LMFDB - Embedded newform 2432.2.c.b.1217.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, 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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (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:	$19.4196177716$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.b.1217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} +2.00000 q^{9} -4.00000i q^{11} +1.00000i q^{13} +2.00000 q^{15} -5.00000 q^{17} +1.00000i q^{19} +1.00000i q^{21} +9.00000 q^{23} +1.00000 q^{25} +5.00000i q^{27} +5.00000i q^{29} +2.00000 q^{31} +4.00000 q^{33} -2.00000i q^{35} -10.0000i q^{37} -1.00000 q^{39} -12.0000 q^{41} -10.0000i q^{43} -4.00000i q^{45} +12.0000 q^{47} -6.00000 q^{49} -5.00000i q^{51} -9.00000i q^{53} -8.00000 q^{55} -1.00000 q^{57} -9.00000i q^{59} -2.00000i q^{61} +2.00000 q^{63} +2.00000 q^{65} +13.0000i q^{67} +9.00000i q^{69} -10.0000 q^{71} +7.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +10.0000i q^{85} -5.00000 q^{87} +6.00000 q^{89} +1.00000i q^{91} +2.00000i q^{93} +2.00000 q^{95} -6.00000 q^{97} -8.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 + 4 * q^9	$ 2 q + 2 q^{7} + 4 q^{9} + 4 q^{15} - 10 q^{17} + 18 q^{23} + 2 q^{25} + 4 q^{31} + 8 q^{33} - 2 q^{39} - 24 q^{41} + 24 q^{47} - 12 q^{49} - 16 q^{55} - 2 q^{57} + 4 q^{63} + 4 q^{65} - 20 q^{71} + 14 q^{73} + 20 q^{79} + 2 q^{81} - 10 q^{87} + 12 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 4 * q^9 + 4 * q^15 - 10 * q^17 + 18 * q^23 + 2 * q^25 + 4 * q^31 + 8 * q^33 - 2 * q^39 - 24 * q^41 + 24 * q^47 - 12 * q^49 - 16 * q^55 - 2 * q^57 + 4 * q^63 + 4 * q^65 - 20 * q^71 + 14 * q^73 + 20 * q^79 + 2 * q^81 - 10 * q^87 + 12 * q^89 + 4 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	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
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	5.00000i	0.928477i	0.885710	+	0.464238i	$0.153672\pi$
$29$	5.00000i	0.928477i	−0.885710	+	0.464238i	$0.846328\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	− 5.00000i	− 0.700140i
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	− 9.00000i	− 1.17170i	−0.810419	−	0.585850i	$-0.800761\pi$
$59$	− 9.00000i	− 1.17170i	0.810419	−	0.585850i	$-0.199239\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	0	0
$69$	9.00000i	1.08347i
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	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
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	10.0000i	1.08465i
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	1.00000i	0.104828i
$92$	0	0
$93$	2.00000i	0.207390i
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	− 8.00000i	− 0.804030i
$100$	0	0
$101$	4.00000i	0.398015i	0.979998	+	0.199007i	$0.0637718\pi$
$101$	4.00000i	0.398015i	−0.979998	+	0.199007i	$0.936228\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	− 11.0000i	− 1.06341i	−0.846930	−	0.531705i	$-0.821551\pi$
$107$	− 11.0000i	− 1.06341i	0.846930	−	0.531705i	$-0.178449\pi$
$108$	0	0
$109$	15.0000i	1.43674i	0.695662	+	0.718370i	$0.255111\pi$
$109$	15.0000i	1.43674i	−0.695662	+	0.718370i	$0.744889\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	− 18.0000i	− 1.67851i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−5.00000	−0.458349
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	− 14.0000i	− 1.22319i	−0.791173	−	0.611593i	$-0.790529\pi$
$131$	− 14.0000i	− 1.22319i	0.791173	−	0.611593i	$-0.209471\pi$
$132$	0	0
$133$	1.00000i	0.0867110i
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	13.0000	1.11066	0.555332	−	0.831628i	$-0.312591\pi$
