LMFDB - Embedded newform 5400.2.f.e.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(649,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5400.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$43.1192170915$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5400.649
Dual form			5400.2.f.e.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{7} +O(q^{10})$	$q-3.00000i q^{7} -4.00000 q^{11} -1.00000i q^{13} +4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} -4.00000 q^{31} -9.00000i q^{37} +8.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000i q^{53} +4.00000 q^{59} -5.00000 q^{61} +11.0000i q^{67} -8.00000 q^{71} -1.00000i q^{73} +12.0000i q^{77} +5.00000 q^{79} +8.00000i q^{83} +12.0000 q^{89} -3.00000 q^{91} +5.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 8 q^{11} + 2 q^{19} - 8 q^{31} - 4 q^{49} + 8 q^{59} - 10 q^{61} - 16 q^{71} + 10 q^{79} + 24 q^{89} - 6 q^{91}+O(q^{100}) $ 2 * q - 8 * q^11 + 2 * q^19 - 8 * q^31 - 4 * q^49 + 8 * q^59 - 10 * q^61 - 16 * q^71 + 10 * q^79 + 24 * q^89 - 6 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$	$2701$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.00000i	− 1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$	− 9.00000i	− 1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	0	0
$39$	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	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.00000i	− 1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$53$	− 8.00000i	− 1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0000i	1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$	11.0000i	1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000i	1.36753i
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.00000i	0.507673i	0.967247	+	0.253837i	$0.0816925\pi$
$97$	5.00000i	0.507673i	−0.967247	+	0.253837i	$0.918307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 1.00000i	− 0.0985329i	−0.998786	−	0.0492665i	$-0.984312\pi$
$103$	− 1.00000i	− 0.0985329i	0.998786	−	0.0492665i	$-0.0156884\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	15.0000i	1.17489i	0.809264	+	0.587445i	$0.199866\pi$
$163$	15.0000i	1.17489i	−0.809264	+	0.587445i	$0.800134\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	21.0000	1.56092	0.780459	−	0.625207i	$-0.214986\pi$
$181$	21.0000	1.56092	0.780459	+	0.625207i	$0.214986\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.0000i	− 1.17004i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	− 23.0000i	− 1.65558i	−0.561041	−	0.827788i	$-0.689599\pi$
$193$	− 23.0000i	− 1.65558i	0.561041	−	0.827788i	$-0.310401\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000i	1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$	16.0000i	1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−15.0000	−0.966235	−0.483117	−	0.875556i	$-0.660496\pi$
$241$	−15.0000	−0.966235	−0.483117	+	0.875556i	$0.660496\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.00000i	− 0.0636285i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	0	0
$259$	−27.0000	−1.67770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	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
$313$	− 3.00000i	− 0.169570i	−0.996399	−	0.0847850i	$-0.972980\pi$
$313$	− 3.00000i	− 0.169570i	0.996399	−	0.0847850i	$-0.0270203\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000i	0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	36.0000	1.98474
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.00000i	− 0.163420i	−0.996656	−	0.0817102i	$-0.973962\pi$
$337$	− 3.00000i	− 0.163420i	0.996656	−	0.0817102i	$-0.0260382\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.0000	0.866449
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	33.0000	1.76645	0.883225	−	0.468950i	$-0.155368\pi$
$349$	33.0000	1.76645	0.883225	+	0.468950i	$0.155368\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 23.0000i	− 1.20059i	−0.799779	−	0.600295i	$-0.795050\pi$
$367$	− 23.0000i	− 1.20059i	0.799779	−	0.600295i	$-0.204950\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	1.00000i	0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$	1.00000i	0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.00000	0.202808	0.101404	−	0.994845i	$-0.467667\pi$
$389$	4.00000	0.202808	0.101404	+	0.994845i	$0.467667\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.0000i	1.78445i
$408$	0	0
$409$	39.0000	1.92843	0.964213	−	0.265129i	$-0.0854146\pi$
$409$	39.0000	1.92843	0.964213	+	0.265129i	$0.0854146\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.0000i	0.725901i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	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
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.00000i	0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$	8.00000i	0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.0000i	1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$	38.0000i	1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	19.0000i	0.883005i	0.897260	+	0.441502i	$0.145554\pi$
