LMFDB - Embedded newform 768.2.d.h.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(385,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("768.385");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.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:	$6.13251087523$
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 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.2.d.h.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} +1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{31} +4.00000 q^{33} -8.00000i q^{35} +2.00000i q^{37} +2.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} +2.00000i q^{45} +8.00000 q^{47} +9.00000 q^{49} -6.00000i q^{51} -10.0000i q^{53} -8.00000 q^{55} +4.00000 q^{57} +4.00000i q^{59} +6.00000i q^{61} -4.00000 q^{63} -4.00000 q^{65} +4.00000i q^{67} +16.0000 q^{71} +6.00000 q^{73} +1.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +12.0000i q^{85} -2.00000 q^{87} -10.0000 q^{89} -8.00000i q^{91} +4.00000i q^{93} -8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 2 q^{9} + 4 q^{15} - 12 q^{17} + 2 q^{25} + 8 q^{31} + 8 q^{33} + 4 q^{39} - 4 q^{41} + 16 q^{47} + 18 q^{49} - 16 q^{55} + 8 q^{57} - 8 q^{63} - 8 q^{65} + 32 q^{71} + 12 q^{73} + 8 q^{79}+ \cdots - 28 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 2 * q^9 + 4 * q^15 - 12 * q^17 + 2 * q^25 + 8 * q^31 + 8 * q^33 + 4 * q^39 - 4 * q^41 + 16 * q^47 + 18 * q^49 - 16 * q^55 + 8 * q^57 - 8 * q^63 - 8 * q^65 + 32 * q^71 + 12 * q^73 + 8 * q^79 + 2 * q^81 - 4 * q^87 - 20 * q^89 - 16 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	− 8.00000i	− 1.35225i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	− 6.00000i	− 0.840168i
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 16.0000i	− 1.82337i
$78$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	12.0000i	1.30158i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	− 8.00000i	− 0.838628i
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	6.00000i	0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$	6.00000i	0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000i	0.349482i	0.984614	+	0.174741i	$0.0559088\pi$
$131$	4.00000i	0.349482i	−0.984614	+	0.174741i	$0.944091\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	20.0000i	1.69638i	0.529694	+	0.848189i	$0.322307\pi$
$139$	20.0000i	1.69638i	−0.529694	+	0.848189i	$0.677693\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 8.00000i	− 0.642575i
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	− 8.00000i	− 0.622799i
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	10.0000i	0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$	10.0000i	0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	24.0000i	1.75505i
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	− 4.00000i	− 0.286446i
$196$	0	0
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	4.00000i	0.279372i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	16.0000i	1.09630i
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	− 16.0000i	− 1.04372i
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 18.0000i	− 1.14998i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	12.0000i	0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$	12.0000i	0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	8.00000i	0.497096i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−20.0000	−1.22859
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	18.0000i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$	18.0000i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 14.0000i	− 0.820695i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 16.0000i	− 0.922225i
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	12.0000i	0.682656i
