LMFDB - Embedded newform 8044.2.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80093 q^{3} -0.853190 q^{5} -2.91337 q^{7} +4.84521 q^{9} +O(q^{10})$	$q-2.80093 q^{3} -0.853190 q^{5} -2.91337 q^{7} +4.84521 q^{9} +2.41038 q^{11} +0.214624 q^{13} +2.38972 q^{15} -4.17497 q^{17} +4.85972 q^{19} +8.16014 q^{21} -3.76647 q^{23} -4.27207 q^{25} -5.16829 q^{27} +7.36453 q^{29} +3.99642 q^{31} -6.75131 q^{33} +2.48566 q^{35} +10.6962 q^{37} -0.601147 q^{39} -10.4291 q^{41} -4.31882 q^{43} -4.13388 q^{45} -0.675134 q^{47} +1.48771 q^{49} +11.6938 q^{51} -4.34852 q^{53} -2.05651 q^{55} -13.6117 q^{57} +13.5578 q^{59} -10.7987 q^{61} -14.1159 q^{63} -0.183115 q^{65} -9.35232 q^{67} +10.5496 q^{69} +6.76464 q^{71} +1.86679 q^{73} +11.9658 q^{75} -7.02233 q^{77} +2.54520 q^{79} -0.0595991 q^{81} +1.70869 q^{83} +3.56204 q^{85} -20.6275 q^{87} -4.73841 q^{89} -0.625279 q^{91} -11.1937 q^{93} -4.14626 q^{95} -14.5077 q^{97} +11.6788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * 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
$3$	−2.80093	−1.61712	−0.808559	−	0.588416i	$-0.799752\pi$
$3$	−2.80093	−1.61712	−0.808559	+	0.588416i	$0.799752\pi$
$4$	0	0
$5$	−0.853190	−0.381558	−0.190779	−	0.981633i	$-0.561101\pi$
$5$	−0.853190	−0.381558	−0.190779	+	0.981633i	$0.561101\pi$
$6$	0	0
$7$	−2.91337	−1.10115	−0.550575	−	0.834786i	$-0.685591\pi$
$7$	−2.91337	−1.10115	−0.550575	+	0.834786i	$0.685591\pi$
$8$	0	0
$9$	4.84521	1.61507
$10$	0	0
$11$	2.41038	0.726758	0.363379	−	0.931641i	$-0.381623\pi$
$11$	2.41038	0.726758	0.363379	+	0.931641i	$0.381623\pi$
$12$	0	0
$13$	0.214624	0.0595260	0.0297630	−	0.999557i	$-0.490525\pi$
$13$	0.214624	0.0595260	0.0297630	+	0.999557i	$0.490525\pi$
$14$	0	0
$15$	2.38972	0.617024
$16$	0	0
$17$	−4.17497	−1.01258	−0.506289	−	0.862364i	$-0.668983\pi$
$17$	−4.17497	−1.01258	−0.506289	+	0.862364i	$0.668983\pi$
$18$	0	0
$19$	4.85972	1.11490	0.557448	−	0.830212i	$-0.311781\pi$
$19$	4.85972	1.11490	0.557448	+	0.830212i	$0.311781\pi$
$20$	0	0
$21$	8.16014	1.78069
$22$	0	0
$23$	−3.76647	−0.785363	−0.392682	−	0.919674i	$-0.628453\pi$
$23$	−3.76647	−0.785363	−0.392682	+	0.919674i	$0.628453\pi$
$24$	0	0
$25$	−4.27207	−0.854413
$26$	0	0
$27$	−5.16829	−0.994638
$28$	0	0
$29$	7.36453	1.36756	0.683780	−	0.729689i	$-0.260335\pi$
$29$	7.36453	1.36756	0.683780	+	0.729689i	$0.260335\pi$
$30$	0	0
$31$	3.99642	0.717778	0.358889	−	0.933380i	$-0.383156\pi$
$31$	3.99642	0.717778	0.358889	+	0.933380i	$0.383156\pi$
$32$	0	0
$33$	−6.75131	−1.17525
$34$	0	0
$35$	2.48566	0.420152
$36$	0	0
$37$	10.6962	1.75844	0.879222	−	0.476412i	$-0.158063\pi$
$37$	10.6962	1.75844	0.879222	+	0.476412i	$0.158063\pi$
$38$	0	0
$39$	−0.601147	−0.0962605
$40$	0	0
$41$	−10.4291	−1.62875	−0.814374	−	0.580340i	$-0.802920\pi$
$41$	−10.4291	−1.62875	−0.814374	+	0.580340i	$0.802920\pi$
$42$	0	0
$43$	−4.31882	−0.658614	−0.329307	−	0.944223i	$-0.606815\pi$
$43$	−4.31882	−0.658614	−0.329307	+	0.944223i	$0.606815\pi$
$44$	0	0
$45$	−4.13388	−0.616242
$46$	0	0
$47$	−0.675134	−0.0984784	−0.0492392	−	0.998787i	$-0.515680\pi$
$47$	−0.675134	−0.0984784	−0.0492392	+	0.998787i	$0.515680\pi$
$48$	0	0
$49$	1.48771	0.212530
$50$	0	0
$51$	11.6938	1.63746
$52$	0	0
$53$	−4.34852	−0.597315	−0.298658	−	0.954360i	$-0.596539\pi$
$53$	−4.34852	−0.597315	−0.298658	+	0.954360i	$0.596539\pi$
$54$	0	0
$55$	−2.05651	−0.277300
$56$	0	0
$57$	−13.6117	−1.80292
$58$	0	0
$59$	13.5578	1.76508	0.882541	−	0.470236i	$-0.155831\pi$
$59$	13.5578	1.76508	0.882541	+	0.470236i	$0.155831\pi$
$60$	0	0
$61$	−10.7987	−1.38264	−0.691319	−	0.722550i	$-0.742970\pi$
$61$	−10.7987	−1.38264	−0.691319	+	0.722550i	$0.742970\pi$
$62$	0	0
$63$	−14.1159	−1.77843
$64$	0	0
$65$	−0.183115	−0.0227126
$66$	0	0
$67$	−9.35232	−1.14257	−0.571284	−	0.820753i	$-0.693554\pi$
$67$	−9.35232	−1.14257	−0.571284	+	0.820753i	$0.693554\pi$
$68$	0	0
$69$	10.5496	1.27002
$70$	0	0
$71$	6.76464	0.802816	0.401408	−	0.915899i	$-0.368521\pi$
$71$	6.76464	0.802816	0.401408	+	0.915899i	$0.368521\pi$
$72$	0	0
$73$	1.86679	0.218492	0.109246	−	0.994015i	$-0.465156\pi$
$73$	1.86679	0.218492	0.109246	+	0.994015i	$0.465156\pi$
$74$	0	0
$75$	11.9658	1.38169
$76$	0	0
$77$	−7.02233	−0.800269
$78$	0	0
$79$	2.54520	0.286357	0.143178	−	0.989697i	$-0.454268\pi$
$79$	2.54520	0.286357	0.143178	+	0.989697i	$0.454268\pi$
$80$	0	0
$81$	−0.0595991	−0.00662212
$82$	0	0
$83$	1.70869	0.187553	0.0937767	−	0.995593i	$-0.470106\pi$
