LMFDB - Embedded newform 5200.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5200,2,Mod(1,5200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5200, 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("5200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.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:	$41.5222090511$
Analytic rank:	$0$
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 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +2.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} +2.00000 q^{7} -2.46410 q^{9} +1.26795 q^{11} -1.00000 q^{13} -3.46410 q^{17} -4.19615 q^{19} -1.46410 q^{21} +4.73205 q^{23} +4.00000 q^{27} -9.46410 q^{29} +0.196152 q^{31} -0.928203 q^{33} +4.00000 q^{37} +0.732051 q^{39} -3.46410 q^{41} +10.1962 q^{43} +6.00000 q^{47} -3.00000 q^{49} +2.53590 q^{51} +10.3923 q^{53} +3.07180 q^{57} +15.1244 q^{59} +12.3923 q^{61} -4.92820 q^{63} -14.3923 q^{67} -3.46410 q^{69} -1.26795 q^{71} +4.00000 q^{73} +2.53590 q^{77} -12.3923 q^{79} +4.46410 q^{81} -6.00000 q^{83} +6.92820 q^{87} +0.928203 q^{89} -2.00000 q^{91} -0.143594 q^{93} -2.00000 q^{97} -3.12436 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{13} + 2 q^{19} + 4 q^{21} + 6 q^{23} + 8 q^{27} - 12 q^{29} - 10 q^{31} + 12 q^{33} + 8 q^{37} - 2 q^{39} + 10 q^{43} + 12 q^{47} - 6 q^{49} + 12 q^{51} + 20 q^{57} + 6 q^{59} + 4 q^{61} + 4 q^{63} - 8 q^{67} - 6 q^{71} + 8 q^{73} + 12 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 12 q^{89} - 4 q^{91} - 28 q^{93} - 4 q^{97} + 18 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^13 + 2 * q^19 + 4 * q^21 + 6 * q^23 + 8 * q^27 - 12 * q^29 - 10 * q^31 + 12 * q^33 + 8 * q^37 - 2 * q^39 + 10 * q^43 + 12 * q^47 - 6 * q^49 + 12 * q^51 + 20 * q^57 + 6 * q^59 + 4 * q^61 + 4 * q^63 - 8 * q^67 - 6 * q^71 + 8 * q^73 + 12 * q^77 - 4 * q^79 + 2 * q^81 - 12 * q^83 - 12 * q^89 - 4 * q^91 - 28 * q^93 - 4 * q^97 + 18 * q^99

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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−4.19615	−0.962663	−0.481332	−	0.876539i	$-0.659847\pi$
$19$	−4.19615	−0.962663	−0.481332	+	0.876539i	$0.659847\pi$
$20$	0	0
$21$	−1.46410	−0.319493
$22$	0	0
$23$	4.73205	0.986701	0.493350	−	0.869831i	$-0.335772\pi$
$23$	4.73205	0.986701	0.493350	+	0.869831i	$0.335772\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	0	0
$31$	0.196152	0.0352300	0.0176150	−	0.999845i	$-0.494393\pi$
$31$	0.196152	0.0352300	0.0176150	+	0.999845i	$0.494393\pi$
$32$	0	0
$33$	−0.928203	−0.161579
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0.732051	0.117222
$40$	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
$43$	10.1962	1.55490	0.777449	−	0.628946i	$-0.216513\pi$
$43$	10.1962	1.55490	0.777449	+	0.628946i	$0.216513\pi$
$44$	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
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.53590	0.355097
$52$	0	0
$53$	10.3923	1.42749	0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923	1.42749	0.713746	+	0.700404i	$0.246997\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.07180	0.406869
$58$	0	0
$59$	15.1244	1.96902	0.984512	−	0.175319i	$-0.0560957\pi$
$59$	15.1244	1.96902	0.984512	+	0.175319i	$0.0560957\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0	0
$63$	−4.92820	−0.620895
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.3923	−1.75830	−0.879150	−	0.476545i	$-0.841889\pi$
$67$	−14.3923	−1.75830	−0.879150	+	0.476545i	$0.841889\pi$
$68$	0	0
$69$	−3.46410	−0.417029
$70$	0	0
$71$	−1.26795	−0.150478	−0.0752389	−	0.997166i	$-0.523972\pi$
$71$	−1.26795	−0.150478	−0.0752389	+	0.997166i	$0.523972\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.53590	0.288992
$78$	0	0
$79$	−12.3923	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$79$	−12.3923	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.92820	0.742781
$88$	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
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−0.143594	−0.0148900
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−3.12436	−0.314010
$100$	0	0
$101$	12.9282	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$101$	12.9282	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$102$	0	0
