LMFDB - Embedded newform 3724.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.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:	$29.7362897127$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} -2.46410 q^{9} +0.464102 q^{11} -4.19615 q^{13} +6.46410 q^{17} -1.00000 q^{19} +6.46410 q^{23} -5.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} -4.19615 q^{31} +0.339746 q^{33} -4.00000 q^{37} -3.07180 q^{39} -3.46410 q^{41} +2.00000 q^{43} +3.92820 q^{47} +4.73205 q^{51} -2.19615 q^{53} -0.732051 q^{57} +1.26795 q^{59} +7.00000 q^{61} -10.0000 q^{67} +4.73205 q^{69} -10.7321 q^{71} -15.3923 q^{73} -3.66025 q^{75} -6.19615 q^{79} +4.46410 q^{81} +9.00000 q^{83} -2.53590 q^{87} +0.928203 q^{89} -3.07180 q^{93} -16.1962 q^{97} -1.14359 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} + 6 q^{23} - 10 q^{25} - 8 q^{27} + 2 q^{31} + 18 q^{33} - 8 q^{37} - 20 q^{39} + 4 q^{43} - 6 q^{47} + 6 q^{51} + 6 q^{53} + 2 q^{57}+ \cdots - 30 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 6 * q^17 - 2 * q^19 + 6 * q^23 - 10 * q^25 - 8 * q^27 + 2 * q^31 + 18 * q^33 - 8 * q^37 - 20 * q^39 + 4 * q^43 - 6 * q^47 + 6 * q^51 + 6 * q^53 + 2 * q^57 + 6 * q^59 + 14 * q^61 - 20 * q^67 + 6 * q^69 - 18 * q^71 - 10 * q^73 + 10 * q^75 - 2 * q^79 + 2 * q^81 + 18 * q^83 - 12 * q^87 - 12 * q^89 - 20 * q^93 - 22 * q^97 - 30 * q^99

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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	0.464102	0.139932	0.0699660	−	0.997549i	$-0.477711\pi$
$11$	0.464102	0.139932	0.0699660	+	0.997549i	$0.477711\pi$
$12$	0	0
$13$	−4.19615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$13$	−4.19615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.46410	1.56777	0.783887	−	0.620903i	$-0.213234\pi$
$17$	6.46410	1.56777	0.783887	+	0.620903i	$0.213234\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.46410	1.34786	0.673929	−	0.738796i	$-0.264605\pi$
$23$	6.46410	1.34786	0.673929	+	0.738796i	$0.264605\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−4.19615	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$31$	−4.19615	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$32$	0	0
$33$	0.339746	0.0591422
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−3.07180	−0.491881
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.92820	0.572987	0.286494	−	0.958082i	$-0.407510\pi$
$47$	3.92820	0.572987	0.286494	+	0.958082i	$0.407510\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.73205	0.662620
$52$	0	0
$53$	−2.19615	−0.301665	−0.150832	−	0.988559i	$-0.548195\pi$
$53$	−2.19615	−0.301665	−0.150832	+	0.988559i	$0.548195\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.732051	−0.0969625
$58$	0	0
$59$	1.26795	0.165073	0.0825365	−	0.996588i	$-0.473698\pi$
$59$	1.26795	0.165073	0.0825365	+	0.996588i	$0.473698\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	4.73205	0.569672
$70$	0	0
$71$	−10.7321	−1.27366	−0.636830	−	0.771004i	$-0.719755\pi$
$71$	−10.7321	−1.27366	−0.636830	+	0.771004i	$0.719755\pi$
$72$	0	0
$73$	−15.3923	−1.80153	−0.900767	−	0.434304i	$-0.856994\pi$
$73$	−15.3923	−1.80153	−0.900767	+	0.434304i	$0.856994\pi$
$74$	0	0
$75$	−3.66025	−0.422650
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.19615	−0.697122	−0.348561	−	0.937286i	$-0.613330\pi$
$79$	−6.19615	−0.697122	−0.348561	+	0.937286i	$0.613330\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.53590	−0.271877
$88$	0	0
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.07180	−0.318530
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.1962	−1.64447	−0.822235	−	0.569148i	$-0.807273\pi$
$97$	−16.1962	−1.64447	−0.822235	+	0.569148i	$0.807273\pi$
$98$	0	0
$99$	−1.14359	−0.114935
$100$	0	0
$101$	−13.3923	−1.33258	−0.666292	−	0.745691i	$-0.732120\pi$
$101$	−13.3923	−1.33258	−0.666292	+	0.745691i	$0.732120\pi$
$102$	0	0
$103$	1.80385	0.177738	0.0888692	−	0.996043i	$-0.471675\pi$
