LMFDB - Embedded newform 1764.4.f.a.881.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(881,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.f (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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{18}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.12
Root			$1.09700 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.a.881.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.54212 q^{5} +O(q^{10})$	$q+8.54212 q^{5} +31.7173i q^{11} +9.92568i q^{13} +127.550 q^{17} +116.535i q^{19} +64.4636i q^{23} -52.0322 q^{25} -113.016i q^{29} +7.31501i q^{31} +369.472 q^{37} -211.959 q^{41} -432.263 q^{43} -400.041 q^{47} +140.722i q^{53} +270.933i q^{55} +518.894 q^{59} -27.2079i q^{61} +84.7863i q^{65} -136.719 q^{67} +604.779i q^{71} -48.4720i q^{73} -831.236 q^{79} +37.2350 q^{83} +1089.55 q^{85} +470.712 q^{89} +995.459i q^{95} +522.691i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 424 * q^25 - 152 * q^37 + 1408 * q^43 + 3056 * q^67 + 728 * q^79 + 7392 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	8.54212	0.764031	0.382015	−	0.924156i	$-0.375230\pi$
$5$	8.54212	0.764031	0.382015	+	0.924156i	$0.375230\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	31.7173i	0.869374i	0.900582	+	0.434687i	$0.143141\pi$
$11$	31.7173i	0.869374i	−0.900582	+	0.434687i	$0.856859\pi$
$12$	0	0
$13$	9.92568i	0.211761i	0.994379	+	0.105880i	$0.0337660\pi$
$13$	9.92568i	0.211761i	−0.994379	+	0.105880i	$0.966234\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	127.550	1.81973	0.909866	−	0.414901i	$-0.136184\pi$
$17$	127.550	1.81973	0.909866	+	0.414901i	$0.136184\pi$
$18$	0	0
$19$	116.535i	1.40711i	0.710642	+	0.703554i	$0.248405\pi$
$19$	116.535i	1.40711i	−0.710642	+	0.703554i	$0.751595\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	64.4636i	0.584417i	0.956355	+	0.292208i	$0.0943901\pi$
$23$	64.4636i	0.584417i	−0.956355	+	0.292208i	$0.905610\pi$
$24$	0	0
$25$	−52.0322	−0.416257
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 113.016i	− 0.723671i	−0.932242	−	0.361836i	$-0.882150\pi$
$29$	− 113.016i	− 0.723671i	0.932242	−	0.361836i	$-0.117850\pi$
$30$	0	0
$31$	7.31501i	0.0423811i	0.999775	+	0.0211906i	$0.00674567\pi$
$31$	7.31501i	0.0423811i	−0.999775	+	0.0211906i	$0.993254\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	369.472	1.64164	0.820822	−	0.571183i	$-0.193515\pi$
$37$	369.472	1.64164	0.820822	+	0.571183i	$0.193515\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−211.959	−0.807376	−0.403688	−	0.914897i	$-0.632272\pi$
$41$	−211.959	−0.807376	−0.403688	+	0.914897i	$0.632272\pi$
$42$	0	0
$43$	−432.263	−1.53301	−0.766506	−	0.642237i	$-0.778007\pi$
$43$	−432.263	−1.53301	−0.766506	+	0.642237i	$0.778007\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−400.041	−1.24153	−0.620766	−	0.783996i	$-0.713178\pi$
$47$	−400.041	−1.24153	−0.620766	+	0.783996i	$0.713178\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	140.722i	0.364712i	0.983233	+	0.182356i	$0.0583723\pi$
$53$	140.722i	0.364712i	−0.983233	+	0.182356i	$0.941628\pi$
$54$	0	0
$55$	270.933i	0.664228i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	518.894	1.14499	0.572493	−	0.819910i	$-0.305976\pi$
$59$	518.894	1.14499	0.572493	+	0.819910i	$0.305976\pi$
$60$	0	0
$61$	− 27.2079i	− 0.0571084i	−0.999592	−	0.0285542i	$-0.990910\pi$
$61$	− 27.2079i	− 0.0571084i	0.999592	−	0.0285542i	$-0.00909031\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	84.7863i	0.161792i
$66$	0	0
$67$	−136.719	−0.249298	−0.124649	−	0.992201i	$-0.539780\pi$
$67$	−136.719	−0.249298	−0.124649	+	0.992201i	$0.539780\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	604.779i	1.01090i	0.862855	+	0.505451i	$0.168674\pi$
$71$	604.779i	1.01090i	−0.862855	+	0.505451i	$0.831326\pi$
$72$	0	0
$73$	− 48.4720i	− 0.0777154i	−0.999245	−	0.0388577i	$-0.987628\pi$
$73$	− 48.4720i	− 0.0777154i	0.999245	−	0.0388577i	$-0.0123719\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−831.236	−1.18381	−0.591907	−	0.806006i	$-0.701625\pi$
$79$	−831.236	−1.18381	−0.591907	+	0.806006i	$0.701625\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	37.2350	0.0492418	0.0246209	−	0.999697i	$-0.492162\pi$
$83$	37.2350	0.0492418	0.0246209	+	0.999697i	$0.492162\pi$
$84$	0	0
$85$	1089.55	1.39033
$86$	0	0
$87$	0	0
$88$	0	0
$89$	470.712	0.560623	0.280311	−	0.959909i	$-0.409562\pi$
