LMFDB - Embedded newform 1872.2.t.o.1153.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(289,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1872.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.t (of order $3$, 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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1153.2
Root			$-1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	1872.1153
Dual form			1872.2.t.o.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{5} +(-1.00000 - 1.73205i) q^{7} +(2.64575 - 4.58258i) q^{11} +(3.50000 + 0.866025i) q^{13} +(-3.96863 - 6.87386i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(-2.64575 + 4.58258i) q^{23} +2.00000 q^{25} +(-1.32288 + 2.29129i) q^{29} -4.00000 q^{31} +(-2.64575 - 4.58258i) q^{35} +(1.50000 - 2.59808i) q^{37} +(-3.96863 + 6.87386i) q^{41} +(-1.00000 - 1.73205i) q^{43} -5.29150 q^{47} +(1.50000 - 2.59808i) q^{49} +7.93725 q^{53} +(7.00000 - 12.1244i) q^{55} +(5.29150 + 9.16515i) q^{59} +(-6.50000 - 11.2583i) q^{61} +(9.26013 + 2.29129i) q^{65} +(-1.00000 + 1.73205i) q^{67} +(2.64575 + 4.58258i) q^{71} -7.00000 q^{73} -10.5830 q^{77} +4.00000 q^{79} +15.8745 q^{83} +(-10.5000 - 18.1865i) q^{85} +(-2.00000 - 6.92820i) q^{91} +(-7.93725 - 13.7477i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 14 q^{13} - 12 q^{19} + 8 q^{25} - 16 q^{31} + 6 q^{37} - 4 q^{43} + 6 q^{49} + 28 q^{55} - 26 q^{61} - 4 q^{67} - 28 q^{73} + 16 q^{79} - 42 q^{85} - 8 q^{91} - 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 14 * q^13 - 12 * q^19 + 8 * q^25 - 16 * q^31 + 6 * q^37 - 4 * q^43 + 6 * q^49 + 28 * q^55 - 26 * q^61 - 4 * q^67 - 28 * q^73 + 16 * q^79 - 42 * q^85 - 8 * q^91 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.64575			1.18322			0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575			1.18322			0.591608	+	0.806226i	$0.298493\pi$
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.64575	−	4.58258i	0.797724	−	1.38170i	−0.123371	−	0.992361i	$-0.539370\pi$
$11$	2.64575	−	4.58258i	0.797724	−	1.38170i	0.921095	−	0.389338i	$-0.127296\pi$
$12$		0			0
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$13$	3.50000	+	0.866025i	0.970725	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.96863	−	6.87386i	−0.962533	−	1.66716i	−0.716101	−	0.697997i	$-0.754075\pi$
$17$	−3.96863	−	6.87386i	−0.962533	−	1.66716i	−0.246433	−	0.969160i	$-0.579258\pi$
$18$		0			0
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	−0.972404	−	0.233301i	$-0.925047\pi$
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	0.284157	−	0.958778i	$-0.408286\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.64575	+	4.58258i	−0.551677	+	0.955533i	0.446476	+	0.894795i	$0.352679\pi$
$23$	−2.64575	+	4.58258i	−0.551677	+	0.955533i	−0.998154	+	0.0607377i	$0.980655\pi$
$24$		0			0
$25$	2.00000			0.400000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.32288	+	2.29129i	−0.245652	+	0.425481i	−0.962315	−	0.271938i	$-0.912335\pi$
$29$	−1.32288	+	2.29129i	−0.245652	+	0.425481i	0.716663	+	0.697420i	$0.245669\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.64575	−	4.58258i	−0.447214	−	0.774597i
$36$		0			0
$37$	1.50000	−	2.59808i	0.246598	−	0.427121i	−0.715981	−	0.698119i	$-0.754020\pi$
$37$	1.50000	−	2.59808i	0.246598	−	0.427121i	0.962580	+	0.270998i	$0.0873538\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.96863	+	6.87386i	−0.619795	+	1.07352i	0.369727	+	0.929140i	$0.379451\pi$
$41$	−3.96863	+	6.87386i	−0.619795	+	1.07352i	−0.989523	+	0.144377i	$0.953882\pi$
$42$		0			0
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	0.779647	−	0.626219i	$-0.215399\pi$
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	−0.932145	+	0.362084i	$0.882065\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.29150			−0.771845			−0.385922	−	0.922531i	$-0.626117\pi$
$47$	−5.29150			−0.771845			−0.385922	+	0.922531i	$0.626117\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.93725			1.09027			0.545133	−	0.838350i	$-0.316479\pi$
$53$	7.93725			1.09027			0.545133	+	0.838350i	$0.316479\pi$
$54$		0			0
$55$	7.00000	−	12.1244i	0.943880	−	1.63485i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.29150	+	9.16515i	0.688895	+	1.19320i	0.972196	+	0.234170i	$0.0752372\pi$
$59$	5.29150	+	9.16515i	0.688895	+	1.19320i	−0.283301	+	0.959031i	$0.591429\pi$
$60$		0			0
$61$	−6.50000	−	11.2583i	−0.832240	−	1.44148i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−6.50000	−	11.2583i	−0.832240	−	1.44148i	0.0640184	−	0.997949i	$-0.479608\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.26013	+	2.29129i	1.14858	+	0.284199i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.64575	+	4.58258i	0.313993	+	0.543852i	0.979223	−	0.202787i	$-0.0649998\pi$
$71$	2.64575	+	4.58258i	0.313993	+	0.543852i	−0.665230	+	0.746639i	$0.731667\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.5830			−1.20605
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.8745			1.74245			0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745			1.74245			0.871227	+	0.490881i	$0.163325\pi$
$84$		0			0
