LMFDB - Embedded newform 8001.2.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.678644$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.678644 q^{2} -1.53944 q^{4} -2.84434 q^{5} -1.00000 q^{7} -2.40202 q^{8} +O(q^{10})$	\(q+0.678644 q^{2} -1.53944 q^{4} -2.84434 q^{5} -1.00000 q^{7} -2.40202 q^{8} -1.93030 q^{10} -1.95936 q^{11} -0.416350 q^{13} -0.678644 q^{14} +1.44876 q^{16} +6.48809 q^{17} -6.07676 q^{19} +4.37870 q^{20} -1.32971 q^{22} +4.22110 q^{23} +3.09028 q^{25} -0.282553 q^{26} +1.53944 q^{28} +10.6915 q^{29} -8.61963 q^{31} +5.78724 q^{32} +4.40311 q^{34} +2.84434 q^{35} +0.268648 q^{37} -4.12396 q^{38} +6.83217 q^{40} +1.71432 q^{41} +7.53157 q^{43} +3.01632 q^{44} +2.86463 q^{46} -3.95287 q^{47} +1.00000 q^{49} +2.09720 q^{50} +0.640946 q^{52} -0.195448 q^{53} +5.57310 q^{55} +2.40202 q^{56} +7.25575 q^{58} -0.232595 q^{59} -2.32325 q^{61} -5.84967 q^{62} +1.02995 q^{64} +1.18424 q^{65} -5.65707 q^{67} -9.98804 q^{68} +1.93030 q^{70} +13.5740 q^{71} +10.5317 q^{73} +0.182317 q^{74} +9.35483 q^{76} +1.95936 q^{77} +9.40508 q^{79} -4.12078 q^{80} +1.16341 q^{82} -12.8480 q^{83} -18.4544 q^{85} +5.11126 q^{86} +4.70643 q^{88} -8.15063 q^{89} +0.416350 q^{91} -6.49814 q^{92} -2.68259 q^{94} +17.2844 q^{95} +0.572444 q^{97} +0.678644 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * 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 $

$2$	0.678644	0.479874	0.239937	−	0.970788i	$-0.422873\pi$
$2$	0.678644	0.479874	0.239937	+	0.970788i	$0.422873\pi$
$3$	0	0
$3$	0	0
$4$	−1.53944	−0.769721
$5$	−2.84434	−1.27203	−0.636014	−	0.771677i	$-0.719418\pi$
$5$	−2.84434	−1.27203	−0.636014	+	0.771677i	$0.719418\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.40202	−0.849243
$9$	0	0
$10$	−1.93030	−0.610413
$11$	−1.95936	−0.590770	−0.295385	−	0.955378i	$-0.595448\pi$
$11$	−1.95936	−0.590770	−0.295385	+	0.955378i	$0.595448\pi$
$12$	0	0
$13$	−0.416350	−0.115475	−0.0577373	−	0.998332i	$-0.518389\pi$
$13$	−0.416350	−0.115475	−0.0577373	+	0.998332i	$0.518389\pi$
$14$	−0.678644	−0.181375
$15$	0	0
$16$	1.44876	0.362191
$17$	6.48809	1.57359	0.786797	−	0.617212i	$-0.211738\pi$
$17$	6.48809	1.57359	0.786797	+	0.617212i	$0.211738\pi$
$18$	0	0
$19$	−6.07676	−1.39411	−0.697053	−	0.717020i	$-0.745506\pi$
$19$	−6.07676	−1.39411	−0.697053	+	0.717020i	$0.745506\pi$
$20$	4.37870	0.979107
$21$	0	0
$22$	−1.32971	−0.283495
$23$	4.22110	0.880161	0.440081	−	0.897958i	$-0.354950\pi$
$23$	4.22110	0.880161	0.440081	+	0.897958i	$0.354950\pi$
$24$	0	0
$25$	3.09028	0.618057
$26$	−0.282553	−0.0554133
$27$	0	0
$28$	1.53944	0.290927
$29$	10.6915	1.98537	0.992684	−	0.120739i	$-0.0385264\pi$
$29$	10.6915	1.98537	0.992684	+	0.120739i	$0.0385264\pi$
$30$	0	0
$31$	−8.61963	−1.54813	−0.774066	−	0.633105i	$-0.781780\pi$
$31$	−8.61963	−1.54813	−0.774066	+	0.633105i	$0.781780\pi$
$32$	5.78724	1.02305
$33$	0	0
$34$	4.40311	0.755127
$35$	2.84434	0.480782
$36$	0	0
$37$	0.268648	0.0441655	0.0220828	−	0.999756i	$-0.492970\pi$
$37$	0.268648	0.0441655	0.0220828	+	0.999756i	$0.492970\pi$
$38$	−4.12396	−0.668995
$39$	0	0
$40$	6.83217	1.08026
$41$	1.71432	0.267731	0.133866	−	0.990999i	$-0.457261\pi$
$41$	1.71432	0.267731	0.133866	+	0.990999i	$0.457261\pi$
$42$	0	0
$43$	7.53157	1.14855	0.574277	−	0.818661i	$-0.305283\pi$
$43$	7.53157	1.14855	0.574277	+	0.818661i	$0.305283\pi$
$44$	3.01632	0.454728
$45$	0	0
$46$	2.86463	0.422366
$47$	−3.95287	−0.576586	−0.288293	−	0.957542i	$-0.593088\pi$
$47$	−3.95287	−0.576586	−0.288293	+	0.957542i	$0.593088\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.09720	0.296589
$51$	0	0
$52$	0.640946	0.0888832
$53$	−0.195448	−0.0268468	−0.0134234	−	0.999910i	$-0.504273\pi$
$53$	−0.195448	−0.0268468	−0.0134234	+	0.999910i	$0.504273\pi$
$54$	0	0
$55$	5.57310	0.751476
$56$	2.40202	0.320984
$57$	0	0
$58$	7.25575	0.952727
$59$	−0.232595	−0.0302812	−0.0151406	−	0.999885i	$-0.504820\pi$
$59$	−0.232595	−0.0302812	−0.0151406	+	0.999885i	$0.504820\pi$
$60$	0	0
$61$	−2.32325	−0.297462	−0.148731	−	0.988878i	$-0.547519\pi$
$61$	−2.32325	−0.297462	−0.148731	+	0.988878i	$0.547519\pi$
$62$	−5.84967	−0.742908
$63$	0	0
$64$	1.02995	0.128744
$65$	1.18424	0.146887
$66$	0	0
$67$	−5.65707	−0.691122	−0.345561	−	0.938396i	$-0.612311\pi$
$67$	−5.65707	−0.691122	−0.345561	+	0.938396i	$0.612311\pi$
$68$	−9.98804	−1.21123
$69$	0	0
$70$	1.93030	0.230715
$71$	13.5740	1.61094	0.805469	−	0.592638i	$-0.201914\pi$
$71$	13.5740	1.61094	0.805469	+	0.592638i	$0.201914\pi$
$72$	0	0
$73$	10.5317	1.23265	0.616323	−	0.787494i	$-0.288622\pi$
$73$	10.5317	1.23265	0.616323	+	0.787494i	$0.288622\pi$
$74$	0.182317	0.0211939
$75$	0	0
$76$	9.35483	1.07307
$77$	1.95936	0.223290
$78$	0	0
$79$	9.40508	1.05815	0.529077	−	0.848574i	$-0.322538\pi$
$79$	9.40508	1.05815	0.529077	+	0.848574i	$0.322538\pi$
$80$	−4.12078	−0.460717
