LMFDB - Embedded newform 450.4.c.a.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.c (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:	$26.5508595026$
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			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.4.c.a.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +1.00000i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} +1.00000i q^{7} +8.00000i q^{8} -42.0000 q^{11} -67.0000i q^{13} +2.00000 q^{14} +16.0000 q^{16} +54.0000i q^{17} +115.000 q^{19} +84.0000i q^{22} +162.000i q^{23} -134.000 q^{26} -4.00000i q^{28} -210.000 q^{29} -193.000 q^{31} -32.0000i q^{32} +108.000 q^{34} +286.000i q^{37} -230.000i q^{38} -12.0000 q^{41} +263.000i q^{43} +168.000 q^{44} +324.000 q^{46} +414.000i q^{47} +342.000 q^{49} +268.000i q^{52} +192.000i q^{53} -8.00000 q^{56} +420.000i q^{58} +690.000 q^{59} -733.000 q^{61} +386.000i q^{62} -64.0000 q^{64} -299.000i q^{67} -216.000i q^{68} +228.000 q^{71} +938.000i q^{73} +572.000 q^{74} -460.000 q^{76} -42.0000i q^{77} +160.000 q^{79} +24.0000i q^{82} +462.000i q^{83} +526.000 q^{86} -336.000i q^{88} -240.000 q^{89} +67.0000 q^{91} -648.000i q^{92} +828.000 q^{94} +511.000i q^{97} -684.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 84 q^{11} + 4 q^{14} + 32 q^{16} + 230 q^{19} - 268 q^{26} - 420 q^{29} - 386 q^{31} + 216 q^{34} - 24 q^{41} + 336 q^{44} + 648 q^{46} + 684 q^{49} - 16 q^{56} + 1380 q^{59} - 1466 q^{61} - 128 q^{64} + 456 q^{71} + 1144 q^{74} - 920 q^{76} + 320 q^{79} + 1052 q^{86} - 480 q^{89} + 134 q^{91} + 1656 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 84 * q^11 + 4 * q^14 + 32 * q^16 + 230 * q^19 - 268 * q^26 - 420 * q^29 - 386 * q^31 + 216 * q^34 - 24 * q^41 + 336 * q^44 + 648 * q^46 + 684 * q^49 - 16 * q^56 + 1380 * q^59 - 1466 * q^61 - 128 * q^64 + 456 * q^71 + 1144 * q^74 - 920 * q^76 + 320 * q^79 + 1052 * q^86 - 480 * q^89 + 134 * q^91 + 1656 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	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	$0.991405\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	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	$-0.746559\pi$
$13$	− 67.0000i	− 1.42942i	0.699421	−	0.714710i	$-0.253441\pi$
$14$	2.00000	0.0381802
$15$	0	0
$16$	16.0000	0.250000
$17$	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	$-0.255721\pi$
$19$	115.000	1.38857	0.694284	+	0.719701i	$0.255721\pi$
$20$	0	0
$21$	0	0
$22$	84.0000i	0.814039i
$23$	162.000i	1.46867i	0.678789	+	0.734333i	$0.262505\pi$
$23$	162.000i	1.46867i	−0.678789	+	0.734333i	$0.737495\pi$
$24$	0	0
$25$	0	0
$26$	−134.000	−1.01075
$27$	0	0
$28$	− 4.00000i	− 0.0269975i
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	−193.000	−1.11819	−0.559094	−	0.829104i	$-0.688851\pi$
$31$	−193.000	−1.11819	−0.559094	+	0.829104i	$0.688851\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	108.000	0.544760
$35$	0	0
$36$	0	0
$37$	286.000i	1.27076i	0.772200	+	0.635380i	$0.219156\pi$
$37$	286.000i	1.27076i	−0.772200	+	0.635380i	$0.780844\pi$
$38$	− 230.000i	− 0.981866i
$39$	0	0
$40$	0	0
$41$	−12.0000	−0.0457094	−0.0228547	−	0.999739i	$-0.507276\pi$
$41$	−12.0000	−0.0457094	−0.0228547	+	0.999739i	$0.507276\pi$
$42$	0	0
$43$	263.000i	0.932724i	0.884594	+	0.466362i	$0.154436\pi$
$43$	263.000i	0.932724i	−0.884594	+	0.466362i	$0.845564\pi$
$44$	168.000	0.575613
$45$	0	0
$46$	324.000	1.03850
$47$	414.000i	1.28485i	0.766347	+	0.642427i	$0.222072\pi$
$47$	414.000i	1.28485i	−0.766347	+	0.642427i	$0.777928\pi$
$48$	0	0
$49$	342.000	0.997085
$50$	0	0
$51$	0	0
$52$	268.000i	0.714710i
$53$	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
$56$	−8.00000	−0.0190901
$57$	0	0
$58$	420.000i	0.950840i
$59$	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	$0.779380\pi$
$62$	386.000i	0.790678i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	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	$-0.0878842\pi$
$68$	− 216.000i	− 0.385204i
$69$	0	0
