LMFDB - Embedded newform 6003.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 $ x^11 - 2x^10 - 18x^9 + 30x^8 + 124x^7 - 152x^6 - 408x^5 + 285x^4 + 634x^3 - 93x^2 - 369x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72479$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72479 q^{2} +5.42450 q^{4} +1.02492 q^{5} +4.32726 q^{7} -9.33107 q^{8} +O(q^{10})$	\(q-2.72479 q^{2} +5.42450 q^{4} +1.02492 q^{5} +4.32726 q^{7} -9.33107 q^{8} -2.79269 q^{10} -2.38338 q^{11} +0.754393 q^{13} -11.7909 q^{14} +14.5762 q^{16} +1.03913 q^{17} -8.17626 q^{19} +5.55967 q^{20} +6.49422 q^{22} -1.00000 q^{23} -3.94954 q^{25} -2.05557 q^{26} +23.4733 q^{28} +1.00000 q^{29} -5.92951 q^{31} -21.0551 q^{32} -2.83143 q^{34} +4.43509 q^{35} +3.05932 q^{37} +22.2786 q^{38} -9.56358 q^{40} -1.01080 q^{41} +8.18673 q^{43} -12.9287 q^{44} +2.72479 q^{46} -8.08697 q^{47} +11.7252 q^{49} +10.7617 q^{50} +4.09221 q^{52} +0.478182 q^{53} -2.44277 q^{55} -40.3780 q^{56} -2.72479 q^{58} +8.09543 q^{59} -14.8897 q^{61} +16.1567 q^{62} +28.2183 q^{64} +0.773191 q^{65} -0.830191 q^{67} +5.63679 q^{68} -12.0847 q^{70} +1.79880 q^{71} +13.2708 q^{73} -8.33601 q^{74} -44.3521 q^{76} -10.3135 q^{77} +2.46332 q^{79} +14.9394 q^{80} +2.75423 q^{82} -4.46805 q^{83} +1.06503 q^{85} -22.3072 q^{86} +22.2395 q^{88} +5.57787 q^{89} +3.26446 q^{91} -5.42450 q^{92} +22.0353 q^{94} -8.37999 q^{95} -0.684597 q^{97} -31.9488 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * 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 $

$2$	−2.72479	−1.92672	−0.963360	−	0.268211i	$-0.913568\pi$
$2$	−2.72479	−1.92672	−0.963360	+	0.268211i	$0.913568\pi$
$3$	0	0
$3$	0	0
$4$	5.42450	2.71225
$5$	1.02492	0.458357	0.229179	−	0.973384i	$-0.426396\pi$
$5$	1.02492	0.458357	0.229179	+	0.973384i	$0.426396\pi$
$6$	0	0
$7$	4.32726	1.63555	0.817776	−	0.575537i	$-0.195207\pi$
$7$	4.32726	1.63555	0.817776	+	0.575537i	$0.195207\pi$
$8$	−9.33107	−3.29903
$9$	0	0
$10$	−2.79269	−0.883126
$11$	−2.38338	−0.718616	−0.359308	−	0.933219i	$-0.616987\pi$
$11$	−2.38338	−0.718616	−0.359308	+	0.933219i	$0.616987\pi$
$12$	0	0
$13$	0.754393	0.209231	0.104616	−	0.994513i	$-0.466639\pi$
$13$	0.754393	0.209231	0.104616	+	0.994513i	$0.466639\pi$
$14$	−11.7909	−3.15125
$15$	0	0
$16$	14.5762	3.64406
$17$	1.03913	0.252027	0.126014	−	0.992029i	$-0.459782\pi$
$17$	1.03913	0.252027	0.126014	+	0.992029i	$0.459782\pi$
$18$	0	0
$19$	−8.17626	−1.87576	−0.937881	−	0.346957i	$-0.887215\pi$
$19$	−8.17626	−1.87576	−0.937881	+	0.346957i	$0.887215\pi$
$20$	5.55967	1.24318
$21$	0	0
$22$	6.49422	1.38457
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.94954	−0.789909
$26$	−2.05557	−0.403130
$27$	0	0
$28$	23.4733	4.43603
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−5.92951	−1.06497	−0.532485	−	0.846439i	$-0.678742\pi$
$31$	−5.92951	−1.06497	−0.532485	+	0.846439i	$0.678742\pi$
$32$	−21.0551	−3.72205
$33$	0	0
$34$	−2.83143	−0.485586
$35$	4.43509	0.749667
$36$	0	0
$37$	3.05932	0.502948	0.251474	−	0.967864i	$-0.419085\pi$
$37$	3.05932	0.502948	0.251474	+	0.967864i	$0.419085\pi$
$38$	22.2786	3.61407
$39$	0	0
$40$	−9.56358	−1.51213
$41$	−1.01080	−0.157861	−0.0789304	−	0.996880i	$-0.525150\pi$
$41$	−1.01080	−0.157861	−0.0789304	+	0.996880i	$0.525150\pi$
$42$	0	0
$43$	8.18673	1.24846	0.624232	−	0.781239i	$-0.285412\pi$
$43$	8.18673	1.24846	0.624232	+	0.781239i	$0.285412\pi$
$44$	−12.9287	−1.94907
$45$	0	0
$46$	2.72479	0.401749
$47$	−8.08697	−1.17961	−0.589803	−	0.807547i	$-0.700795\pi$
$47$	−8.08697	−1.17961	−0.589803	+	0.807547i	$0.700795\pi$
$48$	0	0
$49$	11.7252	1.67503
$50$	10.7617	1.52193
$51$	0	0
$52$	4.09221	0.567487
$53$	0.478182	0.0656834	0.0328417	−	0.999461i	$-0.489544\pi$
$53$	0.478182	0.0656834	0.0328417	+	0.999461i	$0.489544\pi$
$54$	0	0
$55$	−2.44277	−0.329383
$56$	−40.3780	−5.39573
$57$	0	0
$58$	−2.72479	−0.357783
$59$	8.09543	1.05394	0.526968	−	0.849885i	$-0.323329\pi$
$59$	8.09543	1.05394	0.526968	+	0.849885i	$0.323329\pi$
$60$	0	0
$61$	−14.8897	−1.90644	−0.953218	−	0.302285i	$-0.902251\pi$
$61$	−14.8897	−1.90644	−0.953218	+	0.302285i	$0.902251\pi$
$62$	16.1567	2.05190
$63$	0	0
$64$	28.2183	3.52729
$65$	0.773191	0.0959026
$66$	0	0
$67$	−0.830191	−0.101424	−0.0507120	−	0.998713i	$-0.516149\pi$
$67$	−0.830191	−0.101424	−0.0507120	+	0.998713i	$0.516149\pi$
$68$	5.63679	0.683561
$69$	0	0
$70$	−12.0847	−1.44440
$71$	1.79880	0.213478	0.106739	−	0.994287i	$-0.465959\pi$
$71$	1.79880	0.213478	0.106739	+	0.994287i	$0.465959\pi$
$72$	0	0
$73$	13.2708	1.55323	0.776617	−	0.629973i	$-0.216934\pi$
$73$	13.2708	1.55323	0.776617	+	0.629973i	$0.216934\pi$
$74$	−8.33601	−0.969041
$75$	0	0
$76$	−44.3521	−5.08754
$77$	−10.3135	−1.17533
$78$	0	0
$79$	2.46332	0.277145	0.138572	−	0.990352i	$-0.455749\pi$
$79$	2.46332	0.277145	0.138572	+	0.990352i	$0.455749\pi$
