LMFDB - Embedded newform 4368.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.138892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 2x + 12 $ x^4 - x^3 - 10x^2 + 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.21773$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.70948 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.70948 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.51714 q^{11} +1.00000 q^{13} -3.70948 q^{15} -4.95259 q^{17} +3.27403 q^{19} +1.00000 q^{21} +8.14494 q^{23} +8.76025 q^{25} +1.00000 q^{27} -1.27403 q^{29} -10.2266 q^{31} +4.51714 q^{33} -3.70948 q^{35} -4.95259 q^{37} +1.00000 q^{39} -0.435456 q^{41} +8.14494 q^{43} -3.70948 q^{45} -5.24311 q^{47} +1.00000 q^{49} -4.95259 q^{51} -1.74039 q^{53} -16.7562 q^{55} +3.27403 q^{57} -11.4697 q^{59} -2.56791 q^{61} +1.00000 q^{63} -3.70948 q^{65} +12.2899 q^{67} +8.14494 q^{69} +11.4697 q^{71} +14.1449 q^{73} +8.76025 q^{75} +4.51714 q^{77} +3.67857 q^{79} +1.00000 q^{81} -10.1758 q^{83} +18.3716 q^{85} -1.27403 q^{87} +3.24311 q^{89} +1.00000 q^{91} -10.2266 q^{93} -12.1449 q^{95} +9.74039 q^{97} +4.51714 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.70948	−1.65893	−0.829465	−	0.558558i	$-0.811355\pi$
$5$	−3.70948	−1.65893	−0.829465	+	0.558558i	$0.811355\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.51714	1.36197	0.680984	−	0.732298i	$-0.261552\pi$
$11$	4.51714	1.36197	0.680984	+	0.732298i	$0.261552\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.70948	−0.957784
$16$	0	0
$17$	−4.95259	−1.20118	−0.600590	−	0.799557i	$-0.705068\pi$
$17$	−4.95259	−1.20118	−0.600590	+	0.799557i	$0.705068\pi$
$18$	0	0
$19$	3.27403	0.751113	0.375556	−	0.926800i	$-0.377452\pi$
$19$	3.27403	0.751113	0.375556	+	0.926800i	$0.377452\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	8.14494	1.69834	0.849168	−	0.528122i	$-0.177104\pi$
$23$	8.14494	1.69834	0.849168	+	0.528122i	$0.177104\pi$
$24$	0	0
$25$	8.76025	1.75205
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.27403	−0.236581	−0.118290	−	0.992979i	$-0.537741\pi$
$29$	−1.27403	−0.236581	−0.118290	+	0.992979i	$0.537741\pi$
$30$	0	0
$31$	−10.2266	−1.83676	−0.918378	−	0.395705i	$-0.870500\pi$
$31$	−10.2266	−1.83676	−0.918378	+	0.395705i	$0.870500\pi$
$32$	0	0
$33$	4.51714	0.786333
$34$	0	0
$35$	−3.70948	−0.627017
$36$	0	0
$37$	−4.95259	−0.814202	−0.407101	−	0.913383i	$-0.633460\pi$
$37$	−4.95259	−0.814202	−0.407101	+	0.913383i	$0.633460\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−0.435456	−0.0680068	−0.0340034	−	0.999422i	$-0.510826\pi$
$41$	−0.435456	−0.0680068	−0.0340034	+	0.999422i	$0.510826\pi$
$42$	0	0
$43$	8.14494	1.24209	0.621046	−	0.783774i	$-0.286708\pi$
$43$	8.14494	1.24209	0.621046	+	0.783774i	$0.286708\pi$
$44$	0	0
$45$	−3.70948	−0.552977
$46$	0	0
$47$	−5.24311	−0.764787	−0.382393	−	0.924000i	$-0.624900\pi$
$47$	−5.24311	−0.764787	−0.382393	+	0.924000i	$0.624900\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.95259	−0.693502
$52$	0	0
$53$	−1.74039	−0.239061	−0.119531	−	0.992831i	$-0.538139\pi$
$53$	−1.74039	−0.239061	−0.119531	+	0.992831i	$0.538139\pi$
$54$	0	0
$55$	−16.7562	−2.25941
$56$	0	0
$57$	3.27403	0.433655
$58$	0	0
$59$	−11.4697	−1.49323	−0.746616	−	0.665255i	$-0.768323\pi$
$59$	−11.4697	−1.49323	−0.746616	+	0.665255i	$0.768323\pi$
$60$	0	0
$61$	−2.56791	−0.328787	−0.164394	−	0.986395i	$-0.552567\pi$
$61$	−2.56791	−0.328787	−0.164394	+	0.986395i	$0.552567\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.70948	−0.460105
$66$	0	0
$67$	12.2899	1.50145	0.750724	−	0.660616i	$-0.229705\pi$
$67$	12.2899	1.50145	0.750724	+	0.660616i	$0.229705\pi$
$68$	0	0
$69$	8.14494	0.980535
$70$	0	0
$71$	11.4697	1.36121	0.680603	−	0.732652i	$-0.261718\pi$
$71$	11.4697	1.36121	0.680603	+	0.732652i	$0.261718\pi$
$72$	0	0
$73$	14.1449	1.65554	0.827770	−	0.561068i	$-0.189609\pi$
$73$	14.1449	1.65554	0.827770	+	0.561068i	$0.189609\pi$
$74$	0	0
$75$	8.76025	1.01155
$76$	0	0
$77$	4.51714	0.514776
$78$	0	0
$79$	3.67857	0.413871	0.206936	−	0.978355i	$-0.433651\pi$
$79$	3.67857	0.413871	0.206936	+	0.978355i	$0.433651\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.1758	−1.11694	−0.558472	−	0.829523i	$-0.688612\pi$
$83$	−10.1758	−1.11694	−0.558472	+	0.829523i	$0.688612\pi$
$84$	0	0
$85$	18.3716	1.99268
$86$	0	0
$87$	−1.27403	−0.136590
$88$	0	0
$89$	3.24311	0.343769	0.171885	−	0.985117i	$-0.445014\pi$
$89$	3.24311	0.343769	0.171885	+	0.985117i	$0.445014\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−10.2266	−1.06045
