LMFDB - Embedded newform 867.2.d.a.577.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(577,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("867.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (of order $2$, degree $1$, 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:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	867.577
Dual form			867.2.d.a.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000 q^{4} +3.00000i q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000 q^{4} +3.00000i q^{5} +4.00000i q^{7} -1.00000 q^{9} +3.00000i q^{11} -2.00000i q^{12} -1.00000 q^{13} -3.00000 q^{15} +4.00000 q^{16} +1.00000 q^{19} -6.00000i q^{20} -4.00000 q^{21} -9.00000i q^{23} -4.00000 q^{25} -1.00000i q^{27} -8.00000i q^{28} +6.00000i q^{29} +2.00000i q^{31} -3.00000 q^{33} -12.0000 q^{35} +2.00000 q^{36} -4.00000i q^{37} -1.00000i q^{39} +3.00000i q^{41} +7.00000 q^{43} -6.00000i q^{44} -3.00000i q^{45} -6.00000 q^{47} +4.00000i q^{48} -9.00000 q^{49} +2.00000 q^{52} +6.00000 q^{53} -9.00000 q^{55} +1.00000i q^{57} -6.00000 q^{59} +6.00000 q^{60} -8.00000i q^{61} -4.00000i q^{63} -8.00000 q^{64} -3.00000i q^{65} -4.00000 q^{67} +9.00000 q^{69} +12.0000i q^{71} +2.00000i q^{73} -4.00000i q^{75} -2.00000 q^{76} -12.0000 q^{77} +10.0000i q^{79} +12.0000i q^{80} +1.00000 q^{81} +6.00000 q^{83} +8.00000 q^{84} -6.00000 q^{87} -4.00000i q^{91} +18.0000i q^{92} -2.00000 q^{93} +3.00000i q^{95} -16.0000i q^{97} -3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 2 * q^9	$ 2 q - 4 q^{4} - 2 q^{9} - 2 q^{13} - 6 q^{15} + 8 q^{16} + 2 q^{19} - 8 q^{21} - 8 q^{25} - 6 q^{33} - 24 q^{35} + 4 q^{36} + 14 q^{43} - 12 q^{47} - 18 q^{49} + 4 q^{52} + 12 q^{53} - 18 q^{55} - 12 q^{59} + 12 q^{60} - 16 q^{64} - 8 q^{67} + 18 q^{69} - 4 q^{76} - 24 q^{77} + 2 q^{81} + 12 q^{83} + 16 q^{84} - 12 q^{87} - 4 q^{93}+O(q^{100}) $ 2 * q - 4 * q^4 - 2 * q^9 - 2 * q^13 - 6 * q^15 + 8 * q^16 + 2 * q^19 - 8 * q^21 - 8 * q^25 - 6 * q^33 - 24 * q^35 + 4 * q^36 + 14 * q^43 - 12 * q^47 - 18 * q^49 + 4 * q^52 + 12 * q^53 - 18 * q^55 - 12 * q^59 + 12 * q^60 - 16 * q^64 - 8 * q^67 + 18 * q^69 - 4 * q^76 - 24 * q^77 + 2 * q^81 + 12 * q^83 + 16 * q^84 - 12 * q^87 - 4 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.00000	−1.00000
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	− 2.00000i	− 0.577350i
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	4.00000	1.00000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	− 6.00000i	− 1.34164i
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 9.00000i	− 1.87663i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$	− 9.00000i	− 1.87663i	0.345782	−	0.938315i	$-0.387614\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	− 8.00000i	− 1.51186i
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	2.00000i	0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$	2.00000i	0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	−12.0000	−2.02837
$36$	2.00000	0.333333
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	− 1.00000i	− 0.160128i
$40$	0	0
$41$	3.00000i	0.468521i	0.972174	+	0.234261i	$0.0752669\pi$
$41$	3.00000i	0.468521i	−0.972174	+	0.234261i	$0.924733\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	− 6.00000i	− 0.904534i
$45$	− 3.00000i	− 0.447214i
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	4.00000i	0.577350i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−9.00000	−1.21356
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	6.00000	0.774597
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	−8.00000	−1.00000
$65$	− 3.00000i	− 0.372104i
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	9.00000	1.08347
$70$	0	0
$71$	12.0000i	1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$	12.0000i	1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	− 4.00000i	− 0.461880i
$76$	−2.00000	−0.229416
$77$	−12.0000	−1.36753
$78$	0	0
$79$	10.0000i	1.12509i	0.826767	+	0.562544i	$0.190177\pi$