$80$	14.9394	1.67028
$81$	0	0
$82$	2.75423	0.304153
$83$	−4.46805	−0.490432	−0.245216	−	0.969468i	$-0.578859\pi$
$83$	−4.46805	−0.490432	−0.245216	+	0.969468i	$0.578859\pi$
$84$	0	0
$85$	1.06503	0.115518
$86$	−22.3072	−2.40544
$87$	0	0
$88$	22.2395	2.37074
$89$	5.57787	0.591253	0.295626	−	0.955304i	$-0.404472\pi$
$89$	5.57787	0.591253	0.295626	+	0.955304i	$0.404472\pi$
$90$	0	0
$91$	3.26446	0.342208
$92$	−5.42450	−0.565544
$93$	0	0
$94$	22.0353	2.27277
$95$	−8.37999	−0.859769
$96$	0	0
$97$	−0.684597	−0.0695103	−0.0347551	−	0.999396i	$-0.511065\pi$
$97$	−0.684597	−0.0695103	−0.0347551	+	0.999396i	$0.511065\pi$
$98$	−31.9488	−3.22731
$99$	0	0
$100$	−21.4243	−2.14243
$101$	−19.2544	−1.91589	−0.957943	−	0.286958i	$-0.907356\pi$
$101$	−19.2544	−1.91589	−0.957943	+	0.286958i	$0.907356\pi$
$102$	0	0
$103$	−5.09888	−0.502408	−0.251204	−	0.967934i	$-0.580826\pi$
$103$	−5.09888	−0.502408	−0.251204	+	0.967934i	$0.580826\pi$
$104$	−7.03929	−0.690260
$105$	0	0
$106$	−1.30295	−0.126553
$107$	13.7603	1.33026	0.665131	−	0.746727i	$-0.268376\pi$
$107$	13.7603	1.33026	0.665131	+	0.746727i	$0.268376\pi$
$108$	0	0
$109$	−12.0945	−1.15845	−0.579223	−	0.815169i	$-0.696644\pi$
$109$	−12.0945	−1.15845	−0.579223	+	0.815169i	$0.696644\pi$
$110$	6.65604	0.634629
$111$	0	0
$112$	63.0752	5.96005
$113$	−3.66701	−0.344963	−0.172482	−	0.985013i	$-0.555179\pi$
$113$	−3.66701	−0.344963	−0.172482	+	0.985013i	$0.555179\pi$
$114$	0	0
$115$	−1.02492	−0.0955741
$116$	5.42450	0.503653
$117$	0	0
$118$	−22.0584	−2.03064
$119$	4.49661	0.412203
$120$	0	0
$121$	−5.31950	−0.483591
$122$	40.5715	3.67317
$123$	0	0
$124$	−32.1646	−2.88847
$125$	−9.17255	−0.820418
$126$	0	0
$127$	−9.23920	−0.819847	−0.409923	−	0.912120i	$-0.634445\pi$
$127$	−9.23920	−0.819847	−0.409923	+	0.912120i	$0.634445\pi$
$128$	−34.7790	−3.07406
$129$	0	0
$130$	−2.10679	−0.184777
$131$	6.81925	0.595800	0.297900	−	0.954597i	$-0.403714\pi$
$131$	6.81925	0.595800	0.297900	+	0.954597i	$0.403714\pi$
$132$	0	0
$133$	−35.3808	−3.06791
$134$	2.26210	0.195416
$135$	0	0
$136$	−9.69623	−0.831445
$137$	−4.86061	−0.415270	−0.207635	−	0.978206i	$-0.566577\pi$
$137$	−4.86061	−0.415270	−0.207635	+	0.978206i	$0.566577\pi$
$138$	0	0
$139$	−11.5906	−0.983105	−0.491553	−	0.870848i	$-0.663570\pi$
$139$	−11.5906	−0.983105	−0.491553	+	0.870848i	$0.663570\pi$
$140$	24.0582	2.03329
$141$	0	0
$142$	−4.90136	−0.411313
$143$	−1.79801	−0.150357
$144$	0	0
$145$	1.02492	0.0851148
$146$	−36.1603	−2.99265
$147$	0	0
$148$	16.5953	1.36412
$149$	5.07301	0.415597	0.207799	−	0.978172i	$-0.433370\pi$
$149$	5.07301	0.415597	0.207799	+	0.978172i	$0.433370\pi$
$150$	0	0
$151$	9.42334	0.766861	0.383430	−	0.923570i	$-0.374743\pi$
$151$	9.42334	0.766861	0.383430	+	0.923570i	$0.374743\pi$
$152$	76.2932	6.18820
$153$	0	0
$154$	28.1022	2.26454
$155$	−6.07726	−0.488137
$156$	0	0
$157$	−17.9873	−1.43554	−0.717772	−	0.696278i	$-0.754838\pi$
$157$	−17.9873	−1.43554	−0.717772	+	0.696278i	$0.754838\pi$
$158$	−6.71204	−0.533981
$159$	0	0
$160$	−21.5798	−1.70603
$161$	−4.32726	−0.341036
$162$	0	0
$163$	−16.3947	−1.28413	−0.642066	−	0.766649i	$-0.721922\pi$
$163$	−16.3947	−1.28413	−0.642066	+	0.766649i	$0.721922\pi$
$164$	−5.48310	−0.428158
$165$	0	0
$166$	12.1745	0.944925
$167$	−22.7384	−1.75955	−0.879775	−	0.475390i	$-0.842307\pi$
$167$	−22.7384	−1.75955	−0.879775	+	0.475390i	$0.842307\pi$
$168$	0	0
$169$	−12.4309	−0.956222
$170$	−2.90198	−0.222572
$171$	0	0
$172$	44.4090	3.38615
$173$	−21.4566	−1.63132	−0.815659	−	0.578533i	$-0.803625\pi$
$173$	−21.4566	−1.63132	−0.815659	+	0.578533i	$0.803625\pi$
$174$	0	0
$175$	−17.0907	−1.29194
$176$	−34.7407	−2.61868
$177$	0	0
$178$	−15.1985	−1.13918
$179$	14.9868	1.12016	0.560082	−	0.828437i	$-0.310769\pi$
$179$	14.9868	1.12016	0.560082	+	0.828437i	$0.310769\pi$
$180$	0	0
$181$	−17.0940	−1.27059	−0.635295	−	0.772269i	$-0.719122\pi$
$181$	−17.0940	−1.27059	−0.635295	+	0.772269i	$0.719122\pi$
$182$	−8.89498	−0.659340
$183$	0	0
$184$	9.33107	0.687895
$185$	3.13555	0.230530
$186$	0	0
$187$	−2.47665	−0.181111
$188$	−43.8678	−3.19939
$189$	0	0
$190$	22.8338	1.65653
$191$	15.6430	1.13189	0.565944	−	0.824444i	$-0.308512\pi$
$191$	15.6430	1.13189	0.565944	+	0.824444i	$0.308512\pi$
$192$	0	0
$193$	21.1145	1.51985	0.759927	−	0.650009i	$-0.225235\pi$
$193$	21.1145	1.51985	0.759927	+	0.650009i	$0.225235\pi$
$194$	1.86538	0.133927
$195$	0	0
$196$	63.6034	4.54310
$197$	−2.37520	−0.169226	−0.0846128	−	0.996414i	$-0.526965\pi$
$197$	−2.37520	−0.169226	−0.0846128	+	0.996414i	$0.526965\pi$
$198$	0	0
$199$	−14.1714	−1.00459	−0.502294	−	0.864697i	$-0.667510\pi$
$199$	−14.1714	−1.00459	−0.502294	+	0.864697i	$0.667510\pi$
$200$	36.8535	2.60593
$201$	0	0
$202$	52.4643	3.69138
$203$	4.32726	0.303714
$204$	0	0
$205$	−1.03599	−0.0723566
