LMFDB - Embedded newform 8032.2.a.g.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.824754 q^{3} -2.00155 q^{5} -1.90493 q^{7} -2.31978 q^{9} +O(q^{10})$	$q-0.824754 q^{3} -2.00155 q^{5} -1.90493 q^{7} -2.31978 q^{9} +1.84763 q^{11} -3.14987 q^{13} +1.65078 q^{15} -1.88850 q^{17} +5.17623 q^{19} +1.57110 q^{21} +2.13777 q^{23} -0.993813 q^{25} +4.38751 q^{27} +4.63920 q^{29} -1.29111 q^{31} -1.52384 q^{33} +3.81281 q^{35} +2.19298 q^{37} +2.59787 q^{39} +3.50676 q^{41} -7.69041 q^{43} +4.64315 q^{45} +7.52139 q^{47} -3.37122 q^{49} +1.55754 q^{51} +0.239257 q^{53} -3.69812 q^{55} -4.26911 q^{57} -8.06917 q^{59} +15.1119 q^{61} +4.41903 q^{63} +6.30461 q^{65} +1.59776 q^{67} -1.76313 q^{69} +12.4438 q^{71} -5.90259 q^{73} +0.819651 q^{75} -3.51962 q^{77} +2.58948 q^{79} +3.34073 q^{81} -8.52613 q^{83} +3.77991 q^{85} -3.82620 q^{87} +16.4455 q^{89} +6.00030 q^{91} +1.06485 q^{93} -10.3605 q^{95} +8.40781 q^{97} -4.28610 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * 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
$3$	−0.824754	−0.476172	−0.238086	−	0.971244i	$-0.576520\pi$
$3$	−0.824754	−0.476172	−0.238086	+	0.971244i	$0.576520\pi$
$4$	0	0
$5$	−2.00155	−0.895119	−0.447559	−	0.894254i	$-0.647707\pi$
$5$	−2.00155	−0.895119	−0.447559	+	0.894254i	$0.647707\pi$
$6$	0	0
$7$	−1.90493	−0.719998	−0.359999	−	0.932953i	$-0.617223\pi$
$7$	−1.90493	−0.719998	−0.359999	+	0.932953i	$0.617223\pi$
$8$	0	0
$9$	−2.31978	−0.773260
$10$	0	0
$11$	1.84763	0.557082	0.278541	−	0.960424i	$-0.410149\pi$
$11$	1.84763	0.557082	0.278541	+	0.960424i	$0.410149\pi$
$12$	0	0
$13$	−3.14987	−0.873617	−0.436809	−	0.899554i	$-0.643891\pi$
$13$	−3.14987	−0.873617	−0.436809	+	0.899554i	$0.643891\pi$
$14$	0	0
$15$	1.65078	0.426230
$16$	0	0
$17$	−1.88850	−0.458028	−0.229014	−	0.973423i	$-0.573550\pi$
$17$	−1.88850	−0.458028	−0.229014	+	0.973423i	$0.573550\pi$
$18$	0	0
$19$	5.17623	1.18751	0.593754	−	0.804647i	$-0.297645\pi$
$19$	5.17623	1.18751	0.593754	+	0.804647i	$0.297645\pi$
$20$	0	0
$21$	1.57110	0.342843
$22$	0	0
$23$	2.13777	0.445755	0.222878	−	0.974846i	$-0.428455\pi$
$23$	2.13777	0.445755	0.222878	+	0.974846i	$0.428455\pi$
$24$	0	0
$25$	−0.993813	−0.198763
$26$	0	0
$27$	4.38751	0.844377
$28$	0	0
$29$	4.63920	0.861478	0.430739	−	0.902477i	$-0.358253\pi$
$29$	4.63920	0.861478	0.430739	+	0.902477i	$0.358253\pi$
$30$	0	0
$31$	−1.29111	−0.231891	−0.115945	−	0.993256i	$-0.536990\pi$
$31$	−1.29111	−0.231891	−0.115945	+	0.993256i	$0.536990\pi$
$32$	0	0
$33$	−1.52384	−0.265267
$34$	0	0
$35$	3.81281	0.644483
$36$	0	0
$37$	2.19298	0.360524	0.180262	−	0.983619i	$-0.442305\pi$
$37$	2.19298	0.360524	0.180262	+	0.983619i	$0.442305\pi$
$38$	0	0
$39$	2.59787	0.415992
$40$	0	0
$41$	3.50676	0.547664	0.273832	−	0.961778i	$-0.411709\pi$
$41$	3.50676	0.547664	0.273832	+	0.961778i	$0.411709\pi$
$42$	0	0
$43$	−7.69041	−1.17278	−0.586388	−	0.810030i	$-0.699451\pi$
$43$	−7.69041	−1.17278	−0.586388	+	0.810030i	$0.699451\pi$
$44$	0	0
$45$	4.64315	0.692160
$46$	0	0
$47$	7.52139	1.09711	0.548554	−	0.836115i	$-0.315178\pi$
$47$	7.52139	1.09711	0.548554	+	0.836115i	$0.315178\pi$
$48$	0	0
$49$	−3.37122	−0.481604
$50$	0	0
$51$	1.55754	0.218100
$52$	0	0
$53$	0.239257	0.0328644	0.0164322	−	0.999865i	$-0.494769\pi$
$53$	0.239257	0.0328644	0.0164322	+	0.999865i	$0.494769\pi$
$54$	0	0
$55$	−3.69812	−0.498655
$56$	0	0
$57$	−4.26911	−0.565458
$58$	0	0
$59$	−8.06917	−1.05052	−0.525258	−	0.850943i	$-0.676031\pi$
$59$	−8.06917	−1.05052	−0.525258	+	0.850943i	$0.676031\pi$
$60$	0	0
$61$	15.1119	1.93487	0.967437	−	0.253111i	$-0.0814537\pi$
$61$	15.1119	1.93487	0.967437	+	0.253111i	$0.0814537\pi$
$62$	0	0
$63$	4.41903	0.556746
$64$	0	0
$65$	6.30461	0.781991
$66$	0	0
$67$	1.59776	0.195198	0.0975990	−	0.995226i	$-0.468884\pi$
$67$	1.59776	0.195198	0.0975990	+	0.995226i	$0.468884\pi$
$68$	0	0
$69$	−1.76313	−0.212256
$70$	0	0
$71$	12.4438	1.47680	0.738401	−	0.674362i	$-0.235581\pi$
$71$	12.4438	1.47680	0.738401	+	0.674362i	$0.235581\pi$
$72$	0	0
$73$	−5.90259	−0.690846	−0.345423	−	0.938447i	$-0.612265\pi$
$73$	−5.90259	−0.690846	−0.345423	+	0.938447i	$0.612265\pi$
$74$	0	0
$75$	0.819651	0.0946452
$76$	0	0
$77$	−3.51962	−0.401098
$78$	0	0
$79$	2.58948	0.291339	0.145669	−	0.989333i	$-0.453466\pi$
$79$	2.58948	0.291339	0.145669	+	0.989333i	$0.453466\pi$
$80$	0	0
$81$	3.34073	0.371192
$82$	0	0
$83$	−8.52613	−0.935865	−0.467932	−	0.883764i	$-0.655001\pi$
