LMFDB - Embedded newform 4005.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.58455$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58455 q^{2} +4.67989 q^{4} -1.00000 q^{5} +1.54056 q^{7} -6.92630 q^{8} +O(q^{10})$	\(q-2.58455 q^{2} +4.67989 q^{4} -1.00000 q^{5} +1.54056 q^{7} -6.92630 q^{8} +2.58455 q^{10} -3.66358 q^{11} -2.74217 q^{13} -3.98165 q^{14} +8.54157 q^{16} +6.35107 q^{17} +0.0289073 q^{19} -4.67989 q^{20} +9.46869 q^{22} -6.04156 q^{23} +1.00000 q^{25} +7.08726 q^{26} +7.20964 q^{28} -0.507991 q^{29} -0.766003 q^{31} -8.22350 q^{32} -16.4147 q^{34} -1.54056 q^{35} +4.38891 q^{37} -0.0747124 q^{38} +6.92630 q^{40} +1.18586 q^{41} +2.33138 q^{43} -17.1451 q^{44} +15.6147 q^{46} +12.2198 q^{47} -4.62668 q^{49} -2.58455 q^{50} -12.8330 q^{52} +7.61379 q^{53} +3.66358 q^{55} -10.6704 q^{56} +1.31293 q^{58} -2.39526 q^{59} +10.2022 q^{61} +1.97977 q^{62} +4.17088 q^{64} +2.74217 q^{65} -6.72929 q^{67} +29.7223 q^{68} +3.98165 q^{70} -10.8637 q^{71} -3.52821 q^{73} -11.3433 q^{74} +0.135283 q^{76} -5.64396 q^{77} -5.47483 q^{79} -8.54157 q^{80} -3.06490 q^{82} -6.74941 q^{83} -6.35107 q^{85} -6.02557 q^{86} +25.3750 q^{88} +1.00000 q^{89} -4.22447 q^{91} -28.2738 q^{92} -31.5826 q^{94} -0.0289073 q^{95} +12.1041 q^{97} +11.9579 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * 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 $

$2$	−2.58455	−1.82755	−0.913776	−	0.406219i	$-0.866847\pi$
$2$	−2.58455	−1.82755	−0.913776	+	0.406219i	$0.866847\pi$
$3$	0	0
$3$	0	0
$4$	4.67989	2.33994
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.54056	0.582277	0.291138	−	0.956681i	$-0.405966\pi$
$7$	1.54056	0.582277	0.291138	+	0.956681i	$0.405966\pi$
$8$	−6.92630	−2.44882
$9$	0	0
$10$	2.58455	0.817306
$11$	−3.66358	−1.10461	−0.552305	−	0.833642i	$-0.686252\pi$
$11$	−3.66358	−1.10461	−0.552305	+	0.833642i	$0.686252\pi$
$12$	0	0
$13$	−2.74217	−0.760540	−0.380270	−	0.924875i	$-0.624169\pi$
$13$	−2.74217	−0.760540	−0.380270	+	0.924875i	$0.624169\pi$
$14$	−3.98165	−1.06414
$15$	0	0
$16$	8.54157	2.13539
$17$	6.35107	1.54036	0.770181	−	0.637825i	$-0.220166\pi$
$17$	6.35107	1.54036	0.770181	+	0.637825i	$0.220166\pi$
$18$	0	0
$19$	0.0289073	0.00663180	0.00331590	−	0.999995i	$-0.498945\pi$
$19$	0.0289073	0.00663180	0.00331590	+	0.999995i	$0.498945\pi$
$20$	−4.67989	−1.04645
$21$	0	0
$22$	9.46869	2.01873
$23$	−6.04156	−1.25975	−0.629876	−	0.776696i	$-0.716894\pi$
$23$	−6.04156	−1.25975	−0.629876	+	0.776696i	$0.716894\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	7.08726	1.38993
$27$	0	0
$28$	7.20964	1.36249
$29$	−0.507991	−0.0943316	−0.0471658	−	0.998887i	$-0.515019\pi$
$29$	−0.507991	−0.0943316	−0.0471658	+	0.998887i	$0.515019\pi$
$30$	0	0
$31$	−0.766003	−0.137578	−0.0687891	−	0.997631i	$-0.521914\pi$
$31$	−0.766003	−0.137578	−0.0687891	+	0.997631i	$0.521914\pi$
$32$	−8.22350	−1.45372
$33$	0	0
$34$	−16.4147	−2.81509
$35$	−1.54056	−0.260402
$36$	0	0
$37$	4.38891	0.721532	0.360766	−	0.932656i	$-0.382515\pi$
$37$	4.38891	0.721532	0.360766	+	0.932656i	$0.382515\pi$
$38$	−0.0747124	−0.0121199
$39$	0	0
$40$	6.92630	1.09514
$41$	1.18586	0.185200	0.0925998	−	0.995703i	$-0.470482\pi$
$41$	1.18586	0.185200	0.0925998	+	0.995703i	$0.470482\pi$
$42$	0	0
$43$	2.33138	0.355533	0.177766	−	0.984073i	$-0.443113\pi$
$43$	2.33138	0.355533	0.177766	+	0.984073i	$0.443113\pi$
$44$	−17.1451	−2.58473
$45$	0	0
$46$	15.6147	2.30226
$47$	12.2198	1.78244	0.891218	−	0.453575i	$-0.149852\pi$
$47$	12.2198	1.78244	0.891218	+	0.453575i	$0.149852\pi$
$48$	0	0
$49$	−4.62668	−0.660954
$50$	−2.58455	−0.365510
$51$	0	0
$52$	−12.8330	−1.77962
$53$	7.61379	1.04583	0.522917	−	0.852384i	$-0.324844\pi$
$53$	7.61379	1.04583	0.522917	+	0.852384i	$0.324844\pi$
$54$	0	0
$55$	3.66358	0.493997
$56$	−10.6704	−1.42589
$57$	0	0
$58$	1.31293	0.172396
$59$	−2.39526	−0.311836	−0.155918	−	0.987770i	$-0.549834\pi$
$59$	−2.39526	−0.311836	−0.155918	+	0.987770i	$0.549834\pi$
$60$	0	0
$61$	10.2022	1.30626	0.653129	−	0.757247i	$-0.273456\pi$
$61$	10.2022	1.30626	0.653129	+	0.757247i	$0.273456\pi$
$62$	1.97977	0.251431
$63$	0	0
$64$	4.17088	0.521360
$65$	2.74217	0.340124
$66$	0	0
$67$	−6.72929	−0.822114	−0.411057	−	0.911610i	$-0.634840\pi$
$67$	−6.72929	−0.822114	−0.411057	+	0.911610i	$0.634840\pi$
$68$	29.7223	3.60436
$69$	0	0
$70$	3.98165	0.475898
$71$	−10.8637	−1.28929	−0.644645	−	0.764482i	$-0.722995\pi$
$71$	−10.8637	−1.28929	−0.644645	+	0.764482i	$0.722995\pi$
$72$	0	0
$73$	−3.52821	−0.412946	−0.206473	−	0.978452i	$-0.566199\pi$
$73$	−3.52821	−0.412946	−0.206473	+	0.978452i	$0.566199\pi$
$74$	−11.3433	−1.31864
$75$	0	0
$76$	0.135283	0.0155180
$77$	−5.64396	−0.643189
$78$	0	0
$79$	−5.47483	−0.615967	−0.307983	−	0.951392i	$-0.599654\pi$
$79$	−5.47483	−0.615967	−0.307983	+	0.951392i	$0.599654\pi$
