LMFDB - Embedded newform 1503.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.410375$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10637 q^{2} -0.775946 q^{4} +2.19483 q^{5} -4.52759 q^{7} +3.07122 q^{8} +O(q^{10})$	\(q-1.10637 q^{2} -0.775946 q^{4} +2.19483 q^{5} -4.52759 q^{7} +3.07122 q^{8} -2.42829 q^{10} +2.62592 q^{11} +4.50686 q^{13} +5.00918 q^{14} -1.84602 q^{16} -2.14478 q^{17} -6.09764 q^{19} -1.70307 q^{20} -2.90524 q^{22} +6.18843 q^{23} -0.182739 q^{25} -4.98626 q^{26} +3.51316 q^{28} -1.84359 q^{29} +1.20401 q^{31} -4.10007 q^{32} +2.37292 q^{34} -9.93726 q^{35} +10.7075 q^{37} +6.74624 q^{38} +6.74080 q^{40} -0.353073 q^{41} -11.9236 q^{43} -2.03758 q^{44} -6.84670 q^{46} -0.986349 q^{47} +13.4990 q^{49} +0.202177 q^{50} -3.49708 q^{52} +7.36740 q^{53} +5.76345 q^{55} -13.9052 q^{56} +2.03969 q^{58} +4.63958 q^{59} +11.9846 q^{61} -1.33208 q^{62} +8.22822 q^{64} +9.89178 q^{65} -0.652622 q^{67} +1.66424 q^{68} +10.9943 q^{70} +13.2454 q^{71} -7.23916 q^{73} -11.8465 q^{74} +4.73144 q^{76} -11.8891 q^{77} +7.65804 q^{79} -4.05168 q^{80} +0.390629 q^{82} +6.48636 q^{83} -4.70742 q^{85} +13.1919 q^{86} +8.06480 q^{88} +6.00336 q^{89} -20.4052 q^{91} -4.80189 q^{92} +1.09127 q^{94} -13.3833 q^{95} -8.29839 q^{97} -14.9349 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8	$ 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8} + 5 q^{10} + 15 q^{11} + 3 q^{14} + 10 q^{16} + 11 q^{17} - 16 q^{19} + 9 q^{20} + 7 q^{22} + 9 q^{23} + 6 q^{25} - 7 q^{26} + 9 q^{28} + q^{29} - 18 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 7 q^{37} - 20 q^{38} + 32 q^{40} + 10 q^{41} + 6 q^{43} + 5 q^{44} + 13 q^{46} + 7 q^{47} + 11 q^{49} - 3 q^{50} - 6 q^{52} + 9 q^{53} + 17 q^{55} + 15 q^{56} - 19 q^{58} + 37 q^{59} - 2 q^{61} - 18 q^{62} + 14 q^{64} + 16 q^{65} - 26 q^{68} + 7 q^{70} - 13 q^{71} - 6 q^{73} - 13 q^{74} - 50 q^{76} + 2 q^{77} + 14 q^{79} + 10 q^{80} + 39 q^{82} + 5 q^{83} + 29 q^{85} + 37 q^{86} + 48 q^{88} + 30 q^{89} - 33 q^{91} + 31 q^{92} + 34 q^{94} - 43 q^{95} - 9 q^{97} - 36 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8 + 5 * q^10 + 15 * q^11 + 3 * q^14 + 10 * q^16 + 11 * q^17 - 16 * q^19 + 9 * q^20 + 7 * q^22 + 9 * q^23 + 6 * q^25 - 7 * q^26 + 9 * q^28 + q^29 - 18 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 7 * q^37 - 20 * q^38 + 32 * q^40 + 10 * q^41 + 6 * q^43 + 5 * q^44 + 13 * q^46 + 7 * q^47 + 11 * q^49 - 3 * q^50 - 6 * q^52 + 9 * q^53 + 17 * q^55 + 15 * q^56 - 19 * q^58 + 37 * q^59 - 2 * q^61 - 18 * q^62 + 14 * q^64 + 16 * q^65 - 26 * q^68 + 7 * q^70 - 13 * q^71 - 6 * q^73 - 13 * q^74 - 50 * q^76 + 2 * q^77 + 14 * q^79 + 10 * q^80 + 39 * q^82 + 5 * q^83 + 29 * q^85 + 37 * q^86 + 48 * q^88 + 30 * q^89 - 33 * q^91 + 31 * q^92 + 34 * q^94 - 43 * q^95 - 9 * q^97 - 36 * q^98

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 $

$2$	−1.10637	−0.782322	−0.391161	−	0.920322i	$-0.627926\pi$
$2$	−1.10637	−0.782322	−0.391161	+	0.920322i	$0.627926\pi$
$3$	0	0
$3$	0	0
$4$	−0.775946	−0.387973
$5$	2.19483	0.981556	0.490778	−	0.871285i	$-0.336713\pi$
$5$	2.19483	0.981556	0.490778	+	0.871285i	$0.336713\pi$
$6$	0	0
$7$	−4.52759	−1.71127	−0.855633	−	0.517582i	$-0.826832\pi$
$7$	−4.52759	−1.71127	−0.855633	+	0.517582i	$0.826832\pi$
$8$	3.07122	1.08584
$9$	0	0
$10$	−2.42829	−0.767892
$11$	2.62592	0.791746	0.395873	−	0.918305i	$-0.370442\pi$
$11$	2.62592	0.791746	0.395873	+	0.918305i	$0.370442\pi$
$12$	0	0
$13$	4.50686	1.24998	0.624989	−	0.780633i	$-0.285103\pi$
$13$	4.50686	1.24998	0.624989	+	0.780633i	$0.285103\pi$
$14$	5.00918	1.33876
$15$	0	0
$16$	−1.84602	−0.461504
$17$	−2.14478	−0.520186	−0.260093	−	0.965584i	$-0.583753\pi$
$17$	−2.14478	−0.520186	−0.260093	+	0.965584i	$0.583753\pi$
$18$	0	0
$19$	−6.09764	−1.39889	−0.699447	−	0.714684i	$-0.746570\pi$
$19$	−6.09764	−1.39889	−0.699447	+	0.714684i	$0.746570\pi$
$20$	−1.70307	−0.380817
$21$	0	0
$22$	−2.90524	−0.619400
$23$	6.18843	1.29038	0.645189	−	0.764023i	$-0.276779\pi$
$23$	6.18843	1.29038	0.645189	+	0.764023i	$0.276779\pi$
$24$	0	0
$25$	−0.182739	−0.0365478
$26$	−4.98626	−0.977885
$27$	0	0
$28$	3.51316	0.663925
$29$	−1.84359	−0.342345	−0.171173	−	0.985241i	$-0.554756\pi$
$29$	−1.84359	−0.342345	−0.171173	+	0.985241i	$0.554756\pi$
$30$	0	0
$31$	1.20401	0.216247	0.108123	−	0.994137i	$-0.465516\pi$
$31$	1.20401	0.216247	0.108123	+	0.994137i	$0.465516\pi$
$32$	−4.10007	−0.724797
$33$	0	0
$34$	2.37292	0.406953
$35$	−9.93726	−1.67970
$36$	0	0
$37$	10.7075	1.76030	0.880152	−	0.474691i	$-0.157440\pi$
$37$	10.7075	1.76030	0.880152	+	0.474691i	$0.157440\pi$
$38$	6.74624	1.09439
$39$	0	0
$40$	6.74080	1.06581
$41$	−0.353073	−0.0551407	−0.0275703	−	0.999620i	$-0.508777\pi$
$41$	−0.353073	−0.0551407	−0.0275703	+	0.999620i	$0.508777\pi$
$42$	0	0
$43$	−11.9236	−1.81833	−0.909167	−	0.416432i	$-0.863280\pi$
$43$	−11.9236	−1.81833	−0.909167	+	0.416432i	$0.863280\pi$
$44$	−2.03758	−0.307176
$45$	0	0
$46$	−6.84670	−1.00949
$47$	−0.986349	−0.143874	−0.0719369	−	0.997409i	$-0.522918\pi$
$47$	−0.986349	−0.143874	−0.0719369	+	0.997409i	$0.522918\pi$
$48$	0	0
$49$	13.4990	1.92843
$50$	0.202177	0.0285922
$51$	0	0
$52$	−3.49708	−0.484958
