LMFDB - Embedded newform 6003.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.37988$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +2.95969 q^{5} +1.80007 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +2.95969 q^{5} +1.80007 q^{7} +1.40826 q^{11} +1.56788 q^{13} +4.00000 q^{16} +4.88408 q^{17} +5.65189 q^{19} -5.91938 q^{20} -1.00000 q^{23} +3.75976 q^{25} -3.60014 q^{28} -1.00000 q^{29} +7.71944 q^{31} +5.32764 q^{35} +7.59512 q^{37} +7.12771 q^{41} -8.16301 q^{43} -2.81653 q^{44} -2.95969 q^{47} -3.75976 q^{49} -3.13577 q^{52} -7.51951 q^{53} +4.16802 q^{55} +2.88069 q^{59} -10.4873 q^{61} -8.00000 q^{64} +4.64045 q^{65} -3.60014 q^{67} -9.76815 q^{68} +1.79167 q^{71} -7.80007 q^{73} -11.3038 q^{76} +2.53497 q^{77} +9.17141 q^{79} +11.8388 q^{80} -1.74330 q^{83} +14.4553 q^{85} +15.9063 q^{89} +2.82230 q^{91} +2.00000 q^{92} +16.7278 q^{95} +1.35149 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7	$ 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7} + 16 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{25} + 4 q^{28} - 4 q^{29} + 2 q^{31} - 4 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{47} + 4 q^{49} + 8 q^{53} - 8 q^{55} + 22 q^{59} - 16 q^{61} - 32 q^{64} + 10 q^{65} + 4 q^{67} + 4 q^{68} + 22 q^{71} - 22 q^{73} - 8 q^{76} + 8 q^{77} - 20 q^{79} + 8 q^{80} + 10 q^{83} - 2 q^{85} + 14 q^{89} - 26 q^{91} + 8 q^{92} + 14 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7 + 16 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^23 - 4 * q^25 + 4 * q^28 - 4 * q^29 + 2 * q^31 - 4 * q^35 + 4 * q^37 - 6 * q^41 - 2 * q^47 + 4 * q^49 + 8 * q^53 - 8 * q^55 + 22 * q^59 - 16 * q^61 - 32 * q^64 + 10 * q^65 + 4 * q^67 + 4 * q^68 + 22 * q^71 - 22 * q^73 - 8 * q^76 + 8 * q^77 - 20 * q^79 + 8 * q^80 + 10 * q^83 - 2 * q^85 + 14 * q^89 - 26 * q^91 + 8 * q^92 + 14 * q^95 - 8 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	2.95969	1.32361	0.661806	−	0.749675i	$-0.269790\pi$
$5$	2.95969	1.32361	0.661806	+	0.749675i	$0.269790\pi$
$6$	0	0
$7$	1.80007	0.680362	0.340181	−	0.940360i	$-0.389512\pi$
$7$	1.80007	0.680362	0.340181	+	0.940360i	$0.389512\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.40826	0.424607	0.212304	−	0.977204i	$-0.431903\pi$
$11$	1.40826	0.424607	0.212304	+	0.977204i	$0.431903\pi$
$12$	0	0
$13$	1.56788	0.434853	0.217426	−	0.976077i	$-0.430234\pi$
$13$	1.56788	0.434853	0.217426	+	0.976077i	$0.430234\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	4.88408	1.18456	0.592281	−	0.805731i	$-0.298227\pi$
$17$	4.88408	1.18456	0.592281	+	0.805731i	$0.298227\pi$
$18$	0	0
$19$	5.65189	1.29663	0.648317	−	0.761371i	$-0.275473\pi$
$19$	5.65189	1.29663	0.648317	+	0.761371i	$0.275473\pi$
$20$	−5.91938	−1.32361
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.75976	0.751951
$26$	0	0
$27$	0	0
$28$	−3.60014	−0.680362
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.71944	1.38645	0.693227	−	0.720720i	$-0.256189\pi$
$31$	7.71944	1.38645	0.693227	+	0.720720i	$0.256189\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.32764	0.900535
$36$	0	0
$37$	7.59512	1.24863	0.624315	−	0.781172i	$-0.285378\pi$
$37$	7.59512	1.24863	0.624315	+	0.781172i	$0.285378\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.12771	1.11316	0.556580	−	0.830794i	$-0.312113\pi$
$41$	7.12771	1.11316	0.556580	+	0.830794i	$0.312113\pi$
$42$	0	0
$43$	−8.16301	−1.24485	−0.622423	−	0.782681i	$-0.713852\pi$
$43$	−8.16301	−1.24485	−0.622423	+	0.782681i	$0.713852\pi$
$44$	−2.81653	−0.424607
$45$	0	0
$46$	0	0
$47$	−2.95969	−0.431715	−0.215857	−	0.976425i	$-0.569255\pi$
$47$	−2.95969	−0.431715	−0.215857	+	0.976425i	$0.569255\pi$
$48$	0	0
$49$	−3.75976	−0.537108
$50$	0	0
$51$	0	0
$52$	−3.13577	−0.434853
$53$	−7.51951	−1.03288	−0.516442	−	0.856322i	$-0.672744\pi$
$53$	−7.51951	−1.03288	−0.516442	+	0.856322i	$0.672744\pi$
$54$	0	0
$55$	4.16802	0.562016
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.88069	0.375034	0.187517	−	0.982261i	$-0.439956\pi$
$59$	2.88069	0.375034	0.187517	+	0.982261i	$0.439956\pi$
$60$	0	0
$61$	−10.4873	−1.34276	−0.671378	−	0.741115i	$-0.734297\pi$
$61$	−10.4873	−1.34276	−0.671378	+	0.741115i	$0.734297\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	4.64045	0.575577
$66$	0	0
$67$	−3.60014	−0.439827	−0.219913	−	0.975519i	$-0.570577\pi$
$67$	−3.60014	−0.439827	−0.219913	+	0.975519i	$0.570577\pi$
$68$	−9.76815	−1.18456
$69$	0	0
$70$	0	0
$71$	1.79167	0.212632	0.106316	−	0.994332i	$-0.466094\pi$
$71$	1.79167	0.212632	0.106316	+	0.994332i	$0.466094\pi$
$72$	0	0
$73$	−7.80007	−0.912929	−0.456464	−	0.889742i	$-0.650884\pi$