$85$	−10.5000	−	18.1865i	−1.13888	−	1.97261i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−2.00000	−	6.92820i	−0.209657	−	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.93725	−	13.7477i	−0.814345	−	1.41049i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.32288	−	2.29129i	0.131631	−	0.227992i	−0.792674	−	0.609645i	$-0.791312\pi$
$101$	1.32288	−	2.29129i	0.131631	−	0.227992i	0.924305	+	0.381654i	$0.124645\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.64575	+	4.58258i	−0.255774	+	0.443014i	−0.965106	−	0.261861i	$-0.915664\pi$
$107$	−2.64575	+	4.58258i	−0.255774	+	0.443014i	0.709331	+	0.704875i	$0.248997\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.32288	−	2.29129i	−0.124446	−	0.215546i	0.797070	−	0.603886i	$-0.206382\pi$
$113$	−1.32288	−	2.29129i	−0.124446	−	0.215546i	−0.921516	+	0.388340i	$0.873049\pi$
$114$		0			0
$115$	−7.00000	+	12.1244i	−0.652753	+	1.13060i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.93725	+	13.7477i	−0.727607	+	1.26025i
$120$		0			0
$121$	−8.50000	−	14.7224i	−0.772727	−	1.33840i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.93725			−0.709930
$126$		0			0
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	−0.255014	−	0.966937i	$-0.582080\pi$
$127$	8.00000	−	13.8564i	0.709885	−	1.22956i	0.964899	−	0.262620i	$-0.0845865\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.5830			−0.924641			−0.462321	−	0.886713i	$-0.652983\pi$
$131$	−10.5830			−0.924641			−0.462321	+	0.886713i	$0.652983\pi$
$132$		0			0
$133$	−6.00000	+	10.3923i	−0.520266	+	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.96863	+	6.87386i	0.339063	+	0.587274i	0.984257	−	0.176745i	$-0.0565568\pi$
$137$	3.96863	+	6.87386i	0.339063	+	0.587274i	−0.645194	+	0.764019i	$0.723223\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	13.2288	−	13.7477i	1.10624	−	1.14964i
$144$		0			0
$145$	−3.50000	+	6.06218i	−0.290659	+	0.503436i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.96863	+	6.87386i	0.325123	+	0.563129i	0.981537	−	0.191272i	$-0.0612611\pi$
$149$	3.96863	+	6.87386i	0.325123	+	0.563129i	−0.656415	+	0.754400i	$0.727928\pi$
$150$		0			0
$151$	18.0000			1.46482			0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000			1.46482			0.732410	+	0.680864i	$0.238396\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.5830			−0.850047
$156$		0			0
$157$	17.0000			1.35675			0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000			1.35675			0.678374	+	0.734717i	$0.262685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.5830			0.834058
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	11.5000	+	6.06218i	0.884615	+	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.29150	−	9.16515i	−0.402305	−	0.696814i	0.591698	−	0.806160i	$-0.298458\pi$
$173$	−5.29150	−	9.16515i	−0.402305	−	0.696814i	−0.994004	+	0.109346i	$0.965124\pi$
$174$		0			0
$175$	−2.00000	−	3.46410i	−0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.93725	−	13.7477i	0.593258	−	1.02755i	−0.400532	−	0.916283i	$-0.631175\pi$
$179$	7.93725	−	13.7477i	0.593258	−	1.02755i	0.993790	−	0.111271i	$-0.0354920\pi$
$180$		0			0
$181$	−9.00000			−0.668965			−0.334482	−	0.942402i	$-0.608561\pi$
$181$	−9.00000			−0.668965			−0.334482	+	0.942402i	$0.608561\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.96863	−	6.87386i	0.291779	−	0.505376i
$186$		0			0
$187$	−42.0000			−3.07134
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.5830	+	18.3303i	0.765759	+	1.32633i	0.939844	+	0.341603i	$0.110970\pi$
$191$	10.5830	+	18.3303i	0.765759	+	1.32633i	−0.174085	+	0.984731i	$0.555697\pi$
$192$		0			0
$193$	12.5000	−	21.6506i	0.899770	−	1.55845i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	12.5000	−	21.6506i	0.899770	−	1.55845i	0.827788	−	0.561041i	$-0.189599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$	−5.00000	−	8.66025i	−0.354441	−	0.613909i	0.632581	−	0.774494i	$-0.281995\pi$
$199$	−5.00000	−	8.66025i	−0.354441	−	0.613909i	−0.987022	+	0.160585i	$0.948662\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.29150			0.371391
$204$		0			0
$205$	−10.5000	+	18.1865i	−0.733352	+	1.27020i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−31.7490			−2.19613
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.64575	−	4.58258i	−0.180439	−	0.312529i
$216$		0			0
$217$	4.00000	+	6.92820i	0.271538	+	0.470317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.93725	−	27.4955i	−0.533917	−	1.84954i
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.93725	−	13.7477i	−0.526814	−	0.912469i	−0.999512	−	0.0312441i	$-0.990053\pi$
$227$	−7.93725	−	13.7477i	−0.526814	−	0.912469i	0.472698	−	0.881225i	$-0.343280\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.1660			1.38663			0.693316	−	0.720634i	$-0.256149\pi$
$233$	21.1660			1.38663			0.693316	+	0.720634i	$0.256149\pi$
$234$		0			0
$235$	−14.0000			−0.913259
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.8745			−1.02684			−0.513418	−	0.858138i	$-0.671621\pi$
$239$	−15.8745			−1.02684			−0.513418	+	0.858138i	$0.671621\pi$
$240$		0			0
