LMFDB - Embedded newform 8001.2.a.n.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [12,7,0,9,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} + \cdots - 1 $ x^12 - 5x^11 - 3x^10 + 41x^9 - 11x^8 - 123x^7 + 44x^6 + 159x^5 - 39x^4 - 71x^3 + 16x^2 + 7*x - 1
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			1.10
Root			$-1.17133$ 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)

$f(q)$	$=$	\(q+2.17133 q^{2} +2.71466 q^{4} +4.06798 q^{5} +1.00000 q^{7} +1.55176 q^{8} +8.83292 q^{10} -0.266783 q^{11} -1.09460 q^{13} +2.17133 q^{14} -2.05994 q^{16} -2.89261 q^{17} +7.39603 q^{19} +11.0432 q^{20} -0.579273 q^{22} +3.52450 q^{23} +11.5485 q^{25} -2.37673 q^{26} +2.71466 q^{28} +5.16379 q^{29} -8.51653 q^{31} -7.57633 q^{32} -6.28080 q^{34} +4.06798 q^{35} -9.79185 q^{37} +16.0592 q^{38} +6.31253 q^{40} +8.53149 q^{41} +6.33158 q^{43} -0.724225 q^{44} +7.65284 q^{46} +5.55741 q^{47} +1.00000 q^{49} +25.0755 q^{50} -2.97146 q^{52} +9.12252 q^{53} -1.08527 q^{55} +1.55176 q^{56} +11.2123 q^{58} -0.616616 q^{59} +0.338502 q^{61} -18.4922 q^{62} -12.3308 q^{64} -4.45281 q^{65} -2.27396 q^{67} -7.85245 q^{68} +8.83292 q^{70} -4.97702 q^{71} -0.185927 q^{73} -21.2613 q^{74} +20.0777 q^{76} -0.266783 q^{77} -2.63944 q^{79} -8.37981 q^{80} +18.5246 q^{82} +10.6739 q^{83} -11.7671 q^{85} +13.7479 q^{86} -0.413983 q^{88} +7.23902 q^{89} -1.09460 q^{91} +9.56782 q^{92} +12.0670 q^{94} +30.0869 q^{95} -1.68775 q^{97} +2.17133 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31}+ \cdots + 7 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Coefficient data

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

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

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

$2$	2.17133	1.53536	0.767680	−	0.640833i	$-0.221411\pi$
$2$	2.17133	1.53536	0.767680	+	0.640833i	$0.221411\pi$
$3$	0	0
$3$	0	0
$4$	2.71466	1.35733
$5$	4.06798	1.81926	0.909628	−	0.415423i	$-0.136367\pi$
$5$	4.06798	1.81926	0.909628	+	0.415423i	$0.136367\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.55176	0.548630
$9$	0	0
$10$	8.83292	2.79321
$11$	−0.266783	−0.0804381	−0.0402191	−	0.999191i	$-0.512806\pi$
$11$	−0.266783	−0.0804381	−0.0402191	+	0.999191i	$0.512806\pi$
$12$	0	0
$13$	−1.09460	−0.303587	−0.151794	−	0.988412i	$-0.548505\pi$
$13$	−1.09460	−0.303587	−0.151794	+	0.988412i	$0.548505\pi$
$14$	2.17133	0.580311
$15$	0	0
$16$	−2.05994	−0.514986
$17$	−2.89261	−0.701561	−0.350780	−	0.936458i	$-0.614084\pi$
$17$	−2.89261	−0.701561	−0.350780	+	0.936458i	$0.614084\pi$
$18$	0	0
$19$	7.39603	1.69677	0.848383	−	0.529383i	$-0.177577\pi$
$19$	7.39603	1.69677	0.848383	+	0.529383i	$0.177577\pi$
$20$	11.0432	2.46933
$21$	0	0
$22$	−0.579273	−0.123501
$23$	3.52450	0.734909	0.367455	−	0.930041i	$-0.380229\pi$
$23$	3.52450	0.734909	0.367455	+	0.930041i	$0.380229\pi$
$24$	0	0
$25$	11.5485	2.30969
$26$	−2.37673	−0.466116
$27$	0	0
$28$	2.71466	0.513022
$29$	5.16379	0.958893	0.479446	−	0.877571i	$-0.340838\pi$
$29$	5.16379	0.958893	0.479446	+	0.877571i	$0.340838\pi$
$30$	0	0
$31$	−8.51653	−1.52961	−0.764807	−	0.644259i	$-0.777166\pi$
$31$	−8.51653	−1.52961	−0.764807	+	0.644259i	$0.777166\pi$
$32$	−7.57633	−1.33932
$33$	0	0
$34$	−6.28080	−1.07715
$35$	4.06798	0.687614
$36$	0	0
$37$	−9.79185	−1.60977	−0.804885	−	0.593431i	$-0.797773\pi$
$37$	−9.79185	−1.60977	−0.804885	+	0.593431i	$0.797773\pi$
$38$	16.0592	2.60515
$39$	0	0
$40$	6.31253	0.998098
$41$	8.53149	1.33239	0.666197	−	0.745776i	$-0.267921\pi$
$41$	8.53149	1.33239	0.666197	+	0.745776i	$0.267921\pi$
$42$	0	0
$43$	6.33158	0.965556	0.482778	−	0.875743i	$-0.339628\pi$
$43$	6.33158	0.965556	0.482778	+	0.875743i	$0.339628\pi$
$44$	−0.724225	−0.109181
$45$	0	0
$46$	7.65284	1.12835
$47$	5.55741	0.810631	0.405316	−	0.914177i	$-0.367162\pi$
$47$	5.55741	0.810631	0.405316	+	0.914177i	$0.367162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	25.0755	3.54621
$51$	0	0
$52$	−2.97146	−0.412068
$53$	9.12252	1.25307	0.626537	−	0.779392i	$-0.284472\pi$
$53$	9.12252	1.25307	0.626537	+	0.779392i	$0.284472\pi$
$54$	0	0
$55$	−1.08527	−0.146338
$56$	1.55176	0.207363
$57$	0	0
$58$	11.2123	1.47225
$59$	−0.616616	−0.0802766	−0.0401383	−	0.999194i	$-0.512780\pi$
$59$	−0.616616	−0.0802766	−0.0401383	+	0.999194i	$0.512780\pi$
$60$	0	0
$61$	0.338502	0.0433408	0.0216704	−	0.999765i	$-0.493102\pi$
$61$	0.338502	0.0433408	0.0216704	+	0.999765i	$0.493102\pi$
$62$	−18.4922	−2.34851
$63$	0	0
$64$	−12.3308	−1.54135
$65$	−4.45281	−0.552303
$66$	0	0
$67$	−2.27396	−0.277809	−0.138904	−	0.990306i	$-0.544358\pi$
$67$	−2.27396	−0.277809	−0.138904	+	0.990306i	$0.544358\pi$
$68$	−7.85245	−0.952249
$69$	0	0
$70$	8.83292	1.05574
$71$	−4.97702	−0.590663	−0.295332	−	0.955395i	$-0.595430\pi$
$71$	−4.97702	−0.590663	−0.295332	+	0.955395i	$0.595430\pi$
$72$	0	0
$73$	−0.185927	−0.0217611	−0.0108806	−	0.999941i	$-0.503463\pi$