$53$	7.36740	1.01199	0.505995	−	0.862536i	$-0.331125\pi$
$53$	7.36740	1.01199	0.505995	+	0.862536i	$0.331125\pi$
$54$	0	0
$55$	5.76345	0.777143
$56$	−13.9052	−1.85816
$57$	0	0
$58$	2.03969	0.267824
$59$	4.63958	0.604021	0.302011	−	0.953305i	$-0.402342\pi$
$59$	4.63958	0.604021	0.302011	+	0.953305i	$0.402342\pi$
$60$	0	0
$61$	11.9846	1.53447	0.767237	−	0.641363i	$-0.221631\pi$
$61$	11.9846	1.53447	0.767237	+	0.641363i	$0.221631\pi$
$62$	−1.33208	−0.169174
$63$	0	0
$64$	8.22822	1.02853
$65$	9.89178	1.22692
$66$	0	0
$67$	−0.652622	−0.0797304	−0.0398652	−	0.999205i	$-0.512693\pi$
$67$	−0.652622	−0.0797304	−0.0398652	+	0.999205i	$0.512693\pi$
$68$	1.66424	0.201818
$69$	0	0
$70$	10.9943	1.31407
$71$	13.2454	1.57194	0.785972	−	0.618262i	$-0.212163\pi$
$71$	13.2454	1.57194	0.785972	+	0.618262i	$0.212163\pi$
$72$	0	0
$73$	−7.23916	−0.847279	−0.423640	−	0.905831i	$-0.639248\pi$
$73$	−7.23916	−0.847279	−0.423640	+	0.905831i	$0.639248\pi$
$74$	−11.8465	−1.37712
$75$	0	0
$76$	4.73144	0.542733
$77$	−11.8891	−1.35489
$78$	0	0
$79$	7.65804	0.861596	0.430798	−	0.902448i	$-0.358232\pi$
$79$	7.65804	0.861596	0.430798	+	0.902448i	$0.358232\pi$
$80$	−4.05168	−0.452992
$81$	0	0
$82$	0.390629	0.0431377
$83$	6.48636	0.711971	0.355985	−	0.934491i	$-0.384145\pi$
$83$	6.48636	0.711971	0.355985	+	0.934491i	$0.384145\pi$
$84$	0	0
$85$	−4.70742	−0.510592
$86$	13.1919	1.42252
$87$	0	0
$88$	8.06480	0.859711
$89$	6.00336	0.636355	0.318177	−	0.948031i	$-0.396929\pi$
$89$	6.00336	0.636355	0.318177	+	0.948031i	$0.396929\pi$
$90$	0	0
$91$	−20.4052	−2.13905
$92$	−4.80189	−0.500632
$93$	0	0
$94$	1.09127	0.112556
$95$	−13.3833	−1.37309
$96$	0	0
$97$	−8.29839	−0.842573	−0.421287	−	0.906928i	$-0.638421\pi$
$97$	−8.29839	−0.842573	−0.421287	+	0.906928i	$0.638421\pi$
$98$	−14.9349	−1.50866
$99$	0	0
$100$	0.141796	0.0141796
$101$	12.2104	1.21498	0.607489	−	0.794328i	$-0.292177\pi$
$101$	12.2104	1.21498	0.607489	+	0.794328i	$0.292177\pi$
$102$	0	0
$103$	14.8965	1.46780	0.733898	−	0.679260i	$-0.237699\pi$
$103$	14.8965	1.46780	0.733898	+	0.679260i	$0.237699\pi$
$104$	13.8416	1.35728
$105$	0	0
$106$	−8.15107	−0.791702
$107$	−10.0613	−0.972666	−0.486333	−	0.873773i	$-0.661666\pi$
$107$	−10.0613	−0.972666	−0.486333	+	0.873773i	$0.661666\pi$
$108$	0	0
$109$	−10.9585	−1.04964	−0.524819	−	0.851214i	$-0.675867\pi$
$109$	−10.9585	−1.04964	−0.524819	+	0.851214i	$0.675867\pi$
$110$	−6.37650	−0.607976
$111$	0	0
$112$	8.35799	0.789756
$113$	12.0220	1.13093	0.565465	−	0.824772i	$-0.308697\pi$
$113$	12.0220	1.13093	0.565465	+	0.824772i	$0.308697\pi$
$114$	0	0
$115$	13.5825	1.26658
$116$	1.43052	0.132821
$117$	0	0
$118$	−5.13309	−0.472539
$119$	9.71069	0.890177
$120$	0	0
$121$	−4.10452	−0.373138
$122$	−13.2594	−1.20045
$123$	0	0
$124$	−0.934247	−0.0838979
$125$	−11.3752	−1.01743
$126$	0	0
$127$	−2.03046	−0.180174	−0.0900869	−	0.995934i	$-0.528714\pi$
$127$	−2.03046	−0.180174	−0.0900869	+	0.995934i	$0.528714\pi$
$128$	−0.903318	−0.0798428
$129$	0	0
$130$	−10.9440	−0.959849
$131$	−7.73416	−0.675737	−0.337868	−	0.941193i	$-0.609706\pi$
$131$	−7.73416	−0.675737	−0.337868	+	0.941193i	$0.609706\pi$
$132$	0	0
$133$	27.6076	2.39388
$134$	0.722041	0.0623748
$135$	0	0
$136$	−6.58710	−0.564839
$137$	19.1222	1.63372	0.816862	−	0.576833i	$-0.195712\pi$
$137$	19.1222	1.63372	0.816862	+	0.576833i	$0.195712\pi$
$138$	0	0
$139$	−2.90796	−0.246650	−0.123325	−	0.992366i	$-0.539356\pi$
$139$	−2.90796	−0.246650	−0.123325	+	0.992366i	$0.539356\pi$
$140$	7.71078	0.651680
$141$	0	0
$142$	−14.6544	−1.22977
$143$	11.8347	0.989666
$144$	0	0
$145$	−4.04635	−0.336031
$146$	8.00918	0.662845
$147$	0	0
$148$	−8.30845	−0.682951
$149$	13.5856	1.11298	0.556488	−	0.830855i	$-0.312148\pi$
$149$	13.5856	1.11298	0.556488	+	0.830855i	$0.312148\pi$
$150$	0	0
$151$	17.3253	1.40991	0.704955	−	0.709252i	$-0.250967\pi$
$151$	17.3253	1.40991	0.704955	+	0.709252i	$0.250967\pi$
$152$	−18.7272	−1.51898
$153$	0	0
$154$	13.1537	1.05996
$155$	2.64259	0.212258
$156$	0	0
$157$	14.9268	1.19129	0.595646	−	0.803247i	$-0.296896\pi$
$157$	14.9268	1.19129	0.595646	+	0.803247i	$0.296896\pi$
$158$	−8.47262	−0.674045
$159$	0	0
$160$	−8.99894	−0.711429
$161$	−28.0187	−2.20818
$162$	0	0
$163$	18.0795	1.41609	0.708046	−	0.706166i	$-0.249577\pi$
$163$	18.0795	1.41609	0.708046	+	0.706166i	$0.249577\pi$
$164$	0.273965	0.0213931
$165$	0	0
$166$	−7.17632	−0.556990
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	7.31181	0.562447
$170$	5.20815	0.399447
$171$	0	0
$172$	9.25208	0.705465
$173$	17.2480	1.31134	0.655671	−	0.755047i	$-0.272386\pi$
$173$	17.2480	1.31134	0.655671	+	0.755047i	$0.272386\pi$
$174$	0	0
$175$	0.827368	0.0625431
$176$	−4.84750	−0.365394
$177$	0	0
$178$	−6.64193	−0.497834
$179$	13.8251	1.03334	0.516670	−	0.856185i	$-0.327171\pi$
$179$	13.8251	1.03334	0.516670	+	0.856185i	$0.327171\pi$
$180$	0	0
$181$	0.887785	0.0659886	0.0329943	−	0.999456i	$-0.489496\pi$
$181$	0.887785	0.0659886	0.0329943	+	0.999456i	$0.489496\pi$
$182$	22.5757	1.67342
$183$	0	0
$184$	19.0061	1.40115
$185$	23.5011	1.72784
$186$	0	0
$187$	−5.63204	−0.411855
$188$	0.765353	0.0558191
$189$	0	0
$190$	14.8068	1.07420
$191$	−12.8833	−0.932200	−0.466100	−	0.884732i	$-0.654341\pi$
