LMFDB - Embedded newform 8048.2.a.u.1.4

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:	$26$
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.94588 q^{3} -4.09318 q^{5} -3.85397 q^{7} +5.67824 q^{9} +O(q^{10})$	$q-2.94588 q^{3} -4.09318 q^{5} -3.85397 q^{7} +5.67824 q^{9} +2.39680 q^{11} -0.856234 q^{13} +12.0580 q^{15} +8.13515 q^{17} -2.25451 q^{19} +11.3534 q^{21} +5.62077 q^{23} +11.7541 q^{25} -7.88977 q^{27} +3.94542 q^{29} -8.39602 q^{31} -7.06069 q^{33} +15.7750 q^{35} -4.14875 q^{37} +2.52237 q^{39} +5.07170 q^{41} -8.45963 q^{43} -23.2421 q^{45} +6.63115 q^{47} +7.85311 q^{49} -23.9652 q^{51} +2.68574 q^{53} -9.81053 q^{55} +6.64153 q^{57} -2.71535 q^{59} -3.66345 q^{61} -21.8838 q^{63} +3.50472 q^{65} +7.55546 q^{67} -16.5581 q^{69} +7.40365 q^{71} -4.00154 q^{73} -34.6264 q^{75} -9.23719 q^{77} -10.0635 q^{79} +6.20766 q^{81} -8.33576 q^{83} -33.2987 q^{85} -11.6228 q^{87} -13.3808 q^{89} +3.29990 q^{91} +24.7337 q^{93} +9.22813 q^{95} -1.19636 q^{97} +13.6096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	$ 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) $ 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * 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
$3$	−2.94588	−1.70081	−0.850404	−	0.526131i	$-0.823642\pi$
$3$	−2.94588	−1.70081	−0.850404	+	0.526131i	$0.823642\pi$
$4$	0	0
$5$	−4.09318	−1.83053	−0.915264	−	0.402856i	$-0.868018\pi$
$5$	−4.09318	−1.83053	−0.915264	+	0.402856i	$0.868018\pi$
$6$	0	0
$7$	−3.85397	−1.45666	−0.728332	−	0.685224i	$-0.759704\pi$
$7$	−3.85397	−1.45666	−0.728332	+	0.685224i	$0.759704\pi$
$8$	0	0
$9$	5.67824	1.89275
$10$	0	0
$11$	2.39680	0.722662	0.361331	−	0.932438i	$-0.382323\pi$
$11$	2.39680	0.722662	0.361331	+	0.932438i	$0.382323\pi$
$12$	0	0
$13$	−0.856234	−0.237477	−0.118738	−	0.992926i	$-0.537885\pi$
$13$	−0.856234	−0.237477	−0.118738	+	0.992926i	$0.537885\pi$
$14$	0	0
$15$	12.0580	3.11337
$16$	0	0
$17$	8.13515	1.97306	0.986532	−	0.163567i	$-0.0523001\pi$
$17$	8.13515	1.97306	0.986532	+	0.163567i	$0.0523001\pi$
$18$	0	0
$19$	−2.25451	−0.517221	−0.258610	−	0.965982i	$-0.583265\pi$
$19$	−2.25451	−0.517221	−0.258610	+	0.965982i	$0.583265\pi$
$20$	0	0
$21$	11.3534	2.47751
$22$	0	0
$23$	5.62077	1.17201	0.586006	−	0.810307i	$-0.300700\pi$
$23$	5.62077	1.17201	0.586006	+	0.810307i	$0.300700\pi$
$24$	0	0
$25$	11.7541	2.35083
$26$	0	0
$27$	−7.88977	−1.51839
$28$	0	0
$29$	3.94542	0.732646	0.366323	−	0.930488i	$-0.380617\pi$
$29$	3.94542	0.732646	0.366323	+	0.930488i	$0.380617\pi$
$30$	0	0
$31$	−8.39602	−1.50797	−0.753985	−	0.656892i	$-0.771871\pi$
$31$	−8.39602	−1.50797	−0.753985	+	0.656892i	$0.771871\pi$
$32$	0	0
$33$	−7.06069	−1.22911
$34$	0	0