$73$	−0.185927	−0.0217611	−0.0108806	+	0.999941i	$0.503463\pi$
$74$	−21.2613	−2.47158
$75$	0	0
$76$	20.0777	2.30307
$77$	−0.266783	−0.0304028
$78$	0	0
$79$	−2.63944	−0.296960	−0.148480	−	0.988915i	$-0.547438\pi$
$79$	−2.63944	−0.296960	−0.148480	+	0.988915i	$0.547438\pi$
$80$	−8.37981	−0.936891
$81$	0	0
$82$	18.5246	2.04570
$83$	10.6739	1.17162	0.585809	−	0.810449i	$-0.300777\pi$
$83$	10.6739	1.17162	0.585809	+	0.810449i	$0.300777\pi$
$84$	0	0
$85$	−11.7671	−1.27632
$86$	13.7479	1.48248
$87$	0	0
$88$	−0.413983	−0.0441307
$89$	7.23902	0.767334	0.383667	−	0.923471i	$-0.374661\pi$
$89$	7.23902	0.767334	0.383667	+	0.923471i	$0.374661\pi$
$90$	0	0
$91$	−1.09460	−0.114745
$92$	9.56782	0.997514
$93$	0	0
$94$	12.0670	1.24461
$95$	30.0869	3.08685
$96$	0	0
$97$	−1.68775	−0.171365	−0.0856826	−	0.996322i	$-0.527307\pi$
$97$	−1.68775	−0.171365	−0.0856826	+	0.996322i	$0.527307\pi$
$98$	2.17133	0.219337
$99$	0	0
$100$	31.3502	3.13502
$101$	−11.6609	−1.16030	−0.580150	−	0.814509i	$-0.697006\pi$
$101$	−11.6609	−1.16030	−0.580150	+	0.814509i	$0.697006\pi$
$102$	0	0
$103$	6.62298	0.652582	0.326291	−	0.945269i	$-0.394201\pi$
$103$	6.62298	0.652582	0.326291	+	0.945269i	$0.394201\pi$
$104$	−1.69855	−0.166557
$105$	0	0
$106$	19.8080	1.92392
$107$	−19.7612	−1.91038	−0.955191	−	0.295989i	$-0.904351\pi$
$107$	−19.7612	−1.91038	−0.955191	+	0.295989i	$0.904351\pi$
$108$	0	0
$109$	−10.7124	−1.02606	−0.513030	−	0.858371i	$-0.671477\pi$
$109$	−10.7124	−1.02606	−0.513030	+	0.858371i	$0.671477\pi$
$110$	−2.35647	−0.224681
$111$	0	0
$112$	−2.05994	−0.194646
$113$	19.1305	1.79965	0.899823	−	0.436256i	$-0.143696\pi$
$113$	19.1305	1.79965	0.899823	+	0.436256i	$0.143696\pi$
$114$	0	0
$115$	14.3376	1.33699
$116$	14.0179	1.30153
$117$	0	0
$118$	−1.33888	−0.123253
$119$	−2.89261	−0.265165
$120$	0	0
$121$	−10.9288	−0.993530
$122$	0.734999	0.0665437
$123$	0	0
$124$	−23.1195	−2.07619
$125$	26.6390	2.38267
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.6215	−1.02721
$129$	0	0
$130$	−9.66850	−0.847984
$131$	12.8576	1.12337	0.561687	−	0.827350i	$-0.310152\pi$
$131$	12.8576	1.12337	0.561687	+	0.827350i	$0.310152\pi$
$132$	0	0
$133$	7.39603	0.641317
$134$	−4.93752	−0.426537
$135$	0	0
$136$	−4.48863	−0.384897
$137$	10.8694	0.928639	0.464319	−	0.885668i	$-0.346299\pi$
$137$	10.8694	0.928639	0.464319	+	0.885668i	$0.346299\pi$
$138$	0	0
$139$	6.51528	0.552619	0.276310	−	0.961069i	$-0.410888\pi$
$139$	6.51528	0.552619	0.276310	+	0.961069i	$0.410888\pi$
$140$	11.0432	0.933319
$141$	0	0
$142$	−10.8067	−0.906880
$143$	0.292021	0.0244200
$144$	0	0
$145$	21.0062	1.74447
$146$	−0.403708	−0.0334111
$147$	0	0
$148$	−26.5815	−2.18499
$149$	18.2297	1.49344	0.746718	−	0.665140i	$-0.231628\pi$
$149$	18.2297	1.49344	0.746718	+	0.665140i	$0.231628\pi$
$150$	0	0
$151$	7.77110	0.632403	0.316201	−	0.948692i	$-0.397592\pi$
$151$	7.77110	0.632403	0.316201	+	0.948692i	$0.397592\pi$
$152$	11.4769	0.930896
$153$	0	0
$154$	−0.579273	−0.0466792
$155$	−34.6451	−2.78276
$156$	0	0
$157$	−3.14189	−0.250750	−0.125375	−	0.992109i	$-0.540013\pi$
$157$	−3.14189	−0.250750	−0.125375	+	0.992109i	$0.540013\pi$
$158$	−5.73108	−0.455940
$159$	0	0
$160$	−30.8204	−2.43656
$161$	3.52450	0.277770
$162$	0	0
$163$	−11.6797	−0.914825	−0.457413	−	0.889255i	$-0.651224\pi$
$163$	−11.6797	−0.914825	−0.457413	+	0.889255i	$0.651224\pi$
$164$	23.1601	1.80850
$165$	0	0
$166$	23.1766	1.79886
$167$	−23.4356	−1.81350	−0.906749	−	0.421672i	$-0.861444\pi$
$167$	−23.4356	−1.81350	−0.906749	+	0.421672i	$0.861444\pi$
$168$	0	0
$169$	−11.8019	−0.907835
$170$	−25.5502	−1.95961
$171$	0	0
$172$	17.1881	1.31058
$173$	−17.4313	−1.32528	−0.662638	−	0.748940i	$-0.730563\pi$
$173$	−17.4313	−1.32528	−0.662638	+	0.748940i	$0.730563\pi$
$174$	0	0
$175$	11.5485	0.872982
$176$	0.549558	0.0414245
$177$	0	0
$178$	15.7183	1.17813
$179$	−22.0287	−1.64650	−0.823252	−	0.567675i	$-0.807843\pi$
$179$	−22.0287	−1.64650	−0.823252	+	0.567675i	$0.807843\pi$
$180$	0	0
$181$	−11.2226	−0.834169	−0.417085	−	0.908868i	$-0.636948\pi$
$181$	−11.2226	−0.834169	−0.417085	+	0.908868i	$0.636948\pi$
$182$	−2.37673	−0.176175
$183$	0	0
$184$	5.46918	0.403193
$185$	−39.8331	−2.92858
$186$	0	0
$187$	0.771699	0.0564322
$188$	15.0865	1.10029
$189$	0	0
$190$	65.3285	4.73943
$191$	−15.8615	−1.14770	−0.573848	−	0.818962i	$-0.694550\pi$
$191$	−15.8615	−1.14770	−0.573848	+	0.818962i	$0.694550\pi$
$192$	0	0
$193$	−5.38815	−0.387848	−0.193924	−	0.981017i	$-0.562121\pi$
$193$	−5.38815	−0.387848	−0.193924	+	0.981017i	$0.562121\pi$
$194$	−3.66466	−0.263107
$195$	0	0
$196$	2.71466	0.193904
$197$	−18.3913	−1.31033	−0.655163	−	0.755487i	$-0.727400\pi$
$197$	−18.3913	−1.31033	−0.655163	+	0.755487i	$0.727400\pi$
$198$	0	0
$199$	7.96860	0.564879	0.282439	−	0.959285i	$-0.408856\pi$
$199$	7.96860	0.564879	0.282439	+	0.959285i	$0.408856\pi$
$200$	17.9204	1.26717
$201$	0	0
$202$	−25.3196	−1.78148
$203$	5.16379	0.362427
$204$	0	0
$205$	34.7059	2.42397
$206$	14.3807	1.00195
