LMFDB - Embedded newform 4005.2.a.p.1.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.11667$ 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)

$f(q)$	$=$	$q+1.11667 q^{2} -0.753041 q^{4} +1.00000 q^{5} -1.16728 q^{7} -3.07425 q^{8} +O(q^{10})$	\(q+1.11667 q^{2} -0.753041 q^{4} +1.00000 q^{5} -1.16728 q^{7} -3.07425 q^{8} +1.11667 q^{10} +2.19092 q^{11} +1.10132 q^{13} -1.30347 q^{14} -1.92685 q^{16} -7.78009 q^{17} +4.75115 q^{19} -0.753041 q^{20} +2.44654 q^{22} -2.26860 q^{23} +1.00000 q^{25} +1.22981 q^{26} +0.879008 q^{28} +6.01934 q^{29} -1.58614 q^{31} +3.99684 q^{32} -8.68782 q^{34} -1.16728 q^{35} -7.29224 q^{37} +5.30548 q^{38} -3.07425 q^{40} +2.30872 q^{41} -2.34975 q^{43} -1.64985 q^{44} -2.53328 q^{46} +0.949255 q^{47} -5.63746 q^{49} +1.11667 q^{50} -0.829338 q^{52} -8.03974 q^{53} +2.19092 q^{55} +3.58850 q^{56} +6.72163 q^{58} -3.82542 q^{59} -5.72557 q^{61} -1.77120 q^{62} +8.31685 q^{64} +1.10132 q^{65} -15.8843 q^{67} +5.85873 q^{68} -1.30347 q^{70} -2.59018 q^{71} -12.1497 q^{73} -8.14305 q^{74} -3.57781 q^{76} -2.55741 q^{77} +10.7999 q^{79} -1.92685 q^{80} +2.57809 q^{82} -3.60391 q^{83} -7.78009 q^{85} -2.62390 q^{86} -6.73543 q^{88} -1.00000 q^{89} -1.28554 q^{91} +1.70835 q^{92} +1.06001 q^{94} +4.75115 q^{95} -3.17508 q^{97} -6.29520 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.11667	0.789607	0.394804	−	0.918766i	$-0.370813\pi$
$2$	1.11667	0.789607	0.394804	+	0.918766i	$0.370813\pi$
$3$	0	0
$3$	0	0
$4$	−0.753041	−0.376520
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.16728	−0.441190	−0.220595	−	0.975366i	$-0.570800\pi$
$7$	−1.16728	−0.441190	−0.220595	+	0.975366i	$0.570800\pi$
$8$	−3.07425	−1.08691
$9$	0	0
$10$	1.11667	0.353123
$11$	2.19092	0.660587	0.330294	−	0.943878i	$-0.392852\pi$
$11$	2.19092	0.660587	0.330294	+	0.943878i	$0.392852\pi$
$12$	0	0
$13$	1.10132	0.305451	0.152725	−	0.988269i	$-0.451195\pi$
$13$	1.10132	0.305451	0.152725	+	0.988269i	$0.451195\pi$
$14$	−1.30347	−0.348366
$15$	0	0
$16$	−1.92685	−0.481712
$17$	−7.78009	−1.88695	−0.943475	−	0.331445i	$-0.892464\pi$
$17$	−7.78009	−1.88695	−0.943475	+	0.331445i	$0.892464\pi$
$18$	0	0
$19$	4.75115	1.08999	0.544994	−	0.838440i	$-0.316532\pi$
$19$	4.75115	1.08999	0.544994	+	0.838440i	$0.316532\pi$
$20$	−0.753041	−0.168385
$21$	0	0
$22$	2.44654	0.521605
$23$	−2.26860	−0.473035	−0.236517	−	0.971627i	$-0.576006\pi$
$23$	−2.26860	−0.473035	−0.236517	+	0.971627i	$0.576006\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.22981	0.241186
$27$	0	0
$28$	0.879008	0.166117
$29$	6.01934	1.11776	0.558882	−	0.829248i	$-0.311231\pi$
$29$	6.01934	1.11776	0.558882	+	0.829248i	$0.311231\pi$
$30$	0	0
$31$	−1.58614	−0.284879	−0.142440	−	0.989804i	$-0.545495\pi$
$31$	−1.58614	−0.284879	−0.142440	+	0.989804i	$0.545495\pi$
$32$	3.99684	0.706547
$33$	0	0
$34$	−8.68782	−1.48995
$35$	−1.16728	−0.197306
$36$	0	0
$37$	−7.29224	−1.19884	−0.599418	−	0.800436i	$-0.704601\pi$
$37$	−7.29224	−1.19884	−0.599418	+	0.800436i	$0.704601\pi$
$38$	5.30548	0.860663
$39$	0	0
$40$	−3.07425	−0.486081
$41$	2.30872	0.360562	0.180281	−	0.983615i	$-0.442299\pi$
$41$	2.30872	0.360562	0.180281	+	0.983615i	$0.442299\pi$
$42$	0	0
$43$	−2.34975	−0.358333	−0.179167	−	0.983819i	$-0.557340\pi$
$43$	−2.34975	−0.358333	−0.179167	+	0.983819i	$0.557340\pi$
$44$	−1.64985	−0.248725
$45$	0	0
$46$	−2.53328	−0.373512
$47$	0.949255	0.138463	0.0692315	−	0.997601i	$-0.477945\pi$
$47$	0.949255	0.138463	0.0692315	+	0.997601i	$0.477945\pi$
$48$	0	0
$49$	−5.63746	−0.805352
$50$	1.11667	0.157921
$51$	0	0
$52$	−0.829338	−0.115008
$53$	−8.03974	−1.10434	−0.552171	−	0.833731i	$-0.686201\pi$
$53$	−8.03974	−1.10434	−0.552171	+	0.833731i	$0.686201\pi$
$54$	0	0
$55$	2.19092	0.295424
$56$	3.58850	0.479534
$57$	0	0
$58$	6.72163	0.882594
$59$	−3.82542	−0.498027	−0.249013	−	0.968500i	$-0.580106\pi$
$59$	−3.82542	−0.498027	−0.249013	+	0.968500i	$0.580106\pi$
$60$	0	0
$61$	−5.72557	−0.733084	−0.366542	−	0.930402i	$-0.619458\pi$
$61$	−5.72557	−0.733084	−0.366542	+	0.930402i	$0.619458\pi$
$62$	−1.77120	−0.224943
$63$	0	0
$64$	8.31685	1.03961
$65$	1.10132	0.136602
$66$	0	0
$67$	−15.8843	−1.94057	−0.970285	−	0.241964i	$-0.922208\pi$
$67$	−15.8843	−1.94057	−0.970285	+	0.241964i	$0.922208\pi$
$68$	5.85873	0.710475
$69$	0	0
$70$	−1.30347	−0.155794
$71$	−2.59018	−0.307398	−0.153699	−	0.988118i	$-0.549119\pi$
$71$	−2.59018	−0.307398	−0.153699	+	0.988118i	$0.549119\pi$
$72$	0	0
$73$	−12.1497	−1.42202	−0.711010	−	0.703182i	$-0.751762\pi$
$73$	−12.1497	−1.42202	−0.711010	+	0.703182i	$0.751762\pi$
$74$	−8.14305	−0.946610
$75$	0	0
$76$	−3.57781	−0.410403
$77$	−2.55741	−0.291444
$78$	0	0
$79$	10.7999	1.21509	0.607544	−	0.794286i	$-0.292155\pi$
$79$	10.7999	1.21509	0.607544	+	0.794286i	$0.292155\pi$
$80$	−1.92685	−0.215428
$81$	0	0
$82$	2.57809	0.284702