$103$	1.80385	0.177738	0.0888692	+	0.996043i	$0.471675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.8564	−1.91959	−0.959796	−	0.280700i	$-0.909433\pi$
$107$	−19.8564	−1.91959	−0.959796	+	0.280700i	$0.909433\pi$
$108$	0	0
$109$	−2.39230	−0.229141	−0.114571	−	0.993415i	$-0.536549\pi$
$109$	−2.39230	−0.229141	−0.114571	+	0.993415i	$0.536549\pi$
$110$	0	0
$111$	−2.92820	−0.277933
$112$	0	0
$113$	−9.80385	−0.922268	−0.461134	−	0.887330i	$-0.652557\pi$
$113$	−9.80385	−0.922268	−0.461134	+	0.887330i	$0.652557\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	10.3397	0.955910
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.7846	−0.980419
$122$	0	0
$123$	−2.53590	−0.228654
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−20.3923	−1.80952	−0.904762	−	0.425917i	$-0.859952\pi$
$127$	−20.3923	−1.80952	−0.904762	+	0.425917i	$0.859952\pi$
$128$	0	0
$129$	1.46410	0.128907
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.39230	0.631567	0.315784	−	0.948831i	$-0.397733\pi$
$137$	7.39230	0.631567	0.315784	+	0.948831i	$0.397733\pi$
$138$	0	0
$139$	−0.607695	−0.0515440	−0.0257720	−	0.999668i	$-0.508204\pi$
$139$	−0.607695	−0.0515440	−0.0257720	+	0.999668i	$0.508204\pi$
$140$	0	0
$141$	2.87564	0.242173
$142$	0	0
$143$	−1.94744	−0.162853
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.7846	1.70274	0.851371	−	0.524564i	$-0.175772\pi$
$149$	20.7846	1.70274	0.851371	+	0.524564i	$0.175772\pi$
$150$	0	0
$151$	−1.80385	−0.146795	−0.0733975	−	0.997303i	$-0.523384\pi$
$151$	−1.80385	−0.146795	−0.0733975	+	0.997303i	$0.523384\pi$
$152$	0	0
$153$	−15.9282	−1.28772
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.39230	0.430353	0.215176	−	0.976575i	$-0.430967\pi$
$157$	5.39230	0.430353	0.215176	+	0.976575i	$0.430967\pi$
$158$	0	0
$159$	−1.60770	−0.127499
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.0526	1.70648	0.853239	−	0.521520i	$-0.174635\pi$
$167$	22.0526	1.70648	0.853239	+	0.521520i	$0.174635\pi$
$168$	0	0
$169$	4.60770	0.354438
$170$	0	0
$171$	2.46410	0.188435
$172$	0	0
$173$	18.5885	1.41325	0.706627	−	0.707586i	$-0.250216\pi$
$173$	18.5885	1.41325	0.706627	+	0.707586i	$0.250216\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.928203	0.0697680
$178$	0	0
$179$	−15.1244	−1.13045	−0.565224	−	0.824938i	$-0.691210\pi$
$179$	−15.1244	−1.13045	−0.565224	+	0.824938i	$0.691210\pi$
$180$	0	0
$181$	8.39230	0.623795	0.311898	−	0.950116i	$-0.399035\pi$
$181$	8.39230	0.623795	0.311898	+	0.950116i	$0.399035\pi$
$182$	0	0
$183$	5.12436	0.378803
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.928203	0.0671624	0.0335812	−	0.999436i	$-0.489309\pi$
$191$	0.928203	0.0671624	0.0335812	+	0.999436i	$0.489309\pi$
$192$	0	0
$193$	−1.80385	−0.129844	−0.0649219	−	0.997890i	$-0.520680\pi$
$193$	−1.80385	−0.129844	−0.0649219	+	0.997890i	$0.520680\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.3923	−1.80913	−0.904563	−	0.426339i	$-0.859803\pi$
$197$	−25.3923	−1.80913	−0.904563	+	0.426339i	$0.859803\pi$
$198$	0	0
$199$	14.3923	1.02024	0.510122	−	0.860102i	$-0.329600\pi$
$199$	14.3923	1.02024	0.510122	+	0.860102i	$0.329600\pi$
$200$	0	0
$201$	−7.32051	−0.516349
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−15.9282	−1.10709
$208$	0	0
$209$	−0.464102	−0.0321026
$210$	0	0
$211$	26.5885	1.83042	0.915212	−	0.402972i	$-0.132023\pi$
$211$	26.5885	1.83042	0.915212	+	0.402972i	$0.132023\pi$
$212$	0	0
$213$	−7.85641	−0.538312
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.2679	−0.761417
$220$	0	0
$221$	−27.1244	−1.82458
$222$	0	0
$223$	−18.3923	−1.23164	−0.615820	−	0.787887i	$-0.711175\pi$
$223$	−18.3923	−1.23164	−0.615820	+	0.787887i	$0.711175\pi$
$224$	0	0
$225$	12.3205	0.821367
$226$	0	0
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	21.7846	1.43957	0.719784	−	0.694198i	$-0.244241\pi$
$229$	21.7846	1.43957	0.719784	+	0.694198i	$0.244241\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.39230	0.484286	0.242143	−	0.970241i	$-0.422150\pi$