$191$	−12.8833	−0.932200	−0.466100	+	0.884732i	$0.654341\pi$
$192$	0	0
$193$	19.6292	1.41294	0.706470	−	0.707743i	$-0.250286\pi$
$193$	19.6292	1.41294	0.706470	+	0.707743i	$0.250286\pi$
$194$	9.18108	0.659163
$195$	0	0
$196$	−10.4745	−0.748180
$197$	4.98382	0.355082	0.177541	−	0.984113i	$-0.443186\pi$
$197$	4.98382	0.355082	0.177541	+	0.984113i	$0.443186\pi$
$198$	0	0
$199$	−14.6453	−1.03818	−0.519089	−	0.854720i	$-0.673729\pi$
$199$	−14.6453	−1.03818	−0.519089	+	0.854720i	$0.673729\pi$
$200$	−0.561233	−0.0396852
$201$	0	0
$202$	−13.5092	−0.950504
$203$	8.34700	0.585844
$204$	0	0
$205$	−0.774933	−0.0541236
$206$	−16.4810	−1.14829
$207$	0	0
$208$	−8.31974	−0.576870
$209$	−16.0119	−1.10757
$210$	0	0
$211$	3.01322	0.207438	0.103719	−	0.994607i	$-0.466926\pi$
$211$	3.01322	0.207438	0.103719	+	0.994607i	$0.466926\pi$
$212$	−5.71671	−0.392625
$213$	0	0
$214$	11.1316	0.760938
$215$	−26.1703	−1.78480
$216$	0	0
$217$	−5.45126	−0.370056
$218$	12.1242	0.821154
$219$	0	0
$220$	−4.47212	−0.301511
$221$	−9.66624	−0.650222
$222$	0	0
$223$	−9.11072	−0.610099	−0.305050	−	0.952336i	$-0.598673\pi$
$223$	−9.11072	−0.610099	−0.305050	+	0.952336i	$0.598673\pi$
$224$	18.5634	1.24032
$225$	0	0
$226$	−13.3007	−0.884751
$227$	18.3539	1.21819	0.609097	−	0.793096i	$-0.291532\pi$
$227$	18.3539	1.21819	0.609097	+	0.793096i	$0.291532\pi$
$228$	0	0
$229$	5.54404	0.366360	0.183180	−	0.983079i	$-0.441361\pi$
$229$	5.54404	0.366360	0.183180	+	0.983079i	$0.441361\pi$
$230$	−15.0273	−0.990871
$231$	0	0
$232$	−5.66206	−0.371733
$233$	−2.85999	−0.187364	−0.0936821	−	0.995602i	$-0.529864\pi$
$233$	−2.85999	−0.187364	−0.0936821	+	0.995602i	$0.529864\pi$
$234$	0	0
$235$	−2.16486	−0.141220
$236$	−3.60006	−0.234344
$237$	0	0
$238$	−10.7436	−0.696405
$239$	−25.0961	−1.62333	−0.811666	−	0.584122i	$-0.801439\pi$
$239$	−25.0961	−1.62333	−0.811666	+	0.584122i	$0.801439\pi$
$240$	0	0
$241$	−18.3872	−1.18442	−0.592211	−	0.805783i	$-0.701745\pi$
$241$	−18.3872	−1.18442	−0.592211	+	0.805783i	$0.701745\pi$
$242$	4.54112	0.291914
$243$	0	0
$244$	−9.29943	−0.595335
$245$	29.6280	1.89287
$246$	0	0
$247$	−27.4812	−1.74859
$248$	3.69778	0.234809
$249$	0	0
$250$	12.5852	0.795957
$251$	−15.0744	−0.951491	−0.475745	−	0.879583i	$-0.657822\pi$
$251$	−15.0744	−0.951491	−0.475745	+	0.879583i	$0.657822\pi$
$252$	0	0
$253$	16.2504	1.02165
$254$	2.24643	0.140954
$255$	0	0
$256$	−15.4570	−0.966065
$257$	−26.1018	−1.62819	−0.814094	−	0.580733i	$-0.802766\pi$
$257$	−26.1018	−1.62819	−0.814094	+	0.580733i	$0.802766\pi$
$258$	0	0
$259$	−48.4792	−3.01235
$260$	−7.67549	−0.476014
$261$	0	0
$262$	8.55684	0.528643
$263$	17.5960	1.08502	0.542508	−	0.840051i	$-0.317475\pi$
$263$	17.5960	1.08502	0.542508	+	0.840051i	$0.317475\pi$
$264$	0	0
$265$	16.1702	0.993326
$266$	−30.5442	−1.87279
$267$	0	0
$268$	0.506399	0.0309333
$269$	17.8079	1.08577	0.542885	−	0.839807i	$-0.317332\pi$
$269$	17.8079	1.08577	0.542885	+	0.839807i	$0.317332\pi$
$270$	0	0
$271$	−20.5804	−1.25017	−0.625085	−	0.780557i	$-0.714936\pi$
$271$	−20.5804	−1.25017	−0.625085	+	0.780557i	$0.714936\pi$
$272$	3.95930	0.240068
$273$	0	0
$274$	−21.1563	−1.27810
$275$	−0.479860	−0.0289366
$276$	0	0
$277$	4.86009	0.292014	0.146007	−	0.989284i	$-0.453358\pi$
$277$	4.86009	0.292014	0.146007	+	0.989284i	$0.453358\pi$
$278$	3.21728	0.192960
$279$	0	0
$280$	−30.5195	−1.82389
$281$	−0.340342	−0.0203031	−0.0101516	−	0.999948i	$-0.503231\pi$
$281$	−0.340342	−0.0203031	−0.0101516	+	0.999948i	$0.503231\pi$
$282$	0	0
$283$	−10.8080	−0.642469	−0.321235	−	0.947000i	$-0.604098\pi$
$283$	−10.8080	−0.642469	−0.321235	+	0.947000i	$0.604098\pi$
$284$	−10.2777	−0.609872
$285$	0	0
$286$	−13.0935	−0.774237
$287$	1.59857	0.0943604
$288$	0	0
$289$	−12.3999	−0.729406
$290$	4.47676	0.262884
$291$	0	0
$292$	5.61720	0.328722
$293$	−13.0134	−0.760254	−0.380127	−	0.924934i	$-0.624120\pi$
$293$	−13.0134	−0.760254	−0.380127	+	0.924934i	$0.624120\pi$
$294$	0	0
$295$	10.1831	0.592881
$296$	32.8852	1.91141
$297$	0	0
$298$	−15.0307	−0.870706
$299$	27.8904	1.61294
$300$	0	0
$301$	53.9852	3.11165
$302$	−19.1681	−1.10300
$303$	0	0
$304$	11.2563	0.645595
$305$	26.3042	1.50617
$306$	0	0
$307$	10.6004	0.605000	0.302500	−	0.953149i	$-0.402179\pi$
$307$	10.6004	0.605000	0.302500	+	0.953149i	$0.402179\pi$
$308$	9.22530	0.525660
$309$	0	0
$310$	−2.92368	−0.166054
$311$	−23.2712	−1.31959	−0.659795	−	0.751446i	$-0.729357\pi$
$311$	−23.2712	−1.31959	−0.659795	+	0.751446i	$0.729357\pi$
$312$	0	0
$313$	−6.59154	−0.372576	−0.186288	−	0.982495i	$-0.559646\pi$
$313$	−6.59154	−0.372576	−0.186288	+	0.982495i	$0.559646\pi$
$314$	−16.5146	−0.931973
$315$	0	0
$316$	−5.94222	−0.334276
$317$	−8.90860	−0.500357	−0.250178	−	0.968200i	$-0.580489\pi$
$317$	−8.90860	−0.500357	−0.250178	+	0.968200i	$0.580489\pi$
$318$	0	0
$319$	−4.84112	−0.271051
$320$	18.0595	1.00956
$321$	0	0
$322$	30.9990	1.72751
$323$	13.0781	0.727686
$324$	0	0
$325$	−0.823581	−0.0456840
$326$	−20.0026	−1.10784
$327$	0	0
$328$	−1.08436	−0.0598740
$329$	4.46578	0.246206
$330$	0	0
$331$	−11.2953	−0.620845	−0.310422	−	0.950599i	$-0.600470\pi$
$331$	−11.2953	−0.620845	−0.310422	+	0.950599i	$0.600470\pi$
$332$	−5.03307	−0.276226
$333$	0	0
