LMFDB - Embedded newform 8001.2.a.q.1.8

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.146355$ 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)

$f(q)$	$=$	$q-0.146355 q^{2} -1.97858 q^{4} +2.93285 q^{5} -1.00000 q^{7} +0.582285 q^{8} +O(q^{10})$	\(q-0.146355 q^{2} -1.97858 q^{4} +2.93285 q^{5} -1.00000 q^{7} +0.582285 q^{8} -0.429238 q^{10} +1.72763 q^{11} -1.53171 q^{13} +0.146355 q^{14} +3.87194 q^{16} +1.53061 q^{17} +6.29792 q^{19} -5.80289 q^{20} -0.252847 q^{22} +0.209505 q^{23} +3.60163 q^{25} +0.224173 q^{26} +1.97858 q^{28} -9.36234 q^{29} -5.71758 q^{31} -1.73125 q^{32} -0.224013 q^{34} -2.93285 q^{35} -9.72170 q^{37} -0.921731 q^{38} +1.70776 q^{40} -8.99430 q^{41} -11.1715 q^{43} -3.41825 q^{44} -0.0306621 q^{46} -0.690780 q^{47} +1.00000 q^{49} -0.527117 q^{50} +3.03060 q^{52} -6.60824 q^{53} +5.06688 q^{55} -0.582285 q^{56} +1.37022 q^{58} +10.4011 q^{59} +10.7813 q^{61} +0.836796 q^{62} -7.49050 q^{64} -4.49227 q^{65} +3.16071 q^{67} -3.02844 q^{68} +0.429238 q^{70} -7.54917 q^{71} +0.853331 q^{73} +1.42282 q^{74} -12.4609 q^{76} -1.72763 q^{77} +1.08730 q^{79} +11.3558 q^{80} +1.31636 q^{82} -6.30576 q^{83} +4.48907 q^{85} +1.63501 q^{86} +1.00597 q^{88} +0.111705 q^{89} +1.53171 q^{91} -0.414523 q^{92} +0.101099 q^{94} +18.4709 q^{95} +10.1867 q^{97} -0.146355 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

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 $

$2$	−0.146355	−0.103489	−0.0517443	−	0.998660i	$-0.516478\pi$
$2$	−0.146355	−0.103489	−0.0517443	+	0.998660i	$0.516478\pi$
$3$	0	0
$3$	0	0
$4$	−1.97858	−0.989290
$5$	2.93285	1.31161	0.655806	−	0.754929i	$-0.272329\pi$
$5$	2.93285	1.31161	0.655806	+	0.754929i	$0.272329\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.582285	0.205869
$9$	0	0
$10$	−0.429238	−0.135737
$11$	1.72763	0.520900	0.260450	−	0.965487i	$-0.416129\pi$
$11$	1.72763	0.520900	0.260450	+	0.965487i	$0.416129\pi$
$12$	0	0
$13$	−1.53171	−0.424819	−0.212409	−	0.977181i	$-0.568131\pi$
$13$	−1.53171	−0.424819	−0.212409	+	0.977181i	$0.568131\pi$
$14$	0.146355	0.0391150
$15$	0	0
$16$	3.87194	0.967985
$17$	1.53061	0.371228	0.185614	−	0.982623i	$-0.440573\pi$
$17$	1.53061	0.371228	0.185614	+	0.982623i	$0.440573\pi$
$18$	0	0
$19$	6.29792	1.44484	0.722421	−	0.691454i	$-0.243029\pi$
$19$	6.29792	1.44484	0.722421	+	0.691454i	$0.243029\pi$
$20$	−5.80289	−1.29756
$21$	0	0
$22$	−0.252847	−0.0539072
$23$	0.209505	0.0436849	0.0218424	−	0.999761i	$-0.493047\pi$
$23$	0.209505	0.0436849	0.0218424	+	0.999761i	$0.493047\pi$
$24$	0	0
$25$	3.60163	0.720326
$26$	0.224173	0.0439639
$27$	0	0
$28$	1.97858	0.373917
$29$	−9.36234	−1.73854	−0.869271	−	0.494336i	$-0.835411\pi$
$29$	−9.36234	−1.73854	−0.869271	+	0.494336i	$0.835411\pi$
$30$	0	0
$31$	−5.71758	−1.02691	−0.513454	−	0.858117i	$-0.671634\pi$
$31$	−5.71758	−1.02691	−0.513454	+	0.858117i	$0.671634\pi$
$32$	−1.73125	−0.306044
$33$	0	0
$34$	−0.224013	−0.0384179
$35$	−2.93285	−0.495743
$36$	0	0
$37$	−9.72170	−1.59824	−0.799119	−	0.601173i	$-0.794700\pi$
$37$	−9.72170	−1.59824	−0.799119	+	0.601173i	$0.794700\pi$
$38$	−0.921731	−0.149525
$39$	0	0
$40$	1.70776	0.270020
$41$	−8.99430	−1.40467	−0.702337	−	0.711845i	$-0.747860\pi$
$41$	−8.99430	−1.40467	−0.702337	+	0.711845i	$0.747860\pi$
$42$	0	0
$43$	−11.1715	−1.70364	−0.851821	−	0.523833i	$-0.824501\pi$
$43$	−11.1715	−1.70364	−0.851821	+	0.523833i	$0.824501\pi$
$44$	−3.41825	−0.515321
$45$	0	0
$46$	−0.0306621	−0.00452089
$47$	−0.690780	−0.100761	−0.0503803	−	0.998730i	$-0.516043\pi$
$47$	−0.690780	−0.100761	−0.0503803	+	0.998730i	$0.516043\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.527117	−0.0745455
$51$	0	0
$52$	3.03060	0.420269
$53$	−6.60824	−0.907712	−0.453856	−	0.891075i	$-0.649952\pi$
$53$	−6.60824	−0.907712	−0.453856	+	0.891075i	$0.649952\pi$
$54$	0	0
$55$	5.06688	0.683219
$56$	−0.582285	−0.0778111
$57$	0	0
$58$	1.37022	0.179919
$59$	10.4011	1.35411	0.677055	−	0.735932i	$-0.263256\pi$
$59$	10.4011	1.35411	0.677055	+	0.735932i	$0.263256\pi$
$60$	0	0
$61$	10.7813	1.38041	0.690203	−	0.723616i	$-0.257521\pi$
$61$	10.7813	1.38041	0.690203	+	0.723616i	$0.257521\pi$
$62$	0.836796	0.106273
$63$	0	0
$64$	−7.49050	−0.936313
$65$	−4.49227	−0.557197
$66$	0	0
$67$	3.16071	0.386143	0.193071	−	0.981185i	$-0.438155\pi$
$67$	3.16071	0.386143	0.193071	+	0.981185i	$0.438155\pi$
$68$	−3.02844	−0.367253
$69$	0	0
$70$	0.429238	0.0513037
$71$	−7.54917	−0.895921	−0.447961	−	0.894053i	$-0.647849\pi$
$71$	−7.54917	−0.895921	−0.447961	+	0.894053i	$0.647849\pi$
$72$	0	0
$73$	0.853331	0.0998748	0.0499374	−	0.998752i	$-0.484098\pi$
$73$	0.853331	0.0998748	0.0499374	+	0.998752i	$0.484098\pi$
$74$	1.42282	0.165399
$75$	0	0
$76$	−12.4609	−1.42937
$77$	−1.72763	−0.196882
$78$	0	0
$79$	1.08730	0.122331	0.0611655	−	0.998128i	$-0.480518\pi$
$79$	1.08730	0.122331	0.0611655	+	0.998128i	$0.480518\pi$