$81$	0	0
$82$	1.16341	0.128477
$83$	−12.8480	−1.41026	−0.705128	−	0.709080i	$-0.749111\pi$
$83$	−12.8480	−1.41026	−0.705128	+	0.709080i	$0.749111\pi$
$84$	0	0
$85$	−18.4544	−2.00166
$86$	5.11126	0.551161
$87$	0	0
$88$	4.70643	0.501707
$89$	−8.15063	−0.863965	−0.431982	−	0.901882i	$-0.642186\pi$
$89$	−8.15063	−0.863965	−0.431982	+	0.901882i	$0.642186\pi$
$90$	0	0
$91$	0.416350	0.0436453
$92$	−6.49814	−0.677478
$93$	0	0
$94$	−2.68259	−0.276689
$95$	17.2844	1.77334
$96$	0	0
$97$	0.572444	0.0581229	0.0290614	−	0.999578i	$-0.490748\pi$
$97$	0.572444	0.0581229	0.0290614	+	0.999578i	$0.490748\pi$
$98$	0.678644	0.0685534
$99$	0	0
$100$	−4.75731	−0.475731
$101$	18.0627	1.79731	0.898653	−	0.438660i	$-0.144547\pi$
$101$	18.0627	1.79731	0.898653	+	0.438660i	$0.144547\pi$
$102$	0	0
$103$	−10.9118	−1.07517	−0.537587	−	0.843208i	$-0.680664\pi$
$103$	−10.9118	−1.07517	−0.537587	+	0.843208i	$0.680664\pi$
$104$	1.00008	0.0980660
$105$	0	0
$106$	−0.132639	−0.0128831
$107$	6.08048	0.587822	0.293911	−	0.955833i	$-0.405043\pi$
$107$	6.08048	0.587822	0.293911	+	0.955833i	$0.405043\pi$
$108$	0	0
$109$	−7.18280	−0.687988	−0.343994	−	0.938972i	$-0.611780\pi$
$109$	−7.18280	−0.687988	−0.343994	+	0.938972i	$0.611780\pi$
$110$	3.78215	0.360614
$111$	0	0
$112$	−1.44876	−0.136895
$113$	2.05699	0.193505	0.0967527	−	0.995308i	$-0.469154\pi$
$113$	2.05699	0.193505	0.0967527	+	0.995308i	$0.469154\pi$
$114$	0	0
$115$	−12.0063	−1.11959
$116$	−16.4590	−1.52818
$117$	0	0
$118$	−0.157849	−0.0145312
$119$	−6.48809	−0.594762
$120$	0	0
$121$	−7.16090	−0.650991
$122$	−1.57666	−0.142744
$123$	0	0
$124$	13.2694	1.19163
$125$	5.43189	0.485843
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−10.8755	−0.961269
$129$	0	0
$130$	0.803678	0.0704872
$131$	−12.9223	−1.12903	−0.564515	−	0.825423i	$-0.690937\pi$
$131$	−12.9223	−1.12903	−0.564515	+	0.825423i	$0.690937\pi$
$132$	0	0
$133$	6.07676	0.526922
$134$	−3.83914	−0.331651
$135$	0	0
$136$	−15.5845	−1.33636
$137$	−4.41731	−0.377397	−0.188698	−	0.982035i	$-0.560427\pi$
$137$	−4.41731	−0.377397	−0.188698	+	0.982035i	$0.560427\pi$
$138$	0	0
$139$	8.85945	0.751449	0.375724	−	0.926731i	$-0.377394\pi$
$139$	8.85945	0.751449	0.375724	+	0.926731i	$0.377394\pi$
$140$	−4.37870	−0.370068
$141$	0	0
$142$	9.21192	0.773047
$143$	0.815780	0.0682189
$144$	0	0
$145$	−30.4104	−2.52545
$146$	7.14730	0.591515
$147$	0	0
$148$	−0.413568	−0.0339951
$149$	−10.8457	−0.888516	−0.444258	−	0.895899i	$-0.646533\pi$
$149$	−10.8457	−0.888516	−0.444258	+	0.895899i	$0.646533\pi$
$150$	0	0
$151$	−9.96595	−0.811017	−0.405509	−	0.914091i	$-0.632906\pi$
$151$	−9.96595	−0.811017	−0.405509	+	0.914091i	$0.632906\pi$
$152$	14.5965	1.18393
$153$	0	0
$154$	1.32971	0.107151
$155$	24.5172	1.96927
$156$	0	0
$157$	−0.992602	−0.0792182	−0.0396091	−	0.999215i	$-0.512611\pi$
$157$	−0.992602	−0.0792182	−0.0396091	+	0.999215i	$0.512611\pi$
$158$	6.38271	0.507781
$159$	0	0
$160$	−16.4609	−1.30135
$161$	−4.22110	−0.332670
$162$	0	0
$163$	0.690853	0.0541118	0.0270559	−	0.999634i	$-0.491387\pi$
$163$	0.690853	0.0541118	0.0270559	+	0.999634i	$0.491387\pi$
$164$	−2.63909	−0.206078
$165$	0	0
$166$	−8.71925	−0.676745
$167$	−16.5472	−1.28046	−0.640232	−	0.768182i	$-0.721162\pi$
$167$	−16.5472	−1.28046	−0.640232	+	0.768182i	$0.721162\pi$
$168$	0	0
$169$	−12.8267	−0.986666
$170$	−12.5239	−0.960543
$171$	0	0
$172$	−11.5944	−0.884066
$173$	13.3328	1.01367	0.506835	−	0.862043i	$-0.330815\pi$
$173$	13.3328	1.01367	0.506835	+	0.862043i	$0.330815\pi$
$174$	0	0
$175$	−3.09028	−0.233603
$176$	−2.83865	−0.213972
$177$	0	0
$178$	−5.53138	−0.414594
$179$	−9.65507	−0.721654	−0.360827	−	0.932633i	$-0.617505\pi$
$179$	−9.65507	−0.721654	−0.360827	+	0.932633i	$0.617505\pi$
$180$	0	0
$181$	22.5306	1.67469	0.837344	−	0.546676i	$-0.184107\pi$
$181$	22.5306	1.67469	0.837344	+	0.546676i	$0.184107\pi$
$182$	0.282553	0.0209442
$183$	0	0
$184$	−10.1392	−0.747471
$185$	−0.764128	−0.0561798
$186$	0	0
$187$	−12.7125	−0.929632
$188$	6.08522	0.443810
$189$	0	0
$190$	11.7300	0.850981
$191$	−3.32247	−0.240406	−0.120203	−	0.992749i	$-0.538354\pi$
$191$	−3.32247	−0.240406	−0.120203	+	0.992749i	$0.538354\pi$
$192$	0	0
$193$	2.76029	0.198690	0.0993450	−	0.995053i	$-0.468325\pi$
$193$	2.76029	0.198690	0.0993450	+	0.995053i	$0.468325\pi$
$194$	0.388486	0.0278917
$195$	0	0
$196$	−1.53944	−0.109960
$197$	3.45828	0.246392	0.123196	−	0.992382i	$-0.460686\pi$
$197$	3.45828	0.246392	0.123196	+	0.992382i	$0.460686\pi$
$198$	0	0
$199$	−13.2835	−0.941640	−0.470820	−	0.882229i	$-0.656042\pi$
$199$	−13.2835	−0.941640	−0.470820	+	0.882229i	$0.656042\pi$
$200$	−7.42293	−0.524880
$201$	0	0
$202$	12.2582	0.862481
$203$	−10.6915	−0.750399
$204$	0	0
$205$	−4.87610	−0.340562
$206$	−7.40525	−0.515948
$207$	0	0
$208$	−0.603192	−0.0418239
$209$	11.9066	0.823596
$210$	0	0