$70$	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	$0.270886\pi$
$73$	938.000i	1.50390i	−0.659221	+	0.751949i	$0.729114\pi$
$74$	572.000	0.898563
$75$	0	0
$76$	−460.000	−0.694284
$77$	− 42.0000i	− 0.0621603i
$78$	0	0
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	0	0
$82$	24.0000i	0.0323214i
$83$	462.000i	0.610977i	0.952196	+	0.305488i	$0.0988197\pi$
$83$	462.000i	0.610977i	−0.952196	+	0.305488i	$0.901180\pi$
$84$	0	0
$85$	0	0
$86$	526.000	0.659535
$87$	0	0
$88$	− 336.000i	− 0.407020i
$89$	−240.000	−0.285842	−0.142921	−	0.989734i	$-0.545650\pi$
$89$	−240.000	−0.285842	−0.142921	+	0.989734i	$0.545650\pi$
$90$	0	0
$91$	67.0000	0.0771814
$92$	− 648.000i	− 0.734333i
$93$	0	0
$94$	828.000	0.908529
$95$	0	0
$96$	0	0
$97$	511.000i	0.534889i	0.963573	+	0.267444i	$0.0861791\pi$
$97$	511.000i	0.534889i	−0.963573	+	0.267444i	$0.913821\pi$
$98$	− 684.000i	− 0.705045i
$99$	0	0
$100$	0	0
$101$	−912.000	−0.898489	−0.449245	−	0.893409i	$-0.648307\pi$
$101$	−912.000	−0.898489	−0.449245	+	0.893409i	$0.648307\pi$
$102$	0	0
$103$	668.000i	0.639029i	0.947581	+	0.319515i	$0.103520\pi$
$103$	668.000i	0.639029i	−0.947581	+	0.319515i	$0.896480\pi$
$104$	536.000	0.505376
$105$	0	0
$106$	384.000	0.351862
$107$	− 1296.00i	− 1.17093i	−0.810699	−	0.585463i	$-0.800913\pi$
$107$	− 1296.00i	− 1.17093i	0.810699	−	0.585463i	$-0.199087\pi$
$108$	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
$111$	0	0
$112$	16.0000i	0.0134987i
$113$	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
$116$	840.000	0.672345
$117$	0	0
$118$	− 1380.00i	− 1.07660i
$119$	−54.0000	−0.0415981
$120$	0	0
$121$	433.000	0.325319
$122$	1466.00i	1.08791i
$123$	0	0
$124$	772.000	0.559094
$125$	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	$0.998221\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	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
$134$	−598.000	−0.385517
$135$	0	0
$136$	−432.000	−0.272380
$137$	− 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	$-0.845706\pi$
$139$	−2900.00	−1.76960	−0.884801	+	0.465968i	$0.845706\pi$
$140$	0	0
$141$	0	0
$142$	− 456.000i	− 0.269484i
$143$	2814.00i	1.64558i
$144$	0	0
$145$	0	0
$146$	1876.00	1.06342
$147$	0	0
$148$	− 1144.00i	− 0.635380i
$149$	−2070.00	−1.13813	−0.569064	−	0.822293i	$-0.692694\pi$
$149$	−2070.00	−1.13813	−0.569064	+	0.822293i	$0.692694\pi$
$150$	0	0
$151$	2237.00	1.20559	0.602796	−	0.797895i	$-0.294053\pi$
$151$	2237.00	1.20559	0.602796	+	0.797895i	$0.294053\pi$
$152$	920.000i	0.490933i
$153$	0	0
$154$	−84.0000	−0.0439540
$155$	0	0
$156$	0	0
$157$	241.000i	0.122509i	0.998122	+	0.0612544i	$0.0195101\pi$
$157$	241.000i	0.122509i	−0.998122	+	0.0612544i	$0.980490\pi$
$158$	− 320.000i	− 0.161126i
$159$	0	0
$160$	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	$-0.324742\pi$
$164$	48.0000	0.0228547
$165$	0	0
$166$	924.000	0.432026
$167$	984.000i	0.455953i	0.973667	+	0.227977i	$0.0732110\pi$
$167$	984.000i	0.455953i	−0.973667	+	0.227977i	$0.926789\pi$
$168$	0	0
$169$	−2292.00	−1.04324
$170$	0	0
$171$	0	0
$172$	− 1052.00i	− 0.466362i
$173$	− 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
$176$	−672.000	−0.287806
$177$	0	0
$178$	480.000i	0.202121i
$179$	−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	$0.487121\pi$
$182$	− 134.000i	− 0.0545755i
$183$	0	0
$184$	−1296.00	−0.519252
$185$	0	0
$186$	0	0
$187$	− 2268.00i	− 0.886912i
$188$	− 1656.00i	− 0.642427i
$189$	0	0
$190$	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	$0.282916\pi$
$193$	4163.00i	1.55264i	−0.630340	+	0.776319i	$0.717084\pi$
$194$	1022.00	0.378223
$195$	0	0
$196$	−1368.00	−0.498542
$197$	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	$-0.708842\pi$
$199$	−3425.00	−1.22006	−0.610030	+	0.792379i	$0.708842\pi$
$200$	0	0
$201$	0	0
$202$	1824.00i	0.635328i
$203$	− 210.000i	− 0.0726065i
$204$	0	0
$205$	0	0
$206$	1336.00	0.451862
$207$	0	0
$208$	− 1072.00i	− 0.357355i
$209$	−4830.00	−1.59856
$210$	0	0
$211$	−2443.00	−0.797076	−0.398538	−	0.917152i	$-0.630482\pi$
$211$	−2443.00	−0.797076	−0.398538	+	0.917152i	$0.630482\pi$
