LMFDB - Embedded newform 4024.2.a.e.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.795174 q^{3} +3.83818 q^{5} +0.297725 q^{7} -2.36770 q^{9} +O(q^{10})$	$q-0.795174 q^{3} +3.83818 q^{5} +0.297725 q^{7} -2.36770 q^{9} -2.28339 q^{11} -6.87906 q^{13} -3.05202 q^{15} +2.62615 q^{17} -2.89852 q^{19} -0.236743 q^{21} +3.98586 q^{23} +9.73160 q^{25} +4.26825 q^{27} +3.29851 q^{29} -3.05689 q^{31} +1.81569 q^{33} +1.14272 q^{35} +9.12864 q^{37} +5.47005 q^{39} -1.23802 q^{41} -5.27446 q^{43} -9.08764 q^{45} -13.5898 q^{47} -6.91136 q^{49} -2.08825 q^{51} -0.167198 q^{53} -8.76406 q^{55} +2.30483 q^{57} +0.470200 q^{59} -4.07855 q^{61} -0.704922 q^{63} -26.4031 q^{65} -9.63256 q^{67} -3.16945 q^{69} +2.27457 q^{71} -14.3332 q^{73} -7.73831 q^{75} -0.679822 q^{77} -12.7286 q^{79} +3.70909 q^{81} +1.77717 q^{83} +10.0796 q^{85} -2.62289 q^{87} +8.24033 q^{89} -2.04807 q^{91} +2.43076 q^{93} -11.1250 q^{95} -6.79098 q^{97} +5.40638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−0.795174	−0.459094	−0.229547	−	0.973298i	$-0.573724\pi$
$3$	−0.795174	−0.459094	−0.229547	+	0.973298i	$0.573724\pi$
$4$	0	0
$5$	3.83818	1.71648	0.858242	−	0.513245i	$-0.171557\pi$
$5$	3.83818	1.71648	0.858242	+	0.513245i	$0.171557\pi$
$6$	0	0
$7$	0.297725	0.112529	0.0562647	−	0.998416i	$-0.482081\pi$
$7$	0.297725	0.112529	0.0562647	+	0.998416i	$0.482081\pi$
$8$	0	0
$9$	−2.36770	−0.789233
$10$	0	0
$11$	−2.28339	−0.688468	−0.344234	−	0.938884i	$-0.611861\pi$
$11$	−2.28339	−0.688468	−0.344234	+	0.938884i	$0.611861\pi$
$12$	0	0
$13$	−6.87906	−1.90791	−0.953954	−	0.299952i	$-0.903029\pi$
$13$	−6.87906	−1.90791	−0.953954	+	0.299952i	$0.903029\pi$
$14$	0	0
$15$	−3.05202	−0.788028
$16$	0	0
$17$	2.62615	0.636936	0.318468	−	0.947934i	$-0.396832\pi$
$17$	2.62615	0.636936	0.318468	+	0.947934i	$0.396832\pi$
$18$	0	0
$19$	−2.89852	−0.664967	−0.332483	−	0.943109i	$-0.607887\pi$
$19$	−2.89852	−0.664967	−0.332483	+	0.943109i	$0.607887\pi$
$20$	0	0
$21$	−0.236743	−0.0516615
$22$	0	0
$23$	3.98586	0.831109	0.415555	−	0.909568i	$-0.363587\pi$
$23$	3.98586	0.831109	0.415555	+	0.909568i	$0.363587\pi$
$24$	0	0
$25$	9.73160	1.94632
$26$	0	0
$27$	4.26825	0.821426
$28$	0	0
$29$	3.29851	0.612518	0.306259	−	0.951948i	$-0.400923\pi$
$29$	3.29851	0.612518	0.306259	+	0.951948i	$0.400923\pi$
$30$	0	0
$31$	−3.05689	−0.549033	−0.274517	−	0.961582i	$-0.588518\pi$
$31$	−3.05689	−0.549033	−0.274517	+	0.961582i	$0.588518\pi$
$32$	0	0
$33$	1.81569	0.316072
$34$	0	0
$35$	1.14272	0.193155
$36$	0	0
$37$	9.12864	1.50074	0.750370	−	0.661018i	$-0.229875\pi$
$37$	9.12864	1.50074	0.750370	+	0.661018i	$0.229875\pi$
$38$	0	0
$39$	5.47005	0.875909
$40$	0	0
$41$	−1.23802	−0.193346	−0.0966729	−	0.995316i	$-0.530820\pi$
$41$	−1.23802	−0.193346	−0.0966729	+	0.995316i	$0.530820\pi$
$42$	0	0
$43$	−5.27446	−0.804347	−0.402173	−	0.915564i	$-0.631745\pi$
$43$	−5.27446	−0.804347	−0.402173	+	0.915564i	$0.631745\pi$
$44$	0	0
$45$	−9.08764	−1.35471
$46$	0	0
$47$	−13.5898	−1.98228	−0.991141	−	0.132812i	$-0.957599\pi$
$47$	−13.5898	−1.98228	−0.991141	+	0.132812i	$0.957599\pi$
$48$	0	0
$49$	−6.91136	−0.987337
$50$	0	0
$51$	−2.08825	−0.292413
$52$	0	0
$53$	−0.167198	−0.0229665	−0.0114832	−	0.999934i	$-0.503655\pi$
$53$	−0.167198	−0.0229665	−0.0114832	+	0.999934i	$0.503655\pi$
$54$	0	0
$55$	−8.76406	−1.18175
$56$	0	0
$57$	2.30483	0.305282
$58$	0	0
$59$	0.470200	0.0612148	0.0306074	−	0.999531i	$-0.490256\pi$
$59$	0.470200	0.0612148	0.0306074	+	0.999531i	$0.490256\pi$
$60$	0	0
$61$	−4.07855	−0.522205	−0.261102	−	0.965311i	$-0.584086\pi$
$61$	−4.07855	−0.522205	−0.261102	+	0.965311i	$0.584086\pi$
$62$	0	0
$63$	−0.704922	−0.0888119
$64$	0	0
$65$	−26.4031	−3.27490
$66$	0	0
$67$	−9.63256	−1.17681	−0.588403	−	0.808568i	$-0.700243\pi$
$67$	−9.63256	−1.17681	−0.588403	+	0.808568i	$0.700243\pi$
$68$	0	0
$69$	−3.16945	−0.381557
$70$	0	0
$71$	2.27457	0.269942	0.134971	−	0.990850i	$-0.456906\pi$
$71$	2.27457	0.269942	0.134971	+	0.990850i	$0.456906\pi$
$72$	0	0
$73$	−14.3332	−1.67757	−0.838786	−	0.544461i	$-0.816734\pi$
$73$	−14.3332	−1.67757	−0.838786	+	0.544461i	$0.816734\pi$
$74$	0	0
$75$	−7.73831	−0.893543
$76$	0	0
$77$	−0.679822	−0.0774729
$78$	0	0
$79$	−12.7286	−1.43208	−0.716040	−	0.698059i	$-0.754047\pi$
$79$	−12.7286	−1.43208	−0.716040	+	0.698059i	$0.754047\pi$
$80$	0	0
$81$	3.70909	0.412121
$82$	0	0
$83$	1.77717	0.195070	0.0975349	−	0.995232i	$-0.468904\pi$
$83$	1.77717	0.195070	0.0975349	+	0.995232i	$0.468904\pi$
