LMFDB - Embedded newform 2151.2.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.2585660609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 4x^{6} + 15x^{5} + x^{4} - 19x^{3} + 6x^{2} + 3x - 1 $ x^8 - 3x^7 - 4x^6 + 15x^5 + x^4 - 19x^3 + 6x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.74142$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37665 q^{2} +3.64845 q^{4} -0.0257644 q^{5} +0.278803 q^{7} +3.91777 q^{8} +O(q^{10})$	\(q+2.37665 q^{2} +3.64845 q^{4} -0.0257644 q^{5} +0.278803 q^{7} +3.91777 q^{8} -0.0612329 q^{10} +2.03679 q^{11} -5.15341 q^{13} +0.662616 q^{14} +2.01427 q^{16} +6.10504 q^{17} +7.62467 q^{19} -0.0940000 q^{20} +4.84073 q^{22} +9.14155 q^{23} -4.99934 q^{25} -12.2478 q^{26} +1.01720 q^{28} +2.35001 q^{29} +2.04609 q^{31} -3.04835 q^{32} +14.5095 q^{34} -0.00718319 q^{35} -1.68276 q^{37} +18.1211 q^{38} -0.100939 q^{40} +5.01835 q^{41} +6.43995 q^{43} +7.43111 q^{44} +21.7262 q^{46} -6.24342 q^{47} -6.92227 q^{49} -11.8817 q^{50} -18.8019 q^{52} -7.02807 q^{53} -0.0524766 q^{55} +1.09229 q^{56} +5.58513 q^{58} -0.590925 q^{59} +12.7963 q^{61} +4.86283 q^{62} -11.2734 q^{64} +0.132774 q^{65} -13.7116 q^{67} +22.2739 q^{68} -0.0170719 q^{70} +13.1943 q^{71} -15.7242 q^{73} -3.99933 q^{74} +27.8182 q^{76} +0.567863 q^{77} -0.773284 q^{79} -0.0518964 q^{80} +11.9268 q^{82} +0.895907 q^{83} -0.157293 q^{85} +15.3055 q^{86} +7.97967 q^{88} +11.3261 q^{89} -1.43679 q^{91} +33.3524 q^{92} -14.8384 q^{94} -0.196445 q^{95} -7.22776 q^{97} -16.4518 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * 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 $

$2$	2.37665	1.68054	0.840271	−	0.542166i	$-0.182396\pi$
$2$	2.37665	1.68054	0.840271	+	0.542166i	$0.182396\pi$
$3$	0	0
$3$	0	0
$4$	3.64845	1.82422
$5$	−0.0257644	−0.0115222	−0.00576110	−	0.999983i	$-0.501834\pi$
$5$	−0.0257644	−0.0115222	−0.00576110	+	0.999983i	$0.501834\pi$
$6$	0	0
$7$	0.278803	0.105378	0.0526888	−	0.998611i	$-0.483221\pi$
$7$	0.278803	0.105378	0.0526888	+	0.998611i	$0.483221\pi$
$8$	3.91777	1.38514
$9$	0	0
$10$	−0.0612329	−0.0193635
$11$	2.03679	0.614115	0.307057	−	0.951691i	$-0.400656\pi$
$11$	2.03679	0.614115	0.307057	+	0.951691i	$0.400656\pi$
$12$	0	0
$13$	−5.15341	−1.42930	−0.714649	−	0.699483i	$-0.753414\pi$
$13$	−5.15341	−1.42930	−0.714649	+	0.699483i	$0.753414\pi$
$14$	0.662616	0.177092
$15$	0	0
$16$	2.01427	0.503567
$17$	6.10504	1.48069	0.740345	−	0.672227i	$-0.234662\pi$
$17$	6.10504	1.48069	0.740345	+	0.672227i	$0.234662\pi$
$18$	0	0
$19$	7.62467	1.74922	0.874610	−	0.484828i	$-0.161118\pi$
$19$	7.62467	1.74922	0.874610	+	0.484828i	$0.161118\pi$
$20$	−0.0940000	−0.0210190
$21$	0	0
$22$	4.84073	1.03205
$23$	9.14155	1.90614	0.953072	−	0.302744i	$-0.0979026\pi$
$23$	9.14155	1.90614	0.953072	+	0.302744i	$0.0979026\pi$
$24$	0	0
$25$	−4.99934	−0.999867
$26$	−12.2478	−2.40200
$27$	0	0
$28$	1.01720	0.192232
$29$	2.35001	0.436385	0.218193	−	0.975906i	$-0.429984\pi$
$29$	2.35001	0.436385	0.218193	+	0.975906i	$0.429984\pi$
$30$	0	0
$31$	2.04609	0.367488	0.183744	−	0.982974i	$-0.441178\pi$
$31$	2.04609	0.367488	0.183744	+	0.982974i	$0.441178\pi$
$32$	−3.04835	−0.538877
$33$	0	0
$34$	14.5095	2.48836
$35$	−0.00718319	−0.00121418
$36$	0	0
$37$	−1.68276	−0.276644	−0.138322	−	0.990387i	$-0.544171\pi$
$37$	−1.68276	−0.276644	−0.138322	+	0.990387i	$0.544171\pi$
$38$	18.1211	2.93964
$39$	0	0
$40$	−0.100939	−0.0159599
$41$	5.01835	0.783735	0.391867	−	0.920022i	$-0.371829\pi$
$41$	5.01835	0.783735	0.391867	+	0.920022i	$0.371829\pi$
$42$	0	0
$43$	6.43995	0.982083	0.491041	−	0.871136i	$-0.336616\pi$
$43$	6.43995	0.982083	0.491041	+	0.871136i	$0.336616\pi$
$44$	7.43111	1.12028
$45$	0	0
$46$	21.7262	3.20336
$47$	−6.24342	−0.910697	−0.455348	−	0.890313i	$-0.650485\pi$
$47$	−6.24342	−0.910697	−0.455348	+	0.890313i	$0.650485\pi$
$48$	0	0
$49$	−6.92227	−0.988896
$50$	−11.8817	−1.68032
$51$	0	0
$52$	−18.8019	−2.60736
$53$	−7.02807	−0.965380	−0.482690	−	0.875791i	$-0.660340\pi$
$53$	−7.02807	−0.965380	−0.482690	+	0.875791i	$0.660340\pi$
$54$	0	0
$55$	−0.0524766	−0.00707595
$56$	1.09229	0.145963
$57$	0	0
$58$	5.58513	0.733364
$59$	−0.590925	−0.0769319	−0.0384659	−	0.999260i	$-0.512247\pi$
$59$	−0.590925	−0.0769319	−0.0384659	+	0.999260i	$0.512247\pi$
$60$	0	0
$61$	12.7963	1.63840	0.819199	−	0.573509i	$-0.194418\pi$
$61$	12.7963	1.63840	0.819199	+	0.573509i	$0.194418\pi$
$62$	4.86283	0.617580
$63$	0	0
$64$	−11.2734	−1.40917
$65$	0.132774	0.0164686
$66$	0	0
$67$	−13.7116	−1.67514	−0.837568	−	0.546333i	$-0.816023\pi$
$67$	−13.7116	−1.67514	−0.837568	+	0.546333i	$0.816023\pi$
$68$	22.2739	2.70111
$69$	0	0
$70$	−0.0170719	−0.00204048
$71$	13.1943	1.56588	0.782938	−	0.622099i	$-0.213720\pi$
$71$	13.1943	1.56588	0.782938	+	0.622099i	$0.213720\pi$
$72$	0	0
$73$	−15.7242	−1.84038	−0.920190	−	0.391473i	$-0.871966\pi$
