LMFDB - Embedded newform 450.2.c.d.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,2,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, 2, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$3.59326809096$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
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			450.2.c.d.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +6.00000 q^{11} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +4.00000 q^{19} -6.00000i q^{22} +4.00000 q^{26} -2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +8.00000i q^{37} -4.00000i q^{38} -8.00000i q^{43} -6.00000 q^{44} +3.00000 q^{49} -4.00000i q^{52} +6.00000i q^{53} -2.00000 q^{56} -6.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} -12.0000 q^{71} +10.0000i q^{73} +8.00000 q^{74} -4.00000 q^{76} +12.0000i q^{77} +4.00000 q^{79} -12.0000i q^{83} -8.00000 q^{86} +6.00000i q^{88} -12.0000 q^{89} -8.00000 q^{91} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 12 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{26} + 12 q^{29} - 8 q^{31} - 12 q^{34} - 12 q^{44} + 6 q^{49} - 4 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 24 q^{71} + 16 q^{74} - 8 q^{76} + 8 q^{79} - 16 q^{86} - 24 q^{89} - 16 q^{91}+O(q^{100}) $ 2 * q - 2 * q^4 + 12 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^26 + 12 * q^29 - 8 * q^31 - 12 * q^34 - 12 * q^44 + 6 * q^49 - 4 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 - 24 * q^71 + 16 * q^74 - 8 * q^76 + 8 * q^79 - 16 * q^86 - 24 * q^89 - 16 * q^91

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 6.00000i	− 1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	4.00000	0.784465
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 4.00000i	− 0.554700i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−4.00000	−0.458831
$77$	12.0000i	1.36753i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	6.00000i	0.639602i
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	6.00000i	0.552345i
$119$	12.0000	1.10004
$120$	0	0
$121$	25.0000	2.27273
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	24.0000i	2.00698i
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	− 8.00000i	− 0.657596i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000i	0.324443i
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	12.0000i	0.899438i
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	8.00000i	0.592999i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 36.0000i	− 2.63258i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	12.0000i	0.842235i
$204$	0	0
$205$	0	0
$206$	−2.00000	−0.139347
$207$	0	0
$208$	4.00000i	0.277350i
$209$	24.0000	1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	− 12.0000i	− 0.777844i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 25.0000i	− 1.60706i
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	6.00000i	0.370681i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	20.0000i	1.19952i
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	24.0000	1.41915
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	16.0000i	0.920697i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	− 12.0000i	− 0.683763i
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	−20.0000	−1.10770
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	3.00000i	0.163178i
$339$	0	0
$340$	0	0
$341$	−24.0000	−1.29967
$342$	0	0
$343$	20.0000i	1.07990i
$344$	8.00000	0.431331
$345$	0	0
$346$	−18.0000	−0.967686
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	0	0
$352$	− 6.00000i	− 0.319801i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	6.00000i	0.317110i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	0	0
$364$	8.00000	0.419314
$365$	0	0
$366$	0	0
$367$	26.0000i	1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$	26.0000i	1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	28.0000i	1.44979i	0.688862	+	0.724893i	$0.258111\pi$
$373$	28.0000i	1.44979i	−0.688862	+	0.724893i	$0.741889\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	− 12.0000i	− 0.613973i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	− 16.0000i	− 0.803017i	−0.915855	−	0.401508i	$-0.868486\pi$
$397$	− 16.0000i	− 0.803017i	0.915855	−	0.401508i	$-0.131514\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	6.00000	0.298511
$405$	0	0
$406$	12.0000	0.595550
$407$	48.0000i	2.37927i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	2.00000i	0.0985329i
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	− 24.0000i	− 1.17388i
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 38.0000i	− 1.82616i	−0.407777	−	0.913082i	$-0.633696\pi$
$433$	− 38.0000i	− 1.82616i	0.407777	−	0.913082i	$-0.366304\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	− 24.0000i	− 1.14156i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	0	0
$446$	−26.0000	−1.23114
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	0	0
$452$	6.00000i	0.282216i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	34.0000i	1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$	34.0000i	1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	− 48.0000i	− 2.20704i
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	0	0
$478$	− 12.0000i	− 0.548867i
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−32.0000	−1.45907
