LMFDB - Embedded newform 5148.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 572)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302776 q^{7} +O(q^{10})$	$q+0.302776 q^{7} +1.00000 q^{11} +1.00000 q^{13} -2.60555 q^{17} -5.30278 q^{19} +1.69722 q^{23} -5.00000 q^{25} +5.21110 q^{29} +4.60555 q^{31} -9.21110 q^{37} -1.69722 q^{41} -6.60555 q^{43} +6.00000 q^{47} -6.90833 q^{49} +4.69722 q^{53} -14.6056 q^{59} +9.81665 q^{61} -1.39445 q^{67} +3.39445 q^{71} -10.5139 q^{73} +0.302776 q^{77} -15.2111 q^{79} +9.51388 q^{83} +0.302776 q^{91} +2.78890 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{7}+O(q^{10}) $ 2 * q - 3 * q^7	$ 2 q - 3 q^{7} + 2 q^{11} + 2 q^{13} + 2 q^{17} - 7 q^{19} + 7 q^{23} - 10 q^{25} - 4 q^{29} + 2 q^{31} - 4 q^{37} - 7 q^{41} - 6 q^{43} + 12 q^{47} - 3 q^{49} + 13 q^{53} - 22 q^{59} - 2 q^{61} - 10 q^{67} + 14 q^{71} - 3 q^{73} - 3 q^{77} - 16 q^{79} + q^{83} - 3 q^{91} + 20 q^{97}+O(q^{100}) $ 2 * q - 3 * q^7 + 2 * q^11 + 2 * q^13 + 2 * q^17 - 7 * q^19 + 7 * q^23 - 10 * q^25 - 4 * q^29 + 2 * q^31 - 4 * q^37 - 7 * q^41 - 6 * q^43 + 12 * q^47 - 3 * q^49 + 13 * q^53 - 22 * q^59 - 2 * q^61 - 10 * q^67 + 14 * q^71 - 3 * q^73 - 3 * q^77 - 16 * q^79 + q^83 - 3 * q^91 + 20 * 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		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0.302776	0.114438	0.0572192	−	0.998362i	$-0.481777\pi$
$7$	0.302776	0.114438	0.0572192	+	0.998362i	$0.481777\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.60555	−0.631939	−0.315970	−	0.948769i	$-0.602330\pi$
$17$	−2.60555	−0.631939	−0.315970	+	0.948769i	$0.602330\pi$
$18$	0	0
$19$	−5.30278	−1.21654	−0.608270	−	0.793730i	$-0.708136\pi$
$19$	−5.30278	−1.21654	−0.608270	+	0.793730i	$0.708136\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.69722	0.353896	0.176948	−	0.984220i	$-0.443378\pi$
$23$	1.69722	0.353896	0.176948	+	0.984220i	$0.443378\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.21110	0.967677	0.483839	−	0.875157i	$-0.339242\pi$
$29$	5.21110	0.967677	0.483839	+	0.875157i	$0.339242\pi$
$30$	0	0
$31$	4.60555	0.827181	0.413591	−	0.910463i	$-0.364274\pi$
$31$	4.60555	0.827181	0.413591	+	0.910463i	$0.364274\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.21110	−1.51430	−0.757148	−	0.653243i	$-0.773408\pi$
$37$	−9.21110	−1.51430	−0.757148	+	0.653243i	$0.773408\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.69722	−0.265062	−0.132531	−	0.991179i	$-0.542310\pi$
$41$	−1.69722	−0.265062	−0.132531	+	0.991179i	$0.542310\pi$
$42$	0	0
$43$	−6.60555	−1.00734	−0.503669	−	0.863897i	$-0.668017\pi$
$43$	−6.60555	−1.00734	−0.503669	+	0.863897i	$0.668017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.69722	0.645213	0.322607	−	0.946533i	$-0.395441\pi$
$53$	4.69722	0.645213	0.322607	+	0.946533i	$0.395441\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.6056	−1.90148	−0.950740	−	0.309988i	$-0.899675\pi$
$59$	−14.6056	−1.90148	−0.950740	+	0.309988i	$0.899675\pi$
$60$	0	0
$61$	9.81665	1.25689	0.628447	−	0.777853i	$-0.283691\pi$
$61$	9.81665	1.25689	0.628447	+	0.777853i	$0.283691\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.39445	−0.170359	−0.0851795	−	0.996366i	$-0.527146\pi$
$67$	−1.39445	−0.170359	−0.0851795	+	0.996366i	$0.527146\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.39445	0.402847	0.201423	−	0.979504i	$-0.435443\pi$
$71$	3.39445	0.402847	0.201423	+	0.979504i	$0.435443\pi$
$72$	0	0
$73$	−10.5139	−1.23056	−0.615278	−	0.788310i	$-0.710956\pi$
$73$	−10.5139	−1.23056	−0.615278	+	0.788310i	$0.710956\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.302776	0.0345045
$78$	0	0
$79$	−15.2111	−1.71138	−0.855691	−	0.517486i	$-0.826868\pi$
$79$	−15.2111	−1.71138	−0.855691	+	0.517486i	$0.826868\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.51388	1.04428	0.522142	−	0.852859i	$-0.325133\pi$