$83$	1.70869	0.187553	0.0937767	+	0.995593i	$0.470106\pi$
$84$	0	0
$85$	3.56204	0.386357
$86$	0	0
$87$	−20.6275	−2.21150
$88$	0	0
$89$	−4.73841	−0.502270	−0.251135	−	0.967952i	$-0.580804\pi$
$89$	−4.73841	−0.502270	−0.251135	+	0.967952i	$0.580804\pi$
$90$	0	0
$91$	−0.625279	−0.0655470
$92$	0	0
$93$	−11.1937	−1.16073
$94$	0	0
$95$	−4.14626	−0.425397
$96$	0	0
$97$	−14.5077	−1.47303	−0.736516	−	0.676420i	$-0.763530\pi$
$97$	−14.5077	−1.47303	−0.736516	+	0.676420i	$0.763530\pi$
$98$	0	0
$99$	11.6788	1.17376
$100$	0	0
$101$	8.04971	0.800976	0.400488	−	0.916302i	$-0.368841\pi$
$101$	8.04971	0.800976	0.400488	+	0.916302i	$0.368841\pi$
$102$	0	0
$103$	10.8227	1.06640	0.533198	−	0.845990i	$-0.320990\pi$
$103$	10.8227	1.06640	0.533198	+	0.845990i	$0.320990\pi$
$104$	0	0
$105$	−6.96215	−0.679436
$106$	0	0
$107$	−13.9897	−1.35244	−0.676218	−	0.736701i	$-0.736382\pi$
$107$	−13.9897	−1.35244	−0.676218	+	0.736701i	$0.736382\pi$
$108$	0	0
$109$	−4.61370	−0.441913	−0.220956	−	0.975284i	$-0.570918\pi$
$109$	−4.61370	−0.441913	−0.220956	+	0.975284i	$0.570918\pi$
$110$	0	0
$111$	−29.9593	−2.84361
$112$	0	0
$113$	−2.48855	−0.234103	−0.117052	−	0.993126i	$-0.537344\pi$
$113$	−2.48855	−0.234103	−0.117052	+	0.993126i	$0.537344\pi$
$114$	0	0
$115$	3.21351	0.299662
$116$	0	0
$117$	1.03990	0.0961386
$118$	0	0
$119$	12.1632	1.11500
$120$	0	0
$121$	−5.19005	−0.471823
$122$	0	0
$123$	29.2111	2.63388
$124$	0	0
$125$	7.91083	0.707566
$126$	0	0
$127$	−15.5199	−1.37717	−0.688584	−	0.725156i	$-0.741768\pi$
$127$	−15.5199	−1.37717	−0.688584	+	0.725156i	$0.741768\pi$
$128$	0	0
$129$	12.0967	1.06506
$130$	0	0
$131$	−4.59181	−0.401188	−0.200594	−	0.979674i	$-0.564287\pi$
$131$	−4.59181	−0.401188	−0.200594	+	0.979674i	$0.564287\pi$
$132$	0	0
$133$	−14.1581	−1.22767
$134$	0	0
$135$	4.40953	0.379512
$136$	0	0
$137$	−0.0548981	−0.00469026	−0.00234513	−	0.999997i	$-0.500746\pi$
$137$	−0.0548981	−0.00469026	−0.00234513	+	0.999997i	$0.500746\pi$
$138$	0	0
$139$	15.8741	1.34642	0.673211	−	0.739450i	$-0.264914\pi$
$139$	15.8741	1.34642	0.673211	+	0.739450i	$0.264914\pi$
$140$	0	0
$141$	1.89100	0.159251
$142$	0	0
$143$	0.517326	0.0432610
$144$	0	0
$145$	−6.28334	−0.521803
$146$	0	0
$147$	−4.16697	−0.343686
$148$	0	0
$149$	−15.3931	−1.26105	−0.630526	−	0.776168i	$-0.717161\pi$
$149$	−15.3931	−1.26105	−0.630526	+	0.776168i	$0.717161\pi$
$150$	0	0
$151$	−11.4291	−0.930088	−0.465044	−	0.885288i	$-0.653961\pi$
$151$	−11.4291	−0.930088	−0.465044	+	0.885288i	$0.653961\pi$
$152$	0	0
$153$	−20.2286	−1.63538
$154$	0	0
$155$	−3.40970	−0.273874
$156$	0	0
$157$	0.740425	0.0590923	0.0295462	−	0.999563i	$-0.490594\pi$
$157$	0.740425	0.0590923	0.0295462	+	0.999563i	$0.490594\pi$
$158$	0	0
$159$	12.1799	0.965929
$160$	0	0
$161$	10.9731	0.864802
$162$	0	0
$163$	−10.2807	−0.805250	−0.402625	−	0.915365i	$-0.631902\pi$
$163$	−10.2807	−0.805250	−0.402625	+	0.915365i	$0.631902\pi$
$164$	0	0
$165$	5.76015	0.448427
$166$	0	0
$167$	3.66401	0.283530	0.141765	−	0.989900i	$-0.454722\pi$
$167$	3.66401	0.283530	0.141765	+	0.989900i	$0.454722\pi$
$168$	0	0
$169$	−12.9539	−0.996457
$170$	0	0
$171$	23.5463	1.80063
$172$	0	0
$173$	−7.33611	−0.557754	−0.278877	−	0.960327i	$-0.589962\pi$
$173$	−7.33611	−0.557754	−0.278877	+	0.960327i	$0.589962\pi$
$174$	0	0
$175$	12.4461	0.940837
$176$	0	0
$177$	−37.9746	−2.85434
$178$	0	0
$179$	14.1682	1.05898	0.529491	−	0.848315i	$-0.322383\pi$
$179$	14.1682	1.05898	0.529491	+	0.848315i	$0.322383\pi$
$180$	0	0
$181$	1.04792	0.0778914	0.0389457	−	0.999241i	$-0.487600\pi$
$181$	1.04792	0.0778914	0.0389457	+	0.999241i	$0.487600\pi$
$182$	0	0
$183$	30.2465	2.23589
$184$	0	0
$185$	−9.12589	−0.670949
$186$	0	0
$187$	−10.0633	−0.735899
$188$	0	0
$189$	15.0571	1.09525
$190$	0	0
$191$	−2.14255	−0.155029	−0.0775147	−	0.996991i	$-0.524698\pi$
$191$	−2.14255	−0.155029	−0.0775147	+	0.996991i	$0.524698\pi$
$192$	0	0
$193$	7.72244	0.555873	0.277937	−	0.960599i	$-0.410349\pi$
$193$	7.72244	0.555873	0.277937	+	0.960599i	$0.410349\pi$
$194$	0	0
$195$	0.512892	0.0367290
$196$	0	0
$197$	9.13385	0.650760	0.325380	−	0.945583i	$-0.394508\pi$
$197$	9.13385	0.650760	0.325380	+	0.945583i	$0.394508\pi$
$198$	0	0
$199$	26.2910	1.86372	0.931860	−	0.362817i	$-0.118185\pi$
$199$	26.2910	1.86372	0.931860	+	0.362817i	$0.118185\pi$
$200$	0	0
$201$	26.1952	1.84767
$202$	0	0
$203$	−21.4556	−1.50589
$204$	0	0
$205$	8.89798	0.621462
$206$	0	0
$207$	−18.2493	−1.26842
$208$	0	0
$209$	11.7138	0.810259