$79$	10.0000i	1.12509i	−0.826767	+	0.562544i	$0.809823\pi$
$80$	12.0000i	1.34164i
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	− 4.00000i	− 0.419314i
$92$	18.0000i	1.87663i
$93$	−2.00000	−0.207390
$94$	0	0
$95$	3.00000i	0.307794i
$96$	0	0
$97$	− 16.0000i	− 1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$	− 16.0000i	− 1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$	0	0
$99$	− 3.00000i	− 0.301511i
$100$	8.00000	0.800000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	− 12.0000i	− 1.17108i
$106$	0	0
$107$	9.00000i	0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$	9.00000i	0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$	2.00000i	0.192450i
$109$	− 20.0000i	− 1.91565i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	− 20.0000i	− 1.91565i	0.287348	−	0.957826i	$-0.407226\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	16.0000i	1.51186i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	0	0
$115$	27.0000	2.51776
$116$	− 12.0000i	− 1.11417i
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	−3.00000	−0.270501
$124$	− 4.00000i	− 0.359211i
$125$	3.00000i	0.268328i
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	7.00000i	0.616316i
$130$	0	0
$131$	3.00000i	0.262111i	0.991375	+	0.131056i	$0.0418366\pi$
$131$	3.00000i	0.262111i	−0.991375	+	0.131056i	$0.958163\pi$
$132$	6.00000	0.522233
$133$	4.00000i	0.346844i
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	2.00000i	0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$	2.00000i	0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$	24.0000	2.02837
$141$	− 6.00000i	− 0.505291i
$142$	0	0
$143$	− 3.00000i	− 0.250873i
$144$	−4.00000	−0.333333
$145$	−18.0000	−1.49482
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	8.00000i	0.657596i
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	2.00000i	0.160128i
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	6.00000i	0.475831i
$160$	0	0
$161$	36.0000	2.83720
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	− 6.00000i	− 0.468521i
$165$	− 9.00000i	− 0.700649i
$166$	0	0
$167$	21.0000i	1.62503i	0.582941	+	0.812514i	$0.301902\pi$
$167$	21.0000i	1.62503i	−0.582941	+	0.812514i	$0.698098\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−14.0000	−1.06749
$173$	15.0000i	1.14043i	0.821496	+	0.570214i	$0.193140\pi$
$173$	15.0000i	1.14043i	−0.821496	+	0.570214i	$0.806860\pi$
$174$	0	0
$175$	− 16.0000i	− 1.20949i
$176$	12.0000i	0.904534i
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	6.00000i	0.447214i
$181$	− 14.0000i	− 1.04061i	−0.853980	−	0.520306i	$-0.825818\pi$
$181$	− 14.0000i	− 1.04061i	0.853980	−	0.520306i	$-0.174182\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	0	0
$188$	12.0000	0.875190
$189$	4.00000	0.290957
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	− 8.00000i	− 0.577350i
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	18.0000	1.28571
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	0	0
$199$	− 16.0000i	− 1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$	− 16.0000i	− 1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$	0	0
$201$	− 4.00000i	− 0.282138i
$202$	0	0
$203$	−24.0000	−1.68447
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	9.00000i	0.625543i
$208$	−4.00000	−0.277350
$209$	3.00000i	0.207514i
$210$	0	0
$211$	− 2.00000i	− 0.137686i	−0.997628	−	0.0688428i	$-0.978069\pi$
$211$	− 2.00000i	− 0.137686i	0.997628	−	0.0688428i	$-0.0219307\pi$
$212$	−12.0000	−0.824163
$213$	−12.0000	−0.822226
$214$	0	0
$215$	21.0000i	1.43219i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−2.00000	−0.135147
$220$	18.0000	1.21356
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650	0.0334825	−	0.999439i	$-0.489340\pi$
$223$	1.00000	0.0669650	0.0334825	+	0.999439i	$0.489340\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	− 2.00000i	− 0.132453i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	− 12.0000i	− 0.789542i
$232$	0	0
