LMFDB - Embedded newform 8048.2.a.v.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.72455 q^{3} +0.586565 q^{5} +2.46074 q^{7} -0.0259209 q^{9} +O(q^{10})$	$q+1.72455 q^{3} +0.586565 q^{5} +2.46074 q^{7} -0.0259209 q^{9} -2.73095 q^{11} +1.16422 q^{13} +1.01156 q^{15} -3.49409 q^{17} -5.19009 q^{19} +4.24367 q^{21} +1.89484 q^{23} -4.65594 q^{25} -5.21836 q^{27} -3.57220 q^{29} +1.26340 q^{31} -4.70966 q^{33} +1.44338 q^{35} +10.5540 q^{37} +2.00775 q^{39} -4.19222 q^{41} -8.57566 q^{43} -0.0152043 q^{45} -4.29383 q^{47} -0.944777 q^{49} -6.02575 q^{51} -6.91443 q^{53} -1.60188 q^{55} -8.95058 q^{57} +6.43780 q^{59} +5.28560 q^{61} -0.0637845 q^{63} +0.682888 q^{65} -7.31785 q^{67} +3.26775 q^{69} +10.9984 q^{71} -5.43906 q^{73} -8.02941 q^{75} -6.72014 q^{77} -14.0308 q^{79} -8.92157 q^{81} +3.03277 q^{83} -2.04951 q^{85} -6.16044 q^{87} +10.3363 q^{89} +2.86483 q^{91} +2.17879 q^{93} -3.04433 q^{95} -13.9082 q^{97} +0.0707886 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	1.72455	0.995670	0.497835	−	0.867272i	$-0.334128\pi$
$3$	1.72455	0.995670	0.497835	+	0.867272i	$0.334128\pi$
$4$	0	0
$5$	0.586565	0.262320	0.131160	−	0.991361i	$-0.458130\pi$
$5$	0.586565	0.262320	0.131160	+	0.991361i	$0.458130\pi$
$6$	0	0
$7$	2.46074	0.930071	0.465035	−	0.885292i	$-0.346042\pi$
$7$	2.46074	0.930071	0.465035	+	0.885292i	$0.346042\pi$
$8$	0	0
$9$	−0.0259209	−0.00864030
$10$	0	0
$11$	−2.73095	−0.823412	−0.411706	−	0.911317i	$-0.635067\pi$
$11$	−2.73095	−0.823412	−0.411706	+	0.911317i	$0.635067\pi$
$12$	0	0
$13$	1.16422	0.322895	0.161448	−	0.986881i	$-0.448384\pi$
$13$	1.16422	0.322895	0.161448	+	0.986881i	$0.448384\pi$
$14$	0	0
$15$	1.01156	0.261184
$16$	0	0
$17$	−3.49409	−0.847442	−0.423721	−	0.905793i	$-0.639276\pi$
$17$	−3.49409	−0.847442	−0.423721	+	0.905793i	$0.639276\pi$
$18$	0	0
$19$	−5.19009	−1.19069	−0.595344	−	0.803471i	$-0.702984\pi$
$19$	−5.19009	−1.19069	−0.595344	+	0.803471i	$0.702984\pi$
$20$	0	0
$21$	4.24367	0.926044
$22$	0	0
$23$	1.89484	0.395101	0.197550	−	0.980293i	$-0.436701\pi$
$23$	1.89484	0.395101	0.197550	+	0.980293i	$0.436701\pi$
$24$	0	0
$25$	−4.65594	−0.931188
$26$	0	0
$27$	−5.21836	−1.00427
$28$	0	0
$29$	−3.57220	−0.663340	−0.331670	−	0.943395i	$-0.607612\pi$
$29$	−3.57220	−0.663340	−0.331670	+	0.943395i	$0.607612\pi$
$30$	0	0
$31$	1.26340	0.226913	0.113456	−	0.993543i	$-0.463808\pi$
$31$	1.26340	0.226913	0.113456	+	0.993543i	$0.463808\pi$
$32$	0	0
$33$	−4.70966	−0.819847
$34$	0	0
$35$	1.44338	0.243976
$36$	0	0
$37$	10.5540	1.73507	0.867535	−	0.497377i	$-0.165703\pi$
$37$	10.5540	1.73507	0.867535	+	0.497377i	$0.165703\pi$
$38$	0	0
$39$	2.00775	0.321497
$40$	0	0
$41$	−4.19222	−0.654715	−0.327358	−	0.944901i	$-0.606158\pi$
$41$	−4.19222	−0.654715	−0.327358	+	0.944901i	$0.606158\pi$
$42$	0	0
$43$	−8.57566	−1.30778	−0.653888	−	0.756591i	$-0.726863\pi$
$43$	−8.57566	−1.30778	−0.653888	+	0.756591i	$0.726863\pi$
$44$	0	0
$45$	−0.0152043	−0.00226652
$46$	0	0
$47$	−4.29383	−0.626319	−0.313160	−	0.949701i	$-0.601387\pi$
$47$	−4.29383	−0.626319	−0.313160	+	0.949701i	$0.601387\pi$
$48$	0	0
$49$	−0.944777	−0.134968
$50$	0	0
$51$	−6.02575	−0.843773
$52$	0	0
$53$	−6.91443	−0.949771	−0.474885	−	0.880048i	$-0.657510\pi$
$53$	−6.91443	−0.949771	−0.474885	+	0.880048i	$0.657510\pi$
$54$	0	0
$55$	−1.60188	−0.215997
$56$	0	0
$57$	−8.95058	−1.18553
$58$	0	0
$59$	6.43780	0.838130	0.419065	−	0.907956i	$-0.362358\pi$
$59$	6.43780	0.838130	0.419065	+	0.907956i	$0.362358\pi$
$60$	0	0
$61$	5.28560	0.676752	0.338376	−	0.941011i	$-0.390123\pi$
$61$	5.28560	0.676752	0.338376	+	0.941011i	$0.390123\pi$
$62$	0	0
$63$	−0.0637845	−0.00803609
$64$	0	0
$65$	0.682888	0.0847018
$66$	0	0
$67$	−7.31785	−0.894017	−0.447009	−	0.894530i	$-0.647511\pi$
$67$	−7.31785	−0.894017	−0.447009	+	0.894530i	$0.647511\pi$
$68$	0	0
$69$	3.26775	0.393390
$70$	0	0
$71$	10.9984	1.30528	0.652638	−	0.757670i	$-0.273662\pi$
$71$	10.9984	1.30528	0.652638	+	0.757670i	$0.273662\pi$
$72$	0	0
$73$	−5.43906	−0.636593	−0.318297	−	0.947991i	$-0.603111\pi$
$73$	−5.43906	−0.636593	−0.318297	+	0.947991i	$0.603111\pi$
$74$	0	0
$75$	−8.02941	−0.927157
$76$	0	0
$77$	−6.72014	−0.765831
$78$	0	0
$79$	−14.0308	−1.57859	−0.789296	−	0.614013i	$-0.789554\pi$
$79$	−14.0308	−1.57859	−0.789296	+	0.614013i	$0.789554\pi$
$80$	0	0
$81$	−8.92157	−0.991285
$82$	0	0
$83$	3.03277	0.332890	0.166445	−	0.986051i	$-0.446771\pi$
$83$	3.03277	0.332890	0.166445	+	0.986051i	$0.446771\pi$
$84$	0	0
