LMFDB - Embedded newform 6003.2.a.p.1.12

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			1.12
Root			$2.05175$ 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)

$f(q)$	$=$	$q+2.05175 q^{2} +2.20969 q^{4} -3.75134 q^{5} -2.35999 q^{7} +0.430233 q^{8} +O(q^{10})$	\(q+2.05175 q^{2} +2.20969 q^{4} -3.75134 q^{5} -2.35999 q^{7} +0.430233 q^{8} -7.69683 q^{10} -3.27231 q^{11} +4.29495 q^{13} -4.84211 q^{14} -3.53665 q^{16} +5.05332 q^{17} -7.69400 q^{19} -8.28931 q^{20} -6.71396 q^{22} +1.00000 q^{23} +9.07257 q^{25} +8.81218 q^{26} -5.21484 q^{28} +1.00000 q^{29} -8.24396 q^{31} -8.11680 q^{32} +10.3682 q^{34} +8.85312 q^{35} -7.63881 q^{37} -15.7862 q^{38} -1.61395 q^{40} +6.80211 q^{41} +5.31051 q^{43} -7.23078 q^{44} +2.05175 q^{46} +10.1012 q^{47} -1.43046 q^{49} +18.6147 q^{50} +9.49051 q^{52} +8.31770 q^{53} +12.2755 q^{55} -1.01535 q^{56} +2.05175 q^{58} +6.64175 q^{59} -4.78113 q^{61} -16.9146 q^{62} -9.58036 q^{64} -16.1118 q^{65} +15.3363 q^{67} +11.1663 q^{68} +18.1644 q^{70} +11.2487 q^{71} +11.2463 q^{73} -15.6729 q^{74} -17.0014 q^{76} +7.72260 q^{77} +1.65332 q^{79} +13.2672 q^{80} +13.9563 q^{82} -4.90299 q^{83} -18.9567 q^{85} +10.8959 q^{86} -1.40785 q^{88} -8.43075 q^{89} -10.1360 q^{91} +2.20969 q^{92} +20.7251 q^{94} +28.8628 q^{95} +8.12503 q^{97} -2.93494 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.05175	1.45081	0.725404	−	0.688323i	$-0.241653\pi$
$2$	2.05175	1.45081	0.725404	+	0.688323i	$0.241653\pi$
$3$	0	0
$3$	0	0
$4$	2.20969	1.10485
$5$	−3.75134	−1.67765	−0.838826	−	0.544400i	$-0.816757\pi$
$5$	−3.75134	−1.67765	−0.838826	+	0.544400i	$0.816757\pi$
$6$	0	0
$7$	−2.35999	−0.891992	−0.445996	−	0.895035i	$-0.647150\pi$
$7$	−2.35999	−0.891992	−0.445996	+	0.895035i	$0.647150\pi$
$8$	0.430233	0.152110
$9$	0	0
$10$	−7.69683	−2.43395
$11$	−3.27231	−0.986637	−0.493319	−	0.869849i	$-0.664216\pi$
$11$	−3.27231	−0.986637	−0.493319	+	0.869849i	$0.664216\pi$
$12$	0	0
$13$	4.29495	1.19121	0.595603	−	0.803279i	$-0.296913\pi$
$13$	4.29495	1.19121	0.595603	+	0.803279i	$0.296913\pi$
$14$	−4.84211	−1.29411
$15$	0	0
$16$	−3.53665	−0.884162
$17$	5.05332	1.22561	0.612805	−	0.790235i	$-0.290041\pi$
$17$	5.05332	1.22561	0.612805	+	0.790235i	$0.290041\pi$
$18$	0	0
$19$	−7.69400	−1.76513	−0.882563	−	0.470195i	$-0.844184\pi$
$19$	−7.69400	−1.76513	−0.882563	+	0.470195i	$0.844184\pi$
$20$	−8.28931	−1.85355
$21$	0	0
$22$	−6.71396	−1.43142
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	9.07257	1.81451
$26$	8.81218	1.72821
$27$	0	0
$28$	−5.21484	−0.985513
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.24396	−1.48066	−0.740329	−	0.672244i	$-0.765330\pi$
$31$	−8.24396	−1.48066	−0.740329	+	0.672244i	$0.765330\pi$
$32$	−8.11680	−1.43486
$33$	0	0
$34$	10.3682	1.77812
$35$	8.85312	1.49645
$36$	0	0
$37$	−7.63881	−1.25581	−0.627906	−	0.778289i	$-0.716088\pi$
$37$	−7.63881	−1.25581	−0.627906	+	0.778289i	$0.716088\pi$
$38$	−15.7862	−2.56086
$39$	0	0
$40$	−1.61395	−0.255188
$41$	6.80211	1.06231	0.531156	−	0.847274i	$-0.321758\pi$
$41$	6.80211	1.06231	0.531156	+	0.847274i	$0.321758\pi$
$42$	0	0
$43$	5.31051	0.809846	0.404923	−	0.914351i	$-0.367298\pi$
$43$	5.31051	0.809846	0.404923	+	0.914351i	$0.367298\pi$
$44$	−7.23078	−1.09008
$45$	0	0
$46$	2.05175	0.302514
$47$	10.1012	1.47341	0.736704	−	0.676215i	$-0.236381\pi$
$47$	10.1012	1.47341	0.736704	+	0.676215i	$0.236381\pi$
$48$	0	0
$49$	−1.43046	−0.204351
$50$	18.6147	2.63251
$51$	0	0
$52$	9.49051	1.31610
$53$	8.31770	1.14252	0.571262	−	0.820768i	$-0.306454\pi$
$53$	8.31770	1.14252	0.571262	+	0.820768i	$0.306454\pi$
$54$	0	0
$55$	12.2755	1.65523
$56$	−1.01535	−0.135681
$57$	0	0
$58$	2.05175	0.269408
$59$	6.64175	0.864683	0.432341	−	0.901710i	$-0.357688\pi$
$59$	6.64175	0.864683	0.432341	+	0.901710i	$0.357688\pi$
$60$	0	0
$61$	−4.78113	−0.612161	−0.306081	−	0.952006i	$-0.599018\pi$
$61$	−4.78113	−0.612161	−0.306081	+	0.952006i	$0.599018\pi$
$62$	−16.9146	−2.14815
$63$	0	0
$64$	−9.58036	−1.19755
$65$	−16.1118	−1.99843
$66$	0	0
$67$	15.3363	1.87363	0.936815	−	0.349826i	$-0.113759\pi$
$67$	15.3363	1.87363	0.936815	+	0.349826i	$0.113759\pi$
$68$	11.1663	1.35411
$69$	0	0
$70$	18.1644	2.17106
$71$	11.2487	1.33497	0.667487	−	0.744621i	$-0.267370\pi$
$71$	11.2487	1.33497	0.667487	+	0.744621i	$0.267370\pi$
$72$	0	0
$73$	11.2463	1.31627	0.658137	−	0.752898i	$-0.271345\pi$
$73$	11.2463	1.31627	0.658137	+	0.752898i	$0.271345\pi$
$74$	−15.6729	−1.82194
$75$	0	0
$76$	−17.0014	−1.95019
$77$	7.72260	0.880072
$78$	0	0
$79$	1.65332	0.186013	0.0930065	−	0.995665i	$-0.470352\pi$
$79$	1.65332	0.186013	0.0930065	+	0.995665i	$0.470352\pi$
$80$	13.2672	1.48332
$81$	0	0
$82$	13.9563	1.54121
$83$	−4.90299	−0.538173	−0.269086	−	0.963116i	$-0.586722\pi$