$83$	−3.60391	−0.395581	−0.197790	−	0.980244i	$-0.563376\pi$
$83$	−3.60391	−0.395581	−0.197790	+	0.980244i	$0.563376\pi$
$84$	0	0
$85$	−7.78009	−0.843869
$86$	−2.62390	−0.282942
$87$	0	0
$88$	−6.73543	−0.717999
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−1.28554	−0.134762
$92$	1.70835	0.178107
$93$	0	0
$94$	1.06001	0.109331
$95$	4.75115	0.487458
$96$	0	0
$97$	−3.17508	−0.322381	−0.161190	−	0.986923i	$-0.551533\pi$
$97$	−3.17508	−0.322381	−0.161190	+	0.986923i	$0.551533\pi$
$98$	−6.29520	−0.635912
$99$	0	0
$100$	−0.753041	−0.0753041
$101$	6.07838	0.604821	0.302411	−	0.953178i	$-0.402209\pi$
$101$	6.07838	0.604821	0.302411	+	0.953178i	$0.402209\pi$
$102$	0	0
$103$	−15.4733	−1.52463	−0.762315	−	0.647206i	$-0.775937\pi$
$103$	−15.4733	−1.52463	−0.762315	+	0.647206i	$0.775937\pi$
$104$	−3.38572	−0.331998
$105$	0	0
$106$	−8.97776	−0.871997
$107$	−3.75432	−0.362944	−0.181472	−	0.983396i	$-0.558086\pi$
$107$	−3.75432	−0.362944	−0.181472	+	0.983396i	$0.558086\pi$
$108$	0	0
$109$	14.3378	1.37331	0.686654	−	0.726984i	$-0.259079\pi$
$109$	14.3378	1.37331	0.686654	+	0.726984i	$0.259079\pi$
$110$	2.44654	0.233269
$111$	0	0
$112$	2.24917	0.212526
$113$	0.592705	0.0557570	0.0278785	−	0.999611i	$-0.491125\pi$
$113$	0.592705	0.0557570	0.0278785	+	0.999611i	$0.491125\pi$
$114$	0	0
$115$	−2.26860	−0.211548
$116$	−4.53281	−0.420861
$117$	0	0
$118$	−4.27174	−0.393246
$119$	9.08153	0.832502
$120$	0	0
$121$	−6.19987	−0.563624
$122$	−6.39359	−0.578848
$123$	0	0
$124$	1.19443	0.107263
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.3069	−1.53574	−0.767869	−	0.640606i	$-0.778683\pi$
$127$	−17.3069	−1.53574	−0.767869	+	0.640606i	$0.778683\pi$
$128$	1.29354	0.114334
$129$	0	0
$130$	1.22981	0.107862
$131$	−6.52966	−0.570499	−0.285250	−	0.958453i	$-0.592076\pi$
$131$	−6.52966	−0.570499	−0.285250	+	0.958453i	$0.592076\pi$
$132$	0	0
$133$	−5.54591	−0.480891
$134$	−17.7375	−1.53229
$135$	0	0
$136$	23.9179	2.05094
$137$	15.0929	1.28947	0.644737	−	0.764405i	$-0.276967\pi$
$137$	15.0929	1.28947	0.644737	+	0.764405i	$0.276967\pi$
$138$	0	0
$139$	11.1708	0.947492	0.473746	−	0.880662i	$-0.342901\pi$
$139$	11.1708	0.947492	0.473746	+	0.880662i	$0.342901\pi$
$140$	0.879008	0.0742897
$141$	0	0
$142$	−2.89238	−0.242723
$143$	2.41290	0.201777
$144$	0	0
$145$	6.01934	0.499879
$146$	−13.5673	−1.12284
$147$	0	0
$148$	5.49135	0.451387
$149$	−7.56657	−0.619877	−0.309939	−	0.950757i	$-0.600309\pi$
$149$	−7.56657	−0.619877	−0.309939	+	0.950757i	$0.600309\pi$
$150$	0	0
$151$	−4.28422	−0.348645	−0.174322	−	0.984689i	$-0.555773\pi$
$151$	−4.28422	−0.348645	−0.174322	+	0.984689i	$0.555773\pi$
$152$	−14.6062	−1.18472
$153$	0	0
$154$	−2.85579	−0.230126
$155$	−1.58614	−0.127402
$156$	0	0
$157$	−5.39518	−0.430582	−0.215291	−	0.976550i	$-0.569070\pi$
$157$	−5.39518	−0.430582	−0.215291	+	0.976550i	$0.569070\pi$
$158$	12.0600	0.959442
$159$	0	0
$160$	3.99684	0.315978
$161$	2.64808	0.208698
$162$	0	0
$163$	15.7496	1.23360	0.616801	−	0.787119i	$-0.288428\pi$
$163$	15.7496	1.23360	0.616801	+	0.787119i	$0.288428\pi$
$164$	−1.73856	−0.135759
$165$	0	0
$166$	−4.02439	−0.312353
$167$	18.1941	1.40790	0.703951	−	0.710248i	$-0.251417\pi$
$167$	18.1941	1.40790	0.703951	+	0.710248i	$0.251417\pi$
$168$	0	0
$169$	−11.7871	−0.906700
$170$	−8.68782	−0.666325
$171$	0	0
$172$	1.76946	0.134920
$173$	−4.39190	−0.333910	−0.166955	−	0.985964i	$-0.553394\pi$
$173$	−4.39190	−0.333910	−0.166955	+	0.985964i	$0.553394\pi$
$174$	0	0
$175$	−1.16728	−0.0882379
$176$	−4.22157	−0.318213
$177$	0	0
$178$	−1.11667	−0.0836982
$179$	22.7104	1.69745	0.848726	−	0.528833i	$-0.177370\pi$
$179$	22.7104	1.69745	0.848726	+	0.528833i	$0.177370\pi$
$180$	0	0
$181$	−17.2204	−1.27999	−0.639993	−	0.768381i	$-0.721063\pi$
$181$	−17.2204	−1.27999	−0.639993	+	0.768381i	$0.721063\pi$
$182$	−1.43553	−0.106409
$183$	0	0
$184$	6.97422	0.514147
$185$	−7.29224	−0.536136
$186$	0	0
$187$	−17.0456	−1.24649
$188$	−0.714828	−0.0521342
$189$	0	0
$190$	5.30548	0.384900
$191$	−20.5891	−1.48977	−0.744887	−	0.667190i	$-0.767497\pi$
$191$	−20.5891	−1.48977	−0.744887	+	0.667190i	$0.767497\pi$
$192$	0	0
$193$	−9.48692	−0.682884	−0.341442	−	0.939903i	$-0.610915\pi$
$193$	−9.48692	−0.682884	−0.341442	+	0.939903i	$0.610915\pi$
$194$	−3.54553	−0.254554
$195$	0	0
$196$	4.24524	0.303231
$197$	−25.9622	−1.84973	−0.924865	−	0.380297i	$-0.875822\pi$
$197$	−25.9622	−1.84973	−0.924865	+	0.380297i	$0.875822\pi$
$198$	0	0
$199$	−16.4243	−1.16429	−0.582143	−	0.813087i	$-0.697786\pi$
$199$	−16.4243	−1.16429	−0.582143	+	0.813087i	$0.697786\pi$
$200$	−3.07425	−0.217382
$201$	0	0
$202$	6.78756	0.477571
$203$	−7.02624	−0.493145
$204$	0	0
$205$	2.30872	0.161248
$206$	−17.2786	−1.20386
$207$	0	0
$208$	−2.12207	−0.147139
$209$	10.4094	0.720032
$210$	0	0
$211$	9.30596	0.640649	0.320324	−	0.947308i	$-0.396208\pi$