$80$	−8.54157	−0.954976
$81$	0	0
$82$	−3.06490	−0.338462
$83$	−6.74941	−0.740844	−0.370422	−	0.928864i	$-0.620787\pi$
$83$	−6.74941	−0.740844	−0.370422	+	0.928864i	$0.620787\pi$
$84$	0	0
$85$	−6.35107	−0.688871
$86$	−6.02557	−0.649754
$87$	0	0
$88$	25.3750	2.70499
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−4.22447	−0.442845
$92$	−28.2738	−2.94775
$93$	0	0
$94$	−31.5826	−3.25749
$95$	−0.0289073	−0.00296583
$96$	0	0
$97$	12.1041	1.22898	0.614492	−	0.788923i	$-0.289361\pi$
$97$	12.1041	1.22898	0.614492	+	0.788923i	$0.289361\pi$
$98$	11.9579	1.20793
$99$	0	0
$100$	4.67989	0.467989
$101$	−15.2132	−1.51377	−0.756885	−	0.653548i	$-0.773280\pi$
$101$	−15.2132	−1.51377	−0.756885	+	0.653548i	$0.773280\pi$
$102$	0	0
$103$	−10.2630	−1.01124	−0.505620	−	0.862756i	$-0.668736\pi$
$103$	−10.2630	−1.01124	−0.505620	+	0.862756i	$0.668736\pi$
$104$	18.9931	1.86242
$105$	0	0
$106$	−19.6782	−1.91131
$107$	−2.93891	−0.284115	−0.142058	−	0.989858i	$-0.545372\pi$
$107$	−2.93891	−0.284115	−0.142058	+	0.989858i	$0.545372\pi$
$108$	0	0
$109$	−5.87776	−0.562987	−0.281494	−	0.959563i	$-0.590830\pi$
$109$	−5.87776	−0.562987	−0.281494	+	0.959563i	$0.590830\pi$
$110$	−9.46869	−0.902804
$111$	0	0
$112$	13.1588	1.24339
$113$	3.91936	0.368702	0.184351	−	0.982860i	$-0.440982\pi$
$113$	3.91936	0.368702	0.184351	+	0.982860i	$0.440982\pi$
$114$	0	0
$115$	6.04156	0.563378
$116$	−2.37734	−0.220731
$117$	0	0
$118$	6.19066	0.569897
$119$	9.78421	0.896917
$120$	0	0
$121$	2.42180	0.220164
$122$	−26.3681	−2.38725
$123$	0	0
$124$	−3.58481	−0.321925
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.28211	−0.291240	−0.145620	−	0.989341i	$-0.546518\pi$
$127$	−3.28211	−0.291240	−0.145620	+	0.989341i	$0.546518\pi$
$128$	5.66714	0.500909
$129$	0	0
$130$	−7.08726	−0.621594
$131$	10.8390	0.947007	0.473503	−	0.880792i	$-0.342989\pi$
$131$	10.8390	0.947007	0.473503	+	0.880792i	$0.342989\pi$
$132$	0	0
$133$	0.0445334	0.00386154
$134$	17.3922	1.50245
$135$	0	0
$136$	−43.9894	−3.77206
$137$	19.0730	1.62951	0.814757	−	0.579803i	$-0.196870\pi$
$137$	19.0730	1.62951	0.814757	+	0.579803i	$0.196870\pi$
$138$	0	0
$139$	−21.1313	−1.79234	−0.896168	−	0.443715i	$-0.853660\pi$
$139$	−21.1313	−1.79234	−0.896168	+	0.443715i	$0.853660\pi$
$140$	−7.20964	−0.609326
$141$	0	0
$142$	28.0779	2.35624
$143$	10.0461	0.840100
$144$	0	0
$145$	0.507991	0.0421864
$146$	9.11883	0.754680
$147$	0	0
$148$	20.5396	1.68834
$149$	8.99920	0.737243	0.368622	−	0.929579i	$-0.379830\pi$
$149$	8.99920	0.737243	0.368622	+	0.929579i	$0.379830\pi$
$150$	0	0
$151$	3.85805	0.313964	0.156982	−	0.987601i	$-0.449824\pi$
$151$	3.85805	0.313964	0.156982	+	0.987601i	$0.449824\pi$
$152$	−0.200221	−0.0162400
$153$	0	0
$154$	14.5871	1.17546
$155$	0.766003	0.0615269
$156$	0	0
$157$	−8.45176	−0.674524	−0.337262	−	0.941411i	$-0.609501\pi$
$157$	−8.45176	−0.674524	−0.337262	+	0.941411i	$0.609501\pi$
$158$	14.1500	1.12571
$159$	0	0
$160$	8.22350	0.650124
$161$	−9.30738	−0.733524
$162$	0	0
$163$	−13.9653	−1.09385	−0.546923	−	0.837183i	$-0.684201\pi$
$163$	−13.9653	−1.09385	−0.546923	+	0.837183i	$0.684201\pi$
$164$	5.54967	0.433357
$165$	0	0
$166$	17.4442	1.35393
$167$	0.722870	0.0559373	0.0279687	−	0.999609i	$-0.491096\pi$
$167$	0.722870	0.0559373	0.0279687	+	0.999609i	$0.491096\pi$
$168$	0	0
$169$	−5.48052	−0.421579
$170$	16.4147	1.25895
$171$	0	0
$172$	10.9106	0.831927
$173$	−0.762488	−0.0579709	−0.0289854	−	0.999580i	$-0.509228\pi$
$173$	−0.762488	−0.0579709	−0.0289854	+	0.999580i	$0.509228\pi$
$174$	0	0
$175$	1.54056	0.116455
$176$	−31.2927	−2.35878
$177$	0	0
$178$	−2.58455	−0.193720
$179$	−0.191909	−0.0143439	−0.00717197	−	0.999974i	$-0.502283\pi$
$179$	−0.191909	−0.0143439	−0.00717197	+	0.999974i	$0.502283\pi$
$180$	0	0
$181$	−19.6716	−1.46218	−0.731089	−	0.682282i	$-0.760988\pi$
$181$	−19.6716	−1.46218	−0.731089	+	0.682282i	$0.760988\pi$
$182$	10.9183	0.809321
$183$	0	0
$184$	41.8456	3.08490
$185$	−4.38891	−0.322679
$186$	0	0
$187$	−23.2677	−1.70150
$188$	57.1871	4.17080
$189$	0	0
$190$	0.0747124	0.00542020
$191$	2.61269	0.189048	0.0945239	−	0.995523i	$-0.469867\pi$
$191$	2.61269	0.189048	0.0945239	+	0.995523i	$0.469867\pi$
$192$	0	0
$193$	0.384019	0.0276423	0.0138211	−	0.999904i	$-0.495600\pi$
$193$	0.384019	0.0276423	0.0138211	+	0.999904i	$0.495600\pi$
$194$	−31.2836	−2.24603
$195$	0	0
$196$	−21.6523	−1.54659
$197$	−9.41301	−0.670649	−0.335325	−	0.942103i	$-0.608846\pi$
$197$	−9.41301	−0.670649	−0.335325	+	0.942103i	$0.608846\pi$
$198$	0	0
$199$	−4.98426	−0.353325	−0.176662	−	0.984271i	$-0.556530\pi$
$199$	−4.98426	−0.353325	−0.176662	+	0.984271i	$0.556530\pi$
$200$	−6.92630	−0.489763
$201$	0	0
$202$	39.3193	2.76649
$203$	−0.782590	−0.0549271
$204$	0	0
$205$	−1.18586	−0.0828238
$206$	26.5251	1.84809
$207$	0	0
$208$	−23.4224	−1.62405
$209$	−0.105904	−0.00732555