$241$	−10.5000	−	18.1865i	−0.676364	−	1.17150i	−0.976068	−	0.217465i	$-0.930221\pi$
$241$	−10.5000	−	18.1865i	−0.676364	−	1.17150i	0.299704	−	0.954032i	$-0.403112\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.96863	−	6.87386i	0.253546	−	0.439155i
$246$		0			0
$247$	−6.00000	−	20.7846i	−0.381771	−	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$	14.0000	+	24.2487i	0.880172	+	1.52450i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.96863	+	6.87386i	−0.247556	+	0.428780i	−0.962847	−	0.270047i	$-0.912961\pi$
$257$	−3.96863	+	6.87386i	−0.247556	+	0.428780i	0.715291	+	0.698827i	$0.246294\pi$
$258$		0			0
$259$	−6.00000			−0.372822
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.93725	−	13.7477i	0.489432	−	0.847721i	−0.510494	−	0.859881i	$-0.670537\pi$
$263$	7.93725	−	13.7477i	0.489432	−	0.847721i	0.999926	+	0.0121601i	$0.00387079\pi$
$264$		0			0
$265$	21.0000			1.29002
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.8745	+	27.4955i	0.967886	+	1.67643i	0.701653	+	0.712519i	$0.252446\pi$
$269$	15.8745	+	27.4955i	0.967886	+	1.67643i	0.266233	+	0.963909i	$0.414221\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.29150	−	9.16515i	0.319090	−	0.552679i
$276$		0			0
$277$	4.50000	+	7.79423i	0.270379	+	0.468310i	0.968959	−	0.247222i	$-0.0795177\pi$
$277$	4.50000	+	7.79423i	0.270379	+	0.468310i	−0.698580	+	0.715532i	$0.746184\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.2288			0.789161			0.394581	−	0.918861i	$-0.370890\pi$
$281$	13.2288			0.789161			0.394581	+	0.918861i	$0.370890\pi$
$282$		0			0
$283$	−7.00000	+	12.1244i	−0.416107	+	0.720718i	−0.995544	−	0.0942988i	$-0.969939\pi$
$283$	−7.00000	+	12.1244i	−0.416107	+	0.720718i	0.579437	+	0.815017i	$0.303272\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.8745			0.937043
$288$		0			0
$289$	−23.0000	+	39.8372i	−1.35294	+	2.34336i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.26013	+	16.0390i	0.540983	+	0.937009i	0.998848	+	0.0479877i	$0.0152808\pi$
$293$	9.26013	+	16.0390i	0.540983	+	0.937009i	−0.457865	+	0.889022i	$0.651386\pi$
$294$		0			0
$295$	14.0000	+	24.2487i	0.815112	+	1.41181i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.2288	+	13.7477i	−0.765039	+	0.795052i
$300$		0			0
$301$	−2.00000	+	3.46410i	−0.115278	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−17.1974	−	29.7867i	−0.984719	−	1.70558i
$306$		0			0
$307$	−2.00000			−0.114146			−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000			−0.114146			−0.0570730	+	0.998370i	$0.518177\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.8745			0.900161			0.450080	−	0.892988i	$-0.351395\pi$
$311$	15.8745			0.900161			0.450080	+	0.892988i	$0.351395\pi$
$312$		0			0
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.93725			0.445801			0.222900	−	0.974841i	$-0.428448\pi$
$317$	7.93725			0.445801			0.222900	+	0.974841i	$0.428448\pi$
$318$		0			0
$319$	7.00000	+	12.1244i	0.391925	+	0.678834i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−23.8118	+	41.2432i	−1.32492	+	2.29483i
$324$		0			0
$325$	7.00000	+	1.73205i	0.388290	+	0.0960769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.29150	+	9.16515i	0.291730	+	0.505291i
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.64575	+	4.58258i	−0.144553	+	0.250373i
$336$		0			0
$337$	5.00000			0.272367			0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000			0.272367			0.136184	+	0.990684i	$0.456516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.5830	+	18.3303i	−0.573102	+	0.992642i
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.93725	+	13.7477i	0.426094	+	0.738017i	0.996522	−	0.0833311i	$-0.0265559\pi$
$347$	7.93725	+	13.7477i	0.426094	+	0.738017i	−0.570428	+	0.821348i	$0.693223\pi$
$348$		0			0
$349$	1.00000	−	1.73205i	0.0535288	−	0.0927146i	−0.838019	−	0.545640i	$-0.816286\pi$
$349$	1.00000	−	1.73205i	0.0535288	−	0.0927146i	0.891548	+	0.452926i	$0.149620\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.61438	+	11.4564i	−0.352048	+	0.609765i	−0.986608	−	0.163109i	$-0.947848\pi$
$353$	−6.61438	+	11.4564i	−0.352048	+	0.609765i	0.634560	+	0.772873i	$0.281181\pi$
$354$		0			0
$355$	7.00000	+	12.1244i	0.371521	+	0.643494i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.8745			0.837824			0.418912	−	0.908027i	$-0.362411\pi$
$359$	15.8745			0.837824			0.418912	+	0.908027i	$0.362411\pi$
$360$		0			0
$361$	−8.50000	+	14.7224i	−0.447368	+	0.774865i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.5203			−0.969395
$366$		0			0
$367$	7.00000	−	12.1244i	0.365397	−	0.632886i	−0.623443	−	0.781869i	$-0.714267\pi$
$367$	7.00000	−	12.1244i	0.365397	−	0.632886i	0.988840	+	0.148983i	$0.0475999\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−7.93725	−	13.7477i	−0.412082	−	0.713746i
$372$		0			0
$373$	−8.50000	−	14.7224i	−0.440113	−	0.762299i	0.557584	−	0.830120i	$-0.311728\pi$
$373$	−8.50000	−	14.7224i	−0.440113	−	0.762299i	−0.997697	+	0.0678218i	$0.978395\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.61438	+	6.87386i	−0.340658	+	0.354022i
$378$		0			0
$379$	−10.0000	+	17.3205i	−0.513665	+	0.889695i	0.486209	+	0.873843i	$0.338379\pi$
