LMFDB - Embedded newform 768.2.d.c.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 384)
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.c.385.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{13} +4.00000 q^{15} -2.00000 q^{17} -8.00000i q^{19} -2.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} -1.00000i q^{27} -6.00000 q^{31} -4.00000 q^{33} +8.00000i q^{35} -2.00000i q^{37} +2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{45} +4.00000 q^{47} -3.00000 q^{49} -2.00000i q^{51} +16.0000 q^{55} +8.00000 q^{57} -4.00000i q^{59} -14.0000i q^{61} +2.00000 q^{63} -8.00000 q^{65} -4.00000i q^{67} -4.00000i q^{69} -12.0000 q^{71} +10.0000 q^{73} -11.0000i q^{75} -8.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +8.00000i q^{85} +14.0000 q^{89} +4.00000i q^{91} -6.00000i q^{93} -32.0000 q^{95} +10.0000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 2 q^{9} + 8 q^{15} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 12 q^{31} - 8 q^{33} + 4 q^{39} - 12 q^{41} + 8 q^{47} - 6 q^{49} + 32 q^{55} + 16 q^{57} + 4 q^{63} - 16 q^{65} - 24 q^{71} + 20 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 + 8 * q^15 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 12 * q^31 - 8 * q^33 + 4 * q^39 - 12 * q^41 + 8 * q^47 - 6 * q^49 + 32 * q^55 + 16 * q^57 + 4 * q^63 - 16 * q^65 - 24 * q^71 + 20 * q^73 + 20 * q^79 + 2 * q^81 + 28 * q^89 - 64 * q^95 + 20 * 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
$5$	− 4.00000i	− 1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	− 4.00000i	− 1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	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
$15$	4.00000	1.03280
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−11.0000	−2.20000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	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	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	4.00000i	0.596285i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	− 2.00000i	− 0.280056i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	16.0000	2.15744
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	− 14.0000i	− 1.79252i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$	− 14.0000i	− 1.79252i	0.443533	−	0.896258i	$-0.353725\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	− 11.0000i	− 1.27017i
$76$	0	0
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	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
$85$	8.00000i	0.867722i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	4.00000i	0.419314i
$92$	0	0
$93$	− 6.00000i	− 0.622171i
$94$	0	0
$95$	−32.0000	−3.28313
$96$	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
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	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
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	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
$115$	16.0000i	1.49201i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	24.0000i	2.14663i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	4.00000i	0.336861i
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	12.0000i	0.983078i	0.870855	+	0.491539i	$0.163566\pi$
$149$	12.0000i	0.983078i	−0.870855	+	0.491539i	$0.836434\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	24.0000i	1.92773i
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	0	0
$165$	16.0000i	1.24560i
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	8.00000i	0.611775i
$172$	0	0
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	0	0
$175$	22.0000	1.66304
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	− 22.0000i	− 1.63525i	−0.575753	−	0.817624i	$-0.695291\pi$
$181$	− 22.0000i	− 1.63525i	0.575753	−	0.817624i	$-0.304709\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	2.00000i	0.145479i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	− 8.00000i	− 0.572892i
$196$	0	0
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	24.0000i	1.67623i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	− 20.0000i	− 1.37686i	−0.725304	−	0.688428i	$-0.758301\pi$