$207$	0	0
$208$	2.25481	0.156343
$209$	−1.97314	−0.136485
$210$	0	0
$211$	5.02672	0.346054	0.173027	−	0.984917i	$-0.444645\pi$
$211$	5.02672	0.346054	0.173027	+	0.984917i	$0.444645\pi$
$212$	24.7645	1.70084
$213$	0	0
$214$	−42.9079	−2.93313
$215$	25.7567	1.75659
$216$	0	0
$217$	−8.51653	−0.578140
$218$	−23.2601	−1.57537
$219$	0	0
$220$	−2.94613	−0.198628
$221$	3.16625	0.212985
$222$	0	0
$223$	12.5676	0.841591	0.420796	−	0.907156i	$-0.361751\pi$
$223$	12.5676	0.841591	0.420796	+	0.907156i	$0.361751\pi$
$224$	−7.57633	−0.506215
$225$	0	0
$226$	41.5386	2.76310
$227$	8.90570	0.591092	0.295546	−	0.955328i	$-0.404498\pi$
$227$	8.90570	0.591092	0.295546	+	0.955328i	$0.404498\pi$
$228$	0	0
$229$	−20.3430	−1.34430	−0.672151	−	0.740414i	$-0.734629\pi$
$229$	−20.3430	−1.34430	−0.672151	+	0.740414i	$0.734629\pi$
$230$	31.1316	2.05276
$231$	0	0
$232$	8.01297	0.526077
$233$	18.6982	1.22496	0.612480	−	0.790486i	$-0.290172\pi$
$233$	18.6982	1.22496	0.612480	+	0.790486i	$0.290172\pi$
$234$	0	0
$235$	22.6074	1.47475
$236$	−1.67390	−0.108962
$237$	0	0
$238$	−6.28080	−0.407124
$239$	7.69279	0.497605	0.248803	−	0.968554i	$-0.419963\pi$
$239$	7.69279	0.497605	0.248803	+	0.968554i	$0.419963\pi$
$240$	0	0
$241$	−7.23394	−0.465979	−0.232989	−	0.972479i	$-0.574851\pi$
$241$	−7.23394	−0.465979	−0.232989	+	0.972479i	$0.574851\pi$
$242$	−23.7301	−1.52543
$243$	0	0
$244$	0.918919	0.0588277
$245$	4.06798	0.259894
$246$	0	0
$247$	−8.09569	−0.515116
$248$	−13.2156	−0.839192
$249$	0	0
$250$	57.8421	3.65825
$251$	−9.13560	−0.576634	−0.288317	−	0.957535i	$-0.593096\pi$
$251$	−9.13560	−0.576634	−0.288317	+	0.957535i	$0.593096\pi$
$252$	0	0
$253$	−0.940277	−0.0591147
$254$	2.17133	0.136241
$255$	0	0
$256$	−0.572550	−0.0357844
$257$	−8.79482	−0.548606	−0.274303	−	0.961643i	$-0.588447\pi$
$257$	−8.79482	−0.548606	−0.274303	+	0.961643i	$0.588447\pi$
$258$	0	0
$259$	−9.79185	−0.608436
$260$	−12.0879	−0.749657
$261$	0	0
$262$	27.9181	1.72478
$263$	12.0332	0.742002	0.371001	−	0.928633i	$-0.379015\pi$
$263$	12.0332	0.742002	0.371001	+	0.928633i	$0.379015\pi$
$264$	0	0
$265$	37.1102	2.27966
$266$	16.0592	0.984653
$267$	0	0
$268$	−6.17304	−0.377078
$269$	−15.1620	−0.924442	−0.462221	−	0.886765i	$-0.652947\pi$
$269$	−15.1620	−0.924442	−0.462221	+	0.886765i	$0.652947\pi$
$270$	0	0
$271$	−2.25603	−0.137044	−0.0685222	−	0.997650i	$-0.521828\pi$
$271$	−2.25603	−0.137044	−0.0685222	+	0.997650i	$0.521828\pi$
$272$	5.95861	0.361294
$273$	0	0
$274$	23.6011	1.42579
$275$	−3.08094	−0.185787
$276$	0	0
$277$	7.93190	0.476582	0.238291	−	0.971194i	$-0.423413\pi$
$277$	7.93190	0.476582	0.238291	+	0.971194i	$0.423413\pi$
$278$	14.1468	0.848469
$279$	0	0
$280$	6.31253	0.377246
$281$	−24.2357	−1.44578	−0.722892	−	0.690961i	$-0.757187\pi$
$281$	−24.2357	−1.44578	−0.722892	+	0.690961i	$0.757187\pi$
$282$	0	0
$283$	12.1178	0.720327	0.360163	−	0.932889i	$-0.382721\pi$
$283$	12.1178	0.720327	0.360163	+	0.932889i	$0.382721\pi$
$284$	−13.5109	−0.801725
$285$	0	0
$286$	0.634072	0.0374935
$287$	8.53149	0.503598
$288$	0	0
$289$	−8.63281	−0.507812
$290$	45.6114	2.67839
$291$	0	0
$292$	−0.504729	−0.0295370
$293$	−24.3421	−1.42208	−0.711039	−	0.703152i	$-0.751775\pi$
$293$	−24.3421	−1.42208	−0.711039	+	0.703152i	$0.751775\pi$
$294$	0	0
$295$	−2.50838	−0.146044
$296$	−15.1946	−0.883168
$297$	0	0
$298$	39.5827	2.29296
$299$	−3.85791	−0.223109
$300$	0	0
$301$	6.33158	0.364946
$302$	16.8736	0.970966
$303$	0	0
$304$	−15.2354	−0.873810
$305$	1.37702	0.0788480
$306$	0	0
$307$	−2.64544	−0.150983	−0.0754917	−	0.997146i	$-0.524053\pi$
$307$	−2.64544	−0.150983	−0.0754917	+	0.997146i	$0.524053\pi$
$308$	−0.724225	−0.0412666
$309$	0	0
$310$	−75.2258	−4.27254
$311$	0.797374	0.0452149	0.0226075	−	0.999744i	$-0.492803\pi$
$311$	0.797374	0.0452149	0.0226075	+	0.999744i	$0.492803\pi$
$312$	0	0
$313$	13.8241	0.781382	0.390691	−	0.920522i	$-0.372236\pi$
$313$	13.8241	0.781382	0.390691	+	0.920522i	$0.372236\pi$
$314$	−6.82208	−0.384992
$315$	0	0
$316$	−7.16517	−0.403073
$317$	3.66262	0.205713	0.102857	−	0.994696i	$-0.467202\pi$
$317$	3.66262	0.205713	0.102857	+	0.994696i	$0.467202\pi$
$318$	0	0
$319$	−1.37761	−0.0771315
$320$	−50.1614	−2.80411
$321$	0	0
$322$	7.65284	0.426476
$323$	−21.3938	−1.19038
$324$	0	0
$325$	−12.6409	−0.701193
$326$	−25.3605	−1.40459
$327$	0	0
$328$	13.2388	0.730991
$329$	5.55741	0.306390
$330$	0	0
$331$	−21.9622	−1.20715	−0.603576	−	0.797306i	$-0.706258\pi$
$331$	−21.9622	−1.20715	−0.603576	+	0.797306i	$0.706258\pi$
$332$	28.9761	1.59027
$333$	0	0
$334$	−50.8862	−2.78437
$335$	−9.25044	−0.505406
$336$	0	0
$337$	−13.3643	−0.728001	−0.364001	−	0.931399i	$-0.618589\pi$
$337$	−13.3643	−0.728001	−0.364001	+	0.931399i	$0.618589\pi$
$338$	−25.6257	−1.39385
$339$	0	0
$340$	−31.9436	−1.73239
$341$	2.27207	0.123039
$342$	0	0
$343$	1.00000	0.0539949
$344$	9.82508	0.529733
$345$	0	0
$346$	−37.8490	−2.03477
$347$	−31.7118	−1.70238	−0.851190	−	0.524858i	$-0.824119\pi$
