LMFDB - Embedded newform 4004.2.a.h.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 $ x^9 - 3x^8 - 19x^7 + 51x^6 + 116x^5 - 247x^4 - 249x^3 + 288x^2 + 189x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.32951$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32951 q^{3} +2.67963 q^{5} +1.00000 q^{7} +8.08561 q^{9} +O(q^{10})$	$q+3.32951 q^{3} +2.67963 q^{5} +1.00000 q^{7} +8.08561 q^{9} +1.00000 q^{11} -1.00000 q^{13} +8.92185 q^{15} -3.70081 q^{17} -5.63576 q^{19} +3.32951 q^{21} +3.88404 q^{23} +2.18043 q^{25} +16.9326 q^{27} +7.80078 q^{29} -9.28551 q^{31} +3.32951 q^{33} +2.67963 q^{35} -5.64089 q^{37} -3.32951 q^{39} -2.71252 q^{41} +7.67586 q^{43} +21.6665 q^{45} +7.32648 q^{47} +1.00000 q^{49} -12.3219 q^{51} +5.24425 q^{53} +2.67963 q^{55} -18.7643 q^{57} +4.28335 q^{59} -1.00731 q^{61} +8.08561 q^{63} -2.67963 q^{65} -13.1396 q^{67} +12.9319 q^{69} +10.8196 q^{71} +6.70355 q^{73} +7.25975 q^{75} +1.00000 q^{77} -8.98342 q^{79} +32.1203 q^{81} +5.93527 q^{83} -9.91682 q^{85} +25.9727 q^{87} +5.66928 q^{89} -1.00000 q^{91} -30.9162 q^{93} -15.1018 q^{95} -2.80912 q^{97} +8.08561 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9	$ 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * q^99

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	0
$2$	0	0
$3$	3.32951	1.92229	0.961146	−	0.276041i	$-0.0890227\pi$
$3$	3.32951	1.92229	0.961146	+	0.276041i	$0.0890227\pi$
$4$	0	0
$5$	2.67963	1.19837	0.599184	−	0.800611i	$-0.295492\pi$
$5$	2.67963	1.19837	0.599184	+	0.800611i	$0.295492\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.08561	2.69520
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	8.92185	2.30361
$16$	0	0
$17$	−3.70081	−0.897579	−0.448790	−	0.893637i	$-0.648145\pi$
$17$	−3.70081	−0.897579	−0.448790	+	0.893637i	$0.648145\pi$
$18$	0	0
$19$	−5.63576	−1.29293	−0.646466	−	0.762943i	$-0.723754\pi$
$19$	−5.63576	−1.29293	−0.646466	+	0.762943i	$0.723754\pi$
$20$	0	0
$21$	3.32951	0.726558
$22$	0	0
$23$	3.88404	0.809879	0.404939	−	0.914344i	$-0.367293\pi$
$23$	3.88404	0.809879	0.404939	+	0.914344i	$0.367293\pi$
$24$	0	0
$25$	2.18043	0.436086
$26$	0	0
$27$	16.9326	3.25868
$28$	0	0
$29$	7.80078	1.44857	0.724284	−	0.689502i	$-0.242170\pi$
$29$	7.80078	1.44857	0.724284	+	0.689502i	$0.242170\pi$
$30$	0	0
$31$	−9.28551	−1.66773	−0.833863	−	0.551971i	$-0.813876\pi$
$31$	−9.28551	−1.66773	−0.833863	+	0.551971i	$0.813876\pi$
$32$	0	0
$33$	3.32951	0.579593
$34$	0	0
$35$	2.67963	0.452940
$36$	0	0
$37$	−5.64089	−0.927357	−0.463678	−	0.886004i	$-0.653471\pi$
$37$	−5.64089	−0.927357	−0.463678	+	0.886004i	$0.653471\pi$
$38$	0	0
$39$	−3.32951	−0.533148
$40$	0	0
$41$	−2.71252	−0.423624	−0.211812	−	0.977310i	$-0.567936\pi$
$41$	−2.71252	−0.423624	−0.211812	+	0.977310i	$0.567936\pi$
$42$	0	0
$43$	7.67586	1.17056	0.585279	−	0.810832i	$-0.300985\pi$
$43$	7.67586	1.17056	0.585279	+	0.810832i	$0.300985\pi$
$44$	0	0
$45$	21.6665	3.22985
$46$	0	0
$47$	7.32648	1.06868	0.534339	−	0.845270i	$-0.320561\pi$
$47$	7.32648	1.06868	0.534339	+	0.845270i	$0.320561\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.3219	−1.72541
$52$	0	0
$53$	5.24425	0.720353	0.360176	−	0.932884i	$-0.382717\pi$
$53$	5.24425	0.720353	0.360176	+	0.932884i	$0.382717\pi$
$54$	0	0
$55$	2.67963	0.361322
$56$	0	0
$57$	−18.7643	−2.48539
$58$	0	0
$59$	4.28335	0.557645	0.278823	−	0.960343i	$-0.410056\pi$
$59$	4.28335	0.557645	0.278823	+	0.960343i	$0.410056\pi$
$60$	0	0
$61$	−1.00731	−0.128973	−0.0644865	−	0.997919i	$-0.520541\pi$
$61$	−1.00731	−0.128973	−0.0644865	+	0.997919i	$0.520541\pi$
$62$	0	0
$63$	8.08561	1.01869
$64$	0	0
$65$	−2.67963	−0.332367
$66$	0	0
$67$	−13.1396	−1.60526	−0.802631	−	0.596476i	$-0.796567\pi$
$67$	−13.1396	−1.60526	−0.802631	+	0.596476i	$0.796567\pi$
$68$	0	0
$69$	12.9319	1.55682
$70$	0	0
$71$	10.8196	1.28404	0.642022	−	0.766686i	$-0.278096\pi$
$71$	10.8196	1.28404	0.642022	+	0.766686i	$0.278096\pi$
$72$	0	0
$73$	6.70355	0.784591	0.392296	−	0.919839i	$-0.371681\pi$
$73$	6.70355	0.784591	0.392296	+	0.919839i	$0.371681\pi$
$74$	0	0
$75$	7.25975	0.838284
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−8.98342	−1.01071	−0.505357	−	0.862911i	$-0.668639\pi$
$79$	−8.98342	−1.01071	−0.505357	+	0.862911i	$0.668639\pi$
$80$	0	0
$81$	32.1203	3.56892
$82$	0	0
$83$	5.93527	0.651480	0.325740	−	0.945459i	$-0.394387\pi$
$83$	5.93527	0.651480	0.325740	+	0.945459i	$0.394387\pi$
$84$	0	0