$83$	−4.90299	−0.538173	−0.269086	+	0.963116i	$0.586722\pi$
$84$	0	0
$85$	−18.9567	−2.05614
$86$	10.8959	1.17493
$87$	0	0
$88$	−1.40785	−0.150078
$89$	−8.43075	−0.893657	−0.446829	−	0.894620i	$-0.647447\pi$
$89$	−8.43075	−0.893657	−0.446829	+	0.894620i	$0.647447\pi$
$90$	0	0
$91$	−10.1360	−1.06254
$92$	2.20969	0.230376
$93$	0	0
$94$	20.7251	2.13763
$95$	28.8628	2.96127
$96$	0	0
$97$	8.12503	0.824972	0.412486	−	0.910964i	$-0.364661\pi$
$97$	8.12503	0.824972	0.412486	+	0.910964i	$0.364661\pi$
$98$	−2.93494	−0.296474
$99$	0	0
$100$	20.0476	2.00476
$101$	3.48161	0.346433	0.173217	−	0.984884i	$-0.444584\pi$
$101$	3.48161	0.346433	0.173217	+	0.984884i	$0.444584\pi$
$102$	0	0
$103$	−17.8318	−1.75702	−0.878508	−	0.477728i	$-0.841461\pi$
$103$	−17.8318	−1.75702	−0.878508	+	0.477728i	$0.841461\pi$
$104$	1.84783	0.181195
$105$	0	0
$106$	17.0659	1.65758
$107$	−5.22489	−0.505109	−0.252555	−	0.967583i	$-0.581271\pi$
$107$	−5.22489	−0.505109	−0.252555	+	0.967583i	$0.581271\pi$
$108$	0	0
$109$	12.9348	1.23892	0.619462	−	0.785026i	$-0.287351\pi$
$109$	12.9348	1.23892	0.619462	+	0.785026i	$0.287351\pi$
$110$	25.1864	2.40143
$111$	0	0
$112$	8.34645	0.788665
$113$	−9.19380	−0.864880	−0.432440	−	0.901663i	$-0.642347\pi$
$113$	−9.19380	−0.864880	−0.432440	+	0.901663i	$0.642347\pi$
$114$	0	0
$115$	−3.75134	−0.349815
$116$	2.20969	0.205165
$117$	0	0
$118$	13.6272	1.25449
$119$	−11.9258	−1.09323
$120$	0	0
$121$	−0.292011	−0.0265465
$122$	−9.80970	−0.888129
$123$	0	0
$124$	−18.2166	−1.63590
$125$	−15.2776	−1.36647
$126$	0	0
$127$	−9.90131	−0.878599	−0.439299	−	0.898341i	$-0.644773\pi$
$127$	−9.90131	−0.878599	−0.439299	+	0.898341i	$0.644773\pi$
$128$	−3.42295	−0.302549
$129$	0	0
$130$	−33.0575	−2.89933
$131$	0.700093	0.0611674	0.0305837	−	0.999532i	$-0.490263\pi$
$131$	0.700093	0.0611674	0.0305837	+	0.999532i	$0.490263\pi$
$132$	0	0
$133$	18.1578	1.57448
$134$	31.4663	2.71828
$135$	0	0
$136$	2.17410	0.186428
$137$	−10.2345	−0.874393	−0.437197	−	0.899366i	$-0.644029\pi$
$137$	−10.2345	−0.874393	−0.437197	+	0.899366i	$0.644029\pi$
$138$	0	0
$139$	16.2668	1.37973	0.689867	−	0.723936i	$-0.257669\pi$
$139$	16.2668	1.37973	0.689867	+	0.723936i	$0.257669\pi$
$140$	19.5627	1.65335
$141$	0	0
$142$	23.0795	1.93679
$143$	−14.0544	−1.17529
$144$	0	0
$145$	−3.75134	−0.311532
$146$	23.0745	1.90966
$147$	0	0
$148$	−16.8794	−1.38748
$149$	22.1027	1.81072	0.905361	−	0.424643i	$-0.139600\pi$
$149$	22.1027	1.81072	0.905361	+	0.424643i	$0.139600\pi$
$150$	0	0
$151$	−7.14219	−0.581223	−0.290612	−	0.956841i	$-0.593859\pi$
$151$	−7.14219	−0.581223	−0.290612	+	0.956841i	$0.593859\pi$
$152$	−3.31021	−0.268494
$153$	0	0
$154$	15.8449	1.27682
$155$	30.9259	2.48403
$156$	0	0
$157$	7.21404	0.575743	0.287871	−	0.957669i	$-0.407052\pi$
$157$	7.21404	0.575743	0.287871	+	0.957669i	$0.407052\pi$
$158$	3.39220	0.269869
$159$	0	0
$160$	30.4489	2.40720
$161$	−2.35999	−0.185993
$162$	0	0
$163$	−21.6596	−1.69651	−0.848257	−	0.529585i	$-0.822348\pi$
$163$	−21.6596	−1.69651	−0.848257	+	0.529585i	$0.822348\pi$
$164$	15.0306	1.17369
$165$	0	0
$166$	−10.0597	−0.780786
$167$	8.81880	0.682419	0.341210	−	0.939987i	$-0.389163\pi$
$167$	8.81880	0.682419	0.341210	+	0.939987i	$0.389163\pi$
$168$	0	0
$169$	5.44660	0.418969
$170$	−38.8945	−2.98307
$171$	0	0
$172$	11.7346	0.894754
$173$	11.4038	0.867017	0.433509	−	0.901149i	$-0.357275\pi$
$173$	11.4038	0.867017	0.433509	+	0.901149i	$0.357275\pi$
$174$	0	0
$175$	−21.4112	−1.61853
$176$	11.5730	0.872348
$177$	0	0
$178$	−17.2978	−1.29653
$179$	7.78942	0.582208	0.291104	−	0.956691i	$-0.405977\pi$
$179$	7.78942	0.582208	0.291104	+	0.956691i	$0.405977\pi$
$180$	0	0
$181$	7.78005	0.578287	0.289143	−	0.957286i	$-0.406630\pi$
$181$	7.78005	0.578287	0.289143	+	0.957286i	$0.406630\pi$
$182$	−20.7966	−1.54155
$183$	0	0
$184$	0.430233	0.0317172
$185$	28.6558	2.10681
$186$	0	0
$187$	−16.5360	−1.20923
$188$	22.3205	1.62789
$189$	0	0
$190$	59.2194	4.29623
$191$	19.1511	1.38573	0.692863	−	0.721070i	$-0.256349\pi$
$191$	19.1511	1.38573	0.692863	+	0.721070i	$0.256349\pi$
$192$	0	0
$193$	11.4412	0.823552	0.411776	−	0.911285i	$-0.364909\pi$
$193$	11.4412	0.823552	0.411776	+	0.911285i	$0.364909\pi$
$194$	16.6705	1.19688
$195$	0	0
$196$	−3.16087	−0.225776
$197$	8.28753	0.590462	0.295231	−	0.955426i	$-0.404603\pi$
$197$	8.28753	0.590462	0.295231	+	0.955426i	$0.404603\pi$
$198$	0	0
$199$	−5.51469	−0.390926	−0.195463	−	0.980711i	$-0.562621\pi$
$199$	−5.51469	−0.390926	−0.195463	+	0.980711i	$0.562621\pi$
$200$	3.90332	0.276007
$201$	0	0
$202$	7.14340	0.502608
$203$	−2.35999	−0.165639
$204$	0	0
$205$	−25.5170	−1.78219
$206$	−36.5864	−2.54909
$207$	0	0
$208$	−15.1897	−1.05322
$209$	25.1771	1.74154
$210$	0	0
$211$	−19.1656	−1.31941	−0.659706	−	0.751524i	$-0.729319\pi$
$211$	−19.1656	−1.31941	−0.659706	+	0.751524i	$0.729319\pi$