$210$	0	0
$211$	−13.0493	−0.898348	−0.449174	−	0.893444i	$-0.648282\pi$
$211$	−13.0493	−0.898348	−0.449174	+	0.893444i	$0.648282\pi$
$212$	0	0
$213$	−18.9473	−1.29825
$214$	0	0
$215$	3.68477	0.251299
$216$	0	0
$217$	−11.6430	−0.790381
$218$	0	0
$219$	−5.22875	−0.353326
$220$	0	0
$221$	−0.896048	−0.0602747
$222$	0	0
$223$	−1.48046	−0.0991391	−0.0495695	−	0.998771i	$-0.515785\pi$
$223$	−1.48046	−0.0991391	−0.0495695	+	0.998771i	$0.515785\pi$
$224$	0	0
$225$	−20.6990	−1.37994
$226$	0	0
$227$	9.16991	0.608628	0.304314	−	0.952572i	$-0.401573\pi$
$227$	9.16991	0.608628	0.304314	+	0.952572i	$0.401573\pi$
$228$	0	0
$229$	18.2472	1.20581	0.602905	−	0.797813i	$-0.294010\pi$
$229$	18.2472	1.20581	0.602905	+	0.797813i	$0.294010\pi$
$230$	0	0
$231$	19.6691	1.29413
$232$	0	0
$233$	−18.0641	−1.18342	−0.591709	−	0.806152i	$-0.701546\pi$
$233$	−18.0641	−1.18342	−0.591709	+	0.806152i	$0.701546\pi$
$234$	0	0
$235$	0.576017	0.0375752
$236$	0	0
$237$	−7.12891	−0.463073
$238$	0	0
$239$	4.62604	0.299234	0.149617	−	0.988744i	$-0.452196\pi$
$239$	4.62604	0.299234	0.149617	+	0.988744i	$0.452196\pi$
$240$	0	0
$241$	−0.505002	−0.0325301	−0.0162650	−	0.999868i	$-0.505178\pi$
$241$	−0.505002	−0.0325301	−0.0162650	+	0.999868i	$0.505178\pi$
$242$	0	0
$243$	15.6718	1.00535
$244$	0	0
$245$	−1.26930	−0.0810926
$246$	0	0
$247$	1.04301	0.0663653
$248$	0	0
$249$	−4.78593	−0.303296
$250$	0	0
$251$	−9.30990	−0.587636	−0.293818	−	0.955861i	$-0.594926\pi$
$251$	−9.30990	−0.587636	−0.293818	+	0.955861i	$0.594926\pi$
$252$	0	0
$253$	−9.07864	−0.570769
$254$	0	0
$255$	−9.97702	−0.624785
$256$	0	0
$257$	10.9138	0.680781	0.340391	−	0.940284i	$-0.389441\pi$
$257$	10.9138	0.680781	0.340391	+	0.940284i	$0.389441\pi$
$258$	0	0
$259$	−31.1620	−1.93631
$260$	0	0
$261$	35.6827	2.20870
$262$	0	0
$263$	19.1497	1.18082	0.590410	−	0.807103i	$-0.298966\pi$
$263$	19.1497	1.18082	0.590410	+	0.807103i	$0.298966\pi$
$264$	0	0
$265$	3.71011	0.227910
$266$	0	0
$267$	13.2719	0.812229
$268$	0	0
$269$	−30.8838	−1.88302	−0.941509	−	0.336987i	$-0.890592\pi$
$269$	−30.8838	−1.88302	−0.941509	+	0.336987i	$0.890592\pi$
$270$	0	0
$271$	−8.74084	−0.530968	−0.265484	−	0.964115i	$-0.585532\pi$
$271$	−8.74084	−0.530968	−0.265484	+	0.964115i	$0.585532\pi$
$272$	0	0
$273$	1.75136	0.105997
$274$	0	0
$275$	−10.2973	−0.620952
$276$	0	0
$277$	14.7976	0.889103	0.444551	−	0.895753i	$-0.353363\pi$
$277$	14.7976	0.889103	0.444551	+	0.895753i	$0.353363\pi$
$278$	0	0
$279$	19.3635	1.15926
$280$	0	0
$281$	4.09549	0.244317	0.122158	−	0.992511i	$-0.461018\pi$
$281$	4.09549	0.244317	0.122158	+	0.992511i	$0.461018\pi$
$282$	0	0
$283$	−23.1736	−1.37753	−0.688764	−	0.724986i	$-0.741846\pi$
$283$	−23.1736	−1.37753	−0.688764	+	0.724986i	$0.741846\pi$
$284$	0	0
$285$	11.6134	0.687917
$286$	0	0
$287$	30.3837	1.79350
$288$	0	0
$289$	0.430346	0.0253145
$290$	0	0
$291$	40.6350	2.38207
$292$	0	0
$293$	28.8947	1.68804	0.844022	−	0.536308i	$-0.180181\pi$
$293$	28.8947	1.68804	0.844022	+	0.536308i	$0.180181\pi$
$294$	0	0
$295$	−11.5674	−0.673481
$296$	0	0
$297$	−12.4576	−0.722861
$298$	0	0
$299$	−0.808375	−0.0467495
$300$	0	0
$301$	12.5823	0.725232
$302$	0	0
$303$	−22.5467	−1.29527
$304$	0	0
$305$	9.21338	0.527557
$306$	0	0
$307$	6.22324	0.355179	0.177590	−	0.984105i	$-0.443170\pi$
$307$	6.22324	0.355179	0.177590	+	0.984105i	$0.443170\pi$
$308$	0	0
$309$	−30.3137	−1.72449
$310$	0	0
$311$	21.5066	1.21953	0.609764	−	0.792583i	$-0.291264\pi$
$311$	21.5066	1.21953	0.609764	+	0.792583i	$0.291264\pi$
$312$	0	0
$313$	8.82455	0.498793	0.249396	−	0.968401i	$-0.419768\pi$
$313$	8.82455	0.498793	0.249396	+	0.968401i	$0.419768\pi$
$314$	0	0
$315$	12.0435	0.678575
$316$	0	0
$317$	3.11148	0.174758	0.0873792	−	0.996175i	$-0.472151\pi$
$317$	3.11148	0.174758	0.0873792	+	0.996175i	$0.472151\pi$
$318$	0	0
$319$	17.7513	0.993885
$320$	0	0
$321$	39.1842	2.18705
$322$	0	0
$323$	−20.2891	−1.12892
$324$	0	0
$325$	−0.916888	−0.0508598
$326$	0	0
$327$	12.9227	0.714625
$328$	0	0
$329$	1.96691	0.108439
$330$	0	0
$331$	31.8316	1.74963	0.874813	−	0.484462i	$-0.160984\pi$
$331$	31.8316	1.74963	0.874813	+	0.484462i	$0.160984\pi$
$332$	0	0
$333$	51.8253	2.84001
$334$	0	0
$335$	7.97930	0.435956
$336$	0	0
$337$	32.0283	1.74469	0.872345	−	0.488891i	$-0.162598\pi$
$337$	32.0283	1.74469	0.872345	+	0.488891i	$0.162598\pi$
$338$	0	0
$339$	6.97026	0.378572
$340$	0	0
$341$	9.63290	0.521651
$342$	0	0
$343$	16.0593	0.867122
$344$	0	0