$73$	−15.7242	−1.84038	−0.920190	+	0.391473i	$0.871966\pi$
$74$	−3.99933	−0.464912
$75$	0	0
$76$	27.8182	3.19097
$77$	0.567863	0.0647139
$78$	0	0
$79$	−0.773284	−0.0870013	−0.0435006	−	0.999053i	$-0.513851\pi$
$79$	−0.773284	−0.0870013	−0.0435006	+	0.999053i	$0.513851\pi$
$80$	−0.0518964	−0.00580219
$81$	0	0
$82$	11.9268	1.31710
$83$	0.895907	0.0983386	0.0491693	−	0.998790i	$-0.484343\pi$
$83$	0.895907	0.0983386	0.0491693	+	0.998790i	$0.484343\pi$
$84$	0	0
$85$	−0.157293	−0.0170608
$86$	15.3055	1.65043
$87$	0	0
$88$	7.97967	0.850636
$89$	11.3261	1.20056	0.600282	−	0.799789i	$-0.295055\pi$
$89$	11.3261	1.20056	0.600282	+	0.799789i	$0.295055\pi$
$90$	0	0
$91$	−1.43679	−0.150616
$92$	33.3524	3.47723
$93$	0	0
$94$	−14.8384	−1.53046
$95$	−0.196445	−0.0201548
$96$	0	0
$97$	−7.22776	−0.733868	−0.366934	−	0.930247i	$-0.619593\pi$
$97$	−7.22776	−0.733868	−0.366934	+	0.930247i	$0.619593\pi$
$98$	−16.4518	−1.66188
$99$	0	0
$100$	−18.2398	−1.82398
$101$	−16.8027	−1.67194	−0.835968	−	0.548779i	$-0.815093\pi$
$101$	−16.8027	−1.67194	−0.835968	+	0.548779i	$0.815093\pi$
$102$	0	0
$103$	−19.3974	−1.91128	−0.955639	−	0.294539i	$-0.904834\pi$
$103$	−19.3974	−1.91128	−0.955639	+	0.294539i	$0.904834\pi$
$104$	−20.1899	−1.97978
$105$	0	0
$106$	−16.7032	−1.62236
$107$	−6.11779	−0.591429	−0.295715	−	0.955276i	$-0.595558\pi$
$107$	−6.11779	−0.591429	−0.295715	+	0.955276i	$0.595558\pi$
$108$	0	0
$109$	−7.56214	−0.724322	−0.362161	−	0.932116i	$-0.617961\pi$
$109$	−7.56214	−0.724322	−0.362161	+	0.932116i	$0.617961\pi$
$110$	−0.124718	−0.0118914
$111$	0	0
$112$	0.561584	0.0530647
$113$	−15.2645	−1.43596	−0.717982	−	0.696061i	$-0.754934\pi$
$113$	−15.2645	−1.43596	−0.717982	+	0.696061i	$0.754934\pi$
$114$	0	0
$115$	−0.235526	−0.0219630
$116$	8.57387	0.796064
$117$	0	0
$118$	−1.40442	−0.129287
$119$	1.70210	0.156032
$120$	0	0
$121$	−6.85149	−0.622863
$122$	30.4123	2.75340
$123$	0	0
$124$	7.46504	0.670381
$125$	0.257627	0.0230429
$126$	0	0
$127$	12.1796	1.08076	0.540382	−	0.841420i	$-0.318280\pi$
$127$	12.1796	1.08076	0.540382	+	0.841420i	$0.318280\pi$
$128$	−20.6961	−1.82930
$129$	0	0
$130$	0.315558	0.0276763
$131$	19.1767	1.67548	0.837740	−	0.546069i	$-0.183876\pi$
$131$	19.1767	1.67548	0.837740	+	0.546069i	$0.183876\pi$
$132$	0	0
$133$	2.12578	0.184329
$134$	−32.5876	−2.81514
$135$	0	0
$136$	23.9182	2.05097
$137$	−13.4380	−1.14809	−0.574043	−	0.818825i	$-0.694626\pi$
$137$	−13.4380	−1.14809	−0.574043	+	0.818825i	$0.694626\pi$
$138$	0	0
$139$	0.557919	0.0473221	0.0236611	−	0.999720i	$-0.492468\pi$
$139$	0.557919	0.0473221	0.0236611	+	0.999720i	$0.492468\pi$
$140$	−0.0262075	−0.00221494
$141$	0	0
$142$	31.3582	2.63152
$143$	−10.4964	−0.877753
$144$	0	0
$145$	−0.0605465	−0.00502811
$146$	−37.3709	−3.09284
$147$	0	0
$148$	−6.13946	−0.504661
$149$	−12.3376	−1.01074	−0.505369	−	0.862903i	$-0.668644\pi$
$149$	−12.3376	−1.01074	−0.505369	+	0.862903i	$0.668644\pi$
$150$	0	0
$151$	21.2706	1.73098	0.865490	−	0.500926i	$-0.167007\pi$
$151$	21.2706	1.73098	0.865490	+	0.500926i	$0.167007\pi$
$152$	29.8717	2.42292
$153$	0	0
$154$	1.34961	0.108755
$155$	−0.0527162	−0.00423427
$156$	0	0
$157$	5.45605	0.435440	0.217720	−	0.976011i	$-0.430138\pi$
$157$	5.45605	0.435440	0.217720	+	0.976011i	$0.430138\pi$
$158$	−1.83782	−0.146209
$159$	0	0
$160$	0.0785388	0.00620904
$161$	2.54869	0.200865
$162$	0	0
$163$	−14.4183	−1.12933	−0.564664	−	0.825321i	$-0.690994\pi$
$163$	−14.4183	−1.12933	−0.564664	+	0.825321i	$0.690994\pi$
$164$	18.3092	1.42971
$165$	0	0
$166$	2.12925	0.165262
$167$	7.79493	0.603190	0.301595	−	0.953436i	$-0.402481\pi$
$167$	7.79493	0.603190	0.301595	+	0.953436i	$0.402481\pi$
$168$	0	0
$169$	13.5576	1.04289
$170$	−0.373829	−0.0286714
$171$	0	0
$172$	23.4958	1.79154
$173$	6.30699	0.479512	0.239756	−	0.970833i	$-0.422933\pi$
$173$	6.30699	0.479512	0.239756	+	0.970833i	$0.422933\pi$
$174$	0	0
$175$	−1.39383	−0.105364
$176$	4.10264	0.309248
$177$	0	0
$178$	26.9181	2.01760
$179$	−4.21211	−0.314828	−0.157414	−	0.987533i	$-0.550316\pi$
$179$	−4.21211	−0.314828	−0.157414	+	0.987533i	$0.550316\pi$
$180$	0	0
$181$	3.08536	0.229333	0.114666	−	0.993404i	$-0.463420\pi$
$181$	3.08536	0.229333	0.114666	+	0.993404i	$0.463420\pi$
$182$	−3.41473	−0.253117
$183$	0	0
$184$	35.8145	2.64028
$185$	0.0433553	0.00318755
$186$	0	0
$187$	12.4347	0.909314
$188$	−22.7788	−1.66131
$189$	0	0
$190$	−0.466880	−0.0338711
$191$	−15.4370	−1.11698	−0.558491	−	0.829510i	$-0.688620\pi$
$191$	−15.4370	−1.11698	−0.558491	+	0.829510i	$0.688620\pi$
$192$	0	0
$193$	22.0775	1.58917	0.794585	−	0.607153i	$-0.207688\pi$
$193$	22.0775	1.58917	0.794585	+	0.607153i	$0.207688\pi$
$194$	−17.1778	−1.23330
$195$	0	0
$196$	−25.2555	−1.80397
$197$	15.3584	1.09424	0.547120	−	0.837055i	$-0.315724\pi$
$197$	15.3584	1.09424	0.547120	+	0.837055i	$0.315724\pi$
$198$	0	0
$199$	15.0703	1.06830	0.534152	−	0.845388i	$-0.320631\pi$
