LMFDB - Embedded newform 8048.2.a.w.1.3

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:	$0$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.38925 q^{3} -0.218073 q^{5} +2.23687 q^{7} +2.70851 q^{9} +O(q^{10})$	$q-2.38925 q^{3} -0.218073 q^{5} +2.23687 q^{7} +2.70851 q^{9} +2.71275 q^{11} +0.542243 q^{13} +0.521030 q^{15} +6.46238 q^{17} +8.43479 q^{19} -5.34444 q^{21} -4.82169 q^{23} -4.95244 q^{25} +0.696442 q^{27} -1.34395 q^{29} +11.0036 q^{31} -6.48144 q^{33} -0.487801 q^{35} -2.51862 q^{37} -1.29555 q^{39} +5.97613 q^{41} +11.3916 q^{43} -0.590652 q^{45} +9.47139 q^{47} -1.99640 q^{49} -15.4402 q^{51} +13.8300 q^{53} -0.591577 q^{55} -20.1528 q^{57} +5.12310 q^{59} -1.00517 q^{61} +6.05859 q^{63} -0.118248 q^{65} -3.27116 q^{67} +11.5202 q^{69} -9.34078 q^{71} +2.30261 q^{73} +11.8326 q^{75} +6.06808 q^{77} -2.50293 q^{79} -9.78950 q^{81} -1.78482 q^{83} -1.40927 q^{85} +3.21102 q^{87} -8.76836 q^{89} +1.21293 q^{91} -26.2903 q^{93} -1.83940 q^{95} -8.59314 q^{97} +7.34751 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.38925	−1.37943	−0.689717	−	0.724079i	$-0.742265\pi$
$3$	−2.38925	−1.37943	−0.689717	+	0.724079i	$0.742265\pi$
$4$	0	0
$5$	−0.218073	−0.0975251	−0.0487625	−	0.998810i	$-0.515528\pi$
$5$	−0.218073	−0.0975251	−0.0487625	+	0.998810i	$0.515528\pi$
$6$	0	0
$7$	2.23687	0.845458	0.422729	−	0.906256i	$-0.361072\pi$
$7$	2.23687	0.845458	0.422729	+	0.906256i	$0.361072\pi$
$8$	0	0
$9$	2.70851	0.902837
$10$	0	0
$11$	2.71275	0.817925	0.408963	−	0.912551i	$-0.365891\pi$
$11$	2.71275	0.817925	0.408963	+	0.912551i	$0.365891\pi$
$12$	0	0
$13$	0.542243	0.150391	0.0751956	−	0.997169i	$-0.476042\pi$
$13$	0.542243	0.150391	0.0751956	+	0.997169i	$0.476042\pi$
$14$	0	0
$15$	0.521030	0.134529
$16$	0	0
$17$	6.46238	1.56736	0.783678	−	0.621167i	$-0.213341\pi$
$17$	6.46238	1.56736	0.783678	+	0.621167i	$0.213341\pi$
$18$	0	0
$19$	8.43479	1.93507	0.967537	−	0.252729i	$-0.0813282\pi$
$19$	8.43479	1.93507	0.967537	+	0.252729i	$0.0813282\pi$
$20$	0	0
$21$	−5.34444	−1.16625
$22$	0	0
$23$	−4.82169	−1.00539	−0.502696	−	0.864463i	$-0.667658\pi$
$23$	−4.82169	−1.00539	−0.502696	+	0.864463i	$0.667658\pi$
$24$	0	0
$25$	−4.95244	−0.990489
$26$	0	0
$27$	0.696442	0.134030
$28$	0	0
$29$	−1.34395	−0.249565	−0.124782	−	0.992184i	$-0.539823\pi$
$29$	−1.34395	−0.249565	−0.124782	+	0.992184i	$0.539823\pi$
$30$	0	0
$31$	11.0036	1.97630	0.988151	−	0.153484i	$-0.0490494\pi$
$31$	11.0036	1.97630	0.988151	+	0.153484i	$0.0490494\pi$
$32$	0	0
$33$	−6.48144	−1.12827
$34$	0	0
$35$	−0.487801	−0.0824534
$36$	0	0
$37$	−2.51862	−0.414059	−0.207030	−	0.978335i	$-0.566380\pi$
$37$	−2.51862	−0.414059	−0.207030	+	0.978335i	$0.566380\pi$
$38$	0	0
$39$	−1.29555	−0.207455
$40$	0	0
$41$	5.97613	0.933315	0.466658	−	0.884438i	$-0.345458\pi$
$41$	5.97613	0.933315	0.466658	+	0.884438i	$0.345458\pi$
$42$	0	0
$43$	11.3916	1.73721	0.868604	−	0.495507i	$-0.165018\pi$
$43$	11.3916	1.73721	0.868604	+	0.495507i	$0.165018\pi$
$44$	0	0
$45$	−0.590652	−0.0880492
$46$	0	0
$47$	9.47139	1.38154	0.690772	−	0.723072i	$-0.257271\pi$
$47$	9.47139	1.38154	0.690772	+	0.723072i	$0.257271\pi$
$48$	0	0
$49$	−1.99640	−0.285200
$50$	0	0
$51$	−15.4402	−2.16206
$52$	0	0
$53$	13.8300	1.89970	0.949851	−	0.312704i	$-0.101235\pi$
$53$	13.8300	1.89970	0.949851	+	0.312704i	$0.101235\pi$
$54$	0	0
$55$	−0.591577	−0.0797682
$56$	0	0
$57$	−20.1528	−2.66931
$58$	0	0
$59$	5.12310	0.666970	0.333485	−	0.942755i	$-0.391775\pi$
$59$	5.12310	0.666970	0.333485	+	0.942755i	$0.391775\pi$
$60$	0	0
$61$	−1.00517	−0.128699	−0.0643493	−	0.997927i	$-0.520497\pi$
$61$	−1.00517	−0.128699	−0.0643493	+	0.997927i	$0.520497\pi$
$62$	0	0
$63$	6.05859	0.763311
$64$	0	0
$65$	−0.118248	−0.0146669
$66$	0	0
$67$	−3.27116	−0.399636	−0.199818	−	0.979833i	$-0.564035\pi$
$67$	−3.27116	−0.399636	−0.199818	+	0.979833i	$0.564035\pi$
$68$	0	0
$69$	11.5202	1.38687
$70$	0	0
$71$	−9.34078	−1.10855	−0.554273	−	0.832335i	$-0.687004\pi$
$71$	−9.34078	−1.10855	−0.554273	+	0.832335i	$0.687004\pi$
$72$	0	0
$73$	2.30261	0.269501	0.134750	−	0.990880i	$-0.456977\pi$
$73$	2.30261	0.269501	0.134750	+	0.990880i	$0.456977\pi$
$74$	0	0
$75$	11.8326	1.36631
$76$	0	0
$77$	6.06808	0.691522
$78$	0	0
$79$	−2.50293	−0.281602	−0.140801	−	0.990038i	$-0.544968\pi$
$79$	−2.50293	−0.281602	−0.140801	+	0.990038i	$0.544968\pi$
$80$	0	0
$81$	−9.78950	−1.08772
$82$	0	0
$83$	−1.78482	−0.195909	−0.0979547	−	0.995191i	$-0.531230\pi$
