LMFDB - Embedded newform 5148.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.1069869606$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.815952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 9x - 2 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 9x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.201024$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10258 q^{5} +1.32423 q^{7} +O(q^{10})$	$q-1.10258 q^{5} +1.32423 q^{7} +1.00000 q^{11} -1.00000 q^{13} -3.88689 q^{17} +8.25267 q^{19} -8.18538 q^{23} -3.78431 q^{25} +4.09958 q^{29} -9.46506 q^{31} -1.46008 q^{35} +9.58442 q^{37} -5.36547 q^{41} +2.35952 q^{43} +10.2329 q^{47} -5.24641 q^{49} -10.9284 q^{53} -1.10258 q^{55} +1.12532 q^{59} -7.78130 q^{61} +1.10258 q^{65} -5.02176 q^{67} -13.3793 q^{71} +7.73457 q^{73} +1.32423 q^{77} +3.04155 q^{79} +16.5466 q^{83} +4.28563 q^{85} -10.5710 q^{89} -1.32423 q^{91} -9.09927 q^{95} -4.27246 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q - 2 * q^5 - 2 * q^7	$ 5 q - 2 q^{5} - 2 q^{7} + 5 q^{11} - 5 q^{13} + 2 q^{17} - 2 q^{19} - 12 q^{23} - q^{25} - 4 q^{29} - 2 q^{31} + 2 q^{35} - 6 q^{41} + 14 q^{43} - 14 q^{47} - 7 q^{49} - 20 q^{53} - 2 q^{55} - 20 q^{59} + 2 q^{65} + 10 q^{67} - 26 q^{71} + 16 q^{73} - 2 q^{77} + 2 q^{79} - 16 q^{83} - 18 q^{85} - 24 q^{89} + 2 q^{91} - 22 q^{95}+O(q^{100}) $ 5 * q - 2 * q^5 - 2 * q^7 + 5 * q^11 - 5 * q^13 + 2 * q^17 - 2 * q^19 - 12 * q^23 - q^25 - 4 * q^29 - 2 * q^31 + 2 * q^35 - 6 * q^41 + 14 * q^43 - 14 * q^47 - 7 * q^49 - 20 * q^53 - 2 * q^55 - 20 * q^59 + 2 * q^65 + 10 * q^67 - 26 * q^71 + 16 * q^73 - 2 * q^77 + 2 * q^79 - 16 * q^83 - 18 * q^85 - 24 * q^89 + 2 * q^91 - 22 * q^95

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
$5$	−1.10258	−0.493091	−0.246545	−	0.969131i	$-0.579295\pi$
$5$	−1.10258	−0.493091	−0.246545	+	0.969131i	$0.579295\pi$
$6$	0	0
$7$	1.32423	0.500512	0.250256	−	0.968180i	$-0.419485\pi$
$7$	1.32423	0.500512	0.250256	+	0.968180i	$0.419485\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.88689	−0.942710	−0.471355	−	0.881944i	$-0.656235\pi$
$17$	−3.88689	−0.942710	−0.471355	+	0.881944i	$0.656235\pi$
$18$	0	0
$19$	8.25267	1.89329	0.946647	−	0.322273i	$-0.104447\pi$
$19$	8.25267	1.89329	0.946647	+	0.322273i	$0.104447\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.18538	−1.70677	−0.853385	−	0.521281i	$-0.825454\pi$
$23$	−8.18538	−1.70677	−0.853385	+	0.521281i	$0.825454\pi$
$24$	0	0
$25$	−3.78431	−0.756861
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.09958	0.761273	0.380637	−	0.924725i	$-0.375705\pi$
$29$	4.09958	0.761273	0.380637	+	0.924725i	$0.375705\pi$
$30$	0	0
$31$	−9.46506	−1.69997	−0.849987	−	0.526804i	$-0.823390\pi$
$31$	−9.46506	−1.69997	−0.849987	+	0.526804i	$0.823390\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.46008	−0.246798
$36$	0	0
$37$	9.58442	1.57567	0.787835	−	0.615887i	$-0.211202\pi$
$37$	9.58442	1.57567	0.787835	+	0.615887i	$0.211202\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.36547	−0.837946	−0.418973	−	0.907999i	$-0.637610\pi$
$41$	−5.36547	−0.837946	−0.418973	+	0.907999i	$0.637610\pi$
$42$	0	0
$43$	2.35952	0.359824	0.179912	−	0.983683i	$-0.442419\pi$
$43$	2.35952	0.359824	0.179912	+	0.983683i	$0.442419\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.2329	1.49262	0.746310	−	0.665599i	$-0.231824\pi$
$47$	10.2329	1.49262	0.746310	+	0.665599i	$0.231824\pi$
$48$	0	0
$49$	−5.24641	−0.749488
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.9284	−1.50114	−0.750569	−	0.660793i	$-0.770220\pi$
$53$	−10.9284	−1.50114	−0.750569	+	0.660793i	$0.770220\pi$
$54$	0	0
$55$	−1.10258	−0.148672
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.12532	0.146504	0.0732522	−	0.997313i	$-0.476662\pi$
$59$	1.12532	0.146504	0.0732522	+	0.997313i	$0.476662\pi$
$60$	0	0
$61$	−7.78130	−0.996294	−0.498147	−	0.867093i	$-0.665986\pi$
$61$	−7.78130	−0.996294	−0.498147	+	0.867093i	$0.665986\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.10258	0.136759
$66$	0	0
$67$	−5.02176	−0.613506	−0.306753	−	0.951789i	$-0.599243\pi$
$67$	−5.02176	−0.613506	−0.306753	+	0.951789i	$0.599243\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.3793	−1.58783	−0.793913	−	0.608032i	$-0.791959\pi$
$71$	−13.3793	−1.58783	−0.793913	+	0.608032i	$0.791959\pi$
$72$	0	0
$73$	7.73457	0.905263	0.452631	−	0.891698i	$-0.350485\pi$
$73$	7.73457	0.905263	0.452631	+	0.891698i	$0.350485\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.32423	0.150910