$211$	−20.6867	−1.42413	−0.712066	−	0.702112i	$-0.752240\pi$
$211$	−20.6867	−1.42413	−0.712066	+	0.702112i	$0.752240\pi$
$212$	0.300880	0.0206645
$213$	0	0
$214$	4.12648	0.282080
$215$	−21.4224	−1.46099
$216$	0	0
$217$	8.61963	0.585139
$218$	−4.87457	−0.330147
$219$	0	0
$220$	−8.57946	−0.578427
$221$	−2.70131	−0.181710
$222$	0	0
$223$	−14.9115	−0.998545	−0.499273	−	0.866445i	$-0.666399\pi$
$223$	−14.9115	−0.998545	−0.499273	+	0.866445i	$0.666399\pi$
$224$	−5.78724	−0.386676
$225$	0	0
$226$	1.39597	0.0928582
$227$	16.2497	1.07853	0.539264	−	0.842137i	$-0.318702\pi$
$227$	16.2497	1.07853	0.539264	+	0.842137i	$0.318702\pi$
$228$	0	0
$229$	8.89843	0.588025	0.294012	−	0.955802i	$-0.405009\pi$
$229$	8.89843	0.588025	0.294012	+	0.955802i	$0.405009\pi$
$230$	−8.14798	−0.537262
$231$	0	0
$232$	−25.6813	−1.68606
$233$	4.85709	0.318199	0.159099	−	0.987263i	$-0.449141\pi$
$233$	4.85709	0.318199	0.159099	+	0.987263i	$0.449141\pi$
$234$	0	0
$235$	11.2433	0.733434
$236$	0.358066	0.0233081
$237$	0	0
$238$	−4.40311	−0.285411
$239$	−19.9322	−1.28931	−0.644655	−	0.764474i	$-0.722999\pi$
$239$	−19.9322	−1.28931	−0.644655	+	0.764474i	$0.722999\pi$
$240$	0	0
$241$	−2.80065	−0.180406	−0.0902030	−	0.995923i	$-0.528752\pi$
$241$	−2.80065	−0.180406	−0.0902030	+	0.995923i	$0.528752\pi$
$242$	−4.85970	−0.312394
$243$	0	0
$244$	3.57651	0.228963
$245$	−2.84434	−0.181718
$246$	0	0
$247$	2.53006	0.160984
$248$	20.7046	1.31474
$249$	0	0
$250$	3.68632	0.233143
$251$	−22.5830	−1.42542	−0.712712	−	0.701457i	$-0.752533\pi$
$251$	−22.5830	−1.42542	−0.712712	+	0.701457i	$0.752533\pi$
$252$	0	0
$253$	−8.27067	−0.519973
$254$	0.678644	0.0425819
$255$	0	0
$256$	−9.44050	−0.590031
$257$	−15.3995	−0.960596	−0.480298	−	0.877105i	$-0.659472\pi$
$257$	−15.3995	−0.960596	−0.480298	+	0.877105i	$0.659472\pi$
$258$	0	0
$259$	−0.268648	−0.0166930
$260$	−1.82307	−0.113062
$261$	0	0
$262$	−8.76967	−0.541792
$263$	−19.9988	−1.23318	−0.616589	−	0.787285i	$-0.711486\pi$
$263$	−19.9988	−1.23318	−0.616589	+	0.787285i	$0.711486\pi$
$264$	0	0
$265$	0.555920	0.0341499
$266$	4.12396	0.252856
$267$	0	0
$268$	8.70874	0.531971
$269$	16.7251	1.01975	0.509874	−	0.860249i	$-0.329692\pi$
$269$	16.7251	1.01975	0.509874	+	0.860249i	$0.329692\pi$
$270$	0	0
$271$	−11.3979	−0.692371	−0.346186	−	0.938166i	$-0.612523\pi$
$271$	−11.3979	−0.692371	−0.346186	+	0.938166i	$0.612523\pi$
$272$	9.39972	0.569942
$273$	0	0
$274$	−2.99778	−0.181103
$275$	−6.05498	−0.365129
$276$	0	0
$277$	3.79930	0.228278	0.114139	−	0.993465i	$-0.463589\pi$
$277$	3.79930	0.228278	0.114139	+	0.993465i	$0.463589\pi$
$278$	6.01242	0.360601
$279$	0	0
$280$	−6.83217	−0.408300
$281$	7.61849	0.454481	0.227240	−	0.973839i	$-0.427030\pi$
$281$	7.61849	0.454481	0.227240	+	0.973839i	$0.427030\pi$
$282$	0	0
$283$	−11.8094	−0.701995	−0.350997	−	0.936376i	$-0.614157\pi$
$283$	−11.8094	−0.701995	−0.350997	+	0.936376i	$0.614157\pi$
$284$	−20.8964	−1.23997
$285$	0	0
$286$	0.553624	0.0327365
$287$	−1.71432	−0.101193
$288$	0	0
$289$	25.0953	1.47620
$290$	−20.6378	−1.21190
$291$	0	0
$292$	−16.2130	−0.948793
$293$	13.2739	0.775469	0.387734	−	0.921771i	$-0.373258\pi$
$293$	13.2739	0.775469	0.387734	+	0.921771i	$0.373258\pi$
$294$	0	0
$295$	0.661579	0.0385186
$296$	−0.645299	−0.0375073
$297$	0	0
$298$	−7.36039	−0.426376
$299$	−1.75745	−0.101636
$300$	0	0
$301$	−7.53157	−0.434113
$302$	−6.76333	−0.389186
$303$	0	0
$304$	−8.80380	−0.504933
$305$	6.60813	0.378380
$306$	0	0
$307$	−16.1309	−0.920641	−0.460321	−	0.887753i	$-0.652266\pi$
$307$	−16.1309	−0.920641	−0.460321	+	0.887753i	$0.652266\pi$
$308$	−3.01632	−0.171871
$309$	0	0
$310$	16.6385	0.945001
$311$	−5.97244	−0.338666	−0.169333	−	0.985559i	$-0.554161\pi$
$311$	−5.97244	−0.338666	−0.169333	+	0.985559i	$0.554161\pi$
$312$	0	0
$313$	7.75509	0.438344	0.219172	−	0.975686i	$-0.429664\pi$
$313$	7.75509	0.438344	0.219172	+	0.975686i	$0.429664\pi$
$314$	−0.673624	−0.0380148
$315$	0	0
$316$	−14.4786	−0.814484
$317$	10.8814	0.611162	0.305581	−	0.952166i	$-0.401149\pi$
$317$	10.8814	0.611162	0.305581	+	0.952166i	$0.401149\pi$
$318$	0	0
$319$	−20.9486	−1.17290
$320$	−2.92953	−0.163766
$321$	0	0
$322$	−2.86463	−0.159640
$323$	−39.4266	−2.19376
$324$	0	0
$325$	−1.28664	−0.0713698
$326$	0.468844	0.0259669
$327$	0	0
$328$	−4.11783	−0.227369
$329$	3.95287	0.217929
$330$	0	0
$331$	29.2183	1.60598	0.802990	−	0.595992i	$-0.203241\pi$
$331$	29.2183	1.60598	0.802990	+	0.595992i	$0.203241\pi$
$332$	19.7788	1.08550
$333$	0	0
$334$	−11.2297	−0.614461
$335$	16.0907	0.879126
$336$	0	0
$337$	−23.3404	−1.27143	−0.635716	−	0.771923i	$-0.719295\pi$
$337$	−23.3404	−1.27143	−0.635716	+	0.771923i	$0.719295\pi$
$338$	−8.70474	−0.473475
$339$	0	0
$340$	28.4094	1.54072
$341$	16.8890	0.914590
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−18.0910	−0.975401
$345$	0	0