$89$	470.712	0.560623	0.280311	+	0.959909i	$0.409562\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	995.459i	1.07507i
$96$	0	0
$97$	522.691i	0.547126i	0.961854	+	0.273563i	$0.0882022\pi$
$97$	522.691i	0.547126i	−0.961854	+	0.273563i	$0.911798\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1054.72	−1.03910	−0.519549	−	0.854441i	$-0.673900\pi$
$101$	−1054.72	−1.03910	−0.519549	+	0.854441i	$0.673900\pi$
$102$	0	0
$103$	743.616i	0.711366i	0.934607	+	0.355683i	$0.115752\pi$
$103$	743.616i	0.711366i	−0.934607	+	0.355683i	$0.884248\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 236.002i	− 0.213225i	−0.994301	−	0.106613i	$-0.965999\pi$
$107$	− 236.002i	− 0.213225i	0.994301	−	0.106613i	$-0.0340005\pi$
$108$	0	0
$109$	−232.539	−0.204341	−0.102171	−	0.994767i	$-0.532579\pi$
$109$	−232.539	−0.204341	−0.102171	+	0.994767i	$0.532579\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2266.43i	1.88680i	0.331661	+	0.943398i	$0.392391\pi$
$113$	2266.43i	1.88680i	−0.331661	+	0.943398i	$0.607609\pi$
$114$	0	0
$115$	550.656i	0.446512i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	325.016	0.244189
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1512.23	−1.08206
$126$	0	0
$127$	1675.71	1.17083	0.585414	−	0.810735i	$-0.300932\pi$
$127$	1675.71	1.17083	0.585414	+	0.810735i	$0.300932\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1186.21	−0.791139	−0.395570	−	0.918436i	$-0.629453\pi$
$131$	−1186.21	−0.791139	−0.395570	+	0.918436i	$0.629453\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3167.85i	− 1.97553i	−0.155955	−	0.987764i	$-0.549845\pi$
$137$	− 3167.85i	− 1.97553i	0.155955	−	0.987764i	$-0.450155\pi$
$138$	0	0
$139$	− 607.821i	− 0.370897i	−0.982654	−	0.185448i	$-0.940626\pi$
$139$	− 607.821i	− 0.370897i	0.982654	−	0.185448i	$-0.0593738\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−314.815	−0.184099
$144$	0	0
$145$	− 965.393i	− 0.552907i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1290.46i	− 0.709521i	−0.934957	−	0.354760i	$-0.884562\pi$
$149$	− 1290.46i	− 0.709521i	0.934957	−	0.354760i	$-0.115438\pi$
$150$	0	0
$151$	2904.99	1.56559	0.782796	−	0.622278i	$-0.213793\pi$
$151$	2904.99	1.56559	0.782796	+	0.622278i	$0.213793\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	62.4857i	0.0323805i
$156$	0	0
$157$	− 1180.51i	− 0.600094i	−0.953924	−	0.300047i	$-0.902998\pi$
$157$	− 1180.51i	− 0.600094i	0.953924	−	0.300047i	$-0.0970024\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1607.58	0.772488	0.386244	−	0.922397i	$-0.373772\pi$
$163$	1607.58	0.772488	0.386244	+	0.922397i	$0.373772\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2621.57	−1.21475	−0.607374	−	0.794416i	$-0.707777\pi$
$167$	−2621.57	−1.21475	−0.607374	+	0.794416i	$0.707777\pi$
$168$	0	0
$169$	2098.48	0.955157
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2643.57	−1.16177	−0.580887	−	0.813984i	$-0.697294\pi$
$173$	−2643.57	−1.16177	−0.580887	+	0.813984i	$0.697294\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4150.17i	1.73295i	0.499221	+	0.866475i	$0.333620\pi$
$179$	4150.17i	1.73295i	−0.499221	+	0.866475i	$0.666380\pi$
$180$	0	0
$181$	4040.50i	1.65927i	0.558304	+	0.829636i	$0.311452\pi$
$181$	4040.50i	1.65927i	−0.558304	+	0.829636i	$0.688548\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3156.08	1.25427
$186$	0	0
$187$	4045.54i	1.58203i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3726.65i	1.41178i	0.708319	+	0.705892i	$0.249454\pi$
$191$	3726.65i	1.41178i	−0.708319	+	0.705892i	$0.750546\pi$
$192$	0	0
$193$	1336.94	0.498628	0.249314	−	0.968423i	$-0.419795\pi$
$193$	1336.94	0.498628	0.249314	+	0.968423i	$0.419795\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3451.92i	1.24842i	0.781256	+	0.624210i	$0.214579\pi$
$197$	3451.92i	1.24842i	−0.781256	+	0.624210i	$0.785421\pi$
$198$	0	0
$199$	− 5336.30i	− 1.90091i	−0.310869	−	0.950453i	$-0.600620\pi$
$199$	− 5336.30i	− 1.90091i	0.310869	−	0.950453i	$-0.399380\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1810.58	−0.616860
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3696.18	−1.22330
$210$	0	0
$211$	1800.90	0.587578	0.293789	−	0.955870i	$-0.405084\pi$
$211$	1800.90	0.587578	0.293789	+	0.955870i	$0.405084\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3692.45	−1.17127
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1266.02i	0.385348i