$83$	9.51388	1.04428	0.522142	+	0.852859i	$0.325133\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0.302776	0.0317395
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.78890	0.283170	0.141585	−	0.989926i	$-0.454780\pi$
$97$	2.78890	0.283170	0.141585	+	0.989926i	$0.454780\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.8167	−1.37481	−0.687404	−	0.726275i	$-0.741250\pi$
$101$	−13.8167	−1.37481	−0.687404	+	0.726275i	$0.741250\pi$
$102$	0	0
$103$	2.90833	0.286566	0.143283	−	0.989682i	$-0.454234\pi$
$103$	2.90833	0.286566	0.143283	+	0.989682i	$0.454234\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	4.09167	0.391911	0.195956	−	0.980613i	$-0.437219\pi$
$109$	4.09167	0.391911	0.195956	+	0.980613i	$0.437219\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.09167	−0.196768	−0.0983840	−	0.995149i	$-0.531367\pi$
$113$	−2.09167	−0.196768	−0.0983840	+	0.995149i	$0.531367\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.788897	−0.0723181
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.21110	0.455296	0.227648	−	0.973743i	$-0.426896\pi$
$131$	5.21110	0.455296	0.227648	+	0.973743i	$0.426896\pi$
$132$	0	0
$133$	−1.60555	−0.139219
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−18.6056	−1.57810	−0.789051	−	0.614328i	$-0.789427\pi$
$139$	−18.6056	−1.57810	−0.789051	+	0.614328i	$0.789427\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.3305	1.41977	0.709886	−	0.704316i	$-0.248746\pi$
$149$	17.3305	1.41977	0.709886	+	0.704316i	$0.248746\pi$
$150$	0	0
$151$	23.6333	1.92325	0.961626	−	0.274365i	$-0.0884676\pi$
$151$	23.6333	1.92325	0.961626	+	0.274365i	$0.0884676\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.09167	−0.246742	−0.123371	−	0.992361i	$-0.539371\pi$
$157$	−3.09167	−0.246742	−0.123371	+	0.992361i	$0.539371\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.513878	0.0404993
$162$	0	0
$163$	−18.6056	−1.45730	−0.728650	−	0.684887i	$-0.759852\pi$
$163$	−18.6056	−1.45730	−0.728650	+	0.684887i	$0.759852\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.9083	−1.23102	−0.615512	−	0.788128i	$-0.711051\pi$
$167$	−15.9083	−1.23102	−0.615512	+	0.788128i	$0.711051\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.2111	0.852364	0.426182	−	0.904637i	$-0.359858\pi$
$173$	11.2111	0.852364	0.426182	+	0.904637i	$0.359858\pi$
$174$	0	0
$175$	−1.51388	−0.114438
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−8.69722	−0.646460	−0.323230	−	0.946321i	$-0.604769\pi$
$181$	−8.69722	−0.646460	−0.323230	+	0.946321i	$0.604769\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.60555	−0.190537
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.513878	0.0371829	0.0185915	−	0.999827i	$-0.494082\pi$
$191$	0.513878	0.0371829	0.0185915	+	0.999827i	$0.494082\pi$
$192$	0	0
$193$	−17.6972	−1.27387	−0.636937	−	0.770916i	$-0.719799\pi$
$193$	−17.6972	−1.27387	−0.636937	+	0.770916i	$0.719799\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.9083	1.13342	0.566711	−	0.823917i	$-0.308216\pi$
$197$	15.9083	1.13342	0.566711	+	0.823917i	$0.308216\pi$
$198$	0	0
$199$	18.9361	1.34234	0.671172	−	0.741302i	$-0.265791\pi$
$199$	18.9361	1.34234	0.671172	+	0.741302i	$0.265791\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.57779	0.110739
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.30278	−0.366801
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.39445	0.0946613
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.60555	−0.175268
$222$	0	0
$223$	−2.18335	−0.146208	−0.0731038	−	0.997324i	$-0.523290\pi$
$223$	−2.18335	−0.146208	−0.0731038	+	0.997324i	$0.523290\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.1194	−1.40175	−0.700873	−	0.713286i	$-0.747206\pi$
$227$	−21.1194	−1.40175	−0.700873	+	0.713286i	$0.747206\pi$
$228$	0	0
$229$	−15.2111	−1.00518	−0.502589	−	0.864525i	$-0.667619\pi$