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	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
$315$	8.00000i	0.450749i
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	2.00000i	0.108625i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 20.0000i	− 1.07366i	−0.843692	−	0.536828i	$-0.819622\pi$
$347$	− 20.0000i	− 1.07366i	0.843692	−	0.536828i	$-0.180378\pi$
$348$	0	0
$349$	− 26.0000i	− 1.39175i	−0.718164	−	0.695874i	$-0.755017\pi$
$349$	− 26.0000i	− 1.39175i	0.718164	−	0.695874i	$-0.244983\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	− 32.0000i	− 1.69838i
$356$	0	0
$357$	− 24.0000i	− 1.27021i
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	− 40.0000i	− 2.07670i
$372$	0	0
$373$	34.0000i	1.76045i	0.474554	+	0.880227i	$0.342610\pi$
$373$	34.0000i	1.76045i	−0.474554	+	0.880227i	$0.657390\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	28.0000i	1.43826i	0.694874	+	0.719132i	$0.255460\pi$
$379$	28.0000i	1.43826i	−0.694874	+	0.719132i	$0.744540\pi$
$380$	0	0
$381$	− 20.0000i	− 1.02463i
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	−32.0000	−1.63087
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	30.0000i	1.52106i	0.649303	+	0.760530i	$0.275061\pi$
$389$	30.0000i	1.52106i	−0.649303	+	0.760530i	$0.724939\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	− 2.00000i	− 0.0993808i
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	16.0000i	0.787309i
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	− 20.0000i	− 0.977064i	−0.872546	−	0.488532i	$-0.837533\pi$
$419$	− 20.0000i	− 0.977064i	0.872546	−	0.488532i	$-0.162467\pi$
$420$	0	0
$421$	− 38.0000i	− 1.85201i	−0.377515	−	0.926003i	$-0.623221\pi$
$421$	− 38.0000i	− 1.85201i	0.377515	−	0.926003i	$-0.376779\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	24.0000i	1.16144i
$428$	0	0
$429$	− 8.00000i	− 0.386244i
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	4.00000i	0.191785i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	20.0000i	0.948091i
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	8.00000i	0.376705i
$452$	0	0
$453$	− 12.0000i	− 0.563809i
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	6.00000i	0.280056i
$460$	0	0
$461$	− 6.00000i	− 0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$	− 6.00000i	− 0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	16.0000i	0.738811i
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.0000i	1.27141i
$486$	0	0
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 28.0000i	− 1.26362i	−0.775122	−	0.631811i	$-0.782312\pi$
$491$	− 28.0000i	− 1.26362i	0.775122	−	0.631811i	$-0.217688\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	64.0000	2.87079
$498$	0	0
$499$	− 12.0000i	− 0.537194i	−0.963253	−	0.268597i	$-0.913440\pi$
$499$	− 12.0000i	− 0.537194i	0.963253	−	0.268597i	$-0.0865599\pi$
$500$	0	0
$501$	− 8.00000i	− 0.357414i
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	− 24.0000i	− 1.05757i
$516$	0	0
$517$	− 32.0000i	− 1.40736i
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	0	0
$525$	4.00000i	0.174574i
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	− 36.0000i	− 1.55063i
$540$	0	0
$541$	− 2.00000i	− 0.0859867i	−0.999075	−	0.0429934i	$-0.986311\pi$
$541$	− 2.00000i	− 0.0859867i	0.999075	−	0.0429934i	$-0.0136894\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	28.0000	1.19939
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	4.00000i	0.169791i
$556$	0	0
$557$	− 22.0000i	− 0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$	− 22.0000i	− 0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	− 4.00000i	− 0.168281i
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	0	0
