LMFDB - Embedded newform 1200.4.f.c.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$70.8022920069$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.c.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +1.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +1.00000i q^{7} -9.00000 q^{9} -42.0000 q^{11} +67.0000i q^{13} +54.0000i q^{17} -115.000 q^{19} +3.00000 q^{21} -162.000i q^{23} +27.0000i q^{27} +210.000 q^{29} +193.000 q^{31} +126.000i q^{33} -286.000i q^{37} +201.000 q^{39} +12.0000 q^{41} +263.000i q^{43} -414.000i q^{47} +342.000 q^{49} +162.000 q^{51} +192.000i q^{53} +345.000i q^{57} +690.000 q^{59} -733.000 q^{61} -9.00000i q^{63} -299.000i q^{67} -486.000 q^{69} +228.000 q^{71} -938.000i q^{73} -42.0000i q^{77} -160.000 q^{79} +81.0000 q^{81} -462.000i q^{83} -630.000i q^{87} +240.000 q^{89} -67.0000 q^{91} -579.000i q^{93} -511.000i q^{97} +378.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 84 q^{11} - 230 q^{19} + 6 q^{21} + 420 q^{29} + 386 q^{31} + 402 q^{39} + 24 q^{41} + 684 q^{49} + 324 q^{51} + 1380 q^{59} - 1466 q^{61} - 972 q^{69} + 456 q^{71} - 320 q^{79} + 162 q^{81} + 480 q^{89} - 134 q^{91} + 756 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 84 * q^11 - 230 * q^19 + 6 * q^21 + 420 * q^29 + 386 * q^31 + 402 * q^39 + 24 * q^41 + 684 * q^49 + 324 * q^51 + 1380 * q^59 - 1466 * q^61 - 972 * q^69 + 456 * q^71 - 320 * q^79 + 162 * q^81 + 480 * q^89 - 134 * q^91 + 756 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.0539949i	0.999636	+	0.0269975i	$0.00859460\pi$
$7$	1.00000i	0.0539949i	−0.999636	+	0.0269975i	$0.991405\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−42.0000	−1.15123	−0.575613	−	0.817723i	$-0.695236\pi$
$11$	−42.0000	−1.15123	−0.575613	+	0.817723i	$0.695236\pi$
$12$	0	0
$13$	67.0000i	1.42942i	0.699421	+	0.714710i	$0.253441\pi$
$13$	67.0000i	1.42942i	−0.699421	+	0.714710i	$0.746559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	54.0000i	0.770407i	0.922832	+	0.385204i	$0.125869\pi$
$17$	54.0000i	0.770407i	−0.922832	+	0.385204i	$0.874131\pi$
$18$	0	0
$19$	−115.000	−1.38857	−0.694284	−	0.719701i	$-0.744279\pi$
$19$	−115.000	−1.38857	−0.694284	+	0.719701i	$0.744279\pi$
$20$	0	0
$21$	3.00000	0.0311740
$22$	0	0
$23$	− 162.000i	− 1.46867i	−0.678789	−	0.734333i	$-0.737495\pi$
$23$	− 162.000i	− 1.46867i	0.678789	−	0.734333i	$-0.262505\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	0	0
$31$	193.000	1.11819	0.559094	−	0.829104i	$-0.311149\pi$
$31$	193.000	1.11819	0.559094	+	0.829104i	$0.311149\pi$
$32$	0	0
$33$	126.000i	0.664660i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 286.000i	− 1.27076i	−0.772200	−	0.635380i	$-0.780844\pi$
$37$	− 286.000i	− 1.27076i	0.772200	−	0.635380i	$-0.219156\pi$
$38$	0	0
$39$	201.000	0.825276
$40$	0	0
$41$	12.0000	0.0457094	0.0228547	−	0.999739i	$-0.492724\pi$
$41$	12.0000	0.0457094	0.0228547	+	0.999739i	$0.492724\pi$
$42$	0	0
$43$	263.000i	0.932724i	0.884594	+	0.466362i	$0.154436\pi$
$43$	263.000i	0.932724i	−0.884594	+	0.466362i	$0.845564\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 414.000i	− 1.28485i	−0.766347	−	0.642427i	$-0.777928\pi$
$47$	− 414.000i	− 1.28485i	0.766347	−	0.642427i	$-0.222072\pi$
$48$	0	0
$49$	342.000	0.997085
$50$	0	0
$51$	162.000	0.444795
$52$	0	0
$53$	192.000i	0.497608i	0.968554	+	0.248804i	$0.0800375\pi$
$53$	192.000i	0.497608i	−0.968554	+	0.248804i	$0.919962\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	345.000i	0.801691i
$58$	0	0
$59$	690.000	1.52255	0.761274	−	0.648430i	$-0.224574\pi$
$59$	690.000	1.52255	0.761274	+	0.648430i	$0.224574\pi$
$60$	0	0
$61$	−733.000	−1.53854	−0.769271	−	0.638923i	$-0.779380\pi$
$61$	−733.000	−1.53854	−0.769271	+	0.638923i	$0.779380\pi$
$62$	0	0
$63$	− 9.00000i	− 0.0179983i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 299.000i	− 0.545204i	−0.962127	−	0.272602i	$-0.912116\pi$
$67$	− 299.000i	− 0.545204i	0.962127	−	0.272602i	$-0.0878842\pi$
$68$	0	0
$69$	−486.000	−0.847935
$70$	0	0
$71$	228.000	0.381107	0.190554	−	0.981677i	$-0.438972\pi$
$71$	228.000	0.381107	0.190554	+	0.981677i	$0.438972\pi$
$72$	0	0
$73$	− 938.000i	− 1.50390i	−0.659221	−	0.751949i	$-0.729114\pi$
$73$	− 938.000i	− 1.50390i	0.659221	−	0.751949i	$-0.270886\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 42.0000i	− 0.0621603i
$78$	0	0
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 462.000i	− 0.610977i	−0.952196	−	0.305488i	$-0.901180\pi$
$83$	− 462.000i	− 0.610977i	0.952196	−	0.305488i	$-0.0988197\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 630.000i	− 0.776357i
$88$	0	0
$89$	240.000	0.285842	0.142921	−	0.989734i	$-0.454350\pi$