$229$	−15.2111	−1.00518	−0.502589	+	0.864525i	$0.667619\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.39445	0.222378	0.111189	−	0.993799i	$-0.464534\pi$
$233$	3.39445	0.222378	0.111189	+	0.993799i	$0.464534\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−30.5139	−1.97378	−0.986889	−	0.161398i	$-0.948400\pi$
$239$	−30.5139	−1.97378	−0.986889	+	0.161398i	$0.948400\pi$
$240$	0	0
$241$	−9.33053	−0.601032	−0.300516	−	0.953777i	$-0.597159\pi$
$241$	−9.33053	−0.601032	−0.300516	+	0.953777i	$0.597159\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.30278	−0.337408
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.72498	−0.550716	−0.275358	−	0.961342i	$-0.588796\pi$
$251$	−8.72498	−0.550716	−0.275358	+	0.961342i	$0.588796\pi$
$252$	0	0
$253$	1.69722	0.106704
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1194	1.13026	0.565129	−	0.825002i	$-0.308826\pi$
$257$	18.1194	1.13026	0.565129	+	0.825002i	$0.308826\pi$
$258$	0	0
$259$	−2.78890	−0.173294
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.6056	−1.27059	−0.635296	−	0.772268i	$-0.719122\pi$
$263$	−20.6056	−1.27059	−0.635296	+	0.772268i	$0.719122\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.7250	−0.897798	−0.448899	−	0.893583i	$-0.648184\pi$
$269$	−14.7250	−0.897798	−0.448899	+	0.893583i	$0.648184\pi$
$270$	0	0
$271$	−5.30278	−0.322121	−0.161060	−	0.986945i	$-0.551491\pi$
$271$	−5.30278	−0.322121	−0.161060	+	0.986945i	$0.551491\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−16.7889	−1.00875	−0.504374	−	0.863486i	$-0.668277\pi$
$277$	−16.7889	−1.00875	−0.504374	+	0.863486i	$0.668277\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−26.3305	−1.57075	−0.785374	−	0.619022i	$-0.787529\pi$
$281$	−26.3305	−1.57075	−0.785374	+	0.619022i	$0.787529\pi$
$282$	0	0
$283$	−3.21110	−0.190880	−0.0954401	−	0.995435i	$-0.530426\pi$
$283$	−3.21110	−0.190880	−0.0954401	+	0.995435i	$0.530426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.513878	−0.0303333
$288$	0	0
$289$	−10.2111	−0.600653
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.6333	−1.26383	−0.631916	−	0.775037i	$-0.717731\pi$
$293$	−21.6333	−1.26383	−0.631916	+	0.775037i	$0.717731\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.69722	0.0981530
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.4222	1.05141	0.525705	−	0.850667i	$-0.323801\pi$
$307$	18.4222	1.05141	0.525705	+	0.850667i	$0.323801\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.3305	1.83330	0.916648	−	0.399695i	$-0.130884\pi$
$311$	32.3305	1.83330	0.916648	+	0.399695i	$0.130884\pi$
$312$	0	0
$313$	10.7250	0.606212	0.303106	−	0.952957i	$-0.401976\pi$
$313$	10.7250	0.606212	0.303106	+	0.952957i	$0.401976\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.60555	−0.483336	−0.241668	−	0.970359i	$-0.577694\pi$
$317$	−8.60555	−0.483336	−0.241668	+	0.970359i	$0.577694\pi$
$318$	0	0
$319$	5.21110	0.291766
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.8167	0.768779
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.81665	0.100155
$330$	0	0
$331$	10.6056	0.582934	0.291467	−	0.956581i	$-0.405857\pi$
$331$	10.6056	0.582934	0.291467	+	0.956581i	$0.405857\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.8167	0.861588	0.430794	−	0.902450i	$-0.358234\pi$
$337$	15.8167	0.861588	0.430794	+	0.902450i	$0.358234\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.60555	0.249405
$342$	0	0
$343$	−4.21110	−0.227378
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.4500	−1.58096	−0.790478	−	0.612490i	$-0.790168\pi$
$347$	−29.4500	−1.58096	−0.790478	+	0.612490i	$0.790168\pi$
$348$	0	0
$349$	−7.51388	−0.402209	−0.201104	−	0.979570i	$-0.564453\pi$
$349$	−7.51388	−0.402209	−0.201104	+	0.979570i	$0.564453\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.2389	−1.60945	−0.804726	−	0.593646i	$-0.797688\pi$
