LMFDB - Embedded newform 5400.2.f.w.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:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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.w.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{7} +O(q^{10})$	$q-3.00000i q^{7} +4.00000 q^{11} -6.00000i q^{13} +6.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +10.0000 q^{29} +1.00000 q^{31} +11.0000i q^{37} +3.00000i q^{43} +8.00000i q^{47} -2.00000 q^{49} -12.0000i q^{53} +6.00000 q^{59} +5.00000 q^{61} -4.00000i q^{67} -2.00000 q^{71} +9.00000i q^{73} -12.0000i q^{77} +5.00000 q^{79} -18.0000i q^{83} -2.00000 q^{89} -18.0000 q^{91} -5.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 8 q^{11} + 2 q^{19} + 20 q^{29} + 2 q^{31} - 4 q^{49} + 12 q^{59} + 10 q^{61} - 4 q^{71} + 10 q^{79} - 4 q^{89} - 36 q^{91}+O(q^{100}) $ 2 * q + 8 * q^11 + 2 * q^19 + 20 * q^29 + 2 * q^31 - 4 * q^49 + 12 * q^59 + 10 * q^61 - 4 * q^71 + 10 * q^79 - 4 * q^89 - 36 * 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	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	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
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0000i	1.80839i	0.427121	+	0.904194i	$0.359528\pi$
$37$	11.0000i	1.80839i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	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
$43$	3.00000i	0.457496i	0.973486	+	0.228748i	$0.0734631\pi$
$43$	3.00000i	0.457496i	−0.973486	+	0.228748i	$0.926537\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	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	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	9.00000i	1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$	9.00000i	1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$	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
$83$	− 18.0000i	− 1.97576i	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	− 18.0000i	− 1.97576i	0.155230	−	0.987878i	$-0.450388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.00000i	− 0.507673i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$	− 5.00000i	− 0.507673i	0.967247	−	0.253837i	$-0.0816925\pi$
$98$	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
$103$	9.00000i	0.886796i	0.896325	+	0.443398i	$0.146227\pi$
$103$	9.00000i	0.886796i	−0.896325	+	0.443398i	$0.853773\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.0000	1.65006
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000i	0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$	8.00000i	0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	−19.0000	−1.61156	−0.805779	−	0.592216i	$-0.798253\pi$
$139$	−19.0000	−1.61156	−0.805779	+	0.592216i	$0.798253\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 24.0000i	− 2.00698i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.00000i	0.239426i	0.992809	+	0.119713i	$0.0381975\pi$
$157$	3.00000i	0.239426i	−0.992809	+	0.119713i	$0.961803\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	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
$166$	0	0
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.0000i	1.75505i
$188$	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
$193$	7.00000i	0.503871i	0.967744	+	0.251936i	$0.0810671\pi$
$193$	7.00000i	0.503871i	−0.967744	+	0.251936i	$0.918933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.00000i	0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$	8.00000i	0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 30.0000i	− 2.10559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.00000i	− 0.203653i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	36.0000	2.42162
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	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	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	0	0
$253$	24.0000i	1.50887i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.0000i	1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$	28.0000i	1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$	0	0
$259$	33.0000	2.05052
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.00000i	0.300421i	0.988654	+	0.150210i	$0.0479951\pi$
$277$	5.00000i	0.300421i	−0.988654	+	0.150210i	$0.952005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	− 19.0000i	− 1.12943i	−0.825285	−	0.564716i	$-0.808986\pi$
$283$	− 19.0000i	− 1.12943i	0.825285	−	0.564716i	$-0.191014\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.0000i	− 1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$	− 22.0000i	− 1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	36.0000	2.08193
$300$	0	0
$301$	9.00000	0.518751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.00000i	− 0.285365i	−0.989769	−	0.142683i	$-0.954427\pi$
$307$	− 5.00000i	− 0.285365i	0.989769	−	0.142683i	$-0.0455728\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000i	1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$	26.0000i	1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$	0	0
$319$	40.0000	2.23957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.00000i	0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	6.00000i	0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$	0	0
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 8.00000i	− 0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$	− 8.00000i	− 0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−36.0000	−1.86903
$372$	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
$377$	− 60.0000i	− 3.09016i
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 30.0000i	− 1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$	− 30.0000i	− 1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−36.0000	−1.82060
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.0000i	0.953583i	0.879017	+	0.476791i	$0.158200\pi$