$199$	15.0703	1.06830	0.534152	+	0.845388i	$0.320631\pi$
$200$	−19.5863	−1.38496
$201$	0	0
$202$	−39.9342	−2.80976
$203$	0.655189	0.0459852
$204$	0	0
$205$	−0.129295	−0.00903034
$206$	−46.1007	−3.21199
$207$	0	0
$208$	−10.3803	−0.719747
$209$	15.5298	1.07422
$210$	0	0
$211$	−10.7626	−0.740929	−0.370465	−	0.928847i	$-0.620802\pi$
$211$	−10.7626	−0.740929	−0.370465	+	0.928847i	$0.620802\pi$
$212$	−25.6415	−1.76107
$213$	0	0
$214$	−14.5398	−0.993922
$215$	−0.165921	−0.0113157
$216$	0	0
$217$	0.570455	0.0387250
$218$	−17.9725	−1.21725
$219$	0	0
$220$	−0.191458	−0.0129081
$221$	−31.4618	−2.11635
$222$	0	0
$223$	−14.1459	−0.947278	−0.473639	−	0.880719i	$-0.657060\pi$
$223$	−14.1459	−0.947278	−0.473639	+	0.880719i	$0.657060\pi$
$224$	−0.849888	−0.0567855
$225$	0	0
$226$	−36.2783	−2.41320
$227$	12.9750	0.861179	0.430590	−	0.902548i	$-0.358306\pi$
$227$	12.9750	0.861179	0.430590	+	0.902548i	$0.358306\pi$
$228$	0	0
$229$	−6.75261	−0.446225	−0.223112	−	0.974793i	$-0.571622\pi$
$229$	−6.75261	−0.446225	−0.223112	+	0.974793i	$0.571622\pi$
$230$	−0.559763	−0.0369097
$231$	0	0
$232$	9.20679	0.604455
$233$	−6.46492	−0.423531	−0.211766	−	0.977320i	$-0.567921\pi$
$233$	−6.46492	−0.423531	−0.211766	+	0.977320i	$0.567921\pi$
$234$	0	0
$235$	0.160858	0.0104932
$236$	−2.15596	−0.140341
$237$	0	0
$238$	4.04530	0.262218
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	7.00103	0.450976	0.225488	−	0.974246i	$-0.427602\pi$
$241$	7.00103	0.450976	0.225488	+	0.974246i	$0.427602\pi$
$242$	−16.2836	−1.04675
$243$	0	0
$244$	46.6866	2.98880
$245$	0.178348	0.0113942
$246$	0	0
$247$	−39.2930	−2.50016
$248$	8.01611	0.509023
$249$	0	0
$250$	0.612288	0.0387245
$251$	6.65566	0.420102	0.210051	−	0.977690i	$-0.432637\pi$
$251$	6.65566	0.420102	0.210051	+	0.977690i	$0.432637\pi$
$252$	0	0
$253$	18.6194	1.17059
$254$	28.9466	1.81627
$255$	0	0
$256$	−26.6406	−1.66504
$257$	29.2010	1.82151	0.910754	−	0.412950i	$-0.135502\pi$
$257$	29.2010	1.82151	0.910754	+	0.412950i	$0.135502\pi$
$258$	0	0
$259$	−0.469159	−0.0291521
$260$	0.484420	0.0300425
$261$	0	0
$262$	45.5763	2.81572
$263$	−8.29069	−0.511226	−0.255613	−	0.966779i	$-0.582277\pi$
$263$	−8.29069	−0.511226	−0.255613	+	0.966779i	$0.582277\pi$
$264$	0	0
$265$	0.181074	0.0111233
$266$	5.05223	0.309772
$267$	0	0
$268$	−50.0260	−3.05582
$269$	−10.3283	−0.629728	−0.314864	−	0.949137i	$-0.601959\pi$
$269$	−10.3283	−0.629728	−0.314864	+	0.949137i	$0.601959\pi$
$270$	0	0
$271$	−3.51373	−0.213444	−0.106722	−	0.994289i	$-0.534035\pi$
$271$	−3.51373	−0.213444	−0.106722	+	0.994289i	$0.534035\pi$
$272$	12.2972	0.745626
$273$	0	0
$274$	−31.9374	−1.92941
$275$	−10.1826	−0.614033
$276$	0	0
$277$	−2.30574	−0.138539	−0.0692693	−	0.997598i	$-0.522067\pi$
$277$	−2.30574	−0.138539	−0.0692693	+	0.997598i	$0.522067\pi$
$278$	1.32598	0.0795268
$279$	0	0
$280$	−0.0281421	−0.00168181
$281$	−2.37399	−0.141621	−0.0708103	−	0.997490i	$-0.522558\pi$
$281$	−2.37399	−0.141621	−0.0708103	+	0.997490i	$0.522558\pi$
$282$	0	0
$283$	7.74423	0.460347	0.230173	−	0.973150i	$-0.426071\pi$
$283$	7.74423	0.460347	0.230173	+	0.973150i	$0.426071\pi$
$284$	48.1387	2.85651
$285$	0	0
$286$	−24.9462	−1.47510
$287$	1.39913	0.0825881
$288$	0	0
$289$	20.2715	1.19244
$290$	−0.143898	−0.00844996
$291$	0	0
$292$	−57.3689	−3.35726
$293$	−21.6810	−1.26662	−0.633308	−	0.773900i	$-0.718303\pi$
$293$	−21.6810	−1.26662	−0.633308	+	0.773900i	$0.718303\pi$
$294$	0	0
$295$	0.0152248	0.000886424 0
$296$	−6.59268	−0.383192
$297$	0	0
$298$	−29.3222	−1.69859
$299$	−47.1101	−2.72445
$300$	0	0
$301$	1.79548	0.103490
$302$	50.5528	2.90899
$303$	0	0
$304$	15.3581	0.880849
$305$	−0.329689	−0.0188779
$306$	0	0
$307$	7.75733	0.442734	0.221367	−	0.975191i	$-0.428948\pi$
$307$	7.75733	0.442734	0.221367	+	0.975191i	$0.428948\pi$
$308$	2.07182	0.118053
$309$	0	0
$310$	−0.125288	−0.00711587
$311$	−21.5541	−1.22222	−0.611111	−	0.791545i	$-0.709277\pi$
$311$	−21.5541	−1.22222	−0.611111	+	0.791545i	$0.709277\pi$
$312$	0	0
$313$	6.85326	0.387369	0.193685	−	0.981064i	$-0.437956\pi$
$313$	6.85326	0.387369	0.193685	+	0.981064i	$0.437956\pi$
$314$	12.9671	0.731776
$315$	0	0
$316$	−2.82129	−0.158710
$317$	12.3935	0.696089	0.348044	−	0.937478i	$-0.386846\pi$
$317$	12.3935	0.696089	0.348044	+	0.937478i	$0.386846\pi$
$318$	0	0
$319$	4.78646	0.267991
$320$	0.290452	0.0162367
$321$	0	0
$322$	6.05733	0.337562
$323$	46.5489	2.59005
$324$	0	0
$325$	25.7636	1.42911
$326$	−34.2672	−1.89788
$327$	0	0
$328$	19.6608	1.08558
$329$	−1.74068	−0.0959671
$330$	0	0
$331$	−24.9260	−1.37006	−0.685029	−	0.728516i	$-0.740210\pi$
$331$	−24.9260	−1.37006	−0.685029	+	0.728516i	$0.740210\pi$
$332$	3.26867	0.179392
$333$	0	0
$334$	18.5258	1.01369
$335$	0.353271	0.0193012
$336$	0	0
$337$	4.98114	0.271340	0.135670	−	0.990754i	$-0.456681\pi$
$337$	4.98114	0.271340	0.135670	+	0.990754i	$0.456681\pi$
$338$	32.2216	1.75263
$339$	0	0