$94$	0	0
$95$	−12.1449	−1.24604
$96$	0	0
$97$	9.74039	0.988987	0.494494	−	0.869181i	$-0.335354\pi$
$97$	9.74039	0.988987	0.494494	+	0.869181i	$0.335354\pi$
$98$	0	0
$99$	4.51714	0.453990
$100$	0	0
$101$	15.8036	1.57252	0.786261	−	0.617895i	$-0.212014\pi$
$101$	15.8036	1.57252	0.786261	+	0.617895i	$0.212014\pi$
$102$	0	0
$103$	14.3716	1.41607	0.708036	−	0.706177i	$-0.249582\pi$
$103$	14.3716	1.41607	0.708036	+	0.706177i	$0.249582\pi$
$104$	0	0
$105$	−3.70948	−0.362008
$106$	0	0
$107$	−5.90519	−0.570876	−0.285438	−	0.958397i	$-0.592139\pi$
$107$	−5.90519	−0.570876	−0.285438	+	0.958397i	$0.592139\pi$
$108$	0	0
$109$	14.4664	1.38563	0.692813	−	0.721117i	$-0.256371\pi$
$109$	14.4664	1.38563	0.692813	+	0.721117i	$0.256371\pi$
$110$	0	0
$111$	−4.95259	−0.470079
$112$	0	0
$113$	6.61131	0.621939	0.310970	−	0.950420i	$-0.399346\pi$
$113$	6.61131	0.621939	0.310970	+	0.950420i	$0.399346\pi$
$114$	0	0
$115$	−30.2135	−2.81742
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−4.95259	−0.454004
$120$	0	0
$121$	9.40454	0.854958
$122$	0	0
$123$	−0.435456	−0.0392637
$124$	0	0
$125$	−13.9486	−1.24760
$126$	0	0
$127$	−18.7761	−1.66611	−0.833055	−	0.553191i	$-0.813410\pi$
$127$	−18.7761	−1.66611	−0.833055	+	0.553191i	$0.813410\pi$
$128$	0	0
$129$	8.14494	0.717122
$130$	0	0
$131$	10.3716	0.906167	0.453084	−	0.891468i	$-0.350324\pi$
$131$	10.3716	0.906167	0.453084	+	0.891468i	$0.350324\pi$
$132$	0	0
$133$	3.27403	0.283894
$134$	0	0
$135$	−3.70948	−0.319261
$136$	0	0
$137$	−12.2590	−1.04735	−0.523677	−	0.851917i	$-0.675440\pi$
$137$	−12.2590	−1.04735	−0.523677	+	0.851917i	$0.675440\pi$
$138$	0	0
$139$	16.3517	1.38693	0.693467	−	0.720489i	$-0.256083\pi$
$139$	16.3517	1.38693	0.693467	+	0.720489i	$0.256083\pi$
$140$	0	0
$141$	−5.24311	−0.441550
$142$	0	0
$143$	4.51714	0.377742
$144$	0	0
$145$	4.72597	0.392471
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	14.5040	1.18821	0.594107	−	0.804386i	$-0.297505\pi$
$149$	14.5040	1.18821	0.594107	+	0.804386i	$0.297505\pi$
$150$	0	0
$151$	−1.62844	−0.132521	−0.0662604	−	0.997802i	$-0.521107\pi$
$151$	−1.62844	−0.132521	−0.0662604	+	0.997802i	$0.521107\pi$
$152$	0	0
$153$	−4.95259	−0.400394
$154$	0	0
$155$	37.9355	3.04705
$156$	0	0
$157$	3.04741	0.243209	0.121605	−	0.992579i	$-0.461196\pi$
$157$	3.04741	0.243209	0.121605	+	0.992579i	$0.461196\pi$
$158$	0	0
$159$	−1.74039	−0.138022
$160$	0	0
$161$	8.14494	0.641911
$162$	0	0
$163$	13.0343	1.02092	0.510462	−	0.859900i	$-0.329475\pi$
$163$	13.0343	1.02092	0.510462	+	0.859900i	$0.329475\pi$
$164$	0	0
$165$	−16.7562	−1.30447
$166$	0	0
$167$	−3.22325	−0.249423	−0.124711	−	0.992193i	$-0.539801\pi$
$167$	−3.22325	−0.249423	−0.124711	+	0.992193i	$0.539801\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.27403	0.250371
$172$	0	0
$173$	−23.3242	−1.77330	−0.886651	−	0.462439i	$-0.846975\pi$
$173$	−23.3242	−1.77330	−0.886651	+	0.462439i	$0.846975\pi$
$174$	0	0
$175$	8.76025	0.662213
$176$	0	0
$177$	−11.4697	−0.862118
$178$	0	0
$179$	4.61131	0.344665	0.172333	−	0.985039i	$-0.444870\pi$
$179$	4.61131	0.344665	0.172333	+	0.985039i	$0.444870\pi$
$180$	0	0
$181$	20.8379	1.54887	0.774435	−	0.632653i	$-0.218034\pi$
$181$	20.8379	1.54887	0.774435	+	0.632653i	$0.218034\pi$
$182$	0	0
$183$	−2.56791	−0.189825
$184$	0	0
$185$	18.3716	1.35070
$186$	0	0
$187$	−22.3716	−1.63597
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−24.2767	−1.75660	−0.878302	−	0.478107i	$-0.841323\pi$
$191$	−24.2767	−1.75660	−0.878302	+	0.478107i	$0.841323\pi$
$192$	0	0
$193$	23.9052	1.72073	0.860367	−	0.509676i	$-0.170235\pi$
$193$	23.9052	1.72073	0.860367	+	0.509676i	$0.170235\pi$
$194$	0	0
$195$	−3.70948	−0.265641
$196$	0	0
$197$	6.40247	0.456157	0.228079	−	0.973643i	$-0.426756\pi$
$197$	6.40247	0.456157	0.228079	+	0.973643i	$0.426756\pi$
$198$	0	0
$199$	23.4058	1.65920	0.829598	−	0.558361i	$-0.188570\pi$
$199$	23.4058	1.65920	0.829598	+	0.558361i	$0.188570\pi$
$200$	0	0
$201$	12.2899	0.866861
$202$	0	0
$203$	−1.27403	−0.0894191
$204$	0	0
$205$	1.61531	0.112818
$206$	0	0
$207$	8.14494	0.566112
$208$	0	0
$209$	14.7892	1.02299
$210$	0	0
$211$	17.1792	1.18267	0.591333	−	0.806428i	$-0.298602\pi$
$211$	17.1792	1.18267	0.591333	+	0.806428i	$0.298602\pi$
$212$	0	0
$213$	11.4697	0.785893
$214$	0	0
$215$	−30.2135	−2.06054
$216$	0	0
$217$	−10.2266	−0.694228
$218$	0	0
$219$	14.1449	0.955826
$220$	0	0
$221$	−4.95259	−0.333148
$222$	0	0
$223$	−3.67857	−0.246335	−0.123168	−	0.992386i	$-0.539305\pi$
$223$	−3.67857	−0.246335	−0.123168	+	0.992386i	$0.539305\pi$
$224$	0	0