$137$	13.0000	1.11066	0.555332	+	0.831628i	$0.312591\pi$
$138$	0	0
$139$	2.00000i	0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$	2.00000i	0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	− 6.00000i	− 0.494872i
$148$	0	0
$149$	− 8.00000i	− 0.655386i	−0.944784	−	0.327693i	$-0.893729\pi$
$149$	− 8.00000i	− 0.655386i	0.944784	−	0.327693i	$-0.106271\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−10.0000	−0.808452
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	10.0000i	0.783260i	0.920123	+	0.391630i	$0.128089\pi$
$163$	10.0000i	0.783260i	−0.920123	+	0.391630i	$0.871911\pi$
$164$	0	0
$165$	− 8.00000i	− 0.622799i
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	9.00000	0.676481
$178$	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
$181$	− 2.00000i	− 0.148659i	−0.997234	−	0.0743294i	$-0.976318\pi$
$181$	− 2.00000i	− 0.148659i	0.997234	−	0.0743294i	$-0.0236816\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	20.0000i	1.46254i
$188$	0	0
$189$	5.00000i	0.363696i
$190$	0	0
$191$	23.0000	1.66422	0.832111	−	0.554609i	$-0.187132\pi$
$191$	23.0000	1.66422	0.832111	+	0.554609i	$0.187132\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	2.00000i	0.143223i
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	0	0
$201$	−13.0000	−0.916949
$202$	0	0
$203$	5.00000i	0.350931i
$204$	0	0
$205$	24.0000i	1.67623i
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	13.0000i	0.894957i	0.894295	+	0.447478i	$0.147678\pi$
$211$	13.0000i	0.894957i	−0.894295	+	0.447478i	$0.852322\pi$
$212$	0	0
$213$	− 10.0000i	− 0.685189i
$214$	0	0
$215$	−20.0000	−1.36399
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	7.00000i	0.473016i
$220$	0	0
$221$	− 5.00000i	− 0.336336i
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	1.00000i	0.0663723i	0.999449	+	0.0331862i	$0.0105654\pi$
$227$	1.00000i	0.0663723i	−0.999449	+	0.0331862i	$0.989435\pi$
$228$	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
$231$	4.00000	0.263181
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	− 24.0000i	− 1.56559i
$236$	0	0
$237$	10.0000i	0.649570i
$238$	0	0
$239$	19.0000	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$239$	19.0000	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	12.0000i	0.766652i
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	6.00000i	0.378717i	0.981908	+	0.189358i	$0.0606408\pi$
$251$	6.00000i	0.378717i	−0.981908	+	0.189358i	$0.939359\pi$
$252$	0	0
$253$	− 36.0000i	− 2.26330i
$254$	0	0
$255$	−10.0000	−0.626224
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	− 10.0000i	− 0.621370i
$260$	0	0
$261$	10.0000i	0.618984i
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	3.00000	0.182237	0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000	0.182237	0.0911185	+	0.995840i	$0.470956\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	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
$279$	4.00000	0.239474
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	0	0
$285$	2.00000i	0.118470i
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	21.0000i	1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$	21.0000i	1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	9.00000i	0.520483i
$300$	0	0
$301$	− 10.0000i	− 0.576390i
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	−4.00000	−0.229039
$306$	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
$309$	8.00000i	0.455104i
$310$	0	0
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	0	0
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	31.0000i	1.74113i	0.492050	+	0.870567i	$0.336248\pi$