$85$	−2.04951	−0.222301
$86$	0	0
$87$	−6.16044	−0.660468
$88$	0	0
$89$	10.3363	1.09565	0.547823	−	0.836594i	$-0.315457\pi$
$89$	10.3363	1.09565	0.547823	+	0.836594i	$0.315457\pi$
$90$	0	0
$91$	2.86483	0.300316
$92$	0	0
$93$	2.17879	0.225930
$94$	0	0
$95$	−3.04433	−0.312341
$96$	0	0
$97$	−13.9082	−1.41216	−0.706082	−	0.708130i	$-0.749539\pi$
$97$	−13.9082	−1.41216	−0.706082	+	0.708130i	$0.749539\pi$
$98$	0	0
$99$	0.0707886	0.00711452
$100$	0	0
$101$	−5.02750	−0.500255	−0.250127	−	0.968213i	$-0.580473\pi$
$101$	−5.02750	−0.500255	−0.250127	+	0.968213i	$0.580473\pi$
$102$	0	0
$103$	0.956492	0.0942459	0.0471230	−	0.998889i	$-0.484995\pi$
$103$	0.956492	0.0942459	0.0471230	+	0.998889i	$0.484995\pi$
$104$	0	0
$105$	2.48919	0.242920
$106$	0	0
$107$	−11.9511	−1.15536	−0.577680	−	0.816263i	$-0.696042\pi$
$107$	−11.9511	−1.15536	−0.577680	+	0.816263i	$0.696042\pi$
$108$	0	0
$109$	−14.1910	−1.35925	−0.679625	−	0.733559i	$-0.737858\pi$
$109$	−14.1910	−1.35925	−0.679625	+	0.733559i	$0.737858\pi$
$110$	0	0
$111$	18.2009	1.72756
$112$	0	0
$113$	7.20506	0.677795	0.338898	−	0.940823i	$-0.389946\pi$
$113$	7.20506	0.677795	0.338898	+	0.940823i	$0.389946\pi$
$114$	0	0
$115$	1.11145	0.103643
$116$	0	0
$117$	−0.0301775	−0.00278991
$118$	0	0
$119$	−8.59804	−0.788181
$120$	0	0
$121$	−3.54193	−0.321993
$122$	0	0
$123$	−7.22970	−0.651880
$124$	0	0
$125$	−5.66384	−0.506589
$126$	0	0
$127$	18.2133	1.61617	0.808083	−	0.589068i	$-0.200505\pi$
$127$	18.2133	1.61617	0.808083	+	0.589068i	$0.200505\pi$
$128$	0	0
$129$	−14.7892	−1.30211
$130$	0	0
$131$	−4.72366	−0.412708	−0.206354	−	0.978477i	$-0.566160\pi$
$131$	−4.72366	−0.412708	−0.206354	+	0.978477i	$0.566160\pi$
$132$	0	0
$133$	−12.7714	−1.10742
$134$	0	0
$135$	−3.06091	−0.263441
$136$	0	0
$137$	1.70038	0.145274	0.0726368	−	0.997358i	$-0.476859\pi$
$137$	1.70038	0.145274	0.0726368	+	0.997358i	$0.476859\pi$
$138$	0	0
$139$	7.89107	0.669312	0.334656	−	0.942340i	$-0.391380\pi$
$139$	7.89107	0.669312	0.334656	+	0.942340i	$0.391380\pi$
$140$	0	0
$141$	−7.40493	−0.623607
$142$	0	0
$143$	−3.17941	−0.265876
$144$	0	0
$145$	−2.09533	−0.174007
$146$	0	0
$147$	−1.62932	−0.134384
$148$	0	0
$149$	−6.70406	−0.549218	−0.274609	−	0.961556i	$-0.588548\pi$
$149$	−6.70406	−0.549218	−0.274609	+	0.961556i	$0.588548\pi$
$150$	0	0
$151$	21.7904	1.77328	0.886639	−	0.462461i	$-0.153034\pi$
$151$	21.7904	1.77328	0.886639	+	0.462461i	$0.153034\pi$
$152$	0	0
$153$	0.0905700	0.00732215
$154$	0	0
$155$	0.741065	0.0595238
$156$	0	0
$157$	−7.49521	−0.598183	−0.299091	−	0.954224i	$-0.596684\pi$
$157$	−7.49521	−0.598183	−0.299091	+	0.954224i	$0.596684\pi$
$158$	0	0
$159$	−11.9243	−0.945659
$160$	0	0
$161$	4.66270	0.367472
$162$	0	0
$163$	−5.50736	−0.431370	−0.215685	−	0.976463i	$-0.569198\pi$
$163$	−5.50736	−0.431370	−0.215685	+	0.976463i	$0.569198\pi$
$164$	0	0
$165$	−2.76252	−0.215062
$166$	0	0
$167$	−4.04783	−0.313231	−0.156615	−	0.987660i	$-0.550058\pi$
$167$	−4.04783	−0.313231	−0.156615	+	0.987660i	$0.550058\pi$
$168$	0	0
$169$	−11.6446	−0.895739
$170$	0	0
$171$	0.134532	0.0102879
$172$	0	0
$173$	4.84897	0.368660	0.184330	−	0.982864i	$-0.440988\pi$
$173$	4.84897	0.368660	0.184330	+	0.982864i	$0.440988\pi$
$174$	0	0
$175$	−11.4570	−0.866071
$176$	0	0
$177$	11.1023	0.834501
$178$	0	0
$179$	15.0476	1.12471	0.562357	−	0.826895i	$-0.309895\pi$
$179$	15.0476	1.12471	0.562357	+	0.826895i	$0.309895\pi$
$180$	0	0
$181$	−4.49535	−0.334137	−0.167068	−	0.985945i	$-0.553430\pi$
$181$	−4.49535	−0.334137	−0.167068	+	0.985945i	$0.553430\pi$
$182$	0	0
$183$	9.11529	0.673822
$184$	0	0
$185$	6.19061	0.455143
$186$	0	0
$187$	9.54219	0.697794
$188$	0	0
$189$	−12.8410	−0.934045
$190$	0	0
$191$	5.54219	0.401019	0.200509	−	0.979692i	$-0.435740\pi$
$191$	5.54219	0.401019	0.200509	+	0.979692i	$0.435740\pi$
$192$	0	0
$193$	−5.32231	−0.383108	−0.191554	−	0.981482i	$-0.561353\pi$
$193$	−5.32231	−0.383108	−0.191554	+	0.981482i	$0.561353\pi$
$194$	0	0
$195$	1.17768	0.0843351
$196$	0	0
$197$	−1.75087	−0.124744	−0.0623722	−	0.998053i	$-0.519867\pi$
$197$	−1.75087	−0.124744	−0.0623722	+	0.998053i	$0.519867\pi$
$198$	0	0
$199$	−5.09739	−0.361344	−0.180672	−	0.983543i	$-0.557827\pi$
$199$	−5.09739	−0.361344	−0.180672	+	0.983543i	$0.557827\pi$
$200$	0	0
$201$	−12.6200	−0.890147
$202$	0	0
$203$	−8.79023	−0.616953
$204$	0	0
$205$	−2.45901	−0.171745
$206$	0	0
$207$	−0.0491159	−0.00341379
$208$	0	0
$209$	14.1739	0.980427
$210$	0	0