$225$	8.76025	0.584017
$226$	0	0
$227$	20.5040	1.36090	0.680450	−	0.732795i	$-0.261785\pi$
$227$	20.5040	1.36090	0.680450	+	0.732795i	$0.261785\pi$
$228$	0	0
$229$	−16.8379	−1.11268	−0.556341	−	0.830954i	$-0.687795\pi$
$229$	−16.8379	−1.11268	−0.556341	+	0.830954i	$0.687795\pi$
$230$	0	0
$231$	4.51714	0.297206
$232$	0	0
$233$	−17.6786	−1.15816	−0.579081	−	0.815270i	$-0.696588\pi$
$233$	−17.6786	−1.15816	−0.579081	+	0.815270i	$0.696588\pi$
$234$	0	0
$235$	19.4492	1.26873
$236$	0	0
$237$	3.67857	0.238949
$238$	0	0
$239$	−11.3682	−0.735347	−0.367674	−	0.929955i	$-0.619846\pi$
$239$	−11.3682	−0.735347	−0.367674	+	0.929955i	$0.619846\pi$
$240$	0	0
$241$	−19.9355	−1.28416	−0.642078	−	0.766639i	$-0.721927\pi$
$241$	−19.9355	−1.28416	−0.642078	+	0.766639i	$0.721927\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.70948	−0.236990
$246$	0	0
$247$	3.27403	0.208321
$248$	0	0
$249$	−10.1758	−0.644868
$250$	0	0
$251$	12.7562	0.805167	0.402583	−	0.915383i	$-0.368112\pi$
$251$	12.7562	0.805167	0.402583	+	0.915383i	$0.368112\pi$
$252$	0	0
$253$	36.7918	2.31308
$254$	0	0
$255$	18.3716	1.15047
$256$	0	0
$257$	−9.89846	−0.617449	−0.308724	−	0.951152i	$-0.599902\pi$
$257$	−9.89846	−0.617449	−0.308724	+	0.951152i	$0.599902\pi$
$258$	0	0
$259$	−4.95259	−0.307739
$260$	0	0
$261$	−1.27403	−0.0788602
$262$	0	0
$263$	−3.74039	−0.230643	−0.115321	−	0.993328i	$-0.536790\pi$
$263$	−3.74039	−0.230643	−0.115321	+	0.993328i	$0.536790\pi$
$264$	0	0
$265$	6.45596	0.396586
$266$	0	0
$267$	3.24311	0.198475
$268$	0	0
$269$	14.3517	0.875039	0.437519	−	0.899209i	$-0.355857\pi$
$269$	14.3517	0.875039	0.437519	+	0.899209i	$0.355857\pi$
$270$	0	0
$271$	−27.0013	−1.64021	−0.820106	−	0.572212i	$-0.806085\pi$
$271$	−27.0013	−1.64021	−0.820106	+	0.572212i	$0.806085\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	39.5713	2.38624
$276$	0	0
$277$	0.644291	0.0387117	0.0193559	−	0.999813i	$-0.493838\pi$
$277$	0.644291	0.0387117	0.0193559	+	0.999813i	$0.493838\pi$
$278$	0	0
$279$	−10.2266	−0.612252
$280$	0	0
$281$	−8.49728	−0.506905	−0.253453	−	0.967348i	$-0.581566\pi$
$281$	−8.49728	−0.506905	−0.253453	+	0.967348i	$0.581566\pi$
$282$	0	0
$283$	31.8104	1.89093	0.945465	−	0.325723i	$-0.105608\pi$
$283$	31.8104	1.89093	0.945465	+	0.325723i	$0.105608\pi$
$284$	0	0
$285$	−12.1449	−0.719404
$286$	0	0
$287$	−0.435456	−0.0257041
$288$	0	0
$289$	7.52819	0.442835
$290$	0	0
$291$	9.74039	0.570992
$292$	0	0
$293$	21.3366	1.24650	0.623250	−	0.782023i	$-0.285812\pi$
$293$	21.3366	1.24650	0.623250	+	0.782023i	$0.285812\pi$
$294$	0	0
$295$	42.5468	2.47717
$296$	0	0
$297$	4.51714	0.262111
$298$	0	0
$299$	8.14494	0.471034
$300$	0	0
$301$	8.14494	0.469466
$302$	0	0
$303$	15.8036	0.907896
$304$	0	0
$305$	9.52561	0.545435
$306$	0	0
$307$	6.32816	0.361167	0.180584	−	0.983560i	$-0.442201\pi$
$307$	6.32816	0.361167	0.180584	+	0.983560i	$0.442201\pi$
$308$	0	0
$309$	14.3716	0.817569
$310$	0	0
$311$	24.5151	1.39012	0.695061	−	0.718951i	$-0.255377\pi$
$311$	24.5151	1.39012	0.695061	+	0.718951i	$0.255377\pi$
$312$	0	0
$313$	14.1634	0.800561	0.400280	−	0.916393i	$-0.368913\pi$
$313$	14.1634	0.800561	0.400280	+	0.916393i	$0.368913\pi$
$314$	0	0
$315$	−3.70948	−0.209006
$316$	0	0
$317$	3.69105	0.207310	0.103655	−	0.994613i	$-0.466946\pi$
$317$	3.69105	0.207310	0.103655	+	0.994613i	$0.466946\pi$
$318$	0	0
$319$	−5.75495	−0.322215
$320$	0	0
$321$	−5.90519	−0.329596
$322$	0	0
$323$	−16.2149	−0.902222
$324$	0	0
$325$	8.76025	0.485931
$326$	0	0
$327$	14.4664	0.799992
$328$	0	0
$329$	−5.24311	−0.289062
$330$	0	0
$331$	0.479496	0.0263555	0.0131777	−	0.999913i	$-0.495805\pi$
$331$	0.479496	0.0263555	0.0131777	+	0.999913i	$0.495805\pi$
$332$	0	0
$333$	−4.95259	−0.271401
$334$	0	0
$335$	−45.5891	−2.49080
$336$	0	0
$337$	−11.5969	−0.631723	−0.315861	−	0.948805i	$-0.602293\pi$
$337$	−11.5969	−0.631723	−0.315861	+	0.948805i	$0.602293\pi$
$338$	0	0
$339$	6.61131	0.359077
$340$	0	0
$341$	−46.1951	−2.50160
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−30.2135	−1.62664
$346$	0	0
$347$	11.1291	0.597441	0.298720	−	0.954341i	$-0.403440\pi$
$347$	11.1291	0.597441	0.298720	+	0.954341i	$0.403440\pi$
$348$	0	0
$349$	−12.2266	−0.654476	−0.327238	−	0.944942i	$-0.606118\pi$
$349$	−12.2266	−0.654476	−0.327238	+	0.944942i	$0.606118\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	5.46973	0.291125	0.145562	−	0.989349i	$-0.453501\pi$
$353$	5.46973	0.291125	0.145562	+	0.989349i	$0.453501\pi$
$354$	0	0
$355$	−42.5468	−2.25815
$356$	0	0
$357$	−4.95259	−0.262119
$358$	0	0