$35$	15.7750	2.66646
$36$	0	0
$37$	−4.14875	−0.682050	−0.341025	−	0.940054i	$-0.610774\pi$
$37$	−4.14875	−0.682050	−0.341025	+	0.940054i	$0.610774\pi$
$38$	0	0
$39$	2.52237	0.403902
$40$	0	0
$41$	5.07170	0.792066	0.396033	−	0.918236i	$-0.370387\pi$
$41$	5.07170	0.792066	0.396033	+	0.918236i	$0.370387\pi$
$42$	0	0
$43$	−8.45963	−1.29008	−0.645041	−	0.764148i	$-0.723160\pi$
$43$	−8.45963	−1.29008	−0.645041	+	0.764148i	$0.723160\pi$
$44$	0	0
$45$	−23.2421	−3.46472
$46$	0	0
$47$	6.63115	0.967253	0.483627	−	0.875274i	$-0.339319\pi$
$47$	6.63115	0.967253	0.483627	+	0.875274i	$0.339319\pi$
$48$	0	0
$49$	7.85311	1.12187
$50$	0	0
$51$	−23.9652	−3.35580
$52$	0	0
$53$	2.68574	0.368915	0.184458	−	0.982840i	$-0.440947\pi$
$53$	2.68574	0.368915	0.184458	+	0.982840i	$0.440947\pi$
$54$	0	0
$55$	−9.81053	−1.32285
$56$	0	0
$57$	6.64153	0.879693
$58$	0	0
$59$	−2.71535	−0.353508	−0.176754	−	0.984255i	$-0.556560\pi$
$59$	−2.71535	−0.353508	−0.176754	+	0.984255i	$0.556560\pi$
$60$	0	0
$61$	−3.66345	−0.469057	−0.234529	−	0.972109i	$-0.575355\pi$
$61$	−3.66345	−0.469057	−0.234529	+	0.972109i	$0.575355\pi$
$62$	0	0
$63$	−21.8838	−2.75710
$64$	0	0
$65$	3.50472	0.434707
$66$	0	0
$67$	7.55546	0.923046	0.461523	−	0.887128i	$-0.347303\pi$
$67$	7.55546	0.923046	0.461523	+	0.887128i	$0.347303\pi$
$68$	0	0
$69$	−16.5581	−1.99336
$70$	0	0
$71$	7.40365	0.878652	0.439326	−	0.898328i	$-0.355217\pi$
$71$	7.40365	0.878652	0.439326	+	0.898328i	$0.355217\pi$
$72$	0	0
$73$	−4.00154	−0.468344	−0.234172	−	0.972195i	$-0.575238\pi$
$73$	−4.00154	−0.468344	−0.234172	+	0.972195i	$0.575238\pi$
$74$	0	0
$75$	−34.6264	−3.99831
$76$	0	0
$77$	−9.23719	−1.05268
$78$	0	0
$79$	−10.0635	−1.13223	−0.566113	−	0.824327i	$-0.691554\pi$
$79$	−10.0635	−1.13223	−0.566113	+	0.824327i	$0.691554\pi$
$80$	0	0
$81$	6.20766	0.689740
$82$	0	0
$83$	−8.33576	−0.914968	−0.457484	−	0.889218i	$-0.651249\pi$
$83$	−8.33576	−0.914968	−0.457484	+	0.889218i	$0.651249\pi$
$84$	0	0
$85$	−33.2987	−3.61175
$86$	0	0
$87$	−11.6228	−1.24609
$88$	0	0
$89$	−13.3808	−1.41836	−0.709180	−	0.705028i	$-0.750935\pi$
$89$	−13.3808	−1.41836	−0.709180	+	0.705028i	$0.750935\pi$
$90$	0	0
$91$	3.29990	0.345924
$92$	0	0
$93$	24.7337	2.56477
$94$	0	0
$95$	9.22813	0.946786
$96$	0	0
$97$	−1.19636	−0.121472	−0.0607358	−	0.998154i	$-0.519345\pi$
$97$	−1.19636	−0.121472	−0.0607358	+	0.998154i	$0.519345\pi$
$98$	0	0
$99$	13.6096	1.36781
$100$	0	0
$101$	13.8747	1.38058	0.690292	−	0.723531i	$-0.257482\pi$
$101$	13.8747	1.38058	0.690292	+	0.723531i	$0.257482\pi$
$102$	0	0
$103$	9.39098	0.925321	0.462660	−	0.886536i	$-0.346895\pi$
$103$	9.39098	0.925321	0.462660	+	0.886536i	$0.346895\pi$
$104$	0	0
$105$	−46.4714	−4.53514
$106$	0	0