$379$	−10.0000	+	17.3205i	−0.513665	+	0.889695i	−0.999874	+	0.0158521i	$0.994954\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	−28.0000			−1.42701
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.93725			−0.402435			−0.201217	−	0.979547i	$-0.564490\pi$
$389$	−7.93725			−0.402435			−0.201217	+	0.979547i	$0.564490\pi$
$390$		0			0
$391$	42.0000			2.12403
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.5830			0.532489
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.9059	+	20.6216i	−0.594551	+	1.02979i	0.399059	+	0.916925i	$0.369337\pi$
$401$	−11.9059	+	20.6216i	−0.594551	+	1.02979i	−0.993610	+	0.112868i	$0.963996\pi$
$402$		0			0
$403$	−14.0000	−	3.46410i	−0.697390	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.93725	−	13.7477i	−0.393435	−	0.681450i
$408$		0			0
$409$	17.5000	+	30.3109i	0.865319	+	1.49878i	0.866730	+	0.498778i	$0.166218\pi$
$409$	17.5000	+	30.3109i	0.865319	+	1.49878i	−0.00141047	+	0.999999i	$0.500449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10.5830	−	18.3303i	0.520756	−	0.901975i
$414$		0			0
$415$	42.0000			2.06170
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.8745	+	27.4955i	−0.775520	+	1.34324i	0.158981	+	0.987282i	$0.449179\pi$
$419$	−15.8745	+	27.4955i	−0.775520	+	1.34324i	−0.934501	+	0.355959i	$0.884154\pi$
$420$		0			0
$421$	−29.0000			−1.41337			−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000			−1.41337			−0.706687	+	0.707527i	$0.749811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.93725	−	13.7477i	−0.385013	−	0.666863i
$426$		0			0
$427$	−13.0000	+	22.5167i	−0.629114	+	1.08966i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.93725	+	13.7477i	−0.382324	+	0.662205i	−0.991394	−	0.130912i	$-0.958210\pi$
$431$	−7.93725	+	13.7477i	−0.382324	+	0.662205i	0.609070	+	0.793117i	$0.291543\pi$
$432$		0			0
$433$	−9.50000	−	16.4545i	−0.456541	−	0.790752i	0.542234	−	0.840227i	$-0.317578\pi$
$433$	−9.50000	−	16.4545i	−0.456541	−	0.790752i	−0.998775	+	0.0494752i	$0.984245\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	31.7490			1.51876
$438$		0			0
$439$	−11.0000	+	19.0526i	−0.525001	+	0.909329i	0.474575	+	0.880215i	$0.342602\pi$
$439$	−11.0000	+	19.0526i	−0.525001	+	0.909329i	−0.999576	+	0.0291138i	$0.990731\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−31.7490			−1.50844			−0.754221	−	0.656621i	$-0.771985\pi$
$443$	−31.7490			−1.50844			−0.754221	+	0.656621i	$0.771985\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.5830	−	18.3303i	−0.499443	−	0.865060i	0.500557	−	0.865704i	$-0.333129\pi$
$449$	−10.5830	−	18.3303i	−0.499443	−	0.865060i	−1.00000		0.000643168i	$0.999795\pi$
$450$		0			0
$451$	21.0000	+	36.3731i	0.988851	+	1.71274i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.29150	−	18.3303i	−0.248069	−	0.859338i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	0.410184	+	0.912003i	$0.365464\pi$
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	−0.994910	+	0.100771i	$0.967869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11.9059	+	20.6216i	0.554512	+	0.960443i	0.997941	+	0.0641338i	$0.0204284\pi$
$461$	11.9059	+	20.6216i	0.554512	+	0.960443i	−0.443429	+	0.896309i	$0.646238\pi$
$462$		0			0
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5.29150			0.244862			0.122431	−	0.992477i	$-0.460931\pi$
$467$	5.29150			0.244862			0.122431	+	0.992477i	$0.460931\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.5830			−0.486607
$474$		0			0
$475$	−6.00000	−	10.3923i	−0.275299	−	0.476832i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.8745	−	27.4955i	0.725325	−	1.25630i	−0.233515	−	0.972353i	$-0.575023\pi$
$479$	15.8745	−	27.4955i	0.725325	−	1.25630i	0.958840	−	0.283946i	$-0.0916437\pi$
$480$		0			0
$481$	7.50000	−	7.79423i	0.341971	−	0.355386i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.64575	−	4.58258i	−0.120137	−	0.208084i
$486$		0			0
$487$	1.00000	+	1.73205i	0.0453143	+	0.0784867i	0.887793	−	0.460243i	$-0.152238\pi$
$487$	1.00000	+	1.73205i	0.0453143	+	0.0784867i	−0.842479	+	0.538730i	$0.818904\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.5203	−	32.0780i	0.835808	−	1.44766i	−0.0575636	−	0.998342i	$-0.518333\pi$
$491$	18.5203	−	32.0780i	0.835808	−	1.44766i	0.893371	−	0.449319i	$-0.148333\pi$
$492$		0			0
$493$	21.0000			0.945792
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.29150	−	9.16515i	0.237356	−	0.411113i
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13.2288	−	22.9129i	−0.589841	−	1.02163i	−0.994253	−	0.107058i	$-0.965857\pi$
$503$	−13.2288	−	22.9129i	−0.589841	−	1.02163i	0.404412	−	0.914577i	$-0.367476\pi$
$504$		0			0
$505$	3.50000	−	6.06218i	0.155748	−	0.269763i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.1974	−	29.7867i	0.762261	−	1.32027i	−0.179422	−	0.983772i	$-0.557423\pi$
$509$	17.1974	−	29.7867i	0.762261	−	1.32027i	0.941683	−	0.336502i	$-0.109244\pi$
$510$		0			0
$511$	7.00000	+	12.1244i	0.309662	+	0.536350i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	37.0405			1.63220
$516$		0			0