$340$	−0.573874	−0.0311227
$341$	4.16745	0.225680
$342$	0	0
$343$	−3.88157	−0.209585
$344$	25.2303	1.36032
$345$	0	0
$346$	14.9895	0.805840
$347$	−4.09751	−0.219966	−0.109983	−	0.993933i	$-0.535080\pi$
$347$	−4.09751	−0.219966	−0.109983	+	0.993933i	$0.535080\pi$
$348$	0	0
$349$	−28.1421	−1.50641	−0.753207	−	0.657784i	$-0.771494\pi$
$349$	−28.1421	−1.50641	−0.753207	+	0.657784i	$0.771494\pi$
$350$	−3.31264	−0.177068
$351$	0	0
$352$	−6.20884	−0.330932
$353$	−0.0904223	−0.00481269	−0.00240635	−	0.999997i	$-0.500766\pi$
$353$	−0.0904223	−0.00481269	−0.00240635	+	0.999997i	$0.500766\pi$
$354$	0	0
$355$	−0.339944	−0.0180423
$356$	41.3226	2.19009
$357$	0	0
$358$	−10.0107	−0.529081
$359$	−24.8752	−1.31286	−0.656432	−	0.754385i	$-0.727935\pi$
$359$	−24.8752	−1.31286	−0.656432	+	0.754385i	$0.727935\pi$
$360$	0	0
$361$	39.1356	2.05977
$362$	7.33280	0.385403
$363$	0	0
$364$	−5.24203	−0.274757
$365$	0.405125	0.0212052
$366$	0	0
$367$	−29.7572	−1.55331	−0.776657	−	0.629923i	$-0.783086\pi$
$367$	−29.7572	−1.55331	−0.776657	+	0.629923i	$0.783086\pi$
$368$	18.4135	0.959871
$369$	0	0
$370$	0.103040	0.00535681
$371$	−1.95945	−0.101729
$372$	0	0
$373$	−30.5940	−1.58410	−0.792049	−	0.610458i	$-0.790986\pi$
$373$	−30.5940	−1.58410	−0.792049	+	0.610458i	$0.790986\pi$
$374$	29.5528	1.52814
$375$	0	0
$376$	−24.4603	−1.26144
$377$	−12.1105	−0.623724
$378$	0	0
$379$	18.8693	0.969249	0.484625	−	0.874722i	$-0.338956\pi$
$379$	18.8693	0.969249	0.484625	+	0.874722i	$0.338956\pi$
$380$	−0.716719	−0.0367669
$381$	0	0
$382$	−36.6883	−1.87714
$383$	−20.1842	−1.03136	−0.515682	−	0.856780i	$-0.672461\pi$
$383$	−20.1842	−1.03136	−0.515682	+	0.856780i	$0.672461\pi$
$384$	0	0
$385$	−0.0146306	−0.000745647 0
$386$	52.4703	2.67067
$387$	0	0
$388$	−26.3701	−1.33874
$389$	−15.0307	−0.762086	−0.381043	−	0.924557i	$-0.624435\pi$
$389$	−15.0307	−0.762086	−0.381043	+	0.924557i	$0.624435\pi$
$390$	0	0
$391$	55.8095	2.82241
$392$	−27.1199	−1.36976
$393$	0	0
$394$	36.5014	1.83892
$395$	0.0199232	0.00100245
$396$	0	0
$397$	−5.10447	−0.256186	−0.128093	−	0.991762i	$-0.540886\pi$
$397$	−5.10447	−0.256186	−0.128093	+	0.991762i	$0.540886\pi$
$398$	35.8168	1.79533
$399$	0	0
$400$	−10.0700	−0.503500
$401$	5.43949	0.271635	0.135818	−	0.990734i	$-0.456634\pi$
$401$	5.43949	0.271635	0.135818	+	0.990734i	$0.456634\pi$
$402$	0	0
$403$	−10.5443	−0.525250
$404$	−61.3039	−3.04998
$405$	0	0
$406$	1.55715	0.0772801
$407$	−3.42743	−0.169891
$408$	0	0
$409$	24.8260	1.22757	0.613784	−	0.789474i	$-0.289647\pi$
$409$	24.8260	1.22757	0.613784	+	0.789474i	$0.289647\pi$
$410$	−0.307288	−0.0151759
$411$	0	0
$412$	−70.7702	−3.48660
$413$	−0.164752	−0.00810690
$414$	0	0
$415$	−0.0230825	−0.00113308
$416$	15.7094	0.770215
$417$	0	0
$418$	36.9089	1.80527
$419$	8.01165	0.391395	0.195697	−	0.980664i	$-0.437303\pi$
$419$	8.01165	0.391395	0.195697	+	0.980664i	$0.437303\pi$
$420$	0	0
$421$	−18.8614	−0.919250	−0.459625	−	0.888113i	$-0.652016\pi$
$421$	−18.8614	−0.919250	−0.459625	+	0.888113i	$0.652016\pi$
$422$	−25.5789	−1.24516
$423$	0	0
$424$	−27.5344	−1.33719
$425$	−30.5211	−1.48049
$426$	0	0
$427$	3.56765	0.172650
$428$	−22.3204	−1.07890
$429$	0	0
$430$	−0.394336	−0.0190166
$431$	9.12088	0.439337	0.219669	−	0.975575i	$-0.429502\pi$
$431$	9.12088	0.439337	0.219669	+	0.975575i	$0.429502\pi$
$432$	0	0
$433$	19.7465	0.948957	0.474478	−	0.880267i	$-0.342637\pi$
$433$	19.7465	0.948957	0.474478	+	0.880267i	$0.342637\pi$
$434$	1.35577	0.0650791
$435$	0	0
$436$	−27.5901	−1.32132
$437$	69.7013	3.33426
$438$	0	0
$439$	20.6064	0.983489	0.491744	−	0.870740i	$-0.336359\pi$
$439$	20.6064	0.983489	0.491744	+	0.870740i	$0.336359\pi$
$440$	−0.205592	−0.00980119
$441$	0	0
$442$	−74.7735	−3.55661
$443$	−28.8343	−1.36996	−0.684980	−	0.728562i	$-0.740189\pi$
$443$	−28.8343	−1.36996	−0.684980	+	0.728562i	$0.740189\pi$
$444$	0	0
$445$	−0.291810	−0.0138331
$446$	−33.6197	−1.59194
$447$	0	0
$448$	−3.14305	−0.148495
$449$	3.94322	0.186092	0.0930461	−	0.995662i	$-0.470340\pi$
$449$	3.94322	0.186092	0.0930461	+	0.995662i	$0.470340\pi$
$450$	0	0
$451$	10.2213	0.481303
$452$	−55.6918	−2.61952
$453$	0	0
$454$	30.8369	1.44725
$455$	0.0370179	0.00173543
$456$	0	0
$457$	2.01811	0.0944032	0.0472016	−	0.998885i	$-0.484970\pi$
$457$	2.01811	0.0944032	0.0472016	+	0.998885i	$0.484970\pi$
$458$	−16.0486	−0.749900
$459$	0	0
$460$	−0.859306	−0.0400653
$461$	6.74171	0.313993	0.156996	−	0.987599i	$-0.449819\pi$
$461$	6.74171	0.313993	0.156996	+	0.987599i	$0.449819\pi$
$462$	0	0
$463$	−1.18406	−0.0550281	−0.0275141	−	0.999621i	$-0.508759\pi$
$463$	−1.18406	−0.0550281	−0.0275141	+	0.999621i	$0.508759\pi$
$464$	4.73354	0.219749
$465$	0	0
$466$	−15.3648	−0.711762
$467$	19.3721	0.896435	0.448217	−	0.893925i	$-0.352059\pi$
$467$	19.3721	0.896435	0.448217	+	0.893925i	$0.352059\pi$
$468$	0	0
$469$	−3.82283	−0.176522
$470$	0.382303	0.0176343
$471$	0	0
