LMFDB - Embedded newform 8048.2.a.y.1.6

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:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.45195 q^{3} +0.279836 q^{5} -3.64586 q^{7} +3.01204 q^{9} +O(q^{10})$	$q-2.45195 q^{3} +0.279836 q^{5} -3.64586 q^{7} +3.01204 q^{9} -4.16835 q^{11} -0.225891 q^{13} -0.686143 q^{15} +1.76597 q^{17} +2.09795 q^{19} +8.93946 q^{21} +2.12140 q^{23} -4.92169 q^{25} -0.0295247 q^{27} +1.16954 q^{29} -7.55159 q^{31} +10.2206 q^{33} -1.02024 q^{35} -2.81161 q^{37} +0.553872 q^{39} +0.812621 q^{41} -7.36747 q^{43} +0.842877 q^{45} -4.04461 q^{47} +6.29232 q^{49} -4.33006 q^{51} +0.886240 q^{53} -1.16645 q^{55} -5.14406 q^{57} -0.749409 q^{59} -7.77767 q^{61} -10.9815 q^{63} -0.0632124 q^{65} -14.9861 q^{67} -5.20157 q^{69} +6.01214 q^{71} -15.3943 q^{73} +12.0677 q^{75} +15.1972 q^{77} -15.4183 q^{79} -8.96373 q^{81} +3.28502 q^{83} +0.494181 q^{85} -2.86765 q^{87} -3.17270 q^{89} +0.823567 q^{91} +18.5161 q^{93} +0.587082 q^{95} -9.57560 q^{97} -12.5552 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * 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
$3$	−2.45195	−1.41563	−0.707816	−	0.706397i	$-0.750319\pi$
$3$	−2.45195	−1.41563	−0.707816	+	0.706397i	$0.750319\pi$
$4$	0	0
$5$	0.279836	0.125146	0.0625732	−	0.998040i	$-0.480069\pi$
$5$	0.279836	0.125146	0.0625732	+	0.998040i	$0.480069\pi$
$6$	0	0
$7$	−3.64586	−1.37801	−0.689003	−	0.724758i	$-0.741952\pi$
$7$	−3.64586	−1.37801	−0.689003	+	0.724758i	$0.741952\pi$
$8$	0	0
$9$	3.01204	1.00401
$10$	0	0
$11$	−4.16835	−1.25680	−0.628402	−	0.777889i	$-0.716291\pi$
$11$	−4.16835	−1.25680	−0.628402	+	0.777889i	$0.716291\pi$
$12$	0	0
$13$	−0.225891	−0.0626509	−0.0313254	−	0.999509i	$-0.509973\pi$
$13$	−0.225891	−0.0626509	−0.0313254	+	0.999509i	$0.509973\pi$
$14$	0	0
$15$	−0.686143	−0.177161
$16$	0	0
$17$	1.76597	0.428310	0.214155	−	0.976800i	$-0.431300\pi$
$17$	1.76597	0.428310	0.214155	+	0.976800i	$0.431300\pi$
$18$	0	0
$19$	2.09795	0.481303	0.240651	−	0.970612i	$-0.422639\pi$
$19$	2.09795	0.481303	0.240651	+	0.970612i	$0.422639\pi$
$20$	0	0
$21$	8.93946	1.95075
$22$	0	0
$23$	2.12140	0.442343	0.221172	−	0.975235i	$-0.429012\pi$
$23$	2.12140	0.442343	0.221172	+	0.975235i	$0.429012\pi$
$24$	0	0
$25$	−4.92169	−0.984338
$26$	0	0
$27$	−0.0295247	−0.00568202
$28$	0	0
$29$	1.16954	0.217178	0.108589	−	0.994087i	$-0.465367\pi$
$29$	1.16954	0.217178	0.108589	+	0.994087i	$0.465367\pi$
$30$	0	0
$31$	−7.55159	−1.35631	−0.678153	−	0.734921i	$-0.737219\pi$
$31$	−7.55159	−1.35631	−0.678153	+	0.734921i	$0.737219\pi$
$32$	0	0
$33$	10.2206	1.77917
$34$	0	0
$35$	−1.02024	−0.172453
$36$	0	0
$37$	−2.81161	−0.462226	−0.231113	−	0.972927i	$-0.574237\pi$
$37$	−2.81161	−0.462226	−0.231113	+	0.972927i	$0.574237\pi$
$38$	0	0
$39$	0.553872	0.0886906
$40$	0	0
$41$	0.812621	0.126910	0.0634550	−	0.997985i	$-0.479788\pi$
$41$	0.812621	0.126910	0.0634550	+	0.997985i	$0.479788\pi$
$42$	0	0
$43$	−7.36747	−1.12353	−0.561765	−	0.827297i	$-0.689877\pi$
$43$	−7.36747	−1.12353	−0.561765	+	0.827297i	$0.689877\pi$
$44$	0	0
$45$	0.842877	0.125649
$46$	0	0
$47$	−4.04461	−0.589967	−0.294984	−	0.955502i	$-0.595314\pi$
$47$	−4.04461	−0.589967	−0.294984	+	0.955502i	$0.595314\pi$
$48$	0	0
$49$	6.29232	0.898903
$50$	0	0
$51$	−4.33006	−0.606330
$52$	0	0
$53$	0.886240	0.121734	0.0608672	−	0.998146i	$-0.480613\pi$
$53$	0.886240	0.121734	0.0608672	+	0.998146i	$0.480613\pi$
$54$	0	0
$55$	−1.16645	−0.157285
$56$	0	0
$57$	−5.14406	−0.681348
$58$	0	0
$59$	−0.749409	−0.0975647	−0.0487824	−	0.998809i	$-0.515534\pi$
$59$	−0.749409	−0.0975647	−0.0487824	+	0.998809i	$0.515534\pi$
$60$	0	0
$61$	−7.77767	−0.995829	−0.497914	−	0.867226i	$-0.665901\pi$
$61$	−7.77767	−0.995829	−0.497914	+	0.867226i	$0.665901\pi$
$62$	0	0
$63$	−10.9815	−1.38354
$64$	0	0
$65$	−0.0632124	−0.00784053
$66$	0	0
$67$	−14.9861	−1.83084	−0.915420	−	0.402500i	$-0.868141\pi$
$67$	−14.9861	−1.83084	−0.915420	+	0.402500i	$0.868141\pi$
$68$	0	0
$69$	−5.20157	−0.626195
$70$	0	0
$71$	6.01214	0.713510	0.356755	−	0.934198i	$-0.383883\pi$
$71$	6.01214	0.713510	0.356755	+	0.934198i	$0.383883\pi$
$72$	0	0
$73$	−15.3943	−1.80177	−0.900885	−	0.434059i	$-0.857081\pi$
$73$	−15.3943	−1.80177	−0.900885	+	0.434059i	$0.857081\pi$
$74$	0	0
$75$	12.0677	1.39346
$76$	0	0
$77$	15.1972	1.73189
$78$	0	0
$79$	−15.4183	−1.73469	−0.867346	−	0.497705i	$-0.834176\pi$
$79$	−15.4183	−1.73469	−0.867346	+	0.497705i	$0.834176\pi$
$80$	0	0
$81$	−8.96373	−0.995970