$233$	21.0000i	1.37576i	0.725826	+	0.687878i	$0.241458\pi$
$233$	21.0000i	1.37576i	−0.725826	+	0.687878i	$0.758542\pi$
$234$	0	0
$235$	− 18.0000i	− 1.17419i
$236$	12.0000	0.781133
$237$	−10.0000	−0.649570
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−12.0000	−0.774597
$241$	8.00000i	0.515325i	0.966235	+	0.257663i	$0.0829523\pi$
$241$	8.00000i	0.515325i	−0.966235	+	0.257663i	$0.917048\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	16.0000i	1.02430i
$245$	− 27.0000i	− 1.72497i
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	6.00000i	0.380235i
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	8.00000i	0.503953i
$253$	27.0000	1.69748
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	6.00000i	0.372104i
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	18.0000i	1.10573i
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	− 15.0000i	− 0.914566i	−0.889321	−	0.457283i	$-0.848823\pi$
$269$	− 15.0000i	− 0.914566i	0.889321	−	0.457283i	$-0.151177\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	− 12.0000i	− 0.723627i
$276$	−18.0000	−1.08347
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	− 2.00000i	− 0.119737i
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	10.0000i	0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$	10.0000i	0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$	− 24.0000i	− 1.42414i
$285$	−3.00000	−0.177705
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	0	0
$290$	0	0
$291$	16.0000	0.937937
$292$	− 4.00000i	− 0.234082i
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	− 18.0000i	− 1.04800i
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	9.00000i	0.520483i
$300$	8.00000i	0.461880i
$301$	28.0000i	1.61389i
$302$	0	0
$303$	0	0
$304$	4.00000	0.229416
$305$	24.0000	1.37424
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	24.0000	1.36753
$309$	5.00000i	0.284440i
$310$	0	0
$311$	− 24.0000i	− 1.36092i	−0.732787	−	0.680458i	$-0.761781\pi$
$311$	− 24.0000i	− 1.36092i	0.732787	−	0.680458i	$-0.238219\pi$
$312$	0	0
$313$	16.0000i	0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$	16.0000i	0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$	0	0
$315$	12.0000	0.676123
$316$	− 20.0000i	− 1.12509i
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	− 24.0000i	− 1.34164i
$321$	−9.00000	−0.502331
$322$	0	0
$323$	0	0
$324$	−2.00000	−0.111111
$325$	4.00000	0.221880
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	− 24.0000i	− 1.32316i
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	−12.0000	−0.658586
$333$	4.00000i	0.219199i
$334$	0	0
$335$	− 12.0000i	− 0.655630i
$336$	−16.0000	−0.872872
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	27.0000i	1.45363i
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	12.0000	0.643268
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	1.00000i	0.0533761i
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	2.00000i	0.104973i
$364$	8.00000i	0.419314i
$365$	−6.00000	−0.314054
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	− 36.0000i	− 1.87663i
$369$	− 3.00000i	− 0.156174i
$370$	0	0
$371$	24.0000i	1.24602i
$372$	4.00000	0.207390
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	− 6.00000i	− 0.309016i
$378$	0	0
$379$	32.0000i	1.64373i	0.569683	+	0.821865i	$0.307066\pi$
$379$	32.0000i	1.64373i	−0.569683	+	0.821865i	$0.692934\pi$
$380$	− 6.00000i	− 0.307794i
$381$	13.0000i	0.666010i
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	− 36.0000i	− 1.83473i
$386$	0	0
$387$	−7.00000	−0.355830
$388$	32.0000i	1.62455i
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−3.00000	−0.151330
$394$	0	0
$395$	−30.0000	−1.50946
$396$	6.00000i	0.301511i
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	−16.0000	−0.800000
$401$	15.0000i	0.749064i	0.927214	+	0.374532i	$0.122197\pi$
$401$	15.0000i	0.749064i	−0.927214	+	0.374532i	$0.877803\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	0	0
$405$	3.00000i	0.149071i