$517$	−14.0000	+	24.2487i	−0.615719	+	1.06646i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	39.6863			1.73869			0.869344	−	0.494208i	$-0.164542\pi$
$521$	39.6863			1.73869			0.869344	+	0.494208i	$0.164542\pi$
$522$		0			0
$523$	−3.00000	+	5.19615i	−0.131181	+	0.227212i	−0.924132	−	0.382073i	$-0.875210\pi$
$523$	−3.00000	+	5.19615i	−0.131181	+	0.227212i	0.792951	+	0.609285i	$0.208544\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.8745	+	27.4955i	0.691504	+	1.19772i
$528$		0			0
$529$	−2.50000	−	4.33013i	−0.108696	−	0.188266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−19.8431	+	20.6216i	−0.859502	+	0.893220i
$534$		0			0
$535$	−7.00000	+	12.1244i	−0.302636	+	0.524182i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.93725	−	13.7477i	−0.341882	−	0.592157i
$540$		0			0
$541$	−13.0000			−0.558914			−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000			−0.558914			−0.279457	+	0.960158i	$0.590154\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	26.4575			1.13332
$546$		0			0
$547$	−34.0000			−1.45374			−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000			−1.45374			−0.726868	+	0.686778i	$0.759025\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	15.8745			0.676277
$552$		0			0
$553$	−4.00000	−	6.92820i	−0.170097	−	0.294617i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.96863	+	6.87386i	−0.168156	+	0.291255i	−0.937772	−	0.347253i	$-0.887115\pi$
$557$	−3.96863	+	6.87386i	−0.168156	+	0.291255i	0.769615	+	0.638508i	$0.220448\pi$
$558$		0			0
$559$	−2.00000	−	6.92820i	−0.0845910	−	0.293032i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.1660	+	36.6606i	0.892041	+	1.54506i	0.837425	+	0.546553i	$0.184060\pi$
$563$	21.1660	+	36.6606i	0.892041	+	1.54506i	0.0546164	+	0.998507i	$0.482606\pi$
$564$		0			0
$565$	−3.50000	−	6.06218i	−0.147246	−	0.255038i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.29150	−	9.16515i	0.221831	−	0.384223i	−0.733533	−	0.679654i	$-0.762130\pi$
$569$	5.29150	−	9.16515i	0.221831	−	0.384223i	0.955364	+	0.295431i	$0.0954632\pi$
$570$		0			0
$571$	30.0000			1.25546			0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000			1.25546			0.627730	+	0.778431i	$0.283984\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5.29150	+	9.16515i	−0.220671	+	0.382213i
$576$		0			0
$577$	3.00000			0.124892			0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000			0.124892			0.0624458	+	0.998048i	$0.480110\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.8745	−	27.4955i	−0.658586	−	1.14070i
$582$		0			0
$583$	21.0000	−	36.3731i	0.869731	−	1.50642i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.29150	−	9.16515i	0.218404	−	0.378286i	−0.735916	−	0.677072i	$-0.763248\pi$
$587$	5.29150	−	9.16515i	0.218404	−	0.378286i	0.954320	+	0.298786i	$0.0965817\pi$
$588$		0			0
$589$	12.0000	+	20.7846i	0.494451	+	0.856415i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−29.1033			−1.19513			−0.597564	−	0.801821i	$-0.703865\pi$
$593$	−29.1033			−1.19513			−0.597564	+	0.801821i	$0.703865\pi$
$594$		0			0
$595$	−21.0000	+	36.3731i	−0.860916	+	1.49115i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.1660			−0.864820			−0.432410	−	0.901677i	$-0.642337\pi$
$599$	−21.1660			−0.864820			−0.432410	+	0.901677i	$0.642337\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.4889	−	38.9519i	−0.914303	−	1.58362i
$606$		0			0
$607$	−18.0000	−	31.1769i	−0.730597	−	1.26543i	−0.956628	−	0.291312i	$-0.905908\pi$
$607$	−18.0000	−	31.1769i	−0.730597	−	1.26543i	0.226031	−	0.974120i	$-0.427425\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.5203	−	4.58258i	−0.749249	−	0.185391i
$612$		0			0
$613$	−2.50000	+	4.33013i	−0.100974	+	0.174892i	−0.912086	−	0.409998i	$-0.865529\pi$
$613$	−2.50000	+	4.33013i	−0.100974	+	0.174892i	0.811112	+	0.584891i	$0.198863\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.32288	+	2.29129i	0.0532570	+	0.0922438i	0.891425	−	0.453169i	$-0.149706\pi$
$617$	1.32288	+	2.29129i	0.0532570	+	0.0922438i	−0.838168	+	0.545412i	$0.816373\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.8118			−0.949437
$630$		0			0
$631$	−4.00000	−	6.92820i	−0.159237	−	0.275807i	0.775356	−	0.631524i	$-0.217570\pi$
$631$	−4.00000	−	6.92820i	−0.159237	−	0.275807i	−0.934594	+	0.355716i	$0.884237\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	21.1660	−	36.6606i	0.839948	−	1.45483i
$636$		0			0
$637$	7.50000	−	7.79423i	0.297161	−	0.308819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.26013	+	16.0390i	0.365753	+	0.633503i	0.988897	−	0.148604i	$-0.0474781\pi$
$641$	9.26013	+	16.0390i	0.365753	+	0.633503i	−0.623144	+	0.782107i	$0.714145\pi$
$642$		0			0
$643$	−14.0000	−	24.2487i	−0.552106	−	0.956276i	−0.998122	−	0.0612510i	$-0.980491\pi$
$643$	−14.0000	−	24.2487i	−0.552106	−	0.956276i	0.446016	−	0.895025i	$-0.352842\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.1660	+	36.6606i	−0.832122	+	1.44128i	0.0642308	+	0.997935i	$0.479541\pi$
$647$	−21.1660	+	36.6606i	−0.832122	+	1.44128i	−0.896353	+	0.443342i	$0.853793\pi$
$648$		0			0
$649$	56.0000			2.19819
