LMFDB - Embedded newform 624.2.d.d.287.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(287,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.d (of order $2$, degree $1$, 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:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	624.287
Dual form			624.2.d.d.287.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -3.46410i q^{5} -3.46410i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -3.46410i q^{5} -3.46410i q^{7} -3.00000 q^{9} +6.00000 q^{11} -1.00000 q^{13} -6.00000 q^{15} +6.92820i q^{17} -3.46410i q^{19} -6.00000 q^{21} -7.00000 q^{25} +5.19615i q^{27} +6.92820i q^{29} -3.46410i q^{31} -10.3923i q^{33} -12.0000 q^{35} +2.00000 q^{37} +1.73205i q^{39} -3.46410i q^{41} +3.46410i q^{43} +10.3923i q^{45} +6.00000 q^{47} -5.00000 q^{49} +12.0000 q^{51} -6.92820i q^{53} -20.7846i q^{55} -6.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} +10.3923i q^{63} +3.46410i q^{65} +10.3923i q^{67} -6.00000 q^{71} -14.0000 q^{73} +12.1244i q^{75} -20.7846i q^{77} -10.3923i q^{79} +9.00000 q^{81} -6.00000 q^{83} +24.0000 q^{85} +12.0000 q^{87} -10.3923i q^{89} +3.46410i q^{91} -6.00000 q^{93} -12.0000 q^{95} +10.0000 q^{97} -18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} + 12 q^{11} - 2 q^{13} - 12 q^{15} - 12 q^{21} - 14 q^{25} - 24 q^{35} + 4 q^{37} + 12 q^{47} - 10 q^{49} + 24 q^{51} - 12 q^{57} + 12 q^{59} + 4 q^{61} - 12 q^{71} - 28 q^{73} + 18 q^{81} - 12 q^{83} + 48 q^{85} + 24 q^{87} - 12 q^{93} - 24 q^{95} + 20 q^{97} - 36 q^{99}+O(q^{100}) $ 2 * q - 6 * q^9 + 12 * q^11 - 2 * q^13 - 12 * q^15 - 12 * q^21 - 14 * q^25 - 24 * q^35 + 4 * q^37 + 12 * q^47 - 10 * q^49 + 24 * q^51 - 12 * q^57 + 12 * q^59 + 4 * q^61 - 12 * q^71 - 28 * q^73 + 18 * q^81 - 12 * q^83 + 48 * q^85 + 24 * q^87 - 12 * q^93 - 24 * q^95 + 20 * q^97 - 36 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	− 1.73205i	− 1.00000i
$3$	− 1.73205i	− 1.00000i
$4$	0	0
$5$	− 3.46410i	− 1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	− 3.46410i	− 1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$	0	0
$7$	− 3.46410i	− 1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	− 3.46410i	− 1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−6.00000	−1.54919
$16$	0	0
$17$	6.92820i	1.68034i	0.542326	+	0.840168i	$0.317544\pi$
$17$	6.92820i	1.68034i	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	5.19615i	1.00000i
$28$	0	0
$29$	6.92820i	1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$	6.92820i	1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	− 3.46410i	− 0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	− 10.3923i	− 1.80907i
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	1.73205i	0.277350i
$40$	0	0
$41$	− 3.46410i	− 0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$	− 3.46410i	− 0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	0	0
$43$	3.46410i	0.528271i	0.964486	+	0.264135i	$0.0850865\pi$
$43$	3.46410i	0.528271i	−0.964486	+	0.264135i	$0.914913\pi$
$44$	0	0
$45$	10.3923i	1.54919i
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	− 6.92820i	− 0.951662i	−0.879537	−	0.475831i	$-0.842147\pi$
$53$	− 6.92820i	− 0.951662i	0.879537	−	0.475831i	$-0.157853\pi$
$54$	0	0
$55$	− 20.7846i	− 2.80260i
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	10.3923i	1.30931i
$64$	0	0
$65$	3.46410i	0.429669i
$66$	0	0
