LMFDB - Embedded newform 6003.2.a.p.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} + \cdots - 64 $ x^14 - 2x^13 - 18x^12 + 34x^11 + 124x^10 - 216x^9 - 420x^8 + 647x^7 + 750x^6 - 939x^5 - 717x^4 + 604x^3 + 352x^2 - 128*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.35292$ 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)

$f(q)$	$=$	$q+1.35292 q^{2} -0.169600 q^{4} +0.738177 q^{5} +3.44152 q^{7} -2.93530 q^{8} +O(q^{10})$	\(q+1.35292 q^{2} -0.169600 q^{4} +0.738177 q^{5} +3.44152 q^{7} -2.93530 q^{8} +0.998696 q^{10} -1.17333 q^{11} -2.89875 q^{13} +4.65611 q^{14} -3.63203 q^{16} +3.96681 q^{17} -3.02566 q^{19} -0.125195 q^{20} -1.58742 q^{22} +1.00000 q^{23} -4.45509 q^{25} -3.92179 q^{26} -0.583682 q^{28} +1.00000 q^{29} +10.3766 q^{31} +0.956741 q^{32} +5.36678 q^{34} +2.54045 q^{35} +3.14877 q^{37} -4.09349 q^{38} -2.16677 q^{40} +7.62266 q^{41} +6.17008 q^{43} +0.198997 q^{44} +1.35292 q^{46} -11.0057 q^{47} +4.84403 q^{49} -6.02740 q^{50} +0.491629 q^{52} +12.0907 q^{53} -0.866122 q^{55} -10.1019 q^{56} +1.35292 q^{58} -1.75699 q^{59} +2.90124 q^{61} +14.0387 q^{62} +8.55847 q^{64} -2.13979 q^{65} +8.65474 q^{67} -0.672772 q^{68} +3.43703 q^{70} +8.41446 q^{71} +1.23211 q^{73} +4.26004 q^{74} +0.513153 q^{76} -4.03802 q^{77} -15.7529 q^{79} -2.68108 q^{80} +10.3129 q^{82} +9.53737 q^{83} +2.92821 q^{85} +8.34764 q^{86} +3.44407 q^{88} +10.0938 q^{89} -9.97610 q^{91} -0.169600 q^{92} -14.8899 q^{94} -2.23347 q^{95} -12.9789 q^{97} +6.55360 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	$ 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * 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 $

$2$	1.35292	0.956661	0.478330	−	0.878180i	$-0.341242\pi$
$2$	1.35292	0.956661	0.478330	+	0.878180i	$0.341242\pi$
$3$	0	0
$3$	0	0
$4$	−0.169600	−0.0848002
$5$	0.738177	0.330123	0.165061	−	0.986283i	$-0.447218\pi$
$5$	0.738177	0.330123	0.165061	+	0.986283i	$0.447218\pi$
$6$	0	0
$7$	3.44152	1.30077	0.650385	−	0.759604i	$-0.274607\pi$
$7$	3.44152	1.30077	0.650385	+	0.759604i	$0.274607\pi$
$8$	−2.93530	−1.03779
$9$	0	0
$10$	0.998696	0.315816
$11$	−1.17333	−0.353771	−0.176886	−	0.984231i	$-0.556602\pi$
$11$	−1.17333	−0.353771	−0.176886	+	0.984231i	$0.556602\pi$
$12$	0	0
$13$	−2.89875	−0.803969	−0.401984	−	0.915646i	$-0.631679\pi$
$13$	−2.89875	−0.803969	−0.401984	+	0.915646i	$0.631679\pi$
$14$	4.65611	1.24440
$15$	0	0
$16$	−3.63203	−0.908009
$17$	3.96681	0.962092	0.481046	−	0.876695i	$-0.340257\pi$
$17$	3.96681	0.962092	0.481046	+	0.876695i	$0.340257\pi$
$18$	0	0
$19$	−3.02566	−0.694134	−0.347067	−	0.937840i	$-0.612822\pi$
$19$	−3.02566	−0.694134	−0.347067	+	0.937840i	$0.612822\pi$
$20$	−0.125195	−0.0279945
$21$	0	0
$22$	−1.58742	−0.338439
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.45509	−0.891019
$26$	−3.92179	−0.769126
$27$	0	0
$28$	−0.583682	−0.110306
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.3766	1.86369	0.931847	−	0.362852i	$-0.118197\pi$
$31$	10.3766	1.86369	0.931847	+	0.362852i	$0.118197\pi$
$32$	0.956741	0.169129
$33$	0	0
$34$	5.36678	0.920396
$35$	2.54045	0.429414
$36$	0	0
$37$	3.14877	0.517654	0.258827	−	0.965924i	$-0.416664\pi$
$37$	3.14877	0.517654	0.258827	+	0.965924i	$0.416664\pi$
$38$	−4.09349	−0.664051
$39$	0	0
$40$	−2.16677	−0.342597
$41$	7.62266	1.19046	0.595230	−	0.803556i	$-0.297061\pi$
$41$	7.62266	1.19046	0.595230	+	0.803556i	$0.297061\pi$
$42$	0	0
$43$	6.17008	0.940928	0.470464	−	0.882419i	$-0.344086\pi$
$43$	6.17008	0.940928	0.470464	+	0.882419i	$0.344086\pi$
$44$	0.198997	0.0299999
$45$	0	0
$46$	1.35292	0.199478
$47$	−11.0057	−1.60535	−0.802673	−	0.596419i	$-0.796590\pi$
$47$	−11.0057	−1.60535	−0.802673	+	0.596419i	$0.796590\pi$
$48$	0	0
$49$	4.84403	0.692005
$50$	−6.02740	−0.852403
$51$	0	0
$52$	0.491629	0.0681767
$53$	12.0907	1.66079	0.830395	−	0.557176i	$-0.188115\pi$
$53$	12.0907	1.66079	0.830395	+	0.557176i	$0.188115\pi$
$54$	0	0
$55$	−0.866122	−0.116788
$56$	−10.1019	−1.34992
$57$	0	0
$58$	1.35292	0.177647
$59$	−1.75699	−0.228741	−0.114371	−	0.993438i	$-0.536485\pi$
$59$	−1.75699	−0.228741	−0.114371	+	0.993438i	$0.536485\pi$
$60$	0	0
$61$	2.90124	0.371466	0.185733	−	0.982600i	$-0.440534\pi$
$61$	2.90124	0.371466	0.185733	+	0.982600i	$0.440534\pi$
$62$	14.0387	1.78292
$63$	0	0
$64$	8.55847	1.06981
$65$	−2.13979	−0.265408
$66$	0	0
$67$	8.65474	1.05735	0.528673	−	0.848826i	$-0.322690\pi$
$67$	8.65474	1.05735	0.528673	+	0.848826i	$0.322690\pi$
$68$	−0.672772	−0.0815856
$69$	0	0
$70$	3.43703	0.410804
$71$	8.41446	0.998613	0.499306	−	0.866425i	$-0.333588\pi$
$71$	8.41446	0.998613	0.499306	+	0.866425i	$0.333588\pi$
$72$	0	0
$73$	1.23211	0.144207	0.0721037	−	0.997397i	$-0.477029\pi$
$73$	1.23211	0.144207	0.0721037	+	0.997397i	$0.477029\pi$
$74$	4.26004	0.495219
$75$	0	0
$76$	0.513153	0.0588627
$77$	−4.03802	−0.460175
$78$	0	0
$79$	−15.7529	−1.77234	−0.886169	−	0.463363i	$-0.846643\pi$
$79$	−15.7529	−1.77234	−0.886169	+	0.463363i	$0.846643\pi$
$80$	−2.68108	−0.299754
$81$	0	0
$82$	10.3129	1.13887