$233$	7.39230	0.484286	0.242143	+	0.970241i	$0.422150\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.53590	−0.294638
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−12.3923	−0.798259	−0.399129	−	0.916895i	$-0.630688\pi$
$241$	−12.3923	−0.798259	−0.399129	+	0.916895i	$0.630688\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.19615	0.266995
$248$	0	0
$249$	6.58846	0.417527
$250$	0	0
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.26795	−0.453362	−0.226681	−	0.973969i	$-0.572788\pi$
$257$	−7.26795	−0.453362	−0.226681	+	0.973969i	$0.572788\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.53590	0.528359
$262$	0	0
$263$	−16.8564	−1.03941	−0.519705	−	0.854346i	$-0.673958\pi$
$263$	−16.8564	−1.03941	−0.519705	+	0.854346i	$0.673958\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.679492	0.0415842
$268$	0	0
$269$	21.4641	1.30869	0.654345	−	0.756196i	$-0.272945\pi$
$269$	21.4641	1.30869	0.654345	+	0.756196i	$0.272945\pi$
$270$	0	0
$271$	−0.607695	−0.0369149	−0.0184574	−	0.999830i	$-0.505876\pi$
$271$	−0.607695	−0.0369149	−0.0184574	+	0.999830i	$0.505876\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.32051	−0.139932
$276$	0	0
$277$	−15.7846	−0.948405	−0.474203	−	0.880416i	$-0.657264\pi$
$277$	−15.7846	−0.948405	−0.474203	+	0.880416i	$0.657264\pi$
$278$	0	0
$279$	10.3397	0.619024
$280$	0	0
$281$	−2.87564	−0.171547	−0.0857733	−	0.996315i	$-0.527336\pi$
$281$	−2.87564	−0.171547	−0.0857733	+	0.996315i	$0.527336\pi$
$282$	0	0
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	24.7846	1.45792
$290$	0	0
$291$	−11.8564	−0.695035
$292$	0	0
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.85641	−0.107720
$298$	0	0
$299$	−27.1244	−1.56864
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−9.80385	−0.563216
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−22.1962	−1.26680	−0.633401	−	0.773824i	$-0.718341\pi$
$307$	−22.1962	−1.26680	−0.633401	+	0.773824i	$0.718341\pi$
$308$	0	0
$309$	1.32051	0.0751211
$310$	0	0
$311$	−14.0718	−0.797938	−0.398969	−	0.916964i	$-0.630632\pi$
$311$	−14.0718	−0.797938	−0.398969	+	0.916964i	$0.630632\pi$
$312$	0	0
$313$	8.60770	0.486536	0.243268	−	0.969959i	$-0.421781\pi$
$313$	8.60770	0.486536	0.243268	+	0.969959i	$0.421781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.6603	0.991899	0.495949	−	0.868351i	$-0.334820\pi$
$317$	17.6603	0.991899	0.495949	+	0.868351i	$0.334820\pi$
$318$	0	0
$319$	−1.60770	−0.0900136
$320$	0	0
$321$	−14.5359	−0.811315
$322$	0	0
$323$	−6.46410	−0.359672
$324$	0	0
$325$	20.9808	1.16380
$326$	0	0
$327$	−1.75129	−0.0968465
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.1962	−1.32994	−0.664970	−	0.746870i	$-0.731556\pi$
$331$	−24.1962	−1.32994	−0.664970	+	0.746870i	$0.731556\pi$
$332$	0	0
$333$	9.85641	0.540128
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.9808	−1.46974	−0.734868	−	0.678210i	$-0.762756\pi$
$337$	−26.9808	−1.46974	−0.734868	+	0.678210i	$0.762756\pi$
$338$	0	0
$339$	−7.17691	−0.389796
$340$	0	0
$341$	−1.94744	−0.105460
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.92820	0.371925	0.185963	−	0.982557i	$-0.440460\pi$
$347$	6.92820	0.371925	0.185963	+	0.982557i	$0.440460\pi$
$348$	0	0
$349$	21.7846	1.16610	0.583052	−	0.812435i	$-0.301858\pi$
$349$	21.7846	1.16610	0.583052	+	0.812435i	$0.301858\pi$
$350$	0	0
$351$	16.7846	0.895896
$352$	0	0
$353$	14.5359	0.773668	0.386834	−	0.922149i	$-0.373569\pi$
$353$	14.5359	0.773668	0.386834	+	0.922149i	$0.373569\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.89488	−0.414374
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.39230	−0.333676	−0.166838	−	0.985984i	$-0.553356\pi$
$367$	−6.39230	−0.333676	−0.166838	+	0.985984i	$0.553356\pi$
$368$	0	0
$369$	8.53590	0.444361
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.5885	1.06603	0.533015	−	0.846106i	$-0.321059\pi$
$373$	20.5885	1.06603	0.533015	+	0.846106i	$0.321059\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5359	0.748637