$67$	10.3923i	1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$	10.3923i	1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	12.1244i	1.40000i
$76$	0	0
$77$	− 20.7846i	− 2.36863i
$78$	0	0
$79$	− 10.3923i	− 1.16923i	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	− 10.3923i	− 1.16923i	0.811312	−	0.584613i	$-0.198754\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	24.0000	2.60317
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	− 10.3923i	− 1.10158i	−0.834643	−	0.550791i	$-0.814326\pi$
$89$	− 10.3923i	− 1.10158i	0.834643	−	0.550791i	$-0.185674\pi$
$90$	0	0
$91$	3.46410i	0.363137i
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−18.0000	−1.80907
$100$	0	0
$101$	13.8564i	1.37876i	0.724398	+	0.689382i	$0.242118\pi$
$101$	13.8564i	1.37876i	−0.724398	+	0.689382i	$0.757882\pi$
$102$	0	0
$103$	− 3.46410i	− 0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	− 3.46410i	− 0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$	0	0
$105$	20.7846i	2.02837i
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	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
$111$	− 3.46410i	− 0.328798i
$112$	0	0
$113$	13.8564i	1.30350i	0.758433	+	0.651751i	$0.225965\pi$
$113$	13.8564i	1.30350i	−0.758433	+	0.651751i	$0.774035\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.00000	0.277350
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	3.46410i	0.307389i	0.988118	+	0.153695i	$0.0491172\pi$
$127$	3.46410i	0.307389i	−0.988118	+	0.153695i	$0.950883\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	18.0000	1.54919
$136$	0	0
$137$	3.46410i	0.295958i	0.988990	+	0.147979i	$0.0472768\pi$
$137$	3.46410i	0.295958i	−0.988990	+	0.147979i	$0.952723\pi$
$138$	0	0
$139$	− 10.3923i	− 0.881464i	−0.897639	−	0.440732i	$-0.854719\pi$
$139$	− 10.3923i	− 0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$140$	0	0
$141$	− 10.3923i	− 0.875190i
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	24.0000	1.99309
$146$	0	0
$147$	8.66025i	0.714286i
$148$	0	0
$149$	− 3.46410i	− 0.283790i	−0.989882	−	0.141895i	$-0.954680\pi$
$149$	− 3.46410i	− 0.283790i	0.989882	−	0.141895i	$-0.0453196\pi$
$150$	0	0
$151$	− 3.46410i	− 0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$	− 3.46410i	− 0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$	0	0
$153$	− 20.7846i	− 1.68034i
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.46410i	− 0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$	− 3.46410i	− 0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$	0	0
$165$	−36.0000	−2.80260
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.3923i	0.794719i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	24.2487i	1.83303i
$176$	0	0
$177$	− 10.3923i	− 0.781133i
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 3.46410i	− 0.256074i
$184$	0	0
$185$	− 6.92820i	− 0.509372i
$186$	0	0
$187$	41.5692i	3.03984i
$188$	0	0
$189$	18.0000	1.30931
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	10.3923i	0.740421i	0.928948	+	0.370211i	$0.120714\pi$
$197$	10.3923i	0.740421i	−0.928948	+	0.370211i	$0.879286\pi$
$198$	0	0
$199$	− 10.3923i	− 0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$	− 10.3923i	− 0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$	0	0
$201$	18.0000	1.26962
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 20.7846i	− 1.43770i
$210$	0	0
$211$	− 3.46410i	− 0.238479i	−0.992866	−	0.119239i	$-0.961954\pi$
$211$	− 3.46410i	− 0.238479i	0.992866	−	0.119239i	$-0.0380456\pi$
$212$	0	0
$213$	10.3923i	0.712069i
