.38328	−	8.60370i	−0.164850	−	0.323576i
$708$		0			0
$709$	16.5489	+	28.6636i	0.621509	+	1.07648i	0.989205	+	0.146539i	$0.0468132\pi$
$709$	16.5489	+	28.6636i	0.621509	+	1.07648i	−0.367696	+	0.929946i	$0.619853\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	24.5398			0.919022
$714$		0			0
$715$	1.32322			0.0494854
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.6731	−	34.0748i	−0.733684	−	1.27078i	−0.955298	−	0.295643i	$-0.904466\pi$
$719$	−19.6731	−	34.0748i	−0.733684	−	1.27078i	0.221615	−	0.975134i	$-0.428867\pi$
$720$		0			0
$721$	−0.500210	+	9.53580i	−0.0186288	+	0.355132i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−26.7370	−	15.4366i	−0.992987	−	0.573301i
$726$		0			0
$727$			14.7164i			0.545803i	0.962042	+	0.272901i	$0.0879832\pi$
$727$			14.7164i			0.545803i	−0.962042	+	0.272901i	$0.912017\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.43054	+	2.47777i	−0.0529104	+	0.0916435i
$732$		0			0
$733$	−39.3009	+	22.6904i	−1.45161	+	0.838089i	−0.998573	−	0.0534010i	$-0.982994\pi$
$733$	−39.3009	+	22.6904i	−1.45161	+	0.838089i	−0.453040	+	0.891490i	$0.649661\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.8382	−	7.98947i	0.509735	−	0.294296i
$738$		0			0
$739$	−4.90885	+	8.50237i	−0.180575	+	0.312765i	−0.942076	−	0.335398i	$-0.891129\pi$
$739$	−4.90885	+	8.50237i	−0.180575	+	0.312765i	0.761502	+	0.648163i	$0.224462\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			20.2980i			0.744660i	0.928100	+	0.372330i	$0.121441\pi$
$743$			20.2980i			0.744660i	−0.928100	+	0.372330i	$0.878559\pi$
$744$		0			0
$745$	−4.97465	−	2.87212i	−0.182257	−	0.105226i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.94601	−	37.0980i	0.0711057	−	1.35553i
$750$		0			0
$751$	−8.84315	−	15.3168i	−0.322691	−	0.558918i	0.658351	−	0.752711i	$-0.271254\pi$
$751$	−8.84315	−	15.3168i	−0.322691	−	0.558918i	−0.981042	+	0.193793i	$0.937921\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	10.7010			0.389450
$756$		0			0
$757$	−24.2142			−0.880079			−0.440040	−	0.897978i	$-0.645036\pi$
$757$	−24.2142			−0.880079			−0.440040	+	0.897978i	$0.645036\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.664665	+	1.15123i	0.0240941	+	0.0417322i	0.877821	−	0.478989i	$-0.158997\pi$
$761$	0.664665	+	1.15123i	0.0240941	+	0.0417322i	−0.853727	+	0.520721i	$0.825663\pi$
$762$		0			0
$763$	−1.63614	−	3.21149i	−0.0592323	−	0.116264i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.3595	−	5.98107i	−0.374061	−	0.215964i
$768$		0			0
$769$		−	9.99518i		−	0.360435i	−0.983627	−	0.180218i	$-0.942320\pi$
$769$		−	9.99518i		−	0.360435i	0.983627	−	0.180218i	$-0.0576802\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20.3473	−	35.2426i	0.731841	−	1.26759i	−0.224254	−	0.974531i	$-0.571994\pi$
$773$	20.3473	−	35.2426i	0.731841	−	1.26759i	0.956095	−	0.293056i	$-0.0946722\pi$
$774$		0			0
$775$	16.1003	−	9.29552i	0.578340	−	0.333905i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.5092	+	10.6863i	−0.663159	+	0.382875i
$780$		0			0
$781$	−8.19030	+	14.1860i	−0.293072	+	0.507615i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	2.57526i		−	0.0919148i