$378$	0	0
$379$	3.60770	0.185315	0.0926574	−	0.995698i	$-0.470464\pi$
$379$	3.60770	0.185315	0.0926574	+	0.995698i	$0.470464\pi$
$380$	0	0
$381$	−14.9282	−0.764795
$382$	0	0
$383$	3.80385	0.194368	0.0971838	−	0.995266i	$-0.469017\pi$
$383$	3.80385	0.194368	0.0971838	+	0.995266i	$0.469017\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.92820	−0.250515
$388$	0	0
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	41.7846	2.11314
$392$	0	0
$393$	6.58846	0.332344
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.1962	−1.60780	−0.803900	−	0.594765i	$-0.797245\pi$
$401$	−32.1962	−1.60780	−0.803900	+	0.594765i	$0.797245\pi$
$402$	0	0
$403$	17.6077	0.877102
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.85641	−0.0920187
$408$	0	0
$409$	24.7846	1.22552	0.612760	−	0.790269i	$-0.290059\pi$
$409$	24.7846	1.22552	0.612760	+	0.790269i	$0.290059\pi$
$410$	0	0
$411$	5.41154	0.266932
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.444864	−0.0217851
$418$	0	0
$419$	4.60770	0.225101	0.112550	−	0.993646i	$-0.464098\pi$
$419$	4.60770	0.225101	0.112550	+	0.993646i	$0.464098\pi$
$420$	0	0
$421$	28.7846	1.40288	0.701438	−	0.712730i	$-0.252542\pi$
$421$	28.7846	1.40288	0.701438	+	0.712730i	$0.252542\pi$
$422$	0	0
$423$	−9.67949	−0.470633
$424$	0	0
$425$	−32.3205	−1.56777
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.42563	−0.0688299
$430$	0	0
$431$	14.7846	0.712150	0.356075	−	0.934457i	$-0.384115\pi$
$431$	14.7846	0.712150	0.356075	+	0.934457i	$0.384115\pi$
$432$	0	0
$433$	14.3923	0.691650	0.345825	−	0.938299i	$-0.387599\pi$
$433$	14.3923	0.691650	0.345825	+	0.938299i	$0.387599\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.46410	−0.309220
$438$	0	0
$439$	−6.39230	−0.305088	−0.152544	−	0.988297i	$-0.548747\pi$
$439$	−6.39230	−0.305088	−0.152544	+	0.988297i	$0.548747\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.3205	−0.965456	−0.482728	−	0.875770i	$-0.660354\pi$
$443$	−20.3205	−0.965456	−0.482728	+	0.875770i	$0.660354\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.2154	0.719663
$448$	0	0
$449$	8.19615	0.386800	0.193400	−	0.981120i	$-0.438048\pi$
$449$	8.19615	0.386800	0.193400	+	0.981120i	$0.438048\pi$
$450$	0	0
$451$	−1.60770	−0.0757034
$452$	0	0
$453$	−1.32051	−0.0620429
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.7846	0.644817	0.322408	−	0.946601i	$-0.395508\pi$
$457$	13.7846	0.644817	0.322408	+	0.946601i	$0.395508\pi$
$458$	0	0
$459$	−25.8564	−1.20687
$460$	0	0
$461$	−12.4641	−0.580511	−0.290256	−	0.956949i	$-0.593740\pi$
$461$	−12.4641	−0.580511	−0.290256	+	0.956949i	$0.593740\pi$
$462$	0	0
$463$	−27.7846	−1.29126	−0.645630	−	0.763650i	$-0.723405\pi$
$463$	−27.7846	−1.29126	−0.645630	+	0.763650i	$0.723405\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.4641	0.576770	0.288385	−	0.957515i	$-0.406882\pi$
$467$	12.4641	0.576770	0.288385	+	0.957515i	$0.406882\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.94744	0.181888
$472$	0	0
$473$	0.928203	0.0426788
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	5.41154	0.247778
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	16.7846	0.765312
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	36.9808	1.67576	0.837879	−	0.545856i	$-0.183795\pi$
$487$	36.9808	1.67576	0.837879	+	0.545856i	$0.183795\pi$
$488$	0	0
$489$	3.66025	0.165522
$490$	0	0
$491$	−33.9282	−1.53116	−0.765579	−	0.643342i	$-0.777547\pi$
$491$	−33.9282	−1.53116	−0.765579	+	0.643342i	$0.777547\pi$
$492$	0	0
$493$	−22.3923	−1.00850
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.39230	0.286159	0.143079	−	0.989711i	$-0.454300\pi$
$499$	6.39230	0.286159	0.143079	+	0.989711i	$0.454300\pi$
$500$	0	0
$501$	16.1436	0.721243
$502$	0	0
$503$	24.4641	1.09080	0.545400	−	0.838176i	$-0.316378\pi$
$503$	24.4641	1.09080	0.545400	+	0.838176i	$0.316378\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.37307	0.149803
$508$	0	0
$509$	−27.4641	−1.21732	−0.608662	−	0.793429i	$-0.708294\pi$