$359$	−11.3682	−0.599990	−0.299995	−	0.953941i	$-0.596985\pi$
$359$	−11.3682	−0.599990	−0.299995	+	0.953941i	$0.596985\pi$
$360$	0	0
$361$	−8.28076	−0.435829
$362$	0	0
$363$	9.40454	0.493611
$364$	0	0
$365$	−52.4704	−2.74643
$366$	0	0
$367$	−32.1621	−1.67885	−0.839423	−	0.543478i	$-0.817107\pi$
$367$	−32.1621	−1.67885	−0.839423	+	0.543478i	$0.817107\pi$
$368$	0	0
$369$	−0.435456	−0.0226689
$370$	0	0
$371$	−1.74039	−0.0903567
$372$	0	0
$373$	−0.864181	−0.0447456	−0.0223728	−	0.999750i	$-0.507122\pi$
$373$	−0.864181	−0.0447456	−0.0223728	+	0.999750i	$0.507122\pi$
$374$	0	0
$375$	−13.9486	−0.720302
$376$	0	0
$377$	−1.27403	−0.0656157
$378$	0	0
$379$	−20.1621	−1.03566	−0.517828	−	0.855485i	$-0.673259\pi$
$379$	−20.1621	−1.03566	−0.517828	+	0.855485i	$0.673259\pi$
$380$	0	0
$381$	−18.7761	−0.961929
$382$	0	0
$383$	3.77274	0.192778	0.0963889	−	0.995344i	$-0.469271\pi$
$383$	3.77274	0.192778	0.0963889	+	0.995344i	$0.469271\pi$
$384$	0	0
$385$	−16.7562	−0.853977
$386$	0	0
$387$	8.14494	0.414030
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−40.3386	−2.04001
$392$	0	0
$393$	10.3716	0.523176
$394$	0	0
$395$	−13.6456	−0.686584
$396$	0	0
$397$	15.5838	0.782126	0.391063	−	0.920364i	$-0.372107\pi$
$397$	15.5838	0.782126	0.391063	+	0.920364i	$0.372107\pi$
$398$	0	0
$399$	3.27403	0.163906
$400$	0	0
$401$	−31.9560	−1.59580	−0.797902	−	0.602787i	$-0.794057\pi$
$401$	−31.9560	−1.59580	−0.797902	+	0.602787i	$0.794057\pi$
$402$	0	0
$403$	−10.2266	−0.509424
$404$	0	0
$405$	−3.70948	−0.184326
$406$	0	0
$407$	−22.3716	−1.10892
$408$	0	0
$409$	−20.8564	−1.03128	−0.515640	−	0.856805i	$-0.672446\pi$
$409$	−20.8564	−1.03128	−0.515640	+	0.856805i	$0.672446\pi$
$410$	0	0
$411$	−12.2590	−0.604690
$412$	0	0
$413$	−11.4697	−0.564389
$414$	0	0
$415$	37.7471	1.85293
$416$	0	0
$417$	16.3517	0.800746
$418$	0	0
$419$	−3.01442	−0.147264	−0.0736320	−	0.997285i	$-0.523459\pi$
$419$	−3.01442	−0.147264	−0.0736320	+	0.997285i	$0.523459\pi$
$420$	0	0
$421$	18.3517	0.894407	0.447204	−	0.894432i	$-0.352420\pi$
$421$	18.3517	0.894407	0.447204	+	0.894432i	$0.352420\pi$
$422$	0	0
$423$	−5.24311	−0.254929
$424$	0	0
$425$	−43.3860	−2.10453
$426$	0	0
$427$	−2.56791	−0.124270
$428$	0	0
$429$	4.51714	0.218090
$430$	0	0
$431$	−16.8557	−0.811911	−0.405955	−	0.913893i	$-0.633061\pi$
$431$	−16.8557	−0.811911	−0.405955	+	0.913893i	$0.633061\pi$
$432$	0	0
$433$	3.77868	0.181592	0.0907959	−	0.995870i	$-0.471059\pi$
$433$	3.77868	0.181592	0.0907959	+	0.995870i	$0.471059\pi$
$434$	0	0
$435$	4.72597	0.226593
$436$	0	0
$437$	26.6667	1.27564
$438$	0	0
$439$	−25.8921	−1.23576	−0.617880	−	0.786272i	$-0.712008\pi$
$439$	−25.8921	−1.23576	−0.617880	+	0.786272i	$0.712008\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−26.0988	−1.23999	−0.619996	−	0.784605i	$-0.712866\pi$
$443$	−26.0988	−1.23999	−0.619996	+	0.784605i	$0.712866\pi$
$444$	0	0
$445$	−12.0303	−0.570289
$446$	0	0
$447$	14.5040	0.686016
$448$	0	0
$449$	−15.3881	−0.726207	−0.363103	−	0.931749i	$-0.618283\pi$
$449$	−15.3881	−0.726207	−0.363103	+	0.931749i	$0.618283\pi$
$450$	0	0
$451$	−1.96701	−0.0926231
$452$	0	0
$453$	−1.62844	−0.0765109
$454$	0	0
$455$	−3.70948	−0.173903
$456$	0	0
$457$	−37.3530	−1.74730	−0.873650	−	0.486556i	$-0.838253\pi$
$457$	−37.3530	−1.74730	−0.873650	+	0.486556i	$0.838253\pi$
$458$	0	0
$459$	−4.95259	−0.231167
$460$	0	0
$461$	21.1667	0.985833	0.492916	−	0.870077i	$-0.335931\pi$
$461$	21.1667	0.985833	0.492916	+	0.870077i	$0.335931\pi$
$462$	0	0
$463$	13.6272	0.633308	0.316654	−	0.948541i	$-0.397441\pi$
$463$	13.6272	0.633308	0.316654	+	0.948541i	$0.397441\pi$
$464$	0	0
$465$	37.9355	1.75921
$466$	0	0
$467$	6.24505	0.288986	0.144493	−	0.989506i	$-0.453845\pi$
$467$	6.24505	0.288986	0.144493	+	0.989506i	$0.453845\pi$
$468$	0	0
$469$	12.2899	0.567494
$470$	0	0
$471$	3.04741	0.140417
$472$	0	0
$473$	36.7918	1.69169
$474$	0	0
$475$	28.6813	1.31599
$476$	0	0
$477$	−1.74039	−0.0796872
$478$	0	0
$479$	12.3222	0.563016	0.281508	−	0.959559i	$-0.409165\pi$
$479$	12.3222	0.563016	0.281508	+	0.959559i	$0.409165\pi$
$480$	0	0
$481$	−4.95259	−0.225819
$482$	0	0
$483$	8.14494	0.370607
$484$	0	0
$485$	−36.1318	−1.64066
$486$	0	0
$487$	7.25560	0.328782	0.164391	−	0.986395i	$-0.447434\pi$
$487$	7.25560	0.328782	0.164391	+	0.986395i	$0.447434\pi$
$488$	0	0
$489$	13.0343	0.589430
$490$	0	0
$491$	13.9052	0.627532	0.313766	−	0.949500i	$-0.398409\pi$
$491$	13.9052	0.627532	0.313766	+	0.949500i	$0.398409\pi$
$492$	0	0
$493$	6.30973	0.284176
$494$	0	0
$495$	−16.7562	−0.753137
$496$	0	0
$497$	11.4697	0.514488