$83$	−8.52613	−0.935865	−0.467932	+	0.883764i	$0.655001\pi$
$84$	0	0
$85$	3.77991	0.409989
$86$	0	0
$87$	−3.82620	−0.410211
$88$	0	0
$89$	16.4455	1.74322	0.871611	−	0.490199i	$-0.163076\pi$
$89$	16.4455	1.74322	0.871611	+	0.490199i	$0.163076\pi$
$90$	0	0
$91$	6.00030	0.629002
$92$	0	0
$93$	1.06485	0.110420
$94$	0	0
$95$	−10.3605	−1.06296
$96$	0	0
$97$	8.40781	0.853684	0.426842	−	0.904326i	$-0.359626\pi$
$97$	8.40781	0.853684	0.426842	+	0.904326i	$0.359626\pi$
$98$	0	0
$99$	−4.28610	−0.430770
$100$	0	0
$101$	3.12938	0.311385	0.155693	−	0.987806i	$-0.450239\pi$
$101$	3.12938	0.311385	0.155693	+	0.987806i	$0.450239\pi$
$102$	0	0
$103$	−0.634532	−0.0625223	−0.0312611	−	0.999511i	$-0.509952\pi$
$103$	−0.634532	−0.0625223	−0.0312611	+	0.999511i	$0.509952\pi$
$104$	0	0
$105$	−3.14463	−0.306885
$106$	0	0
$107$	−18.8881	−1.82598	−0.912991	−	0.407979i	$-0.866234\pi$
$107$	−18.8881	−1.82598	−0.912991	+	0.407979i	$0.866234\pi$
$108$	0	0
$109$	1.14367	0.109543	0.0547717	−	0.998499i	$-0.482557\pi$
$109$	1.14367	0.109543	0.0547717	+	0.998499i	$0.482557\pi$
$110$	0	0
$111$	−1.80867	−0.171672
$112$	0	0
$113$	−2.13655	−0.200989	−0.100495	−	0.994938i	$-0.532043\pi$
$113$	−2.13655	−0.200989	−0.100495	+	0.994938i	$0.532043\pi$
$114$	0	0
$115$	−4.27884	−0.399004
$116$	0	0
$117$	7.30701	0.675534
$118$	0	0
$119$	3.59746	0.329779
$120$	0	0
$121$	−7.58625	−0.689659
$122$	0	0
$123$	−2.89221	−0.260782
$124$	0	0
$125$	11.9969	1.07303
$126$	0	0
$127$	8.43081	0.748114	0.374057	−	0.927406i	$-0.377966\pi$
$127$	8.43081	0.748114	0.374057	+	0.927406i	$0.377966\pi$
$128$	0	0
$129$	6.34270	0.558443
$130$	0	0
$131$	−15.5781	−1.36106	−0.680532	−	0.732719i	$-0.738251\pi$
$131$	−15.5781	−1.36106	−0.680532	+	0.732719i	$0.738251\pi$
$132$	0	0
$133$	−9.86037	−0.855003
$134$	0	0
$135$	−8.78180	−0.755817
$136$	0	0
$137$	19.0113	1.62425	0.812124	−	0.583484i	$-0.198311\pi$
$137$	19.0113	1.62425	0.812124	+	0.583484i	$0.198311\pi$
$138$	0	0
$139$	−1.55472	−0.131870	−0.0659349	−	0.997824i	$-0.521003\pi$
$139$	−1.55472	−0.131870	−0.0659349	+	0.997824i	$0.521003\pi$
$140$	0	0
$141$	−6.20330	−0.522412
$142$	0	0
$143$	−5.81981	−0.486677
$144$	0	0
$145$	−9.28557	−0.771125
$146$	0	0
$147$	2.78043	0.229326
$148$	0	0
$149$	−12.6169	−1.03362	−0.516810	−	0.856100i	$-0.672881\pi$
$149$	−12.6169	−1.03362	−0.516810	+	0.856100i	$0.672881\pi$
$150$	0	0
$151$	4.58141	0.372830	0.186415	−	0.982471i	$-0.440313\pi$
$151$	4.58141	0.372830	0.186415	+	0.982471i	$0.440313\pi$
$152$	0	0
$153$	4.38090	0.354175
$154$	0	0
$155$	2.58422	0.207570
$156$	0	0
$157$	−13.4963	−1.07713	−0.538563	−	0.842586i	$-0.681032\pi$
$157$	−13.4963	−1.07713	−0.538563	+	0.842586i	$0.681032\pi$
$158$	0	0
$159$	−0.197328	−0.0156491
$160$	0	0
$161$	−4.07231	−0.320943
$162$	0	0
$163$	−19.9214	−1.56036	−0.780181	−	0.625554i	$-0.784873\pi$
$163$	−19.9214	−1.56036	−0.780181	+	0.625554i	$0.784873\pi$
$164$	0	0
$165$	3.05004	0.237445
$166$	0	0
$167$	−5.16552	−0.399720	−0.199860	−	0.979824i	$-0.564049\pi$
$167$	−5.16552	−0.399720	−0.199860	+	0.979824i	$0.564049\pi$
$168$	0	0
$169$	−3.07831	−0.236793
$170$	0	0
$171$	−12.0077	−0.918253
$172$	0	0
$173$	2.34275	0.178116	0.0890581	−	0.996026i	$-0.471614\pi$
$173$	2.34275	0.178116	0.0890581	+	0.996026i	$0.471614\pi$
$174$	0	0
$175$	1.89315	0.143109
$176$	0	0
$177$	6.65507	0.500226
$178$	0	0
$179$	14.7260	1.10067	0.550337	−	0.834943i	$-0.314499\pi$
$179$	14.7260	1.10067	0.550337	+	0.834943i	$0.314499\pi$
$180$	0	0
$181$	−10.3379	−0.768410	−0.384205	−	0.923248i	$-0.625524\pi$
$181$	−10.3379	−0.768410	−0.384205	+	0.923248i	$0.625524\pi$
$182$	0	0
$183$	−12.4636	−0.921333
$184$	0	0
$185$	−4.38936	−0.322712
$186$	0	0
$187$	−3.48925	−0.255159
$188$	0	0
$189$	−8.35792	−0.607949
$190$	0	0
$191$	−14.4060	−1.04238	−0.521189	−	0.853441i	$-0.674511\pi$
$191$	−14.4060	−1.04238	−0.521189	+	0.853441i	$0.674511\pi$
$192$	0	0
$193$	−3.18356	−0.229158	−0.114579	−	0.993414i	$-0.536552\pi$
$193$	−3.18356	−0.229158	−0.114579	+	0.993414i	$0.536552\pi$
$194$	0	0
$195$	−5.19975	−0.372362
$196$	0	0
$197$	−1.89582	−0.135071	−0.0675356	−	0.997717i	$-0.521514\pi$
$197$	−1.89582	−0.135071	−0.0675356	+	0.997717i	$0.521514\pi$
$198$	0	0
$199$	5.09939	0.361486	0.180743	−	0.983530i	$-0.442150\pi$
$199$	5.09939	0.361486	0.180743	+	0.983530i	$0.442150\pi$
$200$	0	0
$201$	−1.31776	−0.0929477
$202$	0	0
$203$	−8.83737	−0.620262
$204$	0	0
$205$	−7.01894	−0.490224
$206$	0	0
$207$	−4.95915	−0.344685
$208$	0	0
$209$	9.56377	0.661540
