LMFDB - Embedded newform 6003.2.a.r.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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.65986$ 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)

$f(q)$	$=$	$q-2.65986 q^{2} +5.07484 q^{4} +2.26251 q^{5} +3.13860 q^{7} -8.17864 q^{8} +O(q^{10})$	\(q-2.65986 q^{2} +5.07484 q^{4} +2.26251 q^{5} +3.13860 q^{7} -8.17864 q^{8} -6.01796 q^{10} -2.75152 q^{11} -4.39658 q^{13} -8.34822 q^{14} +11.6043 q^{16} +5.15020 q^{17} +6.77659 q^{19} +11.4819 q^{20} +7.31866 q^{22} +1.00000 q^{23} +0.118961 q^{25} +11.6943 q^{26} +15.9279 q^{28} +1.00000 q^{29} +2.65285 q^{31} -14.5086 q^{32} -13.6988 q^{34} +7.10112 q^{35} +0.716519 q^{37} -18.0248 q^{38} -18.5043 q^{40} +4.09771 q^{41} +9.18604 q^{43} -13.9635 q^{44} -2.65986 q^{46} +9.21938 q^{47} +2.85080 q^{49} -0.316421 q^{50} -22.3120 q^{52} -7.55372 q^{53} -6.22536 q^{55} -25.6695 q^{56} -2.65986 q^{58} +6.47205 q^{59} -1.02808 q^{61} -7.05619 q^{62} +15.3821 q^{64} -9.94733 q^{65} -8.08875 q^{67} +26.1365 q^{68} -18.8880 q^{70} -4.92645 q^{71} -7.67814 q^{73} -1.90584 q^{74} +34.3901 q^{76} -8.63593 q^{77} -14.3627 q^{79} +26.2549 q^{80} -10.8993 q^{82} +9.71872 q^{83} +11.6524 q^{85} -24.4336 q^{86} +22.5037 q^{88} -4.23269 q^{89} -13.7991 q^{91} +5.07484 q^{92} -24.5222 q^{94} +15.3321 q^{95} +8.66126 q^{97} -7.58272 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * 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 $

$2$	−2.65986	−1.88080	−0.940402	−	0.340066i	$-0.889551\pi$
$2$	−2.65986	−1.88080	−0.940402	+	0.340066i	$0.889551\pi$
$3$	0	0
$3$	0	0
$4$	5.07484	2.53742
$5$	2.26251	1.01183	0.505913	−	0.862584i	$-0.331156\pi$
$5$	2.26251	1.01183	0.505913	+	0.862584i	$0.331156\pi$
$6$	0	0
$7$	3.13860	1.18628	0.593139	−	0.805100i	$-0.297888\pi$
$7$	3.13860	1.18628	0.593139	+	0.805100i	$0.297888\pi$
$8$	−8.17864	−2.89158
$9$	0	0
$10$	−6.01796	−1.90305
$11$	−2.75152	−0.829616	−0.414808	−	0.909909i	$-0.636151\pi$
$11$	−2.75152	−0.829616	−0.414808	+	0.909909i	$0.636151\pi$
$12$	0	0
$13$	−4.39658	−1.21939	−0.609697	−	0.792635i	$-0.708709\pi$
$13$	−4.39658	−1.21939	−0.609697	+	0.792635i	$0.708709\pi$
$14$	−8.34822	−2.23116
$15$	0	0
$16$	11.6043	2.90108
$17$	5.15020	1.24911	0.624554	−	0.780982i	$-0.285281\pi$
$17$	5.15020	1.24911	0.624554	+	0.780982i	$0.285281\pi$
$18$	0	0
$19$	6.77659	1.55466	0.777328	−	0.629095i	$-0.216574\pi$
$19$	6.77659	1.55466	0.777328	+	0.629095i	$0.216574\pi$
$20$	11.4819	2.56743
$21$	0	0
$22$	7.31866	1.56034
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0.118961	0.0237923
$26$	11.6943	2.29344
$27$	0	0
$28$	15.9279	3.01009
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	2.65285	0.476465	0.238233	−	0.971208i	$-0.423432\pi$
$31$	2.65285	0.476465	0.238233	+	0.971208i	$0.423432\pi$
$32$	−14.5086	−2.56478
$33$	0	0
$34$	−13.6988	−2.34933
$35$	7.10112	1.20031
$36$	0	0
$37$	0.716519	0.117795	0.0588975	−	0.998264i	$-0.481241\pi$
$37$	0.716519	0.117795	0.0588975	+	0.998264i	$0.481241\pi$
$38$	−18.0248	−2.92400
$39$	0	0
$40$	−18.5043	−2.92578
$41$	4.09771	0.639955	0.319977	−	0.947425i	$-0.396325\pi$
$41$	4.09771	0.639955	0.319977	+	0.947425i	$0.396325\pi$
$42$	0	0
$43$	9.18604	1.40086	0.700429	−	0.713722i	$-0.252992\pi$
$43$	9.18604	1.40086	0.700429	+	0.713722i	$0.252992\pi$
$44$	−13.9635	−2.10508
$45$	0	0
$46$	−2.65986	−0.392175
$47$	9.21938	1.34478	0.672392	−	0.740195i	$-0.265267\pi$
$47$	9.21938	1.34478	0.672392	+	0.740195i	$0.265267\pi$
$48$	0	0
$49$	2.85080	0.407257
$50$	−0.316421	−0.0447486
$51$	0	0
$52$	−22.3120	−3.09411
$53$	−7.55372	−1.03758	−0.518791	−	0.854901i	$-0.673618\pi$
$53$	−7.55372	−1.03758	−0.518791	+	0.854901i	$0.673618\pi$
$54$	0	0
$55$	−6.22536	−0.839427
$56$	−25.6695	−3.43023
$57$	0	0
$58$	−2.65986	−0.349256
$59$	6.47205	0.842590	0.421295	−	0.906924i	$-0.361576\pi$
$59$	6.47205	0.842590	0.421295	+	0.906924i	$0.361576\pi$
$60$	0	0
$61$	−1.02808	−0.131632	−0.0658158	−	0.997832i	$-0.520965\pi$
$61$	−1.02808	−0.131632	−0.0658158	+	0.997832i	$0.520965\pi$
$62$	−7.05619	−0.896137
$63$	0	0
$64$	15.3821	1.92276
$65$	−9.94733	−1.23381
$66$	0	0
$67$	−8.08875	−0.988198	−0.494099	−	0.869406i	$-0.664502\pi$
$67$	−8.08875	−0.988198	−0.494099	+	0.869406i	$0.664502\pi$
$68$	26.1365	3.16951
$69$	0	0
$70$	−18.8880	−2.25754
$71$	−4.92645	−0.584662	−0.292331	−	0.956317i	$-0.594431\pi$
$71$	−4.92645	−0.584662	−0.292331	+	0.956317i	$0.594431\pi$
$72$	0	0
$73$	−7.67814	−0.898658	−0.449329	−	0.893366i	$-0.648337\pi$
$73$	−7.67814	−0.898658	−0.449329	+	0.893366i	$0.648337\pi$
$74$	−1.90584	−0.221549
$75$	0	0
$76$	34.3901	3.94482
$77$	−8.63593	−0.984155
$78$	0	0
$79$	−14.3627	−1.61593	−0.807964	−	0.589232i	$-0.799430\pi$
$79$	−14.3627	−1.61593	−0.807964	+	0.589232i	$0.799430\pi$