$345$	−9.00082	−0.484588
$346$	0	0
$347$	4.90335	0.263226	0.131613	−	0.991301i	$-0.457984\pi$
$347$	4.90335	0.263226	0.131613	+	0.991301i	$0.457984\pi$
$348$	0	0
$349$	−2.95411	−0.158130	−0.0790650	−	0.996869i	$-0.525193\pi$
$349$	−2.95411	−0.158130	−0.0790650	+	0.996869i	$0.525193\pi$
$350$	0	0
$351$	−1.10924	−0.0592068
$352$	0	0
$353$	−11.3599	−0.604625	−0.302312	−	0.953209i	$-0.597759\pi$
$353$	−11.3599	−0.604625	−0.302312	+	0.953209i	$0.597759\pi$
$354$	0	0
$355$	−5.77153	−0.306321
$356$	0	0
$357$	−34.0683	−1.80309
$358$	0	0
$359$	−18.2835	−0.964968	−0.482484	−	0.875905i	$-0.660265\pi$
$359$	−18.2835	−0.964968	−0.482484	+	0.875905i	$0.660265\pi$
$360$	0	0
$361$	4.61683	0.242991
$362$	0	0
$363$	14.5370	0.762993
$364$	0	0
$365$	−1.59273	−0.0833672
$366$	0	0
$367$	21.4414	1.11923	0.559616	−	0.828752i	$-0.310949\pi$
$367$	21.4414	1.11923	0.559616	+	0.828752i	$0.310949\pi$
$368$	0	0
$369$	−50.5310	−2.63054
$370$	0	0
$371$	12.6688	0.657733
$372$	0	0
$373$	−23.3950	−1.21135	−0.605674	−	0.795713i	$-0.707096\pi$
$373$	−23.3950	−1.21135	−0.605674	+	0.795713i	$0.707096\pi$
$374$	0	0
$375$	−22.1577	−1.14422
$376$	0	0
$377$	1.58061	0.0814053
$378$	0	0
$379$	−4.28442	−0.220076	−0.110038	−	0.993927i	$-0.535097\pi$
$379$	−4.28442	−0.220076	−0.110038	+	0.993927i	$0.535097\pi$
$380$	0	0
$381$	43.4702	2.22704
$382$	0	0
$383$	14.6969	0.750977	0.375488	−	0.926827i	$-0.377475\pi$
$383$	14.6969	0.750977	0.375488	+	0.926827i	$0.377475\pi$
$384$	0	0
$385$	5.99138	0.305349
$386$	0	0
$387$	−20.9256	−1.06371
$388$	0	0
$389$	1.76253	0.0893636	0.0446818	−	0.999001i	$-0.485773\pi$
$389$	1.76253	0.0893636	0.0446818	+	0.999001i	$0.485773\pi$
$390$	0	0
$391$	15.7249	0.795242
$392$	0	0
$393$	12.8613	0.648768
$394$	0	0
$395$	−2.17153	−0.109262
$396$	0	0
$397$	−17.3190	−0.869215	−0.434608	−	0.900620i	$-0.643113\pi$
$397$	−17.3190	−0.869215	−0.434608	+	0.900620i	$0.643113\pi$
$398$	0	0
$399$	39.6559	1.98528
$400$	0	0
$401$	−25.4575	−1.27129	−0.635645	−	0.771982i	$-0.719266\pi$
$401$	−25.4575	−1.27129	−0.635645	+	0.771982i	$0.719266\pi$
$402$	0	0
$403$	0.857727	0.0427264
$404$	0	0
$405$	0.0508493	0.00252672
$406$	0	0
$407$	25.7819	1.27796
$408$	0	0
$409$	23.8156	1.17761	0.588803	−	0.808277i	$-0.299600\pi$
$409$	23.8156	1.17761	0.588803	+	0.808277i	$0.299600\pi$
$410$	0	0
$411$	0.153766	0.00758470
$412$	0	0
$413$	−39.4990	−1.94362
$414$	0	0
$415$	−1.45784	−0.0715625
$416$	0	0
$417$	−44.4622	−2.17732
$418$	0	0
$419$	9.71646	0.474680	0.237340	−	0.971427i	$-0.423724\pi$
$419$	9.71646	0.474680	0.237340	+	0.971427i	$0.423724\pi$
$420$	0	0
$421$	−2.04959	−0.0998908	−0.0499454	−	0.998752i	$-0.515905\pi$
$421$	−2.04959	−0.0998908	−0.0499454	+	0.998752i	$0.515905\pi$
$422$	0	0
$423$	−3.27116	−0.159049
$424$	0	0
$425$	17.8357	0.865160
$426$	0	0
$427$	31.4607	1.52249
$428$	0	0
$429$	−1.44899	−0.0699581
$430$	0	0
$431$	−15.9110	−0.766405	−0.383203	−	0.923664i	$-0.625179\pi$
$431$	−15.9110	−0.766405	−0.383203	+	0.923664i	$0.625179\pi$
$432$	0	0
$433$	18.8717	0.906917	0.453459	−	0.891277i	$-0.350190\pi$
$433$	18.8717	0.906917	0.453459	+	0.891277i	$0.350190\pi$
$434$	0	0
$435$	17.5992	0.843817
$436$	0	0
$437$	−18.3040	−0.875598
$438$	0	0
$439$	−7.21583	−0.344393	−0.172196	−	0.985063i	$-0.555086\pi$
$439$	−7.21583	−0.344393	−0.172196	+	0.985063i	$0.555086\pi$
$440$	0	0
$441$	7.20826	0.343251
$442$	0	0
$443$	−20.9485	−0.995292	−0.497646	−	0.867380i	$-0.665802\pi$
$443$	−20.9485	−0.995292	−0.497646	+	0.867380i	$0.665802\pi$
$444$	0	0
$445$	4.04276	0.191645
$446$	0	0
$447$	43.1150	2.03927
$448$	0	0
$449$	7.32121	0.345509	0.172755	−	0.984965i	$-0.444733\pi$
$449$	7.32121	0.345509	0.172755	+	0.984965i	$0.444733\pi$
$450$	0	0
$451$	−25.1381	−1.18371
$452$	0	0
$453$	32.0121	1.50406
$454$	0	0
$455$	0.533482	0.0250100
$456$	0	0
$457$	33.7292	1.57778	0.788892	−	0.614532i	$-0.210655\pi$
$457$	33.7292	1.57778	0.788892	+	0.614532i	$0.210655\pi$
$458$	0	0
$459$	21.5774	1.00715
$460$	0	0
$461$	17.1927	0.800742	0.400371	−	0.916353i	$-0.368881\pi$
$461$	17.1927	0.800742	0.400371	+	0.916353i	$0.368881\pi$
$462$	0	0
$463$	13.2340	0.615037	0.307518	−	0.951542i	$-0.400501\pi$
$463$	13.2340	0.615037	0.307518	+	0.951542i	$0.400501\pi$
$464$	0	0
$465$	9.55034	0.442886
$466$	0	0
$467$	21.4517	0.992665	0.496333	−	0.868132i	$-0.334680\pi$
$467$	21.4517	0.992665	0.496333	+	0.868132i	$0.334680\pi$
$468$	0	0
$469$	27.2467	1.25814
$470$	0	0
$471$	−2.07388	−0.0955592
$472$	0	0