$211$	8.28571	0.570412	0.285206	−	0.958466i	$-0.407938\pi$
$211$	8.28571	0.570412	0.285206	+	0.958466i	$0.407938\pi$
$212$	0	0
$213$	18.9674	1.29962
$214$	0	0
$215$	−5.03018	−0.343056
$216$	0	0
$217$	3.10889	0.211045
$218$	0	0
$219$	−9.37993	−0.633837
$220$	0	0
$221$	−4.06788	−0.273635
$222$	0	0
$223$	−21.0134	−1.40716	−0.703582	−	0.710614i	$-0.748417\pi$
$223$	−21.0134	−1.40716	−0.703582	+	0.710614i	$0.748417\pi$
$224$	0	0
$225$	0.120686	0.00804575
$226$	0	0
$227$	22.5965	1.49978	0.749890	−	0.661563i	$-0.230107\pi$
$227$	22.5965	1.49978	0.749890	+	0.661563i	$0.230107\pi$
$228$	0	0
$229$	−7.06526	−0.466886	−0.233443	−	0.972371i	$-0.574999\pi$
$229$	−7.06526	−0.466886	−0.233443	+	0.972371i	$0.574999\pi$
$230$	0	0
$231$	−11.5892	−0.762515
$232$	0	0
$233$	−7.68241	−0.503291	−0.251646	−	0.967819i	$-0.580972\pi$
$233$	−7.68241	−0.503291	−0.251646	+	0.967819i	$0.580972\pi$
$234$	0	0
$235$	−2.51861	−0.164296
$236$	0	0
$237$	−24.1969	−1.57176
$238$	0	0
$239$	0.669531	0.0433084	0.0216542	−	0.999766i	$-0.493107\pi$
$239$	0.669531	0.0433084	0.0216542	+	0.999766i	$0.493107\pi$
$240$	0	0
$241$	−10.6397	−0.685361	−0.342681	−	0.939452i	$-0.611335\pi$
$241$	−10.6397	−0.685361	−0.342681	+	0.939452i	$0.611335\pi$
$242$	0	0
$243$	0.269370	0.0172801
$244$	0	0
$245$	−0.554173	−0.0354048
$246$	0	0
$247$	−6.04239	−0.384468
$248$	0	0
$249$	5.23017	0.331449
$250$	0	0
$251$	19.5023	1.23097	0.615486	−	0.788148i	$-0.288960\pi$
$251$	19.5023	1.23097	0.615486	+	0.788148i	$0.288960\pi$
$252$	0	0
$253$	−5.17470	−0.325331
$254$	0	0
$255$	−3.53449	−0.221338
$256$	0	0
$257$	−7.55107	−0.471023	−0.235511	−	0.971872i	$-0.575677\pi$
$257$	−7.55107	−0.471023	−0.235511	+	0.971872i	$0.575677\pi$
$258$	0	0
$259$	25.9706	1.61374
$260$	0	0
$261$	0.0925945	0.00573146
$262$	0	0
$263$	22.6613	1.39736	0.698679	−	0.715435i	$-0.253772\pi$
$263$	22.6613	1.39736	0.698679	+	0.715435i	$0.253772\pi$
$264$	0	0
$265$	−4.05576	−0.249144
$266$	0	0
$267$	17.8255	1.09090
$268$	0	0
$269$	6.83876	0.416966	0.208483	−	0.978026i	$-0.433147\pi$
$269$	6.83876	0.416966	0.208483	+	0.978026i	$0.433147\pi$
$270$	0	0
$271$	19.9514	1.21196	0.605980	−	0.795480i	$-0.292781\pi$
$271$	19.9514	1.21196	0.605980	+	0.795480i	$0.292781\pi$
$272$	0	0
$273$	4.94054	0.299015
$274$	0	0
$275$	12.7151	0.766751
$276$	0	0
$277$	17.8769	1.07412	0.537061	−	0.843544i	$-0.319535\pi$
$277$	17.8769	1.07412	0.537061	+	0.843544i	$0.319535\pi$
$278$	0	0
$279$	−0.0327484	−0.00196060
$280$	0	0
$281$	15.2940	0.912361	0.456180	−	0.889887i	$-0.349217\pi$
$281$	15.2940	0.912361	0.456180	+	0.889887i	$0.349217\pi$
$282$	0	0
$283$	−27.2471	−1.61967	−0.809837	−	0.586654i	$-0.800445\pi$
$283$	−27.2471	−1.61967	−0.809837	+	0.586654i	$0.800445\pi$
$284$	0	0
$285$	−5.25010	−0.310989
$286$	0	0
$287$	−10.3160	−0.608931
$288$	0	0
$289$	−4.79131	−0.281842
$290$	0	0
$291$	−23.9854	−1.40605
$292$	0	0
$293$	−14.2949	−0.835117	−0.417559	−	0.908650i	$-0.637114\pi$
$293$	−14.2949	−0.835117	−0.417559	+	0.908650i	$0.637114\pi$
$294$	0	0
$295$	3.77619	0.219858
$296$	0	0
$297$	14.2511	0.826930
$298$	0	0
$299$	2.20600	0.127576
$300$	0	0
$301$	−21.1024	−1.21632
$302$	0	0
$303$	−8.67018	−0.498089
$304$	0	0
$305$	3.10035	0.177525
$306$	0	0
$307$	−0.433862	−0.0247618	−0.0123809	−	0.999923i	$-0.503941\pi$
$307$	−0.433862	−0.0247618	−0.0123809	+	0.999923i	$0.503941\pi$
$308$	0	0
$309$	1.64952	0.0938379
$310$	0	0
$311$	5.50465	0.312140	0.156070	−	0.987746i	$-0.450117\pi$
$311$	5.50465	0.312140	0.156070	+	0.987746i	$0.450117\pi$
$312$	0	0
$313$	−18.7781	−1.06140	−0.530700	−	0.847560i	$-0.678071\pi$
$313$	−18.7781	−1.06140	−0.530700	+	0.847560i	$0.678071\pi$
$314$	0	0
$315$	−0.0374137	−0.00210803
$316$	0	0
$317$	−29.3977	−1.65114	−0.825571	−	0.564299i	$-0.809147\pi$
$317$	−29.3977	−1.65114	−0.825571	+	0.564299i	$0.809147\pi$
$318$	0	0
$319$	9.75548	0.546202
$320$	0	0
$321$	−20.6104	−1.15036
$322$	0	0
$323$	18.1347	1.00904
$324$	0	0
$325$	−5.42052	−0.300676
$326$	0	0
$327$	−24.4731	−1.35337
$328$	0	0
$329$	−10.5660	−0.582521
$330$	0	0
$331$	−22.6035	−1.24240	−0.621199	−	0.783653i	$-0.713354\pi$
$331$	−22.6035	−1.24240	−0.621199	+	0.783653i	$0.713354\pi$
$332$	0	0
$333$	−0.273569	−0.0149915
$334$	0	0
$335$	−4.29239	−0.234518
$336$	0	0
$337$	24.5776	1.33883	0.669413	−	0.742890i	$-0.266546\pi$
$337$	24.5776	1.33883	0.669413	+	0.742890i	$0.266546\pi$
$338$	0	0
$339$	12.4255	0.674861
$340$	0	0
$341$	−3.45027	−0.186843
$342$	0	0
$343$	−19.5500	−1.05560
$344$	0	0
$345$	1.91674	0.103194