$211$	9.30596	0.640649	0.320324	+	0.947308i	$0.396208\pi$
$212$	6.05425	0.415808
$213$	0	0
$214$	−4.19235	−0.286583
$215$	−2.34975	−0.160251
$216$	0	0
$217$	1.85147	0.125686
$218$	16.0106	1.08437
$219$	0	0
$220$	−1.64985	−0.111233
$221$	−8.56835	−0.576370
$222$	0	0
$223$	5.64930	0.378305	0.189152	−	0.981948i	$-0.439426\pi$
$223$	5.64930	0.378305	0.189152	+	0.981948i	$0.439426\pi$
$224$	−4.66542	−0.311721
$225$	0	0
$226$	0.661858	0.0440261
$227$	−2.69793	−0.179068	−0.0895341	−	0.995984i	$-0.528538\pi$
$227$	−2.69793	−0.179068	−0.0895341	+	0.995984i	$0.528538\pi$
$228$	0	0
$229$	6.50881	0.430115	0.215057	−	0.976601i	$-0.431006\pi$
$229$	6.50881	0.430115	0.215057	+	0.976601i	$0.431006\pi$
$230$	−2.53328	−0.167040
$231$	0	0
$232$	−18.5049	−1.21491
$233$	16.6350	1.08980	0.544899	−	0.838502i	$-0.316568\pi$
$233$	16.6350	1.08980	0.544899	+	0.838502i	$0.316568\pi$
$234$	0	0
$235$	0.949255	0.0619225
$236$	2.88070	0.187517
$237$	0	0
$238$	10.1411	0.657350
$239$	−21.1293	−1.36674	−0.683370	−	0.730072i	$-0.739486\pi$
$239$	−21.1293	−1.36674	−0.683370	+	0.730072i	$0.739486\pi$
$240$	0	0
$241$	−5.60825	−0.361259	−0.180629	−	0.983551i	$-0.557813\pi$
$241$	−5.60825	−0.361259	−0.180629	+	0.983551i	$0.557813\pi$
$242$	−6.92323	−0.445042
$243$	0	0
$244$	4.31159	0.276021
$245$	−5.63746	−0.360164
$246$	0	0
$247$	5.23253	0.332938
$248$	4.87619	0.309638
$249$	0	0
$250$	1.11667	0.0706246
$251$	−15.2914	−0.965187	−0.482593	−	0.875845i	$-0.660305\pi$
$251$	−15.2914	−0.965187	−0.482593	+	0.875845i	$0.660305\pi$
$252$	0	0
$253$	−4.97031	−0.312481
$254$	−19.3262	−1.21263
$255$	0	0
$256$	−15.1892	−0.949328
$257$	−3.40793	−0.212581	−0.106290	−	0.994335i	$-0.533897\pi$
$257$	−3.40793	−0.212581	−0.106290	+	0.994335i	$0.533897\pi$
$258$	0	0
$259$	8.51207	0.528914
$260$	−0.829338	−0.0514333
$261$	0	0
$262$	−7.29150	−0.450470
$263$	−7.66796	−0.472827	−0.236413	−	0.971653i	$-0.575972\pi$
$263$	−7.66796	−0.472827	−0.236413	+	0.971653i	$0.575972\pi$
$264$	0	0
$265$	−8.03974	−0.493877
$266$	−6.19297	−0.379715
$267$	0	0
$268$	11.9615	0.730664
$269$	21.3290	1.30045	0.650227	−	0.759740i	$-0.274674\pi$
$269$	21.3290	1.30045	0.650227	+	0.759740i	$0.274674\pi$
$270$	0	0
$271$	2.10429	0.127827	0.0639134	−	0.997955i	$-0.479642\pi$
$271$	2.10429	0.127827	0.0639134	+	0.997955i	$0.479642\pi$
$272$	14.9910	0.908966
$273$	0	0
$274$	16.8538	1.01818
$275$	2.19092	0.132117
$276$	0	0
$277$	11.9745	0.719477	0.359738	−	0.933053i	$-0.382866\pi$
$277$	11.9745	0.719477	0.359738	+	0.933053i	$0.382866\pi$
$278$	12.4741	0.748147
$279$	0	0
$280$	3.58850	0.214454
$281$	−17.4431	−1.04057	−0.520283	−	0.853994i	$-0.674174\pi$
$281$	−17.4431	−1.04057	−0.520283	+	0.853994i	$0.674174\pi$
$282$	0	0
$283$	22.5162	1.33845	0.669225	−	0.743060i	$-0.266626\pi$
$283$	22.5162	1.33845	0.669225	+	0.743060i	$0.266626\pi$
$284$	1.95051	0.115741
$285$	0	0
$286$	2.69442	0.159324
$287$	−2.69492	−0.159076
$288$	0	0
$289$	43.5298	2.56058
$290$	6.72163	0.394708
$291$	0	0
$292$	9.14925	0.535420
$293$	−10.1887	−0.595229	−0.297615	−	0.954686i	$-0.596191\pi$
$293$	−10.1887	−0.595229	−0.297615	+	0.954686i	$0.596191\pi$
$294$	0	0
$295$	−3.82542	−0.222724
$296$	22.4181	1.30303
$297$	0	0
$298$	−8.44938	−0.489460
$299$	−2.49845	−0.144489
$300$	0	0
$301$	2.74281	0.158093
$302$	−4.78407	−0.275292
$303$	0	0
$304$	−9.15474	−0.525060
$305$	−5.72557	−0.327845
$306$	0	0
$307$	−10.2100	−0.582715	−0.291358	−	0.956614i	$-0.594107\pi$
$307$	−10.2100	−0.582715	−0.291358	+	0.956614i	$0.594107\pi$
$308$	1.92584	0.109735
$309$	0	0
$310$	−1.77120	−0.100597
$311$	−13.3442	−0.756678	−0.378339	−	0.925667i	$-0.623505\pi$
$311$	−13.3442	−0.756678	−0.378339	+	0.925667i	$0.623505\pi$
$312$	0	0
$313$	−13.7779	−0.778772	−0.389386	−	0.921075i	$-0.627313\pi$
$313$	−13.7779	−0.778772	−0.389386	+	0.921075i	$0.627313\pi$
$314$	−6.02465	−0.339991
$315$	0	0
$316$	−8.13279	−0.457505
$317$	32.6813	1.83557	0.917784	−	0.397081i	$-0.129977\pi$
$317$	32.6813	1.83557	0.917784	+	0.397081i	$0.129977\pi$
$318$	0	0
$319$	13.1879	0.738380
$320$	8.31685	0.464926
$321$	0	0
$322$	2.95704	0.164790
$323$	−36.9644	−2.05675
$324$	0	0
$325$	1.10132	0.0610901
$326$	17.5871	0.974061
$327$	0	0
$328$	−7.09759	−0.391899
$329$	−1.10804	−0.0610884
$330$	0	0
$331$	−17.1541	−0.942873	−0.471437	−	0.881900i	$-0.656264\pi$
$331$	−17.1541	−0.942873	−0.471437	+	0.881900i	$0.656264\pi$
$332$	2.71389	0.148944
$333$	0	0
$334$	20.3169	1.11169
$335$	−15.8843	−0.867849
$336$	0	0
$337$	14.8907	0.811149	0.405575	−	0.914062i	$-0.367071\pi$
$337$	14.8907	0.811149	0.405575	+	0.914062i	$0.367071\pi$
$338$	−13.1623	−0.715937
$339$	0	0
$340$	5.85873	0.317734
$341$	−3.47511	−0.188188
$342$	0	0
$343$	14.7514	0.796502
$344$	7.22370	0.389476
$345$	0	0
$346$	−4.90432	−0.263658
$347$	24.1687	1.29745	0.648723	−	0.761024i	$-0.275303\pi$