$84$	0	0
$85$	10.0796	1.09329
$86$	0	0
$87$	−2.62289	−0.281203
$88$	0	0
$89$	8.24033	0.873473	0.436737	−	0.899589i	$-0.356134\pi$
$89$	8.24033	0.873473	0.436737	+	0.899589i	$0.356134\pi$
$90$	0	0
$91$	−2.04807	−0.214696
$92$	0	0
$93$	2.43076	0.252058
$94$	0	0
$95$	−11.1250	−1.14141
$96$	0	0
$97$	−6.79098	−0.689519	−0.344760	−	0.938691i	$-0.612040\pi$
$97$	−6.79098	−0.689519	−0.344760	+	0.938691i	$0.612040\pi$
$98$	0	0
$99$	5.40638	0.543362
$100$	0	0
$101$	10.6439	1.05911	0.529553	−	0.848277i	$-0.322360\pi$
$101$	10.6439	1.05911	0.529553	+	0.848277i	$0.322360\pi$
$102$	0	0
$103$	1.47169	0.145010	0.0725049	−	0.997368i	$-0.476901\pi$
$103$	1.47169	0.145010	0.0725049	+	0.997368i	$0.476901\pi$
$104$	0	0
$105$	−0.908661	−0.0886762
$106$	0	0
$107$	−16.9772	−1.64125	−0.820624	−	0.571468i	$-0.806374\pi$
$107$	−16.9772	−1.64125	−0.820624	+	0.571468i	$0.806374\pi$
$108$	0	0
$109$	−1.21094	−0.115987	−0.0579936	−	0.998317i	$-0.518470\pi$
$109$	−1.21094	−0.115987	−0.0579936	+	0.998317i	$0.518470\pi$
$110$	0	0
$111$	−7.25886	−0.688981
$112$	0	0
$113$	12.1514	1.14310	0.571552	−	0.820566i	$-0.306342\pi$
$113$	12.1514	1.14310	0.571552	+	0.820566i	$0.306342\pi$
$114$	0	0
$115$	15.2984	1.42659
$116$	0	0
$117$	16.2875	1.50578
$118$	0	0
$119$	0.781870	0.0716739
$120$	0	0
$121$	−5.78612	−0.526011
$122$	0	0
$123$	0.984439	0.0887638
$124$	0	0
$125$	18.1607	1.62434
$126$	0	0
$127$	−16.3954	−1.45486	−0.727428	−	0.686184i	$-0.759285\pi$
$127$	−16.3954	−1.45486	−0.727428	+	0.686184i	$0.759285\pi$
$128$	0	0
$129$	4.19411	0.369271
$130$	0	0
$131$	−4.09539	−0.357816	−0.178908	−	0.983866i	$-0.557256\pi$
$131$	−4.09539	−0.357816	−0.178908	+	0.983866i	$0.557256\pi$
$132$	0	0
$133$	−0.862962	−0.0748283
$134$	0	0
$135$	16.3823	1.40996
$136$	0	0
$137$	11.1174	0.949826	0.474913	−	0.880033i	$-0.342480\pi$
$137$	11.1174	0.949826	0.474913	+	0.880033i	$0.342480\pi$
$138$	0	0
$139$	−1.71061	−0.145092	−0.0725459	−	0.997365i	$-0.523112\pi$
$139$	−1.71061	−0.145092	−0.0725459	+	0.997365i	$0.523112\pi$
$140$	0	0
$141$	10.8063	0.910054
$142$	0	0
$143$	15.7076	1.31353
$144$	0	0
$145$	12.6603	1.05138
$146$	0	0
$147$	5.49573	0.453280
$148$	0	0
$149$	−8.73899	−0.715926	−0.357963	−	0.933736i	$-0.616529\pi$
$149$	−8.73899	−0.715926	−0.357963	+	0.933736i	$0.616529\pi$
$150$	0	0
$151$	−19.4556	−1.58327	−0.791635	−	0.610994i	$-0.790770\pi$
$151$	−19.4556	−1.58327	−0.791635	+	0.610994i	$0.790770\pi$
$152$	0	0
$153$	−6.21794	−0.502690
$154$	0	0
$155$	−11.7329	−0.942407
$156$	0	0
$157$	−1.95039	−0.155658	−0.0778291	−	0.996967i	$-0.524799\pi$
$157$	−1.95039	−0.155658	−0.0778291	+	0.996967i	$0.524799\pi$
$158$	0	0
$159$	0.132952	0.0105438
$160$	0	0
$161$	1.18669	0.0935242
$162$	0	0
$163$	−4.01563	−0.314528	−0.157264	−	0.987557i	$-0.550267\pi$
$163$	−4.01563	−0.314528	−0.157264	+	0.987557i	$0.550267\pi$
$164$	0	0
$165$	6.96895	0.542532
$166$	0	0
$167$	7.18233	0.555786	0.277893	−	0.960612i	$-0.410364\pi$
$167$	7.18233	0.555786	0.277893	+	0.960612i	$0.410364\pi$
$168$	0	0
$169$	34.3215	2.64012
$170$	0	0
$171$	6.86283	0.524814
$172$	0	0
$173$	−18.9560	−1.44120	−0.720598	−	0.693353i	$-0.756133\pi$
$173$	−18.9560	−1.44120	−0.720598	+	0.693353i	$0.756133\pi$
$174$	0	0
$175$	2.89734	0.219018
$176$	0	0
$177$	−0.373890	−0.0281033
$178$	0	0
$179$	15.7355	1.17613	0.588065	−	0.808814i	$-0.299890\pi$
$179$	15.7355	1.17613	0.588065	+	0.808814i	$0.299890\pi$
$180$	0	0
$181$	−16.4996	−1.22640	−0.613202	−	0.789926i	$-0.710119\pi$
$181$	−16.4996	−1.22640	−0.613202	+	0.789926i	$0.710119\pi$
$182$	0	0
$183$	3.24316	0.239741
$184$	0	0
$185$	35.0373	2.57600
$186$	0	0
$187$	−5.99653	−0.438510
$188$	0	0
$189$	1.27076	0.0924345
$190$	0	0
$191$	−22.7350	−1.64505	−0.822523	−	0.568732i	$-0.807434\pi$
$191$	−22.7350	−1.64505	−0.822523	+	0.568732i	$0.807434\pi$
$192$	0	0
$193$	−20.3222	−1.46283	−0.731413	−	0.681935i	$-0.761139\pi$
$193$	−20.3222	−1.46283	−0.731413	+	0.681935i	$0.761139\pi$
$194$	0	0
$195$	20.9950	1.50348
$196$	0	0
$197$	−4.38511	−0.312426	−0.156213	−	0.987723i	$-0.549929\pi$
$197$	−4.38511	−0.312426	−0.156213	+	0.987723i	$0.549929\pi$
$198$	0	0
$199$	1.83622	0.130166	0.0650830	−	0.997880i	$-0.479269\pi$
$199$	1.83622	0.130166	0.0650830	+	0.997880i	$0.479269\pi$
$200$	0	0
$201$	7.65956	0.540264
$202$	0	0
$203$	0.982048	0.0689263
$204$	0	0
$205$	−4.75173	−0.331875
$206$	0	0
$207$	−9.43732	−0.655939
$208$	0	0
$209$	6.61846	0.457809
$210$	0	0
