LMFDB - Embedded newform 6012.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0546093$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0546093 q^{5} +3.75281 q^{7} +O(q^{10})$	$q+0.0546093 q^{5} +3.75281 q^{7} -3.94419 q^{11} +6.73983 q^{13} -7.10275 q^{17} -6.19385 q^{19} -0.493660 q^{23} -4.99702 q^{25} +5.62720 q^{29} +3.17610 q^{31} +0.204938 q^{35} +0.462929 q^{37} +8.07740 q^{41} +7.28810 q^{43} +9.82281 q^{47} +7.08356 q^{49} -7.30739 q^{53} -0.215389 q^{55} +2.06719 q^{59} +2.42620 q^{61} +0.368057 q^{65} +9.70533 q^{67} +4.23960 q^{71} +6.05970 q^{73} -14.8018 q^{77} +0.493725 q^{79} +15.9193 q^{83} -0.387876 q^{85} -10.6375 q^{89} +25.2933 q^{91} -0.338242 q^{95} +16.2863 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^5 + 2 * q^7	$ 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) $ 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.0546093	0.0244220	0.0122110	−	0.999925i	$-0.496113\pi$
$5$	0.0546093	0.0244220	0.0122110	+	0.999925i	$0.496113\pi$
$6$	0	0
$7$	3.75281	1.41843	0.709214	−	0.704993i	$-0.249050\pi$
$7$	3.75281	1.41843	0.709214	+	0.704993i	$0.249050\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.94419	−1.18922	−0.594609	−	0.804015i	$-0.702693\pi$
$11$	−3.94419	−1.18922	−0.594609	+	0.804015i	$0.702693\pi$
$12$	0	0
$13$	6.73983	1.86929	0.934646	−	0.355579i	$-0.115716\pi$
$13$	6.73983	1.86929	0.934646	+	0.355579i	$0.115716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.10275	−1.72267	−0.861335	−	0.508038i	$-0.830371\pi$
$17$	−7.10275	−1.72267	−0.861335	+	0.508038i	$0.830371\pi$
$18$	0	0
$19$	−6.19385	−1.42097	−0.710483	−	0.703714i	$-0.751524\pi$
$19$	−6.19385	−1.42097	−0.710483	+	0.703714i	$0.751524\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.493660	−0.102935	−0.0514676	−	0.998675i	$-0.516390\pi$
$23$	−0.493660	−0.102935	−0.0514676	+	0.998675i	$0.516390\pi$
$24$	0	0
$25$	−4.99702	−0.999404
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.62720	1.04495	0.522473	−	0.852656i	$-0.325010\pi$
$29$	5.62720	1.04495	0.522473	+	0.852656i	$0.325010\pi$
$30$	0	0
$31$	3.17610	0.570444	0.285222	−	0.958462i	$-0.407933\pi$
$31$	3.17610	0.570444	0.285222	+	0.958462i	$0.407933\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.204938	0.0346409
$36$	0	0
$37$	0.462929	0.0761050	0.0380525	−	0.999276i	$-0.487885\pi$
$37$	0.462929	0.0761050	0.0380525	+	0.999276i	$0.487885\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.07740	1.26148	0.630739	−	0.775995i	$-0.282752\pi$
$41$	8.07740	1.26148	0.630739	+	0.775995i	$0.282752\pi$
$42$	0	0
$43$	7.28810	1.11142	0.555712	−	0.831375i	$-0.312446\pi$
$43$	7.28810	1.11142	0.555712	+	0.831375i	$0.312446\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.82281	1.43280	0.716402	−	0.697688i	$-0.245788\pi$
$47$	9.82281	1.43280	0.716402	+	0.697688i	$0.245788\pi$
$48$	0	0
$49$	7.08356	1.01194
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.30739	−1.00375	−0.501874	−	0.864941i	$-0.667356\pi$
$53$	−7.30739	−1.00375	−0.501874	+	0.864941i	$0.667356\pi$
$54$	0	0
$55$	−0.215389	−0.0290431
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.06719	0.269125	0.134562	−	0.990905i	$-0.457037\pi$
$59$	2.06719	0.269125	0.134562	+	0.990905i	$0.457037\pi$
$60$	0	0
$61$	2.42620	0.310643	0.155322	−	0.987864i	$-0.450359\pi$
$61$	2.42620	0.310643	0.155322	+	0.987864i	$0.450359\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.368057	0.0456519
$66$	0	0
$67$	9.70533	1.18569	0.592847	−	0.805315i	$-0.298004\pi$
$67$	9.70533	1.18569	0.592847	+	0.805315i	$0.298004\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.23960	0.503148	0.251574	−	0.967838i	$-0.419052\pi$
$71$	4.23960	0.503148	0.251574	+	0.967838i	$0.419052\pi$
$72$	0	0
$73$	6.05970	0.709234	0.354617	−	0.935012i	$-0.384611\pi$
$73$	6.05970	0.709234	0.354617	+	0.935012i	$0.384611\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.8018	−1.68682
$78$	0	0
$79$	0.493725	0.0555484	0.0277742	−	0.999614i	$-0.491158\pi$
$79$	0.493725	0.0555484	0.0277742	+	0.999614i	$0.491158\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.9193	1.74737	0.873686	−	0.486490i	$-0.161723\pi$
$83$	15.9193	1.74737	0.873686	+	0.486490i	$0.161723\pi$
$84$	0	0
$85$	−0.387876	−0.0420710
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.6375	−1.12758	−0.563788	−	0.825919i	$-0.690657\pi$
