LMFDB - Embedded newform 6003.2.a.s.1.8

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.762638$ 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)

$f(q)$	$=$	$q-0.762638 q^{2} -1.41838 q^{4} +4.10409 q^{5} -2.37592 q^{7} +2.60699 q^{8} +O(q^{10})$	\(q-0.762638 q^{2} -1.41838 q^{4} +4.10409 q^{5} -2.37592 q^{7} +2.60699 q^{8} -3.12993 q^{10} +4.76863 q^{11} +5.19051 q^{13} +1.81197 q^{14} +0.848580 q^{16} +2.80912 q^{17} -0.644865 q^{19} -5.82117 q^{20} -3.63674 q^{22} -1.00000 q^{23} +11.8435 q^{25} -3.95848 q^{26} +3.36997 q^{28} -1.00000 q^{29} +0.195443 q^{31} -5.86114 q^{32} -2.14234 q^{34} -9.75098 q^{35} +4.46849 q^{37} +0.491798 q^{38} +10.6993 q^{40} -2.20248 q^{41} +10.7899 q^{43} -6.76374 q^{44} +0.762638 q^{46} +10.3878 q^{47} -1.35501 q^{49} -9.03232 q^{50} -7.36213 q^{52} +3.71349 q^{53} +19.5709 q^{55} -6.19400 q^{56} +0.762638 q^{58} +6.79452 q^{59} +12.5542 q^{61} -0.149052 q^{62} +2.77276 q^{64} +21.3023 q^{65} -2.80619 q^{67} -3.98441 q^{68} +7.43647 q^{70} -14.2651 q^{71} -7.86909 q^{73} -3.40784 q^{74} +0.914665 q^{76} -11.3299 q^{77} -9.62495 q^{79} +3.48264 q^{80} +1.67969 q^{82} -14.6215 q^{83} +11.5289 q^{85} -8.22881 q^{86} +12.4318 q^{88} +8.72169 q^{89} -12.3322 q^{91} +1.41838 q^{92} -7.92214 q^{94} -2.64658 q^{95} -13.7648 q^{97} +1.03338 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * 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 $

$2$	−0.762638	−0.539266	−0.269633	−	0.962963i	$-0.586902\pi$
$2$	−0.762638	−0.539266	−0.269633	+	0.962963i	$0.586902\pi$
$3$	0	0
$3$	0	0
$4$	−1.41838	−0.709192
$5$	4.10409	1.83540	0.917702	−	0.397270i	$-0.130042\pi$
$5$	4.10409	1.83540	0.917702	+	0.397270i	$0.130042\pi$
$6$	0	0
$7$	−2.37592	−0.898013	−0.449007	−	0.893528i	$-0.648222\pi$
$7$	−2.37592	−0.898013	−0.449007	+	0.893528i	$0.648222\pi$
$8$	2.60699	0.921710
$9$	0	0
$10$	−3.12993	−0.989771
$11$	4.76863	1.43780	0.718898	−	0.695116i	$-0.244647\pi$
$11$	4.76863	1.43780	0.718898	+	0.695116i	$0.244647\pi$
$12$	0	0
$13$	5.19051	1.43959	0.719794	−	0.694188i	$-0.244236\pi$
$13$	5.19051	1.43959	0.719794	+	0.694188i	$0.244236\pi$
$14$	1.81197	0.484268
$15$	0	0
$16$	0.848580	0.212145
$17$	2.80912	0.681312	0.340656	−	0.940188i	$-0.389351\pi$
$17$	2.80912	0.681312	0.340656	+	0.940188i	$0.389351\pi$
$18$	0	0
$19$	−0.644865	−0.147942	−0.0739710	−	0.997260i	$-0.523567\pi$
$19$	−0.644865	−0.147942	−0.0739710	+	0.997260i	$0.523567\pi$
$20$	−5.82117	−1.30165
$21$	0	0
$22$	−3.63674	−0.775355
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.8435	2.36871
$26$	−3.95848	−0.776321
$27$	0	0
$28$	3.36997	0.636864
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.195443	0.0351026	0.0175513	−	0.999846i	$-0.494413\pi$
$31$	0.195443	0.0351026	0.0175513	+	0.999846i	$0.494413\pi$
$32$	−5.86114	−1.03611
$33$	0	0
$34$	−2.14234	−0.367409
$35$	−9.75098	−1.64822
$36$	0	0
$37$	4.46849	0.734616	0.367308	−	0.930099i	$-0.380280\pi$
$37$	4.46849	0.734616	0.367308	+	0.930099i	$0.380280\pi$
$38$	0.491798	0.0797802
$39$	0	0
$40$	10.6993	1.69171
$41$	−2.20248	−0.343969	−0.171985	−	0.985100i	$-0.555018\pi$
$41$	−2.20248	−0.343969	−0.171985	+	0.985100i	$0.555018\pi$
$42$	0	0
$43$	10.7899	1.64545	0.822725	−	0.568440i	$-0.192453\pi$
$43$	10.7899	1.64545	0.822725	+	0.568440i	$0.192453\pi$
$44$	−6.76374	−1.01967
$45$	0	0
$46$	0.762638	0.112445
$47$	10.3878	1.51522	0.757610	−	0.652708i	$-0.226367\pi$
$47$	10.3878	1.51522	0.757610	+	0.652708i	$0.226367\pi$
$48$	0	0
$49$	−1.35501	−0.193572
$50$	−9.03232	−1.27736
$51$	0	0
$52$	−7.36213	−1.02094
$53$	3.71349	0.510088	0.255044	−	0.966930i	$-0.417910\pi$
$53$	3.71349	0.510088	0.255044	+	0.966930i	$0.417910\pi$
$54$	0	0
$55$	19.5709	2.63893
$56$	−6.19400	−0.827707
$57$	0	0
$58$	0.762638	0.100139
$59$	6.79452	0.884572	0.442286	−	0.896874i	$-0.354168\pi$
$59$	6.79452	0.884572	0.442286	+	0.896874i	$0.354168\pi$
$60$	0	0
$61$	12.5542	1.60740	0.803698	−	0.595037i	$-0.202863\pi$
$61$	12.5542	1.60740	0.803698	+	0.595037i	$0.202863\pi$
$62$	−0.149052	−0.0189296
$63$	0	0
$64$	2.77276	0.346596
$65$	21.3023	2.64222
$66$	0	0
$67$	−2.80619	−0.342831	−0.171415	−	0.985199i	$-0.554834\pi$
$67$	−2.80619	−0.342831	−0.171415	+	0.985199i	$0.554834\pi$
$68$	−3.98441	−0.483181
$69$	0	0
$70$	7.43647	0.888828
$71$	−14.2651	−1.69295	−0.846476	−	0.532428i	$-0.821280\pi$
$71$	−14.2651	−1.69295	−0.846476	+	0.532428i	$0.821280\pi$
$72$	0	0
$73$	−7.86909	−0.921007	−0.460504	−	0.887658i	$-0.652331\pi$
$73$	−7.86909	−0.921007	−0.460504	+	0.887658i	$0.652331\pi$
$74$	−3.40784	−0.396153
$75$	0	0
$76$	0.914665	0.104919
$77$	−11.3299	−1.29116
$78$	0	0
$79$	−9.62495	−1.08289	−0.541446	−	0.840736i	$-0.682123\pi$
$79$	−9.62495	−1.08289	−0.541446	+	0.840736i	$0.682123\pi$
$80$	3.48264	0.389371
$81$	0	0
$82$	1.67969	0.185491