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	24.2487i	1.63858i
$220$	0	0
$221$	− 6.92820i	− 0.466041i
$222$	0	0
$223$	10.3923i	0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$	10.3923i	0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	21.0000	1.40000
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−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
$231$	−36.0000	−2.36863
$232$	0	0
$233$	20.7846i	1.36165i	0.732448	+	0.680823i	$0.238378\pi$
$233$	20.7846i	1.36165i	−0.732448	+	0.680823i	$0.761622\pi$
$234$	0	0
$235$	− 20.7846i	− 1.35584i
$236$	0	0
$237$	−18.0000	−1.16923
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 15.5885i	− 1.00000i
$244$	0	0
$245$	17.3205i	1.10657i
$246$	0	0
$247$	3.46410i	0.220416i
$248$	0	0
$249$	10.3923i	0.658586i
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 41.5692i	− 2.60317i
$256$	0	0
$257$	6.92820i	0.432169i	0.976375	+	0.216085i	$0.0693287\pi$
$257$	6.92820i	0.432169i	−0.976375	+	0.216085i	$0.930671\pi$
$258$	0	0
$259$	− 6.92820i	− 0.430498i
$260$	0	0
$261$	− 20.7846i	− 1.28654i
$262$	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
$265$	−24.0000	−1.47431
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	− 20.7846i	− 1.26726i	−0.773636	−	0.633630i	$-0.781564\pi$
$269$	− 20.7846i	− 1.26726i	0.773636	−	0.633630i	$-0.218436\pi$
$270$	0	0
$271$	− 17.3205i	− 1.05215i	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	− 17.3205i	− 1.05215i	0.850439	−	0.526073i	$-0.176336\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	−42.0000	−2.53270
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	10.3923i	0.622171i
$280$	0	0
$281$	3.46410i	0.206651i	0.994648	+	0.103325i	$0.0329483\pi$
$281$	3.46410i	0.206651i	−0.994648	+	0.103325i	$0.967052\pi$
$282$	0	0
$283$	− 17.3205i	− 1.02960i	−0.857311	−	0.514799i	$-0.827867\pi$
$283$	− 17.3205i	− 1.02960i	0.857311	−	0.514799i	$-0.172133\pi$
$284$	0	0
$285$	20.7846i	1.23117i
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−31.0000	−1.82353
$290$	0	0
$291$	− 17.3205i	− 1.01535i
$292$	0	0
$293$	17.3205i	1.01187i	0.862570	+	0.505937i	$0.168853\pi$
$293$	17.3205i	1.01187i	−0.862570	+	0.505937i	$0.831147\pi$
$294$	0	0
$295$	− 20.7846i	− 1.21013i
$296$	0	0
$297$	31.1769i	1.80907i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	24.0000	1.37876
$304$	0	0
$305$	− 6.92820i	− 0.396708i
$306$	0	0
$307$	− 17.3205i	− 0.988534i	−0.869310	−	0.494267i	$-0.835437\pi$
$307$	− 17.3205i	− 0.988534i	0.869310	−	0.494267i	$-0.164563\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	36.0000	2.02837
$316$	0	0
$317$	− 17.3205i	− 0.972817i	−0.873732	−	0.486408i	$-0.838307\pi$
$317$	− 17.3205i	− 0.972817i	0.873732	−	0.486408i	$-0.161693\pi$
$318$	0	0
$319$	41.5692i	2.32743i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	7.00000	0.388290
$326$	0	0
$327$	− 17.3205i	− 0.957826i
$328$	0	0
$329$	− 20.7846i	− 1.14589i
$330$	0	0
$331$	10.3923i	0.571213i	0.958347	+	0.285606i	$0.0921950\pi$
$331$	10.3923i	0.571213i	−0.958347	+	0.285606i	$0.907805\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	36.0000	1.96689
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	24.0000	1.30350
$340$	0	0
$341$	− 20.7846i	− 1.12555i
$342$	0	0
$343$	− 6.92820i	− 0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	− 5.19615i	− 0.277350i
$352$	0	0
$353$	− 17.3205i	− 0.921878i	−0.887432	−	0.460939i	$-0.847513\pi$
$353$	− 17.3205i	− 0.921878i	0.887432	−	0.460939i	$-0.152487\pi$