$206$	13.8934	0.968000
$207$	0	0
$208$	10.9962	0.762450
$209$	19.4871	1.34795
$210$	0	0
$211$	12.8837	0.886950	0.443475	−	0.896287i	$-0.353746\pi$
$211$	12.8837	0.886950	0.443475	+	0.896287i	$0.353746\pi$
$212$	2.59390	0.178150
$213$	0	0
$214$	−37.4941	−2.56304
$215$	8.39073	0.572243
$216$	0	0
$217$	−25.6585	−1.74181
$218$	32.9551	2.23200
$219$	0	0
$220$	−13.2508	−0.893369
$221$	0.783916	0.0527319
$222$	0	0
$223$	−3.16955	−0.212249	−0.106124	−	0.994353i	$-0.533844\pi$
$223$	−3.16955	−0.212249	−0.106124	+	0.994353i	$0.533844\pi$
$224$	−91.1109	−6.08761
$225$	0	0
$226$	9.99184	0.664647
$227$	−23.4740	−1.55802	−0.779012	−	0.627009i	$-0.784279\pi$
$227$	−23.4740	−1.55802	−0.779012	+	0.627009i	$0.784279\pi$
$228$	0	0
$229$	12.4848	0.825019	0.412510	−	0.910953i	$-0.364652\pi$
$229$	12.4848	0.825019	0.412510	+	0.910953i	$0.364652\pi$
$230$	2.79269	0.184145
$231$	0	0
$232$	−9.33107	−0.612615
$233$	9.55710	0.626107	0.313053	−	0.949736i	$-0.398648\pi$
$233$	9.55710	0.626107	0.313053	+	0.949736i	$0.398648\pi$
$234$	0	0
$235$	−8.28849	−0.540681
$236$	43.9137	2.85854
$237$	0	0
$238$	−12.2523	−0.794201
$239$	−2.85259	−0.184519	−0.0922593	−	0.995735i	$-0.529409\pi$
$239$	−2.85259	−0.184519	−0.0922593	+	0.995735i	$0.529409\pi$
$240$	0	0
$241$	0.712445	0.0458926	0.0229463	−	0.999737i	$-0.492695\pi$
$241$	0.712445	0.0458926	0.0229463	+	0.999737i	$0.492695\pi$
$242$	14.4946	0.931745
$243$	0	0
$244$	−80.7694	−5.17073
$245$	12.0174	0.767762
$246$	0	0
$247$	−6.16811	−0.392468
$248$	55.3286	3.51337
$249$	0	0
$250$	24.9933	1.58072
$251$	8.60983	0.543447	0.271724	−	0.962375i	$-0.412406\pi$
$251$	8.60983	0.543447	0.271724	+	0.962375i	$0.412406\pi$
$252$	0	0
$253$	2.38338	0.149842
$254$	25.1749	1.57962
$255$	0	0
$256$	38.3289	2.39556
$257$	−21.7102	−1.35425	−0.677123	−	0.735870i	$-0.736773\pi$
$257$	−21.7102	−1.35425	−0.677123	+	0.735870i	$0.736773\pi$
$258$	0	0
$259$	13.2385	0.822598
$260$	4.19418	0.260112
$261$	0	0
$262$	−18.5810	−1.14794
$263$	18.3399	1.13089	0.565443	−	0.824787i	$-0.308705\pi$
$263$	18.3399	1.13089	0.565443	+	0.824787i	$0.308705\pi$
$264$	0	0
$265$	0.490097	0.0301064
$266$	96.4054	5.91100
$267$	0	0
$268$	−4.50337	−0.275087
$269$	12.5342	0.764222	0.382111	−	0.924116i	$-0.375197\pi$
$269$	12.5342	0.764222	0.382111	+	0.924116i	$0.375197\pi$
$270$	0	0
$271$	11.5839	0.703673	0.351837	−	0.936061i	$-0.385557\pi$
$271$	11.5839	0.703673	0.351837	+	0.936061i	$0.385557\pi$
$272$	15.1467	0.918401
$273$	0	0
$274$	13.2442	0.800110
$275$	9.41326	0.567641
$276$	0	0
$277$	−19.8284	−1.19137	−0.595686	−	0.803217i	$-0.703120\pi$
$277$	−19.8284	−1.19137	−0.595686	+	0.803217i	$0.703120\pi$
$278$	31.5821	1.89417
$279$	0	0
$280$	−41.3841	−2.47317
$281$	14.0464	0.837936	0.418968	−	0.908001i	$-0.362392\pi$
$281$	14.0464	0.837936	0.418968	+	0.908001i	$0.362392\pi$
$282$	0	0
$283$	3.86800	0.229929	0.114964	−	0.993370i	$-0.463325\pi$
$283$	3.86800	0.229929	0.114964	+	0.993370i	$0.463325\pi$
$284$	9.75760	0.579007
$285$	0	0
$286$	4.89919	0.289695
$287$	−4.37400	−0.258189
$288$	0	0
$289$	−15.9202	−0.936482
$290$	−2.79269	−0.163992
$291$	0	0
$292$	71.9877	4.21276
$293$	−24.7095	−1.44355	−0.721774	−	0.692129i	$-0.756673\pi$
$293$	−24.7095	−1.44355	−0.721774	+	0.692129i	$0.756673\pi$
$294$	0	0
$295$	8.29715	0.483079
$296$	−28.5467	−1.65924
$297$	0	0
$298$	−13.8229	−0.800740
$299$	−0.754393	−0.0436277
$300$	0	0
$301$	35.4261	2.04193
$302$	−25.6767	−1.47753
$303$	0	0
$304$	−119.179	−6.83539
$305$	−15.2608	−0.873829
$306$	0	0
$307$	28.6139	1.63308	0.816539	−	0.577290i	$-0.195890\pi$
$307$	28.6139	1.63308	0.816539	+	0.577290i	$0.195890\pi$
$308$	−55.9457	−3.18780
$309$	0	0
$310$	16.5593	0.940504
$311$	−9.41777	−0.534033	−0.267016	−	0.963692i	$-0.586038\pi$
$311$	−9.41777	−0.534033	−0.267016	+	0.963692i	$0.586038\pi$
$312$	0	0
$313$	−2.66933	−0.150880	−0.0754398	−	0.997150i	$-0.524036\pi$
$313$	−2.66933	−0.150880	−0.0754398	+	0.997150i	$0.524036\pi$
$314$	49.0118	2.76589
$315$	0	0
$316$	13.3623	0.751687
$317$	−16.2962	−0.915283	−0.457642	−	0.889137i	$-0.651306\pi$
$317$	−16.2962	−0.915283	−0.457642	+	0.889137i	$0.651306\pi$
$318$	0	0
$319$	−2.38338	−0.133444
$320$	28.9215	1.61676
$321$	0	0
$322$	11.7909	0.657081
$323$	−8.49623	−0.472743
$324$	0	0
$325$	−2.97951	−0.165273
$326$	44.6722	2.47416
$327$	0	0
$328$	9.43186	0.520787
$329$	−34.9945	−1.92931
$330$	0	0
$331$	12.7644	0.701597	0.350798	−	0.936451i	$-0.385910\pi$
$331$	12.7644	0.701597	0.350798	+	0.936451i	$0.385910\pi$
$332$	−24.2369	−1.33018
$333$	0	0
$334$	61.9574	3.39016
$335$	−0.850878	−0.0464884
$336$	0	0
$337$	23.1747	1.26240	0.631202	−	0.775618i	$-0.282562\pi$
$337$	23.1747	1.26240	0.631202	+	0.775618i	$0.282562\pi$
$338$	33.8716	1.84237
$339$	0	0
$340$	5.77725	0.313315
$341$	14.1323	0.765305
$342$	0	0