$73$	−7.80007	−0.912929	−0.456464	+	0.889742i	$0.650884\pi$
$74$	0	0
$75$	0	0
$76$	−11.3038	−1.29663
$77$	2.53497	0.288886
$78$	0	0
$79$	9.17141	1.03186	0.515932	−	0.856630i	$-0.327446\pi$
$79$	9.17141	1.03186	0.515932	+	0.856630i	$0.327446\pi$
$80$	11.8388	1.32361
$81$	0	0
$82$	0	0
$83$	−1.74330	−0.191352	−0.0956759	−	0.995413i	$-0.530501\pi$
$83$	−1.74330	−0.191352	−0.0956759	+	0.995413i	$0.530501\pi$
$84$	0	0
$85$	14.4553	1.56790
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.9063	1.68607	0.843033	−	0.537863i	$-0.180768\pi$
$89$	15.9063	1.68607	0.843033	+	0.537863i	$0.180768\pi$
$90$	0	0
$91$	2.82230	0.295857
$92$	2.00000	0.208514
$93$	0	0
$94$	0	0
$95$	16.7278	1.71624
$96$	0	0
$97$	1.35149	0.137223	0.0686117	−	0.997643i	$-0.478143\pi$
$97$	1.35149	0.137223	0.0686117	+	0.997643i	$0.478143\pi$
$98$	0	0
$99$	0	0
$100$	−7.51951	−0.751951
$101$	−3.20833	−0.319241	−0.159620	−	0.987178i	$-0.551027\pi$
$101$	−3.20833	−0.319241	−0.159620	+	0.987178i	$0.551027\pi$
$102$	0	0
$103$	14.6553	1.44403	0.722014	−	0.691879i	$-0.243217\pi$
$103$	14.6553	1.44403	0.722014	+	0.691879i	$0.243217\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.5821	1.69972	0.849861	−	0.527008i	$-0.176686\pi$
$107$	17.5821	1.69972	0.849861	+	0.527008i	$0.176686\pi$
$108$	0	0
$109$	−18.1032	−1.73397	−0.866986	−	0.498333i	$-0.833946\pi$
$109$	−18.1032	−1.73397	−0.866986	+	0.498333i	$0.833946\pi$
$110$	0	0
$111$	0	0
$112$	7.20027	0.680362
$113$	8.16802	0.768383	0.384191	−	0.923254i	$-0.374480\pi$
$113$	8.16802	0.768383	0.384191	+	0.923254i	$0.374480\pi$
$114$	0	0
$115$	−2.95969	−0.275992
$116$	2.00000	0.185695
$117$	0	0
$118$	0	0
$119$	8.79167	0.805931
$120$	0	0
$121$	−9.01680	−0.819709
$122$	0	0
$123$	0	0
$124$	−15.4389	−1.38645
$125$	−3.67074	−0.328321
$126$	0	0
$127$	−21.8875	−1.94220	−0.971099	−	0.238676i	$-0.923286\pi$
$127$	−21.8875	−1.94220	−0.971099	+	0.238676i	$0.923286\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.6224	−1.45230	−0.726151	−	0.687535i	$-0.758693\pi$
$131$	−16.6224	−1.45230	−0.726151	+	0.687535i	$0.758693\pi$
$132$	0	0
$133$	10.1738	0.882179
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.85183	−0.499955	−0.249978	−	0.968252i	$-0.580423\pi$
$137$	−5.85183	−0.499955	−0.249978	+	0.968252i	$0.580423\pi$
$138$	0	0
$139$	−3.57628	−0.303336	−0.151668	−	0.988431i	$-0.548464\pi$
$139$	−3.57628	−0.303336	−0.151668	+	0.988431i	$0.548464\pi$
$140$	−10.6553	−0.900535
$141$	0	0
$142$	0	0
$143$	2.20799	0.184642
$144$	0	0
$145$	−2.95969	−0.245789
$146$	0	0
$147$	0	0
$148$	−15.1902	−1.24863
$149$	−2.08740	−0.171006	−0.0855031	−	0.996338i	$-0.527250\pi$
$149$	−2.08740	−0.171006	−0.0855031	+	0.996338i	$0.527250\pi$
$150$	0	0
$151$	−9.51111	−0.774004	−0.387002	−	0.922079i	$-0.626489\pi$
$151$	−9.51111	−0.774004	−0.387002	+	0.922079i	$0.626489\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	22.8472	1.83513
$156$	0	0
$157$	18.0924	1.44393	0.721966	−	0.691929i	$-0.243239\pi$
$157$	18.0924	1.44393	0.721966	+	0.691929i	$0.243239\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.80007	−0.141865
$162$	0	0
$163$	−4.07060	−0.318834	−0.159417	−	0.987211i	$-0.550961\pi$
$163$	−4.07060	−0.318834	−0.159417	+	0.987211i	$0.550961\pi$
$164$	−14.2554	−1.11316
$165$	0	0
$166$	0	0
$167$	−1.38178	−0.106925	−0.0534627	−	0.998570i	$-0.517026\pi$
$167$	−1.38178	−0.106925	−0.0534627	+	0.998570i	$0.517026\pi$
$168$	0	0
$169$	−10.5417	−0.810903
$170$	0	0
$171$	0	0
$172$	16.3260	1.24485
$173$	14.4231	1.09657	0.548284	−	0.836292i	$-0.315281\pi$
$173$	14.4231	1.09657	0.548284	+	0.836292i	$0.315281\pi$
$174$	0	0
$175$	6.76782	0.511599
$176$	5.63305	0.424607
$177$	0	0
$178$	0	0
$179$	9.12933	0.682358	0.341179	−	0.939998i	$-0.389174\pi$
$179$	9.12933	0.682358	0.341179	+	0.939998i	$0.389174\pi$
$180$	0	0
$181$	3.68076	0.273589	0.136794	−	0.990599i	$-0.456320\pi$
$181$	3.68076	0.273589	0.136794	+	0.990599i	$0.456320\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	22.4792	1.65270
$186$	0	0
$187$	6.87806	0.502974
$188$	5.91938	0.431715
$189$	0	0
$190$	0	0
$191$	−14.1311	−1.02249	−0.511245	−	0.859435i	$-0.670816\pi$
$191$	−14.1311	−1.02249	−0.511245	+	0.859435i	$0.670816\pi$
$192$	0	0
$193$	−17.0390	−1.22650	−0.613248	−	0.789890i	$-0.710137\pi$
$193$	−17.0390	−1.22650	−0.613248	+	0.789890i	$0.710137\pi$
$194$	0	0
$195$	0	0
$196$	7.51951	0.537108
$197$	11.8652	0.845363	0.422681	−	0.906278i	$-0.361089\pi$
$197$	11.8652	0.845363	0.422681	+	0.906278i	$0.361089\pi$
$198$	0	0
$199$	−17.5037	−1.24081	−0.620403	−	0.784283i	$-0.713031\pi$