$210$	0	0
$211$	6.18788	0.425991	0.212996	−	0.977053i	$-0.431678\pi$
$211$	6.18788	0.425991	0.212996	+	0.977053i	$0.431678\pi$
$212$	35.6317	2.44719
$213$	0	0
$214$	7.59576	0.519235
$215$	−2.33138	−0.158999
$216$	0	0
$217$	−1.18007	−0.0801086
$218$	15.1914	1.02889
$219$	0	0
$220$	17.1451	1.15592
$221$	−17.4157	−1.17151
$222$	0	0
$223$	27.6002	1.84825	0.924123	−	0.382094i	$-0.124797\pi$
$223$	27.6002	1.84825	0.924123	+	0.382094i	$0.124797\pi$
$224$	−12.6688	−0.846468
$225$	0	0
$226$	−10.1298	−0.673823
$227$	19.9221	1.32228	0.661140	−	0.750263i	$-0.270073\pi$
$227$	19.9221	1.32228	0.661140	+	0.750263i	$0.270073\pi$
$228$	0	0
$229$	18.6071	1.22959	0.614795	−	0.788687i	$-0.289239\pi$
$229$	18.6071	1.22959	0.614795	+	0.788687i	$0.289239\pi$
$230$	−15.6147	−1.02960
$231$	0	0
$232$	3.51850	0.231001
$233$	−14.1851	−0.929297	−0.464649	−	0.885495i	$-0.653819\pi$
$233$	−14.1851	−0.929297	−0.464649	+	0.885495i	$0.653819\pi$
$234$	0	0
$235$	−12.2198	−0.797130
$236$	−11.2095	−0.729679
$237$	0	0
$238$	−25.2877	−1.63916
$239$	−10.3524	−0.669643	−0.334821	−	0.942282i	$-0.608676\pi$
$239$	−10.3524	−0.669643	−0.334821	+	0.942282i	$0.608676\pi$
$240$	0	0
$241$	29.6050	1.90702	0.953511	−	0.301357i	$-0.0974396\pi$
$241$	29.6050	1.90702	0.953511	+	0.301357i	$0.0974396\pi$
$242$	−6.25926	−0.402360
$243$	0	0
$244$	47.7451	3.05657
$245$	4.62668	0.295588
$246$	0	0
$247$	−0.0792687	−0.00504375
$248$	5.30557	0.336904
$249$	0	0
$250$	2.58455	0.163461
$251$	−12.4055	−0.783030	−0.391515	−	0.920172i	$-0.628049\pi$
$251$	−12.4055	−0.783030	−0.391515	+	0.920172i	$0.628049\pi$
$252$	0	0
$253$	22.1337	1.39153
$254$	8.48277	0.532256
$255$	0	0
$256$	−22.9888	−1.43680
$257$	−27.1084	−1.69098	−0.845488	−	0.533995i	$-0.820690\pi$
$257$	−27.1084	−1.69098	−0.845488	+	0.533995i	$0.820690\pi$
$258$	0	0
$259$	6.76137	0.420131
$260$	12.8330	0.795871
$261$	0	0
$262$	−28.0139	−1.73070
$263$	−2.47834	−0.152821	−0.0764104	−	0.997076i	$-0.524346\pi$
$263$	−2.47834	−0.152821	−0.0764104	+	0.997076i	$0.524346\pi$
$264$	0	0
$265$	−7.61379	−0.467711
$266$	−0.115099	−0.00705716
$267$	0	0
$268$	−31.4923	−1.92370
$269$	−20.7892	−1.26754	−0.633770	−	0.773521i	$-0.718494\pi$
$269$	−20.7892	−1.26754	−0.633770	+	0.773521i	$0.718494\pi$
$270$	0	0
$271$	−19.0355	−1.15632	−0.578161	−	0.815923i	$-0.696229\pi$
$271$	−19.0355	−1.15632	−0.578161	+	0.815923i	$0.696229\pi$
$272$	54.2481	3.28928
$273$	0	0
$274$	−49.2950	−2.97802
$275$	−3.66358	−0.220922
$276$	0	0
$277$	10.4090	0.625414	0.312707	−	0.949850i	$-0.398764\pi$
$277$	10.4090	0.625414	0.312707	+	0.949850i	$0.398764\pi$
$278$	54.6149	3.27559
$279$	0	0
$280$	10.6704	0.637676
$281$	−10.4450	−0.623095	−0.311548	−	0.950231i	$-0.600847\pi$
$281$	−10.4450	−0.623095	−0.311548	+	0.950231i	$0.600847\pi$
$282$	0	0
$283$	−29.0600	−1.72744	−0.863718	−	0.503976i	$-0.831870\pi$
$283$	−29.0600	−1.72744	−0.863718	+	0.503976i	$0.831870\pi$
$284$	−50.8411	−3.01686
$285$	0	0
$286$	−25.9647	−1.53533
$287$	1.82688	0.107837
$288$	0	0
$289$	23.3362	1.37271
$290$	−1.31293	−0.0770978
$291$	0	0
$292$	−16.5116	−0.966270
$293$	16.0102	0.935323	0.467662	−	0.883908i	$-0.345097\pi$
$293$	16.0102	0.935323	0.467662	+	0.883908i	$0.345097\pi$
$294$	0	0
$295$	2.39526	0.139457
$296$	−30.3989	−1.76690
$297$	0	0
$298$	−23.2589	−1.34735
$299$	16.5670	0.958092
$300$	0	0
$301$	3.59164	0.207018
$302$	−9.97132	−0.573785
$303$	0	0
$304$	0.246914	0.0141615
$305$	−10.2022	−0.584176
$306$	0	0
$307$	28.8481	1.64645	0.823223	−	0.567719i	$-0.192174\pi$
$307$	28.8481	1.64645	0.823223	+	0.567719i	$0.192174\pi$
$308$	−26.4131	−1.50502
$309$	0	0
$310$	−1.97977	−0.112444
$311$	−10.6287	−0.602698	−0.301349	−	0.953514i	$-0.597437\pi$
$311$	−10.6287	−0.602698	−0.301349	+	0.953514i	$0.597437\pi$
$312$	0	0
$313$	18.3276	1.03594	0.517969	−	0.855400i	$-0.326688\pi$
$313$	18.3276	1.03594	0.517969	+	0.855400i	$0.326688\pi$
$314$	21.8440	1.23273
$315$	0	0
$316$	−25.6216	−1.44133
$317$	18.3898	1.03287	0.516437	−	0.856325i	$-0.327258\pi$
$317$	18.3898	1.03287	0.516437	+	0.856325i	$0.327258\pi$
$318$	0	0
$319$	1.86107	0.104200
$320$	−4.17088	−0.233159
$321$	0	0
$322$	24.0554	1.34055
$323$	0.183593	0.0102154
$324$	0	0
$325$	−2.74217	−0.152108
$326$	36.0939	1.99906
$327$	0	0
$328$	−8.21359	−0.453520
$329$	18.8253	1.03787
$330$	0	0
$331$	−23.1495	−1.27241	−0.636206	−	0.771519i	$-0.719497\pi$
$331$	−23.1495	−1.27241	−0.636206	+	0.771519i	$0.719497\pi$
$332$	−31.5865	−1.73353
$333$	0	0
$334$	−1.86829	−0.102228
$335$	6.72929	0.367660
$336$	0	0
$337$	−26.9149	−1.46615	−0.733073	−	0.680150i	$-0.761915\pi$
$337$	−26.9149	−1.46615	−0.733073	+	0.680150i	$0.761915\pi$
$338$	14.1647	0.770457
$339$	0	0
$340$	−29.7223	−1.61192
$341$	2.80631	0.151970
$342$	0	0
$343$	−17.9116	−0.967135
$344$	−16.1479	−0.870634
$345$	0	0
$346$	1.97069	0.105945