$571$	− 44.0000i	− 1.84134i	−0.390339	−	0.920671i	$-0.627642\pi$
$571$	− 44.0000i	− 1.84134i	0.390339	−	0.920671i	$-0.372358\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	18.0000i	0.748054i
$580$	0	0
$581$	48.0000i	1.99138i
$582$	0	0
$583$	−40.0000	−1.65663
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	0	0
$589$	− 16.0000i	− 0.659269i
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	48.0000i	1.96781i
$596$	0	0
$597$	− 4.00000i	− 0.163709i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	− 16.0000i	− 0.647291i
$612$	0	0
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	− 12.0000i	− 0.482321i	−0.970485	−	0.241160i	$-0.922472\pi$
$619$	− 12.0000i	− 0.482321i	0.970485	−	0.241160i	$-0.0775280\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−40.0000	−1.60257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	− 12.0000i	− 0.478471i
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	40.0000i	1.58735i
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	−16.0000	−0.632950
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	0	0
$645$	− 8.00000i	− 0.315000i
$646$	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
$649$	16.0000	0.628055
$650$	0	0
$651$	16.0000i	0.627089i
$652$	0	0
$653$	10.0000i	0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$	10.0000i	0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	36.0000i	1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$	36.0000i	1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	0	0
$661$	− 14.0000i	− 0.544537i	−0.962221	−	0.272268i	$-0.912226\pi$
$661$	− 14.0000i	− 0.544537i	0.962221	−	0.272268i	$-0.0877739\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−32.0000	−1.24091
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 4.00000i	− 0.154649i
$670$	0	0
$671$	24.0000	0.926510
$672$	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
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	14.0000i	0.538064i	0.963131	+	0.269032i	$0.0867037\pi$
$677$	14.0000i	0.538064i	−0.963131	+	0.269032i	$0.913296\pi$
$678$	0	0
$679$	−56.0000	−2.14908
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	36.0000i	1.37549i
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	− 52.0000i	− 1.97817i	−0.147335	−	0.989087i	$-0.547070\pi$
$691$	− 52.0000i	− 1.97817i	0.147335	−	0.989087i	$-0.452930\pi$
$692$	0	0
$693$	16.0000i	0.607790i
$694$	0	0
$695$	40.0000	1.51729
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	18.0000i	0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$	18.0000i	0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	− 22.0000i	− 0.826227i	−0.910679	−	0.413114i	$-0.864441\pi$
$709$	− 22.0000i	− 0.826227i	0.910679	−	0.413114i	$-0.135559\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	16.0000i	0.598366i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	48.0000	1.78761
$722$	0	0
$723$	− 14.0000i	− 0.520666i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	− 8.00000i	− 0.293887i
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	16.0000i	0.584627i
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	24.0000i	0.873449i
$756$	0	0
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	0	0
$759$	0	0
$760$	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
$763$	56.0000i	2.02734i
$764$	0	0
$765$	− 12.0000i	− 0.433861i
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	2.00000i	0.0720282i
$772$	0	0
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	8.00000i	0.286630i
$780$	0	0
$781$	− 64.0000i	− 2.29010i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	24.0000i	0.854423i
$790$	0	0
$791$	8.00000	0.284447
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	− 20.0000i	− 0.709327i
$796$	0	0