$89$	240.000	0.285842	0.142921	+	0.989734i	$0.454350\pi$
$90$	0	0
$91$	−67.0000	−0.0771814
$92$	0	0
$93$	− 579.000i	− 0.645586i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 511.000i	− 0.534889i	−0.963573	−	0.267444i	$-0.913821\pi$
$97$	− 511.000i	− 0.534889i	0.963573	−	0.267444i	$-0.0861791\pi$
$98$	0	0
$99$	378.000	0.383742
$100$	0	0
$101$	912.000	0.898489	0.449245	−	0.893409i	$-0.351693\pi$
$101$	912.000	0.898489	0.449245	+	0.893409i	$0.351693\pi$
$102$	0	0
$103$	668.000i	0.639029i	0.947581	+	0.319515i	$0.103520\pi$
$103$	668.000i	0.639029i	−0.947581	+	0.319515i	$0.896480\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1296.00i	1.17093i	0.810699	+	0.585463i	$0.199087\pi$
$107$	1296.00i	1.17093i	−0.810699	+	0.585463i	$0.800913\pi$
$108$	0	0
$109$	1735.00	1.52461	0.762307	−	0.647216i	$-0.224067\pi$
$109$	1735.00	1.52461	0.762307	+	0.647216i	$0.224067\pi$
$110$	0	0
$111$	−858.000	−0.733673
$112$	0	0
$113$	1092.00i	0.909086i	0.890725	+	0.454543i	$0.150197\pi$
$113$	1092.00i	0.909086i	−0.890725	+	0.454543i	$0.849803\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 603.000i	− 0.476473i
$118$	0	0
$119$	−54.0000	−0.0415981
$120$	0	0
$121$	433.000	0.325319
$122$	0	0
$123$	− 36.0000i	− 0.0263903i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	0.0111793i	0.999984	+	0.00558965i	$0.00177925\pi$
$127$	16.0000i	0.0111793i	−0.999984	+	0.00558965i	$0.998221\pi$
$128$	0	0
$129$	789.000	0.538508
$130$	0	0
$131$	−1992.00	−1.32856	−0.664282	−	0.747482i	$-0.731263\pi$
$131$	−1992.00	−1.32856	−0.664282	+	0.747482i	$0.731263\pi$
$132$	0	0
$133$	− 115.000i	− 0.0749757i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2346.00i	− 1.46301i	−0.681836	−	0.731505i	$-0.738818\pi$
$137$	− 2346.00i	− 1.46301i	0.681836	−	0.731505i	$-0.261182\pi$
$138$	0	0
$139$	2900.00	1.76960	0.884801	−	0.465968i	$-0.154294\pi$
$139$	2900.00	1.76960	0.884801	+	0.465968i	$0.154294\pi$
$140$	0	0
$141$	−1242.00	−0.741810
$142$	0	0
$143$	− 2814.00i	− 1.64558i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1026.00i	− 0.575667i
$148$	0	0
$149$	2070.00	1.13813	0.569064	−	0.822293i	$-0.307306\pi$
$149$	2070.00	1.13813	0.569064	+	0.822293i	$0.307306\pi$
$150$	0	0
$151$	−2237.00	−1.20559	−0.602796	−	0.797895i	$-0.705947\pi$
$151$	−2237.00	−1.20559	−0.602796	+	0.797895i	$0.705947\pi$
$152$	0	0
$153$	− 486.000i	− 0.256802i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 241.000i	− 0.122509i	−0.998122	−	0.0612544i	$-0.980490\pi$
$157$	− 241.000i	− 0.122509i	0.998122	−	0.0612544i	$-0.0195101\pi$
$158$	0	0
$159$	576.000	0.287294
$160$	0	0
$161$	162.000	0.0793006
$162$	0	0
$163$	− 3547.00i	− 1.70443i	−0.523190	−	0.852216i	$-0.675258\pi$
$163$	− 3547.00i	− 1.70443i	0.523190	−	0.852216i	$-0.324742\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 984.000i	− 0.455953i	−0.973667	−	0.227977i	$-0.926789\pi$
$167$	− 984.000i	− 0.455953i	0.973667	−	0.227977i	$-0.0732110\pi$
$168$	0	0
$169$	−2292.00	−1.04324
$170$	0	0
$171$	1035.00	0.462856
$172$	0	0
$173$	− 3618.00i	− 1.59001i	−0.606604	−	0.795004i	$-0.707469\pi$
$173$	− 3618.00i	− 1.59001i	0.606604	−	0.795004i	$-0.292531\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2070.00i	− 0.879044i
$178$	0	0
$179$	−150.000	−0.0626342	−0.0313171	−	0.999509i	$-0.509970\pi$
$179$	−150.000	−0.0626342	−0.0313171	+	0.999509i	$0.509970\pi$
$180$	0	0
$181$	197.000	0.0809000	0.0404500	−	0.999182i	$-0.487121\pi$
$181$	197.000	0.0809000	0.0404500	+	0.999182i	$0.487121\pi$
$182$	0	0
$183$	2199.00i	0.888277i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2268.00i	− 0.886912i
$188$	0	0
$189$	−27.0000	−0.0103913
$190$	0	0
$191$	−1302.00	−0.493243	−0.246622	−	0.969112i	$-0.579320\pi$
$191$	−1302.00	−0.493243	−0.246622	+	0.969112i	$0.579320\pi$
$192$	0	0
$193$	− 4163.00i	− 1.55264i	−0.630340	−	0.776319i	$-0.717084\pi$
$193$	− 4163.00i	− 1.55264i	0.630340	−	0.776319i	$-0.282916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3054.00i	1.10451i	0.833675	+	0.552255i	$0.186233\pi$
$197$	3054.00i	1.10451i	−0.833675	+	0.552255i	$0.813767\pi$
$198$	0	0
$199$	3425.00	1.22006	0.610030	−	0.792379i	$-0.291158\pi$
$199$	3425.00	1.22006	0.610030	+	0.792379i	$0.291158\pi$
$200$	0	0
$201$	−897.000	−0.314774
$202$	0	0
$203$	210.000i	0.0726065i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1458.00i	0.489556i
$208$	0	0
$209$	4830.00	1.59856
$210$	0	0
$211$	2443.00	0.797076	0.398538	−	0.917152i	$-0.369518\pi$
$211$	2443.00	0.797076	0.398538	+	0.917152i	$0.369518\pi$
$212$	0	0
$213$	− 684.000i	− 0.220032i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	193.000i	0.0603765i
$218$	0	0
$219$	−2814.00	−0.868276
$220$	0	0
$221$	−3618.00	−1.10124