$103$	10.1962	1.00466	0.502328	−	0.864677i	$-0.332477\pi$
$103$	10.1962	1.00466	0.502328	+	0.864677i	$0.332477\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.339746	0.0328445	0.0164222	−	0.999865i	$-0.494772\pi$
$107$	0.339746	0.0328445	0.0164222	+	0.999865i	$0.494772\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.92820	−0.277933
$112$	0	0
$113$	−15.4641	−1.45474	−0.727370	−	0.686245i	$-0.759258\pi$
$113$	−15.4641	−1.45474	−0.727370	+	0.686245i	$0.759258\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.46410	0.227806
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	2.53590	0.228654
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.80385	0.515008	0.257504	−	0.966277i	$-0.417100\pi$
$127$	5.80385	0.515008	0.257504	+	0.966277i	$0.417100\pi$
$128$	0	0
$129$	−7.46410	−0.657178
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−8.39230	−0.727705
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	−4.39230	−0.369899
$142$	0	0
$143$	−1.26795	−0.106031
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.19615	0.181136
$148$	0	0
$149$	19.8564	1.62670	0.813350	−	0.581775i	$-0.197641\pi$
$149$	19.8564	1.62670	0.813350	+	0.581775i	$0.197641\pi$
$150$	0	0
$151$	12.1962	0.992509	0.496254	−	0.868177i	$-0.334708\pi$
$151$	12.1962	0.992509	0.496254	+	0.868177i	$0.334708\pi$
$152$	0	0
$153$	8.53590	0.690086
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−7.60770	−0.603329
$160$	0	0
$161$	9.46410	0.745876
$162$	0	0
$163$	6.39230	0.500684	0.250342	−	0.968157i	$-0.419457\pi$
$163$	6.39230	0.500684	0.250342	+	0.968157i	$0.419457\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.9282	1.00041	0.500207	−	0.865906i	$-0.333257\pi$
$167$	12.9282	1.00041	0.500207	+	0.865906i	$0.333257\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.3397	0.790700
$172$	0	0
$173$	−15.4641	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$173$	−15.4641	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.0718	−0.832207
$178$	0	0
$179$	5.07180	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$179$	5.07180	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$180$	0	0
$181$	−20.3923	−1.51575	−0.757874	−	0.652401i	$-0.773762\pi$
$181$	−20.3923	−1.51575	−0.757874	+	0.652401i	$0.773762\pi$
$182$	0	0
$183$	−9.07180	−0.670607
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.39230	−0.321197
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.928203	−0.0661317	−0.0330659	−	0.999453i	$-0.510527\pi$
$197$	−0.928203	−0.0661317	−0.0330659	+	0.999453i	$0.510527\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	10.5359	0.743145
$202$	0	0
$203$	−18.9282	−1.32850
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.6603	−0.810444
$208$	0	0
$209$	−5.32051	−0.368027
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0.928203	0.0635994
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.392305	0.0266314
$218$	0	0
$219$	−2.92820	−0.197870
$220$	0	0
$221$	3.46410	0.233021
$222$	0	0
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.46410	0.229920	0.114960	−	0.993370i	$-0.463326\pi$
$227$	3.46410	0.229920	0.114960	+	0.993370i	$0.463326\pi$
$228$	0	0
$229$	−14.3923	−0.951070	−0.475535	−	0.879697i	$-0.657746\pi$
$229$	−14.3923	−0.951070	−0.475535	+	0.879697i	$0.657746\pi$
$230$	0	0
$231$	−1.85641	−0.122143
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.07180	0.589277
$238$	0	0
$239$	3.80385	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$239$	3.80385	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$240$	0	0
$241$	18.3923	1.18475	0.592376	−	0.805661i	$-0.298190\pi$
$241$	18.3923	1.18475	0.592376	+	0.805661i	$0.298190\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.19615	0.266995
$248$	0	0
$249$	4.39230	0.278351
$250$	0	0
$251$	14.5359	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$251$	14.5359	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.85641	0.490069	0.245035	−	0.969514i	$-0.421201\pi$
$257$	7.85641	0.490069	0.245035	+	0.969514i	$0.421201\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	23.3205	1.44350
$262$	0	0
$263$	4.73205	0.291791	0.145895	−	0.989300i	$-0.453394\pi$