$346$	0	0
$347$	9.24264	0.496171	0.248085	−	0.968738i	$-0.420199\pi$
$347$	9.24264	0.496171	0.248085	+	0.968738i	$0.420199\pi$
$348$	0	0
$349$	14.4660	0.774345	0.387172	−	0.922007i	$-0.373452\pi$
$349$	14.4660	0.774345	0.387172	+	0.922007i	$0.373452\pi$
$350$	0	0
$351$	−6.07529	−0.324275
$352$	0	0
$353$	30.4697	1.62174	0.810869	−	0.585227i	$-0.198995\pi$
$353$	30.4697	1.62174	0.810869	+	0.585227i	$0.198995\pi$
$354$	0	0
$355$	6.45130	0.342400
$356$	0	0
$357$	−14.8278	−0.784769
$358$	0	0
$359$	26.9934	1.42466	0.712329	−	0.701846i	$-0.247640\pi$
$359$	26.9934	1.42466	0.712329	+	0.701846i	$0.247640\pi$
$360$	0	0
$361$	7.93706	0.417740
$362$	0	0
$363$	−6.10824	−0.320599
$364$	0	0
$365$	−3.19036	−0.166991
$366$	0	0
$367$	16.3695	0.854481	0.427240	−	0.904138i	$-0.359486\pi$
$367$	16.3695	0.854481	0.427240	+	0.904138i	$0.359486\pi$
$368$	0	0
$369$	0.108666	0.00565693
$370$	0	0
$371$	−17.0146	−0.883354
$372$	0	0
$373$	5.17472	0.267937	0.133968	−	0.990986i	$-0.457228\pi$
$373$	5.17472	0.267937	0.133968	+	0.990986i	$0.457228\pi$
$374$	0	0
$375$	−9.76758	−0.504396
$376$	0	0
$377$	−4.15881	−0.214189
$378$	0	0
$379$	−2.38334	−0.122424	−0.0612120	−	0.998125i	$-0.519497\pi$
$379$	−2.38334	−0.122424	−0.0612120	+	0.998125i	$0.519497\pi$
$380$	0	0
$381$	31.4097	1.60917
$382$	0	0
$383$	−13.5010	−0.689868	−0.344934	−	0.938627i	$-0.612099\pi$
$383$	−13.5010	−0.689868	−0.344934	+	0.938627i	$0.612099\pi$
$384$	0	0
$385$	−3.94180	−0.200893
$386$	0	0
$387$	0.222289	0.0112996
$388$	0	0
$389$	3.97925	0.201756	0.100878	−	0.994899i	$-0.467835\pi$
$389$	3.97925	0.201756	0.100878	+	0.994899i	$0.467835\pi$
$390$	0	0
$391$	−6.62074	−0.334825
$392$	0	0
$393$	−8.14619	−0.410921
$394$	0	0
$395$	−8.22999	−0.414096
$396$	0	0
$397$	−38.2848	−1.92146	−0.960730	−	0.277485i	$-0.910499\pi$
$397$	−38.2848	−1.92146	−0.960730	+	0.277485i	$0.910499\pi$
$398$	0	0
$399$	−22.0250	−1.10263
$400$	0	0
$401$	−24.2381	−1.21039	−0.605196	−	0.796076i	$-0.706905\pi$
$401$	−24.2381	−1.21039	−0.605196	+	0.796076i	$0.706905\pi$
$402$	0	0
$403$	1.47087	0.0732691
$404$	0	0
$405$	−5.23308	−0.260034
$406$	0	0
$407$	−28.8225	−1.42868
$408$	0	0
$409$	−3.53393	−0.174742	−0.0873708	−	0.996176i	$-0.527847\pi$
$409$	−3.53393	−0.174742	−0.0873708	+	0.996176i	$0.527847\pi$
$410$	0	0
$411$	2.93240	0.144645
$412$	0	0
$413$	15.8417	0.779520
$414$	0	0
$415$	1.77892	0.0873236
$416$	0	0
$417$	13.6086	0.666414
$418$	0	0
$419$	−12.4382	−0.607648	−0.303824	−	0.952728i	$-0.598264\pi$
$419$	−12.4382	−0.607648	−0.303824	+	0.952728i	$0.598264\pi$
$420$	0	0
$421$	−6.69903	−0.326491	−0.163245	−	0.986585i	$-0.552196\pi$
$421$	−6.69903	−0.326491	−0.163245	+	0.986585i	$0.552196\pi$
$422$	0	0
$423$	0.111300	0.00541158
$424$	0	0
$425$	16.2683	0.789128
$426$	0	0
$427$	13.0065	0.629427
$428$	0	0
$429$	−5.48306	−0.264725
$430$	0	0
$431$	22.5674	1.08704	0.543518	−	0.839398i	$-0.317092\pi$
$431$	22.5674	1.08704	0.543518	+	0.839398i	$0.317092\pi$
$432$	0	0
$433$	4.92628	0.236742	0.118371	−	0.992969i	$-0.462233\pi$
$433$	4.92628	0.236742	0.118371	+	0.992969i	$0.462233\pi$
$434$	0	0
$435$	−3.61350	−0.173254
$436$	0	0
$437$	−9.83438	−0.470442
$438$	0	0
$439$	33.9326	1.61951	0.809757	−	0.586766i	$-0.199599\pi$
$439$	33.9326	1.61951	0.809757	+	0.586766i	$0.199599\pi$
$440$	0	0
$441$	0.0244895	0.00116617
$442$	0	0
$443$	−5.72745	−0.272119	−0.136060	−	0.990701i	$-0.543444\pi$
$443$	−5.72745	−0.272119	−0.136060	+	0.990701i	$0.543444\pi$
$444$	0	0
$445$	6.06291	0.287410
$446$	0	0
$447$	−11.5615	−0.546840
$448$	0	0
$449$	−18.3283	−0.864964	−0.432482	−	0.901643i	$-0.642362\pi$
$449$	−18.3283	−0.864964	−0.432482	+	0.901643i	$0.642362\pi$
$450$	0	0
$451$	11.4487	0.539100
$452$	0	0
$453$	37.5787	1.76560
$454$	0	0
$455$	1.68041	0.0787787
$456$	0	0
$457$	16.5496	0.774158	0.387079	−	0.922047i	$-0.373484\pi$
$457$	16.5496	0.774158	0.387079	+	0.922047i	$0.373484\pi$
$458$	0	0
$459$	18.2334	0.851064
$460$	0	0
$461$	−14.9454	−0.696078	−0.348039	−	0.937480i	$-0.613152\pi$
$461$	−14.9454	−0.696078	−0.348039	+	0.937480i	$0.613152\pi$
$462$	0	0
$463$	16.6853	0.775432	0.387716	−	0.921779i	$-0.373264\pi$
$463$	16.6853	0.775432	0.387716	+	0.921779i	$0.373264\pi$
$464$	0	0
$465$	1.27800	0.0592660
$466$	0	0
$467$	33.6476	1.55703	0.778513	−	0.627628i	$-0.215974\pi$
$467$	33.6476	1.55703	0.778513	+	0.627628i	$0.215974\pi$
$468$	0	0
$469$	−18.0073	−0.831499
$470$	0	0
$471$	−12.9259	−0.595593
$472$	0	0
$473$	23.4197	1.07684
$474$	0	0
$475$	24.1648	1.10876