$347$	−31.7118	−1.70238	−0.851190	+	0.524858i	$0.824119\pi$
$348$	0	0
$349$	−23.3455	−1.24965	−0.624827	−	0.780764i	$-0.714830\pi$
$349$	−23.3455	−1.24965	−0.624827	+	0.780764i	$0.714830\pi$
$350$	25.0755	1.34034
$351$	0	0
$352$	2.02124	0.107732
$353$	−24.7534	−1.31749	−0.658746	−	0.752365i	$-0.728913\pi$
$353$	−24.7534	−1.31749	−0.658746	+	0.752365i	$0.728913\pi$
$354$	0	0
$355$	−20.2464	−1.07457
$356$	19.6515	1.04153
$357$	0	0
$358$	−47.8316	−2.52798
$359$	12.2004	0.643914	0.321957	−	0.946754i	$-0.395659\pi$
$359$	12.2004	0.643914	0.321957	+	0.946754i	$0.395659\pi$
$360$	0	0
$361$	35.7013	1.87901
$362$	−24.3679	−1.28075
$363$	0	0
$364$	−2.97146	−0.155747
$365$	−0.756348	−0.0395890
$366$	0	0
$367$	22.1639	1.15694	0.578472	−	0.815702i	$-0.303649\pi$
$367$	22.1639	1.15694	0.578472	+	0.815702i	$0.303649\pi$
$368$	−7.26027	−0.378468
$369$	0	0
$370$	−86.4906	−4.49643
$371$	9.12252	0.473618
$372$	0	0
$373$	25.4914	1.31990	0.659948	−	0.751312i	$-0.270578\pi$
$373$	25.4914	1.31990	0.659948	+	0.751312i	$0.270578\pi$
$374$	1.67561	0.0866438
$375$	0	0
$376$	8.62376	0.444737
$377$	−5.65229	−0.291108
$378$	0	0
$379$	−17.0667	−0.876659	−0.438329	−	0.898814i	$-0.644430\pi$
$379$	−17.0667	−0.876659	−0.438329	+	0.898814i	$0.644430\pi$
$380$	81.6757	4.18988
$381$	0	0
$382$	−34.4405	−1.76213
$383$	−20.5994	−1.05258	−0.526291	−	0.850305i	$-0.676418\pi$
$383$	−20.5994	−1.05258	−0.526291	+	0.850305i	$0.676418\pi$
$384$	0	0
$385$	−1.08527	−0.0553104
$386$	−11.6994	−0.595486
$387$	0	0
$388$	−4.58167	−0.232599
$389$	26.1221	1.32444	0.662221	−	0.749309i	$-0.269614\pi$
$389$	26.1221	1.32444	0.662221	+	0.749309i	$0.269614\pi$
$390$	0	0
$391$	−10.1950	−0.515583
$392$	1.55176	0.0783757
$393$	0	0
$394$	−39.9335	−2.01182
$395$	−10.7372	−0.540246
$396$	0	0
$397$	17.7396	0.890324	0.445162	−	0.895450i	$-0.353146\pi$
$397$	17.7396	0.890324	0.445162	+	0.895450i	$0.353146\pi$
$398$	17.3024	0.867292
$399$	0	0
$400$	−23.7892	−1.18946
$401$	17.5413	0.875970	0.437985	−	0.898982i	$-0.355692\pi$
$401$	17.5413	0.875970	0.437985	+	0.898982i	$0.355692\pi$
$402$	0	0
$403$	9.32219	0.464371
$404$	−31.6553	−1.57491
$405$	0	0
$406$	11.2123	0.556456
$407$	2.61230	0.129487
$408$	0	0
$409$	−24.2566	−1.19941	−0.599706	−	0.800220i	$-0.704716\pi$
$409$	−24.2566	−1.19941	−0.599706	+	0.800220i	$0.704716\pi$
$410$	75.3579	3.72166
$411$	0	0
$412$	17.9791	0.885768
$413$	−0.616616	−0.0303417
$414$	0	0
$415$	43.4214	2.13147
$416$	8.29304	0.406600
$417$	0	0
$418$	−4.28432	−0.209553
$419$	0.0194566	0.000950517 0	0.000475259	−	1.00000i	$-0.499849\pi$
$419$	0.0194566	0.000950517 0	0.000475259		1.00000i	$0.499849\pi$
$420$	0	0
$421$	25.4929	1.24245	0.621224	−	0.783633i	$-0.286636\pi$
$421$	25.4929	1.24245	0.621224	+	0.783633i	$0.286636\pi$
$422$	10.9146	0.531317
$423$	0	0
$424$	14.1560	0.687474
$425$	−33.4052	−1.62039
$426$	0	0
$427$	0.338502	0.0163813
$428$	−53.6448	−2.59302
$429$	0	0
$430$	55.9263	2.69700
$431$	13.0434	0.628280	0.314140	−	0.949377i	$-0.398284\pi$
$431$	13.0434	0.628280	0.314140	+	0.949377i	$0.398284\pi$
$432$	0	0
$433$	−40.6553	−1.95377	−0.976884	−	0.213768i	$-0.931426\pi$
$433$	−40.6553	−1.95377	−0.976884	+	0.213768i	$0.931426\pi$
$434$	−18.4922	−0.887653
$435$	0	0
$436$	−29.0805	−1.39270
$437$	26.0673	1.24697
$438$	0	0
$439$	−15.3630	−0.733234	−0.366617	−	0.930372i	$-0.619484\pi$
$439$	−15.3630	−0.733234	−0.366617	+	0.930372i	$0.619484\pi$
$440$	−1.68408	−0.0802851
$441$	0	0
$442$	6.87496	0.327008
$443$	38.4336	1.82604	0.913019	−	0.407918i	$-0.133745\pi$
$443$	38.4336	1.82604	0.913019	+	0.407918i	$0.133745\pi$
$444$	0	0
$445$	29.4482	1.39598
$446$	27.2884	1.29215
$447$	0	0
$448$	−12.3308	−0.582575
$449$	−3.14382	−0.148366	−0.0741829	−	0.997245i	$-0.523635\pi$
$449$	−3.14382	−0.148366	−0.0741829	+	0.997245i	$0.523635\pi$
$450$	0	0
$451$	−2.27606	−0.107175
$452$	51.9328	2.44271
$453$	0	0
$454$	19.3372	0.907539
$455$	−4.45281	−0.208751
$456$	0	0
$457$	−31.2430	−1.46148	−0.730742	−	0.682654i	$-0.760826\pi$
$457$	−31.2430	−1.46148	−0.730742	+	0.682654i	$0.760826\pi$
$458$	−44.1713	−2.06399
$459$	0	0
$460$	38.9217	1.81473
$461$	−14.2643	−0.664356	−0.332178	−	0.943217i	$-0.607784\pi$
$461$	−14.2643	−0.664356	−0.332178	+	0.943217i	$0.607784\pi$
$462$	0	0
$463$	7.98993	0.371324	0.185662	−	0.982614i	$-0.440557\pi$
$463$	7.98993	0.371324	0.185662	+	0.982614i	$0.440557\pi$
$464$	−10.6371	−0.493816
$465$	0	0
$466$	40.5999	1.88076
$467$	3.20601	0.148357	0.0741783	−	0.997245i	$-0.476367\pi$
$467$	3.20601	0.148357	0.0741783	+	0.997245i	$0.476367\pi$
$468$	0	0
$469$	−2.27396	−0.105002
$470$	49.0881	2.26427
$471$	0	0
$472$	−0.956840	−0.0440421
$473$	−1.68916	−0.0776675
$474$	0	0
$475$	85.4128	3.91901
$476$	−7.85245	−0.359916
$477$	0	0
$478$	16.7036	0.764003
$479$	21.4255	0.978957	0.489478	−	0.872015i	$-0.337187\pi$
$479$	21.4255	0.978957	0.489478	+	0.872015i	$0.337187\pi$
$480$	0	0
$481$	10.7182	0.488706
