LMFDB - Embedded newform 1764.4.f.a.881.9

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			881.9
Root			$5.70754 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.a.881.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.82452 q^{5} +O(q^{10})$	$q+6.82452 q^{5} -58.3474i q^{11} -38.5535i q^{13} +32.2519 q^{17} +124.530i q^{19} +201.168i q^{23} -78.4259 q^{25} +104.357i q^{29} +277.990i q^{31} -47.6573 q^{37} +387.272 q^{41} +272.528 q^{43} -163.188 q^{47} -362.422i q^{53} -398.193i q^{55} -211.705 q^{59} -234.310i q^{61} -263.109i q^{65} +524.262 q^{67} -348.689i q^{71} -537.100i q^{73} +725.584 q^{79} +392.121 q^{83} +220.104 q^{85} +860.029 q^{89} +849.857i q^{95} +978.030i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	6.82452	0.610404	0.305202	−	0.952288i	$-0.401276\pi$
$5$	6.82452	0.610404	0.305202	+	0.952288i	$0.401276\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 58.3474i	− 1.59931i	−0.600461	−	0.799654i	$-0.705016\pi$
$11$	− 58.3474i	− 1.59931i	0.600461	−	0.799654i	$-0.294984\pi$
$12$	0	0
$13$	− 38.5535i	− 0.822525i	−0.911517	−	0.411262i	$-0.865088\pi$
$13$	− 38.5535i	− 0.822525i	0.911517	−	0.411262i	$-0.134912\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	32.2519	0.460132	0.230066	−	0.973175i	$-0.426106\pi$
$17$	32.2519	0.460132	0.230066	+	0.973175i	$0.426106\pi$
$18$	0	0
$19$	124.530i	1.50364i	0.659369	+	0.751819i	$0.270823\pi$
$19$	124.530i	1.50364i	−0.659369	+	0.751819i	$0.729177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	201.168i	1.82376i	0.410456	+	0.911880i	$0.365370\pi$
$23$	201.168i	1.82376i	−0.410456	+	0.911880i	$0.634630\pi$
$24$	0	0
$25$	−78.4259	−0.627407
$26$	0	0
$27$	0	0
$28$	0	0
$29$	104.357i	0.668226i	0.942533	+	0.334113i	$0.108437\pi$
$29$	104.357i	0.668226i	−0.942533	+	0.334113i	$0.891563\pi$
$30$	0	0
$31$	277.990i	1.61060i	0.592869	+	0.805299i	$0.297995\pi$
$31$	277.990i	1.61060i	−0.592869	+	0.805299i	$0.702005\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−47.6573	−0.211752	−0.105876	−	0.994379i	$-0.533765\pi$
$37$	−47.6573	−0.211752	−0.105876	+	0.994379i	$0.533765\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	387.272	1.47516	0.737582	−	0.675258i	$-0.235968\pi$
$41$	387.272	1.47516	0.737582	+	0.675258i	$0.235968\pi$
$42$	0	0
$43$	272.528	0.966515	0.483257	−	0.875478i	$-0.339454\pi$
$43$	272.528	0.966515	0.483257	+	0.875478i	$0.339454\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−163.188	−0.506457	−0.253228	−	0.967407i	$-0.581492\pi$
$47$	−163.188	−0.506457	−0.253228	+	0.967407i	$0.581492\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 362.422i	− 0.939293i	−0.882855	−	0.469646i	$-0.844381\pi$
$53$	− 362.422i	− 0.939293i	0.882855	−	0.469646i	$-0.155619\pi$
$54$	0	0
$55$	− 398.193i	− 0.976224i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−211.705	−0.467147	−0.233573	−	0.972339i	$-0.575042\pi$
$59$	−211.705	−0.467147	−0.233573	+	0.972339i	$0.575042\pi$
$60$	0	0
$61$	− 234.310i	− 0.491809i	−0.969294	−	0.245905i	$-0.920915\pi$
$61$	− 234.310i	− 0.491809i	0.969294	−	0.245905i	$-0.0790850\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 263.109i	− 0.502072i
$66$	0	0
$67$	524.262	0.955952	0.477976	−	0.878373i	$-0.341371\pi$
$67$	524.262	0.955952	0.477976	+	0.878373i	$0.341371\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 348.689i	− 0.582842i	−0.956595	−	0.291421i	$-0.905872\pi$
$71$	− 348.689i	− 0.582842i	0.956595	−	0.291421i	$-0.0941280\pi$
$72$	0	0
$73$	− 537.100i	− 0.861135i	−0.902558	−	0.430567i	$-0.858313\pi$
$73$	− 537.100i	− 0.861135i	0.902558	−	0.430567i	$-0.141687\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	725.584	1.03335	0.516675	−	0.856182i	$-0.327170\pi$
$79$	725.584	1.03335	0.516675	+	0.856182i	$0.327170\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	392.121	0.518565	0.259283	−	0.965801i	$-0.416514\pi$
$83$	392.121	0.518565	0.259283	+	0.965801i	$0.416514\pi$
$84$	0	0
$85$	220.104	0.280866
$86$	0	0
$87$	0	0
$88$	0	0
$89$	860.029	1.02430	0.512151	−	0.858895i	$-0.328849\pi$
$89$	860.029	1.02430	0.512151	+	0.858895i	$0.328849\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	849.857i	0.917826i
$96$	0	0
$97$	978.030i	1.02375i	0.859059	+	0.511876i	$0.171049\pi$
$97$	978.030i	1.02375i	−0.859059	+	0.511876i	$0.828951\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	448.301	0.441659	0.220830	−	0.975312i	$-0.429123\pi$