$317$	31.0000i	1.74113i	−0.492050	+	0.870567i	$0.663752\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	11.0000	0.613960
$322$	0	0
$323$	− 5.00000i	− 0.278207i
$324$	0	0
$325$	1.00000i	0.0554700i
$326$	0	0
$327$	−15.0000	−0.829502
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	1.00000i	0.0549650i	0.999622	+	0.0274825i	$0.00874905\pi$
$331$	1.00000i	0.0549650i	−0.999622	+	0.0274825i	$0.991251\pi$
$332$	0	0
$333$	− 20.0000i	− 1.09599i
$334$	0	0
$335$	26.0000	1.42053
$336$	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
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	− 8.00000i	− 0.433224i
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	18.0000	0.969087
$346$	0	0
$347$	22.0000i	1.18102i	0.807030	+	0.590511i	$0.201074\pi$
$347$	22.0000i	1.18102i	−0.807030	+	0.590511i	$0.798926\pi$
$348$	0	0
$349$	26.0000i	1.39175i	0.718164	+	0.695874i	$0.244983\pi$
$349$	26.0000i	1.39175i	−0.718164	+	0.695874i	$0.755017\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−35.0000	−1.86286	−0.931431	−	0.363918i	$-0.881439\pi$
$353$	−35.0000	−1.86286	−0.931431	+	0.363918i	$0.881439\pi$
$354$	0	0
$355$	20.0000i	1.06149i
$356$	0	0
$357$	− 5.00000i	− 0.264628i
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	− 14.0000i	− 0.732793i
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	− 9.00000i	− 0.467257i
$372$	0	0
$373$	13.0000i	0.673114i	0.941663	+	0.336557i	$0.109263\pi$
$373$	13.0000i	0.673114i	−0.941663	+	0.336557i	$0.890737\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	− 17.0000i	− 0.873231i	−0.899648	−	0.436616i	$-0.856177\pi$
$379$	− 17.0000i	− 0.873231i	0.899648	−	0.436616i	$-0.143823\pi$
$380$	0	0
$381$	6.00000i	0.307389i
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	− 20.0000i	− 1.01666i
$388$	0	0
$389$	− 12.0000i	− 0.608424i	−0.952604	−	0.304212i	$-0.901607\pi$
$389$	− 12.0000i	− 0.608424i	0.952604	−	0.304212i	$-0.0983931\pi$
$390$	0	0
$391$	−45.0000	−2.27575
$392$	0	0
$393$	14.0000	0.706207
$394$	0	0
$395$	− 20.0000i	− 1.00631i
$396$	0	0
$397$	− 26.0000i	− 1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$	− 26.0000i	− 1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	2.00000i	0.0996271i
$404$	0	0
$405$	− 2.00000i	− 0.0993808i
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	13.0000i	0.641243i
$412$	0	0
$413$	− 9.00000i	− 0.442861i
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	34.0000i	1.66101i	0.557012	+	0.830504i	$0.311948\pi$
$419$	34.0000i	1.66101i	−0.557012	+	0.830504i	$0.688052\pi$
$420$	0	0
$421$	15.0000i	0.731055i	0.930800	+	0.365528i	$0.119111\pi$
$421$	15.0000i	0.731055i	−0.930800	+	0.365528i	$0.880889\pi$
$422$	0	0
$423$	24.0000	1.16692
$424$	0	0
$425$	−5.00000	−0.242536
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	0	0
$429$	4.00000i	0.193122i
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	10.0000i	0.479463i
$436$	0	0
$437$	9.00000i	0.430528i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	− 12.0000i	− 0.568855i
$446$	0	0
$447$	8.00000	0.378387
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	48.0000i	2.26023i
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	− 25.0000i	− 1.16690i
$460$	0	0
$461$	− 24.0000i	− 1.11779i	−0.829238	−	0.558896i	$-0.811225\pi$
$461$	− 24.0000i	− 1.11779i	0.829238	−	0.558896i	$-0.188775\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	13.0000i	0.600284i
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−40.0000	−1.83920
$474$	0	0
$475$	1.00000i	0.0458831i
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	9.00000i	0.409514i