$482$	−15.7072	−0.715445
$483$	0	0
$484$	−29.6680	−1.34855
$485$	−6.86574	−0.311757
$486$	0	0
$487$	14.0036	0.634563	0.317282	−	0.948331i	$-0.397230\pi$
$487$	14.0036	0.634563	0.317282	+	0.948331i	$0.397230\pi$
$488$	0.525274	0.0237780
$489$	0	0
$490$	8.83292	0.399030
$491$	−8.07691	−0.364506	−0.182253	−	0.983252i	$-0.558339\pi$
$491$	−8.07691	−0.364506	−0.182253	+	0.983252i	$0.558339\pi$
$492$	0	0
$493$	−14.9368	−0.672722
$494$	−17.5784	−0.790889
$495$	0	0
$496$	17.5436	0.787729
$497$	−4.97702	−0.223250
$498$	0	0
$499$	9.80189	0.438793	0.219396	−	0.975636i	$-0.429591\pi$
$499$	9.80189	0.438793	0.219396	+	0.975636i	$0.429591\pi$
$500$	72.3159	3.23407
$501$	0	0
$502$	−19.8364	−0.885340
$503$	−30.8833	−1.37702	−0.688508	−	0.725229i	$-0.741734\pi$
$503$	−30.8833	−1.37702	−0.688508	+	0.725229i	$0.741734\pi$
$504$	0	0
$505$	−47.4362	−2.11088
$506$	−2.04165	−0.0907623
$507$	0	0
$508$	2.71466	0.120444
$509$	34.3483	1.52246	0.761231	−	0.648481i	$-0.224595\pi$
$509$	34.3483	1.52246	0.761231	+	0.648481i	$0.224595\pi$
$510$	0	0
$511$	−0.185927	−0.00822493
$512$	21.9999	0.972266
$513$	0	0
$514$	−19.0964	−0.842307
$515$	26.9422	1.18721
$516$	0	0
$517$	−1.48262	−0.0652057
$518$	−21.2613	−0.934168
$519$	0	0
$520$	−6.90969	−0.303010
$521$	−26.3782	−1.15565	−0.577824	−	0.816161i	$-0.696098\pi$
$521$	−26.3782	−1.15565	−0.577824	+	0.816161i	$0.696098\pi$
$522$	0	0
$523$	29.7002	1.29870	0.649350	−	0.760490i	$-0.275041\pi$
$523$	29.7002	1.29870	0.649350	+	0.760490i	$0.275041\pi$
$524$	34.9040	1.52479
$525$	0	0
$526$	26.1281	1.13924
$527$	24.6350	1.07312
$528$	0	0
$529$	−10.5779	−0.459909
$530$	80.5784	3.50010
$531$	0	0
$532$	20.0777	0.870479
$533$	−9.33856	−0.404498
$534$	0	0
$535$	−80.3880	−3.47548
$536$	−3.52865	−0.152414
$537$	0	0
$538$	−32.9216	−1.41935
$539$	−0.266783	−0.0114912
$540$	0	0
$541$	−22.7056	−0.976190	−0.488095	−	0.872790i	$-0.662308\pi$
$541$	−22.7056	−0.976190	−0.488095	+	0.872790i	$0.662308\pi$
$542$	−4.89859	−0.210412
$543$	0	0
$544$	21.9154	0.939613
$545$	−43.5778	−1.86667
$546$	0	0
$547$	5.19459	0.222105	0.111052	−	0.993815i	$-0.464578\pi$
$547$	5.19459	0.222105	0.111052	+	0.993815i	$0.464578\pi$
$548$	29.5068	1.26047
$549$	0	0
$550$	−6.68972	−0.285251
$551$	38.1916	1.62702
$552$	0	0
$553$	−2.63944	−0.112240
$554$	17.2227	0.731724
$555$	0	0
$556$	17.6868	0.750086
$557$	−8.33926	−0.353346	−0.176673	−	0.984270i	$-0.556533\pi$
$557$	−8.33926	−0.353346	−0.176673	+	0.984270i	$0.556533\pi$
$558$	0	0
$559$	−6.93054	−0.293131
$560$	−8.37981	−0.354111
$561$	0	0
$562$	−52.6237	−2.21980
$563$	−28.1811	−1.18769	−0.593846	−	0.804579i	$-0.702391\pi$
$563$	−28.1811	−1.18769	−0.593846	+	0.804579i	$0.702391\pi$
$564$	0	0
$565$	77.8225	3.27402
$566$	26.3116	1.10596
$567$	0	0
$568$	−7.72313	−0.324055
$569$	9.87355	0.413921	0.206960	−	0.978349i	$-0.433643\pi$
$569$	9.87355	0.413921	0.206960	+	0.978349i	$0.433643\pi$
$570$	0	0
$571$	−1.17178	−0.0490373	−0.0245187	−	0.999699i	$-0.507805\pi$
$571$	−1.17178	−0.0490373	−0.0245187	+	0.999699i	$0.507805\pi$
$572$	0.792736	0.0331460
$573$	0	0
$574$	18.5246	0.773204
$575$	40.7026	1.69741
$576$	0	0
$577$	25.1076	1.04524	0.522621	−	0.852565i	$-0.324954\pi$
$577$	25.1076	1.04524	0.522621	+	0.852565i	$0.324954\pi$
$578$	−18.7447	−0.779675
$579$	0	0
$580$	57.0247	2.36782
$581$	10.6739	0.442830
$582$	0	0
$583$	−2.43373	−0.100795
$584$	−0.288514	−0.0119388
$585$	0	0
$586$	−52.8546	−2.18340
$587$	−38.7751	−1.60042	−0.800210	−	0.599720i	$-0.795279\pi$
$587$	−38.7751	−1.60042	−0.800210	+	0.599720i	$0.795279\pi$
$588$	0	0
$589$	−62.9885	−2.59540
$590$	−5.44652	−0.224230
$591$	0	0
$592$	20.1706	0.829009
$593$	20.4344	0.839141	0.419571	−	0.907723i	$-0.362181\pi$
$593$	20.4344	0.839141	0.419571	+	0.907723i	$0.362181\pi$
$594$	0	0
$595$	−11.7671	−0.482403
$596$	49.4875	2.02709
$597$	0	0
$598$	−8.37679	−0.342553
$599$	22.0441	0.900697	0.450349	−	0.892853i	$-0.351300\pi$
$599$	22.0441	0.900697	0.450349	+	0.892853i	$0.351300\pi$
$600$	0	0
$601$	−18.9878	−0.774529	−0.387265	−	0.921969i	$-0.626580\pi$
$601$	−18.9878	−0.774529	−0.387265	+	0.921969i	$0.626580\pi$
$602$	13.7479	0.560323
$603$	0	0
$604$	21.0959	0.858379
$605$	−44.4583	−1.80749
$606$	0	0
$607$	−17.1567	−0.696370	−0.348185	−	0.937426i	$-0.613202\pi$
$607$	−17.1567	−0.696370	−0.348185	+	0.937426i	$0.613202\pi$
$608$	−56.0347	−2.27251
$609$	0	0
$610$	2.98996	0.121060
$611$	−6.08314	−0.246097
$612$	0	0
$613$	−13.2983	−0.537113	−0.268557	−	0.963264i	$-0.586547\pi$
$613$	−13.2983	−0.537113	−0.268557	+	0.963264i	$0.586547\pi$
$614$	−5.74412	−0.231814
$615$	0	0
$616$	−0.413983	−0.0166799
$617$	35.2566	1.41938	0.709688	−	0.704516i	$-0.248836\pi$
$617$	35.2566	1.41938	0.709688	+	0.704516i	$0.248836\pi$
$618$	0	0
$619$	22.7663	0.915054	0.457527	−	0.889196i	$-0.348735\pi$
$619$	22.7663	0.915054	0.457527	+	0.889196i	$0.348735\pi$
$620$	−94.0496	−3.77712
$621$	0	0
$622$	1.73136	0.0694212
$623$	7.23902	0.290025
$624$	0	0