$334$	1.10637	0.0605379
$335$	−1.43239	−0.0782599
$336$	0	0
$337$	21.4635	1.16919	0.584596	−	0.811325i	$-0.301253\pi$
$337$	21.4635	1.16919	0.584596	+	0.811325i	$0.301253\pi$
$338$	−8.08957	−0.440014
$339$	0	0
$340$	3.65271	0.198096
$341$	3.16164	0.171212
$342$	0	0
$343$	−29.4250	−1.58880
$344$	−36.6201	−1.97442
$345$	0	0
$346$	−19.0827	−1.02589
$347$	−23.3110	−1.25140	−0.625699	−	0.780065i	$-0.715186\pi$
$347$	−23.3110	−1.25140	−0.625699	+	0.780065i	$0.715186\pi$
$348$	0	0
$349$	−22.1588	−1.18613	−0.593066	−	0.805154i	$-0.702083\pi$
$349$	−22.1588	−1.18613	−0.593066	+	0.805154i	$0.702083\pi$
$350$	−0.915375	−0.0489288
$351$	0	0
$352$	−10.7665	−0.573855
$353$	11.5024	0.612209	0.306105	−	0.951998i	$-0.400974\pi$
$353$	11.5024	0.612209	0.306105	+	0.951998i	$0.400974\pi$
$354$	0	0
$355$	29.0714	1.54295
$356$	−4.65828	−0.246888
$357$	0	0
$358$	−15.2957	−0.808404
$359$	−33.4833	−1.76718	−0.883590	−	0.468262i	$-0.844880\pi$
$359$	−33.4833	−1.76718	−0.883590	+	0.468262i	$0.844880\pi$
$360$	0	0
$361$	18.1812	0.956906
$362$	−0.982219	−0.0516243
$363$	0	0
$364$	15.8333	0.829893
$365$	−15.8887	−0.831652
$366$	0	0
$367$	−2.74783	−0.143436	−0.0717179	−	0.997425i	$-0.522848\pi$
$367$	−2.74783	−0.143436	−0.0717179	+	0.997425i	$0.522848\pi$
$368$	−11.4239	−0.595514
$369$	0	0
$370$	−26.0009	−1.35172
$371$	−33.3566	−1.73179
$372$	0	0
$373$	8.10471	0.419646	0.209823	−	0.977739i	$-0.432711\pi$
$373$	8.10471	0.419646	0.209823	+	0.977739i	$0.432711\pi$
$374$	6.23111	0.322203
$375$	0	0
$376$	−3.02930	−0.156224
$377$	−8.30879	−0.427925
$378$	0	0
$379$	−4.29718	−0.220731	−0.110366	−	0.993891i	$-0.535202\pi$
$379$	−4.29718	−0.220731	−0.110366	+	0.993891i	$0.535202\pi$
$380$	10.3847	0.532723
$381$	0	0
$382$	14.2537	0.729280
$383$	27.4329	1.40176	0.700878	−	0.713281i	$-0.252792\pi$
$383$	27.4329	1.40176	0.700878	+	0.713281i	$0.252792\pi$
$384$	0	0
$385$	−26.0945	−1.32990
$386$	−21.7172	−1.10537
$387$	0	0
$388$	6.43910	0.326896
$389$	−1.77324	−0.0899067	−0.0449534	−	0.998989i	$-0.514314\pi$
$389$	−1.77324	−0.0899067	−0.0449534	+	0.998989i	$0.514314\pi$
$390$	0	0
$391$	−13.2728	−0.671236
$392$	41.4586	2.09397
$393$	0	0
$394$	−5.51394	−0.277789
$395$	16.8081	0.845705
$396$	0	0
$397$	−5.70497	−0.286324	−0.143162	−	0.989699i	$-0.545727\pi$
$397$	−5.70497	−0.286324	−0.143162	+	0.989699i	$0.545727\pi$
$398$	16.2031	0.812188
$399$	0	0
$400$	0.337339	0.0168670
$401$	22.7358	1.13537	0.567686	−	0.823245i	$-0.307839\pi$
$401$	22.7358	1.13537	0.567686	+	0.823245i	$0.307839\pi$
$402$	0	0
$403$	5.42631	0.270304
$404$	−9.47460	−0.471379
$405$	0	0
$406$	−9.23486	−0.458319
$407$	28.1171	1.39371
$408$	0	0
$409$	−22.6282	−1.11889	−0.559445	−	0.828867i	$-0.688986\pi$
$409$	−22.6282	−1.11889	−0.559445	+	0.828867i	$0.688986\pi$
$410$	0.857362	0.0423421
$411$	0	0
$412$	−11.5589	−0.569465
$413$	−21.0061	−1.03364
$414$	0	0
$415$	14.2364	0.698839
$416$	−18.4785	−0.905981
$417$	0	0
$418$	17.7151	0.866475
$419$	24.0907	1.17691	0.588453	−	0.808531i	$-0.299737\pi$
$419$	24.0907	1.17691	0.588453	+	0.808531i	$0.299737\pi$
$420$	0	0
$421$	−5.42990	−0.264637	−0.132319	−	0.991207i	$-0.542242\pi$
$421$	−5.42990	−0.264637	−0.132319	+	0.991207i	$0.542242\pi$
$422$	−3.33373	−0.162283
$423$	0	0
$424$	22.6269	1.09886
$425$	0.391936	0.0190117
$426$	0	0
$427$	−54.2614	−2.62590
$428$	7.80706	0.377368
$429$	0	0
$430$	28.9540	1.39628
$431$	10.8167	0.521024	0.260512	−	0.965471i	$-0.416109\pi$
$431$	10.8167	0.521024	0.260512	+	0.965471i	$0.416109\pi$
$432$	0	0
$433$	−17.6776	−0.849530	−0.424765	−	0.905304i	$-0.639643\pi$
$433$	−17.6776	−0.849530	−0.424765	+	0.905304i	$0.639643\pi$
$434$	6.03111	0.289503
$435$	0	0
$436$	8.50323	0.407231
$437$	−37.7348	−1.80510
$438$	0	0
$439$	13.9547	0.666023	0.333011	−	0.942923i	$-0.391935\pi$
$439$	13.9547	0.666023	0.333011	+	0.942923i	$0.391935\pi$
$440$	17.7008	0.843854
$441$	0	0
$442$	10.6944	0.508682
$443$	−41.1782	−1.95643	−0.978217	−	0.207584i	$-0.933440\pi$
$443$	−41.1782	−1.95643	−0.978217	+	0.207584i	$0.933440\pi$
$444$	0	0
$445$	13.1763	0.624618
$446$	10.0798	0.477294
$447$	0	0
$448$	−37.2540	−1.76009
$449$	−33.9009	−1.59988	−0.799942	−	0.600077i	$-0.795137\pi$
$449$	−33.9009	−1.59988	−0.799942	+	0.600077i	$0.795137\pi$
$450$	0	0
$451$	−0.927142	−0.0436574
$452$	−9.32839	−0.438770
$453$	0	0
$454$	−20.3062	−0.953019
$455$	−44.7859	−2.09959
$456$	0	0
$457$	−1.56208	−0.0730708	−0.0365354	−	0.999332i	$-0.511632\pi$
$457$	−1.56208	−0.0730708	−0.0365354	+	0.999332i	$0.511632\pi$
$458$	−6.13375	−0.286612
$459$	0	0
$460$	−10.5393	−0.491398
$461$	−6.34383	−0.295462	−0.147731	−	0.989028i	$-0.547197\pi$
$461$	−6.34383	−0.295462	−0.147731	+	0.989028i	$0.547197\pi$
$462$	0	0
$463$	15.7171	0.730437	0.365218	−	0.930922i	$-0.380994\pi$
$463$	15.7171	0.730437	0.365218	+	0.930922i	$0.380994\pi$
$464$	3.40329	0.157994
$465$	0	0
$466$	3.16421	0.146579
$467$	41.4281	1.91706	0.958532	−	0.284987i	$-0.0919891\pi$
$467$	41.4281	1.91706	0.958532	+	0.284987i	$0.0919891\pi$
$468$	0	0
$469$	2.95480	0.136440
$470$	2.39514	0.110480
$471$	0	0
$472$	14.2492	0.655871
$473$	−31.3105	−1.43966
$474$	0	0
$475$	1.11428	0.0511266
$476$	−7.53497	−0.345365
$477$	0	0
$478$	27.7656	1.26997
