LMFDB - Embedded newform 8001.2.a.w.1.15

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-0.731838$ 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)

$f(q)$	$=$	$q+0.731838 q^{2} -1.46441 q^{4} -3.93264 q^{5} +1.00000 q^{7} -2.53539 q^{8} +O(q^{10})$	\(q+0.731838 q^{2} -1.46441 q^{4} -3.93264 q^{5} +1.00000 q^{7} -2.53539 q^{8} -2.87806 q^{10} -4.56737 q^{11} -3.89555 q^{13} +0.731838 q^{14} +1.07333 q^{16} +6.78176 q^{17} +5.75086 q^{19} +5.75902 q^{20} -3.34257 q^{22} -1.77247 q^{23} +10.4657 q^{25} -2.85091 q^{26} -1.46441 q^{28} +2.04719 q^{29} -4.75876 q^{31} +5.85628 q^{32} +4.96315 q^{34} -3.93264 q^{35} -3.62146 q^{37} +4.20870 q^{38} +9.97078 q^{40} -3.27869 q^{41} +12.5020 q^{43} +6.68851 q^{44} -1.29716 q^{46} -0.353812 q^{47} +1.00000 q^{49} +7.65919 q^{50} +5.70470 q^{52} +8.18257 q^{53} +17.9618 q^{55} -2.53539 q^{56} +1.49821 q^{58} -10.8977 q^{59} +9.82338 q^{61} -3.48264 q^{62} +2.13919 q^{64} +15.3198 q^{65} +6.08050 q^{67} -9.93130 q^{68} -2.87806 q^{70} -11.4162 q^{71} +1.83793 q^{73} -2.65032 q^{74} -8.42164 q^{76} -4.56737 q^{77} -7.38973 q^{79} -4.22104 q^{80} -2.39947 q^{82} +7.42793 q^{83} -26.6703 q^{85} +9.14944 q^{86} +11.5801 q^{88} +6.77417 q^{89} -3.89555 q^{91} +2.59563 q^{92} -0.258933 q^{94} -22.6161 q^{95} +10.9733 q^{97} +0.731838 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	0.731838	0.517488	0.258744	−	0.965946i	$-0.416691\pi$
$2$	0.731838	0.517488	0.258744	+	0.965946i	$0.416691\pi$
$3$	0	0
$3$	0	0
$4$	−1.46441	−0.732207
$5$	−3.93264	−1.75873	−0.879366	−	0.476146i	$-0.842033\pi$
$5$	−3.93264	−1.75873	−0.879366	+	0.476146i	$0.842033\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.53539	−0.896395
$9$	0	0
$10$	−2.87806	−0.910122
$11$	−4.56737	−1.37711	−0.688557	−	0.725183i	$-0.741755\pi$
$11$	−4.56737	−1.37711	−0.688557	+	0.725183i	$0.741755\pi$
$12$	0	0
$13$	−3.89555	−1.08043	−0.540216	−	0.841527i	$-0.681657\pi$
$13$	−3.89555	−1.08043	−0.540216	+	0.841527i	$0.681657\pi$
$14$	0.731838	0.195592
$15$	0	0
$16$	1.07333	0.268333
$17$	6.78176	1.64482	0.822409	−	0.568896i	$-0.192629\pi$
$17$	6.78176	1.64482	0.822409	+	0.568896i	$0.192629\pi$
$18$	0	0
$19$	5.75086	1.31934	0.659669	−	0.751556i	$-0.270696\pi$
$19$	5.75086	1.31934	0.659669	+	0.751556i	$0.270696\pi$
$20$	5.75902	1.28776
$21$	0	0
$22$	−3.34257	−0.712639
$23$	−1.77247	−0.369585	−0.184793	−	0.982778i	$-0.559161\pi$
$23$	−1.77247	−0.369585	−0.184793	+	0.982778i	$0.559161\pi$
$24$	0	0
$25$	10.4657	2.09314
$26$	−2.85091	−0.559110
$27$	0	0
$28$	−1.46441	−0.276748
$29$	2.04719	0.380153	0.190077	−	0.981769i	$-0.439126\pi$
$29$	2.04719	0.380153	0.190077	+	0.981769i	$0.439126\pi$
$30$	0	0
$31$	−4.75876	−0.854698	−0.427349	−	0.904087i	$-0.640553\pi$
$31$	−4.75876	−0.854698	−0.427349	+	0.904087i	$0.640553\pi$
$32$	5.85628	1.03525
$33$	0	0
$34$	4.96315	0.851173
$35$	−3.93264	−0.664738
$36$	0	0
$37$	−3.62146	−0.595365	−0.297683	−	0.954665i	$-0.596214\pi$
$37$	−3.62146	−0.595365	−0.297683	+	0.954665i	$0.596214\pi$
$38$	4.20870	0.682741
$39$	0	0
$40$	9.97078	1.57652
$41$	−3.27869	−0.512045	−0.256023	−	0.966671i	$-0.582412\pi$
$41$	−3.27869	−0.512045	−0.256023	+	0.966671i	$0.582412\pi$
$42$	0	0
$43$	12.5020	1.90654	0.953268	−	0.302125i	$-0.0976960\pi$
$43$	12.5020	1.90654	0.953268	+	0.302125i	$0.0976960\pi$
$44$	6.68851	1.00833
$45$	0	0
$46$	−1.29716	−0.191256
$47$	−0.353812	−0.0516089	−0.0258044	−	0.999667i	$-0.508215\pi$
$47$	−0.353812	−0.0516089	−0.0258044	+	0.999667i	$0.508215\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.65919	1.08317
$51$	0	0
$52$	5.70470	0.791099
$53$	8.18257	1.12396	0.561981	−	0.827150i	$-0.310039\pi$
$53$	8.18257	1.12396	0.561981	+	0.827150i	$0.310039\pi$
$54$	0	0
$55$	17.9618	2.42197
$56$	−2.53539	−0.338806
$57$	0	0
$58$	1.49821	0.196725
$59$	−10.8977	−1.41876	−0.709378	−	0.704828i	$-0.751024\pi$
$59$	−10.8977	−1.41876	−0.709378	+	0.704828i	$0.751024\pi$
$60$	0	0
$61$	9.82338	1.25775	0.628877	−	0.777504i	$-0.283515\pi$
$61$	9.82338	1.25775	0.628877	+	0.777504i	$0.283515\pi$
$62$	−3.48264	−0.442296
$63$	0	0
$64$	2.13919	0.267398
$65$	15.3198	1.90019
$66$	0	0
$67$	6.08050	0.742851	0.371426	−	0.928463i	$-0.378869\pi$
$67$	6.08050	0.742851	0.371426	+	0.928463i	$0.378869\pi$
$68$	−9.93130	−1.20435
$69$	0	0
$70$	−2.87806	−0.343994
$71$	−11.4162	−1.35486	−0.677428	−	0.735589i	$-0.736905\pi$
$71$	−11.4162	−1.35486	−0.677428	+	0.735589i	$0.736905\pi$
$72$	0	0
$73$	1.83793	0.215113	0.107556	−	0.994199i	$-0.465697\pi$
$73$	1.83793	0.215113	0.107556	+	0.994199i	$0.465697\pi$
$74$	−2.65032	−0.308094
$75$	0	0
$76$	−8.42164	−0.966028
$77$	−4.56737	−0.520500
$78$	0	0
$79$	−7.38973	−0.831409	−0.415705	−	0.909500i	$-0.636465\pi$
$79$	−7.38973	−0.831409	−0.415705	+	0.909500i	$0.636465\pi$
$80$	−4.22104	−0.471926
$81$	0	0