$211$	− 20.0000i	− 1.37686i	0.725304	−	0.688428i	$-0.241699\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	10.0000i	0.675737i
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	−18.0000	−1.20537	−0.602685	−	0.797980i	$-0.705902\pi$
$223$	−18.0000	−1.20537	−0.602685	+	0.797980i	$0.705902\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	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
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	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
$237$	10.0000i	0.649570i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	12.0000i	0.766652i
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	0	0
$262$	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
$265$	0	0
$266$	0	0
$267$	14.0000i	0.856786i
$268$	0	0
$269$	− 16.0000i	− 0.975537i	−0.872973	−	0.487769i	$-0.837811\pi$
$269$	− 16.0000i	− 0.975537i	0.872973	−	0.487769i	$-0.162189\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	− 44.0000i	− 2.65330i
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	0	0
$285$	− 32.0000i	− 1.89552i
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	10.0000i	0.586210i
$292$	0	0
$293$	16.0000i	0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$	16.0000i	0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−56.0000	−3.20655
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	10.0000i	0.568880i
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	− 8.00000i	− 0.450749i
$316$	0	0
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	16.0000i	0.890264i
$324$	0	0
$325$	22.0000i	1.22034i
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	2.00000i	0.108625i
$340$	0	0
$341$	− 24.0000i	− 1.29967i
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	2.00000i	0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$	2.00000i	0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	48.0000i	2.54758i
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	− 40.0000i	− 2.09370i
$366$	0	0
$367$	−38.0000	−1.98358	−0.991792	−	0.127862i	$-0.959188\pi$
$367$	−38.0000	−1.98358	−0.991792	+	0.127862i	$0.959188\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 32.0000i	− 1.64373i	−0.569683	−	0.821865i	$-0.692934\pi$
$379$	− 32.0000i	− 1.64373i	0.569683	−	0.821865i	$-0.307066\pi$
$380$	0	0
$381$	− 2.00000i	− 0.102463i
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−32.0000	−1.63087
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 20.0000i	− 1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	− 20.0000i	− 1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	− 40.0000i	− 2.01262i
$396$	0	0
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	− 4.00000i	− 0.198762i
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	2.00000i	0.0986527i
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	4.00000	0.195881
$418$	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
$423$	−4.00000	−0.194487
$424$	0	0
$425$	22.0000	1.06716
$426$	0	0
$427$	28.0000i	1.35501i
$428$	0	0
$429$	8.00000i	0.386244i
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.0000i	1.53077i
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	0	0
$445$	− 56.0000i	− 2.65465i
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	− 10.0000i	− 0.469841i
$454$	0	0
$455$	16.0000	0.750092
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	− 28.0000i	− 1.30409i	−0.758180	−	0.652045i	$-0.773911\pi$
$461$	− 28.0000i	− 1.30409i	0.758180	−	0.652045i	$-0.226089\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	0	0
$469$	8.00000i	0.369406i
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	88.0000i	4.03772i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	− 40.0000i	− 1.81631i
$486$	0	0
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	− 4.00000i	− 0.180517i	−0.995918	−	0.0902587i	$-0.971231\pi$
$491$	− 4.00000i	− 0.180517i	0.995918	−	0.0902587i	$-0.0287694\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	− 16.0000i	− 0.714827i
$502$	0	0
$503$	−44.0000	−1.96186	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000	−1.96186	−0.980932	+	0.194354i	$0.937739\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	8.00000i	0.354594i	0.984157	+	0.177297i	$0.0567353\pi$