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	− 24.0000i	− 0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	0	0
$811$	28.0000i	0.983213i	0.870817	+	0.491606i	$0.163590\pi$
$811$	28.0000i	0.983213i	−0.870817	+	0.491606i	$0.836410\pi$
$812$	0	0
$813$	12.0000i	0.420858i
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−16.0000	−0.559769
$818$	0	0
$819$	8.00000i	0.279543i
$820$	0	0
$821$	− 26.0000i	− 0.907406i	−0.891153	−	0.453703i	$-0.850103\pi$
$821$	− 26.0000i	− 0.907406i	0.891153	−	0.453703i	$-0.149897\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	− 34.0000i	− 1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$	− 34.0000i	− 1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	−54.0000	−1.87099
$834$	0	0
$835$	16.0000i	0.553703i
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	− 18.0000i	− 0.619219i
$846$	0	0
$847$	−20.0000	−0.687208
$848$	0	0
$849$	28.0000	0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	18.0000i	0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$	18.0000i	0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	− 8.00000i	− 0.272639i
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	− 16.0000i	− 0.542763i
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	− 48.0000i	− 1.62270i
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	8.00000i	0.268917i
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−80.0000	−2.68311
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	− 32.0000i	− 1.07084i
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	60.0000i	1.99889i
$902$	0	0
$903$	16.0000	0.532447
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	0	0
$909$	− 6.00000i	− 0.199007i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	12.0000i	0.396708i
$916$	0	0
$917$	16.0000i	0.528367i
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	− 32.0000i	− 1.05329i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	− 36.0000i	− 1.17985i
$932$	0	0
$933$	24.0000i	0.785725i
$934$	0	0
$935$	48.0000	1.56977
$936$	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
$939$	− 10.0000i	− 0.326338i
$940$	0	0
$941$	58.0000i	1.89075i	0.325991	+	0.945373i	$0.394302\pi$
$941$	58.0000i	1.89075i	−0.325991	+	0.945373i	$0.605698\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	− 12.0000i	− 0.389536i
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.00000i	0.258603i
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	− 36.0000i	− 1.15888i
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	− 52.0000i	− 1.66876i	−0.551190	−	0.834380i	$-0.685826\pi$
$971$	− 52.0000i	− 1.66876i	0.551190	−	0.834380i	$-0.314174\pi$
$972$	0	0
$973$	80.0000i	2.56468i
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	40.0000i	1.27841i
$980$	0	0
$981$	− 14.0000i	− 0.446986i
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	44.0000	1.40196
$986$	0	0
$987$	32.0000i	1.01857i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	8.00000i	0.253617i
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.d.h.385.2		2
3.2	odd	2		2304.2.d.s.1153.2		2
4.3	odd	2		768.2.d.a.385.1		2
8.3	odd	2		768.2.d.a.385.2		2
8.5	even	2	inner	768.2.d.h.385.1		2
12.11	even	2		2304.2.d.c.1153.2		2
16.3	odd	4		192.2.a.c.1.1		1
16.5	even	4		96.2.a.b.1.1	yes	1
16.11	odd	4		96.2.a.a.1.1	✓	1
16.13	even	4		192.2.a.a.1.1		1
24.5	odd	2		2304.2.d.s.1153.1		2
24.11	even	2		2304.2.d.c.1153.1		2
48.5	odd	4		288.2.a.b.1.1		1
48.11	even	4		288.2.a.c.1.1		1
48.29	odd	4		576.2.a.g.1.1		1
48.35	even	4		576.2.a.h.1.1		1
80.3	even	4		4800.2.f.bh.3649.2		2
80.13	odd	4		4800.2.f.e.3649.1		2
80.19	odd	4		4800.2.a.f.1.1		1
80.27	even	4		2400.2.f.a.1249.2		2
80.29	even	4		4800.2.a.co.1.1		1
80.37	odd	4		2400.2.f.r.1249.1		2
80.43	even	4		2400.2.f.a.1249.1		2
