LMFDB - Embedded newform 6003.2.a.p.1.2

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			1.2
Root			$-1.98240$ 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)

$f(q)$	$=$	$q-1.98240 q^{2} +1.92989 q^{4} +0.252916 q^{5} -1.21499 q^{7} +0.138982 q^{8} +O(q^{10})$	\(q-1.98240 q^{2} +1.92989 q^{4} +0.252916 q^{5} -1.21499 q^{7} +0.138982 q^{8} -0.501379 q^{10} -2.73986 q^{11} -6.01158 q^{13} +2.40858 q^{14} -4.13530 q^{16} -0.539671 q^{17} +3.87659 q^{19} +0.488100 q^{20} +5.43149 q^{22} +1.00000 q^{23} -4.93603 q^{25} +11.9173 q^{26} -2.34479 q^{28} +1.00000 q^{29} -10.4153 q^{31} +7.91984 q^{32} +1.06984 q^{34} -0.307289 q^{35} -4.48518 q^{37} -7.68494 q^{38} +0.0351509 q^{40} +7.22607 q^{41} +6.16164 q^{43} -5.28764 q^{44} -1.98240 q^{46} -11.5319 q^{47} -5.52381 q^{49} +9.78517 q^{50} -11.6017 q^{52} +3.65289 q^{53} -0.692955 q^{55} -0.168862 q^{56} -1.98240 q^{58} -4.78769 q^{59} -2.30130 q^{61} +20.6472 q^{62} -7.42965 q^{64} -1.52042 q^{65} -13.2382 q^{67} -1.04151 q^{68} +0.609169 q^{70} +10.9508 q^{71} +0.281971 q^{73} +8.89141 q^{74} +7.48140 q^{76} +3.32890 q^{77} -9.98574 q^{79} -1.04588 q^{80} -14.3249 q^{82} +0.778208 q^{83} -0.136491 q^{85} -12.2148 q^{86} -0.380793 q^{88} -8.11723 q^{89} +7.30398 q^{91} +1.92989 q^{92} +22.8608 q^{94} +0.980452 q^{95} +17.8074 q^{97} +10.9504 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.98240	−1.40177	−0.700883	−	0.713277i	$-0.747210\pi$
$2$	−1.98240	−1.40177	−0.700883	+	0.713277i	$0.747210\pi$
$3$	0	0
$3$	0	0
$4$	1.92989	0.964946
$5$	0.252916	0.113107	0.0565537	−	0.998400i	$-0.481989\pi$
$5$	0.252916	0.113107	0.0565537	+	0.998400i	$0.481989\pi$
$6$	0	0
$7$	−1.21499	−0.459222	−0.229611	−	0.973283i	$-0.573745\pi$
$7$	−1.21499	−0.459222	−0.229611	+	0.973283i	$0.573745\pi$
$8$	0.138982	0.0491377
$9$	0	0
$10$	−0.501379	−0.158550
$11$	−2.73986	−0.826100	−0.413050	−	0.910708i	$-0.635536\pi$
$11$	−2.73986	−0.826100	−0.413050	+	0.910708i	$0.635536\pi$
$12$	0	0
$13$	−6.01158	−1.66731	−0.833656	−	0.552285i	$-0.813756\pi$
$13$	−6.01158	−1.66731	−0.833656	+	0.552285i	$0.813756\pi$
$14$	2.40858	0.643721
$15$	0	0
$16$	−4.13530	−1.03383
$17$	−0.539671	−0.130889	−0.0654447	−	0.997856i	$-0.520847\pi$
$17$	−0.539671	−0.130889	−0.0654447	+	0.997856i	$0.520847\pi$
$18$	0	0
$19$	3.87659	0.889351	0.444675	−	0.895692i	$-0.353319\pi$
$19$	3.87659	0.889351	0.444675	+	0.895692i	$0.353319\pi$
$20$	0.488100	0.109143
$21$	0	0
$22$	5.43149	1.15800
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.93603	−0.987207
$26$	11.9173	2.33718
$27$	0	0
$28$	−2.34479	−0.443124
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−10.4153	−1.87064	−0.935320	−	0.353803i	$-0.884888\pi$
$31$	−10.4153	−1.87064	−0.935320	+	0.353803i	$0.884888\pi$
$32$	7.91984	1.40004
$33$	0	0
$34$	1.06984	0.183476
$35$	−0.307289	−0.0519414
$36$	0	0
$37$	−4.48518	−0.737360	−0.368680	−	0.929556i	$-0.620190\pi$
$37$	−4.48518	−0.737360	−0.368680	+	0.929556i	$0.620190\pi$
$38$	−7.68494	−1.24666
$39$	0	0
$40$	0.0351509	0.00555784
$41$	7.22607	1.12852	0.564261	−	0.825596i	$-0.309161\pi$
$41$	7.22607	1.12852	0.564261	+	0.825596i	$0.309161\pi$
$42$	0	0
$43$	6.16164	0.939641	0.469821	−	0.882762i	$-0.344319\pi$
$43$	6.16164	0.939641	0.469821	+	0.882762i	$0.344319\pi$
$44$	−5.28764	−0.797141
$45$	0	0
$46$	−1.98240	−0.292288
$47$	−11.5319	−1.68210	−0.841050	−	0.540957i	$-0.818062\pi$
$47$	−11.5319	−1.68210	−0.841050	+	0.540957i	$0.818062\pi$
$48$	0	0
$49$	−5.52381	−0.789115
$50$	9.78517	1.38383
$51$	0	0
$52$	−11.6017	−1.60887
$53$	3.65289	0.501763	0.250881	−	0.968018i	$-0.419280\pi$
$53$	3.65289	0.501763	0.250881	+	0.968018i	$0.419280\pi$
$54$	0	0
$55$	−0.692955	−0.0934380
$56$	−0.168862	−0.0225651
$57$	0	0
$58$	−1.98240	−0.260301
$59$	−4.78769	−0.623304	−0.311652	−	0.950196i	$-0.600882\pi$
$59$	−4.78769	−0.623304	−0.311652	+	0.950196i	$0.600882\pi$
$60$	0	0
$61$	−2.30130	−0.294651	−0.147325	−	0.989088i	$-0.547066\pi$
$61$	−2.30130	−0.294651	−0.147325	+	0.989088i	$0.547066\pi$
$62$	20.6472	2.62220
$63$	0	0
$64$	−7.42965	−0.928706
$65$	−1.52042	−0.188585
$66$	0	0
$67$	−13.2382	−1.61731	−0.808653	−	0.588286i	$-0.799803\pi$
$67$	−13.2382	−1.61731	−0.808653	+	0.588286i	$0.799803\pi$
$68$	−1.04151	−0.126301
$69$	0	0
$70$	0.609169	0.0728096
$71$	10.9508	1.29962	0.649808	−	0.760098i	$-0.274849\pi$
$71$	10.9508	1.29962	0.649808	+	0.760098i	$0.274849\pi$
$72$	0	0
$73$	0.281971	0.0330022	0.0165011	−	0.999864i	$-0.494747\pi$
$73$	0.281971	0.0330022	0.0165011	+	0.999864i	$0.494747\pi$
$74$	8.89141	1.03361
$75$	0	0
$76$	7.48140	0.858175
$77$	3.32890	0.379363
$78$	0	0
$79$	−9.98574	−1.12348	−0.561742	−	0.827313i	$-0.689868\pi$
$79$	−9.98574	−1.12348	−0.561742	+	0.827313i	$0.689868\pi$
$80$	−1.04588	−0.116933
$81$	0	0
$82$	−14.3249	−1.58192
$83$	0.778208	0.0854194	0.0427097	−	0.999088i	$-0.486401\pi$