$80$	11.3558	1.26962
$81$	0	0
$82$	1.31636	0.145368
$83$	−6.30576	−0.692147	−0.346074	−	0.938207i	$-0.612485\pi$
$83$	−6.30576	−0.692147	−0.346074	+	0.938207i	$0.612485\pi$
$84$	0	0
$85$	4.48907	0.486908
$86$	1.63501	0.176307
$87$	0	0
$88$	1.00597	0.107237
$89$	0.111705	0.0118407	0.00592034	−	0.999982i	$-0.498115\pi$
$89$	0.111705	0.0118407	0.00592034	+	0.999982i	$0.498115\pi$
$90$	0	0
$91$	1.53171	0.160566
$92$	−0.414523	−0.0432170
$93$	0	0
$94$	0.101099	0.0104276
$95$	18.4709	1.89507
$96$	0	0
$97$	10.1867	1.03430	0.517151	−	0.855894i	$-0.326992\pi$
$97$	10.1867	1.03430	0.517151	+	0.855894i	$0.326992\pi$
$98$	−0.146355	−0.0147841
$99$	0	0
$100$	−7.12612	−0.712612
$101$	6.00240	0.597261	0.298630	−	0.954369i	$-0.403470\pi$
$101$	6.00240	0.597261	0.298630	+	0.954369i	$0.403470\pi$
$102$	0	0
$103$	10.6556	1.04993	0.524965	−	0.851124i	$-0.324078\pi$
$103$	10.6556	1.04993	0.524965	+	0.851124i	$0.324078\pi$
$104$	−0.891889	−0.0874569
$105$	0	0
$106$	0.967149	0.0939378
$107$	2.24404	0.216939	0.108470	−	0.994100i	$-0.465405\pi$
$107$	2.24404	0.216939	0.108470	+	0.994100i	$0.465405\pi$
$108$	0	0
$109$	−17.3565	−1.66245	−0.831227	−	0.555933i	$-0.812361\pi$
$109$	−17.3565	−1.66245	−0.831227	+	0.555933i	$0.812361\pi$
$110$	−0.741563	−0.0707053
$111$	0	0
$112$	−3.87194	−0.365864
$113$	18.5223	1.74244	0.871218	−	0.490896i	$-0.163331\pi$
$113$	18.5223	1.74244	0.871218	+	0.490896i	$0.163331\pi$
$114$	0	0
$115$	0.614449	0.0572976
$116$	18.5241	1.71992
$117$	0	0
$118$	−1.52225	−0.140135
$119$	−1.53061	−0.140311
$120$	0	0
$121$	−8.01530	−0.728663
$122$	−1.57790	−0.142856
$123$	0	0
$124$	11.3127	1.01591
$125$	−4.10121	−0.366823
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	4.55877	0.402942
$129$	0	0
$130$	0.657466	0.0576635
$131$	−11.1154	−0.971154	−0.485577	−	0.874194i	$-0.661390\pi$
$131$	−11.1154	−0.971154	−0.485577	+	0.874194i	$0.661390\pi$
$132$	0	0
$133$	−6.29792	−0.546099
$134$	−0.462586	−0.0399613
$135$	0	0
$136$	0.891253	0.0764244
$137$	−5.94963	−0.508311	−0.254156	−	0.967163i	$-0.581798\pi$
$137$	−5.94963	−0.508311	−0.254156	+	0.967163i	$0.581798\pi$
$138$	0	0
$139$	−5.35645	−0.454328	−0.227164	−	0.973856i	$-0.572945\pi$
$139$	−5.35645	−0.454328	−0.227164	+	0.973856i	$0.572945\pi$
$140$	5.80289	0.490433
$141$	0	0
$142$	1.10486	0.0927176
$143$	−2.64622	−0.221288
$144$	0	0
$145$	−27.4584	−2.28029
$146$	−0.124889	−0.0103359
$147$	0	0
$148$	19.2352	1.58112
$149$	5.62398	0.460734	0.230367	−	0.973104i	$-0.426007\pi$
$149$	5.62398	0.460734	0.230367	+	0.973104i	$0.426007\pi$
$150$	0	0
$151$	16.4183	1.33611	0.668053	−	0.744114i	$-0.267128\pi$
$151$	16.4183	1.33611	0.668053	+	0.744114i	$0.267128\pi$
$152$	3.66718	0.297448
$153$	0	0
$154$	0.252847	0.0203750
$155$	−16.7688	−1.34690
$156$	0	0
$157$	12.7126	1.01458	0.507290	−	0.861776i	$-0.330647\pi$
$157$	12.7126	1.01458	0.507290	+	0.861776i	$0.330647\pi$
$158$	−0.159132	−0.0126599
$159$	0	0
$160$	−5.07749	−0.401411
$161$	−0.209505	−0.0165113
$162$	0	0
$163$	−12.0371	−0.942817	−0.471409	−	0.881915i	$-0.656254\pi$
$163$	−12.0371	−0.942817	−0.471409	+	0.881915i	$0.656254\pi$
$164$	17.7959	1.38963
$165$	0	0
$166$	0.922879	0.0716293
$167$	−21.5054	−1.66414	−0.832068	−	0.554674i	$-0.812843\pi$
$167$	−21.5054	−1.66414	−0.832068	+	0.554674i	$0.812843\pi$
$168$	0	0
$169$	−10.6539	−0.819529
$170$	−0.656997	−0.0503894
$171$	0	0
$172$	22.1038	1.68540
$173$	5.47918	0.416575	0.208287	−	0.978068i	$-0.433211\pi$
$173$	5.47918	0.416575	0.208287	+	0.978068i	$0.433211\pi$
$174$	0	0
$175$	−3.60163	−0.272258
$176$	6.68928	0.504223
$177$	0	0
$178$	−0.0163486	−0.00122538
$179$	0.147513	0.0110257	0.00551283	−	0.999985i	$-0.498245\pi$
$179$	0.147513	0.0110257	0.00551283	+	0.999985i	$0.498245\pi$
$180$	0	0
$181$	5.79818	0.430975	0.215488	−	0.976507i	$-0.430866\pi$
$181$	5.79818	0.430975	0.215488	+	0.976507i	$0.430866\pi$
$182$	−0.224173	−0.0166168
$183$	0	0
$184$	0.121992	0.00899335
$185$	−28.5123	−2.09627
$186$	0	0
$187$	2.64433	0.193373
$188$	1.36676	0.0996815
$189$	0	0
$190$	−2.70330	−0.196118
$191$	−17.2762	−1.25006	−0.625031	−	0.780600i	$-0.714914\pi$
$191$	−17.2762	−1.25006	−0.625031	+	0.780600i	$0.714914\pi$
$192$	0	0
$193$	−19.6543	−1.41474	−0.707372	−	0.706841i	$-0.750120\pi$
$193$	−19.6543	−1.41474	−0.707372	+	0.706841i	$0.750120\pi$
$194$	−1.49087	−0.107038
$195$	0	0
$196$	−1.97858	−0.141327
$197$	−16.9881	−1.21035	−0.605177	−	0.796091i	$-0.706898\pi$
$197$	−16.9881	−1.21035	−0.605177	+	0.796091i	$0.706898\pi$
$198$	0	0
$199$	−15.2143	−1.07852	−0.539258	−	0.842140i	$-0.681295\pi$
$199$	−15.2143	−1.07852	−0.539258	+	0.842140i	$0.681295\pi$
$200$	2.09718	0.148293
$201$	0	0
$202$	−0.878480	−0.0618097
$203$	9.36234	0.657107
$204$	0	0
$205$	−26.3790	−1.84239
$206$	−1.55950	−0.108656
$207$	0	0
$208$	−5.93067	−0.411218
$209$	10.8805	0.752618
$210$	0	0