$83$	9.53737	1.04686	0.523431	−	0.852068i	$-0.324652\pi$
$83$	9.53737	1.04686	0.523431	+	0.852068i	$0.324652\pi$
$84$	0	0
$85$	2.92821	0.317609
$86$	8.34764	0.900149
$87$	0	0
$88$	3.44407	0.367139
$89$	10.0938	1.06994	0.534971	−	0.844871i	$-0.320323\pi$
$89$	10.0938	1.06994	0.534971	+	0.844871i	$0.320323\pi$
$90$	0	0
$91$	−9.97610	−1.04578
$92$	−0.169600	−0.0176821
$93$	0	0
$94$	−14.8899	−1.53577
$95$	−2.23347	−0.229150
$96$	0	0
$97$	−12.9789	−1.31780	−0.658902	−	0.752229i	$-0.728979\pi$
$97$	−12.9789	−1.31780	−0.658902	+	0.752229i	$0.728979\pi$
$98$	6.55360	0.662014
$99$	0	0
$100$	0.755586	0.0755586
$101$	6.44364	0.641166	0.320583	−	0.947220i	$-0.396121\pi$
$101$	6.44364	0.641166	0.320583	+	0.947220i	$0.396121\pi$
$102$	0	0
$103$	4.18324	0.412187	0.206094	−	0.978532i	$-0.433925\pi$
$103$	4.18324	0.412187	0.206094	+	0.978532i	$0.433925\pi$
$104$	8.50871	0.834348
$105$	0	0
$106$	16.3578	1.58881
$107$	5.35867	0.518042	0.259021	−	0.965872i	$-0.416600\pi$
$107$	5.35867	0.518042	0.259021	+	0.965872i	$0.416600\pi$
$108$	0	0
$109$	11.3099	1.08329	0.541647	−	0.840606i	$-0.317801\pi$
$109$	11.3099	1.08329	0.541647	+	0.840606i	$0.317801\pi$
$110$	−1.17180	−0.111726
$111$	0	0
$112$	−12.4997	−1.18111
$113$	5.42973	0.510786	0.255393	−	0.966837i	$-0.417795\pi$
$113$	5.42973	0.510786	0.255393	+	0.966837i	$0.417795\pi$
$114$	0	0
$115$	0.738177	0.0688354
$116$	−0.169600	−0.0157470
$117$	0	0
$118$	−2.37708	−0.218828
$119$	13.6518	1.25146
$120$	0	0
$121$	−9.62331	−0.874846
$122$	3.92516	0.355367
$123$	0	0
$124$	−1.75988	−0.158042
$125$	−6.97953	−0.624268
$126$	0	0
$127$	−9.48395	−0.841564	−0.420782	−	0.907162i	$-0.638244\pi$
$127$	−9.48395	−0.841564	−0.420782	+	0.907162i	$0.638244\pi$
$128$	9.66546	0.854314
$129$	0	0
$130$	−2.89497	−0.253906
$131$	0.643336	0.0562085	0.0281042	−	0.999605i	$-0.491053\pi$
$131$	0.643336	0.0562085	0.0281042	+	0.999605i	$0.491053\pi$
$132$	0	0
$133$	−10.4129	−0.902910
$134$	11.7092	1.01152
$135$	0	0
$136$	−11.6438	−0.998446
$137$	15.3086	1.30790	0.653951	−	0.756536i	$-0.273110\pi$
$137$	15.3086	1.30790	0.653951	+	0.756536i	$0.273110\pi$
$138$	0	0
$139$	8.30209	0.704174	0.352087	−	0.935967i	$-0.385472\pi$
$139$	8.30209	0.704174	0.352087	+	0.935967i	$0.385472\pi$
$140$	−0.430861	−0.0364144
$141$	0	0
$142$	11.3841	0.955334
$143$	3.40118	0.284421
$144$	0	0
$145$	0.738177	0.0613023
$146$	1.66695	0.137958
$147$	0	0
$148$	−0.534032	−0.0438972
$149$	−17.6570	−1.44652	−0.723258	−	0.690578i	$-0.757356\pi$
$149$	−17.6570	−1.44652	−0.723258	+	0.690578i	$0.757356\pi$
$150$	0	0
$151$	8.23676	0.670298	0.335149	−	0.942165i	$-0.391213\pi$
$151$	8.23676	0.670298	0.335149	+	0.942165i	$0.391213\pi$
$152$	8.88123	0.720363
$153$	0	0
$154$	−5.46313	−0.440232
$155$	7.65977	0.615248
$156$	0	0
$157$	17.2680	1.37814	0.689068	−	0.724696i	$-0.258020\pi$
$157$	17.2680	1.37814	0.689068	+	0.724696i	$0.258020\pi$
$158$	−21.3124	−1.69553
$159$	0	0
$160$	0.706244	0.0558335
$161$	3.44152	0.271229
$162$	0	0
$163$	−21.4179	−1.67758	−0.838791	−	0.544454i	$-0.816737\pi$
$163$	−21.4179	−1.67758	−0.838791	+	0.544454i	$0.816737\pi$
$164$	−1.29281	−0.100951
$165$	0	0
$166$	12.9033	1.00149
$167$	−1.86970	−0.144682	−0.0723409	−	0.997380i	$-0.523047\pi$
$167$	−1.86970	−0.144682	−0.0723409	+	0.997380i	$0.523047\pi$
$168$	0	0
$169$	−4.59724	−0.353634
$170$	3.96164	0.303844
$171$	0	0
$172$	−1.04645	−0.0797909
$173$	8.89523	0.676292	0.338146	−	0.941094i	$-0.390200\pi$
$173$	8.89523	0.676292	0.338146	+	0.941094i	$0.390200\pi$
$174$	0	0
$175$	−15.3323	−1.15901
$176$	4.26156	0.321227
$177$	0	0
$178$	13.6561	1.02357
$179$	−11.5346	−0.862136	−0.431068	−	0.902319i	$-0.641863\pi$
$179$	−11.5346	−0.862136	−0.431068	+	0.902319i	$0.641863\pi$
$180$	0	0
$181$	4.24247	0.315340	0.157670	−	0.987492i	$-0.449602\pi$
$181$	4.24247	0.315340	0.157670	+	0.987492i	$0.449602\pi$
$182$	−13.4969	−1.00046
$183$	0	0
$184$	−2.93530	−0.216393
$185$	2.32435	0.170889
$186$	0	0
$187$	−4.65436	−0.340360
$188$	1.86657	0.136134
$189$	0	0
$190$	−3.02172	−0.219218
$191$	21.9527	1.58844	0.794222	−	0.607628i	$-0.207879\pi$
$191$	21.9527	1.58844	0.794222	+	0.607628i	$0.207879\pi$
$192$	0	0
$193$	−16.4290	−1.18258	−0.591292	−	0.806457i	$-0.701382\pi$
$193$	−16.4290	−1.18258	−0.591292	+	0.806457i	$0.701382\pi$
$194$	−17.5594	−1.26069
$195$	0	0
$196$	−0.821550	−0.0586821
$197$	−22.2164	−1.58285	−0.791426	−	0.611266i	$-0.790661\pi$
$197$	−22.2164	−1.58285	−0.791426	+	0.611266i	$0.790661\pi$
$198$	0	0
$199$	−16.1955	−1.14807	−0.574033	−	0.818832i	$-0.694622\pi$
$199$	−16.1955	−1.14807	−0.574033	+	0.818832i	$0.694622\pi$
$200$	13.0770	0.924687
$201$	0	0
$202$	8.71775	0.613379
$203$	3.44152	0.241547
$204$	0	0
$205$	5.62687	0.392998
$206$	5.65960	0.394323
$207$	0	0
$208$	10.5284	0.730011
$209$	3.55009	0.245565
$210$	0	0
$211$	19.9907	1.37621	0.688107	−	0.725609i	$-0.258442\pi$
$211$	19.9907	1.37621	0.688107	+	0.725609i	$0.258442\pi$
$212$	−2.05059	−0.140835
$213$	0	0
$214$	7.24987	0.495591
$215$	4.55461	0.310622
$216$	0	0
$217$	35.7113	2.42424
$218$	15.3014	1.03634