$80$	26.2549	2.93539
$81$	0	0
$82$	−10.8993	−1.20363
$83$	9.71872	1.06677	0.533384	−	0.845873i	$-0.320920\pi$
$83$	9.71872	1.06677	0.533384	+	0.845873i	$0.320920\pi$
$84$	0	0
$85$	11.6524	1.26388
$86$	−24.4336	−2.63474
$87$	0	0
$88$	22.5037	2.39890
$89$	−4.23269	−0.448665	−0.224332	−	0.974513i	$-0.572020\pi$
$89$	−4.23269	−0.448665	−0.224332	+	0.974513i	$0.572020\pi$
$90$	0	0
$91$	−13.7991	−1.44654
$92$	5.07484	0.529089
$93$	0	0
$94$	−24.5222	−2.52927
$95$	15.3321	1.57304
$96$	0	0
$97$	8.66126	0.879418	0.439709	−	0.898140i	$-0.355082\pi$
$97$	8.66126	0.879418	0.439709	+	0.898140i	$0.355082\pi$
$98$	−7.58272	−0.765971
$99$	0	0
$100$	0.603710	0.0603710
$101$	−2.93514	−0.292057	−0.146029	−	0.989280i	$-0.546649\pi$
$101$	−2.93514	−0.292057	−0.146029	+	0.989280i	$0.546649\pi$
$102$	0	0
$103$	−1.16031	−0.114328	−0.0571642	−	0.998365i	$-0.518206\pi$
$103$	−1.16031	−0.114328	−0.0571642	+	0.998365i	$0.518206\pi$
$104$	35.9581	3.52598
$105$	0	0
$106$	20.0918	1.95149
$107$	3.72708	0.360311	0.180155	−	0.983638i	$-0.442340\pi$
$107$	3.72708	0.360311	0.180155	+	0.983638i	$0.442340\pi$
$108$	0	0
$109$	11.9328	1.14295	0.571477	−	0.820618i	$-0.306371\pi$
$109$	11.9328	1.14295	0.571477	+	0.820618i	$0.306371\pi$
$110$	16.5586	1.57880
$111$	0	0
$112$	36.4213	3.44149
$113$	1.96344	0.184705	0.0923527	−	0.995726i	$-0.470561\pi$
$113$	1.96344	0.184705	0.0923527	+	0.995726i	$0.470561\pi$
$114$	0	0
$115$	2.26251	0.210980
$116$	5.07484	0.471187
$117$	0	0
$118$	−17.2147	−1.58474
$119$	16.1644	1.48179
$120$	0	0
$121$	−3.42912	−0.311738
$122$	2.73453	0.247573
$123$	0	0
$124$	13.4628	1.20899
$125$	−11.0434	−0.987753
$126$	0	0
$127$	14.4525	1.28245	0.641224	−	0.767354i	$-0.278427\pi$
$127$	14.4525	1.28245	0.641224	+	0.767354i	$0.278427\pi$
$128$	−11.8970	−1.05156
$129$	0	0
$130$	26.4585	2.32056
$131$	10.1767	0.889146	0.444573	−	0.895743i	$-0.353355\pi$
$131$	10.1767	0.889146	0.444573	+	0.895743i	$0.353355\pi$
$132$	0	0
$133$	21.2690	1.84426
$134$	21.5149	1.85861
$135$	0	0
$136$	−42.1216	−3.61190
$137$	15.4872	1.32316	0.661579	−	0.749876i	$-0.269887\pi$
$137$	15.4872	1.32316	0.661579	+	0.749876i	$0.269887\pi$
$138$	0	0
$139$	−6.85193	−0.581173	−0.290587	−	0.956849i	$-0.593850\pi$
$139$	−6.85193	−0.581173	−0.290587	+	0.956849i	$0.593850\pi$
$140$	36.0370	3.04569
$141$	0	0
$142$	13.1037	1.09963
$143$	12.0973	1.01163
$144$	0	0
$145$	2.26251	0.187891
$146$	20.4227	1.69020
$147$	0	0
$148$	3.63622	0.298895
$149$	0.508794	0.0416820	0.0208410	−	0.999783i	$-0.493366\pi$
$149$	0.508794	0.0416820	0.0208410	+	0.999783i	$0.493366\pi$
$150$	0	0
$151$	0.0598146	0.00486764	0.00243382	−	0.999997i	$-0.499225\pi$
$151$	0.0598146	0.00486764	0.00243382	+	0.999997i	$0.499225\pi$
$152$	−55.4233	−4.49542
$153$	0	0
$154$	22.9703	1.85100
$155$	6.00210	0.482100
$156$	0	0
$157$	22.2381	1.77480	0.887398	−	0.461003i	$-0.152510\pi$
$157$	22.2381	1.77480	0.887398	+	0.461003i	$0.152510\pi$
$158$	38.2027	3.03924
$159$	0	0
$160$	−32.8258	−2.59511
$161$	3.13860	0.247356
$162$	0	0
$163$	13.8763	1.08687	0.543437	−	0.839450i	$-0.317123\pi$
$163$	13.8763	1.08687	0.543437	+	0.839450i	$0.317123\pi$
$164$	20.7952	1.62383
$165$	0	0
$166$	−25.8504	−2.00638
$167$	−15.8394	−1.22569	−0.612843	−	0.790204i	$-0.709974\pi$
$167$	−15.8394	−1.22569	−0.612843	+	0.790204i	$0.709974\pi$
$168$	0	0
$169$	6.32995	0.486919
$170$	−30.9937	−2.37711
$171$	0	0
$172$	46.6177	3.55457
$173$	−2.50875	−0.190737	−0.0953683	−	0.995442i	$-0.530403\pi$
$173$	−2.50875	−0.190737	−0.0953683	+	0.995442i	$0.530403\pi$
$174$	0	0
$175$	0.373372	0.0282243
$176$	−31.9296	−2.40678
$177$	0	0
$178$	11.2584	0.843850
$179$	2.55613	0.191054	0.0955269	−	0.995427i	$-0.469546\pi$
$179$	2.55613	0.191054	0.0955269	+	0.995427i	$0.469546\pi$
$180$	0	0
$181$	−22.3089	−1.65821	−0.829104	−	0.559095i	$-0.811149\pi$
$181$	−22.3089	−1.65821	−0.829104	+	0.559095i	$0.811149\pi$
$182$	36.7037	2.72066
$183$	0	0
$184$	−8.17864	−0.602937
$185$	1.62113	0.119188
$186$	0	0
$187$	−14.1709	−1.03628
$188$	46.7869	3.41228
$189$	0	0
$190$	−40.7812	−2.95858
$191$	−23.8617	−1.72657	−0.863287	−	0.504713i	$-0.831598\pi$
$191$	−23.8617	−1.72657	−0.863287	+	0.504713i	$0.831598\pi$
$192$	0	0
$193$	−22.3709	−1.61029	−0.805147	−	0.593076i	$-0.797914\pi$
$193$	−22.3709	−1.61029	−0.805147	+	0.593076i	$0.797914\pi$
$194$	−23.0377	−1.65401
$195$	0	0
$196$	14.4674	1.03338
$197$	10.5354	0.750619	0.375309	−	0.926900i	$-0.377536\pi$
$197$	10.5354	0.750619	0.375309	+	0.926900i	$0.377536\pi$
$198$	0	0
$199$	27.9193	1.97915	0.989575	−	0.144020i	$-0.0460029\pi$
$199$	27.9193	1.97915	0.989575	+	0.144020i	$0.0460029\pi$
$200$	−0.972943	−0.0687974
$201$	0	0
$202$	7.80706	0.549303
$203$	3.13860	0.220286
$204$	0	0
$205$	9.27112	0.647523
$206$	3.08625	0.215029
$207$	0	0
$208$	−51.0194	−3.53756