$263$	4.73205	0.291791	0.145895	+	0.989300i	$0.453394\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.679492	−0.0415842
$268$	0	0
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	0	0
$271$	20.9808	1.27449	0.637245	−	0.770661i	$-0.280074\pi$
$271$	20.9808	1.27449	0.637245	+	0.770661i	$0.280074\pi$
$272$	0	0
$273$	1.46410	0.0886115
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.60770	0.336934	0.168467	−	0.985707i	$-0.446118\pi$
$277$	5.60770	0.336934	0.168467	+	0.985707i	$0.446118\pi$
$278$	0	0
$279$	−0.483340	−0.0289368
$280$	0	0
$281$	1.60770	0.0959071	0.0479535	−	0.998850i	$-0.484730\pi$
$281$	1.60770	0.0959071	0.0479535	+	0.998850i	$0.484730\pi$
$282$	0	0
$283$	1.41154	0.0839075	0.0419538	−	0.999120i	$-0.486642\pi$
$283$	1.41154	0.0839075	0.0419538	+	0.999120i	$0.486642\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.92820	−0.408959
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	1.46410	0.0858272
$292$	0	0
$293$	18.9282	1.10580	0.552899	−	0.833248i	$-0.313522\pi$
$293$	18.9282	1.10580	0.552899	+	0.833248i	$0.313522\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.07180	0.294295
$298$	0	0
$299$	−4.73205	−0.273662
$300$	0	0
$301$	20.3923	1.17539
$302$	0	0
$303$	−9.46410	−0.543698
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.7846	1.30039	0.650193	−	0.759769i	$-0.274688\pi$
$307$	22.7846	1.30039	0.650193	+	0.759769i	$0.274688\pi$
$308$	0	0
$309$	−7.46410	−0.424618
$310$	0	0
$311$	−4.39230	−0.249065	−0.124532	−	0.992216i	$-0.539743\pi$
$311$	−4.39230	−0.249065	−0.124532	+	0.992216i	$0.539743\pi$
$312$	0	0
$313$	−6.39230	−0.361314	−0.180657	−	0.983546i	$-0.557822\pi$
$313$	−6.39230	−0.361314	−0.180657	+	0.983546i	$0.557822\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−0.248711	−0.0138817
$322$	0	0
$323$	14.5359	0.808799
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.46410	−0.0809650
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	28.5885	1.57136	0.785682	−	0.618631i	$-0.212312\pi$
$331$	28.5885	1.57136	0.785682	+	0.618631i	$0.212312\pi$
$332$	0	0
$333$	−9.85641	−0.540128
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.60770	0.305471	0.152735	−	0.988267i	$-0.451192\pi$
$337$	5.60770	0.305471	0.152735	+	0.988267i	$0.451192\pi$
$338$	0	0
$339$	11.3205	0.614846
$340$	0	0
$341$	0.248711	0.0134685
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.6603	0.625955	0.312978	−	0.949761i	$-0.398674\pi$
$347$	11.6603	0.625955	0.312978	+	0.949761i	$0.398674\pi$
$348$	0	0
$349$	6.39230	0.342172	0.171086	−	0.985256i	$-0.445272\pi$
$349$	6.39230	0.342172	0.171086	+	0.985256i	$0.445272\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−27.7128	−1.47500	−0.737502	−	0.675345i	$-0.763995\pi$
$353$	−27.7128	−1.47500	−0.737502	+	0.675345i	$0.763995\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.07180	0.268428
$358$	0	0
$359$	−8.19615	−0.432576	−0.216288	−	0.976330i	$-0.569395\pi$
$359$	−8.19615	−0.432576	−0.216288	+	0.976330i	$0.569395\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	0	0
$363$	6.87564	0.360878
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.1962	1.15863	0.579315	−	0.815104i	$-0.303320\pi$
$367$	22.1962	1.15863	0.579315	+	0.815104i	$0.303320\pi$
$368$	0	0
$369$	8.53590	0.444361
$370$	0	0
$371$	20.7846	1.07908
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.46410	0.487426
$378$	0	0
$379$	32.9808	1.69411	0.847054	−	0.531507i	$-0.178374\pi$
$379$	32.9808	1.69411	0.847054	+	0.531507i	$0.178374\pi$
$380$	0	0
$381$	−4.24871	−0.217668
$382$	0	0
$383$	−0.928203	−0.0474290	−0.0237145	−	0.999719i	$-0.507549\pi$
$383$	−0.928203	−0.0474290	−0.0237145	+	0.999719i	$0.507549\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−25.1244	−1.27714
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−16.3923	−0.828994
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.7846	0.641641	0.320821	−	0.947140i	$-0.396041\pi$
$397$	12.7846	0.641641	0.320821	+	0.947140i	$0.396041\pi$
$398$	0	0
$399$	6.14359	0.307564
$400$	0	0
$401$	−23.0718	−1.15215	−0.576075	−	0.817397i	$-0.695416\pi$
$401$	−23.0718	−1.15215	−0.576075	+	0.817397i	$0.695416\pi$
$402$	0	0
$403$	−0.196152	−0.00977105