$222$	0	0
$223$	23.0000i	0.00690670i	0.999994	+	0.00345335i	$0.00109924\pi$
$223$	23.0000i	0.00690670i	−0.999994	+	0.00345335i	$0.998901\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1956.00i	0.571913i	0.958243	+	0.285957i	$0.0923113\pi$
$227$	1956.00i	0.571913i	−0.958243	+	0.285957i	$0.907689\pi$
$228$	0	0
$229$	−1805.00	−0.520864	−0.260432	−	0.965492i	$-0.583865\pi$
$229$	−1805.00	−0.520864	−0.260432	+	0.965492i	$0.583865\pi$
$230$	0	0
$231$	−126.000	−0.0358883
$232$	0	0
$233$	− 3468.00i	− 0.975091i	−0.873098	−	0.487546i	$-0.837892\pi$
$233$	− 3468.00i	− 0.975091i	0.873098	−	0.487546i	$-0.162108\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	480.000i	0.131558i
$238$	0	0
$239$	2640.00	0.714508	0.357254	−	0.934007i	$-0.383713\pi$
$239$	2640.00	0.714508	0.357254	+	0.934007i	$0.383713\pi$
$240$	0	0
$241$	−5383.00	−1.43879	−0.719397	−	0.694599i	$-0.755582\pi$
$241$	−5383.00	−1.43879	−0.719397	+	0.694599i	$0.755582\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7705.00i	− 1.98485i
$248$	0	0
$249$	−1386.00	−0.352748
$250$	0	0
$251$	5028.00	1.26440	0.632200	−	0.774805i	$-0.282152\pi$
$251$	5028.00	1.26440	0.632200	+	0.774805i	$0.282152\pi$
$252$	0	0
$253$	6804.00i	1.69077i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	564.000i	0.136892i	0.997655	+	0.0684462i	$0.0218042\pi$
$257$	564.000i	0.136892i	−0.997655	+	0.0684462i	$0.978196\pi$
$258$	0	0
$259$	286.000	0.0686146
$260$	0	0
$261$	−1890.00	−0.448230
$262$	0	0
$263$	− 1812.00i	− 0.424839i	−0.977179	−	0.212420i	$-0.931866\pi$
$263$	− 1812.00i	− 0.424839i	0.977179	−	0.212420i	$-0.0681344\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 720.000i	− 0.165031i
$268$	0	0
$269$	5190.00	1.17636	0.588178	−	0.808731i	$-0.299845\pi$
$269$	5190.00	1.17636	0.588178	+	0.808731i	$0.299845\pi$
$270$	0	0
$271$	−4592.00	−1.02931	−0.514657	−	0.857396i	$-0.672081\pi$
$271$	−4592.00	−1.02931	−0.514657	+	0.857396i	$0.672081\pi$
$272$	0	0
$273$	201.000i	0.0445607i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2191.00i	− 0.475251i	−0.971357	−	0.237625i	$-0.923631\pi$
$277$	− 2191.00i	− 0.475251i	0.971357	−	0.237625i	$-0.0763690\pi$
$278$	0	0
$279$	−1737.00	−0.372729
$280$	0	0
$281$	7842.00	1.66482	0.832410	−	0.554160i	$-0.186960\pi$
$281$	7842.00	1.66482	0.832410	+	0.554160i	$0.186960\pi$
$282$	0	0
$283$	− 247.000i	− 0.0518821i	−0.999663	−	0.0259410i	$-0.991742\pi$
$283$	− 247.000i	− 0.0518821i	0.999663	−	0.0259410i	$-0.00825821\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.00246808i
$288$	0	0
$289$	1997.00	0.406473
$290$	0	0
$291$	−1533.00	−0.308818
$292$	0	0
$293$	5442.00i	1.08507i	0.840034	+	0.542534i	$0.182535\pi$
$293$	5442.00i	1.08507i	−0.840034	+	0.542534i	$0.817465\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1134.00i	− 0.221553i
$298$	0	0
$299$	10854.0	2.09934
$300$	0	0
$301$	−263.000	−0.0503624
$302$	0	0
$303$	− 2736.00i	− 0.518743i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3871.00i	0.719641i	0.933022	+	0.359820i	$0.117162\pi$
$307$	3871.00i	0.719641i	−0.933022	+	0.359820i	$0.882838\pi$
$308$	0	0
$309$	2004.00	0.368944
$310$	0	0
$311$	5718.00	1.04257	0.521283	−	0.853384i	$-0.325454\pi$
$311$	5718.00	1.04257	0.521283	+	0.853384i	$0.325454\pi$
$312$	0	0
$313$	3637.00i	0.656790i	0.944540	+	0.328395i	$0.106508\pi$
$313$	3637.00i	0.656790i	−0.944540	+	0.328395i	$0.893492\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1296.00i	− 0.229623i	−0.993387	−	0.114812i	$-0.963374\pi$
$317$	− 1296.00i	− 0.229623i	0.993387	−	0.114812i	$-0.0366265\pi$
$318$	0	0
$319$	−8820.00	−1.54804
$320$	0	0
$321$	3888.00	0.676034
$322$	0	0
$323$	− 6210.00i	− 1.06976i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5205.00i	− 0.880236i
$328$	0	0
$329$	414.000	0.0693756
$330$	0	0
$331$	−5132.00	−0.852206	−0.426103	−	0.904675i	$-0.640114\pi$
$331$	−5132.00	−0.852206	−0.426103	+	0.904675i	$0.640114\pi$
$332$	0	0
$333$	2574.00i	0.423587i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6751.00i	− 1.09125i	−0.838030	−	0.545624i	$-0.816293\pi$
$337$	− 6751.00i	− 1.09125i	0.838030	−	0.545624i	$-0.183707\pi$
$338$	0	0
$339$	3276.00	0.524861
$340$	0	0
$341$	−8106.00	−1.28729
$342$	0	0
$343$	685.000i	0.107832i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5226.00i	0.808491i	0.914651	+	0.404246i	$0.132466\pi$
$347$	5226.00i	0.808491i	−0.914651	+	0.404246i	$0.867534\pi$
$348$	0	0
$349$	6190.00	0.949407	0.474704	−	0.880146i	$-0.342555\pi$
$349$	6190.00	0.949407	0.474704	+	0.880146i	$0.342555\pi$
$350$	0	0
$351$	−1809.00	−0.275092
$352$	0	0
$353$	− 6618.00i	− 0.997849i	−0.866646	−	0.498924i	$-0.833729\pi$