$479$	−21.7486	−0.993720	−0.496860	−	0.867831i	$-0.665514\pi$
$479$	−21.7486	−0.993720	−0.496860	+	0.867831i	$0.665514\pi$
$480$	0	0
$481$	48.2573	2.20034
$482$	20.3430	0.926599
$483$	0	0
$484$	3.18489	0.144768
$485$	−18.2135	−0.827033
$486$	0	0
$487$	−4.22677	−0.191533	−0.0957667	−	0.995404i	$-0.530530\pi$
$487$	−4.22677	−0.191533	−0.0957667	+	0.995404i	$0.530530\pi$
$488$	36.8075	1.66620
$489$	0	0
$490$	−32.7796	−1.48083
$491$	11.2226	0.506469	0.253235	−	0.967405i	$-0.418505\pi$
$491$	11.2226	0.506469	0.253235	+	0.967405i	$0.418505\pi$
$492$	0	0
$493$	3.95409	0.178083
$494$	30.4044	1.36796
$495$	0	0
$496$	−2.22262	−0.0997987
$497$	−59.9699	−2.69002
$498$	0	0
$499$	8.02996	0.359470	0.179735	−	0.983715i	$-0.442476\pi$
$499$	8.02996	0.359470	0.179735	+	0.983715i	$0.442476\pi$
$500$	8.82655	0.394735
$501$	0	0
$502$	16.6779	0.744372
$503$	−35.0823	−1.56424	−0.782121	−	0.623127i	$-0.785862\pi$
$503$	−35.0823	−1.56424	−0.782121	+	0.623127i	$0.785862\pi$
$504$	0	0
$505$	26.7997	1.19257
$506$	−17.9789	−0.799260
$507$	0	0
$508$	1.57552	0.0699026
$509$	0.620513	0.0275038	0.0137519	−	0.999905i	$-0.495623\pi$
$509$	0.620513	0.0275038	0.0137519	+	0.999905i	$0.495623\pi$
$510$	0	0
$511$	32.7759	1.44992
$512$	18.9078	0.835616
$513$	0	0
$514$	28.8783	1.27377
$515$	32.6952	1.44072
$516$	0	0
$517$	−2.59008	−0.113911
$518$	53.6359	2.35663
$519$	0	0
$520$	30.3799	1.33224
$521$	−17.2556	−0.755983	−0.377992	−	0.925809i	$-0.623385\pi$
$521$	−17.2556	−0.755983	−0.377992	+	0.925809i	$0.623385\pi$
$522$	0	0
$523$	27.4817	1.20169	0.600845	−	0.799366i	$-0.294831\pi$
$523$	27.4817	1.20169	0.600845	+	0.799366i	$0.294831\pi$
$524$	6.00129	0.262168
$525$	0	0
$526$	−19.4677	−0.848831
$527$	−2.58234	−0.112488
$528$	0	0
$529$	15.2967	0.665074
$530$	−17.8902	−0.777100
$531$	0	0
$532$	−21.4220	−0.928762
$533$	−1.59125	−0.0689247
$534$	0	0
$535$	−22.0829	−0.954727
$536$	−2.00435	−0.0865746
$537$	0	0
$538$	−19.7022	−0.849421
$539$	35.4475	1.52683
$540$	0	0
$541$	14.2636	0.613239	0.306620	−	0.951832i	$-0.400802\pi$
$541$	14.2636	0.613239	0.306620	+	0.951832i	$0.400802\pi$
$542$	22.7695	0.978035
$543$	0	0
$544$	8.79376	0.377029
$545$	−24.0521	−1.03028
$546$	0	0
$547$	26.1563	1.11836	0.559182	−	0.829045i	$-0.311115\pi$
$547$	26.1563	1.11836	0.559182	+	0.829045i	$0.311115\pi$
$548$	−14.8378	−0.633841
$549$	0	0
$550$	0.530902	0.0226377
$551$	11.2415	0.478905
$552$	0	0
$553$	−34.6724	−1.47442
$554$	−5.37705	−0.228449
$555$	0	0
$556$	2.25642	0.0956936
$557$	22.0575	0.934608	0.467304	−	0.884097i	$-0.345225\pi$
$557$	22.0575	0.934608	0.467304	+	0.884097i	$0.345225\pi$
$558$	0	0
$559$	−53.7381	−2.27288
$560$	18.3443	0.775190
$561$	0	0
$562$	0.376544	0.0158836
$563$	−4.77511	−0.201247	−0.100623	−	0.994925i	$-0.532084\pi$
$563$	−4.77511	−0.201247	−0.100623	+	0.994925i	$0.532084\pi$
$564$	0	0
$565$	26.3861	1.11007
$566$	11.9576	0.502617
$567$	0	0
$568$	40.6797	1.70688
$569$	39.2560	1.64570	0.822850	−	0.568259i	$-0.192383\pi$
$569$	39.2560	1.64570	0.822850	+	0.568259i	$0.192383\pi$
$570$	0	0
$571$	27.3668	1.14527	0.572633	−	0.819812i	$-0.305922\pi$
$571$	27.3668	1.14527	0.572633	+	0.819812i	$0.305922\pi$
$572$	−9.18308	−0.383964
$573$	0	0
$574$	−1.76861	−0.0738202
$575$	−1.13087	−0.0471605
$576$	0	0
$577$	42.7549	1.77991	0.889954	−	0.456050i	$-0.150736\pi$
$577$	42.7549	1.77991	0.889954	+	0.456050i	$0.150736\pi$
$578$	13.7189	0.570630
$579$	0	0
$580$	3.13975	0.130371
$581$	−29.3676	−1.21837
$582$	0	0
$583$	19.3462	0.801240
$584$	−22.2331	−0.920011
$585$	0	0
$586$	14.3977	0.594763
$587$	21.8718	0.902747	0.451374	−	0.892335i	$-0.350934\pi$
$587$	21.8718	0.902747	0.451374	+	0.892335i	$0.350934\pi$
$588$	0	0
$589$	−7.34162	−0.302506
$590$	−11.2662	−0.463823
$591$	0	0
$592$	−19.7662	−0.812387
$593$	0.763853	0.0313677	0.0156839	−	0.999877i	$-0.495007\pi$
$593$	0.763853	0.0313677	0.0156839	+	0.999877i	$0.495007\pi$
$594$	0	0
$595$	21.3133	0.873759
$596$	−10.5417	−0.431805
$597$	0	0
$598$	−30.8571	−1.26184
$599$	−3.83490	−0.156690	−0.0783450	−	0.996926i	$-0.524964\pi$
$599$	−3.83490	−0.156690	−0.0783450	+	0.996926i	$0.524964\pi$
$600$	0	0
$601$	−38.9975	−1.59074	−0.795370	−	0.606124i	$-0.792723\pi$
$601$	−38.9975	−1.59074	−0.795370	+	0.606124i	$0.792723\pi$
$602$	−59.7276	−2.43431
$603$	0	0
$604$	−13.4435	−0.547007
$605$	−9.00870	−0.366256
$606$	0	0
$607$	25.1931	1.02256	0.511279	−	0.859415i	$-0.329172\pi$
$607$	25.1931	1.02256	0.511279	+	0.859415i	$0.329172\pi$
$608$	25.0008	1.01391
$609$	0	0
$610$	−29.1021	−1.17831
$611$	−4.44534	−0.179839
$612$	0	0
$613$	−44.1016	−1.78125	−0.890623	−	0.454742i	$-0.849731\pi$
$613$	−44.1016	−1.78125	−0.890623	+	0.454742i	$0.849731\pi$
$614$	−11.7280	−0.473304
$615$	0	0
$616$	−36.5141	−1.47119
$617$	−3.77330	−0.151908	−0.0759538	−	0.997111i	$-0.524200\pi$
$617$	−3.77330	−0.151908	−0.0759538	+	0.997111i	$0.524200\pi$
$618$	0	0
$619$	−35.1811	−1.41405	−0.707024	−	0.707190i	$-0.749963\pi$
$619$	−35.1811	−1.41405	−0.707024	+	0.707190i	$0.749963\pi$
$620$	−2.05051	−0.0823504
$621$	0	0
$622$	25.7466	1.03234
$623$	−27.1807	−1.08897
$624$	0	0
$625$	−24.0529	−0.962116
$626$	7.29268	0.291474
$627$	0	0
$628$	−11.5824	−0.462189
$629$	−22.9653	−0.915686
$630$	0	0