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.07180	0.251400
$408$	0	0
$409$	−38.3923	−1.89838	−0.949189	−	0.314708i	$-0.898094\pi$
$409$	−38.3923	−1.89838	−0.949189	+	0.314708i	$0.898094\pi$
$410$	0	0
$411$	−9.46410	−0.466830
$412$	0	0
$413$	30.2487	1.48844
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.14359	−0.300853
$418$	0	0
$419$	9.46410	0.462352	0.231176	−	0.972912i	$-0.425743\pi$
$419$	9.46410	0.462352	0.231176	+	0.972912i	$0.425743\pi$
$420$	0	0
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	0	0
$423$	−14.7846	−0.718852
$424$	0	0
$425$	0	0
$426$	0	0
$427$	24.7846	1.19941
$428$	0	0
$429$	0.928203	0.0448141
$430$	0	0
$431$	−19.5167	−0.940084	−0.470042	−	0.882644i	$-0.655761\pi$
$431$	−19.5167	−0.940084	−0.470042	+	0.882644i	$0.655761\pi$
$432$	0	0
$433$	6.78461	0.326048	0.163024	−	0.986622i	$-0.447875\pi$
$433$	6.78461	0.326048	0.163024	+	0.986622i	$0.447875\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−19.8564	−0.949861
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	7.39230	0.352015
$442$	0	0
$443$	34.9808	1.66199	0.830993	−	0.556283i	$-0.187773\pi$
$443$	34.9808	1.66199	0.830993	+	0.556283i	$0.187773\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−14.5359	−0.687524
$448$	0	0
$449$	27.4641	1.29611	0.648056	−	0.761593i	$-0.275582\pi$
$449$	27.4641	1.29611	0.648056	+	0.761593i	$0.275582\pi$
$450$	0	0
$451$	−4.39230	−0.206826
$452$	0	0
$453$	−8.92820	−0.419484
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.7846	1.44004	0.720022	−	0.693952i	$-0.244132\pi$
$457$	30.7846	1.44004	0.720022	+	0.693952i	$0.244132\pi$
$458$	0	0
$459$	−13.8564	−0.646762
$460$	0	0
$461$	3.46410	0.161339	0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410	0.161339	0.0806696	+	0.996741i	$0.474294\pi$
$462$	0	0
$463$	18.3923	0.854763	0.427381	−	0.904071i	$-0.359436\pi$
$463$	18.3923	0.854763	0.427381	+	0.904071i	$0.359436\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38.1962	1.76751	0.883754	−	0.467953i	$-0.155008\pi$
$467$	38.1962	1.76751	0.883754	+	0.467953i	$0.155008\pi$
$468$	0	0
$469$	−28.7846	−1.32915
$470$	0	0
$471$	−7.32051	−0.337311
$472$	0	0
$473$	12.9282	0.594439
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.6077	−1.17250
$478$	0	0
$479$	−18.3397	−0.837964	−0.418982	−	0.907994i	$-0.637613\pi$
$479$	−18.3397	−0.837964	−0.418982	+	0.907994i	$0.637613\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	−6.92820	−0.315244
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.60770	−0.254109	−0.127054	−	0.991896i	$-0.540552\pi$
$487$	−5.60770	−0.254109	−0.127054	+	0.991896i	$0.540552\pi$
$488$	0	0
$489$	−4.67949	−0.211614
$490$	0	0
$491$	9.46410	0.427109	0.213554	−	0.976931i	$-0.431496\pi$
$491$	9.46410	0.427109	0.213554	+	0.976931i	$0.431496\pi$
$492$	0	0
$493$	32.7846	1.47654
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.53590	−0.113751
$498$	0	0
$499$	−12.9808	−0.581099	−0.290549	−	0.956860i	$-0.593838\pi$
$499$	−12.9808	−0.581099	−0.290549	+	0.956860i	$0.593838\pi$
$500$	0	0
$501$	−9.46410	−0.422825
$502$	0	0
$503$	−25.5167	−1.13773	−0.568866	−	0.822430i	$-0.692618\pi$
$503$	−25.5167	−1.13773	−0.568866	+	0.822430i	$0.692618\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.732051	−0.0325115
$508$	0	0
$509$	32.5359	1.44213	0.721064	−	0.692868i	$-0.243653\pi$
$509$	32.5359	1.44213	0.721064	+	0.692868i	$0.243653\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	−16.7846	−0.741059
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.60770	0.334586
$518$	0	0
$519$	11.3205	0.496915
$520$	0	0
$521$	−7.60770	−0.333299	−0.166650	−	0.986016i	$-0.553295\pi$
$521$	−7.60770	−0.333299	−0.166650	+	0.986016i	$0.553295\pi$
$522$	0	0
$523$	−13.8038	−0.603600	−0.301800	−	0.953371i	$-0.597588\pi$
$523$	−13.8038	−0.603600	−0.301800	+	0.953371i	$0.597588\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.679492	−0.0295991
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	0	0
$531$	−37.2679	−1.61729
$532$	0	0
$533$	3.46410	0.150047
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.71281	−0.160220
$538$	0	0
$539$	−3.80385	−0.163843
$540$	0	0
$541$	−5.60770	−0.241094	−0.120547	−	0.992708i	$-0.538465\pi$
$541$	−5.60770	−0.241094	−0.120547	+	0.992708i	$0.538465\pi$