$353$	−30.2389	−1.60945	−0.804726	+	0.593646i	$0.797688\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.30278	0.227092	0.113546	−	0.993533i	$-0.463779\pi$
$359$	4.30278	0.227092	0.113546	+	0.993533i	$0.463779\pi$
$360$	0	0
$361$	9.11943	0.479970
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.5416	−1.54206	−0.771030	−	0.636798i	$-0.780258\pi$
$367$	−29.5416	−1.54206	−0.771030	+	0.636798i	$0.780258\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.42221	0.0738372
$372$	0	0
$373$	34.8444	1.80418	0.902088	−	0.431553i	$-0.142034\pi$
$373$	34.8444	1.80418	0.902088	+	0.431553i	$0.142034\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.21110	0.268385
$378$	0	0
$379$	9.81665	0.504248	0.252124	−	0.967695i	$-0.418871\pi$
$379$	9.81665	0.504248	0.252124	+	0.967695i	$0.418871\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.6333	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$383$	21.6333	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.7250	0.898692	0.449346	−	0.893358i	$-0.351657\pi$
$389$	17.7250	0.898692	0.449346	+	0.893358i	$0.351657\pi$
$390$	0	0
$391$	−4.42221	−0.223641
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.0278	0.754221	0.377111	−	0.926168i	$-0.376918\pi$
$397$	15.0278	0.754221	0.377111	+	0.926168i	$0.376918\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.8167	1.28922	0.644611	−	0.764511i	$-0.277019\pi$
$401$	25.8167	1.28922	0.644611	+	0.764511i	$0.277019\pi$
$402$	0	0
$403$	4.60555	0.229419
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.21110	−0.456577
$408$	0	0
$409$	31.2111	1.54329	0.771645	−	0.636054i	$-0.219434\pi$
$409$	31.2111	1.54329	0.771645	+	0.636054i	$0.219434\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.42221	−0.217602
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	39.7527	1.94205	0.971024	−	0.238981i	$-0.0768135\pi$
$419$	39.7527	1.94205	0.971024	+	0.238981i	$0.0768135\pi$
$420$	0	0
$421$	−29.0278	−1.41473	−0.707363	−	0.706850i	$-0.750115\pi$
$421$	−29.0278	−1.41473	−0.707363	+	0.706850i	$0.750115\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.0278	0.631939
$426$	0	0
$427$	2.97224	0.143837
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.7250	−1.28730	−0.643649	−	0.765321i	$-0.722580\pi$
$431$	−26.7250	−1.28730	−0.643649	+	0.765321i	$0.722580\pi$
$432$	0	0
$433$	−19.1194	−0.918821	−0.459411	−	0.888224i	$-0.651939\pi$
$433$	−19.1194	−0.918821	−0.459411	+	0.888224i	$0.651939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.00000	−0.430528
$438$	0	0
$439$	−27.2111	−1.29872	−0.649358	−	0.760483i	$-0.724962\pi$
$439$	−27.2111	−1.29872	−0.649358	+	0.760483i	$0.724962\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−36.3583	−1.72743	−0.863717	−	0.503977i	$-0.831870\pi$
$443$	−36.3583	−1.72743	−0.863717	+	0.503977i	$0.831870\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.3944	1.00967	0.504833	−	0.863217i	$-0.331554\pi$
$449$	21.3944	1.00967	0.504833	+	0.863217i	$0.331554\pi$
$450$	0	0
$451$	−1.69722	−0.0799192
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.1194	1.92349	0.961743	−	0.273954i	$-0.0883315\pi$
$457$	41.1194	1.92349	0.961743	+	0.273954i	$0.0883315\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.30278	0.0606763	0.0303382	−	0.999540i	$-0.490342\pi$
$461$	1.30278	0.0606763	0.0303382	+	0.999540i	$0.490342\pi$
$462$	0	0
$463$	−34.2389	−1.59121	−0.795607	−	0.605813i	$-0.792848\pi$
$463$	−34.2389	−1.59121	−0.795607	+	0.605813i	$0.792848\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.6333	−1.83401	−0.917005	−	0.398875i	$-0.869401\pi$
$467$	−39.6333	−1.83401	−0.917005	+	0.398875i	$0.869401\pi$
$468$	0	0
$469$	−0.422205	−0.0194956
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.60555	−0.303724
$474$	0	0
$475$	26.5139	1.21654
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.5778	−0.620385	−0.310193	−	0.950674i	$-0.600394\pi$
$479$	−13.5778	−0.620385	−0.310193	+	0.950674i	$0.600394\pi$