$83$	−14.6215	−1.60492	−0.802458	−	0.596708i	$-0.796475\pi$
$83$	−14.6215	−1.60492	−0.802458	+	0.596708i	$0.796475\pi$
$84$	0	0
$85$	11.5289	1.25048
$86$	−8.22881	−0.887336
$87$	0	0
$88$	12.4318	1.32523
$89$	8.72169	0.924497	0.462249	−	0.886750i	$-0.347043\pi$
$89$	8.72169	0.924497	0.462249	+	0.886750i	$0.347043\pi$
$90$	0	0
$91$	−12.3322	−1.29277
$92$	1.41838	0.147877
$93$	0	0
$94$	−7.92214	−0.817107
$95$	−2.64658	−0.271533
$96$	0	0
$97$	−13.7648	−1.39761	−0.698804	−	0.715314i	$-0.746284\pi$
$97$	−13.7648	−1.39761	−0.698804	+	0.715314i	$0.746284\pi$
$98$	1.03338	0.104387
$99$	0	0
$100$	−16.7987	−1.67987
$101$	0.857700	0.0853444	0.0426722	−	0.999089i	$-0.486413\pi$
$101$	0.857700	0.0853444	0.0426722	+	0.999089i	$0.486413\pi$
$102$	0	0
$103$	3.84676	0.379033	0.189516	−	0.981878i	$-0.439308\pi$
$103$	3.84676	0.379033	0.189516	+	0.981878i	$0.439308\pi$
$104$	13.5316	1.32688
$105$	0	0
$106$	−2.83205	−0.275073
$107$	−18.4797	−1.78650	−0.893248	−	0.449565i	$-0.851579\pi$
$107$	−18.4797	−1.78650	−0.893248	+	0.449565i	$0.851579\pi$
$108$	0	0
$109$	−3.88301	−0.371925	−0.185963	−	0.982557i	$-0.559540\pi$
$109$	−3.88301	−0.371925	−0.185963	+	0.982557i	$0.559540\pi$
$110$	−14.9255	−1.42309
$111$	0	0
$112$	−2.01616	−0.190509
$113$	−0.238682	−0.0224533	−0.0112267	−	0.999937i	$-0.503574\pi$
$113$	−0.238682	−0.0224533	−0.0112267	+	0.999937i	$0.503574\pi$
$114$	0	0
$115$	−4.10409	−0.382708
$116$	1.41838	0.131694
$117$	0	0
$118$	−5.18176	−0.477020
$119$	−6.67425	−0.611827
$120$	0	0
$121$	11.7398	1.06726
$122$	−9.57428	−0.866815
$123$	0	0
$124$	−0.277213	−0.0248944
$125$	28.0864	2.51213
$126$	0	0
$127$	6.03868	0.535846	0.267923	−	0.963440i	$-0.413663\pi$
$127$	6.03868	0.535846	0.267923	+	0.963440i	$0.413663\pi$
$128$	9.60766	0.849205
$129$	0	0
$130$	−16.2459	−1.42486
$131$	−16.4899	−1.44073	−0.720364	−	0.693596i	$-0.756025\pi$
$131$	−16.4899	−1.44073	−0.720364	+	0.693596i	$0.756025\pi$
$132$	0	0
$133$	1.53215	0.132854
$134$	2.14011	0.184877
$135$	0	0
$136$	7.32335	0.627972
$137$	−4.51947	−0.386124	−0.193062	−	0.981187i	$-0.561842\pi$
$137$	−4.51947	−0.386124	−0.193062	+	0.981187i	$0.561842\pi$
$138$	0	0
$139$	23.3271	1.97858	0.989288	−	0.145976i	$-0.0466323\pi$
$139$	23.3271	1.97858	0.989288	+	0.145976i	$0.0466323\pi$
$140$	13.8306	1.16890
$141$	0	0
$142$	10.8791	0.912952
$143$	24.7516	2.06983
$144$	0	0
$145$	−4.10409	−0.340826
$146$	6.00127	0.496668
$147$	0	0
$148$	−6.33804	−0.520983
$149$	12.3873	1.01481	0.507403	−	0.861709i	$-0.330605\pi$
$149$	12.3873	1.01481	0.507403	+	0.861709i	$0.330605\pi$
$150$	0	0
$151$	−8.79395	−0.715642	−0.357821	−	0.933790i	$-0.616480\pi$
$151$	−8.79395	−0.715642	−0.357821	+	0.933790i	$0.616480\pi$
$152$	−1.68115	−0.136360
$153$	0	0
$154$	8.64059	0.696279
$155$	0.802114	0.0644274
$156$	0	0
$157$	−12.9451	−1.03313	−0.516564	−	0.856249i	$-0.672789\pi$
$157$	−12.9451	−1.03313	−0.516564	+	0.856249i	$0.672789\pi$
$158$	7.34035	0.583967
$159$	0	0
$160$	−24.0546	−1.90168
$161$	2.37592	0.187249
$162$	0	0
$163$	8.06894	0.632008	0.316004	−	0.948758i	$-0.397659\pi$
$163$	8.06894	0.632008	0.316004	+	0.948758i	$0.397659\pi$
$164$	3.12396	0.243940
$165$	0	0
$166$	11.1509	0.865477
$167$	−17.7452	−1.37317	−0.686584	−	0.727051i	$-0.740891\pi$
$167$	−17.7452	−1.37317	−0.686584	+	0.727051i	$0.740891\pi$
$168$	0	0
$169$	13.9413	1.07241
$170$	−8.79236	−0.674343
$171$	0	0
$172$	−15.3043	−1.16694
$173$	−23.9022	−1.81725	−0.908624	−	0.417615i	$-0.862866\pi$
$173$	−23.9022	−1.81725	−0.908624	+	0.417615i	$0.862866\pi$
$174$	0	0
$175$	−28.1393	−2.12713
$176$	4.04656	0.305021
$177$	0	0
$178$	−6.65149	−0.498550
$179$	17.4110	1.30136	0.650678	−	0.759353i	$-0.274485\pi$
$179$	17.4110	1.30136	0.650678	+	0.759353i	$0.274485\pi$
$180$	0	0
$181$	−8.86252	−0.658746	−0.329373	−	0.944200i	$-0.606837\pi$
$181$	−8.86252	−0.658746	−0.329373	+	0.944200i	$0.606837\pi$
$182$	9.40502	0.697146
$183$	0	0
$184$	−2.60699	−0.192190
$185$	18.3391	1.34832
$186$	0	0
$187$	13.3957	0.979587
$188$	−14.7339	−1.07458
$189$	0	0
$190$	2.01838	0.146429
$191$	−12.9011	−0.933493	−0.466746	−	0.884391i	$-0.654574\pi$
$191$	−12.9011	−0.933493	−0.466746	+	0.884391i	$0.654574\pi$
$192$	0	0
$193$	3.46564	0.249463	0.124731	−	0.992191i	$-0.460193\pi$
$193$	3.46564	0.249463	0.124731	+	0.992191i	$0.460193\pi$
$194$	10.4976	0.753683
$195$	0	0
$196$	1.92192	0.137280
$197$	10.0118	0.713310	0.356655	−	0.934236i	$-0.383917\pi$
$197$	10.0118	0.713310	0.356655	+	0.934236i	$0.383917\pi$
$198$	0	0
$199$	9.14492	0.648266	0.324133	−	0.946012i	$-0.394927\pi$
$199$	9.14492	0.648266	0.324133	+	0.946012i	$0.394927\pi$
$200$	30.8759	2.18326
$201$	0	0
$202$	−0.654115	−0.0460234
$203$	2.37592	0.166757
$204$	0	0
$205$	−9.03916	−0.631322
$206$	−2.93369	−0.204400
$207$	0	0
$208$	4.40456	0.305401
$209$	−3.07512	−0.212710
$210$	0	0