$211$	21.0470	1.44893	0.724467	−	0.689310i	$-0.242086\pi$
$211$	21.0470	1.44893	0.724467	+	0.689310i	$0.242086\pi$
$212$	13.0749	0.897991
$213$	0	0
$214$	−0.328426	−0.0224507
$215$	−32.7644	−2.23452
$216$	0	0
$217$	5.71758	0.388135
$218$	2.54021	0.172045
$219$	0	0
$220$	−10.0252	−0.675901
$221$	−2.34445	−0.157705
$222$	0	0
$223$	6.69129	0.448082	0.224041	−	0.974580i	$-0.428075\pi$
$223$	6.69129	0.448082	0.224041	+	0.974580i	$0.428075\pi$
$224$	1.73125	0.115674
$225$	0	0
$226$	−2.71084	−0.180322
$227$	−25.2721	−1.67737	−0.838684	−	0.544619i	$-0.816674\pi$
$227$	−25.2721	−1.67737	−0.838684	+	0.544619i	$0.816674\pi$
$228$	0	0
$229$	17.8680	1.18075	0.590377	−	0.807128i	$-0.298979\pi$
$229$	17.8680	1.18075	0.590377	+	0.807128i	$0.298979\pi$
$230$	−0.0899276	−0.00592965
$231$	0	0
$232$	−5.45155	−0.357912
$233$	−2.33699	−0.153101	−0.0765505	−	0.997066i	$-0.524391\pi$
$233$	−2.33699	−0.153101	−0.0765505	+	0.997066i	$0.524391\pi$
$234$	0	0
$235$	−2.02596	−0.132159
$236$	−20.5794	−1.33961
$237$	0	0
$238$	0.224013	0.0145206
$239$	−20.9894	−1.35769	−0.678847	−	0.734280i	$-0.737520\pi$
$239$	−20.9894	−1.35769	−0.678847	+	0.734280i	$0.737520\pi$
$240$	0	0
$241$	−8.65032	−0.557216	−0.278608	−	0.960405i	$-0.589873\pi$
$241$	−8.65032	−0.557216	−0.278608	+	0.960405i	$0.589873\pi$
$242$	1.17308	0.0754083
$243$	0	0
$244$	−21.3317	−1.36562
$245$	2.93285	0.187373
$246$	0	0
$247$	−9.64655	−0.613796
$248$	−3.32926	−0.211408
$249$	0	0
$250$	0.600232	0.0379620
$251$	−1.54650	−0.0976144	−0.0488072	−	0.998808i	$-0.515542\pi$
$251$	−1.54650	−0.0976144	−0.0488072	+	0.998808i	$0.515542\pi$
$252$	0	0
$253$	0.361948	0.0227554
$254$	−0.146355	−0.00918312
$255$	0	0
$256$	14.3138	0.894613
$257$	12.6928	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$257$	12.6928	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$258$	0	0
$259$	9.72170	0.604077
$260$	8.88831	0.551230
$261$	0	0
$262$	1.62679	0.100503
$263$	16.3213	1.00642	0.503208	−	0.864165i	$-0.332153\pi$
$263$	16.3213	1.00642	0.503208	+	0.864165i	$0.332153\pi$
$264$	0	0
$265$	−19.3810	−1.19057
$266$	0.921731	0.0565150
$267$	0	0
$268$	−6.25373	−0.382007
$269$	−15.8774	−0.968059	−0.484030	−	0.875052i	$-0.660827\pi$
$269$	−15.8774	−0.968059	−0.484030	+	0.875052i	$0.660827\pi$
$270$	0	0
$271$	−5.24776	−0.318778	−0.159389	−	0.987216i	$-0.550952\pi$
$271$	−5.24776	−0.318778	−0.159389	+	0.987216i	$0.550952\pi$
$272$	5.92645	0.359344
$273$	0	0
$274$	0.870758	0.0526044
$275$	6.22228	0.375218
$276$	0	0
$277$	−8.15843	−0.490193	−0.245096	−	0.969499i	$-0.578820\pi$
$277$	−8.15843	−0.490193	−0.245096	+	0.969499i	$0.578820\pi$
$278$	0.783943	0.0470178
$279$	0	0
$280$	−1.70776	−0.102058
$281$	−28.6692	−1.71026	−0.855130	−	0.518413i	$-0.826523\pi$
$281$	−28.6692	−1.71026	−0.855130	+	0.518413i	$0.826523\pi$
$282$	0	0
$283$	−12.3709	−0.735372	−0.367686	−	0.929950i	$-0.619850\pi$
$283$	−12.3709	−0.735372	−0.367686	+	0.929950i	$0.619850\pi$
$284$	14.9366	0.886326
$285$	0	0
$286$	0.387287	0.0229008
$287$	8.99430	0.530917
$288$	0	0
$289$	−14.6572	−0.862189
$290$	4.01867	0.235984
$291$	0	0
$292$	−1.68838	−0.0988051
$293$	19.5227	1.14053	0.570263	−	0.821462i	$-0.306841\pi$
$293$	19.5227	1.14053	0.570263	+	0.821462i	$0.306841\pi$
$294$	0	0
$295$	30.5050	1.77607
$296$	−5.66080	−0.329027
$297$	0	0
$298$	−0.823097	−0.0476807
$299$	−0.320900	−0.0185582
$300$	0	0
$301$	11.1715	0.643916
$302$	−2.40291	−0.138272
$303$	0	0
$304$	24.3852	1.39858
$305$	31.6200	1.81056
$306$	0	0
$307$	6.42851	0.366895	0.183447	−	0.983030i	$-0.441274\pi$
$307$	6.42851	0.366895	0.183447	+	0.983030i	$0.441274\pi$
$308$	3.41825	0.194773
$309$	0	0
$310$	2.45420	0.139389
$311$	12.3704	0.701459	0.350730	−	0.936477i	$-0.385934\pi$
$311$	12.3704	0.701459	0.350730	+	0.936477i	$0.385934\pi$
$312$	0	0
$313$	29.2201	1.65162	0.825809	−	0.563950i	$-0.190719\pi$
$313$	29.2201	1.65162	0.825809	+	0.563950i	$0.190719\pi$
$314$	−1.86056	−0.104997
$315$	0	0
$316$	−2.15131	−0.121021
$317$	9.33577	0.524349	0.262175	−	0.965020i	$-0.415560\pi$
$317$	9.33577	0.524349	0.262175	+	0.965020i	$0.415560\pi$
$318$	0	0
$319$	−16.1746	−0.905606
$320$	−21.9686	−1.22808
$321$	0	0
$322$	0.0306621	0.00170873
$323$	9.63968	0.536366
$324$	0	0
$325$	−5.51664	−0.306008
$326$	1.76169	0.0975708
$327$	0	0
$328$	−5.23725	−0.289178
$329$	0.690780	0.0380839
$330$	0	0
$331$	15.1011	0.830029	0.415015	−	0.909815i	$-0.363776\pi$
$331$	15.1011	0.830029	0.415015	+	0.909815i	$0.363776\pi$
$332$	12.4765	0.684735
$333$	0	0
$334$	3.14742	0.172219
$335$	9.26991	0.506469
$336$	0	0
$337$	6.43024	0.350278	0.175139	−	0.984544i	$-0.443963\pi$
$337$	6.43024	0.350278	0.175139	+	0.984544i	$0.443963\pi$
$338$	1.55925	0.0848119
$339$	0	0
$340$	−8.88198	−0.481693
$341$	−9.87786	−0.534916
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.50501	−0.350727
$345$	0	0
$346$	−0.801906	−0.0431107