$101$	448.301	0.441659	0.220830	+	0.975312i	$0.429123\pi$
$102$	0	0
$103$	− 1313.55i	− 1.25658i	−0.777980	−	0.628289i	$-0.783756\pi$
$103$	− 1313.55i	− 1.25658i	0.777980	−	0.628289i	$-0.216244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 186.491i	− 0.168493i	−0.996445	−	0.0842466i	$-0.973152\pi$
$107$	− 186.491i	− 0.168493i	0.996445	−	0.0842466i	$-0.0268483\pi$
$108$	0	0
$109$	123.162	0.108227	0.0541137	−	0.998535i	$-0.482767\pi$
$109$	123.162	0.108227	0.0541137	+	0.998535i	$0.482767\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2267.03i	1.88729i	0.330956	+	0.943646i	$0.392629\pi$
$113$	2267.03i	1.88729i	−0.330956	+	0.943646i	$0.607371\pi$
$114$	0	0
$115$	1372.88i	1.11323i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2073.42	−1.55779
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1388.28	−0.993376
$126$	0	0
$127$	839.285	0.586413	0.293207	−	0.956049i	$-0.405278\pi$
$127$	839.285	0.586413	0.293207	+	0.956049i	$0.405278\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2370.67	1.58112	0.790559	−	0.612386i	$-0.209790\pi$
$131$	2370.67	1.58112	0.790559	+	0.612386i	$0.209790\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3067.62i	1.91303i	0.291689	+	0.956513i	$0.405783\pi$
$137$	3067.62i	1.91303i	−0.291689	+	0.956513i	$0.594217\pi$
$138$	0	0
$139$	− 2191.44i	− 1.33723i	−0.743608	−	0.668616i	$-0.766887\pi$
$139$	− 2191.44i	− 1.33723i	0.743608	−	0.668616i	$-0.233113\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2249.50	−1.31547
$144$	0	0
$145$	712.184i	0.407888i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	460.572i	0.253232i	0.991952	+	0.126616i	$0.0404115\pi$
$149$	460.572i	0.253232i	−0.991952	+	0.126616i	$0.959588\pi$
$150$	0	0
$151$	635.105	0.342279	0.171139	−	0.985247i	$-0.445255\pi$
$151$	635.105	0.342279	0.171139	+	0.985247i	$0.445255\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1897.15i	0.983115i
$156$	0	0
$157$	− 553.934i	− 0.281584i	−0.990039	−	0.140792i	$-0.955035\pi$
$157$	− 553.934i	− 0.281584i	0.990039	−	0.140792i	$-0.0449649\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1375.24	0.660842	0.330421	−	0.943834i	$-0.392809\pi$
$163$	1375.24	0.660842	0.330421	+	0.943834i	$0.392809\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3999.79	1.85337	0.926685	−	0.375839i	$-0.122645\pi$
$167$	3999.79	1.85337	0.926685	+	0.375839i	$0.122645\pi$
$168$	0	0
$169$	710.626	0.323453
$170$	0	0
$171$	0	0
$172$	0	0
$173$	451.897	0.198596	0.0992979	−	0.995058i	$-0.468340\pi$
$173$	451.897	0.198596	0.0992979	+	0.995058i	$0.468340\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2262.77i	− 0.944847i	−0.881372	−	0.472424i	$-0.843379\pi$
$179$	− 2262.77i	− 0.944847i	0.881372	−	0.472424i	$-0.156621\pi$
$180$	0	0
$181$	− 3035.53i	− 1.24657i	−0.781994	−	0.623286i	$-0.785797\pi$
$181$	− 3035.53i	− 1.24657i	0.781994	−	0.623286i	$-0.214203\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−325.238	−0.129254
$186$	0	0
$187$	− 1881.81i	− 0.735892i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1936.06i	0.733448i	0.930330	+	0.366724i	$0.119521\pi$
$191$	1936.06i	0.733448i	−0.930330	+	0.366724i	$0.880479\pi$
$192$	0	0
$193$	−2123.99	−0.792168	−0.396084	−	0.918214i	$-0.629631\pi$
$193$	−2123.99	−0.792168	−0.396084	+	0.918214i	$0.629631\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2122.77i	0.767720i	0.923391	+	0.383860i	$0.125406\pi$
$197$	2122.77i	0.767720i	−0.923391	+	0.383860i	$0.874594\pi$
$198$	0	0
$199$	− 1195.32i	− 0.425798i	−0.977074	−	0.212899i	$-0.931709\pi$
$199$	− 1195.32i	− 0.425798i	0.977074	−	0.212899i	$-0.0682905\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2642.95	0.900446
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7265.99	2.40478
$210$	0	0
$211$	188.402	0.0614698	0.0307349	−	0.999528i	$-0.490215\pi$
$211$	188.402	0.0614698	0.0307349	+	0.999528i	$0.490215\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1859.87	0.589964
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1243.42i	− 0.378470i
$222$	0	0
$223$	− 2802.67i	− 0.841618i	−0.907149	−	0.420809i	$-0.861746\pi$
$223$	− 2802.67i	− 0.841618i	0.907149	−	0.420809i	$-0.138254\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1625.18	−0.475185	−0.237593	−	0.971365i	$-0.576358\pi$