$347$	24.1687	1.29745	0.648723	+	0.761024i	$0.275303\pi$
$348$	0	0
$349$	33.4055	1.78816	0.894078	−	0.447910i	$-0.147832\pi$
$349$	33.4055	1.78816	0.894078	+	0.447910i	$0.147832\pi$
$350$	−1.30347	−0.0696733
$351$	0	0
$352$	8.75675	0.466736
$353$	24.5144	1.30477	0.652384	−	0.757889i	$-0.273769\pi$
$353$	24.5144	1.30477	0.652384	+	0.757889i	$0.273769\pi$
$354$	0	0
$355$	−2.59018	−0.137472
$356$	0.753041	0.0399111
$357$	0	0
$358$	25.3601	1.34032
$359$	−11.5135	−0.607657	−0.303829	−	0.952727i	$-0.598265\pi$
$359$	−11.5135	−0.607657	−0.303829	+	0.952727i	$0.598265\pi$
$360$	0	0
$361$	3.57341	0.188074
$362$	−19.2296	−1.01069
$363$	0	0
$364$	0.968067	0.0507405
$365$	−12.1497	−0.635947
$366$	0	0
$367$	−1.52333	−0.0795173	−0.0397587	−	0.999209i	$-0.512659\pi$
$367$	−1.52333	−0.0795173	−0.0397587	+	0.999209i	$0.512659\pi$
$368$	4.37124	0.227867
$369$	0	0
$370$	−8.14305	−0.423337
$371$	9.38461	0.487225
$372$	0	0
$373$	29.7945	1.54270	0.771351	−	0.636410i	$-0.219581\pi$
$373$	29.7945	1.54270	0.771351	+	0.636410i	$0.219581\pi$
$374$	−19.0343	−0.984241
$375$	0	0
$376$	−2.91824	−0.150497
$377$	6.62921	0.341421
$378$	0	0
$379$	19.5814	1.00583	0.502914	−	0.864337i	$-0.332261\pi$
$379$	19.5814	1.00583	0.502914	+	0.864337i	$0.332261\pi$
$380$	−3.57781	−0.183538
$381$	0	0
$382$	−22.9913	−1.17634
$383$	15.6845	0.801440	0.400720	−	0.916201i	$-0.368760\pi$
$383$	15.6845	0.801440	0.400720	+	0.916201i	$0.368760\pi$
$384$	0	0
$385$	−2.55741	−0.130338
$386$	−10.5938	−0.539210
$387$	0	0
$388$	2.39097	0.121383
$389$	−14.7385	−0.747273	−0.373636	−	0.927575i	$-0.621889\pi$
$389$	−14.7385	−0.747273	−0.373636	+	0.927575i	$0.621889\pi$
$390$	0	0
$391$	17.6499	0.892593
$392$	17.3310	0.875345
$393$	0	0
$394$	−28.9913	−1.46056
$395$	10.7999	0.543404
$396$	0	0
$397$	7.30391	0.366573	0.183286	−	0.983060i	$-0.441326\pi$
$397$	7.30391	0.366573	0.183286	+	0.983060i	$0.441326\pi$
$398$	−18.3405	−0.919328
$399$	0	0
$400$	−1.92685	−0.0963424
$401$	7.65061	0.382053	0.191027	−	0.981585i	$-0.438818\pi$
$401$	7.65061	0.382053	0.191027	+	0.981585i	$0.438818\pi$
$402$	0	0
$403$	−1.74684	−0.0870165
$404$	−4.57727	−0.227728
$405$	0	0
$406$	−7.84601	−0.389391
$407$	−15.9767	−0.791936
$408$	0	0
$409$	−23.1681	−1.14559	−0.572793	−	0.819700i	$-0.694140\pi$
$409$	−23.1681	−1.14559	−0.572793	+	0.819700i	$0.694140\pi$
$410$	2.57809	0.127323
$411$	0	0
$412$	11.6520	0.574055
$413$	4.46532	0.219724
$414$	0	0
$415$	−3.60391	−0.176909
$416$	4.40179	0.215815
$417$	0	0
$418$	11.6239	0.568543
$419$	23.7901	1.16222	0.581112	−	0.813824i	$-0.302618\pi$
$419$	23.7901	1.16222	0.581112	+	0.813824i	$0.302618\pi$
$420$	0	0
$421$	10.2706	0.500560	0.250280	−	0.968174i	$-0.419477\pi$
$421$	10.2706	0.500560	0.250280	+	0.968174i	$0.419477\pi$
$422$	10.3917	0.505861
$423$	0	0
$424$	24.7161	1.20032
$425$	−7.78009	−0.377390
$426$	0	0
$427$	6.68333	0.323429
$428$	2.82716	0.136656
$429$	0	0
$430$	−2.62390	−0.126536
$431$	−2.47844	−0.119382	−0.0596911	−	0.998217i	$-0.519012\pi$
$431$	−2.47844	−0.119382	−0.0596911	+	0.998217i	$0.519012\pi$
$432$	0	0
$433$	−13.9612	−0.670931	−0.335465	−	0.942053i	$-0.608894\pi$
$433$	−13.9612	−0.670931	−0.335465	+	0.942053i	$0.608894\pi$
$434$	2.06748	0.0992423
$435$	0	0
$436$	−10.7969	−0.517079
$437$	−10.7784	−0.515603
$438$	0	0
$439$	−3.44298	−0.164324	−0.0821622	−	0.996619i	$-0.526183\pi$
$439$	−3.44298	−0.164324	−0.0821622	+	0.996619i	$0.526183\pi$
$440$	−6.73543	−0.321099
$441$	0	0
$442$	−9.56805	−0.455106
$443$	16.7717	0.796847	0.398424	−	0.917201i	$-0.369557\pi$
$443$	16.7717	0.796847	0.398424	+	0.917201i	$0.369557\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	6.30842	0.298712
$447$	0	0
$448$	−9.70808	−0.458664
$449$	−7.85536	−0.370717	−0.185359	−	0.982671i	$-0.559345\pi$
$449$	−7.85536	−0.370717	−0.185359	+	0.982671i	$0.559345\pi$
$450$	0	0
$451$	5.05823	0.238183
$452$	−0.446331	−0.0209936
$453$	0	0
$454$	−3.01271	−0.141394
$455$	−1.28554	−0.0602672
$456$	0	0
$457$	−14.7105	−0.688129	−0.344064	−	0.938946i	$-0.611804\pi$
$457$	−14.7105	−0.688129	−0.344064	+	0.938946i	$0.611804\pi$
$458$	7.26822	0.339622
$459$	0	0
$460$	1.70835	0.0796520
$461$	9.79268	0.456090	0.228045	−	0.973651i	$-0.426767\pi$
$461$	9.79268	0.456090	0.228045	+	0.973651i	$0.426767\pi$
$462$	0	0
$463$	−12.8936	−0.599216	−0.299608	−	0.954062i	$-0.596856\pi$
$463$	−12.8936	−0.599216	−0.299608	+	0.954062i	$0.596856\pi$
$464$	−11.5983	−0.538440
$465$	0	0
$466$	18.5759	0.860513
$467$	2.35654	0.109048	0.0545239	−	0.998512i	$-0.482636\pi$
$467$	2.35654	0.109048	0.0545239	+	0.998512i	$0.482636\pi$
$468$	0	0
$469$	18.5413	0.856159
$470$	1.06001	0.0488945
$471$	0	0
$472$	11.7603	0.541311
$473$	−5.14811	−0.236710
$474$	0	0
$475$	4.75115	0.217998
$476$	−6.83876	−0.313454
$477$	0	0
$478$	−23.5945	−1.07919
$479$	−10.5084	−0.480142	−0.240071	−	0.970755i	$-0.577171\pi$