$83$	−1.78482	−0.195909	−0.0979547	+	0.995191i	$0.531230\pi$
$84$	0	0
$85$	−1.40927	−0.152857
$86$	0	0
$87$	3.21102	0.344258
$88$	0	0
$89$	−8.76836	−0.929445	−0.464722	−	0.885456i	$-0.653846\pi$
$89$	−8.76836	−0.929445	−0.464722	+	0.885456i	$0.653846\pi$
$90$	0	0
$91$	1.21293	0.127150
$92$	0	0
$93$	−26.2903	−2.72618
$94$	0	0
$95$	−1.83940	−0.188718
$96$	0	0
$97$	−8.59314	−0.872501	−0.436250	−	0.899825i	$-0.643694\pi$
$97$	−8.59314	−0.872501	−0.436250	+	0.899825i	$0.643694\pi$
$98$	0	0
$99$	7.34751	0.738453
$100$	0	0
$101$	17.0369	1.69524	0.847620	−	0.530604i	$-0.178035\pi$
$101$	17.0369	1.69524	0.847620	+	0.530604i	$0.178035\pi$
$102$	0	0
$103$	7.44702	0.733776	0.366888	−	0.930265i	$-0.380423\pi$
$103$	7.44702	0.733776	0.366888	+	0.930265i	$0.380423\pi$
$104$	0	0
$105$	1.16548	0.113739
$106$	0	0
$107$	−8.11678	−0.784679	−0.392340	−	0.919820i	$-0.628334\pi$
$107$	−8.11678	−0.784679	−0.392340	+	0.919820i	$0.628334\pi$
$108$	0	0
$109$	−3.03323	−0.290531	−0.145265	−	0.989393i	$-0.546404\pi$
$109$	−3.03323	−0.290531	−0.145265	+	0.989393i	$0.546404\pi$
$110$	0	0
$111$	6.01762	0.571167
$112$	0	0
$113$	−0.350716	−0.0329926	−0.0164963	−	0.999864i	$-0.505251\pi$
$113$	−0.350716	−0.0329926	−0.0164963	+	0.999864i	$0.505251\pi$
$114$	0	0
$115$	1.05148	0.0980509
$116$	0	0
$117$	1.46867	0.135779
$118$	0	0
$119$	14.4555	1.32513
$120$	0	0
$121$	−3.64098	−0.330998
$122$	0	0
$123$	−14.2785	−1.28745
$124$	0	0
$125$	2.17036	0.194123
$126$	0	0
$127$	−3.52052	−0.312395	−0.156198	−	0.987726i	$-0.549924\pi$
$127$	−3.52052	−0.312395	−0.156198	+	0.987726i	$0.549924\pi$
$128$	0	0
$129$	−27.2174	−2.39636
$130$	0	0
$131$	8.79004	0.767989	0.383995	−	0.923335i	$-0.374548\pi$
$131$	8.79004	0.767989	0.383995	+	0.923335i	$0.374548\pi$
$132$	0	0
$133$	18.8676	1.63602
$134$	0	0
$135$	−0.151875	−0.0130713
$136$	0	0
$137$	−19.9410	−1.70367	−0.851837	−	0.523807i	$-0.824511\pi$
$137$	−19.9410	−1.70367	−0.851837	+	0.523807i	$0.824511\pi$
$138$	0	0
$139$	−9.86515	−0.836751	−0.418376	−	0.908274i	$-0.637400\pi$
$139$	−9.86515	−0.836751	−0.418376	+	0.908274i	$0.637400\pi$
$140$	0	0
$141$	−22.6295	−1.90575
$142$	0	0
$143$	1.47097	0.123009
$144$	0	0
$145$	0.293078	0.0243388
$146$	0	0
$147$	4.76990	0.393415
$148$	0	0
$149$	−12.2993	−1.00759	−0.503797	−	0.863822i	$-0.668064\pi$
$149$	−12.2993	−1.00759	−0.503797	+	0.863822i	$0.668064\pi$
$150$	0	0
$151$	4.93070	0.401255	0.200627	−	0.979668i	$-0.435702\pi$
$151$	4.93070	0.401255	0.200627	+	0.979668i	$0.435702\pi$
$152$	0	0
$153$	17.5034	1.41507
$154$	0	0
$155$	−2.39958	−0.192739
$156$	0	0
$157$	8.83537	0.705139	0.352570	−	0.935786i	$-0.385308\pi$
$157$	8.83537	0.705139	0.352570	+	0.935786i	$0.385308\pi$
$158$	0	0
$159$	−33.0434	−2.62051
$160$	0	0
$161$	−10.7855	−0.850016
$162$	0	0
$163$	9.02636	0.706999	0.353500	−	0.935435i	$-0.384992\pi$
$163$	9.02636	0.706999	0.353500	+	0.935435i	$0.384992\pi$
$164$	0	0
$165$	1.41342	0.110035
$166$	0	0
$167$	−8.27135	−0.640056	−0.320028	−	0.947408i	$-0.603692\pi$
$167$	−8.27135	−0.640056	−0.320028	+	0.947408i	$0.603692\pi$
$168$	0	0
$169$	−12.7060	−0.977382
$170$	0	0
$171$	22.8457	1.74706
$172$	0	0
$173$	−22.1954	−1.68748	−0.843742	−	0.536749i	$-0.819652\pi$
$173$	−22.1954	−1.68748	−0.843742	+	0.536749i	$0.819652\pi$
$174$	0	0
$175$	−11.0780	−0.837417
$176$	0	0
$177$	−12.2404	−0.920041
$178$	0	0
$179$	−9.52771	−0.712135	−0.356067	−	0.934460i	$-0.615883\pi$
$179$	−9.52771	−0.712135	−0.356067	+	0.934460i	$0.615883\pi$
$180$	0	0
$181$	18.0945	1.34496	0.672478	−	0.740117i	$-0.265230\pi$
$181$	18.0945	1.34496	0.672478	+	0.740117i	$0.265230\pi$
$182$	0	0
$183$	2.40160	0.177531
$184$	0	0
$185$	0.549243	0.0403812
$186$	0	0
$187$	17.5308	1.28198
$188$	0	0
$189$	1.55785	0.113317
$190$	0	0
$191$	0.381352	0.0275937	0.0137968	−	0.999905i	$-0.495608\pi$
$191$	0.381352	0.0275937	0.0137968	+	0.999905i	$0.495608\pi$
$192$	0	0
$193$	−8.78307	−0.632219	−0.316109	−	0.948723i	$-0.602377\pi$
$193$	−8.78307	−0.632219	−0.316109	+	0.948723i	$0.602377\pi$
$194$	0	0
$195$	0.282525	0.0202320
$196$	0	0
$197$	−5.52105	−0.393358	−0.196679	−	0.980468i	$-0.563016\pi$
$197$	−5.52105	−0.393358	−0.196679	+	0.980468i	$0.563016\pi$
$198$	0	0
$199$	16.0426	1.13723	0.568614	−	0.822604i	$-0.307480\pi$
$199$	16.0426	1.13723	0.568614	+	0.822604i	$0.307480\pi$
$200$	0	0
$201$	7.81562	0.551272
$202$	0	0
$203$	−3.00624	−0.210996
$204$	0	0
$205$	−1.30323	−0.0910216
$206$	0	0
$207$	−13.0596	−0.907704
$208$	0	0
$209$	22.8815	1.58275