$509$	−27.4641	−1.21732	−0.608662	+	0.793429i	$0.708294\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.82309	0.0801792
$518$	0	0
$519$	13.6077	0.597312
$520$	0	0
$521$	24.3397	1.06634	0.533172	−	0.846007i	$-0.321000\pi$
$521$	24.3397	1.06634	0.533172	+	0.846007i	$0.321000\pi$
$522$	0	0
$523$	35.1769	1.53818	0.769090	−	0.639141i	$-0.220710\pi$
$523$	35.1769	1.53818	0.769090	+	0.639141i	$0.220710\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−27.1244	−1.18156
$528$	0	0
$529$	18.7846	0.816722
$530$	0	0
$531$	−3.12436	−0.135585
$532$	0	0
$533$	14.5359	0.629620
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.0718	−0.477783
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.39230	−0.360813	−0.180407	−	0.983592i	$-0.557741\pi$
$541$	−8.39230	−0.360813	−0.180407	+	0.983592i	$0.557741\pi$
$542$	0	0
$543$	6.14359	0.263647
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.7846	0.974200	0.487100	−	0.873346i	$-0.338055\pi$
$547$	22.7846	0.974200	0.487100	+	0.873346i	$0.338055\pi$
$548$	0	0
$549$	−17.2487	−0.736157
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	−8.39230	−0.354957
$560$	0	0
$561$	2.19615	0.0927216
$562$	0	0
$563$	3.12436	0.131676	0.0658379	−	0.997830i	$-0.479028\pi$
$563$	3.12436	0.131676	0.0658379	+	0.997830i	$0.479028\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.3205	1.48071	0.740356	−	0.672215i	$-0.234657\pi$
$569$	35.3205	1.48071	0.740356	+	0.672215i	$0.234657\pi$
$570$	0	0
$571$	−33.7846	−1.41384	−0.706921	−	0.707293i	$-0.749916\pi$
$571$	−33.7846	−1.41384	−0.706921	+	0.707293i	$0.749916\pi$
$572$	0	0
$573$	0.679492	0.0283862
$574$	0	0
$575$	−32.3205	−1.34786
$576$	0	0
$577$	5.39230	0.224485	0.112242	−	0.993681i	$-0.464197\pi$
$577$	5.39230	0.224485	0.112242	+	0.993681i	$0.464197\pi$
$578$	0	0
$579$	−1.32051	−0.0548784
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.01924	−0.0422125
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.6410	−1.55361	−0.776806	−	0.629741i	$-0.783161\pi$
$587$	−37.6410	−1.55361	−0.776806	+	0.629741i	$0.783161\pi$
$588$	0	0
$589$	4.19615	0.172899
$590$	0	0
$591$	−18.5885	−0.764627
$592$	0	0
$593$	−27.2487	−1.11897	−0.559485	−	0.828840i	$-0.689001\pi$
$593$	−27.2487	−1.11897	−0.559485	+	0.828840i	$0.689001\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.5359	0.431206
$598$	0	0
$599$	25.1769	1.02870	0.514350	−	0.857580i	$-0.328033\pi$
$599$	25.1769	1.02870	0.514350	+	0.857580i	$0.328033\pi$
$600$	0	0
$601$	−28.7846	−1.17415	−0.587074	−	0.809533i	$-0.699720\pi$
$601$	−28.7846	−1.17415	−0.587074	+	0.809533i	$0.699720\pi$
$602$	0	0
$603$	24.6410	1.00346
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−32.5885	−1.32272	−0.661362	−	0.750067i	$-0.730021\pi$
$607$	−32.5885	−1.32272	−0.661362	+	0.750067i	$0.730021\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.4833	−0.666845
$612$	0	0
$613$	46.5692	1.88091	0.940457	−	0.339914i	$-0.110398\pi$
$613$	46.5692	1.88091	0.940457	+	0.339914i	$0.110398\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28.6410	−1.15304	−0.576522	−	0.817082i	$-0.695590\pi$
$617$	−28.6410	−1.15304	−0.576522	+	0.817082i	$0.695590\pi$
$618$	0	0
$619$	−24.1769	−0.971752	−0.485876	−	0.874028i	$-0.661499\pi$
$619$	−24.1769	−0.971752	−0.485876	+	0.874028i	$0.661499\pi$
$620$	0	0
$621$	−25.8564	−1.03758
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−0.339746	−0.0135681
$628$	0	0
$629$	−25.8564	−1.03096
$630$	0	0
$631$	−14.3923	−0.572949	−0.286474	−	0.958088i	$-0.592483\pi$
$631$	−14.3923	−0.572949	−0.286474	+	0.958088i	$0.592483\pi$
$632$	0	0
$633$	19.4641	0.773629
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	26.4449	1.04614
$640$	0	0
$641$	46.7321	1.84581	0.922903	−	0.385034i	$-0.125810\pi$
$641$	46.7321	1.84581	0.922903	+	0.385034i	$0.125810\pi$
$642$	0	0
$643$	−9.39230	−0.370396	−0.185198	−	0.982701i	$-0.559293\pi$
$643$	−9.39230	−0.370396	−0.185198	+	0.982701i	$0.559293\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.9282	1.68768	0.843841	−	0.536593i	$-0.180289\pi$