$472$	−2.31511	−0.106562
$473$	13.1168	0.603112
$474$	0	0
$475$	−38.1183	−1.74899
$476$	6.21003	0.284636
$477$	0	0
$478$	−2.37665	−0.108705
$479$	18.4331	0.842229	0.421114	−	0.907008i	$-0.361639\pi$
$479$	18.4331	0.842229	0.421114	+	0.907008i	$0.361639\pi$
$480$	0	0
$481$	8.67195	0.395407
$482$	16.6390	0.757884
$483$	0	0
$484$	−24.9973	−1.13624
$485$	0.186219	0.00845577
$486$	0	0
$487$	−18.0119	−0.816196	−0.408098	−	0.912938i	$-0.633808\pi$
$487$	−18.0119	−0.816196	−0.408098	+	0.912938i	$0.633808\pi$
$488$	50.1330	2.26941
$489$	0	0
$490$	0.423870	0.0191485
$491$	24.4196	1.10204	0.551022	−	0.834491i	$-0.314238\pi$
$491$	24.4196	1.10204	0.551022	+	0.834491i	$0.314238\pi$
$492$	0	0
$493$	14.3469	0.646151
$494$	−93.3856	−4.20162
$495$	0	0
$496$	4.12137	0.185055
$497$	3.67861	0.165008
$498$	0	0
$499$	−41.9920	−1.87982	−0.939909	−	0.341424i	$-0.889091\pi$
$499$	−41.9920	−1.87982	−0.939909	+	0.341424i	$0.889091\pi$
$500$	0.939938	0.0420353
$501$	0	0
$502$	15.8182	0.705999
$503$	20.7828	0.926660	0.463330	−	0.886186i	$-0.346654\pi$
$503$	20.7828	0.926660	0.463330	+	0.886186i	$0.346654\pi$
$504$	0	0
$505$	0.432913	0.0192644
$506$	44.2517	1.96723
$507$	0	0
$508$	44.4366	1.97155
$509$	−5.99687	−0.265807	−0.132903	−	0.991129i	$-0.542430\pi$
$509$	−5.99687	−0.265807	−0.132903	+	0.991129i	$0.542430\pi$
$510$	0	0
$511$	−4.38396	−0.193935
$512$	−21.9231	−0.968872
$513$	0	0
$514$	69.4004	3.06112
$515$	0.499761	0.0220221
$516$	0	0
$517$	−12.7165	−0.559273
$518$	−1.11502	−0.0489914
$519$	0	0
$520$	0.520180	0.0228114
$521$	−16.5355	−0.724434	−0.362217	−	0.932094i	$-0.617980\pi$
$521$	−16.5355	−0.724434	−0.362217	+	0.932094i	$0.617980\pi$
$522$	0	0
$523$	32.1156	1.40432	0.702159	−	0.712021i	$-0.252220\pi$
$523$	32.1156	1.40432	0.702159	+	0.712021i	$0.252220\pi$
$524$	69.9653	3.05645
$525$	0	0
$526$	−19.7040	−0.859137
$527$	12.4915	0.544136
$528$	0	0
$529$	60.5679	2.63338
$530$	0.430349	0.0186932
$531$	0	0
$532$	7.75579	0.336256
$533$	−25.8616	−1.12019
$534$	0	0
$535$	0.157621	0.00681456
$536$	−53.7189	−2.32030
$537$	0	0
$538$	−24.5467	−1.05828
$539$	−14.0992	−0.607295
$540$	0	0
$541$	5.54845	0.238547	0.119273	−	0.992861i	$-0.461944\pi$
$541$	5.54845	0.238547	0.119273	+	0.992861i	$0.461944\pi$
$542$	−8.35089	−0.358702
$543$	0	0
$544$	−18.6103	−0.797909
$545$	0.194834	0.00834578
$546$	0	0
$547$	−15.5519	−0.664950	−0.332475	−	0.943112i	$-0.607884\pi$
$547$	−15.5519	−0.664950	−0.332475	+	0.943112i	$0.607884\pi$
$548$	−49.0278	−2.09437
$549$	0	0
$550$	−24.2004	−1.03191
$551$	17.9180	0.763333
$552$	0	0
$553$	−0.215594	−0.00916798
$554$	−5.47993	−0.232820
$555$	0	0
$556$	2.03554	0.0863261
$557$	8.63893	0.366043	0.183022	−	0.983109i	$-0.441412\pi$
$557$	8.63893	0.366043	0.183022	+	0.983109i	$0.441412\pi$
$558$	0	0
$559$	−33.1877	−1.40369
$560$	−0.0144689	−0.000611421 0
$561$	0	0
$562$	−5.64214	−0.237999
$563$	0.842054	0.0354883	0.0177442	−	0.999843i	$-0.494352\pi$
$563$	0.842054	0.0354883	0.0177442	+	0.999843i	$0.494352\pi$
$564$	0	0
$565$	0.393281	0.0165455
$566$	18.4053	0.773632
$567$	0	0
$568$	51.6923	2.16896
$569$	21.8149	0.914529	0.457264	−	0.889331i	$-0.348829\pi$
$569$	21.8149	0.914529	0.457264	+	0.889331i	$0.348829\pi$
$570$	0	0
$571$	−42.4539	−1.77664	−0.888320	−	0.459225i	$-0.848127\pi$
$571$	−42.4539	−1.77664	−0.888320	+	0.459225i	$0.848127\pi$
$572$	−38.2956	−1.60122
$573$	0	0
$574$	3.32524	0.138793
$575$	−45.7017	−1.90589
$576$	0	0
$577$	−7.68299	−0.319847	−0.159924	−	0.987129i	$-0.551125\pi$
$577$	−7.68299	−0.319847	−0.159924	+	0.987129i	$0.551125\pi$
$578$	48.1782	2.00395
$579$	0	0
$580$	−0.220901	−0.00917240
$581$	0.249782	0.0103627
$582$	0	0
$583$	−14.3147	−0.592854
$584$	−61.6039	−2.54919
$585$	0	0
$586$	−51.5280	−2.12860
$587$	27.4112	1.13138	0.565691	−	0.824617i	$-0.308609\pi$
$587$	27.4112	1.13138	0.565691	+	0.824617i	$0.308609\pi$
$588$	0	0
$589$	15.6007	0.642818
$590$	0.0361840	0.00148967
$591$	0	0
$592$	−3.38953	−0.139309
$593$	−11.8026	−0.484676	−0.242338	−	0.970192i	$-0.577914\pi$
$593$	−11.8026	−0.484676	−0.242338	+	0.970192i	$0.577914\pi$
$594$	0	0
$595$	−0.0438537	−0.00179783
$596$	−45.0132	−1.84381
$597$	0	0
$598$	−111.964	−4.57855
$599$	−40.5909	−1.65850	−0.829250	−	0.558878i	$-0.811232\pi$
$599$	−40.5909	−1.65850	−0.829250	+	0.558878i	$0.811232\pi$
$600$	0	0
$601$	26.1587	1.06704	0.533518	−	0.845789i	$-0.320870\pi$
$601$	26.1587	1.06704	0.533518	+	0.845789i	$0.320870\pi$
$602$	4.26721	0.173919
$603$	0	0
$604$	77.6048	3.15769
$605$	0.176525	0.00717675
$606$	0	0
$607$	9.11023	0.369773	0.184886	−	0.982760i	$-0.440808\pi$
$607$	9.11023	0.369773	0.184886	+	0.982760i	$0.440808\pi$
$608$	−23.2426	−0.942613
$609$	0	0
$610$	−0.783554	−0.0317252
$611$	32.1749	1.30166
$612$	0	0
$613$	32.7264	1.32181	0.660903	−	0.750471i	$-0.270174\pi$
$613$	32.7264	1.32181	0.660903	+	0.750471i	$0.270174\pi$
$614$	18.4364	0.744034
$615$	0	0