$222$	0	0
$223$	2815.72i	0.845535i	0.906238	+	0.422768i	$0.138941\pi$
$223$	2815.72i	0.845535i	−0.906238	+	0.422768i	$0.861059\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3953.12	1.15585	0.577925	−	0.816090i	$-0.303863\pi$
$227$	3953.12	1.15585	0.577925	+	0.816090i	$0.303863\pi$
$228$	0	0
$229$	2462.59i	0.710622i	0.934748	+	0.355311i	$0.115625\pi$
$229$	2462.59i	0.710622i	−0.934748	+	0.355311i	$0.884375\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5202.31i	1.46272i	0.681989	+	0.731362i	$0.261115\pi$
$233$	5202.31i	1.46272i	−0.681989	+	0.731362i	$0.738885\pi$
$234$	0	0
$235$	−3417.20	−0.948569
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 1510.80i	− 0.408894i	−0.978878	−	0.204447i	$-0.934460\pi$
$239$	− 1510.80i	− 0.408894i	0.978878	−	0.204447i	$-0.0655396\pi$
$240$	0	0
$241$	− 1571.52i	− 0.420043i	−0.977697	−	0.210022i	$-0.932647\pi$
$241$	− 1571.52i	− 0.420043i	0.977697	−	0.210022i	$-0.0673534\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1156.69	−0.297970
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3863.31	−0.971514	−0.485757	−	0.874094i	$-0.661456\pi$
$251$	−3863.31	−0.971514	−0.485757	+	0.874094i	$0.661456\pi$
$252$	0	0
$253$	−2044.61	−0.508077
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5822.19	−1.41314	−0.706572	−	0.707641i	$-0.749760\pi$
$257$	−5822.19	−1.41314	−0.706572	+	0.707641i	$0.749760\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1080.40i	0.253310i	0.991947	+	0.126655i	$0.0404241\pi$
$263$	1080.40i	0.253310i	−0.991947	+	0.126655i	$0.959576\pi$
$264$	0	0
$265$	1202.07i	0.278651i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2285.52	0.518033	0.259016	−	0.965873i	$-0.416602\pi$
$269$	2285.52	0.518033	0.259016	+	0.965873i	$0.416602\pi$
$270$	0	0
$271$	2271.12i	0.509081i	0.967062	+	0.254540i	$0.0819242\pi$
$271$	2271.12i	0.509081i	−0.967062	+	0.254540i	$0.918076\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1650.32i	− 0.361883i
$276$	0	0
$277$	1649.10	0.357707	0.178853	−	0.983876i	$-0.442761\pi$
$277$	1649.10	0.357707	0.178853	+	0.983876i	$0.442761\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7573.33i	1.60778i	0.594776	+	0.803892i	$0.297241\pi$
$281$	7573.33i	1.60778i	−0.594776	+	0.803892i	$0.702759\pi$
$282$	0	0
$283$	5226.18i	1.09775i	0.835903	+	0.548877i	$0.184944\pi$
$283$	5226.18i	1.09775i	−0.835903	+	0.548877i	$0.815056\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	11356.0	2.31143
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4924.11	−0.981808	−0.490904	−	0.871214i	$-0.663333\pi$
$293$	−4924.11	−0.981808	−0.490904	+	0.871214i	$0.663333\pi$
$294$	0	0
$295$	4432.45	0.874805
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−639.845	−0.123756
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 232.413i	− 0.0436325i
$306$	0	0
$307$	10064.5i	1.87105i	0.353256	+	0.935527i	$0.385075\pi$
$307$	10064.5i	1.87105i	−0.353256	+	0.935527i	$0.614925\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2300.23	−0.419403	−0.209701	−	0.977765i	$-0.567249\pi$
$311$	−2300.23	−0.419403	−0.209701	+	0.977765i	$0.567249\pi$
$312$	0	0
$313$	− 1037.23i	− 0.187310i	−0.995605	−	0.0936548i	$-0.970145\pi$
$313$	− 1037.23i	− 0.187310i	0.995605	−	0.0936548i	$-0.0298550\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8331.08i	− 1.47609i	−0.674752	−	0.738044i	$-0.735750\pi$
$317$	− 8331.08i	− 1.47609i	0.674752	−	0.738044i	$-0.264250\pi$
$318$	0	0
$319$	3584.54	0.629141
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14864.1i	2.56056i
$324$	0	0
$325$	− 516.455i	− 0.0881469i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2795.98	−0.464293	−0.232146	−	0.972681i	$-0.574575\pi$
$331$	−2795.98	−0.464293	−0.232146	+	0.972681i	$0.574575\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1167.87	−0.190471
$336$	0	0
$337$	4675.27	0.755721	0.377861	−	0.925863i	$-0.376660\pi$
$337$	4675.27	0.755721	0.377861	+	0.925863i	$0.376660\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−232.012	−0.0368450
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2659.85i	0.411494i	0.978605	+	0.205747i	$0.0659624\pi$
$347$	2659.85i	0.411494i	−0.978605	+	0.205747i	$0.934038\pi$
$348$	0	0
$349$	− 12826.3i	− 1.96727i	−0.180181	−	0.983633i	$-0.557668\pi$
$349$	− 12826.3i	− 1.96727i	0.180181	−	0.983633i	$-0.442332\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8227.53	1.24053	0.620265	−	0.784392i	$-0.287025\pi$