$82$	0	0
$83$	3.28502	0.360578	0.180289	−	0.983614i	$-0.442297\pi$
$83$	3.28502	0.360578	0.180289	+	0.983614i	$0.442297\pi$
$84$	0	0
$85$	0.494181	0.0536015
$86$	0	0
$87$	−2.86765	−0.307445
$88$	0	0
$89$	−3.17270	−0.336306	−0.168153	−	0.985761i	$-0.553780\pi$
$89$	−3.17270	−0.336306	−0.168153	+	0.985761i	$0.553780\pi$
$90$	0	0
$91$	0.823567	0.0863333
$92$	0	0
$93$	18.5161	1.92003
$94$	0	0
$95$	0.587082	0.0602333
$96$	0	0
$97$	−9.57560	−0.972255	−0.486128	−	0.873888i	$-0.661591\pi$
$97$	−9.57560	−0.972255	−0.486128	+	0.873888i	$0.661591\pi$
$98$	0	0
$99$	−12.5552	−1.26185
$100$	0	0
$101$	−13.1773	−1.31119	−0.655594	−	0.755113i	$-0.727582\pi$
$101$	−13.1773	−1.31119	−0.655594	+	0.755113i	$0.727582\pi$
$102$	0	0
$103$	15.0313	1.48108	0.740541	−	0.672012i	$-0.234569\pi$
$103$	15.0313	1.48108	0.740541	+	0.672012i	$0.234569\pi$
$104$	0	0
$105$	2.50158	0.244129
$106$	0	0
$107$	9.74626	0.942206	0.471103	−	0.882078i	$-0.343856\pi$
$107$	9.74626	0.942206	0.471103	+	0.882078i	$0.343856\pi$
$108$	0	0
$109$	−7.14740	−0.684597	−0.342299	−	0.939591i	$-0.611205\pi$
$109$	−7.14740	−0.684597	−0.342299	+	0.939591i	$0.611205\pi$
$110$	0	0
$111$	6.89392	0.654342
$112$	0	0
$113$	14.8344	1.39550	0.697751	−	0.716340i	$-0.254184\pi$
$113$	14.8344	1.39550	0.697751	+	0.716340i	$0.254184\pi$
$114$	0	0
$115$	0.593645	0.0553577
$116$	0	0
$117$	−0.680393	−0.0629023
$118$	0	0
$119$	−6.43848	−0.590214
$120$	0	0
$121$	6.37513	0.579557
$122$	0	0
$123$	−1.99250	−0.179658
$124$	0	0
$125$	−2.77645	−0.248333
$126$	0	0
$127$	−9.44701	−0.838287	−0.419143	−	0.907920i	$-0.637670\pi$
$127$	−9.44701	−0.838287	−0.419143	+	0.907920i	$0.637670\pi$
$128$	0	0
$129$	18.0646	1.59050
$130$	0	0
$131$	9.27582	0.810433	0.405216	−	0.914221i	$-0.367196\pi$
$131$	9.27582	0.810433	0.405216	+	0.914221i	$0.367196\pi$
$132$	0	0
$133$	−7.64884	−0.663239
$134$	0	0
$135$	−0.00826206	−0.000711085 0
$136$	0	0
$137$	−13.6122	−1.16297	−0.581485	−	0.813557i	$-0.697528\pi$
$137$	−13.6122	−1.16297	−0.581485	+	0.813557i	$0.697528\pi$
$138$	0	0
$139$	11.5093	0.976205	0.488102	−	0.872786i	$-0.337689\pi$
$139$	11.5093	0.976205	0.488102	+	0.872786i	$0.337689\pi$
$140$	0	0
$141$	9.91717	0.835177
$142$	0	0
$143$	0.941592	0.0787399
$144$	0	0
$145$	0.327280	0.0271791
$146$	0	0
$147$	−15.4284	−1.27252
$148$	0	0
$149$	14.2726	1.16925	0.584627	−	0.811302i	$-0.301241\pi$
$149$	14.2726	1.16925	0.584627	+	0.811302i	$0.301241\pi$
$150$	0	0
$151$	−13.5122	−1.09960	−0.549802	−	0.835295i	$-0.685297\pi$
$151$	−13.5122	−1.09960	−0.549802	+	0.835295i	$0.685297\pi$
$152$	0	0
$153$	5.31917	0.430029
$154$	0	0
$155$	−2.11321	−0.169737
$156$	0	0
$157$	19.1266	1.52647	0.763233	−	0.646123i	$-0.223611\pi$
$157$	19.1266	1.52647	0.763233	+	0.646123i	$0.223611\pi$
$158$	0	0
$159$	−2.17301	−0.172331
$160$	0	0
$161$	−7.73435	−0.609552
$162$	0	0
$163$	10.7533	0.842262	0.421131	−	0.907000i	$-0.361633\pi$
$163$	10.7533	0.842262	0.421131	+	0.907000i	$0.361633\pi$
$164$	0	0
$165$	2.86008	0.222657
$166$	0	0
$167$	−2.46576	−0.190807	−0.0954033	−	0.995439i	$-0.530414\pi$
$167$	−2.46576	−0.190807	−0.0954033	+	0.995439i	$0.530414\pi$
$168$	0	0
$169$	−12.9490	−0.996075
$170$	0	0
$171$	6.31911	0.483235
$172$	0	0
$173$	−22.4963	−1.71036	−0.855179	−	0.518333i	$-0.826553\pi$
$173$	−22.4963	−1.71036	−0.855179	+	0.518333i	$0.826553\pi$
$174$	0	0
$175$	17.9438	1.35643
$176$	0	0
$177$	1.83751	0.138116
$178$	0	0
$179$	−13.7480	−1.02757	−0.513787	−	0.857918i	$-0.671758\pi$
$179$	−13.7480	−1.02757	−0.513787	+	0.857918i	$0.671758\pi$
$180$	0	0
$181$	17.4413	1.29640	0.648202	−	0.761469i	$-0.275521\pi$
$181$	17.4413	1.29640	0.648202	+	0.761469i	$0.275521\pi$
$182$	0	0
$183$	19.0704	1.40973
$184$	0	0
$185$	−0.786790	−0.0578460
$186$	0	0
$187$	−7.36117	−0.538302
$188$	0	0
$189$	0.107643	0.00782987
$190$	0	0
$191$	−24.4928	−1.77224	−0.886118	−	0.463459i	$-0.846608\pi$
$191$	−24.4928	−1.77224	−0.886118	+	0.463459i	$0.846608\pi$
$192$	0	0
$193$	13.7880	0.992484	0.496242	−	0.868184i	$-0.334713\pi$
$193$	13.7880	0.992484	0.496242	+	0.868184i	$0.334713\pi$
$194$	0	0
$195$	0.154993	0.0110993
$196$	0	0
$197$	−1.93368	−0.137769	−0.0688846	−	0.997625i	$-0.521944\pi$
$197$	−1.93368	−0.137769	−0.0688846	+	0.997625i	$0.521944\pi$
$198$	0	0
$199$	13.3959	0.949613	0.474807	−	0.880090i	$-0.342518\pi$
$199$	13.3959	0.949613	0.474807	+	0.880090i	$0.342518\pi$
$200$	0	0
$201$	36.7450	2.59180
$202$	0	0
$203$	−4.26399	−0.299273
$204$	0	0
$205$	0.227401	0.0158823