$211$	−21.8494	−1.50417	−0.752087	−	0.659064i	$-0.770953\pi$
$211$	−21.8494	−1.50417	−0.752087	+	0.659064i	$0.770953\pi$
$212$	0	0
$213$	−1.80868	−0.123929
$214$	0	0
$215$	−20.2443	−1.38065
$216$	0	0
$217$	−0.910111	−0.0617824
$218$	0	0
$219$	11.3974	0.770163
$220$	0	0
$221$	−18.0655	−1.21521
$222$	0	0
$223$	5.10979	0.342177	0.171088	−	0.985256i	$-0.445272\pi$
$223$	5.10979	0.342177	0.171088	+	0.985256i	$0.445272\pi$
$224$	0	0
$225$	−23.0415	−1.53610
$226$	0	0
$227$	20.0267	1.32922	0.664610	−	0.747191i	$-0.268598\pi$
$227$	20.0267	1.32922	0.664610	+	0.747191i	$0.268598\pi$
$228$	0	0
$229$	28.2650	1.86780	0.933901	−	0.357532i	$-0.116382\pi$
$229$	28.2650	1.86780	0.933901	+	0.357532i	$0.116382\pi$
$230$	0	0
$231$	0.540577	0.0355673
$232$	0	0
$233$	9.96788	0.653017	0.326509	−	0.945194i	$-0.394128\pi$
$233$	9.96788	0.653017	0.326509	+	0.945194i	$0.394128\pi$
$234$	0	0
$235$	−52.1602	−3.40256
$236$	0	0
$237$	10.1215	0.657459
$238$	0	0
$239$	−9.26073	−0.599027	−0.299514	−	0.954092i	$-0.596824\pi$
$239$	−9.26073	−0.599027	−0.299514	+	0.954092i	$0.596824\pi$
$240$	0	0
$241$	−17.6865	−1.13929	−0.569645	−	0.821891i	$-0.692919\pi$
$241$	−17.6865	−1.13929	−0.569645	+	0.821891i	$0.692919\pi$
$242$	0	0
$243$	−15.7541	−1.01063
$244$	0	0
$245$	−26.5270	−1.69475
$246$	0	0
$247$	19.9391	1.26870
$248$	0	0
$249$	−1.41316	−0.0895554
$250$	0	0
$251$	18.4056	1.16175	0.580877	−	0.813992i	$-0.302710\pi$
$251$	18.4056	1.16175	0.580877	+	0.813992i	$0.302710\pi$
$252$	0	0
$253$	−9.10128	−0.572193
$254$	0	0
$255$	−8.01506	−0.501923
$256$	0	0
$257$	24.2339	1.51167	0.755835	−	0.654762i	$-0.227231\pi$
$257$	24.2339	1.51167	0.755835	+	0.654762i	$0.227231\pi$
$258$	0	0
$259$	2.71782	0.168877
$260$	0	0
$261$	−7.80988	−0.483419
$262$	0	0
$263$	−24.2779	−1.49704	−0.748521	−	0.663111i	$-0.769236\pi$
$263$	−24.2779	−1.49704	−0.748521	+	0.663111i	$0.769236\pi$
$264$	0	0
$265$	−0.641737	−0.0394216
$266$	0	0
$267$	−6.55250	−0.401006
$268$	0	0
$269$	18.0146	1.09837	0.549184	−	0.835702i	$-0.314939\pi$
$269$	18.0146	1.09837	0.549184	+	0.835702i	$0.314939\pi$
$270$	0	0
$271$	27.9535	1.69806	0.849028	−	0.528348i	$-0.177188\pi$
$271$	27.9535	1.69806	0.849028	+	0.528348i	$0.177188\pi$
$272$	0	0
$273$	1.62857	0.0985655
$274$	0	0
$275$	−22.2210	−1.33998
$276$	0	0
$277$	−3.92418	−0.235781	−0.117890	−	0.993027i	$-0.537613\pi$
$277$	−3.92418	−0.235781	−0.117890	+	0.993027i	$0.537613\pi$
$278$	0	0
$279$	7.23779	0.433315
$280$	0	0
$281$	17.2583	1.02954	0.514772	−	0.857327i	$-0.327877\pi$
$281$	17.2583	1.02954	0.514772	+	0.857327i	$0.327877\pi$
$282$	0	0
$283$	5.44313	0.323561	0.161780	−	0.986827i	$-0.448276\pi$
$283$	5.44313	0.323561	0.161780	+	0.986827i	$0.448276\pi$
$284$	0	0
$285$	8.84635	0.524012
$286$	0	0
$287$	−0.368588	−0.0217571
$288$	0	0
$289$	−10.1033	−0.594313
$290$	0	0
$291$	5.40001	0.316554
$292$	0	0
$293$	−12.2219	−0.714014	−0.357007	−	0.934102i	$-0.616203\pi$
$293$	−12.2219	−0.714014	−0.357007	+	0.934102i	$0.616203\pi$
$294$	0	0
$295$	1.80471	0.105074
$296$	0	0
$297$	−9.74610	−0.565526
$298$	0	0
$299$	−27.4190	−1.58568
$300$	0	0
$301$	−1.57034	−0.0905126
$302$	0	0
$303$	−8.46375	−0.486230
$304$	0	0
$305$	−15.6542	−0.896357
$306$	0	0
$307$	10.3157	0.588747	0.294373	−	0.955691i	$-0.404889\pi$
$307$	10.3157	0.588747	0.294373	+	0.955691i	$0.404889\pi$
$308$	0	0
$309$	−1.17025	−0.0665731
$310$	0	0
$311$	−6.20464	−0.351833	−0.175916	−	0.984405i	$-0.556289\pi$
$311$	−6.20464	−0.351833	−0.175916	+	0.984405i	$0.556289\pi$
$312$	0	0
$313$	−30.0963	−1.70114	−0.850571	−	0.525861i	$-0.823743\pi$
$313$	−30.0963	−1.70114	−0.850571	+	0.525861i	$0.823743\pi$
$314$	0	0
$315$	−2.70562	−0.152444
$316$	0	0
$317$	18.6041	1.04491	0.522456	−	0.852666i	$-0.325016\pi$
$317$	18.6041	1.04491	0.522456	+	0.852666i	$0.325016\pi$
$318$	0	0
$319$	−7.53179	−0.421699
$320$	0	0
$321$	13.4998	0.753487
$322$	0	0
$323$	−7.61196	−0.423541
$324$	0	0
$325$	−66.9443	−3.71340
$326$	0	0
$327$	0.962909	0.0532490
$328$	0	0
$329$	−4.04603	−0.223065
$330$	0	0
$331$	29.9806	1.64789	0.823943	−	0.566673i	$-0.191770\pi$
$331$	29.9806	1.64789	0.823943	+	0.566673i	$0.191770\pi$
$332$	0	0
$333$	−21.6139	−1.18443
$334$	0	0
$335$	−36.9715	−2.01997
$336$	0	0
$337$	7.82831	0.426435	0.213218	−	0.977005i	$-0.431606\pi$
$337$	7.82831	0.426435	0.213218	+	0.977005i	$0.431606\pi$
$338$	0	0
$339$	−9.66244	−0.524792
$340$	0	0
$341$	6.98007	0.377992
$342$	0	0
$343$	−4.14176	−0.223634
$344$	0	0
$345$	−12.1649	−0.654937