$85$	−9.91682	−1.07563
$86$	0	0
$87$	25.9727	2.78457
$88$	0	0
$89$	5.66928	0.600943	0.300471	−	0.953791i	$-0.402856\pi$
$89$	5.66928	0.600943	0.300471	+	0.953791i	$0.402856\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−30.9162	−3.20586
$94$	0	0
$95$	−15.1018	−1.54941
$96$	0	0
$97$	−2.80912	−0.285223	−0.142611	−	0.989779i	$-0.545550\pi$
$97$	−2.80912	−0.285223	−0.142611	+	0.989779i	$0.545550\pi$
$98$	0	0
$99$	8.08561	0.812635
$100$	0	0
$101$	−8.94008	−0.889572	−0.444786	−	0.895637i	$-0.646720\pi$
$101$	−8.94008	−0.889572	−0.444786	+	0.895637i	$0.646720\pi$
$102$	0	0
$103$	−3.13701	−0.309099	−0.154549	−	0.987985i	$-0.549393\pi$
$103$	−3.13701	−0.309099	−0.154549	+	0.987985i	$0.549393\pi$
$104$	0	0
$105$	8.92185	0.870684
$106$	0	0
$107$	−16.7795	−1.62213	−0.811065	−	0.584956i	$-0.801112\pi$
$107$	−16.7795	−1.62213	−0.811065	+	0.584956i	$0.801112\pi$
$108$	0	0
$109$	−8.03290	−0.769413	−0.384706	−	0.923039i	$-0.625697\pi$
$109$	−8.03290	−0.769413	−0.384706	+	0.923039i	$0.625697\pi$
$110$	0	0
$111$	−18.7814	−1.78265
$112$	0	0
$113$	−0.313852	−0.0295247	−0.0147624	−	0.999891i	$-0.504699\pi$
$113$	−0.313852	−0.0295247	−0.0147624	+	0.999891i	$0.504699\pi$
$114$	0	0
$115$	10.4078	0.970533
$116$	0	0
$117$	−8.08561	−0.747515
$118$	0	0
$119$	−3.70081	−0.339253
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.03135	−0.814329
$124$	0	0
$125$	−7.55542	−0.675777
$126$	0	0
$127$	−6.70802	−0.595241	−0.297620	−	0.954684i	$-0.596193\pi$
$127$	−6.70802	−0.595241	−0.297620	+	0.954684i	$0.596193\pi$
$128$	0	0
$129$	25.5568	2.25015
$130$	0	0
$131$	−9.59484	−0.838305	−0.419153	−	0.907916i	$-0.637673\pi$
$131$	−9.59484	−0.838305	−0.419153	+	0.907916i	$0.637673\pi$
$132$	0	0
$133$	−5.63576	−0.488682
$134$	0	0
$135$	45.3731	3.90509
$136$	0	0
$137$	−5.94039	−0.507522	−0.253761	−	0.967267i	$-0.581668\pi$
$137$	−5.94039	−0.507522	−0.253761	+	0.967267i	$0.581668\pi$
$138$	0	0
$139$	−3.95060	−0.335086	−0.167543	−	0.985865i	$-0.553583\pi$
$139$	−3.95060	−0.335086	−0.167543	+	0.985865i	$0.553583\pi$
$140$	0	0
$141$	24.3936	2.05431
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	20.9032	1.73592
$146$	0	0
$147$	3.32951	0.274613
$148$	0	0
$149$	−21.2846	−1.74370	−0.871851	−	0.489771i	$-0.837080\pi$
$149$	−21.2846	−1.74370	−0.871851	+	0.489771i	$0.837080\pi$
$150$	0	0
$151$	−19.8506	−1.61542	−0.807711	−	0.589579i	$-0.799294\pi$
$151$	−19.8506	−1.61542	−0.807711	+	0.589579i	$0.799294\pi$
$152$	0	0
$153$	−29.9234	−2.41916
$154$	0	0
$155$	−24.8817	−1.99855
$156$	0	0
$157$	24.2532	1.93562	0.967809	−	0.251686i	$-0.0809852\pi$
$157$	24.2532	1.93562	0.967809	+	0.251686i	$0.0809852\pi$
$158$	0	0
$159$	17.4607	1.38473
$160$	0	0
$161$	3.88404	0.306105
$162$	0	0
$163$	19.4726	1.52521	0.762607	−	0.646863i	$-0.223919\pi$
$163$	19.4726	1.52521	0.762607	+	0.646863i	$0.223919\pi$
$164$	0	0
$165$	8.92185	0.694565
$166$	0	0
$167$	17.1008	1.32330	0.661651	−	0.749812i	$-0.269856\pi$
$167$	17.1008	1.32330	0.661651	+	0.749812i	$0.269856\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−45.5686	−3.48472
$172$	0	0
$173$	14.7636	1.12245	0.561227	−	0.827662i	$-0.310330\pi$
$173$	14.7636	1.12245	0.561227	+	0.827662i	$0.310330\pi$
$174$	0	0
$175$	2.18043	0.164825
$176$	0	0
$177$	14.2615	1.07196
$178$	0	0
$179$	−10.6063	−0.792755	−0.396378	−	0.918088i	$-0.629733\pi$
$179$	−10.6063	−0.792755	−0.396378	+	0.918088i	$0.629733\pi$
$180$	0	0
$181$	−0.148220	−0.0110171	−0.00550856	−	0.999985i	$-0.501753\pi$
$181$	−0.148220	−0.0110171	−0.00550856	+	0.999985i	$0.501753\pi$
$182$	0	0
$183$	−3.35385	−0.247924
$184$	0	0
$185$	−15.1155	−1.11131
$186$	0	0
$187$	−3.70081	−0.270630
$188$	0	0
$189$	16.9326	1.23166
$190$	0	0
$191$	−12.8403	−0.929090	−0.464545	−	0.885549i	$-0.653782\pi$
$191$	−12.8403	−0.929090	−0.464545	+	0.885549i	$0.653782\pi$
$192$	0	0
$193$	−4.41284	−0.317643	−0.158822	−	0.987307i	$-0.550770\pi$
$193$	−4.41284	−0.317643	−0.158822	+	0.987307i	$0.550770\pi$
$194$	0	0
$195$	−8.92185	−0.638907
$196$	0	0
$197$	−1.56529	−0.111523	−0.0557613	−	0.998444i	$-0.517759\pi$
$197$	−1.56529	−0.111523	−0.0557613	+	0.998444i	$0.517759\pi$
$198$	0	0
$199$	−14.3258	−1.01553	−0.507765	−	0.861496i	$-0.669528\pi$
$199$	−14.3258	−1.01553	−0.507765	+	0.861496i	$0.669528\pi$
$200$	0	0
$201$	−43.7485	−3.08578
$202$	0	0
$203$	7.80078	0.547507
$204$	0	0
$205$	−7.26855	−0.507658
$206$	0	0
$207$	31.4049	2.18279
$208$	0	0
$209$	−5.63576	−0.389834
$210$	0	0