$354$	0	0
$355$	20.7846i	1.10313i
$356$	0	0
$357$	− 41.5692i	− 2.20008i
$358$	0	0
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	− 43.3013i	− 2.27273i
$364$	0	0
$365$	48.4974i	2.53847i
$366$	0	0
$367$	10.3923i	0.542474i	0.962513	+	0.271237i	$0.0874327\pi$
$367$	10.3923i	0.542474i	−0.962513	+	0.271237i	$0.912567\pi$
$368$	0	0
$369$	10.3923i	0.541002i
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−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
$375$	12.0000	0.619677
$376$	0	0
$377$	− 6.92820i	− 0.356821i
$378$	0	0
$379$	10.3923i	0.533817i	0.963722	+	0.266908i	$0.0860021\pi$
$379$	10.3923i	0.533817i	−0.963722	+	0.266908i	$0.913998\pi$
$380$	0	0
$381$	6.00000	0.307389
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	−72.0000	−3.66946
$386$	0	0
$387$	− 10.3923i	− 0.528271i
$388$	0	0
$389$	27.7128i	1.40510i	0.711637	+	0.702548i	$0.247954\pi$
$389$	27.7128i	1.40510i	−0.711637	+	0.702548i	$0.752046\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−36.0000	−1.81136
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	20.7846i	1.04053i
$400$	0	0
$401$	3.46410i	0.172989i	0.996252	+	0.0864945i	$0.0275665\pi$
$401$	3.46410i	0.172989i	−0.996252	+	0.0864945i	$0.972434\pi$
$402$	0	0
$403$	3.46410i	0.172559i
$404$	0	0
$405$	− 31.1769i	− 1.54919i
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	− 20.7846i	− 1.02274i
$414$	0	0
$415$	20.7846i	1.02028i
$416$	0	0
$417$	−18.0000	−0.881464
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−18.0000	−0.875190
$424$	0	0
$425$	− 48.4974i	− 2.35247i
$426$	0	0
$427$	− 6.92820i	− 0.335279i
$428$	0	0
$429$	10.3923i	0.501745i
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	− 41.5692i	− 1.99309i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	24.2487i	1.15733i	0.815566	+	0.578664i	$0.196426\pi$
$439$	24.2487i	1.15733i	−0.815566	+	0.578664i	$0.803574\pi$
$440$	0	0
$441$	15.0000	0.714286
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	31.1769i	1.47133i	0.677346	+	0.735665i	$0.263130\pi$
$449$	31.1769i	1.47133i	−0.677346	+	0.735665i	$0.736870\pi$
$450$	0	0
$451$	− 20.7846i	− 0.978709i
$452$	0	0
$453$	−6.00000	−0.281905
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	−36.0000	−1.68034
$460$	0	0
$461$	24.2487i	1.12938i	0.825305	+	0.564688i	$0.191003\pi$
$461$	24.2487i	1.12938i	−0.825305	+	0.564688i	$0.808997\pi$
$462$	0	0
$463$	10.3923i	0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$	10.3923i	0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$	0	0
$465$	20.7846i	0.963863i
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	− 24.2487i	− 1.11732i
$472$	0	0
$473$	20.7846i	0.955677i
$474$	0	0
$475$	24.2487i	1.11261i
$476$	0	0
$477$	20.7846i	0.951662i
$478$	0	0
$479$	−42.0000	−1.91903	−0.959514	−	0.281659i	$-0.909115\pi$
$479$	−42.0000	−1.91903	−0.959514	+	0.281659i	$0.909115\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 34.6410i	− 1.57297i
$486$	0	0
$487$	24.2487i	1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$	24.2487i	1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−48.0000	−2.16181
$494$	0	0
$495$	62.3538i	2.80260i
$496$	0	0
$497$	20.7846i	0.932317i
$498$	0	0
$499$	− 31.1769i	− 1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	− 31.1769i	− 1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$	0	0
$501$	31.1769i	1.39288i
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	48.0000	2.13597
$506$	0	0
$507$	− 1.73205i	− 0.0769231i
$508$	0	0