$211$	3.26744	0.224940	0.112470	−	0.993655i	$-0.464124\pi$
$211$	3.26744	0.224940	0.112470	+	0.993655i	$0.464124\pi$
$212$	−5.26716	−0.361750
$213$	0	0
$214$	14.0933	0.963397
$215$	44.2828	3.02006
$216$	0	0
$217$	−0.464356	−0.0315226
$218$	2.96133	0.200567
$219$	0	0
$220$	−27.7590	−1.87151
$221$	14.5808	0.980808
$222$	0	0
$223$	−16.3511	−1.09495	−0.547477	−	0.836821i	$-0.684412\pi$
$223$	−16.3511	−1.09495	−0.547477	+	0.836821i	$0.684412\pi$
$224$	13.9256	0.930442
$225$	0	0
$226$	0.182028	0.0121083
$227$	−6.26229	−0.415643	−0.207822	−	0.978167i	$-0.566637\pi$
$227$	−6.26229	−0.415643	−0.207822	+	0.978167i	$0.566637\pi$
$228$	0	0
$229$	−2.82769	−0.186859	−0.0934293	−	0.995626i	$-0.529783\pi$
$229$	−2.82769	−0.186859	−0.0934293	+	0.995626i	$0.529783\pi$
$230$	3.12993	0.206382
$231$	0	0
$232$	−2.60699	−0.171157
$233$	6.61733	0.433515	0.216758	−	0.976225i	$-0.430452\pi$
$233$	6.61733	0.433515	0.216758	+	0.976225i	$0.430452\pi$
$234$	0	0
$235$	42.6325	2.78104
$236$	−9.63724	−0.627331
$237$	0	0
$238$	5.09003	0.329938
$239$	−17.3956	−1.12523	−0.562615	−	0.826719i	$-0.690204\pi$
$239$	−17.3956	−1.12523	−0.562615	+	0.826719i	$0.690204\pi$
$240$	0	0
$241$	3.26312	0.210196	0.105098	−	0.994462i	$-0.466484\pi$
$241$	3.26312	0.210196	0.105098	+	0.994462i	$0.466484\pi$
$242$	−8.95322	−0.575535
$243$	0	0
$244$	−17.8066	−1.13995
$245$	−5.56106	−0.355283
$246$	0	0
$247$	−3.34717	−0.212976
$248$	0.509517	0.0323544
$249$	0	0
$250$	−21.4198	−1.35471
$251$	23.6292	1.49146	0.745731	−	0.666247i	$-0.232100\pi$
$251$	23.6292	1.49146	0.745731	+	0.666247i	$0.232100\pi$
$252$	0	0
$253$	−4.76863	−0.299801
$254$	−4.60533	−0.288964
$255$	0	0
$256$	−12.8727	−0.804543
$257$	−4.73508	−0.295366	−0.147683	−	0.989035i	$-0.547182\pi$
$257$	−4.73508	−0.295366	−0.147683	+	0.989035i	$0.547182\pi$
$258$	0	0
$259$	−10.6168	−0.659695
$260$	−30.2148	−1.87384
$261$	0	0
$262$	12.5758	0.776937
$263$	−31.4462	−1.93906	−0.969529	−	0.244978i	$-0.921219\pi$
$263$	−31.4462	−1.93906	−0.969529	+	0.244978i	$0.921219\pi$
$264$	0	0
$265$	15.2405	0.936216
$266$	−1.16847	−0.0716437
$267$	0	0
$268$	3.98025	0.243133
$269$	13.8706	0.845704	0.422852	−	0.906199i	$-0.361029\pi$
$269$	13.8706	0.845704	0.422852	+	0.906199i	$0.361029\pi$
$270$	0	0
$271$	14.7222	0.894311	0.447156	−	0.894456i	$-0.352437\pi$
$271$	14.7222	0.894311	0.447156	+	0.894456i	$0.352437\pi$
$272$	2.38376	0.144537
$273$	0	0
$274$	3.44672	0.208224
$275$	56.4774	3.40571
$276$	0	0
$277$	28.6108	1.71906	0.859528	−	0.511088i	$-0.170757\pi$
$277$	28.6108	1.71906	0.859528	+	0.511088i	$0.170757\pi$
$278$	−17.7901	−1.06698
$279$	0	0
$280$	−25.4207	−1.51918
$281$	1.91196	0.114058	0.0570289	−	0.998373i	$-0.481837\pi$
$281$	1.91196	0.114058	0.0570289	+	0.998373i	$0.481837\pi$
$282$	0	0
$283$	−13.5965	−0.808230	−0.404115	−	0.914708i	$-0.632420\pi$
$283$	−13.5965	−0.808230	−0.404115	+	0.914708i	$0.632420\pi$
$284$	20.2333	1.20063
$285$	0	0
$286$	−18.8765	−1.11619
$287$	5.23291	0.308889
$288$	0	0
$289$	−9.10883	−0.535814
$290$	3.12993	0.183796
$291$	0	0
$292$	11.1614	0.653171
$293$	2.08364	0.121727	0.0608637	−	0.998146i	$-0.480614\pi$
$293$	2.08364	0.121727	0.0608637	+	0.998146i	$0.480614\pi$
$294$	0	0
$295$	27.8853	1.62355
$296$	11.6493	0.677102
$297$	0	0
$298$	−9.44702	−0.547251
$299$	−5.19051	−0.300175
$300$	0	0
$301$	−25.6360	−1.47764
$302$	6.70660	0.385921
$303$	0	0
$304$	−0.547219	−0.0313852
$305$	51.5234	2.95022
$306$	0	0
$307$	−30.9157	−1.76445	−0.882227	−	0.470824i	$-0.843957\pi$
$307$	−30.9157	−1.76445	−0.882227	+	0.470824i	$0.843957\pi$
$308$	16.0701	0.915680
$309$	0	0
$310$	−0.611722	−0.0347435
$311$	−12.6680	−0.718334	−0.359167	−	0.933273i	$-0.616939\pi$
$311$	−12.6680	−0.718334	−0.359167	+	0.933273i	$0.616939\pi$
$312$	0	0
$313$	23.5046	1.32856	0.664280	−	0.747484i	$-0.268738\pi$
$313$	23.5046	1.32856	0.664280	+	0.747484i	$0.268738\pi$
$314$	9.87239	0.557131
$315$	0	0
$316$	13.6519	0.767977
$317$	9.06839	0.509332	0.254666	−	0.967029i	$-0.418035\pi$
$317$	9.06839	0.509332	0.254666	+	0.967029i	$0.418035\pi$
$318$	0	0
$319$	−4.76863	−0.266992
$320$	11.3797	0.636143
$321$	0	0
$322$	−1.81197	−0.100977
$323$	−1.81150	−0.100795
$324$	0	0
$325$	61.4739	3.40996
$326$	−6.15368	−0.340821
$327$	0	0
$328$	−5.74183	−0.317040
$329$	−24.6806	−1.36069
$330$	0	0
$331$	−17.1154	−0.940748	−0.470374	−	0.882467i	$-0.655881\pi$
$331$	−17.1154	−0.940748	−0.470374	+	0.882467i	$0.655881\pi$
$332$	20.7389	1.13819
$333$	0	0
$334$	13.5332	0.740503
$335$	−11.5168	−0.629232
$336$	0	0
$337$	9.73964	0.530552	0.265276	−	0.964172i	$-0.414537\pi$
$337$	9.73964	0.530552	0.265276	+	0.964172i	$0.414537\pi$
$338$	−10.6322	−0.578315
$339$	0	0
$340$	−16.3524	−0.886832
$341$	0.931994	0.0504703
$342$	0	0
$343$	19.8508	1.07184
$344$	28.1292	1.51663
$345$	0	0
$346$	18.2287	0.979981