$82$	−2.39947	−0.264977
$83$	7.42793	0.815321	0.407660	−	0.913134i	$-0.366345\pi$
$83$	7.42793	0.815321	0.407660	+	0.913134i	$0.366345\pi$
$84$	0	0
$85$	−26.6703	−2.89280
$86$	9.14944	0.986609
$87$	0	0
$88$	11.5801	1.23444
$89$	6.77417	0.718060	0.359030	−	0.933326i	$-0.383108\pi$
$89$	6.77417	0.718060	0.359030	+	0.933326i	$0.383108\pi$
$90$	0	0
$91$	−3.89555	−0.408365
$92$	2.59563	0.270613
$93$	0	0
$94$	−0.258933	−0.0267069
$95$	−22.6161	−2.32036
$96$	0	0
$97$	10.9733	1.11417	0.557084	−	0.830456i	$-0.311920\pi$
$97$	10.9733	1.11417	0.557084	+	0.830456i	$0.311920\pi$
$98$	0.731838	0.0739268
$99$	0	0
$100$	−15.3261	−1.53261
$101$	−3.77928	−0.376053	−0.188026	−	0.982164i	$-0.560209\pi$
$101$	−3.77928	−0.376053	−0.188026	+	0.982164i	$0.560209\pi$
$102$	0	0
$103$	−8.84662	−0.871683	−0.435842	−	0.900023i	$-0.643549\pi$
$103$	−8.84662	−0.871683	−0.435842	+	0.900023i	$0.643549\pi$
$104$	9.87674	0.968494
$105$	0	0
$106$	5.98832	0.581637
$107$	−11.9793	−1.15808	−0.579040	−	0.815299i	$-0.696573\pi$
$107$	−11.9793	−1.15808	−0.579040	+	0.815299i	$0.696573\pi$
$108$	0	0
$109$	5.50483	0.527267	0.263634	−	0.964623i	$-0.415079\pi$
$109$	5.50483	0.527267	0.263634	+	0.964623i	$0.415079\pi$
$110$	13.1452	1.25334
$111$	0	0
$112$	1.07333	0.101420
$113$	1.11071	0.104487	0.0522435	−	0.998634i	$-0.483363\pi$
$113$	1.11071	0.104487	0.0522435	+	0.998634i	$0.483363\pi$
$114$	0	0
$115$	6.97049	0.650002
$116$	−2.99793	−0.278351
$117$	0	0
$118$	−7.97533	−0.734189
$119$	6.78176	0.621683
$120$	0	0
$121$	9.86085	0.896441
$122$	7.18912	0.650873
$123$	0	0
$124$	6.96879	0.625816
$125$	−21.4946	−1.92254
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.1470	−0.896879
$129$	0	0
$130$	11.2116	0.983324
$131$	11.6742	1.01998	0.509989	−	0.860181i	$-0.329649\pi$
$131$	11.6742	1.01998	0.509989	+	0.860181i	$0.329649\pi$
$132$	0	0
$133$	5.75086	0.498663
$134$	4.44994	0.384416
$135$	0	0
$136$	−17.1944	−1.47441
$137$	−21.9031	−1.87131	−0.935654	−	0.352919i	$-0.885189\pi$
$137$	−21.9031	−1.87131	−0.935654	+	0.352919i	$0.885189\pi$
$138$	0	0
$139$	−19.0623	−1.61684	−0.808421	−	0.588604i	$-0.799678\pi$
$139$	−19.0623	−1.61684	−0.808421	+	0.588604i	$0.799678\pi$
$140$	5.75902	0.486726
$141$	0	0
$142$	−8.35482	−0.701121
$143$	17.7924	1.48788
$144$	0	0
$145$	−8.05086	−0.668587
$146$	1.34506	0.111318
$147$	0	0
$148$	5.30332	0.435930
$149$	1.94463	0.159311	0.0796553	−	0.996822i	$-0.474618\pi$
$149$	1.94463	0.159311	0.0796553	+	0.996822i	$0.474618\pi$
$150$	0	0
$151$	12.3431	1.00447	0.502234	−	0.864732i	$-0.332512\pi$
$151$	12.3431	1.00447	0.502234	+	0.864732i	$0.332512\pi$
$152$	−14.5807	−1.18265
$153$	0	0
$154$	−3.34257	−0.269352
$155$	18.7145	1.50319
$156$	0	0
$157$	3.61611	0.288597	0.144298	−	0.989534i	$-0.453907\pi$
$157$	3.61611	0.288597	0.144298	+	0.989534i	$0.453907\pi$
$158$	−5.40808	−0.430244
$159$	0	0
$160$	−23.0307	−1.82074
$161$	−1.77247	−0.139690
$162$	0	0
$163$	10.0075	0.783847	0.391923	−	0.919998i	$-0.371810\pi$
$163$	10.0075	0.783847	0.391923	+	0.919998i	$0.371810\pi$
$164$	4.80136	0.374923
$165$	0	0
$166$	5.43604	0.421918
$167$	−15.9926	−1.23754	−0.618771	−	0.785572i	$-0.712369\pi$
$167$	−15.9926	−1.23754	−0.618771	+	0.785572i	$0.712369\pi$
$168$	0	0
$169$	2.17531	0.167332
$170$	−19.5183	−1.49699
$171$	0	0
$172$	−18.3081	−1.39598
$173$	−7.76734	−0.590540	−0.295270	−	0.955414i	$-0.595410\pi$
$173$	−7.76734	−0.590540	−0.295270	+	0.955414i	$0.595410\pi$
$174$	0	0
$175$	10.4657	0.791132
$176$	−4.90231	−0.369525
$177$	0	0
$178$	4.95759	0.371587
$179$	23.6261	1.76590	0.882951	−	0.469466i	$-0.155554\pi$
$179$	23.6261	1.76590	0.882951	+	0.469466i	$0.155554\pi$
$180$	0	0
$181$	22.0001	1.63525	0.817626	−	0.575750i	$-0.195290\pi$
$181$	22.0001	1.63525	0.817626	+	0.575750i	$0.195290\pi$
$182$	−2.85091	−0.211324
$183$	0	0
$184$	4.49390	0.331295
$185$	14.2419	1.04709
$186$	0	0
$187$	−30.9748	−2.26510
$188$	0.518128	0.0377883
$189$	0	0
$190$	−16.5513	−1.20076
$191$	−23.6532	−1.71148	−0.855742	−	0.517403i	$-0.826899\pi$
$191$	−23.6532	−1.71148	−0.855742	+	0.517403i	$0.826899\pi$
$192$	0	0
$193$	−14.5476	−1.04716	−0.523581	−	0.851976i	$-0.675404\pi$
$193$	−14.5476	−1.04716	−0.523581	+	0.851976i	$0.675404\pi$
$194$	8.03066	0.576568
$195$	0	0
$196$	−1.46441	−0.104601
$197$	−8.86925	−0.631908	−0.315954	−	0.948775i	$-0.602325\pi$
$197$	−8.86925	−0.631908	−0.315954	+	0.948775i	$0.602325\pi$
$198$	0	0
$199$	−13.8851	−0.984291	−0.492146	−	0.870513i	$-0.663787\pi$
$199$	−13.8851	−0.984291	−0.492146	+	0.870513i	$0.663787\pi$
$200$	−26.5346	−1.87628
$201$	0	0
$202$	−2.76582	−0.194603
$203$	2.04719	0.143684
$204$	0	0
$205$	12.8939	0.900550
$206$	−6.47429	−0.451085
$207$	0	0
$208$	−4.18122	−0.289916
$209$	−26.2663	−1.81688
$210$	0	0
$211$	13.6380	0.938880	0.469440	−	0.882964i	$-0.344456\pi$