$473$	−10.4100	−0.478653
$474$	0	0
$475$	−20.7610	−0.952581
$476$	0	0
$477$	−21.0695	−0.964705
$478$	0	0
$479$	21.0140	0.960153	0.480076	−	0.877227i	$-0.340609\pi$
$479$	21.0140	0.960153	0.480076	+	0.877227i	$0.340609\pi$
$480$	0	0
$481$	2.29566	0.104673
$482$	0	0
$483$	−30.7349	−1.39849
$484$	0	0
$485$	12.3778	0.562047
$486$	0	0
$487$	−8.91259	−0.403868	−0.201934	−	0.979399i	$-0.564723\pi$
$487$	−8.91259	−0.403868	−0.201934	+	0.979399i	$0.564723\pi$
$488$	0	0
$489$	28.7956	1.30218
$490$	0	0
$491$	33.1141	1.49442	0.747210	−	0.664588i	$-0.231393\pi$
$491$	33.1141	1.49442	0.747210	+	0.664588i	$0.231393\pi$
$492$	0	0
$493$	−30.7467	−1.38476
$494$	0	0
$495$	−9.96424	−0.447859
$496$	0	0
$497$	−19.7079	−0.884020
$498$	0	0
$499$	26.0310	1.16531	0.582653	−	0.812721i	$-0.302015\pi$
$499$	26.0310	1.16531	0.582653	+	0.812721i	$0.302015\pi$
$500$	0	0
$501$	−10.2626	−0.458501
$502$	0	0
$503$	28.8720	1.28734	0.643670	−	0.765303i	$-0.277411\pi$
$503$	28.8720	1.28734	0.643670	+	0.765303i	$0.277411\pi$
$504$	0	0
$505$	−6.86793	−0.305619
$506$	0	0
$507$	36.2831	1.61139
$508$	0	0
$509$	−0.294061	−0.0130340	−0.00651701	−	0.999979i	$-0.502074\pi$
$509$	−0.294061	−0.0130340	−0.00651701	+	0.999979i	$0.502074\pi$
$510$	0	0
$511$	−5.43865	−0.240592
$512$	0	0
$513$	−25.1164	−1.10892
$514$	0	0
$515$	−9.23385	−0.406892
$516$	0	0
$517$	−1.62733	−0.0715699
$518$	0	0
$519$	20.5479	0.901954
$520$	0	0
$521$	−0.460514	−0.0201755	−0.0100877	−	0.999949i	$-0.503211\pi$
$521$	−0.460514	−0.0201755	−0.0100877	+	0.999949i	$0.503211\pi$
$522$	0	0
$523$	−1.66950	−0.0730019	−0.0365010	−	0.999334i	$-0.511621\pi$
$523$	−1.66950	−0.0730019	−0.0365010	+	0.999334i	$0.511621\pi$
$524$	0	0
$525$	−34.8607	−1.52144
$526$	0	0
$527$	−16.6849	−0.726806
$528$	0	0
$529$	−8.81371	−0.383205
$530$	0	0
$531$	65.6906	2.85073
$532$	0	0
$533$	−2.23833	−0.0969529
$534$	0	0
$535$	11.9359	0.516033
$536$	0	0
$537$	−39.6842	−1.71250
$538$	0	0
$539$	3.58595	0.154458
$540$	0	0
$541$	20.6663	0.888512	0.444256	−	0.895900i	$-0.353468\pi$
$541$	20.6663	0.888512	0.444256	+	0.895900i	$0.353468\pi$
$542$	0	0
$543$	−2.93516	−0.125960
$544$	0	0
$545$	3.93636	0.168615
$546$	0	0
$547$	15.0071	0.641659	0.320829	−	0.947137i	$-0.396038\pi$
$547$	15.0071	0.641659	0.320829	+	0.947137i	$0.396038\pi$
$548$	0	0
$549$	−52.3222	−2.23306
$550$	0	0
$551$	35.7895	1.52469
$552$	0	0
$553$	−7.41509	−0.315322
$554$	0	0
$555$	25.5610	1.08500
$556$	0	0
$557$	5.89341	0.249712	0.124856	−	0.992175i	$-0.460153\pi$
$557$	5.89341	0.249712	0.124856	+	0.992175i	$0.460153\pi$
$558$	0	0
$559$	−0.926923	−0.0392046
$560$	0	0
$561$	28.1865	1.19004
$562$	0	0
$563$	−13.0725	−0.550942	−0.275471	−	0.961309i	$-0.588834\pi$
$563$	−13.0725	−0.550942	−0.275471	+	0.961309i	$0.588834\pi$
$564$	0	0
$565$	2.12321	0.0893240
$566$	0	0
$567$	0.173634	0.00729194
$568$	0	0
$569$	30.0589	1.26013	0.630067	−	0.776541i	$-0.283027\pi$
$569$	30.0589	1.26013	0.630067	+	0.776541i	$0.283027\pi$
$570$	0	0
$571$	37.6732	1.57657	0.788286	−	0.615308i	$-0.210969\pi$
$571$	37.6732	1.57657	0.788286	+	0.615308i	$0.210969\pi$
$572$	0	0
$573$	6.00113	0.250701
$574$	0	0
$575$	16.0906	0.671025
$576$	0	0
$577$	−8.52522	−0.354909	−0.177455	−	0.984129i	$-0.556786\pi$
$577$	−8.52522	−0.354909	−0.177455	+	0.984129i	$0.556786\pi$
$578$	0	0
$579$	−21.6300	−0.898912
$580$	0	0
$581$	−4.97805	−0.206524
$582$	0	0
$583$	−10.4816	−0.434103
$584$	0	0
$585$	−0.887230	−0.0366825
$586$	0	0
$587$	29.0442	1.19878	0.599392	−	0.800456i	$-0.295409\pi$
$587$	29.0442	1.19878	0.599392	+	0.800456i	$0.295409\pi$
$588$	0	0
$589$	19.4215	0.800247
$590$	0	0
$591$	−25.5833	−1.05235
$592$	0	0
$593$	26.8218	1.10144	0.550720	−	0.834690i	$-0.314353\pi$
$593$	26.8218	1.10144	0.550720	+	0.834690i	$0.314353\pi$
$594$	0	0
$595$	−10.3775	−0.425437
$596$	0	0
$597$	−73.6393	−3.01386
$598$	0	0
$599$	−38.4241	−1.56996	−0.784982	−	0.619518i	$-0.787328\pi$
$599$	−38.4241	−1.56996	−0.784982	+	0.619518i	$0.787328\pi$
$600$	0	0
$601$	18.4269	0.751648	0.375824	−	0.926691i	$-0.377360\pi$
$601$	18.4269	0.751648	0.375824	+	0.926691i	$0.377360\pi$
$602$	0	0
$603$	−45.3139	−1.84532
$604$	0	0
$605$	4.42810	0.180028
$606$	0	0
$607$	−33.1084	−1.34383	−0.671914	−	0.740629i	$-0.734527\pi$
$607$	−33.1084	−1.34383	−0.671914	+	0.740629i	$0.734527\pi$
$608$	0	0
$609$	60.0956	2.43520
$610$	0	0
$611$	−0.144900	−0.00586203
$612$	0	0
$613$	−19.6572	−0.793949	−0.396974	−	0.917830i	$-0.629940\pi$
$613$	−19.6572	−0.793949	−0.396974	+	0.917830i	$0.629940\pi$