$206$	0	0
$207$	6.38975	0.444119
$208$	0	0
$209$	−8.74499	−0.604903
$210$	0	0
$211$	−11.2044	−0.771342	−0.385671	−	0.922636i	$-0.626030\pi$
$211$	−11.2044	−0.771342	−0.385671	+	0.922636i	$0.626030\pi$
$212$	0	0
$213$	−14.7415	−1.01007
$214$	0	0
$215$	−2.06168	−0.140606
$216$	0	0
$217$	27.5321	1.86900
$218$	0	0
$219$	37.7461	2.55064
$220$	0	0
$221$	−0.398916	−0.0268340
$222$	0	0
$223$	23.7475	1.59025	0.795124	−	0.606447i	$-0.207406\pi$
$223$	23.7475	1.59025	0.795124	+	0.606447i	$0.207406\pi$
$224$	0	0
$225$	−14.8243	−0.988289
$226$	0	0
$227$	−22.0382	−1.46273	−0.731364	−	0.681987i	$-0.761116\pi$
$227$	−22.0382	−1.46273	−0.731364	+	0.681987i	$0.761116\pi$
$228$	0	0
$229$	10.5814	0.699238	0.349619	−	0.936892i	$-0.386311\pi$
$229$	10.5814	0.699238	0.349619	+	0.936892i	$0.386311\pi$
$230$	0	0
$231$	−37.2628	−2.45171
$232$	0	0
$233$	1.44152	0.0944370	0.0472185	−	0.998885i	$-0.484964\pi$
$233$	1.44152	0.0944370	0.0472185	+	0.998885i	$0.484964\pi$
$234$	0	0
$235$	−1.13183	−0.0738323
$236$	0	0
$237$	37.8048	2.45569
$238$	0	0
$239$	−4.72999	−0.305957	−0.152979	−	0.988229i	$-0.548887\pi$
$239$	−4.72999	−0.305957	−0.152979	+	0.988229i	$0.548887\pi$
$240$	0	0
$241$	−7.11002	−0.457997	−0.228998	−	0.973427i	$-0.573545\pi$
$241$	−7.11002	−0.457997	−0.228998	+	0.973427i	$0.573545\pi$
$242$	0	0
$243$	22.0672	1.41561
$244$	0	0
$245$	1.76082	0.112495
$246$	0	0
$247$	−0.473908	−0.0301540
$248$	0	0
$249$	−8.05470	−0.510446
$250$	0	0
$251$	10.0189	0.632387	0.316193	−	0.948695i	$-0.397595\pi$
$251$	10.0189	0.632387	0.316193	+	0.948695i	$0.397595\pi$
$252$	0	0
$253$	−8.84275	−0.555939
$254$	0	0
$255$	−1.21171	−0.0758800
$256$	0	0
$257$	−1.56785	−0.0977998	−0.0488999	−	0.998804i	$-0.515572\pi$
$257$	−1.56785	−0.0977998	−0.0488999	+	0.998804i	$0.515572\pi$
$258$	0	0
$259$	10.2508	0.636951
$260$	0	0
$261$	3.52271	0.218050
$262$	0	0
$263$	0.115444	0.00711857	0.00355929	−	0.999994i	$-0.498867\pi$
$263$	0.115444	0.00711857	0.00355929	+	0.999994i	$0.498867\pi$
$264$	0	0
$265$	0.248002	0.0152346
$266$	0	0
$267$	7.77930	0.476085
$268$	0	0
$269$	10.6495	0.649311	0.324656	−	0.945832i	$-0.394752\pi$
$269$	10.6495	0.649311	0.324656	+	0.945832i	$0.394752\pi$
$270$	0	0
$271$	4.22698	0.256771	0.128385	−	0.991724i	$-0.459021\pi$
$271$	4.22698	0.256771	0.128385	+	0.991724i	$0.459021\pi$
$272$	0	0
$273$	−2.01934	−0.122216
$274$	0	0
$275$	20.5153	1.23712
$276$	0	0
$277$	−2.26245	−0.135938	−0.0679688	−	0.997687i	$-0.521652\pi$
$277$	−2.26245	−0.135938	−0.0679688	+	0.997687i	$0.521652\pi$
$278$	0	0
$279$	−22.7457	−1.36175
$280$	0	0
$281$	21.5615	1.28625	0.643125	−	0.765761i	$-0.277638\pi$
$281$	21.5615	1.28625	0.643125	+	0.765761i	$0.277638\pi$
$282$	0	0
$283$	−12.8384	−0.763164	−0.381582	−	0.924335i	$-0.624621\pi$
$283$	−12.8384	−0.763164	−0.381582	+	0.924335i	$0.624621\pi$
$284$	0	0
$285$	−1.43949	−0.0852682
$286$	0	0
$287$	−2.96271	−0.174883
$288$	0	0
$289$	−13.8814	−0.816550
$290$	0	0
$291$	23.4789	1.37636
$292$	0	0
$293$	19.8753	1.16112	0.580562	−	0.814216i	$-0.302833\pi$
$293$	19.8753	1.16112	0.580562	+	0.814216i	$0.302833\pi$
$294$	0	0
$295$	−0.209712	−0.0122099
$296$	0	0
$297$	0.123069	0.00714119
$298$	0	0
$299$	−0.479206	−0.0277132
$300$	0	0
$301$	26.8608	1.54823
$302$	0	0
$303$	32.3100	1.85616
$304$	0	0
$305$	−2.17647	−0.124624
$306$	0	0
$307$	10.5054	0.599575	0.299787	−	0.954006i	$-0.403084\pi$
$307$	10.5054	0.599575	0.299787	+	0.954006i	$0.403084\pi$
$308$	0	0
$309$	−36.8560	−2.09667
$310$	0	0
$311$	−10.5735	−0.599566	−0.299783	−	0.954007i	$-0.596914\pi$
$311$	−10.5735	−0.599566	−0.299783	+	0.954007i	$0.596914\pi$
$312$	0	0
$313$	−0.370212	−0.0209256	−0.0104628	−	0.999945i	$-0.503330\pi$
$313$	−0.370212	−0.0209256	−0.0104628	+	0.999945i	$0.503330\pi$
$314$	0	0
$315$	−3.07302	−0.173145
$316$	0	0
$317$	10.9384	0.614362	0.307181	−	0.951651i	$-0.400614\pi$
$317$	10.9384	0.614362	0.307181	+	0.951651i	$0.400614\pi$
$318$	0	0
$319$	−4.87505	−0.272951
$320$	0	0
$321$	−23.8973	−1.33382
$322$	0	0
$323$	3.70491	0.206147
$324$	0	0
$325$	1.11177	0.0616696
$326$	0	0
$327$	17.5250	0.969137
$328$	0	0
$329$	14.7461	0.812979
$330$	0	0
$331$	−15.4728	−0.850460	−0.425230	−	0.905085i	$-0.639807\pi$
$331$	−15.4728	−0.850460	−0.425230	+	0.905085i	$0.639807\pi$
$332$	0	0
$333$	−8.46869	−0.464081
$334$	0	0
$335$	−4.19364	−0.229123
$336$	0	0
$337$	−6.67136	−0.363412	−0.181706	−	0.983353i	$-0.558162\pi$
$337$	−6.67136	−0.363412	−0.181706	+	0.983353i	$0.558162\pi$
$338$	0	0