$347$	−18.1261	−0.973061	−0.486531	−	0.873664i	$-0.661738\pi$
$347$	−18.1261	−0.973061	−0.486531	+	0.873664i	$0.661738\pi$
$348$	0	0
$349$	19.4709	1.04225	0.521127	−	0.853479i	$-0.325512\pi$
$349$	19.4709	1.04225	0.521127	+	0.853479i	$0.325512\pi$
$350$	−3.98165	−0.212828
$351$	0	0
$352$	30.1274	1.60580
$353$	−33.7488	−1.79627	−0.898134	−	0.439722i	$-0.855077\pi$
$353$	−33.7488	−1.79627	−0.898134	+	0.439722i	$0.855077\pi$
$354$	0	0
$355$	10.8637	0.576588
$356$	4.67989	0.248034
$357$	0	0
$358$	0.495998	0.0262143
$359$	32.6747	1.72450	0.862252	−	0.506479i	$-0.169053\pi$
$359$	32.6747	1.72450	0.862252	+	0.506479i	$0.169053\pi$
$360$	0	0
$361$	−18.9992	−0.999956
$362$	50.8422	2.67221
$363$	0	0
$364$	−19.7700	−1.03623
$365$	3.52821	0.184675
$366$	0	0
$367$	−5.67771	−0.296374	−0.148187	−	0.988959i	$-0.547344\pi$
$367$	−5.67771	−0.296374	−0.148187	+	0.988959i	$0.547344\pi$
$368$	−51.6044	−2.69006
$369$	0	0
$370$	11.3433	0.589713
$371$	11.7295	0.608965
$372$	0	0
$373$	−6.57368	−0.340372	−0.170186	−	0.985412i	$-0.554437\pi$
$373$	−6.57368	−0.340372	−0.170186	+	0.985412i	$0.554437\pi$
$374$	60.1364	3.10958
$375$	0	0
$376$	−84.6377	−4.36486
$377$	1.39300	0.0717430
$378$	0	0
$379$	−3.48353	−0.178937	−0.0894684	−	0.995990i	$-0.528517\pi$
$379$	−3.48353	−0.178937	−0.0894684	+	0.995990i	$0.528517\pi$
$380$	−0.135283	−0.00693987
$381$	0	0
$382$	−6.75263	−0.345495
$383$	−17.6317	−0.900937	−0.450468	−	0.892792i	$-0.648743\pi$
$383$	−17.6317	−0.900937	−0.450468	+	0.892792i	$0.648743\pi$
$384$	0	0
$385$	5.64396	0.287643
$386$	−0.992515	−0.0505177
$387$	0	0
$388$	56.6458	2.87575
$389$	−35.5377	−1.80183	−0.900917	−	0.433992i	$-0.857105\pi$
$389$	−35.5377	−1.80183	−0.900917	+	0.433992i	$0.857105\pi$
$390$	0	0
$391$	−38.3704	−1.94047
$392$	32.0457	1.61855
$393$	0	0
$394$	24.3284	1.22565
$395$	5.47483	0.275469
$396$	0	0
$397$	−3.37989	−0.169632	−0.0848159	−	0.996397i	$-0.527030\pi$
$397$	−3.37989	−0.169632	−0.0848159	+	0.996397i	$0.527030\pi$
$398$	12.8821	0.645719
$399$	0	0
$400$	8.54157	0.427078
$401$	−30.1776	−1.50700	−0.753499	−	0.657450i	$-0.771635\pi$
$401$	−30.1776	−1.50700	−0.753499	+	0.657450i	$0.771635\pi$
$402$	0	0
$403$	2.10051	0.104634
$404$	−71.1961	−3.54214
$405$	0	0
$406$	2.02264	0.100382
$407$	−16.0791	−0.797012
$408$	0	0
$409$	30.6992	1.51798	0.758990	−	0.651102i	$-0.225693\pi$
$409$	30.6992	1.51798	0.758990	+	0.651102i	$0.225693\pi$
$410$	3.06490	0.151365
$411$	0	0
$412$	−48.0295	−2.36624
$413$	−3.69004	−0.181575
$414$	0	0
$415$	6.74941	0.331316
$416$	22.5502	1.10561
$417$	0	0
$418$	0.273715	0.0133878
$419$	−15.8695	−0.775275	−0.387638	−	0.921812i	$-0.626709\pi$
$419$	−15.8695	−0.775275	−0.387638	+	0.921812i	$0.626709\pi$
$420$	0	0
$421$	−4.16569	−0.203024	−0.101512	−	0.994834i	$-0.532368\pi$
$421$	−4.16569	−0.203024	−0.101512	+	0.994834i	$0.532368\pi$
$422$	−15.9929	−0.778521
$423$	0	0
$424$	−52.7353	−2.56105
$425$	6.35107	0.308072
$426$	0	0
$427$	15.7171	0.760603
$428$	−13.7538	−0.664814
$429$	0	0
$430$	6.02557	0.290579
$431$	−33.5279	−1.61498	−0.807492	−	0.589879i	$-0.799175\pi$
$431$	−33.5279	−1.61498	−0.807492	+	0.589879i	$0.799175\pi$
$432$	0	0
$433$	23.3112	1.12026	0.560132	−	0.828404i	$-0.310751\pi$
$433$	23.3112	1.12026	0.560132	+	0.828404i	$0.310751\pi$
$434$	3.04996	0.146403
$435$	0	0
$436$	−27.5073	−1.31736
$437$	−0.174645	−0.00835442
$438$	0	0
$439$	7.71600	0.368264	0.184132	−	0.982901i	$-0.441053\pi$
$439$	7.71600	0.368264	0.184132	+	0.982901i	$0.441053\pi$
$440$	−25.3750	−1.20971
$441$	0	0
$442$	45.0117	2.14099
$443$	−13.0799	−0.621443	−0.310721	−	0.950501i	$-0.600571\pi$
$443$	−13.0799	−0.621443	−0.310721	+	0.950501i	$0.600571\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−71.3341	−3.37777
$447$	0	0
$448$	6.42549	0.303576
$449$	−16.1709	−0.763153	−0.381577	−	0.924337i	$-0.624619\pi$
$449$	−16.1709	−0.763153	−0.381577	+	0.924337i	$0.624619\pi$
$450$	0	0
$451$	−4.34448	−0.204573
$452$	18.3422	0.862743
$453$	0	0
$454$	−51.4897	−2.41653
$455$	4.22447	0.198046
$456$	0	0
$457$	−34.8854	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$457$	−34.8854	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$458$	−48.0909	−2.24714
$459$	0	0
$460$	28.2738	1.31827
$461$	−41.0155	−1.91028	−0.955141	−	0.296152i	$-0.904296\pi$
$461$	−41.0155	−1.91028	−0.955141	+	0.296152i	$0.904296\pi$
$462$	0	0
$463$	−14.3580	−0.667272	−0.333636	−	0.942702i	$-0.608276\pi$
$463$	−14.3580	−0.667272	−0.333636	+	0.942702i	$0.608276\pi$
$464$	−4.33904	−0.201435
$465$	0	0
$466$	36.6621	1.69834
$467$	−16.8986	−0.781974	−0.390987	−	0.920396i	$-0.627866\pi$
$467$	−16.8986	−0.781974	−0.390987	+	0.920396i	$0.627866\pi$
$468$	0	0
$469$	−10.3669	−0.478698
$470$	31.5826	1.45679
$471$	0	0
$472$	16.5903	0.763629
$473$	−8.54121	−0.392725
$474$	0	0
$475$	0.0289073	0.00132636
$476$	45.7890	2.09873
$477$	0	0