$210$	0	0
$211$	−16.7837	−1.15544	−0.577720	−	0.816235i	$-0.696058\pi$
$211$	−16.7837	−1.15544	−0.577720	+	0.816235i	$0.696058\pi$
$212$	0	0
$213$	22.3174	1.52917
$214$	0	0
$215$	−2.48420	−0.169421
$216$	0	0
$217$	24.6136	1.67088
$218$	0	0
$219$	−5.50152	−0.371758
$220$	0	0
$221$	3.50418	0.235717
$222$	0	0
$223$	4.14244	0.277399	0.138699	−	0.990335i	$-0.455708\pi$
$223$	4.14244	0.277399	0.138699	+	0.990335i	$0.455708\pi$
$224$	0	0
$225$	−13.4137	−0.894250
$226$	0	0
$227$	17.6672	1.17261	0.586307	−	0.810089i	$-0.300581\pi$
$227$	17.6672	1.17261	0.586307	+	0.810089i	$0.300581\pi$
$228$	0	0
$229$	−5.50109	−0.363522	−0.181761	−	0.983343i	$-0.558180\pi$
$229$	−5.50109	−0.363522	−0.181761	+	0.983343i	$0.558180\pi$
$230$	0	0
$231$	−14.4981	−0.953908
$232$	0	0
$233$	−25.3781	−1.66257	−0.831287	−	0.555844i	$-0.812395\pi$
$233$	−25.3781	−1.66257	−0.831287	+	0.555844i	$0.812395\pi$
$234$	0	0
$235$	−2.06545	−0.134735
$236$	0	0
$237$	5.98012	0.388451
$238$	0	0
$239$	−9.66828	−0.625389	−0.312695	−	0.949854i	$-0.601232\pi$
$239$	−9.66828	−0.625389	−0.312695	+	0.949854i	$0.601232\pi$
$240$	0	0
$241$	15.6483	1.00800	0.503998	−	0.863705i	$-0.331862\pi$
$241$	15.6483	1.00800	0.503998	+	0.863705i	$0.331862\pi$
$242$	0	0
$243$	21.3002	1.36641
$244$	0	0
$245$	0.435361	0.0278142
$246$	0	0
$247$	4.57371	0.291018
$248$	0	0
$249$	4.26438	0.270244
$250$	0	0
$251$	−23.9099	−1.50918	−0.754590	−	0.656196i	$-0.772164\pi$
$251$	−23.9099	−1.50918	−0.754590	+	0.656196i	$0.772164\pi$
$252$	0	0
$253$	−13.0800	−0.822335
$254$	0	0
$255$	3.36709	0.210855
$256$	0	0
$257$	14.4587	0.901906	0.450953	−	0.892548i	$-0.351084\pi$
$257$	14.4587	0.901906	0.450953	+	0.892548i	$0.351084\pi$
$258$	0	0
$259$	−5.63384	−0.350070
$260$	0	0
$261$	−3.64009	−0.225316
$262$	0	0
$263$	29.1781	1.79920	0.899599	−	0.436717i	$-0.143859\pi$
$263$	29.1781	1.79920	0.899599	+	0.436717i	$0.143859\pi$
$264$	0	0
$265$	−3.01595	−0.185268
$266$	0	0
$267$	20.9498	1.28211
$268$	0	0
$269$	−14.3417	−0.874429	−0.437214	−	0.899357i	$-0.644035\pi$
$269$	−14.3417	−0.874429	−0.437214	+	0.899357i	$0.644035\pi$
$270$	0	0
$271$	−6.89716	−0.418973	−0.209486	−	0.977812i	$-0.567179\pi$
$271$	−6.89716	−0.418973	−0.209486	+	0.977812i	$0.567179\pi$
$272$	0	0
$273$	−2.89799	−0.175394
$274$	0	0
$275$	−13.4348	−0.810146
$276$	0	0
$277$	14.1653	0.851110	0.425555	−	0.904932i	$-0.360079\pi$
$277$	14.1653	0.851110	0.425555	+	0.904932i	$0.360079\pi$
$278$	0	0
$279$	29.8033	1.78428
$280$	0	0
$281$	−15.8379	−0.944813	−0.472406	−	0.881381i	$-0.656615\pi$
$281$	−15.8379	−0.944813	−0.472406	+	0.881381i	$0.656615\pi$
$282$	0	0
$283$	14.1890	0.843449	0.421725	−	0.906724i	$-0.361425\pi$
$283$	14.1890	0.843449	0.421725	+	0.906724i	$0.361425\pi$
$284$	0	0
$285$	4.39478	0.260324
$286$	0	0
$287$	13.3678	0.789079
$288$	0	0
$289$	24.7623	1.45661
$290$	0	0
$291$	20.5311	1.20356
$292$	0	0
$293$	14.2178	0.830613	0.415306	−	0.909682i	$-0.363674\pi$
$293$	14.2178	0.830613	0.415306	+	0.909682i	$0.363674\pi$
$294$	0	0
$295$	−1.11721	−0.0650463
$296$	0	0
$297$	1.88927	0.109627
$298$	0	0
$299$	−2.61453	−0.151202
$300$	0	0
$301$	25.4816	1.46874
$302$	0	0
$303$	−40.7055	−2.33847
$304$	0	0
$305$	0.219200	0.0125513
$306$	0	0
$307$	−13.7858	−0.786797	−0.393399	−	0.919368i	$-0.628701\pi$
$307$	−13.7858	−0.786797	−0.393399	+	0.919368i	$0.628701\pi$
$308$	0	0
$309$	−17.7928	−1.01220
$310$	0	0
$311$	−22.8307	−1.29461	−0.647306	−	0.762231i	$-0.724104\pi$
$311$	−22.8307	−1.29461	−0.647306	+	0.762231i	$0.724104\pi$
$312$	0	0
$313$	31.3637	1.77278	0.886391	−	0.462938i	$-0.153205\pi$
$313$	31.3637	1.77278	0.886391	+	0.462938i	$0.153205\pi$
$314$	0	0
$315$	−1.32121	−0.0744419
$316$	0	0
$317$	24.7552	1.39039	0.695195	−	0.718821i	$-0.255318\pi$
$317$	24.7552	1.39039	0.695195	+	0.718821i	$0.255318\pi$
$318$	0	0
$319$	−3.64579	−0.204125
$320$	0	0
$321$	19.3930	1.08241
$322$	0	0
$323$	54.5088	3.03295
$324$	0	0
$325$	−2.68543	−0.148961
$326$	0	0
$327$	7.24714	0.400768
$328$	0	0
$329$	21.1863	1.16804
$330$	0	0
$331$	4.17935	0.229718	0.114859	−	0.993382i	$-0.463358\pi$
$331$	4.17935	0.229718	0.114859	+	0.993382i	$0.463358\pi$
$332$	0	0
$333$	−6.82172	−0.373828
$334$	0	0
$335$	0.713351	0.0389746
$336$	0	0
$337$	6.64332	0.361885	0.180942	−	0.983494i	$-0.442085\pi$
$337$	6.64332	0.361885	0.180942	+	0.983494i	$0.442085\pi$
$338$	0	0
$339$	0.837948	0.0455111
$340$	0	0
$341$	29.8500	1.61647
$342$	0	0
$343$	−20.1238	−1.08658
$344$	0	0