$542$	0	0
$543$	14.9282	0.640631
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.80385	−0.0771270	−0.0385635	−	0.999256i	$-0.512278\pi$
$547$	−1.80385	−0.0771270	−0.0385635	+	0.999256i	$0.512278\pi$
$548$	0	0
$549$	−30.5359	−1.30324
$550$	0	0
$551$	39.7128	1.69182
$552$	0	0
$553$	−24.7846	−1.05395
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.8564	1.09557	0.547786	−	0.836619i	$-0.315471\pi$
$557$	25.8564	1.09557	0.547786	+	0.836619i	$0.315471\pi$
$558$	0	0
$559$	−10.1962	−0.431251
$560$	0	0
$561$	3.21539	0.135754
$562$	0	0
$563$	16.0526	0.676535	0.338267	−	0.941050i	$-0.390159\pi$
$563$	16.0526	0.676535	0.338267	+	0.941050i	$0.390159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	8.92820	0.374949
$568$	0	0
$569$	−9.46410	−0.396756	−0.198378	−	0.980126i	$-0.563567\pi$
$569$	−9.46410	−0.396756	−0.198378	+	0.980126i	$0.563567\pi$
$570$	0	0
$571$	−15.6077	−0.653162	−0.326581	−	0.945169i	$-0.605897\pi$
$571$	−15.6077	−0.653162	−0.326581	+	0.945169i	$0.605897\pi$
$572$	0	0
$573$	−13.8564	−0.578860
$574$	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
$579$	−7.32051	−0.304230
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	13.1769	0.545732
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.4641	−0.638272	−0.319136	−	0.947709i	$-0.603393\pi$
$587$	−15.4641	−0.638272	−0.319136	+	0.947709i	$0.603393\pi$
$588$	0	0
$589$	−0.823085	−0.0339146
$590$	0	0
$591$	0.679492	0.0279506
$592$	0	0
$593$	14.7846	0.607131	0.303566	−	0.952811i	$-0.401823\pi$
$593$	14.7846	0.607131	0.303566	+	0.952811i	$0.401823\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.6410	0.599217
$598$	0	0
$599$	28.3923	1.16008	0.580039	−	0.814589i	$-0.303037\pi$
$599$	28.3923	1.16008	0.580039	+	0.814589i	$0.303037\pi$
$600$	0	0
$601$	−39.5692	−1.61406	−0.807031	−	0.590509i	$-0.798927\pi$
$601$	−39.5692	−1.61406	−0.807031	+	0.590509i	$0.798927\pi$
$602$	0	0
$603$	35.4641	1.44421
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.9808	−1.09512	−0.547558	−	0.836768i	$-0.684442\pi$
$607$	−26.9808	−1.09512	−0.547558	+	0.836768i	$0.684442\pi$
$608$	0	0
$609$	13.8564	0.561490
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.7128	0.874125	0.437062	−	0.899431i	$-0.356019\pi$
$617$	21.7128	0.874125	0.437062	+	0.899431i	$0.356019\pi$
$618$	0	0
$619$	44.9808	1.80793	0.903965	−	0.427607i	$-0.140643\pi$
$619$	44.9808	1.80793	0.903965	+	0.427607i	$0.140643\pi$
$620$	0	0
$621$	18.9282	0.759563
$622$	0	0
$623$	1.85641	0.0743754
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.89488	0.155547
$628$	0	0
$629$	−13.8564	−0.552491
$630$	0	0
$631$	−16.1962	−0.644759	−0.322379	−	0.946611i	$-0.604483\pi$
$631$	−16.1962	−0.644759	−0.322379	+	0.946611i	$0.604483\pi$
$632$	0	0
$633$	5.85641	0.232771
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	3.12436	0.123598
$640$	0	0
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	0	0
$643$	34.7846	1.37177	0.685886	−	0.727709i	$-0.259415\pi$
$643$	34.7846	1.37177	0.685886	+	0.727709i	$0.259415\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0526	−0.631091	−0.315546	−	0.948910i	$-0.602188\pi$
$647$	−16.0526	−0.631091	−0.315546	+	0.948910i	$0.602188\pi$
$648$	0	0
$649$	19.1769	0.752760
$650$	0	0
$651$	−0.287187	−0.0112557
$652$	0	0
$653$	19.8564	0.777041	0.388521	−	0.921440i	$-0.372986\pi$
$653$	19.8564	0.777041	0.388521	+	0.921440i	$0.372986\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.85641	−0.384535
$658$	0	0
$659$	14.5359	0.566238	0.283119	−	0.959085i	$-0.408631\pi$
$659$	14.5359	0.566238	0.283119	+	0.959085i	$0.408631\pi$
$660$	0	0
$661$	−30.7846	−1.19738	−0.598691	−	0.800980i	$-0.704312\pi$
$661$	−30.7846	−1.19738	−0.598691	+	0.800980i	$0.704312\pi$
$662$	0	0
$663$	−2.53590	−0.0984861
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−44.7846	−1.73407
$668$	0	0
$669$	−1.46410	−0.0566054
$670$	0	0
$671$	15.7128	0.606586
$672$	0	0
$673$	−6.39230	−0.246405	−0.123203	−	0.992382i	$-0.539317\pi$
$673$	−6.39230	−0.246405	−0.123203	+	0.992382i	$0.539317\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.3923	0.399409	0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923	0.399409	0.199704	+	0.979856i	$0.436002\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	−2.53590	−0.0971758
