LMFDB - Embedded newform 8032.2.a.j.1.19

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+1.21978 q^{3} +2.91757 q^{5} +2.62203 q^{7} -1.51215 q^{9} +O(q^{10})$	$q+1.21978 q^{3} +2.91757 q^{5} +2.62203 q^{7} -1.51215 q^{9} -0.848556 q^{11} +3.08779 q^{13} +3.55878 q^{15} +2.86591 q^{17} +1.56214 q^{19} +3.19829 q^{21} -3.64428 q^{23} +3.51223 q^{25} -5.50381 q^{27} -5.07565 q^{29} +7.22203 q^{31} -1.03505 q^{33} +7.64996 q^{35} +3.70707 q^{37} +3.76642 q^{39} -12.6340 q^{41} +4.68466 q^{43} -4.41180 q^{45} +13.0113 q^{47} -0.124958 q^{49} +3.49577 q^{51} +11.3693 q^{53} -2.47572 q^{55} +1.90546 q^{57} +1.66370 q^{59} +5.08991 q^{61} -3.96490 q^{63} +9.00886 q^{65} -9.12908 q^{67} -4.44520 q^{69} +8.23999 q^{71} +10.6448 q^{73} +4.28413 q^{75} -2.22494 q^{77} +5.39490 q^{79} -2.17696 q^{81} -1.93476 q^{83} +8.36151 q^{85} -6.19115 q^{87} +4.39605 q^{89} +8.09629 q^{91} +8.80926 q^{93} +4.55765 q^{95} -10.1659 q^{97} +1.28314 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.21978	0.704238	0.352119	−	0.935955i	$-0.385461\pi$
$3$	1.21978	0.704238	0.352119	+	0.935955i	$0.385461\pi$
$4$	0	0
$5$	2.91757	1.30478	0.652389	−	0.757884i	$-0.273767\pi$
$5$	2.91757	1.30478	0.652389	+	0.757884i	$0.273767\pi$
$6$	0	0
$7$	2.62203	0.991034	0.495517	−	0.868598i	$-0.334979\pi$
$7$	2.62203	0.991034	0.495517	+	0.868598i	$0.334979\pi$
$8$	0	0
$9$	−1.51215	−0.504049
$10$	0	0
$11$	−0.848556	−0.255849	−0.127925	−	0.991784i	$-0.540832\pi$
$11$	−0.848556	−0.255849	−0.127925	+	0.991784i	$0.540832\pi$
$12$	0	0
$13$	3.08779	0.856400	0.428200	−	0.903684i	$-0.359148\pi$
$13$	3.08779	0.856400	0.428200	+	0.903684i	$0.359148\pi$
$14$	0	0
$15$	3.55878	0.918874
$16$	0	0
$17$	2.86591	0.695086	0.347543	−	0.937664i	$-0.387016\pi$
$17$	2.86591	0.695086	0.347543	+	0.937664i	$0.387016\pi$
$18$	0	0
$19$	1.56214	0.358379	0.179189	−	0.983815i	$-0.442652\pi$
$19$	1.56214	0.358379	0.179189	+	0.983815i	$0.442652\pi$
$20$	0	0
$21$	3.19829	0.697924
$22$	0	0
$23$	−3.64428	−0.759884	−0.379942	−	0.925010i	$-0.624056\pi$
$23$	−3.64428	−0.759884	−0.379942	+	0.925010i	$0.624056\pi$
$24$	0	0
$25$	3.51223	0.702446
$26$	0	0
$27$	−5.50381	−1.05921
$28$	0	0
$29$	−5.07565	−0.942525	−0.471262	−	0.881993i	$-0.656201\pi$
$29$	−5.07565	−0.942525	−0.471262	+	0.881993i	$0.656201\pi$
$30$	0	0
$31$	7.22203	1.29712	0.648558	−	0.761166i	$-0.275373\pi$
$31$	7.22203	1.29712	0.648558	+	0.761166i	$0.275373\pi$
$32$	0	0
$33$	−1.03505	−0.180179
$34$	0	0
$35$	7.64996	1.29308
$36$	0	0
$37$	3.70707	0.609439	0.304719	−	0.952442i	$-0.401437\pi$
$37$	3.70707	0.609439	0.304719	+	0.952442i	$0.401437\pi$
$38$	0	0
$39$	3.76642	0.603109
$40$	0	0
$41$	−12.6340	−1.97310	−0.986549	−	0.163464i	$-0.947733\pi$
$41$	−12.6340	−1.97310	−0.986549	+	0.163464i	$0.947733\pi$
$42$	0	0
$43$	4.68466	0.714405	0.357202	−	0.934027i	$-0.383731\pi$
$43$	4.68466	0.714405	0.357202	+	0.934027i	$0.383731\pi$
$44$	0	0
$45$	−4.41180	−0.657673
$46$	0	0
$47$	13.0113	1.89790	0.948948	−	0.315431i	$-0.102149\pi$
$47$	13.0113	1.89790	0.948948	+	0.315431i	$0.102149\pi$
$48$	0	0
$49$	−0.124958	−0.0178512
$50$	0	0
$51$	3.49577	0.489506
$52$	0	0
$53$	11.3693	1.56169	0.780847	−	0.624722i	$-0.214788\pi$
$53$	11.3693	1.56169	0.780847	+	0.624722i	$0.214788\pi$
$54$	0	0
$55$	−2.47572	−0.333827
$56$	0	0
$57$	1.90546	0.252384
$58$	0	0
$59$	1.66370	0.216595	0.108298	−	0.994119i	$-0.465460\pi$
$59$	1.66370	0.216595	0.108298	+	0.994119i	$0.465460\pi$
$60$	0	0
$61$	5.08991	0.651697	0.325848	−	0.945422i	$-0.394350\pi$
$61$	5.08991	0.651697	0.325848	+	0.945422i	$0.394350\pi$
$62$	0	0
$63$	−3.96490	−0.499530
$64$	0	0
$65$	9.00886	1.11741
$66$	0	0
$67$	−9.12908	−1.11529	−0.557647	−	0.830078i	$-0.688296\pi$
$67$	−9.12908	−1.11529	−0.557647	+	0.830078i	$0.688296\pi$
$68$	0	0
$69$	−4.44520	−0.535139
$70$	0	0
$71$	8.23999	0.977907	0.488954	−	0.872310i	$-0.337379\pi$
$71$	8.23999	0.977907	0.488954	+	0.872310i	$0.337379\pi$
$72$	0	0
$73$	10.6448	1.24588	0.622938	−	0.782272i	$-0.285939\pi$
$73$	10.6448	1.24588	0.622938	+	0.782272i	$0.285939\pi$
$74$	0	0
$75$	4.28413	0.494689
$76$	0	0
$77$	−2.22494	−0.253555
$78$	0	0
$79$	5.39490	0.606974	0.303487	−	0.952836i	$-0.401849\pi$
$79$	5.39490	0.606974	0.303487	+	0.952836i	$0.401849\pi$
$80$	0	0
$81$	−2.17696	−0.241885
$82$	0	0
$83$	−1.93476	−0.212368	−0.106184	−	0.994347i	$-0.533863\pi$
$83$	−1.93476	−0.212368	−0.106184	+	0.994347i	$0.533863\pi$
$84$	0	0
$85$	8.36151	0.906933