$616$	2.22476	0.0896380
$617$	−10.9150	−0.439420	−0.219710	−	0.975565i	$-0.570511\pi$
$617$	−10.9150	−0.439420	−0.219710	+	0.975565i	$0.570511\pi$
$618$	0	0
$619$	13.3610	0.537023	0.268512	−	0.963276i	$-0.413468\pi$
$619$	13.3610	0.537023	0.268512	+	0.963276i	$0.413468\pi$
$620$	−0.192332	−0.00772425
$621$	0	0
$622$	−51.2265	−2.05400
$623$	3.15775	0.126512
$624$	0	0
$625$	24.9900	0.999602
$626$	16.2878	0.650990
$627$	0	0
$628$	19.9061	0.794340
$629$	−10.2733	−0.409624
$630$	0	0
$631$	6.65225	0.264822	0.132411	−	0.991195i	$-0.457728\pi$
$631$	6.65225	0.264822	0.132411	+	0.991195i	$0.457728\pi$
$632$	−3.02955	−0.120509
$633$	0	0
$634$	29.4550	1.16981
$635$	−0.313800	−0.0124528
$636$	0	0
$637$	35.6733	1.41343
$638$	11.3757	0.450370
$639$	0	0
$640$	0.533223	0.0210775
$641$	15.5217	0.613072	0.306536	−	0.951859i	$-0.400830\pi$
$641$	15.5217	0.613072	0.306536	+	0.951859i	$0.400830\pi$
$642$	0	0
$643$	11.0238	0.434736	0.217368	−	0.976090i	$-0.430253\pi$
$643$	11.0238	0.434736	0.217368	+	0.976090i	$0.430253\pi$
$644$	9.29876	0.366422
$645$	0	0
$646$	110.630	4.35269
$647$	3.72529	0.146456	0.0732281	−	0.997315i	$-0.476670\pi$
$647$	3.72529	0.146456	0.0732281	+	0.997315i	$0.476670\pi$
$648$	0	0
$649$	−1.20359	−0.0472450
$650$	61.2310	2.40168
$651$	0	0
$652$	−52.6044	−2.06015
$653$	−24.7641	−0.969094	−0.484547	−	0.874765i	$-0.661016\pi$
$653$	−24.7641	−0.969094	−0.484547	+	0.874765i	$0.661016\pi$
$654$	0	0
$655$	−0.494077	−0.0193052
$656$	10.1083	0.394663
$657$	0	0
$658$	−4.13699	−0.161277
$659$	18.3793	0.715955	0.357978	−	0.933730i	$-0.383466\pi$
$659$	18.3793	0.715955	0.357978	+	0.933730i	$0.383466\pi$
$660$	0	0
$661$	9.04414	0.351776	0.175888	−	0.984410i	$-0.443720\pi$
$661$	9.04414	0.351776	0.175888	+	0.984410i	$0.443720\pi$
$662$	−59.2403	−2.30244
$663$	0	0
$664$	3.50996	0.136213
$665$	−0.0547695	−0.00212387
$666$	0	0
$667$	21.4827	0.831813
$668$	28.4394	1.10035
$669$	0	0
$670$	0.839599	0.0324366
$671$	26.0634	1.00616
$672$	0	0
$673$	−8.58994	−0.331118	−0.165559	−	0.986200i	$-0.552943\pi$
$673$	−8.58994	−0.331118	−0.165559	+	0.986200i	$0.552943\pi$
$674$	11.8384	0.455998
$675$	0	0
$676$	49.4642	1.90247
$677$	−25.4361	−0.977589	−0.488795	−	0.872399i	$-0.662563\pi$
$677$	−25.4361	−0.977589	−0.488795	+	0.872399i	$0.662563\pi$
$678$	0	0
$679$	−2.01512	−0.0773333
$680$	−0.616237	−0.0236316
$681$	0	0
$682$	9.90455	0.379265
$683$	19.5583	0.748379	0.374189	−	0.927352i	$-0.377921\pi$
$683$	19.5583	0.748379	0.374189	+	0.927352i	$0.377921\pi$
$684$	0	0
$685$	0.346222	0.0132285
$686$	−9.22512	−0.352217
$687$	0	0
$688$	12.9718	0.494544
$689$	36.2185	1.37982
$690$	0	0
$691$	0.645673	0.0245626	0.0122813	−	0.999925i	$-0.496091\pi$
$691$	0.645673	0.0245626	0.0122813	+	0.999925i	$0.496091\pi$
$692$	23.0107	0.874736
$693$	0	0
$694$	−9.73833	−0.369662
$695$	−0.0143745	−0.000545254 0
$696$	0	0
$697$	30.6372	1.16047
$698$	−66.8839	−2.53159
$699$	0	0
$700$	−5.08531	−0.192207
$701$	−10.2299	−0.386378	−0.193189	−	0.981162i	$-0.561883\pi$
$701$	−10.2299	−0.386378	−0.193189	+	0.981162i	$0.561883\pi$
$702$	0	0
$703$	−12.8305	−0.483911
$704$	−22.9615	−0.865393
$705$	0	0
$706$	−0.214902	−0.00808794
$707$	−4.68465	−0.176185
$708$	0	0
$709$	−24.7768	−0.930512	−0.465256	−	0.885176i	$-0.654038\pi$
$709$	−24.7768	−0.930512	−0.465256	+	0.885176i	$0.654038\pi$
$710$	−0.807926	−0.0303209
$711$	0	0
$712$	44.3730	1.66295
$713$	18.7044	0.700486
$714$	0	0
$715$	0.270433	0.0101136
$716$	−15.3676	−0.574316
$717$	0	0
$718$	−59.1196	−2.20632
$719$	33.8314	1.26170	0.630849	−	0.775906i	$-0.282707\pi$
$719$	33.8314	1.26170	0.630849	+	0.775906i	$0.282707\pi$
$720$	0	0
$721$	−5.40804	−0.201406
$722$	93.0114	3.46153
$723$	0	0
$724$	11.2568	0.418354
$725$	−11.7485	−0.436327
$726$	0	0
$727$	16.1218	0.597926	0.298963	−	0.954265i	$-0.403359\pi$
$727$	16.1218	0.597926	0.298963	+	0.954265i	$0.403359\pi$
$728$	−5.62900	−0.208625
$729$	0	0
$730$	0.962838	0.0356362
$731$	39.3161	1.45416
$732$	0	0
$733$	35.9947	1.32949	0.664746	−	0.747069i	$-0.268540\pi$
$733$	35.9947	1.32949	0.664746	+	0.747069i	$0.268540\pi$
$734$	−70.7224	−2.61041
$735$	0	0
$736$	−27.8666	−1.02718
$737$	−27.9276	−1.02873
$738$	0	0
$739$	−17.9466	−0.660177	−0.330089	−	0.943950i	$-0.607079\pi$
$739$	−17.9466	−0.660177	−0.330089	+	0.943950i	$0.607079\pi$
$740$	0.158180	0.00581480
$741$	0	0
$742$	−4.65691	−0.170961
$743$	16.8853	0.619461	0.309730	−	0.950824i	$-0.399761\pi$
$743$	16.8853	0.619461	0.309730	+	0.950824i	$0.399761\pi$
$744$	0	0
$745$	0.317872	0.0116459
$746$	−72.7111	−2.66214
$747$	0	0
$748$	45.3672	1.65879
$749$	−1.70566	−0.0623234
$750$	0	0
$751$	0.315906	0.0115276	0.00576379	−	0.999983i	$-0.498165\pi$
$751$	0.315906	0.0115276	0.00576379	+	0.999983i	$0.498165\pi$
$752$	−12.5759	−0.458597
$753$	0	0
$754$	−28.7825	−1.04820
$755$	−0.548026	−0.0199447
$756$	0	0
$757$	−15.5642	−0.565691	−0.282846	−	0.959165i	$-0.591278\pi$