$625$	50.6248	2.02499
$626$	30.0166	1.19970
$627$	0	0
$628$	−8.52917	−0.340351
$629$	28.3240	1.12935
$630$	0	0
$631$	−12.5247	−0.498599	−0.249299	−	0.968426i	$-0.580200\pi$
$631$	−12.5247	−0.498599	−0.249299	+	0.968426i	$0.580200\pi$
$632$	−4.09577	−0.162921
$633$	0	0
$634$	7.95275	0.315844
$635$	4.06798	0.161433
$636$	0	0
$637$	−1.09460	−0.0433696
$638$	−2.99125	−0.118425
$639$	0	0
$640$	−47.2762	−1.86875
$641$	36.6628	1.44810	0.724048	−	0.689750i	$-0.242280\pi$
$641$	36.6628	1.44810	0.724048	+	0.689750i	$0.242280\pi$
$642$	0	0
$643$	−18.0203	−0.710653	−0.355326	−	0.934742i	$-0.615630\pi$
$643$	−18.0203	−0.710653	−0.355326	+	0.934742i	$0.615630\pi$
$644$	9.56782	0.377025
$645$	0	0
$646$	−46.4530	−1.82767
$647$	3.01516	0.118538	0.0592691	−	0.998242i	$-0.481123\pi$
$647$	3.01516	0.118538	0.0592691	+	0.998242i	$0.481123\pi$
$648$	0	0
$649$	0.164503	0.00645730
$650$	−27.4476	−1.07658
$651$	0	0
$652$	−31.7064	−1.24172
$653$	−15.6743	−0.613384	−0.306692	−	0.951809i	$-0.599222\pi$
$653$	−15.6743	−0.613384	−0.306692	+	0.951809i	$0.599222\pi$
$654$	0	0
$655$	52.3045	2.04371
$656$	−17.5744	−0.686164
$657$	0	0
$658$	12.0670	0.470419
$659$	39.3096	1.53128	0.765642	−	0.643267i	$-0.222421\pi$
$659$	39.3096	1.53128	0.765642	+	0.643267i	$0.222421\pi$
$660$	0	0
$661$	25.0184	0.973101	0.486551	−	0.873652i	$-0.338255\pi$
$661$	25.0184	0.973101	0.486551	+	0.873652i	$0.338255\pi$
$662$	−47.6871	−1.85341
$663$	0	0
$664$	16.5634	0.642785
$665$	30.0869	1.16672
$666$	0	0
$667$	18.1998	0.704699
$668$	−63.6195	−2.46151
$669$	0	0
$670$	−20.0857	−0.775980
$671$	−0.0903067	−0.00348625
$672$	0	0
$673$	14.3099	0.551605	0.275803	−	0.961214i	$-0.411056\pi$
$673$	14.3099	0.551605	0.275803	+	0.961214i	$0.411056\pi$
$674$	−29.0183	−1.11774
$675$	0	0
$676$	−32.0380	−1.23223
$677$	−16.2084	−0.622941	−0.311471	−	0.950256i	$-0.600822\pi$
$677$	−16.2084	−0.622941	−0.311471	+	0.950256i	$0.600822\pi$
$678$	0	0
$679$	−1.68775	−0.0647700
$680$	−18.2597	−0.700227
$681$	0	0
$682$	4.93340	0.188910
$683$	34.6189	1.32465	0.662327	−	0.749215i	$-0.269569\pi$
$683$	34.6189	1.32465	0.662327	+	0.749215i	$0.269569\pi$
$684$	0	0
$685$	44.2167	1.68943
$686$	2.17133	0.0829016
$687$	0	0
$688$	−13.0427	−0.497248
$689$	−9.98550	−0.380417
$690$	0	0
$691$	−32.8805	−1.25083	−0.625416	−	0.780291i	$-0.715071\pi$
$691$	−32.8805	−1.25083	−0.625416	+	0.780291i	$0.715071\pi$
$692$	−47.3200	−1.79884
$693$	0	0
$694$	−68.8567	−2.61377
$695$	26.5040	1.00536
$696$	0	0
$697$	−24.6783	−0.934756
$698$	−50.6906	−1.91867
$699$	0	0
$700$	31.3502	1.18492
$701$	−15.9695	−0.603160	−0.301580	−	0.953441i	$-0.597514\pi$
$701$	−15.9695	−0.603160	−0.301580	+	0.953441i	$0.597514\pi$
$702$	0	0
$703$	−72.4208	−2.73140
$704$	3.28965	0.123983
$705$	0	0
$706$	−53.7478	−2.02283
$707$	−11.6609	−0.438552
$708$	0	0
$709$	51.9366	1.95052	0.975261	−	0.221058i	$-0.0709509\pi$
$709$	51.9366	1.95052	0.975261	+	0.221058i	$0.0709509\pi$
$710$	−43.9616	−1.64985
$711$	0	0
$712$	11.2332	0.420982
$713$	−30.0165	−1.12413
$714$	0	0
$715$	1.18793	0.0444262
$716$	−59.8005	−2.23485
$717$	0	0
$718$	26.4911	0.988639
$719$	46.3028	1.72680	0.863402	−	0.504516i	$-0.168329\pi$
$719$	46.3028	1.72680	0.863402	+	0.504516i	$0.168329\pi$
$720$	0	0
$721$	6.62298	0.246653
$722$	77.5191	2.88496
$723$	0	0
$724$	−30.4655	−1.13224
$725$	59.6339	2.21475
$726$	0	0
$727$	6.67972	0.247737	0.123868	−	0.992299i	$-0.460470\pi$
$727$	6.67972	0.247737	0.123868	+	0.992299i	$0.460470\pi$
$728$	−1.69855	−0.0629526
$729$	0	0
$730$	−1.64228	−0.0607834
$731$	−18.3148	−0.677397
$732$	0	0
$733$	15.1702	0.560322	0.280161	−	0.959953i	$-0.409612\pi$
$733$	15.1702	0.560322	0.280161	+	0.959953i	$0.409612\pi$
$734$	48.1250	1.77633
$735$	0	0
$736$	−26.7028	−0.984277
$737$	0.606655	0.0223464
$738$	0	0
$739$	−19.5460	−0.719012	−0.359506	−	0.933143i	$-0.617055\pi$
$739$	−19.5460	−0.719012	−0.359506	+	0.933143i	$0.617055\pi$
$740$	−108.133	−3.97505
$741$	0	0
$742$	19.8080	0.727173
$743$	16.3241	0.598873	0.299437	−	0.954116i	$-0.403201\pi$
$743$	16.3241	0.598873	0.299437	+	0.954116i	$0.403201\pi$
$744$	0	0
$745$	74.1581	2.71694
$746$	55.3502	2.02651
$747$	0	0
$748$	2.09490	0.0765972
$749$	−19.7612	−0.722057
$750$	0	0
$751$	28.2667	1.03147	0.515734	−	0.856749i	$-0.327519\pi$
$751$	28.2667	1.03147	0.515734	+	0.856749i	$0.327519\pi$
$752$	−11.4479	−0.417464
$753$	0	0
$754$	−12.2730	−0.446955
$755$	31.6127	1.15050
$756$	0	0
$757$	2.87807	0.104605	0.0523026	−	0.998631i	$-0.483344\pi$
$757$	2.87807	0.104605	0.0523026	+	0.998631i	$0.483344\pi$
$758$	−37.0574	−1.34599
$759$	0	0
$760$	46.6876	1.69354
$761$	−42.4595	−1.53915	−0.769577	−	0.638554i	$-0.779533\pi$
$761$	−42.4595	−1.53915	−0.769577	+	0.638554i	$0.779533\pi$
$762$	0	0
$763$	−10.7124	−0.387814
$764$	−43.0585	−1.55780
$765$	0	0
$766$	−44.7281	−1.61609
$767$	0.674948	0.0243709
$768$	0	0
$769$	−31.3519	−1.13058	−0.565289	−	0.824893i	$-0.691235\pi$
$769$	−31.3519	−1.13058	−0.565289	+	0.824893i	$0.691235\pi$