$83$	0.778208	0.0854194	0.0427097	+	0.999088i	$0.486401\pi$
$84$	0	0
$85$	−0.136491	−0.0148046
$86$	−12.2148	−1.31716
$87$	0	0
$88$	−0.380793	−0.0405927
$89$	−8.11723	−0.860425	−0.430213	−	0.902728i	$-0.641561\pi$
$89$	−8.11723	−0.860425	−0.430213	+	0.902728i	$0.641561\pi$
$90$	0	0
$91$	7.30398	0.765666
$92$	1.92989	0.201205
$93$	0	0
$94$	22.8608	2.35791
$95$	0.980452	0.100592
$96$	0	0
$97$	17.8074	1.80807	0.904033	−	0.427463i	$-0.140593\pi$
$97$	17.8074	1.80807	0.904033	+	0.427463i	$0.140593\pi$
$98$	10.9504	1.10615
$99$	0	0
$100$	−9.52601	−0.952601
$101$	9.49058	0.944348	0.472174	−	0.881505i	$-0.343469\pi$
$101$	9.49058	0.944348	0.472174	+	0.881505i	$0.343469\pi$
$102$	0	0
$103$	20.0335	1.97396	0.986979	−	0.160849i	$-0.0514234\pi$
$103$	20.0335	1.97396	0.986979	+	0.160849i	$0.0514234\pi$
$104$	−0.835504	−0.0819279
$105$	0	0
$106$	−7.24147	−0.703354
$107$	19.9506	1.92870	0.964350	−	0.264630i	$-0.0852498\pi$
$107$	19.9506	1.92870	0.964350	+	0.264630i	$0.0852498\pi$
$108$	0	0
$109$	3.56520	0.341484	0.170742	−	0.985316i	$-0.445384\pi$
$109$	3.56520	0.341484	0.170742	+	0.985316i	$0.445384\pi$
$110$	1.37371	0.130978
$111$	0	0
$112$	5.02434	0.474755
$113$	−5.36104	−0.504324	−0.252162	−	0.967685i	$-0.581142\pi$
$113$	−5.36104	−0.504324	−0.252162	+	0.967685i	$0.581142\pi$
$114$	0	0
$115$	0.252916	0.0235845
$116$	1.92989	0.179186
$117$	0	0
$118$	9.49109	0.873726
$119$	0.655693	0.0601073
$120$	0	0
$121$	−3.49315	−0.317559
$122$	4.56208	0.413031
$123$	0	0
$124$	−20.1004	−1.80507
$125$	−2.51298	−0.224768
$126$	0	0
$127$	0.509875	0.0452441	0.0226220	−	0.999744i	$-0.492799\pi$
$127$	0.509875	0.0452441	0.0226220	+	0.999744i	$0.492799\pi$
$128$	−1.11118	−0.0982151
$129$	0	0
$130$	3.01408	0.264352
$131$	−22.2670	−1.94548	−0.972739	−	0.231901i	$-0.925505\pi$
$131$	−22.2670	−1.94548	−0.972739	+	0.231901i	$0.925505\pi$
$132$	0	0
$133$	−4.71001	−0.408409
$134$	26.2434	2.26708
$135$	0	0
$136$	−0.0750048	−0.00643161
$137$	−12.6651	−1.08206	−0.541028	−	0.841005i	$-0.681965\pi$
$137$	−12.6651	−1.08206	−0.541028	+	0.841005i	$0.681965\pi$
$138$	0	0
$139$	−0.725814	−0.0615627	−0.0307814	−	0.999526i	$-0.509800\pi$
$139$	−0.725814	−0.0615627	−0.0307814	+	0.999526i	$0.509800\pi$
$140$	−0.593035	−0.0501206
$141$	0	0
$142$	−21.7087	−1.82176
$143$	16.4709	1.37737
$144$	0	0
$145$	0.252916	0.0210035
$146$	−0.558978	−0.0462614
$147$	0	0
$148$	−8.65592	−0.711512
$149$	7.39238	0.605607	0.302804	−	0.953053i	$-0.402077\pi$
$149$	7.39238	0.605607	0.302804	+	0.953053i	$0.402077\pi$
$150$	0	0
$151$	2.47925	0.201759	0.100879	−	0.994899i	$-0.467834\pi$
$151$	2.47925	0.201759	0.100879	+	0.994899i	$0.467834\pi$
$152$	0.538778	0.0437007
$153$	0	0
$154$	−6.59919	−0.531778
$155$	−2.63419	−0.211583
$156$	0	0
$157$	10.9461	0.873597	0.436798	−	0.899559i	$-0.356112\pi$
$157$	10.9461	0.873597	0.436798	+	0.899559i	$0.356112\pi$
$158$	19.7957	1.57486
$159$	0	0
$160$	2.00305	0.158355
$161$	−1.21499	−0.0957544
$162$	0	0
$163$	−18.5371	−1.45194	−0.725971	−	0.687726i	$-0.758609\pi$
$163$	−18.5371	−1.45194	−0.725971	+	0.687726i	$0.758609\pi$
$164$	13.9455	1.08896
$165$	0	0
$166$	−1.54272	−0.119738
$167$	8.98050	0.694932	0.347466	−	0.937693i	$-0.387042\pi$
$167$	8.98050	0.694932	0.347466	+	0.937693i	$0.387042\pi$
$168$	0	0
$169$	23.1391	1.77993
$170$	0.270580	0.0207525
$171$	0	0
$172$	11.8913	0.906703
$173$	−4.52117	−0.343738	−0.171869	−	0.985120i	$-0.554981\pi$
$173$	−4.52117	−0.343738	−0.171869	+	0.985120i	$0.554981\pi$
$174$	0	0
$175$	5.99721	0.453347
$176$	11.3302	0.854043
$177$	0	0
$178$	16.0916	1.20611
$179$	19.6654	1.46986	0.734932	−	0.678141i	$-0.237214\pi$
$179$	19.6654	1.46986	0.734932	+	0.678141i	$0.237214\pi$
$180$	0	0
$181$	11.0323	0.820023	0.410011	−	0.912080i	$-0.365525\pi$
$181$	11.0323	0.820023	0.410011	+	0.912080i	$0.365525\pi$
$182$	−14.4794	−1.07328
$183$	0	0
$184$	0.138982	0.0102459
$185$	−1.13437	−0.0834009
$186$	0	0
$187$	1.47862	0.108128
$188$	−22.2553	−1.62314
$189$	0	0
$190$	−1.94364	−0.141007
$191$	−2.35924	−0.170708	−0.0853542	−	0.996351i	$-0.527202\pi$
$191$	−2.35924	−0.170708	−0.0853542	+	0.996351i	$0.527202\pi$
$192$	0	0
$193$	17.0547	1.22763	0.613814	−	0.789451i	$-0.289635\pi$
$193$	17.0547	1.22763	0.613814	+	0.789451i	$0.289635\pi$
$194$	−35.3013	−2.53448
$195$	0	0
$196$	−10.6604	−0.761454
$197$	−5.51284	−0.392774	−0.196387	−	0.980527i	$-0.562921\pi$
$197$	−5.51284	−0.392774	−0.196387	+	0.980527i	$0.562921\pi$
$198$	0	0
$199$	−15.5760	−1.10415	−0.552076	−	0.833794i	$-0.686164\pi$
$199$	−15.5760	−1.10415	−0.552076	+	0.833794i	$0.686164\pi$
$200$	−0.686022	−0.0485091
$201$	0	0
$202$	−18.8141	−1.32375
$203$	−1.21499	−0.0852753
$204$	0	0
$205$	1.82759	0.127644
$206$	−39.7143	−2.76703
$207$	0	0
$208$	24.8597	1.72371
$209$	−10.6213	−0.734693
$210$	0	0