$343$	20.4472	1.10404
$344$	−76.3910	−4.11872
$345$	0	0
$346$	58.4649	3.14309
$347$	7.87073	0.422523	0.211262	−	0.977430i	$-0.432243\pi$
$347$	7.87073	0.422523	0.211262	+	0.977430i	$0.432243\pi$
$348$	0	0
$349$	−20.2614	−1.08457	−0.542284	−	0.840195i	$-0.682440\pi$
$349$	−20.2614	−1.08457	−0.542284	+	0.840195i	$0.682440\pi$
$350$	46.5687	2.48920
$351$	0	0
$352$	50.1823	2.67472
$353$	2.08029	0.110723	0.0553614	−	0.998466i	$-0.482369\pi$
$353$	2.08029	0.110723	0.0553614	+	0.998466i	$0.482369\pi$
$354$	0	0
$355$	1.84362	0.0978494
$356$	30.2572	1.60363
$357$	0	0
$358$	−40.8359	−2.15824
$359$	−24.0681	−1.27027	−0.635133	−	0.772403i	$-0.719055\pi$
$359$	−24.0681	−1.27027	−0.635133	+	0.772403i	$0.719055\pi$
$360$	0	0
$361$	47.8512	2.51848
$362$	46.5778	2.44807
$363$	0	0
$364$	17.7081	0.928155
$365$	13.6015	0.711936
$366$	0	0
$367$	−7.21883	−0.376820	−0.188410	−	0.982090i	$-0.560333\pi$
$367$	−7.21883	−0.376820	−0.188410	+	0.982090i	$0.560333\pi$
$368$	−14.5762	−0.759839
$369$	0	0
$370$	−8.54372	−0.444167
$371$	2.06922	0.107429
$372$	0	0
$373$	−20.7310	−1.07341	−0.536704	−	0.843770i	$-0.680331\pi$
$373$	−20.7310	−1.07341	−0.536704	+	0.843770i	$0.680331\pi$
$374$	6.74837	0.348950
$375$	0	0
$376$	75.4601	3.89156
$377$	0.754393	0.0388532
$378$	0	0
$379$	12.9895	0.667228	0.333614	−	0.942710i	$-0.391732\pi$
$379$	12.9895	0.667228	0.333614	+	0.942710i	$0.391732\pi$
$380$	−45.4573	−2.33191
$381$	0	0
$382$	−42.6240	−2.18083
$383$	35.6863	1.82349	0.911743	−	0.410762i	$-0.134737\pi$
$383$	35.6863	1.82349	0.911743	+	0.410762i	$0.134737\pi$
$384$	0	0
$385$	−10.5705	−0.538723
$386$	−57.5326	−2.92833
$387$	0	0
$388$	−3.71360	−0.188529
$389$	−1.85642	−0.0941243	−0.0470622	−	0.998892i	$-0.514986\pi$
$389$	−1.85642	−0.0941243	−0.0470622	+	0.998892i	$0.514986\pi$
$390$	0	0
$391$	−1.03913	−0.0525513
$392$	−109.409	−5.52597
$393$	0	0
$394$	6.47192	0.326051
$395$	2.52470	0.127031
$396$	0	0
$397$	−18.4801	−0.927489	−0.463745	−	0.885969i	$-0.653494\pi$
$397$	−18.4801	−0.927489	−0.463745	+	0.885969i	$0.653494\pi$
$398$	38.6143	1.93556
$399$	0	0
$400$	−57.5695	−2.87847
$401$	24.3126	1.21411	0.607056	−	0.794659i	$-0.292350\pi$
$401$	24.3126	1.21411	0.607056	+	0.794659i	$0.292350\pi$
$402$	0	0
$403$	−4.47318	−0.222825
$404$	−104.446	−5.19637
$405$	0	0
$406$	−11.7909	−0.585173
$407$	−7.29151	−0.361427
$408$	0	0
$409$	30.3298	1.49971	0.749855	−	0.661602i	$-0.230123\pi$
$409$	30.3298	1.49971	0.749855	+	0.661602i	$0.230123\pi$
$410$	2.82286	0.139411
$411$	0	0
$412$	−27.6589	−1.36266
$413$	35.0311	1.72377
$414$	0	0
$415$	−4.57938	−0.224793
$416$	−15.8838	−0.778769
$417$	0	0
$418$	−53.0984	−2.59713
$419$	−27.8837	−1.36221	−0.681105	−	0.732186i	$-0.738500\pi$
$419$	−27.8837	−1.36221	−0.681105	+	0.732186i	$0.738500\pi$
$420$	0	0
$421$	14.0024	0.682437	0.341219	−	0.939984i	$-0.389160\pi$
$421$	14.0024	0.682437	0.341219	+	0.939984i	$0.389160\pi$
$422$	−35.1054	−1.70890
$423$	0	0
$424$	−4.46195	−0.216691
$425$	−4.10411	−0.199078
$426$	0	0
$427$	−64.4318	−3.11807
$428$	74.6430	3.60801
$429$	0	0
$430$	−22.8630	−1.10255
$431$	−29.8983	−1.44015	−0.720075	−	0.693897i	$-0.755892\pi$
$431$	−29.8983	−1.44015	−0.720075	+	0.693897i	$0.755892\pi$
$432$	0	0
$433$	−37.3902	−1.79686	−0.898428	−	0.439120i	$-0.855290\pi$
$433$	−37.3902	−1.79686	−0.898428	+	0.439120i	$0.855290\pi$
$434$	69.9142	3.35599
$435$	0	0
$436$	−65.6068	−3.14200
$437$	8.17626	0.391123
$438$	0	0
$439$	−27.9734	−1.33510	−0.667549	−	0.744565i	$-0.732657\pi$
$439$	−27.9734	−1.33510	−0.667549	+	0.744565i	$0.732657\pi$
$440$	22.7936	1.08664
$441$	0	0
$442$	−2.13601	−0.101600
$443$	−12.5665	−0.597051	−0.298525	−	0.954402i	$-0.596495\pi$
$443$	−12.5665	−0.597051	−0.298525	+	0.954402i	$0.596495\pi$
$444$	0	0
$445$	5.71686	0.271005
$446$	8.63638	0.408944
$447$	0	0
$448$	122.108	5.76907
$449$	−5.54208	−0.261547	−0.130774	−	0.991412i	$-0.541746\pi$
$449$	−5.54208	−0.261547	−0.130774	+	0.991412i	$0.541746\pi$
$450$	0	0
$451$	2.40912	0.113441
$452$	−19.8917	−0.935627
$453$	0	0
$454$	63.9618	3.00188
$455$	3.34580	0.156854
$456$	0	0
$457$	32.2110	1.50677	0.753383	−	0.657582i	$-0.228421\pi$
$457$	32.2110	1.50677	0.753383	+	0.657582i	$0.228421\pi$
$458$	−34.0185	−1.58958
$459$	0	0
$460$	−5.55967	−0.259221
$461$	22.9127	1.06715	0.533576	−	0.845752i	$-0.320848\pi$
$461$	22.9127	1.06715	0.533576	+	0.845752i	$0.320848\pi$
$462$	0	0
$463$	−1.70309	−0.0791491	−0.0395746	−	0.999217i	$-0.512600\pi$
$463$	−1.70309	−0.0791491	−0.0395746	+	0.999217i	$0.512600\pi$
$464$	14.5762	0.676685
$465$	0	0
$466$	−26.0411	−1.20633
$467$	−28.0758	−1.29919	−0.649596	−	0.760279i	$-0.725062\pi$
$467$	−28.0758	−1.29919	−0.649596	+	0.760279i	$0.725062\pi$
$468$	0	0
$469$	−3.59245	−0.165884
$470$	22.5844	1.04174
$471$	0	0
$472$	−75.5390	−3.47697
$473$	−19.5121	−0.897167
$474$	0	0