$219$	0	0
$220$	0.146895	0.00990364
$221$	−11.4988	−0.773492
$222$	0	0
$223$	8.34982	0.559145	0.279573	−	0.960125i	$-0.409807\pi$
$223$	8.34982	0.559145	0.279573	+	0.960125i	$0.409807\pi$
$224$	3.29264	0.219999
$225$	0	0
$226$	7.34600	0.488649
$227$	7.28325	0.483406	0.241703	−	0.970350i	$-0.422294\pi$
$227$	7.28325	0.483406	0.241703	+	0.970350i	$0.422294\pi$
$228$	0	0
$229$	−16.3948	−1.08340	−0.541700	−	0.840572i	$-0.682219\pi$
$229$	−16.3948	−1.08340	−0.541700	+	0.840572i	$0.682219\pi$
$230$	0.998696	0.0658521
$231$	0	0
$232$	−2.93530	−0.192712
$233$	−17.6144	−1.15396	−0.576980	−	0.816758i	$-0.695769\pi$
$233$	−17.6144	−1.15396	−0.576980	+	0.816758i	$0.695769\pi$
$234$	0	0
$235$	−8.12416	−0.529962
$236$	0.297987	0.0193973
$237$	0	0
$238$	18.4699	1.19722
$239$	20.3195	1.31436	0.657178	−	0.753735i	$-0.271750\pi$
$239$	20.3195	1.31436	0.657178	+	0.753735i	$0.271750\pi$
$240$	0	0
$241$	2.51402	0.161942	0.0809712	−	0.996716i	$-0.474198\pi$
$241$	2.51402	0.161942	0.0809712	+	0.996716i	$0.474198\pi$
$242$	−13.0196	−0.836931
$243$	0	0
$244$	−0.492052	−0.0315004
$245$	3.57575	0.228447
$246$	0	0
$247$	8.77064	0.558062
$248$	−30.4585	−1.93411
$249$	0	0
$250$	−9.44277	−0.597213
$251$	4.97478	0.314005	0.157003	−	0.987598i	$-0.449817\pi$
$251$	4.97478	0.314005	0.157003	+	0.987598i	$0.449817\pi$
$252$	0	0
$253$	−1.17333	−0.0737664
$254$	−12.8310	−0.805092
$255$	0	0
$256$	−4.04031	−0.252519
$257$	14.1760	0.884275	0.442137	−	0.896947i	$-0.354220\pi$
$257$	14.1760	0.884275	0.442137	+	0.896947i	$0.354220\pi$
$258$	0	0
$259$	10.8365	0.673349
$260$	0.362909	0.0225067
$261$	0	0
$262$	0.870383	0.0537725
$263$	18.3150	1.12935	0.564675	−	0.825313i	$-0.309002\pi$
$263$	18.3150	1.12935	0.564675	+	0.825313i	$0.309002\pi$
$264$	0	0
$265$	8.92510	0.548264
$266$	−14.0878	−0.863778
$267$	0	0
$268$	−1.46785	−0.0896631
$269$	12.3846	0.755103	0.377551	−	0.925989i	$-0.376766\pi$
$269$	12.3846	0.755103	0.377551	+	0.925989i	$0.376766\pi$
$270$	0	0
$271$	0.0474151	0.00288026	0.00144013	−	0.999999i	$-0.499542\pi$
$271$	0.0474151	0.00288026	0.00144013	+	0.999999i	$0.499542\pi$
$272$	−14.4076	−0.873588
$273$	0	0
$274$	20.7114	1.25122
$275$	5.22728	0.315217
$276$	0	0
$277$	15.5455	0.934039	0.467020	−	0.884247i	$-0.345328\pi$
$277$	15.5455	0.934039	0.467020	+	0.884247i	$0.345328\pi$
$278$	11.2321	0.673655
$279$	0	0
$280$	−7.45698	−0.445640
$281$	24.5204	1.46277	0.731383	−	0.681967i	$-0.238875\pi$
$281$	24.5204	1.46277	0.731383	+	0.681967i	$0.238875\pi$
$282$	0	0
$283$	24.5260	1.45792	0.728961	−	0.684556i	$-0.240004\pi$
$283$	24.5260	1.45792	0.728961	+	0.684556i	$0.240004\pi$
$284$	−1.42710	−0.0846826
$285$	0	0
$286$	4.60153	0.272094
$287$	26.2335	1.54851
$288$	0	0
$289$	−1.26444	−0.0743786
$290$	0.998696	0.0586455
$291$	0	0
$292$	−0.208966	−0.0122288
$293$	28.1398	1.64394	0.821972	−	0.569528i	$-0.192874\pi$
$293$	28.1398	1.64394	0.821972	+	0.569528i	$0.192874\pi$
$294$	0	0
$295$	−1.29697	−0.0755127
$296$	−9.24258	−0.537214
$297$	0	0
$298$	−23.8885	−1.38383
$299$	−2.89875	−0.167639
$300$	0	0
$301$	21.2344	1.22393
$302$	11.1437	0.641247
$303$	0	0
$304$	10.9893	0.630280
$305$	2.14163	0.122629
$306$	0	0
$307$	−20.2600	−1.15630	−0.578149	−	0.815931i	$-0.696225\pi$
$307$	−20.2600	−1.15630	−0.578149	+	0.815931i	$0.696225\pi$
$308$	0.684850	0.0390229
$309$	0	0
$310$	10.3631	0.588583
$311$	−19.5818	−1.11038	−0.555191	−	0.831723i	$-0.687355\pi$
$311$	−19.5818	−1.11038	−0.555191	+	0.831723i	$0.687355\pi$
$312$	0	0
$313$	−32.6842	−1.84742	−0.923710	−	0.383092i	$-0.874859\pi$
$313$	−32.6842	−1.84742	−0.923710	+	0.383092i	$0.874859\pi$
$314$	23.3623	1.31841
$315$	0	0
$316$	2.67169	0.150295
$317$	13.2909	0.746493	0.373247	−	0.927732i	$-0.378245\pi$
$317$	13.2909	0.746493	0.373247	+	0.927732i	$0.378245\pi$
$318$	0	0
$319$	−1.17333	−0.0656936
$320$	6.31766	0.353168
$321$	0	0
$322$	4.65611	0.259475
$323$	−12.0022	−0.667821
$324$	0	0
$325$	12.9142	0.716352
$326$	−28.9768	−1.60488
$327$	0	0
$328$	−22.3748	−1.23544
$329$	−37.8763	−2.08819
$330$	0	0
$331$	−24.6966	−1.35745	−0.678724	−	0.734394i	$-0.737467\pi$
$331$	−24.6966	−1.35745	−0.678724	+	0.734394i	$0.737467\pi$
$332$	−1.61754	−0.0887742
$333$	0	0
$334$	−2.52956	−0.138411
$335$	6.38873	0.349054
$336$	0	0
$337$	−28.3431	−1.54395	−0.771974	−	0.635654i	$-0.780731\pi$
$337$	−28.3431	−1.54395	−0.771974	+	0.635654i	$0.780731\pi$
$338$	−6.21971	−0.338308
$339$	0	0
$340$	−0.496625	−0.0269333
$341$	−12.1751	−0.659321
$342$	0	0
$343$	−7.41979	−0.400631
$344$	−18.1110	−0.976482
$345$	0	0
$346$	12.0346	0.646982
$347$	7.12832	0.382668	0.191334	−	0.981525i	$-0.438719\pi$
$347$	7.12832	0.382668	0.191334	+	0.981525i	$0.438719\pi$
$348$	0	0
$349$	−20.1149	−1.07673	−0.538364	−	0.842713i	$-0.680957\pi$
$349$	−20.1149	−1.07673	−0.538364	+	0.842713i	$0.680957\pi$
$350$	−20.7434	−1.10878
$351$	0	0
$352$	−1.12257	−0.0598331
$353$	5.22872	0.278297	0.139148	−	0.990272i	$-0.455563\pi$
$353$	5.22872	0.278297	0.139148	+	0.990272i	$0.455563\pi$
$354$	0	0
$355$	6.21136	0.329665
$356$	−1.71191	−0.0907312