$211$	0.675563	0.0465077	0.0232538	−	0.999730i	$-0.492597\pi$
$211$	0.675563	0.0465077	0.0232538	+	0.999730i	$0.492597\pi$
$212$	7.04968	0.484174
$213$	0	0
$214$	−39.5500	−2.70358
$215$	1.55838	0.106280
$216$	0	0
$217$	12.6544	0.859039
$218$	−7.06763	−0.478680
$219$	0	0
$220$	−1.33733	−0.0901626
$221$	3.24427	0.218233
$222$	0	0
$223$	6.68120	0.447407	0.223703	−	0.974657i	$-0.428185\pi$
$223$	6.68120	0.447407	0.223703	+	0.974657i	$0.428185\pi$
$224$	−9.62250	−0.642930
$225$	0	0
$226$	10.6277	0.706944
$227$	1.10336	0.0732325	0.0366163	−	0.999329i	$-0.488342\pi$
$227$	1.10336	0.0732325	0.0366163	+	0.999329i	$0.488342\pi$
$228$	0	0
$229$	−24.1398	−1.59520	−0.797602	−	0.603184i	$-0.793899\pi$
$229$	−24.1398	−1.59520	−0.797602	+	0.603184i	$0.793899\pi$
$230$	−0.501379	−0.0330600
$231$	0	0
$232$	0.138982	0.00912465
$233$	−17.4274	−1.14171	−0.570855	−	0.821051i	$-0.693388\pi$
$233$	−17.4274	−1.14171	−0.570855	+	0.821051i	$0.693388\pi$
$234$	0	0
$235$	−2.91660	−0.190258
$236$	−9.23972	−0.601455
$237$	0	0
$238$	−1.29984	−0.0842563
$239$	−18.1500	−1.17402	−0.587012	−	0.809578i	$-0.699696\pi$
$239$	−18.1500	−1.17402	−0.587012	+	0.809578i	$0.699696\pi$
$240$	0	0
$241$	−21.6333	−1.39353	−0.696763	−	0.717301i	$-0.745377\pi$
$241$	−21.6333	−1.39353	−0.696763	+	0.717301i	$0.745377\pi$
$242$	6.92481	0.445144
$243$	0	0
$244$	−4.44125	−0.284322
$245$	−1.39706	−0.0892548
$246$	0	0
$247$	−23.3044	−1.48282
$248$	−1.44754	−0.0919190
$249$	0	0
$250$	4.98172	0.315072
$251$	6.85515	0.432693	0.216347	−	0.976317i	$-0.430586\pi$
$251$	6.85515	0.432693	0.216347	+	0.976317i	$0.430586\pi$
$252$	0	0
$253$	−2.73986	−0.172254
$254$	−1.01077	−0.0634216
$255$	0	0
$256$	17.0621	1.06638
$257$	20.4624	1.27641	0.638206	−	0.769866i	$-0.279677\pi$
$257$	20.4624	1.27641	0.638206	+	0.769866i	$0.279677\pi$
$258$	0	0
$259$	5.44944	0.338612
$260$	−2.93425	−0.181975
$261$	0	0
$262$	44.1420	2.72710
$263$	2.15812	0.133075	0.0665376	−	0.997784i	$-0.478805\pi$
$263$	2.15812	0.133075	0.0665376	+	0.997784i	$0.478805\pi$
$264$	0	0
$265$	0.923874	0.0567531
$266$	9.33709	0.572494
$267$	0	0
$268$	−25.5483	−1.56061
$269$	−12.6541	−0.771534	−0.385767	−	0.922596i	$-0.626063\pi$
$269$	−12.6541	−0.771534	−0.385767	+	0.922596i	$0.626063\pi$
$270$	0	0
$271$	19.3290	1.17415	0.587075	−	0.809532i	$-0.300279\pi$
$271$	19.3290	1.17415	0.587075	+	0.809532i	$0.300279\pi$
$272$	2.23170	0.135317
$273$	0	0
$274$	25.1073	1.51679
$275$	13.5241	0.815531
$276$	0	0
$277$	29.8131	1.79130	0.895649	−	0.444761i	$-0.146712\pi$
$277$	29.8131	1.79130	0.895649	+	0.444761i	$0.146712\pi$
$278$	1.43885	0.0862965
$279$	0	0
$280$	−0.0427079	−0.00255228
$281$	11.6049	0.692290	0.346145	−	0.938181i	$-0.387491\pi$
$281$	11.6049	0.692290	0.346145	+	0.938181i	$0.387491\pi$
$282$	0	0
$283$	−26.1145	−1.55235	−0.776173	−	0.630520i	$-0.782842\pi$
$283$	−26.1145	−1.55235	−0.776173	+	0.630520i	$0.782842\pi$
$284$	21.1338	1.25406
$285$	0	0
$286$	−32.6518	−1.93074
$287$	−8.77957	−0.518242
$288$	0	0
$289$	−16.7088	−0.982868
$290$	−0.501379	−0.0294420
$291$	0	0
$292$	0.544174	0.0318454
$293$	−20.6073	−1.20389	−0.601945	−	0.798537i	$-0.705608\pi$
$293$	−20.6073	−1.20389	−0.601945	+	0.798537i	$0.705608\pi$
$294$	0	0
$295$	−1.21088	−0.0705003
$296$	−0.623362	−0.0362322
$297$	0	0
$298$	−14.6546	−0.848919
$299$	−6.01158	−0.347658
$300$	0	0
$301$	−7.48631	−0.431504
$302$	−4.91486	−0.282819
$303$	0	0
$304$	−16.0309	−0.919434
$305$	−0.582035	−0.0333272
$306$	0	0
$307$	19.5171	1.11390	0.556950	−	0.830546i	$-0.311972\pi$
$307$	19.5171	1.11390	0.556950	+	0.830546i	$0.311972\pi$
$308$	6.42441	0.366065
$309$	0	0
$310$	5.22201	0.296590
$311$	19.7794	1.12159	0.560794	−	0.827955i	$-0.310496\pi$
$311$	19.7794	1.12159	0.560794	+	0.827955i	$0.310496\pi$
$312$	0	0
$313$	21.3721	1.20802	0.604012	−	0.796975i	$-0.293568\pi$
$313$	21.3721	1.20802	0.604012	+	0.796975i	$0.293568\pi$
$314$	−21.6996	−1.22458
$315$	0	0
$316$	−19.2714	−1.08410
$317$	−6.33500	−0.355809	−0.177905	−	0.984048i	$-0.556932\pi$
$317$	−6.33500	−0.355809	−0.177905	+	0.984048i	$0.556932\pi$
$318$	0	0
$319$	−2.73986	−0.153403
$320$	−1.87908	−0.105044
$321$	0	0
$322$	2.40858	0.134225
$323$	−2.09208	−0.116407
$324$	0	0
$325$	29.6733	1.64598
$326$	36.7479	2.03528
$327$	0	0
$328$	1.00430	0.0554530
$329$	14.0111	0.772457
$330$	0	0
$331$	−2.92635	−0.160847	−0.0804233	−	0.996761i	$-0.525627\pi$
$331$	−2.92635	−0.160847	−0.0804233	+	0.996761i	$0.525627\pi$
$332$	1.50186	0.0824251
$333$	0	0
$334$	−17.8029	−0.974131
$335$	−3.34816	−0.182929
$336$	0	0
$337$	24.2207	1.31938	0.659691	−	0.751537i	$-0.270687\pi$
$337$	24.2207	1.31938	0.659691	+	0.751537i	$0.270687\pi$
$338$	−45.8708	−2.49504
$339$	0	0
$340$	−0.263413	−0.0142856
$341$	28.5364	1.54534
$342$	0	0
$343$	15.2163	0.821601
$344$	0.856360	0.0461718
$345$	0	0
$346$	8.96274	0.481840