$647$	42.9282	1.68768	0.843841	+	0.536593i	$0.180289\pi$
$648$	0	0
$649$	0.588457	0.0230990
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.9282	0.858117	0.429058	−	0.903277i	$-0.358845\pi$
$653$	21.9282	0.858117	0.429058	+	0.903277i	$0.358845\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	37.9282	1.47972
$658$	0	0
$659$	−5.66025	−0.220492	−0.110246	−	0.993904i	$-0.535164\pi$
$659$	−5.66025	−0.220492	−0.110246	+	0.993904i	$0.535164\pi$
$660$	0	0
$661$	−5.80385	−0.225744	−0.112872	−	0.993610i	$-0.536005\pi$
$661$	−5.80385	−0.225744	−0.112872	+	0.993610i	$0.536005\pi$
$662$	0	0
$663$	−19.8564	−0.771159
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−22.3923	−0.867034
$668$	0	0
$669$	−13.4641	−0.520552
$670$	0	0
$671$	3.24871	0.125415
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	21.4641	0.824932	0.412466	−	0.910973i	$-0.364667\pi$
$677$	21.4641	0.824932	0.412466	+	0.910973i	$0.364667\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.53590	−0.0971758
$682$	0	0
$683$	36.8372	1.40954	0.704768	−	0.709438i	$-0.251051\pi$
$683$	36.8372	1.40954	0.704768	+	0.709438i	$0.251051\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.9474	0.608433
$688$	0	0
$689$	9.21539	0.351078
$690$	0	0
$691$	−33.3923	−1.27030	−0.635152	−	0.772388i	$-0.719062\pi$
$691$	−33.3923	−1.27030	−0.635152	+	0.772388i	$0.719062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.3923	−0.848169
$698$	0	0
$699$	5.41154	0.204683
$700$	0	0
$701$	32.0718	1.21134	0.605668	−	0.795718i	$-0.292906\pi$
$701$	32.0718	1.21134	0.605668	+	0.795718i	$0.292906\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.6077	0.473492	0.236746	−	0.971572i	$-0.423919\pi$
$709$	12.6077	0.473492	0.236746	+	0.971572i	$0.423919\pi$
$710$	0	0
$711$	15.2679	0.572593
$712$	0	0
$713$	−27.1244	−1.01582
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.9808	−0.410084
$718$	0	0
$719$	1.39230	0.0519242	0.0259621	−	0.999663i	$-0.491735\pi$
$719$	1.39230	0.0519242	0.0259621	+	0.999663i	$0.491735\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.07180	−0.337384
$724$	0	0
$725$	17.3205	0.643268
$726$	0	0
$727$	5.60770	0.207978	0.103989	−	0.994578i	$-0.466839\pi$
$727$	5.60770	0.207978	0.103989	+	0.994578i	$0.466839\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	12.9282	0.478167
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.64102	−0.170954
$738$	0	0
$739$	−2.60770	−0.0959256	−0.0479628	−	0.998849i	$-0.515273\pi$
$739$	−2.60770	−0.0959256	−0.0479628	+	0.998849i	$0.515273\pi$
$740$	0	0
$741$	3.07180	0.112845
$742$	0	0
$743$	32.4449	1.19029	0.595143	−	0.803620i	$-0.297095\pi$
$743$	32.4449	1.19029	0.595143	+	0.803620i	$0.297095\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.1769	−0.811411
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.1962	1.02889	0.514446	−	0.857523i	$-0.327998\pi$
$751$	28.1962	1.02889	0.514446	+	0.857523i	$0.327998\pi$
$752$	0	0
$753$	2.53590	0.0924133
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.3923	0.559443	0.279721	−	0.960081i	$-0.409758\pi$
$757$	15.3923	0.559443	0.279721	+	0.960081i	$0.409758\pi$
$758$	0	0
$759$	2.19615	0.0797153
$760$	0	0
$761$	−21.7128	−0.787089	−0.393544	−	0.919306i	$-0.628751\pi$
$761$	−21.7128	−0.787089	−0.393544	+	0.919306i	$0.628751\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.32051	−0.192112
$768$	0	0
$769$	24.7846	0.893756	0.446878	−	0.894595i	$-0.352536\pi$
$769$	24.7846	0.893756	0.446878	+	0.894595i	$0.352536\pi$
$770$	0	0
$771$	−5.32051	−0.191613
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	20.9808	0.753651
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.46410	0.124114
$780$	0	0
$781$	−4.98076	−0.178226
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−35.3731	−1.26091	−0.630457	−	0.776224i	$-0.717132\pi$
$787$	−35.3731	−1.26091	−0.630457	+	0.776224i	$0.717132\pi$
$788$	0	0