$86$	0	0
$87$	−6.19115	−0.663761
$88$	0	0
$89$	4.39605	0.465980	0.232990	−	0.972479i	$-0.425149\pi$
$89$	4.39605	0.465980	0.232990	+	0.972479i	$0.425149\pi$
$90$	0	0
$91$	8.09629	0.848722
$92$	0	0
$93$	8.80926	0.913477
$94$	0	0
$95$	4.55765	0.467605
$96$	0	0
$97$	−10.1659	−1.03219	−0.516097	−	0.856530i	$-0.672616\pi$
$97$	−10.1659	−1.03219	−0.516097	+	0.856530i	$0.672616\pi$
$98$	0	0
$99$	1.28314	0.128961
$100$	0	0
$101$	5.42073	0.539383	0.269691	−	0.962947i	$-0.413078\pi$
$101$	5.42073	0.539383	0.269691	+	0.962947i	$0.413078\pi$
$102$	0	0
$103$	−17.6032	−1.73449	−0.867245	−	0.497881i	$-0.834112\pi$
$103$	−17.6032	−1.73449	−0.867245	+	0.497881i	$0.834112\pi$
$104$	0	0
$105$	9.33124	0.910635
$106$	0	0
$107$	4.99018	0.482419	0.241209	−	0.970473i	$-0.422456\pi$
$107$	4.99018	0.482419	0.241209	+	0.970473i	$0.422456\pi$
$108$	0	0
$109$	−13.3045	−1.27434	−0.637172	−	0.770721i	$-0.719896\pi$
$109$	−13.3045	−1.27434	−0.637172	+	0.770721i	$0.719896\pi$
$110$	0	0
$111$	4.52180	0.429190
$112$	0	0
$113$	14.8974	1.40143	0.700714	−	0.713442i	$-0.252865\pi$
$113$	14.8974	1.40143	0.700714	+	0.713442i	$0.252865\pi$
$114$	0	0
$115$	−10.6324	−0.991480
$116$	0	0
$117$	−4.66920	−0.431668
$118$	0	0
$119$	7.51451	0.688854
$120$	0	0
$121$	−10.2800	−0.934541
$122$	0	0
$123$	−15.4106	−1.38953
$124$	0	0
$125$	−4.34068	−0.388242
$126$	0	0
$127$	8.37776	0.743406	0.371703	−	0.928352i	$-0.378774\pi$
$127$	8.37776	0.743406	0.371703	+	0.928352i	$0.378774\pi$
$128$	0	0
$129$	5.71424	0.503111
$130$	0	0
$131$	7.62550	0.666243	0.333122	−	0.942884i	$-0.391898\pi$
$131$	7.62550	0.666243	0.333122	+	0.942884i	$0.391898\pi$
$132$	0	0
$133$	4.09597	0.355166
$134$	0	0
$135$	−16.0578	−1.38203
$136$	0	0
$137$	10.0077	0.855019	0.427510	−	0.904011i	$-0.359391\pi$
$137$	10.0077	0.855019	0.427510	+	0.904011i	$0.359391\pi$
$138$	0	0
$139$	−15.3473	−1.30174	−0.650872	−	0.759187i	$-0.725597\pi$
$139$	−15.3473	−1.30174	−0.650872	+	0.759187i	$0.725597\pi$
$140$	0	0
$141$	15.8709	1.33657
$142$	0	0
$143$	−2.62017	−0.219109
$144$	0	0
$145$	−14.8086	−1.22979
$146$	0	0
$147$	−0.152421	−0.0125715
$148$	0	0
$149$	13.3360	1.09253	0.546265	−	0.837613i	$-0.316049\pi$
$149$	13.3360	1.09253	0.546265	+	0.837613i	$0.316049\pi$
$150$	0	0
$151$	18.1884	1.48015	0.740077	−	0.672522i	$-0.234789\pi$
$151$	18.1884	1.48015	0.740077	+	0.672522i	$0.234789\pi$
$152$	0	0
$153$	−4.33368	−0.350358
$154$	0	0
$155$	21.0708	1.69245
$156$	0	0
$157$	−4.83972	−0.386252	−0.193126	−	0.981174i	$-0.561863\pi$
$157$	−4.83972	−0.386252	−0.193126	+	0.981174i	$0.561863\pi$
$158$	0	0
$159$	13.8680	1.09980
$160$	0	0
$161$	−9.55540	−0.753071
$162$	0	0
$163$	17.9308	1.40445	0.702223	−	0.711957i	$-0.252191\pi$
$163$	17.9308	1.40445	0.702223	+	0.711957i	$0.252191\pi$
$164$	0	0
$165$	−3.01983	−0.235093
$166$	0	0
$167$	17.4135	1.34749	0.673747	−	0.738962i	$-0.264684\pi$
$167$	17.4135	1.34749	0.673747	+	0.738962i	$0.264684\pi$
$168$	0	0
$169$	−3.46553	−0.266579
$170$	0	0
$171$	−2.36218	−0.180641
$172$	0	0
$173$	−7.97419	−0.606267	−0.303133	−	0.952948i	$-0.598033\pi$
$173$	−7.97419	−0.606267	−0.303133	+	0.952948i	$0.598033\pi$
$174$	0	0
$175$	9.20917	0.696148
$176$	0	0
$177$	2.02934	0.152535
$178$	0	0
$179$	3.07336	0.229714	0.114857	−	0.993382i	$-0.463359\pi$
$179$	3.07336	0.229714	0.114857	+	0.993382i	$0.463359\pi$
$180$	0	0
$181$	−16.0579	−1.19358	−0.596788	−	0.802399i	$-0.703556\pi$
$181$	−16.0579	−1.19358	−0.596788	+	0.802399i	$0.703556\pi$
$182$	0	0
$183$	6.20855	0.458949
$184$	0	0
$185$	10.8157	0.795183
$186$	0	0
$187$	−2.43189	−0.177837
$188$	0	0
$189$	−14.4311	−1.04971
$190$	0	0
$191$	12.5494	0.908041	0.454020	−	0.890991i	$-0.349989\pi$
$191$	12.5494	0.908041	0.454020	+	0.890991i	$0.349989\pi$
$192$	0	0
$193$	−20.5115	−1.47645	−0.738224	−	0.674555i	$-0.764335\pi$
$193$	−20.5115	−1.47645	−0.738224	+	0.674555i	$0.764335\pi$
$194$	0	0
$195$	10.9888	0.786924
$196$	0	0
$197$	−9.09868	−0.648254	−0.324127	−	0.946014i	$-0.605071\pi$
$197$	−9.09868	−0.648254	−0.324127	+	0.946014i	$0.605071\pi$
$198$	0	0
$199$	−13.5615	−0.961349	−0.480674	−	0.876899i	$-0.659608\pi$
$199$	−13.5615	−0.961349	−0.480674	+	0.876899i	$0.659608\pi$
$200$	0	0
$201$	−11.1354	−0.785433
$202$	0	0
$203$	−13.3085	−0.934074
$204$	0	0
$205$	−36.8606	−2.57446
$206$	0	0
$207$	5.51069	0.383019
$208$	0	0
$209$	−1.32556	−0.0916910
$210$	0	0
$211$	−2.45107	−0.168738	−0.0843692	−	0.996435i	$-0.526888\pi$