$476$	0	0
$477$	0.179228	0.00820630
$478$	0	0
$479$	−20.5030	−0.936807	−0.468404	−	0.883515i	$-0.655171\pi$
$479$	−20.5030	−0.936807	−0.468404	+	0.883515i	$0.655171\pi$
$480$	0	0
$481$	12.2871	0.560246
$482$	0	0
$483$	8.04106	0.365881
$484$	0	0
$485$	−8.15806	−0.370439
$486$	0	0
$487$	−5.73639	−0.259941	−0.129970	−	0.991518i	$-0.541488\pi$
$487$	−5.73639	−0.259941	−0.129970	+	0.991518i	$0.541488\pi$
$488$	0	0
$489$	−9.49774	−0.429502
$490$	0	0
$491$	25.0417	1.13012	0.565059	−	0.825050i	$-0.308853\pi$
$491$	25.0417	1.13012	0.565059	+	0.825050i	$0.308853\pi$
$492$	0	0
$493$	12.4816	0.562143
$494$	0	0
$495$	0.0415221	0.00186628
$496$	0	0
$497$	27.0643	1.21400
$498$	0	0
$499$	−17.0922	−0.765153	−0.382576	−	0.923924i	$-0.624963\pi$
$499$	−17.0922	−0.765153	−0.382576	+	0.923924i	$0.624963\pi$
$500$	0	0
$501$	−6.98069	−0.311874
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−2.94896	−0.131227
$506$	0	0
$507$	−20.0817	−0.891860
$508$	0	0
$509$	−15.5830	−0.690705	−0.345352	−	0.938473i	$-0.612241\pi$
$509$	−15.5830	−0.690705	−0.345352	+	0.938473i	$0.612241\pi$
$510$	0	0
$511$	−13.3841	−0.592077
$512$	0	0
$513$	27.0838	1.19578
$514$	0	0
$515$	0.561045	0.0247226
$516$	0	0
$517$	11.7262	0.515718
$518$	0	0
$519$	8.36230	0.367064
$520$	0	0
$521$	−31.0136	−1.35873	−0.679364	−	0.733801i	$-0.737744\pi$
$521$	−31.0136	−1.35873	−0.679364	+	0.733801i	$0.737744\pi$
$522$	0	0
$523$	30.9567	1.35364	0.676821	−	0.736147i	$-0.263357\pi$
$523$	30.9567	1.35364	0.676821	+	0.736147i	$0.263357\pi$
$524$	0	0
$525$	−19.7583	−0.862321
$526$	0	0
$527$	−4.41443	−0.192296
$528$	0	0
$529$	−19.4096	−0.843895
$530$	0	0
$531$	−0.166873	−0.00724169
$532$	0	0
$533$	−4.88065	−0.211404
$534$	0	0
$535$	−7.01012	−0.303074
$536$	0	0
$537$	25.9504	1.11984
$538$	0	0
$539$	2.58014	0.111134
$540$	0	0
$541$	−0.609236	−0.0261931	−0.0130965	−	0.999914i	$-0.504169\pi$
$541$	−0.609236	−0.0261931	−0.0130965	+	0.999914i	$0.504169\pi$
$542$	0	0
$543$	−7.75246	−0.332690
$544$	0	0
$545$	−8.32394	−0.356558
$546$	0	0
$547$	−12.2626	−0.524311	−0.262156	−	0.965026i	$-0.584433\pi$
$547$	−12.2626	−0.524311	−0.262156	+	0.965026i	$0.584433\pi$
$548$	0	0
$549$	−0.137007	−0.00584734
$550$	0	0
$551$	18.5400	0.789832
$552$	0	0
$553$	−34.5262	−1.46820
$554$	0	0
$555$	10.6760	0.453172
$556$	0	0
$557$	15.0967	0.639669	0.319834	−	0.947473i	$-0.396373\pi$
$557$	15.0967	0.639669	0.319834	+	0.947473i	$0.396373\pi$
$558$	0	0
$559$	−9.98392	−0.422275
$560$	0	0
$561$	16.4560	0.694773
$562$	0	0
$563$	−5.81953	−0.245264	−0.122632	−	0.992452i	$-0.539133\pi$
$563$	−5.81953	−0.245264	−0.122632	+	0.992452i	$0.539133\pi$
$564$	0	0
$565$	4.22624	0.177799
$566$	0	0
$567$	−21.9536	−0.921965
$568$	0	0
$569$	0.581847	0.0243923	0.0121961	−	0.999926i	$-0.496118\pi$
$569$	0.581847	0.0243923	0.0121961	+	0.999926i	$0.496118\pi$
$570$	0	0
$571$	−35.9394	−1.50402	−0.752009	−	0.659153i	$-0.770915\pi$
$571$	−35.9394	−1.50402	−0.752009	+	0.659153i	$0.770915\pi$
$572$	0	0
$573$	9.55780	0.399283
$574$	0	0
$575$	−8.82225	−0.367913
$576$	0	0
$577$	21.5868	0.898669	0.449335	−	0.893364i	$-0.351661\pi$
$577$	21.5868	0.898669	0.449335	+	0.893364i	$0.351661\pi$
$578$	0	0
$579$	−9.17860	−0.381450
$580$	0	0
$581$	7.46285	0.309611
$582$	0	0
$583$	18.8830	0.782052
$584$	0	0
$585$	−0.0177011	−0.000731849 0
$586$	0	0
$587$	0.825657	0.0340785	0.0170393	−	0.999855i	$-0.494576\pi$
$587$	0.825657	0.0340785	0.0170393	+	0.999855i	$0.494576\pi$
$588$	0	0
$589$	−6.55715	−0.270183
$590$	0	0
$591$	−3.01947	−0.124204
$592$	0	0
$593$	0.459086	0.0188524	0.00942620	−	0.999956i	$-0.497000\pi$
$593$	0.459086	0.0188524	0.00942620	+	0.999956i	$0.497000\pi$
$594$	0	0
$595$	−5.04331	−0.206756
$596$	0	0
$597$	−8.79071	−0.359780
$598$	0	0
$599$	−27.5208	−1.12447	−0.562234	−	0.826978i	$-0.690058\pi$
$599$	−27.5208	−1.12447	−0.562234	+	0.826978i	$0.690058\pi$
$600$	0	0
$601$	−23.4136	−0.955062	−0.477531	−	0.878615i	$-0.658468\pi$
$601$	−23.4136	−0.955062	−0.477531	+	0.878615i	$0.658468\pi$
$602$	0	0
$603$	0.189685	0.00772458
$604$	0	0
$605$	−2.07757	−0.0844653
$606$	0	0
$607$	−14.4146	−0.585070	−0.292535	−	0.956255i	$-0.594499\pi$
$607$	−14.4146	−0.585070	−0.292535	+	0.956255i	$0.594499\pi$
$608$	0	0
$609$	−15.1592	−0.614282
$610$	0	0
$611$	−4.99894	−0.202236
$612$	0	0
$613$	−8.62924	−0.348532	−0.174266	−	0.984699i	$-0.555755\pi$
$613$	−8.62924	−0.348532	−0.174266	+	0.984699i	$0.555755\pi$
$614$	0	0
$615$	−4.24069	−0.171001
$616$	0	0