$757$	−15.5642	−0.565691	−0.282846	+	0.959165i	$0.591278\pi$
$758$	44.8456	1.62886
$759$	0	0
$760$	−0.769627	−0.0279173
$761$	40.5084	1.46843	0.734214	−	0.678918i	$-0.237551\pi$
$761$	40.5084	1.46843	0.734214	+	0.678918i	$0.237551\pi$
$762$	0	0
$763$	−2.10835	−0.0763273
$764$	−56.3211	−2.03763
$765$	0	0
$766$	−47.9707	−1.73325
$767$	3.04528	0.109959
$768$	0	0
$769$	−10.1080	−0.364502	−0.182251	−	0.983252i	$-0.558338\pi$
$769$	−10.1080	−0.364502	−0.182251	+	0.983252i	$0.558338\pi$
$770$	−0.0347719	−0.00125309
$771$	0	0
$772$	80.5484	2.89900
$773$	−11.7846	−0.423862	−0.211931	−	0.977285i	$-0.567975\pi$
$773$	−11.7846	−0.423862	−0.211931	+	0.977285i	$0.567975\pi$
$774$	0	0
$775$	−10.2291	−0.367439
$776$	−28.3167	−1.01651
$777$	0	0
$778$	−35.7226	−1.28072
$779$	38.2633	1.37092
$780$	0	0
$781$	26.8740	0.961628
$782$	132.639	4.74318
$783$	0	0
$784$	−13.9433	−0.497975
$785$	−0.140572	−0.00501723
$786$	0	0
$787$	−21.4867	−0.765919	−0.382959	−	0.923765i	$-0.625095\pi$
$787$	−21.4867	−0.765919	−0.382959	+	0.923765i	$0.625095\pi$
$788$	56.0342	1.99614
$789$	0	0
$790$	0.0473504	0.00168465
$791$	−4.25579	−0.151319
$792$	0	0
$793$	−65.9445	−2.34176
$794$	−12.1315	−0.430531
$795$	0	0
$796$	54.9832	1.94883
$797$	−3.12085	−0.110546	−0.0552731	−	0.998471i	$-0.517603\pi$
$797$	−3.12085	−0.110546	−0.0552731	+	0.998471i	$0.517603\pi$
$798$	0	0
$799$	−38.1164	−1.34846
$800$	15.2397	0.538805
$801$	0	0
$802$	12.9277	0.456494
$803$	−32.0269	−1.13020
$804$	0	0
$805$	−0.0656655	−0.00231440
$806$	−25.0601	−0.882705
$807$	0	0
$808$	−65.8293	−2.31587
$809$	56.3119	1.97982	0.989911	−	0.141692i	$-0.0452543\pi$
$809$	56.3119	1.97982	0.989911	+	0.141692i	$0.0452543\pi$
$810$	0	0
$811$	53.6605	1.88427	0.942137	−	0.335228i	$-0.108813\pi$
$811$	53.6605	1.88427	0.942137	+	0.335228i	$0.108813\pi$
$812$	2.39042	0.0838873
$813$	0	0
$814$	−8.14578	−0.285510
$815$	0.371479	0.0130123
$816$	0	0
$817$	49.1025	1.71788
$818$	59.0027	2.06298
$819$	0	0
$820$	−0.471725	−0.0164734
$821$	18.4668	0.644496	0.322248	−	0.946655i	$-0.395562\pi$
$821$	18.4668	0.644496	0.322248	+	0.946655i	$0.395562\pi$
$822$	0	0
$823$	−17.7034	−0.617103	−0.308551	−	0.951208i	$-0.599844\pi$
$823$	−17.7034	−0.617103	−0.308551	+	0.951208i	$0.599844\pi$
$824$	−75.9945	−2.64739
$825$	0	0
$826$	−0.391556	−0.0136240
$827$	0.788450	0.0274171	0.0137085	−	0.999906i	$-0.495636\pi$
$827$	0.788450	0.0274171	0.0137085	+	0.999906i	$0.495636\pi$
$828$	0	0
$829$	29.0143	1.00771	0.503854	−	0.863789i	$-0.331915\pi$
$829$	29.0143	1.00771	0.503854	+	0.863789i	$0.331915\pi$
$830$	−0.0548590	−0.00190418
$831$	0	0
$832$	58.0963	2.01413
$833$	−42.2607	−1.46425
$834$	0	0
$835$	−0.200832	−0.00695007
$836$	56.6598	1.95962
$837$	0	0
$838$	19.0409	0.657755
$839$	4.66991	0.161223	0.0806115	−	0.996746i	$-0.474313\pi$
$839$	4.66991	0.161223	0.0806115	+	0.996746i	$0.474313\pi$
$840$	0	0
$841$	−23.4775	−0.809568
$842$	−44.8269	−1.54484
$843$	0	0
$844$	−39.2668	−1.35162
$845$	−0.349304	−0.0120164
$846$	0	0
$847$	−1.91022	−0.0656358
$848$	−14.1564	−0.486133
$849$	0	0
$850$	−72.5380	−2.48803
$851$	−15.3830	−0.527324
$852$	0	0
$853$	−16.7142	−0.572285	−0.286142	−	0.958187i	$-0.592373\pi$
$853$	−16.7142	−0.572285	−0.286142	+	0.958187i	$0.592373\pi$
$854$	8.47903	0.290146
$855$	0	0
$856$	−23.9681	−0.819213
$857$	−0.927876	−0.0316956	−0.0158478	−	0.999874i	$-0.505045\pi$
$857$	−0.927876	−0.0316956	−0.0158478	+	0.999874i	$0.505045\pi$
$858$	0	0
$859$	−47.4067	−1.61749	−0.808747	−	0.588156i	$-0.799854\pi$
$859$	−47.4067	−1.61749	−0.808747	+	0.588156i	$0.799854\pi$
$860$	−0.605355	−0.0206424
$861$	0	0
$862$	21.6771	0.738325
$863$	42.6921	1.45326	0.726628	−	0.687031i	$-0.241086\pi$
$863$	42.6921	1.45326	0.726628	+	0.687031i	$0.241086\pi$
$864$	0	0
$865$	−0.162496	−0.00552503
$866$	46.9305	1.59476
$867$	0	0
$868$	2.08128	0.0706431
$869$	−1.57502	−0.0534288
$870$	0	0
$871$	70.6614	2.39427
$872$	−29.6268	−1.00329
$873$	0	0
$874$	165.655	5.60337
$875$	0.0718271	0.00242820
$876$	0	0
$877$	26.8609	0.907029	0.453515	−	0.891249i	$-0.350170\pi$
$877$	26.8609	0.907029	0.453515	+	0.891249i	$0.350170\pi$
$878$	48.9741	1.65279
$879$	0	0
$880$	−0.105702	−0.00356321
$881$	8.49551	0.286221	0.143111	−	0.989707i	$-0.454290\pi$
$881$	8.49551	0.286221	0.143111	+	0.989707i	$0.454290\pi$
$882$	0	0
$883$	−14.3664	−0.483466	−0.241733	−	0.970343i	$-0.577716\pi$
$883$	−14.3664	−0.483466	−0.241733	+	0.970343i	$0.577716\pi$
$884$	−114.787	−3.86069
$885$	0	0
$886$	−68.5290	−2.30228
$887$	33.4301	1.12247	0.561237	−	0.827655i	$-0.310326\pi$
$887$	33.4301	1.12247	0.561237	+	0.827655i	$0.310326\pi$
$888$	0	0
$889$	3.39571	0.113888
$890$	−0.693529	−0.0232471
$891$	0	0
$892$	−51.6105	−1.72805
$893$	−47.6040	−1.59301
$894$	0	0
$895$	0.108522	0.00362751
$896$	−5.77014	−0.192767
$897$	0	0
$898$	9.37165	0.312736
$899$	4.80832	0.160366
$900$	0	0
$901$	−42.9066	−1.42943