$210$	0	0
$211$	−5.49250	−0.378120	−0.189060	−	0.981966i	$-0.560544\pi$
$211$	−5.49250	−0.378120	−0.189060	+	0.981966i	$0.560544\pi$
$212$	0	0
$213$	−10.2630	−0.703212
$214$	0	0
$215$	15.3927	1.04977
$216$	0	0
$217$	2.45948	0.166961
$218$	0	0
$219$	4.86818	0.328961
$220$	0	0
$221$	5.94852	0.400141
$222$	0	0
$223$	−11.7024	−0.783652	−0.391826	−	0.920039i	$-0.628156\pi$
$223$	−11.7024	−0.783652	−0.391826	+	0.920039i	$0.628156\pi$
$224$	0	0
$225$	2.30543	0.153695
$226$	0	0
$227$	−0.431425	−0.0286347	−0.0143174	−	0.999898i	$-0.504558\pi$
$227$	−0.431425	−0.0286347	−0.0143174	+	0.999898i	$0.504558\pi$
$228$	0	0
$229$	22.7121	1.50086	0.750428	−	0.660952i	$-0.229847\pi$
$229$	22.7121	1.50086	0.750428	+	0.660952i	$0.229847\pi$
$230$	0	0
$231$	2.90282	0.190992
$232$	0	0
$233$	8.00471	0.524406	0.262203	−	0.965013i	$-0.415551\pi$
$233$	8.00471	0.524406	0.262203	+	0.965013i	$0.415551\pi$
$234$	0	0
$235$	−15.0544	−0.982042
$236$	0	0
$237$	−2.13568	−0.138727
$238$	0	0
$239$	9.36818	0.605977	0.302989	−	0.952994i	$-0.402016\pi$
$239$	9.36818	0.605977	0.302989	+	0.952994i	$0.402016\pi$
$240$	0	0
$241$	−16.1186	−1.03829	−0.519147	−	0.854685i	$-0.673750\pi$
$241$	−16.1186	−1.03829	−0.519147	+	0.854685i	$0.673750\pi$
$242$	0	0
$243$	−15.9178	−1.02113
$244$	0	0
$245$	6.74766	0.431092
$246$	0	0
$247$	−16.3045	−1.03743
$248$	0	0
$249$	7.03196	0.445632
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	3.94981	0.248322
$254$	0	0
$255$	−3.11750	−0.195225
$256$	0	0
$257$	−17.0405	−1.06296	−0.531479	−	0.847072i	$-0.678363\pi$
$257$	−17.0405	−1.06296	−0.531479	+	0.847072i	$0.678363\pi$
$258$	0	0
$259$	−4.17749	−0.259577
$260$	0	0
$261$	−10.7619	−0.666147
$262$	0	0
$263$	11.9676	0.737956	0.368978	−	0.929438i	$-0.379708\pi$
$263$	11.9676	0.737956	0.368978	+	0.929438i	$0.379708\pi$
$264$	0	0
$265$	−0.478883	−0.0294176
$266$	0	0
$267$	−13.5635	−0.830073
$268$	0	0
$269$	−19.6507	−1.19812	−0.599062	−	0.800703i	$-0.704460\pi$
$269$	−19.6507	−1.19812	−0.599062	+	0.800703i	$0.704460\pi$
$270$	0	0
$271$	3.13363	0.190355	0.0951773	−	0.995460i	$-0.469658\pi$
$271$	3.13363	0.190355	0.0951773	+	0.995460i	$0.469658\pi$
$272$	0	0
$273$	−4.94877	−0.299513
$274$	0	0
$275$	−1.83620	−0.110727
$276$	0	0
$277$	15.3319	0.921208	0.460604	−	0.887606i	$-0.347633\pi$
$277$	15.3319	0.921208	0.460604	+	0.887606i	$0.347633\pi$
$278$	0	0
$279$	2.99510	0.179312
$280$	0	0
$281$	6.02718	0.359551	0.179776	−	0.983708i	$-0.442463\pi$
$281$	6.02718	0.359551	0.179776	+	0.983708i	$0.442463\pi$
$282$	0	0
$283$	−13.6609	−0.812054	−0.406027	−	0.913861i	$-0.633086\pi$
$283$	−13.6609	−0.812054	−0.406027	+	0.913861i	$0.633086\pi$
$284$	0	0
$285$	8.54483	0.506152
$286$	0	0
$287$	−6.68014	−0.394316
$288$	0	0
$289$	−13.4336	−0.790211
$290$	0	0
$291$	−6.93437	−0.406500
$292$	0	0
$293$	−11.2212	−0.655551	−0.327776	−	0.944756i	$-0.606299\pi$
$293$	−11.2212	−0.655551	−0.327776	+	0.944756i	$0.606299\pi$
$294$	0	0
$295$	16.1508	0.940336
$296$	0	0
$297$	8.10651	0.470387
$298$	0	0
$299$	−6.73369	−0.389419
$300$	0	0
$301$	14.6497	0.844397
$302$	0	0
$303$	−2.58097	−0.148273
$304$	0	0
$305$	−30.2471	−1.73194
$306$	0	0
$307$	7.63143	0.435549	0.217774	−	0.975999i	$-0.430120\pi$
$307$	7.63143	0.435549	0.217774	+	0.975999i	$0.430120\pi$
$308$	0	0
$309$	0.523333	0.0297714
$310$	0	0
$311$	−14.4758	−0.820849	−0.410424	−	0.911895i	$-0.634619\pi$
$311$	−14.4758	−0.820849	−0.410424	+	0.911895i	$0.634619\pi$
$312$	0	0
$313$	3.27480	0.185103	0.0925513	−	0.995708i	$-0.470498\pi$
$313$	3.27480	0.185103	0.0925513	+	0.995708i	$0.470498\pi$
$314$	0	0
$315$	−8.84489	−0.498353
$316$	0	0
$317$	−30.9760	−1.73979	−0.869893	−	0.493240i	$-0.835812\pi$
$317$	−30.9760	−1.73979	−0.869893	+	0.493240i	$0.835812\pi$
$318$	0	0
$319$	8.57154	0.479914
$320$	0	0
$321$	15.5780	0.869481
$322$	0	0
$323$	−9.77528	−0.543911
$324$	0	0
$325$	3.13038	0.173642
$326$	0	0
$327$	−0.943244	−0.0521615
$328$	0	0
$329$	−14.3278	−0.789915
$330$	0	0
$331$	22.6554	1.24525	0.622626	−	0.782519i	$-0.286066\pi$
$331$	22.6554	1.24525	0.622626	+	0.782519i	$0.286066\pi$
$332$	0	0
$333$	−5.08724	−0.278779
$334$	0	0
$335$	−3.19800	−0.174725
$336$	0	0
$337$	22.4122	1.22087	0.610434	−	0.792067i	$-0.290995\pi$
$337$	22.4122	1.22087	0.610434	+	0.792067i	$0.290995\pi$
$338$	0	0
$339$	1.76212	0.0957055
$340$	0	0
$341$	−2.38550	−0.129182
$342$	0	0
$343$	19.7565	1.06675
$344$	0	0
$345$	3.52899	0.189994
$346$	0	0
$347$	−29.4861	−1.58290	−0.791449	−	0.611235i	$-0.790673\pi$