$211$	−2.45107	−0.168738	−0.0843692	+	0.996435i	$0.526888\pi$
$212$	0	0
$213$	10.0509	0.688679
$214$	0	0
$215$	13.6678	0.932140
$216$	0	0
$217$	18.9364	1.28549
$218$	0	0
$219$	12.9842	0.877392
$220$	0	0
$221$	8.84935	0.595271
$222$	0	0
$223$	−9.48897	−0.635429	−0.317714	−	0.948187i	$-0.602915\pi$
$223$	−9.48897	−0.635429	−0.317714	+	0.948187i	$0.602915\pi$
$224$	0	0
$225$	−5.31101	−0.354067
$226$	0	0
$227$	−15.1614	−1.00630	−0.503148	−	0.864200i	$-0.667825\pi$
$227$	−15.1614	−1.00630	−0.503148	+	0.864200i	$0.667825\pi$
$228$	0	0
$229$	21.1808	1.39966	0.699832	−	0.714307i	$-0.253258\pi$
$229$	21.1808	1.39966	0.699832	+	0.714307i	$0.253258\pi$
$230$	0	0
$231$	−2.71393	−0.178563
$232$	0	0
$233$	13.5389	0.886963	0.443482	−	0.896283i	$-0.353743\pi$
$233$	13.5389	0.886963	0.443482	+	0.896283i	$0.353743\pi$
$234$	0	0
$235$	37.9615	2.47633
$236$	0	0
$237$	6.58057	0.427454
$238$	0	0
$239$	−1.45540	−0.0941422	−0.0470711	−	0.998892i	$-0.514989\pi$
$239$	−1.45540	−0.0941422	−0.0470711	+	0.998892i	$0.514989\pi$
$240$	0	0
$241$	6.32583	0.407483	0.203741	−	0.979025i	$-0.434690\pi$
$241$	6.32583	0.407483	0.203741	+	0.979025i	$0.434690\pi$
$242$	0	0
$243$	13.8560	0.888864
$244$	0	0
$245$	−0.364575	−0.0232918
$246$	0	0
$247$	4.82356	0.306916
$248$	0	0
$249$	−2.35997	−0.149557
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	3.09237	0.194416
$254$	0	0
$255$	10.1992	0.638696
$256$	0	0
$257$	−5.98563	−0.373373	−0.186687	−	0.982420i	$-0.559775\pi$
$257$	−5.98563	−0.373373	−0.186687	+	0.982420i	$0.559775\pi$
$258$	0	0
$259$	9.72006	0.603975
$260$	0	0
$261$	7.67514	0.475079
$262$	0	0
$263$	−23.7162	−1.46241	−0.731203	−	0.682160i	$-0.761041\pi$
$263$	−23.7162	−1.46241	−0.731203	+	0.682160i	$0.761041\pi$
$264$	0	0
$265$	33.1708	2.03766
$266$	0	0
$267$	5.36219	0.328161
$268$	0	0
$269$	−11.5911	−0.706722	−0.353361	−	0.935487i	$-0.614961\pi$
$269$	−11.5911	−0.706722	−0.353361	+	0.935487i	$0.614961\pi$
$270$	0	0
$271$	10.9315	0.664041	0.332020	−	0.943272i	$-0.392270\pi$
$271$	10.9315	0.664041	0.332020	+	0.943272i	$0.392270\pi$
$272$	0	0
$273$	9.87565	0.597702
$274$	0	0
$275$	−2.98032	−0.179720
$276$	0	0
$277$	−7.08407	−0.425641	−0.212820	−	0.977091i	$-0.568265\pi$
$277$	−7.08407	−0.425641	−0.212820	+	0.977091i	$0.568265\pi$
$278$	0	0
$279$	−10.9208	−0.653810
$280$	0	0
$281$	0.409406	0.0244231	0.0122116	−	0.999925i	$-0.496113\pi$
$281$	0.409406	0.0244231	0.0122116	+	0.999925i	$0.496113\pi$
$282$	0	0
$283$	30.8407	1.83329	0.916644	−	0.399704i	$-0.130887\pi$
$283$	30.8407	1.83329	0.916644	+	0.399704i	$0.130887\pi$
$284$	0	0
$285$	5.55931	0.329305
$286$	0	0
$287$	−33.1267	−1.95541
$288$	0	0
$289$	−8.78655	−0.516856
$290$	0	0
$291$	−12.4002	−0.726910
$292$	0	0
$293$	−10.9141	−0.637609	−0.318805	−	0.947820i	$-0.603281\pi$
$293$	−10.9141	−0.637609	−0.318805	+	0.947820i	$0.603281\pi$
$294$	0	0
$295$	4.85397	0.282609
$296$	0	0
$297$	4.67029	0.270998
$298$	0	0
$299$	−11.2528	−0.650765
$300$	0	0
$301$	12.2833	0.707999
$302$	0	0
$303$	6.61207	0.379854
$304$	0	0
$305$	14.8502	0.850320
$306$	0	0
$307$	−26.6921	−1.52340	−0.761698	−	0.647932i	$-0.775634\pi$
$307$	−26.6921	−1.52340	−0.761698	+	0.647932i	$0.775634\pi$
$308$	0	0
$309$	−21.4719	−1.22149
$310$	0	0
$311$	11.1777	0.633832	0.316916	−	0.948454i	$-0.397353\pi$
$311$	11.1777	0.633832	0.316916	+	0.948454i	$0.397353\pi$
$312$	0	0
$313$	−34.0426	−1.92420	−0.962101	−	0.272692i	$-0.912086\pi$
$313$	−34.0426	−1.92420	−0.962101	+	0.272692i	$0.912086\pi$
$314$	0	0
$315$	−11.5679	−0.651776
$316$	0	0
$317$	5.41404	0.304083	0.152041	−	0.988374i	$-0.451415\pi$
$317$	5.41404	0.304083	0.152041	+	0.988374i	$0.451415\pi$
$318$	0	0
$319$	4.30697	0.241144
$320$	0	0
$321$	6.08689	0.339737
$322$	0	0
$323$	4.47695	0.249104
$324$	0	0
$325$	10.8450	0.601575
$326$	0	0
$327$	−16.2286	−0.897441
$328$	0	0
$329$	34.1161	1.88088
$330$	0	0
$331$	20.6887	1.13715	0.568576	−	0.822631i	$-0.307494\pi$
$331$	20.6887	1.13715	0.568576	+	0.822631i	$0.307494\pi$
$332$	0	0
$333$	−5.60564	−0.307187
$334$	0	0
$335$	−26.6348	−1.45521
$336$	0	0
$337$	22.9656	1.25101	0.625507	−	0.780218i	$-0.284892\pi$
$337$	22.9656	1.25101	0.625507	+	0.780218i	$0.284892\pi$
$338$	0	0
$339$	18.1715	0.986938
$340$	0	0
$341$	−6.12830	−0.331866
$342$	0	0
$343$	−18.6819	−1.00873
$344$	0	0
$345$	−12.9692	−0.698238
$346$	0	0
$347$	2.79436	0.150009	0.0750046	−	0.997183i	$-0.476103\pi$
$347$	2.79436	0.150009	0.0750046	+	0.997183i	$0.476103\pi$