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	0	0
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	2.00000i	0.0905357i
$489$	0	0
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	− 36.0000i	− 1.62136i
$494$	16.0000	0.719874
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 18.0000i	− 0.803379i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	0	0
$518$	16.0000i	0.703000i
$519$	0	0
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−24.0000	−1.04645
$527$	24.0000i	1.04546i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	18.0000i	0.776035i
$539$	18.0000	0.775315
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	− 8.00000i	− 0.343629i
$543$	0	0
$544$	−6.00000	−0.257248
$545$	0	0
$546$	0	0
$547$	− 40.0000i	− 1.71028i	−0.518400	−	0.855138i	$-0.673472\pi$
$547$	− 40.0000i	− 1.71028i	0.518400	−	0.855138i	$-0.326528\pi$
$548$	− 6.00000i	− 0.256307i
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	8.00000i	0.340195i
$554$	8.00000	0.339887
$555$	0	0
$556$	20.0000	0.848189
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	0	0
$562$	− 12.0000i	− 0.506189i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	− 24.0000i	− 1.00349i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.0000i	− 0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$	− 22.0000i	− 0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$	19.0000i	0.790296i
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	36.0000i	1.49097i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	18.0000	0.743573
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	8.00000i	0.328798i
$593$	18.0000i	0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$	18.0000i	0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 16.0000i	− 0.652111i
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	38.0000i	1.54237i	0.636610	+	0.771186i	$0.280336\pi$
$607$	38.0000i	1.54237i	−0.636610	+	0.771186i	$0.719664\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 8.00000i	− 0.323117i	−0.986863	−	0.161558i	$-0.948348\pi$
$613$	− 8.00000i	− 0.323117i	0.986863	−	0.161558i	$-0.0516520\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	−12.0000	−0.483494
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	12.0000i	0.481156i
$623$	− 24.0000i	− 0.961540i
$624$	0	0
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	4.00000i	0.159617i
$629$	48.0000	1.91389
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	− 36.0000i	− 1.42525i
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	− 48.0000i	− 1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$	− 48.0000i	− 1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	24.0000i	0.928588i
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	3.00000	0.115385
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	24.0000i	0.919007i
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 22.0000i	− 0.832712i
$699$	0	0
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	32.0000i	1.20690i
$704$	−6.00000	−0.226134
$705$	0	0
$706$	30.0000	1.12906
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	0	0
$712$	− 12.0000i	− 0.449719i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	24.0000i	0.895672i
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	− 10.0000i	− 0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$	− 10.0000i	− 0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$	− 8.00000i	− 0.296500i
$729$	0	0
$730$	0	0
$731$	−48.0000	−1.77534
$732$	0	0
$733$	− 32.0000i	− 1.18195i	−0.806691	−	0.590973i	$-0.798744\pi$
$733$	− 32.0000i	− 1.18195i	0.806691	−	0.590973i	$-0.201256\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	12.0000i	0.440534i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	28.0000	1.02515
$747$	0	0
$748$	36.0000i	1.31629i
$749$	−24.0000	−0.876941
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	0	0
$754$	24.0000	0.874028
$755$	0	0
$756$	0	0
$757$	− 40.0000i	− 1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$	− 40.0000i	− 1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	0	0
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	− 10.0000i	− 0.359908i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	0	0
$780$	0	0
$781$	−72.0000	−2.57636
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	8.00000i	0.284088i
$794$	−16.0000	−0.567819
$795$	0	0
$796$	−16.0000	−0.567105
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	36.0000i	1.27120i
$803$	60.0000i	2.11735i
$804$	0	0
$805$	0	0
$806$	−16.0000	−0.563576
$807$	0	0
$808$	− 6.00000i	− 0.211079i
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	− 12.0000i	− 0.421117i
$813$	0	0
$814$	48.0000	1.68240
$815$	0	0
$816$	0	0
$817$	− 32.0000i	− 1.11954i
$818$	− 22.0000i	− 0.769212i
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	2.00000	0.0696733
$825$	0	0
$826$	−12.0000	−0.417533
$827$	24.0000i	0.834562i	0.908778	+	0.417281i	$0.137017\pi$
$827$	24.0000i	0.834562i	−0.908778	+	0.417281i	$0.862983\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	0	0
$832$	− 4.00000i	− 0.138675i
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	− 6.00000i	− 0.207267i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 14.0000i	− 0.482472i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	50.0000i	1.71802i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 8.00000i	− 0.273915i	−0.990577	−	0.136957i	$-0.956268\pi$