$346$	9.04820	0.486434
$347$	17.4752	0.938115	0.469058	−	0.883168i	$-0.344594\pi$
$347$	17.4752	0.938115	0.469058	+	0.883168i	$0.344594\pi$
$348$	0	0
$349$	−4.12739	−0.220934	−0.110467	−	0.993880i	$-0.535235\pi$
$349$	−4.12739	−0.220934	−0.110467	+	0.993880i	$0.535235\pi$
$350$	−2.09720	−0.112100
$351$	0	0
$352$	−11.3393	−0.604387
$353$	−0.119097	−0.00633887	−0.00316944	−	0.999995i	$-0.501009\pi$
$353$	−0.119097	−0.00633887	−0.00316944	+	0.999995i	$0.501009\pi$
$354$	0	0
$355$	−38.6091	−2.04916
$356$	12.5474	0.665012
$357$	0	0
$358$	−6.55236	−0.346303
$359$	−7.95213	−0.419697	−0.209849	−	0.977734i	$-0.567297\pi$
$359$	−7.95213	−0.419697	−0.209849	+	0.977734i	$0.567297\pi$
$360$	0	0
$361$	17.9271	0.943530
$362$	15.2903	0.803639
$363$	0	0
$364$	−0.640946	−0.0335947
$365$	−29.9558	−1.56796
$366$	0	0
$367$	2.86667	0.149639	0.0748196	−	0.997197i	$-0.476162\pi$
$367$	2.86667	0.149639	0.0748196	+	0.997197i	$0.476162\pi$
$368$	6.11539	0.318787
$369$	0	0
$370$	−0.518571	−0.0269592
$371$	0.195448	0.0101471
$372$	0	0
$373$	−4.02564	−0.208440	−0.104220	−	0.994554i	$-0.533235\pi$
$373$	−4.02564	−0.208440	−0.104220	+	0.994554i	$0.533235\pi$
$374$	−8.62728	−0.446106
$375$	0	0
$376$	9.49489	0.489661
$377$	−4.45142	−0.229260
$378$	0	0
$379$	10.1545	0.521599	0.260800	−	0.965393i	$-0.416014\pi$
$379$	10.1545	0.521599	0.260800	+	0.965393i	$0.416014\pi$
$380$	−26.6083	−1.36498
$381$	0	0
$382$	−2.25478	−0.115364
$383$	0.429270	0.0219347	0.0109673	−	0.999940i	$-0.496509\pi$
$383$	0.429270	0.0219347	0.0109673	+	0.999940i	$0.496509\pi$
$384$	0	0
$385$	−5.57310	−0.284031
$386$	1.87326	0.0953462
$387$	0	0
$388$	−0.881244	−0.0447384
$389$	−19.7059	−0.999130	−0.499565	−	0.866276i	$-0.666507\pi$
$389$	−19.7059	−0.999130	−0.499565	+	0.866276i	$0.666507\pi$
$390$	0	0
$391$	27.3869	1.38502
$392$	−2.40202	−0.121320
$393$	0	0
$394$	2.34694	0.118237
$395$	−26.7513	−1.34600
$396$	0	0
$397$	−1.29873	−0.0651815	−0.0325908	−	0.999469i	$-0.510376\pi$
$397$	−1.29873	−0.0651815	−0.0325908	+	0.999469i	$0.510376\pi$
$398$	−9.01475	−0.451868
$399$	0	0
$400$	4.47709	0.223855
$401$	32.0953	1.60277	0.801383	−	0.598152i	$-0.204098\pi$
$401$	32.0953	1.60277	0.801383	+	0.598152i	$0.204098\pi$
$402$	0	0
$403$	3.58878	0.178770
$404$	−27.8065	−1.38342
$405$	0	0
$406$	−7.25575	−0.360097
$407$	−0.526379	−0.0260917
$408$	0	0
$409$	9.88425	0.488745	0.244372	−	0.969681i	$-0.421418\pi$
$409$	9.88425	0.488745	0.244372	+	0.969681i	$0.421418\pi$
$410$	−3.30914	−0.163427
$411$	0	0
$412$	16.7981	0.827584
$413$	0.232595	0.0114452
$414$	0	0
$415$	36.5442	1.79389
$416$	−2.40952	−0.118136
$417$	0	0
$418$	8.08034	0.395222
$419$	9.13345	0.446198	0.223099	−	0.974796i	$-0.428383\pi$
$419$	9.13345	0.446198	0.223099	+	0.974796i	$0.428383\pi$
$420$	0	0
$421$	−18.2042	−0.887216	−0.443608	−	0.896221i	$-0.646302\pi$
$421$	−18.2042	−0.887216	−0.443608	+	0.896221i	$0.646302\pi$
$422$	−14.0389	−0.683404
$423$	0	0
$424$	0.469470	0.0227995
$425$	20.0500	0.972570
$426$	0	0
$427$	2.32325	0.112430
$428$	−9.36054	−0.452459
$429$	0	0
$430$	−14.5382	−0.701093
$431$	−20.1347	−0.969856	−0.484928	−	0.874554i	$-0.661154\pi$
$431$	−20.1347	−0.969856	−0.484928	+	0.874554i	$0.661154\pi$
$432$	0	0
$433$	−17.8449	−0.857571	−0.428785	−	0.903406i	$-0.641058\pi$
$433$	−17.8449	−0.857571	−0.428785	+	0.903406i	$0.641058\pi$
$434$	5.84967	0.280793
$435$	0	0
$436$	11.0575	0.529558
$437$	−25.6507	−1.22704
$438$	0	0
$439$	−1.22690	−0.0585568	−0.0292784	−	0.999571i	$-0.509321\pi$
$439$	−1.22690	−0.0585568	−0.0292784	+	0.999571i	$0.509321\pi$
$440$	−13.3867	−0.638186
$441$	0	0
$442$	−1.83323	−0.0871980
$443$	−23.0890	−1.09699	−0.548496	−	0.836153i	$-0.684799\pi$
$443$	−23.0890	−1.09699	−0.548496	+	0.836153i	$0.684799\pi$
$444$	0	0
$445$	23.1832	1.09899
$446$	−10.1196	−0.479176
$447$	0	0
$448$	−1.02995	−0.0486605
$449$	15.5660	0.734606	0.367303	−	0.930101i	$-0.380281\pi$
$449$	15.5660	0.734606	0.367303	+	0.930101i	$0.380281\pi$
$450$	0	0
$451$	−3.35897	−0.158168
$452$	−3.16662	−0.148945
$453$	0	0
$454$	11.0277	0.517558
$455$	−1.18424	−0.0555181
$456$	0	0
$457$	−20.3364	−0.951299	−0.475649	−	0.879635i	$-0.657787\pi$
$457$	−20.3364	−0.951299	−0.475649	+	0.879635i	$0.657787\pi$
$458$	6.03887	0.282178
$459$	0	0
$460$	18.4829	0.861772
$461$	−7.66855	−0.357160	−0.178580	−	0.983925i	$-0.557150\pi$
$461$	−7.66855	−0.357160	−0.178580	+	0.983925i	$0.557150\pi$
$462$	0	0
$463$	40.9301	1.90218	0.951092	−	0.308908i	$-0.0999636\pi$
$463$	40.9301	1.90218	0.951092	+	0.308908i	$0.0999636\pi$
$464$	15.4895	0.719083
$465$	0	0
$466$	3.29624	0.152695
$467$	−17.9516	−0.830699	−0.415349	−	0.909662i	$-0.636341\pi$
$467$	−17.9516	−0.830699	−0.415349	+	0.909662i	$0.636341\pi$
$468$	0	0
$469$	5.65707	0.261219
$470$	7.63022	0.351956
$471$	0	0
$472$	0.558697	0.0257161
$473$	−14.7571	−0.678531
$474$	0	0