$212$	− 768.000i	− 0.248804i
$213$	0	0
$214$	−2592.00	−0.827969
$215$	0	0
$216$	0	0
$217$	− 193.000i	− 0.0603765i
$218$	− 3470.00i	− 1.07806i
$219$	0	0
$220$	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	$0.998901\pi$
$224$	32.0000	0.00954504
$225$	0	0
$226$	2184.00	0.642821
$227$	− 1956.00i	− 0.571913i	−0.958243	−	0.285957i	$-0.907689\pi$
$227$	− 1956.00i	− 0.571913i	0.958243	−	0.285957i	$-0.0923113\pi$
$228$	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
$231$	0	0
$232$	− 1680.00i	− 0.475420i
$233$	− 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
$236$	−2760.00	−0.761274
$237$	0	0
$238$	108.000i	0.0294143i
$239$	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	$0.755582\pi$
$242$	− 866.000i	− 0.230035i
$243$	0	0
$244$	2932.00	0.769271
$245$	0	0
$246$	0	0
$247$	− 7705.00i	− 1.98485i
$248$	− 1544.00i	− 0.395339i
$249$	0	0
$250$	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
$254$	32.0000	0.00790496
$255$	0	0
$256$	256.000	0.0625000
$257$	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
$261$	0	0
$262$	3984.00i	0.939436i
$263$	1812.00i	0.424839i	0.977179	+	0.212420i	$0.0681344\pi$
$263$	1812.00i	0.424839i	−0.977179	+	0.212420i	$0.931866\pi$
$264$	0	0
$265$	0	0
$266$	230.000	0.0530158
$267$	0	0
$268$	1196.00i	0.272602i
$269$	−5190.00	−1.17636	−0.588178	−	0.808731i	$-0.700155\pi$
$269$	−5190.00	−1.17636	−0.588178	+	0.808731i	$0.700155\pi$
$270$	0	0
$271$	4592.00	1.02931	0.514657	−	0.857396i	$-0.327919\pi$
$271$	4592.00	1.02931	0.514657	+	0.857396i	$0.327919\pi$
$272$	864.000i	0.192602i
$273$	0	0
$274$	−4692.00	−1.03450
$275$	0	0
$276$	0	0
$277$	2191.00i	0.475251i	0.971357	+	0.237625i	$0.0763690\pi$
$277$	2191.00i	0.475251i	−0.971357	+	0.237625i	$0.923631\pi$
$278$	5800.00i	1.25130i
$279$	0	0
$280$	0	0
$281$	−7842.00	−1.66482	−0.832410	−	0.554160i	$-0.813040\pi$
$281$	−7842.00	−1.66482	−0.832410	+	0.554160i	$0.813040\pi$
$282$	0	0
$283$	− 247.000i	− 0.0518821i	−0.999663	−	0.0259410i	$-0.991742\pi$
$283$	− 247.000i	− 0.0518821i	0.999663	−	0.0259410i	$-0.00825821\pi$
$284$	−912.000	−0.190554
$285$	0	0
$286$	5628.00	1.16360
$287$	− 12.0000i	− 0.00246808i
$288$	0	0
$289$	1997.00	0.406473
$290$	0	0
$291$	0	0
$292$	− 3752.00i	− 0.751949i
$293$	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
$296$	−2288.00	−0.449281
$297$	0	0
$298$	4140.00i	0.804778i
$299$	10854.0	2.09934
$300$	0	0
$301$	−263.000	−0.0503624
$302$	− 4474.00i	− 0.852483i
$303$	0	0
$304$	1840.00	0.347142
$305$	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	$0.882838\pi$
$308$	168.000i	0.0310802i
$309$	0	0
$310$	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	$-0.893492\pi$
$313$	− 3637.00i	− 0.656790i	0.944540	−	0.328395i	$-0.106508\pi$
$314$	482.000	0.0866269
$315$	0	0
$316$	−640.000	−0.113933
$317$	− 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
$321$	0	0
$322$	324.000i	0.0560740i
$323$	6210.00i	1.06976i
$324$	0	0
$325$	0	0
$326$	−7094.00	−1.20522
$327$	0	0
$328$	− 96.0000i	− 0.0161607i
$329$	−414.000	−0.0693756
$330$	0	0
$331$	5132.00	0.852206	0.426103	−	0.904675i	$-0.359886\pi$
$331$	5132.00	0.852206	0.426103	+	0.904675i	$0.359886\pi$
$332$	− 1848.00i	− 0.305488i
$333$	0	0
$334$	1968.00	0.322408
$335$	0	0
$336$	0	0
$337$	6751.00i	1.09125i	0.838030	+	0.545624i	$0.183707\pi$
$337$	6751.00i	1.09125i	−0.838030	+	0.545624i	$0.816293\pi$
$338$	4584.00i	0.737683i
$339$	0	0
$340$	0	0
$341$	8106.00	1.28729
$342$	0	0
$343$	685.000i	0.107832i
$344$	−2104.00	−0.329768
$345$	0	0
$346$	−7236.00	−1.12431
$347$	− 5226.00i	− 0.808491i	−0.914651	−	0.404246i	$-0.867534\pi$
$347$	− 5226.00i	− 0.808491i	0.914651	−	0.404246i	$-0.132466\pi$
$348$	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
$351$	0	0
$352$	1344.00i	0.203510i
$353$	− 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
$356$	960.000	0.142921
$357$	0	0
$358$	300.000i	0.0442891i
$359$	−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
$362$	− 394.000i	− 0.0572049i
$363$	0	0
