LMFDB - Embedded newform 6300.2.k.h.6049.1

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 2100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{7} -1.00000 q^{11} +4.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} -7.00000i q^{23} -9.00000 q^{29} -2.00000 q^{31} -1.00000i q^{37} -8.00000 q^{41} -9.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +4.00000 q^{59} +4.00000 q^{61} -9.00000i q^{67} -5.00000 q^{71} +10.0000i q^{73} +1.00000i q^{77} +15.0000 q^{79} +6.00000i q^{83} +8.00000 q^{89} +4.00000 q^{91} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 2 q^{11} + 8 q^{19} - 18 q^{29} - 4 q^{31} - 16 q^{41} - 2 q^{49} + 8 q^{59} + 8 q^{61} - 10 q^{71} + 30 q^{79} + 16 q^{89} + 8 q^{91}+O(q^{100}) $ 2 * q - 2 * q^11 + 8 * q^19 - 18 * q^29 - 4 * q^31 - 16 * q^41 - 2 * q^49 + 8 * q^59 + 8 * q^61 - 10 * q^71 + 30 * q^79 + 16 * 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
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	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
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 7.00000i	− 1.45960i	−0.683660	−	0.729800i	$-0.739613\pi$
$23$	− 7.00000i	− 1.45960i	0.683660	−	0.729800i	$-0.260387\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.00000i	− 1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$	− 9.00000i	− 1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00000i	0.113961i
$78$	0	0
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	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
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 13.0000i	− 1.22294i	−0.791269	−	0.611469i	$-0.790579\pi$
$113$	− 13.0000i	− 1.22294i	0.791269	−	0.611469i	$-0.209421\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.00000i	− 0.798621i	−0.916816	−	0.399310i	$-0.869250\pi$
$127$	− 9.00000i	− 0.798621i	0.916816	−	0.399310i	$-0.130750\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	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
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 20.0000i	− 1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$	− 20.0000i	− 1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.00000	−0.551677
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.00000i	− 0.608229i	−0.952636	−	0.304114i	$-0.901639\pi$
$173$	− 8.00000i	− 0.608229i	0.952636	−	0.304114i	$-0.0983605\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	− 17.0000i	− 1.22369i	−0.790979	−	0.611843i	$-0.790428\pi$
$193$	− 17.0000i	− 1.22369i	0.790979	−	0.611843i	$-0.209572\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 15.0000i	− 1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$	− 15.0000i	− 1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.00000i	0.631676i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.00000	−0.276686
$210$	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
$217$	2.00000i	0.135769i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	− 6.00000i	− 0.401790i	−0.979613	−	0.200895i	$-0.935615\pi$
$223$	− 6.00000i	− 0.401790i	0.979613	−	0.200895i	$-0.0643850\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.0000i	− 1.24473i	−0.782727	−	0.622366i	$-0.786172\pi$
$233$	− 19.0000i	− 1.24473i	0.782727	−	0.622366i	$-0.213828\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	7.00000i	0.440086i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.0000i	1.62184i	0.585160	+	0.810918i	$0.301032\pi$
$257$	26.0000i	1.62184i	−0.585160	+	0.810918i	$0.698968\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.0000i	0.924940i	0.886635	+	0.462470i	$0.153037\pi$
$263$	15.0000i	0.924940i	−0.886635	+	0.462470i	$0.846963\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.0000	−1.13344	−0.566722	−	0.823909i	$-0.691789\pi$
$281$	−19.0000	−1.13344	−0.566722	+	0.823909i	$0.691789\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000i	0.472225i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	28.0000	1.61928
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.0000i	1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$	22.0000i	1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	4.00000i	0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$	4.00000i	0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 31.0000i	− 1.74113i	−0.492050	−	0.870567i	$-0.663752\pi$
$317$	− 31.0000i	− 1.74113i	0.492050	−	0.870567i	$-0.336248\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−23.0000	−1.26419	−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000	−1.26419	−0.632097	+	0.774889i	$0.717806\pi$
$332$	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
$341$	2.00000	0.108306
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 27.0000i	− 1.44944i	−0.689046	−	0.724718i	$-0.741970\pi$
$347$	− 27.0000i	− 1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−35.0000	−1.84723	−0.923615	−	0.383322i	$-0.874780\pi$
$359$	−35.0000	−1.84723	−0.923615	+	0.383322i	$0.874780\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	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