$475$	−18.7789	−0.861636
$476$	9.98804	0.457801
$477$	0	0
$478$	−13.5269	−0.618706
$479$	15.7424	0.719291	0.359645	−	0.933089i	$-0.382898\pi$
$479$	15.7424	0.719291	0.359645	+	0.933089i	$0.382898\pi$
$480$	0	0
$481$	−0.111852	−0.00510000
$482$	−1.90065	−0.0865721
$483$	0	0
$484$	11.0238	0.501081
$485$	−1.62823	−0.0739340
$486$	0	0
$487$	−21.8960	−0.992202	−0.496101	−	0.868265i	$-0.665235\pi$
$487$	−21.8960	−0.992202	−0.496101	+	0.868265i	$0.665235\pi$
$488$	5.58051	0.252618
$489$	0	0
$490$	−1.93030	−0.0872019
$491$	43.5036	1.96329	0.981645	−	0.190718i	$-0.0610815\pi$
$491$	43.5036	1.96329	0.981645	+	0.190718i	$0.0610815\pi$
$492$	0	0
$493$	69.3677	3.12416
$494$	1.71701	0.0772519
$495$	0	0
$496$	−12.4878	−0.560720
$497$	−13.5740	−0.608877
$498$	0	0
$499$	29.8312	1.33543	0.667714	−	0.744418i	$-0.267273\pi$
$499$	29.8312	1.33543	0.667714	+	0.744418i	$0.267273\pi$
$500$	−8.36208	−0.373964
$501$	0	0
$502$	−15.3258	−0.684024
$503$	28.4285	1.26756	0.633782	−	0.773512i	$-0.281502\pi$
$503$	28.4285	1.26756	0.633782	+	0.773512i	$0.281502\pi$
$504$	0	0
$505$	−51.3765	−2.28623
$506$	−5.61285	−0.249521
$507$	0	0
$508$	−1.53944	−0.0683017
$509$	−40.2439	−1.78378	−0.891890	−	0.452252i	$-0.850621\pi$
$509$	−40.2439	−1.78378	−0.891890	+	0.452252i	$0.850621\pi$
$510$	0	0
$511$	−10.5317	−0.465896
$512$	15.3443	0.678128
$513$	0	0
$514$	−10.4508	−0.460965
$515$	31.0370	1.36765
$516$	0	0
$517$	7.74511	0.340630
$518$	−0.182317	−0.00801054
$519$	0	0
$520$	−2.84457	−0.124743
$521$	−8.48609	−0.371783	−0.185891	−	0.982570i	$-0.559517\pi$
$521$	−8.48609	−0.371783	−0.185891	+	0.982570i	$0.559517\pi$
$522$	0	0
$523$	10.5402	0.460890	0.230445	−	0.973085i	$-0.425982\pi$
$523$	10.5402	0.460890	0.230445	+	0.973085i	$0.425982\pi$
$524$	19.8932	0.869038
$525$	0	0
$526$	−13.5721	−0.591770
$527$	−55.9250	−2.43613
$528$	0	0
$529$	−5.18228	−0.225316
$530$	0.377272	0.0163876
$531$	0	0
$532$	−9.35483	−0.405583
$533$	−0.713755	−0.0309161
$534$	0	0
$535$	−17.2950	−0.747726
$536$	13.5884	0.586930
$537$	0	0
$538$	11.3504	0.489351
$539$	−1.95936	−0.0843957
$540$	0	0
$541$	−19.7373	−0.848575	−0.424287	−	0.905528i	$-0.639475\pi$
$541$	−19.7373	−0.848575	−0.424287	+	0.905528i	$0.639475\pi$
$542$	−7.73510	−0.332251
$543$	0	0
$544$	37.5482	1.60986
$545$	20.4303	0.875140
$546$	0	0
$547$	−23.8539	−1.01992	−0.509959	−	0.860199i	$-0.670339\pi$
$547$	−23.8539	−1.01992	−0.509959	+	0.860199i	$0.670339\pi$
$548$	6.80020	0.290490
$549$	0	0
$550$	−4.10918	−0.175216
$551$	−64.9700	−2.76781
$552$	0	0
$553$	−9.40508	−0.399945
$554$	2.57838	0.109545
$555$	0	0
$556$	−13.6386	−0.578406
$557$	37.3306	1.58175	0.790875	−	0.611978i	$-0.209626\pi$
$557$	37.3306	1.58175	0.790875	+	0.611978i	$0.209626\pi$
$558$	0	0
$559$	−3.13577	−0.132629
$560$	4.12078	0.174135
$561$	0	0
$562$	5.17025	0.218094
$563$	−33.2716	−1.40223	−0.701116	−	0.713047i	$-0.747314\pi$
$563$	−33.2716	−1.40223	−0.701116	+	0.713047i	$0.747314\pi$
$564$	0	0
$565$	−5.85079	−0.246144
$566$	−8.01437	−0.336869
$567$	0	0
$568$	−32.6051	−1.36808
$569$	−19.7961	−0.829894	−0.414947	−	0.909846i	$-0.636200\pi$
$569$	−19.7961	−0.829894	−0.414947	+	0.909846i	$0.636200\pi$
$570$	0	0
$571$	31.7590	1.32907	0.664536	−	0.747256i	$-0.268629\pi$
$571$	31.7590	1.32907	0.664536	+	0.747256i	$0.268629\pi$
$572$	−1.25585	−0.0525095
$573$	0	0
$574$	−1.16341	−0.0485598
$575$	13.0444	0.543989
$576$	0	0
$577$	−27.9224	−1.16243	−0.581213	−	0.813751i	$-0.697422\pi$
$577$	−27.9224	−1.16243	−0.581213	+	0.813751i	$0.697422\pi$
$578$	17.0308	0.708389
$579$	0	0
$580$	46.8150	1.94389
$581$	12.8480	0.533027
$582$	0	0
$583$	0.382953	0.0158603
$584$	−25.2974	−1.04682
$585$	0	0
$586$	9.00825	0.372127
$587$	9.02241	0.372395	0.186197	−	0.982512i	$-0.440384\pi$
$587$	9.02241	0.372395	0.186197	+	0.982512i	$0.440384\pi$
$588$	0	0
$589$	52.3795	2.15826
$590$	0.448977	0.0184841
$591$	0	0
$592$	0.389208	0.0159964
$593$	−10.7108	−0.439838	−0.219919	−	0.975518i	$-0.570579\pi$
$593$	−10.7108	−0.439838	−0.219919	+	0.975518i	$0.570579\pi$
$594$	0	0
$595$	18.4544	0.756555
$596$	16.6964	0.683909
$597$	0	0
$598$	−1.19269	−0.0487726
$599$	23.0975	0.943739	0.471869	−	0.881668i	$-0.343579\pi$
$599$	23.0975	0.943739	0.471869	+	0.881668i	$0.343579\pi$
$600$	0	0
$601$	−15.3236	−0.625062	−0.312531	−	0.949907i	$-0.601177\pi$
$601$	−15.3236	−0.625062	−0.312531	+	0.949907i	$0.601177\pi$
$602$	−5.11126	−0.208319
$603$	0	0
$604$	15.3420	0.624257
$605$	20.3680	0.828079
$606$	0	0
$607$	−23.0356	−0.934986	−0.467493	−	0.883997i	$-0.654843\pi$
$607$	−23.0356	−0.934986	−0.467493	+	0.883997i	$0.654843\pi$
$608$	−35.1677	−1.42624
$609$	0	0
$610$	4.48457	0.181575
$611$	1.64578	0.0665810
$612$	0	0
$613$	−4.89609	−0.197751	−0.0988757	−	0.995100i	$-0.531525\pi$
$613$	−4.89609	−0.197751	−0.0988757	+	0.995100i	$0.531525\pi$
$614$	−10.9472	−0.441792
$615$	0	0