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	−10.0000	−0.492665
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	18.0000i	0.883585i
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	− 12.0000i	− 0.586238i	−0.956076	−	0.293119i	$-0.905307\pi$
$419$	− 12.0000i	− 0.586238i	0.956076	−	0.293119i	$-0.0946933\pi$
$420$	24.0000i	1.17108i
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.0000	1.54859
$428$	− 18.0000i	− 0.870063i
$429$	3.00000	0.144841
$430$	0	0
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	− 4.00000i	− 0.192450i
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	0	0
$435$	− 18.0000i	− 0.863034i
$436$	40.0000i	1.91565i
$437$	− 9.00000i	− 0.430528i
$438$	0	0
$439$	− 28.0000i	− 1.33637i	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	− 28.0000i	− 1.33637i	0.743996	−	0.668184i	$-0.232928\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	0	0
$447$	− 18.0000i	− 0.851371i
$448$	− 32.0000i	− 1.51186i
$449$	6.00000i	0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$	6.00000i	0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	− 18.0000i	− 0.846649i
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	19.0000	0.888783	0.444391	−	0.895833i	$-0.353420\pi$
$457$	19.0000	0.888783	0.444391	+	0.895833i	$0.353420\pi$
$458$	0	0
$459$	0	0
$460$	−54.0000	−2.51776
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	24.0000i	1.11417i
$465$	− 6.00000i	− 0.278243i
$466$	0	0
$467$	−42.0000	−1.94353	−0.971764	−	0.235954i	$-0.924178\pi$
$467$	−42.0000	−1.94353	−0.971764	+	0.235954i	$0.924178\pi$
$468$	−2.00000	−0.0924500
$469$	− 16.0000i	− 0.738811i
$470$	0	0
$471$	11.0000i	0.506853i
$472$	0	0
$473$	21.0000i	0.965581i
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	− 27.0000i	− 1.23366i	−0.787096	−	0.616831i	$-0.788416\pi$
$479$	− 27.0000i	− 1.23366i	0.787096	−	0.616831i	$-0.211584\pi$
$480$	0	0
$481$	4.00000i	0.182384i
$482$	0	0
$483$	36.0000i	1.63806i
$484$	−4.00000	−0.181818
$485$	48.0000	2.17957
$486$	0	0
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	0	0
$495$	9.00000	0.404520
$496$	8.00000i	0.359211i
$497$	−48.0000	−2.15309
$498$	0	0
$499$	22.0000i	0.984855i	0.870353	+	0.492428i	$0.163890\pi$
$499$	22.0000i	0.984855i	−0.870353	+	0.492428i	$0.836110\pi$
$500$	− 6.00000i	− 0.268328i
$501$	−21.0000	−0.938211
$502$	0	0
$503$	− 15.0000i	− 0.668817i	−0.942428	−	0.334408i	$-0.891463\pi$
$503$	− 15.0000i	− 0.668817i	0.942428	−	0.334408i	$-0.108537\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	−26.0000	−1.15356
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	− 1.00000i	− 0.0441511i
$514$	0	0
$515$	15.0000i	0.660979i
$516$	− 14.0000i	− 0.616316i
$517$	− 18.0000i	− 0.791639i
$518$	0	0
$519$	−15.0000	−0.658427
$520$	0	0
$521$	− 21.0000i	− 0.920027i	−0.887912	−	0.460013i	$-0.847845\pi$
$521$	− 21.0000i	− 0.920027i	0.887912	−	0.460013i	$-0.152155\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	− 6.00000i	− 0.262111i
$525$	16.0000	0.698297
$526$	0	0
$527$	0	0
$528$	−12.0000	−0.522233
$529$	−58.0000	−2.52174
$530$	0	0
$531$	6.00000	0.260378
$532$	− 8.00000i	− 0.346844i
$533$	− 3.00000i	− 0.129944i
$534$	0	0
$535$	−27.0000	−1.16731
$536$	0	0
$537$	6.00000i	0.258919i
$538$	0	0
$539$	− 27.0000i	− 1.16297i
$540$	−6.00000	−0.258199
$541$	− 16.0000i	− 0.687894i	−0.938989	−	0.343947i	$-0.888236\pi$
$541$	− 16.0000i	− 0.687894i	0.938989	−	0.343947i	$-0.111764\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	60.0000	2.57012
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	12.0000	0.512615
$549$	8.00000i	0.341432i
$550$	0	0
$551$	6.00000i	0.255609i
$552$	0	0
$553$	−40.0000	−1.70097
$554$	0	0
$555$	12.0000i	0.509372i
$556$	− 4.00000i	− 0.169638i
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−7.00000	−0.296068
$560$	−48.0000	−2.02837
$561$	0	0
$562$	0	0
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	12.0000i	0.505291i
$565$	−27.0000	−1.13590
$566$	0	0
$567$	4.00000i	0.167984i