$353$	8227.53	1.24053	0.620265	+	0.784392i	$0.287025\pi$
$354$	0	0
$355$	5166.09i	0.772360i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 8.52451i	− 0.00125322i	−1.00000		0.000626611i	$-0.999801\pi$
$359$	− 8.52451i	− 0.00125322i	1.00000		0.000626611i	$-0.000199456\pi$
$360$	0	0
$361$	−6721.49	−0.979952
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 414.054i	− 0.0593769i
$366$	0	0
$367$	7304.60i	1.03896i	0.854484	+	0.519478i	$0.173874\pi$
$367$	7304.60i	1.03896i	−0.854484	+	0.519478i	$0.826126\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8506.79	−1.18087	−0.590436	−	0.807085i	$-0.701044\pi$
$373$	−8506.79	−1.18087	−0.590436	+	0.807085i	$0.701044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1121.76	0.153245
$378$	0	0
$379$	7735.02	1.04834	0.524171	−	0.851613i	$-0.324375\pi$
$379$	7735.02	1.04834	0.524171	+	0.851613i	$0.324375\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7926.86	1.05755	0.528777	−	0.848761i	$-0.322651\pi$
$383$	7926.86	1.05755	0.528777	+	0.848761i	$0.322651\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10719.9i	1.39722i	0.715503	+	0.698609i	$0.246197\pi$
$389$	10719.9i	1.39722i	−0.715503	+	0.698609i	$0.753803\pi$
$390$	0	0
$391$	8222.34i	1.06348i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7100.52	−0.904471
$396$	0	0
$397$	5642.86i	0.713368i	0.934225	+	0.356684i	$0.116093\pi$
$397$	5642.86i	0.713368i	−0.934225	+	0.356684i	$0.883907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 184.679i	− 0.0229986i	−0.999934	−	0.0114993i	$-0.996340\pi$
$401$	− 184.679i	− 0.0229986i	0.999934	−	0.0114993i	$-0.00366042\pi$
$402$	0	0
$403$	−72.6064	−0.00897465
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11718.6i	1.42720i
$408$	0	0
$409$	− 2680.53i	− 0.324067i	−0.986785	−	0.162034i	$-0.948195\pi$
$409$	− 2680.53i	− 0.324067i	0.986785	−	0.162034i	$-0.0518053\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	318.066	0.0376223
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12735.7	−1.48492	−0.742460	−	0.669890i	$-0.766341\pi$
$419$	−12735.7	−1.48492	−0.742460	+	0.669890i	$0.766341\pi$
$420$	0	0
$421$	3317.68	0.384070	0.192035	−	0.981388i	$-0.438491\pi$
$421$	3317.68	0.384070	0.192035	+	0.981388i	$0.438491\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6636.71	−0.757477
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 8947.64i	− 0.999983i	−0.866030	−	0.499992i	$-0.833336\pi$
$431$	− 8947.64i	− 0.999983i	0.866030	−	0.499992i	$-0.166664\pi$
$432$	0	0
$433$	11833.0i	1.31330i	0.754194	+	0.656651i	$0.228028\pi$
$433$	11833.0i	1.31330i	−0.754194	+	0.656651i	$0.771972\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7512.28	−0.822337
$438$	0	0
$439$	− 8517.92i	− 0.926055i	−0.886344	−	0.463028i	$-0.846763\pi$
$439$	− 8517.92i	− 0.926055i	0.886344	−	0.463028i	$-0.153237\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8303.27i	0.890520i	0.895401	+	0.445260i	$0.146889\pi$
$443$	8303.27i	0.890520i	−0.895401	+	0.445260i	$0.853111\pi$
$444$	0	0
$445$	4020.88	0.428333
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 14235.2i	− 1.49622i	−0.663575	−	0.748110i	$-0.730961\pi$
$449$	− 14235.2i	− 1.49622i	0.663575	−	0.748110i	$-0.269039\pi$
$450$	0	0
$451$	− 6722.75i	− 0.701912i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6093.95	−0.623770	−0.311885	−	0.950120i	$-0.600960\pi$
$457$	−6093.95	−0.623770	−0.311885	+	0.950120i	$0.600960\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13335.2	1.34725	0.673625	−	0.739073i	$-0.264736\pi$
$461$	13335.2	1.34725	0.673625	+	0.739073i	$0.264736\pi$
$462$	0	0
$463$	−6596.99	−0.662178	−0.331089	−	0.943600i	$-0.607416\pi$
$463$	−6596.99	−0.662178	−0.331089	+	0.943600i	$0.607416\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4352.69	−0.431303	−0.215652	−	0.976470i	$-0.569188\pi$
$467$	−4352.69	−0.431303	−0.215652	+	0.976470i	$0.569188\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 13710.2i	− 1.33276i
$474$	0	0
$475$	− 6063.59i	− 0.585719i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−319.751	−0.0305006	−0.0152503	−	0.999884i	$-0.504855\pi$
$479$	−319.751	−0.0305006	−0.0152503	+	0.999884i	$0.504855\pi$
$480$	0	0
$481$	3667.26i	0.347636i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4464.89i	0.418021i
$486$	0	0
$487$	5838.97	0.543304	0.271652	−	0.962396i	$-0.412430\pi$