$480$	0	0
$481$	−9.21110	−0.419990
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.02776	0.137201	0.0686004	−	0.997644i	$-0.478147\pi$
$487$	3.02776	0.137201	0.0686004	+	0.997644i	$0.478147\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	42.2389	1.90621	0.953107	−	0.302635i	$-0.0978663\pi$
$491$	42.2389	1.90621	0.953107	+	0.302635i	$0.0978663\pi$
$492$	0	0
$493$	−13.5778	−0.611513
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.02776	0.0461012
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.8167	−0.883581	−0.441790	−	0.897118i	$-0.645657\pi$
$503$	−19.8167	−0.883581	−0.441790	+	0.897118i	$0.645657\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.8167	−0.878358	−0.439179	−	0.898400i	$-0.644731\pi$
$509$	−19.8167	−0.878358	−0.439179	+	0.898400i	$0.644731\pi$
$510$	0	0
$511$	−3.18335	−0.140823
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.0917	1.01166	0.505832	−	0.862632i	$-0.331185\pi$
$521$	23.0917	1.01166	0.505832	+	0.862632i	$0.331185\pi$
$522$	0	0
$523$	−12.6056	−0.551202	−0.275601	−	0.961272i	$-0.588877\pi$
$523$	−12.6056	−0.551202	−0.275601	+	0.961272i	$0.588877\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−20.1194	−0.874758
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.69722	−0.0735149
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.90833	−0.297563
$540$	0	0
$541$	8.11943	0.349082	0.174541	−	0.984650i	$-0.444156\pi$
$541$	8.11943	0.349082	0.174541	+	0.984650i	$0.444156\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.8167	0.676271	0.338136	−	0.941097i	$-0.390204\pi$
$547$	15.8167	0.676271	0.338136	+	0.941097i	$0.390204\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.6333	−1.17722
$552$	0	0
$553$	−4.60555	−0.195848
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.3305	1.24277	0.621387	−	0.783504i	$-0.286569\pi$
$557$	29.3305	1.24277	0.621387	+	0.783504i	$0.286569\pi$
$558$	0	0
$559$	−6.60555	−0.279385
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.1833	−0.934917	−0.467458	−	0.884015i	$-0.654830\pi$
$563$	−22.1833	−0.934917	−0.467458	+	0.884015i	$0.654830\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.3944	−0.896902	−0.448451	−	0.893807i	$-0.648024\pi$
$569$	−21.3944	−0.896902	−0.448451	+	0.893807i	$0.648024\pi$
$570$	0	0
$571$	−11.8167	−0.494512	−0.247256	−	0.968950i	$-0.579529\pi$
$571$	−11.8167	−0.494512	−0.247256	+	0.968950i	$0.579529\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.48612	−0.353896
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.88057	0.119506
$582$	0	0
$583$	4.69722	0.194539
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.8167	−1.06557	−0.532784	−	0.846251i	$-0.678854\pi$
$587$	−25.8167	−1.06557	−0.532784	+	0.846251i	$0.678854\pi$
$588$	0	0
$589$	−24.4222	−1.00630
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.1194	0.990466	0.495233	−	0.868760i	$-0.335083\pi$
$593$	24.1194	0.990466	0.495233	+	0.868760i	$0.335083\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.8806	0.853157	0.426578	−	0.904451i	$-0.359719\pi$
$599$	20.8806	0.853157	0.426578	+	0.904451i	$0.359719\pi$
$600$	0	0
$601$	3.57779	0.145941	0.0729706	−	0.997334i	$-0.476752\pi$
$601$	3.57779	0.145941	0.0729706	+	0.997334i	$0.476752\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.8167	−1.21022	−0.605110	−	0.796142i	$-0.706871\pi$
$607$	−29.8167	−1.21022	−0.605110	+	0.796142i	$0.706871\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	34.3305	1.38660	0.693299	−	0.720650i	$-0.256157\pi$
$613$	34.3305	1.38660	0.693299	+	0.720650i	$0.256157\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.8444	−1.32227	−0.661133	−	0.750269i	$-0.729924\pi$
$617$	−32.8444	−1.32227	−0.661133	+	0.750269i	$0.729924\pi$
$618$	0	0
$619$	−18.8444	−0.757421	−0.378710	−	0.925515i	$-0.623632\pi$