$347$	0.652189	0.0350113	0.0175057	−	0.999847i	$-0.494427\pi$
$347$	0.652189	0.0350113	0.0175057	+	0.999847i	$0.494427\pi$
$348$	0	0
$349$	0.388734	0.0208085	0.0104042	−	0.999946i	$-0.496688\pi$
$349$	0.388734	0.0208085	0.0104042	+	0.999946i	$0.496688\pi$
$350$	21.4601	1.14709
$351$	0	0
$352$	−27.9496	−1.48972
$353$	−33.9129	−1.80500	−0.902501	−	0.430688i	$-0.858271\pi$
$353$	−33.9129	−1.80500	−0.902501	+	0.430688i	$0.858271\pi$
$354$	0	0
$355$	−58.5450	−3.10725
$356$	−12.3707	−0.655646
$357$	0	0
$358$	−13.2783	−0.701778
$359$	−19.5828	−1.03354	−0.516770	−	0.856124i	$-0.672866\pi$
$359$	−19.5828	−1.03354	−0.516770	+	0.856124i	$0.672866\pi$
$360$	0	0
$361$	−18.5841	−0.978113
$362$	6.75890	0.355240
$363$	0	0
$364$	17.4918	0.916821
$365$	−32.2954	−1.69042
$366$	0	0
$367$	11.8792	0.620091	0.310045	−	0.950722i	$-0.399656\pi$
$367$	11.8792	0.620091	0.310045	+	0.950722i	$0.399656\pi$
$368$	−0.848580	−0.0442353
$369$	0	0
$370$	−13.9861	−0.727101
$371$	−8.82296	−0.458065
$372$	0	0
$373$	35.5289	1.83962	0.919808	−	0.392370i	$-0.128345\pi$
$373$	35.5289	1.83962	0.919808	+	0.392370i	$0.128345\pi$
$374$	−10.2160	−0.528258
$375$	0	0
$376$	27.0809	1.39659
$377$	−5.19051	−0.267325
$378$	0	0
$379$	20.2070	1.03796	0.518982	−	0.854785i	$-0.326311\pi$
$379$	20.2070	1.03796	0.518982	+	0.854785i	$0.326311\pi$
$380$	3.75387	0.192569
$381$	0	0
$382$	9.83889	0.503401
$383$	5.97382	0.305248	0.152624	−	0.988284i	$-0.451228\pi$
$383$	5.97382	0.305248	0.152624	+	0.988284i	$0.451228\pi$
$384$	0	0
$385$	−46.4988	−2.36980
$386$	−2.64303	−0.134527
$387$	0	0
$388$	19.5238	0.991172
$389$	10.8957	0.552436	0.276218	−	0.961095i	$-0.410919\pi$
$389$	10.8957	0.552436	0.276218	+	0.961095i	$0.410919\pi$
$390$	0	0
$391$	−2.80912	−0.142063
$392$	−3.53248	−0.178417
$393$	0	0
$394$	−7.63537	−0.384664
$395$	−39.5016	−1.98754
$396$	0	0
$397$	24.4605	1.22764	0.613819	−	0.789447i	$-0.289632\pi$
$397$	24.4605	1.22764	0.613819	+	0.789447i	$0.289632\pi$
$398$	−6.97426	−0.349588
$399$	0	0
$400$	10.0502	0.502509
$401$	−18.6658	−0.932126	−0.466063	−	0.884751i	$-0.654328\pi$
$401$	−18.6658	−0.932126	−0.466063	+	0.884751i	$0.654328\pi$
$402$	0	0
$403$	1.01445	0.0505332
$404$	−1.21655	−0.0605255
$405$	0	0
$406$	−1.81197	−0.0899264
$407$	21.3086	1.05623
$408$	0	0
$409$	23.3006	1.15214	0.576071	−	0.817400i	$-0.304585\pi$
$409$	23.3006	1.15214	0.576071	+	0.817400i	$0.304585\pi$
$410$	6.89360	0.340451
$411$	0	0
$412$	−5.45619	−0.268807
$413$	−16.1432	−0.794357
$414$	0	0
$415$	−60.0078	−2.94567
$416$	−30.4223	−1.49157
$417$	0	0
$418$	2.34520	0.114708
$419$	11.7063	0.571890	0.285945	−	0.958246i	$-0.407693\pi$
$419$	11.7063	0.571890	0.285945	+	0.958246i	$0.407693\pi$
$420$	0	0
$421$	−8.76302	−0.427083	−0.213542	−	0.976934i	$-0.568500\pi$
$421$	−8.76302	−0.427083	−0.213542	+	0.976934i	$0.568500\pi$
$422$	−2.49187	−0.121302
$423$	0	0
$424$	9.68104	0.470153
$425$	33.2699	1.61383
$426$	0	0
$427$	−29.8277	−1.44346
$428$	26.2112	1.26697
$429$	0	0
$430$	−33.7718	−1.62862
$431$	−25.2322	−1.21539	−0.607697	−	0.794169i	$-0.707906\pi$
$431$	−25.2322	−1.21539	−0.607697	+	0.794169i	$0.707906\pi$
$432$	0	0
$433$	24.3713	1.17121	0.585606	−	0.810596i	$-0.300857\pi$
$433$	24.3713	1.17121	0.585606	+	0.810596i	$0.300857\pi$
$434$	0.354136	0.0169991
$435$	0	0
$436$	5.50760	0.263766
$437$	0.644865	0.0308481
$438$	0	0
$439$	40.8264	1.94854	0.974270	−	0.225386i	$-0.0723643\pi$
$439$	40.8264	1.94854	0.974270	+	0.225386i	$0.0723643\pi$
$440$	51.0210	2.43233
$441$	0	0
$442$	−11.1198	−0.528917
$443$	27.5392	1.30842	0.654212	−	0.756311i	$-0.273000\pi$
$443$	27.5392	1.30842	0.654212	+	0.756311i	$0.273000\pi$
$444$	0	0
$445$	35.7946	1.69683
$446$	12.4700	0.590472
$447$	0	0
$448$	−6.58786	−0.311247
$449$	27.1392	1.28078	0.640389	−	0.768051i	$-0.278773\pi$
$449$	27.1392	1.28078	0.640389	+	0.768051i	$0.278773\pi$
$450$	0	0
$451$	−10.5028	−0.494557
$452$	0.338543	0.0159237
$453$	0	0
$454$	4.77586	0.224142
$455$	−50.6125	−2.37275
$456$	0	0
$457$	−11.4419	−0.535230	−0.267615	−	0.963526i	$-0.586236\pi$
$457$	−11.4419	−0.535230	−0.267615	+	0.963526i	$0.586236\pi$
$458$	2.15650	0.100767
$459$	0	0
$460$	5.82117	0.271413
$461$	−18.6583	−0.869004	−0.434502	−	0.900671i	$-0.643076\pi$
$461$	−18.6583	−0.869004	−0.434502	+	0.900671i	$0.643076\pi$
$462$	0	0
$463$	14.6373	0.680254	0.340127	−	0.940380i	$-0.389530\pi$
$463$	14.6373	0.680254	0.340127	+	0.940380i	$0.389530\pi$
$464$	−0.848580	−0.0393943
$465$	0	0
$466$	−5.04662	−0.233780
$467$	−10.6700	−0.493749	−0.246874	−	0.969047i	$-0.579404\pi$
$467$	−10.6700	−0.493749	−0.246874	+	0.969047i	$0.579404\pi$
$468$	0	0
$469$	6.66728	0.307866
$470$	−32.5132	−1.49972
$471$	0	0
$472$	17.7132	0.815318
$473$	51.4532	2.36582
$474$	0	0
$475$	−7.63747	−0.350431
$476$	9.46664	0.433903
$477$	0	0
$478$	13.2666	0.606798