$509$	8.00000i	0.354594i	−0.984157	+	0.177297i	$0.943265\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	−8.00000	−0.353209
$514$	0	0
$515$	− 40.0000i	− 1.76261i
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	22.0000i	0.960159i
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.00000i	0.173585i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	− 12.0000i	− 0.516877i
$540$	0	0
$541$	14.0000i	0.601907i	0.953639	+	0.300954i	$0.0973049\pi$
$541$	14.0000i	0.601907i	−0.953639	+	0.300954i	$0.902695\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	24.0000	1.02805
$546$	0	0
$547$	− 16.0000i	− 0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$	− 16.0000i	− 0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$	0	0
$549$	14.0000i	0.597505i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	− 8.00000i	− 0.339581i
$556$	0	0
$557$	28.0000i	1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$557$	28.0000i	1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	− 8.00000i	− 0.336563i
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−38.0000	−1.59304	−0.796521	−	0.604610i	$-0.793329\pi$
$569$	−38.0000	−1.59304	−0.796521	+	0.604610i	$0.793329\pi$
$570$	0	0
$571$	− 4.00000i	− 0.167395i	−0.996491	−	0.0836974i	$-0.973327\pi$
$571$	− 4.00000i	− 0.167395i	0.996491	−	0.0836974i	$-0.0266729\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	0	0
$575$	44.0000	1.83493
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	− 2.00000i	− 0.0831172i
$580$	0	0
$581$	− 24.0000i	− 0.995688i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	− 4.00000i	− 0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$	− 4.00000i	− 0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	0	0
$589$	48.0000i	1.97781i
$590$	0	0
$591$	8.00000	0.329076
$592$	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
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	− 14.0000i	− 0.572982i
$598$	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
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	20.0000i	0.813116i
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 8.00000i	− 0.323645i
$612$	0	0
$613$	46.0000i	1.85792i	0.370177	+	0.928961i	$0.379297\pi$
$613$	46.0000i	1.85792i	−0.370177	+	0.928961i	$0.620703\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	4.00000i	0.160514i
$622$	0	0
$623$	−28.0000	−1.12180
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	32.0000i	1.27796i
$628$	0	0
$629$	4.00000i	0.159490i
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	8.00000i	0.317470i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	12.0000	0.474713
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	− 48.0000i	− 1.89294i	−0.322799	−	0.946468i	$-0.604624\pi$
$643$	− 48.0000i	− 1.89294i	0.322799	−	0.946468i	$-0.395376\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	12.0000i	0.470317i
$652$	0	0
$653$	12.0000i	0.469596i	0.972044	+	0.234798i	$0.0754429\pi$
$653$	12.0000i	0.469596i	−0.972044	+	0.234798i	$0.924557\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	− 36.0000i	− 1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$	− 36.0000i	− 1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$	0	0
$661$	22.0000i	0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$	22.0000i	0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	64.0000	2.48181
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 18.0000i	− 0.695920i
$670$	0	0
$671$	56.0000	2.16186
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	11.0000i	0.423390i
$676$	0	0
$677$	20.0000i	0.768662i	0.923195	+	0.384331i	$0.125568\pi$
$677$	20.0000i	0.768662i	−0.923195	+	0.384331i	$0.874432\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	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
$685$	− 8.00000i	− 0.305664i
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	0	0
$690$	0	0
$691$	24.0000i	0.913003i	0.889723	+	0.456502i	$0.150898\pi$
$691$	24.0000i	0.913003i	−0.889723	+	0.456502i	$0.849102\pi$
$692$	0	0
$693$	8.00000i	0.303895i