$619$	−18.8444	−0.757421	−0.378710	+	0.925515i	$0.623632\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.0000	0.956943
$630$	0	0
$631$	−36.0555	−1.43535	−0.717674	−	0.696380i	$-0.754793\pi$
$631$	−36.0555	−1.43535	−0.717674	+	0.696380i	$0.754793\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.90833	−0.273718
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−0.908327	−0.0358768	−0.0179384	−	0.999839i	$-0.505710\pi$
$641$	−0.908327	−0.0358768	−0.0179384	+	0.999839i	$0.505710\pi$
$642$	0	0
$643$	9.81665	0.387131	0.193566	−	0.981087i	$-0.437995\pi$
$643$	9.81665	0.387131	0.193566	+	0.981087i	$0.437995\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.9638	1.88565	0.942827	−	0.333284i	$-0.108157\pi$
$647$	47.9638	1.88565	0.942827	+	0.333284i	$0.108157\pi$
$648$	0	0
$649$	−14.6056	−0.573318
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.788897	0.0308719	0.0154360	−	0.999881i	$-0.495086\pi$
$653$	0.788897	0.0308719	0.0154360	+	0.999881i	$0.495086\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	34.8444	1.35529	0.677645	−	0.735389i	$-0.263001\pi$
$661$	34.8444	1.35529	0.677645	+	0.735389i	$0.263001\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.84441	0.342457
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.81665	0.378968
$672$	0	0
$673$	−7.39445	−0.285035	−0.142518	−	0.989792i	$-0.545520\pi$
$673$	−7.39445	−0.285035	−0.142518	+	0.989792i	$0.545520\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.2666	1.66287	0.831436	−	0.555621i	$-0.187520\pi$
$677$	43.2666	1.66287	0.831436	+	0.555621i	$0.187520\pi$
$678$	0	0
$679$	0.844410	0.0324055
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.6333	0.598192	0.299096	−	0.954223i	$-0.403315\pi$
$683$	15.6333	0.598192	0.299096	+	0.954223i	$0.403315\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.69722	0.178950
$690$	0	0
$691$	12.1833	0.463476	0.231738	−	0.972778i	$-0.425559\pi$
$691$	12.1833	0.463476	0.231738	+	0.972778i	$0.425559\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.42221	0.167503
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.0278	−0.492052	−0.246026	−	0.969263i	$-0.579125\pi$
$701$	−13.0278	−0.492052	−0.246026	+	0.969263i	$0.579125\pi$
$702$	0	0
$703$	48.8444	1.84220
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.18335	−0.157331
$708$	0	0
$709$	4.60555	0.172965	0.0864826	−	0.996253i	$-0.472437\pi$
$709$	4.60555	0.172965	0.0864826	+	0.996253i	$0.472437\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.81665	0.292736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34.4222	1.28373	0.641866	−	0.766817i	$-0.278161\pi$
$719$	34.4222	1.28373	0.641866	+	0.766817i	$0.278161\pi$
$720$	0	0
$721$	0.880571	0.0327942
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−26.0555	−0.967677
$726$	0	0
$727$	16.4861	0.611436	0.305718	−	0.952122i	$-0.401103\pi$
$727$	16.4861	0.611436	0.305718	+	0.952122i	$0.401103\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.2111	0.636576
$732$	0	0
$733$	−15.8806	−0.586562	−0.293281	−	0.956026i	$-0.594747\pi$
$733$	−15.8806	−0.586562	−0.293281	+	0.956026i	$0.594747\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.39445	−0.0513652
$738$	0	0
$739$	−25.5139	−0.938543	−0.469272	−	0.883054i	$-0.655483\pi$
$739$	−25.5139	−0.938543	−0.469272	+	0.883054i	$0.655483\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.8444	1.64518	0.822591	−	0.568634i	$-0.192528\pi$
$743$	44.8444	1.64518	0.822591	+	0.568634i	$0.192528\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.81665	−0.0663791
$750$	0	0
$751$	−18.0917	−0.660175	−0.330087	−	0.943950i	$-0.607078\pi$
$751$	−18.0917	−0.660175	−0.330087	+	0.943950i	$0.607078\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−28.9083	−1.05069	−0.525346	−	0.850889i	$-0.676064\pi$
$757$	−28.9083	−1.05069	−0.525346	+	0.850889i	$0.676064\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.7250	−1.18628	−0.593140	−	0.805099i	$-0.702112\pi$