$339$	−36.3731	−1.97552
$340$	0	0
$341$	31.4776	1.70461
$342$	0	0
$343$	2.58010	0.139312
$344$	0	0
$345$	−1.45559	−0.0783661
$346$	0	0
$347$	18.5236	0.994400	0.497200	−	0.867636i	$-0.334362\pi$
$347$	18.5236	0.994400	0.497200	+	0.867636i	$0.334362\pi$
$348$	0	0
$349$	−12.7643	−0.683256	−0.341628	−	0.939835i	$-0.610978\pi$
$349$	−12.7643	−0.683256	−0.341628	+	0.939835i	$0.610978\pi$
$350$	0	0
$351$	0.00666935	0.000355984 0
$352$	0	0
$353$	−17.2783	−0.919633	−0.459816	−	0.888014i	$-0.652085\pi$
$353$	−17.2783	−0.919633	−0.459816	+	0.888014i	$0.652085\pi$
$354$	0	0
$355$	1.68241	0.0892932
$356$	0	0
$357$	15.7868	0.835526
$358$	0	0
$359$	−19.2555	−1.01627	−0.508133	−	0.861279i	$-0.669664\pi$
$359$	−19.2555	−1.01627	−0.508133	+	0.861279i	$0.669664\pi$
$360$	0	0
$361$	−14.5986	−0.768348
$362$	0	0
$363$	−15.6315	−0.820440
$364$	0	0
$365$	−4.30788	−0.225485
$366$	0	0
$367$	13.5451	0.707047	0.353524	−	0.935426i	$-0.384983\pi$
$367$	13.5451	0.707047	0.353524	+	0.935426i	$0.384983\pi$
$368$	0	0
$369$	2.44765	0.127419
$370$	0	0
$371$	−3.23111	−0.167751
$372$	0	0
$373$	11.9860	0.620614	0.310307	−	0.950636i	$-0.399568\pi$
$373$	11.9860	0.620614	0.310307	+	0.950636i	$0.399568\pi$
$374$	0	0
$375$	6.80770	0.351548
$376$	0	0
$377$	−0.264189	−0.0136064
$378$	0	0
$379$	22.3111	1.14604	0.573021	−	0.819540i	$-0.305771\pi$
$379$	22.3111	1.14604	0.573021	+	0.819540i	$0.305771\pi$
$380$	0	0
$381$	23.1636	1.18671
$382$	0	0
$383$	−11.9953	−0.612929	−0.306464	−	0.951882i	$-0.599146\pi$
$383$	−11.9953	−0.612929	−0.306464	+	0.951882i	$0.599146\pi$
$384$	0	0
$385$	4.25273	0.216739
$386$	0	0
$387$	−22.1911	−1.12804
$388$	0	0
$389$	26.1536	1.32604	0.663019	−	0.748602i	$-0.269275\pi$
$389$	26.1536	1.32604	0.663019	+	0.748602i	$0.269275\pi$
$390$	0	0
$391$	3.74633	0.189460
$392$	0	0
$393$	−22.7438	−1.14727
$394$	0	0
$395$	−4.31459	−0.217091
$396$	0	0
$397$	0.895828	0.0449603	0.0224802	−	0.999747i	$-0.492844\pi$
$397$	0.895828	0.0449603	0.0224802	+	0.999747i	$0.492844\pi$
$398$	0	0
$399$	18.7545	0.938902
$400$	0	0
$401$	−15.1250	−0.755307	−0.377654	−	0.925947i	$-0.623269\pi$
$401$	−15.1250	−0.755307	−0.377654	+	0.925947i	$0.623269\pi$
$402$	0	0
$403$	1.70583	0.0849737
$404$	0	0
$405$	−2.50837	−0.124642
$406$	0	0
$407$	11.7198	0.580928
$408$	0	0
$409$	−29.6119	−1.46421	−0.732107	−	0.681190i	$-0.761463\pi$
$409$	−29.6119	−1.46421	−0.732107	+	0.681190i	$0.761463\pi$
$410$	0	0
$411$	33.3764	1.64634
$412$	0	0
$413$	2.73224	0.134445
$414$	0	0
$415$	0.919267	0.0451250
$416$	0	0
$417$	−28.2201	−1.38195
$418$	0	0
$419$	6.05052	0.295587	0.147794	−	0.989018i	$-0.452783\pi$
$419$	6.05052	0.295587	0.147794	+	0.989018i	$0.452783\pi$
$420$	0	0
$421$	−24.0551	−1.17237	−0.586186	−	0.810176i	$-0.699371\pi$
$421$	−24.0551	−1.17237	−0.586186	+	0.810176i	$0.699371\pi$
$422$	0	0
$423$	−12.1825	−0.592335
$424$	0	0
$425$	−8.69155	−0.421602
$426$	0	0
$427$	28.3563	1.37226
$428$	0	0
$429$	−2.30873	−0.111467
$430$	0	0
$431$	23.4569	1.12988	0.564940	−	0.825132i	$-0.308899\pi$
$431$	23.4569	1.12988	0.564940	+	0.825132i	$0.308899\pi$
$432$	0	0
$433$	−18.3380	−0.881266	−0.440633	−	0.897687i	$-0.645246\pi$
$433$	−18.3380	−0.881266	−0.440633	+	0.897687i	$0.645246\pi$
$434$	0	0
$435$	−0.802472	−0.0384756
$436$	0	0
$437$	4.45060	0.212901
$438$	0	0
$439$	−21.7869	−1.03983	−0.519917	−	0.854217i	$-0.674037\pi$
$439$	−21.7869	−1.03983	−0.519917	+	0.854217i	$0.674037\pi$
$440$	0	0
$441$	18.9527	0.902511
$442$	0	0
$443$	34.8636	1.65642	0.828210	−	0.560418i	$-0.189359\pi$
$443$	34.8636	1.65642	0.828210	+	0.560418i	$0.189359\pi$
$444$	0	0
$445$	−0.887836	−0.0420875
$446$	0	0
$447$	−34.9956	−1.65523
$448$	0	0
$449$	34.5288	1.62951	0.814757	−	0.579802i	$-0.196870\pi$
$449$	34.5288	1.62951	0.814757	+	0.579802i	$0.196870\pi$
$450$	0	0
$451$	−3.38729	−0.159501
$452$	0	0
$453$	33.1311	1.55664
$454$	0	0
$455$	0.230464	0.0108043
$456$	0	0
$457$	−27.0723	−1.26639	−0.633195	−	0.773993i	$-0.718257\pi$
$457$	−27.0723	−1.26639	−0.633195	+	0.773993i	$0.718257\pi$
$458$	0	0
$459$	−0.0521396	−0.00243367
$460$	0	0
$461$	−9.06196	−0.422058	−0.211029	−	0.977480i	$-0.567681\pi$
$461$	−9.06196	−0.422058	−0.211029	+	0.977480i	$0.567681\pi$
$462$	0	0
$463$	19.9997	0.929466	0.464733	−	0.885451i	$-0.346150\pi$
$463$	19.9997	0.929466	0.464733	+	0.885451i	$0.346150\pi$
$464$	0	0
$465$	5.18147	0.240285
$466$	0	0
$467$	−9.34435	−0.432405	−0.216202	−	0.976349i	$-0.569367\pi$
$467$	−9.34435	−0.432405	−0.216202	+	0.976349i	$0.569367\pi$