$364$	−268.000	−0.0385907
$365$	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	$0.980270\pi$
$368$	2592.00i	0.367167i
$369$	0	0
$370$	0	0
$371$	−192.000	−0.0268683
$372$	0	0
$373$	6383.00i	0.886057i	0.896508	+	0.443028i	$0.146096\pi$
$373$	6383.00i	0.886057i	−0.896508	+	0.443028i	$0.853904\pi$
$374$	−4536.00	−0.627142
$375$	0	0
$376$	−3312.00	−0.454264
$377$	14070.0i	1.92213i
$378$	0	0
$379$	9865.00	1.33702	0.668511	−	0.743703i	$-0.266932\pi$
$379$	9865.00	1.33702	0.668511	+	0.743703i	$0.266932\pi$
$380$	0	0
$381$	0	0
$382$	2604.00i	0.348775i
$383$	− 9828.00i	− 1.31119i	−0.755111	−	0.655597i	$-0.772417\pi$
$383$	− 9828.00i	− 1.31119i	0.755111	−	0.655597i	$-0.227583\pi$
$384$	0	0
$385$	0	0
$386$	8326.00	1.09788
$387$	0	0
$388$	− 2044.00i	− 0.267444i
$389$	12540.0	1.63446	0.817228	−	0.576315i	$-0.195510\pi$
$389$	12540.0	1.63446	0.817228	+	0.576315i	$0.195510\pi$
$390$	0	0
$391$	−8748.00	−1.13147
$392$	2736.00i	0.352523i
$393$	0	0
$394$	6108.00	0.781007
$395$	0	0
$396$	0	0
$397$	1381.00i	0.174585i	0.996183	+	0.0872927i	$0.0278215\pi$
$397$	1381.00i	0.174585i	−0.996183	+	0.0872927i	$0.972178\pi$
$398$	6850.00i	0.862712i
$399$	0	0
$400$	0	0
$401$	−14232.0	−1.77235	−0.886175	−	0.463351i	$-0.846647\pi$
$401$	−14232.0	−1.77235	−0.886175	+	0.463351i	$0.846647\pi$
$402$	0	0
$403$	12931.0i	1.59836i
$404$	3648.00	0.449245
$405$	0	0
$406$	−420.000	−0.0513405
$407$	− 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
$411$	0	0
$412$	− 2672.00i	− 0.319515i
$413$	690.000i	0.0822099i
$414$	0	0
$415$	0	0
$416$	−2144.00	−0.252688
$417$	0	0
$418$	9660.00i	1.13035i
$419$	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	$0.727867\pi$
$422$	4886.00i	0.563618i
$423$	0	0
$424$	−1536.00	−0.175931
$425$	0	0
$426$	0	0
$427$	− 733.000i	− 0.0830734i
$428$	5184.00i	0.585463i
$429$	0	0
$430$	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	$0.0205575\pi$
$433$	1163.00i	0.129077i	−0.997915	+	0.0645384i	$0.979443\pi$
$434$	−386.000	−0.0426926
$435$	0	0
$436$	−6940.00	−0.762307
$437$	18630.0i	2.03934i
$438$	0	0
$439$	−6695.00	−0.727870	−0.363935	−	0.931424i	$-0.618567\pi$
$439$	−6695.00	−0.727870	−0.363935	+	0.931424i	$0.618567\pi$
$440$	0	0
$441$	0	0
$442$	− 7236.00i	− 0.778691i
$443$	− 16368.0i	− 1.75546i	−0.479159	−	0.877728i	$-0.659058\pi$
$443$	− 16368.0i	− 1.75546i	0.479159	−	0.877728i	$-0.340942\pi$
$444$	0	0
$445$	0	0
$446$	46.0000	0.00488377
$447$	0	0
$448$	− 64.0000i	− 0.00674937i
$449$	16380.0	1.72165	0.860824	−	0.508903i	$-0.169949\pi$
$449$	16380.0	1.72165	0.860824	+	0.508903i	$0.169949\pi$
$450$	0	0
$451$	504.000	0.0526218
$452$	− 4368.00i	− 0.454543i
$453$	0	0
$454$	−3912.00	−0.404404
$455$	0	0
$456$	0	0
$457$	13786.0i	1.41112i	0.708650	+	0.705560i	$0.249304\pi$
$457$	13786.0i	1.41112i	−0.708650	+	0.705560i	$0.750696\pi$
$458$	3610.00i	0.368306i
$459$	0	0
$460$	0	0
$461$	−11832.0	−1.19538	−0.597691	−	0.801726i	$-0.703915\pi$
$461$	−11832.0	−1.19538	−0.597691	+	0.801726i	$0.703915\pi$
$462$	0	0
$463$	3008.00i	0.301930i	0.988539	+	0.150965i	$0.0482381\pi$
$463$	3008.00i	0.301930i	−0.988539	+	0.150965i	$0.951762\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	−6936.00	−0.689494
$467$	4434.00i	0.439360i	0.975572	+	0.219680i	$0.0705013\pi$
$467$	4434.00i	0.439360i	−0.975572	+	0.219680i	$0.929499\pi$
$468$	0	0
$469$	299.000	0.0294382
$470$	0	0
$471$	0	0
$472$	5520.00i	0.538302i
$473$	− 11046.0i	− 1.07378i
$474$	0	0
$475$	0	0
$476$	216.000	0.0207990
$477$	0	0
$478$	− 5280.00i	− 0.505233i
$479$	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
$482$	10766.0i	1.01738i
$483$	0	0
$484$	−1732.00	−0.162660
$485$	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	$0.867825\pi$
$488$	− 5864.00i	− 0.543957i
$489$	0	0
$490$	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
$494$	−15410.0	−1.40350
$495$	0	0
$496$	−3088.00	−0.279547
$497$	228.000i	0.0205779i
$498$	0	0
$499$	8875.00	0.796192	0.398096	−	0.917344i	$-0.369671\pi$
$499$	8875.00	0.796192	0.398096	+	0.917344i	$0.369671\pi$
$500$	0	0
$501$	0	0