$211$	−8.94342	−0.615690	−0.307845	−	0.951436i	$-0.599608\pi$
$211$	−8.94342	−0.615690	−0.307845	+	0.951436i	$0.599608\pi$
$212$	0	0
$213$	36.0238	2.46831
$214$	0	0
$215$	20.5685	1.40276
$216$	0	0
$217$	−9.28551	−0.630341
$218$	0	0
$219$	22.3195	1.50821
$220$	0	0
$221$	3.70081	0.248944
$222$	0	0
$223$	−25.7175	−1.72217	−0.861087	−	0.508458i	$-0.830216\pi$
$223$	−25.7175	−1.72217	−0.861087	+	0.508458i	$0.830216\pi$
$224$	0	0
$225$	17.6301	1.17534
$226$	0	0
$227$	23.7442	1.57596	0.787980	−	0.615701i	$-0.211127\pi$
$227$	23.7442	1.57596	0.787980	+	0.615701i	$0.211127\pi$
$228$	0	0
$229$	−0.261845	−0.0173032	−0.00865160	−	0.999963i	$-0.502754\pi$
$229$	−0.261845	−0.0173032	−0.00865160	+	0.999963i	$0.502754\pi$
$230$	0	0
$231$	3.32951	0.219065
$232$	0	0
$233$	27.3989	1.79496	0.897479	−	0.441056i	$-0.145396\pi$
$233$	27.3989	1.79496	0.897479	+	0.441056i	$0.145396\pi$
$234$	0	0
$235$	19.6323	1.28067
$236$	0	0
$237$	−29.9103	−1.94289
$238$	0	0
$239$	−15.1205	−0.978065	−0.489033	−	0.872266i	$-0.662650\pi$
$239$	−15.1205	−0.978065	−0.489033	+	0.872266i	$0.662650\pi$
$240$	0	0
$241$	−4.87782	−0.314208	−0.157104	−	0.987582i	$-0.550216\pi$
$241$	−4.87782	−0.314208	−0.157104	+	0.987582i	$0.550216\pi$
$242$	0	0
$243$	56.1470	3.60183
$244$	0	0
$245$	2.67963	0.171195
$246$	0	0
$247$	5.63576	0.358595
$248$	0	0
$249$	19.7615	1.25233
$250$	0	0
$251$	17.8066	1.12394	0.561972	−	0.827156i	$-0.310043\pi$
$251$	17.8066	1.12394	0.561972	+	0.827156i	$0.310043\pi$
$252$	0	0
$253$	3.88404	0.244188
$254$	0	0
$255$	−33.0181	−2.06768
$256$	0	0
$257$	−26.2116	−1.63503	−0.817517	−	0.575905i	$-0.804650\pi$
$257$	−26.2116	−1.63503	−0.817517	+	0.575905i	$0.804650\pi$
$258$	0	0
$259$	−5.64089	−0.350508
$260$	0	0
$261$	63.0741	3.90419
$262$	0	0
$263$	−7.57386	−0.467024	−0.233512	−	0.972354i	$-0.575022\pi$
$263$	−7.57386	−0.467024	−0.233512	+	0.972354i	$0.575022\pi$
$264$	0	0
$265$	14.0526	0.863247
$266$	0	0
$267$	18.8759	1.15519
$268$	0	0
$269$	19.3770	1.18144	0.590719	−	0.806877i	$-0.298844\pi$
$269$	19.3770	1.18144	0.590719	+	0.806877i	$0.298844\pi$
$270$	0	0
$271$	−4.51451	−0.274237	−0.137119	−	0.990555i	$-0.543784\pi$
$271$	−4.51451	−0.274237	−0.137119	+	0.990555i	$0.543784\pi$
$272$	0	0
$273$	−3.32951	−0.201511
$274$	0	0
$275$	2.18043	0.131485
$276$	0	0
$277$	−20.8050	−1.25005	−0.625026	−	0.780604i	$-0.714911\pi$
$277$	−20.8050	−1.25005	−0.625026	+	0.780604i	$0.714911\pi$
$278$	0	0
$279$	−75.0790	−4.49486
$280$	0	0
$281$	0.394052	0.0235072	0.0117536	−	0.999931i	$-0.496259\pi$
$281$	0.394052	0.0235072	0.0117536	+	0.999931i	$0.496259\pi$
$282$	0	0
$283$	20.9926	1.24788	0.623942	−	0.781471i	$-0.285530\pi$
$283$	20.9926	1.24788	0.623942	+	0.781471i	$0.285530\pi$
$284$	0	0
$285$	−50.2814	−2.97841
$286$	0	0
$287$	−2.71252	−0.160115
$288$	0	0
$289$	−3.30397	−0.194351
$290$	0	0
$291$	−9.35297	−0.548281
$292$	0	0
$293$	−2.07627	−0.121297	−0.0606483	−	0.998159i	$-0.519317\pi$
$293$	−2.07627	−0.121297	−0.0606483	+	0.998159i	$0.519317\pi$
$294$	0	0
$295$	11.4778	0.668264
$296$	0	0
$297$	16.9326	0.982528
$298$	0	0
$299$	−3.88404	−0.224620
$300$	0	0
$301$	7.67586	0.442429
$302$	0	0
$303$	−29.7661	−1.71002
$304$	0	0
$305$	−2.69922	−0.154557
$306$	0	0
$307$	−19.1790	−1.09460	−0.547301	−	0.836936i	$-0.684345\pi$
$307$	−19.1790	−1.09460	−0.547301	+	0.836936i	$0.684345\pi$
$308$	0	0
$309$	−10.4447	−0.594178
$310$	0	0
$311$	7.39009	0.419053	0.209527	−	0.977803i	$-0.432808\pi$
$311$	7.39009	0.419053	0.209527	+	0.977803i	$0.432808\pi$
$312$	0	0
$313$	29.1450	1.64737	0.823687	−	0.567044i	$-0.191913\pi$
$313$	29.1450	1.64737	0.823687	+	0.567044i	$0.191913\pi$
$314$	0	0
$315$	21.6665	1.22077
$316$	0	0
$317$	−32.4345	−1.82170	−0.910851	−	0.412735i	$-0.864573\pi$
$317$	−32.4345	−1.82170	−0.910851	+	0.412735i	$0.864573\pi$
$318$	0	0
$319$	7.80078	0.436760
$320$	0	0
$321$	−55.8673	−3.11821
$322$	0	0
$323$	20.8569	1.16051
$324$	0	0
$325$	−2.18043	−0.120948
$326$	0	0
$327$	−26.7456	−1.47904
$328$	0	0
$329$	7.32648	0.403922
$330$	0	0
$331$	2.59976	0.142896	0.0714480	−	0.997444i	$-0.477238\pi$
$331$	2.59976	0.142896	0.0714480	+	0.997444i	$0.477238\pi$
$332$	0	0
$333$	−45.6101	−2.49942
$334$	0	0
$335$	−35.2094	−1.92369
$336$	0	0
$337$	−16.1727	−0.880981	−0.440490	−	0.897757i	$-0.645195\pi$
$337$	−16.1727	−0.880981	−0.440490	+	0.897757i	$0.645195\pi$
$338$	0	0
$339$	−1.04497	−0.0567551
$340$	0	0
$341$	−9.28551	−0.502838
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	34.6529	1.86565