$78$	0	0
$79$	3.04155	0.342201	0.171101	−	0.985254i	$-0.445268\pi$
$79$	3.04155	0.342201	0.171101	+	0.985254i	$0.445268\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.5466	1.81622	0.908112	−	0.418727i	$-0.137524\pi$
$83$	16.5466	1.81622	0.908112	+	0.418727i	$0.137524\pi$
$84$	0	0
$85$	4.28563	0.464842
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.5710	−1.12052	−0.560259	−	0.828317i	$-0.689299\pi$
$89$	−10.5710	−1.12052	−0.560259	+	0.828317i	$0.689299\pi$
$90$	0	0
$91$	−1.32423	−0.138817
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−9.09927	−0.933566
$96$	0	0
$97$	−4.27246	−0.433803	−0.216901	−	0.976194i	$-0.569595\pi$
$97$	−4.27246	−0.433803	−0.216901	+	0.976194i	$0.569595\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.09206	−0.606183	−0.303091	−	0.952962i	$-0.598019\pi$
$101$	−6.09206	−0.606183	−0.303091	+	0.952962i	$0.598019\pi$
$102$	0	0
$103$	1.80312	0.177667	0.0888334	−	0.996046i	$-0.471686\pi$
$103$	1.80312	0.177667	0.0888334	+	0.996046i	$0.471686\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.96042	0.382869	0.191434	−	0.981505i	$-0.438686\pi$
$107$	3.96042	0.382869	0.191434	+	0.981505i	$0.438686\pi$
$108$	0	0
$109$	2.10980	0.202082	0.101041	−	0.994882i	$-0.467783\pi$
$109$	2.10980	0.202082	0.101041	+	0.994882i	$0.467783\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.61648	−0.152066	−0.0760329	−	0.997105i	$-0.524225\pi$
$113$	−1.61648	−0.152066	−0.0760329	+	0.997105i	$0.524225\pi$
$114$	0	0
$115$	9.02508	0.841593
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.14714	−0.471838
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.68544	0.866292
$126$	0	0
$127$	−4.36704	−0.387512	−0.193756	−	0.981050i	$-0.562067\pi$
$127$	−4.36704	−0.387512	−0.193756	+	0.981050i	$0.562067\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.6517	−1.71698	−0.858489	−	0.512831i	$-0.828597\pi$
$131$	−19.6517	−1.71698	−0.858489	+	0.512831i	$0.828597\pi$
$132$	0	0
$133$	10.9284	0.947616
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.26741	−0.450025	−0.225012	−	0.974356i	$-0.572242\pi$
$137$	−5.26741	−0.450025	−0.225012	+	0.974356i	$0.572242\pi$
$138$	0	0
$139$	10.3460	0.877536	0.438768	−	0.898600i	$-0.355415\pi$
$139$	10.3460	0.877536	0.438768	+	0.898600i	$0.355415\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−4.52013	−0.375377
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.211430	0.0173211	0.00866053	−	0.999962i	$-0.497243\pi$
$149$	0.211430	0.0173211	0.00866053	+	0.999962i	$0.497243\pi$
$150$	0	0
$151$	21.4275	1.74375	0.871874	−	0.489730i	$-0.162905\pi$
$151$	21.4275	1.74375	0.871874	+	0.489730i	$0.162905\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.4360	0.838242
$156$	0	0
$157$	−23.5990	−1.88340	−0.941702	−	0.336449i	$-0.890774\pi$
$157$	−23.5990	−1.88340	−0.941702	+	0.336449i	$0.890774\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−10.8393	−0.854259
$162$	0	0
$163$	−0.767831	−0.0601411	−0.0300706	−	0.999548i	$-0.509573\pi$
$163$	−0.767831	−0.0601411	−0.0300706	+	0.999548i	$0.509573\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6610	−1.44403	−0.722015	−	0.691877i	$-0.756784\pi$
$167$	−18.6610	−1.44403	−0.722015	+	0.691877i	$0.756784\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.4133	−0.791708	−0.395854	−	0.918314i	$-0.629551\pi$
$173$	−10.4133	−0.791708	−0.395854	+	0.918314i	$0.629551\pi$
$174$	0	0
$175$	−5.01130	−0.378818
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.64469	−0.571391	−0.285695	−	0.958321i	$-0.592225\pi$
$179$	−7.64469	−0.571391	−0.285695	+	0.958321i	$0.592225\pi$
$180$	0	0
$181$	−11.0390	−0.820523	−0.410262	−	0.911968i	$-0.634563\pi$
$181$	−11.0390	−0.820523	−0.410262	+	0.911968i	$0.634563\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.5676	−0.776948
$186$	0	0
$187$	−3.88689	−0.284238
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−23.9812	−1.73522	−0.867611	−	0.497244i	$-0.834345\pi$
$191$	−23.9812	−1.73522	−0.867611	+	0.497244i	$0.834345\pi$
$192$	0	0
$193$	−12.4166	−0.893766	−0.446883	−	0.894592i	$-0.647466\pi$
$193$	−12.4166	−0.893766	−0.446883	+	0.894592i	$0.647466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.04425	−0.644376	−0.322188	−	0.946676i	$-0.604418\pi$
$197$	−9.04425	−0.644376	−0.322188	+	0.946676i	$0.604418\pi$
$198$	0	0
$199$	7.50884	0.532288	0.266144	−	0.963933i	$-0.414250\pi$
$199$	7.50884	0.532288	0.266144	+	0.963933i	$0.414250\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.42879	0.381026
$204$	0	0