$616$	−4.70643	−0.189628
$617$	1.85865	0.0748265	0.0374133	−	0.999300i	$-0.488088\pi$
$617$	1.85865	0.0748265	0.0374133	+	0.999300i	$0.488088\pi$
$618$	0	0
$619$	0.579423	0.0232890	0.0116445	−	0.999932i	$-0.496293\pi$
$619$	0.579423	0.0232890	0.0116445	+	0.999932i	$0.496293\pi$
$620$	−37.7428	−1.51579
$621$	0	0
$622$	−4.05316	−0.162517
$623$	8.15063	0.326548
$624$	0	0
$625$	−30.9016	−1.23606
$626$	5.26295	0.210350
$627$	0	0
$628$	1.52805	0.0609759
$629$	1.74302	0.0694986
$630$	0	0
$631$	2.98548	0.118850	0.0594250	−	0.998233i	$-0.481073\pi$
$631$	2.98548	0.118850	0.0594250	+	0.998233i	$0.481073\pi$
$632$	−22.5912	−0.898631
$633$	0	0
$634$	7.38463	0.293281
$635$	−2.84434	−0.112874
$636$	0	0
$637$	−0.416350	−0.0164964
$638$	−14.2166	−0.562842
$639$	0	0
$640$	30.9337	1.22276
$641$	31.4363	1.24166	0.620830	−	0.783946i	$-0.286796\pi$
$641$	31.4363	1.24166	0.620830	+	0.783946i	$0.286796\pi$
$642$	0	0
$643$	31.5052	1.24245	0.621223	−	0.783634i	$-0.286636\pi$
$643$	31.5052	1.24245	0.621223	+	0.783634i	$0.286636\pi$
$644$	6.49814	0.256063
$645$	0	0
$646$	−26.7567	−1.05273
$647$	−47.5211	−1.86825	−0.934124	−	0.356950i	$-0.883817\pi$
$647$	−47.5211	−1.86825	−0.934124	+	0.356950i	$0.883817\pi$
$648$	0	0
$649$	0.455737	0.0178892
$650$	−0.873170	−0.0342485
$651$	0	0
$652$	−1.06353	−0.0416510
$653$	−48.8029	−1.90980	−0.954902	−	0.296921i	$-0.904040\pi$
$653$	−48.8029	−1.90980	−0.954902	+	0.296921i	$0.904040\pi$
$654$	0	0
$655$	36.7556	1.43616
$656$	2.48364	0.0969699
$657$	0	0
$658$	2.68259	0.104578
$659$	43.0162	1.67567	0.837837	−	0.545921i	$-0.183820\pi$
$659$	43.0162	1.67567	0.837837	+	0.545921i	$0.183820\pi$
$660$	0	0
$661$	21.1904	0.824211	0.412106	−	0.911136i	$-0.364793\pi$
$661$	21.1904	0.824211	0.412106	+	0.911136i	$0.364793\pi$
$662$	19.8288	0.770668
$663$	0	0
$664$	30.8613	1.19765
$665$	−17.2844	−0.670260
$666$	0	0
$667$	45.1301	1.74744
$668$	25.4735	0.985600
$669$	0	0
$670$	10.9198	0.421870
$671$	4.55210	0.175732
$672$	0	0
$673$	−7.73215	−0.298053	−0.149026	−	0.988833i	$-0.547614\pi$
$673$	−7.73215	−0.298053	−0.149026	+	0.988833i	$0.547614\pi$
$674$	−15.8398	−0.610127
$675$	0	0
$676$	19.7459	0.759457
$677$	8.96387	0.344510	0.172255	−	0.985052i	$-0.444895\pi$
$677$	8.96387	0.344510	0.172255	+	0.985052i	$0.444895\pi$
$678$	0	0
$679$	−0.572444	−0.0219684
$680$	44.3278	1.69989
$681$	0	0
$682$	11.4616	0.438888
$683$	−39.3378	−1.50522	−0.752610	−	0.658467i	$-0.771205\pi$
$683$	−39.3378	−1.50522	−0.752610	+	0.658467i	$0.771205\pi$
$684$	0	0
$685$	12.5644	0.480059
$686$	−0.678644	−0.0259108
$687$	0	0
$688$	10.9115	0.415996
$689$	0.0813745	0.00310012
$690$	0	0
$691$	36.3196	1.38166	0.690831	−	0.723016i	$-0.257245\pi$
$691$	36.3196	1.38166	0.690831	+	0.723016i	$0.257245\pi$
$692$	−20.5250	−0.780244
$693$	0	0
$694$	11.8594	0.450177
$695$	−25.1993	−0.955864
$696$	0	0
$697$	11.1226	0.421300
$698$	−2.80103	−0.106021
$699$	0	0
$700$	4.75731	0.179809
$701$	30.3576	1.14659	0.573295	−	0.819349i	$-0.305665\pi$
$701$	30.3576	1.14659	0.573295	+	0.819349i	$0.305665\pi$
$702$	0	0
$703$	−1.63251	−0.0615714
$704$	−2.01804	−0.0760579
$705$	0	0
$706$	−0.0808243	−0.00304186
$707$	−18.0627	−0.679318
$708$	0	0
$709$	0.924053	0.0347035	0.0173518	−	0.999849i	$-0.494476\pi$
$709$	0.924053	0.0347035	0.0173518	+	0.999849i	$0.494476\pi$
$710$	−26.2019	−0.983338
$711$	0	0
$712$	19.5780	0.733716
$713$	−36.3844	−1.36261
$714$	0	0
$715$	−2.32036	−0.0867764
$716$	14.8634	0.555472
$717$	0	0
$718$	−5.39667	−0.201402
$719$	−33.2174	−1.23880	−0.619401	−	0.785075i	$-0.712624\pi$
$719$	−33.2174	−1.23880	−0.619401	+	0.785075i	$0.712624\pi$
$720$	0	0
$721$	10.9118	0.406377
$722$	12.1661	0.452776
$723$	0	0
$724$	−34.6846	−1.28904
$725$	33.0399	1.22707
$726$	0	0
$727$	−43.7282	−1.62179	−0.810894	−	0.585193i	$-0.801019\pi$
$727$	−43.7282	−1.62179	−0.810894	+	0.585193i	$0.801019\pi$
$728$	−1.00008	−0.0370655
$729$	0	0
$730$	−20.3294	−0.752423
$731$	48.8655	1.80736
$732$	0	0
$733$	−8.85549	−0.327085	−0.163543	−	0.986536i	$-0.552292\pi$
$733$	−8.85549	−0.327085	−0.163543	+	0.986536i	$0.552292\pi$
$734$	1.94545	0.0718079
$735$	0	0
$736$	24.4285	0.900448
$737$	11.0843	0.408294
$738$	0	0
$739$	−46.6451	−1.71587	−0.857933	−	0.513761i	$-0.828252\pi$
$739$	−46.6451	−1.71587	−0.857933	+	0.513761i	$0.828252\pi$
$740$	1.17633	0.0432428
$741$	0	0
$742$	0.132639	0.00486935
$743$	−29.9420	−1.09846	−0.549232	−	0.835670i	$-0.685080\pi$
$743$	−29.9420	−1.09846	−0.549232	+	0.835670i	$0.685080\pi$
$744$	0	0
$745$	30.8489	1.13022
$746$	−2.73198	−0.100025
$747$	0	0
$748$	19.5702	0.715557
$749$	−6.08048	−0.222176
$750$	0	0
$751$	23.2599	0.848764	0.424382	−	0.905483i	$-0.360491\pi$
$751$	23.2599	0.848764	0.424382	+	0.905483i	$0.360491\pi$
$752$	−5.72678	−0.208834
$753$	0	0
$754$	−3.02093	−0.110016
$755$	28.3466	1.03164
$756$	0	0