$345$	−2.51224	−0.135255
$346$	0	0
$347$	15.1910	0.815498	0.407749	−	0.913094i	$-0.366314\pi$
$347$	15.1910	0.815498	0.407749	+	0.913094i	$0.366314\pi$
$348$	0	0
$349$	7.56904	0.405161	0.202581	−	0.979266i	$-0.435067\pi$
$349$	7.56904	0.405161	0.202581	+	0.979266i	$0.435067\pi$
$350$	0	0
$351$	0.377641	0.0201570
$352$	0	0
$353$	−33.1275	−1.76320	−0.881600	−	0.471998i	$-0.843533\pi$
$353$	−33.1275	−1.76320	−0.881600	+	0.471998i	$0.843533\pi$
$354$	0	0
$355$	2.03697	0.108111
$356$	0	0
$357$	−34.5378	−1.82794
$358$	0	0
$359$	−5.28154	−0.278749	−0.139375	−	0.990240i	$-0.544509\pi$
$359$	−5.28154	−0.278749	−0.139375	+	0.990240i	$0.544509\pi$
$360$	0	0
$361$	52.1457	2.74451
$362$	0	0
$363$	8.69920	0.456590
$364$	0	0
$365$	−0.502137	−0.0262831
$366$	0	0
$367$	0.613028	0.0319998	0.0159999	−	0.999872i	$-0.494907\pi$
$367$	0.613028	0.0319998	0.0159999	+	0.999872i	$0.494907\pi$
$368$	0	0
$369$	16.1864	0.842631
$370$	0	0
$371$	30.9360	1.60612
$372$	0	0
$373$	−4.48716	−0.232336	−0.116168	−	0.993230i	$-0.537061\pi$
$373$	−4.48716	−0.232336	−0.116168	+	0.993230i	$0.537061\pi$
$374$	0	0
$375$	−5.18552	−0.267779
$376$	0	0
$377$	−0.728746	−0.0375323
$378$	0	0
$379$	1.72605	0.0886615	0.0443307	−	0.999017i	$-0.485884\pi$
$379$	1.72605	0.0886615	0.0443307	+	0.999017i	$0.485884\pi$
$380$	0	0
$381$	8.41139	0.430929
$382$	0	0
$383$	1.11936	0.0571964	0.0285982	−	0.999591i	$-0.490896\pi$
$383$	1.11936	0.0571964	0.0285982	+	0.999591i	$0.490896\pi$
$384$	0	0
$385$	−1.32328	−0.0674407
$386$	0	0
$387$	30.8543	1.56841
$388$	0	0
$389$	−12.3303	−0.625170	−0.312585	−	0.949890i	$-0.601195\pi$
$389$	−12.3303	−0.625170	−0.312585	+	0.949890i	$0.601195\pi$
$390$	0	0
$391$	−31.1596	−1.57581
$392$	0	0
$393$	−21.0016	−1.05939
$394$	0	0
$395$	0.545821	0.0274632
$396$	0	0
$397$	14.1085	0.708087	0.354044	−	0.935229i	$-0.384806\pi$
$397$	14.1085	0.708087	0.354044	+	0.935229i	$0.384806\pi$
$398$	0	0
$399$	−45.0793	−2.25679
$400$	0	0
$401$	9.35145	0.466989	0.233495	−	0.972358i	$-0.424984\pi$
$401$	9.35145	0.466989	0.233495	+	0.972358i	$0.424984\pi$
$402$	0	0
$403$	5.96662	0.297219
$404$	0	0
$405$	2.13482	0.106080
$406$	0	0
$407$	−6.83240	−0.338670
$408$	0	0
$409$	−22.8424	−1.12949	−0.564743	−	0.825267i	$-0.691025\pi$
$409$	−22.8424	−1.12949	−0.564743	+	0.825267i	$0.691025\pi$
$410$	0	0
$411$	47.6440	2.35010
$412$	0	0
$413$	11.4597	0.563896
$414$	0	0
$415$	0.389220	0.0191061
$416$	0	0
$417$	23.5703	1.15424
$418$	0	0
$419$	−2.09518	−0.102356	−0.0511781	−	0.998690i	$-0.516298\pi$
$419$	−2.09518	−0.102356	−0.0511781	+	0.998690i	$0.516298\pi$
$420$	0	0
$421$	29.8188	1.45328	0.726639	−	0.687020i	$-0.241081\pi$
$421$	29.8188	1.45328	0.726639	+	0.687020i	$0.241081\pi$
$422$	0	0
$423$	25.6534	1.24731
$424$	0	0
$425$	−32.0046	−1.55245
$426$	0	0
$427$	−2.24843	−0.108809
$428$	0	0
$429$	−3.51452	−0.169682
$430$	0	0
$431$	−17.9009	−0.862254	−0.431127	−	0.902291i	$-0.641884\pi$
$431$	−17.9009	−0.862254	−0.431127	+	0.902291i	$0.641884\pi$
$432$	0	0
$433$	−31.6146	−1.51930	−0.759650	−	0.650332i	$-0.774630\pi$
$433$	−31.6146	−1.51930	−0.759650	+	0.650332i	$0.774630\pi$
$434$	0	0
$435$	−0.700236	−0.0335738
$436$	0	0
$437$	−40.6699	−1.94551
$438$	0	0
$439$	−18.9158	−0.902801	−0.451400	−	0.892322i	$-0.649075\pi$
$439$	−18.9158	−0.902801	−0.451400	+	0.892322i	$0.649075\pi$
$440$	0	0
$441$	−5.40728	−0.257489
$442$	0	0
$443$	−7.68129	−0.364949	−0.182475	−	0.983211i	$-0.558411\pi$
$443$	−7.68129	−0.364949	−0.182475	+	0.983211i	$0.558411\pi$
$444$	0	0
$445$	1.91214	0.0906441
$446$	0	0
$447$	29.3860	1.38991
$448$	0	0
$449$	22.3537	1.05494	0.527468	−	0.849575i	$-0.323142\pi$
$449$	22.3537	1.05494	0.527468	+	0.849575i	$0.323142\pi$
$450$	0	0
$451$	16.2118	0.763382
$452$	0	0
$453$	−11.7807	−0.553504
$454$	0	0
$455$	−0.264507	−0.0124003
$456$	0	0
$457$	−11.8888	−0.556134	−0.278067	−	0.960562i	$-0.589694\pi$
$457$	−11.8888	−0.556134	−0.278067	+	0.960562i	$0.589694\pi$
$458$	0	0
$459$	4.50067	0.210073
$460$	0	0
$461$	15.1403	0.705155	0.352577	−	0.935783i	$-0.385305\pi$
$461$	15.1403	0.705155	0.352577	+	0.935783i	$0.385305\pi$
$462$	0	0
$463$	−25.1709	−1.16979	−0.584895	−	0.811109i	$-0.698864\pi$
$463$	−25.1709	−1.16979	−0.584895	+	0.811109i	$0.698864\pi$
$464$	0	0
$465$	5.73320	0.265871
$466$	0	0
$467$	−21.8987	−1.01335	−0.506676	−	0.862136i	$-0.669126\pi$
$467$	−21.8987	−1.01335	−0.506676	+	0.862136i	$0.669126\pi$
$468$	0	0
$469$	−7.31717	−0.337876
$470$	0	0
$471$	−21.1099	−0.972693
$472$	0	0