$353$	− 6618.00i	− 0.997849i	0.866646	−	0.498924i	$-0.166271\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	162.000i	0.0240167i
$358$	0	0
$359$	−3420.00	−0.502787	−0.251394	−	0.967885i	$-0.580889\pi$
$359$	−3420.00	−0.502787	−0.251394	+	0.967885i	$0.580889\pi$
$360$	0	0
$361$	6366.00	0.928124
$362$	0	0
$363$	− 1299.00i	− 0.187823i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	871.000i	0.123885i	0.998080	+	0.0619425i	$0.0197296\pi$
$367$	871.000i	0.123885i	−0.998080	+	0.0619425i	$0.980270\pi$
$368$	0	0
$369$	−108.000	−0.0152365
$370$	0	0
$371$	−192.000	−0.0268683
$372$	0	0
$373$	− 6383.00i	− 0.886057i	−0.896508	−	0.443028i	$-0.853904\pi$
$373$	− 6383.00i	− 0.886057i	0.896508	−	0.443028i	$-0.146096\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14070.0i	1.92213i
$378$	0	0
$379$	−9865.00	−1.33702	−0.668511	−	0.743703i	$-0.733068\pi$
$379$	−9865.00	−1.33702	−0.668511	+	0.743703i	$0.733068\pi$
$380$	0	0
$381$	48.0000	0.00645437
$382$	0	0
$383$	9828.00i	1.31119i	0.755111	+	0.655597i	$0.227583\pi$
$383$	9828.00i	1.31119i	−0.755111	+	0.655597i	$0.772417\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2367.00i	− 0.310908i
$388$	0	0
$389$	−12540.0	−1.63446	−0.817228	−	0.576315i	$-0.804490\pi$
$389$	−12540.0	−1.63446	−0.817228	+	0.576315i	$0.804490\pi$
$390$	0	0
$391$	8748.00	1.13147
$392$	0	0
$393$	5976.00i	0.767047i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1381.00i	− 0.174585i	−0.996183	−	0.0872927i	$-0.972178\pi$
$397$	− 1381.00i	− 0.174585i	0.996183	−	0.0872927i	$-0.0278215\pi$
$398$	0	0
$399$	−345.000	−0.0432872
$400$	0	0
$401$	14232.0	1.77235	0.886175	−	0.463351i	$-0.153353\pi$
$401$	14232.0	1.77235	0.886175	+	0.463351i	$0.153353\pi$
$402$	0	0
$403$	12931.0i	1.59836i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12012.0i	1.46293i
$408$	0	0
$409$	−2645.00	−0.319772	−0.159886	−	0.987135i	$-0.551113\pi$
$409$	−2645.00	−0.319772	−0.159886	+	0.987135i	$0.551113\pi$
$410$	0	0
$411$	−7038.00	−0.844669
$412$	0	0
$413$	690.000i	0.0822099i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8700.00i	− 1.02168i
$418$	0	0
$419$	3000.00	0.349784	0.174892	−	0.984588i	$-0.444042\pi$
$419$	3000.00	0.349784	0.174892	+	0.984588i	$0.444042\pi$
$420$	0	0
$421$	−11338.0	−1.31254	−0.656271	−	0.754525i	$-0.727867\pi$
$421$	−11338.0	−1.31254	−0.656271	+	0.754525i	$0.727867\pi$
$422$	0	0
$423$	3726.00i	0.428284i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 733.000i	− 0.0830734i
$428$	0	0
$429$	−8442.00	−0.950078
$430$	0	0
$431$	3258.00	0.364112	0.182056	−	0.983288i	$-0.441725\pi$
$431$	3258.00	0.364112	0.182056	+	0.983288i	$0.441725\pi$
$432$	0	0
$433$	− 1163.00i	− 0.129077i	−0.997915	−	0.0645384i	$-0.979443\pi$
$433$	− 1163.00i	− 0.129077i	0.997915	−	0.0645384i	$-0.0205575\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18630.0i	2.03934i
$438$	0	0
$439$	6695.00	0.727870	0.363935	−	0.931424i	$-0.381433\pi$
$439$	6695.00	0.727870	0.363935	+	0.931424i	$0.381433\pi$
$440$	0	0
$441$	−3078.00	−0.332362
$442$	0	0
$443$	16368.0i	1.75546i	0.479159	+	0.877728i	$0.340942\pi$
$443$	16368.0i	1.75546i	−0.479159	+	0.877728i	$0.659058\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6210.00i	− 0.657098i
$448$	0	0
$449$	−16380.0	−1.72165	−0.860824	−	0.508903i	$-0.830051\pi$
$449$	−16380.0	−1.72165	−0.860824	+	0.508903i	$0.830051\pi$
$450$	0	0
$451$	−504.000	−0.0526218
$452$	0	0
$453$	6711.00i	0.696049i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 13786.0i	− 1.41112i	−0.708650	−	0.705560i	$-0.750696\pi$
$457$	− 13786.0i	− 1.41112i	0.708650	−	0.705560i	$-0.249304\pi$
$458$	0	0
$459$	−1458.00	−0.148265
$460$	0	0
$461$	11832.0	1.19538	0.597691	−	0.801726i	$-0.296085\pi$
$461$	11832.0	1.19538	0.597691	+	0.801726i	$0.296085\pi$
$462$	0	0
$463$	3008.00i	0.301930i	0.988539	+	0.150965i	$0.0482381\pi$
$463$	3008.00i	0.301930i	−0.988539	+	0.150965i	$0.951762\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4434.00i	− 0.439360i	−0.975572	−	0.219680i	$-0.929499\pi$
$467$	− 4434.00i	− 0.439360i	0.975572	−	0.219680i	$-0.0705013\pi$
$468$	0	0
$469$	299.000	0.0294382
$470$	0	0
$471$	−723.000	−0.0707305
$472$	0	0
$473$	− 11046.0i	− 1.07378i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1728.00i	− 0.165869i
$478$	0	0
$479$	7410.00	0.706830	0.353415	−	0.935467i	$-0.385020\pi$
$479$	7410.00	0.706830	0.353415	+	0.935467i	$0.385020\pi$
$480$	0	0
$481$	19162.0	1.81645
$482$	0	0
$483$	− 486.000i	− 0.0457842i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8671.00i	0.806818i	0.915020	+	0.403409i	$0.132175\pi$
$487$	8671.00i	0.806818i	−0.915020	+	0.403409i	$0.867825\pi$
$488$	0	0
$489$	−10641.0	−0.984055
$490$	0	0
$491$	19368.0	1.78017	0.890087	−	0.455790i	$-0.150643\pi$