$205$	5.91589	0.413184
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.25267	0.570849
$210$	0	0
$211$	4.16687	0.286859	0.143430	−	0.989661i	$-0.454187\pi$
$211$	4.16687	0.286859	0.143430	+	0.989661i	$0.454187\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.60157	−0.177426
$216$	0	0
$217$	−12.5339	−0.850858
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.88689	0.261461
$222$	0	0
$223$	−13.7929	−0.923641	−0.461821	−	0.886973i	$-0.652804\pi$
$223$	−13.7929	−0.923641	−0.461821	+	0.886973i	$0.652804\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.92316	−0.525878	−0.262939	−	0.964812i	$-0.584692\pi$
$227$	−7.92316	−0.525878	−0.262939	+	0.964812i	$0.584692\pi$
$228$	0	0
$229$	14.0277	0.926978	0.463489	−	0.886103i	$-0.346597\pi$
$229$	14.0277	0.926978	0.463489	+	0.886103i	$0.346597\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.1946	1.06094	0.530471	−	0.847703i	$-0.322015\pi$
$233$	16.1946	1.06094	0.530471	+	0.847703i	$0.322015\pi$
$234$	0	0
$235$	−11.2826	−0.735997
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.6832	0.949778	0.474889	−	0.880046i	$-0.342488\pi$
$239$	14.6832	0.949778	0.474889	+	0.880046i	$0.342488\pi$
$240$	0	0
$241$	−20.9384	−1.34876	−0.674381	−	0.738384i	$-0.735589\pi$
$241$	−20.9384	−1.34876	−0.674381	+	0.738384i	$0.735589\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.78462	0.369566
$246$	0	0
$247$	−8.25267	−0.525105
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.77882	−0.427875	−0.213938	−	0.976847i	$-0.568629\pi$
$251$	−6.77882	−0.427875	−0.213938	+	0.976847i	$0.568629\pi$
$252$	0	0
$253$	−8.18538	−0.514610
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1177	1.13015	0.565076	−	0.825039i	$-0.308847\pi$
$257$	18.1177	1.13015	0.565076	+	0.825039i	$0.308847\pi$
$258$	0	0
$259$	12.6920	0.788642
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.5533	−1.02072	−0.510361	−	0.859960i	$-0.670488\pi$
$263$	−16.5533	−1.02072	−0.510361	+	0.859960i	$0.670488\pi$
$264$	0	0
$265$	12.0495	0.740197
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.1781	1.10834	0.554170	−	0.832404i	$-0.313036\pi$
$269$	18.1781	1.10834	0.554170	+	0.832404i	$0.313036\pi$
$270$	0	0
$271$	2.43703	0.148039	0.0740195	−	0.997257i	$-0.476417\pi$
$271$	2.43703	0.148039	0.0740195	+	0.997257i	$0.476417\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.78431	−0.228202
$276$	0	0
$277$	−10.9427	−0.657486	−0.328743	−	0.944419i	$-0.606625\pi$
$277$	−10.9427	−0.657486	−0.328743	+	0.944419i	$0.606625\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.24174	0.0740762	0.0370381	−	0.999314i	$-0.488208\pi$
$281$	1.24174	0.0740762	0.0370381	+	0.999314i	$0.488208\pi$
$282$	0	0
$283$	−8.19292	−0.487019	−0.243509	−	0.969899i	$-0.578299\pi$
$283$	−8.19292	−0.487019	−0.243509	+	0.969899i	$0.578299\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.10513	−0.419402
$288$	0	0
$289$	−1.89207	−0.111298
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.9414	1.39867	0.699336	−	0.714793i	$-0.253479\pi$
$293$	23.9414	1.39867	0.699336	+	0.714793i	$0.253479\pi$
$294$	0	0
$295$	−1.24076	−0.0722400
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.18538	0.473373
$300$	0	0
$301$	3.12455	0.180096
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.57955	0.491263
$306$	0	0
$307$	1.37210	0.0783098	0.0391549	−	0.999233i	$-0.487533\pi$
$307$	1.37210	0.0783098	0.0391549	+	0.999233i	$0.487533\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.8953	−1.63850	−0.819251	−	0.573436i	$-0.805610\pi$
$311$	−28.8953	−1.63850	−0.819251	+	0.573436i	$0.805610\pi$
$312$	0	0
$313$	−7.01568	−0.396550	−0.198275	−	0.980146i	$-0.563534\pi$
$313$	−7.01568	−0.396550	−0.198275	+	0.980146i	$0.563534\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.3025	−1.08414	−0.542069	−	0.840334i	$-0.682359\pi$
$317$	−19.3025	−1.08414	−0.542069	+	0.840334i	$0.682359\pi$
$318$	0	0
$319$	4.09958	0.229532
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−32.0773	−1.78483
$324$	0	0
$325$	3.78431	0.209916
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.5507	0.747074
$330$	0	0
$331$	−11.0830	−0.609179	−0.304590	−	0.952484i	$-0.598519\pi$
$331$	−11.0830	−0.609179	−0.304590	+	0.952484i	$0.598519\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.53692	0.302514
$336$	0	0
$337$	−3.40107	−0.185268	−0.0926341	−	0.995700i	$-0.529529\pi$
$337$	−3.40107	−0.185268	−0.0926341	+	0.995700i	$0.529529\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.46506	−0.512561
$342$	0	0