$473$	30.9027	1.42091
$474$	0	0
$475$	−41.7728	−1.91667
$476$	0	0
$477$	37.4588	1.71512
$478$	0	0
$479$	−14.1000	−0.644243	−0.322122	−	0.946698i	$-0.604396\pi$
$479$	−14.1000	−0.644243	−0.322122	+	0.946698i	$0.604396\pi$
$480$	0	0
$481$	−1.36571	−0.0622709
$482$	0	0
$483$	25.7692	1.17254
$484$	0	0
$485$	1.87393	0.0850907
$486$	0	0
$487$	−7.98974	−0.362050	−0.181025	−	0.983479i	$-0.557941\pi$
$487$	−7.98974	−0.362050	−0.181025	+	0.983479i	$0.557941\pi$
$488$	0	0
$489$	−21.5662	−0.975259
$490$	0	0
$491$	6.67695	0.301327	0.150663	−	0.988585i	$-0.451859\pi$
$491$	6.67695	0.301327	0.150663	+	0.988585i	$0.451859\pi$
$492$	0	0
$493$	−8.68509	−0.391157
$494$	0	0
$495$	−1.60229	−0.0720177
$496$	0	0
$497$	−20.8941	−0.937230
$498$	0	0
$499$	−11.5495	−0.517028	−0.258514	−	0.966008i	$-0.583233\pi$
$499$	−11.5495	−0.517028	−0.258514	+	0.966008i	$0.583233\pi$
$500$	0	0
$501$	19.7623	0.882915
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−3.71529	−0.165328
$506$	0	0
$507$	30.3577	1.34823
$508$	0	0
$509$	−5.48482	−0.243110	−0.121555	−	0.992585i	$-0.538788\pi$
$509$	−5.48482	−0.243110	−0.121555	+	0.992585i	$0.538788\pi$
$510$	0	0
$511$	5.15065	0.227851
$512$	0	0
$513$	5.87435	0.259359
$514$	0	0
$515$	−1.62399	−0.0715616
$516$	0	0
$517$	25.6935	1.13000
$518$	0	0
$519$	53.0303	2.32777
$520$	0	0
$521$	35.2848	1.54585	0.772927	−	0.634495i	$-0.218792\pi$
$521$	35.2848	1.54585	0.772927	+	0.634495i	$0.218792\pi$
$522$	0	0
$523$	23.6951	1.03611	0.518056	−	0.855346i	$-0.326656\pi$
$523$	23.6951	1.03611	0.518056	+	0.855346i	$0.326656\pi$
$524$	0	0
$525$	26.4681	1.15516
$526$	0	0
$527$	71.1093	3.09757
$528$	0	0
$529$	0.248673	0.0108119
$530$	0	0
$531$	13.8760	0.602165
$532$	0	0
$533$	3.24052	0.140362
$534$	0	0
$535$	1.77005	0.0765259
$536$	0	0
$537$	22.7641	0.982343
$538$	0	0
$539$	−5.41575	−0.233273
$540$	0	0
$541$	−10.3146	−0.443460	−0.221730	−	0.975108i	$-0.571170\pi$
$541$	−10.3146	−0.443460	−0.221730	+	0.975108i	$0.571170\pi$
$542$	0	0
$543$	−43.2323	−1.85528
$544$	0	0
$545$	0.661464	0.0283340
$546$	0	0
$547$	42.4615	1.81552	0.907761	−	0.419487i	$-0.137790\pi$
$547$	42.4615	1.81552	0.907761	+	0.419487i	$0.137790\pi$
$548$	0	0
$549$	−2.72251	−0.116194
$550$	0	0
$551$	−11.3359	−0.482926
$552$	0	0
$553$	−5.59874	−0.238082
$554$	0	0
$555$	−1.31228	−0.0557031
$556$	0	0
$557$	38.8558	1.64637	0.823186	−	0.567772i	$-0.192195\pi$
$557$	38.8558	1.64637	0.823186	+	0.567772i	$0.192195\pi$
$558$	0	0
$559$	6.17704	0.261261
$560$	0	0
$561$	−41.8855	−1.76841
$562$	0	0
$563$	−33.5964	−1.41592	−0.707960	−	0.706252i	$-0.750384\pi$
$563$	−33.5964	−1.41592	−0.707960	+	0.706252i	$0.750384\pi$
$564$	0	0
$565$	0.0764816	0.00321761
$566$	0	0
$567$	−21.8979	−0.919624
$568$	0	0
$569$	−6.76740	−0.283704	−0.141852	−	0.989888i	$-0.545306\pi$
$569$	−6.76740	−0.283704	−0.141852	+	0.989888i	$0.545306\pi$
$570$	0	0
$571$	45.5563	1.90647	0.953236	−	0.302227i	$-0.0977300\pi$
$571$	45.5563	1.90647	0.953236	+	0.302227i	$0.0977300\pi$
$572$	0	0
$573$	−0.911145	−0.0380636
$574$	0	0
$575$	23.8791	0.995829
$576$	0	0
$577$	1.05510	0.0439242	0.0219621	−	0.999759i	$-0.493009\pi$
$577$	1.05510	0.0439242	0.0219621	+	0.999759i	$0.493009\pi$
$578$	0	0
$579$	20.9849	0.872104
$580$	0	0
$581$	−3.99241	−0.165633
$582$	0	0
$583$	37.5174	1.55381
$584$	0	0
$585$	−0.320277	−0.0132418
$586$	0	0
$587$	−15.0523	−0.621276	−0.310638	−	0.950528i	$-0.600543\pi$
$587$	−15.0523	−0.621276	−0.310638	+	0.950528i	$0.600543\pi$
$588$	0	0
$589$	92.8130	3.82429
$590$	0	0
$591$	13.1912	0.542612
$592$	0	0
$593$	30.9568	1.27124	0.635622	−	0.772000i	$-0.280744\pi$
$593$	30.9568	1.27124	0.635622	+	0.772000i	$0.280744\pi$
$594$	0	0
$595$	−3.15235	−0.129234
$596$	0	0
$597$	−38.3297	−1.56873
$598$	0	0
$599$	26.8859	1.09853	0.549264	−	0.835649i	$-0.314908\pi$
$599$	26.8859	1.09853	0.549264	+	0.835649i	$0.314908\pi$
$600$	0	0
$601$	−16.4725	−0.671928	−0.335964	−	0.941875i	$-0.609062\pi$
$601$	−16.4725	−0.671928	−0.335964	+	0.941875i	$0.609062\pi$
$602$	0	0
$603$	−8.85998	−0.360806
$604$	0	0
$605$	0.793998	0.0322806
$606$	0	0
$607$	44.7949	1.81817	0.909085	−	0.416610i	$-0.136782\pi$
$607$	44.7949	1.81817	0.909085	+	0.416610i	$0.136782\pi$
$608$	0	0
$609$	7.18265	0.291056
$610$	0	0
$611$	5.13580	0.207772
$612$	0	0
$613$	−49.2671	−1.98988	−0.994939	−	0.100482i	$-0.967962\pi$
$613$	−49.2671	−1.98988	−0.994939	+	0.100482i	$0.967962\pi$
$614$	0	0
$615$	3.11374	0.125558
$616$	0	0