$346$	0	0
$347$	−12.2479	−0.657500	−0.328750	−	0.944417i	$-0.606627\pi$
$347$	−12.2479	−0.657500	−0.328750	+	0.944417i	$0.606627\pi$
$348$	0	0
$349$	−22.8626	−1.22381	−0.611904	−	0.790932i	$-0.709596\pi$
$349$	−22.8626	−1.22381	−0.611904	+	0.790932i	$0.709596\pi$
$350$	0	0
$351$	−16.9326	−0.903794
$352$	0	0
$353$	−32.1825	−1.71290	−0.856451	−	0.516228i	$-0.827336\pi$
$353$	−32.1825	−1.71290	−0.856451	+	0.516228i	$0.827336\pi$
$354$	0	0
$355$	28.9924	1.53876
$356$	0	0
$357$	−12.3219	−0.652143
$358$	0	0
$359$	4.49600	0.237290	0.118645	−	0.992937i	$-0.462145\pi$
$359$	4.49600	0.237290	0.118645	+	0.992937i	$0.462145\pi$
$360$	0	0
$361$	12.7618	0.671674
$362$	0	0
$363$	3.32951	0.174754
$364$	0	0
$365$	17.9631	0.940229
$366$	0	0
$367$	0.466331	0.0243423	0.0121711	−	0.999926i	$-0.496126\pi$
$367$	0.466331	0.0243423	0.0121711	+	0.999926i	$0.496126\pi$
$368$	0	0
$369$	−21.9324	−1.14175
$370$	0	0
$371$	5.24425	0.272268
$372$	0	0
$373$	−2.54982	−0.132024	−0.0660122	−	0.997819i	$-0.521028\pi$
$373$	−2.54982	−0.132024	−0.0660122	+	0.997819i	$0.521028\pi$
$374$	0	0
$375$	−25.1558	−1.29904
$376$	0	0
$377$	−7.80078	−0.401760
$378$	0	0
$379$	8.63405	0.443501	0.221751	−	0.975103i	$-0.428823\pi$
$379$	8.63405	0.443501	0.221751	+	0.975103i	$0.428823\pi$
$380$	0	0
$381$	−22.3344	−1.14423
$382$	0	0
$383$	5.09710	0.260450	0.130225	−	0.991484i	$-0.458430\pi$
$383$	5.09710	0.260450	0.130225	+	0.991484i	$0.458430\pi$
$384$	0	0
$385$	2.67963	0.136567
$386$	0	0
$387$	62.0640	3.15489
$388$	0	0
$389$	−11.4209	−0.579064	−0.289532	−	0.957168i	$-0.593500\pi$
$389$	−11.4209	−0.579064	−0.289532	+	0.957168i	$0.593500\pi$
$390$	0	0
$391$	−14.3741	−0.726931
$392$	0	0
$393$	−31.9461	−1.61147
$394$	0	0
$395$	−24.0723	−1.21121
$396$	0	0
$397$	−11.9913	−0.601824	−0.300912	−	0.953652i	$-0.597291\pi$
$397$	−11.9913	−0.601824	−0.300912	+	0.953652i	$0.597291\pi$
$398$	0	0
$399$	−18.7643	−0.939390
$400$	0	0
$401$	25.6053	1.27867	0.639334	−	0.768929i	$-0.279210\pi$
$401$	25.6053	1.27867	0.639334	+	0.768929i	$0.279210\pi$
$402$	0	0
$403$	9.28551	0.462544
$404$	0	0
$405$	86.0706	4.27688
$406$	0	0
$407$	−5.64089	−0.279609
$408$	0	0
$409$	19.8111	0.979595	0.489797	−	0.871836i	$-0.337071\pi$
$409$	19.8111	0.979595	0.489797	+	0.871836i	$0.337071\pi$
$410$	0	0
$411$	−19.7786	−0.975605
$412$	0	0
$413$	4.28335	0.210770
$414$	0	0
$415$	15.9043	0.780713
$416$	0	0
$417$	−13.1536	−0.644133
$418$	0	0
$419$	35.7617	1.74707	0.873537	−	0.486757i	$-0.161820\pi$
$419$	35.7617	1.74707	0.873537	+	0.486757i	$0.161820\pi$
$420$	0	0
$421$	30.9745	1.50960	0.754802	−	0.655952i	$-0.227733\pi$
$421$	30.9745	1.50960	0.754802	+	0.655952i	$0.227733\pi$
$422$	0	0
$423$	59.2391	2.88030
$424$	0	0
$425$	−8.06936	−0.391421
$426$	0	0
$427$	−1.00731	−0.0487472
$428$	0	0
$429$	−3.32951	−0.160750
$430$	0	0
$431$	29.3822	1.41529	0.707645	−	0.706569i	$-0.249758\pi$
$431$	29.3822	1.41529	0.707645	+	0.706569i	$0.249758\pi$
$432$	0	0
$433$	37.5368	1.80391	0.901953	−	0.431835i	$-0.142134\pi$
$433$	37.5368	1.80391	0.901953	+	0.431835i	$0.142134\pi$
$434$	0	0
$435$	69.5974	3.33694
$436$	0	0
$437$	−21.8895	−1.04712
$438$	0	0
$439$	22.1529	1.05730	0.528649	−	0.848840i	$-0.322699\pi$
$439$	22.1529	1.05730	0.528649	+	0.848840i	$0.322699\pi$
$440$	0	0
$441$	8.08561	0.385029
$442$	0	0
$443$	−26.3833	−1.25351	−0.626753	−	0.779218i	$-0.715617\pi$
$443$	−26.3833	−1.25351	−0.626753	+	0.779218i	$0.715617\pi$
$444$	0	0
$445$	15.1916	0.720150
$446$	0	0
$447$	−70.8672	−3.35190
$448$	0	0
$449$	19.2032	0.906257	0.453129	−	0.891445i	$-0.350308\pi$
$449$	19.2032	0.906257	0.453129	+	0.891445i	$0.350308\pi$
$450$	0	0
$451$	−2.71252	−0.127727
$452$	0	0
$453$	−66.0928	−3.10531
$454$	0	0
$455$	−2.67963	−0.125623
$456$	0	0
$457$	19.1971	0.898001	0.449001	−	0.893531i	$-0.351780\pi$
$457$	19.1971	0.898001	0.449001	+	0.893531i	$0.351780\pi$
$458$	0	0
$459$	−62.6644	−2.92492
$460$	0	0
$461$	16.6439	0.775185	0.387593	−	0.921831i	$-0.373307\pi$
$461$	16.6439	0.775185	0.387593	+	0.921831i	$0.373307\pi$
$462$	0	0
$463$	−21.3940	−0.994265	−0.497132	−	0.867675i	$-0.665614\pi$
$463$	−21.3940	−0.994265	−0.497132	+	0.867675i	$0.665614\pi$
$464$	0	0
$465$	−82.8439	−3.84180
$466$	0	0
$467$	3.49880	0.161905	0.0809527	−	0.996718i	$-0.474204\pi$
$467$	3.49880	0.161905	0.0809527	+	0.996718i	$0.474204\pi$
$468$	0	0
$469$	−13.1396	−0.606732
$470$	0	0
$471$	80.7513	3.72082
$472$	0	0