$347$	0.114351	0.00613870	0.00306935	−	0.999995i	$-0.499023\pi$
$347$	0.114351	0.00613870	0.00306935	+	0.999995i	$0.499023\pi$
$348$	0	0
$349$	0.225699	0.0120814	0.00604069	−	0.999982i	$-0.498077\pi$
$349$	0.225699	0.0120814	0.00604069	+	0.999982i	$0.498077\pi$
$350$	0.527117	0.0281756
$351$	0	0
$352$	−2.99095	−0.159418
$353$	−29.3008	−1.55953	−0.779763	−	0.626075i	$-0.784660\pi$
$353$	−29.3008	−1.55953	−0.779763	+	0.626075i	$0.784660\pi$
$354$	0	0
$355$	−22.1406	−1.17510
$356$	−0.221017	−0.0117139
$357$	0	0
$358$	−0.0215893	−0.00114103
$359$	−27.8060	−1.46755	−0.733773	−	0.679395i	$-0.762242\pi$
$359$	−27.8060	−1.46755	−0.733773	+	0.679395i	$0.762242\pi$
$360$	0	0
$361$	20.6638	1.08757
$362$	−0.848592	−0.0446010
$363$	0	0
$364$	−3.03060	−0.158847
$365$	2.50269	0.130997
$366$	0	0
$367$	18.5509	0.968347	0.484173	−	0.874972i	$-0.339120\pi$
$367$	18.5509	0.968347	0.484173	+	0.874972i	$0.339120\pi$
$368$	0.811192	0.0422863
$369$	0	0
$370$	4.17292	0.216940
$371$	6.60824	0.343083
$372$	0	0
$373$	−21.2848	−1.10209	−0.551043	−	0.834477i	$-0.685770\pi$
$373$	−21.2848	−1.10209	−0.551043	+	0.834477i	$0.685770\pi$
$374$	−0.387011	−0.0200119
$375$	0	0
$376$	−0.402231	−0.0207435
$377$	14.3403	0.738565
$378$	0	0
$379$	−10.1819	−0.523010	−0.261505	−	0.965202i	$-0.584219\pi$
$379$	−10.1819	−0.523010	−0.261505	+	0.965202i	$0.584219\pi$
$380$	−36.5461	−1.87478
$381$	0	0
$382$	2.52846	0.129367
$383$	−11.9460	−0.610414	−0.305207	−	0.952286i	$-0.598726\pi$
$383$	−11.9460	−0.610414	−0.305207	+	0.952286i	$0.598726\pi$
$384$	0	0
$385$	−5.06688	−0.258232
$386$	2.87650	0.146410
$387$	0	0
$388$	−20.1552	−1.02323
$389$	0.358811	0.0181925	0.00909623	−	0.999959i	$-0.497105\pi$
$389$	0.358811	0.0181925	0.00909623	+	0.999959i	$0.497105\pi$
$390$	0	0
$391$	0.320672	0.0162171
$392$	0.582285	0.0294098
$393$	0	0
$394$	2.48630	0.125258
$395$	3.18890	0.160451
$396$	0	0
$397$	17.5499	0.880804	0.440402	−	0.897801i	$-0.354836\pi$
$397$	17.5499	0.880804	0.440402	+	0.897801i	$0.354836\pi$
$398$	2.22670	0.111614
$399$	0	0
$400$	13.9453	0.697265
$401$	−4.77067	−0.238236	−0.119118	−	0.992880i	$-0.538007\pi$
$401$	−4.77067	−0.238236	−0.119118	+	0.992880i	$0.538007\pi$
$402$	0	0
$403$	8.75765	0.436250
$404$	−11.8762	−0.590864
$405$	0	0
$406$	−1.37022	−0.0680031
$407$	−16.7955	−0.832522
$408$	0	0
$409$	17.0034	0.840765	0.420383	−	0.907347i	$-0.361896\pi$
$409$	17.0034	0.840765	0.420383	+	0.907347i	$0.361896\pi$
$410$	3.86069	0.190666
$411$	0	0
$412$	−21.0830	−1.03869
$413$	−10.4011	−0.511805
$414$	0	0
$415$	−18.4939	−0.907829
$416$	2.65176	0.130013
$417$	0	0
$418$	−1.59241	−0.0778873
$419$	11.6960	0.571388	0.285694	−	0.958321i	$-0.407776\pi$
$419$	11.6960	0.571388	0.285694	+	0.958321i	$0.407776\pi$
$420$	0	0
$421$	−26.3224	−1.28288	−0.641439	−	0.767174i	$-0.721662\pi$
$421$	−26.3224	−1.28288	−0.641439	+	0.767174i	$0.721662\pi$
$422$	−3.08033	−0.149948
$423$	0	0
$424$	−3.84788	−0.186870
$425$	5.51271	0.267406
$426$	0	0
$427$	−10.7813	−0.521745
$428$	−4.44000	−0.214616
$429$	0	0
$430$	4.79524	0.231247
$431$	−24.0620	−1.15902	−0.579512	−	0.814964i	$-0.696757\pi$
$431$	−24.0620	−1.15902	−0.579512	+	0.814964i	$0.696757\pi$
$432$	0	0
$433$	32.1176	1.54347	0.771736	−	0.635944i	$-0.219389\pi$
$433$	32.1176	1.54347	0.771736	+	0.635944i	$0.219389\pi$
$434$	−0.836796	−0.0401675
$435$	0	0
$436$	34.3413	1.64465
$437$	1.31945	0.0631177
$438$	0	0
$439$	16.4572	0.785461	0.392730	−	0.919654i	$-0.371531\pi$
$439$	16.4572	0.785461	0.392730	+	0.919654i	$0.371531\pi$
$440$	2.95037	0.140653
$441$	0	0
$442$	0.343122	0.0163206
$443$	−19.0353	−0.904393	−0.452196	−	0.891918i	$-0.649359\pi$
$443$	−19.0353	−0.904393	−0.452196	+	0.891918i	$0.649359\pi$
$444$	0	0
$445$	0.327614	0.0155304
$446$	−0.979304	−0.0463714
$447$	0	0
$448$	7.49050	0.353893
$449$	−0.863681	−0.0407596	−0.0203798	−	0.999792i	$-0.506488\pi$
$449$	−0.863681	−0.0407596	−0.0203798	+	0.999792i	$0.506488\pi$
$450$	0	0
$451$	−15.5388	−0.731694
$452$	−36.6479	−1.72377
$453$	0	0
$454$	3.69870	0.173588
$455$	4.49227	0.210601
$456$	0	0
$457$	−4.92462	−0.230364	−0.115182	−	0.993344i	$-0.536745\pi$
$457$	−4.92462	−0.230364	−0.115182	+	0.993344i	$0.536745\pi$
$458$	−2.61508	−0.122194
$459$	0	0
$460$	−1.21574	−0.0566840
$461$	−10.4838	−0.488279	−0.244140	−	0.969740i	$-0.578506\pi$
$461$	−10.4838	−0.488279	−0.244140	+	0.969740i	$0.578506\pi$
$462$	0	0
$463$	−19.8228	−0.921245	−0.460623	−	0.887596i	$-0.652374\pi$
$463$	−19.8228	−0.921245	−0.460623	+	0.887596i	$0.652374\pi$
$464$	−36.2504	−1.68288
$465$	0	0
$466$	0.342029	0.0158442
$467$	5.08243	0.235187	0.117593	−	0.993062i	$-0.462482\pi$
$467$	5.08243	0.235187	0.117593	+	0.993062i	$0.462482\pi$
$468$	0	0
$469$	−3.16071	−0.145948
$470$	0.296509	0.0136769
$471$	0	0
$472$	6.05641	0.278769
$473$	−19.3003	−0.887427
$474$	0	0
$475$	22.6828	1.04076
$476$	3.02844	0.138808
$477$	0	0