$227$	−1625.18	−0.475185	−0.237593	+	0.971365i	$0.576358\pi$
$228$	0	0
$229$	− 6620.04i	− 1.91033i	−0.296082	−	0.955163i	$-0.595680\pi$
$229$	− 6620.04i	− 1.91033i	0.296082	−	0.955163i	$-0.404320\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1566.92i	− 0.440568i	−0.975436	−	0.220284i	$-0.929302\pi$
$233$	− 1566.92i	− 0.440568i	0.975436	−	0.220284i	$-0.0706985\pi$
$234$	0	0
$235$	−1113.68	−0.309143
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5211.23i	1.41040i	0.709006	+	0.705202i	$0.249144\pi$
$239$	5211.23i	1.41040i	−0.709006	+	0.705202i	$0.750856\pi$
$240$	0	0
$241$	3983.50i	1.06473i	0.846515	+	0.532364i	$0.178696\pi$
$241$	3983.50i	1.06473i	−0.846515	+	0.532364i	$0.821304\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4801.07	1.23678
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7086.26	1.78199	0.890997	−	0.454009i	$-0.150007\pi$
$251$	7086.26	1.78199	0.890997	+	0.454009i	$0.150007\pi$
$252$	0	0
$253$	11737.6	2.91676
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2443.64	−0.593113	−0.296556	−	0.955015i	$-0.595838\pi$
$257$	−2443.64	−0.593113	−0.296556	+	0.955015i	$0.595838\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2502.55i	0.586744i	0.955998	+	0.293372i	$0.0947774\pi$
$263$	2502.55i	0.586744i	−0.955998	+	0.293372i	$0.905223\pi$
$264$	0	0
$265$	− 2473.36i	− 0.573348i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8320.36	−1.88588	−0.942939	−	0.332964i	$-0.891951\pi$
$269$	−8320.36	−1.88588	−0.942939	+	0.332964i	$0.891951\pi$
$270$	0	0
$271$	6716.33i	1.50549i	0.658312	+	0.752745i	$0.271271\pi$
$271$	6716.33i	1.50549i	−0.658312	+	0.752745i	$0.728729\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4575.94i	1.00342i
$276$	0	0
$277$	6670.96	1.44700	0.723500	−	0.690325i	$-0.242532\pi$
$277$	6670.96	1.44700	0.723500	+	0.690325i	$0.242532\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5065.50i	1.07538i	0.843142	+	0.537692i	$0.180704\pi$
$281$	5065.50i	1.07538i	−0.843142	+	0.537692i	$0.819296\pi$
$282$	0	0
$283$	2572.57i	0.540365i	0.962809	+	0.270182i	$0.0870840\pi$
$283$	2572.57i	0.540365i	−0.962809	+	0.270182i	$0.912916\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3872.81	−0.788279
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5796.55	−1.15576	−0.577881	−	0.816121i	$-0.696120\pi$
$293$	−5796.55	−1.15576	−0.577881	+	0.816121i	$0.696120\pi$
$294$	0	0
$295$	−1444.79	−0.285148
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7755.75	1.50009
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1599.06i	− 0.300202i
$306$	0	0
$307$	− 753.054i	− 0.139997i	−0.997547	−	0.0699985i	$-0.977701\pi$
$307$	− 753.054i	− 0.139997i	0.997547	−	0.0699985i	$-0.0222994\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5388.91	0.982563	0.491281	−	0.871001i	$-0.336529\pi$
$311$	5388.91	0.982563	0.491281	+	0.871001i	$0.336529\pi$
$312$	0	0
$313$	− 7062.71i	− 1.27542i	−0.770275	−	0.637712i	$-0.779881\pi$
$313$	− 7062.71i	− 1.27542i	0.770275	−	0.637712i	$-0.220119\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2455.53i	0.435068i	0.976053	+	0.217534i	$0.0698013\pi$
$317$	2455.53i	0.435068i	−0.976053	+	0.217534i	$0.930199\pi$
$318$	0	0
$319$	6088.94	1.06870
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4016.33i	0.691872i
$324$	0	0
$325$	3023.59i	0.516058i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8510.25	−1.41319	−0.706594	−	0.707619i	$-0.749769\pi$
$331$	−8510.25	−1.41319	−0.706594	+	0.707619i	$0.749769\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3577.84	0.583517
$336$	0	0
$337$	1803.01	0.291443	0.145722	−	0.989326i	$-0.453450\pi$
$337$	1803.01	0.291443	0.145722	+	0.989326i	$0.453450\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16220.0	2.57584
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8356.67i	1.29282i	0.762989	+	0.646412i	$0.223731\pi$
$347$	8356.67i	1.29282i	−0.762989	+	0.646412i	$0.776269\pi$
$348$	0	0
$349$	− 4977.74i	− 0.763474i	−0.924271	−	0.381737i	$-0.875326\pi$
$349$	− 4977.74i	− 0.763474i	0.924271	−	0.381737i	$-0.124674\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7640.14	1.15197	0.575983	−	0.817462i	$-0.304620\pi$
$353$	7640.14	1.15197	0.575983	+	0.817462i	$0.304620\pi$
$354$	0	0
$355$	− 2379.64i	− 0.355769i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 4158.81i	− 0.611403i	−0.952127	−	0.305701i	$-0.901109\pi$