$487$	5838.97	0.543304	0.271652	+	0.962396i	$0.412430\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12442.2i	− 1.14360i	−0.820392	−	0.571801i	$-0.806245\pi$
$491$	− 12442.2i	− 1.14360i	0.820392	−	0.571801i	$-0.193755\pi$
$492$	0	0
$493$	− 14415.2i	− 1.31689i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16696.8	1.49790	0.748949	−	0.662627i	$-0.230559\pi$
$499$	16696.8	1.49790	0.748949	+	0.662627i	$0.230559\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4435.36	−0.393167	−0.196583	−	0.980487i	$-0.562985\pi$
$503$	−4435.36	−0.393167	−0.196583	+	0.980487i	$0.562985\pi$
$504$	0	0
$505$	−9009.57	−0.793902
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22414.8	−1.95190	−0.975951	−	0.217990i	$-0.930050\pi$
$509$	−22414.8	−1.95190	−0.975951	+	0.217990i	$0.930050\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6352.06i	0.543505i
$516$	0	0
$517$	− 12688.2i	− 1.07936i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19122.3	1.60799	0.803996	−	0.594635i	$-0.202703\pi$
$521$	19122.3	1.60799	0.803996	+	0.594635i	$0.202703\pi$
$522$	0	0
$523$	− 21793.1i	− 1.82207i	−0.412325	−	0.911037i	$-0.635283\pi$
$523$	− 21793.1i	− 1.82207i	0.412325	−	0.911037i	$-0.364717\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	933.031i	0.0771223i
$528$	0	0
$529$	8011.45	0.658457
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 2103.84i	− 0.170970i
$534$	0	0
$535$	− 2015.95i	− 0.162911i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−202.214	−0.0160700	−0.00803498	−	0.999968i	$-0.502558\pi$
$541$	−202.214	−0.0160700	−0.00803498	+	0.999968i	$0.502558\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1986.38	−0.156123
$546$	0	0
$547$	1510.85	0.118097	0.0590486	−	0.998255i	$-0.481193\pi$
$547$	1510.85	0.118097	0.0590486	+	0.998255i	$0.481193\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13170.3	1.01828
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3771.44i	− 0.286896i	−0.989658	−	0.143448i	$-0.954181\pi$
$557$	− 3771.44i	− 0.286896i	0.989658	−	0.143448i	$-0.0458189\pi$
$558$	0	0
$559$	− 4290.51i	− 0.324632i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7833.11	−0.586370	−0.293185	−	0.956056i	$-0.594715\pi$
$563$	−7833.11	−0.586370	−0.293185	+	0.956056i	$0.594715\pi$
$564$	0	0
$565$	19360.1i	1.44157i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5706.24i	0.420418i	0.977656	+	0.210209i	$0.0674145\pi$
$569$	5706.24i	0.420418i	−0.977656	+	0.210209i	$0.932586\pi$
$570$	0	0
$571$	7137.66	0.523120	0.261560	−	0.965187i	$-0.415763\pi$
$571$	7137.66	0.523120	0.261560	+	0.965187i	$0.415763\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 3354.18i	− 0.243268i
$576$	0	0
$577$	− 7597.05i	− 0.548127i	−0.961712	−	0.274064i	$-0.911632\pi$
$577$	− 7597.05i	− 0.548127i	0.961712	−	0.274064i	$-0.0883679\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4463.33	−0.317071
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8212.13	−0.577429	−0.288714	−	0.957415i	$-0.593228\pi$
$587$	−8212.13	−0.577429	−0.288714	+	0.957415i	$0.593228\pi$
$588$	0	0
$589$	−852.457	−0.0596348
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18980.2	1.31437	0.657186	−	0.753729i	$-0.271747\pi$
$593$	18980.2	1.31437	0.657186	+	0.753729i	$0.271747\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20547.0i	1.40155i	0.713382	+	0.700776i	$0.247163\pi$
$599$	20547.0i	1.40155i	−0.713382	+	0.700776i	$0.752837\pi$
$600$	0	0
$601$	15444.1i	1.04821i	0.851652	+	0.524107i	$0.175601\pi$
$601$	15444.1i	1.04821i	−0.851652	+	0.524107i	$0.824399\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2776.33	0.186568
$606$	0	0
$607$	− 1524.98i	− 0.101972i	−0.998699	−	0.0509859i	$-0.983764\pi$
$607$	− 1524.98i	− 0.101972i	0.998699	−	0.0509859i	$-0.0162364\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3970.68i	− 0.262908i
$612$	0	0
$613$	−17767.0	−1.17064	−0.585319	−	0.810803i	$-0.699031\pi$
$613$	−17767.0	−1.17064	−0.585319	+	0.810803i	$0.699031\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21575.8i	− 1.40779i	−0.710302	−	0.703897i	$-0.751442\pi$
$617$	− 21575.8i	− 1.40779i	0.710302	−	0.703897i	$-0.248558\pi$
$618$	0	0
$619$	8601.02i	0.558488i	0.960220	+	0.279244i	$0.0900839\pi$
$619$	8601.02i	0.558488i	−0.960220	+	0.279244i	$0.909916\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6413.63	−0.410473
$626$	0	0
$627$	0	0
$628$	0	0
$629$	47126.2	2.98736
$630$	0	0
