LMFDB - Embedded newform 2400.2.d.c.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,2,Mod(49,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.49"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,2,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1640964851$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2400.49
Dual form			2400.2.d.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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000i q^{7} +1.00000 q^{9} -4.00000 q^{13} +2.00000i q^{17} -4.00000i q^{19} -2.00000i q^{21} -4.00000i q^{23} +1.00000 q^{27} -6.00000i q^{29} -2.00000 q^{31} -8.00000 q^{37} -4.00000 q^{39} +2.00000 q^{41} +4.00000 q^{43} -12.0000i q^{47} +3.00000 q^{49} +2.00000i q^{51} +6.00000 q^{53} -4.00000i q^{57} -4.00000i q^{59} -2.00000i q^{63} -12.0000 q^{67} -4.00000i q^{69} -12.0000 q^{71} -6.00000i q^{73} +10.0000 q^{79} +1.00000 q^{81} -16.0000 q^{83} -6.00000i q^{87} +10.0000 q^{89} +8.00000i q^{91} -2.00000 q^{93} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9} - 8 q^{13} + 2 q^{27} - 4 q^{31} - 16 q^{37} - 8 q^{39} + 4 q^{41} + 8 q^{43} + 6 q^{49} + 12 q^{53} - 24 q^{67} - 24 q^{71} + 20 q^{79} + 2 q^{81} - 32 q^{83} + 20 q^{89} - 4 q^{93}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 8 * q^13 + 2 * q^27 - 4 * q^31 - 16 * q^37 - 8 * q^39 + 4 * q^41 + 8 * q^43 + 6 * q^49 + 12 * q^53 - 24 * q^67 - 24 * q^71 + 20 * q^79 + 2 * q^81 - 32 * q^83 + 20 * q^89 - 4 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	2.00000i	0.280056i
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	− 4.00000i	− 0.481543i
$70$	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
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	8.00000i	0.838628i
$92$	0	0
$93$	−2.00000	−0.207390
$94$	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	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.0000i	0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$	10.0000i	0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	4.00000i	0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$	4.00000i	0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	− 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
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	− 20.0000i	− 1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$	− 20.0000i	− 1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	− 4.00000i	− 0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$	− 4.00000i	− 0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	0	0
$141$	− 12.0000i	− 1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	− 4.00000i	− 0.298974i	−0.988764	−	0.149487i	$-0.952238\pi$
$179$	− 4.00000i	− 0.298974i	0.988764	−	0.149487i	$-0.0477622\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 2.00000i	− 0.145479i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000i	1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$	20.0000i	1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	16.0000i	0.994192i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.00000i	− 0.236113i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	2.00000i	0.117242i
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.0000i	0.925304i
$300$	0	0
$301$	− 8.00000i	− 0.461112i
$302$	0	0
$303$	10.0000i	0.574485i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	6.00000i	0.341328i
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.00000i	0.221201i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	20.0000i	1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$	20.0000i	1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	− 16.0000i	− 0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$	− 16.0000i	− 0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.0000i	0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$	18.0000i	0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	− 12.0000i	− 0.623009i
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	− 2.00000i	− 0.102463i
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	34.0000i	1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$	34.0000i	1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	− 20.0000i	− 1.00887i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	0	0
