LMFDB - Embedded newform 6300.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1,6300)]

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, 0, 0]))

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

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

chi := DirichletCharacter("6300.1");

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.a (trivial)

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:	yes
Analytic conductor:	$50.3057532734$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	6300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{7} -2.64575 q^{11} -5.29150 q^{17} +7.93725 q^{23} +7.93725 q^{29} -10.0000 q^{31} +11.0000 q^{37} +10.5830 q^{41} +1.00000 q^{43} +10.5830 q^{47} +1.00000 q^{49} -10.5830 q^{53} -10.5830 q^{59} -8.00000 q^{61} -3.00000 q^{67} +2.64575 q^{71} -10.0000 q^{73} +2.64575 q^{77} -11.0000 q^{79} -5.29150 q^{83} -10.5830 q^{89} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^7	$ 2 q - 2 q^{7} - 20 q^{31} + 22 q^{37} + 2 q^{43} + 2 q^{49} - 16 q^{61} - 6 q^{67} - 20 q^{73} - 22 q^{79} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 20 * q^31 + 22 * q^37 + 2 * q^43 + 2 * q^49 - 16 * q^61 - 6 * q^67 - 20 * q^73 - 22 * q^79 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.64575	−0.797724	−0.398862	−	0.917011i	$-0.630595\pi$
$11$	−2.64575	−0.797724	−0.398862	+	0.917011i	$0.630595\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.29150	−1.28338	−0.641689	−	0.766965i	$-0.721766\pi$
$17$	−5.29150	−1.28338	−0.641689	+	0.766965i	$0.721766\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.93725	1.65503	0.827516	−	0.561442i	$-0.189753\pi$
$23$	7.93725	1.65503	0.827516	+	0.561442i	$0.189753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.93725	1.47391	0.736956	−	0.675941i	$-0.236263\pi$
$29$	7.93725	1.47391	0.736956	+	0.675941i	$0.236263\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.5830	1.65279	0.826394	−	0.563093i	$-0.190389\pi$
$41$	10.5830	1.65279	0.826394	+	0.563093i	$0.190389\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.5830	1.54369	0.771845	−	0.635811i	$-0.219334\pi$
$47$	10.5830	1.54369	0.771845	+	0.635811i	$0.219334\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.5830	−1.45369	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	−10.5830	−1.45369	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.5830	−1.37779	−0.688895	−	0.724861i	$-0.741904\pi$
$59$	−10.5830	−1.37779	−0.688895	+	0.724861i	$0.741904\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.64575	0.313993	0.156996	−	0.987599i	$-0.449819\pi$
$71$	2.64575	0.313993	0.156996	+	0.987599i	$0.449819\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.64575	0.301511
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.29150	−0.580818	−0.290409	−	0.956903i	$-0.593791\pi$
$83$	−5.29150	−0.580818	−0.290409	+	0.956903i	$0.593791\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.5830	−1.12180	−0.560898	−	0.827885i	$-0.689544\pi$
$89$	−10.5830	−1.12180	−0.560898	+	0.827885i	$0.689544\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.8745	−1.57957	−0.789786	−	0.613382i	$-0.789809\pi$
$101$	−15.8745	−1.57957	−0.789786	+	0.613382i	$0.789809\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.8745	−1.53465	−0.767323	−	0.641260i	$-0.778412\pi$
$107$	−15.8745	−1.53465	−0.767323	+	0.641260i	$0.778412\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.64575	0.248891	0.124446	−	0.992226i	$-0.460285\pi$
$113$	2.64575	0.248891	0.124446	+	0.992226i	$0.460285\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.29150	0.485071
$120$	0	0
$121$	−4.00000	−0.363636
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.29150	−0.462321	−0.231160	−	0.972916i	$-0.574252\pi$
$131$	−5.29150	−0.462321	−0.231160	+	0.972916i	$0.574252\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.93725	0.650245	0.325123	−	0.945672i	$-0.394594\pi$
$149$	7.93725	0.650245	0.325123	+	0.945672i	$0.394594\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.93725	−0.625543