$211$	13.6380	0.938880	0.469440	+	0.882964i	$0.344456\pi$
$212$	−11.9827	−0.822973
$213$	0	0
$214$	−8.76689	−0.599292
$215$	−49.1659	−3.35309
$216$	0	0
$217$	−4.75876	−0.323046
$218$	4.02864	0.272854
$219$	0	0
$220$	−26.3035	−1.77338
$221$	−26.4187	−1.77711
$222$	0	0
$223$	−9.12924	−0.611339	−0.305670	−	0.952138i	$-0.598880\pi$
$223$	−9.12924	−0.611339	−0.305670	+	0.952138i	$0.598880\pi$
$224$	5.85628	0.391289
$225$	0	0
$226$	0.812862	0.0540708
$227$	−25.9253	−1.72072	−0.860362	−	0.509683i	$-0.829763\pi$
$227$	−25.9253	−1.72072	−0.860362	+	0.509683i	$0.829763\pi$
$228$	0	0
$229$	6.02046	0.397843	0.198922	−	0.980015i	$-0.436256\pi$
$229$	6.02046	0.397843	0.198922	+	0.980015i	$0.436256\pi$
$230$	5.10127	0.336368
$231$	0	0
$232$	−5.19042	−0.340767
$233$	2.47265	0.161989	0.0809943	−	0.996715i	$-0.474190\pi$
$233$	2.47265	0.161989	0.0809943	+	0.996715i	$0.474190\pi$
$234$	0	0
$235$	1.39142	0.0907661
$236$	15.9587	1.03882
$237$	0	0
$238$	4.96315	0.321713
$239$	−21.8465	−1.41313	−0.706567	−	0.707646i	$-0.749757\pi$
$239$	−21.8465	−1.41313	−0.706567	+	0.707646i	$0.749757\pi$
$240$	0	0
$241$	−7.16053	−0.461250	−0.230625	−	0.973043i	$-0.574077\pi$
$241$	−7.16053	−0.461250	−0.230625	+	0.973043i	$0.574077\pi$
$242$	7.21654	0.463897
$243$	0	0
$244$	−14.3855	−0.920936
$245$	−3.93264	−0.251247
$246$	0	0
$247$	−22.4028	−1.42545
$248$	12.0653	0.766148
$249$	0	0
$250$	−15.7306	−0.994889
$251$	10.6859	0.674488	0.337244	−	0.941417i	$-0.390505\pi$
$251$	10.6859	0.674488	0.337244	+	0.941417i	$0.390505\pi$
$252$	0	0
$253$	8.09552	0.508961
$254$	−0.731838	−0.0459196
$255$	0	0
$256$	−11.7044	−0.731522
$257$	16.4530	1.02631	0.513155	−	0.858296i	$-0.328477\pi$
$257$	16.4530	1.02631	0.513155	+	0.858296i	$0.328477\pi$
$258$	0	0
$259$	−3.62146	−0.225027
$260$	−22.4345	−1.39133
$261$	0	0
$262$	8.54361	0.527826
$263$	−5.82474	−0.359169	−0.179584	−	0.983743i	$-0.557475\pi$
$263$	−5.82474	−0.359169	−0.179584	+	0.983743i	$0.557475\pi$
$264$	0	0
$265$	−32.1792	−1.97675
$266$	4.20870	0.258052
$267$	0	0
$268$	−8.90437	−0.543921
$269$	15.3418	0.935406	0.467703	−	0.883886i	$-0.345082\pi$
$269$	15.3418	0.935406	0.467703	+	0.883886i	$0.345082\pi$
$270$	0	0
$271$	24.0137	1.45873	0.729365	−	0.684125i	$-0.239816\pi$
$271$	24.0137	1.45873	0.729365	+	0.684125i	$0.239816\pi$
$272$	7.27909	0.441359
$273$	0	0
$274$	−16.0295	−0.968378
$275$	−47.8007	−2.88249
$276$	0	0
$277$	−4.16090	−0.250004	−0.125002	−	0.992156i	$-0.539894\pi$
$277$	−4.16090	−0.250004	−0.125002	+	0.992156i	$0.539894\pi$
$278$	−13.9505	−0.836696
$279$	0	0
$280$	9.97078	0.595868
$281$	14.1868	0.846312	0.423156	−	0.906057i	$-0.360922\pi$
$281$	14.1868	0.846312	0.423156	+	0.906057i	$0.360922\pi$
$282$	0	0
$283$	−15.9011	−0.945225	−0.472612	−	0.881270i	$-0.656689\pi$
$283$	−15.9011	−0.945225	−0.472612	+	0.881270i	$0.656689\pi$
$284$	16.7181	0.992034
$285$	0	0
$286$	13.0212	0.769957
$287$	−3.27869	−0.193535
$288$	0	0
$289$	28.9923	1.70543
$290$	−5.89192	−0.345986
$291$	0	0
$292$	−2.69148	−0.157507
$293$	−7.33737	−0.428654	−0.214327	−	0.976762i	$-0.568756\pi$
$293$	−7.33737	−0.428654	−0.214327	+	0.976762i	$0.568756\pi$
$294$	0	0
$295$	42.8567	2.49521
$296$	9.18182	0.533682
$297$	0	0
$298$	1.42316	0.0824413
$299$	6.90474	0.399312
$300$	0	0
$301$	12.5020	0.720603
$302$	9.03315	0.519799
$303$	0	0
$304$	6.17259	0.354022
$305$	−38.6319	−2.21205
$306$	0	0
$307$	−25.7027	−1.46693	−0.733465	−	0.679727i	$-0.762098\pi$
$307$	−25.7027	−1.46693	−0.733465	+	0.679727i	$0.762098\pi$
$308$	6.68851	0.381113
$309$	0	0
$310$	13.6960	0.777880
$311$	−1.41386	−0.0801724	−0.0400862	−	0.999196i	$-0.512763\pi$
$311$	−1.41386	−0.0801724	−0.0400862	+	0.999196i	$0.512763\pi$
$312$	0	0
$313$	−11.7255	−0.662762	−0.331381	−	0.943497i	$-0.607514\pi$
$313$	−11.7255	−0.662762	−0.331381	+	0.943497i	$0.607514\pi$
$314$	2.64641	0.149345
$315$	0	0
$316$	10.8216	0.608763
$317$	9.44210	0.530321	0.265161	−	0.964204i	$-0.414575\pi$
$317$	9.44210	0.530321	0.265161	+	0.964204i	$0.414575\pi$
$318$	0	0
$319$	−9.35026	−0.523514
$320$	−8.41265	−0.470282
$321$	0	0
$322$	−1.29716	−0.0722879
$323$	39.0010	2.17007
$324$	0	0
$325$	−40.7696	−2.26149
$326$	7.32386	0.405631
$327$	0	0
$328$	8.31275	0.458995
$329$	−0.353812	−0.0195063
$330$	0	0
$331$	1.56053	0.0857743	0.0428871	−	0.999080i	$-0.486344\pi$
$331$	1.56053	0.0857743	0.0428871	+	0.999080i	$0.486344\pi$
$332$	−10.8776	−0.596983
$333$	0	0
$334$	−11.7040	−0.640412
$335$	−23.9124	−1.30648
$336$	0	0
$337$	−4.82064	−0.262597	−0.131298	−	0.991343i	$-0.541915\pi$
$337$	−4.82064	−0.262597	−0.131298	+	0.991343i	$0.541915\pi$
$338$	1.59198	0.0865922
$339$	0	0
$340$	39.0563	2.11812
$341$	21.7350	1.17702
$342$	0	0
$343$	1.00000	0.0539949
$344$	−31.6974	−1.70901
$345$	0	0
$346$	−5.68443	−0.305597
$347$	7.89995	0.424092	0.212046	−	0.977260i	$-0.431987\pi$