$479$	−32.0589	−1.46481	−0.732405	−	0.680870i	$-0.761602\pi$
$479$	−32.0589	−1.46481	−0.732405	+	0.680870i	$0.761602\pi$
$480$	0	0
$481$	23.1937	1.05754
$482$	−2.48857	−0.113351
$483$	0	0
$484$	−16.6515	−0.756889
$485$	−56.4921	−2.56517
$486$	0	0
$487$	−23.8137	−1.07910	−0.539551	−	0.841953i	$-0.681406\pi$
$487$	−23.8137	−1.07910	−0.539551	+	0.841953i	$0.681406\pi$
$488$	32.7286	1.48155
$489$	0	0
$490$	4.24107	0.191592
$491$	18.2787	0.824905	0.412453	−	0.910979i	$-0.364672\pi$
$491$	18.2787	0.824905	0.412453	+	0.910979i	$0.364672\pi$
$492$	0	0
$493$	−2.80912	−0.126516
$494$	2.55268	0.114851
$495$	0	0
$496$	0.165849	0.00744683
$497$	33.8926	1.52029
$498$	0	0
$499$	4.85298	0.217249	0.108625	−	0.994083i	$-0.465355\pi$
$499$	4.85298	0.217249	0.108625	+	0.994083i	$0.465355\pi$
$500$	−39.8373	−1.78158
$501$	0	0
$502$	−18.0205	−0.804295
$503$	12.1670	0.542499	0.271249	−	0.962509i	$-0.412563\pi$
$503$	12.1670	0.542499	0.271249	+	0.962509i	$0.412563\pi$
$504$	0	0
$505$	3.52008	0.156641
$506$	3.63674	0.161673
$507$	0	0
$508$	−8.56517	−0.380018
$509$	−13.0048	−0.576429	−0.288214	−	0.957566i	$-0.593062\pi$
$509$	−13.0048	−0.576429	−0.288214	+	0.957566i	$0.593062\pi$
$510$	0	0
$511$	18.6963	0.827077
$512$	−9.39811	−0.415342
$513$	0	0
$514$	3.61115	0.159281
$515$	15.7874	0.695678
$516$	0	0
$517$	49.5356	2.17857
$518$	8.09676	0.355751
$519$	0	0
$520$	55.5348	2.43536
$521$	24.7847	1.08584	0.542919	−	0.839785i	$-0.317319\pi$
$521$	24.7847	1.08584	0.542919	+	0.839785i	$0.317319\pi$
$522$	0	0
$523$	1.69891	0.0742879	0.0371440	−	0.999310i	$-0.488174\pi$
$523$	1.69891	0.0742879	0.0371440	+	0.999310i	$0.488174\pi$
$524$	23.3890	1.02175
$525$	0	0
$526$	23.9821	1.04567
$527$	0.549023	0.0239158
$528$	0	0
$529$	1.00000	0.0434783
$530$	−11.6230	−0.504870
$531$	0	0
$532$	−2.17317	−0.0942189
$533$	−11.4320	−0.495174
$534$	0	0
$535$	−75.8421	−3.27894
$536$	−7.31570	−0.315990
$537$	0	0
$538$	−10.5782	−0.456060
$539$	−6.46152	−0.278317
$540$	0	0
$541$	−24.7795	−1.06535	−0.532677	−	0.846319i	$-0.678814\pi$
$541$	−24.7795	−1.06535	−0.532677	+	0.846319i	$0.678814\pi$
$542$	−11.2277	−0.482272
$543$	0	0
$544$	−16.4646	−0.705916
$545$	−15.9362	−0.682633
$546$	0	0
$547$	−24.2508	−1.03689	−0.518444	−	0.855112i	$-0.673489\pi$
$547$	−24.2508	−1.03689	−0.518444	+	0.855112i	$0.673489\pi$
$548$	6.41034	0.273836
$549$	0	0
$550$	−43.0718	−1.83659
$551$	0.644865	0.0274722
$552$	0	0
$553$	22.8681	0.972451
$554$	−21.8197	−0.927029
$555$	0	0
$556$	−33.0867	−1.40319
$557$	21.1596	0.896559	0.448280	−	0.893893i	$-0.352037\pi$
$557$	21.1596	0.896559	0.448280	+	0.893893i	$0.352037\pi$
$558$	0	0
$559$	56.0052	2.36877
$560$	−8.27448	−0.349661
$561$	0	0
$562$	−1.45813	−0.0615075
$563$	47.1640	1.98773	0.993864	−	0.110613i	$-0.0352814\pi$
$563$	47.1640	1.98773	0.993864	+	0.110613i	$0.0352814\pi$
$564$	0	0
$565$	−0.979572	−0.0412109
$566$	10.3692	0.435851
$567$	0	0
$568$	−37.1888	−1.56041
$569$	11.1557	0.467673	0.233836	−	0.972276i	$-0.424872\pi$
$569$	11.1557	0.467673	0.233836	+	0.972276i	$0.424872\pi$
$570$	0	0
$571$	16.5122	0.691013	0.345506	−	0.938416i	$-0.387707\pi$
$571$	16.5122	0.691013	0.345506	+	0.938416i	$0.387707\pi$
$572$	−35.1072	−1.46791
$573$	0	0
$574$	−3.99082	−0.166573
$575$	−11.8435	−0.493909
$576$	0	0
$577$	−6.85661	−0.285444	−0.142722	−	0.989763i	$-0.545586\pi$
$577$	−6.85661	−0.285444	−0.142722	+	0.989763i	$0.545586\pi$
$578$	6.94674	0.288946
$579$	0	0
$580$	5.82117	0.241711
$581$	34.7395	1.44124
$582$	0	0
$583$	17.7083	0.733402
$584$	−20.5146	−0.848901
$585$	0	0
$586$	−1.58906	−0.0656435
$587$	16.3914	0.676545	0.338273	−	0.941048i	$-0.390157\pi$
$587$	16.3914	0.676545	0.338273	+	0.941048i	$0.390157\pi$
$588$	0	0
$589$	−0.126034	−0.00519315
$590$	−21.2664	−0.875524
$591$	0	0
$592$	3.79187	0.155845
$593$	40.1902	1.65041	0.825207	−	0.564830i	$-0.191058\pi$
$593$	40.1902	1.65041	0.825207	+	0.564830i	$0.191058\pi$
$594$	0	0
$595$	−27.3917	−1.12295
$596$	−17.5699	−0.719693
$597$	0	0
$598$	3.95848	0.161874
$599$	−28.8613	−1.17924	−0.589621	−	0.807680i	$-0.700723\pi$
$599$	−28.8613	−1.17924	−0.589621	+	0.807680i	$0.700723\pi$
$600$	0	0
$601$	−24.5658	−1.00206	−0.501031	−	0.865429i	$-0.667046\pi$
$601$	−24.5658	−1.00206	−0.501031	+	0.865429i	$0.667046\pi$
$602$	19.5510	0.796839
$603$	0	0
$604$	12.4732	0.507527
$605$	48.1812	1.95884
$606$	0	0
$607$	29.3832	1.19263	0.596314	−	0.802751i	$-0.296631\pi$
$607$	29.3832	1.19263	0.596314	+	0.802751i	$0.296631\pi$
$608$	3.77964	0.153285
$609$	0	0
$610$	−39.2937	−1.59095
$611$	53.9180	2.18129
$612$	0	0
$613$	22.8059	0.921122	0.460561	−	0.887628i	$-0.347648\pi$
$613$	22.8059	0.921122	0.460561	+	0.887628i	$0.347648\pi$
$614$	23.5775	0.951511
$615$	0	0
$616$	−29.5369	−1.19007
$617$	−20.2767	−0.816309	−0.408154	−	0.912913i	$-0.633828\pi$