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	40.0000i	1.51078i	0.655276	+	0.755390i	$0.272552\pi$
$701$	40.0000i	1.51078i	−0.655276	+	0.755390i	$0.727448\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.00000i	0.0751116i	0.999295	+	0.0375558i	$0.0119572\pi$
$709$	2.00000i	0.0751116i	−0.999295	+	0.0375558i	$0.988043\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	− 32.0000i	− 1.19673i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 42.0000i	− 1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$	− 42.0000i	− 1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	− 8.00000i	− 0.292314i
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	40.0000i	1.45575i
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	− 12.0000i	− 0.434429i
$764$	0	0
$765$	− 8.00000i	− 0.289241i
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	18.0000i	0.648254i
$772$	0	0
$773$	24.0000i	0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$	24.0000i	0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$	0	0
$775$	66.0000	2.37079
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	48.0000i	1.71978i
$780$	0	0
$781$	− 48.0000i	− 1.71758i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	72.0000	2.56979
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	0	0
$789$	24.0000i	0.854423i
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	−28.0000	−0.994309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	40.0000i	1.41157i
$804$	0	0
$805$	− 32.0000i	− 1.12785i
$806$	0	0
$807$	16.0000	0.563227
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	24.0000i	0.842754i	0.906886	+	0.421377i	$0.138453\pi$
$811$	24.0000i	0.842754i	−0.906886	+	0.421377i	$0.861547\pi$
$812$	0	0
$813$	14.0000i	0.491001i
$814$	0	0
$815$	−32.0000	−1.12091
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 4.00000i	− 0.139771i
$820$	0	0
$821$	− 8.00000i	− 0.279202i	−0.990208	−	0.139601i	$-0.955418\pi$
$821$	− 8.00000i	− 0.279202i	0.990208	−	0.139601i	$-0.0445820\pi$
$822$	0	0
$823$	22.0000	0.766872	0.383436	−	0.923567i	$-0.374741\pi$
$823$	22.0000	0.766872	0.383436	+	0.923567i	$0.374741\pi$
$824$	0	0
$825$	44.0000	1.53188
$826$	0	0
$827$	− 44.0000i	− 1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	− 44.0000i	− 1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$	0	0
$829$	38.0000i	1.31979i	0.751356	+	0.659897i	$0.229400\pi$
$829$	38.0000i	1.31979i	−0.751356	+	0.659897i	$0.770600\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	64.0000i	2.21481i
$836$	0	0
$837$	6.00000i	0.207390i
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	− 36.0000i	− 1.23844i
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	8.00000i	0.274236i
$852$	0	0
$853$	38.0000i	1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$	38.0000i	1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$	0	0
$855$	32.0000	1.09438
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	− 8.00000i	− 0.272956i	−0.990643	−	0.136478i	$-0.956422\pi$
$859$	− 8.00000i	− 0.272956i	0.990643	−	0.136478i	$-0.0435784\pi$
$860$	0	0
$861$	12.0000i	0.408959i
$862$	0	0
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	0	0
$865$	−48.0000	−1.63205
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	40.0000i	1.35691i
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	− 48.0000i	− 1.62270i
$876$	0	0
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	0	0
$879$	−16.0000	−0.539667
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	− 40.0000i	− 1.34611i	−0.739594	−	0.673054i	$-0.764982\pi$
$883$	− 40.0000i	− 1.34611i	0.739594	−	0.673054i	$-0.235018\pi$
$884$	0	0
$885$	− 16.0000i	− 0.537834i
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	− 32.0000i	− 1.07084i
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−88.0000	−2.92522
$906$	0	0
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	− 56.0000i	− 1.85130i
$916$	0	0
$917$	24.0000i	0.792550i
$918$	0	0
$919$	6.00000	0.197922	0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000	0.197922	0.0989609	+	0.995091i	$0.468448\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	22.0000i	0.723356i