$107$	12.1956	1.17899	0.589495	−	0.807772i	$-0.299327\pi$
$107$	12.1956	1.17899	0.589495	+	0.807772i	$0.299327\pi$
$108$	0	0
$109$	−5.58541	−0.534985	−0.267493	−	0.963560i	$-0.586195\pi$
$109$	−5.58541	−0.534985	−0.267493	+	0.963560i	$0.586195\pi$
$110$	0	0
$111$	12.2217	1.16004
$112$	0	0
$113$	4.09695	0.385408	0.192704	−	0.981257i	$-0.438274\pi$
$113$	4.09695	0.385408	0.192704	+	0.981257i	$0.438274\pi$
$114$	0	0
$115$	−23.0068	−2.14540
$116$	0	0
$117$	−4.86190	−0.449483
$118$	0	0
$119$	−31.3527	−2.87409
$120$	0	0
$121$	−5.25536	−0.477760
$122$	0	0
$123$	−14.9406	−1.34715
$124$	0	0
$125$	−27.6460	−2.47273
$126$	0	0
$127$	0.349069	0.0309748	0.0154874	−	0.999880i	$-0.495070\pi$
$127$	0.349069	0.0309748	0.0154874	+	0.999880i	$0.495070\pi$
$128$	0	0
$129$	24.9211	2.19418
$130$	0	0
$131$	−3.14616	−0.274881	−0.137440	−	0.990510i	$-0.543888\pi$
$131$	−3.14616	−0.274881	−0.137440	+	0.990510i	$0.543888\pi$
$132$	0	0
$133$	8.68883	0.753417
$134$	0	0
$135$	32.2943	2.77945
$136$	0	0
$137$	−6.94386	−0.593254	−0.296627	−	0.954993i	$-0.595862\pi$
$137$	−6.94386	−0.593254	−0.296627	+	0.954993i	$0.595862\pi$
$138$	0	0
$139$	13.8252	1.17264	0.586320	−	0.810080i	$-0.300576\pi$
$139$	13.8252	1.17264	0.586320	+	0.810080i	$0.300576\pi$
$140$	0	0
$141$	−19.5346	−1.64511
$142$	0	0
$143$	−2.05222	−0.171615
$144$	0	0
$145$	−16.1493	−1.34113
$146$	0	0
$147$	−23.1344	−1.90809
$148$	0	0
$149$	−5.43773	−0.445476	−0.222738	−	0.974878i	$-0.571499\pi$
$149$	−5.43773	−0.445476	−0.222738	+	0.974878i	$0.571499\pi$
$150$	0	0
$151$	9.88313	0.804277	0.402139	−	0.915579i	$-0.368267\pi$
$151$	9.88313	0.804277	0.402139	+	0.915579i	$0.368267\pi$
$152$	0	0
$153$	46.1933	3.73451
$154$	0	0
$155$	34.3664	2.76038
$156$	0	0
$157$	−13.6155	−1.08664	−0.543318	−	0.839527i	$-0.682832\pi$
$157$	−13.6155	−1.08664	−0.543318	+	0.839527i	$0.682832\pi$
$158$	0	0
$159$	−7.91189	−0.627453
$160$	0	0
$161$	−21.6623	−1.70723
$162$	0	0
$163$	11.4139	0.894007	0.447004	−	0.894532i	$-0.352491\pi$
$163$	11.4139	0.894007	0.447004	+	0.894532i	$0.352491\pi$
$164$	0	0
$165$	28.9007	2.24992
$166$	0	0
$167$	−1.86798	−0.144549	−0.0722745	−	0.997385i	$-0.523026\pi$
$167$	−1.86798	−0.144549	−0.0722745	+	0.997385i	$0.523026\pi$
$168$	0	0
$169$	−12.2669	−0.943605
$170$	0	0
$171$	−12.8017	−0.978967
$172$	0	0
$173$	17.8382	1.35621	0.678105	−	0.734965i	$-0.262801\pi$
$173$	17.8382	1.35621	0.678105	+	0.734965i	$0.262801\pi$
$174$	0	0
$175$	−45.3002	−3.42437
$176$	0	0
$177$	7.99910	0.601249
$178$	0	0
$179$	−2.82724	−0.211318	−0.105659	−	0.994402i	$-0.533695\pi$
$179$	−2.82724	−0.211318	−0.105659	+	0.994402i	$0.533695\pi$
$180$	0	0
$181$	−11.9166	−0.885756	−0.442878	−	0.896582i	$-0.646043\pi$
$181$	−11.9166	−0.885756	−0.442878	+	0.896582i	$0.646043\pi$