$348$	0	0
$349$	−13.8025	−0.738830	−0.369415	−	0.929264i	$-0.620442\pi$
$349$	−13.8025	−0.738830	−0.369415	+	0.929264i	$0.620442\pi$
$350$	0	0
$351$	−16.9946	−0.907106
$352$	0	0
$353$	−8.67729	−0.461846	−0.230923	−	0.972972i	$-0.574174\pi$
$353$	−8.67729	−0.461846	−0.230923	+	0.972972i	$0.574174\pi$
$354$	0	0
$355$	24.0408	1.27595
$356$	0	0
$357$	9.16601	0.485117
$358$	0	0
$359$	12.6902	0.669764	0.334882	−	0.942260i	$-0.391303\pi$
$359$	12.6902	0.669764	0.334882	+	0.942260i	$0.391303\pi$
$360$	0	0
$361$	−16.5597	−0.871565
$362$	0	0
$363$	−12.5392	−0.658139
$364$	0	0
$365$	31.0569	1.62559
$366$	0	0
$367$	−14.5537	−0.759696	−0.379848	−	0.925049i	$-0.624024\pi$
$367$	−14.5537	−0.759696	−0.379848	+	0.925049i	$0.624024\pi$
$368$	0	0
$369$	19.1045	0.994539
$370$	0	0
$371$	29.8107	1.54769
$372$	0	0
$373$	−21.4985	−1.11315	−0.556575	−	0.830797i	$-0.687885\pi$
$373$	−21.4985	−1.11315	−0.556575	+	0.830797i	$0.687885\pi$
$374$	0	0
$375$	−5.29465	−0.273415
$376$	0	0
$377$	−15.6726	−0.807178
$378$	0	0
$379$	−0.935896	−0.0480738	−0.0240369	−	0.999711i	$-0.507652\pi$
$379$	−0.935896	−0.0480738	−0.0240369	+	0.999711i	$0.507652\pi$
$380$	0	0
$381$	10.2190	0.523535
$382$	0	0
$383$	4.26819	0.218094	0.109047	−	0.994037i	$-0.465220\pi$
$383$	4.26819	0.218094	0.109047	+	0.994037i	$0.465220\pi$
$384$	0	0
$385$	−6.49142	−0.330833
$386$	0	0
$387$	−7.08391	−0.360095
$388$	0	0
$389$	−26.1354	−1.32512	−0.662560	−	0.749009i	$-0.730530\pi$
$389$	−26.1354	−1.32512	−0.662560	+	0.749009i	$0.730530\pi$
$390$	0	0
$391$	−10.4442	−0.528185
$392$	0	0
$393$	9.30140	0.469193
$394$	0	0
$395$	15.7400	0.791966
$396$	0	0
$397$	−2.78503	−0.139777	−0.0698884	−	0.997555i	$-0.522264\pi$
$397$	−2.78503	−0.139777	−0.0698884	+	0.997555i	$0.522264\pi$
$398$	0	0
$399$	4.99617	0.250121
$400$	0	0
$401$	−14.1172	−0.704979	−0.352490	−	0.935816i	$-0.614665\pi$
$401$	−14.1172	−0.704979	−0.352490	+	0.935816i	$0.614665\pi$
$402$	0	0
$403$	22.3001	1.11085
$404$	0	0
$405$	−6.35145	−0.315606
$406$	0	0
$407$	−3.14566	−0.155925
$408$	0	0
$409$	−12.8988	−0.637805	−0.318902	−	0.947788i	$-0.603314\pi$
$409$	−12.8988	−0.637805	−0.318902	+	0.947788i	$0.603314\pi$
$410$	0	0
$411$	12.2072	0.602137
$412$	0	0
$413$	4.36227	0.214653
$414$	0	0
$415$	−5.64480	−0.277093
$416$	0	0
$417$	−18.7203	−0.916738
$418$	0	0
$419$	17.5317	0.856479	0.428240	−	0.903665i	$-0.359134\pi$
$419$	17.5317	0.856479	0.428240	+	0.903665i	$0.359134\pi$
$420$	0	0
$421$	−40.0824	−1.95350	−0.976748	−	0.214392i	$-0.931223\pi$
$421$	−40.0824	−1.95350	−0.976748	+	0.214392i	$0.931223\pi$
$422$	0	0
$423$	−19.6751	−0.956634
$424$	0	0
$425$	10.0657	0.488260
$426$	0	0
$427$	13.3459	0.645854
$428$	0	0
$429$	−3.19601	−0.154305
$430$	0	0
$431$	−31.4802	−1.51635	−0.758174	−	0.652053i	$-0.773908\pi$
$431$	−31.4802	−1.51635	−0.758174	+	0.652053i	$0.773908\pi$
$432$	0	0
$433$	−32.8773	−1.57998	−0.789992	−	0.613118i	$-0.789915\pi$
$433$	−32.8773	−1.57998	−0.789992	+	0.613118i	$0.789915\pi$
$434$	0	0
$435$	−18.0631	−0.866061
$436$	0	0
$437$	−5.69286	−0.272326
$438$	0	0
$439$	−23.9456	−1.14286	−0.571430	−	0.820651i	$-0.693611\pi$
$439$	−23.9456	−1.14286	−0.571430	+	0.820651i	$0.693611\pi$
$440$	0	0
$441$	0.188955	0.00899787
$442$	0	0
$443$	29.5225	1.40266	0.701329	−	0.712838i	$-0.252591\pi$
$443$	29.5225	1.40266	0.701329	+	0.712838i	$0.252591\pi$
$444$	0	0
$445$	12.8258	0.608001
$446$	0	0
$447$	16.2670	0.769400
$448$	0	0
$449$	16.3127	0.769844	0.384922	−	0.922949i	$-0.374228\pi$
$449$	16.3127	0.769844	0.384922	+	0.922949i	$0.374228\pi$
$450$	0	0
$451$	10.7207	0.504816
$452$	0	0
$453$	22.1858	1.04238
$454$	0	0
$455$	23.6215	1.10739
$456$	0	0
$457$	−8.97240	−0.419711	−0.209856	−	0.977732i	$-0.567299\pi$
$457$	−8.97240	−0.419711	−0.209856	+	0.977732i	$0.567299\pi$
$458$	0	0
$459$	−15.7734	−0.736240
$460$	0	0
$461$	−37.5557	−1.74914	−0.874571	−	0.484898i	$-0.838857\pi$
$461$	−37.5557	−1.74914	−0.874571	+	0.484898i	$0.838857\pi$
$462$	0	0
$463$	39.1052	1.81737	0.908687	−	0.417478i	$-0.137086\pi$
$463$	39.1052	1.81737	0.908687	+	0.417478i	$0.137086\pi$
$464$	0	0
$465$	25.7016	1.19189
$466$	0	0
$467$	5.39418	0.249613	0.124806	−	0.992181i	$-0.460169\pi$
$467$	5.39418	0.249613	0.124806	+	0.992181i	$0.460169\pi$
$468$	0	0
$469$	−23.9367	−1.10530
$470$	0	0
$471$	−5.90337	−0.272013
$472$	0	0
$473$	−3.97520	−0.182780
$474$	0	0
$475$	5.48659	0.251742
$476$	0	0
$477$	−17.1921	−0.787171
$478$	0	0