$475$	32.2925	1.48168
$476$	24.3919	1.11800
$477$	0	0
$478$	7.77272	0.355516
$479$	25.5753	1.16856	0.584282	−	0.811551i	$-0.301376\pi$
$479$	25.5753	1.16856	0.584282	+	0.811551i	$0.301376\pi$
$480$	0	0
$481$	2.30793	0.105232
$482$	−1.94127	−0.0884223
$483$	0	0
$484$	−28.8557	−1.31162
$485$	−0.701655	−0.0318605
$486$	0	0
$487$	34.5635	1.56622	0.783112	−	0.621881i	$-0.213631\pi$
$487$	34.5635	1.56622	0.783112	+	0.621881i	$0.213631\pi$
$488$	138.937	6.28939
$489$	0	0
$490$	−32.7449	−1.47926
$491$	−14.9939	−0.676664	−0.338332	−	0.941027i	$-0.609863\pi$
$491$	−14.9939	−0.676664	−0.338332	+	0.941027i	$0.609863\pi$
$492$	0	0
$493$	1.03913	0.0468003
$494$	16.8068	0.756175
$495$	0	0
$496$	−86.4299	−3.88082
$497$	7.78388	0.349155
$498$	0	0
$499$	1.58115	0.0707819	0.0353909	−	0.999374i	$-0.488732\pi$
$499$	1.58115	0.0707819	0.0353909	+	0.999374i	$0.488732\pi$
$500$	−49.7565	−2.22518
$501$	0	0
$502$	−23.4600	−1.04707
$503$	−38.2023	−1.70336	−0.851679	−	0.524064i	$-0.824415\pi$
$503$	−38.2023	−1.70336	−0.851679	+	0.524064i	$0.824415\pi$
$504$	0	0
$505$	−19.7342	−0.878161
$506$	−6.49422	−0.288703
$507$	0	0
$508$	−50.1181	−2.22363
$509$	−0.866481	−0.0384061	−0.0192030	−	0.999816i	$-0.506113\pi$
$509$	−0.866481	−0.0384061	−0.0192030	+	0.999816i	$0.506113\pi$
$510$	0	0
$511$	57.4264	2.54039
$512$	−34.8804	−1.54151
$513$	0	0
$514$	59.1558	2.60925
$515$	−5.22594	−0.230282
$516$	0	0
$517$	19.2743	0.847684
$518$	−36.0721	−1.58492
$519$	0	0
$520$	−7.21470	−0.316386
$521$	−3.68724	−0.161541	−0.0807704	−	0.996733i	$-0.525738\pi$
$521$	−3.68724	−0.161541	−0.0807704	+	0.996733i	$0.525738\pi$
$522$	0	0
$523$	−41.5135	−1.81526	−0.907628	−	0.419775i	$-0.862109\pi$
$523$	−41.5135	−1.81526	−0.907628	+	0.419775i	$0.862109\pi$
$524$	36.9910	1.61596
$525$	0	0
$526$	−49.9724	−2.17890
$527$	−6.16155	−0.268401
$528$	0	0
$529$	1.00000	0.0434783
$530$	−1.33541	−0.0580067
$531$	0	0
$532$	−191.923	−8.32093
$533$	−0.762542	−0.0330294
$534$	0	0
$535$	14.1032	0.609735
$536$	7.74657	0.334601
$537$	0	0
$538$	−34.1530	−1.47244
$539$	−27.9456	−1.20370
$540$	0	0
$541$	−25.7510	−1.10712	−0.553561	−	0.832809i	$-0.686731\pi$
$541$	−25.7510	−1.10712	−0.553561	+	0.832809i	$0.686731\pi$
$542$	−31.5638	−1.35578
$543$	0	0
$544$	−21.8791	−0.938058
$545$	−12.3959	−0.530982
$546$	0	0
$547$	28.4992	1.21854	0.609268	−	0.792964i	$-0.291463\pi$
$547$	28.4992	1.21854	0.609268	+	0.792964i	$0.291463\pi$
$548$	−26.3664	−1.12632
$549$	0	0
$550$	−25.6492	−1.09369
$551$	−8.17626	−0.348320
$552$	0	0
$553$	10.6594	0.453285
$554$	54.0283	2.29544
$555$	0	0
$556$	−62.8735	−2.66643
$557$	29.0231	1.22975	0.614873	−	0.788626i	$-0.289207\pi$
$557$	29.0231	1.22975	0.614873	+	0.788626i	$0.289207\pi$
$558$	0	0
$559$	6.17602	0.261218
$560$	64.6469	2.73183
$561$	0	0
$562$	−38.2735	−1.61447
$563$	30.2427	1.27458	0.637288	−	0.770626i	$-0.280056\pi$
$563$	30.2427	1.27458	0.637288	+	0.770626i	$0.280056\pi$
$564$	0	0
$565$	−3.75838	−0.158116
$566$	−10.5395	−0.443009
$567$	0	0
$568$	−16.7847	−0.704272
$569$	34.0959	1.42937	0.714687	−	0.699444i	$-0.246569\pi$
$569$	34.0959	1.42937	0.714687	+	0.699444i	$0.246569\pi$
$570$	0	0
$571$	−7.96785	−0.333444	−0.166722	−	0.986004i	$-0.553318\pi$
$571$	−7.96785	−0.333444	−0.166722	+	0.986004i	$0.553318\pi$
$572$	−9.75329	−0.407805
$573$	0	0
$574$	11.9183	0.497459
$575$	3.94954	0.164707
$576$	0	0
$577$	−35.2407	−1.46709	−0.733545	−	0.679640i	$-0.762136\pi$
$577$	−35.2407	−1.46709	−0.733545	+	0.679640i	$0.762136\pi$
$578$	43.3793	1.80434
$579$	0	0
$580$	5.55967	0.230853
$581$	−19.3344	−0.802127
$582$	0	0
$583$	−1.13969	−0.0472011
$584$	−123.831	−5.12416
$585$	0	0
$586$	67.3284	2.78131
$587$	6.85699	0.283018	0.141509	−	0.989937i	$-0.454805\pi$
$587$	6.85699	0.283018	0.141509	+	0.989937i	$0.454805\pi$
$588$	0	0
$589$	48.4812	1.99763
$590$	−22.6080	−0.930758
$591$	0	0
$592$	44.5933	1.83277
$593$	−36.7362	−1.50858	−0.754288	−	0.656543i	$-0.772018\pi$
$593$	−36.7362	−1.50858	−0.754288	+	0.656543i	$0.772018\pi$
$594$	0	0
$595$	4.60865	0.188936
$596$	27.5186	1.12720
$597$	0	0
$598$	2.05557	0.0840584
$599$	−39.6697	−1.62086	−0.810430	−	0.585836i	$-0.800766\pi$
$599$	−39.6697	−1.62086	−0.810430	+	0.585836i	$0.800766\pi$
$600$	0	0
$601$	−13.4001	−0.546601	−0.273300	−	0.961929i	$-0.588115\pi$
$601$	−13.4001	−0.546601	−0.273300	+	0.961929i	$0.588115\pi$
$602$	−96.5289	−3.93423
$603$	0	0
$604$	51.1169	2.07992
$605$	−5.45205	−0.221658
$606$	0	0
$607$	10.1830	0.413313	0.206657	−	0.978414i	$-0.433742\pi$
$607$	10.1830	0.413313	0.206657	+	0.978414i	$0.433742\pi$
$608$	172.152	6.98168
$609$	0	0
$610$	41.5824	1.68362
$611$	−6.10076	−0.246810
$612$	0	0
$613$	−9.64274	−0.389467	−0.194733	−	0.980856i	$-0.562384\pi$
$613$	−9.64274	−0.389467	−0.194733	+	0.980856i	$0.562384\pi$
$614$	−77.9669	−3.14649
$615$	0	0