$343$	−16.2171	−0.875640
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.8468	−1.49489	−0.747447	−	0.664321i	$-0.768721\pi$
$347$	−27.8468	−1.49489	−0.747447	+	0.664321i	$0.768721\pi$
$348$	0	0
$349$	−0.211918	−0.0113437	−0.00567185	−	0.999984i	$-0.501805\pi$
$349$	−0.211918	−0.0113437	−0.00567185	+	0.999984i	$0.501805\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.1765	−1.23356	−0.616782	−	0.787134i	$-0.711564\pi$
$353$	−23.1765	−1.23356	−0.616782	+	0.787134i	$0.711564\pi$
$354$	0	0
$355$	14.7518	0.782942
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.6605	−0.932085	−0.466043	−	0.884762i	$-0.654321\pi$
$359$	−17.6605	−0.932085	−0.466043	+	0.884762i	$0.654321\pi$
$360$	0	0
$361$	49.1066	2.58456
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.52802	−0.446377
$366$	0	0
$367$	−25.4153	−1.32667	−0.663333	−	0.748324i	$-0.730859\pi$
$367$	−25.4153	−1.32667	−0.663333	+	0.748324i	$0.730859\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.4718	−0.751337
$372$	0	0
$373$	9.43464	0.488507	0.244254	−	0.969711i	$-0.421457\pi$
$373$	9.43464	0.488507	0.244254	+	0.969711i	$0.421457\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.09958	−0.211139
$378$	0	0
$379$	14.5560	0.747693	0.373847	−	0.927491i	$-0.378039\pi$
$379$	14.5560	0.747693	0.373847	+	0.927491i	$0.378039\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.8629	−1.01495	−0.507473	−	0.861667i	$-0.669420\pi$
$383$	−19.8629	−1.01495	−0.507473	+	0.861667i	$0.669420\pi$
$384$	0	0
$385$	−1.46008	−0.0744124
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.2456	−1.12789	−0.563947	−	0.825811i	$-0.690718\pi$
$389$	−22.2456	−1.12789	−0.563947	+	0.825811i	$0.690718\pi$
$390$	0	0
$391$	31.8157	1.60899
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.35357	−0.168736
$396$	0	0
$397$	29.1470	1.46285	0.731424	−	0.681923i	$-0.238856\pi$
$397$	29.1470	1.46285	0.731424	+	0.681923i	$0.238856\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.65929	0.432424	0.216212	−	0.976346i	$-0.430630\pi$
$401$	8.65929	0.432424	0.216212	+	0.976346i	$0.430630\pi$
$402$	0	0
$403$	9.46506	0.471488
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.58442	0.475082
$408$	0	0
$409$	−8.93242	−0.441680	−0.220840	−	0.975310i	$-0.570880\pi$
$409$	−8.93242	−0.441680	−0.220840	+	0.975310i	$0.570880\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.49019	0.0733273
$414$	0	0
$415$	−18.2440	−0.895564
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.16205	−0.203329	−0.101665	−	0.994819i	$-0.532417\pi$
$419$	−4.16205	−0.203329	−0.101665	+	0.994819i	$0.532417\pi$
$420$	0	0
$421$	31.2153	1.52134	0.760669	−	0.649140i	$-0.224871\pi$
$421$	31.2153	1.52134	0.760669	+	0.649140i	$0.224871\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14.7092	0.713501
$426$	0	0
$427$	−10.3042	−0.498657
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.5114	−0.698987	−0.349494	−	0.936939i	$-0.613646\pi$
$431$	−14.5114	−0.698987	−0.349494	+	0.936939i	$0.613646\pi$
$432$	0	0
$433$	−3.58771	−0.172414	−0.0862072	−	0.996277i	$-0.527475\pi$
$433$	−3.58771	−0.172414	−0.0862072	+	0.996277i	$0.527475\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−67.5513	−3.23142
$438$	0	0
$439$	−12.3469	−0.589285	−0.294642	−	0.955608i	$-0.595201\pi$
$439$	−12.3469	−0.589285	−0.294642	+	0.955608i	$0.595201\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.2487	−1.34214	−0.671068	−	0.741396i	$-0.734164\pi$
$443$	−28.2487	−1.34214	−0.671068	+	0.741396i	$0.734164\pi$
$444$	0	0
$445$	11.6554	0.552518
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.0579	1.56010	0.780050	−	0.625717i	$-0.215194\pi$
$449$	33.0579	1.56010	0.780050	+	0.625717i	$0.215194\pi$
$450$	0	0
$451$	−5.36547	−0.252650
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.46008	0.0684494
$456$	0	0
$457$	26.3999	1.23493	0.617467	−	0.786597i	$-0.288159\pi$
$457$	26.3999	1.23493	0.617467	+	0.786597i	$0.288159\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.7092	1.75629	0.878146	−	0.478393i	$-0.158781\pi$
$461$	37.7092	1.75629	0.878146	+	0.478393i	$0.158781\pi$
$462$	0	0
$463$	−19.5759	−0.909768	−0.454884	−	0.890551i	$-0.650319\pi$
$463$	−19.5759	−0.909768	−0.454884	+	0.890551i	$0.650319\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.58779	−0.443670	−0.221835	−	0.975084i	$-0.571205\pi$
$467$	−9.58779	−0.443670	−0.221835	+	0.975084i	$0.571205\pi$
$468$	0	0
$469$	−6.64997	−0.307067
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.35952	0.108491
$474$	0	0