$770$	−2.35647	−0.0849214
$771$	0	0
$772$	−14.6270	−0.526437
$773$	16.4511	0.591705	0.295852	−	0.955234i	$-0.404396\pi$
$773$	16.4511	0.591705	0.295852	+	0.955234i	$0.404396\pi$
$774$	0	0
$775$	−98.3529	−3.53294
$776$	−2.61899	−0.0940161
$777$	0	0
$778$	56.7196	2.03350
$779$	63.0991	2.26076
$780$	0	0
$781$	1.32778	0.0475118
$782$	−22.1367	−0.791606
$783$	0	0
$784$	−2.05994	−0.0735694
$785$	−12.7812	−0.456179
$786$	0	0
$787$	−3.60308	−0.128436	−0.0642180	−	0.997936i	$-0.520455\pi$
$787$	−3.60308	−0.128436	−0.0642180	+	0.997936i	$0.520455\pi$
$788$	−49.9261	−1.77855
$789$	0	0
$790$	−23.3139	−0.829472
$791$	19.1305	0.680202
$792$	0	0
$793$	−0.370524	−0.0131577
$794$	38.5184	1.36697
$795$	0	0
$796$	21.6320	0.766727
$797$	−14.5000	−0.513617	−0.256809	−	0.966462i	$-0.582671\pi$
$797$	−14.5000	−0.513617	−0.256809	+	0.966462i	$0.582671\pi$
$798$	0	0
$799$	−16.0754	−0.568707
$800$	−87.4950	−3.09341
$801$	0	0
$802$	38.0878	1.34493
$803$	0.0496022	0.00175042
$804$	0	0
$805$	14.3376	0.505334
$806$	20.2415	0.712977
$807$	0	0
$808$	−18.0949	−0.636575
$809$	−11.3513	−0.399090	−0.199545	−	0.979889i	$-0.563946\pi$
$809$	−11.3513	−0.399090	−0.199545	+	0.979889i	$0.563946\pi$
$810$	0	0
$811$	20.8018	0.730452	0.365226	−	0.930919i	$-0.380992\pi$
$811$	20.8018	0.730452	0.365226	+	0.930919i	$0.380992\pi$
$812$	14.0179	0.491933
$813$	0	0
$814$	5.67216	0.198809
$815$	−47.5128	−1.66430
$816$	0	0
$817$	46.8285	1.63832
$818$	−52.6690	−1.84153
$819$	0	0
$820$	94.2148	3.29012
$821$	13.7513	0.479922	0.239961	−	0.970782i	$-0.422865\pi$
$821$	13.7513	0.479922	0.239961	+	0.970782i	$0.422865\pi$
$822$	0	0
$823$	−31.4826	−1.09741	−0.548707	−	0.836015i	$-0.684880\pi$
$823$	−31.4826	−1.09741	−0.548707	+	0.836015i	$0.684880\pi$
$824$	10.2773	0.358026
$825$	0	0
$826$	−1.33888	−0.0465854
$827$	−23.3852	−0.813183	−0.406592	−	0.913610i	$-0.633283\pi$
$827$	−23.3852	−0.813183	−0.406592	+	0.913610i	$0.633283\pi$
$828$	0	0
$829$	8.95837	0.311137	0.155569	−	0.987825i	$-0.450279\pi$
$829$	8.95837	0.311137	0.155569	+	0.987825i	$0.450279\pi$
$830$	94.2821	3.27258
$831$	0	0
$832$	13.4973	0.467934
$833$	−2.89261	−0.100223
$834$	0	0
$835$	−95.3354	−3.29922
$836$	−5.35639	−0.185255
$837$	0	0
$838$	0.0422466	0.00145939
$839$	−41.7392	−1.44100	−0.720498	−	0.693457i	$-0.756087\pi$
$839$	−41.7392	−1.44100	−0.720498	+	0.693457i	$0.756087\pi$
$840$	0	0
$841$	−2.33522	−0.0805249
$842$	55.3534	1.90760
$843$	0	0
$844$	13.6458	0.469709
$845$	−48.0097	−1.65158
$846$	0	0
$847$	−10.9288	−0.375519
$848$	−18.7919	−0.645315
$849$	0	0
$850$	−72.5336	−2.48788
$851$	−34.5114	−1.18303
$852$	0	0
$853$	18.1287	0.620716	0.310358	−	0.950620i	$-0.399551\pi$
$853$	18.1287	0.620716	0.310358	+	0.950620i	$0.399551\pi$
$854$	0.734999	0.0251512
$855$	0	0
$856$	−30.6646	−1.04809
$857$	30.3642	1.03722	0.518610	−	0.855011i	$-0.326450\pi$
$857$	30.3642	1.03722	0.518610	+	0.855011i	$0.326450\pi$
$858$	0	0
$859$	16.0016	0.545968	0.272984	−	0.962019i	$-0.411989\pi$
$859$	16.0016	0.545968	0.272984	+	0.962019i	$0.411989\pi$
$860$	69.9208	2.38428
$861$	0	0
$862$	28.3215	0.964635
$863$	−39.0527	−1.32937	−0.664684	−	0.747124i	$-0.731434\pi$
$863$	−39.0527	−1.32937	−0.664684	+	0.747124i	$0.731434\pi$
$864$	0	0
$865$	−70.9101	−2.41102
$866$	−88.2759	−2.99974
$867$	0	0
$868$	−23.1195	−0.784726
$869$	0.704157	0.0238869
$870$	0	0
$871$	2.48908	0.0843392
$872$	−16.6230	−0.562927
$873$	0	0
$874$	56.6006	1.91455
$875$	26.6390	0.900564
$876$	0	0
$877$	53.1728	1.79552	0.897758	−	0.440488i	$-0.145195\pi$
$877$	53.1728	1.79552	0.897758	+	0.440488i	$0.145195\pi$
$878$	−33.3580	−1.12578
$879$	0	0
$880$	2.23559	0.0753617
$881$	23.5411	0.793120	0.396560	−	0.918009i	$-0.370204\pi$
$881$	23.5411	0.793120	0.396560	+	0.918009i	$0.370204\pi$
$882$	0	0
$883$	−35.3002	−1.18795	−0.593973	−	0.804485i	$-0.702441\pi$
$883$	−35.3002	−1.18795	−0.593973	+	0.804485i	$0.702441\pi$
$884$	8.59529	0.289091
$885$	0	0
$886$	83.4520	2.80362
$887$	22.2206	0.746094	0.373047	−	0.927812i	$-0.378313\pi$
$887$	22.2206	0.746094	0.373047	+	0.927812i	$0.378313\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	63.9416	2.14333
$891$	0	0
$892$	34.1169	1.14232
$893$	41.1028	1.37545
$894$	0	0
$895$	−89.6125	−2.99541
$896$	−11.6215	−0.388248
$897$	0	0
$898$	−6.82625	−0.227795
$899$	−43.9776	−1.46674
$900$	0	0
$901$	−26.3879	−0.879108
$902$	−4.94206	−0.164553
$903$	0	0
$904$	29.6859	0.987339
$905$	−45.6533	−1.51757
$906$	0	0
$907$	13.3732	0.444050	0.222025	−	0.975041i	$-0.428733\pi$
$907$	13.3732	0.444050	0.222025	+	0.975041i	$0.428733\pi$
$908$	24.1760	0.802307
$909$	0	0
$910$	−9.66850	−0.320508
$911$	37.2407	1.23384	0.616919	−	0.787026i	$-0.288381\pi$
$911$	37.2407	1.23384	0.616919	+	0.787026i	$0.288381\pi$
$912$	0	0
$913$	−2.84763	−0.0942428
$914$	−67.8387	−2.24390
$915$	0	0
$916$	−55.2243	−1.82466
$917$	12.8576	0.424596
$918$	0	0
$919$	48.0271	1.58427	0.792134	−	0.610347i	$-0.208970\pi$