$617$	−2.19447	−0.0883461	−0.0441730	−	0.999024i	$-0.514065\pi$
$617$	−2.19447	−0.0883461	−0.0441730	+	0.999024i	$0.514065\pi$
$618$	0	0
$619$	−30.7868	−1.23743	−0.618714	−	0.785617i	$-0.712346\pi$
$619$	−30.7868	−1.23743	−0.618714	+	0.785617i	$0.712346\pi$
$620$	0	0
$621$	−3.35803	−0.134753
$622$	0	0
$623$	−19.6137	−0.785807
$624$	0	0
$625$	24.2889	0.971557
$626$	0	0
$627$	−54.6696	−2.18329
$628$	0	0
$629$	−16.2763	−0.648979
$630$	0	0
$631$	−1.59503	−0.0634971	−0.0317486	−	0.999496i	$-0.510108\pi$
$631$	−1.59503	−0.0634971	−0.0317486	+	0.999496i	$0.510108\pi$
$632$	0	0
$633$	40.1005	1.59385
$634$	0	0
$635$	0.767728	0.0304664
$636$	0	0
$637$	−1.08254	−0.0428917
$638$	0	0
$639$	−25.2996	−1.00084
$640$	0	0
$641$	−22.5195	−0.889467	−0.444734	−	0.895663i	$-0.646702\pi$
$641$	−22.5195	−0.889467	−0.444734	+	0.895663i	$0.646702\pi$
$642$	0	0
$643$	25.4511	1.00369	0.501847	−	0.864956i	$-0.332654\pi$
$643$	25.4511	1.00369	0.501847	+	0.864956i	$0.332654\pi$
$644$	0	0
$645$	5.93538	0.233705
$646$	0	0
$647$	21.2329	0.834751	0.417375	−	0.908734i	$-0.362950\pi$
$647$	21.2329	0.834751	0.417375	+	0.908734i	$0.362950\pi$
$648$	0	0
$649$	13.8977	0.545532
$650$	0	0
$651$	−58.8080	−2.30487
$652$	0	0
$653$	38.7630	1.51692	0.758458	−	0.651722i	$-0.225953\pi$
$653$	38.7630	1.51692	0.758458	+	0.651722i	$0.225953\pi$
$654$	0	0
$655$	−1.91687	−0.0748982
$656$	0	0
$657$	6.23665	0.243315
$658$	0	0
$659$	20.2316	0.788110	0.394055	−	0.919087i	$-0.371072\pi$
$659$	20.2316	0.788110	0.394055	+	0.919087i	$0.371072\pi$
$660$	0	0
$661$	5.60884	0.218159	0.109079	−	0.994033i	$-0.465210\pi$
$661$	5.60884	0.218159	0.109079	+	0.994033i	$0.465210\pi$
$662$	0	0
$663$	−8.37236	−0.325156
$664$	0	0
$665$	−4.11450	−0.159553
$666$	0	0
$667$	6.48009	0.250910
$668$	0	0
$669$	−9.89733	−0.382653
$670$	0	0
$671$	−2.72677	−0.105266
$672$	0	0
$673$	−14.9711	−0.577094	−0.288547	−	0.957466i	$-0.593172\pi$
$673$	−14.9711	−0.577094	−0.288547	+	0.957466i	$0.593172\pi$
$674$	0	0
$675$	−3.44909	−0.132756
$676$	0	0
$677$	38.4961	1.47953	0.739763	−	0.672868i	$-0.234938\pi$
$677$	38.4961	1.47953	0.739763	+	0.672868i	$0.234938\pi$
$678$	0	0
$679$	−19.2217	−0.737663
$680$	0	0
$681$	−42.2114	−1.61754
$682$	0	0
$683$	−32.6641	−1.24986	−0.624929	−	0.780682i	$-0.714872\pi$
$683$	−32.6641	−1.24986	−0.624929	+	0.780682i	$0.714872\pi$
$684$	0	0
$685$	4.34858	0.166151
$686$	0	0
$687$	13.1435	0.501455
$688$	0	0
$689$	7.49924	0.285698
$690$	0	0
$691$	−10.5386	−0.400908	−0.200454	−	0.979703i	$-0.564242\pi$
$691$	−10.5386	−0.400908	−0.200454	+	0.979703i	$0.564242\pi$
$692$	0	0
$693$	16.4355	0.624331
$694$	0	0
$695$	2.15132	0.0816042
$696$	0	0
$697$	38.6200	1.46284
$698$	0	0
$699$	60.6346	2.29341
$700$	0	0
$701$	−0.339217	−0.0128120	−0.00640602	−	0.999979i	$-0.502039\pi$
$701$	−0.339217	−0.0128120	−0.00640602	+	0.999979i	$0.502039\pi$
$702$	0	0
$703$	−21.2441	−0.801236
$704$	0	0
$705$	4.93488	0.185858
$706$	0	0
$707$	38.1095	1.43325
$708$	0	0
$709$	−32.5979	−1.22424	−0.612120	−	0.790765i	$-0.709683\pi$
$709$	−32.5979	−1.22424	−0.612120	+	0.790765i	$0.709683\pi$
$710$	0	0
$711$	−6.77921	−0.254240
$712$	0	0
$713$	−53.0559	−1.98696
$714$	0	0
$715$	−0.320779	−0.0119964
$716$	0	0
$717$	23.0999	0.862683
$718$	0	0
$719$	12.5086	0.466492	0.233246	−	0.972418i	$-0.425065\pi$
$719$	12.5086	0.466492	0.233246	+	0.972418i	$0.425065\pi$
$720$	0	0
$721$	16.6580	0.620377
$722$	0	0
$723$	−37.3877	−1.39046
$724$	0	0
$725$	6.65582	0.247191
$726$	0	0
$727$	25.4282	0.943080	0.471540	−	0.881845i	$-0.343698\pi$
$727$	25.4282	0.943080	0.471540	+	0.881845i	$0.343698\pi$
$728$	0	0
$729$	−21.5230	−0.797150
$730$	0	0
$731$	73.6170	2.72282
$732$	0	0
$733$	−41.9058	−1.54783	−0.773913	−	0.633291i	$-0.781703\pi$
$733$	−41.9058	−1.54783	−0.773913	+	0.633291i	$0.781703\pi$
$734$	0	0
$735$	−1.04019	−0.0383678
$736$	0	0
$737$	−8.87385	−0.326873
$738$	0	0
$739$	5.41764	0.199291	0.0996455	−	0.995023i	$-0.468229\pi$
$739$	5.41764	0.199291	0.0996455	+	0.995023i	$0.468229\pi$
$740$	0	0
$741$	−10.9277	−0.401440
$742$	0	0
$743$	−21.9399	−0.804896	−0.402448	−	0.915443i	$-0.631841\pi$
$743$	−21.9399	−0.804896	−0.402448	+	0.915443i	$0.631841\pi$
$744$	0	0
$745$	2.68213	0.0982657
$746$	0	0
$747$	−4.83420	−0.176874
$748$	0	0
$749$	−18.1562	−0.663413
$750$	0	0
$751$	24.9199	0.909342	0.454671	−	0.890660i	$-0.349757\pi$
$751$	24.9199	0.909342	0.454671	+	0.890660i	$0.349757\pi$
$752$	0	0
$753$	57.1268	2.08181
$754$	0	0
$755$	−1.07525	−0.0391324
$756$	0	0