$757$	−2.68171	−0.0974683	−0.0487342	−	0.998812i	$-0.515519\pi$
$757$	−2.68171	−0.0974683	−0.0487342	+	0.998812i	$0.515519\pi$
$758$	6.89126	0.250302
$759$	0	0
$760$	−41.5175	−1.50600
$761$	−19.3399	−0.701070	−0.350535	−	0.936550i	$-0.614000\pi$
$761$	−19.3399	−0.701070	−0.350535	+	0.936550i	$0.614000\pi$
$762$	0	0
$763$	7.18280	0.260035
$764$	5.11475	0.185045
$765$	0	0
$766$	0.291322	0.0105259
$767$	0.0968407	0.00349671
$768$	0	0
$769$	−44.8959	−1.61899	−0.809494	−	0.587128i	$-0.800258\pi$
$769$	−44.8959	−1.61899	−0.809494	+	0.587128i	$0.800258\pi$
$770$	−3.78215	−0.136299
$771$	0	0
$772$	−4.24931	−0.152936
$773$	24.8855	0.895068	0.447534	−	0.894267i	$-0.352302\pi$
$773$	24.8855	0.895068	0.447534	+	0.894267i	$0.352302\pi$
$774$	0	0
$775$	−26.6371	−0.956833
$776$	−1.37502	−0.0493605
$777$	0	0
$778$	−13.3733	−0.479456
$779$	−10.4175	−0.373245
$780$	0	0
$781$	−26.5964	−0.951693
$782$	18.5860	0.664633
$783$	0	0
$784$	1.44876	0.0517416
$785$	2.82330	0.100768
$786$	0	0
$787$	40.0649	1.42816	0.714080	−	0.700065i	$-0.246845\pi$
$787$	40.0649	1.42816	0.714080	+	0.700065i	$0.246845\pi$
$788$	−5.32382	−0.189653
$789$	0	0
$790$	−18.1546	−0.645912
$791$	−2.05699	−0.0731382
$792$	0	0
$793$	0.967286	0.0343493
$794$	−0.881378	−0.0312789
$795$	0	0
$796$	20.4491	0.724800
$797$	−30.3698	−1.07575	−0.537876	−	0.843024i	$-0.680773\pi$
$797$	−30.3698	−1.07575	−0.537876	+	0.843024i	$0.680773\pi$
$798$	0	0
$799$	−25.6466	−0.907312
$800$	17.8842	0.632302
$801$	0	0
$802$	21.7813	0.769125
$803$	−20.6355	−0.728210
$804$	0	0
$805$	12.0063	0.423165
$806$	2.43551	0.0857871
$807$	0	0
$808$	−43.3870	−1.52635
$809$	−18.9059	−0.664695	−0.332348	−	0.943157i	$-0.607841\pi$
$809$	−18.9059	−0.664695	−0.332348	+	0.943157i	$0.607841\pi$
$810$	0	0
$811$	0.513163	0.0180196	0.00900980	−	0.999959i	$-0.497132\pi$
$811$	0.513163	0.0180196	0.00900980	+	0.999959i	$0.497132\pi$
$812$	16.4590	0.577598
$813$	0	0
$814$	−0.357224	−0.0125207
$815$	−1.96502	−0.0688318
$816$	0	0
$817$	−45.7676	−1.60121
$818$	6.70789	0.234536
$819$	0	0
$820$	7.50647	0.262137
$821$	−15.9543	−0.556810	−0.278405	−	0.960464i	$-0.589806\pi$
$821$	−15.9543	−0.556810	−0.278405	+	0.960464i	$0.589806\pi$
$822$	0	0
$823$	−37.0876	−1.29279	−0.646397	−	0.763001i	$-0.723725\pi$
$823$	−37.0876	−1.29279	−0.646397	+	0.763001i	$0.723725\pi$
$824$	26.2104	0.913084
$825$	0	0
$826$	0.157849	0.00549227
$827$	23.0458	0.801381	0.400691	−	0.916213i	$-0.368770\pi$
$827$	23.0458	0.801381	0.400691	+	0.916213i	$0.368770\pi$
$828$	0	0
$829$	−40.0594	−1.39132	−0.695660	−	0.718371i	$-0.744888\pi$
$829$	−40.0594	−1.39132	−0.695660	+	0.718371i	$0.744888\pi$
$830$	24.8005	0.860839
$831$	0	0
$832$	−0.428819	−0.0148666
$833$	6.48809	0.224799
$834$	0	0
$835$	47.0660	1.62879
$836$	−18.3295	−0.633939
$837$	0	0
$838$	6.19837	0.214119
$839$	−19.7075	−0.680380	−0.340190	−	0.940357i	$-0.610491\pi$
$839$	−19.7075	−0.680380	−0.340190	+	0.940357i	$0.610491\pi$
$840$	0	0
$841$	85.3090	2.94169
$842$	−12.3542	−0.425752
$843$	0	0
$844$	31.8460	1.09618
$845$	36.4834	1.25507
$846$	0	0
$847$	7.16090	0.246051
$848$	−0.283158	−0.00972367
$849$	0	0
$850$	13.6068	0.466711
$851$	1.13399	0.0388728
$852$	0	0
$853$	−7.69175	−0.263361	−0.131680	−	0.991292i	$-0.542037\pi$
$853$	−7.69175	−0.263361	−0.131680	+	0.991292i	$0.542037\pi$
$854$	1.57666	0.0539523
$855$	0	0
$856$	−14.6054	−0.499204
$857$	13.8104	0.471756	0.235878	−	0.971783i	$-0.424203\pi$
$857$	13.8104	0.471756	0.235878	+	0.971783i	$0.424203\pi$
$858$	0	0
$859$	53.9358	1.84027	0.920133	−	0.391605i	$-0.128080\pi$
$859$	53.9358	1.84027	0.920133	+	0.391605i	$0.128080\pi$
$860$	32.9785	1.12456
$861$	0	0
$862$	−13.6643	−0.465409
$863$	5.75913	0.196043	0.0980215	−	0.995184i	$-0.468749\pi$
$863$	5.75913	0.196043	0.0980215	+	0.995184i	$0.468749\pi$
$864$	0	0
$865$	−37.9229	−1.28942
$866$	−12.1103	−0.411526
$867$	0	0
$868$	−13.2694	−0.450394
$869$	−18.4280	−0.625126
$870$	0	0
$871$	2.35532	0.0798070
$872$	17.2532	0.584269
$873$	0	0
$874$	−17.4077	−0.588823
$875$	−5.43189	−0.183631
$876$	0	0
$877$	−30.9006	−1.04344	−0.521719	−	0.853117i	$-0.674709\pi$
$877$	−30.9006	−1.04344	−0.521719	+	0.853117i	$0.674709\pi$
$878$	−0.832630	−0.0280999
$879$	0	0
$880$	8.07410	0.272178
$881$	3.68840	0.124265	0.0621327	−	0.998068i	$-0.480210\pi$
$881$	3.68840	0.124265	0.0621327	+	0.998068i	$0.480210\pi$
$882$	0	0
$883$	6.59636	0.221985	0.110993	−	0.993821i	$-0.464597\pi$
$883$	6.59636	0.221985	0.110993	+	0.993821i	$0.464597\pi$
$884$	4.15852	0.139866
$885$	0	0
$886$	−15.6692	−0.526418
$887$	9.91206	0.332814	0.166407	−	0.986057i	$-0.446783\pi$
$887$	9.91206	0.332814	0.166407	+	0.986057i	$0.446783\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	15.7331	0.527376
$891$	0	0
$892$	22.9553	0.768601
$893$	24.0207	0.803821
$894$	0	0
$895$	27.4623	0.917964
$896$	10.8755	0.363325
$897$	0	0