$397$	19.0000i	0.953583i	−0.879017	+	0.476791i	$0.841800\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.0000i	2.18100i
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	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
$431$	22.0000	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$431$	22.0000	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$432$	0	0
$433$	− 21.0000i	− 1.00920i	−0.863355	−	0.504598i	$-0.831641\pi$
$433$	− 21.0000i	− 1.00920i	0.863355	−	0.504598i	$-0.168359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000i	0.287019i
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.0000i	1.04525i	0.852562	+	0.522626i	$0.175047\pi$
$443$	22.0000i	1.04525i	−0.852562	+	0.522626i	$0.824953\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	29.0000i	1.34774i	0.738848	+	0.673872i	$0.235370\pi$
$463$	29.0000i	1.34774i	−0.738848	+	0.673872i	$0.764630\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.0000i	− 1.01804i	−0.860755	−	0.509019i	$-0.830008\pi$
$467$	− 22.0000i	− 1.01804i	0.860755	−	0.509019i	$-0.169992\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	66.0000	3.00934
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	60.0000i	2.70226i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000i	0.269137i
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.0000i	1.40736i
$518$	0	0
$519$	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
$523$	− 1.00000i	− 0.0437269i	−0.999761	−	0.0218635i	$-0.993040\pi$
$523$	− 1.00000i	− 0.0437269i	0.999761	−	0.0218635i	$-0.00695991\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000i	0.261364i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	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
$541$	−21.0000	−0.902861	−0.451430	−	0.892306i	$-0.649086\pi$
$541$	−21.0000	−0.902861	−0.451430	+	0.892306i	$0.649086\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 27.0000i	− 1.15444i	−0.816590	−	0.577218i	$-0.804138\pi$
$547$	− 27.0000i	− 1.15444i	0.816590	−	0.577218i	$-0.195862\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.0000	0.426014
$552$	0	0
$553$	− 15.0000i	− 0.637865i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	18.0000	0.761319
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.0000i	1.34864i	0.738440	+	0.674320i	$0.235563\pi$
$563$	32.0000i	1.34864i	−0.738440	+	0.674320i	$0.764437\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	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
$577$	− 23.0000i	− 0.957503i	−0.877951	−	0.478751i	$-0.841090\pi$
$577$	− 23.0000i	− 0.957503i	0.877951	−	0.478751i	$-0.158910\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−54.0000	−2.24030
$582$	0	0
$583$	− 48.0000i	− 1.98796i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.0000i	1.56843i	0.620491	+	0.784214i	$0.286934\pi$
$587$	38.0000i	1.56843i	−0.620491	+	0.784214i	$0.713066\pi$
$588$	0	0
$589$	1.00000	0.0412043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 25.0000i	− 1.01472i	−0.861735	−	0.507359i	$-0.830622\pi$
$607$	− 25.0000i	− 1.01472i	0.861735	−	0.507359i	$-0.169378\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	25.0000i	1.00974i	0.863195	+	0.504870i	$0.168460\pi$
$613$	25.0000i	1.00974i	−0.863195	+	0.504870i	$0.831540\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.00000i	− 0.161034i	−0.996753	−	0.0805170i	$-0.974343\pi$
$617$	− 4.00000i	− 0.161034i	0.996753	−	0.0805170i	$-0.0256571\pi$
$618$	0	0
$619$	37.0000	1.48716	0.743578	−	0.668649i	$-0.233127\pi$
$619$	37.0000	1.48716	0.743578	+	0.668649i	$0.233127\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000i	0.240385i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−66.0000	−2.63159
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	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
$646$	0	0
$647$	− 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
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.0000i	0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$	16.0000i	0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.0000	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	60.0000i	2.32321i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20.0000	0.772091
$672$	0	0
$673$	− 29.0000i	− 1.11787i	−0.829212	−	0.558934i	$-0.811211\pi$
$673$	− 29.0000i	− 1.11787i	0.829212	−	0.558934i	$-0.188789\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.00000i	− 0.153732i	−0.997041	−	0.0768662i	$-0.975509\pi$
$677$	− 4.00000i	− 0.153732i	0.997041	−	0.0768662i	$-0.0244914\pi$
$678$	0	0
$679$	−15.0000	−0.575647
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 46.0000i	− 1.76014i	−0.474843	−	0.880071i	$-0.657495\pi$
$683$	− 46.0000i	− 1.76014i	0.474843	−	0.880071i	$-0.342505\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−72.0000	−2.74298
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	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
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	11.0000i	0.414873i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.0000	1.38956	0.694782	−	0.719220i	$-0.255501\pi$
$709$	37.0000	1.38956	0.694782	+	0.719220i	$0.255501\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.00000i	0.224702i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	27.0000	1.00553
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	47.0000i	1.74313i	0.490277	+	0.871567i	$0.336896\pi$
$727$	47.0000i	1.74313i	−0.490277	+	0.871567i	$0.663104\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18.0000	−0.665754
$732$	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
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	−37.0000	−1.35015	−0.675075	−	0.737749i	$-0.735889\pi$