$347$	7.89995	0.424092	0.212046	+	0.977260i	$0.431987\pi$
$348$	0	0
$349$	−25.0702	−1.34198	−0.670988	−	0.741468i	$-0.734130\pi$
$349$	−25.0702	−1.34198	−0.670988	+	0.741468i	$0.734130\pi$
$350$	7.65919	0.409401
$351$	0	0
$352$	−26.7478	−1.42566
$353$	3.30600	0.175961	0.0879804	−	0.996122i	$-0.471959\pi$
$353$	3.30600	0.175961	0.0879804	+	0.996122i	$0.471959\pi$
$354$	0	0
$355$	44.8959	2.38283
$356$	−9.92018	−0.525768
$357$	0	0
$358$	17.2905	0.913832
$359$	−25.7381	−1.35840	−0.679202	−	0.733952i	$-0.737674\pi$
$359$	−25.7381	−1.35840	−0.679202	+	0.733952i	$0.737674\pi$
$360$	0	0
$361$	14.0724	0.740654
$362$	16.1005	0.846223
$363$	0	0
$364$	5.70470	0.299007
$365$	−7.22791	−0.378326
$366$	0	0
$367$	−0.640561	−0.0334370	−0.0167185	−	0.999860i	$-0.505322\pi$
$367$	−0.640561	−0.0334370	−0.0167185	+	0.999860i	$0.505322\pi$
$368$	−1.90245	−0.0991720
$369$	0	0
$370$	10.4228	0.541855
$371$	8.18257	0.424818
$372$	0	0
$373$	5.16876	0.267628	0.133814	−	0.991006i	$-0.457277\pi$
$373$	5.16876	0.267628	0.133814	+	0.991006i	$0.457277\pi$
$374$	−22.6685	−1.17216
$375$	0	0
$376$	0.897052	0.0462619
$377$	−7.97492	−0.410729
$378$	0	0
$379$	−33.1015	−1.70031	−0.850156	−	0.526531i	$-0.823493\pi$
$379$	−33.1015	−1.70031	−0.850156	+	0.526531i	$0.823493\pi$
$380$	33.1193	1.69898
$381$	0	0
$382$	−17.3103	−0.885672
$383$	22.3539	1.14223	0.571116	−	0.820869i	$-0.306510\pi$
$383$	22.3539	1.14223	0.571116	+	0.820869i	$0.306510\pi$
$384$	0	0
$385$	17.9618	0.915420
$386$	−10.6465	−0.541893
$387$	0	0
$388$	−16.0694	−0.815801
$389$	13.3831	0.678551	0.339275	−	0.940687i	$-0.389818\pi$
$389$	13.3831	0.678551	0.339275	+	0.940687i	$0.389818\pi$
$390$	0	0
$391$	−12.0205	−0.607901
$392$	−2.53539	−0.128056
$393$	0	0
$394$	−6.49085	−0.327005
$395$	29.0612	1.46223
$396$	0	0
$397$	18.3190	0.919404	0.459702	−	0.888073i	$-0.347956\pi$
$397$	18.3190	0.919404	0.459702	+	0.888073i	$0.347956\pi$
$398$	−10.1617	−0.509359
$399$	0	0
$400$	11.2332	0.561658
$401$	−26.1476	−1.30575	−0.652873	−	0.757467i	$-0.726437\pi$
$401$	−26.1476	−1.30575	−0.652873	+	0.757467i	$0.726437\pi$
$402$	0	0
$403$	18.5380	0.923443
$404$	5.53443	0.275348
$405$	0	0
$406$	1.49821	0.0743549
$407$	16.5406	0.819885
$408$	0	0
$409$	20.8085	1.02891	0.514457	−	0.857516i	$-0.327994\pi$
$409$	20.8085	1.02891	0.514457	+	0.857516i	$0.327994\pi$
$410$	9.43626	0.466024
$411$	0	0
$412$	12.9551	0.638252
$413$	−10.8977	−0.536240
$414$	0	0
$415$	−29.2114	−1.43393
$416$	−22.8134	−1.11852
$417$	0	0
$418$	−19.2227	−0.940212
$419$	37.7760	1.84548	0.922740	−	0.385423i	$-0.125944\pi$
$419$	37.7760	1.84548	0.922740	+	0.385423i	$0.125944\pi$
$420$	0	0
$421$	−1.29583	−0.0631551	−0.0315776	−	0.999501i	$-0.510053\pi$
$421$	−1.29583	−0.0631551	−0.0315776	+	0.999501i	$0.510053\pi$
$422$	9.98082	0.485859
$423$	0	0
$424$	−20.7460	−1.00752
$425$	70.9758	3.44283
$426$	0	0
$427$	9.82338	0.475387
$428$	17.5426	0.847954
$429$	0	0
$430$	−35.9815	−1.73518
$431$	10.8722	0.523694	0.261847	−	0.965109i	$-0.415668\pi$
$431$	10.8722	0.523694	0.261847	+	0.965109i	$0.415668\pi$
$432$	0	0
$433$	−11.3839	−0.547075	−0.273538	−	0.961861i	$-0.588194\pi$
$433$	−11.3839	−0.547075	−0.273538	+	0.961861i	$0.588194\pi$
$434$	−3.48264	−0.167172
$435$	0	0
$436$	−8.06135	−0.386069
$437$	−10.1932	−0.487608
$438$	0	0
$439$	8.55817	0.408459	0.204230	−	0.978923i	$-0.434531\pi$
$439$	8.55817	0.408459	0.204230	+	0.978923i	$0.434531\pi$
$440$	−45.5402	−2.17105
$441$	0	0
$442$	−19.3342	−0.919634
$443$	20.8475	0.990493	0.495246	−	0.868753i	$-0.335078\pi$
$443$	20.8475	0.990493	0.495246	+	0.868753i	$0.335078\pi$
$444$	0	0
$445$	−26.6404	−1.26288
$446$	−6.68112	−0.316360
$447$	0	0
$448$	2.13919	0.101067
$449$	24.0389	1.13447	0.567233	−	0.823558i	$-0.308014\pi$
$449$	24.0389	1.13447	0.567233	+	0.823558i	$0.308014\pi$
$450$	0	0
$451$	14.9750	0.705144
$452$	−1.62654	−0.0765061
$453$	0	0
$454$	−18.9731	−0.890454
$455$	15.3198	0.718204
$456$	0	0
$457$	−16.8458	−0.788013	−0.394007	−	0.919108i	$-0.628911\pi$
$457$	−16.8458	−0.788013	−0.394007	+	0.919108i	$0.628911\pi$
$458$	4.40600	0.205879
$459$	0	0
$460$	−10.2077	−0.475935
$461$	2.07046	0.0964310	0.0482155	−	0.998837i	$-0.484647\pi$
$461$	2.07046	0.0964310	0.0482155	+	0.998837i	$0.484647\pi$
$462$	0	0
$463$	−4.79150	−0.222680	−0.111340	−	0.993782i	$-0.535514\pi$
$463$	−4.79150	−0.222680	−0.111340	+	0.993782i	$0.535514\pi$
$464$	2.19731	0.102008
$465$	0	0
$466$	1.80958	0.0838271
$467$	16.9848	0.785961	0.392980	−	0.919547i	$-0.371444\pi$
$467$	16.9848	0.785961	0.392980	+	0.919547i	$0.371444\pi$
$468$	0	0
$469$	6.08050	0.280771
$470$	1.01829	0.0469704
$471$	0	0
$472$	27.6299	1.27177
$473$	−57.1012	−2.62552
$474$	0	0
$475$	60.1868	2.76156
$476$	−9.93130	−0.455200
$477$	0	0
$478$	−15.9881	−0.731279
$479$	−19.0396	−0.869942	−0.434971	−	0.900444i	$-0.643241\pi$