$371$	−6.00000	−0.311504
$372$	0	0
$373$	11.0000i	0.569558i	0.958593	+	0.284779i	$0.0919203\pi$
$373$	11.0000i	0.569558i	−0.958593	+	0.284779i	$0.908080\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 36.0000i	− 1.85409i
$378$	0	0
$379$	23.0000	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.0000	1.36895	0.684477	−	0.729034i	$-0.260031\pi$
$389$	27.0000	1.36895	0.684477	+	0.729034i	$0.260031\pi$
$390$	0	0
$391$	14.0000	0.708010
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.00000i	0.0495682i
$408$	0	0
$409$	−40.0000	−1.97787	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000	−1.97787	−0.988936	+	0.148340i	$0.952607\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.00000i	− 0.193574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 28.0000i	− 1.33942i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.0000	0.613508	0.306754	−	0.951789i	$-0.400757\pi$
$449$	13.0000	0.613508	0.306754	+	0.951789i	$0.400757\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.00000i	0.233890i	0.993138	+	0.116945i	$0.0373101\pi$
$457$	5.00000i	0.233890i	−0.993138	+	0.116945i	$0.962690\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 38.0000i	− 1.75843i	−0.476425	−	0.879215i	$-0.658068\pi$
$467$	− 38.0000i	− 1.75843i	0.476425	−	0.879215i	$-0.341932\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.00000i	0.413820i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.00000i	− 0.135943i	−0.997687	−	0.0679715i	$-0.978347\pi$
$487$	− 3.00000i	− 0.135943i	0.997687	−	0.0679715i	$-0.0216527\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.00000i	0.224281i
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.0000i	0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$	14.0000i	0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.00000i	− 0.175920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 40.0000i	− 1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$523$	− 40.0000i	− 1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	−26.0000	−1.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 32.0000i	− 1.38607i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.0000i	1.49649i	0.663421	+	0.748246i	$0.269104\pi$
$547$	35.0000i	1.49649i	−0.663421	+	0.748246i	$0.730896\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	− 15.0000i	− 0.637865i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.00000i	0.0423714i	0.999776	+	0.0211857i	$0.00674412\pi$
$557$	1.00000i	0.0423714i	−0.999776	+	0.0211857i	$0.993256\pi$
$558$	0	0
$559$	36.0000	1.52264
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.0000i	1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$	30.0000i	1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−45.0000	−1.88650	−0.943249	−	0.332086i	$-0.892248\pi$
$569$	−45.0000	−1.88650	−0.943249	+	0.332086i	$0.892248\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.00000i	− 0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$	− 8.00000i	− 0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	29.0000i	1.17130i	0.810564	+	0.585649i	$0.199160\pi$
$613$	29.0000i	1.17130i	−0.810564	+	0.585649i	$0.800840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.0000i	1.24801i	0.781419	+	0.624007i	$0.214496\pi$
$617$	31.0000i	1.24801i	−0.781419	+	0.624007i	$0.785504\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	−27.0000	−1.07485	−0.537427	−	0.843311i	$-0.680603\pi$
$631$	−27.0000	−1.07485	−0.537427	+	0.843311i	$0.680603\pi$
$632$	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
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.0000i	1.80845i	0.427060	+	0.904223i	$0.359549\pi$
$647$	46.0000i	1.80845i	−0.427060	+	0.904223i	$0.640451\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	63.0000i	2.43937i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.0000i	0.727015i	0.931591	+	0.363507i	$0.118421\pi$
$683$	19.0000i	0.727015i	−0.931591	+	0.363507i	$0.881579\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 16.0000i	− 0.606043i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.0000i	0.524304i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 30.0000i	− 1.11264i	−0.830969	−	0.556319i	$-0.812213\pi$
$727$	− 30.0000i	− 1.11264i	0.830969	−	0.556319i	$-0.187787\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.0000	0.665754
$732$	0	0
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.00000i	0.331519i
$738$	0	0
$739$	9.00000	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.00000i	0.254419i	0.991876	+	0.127210i	$0.0406021\pi$
$757$	7.00000i	0.254419i	−0.991876	+	0.127210i	$0.959398\pi$
$758$	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
$763$	1.00000i	0.0362024i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.0000i	0.577727i
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	5.00000	0.178914
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 26.0000i	− 0.926800i	−0.886149	−	0.463400i	$-0.846629\pi$
$787$	− 26.0000i	− 0.926800i	0.886149	−	0.463400i	$-0.153371\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.0000	−0.462227