$761$	−32.7250	−1.18628	−0.593140	+	0.805099i	$0.702112\pi$
$762$	0	0
$763$	1.23886	0.0448497
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.6056	−0.527376
$768$	0	0
$769$	1.09167	0.0393667	0.0196834	−	0.999806i	$-0.493734\pi$
$769$	1.09167	0.0393667	0.0196834	+	0.999806i	$0.493734\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.2389	1.51923	0.759613	−	0.650375i	$-0.225388\pi$
$773$	42.2389	1.51923	0.759613	+	0.650375i	$0.225388\pi$
$774$	0	0
$775$	−23.0278	−0.827181
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.00000	0.322458
$780$	0	0
$781$	3.39445	0.121463
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.0917	−0.965714	−0.482857	−	0.875699i	$-0.660401\pi$
$787$	−27.0917	−0.965714	−0.482857	+	0.875699i	$0.660401\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.633308	−0.0225178
$792$	0	0
$793$	9.81665	0.348600
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−15.6333	−0.553067
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.5139	−0.371027
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	15.6972	0.551204	0.275602	−	0.961272i	$-0.411123\pi$
$811$	15.6972	0.551204	0.275602	+	0.961272i	$0.411123\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	35.0278	1.22547
$818$	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
$823$	−12.4861	−0.435239	−0.217619	−	0.976034i	$-0.569829\pi$
$823$	−12.4861	−0.435239	−0.217619	+	0.976034i	$0.569829\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.9361	−1.00621	−0.503103	−	0.864226i	$-0.667808\pi$
$827$	−28.9361	−1.00621	−0.503103	+	0.864226i	$0.667808\pi$
$828$	0	0
$829$	−6.72498	−0.233568	−0.116784	−	0.993157i	$-0.537259\pi$
$829$	−6.72498	−0.233568	−0.116784	+	0.993157i	$0.537259\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.57779	−0.261615	−0.130807	−	0.991408i	$-0.541757\pi$
$839$	−7.57779	−0.261615	−0.130807	+	0.991408i	$0.541757\pi$
$840$	0	0
$841$	−1.84441	−0.0636004
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.302776	0.0104035
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.6333	−0.535903
$852$	0	0
$853$	23.9083	0.818606	0.409303	−	0.912399i	$-0.365772\pi$
$853$	23.9083	0.818606	0.409303	+	0.912399i	$0.365772\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.0278	1.67476	0.837378	−	0.546624i	$-0.184087\pi$
$857$	49.0278	1.67476	0.837378	+	0.546624i	$0.184087\pi$
$858$	0	0
$859$	19.4861	0.664858	0.332429	−	0.943128i	$-0.392132\pi$
$859$	19.4861	0.664858	0.332429	+	0.943128i	$0.392132\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.3944	1.13676	0.568380	−	0.822766i	$-0.307570\pi$
$863$	33.3944	1.13676	0.568380	+	0.822766i	$0.307570\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.2111	−0.516001
$870$	0	0
$871$	−1.39445	−0.0472491
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.11943	−0.0378004	−0.0189002	−	0.999821i	$-0.506016\pi$
$877$	−1.11943	−0.0378004	−0.0189002	+	0.999821i	$0.506016\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	51.1194	1.72226	0.861129	−	0.508387i	$-0.169758\pi$
$881$	51.1194	1.72226	0.861129	+	0.508387i	$0.169758\pi$
$882$	0	0
$883$	−14.9361	−0.502639	−0.251320	−	0.967904i	$-0.580865\pi$
$883$	−14.9361	−0.502639	−0.251320	+	0.967904i	$0.580865\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.7889	1.03379	0.516895	−	0.856049i	$-0.327088\pi$
$887$	30.7889	1.03379	0.516895	+	0.856049i	$0.327088\pi$
$888$	0	0
$889$	−3.02776	−0.101548
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−31.8167	−1.06470
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	−12.2389	−0.407736
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.1194	−0.535237	−0.267618	−	0.963525i	$-0.586237\pi$
$907$	−16.1194	−0.535237	−0.267618	+	0.963525i	$0.586237\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.69722	0.0562316	0.0281158	−	0.999605i	$-0.491049\pi$