$479$	−19.0396	−0.869942	−0.434971	+	0.900444i	$0.643241\pi$
$480$	0	0
$481$	14.1076	0.643251
$482$	−5.24035	−0.238691
$483$	0	0
$484$	−14.4404	−0.656380
$485$	−43.1540	−1.95952
$486$	0	0
$487$	−23.9169	−1.08378	−0.541890	−	0.840450i	$-0.682291\pi$
$487$	−23.9169	−1.08378	−0.541890	+	0.840450i	$0.682291\pi$
$488$	−24.9061	−1.12745
$489$	0	0
$490$	−2.87806	−0.130017
$491$	−38.6421	−1.74390	−0.871948	−	0.489599i	$-0.837143\pi$
$491$	−38.6421	−1.74390	−0.871948	+	0.489599i	$0.837143\pi$
$492$	0	0
$493$	13.8835	0.625283
$494$	−16.3952	−0.737655
$495$	0	0
$496$	−5.10773	−0.229344
$497$	−11.4162	−0.512087
$498$	0	0
$499$	28.3037	1.26705	0.633523	−	0.773724i	$-0.281608\pi$
$499$	28.3037	1.26705	0.633523	+	0.773724i	$0.281608\pi$
$500$	31.4770	1.40769
$501$	0	0
$502$	7.82034	0.349039
$503$	−17.3592	−0.774010	−0.387005	−	0.922078i	$-0.626490\pi$
$503$	−17.3592	−0.774010	−0.387005	+	0.922078i	$0.626490\pi$
$504$	0	0
$505$	14.8626	0.661376
$506$	5.92461	0.263381
$507$	0	0
$508$	1.46441	0.0649728
$509$	30.8979	1.36952	0.684762	−	0.728767i	$-0.259906\pi$
$509$	30.8979	1.36952	0.684762	+	0.728767i	$0.259906\pi$
$510$	0	0
$511$	1.83793	0.0813051
$512$	11.7284	0.518326
$513$	0	0
$514$	12.0409	0.531103
$515$	34.7906	1.53306
$516$	0	0
$517$	1.61599	0.0710712
$518$	−2.65032	−0.116449
$519$	0	0
$520$	−38.8417	−1.70332
$521$	−8.47859	−0.371454	−0.185727	−	0.982601i	$-0.559464\pi$
$521$	−8.47859	−0.371454	−0.185727	+	0.982601i	$0.559464\pi$
$522$	0	0
$523$	29.9260	1.30857	0.654286	−	0.756247i	$-0.272969\pi$
$523$	29.9260	1.30857	0.654286	+	0.756247i	$0.272969\pi$
$524$	−17.0958	−0.746835
$525$	0	0
$526$	−4.26276	−0.185865
$527$	−32.2728	−1.40582
$528$	0	0
$529$	−19.8584	−0.863407
$530$	−23.5499	−1.02294
$531$	0	0
$532$	−8.42164	−0.365124
$533$	12.7723	0.553230
$534$	0	0
$535$	47.1102	2.03675
$536$	−15.4164	−0.665889
$537$	0	0
$538$	11.2277	0.484061
$539$	−4.56737	−0.196730
$540$	0	0
$541$	−10.1813	−0.437728	−0.218864	−	0.975755i	$-0.570235\pi$
$541$	−10.1813	−0.437728	−0.218864	+	0.975755i	$0.570235\pi$
$542$	17.5742	0.754875
$543$	0	0
$544$	39.7159	1.70281
$545$	−21.6485	−0.927322
$546$	0	0
$547$	−29.5473	−1.26335	−0.631675	−	0.775233i	$-0.717632\pi$
$547$	−29.5473	−1.26335	−0.631675	+	0.775233i	$0.717632\pi$
$548$	32.0752	1.37018
$549$	0	0
$550$	−34.9823	−1.49165
$551$	11.7731	0.501551
$552$	0	0
$553$	−7.38973	−0.314243
$554$	−3.04510	−0.129374
$555$	0	0
$556$	27.9151	1.18386
$557$	−15.4341	−0.653963	−0.326981	−	0.945031i	$-0.606031\pi$
$557$	−15.4341	−0.653963	−0.326981	+	0.945031i	$0.606031\pi$
$558$	0	0
$559$	−48.7022	−2.05988
$560$	−4.22104	−0.178371
$561$	0	0
$562$	10.3824	0.437956
$563$	41.3758	1.74378	0.871890	−	0.489702i	$-0.162894\pi$
$563$	41.3758	1.74378	0.871890	+	0.489702i	$0.162894\pi$
$564$	0	0
$565$	−4.36804	−0.183765
$566$	−11.6371	−0.489142
$567$	0	0
$568$	28.9445	1.21449
$569$	13.4486	0.563796	0.281898	−	0.959444i	$-0.409036\pi$
$569$	13.4486	0.563796	0.281898	+	0.959444i	$0.409036\pi$
$570$	0	0
$571$	−10.6397	−0.445258	−0.222629	−	0.974903i	$-0.571464\pi$
$571$	−10.6397	−0.445258	−0.222629	+	0.974903i	$0.571464\pi$
$572$	−26.0554	−1.08943
$573$	0	0
$574$	−2.39947	−0.100152
$575$	−18.5501	−0.773593
$576$	0	0
$577$	9.16461	0.381528	0.190764	−	0.981636i	$-0.438904\pi$
$577$	9.16461	0.381528	0.190764	+	0.981636i	$0.438904\pi$
$578$	21.2177	0.882538
$579$	0	0
$580$	11.7898	0.489544
$581$	7.42793	0.308162
$582$	0	0
$583$	−37.3728	−1.54782
$584$	−4.65986	−0.192826
$585$	0	0
$586$	−5.36977	−0.221823
$587$	−8.98235	−0.370741	−0.185371	−	0.982669i	$-0.559349\pi$
$587$	−8.98235	−0.370741	−0.185371	+	0.982669i	$0.559349\pi$
$588$	0	0
$589$	−27.3670	−1.12764
$590$	31.3642	1.29124
$591$	0	0
$592$	−3.88704	−0.159756
$593$	10.5609	0.433683	0.216842	−	0.976207i	$-0.430425\pi$
$593$	10.5609	0.433683	0.216842	+	0.976207i	$0.430425\pi$
$594$	0	0
$595$	−26.6703	−1.09337
$596$	−2.84775	−0.116648
$597$	0	0
$598$	5.05315	0.206639
$599$	−12.1190	−0.495169	−0.247585	−	0.968866i	$-0.579637\pi$
$599$	−12.1190	−0.495169	−0.247585	+	0.968866i	$0.579637\pi$
$600$	0	0
$601$	−29.2199	−1.19190	−0.595952	−	0.803020i	$-0.703225\pi$
$601$	−29.2199	−1.19190	−0.595952	+	0.803020i	$0.703225\pi$
$602$	9.14944	0.372903
$603$	0	0
$604$	−18.0754	−0.735477
$605$	−38.7792	−1.57660
$606$	0	0
$607$	−23.3541	−0.947916	−0.473958	−	0.880548i	$-0.657175\pi$
$607$	−23.3541	−0.947916	−0.473958	+	0.880548i	$0.657175\pi$
$608$	33.6787	1.36585
$609$	0	0
$610$	−28.2723	−1.14471
$611$	1.37829	0.0557598
$612$	0	0
$613$	0.913187	0.0368833	0.0184416	−	0.999830i	$-0.494130\pi$
$613$	0.913187	0.0368833	0.0184416	+	0.999830i	$0.494130\pi$
$614$	−18.8102	−0.759118
$615$	0	0
$616$	11.5801	0.466574
$617$	5.60825	0.225780	0.112890	−	0.993608i	$-0.463989\pi$