$478$	26.7563	1.22381
$479$	−11.0240	−0.503699	−0.251850	−	0.967766i	$-0.581039\pi$
$479$	−11.0240	−0.503699	−0.251850	+	0.967766i	$0.581039\pi$
$480$	0	0
$481$	−12.0351	−0.548754
$482$	−76.5154	−3.48518
$483$	0	0
$484$	11.3337	0.515170
$485$	−12.1041	−0.549618
$486$	0	0
$487$	24.9199	1.12923	0.564615	−	0.825354i	$-0.309025\pi$
$487$	24.9199	1.12923	0.564615	+	0.825354i	$0.309025\pi$
$488$	−70.6634	−3.19878
$489$	0	0
$490$	−11.9579	−0.540201
$491$	−0.0211847	−0.000956051 0	−0.000478026	−	1.00000i	$-0.500152\pi$
$491$	−0.0211847	−0.000956051 0	−0.000478026		1.00000i	$0.500152\pi$
$492$	0	0
$493$	−3.22629	−0.145305
$494$	0.204874	0.00921770
$495$	0	0
$496$	−6.54287	−0.293783
$497$	−16.7362	−0.750723
$498$	0	0
$499$	−38.2628	−1.71288	−0.856439	−	0.516249i	$-0.827328\pi$
$499$	−38.2628	−1.71288	−0.856439	+	0.516249i	$0.827328\pi$
$500$	−4.67989	−0.209291
$501$	0	0
$502$	32.0627	1.43103
$503$	−14.0692	−0.627316	−0.313658	−	0.949536i	$-0.601555\pi$
$503$	−14.0692	−0.627316	−0.313658	+	0.949536i	$0.601555\pi$
$504$	0	0
$505$	15.2132	0.676979
$506$	−57.2056	−2.54310
$507$	0	0
$508$	−15.3599	−0.681485
$509$	22.4452	0.994866	0.497433	−	0.867502i	$-0.334276\pi$
$509$	22.4452	0.994866	0.497433	+	0.867502i	$0.334276\pi$
$510$	0	0
$511$	−5.43542	−0.240449
$512$	48.0813	2.12491
$513$	0	0
$514$	70.0629	3.09034
$515$	10.2630	0.452240
$516$	0	0
$517$	−44.7681	−1.96890
$518$	−17.4751	−0.767812
$519$	0	0
$520$	−18.9931	−0.832900
$521$	10.0278	0.439328	0.219664	−	0.975576i	$-0.429504\pi$
$521$	10.0278	0.439328	0.219664	+	0.975576i	$0.429504\pi$
$522$	0	0
$523$	−22.5646	−0.986682	−0.493341	−	0.869836i	$-0.664225\pi$
$523$	−22.5646	−0.986682	−0.493341	+	0.869836i	$0.664225\pi$
$524$	50.7252	2.21594
$525$	0	0
$526$	6.40538	0.279288
$527$	−4.86495	−0.211920
$528$	0	0
$529$	13.5004	0.586975
$530$	19.6782	0.854766
$531$	0	0
$532$	0.208411	0.00903578
$533$	−3.25181	−0.140852
$534$	0	0
$535$	2.93891	0.127060
$536$	46.6091	2.01320
$537$	0	0
$538$	53.7307	2.31650
$539$	16.9502	0.730096
$540$	0	0
$541$	−7.91795	−0.340419	−0.170210	−	0.985408i	$-0.554444\pi$
$541$	−7.91795	−0.340419	−0.170210	+	0.985408i	$0.554444\pi$
$542$	49.1981	2.11324
$543$	0	0
$544$	−52.2280	−2.23926
$545$	5.87776	0.251776
$546$	0	0
$547$	26.4211	1.12969	0.564843	−	0.825199i	$-0.308937\pi$
$547$	26.4211	1.12969	0.564843	+	0.825199i	$0.308937\pi$
$548$	89.2593	3.81297
$549$	0	0
$550$	9.46869	0.403746
$551$	−0.0146847	−0.000625588 0
$552$	0	0
$553$	−8.43430	−0.358663
$554$	−26.9024	−1.14298
$555$	0	0
$556$	−98.8922	−4.19396
$557$	−36.9873	−1.56720	−0.783601	−	0.621264i	$-0.786619\pi$
$557$	−36.9873	−1.56720	−0.783601	+	0.621264i	$0.786619\pi$
$558$	0	0
$559$	−6.39304	−0.270397
$560$	−13.1588	−0.556060
$561$	0	0
$562$	26.9955	1.13874
$563$	28.2903	1.19229	0.596147	−	0.802876i	$-0.296698\pi$
$563$	28.2903	1.19229	0.596147	+	0.802876i	$0.296698\pi$
$564$	0	0
$565$	−3.91936	−0.164889
$566$	75.1069	3.15698
$567$	0	0
$568$	75.2455	3.15723
$569$	−12.4541	−0.522102	−0.261051	−	0.965325i	$-0.584069\pi$
$569$	−12.4541	−0.522102	−0.261051	+	0.965325i	$0.584069\pi$
$570$	0	0
$571$	17.9970	0.753153	0.376577	−	0.926386i	$-0.377101\pi$
$571$	17.9970	0.753153	0.376577	+	0.926386i	$0.377101\pi$
$572$	47.0148	1.96579
$573$	0	0
$574$	−4.72166	−0.197078
$575$	−6.04156	−0.251950
$576$	0	0
$577$	−18.3365	−0.763358	−0.381679	−	0.924295i	$-0.624654\pi$
$577$	−18.3365	−0.763358	−0.381679	+	0.924295i	$0.624654\pi$
$578$	−60.3134	−2.50871
$579$	0	0
$580$	2.37734	0.0987137
$581$	−10.3979	−0.431376
$582$	0	0
$583$	−27.8937	−1.15524
$584$	24.4374	1.01123
$585$	0	0
$586$	−41.3790	−1.70935
$587$	10.6273	0.438637	0.219319	−	0.975653i	$-0.429617\pi$
$587$	10.6273	0.438637	0.219319	+	0.975653i	$0.429617\pi$
$588$	0	0
$589$	−0.0221431	−0.000912391 0
$590$	−6.19066	−0.254866
$591$	0	0
$592$	37.4882	1.54075
$593$	2.64240	0.108510	0.0542552	−	0.998527i	$-0.482722\pi$
$593$	2.64240	0.108510	0.0542552	+	0.998527i	$0.482722\pi$
$594$	0	0
$595$	−9.78421	−0.401113
$596$	42.1153	1.72511
$597$	0	0
$598$	−42.8181	−1.75096
$599$	3.90858	0.159700	0.0798501	−	0.996807i	$-0.474556\pi$
$599$	3.90858	0.159700	0.0798501	+	0.996807i	$0.474556\pi$
$600$	0	0
$601$	25.2531	1.03009	0.515047	−	0.857162i	$-0.327774\pi$
$601$	25.2531	1.03009	0.515047	+	0.857162i	$0.327774\pi$
$602$	−9.28275	−0.378337
$603$	0	0
$604$	18.0553	0.734658
$605$	−2.42180	−0.0984601
$606$	0	0
$607$	−21.4747	−0.871633	−0.435817	−	0.900036i	$-0.643540\pi$
$607$	−21.4747	−0.871633	−0.435817	+	0.900036i	$0.643540\pi$
$608$	−0.237719	−0.00964079
$609$	0	0
$610$	26.3681	1.06761
$611$	−33.5086	−1.35561
$612$	0	0
$613$	−26.4183	−1.06703	−0.533513	−	0.845792i	$-0.679128\pi$
$613$	−26.4183	−1.06703	−0.533513	+	0.845792i	$0.679128\pi$
$614$	−74.5592	−3.00896
$615$	0	0
$616$	39.0917	1.57505
$617$	22.7901	0.917496	0.458748	−	0.888566i	$-0.348298\pi$