$502$	− 10056.0i	− 0.894066i
$503$	10452.0i	0.926504i	0.886227	+	0.463252i	$0.153318\pi$
$503$	10452.0i	0.926504i	−0.886227	+	0.463252i	$0.846682\pi$
$504$	0	0
$505$	0	0
$506$	−13608.0	−1.19555
$507$	0	0
$508$	− 64.0000i	− 0.00558965i
$509$	−19770.0	−1.72159	−0.860796	−	0.508951i	$-0.830033\pi$
$509$	−19770.0	−1.72159	−0.860796	+	0.508951i	$0.830033\pi$
$510$	0	0
$511$	−938.000	−0.0812029
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	1128.00	0.0967976
$515$	0	0
$516$	0	0
$517$	− 17388.0i	− 1.47916i
$518$	572.000i	0.0485178i
$519$	0	0
$520$	0	0
$521$	11238.0	0.945001	0.472501	−	0.881330i	$-0.343351\pi$
$521$	11238.0	0.945001	0.472501	+	0.881330i	$0.343351\pi$
$522$	0	0
$523$	− 7447.00i	− 0.622628i	−0.950307	−	0.311314i	$-0.899231\pi$
$523$	− 7447.00i	− 0.622628i	0.950307	−	0.311314i	$-0.100769\pi$
$524$	7968.00	0.664282
$525$	0	0
$526$	3624.00	0.300407
$527$	− 10422.0i	− 0.861460i
$528$	0	0
$529$	−14077.0	−1.15698
$530$	0	0
$531$	0	0
$532$	− 460.000i	− 0.0374878i
$533$	804.000i	0.0653379i
$534$	0	0
$535$	0	0
$536$	2392.00	0.192759
$537$	0	0
$538$	10380.0i	0.831810i
$539$	−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	$0.746929\pi$
$542$	− 9184.00i	− 0.727835i
$543$	0	0
$544$	1728.00	0.136190
$545$	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	$0.870889\pi$
$548$	9384.00i	0.731505i
$549$	0	0
$550$	0	0
$551$	−24150.0	−1.86720
$552$	0	0
$553$	160.000i	0.0123036i
$554$	4382.00	0.336053
$555$	0	0
$556$	11600.0	0.884801
$557$	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
$561$	0	0
$562$	15684.0i	1.17721i
$563$	10242.0i	0.766694i	0.923604	+	0.383347i	$0.125229\pi$
$563$	10242.0i	0.766694i	−0.923604	+	0.383347i	$0.874771\pi$
$564$	0	0
$565$	0	0
$566$	−494.000	−0.0366862
$567$	0	0
$568$	1824.00i	0.134742i
$569$	−6750.00	−0.497319	−0.248660	−	0.968591i	$-0.579990\pi$
$569$	−6750.00	−0.497319	−0.248660	+	0.968591i	$0.579990\pi$
$570$	0	0
$571$	17117.0	1.25451	0.627254	−	0.778815i	$-0.284179\pi$
$571$	17117.0	1.25451	0.627254	+	0.778815i	$0.284179\pi$
$572$	− 11256.0i	− 0.822792i
$573$	0	0
$574$	−24.0000	−0.00174519
$575$	0	0
$576$	0	0
$577$	301.000i	0.0217171i	0.999941	+	0.0108586i	$0.00345646\pi$
$577$	301.000i	0.0217171i	−0.999941	+	0.0108586i	$0.996544\pi$
$578$	− 3994.00i	− 0.287420i
$579$	0	0
$580$	0	0
$581$	−462.000	−0.0329897
$582$	0	0
$583$	− 8064.00i	− 0.572859i
$584$	−7504.00	−0.531708
$585$	0	0
$586$	10884.0	0.767259
$587$	− 15456.0i	− 1.08678i	−0.839482	−	0.543388i	$-0.817141\pi$
$587$	− 15456.0i	− 1.08678i	0.839482	−	0.543388i	$-0.182859\pi$
$588$	0	0
$589$	−22195.0	−1.55268
$590$	0	0
$591$	0	0
$592$	4576.00i	0.317690i
$593$	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
$596$	8280.00	0.569064
$597$	0	0
$598$	− 21708.0i	− 1.48446i
$599$	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	$0.334663\pi$
$602$	526.000i	0.0356116i
$603$	0	0
$604$	−8948.00	−0.602796
$605$	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	$-0.181990\pi$
$608$	− 3680.00i	− 0.245467i
$609$	0	0
$610$	0	0
$611$	27738.0	1.83659
$612$	0	0
$613$	− 18502.0i	− 1.21907i	−0.792760	−	0.609534i	$-0.791357\pi$
$613$	− 18502.0i	− 1.21907i	0.792760	−	0.609534i	$-0.208643\pi$
$614$	7742.00	0.508863
$615$	0	0
$616$	336.000	0.0219770
$617$	− 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	$-0.395895\pi$
$619$	9895.00	0.642510	0.321255	+	0.946993i	$0.395895\pi$
$620$	0	0
$621$	0	0
$622$	− 11436.0i	− 0.737206i
$623$	− 240.000i	− 0.0154340i
$624$	0	0
$625$	0	0
$626$	−7274.00	−0.464421
$627$	0	0
$628$	− 964.000i	− 0.0612544i
$629$	−15444.0	−0.979003
$630$	0	0
$631$	467.000	0.0294627	0.0147314	−	0.999891i	$-0.495311\pi$
$631$	467.000	0.0294627	0.0147314	+	0.999891i	$0.495311\pi$
$632$	1280.00i	0.0805628i
$633$	0	0
$634$	−2592.00	−0.162368
$635$	0	0
$636$	0	0
$637$	− 22914.0i	− 1.42525i
$638$	− 17640.0i	− 1.09463i
$639$	0	0
$640$	0	0
$641$	−30612.0	−1.88627	−0.943137	−	0.332405i	$-0.892140\pi$
$641$	−30612.0	−1.88627	−0.943137	+	0.332405i	$0.892140\pi$
$642$	0	0