$347$	−29.4861	−1.58290	−0.791449	+	0.611235i	$0.790673\pi$
$348$	0	0
$349$	−8.18147	−0.437944	−0.218972	−	0.975731i	$-0.570270\pi$
$349$	−8.18147	−0.437944	−0.218972	+	0.975731i	$0.570270\pi$
$350$	0	0
$351$	−13.8201	−0.737662
$352$	0	0
$353$	−6.34108	−0.337501	−0.168751	−	0.985659i	$-0.553973\pi$
$353$	−6.34108	−0.337501	−0.168751	+	0.985659i	$0.553973\pi$
$354$	0	0
$355$	−24.9068	−1.32191
$356$	0	0
$357$	−2.96702	−0.157031
$358$	0	0
$359$	3.89544	0.205594	0.102797	−	0.994702i	$-0.467221\pi$
$359$	3.89544	0.205594	0.102797	+	0.994702i	$0.467221\pi$
$360$	0	0
$361$	7.79333	0.410175
$362$	0	0
$363$	6.25679	0.328396
$364$	0	0
$365$	11.8143	0.618389
$366$	0	0
$367$	−4.52504	−0.236205	−0.118103	−	0.993001i	$-0.537681\pi$
$367$	−4.52504	−0.236205	−0.118103	+	0.993001i	$0.537681\pi$
$368$	0	0
$369$	−8.13491	−0.423487
$370$	0	0
$371$	−0.455768	−0.0236623
$372$	0	0
$373$	1.28919	0.0667518	0.0333759	−	0.999443i	$-0.489374\pi$
$373$	1.28919	0.0667518	0.0333759	+	0.999443i	$0.489374\pi$
$374$	0	0
$375$	−9.89448	−0.510949
$376$	0	0
$377$	−14.6129	−0.752602
$378$	0	0
$379$	2.65784	0.136524	0.0682621	−	0.997667i	$-0.478255\pi$
$379$	2.65784	0.136524	0.0682621	+	0.997667i	$0.478255\pi$
$380$	0	0
$381$	−6.95335	−0.356231
$382$	0	0
$383$	−26.9687	−1.37804	−0.689018	−	0.724744i	$-0.741958\pi$
$383$	−26.9687	−1.37804	−0.689018	+	0.724744i	$0.741958\pi$
$384$	0	0
$385$	7.04468	0.359030
$386$	0	0
$387$	17.8401	0.906862
$388$	0	0
$389$	10.9742	0.556414	0.278207	−	0.960521i	$-0.410260\pi$
$389$	10.9742	0.556414	0.278207	+	0.960521i	$0.410260\pi$
$390$	0	0
$391$	−4.03716	−0.204168
$392$	0	0
$393$	12.8481	0.648100
$394$	0	0
$395$	−5.18295	−0.260783
$396$	0	0
$397$	−19.2717	−0.967220	−0.483610	−	0.875284i	$-0.660675\pi$
$397$	−19.2717	−0.967220	−0.483610	+	0.875284i	$0.660675\pi$
$398$	0	0
$399$	8.13238	0.407128
$400$	0	0
$401$	2.59530	0.129603	0.0648015	−	0.997898i	$-0.479359\pi$
$401$	2.59530	0.129603	0.0648015	+	0.997898i	$0.479359\pi$
$402$	0	0
$403$	4.06684	0.202584
$404$	0	0
$405$	−6.68662	−0.332261
$406$	0	0
$407$	4.05183	0.200842
$408$	0	0
$409$	23.2613	1.15020	0.575098	−	0.818085i	$-0.304964\pi$
$409$	23.2613	1.15020	0.575098	+	0.818085i	$0.304964\pi$
$410$	0	0
$411$	−15.6797	−0.773422
$412$	0	0
$413$	15.3712	0.756369
$414$	0	0
$415$	17.0654	0.837710
$416$	0	0
$417$	1.28226	0.0627926
$418$	0	0
$419$	0.200470	0.00979362	0.00489681	−	0.999988i	$-0.498441\pi$
$419$	0.200470	0.00979362	0.00489681	+	0.999988i	$0.498441\pi$
$420$	0	0
$421$	−4.55368	−0.221933	−0.110966	−	0.993824i	$-0.535395\pi$
$421$	−4.55368	−0.221933	−0.110966	+	0.993824i	$0.535395\pi$
$422$	0	0
$423$	−17.4480	−0.848350
$424$	0	0
$425$	1.87681	0.0910388
$426$	0	0
$427$	−28.7871	−1.39311
$428$	0	0
$429$	4.79991	0.231742
$430$	0	0
$431$	−32.5518	−1.56797	−0.783983	−	0.620783i	$-0.786815\pi$
$431$	−32.5518	−1.56797	−0.783983	+	0.620783i	$0.786815\pi$
$432$	0	0
$433$	−8.78352	−0.422109	−0.211055	−	0.977474i	$-0.567690\pi$
$433$	−8.78352	−0.422109	−0.211055	+	0.977474i	$0.567690\pi$
$434$	0	0
$435$	7.65831	0.367188
$436$	0	0
$437$	11.0656	0.529338
$438$	0	0
$439$	−0.564800	−0.0269565	−0.0134782	−	0.999909i	$-0.504290\pi$
$439$	−0.564800	−0.0269565	−0.0134782	+	0.999909i	$0.504290\pi$
$440$	0	0
$441$	7.82050	0.372405
$442$	0	0
$443$	10.5664	0.502023	0.251011	−	0.967984i	$-0.419237\pi$
$443$	10.5664	0.502023	0.251011	+	0.967984i	$0.419237\pi$
$444$	0	0
$445$	−32.9165	−1.56039
$446$	0	0
$447$	10.4059	0.492181
$448$	0	0
$449$	16.6579	0.786135	0.393067	−	0.919510i	$-0.371414\pi$
$449$	16.6579	0.786135	0.393067	+	0.919510i	$0.371414\pi$
$450$	0	0
$451$	6.47920	0.305094
$452$	0	0
$453$	−3.77854	−0.177531
$454$	0	0
$455$	−12.0099	−0.563032
$456$	0	0
$457$	5.90911	0.276417	0.138208	−	0.990403i	$-0.455866\pi$
$457$	5.90911	0.276417	0.138208	+	0.990403i	$0.455866\pi$
$458$	0	0
$459$	−8.28579	−0.386748
$460$	0	0
$461$	35.8482	1.66962	0.834808	−	0.550541i	$-0.185579\pi$
$461$	35.8482	1.66962	0.834808	+	0.550541i	$0.185579\pi$
$462$	0	0
$463$	−14.0275	−0.651911	−0.325956	−	0.945385i	$-0.605686\pi$
$463$	−14.0275	−0.651911	−0.325956	+	0.945385i	$0.605686\pi$
$464$	0	0
$465$	−2.13135	−0.0988388
$466$	0	0
$467$	2.14647	0.0993267	0.0496633	−	0.998766i	$-0.484185\pi$
$467$	2.14647	0.0993267	0.0496633	+	0.998766i	$0.484185\pi$
$468$	0	0
$469$	−3.04364	−0.140542
$470$	0	0
$471$	11.1312	0.512897
$472$	0	0
$473$	−14.2091	−0.653333
$474$	0	0
$475$	−5.14420	−0.236032