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	− 8.00000i	− 0.334790i	−0.985890	−	0.167395i	$-0.946465\pi$
$571$	− 8.00000i	− 0.334790i	0.985890	−	0.167395i	$-0.0535355\pi$
$572$	6.00000i	0.250873i
$573$	18.0000i	0.751961i
$574$	0	0
$575$	36.0000i	1.50130i
$576$	8.00000	0.333333
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	−22.0000	−0.914289
$580$	36.0000	1.49482
$581$	24.0000i	0.995688i
$582$	0	0
$583$	18.0000i	0.745484i
$584$	0	0
$585$	3.00000i	0.124035i
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	18.0000i	0.742307i
$589$	2.00000i	0.0824086i
$590$	0	0
$591$	3.00000	0.123404
$592$	− 16.0000i	− 0.657596i
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	36.0000	1.47462
$597$	16.0000	0.654836
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	− 38.0000i	− 1.55005i	−0.631929	−	0.775026i	$-0.717737\pi$
$601$	− 38.0000i	− 1.55005i	0.631929	−	0.775026i	$-0.282263\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	16.0000	0.651031
$605$	6.00000i	0.243935i
$606$	0	0
$607$	38.0000i	1.54237i	0.636610	+	0.771186i	$0.280336\pi$
$607$	38.0000i	1.54237i	−0.636610	+	0.771186i	$0.719664\pi$
$608$	0	0
$609$	− 24.0000i	− 0.972529i
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	0	0
$615$	− 9.00000i	− 0.362915i
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	10.0000i	0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$	10.0000i	0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$	12.0000	0.481932
$621$	−9.00000	−0.361158
$622$	0	0
$623$	0	0
$624$	− 4.00000i	− 0.160128i
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−3.00000	−0.119808
$628$	−22.0000	−0.877896
$629$	0	0
$630$	0	0
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	0	0
$633$	2.00000	0.0794929
$634$	0	0
$635$	39.0000i	1.54767i
$636$	− 12.0000i	− 0.475831i
$637$	9.00000	0.356593
$638$	0	0
$639$	− 12.0000i	− 0.474713i
$640$	0	0
$641$	− 33.0000i	− 1.30342i	−0.758468	−	0.651711i	$-0.774052\pi$
$641$	− 33.0000i	− 1.30342i	0.758468	−	0.651711i	$-0.225948\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	−72.0000	−2.83720
$645$	−21.0000	−0.826874
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	− 18.0000i	− 0.706562i
$650$	0	0
$651$	− 8.00000i	− 0.313545i
$652$	4.00000i	0.156652i
$653$	− 27.0000i	− 1.05659i	−0.849060	−	0.528296i	$-0.822831\pi$
$653$	− 27.0000i	− 1.05659i	0.849060	−	0.528296i	$-0.177169\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	12.0000i	0.468521i
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	18.0000i	0.700649i
$661$	31.0000	1.20576	0.602880	−	0.797832i	$-0.294020\pi$
$661$	31.0000	1.20576	0.602880	+	0.797832i	$0.294020\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	54.0000	2.09089
$668$	− 42.0000i	− 1.62503i
$669$	1.00000i	0.0386622i
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	22.0000i	0.848038i	0.905653	+	0.424019i	$0.139381\pi$
$673$	22.0000i	0.848038i	−0.905653	+	0.424019i	$0.860619\pi$
$674$	0	0
$675$	4.00000i	0.153960i
$676$	24.0000	0.923077
$677$	− 27.0000i	− 1.03769i	−0.854867	−	0.518847i	$-0.826361\pi$
$677$	− 27.0000i	− 1.03769i	0.854867	−	0.518847i	$-0.173639\pi$
$678$	0	0
$679$	64.0000	2.45609
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	3.00000i	0.114792i	0.998351	+	0.0573959i	$0.0182797\pi$
$683$	3.00000i	0.114792i	−0.998351	+	0.0573959i	$0.981720\pi$
$684$	2.00000	0.0764719
$685$	− 18.0000i	− 0.687745i
$686$	0	0
$687$	− 14.0000i	− 0.534133i
$688$	28.0000	1.06749
$689$	−6.00000	−0.228582
$690$	0	0
$691$	− 20.0000i	− 0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$	− 20.0000i	− 0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	− 30.0000i	− 1.14043i
$693$	12.0000	0.455842
$694$	0	0
$695$	−6.00000	−0.227593
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−21.0000	−0.794293
$700$	32.0000i	1.20949i
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	− 24.0000i	− 0.904534i
$705$	18.0000	0.677919
$706$	0	0
$707$	0	0
$708$	12.0000i	0.450988i