$209$	−18.6460	−1.28977
$210$	0	0
$211$	−22.0925	−1.52091	−0.760456	−	0.649389i	$-0.775025\pi$
$211$	−22.0925	−1.52091	−0.760456	+	0.649389i	$0.775025\pi$
$212$	−38.3339	−2.63278
$213$	0	0
$214$	−9.91351	−0.677674
$215$	20.7835	1.41743
$216$	0	0
$217$	8.32622	0.565221
$218$	−31.7395	−2.14967
$219$	0	0
$220$	−31.5927	−2.12998
$221$	−22.6433	−1.52315
$222$	0	0
$223$	2.05410	0.137553	0.0687763	−	0.997632i	$-0.478091\pi$
$223$	2.05410	0.137553	0.0687763	+	0.997632i	$0.478091\pi$
$224$	−45.5366	−3.04254
$225$	0	0
$226$	−5.22248	−0.347394
$227$	4.41425	0.292984	0.146492	−	0.989212i	$-0.453202\pi$
$227$	4.41425	0.292984	0.146492	+	0.989212i	$0.453202\pi$
$228$	0	0
$229$	−15.6794	−1.03612	−0.518062	−	0.855343i	$-0.673346\pi$
$229$	−15.6794	−1.03612	−0.518062	+	0.855343i	$0.673346\pi$
$230$	−6.01796	−0.396812
$231$	0	0
$232$	−8.17864	−0.536954
$233$	−11.0992	−0.727133	−0.363567	−	0.931568i	$-0.618441\pi$
$233$	−11.0992	−0.727133	−0.363567	+	0.931568i	$0.618441\pi$
$234$	0	0
$235$	20.8590	1.36069
$236$	32.8446	2.13800
$237$	0	0
$238$	−42.9950	−2.78696
$239$	26.8994	1.73997	0.869987	−	0.493074i	$-0.164127\pi$
$239$	26.8994	1.73997	0.869987	+	0.493074i	$0.164127\pi$
$240$	0	0
$241$	18.6629	1.20218	0.601090	−	0.799181i	$-0.294733\pi$
$241$	18.6629	1.20218	0.601090	+	0.799181i	$0.294733\pi$
$242$	9.12096	0.586317
$243$	0	0
$244$	−5.21732	−0.334005
$245$	6.44997	0.412073
$246$	0	0
$247$	−29.7939	−1.89574
$248$	−21.6967	−1.37774
$249$	0	0
$250$	29.3739	1.85777
$251$	26.5817	1.67782	0.838909	−	0.544271i	$-0.183194\pi$
$251$	26.5817	1.67782	0.838909	+	0.544271i	$0.183194\pi$
$252$	0	0
$253$	−2.75152	−0.172987
$254$	−38.4415	−2.41203
$255$	0	0
$256$	0.880170	0.0550106
$257$	17.3122	1.07991	0.539953	−	0.841695i	$-0.318442\pi$
$257$	17.3122	1.07991	0.539953	+	0.841695i	$0.318442\pi$
$258$	0	0
$259$	2.24886	0.139738
$260$	−50.4811	−3.13070
$261$	0	0
$262$	−27.0687	−1.67231
$263$	12.5229	0.772197	0.386099	−	0.922458i	$-0.373822\pi$
$263$	12.5229	0.772197	0.386099	+	0.922458i	$0.373822\pi$
$264$	0	0
$265$	−17.0904	−1.04985
$266$	−56.5725	−3.46868
$267$	0	0
$268$	−41.0491	−2.50747
$269$	14.9810	0.913410	0.456705	−	0.889618i	$-0.349030\pi$
$269$	14.9810	0.913410	0.456705	+	0.889618i	$0.349030\pi$
$270$	0	0
$271$	7.41476	0.450414	0.225207	−	0.974311i	$-0.427694\pi$
$271$	7.41476	0.450414	0.225207	+	0.974311i	$0.427694\pi$
$272$	59.7646	3.62376
$273$	0	0
$274$	−41.1936	−2.48860
$275$	−0.327325	−0.0197385
$276$	0	0
$277$	9.25490	0.556073	0.278037	−	0.960570i	$-0.410316\pi$
$277$	9.25490	0.556073	0.278037	+	0.960570i	$0.410316\pi$
$278$	18.2252	1.09307
$279$	0	0
$280$	−58.0775	−3.47079
$281$	−15.6501	−0.933607	−0.466803	−	0.884361i	$-0.654594\pi$
$281$	−15.6501	−0.933607	−0.466803	+	0.884361i	$0.654594\pi$
$282$	0	0
$283$	−27.7066	−1.64699	−0.823493	−	0.567326i	$-0.807978\pi$
$283$	−27.7066	−1.64699	−0.823493	+	0.567326i	$0.807978\pi$
$284$	−25.0010	−1.48353
$285$	0	0
$286$	−32.1771	−1.90267
$287$	12.8611	0.759164
$288$	0	0
$289$	9.52459	0.560270
$290$	−6.01796	−0.353387
$291$	0	0
$292$	−38.9653	−2.28027
$293$	−13.6958	−0.800115	−0.400057	−	0.916490i	$-0.631010\pi$
$293$	−13.6958	−0.800115	−0.400057	+	0.916490i	$0.631010\pi$
$294$	0	0
$295$	14.6431	0.852554
$296$	−5.86015	−0.340614
$297$	0	0
$298$	−1.35332	−0.0783957
$299$	−4.39658	−0.254261
$300$	0	0
$301$	28.8313	1.66181
$302$	−0.159098	−0.00915508
$303$	0	0
$304$	78.6378	4.51019
$305$	−2.32603	−0.133188
$306$	0	0
$307$	−20.1645	−1.15085	−0.575424	−	0.817855i	$-0.695163\pi$
$307$	−20.1645	−1.15085	−0.575424	+	0.817855i	$0.695163\pi$
$308$	−43.8260	−2.49722
$309$	0	0
$310$	−15.9647	−0.906735
$311$	−15.6566	−0.887804	−0.443902	−	0.896075i	$-0.646406\pi$
$311$	−15.6566	−0.887804	−0.443902	+	0.896075i	$0.646406\pi$
$312$	0	0
$313$	−16.9031	−0.955417	−0.477709	−	0.878518i	$-0.658533\pi$
$313$	−16.9031	−0.955417	−0.477709	+	0.878518i	$0.658533\pi$
$314$	−59.1503	−3.33804
$315$	0	0
$316$	−72.8883	−4.10029
$317$	15.6646	0.879811	0.439905	−	0.898044i	$-0.355012\pi$
$317$	15.6646	0.879811	0.439905	+	0.898044i	$0.355012\pi$
$318$	0	0
$319$	−2.75152	−0.154056
$320$	34.8022	1.94550
$321$	0	0
$322$	−8.34822	−0.465228
$323$	34.9008	1.94193
$324$	0	0
$325$	−0.523024	−0.0290122
$326$	−36.9089	−2.04420
$327$	0	0
$328$	−33.5137	−1.85048
$329$	28.9359	1.59529
$330$	0	0
$331$	16.9341	0.930783	0.465392	−	0.885105i	$-0.345913\pi$
$331$	16.9341	0.930783	0.465392	+	0.885105i	$0.345913\pi$
$332$	49.3209	2.70684
$333$	0	0
$334$	42.1304	2.30527
$335$	−18.3009	−0.999885
$336$	0	0
$337$	−7.35219	−0.400499	−0.200250	−	0.979745i	$-0.564175\pi$
$337$	−7.35219	−0.400499	−0.200250	+	0.979745i	$0.564175\pi$
$338$	−16.8368	−0.915800
$339$	0	0
$340$	59.1341	3.20699
$341$	−7.29937	−0.395283
$342$	0	0
$343$	−13.0227	−0.703158