$786$		0			0
$787$	−16.0048	−	9.24038i	−0.570510	−	0.329384i	0.186843	−	0.982390i	$-0.440174\pi$
$787$	−16.0048	−	9.24038i	−0.570510	−	0.329384i	−0.757353	+	0.653006i	$0.773508\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	35.7825	+	23.2399i	1.27228	+	0.826314i
$792$		0			0
$793$	−0.335563	−	0.581212i	−0.0119162	−	0.0206394i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	35.8741			1.27073			0.635363	−	0.772213i	$-0.280850\pi$
$797$	35.8741			1.27073			0.635363	+	0.772213i	$0.280850\pi$
$798$		0			0
$799$	−3.33000			−0.117807
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.77758	+	4.81091i	0.0980187	+	0.169773i
$804$		0			0
$805$	8.91026	−	4.53946i	0.314045	−	0.159995i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	36.3089	+	20.9630i	1.27655	+	0.737018i	0.976213	−	0.216814i	$-0.0695665\pi$
$809$	36.3089	+	20.9630i	1.27655	+	0.737018i	0.300340	+	0.953832i	$0.402900\pi$
$810$		0			0
$811$		−	34.1402i		−	1.19883i	−0.800440	−	0.599413i	$-0.795401\pi$
$811$		−	34.1402i		−	1.19883i	0.800440	−	0.599413i	$-0.204599\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.95982	+	6.85861i	−0.138706	+	0.240247i
$816$		0			0
$817$	3.45747	−	1.99617i	0.120962	−	0.0698372i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	27.2818	−	15.7512i	0.952143	−	0.549720i	0.0583971	−	0.998293i	$-0.481401\pi$
$821$	27.2818	−	15.7512i	0.952143	−	0.549720i	0.893746	+	0.448573i	$0.148068\pi$
$822$		0			0
$823$	−8.47768	+	14.6838i	−0.295513	+	0.511844i	−0.975104	−	0.221747i	$-0.928824\pi$
$823$	−8.47768	+	14.6838i	−0.295513	+	0.511844i	0.679591	+	0.733591i	$0.262157\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	43.7684i		−	1.52198i	−0.648766	−	0.760988i	$-0.724715\pi$
$827$		−	43.7684i		−	1.52198i	0.648766	−	0.760988i	$-0.275285\pi$
$828$		0			0
$829$	4.86284	+	2.80756i	0.168893	+	0.0975107i	0.582064	−	0.813143i	$-0.302245\pi$
$829$	4.86284	+	2.80756i	0.168893	+	0.0975107i	−0.413171	+	0.910654i	$0.635579\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−9.09859	−	0.957185i	−0.315247	−	0.0331645i
$834$		0			0
$835$	−6.36228	−	11.0198i	−0.220176	−	0.381356i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36.7401			1.26841			0.634204	−	0.773166i	$-0.281328\pi$
$839$	36.7401			1.26841			0.634204	+	0.773166i	$0.281328\pi$
$840$		0			0
$841$	−15.7520			−0.543172
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.74481	−	6.48620i	−0.128825	−	0.223132i
$846$		0			0
$847$	16.1242	+	0.845809i	0.554033	+	0.0290623i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.74519	+	1.00759i	0.0598245	+	0.0345397i
$852$		0			0
$853$		−	35.6583i		−	1.22092i	−0.792048	−	0.610459i	$-0.790985\pi$
$853$		−	35.6583i		−	1.22092i	0.792048	−	0.610459i	$-0.209015\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.444720	+	0.770278i	−0.0151914	+	0.0263122i	−0.873521	−	0.486786i	$-0.838169\pi$
$857$	−0.444720	+	0.770278i	−0.0151914	+	0.0263122i	0.858330	+	0.513098i	$0.171502\pi$
$858$		0			0
$859$	20.1560	−	11.6371i	0.687715	−	0.397052i	−0.115040	−	0.993361i	$-0.536700\pi$