$614$	0	0
$615$	−24.9226	−1.00498
$616$	0	0
$617$	−11.3550	−0.457135	−0.228567	−	0.973528i	$-0.573404\pi$
$617$	−11.3550	−0.457135	−0.228567	+	0.973528i	$0.573404\pi$
$618$	0	0
$619$	8.01652	0.322211	0.161105	−	0.986937i	$-0.448494\pi$
$619$	8.01652	0.322211	0.161105	+	0.986937i	$0.448494\pi$
$620$	0	0
$621$	19.4662	0.781152
$622$	0	0
$623$	13.8047	0.553074
$624$	0	0
$625$	14.6109	0.584436
$626$	0	0
$627$	−32.8095	−1.31028
$628$	0	0
$629$	−44.6563	−1.78056
$630$	0	0
$631$	−12.5734	−0.500538	−0.250269	−	0.968176i	$-0.580519\pi$
$631$	−12.5734	−0.500538	−0.250269	+	0.968176i	$0.580519\pi$
$632$	0	0
$633$	36.5501	1.45273
$634$	0	0
$635$	13.2414	0.525470
$636$	0	0
$637$	0.319299	0.0126511
$638$	0	0
$639$	32.7761	1.29660
$640$	0	0
$641$	−11.4401	−0.451858	−0.225929	−	0.974144i	$-0.572542\pi$
$641$	−11.4401	−0.451858	−0.225929	+	0.974144i	$0.572542\pi$
$642$	0	0
$643$	36.8616	1.45368	0.726839	−	0.686808i	$-0.240989\pi$
$643$	36.8616	1.45368	0.726839	+	0.686808i	$0.240989\pi$
$644$	0	0
$645$	−10.3208	−0.406381
$646$	0	0
$647$	−3.23085	−0.127018	−0.0635090	−	0.997981i	$-0.520229\pi$
$647$	−3.23085	−0.127018	−0.0635090	+	0.997981i	$0.520229\pi$
$648$	0	0
$649$	32.6796	1.28279
$650$	0	0
$651$	32.6113	1.27814
$652$	0	0
$653$	12.1979	0.477340	0.238670	−	0.971101i	$-0.423289\pi$
$653$	12.1979	0.477340	0.238670	+	0.971101i	$0.423289\pi$
$654$	0	0
$655$	3.91768	0.153076
$656$	0	0
$657$	9.04499	0.352879
$658$	0	0
$659$	5.81187	0.226398	0.113199	−	0.993572i	$-0.463890\pi$
$659$	5.81187	0.226398	0.113199	+	0.993572i	$0.463890\pi$
$660$	0	0
$661$	25.7247	1.00058	0.500288	−	0.865859i	$-0.333227\pi$
$661$	25.7247	1.00058	0.500288	+	0.865859i	$0.333227\pi$
$662$	0	0
$663$	2.50977	0.0974713
$664$	0	0
$665$	12.0796	0.468426
$666$	0	0
$667$	−27.7383	−1.07403
$668$	0	0
$669$	4.14667	0.160320
$670$	0	0
$671$	−26.0291	−1.00484
$672$	0	0
$673$	−39.4474	−1.52058	−0.760292	−	0.649581i	$-0.774944\pi$
$673$	−39.4474	−1.52058	−0.760292	+	0.649581i	$0.774944\pi$
$674$	0	0
$675$	22.0793	0.849832
$676$	0	0
$677$	−13.4152	−0.515587	−0.257793	−	0.966200i	$-0.582995\pi$
$677$	−13.4152	−0.515587	−0.257793	+	0.966200i	$0.582995\pi$
$678$	0	0
$679$	42.2662	1.62203
$680$	0	0
$681$	−25.6843	−0.984223
$682$	0	0
$683$	20.5422	0.786027	0.393014	−	0.919533i	$-0.371433\pi$
$683$	20.5422	0.786027	0.393014	+	0.919533i	$0.371433\pi$
$684$	0	0
$685$	0.0468385	0.00178961
$686$	0	0
$687$	−51.1092	−1.94994
$688$	0	0
$689$	−0.933297	−0.0355558
$690$	0	0
$691$	−27.5988	−1.04991	−0.524955	−	0.851130i	$-0.675918\pi$
$691$	−27.5988	−1.04991	−0.524955	+	0.851130i	$0.675918\pi$
$692$	0	0
$693$	−34.0246	−1.29249
$694$	0	0
$695$	−13.5436	−0.513739
$696$	0	0
$697$	43.5411	1.64923
$698$	0	0
$699$	50.5962	1.91372
$700$	0	0
$701$	−46.9702	−1.77404	−0.887020	−	0.461730i	$-0.847229\pi$
$701$	−46.9702	−1.77404	−0.887020	+	0.461730i	$0.847229\pi$
$702$	0	0
$703$	51.9805	1.96048
$704$	0	0
$705$	−1.61338	−0.0607635
$706$	0	0
$707$	−23.4518	−0.881994
$708$	0	0
$709$	−32.6609	−1.22661	−0.613303	−	0.789847i	$-0.710160\pi$
$709$	−32.6609	−1.22661	−0.613303	+	0.789847i	$0.710160\pi$
$710$	0	0
$711$	12.3320	0.462486
$712$	0	0
$713$	−15.0524	−0.563716
$714$	0	0
$715$	−0.441377	−0.0165066
$716$	0	0
$717$	−12.9572	−0.483896
$718$	0	0
$719$	1.34533	0.0501724	0.0250862	−	0.999685i	$-0.492014\pi$
$719$	1.34533	0.0501724	0.0250862	+	0.999685i	$0.492014\pi$
$720$	0	0
$721$	−31.5306	−1.17426
$722$	0	0
$723$	1.41448	0.0526049
$724$	0	0
$725$	−31.4618	−1.16846
$726$	0	0
$727$	17.3657	0.644057	0.322028	−	0.946730i	$-0.395635\pi$
$727$	17.3657	0.644057	0.322028	+	0.946730i	$0.395635\pi$
$728$	0	0
$729$	−43.7168	−1.61914
$730$	0	0
$731$	18.0309	0.666898
$732$	0	0
$733$	21.2269	0.784035	0.392017	−	0.919958i	$-0.371777\pi$
$733$	21.2269	0.784035	0.392017	+	0.919958i	$0.371777\pi$
$734$	0	0
$735$	3.55522	0.131136
$736$	0	0
$737$	−22.5427	−0.830370
$738$	0	0
$739$	22.7492	0.836841	0.418421	−	0.908253i	$-0.362584\pi$
$739$	22.7492	0.836841	0.418421	+	0.908253i	$0.362584\pi$
$740$	0	0
$741$	−2.92140	−0.107320
$742$	0	0
$743$	38.7947	1.42324	0.711620	−	0.702565i	$-0.247962\pi$
$743$	38.7947	1.42324	0.711620	+	0.702565i	$0.247962\pi$
$744$	0	0
$745$	13.1332	0.481164
$746$	0	0
$747$	8.27897	0.302912
$748$	0	0
$749$	40.7572	1.48924
$750$	0	0
$751$	−19.5353	−0.712853	−0.356427	−	0.934323i	$-0.616005\pi$
$751$	−19.5353	−0.712853	−0.356427	+	0.934323i	$0.616005\pi$
$752$	0	0