$478$	3.07191	0.140506
$479$	−35.0791	−1.60280	−0.801402	−	0.598125i	$-0.795912\pi$
$479$	−35.0791	−1.60280	−0.801402	+	0.598125i	$0.795912\pi$
$480$	0	0
$481$	14.8908	0.678961
$482$	1.26602	0.0576655
$483$	0	0
$484$	15.8589	0.720859
$485$	29.8761	1.35660
$486$	0	0
$487$	−4.22374	−0.191396	−0.0956980	−	0.995410i	$-0.530508\pi$
$487$	−4.22374	−0.191396	−0.0956980	+	0.995410i	$0.530508\pi$
$488$	6.27780	0.284183
$489$	0	0
$490$	−0.429238	−0.0193910
$491$	−23.9699	−1.08174	−0.540872	−	0.841105i	$-0.681906\pi$
$491$	−23.9699	−1.08174	−0.540872	+	0.841105i	$0.681906\pi$
$492$	0	0
$493$	−14.3301	−0.645396
$494$	1.41182	0.0635208
$495$	0	0
$496$	−22.1381	−0.994031
$497$	7.54917	0.338626
$498$	0	0
$499$	7.84954	0.351394	0.175697	−	0.984444i	$-0.443782\pi$
$499$	7.84954	0.351394	0.175697	+	0.984444i	$0.443782\pi$
$500$	8.11457	0.362895
$501$	0	0
$502$	0.226338	0.0101020
$503$	32.3141	1.44081	0.720407	−	0.693551i	$-0.243955\pi$
$503$	32.3141	1.44081	0.720407	+	0.693551i	$0.243955\pi$
$504$	0	0
$505$	17.6042	0.783374
$506$	−0.0529728	−0.00235493
$507$	0	0
$508$	−1.97858	−0.0877853
$509$	6.58575	0.291908	0.145954	−	0.989291i	$-0.453375\pi$
$509$	6.58575	0.291908	0.145954	+	0.989291i	$0.453375\pi$
$510$	0	0
$511$	−0.853331	−0.0377491
$512$	−11.2124	−0.495524
$513$	0	0
$514$	−1.85766	−0.0819378
$515$	31.2514	1.37710
$516$	0	0
$517$	−1.19341	−0.0524862
$518$	−1.42282	−0.0625151
$519$	0	0
$520$	−2.61578	−0.114710
$521$	−4.25977	−0.186624	−0.0933119	−	0.995637i	$-0.529745\pi$
$521$	−4.25977	−0.186624	−0.0933119	+	0.995637i	$0.529745\pi$
$522$	0	0
$523$	−37.7355	−1.65006	−0.825029	−	0.565090i	$-0.808841\pi$
$523$	−37.7355	−1.65006	−0.825029	+	0.565090i	$0.808841\pi$
$524$	21.9926	0.960753
$525$	0	0
$526$	−2.38871	−0.104153
$527$	−8.75141	−0.381217
$528$	0	0
$529$	−22.9561	−0.998092
$530$	2.83651	0.123210
$531$	0	0
$532$	12.4609	0.540250
$533$	13.7766	0.596732
$534$	0	0
$535$	6.58143	0.284540
$536$	1.84044	0.0794947
$537$	0	0
$538$	2.32373	0.100183
$539$	1.72763	0.0744143
$540$	0	0
$541$	−34.2660	−1.47321	−0.736606	−	0.676322i	$-0.763573\pi$
$541$	−34.2660	−1.47321	−0.736606	+	0.676322i	$0.763573\pi$
$542$	0.768035	0.0329899
$543$	0	0
$544$	−2.64987	−0.113612
$545$	−50.9042	−2.18049
$546$	0	0
$547$	−24.1638	−1.03317	−0.516584	−	0.856237i	$-0.672797\pi$
$547$	−24.1638	−1.03317	−0.516584	+	0.856237i	$0.672797\pi$
$548$	11.7718	0.502867
$549$	0	0
$550$	−0.910662	−0.0388308
$551$	−58.9632	−2.51192
$552$	0	0
$553$	−1.08730	−0.0462367
$554$	1.19403	0.0507294
$555$	0	0
$556$	10.5982	0.449462
$557$	28.0877	1.19012	0.595058	−	0.803683i	$-0.297129\pi$
$557$	28.0877	1.19012	0.595058	+	0.803683i	$0.297129\pi$
$558$	0	0
$559$	17.1115	0.723739
$560$	−11.3558	−0.479872
$561$	0	0
$562$	4.19588	0.176992
$563$	7.52065	0.316958	0.158479	−	0.987362i	$-0.449341\pi$
$563$	7.52065	0.316958	0.158479	+	0.987362i	$0.449341\pi$
$564$	0	0
$565$	54.3233	2.28540
$566$	1.81054	0.0761026
$567$	0	0
$568$	−4.39576	−0.184442
$569$	34.5842	1.44984	0.724922	−	0.688831i	$-0.241876\pi$
$569$	34.5842	1.44984	0.724922	+	0.688831i	$0.241876\pi$
$570$	0	0
$571$	45.2290	1.89277	0.946387	−	0.323035i	$-0.104703\pi$
$571$	45.2290	1.89277	0.946387	+	0.323035i	$0.104703\pi$
$572$	5.23576	0.218918
$573$	0	0
$574$	−1.31636	−0.0549438
$575$	0.754561	0.0314674
$576$	0	0
$577$	−10.2541	−0.426882	−0.213441	−	0.976956i	$-0.568467\pi$
$577$	−10.2541	−0.426882	−0.213441	+	0.976956i	$0.568467\pi$
$578$	2.14516	0.0892267
$579$	0	0
$580$	54.3286	2.25587
$581$	6.30576	0.261607
$582$	0	0
$583$	−11.4166	−0.472827
$584$	0.496881	0.0205611
$585$	0	0
$586$	−2.85724	−0.118031
$587$	20.0669	0.828250	0.414125	−	0.910220i	$-0.364088\pi$
$587$	20.0669	0.828250	0.414125	+	0.910220i	$0.364088\pi$
$588$	0	0
$589$	−36.0089	−1.48372
$590$	−4.46455	−0.183803
$591$	0	0
$592$	−37.6419	−1.54707
$593$	−13.7167	−0.563279	−0.281640	−	0.959520i	$-0.590878\pi$
$593$	−13.7167	−0.563279	−0.281640	+	0.959520i	$0.590878\pi$
$594$	0	0
$595$	−4.48907	−0.184034
$596$	−11.1275	−0.455800
$597$	0	0
$598$	0.0469654	0.00192056
$599$	−33.9603	−1.38758	−0.693791	−	0.720176i	$-0.744061\pi$
$599$	−33.9603	−1.38758	−0.693791	+	0.720176i	$0.744061\pi$
$600$	0	0
$601$	12.2654	0.500317	0.250159	−	0.968205i	$-0.419517\pi$
$601$	12.2654	0.500317	0.250159	+	0.968205i	$0.419517\pi$
$602$	−1.63501	−0.0666379
$603$	0	0
$604$	−32.4850	−1.32180
$605$	−23.5077	−0.955724
$606$	0	0
$607$	42.8340	1.73858	0.869288	−	0.494305i	$-0.164578\pi$
$607$	42.8340	1.73858	0.869288	+	0.494305i	$0.164578\pi$
$608$	−10.9033	−0.442185
$609$	0	0
$610$	−4.62775	−0.187372
$611$	1.05807	0.0428050
$612$	0	0
$613$	33.7232	1.36207	0.681034	−	0.732252i	$-0.261531\pi$
$613$	33.7232	1.36207	0.681034	+	0.732252i	$0.261531\pi$
$614$	−0.940845	−0.0379694
$615$	0	0
$616$	−1.00597	−0.0405318
$617$	25.9135	1.04324	0.521618	−	0.853179i	$-0.325329\pi$