$859$	20.1560	−	11.6371i	0.687715	−	0.397052i	0.802755	+	0.596308i	$0.203366\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.72738	−	2.15201i	0.126882	−	0.0732551i	−0.435216	−	0.900326i	$-0.643328\pi$
$863$	3.72738	−	2.15201i	0.126882	−	0.0732551i	0.562097	+	0.827071i	$0.309995\pi$
$864$		0			0
$865$	2.36446	−	4.09537i	0.0803941	−	0.139247i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	20.0243i		−	0.679278i
$870$		0			0
$871$	6.02627	+	3.47927i	0.204192	+	0.117891i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.59702	−	13.2369i	0.290632	−	0.447488i
$876$		0			0
$877$	29.1826	+	50.5457i	0.985425	+	1.70681i	0.640034	+	0.768347i	$0.278920\pi$
$877$	29.1826	+	50.5457i	0.985425	+	1.70681i	0.345391	+	0.938459i	$0.387746\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13.9914			0.471381			0.235691	−	0.971828i	$-0.424265\pi$
$881$	13.9914			0.471381			0.235691	+	0.971828i	$0.424265\pi$
$882$		0			0
$883$	−1.64531			−0.0553691			−0.0276845	−	0.999617i	$-0.508813\pi$
$883$	−1.64531			−0.0553691			−0.0276845	+	0.999617i	$0.508813\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.976428	−	1.69122i	−0.0327852	−	0.0567857i	0.849167	−	0.528124i	$-0.177104\pi$
$887$	−0.976428	−	1.69122i	−0.0327852	−	0.0567857i	−0.881952	+	0.471338i	$0.843771\pi$
$888$		0			0
$889$	−11.7685	+	18.1200i	−0.394703	+	0.607725i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.02414	+	2.32334i	0.134663	+	0.0777475i
$894$		0			0
$895$			4.14642i			0.138599i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.4742	−	23.3381i	0.449391	−	0.778368i
$900$		0			0
$901$	1.28844	−	0.743879i	0.0429240	−	0.0247822i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.84130	+	1.64042i	−0.0944479	+	0.0545295i
$906$		0			0
$907$	−6.98547	+	12.0992i	−0.231949	+	0.401747i	−0.958382	−	0.285491i	$-0.907843\pi$
$907$	−6.98547	+	12.0992i	−0.231949	+	0.401747i	0.726433	+	0.687237i	$0.241177\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	6.79122i		−	0.225003i	−0.993652	−	0.112502i	$-0.964114\pi$
$911$		−	6.79122i		−	0.225003i	0.993652	−	0.112502i	$-0.0358863\pi$
$912$		0			0
$913$	−16.9717	−	9.79864i	−0.561683	−	0.324288i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13.2312	+	0.694058i	0.436934	+	0.0229198i
$918$		0			0
$919$	0.292791	+	0.507128i	0.00965828	+	0.0167286i	0.870814	−	0.491612i	$-0.163592\pi$
$919$	0.292791	+	0.507128i	0.00965828	+	0.0167286i	−0.861156	+	0.508341i	$0.830259\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.13345			−0.234800
$924$		0			0
$925$	1.52667			0.0501967
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.4832	−	32.0139i	−0.606415	−	1.05034i	−0.991826	−	0.127596i	$-0.959274\pi$
$929$	−18.4832	−	32.0139i	−0.606415	−	1.05034i	0.385411	−	0.922745i	$-0.374060\pi$
$930$		0			0
$931$	10.3274	+	7.50479i	0.338466	+	0.245960i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.55411	+	0.897263i	0.0508247	+	0.0293436i
$936$		0			0
$937$			33.9658i			1.10961i	0.831979	+	0.554807i	$0.187208\pi$