$199$	−17.5037	−1.24081	−0.620403	+	0.784283i	$0.713031\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.80007	−0.126340
$204$	0	0
$205$	21.0958	1.47339
$206$	0	0
$207$	0	0
$208$	6.27153	0.434853
$209$	7.95935	0.550560
$210$	0	0
$211$	10.9197	0.751744	0.375872	−	0.926672i	$-0.377343\pi$
$211$	10.9197	0.751744	0.375872	+	0.926672i	$0.377343\pi$
$212$	15.0390	1.03288
$213$	0	0
$214$	0	0
$215$	−24.1600	−1.64770
$216$	0	0
$217$	13.8955	0.943290
$218$	0	0
$219$	0	0
$220$	−8.33604	−0.562016
$221$	7.65766	0.515110
$222$	0	0
$223$	10.5434	0.706036	0.353018	−	0.935617i	$-0.385155\pi$
$223$	10.5434	0.706036	0.353018	+	0.935617i	$0.385155\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.8932	−1.45311	−0.726553	−	0.687111i	$-0.758879\pi$
$227$	−21.8932	−1.45311	−0.726553	+	0.687111i	$0.758879\pi$
$228$	0	0
$229$	25.4678	1.68296	0.841478	−	0.540291i	$-0.181686\pi$
$229$	25.4678	1.68296	0.841478	+	0.540291i	$0.181686\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.02486	−0.198165	−0.0990824	−	0.995079i	$-0.531591\pi$
$233$	−3.02486	−0.198165	−0.0990824	+	0.995079i	$0.531591\pi$
$234$	0	0
$235$	−8.75976	−0.571423
$236$	−5.76138	−0.375034
$237$	0	0
$238$	0	0
$239$	−15.2373	−0.985621	−0.492811	−	0.870137i	$-0.664030\pi$
$239$	−15.2373	−0.985621	−0.492811	+	0.870137i	$0.664030\pi$
$240$	0	0
$241$	3.70365	0.238573	0.119287	−	0.992860i	$-0.461939\pi$
$241$	3.70365	0.238573	0.119287	+	0.992860i	$0.461939\pi$
$242$	0	0
$243$	0	0
$244$	20.9745	1.34276
$245$	−11.1277	−0.710923
$246$	0	0
$247$	8.86151	0.563844
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.30780	0.461264	0.230632	−	0.973041i	$-0.425921\pi$
$251$	7.30780	0.461264	0.230632	+	0.973041i	$0.425921\pi$
$252$	0	0
$253$	−1.40826	−0.0885367
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	21.8314	1.36180	0.680901	−	0.732375i	$-0.261588\pi$
$257$	21.8314	1.36180	0.680901	+	0.732375i	$0.261588\pi$
$258$	0	0
$259$	13.6717	0.849520
$260$	−9.28089	−0.575577
$261$	0	0
$262$	0	0
$263$	−8.34987	−0.514875	−0.257437	−	0.966295i	$-0.582878\pi$
$263$	−8.34987	−0.514875	−0.257437	+	0.966295i	$0.582878\pi$
$264$	0	0
$265$	−22.2554	−1.36714
$266$	0	0
$267$	0	0
$268$	7.20027	0.439827
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−28.1319	−1.70889	−0.854444	−	0.519543i	$-0.826102\pi$
$271$	−28.1319	−1.70889	−0.854444	+	0.519543i	$0.826102\pi$
$272$	19.5363	1.18456
$273$	0	0
$274$	0	0
$275$	5.29472	0.319284
$276$	0	0
$277$	32.0958	1.92845	0.964225	−	0.265086i	$-0.0854005\pi$
$277$	32.0958	1.92845	0.964225	+	0.265086i	$0.0854005\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.59011	0.572098	0.286049	−	0.958215i	$-0.407658\pi$
$281$	9.59011	0.572098	0.286049	+	0.958215i	$0.407658\pi$
$282$	0	0
$283$	−17.3196	−1.02954	−0.514771	−	0.857328i	$-0.672123\pi$
$283$	−17.3196	−1.02954	−0.514771	+	0.857328i	$0.672123\pi$
$284$	−3.58334	−0.212632
$285$	0	0
$286$	0	0
$287$	12.8304	0.757352
$288$	0	0
$289$	6.85421	0.403189
$290$	0	0
$291$	0	0
$292$	15.6001	0.912929
$293$	−13.3007	−0.777037	−0.388519	−	0.921441i	$-0.627013\pi$
$293$	−13.3007	−0.777037	−0.388519	+	0.921441i	$0.627013\pi$
$294$	0	0
$295$	8.52595	0.496400
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.56788	−0.0906730
$300$	0	0
$301$	−14.6940	−0.846946
$302$	0	0
$303$	0	0
$304$	22.6076	1.29663
$305$	−31.0390	−1.77729
$306$	0	0
$307$	−4.75779	−0.271542	−0.135771	−	0.990740i	$-0.543351\pi$
$307$	−4.75779	−0.271542	−0.135771	+	0.990740i	$0.543351\pi$
$308$	−5.06994	−0.288886
$309$	0	0
$310$	0	0
$311$	9.19025	0.521131	0.260566	−	0.965456i	$-0.416091\pi$
$311$	9.19025	0.521131	0.260566	+	0.965456i	$0.416091\pi$
$312$	0	0
$313$	−12.5427	−0.708958	−0.354479	−	0.935064i	$-0.615342\pi$
$313$	−12.5427	−0.708958	−0.354479	+	0.935064i	$0.615342\pi$
$314$	0	0
$315$	0	0
$316$	−18.3428	−1.03186
$317$	32.3609	1.81757	0.908784	−	0.417266i	$-0.137012\pi$
$317$	32.3609	1.81757	0.908784	+	0.417266i	$0.137012\pi$
$318$	0	0
$319$	−1.40826	−0.0788476
$320$	−23.6775	−1.32361
$321$	0	0
$322$	0	0
$323$	27.6043	1.53594
$324$	0	0
$325$	5.89486	0.326988
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.32764	−0.293722
$330$	0	0
$331$	29.5363	1.62346	0.811731	−	0.584031i	$-0.198525\pi$
$331$	29.5363	1.62346	0.811731	+	0.584031i	$0.198525\pi$
$332$	3.48660	0.191352
$333$	0	0
$334$	0	0
$335$	−10.6553	−0.582160
$336$	0	0
$337$	27.4836	1.49713	0.748563	−	0.663063i	$-0.230744\pi$
$337$	27.4836	1.49713	0.748563	+	0.663063i	$0.230744\pi$
$338$	0	0
$339$	0	0
$340$	−28.9107	−1.56790
$341$	10.8710	0.588698
$342$	0	0
$343$	−19.3683	−1.04579