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	−3.22325	−0.144004
$502$	0	0
$503$	−1.35041	−0.0602117	−0.0301058	−	0.999547i	$-0.509584\pi$
$503$	−1.35041	−0.0602117	−0.0301058	+	0.999547i	$0.509584\pi$
$504$	0	0
$505$	−58.6233	−2.60870
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−10.5737	−0.468669	−0.234335	−	0.972156i	$-0.575291\pi$
$509$	−10.5737	−0.468669	−0.234335	+	0.972156i	$0.575291\pi$
$510$	0	0
$511$	14.1449	0.625735
$512$	0	0
$513$	3.27403	0.144552
$514$	0	0
$515$	−53.3110	−2.34916
$516$	0	0
$517$	−23.6839	−1.04162
$518$	0	0
$519$	−23.3242	−1.02382
$520$	0	0
$521$	25.1778	1.10306	0.551529	−	0.834155i	$-0.314044\pi$
$521$	25.1778	1.10306	0.551529	+	0.834155i	$0.314044\pi$
$522$	0	0
$523$	−20.1621	−0.881626	−0.440813	−	0.897599i	$-0.645310\pi$
$523$	−20.1621	−0.881626	−0.440813	+	0.897599i	$0.645310\pi$
$524$	0	0
$525$	8.76025	0.382329
$526$	0	0
$527$	50.6483	2.20627
$528$	0	0
$529$	43.3400	1.88435
$530$	0	0
$531$	−11.4697	−0.497744
$532$	0	0
$533$	−0.435456	−0.0188617
$534$	0	0
$535$	21.9052	0.947044
$536$	0	0
$537$	4.61131	0.198993
$538$	0	0
$539$	4.51714	0.194567
$540$	0	0
$541$	−1.01442	−0.0436133	−0.0218066	−	0.999762i	$-0.506942\pi$
$541$	−1.01442	−0.0436133	−0.0218066	+	0.999762i	$0.506942\pi$
$542$	0	0
$543$	20.8379	0.894241
$544$	0	0
$545$	−53.6627	−2.29866
$546$	0	0
$547$	27.4243	1.17258	0.586288	−	0.810102i	$-0.300589\pi$
$547$	27.4243	1.17258	0.586288	+	0.810102i	$0.300589\pi$
$548$	0	0
$549$	−2.56791	−0.109596
$550$	0	0
$551$	−4.17119	−0.177699
$552$	0	0
$553$	3.67857	0.156429
$554$	0	0
$555$	18.3716	0.779829
$556$	0	0
$557$	7.02322	0.297584	0.148792	−	0.988869i	$-0.452462\pi$
$557$	7.02322	0.297584	0.148792	+	0.988869i	$0.452462\pi$
$558$	0	0
$559$	8.14494	0.344494
$560$	0	0
$561$	−22.3716	−0.944528
$562$	0	0
$563$	23.3661	0.984764	0.492382	−	0.870379i	$-0.336126\pi$
$563$	23.3661	0.984764	0.492382	+	0.870379i	$0.336126\pi$
$564$	0	0
$565$	−24.5245	−1.03175
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	45.1331	1.89208	0.946039	−	0.324053i	$-0.105046\pi$
$569$	45.1331	1.89208	0.946039	+	0.324053i	$0.105046\pi$
$570$	0	0
$571$	−42.0370	−1.75919	−0.879597	−	0.475720i	$-0.842188\pi$
$571$	−42.0370	−1.75919	−0.879597	+	0.475720i	$0.842188\pi$
$572$	0	0
$573$	−24.2767	−1.01418
$574$	0	0
$575$	71.3517	2.97557
$576$	0	0
$577$	16.0330	0.667462	0.333731	−	0.942668i	$-0.391692\pi$
$577$	16.0330	0.667462	0.333731	+	0.942668i	$0.391692\pi$
$578$	0	0
$579$	23.9052	0.993466
$580$	0	0
$581$	−10.1758	−0.422165
$582$	0	0
$583$	−7.86160	−0.325594
$584$	0	0
$585$	−3.70948	−0.153368
$586$	0	0
$587$	−13.6595	−0.563788	−0.281894	−	0.959446i	$-0.590963\pi$
$587$	−13.6595	−0.563788	−0.281894	+	0.959446i	$0.590963\pi$
$588$	0	0
$589$	−33.4822	−1.37961
$590$	0	0
$591$	6.40247	0.263362
$592$	0	0
$593$	−3.53428	−0.145135	−0.0725677	−	0.997363i	$-0.523119\pi$
$593$	−3.53428	−0.145135	−0.0725677	+	0.997363i	$0.523119\pi$
$594$	0	0
$595$	18.3716	0.753160
$596$	0	0
$597$	23.4058	0.957937
$598$	0	0
$599$	17.2808	0.706073	0.353036	−	0.935610i	$-0.385149\pi$
$599$	17.2808	0.706073	0.353036	+	0.935610i	$0.385149\pi$
$600$	0	0
$601$	17.0673	0.696188	0.348094	−	0.937460i	$-0.386829\pi$
$601$	17.0673	0.696188	0.348094	+	0.937460i	$0.386829\pi$
$602$	0	0
$603$	12.2899	0.500482
$604$	0	0
$605$	−34.8860	−1.41832
$606$	0	0
$607$	16.9593	0.688358	0.344179	−	0.938904i	$-0.388157\pi$
$607$	16.9593	0.688358	0.344179	+	0.938904i	$0.388157\pi$
$608$	0	0
$609$	−1.27403	−0.0516261
$610$	0	0
$611$	−5.24311	−0.212114
$612$	0	0
$613$	−13.7617	−0.555829	−0.277915	−	0.960606i	$-0.589643\pi$
$613$	−13.7617	−0.555829	−0.277915	+	0.960606i	$0.589643\pi$
$614$	0	0
$615$	1.61531	0.0651358
$616$	0	0
$617$	−3.76600	−0.151614	−0.0758068	−	0.997123i	$-0.524153\pi$
$617$	−3.76600	−0.151614	−0.0758068	+	0.997123i	$0.524153\pi$
$618$	0	0
$619$	−24.1752	−0.971684	−0.485842	−	0.874047i	$-0.661487\pi$
$619$	−24.1752	−0.971684	−0.485842	+	0.874047i	$0.661487\pi$
$620$	0	0
$621$	8.14494	0.326845
$622$	0	0
$623$	3.24311	0.129933
$624$	0	0
$625$	7.94076	0.317630
$626$	0	0
$627$	14.7892	0.590625
$628$	0	0
$629$	24.5282	0.978003
$630$	0	0
$631$	−7.24376	−0.288369	−0.144185	−	0.989551i	$-0.546056\pi$
$631$	−7.24376	−0.288369	−0.144185	+	0.989551i	$0.546056\pi$
$632$	0	0
$633$	17.1792	0.682812
$634$	0	0
$635$	69.6496	2.76396
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	11.4697	0.453736
$640$	0	0
$641$	36.3887	1.43727	0.718634	−	0.695389i	$-0.244768\pi$
$641$	36.3887	1.43727	0.718634	+	0.695389i	$0.244768\pi$