$344$	−75.1293	−4.05070
$345$	0	0
$346$	6.67291	0.358738
$347$	−25.9400	−1.39253	−0.696267	−	0.717783i	$-0.745157\pi$
$347$	−25.9400	−1.39253	−0.696267	+	0.717783i	$0.745157\pi$
$348$	0	0
$349$	27.9065	1.49380	0.746901	−	0.664935i	$-0.231541\pi$
$349$	27.9065	1.49380	0.746901	+	0.664935i	$0.231541\pi$
$350$	−0.993117	−0.0530843
$351$	0	0
$352$	39.9207	2.12778
$353$	−1.17625	−0.0626056	−0.0313028	−	0.999510i	$-0.509966\pi$
$353$	−1.17625	−0.0626056	−0.0313028	+	0.999510i	$0.509966\pi$
$354$	0	0
$355$	−11.1462	−0.591576
$356$	−21.4802	−1.13845
$357$	0	0
$358$	−6.79893	−0.359335
$359$	−22.1702	−1.17010	−0.585049	−	0.810998i	$-0.698925\pi$
$359$	−22.1702	−1.17010	−0.585049	+	0.810998i	$0.698925\pi$
$360$	0	0
$361$	26.9222	1.41696
$362$	59.3385	3.11876
$363$	0	0
$364$	−70.0283	−3.67048
$365$	−17.3719	−0.909286
$366$	0	0
$367$	13.8124	0.721001	0.360501	−	0.932759i	$-0.382606\pi$
$367$	13.8124	0.721001	0.360501	+	0.932759i	$0.382606\pi$
$368$	11.6043	0.604917
$369$	0	0
$370$	−4.31198	−0.224169
$371$	−23.7081	−1.23086
$372$	0	0
$373$	24.7713	1.28261	0.641306	−	0.767286i	$-0.278393\pi$
$373$	24.7713	1.28261	0.641306	+	0.767286i	$0.278393\pi$
$374$	37.6926	1.94904
$375$	0	0
$376$	−75.4019	−3.88856
$377$	−4.39658	−0.226436
$378$	0	0
$379$	33.4991	1.72073	0.860367	−	0.509675i	$-0.170234\pi$
$379$	33.4991	1.72073	0.860367	+	0.509675i	$0.170234\pi$
$380$	77.8081	3.99147
$381$	0	0
$382$	63.4688	3.24735
$383$	15.4759	0.790781	0.395390	−	0.918513i	$-0.370609\pi$
$383$	15.4759	0.790781	0.395390	+	0.918513i	$0.370609\pi$
$384$	0	0
$385$	−19.5389	−0.995794
$386$	59.5034	3.02865
$387$	0	0
$388$	43.9545	2.23145
$389$	1.64191	0.0832484	0.0416242	−	0.999133i	$-0.486747\pi$
$389$	1.64191	0.0832484	0.0416242	+	0.999133i	$0.486747\pi$
$390$	0	0
$391$	5.15020	0.260457
$392$	−23.3157	−1.17762
$393$	0	0
$394$	−28.0228	−1.41177
$395$	−32.4957	−1.63504
$396$	0	0
$397$	36.6980	1.84182	0.920911	−	0.389773i	$-0.127447\pi$
$397$	36.6980	1.84182	0.920911	+	0.389773i	$0.127447\pi$
$398$	−74.2615	−3.72239
$399$	0	0
$400$	1.38047	0.0690234
$401$	1.23031	0.0614389	0.0307194	−	0.999528i	$-0.490220\pi$
$401$	1.23031	0.0614389	0.0307194	+	0.999528i	$0.490220\pi$
$402$	0	0
$403$	−11.6635	−0.580999
$404$	−14.8954	−0.741073
$405$	0	0
$406$	−8.34822	−0.414315
$407$	−1.97152	−0.0977245
$408$	0	0
$409$	20.5385	1.01556	0.507781	−	0.861486i	$-0.330466\pi$
$409$	20.5385	1.01556	0.507781	+	0.861486i	$0.330466\pi$
$410$	−24.6598	−1.21786
$411$	0	0
$412$	−5.88837	−0.290099
$413$	20.3132	0.999546
$414$	0	0
$415$	21.9887	1.07938
$416$	63.7882	3.12747
$417$	0	0
$418$	49.5956	2.42580
$419$	−2.61132	−0.127571	−0.0637857	−	0.997964i	$-0.520317\pi$
$419$	−2.61132	−0.127571	−0.0637857	+	0.997964i	$0.520317\pi$
$420$	0	0
$421$	−1.79880	−0.0876680	−0.0438340	−	0.999039i	$-0.513957\pi$
$421$	−1.79880	−0.0876680	−0.0438340	+	0.999039i	$0.513957\pi$
$422$	58.7630	2.86054
$423$	0	0
$424$	61.7791	3.00026
$425$	0.612676	0.0297191
$426$	0	0
$427$	−3.22672	−0.156152
$428$	18.9144	0.914260
$429$	0	0
$430$	−55.2812	−2.66590
$431$	−33.3744	−1.60759	−0.803794	−	0.594908i	$-0.797188\pi$
$431$	−33.3744	−1.60759	−0.803794	+	0.594908i	$0.797188\pi$
$432$	0	0
$433$	10.1052	0.485627	0.242813	−	0.970073i	$-0.421930\pi$
$433$	10.1052	0.485627	0.242813	+	0.970073i	$0.421930\pi$
$434$	−22.1466	−1.06307
$435$	0	0
$436$	60.5570	2.90015
$437$	6.77659	0.324168
$438$	0	0
$439$	27.6169	1.31808	0.659042	−	0.752106i	$-0.270962\pi$
$439$	27.6169	1.31808	0.659042	+	0.752106i	$0.270962\pi$
$440$	50.9149	2.42727
$441$	0	0
$442$	60.2280	2.86475
$443$	16.2505	0.772085	0.386042	−	0.922481i	$-0.373842\pi$
$443$	16.2505	0.772085	0.386042	+	0.922481i	$0.373842\pi$
$444$	0	0
$445$	−9.57652	−0.453971
$446$	−5.46361	−0.258710
$447$	0	0
$448$	48.2782	2.28093
$449$	1.13164	0.0534055	0.0267028	−	0.999643i	$-0.491499\pi$
$449$	1.13164	0.0534055	0.0267028	+	0.999643i	$0.491499\pi$
$450$	0	0
$451$	−11.2749	−0.530916
$452$	9.96417	0.468675
$453$	0	0
$454$	−11.7413	−0.551045
$455$	−31.2207	−1.46365
$456$	0	0
$457$	25.6079	1.19788	0.598942	−	0.800792i	$-0.295588\pi$
$457$	25.6079	1.19788	0.598942	+	0.800792i	$0.295588\pi$
$458$	41.7050	1.94875
$459$	0	0
$460$	11.4819	0.535346
$461$	5.01815	0.233718	0.116859	−	0.993148i	$-0.462717\pi$
$461$	5.01815	0.233718	0.116859	+	0.993148i	$0.462717\pi$
$462$	0	0
$463$	−23.3885	−1.08696	−0.543479	−	0.839423i	$-0.682893\pi$
$463$	−23.3885	−1.08696	−0.543479	+	0.839423i	$0.682893\pi$
$464$	11.6043	0.538717
$465$	0	0
$466$	29.5223	1.36759
$467$	−33.4940	−1.54992	−0.774959	−	0.632011i	$-0.782230\pi$
$467$	−33.4940	−1.54992	−0.774959	+	0.632011i	$0.782230\pi$
$468$	0	0
$469$	−25.3873	−1.17228
$470$	−55.4818	−2.55919
$471$	0	0
$472$	−52.9326	−2.43642
$473$	−25.2756	−1.16217
$474$	0	0