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.8745	+	27.4955i	−0.621217	+	1.07598i	0.368042	+	0.929809i	$0.380028\pi$
$653$	−15.8745	+	27.4955i	−0.621217	+	1.07598i	−0.989259	+	0.146171i	$0.953305\pi$
$654$		0			0
$655$	−28.0000			−1.09405
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.8745	−	27.4955i	−0.618383	−	1.07107i	−0.989781	−	0.142597i	$-0.954455\pi$
$659$	−15.8745	−	27.4955i	−0.618383	−	1.07107i	0.371398	−	0.928474i	$-0.378879\pi$
$660$		0			0
$661$	15.5000	−	26.8468i	0.602880	−	1.04422i	−0.389503	−	0.921025i	$-0.627353\pi$
$661$	15.5000	−	26.8468i	0.602880	−	1.04422i	0.992383	−	0.123194i	$-0.0393136\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−15.8745	+	27.4955i	−0.615587	+	1.06623i
$666$		0			0
$667$	−7.00000	−	12.1244i	−0.271041	−	0.469457i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−68.7895			−2.65559
$672$		0			0
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	−0.988968	−	0.148132i	$-0.952674\pi$
$673$	−9.50000	+	16.4545i	−0.366198	+	0.634274i	0.622770	+	0.782405i	$0.286007\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.7490			1.22021			0.610107	−	0.792319i	$-0.291126\pi$
$677$	31.7490			1.22021			0.610107	+	0.792319i	$0.291126\pi$
$678$		0			0
$679$	−2.00000	+	3.46410i	−0.0767530	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.1660	−	36.6606i	−0.809895	−	1.40278i	−0.912936	−	0.408102i	$-0.866191\pi$
$683$	−21.1660	−	36.6606i	−0.809895	−	1.40278i	0.103041	−	0.994677i	$-0.467143\pi$
$684$		0			0
$685$	10.5000	+	18.1865i	0.401184	+	0.694872i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	27.7804	+	6.87386i	1.05835	+	0.261873i
$690$		0			0
$691$	−7.00000	+	12.1244i	−0.266293	+	0.461232i	−0.967901	−	0.251330i	$-0.919132\pi$
$691$	−7.00000	+	12.1244i	−0.266293	+	0.461232i	0.701609	+	0.712562i	$0.252465\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.1660	+	36.6606i	0.802873	+	1.39062i
$696$		0			0
$697$	63.0000			2.38630
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31.7490			−1.19914			−0.599572	−	0.800321i	$-0.704662\pi$
$701$	−31.7490			−1.19914			−0.599572	+	0.800321i	$0.704662\pi$
$702$		0			0
$703$	−18.0000			−0.678883
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.29150			−0.199007
$708$		0			0
$709$	−5.50000	−	9.52628i	−0.206557	−	0.357767i	0.744071	−	0.668101i	$-0.232892\pi$
$709$	−5.50000	−	9.52628i	−0.206557	−	0.357767i	−0.950628	+	0.310334i	$0.899559\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.5830	−	18.3303i	0.396337	−	0.686475i
$714$		0			0
$715$	35.0000	−	36.3731i	1.30893	−	1.36028i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.29150	+	9.16515i	0.197340	+	0.341802i	0.947665	−	0.319266i	$-0.103436\pi$
$719$	5.29150	+	9.16515i	0.197340	+	0.341802i	−0.750325	+	0.661069i	$0.770103\pi$
$720$		0			0
$721$	−14.0000	−	24.2487i	−0.521387	−	0.903069i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.64575	+	4.58258i	−0.0982607	+	0.170193i
$726$		0			0
$727$	2.00000			0.0741759			0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000			0.0741759			0.0370879	+	0.999312i	$0.488192\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−7.93725	+	13.7477i	−0.293570	+	0.508478i
$732$		0			0
$733$	−1.00000			−0.0369358			−0.0184679	−	0.999829i	$-0.505879\pi$
$733$	−1.00000			−0.0369358			−0.0184679	+	0.999829i	$0.505879\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.29150	+	9.16515i	0.194915	+	0.337603i
$738$		0			0
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	−0.734311	−	0.678813i	$-0.762495\pi$
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	0.955025	+	0.296526i	$0.0958281\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	10.5830	−	18.3303i	0.388253	−	0.672474i	−0.603962	−	0.797013i	$-0.706412\pi$
$743$	10.5830	−	18.3303i	0.388253	−	0.672474i	0.992215	+	0.124540i	$0.0397454\pi$
$744$		0			0
$745$	10.5000	+	18.1865i	0.384690	+	0.666303i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.5830			0.386695
$750$		0			0
$751$	9.00000	−	15.5885i	0.328415	−	0.568831i	−0.653783	−	0.756682i	$-0.726819\pi$
$751$	9.00000	−	15.5885i	0.328415	−	0.568831i	0.982197	+	0.187851i	$0.0601523\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	47.6235			1.73320
$756$		0			0
$757$	15.0000	−	25.9808i	0.545184	−	0.944287i	−0.453411	−	0.891302i	$-0.649793\pi$
$757$	15.0000	−	25.9808i	0.545184	−	0.944287i	0.998595	−	0.0529853i	$-0.0168737\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−10.0000	−	17.3205i	−0.362024	−	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.5830	+	36.6606i	0.382130	+	1.32374i
$768$		0			0
$769$	−15.0000	+	25.9808i	−0.540914	+	0.936890i	0.457938	+	0.888984i	$0.348588\pi$
$769$	−15.0000	+	25.9808i	−0.540914	+	0.936890i	−0.998852	+	0.0479061i	$0.984745\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.29150	−	9.16515i	−0.190322	−	0.329648i	0.755035	−	0.655685i	$-0.227620\pi$
$773$	−5.29150	−	9.16515i	−0.190322	−	0.329648i	−0.945357	+	0.326037i	$0.894287\pi$
$774$		0			0
$775$	−8.00000			−0.287368
$776$		0			0
$777$		0			0
$778$		0			0
$779$	47.6235			1.70629
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$		0			0
