LMFDB - Embedded newform 6300.2.k.e.6049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(6049,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6300.6049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.k (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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6049.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	6300.6049
Dual form			6300.2.k.e.6049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{7} +O(q^{10})$	$q+1.00000i q^{7} -2.00000 q^{11} +4.00000i q^{13} +6.00000i q^{17} -6.00000 q^{19} +8.00000i q^{23} -2.00000 q^{29} +10.0000 q^{31} -2.00000i q^{37} -10.0000 q^{41} -4.00000i q^{43} -8.00000i q^{47} -1.00000 q^{49} -4.00000i q^{53} -8.00000 q^{59} +6.00000 q^{61} -12.0000i q^{67} +6.00000 q^{71} -12.0000i q^{73} -2.00000i q^{77} +8.00000 q^{79} +4.00000i q^{83} -10.0000 q^{89} -4.00000 q^{91} -8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 4 q^{11} - 12 q^{19} - 4 q^{29} + 20 q^{31} - 20 q^{41} - 2 q^{49} - 16 q^{59} + 12 q^{61} + 12 q^{71} + 16 q^{79} - 20 q^{89} - 8 q^{91}+O(q^{100}) $ 2 * q - 4 * q^11 - 12 * q^19 - 4 * q^29 + 20 * q^31 - 20 * q^41 - 2 * q^49 - 16 * q^59 + 12 * q^61 + 12 * q^71 + 16 * q^79 - 20 * q^89 - 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	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
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	− 12.0000i	− 1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$	− 12.0000i	− 1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 8.00000i	− 0.812277i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	− 8.00000i	− 0.812277i	0.913812	−	0.406138i	$-0.133125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.0000i	1.27694i	0.769647	+	0.638470i	$0.220432\pi$
$157$	16.0000i	1.27694i	−0.769647	+	0.638470i	$0.779568\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	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
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.0000i	0.678844i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	− 12.0000i	− 0.803579i	−0.915732	−	0.401790i	$-0.868388\pi$
$223$	− 12.0000i	− 0.803579i	0.915732	−	0.401790i	$-0.131612\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.0000i	− 1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$	− 24.0000i	− 1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 24.0000i	− 1.52708i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	− 16.0000i	− 1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.0000i	− 0.590281i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−32.0000	−1.85061
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 36.0000i	− 2.00309i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.00000i	0.326841i	0.986557	+	0.163420i	$0.0522527\pi$
$337$	6.00000i	0.326841i	−0.986557	+	0.163420i	$0.947747\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.00000	−0.105556	−0.0527780	−	0.998606i	$-0.516808\pi$
$359$	−2.00000	−0.105556	−0.0527780	+	0.998606i	$0.516808\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.00000i	− 0.412021i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.0000i	1.40528i	0.711546	+	0.702640i	$0.247995\pi$
$397$	28.0000i	1.40528i	−0.711546	+	0.702640i	$0.752005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	40.0000i	1.99254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	16.0000i	0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$	16.0000i	0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 48.0000i	− 2.29615i
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 28.0000i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$443$	− 28.0000i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.0000	1.22702	0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000	1.22702	0.613508	+	0.789689i	$0.289758\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 30.0000i	− 1.40334i	−0.712502	−	0.701670i	$-0.752438\pi$
$457$	− 30.0000i	− 1.40334i	0.712502	−	0.701670i	$-0.247562\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	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
$469$	12.0000	0.554109
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000i	0.269137i
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	32.0000i	1.39926i	0.714504	+	0.699631i	$0.246652\pi$
$523$	32.0000i	1.39926i	−0.714504	+	0.699631i	$0.753348\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.0000i	2.61364i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	8.00000i	0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 28.0000i	− 1.18006i	−0.807382	−	0.590030i	$-0.799116\pi$
$563$	− 28.0000i	− 1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.00000i	0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$	8.00000i	0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	8.00000i	0.331326i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 22.0000i	− 0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$	− 22.0000i	− 0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−34.0000	−1.38920	−0.694601	−	0.719395i	$-0.744419\pi$
$599$	−34.0000	−1.38920	−0.694601	+	0.719395i	$0.744419\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 20.0000i	− 0.811775i	−0.913923	−	0.405887i	$-0.866962\pi$
$607$	− 20.0000i	− 0.811775i	0.913923	−	0.405887i	$-0.133038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.0000	1.29458