$757$	37.6505	1.36843	0.684215	−	0.729280i	$-0.260145\pi$
$757$	37.6505	1.36843	0.684215	+	0.729280i	$0.260145\pi$
$758$	0	0
$759$	31.2515	1.13436
$760$	0	0
$761$	−37.7080	−1.36691	−0.683456	−	0.729991i	$-0.739524\pi$
$761$	−37.7080	−1.36691	−0.683456	+	0.729991i	$0.739524\pi$
$762$	0	0
$763$	−6.78494	−0.245631
$764$	0	0
$765$	−3.81702	−0.138005
$766$	0	0
$767$	2.77797	0.100307
$768$	0	0
$769$	20.4708	0.738196	0.369098	−	0.929391i	$-0.379667\pi$
$769$	20.4708	0.738196	0.369098	+	0.929391i	$0.379667\pi$
$770$	0	0
$771$	−34.5453	−1.24412
$772$	0	0
$773$	33.8734	1.21834	0.609172	−	0.793038i	$-0.291502\pi$
$773$	33.8734	1.21834	0.609172	+	0.793038i	$0.291502\pi$
$774$	0	0
$775$	−54.4946	−1.95751
$776$	0	0
$777$	13.4606	0.482898
$778$	0	0
$779$	50.4075	1.80603
$780$	0	0
$781$	−25.3392	−0.906708
$782$	0	0
$783$	−0.935981	−0.0334492
$784$	0	0
$785$	−1.92675	−0.0687688
$786$	0	0
$787$	15.4981	0.552446	0.276223	−	0.961094i	$-0.410917\pi$
$787$	15.4981	0.552446	0.276223	+	0.961094i	$0.410917\pi$
$788$	0	0
$789$	−69.7137	−2.48187
$790$	0	0
$791$	−0.784507	−0.0278939
$792$	0	0
$793$	−0.545046	−0.0193551
$794$	0	0
$795$	7.20586	0.255566
$796$	0	0
$797$	4.04964	0.143446	0.0717229	−	0.997425i	$-0.477150\pi$
$797$	4.04964	0.143446	0.0717229	+	0.997425i	$0.477150\pi$
$798$	0	0
$799$	61.2077	2.16537
$800$	0	0
$801$	−23.7492	−0.839137
$802$	0	0
$803$	6.24642	0.220431
$804$	0	0
$805$	2.35202	0.0828979
$806$	0	0
$807$	34.2659	1.20622
$808$	0	0
$809$	−13.6925	−0.481404	−0.240702	−	0.970599i	$-0.577378\pi$
$809$	−13.6925	−0.481404	−0.240702	+	0.970599i	$0.577378\pi$
$810$	0	0
$811$	6.21244	0.218148	0.109074	−	0.994034i	$-0.465211\pi$
$811$	6.21244	0.218148	0.109074	+	0.994034i	$0.465211\pi$
$812$	0	0
$813$	16.4790	0.577945
$814$	0	0
$815$	−1.96840	−0.0689502
$816$	0	0
$817$	96.0861	3.36163
$818$	0	0
$819$	3.28523	0.114795
$820$	0	0
$821$	−9.39213	−0.327788	−0.163894	−	0.986478i	$-0.552405\pi$
$821$	−9.39213	−0.327788	−0.163894	+	0.986478i	$0.552405\pi$
$822$	0	0
$823$	27.2775	0.950836	0.475418	−	0.879760i	$-0.342297\pi$
$823$	27.2775	0.950836	0.475418	+	0.879760i	$0.342297\pi$
$824$	0	0
$825$	32.0990	1.11754
$826$	0	0
$827$	−1.69868	−0.0590688	−0.0295344	−	0.999564i	$-0.509402\pi$
$827$	−1.69868	−0.0590688	−0.0295344	+	0.999564i	$0.509402\pi$
$828$	0	0
$829$	−53.0261	−1.84167	−0.920836	−	0.389951i	$-0.872492\pi$
$829$	−53.0261	−1.84167	−0.920836	+	0.389951i	$0.872492\pi$
$830$	0	0
$831$	−33.8444	−1.17405
$832$	0	0
$833$	−12.9015	−0.447011
$834$	0	0
$835$	1.80376	0.0624215
$836$	0	0
$837$	7.66336	0.264884
$838$	0	0
$839$	48.2790	1.66678	0.833389	−	0.552687i	$-0.186398\pi$
$839$	48.2790	1.66678	0.833389	+	0.552687i	$0.186398\pi$
$840$	0	0
$841$	−27.1938	−0.937718
$842$	0	0
$843$	37.8408	1.30331
$844$	0	0
$845$	2.77083	0.0953193
$846$	0	0
$847$	−8.14440	−0.279845
$848$	0	0
$849$	−33.9011	−1.16348
$850$	0	0
$851$	12.1440	0.416292
$852$	0	0
$853$	−4.57169	−0.156532	−0.0782658	−	0.996933i	$-0.524938\pi$
$853$	−4.57169	−0.156532	−0.0782658	+	0.996933i	$0.524938\pi$
$854$	0	0
$855$	−4.98203	−0.170382
$856$	0	0
$857$	−27.5715	−0.941825	−0.470913	−	0.882180i	$-0.656075\pi$
$857$	−27.5715	−0.941825	−0.470913	+	0.882180i	$0.656075\pi$
$858$	0	0
$859$	4.72752	0.161301	0.0806505	−	0.996742i	$-0.474300\pi$
$859$	4.72752	0.161301	0.0806505	+	0.996742i	$0.474300\pi$
$860$	0	0
$861$	−31.9391	−1.08848
$862$	0	0
$863$	−21.7269	−0.739591	−0.369796	−	0.929113i	$-0.620572\pi$
$863$	−21.7269	−0.739591	−0.369796	+	0.929113i	$0.620572\pi$
$864$	0	0
$865$	4.84021	0.164572
$866$	0	0
$867$	−59.1634	−2.00929
$868$	0	0
$869$	−6.78983	−0.230329
$870$	0	0
$871$	−1.77377	−0.0601018
$872$	0	0
$873$	−23.2746	−0.787726
$874$	0	0
$875$	4.85481	0.164122
$876$	0	0
$877$	−46.3879	−1.56641	−0.783204	−	0.621765i	$-0.786416\pi$
$877$	−46.3879	−1.56641	−0.783204	+	0.621765i	$0.786416\pi$
$878$	0	0
$879$	−33.9699	−1.14578
$880$	0	0
$881$	3.92614	0.132275	0.0661376	−	0.997811i	$-0.478932\pi$
$881$	3.92614	0.132275	0.0661376	+	0.997811i	$0.478932\pi$
$882$	0	0
$883$	−40.8041	−1.37317	−0.686583	−	0.727051i	$-0.740890\pi$
$883$	−40.8041	−1.37317	−0.686583	+	0.727051i	$0.740890\pi$
$884$	0	0
$885$	2.66929	0.0897271
$886$	0	0
$887$	−14.7681	−0.495865	−0.247932	−	0.968777i	$-0.579751\pi$
$887$	−14.7681	−0.495865	−0.247932	+	0.968777i	$0.579751\pi$
$888$	0	0
$889$	−7.87494	−0.264117
$890$	0	0
$891$	−26.5565	−0.889676
$892$	0	0
$893$	79.8892	2.67339
$894$	0	0
$895$	2.07773	0.0694510