$479$	−10.5084	−0.480142	−0.240071	+	0.970755i	$0.577171\pi$
$480$	0	0
$481$	−8.03107	−0.366185
$482$	−6.26258	−0.285253
$483$	0	0
$484$	4.66875	0.212216
$485$	−3.17508	−0.144173
$486$	0	0
$487$	−22.3713	−1.01374	−0.506870	−	0.862023i	$-0.669198\pi$
$487$	−22.3713	−1.01374	−0.506870	+	0.862023i	$0.669198\pi$
$488$	17.6018	0.796796
$489$	0	0
$490$	−6.29520	−0.284388
$491$	−9.72458	−0.438864	−0.219432	−	0.975628i	$-0.570420\pi$
$491$	−9.72458	−0.438864	−0.219432	+	0.975628i	$0.570420\pi$
$492$	0	0
$493$	−46.8310	−2.10916
$494$	5.84302	0.262890
$495$	0	0
$496$	3.05625	0.137230
$497$	3.02346	0.135621
$498$	0	0
$499$	21.5022	0.962570	0.481285	−	0.876564i	$-0.340170\pi$
$499$	21.5022	0.962570	0.481285	+	0.876564i	$0.340170\pi$
$500$	−0.753041	−0.0336770
$501$	0	0
$502$	−17.0755	−0.762118
$503$	−31.3622	−1.39837	−0.699186	−	0.714940i	$-0.746454\pi$
$503$	−31.3622	−1.39837	−0.699186	+	0.714940i	$0.746454\pi$
$504$	0	0
$505$	6.07838	0.270484
$506$	−5.55022	−0.246737
$507$	0	0
$508$	13.0328	0.578237
$509$	18.7676	0.831859	0.415929	−	0.909397i	$-0.363456\pi$
$509$	18.7676	0.831859	0.415929	+	0.909397i	$0.363456\pi$
$510$	0	0
$511$	14.1821	0.627380
$512$	−19.5485	−0.863930
$513$	0	0
$514$	−3.80554	−0.167855
$515$	−15.4733	−0.681836
$516$	0	0
$517$	2.07974	0.0914669
$518$	9.50520	0.417634
$519$	0	0
$520$	−3.38572	−0.148474
$521$	−21.5513	−0.944180	−0.472090	−	0.881550i	$-0.656500\pi$
$521$	−21.5513	−0.944180	−0.472090	+	0.881550i	$0.656500\pi$
$522$	0	0
$523$	−8.72082	−0.381335	−0.190667	−	0.981655i	$-0.561065\pi$
$523$	−8.72082	−0.381335	−0.190667	+	0.981655i	$0.561065\pi$
$524$	4.91710	0.214805
$525$	0	0
$526$	−8.56260	−0.373347
$527$	12.3403	0.537552
$528$	0	0
$529$	−17.8535	−0.776238
$530$	−8.97776	−0.389969
$531$	0	0
$532$	4.17630	0.181065
$533$	2.54264	0.110134
$534$	0	0
$535$	−3.75432	−0.162313
$536$	48.8321	2.10923
$537$	0	0
$538$	23.8176	1.02685
$539$	−12.3512	−0.532005
$540$	0	0
$541$	−35.1880	−1.51285	−0.756424	−	0.654081i	$-0.773055\pi$
$541$	−35.1880	−1.51285	−0.756424	+	0.654081i	$0.773055\pi$
$542$	2.34981	0.100933
$543$	0	0
$544$	−31.0957	−1.33322
$545$	14.3378	0.614162
$546$	0	0
$547$	21.7368	0.929399	0.464700	−	0.885468i	$-0.346162\pi$
$547$	21.7368	0.929399	0.464700	+	0.885468i	$0.346162\pi$
$548$	−11.3656	−0.485513
$549$	0	0
$550$	2.44654	0.104321
$551$	28.5988	1.21835
$552$	0	0
$553$	−12.6065	−0.536084
$554$	13.3716	0.568104
$555$	0	0
$556$	−8.41204	−0.356750
$557$	35.5213	1.50509	0.752544	−	0.658542i	$-0.228827\pi$
$557$	35.5213	1.50509	0.752544	+	0.658542i	$0.228827\pi$
$558$	0	0
$559$	−2.58782	−0.109453
$560$	2.24917	0.0950446
$561$	0	0
$562$	−19.4782	−0.821639
$563$	−24.4265	−1.02946	−0.514728	−	0.857353i	$-0.672107\pi$
$563$	−24.4265	−1.02946	−0.514728	+	0.857353i	$0.672107\pi$
$564$	0	0
$565$	0.592705	0.0249353
$566$	25.1432	1.05685
$567$	0	0
$568$	7.96285	0.334114
$569$	4.52474	0.189687	0.0948435	−	0.995492i	$-0.469765\pi$
$569$	4.52474	0.189687	0.0948435	+	0.995492i	$0.469765\pi$
$570$	0	0
$571$	20.6420	0.863842	0.431921	−	0.901911i	$-0.357836\pi$
$571$	20.6420	0.863842	0.431921	+	0.901911i	$0.357836\pi$
$572$	−1.81701	−0.0759731
$573$	0	0
$574$	−3.00935	−0.125608
$575$	−2.26860	−0.0946070
$576$	0	0
$577$	29.9774	1.24798	0.623989	−	0.781433i	$-0.285511\pi$
$577$	29.9774	1.24798	0.623989	+	0.781433i	$0.285511\pi$
$578$	48.6086	2.02185
$579$	0	0
$580$	−4.53281	−0.188215
$581$	4.20676	0.174526
$582$	0	0
$583$	−17.6144	−0.729515
$584$	37.3513	1.54561
$585$	0	0
$586$	−11.3774	−0.469997
$587$	−33.2868	−1.37389	−0.686947	−	0.726708i	$-0.741050\pi$
$587$	−33.2868	−1.37389	−0.686947	+	0.726708i	$0.741050\pi$
$588$	0	0
$589$	−7.53599	−0.310515
$590$	−4.27174	−0.175865
$591$	0	0
$592$	14.0510	0.577494
$593$	29.9369	1.22936	0.614681	−	0.788776i	$-0.289285\pi$
$593$	29.9369	1.22936	0.614681	+	0.788776i	$0.289285\pi$
$594$	0	0
$595$	9.08153	0.372306
$596$	5.69793	0.233396
$597$	0	0
$598$	−2.78995	−0.114089
$599$	8.69422	0.355236	0.177618	−	0.984099i	$-0.443161\pi$
$599$	8.69422	0.355236	0.177618	+	0.984099i	$0.443161\pi$
$600$	0	0
$601$	−3.33351	−0.135977	−0.0679884	−	0.997686i	$-0.521658\pi$
$601$	−3.33351	−0.135977	−0.0679884	+	0.997686i	$0.521658\pi$
$602$	3.06282	0.124831
$603$	0	0
$604$	3.22619	0.131272
$605$	−6.19987	−0.252060
$606$	0	0
$607$	19.4556	0.789679	0.394839	−	0.918750i	$-0.370800\pi$
$607$	19.4556	0.789679	0.394839	+	0.918750i	$0.370800\pi$
$608$	18.9896	0.770128
$609$	0	0
$610$	−6.39359	−0.258869
$611$	1.04543	0.0422936
$612$	0	0
$613$	−7.99162	−0.322778	−0.161389	−	0.986891i	$-0.551597\pi$
$613$	−7.99162	−0.322778	−0.161389	+	0.986891i	$0.551597\pi$
$614$	−11.4012	−0.460116
$615$	0	0
$616$	7.86212	0.316774
$617$	−16.9756	−0.683414	−0.341707	−	0.939807i	$-0.611005\pi$
$617$	−16.9756	−0.683414	−0.341707	+	0.939807i	$0.611005\pi$