$475$	0.806153	0.0369888
$476$	82.0319	3.75992
$477$	0	0
$478$	−71.5484	−3.27255
$479$	−4.25385	−0.194364	−0.0971818	−	0.995267i	$-0.530983\pi$
$479$	−4.25385	−0.194364	−0.0971818	+	0.995267i	$0.530983\pi$
$480$	0	0
$481$	−3.15023	−0.143638
$482$	−49.6406	−2.26107
$483$	0	0
$484$	−17.4022	−0.791010
$485$	19.5962	0.889818
$486$	0	0
$487$	9.76671	0.442572	0.221286	−	0.975209i	$-0.428975\pi$
$487$	9.76671	0.442572	0.221286	+	0.975209i	$0.428975\pi$
$488$	8.40826	0.380624
$489$	0	0
$490$	−17.1560	−0.775029
$491$	−17.6398	−0.796074	−0.398037	−	0.917369i	$-0.630308\pi$
$491$	−17.6398	−0.796074	−0.398037	+	0.917369i	$0.630308\pi$
$492$	0	0
$493$	5.15020	0.231953
$494$	79.2474	3.56551
$495$	0	0
$496$	30.7845	1.38226
$497$	−15.4621	−0.693572
$498$	0	0
$499$	28.2525	1.26476	0.632379	−	0.774659i	$-0.282079\pi$
$499$	28.2525	1.26476	0.632379	+	0.774659i	$0.282079\pi$
$500$	−56.0435	−2.50634
$501$	0	0
$502$	−70.7034	−3.15565
$503$	27.9559	1.24649	0.623246	−	0.782026i	$-0.285814\pi$
$503$	27.9559	1.24649	0.623246	+	0.782026i	$0.285814\pi$
$504$	0	0
$505$	−6.64079	−0.295511
$506$	7.31866	0.325354
$507$	0	0
$508$	73.3439	3.25411
$509$	−43.0991	−1.91033	−0.955166	−	0.296069i	$-0.904324\pi$
$509$	−43.0991	−1.91033	−0.955166	+	0.296069i	$0.904324\pi$
$510$	0	0
$511$	−24.0986	−1.06606
$512$	21.4529	0.948093
$513$	0	0
$514$	−46.0480	−2.03109
$515$	−2.62521	−0.115680
$516$	0	0
$517$	−25.3673	−1.11565
$518$	−5.98166	−0.262819
$519$	0	0
$520$	81.3556	3.56768
$521$	−28.7125	−1.25792	−0.628959	−	0.777439i	$-0.716519\pi$
$521$	−28.7125	−1.25792	−0.628959	+	0.777439i	$0.716519\pi$
$522$	0	0
$523$	−22.2348	−0.972259	−0.486130	−	0.873887i	$-0.661592\pi$
$523$	−22.2348	−0.972259	−0.486130	+	0.873887i	$0.661592\pi$
$524$	51.6453	2.25614
$525$	0	0
$526$	−33.3092	−1.45235
$527$	13.6627	0.595157
$528$	0	0
$529$	1.00000	0.0434783
$530$	45.4580	1.97457
$531$	0	0
$532$	107.937	4.67965
$533$	−18.0159	−0.780356
$534$	0	0
$535$	8.43257	0.364572
$536$	66.1550	2.85746
$537$	0	0
$538$	−39.8474	−1.71794
$539$	−7.84405	−0.337867
$540$	0	0
$541$	−11.0404	−0.474662	−0.237331	−	0.971429i	$-0.576273\pi$
$541$	−11.0404	−0.474662	−0.237331	+	0.971429i	$0.576273\pi$
$542$	−19.7222	−0.847141
$543$	0	0
$544$	−74.7221	−3.20368
$545$	26.9981	1.15647
$546$	0	0
$547$	26.4815	1.13227	0.566133	−	0.824314i	$-0.308439\pi$
$547$	26.4815	1.13227	0.566133	+	0.824314i	$0.308439\pi$
$548$	78.5949	3.35741
$549$	0	0
$550$	0.870639	0.0371242
$551$	6.77659	0.288692
$552$	0	0
$553$	−45.0787	−1.91694
$554$	−24.6167	−1.04586
$555$	0	0
$556$	−34.7725	−1.47468
$557$	42.2594	1.79059	0.895293	−	0.445478i	$-0.146966\pi$
$557$	42.2594	1.79059	0.895293	+	0.445478i	$0.146966\pi$
$558$	0	0
$559$	−40.3872	−1.70820
$560$	82.4037	3.48219
$561$	0	0
$562$	41.6270	1.75593
$563$	17.9672	0.757226	0.378613	−	0.925555i	$-0.376401\pi$
$563$	17.9672	0.757226	0.378613	+	0.925555i	$0.376401\pi$
$564$	0	0
$565$	4.44232	0.186890
$566$	73.6956	3.09766
$567$	0	0
$568$	40.2916	1.69060
$569$	27.4987	1.15281	0.576403	−	0.817165i	$-0.304456\pi$
$569$	27.4987	1.15281	0.576403	+	0.817165i	$0.304456\pi$
$570$	0	0
$571$	11.2951	0.472685	0.236342	−	0.971670i	$-0.424051\pi$
$571$	11.2951	0.472685	0.236342	+	0.971670i	$0.424051\pi$
$572$	61.3919	2.56692
$573$	0	0
$574$	−34.2086	−1.42784
$575$	0.118961	0.00496104
$576$	0	0
$577$	15.5449	0.647142	0.323571	−	0.946204i	$-0.395117\pi$
$577$	15.5449	0.647142	0.323571	+	0.946204i	$0.395117\pi$
$578$	−25.3341	−1.05376
$579$	0	0
$580$	11.4819	0.476759
$581$	30.5032	1.26548
$582$	0	0
$583$	20.7842	0.860795
$584$	62.7967	2.59855
$585$	0	0
$586$	36.4288	1.50486
$587$	−45.1478	−1.86345	−0.931725	−	0.363165i	$-0.881696\pi$
$587$	−45.1478	−1.86345	−0.931725	+	0.363165i	$0.881696\pi$
$588$	0	0
$589$	17.9773	0.740740
$590$	−38.9486	−1.60349
$591$	0	0
$592$	8.31472	0.341733
$593$	−41.3729	−1.69898	−0.849490	−	0.527605i	$-0.823090\pi$
$593$	−41.3729	−1.69898	−0.849490	+	0.527605i	$0.823090\pi$
$594$	0	0
$595$	36.5722	1.49931
$596$	2.58205	0.105765
$597$	0	0
$598$	11.6943	0.478215
$599$	−8.78420	−0.358913	−0.179456	−	0.983766i	$-0.557434\pi$
$599$	−8.78420	−0.358913	−0.179456	+	0.983766i	$0.557434\pi$
$600$	0	0
$601$	35.7677	1.45899	0.729497	−	0.683984i	$-0.239754\pi$
$601$	35.7677	1.45899	0.729497	+	0.683984i	$0.239754\pi$
$602$	−76.6871	−3.12553
$603$	0	0
$604$	0.303550	0.0123513
$605$	−7.75842	−0.315424
$606$	0	0
$607$	−25.2442	−1.02463	−0.512316	−	0.858797i	$-0.671212\pi$
$607$	−25.2442	−1.02463	−0.512316	+	0.858797i	$0.671212\pi$
$608$	−98.3187	−3.98735
$609$	0	0
$610$	6.18692	0.250501
$611$	−40.5338	−1.63982
$612$	0	0
$613$	18.8034	0.759461	0.379730	−	0.925097i	$-0.376017\pi$
$613$	18.8034	0.759461	0.379730	+	0.925097i	$0.376017\pi$
$614$	53.6346	2.16452
$615$	0	0
$616$	70.6301	2.84577