$484$	0	0
$485$	12.0000i	0.544892i
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	− 25.0000i	− 1.12594i
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	−10.0000	−0.448561
$498$	0	0
$499$	10.0000i	0.447661i	0.974628	+	0.223831i	$0.0718563\pi$
$499$	10.0000i	0.447661i	−0.974628	+	0.223831i	$0.928144\pi$
$500$	0	0
$501$	− 24.0000i	− 1.07224i
$502$	0	0
$503$	−11.0000	−0.490466	−0.245233	−	0.969464i	$-0.578864\pi$
$503$	−11.0000	−0.490466	−0.245233	+	0.969464i	$0.578864\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	12.0000i	0.532939i
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	0	0
$523$	17.0000i	0.743358i	0.928361	+	0.371679i	$0.121218\pi$
$523$	17.0000i	0.743358i	−0.928361	+	0.371679i	$0.878782\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	−10.0000	−0.435607
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	− 18.0000i	− 0.781133i
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	−22.0000	−0.951143
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	24.0000i	1.03375i
$540$	0	0
$541$	4.00000i	0.171973i	0.996296	+	0.0859867i	$0.0274043\pi$
$541$	4.00000i	0.171973i	−0.996296	+	0.0859867i	$0.972596\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	30.0000	1.28506
$546$	0	0
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	0	0
$549$	− 4.00000i	− 0.170716i
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	− 20.0000i	− 0.848953i
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	−20.0000	−0.844401
$562$	0	0
$563$	20.0000i	0.842900i	0.906852	+	0.421450i	$0.138479\pi$
$563$	20.0000i	0.842900i	−0.906852	+	0.421450i	$0.861521\pi$
$564$	0	0
$565$	12.0000i	0.504844i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	20.0000i	0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$	20.0000i	0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$	0	0
$573$	23.0000i	0.960839i
$574$	0	0
$575$	9.00000	0.375326
$576$	0	0
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	0	0
$579$	− 4.00000i	− 0.166234i
$580$	0	0
$581$	− 8.00000i	− 0.331896i
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	− 48.0000i	− 1.98117i	−0.136892	−	0.990586i	$-0.543711\pi$
$587$	− 48.0000i	− 1.98117i	0.136892	−	0.990586i	$-0.456289\pi$
$588$	0	0
$589$	2.00000i	0.0824086i
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	10.0000i	0.409960i
$596$	0	0
$597$	5.00000i	0.204636i
$598$	0	0
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	26.0000i	1.05880i
$604$	0	0
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	12.0000i	0.485468i
$612$	0	0
$613$	12.0000i	0.484675i	0.970192	+	0.242338i	$0.0779142\pi$
$613$	12.0000i	0.484675i	−0.970192	+	0.242338i	$0.922086\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	10.0000i	0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$	10.0000i	0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$	0	0
$621$	45.0000i	1.80579i
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	50.0000i	1.99363i
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	− 12.0000i	− 0.476205i
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	−20.0000	−0.791188
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	0	0
$645$	− 20.0000i	− 0.787499i
$646$	0	0
$647$	37.0000	1.45462	0.727310	−	0.686309i	$-0.240770\pi$
$647$	37.0000	1.45462	0.727310	+	0.686309i	$0.240770\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	2.00000i	0.0783862i
$652$	0	0
$653$	46.0000i	1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$	46.0000i	1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	19.0000i	0.740135i	0.929005	+	0.370067i	$0.120665\pi$