$476$	0	0
$477$	−0.555023	−0.0254128
$478$	0	0
$479$	−22.3774	−1.02245	−0.511224	−	0.859447i	$-0.670808\pi$
$479$	−22.3774	−1.02245	−0.511224	+	0.859447i	$0.670808\pi$
$480$	0	0
$481$	−6.90762	−0.314960
$482$	0	0
$483$	3.35865	0.152824
$484$	0	0
$485$	−16.8286	−0.764148
$486$	0	0
$487$	31.7069	1.43677	0.718387	−	0.695643i	$-0.244881\pi$
$487$	31.7069	1.43677	0.718387	+	0.695643i	$0.244881\pi$
$488$	0	0
$489$	16.4302	0.743001
$490$	0	0
$491$	−35.9752	−1.62354	−0.811769	−	0.583979i	$-0.801495\pi$
$491$	−35.9752	−1.62354	−0.811769	+	0.583979i	$0.801495\pi$
$492$	0	0
$493$	−8.76111	−0.394580
$494$	0	0
$495$	8.57884	0.385590
$496$	0	0
$497$	−23.7045	−1.06329
$498$	0	0
$499$	−23.1838	−1.03785	−0.518925	−	0.854820i	$-0.673668\pi$
$499$	−23.1838	−1.03785	−0.518925	+	0.854820i	$0.673668\pi$
$500$	0	0
$501$	4.26028	0.190335
$502$	0	0
$503$	−6.50004	−0.289822	−0.144911	−	0.989445i	$-0.546290\pi$
$503$	−6.50004	−0.289822	−0.144911	+	0.989445i	$0.546290\pi$
$504$	0	0
$505$	−6.26360	−0.278727
$506$	0	0
$507$	2.53885	0.112754
$508$	0	0
$509$	−12.4062	−0.549897	−0.274948	−	0.961459i	$-0.588661\pi$
$509$	−12.4062	−0.549897	−0.274948	+	0.961459i	$0.588661\pi$
$510$	0	0
$511$	11.2441	0.497408
$512$	0	0
$513$	22.7107	1.00270
$514$	0	0
$515$	1.27004	0.0559649
$516$	0	0
$517$	13.8968	0.611180
$518$	0	0
$519$	−1.93219	−0.0848139
$520$	0	0
$521$	18.4949	0.810276	0.405138	−	0.914256i	$-0.367223\pi$
$521$	18.4949	0.810276	0.405138	+	0.914256i	$0.367223\pi$
$522$	0	0
$523$	−28.3658	−1.24035	−0.620174	−	0.784464i	$-0.712938\pi$
$523$	−28.3658	−1.24035	−0.620174	+	0.784464i	$0.712938\pi$
$524$	0	0
$525$	−1.56138	−0.0681443
$526$	0	0
$527$	2.43826	0.106212
$528$	0	0
$529$	−18.4300	−0.801302
$530$	0	0
$531$	18.7187	0.812322
$532$	0	0
$533$	−11.0458	−0.478448
$534$	0	0
$535$	37.8054	1.63447
$536$	0	0
$537$	−12.1453	−0.524110
$538$	0	0
$539$	−6.22879	−0.268293
$540$	0	0
$541$	10.2867	0.442259	0.221129	−	0.975244i	$-0.429026\pi$
$541$	10.2867	0.442259	0.221129	+	0.975244i	$0.429026\pi$
$542$	0	0
$543$	8.52623	0.365895
$544$	0	0
$545$	−2.28910	−0.0980544
$546$	0	0
$547$	−13.1765	−0.563385	−0.281693	−	0.959505i	$-0.590896\pi$
$547$	−13.1765	−0.563385	−0.281693	+	0.959505i	$0.590896\pi$
$548$	0	0
$549$	−35.0562	−1.49616
$550$	0	0
$551$	24.0135	1.02301
$552$	0	0
$553$	−4.93278	−0.209763
$554$	0	0
$555$	3.62014	0.153666
$556$	0	0
$557$	−10.0564	−0.426102	−0.213051	−	0.977041i	$-0.568340\pi$
$557$	−10.0564	−0.426102	−0.213051	+	0.977041i	$0.568340\pi$
$558$	0	0
$559$	24.2238	1.02456
$560$	0	0
$561$	2.87777	0.121500
$562$	0	0
$563$	35.2554	1.48584	0.742918	−	0.669382i	$-0.233441\pi$
$563$	35.2554	1.48584	0.742918	+	0.669382i	$0.233441\pi$
$564$	0	0
$565$	4.27640	0.179909
$566$	0	0
$567$	−6.36387	−0.267257
$568$	0	0
$569$	−5.69642	−0.238806	−0.119403	−	0.992846i	$-0.538098\pi$
$569$	−5.69642	−0.238806	−0.119403	+	0.992846i	$0.538098\pi$
$570$	0	0
$571$	−36.5714	−1.53046	−0.765232	−	0.643754i	$-0.777376\pi$
$571$	−36.5714	−1.53046	−0.765232	+	0.643754i	$0.777376\pi$
$572$	0	0
$573$	11.8814	0.496351
$574$	0	0
$575$	−2.12454	−0.0885995
$576$	0	0
$577$	−18.8861	−0.786238	−0.393119	−	0.919488i	$-0.628604\pi$
$577$	−18.8861	−0.786238	−0.393119	+	0.919488i	$0.628604\pi$
$578$	0	0
$579$	2.62565	0.109118
$580$	0	0
$581$	16.2417	0.673820
$582$	0	0
$583$	0.442058	0.0183082
$584$	0	0
$585$	−14.6253	−0.604683
$586$	0	0
$587$	−31.8806	−1.31585	−0.657927	−	0.753082i	$-0.728566\pi$
$587$	−31.8806	−1.31585	−0.657927	+	0.753082i	$0.728566\pi$
$588$	0	0
$589$	−6.68309	−0.275372
$590$	0	0
$591$	1.56358	0.0643171
$592$	0	0
$593$	37.0775	1.52259	0.761296	−	0.648405i	$-0.224563\pi$
$593$	37.0775	1.52259	0.761296	+	0.648405i	$0.224563\pi$
$594$	0	0
$595$	−7.20048	−0.295191
$596$	0	0
$597$	−4.20574	−0.172130
$598$	0	0
$599$	−16.4725	−0.673050	−0.336525	−	0.941675i	$-0.609252\pi$
$599$	−16.4725	−0.673050	−0.336525	+	0.941675i	$0.609252\pi$
$600$	0	0
$601$	20.8475	0.850387	0.425193	−	0.905103i	$-0.360206\pi$
$601$	20.8475	0.850387	0.425193	+	0.905103i	$0.360206\pi$
$602$	0	0
$603$	−3.70646	−0.150939
$604$	0	0
$605$	15.1842	0.617327
$606$	0	0
$607$	33.5978	1.36369	0.681847	−	0.731495i	$-0.261177\pi$
$607$	33.5978	1.36369	0.681847	+	0.731495i	$0.261177\pi$
$608$	0	0
$609$	7.28865	0.295351
$610$	0	0
$611$	−23.6914	−0.958453
$612$	0	0
$613$	−21.3054	−0.860519	−0.430259	−	0.902705i	$-0.641578\pi$
$613$	−21.3054	−0.860519	−0.430259	+	0.902705i	$0.641578\pi$
$614$	0	0