$642$	0	0
$643$	−18.4731	−0.728508	−0.364254	−	0.931300i	$-0.618676\pi$
$643$	−18.4731	−0.728508	−0.364254	+	0.931300i	$0.618676\pi$
$644$	0	0
$645$	−30.2135	−1.18966
$646$	0	0
$647$	11.2912	0.443902	0.221951	−	0.975058i	$-0.428758\pi$
$647$	11.2912	0.443902	0.221951	+	0.975058i	$0.428758\pi$
$648$	0	0
$649$	−51.8104	−2.03374
$650$	0	0
$651$	−10.2266	−0.400813
$652$	0	0
$653$	−18.7931	−0.735431	−0.367715	−	0.929938i	$-0.619860\pi$
$653$	−18.7931	−0.735431	−0.367715	+	0.929938i	$0.619860\pi$
$654$	0	0
$655$	−38.4731	−1.50327
$656$	0	0
$657$	14.1449	0.551847
$658$	0	0
$659$	7.80508	0.304043	0.152021	−	0.988377i	$-0.451422\pi$
$659$	7.80508	0.304043	0.152021	+	0.988377i	$0.451422\pi$
$660$	0	0
$661$	−25.1991	−0.980130	−0.490065	−	0.871686i	$-0.663027\pi$
$661$	−25.1991	−0.980130	−0.490065	+	0.871686i	$0.663027\pi$
$662$	0	0
$663$	−4.95259	−0.192343
$664$	0	0
$665$	−12.1449	−0.470960
$666$	0	0
$667$	−10.3769	−0.401794
$668$	0	0
$669$	−3.67857	−0.142222
$670$	0	0
$671$	−11.5996	−0.447798
$672$	0	0
$673$	17.9553	0.692127	0.346063	−	0.938211i	$-0.387518\pi$
$673$	17.9553	0.692127	0.346063	+	0.938211i	$0.387518\pi$
$674$	0	0
$675$	8.76025	0.337182
$676$	0	0
$677$	−2.97245	−0.114241	−0.0571203	−	0.998367i	$-0.518192\pi$
$677$	−2.97245	−0.114241	−0.0571203	+	0.998367i	$0.518192\pi$
$678$	0	0
$679$	9.74039	0.373802
$680$	0	0
$681$	20.5040	0.785715
$682$	0	0
$683$	−8.86884	−0.339357	−0.169678	−	0.985500i	$-0.554273\pi$
$683$	−8.86884	−0.339357	−0.169678	+	0.985500i	$0.554273\pi$
$684$	0	0
$685$	45.4744	1.73749
$686$	0	0
$687$	−16.8379	−0.642407
$688$	0	0
$689$	−1.74039	−0.0663037
$690$	0	0
$691$	−5.22920	−0.198928	−0.0994641	−	0.995041i	$-0.531713\pi$
$691$	−5.22920	−0.198928	−0.0994641	+	0.995041i	$0.531713\pi$
$692$	0	0
$693$	4.51714	0.171592
$694$	0	0
$695$	−60.6563	−2.30083
$696$	0	0
$697$	2.15664	0.0816884
$698$	0	0
$699$	−17.6786	−0.668665
$700$	0	0
$701$	5.41366	0.204471	0.102236	−	0.994760i	$-0.467400\pi$
$701$	5.41366	0.204471	0.102236	+	0.994760i	$0.467400\pi$
$702$	0	0
$703$	−16.2149	−0.611557
$704$	0	0
$705$	19.4492	0.732500
$706$	0	0
$707$	15.8036	0.594357
$708$	0	0
$709$	29.0631	1.09149	0.545744	−	0.837952i	$-0.316247\pi$
$709$	29.0631	1.09149	0.545744	+	0.837952i	$0.316247\pi$
$710$	0	0
$711$	3.67857	0.137957
$712$	0	0
$713$	−83.2952	−3.11943
$714$	0	0
$715$	−16.7562	−0.626648
$716$	0	0
$717$	−11.3682	−0.424553
$718$	0	0
$719$	27.9987	1.04418	0.522088	−	0.852892i	$-0.325153\pi$
$719$	27.9987	1.04418	0.522088	+	0.852892i	$0.325153\pi$
$720$	0	0
$721$	14.3716	0.535225
$722$	0	0
$723$	−19.9355	−0.741408
$724$	0	0
$725$	−11.1608	−0.414501
$726$	0	0
$727$	44.4651	1.64912	0.824559	−	0.565776i	$-0.191423\pi$
$727$	44.4651	1.64912	0.824559	+	0.565776i	$0.191423\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.3386	−1.49198
$732$	0	0
$733$	−29.8406	−1.10219	−0.551095	−	0.834443i	$-0.685790\pi$
$733$	−29.8406	−1.10219	−0.551095	+	0.834443i	$0.685790\pi$
$734$	0	0
$735$	−3.70948	−0.136826
$736$	0	0
$737$	55.5151	2.04492
$738$	0	0
$739$	−15.3174	−0.563460	−0.281730	−	0.959494i	$-0.590908\pi$
$739$	−15.3174	−0.563460	−0.281730	+	0.959494i	$0.590908\pi$
$740$	0	0
$741$	3.27403	0.120274
$742$	0	0
$743$	−1.27338	−0.0467158	−0.0233579	−	0.999727i	$-0.507436\pi$
$743$	−1.27338	−0.0467158	−0.0233579	+	0.999727i	$0.507436\pi$
$744$	0	0
$745$	−53.8024	−1.97117
$746$	0	0
$747$	−10.1758	−0.372315
$748$	0	0
$749$	−5.90519	−0.215771
$750$	0	0
$751$	−5.35571	−0.195433	−0.0977163	−	0.995214i	$-0.531154\pi$
$751$	−5.35571	−0.195433	−0.0977163	+	0.995214i	$0.531154\pi$
$752$	0	0
$753$	12.7562	0.464863
$754$	0	0
$755$	6.04068	0.219843
$756$	0	0
$757$	38.5232	1.40015	0.700075	−	0.714069i	$-0.253150\pi$
$757$	38.5232	1.40015	0.700075	+	0.714069i	$0.253150\pi$
$758$	0	0
$759$	36.7918	1.33546
$760$	0	0
$761$	15.2418	0.552516	0.276258	−	0.961084i	$-0.410906\pi$
$761$	15.2418	0.552516	0.276258	+	0.961084i	$0.410906\pi$
$762$	0	0
$763$	14.4664	0.523718
$764$	0	0
$765$	18.3716	0.664225
$766$	0	0
$767$	−11.4697	−0.414148
$768$	0	0
$769$	−4.31503	−0.155604	−0.0778020	−	0.996969i	$-0.524790\pi$
$769$	−4.31503	−0.155604	−0.0778020	+	0.996969i	$0.524790\pi$
$770$	0	0
$771$	−9.89846	−0.356484
$772$	0	0
$773$	−13.0034	−0.467699	−0.233849	−	0.972273i	$-0.575132\pi$
$773$	−13.0034	−0.467699	−0.233849	+	0.972273i	$0.575132\pi$
$774$	0	0
$775$	−89.5878	−3.21809
$776$	0	0
$777$	−4.95259	−0.177673
$778$	0	0
$779$	−1.42569	−0.0510808
$780$	0	0
$781$	51.8104	1.85392
$782$	0	0
$783$	−1.27403	−0.0455300
$784$	0	0
$785$	−11.3043	−0.403468