$618$	0	0
$619$	−30.8826	−1.24128	−0.620638	−	0.784097i	$-0.713126\pi$
$619$	−30.8826	−1.24128	−0.620638	+	0.784097i	$0.713126\pi$
$620$	1.19443	0.0479694
$621$	0	0
$622$	−14.9011	−0.597478
$623$	1.16728	0.0467660
$624$	0	0
$625$	1.00000	0.0400000
$626$	−15.3854	−0.614924
$627$	0	0
$628$	4.06279	0.162123
$629$	56.7343	2.26214
$630$	0	0
$631$	21.6472	0.861763	0.430881	−	0.902409i	$-0.358203\pi$
$631$	21.6472	0.861763	0.430881	+	0.902409i	$0.358203\pi$
$632$	−33.2017	−1.32069
$633$	0	0
$634$	36.4944	1.44938
$635$	−17.3069	−0.686803
$636$	0	0
$637$	−6.20864	−0.245995
$638$	14.7266	0.583030
$639$	0	0
$640$	1.29354	0.0511316
$641$	−39.4212	−1.55704	−0.778522	−	0.627617i	$-0.784030\pi$
$641$	−39.4212	−1.55704	−0.778522	+	0.627617i	$0.784030\pi$
$642$	0	0
$643$	42.5193	1.67680	0.838398	−	0.545058i	$-0.183492\pi$
$643$	42.5193	1.67680	0.838398	+	0.545058i	$0.183492\pi$
$644$	−1.99411	−0.0785791
$645$	0	0
$646$	−41.2771	−1.62403
$647$	27.5508	1.08313	0.541566	−	0.840658i	$-0.317831\pi$
$647$	27.5508	1.08313	0.541566	+	0.840658i	$0.317831\pi$
$648$	0	0
$649$	−8.38118	−0.328990
$650$	1.22981	0.0482372
$651$	0	0
$652$	−11.8601	−0.464476
$653$	7.69506	0.301131	0.150566	−	0.988600i	$-0.451891\pi$
$653$	7.69506	0.301131	0.150566	+	0.988600i	$0.451891\pi$
$654$	0	0
$655$	−6.52966	−0.255135
$656$	−4.44856	−0.173687
$657$	0	0
$658$	−1.23732	−0.0482359
$659$	−23.7448	−0.924965	−0.462482	−	0.886628i	$-0.653041\pi$
$659$	−23.7448	−0.924965	−0.462482	+	0.886628i	$0.653041\pi$
$660$	0	0
$661$	−13.5532	−0.527159	−0.263580	−	0.964638i	$-0.584903\pi$
$661$	−13.5532	−0.527159	−0.263580	+	0.964638i	$0.584903\pi$
$662$	−19.1555	−0.744499
$663$	0	0
$664$	11.0793	0.429961
$665$	−5.54591	−0.215061
$666$	0	0
$667$	−13.6554	−0.528741
$668$	−13.7009	−0.530104
$669$	0	0
$670$	−17.7375	−0.685260
$671$	−12.5443	−0.484266
$672$	0	0
$673$	34.2943	1.32195	0.660973	−	0.750409i	$-0.270144\pi$
$673$	34.2943	1.32195	0.660973	+	0.750409i	$0.270144\pi$
$674$	16.6281	0.640489
$675$	0	0
$676$	8.87617	0.341391
$677$	43.3280	1.66523	0.832616	−	0.553851i	$-0.186842\pi$
$677$	43.3280	1.66523	0.832616	+	0.553851i	$0.186842\pi$
$678$	0	0
$679$	3.70620	0.142231
$680$	23.9179	0.917210
$681$	0	0
$682$	−3.88056	−0.148594
$683$	11.4828	0.439378	0.219689	−	0.975570i	$-0.429496\pi$
$683$	11.4828	0.439378	0.219689	+	0.975570i	$0.429496\pi$
$684$	0	0
$685$	15.0929	0.576670
$686$	16.4725	0.628924
$687$	0	0
$688$	4.52760	0.172613
$689$	−8.85431	−0.337322
$690$	0	0
$691$	−6.59111	−0.250738	−0.125369	−	0.992110i	$-0.540011\pi$
$691$	−6.59111	−0.250738	−0.125369	+	0.992110i	$0.540011\pi$
$692$	3.30728	0.125724
$693$	0	0
$694$	26.9886	1.02447
$695$	11.1708	0.423731
$696$	0	0
$697$	−17.9621	−0.680362
$698$	37.3031	1.41194
$699$	0	0
$700$	0.879008	0.0332234
$701$	25.5796	0.966126	0.483063	−	0.875586i	$-0.339524\pi$
$701$	25.5796	0.966126	0.483063	+	0.875586i	$0.339524\pi$
$702$	0	0
$703$	−34.6465	−1.30672
$704$	18.2216	0.686751
$705$	0	0
$706$	27.3745	1.03025
$707$	−7.09516	−0.266841
$708$	0	0
$709$	33.3520	1.25256	0.626280	−	0.779598i	$-0.284577\pi$
$709$	33.3520	1.25256	0.626280	+	0.779598i	$0.284577\pi$
$710$	−2.89238	−0.108549
$711$	0	0
$712$	3.07425	0.115212
$713$	3.59831	0.134758
$714$	0	0
$715$	2.41290	0.0902373
$716$	−17.1018	−0.639126
$717$	0	0
$718$	−12.8568	−0.479811
$719$	−35.2920	−1.31617	−0.658084	−	0.752944i	$-0.728633\pi$
$719$	−35.2920	−1.31617	−0.658084	+	0.752944i	$0.728633\pi$
$720$	0	0
$721$	18.0617	0.672651
$722$	3.99033	0.148505
$723$	0	0
$724$	12.9677	0.481941
$725$	6.01934	0.223553
$726$	0	0
$727$	−28.5760	−1.05982	−0.529912	−	0.848053i	$-0.677775\pi$
$727$	−28.5760	−1.05982	−0.529912	+	0.848053i	$0.677775\pi$
$728$	3.95208	0.146474
$729$	0	0
$730$	−13.5673	−0.502148
$731$	18.2812	0.676156
$732$	0	0
$733$	40.9855	1.51384	0.756918	−	0.653510i	$-0.226704\pi$
$733$	40.9855	1.51384	0.756918	+	0.653510i	$0.226704\pi$
$734$	−1.70106	−0.0627874
$735$	0	0
$736$	−9.06720	−0.334222
$737$	−34.8011	−1.28192
$738$	0	0
$739$	−12.0594	−0.443613	−0.221807	−	0.975091i	$-0.571195\pi$
$739$	−12.0594	−0.443613	−0.221807	+	0.975091i	$0.571195\pi$
$740$	5.49135	0.201866
$741$	0	0
$742$	10.4795	0.384716
$743$	33.2440	1.21961	0.609803	−	0.792553i	$-0.291249\pi$
$743$	33.2440	1.21961	0.609803	+	0.792553i	$0.291249\pi$
$744$	0	0
$745$	−7.56657	−0.277218
$746$	33.2707	1.21813
$747$	0	0
$748$	12.8360	0.469331
$749$	4.38234	0.160127
$750$	0	0
$751$	17.5671	0.641033	0.320516	−	0.947243i	$-0.396144\pi$
$751$	17.5671	0.641033	0.320516	+	0.947243i	$0.396144\pi$
$752$	−1.82907	−0.0666993
$753$	0	0
$754$	7.40266	0.269589
$755$	−4.28422	−0.155919
$756$	0	0
$757$	8.12514	0.295313	0.147657	−	0.989039i	$-0.452827\pi$
$757$	8.12514	0.295313	0.147657	+	0.989039i	$0.452827\pi$
$758$	21.8660	0.794208
$759$	0	0
$760$	−14.6062	−0.529823