$479$	−40.5879	−1.85451	−0.927254	−	0.374432i	$-0.877838\pi$
$479$	−40.5879	−1.85451	−0.927254	+	0.374432i	$0.877838\pi$
$480$	0	0
$481$	11.4467	0.521924
$482$	0	0
$483$	−11.6554	−0.530341
$484$	0	0
$485$	−29.6599	−1.34678
$486$	0	0
$487$	−20.5232	−0.929996	−0.464998	−	0.885312i	$-0.653945\pi$
$487$	−20.5232	−0.929996	−0.464998	+	0.885312i	$0.653945\pi$
$488$	0	0
$489$	21.8715	0.989064
$490$	0	0
$491$	−25.1872	−1.13668	−0.568342	−	0.822792i	$-0.692415\pi$
$491$	−25.1872	−1.13668	−0.568342	+	0.822792i	$0.692415\pi$
$492$	0	0
$493$	−14.5464	−0.655135
$494$	0	0
$495$	3.74366	0.168265
$496$	0	0
$497$	21.6055	0.969139
$498$	0	0
$499$	35.9379	1.60880	0.804401	−	0.594087i	$-0.202486\pi$
$499$	35.9379	1.60880	0.804401	+	0.594087i	$0.202486\pi$
$500$	0	0
$501$	21.2405	0.948956
$502$	0	0
$503$	32.8952	1.46672	0.733362	−	0.679838i	$-0.237950\pi$
$503$	32.8952	1.46672	0.733362	+	0.679838i	$0.237950\pi$
$504$	0	0
$505$	15.8154	0.703775
$506$	0	0
$507$	−4.22716	−0.187735
$508$	0	0
$509$	26.8175	1.18866	0.594331	−	0.804220i	$-0.297417\pi$
$509$	26.8175	1.18866	0.594331	+	0.804220i	$0.297417\pi$
$510$	0	0
$511$	27.9109	1.23470
$512$	0	0
$513$	−8.59770	−0.379598
$514$	0	0
$515$	−51.3585	−2.26313
$516$	0	0
$517$	−11.0408	−0.485575
$518$	0	0
$519$	−9.72672	−0.426956
$520$	0	0
$521$	9.86172	0.432050	0.216025	−	0.976388i	$-0.430691\pi$
$521$	9.86172	0.432050	0.216025	+	0.976388i	$0.430691\pi$
$522$	0	0
$523$	−23.7997	−1.04069	−0.520345	−	0.853956i	$-0.674197\pi$
$523$	−23.7997	−1.04069	−0.520345	+	0.853956i	$0.674197\pi$
$524$	0	0
$525$	11.2331	0.490254
$526$	0	0
$527$	20.6977	0.901606
$528$	0	0
$529$	−9.71925	−0.422576
$530$	0	0
$531$	−2.51576	−0.109175
$532$	0	0
$533$	−39.0112	−1.68976
$534$	0	0
$535$	14.5592	0.629449
$536$	0	0
$537$	3.74881	0.161773
$538$	0	0
$539$	0.106034	0.00456721
$540$	0	0
$541$	37.0695	1.59374	0.796871	−	0.604149i	$-0.206487\pi$
$541$	37.0695	1.59374	0.796871	+	0.604149i	$0.206487\pi$
$542$	0	0
$543$	−19.5870	−0.840560
$544$	0	0
$545$	−38.8170	−1.66274
$546$	0	0
$547$	−39.5633	−1.69160	−0.845802	−	0.533496i	$-0.820878\pi$
$547$	−39.5633	−1.69160	−0.845802	+	0.533496i	$0.820878\pi$
$548$	0	0
$549$	−7.69670	−0.328487
$550$	0	0
$551$	−7.92886	−0.337781
$552$	0	0
$553$	14.1456	0.601532
$554$	0	0
$555$	13.1927	0.559998
$556$	0	0
$557$	−25.9243	−1.09845	−0.549224	−	0.835675i	$-0.685077\pi$
$557$	−25.9243	−1.09845	−0.549224	+	0.835675i	$0.685077\pi$
$558$	0	0
$559$	14.4653	0.611816
$560$	0	0
$561$	−2.96636	−0.125240
$562$	0	0
$563$	19.4559	0.819969	0.409984	−	0.912092i	$-0.365534\pi$
$563$	19.4559	0.819969	0.409984	+	0.912092i	$0.365534\pi$
$564$	0	0
$565$	43.4642	1.82855
$566$	0	0
$567$	−5.70806	−0.239716
$568$	0	0
$569$	17.1198	0.717700	0.358850	−	0.933395i	$-0.383169\pi$
$569$	17.1198	0.717700	0.358850	+	0.933395i	$0.383169\pi$
$570$	0	0
$571$	5.36580	0.224552	0.112276	−	0.993677i	$-0.464186\pi$
$571$	5.36580	0.224552	0.112276	+	0.993677i	$0.464186\pi$
$572$	0	0
$573$	15.3074	0.639477
$574$	0	0
$575$	−12.7995	−0.533778
$576$	0	0
$577$	−28.8690	−1.20183	−0.600917	−	0.799312i	$-0.705198\pi$
$577$	−28.8690	−1.20183	−0.600917	+	0.799312i	$0.705198\pi$
$578$	0	0
$579$	−25.0194	−1.03977
$580$	0	0
$581$	−5.07300	−0.210464
$582$	0	0
$583$	−9.64749	−0.399558
$584$	0	0
$585$	−13.6227	−0.563231
$586$	0	0
$587$	2.36604	0.0976567	0.0488284	−	0.998807i	$-0.484451\pi$
$587$	2.36604	0.0976567	0.0488284	+	0.998807i	$0.484451\pi$
$588$	0	0
$589$	11.2818	0.464859
$590$	0	0
$591$	−11.0983	−0.456525
$592$	0	0
$593$	−5.91296	−0.242816	−0.121408	−	0.992603i	$-0.538741\pi$
$593$	−5.91296	−0.242816	−0.121408	+	0.992603i	$0.538741\pi$
$594$	0	0
$595$	21.9241	0.898801
$596$	0	0
$597$	−16.5420	−0.677018
$598$	0	0
$599$	18.4724	0.754762	0.377381	−	0.926058i	$-0.376825\pi$
$599$	18.4724	0.754762	0.377381	+	0.926058i	$0.376825\pi$
$600$	0	0
$601$	32.3626	1.32010	0.660049	−	0.751223i	$-0.270536\pi$
$601$	32.3626	1.32010	0.660049	+	0.751223i	$0.270536\pi$
$602$	0	0
$603$	13.8045	0.562164
$604$	0	0
$605$	−29.9925	−1.21937
$606$	0	0
$607$	26.9946	1.09568	0.547839	−	0.836584i	$-0.315451\pi$
$607$	26.9946	1.09568	0.547839	+	0.836584i	$0.315451\pi$
$608$	0	0
$609$	−16.2334	−0.657810
$610$	0	0
$611$	40.1763	1.62536
$612$	0	0
$613$	−18.9353	−0.764789	−0.382395	−	0.923999i	$-0.624901\pi$
$613$	−18.9353	−0.764789	−0.382395	+	0.923999i	$0.624901\pi$
$614$	0	0
$615$	−44.9617	−1.81303
$616$	0	0