$89$	−10.6375	−1.12758	−0.563788	+	0.825919i	$0.690657\pi$
$90$	0	0
$91$	25.2933	2.65146
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.338242	−0.0347029
$96$	0	0
$97$	16.2863	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$97$	16.2863	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.1169	1.10617	0.553087	−	0.833124i	$-0.313450\pi$
$101$	11.1169	1.10617	0.553087	+	0.833124i	$0.313450\pi$
$102$	0	0
$103$	15.0378	1.48171	0.740857	−	0.671663i	$-0.234419\pi$
$103$	15.0378	1.48171	0.740857	+	0.671663i	$0.234419\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.77010	0.267796	0.133898	−	0.990995i	$-0.457251\pi$
$107$	2.77010	0.267796	0.133898	+	0.990995i	$0.457251\pi$
$108$	0	0
$109$	−15.9154	−1.52442	−0.762210	−	0.647330i	$-0.775886\pi$
$109$	−15.9154	−1.52442	−0.762210	+	0.647330i	$0.775886\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.18265	0.299399	0.149699	−	0.988732i	$-0.452169\pi$
$113$	3.18265	0.299399	0.149699	+	0.988732i	$0.452169\pi$
$114$	0	0
$115$	−0.0269584	−0.00251388
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.6552	−2.44348
$120$	0	0
$121$	4.55662	0.414238
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.545930	−0.0488294
$126$	0	0
$127$	−17.0341	−1.51154	−0.755768	−	0.654840i	$-0.772736\pi$
$127$	−17.0341	−1.51154	−0.755768	+	0.654840i	$0.772736\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.17786	0.277651	0.138825	−	0.990317i	$-0.455667\pi$
$131$	3.17786	0.277651	0.138825	+	0.990317i	$0.455667\pi$
$132$	0	0
$133$	−23.2443	−2.01554
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1693	−1.29600	−0.648002	−	0.761638i	$-0.724395\pi$
$137$	−15.1693	−1.29600	−0.648002	+	0.761638i	$0.724395\pi$
$138$	0	0
$139$	−23.4814	−1.99167	−0.995835	−	0.0911772i	$-0.970937\pi$
$139$	−23.4814	−1.99167	−0.995835	+	0.0911772i	$0.970937\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.5832	−2.22300
$144$	0	0
$145$	0.307298	0.0255197
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.2268	1.73897	0.869484	−	0.493961i	$-0.164451\pi$
$149$	21.2268	1.73897	0.869484	+	0.493961i	$0.164451\pi$
$150$	0	0
$151$	12.1824	0.991389	0.495695	−	0.868497i	$-0.334914\pi$
$151$	12.1824	0.991389	0.495695	+	0.868497i	$0.334914\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.173444	0.0139314
$156$	0	0
$157$	15.2722	1.21886	0.609428	−	0.792842i	$-0.291399\pi$
$157$	15.2722	1.21886	0.609428	+	0.792842i	$0.291399\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.85261	−0.146006
$162$	0	0
$163$	−12.0331	−0.942507	−0.471253	−	0.881998i	$-0.656198\pi$
$163$	−12.0331	−0.942507	−0.471253	+	0.881998i	$0.656198\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	32.4253	2.49425
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.28616	−0.553957	−0.276978	−	0.960876i	$-0.589333\pi$
$173$	−7.28616	−0.553957	−0.276978	+	0.960876i	$0.589333\pi$
$174$	0	0
$175$	−18.7528	−1.41758
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.97190	0.147387	0.0736933	−	0.997281i	$-0.476521\pi$
$179$	1.97190	0.147387	0.0736933	+	0.997281i	$0.476521\pi$
$180$	0	0
$181$	3.75141	0.278840	0.139420	−	0.990233i	$-0.455476\pi$
$181$	3.75141	0.278840	0.139420	+	0.990233i	$0.455476\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.0252802	0.00185864
$186$	0	0
$187$	28.0146	2.04863
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.75171	−0.343822	−0.171911	−	0.985113i	$-0.554994\pi$
$191$	−4.75171	−0.343822	−0.171911	+	0.985113i	$0.554994\pi$
$192$	0	0
$193$	−2.80365	−0.201811	−0.100906	−	0.994896i	$-0.532174\pi$
$193$	−2.80365	−0.201811	−0.100906	+	0.994896i	$0.532174\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.5221	−1.10590	−0.552952	−	0.833213i	$-0.686499\pi$
$197$	−15.5221	−1.10590	−0.552952	+	0.833213i	$0.686499\pi$
$198$	0	0
$199$	2.82469	0.200237	0.100118	−	0.994976i	$-0.468078\pi$
$199$	2.82469	0.200237	0.100118	+	0.994976i	$0.468078\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	21.1178	1.48218
$204$	0	0
$205$	0.441101	0.0308078
$206$	0	0
$207$	0	0
$208$	0	0
$209$	24.4297	1.68984
$210$	0	0
$211$	7.38226	0.508216	0.254108	−	0.967176i	$-0.418218\pi$
$211$	7.38226	0.508216	0.254108	+	0.967176i	$0.418218\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.397998	0.0271432
$216$	0	0
$217$	11.9193	0.809133
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−47.8713	−3.22017
$222$	0	0