80.53	odd	4		2400.2.f.r.1249.2		2
80.59	odd	4		2400.2.a.r.1.1		1
80.67	even	4		4800.2.f.bh.3649.1		2
80.69	even	4		2400.2.a.q.1.1		1
80.77	odd	4		4800.2.f.e.3649.2		2
112.13	odd	4		9408.2.a.ct.1.1		1
112.27	even	4		4704.2.a.t.1.1		1
112.69	odd	4		4704.2.a.e.1.1		1
112.83	even	4		9408.2.a.bj.1.1		1
144.5	odd	12		2592.2.i.w.865.1		2
144.11	even	12		2592.2.i.q.1729.1		2
144.43	odd	12		2592.2.i.b.1729.1		2
144.59	even	12		2592.2.i.q.865.1		2
144.85	even	12		2592.2.i.h.865.1		2
144.101	odd	12		2592.2.i.w.1729.1		2
144.133	even	12		2592.2.i.h.1729.1		2
144.139	odd	12		2592.2.i.b.865.1		2
240.53	even	4		7200.2.f.f.6049.2		2
240.59	even	4		7200.2.a.e.1.1		1
240.107	odd	4		7200.2.f.x.6049.2		2
240.149	odd	4		7200.2.a.bx.1.1		1
240.197	even	4		7200.2.f.f.6049.1		2
240.203	odd	4		7200.2.f.x.6049.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.a.a.1.1	✓	1	16.11	odd	4
96.2.a.b.1.1	yes	1	16.5	even	4
192.2.a.a.1.1		1	16.13	even	4
192.2.a.c.1.1		1	16.3	odd	4
288.2.a.b.1.1		1	48.5	odd	4
288.2.a.c.1.1		1	48.11	even	4
576.2.a.g.1.1		1	48.29	odd	4
576.2.a.h.1.1		1	48.35	even	4
768.2.d.a.385.1		2	4.3	odd	2
768.2.d.a.385.2		2	8.3	odd	2
768.2.d.h.385.1		2	8.5	even	2	inner
768.2.d.h.385.2		2	1.1	even	1	trivial
2304.2.d.c.1153.1		2	24.11	even	2
2304.2.d.c.1153.2		2	12.11	even	2
2304.2.d.s.1153.1		2	24.5	odd	2
2304.2.d.s.1153.2		2	3.2	odd	2
2400.2.a.q.1.1		1	80.69	even	4
2400.2.a.r.1.1		1	80.59	odd	4
2400.2.f.a.1249.1		2	80.43	even	4
2400.2.f.a.1249.2		2	80.27	even	4
2400.2.f.r.1249.1		2	80.37	odd	4
2400.2.f.r.1249.2		2	80.53	odd	4
2592.2.i.b.865.1		2	144.139	odd	12
2592.2.i.b.1729.1		2	144.43	odd	12
2592.2.i.h.865.1		2	144.85	even	12
2592.2.i.h.1729.1		2	144.133	even	12
2592.2.i.q.865.1		2	144.59	even	12
2592.2.i.q.1729.1		2	144.11	even	12
2592.2.i.w.865.1		2	144.5	odd	12
2592.2.i.w.1729.1		2	144.101	odd	12
4704.2.a.e.1.1		1	112.69	odd	4
4704.2.a.t.1.1		1	112.27	even	4
4800.2.a.f.1.1		1	80.19	odd	4
4800.2.a.co.1.1		1	80.29	even	4
4800.2.f.e.3649.1		2	80.13	odd	4
4800.2.f.e.3649.2		2	80.77	odd	4
4800.2.f.bh.3649.1		2	80.67	even	4
4800.2.f.bh.3649.2		2	80.3	even	4
7200.2.a.e.1.1		1	240.59	even	4
7200.2.a.bx.1.1		1	240.149	odd	4
7200.2.f.f.6049.1		2	240.197	even	4
7200.2.f.f.6049.2		2	240.53	even	4
7200.2.f.x.6049.1		2	240.203	odd	4
7200.2.f.x.6049.2		2	240.107	odd	4
9408.2.a.bj.1.1		1	112.83	even	4
9408.2.a.ct.1.1		1	112.13	odd	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000i q^{13} +2.00000 q^{15} -6.00000 q^{17} -4.00000i q^{19} +4.00000i q^{21} +1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +4.00000 q^{31} +4.00000 q^{33} -8.00000i q^{35} +2.00000i q^{37} +2.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} +2.00000i q^{45} +8.00000 q^{47} +9.00000 q^{49} -6.00000i q^{51} -10.0000i q^{53} -8.00000 q^{55} +4.00000 q^{57} +4.00000i q^{59} +6.00000i q^{61} -4.00000 q^{63} -4.00000 q^{65} +4.00000i q^{67} +16.0000 q^{71} +6.00000 q^{73} +1.00000i q^{75} -16.0000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +12.0000i q^{85} -2.00000 q^{87} -10.0000 q^{89} -8.00000i q^{91} +4.00000i q^{93} -8.00000 q^{95} -14.0000 q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9} + 4 q^{15} - 12 q^{17} + 2 q^{25} + 8 q^{31} + 8 q^{33} + 4 q^{39} - 4 q^{41} + 16 q^{47} + 18 q^{49} - 16 q^{55} + 8 q^{57} - 8 q^{63} - 8 q^{65} + 32 q^{71} + 12 q^{73} + 8 q^{79}+ \cdots - 28 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 2 * q^9 + 4 * q^15 - 12 * q^17 + 2 * q^25 + 8 * q^31 + 8 * q^33 + 4 * q^39 - 4 * q^41 + 16 * q^47 + 18 * q^49 - 16 * q^55 + 8 * q^57 - 8 * q^63 - 8 * q^65 + 32 * q^71 + 12 * q^73 + 8 * q^79 + 2 * q^81 - 4 * q^87 - 20 * q^89 - 16 * q^95 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more