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.5830	0.804611	0.402305	−	0.915505i	$-0.368209\pi$
$173$	10.5830	0.804611	0.402305	+	0.915505i	$0.368209\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.8745	1.18652	0.593258	−	0.805012i	$-0.297841\pi$
$179$	15.8745	1.18652	0.593258	+	0.805012i	$0.297841\pi$
$180$	0	0
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.0000	1.02378
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.8745	1.14864	0.574320	−	0.818631i	$-0.305267\pi$
$191$	15.8745	1.14864	0.574320	+	0.818631i	$0.305267\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.5203	1.31951	0.659757	−	0.751479i	$-0.270659\pi$
$197$	18.5203	1.31951	0.659757	+	0.751479i	$0.270659\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.93725	−0.557086
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.5830	0.702419	0.351209	−	0.936297i	$-0.385771\pi$
$227$	10.5830	0.702419	0.351209	+	0.936297i	$0.385771\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.93725	−0.519987	−0.259993	−	0.965610i	$-0.583720\pi$
$233$	−7.93725	−0.519987	−0.259993	+	0.965610i	$0.583720\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.4575	−1.71139	−0.855697	−	0.517477i	$-0.826871\pi$
$239$	−26.4575	−1.71139	−0.855697	+	0.517477i	$0.826871\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.8745	−1.00199	−0.500995	−	0.865450i	$-0.667033\pi$
$251$	−15.8745	−1.00199	−0.500995	+	0.865450i	$0.667033\pi$
$252$	0	0
$253$	−21.0000	−1.32026
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.29150	−0.330075	−0.165037	−	0.986287i	$-0.552774\pi$
$257$	−5.29150	−0.330075	−0.165037	+	0.986287i	$0.552774\pi$
$258$	0	0
$259$	−11.0000	−0.683507
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.5203	1.14201	0.571004	−	0.820947i	$-0.306554\pi$
$263$	18.5203	1.14201	0.571004	+	0.820947i	$0.306554\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.8745	0.967886	0.483943	−	0.875100i	$-0.339204\pi$
$269$	15.8745	0.967886	0.483943	+	0.875100i	$0.339204\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.5203	1.10483	0.552413	−	0.833571i	$-0.313707\pi$
$281$	18.5203	1.10483	0.552413	+	0.833571i	$0.313707\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.5830	−0.624695
$288$	0	0
$289$	11.0000	0.647059
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.1660	1.23653	0.618266	−	0.785969i	$-0.287836\pi$
$293$	21.1660	1.23653	0.618266	+	0.785969i	$0.287836\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.8745	0.900161	0.450080	−	0.892988i	$-0.351395\pi$
$311$	15.8745	0.900161	0.450080	+	0.892988i	$0.351395\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.64575	−0.148600	−0.0743001	−	0.997236i	$-0.523672\pi$
$317$	−2.64575	−0.148600	−0.0743001	+	0.997236i	$0.523672\pi$
$318$	0	0
$319$	−21.0000	−1.17577
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.5830	−0.583460
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.4575	1.43275
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.5203	−0.994220	−0.497110	−	0.867688i	$-0.665605\pi$
$347$	−18.5203	−0.994220	−0.497110	+	0.867688i	$0.665605\pi$
$348$	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
$353$	−15.8745	−0.844915	−0.422457	−	0.906383i	$-0.638832\pi$
$353$	−15.8745	−0.844915	−0.422457	+	0.906383i	$0.638832\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.64575	0.139637	0.0698187	−	0.997560i	$-0.477758\pi$
$359$	2.64575	0.139637	0.0698187	+	0.997560i	$0.477758\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.5830	0.549442
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.7490	−1.62230	−0.811149	−	0.584839i	$-0.801158\pi$
$383$	−31.7490	−1.62230	−0.811149	+	0.584839i	$0.801158\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.1033	−1.47559	−0.737797	−	0.675023i	$-0.764134\pi$
$389$	−29.1033	−1.47559	−0.737797	+	0.675023i	$0.764134\pi$
$390$	0	0
$391$	−42.0000	−2.12403
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.3948	−1.71759	−0.858796	−	0.512317i	$-0.828787\pi$