$784$		0			0
$785$	44.9778			1.60533
$786$		0			0
$787$	−26.0000	−	45.0333i	−0.926800	−	1.60526i	−0.788641	−	0.614855i	$-0.789215\pi$
$787$	−26.0000	−	45.0333i	−0.926800	−	1.60526i	−0.138159	−	0.990410i	$-0.544119\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.64575	+	4.58258i	−0.0940721	+	0.162938i
$792$		0			0
$793$	−13.0000	−	45.0333i	−0.461644	−	1.59918i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$	21.0000	+	36.3731i	0.742927	+	1.28679i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.5203	+	32.0780i	−0.653566	+	1.13201i
$804$		0			0
$805$	28.0000			0.986870
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.9059	+	20.6216i	−0.418588	+	0.725017i	−0.995798	−	0.0915798i	$-0.970808\pi$
$809$	−11.9059	+	20.6216i	−0.418588	+	0.725017i	0.577209	+	0.816596i	$0.304142\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.29150	−	9.16515i	−0.185353	−	0.321041i
$816$		0			0
$817$	−6.00000	+	10.3923i	−0.209913	+	0.363581i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.29150	+	9.16515i	−0.184675	+	0.319866i	−0.943467	−	0.331467i	$-0.892456\pi$
$821$	−5.29150	+	9.16515i	−0.184675	+	0.319866i	0.758792	+	0.651333i	$0.225790\pi$
$822$		0			0
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	0.577474	−	0.816409i	$-0.304038\pi$
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	−0.995768	+	0.0919029i	$0.970705\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−3.50000	+	6.06218i	−0.121560	+	0.210548i	−0.920383	−	0.391018i	$-0.872123\pi$
$829$	−3.50000	+	6.06218i	−0.121560	+	0.210548i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23.8118			−0.825029
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.29150	+	9.16515i	0.182683	+	0.316416i	0.942793	−	0.333378i	$-0.108188\pi$
$839$	5.29150	+	9.16515i	0.182683	+	0.316416i	−0.760110	+	0.649794i	$0.774855\pi$
$840$		0			0
$841$	11.0000	+	19.0526i	0.379310	+	0.656985i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	30.4261	+	16.0390i	1.04669	+	0.551759i
$846$		0			0
$847$	−17.0000	+	29.4449i	−0.584127	+	1.01174i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.93725	+	13.7477i	0.272086	+	0.471266i
$852$		0			0
$853$	−47.0000			−1.60925			−0.804625	−	0.593784i	$-0.797633\pi$
$853$	−47.0000			−1.60925			−0.804625	+	0.593784i	$0.797633\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−34.3948			−1.17490			−0.587451	−	0.809259i	$-0.699869\pi$
$857$	−34.3948			−1.17490			−0.587451	+	0.809259i	$0.699869\pi$
$858$		0			0
$859$	22.0000			0.750630			0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000			0.750630			0.375315	+	0.926897i	$0.377534\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.29150			−0.180125			−0.0900624	−	0.995936i	$-0.528707\pi$
$863$	−5.29150			−0.180125			−0.0900624	+	0.995936i	$0.528707\pi$
$864$		0			0
$865$	−14.0000	−	24.2487i	−0.476014	−	0.824481i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.5830	−	18.3303i	0.359004	−	0.621813i
$870$		0			0
$871$	−5.00000	+	5.19615i	−0.169419	+	0.176065i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.93725	+	13.7477i	0.268328	+	0.464758i
$876$		0			0
$877$	13.5000	+	23.3827i	0.455863	+	0.789577i	0.998737	−	0.0502365i	$-0.0159975\pi$
$877$	13.5000	+	23.3827i	0.455863	+	0.789577i	−0.542875	+	0.839814i	$0.682664\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.96863	−	6.87386i	0.133706	−	0.231586i	−0.791396	−	0.611304i	$-0.790645\pi$
$881$	3.96863	−	6.87386i	0.133706	−	0.231586i	0.925103	+	0.379717i	$0.123979\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	26.4575	−	45.8258i	0.888356	−	1.53868i	0.0465386	−	0.998916i	$-0.485181\pi$
$887$	26.4575	−	45.8258i	0.888356	−	1.53868i	0.841818	−	0.539762i	$-0.181486\pi$
$888$		0			0
$889$	−32.0000			−1.07325
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.8745	+	27.4955i	0.531220	+	0.920100i
$894$		0			0
$895$	21.0000	−	36.3731i	0.701953	−	1.21582i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.29150	−	9.16515i	0.176481	−	0.305675i
$900$		0			0
$901$	−31.5000	−	54.5596i	−1.04942	−	1.81764i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23.8118			−0.791530
$906$		0			0
$907$	−6.00000	+	10.3923i	−0.199227	+	0.345071i	−0.948278	−	0.317441i	$-0.897176\pi$
$907$	−6.00000	+	10.3923i	−0.199227	+	0.345071i	0.749051	+	0.662512i	$0.230510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	31.7490			1.05189			0.525946	−	0.850518i	$-0.323711\pi$
$911$	31.7490			1.05189			0.525946	+	0.850518i	$0.323711\pi$
$912$		0			0
$913$	42.0000	−	72.7461i	1.39000	−	2.40755i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.5830	+	18.3303i	0.349482	+	0.605320i
$918$		0			0
$919$	−12.0000	−	20.7846i	−0.395843	−	0.685621i	0.597365	−	0.801970i	$-0.296214\pi$
$919$	−12.0000	−	20.7846i	−0.395843	−	0.685621i	−0.993208	+	0.116348i	$0.962881\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.29150	+	18.3303i	0.174172	+	0.603349i
$924$		0			0
$925$	3.00000	−	5.19615i	0.0986394	−	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.96863	+	6.87386i	0.130206	+	0.225524i	0.923756	−	0.382981i	$-0.125103\pi$