$911$	1.69722	0.0562316	0.0281158	+	0.999605i	$0.491049\pi$
$912$	0	0
$913$	9.51388	0.314863
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.57779	0.0521034
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.39445	0.111730
$924$	0	0
$925$	46.0555	1.51430
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.2389	0.795251	0.397626	−	0.917548i	$-0.369834\pi$
$929$	24.2389	0.795251	0.397626	+	0.917548i	$0.369834\pi$
$930$	0	0
$931$	36.6333	1.20061
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.5416	−1.02823	−0.514114	−	0.857722i	$-0.671879\pi$
$941$	−31.5416	−1.02823	−0.514114	+	0.857722i	$0.671879\pi$
$942$	0	0
$943$	−2.88057	−0.0938043
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.7889	0.610557	0.305279	−	0.952263i	$-0.401250\pi$
$947$	18.7889	0.610557	0.305279	+	0.952263i	$0.401250\pi$
$948$	0	0
$949$	−10.5139	−0.341295
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.6333	0.895131	0.447565	−	0.894251i	$-0.352291\pi$
$953$	27.6333	0.895131	0.447565	+	0.894251i	$0.352291\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.81665	0.0586628
$960$	0	0
$961$	−9.78890	−0.315771
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.6695	−0.439580	−0.219790	−	0.975547i	$-0.570537\pi$
$967$	−13.6695	−0.439580	−0.219790	+	0.975547i	$0.570537\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24.5139	−0.786688	−0.393344	−	0.919391i	$-0.628682\pi$
$971$	−24.5139	−0.786688	−0.393344	+	0.919391i	$0.628682\pi$
$972$	0	0
$973$	−5.63331	−0.180596
$974$	0	0
$975$	0	0
$976$	0	0
$977$	34.4222	1.10126	0.550632	−	0.834748i	$-0.314387\pi$
$977$	34.4222	1.10126	0.550632	+	0.834748i	$0.314387\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16.4222	0.523787	0.261893	−	0.965097i	$-0.415653\pi$
$983$	16.4222	0.523787	0.261893	+	0.965097i	$0.415653\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.2111	−0.356492
$990$	0	0
$991$	41.9083	1.33126	0.665631	−	0.746281i	$-0.268163\pi$
$991$	41.9083	1.33126	0.665631	+	0.746281i	$0.268163\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.6056	0.525903	0.262952	−	0.964809i	$-0.415304\pi$
$997$	16.6056	0.525903	0.262952	+	0.964809i	$0.415304\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.h.1.2		2
3.2	odd	2		572.2.a.b.1.1	✓	2
12.11	even	2		2288.2.a.r.1.2		2
24.5	odd	2		9152.2.a.bq.1.2		2
24.11	even	2		9152.2.a.bg.1.1		2
33.32	even	2		6292.2.a.k.1.1		2
39.38	odd	2		7436.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.a.b.1.1	✓	2	3.2	odd	2
2288.2.a.r.1.2		2	12.11	even	2
5148.2.a.h.1.2		2	1.1	even	1	trivial
6292.2.a.k.1.1		2	33.32	even	2
7436.2.a.e.1.1		2	39.38	odd	2
9152.2.a.bg.1.1		2	24.11	even	2
9152.2.a.bq.1.2		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q+0.302776 q^{7} +O(q^{10})\)	\(q+0.302776 q^{7} +1.00000 q^{11} +1.00000 q^{13} -2.60555 q^{17} -5.30278 q^{19} +1.69722 q^{23} -5.00000 q^{25} +5.21110 q^{29} +4.60555 q^{31} -9.21110 q^{37} -1.69722 q^{41} -6.60555 q^{43} +6.00000 q^{47} -6.90833 q^{49} +4.69722 q^{53} -14.6056 q^{59} +9.81665 q^{61} -1.39445 q^{67} +3.39445 q^{71} -10.5139 q^{73} +0.302776 q^{77} -15.2111 q^{79} +9.51388 q^{83} +0.302776 q^{91} +2.78890 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{7}+O(q^{10}) \) 2 * q - 3 * q^7	\( 2 q - 3 q^{7} + 2 q^{11} + 2 q^{13} + 2 q^{17} - 7 q^{19} + 7 q^{23} - 10 q^{25} - 4 q^{29} + 2 q^{31} - 4 q^{37} - 7 q^{41} - 6 q^{43} + 12 q^{47} - 3 q^{49} + 13 q^{53} - 22 q^{59} - 2 q^{61} - 10 q^{67} + 14 q^{71} - 3 q^{73} - 3 q^{77} - 16 q^{79} + q^{83} - 3 q^{91} + 20 q^{97}+O(q^{100}) \) 2 * q - 3 * q^7 + 2 * q^11 + 2 * q^13 + 2 * q^17 - 7 * q^19 + 7 * q^23 - 10 * q^25 - 4 * q^29 + 2 * q^31 - 4 * q^37 - 7 * q^41 - 6 * q^43 + 12 * q^47 - 3 * q^49 + 13 * q^53 - 22 * q^59 - 2 * q^61 - 10 * q^67 + 14 * q^71 - 3 * q^73 - 3 * q^77 - 16 * q^79 + q^83 - 3 * q^91 + 20 * q^97

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