$401$	−34.3948	−1.71759	−0.858796	+	0.512317i	$0.828787\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.1033	−1.44260
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.5830	0.520756
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15.8745	0.775520	0.387760	−	0.921760i	$-0.373249\pi$
$419$	15.8745	0.775520	0.387760	+	0.921760i	$0.373249\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.8745	−0.764648	−0.382324	−	0.924028i	$-0.624876\pi$
$431$	−15.8745	−0.764648	−0.382324	+	0.924028i	$0.624876\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	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
$443$	−26.4575	−1.25703	−0.628517	−	0.777796i	$-0.716338\pi$
$443$	−26.4575	−1.25703	−0.628517	+	0.777796i	$0.716338\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.93725	0.374582	0.187291	−	0.982304i	$-0.440029\pi$
$449$	7.93725	0.374582	0.187291	+	0.982304i	$0.440029\pi$
$450$	0	0
$451$	−28.0000	−1.31847
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.5830	−0.492900	−0.246450	−	0.969156i	$-0.579264\pi$
$461$	−10.5830	−0.492900	−0.246450	+	0.969156i	$0.579264\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4575	1.22431	0.612154	−	0.790739i	$-0.290303\pi$
$467$	26.4575	1.22431	0.612154	+	0.790739i	$0.290303\pi$
$468$	0	0
$469$	3.00000	0.138527
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.64575	−0.121652
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.7490	1.45065	0.725325	−	0.688407i	$-0.241690\pi$
$479$	31.7490	1.45065	0.725325	+	0.688407i	$0.241690\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.6863	−1.79102	−0.895508	−	0.445045i	$-0.853188\pi$
$491$	−39.6863	−1.79102	−0.895508	+	0.445045i	$0.853188\pi$
$492$	0	0
$493$	−42.0000	−1.89158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.64575	−0.118678
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.8745	0.707809	0.353905	−	0.935282i	$-0.384854\pi$
$503$	15.8745	0.707809	0.353905	+	0.935282i	$0.384854\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.1660	0.938167	0.469083	−	0.883154i	$-0.344584\pi$
$509$	21.1660	0.938167	0.469083	+	0.883154i	$0.344584\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−28.0000	−1.23144
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.4575	1.15912	0.579562	−	0.814928i	$-0.303224\pi$
$521$	26.4575	1.15912	0.579562	+	0.814928i	$0.303224\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	52.9150	2.30501
$528$	0	0
$529$	40.0000	1.73913
$530$	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
$539$	−2.64575	−0.113961
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.00000	0.384812	0.192406	−	0.981315i	$-0.438371\pi$
$547$	9.00000	0.384812	0.192406	+	0.981315i	$0.438371\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	11.0000	0.467768
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.2288	0.560520	0.280260	−	0.959924i	$-0.409579\pi$
$557$	13.2288	0.560520	0.280260	+	0.959924i	$0.409579\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.4575	1.11505	0.557526	−	0.830160i	$-0.311751\pi$
$563$	26.4575	1.11505	0.557526	+	0.830160i	$0.311751\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.8118	−0.998241	−0.499120	−	0.866533i	$-0.666343\pi$
$569$	−23.8118	−0.998241	−0.499120	+	0.866533i	$0.666343\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.29150	0.219529
$582$	0	0
$583$	28.0000	1.15964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.4575	1.09202	0.546009	−	0.837779i	$-0.316146\pi$
$587$	26.4575	1.09202	0.546009	+	0.837779i	$0.316146\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.7490	1.30378	0.651888	−	0.758315i	$-0.273977\pi$
$593$	31.7490	1.30378	0.651888	+	0.758315i	$0.273977\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.93725	−0.324307	−0.162154	−	0.986766i	$-0.551844\pi$
$599$	−7.93725	−0.324307	−0.162154	+	0.986766i	$0.551844\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−3.00000	−0.121169	−0.0605844	−	0.998163i	$-0.519296\pi$