$617$	22.7901	0.917496	0.458748	+	0.888566i	$0.348298\pi$
$618$	0	0
$619$	−27.7216	−1.11423	−0.557113	−	0.830437i	$-0.688091\pi$
$619$	−27.7216	−1.11423	−0.557113	+	0.830437i	$0.688091\pi$
$620$	3.58481	0.143969
$621$	0	0
$622$	27.4704	1.10146
$623$	1.54056	0.0617212
$624$	0	0
$625$	1.00000	0.0400000
$626$	−47.3686	−1.89323
$627$	0	0
$628$	−39.5533	−1.57835
$629$	27.8743	1.11142
$630$	0	0
$631$	47.1147	1.87560	0.937802	−	0.347169i	$-0.112857\pi$
$631$	47.1147	1.87560	0.937802	+	0.347169i	$0.112857\pi$
$632$	37.9203	1.50839
$633$	0	0
$634$	−47.5293	−1.88763
$635$	3.28211	0.130247
$636$	0	0
$637$	12.6871	0.502682
$638$	−4.81001	−0.190430
$639$	0	0
$640$	−5.66714	−0.224014
$641$	−34.7609	−1.37297	−0.686487	−	0.727142i	$-0.740848\pi$
$641$	−34.7609	−1.37297	−0.686487	+	0.727142i	$0.740848\pi$
$642$	0	0
$643$	−1.15764	−0.0456530	−0.0228265	−	0.999739i	$-0.507267\pi$
$643$	−1.15764	−0.0456530	−0.0228265	+	0.999739i	$0.507267\pi$
$644$	−43.5575	−1.71640
$645$	0	0
$646$	−0.474504	−0.0186691
$647$	40.7772	1.60312	0.801559	−	0.597916i	$-0.204004\pi$
$647$	40.7772	1.60312	0.801559	+	0.597916i	$0.204004\pi$
$648$	0	0
$649$	8.77522	0.344458
$650$	7.08726	0.277985
$651$	0	0
$652$	−65.3559	−2.55954
$653$	10.4492	0.408910	0.204455	−	0.978876i	$-0.434458\pi$
$653$	10.4492	0.408910	0.204455	+	0.978876i	$0.434458\pi$
$654$	0	0
$655$	−10.8390	−0.423514
$656$	10.1291	0.395474
$657$	0	0
$658$	−48.6548	−1.89676
$659$	−40.0138	−1.55872	−0.779358	−	0.626579i	$-0.784454\pi$
$659$	−40.0138	−1.55872	−0.779358	+	0.626579i	$0.784454\pi$
$660$	0	0
$661$	16.5075	0.642068	0.321034	−	0.947068i	$-0.395970\pi$
$661$	16.5075	0.642068	0.321034	+	0.947068i	$0.395970\pi$
$662$	59.8310	2.32540
$663$	0	0
$664$	46.7484	1.81419
$665$	−0.0445334	−0.00172693
$666$	0	0
$667$	3.06906	0.118834
$668$	3.38295	0.130890
$669$	0	0
$670$	−17.3922	−0.671918
$671$	−37.3765	−1.44291
$672$	0	0
$673$	−26.9272	−1.03797	−0.518984	−	0.854784i	$-0.673690\pi$
$673$	−26.9272	−1.03797	−0.518984	+	0.854784i	$0.673690\pi$
$674$	69.5627	2.67946
$675$	0	0
$676$	−25.6482	−0.986470
$677$	−28.8657	−1.10940	−0.554699	−	0.832051i	$-0.687167\pi$
$677$	−28.8657	−1.10940	−0.554699	+	0.832051i	$0.687167\pi$
$678$	0	0
$679$	18.6471	0.715609
$680$	43.9894	1.68692
$681$	0	0
$682$	−7.25305	−0.277734
$683$	−23.1581	−0.886119	−0.443059	−	0.896492i	$-0.646107\pi$
$683$	−23.1581	−0.886119	−0.443059	+	0.896492i	$0.646107\pi$
$684$	0	0
$685$	−19.0730	−0.728741
$686$	46.2933	1.76749
$687$	0	0
$688$	19.9137	0.759202
$689$	−20.8783	−0.795399
$690$	0	0
$691$	8.71549	0.331553	0.165776	−	0.986163i	$-0.446987\pi$
$691$	8.71549	0.331553	0.165776	+	0.986163i	$0.446987\pi$
$692$	−3.56836	−0.135649
$693$	0	0
$694$	46.8478	1.77832
$695$	21.1313	0.801557
$696$	0	0
$697$	7.53146	0.285274
$698$	−50.3235	−1.90477
$699$	0	0
$700$	7.20964	0.272499
$701$	9.18402	0.346876	0.173438	−	0.984845i	$-0.444512\pi$
$701$	9.18402	0.346876	0.173438	+	0.984845i	$0.444512\pi$
$702$	0	0
$703$	0.126872	0.00478505
$704$	−15.2804	−0.575900
$705$	0	0
$706$	87.2254	3.28277
$707$	−23.4368	−0.881433
$708$	0	0
$709$	−22.9987	−0.863734	−0.431867	−	0.901937i	$-0.642145\pi$
$709$	−22.9987	−0.863734	−0.431867	+	0.901937i	$0.642145\pi$
$710$	−28.0779	−1.05374
$711$	0	0
$712$	−6.92630	−0.259574
$713$	4.62785	0.173315
$714$	0	0
$715$	−10.0461	−0.375704
$716$	−0.898112	−0.0335640
$717$	0	0
$718$	−84.4493	−3.15162
$719$	17.2899	0.644805	0.322403	−	0.946603i	$-0.395510\pi$
$719$	17.2899	0.644805	0.322403	+	0.946603i	$0.395510\pi$
$720$	0	0
$721$	−15.8107	−0.588821
$722$	49.1042	1.82747
$723$	0	0
$724$	−92.0609	−3.42142
$725$	−0.507991	−0.0188663
$726$	0	0
$727$	−13.8028	−0.511918	−0.255959	−	0.966688i	$-0.582391\pi$
$727$	−13.8028	−0.511918	−0.255959	+	0.966688i	$0.582391\pi$
$728$	29.2599	1.08444
$729$	0	0
$730$	−9.11883	−0.337503
$731$	14.8068	0.547649
$732$	0	0
$733$	−12.3777	−0.457182	−0.228591	−	0.973523i	$-0.573412\pi$
$733$	−12.3777	−0.457182	−0.228591	+	0.973523i	$0.573412\pi$
$734$	14.6743	0.541639
$735$	0	0
$736$	49.6827	1.83133
$737$	24.6533	0.908115
$738$	0	0
$739$	−15.7130	−0.578014	−0.289007	−	0.957327i	$-0.593325\pi$
$739$	−15.7130	−0.578014	−0.289007	+	0.957327i	$0.593325\pi$
$740$	−20.5396	−0.755051
$741$	0	0
$742$	−30.3154	−1.11291
$743$	−12.1302	−0.445013	−0.222506	−	0.974931i	$-0.571424\pi$
$743$	−12.1302	−0.445013	−0.222506	+	0.974931i	$0.571424\pi$
$744$	0	0
$745$	−8.99920	−0.329705
$746$	16.9900	0.622048
$747$	0	0
$748$	−108.890	−3.98141
$749$	−4.52757	−0.165434
$750$	0	0
$751$	50.7512	1.85194	0.925969	−	0.377599i	$-0.123250\pi$
$751$	50.7512	1.85194	0.925969	+	0.377599i	$0.123250\pi$
$752$	104.376	3.80620
$753$	0	0
$754$	−3.60027	−0.131114
$755$	−3.85805	−0.140409
$756$	0	0
$757$	17.3342	0.630022	0.315011	−	0.949088i	$-0.397992\pi$