$223$	−5.69989	−0.381693	−0.190847	−	0.981620i	$-0.561123\pi$
$223$	−5.69989	−0.381693	−0.190847	+	0.981620i	$0.561123\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.2358	0.745749	0.372874	−	0.927882i	$-0.378372\pi$
$227$	11.2358	0.745749	0.372874	+	0.927882i	$0.378372\pi$
$228$	0	0
$229$	1.57960	0.104383	0.0521914	−	0.998637i	$-0.483379\pi$
$229$	1.57960	0.104383	0.0521914	+	0.998637i	$0.483379\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.2751	−1.39378	−0.696890	−	0.717178i	$-0.745433\pi$
$233$	−21.2751	−1.39378	−0.696890	+	0.717178i	$0.745433\pi$
$234$	0	0
$235$	0.536417	0.0349920
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.5531	1.45884	0.729420	−	0.684066i	$-0.239790\pi$
$239$	22.5531	1.45884	0.729420	+	0.684066i	$0.239790\pi$
$240$	0	0
$241$	20.3851	1.31312	0.656560	−	0.754274i	$-0.272011\pi$
$241$	20.3851	1.31312	0.656560	+	0.754274i	$0.272011\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.386828	0.0247135
$246$	0	0
$247$	−41.7455	−2.65620
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.6282	1.17580	0.587900	−	0.808934i	$-0.299955\pi$
$251$	18.6282	1.17580	0.587900	+	0.808934i	$0.299955\pi$
$252$	0	0
$253$	1.94709	0.122412
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.42543	0.213672	0.106836	−	0.994277i	$-0.465928\pi$
$257$	3.42543	0.213672	0.106836	+	0.994277i	$0.465928\pi$
$258$	0	0
$259$	1.73728	0.107949
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.0393	0.927362	0.463681	−	0.886002i	$-0.346528\pi$
$263$	15.0393	0.927362	0.463681	+	0.886002i	$0.346528\pi$
$264$	0	0
$265$	−0.399051	−0.0245135
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.58206	0.523257	0.261629	−	0.965169i	$-0.415740\pi$
$269$	8.58206	0.523257	0.261629	+	0.965169i	$0.415740\pi$
$270$	0	0
$271$	6.35340	0.385942	0.192971	−	0.981204i	$-0.438188\pi$
$271$	6.35340	0.385942	0.192971	+	0.981204i	$0.438188\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19.7092	1.18851
$276$	0	0
$277$	−2.10298	−0.126356	−0.0631780	−	0.998002i	$-0.520124\pi$
$277$	−2.10298	−0.126356	−0.0631780	+	0.998002i	$0.520124\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.08153	0.0645187	0.0322593	−	0.999480i	$-0.489730\pi$
$281$	1.08153	0.0645187	0.0322593	+	0.999480i	$0.489730\pi$
$282$	0	0
$283$	7.76067	0.461324	0.230662	−	0.973034i	$-0.425911\pi$
$283$	7.76067	0.461324	0.230662	+	0.973034i	$0.425911\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.3129	1.78931
$288$	0	0
$289$	33.4490	1.96759
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−16.5238	−0.965330	−0.482665	−	0.875805i	$-0.660331\pi$
$293$	−16.5238	−0.965330	−0.482665	+	0.875805i	$0.660331\pi$
$294$	0	0
$295$	0.112888	0.00657257
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.32718	−0.192416
$300$	0	0
$301$	27.3508	1.57648
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.132493	0.00758654
$306$	0	0
$307$	12.8192	0.731631	0.365816	−	0.930687i	$-0.380790\pi$
$307$	12.8192	0.731631	0.365816	+	0.930687i	$0.380790\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.2219	−0.863153	−0.431576	−	0.902076i	$-0.642042\pi$
$311$	−15.2219	−0.863153	−0.431576	+	0.902076i	$0.642042\pi$
$312$	0	0
$313$	−15.5788	−0.880564	−0.440282	−	0.897859i	$-0.645122\pi$
$313$	−15.5788	−0.880564	−0.440282	+	0.897859i	$0.645122\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.1856	1.18990	0.594952	−	0.803761i	$-0.297171\pi$
$317$	21.1856	1.18990	0.594952	+	0.803761i	$0.297171\pi$
$318$	0	0
$319$	−22.1948	−1.24267
$320$	0	0
$321$	0	0
$322$	0	0
$323$	43.9934	2.44786
$324$	0	0
$325$	−33.6790	−1.86818
$326$	0	0
$327$	0	0
$328$	0	0
$329$	36.8631	2.03233
$330$	0	0
$331$	−27.5518	−1.51438	−0.757192	−	0.653192i	$-0.773430\pi$
$331$	−27.5518	−1.51438	−0.757192	+	0.653192i	$0.773430\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.530001	0.0289570
$336$	0	0
$337$	22.2662	1.21292	0.606459	−	0.795114i	$-0.292589\pi$
$337$	22.2662	1.21292	0.606459	+	0.795114i	$0.292589\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.5271	−0.678382
$342$	0	0
$343$	0.313587	0.0169321
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.286340	0.0153715	0.00768576	−	0.999970i	$-0.497554\pi$
$347$	0.286340	0.0153715	0.00768576	+	0.999970i	$0.497554\pi$
$348$	0	0
$349$	−20.2412	−1.08349	−0.541744	−	0.840544i	$-0.682236\pi$