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.19993	0.0644157	0.0322079	−	0.999481i	$-0.489746\pi$
$347$	1.19993	0.0644157	0.0322079	+	0.999481i	$0.489746\pi$
$348$	0	0
$349$	−3.71911	−0.199079	−0.0995396	−	0.995034i	$-0.531737\pi$
$349$	−3.71911	−0.199079	−0.0995396	+	0.995034i	$0.531737\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.39214	−0.286995	−0.143497	−	0.989651i	$-0.545835\pi$
$353$	−5.39214	−0.286995	−0.143497	+	0.989651i	$0.545835\pi$
$354$	0	0
$355$	5.30278	0.281443
$356$	−31.8126	−1.68607
$357$	0	0
$358$	0	0
$359$	27.1465	1.43274	0.716370	−	0.697721i	$-0.245802\pi$
$359$	27.1465	1.43274	0.716370	+	0.697721i	$0.245802\pi$
$360$	0	0
$361$	12.9439	0.681258
$362$	0	0
$363$	0	0
$364$	−5.64459	−0.295857
$365$	−23.0858	−1.20836
$366$	0	0
$367$	−10.7359	−0.560410	−0.280205	−	0.959940i	$-0.590402\pi$
$367$	−10.7359	−0.560410	−0.280205	+	0.959940i	$0.590402\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−13.5356	−0.702735
$372$	0	0
$373$	3.30278	0.171012	0.0855058	−	0.996338i	$-0.472749\pi$
$373$	3.30278	0.171012	0.0855058	+	0.996338i	$0.472749\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.56788	−0.0807501
$378$	0	0
$379$	−12.5967	−0.647052	−0.323526	−	0.946219i	$-0.604868\pi$
$379$	−12.5967	−0.647052	−0.323526	+	0.946219i	$0.604868\pi$
$380$	−33.4557	−1.71624
$381$	0	0
$382$	0	0
$383$	35.7420	1.82633	0.913166	−	0.407588i	$-0.133630\pi$
$383$	35.7420	1.82633	0.913166	+	0.407588i	$0.133630\pi$
$384$	0	0
$385$	7.50272	0.382374
$386$	0	0
$387$	0	0
$388$	−2.70299	−0.137223
$389$	4.50807	0.228568	0.114284	−	0.993448i	$-0.463543\pi$
$389$	4.50807	0.228568	0.114284	+	0.993448i	$0.463543\pi$
$390$	0	0
$391$	−4.88408	−0.246998
$392$	0	0
$393$	0	0
$394$	0	0
$395$	27.1445	1.36579
$396$	0	0
$397$	−3.38212	−0.169744	−0.0848719	−	0.996392i	$-0.527048\pi$
$397$	−3.38212	−0.169744	−0.0848719	+	0.996392i	$0.527048\pi$
$398$	0	0
$399$	0	0
$400$	15.0390	0.751951
$401$	6.15190	0.307211	0.153606	−	0.988132i	$-0.450911\pi$
$401$	6.15190	0.307211	0.153606	+	0.988132i	$0.450911\pi$
$402$	0	0
$403$	12.1032	0.602903
$404$	6.41666	0.319241
$405$	0	0
$406$	0	0
$407$	10.6959	0.530178
$408$	0	0
$409$	9.43789	0.466674	0.233337	−	0.972396i	$-0.425036\pi$
$409$	9.43789	0.466674	0.233337	+	0.972396i	$0.425036\pi$
$410$	0	0
$411$	0	0
$412$	−29.3106	−1.44403
$413$	5.18544	0.255159
$414$	0	0
$415$	−5.15962	−0.253276
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.3260	1.09070	0.545349	−	0.838209i	$-0.316397\pi$
$419$	22.3260	1.09070	0.545349	+	0.838209i	$0.316397\pi$
$420$	0	0
$421$	−24.4094	−1.18964	−0.594820	−	0.803859i	$-0.702777\pi$
$421$	−24.4094	−1.18964	−0.594820	+	0.803859i	$0.702777\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.3629	0.890733
$426$	0	0
$427$	−18.8778	−0.913560
$428$	−35.1641	−1.69972
$429$	0	0
$430$	0	0
$431$	−36.9752	−1.78103	−0.890516	−	0.454951i	$-0.849657\pi$
$431$	−36.9752	−1.78103	−0.890516	+	0.454951i	$0.849657\pi$
$432$	0	0
$433$	−34.9446	−1.67933	−0.839664	−	0.543106i	$-0.817248\pi$
$433$	−34.9446	−1.67933	−0.839664	+	0.543106i	$0.817248\pi$
$434$	0	0
$435$	0	0
$436$	36.2064	1.73397
$437$	−5.65189	−0.270367
$438$	0	0
$439$	−6.81490	−0.325257	−0.162629	−	0.986687i	$-0.551997\pi$
$439$	−6.81490	−0.325257	−0.162629	+	0.986687i	$0.551997\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.7669	1.22422	0.612110	−	0.790772i	$-0.290321\pi$
$443$	25.7669	1.22422	0.612110	+	0.790772i	$0.290321\pi$
$444$	0	0
$445$	47.0777	2.23170
$446$	0	0
$447$	0	0
$448$	−14.4005	−0.680362
$449$	32.8133	1.54855	0.774277	−	0.632847i	$-0.218114\pi$
$449$	32.8133	1.54855	0.774277	+	0.632847i	$0.218114\pi$
$450$	0	0
$451$	10.0377	0.472656
$452$	−16.3360	−0.768383
$453$	0	0
$454$	0	0
$455$	8.35312	0.391600
$456$	0	0
$457$	6.36795	0.297880	0.148940	−	0.988846i	$-0.452414\pi$
$457$	6.36795	0.297880	0.148940	+	0.988846i	$0.452414\pi$
$458$	0	0
$459$	0	0
$460$	5.91938	0.275992
$461$	33.1970	1.54614	0.773070	−	0.634321i	$-0.218720\pi$
$461$	33.1970	1.54614	0.773070	+	0.634321i	$0.218720\pi$
$462$	0	0
$463$	−12.9332	−0.601057	−0.300529	−	0.953773i	$-0.597163\pi$
$463$	−12.9332	−0.601057	−0.300529	+	0.953773i	$0.597163\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	0	0
$467$	−2.47648	−0.114598	−0.0572989	−	0.998357i	$-0.518249\pi$
$467$	−2.47648	−0.114598	−0.0572989	+	0.998357i	$0.518249\pi$
$468$	0	0
$469$	−6.48049	−0.299241
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.4957	−0.528571
$474$	0	0
$475$	21.2497	0.975005
$476$	−17.5833	−0.805931
$477$	0	0
$478$	0	0
$479$	17.6192	0.805043	0.402521	−	0.915411i	$-0.368134\pi$