$347$	−1.43278	−0.0769156	−0.0384578	−	0.999260i	$-0.512245\pi$
$347$	−1.43278	−0.0769156	−0.0384578	+	0.999260i	$0.512245\pi$
$348$	0	0
$349$	−19.1966	−1.02757	−0.513786	−	0.857919i	$-0.671757\pi$
$349$	−19.1966	−1.02757	−0.513786	+	0.857919i	$0.671757\pi$
$350$	−11.8888	−0.635486
$351$	0	0
$352$	−21.6993	−1.15657
$353$	−10.2050	−0.543155	−0.271577	−	0.962417i	$-0.587545\pi$
$353$	−10.2050	−0.543155	−0.271577	+	0.962417i	$0.587545\pi$
$354$	0	0
$355$	2.76962	0.146996
$356$	−15.6654	−0.830264
$357$	0	0
$358$	−38.9847	−2.06040
$359$	23.1960	1.22424	0.612119	−	0.790766i	$-0.290317\pi$
$359$	23.1960	1.22424	0.612119	+	0.790766i	$0.290317\pi$
$360$	0	0
$361$	−3.97204	−0.209055
$362$	−21.8703	−1.14948
$363$	0	0
$364$	14.0959	0.738826
$365$	0.0713150	0.00373280
$366$	0	0
$367$	25.1544	1.31305	0.656524	−	0.754305i	$-0.272026\pi$
$367$	25.1544	1.31305	0.656524	+	0.754305i	$0.272026\pi$
$368$	−4.13530	−0.215567
$369$	0	0
$370$	2.24878	0.116908
$371$	−4.43821	−0.230420
$372$	0	0
$373$	−28.6093	−1.48133	−0.740667	−	0.671872i	$-0.765491\pi$
$373$	−28.6093	−1.48133	−0.740667	+	0.671872i	$0.765491\pi$
$374$	−2.93122	−0.151570
$375$	0	0
$376$	−1.60273	−0.0826546
$377$	−6.01158	−0.309612
$378$	0	0
$379$	31.9594	1.64165	0.820823	−	0.571183i	$-0.193515\pi$
$379$	31.9594	1.64165	0.820823	+	0.571183i	$0.193515\pi$
$380$	1.89217	0.0970660
$381$	0	0
$382$	4.67694	0.239293
$383$	23.7993	1.21609	0.608043	−	0.793904i	$-0.291955\pi$
$383$	23.7993	1.21609	0.608043	+	0.793904i	$0.291955\pi$
$384$	0	0
$385$	0.841931	0.0429088
$386$	−33.8092	−1.72085
$387$	0	0
$388$	34.3663	1.74468
$389$	22.9294	1.16257	0.581283	−	0.813701i	$-0.302551\pi$
$389$	22.9294	1.16257	0.581283	+	0.813701i	$0.302551\pi$
$390$	0	0
$391$	−0.539671	−0.0272923
$392$	−0.767713	−0.0387753
$393$	0	0
$394$	10.9286	0.550576
$395$	−2.52555	−0.127074
$396$	0	0
$397$	−24.5659	−1.23293	−0.616465	−	0.787382i	$-0.711436\pi$
$397$	−24.5659	−1.23293	−0.616465	+	0.787382i	$0.711436\pi$
$398$	30.8778	1.54776
$399$	0	0
$400$	20.4120	1.02060
$401$	10.0873	0.503736	0.251868	−	0.967762i	$-0.418955\pi$
$401$	10.0873	0.503736	0.251868	+	0.967762i	$0.418955\pi$
$402$	0	0
$403$	62.6123	3.11894
$404$	18.3158	0.911245
$405$	0	0
$406$	2.40858	0.119536
$407$	12.2888	0.609133
$408$	0	0
$409$	−2.20177	−0.108871	−0.0544353	−	0.998517i	$-0.517336\pi$
$409$	−2.20177	−0.108871	−0.0544353	+	0.998517i	$0.517336\pi$
$410$	−3.62300	−0.178927
$411$	0	0
$412$	38.6625	1.90476
$413$	5.81698	0.286235
$414$	0	0
$415$	0.196821	0.00966157
$416$	−47.6107	−2.33431
$417$	0	0
$418$	21.0557	1.02987
$419$	−29.3618	−1.43442	−0.717209	−	0.696858i	$-0.754581\pi$
$419$	−29.3618	−1.43442	−0.717209	+	0.696858i	$0.754581\pi$
$420$	0	0
$421$	15.6470	0.762587	0.381293	−	0.924454i	$-0.375479\pi$
$421$	15.6470	0.762587	0.381293	+	0.924454i	$0.375479\pi$
$422$	−1.33923	−0.0651928
$423$	0	0
$424$	0.507688	0.0246555
$425$	2.66383	0.129215
$426$	0	0
$427$	2.79604	0.135310
$428$	38.5026	1.86109
$429$	0	0
$430$	−3.08932	−0.148980
$431$	38.3036	1.84502	0.922509	−	0.385976i	$-0.126135\pi$
$431$	38.3036	1.84502	0.922509	+	0.385976i	$0.126135\pi$
$432$	0	0
$433$	8.99905	0.432467	0.216233	−	0.976342i	$-0.430623\pi$
$433$	8.99905	0.432467	0.216233	+	0.976342i	$0.430623\pi$
$434$	−25.0861	−1.20417
$435$	0	0
$436$	6.88044	0.329513
$437$	3.87659	0.185442
$438$	0	0
$439$	−8.33310	−0.397717	−0.198859	−	0.980028i	$-0.563723\pi$
$439$	−8.33310	−0.397717	−0.198859	+	0.980028i	$0.563723\pi$
$440$	−0.0963086	−0.00459133
$441$	0	0
$442$	−6.43143	−0.305912
$443$	−14.4427	−0.686192	−0.343096	−	0.939300i	$-0.611476\pi$
$443$	−14.4427	−0.686192	−0.343096	+	0.939300i	$0.611476\pi$
$444$	0	0
$445$	−2.05298	−0.0973205
$446$	−13.2448	−0.627159
$447$	0	0
$448$	9.02692	0.426482
$449$	−32.1523	−1.51736	−0.758682	−	0.651462i	$-0.774156\pi$
$449$	−32.1523	−1.51736	−0.758682	+	0.651462i	$0.774156\pi$
$450$	0	0
$451$	−19.7984	−0.932272
$452$	−10.3462	−0.486645
$453$	0	0
$454$	−2.18729	−0.102655
$455$	1.84729	0.0866025
$456$	0	0
$457$	−6.45644	−0.302020	−0.151010	−	0.988532i	$-0.548252\pi$
$457$	−6.45644	−0.302020	−0.151010	+	0.988532i	$0.548252\pi$
$458$	47.8547	2.23610
$459$	0	0
$460$	0.488100	0.0227578
$461$	17.1311	0.797876	0.398938	−	0.916978i	$-0.369379\pi$
$461$	17.1311	0.797876	0.398938	+	0.916978i	$0.369379\pi$
$462$	0	0
$463$	−16.2500	−0.755200	−0.377600	−	0.925969i	$-0.623251\pi$
$463$	−16.2500	−0.755200	−0.377600	+	0.925969i	$0.623251\pi$
$464$	−4.13530	−0.191977
$465$	0	0
$466$	34.5481	1.60041
$467$	−2.20551	−0.102059	−0.0510295	−	0.998697i	$-0.516250\pi$
$467$	−2.20551	−0.102059	−0.0510295	+	0.998697i	$0.516250\pi$
$468$	0	0
$469$	16.0843	0.742702
$470$	5.78186	0.266697
$471$	0	0
$472$	−0.665405	−0.0306278
$473$	−16.8820	−0.776237
$474$	0	0
$475$	−19.1350	−0.877973
$476$	1.26542	0.0580002
$477$	0	0
$478$	35.9804	1.64571