$898$	10.5638	0.352518
$899$	−92.1571	−3.07361
$900$	0	0
$901$	−1.26808	−0.0422459
$902$	−2.27954	−0.0759005
$903$	0	0
$904$	−4.94094	−0.164333
$905$	−64.0848	−2.13025
$906$	0	0
$907$	28.2395	0.937677	0.468839	−	0.883284i	$-0.344673\pi$
$907$	28.2395	0.937677	0.468839	+	0.883284i	$0.344673\pi$
$908$	−25.0154	−0.830166
$909$	0	0
$910$	−0.803678	−0.0266417
$911$	21.2259	0.703246	0.351623	−	0.936142i	$-0.385630\pi$
$911$	21.2259	0.703246	0.351623	+	0.936142i	$0.385630\pi$
$912$	0	0
$913$	25.1740	0.833137
$914$	−13.8012	−0.456504
$915$	0	0
$916$	−13.6986	−0.452615
$917$	12.9223	0.426733
$918$	0	0
$919$	−32.0624	−1.05764	−0.528821	−	0.848733i	$-0.677366\pi$
$919$	−32.0624	−1.05764	−0.528821	+	0.848733i	$0.677366\pi$
$920$	28.8393	0.950804
$921$	0	0
$922$	−5.20422	−0.171392
$923$	−5.65153	−0.186022
$924$	0	0
$925$	0.830199	0.0272968
$926$	27.7770	0.912809
$927$	0	0
$928$	61.8745	2.03113
$929$	−36.1749	−1.18686	−0.593431	−	0.804885i	$-0.702227\pi$
$929$	−36.1749	−1.18686	−0.593431	+	0.804885i	$0.702227\pi$
$930$	0	0
$931$	−6.07676	−0.199158
$932$	−7.47721	−0.244924
$933$	0	0
$934$	−12.1827	−0.398631
$935$	36.1588	1.18252
$936$	0	0
$937$	1.42584	0.0465801	0.0232900	−	0.999729i	$-0.492586\pi$
$937$	1.42584	0.0465801	0.0232900	+	0.999729i	$0.492586\pi$
$938$	3.83914	0.125352
$939$	0	0
$940$	−17.3084	−0.564539
$941$	5.92114	0.193024	0.0965118	−	0.995332i	$-0.469231\pi$
$941$	5.92114	0.193024	0.0965118	+	0.995332i	$0.469231\pi$
$942$	0	0
$943$	7.23631	0.235647
$944$	−0.336975	−0.0109676
$945$	0	0
$946$	−10.0148	−0.325609
$947$	16.5305	0.537169	0.268585	−	0.963256i	$-0.413444\pi$
$947$	16.5305	0.537169	0.268585	+	0.963256i	$0.413444\pi$
$948$	0	0
$949$	−4.38488	−0.142339
$950$	−12.7442	−0.413477
$951$	0	0
$952$	15.5845	0.505098
$953$	53.4304	1.73078	0.865390	−	0.501099i	$-0.167071\pi$
$953$	53.4304	1.73078	0.865390	+	0.501099i	$0.167071\pi$
$954$	0	0
$955$	9.45025	0.305803
$956$	30.6845	0.992409
$957$	0	0
$958$	10.6835	0.345169
$959$	4.41731	0.142642
$960$	0	0
$961$	43.2981	1.39671
$962$	−0.0759075	−0.00244736
$963$	0	0
$964$	4.31144	0.138862
$965$	−7.85121	−0.252739
$966$	0	0
$967$	52.0476	1.67374	0.836869	−	0.547403i	$-0.184383\pi$
$967$	52.0476	1.67374	0.836869	+	0.547403i	$0.184383\pi$
$968$	17.2006	0.552850
$969$	0	0
$970$	−1.10499	−0.0354790
$971$	−11.2656	−0.361531	−0.180766	−	0.983526i	$-0.557858\pi$
$971$	−11.2656	−0.361531	−0.180766	+	0.983526i	$0.557858\pi$
$972$	0	0
$973$	−8.85945	−0.284021
$974$	−14.8596	−0.476132
$975$	0	0
$976$	−3.36585	−0.107738
$977$	27.7260	0.887033	0.443516	−	0.896266i	$-0.353731\pi$
$977$	27.7260	0.887033	0.443516	+	0.896266i	$0.353731\pi$
$978$	0	0
$979$	15.9700	0.510404
$980$	4.37870	0.139872
$981$	0	0
$982$	29.5235	0.942132
$983$	−47.8594	−1.52648	−0.763239	−	0.646117i	$-0.776392\pi$
$983$	−47.8594	−1.52648	−0.763239	+	0.646117i	$0.776392\pi$
$984$	0	0
$985$	−9.83653	−0.313418
$986$	47.0760	1.49920
$987$	0	0
$988$	−3.89488	−0.123913
$989$	31.7915	1.01091
$990$	0	0
$991$	9.47275	0.300912	0.150456	−	0.988617i	$-0.451926\pi$
$991$	9.47275	0.300912	0.150456	+	0.988617i	$0.451926\pi$
$992$	−49.8839	−1.58382
$993$	0	0
$994$	−9.21192	−0.292184
$995$	37.7827	1.19779
$996$	0	0
$997$	−3.49551	−0.110704	−0.0553520	−	0.998467i	$-0.517628\pi$
$997$	−3.49551	−0.110704	−0.0553520	+	0.998467i	$0.517628\pi$
$998$	20.2448	0.640837
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.7		11
3.2	odd	2		2667.2.a.k.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.5	✓	11	3.2	odd	2
8001.2.a.m.1.7		11	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.678644 q^{2} -1.53944 q^{4} -2.84434 q^{5} -1.00000 q^{7} -2.40202 q^{8} +O(q^{10})\)	\(q+0.678644 q^{2} -1.53944 q^{4} -2.84434 q^{5} -1.00000 q^{7} -2.40202 q^{8} -1.93030 q^{10} -1.95936 q^{11} -0.416350 q^{13} -0.678644 q^{14} +1.44876 q^{16} +6.48809 q^{17} -6.07676 q^{19} +4.37870 q^{20} -1.32971 q^{22} +4.22110 q^{23} +3.09028 q^{25} -0.282553 q^{26} +1.53944 q^{28} +10.6915 q^{29} -8.61963 q^{31} +5.78724 q^{32} +4.40311 q^{34} +2.84434 q^{35} +0.268648 q^{37} -4.12396 q^{38} +6.83217 q^{40} +1.71432 q^{41} +7.53157 q^{43} +3.01632 q^{44} +2.86463 q^{46} -3.95287 q^{47} +1.00000 q^{49} +2.09720 q^{50} +0.640946 q^{52} -0.195448 q^{53} +5.57310 q^{55} +2.40202 q^{56} +7.25575 q^{58} -0.232595 q^{59} -2.32325 q^{61} -5.84967 q^{62} +1.02995 q^{64} +1.18424 q^{65} -5.65707 q^{67} -9.98804 q^{68} +1.93030 q^{70} +13.5740 q^{71} +10.5317 q^{73} +0.182317 q^{74} +9.35483 q^{76} +1.95936 q^{77} +9.40508 q^{79} -4.12078 q^{80} +1.16341 q^{82} -12.8480 q^{83} -18.4544 q^{85} +5.11126 q^{86} +4.70643 q^{88} -8.15063 q^{89} +0.416350 q^{91} -6.49814 q^{92} -2.68259 q^{94} +17.2844 q^{95} +0.572444 q^{97} +0.678644 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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