$613$	−3.00000	−0.121169	−0.0605844	+	0.998163i	$0.519296\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.2288	−0.532570	−0.266285	−	0.963894i	$-0.585796\pi$
$617$	−13.2288	−0.532570	−0.266285	+	0.963894i	$0.585796\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.5830	0.423999
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−58.2065	−2.32085
$630$	0	0
$631$	41.0000	1.63218	0.816092	−	0.577922i	$-0.196136\pi$
$631$	41.0000	1.63218	0.816092	+	0.577922i	$0.196136\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.64575	0.104501	0.0522504	−	0.998634i	$-0.483361\pi$
$641$	2.64575	0.104501	0.0522504	+	0.998634i	$0.483361\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.4575	−1.04015	−0.520076	−	0.854120i	$-0.674096\pi$
$647$	−26.4575	−1.04015	−0.520076	+	0.854120i	$0.674096\pi$
$648$	0	0
$649$	28.0000	1.09910
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.5830	0.414145	0.207072	−	0.978326i	$-0.433606\pi$
$653$	10.5830	0.414145	0.207072	+	0.978326i	$0.433606\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.29150	0.206128	0.103064	−	0.994675i	$-0.467135\pi$
$659$	5.29150	0.206128	0.103064	+	0.994675i	$0.467135\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	63.0000	2.43937
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.1660	0.817105
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.6235	−1.83032	−0.915160	−	0.403090i	$-0.867936\pi$
$677$	−47.6235	−1.83032	−0.915160	+	0.403090i	$0.867936\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.64575	0.101237	0.0506184	−	0.998718i	$-0.483881\pi$
$683$	2.64575	0.101237	0.0506184	+	0.998718i	$0.483881\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	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
$697$	−56.0000	−2.12115
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10.5830	−0.399715	−0.199857	−	0.979825i	$-0.564048\pi$
$701$	−10.5830	−0.399715	−0.199857	+	0.979825i	$0.564048\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.8745	0.597022
$708$	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
$713$	−79.3725	−2.97252
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.29150	−0.195713
$732$	0	0
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.93725	0.292373
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.8745	−0.582379	−0.291190	−	0.956665i	$-0.594051\pi$
$743$	−15.8745	−0.582379	−0.291190	+	0.956665i	$0.594051\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.8745	0.580042
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.0000	1.56286	0.781431	−	0.623992i	$-0.214490\pi$
$757$	43.0000	1.56286	0.781431	+	0.623992i	$0.214490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.7490	−1.15090	−0.575450	−	0.817837i	$-0.695173\pi$
$761$	−31.7490	−1.15090	−0.575450	+	0.817837i	$0.695173\pi$
$762$	0	0
$763$	−3.00000	−0.108607
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15.8745	−0.570966	−0.285483	−	0.958384i	$-0.592154\pi$
$773$	−15.8745	−0.570966	−0.285483	+	0.958384i	$0.592154\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−7.00000	−0.250480
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.64575	−0.0940721
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−47.6235	−1.68691	−0.843456	−	0.537198i	$-0.819483\pi$
$797$	−47.6235	−1.68691	−0.843456	+	0.537198i	$0.819483\pi$
$798$	0	0
$799$	−56.0000	−1.98114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26.4575	0.933665
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	55.5608	1.95341	0.976706	−	0.214580i	$-0.0688382\pi$
$809$	55.5608	1.95341	0.976706	+	0.214580i	$0.0688382\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.5203	0.644013	0.322006	−	0.946738i	$-0.395643\pi$
$827$	18.5203	0.644013	0.322006	+	0.946738i	$0.395643\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.29150	−0.183340
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.4575	0.913415	0.456707	−	0.889617i	$-0.349029\pi$
$839$	26.4575	0.913415	0.456707	+	0.889617i	$0.349029\pi$
$840$	0	0
$841$	34.0000	1.17241
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.00000	0.137442