$617$	5.84746	0.235410	0.117705	−	0.993049i	$-0.462446\pi$
$617$	5.84746	0.235410	0.117705	+	0.993049i	$0.462446\pi$
$618$	0	0
$619$	34.5318	1.38795	0.693976	−	0.719998i	$-0.255857\pi$
$619$	34.5318	1.38795	0.693976	+	0.719998i	$0.255857\pi$
$620$	0	0
$621$	20.0574	0.804875
$622$	0	0
$623$	11.5266	0.461802
$624$	0	0
$625$	−30.2254	−1.20902
$626$	0	0
$627$	−1.61689	−0.0645722
$628$	0	0
$629$	10.6241	0.423612
$630$	0	0
$631$	0.401992	0.0160030	0.00800152	−	0.999968i	$-0.497453\pi$
$631$	0.401992	0.0160030	0.00800152	+	0.999968i	$0.497453\pi$
$632$	0	0
$633$	−2.98975	−0.118832
$634$	0	0
$635$	24.4427	0.969980
$636$	0	0
$637$	−0.385845	−0.0152877
$638$	0	0
$639$	−12.4601	−0.492913
$640$	0	0
$641$	−11.5139	−0.454773	−0.227387	−	0.973805i	$-0.573018\pi$
$641$	−11.5139	−0.454773	−0.227387	+	0.973805i	$0.573018\pi$
$642$	0	0
$643$	32.6139	1.28617	0.643084	−	0.765796i	$-0.277655\pi$
$643$	32.6139	1.28617	0.643084	+	0.765796i	$0.277655\pi$
$644$	0	0
$645$	16.6717	0.656448
$646$	0	0
$647$	−22.1974	−0.872671	−0.436335	−	0.899784i	$-0.643724\pi$
$647$	−22.1974	−0.872671	−0.436335	+	0.899784i	$0.643724\pi$
$648$	0	0
$649$	−1.41174	−0.0554157
$650$	0	0
$651$	23.0981	0.905287
$652$	0	0
$653$	17.2601	0.675441	0.337720	−	0.941246i	$-0.390344\pi$
$653$	17.2601	0.675441	0.337720	+	0.941246i	$0.390344\pi$
$654$	0	0
$655$	22.2480	0.869299
$656$	0	0
$657$	−16.0965	−0.627983
$658$	0	0
$659$	1.20019	0.0467526	0.0233763	−	0.999727i	$-0.492558\pi$
$659$	1.20019	0.0467526	0.0233763	+	0.999727i	$0.492558\pi$
$660$	0	0
$661$	15.5852	0.606194	0.303097	−	0.952960i	$-0.401979\pi$
$661$	15.5852	0.606194	0.303097	+	0.952960i	$0.401979\pi$
$662$	0	0
$663$	10.7942	0.419213
$664$	0	0
$665$	11.9503	0.463413
$666$	0	0
$667$	18.4971	0.716210
$668$	0	0
$669$	−11.5744	−0.447493
$670$	0	0
$671$	−4.31908	−0.166736
$672$	0	0
$673$	38.5133	1.48458	0.742290	−	0.670079i	$-0.233740\pi$
$673$	38.5133	1.48458	0.742290	+	0.670079i	$0.233740\pi$
$674$	0	0
$675$	−19.3306	−0.744037
$676$	0	0
$677$	3.72315	0.143092	0.0715462	−	0.997437i	$-0.477207\pi$
$677$	3.72315	0.143092	0.0715462	+	0.997437i	$0.477207\pi$
$678$	0	0
$679$	−26.6554	−1.02294
$680$	0	0
$681$	−18.4935	−0.708671
$682$	0	0
$683$	37.0258	1.41675	0.708377	−	0.705834i	$-0.249428\pi$
$683$	37.0258	1.41675	0.708377	+	0.705834i	$0.249428\pi$
$684$	0	0
$685$	29.1983	1.11561
$686$	0	0
$687$	25.8358	0.985697
$688$	0	0
$689$	35.1061	1.33744
$690$	0	0
$691$	−5.98996	−0.227869	−0.113934	−	0.993488i	$-0.536345\pi$
$691$	−5.98996	−0.227869	−0.113934	+	0.993488i	$0.536345\pi$
$692$	0	0
$693$	3.36444	0.127804
$694$	0	0
$695$	−44.7770	−1.69849
$696$	0	0
$697$	−36.2079	−1.37147
$698$	0	0
$699$	16.5144	0.624633
$700$	0	0
$701$	29.6956	1.12159	0.560793	−	0.827956i	$-0.310496\pi$
$701$	29.6956	1.12159	0.560793	+	0.827956i	$0.310496\pi$
$702$	0	0
$703$	5.79096	0.218410
$704$	0	0
$705$	46.3045	1.74393
$706$	0	0
$707$	14.2133	0.534547
$708$	0	0
$709$	44.9926	1.68973	0.844866	−	0.534978i	$-0.179680\pi$
$709$	44.9926	1.68973	0.844866	+	0.534978i	$0.179680\pi$
$710$	0	0
$711$	−8.15789	−0.305945
$712$	0	0
$713$	−26.3191	−0.985657
$714$	0	0
$715$	−7.64453	−0.285889
$716$	0	0
$717$	−1.77526	−0.0662985
$718$	0	0
$719$	12.1953	0.454809	0.227404	−	0.973800i	$-0.426976\pi$
$719$	12.1953	0.454809	0.227404	+	0.973800i	$0.426976\pi$
$720$	0	0
$721$	−46.1560	−1.71894
$722$	0	0
$723$	7.71610	0.286965
$724$	0	0
$725$	−17.8269	−0.662073
$726$	0	0
$727$	6.92138	0.256700	0.128350	−	0.991729i	$-0.459032\pi$
$727$	6.92138	0.256700	0.128350	+	0.991729i	$0.459032\pi$
$728$	0	0
$729$	23.4321	0.867856
$730$	0	0
$731$	13.4258	0.496572
$732$	0	0
$733$	−16.3036	−0.602187	−0.301094	−	0.953595i	$-0.597352\pi$
$733$	−16.3036	−0.602187	−0.301094	+	0.953595i	$0.597352\pi$
$734$	0	0
$735$	−0.444699	−0.0164030
$736$	0	0
$737$	7.74654	0.285347
$738$	0	0
$739$	34.7044	1.27662	0.638312	−	0.769778i	$-0.279633\pi$
$739$	34.7044	1.27662	0.638312	+	0.769778i	$0.279633\pi$
$740$	0	0
$741$	5.88366	0.216142
$742$	0	0
$743$	−43.7253	−1.60413	−0.802063	−	0.597239i	$-0.796264\pi$
$743$	−43.7253	−1.60413	−0.802063	+	0.597239i	$0.796264\pi$
$744$	0	0
$745$	38.9088	1.42551
$746$	0	0
$747$	2.92564	0.107044
$748$	0	0
$749$	13.0844	0.478093
$750$	0	0
$751$	−30.5956	−1.11645	−0.558224	−	0.829690i	$-0.688517\pi$
$751$	−30.5956	−1.11645	−0.558224	+	0.829690i	$0.688517\pi$
$752$	0	0
$753$	−1.21978	−0.0444511
$754$	0	0
$755$	53.0661	1.93127
$756$	0	0