$212$	18.3795	1.26231
$213$	0	0
$214$	−10.7202	−0.732817
$215$	−19.9216	−1.35864
$216$	0	0
$217$	19.4556	1.32074
$218$	26.5389	1.79744
$219$	0	0
$220$	27.1252	1.82878
$221$	21.7037	1.45995
$222$	0	0
$223$	2.24920	0.150617	0.0753087	−	0.997160i	$-0.476006\pi$
$223$	2.24920	0.150617	0.0753087	+	0.997160i	$0.476006\pi$
$224$	19.1555	1.27988
$225$	0	0
$226$	−18.8634	−1.25477
$227$	−22.0599	−1.46417	−0.732084	−	0.681214i	$-0.761452\pi$
$227$	−22.0599	−1.46417	−0.732084	+	0.681214i	$0.761452\pi$
$228$	0	0
$229$	11.0208	0.728278	0.364139	−	0.931345i	$-0.381363\pi$
$229$	11.0208	0.728278	0.364139	+	0.931345i	$0.381363\pi$
$230$	−7.69683	−0.507514
$231$	0	0
$232$	0.430233	0.0282462
$233$	−16.8106	−1.10130	−0.550650	−	0.834737i	$-0.685620\pi$
$233$	−16.8106	−1.10130	−0.550650	+	0.834737i	$0.685620\pi$
$234$	0	0
$235$	−37.8930	−2.47187
$236$	14.6762	0.955341
$237$	0	0
$238$	−24.4687	−1.58607
$239$	−0.695292	−0.0449747	−0.0224873	−	0.999747i	$-0.507159\pi$
$239$	−0.695292	−0.0449747	−0.0224873	+	0.999747i	$0.507159\pi$
$240$	0	0
$241$	−17.3711	−1.11897	−0.559485	−	0.828840i	$-0.689001\pi$
$241$	−17.3711	−1.11897	−0.559485	+	0.828840i	$0.689001\pi$
$242$	−0.599135	−0.0385139
$243$	0	0
$244$	−10.5648	−0.676343
$245$	5.36613	0.342830
$246$	0	0
$247$	−33.0454	−2.10263
$248$	−3.54682	−0.225224
$249$	0	0
$250$	−31.3459	−1.98249
$251$	22.7680	1.43710	0.718551	−	0.695474i	$-0.244806\pi$
$251$	22.7680	1.43710	0.718551	+	0.695474i	$0.244806\pi$
$252$	0	0
$253$	−3.27231	−0.205728
$254$	−20.3150	−1.27468
$255$	0	0
$256$	12.1377	0.758605
$257$	−14.0147	−0.874211	−0.437105	−	0.899410i	$-0.643996\pi$
$257$	−14.0147	−0.874211	−0.437105	+	0.899410i	$0.643996\pi$
$258$	0	0
$259$	18.0275	1.12017
$260$	−35.6022	−2.20795
$261$	0	0
$262$	1.43642	0.0887422
$263$	15.9795	0.985341	0.492671	−	0.870216i	$-0.336021\pi$
$263$	15.9795	0.985341	0.492671	+	0.870216i	$0.336021\pi$
$264$	0	0
$265$	−31.2026	−1.91676
$266$	37.2552	2.28426
$267$	0	0
$268$	33.8885	2.07007
$269$	−6.05875	−0.369409	−0.184704	−	0.982794i	$-0.559133\pi$
$269$	−6.05875	−0.369409	−0.184704	+	0.982794i	$0.559133\pi$
$270$	0	0
$271$	22.0833	1.34146	0.670731	−	0.741700i	$-0.265980\pi$
$271$	22.0833	1.34146	0.670731	+	0.741700i	$0.265980\pi$
$272$	−17.8718	−1.08364
$273$	0	0
$274$	−20.9987	−1.26858
$275$	−29.6882	−1.79027
$276$	0	0
$277$	1.07848	0.0647993	0.0323997	−	0.999475i	$-0.489685\pi$
$277$	1.07848	0.0647993	0.0323997	+	0.999475i	$0.489685\pi$
$278$	33.3755	2.00173
$279$	0	0
$280$	3.80891	0.227626
$281$	19.6635	1.17303	0.586514	−	0.809939i	$-0.300500\pi$
$281$	19.6635	1.17303	0.586514	+	0.809939i	$0.300500\pi$
$282$	0	0
$283$	19.6493	1.16803	0.584014	−	0.811743i	$-0.301481\pi$
$283$	19.6493	1.16803	0.584014	+	0.811743i	$0.301481\pi$
$284$	24.8561	1.47494
$285$	0	0
$286$	−28.8361	−1.70512
$287$	−16.0529	−0.947573
$288$	0	0
$289$	8.53600	0.502117
$290$	−7.69683	−0.451973
$291$	0	0
$292$	24.8507	1.45428
$293$	19.6322	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$293$	19.6322	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$294$	0	0
$295$	−24.9155	−1.45064
$296$	−3.28647	−0.191022
$297$	0	0
$298$	45.3492	2.62701
$299$	4.29495	0.248383
$300$	0	0
$301$	−12.5327	−0.722376
$302$	−14.6540	−0.843244
$303$	0	0
$304$	27.2110	1.56066
$305$	17.9357	1.02699
$306$	0	0
$307$	−7.90639	−0.451242	−0.225621	−	0.974215i	$-0.572441\pi$
$307$	−7.90639	−0.451242	−0.225621	+	0.974215i	$0.572441\pi$
$308$	17.0646	0.972344
$309$	0	0
$310$	63.4523	3.60385
$311$	−22.5093	−1.27638	−0.638192	−	0.769878i	$-0.720317\pi$
$311$	−22.5093	−1.27638	−0.638192	+	0.769878i	$0.720317\pi$
$312$	0	0
$313$	10.5353	0.595488	0.297744	−	0.954646i	$-0.403766\pi$
$313$	10.5353	0.595488	0.297744	+	0.954646i	$0.403766\pi$
$314$	14.8014	0.835293
$315$	0	0
$316$	3.65333	0.205516
$317$	0.622592	0.0349683	0.0174841	−	0.999847i	$-0.494434\pi$
$317$	0.622592	0.0349683	0.0174841	+	0.999847i	$0.494434\pi$
$318$	0	0
$319$	−3.27231	−0.183214
$320$	35.9392	2.00906
$321$	0	0
$322$	−4.84211	−0.269840
$323$	−38.8802	−2.16335
$324$	0	0
$325$	38.9663	2.16146
$326$	−44.4402	−2.46132
$327$	0	0
$328$	2.92649	0.161589
$329$	−23.8387	−1.31427
$330$	0	0
$331$	2.04097	0.112182	0.0560909	−	0.998426i	$-0.482136\pi$
$331$	2.04097	0.112182	0.0560909	+	0.998426i	$0.482136\pi$
$332$	−10.8341	−0.594598
$333$	0	0
$334$	18.0940	0.990060
$335$	−57.5318	−3.14330
$336$	0	0
$337$	−7.27857	−0.396489	−0.198245	−	0.980153i	$-0.563524\pi$
$337$	−7.27857	−0.396489	−0.198245	+	0.980153i	$0.563524\pi$
$338$	11.1751	0.607844
$339$	0	0
$340$	−41.8885	−2.27172
$341$	26.9768	1.46087
$342$	0	0
$343$	19.8958	1.07427
$344$	2.28476	0.123186
$345$	0	0
$346$	23.3978	1.25788
$347$	4.86768	0.261311	0.130655	−	0.991428i	$-0.458292\pi$