$617$	−20.2767	−0.816309	−0.408154	+	0.912913i	$0.633828\pi$
$618$	0	0
$619$	−20.2300	−0.813113	−0.406556	−	0.913626i	$-0.633271\pi$
$619$	−20.2300	−0.813113	−0.406556	+	0.913626i	$0.633271\pi$
$620$	−1.13771	−0.0456914
$621$	0	0
$622$	9.66107	0.387374
$623$	−20.7220	−0.830211
$624$	0	0
$625$	56.0515	2.24206
$626$	−17.9255	−0.716448
$627$	0	0
$628$	18.3611	0.732686
$629$	12.5525	0.500503
$630$	0	0
$631$	22.9942	0.915385	0.457693	−	0.889110i	$-0.348676\pi$
$631$	22.9942	0.915385	0.457693	+	0.889110i	$0.348676\pi$
$632$	−25.0921	−0.998111
$633$	0	0
$634$	−6.91590	−0.274665
$635$	24.7833	0.983494
$636$	0	0
$637$	−7.03316	−0.278664
$638$	3.63674	0.143980
$639$	0	0
$640$	39.4307	1.55863
$641$	−30.1570	−1.19113	−0.595565	−	0.803307i	$-0.703072\pi$
$641$	−30.1570	−1.19113	−0.595565	+	0.803307i	$0.703072\pi$
$642$	0	0
$643$	−0.607328	−0.0239507	−0.0119753	−	0.999928i	$-0.503812\pi$
$643$	−0.607328	−0.0239507	−0.0119753	+	0.999928i	$0.503812\pi$
$644$	−3.36997	−0.132795
$645$	0	0
$646$	1.38152	0.0543552
$647$	−11.4460	−0.449989	−0.224994	−	0.974360i	$-0.572236\pi$
$647$	−11.4460	−0.449989	−0.224994	+	0.974360i	$0.572236\pi$
$648$	0	0
$649$	32.4006	1.27183
$650$	−46.8823	−1.83888
$651$	0	0
$652$	−11.4449	−0.448215
$653$	24.7401	0.968153	0.484077	−	0.875026i	$-0.339156\pi$
$653$	24.7401	0.968153	0.484077	+	0.875026i	$0.339156\pi$
$654$	0	0
$655$	−67.6760	−2.64432
$656$	−1.86898	−0.0729713
$657$	0	0
$658$	18.8224	0.733773
$659$	−18.8865	−0.735715	−0.367858	−	0.929882i	$-0.619909\pi$
$659$	−18.8865	−0.735715	−0.367858	+	0.929882i	$0.619909\pi$
$660$	0	0
$661$	−0.433103	−0.0168457	−0.00842287	−	0.999965i	$-0.502681\pi$
$661$	−0.433103	−0.0168457	−0.00842287	+	0.999965i	$0.502681\pi$
$662$	13.0529	0.507314
$663$	0	0
$664$	−38.1180	−1.47927
$665$	6.28806	0.243841
$666$	0	0
$667$	1.00000	0.0387202
$668$	25.1696	0.973839
$669$	0	0
$670$	8.78318	0.339324
$671$	59.8661	2.31111
$672$	0	0
$673$	7.37791	0.284398	0.142199	−	0.989838i	$-0.454583\pi$
$673$	7.37791	0.284398	0.142199	+	0.989838i	$0.454583\pi$
$674$	−7.42782	−0.286109
$675$	0	0
$676$	−19.7742	−0.760545
$677$	38.0288	1.46156	0.730782	−	0.682611i	$-0.239156\pi$
$677$	38.0288	1.46156	0.730782	+	0.682611i	$0.239156\pi$
$678$	0	0
$679$	32.7041	1.25507
$680$	30.0557	1.15258
$681$	0	0
$682$	−0.710774	−0.0272169
$683$	−2.89906	−0.110929	−0.0554647	−	0.998461i	$-0.517664\pi$
$683$	−2.89906	−0.110929	−0.0554647	+	0.998461i	$0.517664\pi$
$684$	0	0
$685$	−18.5483	−0.708694
$686$	−15.1390	−0.578009
$687$	0	0
$688$	9.15612	0.349074
$689$	19.2749	0.734316
$690$	0	0
$691$	2.97532	0.113186	0.0565932	−	0.998397i	$-0.481976\pi$
$691$	2.97532	0.113186	0.0565932	+	0.998397i	$0.481976\pi$
$692$	33.9024	1.28878
$693$	0	0
$694$	−0.497384	−0.0188804
$695$	95.7363	3.63149
$696$	0	0
$697$	−6.18703	−0.234350
$698$	−0.296463	−0.0112213
$699$	0	0
$700$	39.9123	1.50854
$701$	39.2450	1.48226	0.741132	−	0.671359i	$-0.234289\pi$
$701$	39.2450	1.48226	0.741132	+	0.671359i	$0.234289\pi$
$702$	0	0
$703$	−2.88157	−0.108681
$704$	13.2223	0.498333
$705$	0	0
$706$	25.8633	0.973377
$707$	−2.03783	−0.0766404
$708$	0	0
$709$	21.1537	0.794445	0.397223	−	0.917722i	$-0.369974\pi$
$709$	21.1537	0.794445	0.397223	+	0.917722i	$0.369974\pi$
$710$	44.6487	1.67563
$711$	0	0
$712$	22.7373	0.852118
$713$	−0.195443	−0.00731939
$714$	0	0
$715$	101.583	3.79898
$716$	−24.6954	−0.922912
$717$	0	0
$718$	14.9346	0.557353
$719$	28.2170	1.05232	0.526159	−	0.850386i	$-0.323632\pi$
$719$	28.2170	1.05232	0.526159	+	0.850386i	$0.323632\pi$
$720$	0	0
$721$	−9.13960	−0.340377
$722$	14.1730	0.527463
$723$	0	0
$724$	12.5705	0.467177
$725$	−11.8435	−0.439858
$726$	0	0
$727$	−1.17513	−0.0435833	−0.0217917	−	0.999763i	$-0.506937\pi$
$727$	−1.17513	−0.0435833	−0.0217917	+	0.999763i	$0.506937\pi$
$728$	−32.1500	−1.19156
$729$	0	0
$730$	24.6297	0.911587
$731$	30.3103	1.12107
$732$	0	0
$733$	−30.9783	−1.14421	−0.572105	−	0.820181i	$-0.693873\pi$
$733$	−30.9783	−1.14421	−0.572105	+	0.820181i	$0.693873\pi$
$734$	−9.05955	−0.334394
$735$	0	0
$736$	5.86114	0.216044
$737$	−13.3817	−0.492920
$738$	0	0
$739$	4.13263	0.152021	0.0760106	−	0.997107i	$-0.475782\pi$
$739$	4.13263	0.152021	0.0760106	+	0.997107i	$0.475782\pi$
$740$	−26.0119	−0.956215
$741$	0	0
$742$	6.72872	0.247019
$743$	−29.3385	−1.07633	−0.538163	−	0.842841i	$-0.680881\pi$
$743$	−29.3385	−1.07633	−0.538163	+	0.842841i	$0.680881\pi$
$744$	0	0
$745$	50.8385	1.86258
$746$	−27.0957	−0.992043
$747$	0	0
$748$	−19.0002	−0.694715
$749$	43.9062	1.60430
$750$	0	0
$751$	−31.0677	−1.13368	−0.566838	−	0.823830i	$-0.691833\pi$
$751$	−31.0677	−1.13368	−0.566838	+	0.823830i	$0.691833\pi$
$752$	8.81489	0.321446
$753$	0	0
$754$	3.95848	0.144159
$755$	−36.0911	−1.31349
$756$	0	0
$757$	−34.1761	−1.24215	−0.621076	−	0.783750i	$-0.713304\pi$