$631$	−5922.32	−0.373635	−0.186817	−	0.982395i	$-0.559817\pi$
$631$	−5922.32	−0.373635	−0.186817	+	0.982395i	$0.559817\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14314.1	0.894548
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 22399.9i	− 1.38025i	−0.723689	−	0.690127i	$-0.757555\pi$
$641$	− 22399.9i	− 1.38025i	0.723689	−	0.690127i	$-0.242445\pi$
$642$	0	0
$643$	− 19715.7i	− 1.20919i	−0.796533	−	0.604595i	$-0.793335\pi$
$643$	− 19715.7i	− 1.20919i	0.796533	−	0.604595i	$-0.206665\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7527.37	−0.457390	−0.228695	−	0.973498i	$-0.573446\pi$
$647$	−7527.37	−0.457390	−0.228695	+	0.973498i	$0.573446\pi$
$648$	0	0
$649$	16457.9i	0.995421i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 16771.3i	− 1.00507i	−0.864556	−	0.502536i	$-0.832401\pi$
$653$	− 16771.3i	− 1.00507i	0.864556	−	0.502536i	$-0.167599\pi$
$654$	0	0
$655$	−10132.7	−0.604455
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 20834.0i	− 1.23153i	−0.787930	−	0.615765i	$-0.788847\pi$
$659$	− 20834.0i	− 1.23153i	0.787930	−	0.615765i	$-0.211153\pi$
$660$	0	0
$661$	26371.4i	1.55178i	0.630866	+	0.775892i	$0.282700\pi$
$661$	26371.4i	1.55178i	−0.630866	+	0.775892i	$0.717300\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7285.39	0.422926
$668$	0	0
$669$	0	0
$670$	0	0
$671$	862.959	0.0496485
$672$	0	0
$673$	−18150.5	−1.03960	−0.519800	−	0.854288i	$-0.673993\pi$
$673$	−18150.5	−1.03960	−0.519800	+	0.854288i	$0.673993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22715.5	1.28955	0.644776	−	0.764371i	$-0.276950\pi$
$677$	22715.5	1.28955	0.644776	+	0.764371i	$0.276950\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 13611.8i	− 0.762581i	−0.924455	−	0.381290i	$-0.875480\pi$
$683$	− 13611.8i	− 0.762581i	0.924455	−	0.381290i	$-0.124520\pi$
$684$	0	0
$685$	− 27060.1i	− 1.50936i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1396.77	−0.0772315
$690$	0	0
$691$	− 28365.9i	− 1.56163i	−0.624760	−	0.780817i	$-0.714803\pi$
$691$	− 28365.9i	− 1.56163i	0.624760	−	0.780817i	$-0.285197\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 5192.08i	− 0.283377i
$696$	0	0
$697$	−27035.4	−1.46921
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9203.01i	0.495853i	0.968779	+	0.247927i	$0.0797492\pi$
$701$	9203.01i	0.495853i	−0.968779	+	0.247927i	$0.920251\pi$
$702$	0	0
$703$	43056.6i	2.30997i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−21951.4	−1.16277	−0.581383	−	0.813630i	$-0.697488\pi$
$709$	−21951.4	−1.16277	−0.581383	+	0.813630i	$0.697488\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−471.552	−0.0247682
$714$	0	0
$715$	−2689.19	−0.140657
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10180.1	−0.528032	−0.264016	−	0.964518i	$-0.585047\pi$
$719$	−10180.1	−0.528032	−0.264016	+	0.964518i	$0.585047\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5880.45i	0.301234i
$726$	0	0
$727$	19192.2i	0.979093i	0.871977	+	0.489546i	$0.162838\pi$
$727$	19192.2i	0.979093i	−0.871977	+	0.489546i	$0.837162\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−55135.3	−2.78967
$732$	0	0
$733$	− 35699.2i	− 1.79888i	−0.437040	−	0.899442i	$-0.643973\pi$
$733$	− 35699.2i	− 1.79888i	0.437040	−	0.899442i	$-0.356027\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4336.37i	− 0.216733i
$738$	0	0
$739$	27743.9	1.38102	0.690511	−	0.723322i	$-0.257386\pi$
$739$	27743.9	1.38102	0.690511	+	0.723322i	$0.257386\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25212.0i	1.24487i	0.782671	+	0.622435i	$0.213857\pi$
$743$	25212.0i	1.24487i	−0.782671	+	0.622435i	$0.786143\pi$
$744$	0	0
$745$	− 11023.3i	− 0.542096i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12958.4	0.629637	0.314818	−	0.949152i	$-0.398056\pi$
$751$	12958.4	0.629637	0.314818	+	0.949152i	$0.398056\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24814.7	1.19616
$756$	0	0
$757$	−1671.41	−0.0802488	−0.0401244	−	0.999195i	$-0.512775\pi$
$757$	−1671.41	−0.0802488	−0.0401244	+	0.999195i	$0.512775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27967.9	1.33224	0.666120	−	0.745845i	$-0.267954\pi$
$761$	27967.9	1.33224	0.666120	+	0.745845i	$0.267954\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5150.37i	0.242463i
$768$	0	0
$769$	− 22787.5i	− 1.06858i	−0.845301	−	0.534290i	$-0.820579\pi$