$479$	17.6192	0.805043	0.402521	+	0.915411i	$0.368134\pi$
$480$	0	0
$481$	11.9083	0.542970
$482$	0	0
$483$	0	0
$484$	18.0336	0.819709
$485$	4.00000	0.181631
$486$	0	0
$487$	−0.0715614	−0.00324276	−0.00162138	−	0.999999i	$-0.500516\pi$
$487$	−0.0715614	−0.00324276	−0.00162138	+	0.999999i	$0.500516\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.2306	−0.822734	−0.411367	−	0.911470i	$-0.634948\pi$
$491$	−18.2306	−0.822734	−0.411367	+	0.911470i	$0.634948\pi$
$492$	0	0
$493$	−4.88408	−0.219968
$494$	0	0
$495$	0	0
$496$	30.8778	1.38645
$497$	3.22513	0.144667
$498$	0	0
$499$	1.28929	0.0577166	0.0288583	−	0.999584i	$-0.490813\pi$
$499$	1.28929	0.0577166	0.0288583	+	0.999584i	$0.490813\pi$
$500$	7.34147	0.328321
$501$	0	0
$502$	0	0
$503$	41.4188	1.84677	0.923386	−	0.383874i	$-0.125410\pi$
$503$	41.4188	1.84677	0.923386	+	0.383874i	$0.125410\pi$
$504$	0	0
$505$	−9.49566	−0.422551
$506$	0	0
$507$	0	0
$508$	43.7749	1.94220
$509$	16.4698	0.730013	0.365006	−	0.931005i	$-0.381067\pi$
$509$	16.4698	0.730013	0.365006	+	0.931005i	$0.381067\pi$
$510$	0	0
$511$	−14.0407	−0.621122
$512$	0	0
$513$	0	0
$514$	0	0
$515$	43.3751	1.91133
$516$	0	0
$517$	−4.16802	−0.183309
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.7898	−1.48036	−0.740178	−	0.672411i	$-0.765259\pi$
$521$	−33.7898	−1.48036	−0.740178	+	0.672411i	$0.765259\pi$
$522$	0	0
$523$	36.6940	1.60452	0.802258	−	0.596978i	$-0.203632\pi$
$523$	36.6940	1.60452	0.802258	+	0.596978i	$0.203632\pi$
$524$	33.2447	1.45230
$525$	0	0
$526$	0	0
$527$	37.7024	1.64234
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−20.3476	−0.882179
$533$	11.1754	0.484061
$534$	0	0
$535$	52.0374	2.24977
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.29472	−0.228060
$540$	0	0
$541$	−7.51884	−0.323260	−0.161630	−	0.986851i	$-0.551675\pi$
$541$	−7.51884	−0.323260	−0.161630	+	0.986851i	$0.551675\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−53.5798	−2.29511
$546$	0	0
$547$	−34.7182	−1.48444	−0.742221	−	0.670156i	$-0.766227\pi$
$547$	−34.7182	−1.48444	−0.742221	+	0.670156i	$0.766227\pi$
$548$	11.7037	0.499955
$549$	0	0
$550$	0	0
$551$	−5.65189	−0.240779
$552$	0	0
$553$	16.5092	0.702041
$554$	0	0
$555$	0	0
$556$	7.15256	0.303336
$557$	−2.62433	−0.111196	−0.0555982	−	0.998453i	$-0.517707\pi$
$557$	−2.62433	−0.111196	−0.0555982	+	0.998453i	$0.517707\pi$
$558$	0	0
$559$	−12.7986	−0.541325
$560$	21.3106	0.900535
$561$	0	0
$562$	0	0
$563$	−39.9705	−1.68456	−0.842278	−	0.539043i	$-0.818786\pi$
$563$	−39.9705	−1.68456	−0.842278	+	0.539043i	$0.818786\pi$
$564$	0	0
$565$	24.1748	1.01704
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.4850	0.523397	0.261699	−	0.965150i	$-0.415717\pi$
$569$	12.4850	0.523397	0.261699	+	0.965150i	$0.415717\pi$
$570$	0	0
$571$	4.58368	0.191821	0.0959105	−	0.995390i	$-0.469424\pi$
$571$	4.58368	0.191821	0.0959105	+	0.995390i	$0.469424\pi$
$572$	−4.41598	−0.184642
$573$	0	0
$574$	0	0
$575$	−3.75976	−0.156793
$576$	0	0
$577$	6.00902	0.250159	0.125079	−	0.992147i	$-0.460081\pi$
$577$	6.00902	0.250159	0.125079	+	0.992147i	$0.460081\pi$
$578$	0	0
$579$	0	0
$580$	5.91938	0.245789
$581$	−3.13806	−0.130188
$582$	0	0
$583$	−10.5894	−0.438570
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.7837	−0.486363	−0.243182	−	0.969981i	$-0.578191\pi$
$587$	−11.7837	−0.486363	−0.243182	+	0.969981i	$0.578191\pi$
$588$	0	0
$589$	43.6295	1.79772
$590$	0	0
$591$	0	0
$592$	30.3805	1.24863
$593$	−29.8559	−1.22603	−0.613017	−	0.790070i	$-0.710044\pi$
$593$	−29.8559	−1.22603	−0.613017	+	0.790070i	$0.710044\pi$
$594$	0	0
$595$	26.0206	1.06674
$596$	4.17479	0.171006
$597$	0	0
$598$	0	0
$599$	0.865519	0.0353642	0.0176821	−	0.999844i	$-0.494371\pi$
$599$	0.865519	0.0353642	0.0176821	+	0.999844i	$0.494371\pi$
$600$	0	0
$601$	39.3321	1.60439	0.802195	−	0.597062i	$-0.203665\pi$
$601$	39.3321	1.60439	0.802195	+	0.597062i	$0.203665\pi$
$602$	0	0
$603$	0	0
$604$	19.0222	0.774004
$605$	−26.6869	−1.08498
$606$	0	0
$607$	3.35250	0.136074	0.0680368	−	0.997683i	$-0.478326\pi$
$607$	3.35250	0.136074	0.0680368	+	0.997683i	$0.478326\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.64045	−0.187732
$612$	0	0
$613$	10.6930	0.431885	0.215942	−	0.976406i	$-0.430718\pi$
$613$	10.6930	0.431885	0.215942	+	0.976406i	$0.430718\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.0245	−0.564606	−0.282303	−	0.959325i	$-0.591098\pi$
$617$	−14.0245	−0.564606	−0.282303	+	0.959325i	$0.591098\pi$
$618$	0	0
$619$	−21.1408	−0.849720	−0.424860	−	0.905259i	$-0.639677\pi$
$619$	−21.1408	−0.849720	−0.424860	+	0.905259i	$0.639677\pi$
$620$	−45.6943	−1.83513
$621$	0	0