$357$	0	0
$358$	−15.6054	−0.824772
$359$	6.09563	0.321715	0.160857	−	0.986978i	$-0.448574\pi$
$359$	6.09563	0.321715	0.160857	+	0.986978i	$0.448574\pi$
$360$	0	0
$361$	−9.84537	−0.518178
$362$	5.73973	0.301674
$363$	0	0
$364$	1.69195	0.0886823
$365$	0.909514	0.0476061
$366$	0	0
$367$	24.0718	1.25654	0.628268	−	0.777997i	$-0.283764\pi$
$367$	24.0718	1.25654	0.628268	+	0.777997i	$0.283764\pi$
$368$	−3.63203	−0.189333
$369$	0	0
$370$	3.14466	0.163483
$371$	41.6104	2.16031
$372$	0	0
$373$	8.81626	0.456488	0.228244	−	0.973604i	$-0.426702\pi$
$373$	8.81626	0.456488	0.228244	+	0.973604i	$0.426702\pi$
$374$	−6.29699	−0.325609
$375$	0	0
$376$	32.3051	1.66601
$377$	−2.89875	−0.149293
$378$	0	0
$379$	16.9793	0.872169	0.436085	−	0.899906i	$-0.356365\pi$
$379$	16.9793	0.872169	0.436085	+	0.899906i	$0.356365\pi$
$380$	0.378798	0.0194319
$381$	0	0
$382$	29.7003	1.51960
$383$	−20.0054	−1.02223	−0.511114	−	0.859513i	$-0.670767\pi$
$383$	−20.0054	−1.02223	−0.511114	+	0.859513i	$0.670767\pi$
$384$	0	0
$385$	−2.98077	−0.151914
$386$	−22.2271	−1.13133
$387$	0	0
$388$	2.20122	0.111750
$389$	−2.36742	−0.120033	−0.0600164	−	0.998197i	$-0.519115\pi$
$389$	−2.36742	−0.120033	−0.0600164	+	0.998197i	$0.519115\pi$
$390$	0	0
$391$	3.96681	0.200610
$392$	−14.2187	−0.718153
$393$	0	0
$394$	−30.0570	−1.51425
$395$	−11.6284	−0.585089
$396$	0	0
$397$	−22.4601	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$397$	−22.4601	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$398$	−21.9112	−1.09831
$399$	0	0
$400$	16.1811	0.809053
$401$	38.1990	1.90757	0.953783	−	0.300498i	$-0.0971528\pi$
$401$	38.1990	1.90757	0.953783	+	0.300498i	$0.0971528\pi$
$402$	0	0
$403$	−30.0792	−1.49835
$404$	−1.09284	−0.0543710
$405$	0	0
$406$	4.65611	0.231079
$407$	−3.69453	−0.183131
$408$	0	0
$409$	−26.4483	−1.30778	−0.653892	−	0.756588i	$-0.726865\pi$
$409$	−26.4483	−1.30778	−0.653892	+	0.756588i	$0.726865\pi$
$410$	7.61272	0.375965
$411$	0	0
$412$	−0.709480	−0.0349536
$413$	−6.04672	−0.297540
$414$	0	0
$415$	7.04027	0.345593
$416$	−2.77335	−0.135975
$417$	0	0
$418$	4.80299	0.234922
$419$	18.1958	0.888925	0.444462	−	0.895798i	$-0.353395\pi$
$419$	18.1958	0.888925	0.444462	+	0.895798i	$0.353395\pi$
$420$	0	0
$421$	17.3568	0.845920	0.422960	−	0.906148i	$-0.360991\pi$
$421$	17.3568	0.845920	0.422960	+	0.906148i	$0.360991\pi$
$422$	27.0458	1.31657
$423$	0	0
$424$	−35.4899	−1.72354
$425$	−17.6725	−0.857242
$426$	0	0
$427$	9.98468	0.483192
$428$	−0.908833	−0.0439301
$429$	0	0
$430$	6.16204	0.297160
$431$	−6.91304	−0.332990	−0.166495	−	0.986042i	$-0.553245\pi$
$431$	−6.91304	−0.332990	−0.166495	+	0.986042i	$0.553245\pi$
$432$	0	0
$433$	14.4908	0.696383	0.348192	−	0.937423i	$-0.386796\pi$
$433$	14.4908	0.696383	0.348192	+	0.937423i	$0.386796\pi$
$434$	48.3146	2.31917
$435$	0	0
$436$	−1.91817	−0.0918635
$437$	−3.02566	−0.144737
$438$	0	0
$439$	−23.0948	−1.10225	−0.551127	−	0.834421i	$-0.685802\pi$
$439$	−23.0948	−1.10225	−0.551127	+	0.834421i	$0.685802\pi$
$440$	2.54233	0.121201
$441$	0	0
$442$	−15.5570	−0.739970
$443$	−3.39253	−0.161184	−0.0805920	−	0.996747i	$-0.525681\pi$
$443$	−3.39253	−0.161184	−0.0805920	+	0.996747i	$0.525681\pi$
$444$	0	0
$445$	7.45102	0.353212
$446$	11.2967	0.534912
$447$	0	0
$448$	29.4541	1.39158
$449$	8.19587	0.386787	0.193394	−	0.981121i	$-0.438051\pi$
$449$	8.19587	0.386787	0.193394	+	0.981121i	$0.438051\pi$
$450$	0	0
$451$	−8.94386	−0.421150
$452$	−0.920884	−0.0433147
$453$	0	0
$454$	9.85367	0.462456
$455$	−7.36413	−0.345236
$456$	0	0
$457$	39.5184	1.84859	0.924297	−	0.381674i	$-0.124652\pi$
$457$	39.5184	1.84859	0.924297	+	0.381674i	$0.124652\pi$
$458$	−22.1809	−1.03645
$459$	0	0
$460$	−0.125195	−0.00583725
$461$	−25.3442	−1.18040	−0.590200	−	0.807257i	$-0.700951\pi$
$461$	−25.3442	−1.18040	−0.590200	+	0.807257i	$0.700951\pi$
$462$	0	0
$463$	18.3101	0.850943	0.425471	−	0.904972i	$-0.360108\pi$
$463$	18.3101	0.850943	0.425471	+	0.904972i	$0.360108\pi$
$464$	−3.63203	−0.168613
$465$	0	0
$466$	−23.8310	−1.10395
$467$	4.56418	0.211205	0.105603	−	0.994408i	$-0.466323\pi$
$467$	4.56418	0.211205	0.105603	+	0.994408i	$0.466323\pi$
$468$	0	0
$469$	29.7854	1.37536
$470$	−10.9914	−0.506993
$471$	0	0
$472$	5.15731	0.237384
$473$	−7.23952	−0.332873
$474$	0	0
$475$	13.4796	0.618487
$476$	−2.31536	−0.106124
$477$	0	0
$478$	27.4907	1.25739
$479$	−15.6660	−0.715797	−0.357898	−	0.933761i	$-0.616507\pi$
$479$	−15.6660	−0.715797	−0.357898	+	0.933761i	$0.616507\pi$
$480$	0	0
$481$	−9.12749	−0.416178
$482$	3.40128	0.154924
$483$	0	0
$484$	1.63212	0.0741871
$485$	−9.58070	−0.435037
$486$	0	0
$487$	13.9794	0.633468	0.316734	−	0.948514i	$-0.397414\pi$
$487$	13.9794	0.633468	0.316734	+	0.948514i	$0.397414\pi$
$488$	−8.51603	−0.385502
$489$	0	0
$490$	4.83772	0.218546
$491$	41.1904	1.85890	0.929449	−	0.368951i	$-0.120283\pi$
$491$	41.1904	1.85890	0.929449	+	0.368951i	$0.120283\pi$
$492$	0	0
$493$	3.96681	0.178656
$494$	11.8660	0.533876
$495$	0	0
$496$	−37.6882	−1.69225
$497$	28.9585	1.29897
$498$	0	0