$475$	−31.2307	−1.43296
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.59654	−0.347095	−0.173547	−	0.984826i	$-0.555523\pi$
$479$	−7.59654	−0.347095	−0.173547	+	0.984826i	$0.555523\pi$
$480$	0	0
$481$	−9.58442	−0.437012
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.71075	0.213904
$486$	0	0
$487$	−12.6332	−0.572467	−0.286233	−	0.958160i	$-0.592403\pi$
$487$	−12.6332	−0.572467	−0.286233	+	0.958160i	$0.592403\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.4330	1.14778	0.573888	−	0.818934i	$-0.305434\pi$
$491$	25.4330	1.14778	0.573888	+	0.818934i	$0.305434\pi$
$492$	0	0
$493$	−15.9346	−0.717659
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.7172	−0.794726
$498$	0	0
$499$	−5.08331	−0.227560	−0.113780	−	0.993506i	$-0.536296\pi$
$499$	−5.08331	−0.227560	−0.113780	+	0.993506i	$0.536296\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.1657	1.47879	0.739393	−	0.673274i	$-0.235113\pi$
$503$	33.1657	1.47879	0.739393	+	0.673274i	$0.235113\pi$
$504$	0	0
$505$	6.71701	0.298903
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.5979	−0.912986	−0.456493	−	0.889727i	$-0.650895\pi$
$509$	−20.5979	−0.912986	−0.456493	+	0.889727i	$0.650895\pi$
$510$	0	0
$511$	10.2424	0.453095
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.98809	−0.0876059
$516$	0	0
$517$	10.2329	0.450042
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.5456	0.768688	0.384344	−	0.923190i	$-0.374428\pi$
$521$	17.5456	0.768688	0.384344	+	0.923190i	$0.374428\pi$
$522$	0	0
$523$	12.2014	0.533529	0.266764	−	0.963762i	$-0.414045\pi$
$523$	12.2014	0.533529	0.266764	+	0.963762i	$0.414045\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.7896	1.60258
$528$	0	0
$529$	44.0005	1.91306
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.36547	0.232405
$534$	0	0
$535$	−4.36670	−0.188789
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.24641	−0.225979
$540$	0	0
$541$	−35.0641	−1.50752	−0.753761	−	0.657148i	$-0.771763\pi$
$541$	−35.0641	−1.50752	−0.753761	+	0.657148i	$0.771763\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.32623	−0.0996447
$546$	0	0
$547$	9.21661	0.394074	0.197037	−	0.980396i	$-0.436868\pi$
$547$	9.21661	0.394074	0.197037	+	0.980396i	$0.436868\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	33.8325	1.44131
$552$	0	0
$553$	4.02772	0.171276
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0137	0.932750	0.466375	−	0.884587i	$-0.345560\pi$
$557$	22.0137	0.932750	0.466375	+	0.884587i	$0.345560\pi$
$558$	0	0
$559$	−2.35952	−0.0997971
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.2362	−0.684273	−0.342137	−	0.939650i	$-0.611151\pi$
$563$	−16.2362	−0.684273	−0.342137	+	0.939650i	$0.611151\pi$
$564$	0	0
$565$	1.78231	0.0749823
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.4270	0.479046	0.239523	−	0.970891i	$-0.423009\pi$
$569$	11.4270	0.479046	0.239523	+	0.970891i	$0.423009\pi$
$570$	0	0
$571$	32.6051	1.36448	0.682240	−	0.731128i	$-0.261006\pi$
$571$	32.6051	1.36448	0.682240	+	0.731128i	$0.261006\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	30.9760	1.29179
$576$	0	0
$577$	−10.0059	−0.416551	−0.208275	−	0.978070i	$-0.566785\pi$
$577$	−10.0059	−0.416551	−0.208275	+	0.978070i	$0.566785\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	21.9115	0.909042
$582$	0	0
$583$	−10.9284	−0.452610
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.0065	0.578111	0.289055	−	0.957312i	$-0.406659\pi$
$587$	14.0065	0.578111	0.289055	+	0.957312i	$0.406659\pi$
$588$	0	0
$589$	−78.1120	−3.21855
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.14263	0.170117	0.0850586	−	0.996376i	$-0.472892\pi$
$593$	4.14263	0.170117	0.0850586	+	0.996376i	$0.472892\pi$
$594$	0	0
$595$	5.67516	0.232659
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.0382	1.79935	0.899676	−	0.436558i	$-0.143803\pi$
$599$	44.0382	1.79935	0.899676	+	0.436558i	$0.143803\pi$
$600$	0	0
$601$	15.6222	0.637243	0.318622	−	0.947882i	$-0.396780\pi$
$601$	15.6222	0.637243	0.318622	+	0.947882i	$0.396780\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.10258	−0.0448264
$606$	0	0
$607$	19.5704	0.794337	0.397169	−	0.917746i	$-0.369993\pi$
$607$	19.5704	0.794337	0.397169	+	0.917746i	$0.369993\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.2329	−0.413978
$612$	0	0
$613$	−16.3975	−0.662289	−0.331144	−	0.943580i	$-0.607435\pi$
$613$	−16.3975	−0.662289	−0.331144	+	0.943580i	$0.607435\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.50618	−0.100895	−0.0504474	−	0.998727i	$-0.516065\pi$