$182$	0	0
$183$	10.7921	0.797776
$184$	0	0
$185$	16.9816	1.24851
$186$	0	0
$187$	19.4983	1.42586
$188$	0	0
$189$	30.4070	2.21178
$190$	0	0
$191$	0.142126	0.0102839	0.00514193	−	0.999987i	$-0.498363\pi$
$191$	0.142126	0.0102839	0.00514193	+	0.999987i	$0.498363\pi$
$192$	0	0
$193$	7.47098	0.537773	0.268887	−	0.963172i	$-0.413344\pi$
$193$	7.47098	0.537773	0.268887	+	0.963172i	$0.413344\pi$
$194$	0	0
$195$	−10.3245	−0.739353
$196$	0	0
$197$	0.876337	0.0624364	0.0312182	−	0.999513i	$-0.490061\pi$
$197$	0.876337	0.0624364	0.0312182	+	0.999513i	$0.490061\pi$
$198$	0	0
$199$	1.44207	0.102226	0.0511129	−	0.998693i	$-0.483723\pi$
$199$	1.44207	0.102226	0.0511129	+	0.998693i	$0.483723\pi$
$200$	0	0
$201$	−22.2575	−1.56992
$202$	0	0
$203$	−15.2055	−1.06722
$204$	0	0
$205$	−20.7594	−1.44990
$206$	0	0
$207$	31.9160	2.21832
$208$	0	0
$209$	−5.40361	−0.373775
$210$	0	0
$211$	4.83092	0.332574	0.166287	−	0.986077i	$-0.446822\pi$
$211$	4.83092	0.332574	0.166287	+	0.986077i	$0.446822\pi$
$212$	0	0
$213$	−21.8103	−1.49442
$214$	0	0
$215$	34.6268	2.36153
$216$	0	0
$217$	32.3580	2.19661
$218$	0	0
$219$	11.7881	0.796563
$220$	0	0
$221$	−6.96559	−0.468557
$222$	0	0
$223$	−28.1116	−1.88249	−0.941245	−	0.337725i	$-0.890342\pi$
$223$	−28.1116	−1.88249	−0.941245	+	0.337725i	$0.890342\pi$
$224$	0	0
$225$	66.7428	4.44952
$226$	0	0
$227$	19.3702	1.28564	0.642821	−	0.766016i	$-0.277764\pi$
$227$	19.3702	1.28564	0.642821	+	0.766016i	$0.277764\pi$
$228$	0	0
$229$	−19.6411	−1.29792	−0.648959	−	0.760823i	$-0.724796\pi$
$229$	−19.6411	−1.29792	−0.648959	+	0.760823i	$0.724796\pi$
$230$	0	0
$231$	27.2117	1.79040
$232$	0	0
$233$	6.85427	0.449038	0.224519	−	0.974470i	$-0.427919\pi$
$233$	6.85427	0.449038	0.224519	+	0.974470i	$0.427919\pi$
$234$	0	0
$235$	−27.1425	−1.77058
$236$	0	0
$237$	29.6458	1.92570
$238$	0	0
$239$	10.1902	0.659151	0.329576	−	0.944129i	$-0.393094\pi$
$239$	10.1902	0.659151	0.329576	+	0.944129i	$0.393094\pi$
$240$	0	0
$241$	13.1938	0.849887	0.424944	−	0.905220i	$-0.360294\pi$
$241$	13.1938	0.849887	0.424944	+	0.905220i	$0.360294\pi$
$242$	0	0
$243$	5.38228	0.345273
$244$	0	0
$245$	−32.1442	−2.05362
$246$	0	0
$247$	1.93039	0.122828
$248$	0	0
$249$	24.5562	1.55618
$250$	0	0
$251$	−26.0784	−1.64605	−0.823027	−	0.568003i	$-0.807716\pi$
$251$	−26.0784	−1.64605	−0.823027	+	0.568003i	$0.807716\pi$
$252$	0	0
$253$	13.4718	0.846967
$254$	0	0
$255$	98.0940	6.14289
$256$	0	0
$257$	6.85607	0.427670	0.213835	−	0.976870i	$-0.431405\pi$
$257$	6.85607	0.427670	0.213835	+	0.976870i	$0.431405\pi$
$258$	0	0
$259$	15.9892	0.993519
$260$	0	0
$261$	22.4030	1.38671
$262$	0	0
$263$	12.2008	0.752334	0.376167	−	0.926552i	$-0.377242\pi$