$617$	−9.10900	−0.366715	−0.183357	−	0.983046i	$-0.558697\pi$
$617$	−9.10900	−0.366715	−0.183357	+	0.983046i	$0.558697\pi$
$618$	0	0
$619$	−2.64073	−0.106140	−0.0530699	−	0.998591i	$-0.516901\pi$
$619$	−2.64073	−0.106140	−0.0530699	+	0.998591i	$0.516901\pi$
$620$	30.4597	1.22329
$621$	0	0
$622$	41.6443	1.66979
$623$	−13.2847	−0.532241
$624$	0	0
$625$	−25.5807	−1.02323
$626$	44.9597	1.79695
$627$	0	0
$628$	112.855	4.50341
$629$	3.69022	0.147139
$630$	0	0
$631$	40.0674	1.59506	0.797529	−	0.603281i	$-0.206140\pi$
$631$	40.0674	1.59506	0.797529	+	0.603281i	$0.206140\pi$
$632$	117.467	4.67259
$633$	0	0
$634$	−41.6655	−1.65475
$635$	32.6989	1.29761
$636$	0	0
$637$	−12.5338	−0.496607
$638$	7.31866	0.289749
$639$	0	0
$640$	−26.9171	−1.06399
$641$	4.01980	0.158773	0.0793863	−	0.996844i	$-0.474704\pi$
$641$	4.01980	0.158773	0.0793863	+	0.996844i	$0.474704\pi$
$642$	0	0
$643$	−23.5020	−0.926827	−0.463414	−	0.886142i	$-0.653376\pi$
$643$	−23.5020	−0.926827	−0.463414	+	0.886142i	$0.653376\pi$
$644$	15.9279	0.627647
$645$	0	0
$646$	−92.8312	−3.65239
$647$	46.8966	1.84369	0.921847	−	0.387554i	$-0.126680\pi$
$647$	46.8966	1.84369	0.921847	+	0.387554i	$0.126680\pi$
$648$	0	0
$649$	−17.8080	−0.699026
$650$	1.39117	0.0545662
$651$	0	0
$652$	70.4199	2.75786
$653$	5.27270	0.206337	0.103168	−	0.994664i	$-0.467102\pi$
$653$	5.27270	0.206337	0.103168	+	0.994664i	$0.467102\pi$
$654$	0	0
$655$	23.0250	0.899661
$656$	47.5511	1.85656
$657$	0	0
$658$	−76.9654	−3.00042
$659$	−1.95551	−0.0761760	−0.0380880	−	0.999274i	$-0.512127\pi$
$659$	−1.95551	−0.0761760	−0.0380880	+	0.999274i	$0.512127\pi$
$660$	0	0
$661$	−16.2707	−0.632859	−0.316429	−	0.948616i	$-0.602484\pi$
$661$	−16.2707	−0.632859	−0.316429	+	0.948616i	$0.602484\pi$
$662$	−45.0423	−1.75062
$663$	0	0
$664$	−79.4859	−3.08465
$665$	48.1214	1.86607
$666$	0	0
$667$	1.00000	0.0387202
$668$	−80.3822	−3.11008
$669$	0	0
$670$	48.6778	1.88059
$671$	2.82878	0.109204
$672$	0	0
$673$	−15.0247	−0.579160	−0.289580	−	0.957154i	$-0.593516\pi$
$673$	−15.0247	−0.579160	−0.289580	+	0.957154i	$0.593516\pi$
$674$	19.5558	0.753261
$675$	0	0
$676$	32.1235	1.23552
$677$	−19.2334	−0.739199	−0.369599	−	0.929191i	$-0.620505\pi$
$677$	−19.2334	−0.739199	−0.369599	+	0.929191i	$0.620505\pi$
$678$	0	0
$679$	27.1842	1.04323
$680$	−95.3007	−3.65462
$681$	0	0
$682$	19.4153	0.743450
$683$	10.2414	0.391878	0.195939	−	0.980616i	$-0.437225\pi$
$683$	10.2414	0.391878	0.195939	+	0.980616i	$0.437225\pi$
$684$	0	0
$685$	35.0399	1.33881
$686$	34.6384	1.32250
$687$	0	0
$688$	106.598	4.06400
$689$	33.2106	1.26522
$690$	0	0
$691$	−27.2229	−1.03561	−0.517804	−	0.855499i	$-0.673250\pi$
$691$	−27.2229	−1.03561	−0.517804	+	0.855499i	$0.673250\pi$
$692$	−12.7315	−0.483979
$693$	0	0
$694$	68.9968	2.61908
$695$	−15.5026	−0.588046
$696$	0	0
$697$	21.1040	0.799372
$698$	−74.2274	−2.80955
$699$	0	0
$700$	1.89480	0.0716169
$701$	22.4181	0.846719	0.423360	−	0.905962i	$-0.360851\pi$
$701$	22.4181	0.846719	0.423360	+	0.905962i	$0.360851\pi$
$702$	0	0
$703$	4.85555	0.183131
$704$	−42.3242	−1.59515
$705$	0	0
$706$	3.12866	0.117749
$707$	−9.21223	−0.346462
$708$	0	0
$709$	12.3777	0.464855	0.232427	−	0.972614i	$-0.425333\pi$
$709$	12.3777	0.464855	0.232427	+	0.972614i	$0.425333\pi$
$710$	29.6472	1.11264
$711$	0	0
$712$	34.6177	1.29735
$713$	2.65285	0.0993499
$714$	0	0
$715$	27.3703	1.02359
$716$	12.9719	0.484784
$717$	0	0
$718$	58.9696	2.20073
$719$	−21.8368	−0.814376	−0.407188	−	0.913344i	$-0.633491\pi$
$719$	−21.8368	−0.814376	−0.407188	+	0.913344i	$0.633491\pi$
$720$	0	0
$721$	−3.64173	−0.135625
$722$	−71.6092	−2.66502
$723$	0	0
$724$	−113.214	−4.20757
$725$	0.118961	0.00441812
$726$	0	0
$727$	21.5088	0.797717	0.398858	−	0.917013i	$-0.369407\pi$
$727$	21.5088	0.797717	0.398858	+	0.917013i	$0.369407\pi$
$728$	112.858	4.18279
$729$	0	0
$730$	46.2067	1.71019
$731$	47.3100	1.74982
$732$	0	0
$733$	17.4284	0.643734	0.321867	−	0.946785i	$-0.395690\pi$
$733$	17.4284	0.643734	0.321867	+	0.946785i	$0.395690\pi$
$734$	−36.7390	−1.35606
$735$	0	0
$736$	−14.5086	−0.534793
$737$	22.2564	0.819825
$738$	0	0
$739$	−50.8593	−1.87089	−0.935445	−	0.353471i	$-0.885001\pi$
$739$	−50.8593	−1.87089	−0.935445	+	0.353471i	$0.885001\pi$
$740$	8.22699	0.302430
$741$	0	0
$742$	63.0601	2.31501
$743$	−45.6613	−1.67515	−0.837575	−	0.546323i	$-0.816027\pi$
$743$	−45.6613	−1.67515	−0.837575	+	0.546323i	$0.816027\pi$
$744$	0	0
$745$	1.15115	0.0421750
$746$	−65.8882	−2.41234
$747$	0	0
$748$	−71.9151	−2.62948
$749$	11.6978	0.427429
$750$	0	0
$751$	35.9889	1.31326	0.656628	−	0.754215i	$-0.271982\pi$
$751$	35.9889	1.31326	0.656628	+	0.754215i	$0.271982\pi$
$752$	106.985	3.90133
$753$	0	0
$754$	11.6943	0.425881
$755$	0.135331	0.00492521
$756$	0	0
$757$	26.0191	0.945679	0.472839	−	0.881149i	$-0.343229\pi$