$617$	−2.50618	−0.100895	−0.0504474	+	0.998727i	$0.516065\pi$
$618$	0	0
$619$	26.2218	1.05394	0.526972	−	0.849883i	$-0.323327\pi$
$619$	26.2218	1.05394	0.526972	+	0.849883i	$0.323327\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.9984	−0.560833
$624$	0	0
$625$	8.24251	0.329701
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.2536	−1.48540
$630$	0	0
$631$	−16.3550	−0.651081	−0.325541	−	0.945528i	$-0.605546\pi$
$631$	−16.3550	−0.651081	−0.325541	+	0.945528i	$0.605546\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.81503	0.191079
$636$	0	0
$637$	5.24641	0.207870
$638$	0	0
$639$	0	0
$640$	0	0
$641$	31.3964	1.24008	0.620042	−	0.784569i	$-0.287116\pi$
$641$	31.3964	1.24008	0.620042	+	0.784569i	$0.287116\pi$
$642$	0	0
$643$	26.8416	1.05853	0.529265	−	0.848456i	$-0.322468\pi$
$643$	26.8416	1.05853	0.529265	+	0.848456i	$0.322468\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.44232	−0.213960	−0.106980	−	0.994261i	$-0.534118\pi$
$647$	−5.44232	−0.213960	−0.106980	+	0.994261i	$0.534118\pi$
$648$	0	0
$649$	1.12532	0.0441728
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.16523	−0.358663	−0.179332	−	0.983789i	$-0.557394\pi$
$653$	−9.16523	−0.358663	−0.179332	+	0.983789i	$0.557394\pi$
$654$	0	0
$655$	21.6677	0.846626
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.8455	0.656207	0.328104	−	0.944642i	$-0.393590\pi$
$659$	16.8455	0.656207	0.328104	+	0.944642i	$0.393590\pi$
$660$	0	0
$661$	−14.1775	−0.551441	−0.275720	−	0.961238i	$-0.588916\pi$
$661$	−14.1775	−0.551441	−0.275720	+	0.961238i	$0.588916\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0495	−0.467261
$666$	0	0
$667$	−33.5566	−1.29932
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.78130	−0.300394
$672$	0	0
$673$	−49.1337	−1.89396	−0.946982	−	0.321288i	$-0.895884\pi$
$673$	−49.1337	−1.89396	−0.946982	+	0.321288i	$0.895884\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.7935	−0.876026	−0.438013	−	0.898969i	$-0.644318\pi$
$677$	−22.7935	−0.876026	−0.438013	+	0.898969i	$0.644318\pi$
$678$	0	0
$679$	−5.65773	−0.217124
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.3344	−1.23724	−0.618621	−	0.785690i	$-0.712308\pi$
$683$	−32.3344	−1.23724	−0.618621	+	0.785690i	$0.712308\pi$
$684$	0	0
$685$	5.80776	0.221903
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.9284	0.416341
$690$	0	0
$691$	49.0679	1.86663	0.933315	−	0.359058i	$-0.116902\pi$
$691$	49.0679	1.86663	0.933315	+	0.359058i	$0.116902\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.4073	−0.432705
$696$	0	0
$697$	20.8550	0.789940
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.51682	−0.321676	−0.160838	−	0.986981i	$-0.551420\pi$
$701$	−8.51682	−0.321676	−0.160838	+	0.986981i	$0.551420\pi$
$702$	0	0
$703$	79.0971	2.98320
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.06729	−0.303402
$708$	0	0
$709$	−14.1294	−0.530639	−0.265319	−	0.964161i	$-0.585477\pi$
$709$	−14.1294	−0.530639	−0.265319	+	0.964161i	$0.585477\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	77.4751	2.90146
$714$	0	0
$715$	1.10258	0.0412343
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.5799	1.51337	0.756687	−	0.653777i	$-0.226817\pi$
$719$	40.5799	1.51337	0.756687	+	0.653777i	$0.226817\pi$
$720$	0	0
$721$	2.38775	0.0889244
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−15.5141	−0.576178
$726$	0	0
$727$	40.2421	1.49250	0.746248	−	0.665668i	$-0.231853\pi$
$727$	40.2421	1.49250	0.746248	+	0.665668i	$0.231853\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.17121	−0.339209
$732$	0	0
$733$	−9.30724	−0.343771	−0.171885	−	0.985117i	$-0.554986\pi$
$733$	−9.30724	−0.343771	−0.171885	+	0.985117i	$0.554986\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.02176	−0.184979
$738$	0	0
$739$	−45.9562	−1.69053	−0.845263	−	0.534350i	$-0.820556\pi$
$739$	−45.9562	−1.69053	−0.845263	+	0.534350i	$0.820556\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.4726	−0.384204	−0.192102	−	0.981375i	$-0.561530\pi$
$743$	−10.4726	−0.384204	−0.192102	+	0.981375i	$0.561530\pi$
$744$	0	0
$745$	−0.233120	−0.00854085
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.24451	0.191630
$750$	0	0
$751$	−13.7511	−0.501784	−0.250892	−	0.968015i	$-0.580724\pi$
$751$	−13.7511	−0.501784	−0.250892	+	0.968015i	$0.580724\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23.6257	−0.859826
$756$	0	0
$757$	25.4242	0.924060	0.462030	−	0.886864i	$-0.347121\pi$