$473$	7.67586	0.352936
$474$	0	0
$475$	−12.2884	−0.563829
$476$	0	0
$477$	42.4029	1.94150
$478$	0	0
$479$	14.7128	0.672244	0.336122	−	0.941818i	$-0.390885\pi$
$479$	14.7128	0.672244	0.336122	+	0.941818i	$0.390885\pi$
$480$	0	0
$481$	5.64089	0.257202
$482$	0	0
$483$	12.9319	0.588424
$484$	0	0
$485$	−7.52740	−0.341802
$486$	0	0
$487$	−24.5098	−1.11065	−0.555323	−	0.831635i	$-0.687405\pi$
$487$	−24.5098	−1.11065	−0.555323	+	0.831635i	$0.687405\pi$
$488$	0	0
$489$	64.8342	2.93190
$490$	0	0
$491$	−32.7641	−1.47863	−0.739313	−	0.673362i	$-0.764849\pi$
$491$	−32.7641	−1.47863	−0.739313	+	0.673362i	$0.764849\pi$
$492$	0	0
$493$	−28.8692	−1.30020
$494$	0	0
$495$	21.6665	0.973835
$496$	0	0
$497$	10.8196	0.485323
$498$	0	0
$499$	13.7627	0.616105	0.308052	−	0.951369i	$-0.400323\pi$
$499$	13.7627	0.616105	0.308052	+	0.951369i	$0.400323\pi$
$500$	0	0
$501$	56.9373	2.54377
$502$	0	0
$503$	32.3156	1.44088	0.720440	−	0.693518i	$-0.243940\pi$
$503$	32.3156	1.44088	0.720440	+	0.693518i	$0.243940\pi$
$504$	0	0
$505$	−23.9561	−1.06603
$506$	0	0
$507$	3.32951	0.147869
$508$	0	0
$509$	−13.7088	−0.607634	−0.303817	−	0.952730i	$-0.598261\pi$
$509$	−13.7088	−0.607634	−0.303817	+	0.952730i	$0.598261\pi$
$510$	0	0
$511$	6.70355	0.296548
$512$	0	0
$513$	−95.4280	−4.21325
$514$	0	0
$515$	−8.40603	−0.370414
$516$	0	0
$517$	7.32648	0.322218
$518$	0	0
$519$	49.1554	2.15768
$520$	0	0
$521$	5.11723	0.224190	0.112095	−	0.993698i	$-0.464244\pi$
$521$	5.11723	0.224190	0.112095	+	0.993698i	$0.464244\pi$
$522$	0	0
$523$	3.86684	0.169085	0.0845425	−	0.996420i	$-0.473057\pi$
$523$	3.86684	0.169085	0.0845425	+	0.996420i	$0.473057\pi$
$524$	0	0
$525$	7.25975	0.316841
$526$	0	0
$527$	34.3639	1.49692
$528$	0	0
$529$	−7.91421	−0.344096
$530$	0	0
$531$	34.6335	1.50297
$532$	0	0
$533$	2.71252	0.117492
$534$	0	0
$535$	−44.9628	−1.94391
$536$	0	0
$537$	−35.3139	−1.52391
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−16.7912	−0.721912	−0.360956	−	0.932583i	$-0.617550\pi$
$541$	−16.7912	−0.721912	−0.360956	+	0.932583i	$0.617550\pi$
$542$	0	0
$543$	−0.493500	−0.0211781
$544$	0	0
$545$	−21.5252	−0.922039
$546$	0	0
$547$	−24.4429	−1.04510	−0.522551	−	0.852608i	$-0.675020\pi$
$547$	−24.4429	−1.04510	−0.522551	+	0.852608i	$0.675020\pi$
$548$	0	0
$549$	−8.14473	−0.347609
$550$	0	0
$551$	−43.9633	−1.87290
$552$	0	0
$553$	−8.98342	−0.382014
$554$	0	0
$555$	−50.3272	−2.13627
$556$	0	0
$557$	22.5170	0.954077	0.477038	−	0.878882i	$-0.341710\pi$
$557$	22.5170	0.954077	0.477038	+	0.878882i	$0.341710\pi$
$558$	0	0
$559$	−7.67586	−0.324654
$560$	0	0
$561$	−12.3219	−0.520230
$562$	0	0
$563$	−5.18543	−0.218540	−0.109270	−	0.994012i	$-0.534851\pi$
$563$	−5.18543	−0.218540	−0.109270	+	0.994012i	$0.534851\pi$
$564$	0	0
$565$	−0.841008	−0.0353815
$566$	0	0
$567$	32.1203	1.34893
$568$	0	0
$569$	−5.90356	−0.247490	−0.123745	−	0.992314i	$-0.539491\pi$
$569$	−5.90356	−0.247490	−0.123745	+	0.992314i	$0.539491\pi$
$570$	0	0
$571$	27.2888	1.14200	0.571001	−	0.820949i	$-0.306555\pi$
$571$	27.2888	1.14200	0.571001	+	0.820949i	$0.306555\pi$
$572$	0	0
$573$	−42.7518	−1.78598
$574$	0	0
$575$	8.46888	0.353177
$576$	0	0
$577$	11.1003	0.462112	0.231056	−	0.972940i	$-0.425782\pi$
$577$	11.1003	0.462112	0.231056	+	0.972940i	$0.425782\pi$
$578$	0	0
$579$	−14.6926	−0.610603
$580$	0	0
$581$	5.93527	0.246236
$582$	0	0
$583$	5.24425	0.217194
$584$	0	0
$585$	−21.6665	−0.895798
$586$	0	0
$587$	−23.5968	−0.973946	−0.486973	−	0.873417i	$-0.661899\pi$
$587$	−23.5968	−0.973946	−0.486973	+	0.873417i	$0.661899\pi$
$588$	0	0
$589$	52.3309	2.15626
$590$	0	0
$591$	−5.21166	−0.214379
$592$	0	0
$593$	−11.7002	−0.480471	−0.240235	−	0.970715i	$-0.577225\pi$
$593$	−11.7002	−0.480471	−0.240235	+	0.970715i	$0.577225\pi$
$594$	0	0
$595$	−9.91682	−0.406550
$596$	0	0
$597$	−47.6979	−1.95214
$598$	0	0
$599$	−12.5714	−0.513654	−0.256827	−	0.966457i	$-0.582677\pi$
$599$	−12.5714	−0.513654	−0.256827	+	0.966457i	$0.582677\pi$
$600$	0	0
$601$	−7.28812	−0.297289	−0.148644	−	0.988891i	$-0.547491\pi$
$601$	−7.28812	−0.297289	−0.148644	+	0.988891i	$0.547491\pi$
$602$	0	0
$603$	−106.242	−4.32651
$604$	0	0
$605$	2.67963	0.108943
$606$	0	0
$607$	37.7274	1.53131	0.765653	−	0.643253i	$-0.222416\pi$
$607$	37.7274	1.53131	0.765653	+	0.643253i	$0.222416\pi$
$608$	0	0
$609$	25.9727	1.05247
$610$	0	0
$611$	−7.32648	−0.296398
$612$	0	0
$613$	12.6653	0.511546	0.255773	−	0.966737i	$-0.417670\pi$