$757$	26.0191	0.945679	0.472839	+	0.881149i	$0.343229\pi$
$758$	−89.1029	−3.23636
$759$	0	0
$760$	−125.396	−4.54859
$761$	−43.2239	−1.56687	−0.783433	−	0.621476i	$-0.786533\pi$
$761$	−43.2239	−1.56687	−0.783433	+	0.621476i	$0.786533\pi$
$762$	0	0
$763$	37.4522	1.35586
$764$	−121.094	−4.38105
$765$	0	0
$766$	−41.1636	−1.48730
$767$	−28.4549	−1.02745
$768$	0	0
$769$	−19.0024	−0.685243	−0.342621	−	0.939474i	$-0.611315\pi$
$769$	−19.0024	−0.685243	−0.342621	+	0.939474i	$0.611315\pi$
$770$	51.9707	1.87289
$771$	0	0
$772$	−113.529	−4.08599
$773$	26.9762	0.970266	0.485133	−	0.874440i	$-0.338771\pi$
$773$	26.9762	0.970266	0.485133	+	0.874440i	$0.338771\pi$
$774$	0	0
$775$	0.315587	0.0113362
$776$	−70.8373	−2.54291
$777$	0	0
$778$	−4.36726	−0.156574
$779$	27.7685	0.994910
$780$	0	0
$781$	13.5552	0.485045
$782$	−13.6988	−0.489868
$783$	0	0
$784$	33.0816	1.18149
$785$	50.3141	1.79579
$786$	0	0
$787$	22.8508	0.814544	0.407272	−	0.913307i	$-0.366480\pi$
$787$	22.8508	0.814544	0.407272	+	0.913307i	$0.366480\pi$
$788$	53.4657	1.90464
$789$	0	0
$790$	86.4340	3.07518
$791$	6.16246	0.219112
$792$	0	0
$793$	4.52002	0.160511
$794$	−97.6116	−3.46410
$795$	0	0
$796$	141.686	5.02193
$797$	48.5677	1.72036	0.860178	−	0.509993i	$-0.170352\pi$
$797$	48.5677	1.72036	0.860178	+	0.509993i	$0.170352\pi$
$798$	0	0
$799$	47.4817	1.67978
$800$	−1.72596	−0.0610220
$801$	0	0
$802$	−3.27246	−0.115554
$803$	21.1266	0.745541
$804$	0	0
$805$	7.10112	0.250281
$806$	31.0231	1.09274
$807$	0	0
$808$	24.0055	0.844509
$809$	−46.4906	−1.63452	−0.817261	−	0.576268i	$-0.804509\pi$
$809$	−46.4906	−1.63452	−0.817261	+	0.576268i	$0.804509\pi$
$810$	0	0
$811$	−27.4842	−0.965099	−0.482550	−	0.875869i	$-0.660289\pi$
$811$	−27.4842	−0.965099	−0.482550	+	0.875869i	$0.660289\pi$
$812$	15.9279	0.558959
$813$	0	0
$814$	5.24396	0.183801
$815$	31.3953	1.09973
$816$	0	0
$817$	62.2500	2.17785
$818$	−54.6294	−1.91007
$819$	0	0
$820$	47.0494	1.64304
$821$	−31.8446	−1.11139	−0.555693	−	0.831388i	$-0.687547\pi$
$821$	−31.8446	−1.11139	−0.555693	+	0.831388i	$0.687547\pi$
$822$	0	0
$823$	−20.0814	−0.699993	−0.349996	−	0.936751i	$-0.613817\pi$
$823$	−20.0814	−0.699993	−0.349996	+	0.936751i	$0.613817\pi$
$824$	9.48972	0.330590
$825$	0	0
$826$	−54.0301	−1.87995
$827$	41.1649	1.43145	0.715723	−	0.698385i	$-0.246097\pi$
$827$	41.1649	1.43145	0.715723	+	0.698385i	$0.246097\pi$
$828$	0	0
$829$	41.8877	1.45482	0.727410	−	0.686203i	$-0.240724\pi$
$829$	41.8877	1.45482	0.727410	+	0.686203i	$0.240724\pi$
$830$	−58.4868	−2.03011
$831$	0	0
$832$	−67.6287	−2.34460
$833$	14.6822	0.508708
$834$	0	0
$835$	−35.8367	−1.24018
$836$	−94.6252	−3.27268
$837$	0	0
$838$	6.94574	0.239937
$839$	−37.1691	−1.28322	−0.641609	−	0.767031i	$-0.721733\pi$
$839$	−37.1691	−1.28322	−0.641609	+	0.767031i	$0.721733\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	4.78454	0.164886
$843$	0	0
$844$	−112.116	−3.85919
$845$	14.3216	0.492678
$846$	0	0
$847$	−10.7626	−0.369808
$848$	−87.6558	−3.01011
$849$	0	0
$850$	−1.62963	−0.0558958
$851$	0.716519	0.0245619
$852$	0	0
$853$	−25.3088	−0.866558	−0.433279	−	0.901260i	$-0.642644\pi$
$853$	−25.3088	−0.866558	−0.433279	+	0.901260i	$0.642644\pi$
$854$	8.58261	0.293691
$855$	0	0
$856$	−30.4825	−1.04187
$857$	22.6996	0.775403	0.387702	−	0.921785i	$-0.373269\pi$
$857$	22.6996	0.775403	0.387702	+	0.921785i	$0.373269\pi$
$858$	0	0
$859$	−50.1959	−1.71266	−0.856331	−	0.516428i	$-0.827262\pi$
$859$	−50.1959	−1.71266	−0.856331	+	0.516428i	$0.827262\pi$
$860$	105.473	3.59660
$861$	0	0
$862$	88.7711	3.02355
$863$	−22.0917	−0.752009	−0.376004	−	0.926618i	$-0.622702\pi$
$863$	−22.0917	−0.752009	−0.376004	+	0.926618i	$0.622702\pi$
$864$	0	0
$865$	−5.67607	−0.192992
$866$	−26.8785	−0.913369
$867$	0	0
$868$	42.2542	1.43420
$869$	39.5193	1.34060
$870$	0	0
$871$	35.5629	1.20500
$872$	−97.5939	−3.30495
$873$	0	0
$874$	−18.0248	−0.609697
$875$	−34.6608	−1.17175
$876$	0	0
$877$	−7.93433	−0.267923	−0.133962	−	0.990987i	$-0.542770\pi$
$877$	−7.93433	−0.267923	−0.133962	+	0.990987i	$0.542770\pi$
$878$	−73.4571	−2.47906
$879$	0	0
$880$	−72.2411	−2.43525
$881$	−42.9553	−1.44720	−0.723601	−	0.690219i	$-0.757514\pi$
$881$	−42.9553	−1.44720	−0.723601	+	0.690219i	$0.757514\pi$
$882$	0	0
$883$	50.9072	1.71316	0.856581	−	0.516012i	$-0.172584\pi$
$883$	50.9072	1.71316	0.856581	+	0.516012i	$0.172584\pi$
$884$	−114.911	−3.86488
$885$	0	0
$886$	−43.2240	−1.45214
$887$	12.4970	0.419609	0.209804	−	0.977743i	$-0.432717\pi$
$887$	12.4970	0.419609	0.209804	+	0.977743i	$0.432717\pi$
$888$	0	0
$889$	45.3605	1.52134
$890$	25.4722	0.853829
$891$	0	0
$892$	10.4242	0.349029
$893$	62.4759	2.09068
$894$	0	0
$895$	5.78327	0.193313
$896$	−37.3399	−1.24744
$897$	0	0
$898$	−3.01001	−0.100445
$899$	2.65285	0.0884774