$757$	36.1154	1.31264	0.656319	−	0.754484i	$-0.272113\pi$
$757$	36.1154	1.31264	0.656319	+	0.754484i	$0.272113\pi$
$758$	0	0
$759$	3.77200	0.136915
$760$	0	0
$761$	−19.6901	−0.713767	−0.356884	−	0.934149i	$-0.616161\pi$
$761$	−19.6901	−0.713767	−0.356884	+	0.934149i	$0.616161\pi$
$762$	0	0
$763$	−34.8849	−1.26292
$764$	0	0
$765$	−12.6438	−0.457139
$766$	0	0
$767$	5.13716	0.185492
$768$	0	0
$769$	−9.10307	−0.328265	−0.164133	−	0.986438i	$-0.552482\pi$
$769$	−9.10307	−0.328265	−0.164133	+	0.986438i	$0.552482\pi$
$770$	0	0
$771$	−7.30112	−0.262944
$772$	0	0
$773$	23.3743	0.840714	0.420357	−	0.907359i	$-0.361905\pi$
$773$	23.3743	0.840714	0.420357	+	0.907359i	$0.361905\pi$
$774$	0	0
$775$	25.3654	0.911153
$776$	0	0
$777$	11.8563	0.425342
$778$	0	0
$779$	−19.7360	−0.707117
$780$	0	0
$781$	−6.99209	−0.250197
$782$	0	0
$783$	27.9354	0.998330
$784$	0	0
$785$	−14.1202	−0.503973
$786$	0	0
$787$	−0.146729	−0.00523031	−0.00261516	−	0.999997i	$-0.500832\pi$
$787$	−0.146729	−0.00523031	−0.00261516	+	0.999997i	$0.500832\pi$
$788$	0	0
$789$	−28.9285	−1.02988
$790$	0	0
$791$	39.0614	1.38886
$792$	0	0
$793$	15.7166	0.558113
$794$	0	0
$795$	40.4609	1.43500
$796$	0	0
$797$	13.4348	0.475886	0.237943	−	0.971279i	$-0.423527\pi$
$797$	13.4348	0.475886	0.237943	+	0.971279i	$0.423527\pi$
$798$	0	0
$799$	37.2893	1.31920
$800$	0	0
$801$	−6.64748	−0.234877
$802$	0	0
$803$	−9.03268	−0.318756
$804$	0	0
$805$	−27.8786	−0.982591
$806$	0	0
$807$	−14.1385	−0.497700
$808$	0	0
$809$	10.2301	0.359670	0.179835	−	0.983697i	$-0.442444\pi$
$809$	10.2301	0.359670	0.179835	+	0.983697i	$0.442444\pi$
$810$	0	0
$811$	25.4550	0.893847	0.446924	−	0.894572i	$-0.352520\pi$
$811$	25.4550	0.893847	0.446924	+	0.894572i	$0.352520\pi$
$812$	0	0
$813$	13.3340	0.467642
$814$	0	0
$815$	52.3143	1.83249
$816$	0	0
$817$	7.31809	0.256028
$818$	0	0
$819$	−12.2428	−0.427798
$820$	0	0
$821$	44.5755	1.55570	0.777848	−	0.628452i	$-0.216311\pi$
$821$	44.5755	1.55570	0.777848	+	0.628452i	$0.216311\pi$
$822$	0	0
$823$	−29.8394	−1.04014	−0.520068	−	0.854125i	$-0.674094\pi$
$823$	−29.8394	−1.04014	−0.520068	+	0.854125i	$0.674094\pi$
$824$	0	0
$825$	−3.63533	−0.126566
$826$	0	0
$827$	6.70942	0.233309	0.116655	−	0.993173i	$-0.462783\pi$
$827$	6.70942	0.233309	0.116655	+	0.993173i	$0.462783\pi$
$828$	0	0
$829$	−47.7338	−1.65786	−0.828932	−	0.559349i	$-0.811051\pi$
$829$	−47.7338	−1.65786	−0.828932	+	0.559349i	$0.811051\pi$
$830$	0	0
$831$	−8.64098	−0.299752
$832$	0	0
$833$	−0.358119	−0.0124081
$834$	0	0
$835$	50.8050	1.75818
$836$	0	0
$837$	−39.7487	−1.37391
$838$	0	0
$839$	18.8346	0.650241	0.325121	−	0.945673i	$-0.394595\pi$
$839$	18.8346	0.650241	0.325121	+	0.945673i	$0.394595\pi$
$840$	0	0
$841$	−3.23777	−0.111647
$842$	0	0
$843$	0.499384	0.0171997
$844$	0	0
$845$	−10.1109	−0.347826
$846$	0	0
$847$	−26.9543	−0.926162
$848$	0	0
$849$	37.6187	1.29107
$850$	0	0
$851$	−13.5096	−0.463103
$852$	0	0
$853$	18.8435	0.645188	0.322594	−	0.946537i	$-0.395445\pi$
$853$	18.8435	0.645188	0.322594	+	0.946537i	$0.395445\pi$
$854$	0	0
$855$	−6.89184	−0.235696
$856$	0	0
$857$	7.60727	0.259859	0.129930	−	0.991523i	$-0.458525\pi$
$857$	7.60727	0.259859	0.129930	+	0.991523i	$0.458525\pi$
$858$	0	0
$859$	−42.0731	−1.43552	−0.717758	−	0.696293i	$-0.754831\pi$
$859$	−42.0731	−1.43552	−0.717758	+	0.696293i	$0.754831\pi$
$860$	0	0
$861$	−40.4072	−1.37707
$862$	0	0
$863$	−17.6192	−0.599765	−0.299882	−	0.953976i	$-0.596947\pi$
$863$	−17.6192	−0.599765	−0.299882	+	0.953976i	$0.596947\pi$
$864$	0	0
$865$	−23.2653	−0.791043
$866$	0	0
$867$	−10.7176	−0.363989
$868$	0	0
$869$	−4.57788	−0.155294
$870$	0	0
$871$	−28.1887	−0.955139
$872$	0	0
$873$	15.3724	0.520277
$874$	0	0
$875$	−11.3814	−0.384761
$876$	0	0
$877$	3.52324	0.118971	0.0594857	−	0.998229i	$-0.481054\pi$
$877$	3.52324	0.118971	0.0594857	+	0.998229i	$0.481054\pi$
$878$	0	0
$879$	−13.3128	−0.449028
$880$	0	0
$881$	−17.0600	−0.574768	−0.287384	−	0.957815i	$-0.592786\pi$
$881$	−17.0600	−0.574768	−0.287384	+	0.957815i	$0.592786\pi$
$882$	0	0
$883$	26.2038	0.881826	0.440913	−	0.897550i	$-0.354655\pi$
$883$	26.2038	0.881826	0.440913	+	0.897550i	$0.354655\pi$
$884$	0	0
$885$	5.92075	0.199024
$886$	0	0
$887$	−31.5443	−1.05915	−0.529577	−	0.848262i	$-0.677649\pi$
$887$	−31.5443	−1.05915	−0.529577	+	0.848262i	$0.677649\pi$
$888$	0	0
$889$	21.9667	0.736741
$890$	0	0
$891$	1.84728	0.0618861
$892$	0	0
$893$	20.3255	0.680166
$894$	0	0
$895$	8.96675	0.299725