$617$	5.60825	0.225780	0.112890	+	0.993608i	$0.463989\pi$
$618$	0	0
$619$	−4.38738	−0.176344	−0.0881718	−	0.996105i	$-0.528102\pi$
$619$	−4.38738	−0.176344	−0.0881718	+	0.996105i	$0.528102\pi$
$620$	−27.4058	−1.10064
$621$	0	0
$622$	−1.03471	−0.0414882
$623$	6.77417	0.271401
$624$	0	0
$625$	32.2022	1.28809
$626$	−8.58113	−0.342971
$627$	0	0
$628$	−5.29548	−0.211313
$629$	−24.5599	−0.979268
$630$	0	0
$631$	−6.10432	−0.243009	−0.121505	−	0.992591i	$-0.538772\pi$
$631$	−6.10432	−0.243009	−0.121505	+	0.992591i	$0.538772\pi$
$632$	18.7358	0.745271
$633$	0	0
$634$	6.91009	0.274435
$635$	3.93264	0.156062
$636$	0	0
$637$	−3.89555	−0.154347
$638$	−6.84287	−0.270912
$639$	0	0
$640$	39.9047	1.57737
$641$	−23.1532	−0.914496	−0.457248	−	0.889339i	$-0.651165\pi$
$641$	−23.1532	−0.914496	−0.457248	+	0.889339i	$0.651165\pi$
$642$	0	0
$643$	7.44468	0.293589	0.146795	−	0.989167i	$-0.453104\pi$
$643$	7.44468	0.293589	0.146795	+	0.989167i	$0.453104\pi$
$644$	2.59563	0.102282
$645$	0	0
$646$	28.5424	1.12299
$647$	19.7448	0.776248	0.388124	−	0.921607i	$-0.373123\pi$
$647$	19.7448	0.776248	0.388124	+	0.921607i	$0.373123\pi$
$648$	0	0
$649$	49.7737	1.95379
$650$	−29.8368	−1.17029
$651$	0	0
$652$	−14.6551	−0.573938
$653$	−4.45874	−0.174484	−0.0872419	−	0.996187i	$-0.527805\pi$
$653$	−4.45874	−0.174484	−0.0872419	+	0.996187i	$0.527805\pi$
$654$	0	0
$655$	−45.9104	−1.79387
$656$	−3.51912	−0.137399
$657$	0	0
$658$	−0.258933	−0.0100943
$659$	−20.3543	−0.792890	−0.396445	−	0.918059i	$-0.629756\pi$
$659$	−20.3543	−0.792890	−0.396445	+	0.918059i	$0.629756\pi$
$660$	0	0
$661$	−34.8480	−1.35543	−0.677715	−	0.735325i	$-0.737030\pi$
$661$	−34.8480	−1.35543	−0.677715	+	0.735325i	$0.737030\pi$
$662$	1.14205	0.0443871
$663$	0	0
$664$	−18.8327	−0.730850
$665$	−22.6161	−0.877015
$666$	0	0
$667$	−3.62858	−0.140499
$668$	23.4197	0.906136
$669$	0	0
$670$	−17.5000	−0.676085
$671$	−44.8670	−1.73207
$672$	0	0
$673$	−4.10381	−0.158190	−0.0790952	−	0.996867i	$-0.525203\pi$
$673$	−4.10381	−0.158190	−0.0790952	+	0.996867i	$0.525203\pi$
$674$	−3.52793	−0.135891
$675$	0	0
$676$	−3.18556	−0.122522
$677$	−8.02022	−0.308242	−0.154121	−	0.988052i	$-0.549255\pi$
$677$	−8.02022	−0.308242	−0.154121	+	0.988052i	$0.549255\pi$
$678$	0	0
$679$	10.9733	0.421116
$680$	67.6195	2.59309
$681$	0	0
$682$	15.9065	0.609091
$683$	3.56422	0.136381	0.0681906	−	0.997672i	$-0.478277\pi$
$683$	3.56422	0.136381	0.0681906	+	0.997672i	$0.478277\pi$
$684$	0	0
$685$	86.1371	3.29113
$686$	0.731838	0.0279417
$687$	0	0
$688$	13.4188	0.511587
$689$	−31.8756	−1.21436
$690$	0	0
$691$	30.6852	1.16732	0.583660	−	0.811998i	$-0.301620\pi$
$691$	30.6852	1.16732	0.583660	+	0.811998i	$0.301620\pi$
$692$	11.3746	0.432397
$693$	0	0
$694$	5.78149	0.219462
$695$	74.9652	2.84359
$696$	0	0
$697$	−22.2353	−0.842222
$698$	−18.3473	−0.694456
$699$	0	0
$700$	−15.3261	−0.579272
$701$	−34.2834	−1.29487	−0.647434	−	0.762122i	$-0.724158\pi$
$701$	−34.2834	−1.29487	−0.647434	+	0.762122i	$0.724158\pi$
$702$	0	0
$703$	−20.8265	−0.785488
$704$	−9.77045	−0.368238
$705$	0	0
$706$	2.41946	0.0910575
$707$	−3.77928	−0.142135
$708$	0	0
$709$	−50.8968	−1.91147	−0.955735	−	0.294229i	$-0.904937\pi$
$709$	−50.8968	−1.91147	−0.955735	+	0.294229i	$0.904937\pi$
$710$	32.8565	1.23308
$711$	0	0
$712$	−17.1751	−0.643666
$713$	8.43475	0.315884
$714$	0	0
$715$	−69.9712	−2.61678
$716$	−34.5984	−1.29300
$717$	0	0
$718$	−18.8361	−0.702957
$719$	−4.88651	−0.182236	−0.0911181	−	0.995840i	$-0.529044\pi$
$719$	−4.88651	−0.182236	−0.0911181	+	0.995840i	$0.529044\pi$
$720$	0	0
$721$	−8.84662	−0.329465
$722$	10.2987	0.383279
$723$	0	0
$724$	−32.2172	−1.19734
$725$	21.4252	0.795713
$726$	0	0
$727$	−17.8485	−0.661963	−0.330981	−	0.943637i	$-0.607380\pi$
$727$	−17.8485	−0.661963	−0.330981	+	0.943637i	$0.607380\pi$
$728$	9.87674	0.366056
$729$	0	0
$730$	−5.28966	−0.195779
$731$	84.7856	3.13591
$732$	0	0
$733$	50.4742	1.86431	0.932153	−	0.362065i	$-0.117928\pi$
$733$	50.4742	1.86431	0.932153	+	0.362065i	$0.117928\pi$
$734$	−0.468787	−0.0173032
$735$	0	0
$736$	−10.3801	−0.382615
$737$	−27.7719	−1.02299
$738$	0	0
$739$	−41.4949	−1.52642	−0.763208	−	0.646153i	$-0.776377\pi$
$739$	−41.4949	−1.52642	−0.763208	+	0.646153i	$0.776377\pi$
$740$	−20.8561	−0.766684
$741$	0	0
$742$	5.98832	0.219838
$743$	−48.8699	−1.79286	−0.896431	−	0.443183i	$-0.853849\pi$
$743$	−48.8699	−1.79286	−0.896431	+	0.443183i	$0.853849\pi$
$744$	0	0
$745$	−7.64755	−0.280185
$746$	3.78269	0.138494
$747$	0	0
$748$	45.3599	1.65852
$749$	−11.9793	−0.437713
$750$	0	0
$751$	28.2391	1.03046	0.515230	−	0.857052i	$-0.327706\pi$
$751$	28.2391	1.03046	0.515230	+	0.857052i	$0.327706\pi$
$752$	−0.379758	−0.0138484
$753$	0	0
$754$	−5.83635	−0.212547
$755$	−48.5410	−1.76659
$756$	0	0
$757$	−26.4750	−0.962249	−0.481124	−	0.876652i	$-0.659771\pi$