$468$	0	0
$469$	54.6372	2.52291
$470$	0	0
$471$	−46.8973	−2.16092
$472$	0	0
$473$	30.7102	1.41206
$474$	0	0
$475$	−10.3255	−0.473765
$476$	0	0
$477$	2.66939	0.122223
$478$	0	0
$479$	−1.81702	−0.0830215	−0.0415108	−	0.999138i	$-0.513217\pi$
$479$	−1.81702	−0.0830215	−0.0415108	+	0.999138i	$0.513217\pi$
$480$	0	0
$481$	0.635118	0.0289589
$482$	0	0
$483$	18.9642	0.862901
$484$	0	0
$485$	−2.67960	−0.121674
$486$	0	0
$487$	18.0300	0.817019	0.408510	−	0.912754i	$-0.366049\pi$
$487$	18.0300	0.817019	0.408510	+	0.912754i	$0.366049\pi$
$488$	0	0
$489$	−26.3665	−1.19233
$490$	0	0
$491$	34.2552	1.54591	0.772957	−	0.634458i	$-0.218777\pi$
$491$	34.2552	1.54591	0.772957	+	0.634458i	$0.218777\pi$
$492$	0	0
$493$	2.06537	0.0930197
$494$	0	0
$495$	−3.51341	−0.157916
$496$	0	0
$497$	−21.9195	−0.983222
$498$	0	0
$499$	9.11779	0.408168	0.204084	−	0.978953i	$-0.434578\pi$
$499$	9.11779	0.408168	0.204084	+	0.978953i	$0.434578\pi$
$500$	0	0
$501$	6.04592	0.270112
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−3.68748	−0.164091
$506$	0	0
$507$	31.7502	1.41008
$508$	0	0
$509$	−10.6683	−0.472862	−0.236431	−	0.971648i	$-0.575978\pi$
$509$	−10.6683	−0.472862	−0.236431	+	0.971648i	$0.575978\pi$
$510$	0	0
$511$	56.1256	2.48285
$512$	0	0
$513$	−0.0619413	−0.00273477
$514$	0	0
$515$	4.20631	0.185352
$516$	0	0
$517$	16.8594	0.741473
$518$	0	0
$519$	55.1596	2.42124
$520$	0	0
$521$	−9.57053	−0.419293	−0.209646	−	0.977777i	$-0.567231\pi$
$521$	−9.57053	−0.419293	−0.209646	+	0.977777i	$0.567231\pi$
$522$	0	0
$523$	20.3693	0.890689	0.445344	−	0.895359i	$-0.353081\pi$
$523$	20.3693	0.890689	0.445344	+	0.895359i	$0.353081\pi$
$524$	0	0
$525$	−43.9973	−1.92020
$526$	0	0
$527$	−13.3359	−0.580919
$528$	0	0
$529$	−18.4996	−0.804333
$530$	0	0
$531$	−2.25725	−0.0979563
$532$	0	0
$533$	−0.183564	−0.00795103
$534$	0	0
$535$	2.72735	0.117914
$536$	0	0
$537$	33.7094	1.45467
$538$	0	0
$539$	−26.2286	−1.12975
$540$	0	0
$541$	−31.8467	−1.36920	−0.684599	−	0.728920i	$-0.740023\pi$
$541$	−31.8467	−1.36920	−0.684599	+	0.728920i	$0.740023\pi$
$542$	0	0
$543$	−42.7652	−1.83523
$544$	0	0
$545$	−2.00010	−0.0856749
$546$	0	0
$547$	27.4629	1.17423	0.587114	−	0.809505i	$-0.300264\pi$
$547$	27.4629	1.17423	0.587114	+	0.809505i	$0.300264\pi$
$548$	0	0
$549$	−23.4267	−0.999826
$550$	0	0
$551$	2.45364	0.104529
$552$	0	0
$553$	56.2130	2.39042
$554$	0	0
$555$	1.92917	0.0818886
$556$	0	0
$557$	35.3783	1.49903	0.749514	−	0.661989i	$-0.230287\pi$
$557$	35.3783	1.49903	0.749514	+	0.661989i	$0.230287\pi$
$558$	0	0
$559$	1.66425	0.0703901
$560$	0	0
$561$	18.0492	0.762038
$562$	0	0
$563$	25.1900	1.06163	0.530816	−	0.847487i	$-0.321885\pi$
$563$	25.1900	1.06163	0.530816	+	0.847487i	$0.321885\pi$
$564$	0	0
$565$	4.15120	0.174642
$566$	0	0
$567$	32.6805	1.37245
$568$	0	0
$569$	13.1636	0.551846	0.275923	−	0.961180i	$-0.411017\pi$
$569$	13.1636	0.551846	0.275923	+	0.961180i	$0.411017\pi$
$570$	0	0
$571$	−9.95587	−0.416640	−0.208320	−	0.978061i	$-0.566800\pi$
$571$	−9.95587	−0.416640	−0.208320	+	0.978061i	$0.566800\pi$
$572$	0	0
$573$	60.0550	2.50884
$574$	0	0
$575$	−10.4409	−0.435415
$576$	0	0
$577$	21.1666	0.881176	0.440588	−	0.897710i	$-0.354770\pi$
$577$	21.1666	0.881176	0.440588	+	0.897710i	$0.354770\pi$
$578$	0	0
$579$	−33.8075	−1.40499
$580$	0	0
$581$	−11.9767	−0.496879
$582$	0	0
$583$	−3.69416	−0.152996
$584$	0	0
$585$	−0.190398	−0.00787200
$586$	0	0
$587$	−13.3303	−0.550202	−0.275101	−	0.961415i	$-0.588711\pi$
$587$	−13.3303	−0.550202	−0.275101	+	0.961415i	$0.588711\pi$
$588$	0	0
$589$	−15.8429	−0.652794
$590$	0	0
$591$	4.74129	0.195030
$592$	0	0
$593$	9.44323	0.387787	0.193893	−	0.981023i	$-0.437888\pi$
$593$	9.44323	0.387787	0.193893	+	0.981023i	$0.437888\pi$
$594$	0	0
$595$	−1.80172	−0.0738632
$596$	0	0
$597$	−32.8461	−1.34430
$598$	0	0
$599$	−24.0118	−0.981097	−0.490549	−	0.871414i	$-0.663204\pi$
$599$	−24.0118	−0.981097	−0.490549	+	0.871414i	$0.663204\pi$
$600$	0	0
$601$	45.0203	1.83641	0.918207	−	0.396100i	$-0.129637\pi$
$601$	45.0203	1.83641	0.918207	+	0.396100i	$0.129637\pi$
$602$	0	0
$603$	−45.1387	−1.83819
$604$	0	0
$605$	1.78399	0.0725295
$606$	0	0
$607$	−7.44273	−0.302091	−0.151046	−	0.988527i	$-0.548264\pi$
$607$	−7.44273	−0.302091	−0.151046	+	0.988527i	$0.548264\pi$
$608$	0	0
$609$	10.4551	0.423661
$610$	0	0
$611$	0.913641	0.0369620
$612$	0	0
$613$	−28.4172	−1.14776	−0.573879	−	0.818940i	$-0.694562\pi$
$613$	−28.4172	−1.14776	−0.573879	+	0.818940i	$0.694562\pi$