$789$	−12.3397	−0.439307
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−29.3731	−1.04307
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−50.7846	−1.79888	−0.899442	−	0.437041i	$-0.856026\pi$
$797$	−50.7846	−1.79888	−0.899442	+	0.437041i	$0.856026\pi$
$798$	0	0
$799$	25.3923	0.898315
$800$	0	0
$801$	−2.28719	−0.0808138
$802$	0	0
$803$	−7.14359	−0.252092
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.7128	0.553117
$808$	0	0
$809$	−30.9282	−1.08738	−0.543689	−	0.839287i	$-0.682973\pi$
$809$	−30.9282	−1.08738	−0.543689	+	0.839287i	$0.682973\pi$
$810$	0	0
$811$	50.3923	1.76951	0.884757	−	0.466053i	$-0.154325\pi$
$811$	50.3923	1.76951	0.884757	+	0.466053i	$0.154325\pi$
$812$	0	0
$813$	−0.444864	−0.0156021
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.3205	−0.918592	−0.459296	−	0.888283i	$-0.651898\pi$
$821$	−26.3205	−0.918592	−0.459296	+	0.888283i	$0.651898\pi$
$822$	0	0
$823$	−20.6077	−0.718339	−0.359170	−	0.933272i	$-0.616940\pi$
$823$	−20.6077	−0.718339	−0.359170	+	0.933272i	$0.616940\pi$
$824$	0	0
$825$	−1.69873	−0.0591422
$826$	0	0
$827$	31.5167	1.09594	0.547971	−	0.836497i	$-0.315400\pi$
$827$	31.5167	1.09594	0.547971	+	0.836497i	$0.315400\pi$
$828$	0	0
$829$	20.9808	0.728692	0.364346	−	0.931264i	$-0.381293\pi$
$829$	20.9808	0.728692	0.364346	+	0.931264i	$0.381293\pi$
$830$	0	0
$831$	−11.5551	−0.400843
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.7846	0.580161
$838$	0	0
$839$	6.92820	0.239188	0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820	0.239188	0.119594	+	0.992823i	$0.461841\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−2.10512	−0.0725041
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.46410	−0.0502478
$850$	0	0
$851$	−25.8564	−0.886346
$852$	0	0
$853$	27.7846	0.951327	0.475663	−	0.879627i	$-0.342208\pi$
$853$	27.7846	0.951327	0.475663	+	0.879627i	$0.342208\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.0526	1.57313	0.786563	−	0.617510i	$-0.211859\pi$
$857$	46.0526	1.57313	0.786563	+	0.617510i	$0.211859\pi$
$858$	0	0
$859$	−46.7846	−1.59627	−0.798135	−	0.602478i	$-0.794180\pi$
$859$	−46.7846	−1.59627	−0.798135	+	0.602478i	$0.794180\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−51.0333	−1.73719	−0.868597	−	0.495519i	$-0.834978\pi$
$863$	−51.0333	−1.73719	−0.868597	+	0.495519i	$0.834978\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18.1436	0.616189
$868$	0	0
$869$	−2.87564	−0.0975496
$870$	0	0
$871$	41.9615	1.42181
$872$	0	0
$873$	39.9090	1.35071
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.7846	0.566776	0.283388	−	0.959005i	$-0.408542\pi$
$877$	16.7846	0.566776	0.283388	+	0.959005i	$0.408542\pi$
$878$	0	0
$879$	−21.9615	−0.740744
$880$	0	0
$881$	−28.6077	−0.963818	−0.481909	−	0.876221i	$-0.660056\pi$
$881$	−28.6077	−0.963818	−0.481909	+	0.876221i	$0.660056\pi$
$882$	0	0
$883$	47.0000	1.58168	0.790838	−	0.612026i	$-0.209645\pi$
$883$	47.0000	1.58168	0.790838	+	0.612026i	$0.209645\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.0718	−0.371755	−0.185877	−	0.982573i	$-0.559513\pi$
$887$	−11.0718	−0.371755	−0.185877	+	0.982573i	$0.559513\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.07180	0.0694078
$892$	0	0
$893$	−3.92820	−0.131452
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−19.8564	−0.662986
$898$	0	0
$899$	14.5359	0.484799
$900$	0	0
$901$	−14.1962	−0.472942
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.1962	0.537784	0.268892	−	0.963170i	$-0.413342\pi$
$907$	16.1962	0.537784	0.268892	+	0.963170i	$0.413342\pi$
$908$	0	0
$909$	33.0000	1.09454
$910$	0	0
$911$	−23.4115	−0.775659	−0.387830	−	0.921731i	$-0.626775\pi$
$911$	−23.4115	−0.775659	−0.387830	+	0.921731i	$0.626775\pi$
$912$	0	0
$913$	4.17691	0.138236
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	33.3923	1.10151	0.550755	−	0.834667i	$-0.314340\pi$
$919$	33.3923	1.10151	0.550755	+	0.834667i	$0.314340\pi$
$920$	0	0
$921$	−16.2487	−0.535413
$922$	0	0
$923$	45.0333	1.48229