$499$	−41.4979	−1.85770	−0.928851	−	0.370454i	$-0.879202\pi$
$499$	−41.4979	−1.85770	−0.928851	+	0.370454i	$0.879202\pi$
$500$	1.18373	0.0529381
$501$	0	0
$502$	6.73050	0.300397
$503$	−16.9864	−0.757384	−0.378692	−	0.925523i	$-0.623626\pi$
$503$	−16.9864	−0.757384	−0.378692	+	0.925523i	$0.623626\pi$
$504$	0	0
$505$	4.75655	0.211664
$506$	−1.58742	−0.0705694
$507$	0	0
$508$	1.60848	0.0713648
$509$	−27.3370	−1.21169	−0.605846	−	0.795582i	$-0.707165\pi$
$509$	−27.3370	−1.21169	−0.605846	+	0.795582i	$0.707165\pi$
$510$	0	0
$511$	4.24032	0.187581
$512$	−24.7972	−1.09589
$513$	0	0
$514$	19.1790	0.845951
$515$	3.08797	0.136072
$516$	0	0
$517$	12.9133	0.567925
$518$	14.6610	0.644167
$519$	0	0
$520$	6.28093	0.275437
$521$	11.3336	0.496535	0.248268	−	0.968691i	$-0.420139\pi$
$521$	11.3336	0.496535	0.248268	+	0.968691i	$0.420139\pi$
$522$	0	0
$523$	−3.81341	−0.166749	−0.0833743	−	0.996518i	$-0.526570\pi$
$523$	−3.81341	−0.166749	−0.0833743	+	0.996518i	$0.526570\pi$
$524$	−0.109110	−0.00476649
$525$	0	0
$526$	24.7787	1.08040
$527$	41.1620	1.79304
$528$	0	0
$529$	1.00000	0.0434783
$530$	12.0750	0.524503
$531$	0	0
$532$	1.76603	0.0765669
$533$	−22.0962	−0.957092
$534$	0	0
$535$	3.95565	0.171018
$536$	−25.4043	−1.09730
$537$	0	0
$538$	16.7554	0.722377
$539$	−5.68363	−0.244811
$540$	0	0
$541$	−12.8352	−0.551827	−0.275913	−	0.961183i	$-0.588980\pi$
$541$	−12.8352	−0.551827	−0.275913	+	0.961183i	$0.588980\pi$
$542$	0.0641489	0.00275543
$543$	0	0
$544$	3.79521	0.162718
$545$	8.34872	0.357620
$546$	0	0
$547$	25.7312	1.10019	0.550094	−	0.835103i	$-0.314592\pi$
$547$	25.7312	1.10019	0.550094	+	0.835103i	$0.314592\pi$
$548$	−2.59635	−0.110910
$549$	0	0
$550$	7.07210	0.301556
$551$	−3.02566	−0.128898
$552$	0	0
$553$	−54.2138	−2.30540
$554$	21.0319	0.893559
$555$	0	0
$556$	−1.40804	−0.0597141
$557$	−34.7825	−1.47378	−0.736890	−	0.676012i	$-0.763707\pi$
$557$	−34.7825	−1.47378	−0.736890	+	0.676012i	$0.763707\pi$
$558$	0	0
$559$	−17.8855	−0.756477
$560$	−9.22700	−0.389912
$561$	0	0
$562$	33.1742	1.39937
$563$	−16.3561	−0.689326	−0.344663	−	0.938727i	$-0.612007\pi$
$563$	−16.3561	−0.689326	−0.344663	+	0.938727i	$0.612007\pi$
$564$	0	0
$565$	4.00810	0.168622
$566$	33.1818	1.39474
$567$	0	0
$568$	−24.6990	−1.03635
$569$	−10.5054	−0.440407	−0.220204	−	0.975454i	$-0.570672\pi$
$569$	−10.5054	−0.440407	−0.220204	+	0.975454i	$0.570672\pi$
$570$	0	0
$571$	−21.7569	−0.910498	−0.455249	−	0.890364i	$-0.650450\pi$
$571$	−21.7569	−0.910498	−0.455249	+	0.890364i	$0.650450\pi$
$572$	−0.576841	−0.0241190
$573$	0	0
$574$	35.4919	1.48140
$575$	−4.45509	−0.185790
$576$	0	0
$577$	31.5980	1.31544	0.657721	−	0.753262i	$-0.271521\pi$
$577$	31.5980	1.31544	0.657721	+	0.753262i	$0.271521\pi$
$578$	−1.71068	−0.0711551
$579$	0	0
$580$	−0.125195	−0.00519844
$581$	32.8230	1.36173
$582$	0	0
$583$	−14.1864	−0.587539
$584$	−3.61661	−0.149656
$585$	0	0
$586$	38.0710	1.57270
$587$	−38.0454	−1.57030	−0.785151	−	0.619304i	$-0.787415\pi$
$587$	−38.0454	−1.57030	−0.785151	+	0.619304i	$0.787415\pi$
$588$	0	0
$589$	−31.3961	−1.29365
$590$	−1.75470	−0.0722400
$591$	0	0
$592$	−11.4364	−0.470034
$593$	−18.4978	−0.759612	−0.379806	−	0.925066i	$-0.624009\pi$
$593$	−18.4978	−0.759612	−0.379806	+	0.925066i	$0.624009\pi$
$594$	0	0
$595$	10.0775	0.413136
$596$	2.99463	0.122665
$597$	0	0
$598$	−3.92179	−0.160374
$599$	0.949485	0.0387949	0.0193975	−	0.999812i	$-0.493825\pi$
$599$	0.949485	0.0387949	0.0193975	+	0.999812i	$0.493825\pi$
$600$	0	0
$601$	30.9918	1.26418	0.632091	−	0.774894i	$-0.282197\pi$
$601$	30.9918	1.26418	0.632091	+	0.774894i	$0.282197\pi$
$602$	28.7285	1.17089
$603$	0	0
$604$	−1.39696	−0.0568414
$605$	−7.10370	−0.288807
$606$	0	0
$607$	−23.3743	−0.948734	−0.474367	−	0.880327i	$-0.657323\pi$
$607$	−23.3743	−0.948734	−0.474367	+	0.880327i	$0.657323\pi$
$608$	−2.89477	−0.117399
$609$	0	0
$610$	2.89746	0.117315
$611$	31.9028	1.29065
$612$	0	0
$613$	−7.83570	−0.316481	−0.158241	−	0.987401i	$-0.550582\pi$
$613$	−7.83570	−0.316481	−0.158241	+	0.987401i	$0.550582\pi$
$614$	−27.4102	−1.10618
$615$	0	0
$616$	11.8528	0.477563
$617$	−1.46753	−0.0590806	−0.0295403	−	0.999564i	$-0.509404\pi$
$617$	−1.46753	−0.0590806	−0.0295403	+	0.999564i	$0.509404\pi$
$618$	0	0
$619$	27.2490	1.09523	0.547614	−	0.836731i	$-0.315536\pi$
$619$	27.2490	1.09523	0.547614	+	0.836731i	$0.315536\pi$
$620$	−1.29910	−0.0521731
$621$	0	0
$622$	−26.4927	−1.06226
$623$	34.7380	1.39175
$624$	0	0
$625$	17.1233	0.684934
$626$	−44.2192	−1.76735
$627$	0	0
$628$	−2.92866	−0.116866
$629$	12.4906	0.498031
$630$	0	0
$631$	−5.06096	−0.201474	−0.100737	−	0.994913i	$-0.532120\pi$
$631$	−5.06096	−0.201474	−0.100737	+	0.994913i	$0.532120\pi$
$632$	46.2394	1.83931
$633$	0	0
$634$	17.9816	0.714141
$635$	−7.00083	−0.277820
$636$	0	0
$637$	−14.0416	−0.556350
$638$	−1.58742	−0.0628465
$639$	0	0
$640$	7.13482	0.282029
$641$	−10.4401	−0.412358	−0.206179	−	0.978514i	$-0.566103\pi$
$641$	−10.4401	−0.412358	−0.206179	+	0.978514i	$0.566103\pi$
$642$	0	0
$643$	12.1008	0.477209	0.238605	−	0.971117i	$-0.423310\pi$