$615$	5.78890	0.233431
$616$	0	0
$617$	45.8431	1.84557	0.922787	−	0.385309i	$-0.125905\pi$
$617$	45.8431	1.84557	0.922787	+	0.385309i	$0.125905\pi$
$618$	0	0
$619$	−14.0980	−0.566647	−0.283323	−	0.959024i	$-0.591437\pi$
$619$	−14.0980	−0.566647	−0.283323	+	0.959024i	$0.591437\pi$
$620$	0	0
$621$	9.37947	0.376385
$622$	0	0
$623$	−31.3276	−1.25512
$624$	0	0
$625$	−19.0433	−0.761731
$626$	0	0
$627$	−7.88775	−0.315007
$628$	0	0
$629$	−4.14144	−0.165130
$630$	0	0
$631$	−31.3536	−1.24817	−0.624085	−	0.781357i	$-0.714528\pi$
$631$	−31.3536	−1.24817	−0.624085	+	0.781357i	$0.714528\pi$
$632$	0	0
$633$	4.52996	0.180050
$634$	0	0
$635$	−16.8747	−0.669651
$636$	0	0
$637$	10.6189	0.420737
$638$	0	0
$639$	−28.8668	−1.14195
$640$	0	0
$641$	−41.5589	−1.64148	−0.820740	−	0.571302i	$-0.806438\pi$
$641$	−41.5589	−1.64148	−0.820740	+	0.571302i	$0.806438\pi$
$642$	0	0
$643$	14.9441	0.589336	0.294668	−	0.955600i	$-0.404791\pi$
$643$	14.9441	0.589336	0.294668	+	0.955600i	$0.404791\pi$
$644$	0	0
$645$	−12.6952	−0.499873
$646$	0	0
$647$	−32.1321	−1.26324	−0.631622	−	0.775276i	$-0.717611\pi$
$647$	−32.1321	−1.26324	−0.631622	+	0.775276i	$0.717611\pi$
$648$	0	0
$649$	−14.9089	−0.585224
$650$	0	0
$651$	−2.02847	−0.0795020
$652$	0	0
$653$	−20.0770	−0.785675	−0.392837	−	0.919608i	$-0.628506\pi$
$653$	−20.0770	−0.785675	−0.392837	+	0.919608i	$0.628506\pi$
$654$	0	0
$655$	31.1802	1.21831
$656$	0	0
$657$	13.6927	0.534204
$658$	0	0
$659$	−27.1911	−1.05922	−0.529608	−	0.848242i	$-0.677661\pi$
$659$	−27.1911	−1.05922	−0.529608	+	0.848242i	$0.677661\pi$
$660$	0	0
$661$	24.2797	0.944372	0.472186	−	0.881499i	$-0.343465\pi$
$661$	24.2797	0.944372	0.472186	+	0.881499i	$0.343465\pi$
$662$	0	0
$663$	−4.90606	−0.190536
$664$	0	0
$665$	19.7360	0.765329
$666$	0	0
$667$	9.91753	0.384008
$668$	0	0
$669$	9.65161	0.373153
$670$	0	0
$671$	27.9212	1.07788
$672$	0	0
$673$	15.6721	0.604113	0.302057	−	0.953290i	$-0.402327\pi$
$673$	15.6721	0.604113	0.302057	+	0.953290i	$0.402327\pi$
$674$	0	0
$675$	−4.36036	−0.167831
$676$	0	0
$677$	−5.73143	−0.220277	−0.110138	−	0.993916i	$-0.535129\pi$
$677$	−5.73143	−0.220277	−0.110138	+	0.993916i	$0.535129\pi$
$678$	0	0
$679$	−16.0163	−0.614650
$680$	0	0
$681$	0.355820	0.0136350
$682$	0	0
$683$	1.02589	0.0392544	0.0196272	−	0.999807i	$-0.493752\pi$
$683$	1.02589	0.0392544	0.0196272	+	0.999807i	$0.493752\pi$
$684$	0	0
$685$	−38.0521	−1.45390
$686$	0	0
$687$	−18.7319	−0.714666
$688$	0	0
$689$	−0.753628	−0.0287109
$690$	0	0
$691$	37.8509	1.43992	0.719958	−	0.694018i	$-0.244161\pi$
$691$	37.8509	1.43992	0.719958	+	0.694018i	$0.244161\pi$
$692$	0	0
$693$	8.16475	0.310153
$694$	0	0
$695$	3.11185	0.118039
$696$	0	0
$697$	−6.62250	−0.250845
$698$	0	0
$699$	−6.60192	−0.249707
$700$	0	0
$701$	31.5883	1.19307	0.596536	−	0.802586i	$-0.296543\pi$
$701$	31.5883	1.19307	0.596536	+	0.802586i	$0.296543\pi$
$702$	0	0
$703$	11.3514	0.428126
$704$	0	0
$705$	12.4162	0.467621
$706$	0	0
$707$	−5.96127	−0.224197
$708$	0	0
$709$	−27.2859	−1.02474	−0.512372	−	0.858763i	$-0.671233\pi$
$709$	−27.2859	−1.02474	−0.512372	+	0.858763i	$0.671233\pi$
$710$	0	0
$711$	−6.00702	−0.225281
$712$	0	0
$713$	−2.76010	−0.103366
$714$	0	0
$715$	11.6486	0.435633
$716$	0	0
$717$	−7.72645	−0.288549
$718$	0	0
$719$	12.2004	0.454999	0.227499	−	0.973778i	$-0.426945\pi$
$719$	12.2004	0.454999	0.227499	+	0.973778i	$0.426945\pi$
$720$	0	0
$721$	1.20874	0.0450159
$722$	0	0
$723$	13.2939	0.494406
$724$	0	0
$725$	−4.61050	−0.171230
$726$	0	0
$727$	36.4970	1.35360	0.676800	−	0.736167i	$-0.263366\pi$
$727$	36.4970	1.35360	0.676800	+	0.736167i	$0.263366\pi$
$728$	0	0
$729$	3.10609	0.115040
$730$	0	0
$731$	14.5233	0.537164
$732$	0	0
$733$	35.8715	1.32494	0.662472	−	0.749087i	$-0.269507\pi$
$733$	35.8715	1.32494	0.662472	+	0.749087i	$0.269507\pi$
$734$	0	0
$735$	−5.56516	−0.205274
$736$	0	0
$737$	2.95208	0.108741
$738$	0	0
$739$	48.3836	1.77982	0.889910	−	0.456136i	$-0.150767\pi$
$739$	48.3836	1.77982	0.889910	+	0.456136i	$0.150767\pi$
$740$	0	0
$741$	13.4472	0.493994
$742$	0	0
$743$	−20.3152	−0.745293	−0.372647	−	0.927973i	$-0.621550\pi$
$743$	−20.3152	−0.745293	−0.372647	+	0.927973i	$0.621550\pi$
$744$	0	0
$745$	25.2534	0.925213
$746$	0	0
$747$	19.7788	0.723667
$748$	0	0
$749$	35.9806	1.31470
$750$	0	0
$751$	−2.55389	−0.0931929	−0.0465965	−	0.998914i	$-0.514837\pi$
$751$	−2.55389	−0.0931929	−0.0465965	+	0.998914i	$0.514837\pi$
$752$	0	0
$753$	−0.824754	−0.0300557
$754$	0	0