$509$	24.2487i	1.07481i	0.843326	+	0.537403i	$0.180594\pi$
$509$	24.2487i	1.07481i	−0.843326	+	0.537403i	$0.819406\pi$
$510$	0	0
$511$	48.4974i	2.14540i
$512$	0	0
$513$	18.0000	0.794719
$514$	0	0
$515$	−12.0000	−0.528783
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 13.8564i	− 0.607060i	−0.952822	−	0.303530i	$-0.901835\pi$
$521$	− 13.8564i	− 0.607060i	0.952822	−	0.303530i	$-0.0981653\pi$
$522$	0	0
$523$	17.3205i	0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$	17.3205i	0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$	0	0
$525$	42.0000	1.83303
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	3.46410i	0.150047i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 20.7846i	− 0.896922i
$538$	0	0
$539$	−30.0000	−1.29219
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	3.46410i	0.148659i
$544$	0	0
$545$	− 34.6410i	− 1.48386i
$546$	0	0
$547$	31.1769i	1.33303i	0.745492	+	0.666514i	$0.232214\pi$
$547$	31.1769i	1.33303i	−0.745492	+	0.666514i	$0.767786\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	3.46410i	0.146779i	0.997303	+	0.0733893i	$0.0233816\pi$
$557$	3.46410i	0.146779i	−0.997303	+	0.0733893i	$0.976618\pi$
$558$	0	0
$559$	− 3.46410i	− 0.146516i
$560$	0	0
$561$	72.0000	3.03984
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	48.0000	2.01938
$566$	0	0
$567$	− 31.1769i	− 1.30931i
$568$	0	0
$569$	20.7846i	0.871336i	0.900107	+	0.435668i	$0.143488\pi$
$569$	20.7846i	0.871336i	−0.900107	+	0.435668i	$0.856512\pi$
$570$	0	0
$571$	38.1051i	1.59465i	0.603550	+	0.797325i	$0.293752\pi$
$571$	38.1051i	1.59465i	−0.603550	+	0.797325i	$0.706248\pi$
$572$	0	0
$573$	− 20.7846i	− 0.868290i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	38.1051i	1.58359i
$580$	0	0
$581$	20.7846i	0.862291i
$582$	0	0
$583$	− 41.5692i	− 1.72162i
$584$	0	0
$585$	− 10.3923i	− 0.429669i
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	3.46410i	0.142254i	0.997467	+	0.0711268i	$0.0226595\pi$
$593$	3.46410i	0.142254i	−0.997467	+	0.0711268i	$0.977341\pi$
$594$	0	0
$595$	− 83.1384i	− 3.40834i
$596$	0	0
$597$	−18.0000	−0.736691
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	− 31.1769i	− 1.26962i
$604$	0	0
$605$	− 86.6025i	− 3.52089i
$606$	0	0
$607$	− 17.3205i	− 0.703018i	−0.936185	−	0.351509i	$-0.885669\pi$
$607$	− 17.3205i	− 0.703018i	0.936185	−	0.351509i	$-0.114331\pi$
$608$	0	0
$609$	− 41.5692i	− 1.68447i
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	20.7846i	0.838116i
$616$	0	0
$617$	− 3.46410i	− 0.139459i	−0.997566	−	0.0697297i	$-0.977786\pi$
$617$	− 3.46410i	− 0.139459i	0.997566	−	0.0697297i	$-0.0222137\pi$
$618$	0	0
$619$	24.2487i	0.974638i	0.873224	+	0.487319i	$0.162025\pi$
$619$	24.2487i	0.974638i	−0.873224	+	0.487319i	$0.837975\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−36.0000	−1.44231
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−36.0000	−1.43770
$628$	0	0
$629$	13.8564i	0.552491i
$630$	0	0
$631$	10.3923i	0.413711i	0.978371	+	0.206856i	$0.0663230\pi$
$631$	10.3923i	0.413711i	−0.978371	+	0.206856i	$0.933677\pi$
$632$	0	0
$633$	−6.00000	−0.238479
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	18.0000	0.712069
$640$	0	0
$641$	34.6410i	1.36824i	0.729370	+	0.684119i	$0.239813\pi$
$641$	34.6410i	1.36824i	−0.729370	+	0.684119i	$0.760187\pi$
$642$	0	0
$643$	− 17.3205i	− 0.683054i	−0.939872	−	0.341527i	$-0.889056\pi$
$643$	− 17.3205i	− 0.683054i	0.939872	−	0.341527i	$-0.110944\pi$
$644$	0	0
$645$	− 20.7846i	− 0.818393i