$792$	0	0
$793$	16.0000i	0.568177i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 54.0000i	− 1.91278i	−0.292096	−	0.956389i	$-0.594353\pi$
$797$	− 54.0000i	− 1.91278i	0.292096	−	0.956389i	$-0.405647\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.0000	−0.808637	−0.404318	−	0.914618i	$-0.632491\pi$
$809$	−23.0000	−0.808637	−0.404318	+	0.914618i	$0.632491\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 36.0000i	− 1.25948i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	− 11.0000i	− 0.383436i	−0.981450	−	0.191718i	$-0.938594\pi$
$823$	− 11.0000i	− 0.383436i	0.981450	−	0.191718i	$-0.0614059\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.0000i	1.14752i	0.819023	+	0.573761i	$0.194516\pi$
$827$	33.0000i	1.14752i	−0.819023	+	0.573761i	$0.805484\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	28.0000i	0.958702i	0.877623	+	0.479351i	$0.159128\pi$
$853$	28.0000i	0.958702i	−0.877623	+	0.479351i	$0.840872\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 4.00000i	− 0.136637i	−0.997664	−	0.0683187i	$-0.978237\pi$
$857$	− 4.00000i	− 0.136637i	0.997664	−	0.0683187i	$-0.0217635\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 33.0000i	− 1.12333i	−0.827364	−	0.561667i	$-0.810160\pi$
$863$	− 33.0000i	− 1.12333i	0.827364	−	0.561667i	$-0.189840\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	36.0000	1.21981
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	1.00000i	0.0336527i	0.999858	+	0.0168263i	$0.00535624\pi$
$883$	1.00000i	0.0336527i	−0.999858	+	0.0168263i	$0.994644\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.0000i	1.74599i	0.487730	+	0.872995i	$0.337825\pi$
$887$	52.0000i	1.74599i	−0.487730	+	0.872995i	$0.662175\pi$
$888$	0	0
$889$	−9.00000	−0.301850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.0000	0.600334
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 24.0000i	− 0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$	− 24.0000i	− 0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.0000i	0.594412i
$918$	0	0
$919$	59.0000	1.94623	0.973115	−	0.230319i	$-0.0739769\pi$
$919$	59.0000	1.94623	0.973115	+	0.230319i	$0.0739769\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 20.0000i	− 0.658308i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 36.0000i	− 1.17607i	−0.808836	−	0.588034i	$-0.799902\pi$
$937$	− 36.0000i	− 1.17607i	0.808836	−	0.588034i	$-0.200098\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	0	0
$943$	56.0000i	1.82361i
$944$	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
$949$	−40.0000	−1.29845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 21.0000i	− 0.680257i	−0.940379	−	0.340128i	$-0.889529\pi$
$953$	− 21.0000i	− 0.680257i	0.940379	−	0.340128i	$-0.110471\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	−27.0000	−0.870968
$962$	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
$973$	− 10.0000i	− 0.320585i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.0000i	− 1.05576i	−0.849318	−	0.527882i	$-0.822986\pi$
$977$	− 33.0000i	− 1.05576i	0.849318	−	0.527882i	$-0.177014\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	0	0
$983$	58.0000i	1.84991i	0.380073	+	0.924956i	$0.375899\pi$
$983$	58.0000i	1.84991i	−0.380073	+	0.924956i	$0.624101\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−63.0000	−2.00328
$990$	0	0
$991$	37.0000	1.17534	0.587672	−	0.809099i	$-0.300045\pi$
$991$	37.0000	1.17534	0.587672	+	0.809099i	$0.300045\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.k.h.6049.1		2
3.2	odd	2		2100.2.k.f.1849.1		2
5.2	odd	4		6300.2.a.x.1.1		1
5.3	odd	4		6300.2.a.f.1.1		1
5.4	even	2	inner	6300.2.k.h.6049.2		2
15.2	even	4		2100.2.a.g.1.1	✓	1
15.8	even	4		2100.2.a.m.1.1	yes	1
15.14	odd	2		2100.2.k.f.1849.2		2
60.23	odd	4		8400.2.a.x.1.1		1
60.47	odd	4		8400.2.a.bv.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.a.g.1.1	✓	1	15.2	even	4
2100.2.a.m.1.1	yes	1	15.8	even	4
2100.2.k.f.1849.1		2	3.2	odd	2
2100.2.k.f.1849.2		2	15.14	odd	2
6300.2.a.f.1.1		1	5.3	odd	4
6300.2.a.x.1.1		1	5.2	odd	4
6300.2.k.h.6049.1		2	1.1	even	1	trivial
6300.2.k.h.6049.2		2	5.4	even	2	inner
8400.2.a.x.1.1		1	60.23	odd	4
8400.2.a.bv.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{7} -1.00000 q^{11} +4.00000i q^{13} +2.00000i q^{17} +4.00000 q^{19} -7.00000i q^{23} -9.00000 q^{29} -2.00000 q^{31} -1.00000i q^{37} -8.00000 q^{41} -9.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +4.00000 q^{59} +4.00000 q^{61} -9.00000i q^{67} -5.00000 q^{71} +10.0000i q^{73} +1.00000i q^{77} +15.0000 q^{79} +6.00000i q^{83} +8.00000 q^{89} +4.00000 q^{91} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 2 q^{11} + 8 q^{19} - 18 q^{29} - 4 q^{31} - 16 q^{41} - 2 q^{49} + 8 q^{59} + 8 q^{61} - 10 q^{71} + 30 q^{79} + 16 q^{89} + 8 q^{91}+O(q^{100}) \) 2 * q - 2 * q^11 + 8 * q^19 - 18 * q^29 - 4 * q^31 - 16 * q^41 - 2 * q^49 + 8 * q^59 + 8 * q^61 - 10 * q^71 + 30 * q^79 + 16 * q^89 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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