$491$	19368.0	1.78017	0.890087	+	0.455790i	$0.150643\pi$
$492$	0	0
$493$	11340.0i	1.03596i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	228.000i	0.0205779i
$498$	0	0
$499$	−8875.00	−0.796192	−0.398096	−	0.917344i	$-0.630329\pi$
$499$	−8875.00	−0.796192	−0.398096	+	0.917344i	$0.630329\pi$
$500$	0	0
$501$	−2952.00	−0.263245
$502$	0	0
$503$	− 10452.0i	− 0.926504i	−0.886227	−	0.463252i	$-0.846682\pi$
$503$	− 10452.0i	− 0.926504i	0.886227	−	0.463252i	$-0.153318\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6876.00i	0.602315i
$508$	0	0
$509$	19770.0	1.72159	0.860796	−	0.508951i	$-0.169967\pi$
$509$	19770.0	1.72159	0.860796	+	0.508951i	$0.169967\pi$
$510$	0	0
$511$	938.000	0.0812029
$512$	0	0
$513$	− 3105.00i	− 0.267230i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17388.0i	1.47916i
$518$	0	0
$519$	−10854.0	−0.917992
$520$	0	0
$521$	−11238.0	−0.945001	−0.472501	−	0.881330i	$-0.656649\pi$
$521$	−11238.0	−0.945001	−0.472501	+	0.881330i	$0.656649\pi$
$522$	0	0
$523$	− 7447.00i	− 0.622628i	−0.950307	−	0.311314i	$-0.899231\pi$
$523$	− 7447.00i	− 0.622628i	0.950307	−	0.311314i	$-0.100769\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10422.0i	0.861460i
$528$	0	0
$529$	−14077.0	−1.15698
$530$	0	0
$531$	−6210.00	−0.507516
$532$	0	0
$533$	804.000i	0.0653379i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	450.000i	0.0361619i
$538$	0	0
$539$	−14364.0	−1.14787
$540$	0	0
$541$	−17623.0	−1.40050	−0.700251	−	0.713896i	$-0.746929\pi$
$541$	−17623.0	−1.40050	−0.700251	+	0.713896i	$0.746929\pi$
$542$	0	0
$543$	− 591.000i	− 0.0467076i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10096.0i	0.789166i	0.918860	+	0.394583i	$0.129111\pi$
$547$	10096.0i	0.789166i	−0.918860	+	0.394583i	$0.870889\pi$
$548$	0	0
$549$	6597.00	0.512847
$550$	0	0
$551$	−24150.0	−1.86720
$552$	0	0
$553$	− 160.000i	− 0.0123036i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14514.0i	1.10409i	0.833814	+	0.552045i	$0.186152\pi$
$557$	14514.0i	1.10409i	−0.833814	+	0.552045i	$0.813848\pi$
$558$	0	0
$559$	−17621.0	−1.33325
$560$	0	0
$561$	−6804.00	−0.512059
$562$	0	0
$563$	− 10242.0i	− 0.766694i	−0.923604	−	0.383347i	$-0.874771\pi$
$563$	− 10242.0i	− 0.766694i	0.923604	−	0.383347i	$-0.125229\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	81.0000i	0.00599944i
$568$	0	0
$569$	6750.00	0.497319	0.248660	−	0.968591i	$-0.420010\pi$
$569$	6750.00	0.497319	0.248660	+	0.968591i	$0.420010\pi$
$570$	0	0
$571$	−17117.0	−1.25451	−0.627254	−	0.778815i	$-0.715821\pi$
$571$	−17117.0	−1.25451	−0.627254	+	0.778815i	$0.715821\pi$
$572$	0	0
$573$	3906.00i	0.284774i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 301.000i	− 0.0217171i	−0.999941	−	0.0108586i	$-0.996544\pi$
$577$	− 301.000i	− 0.0217171i	0.999941	−	0.0108586i	$-0.00345646\pi$
$578$	0	0
$579$	−12489.0	−0.896416
$580$	0	0
$581$	462.000	0.0329897
$582$	0	0
$583$	− 8064.00i	− 0.572859i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15456.0i	1.08678i	0.839482	+	0.543388i	$0.182859\pi$
$587$	15456.0i	1.08678i	−0.839482	+	0.543388i	$0.817141\pi$
$588$	0	0
$589$	−22195.0	−1.55268
$590$	0	0
$591$	9162.00	0.637689
$592$	0	0
$593$	9492.00i	0.657318i	0.944449	+	0.328659i	$0.106597\pi$
$593$	9492.00i	0.657318i	−0.944449	+	0.328659i	$0.893403\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10275.0i	− 0.704402i
$598$	0	0
$599$	1500.00	0.102318	0.0511589	−	0.998691i	$-0.483709\pi$
$599$	1500.00	0.102318	0.0511589	+	0.998691i	$0.483709\pi$
$600$	0	0
$601$	14627.0	0.992758	0.496379	−	0.868106i	$-0.334663\pi$
$601$	14627.0	0.992758	0.496379	+	0.868106i	$0.334663\pi$
$602$	0	0
$603$	2691.00i	0.181735i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16184.0i	− 1.08219i	−0.840962	−	0.541094i	$-0.818010\pi$
$607$	− 16184.0i	− 1.08219i	0.840962	−	0.541094i	$-0.181990\pi$
$608$	0	0
$609$	630.000	0.0419194
$610$	0	0
$611$	27738.0	1.83659
$612$	0	0
$613$	18502.0i	1.21907i	0.792760	+	0.609534i	$0.208643\pi$
$613$	18502.0i	1.21907i	−0.792760	+	0.609534i	$0.791357\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13896.0i	− 0.906697i	−0.891333	−	0.453348i	$-0.850229\pi$
$617$	− 13896.0i	− 0.906697i	0.891333	−	0.453348i	$-0.149771\pi$
$618$	0	0
$619$	−9895.00	−0.642510	−0.321255	−	0.946993i	$-0.604105\pi$
$619$	−9895.00	−0.642510	−0.321255	+	0.946993i	$0.604105\pi$
$620$	0	0
$621$	4374.00	0.282645
$622$	0	0
$623$	240.000i	0.0154340i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 14490.0i	− 0.922926i
$628$	0	0
$629$	15444.0	0.979003
$630$	0	0
$631$	−467.000	−0.0294627	−0.0147314	−	0.999891i	$-0.504689\pi$
$631$	−467.000	−0.0294627	−0.0147314	+	0.999891i	$0.504689\pi$
$632$	0	0