$347$	4.86768	0.261311	0.130655	+	0.991428i	$0.458292\pi$
$348$	0	0
$349$	16.2572	0.870228	0.435114	−	0.900375i	$-0.356708\pi$
$349$	16.2572	0.870228	0.435114	+	0.900375i	$0.356708\pi$
$350$	−43.9304	−2.34818
$351$	0	0
$352$	26.5606	1.41569
$353$	−13.1198	−0.698295	−0.349148	−	0.937068i	$-0.613529\pi$
$353$	−13.1198	−0.698295	−0.349148	+	0.937068i	$0.613529\pi$
$354$	0	0
$355$	−42.1977	−2.23962
$356$	−18.6293	−0.987353
$357$	0	0
$358$	15.9820	0.844673
$359$	−31.5273	−1.66395	−0.831974	−	0.554814i	$-0.812789\pi$
$359$	−31.5273	−1.66395	−0.831974	+	0.554814i	$0.812789\pi$
$360$	0	0
$361$	40.1977	2.11567
$362$	15.9627	0.838983
$363$	0	0
$364$	−22.3975	−1.17395
$365$	−42.1885	−2.20825
$366$	0	0
$367$	23.9006	1.24760	0.623800	−	0.781584i	$-0.285588\pi$
$367$	23.9006	1.24760	0.623800	+	0.781584i	$0.285588\pi$
$368$	−3.53665	−0.184361
$369$	0	0
$370$	58.7946	3.05659
$371$	−19.6297	−1.01912
$372$	0	0
$373$	−5.72709	−0.296538	−0.148269	−	0.988947i	$-0.547370\pi$
$373$	−5.72709	−0.296538	−0.148269	+	0.988947i	$0.547370\pi$
$374$	−33.9278	−1.75436
$375$	0	0
$376$	4.34586	0.224121
$377$	4.29495	0.221201
$378$	0	0
$379$	−34.1260	−1.75293	−0.876467	−	0.481462i	$-0.840106\pi$
$379$	−34.1260	−1.75293	−0.876467	+	0.481462i	$0.840106\pi$
$380$	63.7779	3.27174
$381$	0	0
$382$	39.2933	2.01042
$383$	−28.4932	−1.45593	−0.727967	−	0.685613i	$-0.759534\pi$
$383$	−28.4932	−1.45593	−0.727967	+	0.685613i	$0.759534\pi$
$384$	0	0
$385$	−28.9701	−1.47645
$386$	23.4744	1.19482
$387$	0	0
$388$	17.9538	0.911466
$389$	34.1307	1.73050	0.865248	−	0.501344i	$-0.167161\pi$
$389$	34.1307	1.73050	0.865248	+	0.501344i	$0.167161\pi$
$390$	0	0
$391$	5.05332	0.255557
$392$	−0.615430	−0.0310839
$393$	0	0
$394$	17.0040	0.856648
$395$	−6.20217	−0.312065
$396$	0	0
$397$	−0.398698	−0.0200101	−0.0100050	−	0.999950i	$-0.503185\pi$
$397$	−0.398698	−0.0200101	−0.0100050	+	0.999950i	$0.503185\pi$
$398$	−11.3148	−0.567158
$399$	0	0
$400$	−32.0865	−1.60433
$401$	−18.7059	−0.934129	−0.467064	−	0.884223i	$-0.654688\pi$
$401$	−18.7059	−0.934129	−0.467064	+	0.884223i	$0.654688\pi$
$402$	0	0
$403$	−35.4074	−1.76377
$404$	7.69328	0.382755
$405$	0	0
$406$	−4.84211	−0.240310
$407$	24.9965	1.23903
$408$	0	0
$409$	13.7661	0.680688	0.340344	−	0.940301i	$-0.389456\pi$
$409$	13.7661	0.680688	0.340344	+	0.940301i	$0.389456\pi$
$410$	−52.3547	−2.58561
$411$	0	0
$412$	−39.4027	−1.94123
$413$	−15.6745	−0.771290
$414$	0	0
$415$	18.3928	0.902866
$416$	−34.8612	−1.70921
$417$	0	0
$418$	51.6573	2.52664
$419$	7.80433	0.381266	0.190633	−	0.981661i	$-0.438946\pi$
$419$	7.80433	0.381266	0.190633	+	0.981661i	$0.438946\pi$
$420$	0	0
$421$	4.95845	0.241660	0.120830	−	0.992673i	$-0.461444\pi$
$421$	4.95845	0.241660	0.120830	+	0.992673i	$0.461444\pi$
$422$	−39.3230	−1.91421
$423$	0	0
$424$	3.57855	0.173790
$425$	45.8466	2.22389
$426$	0	0
$427$	11.2834	0.546043
$428$	−11.5454	−0.558067
$429$	0	0
$430$	−40.8741	−1.97113
$431$	−14.3407	−0.690769	−0.345384	−	0.938461i	$-0.612251\pi$
$431$	−14.3407	−0.690769	−0.345384	+	0.938461i	$0.612251\pi$
$432$	0	0
$433$	−20.2753	−0.974371	−0.487185	−	0.873299i	$-0.661976\pi$
$433$	−20.2753	−0.974371	−0.487185	+	0.873299i	$0.661976\pi$
$434$	39.9182	1.91613
$435$	0	0
$436$	28.5818	1.36882
$437$	−7.69400	−0.368054
$438$	0	0
$439$	11.2623	0.537523	0.268761	−	0.963207i	$-0.413386\pi$
$439$	11.2623	0.537523	0.268761	+	0.963207i	$0.413386\pi$
$440$	5.28135	0.251778
$441$	0	0
$442$	44.5307	2.11811
$443$	8.69105	0.412924	0.206462	−	0.978455i	$-0.433805\pi$
$443$	8.69105	0.412924	0.206462	+	0.978455i	$0.433805\pi$
$444$	0	0
$445$	31.6266	1.49925
$446$	4.61480	0.218517
$447$	0	0
$448$	22.6095	1.06820
$449$	26.7449	1.26217	0.631085	−	0.775714i	$-0.282610\pi$
$449$	26.7449	1.26217	0.631085	+	0.775714i	$0.282610\pi$
$450$	0	0
$451$	−22.2586	−1.04812
$452$	−20.3154	−0.955558
$453$	0	0
$454$	−45.2615	−2.12423
$455$	38.0237	1.78258
$456$	0	0
$457$	−18.2879	−0.855473	−0.427736	−	0.903904i	$-0.640689\pi$
$457$	−18.2879	−0.855473	−0.427736	+	0.903904i	$0.640689\pi$
$458$	22.6120	1.05659
$459$	0	0
$460$	−8.28931	−0.386491
$461$	−0.584892	−0.0272411	−0.0136206	−	0.999907i	$-0.504336\pi$
$461$	−0.584892	−0.0272411	−0.0136206	+	0.999907i	$0.504336\pi$
$462$	0	0
$463$	−37.2767	−1.73240	−0.866198	−	0.499701i	$-0.833443\pi$
$463$	−37.2767	−1.73240	−0.866198	+	0.499701i	$0.833443\pi$
$464$	−3.53665	−0.164185
$465$	0	0
$466$	−34.4912	−1.59777
$467$	29.1928	1.35088	0.675440	−	0.737415i	$-0.263954\pi$
$467$	29.1928	1.35088	0.675440	+	0.737415i	$0.263954\pi$
$468$	0	0
$469$	−36.1935	−1.67126
$470$	−77.7471	−3.58621
$471$	0	0
$472$	2.85750	0.131527
$473$	−17.3776	−0.799024
$474$	0	0
$475$	−69.8044	−3.20285
$476$	−26.3522	−1.20785
$477$	0	0
$478$	−1.42657	−0.0652496
$479$	−31.5875	−1.44327	−0.721636	−	0.692273i	$-0.756609\pi$