$757$	−34.1761	−1.24215	−0.621076	+	0.783750i	$0.713304\pi$
$758$	−15.4106	−0.559739
$759$	0	0
$760$	−6.89960	−0.250275
$761$	−22.4770	−0.814791	−0.407395	−	0.913252i	$-0.633563\pi$
$761$	−22.4770	−0.814791	−0.407395	+	0.913252i	$0.633563\pi$
$762$	0	0
$763$	9.22573	0.333994
$764$	18.2987	0.662025
$765$	0	0
$766$	−4.55586	−0.164610
$767$	35.2670	1.27342
$768$	0	0
$769$	18.3350	0.661175	0.330588	−	0.943775i	$-0.392753\pi$
$769$	18.3350	0.661175	0.330588	+	0.943775i	$0.392753\pi$
$770$	35.4617	1.27795
$771$	0	0
$772$	−4.91561	−0.176917
$773$	−20.4086	−0.734046	−0.367023	−	0.930212i	$-0.619623\pi$
$773$	−20.4086	−0.734046	−0.367023	+	0.930212i	$0.619623\pi$
$774$	0	0
$775$	2.31473	0.0831476
$776$	−35.8848	−1.28819
$777$	0	0
$778$	−8.30950	−0.297910
$779$	1.42030	0.0508875
$780$	0	0
$781$	−68.0248	−2.43412
$782$	2.14234	0.0766100
$783$	0	0
$784$	−1.14983	−0.0410654
$785$	−53.1276	−1.89621
$786$	0	0
$787$	−2.03075	−0.0723884	−0.0361942	−	0.999345i	$-0.511523\pi$
$787$	−2.03075	−0.0723884	−0.0361942	+	0.999345i	$0.511523\pi$
$788$	−14.2006	−0.505874
$789$	0	0
$790$	30.1254	1.07181
$791$	0.567090	0.0201634
$792$	0	0
$793$	65.1625	2.31399
$794$	−18.6545	−0.662024
$795$	0	0
$796$	−12.9710	−0.459745
$797$	−41.4799	−1.46929	−0.734647	−	0.678450i	$-0.762652\pi$
$797$	−41.4799	−1.46929	−0.734647	+	0.678450i	$0.762652\pi$
$798$	0	0
$799$	29.1806	1.03234
$800$	−69.4165	−2.45424
$801$	0	0
$802$	14.2353	0.502664
$803$	−37.5248	−1.32422
$804$	0	0
$805$	9.75098	0.343677
$806$	−0.773655	−0.0272508
$807$	0	0
$808$	2.23602	0.0786627
$809$	53.1717	1.86942	0.934710	−	0.355412i	$-0.115660\pi$
$809$	53.1717	1.86942	0.934710	+	0.355412i	$0.115660\pi$
$810$	0	0
$811$	−52.5852	−1.84652	−0.923258	−	0.384181i	$-0.874484\pi$
$811$	−52.5852	−1.84652	−0.923258	+	0.384181i	$0.874484\pi$
$812$	−3.36997	−0.118263
$813$	0	0
$814$	−16.2507	−0.569588
$815$	33.1156	1.15999
$816$	0	0
$817$	−6.95805	−0.243431
$818$	−17.7699	−0.621311
$819$	0	0
$820$	12.8210	0.447729
$821$	−47.8814	−1.67107	−0.835536	−	0.549436i	$-0.814843\pi$
$821$	−47.8814	−1.67107	−0.835536	+	0.549436i	$0.814843\pi$
$822$	0	0
$823$	38.1344	1.32928	0.664641	−	0.747162i	$-0.268584\pi$
$823$	38.1344	1.32928	0.664641	+	0.747162i	$0.268584\pi$
$824$	10.0285	0.349358
$825$	0	0
$826$	12.3114	0.428370
$827$	−43.3718	−1.50818	−0.754092	−	0.656769i	$-0.771923\pi$
$827$	−43.3718	−1.50818	−0.754092	+	0.656769i	$0.771923\pi$
$828$	0	0
$829$	−17.1770	−0.596581	−0.298290	−	0.954475i	$-0.596416\pi$
$829$	−17.1770	−0.596581	−0.298290	+	0.954475i	$0.596416\pi$
$830$	45.7642	1.58850
$831$	0	0
$832$	14.3920	0.498954
$833$	−3.80638	−0.131883
$834$	0	0
$835$	−72.8280	−2.52032
$836$	4.36170	0.150853
$837$	0	0
$838$	−8.92766	−0.308401
$839$	−50.2700	−1.73551	−0.867756	−	0.496991i	$-0.834438\pi$
$839$	−50.2700	−1.73551	−0.867756	+	0.496991i	$0.834438\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	6.68301	0.230312
$843$	0	0
$844$	−4.63448	−0.159525
$845$	57.2165	1.96831
$846$	0	0
$847$	−27.8928	−0.958409
$848$	3.15120	0.108212
$849$	0	0
$850$	−25.3729	−0.870283
$851$	−4.46849	−0.153178
$852$	0	0
$853$	18.6995	0.640258	0.320129	−	0.947374i	$-0.396274\pi$
$853$	18.6995	0.640258	0.320129	+	0.947374i	$0.396274\pi$
$854$	22.7477	0.778411
$855$	0	0
$856$	−48.1762	−1.64663
$857$	−6.86614	−0.234543	−0.117271	−	0.993100i	$-0.537415\pi$
$857$	−6.86614	−0.234543	−0.117271	+	0.993100i	$0.537415\pi$
$858$	0	0
$859$	−30.9629	−1.05644	−0.528220	−	0.849108i	$-0.677140\pi$
$859$	−30.9629	−1.05644	−0.528220	+	0.849108i	$0.677140\pi$
$860$	−62.8101	−2.14181
$861$	0	0
$862$	19.2430	0.655421
$863$	−34.0856	−1.16029	−0.580143	−	0.814515i	$-0.697003\pi$
$863$	−34.0856	−1.16029	−0.580143	+	0.814515i	$0.697003\pi$
$864$	0	0
$865$	−98.0966	−3.33538
$866$	−18.5865	−0.631595
$867$	0	0
$868$	0.658635	0.0223555
$869$	−45.8978	−1.55698
$870$	0	0
$871$	−14.5655	−0.493535
$872$	−10.1230	−0.342807
$873$	0	0
$874$	−0.491798	−0.0166353
$875$	−66.7311	−2.25592
$876$	0	0
$877$	14.0788	0.475406	0.237703	−	0.971338i	$-0.423605\pi$
$877$	14.0788	0.475406	0.237703	+	0.971338i	$0.423605\pi$
$878$	−31.1358	−1.05078
$879$	0	0
$880$	16.6074	0.559836
$881$	24.2508	0.817030	0.408515	−	0.912751i	$-0.366047\pi$
$881$	24.2508	0.817030	0.408515	+	0.912751i	$0.366047\pi$
$882$	0	0
$883$	52.0180	1.75054	0.875272	−	0.483631i	$-0.160682\pi$
$883$	52.0180	1.75054	0.875272	+	0.483631i	$0.160682\pi$
$884$	−20.6811	−0.695581
$885$	0	0
$886$	−21.0024	−0.705589
$887$	−10.3246	−0.346665	−0.173333	−	0.984863i	$-0.555454\pi$
$887$	−10.3246	−0.346665	−0.173333	+	0.984863i	$0.555454\pi$
$888$	0	0
$889$	−14.3474	−0.481197
$890$	−27.2983	−0.915041
$891$	0	0
$892$	23.1922	0.776532
$893$	−6.69874	−0.224165
$894$	0	0
$895$	71.4561	2.38851
$896$	−22.8270	−0.762597
$897$	0	0
$898$	−20.6974	−0.690680
$899$	−0.195443	−0.00651838