$399$	−8.00000	−0.400501
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	− 18.0000i	− 0.887875i
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.00000i	− 0.195881i
$418$	0	0
$419$	− 24.0000i	− 1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$	− 24.0000i	− 1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$	0	0
$421$	20.0000i	0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$	20.0000i	0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$	0	0
$423$	− 12.0000i	− 0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−30.0000	−1.43182	−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000	−1.43182	−0.715911	+	0.698192i	$0.753988\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 6.00000i	− 0.283790i
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	18.0000	0.845714
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	0	0
$459$	2.00000i	0.0933520i
$460$	0	0
$461$	− 30.0000i	− 1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$	− 30.0000i	− 1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$	0	0
$463$	26.0000i	1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$	26.0000i	1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	24.0000i	1.10822i
$470$	0	0
$471$	−8.00000	−0.368621
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	−8.00000	−0.364013
$484$	0	0
$485$	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	$0.669856\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000i	1.07655i
$498$	0	0
$499$	36.0000i	1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$	36.0000i	1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$	0	0
$501$	8.00000i	0.357414i
$502$	0	0
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.00000i	− 0.172613i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 20.0000i	− 0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	− 20.0000i	− 0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	0	0
$543$	20.0000i	0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	− 20.0000i	− 0.850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	0	0
$576$	0	0
$577$	42.0000i	1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$	42.0000i	1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$	0	0
$579$	− 6.00000i	− 0.249351i
$580$	0	0
$581$	32.0000i	1.32758i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.0000	0.409273
$598$	0	0
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.00000i	− 0.0811775i	−0.999176	−	0.0405887i	$-0.987077\pi$
$607$	− 2.00000i	− 0.0811775i	0.999176	−	0.0405887i	$-0.0129233\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	48.0000i	1.94187i
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	36.0000i	1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$	36.0000i	1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	− 20.0000i	− 0.801283i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.0000i	− 0.637962i
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	0	0
$633$	20.0000i	0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.00000i	0.156772i
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	36.0000i	1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$	36.0000i	1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	0	0
$661$	40.0000i	1.55582i	0.628376	+	0.777910i	$0.283720\pi$
$661$	40.0000i	1.55582i	−0.628376	+	0.777910i	$0.716280\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	− 14.0000i	− 0.541271i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.00000i	0.152610i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	− 20.0000i	− 0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$	− 20.0000i	− 0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	14.0000i	0.529529i
$700$	0	0
$701$	− 50.0000i	− 1.88847i	−0.329267	−	0.944237i	$-0.606802\pi$
$701$	− 50.0000i	− 1.88847i	0.329267	−	0.944237i	$-0.393198\pi$
$702$	0	0
$703$	32.0000i	1.20690i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.0000	0.752177
$708$	0	0
$709$	4.00000i	0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$	4.00000i	0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	2.00000	0.0743808