$349$	−20.2412	−1.08349	−0.541744	+	0.840544i	$0.682236\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−25.8071	−1.37358	−0.686788	−	0.726858i	$-0.740980\pi$
$353$	−25.8071	−1.37358	−0.686788	+	0.726858i	$0.740980\pi$
$354$	0	0
$355$	0.231521	0.0122879
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.15722	−0.377744	−0.188872	−	0.982002i	$-0.560483\pi$
$359$	−7.15722	−0.377744	−0.188872	+	0.982002i	$0.560483\pi$
$360$	0	0
$361$	19.3638	1.01915
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.330916	0.0173209
$366$	0	0
$367$	−21.1432	−1.10367	−0.551833	−	0.833955i	$-0.686071\pi$
$367$	−21.1432	−1.10367	−0.551833	+	0.833955i	$0.686071\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−27.4232	−1.42374
$372$	0	0
$373$	12.2934	0.636529	0.318264	−	0.948002i	$-0.396900\pi$
$373$	12.2934	0.636529	0.318264	+	0.948002i	$0.396900\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	37.9264	1.95331
$378$	0	0
$379$	17.0153	0.874017	0.437009	−	0.899457i	$-0.356038\pi$
$379$	17.0153	0.874017	0.437009	+	0.899457i	$0.356038\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.38046	−0.274929	−0.137464	−	0.990507i	$-0.543895\pi$
$383$	−5.38046	−0.274929	−0.137464	+	0.990507i	$0.543895\pi$
$384$	0	0
$385$	−0.808314	−0.0411955
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.63501	0.285706	0.142853	−	0.989744i	$-0.454372\pi$
$389$	5.63501	0.285706	0.142853	+	0.989744i	$0.454372\pi$
$390$	0	0
$391$	3.50634	0.177323
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.0269620	0.00135660
$396$	0	0
$397$	4.73695	0.237741	0.118870	−	0.992910i	$-0.462073\pi$
$397$	4.73695	0.237741	0.118870	+	0.992910i	$0.462073\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.6638	1.18171	0.590857	−	0.806776i	$-0.298790\pi$
$401$	23.6638	1.18171	0.590857	+	0.806776i	$0.298790\pi$
$402$	0	0
$403$	21.4063	1.06633
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.82588	−0.0905054
$408$	0	0
$409$	−28.3500	−1.40182	−0.700910	−	0.713250i	$-0.747222\pi$
$409$	−28.3500	−1.40182	−0.700910	+	0.713250i	$0.747222\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.75775	0.381734
$414$	0	0
$415$	0.869342	0.0426743
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.6136	1.69098	0.845491	−	0.533989i	$-0.179308\pi$
$419$	34.6136	1.69098	0.845491	+	0.533989i	$0.179308\pi$
$420$	0	0
$421$	−8.46717	−0.412665	−0.206332	−	0.978482i	$-0.566153\pi$
$421$	−8.46717	−0.412665	−0.206332	+	0.978482i	$0.566153\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	35.4926	1.72164
$426$	0	0
$427$	9.10507	0.440625
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.64958	−0.0794576	−0.0397288	−	0.999210i	$-0.512649\pi$
$431$	−1.64958	−0.0794576	−0.0397288	+	0.999210i	$0.512649\pi$
$432$	0	0
$433$	−18.2488	−0.876983	−0.438492	−	0.898735i	$-0.644487\pi$
$433$	−18.2488	−0.876983	−0.438492	+	0.898735i	$0.644487\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.05765	0.146267
$438$	0	0
$439$	25.0738	1.19671	0.598354	−	0.801232i	$-0.295822\pi$
$439$	25.0738	1.19671	0.598354	+	0.801232i	$0.295822\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−40.0318	−1.90197	−0.950985	−	0.309238i	$-0.899926\pi$
$443$	−40.0318	−1.90197	−0.950985	+	0.309238i	$0.899926\pi$
$444$	0	0
$445$	−0.580908	−0.0275377
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.4053	−0.727020	−0.363510	−	0.931590i	$-0.618422\pi$
$449$	−15.4053	−0.727020	−0.363510	+	0.931590i	$0.618422\pi$
$450$	0	0
$451$	−31.8588	−1.50017
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.38125	0.0647539
$456$	0	0
$457$	14.7340	0.689227	0.344613	−	0.938745i	$-0.388010\pi$
$457$	14.7340	0.689227	0.344613	+	0.938745i	$0.388010\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−33.1458	−1.54375	−0.771877	−	0.635772i	$-0.780682\pi$
$461$	−33.1458	−1.54375	−0.771877	+	0.635772i	$0.780682\pi$
$462$	0	0
$463$	−2.86786	−0.133281	−0.0666404	−	0.997777i	$-0.521228\pi$
$463$	−2.86786	−0.133281	−0.0666404	+	0.997777i	$0.521228\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.3210	1.26426	0.632132	−	0.774861i	$-0.282180\pi$
$467$	27.3210	1.26426	0.632132	+	0.774861i	$0.282180\pi$
$468$	0	0
$469$	36.4222	1.68182
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28.7456	−1.32173
$474$	0	0
$475$	30.9508	1.42012
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.84430	−0.404107	−0.202053	−	0.979375i	$-0.564761\pi$
$479$	−8.84430	−0.404107	−0.202053	+	0.979375i	$0.564761\pi$