$757$	−26.4750	−0.962249	−0.481124	+	0.876652i	$0.659771\pi$
$758$	−24.2250	−0.879890
$759$	0	0
$760$	57.3406	2.07996
$761$	7.30535	0.264819	0.132409	−	0.991195i	$-0.457729\pi$
$761$	7.30535	0.264819	0.132409	+	0.991195i	$0.457729\pi$
$762$	0	0
$763$	5.50483	0.199288
$764$	34.6380	1.25316
$765$	0	0
$766$	16.3595	0.591091
$767$	42.4525	1.53287
$768$	0	0
$769$	32.7720	1.18179	0.590894	−	0.806749i	$-0.298775\pi$
$769$	32.7720	1.18179	0.590894	+	0.806749i	$0.298775\pi$
$770$	13.1452	0.473718
$771$	0	0
$772$	21.3037	0.766739
$773$	−16.5359	−0.594754	−0.297377	−	0.954760i	$-0.596112\pi$
$773$	−16.5359	−0.594754	−0.297377	+	0.954760i	$0.596112\pi$
$774$	0	0
$775$	−49.8037	−1.78900
$776$	−27.8215	−0.998735
$777$	0	0
$778$	9.79427	0.351141
$779$	−18.8553	−0.675561
$780$	0	0
$781$	52.1420	1.86579
$782$	−8.79703	−0.314581
$783$	0	0
$784$	1.07333	0.0383333
$785$	−14.2209	−0.507565
$786$	0	0
$787$	−13.9918	−0.498753	−0.249376	−	0.968407i	$-0.580226\pi$
$787$	−13.9918	−0.498753	−0.249376	+	0.968407i	$0.580226\pi$
$788$	12.9882	0.462687
$789$	0	0
$790$	21.2681	0.756684
$791$	1.11071	0.0394924
$792$	0	0
$793$	−38.2675	−1.35892
$794$	13.4065	0.475780
$795$	0	0
$796$	20.3336	0.720705
$797$	−3.67638	−0.130224	−0.0651121	−	0.997878i	$-0.520741\pi$
$797$	−3.67638	−0.130224	−0.0651121	+	0.997878i	$0.520741\pi$
$798$	0	0
$799$	−2.39947	−0.0848872
$800$	61.2901	2.16693
$801$	0	0
$802$	−19.1358	−0.675708
$803$	−8.39449	−0.296235
$804$	0	0
$805$	6.97049	0.245677
$806$	13.5668	0.477870
$807$	0	0
$808$	9.58195	0.337092
$809$	17.2258	0.605628	0.302814	−	0.953050i	$-0.402074\pi$
$809$	17.2258	0.605628	0.302814	+	0.953050i	$0.402074\pi$
$810$	0	0
$811$	−22.5686	−0.792491	−0.396245	−	0.918145i	$-0.629687\pi$
$811$	−22.5686	−0.792491	−0.396245	+	0.918145i	$0.629687\pi$
$812$	−2.99793	−0.105207
$813$	0	0
$814$	12.1050	0.424280
$815$	−39.3559	−1.37858
$816$	0	0
$817$	71.8973	2.51537
$818$	15.2284	0.532450
$819$	0	0
$820$	−18.8820	−0.659389
$821$	41.6913	1.45504	0.727518	−	0.686089i	$-0.240674\pi$
$821$	41.6913	1.45504	0.727518	+	0.686089i	$0.240674\pi$
$822$	0	0
$823$	43.0148	1.49940	0.749701	−	0.661777i	$-0.230197\pi$
$823$	43.0148	1.49940	0.749701	+	0.661777i	$0.230197\pi$
$824$	22.4296	0.781373
$825$	0	0
$826$	−7.97533	−0.277497
$827$	37.9685	1.32029	0.660147	−	0.751136i	$-0.270494\pi$
$827$	37.9685	1.32029	0.660147	+	0.751136i	$0.270494\pi$
$828$	0	0
$829$	2.20643	0.0766324	0.0383162	−	0.999266i	$-0.487801\pi$
$829$	2.20643	0.0766324	0.0383162	+	0.999266i	$0.487801\pi$
$830$	−21.3780	−0.742041
$831$	0	0
$832$	−8.33330	−0.288905
$833$	6.78176	0.234974
$834$	0	0
$835$	62.8931	2.17650
$836$	38.4647	1.33033
$837$	0	0
$838$	27.6459	0.955013
$839$	−45.0290	−1.55457	−0.777287	−	0.629147i	$-0.783405\pi$
$839$	−45.0290	−1.55457	−0.777287	+	0.629147i	$0.783405\pi$
$840$	0	0
$841$	−24.8090	−0.855484
$842$	−0.948341	−0.0326820
$843$	0	0
$844$	−19.9717	−0.687454
$845$	−8.55474	−0.294292
$846$	0	0
$847$	9.86085	0.338823
$848$	8.78262	0.301597
$849$	0	0
$850$	51.9428	1.78162
$851$	6.41893	0.220038
$852$	0	0
$853$	−34.0997	−1.16755	−0.583776	−	0.811915i	$-0.698425\pi$
$853$	−34.0997	−1.16755	−0.583776	+	0.811915i	$0.698425\pi$
$854$	7.18912	0.246007
$855$	0	0
$856$	30.3721	1.03810
$857$	40.1638	1.37197	0.685984	−	0.727617i	$-0.259372\pi$
$857$	40.1638	1.37197	0.685984	+	0.727617i	$0.259372\pi$
$858$	0	0
$859$	−32.9917	−1.12566	−0.562831	−	0.826572i	$-0.690288\pi$
$859$	−32.9917	−1.12566	−0.562831	+	0.826572i	$0.690288\pi$
$860$	71.9992	2.45515
$861$	0	0
$862$	7.95667	0.271005
$863$	−17.4516	−0.594060	−0.297030	−	0.954868i	$-0.595996\pi$
$863$	−17.4516	−0.594060	−0.297030	+	0.954868i	$0.595996\pi$
$864$	0	0
$865$	30.5462	1.03860
$866$	−8.33117	−0.283105
$867$	0	0
$868$	6.96879	0.236536
$869$	33.7516	1.14494
$870$	0	0
$871$	−23.6869	−0.802600
$872$	−13.9569	−0.472640
$873$	0	0
$874$	−7.45979	−0.252331
$875$	−21.4946	−0.726651
$876$	0	0
$877$	−48.5747	−1.64025	−0.820125	−	0.572184i	$-0.806096\pi$
$877$	−48.5747	−1.64025	−0.820125	+	0.572184i	$0.806096\pi$
$878$	6.26320	0.211373
$879$	0	0
$880$	19.2790	0.649896
$881$	13.9540	0.470123	0.235062	−	0.971980i	$-0.424471\pi$
$881$	13.9540	0.470123	0.235062	+	0.971980i	$0.424471\pi$
$882$	0	0
$883$	−12.7927	−0.430509	−0.215255	−	0.976558i	$-0.569058\pi$
$883$	−12.7927	−0.430509	−0.215255	+	0.976558i	$0.569058\pi$
$884$	38.6879	1.30121
$885$	0	0
$886$	15.2570	0.512568
$887$	18.7133	0.628332	0.314166	−	0.949368i	$-0.398275\pi$
$887$	18.7133	0.628332	0.314166	+	0.949368i	$0.398275\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−19.4964	−0.653522
$891$	0	0
$892$	13.3690	0.447627
$893$	−2.03473	−0.0680895
$894$	0	0
$895$	−92.9132	−3.10575
$896$	−10.1470	−0.338988
$897$	0	0
$898$	17.5926	0.587072
$899$	−9.74207	−0.324916
$900$	0	0
$901$	55.4923	1.84872