$757$	17.3342	0.630022	0.315011	+	0.949088i	$0.397992\pi$
$758$	9.00334	0.327016
$759$	0	0
$760$	0.200221	0.00726277
$761$	−17.8283	−0.646275	−0.323137	−	0.946352i	$-0.604738\pi$
$761$	−17.8283	−0.646275	−0.323137	+	0.946352i	$0.604738\pi$
$762$	0	0
$763$	−9.05504	−0.327814
$764$	12.2271	0.442361
$765$	0	0
$766$	45.5699	1.64651
$767$	6.56820	0.237164
$768$	0	0
$769$	42.6586	1.53831	0.769154	−	0.639064i	$-0.220678\pi$
$769$	42.6586	1.53831	0.769154	+	0.639064i	$0.220678\pi$
$770$	−14.5871	−0.525682
$771$	0	0
$772$	1.79716	0.0646814
$773$	9.23632	0.332207	0.166104	−	0.986108i	$-0.446881\pi$
$773$	9.23632	0.332207	0.166104	+	0.986108i	$0.446881\pi$
$774$	0	0
$775$	−0.766003	−0.0275157
$776$	−83.8365	−3.00956
$777$	0	0
$778$	91.8489	3.29294
$779$	0.0342799	0.00122821
$780$	0	0
$781$	39.8002	1.42416
$782$	99.1701	3.54632
$783$	0	0
$784$	−39.5191	−1.41140
$785$	8.45176	0.301656
$786$	0	0
$787$	−43.6355	−1.55544	−0.777719	−	0.628612i	$-0.783623\pi$
$787$	−43.6355	−1.55544	−0.777719	+	0.628612i	$0.783623\pi$
$788$	−44.0518	−1.56928
$789$	0	0
$790$	−14.1500	−0.503433
$791$	6.03801	0.214687
$792$	0	0
$793$	−27.9761	−0.993461
$794$	8.73548	0.310011
$795$	0	0
$796$	−23.3258	−0.826760
$797$	−16.7067	−0.591780	−0.295890	−	0.955222i	$-0.595616\pi$
$797$	−16.7067	−0.591780	−0.295890	+	0.955222i	$0.595616\pi$
$798$	0	0
$799$	77.6086	2.74560
$800$	−8.22350	−0.290744
$801$	0	0
$802$	77.9954	2.75411
$803$	12.9259	0.456144
$804$	0	0
$805$	9.30738	0.328042
$806$	−5.42887	−0.191224
$807$	0	0
$808$	105.371	3.70695
$809$	42.8086	1.50507	0.752536	−	0.658551i	$-0.228830\pi$
$809$	42.8086	1.50507	0.752536	+	0.658551i	$0.228830\pi$
$810$	0	0
$811$	−24.5266	−0.861245	−0.430623	−	0.902532i	$-0.641706\pi$
$811$	−24.5266	−0.861245	−0.430623	+	0.902532i	$0.641706\pi$
$812$	−3.66243	−0.128526
$813$	0	0
$814$	41.5572	1.45658
$815$	13.9653	0.489182
$816$	0	0
$817$	0.0673941	0.00235782
$818$	−79.3437	−2.77419
$819$	0	0
$820$	−5.54967	−0.193803
$821$	−10.5364	−0.367724	−0.183862	−	0.982952i	$-0.558860\pi$
$821$	−10.5364	−0.367724	−0.183862	+	0.982952i	$0.558860\pi$
$822$	0	0
$823$	2.85477	0.0995111	0.0497555	−	0.998761i	$-0.484156\pi$
$823$	2.85477	0.0995111	0.0497555	+	0.998761i	$0.484156\pi$
$824$	71.0843	2.47634
$825$	0	0
$826$	9.53708	0.331838
$827$	3.84473	0.133694	0.0668471	−	0.997763i	$-0.478706\pi$
$827$	3.84473	0.133694	0.0668471	+	0.997763i	$0.478706\pi$
$828$	0	0
$829$	−35.3495	−1.22774	−0.613870	−	0.789407i	$-0.710388\pi$
$829$	−35.3495	−1.22774	−0.613870	+	0.789407i	$0.710388\pi$
$830$	−17.4442	−0.605496
$831$	0	0
$832$	−11.4373	−0.396516
$833$	−29.3844	−1.01811
$834$	0	0
$835$	−0.722870	−0.0250159
$836$	−0.495620	−0.0171414
$837$	0	0
$838$	41.0155	1.41686
$839$	43.5504	1.50353	0.751763	−	0.659434i	$-0.229204\pi$
$839$	43.5504	1.50353	0.751763	+	0.659434i	$0.229204\pi$
$840$	0	0
$841$	−28.7419	−0.991102
$842$	10.7664	0.371036
$843$	0	0
$844$	28.9586	0.996796
$845$	5.48052	0.188536
$846$	0	0
$847$	3.73092	0.128196
$848$	65.0337	2.23326
$849$	0	0
$850$	−16.4147	−0.563018
$851$	−26.5159	−0.908952
$852$	0	0
$853$	−31.4116	−1.07551	−0.537757	−	0.843100i	$-0.680728\pi$
$853$	−31.4116	−1.07551	−0.537757	+	0.843100i	$0.680728\pi$
$854$	−40.6216	−1.39004
$855$	0	0
$856$	20.3558	0.695746
$857$	−29.5222	−1.00846	−0.504229	−	0.863570i	$-0.668223\pi$
$857$	−29.5222	−1.00846	−0.504229	+	0.863570i	$0.668223\pi$
$858$	0	0
$859$	21.1204	0.720619	0.360309	−	0.932833i	$-0.382671\pi$
$859$	21.1204	0.720619	0.360309	+	0.932833i	$0.382671\pi$
$860$	−10.9106	−0.372049
$861$	0	0
$862$	86.6545	2.95146
$863$	42.7945	1.45674	0.728370	−	0.685184i	$-0.240278\pi$
$863$	42.7945	1.45674	0.728370	+	0.685184i	$0.240278\pi$
$864$	0	0
$865$	0.762488	0.0259254
$866$	−60.2488	−2.04734
$867$	0	0
$868$	−5.52261	−0.187450
$869$	20.0575	0.680403
$870$	0	0
$871$	18.4528	0.625250
$872$	40.7111	1.37865
$873$	0	0
$874$	0.451379	0.0152681
$875$	−1.54056	−0.0520804
$876$	0	0
$877$	−42.4645	−1.43392	−0.716962	−	0.697112i	$-0.754468\pi$
$877$	−42.4645	−1.43392	−0.716962	+	0.697112i	$0.754468\pi$
$878$	−19.9424	−0.673022
$879$	0	0
$880$	31.2927	1.05488
$881$	45.5687	1.53525	0.767623	−	0.640901i	$-0.221439\pi$
$881$	45.5687	1.53525	0.767623	+	0.640901i	$0.221439\pi$
$882$	0	0
$883$	25.2412	0.849435	0.424718	−	0.905326i	$-0.360373\pi$
$883$	25.2412	0.849435	0.424718	+	0.905326i	$0.360373\pi$
$884$	−81.5035	−2.74126
$885$	0	0
$886$	33.8055	1.13572
$887$	8.02794	0.269552	0.134776	−	0.990876i	$-0.456969\pi$
$887$	8.02794	0.269552	0.134776	+	0.990876i	$0.456969\pi$
$888$	0	0
$889$	−5.05628	−0.169582
$890$	2.58455	0.0866342
$891$	0	0
$892$	129.166	4.32479
$893$	0.353241	0.0118207
$894$	0	0
$895$	0.191909	0.00641481
$896$	8.73057	0.291668
$897$	0	0
$898$	41.7945	1.39470
$899$	0.389123	0.0129780
$900$	0	0
$901$	48.3557	1.61096
$902$	11.2285	0.373868
$903$	0	0