$480$	0	0
$481$	3.12006	0.142262
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.889384	0.0403848
$486$	0	0
$487$	−30.6735	−1.38995	−0.694974	−	0.719035i	$-0.744584\pi$
$487$	−30.6735	−1.38995	−0.694974	+	0.719035i	$0.744584\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.22374	−0.145485	−0.0727426	−	0.997351i	$-0.523175\pi$
$491$	−3.22374	−0.145485	−0.0727426	+	0.997351i	$0.523175\pi$
$492$	0	0
$493$	−39.9686	−1.80010
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.9104	0.713679
$498$	0	0
$499$	−3.78343	−0.169370	−0.0846849	−	0.996408i	$-0.526988\pi$
$499$	−3.78343	−0.169370	−0.0846849	+	0.996408i	$0.526988\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.6873	1.50204	0.751021	−	0.660279i	$-0.229562\pi$
$503$	33.6873	1.50204	0.751021	+	0.660279i	$0.229562\pi$
$504$	0	0
$505$	0.607086	0.0270150
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.7358	−0.608827	−0.304413	−	0.952540i	$-0.598460\pi$
$509$	−13.7358	−0.608827	−0.304413	+	0.952540i	$0.598460\pi$
$510$	0	0
$511$	22.7409	1.00600
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.821201	0.0361864
$516$	0	0
$517$	−38.7430	−1.70392
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.6156	0.990806	0.495403	−	0.868663i	$-0.335020\pi$
$521$	22.6156	0.990806	0.495403	+	0.868663i	$0.335020\pi$
$522$	0	0
$523$	3.61400	0.158029	0.0790146	−	0.996873i	$-0.474823\pi$
$523$	3.61400	0.158029	0.0790146	+	0.996873i	$0.474823\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.5590	−0.982686
$528$	0	0
$529$	−22.7563	−0.989404
$530$	0	0
$531$	0	0
$532$	0	0
$533$	54.4403	2.35807
$534$	0	0
$535$	0.151273	0.00654011
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−27.9389	−1.20341
$540$	0	0
$541$	12.6414	0.543496	0.271748	−	0.962368i	$-0.412398\pi$
$541$	12.6414	0.543496	0.271748	+	0.962368i	$0.412398\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.869129	−0.0372294
$546$	0	0
$547$	−11.6254	−0.497065	−0.248532	−	0.968624i	$-0.579948\pi$
$547$	−11.6254	−0.497065	−0.248532	+	0.968624i	$0.579948\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.8541	−1.48483
$552$	0	0
$553$	1.85286	0.0787914
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.0330	−1.31491	−0.657454	−	0.753494i	$-0.728367\pi$
$557$	−31.0330	−1.31491	−0.657454	+	0.753494i	$0.728367\pi$
$558$	0	0
$559$	49.1205	2.07758
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.6732	1.29272	0.646362	−	0.763031i	$-0.276290\pi$
$563$	30.6732	1.29272	0.646362	+	0.763031i	$0.276290\pi$
$564$	0	0
$565$	0.173802	0.00731192
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.85125	−0.412986	−0.206493	−	0.978448i	$-0.566205\pi$
$569$	−9.85125	−0.412986	−0.206493	+	0.978448i	$0.566205\pi$
$570$	0	0
$571$	−10.5595	−0.441903	−0.220952	−	0.975285i	$-0.570916\pi$
$571$	−10.5595	−0.441903	−0.220952	+	0.975285i	$0.570916\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.46683	0.102874
$576$	0	0
$577$	−39.4284	−1.64142	−0.820712	−	0.571342i	$-0.806423\pi$
$577$	−39.4284	−1.64142	−0.820712	+	0.571342i	$0.806423\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	59.7421	2.47852
$582$	0	0
$583$	28.8217	1.19367
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.7871	−1.27072	−0.635359	−	0.772217i	$-0.719148\pi$
$587$	−30.7871	−1.27072	−0.635359	+	0.772217i	$0.719148\pi$
$588$	0	0
$589$	−19.6723	−0.810582
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20.1759	−0.828525	−0.414262	−	0.910157i	$-0.635960\pi$
$593$	−20.1759	−0.828525	−0.414262	+	0.910157i	$0.635960\pi$
$594$	0	0
$595$	−1.45562	−0.0596747
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.10179	0.0858768	0.0429384	−	0.999078i	$-0.486328\pi$
$599$	2.10179	0.0858768	0.0429384	+	0.999078i	$0.486328\pi$
$600$	0	0
$601$	16.4395	0.670582	0.335291	−	0.942115i	$-0.391165\pi$
$601$	16.4395	0.670582	0.335291	+	0.942115i	$0.391165\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.248834	0.0101165
$606$	0	0
$607$	−23.3144	−0.946303	−0.473152	−	0.880981i	$-0.656884\pi$
$607$	−23.3144	−0.946303	−0.473152	+	0.880981i	$0.656884\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	66.2041	2.67833
$612$	0	0
$613$	−16.6836	−0.673846	−0.336923	−	0.941532i	$-0.609386\pi$
$613$	−16.6836	−0.673846	−0.336923	+	0.941532i	$0.609386\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.2179	−0.491874	−0.245937	−	0.969286i	$-0.579096\pi$