$900$	0	0
$901$	−38.9032	−1.29605
$902$	29.9897	0.998549
$903$	0	0
$904$	−16.0583	−0.534091
$905$	−50.4742	−1.67782
$906$	0	0
$907$	1.42754	0.0474008	0.0237004	−	0.999719i	$-0.492455\pi$
$907$	1.42754	0.0474008	0.0237004	+	0.999719i	$0.492455\pi$
$908$	22.4016	0.743423
$909$	0	0
$910$	83.0425	2.75283
$911$	3.43971	0.113963	0.0569813	−	0.998375i	$-0.481852\pi$
$911$	3.43971	0.113963	0.0569813	+	0.998375i	$0.481852\pi$
$912$	0	0
$913$	−26.7413	−0.885007
$914$	−68.1132	−2.25299
$915$	0	0
$916$	−79.5705	−2.62908
$917$	31.9407	1.05477
$918$	0	0
$919$	−57.5205	−1.89743	−0.948713	−	0.316140i	$-0.897613\pi$
$919$	−57.5205	−1.89743	−0.948713	+	0.316140i	$0.897613\pi$
$920$	−18.5043	−0.610068
$921$	0	0
$922$	−13.3476	−0.439578
$923$	21.6596	0.712933
$924$	0	0
$925$	0.0852381	0.00280261
$926$	62.2102	2.04435
$927$	0	0
$928$	−14.5086	−0.476267
$929$	27.5561	0.904088	0.452044	−	0.891996i	$-0.350695\pi$
$929$	27.5561	0.904088	0.452044	+	0.891996i	$0.350695\pi$
$930$	0	0
$931$	19.3187	0.633145
$932$	−56.3267	−1.84504
$933$	0	0
$934$	89.0893	2.91509
$935$	−32.0619	−1.04853
$936$	0	0
$937$	47.8901	1.56450	0.782251	−	0.622963i	$-0.214071\pi$
$937$	47.8901	1.56450	0.782251	+	0.622963i	$0.214071\pi$
$938$	67.5267	2.20482
$939$	0	0
$940$	105.856	3.45264
$941$	9.91312	0.323158	0.161579	−	0.986860i	$-0.448341\pi$
$941$	9.91312	0.323158	0.161579	+	0.986860i	$0.448341\pi$
$942$	0	0
$943$	4.09771	0.133440
$944$	75.1038	2.44442
$945$	0	0
$946$	67.2295	2.18582
$947$	7.68531	0.249739	0.124869	−	0.992173i	$-0.460149\pi$
$947$	7.68531	0.249739	0.124869	+	0.992173i	$0.460149\pi$
$948$	0	0
$949$	33.7576	1.09582
$950$	−2.14425	−0.0695687
$951$	0	0
$952$	−132.203	−4.28472
$953$	10.7790	0.349166	0.174583	−	0.984642i	$-0.444142\pi$
$953$	10.7790	0.349166	0.174583	+	0.984642i	$0.444142\pi$
$954$	0	0
$955$	−53.9875	−1.74699
$956$	136.510	4.41505
$957$	0	0
$958$	11.3146	0.365560
$959$	48.6080	1.56963
$960$	0	0
$961$	−23.9624	−0.772981
$962$	8.37917	0.270155
$963$	0	0
$964$	94.7111	3.05044
$965$	−50.6145	−1.62934
$966$	0	0
$967$	57.1901	1.83911	0.919555	−	0.392961i	$-0.128549\pi$
$967$	57.1901	1.83911	0.919555	+	0.392961i	$0.128549\pi$
$968$	28.0455	0.901416
$969$	0	0
$970$	−52.1231	−1.67357
$971$	−2.44663	−0.0785162	−0.0392581	−	0.999229i	$-0.512499\pi$
$971$	−2.44663	−0.0785162	−0.0392581	+	0.999229i	$0.512499\pi$
$972$	0	0
$973$	−21.5055	−0.689434
$974$	−25.9781	−0.832391
$975$	0	0
$976$	−11.9301	−0.381874
$977$	23.1260	0.739868	0.369934	−	0.929058i	$-0.379380\pi$
$977$	23.1260	0.739868	0.369934	+	0.929058i	$0.379380\pi$
$978$	0	0
$979$	11.6464	0.372219
$980$	32.7326	1.04560
$981$	0	0
$982$	46.9194	1.49726
$983$	52.2201	1.66556	0.832781	−	0.553603i	$-0.186747\pi$
$983$	52.2201	1.66556	0.832781	+	0.553603i	$0.186747\pi$
$984$	0	0
$985$	23.8366	0.759496
$986$	−13.6988	−0.436259
$987$	0	0
$988$	−151.199	−4.81028
$989$	9.18604	0.292099
$990$	0	0
$991$	−44.7304	−1.42091	−0.710454	−	0.703744i	$-0.751510\pi$
$991$	−44.7304	−1.42091	−0.710454	+	0.703744i	$0.751510\pi$
$992$	−38.4890	−1.22203
$993$	0	0
$994$	41.1271	1.30447
$995$	63.1679	2.00256
$996$	0	0
$997$	−36.2007	−1.14649	−0.573244	−	0.819385i	$-0.694315\pi$
$997$	−36.2007	−1.14649	−0.573244	+	0.819385i	$0.694315\pi$
$998$	−75.1477	−2.37876
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.1		16
3.2	odd	2		2001.2.a.n.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.16	✓	16	3.2	odd	2
6003.2.a.r.1.1		16	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)

\(f(q)\)	\(=\)	\(q-2.65986 q^{2} +5.07484 q^{4} +2.26251 q^{5} +3.13860 q^{7} -8.17864 q^{8} +O(q^{10})\)	\(q-2.65986 q^{2} +5.07484 q^{4} +2.26251 q^{5} +3.13860 q^{7} -8.17864 q^{8} -6.01796 q^{10} -2.75152 q^{11} -4.39658 q^{13} -8.34822 q^{14} +11.6043 q^{16} +5.15020 q^{17} +6.77659 q^{19} +11.4819 q^{20} +7.31866 q^{22} +1.00000 q^{23} +0.118961 q^{25} +11.6943 q^{26} +15.9279 q^{28} +1.00000 q^{29} +2.65285 q^{31} -14.5086 q^{32} -13.6988 q^{34} +7.10112 q^{35} +0.716519 q^{37} -18.0248 q^{38} -18.5043 q^{40} +4.09771 q^{41} +9.18604 q^{43} -13.9635 q^{44} -2.65986 q^{46} +9.21938 q^{47} +2.85080 q^{49} -0.316421 q^{50} -22.3120 q^{52} -7.55372 q^{53} -6.22536 q^{55} -25.6695 q^{56} -2.65986 q^{58} +6.47205 q^{59} -1.02808 q^{61} -7.05619 q^{62} +15.3821 q^{64} -9.94733 q^{65} -8.08875 q^{67} +26.1365 q^{68} -18.8880 q^{70} -4.92645 q^{71} -7.67814 q^{73} -1.90584 q^{74} +34.3901 q^{76} -8.63593 q^{77} -14.3627 q^{79} +26.2549 q^{80} -10.8993 q^{82} +9.71872 q^{83} +11.6524 q^{85} -24.4336 q^{86} +22.5037 q^{88} -4.23269 q^{89} -13.7991 q^{91} +5.07484 q^{92} -24.5222 q^{94} +15.3321 q^{95} +8.66126 q^{97} -7.58272 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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