$616$	96.2361	3.87746
$617$	1.00133	0.0403122	0.0201561	−	0.999797i	$-0.493584\pi$
$617$	1.00133	0.0403122	0.0201561	+	0.999797i	$0.493584\pi$
$618$	0	0
$619$	29.4218	1.18256	0.591281	−	0.806466i	$-0.298622\pi$
$619$	29.4218	1.18256	0.591281	+	0.806466i	$0.298622\pi$
$620$	−32.9661	−1.32395
$621$	0	0
$622$	25.6615	1.02893
$623$	24.1369	0.967025
$624$	0	0
$625$	10.3466	0.413864
$626$	7.27338	0.290703
$627$	0	0
$628$	−97.5723	−3.89356
$629$	3.17904	0.126757
$630$	0	0
$631$	−46.2853	−1.84259	−0.921294	−	0.388866i	$-0.872867\pi$
$631$	−46.2853	−1.84259	−0.921294	+	0.388866i	$0.872867\pi$
$632$	−22.9854	−0.914310
$633$	0	0
$634$	44.4037	1.76349
$635$	−9.46943	−0.375783
$636$	0	0
$637$	8.84541	0.350468
$638$	6.49422	0.257109
$639$	0	0
$640$	−35.6456	−1.40902
$641$	−29.5602	−1.16756	−0.583780	−	0.811912i	$-0.698427\pi$
$641$	−29.5602	−1.16756	−0.583780	+	0.811912i	$0.698427\pi$
$642$	0	0
$643$	22.8121	0.899621	0.449810	−	0.893124i	$-0.351492\pi$
$643$	22.8121	0.899621	0.449810	+	0.893124i	$0.351492\pi$
$644$	−23.4733	−0.924976
$645$	0	0
$646$	23.1505	0.910843
$647$	−27.7127	−1.08950	−0.544749	−	0.838599i	$-0.683375\pi$
$647$	−27.7127	−1.08950	−0.544749	+	0.838599i	$0.683375\pi$
$648$	0	0
$649$	−19.2945	−0.757375
$650$	8.11855	0.318436
$651$	0	0
$652$	−88.9331	−3.48289
$653$	18.2397	0.713774	0.356887	−	0.934148i	$-0.383838\pi$
$653$	18.2397	0.713774	0.356887	+	0.934148i	$0.383838\pi$
$654$	0	0
$655$	6.98917	0.273089
$656$	−14.7337	−0.575254
$657$	0	0
$658$	95.3527	3.71724
$659$	−18.0505	−0.703149	−0.351574	−	0.936160i	$-0.614354\pi$
$659$	−18.0505	−0.703149	−0.351574	+	0.936160i	$0.614354\pi$
$660$	0	0
$661$	−9.33849	−0.363225	−0.181613	−	0.983370i	$-0.558132\pi$
$661$	−9.33849	−0.363225	−0.181613	+	0.983370i	$0.558132\pi$
$662$	−34.7805	−1.35178
$663$	0	0
$664$	41.6917	1.61795
$665$	−36.2624	−1.40620
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−123.345	−4.77234
$669$	0	0
$670$	2.31847	0.0895702
$671$	35.4879	1.36999
$672$	0	0
$673$	12.8952	0.497074	0.248537	−	0.968622i	$-0.420050\pi$
$673$	12.8952	0.497074	0.248537	+	0.968622i	$0.420050\pi$
$674$	−63.1462	−2.43230
$675$	0	0
$676$	−67.4314	−2.59352
$677$	−45.6983	−1.75633	−0.878165	−	0.478358i	$-0.841232\pi$
$677$	−45.6983	−1.75633	−0.878165	+	0.478358i	$0.841232\pi$
$678$	0	0
$679$	−2.96243	−0.113688
$680$	−9.93784	−0.381099
$681$	0	0
$682$	−38.5075	−1.47453
$683$	21.1386	0.808847	0.404423	−	0.914572i	$-0.367472\pi$
$683$	21.1386	0.808847	0.404423	+	0.914572i	$0.367472\pi$
$684$	0	0
$685$	−4.98173	−0.190342
$686$	−55.7144	−2.12719
$687$	0	0
$688$	119.332	4.54948
$689$	0.360737	0.0137430
$690$	0	0
$691$	10.9159	0.415261	0.207630	−	0.978207i	$-0.433425\pi$
$691$	10.9159	0.415261	0.207630	+	0.978207i	$0.433425\pi$
$692$	−116.392	−4.42455
$693$	0	0
$694$	−21.4461	−0.814084
$695$	−11.8795	−0.450614
$696$	0	0
$697$	−1.05036	−0.0397852
$698$	55.2081	2.08966
$699$	0	0
$700$	−92.7086	−3.50406
$701$	12.4276	0.469384	0.234692	−	0.972070i	$-0.424592\pi$
$701$	12.4276	0.469384	0.234692	+	0.972070i	$0.424592\pi$
$702$	0	0
$703$	−25.0138	−0.943411
$704$	−67.2550	−2.53477
$705$	0	0
$706$	−5.66837	−0.213332
$707$	−83.3189	−3.13353
$708$	0	0
$709$	−45.4539	−1.70706	−0.853529	−	0.521046i	$-0.825542\pi$
$709$	−45.4539	−1.70706	−0.853529	+	0.521046i	$0.825542\pi$
$710$	−5.02350	−0.188528
$711$	0	0
$712$	−52.0475	−1.95056
$713$	5.92951	0.222062
$714$	0	0
$715$	−1.84281	−0.0689171
$716$	81.2959	3.03817
$717$	0	0
$718$	65.5806	2.44745
$719$	30.5357	1.13879	0.569394	−	0.822065i	$-0.307178\pi$
$719$	30.5357	1.13879	0.569394	+	0.822065i	$0.307178\pi$
$720$	0	0
$721$	−22.0642	−0.821714
$722$	−130.385	−4.85241
$723$	0	0
$724$	−92.7267	−3.44616
$725$	−3.94954	−0.146682
$726$	0	0
$727$	13.9153	0.516091	0.258045	−	0.966133i	$-0.416922\pi$
$727$	13.9153	0.516091	0.258045	+	0.966133i	$0.416922\pi$
$728$	−30.4609	−1.12896
$729$	0	0
$730$	−37.0613	−1.37170
$731$	8.50711	0.314647
$732$	0	0
$733$	7.18360	0.265332	0.132666	−	0.991161i	$-0.457646\pi$
$733$	7.18360	0.265332	0.132666	+	0.991161i	$0.457646\pi$
$734$	19.6698	0.726027
$735$	0	0
$736$	21.0551	0.776101
$737$	1.97866	0.0728849
$738$	0	0
$739$	3.44007	0.126545	0.0632725	−	0.997996i	$-0.479846\pi$
$739$	3.44007	0.126545	0.0632725	+	0.997996i	$0.479846\pi$
$740$	17.0088	0.625256
$741$	0	0
$742$	−5.63820	−0.206985
$743$	6.89471	0.252942	0.126471	−	0.991970i	$-0.459635\pi$
$743$	6.89471	0.252942	0.126471	+	0.991970i	$0.459635\pi$
$744$	0	0
$745$	5.19942	0.190492
$746$	56.4876	2.06816
$747$	0	0
$748$	−13.4346	−0.491218
$749$	59.5446	2.17571
$750$	0	0
$751$	44.3395	1.61797	0.808985	−	0.587829i	$-0.200017\pi$
$751$	44.3395	1.61797	0.808985	+	0.587829i	$0.200017\pi$
$752$	−117.878	−4.29855
$753$	0	0
$754$	−2.05557	−0.0748593
$755$	9.65815	0.351496
$756$	0	0