$359$	− 4158.81i	− 0.611403i	0.952127	−	0.305701i	$-0.0988909\pi$
$360$	0	0
$361$	−8648.70	−1.26093
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3665.45i	− 0.525640i
$366$	0	0
$367$	2177.27i	0.309680i	0.987940	+	0.154840i	$0.0494862\pi$
$367$	2177.27i	0.309680i	−0.987940	+	0.154840i	$0.950514\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−380.271	−0.0527874	−0.0263937	−	0.999652i	$-0.508402\pi$
$373$	−380.271	−0.0527874	−0.0263937	+	0.999652i	$0.508402\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4023.32	0.549632
$378$	0	0
$379$	6918.48	0.937673	0.468837	−	0.883285i	$-0.344673\pi$
$379$	6918.48	0.937673	0.468837	+	0.883285i	$0.344673\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	126.823	0.0169200	0.00846002	−	0.999964i	$-0.497307\pi$
$383$	126.823	0.0169200	0.00846002	+	0.999964i	$0.497307\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1475.93i	0.192371i	0.995363	+	0.0961856i	$0.0306642\pi$
$389$	1475.93i	0.192371i	−0.995363	+	0.0961856i	$0.969336\pi$
$390$	0	0
$391$	6488.06i	0.839170i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4951.76	0.630760
$396$	0	0
$397$	867.816i	0.109709i	0.998494	+	0.0548545i	$0.0174695\pi$
$397$	867.816i	0.109709i	−0.998494	+	0.0548545i	$0.982531\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	697.245i	0.0868298i	0.999057	+	0.0434149i	$0.0138237\pi$
$401$	697.245i	0.0868298i	−0.999057	+	0.0434149i	$0.986176\pi$
$402$	0	0
$403$	10717.5	1.32476
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2780.68i	0.338656i
$408$	0	0
$409$	6189.18i	0.748252i	0.927378	+	0.374126i	$0.122057\pi$
$409$	6189.18i	0.748252i	−0.927378	+	0.374126i	$0.877943\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2676.04	0.316534
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1443.53	−0.168308	−0.0841538	−	0.996453i	$-0.526819\pi$
$419$	−1443.53	−0.168308	−0.0841538	+	0.996453i	$0.526819\pi$
$420$	0	0
$421$	−15750.0	−1.82330	−0.911648	−	0.410971i	$-0.865190\pi$
$421$	−15750.0	−1.82330	−0.911648	+	0.410971i	$0.865190\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2529.39	−0.288690
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11307.1i	1.26367i	0.775101	+	0.631837i	$0.217699\pi$
$431$	11307.1i	1.26367i	−0.775101	+	0.631837i	$0.782301\pi$
$432$	0	0
$433$	− 2318.26i	− 0.257295i	−0.991690	−	0.128647i	$-0.958936\pi$
$433$	− 2318.26i	− 0.257295i	0.991690	−	0.128647i	$-0.0410635\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−25051.5	−2.74228
$438$	0	0
$439$	− 7712.06i	− 0.838443i	−0.907884	−	0.419222i	$-0.862303\pi$
$439$	− 7712.06i	− 0.838443i	0.907884	−	0.419222i	$-0.137697\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12884.3i	− 1.38183i	−0.722935	−	0.690916i	$-0.757208\pi$
$443$	− 12884.3i	− 1.38183i	0.722935	−	0.690916i	$-0.242792\pi$
$444$	0	0
$445$	5869.29	0.625238
$446$	0	0
$447$	0	0
$448$	0	0
$449$	319.989i	0.0336330i	0.999859	+	0.0168165i	$0.00535311\pi$
$449$	319.989i	0.0336330i	−0.999859	+	0.0168165i	$0.994647\pi$
$450$	0	0
$451$	− 22596.3i	− 2.35924i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7794.97	0.797885	0.398942	−	0.916976i	$-0.369377\pi$
$457$	7794.97	0.797885	0.398942	+	0.916976i	$0.369377\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4610.97	0.465844	0.232922	−	0.972495i	$-0.425171\pi$
$461$	4610.97	0.465844	0.232922	+	0.972495i	$0.425171\pi$
$462$	0	0
$463$	15203.3	1.52604	0.763022	−	0.646373i	$-0.223715\pi$
$463$	15203.3	1.52604	0.763022	+	0.646373i	$0.223715\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−16120.1	−1.59733	−0.798663	−	0.601779i	$-0.794459\pi$
$467$	−16120.1	−1.59733	−0.798663	+	0.601779i	$0.794459\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 15901.3i	− 1.54576i
$474$	0	0
$475$	− 9766.37i	− 0.943393i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6723.98	−0.641391	−0.320696	−	0.947182i	$-0.603917\pi$
$479$	−6723.98	−0.641391	−0.320696	+	0.947182i	$0.603917\pi$
$480$	0	0
$481$	1837.36i	0.174171i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6674.59i	0.624902i
$486$	0	0
$487$	−10059.0	−0.935972	−0.467986	−	0.883736i	$-0.655020\pi$
$487$	−10059.0	−0.935972	−0.467986	+	0.883736i	$0.655020\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18097.6i	1.66340i	0.555222	+	0.831702i	$0.312633\pi$