$633$	− 7329.00i	− 0.460192i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22914.0i	1.42525i
$638$	0	0
$639$	−2052.00	−0.127036
$640$	0	0
$641$	30612.0	1.88627	0.943137	−	0.332405i	$-0.107860\pi$
$641$	30612.0	1.88627	0.943137	+	0.332405i	$0.107860\pi$
$642$	0	0
$643$	− 1852.00i	− 0.113586i	−0.998386	−	0.0567930i	$-0.981913\pi$
$643$	− 1852.00i	− 0.113586i	0.998386	−	0.0567930i	$-0.0180875\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21156.0i	1.28551i	0.766070	+	0.642757i	$0.222210\pi$
$647$	21156.0i	1.28551i	−0.766070	+	0.642757i	$0.777790\pi$
$648$	0	0
$649$	−28980.0	−1.75280
$650$	0	0
$651$	579.000	0.0348584
$652$	0	0
$653$	9702.00i	0.581422i	0.956811	+	0.290711i	$0.0938918\pi$
$653$	9702.00i	0.581422i	−0.956811	+	0.290711i	$0.906108\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8442.00i	0.501300i
$658$	0	0
$659$	1980.00	0.117041	0.0585204	−	0.998286i	$-0.481362\pi$
$659$	1980.00	0.117041	0.0585204	+	0.998286i	$0.481362\pi$
$660$	0	0
$661$	−20158.0	−1.18617	−0.593083	−	0.805142i	$-0.702089\pi$
$661$	−20158.0	−1.18617	−0.593083	+	0.805142i	$0.702089\pi$
$662$	0	0
$663$	10854.0i	0.635799i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 34020.0i	− 1.97490i
$668$	0	0
$669$	69.0000	0.00398758
$670$	0	0
$671$	30786.0	1.77121
$672$	0	0
$673$	16882.0i	0.966944i	0.875360	+	0.483472i	$0.160624\pi$
$673$	16882.0i	0.966944i	−0.875360	+	0.483472i	$0.839376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20934.0i	1.18842i	0.804311	+	0.594209i	$0.202535\pi$
$677$	20934.0i	1.18842i	−0.804311	+	0.594209i	$0.797465\pi$
$678$	0	0
$679$	511.000	0.0288813
$680$	0	0
$681$	5868.00	0.330194
$682$	0	0
$683$	− 8712.00i	− 0.488075i	−0.969766	−	0.244038i	$-0.921528\pi$
$683$	− 8712.00i	− 0.488075i	0.969766	−	0.244038i	$-0.0784720\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5415.00i	0.300721i
$688$	0	0
$689$	−12864.0	−0.711291
$690$	0	0
$691$	14128.0	0.777792	0.388896	−	0.921282i	$-0.372856\pi$
$691$	14128.0	0.777792	0.388896	+	0.921282i	$0.372856\pi$
$692$	0	0
$693$	378.000i	0.0207201i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	648.000i	0.0352148i
$698$	0	0
$699$	−10404.0	−0.562969
$700$	0	0
$701$	−28278.0	−1.52360	−0.761801	−	0.647811i	$-0.775685\pi$
$701$	−28278.0	−1.52360	−0.761801	+	0.647811i	$0.775685\pi$
$702$	0	0
$703$	32890.0i	1.76454i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	912.000i	0.0485138i
$708$	0	0
$709$	−8885.00	−0.470639	−0.235320	−	0.971918i	$-0.575614\pi$
$709$	−8885.00	−0.470639	−0.235320	+	0.971918i	$0.575614\pi$
$710$	0	0
$711$	1440.00	0.0759553
$712$	0	0
$713$	− 31266.0i	− 1.64225i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7920.00i	− 0.412521i
$718$	0	0
$719$	−7530.00	−0.390572	−0.195286	−	0.980746i	$-0.562564\pi$
$719$	−7530.00	−0.390572	−0.195286	+	0.980746i	$0.562564\pi$
$720$	0	0
$721$	−668.000	−0.0345043
$722$	0	0
$723$	16149.0i	0.830688i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1801.00i	0.0918781i	0.998944	+	0.0459391i	$0.0146280\pi$
$727$	1801.00i	0.0918781i	−0.998944	+	0.0459391i	$0.985372\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−14202.0	−0.718577
$732$	0	0
$733$	7882.00i	0.397174i	0.980083	+	0.198587i	$0.0636352\pi$
$733$	7882.00i	0.397174i	−0.980083	+	0.198587i	$0.936365\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12558.0i	0.627652i
$738$	0	0
$739$	33860.0	1.68547	0.842734	−	0.538331i	$-0.180945\pi$
$739$	33860.0	1.68547	0.842734	+	0.538331i	$0.180945\pi$
$740$	0	0
$741$	−23115.0	−1.14595
$742$	0	0
$743$	− 20652.0i	− 1.01972i	−0.860259	−	0.509858i	$-0.829698\pi$
$743$	− 20652.0i	− 1.01972i	0.860259	−	0.509858i	$-0.170302\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4158.00i	0.203659i
$748$	0	0
$749$	−1296.00	−0.0632240
$750$	0	0
$751$	−7472.00	−0.363059	−0.181529	−	0.983386i	$-0.558105\pi$
$751$	−7472.00	−0.363059	−0.181529	+	0.983386i	$0.558105\pi$
$752$	0	0
$753$	− 15084.0i	− 0.730002i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32251.0i	− 1.54846i	−0.632906	−	0.774229i	$-0.718138\pi$
$757$	− 32251.0i	− 1.54846i	0.632906	−	0.774229i	$-0.281862\pi$
$758$	0	0
$759$	20412.0	0.976164
$760$	0	0
$761$	16812.0	0.800834	0.400417	−	0.916333i	$-0.368865\pi$
$761$	16812.0	0.800834	0.400417	+	0.916333i	$0.368865\pi$
$762$	0	0
$763$	1735.00i	0.0823214i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46230.0i	2.17636i
$768$	0	0
$769$	34645.0	1.62462	0.812309	−	0.583228i	$-0.198211\pi$
$769$	34645.0	1.62462	0.812309	+	0.583228i	$0.198211\pi$
$770$	0	0
$771$	1692.00	0.0790349
$772$	0	0
$773$	8412.00i	0.391408i	0.980663	+	0.195704i	$0.0626992\pi$
$773$	8412.00i	0.391408i	−0.980663	+	0.195704i	$0.937301\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 858.000i	− 0.0396146i