$263$	12.2008	0.752334	0.376167	+	0.926552i	$0.377242\pi$
$264$	0	0
$265$	−10.9932	−0.675309
$266$	0	0
$267$	39.4182	2.41236
$268$	0	0
$269$	−20.3164	−1.23871	−0.619356	−	0.785111i	$-0.712606\pi$
$269$	−20.3164	−1.23871	−0.619356	+	0.785111i	$0.712606\pi$
$270$	0	0
$271$	−29.4168	−1.78695	−0.893473	−	0.449117i	$-0.851739\pi$
$271$	−29.4168	−1.78695	−0.893473	+	0.449117i	$0.851739\pi$
$272$	0	0
$273$	−9.72113	−0.588350
$274$	0	0
$275$	28.1723	1.69885
$276$	0	0
$277$	7.50942	0.451197	0.225599	−	0.974220i	$-0.427566\pi$
$277$	7.50942	0.451197	0.225599	+	0.974220i	$0.427566\pi$
$278$	0	0
$279$	−47.6746	−2.85420
$280$	0	0
$281$	1.25547	0.0748951	0.0374475	−	0.999299i	$-0.488077\pi$
$281$	1.25547	0.0748951	0.0374475	+	0.999299i	$0.488077\pi$
$282$	0	0
$283$	13.4127	0.797305	0.398653	−	0.917102i	$-0.369478\pi$
$283$	13.4127	0.797305	0.398653	+	0.917102i	$0.369478\pi$
$284$	0	0
$285$	−27.1850	−1.61030
$286$	0	0
$287$	−19.5462	−1.15378
$288$	0	0
$289$	49.1807	2.89298
$290$	0	0
$291$	3.52433	0.206600
$292$	0	0
$293$	−14.7688	−0.862801	−0.431401	−	0.902160i	$-0.641980\pi$
$293$	−14.7688	−0.862801	−0.431401	+	0.902160i	$0.641980\pi$
$294$	0	0
$295$	11.1144	0.647106
$296$	0	0
$297$	−18.9102	−1.09728
$298$	0	0
$299$	−4.81269	−0.278325
$300$	0	0
$301$	32.6032	1.87922
$302$	0	0
$303$	−40.8733	−2.34811
$304$	0	0
$305$	14.9952	0.858622
$306$	0	0
$307$	7.69171	0.438989	0.219495	−	0.975614i	$-0.429559\pi$
$307$	7.69171	0.438989	0.219495	+	0.975614i	$0.429559\pi$
$308$	0	0
$309$	−27.6648	−1.57379
$310$	0	0
$311$	−10.1033	−0.572904	−0.286452	−	0.958095i	$-0.592476\pi$
$311$	−10.1033	−0.572904	−0.286452	+	0.958095i	$0.592476\pi$
$312$	0	0
$313$	27.6938	1.56535	0.782673	−	0.622433i	$-0.213856\pi$
$313$	27.6938	1.56535	0.782673	+	0.622433i	$0.213856\pi$
$314$	0	0
$315$	89.5743	5.04694
$316$	0	0
$317$	−5.89330	−0.331001	−0.165500	−	0.986210i	$-0.552924\pi$
$317$	−5.89330	−0.331001	−0.165500	+	0.986210i	$0.552924\pi$
$318$	0	0
$319$	9.45637	0.529455
$320$	0	0
$321$	−35.9267	−2.00523
$322$	0	0
$323$	−18.3408	−1.02051
$324$	0	0
$325$	−10.0643	−0.558267
$326$	0	0
$327$	16.4540	0.909907
$328$	0	0
$329$	−25.5563	−1.40896
$330$	0	0
$331$	2.56187	0.140813	0.0704066	−	0.997518i	$-0.477570\pi$
$331$	2.56187	0.140813	0.0704066	+	0.997518i	$0.477570\pi$
$332$	0	0
$333$	−23.5576	−1.29095
$334$	0	0
$335$	−30.9259	−1.68966
$336$	0	0
$337$	30.8882	1.68259	0.841294	−	0.540579i	$-0.181795\pi$
$337$	30.8882	1.68259	0.841294	+	0.540579i	$0.181795\pi$
$338$	0	0
$339$	−12.0691	−0.655505
$340$	0	0
$341$	−20.1236	−1.08975
$342$	0	0
$343$	−3.28786	−0.177528
$344$	0	0
$345$	67.7755	3.64891
$346$	0	0
$347$	3.64728	0.195796	0.0978980	−	0.995196i	$-0.468788\pi$