$682$	0	0
$683$	39.4641	1.51005	0.755026	−	0.655695i	$-0.227624\pi$
$683$	39.4641	1.51005	0.755026	+	0.655695i	$0.227624\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.5359	0.401970
$688$	0	0
$689$	−10.3923	−0.395915
$690$	0	0
$691$	−45.7654	−1.74100	−0.870498	−	0.492171i	$-0.836203\pi$
$691$	−45.7654	−1.74100	−0.870498	+	0.492171i	$0.836203\pi$
$692$	0	0
$693$	−6.24871	−0.237369
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	−4.39230	−0.166132
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−16.7846	−0.633044
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.8564	0.972430
$708$	0	0
$709$	9.60770	0.360825	0.180412	−	0.983591i	$-0.442257\pi$
$709$	9.60770	0.360825	0.180412	+	0.983591i	$0.442257\pi$
$710$	0	0
$711$	30.5359	1.14519
$712$	0	0
$713$	0.928203	0.0347615
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.78461	−0.103993
$718$	0	0
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	0	0
$721$	20.3923	0.759449
$722$	0	0
$723$	−13.4641	−0.500735
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.4115	0.497407	0.248703	−	0.968580i	$-0.419996\pi$
$727$	13.4115	0.497407	0.248703	+	0.968580i	$0.419996\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−35.3205	−1.30638
$732$	0	0
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.2487	−0.672200
$738$	0	0
$739$	7.80385	0.287069	0.143535	−	0.989645i	$-0.454153\pi$
$739$	7.80385	0.287069	0.143535	+	0.989645i	$0.454153\pi$
$740$	0	0
$741$	−3.07180	−0.112845
$742$	0	0
$743$	−43.8564	−1.60894	−0.804468	−	0.593996i	$-0.797549\pi$
$743$	−43.8564	−1.60894	−0.804468	+	0.593996i	$0.797549\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.7846	0.540941
$748$	0	0
$749$	0.679492	0.0248281
$750$	0	0
$751$	−15.6077	−0.569533	−0.284766	−	0.958597i	$-0.591916\pi$
$751$	−15.6077	−0.569533	−0.284766	+	0.958597i	$0.591916\pi$
$752$	0	0
$753$	−10.6410	−0.387780
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.3923	−0.668480	−0.334240	−	0.942488i	$-0.608480\pi$
$757$	−18.3923	−0.668480	−0.334240	+	0.942488i	$0.608480\pi$
$758$	0	0
$759$	−4.39230	−0.159431
$760$	0	0
$761$	7.85641	0.284795	0.142397	−	0.989810i	$-0.454519\pi$
$761$	7.85641	0.284795	0.142397	+	0.989810i	$0.454519\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.1244	−0.546109
$768$	0	0
$769$	−6.78461	−0.244659	−0.122330	−	0.992490i	$-0.539037\pi$
$769$	−6.78461	−0.244659	−0.122330	+	0.992490i	$0.539037\pi$
$770$	0	0
$771$	−5.75129	−0.207128
$772$	0	0
$773$	6.92820	0.249190	0.124595	−	0.992208i	$-0.460237\pi$
$773$	6.92820	0.249190	0.124595	+	0.992208i	$0.460237\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−5.85641	−0.210097
$778$	0	0
$779$	14.5359	0.520803
$780$	0	0
$781$	−1.60770	−0.0575279
$782$	0	0
$783$	−37.8564	−1.35288
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−51.5692	−1.83824	−0.919122	−	0.393973i	$-0.871100\pi$
$787$	−51.5692	−1.83824	−0.919122	+	0.393973i	$0.871100\pi$
$788$	0	0
$789$	−3.46410	−0.123325
$790$	0	0
$791$	−30.9282	−1.09968
$792$	0	0
$793$	−12.3923	−0.440064
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.6410	−1.01452	−0.507258	−	0.861794i	$-0.669341\pi$
$797$	−28.6410	−1.01452	−0.507258	+	0.861794i	$0.669341\pi$
$798$	0	0
$799$	−20.7846	−0.735307
$800$	0	0
$801$	−2.28719	−0.0808138
$802$	0	0
$803$	5.07180	0.178980
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.75129	−0.202455
$808$	0	0
$809$	−9.46410	−0.332740	−0.166370	−	0.986063i	$-0.553205\pi$
$809$	−9.46410	−0.332740	−0.166370	+	0.986063i	$0.553205\pi$
$810$	0	0
$811$	−28.1962	−0.990101	−0.495050	−	0.868864i	$-0.664850\pi$
$811$	−28.1962	−0.990101	−0.495050	+	0.868864i	$0.664850\pi$
$812$	0	0
$813$	−15.3590	−0.538663
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−42.7846	−1.49684
$818$	0	0
$819$	4.92820	0.172205
$820$	0	0
$821$	−40.6410	−1.41838	−0.709191	−	0.705017i	$-0.750939\pi$
$821$	−40.6410	−1.41838	−0.709191	+	0.705017i	$0.750939\pi$
$822$	0	0
$823$	−46.5885	−1.62397	−0.811986	−	0.583677i	$-0.801613\pi$
$823$	−46.5885	−1.62397	−0.811986	+	0.583677i	$0.801613\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18.0000	0.625921	0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000	0.625921	0.312961	+	0.949766i	$0.398679\pi$