$643$	− 1852.00i	− 0.113586i	−0.998386	−	0.0567930i	$-0.981913\pi$
$643$	− 1852.00i	− 0.113586i	0.998386	−	0.0567930i	$-0.0180875\pi$
$644$	648.000	0.0396503
$645$	0	0
$646$	12420.0	0.756437
$647$	− 21156.0i	− 1.28551i	−0.766070	−	0.642757i	$-0.777790\pi$
$647$	− 21156.0i	− 1.28551i	0.766070	−	0.642757i	$-0.222210\pi$
$648$	0	0
$649$	−28980.0	−1.75280
$650$	0	0
$651$	0	0
$652$	14188.0i	0.852216i
$653$	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
$656$	−192.000	−0.0114273
$657$	0	0
$658$	828.000i	0.0490559i
$659$	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	$0.702089\pi$
$662$	− 10264.0i	− 0.602601i
$663$	0	0
$664$	−3696.00	−0.216013
$665$	0	0
$666$	0	0
$667$	− 34020.0i	− 1.97490i
$668$	− 3936.00i	− 0.227977i
$669$	0	0
$670$	0	0
$671$	30786.0	1.77121
$672$	0	0
$673$	− 16882.0i	− 0.966944i	−0.875360	−	0.483472i	$-0.839376\pi$
$673$	− 16882.0i	− 0.966944i	0.875360	−	0.483472i	$-0.160624\pi$
$674$	13502.0	0.771628
$675$	0	0
$676$	9168.00	0.521620
$677$	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
$681$	0	0
$682$	− 16212.0i	− 0.910249i
$683$	8712.00i	0.488075i	0.969766	+	0.244038i	$0.0784720\pi$
$683$	8712.00i	0.488075i	−0.969766	+	0.244038i	$0.921528\pi$
$684$	0	0
$685$	0	0
$686$	1370.00	0.0762490
$687$	0	0
$688$	4208.00i	0.233181i
$689$	12864.0	0.711291
$690$	0	0
$691$	−14128.0	−0.777792	−0.388896	−	0.921282i	$-0.627144\pi$
$691$	−14128.0	−0.777792	−0.388896	+	0.921282i	$0.627144\pi$
$692$	14472.0i	0.795004i
$693$	0	0
$694$	−10452.0	−0.571689
$695$	0	0
$696$	0	0
$697$	− 648.000i	− 0.0352148i
$698$	− 12380.0i	− 0.671332i
$699$	0	0
$700$	0	0
$701$	28278.0	1.52360	0.761801	−	0.647811i	$-0.224315\pi$
$701$	28278.0	1.52360	0.761801	+	0.647811i	$0.224315\pi$
$702$	0	0
$703$	32890.0i	1.76454i
$704$	2688.00	0.143903
$705$	0	0
$706$	−13236.0	−0.705586
$707$	− 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
$711$	0	0
$712$	− 1920.00i	− 0.101060i
$713$	− 31266.0i	− 1.64225i
$714$	0	0
$715$	0	0
$716$	600.000	0.0313171
$717$	0	0
$718$	6840.00i	0.355524i
$719$	−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
$722$	− 12732.0i	− 0.656283i
$723$	0	0
$724$	−788.000	−0.0404500
$725$	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	$0.985372\pi$
$728$	536.000i	0.0272877i
$729$	0	0
$730$	0	0
$731$	−14202.0	−0.718577
$732$	0	0
$733$	− 7882.00i	− 0.397174i	−0.980083	−	0.198587i	$-0.936365\pi$
$733$	− 7882.00i	− 0.397174i	0.980083	−	0.198587i	$-0.0636352\pi$
$734$	1742.00	0.0876000
$735$	0	0
$736$	5184.00	0.259626
$737$	12558.0i	0.627652i
$738$	0	0
$739$	−33860.0	−1.68547	−0.842734	−	0.538331i	$-0.819055\pi$
$739$	−33860.0	−1.68547	−0.842734	+	0.538331i	$0.819055\pi$
$740$	0	0
$741$	0	0
$742$	384.000i	0.0189988i
$743$	20652.0i	1.01972i	0.860259	+	0.509858i	$0.170302\pi$
$743$	20652.0i	1.01972i	−0.860259	+	0.509858i	$0.829698\pi$
$744$	0	0
$745$	0	0
$746$	12766.0	0.626537
$747$	0	0
$748$	9072.00i	0.443456i
$749$	1296.00	0.0632240
$750$	0	0
$751$	7472.00	0.363059	0.181529	−	0.983386i	$-0.441895\pi$
$751$	7472.00	0.363059	0.181529	+	0.983386i	$0.441895\pi$
$752$	6624.00i	0.321213i
$753$	0	0
$754$	28140.0	1.35915
$755$	0	0
$756$	0	0
$757$	32251.0i	1.54846i	0.632906	+	0.774229i	$0.281862\pi$
$757$	32251.0i	1.54846i	−0.632906	+	0.774229i	$0.718138\pi$
$758$	− 19730.0i	− 0.945417i
$759$	0	0
$760$	0	0
$761$	−16812.0	−0.800834	−0.400417	−	0.916333i	$-0.631135\pi$
$761$	−16812.0	−0.800834	−0.400417	+	0.916333i	$0.631135\pi$
$762$	0	0
$763$	1735.00i	0.0823214i
$764$	5208.00	0.246622
$765$	0	0
$766$	−19656.0	−0.927154
$767$	− 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
$771$	0	0
$772$	− 16652.0i	− 0.776319i
$773$	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
$776$	−4088.00	−0.189112
$777$	0	0
$778$	− 25080.0i	− 1.15573i
$779$	−1380.00	−0.0634706
$780$	0	0
$781$	−9576.00	−0.438740
$782$	17496.0i	0.800071i
$783$	0	0
$784$	5472.00	0.249271
$785$	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	$-0.136251\pi$
$788$	− 12216.0i	− 0.552255i
$789$	0	0
$790$	0	0
$791$	−1092.00	−0.0490860