$617$	−12.2179	−0.491874	−0.245937	+	0.969286i	$0.579096\pi$
$618$	0	0
$619$	28.1276	1.13054	0.565272	−	0.824904i	$-0.308771\pi$
$619$	28.1276	1.13054	0.565272	+	0.824904i	$0.308771\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−39.9206	−1.59939
$624$	0	0
$625$	24.9553	0.998211
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.28806	−0.131104
$630$	0	0
$631$	20.1099	0.800563	0.400282	−	0.916392i	$-0.368912\pi$
$631$	20.1099	0.800563	0.400282	+	0.916392i	$0.368912\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.930222	−0.0369147
$636$	0	0
$637$	47.7420	1.89161
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.52727	−0.297309	−0.148655	−	0.988889i	$-0.547494\pi$
$641$	−7.52727	−0.297309	−0.148655	+	0.988889i	$0.547494\pi$
$642$	0	0
$643$	−18.0426	−0.711530	−0.355765	−	0.934575i	$-0.615780\pi$
$643$	−18.0426	−0.711530	−0.355765	+	0.934575i	$0.615780\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.6809	1.28482	0.642410	−	0.766361i	$-0.277935\pi$
$647$	32.6809	1.28482	0.642410	+	0.766361i	$0.277935\pi$
$648$	0	0
$649$	−8.15337	−0.320048
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.3658	1.11004	0.555020	−	0.831837i	$-0.312711\pi$
$653$	28.3658	1.11004	0.555020	+	0.831837i	$0.312711\pi$
$654$	0	0
$655$	0.173541	0.00678079
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.6475	1.46654	0.733269	−	0.679939i	$-0.237994\pi$
$659$	37.6475	1.46654	0.733269	+	0.679939i	$0.237994\pi$
$660$	0	0
$661$	32.2771	1.25543	0.627716	−	0.778442i	$-0.283990\pi$
$661$	32.2771	1.25543	0.627716	+	0.778442i	$0.283990\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.26936	−0.0492235
$666$	0	0
$667$	−2.77792	−0.107562
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.56940	−0.369423
$672$	0	0
$673$	−20.1631	−0.777229	−0.388615	−	0.921400i	$-0.627046\pi$
$673$	−20.1631	−0.777229	−0.388615	+	0.921400i	$0.627046\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.3795	1.09071	0.545357	−	0.838204i	$-0.316394\pi$
$677$	28.3795	1.09071	0.545357	+	0.838204i	$0.316394\pi$
$678$	0	0
$679$	61.1194	2.34555
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.9096	1.10620	0.553098	−	0.833116i	$-0.313446\pi$
$683$	28.9096	1.10620	0.553098	+	0.833116i	$0.313446\pi$
$684$	0	0
$685$	−0.828387	−0.0316510
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−49.2506	−1.87630
$690$	0	0
$691$	41.5412	1.58030	0.790151	−	0.612912i	$-0.210002\pi$
$691$	41.5412	1.58030	0.790151	+	0.612912i	$0.210002\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.28230	−0.0486406
$696$	0	0
$697$	−57.3717	−2.17311
$698$	0	0
$699$	0	0
$700$	0	0
$701$	45.5523	1.72049	0.860243	−	0.509884i	$-0.170312\pi$
$701$	45.5523	1.72049	0.860243	+	0.509884i	$0.170312\pi$
$702$	0	0
$703$	−2.86731	−0.108143
$704$	0	0
$705$	0	0
$706$	0	0
$707$	41.7196	1.56903
$708$	0	0
$709$	−44.8662	−1.68499	−0.842493	−	0.538707i	$-0.818913\pi$
$709$	−44.8662	−1.68499	−0.842493	+	0.538707i	$0.818913\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.56791	−0.0587187
$714$	0	0
$715$	−1.45169	−0.0542900
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.9351	0.407809	0.203905	−	0.978991i	$-0.434637\pi$
$719$	10.9351	0.407809	0.203905	+	0.978991i	$0.434637\pi$
$720$	0	0
$721$	56.4338	2.10170
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−28.1192	−1.04432
$726$	0	0
$727$	20.5083	0.760612	0.380306	−	0.924861i	$-0.375819\pi$
$727$	20.5083	0.760612	0.380306	+	0.924861i	$0.375819\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−51.7655	−1.91462
$732$	0	0
$733$	−18.5987	−0.686958	−0.343479	−	0.939160i	$-0.611606\pi$
$733$	−18.5987	−0.686958	−0.343479	+	0.939160i	$0.611606\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.2796	−1.41005
$738$	0	0
$739$	−47.9696	−1.76459	−0.882295	−	0.470696i	$-0.844003\pi$
$739$	−47.9696	−1.76459	−0.882295	+	0.470696i	$0.844003\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.99821	0.109993	0.0549967	−	0.998487i	$-0.482485\pi$
$743$	2.99821	0.109993	0.0549967	+	0.998487i	$0.482485\pi$
$744$	0	0
$745$	1.15918	0.0424691
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.3957	0.379849
$750$	0	0
$751$	15.4249	0.562861	0.281431	−	0.959582i	$-0.409191\pi$
$751$	15.4249	0.562861	0.281431	+	0.959582i	$0.409191\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0.665272	0.0242117
$756$	0	0