$896$	0	0
$897$	−13.7259	−0.458293
$898$	0	0
$899$	−36.6565	−1.22256
$900$	0	0
$901$	32.5834	1.08551
$902$	0	0
$903$	14.9829	0.498600
$904$	0	0
$905$	−46.8501	−1.55735
$906$	0	0
$907$	−29.7760	−0.988695	−0.494347	−	0.869264i	$-0.664593\pi$
$907$	−29.7760	−0.988695	−0.494347	+	0.869264i	$0.664593\pi$
$908$	0	0
$909$	−8.19695	−0.271876
$910$	0	0
$911$	−22.0467	−0.730440	−0.365220	−	0.930921i	$-0.619006\pi$
$911$	−22.0467	−0.730440	−0.365220	+	0.930921i	$0.619006\pi$
$912$	0	0
$913$	1.64175	0.0543341
$914$	0	0
$915$	18.1139	0.598827
$916$	0	0
$917$	19.9943	0.660270
$918$	0	0
$919$	−52.2494	−1.72355	−0.861775	−	0.507290i	$-0.830647\pi$
$919$	−52.2494	−1.72355	−0.861775	+	0.507290i	$0.830647\pi$
$920$	0	0
$921$	−32.5583	−1.07283
$922$	0	0
$923$	25.4434	0.837480
$924$	0	0
$925$	13.0201	0.428098
$926$	0	0
$927$	26.6186	0.874269
$928$	0	0
$929$	−14.4871	−0.475305	−0.237653	−	0.971350i	$-0.576378\pi$
$929$	−14.4871	−0.475305	−0.237653	+	0.971350i	$0.576378\pi$
$930$	0	0
$931$	−0.195202	−0.00639748
$932$	0	0
$933$	13.6343	0.446368
$934$	0	0
$935$	−7.09521	−0.232038
$936$	0	0
$937$	28.6549	0.936115	0.468058	−	0.883698i	$-0.344954\pi$
$937$	28.6549	0.936115	0.468058	+	0.883698i	$0.344954\pi$
$938$	0	0
$939$	−41.5244	−1.35510
$940$	0	0
$941$	9.91272	0.323145	0.161573	−	0.986861i	$-0.448343\pi$
$941$	9.91272	0.323145	0.161573	+	0.986861i	$0.448343\pi$
$942$	0	0
$943$	46.0418	1.49933
$944$	0	0
$945$	−42.1039	−1.36964
$946$	0	0
$947$	49.0350	1.59342	0.796711	−	0.604360i	$-0.206571\pi$
$947$	49.0350	1.59342	0.796711	+	0.604360i	$0.206571\pi$
$948$	0	0
$949$	32.8688	1.06697
$950$	0	0
$951$	6.60391	0.214146
$952$	0	0
$953$	−50.6006	−1.63911	−0.819557	−	0.572998i	$-0.805780\pi$
$953$	−50.6006	−1.63911	−0.819557	+	0.572998i	$0.805780\pi$
$954$	0	0
$955$	36.6137	1.18479
$956$	0	0
$957$	5.25354	0.169823
$958$	0	0
$959$	26.2406	0.847353
$960$	0	0
$961$	21.1577	0.682508
$962$	0	0
$963$	−7.54589	−0.243163
$964$	0	0
$965$	−59.8437	−1.92644
$966$	0	0
$967$	−28.3933	−0.913066	−0.456533	−	0.889706i	$-0.650909\pi$
$967$	−28.3933	−0.913066	−0.456533	+	0.889706i	$0.650909\pi$
$968$	0	0
$969$	5.46087	0.175428
$970$	0	0
$971$	11.2456	0.360890	0.180445	−	0.983585i	$-0.442246\pi$
$971$	11.2456	0.360890	0.180445	+	0.983585i	$0.442246\pi$
$972$	0	0
$973$	−40.2412	−1.29007
$974$	0	0
$975$	13.2285	0.423652
$976$	0	0
$977$	−60.0821	−1.92220	−0.961099	−	0.276206i	$-0.910923\pi$
$977$	−60.0821	−1.92220	−0.961099	+	0.276206i	$0.910923\pi$
$978$	0	0
$979$	−3.73029	−0.119221
$980$	0	0
$981$	20.1184	0.642333
$982$	0	0
$983$	−6.87900	−0.219406	−0.109703	−	0.993964i	$-0.534990\pi$
$983$	−6.87900	−0.219406	−0.109703	+	0.993964i	$0.534990\pi$
$984$	0	0
$985$	−26.5461	−0.845828
$986$	0	0
$987$	41.6140	1.32459
$988$	0	0
$989$	−17.0722	−0.542865
$990$	0	0
$991$	28.3645	0.901028	0.450514	−	0.892769i	$-0.351241\pi$
$991$	28.3645	0.901028	0.450514	+	0.892769i	$0.351241\pi$
$992$	0	0
$993$	25.2355	0.800826
$994$	0	0
$995$	−39.5666	−1.25435
$996$	0	0
$997$	−8.26611	−0.261790	−0.130895	−	0.991396i	$-0.541785\pi$
$997$	−8.26611	−0.261790	−0.130895	+	0.991396i	$0.541785\pi$
$998$	0	0
$999$	−20.4030	−0.645523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.19	yes	30
4.3	odd	2		8032.2.a.g.1.12	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.12	✓	30	4.3	odd	2
8032.2.a.j.1.19	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21978 q^{3} +2.91757 q^{5} +2.62203 q^{7} -1.51215 q^{9} +O(q^{10})\)	\(q+1.21978 q^{3} +2.91757 q^{5} +2.62203 q^{7} -1.51215 q^{9} -0.848556 q^{11} +3.08779 q^{13} +3.55878 q^{15} +2.86591 q^{17} +1.56214 q^{19} +3.19829 q^{21} -3.64428 q^{23} +3.51223 q^{25} -5.50381 q^{27} -5.07565 q^{29} +7.22203 q^{31} -1.03505 q^{33} +7.64996 q^{35} +3.70707 q^{37} +3.76642 q^{39} -12.6340 q^{41} +4.68466 q^{43} -4.41180 q^{45} +13.0113 q^{47} -0.124958 q^{49} +3.49577 q^{51} +11.3693 q^{53} -2.47572 q^{55} +1.90546 q^{57} +1.66370 q^{59} +5.08991 q^{61} -3.96490 q^{63} +9.00886 q^{65} -9.12908 q^{67} -4.44520 q^{69} +8.23999 q^{71} +10.6448 q^{73} +4.28413 q^{75} -2.22494 q^{77} +5.39490 q^{79} -2.17696 q^{81} -1.93476 q^{83} +8.36151 q^{85} -6.19115 q^{87} +4.39605 q^{89} +8.09629 q^{91} +8.80926 q^{93} +4.55765 q^{95} -10.1659 q^{97} +1.28314 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 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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