$786$	0	0
$787$	0.268587	0.00957409	0.00478704	−	0.999989i	$-0.498476\pi$
$787$	0.268587	0.00957409	0.00478704	+	0.999989i	$0.498476\pi$
$788$	0	0
$789$	−3.74039	−0.133162
$790$	0	0
$791$	6.61131	0.235071
$792$	0	0
$793$	−2.56791	−0.0911891
$794$	0	0
$795$	6.45596	0.228969
$796$	0	0
$797$	29.5810	1.04781	0.523907	−	0.851775i	$-0.324474\pi$
$797$	29.5810	1.04781	0.523907	+	0.851775i	$0.324474\pi$
$798$	0	0
$799$	25.9670	0.918647
$800$	0	0
$801$	3.24311	0.114590
$802$	0	0
$803$	63.8946	2.25479
$804$	0	0
$805$	−30.2135	−1.06489
$806$	0	0
$807$	14.3517	0.505204
$808$	0	0
$809$	−26.0921	−0.917349	−0.458675	−	0.888604i	$-0.651676\pi$
$809$	−26.0921	−0.917349	−0.458675	+	0.888604i	$0.651676\pi$
$810$	0	0
$811$	−37.0461	−1.30087	−0.650433	−	0.759564i	$-0.725412\pi$
$811$	−37.0461	−1.30087	−0.650433	+	0.759564i	$0.725412\pi$
$812$	0	0
$813$	−27.0013	−0.946977
$814$	0	0
$815$	−48.3504	−1.69364
$816$	0	0
$817$	26.6667	0.932951
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−31.3749	−1.09499	−0.547496	−	0.836808i	$-0.684419\pi$
$821$	−31.3749	−1.09499	−0.547496	+	0.836808i	$0.684419\pi$
$822$	0	0
$823$	−17.8655	−0.622751	−0.311376	−	0.950287i	$-0.600790\pi$
$823$	−17.8655	−0.622751	−0.311376	+	0.950287i	$0.600790\pi$
$824$	0	0
$825$	39.5713	1.37769
$826$	0	0
$827$	−6.19428	−0.215396	−0.107698	−	0.994184i	$-0.534348\pi$
$827$	−6.19428	−0.215396	−0.107698	+	0.994184i	$0.534348\pi$
$828$	0	0
$829$	0.662720	0.0230172	0.0115086	−	0.999934i	$-0.496337\pi$
$829$	0.662720	0.0230172	0.0115086	+	0.999934i	$0.496337\pi$
$830$	0	0
$831$	0.644291	0.0223502
$832$	0	0
$833$	−4.95259	−0.171597
$834$	0	0
$835$	11.9566	0.413775
$836$	0	0
$837$	−10.2266	−0.353484
$838$	0	0
$839$	−54.3077	−1.87491	−0.937454	−	0.348108i	$-0.886824\pi$
$839$	−54.3077	−1.87491	−0.937454	+	0.348108i	$0.886824\pi$
$840$	0	0
$841$	−27.3769	−0.944030
$842$	0	0
$843$	−8.49728	−0.292662
$844$	0	0
$845$	−3.70948	−0.127610
$846$	0	0
$847$	9.40454	0.323144
$848$	0	0
$849$	31.8104	1.09173
$850$	0	0
$851$	−40.3386	−1.38279
$852$	0	0
$853$	−30.0053	−1.02736	−0.513681	−	0.857981i	$-0.671719\pi$
$853$	−30.0053	−1.02736	−0.513681	+	0.857981i	$0.671719\pi$
$854$	0	0
$855$	−12.1449	−0.415348
$856$	0	0
$857$	20.7364	0.708341	0.354171	−	0.935181i	$-0.384763\pi$
$857$	20.7364	0.708341	0.354171	+	0.935181i	$0.384763\pi$
$858$	0	0
$859$	3.68387	0.125692	0.0628460	−	0.998023i	$-0.479982\pi$
$859$	3.68387	0.125692	0.0628460	+	0.998023i	$0.479982\pi$
$860$	0	0
$861$	−0.435456	−0.0148403
$862$	0	0
$863$	16.1113	0.548435	0.274218	−	0.961668i	$-0.411581\pi$
$863$	16.1113	0.548435	0.274218	+	0.961668i	$0.411581\pi$
$864$	0	0
$865$	86.5205	2.94179
$866$	0	0
$867$	7.52819	0.255671
$868$	0	0
$869$	16.6166	0.563680
$870$	0	0
$871$	12.2899	0.416427
$872$	0	0
$873$	9.74039	0.329662
$874$	0	0
$875$	−13.9486	−0.471548
$876$	0	0
$877$	−4.38469	−0.148060	−0.0740301	−	0.997256i	$-0.523586\pi$
$877$	−4.38469	−0.148060	−0.0740301	+	0.997256i	$0.523586\pi$
$878$	0	0
$879$	21.3366	0.719667
$880$	0	0
$881$	51.4126	1.73213	0.866067	−	0.499929i	$-0.166640\pi$
$881$	51.4126	1.73213	0.866067	+	0.499929i	$0.166640\pi$
$882$	0	0
$883$	12.6957	0.427245	0.213622	−	0.976916i	$-0.431474\pi$
$883$	12.6957	0.427245	0.213622	+	0.976916i	$0.431474\pi$
$884$	0	0
$885$	42.5468	1.43019
$886$	0	0
$887$	−32.7828	−1.10074	−0.550370	−	0.834921i	$-0.685513\pi$
$887$	−32.7828	−1.10074	−0.550370	+	0.834921i	$0.685513\pi$
$888$	0	0
$889$	−18.7761	−0.629730
$890$	0	0
$891$	4.51714	0.151330
$892$	0	0
$893$	−17.1661	−0.574441
$894$	0	0
$895$	−17.1056	−0.571776
$896$	0	0
$897$	8.14494	0.271952
$898$	0	0
$899$	13.0290	0.434541
$900$	0	0
$901$	8.61946	0.287156
$902$	0	0
$903$	8.14494	0.271047
$904$	0	0
$905$	−77.2979	−2.56947
$906$	0	0
$907$	−6.02354	−0.200008	−0.100004	−	0.994987i	$-0.531886\pi$
$907$	−6.02354	−0.200008	−0.100004	+	0.994987i	$0.531886\pi$
$908$	0	0
$909$	15.8036	0.524174
$910$	0	0
$911$	−26.1120	−0.865128	−0.432564	−	0.901603i	$-0.642391\pi$
$911$	−26.1120	−0.865128	−0.432564	+	0.901603i	$0.642391\pi$
$912$	0	0
$913$	−45.9657	−1.52124
$914$	0	0
$915$	9.52561	0.314907
$916$	0	0
$917$	10.3716	0.342499
$918$	0	0
$919$	−6.13038	−0.202223	−0.101111	−	0.994875i	$-0.532240\pi$
$919$	−6.13038	−0.202223	−0.101111	+	0.994875i	$0.532240\pi$
$920$	0	0
$921$	6.32816	0.208520
$922$	0	0
$923$	11.4697	0.377531
$924$	0	0
$925$	−43.3860	−1.42652
$926$	0	0
$927$	14.3716	0.472024
$928$	0	0
$929$	−55.2169	−1.81161	−0.905803	−	0.423699i	$-0.860732\pi$
$929$	−55.2169	−1.81161	−0.905803	+	0.423699i	$0.860732\pi$