$757$	25.4242	0.924060	0.462030	+	0.886864i	$0.347121\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.6491	1.29228	0.646139	−	0.763220i	$-0.276383\pi$
$761$	35.6491	1.29228	0.646139	+	0.763220i	$0.276383\pi$
$762$	0	0
$763$	2.79386	0.101144
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.12532	−0.0406330
$768$	0	0
$769$	−3.24231	−0.116921	−0.0584603	−	0.998290i	$-0.518619\pi$
$769$	−3.24231	−0.116921	−0.0584603	+	0.998290i	$0.518619\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.71989	−0.349600	−0.174800	−	0.984604i	$-0.555928\pi$
$773$	−9.71989	−0.349600	−0.174800	+	0.984604i	$0.555928\pi$
$774$	0	0
$775$	35.8187	1.28664
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−44.2795	−1.58648
$780$	0	0
$781$	−13.3793	−0.478747
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.0199	0.928689
$786$	0	0
$787$	−41.6830	−1.48584	−0.742919	−	0.669382i	$-0.766559\pi$
$787$	−41.6830	−1.48584	−0.742919	+	0.669382i	$0.766559\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.14060	−0.0761108
$792$	0	0
$793$	7.78130	0.276322
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.2473	1.00057	0.500286	−	0.865860i	$-0.333228\pi$
$797$	28.2473	1.00057	0.500286	+	0.865860i	$0.333228\pi$
$798$	0	0
$799$	−39.7741	−1.40711
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.73457	0.272947
$804$	0	0
$805$	11.9513	0.421227
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.8947	1.43778	0.718891	−	0.695123i	$-0.244650\pi$
$809$	40.8947	1.43778	0.718891	+	0.695123i	$0.244650\pi$
$810$	0	0
$811$	−48.4263	−1.70048	−0.850238	−	0.526398i	$-0.823542\pi$
$811$	−48.4263	−1.70048	−0.850238	+	0.526398i	$0.823542\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.846598	0.0296550
$816$	0	0
$817$	19.4724	0.681252
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.0195	1.08259	0.541294	−	0.840834i	$-0.317935\pi$
$821$	31.0195	1.08259	0.541294	+	0.840834i	$0.317935\pi$
$822$	0	0
$823$	−49.0332	−1.70919	−0.854595	−	0.519296i	$-0.826194\pi$
$823$	−49.0332	−1.70919	−0.854595	+	0.519296i	$0.826194\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.72194	0.129424	0.0647122	−	0.997904i	$-0.479387\pi$
$827$	3.72194	0.129424	0.0647122	+	0.997904i	$0.479387\pi$
$828$	0	0
$829$	−22.3682	−0.776881	−0.388441	−	0.921474i	$-0.626986\pi$
$829$	−22.3682	−0.776881	−0.388441	+	0.921474i	$0.626986\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20.3922	0.706549
$834$	0	0
$835$	20.5753	0.712038
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.2968	0.355487	0.177743	−	0.984077i	$-0.443120\pi$
$839$	10.2968	0.355487	0.177743	+	0.984077i	$0.443120\pi$
$840$	0	0
$841$	−12.1934	−0.420463
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.10258	−0.0379301
$846$	0	0
$847$	1.32423	0.0455011
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−78.4522	−2.68931
$852$	0	0
$853$	−54.3841	−1.86208	−0.931038	−	0.364922i	$-0.881096\pi$
$853$	−54.3841	−1.86208	−0.931038	+	0.364922i	$0.881096\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.04679	0.240714	0.120357	−	0.992731i	$-0.461596\pi$
$857$	7.04679	0.240714	0.120357	+	0.992731i	$0.461596\pi$
$858$	0	0
$859$	−17.6623	−0.602630	−0.301315	−	0.953525i	$-0.597426\pi$
$859$	−17.6623	−0.602630	−0.301315	+	0.953525i	$0.597426\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40.6583	−1.38402	−0.692012	−	0.721886i	$-0.743276\pi$
$863$	−40.6583	−1.38402	−0.692012	+	0.721886i	$0.743276\pi$
$864$	0	0
$865$	11.4815	0.390384
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.04155	0.103178
$870$	0	0
$871$	5.02176	0.170156
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.8258	0.433590
$876$	0	0
$877$	16.2002	0.547042	0.273521	−	0.961866i	$-0.411812\pi$
$877$	16.2002	0.547042	0.273521	+	0.961866i	$0.411812\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45.2262	1.52371	0.761855	−	0.647747i	$-0.224289\pi$
$881$	45.2262	1.52371	0.761855	+	0.647747i	$0.224289\pi$
$882$	0	0
$883$	−42.8672	−1.44260	−0.721298	−	0.692625i	$-0.756454\pi$
$883$	−42.8672	−1.44260	−0.721298	+	0.692625i	$0.756454\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.2010	1.08120	0.540602	−	0.841278i	$-0.318196\pi$
$887$	32.2010	1.08120	0.540602	+	0.841278i	$0.318196\pi$
$888$	0	0
$889$	−5.78297	−0.193955
$890$	0	0
$891$	0	0
$892$	0	0
$893$	84.4487	2.82597
$894$	0	0
$895$	8.42891	0.281748
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38.8028	−1.29414
$900$	0	0
$901$	42.4777	1.41514
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.1714	0.404593
$906$	0	0
$907$	−26.7097	−0.886880	−0.443440	−	0.896304i	$-0.646242\pi$