$709$	14.0000i	0.525781i	0.964826	+	0.262891i	$0.0846758\pi$
$709$	14.0000i	0.525781i	−0.964826	+	0.262891i	$0.915324\pi$
$710$	0	0
$711$	− 10.0000i	− 0.375029i
$712$	0	0
$713$	18.0000	0.674105
$714$	0	0
$715$	9.00000	0.336581
$716$	−12.0000	−0.448461
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	3.00000i	0.111881i	0.998434	+	0.0559406i	$0.0178157\pi$
$719$	3.00000i	0.111881i	−0.998434	+	0.0559406i	$0.982184\pi$
$720$	− 12.0000i	− 0.447214i
$721$	20.0000i	0.744839i
$722$	0	0
$723$	−8.00000	−0.297523
$724$	28.0000i	1.04061i
$725$	− 24.0000i	− 0.891338i
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	−16.0000	−0.591377
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	27.0000	0.995910
$736$	0	0
$737$	− 12.0000i	− 0.442026i
$738$	0	0
$739$	1.00000	0.0367856	0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000	0.0367856	0.0183928	+	0.999831i	$0.494145\pi$
$740$	−24.0000	−0.882258
$741$	− 1.00000i	− 0.0367359i
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	− 54.0000i	− 1.97841i
$746$	0	0
$747$	−6.00000	−0.219529
$748$	0	0
$749$	−36.0000	−1.31541
$750$	0	0
$751$	− 46.0000i	− 1.67856i	−0.543696	−	0.839282i	$-0.682976\pi$
$751$	− 46.0000i	− 1.67856i	0.543696	−	0.839282i	$-0.317024\pi$
$752$	−24.0000	−0.875190
$753$	24.0000i	0.874609i
$754$	0	0
$755$	− 24.0000i	− 0.873449i
$756$	−8.00000	−0.290957
$757$	43.0000	1.56286	0.781431	−	0.623992i	$-0.214490\pi$
$757$	43.0000	1.56286	0.781431	+	0.623992i	$0.214490\pi$
$758$	0	0
$759$	27.0000i	0.980038i
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	80.0000	2.89619
$764$	−36.0000	−1.30243
$765$	0	0
$766$	0	0
$767$	6.00000	0.216647
$768$	16.0000i	0.577350i
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	− 12.0000i	− 0.432169i
$772$	− 44.0000i	− 1.58359i
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	− 8.00000i	− 0.287368i
$776$	0	0
$777$	16.0000i	0.573997i
$778$	0	0
$779$	3.00000i	0.107486i
$780$	−6.00000	−0.214834
$781$	−36.0000	−1.28818
$782$	0	0
$783$	6.00000	0.214423
$784$	−36.0000	−1.28571
$785$	33.0000i	1.17782i
$786$	0	0
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	6.00000i	0.213741i
$789$	− 12.0000i	− 0.427211i
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	8.00000i	0.284088i
$794$	0	0
$795$	−18.0000	−0.638394
$796$	32.0000i	1.13421i
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	8.00000i	0.282138i
$805$	108.000i	3.80650i
$806$	0	0
$807$	15.0000	0.528025
$808$	0	0
$809$	51.0000i	1.79306i	0.442978	+	0.896532i	$0.353922\pi$
$809$	51.0000i	1.79306i	−0.442978	+	0.896532i	$0.646078\pi$
$810$	0	0
$811$	− 10.0000i	− 0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$	− 10.0000i	− 0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$	48.0000	1.68447
$813$	11.0000i	0.385787i
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	7.00000	0.244899
$818$	0	0
$819$	4.00000i	0.139771i
$820$	18.0000	0.628587
$821$	− 21.0000i	− 0.732905i	−0.930437	−	0.366453i	$-0.880572\pi$
$821$	− 21.0000i	− 0.732905i	0.930437	−	0.366453i	$-0.119428\pi$
$822$	0	0
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	21.0000i	0.730242i	0.930960	+	0.365121i	$0.118972\pi$
$827$	21.0000i	0.730242i	−0.930960	+	0.365121i	$0.881028\pi$
$828$	− 18.0000i	− 0.625543i
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	8.00000	0.277350
$833$	0	0
$834$	0	0
$835$	−63.0000	−2.18020
$836$	− 6.00000i	− 0.207514i
$837$	2.00000	0.0691301
$838$	0	0
$839$	57.0000i	1.96786i	0.178559	+	0.983929i	$0.442857\pi$
$839$	57.0000i	1.96786i	−0.178559	+	0.983929i	$0.557143\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	− 12.0000i	− 0.413302i
$844$	4.00000i	0.137686i
$845$	− 36.0000i	− 1.23844i
$846$	0	0
$847$	8.00000i	0.274883i
$848$	24.0000	0.824163
$849$	−10.0000	−0.343199
$850$	0	0
$851$	−36.0000	−1.23406
$852$	24.0000	0.822226
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	− 3.00000i	− 0.102598i
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	− 42.0000i	− 1.43219i