$479$	42.1804	1.92727	0.963637	−	0.267216i	$-0.0861037\pi$
$479$	42.1804	1.92727	0.963637	+	0.267216i	$0.0861037\pi$
$480$	0	0
$481$	26.9630	1.22941
$482$	42.8858	1.95340
$483$	0	0
$484$	−6.74141	−0.306428
$485$	4.50377	0.204506
$486$	0	0
$487$	−40.8033	−1.84897	−0.924486	−	0.381216i	$-0.875506\pi$
$487$	−40.8033	−1.84897	−0.924486	+	0.381216i	$0.875506\pi$
$488$	−0.319840	−0.0144785
$489$	0	0
$490$	2.76952	0.125114
$491$	2.69191	0.121484	0.0607421	−	0.998153i	$-0.480653\pi$
$491$	2.69191	0.121484	0.0607421	+	0.998153i	$0.480653\pi$
$492$	0	0
$493$	−0.539671	−0.0243056
$494$	46.1986	2.07857
$495$	0	0
$496$	43.0703	1.93392
$497$	−13.3050	−0.596812
$498$	0	0
$499$	−33.6818	−1.50780	−0.753902	−	0.656987i	$-0.771831\pi$
$499$	−33.6818	−1.50780	−0.753902	+	0.656987i	$0.771831\pi$
$500$	−4.84978	−0.216889
$501$	0	0
$502$	−13.5896	−0.606534
$503$	31.2381	1.39284	0.696418	−	0.717636i	$-0.254776\pi$
$503$	31.2381	1.39284	0.696418	+	0.717636i	$0.254776\pi$
$504$	0	0
$505$	2.40032	0.106813
$506$	5.43149	0.241459
$507$	0	0
$508$	0.984004	0.0436581
$509$	20.3750	0.903107	0.451553	−	0.892244i	$-0.350870\pi$
$509$	20.3750	0.903107	0.451553	+	0.892244i	$0.350870\pi$
$510$	0	0
$511$	−0.342591	−0.0151553
$512$	−31.6014	−1.39660
$513$	0	0
$514$	−40.5646	−1.78923
$515$	5.06679	0.223269
$516$	0	0
$517$	31.5958	1.38958
$518$	−10.8029	−0.474654
$519$	0	0
$520$	−0.211312	−0.00926666
$521$	−23.0113	−1.00814	−0.504071	−	0.863662i	$-0.668165\pi$
$521$	−23.0113	−1.00814	−0.504071	+	0.863662i	$0.668165\pi$
$522$	0	0
$523$	31.1568	1.36239	0.681197	−	0.732100i	$-0.261460\pi$
$523$	31.1568	1.36239	0.681197	+	0.732100i	$0.261460\pi$
$524$	−42.9729	−1.87728
$525$	0	0
$526$	−4.27824	−0.186540
$527$	5.62082	0.244847
$528$	0	0
$529$	1.00000	0.0434783
$530$	−1.83148	−0.0795545
$531$	0	0
$532$	−9.08980	−0.394093
$533$	−43.4401	−1.88160
$534$	0	0
$535$	5.04583	0.218150
$536$	−1.83988	−0.0794707
$537$	0	0
$538$	25.0854	1.08151
$539$	15.1345	0.651888
$540$	0	0
$541$	4.06950	0.174962	0.0874808	−	0.996166i	$-0.472118\pi$
$541$	4.06950	0.174962	0.0874808	+	0.996166i	$0.472118\pi$
$542$	−38.3176	−1.64588
$543$	0	0
$544$	−4.27411	−0.183251
$545$	0.901695	0.0386244
$546$	0	0
$547$	−29.2713	−1.25155	−0.625776	−	0.780003i	$-0.715217\pi$
$547$	−29.2713	−1.25155	−0.625776	+	0.780003i	$0.715217\pi$
$548$	−24.4423	−1.04412
$549$	0	0
$550$	−26.8100	−1.14318
$551$	3.87659	0.165148
$552$	0	0
$553$	12.1325	0.515928
$554$	−59.1014	−2.51098
$555$	0	0
$556$	−1.40074	−0.0594047
$557$	14.7898	0.626664	0.313332	−	0.949644i	$-0.398555\pi$
$557$	14.7898	0.626664	0.313332	+	0.949644i	$0.398555\pi$
$558$	0	0
$559$	−37.0412	−1.56667
$560$	1.27073	0.0536983
$561$	0	0
$562$	−23.0055	−0.970428
$563$	−22.0614	−0.929776	−0.464888	−	0.885369i	$-0.653905\pi$
$563$	−22.0614	−0.929776	−0.464888	+	0.885369i	$0.653905\pi$
$564$	0	0
$565$	−1.35589	−0.0570428
$566$	51.7693	2.17602
$567$	0	0
$568$	1.52196	0.0638602
$569$	−26.5470	−1.11291	−0.556454	−	0.830879i	$-0.687838\pi$
$569$	−26.5470	−1.11291	−0.556454	+	0.830879i	$0.687838\pi$
$570$	0	0
$571$	10.2014	0.426914	0.213457	−	0.976952i	$-0.431528\pi$
$571$	10.2014	0.426914	0.213457	+	0.976952i	$0.431528\pi$
$572$	31.7870	1.32908
$573$	0	0
$574$	17.4046	0.726454
$575$	−4.93603	−0.205847
$576$	0	0
$577$	1.63413	0.0680295	0.0340148	−	0.999421i	$-0.489171\pi$
$577$	1.63413	0.0680295	0.0340148	+	0.999421i	$0.489171\pi$
$578$	33.1234	1.37775
$579$	0	0
$580$	0.488100	0.0202673
$581$	−0.945512	−0.0392264
$582$	0	0
$583$	−10.0084	−0.414506
$584$	0.0391891	0.00162165
$585$	0	0
$586$	40.8518	1.68757
$587$	−15.9913	−0.660032	−0.330016	−	0.943975i	$-0.607054\pi$
$587$	−15.9913	−0.660032	−0.330016	+	0.943975i	$0.607054\pi$
$588$	0	0
$589$	−40.3758	−1.66366
$590$	2.40045	0.0988249
$591$	0	0
$592$	18.5476	0.762301
$593$	44.3566	1.82151	0.910753	−	0.412952i	$-0.135502\pi$
$593$	44.3566	1.82151	0.910753	+	0.412952i	$0.135502\pi$
$594$	0	0
$595$	0.165835	0.00679858
$596$	14.2665	0.584378
$597$	0	0
$598$	11.9173	0.487336
$599$	28.0776	1.14722	0.573610	−	0.819129i	$-0.305543\pi$
$599$	28.0776	1.14722	0.573610	+	0.819129i	$0.305543\pi$
$600$	0	0
$601$	−24.0537	−0.981171	−0.490586	−	0.871393i	$-0.663217\pi$
$601$	−24.0537	−0.981171	−0.490586	+	0.871393i	$0.663217\pi$
$602$	14.8408	0.604867
$603$	0	0
$604$	4.78469	0.194686
$605$	−0.883474	−0.0359183
$606$	0	0
$607$	42.6340	1.73046	0.865229	−	0.501376i	$-0.167173\pi$
$607$	42.6340	1.73046	0.865229	+	0.501376i	$0.167173\pi$
$608$	30.7020	1.24513
$609$	0	0
$610$	1.15382	0.0467169
$611$	69.3249	2.80459
$612$	0	0
$613$	35.6653	1.44051	0.720254	−	0.693711i	$-0.244025\pi$
$613$	35.6653	1.44051	0.720254	+	0.693711i	$0.244025\pi$
$614$	−38.6906	−1.56143
$615$	0	0
$616$	0.462658	0.0186410
$617$	21.7432	0.875350	0.437675	−	0.899133i	$-0.355802\pi$