$896$	0	0
$897$	6.24676	0.208573
$898$	0	0
$899$	−14.7882	−0.493215
$900$	0	0
$901$	89.3749	2.97751
$902$	0	0
$903$	−60.8819	−2.02602
$904$	0	0
$905$	−3.94592	−0.131167
$906$	0	0
$907$	35.9024	1.19212	0.596060	−	0.802940i	$-0.296732\pi$
$907$	35.9024	1.19212	0.596060	+	0.802940i	$0.296732\pi$
$908$	0	0
$909$	46.1447	1.53052
$910$	0	0
$911$	−39.2080	−1.29902	−0.649509	−	0.760354i	$-0.725026\pi$
$911$	−39.2080	−1.29902	−0.649509	+	0.760354i	$0.725026\pi$
$912$	0	0
$913$	−4.84177	−0.160239
$914$	0	0
$915$	−0.523722	−0.0173137
$916$	0	0
$917$	19.6622	0.649303
$918$	0	0
$919$	−25.8981	−0.854298	−0.427149	−	0.904181i	$-0.640482\pi$
$919$	−25.8981	−0.854298	−0.427149	+	0.904181i	$0.640482\pi$
$920$	0	0
$921$	32.9377	1.08533
$922$	0	0
$923$	−5.06497	−0.166716
$924$	0	0
$925$	12.4733	0.410121
$926$	0	0
$927$	20.1703	0.662480
$928$	0	0
$929$	27.8765	0.914598	0.457299	−	0.889313i	$-0.348817\pi$
$929$	27.8765	0.914598	0.457299	+	0.889313i	$0.348817\pi$
$930$	0	0
$931$	−16.8392	−0.551884
$932$	0	0
$933$	54.5483	1.78583
$934$	0	0
$935$	−3.82299	−0.125025
$936$	0	0
$937$	4.27517	0.139664	0.0698318	−	0.997559i	$-0.477754\pi$
$937$	4.27517	0.139664	0.0698318	+	0.997559i	$0.477754\pi$
$938$	0	0
$939$	−74.9357	−2.44543
$940$	0	0
$941$	49.0169	1.59790	0.798952	−	0.601395i	$-0.205388\pi$
$941$	49.0169	1.59790	0.798952	+	0.601395i	$0.205388\pi$
$942$	0	0
$943$	−28.8151	−0.938347
$944$	0	0
$945$	−0.339725	−0.0110513
$946$	0	0
$947$	−56.6767	−1.84175	−0.920873	−	0.389863i	$-0.872522\pi$
$947$	−56.6767	−1.84175	−0.920873	+	0.389863i	$0.872522\pi$
$948$	0	0
$949$	1.24858	0.0405305
$950$	0	0
$951$	−59.1463	−1.91795
$952$	0	0
$953$	−2.11021	−0.0683563	−0.0341781	−	0.999416i	$-0.510881\pi$
$953$	−2.11021	−0.0683563	−0.0341781	+	0.999416i	$0.510881\pi$
$954$	0	0
$955$	−0.0831625	−0.00269107
$956$	0	0
$957$	8.71071	0.281577
$958$	0	0
$959$	−44.6054	−1.44038
$960$	0	0
$961$	90.0789	2.90577
$962$	0	0
$963$	−21.9844	−0.708437
$964$	0	0
$965$	1.91535	0.0616572
$966$	0	0
$967$	26.1472	0.840838	0.420419	−	0.907330i	$-0.361883\pi$
$967$	26.1472	0.840838	0.420419	+	0.907330i	$0.361883\pi$
$968$	0	0
$969$	−130.235	−4.18376
$970$	0	0
$971$	−31.7226	−1.01803	−0.509013	−	0.860759i	$-0.669989\pi$
$971$	−31.7226	−1.01803	−0.509013	+	0.860759i	$0.669989\pi$
$972$	0	0
$973$	−22.0671	−0.707438
$974$	0	0
$975$	6.41616	0.205482
$976$	0	0
$977$	−18.0973	−0.578982	−0.289491	−	0.957181i	$-0.593486\pi$
$977$	−18.0973	−0.578982	−0.289491	+	0.957181i	$0.593486\pi$
$978$	0	0
$979$	−23.7864	−0.760216
$980$	0	0
$981$	−8.21553	−0.262302
$982$	0	0
$983$	10.2900	0.328201	0.164100	−	0.986444i	$-0.447528\pi$
$983$	10.2900	0.328201	0.164100	+	0.986444i	$0.447528\pi$
$984$	0	0
$985$	1.20399	0.0383623
$986$	0	0
$987$	−50.6193	−1.61123
$988$	0	0
$989$	−54.9269	−1.74657
$990$	0	0
$991$	−23.6683	−0.751848	−0.375924	−	0.926651i	$-0.622675\pi$
$991$	−23.6683	−0.751848	−0.375924	+	0.926651i	$0.622675\pi$
$992$	0	0
$993$	−9.98551	−0.316880
$994$	0	0
$995$	−3.49845	−0.110908
$996$	0	0
$997$	9.00040	0.285046	0.142523	−	0.989792i	$-0.454479\pi$
$997$	9.00040	0.285046	0.142523	+	0.989792i	$0.454479\pi$
$998$	0	0
$999$	−1.75408	−0.0554965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.3		29
4.3	odd	2		4024.2.a.e.1.27	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.27	✓	29	4.3	odd	2
8048.2.a.w.1.3		29	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)

\(f(q)\)	\(=\)	\(q-2.38925 q^{3} -0.218073 q^{5} +2.23687 q^{7} +2.70851 q^{9} +O(q^{10})\)	\(q-2.38925 q^{3} -0.218073 q^{5} +2.23687 q^{7} +2.70851 q^{9} +2.71275 q^{11} +0.542243 q^{13} +0.521030 q^{15} +6.46238 q^{17} +8.43479 q^{19} -5.34444 q^{21} -4.82169 q^{23} -4.95244 q^{25} +0.696442 q^{27} -1.34395 q^{29} +11.0036 q^{31} -6.48144 q^{33} -0.487801 q^{35} -2.51862 q^{37} -1.29555 q^{39} +5.97613 q^{41} +11.3916 q^{43} -0.590652 q^{45} +9.47139 q^{47} -1.99640 q^{49} -15.4402 q^{51} +13.8300 q^{53} -0.591577 q^{55} -20.1528 q^{57} +5.12310 q^{59} -1.00517 q^{61} +6.05859 q^{63} -0.118248 q^{65} -3.27116 q^{67} +11.5202 q^{69} -9.34078 q^{71} +2.30261 q^{73} +11.8326 q^{75} +6.06808 q^{77} -2.50293 q^{79} -9.78950 q^{81} -1.78482 q^{83} -1.40927 q^{85} +3.21102 q^{87} -8.76836 q^{89} +1.21293 q^{91} -26.2903 q^{93} -1.83940 q^{95} -8.59314 q^{97} +7.34751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 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

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