$346$	0	0
$347$	10.6775	0.573197	0.286599	−	0.958051i	$-0.407475\pi$
$347$	10.6775	0.573197	0.286599	+	0.958051i	$0.407475\pi$
$348$	0	0
$349$	−24.0258	−1.28607	−0.643035	−	0.765837i	$-0.722325\pi$
$349$	−24.0258	−1.28607	−0.643035	+	0.765837i	$0.722325\pi$
$350$	0	0
$351$	−29.3616	−1.56721
$352$	0	0
$353$	28.5933	1.52187	0.760935	−	0.648828i	$-0.224741\pi$
$353$	28.5933	1.52187	0.760935	+	0.648828i	$0.224741\pi$
$354$	0	0
$355$	8.73019	0.463351
$356$	0	0
$357$	−0.621723	−0.0329051
$358$	0	0
$359$	33.0259	1.74304	0.871520	−	0.490360i	$-0.163135\pi$
$359$	33.0259	1.74304	0.871520	+	0.490360i	$0.163135\pi$
$360$	0	0
$361$	−10.5986	−0.557819
$362$	0	0
$363$	4.60097	0.241489
$364$	0	0
$365$	−55.0133	−2.87953
$366$	0	0
$367$	−11.4028	−0.595223	−0.297612	−	0.954687i	$-0.596190\pi$
$367$	−11.4028	−0.595223	−0.297612	+	0.954687i	$0.596190\pi$
$368$	0	0
$369$	2.93125	0.152595
$370$	0	0
$371$	−0.0497791	−0.00258440
$372$	0	0
$373$	29.3841	1.52145	0.760726	−	0.649073i	$-0.224843\pi$
$373$	29.3841	1.52145	0.760726	+	0.649073i	$0.224843\pi$
$374$	0	0
$375$	−14.4409	−0.745726
$376$	0	0
$377$	−22.6907	−1.16863
$378$	0	0
$379$	28.8597	1.48242	0.741212	−	0.671271i	$-0.234251\pi$
$379$	28.8597	1.48242	0.741212	+	0.671271i	$0.234251\pi$
$380$	0	0
$381$	13.0372	0.667916
$382$	0	0
$383$	16.5216	0.844214	0.422107	−	0.906546i	$-0.361291\pi$
$383$	16.5216	0.844214	0.422107	+	0.906546i	$0.361291\pi$
$384$	0	0
$385$	−2.60928	−0.132981
$386$	0	0
$387$	12.4883	0.634817
$388$	0	0
$389$	−0.471807	−0.0239216	−0.0119608	−	0.999928i	$-0.503807\pi$
$389$	−0.471807	−0.0239216	−0.0119608	+	0.999928i	$0.503807\pi$
$390$	0	0
$391$	10.4675	0.529363
$392$	0	0
$393$	3.25655	0.164271
$394$	0	0
$395$	−48.8546	−2.45814
$396$	0	0
$397$	−20.1239	−1.00999	−0.504994	−	0.863123i	$-0.668505\pi$
$397$	−20.1239	−1.00999	−0.504994	+	0.863123i	$0.668505\pi$
$398$	0	0
$399$	0.686205	0.0343532
$400$	0	0
$401$	14.1759	0.707913	0.353956	−	0.935262i	$-0.384836\pi$
$401$	14.1759	0.707913	0.353956	+	0.935262i	$0.384836\pi$
$402$	0	0
$403$	21.0285	1.04751
$404$	0	0
$405$	14.2361	0.707400
$406$	0	0
$407$	−20.8443	−1.03321
$408$	0	0
$409$	5.72639	0.283152	0.141576	−	0.989927i	$-0.454783\pi$
$409$	5.72639	0.283152	0.141576	+	0.989927i	$0.454783\pi$
$410$	0	0
$411$	−8.84029	−0.436059
$412$	0	0
$413$	0.139990	0.00688846
$414$	0	0
$415$	6.82109	0.334834
$416$	0	0
$417$	1.36023	0.0666107
$418$	0	0
$419$	3.80700	0.185984	0.0929921	−	0.995667i	$-0.470357\pi$
$419$	3.80700	0.185984	0.0929921	+	0.995667i	$0.470357\pi$
$420$	0	0
$421$	−20.4435	−0.996357	−0.498179	−	0.867074i	$-0.665998\pi$
$421$	−20.4435	−0.996357	−0.498179	+	0.867074i	$0.665998\pi$
$422$	0	0
$423$	32.1767	1.56448
$424$	0	0
$425$	25.5567	1.23968
$426$	0	0
$427$	−1.21429	−0.0587634
$428$	0	0
$429$	−12.4903	−0.603036
$430$	0	0
$431$	11.4120	0.549699	0.274849	−	0.961487i	$-0.411372\pi$
$431$	11.4120	0.549699	0.274849	+	0.961487i	$0.411372\pi$
$432$	0	0
$433$	9.00407	0.432708	0.216354	−	0.976315i	$-0.430583\pi$
$433$	9.00407	0.432708	0.216354	+	0.976315i	$0.430583\pi$
$434$	0	0
$435$	−10.0671	−0.482681
$436$	0	0
$437$	−11.5531	−0.552660
$438$	0	0
$439$	28.5676	1.36346	0.681730	−	0.731604i	$-0.261228\pi$
$439$	28.5676	1.36346	0.681730	+	0.731604i	$0.261228\pi$
$440$	0	0
$441$	16.3640	0.779239
$442$	0	0
$443$	−13.9781	−0.664121	−0.332060	−	0.943258i	$-0.607744\pi$
$443$	−13.9781	−0.664121	−0.332060	+	0.943258i	$0.607744\pi$
$444$	0	0
$445$	31.6278	1.49930
$446$	0	0
$447$	6.94902	0.328677
$448$	0	0
$449$	5.96977	0.281731	0.140865	−	0.990029i	$-0.455012\pi$
$449$	5.96977	0.281731	0.140865	+	0.990029i	$0.455012\pi$
$450$	0	0
$451$	2.82688	0.133112
$452$	0	0
$453$	15.4706	0.726870
$454$	0	0
$455$	−7.86084	−0.368522
$456$	0	0
$457$	−7.68190	−0.359344	−0.179672	−	0.983727i	$-0.557504\pi$
$457$	−7.68190	−0.359344	−0.179672	+	0.983727i	$0.557504\pi$
$458$	0	0
$459$	11.2091	0.523195
$460$	0	0
$461$	4.84480	0.225645	0.112822	−	0.993615i	$-0.464011\pi$
$461$	4.84480	0.225645	0.112822	+	0.993615i	$0.464011\pi$
$462$	0	0
$463$	0.0865042	0.00402019	0.00201010	−	0.999998i	$-0.499360\pi$
$463$	0.0865042	0.00402019	0.00201010	+	0.999998i	$0.499360\pi$
$464$	0	0
$465$	9.32968	0.432653
$466$	0	0
$467$	−4.05869	−0.187814	−0.0939068	−	0.995581i	$-0.529936\pi$
$467$	−4.05869	−0.187814	−0.0939068	+	0.995581i	$0.529936\pi$
$468$	0	0
$469$	−2.86785	−0.132425
$470$	0	0
$471$	1.55090	0.0714617
$472$	0	0
$473$	12.0436	0.553767