$479$	−31.5875	−1.44327	−0.721636	+	0.692273i	$0.756609\pi$
$480$	0	0
$481$	−32.8083	−1.49593
$482$	−35.6412	−1.62341
$483$	0	0
$484$	−0.645255	−0.0293298
$485$	−30.4798	−1.38401
$486$	0	0
$487$	−25.0991	−1.13735	−0.568674	−	0.822563i	$-0.692543\pi$
$487$	−25.0991	−1.13735	−0.568674	+	0.822563i	$0.692543\pi$
$488$	−2.05700	−0.0931161
$489$	0	0
$490$	11.0100	0.497380
$491$	10.8979	0.491815	0.245908	−	0.969293i	$-0.420914\pi$
$491$	10.8979	0.491815	0.245908	+	0.969293i	$0.420914\pi$
$492$	0	0
$493$	5.05332	0.227590
$494$	−67.8009	−3.05051
$495$	0	0
$496$	29.1560	1.30914
$497$	−26.5468	−1.19079
$498$	0	0
$499$	−23.4104	−1.04799	−0.523997	−	0.851720i	$-0.675560\pi$
$499$	−23.4104	−1.04799	−0.523997	+	0.851720i	$0.675560\pi$
$500$	−33.7588	−1.50974
$501$	0	0
$502$	46.7143	2.08496
$503$	11.4338	0.509807	0.254903	−	0.966966i	$-0.417956\pi$
$503$	11.4338	0.509807	0.254903	+	0.966966i	$0.417956\pi$
$504$	0	0
$505$	−13.0607	−0.581194
$506$	−6.71396	−0.298472
$507$	0	0
$508$	−21.8788	−0.970716
$509$	0.942475	0.0417745	0.0208872	−	0.999782i	$-0.493351\pi$
$509$	0.942475	0.0417745	0.0208872	+	0.999782i	$0.493351\pi$
$510$	0	0
$511$	−26.5410	−1.17411
$512$	31.7494	1.40314
$513$	0	0
$514$	−28.7546	−1.26831
$515$	66.8931	2.94766
$516$	0	0
$517$	−33.0542	−1.45372
$518$	36.9880	1.62516
$519$	0	0
$520$	−6.93184	−0.303981
$521$	−12.0132	−0.526307	−0.263154	−	0.964754i	$-0.584763\pi$
$521$	−12.0132	−0.526307	−0.263154	+	0.964754i	$0.584763\pi$
$522$	0	0
$523$	24.2290	1.05946	0.529731	−	0.848166i	$-0.322293\pi$
$523$	24.2290	1.05946	0.529731	+	0.848166i	$0.322293\pi$
$524$	1.54699	0.0675806
$525$	0	0
$526$	32.7861	1.42954
$527$	−41.6593	−1.81471
$528$	0	0
$529$	1.00000	0.0434783
$530$	−64.0199	−2.78085
$531$	0	0
$532$	40.1230	1.73955
$533$	29.2147	1.26543
$534$	0	0
$535$	19.6003	0.847397
$536$	6.59819	0.284998
$537$	0	0
$538$	−12.4311	−0.535941
$539$	4.68089	0.201620
$540$	0	0
$541$	25.5374	1.09794	0.548969	−	0.835842i	$-0.315020\pi$
$541$	25.5374	1.09794	0.548969	+	0.835842i	$0.315020\pi$
$542$	45.3094	1.94621
$543$	0	0
$544$	−41.0167	−1.75858
$545$	−48.5227	−2.07848
$546$	0	0
$547$	12.3290	0.527150	0.263575	−	0.964639i	$-0.415098\pi$
$547$	12.3290	0.527150	0.263575	+	0.964639i	$0.415098\pi$
$548$	−22.6151	−0.966069
$549$	0	0
$550$	−60.9129	−2.59734
$551$	−7.69400	−0.327776
$552$	0	0
$553$	−3.90182	−0.165922
$554$	2.21277	0.0940114
$555$	0	0
$556$	35.9446	1.52439
$557$	41.4675	1.75703	0.878516	−	0.477712i	$-0.158534\pi$
$557$	41.4675	1.75703	0.878516	+	0.477712i	$0.158534\pi$
$558$	0	0
$559$	22.8084	0.964692
$560$	−31.3104	−1.32311
$561$	0	0
$562$	40.3447	1.70184
$563$	−7.39212	−0.311541	−0.155770	−	0.987793i	$-0.549786\pi$
$563$	−7.39212	−0.311541	−0.155770	+	0.987793i	$0.549786\pi$
$564$	0	0
$565$	34.4891	1.45097
$566$	40.3155	1.69459
$567$	0	0
$568$	4.83956	0.203063
$569$	6.57511	0.275643	0.137821	−	0.990457i	$-0.455990\pi$
$569$	6.57511	0.275643	0.137821	+	0.990457i	$0.455990\pi$
$570$	0	0
$571$	39.1977	1.64037	0.820186	−	0.572097i	$-0.193870\pi$
$571$	39.1977	1.64037	0.820186	+	0.572097i	$0.193870\pi$
$572$	−31.0559	−1.29851
$573$	0	0
$574$	−32.9366	−1.37475
$575$	9.07257	0.378352
$576$	0	0
$577$	−8.73676	−0.363716	−0.181858	−	0.983325i	$-0.558211\pi$
$577$	−8.73676	−0.363716	−0.181858	+	0.983325i	$0.558211\pi$
$578$	17.5138	0.728476
$579$	0	0
$580$	−8.28931	−0.344195
$581$	11.5710	0.480046
$582$	0	0
$583$	−27.2181	−1.12726
$584$	4.83851	0.200219
$585$	0	0
$586$	40.2804	1.66397
$587$	12.6202	0.520893	0.260447	−	0.965488i	$-0.416130\pi$
$587$	12.6202	0.520893	0.260447	+	0.965488i	$0.416130\pi$
$588$	0	0
$589$	63.4290	2.61355
$590$	−51.1204	−2.10460
$591$	0	0
$592$	27.0158	1.11034
$593$	−25.6149	−1.05188	−0.525939	−	0.850522i	$-0.676286\pi$
$593$	−25.6149	−1.05188	−0.525939	+	0.850522i	$0.676286\pi$
$594$	0	0
$595$	44.7376	1.83406
$596$	48.8401	2.00057
$597$	0	0
$598$	8.81218	0.360357
$599$	25.0727	1.02444	0.512222	−	0.858853i	$-0.328823\pi$
$599$	25.0727	1.02444	0.512222	+	0.858853i	$0.328823\pi$
$600$	0	0
$601$	14.8990	0.607742	0.303871	−	0.952713i	$-0.401721\pi$
$601$	14.8990	0.607742	0.303871	+	0.952713i	$0.401721\pi$
$602$	−25.7141	−1.04803
$603$	0	0
$604$	−15.7820	−0.642162
$605$	1.09543	0.0445358
$606$	0	0
$607$	24.3673	0.989036	0.494518	−	0.869167i	$-0.335345\pi$
$607$	24.3673	0.989036	0.494518	+	0.869167i	$0.335345\pi$
$608$	62.4507	2.53271
$609$	0	0
$610$	36.7996	1.48997
$611$	43.3841	1.75513
$612$	0	0
$613$	−27.3595	−1.10504	−0.552521	−	0.833499i	$-0.686334\pi$
$613$	−27.3595	−1.10504	−0.552521	+	0.833499i	$0.686334\pi$
$614$	−16.2220	−0.654665
$615$	0	0
$616$	3.32252	0.133868
$617$	−14.1553	−0.569869	−0.284935	−	0.958547i	$-0.591972\pi$
$617$	−14.1553	−0.569869	−0.284935	+	0.958547i	$0.591972\pi$
$618$	0	0