$769$	− 22787.5i	− 1.06858i	0.845301	−	0.534290i	$-0.179421\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1509.61	0.0702416	0.0351208	−	0.999383i	$-0.488818\pi$
$773$	1509.61	0.0702416	0.0351208	+	0.999383i	$0.488818\pi$
$774$	0	0
$775$	− 380.616i	− 0.0176414i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 24700.7i	− 1.13606i
$780$	0	0
$781$	−19181.9	−0.878852
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 10084.0i	− 0.458490i
$786$	0	0
$787$	− 5521.12i	− 0.250072i	−0.992152	−	0.125036i	$-0.960095\pi$
$787$	− 5521.12i	− 0.250072i	0.992152	−	0.125036i	$-0.0399046\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	270.057	0.0120933
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32608.8	1.44926	0.724631	−	0.689137i	$-0.242010\pi$
$797$	32608.8	1.44926	0.724631	+	0.689137i	$0.242010\pi$
$798$	0	0
$799$	−51025.3	−2.25926
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1537.40	0.0675637
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 9411.18i	− 0.408998i	−0.978867	−	0.204499i	$-0.934443\pi$
$809$	− 9411.18i	− 0.408998i	0.978867	−	0.204499i	$-0.0655565\pi$
$810$	0	0
$811$	− 26945.2i	− 1.16668i	−0.812230	−	0.583338i	$-0.801746\pi$
$811$	− 26945.2i	− 1.16668i	0.812230	−	0.583338i	$-0.198254\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13732.2	0.590204
$816$	0	0
$817$	− 50374.0i	− 2.15711i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 9963.93i	− 0.423561i	−0.977317	−	0.211781i	$-0.932074\pi$
$821$	− 9963.93i	− 0.423561i	0.977317	−	0.211781i	$-0.0679262\pi$
$822$	0	0
$823$	77.0563	0.00326369	0.00163184	−	0.999999i	$-0.499481\pi$
$823$	77.0563	0.00326369	0.00163184	+	0.999999i	$0.499481\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17261.3i	− 0.725797i	−0.931829	−	0.362898i	$-0.881787\pi$
$827$	− 17261.3i	− 0.725797i	0.931829	−	0.362898i	$-0.118213\pi$
$828$	0	0
$829$	− 1658.03i	− 0.0694640i	−0.999397	−	0.0347320i	$-0.988942\pi$
$829$	− 1658.03i	− 0.0694640i	0.999397	−	0.0347320i	$-0.0110578\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−22393.8	−0.928105
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31632.6	1.30164	0.650821	−	0.759231i	$-0.274425\pi$
$839$	31632.6	1.30164	0.650821	+	0.759231i	$0.274425\pi$
$840$	0	0
$841$	11616.5	0.476300
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17925.5	0.729769
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	23817.5i	0.959405i
$852$	0	0
$853$	− 7848.64i	− 0.315044i	−0.987516	−	0.157522i	$-0.949650\pi$
$853$	− 7848.64i	− 0.315044i	0.987516	−	0.157522i	$-0.0503505\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8581.68	0.342059	0.171030	−	0.985266i	$-0.445291\pi$
$857$	8581.68	0.342059	0.171030	+	0.985266i	$0.445291\pi$
$858$	0	0
$859$	41449.7i	1.64638i	0.567763	+	0.823192i	$0.307809\pi$
$859$	41449.7i	1.64638i	−0.567763	+	0.823192i	$0.692191\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6541.54i	0.258026i	0.991643	+	0.129013i	$0.0411809\pi$
$863$	6541.54i	0.258026i	−0.991643	+	0.129013i	$0.958819\pi$
$864$	0	0
$865$	−22581.7	−0.887631
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 26364.5i	− 1.02918i
$870$	0	0
$871$	− 1357.03i	− 0.0527914i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15706.5	0.604757	0.302378	−	0.953188i	$-0.402219\pi$
$877$	15706.5	0.604757	0.302378	+	0.953188i	$0.402219\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−15432.4	−0.590161	−0.295080	−	0.955472i	$-0.595346\pi$
$881$	−15432.4	−0.590161	−0.295080	+	0.955472i	$0.595346\pi$
$882$	0	0
$883$	17372.6	0.662101	0.331050	−	0.943613i	$-0.392597\pi$
$883$	17372.6	0.662101	0.331050	+	0.943613i	$0.392597\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31885.1	1.20699	0.603493	−	0.797369i	$-0.293775\pi$
$887$	31885.1	1.20699	0.603493	+	0.797369i	$0.293775\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 46619.0i	− 1.74697i
$894$	0	0
$895$	35451.2i	1.32403i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	826.710	0.0306700
$900$	0	0
$901$	17949.2i	0.663677i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	34514.5i	1.26773i
$906$	0	0
$907$	8034.59	0.294139	0.147070	−	0.989126i	$-0.453016\pi$
$907$	8034.59	0.294139	0.147070	+	0.989126i	$0.453016\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 51249.1i	− 1.86384i	−0.362662	−	0.931921i	$-0.618132\pi$
$911$	− 51249.1i	− 1.86384i	0.362662	−	0.931921i	$-0.381868\pi$
$912$	0	0
$913$	1180.99i	0.0428096i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−37985.3	−1.36346	−0.681730	−	0.731604i	$-0.738772\pi$