$474$	0	0
$475$	−28.2073	−1.29424
$476$	0	0
$477$	0.395875	0.0181259
$478$	0	0
$479$	−7.26030	−0.331732	−0.165866	−	0.986148i	$-0.553042\pi$
$479$	−7.26030	−0.331732	−0.165866	+	0.986148i	$0.553042\pi$
$480$	0	0
$481$	−62.7965	−2.86327
$482$	0	0
$483$	−0.943624	−0.0429364
$484$	0	0
$485$	−26.0650	−1.18355
$486$	0	0
$487$	−26.2941	−1.19150	−0.595750	−	0.803170i	$-0.703145\pi$
$487$	−26.2941	−1.19150	−0.595750	+	0.803170i	$0.703145\pi$
$488$	0	0
$489$	3.19312	0.144398
$490$	0	0
$491$	−38.1973	−1.72382	−0.861911	−	0.507060i	$-0.830732\pi$
$491$	−38.1973	−1.72382	−0.861911	+	0.507060i	$0.830732\pi$
$492$	0	0
$493$	8.66239	0.390134
$494$	0	0
$495$	20.7506	0.932672
$496$	0	0
$497$	0.677195	0.0303764
$498$	0	0
$499$	8.88985	0.397964	0.198982	−	0.980003i	$-0.436236\pi$
$499$	8.88985	0.397964	0.198982	+	0.980003i	$0.436236\pi$
$500$	0	0
$501$	−5.71120	−0.255158
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	40.8531	1.81794
$506$	0	0
$507$	−27.2916	−1.21206
$508$	0	0
$509$	−13.7743	−0.610537	−0.305268	−	0.952266i	$-0.598746\pi$
$509$	−13.7743	−0.610537	−0.305268	+	0.952266i	$0.598746\pi$
$510$	0	0
$511$	−4.26734	−0.188776
$512$	0	0
$513$	−12.3716	−0.546221
$514$	0	0
$515$	5.64860	0.248907
$516$	0	0
$517$	31.0309	1.36474
$518$	0	0
$519$	15.0733	0.661644
$520$	0	0
$521$	−32.9069	−1.44168	−0.720838	−	0.693103i	$-0.756243\pi$
$521$	−32.9069	−1.44168	−0.720838	+	0.693103i	$0.756243\pi$
$522$	0	0
$523$	−12.3292	−0.539119	−0.269560	−	0.962984i	$-0.586878\pi$
$523$	−12.3292	−0.539119	−0.269560	+	0.962984i	$0.586878\pi$
$524$	0	0
$525$	−2.30389	−0.100550
$526$	0	0
$527$	−8.02786	−0.349699
$528$	0	0
$529$	−7.11291	−0.309257
$530$	0	0
$531$	−1.11329	−0.0483127
$532$	0	0
$533$	8.51639	0.368886
$534$	0	0
$535$	−65.1615	−2.81718
$536$	0	0
$537$	−12.5125	−0.539954
$538$	0	0
$539$	15.7813	0.679751
$540$	0	0
$541$	−5.09608	−0.219098	−0.109549	−	0.993981i	$-0.534941\pi$
$541$	−5.09608	−0.219098	−0.109549	+	0.993981i	$0.534941\pi$
$542$	0	0
$543$	13.1200	0.563035
$544$	0	0
$545$	−4.64781	−0.199090
$546$	0	0
$547$	40.3351	1.72461	0.862303	−	0.506393i	$-0.169022\pi$
$547$	40.3351	1.72461	0.862303	+	0.506393i	$0.169022\pi$
$548$	0	0
$549$	9.65678	0.412141
$550$	0	0
$551$	−9.56081	−0.407304
$552$	0	0
$553$	−3.78962	−0.161151
$554$	0	0
$555$	−27.8608	−1.18262
$556$	0	0
$557$	−7.77414	−0.329401	−0.164700	−	0.986344i	$-0.552666\pi$
$557$	−7.77414	−0.329401	−0.164700	+	0.986344i	$0.552666\pi$
$558$	0	0
$559$	36.2833	1.53462
$560$	0	0
$561$	4.76829	0.201317
$562$	0	0
$563$	45.9754	1.93763	0.968817	−	0.247779i	$-0.0797007\pi$
$563$	45.9754	1.93763	0.968817	+	0.247779i	$0.0797007\pi$
$564$	0	0
$565$	46.6390	1.96212
$566$	0	0
$567$	1.10429	0.0463757
$568$	0	0
$569$	−13.4480	−0.563770	−0.281885	−	0.959448i	$-0.590960\pi$
$569$	−13.4480	−0.563770	−0.281885	+	0.959448i	$0.590960\pi$
$570$	0	0
$571$	19.6148	0.820853	0.410426	−	0.911894i	$-0.365380\pi$
$571$	19.6148	0.820853	0.410426	+	0.911894i	$0.365380\pi$
$572$	0	0
$573$	18.0783	0.755231
$574$	0	0
$575$	38.7888	1.61760
$576$	0	0
$577$	−17.0503	−0.709814	−0.354907	−	0.934902i	$-0.615487\pi$
$577$	−17.0503	−0.709814	−0.354907	+	0.934902i	$0.615487\pi$
$578$	0	0
$579$	16.1597	0.671575
$580$	0	0
$581$	0.529108	0.0219511
$582$	0	0
$583$	0.381779	0.0158117
$584$	0	0
$585$	62.5145	2.58466
$586$	0	0
$587$	−27.6051	−1.13938	−0.569692	−	0.821858i	$-0.692937\pi$
$587$	−27.6051	−1.13938	−0.569692	+	0.821858i	$0.692937\pi$
$588$	0	0
$589$	8.86046	0.365089
$590$	0	0
$591$	3.48693	0.143433
$592$	0	0
$593$	5.63771	0.231513	0.115757	−	0.993278i	$-0.463071\pi$
$593$	5.63771	0.231513	0.115757	+	0.993278i	$0.463071\pi$
$594$	0	0
$595$	3.00096	0.123027
$596$	0	0
$597$	−1.46011	−0.0597584
$598$	0	0
$599$	−46.4484	−1.89783	−0.948914	−	0.315534i	$-0.897816\pi$
$599$	−46.4484	−1.89783	−0.948914	+	0.315534i	$0.897816\pi$
$600$	0	0
$601$	−36.4500	−1.48683	−0.743413	−	0.668832i	$-0.766794\pi$
$601$	−36.4500	−1.48683	−0.743413	+	0.668832i	$0.766794\pi$
$602$	0	0
$603$	22.8070	0.928773
$604$	0	0
$605$	−22.2082	−0.902890
$606$	0	0
$607$	37.1707	1.50871	0.754357	−	0.656465i	$-0.227949\pi$
$607$	37.1707	1.50871	0.754357	+	0.656465i	$0.227949\pi$
$608$	0	0
$609$	−0.780899	−0.0316436
$610$	0	0
$611$	93.4854	3.78201
$612$	0	0
$613$	−10.1867	−0.411438	−0.205719	−	0.978611i	$-0.565953\pi$
$613$	−10.1867	−0.411438	−0.205719	+	0.978611i	$0.565953\pi$
$614$	0	0
$615$	3.77845	0.152362
$616$	0	0