$631$	−15.6390	−0.622579	−0.311289	−	0.950315i	$-0.600761\pi$
$631$	−15.6390	−0.622579	−0.311289	+	0.950315i	$0.600761\pi$
$632$	23.5195	0.935557
$633$	0	0
$634$	9.85621	0.391440
$635$	−4.45650	−0.176851
$636$	0	0
$637$	60.8383	2.41050
$638$	5.35607	0.212049
$639$	0	0
$640$	−1.98263	−0.0783702
$641$	−24.1866	−0.955313	−0.477657	−	0.878547i	$-0.658514\pi$
$641$	−24.1866	−0.955313	−0.477657	+	0.878547i	$0.658514\pi$
$642$	0	0
$643$	−24.7104	−0.974483	−0.487242	−	0.873267i	$-0.661997\pi$
$643$	−24.7104	−0.974483	−0.487242	+	0.873267i	$0.661997\pi$
$644$	21.7410	0.856714
$645$	0	0
$646$	−14.4692	−0.569284
$647$	−5.00060	−0.196594	−0.0982969	−	0.995157i	$-0.531339\pi$
$647$	−5.00060	−0.196594	−0.0982969	+	0.995157i	$0.531339\pi$
$648$	0	0
$649$	12.1832	0.478232
$650$	0.911185	0.0357396
$651$	0	0
$652$	−14.0287	−0.549406
$653$	−14.9748	−0.586009	−0.293005	−	0.956111i	$-0.594655\pi$
$653$	−14.9748	−0.586009	−0.293005	+	0.956111i	$0.594655\pi$
$654$	0	0
$655$	−16.9751	−0.663273
$656$	0.651777	0.0254476
$657$	0	0
$658$	−4.94080	−0.192613
$659$	33.2094	1.29366	0.646828	−	0.762636i	$-0.276095\pi$
$659$	33.2094	1.29366	0.646828	+	0.762636i	$0.276095\pi$
$660$	0	0
$661$	−5.56439	−0.216430	−0.108215	−	0.994128i	$-0.534513\pi$
$661$	−5.56439	−0.216430	−0.108215	+	0.994128i	$0.534513\pi$
$662$	12.4968	0.485700
$663$	0	0
$664$	19.9211	0.773087
$665$	60.5939	2.34973
$666$	0	0
$667$	−11.4089	−0.441755
$668$	0.775946	0.0300223
$669$	0	0
$670$	1.58475	0.0612244
$671$	31.4707	1.21491
$672$	0	0
$673$	42.9558	1.65582	0.827911	−	0.560859i	$-0.189529\pi$
$673$	42.9558	1.65582	0.827911	+	0.560859i	$0.189529\pi$
$674$	−23.7466	−0.914684
$675$	0	0
$676$	−5.67357	−0.218214
$677$	12.7007	0.488126	0.244063	−	0.969759i	$-0.421520\pi$
$677$	12.7007	0.488126	0.244063	+	0.969759i	$0.421520\pi$
$678$	0	0
$679$	37.5717	1.44187
$680$	−14.4575	−0.554422
$681$	0	0
$682$	−3.49794	−0.133943
$683$	29.4729	1.12775	0.563874	−	0.825861i	$-0.309310\pi$
$683$	29.4729	1.12775	0.563874	+	0.825861i	$0.309310\pi$
$684$	0	0
$685$	41.9700	1.60359
$686$	32.5549	1.24295
$687$	0	0
$688$	22.0112	0.839168
$689$	33.2039	1.26497
$690$	0	0
$691$	−13.4746	−0.512599	−0.256300	−	0.966597i	$-0.582503\pi$
$691$	−13.4746	−0.512599	−0.256300	+	0.966597i	$0.582503\pi$
$692$	−13.3835	−0.508765
$693$	0	0
$694$	25.7905	0.978995
$695$	−6.38247	−0.242101
$696$	0	0
$697$	0.757264	0.0286834
$698$	24.5158	0.927937
$699$	0	0
$700$	−0.641993	−0.0242650
$701$	32.7670	1.23759	0.618796	−	0.785552i	$-0.287621\pi$
$701$	32.7670	1.23759	0.618796	+	0.785552i	$0.287621\pi$
$702$	0	0
$703$	−65.2906	−2.46248
$704$	21.6067	0.814333
$705$	0	0
$706$	−12.7259	−0.478944
$707$	−55.2836	−2.07915
$708$	0	0
$709$	−34.9670	−1.31321	−0.656607	−	0.754233i	$-0.728009\pi$
$709$	−34.9670	−1.31321	−0.656607	+	0.754233i	$0.728009\pi$
$710$	−32.1638	−1.20708
$711$	0	0
$712$	18.4376	0.690980
$713$	7.45094	0.279040
$714$	0	0
$715$	25.9751	0.971413
$716$	−10.7276	−0.400908
$717$	0	0
$718$	37.0449	1.38250
$719$	9.48429	0.353704	0.176852	−	0.984237i	$-0.443409\pi$
$719$	9.48429	0.353704	0.176852	+	0.984237i	$0.443409\pi$
$720$	0	0
$721$	−67.4452	−2.51179
$722$	−20.1152	−0.748608
$723$	0	0
$724$	−0.688874	−0.0256018
$725$	0.336896	0.0125120
$726$	0	0
$727$	−14.6705	−0.544098	−0.272049	−	0.962283i	$-0.587701\pi$
$727$	−14.6705	−0.544098	−0.272049	+	0.962283i	$0.587701\pi$
$728$	−62.6689	−2.32267
$729$	0	0
$730$	17.5788	0.650619
$731$	25.5736	0.945872
$732$	0	0
$733$	1.04809	0.0387121	0.0193560	−	0.999813i	$-0.493838\pi$
$733$	1.04809	0.0387121	0.0193560	+	0.999813i	$0.493838\pi$
$734$	3.04012	0.112213
$735$	0	0
$736$	−25.3730	−0.935262
$737$	−1.71374	−0.0631263
$738$	0	0
$739$	11.8911	0.437422	0.218711	−	0.975790i	$-0.429815\pi$
$739$	11.8911	0.437422	0.218711	+	0.975790i	$0.429815\pi$
$740$	−18.2356	−0.670354
$741$	0	0
$742$	36.9047	1.35481
$743$	−36.4383	−1.33679	−0.668396	−	0.743806i	$-0.733019\pi$
$743$	−36.4383	−1.33679	−0.668396	+	0.743806i	$0.733019\pi$
$744$	0	0
$745$	29.8181	1.09245
$746$	−8.96680	−0.328298
$747$	0	0
$748$	4.37016	0.159789
$749$	45.5536	1.66449
$750$	0	0
$751$	−17.9405	−0.654659	−0.327330	−	0.944910i	$-0.606149\pi$
$751$	−17.9405	−0.654659	−0.327330	+	0.944910i	$0.606149\pi$
$752$	1.82081	0.0663983
$753$	0	0
$754$	9.19260	0.334775
$755$	38.0259	1.38391
$756$	0	0
$757$	−12.8400	−0.466679	−0.233339	−	0.972395i	$-0.574965\pi$
$757$	−12.8400	−0.466679	−0.233339	+	0.972395i	$0.574965\pi$
$758$	4.75427	0.172683
$759$	0	0
$760$	−41.1030	−1.49096
$761$	51.8862	1.88087	0.940437	−	0.339968i	$-0.110416\pi$
$761$	51.8862	1.88087	0.940437	+	0.339968i	$0.110416\pi$
$762$	0	0
$763$	49.6157	1.79621
$764$	9.99672	0.361669
$765$	0	0
$766$	−30.3509	−1.09662
$767$	20.9099	0.755014
$768$	0	0
$769$	−28.7814	−1.03789	−0.518943	−	0.854809i	$-0.673674\pi$
$769$	−28.7814	−1.03789	−0.518943	+	0.854809i	$0.673674\pi$
$770$	28.8702	1.04041
$771$	0	0
$772$	−15.2312	−0.548183
$773$	6.52345	0.234632	0.117316	−	0.993095i	$-0.462571\pi$
$773$	6.52345	0.234632	0.117316	+	0.993095i	$0.462571\pi$
$774$	0	0
$775$	−0.220020	−0.00790335
$776$	−25.4862	−0.914901
$777$	0	0
$778$	1.96186	0.0703360
$779$	2.15291	0.0771360
$780$	0	0
$781$	34.7815	1.24458
$782$	14.6847	0.525123