$778$	0	0
$779$	−1380.00	−0.0634706
$780$	0	0
$781$	−9576.00	−0.438740
$782$	0	0
$783$	5670.00i	0.258786i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18329.0i	− 0.830188i	−0.909778	−	0.415094i	$-0.863749\pi$
$787$	− 18329.0i	− 0.830188i	0.909778	−	0.415094i	$-0.136251\pi$
$788$	0	0
$789$	−5436.00	−0.245281
$790$	0	0
$791$	−1092.00	−0.0490860
$792$	0	0
$793$	− 49111.0i	− 2.19922i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16044.0i	0.713059i	0.934284	+	0.356529i	$0.116040\pi$
$797$	16044.0i	0.713059i	−0.934284	+	0.356529i	$0.883960\pi$
$798$	0	0
$799$	22356.0	0.989860
$800$	0	0
$801$	−2160.00	−0.0952807
$802$	0	0
$803$	39396.0i	1.73133i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 15570.0i	− 0.679170i
$808$	0	0
$809$	24000.0	1.04301	0.521505	−	0.853248i	$-0.325371\pi$
$809$	24000.0	1.04301	0.521505	+	0.853248i	$0.325371\pi$
$810$	0	0
$811$	−5117.00	−0.221556	−0.110778	−	0.993845i	$-0.535334\pi$
$811$	−5117.00	−0.221556	−0.110778	+	0.993845i	$0.535334\pi$
$812$	0	0
$813$	13776.0i	0.594275i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 30245.0i	− 1.29515i
$818$	0	0
$819$	603.000	0.0257271
$820$	0	0
$821$	13542.0	0.575663	0.287831	−	0.957681i	$-0.407066\pi$
$821$	13542.0	0.575663	0.287831	+	0.957681i	$0.407066\pi$
$822$	0	0
$823$	1283.00i	0.0543409i	0.999631	+	0.0271705i	$0.00864969\pi$
$823$	1283.00i	0.0543409i	−0.999631	+	0.0271705i	$0.991350\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 16344.0i	− 0.687227i	−0.939111	−	0.343613i	$-0.888349\pi$
$827$	− 16344.0i	− 0.687227i	0.939111	−	0.343613i	$-0.111651\pi$
$828$	0	0
$829$	790.000	0.0330975	0.0165488	−	0.999863i	$-0.494732\pi$
$829$	790.000	0.0330975	0.0165488	+	0.999863i	$0.494732\pi$
$830$	0	0
$831$	−6573.00	−0.274386
$832$	0	0
$833$	18468.0i	0.768161i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5211.00i	0.215195i
$838$	0	0
$839$	−9990.00	−0.411076	−0.205538	−	0.978649i	$-0.565894\pi$
$839$	−9990.00	−0.411076	−0.205538	+	0.978649i	$0.565894\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	0	0
$843$	− 23526.0i	− 0.961184i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	433.000i	0.0175656i
$848$	0	0
$849$	−741.000	−0.0299541
$850$	0	0
$851$	−46332.0	−1.86632
$852$	0	0
$853$	− 24743.0i	− 0.993182i	−0.867985	−	0.496591i	$-0.834585\pi$
$853$	− 24743.0i	− 0.993182i	0.867985	−	0.496591i	$-0.165415\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 23556.0i	− 0.938924i	−0.882953	−	0.469462i	$-0.844448\pi$
$857$	− 23556.0i	− 0.938924i	0.882953	−	0.469462i	$-0.155552\pi$
$858$	0	0
$859$	−34000.0	−1.35048	−0.675242	−	0.737597i	$-0.735961\pi$
$859$	−34000.0	−1.35048	−0.675242	+	0.737597i	$0.735961\pi$
$860$	0	0
$861$	36.0000	0.00142494
$862$	0	0
$863$	− 37032.0i	− 1.46070i	−0.683073	−	0.730350i	$-0.739357\pi$
$863$	− 37032.0i	− 1.46070i	0.683073	−	0.730350i	$-0.260643\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5991.00i	− 0.234677i
$868$	0	0
$869$	6720.00	0.262325
$870$	0	0
$871$	20033.0	0.779325
$872$	0	0
$873$	4599.00i	0.178296i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2519.00i	0.0969904i	0.998823	+	0.0484952i	$0.0154426\pi$
$877$	2519.00i	0.0969904i	−0.998823	+	0.0484952i	$0.984557\pi$
$878$	0	0
$879$	16326.0	0.626465
$880$	0	0
$881$	43992.0	1.68232	0.841162	−	0.540783i	$-0.181872\pi$
$881$	43992.0	1.68232	0.841162	+	0.540783i	$0.181872\pi$
$882$	0	0
$883$	− 19177.0i	− 0.730869i	−0.930837	−	0.365435i	$-0.880920\pi$
$883$	− 19177.0i	− 0.730869i	0.930837	−	0.365435i	$-0.119080\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 44994.0i	− 1.70321i	−0.524181	−	0.851607i	$-0.675628\pi$
$887$	− 44994.0i	− 1.70321i	0.524181	−	0.851607i	$-0.324372\pi$
$888$	0	0
$889$	−16.0000	−0.000603625 0
$890$	0	0
$891$	−3402.00	−0.127914
$892$	0	0
$893$	47610.0i	1.78411i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 32562.0i	− 1.21206i
$898$	0	0
$899$	40530.0	1.50362
$900$	0	0
$901$	−10368.0	−0.383361
$902$	0	0
$903$	789.000i	0.0290767i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52396.0i	1.91817i	0.283117	+	0.959085i	$0.408631\pi$
$907$	52396.0i	1.91817i	−0.283117	+	0.959085i	$0.591369\pi$
$908$	0	0
$909$	−8208.00	−0.299496
$910$	0	0
$911$	−7242.00	−0.263379	−0.131689	−	0.991291i	$-0.542040\pi$
$911$	−7242.00	−0.263379	−0.131689	+	0.991291i	$0.542040\pi$
$912$	0	0
$913$	19404.0i	0.703372i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1992.00i	− 0.0717357i
$918$	0	0
$919$	4085.00	0.146629	0.0733143	−	0.997309i	$-0.476642\pi$
$919$	4085.00	0.146629	0.0733143	+	0.997309i	$0.476642\pi$
$920$	0	0
$921$	11613.0	0.415485
$922$	0	0