$755$	−9.16991	−0.333727
$756$	0	0
$757$	−4.98278	−0.181102	−0.0905510	−	0.995892i	$-0.528863\pi$
$757$	−4.98278	−0.181102	−0.0905510	+	0.995892i	$0.528863\pi$
$758$	0	0
$759$	−3.25762	−0.118244
$760$	0	0
$761$	−26.5510	−0.962474	−0.481237	−	0.876591i	$-0.659812\pi$
$761$	−26.5510	−0.962474	−0.481237	+	0.876591i	$0.659812\pi$
$762$	0	0
$763$	−2.17861	−0.0788710
$764$	0	0
$765$	−8.76857	−0.317028
$766$	0	0
$767$	25.4168	0.917749
$768$	0	0
$769$	−36.2324	−1.30657	−0.653287	−	0.757110i	$-0.726610\pi$
$769$	−36.2324	−1.30657	−0.653287	+	0.757110i	$0.726610\pi$
$770$	0	0
$771$	14.0542	0.506150
$772$	0	0
$773$	−19.5563	−0.703392	−0.351696	−	0.936114i	$-0.614395\pi$
$773$	−19.5563	−0.703392	−0.351696	+	0.936114i	$0.614395\pi$
$774$	0	0
$775$	1.28312	0.0460912
$776$	0	0
$777$	3.44540	0.123603
$778$	0	0
$779$	18.1518	0.650355
$780$	0	0
$781$	22.9915	0.822701
$782$	0	0
$783$	20.3545	0.727412
$784$	0	0
$785$	27.0135	0.964155
$786$	0	0
$787$	−1.45338	−0.0518074	−0.0259037	−	0.999664i	$-0.508246\pi$
$787$	−1.45338	−0.0518074	−0.0259037	+	0.999664i	$0.508246\pi$
$788$	0	0
$789$	−9.87036	−0.351394
$790$	0	0
$791$	4.06998	0.144712
$792$	0	0
$793$	−47.6004	−1.69034
$794$	0	0
$795$	0.394961	0.0140078
$796$	0	0
$797$	−42.2752	−1.49746	−0.748732	−	0.662873i	$-0.769337\pi$
$797$	−42.2752	−1.49746	−0.748732	+	0.662873i	$0.769337\pi$
$798$	0	0
$799$	−14.2041	−0.502506
$800$	0	0
$801$	−38.1500	−1.34796
$802$	0	0
$803$	−10.9058	−0.384858
$804$	0	0
$805$	8.15091	0.287282
$806$	0	0
$807$	16.2070	0.570513
$808$	0	0
$809$	44.4367	1.56231	0.781155	−	0.624337i	$-0.214631\pi$
$809$	44.4367	1.56231	0.781155	+	0.624337i	$0.214631\pi$
$810$	0	0
$811$	2.63079	0.0923795	0.0461898	−	0.998933i	$-0.485292\pi$
$811$	2.63079	0.0923795	0.0461898	+	0.998933i	$0.485292\pi$
$812$	0	0
$813$	−2.58448	−0.0906415
$814$	0	0
$815$	39.8736	1.39671
$816$	0	0
$817$	−39.8073	−1.39268
$818$	0	0
$819$	−13.9194	−0.486383
$820$	0	0
$821$	35.7655	1.24822	0.624112	−	0.781335i	$-0.285461\pi$
$821$	35.7655	1.24822	0.624112	+	0.781335i	$0.285461\pi$
$822$	0	0
$823$	−22.5838	−0.787223	−0.393611	−	0.919277i	$-0.628774\pi$
$823$	−22.5838	−0.787223	−0.393611	+	0.919277i	$0.628774\pi$
$824$	0	0
$825$	1.51441	0.0527251
$826$	0	0
$827$	−21.5831	−0.750516	−0.375258	−	0.926920i	$-0.622446\pi$
$827$	−21.5831	−0.750516	−0.375258	+	0.926920i	$0.622446\pi$
$828$	0	0
$829$	25.7936	0.895847	0.447924	−	0.894072i	$-0.352164\pi$
$829$	25.7936	0.895847	0.447924	+	0.894072i	$0.352164\pi$
$830$	0	0
$831$	−12.6451	−0.438653
$832$	0	0
$833$	6.36654	0.220588
$834$	0	0
$835$	10.3390	0.357797
$836$	0	0
$837$	−5.66477	−0.195803
$838$	0	0
$839$	22.8384	0.788469	0.394234	−	0.919010i	$-0.371010\pi$
$839$	22.8384	0.788469	0.394234	+	0.919010i	$0.371010\pi$
$840$	0	0
$841$	−7.47784	−0.257856
$842$	0	0
$843$	−4.97094	−0.171208
$844$	0	0
$845$	6.16138	0.211958
$846$	0	0
$847$	14.4513	0.496553
$848$	0	0
$849$	11.2668	0.386677
$850$	0	0
$851$	4.68809	0.160706
$852$	0	0
$853$	40.0666	1.37185	0.685927	−	0.727670i	$-0.259397\pi$
$853$	40.0666	1.37185	0.685927	+	0.727670i	$0.259397\pi$
$854$	0	0
$855$	24.0340	0.821945
$856$	0	0
$857$	−38.3658	−1.31055	−0.655275	−	0.755390i	$-0.727447\pi$
$857$	−38.3658	−1.31055	−0.655275	+	0.755390i	$0.727447\pi$
$858$	0	0
$859$	2.21849	0.0756937	0.0378469	−	0.999284i	$-0.487950\pi$
$859$	2.21849	0.0756937	0.0378469	+	0.999284i	$0.487950\pi$
$860$	0	0
$861$	5.50947	0.187762
$862$	0	0
$863$	18.8604	0.642016	0.321008	−	0.947077i	$-0.395978\pi$
$863$	18.8604	0.642016	0.321008	+	0.947077i	$0.395978\pi$
$864$	0	0
$865$	−4.68913	−0.159435
$866$	0	0
$867$	11.0794	0.376276
$868$	0	0
$869$	4.78440	0.162300
$870$	0	0
$871$	−5.03275	−0.170528
$872$	0	0
$873$	−19.5043	−0.660120
$874$	0	0
$875$	−22.8533	−0.772582
$876$	0	0
$877$	−45.4062	−1.53326	−0.766629	−	0.642091i	$-0.778067\pi$
$877$	−45.4062	−1.53326	−0.766629	+	0.642091i	$0.778067\pi$
$878$	0	0
$879$	9.25475	0.312155
$880$	0	0
$881$	−11.3404	−0.382069	−0.191034	−	0.981583i	$-0.561184\pi$
$881$	−11.3404	−0.382069	−0.191034	+	0.981583i	$0.561184\pi$
$882$	0	0
$883$	2.53879	0.0854370	0.0427185	−	0.999087i	$-0.486398\pi$
$883$	2.53879	0.0854370	0.0427185	+	0.999087i	$0.486398\pi$
$884$	0	0
$885$	−13.3204	−0.447762
$886$	0	0
$887$	29.6869	0.996789	0.498395	−	0.866950i	$-0.333923\pi$
$887$	29.6869	0.996789	0.498395	+	0.866950i	$0.333923\pi$
$888$	0	0
$889$	−16.0601	−0.538640
$890$	0	0
$891$	6.17244	0.206785
$892$	0	0
$893$	38.9324	1.30282