$907$	−26.7097	−0.886880	−0.443440	+	0.896304i	$0.646242\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.5096	0.414461	0.207230	−	0.978292i	$-0.433555\pi$
$911$	12.5096	0.414461	0.207230	+	0.978292i	$0.433555\pi$
$912$	0	0
$913$	16.5466	0.547612
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.0234	−0.859369
$918$	0	0
$919$	−10.8589	−0.358203	−0.179102	−	0.983831i	$-0.557319\pi$
$919$	−10.8589	−0.358203	−0.179102	+	0.983831i	$0.557319\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.3793	0.440384
$924$	0	0
$925$	−36.2704	−1.19256
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.1441	−0.923379	−0.461690	−	0.887042i	$-0.652757\pi$
$929$	−28.1441	−0.923379	−0.461690	+	0.887042i	$0.652757\pi$
$930$	0	0
$931$	−43.2969	−1.41900
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.28563	0.140155
$936$	0	0
$937$	−32.4153	−1.05896	−0.529482	−	0.848321i	$-0.677613\pi$
$937$	−32.4153	−1.05896	−0.529482	+	0.848321i	$0.677613\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55.1473	1.79775	0.898875	−	0.438206i	$-0.144386\pi$
$941$	55.1473	1.79775	0.898875	+	0.438206i	$0.144386\pi$
$942$	0	0
$943$	43.9185	1.43018
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.85366	−0.0602359	−0.0301180	−	0.999546i	$-0.509588\pi$
$947$	−1.85366	−0.0602359	−0.0301180	+	0.999546i	$0.509588\pi$
$948$	0	0
$949$	−7.73457	−0.251075
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33.8227	−1.09562	−0.547812	−	0.836601i	$-0.684539\pi$
$953$	−33.8227	−1.09562	−0.547812	+	0.836601i	$0.684539\pi$
$954$	0	0
$955$	26.4413	0.855622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.97526	−0.225243
$960$	0	0
$961$	58.5873	1.88991
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.6904	0.440708
$966$	0	0
$967$	46.0634	1.48130	0.740649	−	0.671892i	$-0.234518\pi$
$967$	46.0634	1.48130	0.740649	+	0.671892i	$0.234518\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.1617	1.25676	0.628379	−	0.777907i	$-0.283719\pi$
$971$	39.1617	1.25676	0.628379	+	0.777907i	$0.283719\pi$
$972$	0	0
$973$	13.7005	0.439217
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.60139	0.211197	0.105599	−	0.994409i	$-0.466324\pi$
$977$	6.60139	0.211197	0.105599	+	0.994409i	$0.466324\pi$
$978$	0	0
$979$	−10.5710	−0.337849
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32.9253	−1.05016	−0.525078	−	0.851054i	$-0.675964\pi$
$983$	−32.9253	−1.05016	−0.525078	+	0.851054i	$0.675964\pi$
$984$	0	0
$985$	9.97205	0.317736
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.3136	−0.614136
$990$	0	0
$991$	23.8409	0.757332	0.378666	−	0.925533i	$-0.376383\pi$
$991$	23.8409	0.757332	0.378666	+	0.925533i	$0.376383\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.27913	−0.262466
$996$	0	0
$997$	34.6716	1.09806	0.549030	−	0.835803i	$-0.314997\pi$
$997$	34.6716	1.09806	0.549030	+	0.835803i	$0.314997\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.s.1.2	✓	5
3.2	odd	2		5148.2.a.t.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5148.2.a.s.1.2	✓	5	1.1	even	1	trivial
5148.2.a.t.1.4	yes	5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-1.10258 q^{5} +1.32423 q^{7} +O(q^{10})\)	\(q-1.10258 q^{5} +1.32423 q^{7} +1.00000 q^{11} -1.00000 q^{13} -3.88689 q^{17} +8.25267 q^{19} -8.18538 q^{23} -3.78431 q^{25} +4.09958 q^{29} -9.46506 q^{31} -1.46008 q^{35} +9.58442 q^{37} -5.36547 q^{41} +2.35952 q^{43} +10.2329 q^{47} -5.24641 q^{49} -10.9284 q^{53} -1.10258 q^{55} +1.12532 q^{59} -7.78130 q^{61} +1.10258 q^{65} -5.02176 q^{67} -13.3793 q^{71} +7.73457 q^{73} +1.32423 q^{77} +3.04155 q^{79} +16.5466 q^{83} +4.28563 q^{85} -10.5710 q^{89} -1.32423 q^{91} -9.09927 q^{95} -4.27246 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q - 2 * q^5 - 2 * q^7	\( 5 q - 2 q^{5} - 2 q^{7} + 5 q^{11} - 5 q^{13} + 2 q^{17} - 2 q^{19} - 12 q^{23} - q^{25} - 4 q^{29} - 2 q^{31} + 2 q^{35} - 6 q^{41} + 14 q^{43} - 14 q^{47} - 7 q^{49} - 20 q^{53} - 2 q^{55} - 20 q^{59} + 2 q^{65} + 10 q^{67} - 26 q^{71} + 16 q^{73} - 2 q^{77} + 2 q^{79} - 16 q^{83} - 18 q^{85} - 24 q^{89} + 2 q^{91} - 22 q^{95}+O(q^{100}) \) 5 * q - 2 * q^5 - 2 * q^7 + 5 * q^11 - 5 * q^13 + 2 * q^17 - 2 * q^19 - 12 * q^23 - q^25 - 4 * q^29 - 2 * q^31 + 2 * q^35 - 6 * q^41 + 14 * q^43 - 14 * q^47 - 7 * q^49 - 20 * q^53 - 2 * q^55 - 20 * q^59 + 2 * q^65 + 10 * q^67 - 26 * q^71 + 16 * q^73 - 2 * q^77 + 2 * q^79 - 16 * q^83 - 18 * q^85 - 24 * q^89 + 2 * q^91 - 22 * q^95

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