$622$	0	0
$623$	28.6324	1.14713
$624$	0	0
$625$	−29.6630	−1.18652
$626$	0	0
$627$	0	0
$628$	−36.1848	−1.44393
$629$	37.0952	1.47908
$630$	0	0
$631$	−39.9262	−1.58943	−0.794717	−	0.606980i	$-0.792381\pi$
$631$	−39.9262	−1.58943	−0.794717	+	0.606980i	$0.792381\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−64.7801	−2.57072
$636$	0	0
$637$	−5.89486	−0.233563
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.2265	−0.640909	−0.320454	−	0.947264i	$-0.603836\pi$
$641$	−16.2265	−0.640909	−0.320454	+	0.947264i	$0.603836\pi$
$642$	0	0
$643$	−45.1332	−1.77988	−0.889940	−	0.456078i	$-0.849254\pi$
$643$	−45.1332	−1.77988	−0.889940	+	0.456078i	$0.849254\pi$
$644$	3.60014	0.141865
$645$	0	0
$646$	0	0
$647$	−9.90196	−0.389286	−0.194643	−	0.980874i	$-0.562355\pi$
$647$	−9.90196	−0.389286	−0.194643	+	0.980874i	$0.562355\pi$
$648$	0	0
$649$	4.05677	0.159242
$650$	0	0
$651$	0	0
$652$	8.14120	0.318834
$653$	−18.8730	−0.738556	−0.369278	−	0.929319i	$-0.620395\pi$
$653$	−18.8730	−0.738556	−0.369278	+	0.929319i	$0.620395\pi$
$654$	0	0
$655$	−49.1970	−1.92229
$656$	28.5108	1.11316
$657$	0	0
$658$	0	0
$659$	−1.95534	−0.0761692	−0.0380846	−	0.999275i	$-0.512126\pi$
$659$	−1.95534	−0.0761692	−0.0380846	+	0.999275i	$0.512126\pi$
$660$	0	0
$661$	31.8230	1.23777	0.618885	−	0.785482i	$-0.287585\pi$
$661$	31.8230	1.23777	0.618885	+	0.785482i	$0.287585\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	30.1112	1.16766
$666$	0	0
$667$	1.00000	0.0387202
$668$	2.76356	0.106925
$669$	0	0
$670$	0	0
$671$	−14.7688	−0.570144
$672$	0	0
$673$	−16.5833	−0.639241	−0.319620	−	0.947546i	$-0.603555\pi$
$673$	−16.5833	−0.639241	−0.319620	+	0.947546i	$0.603555\pi$
$674$	0	0
$675$	0	0
$676$	21.0835	0.810903
$677$	−34.6153	−1.33037	−0.665187	−	0.746677i	$-0.731648\pi$
$677$	−34.6153	−1.33037	−0.665187	+	0.746677i	$0.731648\pi$
$678$	0	0
$679$	2.43278	0.0933615
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.5660	1.05478	0.527391	−	0.849622i	$-0.323170\pi$
$683$	27.5660	1.05478	0.527391	+	0.849622i	$0.323170\pi$
$684$	0	0
$685$	−17.3196	−0.661747
$686$	0	0
$687$	0	0
$688$	−32.6520	−1.24485
$689$	−11.7897	−0.449153
$690$	0	0
$691$	24.4211	0.929024	0.464512	−	0.885567i	$-0.346230\pi$
$691$	24.4211	0.929024	0.464512	+	0.885567i	$0.346230\pi$
$692$	−28.8462	−1.09657
$693$	0	0
$694$	0	0
$695$	−10.5847	−0.401500
$696$	0	0
$697$	34.8123	1.31861
$698$	0	0
$699$	0	0
$700$	−13.5356	−0.511599
$701$	30.8811	1.16636	0.583181	−	0.812342i	$-0.301808\pi$
$701$	30.8811	1.16636	0.583181	+	0.812342i	$0.301808\pi$
$702$	0	0
$703$	42.9268	1.61902
$704$	−11.2661	−0.424607
$705$	0	0
$706$	0	0
$707$	−5.77521	−0.217199
$708$	0	0
$709$	−11.1980	−0.420551	−0.210275	−	0.977642i	$-0.567436\pi$
$709$	−11.1980	−0.420551	−0.210275	+	0.977642i	$0.567436\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.71944	−0.289095
$714$	0	0
$715$	6.53497	0.244394
$716$	−18.2587	−0.682358
$717$	0	0
$718$	0	0
$719$	3.74845	0.139793	0.0698967	−	0.997554i	$-0.477733\pi$
$719$	3.74845	0.139793	0.0698967	+	0.997554i	$0.477733\pi$
$720$	0	0
$721$	26.3805	0.982461
$722$	0	0
$723$	0	0
$724$	−7.36152	−0.273589
$725$	−3.75976	−0.139634
$726$	0	0
$727$	14.7321	0.546383	0.273192	−	0.961960i	$-0.411921\pi$
$727$	14.7321	0.546383	0.273192	+	0.961960i	$0.411921\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−39.8688	−1.47460
$732$	0	0
$733$	13.3038	0.491387	0.245693	−	0.969348i	$-0.420984\pi$
$733$	13.3038	0.491387	0.245693	+	0.969348i	$0.420984\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.06994	−0.186754
$738$	0	0
$739$	−29.2347	−1.07542	−0.537708	−	0.843131i	$-0.680710\pi$
$739$	−29.2347	−1.07542	−0.537708	+	0.843131i	$0.680710\pi$
$740$	−44.9584	−1.65270
$741$	0	0
$742$	0	0
$743$	33.0818	1.21365	0.606826	−	0.794835i	$-0.292442\pi$
$743$	33.0818	1.21365	0.606826	+	0.794835i	$0.292442\pi$
$744$	0	0
$745$	−6.17804	−0.226346
$746$	0	0
$747$	0	0
$748$	−13.7561	−0.502974
$749$	31.6489	1.15643
$750$	0	0
$751$	−29.5451	−1.07812	−0.539059	−	0.842268i	$-0.681220\pi$
$751$	−29.5451	−1.07812	−0.539059	+	0.842268i	$0.681220\pi$
$752$	−11.8388	−0.431715
$753$	0	0
$754$	0	0
$755$	−28.1499	−1.02448
$756$	0	0
$757$	−6.44456	−0.234232	−0.117116	−	0.993118i	$-0.537365\pi$
$757$	−6.44456	−0.234232	−0.117116	+	0.993118i	$0.537365\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.9449	1.04925	0.524626	−	0.851333i	$-0.324205\pi$
$761$	28.9449	1.04925	0.524626	+	0.851333i	$0.324205\pi$
$762$	0	0
$763$	−32.5870	−1.17973
$764$	28.2622	1.02249
$765$	0	0
$766$	0	0
$767$	4.51659	0.163085
$768$	0	0
$769$	−16.3478	−0.589518	−0.294759	−	0.955572i	$-0.595239\pi$