$894$	0	0
$895$	−29.4748	−0.985233
$896$	0	0
$897$	5.55364	0.185431
$898$	0	0
$899$	−5.98973	−0.199769
$900$	0	0
$901$	−0.451835	−0.0150528
$902$	0	0
$903$	−12.0824	−0.402078
$904$	0	0
$905$	20.6918	0.687819
$906$	0	0
$907$	−42.3559	−1.40640	−0.703202	−	0.710990i	$-0.748247\pi$
$907$	−42.3559	−1.40640	−0.703202	+	0.710990i	$0.748247\pi$
$908$	0	0
$909$	−7.25948	−0.240782
$910$	0	0
$911$	−47.8348	−1.58484	−0.792420	−	0.609976i	$-0.791179\pi$
$911$	−47.8348	−1.58484	−0.792420	+	0.609976i	$0.791179\pi$
$912$	0	0
$913$	−15.7532	−0.521354
$914$	0	0
$915$	24.9464	0.824702
$916$	0	0
$917$	29.6752	0.979962
$918$	0	0
$919$	−42.2790	−1.39466	−0.697328	−	0.716752i	$-0.745628\pi$
$919$	−42.2790	−1.39466	−0.697328	+	0.716752i	$0.745628\pi$
$920$	0	0
$921$	−6.29405	−0.207396
$922$	0	0
$923$	−39.1962	−1.29016
$924$	0	0
$925$	−2.17942	−0.0716588
$926$	0	0
$927$	1.47198	0.0483460
$928$	0	0
$929$	15.7514	0.516787	0.258394	−	0.966040i	$-0.416807\pi$
$929$	15.7514	0.516787	0.258394	+	0.966040i	$0.416807\pi$
$930$	0	0
$931$	−17.4502	−0.571908
$932$	0	0
$933$	11.9390	0.390865
$934$	0	0
$935$	6.98389	0.228398
$936$	0	0
$937$	−36.7321	−1.19999	−0.599993	−	0.800005i	$-0.704830\pi$
$937$	−36.7321	−1.19999	−0.599993	+	0.800005i	$0.704830\pi$
$938$	0	0
$939$	−2.70090	−0.0881406
$940$	0	0
$941$	−18.9523	−0.617827	−0.308913	−	0.951090i	$-0.599965\pi$
$941$	−18.9523	−0.617827	−0.308913	+	0.951090i	$0.599965\pi$
$942$	0	0
$943$	7.49663	0.244124
$944$	0	0
$945$	16.7288	0.544187
$946$	0	0
$947$	0.100199	0.00325604	0.00162802	−	0.999999i	$-0.499482\pi$
$947$	0.100199	0.00325604	0.00162802	+	0.999999i	$0.499482\pi$
$948$	0	0
$949$	18.5924	0.603535
$950$	0	0
$951$	25.5476	0.828437
$952$	0	0
$953$	−40.8098	−1.32196	−0.660979	−	0.750404i	$-0.729859\pi$
$953$	−40.8098	−1.32196	−0.660979	+	0.750404i	$0.729859\pi$
$954$	0	0
$955$	28.8342	0.933053
$956$	0	0
$957$	−7.06941	−0.228521
$958$	0	0
$959$	−36.2154	−1.16946
$960$	0	0
$961$	−29.3330	−0.946227
$962$	0	0
$963$	43.8163	1.41196
$964$	0	0
$965$	6.37205	0.205123
$966$	0	0
$967$	42.7152	1.37363	0.686813	−	0.726834i	$-0.259009\pi$
$967$	42.7152	1.37363	0.686813	+	0.726834i	$0.259009\pi$
$968$	0	0
$969$	8.06220	0.258995
$970$	0	0
$971$	45.7048	1.46674	0.733368	−	0.679832i	$-0.237947\pi$
$971$	45.7048	1.46674	0.733368	+	0.679832i	$0.237947\pi$
$972$	0	0
$973$	2.96164	0.0949459
$974$	0	0
$975$	−2.58180	−0.0826836
$976$	0	0
$977$	39.8377	1.27452	0.637260	−	0.770648i	$-0.280068\pi$
$977$	39.8377	1.27452	0.637260	+	0.770648i	$0.280068\pi$
$978$	0	0
$979$	30.3853	0.971118
$980$	0	0
$981$	−2.65306	−0.0847056
$982$	0	0
$983$	−45.7498	−1.45919	−0.729596	−	0.683878i	$-0.760292\pi$
$983$	−45.7498	−1.45919	−0.729596	+	0.683878i	$0.760292\pi$
$984$	0	0
$985$	3.79456	0.120905
$986$	0	0
$987$	11.8169	0.376135
$988$	0	0
$989$	−16.4403	−0.522771
$990$	0	0
$991$	−49.0035	−1.55665	−0.778324	−	0.627863i	$-0.783930\pi$
$991$	−49.0035	−1.55665	−0.778324	+	0.627863i	$0.783930\pi$
$992$	0	0
$993$	−18.6851	−0.592954
$994$	0	0
$995$	−10.2067	−0.323573
$996$	0	0
$997$	23.0059	0.728604	0.364302	−	0.931281i	$-0.381308\pi$
$997$	23.0059	0.728604	0.364302	+	0.931281i	$0.381308\pi$
$998$	0	0
$999$	9.62174	0.304418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.g.1.14	✓	30
4.3	odd	2		8032.2.a.j.1.17	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.14	✓	30	1.1	even	1	trivial
8032.2.a.j.1.17	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-0.824754 q^{3} -2.00155 q^{5} -1.90493 q^{7} -2.31978 q^{9} +O(q^{10})\)	\(q-0.824754 q^{3} -2.00155 q^{5} -1.90493 q^{7} -2.31978 q^{9} +1.84763 q^{11} -3.14987 q^{13} +1.65078 q^{15} -1.88850 q^{17} +5.17623 q^{19} +1.57110 q^{21} +2.13777 q^{23} -0.993813 q^{25} +4.38751 q^{27} +4.63920 q^{29} -1.29111 q^{31} -1.52384 q^{33} +3.81281 q^{35} +2.19298 q^{37} +2.59787 q^{39} +3.50676 q^{41} -7.69041 q^{43} +4.64315 q^{45} +7.52139 q^{47} -3.37122 q^{49} +1.55754 q^{51} +0.239257 q^{53} -3.69812 q^{55} -4.26911 q^{57} -8.06917 q^{59} +15.1119 q^{61} +4.41903 q^{63} +6.30461 q^{65} +1.59776 q^{67} -1.76313 q^{69} +12.4438 q^{71} -5.90259 q^{73} +0.819651 q^{75} -3.51962 q^{77} +2.58948 q^{79} +3.34073 q^{81} -8.52613 q^{83} +3.77991 q^{85} -3.82620 q^{87} +16.4455 q^{89} +6.00030 q^{91} +1.06485 q^{93} -10.3605 q^{95} +8.40781 q^{97} -4.28610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * q^99

Introduction

L-functions

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