$930$	0	0
$931$	3.27403	0.107302
$932$	0	0
$933$	24.5151	0.802587
$934$	0	0
$935$	82.9869	2.71396
$936$	0	0
$937$	48.1951	1.57446	0.787232	−	0.616657i	$-0.211513\pi$
$937$	48.1951	1.57446	0.787232	+	0.616657i	$0.211513\pi$
$938$	0	0
$939$	14.1634	0.462204
$940$	0	0
$941$	50.9520	1.66099	0.830493	−	0.557029i	$-0.188059\pi$
$941$	50.9520	1.66099	0.830493	+	0.557029i	$0.188059\pi$
$942$	0	0
$943$	−3.54676	−0.115498
$944$	0	0
$945$	−3.70948	−0.120669
$946$	0	0
$947$	3.12987	0.101707	0.0508536	−	0.998706i	$-0.483806\pi$
$947$	3.12987	0.101707	0.0508536	+	0.998706i	$0.483806\pi$
$948$	0	0
$949$	14.1449	0.459164
$950$	0	0
$951$	3.69105	0.119691
$952$	0	0
$953$	−27.4835	−0.890278	−0.445139	−	0.895461i	$-0.646846\pi$
$953$	−27.4835	−0.890278	−0.445139	+	0.895461i	$0.646846\pi$
$954$	0	0
$955$	90.0541	2.91408
$956$	0	0
$957$	−5.75495	−0.186031
$958$	0	0
$959$	−12.2590	−0.395863
$960$	0	0
$961$	73.5838	2.37367
$962$	0	0
$963$	−5.90519	−0.190292
$964$	0	0
$965$	−88.6759	−2.85458
$966$	0	0
$967$	−42.2450	−1.35851	−0.679255	−	0.733903i	$-0.737697\pi$
$967$	−42.2450	−1.35851	−0.679255	+	0.733903i	$0.737697\pi$
$968$	0	0
$969$	−16.2149	−0.520898
$970$	0	0
$971$	−53.5493	−1.71848	−0.859240	−	0.511573i	$-0.829063\pi$
$971$	−53.5493	−1.71848	−0.859240	+	0.511573i	$0.829063\pi$
$972$	0	0
$973$	16.3517	0.524211
$974$	0	0
$975$	8.76025	0.280553
$976$	0	0
$977$	−15.0982	−0.483033	−0.241517	−	0.970397i	$-0.577645\pi$
$977$	−15.0982	−0.483033	−0.241517	+	0.970397i	$0.577645\pi$
$978$	0	0
$979$	14.6496	0.468203
$980$	0	0
$981$	14.4664	0.461876
$982$	0	0
$983$	−22.1230	−0.705614	−0.352807	−	0.935696i	$-0.614773\pi$
$983$	−22.1230	−0.705614	−0.352807	+	0.935696i	$0.614773\pi$
$984$	0	0
$985$	−23.7498	−0.756733
$986$	0	0
$987$	−5.24311	−0.166890
$988$	0	0
$989$	66.3400	2.10949
$990$	0	0
$991$	−55.9737	−1.77806	−0.889032	−	0.457844i	$-0.848622\pi$
$991$	−55.9737	−1.77806	−0.889032	+	0.457844i	$0.848622\pi$
$992$	0	0
$993$	0.479496	0.0152163
$994$	0	0
$995$	−86.8235	−2.75249
$996$	0	0
$997$	12.3187	0.390138	0.195069	−	0.980790i	$-0.437507\pi$
$997$	12.3187	0.390138	0.195069	+	0.980790i	$0.437507\pi$
$998$	0	0
$999$	−4.95259	−0.156693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bs.1.1		4
4.3	odd	2		2184.2.a.v.1.1	✓	4
12.11	even	2		6552.2.a.bs.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.v.1.1	✓	4	4.3	odd	2
4368.2.a.bs.1.1		4	1.1	even	1	trivial
6552.2.a.bs.1.4		4	12.11	even	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.70948 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.70948 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.51714 q^{11} +1.00000 q^{13} -3.70948 q^{15} -4.95259 q^{17} +3.27403 q^{19} +1.00000 q^{21} +8.14494 q^{23} +8.76025 q^{25} +1.00000 q^{27} -1.27403 q^{29} -10.2266 q^{31} +4.51714 q^{33} -3.70948 q^{35} -4.95259 q^{37} +1.00000 q^{39} -0.435456 q^{41} +8.14494 q^{43} -3.70948 q^{45} -5.24311 q^{47} +1.00000 q^{49} -4.95259 q^{51} -1.74039 q^{53} -16.7562 q^{55} +3.27403 q^{57} -11.4697 q^{59} -2.56791 q^{61} +1.00000 q^{63} -3.70948 q^{65} +12.2899 q^{67} +8.14494 q^{69} +11.4697 q^{71} +14.1449 q^{73} +8.76025 q^{75} +4.51714 q^{77} +3.67857 q^{79} +1.00000 q^{81} -10.1758 q^{83} +18.3716 q^{85} -1.27403 q^{87} +3.24311 q^{89} +1.00000 q^{91} -10.2266 q^{93} -12.1449 q^{95} +9.74039 q^{97} +4.51714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 3 q^{11} + 4 q^{13} + 2 q^{15} + 3 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 14 q^{25} + 4 q^{27} + 4 q^{29} - 9 q^{31} + 3 q^{33} + 2 q^{35} + 3 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 2 q^{45} - 15 q^{47} + 4 q^{49} + 3 q^{51} + 13 q^{53} - 7 q^{55} + 4 q^{57} - 8 q^{59} + 9 q^{61} + 4 q^{63} + 2 q^{65} + 8 q^{69} + 8 q^{71} + 32 q^{73} + 14 q^{75} + 3 q^{77} + q^{79} + 4 q^{81} - 13 q^{83} + 17 q^{85} + 4 q^{87} + 7 q^{89} + 4 q^{91} - 9 q^{93} - 24 q^{95} + 19 q^{97} + 3 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 3 * q^11 + 4 * q^13 + 2 * q^15 + 3 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 14 * q^25 + 4 * q^27 + 4 * q^29 - 9 * q^31 + 3 * q^33 + 2 * q^35 + 3 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 2 * q^45 - 15 * q^47 + 4 * q^49 + 3 * q^51 + 13 * q^53 - 7 * q^55 + 4 * q^57 - 8 * q^59 + 9 * q^61 + 4 * q^63 + 2 * q^65 + 8 * q^69 + 8 * q^71 + 32 * q^73 + 14 * q^75 + 3 * q^77 + q^79 + 4 * q^81 - 13 * q^83 + 17 * q^85 + 4 * q^87 + 7 * q^89 + 4 * q^91 - 9 * q^93 - 24 * q^95 + 19 * q^97 + 3 * q^99

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