$919$	−37985.3	−1.36346	−0.681730	+	0.731604i	$0.738772\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6002.84	−0.214069
$924$	0	0
$925$	−19224.4	−0.683347
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26018.7	−0.918886	−0.459443	−	0.888207i	$-0.651951\pi$
$929$	−26018.7	−0.918886	−0.459443	+	0.888207i	$0.651951\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	34557.5i	1.20872i
$936$	0	0
$937$	− 44218.3i	− 1.54168i	−0.637031	−	0.770838i	$-0.719838\pi$
$937$	− 44218.3i	− 1.54168i	0.637031	−	0.770838i	$-0.280162\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6206.92	0.215027	0.107513	−	0.994204i	$-0.465711\pi$
$941$	6206.92	0.215027	0.107513	+	0.994204i	$0.465711\pi$
$942$	0	0
$943$	− 13663.6i	− 0.471844i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39580.0i	1.35816i	0.734065	+	0.679079i	$0.237621\pi$
$947$	39580.0i	1.35816i	−0.734065	+	0.679079i	$0.762379\pi$
$948$	0	0
$949$	481.118	0.0164571
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23083.3i	− 0.784619i	−0.919833	−	0.392310i	$-0.871676\pi$
$953$	− 23083.3i	− 0.784619i	0.919833	−	0.392310i	$-0.128324\pi$
$954$	0	0
$955$	31833.5i	1.07865i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29737.5	0.998204
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11420.3	0.380967
$966$	0	0
$967$	−34489.9	−1.14697	−0.573486	−	0.819215i	$-0.694409\pi$
$967$	−34489.9	−1.14697	−0.573486	+	0.819215i	$0.694409\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5002.61	0.165336	0.0826681	−	0.996577i	$-0.473656\pi$
$971$	5002.61	0.165336	0.0826681	+	0.996577i	$0.473656\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 48124.1i	− 1.57587i	−0.615758	−	0.787936i	$-0.711150\pi$
$977$	− 48124.1i	− 1.57587i	0.615758	−	0.787936i	$-0.288850\pi$
$978$	0	0
$979$	14929.7i	0.487391i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20789.6	−0.674552	−0.337276	−	0.941406i	$-0.609506\pi$
$983$	−20789.6	−0.674552	−0.337276	+	0.941406i	$0.609506\pi$
$984$	0	0
$985$	29486.7i	0.953832i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 27865.2i	− 0.895918i
$990$	0	0
$991$	49403.5	1.58361	0.791803	−	0.610776i	$-0.209143\pi$
$991$	49403.5	1.58361	0.791803	+	0.610776i	$0.209143\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 45583.3i	− 1.45235i
$996$	0	0
$997$	− 42690.6i	− 1.35609i	−0.735019	−	0.678047i	$-0.762827\pi$
$997$	− 42690.6i	− 1.35609i	0.735019	−	0.678047i	$-0.237173\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.f.a.881.12		16
3.2	odd	2	inner	1764.4.f.a.881.5		16
7.2	even	3		1764.4.t.b.521.3		16
7.3	odd	6		1764.4.t.b.1097.6		16
7.4	even	3		252.4.t.a.89.3	yes	16
7.5	odd	6		252.4.t.a.17.6	yes	16
7.6	odd	2	inner	1764.4.f.a.881.6		16
21.2	odd	6		1764.4.t.b.521.6		16
21.5	even	6		252.4.t.a.17.3	✓	16
21.11	odd	6		252.4.t.a.89.6	yes	16
21.17	even	6		1764.4.t.b.1097.3		16
21.20	even	2	inner	1764.4.f.a.881.11		16
28.11	odd	6		1008.4.bt.b.593.3		16
28.19	even	6		1008.4.bt.b.17.6		16
84.11	even	6		1008.4.bt.b.593.6		16
84.47	odd	6		1008.4.bt.b.17.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.3	✓	16	21.5	even	6
252.4.t.a.17.6	yes	16	7.5	odd	6
252.4.t.a.89.3	yes	16	7.4	even	3
252.4.t.a.89.6	yes	16	21.11	odd	6
1008.4.bt.b.17.3		16	84.47	odd	6
1008.4.bt.b.17.6		16	28.19	even	6
1008.4.bt.b.593.3		16	28.11	odd	6
1008.4.bt.b.593.6		16	84.11	even	6
1764.4.f.a.881.5		16	3.2	odd	2	inner
1764.4.f.a.881.6		16	7.6	odd	2	inner
1764.4.f.a.881.11		16	21.20	even	2	inner
1764.4.f.a.881.12		16	1.1	even	1	trivial
1764.4.t.b.521.3		16	7.2	even	3
1764.4.t.b.521.6		16	21.2	odd	6
1764.4.t.b.1097.3		16	21.17	even	6
1764.4.t.b.1097.6		16	7.3	odd	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.54212 q^{5} +O(q^{10})\)	\(q+8.54212 q^{5} +31.7173i q^{11} +9.92568i q^{13} +127.550 q^{17} +116.535i q^{19} +64.4636i q^{23} -52.0322 q^{25} -113.016i q^{29} +7.31501i q^{31} +369.472 q^{37} -211.959 q^{41} -432.263 q^{43} -400.041 q^{47} +140.722i q^{53} +270.933i q^{55} +518.894 q^{59} -27.2079i q^{61} +84.7863i q^{65} -136.719 q^{67} +604.779i q^{71} -48.4720i q^{73} -831.236 q^{79} +37.2350 q^{83} +1089.55 q^{85} +470.712 q^{89} +995.459i q^{95} +522.691i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 424 * q^25 - 152 * q^37 + 1408 * q^43 + 3056 * q^67 + 728 * q^79 + 7392 * q^85

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