$757$	−33.1775	−1.20586	−0.602929	−	0.797795i	$-0.706000\pi$
$757$	−33.1775	−1.20586	−0.602929	+	0.797795i	$0.706000\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−40.3927	−1.46423	−0.732117	−	0.681179i	$-0.761468\pi$
$761$	−40.3927	−1.46423	−0.732117	+	0.681179i	$0.761468\pi$
$762$	0	0
$763$	−59.7275	−2.16228
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.9325	0.503073
$768$	0	0
$769$	1.39904	0.0504507	0.0252253	−	0.999682i	$-0.491970\pi$
$769$	1.39904	0.0504507	0.0252253	+	0.999682i	$0.491970\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−21.0215	−0.756092	−0.378046	−	0.925787i	$-0.623404\pi$
$773$	−21.0215	−0.756092	−0.378046	+	0.925787i	$0.623404\pi$
$774$	0	0
$775$	−15.8710	−0.570104
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−50.0302	−1.79252
$780$	0	0
$781$	−16.7218	−0.598352
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.834005	0.0297669
$786$	0	0
$787$	22.0428	0.785740	0.392870	−	0.919594i	$-0.371482\pi$
$787$	22.0428	0.785740	0.392870	+	0.919594i	$0.371482\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.9439	0.424676
$792$	0	0
$793$	16.3522	0.580683
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8.05010	−0.285149	−0.142575	−	0.989784i	$-0.545538\pi$
$797$	−8.05010	−0.285149	−0.142575	+	0.989784i	$0.545538\pi$
$798$	0	0
$799$	−69.7690	−2.46825
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.9006	−0.843434
$804$	0	0
$805$	−0.101170	−0.00356576
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.69624	−0.0596366	−0.0298183	−	0.999555i	$-0.509493\pi$
$809$	−1.69624	−0.0596366	−0.0298183	+	0.999555i	$0.509493\pi$
$810$	0	0
$811$	38.0615	1.33652	0.668259	−	0.743928i	$-0.267040\pi$
$811$	38.0615	1.33652	0.668259	+	0.743928i	$0.267040\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.657120	−0.0230179
$816$	0	0
$817$	−45.1414	−1.57930
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.00466	−0.0350628	−0.0175314	−	0.999846i	$-0.505581\pi$
$821$	−1.00466	−0.0350628	−0.0175314	+	0.999846i	$0.505581\pi$
$822$	0	0
$823$	14.6633	0.511132	0.255566	−	0.966792i	$-0.417738\pi$
$823$	14.6633	0.511132	0.255566	+	0.966792i	$0.417738\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.5311	0.922575	0.461288	−	0.887251i	$-0.347388\pi$
$827$	26.5311	0.922575	0.461288	+	0.887251i	$0.347388\pi$
$828$	0	0
$829$	−2.23493	−0.0776223	−0.0388112	−	0.999247i	$-0.512357\pi$
$829$	−2.23493	−0.0776223	−0.0388112	+	0.999247i	$0.512357\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−50.3127	−1.74323
$834$	0	0
$835$	0.0546093	0.00188983
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−52.6946	−1.81922	−0.909610	−	0.415462i	$-0.863620\pi$
$839$	−52.6946	−1.81922	−0.909610	+	0.415462i	$0.863620\pi$
$840$	0	0
$841$	2.66543	0.0919114
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.77072	0.0609147
$846$	0	0
$847$	17.1001	0.587567
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.228529	−0.00783388
$852$	0	0
$853$	19.1347	0.655160	0.327580	−	0.944824i	$-0.393767\pi$
$853$	19.1347	0.655160	0.327580	+	0.944824i	$0.393767\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.5050	−1.69106	−0.845529	−	0.533930i	$-0.820715\pi$
$857$	−49.5050	−1.69106	−0.845529	+	0.533930i	$0.820715\pi$
$858$	0	0
$859$	9.92447	0.338619	0.169309	−	0.985563i	$-0.445846\pi$
$859$	9.92447	0.338619	0.169309	+	0.985563i	$0.445846\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−50.1289	−1.70641	−0.853203	−	0.521578i	$-0.825343\pi$
$863$	−50.1289	−1.70641	−0.853203	+	0.521578i	$0.825343\pi$
$864$	0	0
$865$	−0.397892	−0.0135287
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.94735	−0.0660592
$870$	0	0
$871$	65.4123	2.21641
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.04877	−0.0692610
$876$	0	0
$877$	21.6006	0.729400	0.364700	−	0.931125i	$-0.381172\pi$
$877$	21.6006	0.729400	0.364700	+	0.931125i	$0.381172\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.44439	−0.149735	−0.0748677	−	0.997193i	$-0.523853\pi$
$881$	−4.44439	−0.149735	−0.0748677	+	0.997193i	$0.523853\pi$
$882$	0	0
$883$	34.5465	1.16258	0.581291	−	0.813696i	$-0.302548\pi$
$883$	34.5465	1.16258	0.581291	+	0.813696i	$0.302548\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.0973	−1.24561	−0.622803	−	0.782379i	$-0.714006\pi$
$887$	−37.0973	−1.24561	−0.622803	+	0.782379i	$0.714006\pi$
$888$	0	0
$889$	−63.9259	−2.14400