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 42.0000i	− 1.55769i	−0.627214	−	0.778847i	$-0.715805\pi$
$727$	− 42.0000i	− 1.55769i	0.627214	−	0.778847i	$-0.284195\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000i	0.295891i
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 44.0000i	− 1.61857i	−0.587419	−	0.809283i	$-0.699856\pi$
$739$	− 44.0000i	− 1.61857i	0.587419	−	0.809283i	$-0.300144\pi$
$740$	0	0
$741$	16.0000i	0.587775i
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−16.0000	−0.585409
$748$	0	0
$749$	24.0000i	0.876941i
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.0000i	0.577727i
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	− 18.0000i	− 0.648254i
$772$	0	0
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.0000i	0.573997i
$778$	0	0
$779$	− 8.00000i	− 0.286630i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 6.00000i	− 0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	0	0
$789$	16.0000i	0.569615i
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.00000i	− 0.211210i
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	18.0000	0.631288
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	8.00000i	0.279543i
$820$	0	0
$821$	10.0000i	0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$	10.0000i	0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$	0	0
$823$	− 14.0000i	− 0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$	− 14.0000i	− 0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	4.00000i	0.138926i	0.997585	+	0.0694629i	$0.0221285\pi$
$829$	4.00000i	0.138926i	−0.997585	+	0.0694629i	$0.977871\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	−18.0000	−0.619953
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	32.0000i	1.09695i
$852$	0	0
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	− 44.0000i	− 1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$	− 44.0000i	− 1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$	0	0
$861$	− 4.00000i	− 0.136320i
$862$	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
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	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
$892$	0	0
$893$	−48.0000	−1.60626
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000i	0.534224i
$898$	0	0
$899$	12.0000i	0.400222i
$900$	0	0
$901$	12.0000i	0.399778i
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−40.0000	−1.32092
$918$	0	0
$919$	30.0000	0.989609	0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000	0.989609	0.494804	+	0.869004i	$0.335240\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	48.0000	1.57994
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.00000i	0.197066i
$928$	0	0
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	0	0
$939$	14.0000i	0.456873i
$940$	0	0
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	24.0000i	0.779073i
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.0000	−1.16250
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 22.0000i	− 0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$	− 22.0000i	− 0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.00000i	0.127710i
$982$	0	0
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−24.0000	−0.763928
$988$	0	0
$989$	− 16.0000i	− 0.508770i
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.d.c.49.1		2
3.2	odd	2		7200.2.d.d.2449.1		2
4.3	odd	2		600.2.d.c.349.2		2
5.2	odd	4		96.2.d.a.49.1		2
5.3	odd	4		2400.2.k.a.1201.2		2
5.4	even	2		2400.2.d.b.49.2		2
8.3	odd	2		600.2.d.b.349.2		2
8.5	even	2		2400.2.d.b.49.1		2
12.11	even	2		1800.2.d.b.1549.1		2
15.2	even	4		288.2.d.b.145.2		2
15.8	even	4		7200.2.k.d.3601.1		2
15.14	odd	2		7200.2.d.g.2449.2		2
20.3	even	4		600.2.k.b.301.1		2
20.7	even	4		24.2.d.a.13.2	yes	2
20.19	odd	2		600.2.d.b.349.1		2
24.5	odd	2		7200.2.d.g.2449.1		2
24.11	even	2		1800.2.d.i.1549.1		2
35.27	even	4		4704.2.c.a.2353.2		2
40.3	even	4		600.2.k.b.301.2		2
40.13	odd	4		2400.2.k.a.1201.1		2
40.19	odd	2		600.2.d.c.349.1		2
40.27	even	4		24.2.d.a.13.1	✓	2
40.29	even	2	inner	2400.2.d.c.49.2		2
40.37	odd	4		96.2.d.a.49.2		2
45.2	even	12		2592.2.r.g.433.1		4
45.7	odd	12		2592.2.r.f.433.2		4
45.22	odd	12		2592.2.r.f.2161.1		4
45.32	even	12		2592.2.r.g.2161.2		4
60.23	odd	4		1800.2.k.a.901.2		2