$659$	19.0000i	0.740135i	−0.929005	+	0.370067i	$0.879335\pi$
$660$	0	0
$661$	− 15.0000i	− 0.583432i	−0.956505	−	0.291716i	$-0.905774\pi$
$661$	− 15.0000i	− 0.583432i	0.956505	−	0.291716i	$-0.0942263\pi$
$662$	0	0
$663$	5.00000	0.194184
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	45.0000i	1.74241i
$668$	0	0
$669$	− 16.0000i	− 0.618596i
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−20.0000	−0.770943	−0.385472	−	0.922720i	$-0.625961\pi$
$673$	−20.0000	−0.770943	−0.385472	+	0.922720i	$0.625961\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	0	0
$677$	− 15.0000i	− 0.576497i	−0.957556	−	0.288248i	$-0.906927\pi$
$677$	− 15.0000i	− 0.576497i	0.957556	−	0.288248i	$-0.0930729\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−1.00000	−0.0383201
$682$	0	0
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	0	0
$685$	− 26.0000i	− 0.993409i
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	30.0000i	1.14125i	0.821209	+	0.570627i	$0.193300\pi$
$691$	30.0000i	1.14125i	−0.821209	+	0.570627i	$0.806700\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	60.0000	2.27266
$698$	0	0
$699$	10.0000i	0.378235i
$700$	0	0
$701$	20.0000i	0.755390i	0.925930	+	0.377695i	$0.123283\pi$
$701$	20.0000i	0.755390i	−0.925930	+	0.377695i	$0.876717\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	4.00000i	0.150435i
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	18.0000	0.674105
$714$	0	0
$715$	− 8.00000i	− 0.299183i
$716$	0	0
$717$	19.0000i	0.709568i
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 16.0000i	− 0.595046i
$724$	0	0
$725$	5.00000i	0.185695i
$726$	0	0
$727$	29.0000	1.07555	0.537775	−	0.843088i	$-0.319265\pi$
$727$	29.0000	1.07555	0.537775	+	0.843088i	$0.319265\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	50.0000i	1.84932i
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	52.0000	1.91544
$738$	0	0
$739$	− 20.0000i	− 0.735712i	−0.929883	−	0.367856i	$-0.880092\pi$
$739$	− 20.0000i	− 0.735712i	0.929883	−	0.367856i	$-0.119908\pi$
$740$	0	0
$741$	− 1.00000i	− 0.0367359i
$742$	0	0
$743$	50.0000	1.83432	0.917161	−	0.398517i	$-0.130475\pi$
$743$	50.0000	1.83432	0.917161	+	0.398517i	$0.130475\pi$
$744$	0	0
$745$	−16.0000	−0.586195
$746$	0	0
$747$	− 16.0000i	− 0.585409i
$748$	0	0
$749$	− 11.0000i	− 0.401931i
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	− 16.0000i	− 0.582300i
$756$	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
$759$	36.0000	1.30672
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	15.0000i	0.543036i
$764$	0	0
$765$	20.0000i	0.723102i
$766$	0	0
$767$	9.00000	0.324971
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	9.00000i	0.323708i	0.986815	+	0.161854i	$0.0517473\pi$
$773$	9.00000i	0.323708i	−0.986815	+	0.161854i	$0.948253\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	− 12.0000i	− 0.429945i
$780$	0	0
$781$	40.0000i	1.43131i
$782$	0	0
$783$	−25.0000	−0.893427
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	− 11.0000i	− 0.392108i	−0.980593	−	0.196054i	$-0.937187\pi$
$787$	− 11.0000i	− 0.392108i	0.980593	−	0.196054i	$-0.0628127\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	− 18.0000i	− 0.638394i
$796$	0	0
$797$	5.00000i	0.177109i	0.996071	+	0.0885545i	$0.0282248\pi$
$797$	5.00000i	0.177109i	−0.996071	+	0.0885545i	$0.971775\pi$
$798$	0	0
$799$	−60.0000	−2.12265
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	− 28.0000i	− 0.988099i
$804$	0	0
$805$	− 18.0000i	− 0.634417i
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	9.00000i	0.316033i	0.987436	+	0.158016i	$0.0505099\pi$