$792$	0	0
$793$	49111.0i	2.19922i
$794$	2762.00	0.123451
$795$	0	0
$796$	13700.0	0.610030
$797$	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
$801$	0	0
$802$	28464.0i	1.25324i
$803$	− 39396.0i	− 1.73133i
$804$	0	0
$805$	0	0
$806$	25862.0	1.13021
$807$	0	0
$808$	− 7296.00i	− 0.317664i
$809$	−24000.0	−1.04301	−0.521505	−	0.853248i	$-0.674629\pi$
$809$	−24000.0	−1.04301	−0.521505	+	0.853248i	$0.674629\pi$
$810$	0	0
$811$	5117.00	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$811$	5117.00	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$812$	840.000i	0.0363032i
$813$	0	0
$814$	−24024.0	−1.03445
$815$	0	0
$816$	0	0
$817$	30245.0i	1.29515i
$818$	5290.00i	0.226113i
$819$	0	0
$820$	0	0
$821$	−13542.0	−0.575663	−0.287831	−	0.957681i	$-0.592934\pi$
$821$	−13542.0	−0.575663	−0.287831	+	0.957681i	$0.592934\pi$
$822$	0	0
$823$	1283.00i	0.0543409i	0.999631	+	0.0271705i	$0.00864969\pi$
$823$	1283.00i	0.0543409i	−0.999631	+	0.0271705i	$0.991350\pi$
$824$	−5344.00	−0.225931
$825$	0	0
$826$	1380.00	0.0581312
$827$	16344.0i	0.687227i	0.939111	+	0.343613i	$0.111651\pi$
$827$	16344.0i	0.687227i	−0.939111	+	0.343613i	$0.888349\pi$
$828$	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
$831$	0	0
$832$	4288.00i	0.178677i
$833$	18468.0i	0.768161i
$834$	0	0
$835$	0	0
$836$	19320.0	0.799278
$837$	0	0
$838$	− 6000.00i	− 0.247335i
$839$	−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
$842$	22676.0i	0.928108i
$843$	0	0
$844$	9772.00	0.398538
$845$	0	0
$846$	0	0
$847$	433.000i	0.0175656i
$848$	3072.00i	0.124402i
$849$	0	0
$850$	0	0
$851$	−46332.0	−1.86632
$852$	0	0
$853$	24743.0i	0.993182i	0.867985	+	0.496591i	$0.165415\pi$
$853$	24743.0i	0.993182i	−0.867985	+	0.496591i	$0.834585\pi$
$854$	−1466.00	−0.0587418
$855$	0	0
$856$	10368.0	0.413985
$857$	− 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	$-0.264039\pi$
$859$	34000.0	1.35048	0.675242	+	0.737597i	$0.264039\pi$
$860$	0	0
$861$	0	0
$862$	− 6516.00i	− 0.257466i
$863$	37032.0i	1.46070i	0.683073	+	0.730350i	$0.260643\pi$
$863$	37032.0i	1.46070i	−0.683073	+	0.730350i	$0.739357\pi$
$864$	0	0
$865$	0	0
$866$	2326.00	0.0912710
$867$	0	0
$868$	772.000i	0.0301882i
$869$	−6720.00	−0.262325
$870$	0	0
$871$	−20033.0	−0.779325
$872$	13880.0i	0.539032i
$873$	0	0
$874$	37260.0	1.44203
$875$	0	0
$876$	0	0
$877$	− 2519.00i	− 0.0969904i	−0.998823	−	0.0484952i	$-0.984557\pi$
$877$	− 2519.00i	− 0.0969904i	0.998823	−	0.0484952i	$-0.0154426\pi$
$878$	13390.0i	0.514682i
$879$	0	0
$880$	0	0
$881$	−43992.0	−1.68232	−0.841162	−	0.540783i	$-0.818128\pi$
$881$	−43992.0	−1.68232	−0.841162	+	0.540783i	$0.818128\pi$
$882$	0	0
$883$	− 19177.0i	− 0.730869i	−0.930837	−	0.365435i	$-0.880920\pi$
$883$	− 19177.0i	− 0.730869i	0.930837	−	0.365435i	$-0.119080\pi$
$884$	−14472.0	−0.550618
$885$	0	0
$886$	−32736.0	−1.24130
$887$	44994.0i	1.70321i	0.524181	+	0.851607i	$0.324372\pi$
$887$	44994.0i	1.70321i	−0.524181	+	0.851607i	$0.675628\pi$
$888$	0	0
$889$	−16.0000	−0.000603625 0
$890$	0	0
$891$	0	0
$892$	− 92.0000i	− 0.00345335i
$893$	47610.0i	1.78411i
$894$	0	0
$895$	0	0
$896$	−128.000	−0.00477252
$897$	0	0
$898$	− 32760.0i	− 1.21739i
$899$	40530.0	1.50362
$900$	0	0
$901$	−10368.0	−0.383361
$902$	− 1008.00i	− 0.0372092i
$903$	0	0
$904$	−8736.00	−0.321410
$905$	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	$0.591369\pi$
$908$	7824.00i	0.285957i
$909$	0	0
$910$	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
$914$	27572.0	0.997813
$915$	0	0
$916$	7220.00	0.260432
$917$	− 1992.00i	− 0.0717357i
$918$	0	0
$919$	−4085.00	−0.146629	−0.0733143	−	0.997309i	$-0.523358\pi$
$919$	−4085.00	−0.146629	−0.0733143	+	0.997309i	$0.523358\pi$
$920$	0	0
$921$	0	0
$922$	23664.0i	0.845263i
$923$	− 15276.0i	− 0.544762i
$924$	0	0
$925$	0	0
$926$	6016.00	0.213497
$927$	0	0
$928$	6720.00i	0.237710i
$929$	−3030.00	−0.107009	−0.0535043	−	0.998568i	$-0.517039\pi$
$929$	−3030.00	−0.107009	−0.0535043	+	0.998568i	$0.517039\pi$
$930$	0	0
$931$	39330.0	1.38452
$932$	13872.0i	0.487546i
$933$	0	0