$761$	1.67104	0.0605750	0.0302875	−	0.999541i	$-0.490358\pi$
$761$	1.67104	0.0605750	0.0302875	+	0.999541i	$0.490358\pi$
$762$	0	0
$763$	−16.7362	−0.605890
$764$	15.5044	0.560931
$765$	0	0
$766$	17.5144	0.632823
$767$	−4.21300	−0.152123
$768$	0	0
$769$	−16.2382	−0.585563	−0.292782	−	0.956179i	$-0.594581\pi$
$769$	−16.2382	−0.585563	−0.292782	+	0.956179i	$0.594581\pi$
$770$	−2.85579	−0.102916
$771$	0	0
$772$	7.14404	0.257120
$773$	20.2377	0.727898	0.363949	−	0.931419i	$-0.381428\pi$
$773$	20.2377	0.727898	0.363949	+	0.931419i	$0.381428\pi$
$774$	0	0
$775$	−1.58614	−0.0569758
$776$	9.76099	0.350399
$777$	0	0
$778$	−16.4581	−0.590052
$779$	10.9691	0.393008
$780$	0	0
$781$	−5.67487	−0.203063
$782$	19.7092	0.704798
$783$	0	0
$784$	10.8625	0.387948
$785$	−5.39518	−0.192562
$786$	0	0
$787$	−19.8995	−0.709340	−0.354670	−	0.934991i	$-0.615407\pi$
$787$	−19.8995	−0.709340	−0.354670	+	0.934991i	$0.615407\pi$
$788$	19.5506	0.696461
$789$	0	0
$790$	12.0600	0.429076
$791$	−0.691851	−0.0245994
$792$	0	0
$793$	−6.30567	−0.223921
$794$	8.15609	0.289449
$795$	0	0
$796$	12.3681	0.438377
$797$	45.1119	1.59795	0.798973	−	0.601366i	$-0.205377\pi$
$797$	45.1119	1.59795	0.798973	+	0.601366i	$0.205377\pi$
$798$	0	0
$799$	−7.38529	−0.261273
$800$	3.99684	0.141309
$801$	0	0
$802$	8.54324	0.301672
$803$	−26.6191	−0.939368
$804$	0	0
$805$	2.64808	0.0933326
$806$	−1.95065	−0.0687089
$807$	0	0
$808$	−18.6864	−0.657387
$809$	10.7813	0.379049	0.189525	−	0.981876i	$-0.439305\pi$
$809$	10.7813	0.379049	0.189525	+	0.981876i	$0.439305\pi$
$810$	0	0
$811$	10.2037	0.358301	0.179150	−	0.983822i	$-0.442665\pi$
$811$	10.2037	0.358301	0.179150	+	0.983822i	$0.442665\pi$
$812$	5.29105	0.185679
$813$	0	0
$814$	−17.8408	−0.625319
$815$	15.7496	0.551683
$816$	0	0
$817$	−11.1640	−0.390579
$818$	−25.8711	−0.904564
$819$	0	0
$820$	−1.73856	−0.0607133
$821$	17.4466	0.608891	0.304445	−	0.952530i	$-0.401529\pi$
$821$	17.4466	0.608891	0.304445	+	0.952530i	$0.401529\pi$
$822$	0	0
$823$	1.89092	0.0659133	0.0329566	−	0.999457i	$-0.489508\pi$
$823$	1.89092	0.0659133	0.0329566	+	0.999457i	$0.489508\pi$
$824$	47.5688	1.65714
$825$	0	0
$826$	4.98631	0.173496
$827$	10.1852	0.354173	0.177086	−	0.984195i	$-0.443333\pi$
$827$	10.1852	0.354173	0.177086	+	0.984195i	$0.443333\pi$
$828$	0	0
$829$	−4.41864	−0.153466	−0.0767328	−	0.997052i	$-0.524449\pi$
$829$	−4.41864	−0.153466	−0.0767328	+	0.997052i	$0.524449\pi$
$830$	−4.02439	−0.139689
$831$	0	0
$832$	9.15950	0.317549
$833$	43.8600	1.51966
$834$	0	0
$835$	18.1941	0.629633
$836$	−7.83869	−0.271107
$837$	0	0
$838$	26.5658	0.917700
$839$	−10.8397	−0.374227	−0.187113	−	0.982338i	$-0.559913\pi$
$839$	−10.8397	−0.374227	−0.187113	+	0.982338i	$0.559913\pi$
$840$	0	0
$841$	7.23243	0.249394
$842$	11.4689	0.395246
$843$	0	0
$844$	−7.00777	−0.241217
$845$	−11.7871	−0.405489
$846$	0	0
$847$	7.23697	0.248665
$848$	15.4913	0.531975
$849$	0	0
$850$	−8.68782	−0.297990
$851$	16.5431	0.567092
$852$	0	0
$853$	38.3227	1.31214	0.656072	−	0.754698i	$-0.272217\pi$
$853$	38.3227	1.31214	0.656072	+	0.754698i	$0.272217\pi$
$854$	7.46309	0.255382
$855$	0	0
$856$	11.5417	0.394488
$857$	10.9435	0.373822	0.186911	−	0.982377i	$-0.440152\pi$
$857$	10.9435	0.373822	0.186911	+	0.982377i	$0.440152\pi$
$858$	0	0
$859$	−11.3613	−0.387643	−0.193821	−	0.981037i	$-0.562088\pi$
$859$	−11.3613	−0.387643	−0.193821	+	0.981037i	$0.562088\pi$
$860$	1.76946	0.0603379
$861$	0	0
$862$	−2.76761	−0.0942651
$863$	−33.9378	−1.15526	−0.577628	−	0.816300i	$-0.696022\pi$
$863$	−33.9378	−1.15526	−0.577628	+	0.816300i	$0.696022\pi$
$864$	0	0
$865$	−4.39190	−0.149329
$866$	−15.5901	−0.529772
$867$	0	0
$868$	−1.39423	−0.0473232
$869$	23.6618	0.802672
$870$	0	0
$871$	−17.4936	−0.592748
$872$	−44.0778	−1.49266
$873$	0	0
$874$	−12.0360	−0.407123
$875$	−1.16728	−0.0394612
$876$	0	0
$877$	30.3020	1.02322	0.511612	−	0.859216i	$-0.329048\pi$
$877$	30.3020	1.02322	0.511612	+	0.859216i	$0.329048\pi$
$878$	−3.84468	−0.129752
$879$	0	0
$880$	−4.22157	−0.142309
$881$	0.576968	0.0194385	0.00971927	−	0.999953i	$-0.496906\pi$
$881$	0.576968	0.0194385	0.00971927	+	0.999953i	$0.496906\pi$
$882$	0	0
$883$	24.3155	0.818282	0.409141	−	0.912471i	$-0.365828\pi$
$883$	24.3155	0.818282	0.409141	+	0.912471i	$0.365828\pi$
$884$	6.45232	0.217015
$885$	0	0
$886$	18.7285	0.629196
$887$	−36.5382	−1.22683	−0.613416	−	0.789760i	$-0.710205\pi$
$887$	−36.5382	−1.22683	−0.613416	+	0.789760i	$0.710205\pi$
$888$	0	0
$889$	20.2020	0.677552
$890$	−1.11667	−0.0374310
$891$	0	0
$892$	−4.25415	−0.142440
$893$	4.51005	0.150923
$894$	0	0
$895$	22.7104	0.759124
$896$	−1.50992	−0.0504428
$897$	0	0
$898$	−8.77187	−0.292721
$899$	−9.54751	−0.318427
$900$	0	0
$901$	62.5499	2.08384
$902$	5.64839	0.188071
$903$	0	0
$904$	−1.82212	−0.0606029
$905$	−17.2204	−0.572427
$906$	0	0