$617$	10.4330	0.420016	0.210008	−	0.977700i	$-0.432651\pi$
$617$	10.4330	0.420016	0.210008	+	0.977700i	$0.432651\pi$
$618$	0	0
$619$	29.7930	1.19748	0.598742	−	0.800942i	$-0.295668\pi$
$619$	29.7930	1.19748	0.598742	+	0.800942i	$0.295668\pi$
$620$	0	0
$621$	17.0127	0.682695
$622$	0	0
$623$	2.45335	0.0982914
$624$	0	0
$625$	21.0460	0.841839
$626$	0	0
$627$	−5.26283	−0.210177
$628$	0	0
$629$	23.9732	0.955875
$630$	0	0
$631$	−21.2357	−0.845379	−0.422690	−	0.906275i	$-0.638914\pi$
$631$	−21.2357	−0.845379	−0.422690	+	0.906275i	$0.638914\pi$
$632$	0	0
$633$	17.3741	0.690557
$634$	0	0
$635$	−62.9284	−2.49724
$636$	0	0
$637$	47.5437	1.88375
$638$	0	0
$639$	−5.38549	−0.213047
$640$	0	0
$641$	−2.46028	−0.0971752	−0.0485876	−	0.998819i	$-0.515472\pi$
$641$	−2.46028	−0.0971752	−0.0485876	+	0.998819i	$0.515472\pi$
$642$	0	0
$643$	12.6455	0.498688	0.249344	−	0.968415i	$-0.419785\pi$
$643$	12.6455	0.498688	0.249344	+	0.968415i	$0.419785\pi$
$644$	0	0
$645$	16.0977	0.633848
$646$	0	0
$647$	27.5280	1.08224	0.541119	−	0.840946i	$-0.318001\pi$
$647$	27.5280	1.08224	0.541119	+	0.840946i	$0.318001\pi$
$648$	0	0
$649$	−1.07365	−0.0421444
$650$	0	0
$651$	0.723697	0.0283639
$652$	0	0
$653$	−2.36425	−0.0925202	−0.0462601	−	0.998929i	$-0.514730\pi$
$653$	−2.36425	−0.0925202	−0.0462601	+	0.998929i	$0.514730\pi$
$654$	0	0
$655$	−15.7188	−0.614186
$656$	0	0
$657$	33.9367	1.32399
$658$	0	0
$659$	−22.7471	−0.886103	−0.443051	−	0.896496i	$-0.646104\pi$
$659$	−22.7471	−0.886103	−0.443051	+	0.896496i	$0.646104\pi$
$660$	0	0
$661$	−19.1469	−0.744729	−0.372365	−	0.928087i	$-0.621453\pi$
$661$	−19.1469	−0.744729	−0.372365	+	0.928087i	$0.621453\pi$
$662$	0	0
$663$	14.3652	0.557898
$664$	0	0
$665$	−3.31220	−0.128442
$666$	0	0
$667$	13.1474	0.509069
$668$	0	0
$669$	−4.06317	−0.157091
$670$	0	0
$671$	9.31293	0.359522
$672$	0	0
$673$	49.8662	1.92220	0.961100	−	0.276200i	$-0.0890752\pi$
$673$	49.8662	1.92220	0.961100	+	0.276200i	$0.0890752\pi$
$674$	0	0
$675$	41.5369	1.59876
$676$	0	0
$677$	−2.02041	−0.0776508	−0.0388254	−	0.999246i	$-0.512362\pi$
$677$	−2.02041	−0.0776508	−0.0388254	+	0.999246i	$0.512362\pi$
$678$	0	0
$679$	−2.02184	−0.0775912
$680$	0	0
$681$	−15.9247	−0.610237
$682$	0	0
$683$	−36.5216	−1.39746	−0.698731	−	0.715385i	$-0.746251\pi$
$683$	−36.5216	−1.39746	−0.698731	+	0.715385i	$0.746251\pi$
$684$	0	0
$685$	42.6706	1.63036
$686$	0	0
$687$	−22.4756	−0.857496
$688$	0	0
$689$	1.15017	0.0438179
$690$	0	0
$691$	−2.28799	−0.0870394	−0.0435197	−	0.999053i	$-0.513857\pi$
$691$	−2.28799	−0.0870394	−0.0435197	+	0.999053i	$0.513857\pi$
$692$	0	0
$693$	1.60961	0.0611442
$694$	0	0
$695$	−6.56561	−0.249048
$696$	0	0
$697$	−3.25122	−0.123149
$698$	0	0
$699$	−7.92620	−0.299796
$700$	0	0
$701$	33.5767	1.26818	0.634088	−	0.773261i	$-0.281376\pi$
$701$	33.5767	1.26818	0.634088	+	0.773261i	$0.281376\pi$
$702$	0	0
$703$	−26.4596	−0.997942
$704$	0	0
$705$	41.4764	1.56209
$706$	0	0
$707$	3.16895	0.119181
$708$	0	0
$709$	30.4792	1.14467	0.572335	−	0.820020i	$-0.306038\pi$
$709$	30.4792	1.14467	0.572335	+	0.820020i	$0.306038\pi$
$710$	0	0
$711$	30.1375	1.13024
$712$	0	0
$713$	−12.1843	−0.456307
$714$	0	0
$715$	60.2885	2.25466
$716$	0	0
$717$	7.36389	0.275010
$718$	0	0
$719$	−8.73363	−0.325709	−0.162855	−	0.986650i	$-0.552070\pi$
$719$	−8.73363	−0.325709	−0.162855	+	0.986650i	$0.552070\pi$
$720$	0	0
$721$	0.438158	0.0163179
$722$	0	0
$723$	14.0639	0.523041
$724$	0	0
$725$	32.0998	1.19216
$726$	0	0
$727$	−52.6355	−1.95214	−0.976071	−	0.217451i	$-0.930226\pi$
$727$	−52.6355	−1.95214	−0.976071	+	0.217451i	$0.930226\pi$
$728$	0	0
$729$	1.40000	0.0518520
$730$	0	0
$731$	−13.8515	−0.512317
$732$	0	0
$733$	0.238594	0.00881268	0.00440634	−	0.999990i	$-0.498597\pi$
$733$	0.238594	0.00881268	0.00440634	+	0.999990i	$0.498597\pi$
$734$	0	0
$735$	21.0936	0.778049
$736$	0	0
$737$	21.9949	0.810193
$738$	0	0
$739$	−43.7746	−1.61028	−0.805138	−	0.593088i	$-0.797909\pi$
$739$	−43.7746	−1.61028	−0.805138	+	0.593088i	$0.797909\pi$
$740$	0	0
$741$	−15.8551	−0.582451
$742$	0	0
$743$	23.6689	0.868329	0.434164	−	0.900834i	$-0.357044\pi$
$743$	23.6689	0.868329	0.434164	+	0.900834i	$0.357044\pi$
$744$	0	0
$745$	−33.5418	−1.22888
$746$	0	0
$747$	−4.20780	−0.153955
$748$	0	0
$749$	−5.05453	−0.184689
$750$	0	0
$751$	−15.5827	−0.568621	−0.284311	−	0.958732i	$-0.591765\pi$
$751$	−15.5827	−0.568621	−0.284311	+	0.958732i	$0.591765\pi$
$752$	0	0
$753$	−14.6357	−0.533354
$754$	0	0
$755$	−74.6739	−2.71766