$614$	0	0
$615$	−0.557574	−0.0224836
$616$	0	0
$617$	26.2593	1.05716	0.528579	−	0.848884i	$-0.322725\pi$
$617$	26.2593	1.05716	0.528579	+	0.848884i	$0.322725\pi$
$618$	0	0
$619$	−18.0880	−0.727019	−0.363509	−	0.931590i	$-0.618422\pi$
$619$	−18.0880	−0.727019	−0.363509	+	0.931590i	$0.618422\pi$
$620$	0	0
$621$	−0.0626337	−0.00251340
$622$	0	0
$623$	11.5672	0.463432
$624$	0	0
$625$	23.8315	0.953260
$626$	0	0
$627$	21.4422	0.856321
$628$	0	0
$629$	−4.96522	−0.197976
$630$	0	0
$631$	26.4988	1.05490	0.527451	−	0.849586i	$-0.323148\pi$
$631$	26.4988	1.05490	0.527451	+	0.849586i	$0.323148\pi$
$632$	0	0
$633$	27.4726	1.09194
$634$	0	0
$635$	−2.64361	−0.104909
$636$	0	0
$637$	−1.42138	−0.0563171
$638$	0	0
$639$	18.1088	0.716374
$640$	0	0
$641$	36.9823	1.46071	0.730357	−	0.683066i	$-0.239354\pi$
$641$	36.9823	1.46071	0.730357	+	0.683066i	$0.239354\pi$
$642$	0	0
$643$	8.21713	0.324052	0.162026	−	0.986787i	$-0.448197\pi$
$643$	8.21713	0.324052	0.162026	+	0.986787i	$0.448197\pi$
$644$	0	0
$645$	5.05514	0.199046
$646$	0	0
$647$	−37.1321	−1.45982	−0.729908	−	0.683546i	$-0.760437\pi$
$647$	−37.1321	−1.45982	−0.729908	+	0.683546i	$0.760437\pi$
$648$	0	0
$649$	3.12380	0.122620
$650$	0	0
$651$	−67.5071	−2.64581
$652$	0	0
$653$	10.4087	0.407325	0.203662	−	0.979041i	$-0.434716\pi$
$653$	10.4087	0.407325	0.203662	+	0.979041i	$0.434716\pi$
$654$	0	0
$655$	2.59571	0.101423
$656$	0	0
$657$	−46.3683	−1.80900
$658$	0	0
$659$	4.11547	0.160316	0.0801579	−	0.996782i	$-0.474458\pi$
$659$	4.11547	0.160316	0.0801579	+	0.996782i	$0.474458\pi$
$660$	0	0
$661$	−6.06571	−0.235929	−0.117964	−	0.993018i	$-0.537637\pi$
$661$	−6.06571	−0.235929	−0.117964	+	0.993018i	$0.537637\pi$
$662$	0	0
$663$	0.978121	0.0379871
$664$	0	0
$665$	−2.14042	−0.0830019
$666$	0	0
$667$	2.48107	0.0960673
$668$	0	0
$669$	−58.2275	−2.25121
$670$	0	0
$671$	32.4200	1.25156
$672$	0	0
$673$	28.4558	1.09689	0.548445	−	0.836187i	$-0.315220\pi$
$673$	28.4558	1.09689	0.548445	+	0.836187i	$0.315220\pi$
$674$	0	0
$675$	0.145311	0.00559304
$676$	0	0
$677$	−7.44305	−0.286060	−0.143030	−	0.989718i	$-0.545684\pi$
$677$	−7.44305	−0.286060	−0.143030	+	0.989718i	$0.545684\pi$
$678$	0	0
$679$	34.9113	1.33977
$680$	0	0
$681$	54.0365	2.07068
$682$	0	0
$683$	0.0920853	0.00352354	0.00176177	−	0.999998i	$-0.499439\pi$
$683$	0.0920853	0.00352354	0.00176177	+	0.999998i	$0.499439\pi$
$684$	0	0
$685$	−3.80919	−0.145542
$686$	0	0
$687$	−25.9450	−0.989863
$688$	0	0
$689$	−0.200194	−0.00762677
$690$	0	0
$691$	25.4439	0.967932	0.483966	−	0.875087i	$-0.339196\pi$
$691$	25.4439	0.967932	0.483966	+	0.875087i	$0.339196\pi$
$692$	0	0
$693$	45.7747	1.73884
$694$	0	0
$695$	3.22071	0.122169
$696$	0	0
$697$	1.43506	0.0543569
$698$	0	0
$699$	−3.53452	−0.133688
$700$	0	0
$701$	−5.52065	−0.208512	−0.104256	−	0.994550i	$-0.533246\pi$
$701$	−5.52065	−0.208512	−0.104256	+	0.994550i	$0.533246\pi$
$702$	0	0
$703$	−5.89862	−0.222471
$704$	0	0
$705$	2.77518	0.104519
$706$	0	0
$707$	48.0426	1.80683
$708$	0	0
$709$	25.5373	0.959074	0.479537	−	0.877522i	$-0.340805\pi$
$709$	25.5373	0.959074	0.479537	+	0.877522i	$0.340805\pi$
$710$	0	0
$711$	−46.4405	−1.74166
$712$	0	0
$713$	−16.0200	−0.599952
$714$	0	0
$715$	0.263491	0.00985401
$716$	0	0
$717$	11.5977	0.433123
$718$	0	0
$719$	28.5238	1.06376	0.531880	−	0.846820i	$-0.321486\pi$
$719$	28.5238	1.06376	0.531880	+	0.846820i	$0.321486\pi$
$720$	0	0
$721$	−54.8022	−2.04094
$722$	0	0
$723$	17.4334	0.648355
$724$	0	0
$725$	−5.75612	−0.213777
$726$	0	0
$727$	18.4473	0.684171	0.342086	−	0.939669i	$-0.388867\pi$
$727$	18.4473	0.684171	0.342086	+	0.939669i	$0.388867\pi$
$728$	0	0
$729$	−27.2163	−1.00801
$730$	0	0
$731$	−13.0107	−0.481219
$732$	0	0
$733$	11.9309	0.440677	0.220338	−	0.975424i	$-0.429284\pi$
$733$	11.9309	0.440677	0.220338	+	0.975424i	$0.429284\pi$
$734$	0	0
$735$	−4.31743	−0.159251
$736$	0	0
$737$	62.4672	2.30101
$738$	0	0
$739$	14.6370	0.538431	0.269215	−	0.963080i	$-0.413236\pi$
$739$	14.6370	0.538431	0.269215	+	0.963080i	$0.413236\pi$
$740$	0	0
$741$	1.16200	0.0426870
$742$	0	0
$743$	21.3801	0.784360	0.392180	−	0.919889i	$-0.371721\pi$
$743$	21.3801	0.784360	0.392180	+	0.919889i	$0.371721\pi$
$744$	0	0
$745$	3.99398	0.146328
$746$	0	0
$747$	9.89462	0.362025
$748$	0	0
$749$	−35.5335	−1.29837
$750$	0	0
$751$	3.67550	0.134121	0.0670605	−	0.997749i	$-0.478638\pi$
$751$	3.67550	0.134121	0.0670605	+	0.997749i	$0.478638\pi$
$752$	0	0
$753$	−24.5658	−0.895227
$754$	0	0