$613$	12.6653	0.511546	0.255773	+	0.966737i	$0.417670\pi$
$614$	0	0
$615$	−24.2007	−0.975866
$616$	0	0
$617$	44.4296	1.78867	0.894335	−	0.447399i	$-0.147650\pi$
$617$	44.4296	1.78867	0.894335	+	0.447399i	$0.147650\pi$
$618$	0	0
$619$	34.3514	1.38070	0.690350	−	0.723475i	$-0.257456\pi$
$619$	34.3514	1.38070	0.690350	+	0.723475i	$0.257456\pi$
$620$	0	0
$621$	65.7669	2.63913
$622$	0	0
$623$	5.66928	0.227135
$624$	0	0
$625$	−31.1479	−1.24591
$626$	0	0
$627$	−18.7643	−0.749374
$628$	0	0
$629$	20.8759	0.832376
$630$	0	0
$631$	−28.5043	−1.13474	−0.567370	−	0.823463i	$-0.692039\pi$
$631$	−28.5043	−1.13474	−0.567370	+	0.823463i	$0.692039\pi$
$632$	0	0
$633$	−29.7772	−1.18354
$634$	0	0
$635$	−17.9750	−0.713317
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	87.4827	3.46076
$640$	0	0
$641$	19.3930	0.765977	0.382989	−	0.923753i	$-0.374895\pi$
$641$	19.3930	0.765977	0.382989	+	0.923753i	$0.374895\pi$
$642$	0	0
$643$	15.2063	0.599677	0.299839	−	0.953990i	$-0.403067\pi$
$643$	15.2063	0.599677	0.299839	+	0.953990i	$0.403067\pi$
$644$	0	0
$645$	68.4829	2.69651
$646$	0	0
$647$	−5.20796	−0.204746	−0.102373	−	0.994746i	$-0.532644\pi$
$647$	−5.20796	−0.204746	−0.102373	+	0.994746i	$0.532644\pi$
$648$	0	0
$649$	4.28335	0.168136
$650$	0	0
$651$	−30.9162	−1.21170
$652$	0	0
$653$	1.67152	0.0654118	0.0327059	−	0.999465i	$-0.489588\pi$
$653$	1.67152	0.0654118	0.0327059	+	0.999465i	$0.489588\pi$
$654$	0	0
$655$	−25.7106	−1.00460
$656$	0	0
$657$	54.2023	2.11463
$658$	0	0
$659$	43.9094	1.71047	0.855234	−	0.518241i	$-0.173413\pi$
$659$	43.9094	1.71047	0.855234	+	0.518241i	$0.173413\pi$
$660$	0	0
$661$	−6.04920	−0.235286	−0.117643	−	0.993056i	$-0.537534\pi$
$661$	−6.04920	−0.235286	−0.117643	+	0.993056i	$0.537534\pi$
$662$	0	0
$663$	12.3219	0.478542
$664$	0	0
$665$	−15.1018	−0.585621
$666$	0	0
$667$	30.2985	1.17316
$668$	0	0
$669$	−85.6267	−3.31052
$670$	0	0
$671$	−1.00731	−0.0388868
$672$	0	0
$673$	25.6959	0.990505	0.495252	−	0.868749i	$-0.335076\pi$
$673$	25.6959	0.990505	0.495252	+	0.868749i	$0.335076\pi$
$674$	0	0
$675$	36.9203	1.42106
$676$	0	0
$677$	42.1063	1.61828	0.809139	−	0.587617i	$-0.199934\pi$
$677$	42.1063	1.61828	0.809139	+	0.587617i	$0.199934\pi$
$678$	0	0
$679$	−2.80912	−0.107804
$680$	0	0
$681$	79.0566	3.02945
$682$	0	0
$683$	−24.9902	−0.956224	−0.478112	−	0.878299i	$-0.658679\pi$
$683$	−24.9902	−0.956224	−0.478112	+	0.878299i	$0.658679\pi$
$684$	0	0
$685$	−15.9181	−0.608198
$686$	0	0
$687$	−0.871815	−0.0332618
$688$	0	0
$689$	−5.24425	−0.199790
$690$	0	0
$691$	−16.5206	−0.628473	−0.314237	−	0.949345i	$-0.601748\pi$
$691$	−16.5206	−0.628473	−0.314237	+	0.949345i	$0.601748\pi$
$692$	0	0
$693$	8.08561	0.307147
$694$	0	0
$695$	−10.5862	−0.401556
$696$	0	0
$697$	10.0385	0.380236
$698$	0	0
$699$	91.2247	3.45043
$700$	0	0
$701$	19.0747	0.720443	0.360221	−	0.932867i	$-0.382701\pi$
$701$	19.0747	0.720443	0.360221	+	0.932867i	$0.382701\pi$
$702$	0	0
$703$	31.7907	1.19901
$704$	0	0
$705$	65.3658	2.46182
$706$	0	0
$707$	−8.94008	−0.336226
$708$	0	0
$709$	−39.5225	−1.48430	−0.742149	−	0.670235i	$-0.766193\pi$
$709$	−39.5225	−1.48430	−0.742149	+	0.670235i	$0.766193\pi$
$710$	0	0
$711$	−72.6364	−2.72408
$712$	0	0
$713$	−36.0653	−1.35066
$714$	0	0
$715$	−2.67963	−0.100213
$716$	0	0
$717$	−50.3439	−1.88013
$718$	0	0
$719$	−33.8137	−1.26104	−0.630519	−	0.776174i	$-0.717158\pi$
$719$	−33.8137	−1.26104	−0.630519	+	0.776174i	$0.717158\pi$
$720$	0	0
$721$	−3.13701	−0.116828
$722$	0	0
$723$	−16.2407	−0.603999
$724$	0	0
$725$	17.0090	0.631700
$726$	0	0
$727$	−3.17123	−0.117615	−0.0588073	−	0.998269i	$-0.518730\pi$
$727$	−3.17123	−0.117615	−0.0588073	+	0.998269i	$0.518730\pi$
$728$	0	0
$729$	90.5809	3.35485
$730$	0	0
$731$	−28.4069	−1.05067
$732$	0	0
$733$	−3.15545	−0.116549	−0.0582746	−	0.998301i	$-0.518560\pi$
$733$	−3.15545	−0.116549	−0.0582746	+	0.998301i	$0.518560\pi$
$734$	0	0
$735$	8.92185	0.329087
$736$	0	0
$737$	−13.1396	−0.484005
$738$	0	0
$739$	−37.9978	−1.39777	−0.698886	−	0.715233i	$-0.746321\pi$
$739$	−37.9978	−1.39777	−0.698886	+	0.715233i	$0.746321\pi$
$740$	0	0
$741$	18.7643	0.689324
$742$	0	0
$743$	36.4939	1.33883	0.669416	−	0.742888i	$-0.266544\pi$
$743$	36.4939	1.33883	0.669416	+	0.742888i	$0.266544\pi$
$744$	0	0
$745$	−57.0349	−2.08960
$746$	0	0
$747$	47.9903	1.75587
$748$	0	0
$749$	−16.7795	−0.613108
$750$	0	0
$751$	−31.8986	−1.16400	−0.581998	−	0.813190i	$-0.697729\pi$
$751$	−31.8986	−1.16400	−0.581998	+	0.813190i	$0.697729\pi$