$811$	9.00000i	0.316033i	−0.987436	+	0.158016i	$0.949490\pi$
$812$	0	0
$813$	3.00000i	0.105215i
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	10.0000	0.349856
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	− 50.0000i	− 1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	− 50.0000i	− 1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$	0	0
$823$	3.00000	0.104573	0.0522867	−	0.998632i	$-0.483349\pi$
$823$	3.00000	0.104573	0.0522867	+	0.998632i	$0.483349\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	19.0000i	0.660695i	0.943859	+	0.330347i	$0.107166\pi$
$827$	19.0000i	0.660695i	−0.943859	+	0.330347i	$0.892834\pi$
$828$	0	0
$829$	7.00000i	0.243120i	0.992584	+	0.121560i	$0.0387897\pi$
$829$	7.00000i	0.243120i	−0.992584	+	0.121560i	$0.961210\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	48.0000i	1.66111i
$836$	0	0
$837$	10.0000i	0.345651i
$838$	0	0
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	0	0
$841$	4.00000	0.137931
$842$	0	0
$843$	20.0000i	0.688837i
$844$	0	0
$845$	− 24.0000i	− 0.825625i
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−24.0000	−0.823678
$850$	0	0
$851$	− 90.0000i	− 3.08516i
$852$	0	0
$853$	40.0000i	1.36957i	0.728743	+	0.684787i	$0.240105\pi$
$853$	40.0000i	1.36957i	−0.728743	+	0.684787i	$0.759895\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−12.0000	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$857$	−12.0000	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$858$	0	0
$859$	16.0000i	0.545913i	0.962026	+	0.272956i	$0.0880015\pi$
$859$	16.0000i	0.545913i	−0.962026	+	0.272956i	$0.911998\pi$
$860$	0	0
$861$	− 12.0000i	− 0.408959i
$862$	0	0
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	−28.0000	−0.952029
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	− 40.0000i	− 1.35691i
$870$	0	0
$871$	−13.0000	−0.440488
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	− 12.0000i	− 0.405674i
$876$	0	0
$877$	− 41.0000i	− 1.38447i	−0.721671	−	0.692236i	$-0.756626\pi$
$877$	− 41.0000i	− 1.38447i	0.721671	−	0.692236i	$-0.243374\pi$
$878$	0	0
$879$	−21.0000	−0.708312
$880$	0	0
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	0	0
$883$	− 26.0000i	− 0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$	− 26.0000i	− 0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$	0	0
$885$	− 18.0000i	− 0.605063i
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	6.00000	0.201234
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−9.00000	−0.300501
$898$	0	0
$899$	10.0000i	0.333519i
$900$	0	0
$901$	45.0000i	1.49917i
$902$	0	0
$903$	10.0000	0.332779
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	− 51.0000i	− 1.69343i	−0.532049	−	0.846714i	$-0.678578\pi$
$907$	− 51.0000i	− 1.69343i	0.532049	−	0.846714i	$-0.321422\pi$
$908$	0	0
$909$	8.00000i	0.265343i
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	0	0
$915$	− 4.00000i	− 0.132236i
$916$	0	0
$917$	− 14.0000i	− 0.462321i
$918$	0	0
$919$	43.0000	1.41844	0.709220	−	0.704988i	$-0.249047\pi$
$919$	43.0000	1.41844	0.709220	+	0.704988i	$0.249047\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	− 10.0000i	− 0.329154i
$924$	0	0
$925$	− 10.0000i	− 0.328798i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	13.0000	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$929$	13.0000	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$930$	0	0
$931$	− 6.00000i	− 0.196642i
$932$	0	0
$933$	33.0000i	1.08037i
$934$	0	0
$935$	40.0000	1.30814
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	− 21.0000i	− 0.685309i
$940$	0	0
$941$	− 3.00000i	− 0.0977972i	−0.998804	−	0.0488986i	$-0.984429\pi$