$463$	19.0000i	0.883005i	−0.897260	+	0.441502i	$0.854446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	33.0000	1.52380
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 32.0000i	− 1.47136i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000i	1.07655i
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 48.0000i	− 2.11104i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.0000	1.22670	0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000	1.22670	0.613351	+	0.789810i	$0.289821\pi$
$522$	0	0
$523$	29.0000i	1.26808i	0.773300	+	0.634041i	$0.218605\pi$
$523$	29.0000i	1.26808i	−0.773300	+	0.634041i	$0.781395\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.00000	0.344584
$540$	0	0
$541$	9.00000	0.386940	0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000	0.386940	0.193470	+	0.981106i	$0.438026\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 7.00000i	− 0.299298i	−0.988739	−	0.149649i	$-0.952186\pi$
$547$	− 7.00000i	− 0.299298i	0.988739	−	0.149649i	$-0.0478144\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 15.0000i	− 0.637865i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 28.0000i	− 1.18640i	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	− 28.0000i	− 1.18640i	0.805056	−	0.593199i	$-0.202135\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 32.0000i	− 1.34864i	−0.738440	−	0.674320i	$-0.764437\pi$
$563$	− 32.0000i	− 1.34864i	0.738440	−	0.674320i	$-0.235563\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	−33.0000	−1.38101	−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000	−1.38101	−0.690504	+	0.723329i	$0.742611\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 13.0000i	− 0.541197i	−0.962692	−	0.270599i	$-0.912778\pi$
$577$	− 13.0000i	− 0.541197i	0.962692	−	0.270599i	$-0.0872216\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	32.0000i	1.32530i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	40.0000i	1.64260i	0.570494	+	0.821302i	$0.306752\pi$
$593$	40.0000i	1.64260i	−0.570494	+	0.821302i	$0.693248\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.00000i	0.202944i	0.994838	+	0.101472i	$0.0323552\pi$
$607$	5.00000i	0.202944i	−0.994838	+	0.101472i	$0.967645\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	− 35.0000i	− 1.41364i	−0.707395	−	0.706818i	$-0.750130\pi$
$613$	− 35.0000i	− 1.41364i	0.707395	−	0.706818i	$-0.249870\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 36.0000i	− 1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$	− 36.0000i	− 1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 36.0000i	− 1.44231i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.0000	1.43541
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	24.0000i	0.946468i	0.880937	+	0.473234i	$0.156913\pi$
$643$	24.0000i	0.946468i	−0.880937	+	0.473234i	$0.843087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 32.0000i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$	− 32.0000i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	− 19.0000i	− 0.732396i	−0.930537	−	0.366198i	$-0.880659\pi$
$673$	− 19.0000i	− 0.732396i	0.930537	−	0.366198i	$-0.119341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.0000i	1.69106i	0.533930	+	0.845529i	$0.320715\pi$
$677$	44.0000i	1.69106i	−0.533930	+	0.845529i	$0.679285\pi$
$678$	0	0
$679$	15.0000	0.575647
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	− 9.00000i	− 0.339441i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.00000	0.262891	0.131445	−	0.991323i	$-0.458038\pi$
$709$	7.00000	0.262891	0.131445	+	0.991323i	$0.458038\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12.0000i	0.445055i	0.974926	+	0.222528i	$0.0714308\pi$
$727$	12.0000i	0.445055i	−0.974926	+	0.222528i	$0.928569\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 44.0000i	− 1.62076i
$738$	0	0
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−36.0000	−1.31541
$750$	0	0
$751$	3.00000	0.109472	0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000	0.109472	0.0547358	+	0.998501i	$0.482568\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 31.0000i	− 1.12671i	−0.826214	−	0.563357i	$-0.809510\pi$
$757$	− 31.0000i	− 1.12671i	0.826214	−	0.563357i	$-0.190490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.00000	0.145000	0.0724999	−	0.997368i	$-0.476902\pi$
$761$	4.00000	0.145000	0.0724999	+	0.997368i	$0.476902\pi$
$762$	0	0
$763$	− 42.0000i	− 1.52050i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.00000i	− 0.144432i
$768$	0	0
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	36.0000	1.28001
$792$	0	0
$793$	5.00000i	0.177555i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.0000i	− 0.566749i	−0.959009	−	0.283375i	$-0.908546\pi$
$797$	− 16.0000i	− 0.566749i	0.959009	−	0.283375i	$-0.0914540\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.0000	−1.40633	−0.703163	−	0.711029i	$-0.748229\pi$
$809$	−40.0000	−1.40633	−0.703163	+	0.711029i	$0.748229\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000i	0.279885i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	0	0
$823$	25.0000i	0.871445i	0.900081	+	0.435723i	$0.143507\pi$