$753$	26.0764	0.950276
$754$	0	0
$755$	9.75120	0.354883
$756$	0	0
$757$	−33.5626	−1.21985	−0.609927	−	0.792458i	$-0.708801\pi$
$757$	−33.5626	−1.21985	−0.609927	+	0.792458i	$0.708801\pi$
$758$	0	0
$759$	25.4286	0.923000
$760$	0	0
$761$	18.5635	0.672926	0.336463	−	0.941697i	$-0.390769\pi$
$761$	18.5635	0.672926	0.336463	+	0.941697i	$0.390769\pi$
$762$	0	0
$763$	13.4414	0.486612
$764$	0	0
$765$	17.2588	0.623994
$766$	0	0
$767$	2.90984	0.105068
$768$	0	0
$769$	−18.3170	−0.660528	−0.330264	−	0.943889i	$-0.607138\pi$
$769$	−18.3170	−0.660528	−0.330264	+	0.943889i	$0.607138\pi$
$770$	0	0
$771$	−30.5687	−1.10090
$772$	0	0
$773$	−21.8345	−0.785333	−0.392667	−	0.919681i	$-0.628447\pi$
$773$	−21.8345	−0.785333	−0.392667	+	0.919681i	$0.628447\pi$
$774$	0	0
$775$	−17.0730	−0.613279
$776$	0	0
$777$	87.2825	3.13124
$778$	0	0
$779$	−50.6824	−1.81588
$780$	0	0
$781$	16.3054	0.583453
$782$	0	0
$783$	−38.0620	−1.36023
$784$	0	0
$785$	−0.631723	−0.0225471
$786$	0	0
$787$	29.2142	1.04137	0.520686	−	0.853748i	$-0.325676\pi$
$787$	29.2142	1.04137	0.520686	+	0.853748i	$0.325676\pi$
$788$	0	0
$789$	−53.6369	−1.90953
$790$	0	0
$791$	7.25006	0.257783
$792$	0	0
$793$	−2.31767	−0.0823029
$794$	0	0
$795$	−10.3918	−0.368558
$796$	0	0
$797$	13.9571	0.494387	0.247193	−	0.968966i	$-0.420492\pi$
$797$	13.9571	0.494387	0.247193	+	0.968966i	$0.420492\pi$
$798$	0	0
$799$	2.81866	0.0997171
$800$	0	0
$801$	−22.9585	−0.811200
$802$	0	0
$803$	4.49969	0.158790
$804$	0	0
$805$	−9.36215	−0.329972
$806$	0	0
$807$	86.5033	3.04506
$808$	0	0
$809$	30.4703	1.07128	0.535639	−	0.844447i	$-0.320071\pi$
$809$	30.4703	1.07128	0.535639	+	0.844447i	$0.320071\pi$
$810$	0	0
$811$	6.12517	0.215084	0.107542	−	0.994201i	$-0.465702\pi$
$811$	6.12517	0.215084	0.107542	+	0.994201i	$0.465702\pi$
$812$	0	0
$813$	24.4825	0.858638
$814$	0	0
$815$	8.77143	0.307250
$816$	0	0
$817$	−20.9882	−0.734285
$818$	0	0
$819$	−3.02960	−0.105863
$820$	0	0
$821$	−16.7013	−0.582878	−0.291439	−	0.956589i	$-0.594134\pi$
$821$	−16.7013	−0.582878	−0.291439	+	0.956589i	$0.594134\pi$
$822$	0	0
$823$	−46.3908	−1.61708	−0.808541	−	0.588439i	$-0.799743\pi$
$823$	−46.3908	−1.61708	−0.808541	+	0.588439i	$0.799743\pi$
$824$	0	0
$825$	28.8421	1.00415
$826$	0	0
$827$	31.8059	1.10600	0.553000	−	0.833181i	$-0.313483\pi$
$827$	31.8059	1.10600	0.553000	+	0.833181i	$0.313483\pi$
$828$	0	0
$829$	20.9702	0.728326	0.364163	−	0.931335i	$-0.381355\pi$
$829$	20.9702	0.728326	0.364163	+	0.931335i	$0.381355\pi$
$830$	0	0
$831$	−41.4471	−1.43778
$832$	0	0
$833$	−6.21114	−0.215203
$834$	0	0
$835$	−3.12610	−0.108183
$836$	0	0
$837$	−20.6546	−0.713929
$838$	0	0
$839$	−35.4801	−1.22491	−0.612455	−	0.790505i	$-0.709818\pi$
$839$	−35.4801	−1.22491	−0.612455	+	0.790505i	$0.709818\pi$
$840$	0	0
$841$	25.2363	0.870218
$842$	0	0
$843$	−11.4712	−0.395089
$844$	0	0
$845$	11.0522	0.380206
$846$	0	0
$847$	15.1205	0.519548
$848$	0	0
$849$	64.9076	2.22762
$850$	0	0
$851$	−40.2869	−1.38102
$852$	0	0
$853$	36.5371	1.25101	0.625503	−	0.780222i	$-0.284894\pi$
$853$	36.5371	1.25101	0.625503	+	0.780222i	$0.284894\pi$
$854$	0	0
$855$	−20.0895	−0.687046
$856$	0	0
$857$	−25.9744	−0.887268	−0.443634	−	0.896208i	$-0.646311\pi$
$857$	−25.9744	−0.887268	−0.443634	+	0.896208i	$0.646311\pi$
$858$	0	0
$859$	−37.0257	−1.26330	−0.631650	−	0.775253i	$-0.717622\pi$
$859$	−37.0257	−1.26330	−0.631650	+	0.775253i	$0.717622\pi$
$860$	0	0
$861$	−85.1027	−2.90029
$862$	0	0
$863$	11.1863	0.380786	0.190393	−	0.981708i	$-0.439024\pi$
$863$	11.1863	0.380786	0.190393	+	0.981708i	$0.439024\pi$
$864$	0	0
$865$	6.25910	0.212816
$866$	0	0
$867$	−1.20537	−0.0409365
$868$	0	0
$869$	6.13490	0.208112
$870$	0	0
$871$	−2.00723	−0.0680125
$872$	0	0
$873$	−70.2927	−2.37905
$874$	0	0
$875$	−23.0472	−0.779136
$876$	0	0
$877$	−27.3432	−0.923316	−0.461658	−	0.887058i	$-0.652745\pi$
$877$	−27.3432	−0.923316	−0.461658	+	0.887058i	$0.652745\pi$
$878$	0	0
$879$	−80.9319	−2.72977
$880$	0	0
$881$	15.9311	0.536731	0.268365	−	0.963317i	$-0.413517\pi$
$881$	15.9311	0.536731	0.268365	+	0.963317i	$0.413517\pi$
$882$	0	0
$883$	−37.6214	−1.26606	−0.633030	−	0.774127i	$-0.718189\pi$
$883$	−37.6214	−1.26606	−0.633030	+	0.774127i	$0.718189\pi$
$884$	0	0
$885$	32.3995	1.08910
$886$	0	0
$887$	−7.55233	−0.253582	−0.126791	−	0.991929i	$-0.540468\pi$
$887$	−7.55233	−0.253582	−0.126791	+	0.991929i	$0.540468\pi$
$888$	0	0
$889$	45.2152	1.51647
$890$	0	0
$891$	−0.143657	−0.00481268
$892$	0	0