$617$	−28.5041	−1.14753	−0.573765	−	0.819020i	$-0.694518\pi$
$617$	−28.5041	−1.14753	−0.573765	+	0.819020i	$0.694518\pi$
$618$	0	0
$619$	20.8807	0.839266	0.419633	−	0.907694i	$-0.362159\pi$
$619$	20.8807	0.839266	0.419633	+	0.907694i	$0.362159\pi$
$620$	0	0
$621$	−9.88794	−0.396789
$622$	0	0
$623$	25.4349	1.01903
$624$	0	0
$625$	19.9575	0.798300
$626$	0	0
$627$	24.4436	0.976182
$628$	0	0
$629$	−36.8767	−1.47037
$630$	0	0
$631$	−10.2528	−0.408157	−0.204078	−	0.978955i	$-0.565420\pi$
$631$	−10.2528	−0.408157	−0.204078	+	0.978955i	$0.565420\pi$
$632$	0	0
$633$	14.2891	0.567942
$634$	0	0
$635$	10.6833	0.423953
$636$	0	0
$637$	−1.09992	−0.0435806
$638$	0	0
$639$	−0.285090	−0.0112780
$640$	0	0
$641$	−14.1573	−0.559181	−0.279591	−	0.960119i	$-0.590199\pi$
$641$	−14.1573	−0.559181	−0.279591	+	0.960119i	$0.590199\pi$
$642$	0	0
$643$	−39.3320	−1.55110	−0.775551	−	0.631285i	$-0.782528\pi$
$643$	−39.3320	−1.55110	−0.775551	+	0.631285i	$0.782528\pi$
$644$	0	0
$645$	−8.67481	−0.341570
$646$	0	0
$647$	−46.4027	−1.82428	−0.912139	−	0.409881i	$-0.865570\pi$
$647$	−46.4027	−1.82428	−0.912139	+	0.409881i	$0.865570\pi$
$648$	0	0
$649$	−17.5813	−0.690126
$650$	0	0
$651$	5.36144	0.210131
$652$	0	0
$653$	19.1450	0.749202	0.374601	−	0.927186i	$-0.377780\pi$
$653$	19.1450	0.749202	0.374601	+	0.927186i	$0.377780\pi$
$654$	0	0
$655$	−2.77073	−0.108261
$656$	0	0
$657$	0.140985	0.00550036
$658$	0	0
$659$	−2.37617	−0.0925623	−0.0462811	−	0.998928i	$-0.514737\pi$
$659$	−2.37617	−0.0925623	−0.0462811	+	0.998928i	$0.514737\pi$
$660$	0	0
$661$	23.2340	0.903696	0.451848	−	0.892095i	$-0.350765\pi$
$661$	23.2340	0.903696	0.451848	+	0.892095i	$0.350765\pi$
$662$	0	0
$663$	−7.01527	−0.272450
$664$	0	0
$665$	−7.49128	−0.290499
$666$	0	0
$667$	−6.76873	−0.262086
$668$	0	0
$669$	−36.2388	−1.40107
$670$	0	0
$671$	−14.4347	−0.557245
$672$	0	0
$673$	25.5526	0.984979	0.492489	−	0.870318i	$-0.336087\pi$
$673$	25.5526	0.984979	0.492489	+	0.870318i	$0.336087\pi$
$674$	0	0
$675$	24.2964	0.935168
$676$	0	0
$677$	24.0205	0.923182	0.461591	−	0.887093i	$-0.347279\pi$
$677$	24.0205	0.923182	0.461591	+	0.887093i	$0.347279\pi$
$678$	0	0
$679$	−34.2244	−1.31341
$680$	0	0
$681$	38.9688	1.49329
$682$	0	0
$683$	−21.8018	−0.834224	−0.417112	−	0.908855i	$-0.636958\pi$
$683$	−21.8018	−0.834224	−0.417112	+	0.908855i	$0.636958\pi$
$684$	0	0
$685$	0.997386	0.0381082
$686$	0	0
$687$	−12.1844	−0.464864
$688$	0	0
$689$	−8.04989	−0.306677
$690$	0	0
$691$	−11.0933	−0.422008	−0.211004	−	0.977485i	$-0.567673\pi$
$691$	−11.0933	−0.422008	−0.211004	+	0.977485i	$0.567673\pi$
$692$	0	0
$693$	0.174192	0.00661701
$694$	0	0
$695$	4.62863	0.175574
$696$	0	0
$697$	14.6480	0.554833
$698$	0	0
$699$	−13.2487	−0.501112
$700$	0	0
$701$	49.1729	1.85723	0.928617	−	0.371039i	$-0.120998\pi$
$701$	49.1729	1.85723	0.928617	+	0.371039i	$0.120998\pi$
$702$	0	0
$703$	−54.7763	−2.06593
$704$	0	0
$705$	−4.34347	−0.163585
$706$	0	0
$707$	−12.3714	−0.465273
$708$	0	0
$709$	−25.8276	−0.969975	−0.484987	−	0.874521i	$-0.661176\pi$
$709$	−25.8276	−0.969975	−0.484987	+	0.874521i	$0.661176\pi$
$710$	0	0
$711$	0.363692	0.0136395
$712$	0	0
$713$	2.39393	0.0896535
$714$	0	0
$715$	−1.86493	−0.0697445
$716$	0	0
$717$	1.15464	0.0431209
$718$	0	0
$719$	1.62825	0.0607236	0.0303618	−	0.999539i	$-0.490334\pi$
$719$	1.62825	0.0607236	0.0303618	+	0.999539i	$0.490334\pi$
$720$	0	0
$721$	2.35367	0.0876554
$722$	0	0
$723$	−18.3487	−0.682394
$724$	0	0
$725$	16.6319	0.617695
$726$	0	0
$727$	38.4256	1.42513	0.712564	−	0.701607i	$-0.247534\pi$
$727$	38.4256	1.42513	0.712564	+	0.701607i	$0.247534\pi$
$728$	0	0
$729$	27.2292	1.00849
$730$	0	0
$731$	29.9642	1.10826
$732$	0	0
$733$	14.5046	0.535739	0.267870	−	0.963455i	$-0.413680\pi$
$733$	14.5046	0.535739	0.267870	+	0.963455i	$0.413680\pi$
$734$	0	0
$735$	−0.955700	−0.0352515
$736$	0	0
$737$	19.9847	0.736144
$738$	0	0
$739$	−35.2529	−1.29680	−0.648399	−	0.761300i	$-0.724561\pi$
$739$	−35.2529	−1.29680	−0.648399	+	0.761300i	$0.724561\pi$
$740$	0	0
$741$	−10.4204	−0.382803
$742$	0	0
$743$	26.3936	0.968286	0.484143	−	0.874989i	$-0.339131\pi$
$743$	26.3936	0.968286	0.484143	+	0.874989i	$0.339131\pi$
$744$	0	0
$745$	−3.93236	−0.144071
$746$	0	0
$747$	−0.0786121	−0.00287627
$748$	0	0
$749$	−29.4086	−1.07457
$750$	0	0
$751$	−46.0714	−1.68117	−0.840585	−	0.541680i	$-0.817789\pi$
$751$	−46.0714	−1.68117	−0.840585	+	0.541680i	$0.817789\pi$
$752$	0	0
$753$	33.6327	1.22564
$754$	0	0
$755$	12.7815	0.465166
$756$	0	0