$902$	24.2925	0.808850
$903$	0	0
$904$	−59.8029	−1.98902
$905$	−0.0794923	−0.00264241
$906$	0	0
$907$	35.8979	1.19197	0.595985	−	0.802995i	$-0.296762\pi$
$907$	35.8979	1.19197	0.595985	+	0.802995i	$0.296762\pi$
$908$	47.3385	1.57098
$909$	0	0
$910$	0.0879785	0.00291646
$911$	7.57044	0.250820	0.125410	−	0.992105i	$-0.459975\pi$
$911$	7.57044	0.250820	0.125410	+	0.992105i	$0.459975\pi$
$912$	0	0
$913$	1.82477	0.0603912
$914$	4.79633	0.158649
$915$	0	0
$916$	−24.6365	−0.814014
$917$	5.34653	0.176558
$918$	0	0
$919$	−54.7630	−1.80647	−0.903233	−	0.429151i	$-0.858813\pi$
$919$	−54.7630	−1.80647	−0.903233	+	0.429151i	$0.858813\pi$
$920$	−0.922739	−0.0304218
$921$	0	0
$922$	16.0227	0.527678
$923$	−67.9957	−2.23810
$924$	0	0
$925$	8.41269	0.276608
$926$	−2.81410	−0.0924771
$927$	0	0
$928$	−7.16363	−0.235158
$929$	24.6234	0.807868	0.403934	−	0.914788i	$-0.367642\pi$
$929$	24.6234	0.807868	0.403934	+	0.914788i	$0.367642\pi$
$930$	0	0
$931$	−52.7800	−1.72980
$932$	−23.5869	−0.772615
$933$	0	0
$934$	46.0407	1.50650
$935$	−0.320372	−0.0104773
$936$	0	0
$937$	50.6151	1.65352	0.826762	−	0.562552i	$-0.190180\pi$
$937$	50.6151	1.65352	0.826762	+	0.562552i	$0.190180\pi$
$938$	−9.08551	−0.296652
$939$	0	0
$940$	0.586882	0.0191420
$941$	−45.9093	−1.49660	−0.748299	−	0.663361i	$-0.769129\pi$
$941$	−45.9093	−1.49660	−0.748299	+	0.663361i	$0.769129\pi$
$942$	0	0
$943$	45.8755	1.49391
$944$	−1.19028	−0.0387403
$945$	0	0
$946$	31.1740	1.01355
$947$	59.4856	1.93302	0.966511	−	0.256624i	$-0.0826101\pi$
$947$	59.4856	1.93302	0.966511	+	0.256624i	$0.0826101\pi$
$948$	0	0
$949$	81.0333	2.63045
$950$	−90.5937	−2.93925
$951$	0	0
$952$	6.66845	0.216126
$953$	17.3642	0.562482	0.281241	−	0.959637i	$-0.409254\pi$
$953$	17.3642	0.562482	0.281241	+	0.959637i	$0.409254\pi$
$954$	0	0
$955$	0.397725	0.0128701
$956$	−3.64845	−0.117999
$957$	0	0
$958$	43.8089	1.41540
$959$	−3.74656	−0.120983
$960$	0	0
$961$	−26.8135	−0.864952
$962$	20.6102	0.664498
$963$	0	0
$964$	25.5429	0.822680
$965$	−0.568812	−0.0183107
$966$	0	0
$967$	−23.7018	−0.762198	−0.381099	−	0.924534i	$-0.624454\pi$
$967$	−23.7018	−0.762198	−0.381099	+	0.924534i	$0.624454\pi$
$968$	−26.8426	−0.862754
$969$	0	0
$970$	0.442577	0.0142103
$971$	−49.1695	−1.57792	−0.788961	−	0.614443i	$-0.789381\pi$
$971$	−49.1695	−1.57792	−0.788961	+	0.614443i	$0.789381\pi$
$972$	0	0
$973$	0.155550	0.00498669
$974$	−42.8079	−1.37165
$975$	0	0
$976$	25.7752	0.825043
$977$	13.2514	0.423950	0.211975	−	0.977275i	$-0.432010\pi$
$977$	13.2514	0.423950	0.211975	+	0.977275i	$0.432010\pi$
$978$	0	0
$979$	23.0688	0.737284
$980$	0.650694	0.0207856
$981$	0	0
$982$	58.0369	1.85203
$983$	−37.4907	−1.19577	−0.597884	−	0.801582i	$-0.703992\pi$
$983$	−37.4907	−1.19577	−0.597884	+	0.801582i	$0.703992\pi$
$984$	0	0
$985$	−0.395699	−0.0126080
$986$	34.0975	1.08588
$987$	0	0
$988$	−143.358	−4.56084
$989$	58.8711	1.87199
$990$	0	0
$991$	−2.23181	−0.0708957	−0.0354479	−	0.999372i	$-0.511286\pi$
$991$	−2.23181	−0.0708957	−0.0354479	+	0.999372i	$0.511286\pi$
$992$	−6.23718	−0.198031
$993$	0	0
$994$	8.74276	0.277304
$995$	−0.388277	−0.0123092
$996$	0	0
$997$	−39.8110	−1.26083	−0.630414	−	0.776259i	$-0.717115\pi$
$997$	−39.8110	−1.26083	−0.630414	+	0.776259i	$0.717115\pi$
$998$	−99.8001	−3.15912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.g.1.7		8
3.2	odd	2		717.2.a.f.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.f.1.2	✓	8	3.2	odd	2
2151.2.a.g.1.7		8	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37665 q^{2} +3.64845 q^{4} -0.0257644 q^{5} +0.278803 q^{7} +3.91777 q^{8} +O(q^{10})\)	\(q+2.37665 q^{2} +3.64845 q^{4} -0.0257644 q^{5} +0.278803 q^{7} +3.91777 q^{8} -0.0612329 q^{10} +2.03679 q^{11} -5.15341 q^{13} +0.662616 q^{14} +2.01427 q^{16} +6.10504 q^{17} +7.62467 q^{19} -0.0940000 q^{20} +4.84073 q^{22} +9.14155 q^{23} -4.99934 q^{25} -12.2478 q^{26} +1.01720 q^{28} +2.35001 q^{29} +2.04609 q^{31} -3.04835 q^{32} +14.5095 q^{34} -0.00718319 q^{35} -1.68276 q^{37} +18.1211 q^{38} -0.100939 q^{40} +5.01835 q^{41} +6.43995 q^{43} +7.43111 q^{44} +21.7262 q^{46} -6.24342 q^{47} -6.92227 q^{49} -11.8817 q^{50} -18.8019 q^{52} -7.02807 q^{53} -0.0524766 q^{55} +1.09229 q^{56} +5.58513 q^{58} -0.590925 q^{59} +12.7963 q^{61} +4.86283 q^{62} -11.2734 q^{64} +0.132774 q^{65} -13.7116 q^{67} +22.2739 q^{68} -0.0170719 q^{70} +13.1943 q^{71} -15.7242 q^{73} -3.99933 q^{74} +27.8182 q^{76} +0.567863 q^{77} -0.773284 q^{79} -0.0518964 q^{80} +11.9268 q^{82} +0.895907 q^{83} -0.157293 q^{85} +15.3055 q^{86} +7.97967 q^{88} +11.3261 q^{89} -1.43679 q^{91} +33.3524 q^{92} -14.8384 q^{94} -0.196445 q^{95} -7.22776 q^{97} -16.4518 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * q^98

Introduction

L-functions

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Database

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