$617$	21.7432	0.875350	0.437675	+	0.899133i	$0.355802\pi$
$618$	0	0
$619$	−19.1222	−0.768588	−0.384294	−	0.923211i	$-0.625555\pi$
$619$	−19.1222	−0.768588	−0.384294	+	0.923211i	$0.625555\pi$
$620$	−5.08370	−0.204166
$621$	0	0
$622$	−39.2106	−1.57220
$623$	9.86233	0.395126
$624$	0	0
$625$	24.0446	0.961784
$626$	−42.3680	−1.69337
$627$	0	0
$628$	21.1249	0.842973
$629$	2.42052	0.0965126
$630$	0	0
$631$	−6.64264	−0.264439	−0.132220	−	0.991220i	$-0.542210\pi$
$631$	−6.64264	−0.264439	−0.132220	+	0.991220i	$0.542210\pi$
$632$	−1.38784	−0.0552054
$633$	0	0
$634$	12.5585	0.498761
$635$	0.128956	0.00511744
$636$	0	0
$637$	33.2068	1.31570
$638$	5.43149	0.215035
$639$	0	0
$640$	−0.281034	−0.0111089
$641$	40.3737	1.59466	0.797332	−	0.603541i	$-0.206244\pi$
$641$	40.3737	1.59466	0.797332	+	0.603541i	$0.206244\pi$
$642$	0	0
$643$	20.0493	0.790666	0.395333	−	0.918538i	$-0.370629\pi$
$643$	20.0493	0.790666	0.395333	+	0.918538i	$0.370629\pi$
$644$	−2.34479	−0.0923978
$645$	0	0
$646$	4.14734	0.163175
$647$	40.1626	1.57896	0.789478	−	0.613779i	$-0.210351\pi$
$647$	40.1626	1.57896	0.789478	+	0.613779i	$0.210351\pi$
$648$	0	0
$649$	13.1176	0.514911
$650$	−58.8243	−2.30728
$651$	0	0
$652$	−35.7747	−1.40104
$653$	−12.0476	−0.471457	−0.235729	−	0.971819i	$-0.575748\pi$
$653$	−12.0476	−0.471457	−0.235729	+	0.971819i	$0.575748\pi$
$654$	0	0
$655$	−5.63168	−0.220048
$656$	−29.8820	−1.16669
$657$	0	0
$658$	−27.7755	−1.08280
$659$	−3.01463	−0.117433	−0.0587166	−	0.998275i	$-0.518701\pi$
$659$	−3.01463	−0.117433	−0.0587166	+	0.998275i	$0.518701\pi$
$660$	0	0
$661$	14.9576	0.581781	0.290891	−	0.956756i	$-0.406048\pi$
$661$	14.9576	0.581781	0.290891	+	0.956756i	$0.406048\pi$
$662$	5.80118	0.225469
$663$	0	0
$664$	0.108157	0.00419732
$665$	−1.19124	−0.0461941
$666$	0	0
$667$	1.00000	0.0387202
$668$	17.3314	0.670572
$669$	0	0
$670$	6.63737	0.256424
$671$	6.30524	0.243411
$672$	0	0
$673$	−16.2877	−0.627845	−0.313922	−	0.949449i	$-0.601643\pi$
$673$	−16.2877	−0.627845	−0.313922	+	0.949449i	$0.601643\pi$
$674$	−48.0149	−1.84947
$675$	0	0
$676$	44.6559	1.71753
$677$	40.9529	1.57395	0.786974	−	0.616986i	$-0.211646\pi$
$677$	40.9529	1.57395	0.786974	+	0.616986i	$0.211646\pi$
$678$	0	0
$679$	−21.6357	−0.830303
$680$	−0.0189699	−0.000727463 0
$681$	0	0
$682$	−56.5705	−2.16620
$683$	−10.0867	−0.385957	−0.192979	−	0.981203i	$-0.561815\pi$
$683$	−10.0867	−0.385957	−0.192979	+	0.981203i	$0.561815\pi$
$684$	0	0
$685$	−3.20321	−0.122389
$686$	−30.1646	−1.15169
$687$	0	0
$688$	−25.4802	−0.971425
$689$	−21.9596	−0.836595
$690$	0	0
$691$	−6.48669	−0.246765	−0.123383	−	0.992359i	$-0.539374\pi$
$691$	−6.48669	−0.246765	−0.123383	+	0.992359i	$0.539374\pi$
$692$	−8.72537	−0.331689
$693$	0	0
$694$	2.84034	0.107818
$695$	−0.183570	−0.00696320
$696$	0	0
$697$	−3.89970	−0.147712
$698$	38.0553	1.44041
$699$	0	0
$700$	11.5740	0.437455
$701$	14.3536	0.542129	0.271064	−	0.962561i	$-0.412624\pi$
$701$	14.3536	0.542129	0.271064	+	0.962561i	$0.412624\pi$
$702$	0	0
$703$	−17.3872	−0.655772
$704$	20.3562	0.767204
$705$	0	0
$706$	20.2303	0.761376
$707$	−11.5309	−0.433665
$708$	0	0
$709$	36.9482	1.38762	0.693808	−	0.720160i	$-0.255931\pi$
$709$	36.9482	1.38762	0.693808	+	0.720160i	$0.255931\pi$
$710$	−5.49049	−0.206054
$711$	0	0
$712$	−1.12815	−0.0422793
$713$	−10.4153	−0.390055
$714$	0	0
$715$	4.16575	0.155790
$716$	37.9521	1.41834
$717$	0	0
$718$	−45.9836	−1.71609
$719$	17.1679	0.640256	0.320128	−	0.947374i	$-0.396274\pi$
$719$	17.1679	0.640256	0.320128	+	0.947374i	$0.396274\pi$
$720$	0	0
$721$	−24.3404	−0.906484
$722$	7.87416	0.293046
$723$	0	0
$724$	21.2911	0.791277
$725$	−4.93603	−0.183320
$726$	0	0
$727$	−23.1921	−0.860146	−0.430073	−	0.902794i	$-0.641512\pi$
$727$	−23.1921	−0.860146	−0.430073	+	0.902794i	$0.641512\pi$
$728$	1.01513	0.0376231
$729$	0	0
$730$	−0.141375	−0.00523251
$731$	−3.32526	−0.122989
$732$	0	0
$733$	46.3741	1.71287	0.856434	−	0.516257i	$-0.172675\pi$
$733$	46.3741	1.71287	0.856434	+	0.516257i	$0.172675\pi$
$734$	−49.8659	−1.84059
$735$	0	0
$736$	7.91984	0.291929
$737$	36.2709	1.33606
$738$	0	0
$739$	−7.51225	−0.276342	−0.138171	−	0.990408i	$-0.544122\pi$
$739$	−7.51225	−0.276342	−0.138171	+	0.990408i	$0.544122\pi$
$740$	−2.18922	−0.0804773
$741$	0	0
$742$	8.79829	0.322995
$743$	19.5439	0.716997	0.358499	−	0.933530i	$-0.383289\pi$
$743$	19.5439	0.716997	0.358499	+	0.933530i	$0.383289\pi$
$744$	0	0
$745$	1.86965	0.0684987
$746$	56.7150	2.07648
$747$	0	0
$748$	2.85358	0.104337
$749$	−24.2397	−0.885701
$750$	0	0
$751$	−23.4707	−0.856457	−0.428228	−	0.903671i	$-0.640862\pi$
$751$	−23.4707	−0.856457	−0.428228	+	0.903671i	$0.640862\pi$
$752$	47.6879	1.73900
$753$	0	0
$754$	11.9173	0.434003
$755$	0.627043	0.0228204
$756$	0	0
$757$	53.8600	1.95757	0.978787	−	0.204880i	$-0.0656803\pi$
$757$	53.8600	1.95757	0.978787	+	0.204880i	$0.0656803\pi$