$491$	18097.6i	1.66340i	−0.555222	+	0.831702i	$0.687367\pi$
$492$	0	0
$493$	3365.70i	0.307472i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11188.5	1.00374	0.501870	−	0.864943i	$-0.332646\pi$
$499$	11188.5	1.00374	0.501870	+	0.864943i	$0.332646\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2912.13	0.258142	0.129071	−	0.991635i	$-0.458800\pi$
$503$	2912.13	0.258142	0.129071	+	0.991635i	$0.458800\pi$
$504$	0	0
$505$	3059.44	0.269591
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4800.96	0.418072	0.209036	−	0.977908i	$-0.432967\pi$
$509$	4800.96	0.418072	0.209036	+	0.977908i	$0.432967\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 8964.32i	− 0.767020i
$516$	0	0
$517$	9521.61i	0.809980i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6513.26	0.547699	0.273849	−	0.961773i	$-0.411703\pi$
$521$	6513.26	0.547699	0.273849	+	0.961773i	$0.411703\pi$
$522$	0	0
$523$	− 4453.23i	− 0.372326i	−0.982519	−	0.186163i	$-0.940395\pi$
$523$	− 4453.23i	− 0.372326i	0.982519	−	0.186163i	$-0.0596052\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8965.72i	0.741087i
$528$	0	0
$529$	−28301.7	−2.32610
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 14930.7i	− 1.21336i
$534$	0	0
$535$	− 1272.71i	− 0.102849i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7770.64	0.617534	0.308767	−	0.951138i	$-0.400084\pi$
$541$	7770.64	0.617534	0.308767	+	0.951138i	$0.400084\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	840.523	0.0660624
$546$	0	0
$547$	−4095.62	−0.320139	−0.160069	−	0.987106i	$-0.551172\pi$
$547$	−4095.62	−0.320139	−0.160069	+	0.987106i	$0.551172\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12995.5	−1.00477
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 253.189i	− 0.0192602i	−0.999954	−	0.00963012i	$-0.996935\pi$
$557$	− 253.189i	− 0.0192602i	0.999954	−	0.00963012i	$-0.00306541\pi$
$558$	0	0
$559$	− 10506.9i	− 0.794982i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20504.7	1.53494	0.767469	−	0.641086i	$-0.221516\pi$
$563$	20504.7	1.53494	0.767469	+	0.641086i	$0.221516\pi$
$564$	0	0
$565$	15471.4i	1.15201i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13564.1i	0.999358i	0.866211	+	0.499679i	$0.166549\pi$
$569$	13564.1i	0.999358i	−0.866211	+	0.499679i	$0.833451\pi$
$570$	0	0
$571$	−4701.92	−0.344605	−0.172302	−	0.985044i	$-0.555121\pi$
$571$	−4701.92	−0.344605	−0.172302	+	0.985044i	$0.555121\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 15776.8i	− 1.14424i
$576$	0	0
$577$	3947.34i	0.284800i	0.989809	+	0.142400i	$0.0454820\pi$
$577$	3947.34i	0.284800i	−0.989809	+	0.142400i	$0.954518\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−21146.4	−1.50222
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12531.1	−0.881116	−0.440558	−	0.897724i	$-0.645219\pi$
$587$	−12531.1	−0.881116	−0.440558	+	0.897724i	$0.645219\pi$
$588$	0	0
$589$	−34618.1	−2.42176
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12950.8	−0.896840	−0.448420	−	0.893823i	$-0.648013\pi$
$593$	−12950.8	−0.896840	−0.448420	+	0.893823i	$0.648013\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	140.626i	0.00959235i	0.999988	+	0.00479618i	$0.00152668\pi$
$599$	140.626i	0.00959235i	−0.999988	+	0.00479618i	$0.998473\pi$
$600$	0	0
$601$	19220.3i	1.30451i	0.758000	+	0.652255i	$0.226177\pi$
$601$	19220.3i	1.30451i	−0.758000	+	0.652255i	$0.773823\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−14150.1	−0.950879
$606$	0	0
$607$	17032.6i	1.13893i	0.822015	+	0.569466i	$0.192850\pi$
$607$	17032.6i	1.13893i	−0.822015	+	0.569466i	$0.807150\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6291.48i	0.416573i
$612$	0	0
$613$	6493.98	0.427878	0.213939	−	0.976847i	$-0.431371\pi$
$613$	6493.98	0.427878	0.213939	+	0.976847i	$0.431371\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12338.5i	0.805074i	0.915404	+	0.402537i	$0.131871\pi$
$617$	12338.5i	0.805074i	−0.915404	+	0.402537i	$0.868129\pi$
$618$	0	0
$619$	10354.8i	0.672366i	0.941797	+	0.336183i	$0.109136\pi$
$619$	10354.8i	0.672366i	−0.941797	+	0.336183i	$0.890864\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	328.860	0.0210471
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1537.04	−0.0974336
$630$	0	0
$631$	−4917.41	−0.310236	−0.155118	−	0.987896i	$-0.549576\pi$