$646$	0	0
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	20.7846i	0.814613i
$652$	0	0
$653$	− 20.7846i	− 0.813365i	−0.913570	−	0.406682i	$-0.866686\pi$
$653$	− 20.7846i	− 0.813365i	0.913570	−	0.406682i	$-0.133314\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	42.0000	1.63858
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	50.0000	1.94477	0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000	1.94477	0.972387	+	0.233373i	$0.0749763\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	41.5692i	1.61199i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.0000	0.695920
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	− 36.3731i	− 1.40000i
$676$	0	0
$677$	20.7846i	0.798817i	0.916773	+	0.399409i	$0.130785\pi$
$677$	20.7846i	0.798817i	−0.916773	+	0.399409i	$0.869215\pi$
$678$	0	0
$679$	− 34.6410i	− 1.32940i
$680$	0	0
$681$	− 31.1769i	− 1.19470i
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	24.2487i	0.925146i
$688$	0	0
$689$	6.92820i	0.263944i
$690$	0	0
$691$	− 3.46410i	− 0.131781i	−0.997827	−	0.0658903i	$-0.979011\pi$
$691$	− 3.46410i	− 0.131781i	0.997827	−	0.0658903i	$-0.0209887\pi$
$692$	0	0
$693$	62.3538i	2.36863i
$694$	0	0
$695$	−36.0000	−1.36556
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	36.0000	1.36165
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 6.92820i	− 0.261302i
$704$	0	0
$705$	−36.0000	−1.35584
$706$	0	0
$707$	48.0000	1.80523
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	31.1769i	1.16923i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	20.7846i	0.777300i
$716$	0	0
$717$	10.3923i	0.388108i
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	− 17.3205i	− 0.644157i
$724$	0	0
$725$	− 48.4974i	− 1.80115i
$726$	0	0
$727$	45.0333i	1.67019i	0.550103	+	0.835097i	$0.314588\pi$
$727$	45.0333i	1.67019i	−0.550103	+	0.835097i	$0.685412\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	30.0000	1.10657
$736$	0	0
$737$	62.3538i	2.29683i
$738$	0	0
$739$	− 31.1769i	− 1.14686i	−0.819254	−	0.573431i	$-0.805612\pi$
$739$	− 31.1769i	− 1.14686i	0.819254	−	0.573431i	$-0.194388\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	18.0000	0.658586
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38.1051i	1.39048i	0.718780	+	0.695238i	$0.244701\pi$
$751$	38.1051i	1.39048i	−0.718780	+	0.695238i	$0.755299\pi$
$752$	0	0
$753$	− 20.7846i	− 0.757433i
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 10.3923i	− 0.376721i	−0.982100	−	0.188360i	$-0.939683\pi$
$761$	− 10.3923i	− 0.376721i	0.982100	−	0.188360i	$-0.0603173\pi$
$762$	0	0
$763$	− 34.6410i	− 1.25409i
$764$	0	0
$765$	−72.0000	−2.60317
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	10.3923i	0.373785i	0.982380	+	0.186893i	$0.0598416\pi$
$773$	10.3923i	0.373785i	−0.982380	+	0.186893i	$0.940158\pi$
$774$	0	0
$775$	24.2487i	0.871039i
$776$	0	0
$777$	−12.0000	−0.430498
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	−36.0000	−1.28654
$784$	0	0
$785$	− 48.4974i	− 1.73095i
$786$	0	0
$787$	10.3923i	0.370446i	0.982697	+	0.185223i	$0.0593007\pi$
$787$	10.3923i	0.370446i	−0.982697	+	0.185223i	$0.940699\pi$
$788$	0	0
$789$	20.7846i	0.739952i
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	41.5692i	1.47431i
$796$	0	0
$797$	41.5692i	1.47246i	0.676733	+	0.736229i	$0.263395\pi$
$797$	41.5692i	1.47246i	−0.676733	+	0.736229i	$0.736605\pi$
$798$	0	0
$799$	41.5692i	1.47061i
$800$	0	0
$801$	31.1769i	1.10158i
$802$	0	0