$893$	−3.28096	−0.109793
$894$	0	0
$895$	−12.0882	−0.404063
$896$	0	0
$897$	2.26420	0.0755995
$898$	0	0
$899$	29.4317	0.981604
$900$	0	0
$901$	18.1549	0.604828
$902$	0	0
$903$	−35.2422	−1.17279
$904$	0	0
$905$	−0.894076	−0.0297201
$906$	0	0
$907$	22.5884	0.750035	0.375017	−	0.927018i	$-0.377637\pi$
$907$	22.5884	0.750035	0.375017	+	0.927018i	$0.377637\pi$
$908$	0	0
$909$	39.0025	1.29363
$910$	0	0
$911$	51.0206	1.69039	0.845194	−	0.534460i	$-0.179485\pi$
$911$	51.0206	1.69039	0.845194	+	0.534460i	$0.179485\pi$
$912$	0	0
$913$	4.11860	0.136306
$914$	0	0
$915$	−25.8060	−0.853121
$916$	0	0
$917$	13.3776	0.441768
$918$	0	0
$919$	−8.54415	−0.281846	−0.140923	−	0.990021i	$-0.545007\pi$
$919$	−8.54415	−0.281846	−0.140923	+	0.990021i	$0.545007\pi$
$920$	0	0
$921$	−17.4309	−0.574366
$922$	0	0
$923$	1.45186	0.0477884
$924$	0	0
$925$	−45.6949	−1.50244
$926$	0	0
$927$	52.4384	1.72230
$928$	0	0
$929$	27.6881	0.908416	0.454208	−	0.890896i	$-0.349922\pi$
$929$	27.6881	0.908416	0.454208	+	0.890896i	$0.349922\pi$
$930$	0	0
$931$	7.22985	0.236949
$932$	0	0
$933$	−60.2385	−1.97212
$934$	0	0
$935$	8.58588	0.280788
$936$	0	0
$937$	−53.9561	−1.76267	−0.881334	−	0.472493i	$-0.843354\pi$
$937$	−53.9561	−1.76267	−0.881334	+	0.472493i	$0.843354\pi$
$938$	0	0
$939$	−24.7169	−0.806607
$940$	0	0
$941$	7.30751	0.238218	0.119109	−	0.992881i	$-0.461996\pi$
$941$	7.30751	0.238218	0.119109	+	0.992881i	$0.461996\pi$
$942$	0	0
$943$	39.2808	1.27916
$944$	0	0
$945$	−12.8466	−0.417900
$946$	0	0
$947$	−40.9694	−1.33133	−0.665663	−	0.746252i	$-0.731851\pi$
$947$	−40.9694	−1.33133	−0.665663	+	0.746252i	$0.731851\pi$
$948$	0	0
$949$	0.400659	0.0130059
$950$	0	0
$951$	−8.71505	−0.282605
$952$	0	0
$953$	51.5941	1.67130	0.835648	−	0.549266i	$-0.185092\pi$
$953$	51.5941	1.67130	0.835648	+	0.549266i	$0.185092\pi$
$954$	0	0
$955$	1.82800	0.0591527
$956$	0	0
$957$	−49.7203	−1.60723
$958$	0	0
$959$	0.159938	0.00516468
$960$	0	0
$961$	−15.0286	−0.484795
$962$	0	0
$963$	−67.7831	−2.18428
$964$	0	0
$965$	−6.58871	−0.212098
$966$	0	0
$967$	10.6517	0.342535	0.171268	−	0.985225i	$-0.445214\pi$
$967$	10.6517	0.342535	0.171268	+	0.985225i	$0.445214\pi$
$968$	0	0
$969$	56.8285	1.82559
$970$	0	0
$971$	−5.30497	−0.170244	−0.0851222	−	0.996371i	$-0.527128\pi$
$971$	−5.30497	−0.170244	−0.0851222	+	0.996371i	$0.527128\pi$
$972$	0	0
$973$	−46.2471	−1.48261
$974$	0	0
$975$	2.56814	0.0822463
$976$	0	0
$977$	43.3262	1.38613	0.693064	−	0.720876i	$-0.256260\pi$
$977$	43.3262	1.38613	0.693064	+	0.720876i	$0.256260\pi$
$978$	0	0
$979$	−11.4214	−0.365029
$980$	0	0
$981$	−22.3543	−0.713719
$982$	0	0
$983$	−39.3646	−1.25554	−0.627768	−	0.778401i	$-0.716031\pi$
$983$	−39.3646	−1.25554	−0.627768	+	0.778401i	$0.716031\pi$
$984$	0	0
$985$	−7.79291	−0.248303
$986$	0	0
$987$	−5.50918	−0.175359
$988$	0	0
$989$	16.2667	0.517251
$990$	0	0
$991$	−13.9311	−0.442536	−0.221268	−	0.975213i	$-0.571020\pi$
$991$	−13.9311	−0.442536	−0.221268	+	0.975213i	$0.571020\pi$
$992$	0	0
$993$	−89.1582	−2.82935
$994$	0	0
$995$	−22.4312	−0.711118
$996$	0	0
$997$	−27.4237	−0.868519	−0.434259	−	0.900788i	$-0.642990\pi$
$997$	−27.4237	−0.868519	−0.434259	+	0.900788i	$0.642990\pi$
$998$	0	0
$999$	−55.2811	−1.74902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.7	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.7	✓	87	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:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80093 q^{3} -0.853190 q^{5} -2.91337 q^{7} +4.84521 q^{9} +O(q^{10})\)	\(q-2.80093 q^{3} -0.853190 q^{5} -2.91337 q^{7} +4.84521 q^{9} +2.41038 q^{11} +0.214624 q^{13} +2.38972 q^{15} -4.17497 q^{17} +4.85972 q^{19} +8.16014 q^{21} -3.76647 q^{23} -4.27207 q^{25} -5.16829 q^{27} +7.36453 q^{29} +3.99642 q^{31} -6.75131 q^{33} +2.48566 q^{35} +10.6962 q^{37} -0.601147 q^{39} -10.4291 q^{41} -4.31882 q^{43} -4.13388 q^{45} -0.675134 q^{47} +1.48771 q^{49} +11.6938 q^{51} -4.34852 q^{53} -2.05651 q^{55} -13.6117 q^{57} +13.5578 q^{59} -10.7987 q^{61} -14.1159 q^{63} -0.183115 q^{65} -9.35232 q^{67} +10.5496 q^{69} +6.76464 q^{71} +1.86679 q^{73} +11.9658 q^{75} -7.02233 q^{77} +2.54520 q^{79} -0.0595991 q^{81} +1.70869 q^{83} +3.56204 q^{85} -20.6275 q^{87} -4.73841 q^{89} -0.625279 q^{91} -11.1937 q^{93} -4.14626 q^{95} -14.5077 q^{97} +11.6788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * q^99

Introduction

L-functions

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