60.47	odd	4		72.2.d.b.37.1		2
60.59	even	2		1800.2.d.i.1549.2		2
80.27	even	4		768.2.a.h.1.1		1
80.37	odd	4		768.2.a.d.1.1		1
80.67	even	4		768.2.a.a.1.1		1
80.77	odd	4		768.2.a.e.1.1		1
120.29	odd	2		7200.2.d.d.2449.2		2
120.53	even	4		7200.2.k.d.3601.2		2
120.59	even	2		1800.2.d.b.1549.2		2
120.77	even	4		288.2.d.b.145.1		2
120.83	odd	4		1800.2.k.a.901.1		2
120.107	odd	4		72.2.d.b.37.2		2
140.27	odd	4		1176.2.c.a.589.2		2
180.7	even	12		648.2.n.k.109.2		4
180.47	odd	12		648.2.n.c.109.1		4
180.67	even	12		648.2.n.k.541.1		4
180.167	odd	12		648.2.n.c.541.2		4
240.77	even	4		2304.2.a.l.1.1		1
240.107	odd	4		2304.2.a.e.1.1		1
240.197	even	4		2304.2.a.b.1.1		1
240.227	odd	4		2304.2.a.o.1.1		1
280.27	odd	4		1176.2.c.a.589.1		2
280.237	even	4		4704.2.c.a.2353.1		2
360.67	even	12		648.2.n.k.541.2		4
360.77	even	12		2592.2.r.g.2161.1		4
360.157	odd	12		2592.2.r.f.2161.2		4
360.187	even	12		648.2.n.k.109.1		4
360.227	odd	12		648.2.n.c.109.2		4
360.277	odd	12		2592.2.r.f.433.1		4
360.317	even	12		2592.2.r.g.433.2		4
360.347	odd	12		648.2.n.c.541.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.d.a.13.1	✓	2	40.27	even	4
24.2.d.a.13.2	yes	2	20.7	even	4
72.2.d.b.37.1		2	60.47	odd	4
72.2.d.b.37.2		2	120.107	odd	4
96.2.d.a.49.1		2	5.2	odd	4
96.2.d.a.49.2		2	40.37	odd	4
288.2.d.b.145.1		2	120.77	even	4
288.2.d.b.145.2		2	15.2	even	4
600.2.d.b.349.1		2	20.19	odd	2
600.2.d.b.349.2		2	8.3	odd	2
600.2.d.c.349.1		2	40.19	odd	2
600.2.d.c.349.2		2	4.3	odd	2
600.2.k.b.301.1		2	20.3	even	4
600.2.k.b.301.2		2	40.3	even	4
648.2.n.c.109.1		4	180.47	odd	12
648.2.n.c.109.2		4	360.227	odd	12
648.2.n.c.541.1		4	360.347	odd	12
648.2.n.c.541.2		4	180.167	odd	12
648.2.n.k.109.1		4	360.187	even	12
648.2.n.k.109.2		4	180.7	even	12
648.2.n.k.541.1		4	180.67	even	12
648.2.n.k.541.2		4	360.67	even	12
768.2.a.a.1.1		1	80.67	even	4
768.2.a.d.1.1		1	80.37	odd	4
768.2.a.e.1.1		1	80.77	odd	4
768.2.a.h.1.1		1	80.27	even	4
1176.2.c.a.589.1		2	280.27	odd	4
1176.2.c.a.589.2		2	140.27	odd	4
1800.2.d.b.1549.1		2	12.11	even	2
1800.2.d.b.1549.2		2	120.59	even	2
1800.2.d.i.1549.1		2	24.11	even	2
1800.2.d.i.1549.2		2	60.59	even	2
1800.2.k.a.901.1		2	120.83	odd	4
1800.2.k.a.901.2		2	60.23	odd	4
2304.2.a.b.1.1		1	240.197	even	4
2304.2.a.e.1.1		1	240.107	odd	4
2304.2.a.l.1.1		1	240.77	even	4
2304.2.a.o.1.1		1	240.227	odd	4
2400.2.d.b.49.1		2	8.5	even	2
2400.2.d.b.49.2		2	5.4	even	2
2400.2.d.c.49.1		2	1.1	even	1	trivial
2400.2.d.c.49.2		2	40.29	even	2	inner
2400.2.k.a.1201.1		2	40.13	odd	4
2400.2.k.a.1201.2		2	5.3	odd	4
2592.2.r.f.433.1		4	360.277	odd	12
2592.2.r.f.433.2		4	45.7	odd	12
2592.2.r.f.2161.1		4	45.22	odd	12
2592.2.r.f.2161.2		4	360.157	odd	12
2592.2.r.g.433.1		4	45.2	even	12
2592.2.r.g.433.2		4	360.317	even	12
2592.2.r.g.2161.1		4	360.77	even	12
2592.2.r.g.2161.2		4	45.32	even	12
4704.2.c.a.2353.1		2	280.237	even	4
4704.2.c.a.2353.2		2	35.27	even	4
7200.2.d.d.2449.1		2	3.2	odd	2
7200.2.d.d.2449.2		2	120.29	odd	2
7200.2.d.g.2449.1		2	24.5	odd	2
7200.2.d.g.2449.2		2	15.14	odd	2
7200.2.k.d.3601.1		2	15.8	even	4
7200.2.k.d.3601.2		2	120.53	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000i q^{7} +1.00000 q^{9} -4.00000 q^{13} +2.00000i q^{17} -4.00000i q^{19} -2.00000i q^{21} -4.00000i q^{23} +1.00000 q^{27} -6.00000i q^{29} -2.00000 q^{31} -8.00000 q^{37} -4.00000 q^{39} +2.00000 q^{41} +4.00000 q^{43} -12.0000i q^{47} +3.00000 q^{49} +2.00000i q^{51} +6.00000 q^{53} -4.00000i q^{57} -4.00000i q^{59} -2.00000i q^{63} -12.0000 q^{67} -4.00000i q^{69} -12.0000 q^{71} -6.00000i q^{73} +10.0000 q^{79} +1.00000 q^{81} -16.0000 q^{83} -6.00000i q^{87} +10.0000 q^{89} +8.00000i q^{91} -2.00000 q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9} - 8 q^{13} + 2 q^{27} - 4 q^{31} - 16 q^{37} - 8 q^{39} + 4 q^{41} + 8 q^{43} + 6 q^{49} + 12 q^{53} - 24 q^{67} - 24 q^{71} + 20 q^{79} + 2 q^{81} - 32 q^{83} + 20 q^{89} - 4 q^{93}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 8 * q^13 + 2 * q^27 - 4 * q^31 - 16 * q^37 - 8 * q^39 + 4 * q^41 + 8 * q^43 + 6 * q^49 + 12 * q^53 - 24 * q^67 - 24 * q^71 + 20 * q^79 + 2 * q^81 - 32 * q^83 + 20 * q^89 - 4 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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