$823$	25.0000i	0.871445i	−0.900081	+	0.435723i	$0.856493\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	35.0000	1.21560	0.607800	−	0.794090i	$-0.292052\pi$
$829$	35.0000	1.21560	0.607800	+	0.794090i	$0.292052\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 8.00000i	− 0.277184i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 15.0000i	− 0.515406i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	36.0000	1.23406
$852$	0	0
$853$	1.00000i	0.0342393i	0.999853	+	0.0171197i	$0.00544963\pi$
$853$	1.00000i	0.0342393i	−0.999853	+	0.0171197i	$0.994550\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	27.0000	0.921228	0.460614	−	0.887601i	$-0.347629\pi$
$859$	27.0000	0.921228	0.460614	+	0.887601i	$0.347629\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 44.0000i	− 1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$	− 44.0000i	− 1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	11.0000	0.372721
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 33.0000i	− 1.11433i	−0.830402	−	0.557165i	$-0.811889\pi$
$877$	− 33.0000i	− 1.11433i	0.830402	−	0.557165i	$-0.188111\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	0	0
$883$	7.00000i	0.235569i	0.993039	+	0.117784i	$0.0375792\pi$
$883$	7.00000i	0.235569i	−0.993039	+	0.117784i	$0.962421\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 56.0000i	− 1.88030i	−0.340766	−	0.940148i	$-0.610687\pi$
$887$	− 56.0000i	− 1.88030i	0.340766	−	0.940148i	$-0.389313\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	32.0000	1.06607
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.00000i	− 0.166022i	−0.996549	−	0.0830111i	$-0.973546\pi$
$907$	− 5.00000i	− 0.166022i	0.996549	−	0.0830111i	$-0.0264537\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	− 32.0000i	− 1.05905i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.0000i	− 1.58510i
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.00000i	0.263323i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.0000i	1.66610i	0.553200	+	0.833049i	$0.313407\pi$
$937$	51.0000i	1.66610i	−0.553200	+	0.833049i	$0.686593\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	0	0
$949$	−1.00000	−0.0324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	59.0000i	1.89731i	0.316310	+	0.948656i	$0.397556\pi$
$967$	59.0000i	1.89731i	−0.316310	+	0.948656i	$0.602444\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	27.0000i	0.865580i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.0000i	0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$	24.0000i	0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$	0	0
$979$	−48.0000	−1.53409
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	25.0000	0.794151	0.397076	−	0.917786i	$-0.370025\pi$
$991$	25.0000	0.794151	0.397076	+	0.917786i	$0.370025\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.f.e.649.1		2
3.2	odd	2		5400.2.f.v.649.1		2
5.2	odd	4		5400.2.a.bn.1.1		1
5.3	odd	4		216.2.a.a.1.1	✓	1
5.4	even	2	inner	5400.2.f.e.649.2		2
15.2	even	4		5400.2.a.bp.1.1		1
15.8	even	4		216.2.a.d.1.1	yes	1
15.14	odd	2		5400.2.f.v.649.2		2
20.3	even	4		432.2.a.a.1.1		1
40.3	even	4		1728.2.a.bb.1.1		1
40.13	odd	4		1728.2.a.ba.1.1		1
45.13	odd	12		648.2.i.h.217.1		2
45.23	even	12		648.2.i.a.217.1		2
45.38	even	12		648.2.i.a.433.1		2
45.43	odd	12		648.2.i.h.433.1		2
60.23	odd	4		432.2.a.h.1.1		1
120.53	even	4		1728.2.a.a.1.1		1
120.83	odd	4		1728.2.a.b.1.1		1
180.23	odd	12		1296.2.i.a.865.1		2
180.43	even	12		1296.2.i.q.433.1		2
180.83	odd	12		1296.2.i.a.433.1		2
180.103	even	12		1296.2.i.q.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.a.a.1.1	✓	1	5.3	odd	4
216.2.a.d.1.1	yes	1	15.8	even	4
432.2.a.a.1.1		1	20.3	even	4
432.2.a.h.1.1		1	60.23	odd	4
648.2.i.a.217.1		2	45.23	even	12
648.2.i.a.433.1		2	45.38	even	12
648.2.i.h.217.1		2	45.13	odd	12
648.2.i.h.433.1		2	45.43	odd	12
1296.2.i.a.433.1		2	180.83	odd	12
1296.2.i.a.865.1		2	180.23	odd	12
1296.2.i.q.433.1		2	180.43	even	12
1296.2.i.q.865.1		2	180.103	even	12
1728.2.a.a.1.1		1	120.53	even	4
1728.2.a.b.1.1		1	120.83	odd	4
1728.2.a.ba.1.1		1	40.13	odd	4
1728.2.a.bb.1.1		1	40.3	even	4
5400.2.a.bn.1.1		1	5.2	odd	4
5400.2.a.bp.1.1		1	15.2	even	4
5400.2.f.e.649.1		2	1.1	even	1	trivial
5400.2.f.e.649.2		2	5.4	even	2	inner
5400.2.f.v.649.1		2	3.2	odd	2
5400.2.f.v.649.2		2	15.14	odd	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:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{7} +O(q^{10})\)	\(q-3.00000i q^{7} -4.00000 q^{11} -1.00000i q^{13} +4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} -4.00000 q^{31} -9.00000i q^{37} +8.00000i q^{43} +12.0000i q^{47} -2.00000 q^{49} -8.00000i q^{53} +4.00000 q^{59} -5.00000 q^{61} +11.0000i q^{67} -8.00000 q^{71} -1.00000i q^{73} +12.0000i q^{77} +5.00000 q^{79} +8.00000i q^{83} +12.0000 q^{89} -3.00000 q^{91} +5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 8 q^{11} + 2 q^{19} - 8 q^{31} - 4 q^{49} + 8 q^{59} - 10 q^{61} - 16 q^{71} + 10 q^{79} + 24 q^{89} - 6 q^{91}+O(q^{100}) \) 2 * q - 8 * q^11 + 2 * q^19 - 8 * q^31 - 4 * q^49 + 8 * q^59 - 10 * q^61 - 16 * q^71 + 10 * q^79 + 24 * q^89 - 6 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more