$617$	25.9135	1.04324	0.521618	+	0.853179i	$0.325329\pi$
$618$	0	0
$619$	15.2488	0.612902	0.306451	−	0.951886i	$-0.400858\pi$
$619$	15.2488	0.612902	0.306451	+	0.951886i	$0.400858\pi$
$620$	33.1785	1.33248
$621$	0	0
$622$	−1.81046	−0.0725930
$623$	−0.111705	−0.00447536
$624$	0	0
$625$	−30.0364	−1.20146
$626$	−4.27651	−0.170924
$627$	0	0
$628$	−25.1530	−1.00371
$629$	−14.8802	−0.593312
$630$	0	0
$631$	−21.0185	−0.836734	−0.418367	−	0.908278i	$-0.637397\pi$
$631$	−21.0185	−0.836734	−0.418367	+	0.908278i	$0.637397\pi$
$632$	0.633119	0.0251841
$633$	0	0
$634$	−1.36634	−0.0542641
$635$	2.93285	0.116387
$636$	0	0
$637$	−1.53171	−0.0606884
$638$	2.36724	0.0937199
$639$	0	0
$640$	13.3702	0.528503
$641$	−42.5009	−1.67869	−0.839343	−	0.543602i	$-0.817060\pi$
$641$	−42.5009	−1.67869	−0.839343	+	0.543602i	$0.817060\pi$
$642$	0	0
$643$	−4.10162	−0.161752	−0.0808760	−	0.996724i	$-0.525772\pi$
$643$	−4.10162	−0.161752	−0.0808760	+	0.996724i	$0.525772\pi$
$644$	0.414523	0.0163345
$645$	0	0
$646$	−1.41082	−0.0555078
$647$	−17.4586	−0.686370	−0.343185	−	0.939268i	$-0.611506\pi$
$647$	−17.4586	−0.686370	−0.343185	+	0.939268i	$0.611506\pi$
$648$	0	0
$649$	17.9693	0.705356
$650$	0.807387	0.0316683
$651$	0	0
$652$	23.8163	0.932720
$653$	−49.1756	−1.92439	−0.962195	−	0.272363i	$-0.912195\pi$
$653$	−49.1756	−1.92439	−0.962195	+	0.272363i	$0.912195\pi$
$654$	0	0
$655$	−32.5997	−1.27378
$656$	−34.8254	−1.35970
$657$	0	0
$658$	−0.101099	−0.00394125
$659$	24.2280	0.943789	0.471895	−	0.881655i	$-0.343570\pi$
$659$	24.2280	0.943789	0.471895	+	0.881655i	$0.343570\pi$
$660$	0	0
$661$	−21.4505	−0.834328	−0.417164	−	0.908831i	$-0.636976\pi$
$661$	−21.4505	−0.834328	−0.417164	+	0.908831i	$0.636976\pi$
$662$	−2.21011	−0.0858986
$663$	0	0
$664$	−3.67175	−0.142492
$665$	−18.4709	−0.716270
$666$	0	0
$667$	−1.96146	−0.0759480
$668$	42.5501	1.64631
$669$	0	0
$670$	−1.35670	−0.0524138
$671$	18.6261	0.719053
$672$	0	0
$673$	19.3388	0.745456	0.372728	−	0.927941i	$-0.378422\pi$
$673$	19.3388	0.745456	0.372728	+	0.927941i	$0.378422\pi$
$674$	−0.941098	−0.0362497
$675$	0	0
$676$	21.0796	0.810752
$677$	32.4564	1.24740	0.623701	−	0.781663i	$-0.285628\pi$
$677$	32.4564	1.24740	0.623701	+	0.781663i	$0.285628\pi$
$678$	0	0
$679$	−10.1867	−0.390930
$680$	2.61392	0.100239
$681$	0	0
$682$	1.44567	0.0553577
$683$	−5.34403	−0.204484	−0.102242	−	0.994760i	$-0.532602\pi$
$683$	−5.34403	−0.204484	−0.102242	+	0.994760i	$0.532602\pi$
$684$	0	0
$685$	−17.4494	−0.666707
$686$	0.146355	0.00558786
$687$	0	0
$688$	−43.2555	−1.64910
$689$	10.1219	0.385613
$690$	0	0
$691$	−30.3520	−1.15464	−0.577322	−	0.816516i	$-0.695902\pi$
$691$	−30.3520	−1.15464	−0.577322	+	0.816516i	$0.695902\pi$
$692$	−10.8410	−0.412113
$693$	0	0
$694$	−0.0167359	−0.000635286 0
$695$	−15.7097	−0.595902
$696$	0	0
$697$	−13.7668	−0.521455
$698$	−0.0330321	−0.00125028
$699$	0	0
$700$	7.12612	0.269342
$701$	20.2251	0.763891	0.381946	−	0.924185i	$-0.375254\pi$
$701$	20.2251	0.763891	0.381946	+	0.924185i	$0.375254\pi$
$702$	0	0
$703$	−61.2265	−2.30920
$704$	−12.9408	−0.487725
$705$	0	0
$706$	4.28832	0.161393
$707$	−6.00240	−0.225743
$708$	0	0
$709$	−24.9412	−0.936685	−0.468343	−	0.883547i	$-0.655149\pi$
$709$	−24.9412	−0.936685	−0.468343	+	0.883547i	$0.655149\pi$
$710$	3.24039	0.121610
$711$	0	0
$712$	0.0650440	0.00243763
$713$	−1.19786	−0.0448604
$714$	0	0
$715$	−7.76097	−0.290244
$716$	−0.291867	−0.0109076
$717$	0	0
$718$	4.06955	0.151874
$719$	−48.3548	−1.80333	−0.901666	−	0.432434i	$-0.857655\pi$
$719$	−48.3548	−1.80333	−0.901666	+	0.432434i	$0.857655\pi$
$720$	0	0
$721$	−10.6556	−0.396836
$722$	−3.02424	−0.112551
$723$	0	0
$724$	−11.4722	−0.426359
$725$	−33.7197	−1.25232
$726$	0	0
$727$	45.4710	1.68642	0.843212	−	0.537581i	$-0.180662\pi$
$727$	45.4710	1.68642	0.843212	+	0.537581i	$0.180662\pi$
$728$	0.891889	0.0330556
$729$	0	0
$730$	−0.366282	−0.0135567
$731$	−17.0993	−0.632440
$732$	0	0
$733$	−38.6967	−1.42930	−0.714648	−	0.699484i	$-0.753413\pi$
$733$	−38.6967	−1.42930	−0.714648	+	0.699484i	$0.753413\pi$
$734$	−2.71501	−0.100213
$735$	0	0
$736$	−0.362706	−0.0133695
$737$	5.46054	0.201142
$738$	0	0
$739$	16.2232	0.596781	0.298390	−	0.954444i	$-0.403550\pi$
$739$	16.2232	0.596781	0.298390	+	0.954444i	$0.403550\pi$
$740$	56.4139	2.07382
$741$	0	0
$742$	−0.967149	−0.0355052
$743$	−14.7724	−0.541948	−0.270974	−	0.962587i	$-0.587346\pi$
$743$	−14.7724	−0.541948	−0.270974	+	0.962587i	$0.587346\pi$
$744$	0	0
$745$	16.4943	0.604305
$746$	3.11514	0.114053
$747$	0	0
$748$	−5.23203	−0.191302
$749$	−2.24404	−0.0819953
$750$	0	0
$751$	44.1726	1.61188	0.805941	−	0.591996i	$-0.201660\pi$
$751$	44.1726	1.61188	0.805941	+	0.591996i	$0.201660\pi$
$752$	−2.67466	−0.0975348
$753$	0	0
$754$	−2.09878	−0.0764330
$755$	48.1526	1.75245
$756$	0	0
$757$	6.55045	0.238080	0.119040	−	0.992889i	$-0.462018\pi$