$783$	0	0
$784$	−24.9194	−0.889980
$785$	32.7618	1.16932
$786$	0	0
$787$	27.3603	0.975290	0.487645	−	0.873042i	$-0.337856\pi$
$787$	27.3603	0.975290	0.487645	+	0.873042i	$0.337856\pi$
$788$	−3.86717	−0.137762
$789$	0	0
$790$	−18.5959	−0.661613
$791$	−54.4304	−1.93532
$792$	0	0
$793$	54.0131	1.91806
$794$	6.31181	0.223998
$795$	0	0
$796$	11.3640	0.402785
$797$	−16.1698	−0.572764	−0.286382	−	0.958116i	$-0.592453\pi$
$797$	−16.1698	−0.572764	−0.286382	+	0.958116i	$0.592453\pi$
$798$	0	0
$799$	2.11550	0.0748411
$800$	0.749244	0.0264898
$801$	0	0
$802$	−25.1542	−0.888226
$803$	−19.0095	−0.670830
$804$	0	0
$805$	−61.4961	−2.16745
$806$	−6.00350	−0.211464
$807$	0	0
$808$	37.5008	1.31927
$809$	9.51312	0.334464	0.167232	−	0.985918i	$-0.446517\pi$
$809$	9.51312	0.334464	0.167232	+	0.985918i	$0.446517\pi$
$810$	0	0
$811$	−3.25453	−0.114282	−0.0571410	−	0.998366i	$-0.518198\pi$
$811$	−3.25453	−0.114282	−0.0571410	+	0.998366i	$0.518198\pi$
$812$	−6.47682	−0.227292
$813$	0	0
$814$	−31.1079	−1.09033
$815$	39.6813	1.38997
$816$	0	0
$817$	72.7059	2.54366
$818$	25.0351	0.875332
$819$	0	0
$820$	0.601306	0.0209985
$821$	−14.1377	−0.493408	−0.246704	−	0.969091i	$-0.579348\pi$
$821$	−14.1377	−0.493408	−0.246704	+	0.969091i	$0.579348\pi$
$822$	0	0
$823$	−4.99950	−0.174272	−0.0871358	−	0.996196i	$-0.527771\pi$
$823$	−4.99950	−0.174272	−0.0871358	+	0.996196i	$0.527771\pi$
$824$	45.7505	1.59379
$825$	0	0
$826$	23.2405	0.808640
$827$	30.7739	1.07011	0.535056	−	0.844816i	$-0.320290\pi$
$827$	30.7739	1.07011	0.535056	+	0.844816i	$0.320290\pi$
$828$	0	0
$829$	11.7810	0.409171	0.204586	−	0.978849i	$-0.434415\pi$
$829$	11.7810	0.409171	0.204586	+	0.978849i	$0.434415\pi$
$830$	−15.7508	−0.546717
$831$	0	0
$832$	37.0835	1.28564
$833$	−28.9525	−1.00314
$834$	0	0
$835$	−2.19483	−0.0759551
$836$	12.4244	0.429707
$837$	0	0
$838$	−26.6532	−0.920719
$839$	27.5577	0.951398	0.475699	−	0.879608i	$-0.342195\pi$
$839$	27.5577	0.951398	0.475699	+	0.879608i	$0.342195\pi$
$840$	0	0
$841$	−25.6012	−0.882800
$842$	6.00747	0.207031
$843$	0	0
$844$	−2.33809	−0.0804805
$845$	16.0482	0.552073
$846$	0	0
$847$	18.5836	0.638539
$848$	−13.6003	−0.467038
$849$	0	0
$850$	−0.433626	−0.0148732
$851$	66.2627	2.27146
$852$	0	0
$853$	11.9459	0.409020	0.204510	−	0.978864i	$-0.434440\pi$
$853$	11.9459	0.409020	0.204510	+	0.978864i	$0.434440\pi$
$854$	60.0332	2.05429
$855$	0	0
$856$	−30.9006	−1.05616
$857$	41.7299	1.42547	0.712734	−	0.701435i	$-0.247457\pi$
$857$	41.7299	1.42547	0.712734	+	0.701435i	$0.247457\pi$
$858$	0	0
$859$	24.4056	0.832708	0.416354	−	0.909203i	$-0.363308\pi$
$859$	24.4056	0.832708	0.416354	+	0.909203i	$0.363308\pi$
$860$	20.3067	0.692453
$861$	0	0
$862$	−11.9673	−0.407609
$863$	6.92599	0.235763	0.117882	−	0.993028i	$-0.462390\pi$
$863$	6.92599	0.235763	0.117882	+	0.993028i	$0.462390\pi$
$864$	0	0
$865$	37.8564	1.28715
$866$	19.5579	0.664606
$867$	0	0
$868$	4.22988	0.143572
$869$	20.1094	0.682166
$870$	0	0
$871$	−2.94128	−0.0996614
$872$	−33.6561	−1.13974
$873$	0	0
$874$	41.7487	1.41217
$875$	51.5022	1.74109
$876$	0	0
$877$	28.7443	0.970625	0.485312	−	0.874341i	$-0.338706\pi$
$877$	28.7443	0.970625	0.485312	+	0.874341i	$0.338706\pi$
$878$	−15.4391	−0.521044
$879$	0	0
$880$	−10.6394	−0.358655
$881$	−39.8532	−1.34269	−0.671343	−	0.741146i	$-0.734282\pi$
$881$	−39.8532	−1.34269	−0.671343	+	0.741146i	$0.734282\pi$
$882$	0	0
$883$	−32.7450	−1.10196	−0.550979	−	0.834519i	$-0.685745\pi$
$883$	−32.7450	−1.10196	−0.550979	+	0.834519i	$0.685745\pi$
$884$	7.50048	0.252268
$885$	0	0
$886$	45.5583	1.53056
$887$	−34.6559	−1.16363	−0.581815	−	0.813321i	$-0.697657\pi$
$887$	−34.6559	−1.16363	−0.581815	+	0.813321i	$0.697657\pi$
$888$	0	0
$889$	9.19306	0.308325
$890$	−14.5779	−0.488652
$891$	0	0
$892$	7.06943	0.236702
$893$	6.01440	0.201264
$894$	0	0
$895$	30.3438	1.01428
$896$	4.08985	0.136632
$897$	0	0
$898$	37.5070	1.25162
$899$	−2.21970	−0.0740310
$900$	0	0
$901$	−15.8015	−0.526423
$902$	1.02576	0.0341541
$903$	0	0
$904$	36.9221	1.22801
$905$	1.94853	0.0647715
$906$	0	0
$907$	−12.3392	−0.409717	−0.204858	−	0.978792i	$-0.565673\pi$
$907$	−12.3392	−0.409717	−0.204858	+	0.978792i	$0.565673\pi$
$908$	−14.2417	−0.472626
$909$	0	0
$910$	49.5497	1.64256
$911$	−52.4048	−1.73625	−0.868124	−	0.496348i	$-0.834674\pi$
$911$	−52.4048	−1.73625	−0.868124	+	0.496348i	$0.834674\pi$
$912$	0	0
$913$	17.0327	0.563700
$914$	1.72823	0.0571649
$915$	0	0
$916$	−4.30187	−0.142138
$917$	35.0171	1.15637
$918$	0	0
$919$	−34.8078	−1.14820	−0.574101	−	0.818784i	$-0.694648\pi$
$919$	−34.8078	−1.14820	−0.574101	+	0.818784i	$0.694648\pi$
$920$	41.7150	1.37530
$921$	0	0
$922$	7.01862	0.231146
$923$	59.6954	1.96490
$924$	0	0
$925$	−1.95668	−0.0643354
$926$	−17.3890	−0.571436
$927$	0	0
$928$	7.55883	0.248131
$929$	−50.9235	−1.67074	−0.835372	−	0.549685i	$-0.814748\pi$
$929$	−50.9235	−1.67074	−0.835372	+	0.549685i	$0.814748\pi$
$930$	0	0
$931$	−82.3123	−2.69768
$932$	2.21920	0.0726922
$933$	0	0
$934$	−45.8348	−1.49976
$935$	−12.3613	−0.404259
$936$	0	0
$937$	23.9406	0.782104	0.391052	−	0.920369i	$-0.372111\pi$
$937$	23.9406	0.782104	0.391052	+	0.920369i	$0.372111\pi$
$938$	−3.26910	−0.106740
$939$	0	0
$940$	1.67982	0.0547896