$619$	25.6702	1.03177	0.515886	−	0.856657i	$-0.327463\pi$
$619$	25.6702	1.03177	0.515886	+	0.856657i	$0.327463\pi$
$620$	68.3367	2.74447
$621$	0	0
$622$	−46.1834	−1.85179
$623$	19.8965	0.797135
$624$	0	0
$625$	11.9487	0.477949
$626$	21.6158	0.863940
$627$	0	0
$628$	15.9408	0.636107
$629$	−38.6013	−1.53913
$630$	0	0
$631$	−36.6673	−1.45970	−0.729851	−	0.683606i	$-0.760411\pi$
$631$	−36.6673	−1.45970	−0.729851	+	0.683606i	$0.760411\pi$
$632$	0.711313	0.0282945
$633$	0	0
$634$	1.27741	0.0507323
$635$	37.1432	1.47398
$636$	0	0
$637$	−6.14374	−0.243424
$638$	−6.71396	−0.265808
$639$	0	0
$640$	12.8407	0.507571
$641$	7.88987	0.311631	0.155816	−	0.987786i	$-0.450199\pi$
$641$	7.88987	0.311631	0.155816	+	0.987786i	$0.450199\pi$
$642$	0	0
$643$	−15.0059	−0.591774	−0.295887	−	0.955223i	$-0.595615\pi$
$643$	−15.0059	−0.591774	−0.295887	+	0.955223i	$0.595615\pi$
$644$	−5.21484	−0.205494
$645$	0	0
$646$	−79.7726	−3.13861
$647$	−36.5839	−1.43826	−0.719131	−	0.694875i	$-0.755460\pi$
$647$	−36.5839	−1.43826	−0.719131	+	0.694875i	$0.755460\pi$
$648$	0	0
$649$	−21.7339	−0.853128
$650$	79.9491	3.13586
$651$	0	0
$652$	−47.8611	−1.87439
$653$	21.5028	0.841470	0.420735	−	0.907184i	$-0.361772\pi$
$653$	21.5028	0.841470	0.420735	+	0.907184i	$0.361772\pi$
$654$	0	0
$655$	−2.62629	−0.102618
$656$	−24.0567	−0.939255
$657$	0	0
$658$	−48.9111	−1.90675
$659$	20.9402	0.815716	0.407858	−	0.913045i	$-0.366276\pi$
$659$	20.9402	0.815716	0.407858	+	0.913045i	$0.366276\pi$
$660$	0	0
$661$	−29.4919	−1.14710	−0.573551	−	0.819170i	$-0.694434\pi$
$661$	−29.4919	−1.14710	−0.573551	+	0.819170i	$0.694434\pi$
$662$	4.18756	0.162754
$663$	0	0
$664$	−2.10943	−0.0818617
$665$	−68.1160	−2.64142
$666$	0	0
$667$	1.00000	0.0387202
$668$	19.4868	0.753968
$669$	0	0
$670$	−118.041	−4.56032
$671$	15.6453	0.603981
$672$	0	0
$673$	33.3572	1.28583	0.642913	−	0.765939i	$-0.277726\pi$
$673$	33.3572	1.28583	0.642913	+	0.765939i	$0.277726\pi$
$674$	−14.9338	−0.575230
$675$	0	0
$676$	12.0353	0.462896
$677$	−17.4062	−0.668975	−0.334488	−	0.942400i	$-0.608563\pi$
$677$	−17.4062	−0.668975	−0.334488	+	0.942400i	$0.608563\pi$
$678$	0	0
$679$	−19.1750	−0.735868
$680$	−8.15581	−0.312761
$681$	0	0
$682$	55.3497	2.11945
$683$	−34.7410	−1.32933	−0.664664	−	0.747142i	$-0.731425\pi$
$683$	−34.7410	−1.32933	−0.664664	+	0.747142i	$0.731425\pi$
$684$	0	0
$685$	38.3932	1.46693
$686$	40.8212	1.55856
$687$	0	0
$688$	−18.7814	−0.716035
$689$	35.7241	1.36098
$690$	0	0
$691$	−12.4183	−0.472416	−0.236208	−	0.971703i	$-0.575905\pi$
$691$	−12.4183	−0.472416	−0.236208	+	0.971703i	$0.575905\pi$
$692$	25.1989	0.957920
$693$	0	0
$694$	9.98728	0.379112
$695$	−61.0224	−2.31471
$696$	0	0
$697$	34.3732	1.30198
$698$	33.3558	1.26253
$699$	0	0
$700$	−47.3120	−1.78823
$701$	−31.4022	−1.18605	−0.593023	−	0.805186i	$-0.702066\pi$
$701$	−31.4022	−1.18605	−0.593023	+	0.805186i	$0.702066\pi$
$702$	0	0
$703$	58.7730	2.21667
$704$	31.3499	1.18154
$705$	0	0
$706$	−26.9185	−1.01309
$707$	−8.21656	−0.309015
$708$	0	0
$709$	−47.4665	−1.78264	−0.891322	−	0.453372i	$-0.850221\pi$
$709$	−47.4665	−1.78264	−0.891322	+	0.453372i	$0.850221\pi$
$710$	−86.5793	−3.24926
$711$	0	0
$712$	−3.62719	−0.135935
$713$	−8.24396	−0.308739
$714$	0	0
$715$	52.7229	1.97172
$716$	17.2122	0.643250
$717$	0	0
$718$	−64.6863	−2.41407
$719$	30.3339	1.13126	0.565632	−	0.824658i	$-0.308632\pi$
$719$	30.3339	1.13126	0.565632	+	0.824658i	$0.308632\pi$
$720$	0	0
$721$	42.0827	1.56724
$722$	82.4757	3.06943
$723$	0	0
$724$	17.1915	0.638917
$725$	9.07257	0.336947
$726$	0	0
$727$	37.0949	1.37577	0.687887	−	0.725818i	$-0.258538\pi$
$727$	37.0949	1.37577	0.687887	+	0.725818i	$0.258538\pi$
$728$	−4.36086	−0.161624
$729$	0	0
$730$	−86.5605	−3.20375
$731$	26.8357	0.992554
$732$	0	0
$733$	39.3141	1.45210	0.726049	−	0.687643i	$-0.241354\pi$
$733$	39.3141	1.45210	0.726049	+	0.687643i	$0.241354\pi$
$734$	49.0381	1.81003
$735$	0	0
$736$	−8.11680	−0.299189
$737$	−50.1851	−1.84859
$738$	0	0
$739$	15.6659	0.576280	0.288140	−	0.957588i	$-0.406963\pi$
$739$	15.6659	0.576280	0.288140	+	0.957588i	$0.406963\pi$
$740$	63.3204	2.32770
$741$	0	0
$742$	−40.2753	−1.47855
$743$	−27.8413	−1.02140	−0.510698	−	0.859760i	$-0.670613\pi$
$743$	−27.8413	−1.02140	−0.510698	+	0.859760i	$0.670613\pi$
$744$	0	0
$745$	−82.9147	−3.03776
$746$	−11.7506	−0.430219
$747$	0	0
$748$	−36.5394	−1.33601
$749$	12.3307	0.450553
$750$	0	0
$751$	−38.0674	−1.38910	−0.694550	−	0.719444i	$-0.744397\pi$
$751$	−38.0674	−1.38910	−0.694550	+	0.719444i	$0.744397\pi$
$752$	−35.7243	−1.30273
$753$	0	0
$754$	8.81218	0.320921
$755$	26.7928	0.975090
$756$	0	0
$757$	−3.87374	−0.140793	−0.0703967	−	0.997519i	$-0.522427\pi$
$757$	−3.87374	−0.140793	−0.0703967	+	0.997519i	$0.522427\pi$
$758$	−70.0181	−2.54317
$759$	0	0