$769$	−16.3478	−0.589518	−0.294759	+	0.955572i	$0.595239\pi$
$770$	0	0
$771$	0	0
$772$	34.0780	1.22650
$773$	42.2612	1.52003	0.760015	−	0.649905i	$-0.225191\pi$
$773$	42.2612	1.52003	0.760015	+	0.649905i	$0.225191\pi$
$774$	0	0
$775$	29.0232	1.04255
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.2850	1.44336
$780$	0	0
$781$	2.52314	0.0902851
$782$	0	0
$783$	0	0
$784$	−15.0390	−0.537108
$785$	53.5479	1.91121
$786$	0	0
$787$	−28.8140	−1.02711	−0.513553	−	0.858058i	$-0.671671\pi$
$787$	−28.8140	−1.02711	−0.513553	+	0.858058i	$0.671671\pi$
$788$	−23.7305	−0.845363
$789$	0	0
$790$	0	0
$791$	14.7030	0.522778
$792$	0	0
$793$	−16.4428	−0.583901
$794$	0	0
$795$	0	0
$796$	35.0074	1.24081
$797$	−49.8117	−1.76442	−0.882210	−	0.470857i	$-0.843945\pi$
$797$	−49.8117	−1.76442	−0.882210	+	0.470857i	$0.843945\pi$
$798$	0	0
$799$	−14.4553	−0.511393
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.9845	−0.387636
$804$	0	0
$805$	−5.32764	−0.187775
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.7527	0.729626	0.364813	−	0.931081i	$-0.381133\pi$
$809$	20.7527	0.729626	0.364813	+	0.931081i	$0.381133\pi$
$810$	0	0
$811$	35.0287	1.23002	0.615012	−	0.788518i	$-0.289151\pi$
$811$	35.0287	1.23002	0.615012	+	0.788518i	$0.289151\pi$
$812$	3.60014	0.126340
$813$	0	0
$814$	0	0
$815$	−12.0477	−0.422013
$816$	0	0
$817$	−46.1364	−1.61411
$818$	0	0
$819$	0	0
$820$	−42.1916	−1.47339
$821$	−19.3895	−0.676699	−0.338349	−	0.941021i	$-0.609869\pi$
$821$	−19.3895	−0.676699	−0.338349	+	0.941021i	$0.609869\pi$
$822$	0	0
$823$	8.57498	0.298905	0.149453	−	0.988769i	$-0.452249\pi$
$823$	8.57498	0.298905	0.149453	+	0.988769i	$0.452249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.7518	0.478198	0.239099	−	0.970995i	$-0.423148\pi$
$827$	13.7518	0.478198	0.239099	+	0.970995i	$0.423148\pi$
$828$	0	0
$829$	−39.0142	−1.35502	−0.677510	−	0.735514i	$-0.736941\pi$
$829$	−39.0142	−1.35502	−0.677510	+	0.735514i	$0.736941\pi$
$830$	0	0
$831$	0	0
$832$	−12.5431	−0.434853
$833$	−18.3629	−0.636238
$834$	0	0
$835$	−4.08964	−0.141528
$836$	−15.9187	−0.550560
$837$	0	0
$838$	0	0
$839$	−54.9242	−1.89620	−0.948098	−	0.317979i	$-0.896996\pi$
$839$	−54.9242	−1.89620	−0.948098	+	0.317979i	$0.896996\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	−21.8394	−0.751744
$845$	−31.2003	−1.07332
$846$	0	0
$847$	−16.2308	−0.557698
$848$	−30.0780	−1.03288
$849$	0	0
$850$	0	0
$851$	−7.59512	−0.260357
$852$	0	0
$853$	−1.22793	−0.0420436	−0.0210218	−	0.999779i	$-0.506692\pi$
$853$	−1.22793	−0.0420436	−0.0210218	+	0.999779i	$0.506692\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.19091	−0.211477	−0.105739	−	0.994394i	$-0.533721\pi$
$857$	−6.19091	−0.211477	−0.105739	+	0.994394i	$0.533721\pi$
$858$	0	0
$859$	−22.0884	−0.753646	−0.376823	−	0.926285i	$-0.622984\pi$
$859$	−22.0884	−0.753646	−0.376823	+	0.926285i	$0.622984\pi$
$860$	48.3199	1.64770
$861$	0	0
$862$	0	0
$863$	−24.0297	−0.817979	−0.408990	−	0.912539i	$-0.634119\pi$
$863$	−24.0297	−0.817979	−0.408990	+	0.912539i	$0.634119\pi$
$864$	0	0
$865$	42.6879	1.45143
$866$	0	0
$867$	0	0
$868$	−27.7910	−0.943290
$869$	12.9157	0.438137
$870$	0	0
$871$	−5.64459	−0.191260
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.60757	−0.223377
$876$	0	0
$877$	16.5733	0.559641	0.279821	−	0.960052i	$-0.409725\pi$
$877$	16.5733	0.559641	0.279821	+	0.960052i	$0.409725\pi$
$878$	0	0
$879$	0	0
$880$	16.6721	0.562016
$881$	8.05304	0.271314	0.135657	−	0.990756i	$-0.456685\pi$
$881$	8.05304	0.271314	0.135657	+	0.990756i	$0.456685\pi$
$882$	0	0
$883$	−0.989977	−0.0333154	−0.0166577	−	0.999861i	$-0.505303\pi$
$883$	−0.989977	−0.0333154	−0.0166577	+	0.999861i	$0.505303\pi$
$884$	−15.3153	−0.515110
$885$	0	0
$886$	0	0
$887$	−5.23447	−0.175756	−0.0878782	−	0.996131i	$-0.528009\pi$
$887$	−5.23447	−0.175756	−0.0878782	+	0.996131i	$0.528009\pi$
$888$	0	0
$889$	−39.3989	−1.32140
$890$	0	0
$891$	0	0
$892$	−21.0867	−0.706036
$893$	−16.7278	−0.559776
$894$	0	0
$895$	27.0200	0.903178
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.71944	−0.257458
$900$	0	0
$901$	−36.7259	−1.22352
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10.8939	0.362125
$906$	0	0
$907$	34.3798	1.14156	0.570781	−	0.821102i	$-0.306641\pi$
$907$	34.3798	1.14156	0.570781	+	0.821102i	$0.306641\pi$
$908$	43.7865	1.45311
$909$	0	0
$910$	0	0
$911$	−0.150463	−0.00498507	−0.00249253	−	0.999997i	$-0.500793\pi$
$911$	−0.150463	−0.00498507	−0.00249253	+	0.999997i	$0.500793\pi$
$912$	0	0
$913$	−2.45502	−0.0812494
$914$	0	0
$915$	0	0
$916$	−50.9355	−1.68296