$907$	−34.4021	−1.14230	−0.571152	−	0.820845i	$-0.693503\pi$
$907$	−34.4021	−1.14230	−0.571152	+	0.820845i	$0.693503\pi$
$908$	2.03165	0.0674228
$909$	0	0
$910$	−1.43553	−0.0475874
$911$	−31.0883	−1.03000	−0.515001	−	0.857189i	$-0.672209\pi$
$911$	−31.0883	−1.03000	−0.515001	+	0.857189i	$0.672209\pi$
$912$	0	0
$913$	−7.89588	−0.261315
$914$	−16.4268	−0.543351
$915$	0	0
$916$	−4.90140	−0.161947
$917$	7.62193	0.251698
$918$	0	0
$919$	−12.3439	−0.407188	−0.203594	−	0.979055i	$-0.565262\pi$
$919$	−12.3439	−0.407188	−0.203594	+	0.979055i	$0.565262\pi$
$920$	6.97422	0.229933
$921$	0	0
$922$	10.9352	0.360132
$923$	−2.85261	−0.0938948
$924$	0	0
$925$	−7.29224	−0.239767
$926$	−14.3979	−0.473145
$927$	0	0
$928$	24.0583	0.789752
$929$	5.59982	0.183724	0.0918621	−	0.995772i	$-0.470718\pi$
$929$	5.59982	0.183724	0.0918621	+	0.995772i	$0.470718\pi$
$930$	0	0
$931$	−26.7844	−0.877824
$932$	−12.5269	−0.410331
$933$	0	0
$934$	2.63149	0.0861050
$935$	−17.0456	−0.557449
$936$	0	0
$937$	−18.6418	−0.609002	−0.304501	−	0.952512i	$-0.598490\pi$
$937$	−18.6418	−0.609002	−0.304501	+	0.952512i	$0.598490\pi$
$938$	20.7046	0.676030
$939$	0	0
$940$	−0.714828	−0.0233151
$941$	−2.41923	−0.0788647	−0.0394324	−	0.999222i	$-0.512555\pi$
$941$	−2.41923	−0.0788647	−0.0394324	+	0.999222i	$0.512555\pi$
$942$	0	0
$943$	−5.23756	−0.170558
$944$	7.37099	0.239905
$945$	0	0
$946$	−5.74876	−0.186908
$947$	−42.9867	−1.39688	−0.698440	−	0.715669i	$-0.746122\pi$
$947$	−42.9867	−1.39688	−0.698440	+	0.715669i	$0.746122\pi$
$948$	0	0
$949$	−13.3807	−0.434357
$950$	5.30548	0.172133
$951$	0	0
$952$	−27.9189	−0.904856
$953$	−1.84709	−0.0598332	−0.0299166	−	0.999552i	$-0.509524\pi$
$953$	−1.84709	−0.0598332	−0.0299166	+	0.999552i	$0.509524\pi$
$954$	0	0
$955$	−20.5891	−0.666247
$956$	15.9112	0.514606
$957$	0	0
$958$	−11.7345	−0.379124
$959$	−17.6176	−0.568902
$960$	0	0
$961$	−28.4842	−0.918844
$962$	−8.96808	−0.289143
$963$	0	0
$964$	4.22324	0.136021
$965$	−9.48692	−0.305395
$966$	0	0
$967$	−3.91413	−0.125870	−0.0629350	−	0.998018i	$-0.520046\pi$
$967$	−3.91413	−0.125870	−0.0629350	+	0.998018i	$0.520046\pi$
$968$	19.0599	0.612609
$969$	0	0
$970$	−3.54553	−0.113840
$971$	−0.764398	−0.0245307	−0.0122653	−	0.999925i	$-0.503904\pi$
$971$	−0.764398	−0.0245307	−0.0122653	+	0.999925i	$0.503904\pi$
$972$	0	0
$973$	−13.0394	−0.418024
$974$	−24.9814	−0.800456
$975$	0	0
$976$	11.0323	0.353135
$977$	−23.1991	−0.742204	−0.371102	−	0.928592i	$-0.621020\pi$
$977$	−23.1991	−0.742204	−0.371102	+	0.928592i	$0.621020\pi$
$978$	0	0
$979$	−2.19092	−0.0700221
$980$	4.24524	0.135609
$981$	0	0
$982$	−10.8592	−0.346530
$983$	−58.1928	−1.85606	−0.928031	−	0.372503i	$-0.878500\pi$
$983$	−58.1928	−1.85606	−0.928031	+	0.372503i	$0.878500\pi$
$984$	0	0
$985$	−25.9622	−0.827224
$986$	−52.2949	−1.66541
$987$	0	0
$988$	−3.94031	−0.125358
$989$	5.33063	0.169504
$990$	0	0
$991$	−29.7580	−0.945295	−0.472648	−	0.881252i	$-0.656702\pi$
$991$	−29.7580	−0.945295	−0.472648	+	0.881252i	$0.656702\pi$
$992$	−6.33954	−0.201281
$993$	0	0
$994$	3.37621	0.107087
$995$	−16.4243	−0.520684
$996$	0	0
$997$	30.6497	0.970687	0.485343	−	0.874324i	$-0.338695\pi$
$997$	30.6497	0.970687	0.485343	+	0.874324i	$0.338695\pi$
$998$	24.0109	0.760053
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.p.1.6		8
3.2	odd	2		445.2.a.g.1.3	✓	8
12.11	even	2		7120.2.a.bk.1.1		8
15.14	odd	2		2225.2.a.l.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.3	✓	8	3.2	odd	2
2225.2.a.l.1.6		8	15.14	odd	2
4005.2.a.p.1.6		8	1.1	even	1	trivial
7120.2.a.bk.1.1		8	12.11	even	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)

\(f(q)\)	\(=\)	\(q+1.11667 q^{2} -0.753041 q^{4} +1.00000 q^{5} -1.16728 q^{7} -3.07425 q^{8} +O(q^{10})\)	\(q+1.11667 q^{2} -0.753041 q^{4} +1.00000 q^{5} -1.16728 q^{7} -3.07425 q^{8} +1.11667 q^{10} +2.19092 q^{11} +1.10132 q^{13} -1.30347 q^{14} -1.92685 q^{16} -7.78009 q^{17} +4.75115 q^{19} -0.753041 q^{20} +2.44654 q^{22} -2.26860 q^{23} +1.00000 q^{25} +1.22981 q^{26} +0.879008 q^{28} +6.01934 q^{29} -1.58614 q^{31} +3.99684 q^{32} -8.68782 q^{34} -1.16728 q^{35} -7.29224 q^{37} +5.30548 q^{38} -3.07425 q^{40} +2.30872 q^{41} -2.34975 q^{43} -1.64985 q^{44} -2.53328 q^{46} +0.949255 q^{47} -5.63746 q^{49} +1.11667 q^{50} -0.829338 q^{52} -8.03974 q^{53} +2.19092 q^{55} +3.58850 q^{56} +6.72163 q^{58} -3.82542 q^{59} -5.72557 q^{61} -1.77120 q^{62} +8.31685 q^{64} +1.10132 q^{65} -15.8843 q^{67} +5.85873 q^{68} -1.30347 q^{70} -2.59018 q^{71} -12.1497 q^{73} -8.14305 q^{74} -3.57781 q^{76} -2.55741 q^{77} +10.7999 q^{79} -1.92685 q^{80} +2.57809 q^{82} -3.60391 q^{83} -7.78009 q^{85} -2.62390 q^{86} -6.73543 q^{88} -1.00000 q^{89} -1.28554 q^{91} +1.70835 q^{92} +1.06001 q^{94} +4.75115 q^{95} -3.17508 q^{97} -6.29520 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more