$861$	− 12.0000i	− 0.408959i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−45.0000	−1.53005
$866$	0	0
$867$	0	0
$868$	16.0000	0.543075
$869$	−30.0000	−1.01768
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	16.0000i	0.541518i
$874$	0	0
$875$	−12.0000	−0.405674
$876$	4.00000	0.135147
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	− 24.0000i	− 0.809500i
$880$	−36.0000	−1.21356
$881$	18.0000i	0.606435i	0.952921	+	0.303218i	$0.0980609\pi$
$881$	18.0000i	0.606435i	−0.952921	+	0.303218i	$0.901939\pi$
$882$	0	0
$883$	11.0000	0.370179	0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000	0.370179	0.185090	+	0.982722i	$0.440742\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	39.0000i	1.30949i	0.755849	+	0.654746i	$0.227224\pi$
$887$	39.0000i	1.30949i	−0.755849	+	0.654746i	$0.772776\pi$
$888$	0	0
$889$	52.0000i	1.74402i
$890$	0	0
$891$	3.00000i	0.100504i
$892$	−2.00000	−0.0669650
$893$	−6.00000	−0.200782
$894$	0	0
$895$	18.0000i	0.601674i
$896$	0	0
$897$	−9.00000	−0.300501
$898$	0	0
$899$	−12.0000	−0.400222
$900$	−8.00000	−0.266667
$901$	0	0
$902$	0	0
$903$	−28.0000	−0.931782
$904$	0	0
$905$	42.0000	1.39613
$906$	0	0
$907$	− 2.00000i	− 0.0664089i	−0.999449	−	0.0332045i	$-0.989429\pi$
$907$	− 2.00000i	− 0.0664089i	0.999449	−	0.0332045i	$-0.0105712\pi$
$908$	6.00000i	0.199117i
$909$	0	0
$910$	0	0
$911$	27.0000i	0.894550i	0.894397	+	0.447275i	$0.147605\pi$
$911$	27.0000i	0.894550i	−0.894397	+	0.447275i	$0.852395\pi$
$912$	4.00000i	0.132453i
$913$	18.0000i	0.595713i
$914$	0	0
$915$	24.0000i	0.793416i
$916$	28.0000	0.925146
$917$	−12.0000	−0.396275
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	20.0000i	0.659022i
$922$	0	0
$923$	− 12.0000i	− 0.394985i
$924$	24.0000i	0.789542i
$925$	16.0000i	0.526077i
$926$	0	0
$927$	−5.00000	−0.164222
$928$	0	0
$929$	− 15.0000i	− 0.492134i	−0.969253	−	0.246067i	$-0.920862\pi$
$929$	− 15.0000i	− 0.492134i	0.969253	−	0.246067i	$-0.0791383\pi$
$930$	0	0
$931$	−9.00000	−0.294963
$932$	− 42.0000i	− 1.37576i
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	36.0000i	1.17419i
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	27.0000	0.879241
$944$	−24.0000	−0.781133
$945$	12.0000i	0.390360i
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	20.0000	0.649570
$949$	− 2.00000i	− 0.0649227i
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	54.0000i	1.74740i
$956$	24.0000	0.776215
$957$	− 18.0000i	− 0.581857i
$958$	0	0
$959$	− 24.0000i	− 0.775000i
$960$	24.0000	0.774597
$961$	27.0000	0.870968
$962$	0	0
$963$	− 9.00000i	− 0.290021i
$964$	− 16.0000i	− 0.515325i
$965$	−66.0000	−2.12462
$966$	0	0
$967$	−41.0000	−1.31847	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	−41.0000	−1.31847	−0.659236	+	0.751936i	$0.729120\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	− 2.00000i	− 0.0641500i
$973$	−8.00000	−0.256468
$974$	0	0
$975$	4.00000i	0.128103i
$976$	− 32.0000i	− 1.02430i
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	0	0
$980$	54.0000i	1.72497i
$981$	20.0000i	0.638551i
$982$	0	0
$983$	− 9.00000i	− 0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$	− 9.00000i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$	0	0
$985$	9.00000	0.286764
$986$	0	0
$987$	24.0000	0.763928
$988$	2.00000	0.0636285
$989$	− 63.0000i	− 2.00328i
$990$	0	0
$991$	− 52.0000i	− 1.65183i	−0.563791	−	0.825917i	$-0.690658\pi$
$991$	− 52.0000i	− 1.65183i	0.563791	−	0.825917i	$-0.309342\pi$
$992$	0	0
$993$	13.0000i	0.412543i
$994$	0	0
$995$	48.0000	1.52170
$996$	− 12.0000i	− 0.380235i
$997$	− 62.0000i	− 1.96356i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	− 62.0000i	− 1.96356i	0.190022	−	0.981780i	$-0.439144\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.d.a.577.2		2
17.2	even	8		867.2.e.e.829.2		4
17.3	odd	16		867.2.h.c.688.1		8
17.4	even	4		51.2.a.a.1.1	✓	1
17.5	odd	16		867.2.h.c.757.1		8
17.6	odd	16		867.2.h.c.712.2		8
17.7	odd	16		867.2.h.c.733.2		8
17.8	even	8		867.2.e.e.616.2		4
17.9	even	8		867.2.e.e.616.1		4