$757$	35.8348	1.30244	0.651219	−	0.758890i	$-0.274258\pi$
$757$	35.8348	1.30244	0.651219	+	0.758890i	$0.274258\pi$
$758$	0	0
$759$	−8.92404	−0.323922
$760$	0	0
$761$	24.1420	0.875148	0.437574	−	0.899182i	$-0.355838\pi$
$761$	24.1420	0.875148	0.437574	+	0.899182i	$0.355838\pi$
$762$	0	0
$763$	−34.9203	−1.26420
$764$	0	0
$765$	0.0531252	0.00192075
$766$	0	0
$767$	7.49499	0.270628
$768$	0	0
$769$	20.0154	0.721773	0.360887	−	0.932610i	$-0.382474\pi$
$769$	20.0154	0.721773	0.360887	+	0.932610i	$0.382474\pi$
$770$	0	0
$771$	−13.0222	−0.468984
$772$	0	0
$773$	49.8719	1.79377	0.896884	−	0.442265i	$-0.145825\pi$
$773$	49.8719	1.79377	0.896884	+	0.442265i	$0.145825\pi$
$774$	0	0
$775$	−5.88231	−0.211299
$776$	0	0
$777$	44.7877	1.60675
$778$	0	0
$779$	21.7580	0.779562
$780$	0	0
$781$	−30.0362	−1.07478
$782$	0	0
$783$	18.6410	0.666175
$784$	0	0
$785$	−4.39643	−0.156915
$786$	0	0
$787$	−14.7252	−0.524895	−0.262447	−	0.964946i	$-0.584530\pi$
$787$	−14.7252	−0.524895	−0.262447	+	0.964946i	$0.584530\pi$
$788$	0	0
$789$	39.0806	1.39131
$790$	0	0
$791$	17.7298	0.630398
$792$	0	0
$793$	6.15358	0.218520
$794$	0	0
$795$	−6.99438	−0.248065
$796$	0	0
$797$	−19.4782	−0.689955	−0.344977	−	0.938611i	$-0.612113\pi$
$797$	−19.4782	−0.689955	−0.344977	+	0.938611i	$0.612113\pi$
$798$	0	0
$799$	15.0030	0.530769
$800$	0	0
$801$	−0.267926	−0.00946671
$802$	0	0
$803$	14.8538	0.524178
$804$	0	0
$805$	2.73497	0.0963951
$806$	0	0
$807$	11.7938	0.415161
$808$	0	0
$809$	−53.2703	−1.87288	−0.936442	−	0.350823i	$-0.885902\pi$
$809$	−53.2703	−1.87288	−0.936442	+	0.350823i	$0.885902\pi$
$810$	0	0
$811$	−37.3268	−1.31072	−0.655360	−	0.755317i	$-0.727483\pi$
$811$	−37.3268	−1.31072	−0.655360	+	0.755317i	$0.727483\pi$
$812$	0	0
$813$	34.4072	1.20671
$814$	0	0
$815$	−3.23043	−0.113157
$816$	0	0
$817$	44.5085	1.55715
$818$	0	0
$819$	−0.0742589	−0.00259482
$820$	0	0
$821$	0.714503	0.0249363	0.0124682	−	0.999922i	$-0.496031\pi$
$821$	0.714503	0.0249363	0.0124682	+	0.999922i	$0.496031\pi$
$822$	0	0
$823$	−34.6220	−1.20685	−0.603424	−	0.797421i	$-0.706197\pi$
$823$	−34.6220	−1.20685	−0.603424	+	0.797421i	$0.706197\pi$
$824$	0	0
$825$	21.9279	0.763432
$826$	0	0
$827$	−26.9252	−0.936281	−0.468141	−	0.883654i	$-0.655076\pi$
$827$	−26.9252	−0.936281	−0.468141	+	0.883654i	$0.655076\pi$
$828$	0	0
$829$	38.5396	1.33854	0.669268	−	0.743021i	$-0.266608\pi$
$829$	38.5396	1.33854	0.669268	+	0.743021i	$0.266608\pi$
$830$	0	0
$831$	30.8297	1.06947
$832$	0	0
$833$	3.30114	0.114378
$834$	0	0
$835$	−2.37432	−0.0821666
$836$	0	0
$837$	−6.59286	−0.227883
$838$	0	0
$839$	20.3261	0.701735	0.350868	−	0.936425i	$-0.385887\pi$
$839$	20.3261	0.701735	0.350868	+	0.936425i	$0.385887\pi$
$840$	0	0
$841$	−16.2394	−0.559980
$842$	0	0
$843$	26.3752	0.908411
$844$	0	0
$845$	−6.83031	−0.234970
$846$	0	0
$847$	−8.71575	−0.299477
$848$	0	0
$849$	−46.9891	−1.61266
$850$	0	0
$851$	19.9981	0.685527
$852$	0	0
$853$	−18.6996	−0.640261	−0.320130	−	0.947373i	$-0.603727\pi$
$853$	−18.6996	−0.640261	−0.320130	+	0.947373i	$0.603727\pi$
$854$	0	0
$855$	0.0789117	0.00269872
$856$	0	0
$857$	58.1623	1.98679	0.993394	−	0.114754i	$-0.0366078\pi$
$857$	58.1623	1.98679	0.993394	+	0.114754i	$0.0366078\pi$
$858$	0	0
$859$	−36.7245	−1.25302	−0.626512	−	0.779412i	$-0.715518\pi$
$859$	−36.7245	−1.25302	−0.626512	+	0.779412i	$0.715518\pi$
$860$	0	0
$861$	−17.7904	−0.606295
$862$	0	0
$863$	−14.8425	−0.505245	−0.252623	−	0.967565i	$-0.581293\pi$
$863$	−14.8425	−0.505245	−0.252623	+	0.967565i	$0.581293\pi$
$864$	0	0
$865$	2.84424	0.0967069
$866$	0	0
$867$	−8.26286	−0.280621
$868$	0	0
$869$	38.3175	1.29983
$870$	0	0
$871$	−8.51955	−0.288674
$872$	0	0
$873$	0.360513	0.0122015
$874$	0	0
$875$	−13.9372	−0.471164
$876$	0	0
$877$	−48.5667	−1.63998	−0.819990	−	0.572378i	$-0.806021\pi$
$877$	−48.5667	−1.63998	−0.819990	+	0.572378i	$0.806021\pi$
$878$	0	0
$879$	−24.6523	−0.831502
$880$	0	0
$881$	27.4395	0.924459	0.462230	−	0.886760i	$-0.347050\pi$
$881$	27.4395	0.924459	0.462230	+	0.886760i	$0.347050\pi$
$882$	0	0
$883$	−0.411016	−0.0138318	−0.00691590	−	0.999976i	$-0.502201\pi$
$883$	−0.411016	−0.0138318	−0.00691590	+	0.999976i	$0.502201\pi$
$884$	0	0
$885$	6.51223	0.218906
$886$	0	0
$887$	22.5908	0.758524	0.379262	−	0.925289i	$-0.376178\pi$
$887$	22.5908	0.758524	0.379262	+	0.925289i	$0.376178\pi$
$888$	0	0
$889$	44.8181	1.50315
$890$	0	0
$891$	24.3643	0.816236
$892$	0	0
$893$	22.2854	0.745751
$894$	0	0
$895$	8.82642	0.295035
$896$	0	0