$848$	0	0
$849$	0	0
$850$	0	0
$851$	87.3098	2.99294
$852$	0	0
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.7490	1.08453	0.542263	−	0.840209i	$-0.317568\pi$
$857$	31.7490	1.08453	0.542263	+	0.840209i	$0.317568\pi$
$858$	0	0
$859$	−42.0000	−1.43302	−0.716511	−	0.697576i	$-0.754262\pi$
$859$	−42.0000	−1.43302	−0.716511	+	0.697576i	$0.754262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.64575	−0.0900624	−0.0450312	−	0.998986i	$-0.514339\pi$
$863$	−2.64575	−0.0900624	−0.0450312	+	0.998986i	$0.514339\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	29.1033	0.987261
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.29150	−0.178275	−0.0891376	−	0.996019i	$-0.528411\pi$
$881$	−5.29150	−0.178275	−0.0891376	+	0.996019i	$0.528411\pi$
$882$	0	0
$883$	−9.00000	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−52.9150	−1.77671	−0.888356	−	0.459155i	$-0.848152\pi$
$887$	−52.9150	−1.77671	−0.888356	+	0.459155i	$0.848152\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−79.3725	−2.64722
$900$	0	0
$901$	56.0000	1.86563
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.8118	0.788919	0.394459	−	0.918913i	$-0.370932\pi$
$911$	23.8118	0.788919	0.394459	+	0.918913i	$0.370932\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.29150	0.174741
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.29150	0.173609	0.0868043	−	0.996225i	$-0.472335\pi$
$929$	5.29150	0.173609	0.0868043	+	0.996225i	$0.472335\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−40.0000	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.29150	−0.172498	−0.0862490	−	0.996274i	$-0.527488\pi$
$941$	−5.29150	−0.172498	−0.0862490	+	0.996274i	$0.527488\pi$
$942$	0	0
$943$	84.0000	2.73542
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.0405	−1.20366	−0.601828	−	0.798626i	$-0.705561\pi$
$947$	−37.0405	−1.20366	−0.601828	+	0.798626i	$0.705561\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.8118	0.771339	0.385669	−	0.922637i	$-0.373971\pi$
$953$	23.8118	0.771339	0.385669	+	0.922637i	$0.373971\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.8118	0.761806	0.380903	−	0.924615i	$-0.375613\pi$
$977$	23.8118	0.761806	0.380903	+	0.924615i	$0.375613\pi$
$978$	0	0
$979$	28.0000	0.894884
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.4575	0.843864	0.421932	−	0.906628i	$-0.361352\pi$
$983$	26.4575	0.843864	0.421932	+	0.906628i	$0.361352\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.93725	0.252390
$990$	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
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.a.bh.1.1	✓	2
3.2	odd	2	inner	6300.2.a.bh.1.2	yes	2
5.2	odd	4		6300.2.k.t.6049.1		4
5.3	odd	4		6300.2.k.t.6049.3		4
5.4	even	2		6300.2.a.bi.1.1	yes	2
15.2	even	4		6300.2.k.t.6049.2		4
15.8	even	4		6300.2.k.t.6049.4		4
15.14	odd	2		6300.2.a.bi.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6300.2.a.bh.1.1	✓	2	1.1	even	1	trivial
6300.2.a.bh.1.2	yes	2	3.2	odd	2	inner
6300.2.a.bi.1.1	yes	2	5.4	even	2
6300.2.a.bi.1.2	yes	2	15.14	odd	2
6300.2.k.t.6049.1		4	5.2	odd	4
6300.2.k.t.6049.2		4	15.2	even	4
6300.2.k.t.6049.3		4	5.3	odd	4
6300.2.k.t.6049.4		4	15.8	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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{7} -2.64575 q^{11} -5.29150 q^{17} +7.93725 q^{23} +7.93725 q^{29} -10.0000 q^{31} +11.0000 q^{37} +10.5830 q^{41} +1.00000 q^{43} +10.5830 q^{47} +1.00000 q^{49} -10.5830 q^{53} -10.5830 q^{59} -8.00000 q^{61} -3.00000 q^{67} +2.64575 q^{71} -10.0000 q^{73} +2.64575 q^{77} -11.0000 q^{79} -5.29150 q^{83} -10.5830 q^{89} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^7	\( 2 q - 2 q^{7} - 20 q^{31} + 22 q^{37} + 2 q^{43} + 2 q^{49} - 16 q^{61} - 6 q^{67} - 20 q^{73} - 22 q^{79} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 20 * q^31 + 22 * q^37 + 2 * q^43 + 2 * q^49 - 16 * q^61 - 6 * q^67 - 20 * q^73 - 22 * q^79 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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