17.10	odd	16		867.2.h.c.733.1		8
17.11	odd	16		867.2.h.c.712.1		8
17.12	odd	16		867.2.h.c.757.2		8
17.13	even	4		867.2.a.c.1.1		1
17.14	odd	16		867.2.h.c.688.2		8
17.15	even	8		867.2.e.e.829.1		4
17.16	even	2	inner	867.2.d.a.577.1		2
51.38	odd	4		153.2.a.b.1.1		1
51.47	odd	4		2601.2.a.f.1.1		1
68.55	odd	4		816.2.a.g.1.1		1
85.4	even	4		1275.2.a.d.1.1		1
85.38	odd	4		1275.2.b.b.1174.2		2
85.72	odd	4		1275.2.b.b.1174.1		2
119.55	odd	4		2499.2.a.d.1.1		1
136.21	even	4		3264.2.a.a.1.1		1
136.123	odd	4		3264.2.a.r.1.1		1
187.21	odd	4		6171.2.a.e.1.1		1
204.191	even	4		2448.2.a.c.1.1		1
221.38	even	4		8619.2.a.g.1.1		1
255.89	odd	4		3825.2.a.i.1.1		1
357.293	even	4		7497.2.a.j.1.1		1
408.293	odd	4		9792.2.a.by.1.1		1
408.395	even	4		9792.2.a.cd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.a.1.1	✓	1	17.4	even	4
153.2.a.b.1.1		1	51.38	odd	4
816.2.a.g.1.1		1	68.55	odd	4
867.2.a.c.1.1		1	17.13	even	4
867.2.d.a.577.1		2	17.16	even	2	inner
867.2.d.a.577.2		2	1.1	even	1	trivial
867.2.e.e.616.1		4	17.9	even	8
867.2.e.e.616.2		4	17.8	even	8
867.2.e.e.829.1		4	17.15	even	8
867.2.e.e.829.2		4	17.2	even	8
867.2.h.c.688.1		8	17.3	odd	16
867.2.h.c.688.2		8	17.14	odd	16
867.2.h.c.712.1		8	17.11	odd	16
867.2.h.c.712.2		8	17.6	odd	16
867.2.h.c.733.1		8	17.10	odd	16
867.2.h.c.733.2		8	17.7	odd	16
867.2.h.c.757.1		8	17.5	odd	16
867.2.h.c.757.2		8	17.12	odd	16
1275.2.a.d.1.1		1	85.4	even	4
1275.2.b.b.1174.1		2	85.72	odd	4
1275.2.b.b.1174.2		2	85.38	odd	4
2448.2.a.c.1.1		1	204.191	even	4
2499.2.a.d.1.1		1	119.55	odd	4
2601.2.a.f.1.1		1	51.47	odd	4
3264.2.a.a.1.1		1	136.21	even	4
3264.2.a.r.1.1		1	136.123	odd	4
3825.2.a.i.1.1		1	255.89	odd	4
6171.2.a.e.1.1		1	187.21	odd	4
7497.2.a.j.1.1		1	357.293	even	4
8619.2.a.g.1.1		1	221.38	even	4
9792.2.a.by.1.1		1	408.293	odd	4
9792.2.a.cd.1.1		1	408.395	even	4

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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000 q^{4} +3.00000i q^{5} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000 q^{4} +3.00000i q^{5} +4.00000i q^{7} -1.00000 q^{9} +3.00000i q^{11} -2.00000i q^{12} -1.00000 q^{13} -3.00000 q^{15} +4.00000 q^{16} +1.00000 q^{19} -6.00000i q^{20} -4.00000 q^{21} -9.00000i q^{23} -4.00000 q^{25} -1.00000i q^{27} -8.00000i q^{28} +6.00000i q^{29} +2.00000i q^{31} -3.00000 q^{33} -12.0000 q^{35} +2.00000 q^{36} -4.00000i q^{37} -1.00000i q^{39} +3.00000i q^{41} +7.00000 q^{43} -6.00000i q^{44} -3.00000i q^{45} -6.00000 q^{47} +4.00000i q^{48} -9.00000 q^{49} +2.00000 q^{52} +6.00000 q^{53} -9.00000 q^{55} +1.00000i q^{57} -6.00000 q^{59} +6.00000 q^{60} -8.00000i q^{61} -4.00000i q^{63} -8.00000 q^{64} -3.00000i q^{65} -4.00000 q^{67} +9.00000 q^{69} +12.0000i q^{71} +2.00000i q^{73} -4.00000i q^{75} -2.00000 q^{76} -12.0000 q^{77} +10.0000i q^{79} +12.0000i q^{80} +1.00000 q^{81} +6.00000 q^{83} +8.00000 q^{84} -6.00000 q^{87} -4.00000i q^{91} +18.0000i q^{92} -2.00000 q^{93} +3.00000i q^{95} -16.0000i q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 2 * q^9	\( 2 q - 4 q^{4} - 2 q^{9} - 2 q^{13} - 6 q^{15} + 8 q^{16} + 2 q^{19} - 8 q^{21} - 8 q^{25} - 6 q^{33} - 24 q^{35} + 4 q^{36} + 14 q^{43} - 12 q^{47} - 18 q^{49} + 4 q^{52} + 12 q^{53} - 18 q^{55} - 12 q^{59} + 12 q^{60} - 16 q^{64} - 8 q^{67} + 18 q^{69} - 4 q^{76} - 24 q^{77} + 2 q^{81} + 12 q^{83} + 16 q^{84} - 12 q^{87} - 4 q^{93}+O(q^{100}) \) 2 * q - 4 * q^4 - 2 * q^9 - 2 * q^13 - 6 * q^15 + 8 * q^16 + 2 * q^19 - 8 * q^21 - 8 * q^25 - 6 * q^33 - 24 * q^35 + 4 * q^36 + 14 * q^43 - 12 * q^47 - 18 * q^49 + 4 * q^52 + 12 * q^53 - 18 * q^55 - 12 * q^59 + 12 * q^60 - 16 * q^64 - 8 * q^67 + 18 * q^69 - 4 * q^76 - 24 * q^77 + 2 * q^81 + 12 * q^83 + 16 * q^84 - 12 * q^87 - 4 * q^93

Introduction

L-functions

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Representations

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Database

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