$803$	−84.0000	−2.96430
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36.0000	−1.26726
$808$	0	0
$809$	13.8564i	0.487165i	0.969880	+	0.243583i	$0.0783227\pi$
$809$	13.8564i	0.487165i	−0.969880	+	0.243583i	$0.921677\pi$
$810$	0	0
$811$	− 3.46410i	− 0.121641i	−0.998149	−	0.0608205i	$-0.980628\pi$
$811$	− 3.46410i	− 0.121641i	0.998149	−	0.0608205i	$-0.0193717\pi$
$812$	0	0
$813$	−30.0000	−1.05215
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	12.0000	0.419827
$818$	0	0
$819$	− 10.3923i	− 0.363137i
$820$	0	0
$821$	− 31.1769i	− 1.08808i	−0.839059	−	0.544041i	$-0.816894\pi$
$821$	− 31.1769i	− 1.08808i	0.839059	−	0.544041i	$-0.183106\pi$
$822$	0	0
$823$	− 3.46410i	− 0.120751i	−0.998176	−	0.0603755i	$-0.980770\pi$
$823$	− 3.46410i	− 0.120751i	0.998176	−	0.0603755i	$-0.0192298\pi$
$824$	0	0
$825$	72.7461i	2.53270i
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	3.46410i	0.120168i
$832$	0	0
$833$	− 34.6410i	− 1.20024i
$834$	0	0
$835$	62.3538i	2.15784i
$836$	0	0
$837$	18.0000	0.622171
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−19.0000	−0.655172
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	− 3.46410i	− 0.119169i
$846$	0	0
$847$	− 86.6025i	− 2.97570i
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	0	0
$857$	− 34.6410i	− 1.18331i	−0.806190	−	0.591657i	$-0.798474\pi$
$857$	− 34.6410i	− 1.18331i	0.806190	−	0.591657i	$-0.201526\pi$
$858$	0	0
$859$	− 45.0333i	− 1.53652i	−0.640140	−	0.768259i	$-0.721124\pi$
$859$	− 45.0333i	− 1.53652i	0.640140	−	0.768259i	$-0.278876\pi$
$860$	0	0
$861$	20.7846i	0.708338i
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	53.6936i	1.82353i
$868$	0	0
$869$	− 62.3538i	− 2.11521i
$870$	0	0
$871$	− 10.3923i	− 0.352130i
$872$	0	0
$873$	−30.0000	−1.01535
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	− 6.92820i	− 0.233417i	−0.993166	−	0.116709i	$-0.962766\pi$
$881$	− 6.92820i	− 0.233417i	0.993166	−	0.116709i	$-0.0372343\pi$
$882$	0	0
$883$	− 31.1769i	− 1.04919i	−0.851353	−	0.524593i	$-0.824217\pi$
$883$	− 31.1769i	− 1.04919i	0.851353	−	0.524593i	$-0.175783\pi$
$884$	0	0
$885$	−36.0000	−1.21013
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	54.0000	1.80907
$892$	0	0
$893$	− 20.7846i	− 0.695530i
$894$	0	0
$895$	− 41.5692i	− 1.38951i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	48.0000	1.59911
$902$	0	0
$903$	− 20.7846i	− 0.691669i
$904$	0	0
$905$	6.92820i	0.230301i
$906$	0	0
$907$	− 45.0333i	− 1.49531i	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	− 45.0333i	− 1.49531i	0.664089	−	0.747653i	$-0.268820\pi$
$908$	0	0
$909$	− 41.5692i	− 1.37876i
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	−12.0000	−0.396708
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 51.9615i	− 1.71405i	−0.515273	−	0.857026i	$-0.672309\pi$
$919$	− 51.9615i	− 1.71405i	0.515273	−	0.857026i	$-0.327691\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	0	0
$923$	6.00000	0.197492
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	10.3923i	0.341328i
$928$	0	0
$929$	− 17.3205i	− 0.568267i	−0.958785	−	0.284134i	$-0.908294\pi$
$929$	− 17.3205i	− 0.568267i	0.958785	−	0.284134i	$-0.0917060\pi$
$930$	0	0
$931$	17.3205i	0.567657i
$932$	0	0
$933$	41.5692i	1.36092i
$934$	0	0
$935$	144.000	4.70930
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	17.3205i	0.565233i
$940$	0	0
$941$	51.9615i	1.69390i	0.531675	+	0.846949i	$0.321563\pi$
$941$	51.9615i	1.69390i	−0.531675	+	0.846949i	$0.678437\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	− 62.3538i	− 2.02837i