$643$	12.1008	0.477209	0.238605	+	0.971117i	$0.423310\pi$
$644$	−0.583682	−0.0230003
$645$	0	0
$646$	−16.2381	−0.638878
$647$	−34.6932	−1.36393	−0.681966	−	0.731384i	$-0.738875\pi$
$647$	−34.6932	−1.36393	−0.681966	+	0.731384i	$0.738875\pi$
$648$	0	0
$649$	2.06153	0.0809220
$650$	17.4719	0.685305
$651$	0	0
$652$	3.63249	0.142259
$653$	−10.2303	−0.400342	−0.200171	−	0.979761i	$-0.564150\pi$
$653$	−10.2303	−0.400342	−0.200171	+	0.979761i	$0.564150\pi$
$654$	0	0
$655$	0.474896	0.0185557
$656$	−27.6858	−1.08095
$657$	0	0
$658$	−51.2437	−1.99769
$659$	13.9869	0.544853	0.272427	−	0.962177i	$-0.412174\pi$
$659$	13.9869	0.544853	0.272427	+	0.962177i	$0.412174\pi$
$660$	0	0
$661$	35.3316	1.37424	0.687120	−	0.726544i	$-0.258875\pi$
$661$	35.3316	1.37424	0.687120	+	0.726544i	$0.258875\pi$
$662$	−33.4126	−1.29862
$663$	0	0
$664$	−27.9951	−1.08642
$665$	−7.68654	−0.298071
$666$	0	0
$667$	1.00000	0.0387202
$668$	0.317102	0.0122690
$669$	0	0
$670$	8.64346	0.333926
$671$	−3.40411	−0.131414
$672$	0	0
$673$	35.8417	1.38160	0.690799	−	0.723047i	$-0.257259\pi$
$673$	35.8417	1.38160	0.690799	+	0.723047i	$0.257259\pi$
$674$	−38.3461	−1.47703
$675$	0	0
$676$	0.779694	0.0299882
$677$	−18.3954	−0.706992	−0.353496	−	0.935436i	$-0.615007\pi$
$677$	−18.3954	−0.706992	−0.353496	+	0.935436i	$0.615007\pi$
$678$	0	0
$679$	−44.6670	−1.71416
$680$	−8.59517	−0.329610
$681$	0	0
$682$	−16.4720	−0.630746
$683$	−14.3777	−0.550146	−0.275073	−	0.961423i	$-0.588702\pi$
$683$	−14.3777	−0.550146	−0.275073	+	0.961423i	$0.588702\pi$
$684$	0	0
$685$	11.3005	0.431769
$686$	−10.0384	−0.383268
$687$	0	0
$688$	−22.4099	−0.854371
$689$	−35.0480	−1.33522
$690$	0	0
$691$	−35.0831	−1.33462	−0.667311	−	0.744779i	$-0.732555\pi$
$691$	−35.0831	−1.33462	−0.667311	+	0.744779i	$0.732555\pi$
$692$	−1.50864	−0.0573497
$693$	0	0
$694$	9.64406	0.366084
$695$	6.12841	0.232464
$696$	0	0
$697$	30.2376	1.14533
$698$	−27.2139	−1.03006
$699$	0	0
$700$	2.60036	0.0982844
$701$	30.9498	1.16896	0.584480	−	0.811408i	$-0.301299\pi$
$701$	30.9498	1.16896	0.584480	+	0.811408i	$0.301299\pi$
$702$	0	0
$703$	−9.52710	−0.359322
$704$	−10.0419	−0.378467
$705$	0	0
$706$	7.07406	0.266236
$707$	22.1759	0.834010
$708$	0	0
$709$	−11.5484	−0.433708	−0.216854	−	0.976204i	$-0.569580\pi$
$709$	−11.5484	−0.433708	−0.216854	+	0.976204i	$0.569580\pi$
$710$	8.40349	0.315377
$711$	0	0
$712$	−29.6284	−1.11037
$713$	10.3766	0.388607
$714$	0	0
$715$	2.51067	0.0938939
$716$	1.95627	0.0731093
$717$	0	0
$718$	8.24691	0.307772
$719$	−21.9392	−0.818192	−0.409096	−	0.912491i	$-0.634156\pi$
$719$	−21.9392	−0.818192	−0.409096	+	0.912491i	$0.634156\pi$
$720$	0	0
$721$	14.3967	0.536161
$722$	−13.3200	−0.495720
$723$	0	0
$724$	−0.719524	−0.0267409
$725$	−4.45509	−0.165458
$726$	0	0
$727$	−0.989895	−0.0367132	−0.0183566	−	0.999832i	$-0.505843\pi$
$727$	−0.989895	−0.0367132	−0.0183566	+	0.999832i	$0.505843\pi$
$728$	29.2829	1.08529
$729$	0	0
$730$	1.23050	0.0455429
$731$	24.4755	0.905260
$732$	0	0
$733$	−17.2010	−0.635334	−0.317667	−	0.948202i	$-0.602899\pi$
$733$	−17.2010	−0.635334	−0.317667	+	0.948202i	$0.602899\pi$
$734$	32.5672	1.20208
$735$	0	0
$736$	0.956741	0.0352659
$737$	−10.1548	−0.374058
$738$	0	0
$739$	−50.3396	−1.85177	−0.925885	−	0.377804i	$-0.876679\pi$
$739$	−50.3396	−1.85177	−0.925885	+	0.377804i	$0.876679\pi$
$740$	−0.394210	−0.0144915
$741$	0	0
$742$	56.2957	2.06668
$743$	−19.7654	−0.725123	−0.362562	−	0.931960i	$-0.618098\pi$
$743$	−19.7654	−0.725123	−0.362562	+	0.931960i	$0.618098\pi$
$744$	0	0
$745$	−13.0340	−0.477528
$746$	11.9277	0.436705
$747$	0	0
$748$	0.789381	0.0288626
$749$	18.4420	0.673854
$750$	0	0
$751$	5.73724	0.209355	0.104677	−	0.994506i	$-0.466619\pi$
$751$	5.73724	0.209355	0.104677	+	0.994506i	$0.466619\pi$
$752$	39.9731	1.45767
$753$	0	0
$754$	−3.92179	−0.142823
$755$	6.08018	0.221281
$756$	0	0
$757$	−16.1728	−0.587811	−0.293905	−	0.955834i	$-0.594955\pi$
$757$	−16.1728	−0.587811	−0.293905	+	0.955834i	$0.594955\pi$
$758$	22.9717	0.834370
$759$	0	0
$760$	6.55592	0.237808
$761$	−47.5884	−1.72508	−0.862538	−	0.505992i	$-0.831127\pi$
$761$	−47.5884	−1.72508	−0.862538	+	0.505992i	$0.831127\pi$
$762$	0	0
$763$	38.9233	1.40912
$764$	−3.72319	−0.134700
$765$	0	0
$766$	−27.0657	−0.977925
$767$	5.09309	0.183901
$768$	0	0
$769$	−26.6504	−0.961038	−0.480519	−	0.876984i	$-0.659552\pi$
$769$	−26.6504	−0.961038	−0.480519	+	0.876984i	$0.659552\pi$
$770$	−4.03276	−0.145330
$771$	0	0
$772$	2.78636	0.100283
$773$	12.8708	0.462932	0.231466	−	0.972843i	$-0.425648\pi$
$773$	12.8708	0.462932	0.231466	+	0.972843i	$0.425648\pi$
$774$	0	0
$775$	−46.2288	−1.66059
$776$	38.0969	1.36760
$777$	0	0
$778$	−3.20293	−0.114831
$779$	−23.0636	−0.826338
$780$	0	0
$781$	−9.87291	−0.353280
$782$	5.36678	0.191916
$783$	0	0
$784$	−17.5937	−0.628346
$785$	12.7468	0.454954
$786$	0	0
$787$	45.1481	1.60936	0.804678	−	0.593712i	$-0.202338\pi$
$787$	45.1481	1.60936	0.804678	+	0.593712i	$0.202338\pi$
$788$	3.76791	0.134226
$789$	0	0
$790$	−15.7323	−0.559732
$791$	18.6865	0.664415
$792$	0	0