$756$	0	0
$757$	29.6465	1.07752	0.538759	−	0.842460i	$-0.318893\pi$
$757$	29.6465	1.07752	0.538759	+	0.842460i	$0.318893\pi$
$758$	0	0
$759$	7.23710	0.262690
$760$	0	0
$761$	0.954696	0.0346077	0.0173039	−	0.999850i	$-0.494492\pi$
$761$	0.954696	0.0346077	0.0173039	+	0.999850i	$0.494492\pi$
$762$	0	0
$763$	−0.360527	−0.0130520
$764$	0	0
$765$	−23.8655	−0.862860
$766$	0	0
$767$	−3.23453	−0.116792
$768$	0	0
$769$	−0.976425	−0.0352108	−0.0176054	−	0.999845i	$-0.505604\pi$
$769$	−0.976425	−0.0352108	−0.0176054	+	0.999845i	$0.505604\pi$
$770$	0	0
$771$	−19.2702	−0.693999
$772$	0	0
$773$	−13.3884	−0.481549	−0.240774	−	0.970581i	$-0.577401\pi$
$773$	−13.3884	−0.481549	−0.240774	+	0.970581i	$0.577401\pi$
$774$	0	0
$775$	−29.7484	−1.06859
$776$	0	0
$777$	−2.16114	−0.0775305
$778$	0	0
$779$	3.58842	0.128569
$780$	0	0
$781$	−5.19373	−0.185846
$782$	0	0
$783$	14.0789	0.503138
$784$	0	0
$785$	−7.48594	−0.267185
$786$	0	0
$787$	−9.21648	−0.328532	−0.164266	−	0.986416i	$-0.552526\pi$
$787$	−9.21648	−0.328532	−0.164266	+	0.986416i	$0.552526\pi$
$788$	0	0
$789$	19.3052	0.687283
$790$	0	0
$791$	3.61776	0.128633
$792$	0	0
$793$	28.0566	0.996319
$794$	0	0
$795$	0.510292	0.0180982
$796$	0	0
$797$	27.5036	0.974229	0.487114	−	0.873338i	$-0.338050\pi$
$797$	27.5036	0.974229	0.487114	+	0.873338i	$0.338050\pi$
$798$	0	0
$799$	−35.6890	−1.26259
$800$	0	0
$801$	−19.5106	−0.689374
$802$	0	0
$803$	32.7283	1.15496
$804$	0	0
$805$	4.55472	0.160533
$806$	0	0
$807$	−14.3247	−0.504254
$808$	0	0
$809$	−15.0859	−0.530392	−0.265196	−	0.964194i	$-0.585437\pi$
$809$	−15.0859	−0.530392	−0.265196	+	0.964194i	$0.585437\pi$
$810$	0	0
$811$	−19.6225	−0.689038	−0.344519	−	0.938779i	$-0.611958\pi$
$811$	−19.6225	−0.689038	−0.344519	+	0.938779i	$0.611958\pi$
$812$	0	0
$813$	−22.2279	−0.779567
$814$	0	0
$815$	−15.4127	−0.539883
$816$	0	0
$817$	15.2881	0.534864
$818$	0	0
$819$	4.84920	0.169445
$820$	0	0
$821$	−21.8573	−0.762826	−0.381413	−	0.924405i	$-0.624562\pi$
$821$	−21.8573	−0.762826	−0.381413	+	0.924405i	$0.624562\pi$
$822$	0	0
$823$	45.2689	1.57797	0.788987	−	0.614410i	$-0.210606\pi$
$823$	45.2689	1.57797	0.788987	+	0.614410i	$0.210606\pi$
$824$	0	0
$825$	17.6696	0.615176
$826$	0	0
$827$	6.02831	0.209625	0.104812	−	0.994492i	$-0.466576\pi$
$827$	6.02831	0.209625	0.104812	+	0.994492i	$0.466576\pi$
$828$	0	0
$829$	−52.8608	−1.83593	−0.917965	−	0.396661i	$-0.870169\pi$
$829$	−52.8608	−1.83593	−0.917965	+	0.396661i	$0.870169\pi$
$830$	0	0
$831$	3.12040	0.108246
$832$	0	0
$833$	−18.1503	−0.628870
$834$	0	0
$835$	27.5671	0.953997
$836$	0	0
$837$	−13.0476	−0.450990
$838$	0	0
$839$	33.4463	1.15470	0.577348	−	0.816498i	$-0.304088\pi$
$839$	33.4463	1.15470	0.577348	+	0.816498i	$0.304088\pi$
$840$	0	0
$841$	−18.1198	−0.624822
$842$	0	0
$843$	−13.7234	−0.472657
$844$	0	0
$845$	131.732	4.53172
$846$	0	0
$847$	−1.72267	−0.0591917
$848$	0	0
$849$	−4.32824	−0.148545
$850$	0	0
$851$	36.3855	1.24728
$852$	0	0
$853$	55.1546	1.88846	0.944229	−	0.329289i	$-0.106809\pi$
$853$	55.1546	1.88846	0.944229	+	0.329289i	$0.106809\pi$
$854$	0	0
$855$	26.3407	0.900835
$856$	0	0
$857$	41.9868	1.43424	0.717120	−	0.696949i	$-0.245460\pi$
$857$	41.9868	1.43424	0.717120	+	0.696949i	$0.245460\pi$
$858$	0	0
$859$	31.1083	1.06140	0.530701	−	0.847559i	$-0.321929\pi$
$859$	31.1083	1.06140	0.530701	+	0.847559i	$0.321929\pi$
$860$	0	0
$861$	0.293092	0.00998854
$862$	0	0
$863$	−14.0355	−0.477775	−0.238887	−	0.971047i	$-0.576783\pi$
$863$	−14.0355	−0.477775	−0.238887	+	0.971047i	$0.576783\pi$
$864$	0	0
$865$	−72.7563	−2.47379
$866$	0	0
$867$	8.03390	0.272846
$868$	0	0
$869$	29.0644	0.985942
$870$	0	0
$871$	66.2630	2.24524
$872$	0	0
$873$	16.0790	0.544191
$874$	0	0
$875$	5.40689	0.182786
$876$	0	0
$877$	22.1350	0.747445	0.373722	−	0.927541i	$-0.378081\pi$
$877$	22.1350	0.747445	0.373722	+	0.927541i	$0.378081\pi$
$878$	0	0
$879$	9.71857	0.327799
$880$	0	0
$881$	0.396785	0.0133680	0.00668401	−	0.999978i	$-0.497872\pi$
$881$	0.396785	0.0133680	0.00668401	+	0.999978i	$0.497872\pi$
$882$	0	0
$883$	−3.00607	−0.101162	−0.0505811	−	0.998720i	$-0.516107\pi$
$883$	−3.00607	−0.101162	−0.0505811	+	0.998720i	$0.516107\pi$
$884$	0	0
$885$	−1.43506	−0.0482389
$886$	0	0
$887$	−7.43501	−0.249643	−0.124822	−	0.992179i	$-0.539836\pi$
$887$	−7.43501	−0.249643	−0.124822	+	0.992179i	$0.539836\pi$
$888$	0	0
$889$	−4.88132	−0.163714
$890$	0	0
$891$	−8.46931	−0.283732
$892$	0	0
$893$	39.3905	1.31815
$894$	0	0