$937$			33.9658i			1.10961i	−0.831979	+	0.554807i	$0.812792\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.0968	+	40.0048i	−0.752933	+	1.30412i	0.193462	+	0.981108i	$0.438028\pi$
$941$	−23.0968	+	40.0048i	−0.752933	+	1.30412i	−0.946395	+	0.323011i	$0.895305\pi$
$942$		0			0
$943$	61.8251	−	35.6947i	2.01330	−	1.16238i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.9927	−	12.1202i	0.682173	−	0.393853i	−0.118500	−	0.992954i	$-0.537809\pi$
$947$	20.9927	−	12.1202i	0.682173	−	0.393853i	0.800673	+	0.599101i	$0.204475\pi$
$948$		0			0
$949$	−1.20959	+	2.09506i	−0.0392648	+	0.0680087i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			27.7820i			0.899947i	0.893042	+	0.449974i	$0.148567\pi$
$953$			27.7820i			0.899947i	−0.893042	+	0.449974i	$0.851433\pi$
$954$		0			0
$955$	4.00460	+	2.31206i	0.129586	+	0.0748164i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	11.1091	−	5.65971i	0.358733	−	0.182762i
$960$		0			0
$961$	−7.38617	−	12.7932i	−0.238263	−	0.412684i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.0928			−0.389283
$966$		0			0
$967$	7.92003			0.254691			0.127345	−	0.991858i	$-0.459354\pi$
$967$	7.92003			0.254691			0.127345	+	0.991858i	$0.459354\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.806196	−	1.39637i	−0.0258720	−	0.0448117i	0.852799	−	0.522238i	$-0.174903\pi$
$971$	−0.806196	−	1.39637i	−0.0258720	−	0.0448117i	−0.878672	+	0.477427i	$0.841570\pi$
$972$		0			0
$973$	−23.0130	−	14.9464i	−0.737763	−	0.479159i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.4103	−	6.58776i	−0.365049	−	0.210761i	0.306244	−	0.951953i	$-0.400928\pi$
$977$	−11.4103	−	6.58776i	−0.365049	−	0.210761i	−0.671293	+	0.741192i	$0.734261\pi$
$978$		0			0
$979$			18.6301i			0.595420i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.52198	+	2.63615i	−0.0485437	+	0.0840802i	−0.889276	−	0.457370i	$-0.848791\pi$
$983$	−1.52198	+	2.63615i	−0.0485437	+	0.0840802i	0.840733	+	0.541451i	$0.182125\pi$
$984$		0			0
$985$	−1.40812	+	0.812981i	−0.0448665	+	0.0259037i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.5488	+	6.66770i	−0.367230	+	0.212021i
$990$		0			0
$991$	−12.7482	+	22.0806i	−0.404961	+	0.701414i	−0.994317	−	0.106460i	$-0.966048\pi$
$991$	−12.7482	+	22.0806i	−0.404961	+	0.701414i	0.589356	+	0.807874i	$0.299382\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.08691i		−	0.161266i
$996$		0			0
$997$	−5.34395	−	3.08533i	−0.169245	−	0.0977135i	0.412985	−	0.910738i	$-0.364486\pi$
$997$	−5.34395	−	3.08533i	−0.169245	−	0.0977135i	−0.582230	+	0.813024i	$0.697820\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.bl.d.1025.4	yes	16
3.2	odd	2	inner	1512.2.bl.d.1025.5	yes	16
7.5	odd	6	inner	1512.2.bl.d.593.5	yes	16
21.5	even	6	inner	1512.2.bl.d.593.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.bl.d.593.4	✓	16	21.5	even	6	inner
1512.2.bl.d.593.5	yes	16	7.5	odd	6	inner
1512.2.bl.d.1025.4	yes	16	1.1	even	1	trivial
1512.2.bl.d.1025.5	yes	16	3.2	odd	2	inner

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1512.2.bl.d.1025.4

Level	$1512$
Weight	$2$
Character	1512.1025
Analytic conductor	$12.073$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$