$793$	−8.40999	−0.298647
$794$	−30.3867	−1.07838
$795$	0	0
$796$	2.74676	0.0973562
$797$	−53.6633	−1.90085	−0.950427	−	0.310949i	$-0.899353\pi$
$797$	−53.6633	−1.90085	−0.950427	+	0.310949i	$0.899353\pi$
$798$	0	0
$799$	−43.6575	−1.54449
$800$	−4.26237	−0.150698
$801$	0	0
$802$	51.6802	1.82489
$803$	−1.44566	−0.0510164
$804$	0	0
$805$	2.54045	0.0895390
$806$	−40.6948	−1.43341
$807$	0	0
$808$	−18.9140	−0.665393
$809$	1.77290	0.0623319	0.0311660	−	0.999514i	$-0.490078\pi$
$809$	1.77290	0.0623319	0.0311660	+	0.999514i	$0.490078\pi$
$810$	0	0
$811$	−1.70116	−0.0597359	−0.0298680	−	0.999554i	$-0.509509\pi$
$811$	−1.70116	−0.0597359	−0.0298680	+	0.999554i	$0.509509\pi$
$812$	−0.583682	−0.0204832
$813$	0	0
$814$	−4.99841	−0.175194
$815$	−15.8102	−0.553808
$816$	0	0
$817$	−18.6686	−0.653131
$818$	−35.7825	−1.25111
$819$	0	0
$820$	−0.954319	−0.0333263
$821$	43.2876	1.51075	0.755374	−	0.655294i	$-0.227455\pi$
$821$	43.2876	1.51075	0.755374	+	0.655294i	$0.227455\pi$
$822$	0	0
$823$	−40.8222	−1.42297	−0.711485	−	0.702701i	$-0.751977\pi$
$823$	−40.8222	−1.42297	−0.711485	+	0.702701i	$0.751977\pi$
$824$	−12.2791	−0.427762
$825$	0	0
$826$	−8.18075	−0.284645
$827$	−40.1223	−1.39519	−0.697595	−	0.716493i	$-0.745746\pi$
$827$	−40.1223	−1.39519	−0.697595	+	0.716493i	$0.745746\pi$
$828$	0	0
$829$	−14.4375	−0.501437	−0.250718	−	0.968060i	$-0.580667\pi$
$829$	−14.4375	−0.501437	−0.250718	+	0.968060i	$0.580667\pi$
$830$	9.52494	0.330616
$831$	0	0
$832$	−24.8089	−0.860093
$833$	19.2153	0.665772
$834$	0	0
$835$	−1.38017	−0.0477627
$836$	−0.602096	−0.0208239
$837$	0	0
$838$	24.6176	0.850399
$839$	50.2263	1.73401	0.867003	−	0.498303i	$-0.166043\pi$
$839$	50.2263	1.73401	0.867003	+	0.498303i	$0.166043\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	23.4824	0.809258
$843$	0	0
$844$	−3.39042	−0.116703
$845$	−3.39358	−0.116743
$846$	0	0
$847$	−33.1188	−1.13797
$848$	−43.9139	−1.50801
$849$	0	0
$850$	−23.9095	−0.820090
$851$	3.14877	0.107938
$852$	0	0
$853$	−29.4488	−1.00831	−0.504154	−	0.863614i	$-0.668196\pi$
$853$	−29.4488	−1.00831	−0.504154	+	0.863614i	$0.668196\pi$
$854$	13.5085	0.462251
$855$	0	0
$856$	−15.7293	−0.537617
$857$	−3.87360	−0.132320	−0.0661598	−	0.997809i	$-0.521075\pi$
$857$	−3.87360	−0.132320	−0.0661598	+	0.997809i	$0.521075\pi$
$858$	0	0
$859$	24.7385	0.844068	0.422034	−	0.906580i	$-0.361316\pi$
$859$	24.7385	0.844068	0.422034	+	0.906580i	$0.361316\pi$
$860$	−0.772464	−0.0263408
$861$	0	0
$862$	−9.35281	−0.318558
$863$	14.4203	0.490873	0.245437	−	0.969413i	$-0.421069\pi$
$863$	14.4203	0.490873	0.245437	+	0.969413i	$0.421069\pi$
$864$	0	0
$865$	6.56626	0.223259
$866$	19.6049	0.666202
$867$	0	0
$868$	−6.05664	−0.205576
$869$	18.4833	0.627002
$870$	0	0
$871$	−25.0879	−0.850073
$872$	−33.1980	−1.12423
$873$	0	0
$874$	−4.09349	−0.138464
$875$	−24.0202	−0.812030
$876$	0	0
$877$	24.0202	0.811103	0.405552	−	0.914072i	$-0.367079\pi$
$877$	24.0202	0.811103	0.405552	+	0.914072i	$0.367079\pi$
$878$	−31.2455	−1.05448
$879$	0	0
$880$	3.14579	0.106044
$881$	−37.0159	−1.24710	−0.623549	−	0.781785i	$-0.714310\pi$
$881$	−37.0159	−1.24710	−0.623549	+	0.781785i	$0.714310\pi$
$882$	0	0
$883$	−23.0023	−0.774089	−0.387045	−	0.922061i	$-0.626504\pi$
$883$	−23.0023	−0.774089	−0.387045	+	0.922061i	$0.626504\pi$
$884$	1.95020	0.0655923
$885$	0	0
$886$	−4.58983	−0.154198
$887$	−41.1958	−1.38322	−0.691610	−	0.722271i	$-0.743098\pi$
$887$	−41.1958	−1.38322	−0.691610	+	0.722271i	$0.743098\pi$
$888$	0	0
$889$	−32.6392	−1.09468
$890$	10.0806	0.337904
$891$	0	0
$892$	−1.41613	−0.0474156
$893$	33.2995	1.11433
$894$	0	0
$895$	−8.51457	−0.284611
$896$	33.2638	1.11127
$897$	0	0
$898$	11.0884	0.370024
$899$	10.3766	0.346079
$900$	0	0
$901$	47.9616	1.59783
$902$	−12.1004	−0.402898
$903$	0	0
$904$	−15.9379	−0.530086
$905$	3.13169	0.104101
$906$	0	0
$907$	−19.8581	−0.659379	−0.329689	−	0.944089i	$-0.606944\pi$
$907$	−19.8581	−0.659379	−0.329689	+	0.944089i	$0.606944\pi$
$908$	−1.23524	−0.0409929
$909$	0	0
$910$	−9.96309	−0.330273
$911$	−0.186985	−0.00619509	−0.00309755	−	0.999995i	$-0.500986\pi$
$911$	−0.186985	−0.00619509	−0.00309755	+	0.999995i	$0.500986\pi$
$912$	0	0
$913$	−11.1904	−0.370350
$914$	53.4654	1.76848
$915$	0	0
$916$	2.78057	0.0918725
$917$	2.21405	0.0731144
$918$	0	0
$919$	10.2250	0.337293	0.168646	−	0.985677i	$-0.446060\pi$
$919$	10.2250	0.337293	0.168646	+	0.985677i	$0.446060\pi$
$920$	−2.16677	−0.0714364
$921$	0	0
$922$	−34.2888	−1.12924
$923$	−24.3914	−0.802854
$924$	0	0
$925$	−14.0281	−0.461240
$926$	24.7722	0.814064
$927$	0	0
$928$	0.956741	0.0314065
$929$	5.47909	0.179763	0.0898815	−	0.995952i	$-0.471351\pi$
$929$	5.47909	0.179763	0.0898815	+	0.995952i	$0.471351\pi$
$930$	0	0
$931$	−14.6564	−0.480344
$932$	2.98741	0.0978560
$933$	0	0
$934$	6.17498	0.202052
$935$	−3.43574	−0.112361
$936$	0	0
$937$	−12.6673	−0.413822	−0.206911	−	0.978360i	$-0.566341\pi$
$937$	−12.6673	−0.413822	−0.206911	+	0.978360i	$0.566341\pi$
$938$	40.2974	1.31576
$939$	0	0
$940$	1.37786	0.0449408
$941$	29.4280	0.959325	0.479663	−	0.877453i	$-0.340759\pi$