$631$	−4917.41	−0.310236	−0.155118	+	0.987896i	$0.549576\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5727.72	0.357949
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 17099.5i	− 1.05365i	−0.849975	−	0.526824i	$-0.823383\pi$
$641$	− 17099.5i	− 1.05365i	0.849975	−	0.526824i	$-0.176617\pi$
$642$	0	0
$643$	− 14631.4i	− 0.897365i	−0.893691	−	0.448683i	$-0.851893\pi$
$643$	− 14631.4i	− 0.897365i	0.893691	−	0.448683i	$-0.148107\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29617.5	1.79967	0.899833	−	0.436234i	$-0.143688\pi$
$647$	29617.5	1.79967	0.899833	+	0.436234i	$0.143688\pi$
$648$	0	0
$649$	12352.4i	0.747112i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 28171.3i	− 1.68825i	−0.536144	−	0.844127i	$-0.680120\pi$
$653$	− 28171.3i	− 1.68825i	0.536144	−	0.844127i	$-0.319880\pi$
$654$	0	0
$655$	16178.7	0.965120
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 25287.9i	− 1.49480i	−0.664372	−	0.747402i	$-0.731301\pi$
$659$	− 25287.9i	− 1.49480i	0.664372	−	0.747402i	$-0.268699\pi$
$660$	0	0
$661$	− 9862.71i	− 0.580355i	−0.956973	−	0.290178i	$-0.906286\pi$
$661$	− 9862.71i	− 0.580355i	0.956973	−	0.290178i	$-0.0937144\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20993.3	−1.21868
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13671.4	−0.786555
$672$	0	0
$673$	21145.6	1.21115	0.605574	−	0.795789i	$-0.292943\pi$
$673$	21145.6	1.21115	0.605574	+	0.795789i	$0.292943\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24124.8	1.36956	0.684779	−	0.728750i	$-0.259899\pi$
$677$	24124.8	1.36956	0.684779	+	0.728750i	$0.259899\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	30922.7i	1.73239i	0.499703	+	0.866197i	$0.333442\pi$
$683$	30922.7i	1.73239i	−0.499703	+	0.866197i	$0.666558\pi$
$684$	0	0
$685$	20935.1i	1.16772i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13972.7	−0.772592
$690$	0	0
$691$	12639.8i	0.695864i	0.937520	+	0.347932i	$0.113116\pi$
$691$	12639.8i	0.695864i	−0.937520	+	0.347932i	$0.886884\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 14955.5i	− 0.816252i
$696$	0	0
$697$	12490.3	0.678770
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24051.5i	− 1.29588i	−0.761690	−	0.647942i	$-0.775630\pi$
$701$	− 24051.5i	− 1.29588i	0.761690	−	0.647942i	$-0.224370\pi$
$702$	0	0
$703$	− 5934.76i	− 0.318398i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10876.3	−0.576118	−0.288059	−	0.957613i	$-0.593010\pi$
$709$	−10876.3	−0.576118	−0.288059	+	0.957613i	$0.593010\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−55922.8	−2.93735
$714$	0	0
$715$	−15351.7	−0.802968
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20913.5	−1.08476	−0.542381	−	0.840133i	$-0.682477\pi$
$719$	−20913.5	−1.08476	−0.542381	+	0.840133i	$0.682477\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8184.27i	− 0.419250i
$726$	0	0
$727$	22897.9i	1.16814i	0.811704	+	0.584068i	$0.198540\pi$
$727$	22897.9i	1.16814i	−0.811704	+	0.584068i	$0.801460\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8789.55	0.444724
$732$	0	0
$733$	− 31061.3i	− 1.56518i	−0.622537	−	0.782590i	$-0.713898\pi$
$733$	− 31061.3i	− 1.56518i	0.622537	−	0.782590i	$-0.286102\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 30589.3i	− 1.52886i
$738$	0	0
$739$	20341.7	1.01256	0.506281	−	0.862369i	$-0.331020\pi$
$739$	20341.7	1.01256	0.506281	+	0.862369i	$0.331020\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 31458.9i	− 1.55332i	−0.629921	−	0.776660i	$-0.716913\pi$
$743$	− 31458.9i	− 1.55332i	0.629921	−	0.776660i	$-0.283087\pi$
$744$	0	0
$745$	3143.18i	0.154574i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19307.6	0.938143	0.469071	−	0.883160i	$-0.344589\pi$
$751$	19307.6	0.938143	0.469071	+	0.883160i	$0.344589\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4334.29	0.208928
$756$	0	0
$757$	−18791.2	−0.902216	−0.451108	−	0.892469i	$-0.648971\pi$
$757$	−18791.2	−0.902216	−0.451108	+	0.892469i	$0.648971\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26499.4	−1.26229	−0.631145	−	0.775665i	$-0.717415\pi$
$761$	−26499.4	−1.26229	−0.631145	+	0.775665i	$0.717415\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8161.98i	0.384240i
$768$	0	0
$769$	29077.1i	1.36352i	0.731575	+	0.681761i	$0.238785\pi$
$769$	29077.1i	1.36352i	−0.731575	+	0.681761i	$0.761215\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29210.6	1.35916	0.679582	−	0.733600i	$-0.262161\pi$