$919$	48.0271	1.58427	0.792134	+	0.610347i	$0.208970\pi$
$920$	22.2485	0.733511
$921$	0	0
$922$	−30.9725	−1.02003
$923$	5.44784	0.179318
$924$	0	0
$925$	−113.081	−3.71808
$926$	17.3488	0.570115
$927$	0	0
$928$	−39.1226	−1.28426
$929$	38.7188	1.27032	0.635161	−	0.772379i	$-0.280934\pi$
$929$	38.7188	1.27032	0.635161	+	0.772379i	$0.280934\pi$
$930$	0	0
$931$	7.39603	0.242395
$932$	50.7593	1.66268
$933$	0	0
$934$	6.96130	0.227781
$935$	3.13926	0.102665
$936$	0	0
$937$	−49.3544	−1.61234	−0.806168	−	0.591686i	$-0.798462\pi$
$937$	−49.3544	−1.61234	−0.806168	+	0.591686i	$0.798462\pi$
$938$	−4.93752	−0.161216
$939$	0	0
$940$	61.3715	2.00172
$941$	6.75627	0.220248	0.110124	−	0.993918i	$-0.464875\pi$
$941$	6.75627	0.220248	0.110124	+	0.993918i	$0.464875\pi$
$942$	0	0
$943$	30.0692	0.979189
$944$	1.27019	0.0413413
$945$	0	0
$946$	−3.66771	−0.119248
$947$	−22.6244	−0.735193	−0.367596	−	0.929985i	$-0.619819\pi$
$947$	−22.6244	−0.735193	−0.367596	+	0.929985i	$0.619819\pi$
$948$	0	0
$949$	0.203516	0.00660640
$950$	185.459	6.01709
$951$	0	0
$952$	−4.48863	−0.145477
$953$	13.5093	0.437608	0.218804	−	0.975769i	$-0.429785\pi$
$953$	13.5093	0.437608	0.218804	+	0.975769i	$0.429785\pi$
$954$	0	0
$955$	−64.5242	−2.08795
$956$	20.8833	0.675415
$957$	0	0
$958$	46.5218	1.50305
$959$	10.8694	0.350992
$960$	0	0
$961$	41.5313	1.33972
$962$	23.2726	0.750339
$963$	0	0
$964$	−19.6377	−0.632487
$965$	−21.9189	−0.705594
$966$	0	0
$967$	−44.4063	−1.42801	−0.714006	−	0.700140i	$-0.753121\pi$
$967$	−44.4063	−1.42801	−0.714006	+	0.700140i	$0.753121\pi$
$968$	−16.9589	−0.545080
$969$	0	0
$970$	−14.9078	−0.478660
$971$	−15.5017	−0.497474	−0.248737	−	0.968571i	$-0.580016\pi$
$971$	−15.5017	−0.497474	−0.248737	+	0.968571i	$0.580016\pi$
$972$	0	0
$973$	6.51528	0.208870
$974$	30.4064	0.974283
$975$	0	0
$976$	−0.697296	−0.0223199
$977$	−51.7424	−1.65539	−0.827693	−	0.561181i	$-0.810347\pi$
$977$	−51.7424	−1.65539	−0.827693	+	0.561181i	$0.810347\pi$
$978$	0	0
$979$	−1.93125	−0.0617229
$980$	11.0432	0.352762
$981$	0	0
$982$	−17.5376	−0.559647
$983$	−2.48511	−0.0792627	−0.0396313	−	0.999214i	$-0.512618\pi$
$983$	−2.48511	−0.0792627	−0.0396313	+	0.999214i	$0.512618\pi$
$984$	0	0
$985$	−74.8155	−2.38382
$986$	−32.4328	−1.03287
$987$	0	0
$988$	−21.9770	−0.699183
$989$	22.3156	0.709596
$990$	0	0
$991$	49.1615	1.56167	0.780833	−	0.624740i	$-0.214795\pi$
$991$	49.1615	1.56167	0.780833	+	0.624740i	$0.214795\pi$
$992$	64.5240	2.04864
$993$	0	0
$994$	−10.8067	−0.342769
$995$	32.4161	1.02766
$996$	0	0
$997$	−14.0846	−0.446063	−0.223032	−	0.974811i	$-0.571595\pi$
$997$	−14.0846	−0.446063	−0.223032	+	0.974811i	$0.571595\pi$
$998$	21.2831	0.673705
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.n.1.10		12
3.2	odd	2		889.2.a.a.1.3	✓	12
21.20	even	2		6223.2.a.i.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.a.1.3	✓	12	3.2	odd	2
6223.2.a.i.1.3		12	21.20	even	2
8001.2.a.n.1.10		12	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.17133 q^{2} +2.71466 q^{4} +4.06798 q^{5} +1.00000 q^{7} +1.55176 q^{8} +8.83292 q^{10} -0.266783 q^{11} -1.09460 q^{13} +2.17133 q^{14} -2.05994 q^{16} -2.89261 q^{17} +7.39603 q^{19} +11.0432 q^{20} -0.579273 q^{22} +3.52450 q^{23} +11.5485 q^{25} -2.37673 q^{26} +2.71466 q^{28} +5.16379 q^{29} -8.51653 q^{31} -7.57633 q^{32} -6.28080 q^{34} +4.06798 q^{35} -9.79185 q^{37} +16.0592 q^{38} +6.31253 q^{40} +8.53149 q^{41} +6.33158 q^{43} -0.724225 q^{44} +7.65284 q^{46} +5.55741 q^{47} +1.00000 q^{49} +25.0755 q^{50} -2.97146 q^{52} +9.12252 q^{53} -1.08527 q^{55} +1.55176 q^{56} +11.2123 q^{58} -0.616616 q^{59} +0.338502 q^{61} -18.4922 q^{62} -12.3308 q^{64} -4.45281 q^{65} -2.27396 q^{67} -7.85245 q^{68} +8.83292 q^{70} -4.97702 q^{71} -0.185927 q^{73} -21.2613 q^{74} +20.0777 q^{76} -0.266783 q^{77} -2.63944 q^{79} -8.37981 q^{80} +18.5246 q^{82} +10.6739 q^{83} -11.7671 q^{85} +13.7479 q^{86} -0.413983 q^{88} +7.23902 q^{89} -1.09460 q^{91} +9.56782 q^{92} +12.0670 q^{94} +30.0869 q^{95} -1.68775 q^{97} +2.17133 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31}+ \cdots + 7 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 9 * q^4 + 7 * q^5 + 12 * q^7 + 15 * q^8 - 2 * q^10 + 22 * q^11 + 7 * q^14 + 7 * q^16 + 6 * q^17 - 7 * q^19 + 8 * q^20 + 13 * q^22 + 29 * q^23 + 3 * q^25 + 9 * q^28 + 22 * q^29 - 16 * q^31 + 27 * q^32 - 5 * q^34 + 7 * q^35 - 4 * q^37 - 2 * q^38 + 16 * q^40 + 21 * q^41 + 11 * q^43 + 11 * q^44 + 31 * q^47 + 12 * q^49 + 21 * q^50 + 3 * q^52 + 38 * q^53 - 11 * q^55 + 15 * q^56 + 20 * q^58 + 15 * q^59 - 3 * q^61 + 4 * q^62 + 29 * q^64 + 32 * q^65 - q^67 - 17 * q^68 - 2 * q^70 + 57 * q^71 - 7 * q^73 + 42 * q^74 - 44 * q^76 + 22 * q^77 - 18 * q^79 - q^80 + 56 * q^82 + 21 * q^83 - 5 * q^85 + 32 * q^86 - 10 * q^88 - 6 * q^89 + 15 * q^92 + 35 * q^94 + 57 * q^95 + 4 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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