$853$	− 8.00000i	− 0.273915i	0.990577	−	0.136957i	$-0.0437323\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	−12.0000	−0.410152
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	−38.0000	−1.29129
$867$	0	0
$868$	8.00000i	0.271538i
$869$	24.0000	0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 28.0000i	− 0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$	− 28.0000i	− 0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	26.0000i	0.870544i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	− 12.0000i	− 0.400445i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	0	0
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	− 72.0000i	− 2.38285i
$914$	26.0000	0.860004
$915$	0	0
$916$	14.0000	0.462573
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	0	0
$922$	18.0000i	0.592798i
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	34.0000	1.11731
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	− 8.00000i	− 0.261209i
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−48.0000	−1.56061
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	0	0
$949$	−40.0000	−1.29845
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	− 30.0000i	− 0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$	− 30.0000i	− 0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$	0	0
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	36.0000i	1.16311i
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	32.0000i	1.03172i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	25.0000i	0.803530i
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	− 40.0000i	− 1.28234i
$974$	38.0000	1.21760
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	0	0
$982$	6.00000i	0.191468i
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	0	0
$985$	0	0
$986$	−36.0000	−1.14647
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	−24.0000	−0.761234
$995$	0	0
$996$	0	0
$997$	44.0000i	1.39349i	0.717317	+	0.696747i	$0.245370\pi$
$997$	44.0000i	1.39349i	−0.717317	+	0.696747i	$0.754630\pi$
$998$	− 4.00000i	− 0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.2.c.d.199.1		2
3.2	odd	2		450.2.c.a.199.2		2
4.3	odd	2		3600.2.f.a.2449.1		2
5.2	odd	4		450.2.a.e.1.1		1
5.3	odd	4		90.2.a.a.1.1	✓	1
5.4	even	2	inner	450.2.c.d.199.2		2
12.11	even	2		3600.2.f.u.2449.1		2
15.2	even	4		450.2.a.a.1.1		1
15.8	even	4		90.2.a.b.1.1	yes	1
15.14	odd	2		450.2.c.a.199.1		2
20.3	even	4		720.2.a.g.1.1		1
20.7	even	4		3600.2.a.ba.1.1		1
20.19	odd	2		3600.2.f.a.2449.2		2
35.13	even	4		4410.2.a.k.1.1		1
40.3	even	4		2880.2.a.h.1.1		1
40.13	odd	4		2880.2.a.k.1.1		1
45.13	odd	12		810.2.e.h.541.1		2
45.23	even	12		810.2.e.e.541.1		2
45.38	even	12		810.2.e.e.271.1		2
45.43	odd	12		810.2.e.h.271.1		2
60.23	odd	4		720.2.a.b.1.1		1
60.47	odd	4		3600.2.a.bj.1.1		1
60.59	even	2		3600.2.f.u.2449.2		2
105.83	odd	4		4410.2.a.bf.1.1		1
120.53	even	4		2880.2.a.bf.1.1		1
120.83	odd	4		2880.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.a.a.1.1	✓	1	5.3	odd	4
90.2.a.b.1.1	yes	1	15.8	even	4
450.2.a.a.1.1		1	15.2	even	4
450.2.a.e.1.1		1	5.2	odd	4
450.2.c.a.199.1		2	15.14	odd	2
450.2.c.a.199.2		2	3.2	odd	2
450.2.c.d.199.1		2	1.1	even	1	trivial
450.2.c.d.199.2		2	5.4	even	2	inner
720.2.a.b.1.1		1	60.23	odd	4
720.2.a.g.1.1		1	20.3	even	4
810.2.e.e.271.1		2	45.38	even	12
810.2.e.e.541.1		2	45.23	even	12
810.2.e.h.271.1		2	45.43	odd	12
810.2.e.h.541.1		2	45.13	odd	12
2880.2.a.h.1.1		1	40.3	even	4
2880.2.a.k.1.1		1	40.13	odd	4
2880.2.a.u.1.1		1	120.83	odd	4
2880.2.a.bf.1.1		1	120.53	even	4
3600.2.a.ba.1.1		1	20.7	even	4
3600.2.a.bj.1.1		1	60.47	odd	4
3600.2.f.a.2449.1		2	4.3	odd	2
3600.2.f.a.2449.2		2	20.19	odd	2
3600.2.f.u.2449.1		2	12.11	even	2
3600.2.f.u.2449.2		2	60.59	even	2
4410.2.a.k.1.1		1	35.13	even	4
4410.2.a.bf.1.1		1	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +6.00000 q^{11} +4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +4.00000 q^{19} -6.00000i q^{22} +4.00000 q^{26} -2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -6.00000 q^{34} +8.00000i q^{37} -4.00000i q^{38} -8.00000i q^{43} -6.00000 q^{44} +3.00000 q^{49} -4.00000i q^{52} +6.00000i q^{53} -2.00000 q^{56} -6.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +6.00000i q^{68} -12.0000 q^{71} +10.0000i q^{73} +8.00000 q^{74} -4.00000 q^{76} +12.0000i q^{77} +4.00000 q^{79} -12.0000i q^{83} -8.00000 q^{86} +6.00000i q^{88} -12.0000 q^{89} -8.00000 q^{91} +2.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 12 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{26} + 12 q^{29} - 8 q^{31} - 12 q^{34} - 12 q^{44} + 6 q^{49} - 4 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 24 q^{71} + 16 q^{74} - 8 q^{76} + 8 q^{79} - 16 q^{86} - 24 q^{89} - 16 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 + 12 * q^11 + 4 * q^14 + 2 * q^16 + 8 * q^19 + 8 * q^26 + 12 * q^29 - 8 * q^31 - 12 * q^34 - 12 * q^44 + 6 * q^49 - 4 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 - 24 * q^71 + 16 * q^74 - 8 * q^76 + 8 * q^79 - 16 * q^86 - 24 * q^89 - 16 * q^91

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