$760$	12.4178	0.450439
$761$	11.0651	0.401109	0.200554	−	0.979683i	$-0.435726\pi$
$761$	11.0651	0.401109	0.200554	+	0.979683i	$0.435726\pi$
$762$	0	0
$763$	−30.5259	−1.10511
$764$	42.3180	1.53101
$765$	0	0
$766$	−58.4610	−2.11228
$767$	28.5260	1.03001
$768$	0	0
$769$	36.4401	1.31406	0.657032	−	0.753862i	$-0.271812\pi$
$769$	36.4401	1.31406	0.657032	+	0.753862i	$0.271812\pi$
$770$	−59.4396	−2.14205
$771$	0	0
$772$	25.2814	0.909898
$773$	43.9907	1.58224	0.791118	−	0.611663i	$-0.209499\pi$
$773$	43.9907	1.58224	0.791118	+	0.611663i	$0.209499\pi$
$774$	0	0
$775$	−74.7939	−2.68668
$776$	3.49566	0.125487
$777$	0	0
$778$	70.0278	2.51062
$779$	−52.3355	−1.87511
$780$	0	0
$781$	−36.8092	−1.31714
$782$	10.3682	0.370764
$783$	0	0
$784$	5.05902	0.180679
$785$	−27.0623	−0.965896
$786$	0	0
$787$	−8.32724	−0.296834	−0.148417	−	0.988925i	$-0.547418\pi$
$787$	−8.32724	−0.296834	−0.148417	+	0.988925i	$0.547418\pi$
$788$	18.3129	0.652370
$789$	0	0
$790$	−12.7253	−0.452747
$791$	21.6972	0.771465
$792$	0	0
$793$	−20.5347	−0.729209
$794$	−0.818029	−0.0290308
$795$	0	0
$796$	−12.1858	−0.431913
$797$	39.5518	1.40100	0.700499	−	0.713653i	$-0.252961\pi$
$797$	39.5518	1.40100	0.700499	+	0.713653i	$0.252961\pi$
$798$	0	0
$799$	51.0445	1.80582
$800$	−73.6402	−2.60358
$801$	0	0
$802$	−38.3799	−1.35524
$803$	−36.8012	−1.29869
$804$	0	0
$805$	8.85312	0.312032
$806$	−72.6472	−2.55889
$807$	0	0
$808$	1.49790	0.0526961
$809$	4.41022	0.155055	0.0775275	−	0.996990i	$-0.475297\pi$
$809$	4.41022	0.155055	0.0775275	+	0.996990i	$0.475297\pi$
$810$	0	0
$811$	−10.0306	−0.352223	−0.176111	−	0.984370i	$-0.556352\pi$
$811$	−10.0306	−0.352223	−0.176111	+	0.984370i	$0.556352\pi$
$812$	−5.21484	−0.183005
$813$	0	0
$814$	51.2867	1.79760
$815$	81.2528	2.84616
$816$	0	0
$817$	−40.8591	−1.42948
$818$	28.2446	0.987548
$819$	0	0
$820$	−56.3848	−1.96904
$821$	25.8478	0.902094	0.451047	−	0.892500i	$-0.351051\pi$
$821$	25.8478	0.902094	0.451047	+	0.892500i	$0.351051\pi$
$822$	0	0
$823$	13.8287	0.482037	0.241018	−	0.970521i	$-0.422519\pi$
$823$	13.8287	0.482037	0.241018	+	0.970521i	$0.422519\pi$
$824$	−7.67182	−0.267260
$825$	0	0
$826$	−32.1601	−1.11899
$827$	−12.5175	−0.435276	−0.217638	−	0.976030i	$-0.569835\pi$
$827$	−12.5175	−0.435276	−0.217638	+	0.976030i	$0.569835\pi$
$828$	0	0
$829$	34.8545	1.21055	0.605273	−	0.796018i	$-0.293064\pi$
$829$	34.8545	1.21055	0.605273	+	0.796018i	$0.293064\pi$
$830$	37.7375	1.30989
$831$	0	0
$832$	−41.1472	−1.42652
$833$	−7.22855	−0.250454
$834$	0	0
$835$	−33.0823	−1.14486
$836$	55.6337	1.92413
$837$	0	0
$838$	16.0126	0.553144
$839$	30.4417	1.05097	0.525483	−	0.850804i	$-0.323885\pi$
$839$	30.4417	1.05097	0.525483	+	0.850804i	$0.323885\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	10.1735	0.350602
$843$	0	0
$844$	−42.3500	−1.45775
$845$	−20.4321	−0.702885
$846$	0	0
$847$	0.689143	0.0236792
$848$	−29.4168	−1.01018
$849$	0	0
$850$	94.0659	3.22643
$851$	−7.63881	−0.261855
$852$	0	0
$853$	7.79943	0.267047	0.133524	−	0.991046i	$-0.457371\pi$
$853$	7.79943	0.267047	0.133524	+	0.991046i	$0.457371\pi$
$854$	23.1508	0.792203
$855$	0	0
$856$	−2.24792	−0.0768323
$857$	−15.8702	−0.542115	−0.271058	−	0.962563i	$-0.587373\pi$
$857$	−15.8702	−0.542115	−0.271058	+	0.962563i	$0.587373\pi$
$858$	0	0
$859$	7.93953	0.270893	0.135447	−	0.990785i	$-0.456753\pi$
$859$	7.93953	0.270893	0.135447	+	0.990785i	$0.456753\pi$
$860$	−44.0205	−1.50109
$861$	0	0
$862$	−29.4236	−1.00217
$863$	−36.0705	−1.22786	−0.613928	−	0.789362i	$-0.710411\pi$
$863$	−36.0705	−1.22786	−0.613928	+	0.789362i	$0.710411\pi$
$864$	0	0
$865$	−42.7797	−1.45455
$866$	−41.6000	−1.41363
$867$	0	0
$868$	42.9910	1.45921
$869$	−5.41017	−0.183527
$870$	0	0
$871$	65.8687	2.23188
$872$	5.56496	0.188453
$873$	0	0
$874$	−15.7862	−0.533976
$875$	36.0550	1.21888
$876$	0	0
$877$	−9.29593	−0.313901	−0.156951	−	0.987606i	$-0.550166\pi$
$877$	−9.29593	−0.313901	−0.156951	+	0.987606i	$0.550166\pi$
$878$	23.1076	0.779842
$879$	0	0
$880$	−43.4143	−1.46350
$881$	−19.7875	−0.666657	−0.333329	−	0.942811i	$-0.608172\pi$
$881$	−19.7875	−0.666657	−0.333329	+	0.942811i	$0.608172\pi$
$882$	0	0
$883$	36.9124	1.24220	0.621101	−	0.783731i	$-0.286686\pi$
$883$	36.9124	1.24220	0.621101	+	0.783731i	$0.286686\pi$
$884$	47.9586	1.61302
$885$	0	0
$886$	17.8319	0.599074
$887$	5.46715	0.183569	0.0917845	−	0.995779i	$-0.470743\pi$
$887$	5.46715	0.183569	0.0917845	+	0.995779i	$0.470743\pi$
$888$	0	0
$889$	23.3670	0.783703
$890$	64.8900	2.17512
$891$	0	0
$892$	4.97003	0.166409
$893$	−77.7185	−2.60075
$894$	0	0
$895$	−29.2208	−0.976743
$896$	8.07812	0.269871
$897$	0	0
$898$	54.8739	1.83117
$899$	−8.24396	−0.274951
$900$	0	0
$901$	42.0320	1.40029
$902$	−45.6691	−1.52062
$903$	0	0
$904$	−3.95548	−0.131557
$905$	−29.1856	−0.970164
$906$	0	0