$923$	15276.0i	0.544762i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6012.00i	− 0.213010i
$928$	0	0
$929$	3030.00	0.107009	0.0535043	−	0.998568i	$-0.482961\pi$
$929$	3030.00	0.107009	0.0535043	+	0.998568i	$0.482961\pi$
$930$	0	0
$931$	−39330.0	−1.38452
$932$	0	0
$933$	− 17154.0i	− 0.601926i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5759.00i	0.200788i	0.994948	+	0.100394i	$0.0320103\pi$
$937$	5759.00i	0.200788i	−0.994948	+	0.100394i	$0.967990\pi$
$938$	0	0
$939$	10911.0	0.379198
$940$	0	0
$941$	−258.000	−0.00893790	−0.00446895	−	0.999990i	$-0.501423\pi$
$941$	−258.000	−0.00893790	−0.00446895	+	0.999990i	$0.501423\pi$
$942$	0	0
$943$	− 1944.00i	− 0.0671319i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1374.00i	− 0.0471478i	−0.999722	−	0.0235739i	$-0.992495\pi$
$947$	− 1374.00i	− 0.0471478i	0.999722	−	0.0235739i	$-0.00750451\pi$
$948$	0	0
$949$	62846.0	2.14970
$950$	0	0
$951$	−3888.00	−0.132573
$952$	0	0
$953$	− 9288.00i	− 0.315706i	−0.987463	−	0.157853i	$-0.949543\pi$
$953$	− 9288.00i	− 0.315706i	0.987463	−	0.157853i	$-0.0504572\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26460.0i	0.893762i
$958$	0	0
$959$	2346.00	0.0789951
$960$	0	0
$961$	7458.00	0.250344
$962$	0	0
$963$	− 11664.0i	− 0.390309i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21616.0i	0.718846i	0.933175	+	0.359423i	$0.117026\pi$
$967$	21616.0i	0.718846i	−0.933175	+	0.359423i	$0.882974\pi$
$968$	0	0
$969$	−18630.0	−0.617628
$970$	0	0
$971$	19098.0	0.631188	0.315594	−	0.948894i	$-0.397796\pi$
$971$	19098.0	0.631188	0.315594	+	0.948894i	$0.397796\pi$
$972$	0	0
$973$	2900.00i	0.0955496i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18246.0i	− 0.597483i	−0.954334	−	0.298742i	$-0.903433\pi$
$977$	− 18246.0i	− 0.597483i	0.954334	−	0.298742i	$-0.0965669\pi$
$978$	0	0
$979$	−10080.0	−0.329069
$980$	0	0
$981$	−15615.0	−0.508204
$982$	0	0
$983$	− 38772.0i	− 1.25802i	−0.777397	−	0.629011i	$-0.783460\pi$
$983$	− 38772.0i	− 1.25802i	0.777397	−	0.629011i	$-0.216540\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1242.00i	− 0.0400540i
$988$	0	0
$989$	42606.0	1.36986
$990$	0	0
$991$	23053.0	0.738953	0.369477	−	0.929240i	$-0.379537\pi$
$991$	23053.0	0.738953	0.369477	+	0.929240i	$0.379537\pi$
$992$	0	0
$993$	15396.0i	0.492021i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 10366.0i	− 0.329282i	−0.986354	−	0.164641i	$-0.947353\pi$
$997$	− 10366.0i	− 0.329282i	0.986354	−	0.164641i	$-0.0526466\pi$
$998$	0	0
$999$	7722.00	0.244558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.c.49.1		2
4.3	odd	2		150.4.c.e.49.1		2
5.2	odd	4		1200.4.a.i.1.1		1
5.3	odd	4		1200.4.a.bb.1.1		1
5.4	even	2	inner	1200.4.f.c.49.2		2
12.11	even	2		450.4.c.a.199.2		2
20.3	even	4		150.4.a.a.1.1	✓	1
20.7	even	4		150.4.a.h.1.1	yes	1
20.19	odd	2		150.4.c.e.49.2		2
60.23	odd	4		450.4.a.o.1.1		1
60.47	odd	4		450.4.a.f.1.1		1
60.59	even	2		450.4.c.a.199.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.a.1.1	✓	1	20.3	even	4
150.4.a.h.1.1	yes	1	20.7	even	4
150.4.c.e.49.1		2	4.3	odd	2
150.4.c.e.49.2		2	20.19	odd	2
450.4.a.f.1.1		1	60.47	odd	4
450.4.a.o.1.1		1	60.23	odd	4
450.4.c.a.199.1		2	60.59	even	2
450.4.c.a.199.2		2	12.11	even	2
1200.4.a.i.1.1		1	5.2	odd	4
1200.4.a.bb.1.1		1	5.3	odd	4
1200.4.f.c.49.1		2	1.1	even	1	trivial
1200.4.f.c.49.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +1.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +1.00000i q^{7} -9.00000 q^{9} -42.0000 q^{11} +67.0000i q^{13} +54.0000i q^{17} -115.000 q^{19} +3.00000 q^{21} -162.000i q^{23} +27.0000i q^{27} +210.000 q^{29} +193.000 q^{31} +126.000i q^{33} -286.000i q^{37} +201.000 q^{39} +12.0000 q^{41} +263.000i q^{43} -414.000i q^{47} +342.000 q^{49} +162.000 q^{51} +192.000i q^{53} +345.000i q^{57} +690.000 q^{59} -733.000 q^{61} -9.00000i q^{63} -299.000i q^{67} -486.000 q^{69} +228.000 q^{71} -938.000i q^{73} -42.0000i q^{77} -160.000 q^{79} +81.0000 q^{81} -462.000i q^{83} -630.000i q^{87} +240.000 q^{89} -67.0000 q^{91} -579.000i q^{93} -511.000i q^{97} +378.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 84 q^{11} - 230 q^{19} + 6 q^{21} + 420 q^{29} + 386 q^{31} + 402 q^{39} + 24 q^{41} + 684 q^{49} + 324 q^{51} + 1380 q^{59} - 1466 q^{61} - 972 q^{69} + 456 q^{71} - 320 q^{79} + 162 q^{81} + 480 q^{89} - 134 q^{91} + 756 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 84 * q^11 - 230 * q^19 + 6 * q^21 + 420 * q^29 + 386 * q^31 + 402 * q^39 + 24 * q^41 + 684 * q^49 + 324 * q^51 + 1380 * q^59 - 1466 * q^61 - 972 * q^69 + 456 * q^71 - 320 * q^79 + 162 * q^81 + 480 * q^89 - 134 * q^91 + 756 * q^99

Introduction

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