$757$	6.55045	0.238080	0.119040	+	0.992889i	$0.462018\pi$
$758$	1.49017	0.0541255
$759$	0	0
$760$	10.7553	0.390136
$761$	22.3490	0.810150	0.405075	−	0.914283i	$-0.367245\pi$
$761$	22.3490	0.810150	0.405075	+	0.914283i	$0.367245\pi$
$762$	0	0
$763$	17.3565	0.628348
$764$	34.1823	1.23667
$765$	0	0
$766$	1.74836	0.0631709
$767$	−15.9314	−0.575251
$768$	0	0
$769$	−36.5334	−1.31743	−0.658713	−	0.752394i	$-0.728899\pi$
$769$	−36.5334	−1.31743	−0.658713	+	0.752394i	$0.728899\pi$
$770$	0.741563	0.0267241
$771$	0	0
$772$	38.8875	1.39959
$773$	44.2156	1.59033	0.795163	−	0.606396i	$-0.207385\pi$
$773$	44.2156	1.59033	0.795163	+	0.606396i	$0.207385\pi$
$774$	0	0
$775$	−20.5926	−0.739709
$776$	5.93156	0.212931
$777$	0	0
$778$	−0.0525138	−0.00188271
$779$	−56.6454	−2.02953
$780$	0	0
$781$	−13.0422	−0.466685
$782$	−0.0469319	−0.00167828
$783$	0	0
$784$	3.87194	0.138284
$785$	37.2843	1.33073
$786$	0	0
$787$	−34.0132	−1.21244	−0.606220	−	0.795297i	$-0.707315\pi$
$787$	−34.0132	−1.21244	−0.606220	+	0.795297i	$0.707315\pi$
$788$	33.6124	1.19739
$789$	0	0
$790$	−0.466711	−0.0166048
$791$	−18.5223	−0.658579
$792$	0	0
$793$	−16.5138	−0.586422
$794$	−2.56851	−0.0911531
$795$	0	0
$796$	30.1028	1.06697
$797$	44.6751	1.58247	0.791237	−	0.611510i	$-0.209438\pi$
$797$	44.6751	1.58247	0.791237	+	0.611510i	$0.209438\pi$
$798$	0	0
$799$	−1.05732	−0.0374052
$800$	−6.23531	−0.220452
$801$	0	0
$802$	0.698211	0.0246547
$803$	1.47424	0.0520248
$804$	0	0
$805$	−0.614449	−0.0216565
$806$	−1.28173	−0.0451468
$807$	0	0
$808$	3.49510	0.122957
$809$	32.0310	1.12615	0.563076	−	0.826405i	$-0.309618\pi$
$809$	32.0310	1.12615	0.563076	+	0.826405i	$0.309618\pi$
$810$	0	0
$811$	−16.9145	−0.593949	−0.296975	−	0.954885i	$-0.595978\pi$
$811$	−16.9145	−0.593949	−0.296975	+	0.954885i	$0.595978\pi$
$812$	−18.5241	−0.650070
$813$	0	0
$814$	2.45810	0.0861565
$815$	−35.3030	−1.23661
$816$	0	0
$817$	−70.3573	−2.46149
$818$	−2.48854	−0.0870096
$819$	0	0
$820$	52.1929	1.82266
$821$	−26.5144	−0.925360	−0.462680	−	0.886525i	$-0.653112\pi$
$821$	−26.5144	−0.925360	−0.462680	+	0.886525i	$0.653112\pi$
$822$	0	0
$823$	−17.0148	−0.593098	−0.296549	−	0.955018i	$-0.595836\pi$
$823$	−17.0148	−0.593098	−0.296549	+	0.955018i	$0.595836\pi$
$824$	6.20461	0.216148
$825$	0	0
$826$	1.52225	0.0529660
$827$	24.2581	0.843536	0.421768	−	0.906704i	$-0.361410\pi$
$827$	24.2581	0.843536	0.421768	+	0.906704i	$0.361410\pi$
$828$	0	0
$829$	49.8459	1.73122	0.865609	−	0.500721i	$-0.166932\pi$
$829$	49.8459	1.73122	0.865609	+	0.500721i	$0.166932\pi$
$830$	2.70667	0.0939499
$831$	0	0
$832$	11.4732	0.397763
$833$	1.53061	0.0530326
$834$	0	0
$835$	−63.0721	−2.18270
$836$	−21.5279	−0.744557
$837$	0	0
$838$	−1.71177	−0.0591321
$839$	−45.6299	−1.57532	−0.787660	−	0.616111i	$-0.788707\pi$
$839$	−45.6299	−1.57532	−0.787660	+	0.616111i	$0.788707\pi$
$840$	0	0
$841$	58.6533	2.02253
$842$	3.85242	0.132763
$843$	0	0
$844$	−41.6431	−1.43342
$845$	−31.2463	−1.07490
$846$	0	0
$847$	8.01530	0.275409
$848$	−25.5867	−0.878652
$849$	0	0
$850$	−0.806812	−0.0276734
$851$	−2.03675	−0.0698188
$852$	0	0
$853$	−35.8800	−1.22851	−0.614254	−	0.789108i	$-0.710543\pi$
$853$	−35.8800	−1.22851	−0.614254	+	0.789108i	$0.710543\pi$
$854$	1.57790	0.0539946
$855$	0	0
$856$	1.30667	0.0446610
$857$	44.3582	1.51525	0.757624	−	0.652692i	$-0.226360\pi$
$857$	44.3582	1.51525	0.757624	+	0.652692i	$0.226360\pi$
$858$	0	0
$859$	4.72367	0.161170	0.0805848	−	0.996748i	$-0.474321\pi$
$859$	4.72367	0.161170	0.0805848	+	0.996748i	$0.474321\pi$
$860$	64.8271	2.21059
$861$	0	0
$862$	3.52159	0.119946
$863$	−28.8107	−0.980729	−0.490364	−	0.871518i	$-0.663136\pi$
$863$	−28.8107	−0.980729	−0.490364	+	0.871518i	$0.663136\pi$
$864$	0	0
$865$	16.0696	0.546384
$866$	−4.70056	−0.159732
$867$	0	0
$868$	−11.3127	−0.383978
$869$	1.87845	0.0637222
$870$	0	0
$871$	−4.84128	−0.164041
$872$	−10.1064	−0.342247
$873$	0	0
$874$	−0.193108	−0.00653196
$875$	4.10121	0.138646
$876$	0	0
$877$	6.40390	0.216244	0.108122	−	0.994138i	$-0.465516\pi$
$877$	6.40390	0.216244	0.108122	+	0.994138i	$0.465516\pi$
$878$	−2.40860	−0.0812862
$879$	0	0
$880$	19.6187	0.661345
$881$	55.1594	1.85837	0.929183	−	0.369620i	$-0.120512\pi$
$881$	55.1594	1.85837	0.929183	+	0.369620i	$0.120512\pi$
$882$	0	0
$883$	−47.1442	−1.58653	−0.793264	−	0.608877i	$-0.791620\pi$
$883$	−47.1442	−1.58653	−0.793264	+	0.608877i	$0.791620\pi$
$884$	4.63868	0.156016
$885$	0	0
$886$	2.78591	0.0935943
$887$	24.8385	0.833994	0.416997	−	0.908908i	$-0.363083\pi$
$887$	24.8385	0.833994	0.416997	+	0.908908i	$0.363083\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−0.0479479	−0.00160722
$891$	0	0
$892$	−13.2393	−0.443283
$893$	−4.35048	−0.145583
$894$	0	0
$895$	0.432635	0.0144614
$896$	−4.55877	−0.152298
$897$	0	0
$898$	0.126404	0.00421816
$899$	53.5299	1.78532
$900$	0	0
$901$	−10.1147	−0.336969
$902$	2.27418	0.0757220