$929$	3.96863	+	6.87386i	0.130206	+	0.225524i	−0.793550	+	0.608506i	$0.791769\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−111.122			−3.63406
$936$		0			0
$937$	−9.00000			−0.294017			−0.147009	−	0.989135i	$-0.546964\pi$
$937$	−9.00000			−0.294017			−0.147009	+	0.989135i	$0.546964\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.5830			−0.344996			−0.172498	−	0.985010i	$-0.555184\pi$
$941$	−10.5830			−0.344996			−0.172498	+	0.985010i	$0.555184\pi$
$942$		0			0
$943$	−21.0000	−	36.3731i	−0.683854	−	1.18447i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	−24.5000	−	6.06218i	−0.795304	−	0.196787i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.29150	−	9.16515i	−0.171409	−	0.296888i	0.767504	−	0.641044i	$-0.221498\pi$
$953$	−5.29150	−	9.16515i	−0.171409	−	0.296888i	−0.938913	+	0.344156i	$0.888165\pi$
$954$		0			0
$955$	28.0000	+	48.4974i	0.906059	+	1.56934i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.93725	−	13.7477i	0.256307	−	0.443937i
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	33.0719	−	57.2822i	1.06462	−	1.84398i
$966$		0			0
$967$	22.0000			0.707472			0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000			0.707472			0.353736	+	0.935345i	$0.384911\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15.8745	+	27.4955i	0.509437	+	0.882371i	0.999940	+	0.0109316i	$0.00347972\pi$
$971$	15.8745	+	27.4955i	0.509437	+	0.882371i	−0.490503	+	0.871439i	$0.663187\pi$
$972$		0			0
$973$	16.0000	−	27.7128i	0.512936	−	0.888432i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.8431	−	34.3693i	0.634838	−	1.09957i	−0.351711	−	0.936109i	$-0.614400\pi$
$977$	19.8431	−	34.3693i	0.634838	−	1.09957i	0.986549	−	0.163463i	$-0.0522666\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−10.5830			−0.337545			−0.168773	−	0.985655i	$-0.553980\pi$
$983$	−10.5830			−0.337545			−0.168773	+	0.985655i	$0.553980\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	10.5830			0.336520
$990$		0			0
$991$	−27.0000	+	46.7654i	−0.857683	+	1.48555i	0.0164499	+	0.999865i	$0.494764\pi$
$991$	−27.0000	+	46.7654i	−0.857683	+	1.48555i	−0.874133	+	0.485686i	$0.838570\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13.2288	−	22.9129i	−0.419380	−	0.726387i
$996$		0			0
$997$	3.50000	+	6.06218i	0.110846	+	0.191991i	0.916112	−	0.400923i	$-0.131311\pi$
$997$	3.50000	+	6.06218i	0.110846	+	0.191991i	−0.805266	+	0.592914i	$0.797977\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.t.o.1153.2		4
3.2	odd	2	inner	1872.2.t.o.1153.1		4
4.3	odd	2		468.2.l.e.217.2	yes	4
12.11	even	2		468.2.l.e.217.1	✓	4
13.3	even	3	inner	1872.2.t.o.289.2		4
39.29	odd	6	inner	1872.2.t.o.289.1		4
52.3	odd	6		468.2.l.e.289.2	yes	4
52.7	even	12		6084.2.b.n.4393.4		4
52.19	even	12		6084.2.b.n.4393.1		4
52.35	odd	6		6084.2.a.q.1.2		2
52.43	odd	6		6084.2.a.w.1.1		2
156.35	even	6		6084.2.a.q.1.1		2
156.59	odd	12		6084.2.b.n.4393.2		4
156.71	odd	12		6084.2.b.n.4393.3		4
156.95	even	6		6084.2.a.w.1.2		2
156.107	even	6		468.2.l.e.289.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.l.e.217.1	✓	4	12.11	even	2
468.2.l.e.217.2	yes	4	4.3	odd	2
468.2.l.e.289.1	yes	4	156.107	even	6
468.2.l.e.289.2	yes	4	52.3	odd	6
1872.2.t.o.289.1		4	39.29	odd	6	inner
1872.2.t.o.289.2		4	13.3	even	3	inner
1872.2.t.o.1153.1		4	3.2	odd	2	inner
1872.2.t.o.1153.2		4	1.1	even	1	trivial
6084.2.a.q.1.1		2	156.35	even	6
6084.2.a.q.1.2		2	52.35	odd	6
6084.2.a.w.1.1		2	52.43	odd	6
6084.2.a.w.1.2		2	156.95	even	6
6084.2.b.n.4393.1		4	52.19	even	12
6084.2.b.n.4393.2		4	156.59	odd	12
6084.2.b.n.4393.3		4	156.71	odd	12
6084.2.b.n.4393.4		4	52.7	even	12

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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.t (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+2.64575 q^{5} +(-1.00000 - 1.73205i) q^{7} +(2.64575 - 4.58258i) q^{11} +(3.50000 + 0.866025i) q^{13} +(-3.96863 - 6.87386i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(-2.64575 + 4.58258i) q^{23} +2.00000 q^{25} +(-1.32288 + 2.29129i) q^{29} -4.00000 q^{31} +(-2.64575 - 4.58258i) q^{35} +(1.50000 - 2.59808i) q^{37} +(-3.96863 + 6.87386i) q^{41} +(-1.00000 - 1.73205i) q^{43} -5.29150 q^{47} +(1.50000 - 2.59808i) q^{49} +7.93725 q^{53} +(7.00000 - 12.1244i) q^{55} +(5.29150 + 9.16515i) q^{59} +(-6.50000 - 11.2583i) q^{61} +(9.26013 + 2.29129i) q^{65} +(-1.00000 + 1.73205i) q^{67} +(2.64575 + 4.58258i) q^{71} -7.00000 q^{73} -10.5830 q^{77} +4.00000 q^{79} +15.8745 q^{83} +(-10.5000 - 18.1865i) q^{85} +(-2.00000 - 6.92820i) q^{91} +(-7.93725 - 13.7477i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 14 q^{13} - 12 q^{19} + 8 q^{25} - 16 q^{31} + 6 q^{37} - 4 q^{43} + 6 q^{49} + 28 q^{55} - 26 q^{61} - 4 q^{67} - 28 q^{73} + 16 q^{79} - 42 q^{85} - 8 q^{91} - 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 14 * q^13 - 12 * q^19 + 8 * q^25 - 16 * q^31 + 6 * q^37 - 4 * q^43 + 6 * q^49 + 28 * q^55 - 26 * q^61 - 4 * q^67 - 28 * q^73 + 16 * q^79 - 42 * q^85 - 8 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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