$934$	8868.00	0.310674
$935$	0	0
$936$	0	0
$937$	− 5759.00i	− 0.200788i	−0.994948	−	0.100394i	$-0.967990\pi$
$937$	− 5759.00i	− 0.200788i	0.994948	−	0.100394i	$-0.0320103\pi$
$938$	− 598.000i	− 0.0208160i
$939$	0	0
$940$	0	0
$941$	258.000	0.00893790	0.00446895	−	0.999990i	$-0.498577\pi$
$941$	258.000	0.00893790	0.00446895	+	0.999990i	$0.498577\pi$
$942$	0	0
$943$	− 1944.00i	− 0.0671319i
$944$	11040.0	0.380637
$945$	0	0
$946$	−22092.0	−0.759274
$947$	1374.00i	0.0471478i	0.999722	+	0.0235739i	$0.00750451\pi$
$947$	1374.00i	0.0471478i	−0.999722	+	0.0235739i	$0.992495\pi$
$948$	0	0
$949$	62846.0	2.14970
$950$	0	0
$951$	0	0
$952$	− 432.000i	− 0.0147071i
$953$	− 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
$956$	−10560.0	−0.357254
$957$	0	0
$958$	− 14820.0i	− 0.499804i
$959$	2346.00	0.0789951
$960$	0	0
$961$	7458.00	0.250344
$962$	− 38324.0i	− 1.28442i
$963$	0	0
$964$	21532.0	0.719397
$965$	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	$0.882974\pi$
$968$	3464.00i	0.115018i
$969$	0	0
$970$	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
$974$	17342.0	0.570507
$975$	0	0
$976$	−11728.0	−0.384635
$977$	− 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
$981$	0	0
$982$	− 38736.0i	− 1.25877i
$983$	38772.0i	1.25802i	0.777397	+	0.629011i	$0.216540\pi$
$983$	38772.0i	1.25802i	−0.777397	+	0.629011i	$0.783460\pi$
$984$	0	0
$985$	0	0
$986$	−22680.0	−0.732534
$987$	0	0
$988$	30820.0i	0.992424i
$989$	−42606.0	−1.36986
$990$	0	0
$991$	−23053.0	−0.738953	−0.369477	−	0.929240i	$-0.620463\pi$
$991$	−23053.0	−0.738953	−0.369477	+	0.929240i	$0.620463\pi$
$992$	6176.00i	0.197670i
$993$	0	0
$994$	456.000	0.0145507
$995$	0	0
$996$	0	0
$997$	10366.0i	0.329282i	0.986354	+	0.164641i	$0.0526466\pi$
$997$	10366.0i	0.329282i	−0.986354	+	0.164641i	$0.947353\pi$
$998$	− 17750.0i	− 0.562992i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.a.1.1	✓	1	15.2	even	4
150.4.a.h.1.1	yes	1	15.8	even	4
150.4.c.e.49.1		2	15.14	odd	2
150.4.c.e.49.2		2	3.2	odd	2
450.4.a.f.1.1		1	5.3	odd	4
450.4.a.o.1.1		1	5.2	odd	4
450.4.c.a.199.1		2	1.1	even	1	trivial
450.4.c.a.199.2		2	5.4	even	2	inner
1200.4.a.i.1.1		1	60.23	odd	4
1200.4.a.bb.1.1		1	60.47	odd	4
1200.4.f.c.49.1		2	60.59	even	2
1200.4.f.c.49.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +1.00000i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +1.00000i q^{7} +8.00000i q^{8} -42.0000 q^{11} -67.0000i q^{13} +2.00000 q^{14} +16.0000 q^{16} +54.0000i q^{17} +115.000 q^{19} +84.0000i q^{22} +162.000i q^{23} -134.000 q^{26} -4.00000i q^{28} -210.000 q^{29} -193.000 q^{31} -32.0000i q^{32} +108.000 q^{34} +286.000i q^{37} -230.000i q^{38} -12.0000 q^{41} +263.000i q^{43} +168.000 q^{44} +324.000 q^{46} +414.000i q^{47} +342.000 q^{49} +268.000i q^{52} +192.000i q^{53} -8.00000 q^{56} +420.000i q^{58} +690.000 q^{59} -733.000 q^{61} +386.000i q^{62} -64.0000 q^{64} -299.000i q^{67} -216.000i q^{68} +228.000 q^{71} +938.000i q^{73} +572.000 q^{74} -460.000 q^{76} -42.0000i q^{77} +160.000 q^{79} +24.0000i q^{82} +462.000i q^{83} +526.000 q^{86} -336.000i q^{88} -240.000 q^{89} +67.0000 q^{91} -648.000i q^{92} +828.000 q^{94} +511.000i q^{97} -684.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 84 q^{11} + 4 q^{14} + 32 q^{16} + 230 q^{19} - 268 q^{26} - 420 q^{29} - 386 q^{31} + 216 q^{34} - 24 q^{41} + 336 q^{44} + 648 q^{46} + 684 q^{49} - 16 q^{56} + 1380 q^{59} - 1466 q^{61} - 128 q^{64} + 456 q^{71} + 1144 q^{74} - 920 q^{76} + 320 q^{79} + 1052 q^{86} - 480 q^{89} + 134 q^{91} + 1656 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 84 * q^11 + 4 * q^14 + 32 * q^16 + 230 * q^19 - 268 * q^26 - 420 * q^29 - 386 * q^31 + 216 * q^34 - 24 * q^41 + 336 * q^44 + 648 * q^46 + 684 * q^49 - 16 * q^56 + 1380 * q^59 - 1466 * q^61 - 128 * q^64 + 456 * q^71 + 1144 * q^74 - 920 * q^76 + 320 * q^79 + 1052 * q^86 - 480 * q^89 + 134 * q^91 + 1656 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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