$828$	0	0
$829$	−20.3923	−0.708254	−0.354127	−	0.935197i	$-0.615222\pi$
$829$	−20.3923	−0.708254	−0.354127	+	0.935197i	$0.615222\pi$
$830$	0	0
$831$	−4.10512	−0.142405
$832$	0	0
$833$	10.3923	0.360072
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.784610	0.0271201
$838$	0	0
$839$	−17.6603	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$839$	−17.6603	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	0	0
$843$	−1.17691	−0.0405351
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−18.7846	−0.645447
$848$	0	0
$849$	−1.03332	−0.0354635
$850$	0	0
$851$	18.9282	0.648850
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.5692	−1.62493	−0.812467	−	0.583007i	$-0.801876\pi$
$857$	−47.5692	−1.62493	−0.812467	+	0.583007i	$0.801876\pi$
$858$	0	0
$859$	−45.1769	−1.54142	−0.770708	−	0.637188i	$-0.780097\pi$
$859$	−45.1769	−1.54142	−0.770708	+	0.637188i	$0.780097\pi$
$860$	0	0
$861$	5.07180	0.172846
$862$	0	0
$863$	2.78461	0.0947892	0.0473946	−	0.998876i	$-0.484908\pi$
$863$	2.78461	0.0947892	0.0473946	+	0.998876i	$0.484908\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.66025	0.124309
$868$	0	0
$869$	−15.7128	−0.533021
$870$	0	0
$871$	14.3923	0.487665
$872$	0	0
$873$	4.92820	0.166794
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	0	0
$879$	−13.8564	−0.467365
$880$	0	0
$881$	−12.6795	−0.427183	−0.213591	−	0.976923i	$-0.568516\pi$
$881$	−12.6795	−0.427183	−0.213591	+	0.976923i	$0.568516\pi$
$882$	0	0
$883$	34.1962	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$883$	34.1962	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.9090	0.601324	0.300662	−	0.953731i	$-0.402792\pi$
$887$	17.9090	0.601324	0.300662	+	0.953731i	$0.402792\pi$
$888$	0	0
$889$	11.6077	0.389310
$890$	0	0
$891$	5.66025	0.189626
$892$	0	0
$893$	−25.1769	−0.842513
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.46410	0.115663
$898$	0	0
$899$	−1.85641	−0.0619146
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	−14.9282	−0.496779
$904$	0	0
$905$	0	0
$906$	0	0
$907$	39.7654	1.32039	0.660194	−	0.751095i	$-0.270474\pi$
$907$	39.7654	1.32039	0.660194	+	0.751095i	$0.270474\pi$
$908$	0	0
$909$	−31.8564	−1.05661
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	−7.60770	−0.251778
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	53.1769	1.75414	0.877072	−	0.480358i	$-0.159493\pi$
$919$	53.1769	1.75414	0.877072	+	0.480358i	$0.159493\pi$
$920$	0	0
$921$	−16.6795	−0.549608
$922$	0	0
$923$	1.26795	0.0417351
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−25.1244	−0.825192
$928$	0	0
$929$	51.4641	1.68848	0.844241	−	0.535963i	$-0.180052\pi$
$929$	51.4641	1.68848	0.844241	+	0.535963i	$0.180052\pi$
$930$	0	0
$931$	12.5885	0.412570
$932$	0	0
$933$	3.21539	0.105267
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.78461	0.221644	0.110822	−	0.993840i	$-0.464652\pi$
$937$	6.78461	0.221644	0.110822	+	0.993840i	$0.464652\pi$
$938$	0	0
$939$	4.67949	0.152709
$940$	0	0
$941$	−31.1769	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$941$	−31.1769	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$942$	0	0
$943$	−16.3923	−0.533807
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.6410	−0.930708	−0.465354	−	0.885125i	$-0.654073\pi$
$947$	−28.6410	−0.930708	−0.465354	+	0.885125i	$0.654073\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	17.5692	0.569721
$952$	0	0
$953$	12.9282	0.418786	0.209393	−	0.977832i	$-0.432851\pi$
$953$	12.9282	0.418786	0.209393	+	0.977832i	$0.432851\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.78461	0.283966
$958$	0	0
$959$	25.8564	0.834947
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	−0.837169	−0.0269774
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.6077	−0.952119	−0.476060	−	0.879413i	$-0.657935\pi$
$967$	−29.6077	−0.952119	−0.476060	+	0.879413i	$0.657935\pi$
$968$	0	0
$969$	−10.6410	−0.341839
$970$	0	0
$971$	−5.07180	−0.162762	−0.0813809	−	0.996683i	$-0.525933\pi$
$971$	−5.07180	−0.162762	−0.0813809	+	0.996683i	$0.525933\pi$
$972$	0	0
$973$	16.7846	0.538090