$946$	0	0
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	20.7846i	0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$	20.7846i	0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$	0	0
$955$	− 41.5692i	− 1.34515i
$956$	0	0
$957$	72.0000	2.32743
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	76.2102i	2.45329i
$966$	0	0
$967$	38.1051i	1.22538i	0.790324	+	0.612689i	$0.209912\pi$
$967$	38.1051i	1.22538i	−0.790324	+	0.612689i	$0.790088\pi$
$968$	0	0
$969$	− 41.5692i	− 1.33540i
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	−36.0000	−1.15411
$974$	0	0
$975$	− 12.1244i	− 0.388290i
$976$	0	0
$977$	− 58.8897i	− 1.88405i	−0.335544	−	0.942025i	$-0.608920\pi$
$977$	− 58.8897i	− 1.88405i	0.335544	−	0.942025i	$-0.391080\pi$
$978$	0	0
$979$	− 62.3538i	− 1.99284i
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	−36.0000	−1.14589
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 17.3205i	− 0.550204i	−0.961415	−	0.275102i	$-0.911288\pi$
$991$	− 17.3205i	− 0.550204i	0.961415	−	0.275102i	$-0.0887116\pi$
$992$	0	0
$993$	18.0000	0.571213
$994$	0	0
$995$	−36.0000	−1.14128
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	10.3923i	0.328798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.d.d.287.1	yes	2
3.2	odd	2		624.2.d.c.287.1	✓	2
4.3	odd	2		624.2.d.c.287.2	yes	2
8.3	odd	2		2496.2.d.e.1535.1		2
8.5	even	2		2496.2.d.d.1535.2		2
12.11	even	2	inner	624.2.d.d.287.2	yes	2
24.5	odd	2		2496.2.d.e.1535.2		2
24.11	even	2		2496.2.d.d.1535.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.2.d.c.287.1	✓	2	3.2	odd	2
624.2.d.c.287.2	yes	2	4.3	odd	2
624.2.d.d.287.1	yes	2	1.1	even	1	trivial
624.2.d.d.287.2	yes	2	12.11	even	2	inner
2496.2.d.d.1535.1		2	24.11	even	2
2496.2.d.d.1535.2		2	8.5	even	2
2496.2.d.e.1535.1		2	8.3	odd	2
2496.2.d.e.1535.2		2	24.5	odd	2

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -3.46410i q^{5} -3.46410i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -3.46410i q^{5} -3.46410i q^{7} -3.00000 q^{9} +6.00000 q^{11} -1.00000 q^{13} -6.00000 q^{15} +6.92820i q^{17} -3.46410i q^{19} -6.00000 q^{21} -7.00000 q^{25} +5.19615i q^{27} +6.92820i q^{29} -3.46410i q^{31} -10.3923i q^{33} -12.0000 q^{35} +2.00000 q^{37} +1.73205i q^{39} -3.46410i q^{41} +3.46410i q^{43} +10.3923i q^{45} +6.00000 q^{47} -5.00000 q^{49} +12.0000 q^{51} -6.92820i q^{53} -20.7846i q^{55} -6.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} +10.3923i q^{63} +3.46410i q^{65} +10.3923i q^{67} -6.00000 q^{71} -14.0000 q^{73} +12.1244i q^{75} -20.7846i q^{77} -10.3923i q^{79} +9.00000 q^{81} -6.00000 q^{83} +24.0000 q^{85} +12.0000 q^{87} -10.3923i q^{89} +3.46410i q^{91} -6.00000 q^{93} -12.0000 q^{95} +10.0000 q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} + 12 q^{11} - 2 q^{13} - 12 q^{15} - 12 q^{21} - 14 q^{25} - 24 q^{35} + 4 q^{37} + 12 q^{47} - 10 q^{49} + 24 q^{51} - 12 q^{57} + 12 q^{59} + 4 q^{61} - 12 q^{71} - 28 q^{73} + 18 q^{81} - 12 q^{83} + 48 q^{85} + 24 q^{87} - 12 q^{93} - 24 q^{95} + 20 q^{97} - 36 q^{99}+O(q^{100}) \) 2 * q - 6 * q^9 + 12 * q^11 - 2 * q^13 - 12 * q^15 - 12 * q^21 - 14 * q^25 - 24 * q^35 + 4 * q^37 + 12 * q^47 - 10 * q^49 + 24 * q^51 - 12 * q^57 + 12 * q^59 + 4 * q^61 - 12 * q^71 - 28 * q^73 + 18 * q^81 - 12 * q^83 + 48 * q^85 + 24 * q^87 - 12 * q^93 - 24 * q^95 + 20 * q^97 - 36 * q^99

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