$904$	−27.1467	−0.902884
$905$	19.6716	0.653906
$906$	0	0
$907$	19.8045	0.657597	0.328798	−	0.944400i	$-0.393356\pi$
$907$	19.8045	0.657597	0.328798	+	0.944400i	$0.393356\pi$
$908$	93.2334	3.09406
$909$	0	0
$910$	−10.9183	−0.361940
$911$	−38.7415	−1.28356	−0.641781	−	0.766888i	$-0.721804\pi$
$911$	−38.7415	−1.28356	−0.641781	+	0.766888i	$0.721804\pi$
$912$	0	0
$913$	24.7270	0.818344
$914$	90.1630	2.98233
$915$	0	0
$916$	87.0791	2.87717
$917$	16.6981	0.551420
$918$	0	0
$919$	38.2805	1.26276	0.631378	−	0.775475i	$-0.282489\pi$
$919$	38.2805	1.26276	0.631378	+	0.775475i	$0.282489\pi$
$920$	−41.8456	−1.37961
$921$	0	0
$922$	106.006	3.49114
$923$	29.7902	0.980556
$924$	0	0
$925$	4.38891	0.144306
$926$	37.1089	1.21947
$927$	0	0
$928$	4.17746	0.137132
$929$	43.3772	1.42316	0.711580	−	0.702606i	$-0.247980\pi$
$929$	43.3772	1.42316	0.711580	+	0.702606i	$0.247980\pi$
$930$	0	0
$931$	−0.133745	−0.00438331
$932$	−66.3847	−2.17450
$933$	0	0
$934$	43.6752	1.42910
$935$	23.2677	0.760934
$936$	0	0
$937$	47.5824	1.55445	0.777224	−	0.629224i	$-0.216627\pi$
$937$	47.5824	1.55445	0.777224	+	0.629224i	$0.216627\pi$
$938$	26.7937	0.874844
$939$	0	0
$940$	−57.1871	−1.86524
$941$	−8.72392	−0.284392	−0.142196	−	0.989839i	$-0.545416\pi$
$941$	−8.72392	−0.284392	−0.142196	+	0.989839i	$0.545416\pi$
$942$	0	0
$943$	−7.16442	−0.233306
$944$	−20.4593	−0.665893
$945$	0	0
$946$	22.0752	0.717725
$947$	−36.8499	−1.19746	−0.598730	−	0.800951i	$-0.704328\pi$
$947$	−36.8499	−1.19746	−0.598730	+	0.800951i	$0.704328\pi$
$948$	0	0
$949$	9.67494	0.314062
$950$	−0.0747124	−0.00242399
$951$	0	0
$952$	−67.7683	−2.19638
$953$	−40.8850	−1.32440	−0.662198	−	0.749329i	$-0.730376\pi$
$953$	−40.8850	−1.32440	−0.662198	+	0.749329i	$0.730376\pi$
$954$	0	0
$955$	−2.61269	−0.0845448
$956$	−48.4482	−1.56693
$957$	0	0
$958$	28.4921	0.920536
$959$	29.3830	0.948828
$960$	0	0
$961$	−30.4132	−0.981072
$962$	31.1053	1.00288
$963$	0	0
$964$	138.548	4.46233
$965$	−0.384019	−0.0123620
$966$	0	0
$967$	−4.95858	−0.159457	−0.0797287	−	0.996817i	$-0.525405\pi$
$967$	−4.95858	−0.159457	−0.0797287	+	0.996817i	$0.525405\pi$
$968$	−16.7741	−0.539140
$969$	0	0
$970$	31.2836	1.00446
$971$	24.0387	0.771439	0.385720	−	0.922616i	$-0.373953\pi$
$971$	24.0387	0.771439	0.385720	+	0.922616i	$0.373953\pi$
$972$	0	0
$973$	−32.5541	−1.04364
$974$	−64.4068	−2.06373
$975$	0	0
$976$	87.1428	2.78937
$977$	−22.5308	−0.720824	−0.360412	−	0.932793i	$-0.617364\pi$
$977$	−22.5308	−0.720824	−0.360412	+	0.932793i	$0.617364\pi$
$978$	0	0
$979$	−3.66358	−0.117088
$980$	21.6523	0.691658
$981$	0	0
$982$	0.0547528	0.00174723
$983$	25.5614	0.815283	0.407642	−	0.913142i	$-0.366351\pi$
$983$	25.5614	0.815283	0.407642	+	0.913142i	$0.366351\pi$
$984$	0	0
$985$	9.41301	0.299923
$986$	8.33850	0.265552
$987$	0	0
$988$	−0.370969	−0.0118021
$989$	−14.0852	−0.447883
$990$	0	0
$991$	11.5115	0.365675	0.182838	−	0.983143i	$-0.441472\pi$
$991$	11.5115	0.365675	0.182838	+	0.983143i	$0.441472\pi$
$992$	6.29923	0.200001
$993$	0	0
$994$	43.2556	1.37198
$995$	4.98426	0.158012
$996$	0	0
$997$	−44.8905	−1.42170	−0.710848	−	0.703346i	$-0.751688\pi$
$997$	−44.8905	−1.42170	−0.710848	+	0.703346i	$0.751688\pi$
$998$	98.8920	3.13037
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.1		10
3.2	odd	2		1335.2.a.j.1.10	✓	10
15.14	odd	2		6675.2.a.z.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.10	✓	10	3.2	odd	2
4005.2.a.s.1.1		10	1.1	even	1	trivial
6675.2.a.z.1.1		10	15.14	odd	2

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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58455 q^{2} +4.67989 q^{4} -1.00000 q^{5} +1.54056 q^{7} -6.92630 q^{8} +O(q^{10})\)	\(q-2.58455 q^{2} +4.67989 q^{4} -1.00000 q^{5} +1.54056 q^{7} -6.92630 q^{8} +2.58455 q^{10} -3.66358 q^{11} -2.74217 q^{13} -3.98165 q^{14} +8.54157 q^{16} +6.35107 q^{17} +0.0289073 q^{19} -4.67989 q^{20} +9.46869 q^{22} -6.04156 q^{23} +1.00000 q^{25} +7.08726 q^{26} +7.20964 q^{28} -0.507991 q^{29} -0.766003 q^{31} -8.22350 q^{32} -16.4147 q^{34} -1.54056 q^{35} +4.38891 q^{37} -0.0747124 q^{38} +6.92630 q^{40} +1.18586 q^{41} +2.33138 q^{43} -17.1451 q^{44} +15.6147 q^{46} +12.2198 q^{47} -4.62668 q^{49} -2.58455 q^{50} -12.8330 q^{52} +7.61379 q^{53} +3.66358 q^{55} -10.6704 q^{56} +1.31293 q^{58} -2.39526 q^{59} +10.2022 q^{61} +1.97977 q^{62} +4.17088 q^{64} +2.74217 q^{65} -6.72929 q^{67} +29.7223 q^{68} +3.98165 q^{70} -10.8637 q^{71} -3.52821 q^{73} -11.3433 q^{74} +0.135283 q^{76} -5.64396 q^{77} -5.47483 q^{79} -8.54157 q^{80} -3.06490 q^{82} -6.74941 q^{83} -6.35107 q^{85} -6.02557 q^{86} +25.3750 q^{88} +1.00000 q^{89} -4.22447 q^{91} -28.2738 q^{92} -31.5826 q^{94} -0.0289073 q^{95} +12.1041 q^{97} +11.9579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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