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.7128	−1.27053	−0.635263	−	0.772296i	$-0.719108\pi$
$977$	−39.7128	−1.27053	−0.635263	+	0.772296i	$0.719108\pi$
$978$	0	0
$979$	1.17691	0.0376144
$980$	0	0
$981$	−4.92820	−0.157345
$982$	0	0
$983$	−13.6077	−0.434018	−0.217009	−	0.976170i	$-0.569630\pi$
$983$	−13.6077	−0.434018	−0.217009	+	0.976170i	$0.569630\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.78461	−0.279617
$988$	0	0
$989$	48.2487	1.53422
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	−20.9282	−0.664136
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54.3923	−1.72262	−0.861311	−	0.508078i	$-0.830356\pi$
$997$	−54.3923	−1.72262	−0.861311	+	0.508078i	$0.830356\pi$
$998$	0	0
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.ca.1.1		2
4.3	odd	2		325.2.a.g.1.1		2
5.4	even	2		1040.2.a.h.1.2		2
12.11	even	2		2925.2.a.z.1.2		2
15.14	odd	2		9360.2.a.cm.1.2		2
20.3	even	4		325.2.b.e.274.3		4
20.7	even	4		325.2.b.e.274.2		4
20.19	odd	2		65.2.a.c.1.2	✓	2
40.19	odd	2		4160.2.a.y.1.2		2
40.29	even	2		4160.2.a.bj.1.1		2
52.51	odd	2		4225.2.a.w.1.2		2
60.23	odd	4		2925.2.c.v.2224.2		4
60.47	odd	4		2925.2.c.v.2224.3		4
60.59	even	2		585.2.a.k.1.1		2
140.139	even	2		3185.2.a.k.1.2		2
220.219	even	2		7865.2.a.h.1.1		2
260.19	even	12		845.2.m.a.361.2		4
260.59	even	12		845.2.m.c.361.2		4
260.99	even	4		845.2.c.e.506.3		4
260.119	even	12		845.2.m.c.316.2		4
260.139	odd	6		845.2.e.e.146.1		4
260.159	odd	6		845.2.e.e.191.1		4
260.179	odd	6		845.2.e.f.191.2		4
260.199	odd	6		845.2.e.f.146.2		4
260.219	even	12		845.2.m.a.316.2		4
260.239	even	4		845.2.c.e.506.1		4
260.259	odd	2		845.2.a.d.1.1		2
780.779	even	2		7605.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.2	✓	2	20.19	odd	2
325.2.a.g.1.1		2	4.3	odd	2
325.2.b.e.274.2		4	20.7	even	4
325.2.b.e.274.3		4	20.3	even	4
585.2.a.k.1.1		2	60.59	even	2
845.2.a.d.1.1		2	260.259	odd	2
845.2.c.e.506.1		4	260.239	even	4
845.2.c.e.506.3		4	260.99	even	4
845.2.e.e.146.1		4	260.139	odd	6
845.2.e.e.191.1		4	260.159	odd	6
845.2.e.f.146.2		4	260.199	odd	6
845.2.e.f.191.2		4	260.179	odd	6
845.2.m.a.316.2		4	260.219	even	12
845.2.m.a.361.2		4	260.19	even	12
845.2.m.c.316.2		4	260.119	even	12
845.2.m.c.361.2		4	260.59	even	12
1040.2.a.h.1.2		2	5.4	even	2
2925.2.a.z.1.2		2	12.11	even	2
2925.2.c.v.2224.2		4	60.23	odd	4
2925.2.c.v.2224.3		4	60.47	odd	4
3185.2.a.k.1.2		2	140.139	even	2
4160.2.a.y.1.2		2	40.19	odd	2
4160.2.a.bj.1.1		2	40.29	even	2
4225.2.a.w.1.2		2	52.51	odd	2
5200.2.a.ca.1.1		2	1.1	even	1	trivial
7605.2.a.be.1.2		2	780.779	even	2
7865.2.a.h.1.1		2	220.219	even	2
9360.2.a.cm.1.2		2	15.14	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:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +2.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} +2.00000 q^{7} -2.46410 q^{9} +1.26795 q^{11} -1.00000 q^{13} -3.46410 q^{17} -4.19615 q^{19} -1.46410 q^{21} +4.73205 q^{23} +4.00000 q^{27} -9.46410 q^{29} +0.196152 q^{31} -0.928203 q^{33} +4.00000 q^{37} +0.732051 q^{39} -3.46410 q^{41} +10.1962 q^{43} +6.00000 q^{47} -3.00000 q^{49} +2.53590 q^{51} +10.3923 q^{53} +3.07180 q^{57} +15.1244 q^{59} +12.3923 q^{61} -4.92820 q^{63} -14.3923 q^{67} -3.46410 q^{69} -1.26795 q^{71} +4.00000 q^{73} +2.53590 q^{77} -12.3923 q^{79} +4.46410 q^{81} -6.00000 q^{83} +6.92820 q^{87} +0.928203 q^{89} -2.00000 q^{91} -0.143594 q^{93} -2.00000 q^{97} -3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{13} + 2 q^{19} + 4 q^{21} + 6 q^{23} + 8 q^{27} - 12 q^{29} - 10 q^{31} + 12 q^{33} + 8 q^{37} - 2 q^{39} + 10 q^{43} + 12 q^{47} - 6 q^{49} + 12 q^{51} + 20 q^{57} + 6 q^{59} + 4 q^{61} + 4 q^{63} - 8 q^{67} - 6 q^{71} + 8 q^{73} + 12 q^{77} - 4 q^{79} + 2 q^{81} - 12 q^{83} - 12 q^{89} - 4 q^{91} - 28 q^{93} - 4 q^{97} + 18 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^13 + 2 * q^19 + 4 * q^21 + 6 * q^23 + 8 * q^27 - 12 * q^29 - 10 * q^31 + 12 * q^33 + 8 * q^37 - 2 * q^39 + 10 * q^43 + 12 * q^47 - 6 * q^49 + 12 * q^51 + 20 * q^57 + 6 * q^59 + 4 * q^61 + 4 * q^63 - 8 * q^67 - 6 * q^71 + 8 * q^73 + 12 * q^77 - 4 * q^79 + 2 * q^81 - 12 * q^83 - 12 * q^89 - 4 * q^91 - 28 * q^93 - 4 * q^97 + 18 * q^99

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