$773$	29210.6	1.35916	0.679582	+	0.733600i	$0.262161\pi$
$774$	0	0
$775$	− 21801.6i	− 1.01050i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48226.9i	2.21811i
$780$	0	0
$781$	−20345.1	−0.932144
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3780.33i	− 0.171880i
$786$	0	0
$787$	− 34510.8i	− 1.56312i	−0.623829	−	0.781561i	$-0.714424\pi$
$787$	− 34510.8i	− 1.56312i	0.623829	−	0.781561i	$-0.285576\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9033.49	−0.404525
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−40125.5	−1.78333	−0.891667	−	0.452692i	$-0.850464\pi$
$797$	−40125.5	−1.78333	−0.891667	+	0.452692i	$0.850464\pi$
$798$	0	0
$799$	−5263.13	−0.233037
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−31338.4	−1.37722
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 25510.4i	− 1.10865i	−0.832300	−	0.554326i	$-0.812976\pi$
$809$	− 25510.4i	− 1.10865i	0.832300	−	0.554326i	$-0.187024\pi$
$810$	0	0
$811$	34785.8i	1.50616i	0.657930	+	0.753079i	$0.271432\pi$
$811$	34785.8i	1.50616i	−0.657930	+	0.753079i	$0.728568\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9385.37	0.403380
$816$	0	0
$817$	33937.9i	1.45329i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 22602.9i	− 0.960836i	−0.877040	−	0.480418i	$-0.840485\pi$
$821$	− 22602.9i	− 0.960836i	0.877040	−	0.480418i	$-0.159515\pi$
$822$	0	0
$823$	−19108.6	−0.809338	−0.404669	−	0.914463i	$-0.632613\pi$
$823$	−19108.6	−0.809338	−0.404669	+	0.914463i	$0.632613\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7545.77i	− 0.317282i	−0.987336	−	0.158641i	$-0.949289\pi$
$827$	− 7545.77i	− 0.317282i	0.987336	−	0.158641i	$-0.0507112\pi$
$828$	0	0
$829$	32019.5i	1.34148i	0.741694	+	0.670738i	$0.234023\pi$
$829$	32019.5i	1.34148i	−0.741694	+	0.670738i	$0.765977\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	27296.6	1.13130
$836$	0	0
$837$	0	0
$838$	0	0
$839$	33582.9	1.38190	0.690948	−	0.722905i	$-0.257194\pi$
$839$	33582.9	1.38190	0.690948	+	0.722905i	$0.257194\pi$
$840$	0	0
$841$	13498.7	0.553474
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4849.68	0.197437
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 9587.13i	− 0.386184i
$852$	0	0
$853$	119.927i	0.00481387i	0.999997	+	0.00240693i	$0.000766151\pi$
$853$	119.927i	0.00481387i	−0.999997	+	0.00240693i	$0.999234\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5212.09	0.207750	0.103875	−	0.994590i	$-0.466876\pi$
$857$	5212.09	0.207750	0.103875	+	0.994590i	$0.466876\pi$
$858$	0	0
$859$	19056.6i	0.756929i	0.925616	+	0.378464i	$0.123548\pi$
$859$	19056.6i	0.756929i	−0.925616	+	0.378464i	$0.876452\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3985.34i	− 0.157199i	−0.996906	−	0.0785994i	$-0.974955\pi$
$863$	− 3985.34i	− 0.157199i	0.996906	−	0.0785994i	$-0.0250448\pi$
$864$	0	0
$865$	3083.98	0.121224
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 42335.9i	− 1.65264i
$870$	0	0
$871$	− 20212.1i	− 0.786294i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31002.3	−1.19370	−0.596848	−	0.802354i	$-0.703581\pi$
$877$	−31002.3	−1.19370	−0.596848	+	0.802354i	$0.703581\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29240.6	−1.11821	−0.559103	−	0.829098i	$-0.688854\pi$
$881$	−29240.6	−1.11821	−0.559103	+	0.829098i	$0.688854\pi$
$882$	0	0
$883$	−42056.9	−1.60286	−0.801431	−	0.598088i	$-0.795927\pi$
$883$	−42056.9	−1.60286	−0.801431	+	0.598088i	$0.795927\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2432.31	−0.0920734	−0.0460367	−	0.998940i	$-0.514659\pi$
$887$	−2432.31	−0.0920734	−0.0460367	+	0.998940i	$0.514659\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 20321.8i	− 0.761527i
$894$	0	0
$895$	− 15442.4i	− 0.576738i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−29010.1	−1.07624
$900$	0	0
$901$	− 11688.8i	− 0.432198i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 20716.1i	− 0.760912i
$906$	0	0
$907$	−6753.40	−0.247236	−0.123618	−	0.992330i	$-0.539450\pi$
$907$	−6753.40	−0.247236	−0.123618	+	0.992330i	$0.539450\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13545.2i	0.492615i	0.969192	+	0.246308i	$0.0792174\pi$
$911$	13545.2i	0.492615i	−0.969192	+	0.246308i	$0.920783\pi$
$912$	0	0