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.0000i	− 1.28827i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	− 32.0000i	− 1.28827i	0.764911	−	0.644136i	$-0.222783\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 10.0000i	− 0.400642i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 16.0000i	− 0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$	− 16.0000i	− 0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34.0000i	− 1.30673i	−0.757045	−	0.653363i	$-0.773358\pi$
$677$	− 34.0000i	− 1.30673i	0.757045	−	0.653363i	$-0.226642\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.0000	0.609551
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 60.0000i	− 2.27266i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	80.0000i	2.99602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	44.0000i	1.62518i	0.582838	+	0.812589i	$0.301942\pi$
$733$	44.0000i	1.62518i	−0.582838	+	0.812589i	$0.698058\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.0000i	0.884051i
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.0000i	0.733729i	0.930274	+	0.366864i	$0.119569\pi$
$743$	20.0000i	0.733729i	−0.930274	+	0.366864i	$0.880431\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 18.0000i	− 0.651644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 32.0000i	− 1.15545i
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	60.0000	2.14972
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24.0000i	0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.0000i	0.839654i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.00000i	0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$	8.00000i	0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.00000i	− 0.240523i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16.0000	0.548473
$852$	0	0
$853$	4.00000i	0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$	4.00000i	0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38.0000i	− 1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$	− 38.0000i	− 1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$	0	0
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 26.0000i	− 0.877958i	−0.898497	−	0.438979i	$-0.855340\pi$
$877$	− 26.0000i	− 0.877958i	0.898497	−	0.438979i	$-0.144660\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	0	0
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.0000	−0.667037
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.0000	−0.728893	−0.364446	−	0.931224i	$-0.618742\pi$
$911$	−22.0000	−0.728893	−0.364446	+	0.931224i	$0.618742\pi$
$912$	0	0
$913$	− 8.00000i	− 0.264761i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 12.0000i	− 0.392023i	−0.980602	−	0.196011i	$-0.937201\pi$
$937$	− 12.0000i	− 0.392023i	0.980602	−	0.196011i	$-0.0627990\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	− 80.0000i	− 2.60516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	0	0
$949$	48.0000	1.55815
$950$	0	0
$951$	0	0
$952$	0	0
$953$	32.0000i	1.03658i	0.855204	+	0.518291i	$0.173432\pi$
$953$	32.0000i	1.03658i	−0.855204	+	0.518291i	$0.826568\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.00000i	0.255160i	0.991828	+	0.127580i	$0.0407210\pi$
$983$	8.00000i	0.255160i	−0.991828	+	0.127580i	$0.959279\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000	1.01754
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.0000i	1.64686i	0.567420	+	0.823428i	$0.307941\pi$
$997$	52.0000i	1.64686i	−0.567420	+	0.823428i	$0.692059\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.k.e.6049.2		2
3.2	odd	2		2100.2.k.g.1849.1		2
5.2	odd	4		1260.2.a.g.1.1		1
5.3	odd	4		6300.2.a.v.1.1		1
5.4	even	2	inner	6300.2.k.e.6049.1		2
15.2	even	4		420.2.a.a.1.1	✓	1
15.8	even	4		2100.2.a.q.1.1		1
15.14	odd	2		2100.2.k.g.1849.2		2
20.7	even	4		5040.2.a.bl.1.1		1
35.27	even	4		8820.2.a.f.1.1		1
60.23	odd	4		8400.2.a.c.1.1		1
60.47	odd	4		1680.2.a.n.1.1		1
105.2	even	12		2940.2.q.m.361.1		2
105.17	odd	12		2940.2.q.a.961.1		2
105.32	even	12		2940.2.q.m.961.1		2
105.47	odd	12		2940.2.q.a.361.1		2
105.62	odd	4		2940.2.a.l.1.1		1
120.77	even	4		6720.2.a.cb.1.1		1
120.107	odd	4		6720.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.a.a.1.1	✓	1	15.2	even	4
1260.2.a.g.1.1		1	5.2	odd	4
1680.2.a.n.1.1		1	60.47	odd	4
2100.2.a.q.1.1		1	15.8	even	4
2100.2.k.g.1849.1		2	3.2	odd	2
2100.2.k.g.1849.2		2	15.14	odd	2
2940.2.a.l.1.1		1	105.62	odd	4
2940.2.q.a.361.1		2	105.47	odd	12
2940.2.q.a.961.1		2	105.17	odd	12
2940.2.q.m.361.1		2	105.2	even	12
2940.2.q.m.961.1		2	105.32	even	12
5040.2.a.bl.1.1		1	20.7	even	4
6300.2.a.v.1.1		1	5.3	odd	4
6300.2.k.e.6049.1		2	5.4	even	2	inner
6300.2.k.e.6049.2		2	1.1	even	1	trivial
6720.2.a.bd.1.1		1	120.107	odd	4
6720.2.a.cb.1.1		1	120.77	even	4
8400.2.a.c.1.1		1	60.23	odd	4
8820.2.a.f.1.1		1	35.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{7} +O(q^{10})\)	\(q+1.00000i q^{7} -2.00000 q^{11} +4.00000i q^{13} +6.00000i q^{17} -6.00000 q^{19} +8.00000i q^{23} -2.00000 q^{29} +10.0000 q^{31} -2.00000i q^{37} -10.0000 q^{41} -4.00000i q^{43} -8.00000i q^{47} -1.00000 q^{49} -4.00000i q^{53} -8.00000 q^{59} +6.00000 q^{61} -12.0000i q^{67} +6.00000 q^{71} -12.0000i q^{73} -2.00000i q^{77} +8.00000 q^{79} +4.00000i q^{83} -10.0000 q^{89} -4.00000 q^{91} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 4 q^{11} - 12 q^{19} - 4 q^{29} + 20 q^{31} - 20 q^{41} - 2 q^{49} - 16 q^{59} + 12 q^{61} + 12 q^{71} + 16 q^{79} - 20 q^{89} - 8 q^{91}+O(q^{100}) \) 2 * q - 4 * q^11 - 12 * q^19 - 4 * q^29 + 20 * q^31 - 20 * q^41 - 2 * q^49 - 16 * q^59 + 12 * q^61 + 12 * q^71 + 16 * q^79 - 20 * q^89 - 8 * q^91

Introduction

L-functions

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