$941$	− 3.00000i	− 0.0977972i	0.998804	−	0.0488986i	$-0.0155711\pi$
$942$	0	0
$943$	−108.000	−3.51696
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	38.0000i	1.23483i	0.786636	+	0.617417i	$0.211821\pi$
$947$	38.0000i	1.23483i	−0.786636	+	0.617417i	$0.788179\pi$
$948$	0	0
$949$	7.00000i	0.227230i
$950$	0	0
$951$	−31.0000	−1.00524
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	− 46.0000i	− 1.48853i
$956$	0	0
$957$	20.0000i	0.646508i
$958$	0	0
$959$	13.0000	0.419792
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	− 22.0000i	− 0.708940i
$964$	0	0
$965$	8.00000i	0.257529i
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	12.0000i	0.385098i	0.981287	+	0.192549i	$0.0616755\pi$
$971$	12.0000i	0.385098i	−0.981287	+	0.192549i	$0.938325\pi$
$972$	0	0
$973$	2.00000i	0.0641171i
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	− 24.0000i	− 0.767043i
$980$	0	0
$981$	30.0000i	0.957826i
$982$	0	0
$983$	−2.00000	−0.0637901	−0.0318950	−	0.999491i	$-0.510154\pi$
$983$	−2.00000	−0.0637901	−0.0318950	+	0.999491i	$0.510154\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	− 90.0000i	− 2.86183i
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	0	0
$993$	−1.00000	−0.0317340
$994$	0	0
$995$	− 10.0000i	− 0.317021i
$996$	0	0
$997$	50.0000i	1.58352i	0.610835	+	0.791758i	$0.290834\pi$
$997$	50.0000i	1.58352i	−0.610835	+	0.791758i	$0.709166\pi$
$998$	0	0
$999$	50.0000	1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.b.1217.2	yes	2
4.3	odd	2		2432.2.c.a.1217.1	✓	2
8.3	odd	2		2432.2.c.a.1217.2	yes	2
8.5	even	2	inner	2432.2.c.b.1217.1	yes	2
16.3	odd	4		4864.2.a.k.1.1		1
16.5	even	4		4864.2.a.n.1.1		1
16.11	odd	4		4864.2.a.f.1.1		1
16.13	even	4		4864.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.a.1217.1	✓	2	4.3	odd	2
2432.2.c.a.1217.2	yes	2	8.3	odd	2
2432.2.c.b.1217.1	yes	2	8.5	even	2	inner
2432.2.c.b.1217.2	yes	2	1.1	even	1	trivial
4864.2.a.c.1.1		1	16.13	even	4
4864.2.a.f.1.1		1	16.11	odd	4
4864.2.a.k.1.1		1	16.3	odd	4
4864.2.a.n.1.1		1	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{5} +1.00000 q^{7} +2.00000 q^{9} -4.00000i q^{11} +1.00000i q^{13} +2.00000 q^{15} -5.00000 q^{17} +1.00000i q^{19} +1.00000i q^{21} +9.00000 q^{23} +1.00000 q^{25} +5.00000i q^{27} +5.00000i q^{29} +2.00000 q^{31} +4.00000 q^{33} -2.00000i q^{35} -10.0000i q^{37} -1.00000 q^{39} -12.0000 q^{41} -10.0000i q^{43} -4.00000i q^{45} +12.0000 q^{47} -6.00000 q^{49} -5.00000i q^{51} -9.00000i q^{53} -8.00000 q^{55} -1.00000 q^{57} -9.00000i q^{59} -2.00000i q^{61} +2.00000 q^{63} +2.00000 q^{65} +13.0000i q^{67} +9.00000i q^{69} -10.0000 q^{71} +7.00000 q^{73} +1.00000i q^{75} -4.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.00000i q^{83} +10.0000i q^{85} -5.00000 q^{87} +6.00000 q^{89} +1.00000i q^{91} +2.00000i q^{93} +2.00000 q^{95} -6.00000 q^{97} -8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 + 4 * q^9	\( 2 q + 2 q^{7} + 4 q^{9} + 4 q^{15} - 10 q^{17} + 18 q^{23} + 2 q^{25} + 4 q^{31} + 8 q^{33} - 2 q^{39} - 24 q^{41} + 24 q^{47} - 12 q^{49} - 16 q^{55} - 2 q^{57} + 4 q^{63} + 4 q^{65} - 20 q^{71} + 14 q^{73} + 20 q^{79} + 2 q^{81} - 10 q^{87} + 12 q^{89} + 4 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 4 * q^9 + 4 * q^15 - 10 * q^17 + 18 * q^23 + 2 * q^25 + 4 * q^31 + 8 * q^33 - 2 * q^39 - 24 * q^41 + 24 * q^47 - 12 * q^49 - 16 * q^55 - 2 * q^57 + 4 * q^63 + 4 * q^65 - 20 * q^71 + 14 * q^73 + 20 * q^79 + 2 * q^81 - 10 * q^87 + 12 * q^89 + 4 * q^95 - 12 * q^97

Introduction

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