$902$	10.9593	0.364903
$903$	0	0
$904$	−2.81609	−0.0936617
$905$	−86.5184	−2.87597
$906$	0	0
$907$	−10.2333	−0.339791	−0.169896	−	0.985462i	$-0.554343\pi$
$907$	−10.2333	−0.339791	−0.169896	+	0.985462i	$0.554343\pi$
$908$	37.9654	1.25993
$909$	0	0
$910$	11.2116	0.371662
$911$	−6.73786	−0.223235	−0.111618	−	0.993751i	$-0.535603\pi$
$911$	−6.73786	−0.223235	−0.111618	+	0.993751i	$0.535603\pi$
$912$	0	0
$913$	−33.9261	−1.12279
$914$	−12.3284	−0.407787
$915$	0	0
$916$	−8.81645	−0.291304
$917$	11.6742	0.385516
$918$	0	0
$919$	14.4499	0.476658	0.238329	−	0.971184i	$-0.423400\pi$
$919$	14.4499	0.476658	0.238329	+	0.971184i	$0.423400\pi$
$920$	−17.6729	−0.582658
$921$	0	0
$922$	1.51524	0.0499019
$923$	44.4724	1.46383
$924$	0	0
$925$	−37.9011	−1.24618
$926$	−3.50660	−0.115234
$927$	0	0
$928$	11.9889	0.393555
$929$	11.8627	0.389203	0.194601	−	0.980882i	$-0.437659\pi$
$929$	11.8627	0.389203	0.194601	+	0.980882i	$0.437659\pi$
$930$	0	0
$931$	5.75086	0.188477
$932$	−3.62098	−0.118609
$933$	0	0
$934$	12.4301	0.406725
$935$	121.813	3.98371
$936$	0	0
$937$	−9.41053	−0.307429	−0.153714	−	0.988115i	$-0.549124\pi$
$937$	−9.41053	−0.307429	−0.153714	+	0.988115i	$0.549124\pi$
$938$	4.44994	0.145296
$939$	0	0
$940$	−2.03761	−0.0664596
$941$	17.2696	0.562974	0.281487	−	0.959565i	$-0.409172\pi$
$941$	17.2696	0.562974	0.281487	+	0.959565i	$0.409172\pi$
$942$	0	0
$943$	5.81138	0.189244
$944$	−11.6968	−0.380700
$945$	0	0
$946$	−41.7888	−1.35867
$947$	−12.8026	−0.416030	−0.208015	−	0.978126i	$-0.566700\pi$
$947$	−12.8026	−0.416030	−0.208015	+	0.978126i	$0.566700\pi$
$948$	0	0
$949$	−7.15973	−0.232415
$950$	44.0470	1.42907
$951$	0	0
$952$	−17.1944	−0.557274
$953$	−33.1957	−1.07531	−0.537657	−	0.843164i	$-0.680690\pi$
$953$	−33.1957	−1.07531	−0.537657	+	0.843164i	$0.680690\pi$
$954$	0	0
$955$	93.0195	3.01004
$956$	31.9923	1.03471
$957$	0	0
$958$	−13.9339	−0.450184
$959$	−21.9031	−0.707288
$960$	0	0
$961$	−8.35421	−0.269490
$962$	10.3245	0.332874
$963$	0	0
$964$	10.4860	0.337731
$965$	57.2107	1.84168
$966$	0	0
$967$	41.4000	1.33133	0.665667	−	0.746249i	$-0.268147\pi$
$967$	41.4000	1.33133	0.665667	+	0.746249i	$0.268147\pi$
$968$	−25.0011	−0.803566
$969$	0	0
$970$	−31.5817	−1.01403
$971$	−28.4833	−0.914072	−0.457036	−	0.889448i	$-0.651089\pi$
$971$	−28.4833	−0.914072	−0.457036	+	0.889448i	$0.651089\pi$
$972$	0	0
$973$	−19.0623	−0.611109
$974$	−17.5033	−0.560842
$975$	0	0
$976$	10.5438	0.337497
$977$	−43.3674	−1.38745	−0.693723	−	0.720242i	$-0.744031\pi$
$977$	−43.3674	−1.38745	−0.693723	+	0.720242i	$0.744031\pi$
$978$	0	0
$979$	−30.9401	−0.988850
$980$	5.75902	0.183965
$981$	0	0
$982$	−28.2798	−0.902444
$983$	−18.7401	−0.597715	−0.298857	−	0.954298i	$-0.596606\pi$
$983$	−18.7401	−0.597715	−0.298857	+	0.954298i	$0.596606\pi$
$984$	0	0
$985$	34.8796	1.11136
$986$	10.1605	0.323576
$987$	0	0
$988$	32.8069	1.04373
$989$	−22.1594	−0.704628
$990$	0	0
$991$	45.6186	1.44912	0.724561	−	0.689211i	$-0.242043\pi$
$991$	45.6186	1.44912	0.724561	+	0.689211i	$0.242043\pi$
$992$	−27.8686	−0.884830
$993$	0	0
$994$	−8.35482	−0.264999
$995$	54.6053	1.73110
$996$	0	0
$997$	33.1661	1.05038	0.525190	−	0.850985i	$-0.323994\pi$
$997$	33.1661	1.05038	0.525190	+	0.850985i	$0.323994\pi$
$998$	20.7137	0.655681
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.15		20
3.2	odd	2		889.2.a.d.1.6	✓	20
21.20	even	2		6223.2.a.l.1.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.6	✓	20	3.2	odd	2
6223.2.a.l.1.6		20	21.20	even	2
8001.2.a.w.1.15		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.731838 q^{2} -1.46441 q^{4} -3.93264 q^{5} +1.00000 q^{7} -2.53539 q^{8} +O(q^{10})\)	\(q+0.731838 q^{2} -1.46441 q^{4} -3.93264 q^{5} +1.00000 q^{7} -2.53539 q^{8} -2.87806 q^{10} -4.56737 q^{11} -3.89555 q^{13} +0.731838 q^{14} +1.07333 q^{16} +6.78176 q^{17} +5.75086 q^{19} +5.75902 q^{20} -3.34257 q^{22} -1.77247 q^{23} +10.4657 q^{25} -2.85091 q^{26} -1.46441 q^{28} +2.04719 q^{29} -4.75876 q^{31} +5.85628 q^{32} +4.96315 q^{34} -3.93264 q^{35} -3.62146 q^{37} +4.20870 q^{38} +9.97078 q^{40} -3.27869 q^{41} +12.5020 q^{43} +6.68851 q^{44} -1.29716 q^{46} -0.353812 q^{47} +1.00000 q^{49} +7.65919 q^{50} +5.70470 q^{52} +8.18257 q^{53} +17.9618 q^{55} -2.53539 q^{56} +1.49821 q^{58} -10.8977 q^{59} +9.82338 q^{61} -3.48264 q^{62} +2.13919 q^{64} +15.3198 q^{65} +6.08050 q^{67} -9.93130 q^{68} -2.87806 q^{70} -11.4162 q^{71} +1.83793 q^{73} -2.65032 q^{74} -8.42164 q^{76} -4.56737 q^{77} -7.38973 q^{79} -4.22104 q^{80} -2.39947 q^{82} +7.42793 q^{83} -26.6703 q^{85} +9.14944 q^{86} +11.5801 q^{88} +6.77417 q^{89} -3.89555 q^{91} +2.59563 q^{92} -0.258933 q^{94} -22.6161 q^{95} +10.9733 q^{97} +0.731838 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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