$913$	− 22879.2i	− 0.829346i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	39397.6	1.41415	0.707077	−	0.707137i	$-0.250014\pi$
$919$	39397.6	1.41415	0.707077	+	0.707137i	$0.250014\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−13443.2	−0.479402
$924$	0	0
$925$	3737.56	0.132854
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11148.2	−0.393713	−0.196857	−	0.980432i	$-0.563073\pi$
$929$	−11148.2	−0.393713	−0.196857	+	0.980432i	$0.563073\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 12842.5i	− 0.449192i
$936$	0	0
$937$	− 22882.5i	− 0.797800i	−0.916994	−	0.398900i	$-0.869392\pi$
$937$	− 22882.5i	− 0.797800i	0.916994	−	0.398900i	$-0.130608\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5650.55	0.195752	0.0978761	−	0.995199i	$-0.468795\pi$
$941$	5650.55	0.195752	0.0978761	+	0.995199i	$0.468795\pi$
$942$	0	0
$943$	77906.9i	2.69035i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30477.8i	1.04582i	0.852387	+	0.522911i	$0.175154\pi$
$947$	30477.8i	1.04582i	−0.852387	+	0.522911i	$0.824846\pi$
$948$	0	0
$949$	−20707.1	−0.708305
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24139.2i	0.820511i	0.911971	+	0.410255i	$0.134560\pi$
$953$	24139.2i	0.820511i	−0.911971	+	0.410255i	$0.865440\pi$
$954$	0	0
$955$	13212.7i	0.447699i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−47487.6	−1.59403
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14495.2	−0.483542
$966$	0	0
$967$	11475.1	0.381609	0.190804	−	0.981628i	$-0.438890\pi$
$967$	11475.1	0.381609	0.190804	+	0.981628i	$0.438890\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5038.83	−0.166533	−0.0832667	−	0.996527i	$-0.526535\pi$
$971$	−5038.83	−0.166533	−0.0832667	+	0.996527i	$0.526535\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13377.5i	− 0.438059i	−0.975718	−	0.219029i	$-0.929711\pi$
$977$	− 13377.5i	− 0.438059i	0.975718	−	0.219029i	$-0.0702891\pi$
$978$	0	0
$979$	− 50180.4i	− 1.63818i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11570.2	0.375413	0.187707	−	0.982225i	$-0.439895\pi$
$983$	11570.2	0.375413	0.187707	+	0.982225i	$0.439895\pi$
$984$	0	0
$985$	14486.9i	0.468619i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	54824.0i	1.76269i
$990$	0	0
$991$	20059.1	0.642986	0.321493	−	0.946912i	$-0.395815\pi$
$991$	20059.1	0.642986	0.321493	+	0.946912i	$0.395815\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8157.47i	− 0.259909i
$996$	0	0
$997$	34082.1i	1.08264i	0.840817	+	0.541319i	$0.182075\pi$
$997$	34082.1i	1.08264i	−0.840817	+	0.541319i	$0.817925\pi$
$998$	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.881.9		16
3.2	odd	2	inner	1764.4.f.a.881.8		16
7.2	even	3		252.4.t.a.17.4	✓	16
7.3	odd	6		252.4.t.a.89.5	yes	16
7.4	even	3		1764.4.t.b.1097.4		16
7.5	odd	6		1764.4.t.b.521.5		16
7.6	odd	2	inner	1764.4.f.a.881.7		16
21.2	odd	6		252.4.t.a.17.5	yes	16
21.5	even	6		1764.4.t.b.521.4		16
21.11	odd	6		1764.4.t.b.1097.5		16
21.17	even	6		252.4.t.a.89.4	yes	16
21.20	even	2	inner	1764.4.f.a.881.10		16
28.3	even	6		1008.4.bt.b.593.5		16
28.23	odd	6		1008.4.bt.b.17.4		16
84.23	even	6		1008.4.bt.b.17.5		16
84.59	odd	6		1008.4.bt.b.593.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.4	✓	16	7.2	even	3
252.4.t.a.17.5	yes	16	21.2	odd	6
252.4.t.a.89.4	yes	16	21.17	even	6
252.4.t.a.89.5	yes	16	7.3	odd	6
1008.4.bt.b.17.4		16	28.23	odd	6
1008.4.bt.b.17.5		16	84.23	even	6
1008.4.bt.b.593.4		16	84.59	odd	6
1008.4.bt.b.593.5		16	28.3	even	6
1764.4.f.a.881.7		16	7.6	odd	2	inner
1764.4.f.a.881.8		16	3.2	odd	2	inner
1764.4.f.a.881.9		16	1.1	even	1	trivial
1764.4.f.a.881.10		16	21.20	even	2	inner
1764.4.t.b.521.4		16	21.5	even	6
1764.4.t.b.521.5		16	7.5	odd	6
1764.4.t.b.1097.4		16	7.4	even	3
1764.4.t.b.1097.5		16	21.11	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)

\(f(q)\)	\(=\)	\(q+6.82452 q^{5} +O(q^{10})\)	\(q+6.82452 q^{5} -58.3474i q^{11} -38.5535i q^{13} +32.2519 q^{17} +124.530i q^{19} +201.168i q^{23} -78.4259 q^{25} +104.357i q^{29} +277.990i q^{31} -47.6573 q^{37} +387.272 q^{41} +272.528 q^{43} -163.188 q^{47} -362.422i q^{53} -398.193i q^{55} -211.705 q^{59} -234.310i q^{61} -263.109i q^{65} +524.262 q^{67} -348.689i q^{71} -537.100i q^{73} +725.584 q^{79} +392.121 q^{83} +220.104 q^{85} +860.029 q^{89} +849.857i q^{95} +978.030i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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