$941$	58.0602	1.89271	0.946355	−	0.323130i	$-0.104735\pi$
$941$	58.0602	1.89271	0.946355	+	0.323130i	$0.104735\pi$
$942$	0	0
$943$	−2.18497	−0.0711523
$944$	−8.56473	−0.278758
$945$	0	0
$946$	34.6410	1.12628
$947$	−32.0239	−1.04064	−0.520319	−	0.853972i	$-0.674187\pi$
$947$	−32.0239	−1.04064	−0.520319	+	0.853972i	$0.674187\pi$
$948$	0	0
$949$	−32.6259	−1.05908
$950$	−1.23280	−0.0399974
$951$	0	0
$952$	29.8237	0.966591
$953$	−34.0370	−1.10257	−0.551283	−	0.834318i	$-0.685862\pi$
$953$	−34.0370	−1.10257	−0.551283	+	0.834318i	$0.685862\pi$
$954$	0	0
$955$	−28.2765	−0.915007
$956$	19.4732	0.629809
$957$	0	0
$958$	24.0620	0.777408
$959$	−86.5776	−2.79574
$960$	0	0
$961$	−29.5504	−0.953237
$962$	−53.3904	−1.72138
$963$	0	0
$964$	14.2675	0.459524
$965$	43.0827	1.38688
$966$	0	0
$967$	45.0634	1.44914	0.724571	−	0.689200i	$-0.242038\pi$
$967$	45.0634	1.44914	0.724571	+	0.689200i	$0.242038\pi$
$968$	−12.6059	−0.405169
$969$	0	0
$970$	20.1509	0.647006
$971$	47.3027	1.51802	0.759008	−	0.651082i	$-0.225684\pi$
$971$	47.3027	1.51802	0.759008	+	0.651082i	$0.225684\pi$
$972$	0	0
$973$	13.1660	0.422084
$974$	4.67637	0.149841
$975$	0	0
$976$	−22.1238	−0.708166
$977$	16.0859	0.514635	0.257317	−	0.966327i	$-0.417161\pi$
$977$	16.0859	0.514635	0.257317	+	0.966327i	$0.417161\pi$
$978$	0	0
$979$	15.7644	0.503831
$980$	−22.9898	−0.734381
$981$	0	0
$982$	−12.4164	−0.396222
$983$	5.84308	0.186365	0.0931826	−	0.995649i	$-0.470296\pi$
$983$	5.84308	0.186365	0.0931826	+	0.995649i	$0.470296\pi$
$984$	0	0
$985$	10.9386	0.348533
$986$	−4.37469	−0.139318
$987$	0	0
$988$	21.3240	0.678405
$989$	−73.7885	−2.34634
$990$	0	0
$991$	14.7445	0.468376	0.234188	−	0.972191i	$-0.424757\pi$
$991$	14.7445	0.468376	0.234188	+	0.972191i	$0.424757\pi$
$992$	−4.93653	−0.156735
$993$	0	0
$994$	66.3488	2.10446
$995$	−32.1439	−1.01903
$996$	0	0
$997$	−41.7942	−1.32364	−0.661818	−	0.749664i	$-0.730215\pi$
$997$	−41.7942	−1.32364	−0.661818	+	0.749664i	$0.730215\pi$
$998$	−8.88411	−0.281221
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.d.1.1		5
3.2	odd	2		501.2.a.b.1.5	✓	5
12.11	even	2		8016.2.a.p.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.b.1.5	✓	5	3.2	odd	2
1503.2.a.d.1.1		5	1.1	even	1	trivial
8016.2.a.p.1.3		5	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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q-1.10637 q^{2} -0.775946 q^{4} +2.19483 q^{5} -4.52759 q^{7} +3.07122 q^{8} +O(q^{10})\)	\(q-1.10637 q^{2} -0.775946 q^{4} +2.19483 q^{5} -4.52759 q^{7} +3.07122 q^{8} -2.42829 q^{10} +2.62592 q^{11} +4.50686 q^{13} +5.00918 q^{14} -1.84602 q^{16} -2.14478 q^{17} -6.09764 q^{19} -1.70307 q^{20} -2.90524 q^{22} +6.18843 q^{23} -0.182739 q^{25} -4.98626 q^{26} +3.51316 q^{28} -1.84359 q^{29} +1.20401 q^{31} -4.10007 q^{32} +2.37292 q^{34} -9.93726 q^{35} +10.7075 q^{37} +6.74624 q^{38} +6.74080 q^{40} -0.353073 q^{41} -11.9236 q^{43} -2.03758 q^{44} -6.84670 q^{46} -0.986349 q^{47} +13.4990 q^{49} +0.202177 q^{50} -3.49708 q^{52} +7.36740 q^{53} +5.76345 q^{55} -13.9052 q^{56} +2.03969 q^{58} +4.63958 q^{59} +11.9846 q^{61} -1.33208 q^{62} +8.22822 q^{64} +9.89178 q^{65} -0.652622 q^{67} +1.66424 q^{68} +10.9943 q^{70} +13.2454 q^{71} -7.23916 q^{73} -11.8465 q^{74} +4.73144 q^{76} -11.8891 q^{77} +7.65804 q^{79} -4.05168 q^{80} +0.390629 q^{82} +6.48636 q^{83} -4.70742 q^{85} +13.1919 q^{86} +8.06480 q^{88} +6.00336 q^{89} -20.4052 q^{91} -4.80189 q^{92} +1.09127 q^{94} -13.3833 q^{95} -8.29839 q^{97} -14.9349 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8	\( 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8} + 5 q^{10} + 15 q^{11} + 3 q^{14} + 10 q^{16} + 11 q^{17} - 16 q^{19} + 9 q^{20} + 7 q^{22} + 9 q^{23} + 6 q^{25} - 7 q^{26} + 9 q^{28} + q^{29} - 18 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 7 q^{37} - 20 q^{38} + 32 q^{40} + 10 q^{41} + 6 q^{43} + 5 q^{44} + 13 q^{46} + 7 q^{47} + 11 q^{49} - 3 q^{50} - 6 q^{52} + 9 q^{53} + 17 q^{55} + 15 q^{56} - 19 q^{58} + 37 q^{59} - 2 q^{61} - 18 q^{62} + 14 q^{64} + 16 q^{65} - 26 q^{68} + 7 q^{70} - 13 q^{71} - 6 q^{73} - 13 q^{74} - 50 q^{76} + 2 q^{77} + 14 q^{79} + 10 q^{80} + 39 q^{82} + 5 q^{83} + 29 q^{85} + 37 q^{86} + 48 q^{88} + 30 q^{89} - 33 q^{91} + 31 q^{92} + 34 q^{94} - 43 q^{95} - 9 q^{97} - 36 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8 + 5 * q^10 + 15 * q^11 + 3 * q^14 + 10 * q^16 + 11 * q^17 - 16 * q^19 + 9 * q^20 + 7 * q^22 + 9 * q^23 + 6 * q^25 - 7 * q^26 + 9 * q^28 + q^29 - 18 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 7 * q^37 - 20 * q^38 + 32 * q^40 + 10 * q^41 + 6 * q^43 + 5 * q^44 + 13 * q^46 + 7 * q^47 + 11 * q^49 - 3 * q^50 - 6 * q^52 + 9 * q^53 + 17 * q^55 + 15 * q^56 - 19 * q^58 + 37 * q^59 - 2 * q^61 - 18 * q^62 + 14 * q^64 + 16 * q^65 - 26 * q^68 + 7 * q^70 - 13 * q^71 - 6 * q^73 - 13 * q^74 - 50 * q^76 + 2 * q^77 + 14 * q^79 + 10 * q^80 + 39 * q^82 + 5 * q^83 + 29 * q^85 + 37 * q^86 + 48 * q^88 + 30 * q^89 - 33 * q^91 + 31 * q^92 + 34 * q^94 - 43 * q^95 - 9 * q^97 - 36 * q^98

Introduction

L-functions

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Database

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