$900$	0	0
$901$	10.4317	0.347529
$902$	8.00983	0.266698
$903$	0	0
$904$	−0.622242	−0.0206955
$905$	−36.3726	−1.20907
$906$	0	0
$907$	−36.6028	−1.21538	−0.607689	−	0.794175i	$-0.707903\pi$
$907$	−36.6028	−1.21538	−0.607689	+	0.794175i	$0.707903\pi$
$908$	8.88234	0.294771
$909$	0	0
$910$	38.5990	1.27954
$911$	25.3597	0.840205	0.420102	−	0.907477i	$-0.361994\pi$
$911$	25.3597	0.840205	0.420102	+	0.907477i	$0.361994\pi$
$912$	0	0
$913$	−69.7244	−2.30754
$914$	8.72604	0.288632
$915$	0	0
$916$	4.01074	0.132519
$917$	39.1787	1.29379
$918$	0	0
$919$	−9.02212	−0.297612	−0.148806	−	0.988866i	$-0.547543\pi$
$919$	−9.02212	−0.297612	−0.148806	+	0.988866i	$0.547543\pi$
$920$	−10.6993	−0.352746
$921$	0	0
$922$	14.2295	0.468624
$923$	−74.0429	−2.43715
$924$	0	0
$925$	52.9227	1.74009
$926$	−11.1630	−0.366838
$927$	0	0
$928$	5.86114	0.192401
$929$	−34.1265	−1.11965	−0.559827	−	0.828610i	$-0.689132\pi$
$929$	−34.1265	−1.11965	−0.559827	+	0.828610i	$0.689132\pi$
$930$	0	0
$931$	0.873795	0.0286375
$932$	−9.38591	−0.307446
$933$	0	0
$934$	8.13735	0.266262
$935$	54.9769	1.79794
$936$	0	0
$937$	−34.4584	−1.12571	−0.562854	−	0.826557i	$-0.690297\pi$
$937$	−34.4584	−1.12571	−0.562854	+	0.826557i	$0.690297\pi$
$938$	−5.08472	−0.166022
$939$	0	0
$940$	−60.4693	−1.97229
$941$	−9.00188	−0.293453	−0.146727	−	0.989177i	$-0.546874\pi$
$941$	−9.00188	−0.293453	−0.146727	+	0.989177i	$0.546874\pi$
$942$	0	0
$943$	2.20248	0.0717225
$944$	5.76570	0.187657
$945$	0	0
$946$	−39.2401	−1.27581
$947$	−51.1277	−1.66143	−0.830713	−	0.556701i	$-0.812067\pi$
$947$	−51.1277	−1.66143	−0.830713	+	0.556701i	$0.812067\pi$
$948$	0	0
$949$	−40.8446	−1.32587
$950$	5.82462	0.188976
$951$	0	0
$952$	−17.3997	−0.563927
$953$	−7.73247	−0.250479	−0.125240	−	0.992127i	$-0.539970\pi$
$953$	−7.73247	−0.250479	−0.125240	+	0.992127i	$0.539970\pi$
$954$	0	0
$955$	−52.9473	−1.71334
$956$	24.6737	0.798003
$957$	0	0
$958$	24.4493	0.789922
$959$	10.7379	0.346745
$960$	0	0
$961$	−30.9618	−0.998768
$962$	−17.6884	−0.570297
$963$	0	0
$964$	−4.62835	−0.149069
$965$	14.2233	0.457864
$966$	0	0
$967$	15.6598	0.503586	0.251793	−	0.967781i	$-0.418980\pi$
$967$	15.6598	0.503586	0.251793	+	0.967781i	$0.418980\pi$
$968$	30.6055	0.983699
$969$	0	0
$970$	43.0830	1.38331
$971$	21.6197	0.693810	0.346905	−	0.937900i	$-0.387233\pi$
$971$	21.6197	0.693810	0.346905	+	0.937900i	$0.387233\pi$
$972$	0	0
$973$	−55.4232	−1.77679
$974$	18.1612	0.581924
$975$	0	0
$976$	10.6532	0.341001
$977$	−5.57404	−0.178329	−0.0891647	−	0.996017i	$-0.528420\pi$
$977$	−5.57404	−0.178329	−0.0891647	+	0.996017i	$0.528420\pi$
$978$	0	0
$979$	41.5905	1.32924
$980$	7.88772	0.251964
$981$	0	0
$982$	−13.9400	−0.444843
$983$	43.2735	1.38021	0.690105	−	0.723710i	$-0.257565\pi$
$983$	43.2735	1.38021	0.690105	+	0.723710i	$0.257565\pi$
$984$	0	0
$985$	41.0892	1.30921
$986$	2.14234	0.0682261
$987$	0	0
$988$	4.74758	0.151041
$989$	−10.7899	−0.343100
$990$	0	0
$991$	9.95405	0.316201	0.158100	−	0.987423i	$-0.449463\pi$
$991$	9.95405	0.316201	0.158100	+	0.987423i	$0.449463\pi$
$992$	−1.14552	−0.0363702
$993$	0	0
$994$	−25.8478	−0.819843
$995$	37.5315	1.18983
$996$	0	0
$997$	9.24724	0.292863	0.146431	−	0.989221i	$-0.453221\pi$
$997$	9.24724	0.292863	0.146431	+	0.989221i	$0.453221\pi$
$998$	−3.70106	−0.117155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.8		20
3.2	odd	2		2001.2.a.o.1.13	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.13	✓	20	3.2	odd	2
6003.2.a.s.1.8		20	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)

\(f(q)\)	\(=\)	\(q-0.762638 q^{2} -1.41838 q^{4} +4.10409 q^{5} -2.37592 q^{7} +2.60699 q^{8} +O(q^{10})\)	\(q-0.762638 q^{2} -1.41838 q^{4} +4.10409 q^{5} -2.37592 q^{7} +2.60699 q^{8} -3.12993 q^{10} +4.76863 q^{11} +5.19051 q^{13} +1.81197 q^{14} +0.848580 q^{16} +2.80912 q^{17} -0.644865 q^{19} -5.82117 q^{20} -3.63674 q^{22} -1.00000 q^{23} +11.8435 q^{25} -3.95848 q^{26} +3.36997 q^{28} -1.00000 q^{29} +0.195443 q^{31} -5.86114 q^{32} -2.14234 q^{34} -9.75098 q^{35} +4.46849 q^{37} +0.491798 q^{38} +10.6993 q^{40} -2.20248 q^{41} +10.7899 q^{43} -6.76374 q^{44} +0.762638 q^{46} +10.3878 q^{47} -1.35501 q^{49} -9.03232 q^{50} -7.36213 q^{52} +3.71349 q^{53} +19.5709 q^{55} -6.19400 q^{56} +0.762638 q^{58} +6.79452 q^{59} +12.5542 q^{61} -0.149052 q^{62} +2.77276 q^{64} +21.3023 q^{65} -2.80619 q^{67} -3.98441 q^{68} +7.43647 q^{70} -14.2651 q^{71} -7.86909 q^{73} -3.40784 q^{74} +0.914665 q^{76} -11.3299 q^{77} -9.62495 q^{79} +3.48264 q^{80} +1.67969 q^{82} -14.6215 q^{83} +11.5289 q^{85} -8.22881 q^{86} +12.4318 q^{88} +8.72169 q^{89} -12.3322 q^{91} +1.41838 q^{92} -7.92214 q^{94} -2.64658 q^{95} -13.7648 q^{97} +1.03338 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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