$757$	−30.4467	−1.10660	−0.553302	−	0.832981i	$-0.686633\pi$
$757$	−30.4467	−1.10660	−0.553302	+	0.832981i	$0.686633\pi$
$758$	−35.3938	−1.28556
$759$	0	0
$760$	78.1943	2.83640
$761$	−34.4231	−1.24784	−0.623919	−	0.781489i	$-0.714460\pi$
$761$	−34.4231	−1.24784	−0.623919	+	0.781489i	$0.714460\pi$
$762$	0	0
$763$	−52.3362	−1.89470
$764$	84.8555	3.06997
$765$	0	0
$766$	−97.2378	−3.51335
$767$	6.10714	0.220516
$768$	0	0
$769$	−34.2773	−1.23607	−0.618035	−	0.786151i	$-0.712071\pi$
$769$	−34.2773	−1.23607	−0.618035	+	0.786151i	$0.712071\pi$
$770$	28.8024	1.03797
$771$	0	0
$772$	114.536	4.12223
$773$	10.4729	0.376682	0.188341	−	0.982104i	$-0.439689\pi$
$773$	10.4729	0.376682	0.188341	+	0.982104i	$0.439689\pi$
$774$	0	0
$775$	23.4188	0.841229
$776$	6.38802	0.229316
$777$	0	0
$778$	5.05837	0.181351
$779$	8.26457	0.296109
$780$	0	0
$781$	−4.28723	−0.153409
$782$	2.83143	0.101252
$783$	0	0
$784$	170.909	6.10390
$785$	−18.4355	−0.657993
$786$	0	0
$787$	43.2021	1.53999	0.769995	−	0.638050i	$-0.220259\pi$
$787$	43.2021	1.53999	0.769995	+	0.638050i	$0.220259\pi$
$788$	−12.8843	−0.458983
$789$	0	0
$790$	−6.87929	−0.244754
$791$	−15.8681	−0.564205
$792$	0	0
$793$	−11.2327	−0.398885
$794$	50.3544	1.78701
$795$	0	0
$796$	−76.8731	−2.72469
$797$	22.4875	0.796549	0.398274	−	0.917266i	$-0.369609\pi$
$797$	22.4875	0.796549	0.398274	+	0.917266i	$0.369609\pi$
$798$	0	0
$799$	−8.40345	−0.297293
$800$	83.1580	2.94008
$801$	0	0
$802$	−66.2468	−2.33926
$803$	−31.6294	−1.11618
$804$	0	0
$805$	−4.43509	−0.156316
$806$	12.1885	0.429321
$807$	0	0
$808$	179.664	6.32057
$809$	−14.7365	−0.518108	−0.259054	−	0.965863i	$-0.583411\pi$
$809$	−14.7365	−0.518108	−0.259054	+	0.965863i	$0.583411\pi$
$810$	0	0
$811$	53.4867	1.87817	0.939086	−	0.343682i	$-0.111674\pi$
$811$	53.4867	1.87817	0.939086	+	0.343682i	$0.111674\pi$
$812$	23.4733	0.823750
$813$	0	0
$814$	19.8679	0.696368
$815$	−16.8032	−0.588591
$816$	0	0
$817$	−66.9368	−2.34182
$818$	−82.6423	−2.88952
$819$	0	0
$820$	−5.61973	−0.196249
$821$	−21.2503	−0.741640	−0.370820	−	0.928705i	$-0.620923\pi$
$821$	−21.2503	−0.741640	−0.370820	+	0.928705i	$0.620923\pi$
$822$	0	0
$823$	−11.6892	−0.407459	−0.203730	−	0.979027i	$-0.565306\pi$
$823$	−11.6892	−0.407459	−0.203730	+	0.979027i	$0.565306\pi$
$824$	47.5780	1.65746
$825$	0	0
$826$	−95.4524	−3.32121
$827$	36.0442	1.25338	0.626690	−	0.779269i	$-0.284409\pi$
$827$	36.0442	1.25338	0.626690	+	0.779269i	$0.284409\pi$
$828$	0	0
$829$	11.0806	0.384846	0.192423	−	0.981312i	$-0.438366\pi$
$829$	11.0806	0.384846	0.192423	+	0.981312i	$0.438366\pi$
$830$	12.4779	0.433113
$831$	0	0
$832$	21.2877	0.738019
$833$	12.1841	0.422153
$834$	0	0
$835$	−23.3050	−0.806503
$836$	105.708	3.65599
$837$	0	0
$838$	75.9775	2.62460
$839$	−40.9744	−1.41459	−0.707296	−	0.706917i	$-0.750085\pi$
$839$	−40.9744	−1.41459	−0.707296	+	0.706917i	$0.750085\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−38.1538	−1.31487
$843$	0	0
$844$	69.8876	2.40563
$845$	−12.7406	−0.438292
$846$	0	0
$847$	−23.0189	−0.790938
$848$	6.97009	0.239354
$849$	0	0
$850$	11.1828	0.383568
$851$	−3.05932	−0.104872
$852$	0	0
$853$	20.2611	0.693726	0.346863	−	0.937916i	$-0.387247\pi$
$853$	20.2611	0.693726	0.346863	+	0.937916i	$0.387247\pi$
$854$	175.563	6.00766
$855$	0	0
$856$	−128.399	−4.38858
$857$	−12.5526	−0.428789	−0.214395	−	0.976747i	$-0.568778\pi$
$857$	−12.5526	−0.428789	−0.214395	+	0.976747i	$0.568778\pi$
$858$	0	0
$859$	−49.7481	−1.69738	−0.848692	−	0.528887i	$-0.822609\pi$
$859$	−49.7481	−1.69738	−0.848692	+	0.528887i	$0.822609\pi$
$860$	45.5155	1.55207
$861$	0	0
$862$	81.4666	2.77476
$863$	−29.3777	−1.00003	−0.500015	−	0.866017i	$-0.666672\pi$
$863$	−29.3777	−1.00003	−0.500015	+	0.866017i	$0.666672\pi$
$864$	0	0
$865$	−21.9913	−0.747727
$866$	101.881	3.46204
$867$	0	0
$868$	−139.185	−4.72424
$869$	−5.87102	−0.199161
$870$	0	0
$871$	−0.626290	−0.0212210
$872$	112.855	3.82175
$873$	0	0
$874$	−22.2786	−0.753585
$875$	−39.6920	−1.34184
$876$	0	0
$877$	47.3882	1.60019	0.800093	−	0.599876i	$-0.204783\pi$
$877$	47.3882	1.60019	0.800093	+	0.599876i	$0.204783\pi$
$878$	76.2218	2.57236
$879$	0	0
$880$	−35.6064	−1.20029
$881$	46.1479	1.55476	0.777382	−	0.629029i	$-0.216547\pi$
$881$	46.1479	1.55476	0.777382	+	0.629029i	$0.216547\pi$
$882$	0	0
$883$	−17.8622	−0.601111	−0.300556	−	0.953764i	$-0.597172\pi$
$883$	−17.8622	−0.601111	−0.300556	+	0.953764i	$0.597172\pi$
$884$	4.25235	0.143022
$885$	0	0
$886$	34.2410	1.15035
$887$	−34.0136	−1.14207	−0.571033	−	0.820927i	$-0.693457\pi$
$887$	−34.0136	−1.14207	−0.571033	+	0.820927i	$0.693457\pi$
$888$	0	0
$889$	−39.9805	−1.34090
$890$	−15.5773	−0.522151
$891$	0	0
$892$	−17.1933	−0.575673
$893$	66.1212	2.21266
$894$	0	0
$895$	15.3602	0.513436
$896$	−150.498	−5.02778
$897$	0	0
$898$	15.1010	0.503928
$899$	−5.92951	−0.197760