$758$	−63.3562	−2.30120
$759$	0	0
$760$	0.136266	0.00494287
$761$	−3.33811	−0.121006	−0.0605032	−	0.998168i	$-0.519271\pi$
$761$	−3.33811	−0.121006	−0.0605032	+	0.998168i	$0.519271\pi$
$762$	0	0
$763$	−4.33166	−0.156817
$764$	−4.55307	−0.164724
$765$	0	0
$766$	−47.1796	−1.70467
$767$	28.7816	1.03924
$768$	0	0
$769$	−16.2700	−0.586712	−0.293356	−	0.956003i	$-0.594772\pi$
$769$	−16.2700	−0.586712	−0.293356	+	0.956003i	$0.594772\pi$
$770$	−1.66904	−0.0601480
$771$	0	0
$772$	32.9138	1.18459
$773$	36.7642	1.32232	0.661159	−	0.750246i	$-0.270065\pi$
$773$	36.7642	1.32232	0.661159	+	0.750246i	$0.270065\pi$
$774$	0	0
$775$	51.4102	1.84671
$776$	2.47491	0.0888442
$777$	0	0
$778$	−45.4551	−1.62965
$779$	28.0125	1.00365
$780$	0	0
$781$	−30.0036	−1.07361
$782$	1.06984	0.0382574
$783$	0	0
$784$	22.8426	0.815808
$785$	2.76845	0.0988103
$786$	0	0
$787$	−21.2569	−0.757727	−0.378863	−	0.925453i	$-0.623685\pi$
$787$	−21.2569	−0.757727	−0.378863	+	0.925453i	$0.623685\pi$
$788$	−10.6392	−0.379005
$789$	0	0
$790$	5.00664	0.178128
$791$	6.51359	0.231597
$792$	0	0
$793$	13.8344	0.491275
$794$	48.6994	1.72828
$795$	0	0
$796$	−30.0600	−1.06545
$797$	47.3125	1.67589	0.837947	−	0.545752i	$-0.183756\pi$
$797$	47.3125	1.67589	0.837947	+	0.545752i	$0.183756\pi$
$798$	0	0
$799$	6.22343	0.220169
$800$	−39.0926	−1.38213
$801$	0	0
$802$	−19.9970	−0.706120
$803$	−0.772562	−0.0272631
$804$	0	0
$805$	−0.307289	−0.0108305
$806$	−124.122	−4.37202
$807$	0	0
$808$	1.31902	0.0464031
$809$	−7.50118	−0.263727	−0.131864	−	0.991268i	$-0.542096\pi$
$809$	−7.50118	−0.263727	−0.131864	+	0.991268i	$0.542096\pi$
$810$	0	0
$811$	28.5896	1.00392	0.501959	−	0.864891i	$-0.332613\pi$
$811$	28.5896	1.00392	0.501959	+	0.864891i	$0.332613\pi$
$812$	−2.34479	−0.0822861
$813$	0	0
$814$	−24.3612	−0.853861
$815$	−4.68834	−0.164225
$816$	0	0
$817$	23.8862	0.835671
$818$	4.36478	0.152611
$819$	0	0
$820$	3.52705	0.123170
$821$	27.1303	0.946853	0.473426	−	0.880833i	$-0.343017\pi$
$821$	27.1303	0.946853	0.473426	+	0.880833i	$0.343017\pi$
$822$	0	0
$823$	5.22868	0.182260	0.0911301	−	0.995839i	$-0.470952\pi$
$823$	5.22868	0.182260	0.0911301	+	0.995839i	$0.470952\pi$
$824$	2.78430	0.0969958
$825$	0	0
$826$	−11.5316	−0.401234
$827$	−7.46403	−0.259550	−0.129775	−	0.991543i	$-0.541425\pi$
$827$	−7.46403	−0.259550	−0.129775	+	0.991543i	$0.541425\pi$
$828$	0	0
$829$	6.68393	0.232143	0.116071	−	0.993241i	$-0.462970\pi$
$829$	6.68393	0.232143	0.116071	+	0.993241i	$0.462970\pi$
$830$	−0.390177	−0.0135433
$831$	0	0
$832$	44.6639	1.54844
$833$	2.98104	0.103287
$834$	0	0
$835$	2.27131	0.0786020
$836$	−20.4980	−0.708938
$837$	0	0
$838$	58.2067	2.01072
$839$	23.6750	0.817352	0.408676	−	0.912680i	$-0.365991\pi$
$839$	23.6750	0.817352	0.408676	+	0.912680i	$0.365991\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−31.0185	−1.06897
$843$	0	0
$844$	1.30376	0.0448774
$845$	5.85224	0.201323
$846$	0	0
$847$	4.24413	0.145830
$848$	−15.1058	−0.518735
$849$	0	0
$850$	−5.28077	−0.181129
$851$	−4.48518	−0.153750
$852$	0	0
$853$	52.3533	1.79254	0.896271	−	0.443507i	$-0.146266\pi$
$853$	52.3533	1.79254	0.896271	+	0.443507i	$0.146266\pi$
$854$	−5.54287	−0.189673
$855$	0	0
$856$	2.77279	0.0947720
$857$	52.5066	1.79359	0.896796	−	0.442444i	$-0.145888\pi$
$857$	52.5066	1.79359	0.896796	+	0.442444i	$0.145888\pi$
$858$	0	0
$859$	−19.4193	−0.662577	−0.331288	−	0.943530i	$-0.607483\pi$
$859$	−19.4193	−0.662577	−0.331288	+	0.943530i	$0.607483\pi$
$860$	3.00750	0.102555
$861$	0	0
$862$	−75.9328	−2.58628
$863$	−25.1140	−0.854891	−0.427445	−	0.904041i	$-0.640586\pi$
$863$	−25.1140	−0.854891	−0.427445	+	0.904041i	$0.640586\pi$
$864$	0	0
$865$	−1.14348	−0.0388793
$866$	−17.8397	−0.606217
$867$	0	0
$868$	24.4217	0.828926
$869$	27.3595	0.928109
$870$	0	0
$871$	79.5825	2.69655
$872$	0.495500	0.0167797
$873$	0	0
$874$	−7.68494	−0.259947
$875$	3.05324	0.103218
$876$	0	0
$877$	−38.4365	−1.29791	−0.648954	−	0.760828i	$-0.724793\pi$
$877$	−38.4365	−1.29791	−0.648954	+	0.760828i	$0.724793\pi$
$878$	16.5195	0.557506
$879$	0	0
$880$	2.86558	0.0965986
$881$	−10.3064	−0.347230	−0.173615	−	0.984814i	$-0.555545\pi$
$881$	−10.3064	−0.347230	−0.173615	+	0.984814i	$0.555545\pi$
$882$	0	0
$883$	−26.0404	−0.876328	−0.438164	−	0.898895i	$-0.644371\pi$
$883$	−26.0404	−0.876328	−0.438164	+	0.898895i	$0.644371\pi$
$884$	6.26109	0.210583
$885$	0	0
$886$	28.6311	0.961880
$887$	1.08373	0.0363882	0.0181941	−	0.999834i	$-0.494208\pi$
$887$	1.08373	0.0363882	0.0181941	+	0.999834i	$0.494208\pi$
$888$	0	0
$889$	−0.619491	−0.0207771
$890$	4.06981	0.136420
$891$	0	0
$892$	12.8940	0.431723
$893$	−44.7045	−1.49598
$894$	0	0
$895$	4.97370	0.166252
$896$	1.35006	0.0451025
$897$	0	0
$898$	63.7387	2.12699
$899$	−10.4153	−0.347369
$900$	0	0
$901$	−1.97136	−0.0656754
$902$	39.2483	1.30683