$752$	0	0
$753$	59.2873	2.16055
$754$	0	0
$755$	−53.1924	−1.93587
$756$	0	0
$757$	18.1091	0.658186	0.329093	−	0.944298i	$-0.393257\pi$
$757$	18.1091	0.658186	0.329093	+	0.944298i	$0.393257\pi$
$758$	0	0
$759$	12.9319	0.469400
$760$	0	0
$761$	13.7067	0.496867	0.248433	−	0.968649i	$-0.420084\pi$
$761$	13.7067	0.496867	0.248433	+	0.968649i	$0.420084\pi$
$762$	0	0
$763$	−8.03290	−0.290811
$764$	0	0
$765$	−80.1836	−2.89904
$766$	0	0
$767$	−4.28335	−0.154663
$768$	0	0
$769$	−17.7067	−0.638520	−0.319260	−	0.947667i	$-0.603434\pi$
$769$	−17.7067	−0.638520	−0.319260	+	0.947667i	$0.603434\pi$
$770$	0	0
$771$	−87.2716	−3.14301
$772$	0	0
$773$	−44.1419	−1.58767	−0.793837	−	0.608130i	$-0.791920\pi$
$773$	−44.1419	−1.58767	−0.793837	+	0.608130i	$0.791920\pi$
$774$	0	0
$775$	−20.2464	−0.727272
$776$	0	0
$777$	−18.7814	−0.673778
$778$	0	0
$779$	15.2871	0.547717
$780$	0	0
$781$	10.8196	0.387154
$782$	0	0
$783$	132.087	4.72041
$784$	0	0
$785$	64.9897	2.31958
$786$	0	0
$787$	21.0126	0.749019	0.374510	−	0.927223i	$-0.377811\pi$
$787$	21.0126	0.749019	0.374510	+	0.927223i	$0.377811\pi$
$788$	0	0
$789$	−25.2172	−0.897756
$790$	0	0
$791$	−0.313852	−0.0111593
$792$	0	0
$793$	1.00731	0.0357707
$794$	0	0
$795$	46.7884	1.65941
$796$	0	0
$797$	−2.76240	−0.0978492	−0.0489246	−	0.998802i	$-0.515579\pi$
$797$	−2.76240	−0.0978492	−0.0489246	+	0.998802i	$0.515579\pi$
$798$	0	0
$799$	−27.1140	−0.959223
$800$	0	0
$801$	45.8396	1.61966
$802$	0	0
$803$	6.70355	0.236563
$804$	0	0
$805$	10.4078	0.366827
$806$	0	0
$807$	64.5159	2.27107
$808$	0	0
$809$	50.8129	1.78649	0.893243	−	0.449574i	$-0.148424\pi$
$809$	50.8129	1.78649	0.893243	+	0.449574i	$0.148424\pi$
$810$	0	0
$811$	−13.6550	−0.479492	−0.239746	−	0.970836i	$-0.577064\pi$
$811$	−13.6550	−0.479492	−0.239746	+	0.970836i	$0.577064\pi$
$812$	0	0
$813$	−15.0311	−0.527164
$814$	0	0
$815$	52.1795	1.82777
$816$	0	0
$817$	−43.2593	−1.51345
$818$	0	0
$819$	−8.08561	−0.282534
$820$	0	0
$821$	−35.0989	−1.22496	−0.612480	−	0.790486i	$-0.709828\pi$
$821$	−35.0989	−1.22496	−0.612480	+	0.790486i	$0.709828\pi$
$822$	0	0
$823$	−47.0382	−1.63965	−0.819825	−	0.572615i	$-0.805929\pi$
$823$	−47.0382	−1.63965	−0.819825	+	0.572615i	$0.805929\pi$
$824$	0	0
$825$	7.25975	0.252752
$826$	0	0
$827$	−17.6689	−0.614409	−0.307205	−	0.951643i	$-0.599394\pi$
$827$	−17.6689	−0.614409	−0.307205	+	0.951643i	$0.599394\pi$
$828$	0	0
$829$	22.1666	0.769878	0.384939	−	0.922942i	$-0.374222\pi$
$829$	22.1666	0.769878	0.384939	+	0.922942i	$0.374222\pi$
$830$	0	0
$831$	−69.2704	−2.40296
$832$	0	0
$833$	−3.70081	−0.128226
$834$	0	0
$835$	45.8239	1.58580
$836$	0	0
$837$	−157.228	−5.43458
$838$	0	0
$839$	−19.6848	−0.679594	−0.339797	−	0.940499i	$-0.610358\pi$
$839$	−19.6848	−0.679594	−0.339797	+	0.940499i	$0.610358\pi$
$840$	0	0
$841$	31.8521	1.09835
$842$	0	0
$843$	1.31200	0.0451876
$844$	0	0
$845$	2.67963	0.0921821
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	69.8951	2.39880
$850$	0	0
$851$	−21.9095	−0.751047
$852$	0	0
$853$	25.6203	0.877223	0.438611	−	0.898677i	$-0.355470\pi$
$853$	25.6203	0.877223	0.438611	+	0.898677i	$0.355470\pi$
$854$	0	0
$855$	−122.107	−4.17597
$856$	0	0
$857$	40.4145	1.38053	0.690266	−	0.723556i	$-0.257494\pi$
$857$	40.4145	1.38053	0.690266	+	0.723556i	$0.257494\pi$
$858$	0	0
$859$	33.7271	1.15075	0.575377	−	0.817888i	$-0.304855\pi$
$859$	33.7271	1.15075	0.575377	+	0.817888i	$0.304855\pi$
$860$	0	0
$861$	−9.03135	−0.307787
$862$	0	0
$863$	10.8494	0.369319	0.184660	−	0.982803i	$-0.440882\pi$
$863$	10.8494	0.369319	0.184660	+	0.982803i	$0.440882\pi$
$864$	0	0
$865$	39.5609	1.34511
$866$	0	0
$867$	−11.0006	−0.373599
$868$	0	0
$869$	−8.98342	−0.304742
$870$	0	0
$871$	13.1396	0.445220
$872$	0	0
$873$	−22.7134	−0.768733
$874$	0	0
$875$	−7.55542	−0.255420
$876$	0	0
$877$	41.2764	1.39381	0.696903	−	0.717165i	$-0.254561\pi$
$877$	41.2764	1.39381	0.696903	+	0.717165i	$0.254561\pi$
$878$	0	0
$879$	−6.91294	−0.233168
$880$	0	0
$881$	20.2870	0.683486	0.341743	−	0.939794i	$-0.388983\pi$
$881$	20.2870	0.683486	0.341743	+	0.939794i	$0.388983\pi$
$882$	0	0
$883$	29.1177	0.979890	0.489945	−	0.871753i	$-0.337017\pi$
$883$	29.1177	0.979890	0.489945	+	0.871753i	$0.337017\pi$
$884$	0	0
$885$	38.2155	1.28460
$886$	0	0
$887$	−14.8799	−0.499618	−0.249809	−	0.968295i	$-0.580368\pi$
$887$	−14.8799	−0.499618	−0.249809	+	0.968295i	$0.580368\pi$
$888$	0	0
$889$	−6.70802	−0.224980
$890$	0	0