$895$	60.3958	2.01881
$896$	0	0
$897$	21.8029	0.727976
$898$	0	0
$899$	−10.0832	−0.336293
$900$	0	0
$901$	−0.439088	−0.0146282
$902$	0	0
$903$	1.24869	0.0415538
$904$	0	0
$905$	−63.3283	−2.10510
$906$	0	0
$907$	−9.15606	−0.304022	−0.152011	−	0.988379i	$-0.548575\pi$
$907$	−9.15606	−0.304022	−0.152011	+	0.988379i	$0.548575\pi$
$908$	0	0
$909$	−25.2015	−0.835882
$910$	0	0
$911$	−38.2328	−1.26671	−0.633355	−	0.773861i	$-0.718323\pi$
$911$	−38.2328	−1.26671	−0.633355	+	0.773861i	$0.718323\pi$
$912$	0	0
$913$	−4.05798	−0.134299
$914$	0	0
$915$	12.4478	0.411512
$916$	0	0
$917$	−1.21930	−0.0402648
$918$	0	0
$919$	33.2511	1.09685	0.548427	−	0.836198i	$-0.315227\pi$
$919$	33.2511	1.09685	0.548427	+	0.836198i	$0.315227\pi$
$920$	0	0
$921$	−8.20275	−0.270290
$922$	0	0
$923$	−15.6469	−0.515024
$924$	0	0
$925$	88.8363	2.92092
$926$	0	0
$927$	−3.48451	−0.114446
$928$	0	0
$929$	−18.0213	−0.591260	−0.295630	−	0.955303i	$-0.595529\pi$
$929$	−18.0213	−0.591260	−0.295630	+	0.955303i	$0.595529\pi$
$930$	0	0
$931$	20.0327	0.656547
$932$	0	0
$933$	4.93377	0.161524
$934$	0	0
$935$	−23.0158	−0.752696
$936$	0	0
$937$	−52.3452	−1.71004	−0.855022	−	0.518592i	$-0.826456\pi$
$937$	−52.3452	−1.71004	−0.855022	+	0.518592i	$0.826456\pi$
$938$	0	0
$939$	23.9318	0.780984
$940$	0	0
$941$	24.1500	0.787266	0.393633	−	0.919268i	$-0.371218\pi$
$941$	24.1500	0.787266	0.393633	+	0.919268i	$0.371218\pi$
$942$	0	0
$943$	−4.93456	−0.160691
$944$	0	0
$945$	4.87742	0.158662
$946$	0	0
$947$	33.4604	1.08732	0.543658	−	0.839307i	$-0.317039\pi$
$947$	33.4604	1.08732	0.543658	+	0.839307i	$0.317039\pi$
$948$	0	0
$949$	98.5988	3.20065
$950$	0	0
$951$	−14.7935	−0.479713
$952$	0	0
$953$	26.5677	0.860613	0.430306	−	0.902683i	$-0.358406\pi$
$953$	26.5677	0.860613	0.430306	+	0.902683i	$0.358406\pi$
$954$	0	0
$955$	−87.2609	−2.82370
$956$	0	0
$957$	5.98908	0.193600
$958$	0	0
$959$	3.30993	0.106883
$960$	0	0
$961$	−21.6554	−0.698562
$962$	0	0
$963$	40.1969	1.29533
$964$	0	0
$965$	−78.0003	−2.51092
$966$	0	0
$967$	4.03416	0.129730	0.0648649	−	0.997894i	$-0.479338\pi$
$967$	4.03416	0.129730	0.0648649	+	0.997894i	$0.479338\pi$
$968$	0	0
$969$	6.05284	0.194445
$970$	0	0
$971$	45.0210	1.44479	0.722397	−	0.691479i	$-0.243040\pi$
$971$	45.0210	1.44479	0.722397	+	0.691479i	$0.243040\pi$
$972$	0	0
$973$	−0.509290	−0.0163271
$974$	0	0
$975$	53.2323	1.70480
$976$	0	0
$977$	−28.3535	−0.907109	−0.453554	−	0.891229i	$-0.649844\pi$
$977$	−28.3535	−0.907109	−0.453554	+	0.891229i	$0.649844\pi$
$978$	0	0
$979$	−18.8159	−0.601359
$980$	0	0
$981$	2.86714	0.0915409
$982$	0	0
$983$	19.0212	0.606683	0.303342	−	0.952882i	$-0.401898\pi$
$983$	19.0212	0.606683	0.303342	+	0.952882i	$0.401898\pi$
$984$	0	0
$985$	−16.8308	−0.536275
$986$	0	0
$987$	3.21730	0.102408
$988$	0	0
$989$	−21.0232	−0.668500
$990$	0	0
$991$	−29.0611	−0.923157	−0.461579	−	0.887099i	$-0.652717\pi$
$991$	−29.0611	−0.923157	−0.461579	+	0.887099i	$0.652717\pi$
$992$	0	0
$993$	−23.8398	−0.756534
$994$	0	0
$995$	7.04773	0.223428
$996$	0	0
$997$	57.9468	1.83520	0.917598	−	0.397511i	$-0.130126\pi$
$997$	57.9468	1.83520	0.917598	+	0.397511i	$0.130126\pi$
$998$	0	0
$999$	38.9634	1.23275

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.12	✓	29
4.3	odd	2		8048.2.a.w.1.18		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.12	✓	29	1.1	even	1	trivial
8048.2.a.w.1.18		29	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-0.795174 q^{3} +3.83818 q^{5} +0.297725 q^{7} -2.36770 q^{9} +O(q^{10})\)	\(q-0.795174 q^{3} +3.83818 q^{5} +0.297725 q^{7} -2.36770 q^{9} -2.28339 q^{11} -6.87906 q^{13} -3.05202 q^{15} +2.62615 q^{17} -2.89852 q^{19} -0.236743 q^{21} +3.98586 q^{23} +9.73160 q^{25} +4.26825 q^{27} +3.29851 q^{29} -3.05689 q^{31} +1.81569 q^{33} +1.14272 q^{35} +9.12864 q^{37} +5.47005 q^{39} -1.23802 q^{41} -5.27446 q^{43} -9.08764 q^{45} -13.5898 q^{47} -6.91136 q^{49} -2.08825 q^{51} -0.167198 q^{53} -8.76406 q^{55} +2.30483 q^{57} +0.470200 q^{59} -4.07855 q^{61} -0.704922 q^{63} -26.4031 q^{65} -9.63256 q^{67} -3.16945 q^{69} +2.27457 q^{71} -14.3332 q^{73} -7.73831 q^{75} -0.679822 q^{77} -12.7286 q^{79} +3.70909 q^{81} +1.77717 q^{83} +10.0796 q^{85} -2.62289 q^{87} +8.24033 q^{89} -2.04807 q^{91} +2.43076 q^{93} -11.1250 q^{95} -6.79098 q^{97} +5.40638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

Introduction

L-functions

Modular forms

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