$900$	0	0
$901$	0.496895	0.0165540
$902$	−6.56437	−0.218570
$903$	0	0
$904$	34.2171	1.13804
$905$	−17.5200	−0.582385
$906$	0	0
$907$	1.99413	0.0662141	0.0331070	−	0.999452i	$-0.489460\pi$
$907$	1.99413	0.0662141	0.0331070	+	0.999452i	$0.489460\pi$
$908$	−127.335	−4.22575
$909$	0	0
$910$	−9.11662	−0.302213
$911$	−3.41787	−0.113239	−0.0566196	−	0.998396i	$-0.518032\pi$
$911$	−3.41787	−0.113239	−0.0566196	+	0.998396i	$0.518032\pi$
$912$	0	0
$913$	10.6491	0.352432
$914$	−87.7683	−2.90312
$915$	0	0
$916$	67.7239	2.23766
$917$	29.5087	0.974462
$918$	0	0
$919$	5.56763	0.183659	0.0918296	−	0.995775i	$-0.470729\pi$
$919$	5.56763	0.183659	0.0918296	+	0.995775i	$0.470729\pi$
$920$	9.56358	0.315302
$921$	0	0
$922$	−62.4325	−2.05610
$923$	1.35700	0.0446663
$924$	0	0
$925$	−12.0829	−0.397283
$926$	4.64056	0.152498
$927$	0	0
$928$	−21.0551	−0.691167
$929$	32.3923	1.06276	0.531379	−	0.847134i	$-0.321674\pi$
$929$	32.3923	1.06276	0.531379	+	0.847134i	$0.321674\pi$
$930$	0	0
$931$	−95.8683	−3.14196
$932$	51.8425	1.69816
$933$	0	0
$934$	76.5007	2.50318
$935$	−2.53836	−0.0830134
$936$	0	0
$937$	−1.53335	−0.0500925	−0.0250462	−	0.999686i	$-0.507973\pi$
$937$	−1.53335	−0.0500925	−0.0250462	+	0.999686i	$0.507973\pi$
$938$	9.78870	0.319612
$939$	0	0
$940$	−44.9609	−1.46646
$941$	4.27935	0.139503	0.0697514	−	0.997564i	$-0.477779\pi$
$941$	4.27935	0.139503	0.0697514	+	0.997564i	$0.477779\pi$
$942$	0	0
$943$	1.01080	0.0329162
$944$	118.001	3.84060
$945$	0	0
$946$	53.1664	1.72859
$947$	−55.7160	−1.81053	−0.905263	−	0.424851i	$-0.860327\pi$
$947$	−55.7160	−1.81053	−0.905263	+	0.424851i	$0.860327\pi$
$948$	0	0
$949$	10.0114	0.324985
$950$	−87.9903	−2.85478
$951$	0	0
$952$	−41.9581	−1.35987
$953$	−60.9052	−1.97291	−0.986457	−	0.164019i	$-0.947554\pi$
$953$	−60.9052	−1.97291	−0.986457	+	0.164019i	$0.947554\pi$
$954$	0	0
$955$	16.0328	0.518809
$956$	−15.4739	−0.500461
$957$	0	0
$958$	−69.6873	−2.25150
$959$	−21.0332	−0.679196
$960$	0	0
$961$	4.15904	0.134163
$962$	−6.28863	−0.202753
$963$	0	0
$964$	3.86466	0.124472
$965$	21.6406	0.696636
$966$	0	0
$967$	27.2667	0.876839	0.438420	−	0.898770i	$-0.355538\pi$
$967$	27.2667	0.876839	0.438420	+	0.898770i	$0.355538\pi$
$968$	49.6366	1.59538
$969$	0	0
$970$	1.91187	0.0613863
$971$	−44.4273	−1.42574	−0.712870	−	0.701297i	$-0.752605\pi$
$971$	−44.4273	−1.42574	−0.712870	+	0.701297i	$0.752605\pi$
$972$	0	0
$973$	−50.1558	−1.60792
$974$	−94.1785	−3.01767
$975$	0	0
$976$	−217.036	−6.94716
$977$	−0.396649	−0.0126899	−0.00634496	−	0.999980i	$-0.502020\pi$
$977$	−0.396649	−0.0126899	−0.00634496	+	0.999980i	$0.502020\pi$
$978$	0	0
$979$	−13.2942	−0.424884
$980$	65.1883	2.08236
$981$	0	0
$982$	40.8552	1.30374
$983$	2.59076	0.0826323	0.0413161	−	0.999146i	$-0.486845\pi$
$983$	2.59076	0.0826323	0.0413161	+	0.999146i	$0.486845\pi$
$984$	0	0
$985$	−2.43438	−0.0775658
$986$	−2.83143	−0.0901710
$987$	0	0
$988$	−33.4590	−1.06447
$989$	−8.18673	−0.260323
$990$	0	0
$991$	55.7666	1.77148	0.885742	−	0.464178i	$-0.153650\pi$
$991$	55.7666	1.77148	0.885742	+	0.464178i	$0.153650\pi$
$992$	124.846	3.96388
$993$	0	0
$994$	−21.2095	−0.672724
$995$	−14.5246	−0.460460
$996$	0	0
$997$	−4.99638	−0.158237	−0.0791184	−	0.996865i	$-0.525211\pi$
$997$	−4.99638	−0.158237	−0.0791184	+	0.996865i	$0.525211\pi$
$998$	−4.30830	−0.136377
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.1		11
3.2	odd	2		2001.2.a.l.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.11	✓	11	3.2	odd	2
6003.2.a.m.1.1		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.72479 q^{2} +5.42450 q^{4} +1.02492 q^{5} +4.32726 q^{7} -9.33107 q^{8} +O(q^{10})\)	\(q-2.72479 q^{2} +5.42450 q^{4} +1.02492 q^{5} +4.32726 q^{7} -9.33107 q^{8} -2.79269 q^{10} -2.38338 q^{11} +0.754393 q^{13} -11.7909 q^{14} +14.5762 q^{16} +1.03913 q^{17} -8.17626 q^{19} +5.55967 q^{20} +6.49422 q^{22} -1.00000 q^{23} -3.94954 q^{25} -2.05557 q^{26} +23.4733 q^{28} +1.00000 q^{29} -5.92951 q^{31} -21.0551 q^{32} -2.83143 q^{34} +4.43509 q^{35} +3.05932 q^{37} +22.2786 q^{38} -9.56358 q^{40} -1.01080 q^{41} +8.18673 q^{43} -12.9287 q^{44} +2.72479 q^{46} -8.08697 q^{47} +11.7252 q^{49} +10.7617 q^{50} +4.09221 q^{52} +0.478182 q^{53} -2.44277 q^{55} -40.3780 q^{56} -2.72479 q^{58} +8.09543 q^{59} -14.8897 q^{61} +16.1567 q^{62} +28.2183 q^{64} +0.773191 q^{65} -0.830191 q^{67} +5.63679 q^{68} -12.0847 q^{70} +1.79880 q^{71} +13.2708 q^{73} -8.33601 q^{74} -44.3521 q^{76} -10.3135 q^{77} +2.46332 q^{79} +14.9394 q^{80} +2.75423 q^{82} -4.46805 q^{83} +1.06503 q^{85} -22.3072 q^{86} +22.2395 q^{88} +5.57787 q^{89} +3.26446 q^{91} -5.42450 q^{92} +22.0353 q^{94} -8.37999 q^{95} -0.684597 q^{97} -31.9488 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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