$907$	1.43927	0.0477903	0.0238952	−	0.999714i	$-0.492393\pi$
$907$	1.43927	0.0477903	0.0238952	+	0.999714i	$0.492393\pi$
$908$	−48.7456	−1.61768
$909$	0	0
$910$	78.0153	2.58618
$911$	45.8742	1.51988	0.759941	−	0.649993i	$-0.225228\pi$
$911$	45.8742	1.51988	0.759941	+	0.649993i	$0.225228\pi$
$912$	0	0
$913$	16.0441	0.530981
$914$	−37.5223	−1.24113
$915$	0	0
$916$	24.3526	0.804634
$917$	−1.65221	−0.0545608
$918$	0	0
$919$	22.7859	0.751637	0.375818	−	0.926693i	$-0.377362\pi$
$919$	22.7859	0.751637	0.375818	+	0.926693i	$0.377362\pi$
$920$	−1.61395	−0.0532104
$921$	0	0
$922$	−1.20005	−0.0395217
$923$	48.3126	1.59023
$924$	0	0
$925$	−69.3036	−2.27869
$926$	−76.4826	−2.51337
$927$	0	0
$928$	−8.11680	−0.266447
$929$	−27.0810	−0.888499	−0.444250	−	0.895903i	$-0.646530\pi$
$929$	−27.0810	−0.888499	−0.444250	+	0.895903i	$0.646530\pi$
$930$	0	0
$931$	11.0059	0.360705
$932$	−37.1462	−1.21677
$933$	0	0
$934$	59.8964	1.95987
$935$	62.0322	2.02867
$936$	0	0
$937$	−39.3374	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$937$	−39.3374	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$938$	−74.2602	−2.42468
$939$	0	0
$940$	−83.7318	−2.73103
$941$	54.4213	1.77408	0.887041	−	0.461690i	$-0.152757\pi$
$941$	54.4213	1.77408	0.887041	+	0.461690i	$0.152757\pi$
$942$	0	0
$943$	6.80211	0.221507
$944$	−23.4896	−0.764520
$945$	0	0
$946$	−35.6546	−1.15923
$947$	30.7692	0.999866	0.499933	−	0.866064i	$-0.333358\pi$
$947$	30.7692	0.999866	0.499933	+	0.866064i	$0.333358\pi$
$948$	0	0
$949$	48.3021	1.56795
$950$	−143.221	−4.64672
$951$	0	0
$952$	−5.13086	−0.166292
$953$	7.40619	0.239910	0.119955	−	0.992779i	$-0.461725\pi$
$953$	7.40619	0.239910	0.119955	+	0.992779i	$0.461725\pi$
$954$	0	0
$955$	−71.8424	−2.32476
$956$	−1.53638	−0.0496901
$957$	0	0
$958$	−64.8098	−2.09391
$959$	24.1533	0.779952
$960$	0	0
$961$	36.9629	1.19235
$962$	−67.3145	−2.17031
$963$	0	0
$964$	−38.3848	−1.23629
$965$	−42.9197	−1.38163
$966$	0	0
$967$	−48.3624	−1.55523	−0.777614	−	0.628742i	$-0.783570\pi$
$967$	−48.3624	−1.55523	−0.777614	+	0.628742i	$0.783570\pi$
$968$	−0.125633	−0.00403800
$969$	0	0
$970$	−62.5369	−2.00794
$971$	−12.6667	−0.406494	−0.203247	−	0.979127i	$-0.565150\pi$
$971$	−12.6667	−0.406494	−0.203247	+	0.979127i	$0.565150\pi$
$972$	0	0
$973$	−38.3895	−1.23071
$974$	−51.4971	−1.65007
$975$	0	0
$976$	16.9092	0.541250
$977$	55.1668	1.76494	0.882472	−	0.470365i	$-0.155878\pi$
$977$	55.1668	1.76494	0.882472	+	0.470365i	$0.155878\pi$
$978$	0	0
$979$	27.5880	0.881716
$980$	11.8575	0.378774
$981$	0	0
$982$	22.3598	0.713530
$983$	50.7397	1.61834	0.809172	−	0.587571i	$-0.199916\pi$
$983$	50.7397	1.61834	0.809172	+	0.587571i	$0.199916\pi$
$984$	0	0
$985$	−31.0894	−0.990590
$986$	10.3682	0.330189
$987$	0	0
$988$	−73.0200	−2.32308
$989$	5.31051	0.168865
$990$	0	0
$991$	15.7634	0.500741	0.250371	−	0.968150i	$-0.419448\pi$
$991$	15.7634	0.500741	0.250371	+	0.968150i	$0.419448\pi$
$992$	66.9145	2.12454
$993$	0	0
$994$	−54.4674	−1.72760
$995$	20.6875	0.655837
$996$	0	0
$997$	43.2959	1.37119	0.685597	−	0.727981i	$-0.259541\pi$
$997$	43.2959	1.37119	0.685597	+	0.727981i	$0.259541\pi$
$998$	−48.0324	−1.52044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.p.1.12		14
3.2	odd	2		2001.2.a.m.1.3	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.m.1.3	✓	14	3.2	odd	2
6003.2.a.p.1.12		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)

\(f(q)\)	\(=\)	\(q+2.05175 q^{2} +2.20969 q^{4} -3.75134 q^{5} -2.35999 q^{7} +0.430233 q^{8} +O(q^{10})\)	\(q+2.05175 q^{2} +2.20969 q^{4} -3.75134 q^{5} -2.35999 q^{7} +0.430233 q^{8} -7.69683 q^{10} -3.27231 q^{11} +4.29495 q^{13} -4.84211 q^{14} -3.53665 q^{16} +5.05332 q^{17} -7.69400 q^{19} -8.28931 q^{20} -6.71396 q^{22} +1.00000 q^{23} +9.07257 q^{25} +8.81218 q^{26} -5.21484 q^{28} +1.00000 q^{29} -8.24396 q^{31} -8.11680 q^{32} +10.3682 q^{34} +8.85312 q^{35} -7.63881 q^{37} -15.7862 q^{38} -1.61395 q^{40} +6.80211 q^{41} +5.31051 q^{43} -7.23078 q^{44} +2.05175 q^{46} +10.1012 q^{47} -1.43046 q^{49} +18.6147 q^{50} +9.49051 q^{52} +8.31770 q^{53} +12.2755 q^{55} -1.01535 q^{56} +2.05175 q^{58} +6.64175 q^{59} -4.78113 q^{61} -16.9146 q^{62} -9.58036 q^{64} -16.1118 q^{65} +15.3363 q^{67} +11.1663 q^{68} +18.1644 q^{70} +11.2487 q^{71} +11.2463 q^{73} -15.6729 q^{74} -17.0014 q^{76} +7.72260 q^{77} +1.65332 q^{79} +13.2672 q^{80} +13.9563 q^{82} -4.90299 q^{83} -18.9567 q^{85} +10.8959 q^{86} -1.40785 q^{88} -8.43075 q^{89} -10.1360 q^{91} +2.20969 q^{92} +20.7251 q^{94} +28.8628 q^{95} +8.12503 q^{97} -2.93494 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 12 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 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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