$903$	0	0
$904$	10.7853	0.358713
$905$	17.0052	0.565272
$906$	0	0
$907$	15.1021	0.501457	0.250728	−	0.968057i	$-0.419330\pi$
$907$	15.1021	0.501457	0.250728	+	0.968057i	$0.419330\pi$
$908$	50.0029	1.65940
$909$	0	0
$910$	−0.657466	−0.0217948
$911$	9.32638	0.308997	0.154498	−	0.987993i	$-0.450624\pi$
$911$	9.32638	0.308997	0.154498	+	0.987993i	$0.450624\pi$
$912$	0	0
$913$	−10.8940	−0.360539
$914$	0.720743	0.0238401
$915$	0	0
$916$	−35.3534	−1.16811
$917$	11.1154	0.367062
$918$	0	0
$919$	−42.1214	−1.38946	−0.694729	−	0.719271i	$-0.744476\pi$
$919$	−42.1214	−1.38946	−0.694729	+	0.719271i	$0.744476\pi$
$920$	0.357784	0.0117958
$921$	0	0
$922$	1.53436	0.0505313
$923$	11.5631	0.380604
$924$	0	0
$925$	−35.0140	−1.15125
$926$	2.90117	0.0953383
$927$	0	0
$928$	16.2085	0.532071
$929$	27.6162	0.906058	0.453029	−	0.891496i	$-0.350343\pi$
$929$	27.6162	0.906058	0.453029	+	0.891496i	$0.350343\pi$
$930$	0	0
$931$	6.29792	0.206406
$932$	4.62391	0.151461
$933$	0	0
$934$	−0.743839	−0.0243392
$935$	7.75545	0.253630
$936$	0	0
$937$	−4.20843	−0.137483	−0.0687416	−	0.997634i	$-0.521898\pi$
$937$	−4.20843	−0.137483	−0.0687416	+	0.997634i	$0.521898\pi$
$938$	0.462586	0.0151040
$939$	0	0
$940$	4.00852	0.130743
$941$	−18.5529	−0.604807	−0.302404	−	0.953180i	$-0.597789\pi$
$941$	−18.5529	−0.604807	−0.302404	+	0.953180i	$0.597789\pi$
$942$	0	0
$943$	−1.88435	−0.0613630
$944$	40.2725	1.31076
$945$	0	0
$946$	2.82469	0.0918385
$947$	13.4123	0.435842	0.217921	−	0.975966i	$-0.430072\pi$
$947$	13.4123	0.435842	0.217921	+	0.975966i	$0.430072\pi$
$948$	0	0
$949$	−1.30705	−0.0424287
$950$	−3.31974	−0.107706
$951$	0	0
$952$	−0.891253	−0.0288857
$953$	−29.5134	−0.956033	−0.478016	−	0.878351i	$-0.658644\pi$
$953$	−29.5134	−0.956033	−0.478016	+	0.878351i	$0.658644\pi$
$954$	0	0
$955$	−50.6685	−1.63960
$956$	41.5293	1.34315
$957$	0	0
$958$	5.13400	0.165872
$959$	5.94963	0.192124
$960$	0	0
$961$	1.69073	0.0545397
$962$	−2.17934	−0.0702647
$963$	0	0
$964$	17.1154	0.551248
$965$	−57.6431	−1.85560
$966$	0	0
$967$	−13.3753	−0.430120	−0.215060	−	0.976601i	$-0.568995\pi$
$967$	−13.3753	−0.430120	−0.215060	+	0.976601i	$0.568995\pi$
$968$	−4.66719	−0.150009
$969$	0	0
$970$	−4.37251	−0.140393
$971$	43.5573	1.39782	0.698911	−	0.715209i	$-0.253669\pi$
$971$	43.5573	1.39782	0.698911	+	0.715209i	$0.253669\pi$
$972$	0	0
$973$	5.35645	0.171720
$974$	0.618166	0.0198073
$975$	0	0
$976$	41.7446	1.33621
$977$	−17.6601	−0.564997	−0.282499	−	0.959268i	$-0.591163\pi$
$977$	−17.6601	−0.564997	−0.282499	+	0.959268i	$0.591163\pi$
$978$	0	0
$979$	0.192985	0.00616781
$980$	−5.80289	−0.185366
$981$	0	0
$982$	3.50811	0.111948
$983$	−58.8506	−1.87704	−0.938521	−	0.345221i	$-0.887804\pi$
$983$	−58.8506	−1.87704	−0.938521	+	0.345221i	$0.887804\pi$
$984$	0	0
$985$	−49.8237	−1.58752
$986$	2.09728	0.0667911
$987$	0	0
$988$	19.0865	0.607222
$989$	−2.34049	−0.0744234
$990$	0	0
$991$	−0.927443	−0.0294612	−0.0147306	−	0.999891i	$-0.504689\pi$
$991$	−0.927443	−0.0294612	−0.0147306	+	0.999891i	$0.504689\pi$
$992$	9.89855	0.314279
$993$	0	0
$994$	−1.10486	−0.0350440
$995$	−44.6215	−1.41460
$996$	0	0
$997$	33.7255	1.06810	0.534048	−	0.845454i	$-0.320670\pi$
$997$	33.7255	1.06810	0.534048	+	0.845454i	$0.320670\pi$
$998$	−1.14882	−0.0363652
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.8		15
3.2	odd	2		889.2.a.b.1.8	✓	15
21.20	even	2		6223.2.a.j.1.8		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.8	✓	15	3.2	odd	2
6223.2.a.j.1.8		15	21.20	even	2
8001.2.a.q.1.8		15	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)

\(f(q)\)	\(=\)	\(q-0.146355 q^{2} -1.97858 q^{4} +2.93285 q^{5} -1.00000 q^{7} +0.582285 q^{8} +O(q^{10})\)	\(q-0.146355 q^{2} -1.97858 q^{4} +2.93285 q^{5} -1.00000 q^{7} +0.582285 q^{8} -0.429238 q^{10} +1.72763 q^{11} -1.53171 q^{13} +0.146355 q^{14} +3.87194 q^{16} +1.53061 q^{17} +6.29792 q^{19} -5.80289 q^{20} -0.252847 q^{22} +0.209505 q^{23} +3.60163 q^{25} +0.224173 q^{26} +1.97858 q^{28} -9.36234 q^{29} -5.71758 q^{31} -1.73125 q^{32} -0.224013 q^{34} -2.93285 q^{35} -9.72170 q^{37} -0.921731 q^{38} +1.70776 q^{40} -8.99430 q^{41} -11.1715 q^{43} -3.41825 q^{44} -0.0306621 q^{46} -0.690780 q^{47} +1.00000 q^{49} -0.527117 q^{50} +3.03060 q^{52} -6.60824 q^{53} +5.06688 q^{55} -0.582285 q^{56} +1.37022 q^{58} +10.4011 q^{59} +10.7813 q^{61} +0.836796 q^{62} -7.49050 q^{64} -4.49227 q^{65} +3.16071 q^{67} -3.02844 q^{68} +0.429238 q^{70} -7.54917 q^{71} +0.853331 q^{73} +1.42282 q^{74} -12.4609 q^{76} -1.72763 q^{77} +1.08730 q^{79} +11.3558 q^{80} +1.31636 q^{82} -6.30576 q^{83} +4.48907 q^{85} +1.63501 q^{86} +1.00597 q^{88} +0.111705 q^{89} +1.53171 q^{91} -0.414523 q^{92} +0.101099 q^{94} +18.4709 q^{95} +10.1867 q^{97} -0.146355 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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