$751$	−37.0000	−1.35015	−0.675075	+	0.737749i	$0.735889\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.0000i	1.05402i	0.849858	+	0.527011i	$0.176688\pi$
$757$	29.0000i	1.05402i	−0.849858	+	0.527011i	$0.823312\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.0000	0.580000	0.290000	−	0.957027i	$-0.406345\pi$
$761$	16.0000	0.580000	0.290000	+	0.957027i	$0.406345\pi$
$762$	0	0
$763$	− 27.0000i	− 0.977466i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 36.0000i	− 1.29988i
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 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
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 43.0000i	− 1.53278i	−0.642373	−	0.766392i	$-0.722050\pi$
$787$	− 43.0000i	− 1.53278i	0.642373	−	0.766392i	$-0.277950\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	− 30.0000i	− 1.06533i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44.0000i	− 1.55856i	−0.626676	−	0.779280i	$-0.715585\pi$
$797$	− 44.0000i	− 1.55856i	0.626676	−	0.779280i	$-0.284415\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	0	0
$802$	0	0
$803$	36.0000i	1.27041i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	5.00000	0.175574	0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000	0.175574	0.0877869	+	0.996139i	$0.472021\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.00000i	0.104957i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	5.00000	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 12.0000i	− 0.415775i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	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
$851$	−66.0000	−2.26245
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.0000i	1.50301i	0.659727	+	0.751506i	$0.270672\pi$
$857$	44.0000i	1.50301i	−0.659727	+	0.751506i	$0.729328\pi$
$858$	0	0
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.0000i	1.15737i	0.815550	+	0.578687i	$0.196435\pi$
$863$	34.0000i	1.15737i	−0.815550	+	0.578687i	$0.803565\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.0000	0.678454
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 13.0000i	− 0.438979i	−0.975615	−	0.219489i	$-0.929561\pi$
$877$	− 13.0000i	− 0.438979i	0.975615	−	0.219489i	$-0.0704391\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	0	0
$883$	37.0000i	1.24515i	0.782560	+	0.622575i	$0.213913\pi$
$883$	37.0000i	1.24515i	−0.782560	+	0.622575i	$0.786087\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.00000i	0.267710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.0000	0.333519
$900$	0	0
$901$	72.0000	2.39867
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 25.0000i	− 0.830111i	−0.909796	−	0.415056i	$-0.863762\pi$
$907$	− 25.0000i	− 0.830111i	0.909796	−	0.415056i	$-0.136238\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−54.0000	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$912$	0	0
$913$	− 72.0000i	− 2.38285i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 42.0000i	− 1.38696i
$918$	0	0
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 39.0000i	− 1.27407i	−0.770833	−	0.637037i	$-0.780160\pi$
$937$	− 39.0000i	− 1.27407i	0.770833	−	0.637037i	$-0.219840\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.00000i	− 0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$	− 6.00000i	− 0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$	0	0
$949$	54.0000	1.75291
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 4.00000i	− 0.129573i	−0.997899	−	0.0647864i	$-0.979363\pi$
$953$	− 4.00000i	− 0.129573i	0.997899	−	0.0647864i	$-0.0206366\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9.00000i	0.289420i	0.989474	+	0.144710i	$0.0462250\pi$
$967$	9.00000i	0.289420i	−0.989474	+	0.144710i	$0.953775\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	57.0000i	1.82734i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 14.0000i	− 0.447900i	−0.974601	−	0.223950i	$-0.928105\pi$
$977$	− 14.0000i	− 0.447900i	0.974601	−	0.223950i	$-0.0718952\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.0000i	0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$	28.0000i	0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.0000	−0.572367
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 6.00000i	− 0.190022i	−0.995476	−	0.0950110i	$-0.969711\pi$
$997$	− 6.00000i	− 0.190022i	0.995476	−	0.0950110i	$-0.0302886\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.f.w.649.1		2
3.2	odd	2		5400.2.f.d.649.1		2
5.2	odd	4		5400.2.a.bo.1.1	yes	1
5.3	odd	4		5400.2.a.g.1.1	yes	1
5.4	even	2	inner	5400.2.f.w.649.2		2
15.2	even	4		5400.2.a.bm.1.1	yes	1
15.8	even	4		5400.2.a.f.1.1	✓	1
15.14	odd	2		5400.2.f.d.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5400.2.a.f.1.1	✓	1	15.8	even	4
5400.2.a.g.1.1	yes	1	5.3	odd	4
5400.2.a.bm.1.1	yes	1	15.2	even	4
5400.2.a.bo.1.1	yes	1	5.2	odd	4
5400.2.f.d.649.1		2	3.2	odd	2
5400.2.f.d.649.2		2	15.14	odd	2
5400.2.f.w.649.1		2	1.1	even	1	trivial
5400.2.f.w.649.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.00000i q^{7} +O(q^{10})\)	\(q-3.00000i q^{7} +4.00000 q^{11} -6.00000i q^{13} +6.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +10.0000 q^{29} +1.00000 q^{31} +11.0000i q^{37} +3.00000i q^{43} +8.00000i q^{47} -2.00000 q^{49} -12.0000i q^{53} +6.00000 q^{59} +5.00000 q^{61} -4.00000i q^{67} -2.00000 q^{71} +9.00000i q^{73} -12.0000i q^{77} +5.00000 q^{79} -18.0000i q^{83} -2.00000 q^{89} -18.0000 q^{91} -5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 8 q^{11} + 2 q^{19} + 20 q^{29} + 2 q^{31} - 4 q^{49} + 12 q^{59} + 10 q^{61} - 4 q^{71} + 10 q^{79} - 4 q^{89} - 36 q^{91}+O(q^{100}) \) 2 * q + 8 * q^11 + 2 * q^19 + 20 * q^29 + 2 * q^31 - 4 * q^49 + 12 * q^59 + 10 * q^61 - 4 * q^71 + 10 * q^79 - 4 * q^89 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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