$941$	29.4280	0.959325	0.479663	+	0.877453i	$0.340759\pi$
$942$	0	0
$943$	7.62266	0.248228
$944$	6.38146	0.207699
$945$	0	0
$946$	−9.79451	−0.318447
$947$	26.1468	0.849656	0.424828	−	0.905274i	$-0.360335\pi$
$947$	26.1468	0.849656	0.424828	+	0.905274i	$0.360335\pi$
$948$	0	0
$949$	−3.57158	−0.115938
$950$	18.2369	0.591682
$951$	0	0
$952$	−40.0722	−1.29875
$953$	−8.31667	−0.269403	−0.134702	−	0.990886i	$-0.543008\pi$
$953$	−8.31667	−0.269403	−0.134702	+	0.990886i	$0.543008\pi$
$954$	0	0
$955$	16.2050	0.524381
$956$	−3.44619	−0.111458
$957$	0	0
$958$	−21.1949	−0.684775
$959$	52.6848	1.70128
$960$	0	0
$961$	76.6739	2.47335
$962$	−12.3488	−0.398141
$963$	0	0
$964$	−0.426379	−0.0137327
$965$	−12.1275	−0.390398
$966$	0	0
$967$	−53.2301	−1.71176	−0.855882	−	0.517171i	$-0.826985\pi$
$967$	−53.2301	−1.71176	−0.855882	+	0.517171i	$0.826985\pi$
$968$	28.2473	0.907903
$969$	0	0
$970$	−12.9619	−0.416183
$971$	59.8856	1.92182	0.960910	−	0.276861i	$-0.0892942\pi$
$971$	59.8856	1.92182	0.960910	+	0.276861i	$0.0892942\pi$
$972$	0	0
$973$	28.5718	0.915969
$974$	18.9131	0.606014
$975$	0	0
$976$	−10.5374	−0.337295
$977$	22.6380	0.724253	0.362126	−	0.932129i	$-0.382051\pi$
$977$	22.6380	0.724253	0.362126	+	0.932129i	$0.382051\pi$
$978$	0	0
$979$	−11.8433	−0.378514
$980$	−0.606449	−0.0193723
$981$	0	0
$982$	55.7275	1.77833
$983$	6.36352	0.202965	0.101482	−	0.994837i	$-0.467641\pi$
$983$	6.36352	0.202965	0.101482	+	0.994837i	$0.467641\pi$
$984$	0	0
$985$	−16.3996	−0.522535
$986$	5.36678	0.170913
$987$	0	0
$988$	−1.48750	−0.0473238
$989$	6.17008	0.196197
$990$	0	0
$991$	−7.24797	−0.230239	−0.115120	−	0.993352i	$-0.536725\pi$
$991$	−7.24797	−0.230239	−0.115120	+	0.993352i	$0.536725\pi$
$992$	9.92772	0.315205
$993$	0	0
$994$	39.1786	1.24267
$995$	−11.9551	−0.379003
$996$	0	0
$997$	32.7559	1.03739	0.518694	−	0.854960i	$-0.326418\pi$
$997$	32.7559	1.03739	0.518694	+	0.854960i	$0.326418\pi$
$998$	−56.1435	−1.77719
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.10		14
3.2	odd	2		2001.2.a.m.1.5	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.5	✓	14	3.2	odd	2
6003.2.a.p.1.10		14	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)

\(f(q)\)	\(=\)	\(q+1.35292 q^{2} -0.169600 q^{4} +0.738177 q^{5} +3.44152 q^{7} -2.93530 q^{8} +O(q^{10})\)	\(q+1.35292 q^{2} -0.169600 q^{4} +0.738177 q^{5} +3.44152 q^{7} -2.93530 q^{8} +0.998696 q^{10} -1.17333 q^{11} -2.89875 q^{13} +4.65611 q^{14} -3.63203 q^{16} +3.96681 q^{17} -3.02566 q^{19} -0.125195 q^{20} -1.58742 q^{22} +1.00000 q^{23} -4.45509 q^{25} -3.92179 q^{26} -0.583682 q^{28} +1.00000 q^{29} +10.3766 q^{31} +0.956741 q^{32} +5.36678 q^{34} +2.54045 q^{35} +3.14877 q^{37} -4.09349 q^{38} -2.16677 q^{40} +7.62266 q^{41} +6.17008 q^{43} +0.198997 q^{44} +1.35292 q^{46} -11.0057 q^{47} +4.84403 q^{49} -6.02740 q^{50} +0.491629 q^{52} +12.0907 q^{53} -0.866122 q^{55} -10.1019 q^{56} +1.35292 q^{58} -1.75699 q^{59} +2.90124 q^{61} +14.0387 q^{62} +8.55847 q^{64} -2.13979 q^{65} +8.65474 q^{67} -0.672772 q^{68} +3.43703 q^{70} +8.41446 q^{71} +1.23211 q^{73} +4.26004 q^{74} +0.513153 q^{76} -4.03802 q^{77} -15.7529 q^{79} -2.68108 q^{80} +10.3129 q^{82} +9.53737 q^{83} +2.92821 q^{85} +8.34764 q^{86} +3.44407 q^{88} +10.0938 q^{89} -9.97610 q^{91} -0.169600 q^{92} -14.8899 q^{94} -2.23347 q^{95} -12.9789 q^{97} +6.55360 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 5 q^{10} + 12 q^{11} + 13 q^{13} + 9 q^{14} + 14 q^{17} - 9 q^{19} + 2 q^{20} - 9 q^{22} + 14 q^{23} + 13 q^{25} + 16 q^{26} + 3 q^{28} + 14 q^{29} - 28 q^{31} + 4 q^{32} + 14 q^{34} + 9 q^{35} - 12 q^{37} - 2 q^{38} - 20 q^{40} + 25 q^{41} + 5 q^{43} + 37 q^{44} + 2 q^{46} + 17 q^{47} + 17 q^{49} + 44 q^{50} + 25 q^{52} + 17 q^{53} + q^{55} + 54 q^{56} + 2 q^{58} + 18 q^{59} - 13 q^{61} + 8 q^{62} + 20 q^{64} + 16 q^{65} + 2 q^{67} + 19 q^{68} + 14 q^{70} + 55 q^{71} + 19 q^{73} - 4 q^{74} - 32 q^{76} + 19 q^{77} - 68 q^{79} + 2 q^{80} - 12 q^{82} + 21 q^{83} + 16 q^{85} + 22 q^{86} - 25 q^{88} + 17 q^{89} - 30 q^{91} + 12 q^{92} + 16 q^{94} + 55 q^{95} + 25 q^{97} + 31 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 + 12 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 5 * q^10 + 12 * q^11 + 13 * q^13 + 9 * q^14 + 14 * q^17 - 9 * q^19 + 2 * q^20 - 9 * q^22 + 14 * q^23 + 13 * q^25 + 16 * q^26 + 3 * q^28 + 14 * q^29 - 28 * q^31 + 4 * q^32 + 14 * q^34 + 9 * q^35 - 12 * q^37 - 2 * q^38 - 20 * q^40 + 25 * q^41 + 5 * q^43 + 37 * q^44 + 2 * q^46 + 17 * q^47 + 17 * q^49 + 44 * q^50 + 25 * q^52 + 17 * q^53 + q^55 + 54 * q^56 + 2 * q^58 + 18 * q^59 - 13 * q^61 + 8 * q^62 + 20 * q^64 + 16 * q^65 + 2 * q^67 + 19 * q^68 + 14 * q^70 + 55 * q^71 + 19 * q^73 - 4 * q^74 - 32 * q^76 + 19 * q^77 - 68 * q^79 + 2 * q^80 - 12 * q^82 + 21 * q^83 + 16 * q^85 + 22 * q^86 - 25 * q^88 + 17 * q^89 - 30 * q^91 + 12 * q^92 + 16 * q^94 + 55 * q^95 + 25 * q^97 + 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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