$924$	0	0
$925$	20.0000	0.657596
$926$	0	0
$927$	−4.44486	−0.145988
$928$	0	0
$929$	32.5692	1.06856	0.534281	−	0.845307i	$-0.320583\pi$
$929$	32.5692	1.06856	0.534281	+	0.845307i	$0.320583\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.3013	−0.337248
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.60770	−0.117858	−0.0589291	−	0.998262i	$-0.518769\pi$
$937$	−3.60770	−0.117858	−0.0589291	+	0.998262i	$0.518769\pi$
$938$	0	0
$939$	6.30127	0.205634
$940$	0	0
$941$	28.9808	0.944746	0.472373	−	0.881399i	$-0.343398\pi$
$941$	28.9808	0.944746	0.472373	+	0.881399i	$0.343398\pi$
$942$	0	0
$943$	−22.3923	−0.729194
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.3923	0.337705	0.168852	−	0.985641i	$-0.445994\pi$
$947$	10.3923	0.337705	0.168852	+	0.985641i	$0.445994\pi$
$948$	0	0
$949$	64.5885	2.09663
$950$	0	0
$951$	12.9282	0.419226
$952$	0	0
$953$	7.85641	0.254494	0.127247	−	0.991871i	$-0.459386\pi$
$953$	7.85641	0.254494	0.127247	+	0.991871i	$0.459386\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.17691	−0.0380442
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.3923	−0.432010
$962$	0	0
$963$	48.9282	1.57669
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.7846	0.732704	0.366352	−	0.930476i	$-0.380607\pi$
$967$	22.7846	0.732704	0.366352	+	0.930476i	$0.380607\pi$
$968$	0	0
$969$	−4.73205	−0.152015
$970$	0	0
$971$	−26.7846	−0.859559	−0.429780	−	0.902934i	$-0.641409\pi$
$971$	−26.7846	−0.859559	−0.429780	+	0.902934i	$0.641409\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	15.3590	0.491881
$976$	0	0
$977$	−16.7321	−0.535306	−0.267653	−	0.963515i	$-0.586248\pi$
$977$	−16.7321	−0.535306	−0.267653	+	0.963515i	$0.586248\pi$
$978$	0	0
$979$	0.430781	0.0137678
$980$	0	0
$981$	5.89488	0.188209
$982$	0	0
$983$	−3.71281	−0.118420	−0.0592102	−	0.998246i	$-0.518858\pi$
$983$	−3.71281	−0.118420	−0.0592102	+	0.998246i	$0.518858\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.9282	0.411093
$990$	0	0
$991$	12.9808	0.412347	0.206174	−	0.978515i	$-0.433899\pi$
$991$	12.9808	0.412347	0.206174	+	0.978515i	$0.433899\pi$
$992$	0	0
$993$	−17.7128	−0.562099
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−45.3923	−1.43759	−0.718794	−	0.695223i	$-0.755306\pi$
$997$	−45.3923	−1.43759	−0.718794	+	0.695223i	$0.755306\pi$
$998$	0	0
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.a.c.1.2		2
7.3	odd	6		532.2.i.a.457.2	yes	4
7.5	odd	6		532.2.i.a.305.2	✓	4
7.6	odd	2		3724.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.i.a.305.2	✓	4	7.5	odd	6
532.2.i.a.457.2	yes	4	7.3	odd	6
3724.2.a.c.1.2		2	1.1	even	1	trivial
3724.2.a.f.1.1		2	7.6	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:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -2.46410 q^{9} +0.464102 q^{11} -4.19615 q^{13} +6.46410 q^{17} -1.00000 q^{19} +6.46410 q^{23} -5.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} -4.19615 q^{31} +0.339746 q^{33} -4.00000 q^{37} -3.07180 q^{39} -3.46410 q^{41} +2.00000 q^{43} +3.92820 q^{47} +4.73205 q^{51} -2.19615 q^{53} -0.732051 q^{57} +1.26795 q^{59} +7.00000 q^{61} -10.0000 q^{67} +4.73205 q^{69} -10.7321 q^{71} -15.3923 q^{73} -3.66025 q^{75} -6.19615 q^{79} +4.46410 q^{81} +9.00000 q^{83} -2.53590 q^{87} +0.928203 q^{89} -3.07180 q^{93} -16.1962 q^{97} -1.14359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} + 6 q^{23} - 10 q^{25} - 8 q^{27} + 2 q^{31} + 18 q^{33} - 8 q^{37} - 20 q^{39} + 4 q^{43} - 6 q^{47} + 6 q^{51} + 6 q^{53} + 2 q^{57}+ \cdots - 30 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^11 + 2 * q^13 + 6 * q^17 - 2 * q^19 + 6 * q^23 - 10 * q^25 - 8 * q^27 + 2 * q^31 + 18 * q^33 - 8 * q^37 - 20 * q^39 + 4 * q^43 - 6 * q^47 + 6 * q^51 + 6 * q^53 + 2 * q^57 + 6 * q^59 + 14 * q^61 - 20 * q^67 + 6 * q^69 - 18 * q^71 - 10 * q^73 + 10 * q^75 - 2 * q^79 + 2 * q^81 + 18 * q^83 - 12 * q^87 - 12 * q^89 - 20 * q^93 - 22 * q^97 - 30 * q^99

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