$917$	−29.9214	−0.988091
$918$	0	0
$919$	−5.26410	−0.173647	−0.0868233	−	0.996224i	$-0.527672\pi$
$919$	−5.26410	−0.173647	−0.0868233	+	0.996224i	$0.527672\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.80913	0.0924636
$924$	0	0
$925$	28.5558	0.938909
$926$	0	0
$927$	0	0
$928$	0	0
$929$	50.4457	1.65507	0.827535	−	0.561414i	$-0.189743\pi$
$929$	50.4457	1.65507	0.827535	+	0.561414i	$0.189743\pi$
$930$	0	0
$931$	−21.2497	−0.696432
$932$	6.04971	0.198165
$933$	0	0
$934$	0	0
$935$	20.3569	0.665743
$936$	0	0
$937$	−6.09349	−0.199066	−0.0995329	−	0.995034i	$-0.531735\pi$
$937$	−6.09349	−0.199066	−0.0995329	+	0.995034i	$0.531735\pi$
$938$	0	0
$939$	0	0
$940$	17.5195	0.571423
$941$	−30.1142	−0.981695	−0.490848	−	0.871245i	$-0.663313\pi$
$941$	−30.1142	−0.981695	−0.490848	+	0.871245i	$0.663313\pi$
$942$	0	0
$943$	−7.12771	−0.232110
$944$	11.5228	0.375034
$945$	0	0
$946$	0	0
$947$	8.82454	0.286759	0.143380	−	0.989668i	$-0.454203\pi$
$947$	8.82454	0.286759	0.143380	+	0.989668i	$0.454203\pi$
$948$	0	0
$949$	−12.2296	−0.396990
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−43.8375	−1.42003	−0.710017	−	0.704184i	$-0.751313\pi$
$953$	−43.8375	−1.42003	−0.710017	+	0.704184i	$0.751313\pi$
$954$	0	0
$955$	−41.8236	−1.35338
$956$	30.4747	0.985621
$957$	0	0
$958$	0	0
$959$	−10.5337	−0.340150
$960$	0	0
$961$	28.5898	0.922253
$962$	0	0
$963$	0	0
$964$	−7.40730	−0.238573
$965$	−50.4302	−1.62341
$966$	0	0
$967$	6.12933	0.197106	0.0985530	−	0.995132i	$-0.468579\pi$
$967$	6.12933	0.197106	0.0985530	+	0.995132i	$0.468579\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.73905	−0.248358	−0.124179	−	0.992260i	$-0.539630\pi$
$971$	−7.73905	−0.248358	−0.124179	+	0.992260i	$0.539630\pi$
$972$	0	0
$973$	−6.43755	−0.206378
$974$	0	0
$975$	0	0
$976$	−41.9490	−1.34276
$977$	17.0564	0.545684	0.272842	−	0.962059i	$-0.412036\pi$
$977$	17.0564	0.545684	0.272842	+	0.962059i	$0.412036\pi$
$978$	0	0
$979$	22.4003	0.715915
$980$	22.2554	0.710923
$981$	0	0
$982$	0	0
$983$	35.9944	1.14804	0.574021	−	0.818840i	$-0.305383\pi$
$983$	35.9944	1.14804	0.574021	+	0.818840i	$0.305383\pi$
$984$	0	0
$985$	35.1174	1.11893
$986$	0	0
$987$	0	0
$988$	−17.7230	−0.563844
$989$	8.16301	0.259569
$990$	0	0
$991$	−21.8078	−0.692749	−0.346374	−	0.938096i	$-0.612587\pi$
$991$	−21.8078	−0.692749	−0.346374	+	0.938096i	$0.612587\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−51.8056	−1.64235
$996$	0	0
$997$	−45.7011	−1.44737	−0.723684	−	0.690132i	$-0.757553\pi$
$997$	−45.7011	−1.44737	−0.723684	+	0.690132i	$0.757553\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.g.1.4		4
3.2	odd	2		2001.2.a.g.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.g.1.1	✓	4	3.2	odd	2
6003.2.a.g.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +2.95969 q^{5} +1.80007 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +2.95969 q^{5} +1.80007 q^{7} +1.40826 q^{11} +1.56788 q^{13} +4.00000 q^{16} +4.88408 q^{17} +5.65189 q^{19} -5.91938 q^{20} -1.00000 q^{23} +3.75976 q^{25} -3.60014 q^{28} -1.00000 q^{29} +7.71944 q^{31} +5.32764 q^{35} +7.59512 q^{37} +7.12771 q^{41} -8.16301 q^{43} -2.81653 q^{44} -2.95969 q^{47} -3.75976 q^{49} -3.13577 q^{52} -7.51951 q^{53} +4.16802 q^{55} +2.88069 q^{59} -10.4873 q^{61} -8.00000 q^{64} +4.64045 q^{65} -3.60014 q^{67} -9.76815 q^{68} +1.79167 q^{71} -7.80007 q^{73} -11.3038 q^{76} +2.53497 q^{77} +9.17141 q^{79} +11.8388 q^{80} -1.74330 q^{83} +14.4553 q^{85} +15.9063 q^{89} +2.82230 q^{91} +2.00000 q^{92} +16.7278 q^{95} +1.35149 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7	\( 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7} + 16 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{25} + 4 q^{28} - 4 q^{29} + 2 q^{31} - 4 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{47} + 4 q^{49} + 8 q^{53} - 8 q^{55} + 22 q^{59} - 16 q^{61} - 32 q^{64} + 10 q^{65} + 4 q^{67} + 4 q^{68} + 22 q^{71} - 22 q^{73} - 8 q^{76} + 8 q^{77} - 20 q^{79} + 8 q^{80} + 10 q^{83} - 2 q^{85} + 14 q^{89} - 26 q^{91} + 8 q^{92} + 14 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7 + 16 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^23 - 4 * q^25 + 4 * q^28 - 4 * q^29 + 2 * q^31 - 4 * q^35 + 4 * q^37 - 6 * q^41 - 2 * q^47 + 4 * q^49 + 8 * q^53 - 8 * q^55 + 22 * q^59 - 16 * q^61 - 32 * q^64 + 10 * q^65 + 4 * q^67 + 4 * q^68 + 22 * q^71 - 22 * q^73 - 8 * q^76 + 8 * q^77 - 20 * q^79 + 8 * q^80 + 10 * q^83 - 2 * q^85 + 14 * q^89 - 26 * q^91 + 8 * q^92 + 14 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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