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−60.8410	−2.03597
$894$	0	0
$895$	0.107684	0.00359948
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.8725	0.596083
$900$	0	0
$901$	51.9025	1.72912
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.204862	0.00680983
$906$	0	0
$907$	−36.1679	−1.20094	−0.600468	−	0.799649i	$-0.705019\pi$
$907$	−36.1679	−1.20094	−0.600468	+	0.799649i	$0.705019\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.6331	−1.54502	−0.772512	−	0.635000i	$-0.781000\pi$
$911$	−46.6331	−1.54502	−0.772512	+	0.635000i	$0.781000\pi$
$912$	0	0
$913$	−62.7888	−2.07801
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.9259	0.393828
$918$	0	0
$919$	−0.354413	−0.0116910	−0.00584549	−	0.999983i	$-0.501861\pi$
$919$	−0.354413	−0.0116910	−0.00584549	+	0.999983i	$0.501861\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.5742	0.940530
$924$	0	0
$925$	−2.31326	−0.0760596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	49.5482	1.62562	0.812812	−	0.582526i	$-0.197935\pi$
$929$	49.5482	1.62562	0.812812	+	0.582526i	$0.197935\pi$
$930$	0	0
$931$	−43.8745	−1.43793
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.52986	0.0500316
$936$	0	0
$937$	−53.7320	−1.75535	−0.877674	−	0.479259i	$-0.840905\pi$
$937$	−53.7320	−1.75535	−0.877674	+	0.479259i	$0.840905\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34.9369	1.13891	0.569455	−	0.822023i	$-0.307154\pi$
$941$	34.9369	1.13891	0.569455	+	0.822023i	$0.307154\pi$
$942$	0	0
$943$	−3.98749	−0.129850
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.195497	0.00635280	0.00317640	−	0.999995i	$-0.498989\pi$
$947$	0.195497	0.00635280	0.00317640	+	0.999995i	$0.498989\pi$
$948$	0	0
$949$	40.8413	1.32577
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.9547	1.45622	0.728112	−	0.685458i	$-0.240398\pi$
$953$	44.9547	1.45622	0.728112	+	0.685458i	$0.240398\pi$
$954$	0	0
$955$	−0.259487	−0.00839681
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−56.9276	−1.83829
$960$	0	0
$961$	−20.9124	−0.674594
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.153105	−0.00492863
$966$	0	0
$967$	36.1803	1.16348	0.581741	−	0.813374i	$-0.302372\pi$
$967$	36.1803	1.16348	0.581741	+	0.813374i	$0.302372\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.9402	0.351088	0.175544	−	0.984472i	$-0.443832\pi$
$971$	10.9402	0.351088	0.175544	+	0.984472i	$0.443832\pi$
$972$	0	0
$973$	−88.1213	−2.82504
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.3160	−0.426018	−0.213009	−	0.977050i	$-0.568326\pi$
$977$	−13.3160	−0.426018	−0.213009	+	0.977050i	$0.568326\pi$
$978$	0	0
$979$	41.9565	1.34093
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.47777	−0.174714	−0.0873568	−	0.996177i	$-0.527842\pi$
$983$	−5.47777	−0.174714	−0.0873568	+	0.996177i	$0.527842\pi$
$984$	0	0
$985$	−0.847650	−0.0270084
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.59784	−0.114405
$990$	0	0
$991$	31.0728	0.987059	0.493530	−	0.869729i	$-0.335706\pi$
$991$	31.0728	0.987059	0.493530	+	0.869729i	$0.335706\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.154254	0.00489018
$996$	0	0
$997$	−43.7477	−1.38550	−0.692752	−	0.721175i	$-0.743602\pi$
$997$	−43.7477	−1.38550	−0.692752	+	0.721175i	$0.743602\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.i.1.5		9
3.2	odd	2		2004.2.a.c.1.5	✓	9
12.11	even	2		8016.2.a.bc.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.5	✓	9	3.2	odd	2
6012.2.a.i.1.5		9	1.1	even	1	trivial
8016.2.a.bc.1.5		9	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0546093 q^{5} +3.75281 q^{7} +O(q^{10})\)	\(q+0.0546093 q^{5} +3.75281 q^{7} -3.94419 q^{11} +6.73983 q^{13} -7.10275 q^{17} -6.19385 q^{19} -0.493660 q^{23} -4.99702 q^{25} +5.62720 q^{29} +3.17610 q^{31} +0.204938 q^{35} +0.462929 q^{37} +8.07740 q^{41} +7.28810 q^{43} +9.82281 q^{47} +7.08356 q^{49} -7.30739 q^{53} -0.215389 q^{55} +2.06719 q^{59} +2.42620 q^{61} +0.368057 q^{65} +9.70533 q^{67} +4.23960 q^{71} +6.05970 q^{73} -14.8018 q^{77} +0.493725 q^{79} +15.9193 q^{83} -0.387876 q^{85} -10.6375 q^{89} +25.2933 q^{91} -0.338242 q^{95} +16.2863 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^5 + 2 * q^7	\( 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) \) 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

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