$755$	−3.78119	−0.137612
$756$	0	0
$757$	32.4950	1.18105	0.590525	−	0.807019i	$-0.298921\pi$
$757$	32.4950	1.18105	0.590525	+	0.807019i	$0.298921\pi$
$758$	0	0
$759$	21.6819	0.787005
$760$	0	0
$761$	−30.0784	−1.09034	−0.545170	−	0.838326i	$-0.683535\pi$
$761$	−30.0784	−1.09034	−0.545170	+	0.838326i	$0.683535\pi$
$762$	0	0
$763$	26.0585	0.943380
$764$	0	0
$765$	1.48849	0.0538166
$766$	0	0
$767$	0.169285	0.00611251
$768$	0	0
$769$	−41.8504	−1.50917	−0.754583	−	0.656205i	$-0.772161\pi$
$769$	−41.8504	−1.50917	−0.754583	+	0.656205i	$0.772161\pi$
$770$	0	0
$771$	3.84429	0.138449
$772$	0	0
$773$	41.1452	1.47989	0.739945	−	0.672668i	$-0.234852\pi$
$773$	41.1452	1.47989	0.739945	+	0.672668i	$0.234852\pi$
$774$	0	0
$775$	37.1666	1.33506
$776$	0	0
$777$	−25.1343	−0.901688
$778$	0	0
$779$	1.70484	0.0610822
$780$	0	0
$781$	−25.0607	−0.896743
$782$	0	0
$783$	−0.0345303	−0.00123401
$784$	0	0
$785$	5.35230	0.191032
$786$	0	0
$787$	−27.9028	−0.994627	−0.497313	−	0.867571i	$-0.665680\pi$
$787$	−27.9028	−0.994627	−0.497313	+	0.867571i	$0.665680\pi$
$788$	0	0
$789$	−0.283062	−0.0100773
$790$	0	0
$791$	−54.0842	−1.92301
$792$	0	0
$793$	1.75691	0.0623895
$794$	0	0
$795$	−0.608087	−0.0215666
$796$	0	0
$797$	−3.48784	−0.123545	−0.0617727	−	0.998090i	$-0.519675\pi$
$797$	−3.48784	−0.123545	−0.0617727	+	0.998090i	$0.519675\pi$
$798$	0	0
$799$	−7.14266	−0.252689
$800$	0	0
$801$	−9.55631	−0.337656
$802$	0	0
$803$	64.1689	2.26447
$804$	0	0
$805$	−2.16435	−0.0762832
$806$	0	0
$807$	−26.1120	−0.919186
$808$	0	0
$809$	−22.9333	−0.806291	−0.403146	−	0.915136i	$-0.632083\pi$
$809$	−22.9333	−0.806291	−0.403146	+	0.915136i	$0.632083\pi$
$810$	0	0
$811$	15.1255	0.531128	0.265564	−	0.964093i	$-0.414442\pi$
$811$	15.1255	0.531128	0.265564	+	0.964093i	$0.414442\pi$
$812$	0	0
$813$	−10.3643	−0.363493
$814$	0	0
$815$	3.00916	0.105406
$816$	0	0
$817$	−15.4566	−0.540758
$818$	0	0
$819$	2.48062	0.0866798
$820$	0	0
$821$	5.73331	0.200094	0.100047	−	0.994983i	$-0.468101\pi$
$821$	5.73331	0.200094	0.100047	+	0.994983i	$0.468101\pi$
$822$	0	0
$823$	6.64174	0.231516	0.115758	−	0.993277i	$-0.463070\pi$
$823$	6.64174	0.231516	0.115758	+	0.993277i	$0.463070\pi$
$824$	0	0
$825$	−50.3025	−1.75131
$826$	0	0
$827$	28.9616	1.00709	0.503547	−	0.863968i	$-0.332028\pi$
$827$	28.9616	1.00709	0.503547	+	0.863968i	$0.332028\pi$
$828$	0	0
$829$	22.9043	0.795498	0.397749	−	0.917494i	$-0.369791\pi$
$829$	22.9043	0.795498	0.397749	+	0.917494i	$0.369791\pi$
$830$	0	0
$831$	5.54741	0.192438
$832$	0	0
$833$	11.1120	0.385009
$834$	0	0
$835$	−0.690009	−0.0238788
$836$	0	0
$837$	0.222958	0.00770656
$838$	0	0
$839$	20.7724	0.717144	0.358572	−	0.933502i	$-0.383264\pi$
$839$	20.7724	0.717144	0.358572	+	0.933502i	$0.383264\pi$
$840$	0	0
$841$	−27.6322	−0.952834
$842$	0	0
$843$	−52.8676	−1.82086
$844$	0	0
$845$	−3.62359	−0.124655
$846$	0	0
$847$	−23.2429	−0.798634
$848$	0	0
$849$	31.4791	1.08036
$850$	0	0
$851$	−5.96456	−0.204463
$852$	0	0
$853$	−6.17843	−0.211546	−0.105773	−	0.994390i	$-0.533732\pi$
$853$	−6.17843	−0.211546	−0.105773	+	0.994390i	$0.533732\pi$
$854$	0	0
$855$	1.76831	0.0604751
$856$	0	0
$857$	−30.2746	−1.03416	−0.517080	−	0.855937i	$-0.672981\pi$
$857$	−30.2746	−1.03416	−0.517080	+	0.855937i	$0.672981\pi$
$858$	0	0
$859$	−15.8620	−0.541206	−0.270603	−	0.962691i	$-0.587223\pi$
$859$	−15.8620	−0.541206	−0.270603	+	0.962691i	$0.587223\pi$
$860$	0	0
$861$	7.26440	0.247570
$862$	0	0
$863$	−26.0206	−0.885753	−0.442877	−	0.896583i	$-0.646042\pi$
$863$	−26.0206	−0.885753	−0.442877	+	0.896583i	$0.646042\pi$
$864$	0	0
$865$	−6.29526	−0.214045
$866$	0	0
$867$	34.0363	1.15593
$868$	0	0
$869$	64.2688	2.18017
$870$	0	0
$871$	3.38522	0.114704
$872$	0	0
$873$	−28.8421	−0.976158
$874$	0	0
$875$	10.1225	0.342204
$876$	0	0
$877$	−28.1487	−0.950515	−0.475257	−	0.879847i	$-0.657645\pi$
$877$	−28.1487	−0.950515	−0.475257	+	0.879847i	$0.657645\pi$
$878$	0	0
$879$	−48.7331	−1.64373
$880$	0	0
$881$	28.2636	0.952226	0.476113	−	0.879384i	$-0.342045\pi$
$881$	28.2636	0.952226	0.476113	+	0.879384i	$0.342045\pi$
$882$	0	0
$883$	−36.1411	−1.21625	−0.608123	−	0.793843i	$-0.708077\pi$
$883$	−36.1411	−1.21625	−0.608123	+	0.793843i	$0.708077\pi$
$884$	0	0
$885$	0.514201	0.0172847
$886$	0	0
$887$	−58.0021	−1.94752	−0.973760	−	0.227579i	$-0.926919\pi$
$887$	−58.0021	−1.94752	−0.973760	+	0.227579i	$0.926919\pi$
$888$	0	0
$889$	34.4425	1.15517
$890$	0	0
$891$	37.3640	1.25174
$892$	0	0
$893$	−8.48540	−0.283953
$894$	0	0