$903$	0	0
$904$	−0.745090	−0.0247813
$905$	2.79024	0.0927507
$906$	0	0
$907$	−28.1863	−0.935910	−0.467955	−	0.883752i	$-0.655009\pi$
$907$	−28.1863	−0.935910	−0.467955	+	0.883752i	$0.655009\pi$
$908$	2.12936	0.0706654
$909$	0	0
$910$	−3.66207	−0.121396
$911$	36.4596	1.20796	0.603980	−	0.796999i	$-0.293581\pi$
$911$	36.4596	1.20796	0.603980	+	0.796999i	$0.293581\pi$
$912$	0	0
$913$	−2.13218	−0.0705649
$914$	12.7992	0.423361
$915$	0	0
$916$	−46.5872	−1.53929
$917$	27.0541	0.893406
$918$	0	0
$919$	−46.5156	−1.53441	−0.767204	−	0.641403i	$-0.778353\pi$
$919$	−46.5156	−1.53441	−0.767204	+	0.641403i	$0.778353\pi$
$920$	0.0351509	0.00115889
$921$	0	0
$922$	−33.9607	−1.11844
$923$	−65.8314	−2.16687
$924$	0	0
$925$	22.1390	0.727927
$926$	32.2139	1.05861
$927$	0	0
$928$	7.91984	0.259981
$929$	−1.73608	−0.0569591	−0.0284795	−	0.999594i	$-0.509067\pi$
$929$	−1.73608	−0.0569591	−0.0284795	+	0.999594i	$0.509067\pi$
$930$	0	0
$931$	−21.4135	−0.701801
$932$	−33.6331	−1.10169
$933$	0	0
$934$	4.37220	0.143063
$935$	0.373968	0.0122300
$936$	0	0
$937$	−8.65638	−0.282792	−0.141396	−	0.989953i	$-0.545159\pi$
$937$	−8.65638	−0.282792	−0.141396	+	0.989953i	$0.545159\pi$
$938$	−31.8853	−1.04109
$939$	0	0
$940$	−5.62872	−0.183589
$941$	−15.3813	−0.501417	−0.250708	−	0.968063i	$-0.580664\pi$
$941$	−15.3813	−0.501417	−0.250708	+	0.968063i	$0.580664\pi$
$942$	0	0
$943$	7.22607	0.235313
$944$	19.7985	0.644388
$945$	0	0
$946$	33.4669	1.08810
$947$	−48.5352	−1.57718	−0.788591	−	0.614918i	$-0.789189\pi$
$947$	−48.5352	−1.57718	−0.788591	+	0.614918i	$0.789189\pi$
$948$	0	0
$949$	−1.69509	−0.0550250
$950$	37.9331	1.23071
$951$	0	0
$952$	0.0911298	0.00295353
$953$	46.7032	1.51286	0.756432	−	0.654072i	$-0.226941\pi$
$953$	46.7032	1.51286	0.756432	+	0.654072i	$0.226941\pi$
$954$	0	0
$955$	−0.596689	−0.0193084
$956$	−35.0275	−1.13287
$957$	0	0
$958$	−83.6183	−2.70158
$959$	15.3880	0.496903
$960$	0	0
$961$	77.4781	2.49929
$962$	−53.4514	−1.72334
$963$	0	0
$964$	−41.7500	−1.34468
$965$	4.31342	0.138854
$966$	0	0
$967$	−26.5967	−0.855291	−0.427645	−	0.903947i	$-0.640657\pi$
$967$	−26.5967	−0.855291	−0.427645	+	0.903947i	$0.640657\pi$
$968$	−0.485487	−0.0156041
$969$	0	0
$970$	−8.92825	−0.286669
$971$	−44.5888	−1.43092	−0.715462	−	0.698652i	$-0.753784\pi$
$971$	−44.5888	−1.43092	−0.715462	+	0.698652i	$0.753784\pi$
$972$	0	0
$973$	0.881854	0.0282709
$974$	80.8882	2.59183
$975$	0	0
$976$	9.51656	0.304618
$977$	45.3191	1.44989	0.724944	−	0.688808i	$-0.241866\pi$
$977$	45.3191	1.44989	0.724944	+	0.688808i	$0.241866\pi$
$978$	0	0
$979$	22.2401	0.710797
$980$	−2.69617	−0.0861261
$981$	0	0
$982$	−5.33643	−0.170292
$983$	−31.5879	−1.00750	−0.503749	−	0.863850i	$-0.668046\pi$
$983$	−31.5879	−1.00750	−0.503749	+	0.863850i	$0.668046\pi$
$984$	0	0
$985$	−1.39428	−0.0444256
$986$	1.06984	0.0340707
$987$	0	0
$988$	−44.9750	−1.43085
$989$	6.16164	0.195929
$990$	0	0
$991$	−45.9658	−1.46015	−0.730076	−	0.683366i	$-0.760515\pi$
$991$	−45.9658	−1.46015	−0.730076	+	0.683366i	$0.760515\pi$
$992$	−82.4874	−2.61898
$993$	0	0
$994$	26.3758	0.836591
$995$	−3.93942	−0.124888
$996$	0	0
$997$	−34.2525	−1.08479	−0.542394	−	0.840125i	$-0.682482\pi$
$997$	−34.2525	−1.08479	−0.542394	+	0.840125i	$0.682482\pi$
$998$	66.7706	2.11359
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.2		14
3.2	odd	2		2001.2.a.m.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.13	✓	14	3.2	odd	2
6003.2.a.p.1.2		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)

\(f(q)\)	\(=\)	\(q-1.98240 q^{2} +1.92989 q^{4} +0.252916 q^{5} -1.21499 q^{7} +0.138982 q^{8} +O(q^{10})\)	\(q-1.98240 q^{2} +1.92989 q^{4} +0.252916 q^{5} -1.21499 q^{7} +0.138982 q^{8} -0.501379 q^{10} -2.73986 q^{11} -6.01158 q^{13} +2.40858 q^{14} -4.13530 q^{16} -0.539671 q^{17} +3.87659 q^{19} +0.488100 q^{20} +5.43149 q^{22} +1.00000 q^{23} -4.93603 q^{25} +11.9173 q^{26} -2.34479 q^{28} +1.00000 q^{29} -10.4153 q^{31} +7.91984 q^{32} +1.06984 q^{34} -0.307289 q^{35} -4.48518 q^{37} -7.68494 q^{38} +0.0351509 q^{40} +7.22607 q^{41} +6.16164 q^{43} -5.28764 q^{44} -1.98240 q^{46} -11.5319 q^{47} -5.52381 q^{49} +9.78517 q^{50} -11.6017 q^{52} +3.65289 q^{53} -0.692955 q^{55} -0.168862 q^{56} -1.98240 q^{58} -4.78769 q^{59} -2.30130 q^{61} +20.6472 q^{62} -7.42965 q^{64} -1.52042 q^{65} -13.2382 q^{67} -1.04151 q^{68} +0.609169 q^{70} +10.9508 q^{71} +0.281971 q^{73} +8.89141 q^{74} +7.48140 q^{76} +3.32890 q^{77} -9.98574 q^{79} -1.04588 q^{80} -14.3249 q^{82} +0.778208 q^{83} -0.136491 q^{85} -12.2148 q^{86} -0.380793 q^{88} -8.11723 q^{89} +7.30398 q^{91} +1.92989 q^{92} +22.8608 q^{94} +0.980452 q^{95} +17.8074 q^{97} +10.9504 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 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

Related objects

Downloads

Learn more