$897$	3.80436	0.127024
$898$	0	0
$899$	−4.51310	−0.150520
$900$	0	0
$901$	24.1597	0.804876
$902$	0	0
$903$	−36.3923	−1.21106
$904$	0	0
$905$	−2.63681	−0.0876506
$906$	0	0
$907$	58.2378	1.93376	0.966878	−	0.255240i	$-0.0821545\pi$
$907$	58.2378	1.93376	0.966878	+	0.255240i	$0.0821545\pi$
$908$	0	0
$909$	0.130317	0.00432235
$910$	0	0
$911$	26.7074	0.884855	0.442428	−	0.896804i	$-0.354117\pi$
$911$	26.7074	0.884855	0.442428	+	0.896804i	$0.354117\pi$
$912$	0	0
$913$	−8.28234	−0.274105
$914$	0	0
$915$	5.34671	0.176757
$916$	0	0
$917$	−11.6237	−0.383847
$918$	0	0
$919$	−4.37474	−0.144309	−0.0721546	−	0.997393i	$-0.522987\pi$
$919$	−4.37474	−0.144309	−0.0721546	+	0.997393i	$0.522987\pi$
$920$	0	0
$921$	−0.748218	−0.0246546
$922$	0	0
$923$	12.8046	0.421467
$924$	0	0
$925$	−49.1389	−1.61568
$926$	0	0
$927$	−0.0247931	−0.000814313 0
$928$	0	0
$929$	44.5806	1.46264	0.731321	−	0.682034i	$-0.238904\pi$
$929$	44.5806	1.46264	0.731321	+	0.682034i	$0.238904\pi$
$930$	0	0
$931$	4.90348	0.160705
$932$	0	0
$933$	9.49305	0.310789
$934$	0	0
$935$	5.59711	0.183045
$936$	0	0
$937$	−20.8335	−0.680601	−0.340301	−	0.940317i	$-0.610529\pi$
$937$	−20.8335	−0.680601	−0.340301	+	0.940317i	$0.610529\pi$
$938$	0	0
$939$	−32.3838	−1.05680
$940$	0	0
$941$	−24.6137	−0.802383	−0.401192	−	0.915994i	$-0.631404\pi$
$941$	−24.6137	−0.802383	−0.401192	+	0.915994i	$0.631404\pi$
$942$	0	0
$943$	−7.94358	−0.258679
$944$	0	0
$945$	−7.53208	−0.245019
$946$	0	0
$947$	−20.2229	−0.657157	−0.328579	−	0.944477i	$-0.606570\pi$
$947$	−20.2229	−0.657157	−0.328579	+	0.944477i	$0.606570\pi$
$948$	0	0
$949$	−6.33223	−0.205553
$950$	0	0
$951$	−50.6979	−1.64399
$952$	0	0
$953$	4.43796	0.143760	0.0718799	−	0.997413i	$-0.477100\pi$
$953$	4.43796	0.143760	0.0718799	+	0.997413i	$0.477100\pi$
$954$	0	0
$955$	3.25086	0.105195
$956$	0	0
$957$	16.8238	0.543837
$958$	0	0
$959$	4.18420	0.135115
$960$	0	0
$961$	−29.4038	−0.948511
$962$	0	0
$963$	0.309784	0.00998266
$964$	0	0
$965$	−3.12188	−0.100497
$966$	0	0
$967$	−33.5873	−1.08010	−0.540048	−	0.841634i	$-0.681594\pi$
$967$	−33.5873	−1.08010	−0.540048	+	0.841634i	$0.681594\pi$
$968$	0	0
$969$	31.2742	1.00467
$970$	0	0
$971$	59.7011	1.91590	0.957950	−	0.286934i	$-0.0926359\pi$
$971$	59.7011	1.91590	0.957950	+	0.286934i	$0.0926359\pi$
$972$	0	0
$973$	19.4179	0.622508
$974$	0	0
$975$	−9.34797	−0.299375
$976$	0	0
$977$	−45.5376	−1.45688	−0.728438	−	0.685112i	$-0.759753\pi$
$977$	−45.5376	−1.45688	−0.728438	+	0.685112i	$0.759753\pi$
$978$	0	0
$979$	−28.2279	−0.902167
$980$	0	0
$981$	0.367843	0.0117443
$982$	0	0
$983$	−28.5375	−0.910206	−0.455103	−	0.890439i	$-0.650398\pi$
$983$	−28.5375	−0.910206	−0.455103	+	0.890439i	$0.650398\pi$
$984$	0	0
$985$	−1.02700	−0.0327229
$986$	0	0
$987$	−18.2216	−0.579999
$988$	0	0
$989$	−16.2495	−0.516704
$990$	0	0
$991$	42.9458	1.36422	0.682109	−	0.731251i	$-0.261063\pi$
$991$	42.9458	1.36422	0.682109	+	0.731251i	$0.261063\pi$
$992$	0	0
$993$	−38.9808	−1.23702
$994$	0	0
$995$	−2.98995	−0.0947877
$996$	0	0
$997$	−20.5938	−0.652212	−0.326106	−	0.945333i	$-0.605737\pi$
$997$	−20.5938	−0.652212	−0.326106	+	0.945333i	$0.605737\pi$
$998$	0	0
$999$	−55.0746	−1.74248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.21		28
4.3	odd	2		4024.2.a.d.1.8	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.8	✓	28	4.3	odd	2
8048.2.a.v.1.21		28	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72455 q^{3} +0.586565 q^{5} +2.46074 q^{7} -0.0259209 q^{9} +O(q^{10})\)	\(q+1.72455 q^{3} +0.586565 q^{5} +2.46074 q^{7} -0.0259209 q^{9} -2.73095 q^{11} +1.16422 q^{13} +1.01156 q^{15} -3.49409 q^{17} -5.19009 q^{19} +4.24367 q^{21} +1.89484 q^{23} -4.65594 q^{25} -5.21836 q^{27} -3.57220 q^{29} +1.26340 q^{31} -4.70966 q^{33} +1.44338 q^{35} +10.5540 q^{37} +2.00775 q^{39} -4.19222 q^{41} -8.57566 q^{43} -0.0152043 q^{45} -4.29383 q^{47} -0.944777 q^{49} -6.02575 q^{51} -6.91443 q^{53} -1.60188 q^{55} -8.95058 q^{57} +6.43780 q^{59} +5.28560 q^{61} -0.0637845 q^{63} +0.682888 q^{65} -7.31785 q^{67} +3.26775 q^{69} +10.9984 q^{71} -5.43906 q^{73} -8.02941 q^{75} -6.72014 q^{77} -14.0308 q^{79} -8.92157 q^{81} +3.03277 q^{83} -2.04951 q^{85} -6.16044 q^{87} +10.3363 q^{89} +2.86483 q^{91} +2.17879 q^{93} -3.04433 q^{95} -13.9082 q^{97} +0.0707886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

L-functions

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Database

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