$895$	−3.84718	−0.128597
$896$	0	0
$897$	1.17499	0.0392317
$898$	0	0
$899$	−8.83189	−0.294560
$900$	0	0
$901$	1.56507	0.0521401
$902$	0	0
$903$	−65.8612	−2.19173
$904$	0	0
$905$	4.88071	0.162240
$906$	0	0
$907$	0.593648	0.0197117	0.00985587	−	0.999951i	$-0.496863\pi$
$907$	0.593648	0.0197117	0.00985587	+	0.999951i	$0.496863\pi$
$908$	0	0
$909$	−39.6905	−1.31645
$910$	0	0
$911$	−49.4928	−1.63977	−0.819885	−	0.572529i	$-0.805963\pi$
$911$	−49.4928	−1.63977	−0.819885	+	0.572529i	$0.805963\pi$
$912$	0	0
$913$	−13.6931	−0.453176
$914$	0	0
$915$	5.33659	0.176422
$916$	0	0
$917$	−33.8184	−1.11678
$918$	0	0
$919$	−28.9431	−0.954746	−0.477373	−	0.878701i	$-0.658411\pi$
$919$	−28.9431	−0.954746	−0.477373	+	0.878701i	$0.658411\pi$
$920$	0	0
$921$	−25.7587	−0.848777
$922$	0	0
$923$	−1.35809	−0.0447020
$924$	0	0
$925$	13.8379	0.454987
$926$	0	0
$927$	45.2750	1.48703
$928$	0	0
$929$	−2.16662	−0.0710845	−0.0355423	−	0.999368i	$-0.511316\pi$
$929$	−2.16662	−0.0710845	−0.0355423	+	0.999368i	$0.511316\pi$
$930$	0	0
$931$	13.2010	0.432645
$932$	0	0
$933$	25.9255	0.848764
$934$	0	0
$935$	−2.05992	−0.0673666
$936$	0	0
$937$	48.0007	1.56811	0.784057	−	0.620689i	$-0.213147\pi$
$937$	48.0007	1.56811	0.784057	+	0.620689i	$0.213147\pi$
$938$	0	0
$939$	0.907741	0.0296230
$940$	0	0
$941$	−35.4555	−1.15581	−0.577907	−	0.816102i	$-0.696131\pi$
$941$	−35.4555	−1.15581	−0.577907	+	0.816102i	$0.696131\pi$
$942$	0	0
$943$	1.72390	0.0561378
$944$	0	0
$945$	0.0301224	0.000979880 0
$946$	0	0
$947$	−26.4416	−0.859236	−0.429618	−	0.903011i	$-0.641352\pi$
$947$	−26.4416	−0.859236	−0.429618	+	0.903011i	$0.641352\pi$
$948$	0	0
$949$	3.47744	0.112882
$950$	0	0
$951$	−26.8204	−0.869711
$952$	0	0
$953$	−55.7670	−1.80647	−0.903236	−	0.429144i	$-0.858815\pi$
$953$	−55.7670	−1.80647	−0.903236	+	0.429144i	$0.858815\pi$
$954$	0	0
$955$	−6.85397	−0.221789
$956$	0	0
$957$	11.9534	0.386398
$958$	0	0
$959$	49.6283	1.60258
$960$	0	0
$961$	26.0265	0.839564
$962$	0	0
$963$	29.3561	0.945988
$964$	0	0
$965$	3.85839	0.124206
$966$	0	0
$967$	7.42604	0.238805	0.119403	−	0.992846i	$-0.461902\pi$
$967$	7.42604	0.238805	0.119403	+	0.992846i	$0.461902\pi$
$968$	0	0
$969$	−9.08425	−0.291828
$970$	0	0
$971$	−0.333254	−0.0106946	−0.00534731	−	0.999986i	$-0.501702\pi$
$971$	−0.333254	−0.0106946	−0.00534731	+	0.999986i	$0.501702\pi$
$972$	0	0
$973$	−41.9613	−1.34522
$974$	0	0
$975$	−2.72599	−0.0873015
$976$	0	0
$977$	−45.3521	−1.45094	−0.725470	−	0.688253i	$-0.758378\pi$
$977$	−45.3521	−1.45094	−0.725470	+	0.688253i	$0.758378\pi$
$978$	0	0
$979$	13.2249	0.422671
$980$	0	0
$981$	−21.5283	−0.687345
$982$	0	0
$983$	−21.6324	−0.689965	−0.344983	−	0.938609i	$-0.612115\pi$
$983$	−21.6324	−0.689965	−0.344983	+	0.938609i	$0.612115\pi$
$984$	0	0
$985$	−0.541114	−0.0172413
$986$	0	0
$987$	−36.1567	−1.15088
$988$	0	0
$989$	−15.6294	−0.496985
$990$	0	0
$991$	41.5815	1.32088	0.660441	−	0.750878i	$-0.270370\pi$
$991$	41.5815	1.32088	0.660441	+	0.750878i	$0.270370\pi$
$992$	0	0
$993$	37.9384	1.20394
$994$	0	0
$995$	3.74867	0.118841
$996$	0	0
$997$	22.3236	0.706995	0.353498	−	0.935435i	$-0.384992\pi$
$997$	22.3236	0.706995	0.353498	+	0.935435i	$0.384992\pi$
$998$	0	0
$999$	0.0830119	0.00262638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.6		33
4.3	odd	2		4024.2.a.f.1.28	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.28	✓	33	4.3	odd	2
8048.2.a.y.1.6		33	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)

\(f(q)\)	\(=\)	\(q-2.45195 q^{3} +0.279836 q^{5} -3.64586 q^{7} +3.01204 q^{9} +O(q^{10})\)	\(q-2.45195 q^{3} +0.279836 q^{5} -3.64586 q^{7} +3.01204 q^{9} -4.16835 q^{11} -0.225891 q^{13} -0.686143 q^{15} +1.76597 q^{17} +2.09795 q^{19} +8.93946 q^{21} +2.12140 q^{23} -4.92169 q^{25} -0.0295247 q^{27} +1.16954 q^{29} -7.55159 q^{31} +10.2206 q^{33} -1.02024 q^{35} -2.81161 q^{37} +0.553872 q^{39} +0.812621 q^{41} -7.36747 q^{43} +0.842877 q^{45} -4.04461 q^{47} +6.29232 q^{49} -4.33006 q^{51} +0.886240 q^{53} -1.16645 q^{55} -5.14406 q^{57} -0.749409 q^{59} -7.77767 q^{61} -10.9815 q^{63} -0.0632124 q^{65} -14.9861 q^{67} -5.20157 q^{69} +6.01214 q^{71} -15.3943 q^{73} +12.0677 q^{75} +15.1972 q^{77} -15.4183 q^{79} -8.96373 q^{81} +3.28502 q^{83} +0.494181 q^{85} -2.86765 q^{87} -3.17270 q^{89} +0.823567 q^{91} +18.5161 q^{93} +0.587082 q^{95} -9.57560 q^{97} -12.5552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

Introduction

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