$926$	0	0
$927$	−10.0000	−0.328443
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	24.0000i	0.786568i
$932$	0	0
$933$	16.0000i	0.523816i
$934$	0	0
$935$	−32.0000	−1.04651
$936$	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
$939$	26.0000i	0.848478i
$940$	0	0
$941$	− 28.0000i	− 0.912774i	−0.889781	−	0.456387i	$-0.849143\pi$
$941$	− 28.0000i	− 0.912774i	0.889781	−	0.456387i	$-0.150857\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	0	0
$949$	− 20.0000i	− 0.649227i
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	− 32.0000i	− 1.03550i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.00000	−0.129167
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	8.00000i	0.257529i
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	− 4.00000i	− 0.128366i	−0.997938	−	0.0641831i	$-0.979556\pi$
$971$	− 4.00000i	− 0.128366i	0.997938	−	0.0641831i	$-0.0204442\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	0	0
$975$	−22.0000	−0.704564
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	56.0000i	1.78977i
$980$	0	0
$981$	− 6.00000i	− 0.191565i
$982$	0	0
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	0	0
$985$	−32.0000	−1.01960
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	56.0000i	1.77532i
$996$	0	0
$997$	54.0000i	1.71020i	0.518465	+	0.855099i	$0.326503\pi$
$997$	54.0000i	1.71020i	−0.518465	+	0.855099i	$0.673497\pi$
$998$	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.c.385.2		2
3.2	odd	2		2304.2.d.f.1153.2		2
4.3	odd	2		768.2.d.f.385.1		2
8.3	odd	2		768.2.d.f.385.2		2
8.5	even	2	inner	768.2.d.c.385.1		2
12.11	even	2		2304.2.d.o.1153.2		2
16.3	odd	4		384.2.a.e.1.1	yes	1
16.5	even	4		384.2.a.h.1.1	yes	1
16.11	odd	4		384.2.a.d.1.1	yes	1
16.13	even	4		384.2.a.a.1.1	✓	1
24.5	odd	2		2304.2.d.f.1153.1		2
24.11	even	2		2304.2.d.o.1153.1		2
48.5	odd	4		1152.2.a.b.1.1		1
48.11	even	4		1152.2.a.a.1.1		1
48.29	odd	4		1152.2.a.t.1.1		1
48.35	even	4		1152.2.a.s.1.1		1
80.19	odd	4		9600.2.a.t.1.1		1
80.29	even	4		9600.2.a.bk.1.1		1
80.59	odd	4		9600.2.a.bz.1.1		1
80.69	even	4		9600.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.a.a.1.1	✓	1	16.13	even	4
384.2.a.d.1.1	yes	1	16.11	odd	4
384.2.a.e.1.1	yes	1	16.3	odd	4
384.2.a.h.1.1	yes	1	16.5	even	4
768.2.d.c.385.1		2	8.5	even	2	inner
768.2.d.c.385.2		2	1.1	even	1	trivial
768.2.d.f.385.1		2	4.3	odd	2
768.2.d.f.385.2		2	8.3	odd	2
1152.2.a.a.1.1		1	48.11	even	4
1152.2.a.b.1.1		1	48.5	odd	4
1152.2.a.s.1.1		1	48.35	even	4
1152.2.a.t.1.1		1	48.29	odd	4
2304.2.d.f.1153.1		2	24.5	odd	2
2304.2.d.f.1153.2		2	3.2	odd	2
2304.2.d.o.1153.1		2	24.11	even	2
2304.2.d.o.1153.2		2	12.11	even	2
9600.2.a.e.1.1		1	80.69	even	4
9600.2.a.t.1.1		1	80.19	odd	4
9600.2.a.bk.1.1		1	80.29	even	4
9600.2.a.bz.1.1		1	80.59	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{13} +4.00000 q^{15} -2.00000 q^{17} -8.00000i q^{19} -2.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} -1.00000i q^{27} -6.00000 q^{31} -4.00000 q^{33} +8.00000i q^{35} -2.00000i q^{37} +2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{45} +4.00000 q^{47} -3.00000 q^{49} -2.00000i q^{51} +16.0000 q^{55} +8.00000 q^{57} -4.00000i q^{59} -14.0000i q^{61} +2.00000 q^{63} -8.00000 q^{65} -4.00000i q^{67} -4.00000i q^{69} -12.0000 q^{71} +10.0000 q^{73} -11.0000i q^{75} -8.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +8.00000i q^{85} +14.0000 q^{89} +4.00000i q^{91} -6.00000i q^{93} -32.0000 q^{95} +10.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9} + 8 q^{15} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 12 q^{31} - 8 q^{33} + 4 q^{39} - 12 q^{41} + 8 q^{47} - 6 q^{49} + 32 q^{55} + 16 q^{57} + 4 q^{63} - 16 q^{65} - 24 q^{71} + 20 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 + 8 * q^15 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 12 * q^31 - 8 * q^33 + 4 * q^39 - 12 * q^41 + 8 * q^47 - 6 * q^49 + 32 * q^55 + 16 * q^57 + 4 * q^63 - 16 * q^65 - 24 * q^71 + 20 * q^73 + 20 * q^79 + 2 * q^81 + 28 * q^89 - 64 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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