$891$	32.1203	1.07607
$892$	0	0
$893$	−41.2903	−1.38173
$894$	0	0
$895$	−28.4211	−0.950013
$896$	0	0
$897$	−12.9319	−0.431785
$898$	0	0
$899$	−72.4342	−2.41581
$900$	0	0
$901$	−19.4080	−0.646574
$902$	0	0
$903$	25.5568	0.850478
$904$	0	0
$905$	−0.397175	−0.0132026
$906$	0	0
$907$	−56.9595	−1.89131	−0.945655	−	0.325172i	$-0.894578\pi$
$907$	−56.9595	−1.89131	−0.945655	+	0.325172i	$0.894578\pi$
$908$	0	0
$909$	−72.2861	−2.39758
$910$	0	0
$911$	15.9923	0.529850	0.264925	−	0.964269i	$-0.414653\pi$
$911$	15.9923	0.529850	0.264925	+	0.964269i	$0.414653\pi$
$912$	0	0
$913$	5.93527	0.196429
$914$	0	0
$915$	−8.98708	−0.297104
$916$	0	0
$917$	−9.59484	−0.316850
$918$	0	0
$919$	6.57596	0.216921	0.108460	−	0.994101i	$-0.465408\pi$
$919$	6.57596	0.216921	0.108460	+	0.994101i	$0.465408\pi$
$920$	0	0
$921$	−63.8566	−2.10415
$922$	0	0
$923$	−10.8196	−0.356130
$924$	0	0
$925$	−12.2996	−0.404407
$926$	0	0
$927$	−25.3646	−0.833084
$928$	0	0
$929$	15.8787	0.520964	0.260482	−	0.965479i	$-0.416118\pi$
$929$	15.8787	0.520964	0.260482	+	0.965479i	$0.416118\pi$
$930$	0	0
$931$	−5.63576	−0.184705
$932$	0	0
$933$	24.6053	0.805543
$934$	0	0
$935$	−9.91682	−0.324315
$936$	0	0
$937$	50.0269	1.63431	0.817154	−	0.576419i	$-0.195550\pi$
$937$	50.0269	1.63431	0.817154	+	0.576419i	$0.195550\pi$
$938$	0	0
$939$	97.0386	3.16673
$940$	0	0
$941$	−26.1797	−0.853434	−0.426717	−	0.904385i	$-0.640330\pi$
$941$	−26.1797	−0.853434	−0.426717	+	0.904385i	$0.640330\pi$
$942$	0	0
$943$	−10.5355	−0.343084
$944$	0	0
$945$	45.3731	1.47599
$946$	0	0
$947$	7.46780	0.242671	0.121335	−	0.992612i	$-0.461282\pi$
$947$	7.46780	0.242671	0.121335	+	0.992612i	$0.461282\pi$
$948$	0	0
$949$	−6.70355	−0.217607
$950$	0	0
$951$	−107.991	−3.50184
$952$	0	0
$953$	25.4584	0.824680	0.412340	−	0.911030i	$-0.364712\pi$
$953$	25.4584	0.824680	0.412340	+	0.911030i	$0.364712\pi$
$954$	0	0
$955$	−34.4072	−1.11339
$956$	0	0
$957$	25.9727	0.839579
$958$	0	0
$959$	−5.94039	−0.191825
$960$	0	0
$961$	55.2207	1.78131
$962$	0	0
$963$	−135.672	−4.37197
$964$	0	0
$965$	−11.8248	−0.380654
$966$	0	0
$967$	17.3896	0.559213	0.279606	−	0.960115i	$-0.409796\pi$
$967$	17.3896	0.559213	0.279606	+	0.960115i	$0.409796\pi$
$968$	0	0
$969$	69.4432	2.23084
$970$	0	0
$971$	42.8134	1.37395	0.686974	−	0.726682i	$-0.258939\pi$
$971$	42.8134	1.37395	0.686974	+	0.726682i	$0.258939\pi$
$972$	0	0
$973$	−3.95060	−0.126651
$974$	0	0
$975$	−7.25975	−0.232498
$976$	0	0
$977$	0.569032	0.0182049	0.00910247	−	0.999959i	$-0.497103\pi$
$977$	0.569032	0.0182049	0.00910247	+	0.999959i	$0.497103\pi$
$978$	0	0
$979$	5.66928	0.181191
$980$	0	0
$981$	−64.9510	−2.07372
$982$	0	0
$983$	−53.8814	−1.71855	−0.859275	−	0.511514i	$-0.829085\pi$
$983$	−53.8814	−1.71855	−0.859275	+	0.511514i	$0.829085\pi$
$984$	0	0
$985$	−4.19441	−0.133645
$986$	0	0
$987$	24.3936	0.776456
$988$	0	0
$989$	29.8134	0.948010
$990$	0	0
$991$	−22.1837	−0.704688	−0.352344	−	0.935871i	$-0.614615\pi$
$991$	−22.1837	−0.704688	−0.352344	+	0.935871i	$0.614615\pi$
$992$	0	0
$993$	8.65593	0.274688
$994$	0	0
$995$	−38.3879	−1.21698
$996$	0	0
$997$	47.0721	1.49079	0.745394	−	0.666624i	$-0.232262\pi$
$997$	47.0721	1.49079	0.745394	+	0.666624i	$0.232262\pi$
$998$	0	0
$999$	−95.5148	−3.02196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.h.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.h.1.8	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32951 q^{3} +2.67963 q^{5} +1.00000 q^{7} +8.08561 q^{9} +O(q^{10})\)	\(q+3.32951 q^{3} +2.67963 q^{5} +1.00000 q^{7} +8.08561 q^{9} +1.00000 q^{11} -1.00000 q^{13} +8.92185 q^{15} -3.70081 q^{17} -5.63576 q^{19} +3.32951 q^{21} +3.88404 q^{23} +2.18043 q^{25} +16.9326 q^{27} +7.80078 q^{29} -9.28551 q^{31} +3.32951 q^{33} +2.67963 q^{35} -5.64089 q^{37} -3.32951 q^{39} -2.71252 q^{41} +7.67586 q^{43} +21.6665 q^{45} +7.32648 q^{47} +1.00000 q^{49} -12.3219 q^{51} +5.24425 q^{53} +2.67963 q^{55} -18.7643 q^{57} +4.28335 q^{59} -1.00731 q^{61} +8.08561 q^{63} -2.67963 q^{65} -13.1396 q^{67} +12.9319 q^{69} +10.8196 q^{71} +6.70355 q^{73} +7.25975 q^{75} +1.00000 q^{77} -8.98342 q^{79} +32.1203 q^{81} +5.93527 q^{83} -9.91682 q^{85} +25.9727 q^{87} +5.66928 q^{89} -1.00000 q^{91} -30.9162 q^{93} -15.1018 q^{95} -2.80912 q^{97} +8.08561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9	\( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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