$347$	3.64728	0.195796	0.0978980	+	0.995196i	$0.468788\pi$
$348$	0	0
$349$	−9.63208	−0.515593	−0.257797	−	0.966199i	$-0.582996\pi$
$349$	−9.63208	−0.515593	−0.257797	+	0.966199i	$0.582996\pi$
$350$	0	0
$351$	6.75549	0.360582
$352$	0	0
$353$	3.99670	0.212723	0.106362	−	0.994328i	$-0.466080\pi$
$353$	3.99670	0.212723	0.106362	+	0.994328i	$0.466080\pi$
$354$	0	0
$355$	−30.3045	−1.60840
$356$	0	0
$357$	92.3613	4.88828
$358$	0	0
$359$	−12.7161	−0.671128	−0.335564	−	0.942017i	$-0.608927\pi$
$359$	−12.7161	−0.671128	−0.335564	+	0.942017i	$0.608927\pi$
$360$	0	0
$361$	−13.9172	−0.732483
$362$	0	0
$363$	15.4817	0.812578
$364$	0	0
$365$	16.3790	0.857317
$366$	0	0
$367$	23.8629	1.24564	0.622818	−	0.782367i	$-0.285988\pi$
$367$	23.8629	1.24564	0.622818	+	0.782367i	$0.285988\pi$
$368$	0	0
$369$	28.7983	1.49918
$370$	0	0
$371$	−10.3508	−0.537386
$372$	0	0
$373$	−36.3733	−1.88334	−0.941669	−	0.336539i	$-0.890744\pi$
$373$	−36.3733	−1.88334	−0.941669	+	0.336539i	$0.890744\pi$
$374$	0	0
$375$	81.4418	4.20564
$376$	0	0
$377$	−3.37820	−0.173986
$378$	0	0
$379$	−28.6869	−1.47355	−0.736774	−	0.676139i	$-0.763652\pi$
$379$	−28.6869	−1.47355	−0.736774	+	0.676139i	$0.763652\pi$
$380$	0	0
$381$	−1.02832	−0.0526822
$382$	0	0
$383$	−23.9107	−1.22178	−0.610890	−	0.791716i	$-0.709188\pi$
$383$	−23.9107	−1.22178	−0.610890	+	0.791716i	$0.709188\pi$
$384$	0	0
$385$	37.8095	1.92695
$386$	0	0
$387$	−48.0358	−2.44180
$388$	0	0
$389$	−7.02093	−0.355975	−0.177988	−	0.984033i	$-0.556959\pi$
$389$	−7.02093	−0.355975	−0.177988	+	0.984033i	$0.556959\pi$
$390$	0	0
$391$	45.7258	2.31245
$392$	0	0
$393$	9.26821	0.467519
$394$	0	0
$395$	41.1916	2.07257
$396$	0	0
$397$	10.9725	0.550693	0.275346	−	0.961345i	$-0.411207\pi$
$397$	10.9725	0.550693	0.275346	+	0.961345i	$0.411207\pi$
$398$	0	0
$399$	−25.5963	−1.28142
$400$	0	0
$401$	20.0329	1.00040	0.500198	−	0.865911i	$-0.333261\pi$
$401$	20.0329	1.00040	0.500198	+	0.865911i	$0.333261\pi$
$402$	0	0
$403$	7.18896	0.358107
$404$	0	0
$405$	−25.4091	−1.26259
$406$	0	0
$407$	−9.94371	−0.492892
$408$	0	0
$409$	−22.7538	−1.12510	−0.562551	−	0.826763i	$-0.690180\pi$
$409$	−22.7538	−1.12510	−0.562551	+	0.826763i	$0.690180\pi$
$410$	0	0
$411$	20.4558	1.00901
$412$	0	0
$413$	10.4649	0.514943
$414$	0	0
$415$	34.1198	1.67487
$416$	0	0
$417$	−40.7275	−1.99443
$418$	0	0
$419$	−9.39083	−0.458772	−0.229386	−	0.973336i	$-0.573672\pi$
$419$	−9.39083	−0.458772	−0.229386	+	0.973336i	$0.573672\pi$
$420$	0	0
$421$	2.58953	0.126206	0.0631030	−	0.998007i	$-0.479900\pi$
$421$	2.58953	0.126206	0.0631030	+	0.998007i	$0.479900\pi$
$422$	0	0
$423$	37.6533	1.83076
$424$	0	0
$425$	95.6218	4.63834
$426$	0	0
$427$	14.1189	0.683259
$428$	0	0
$429$	6.04560	0.291884
$430$	0	0
$431$	6.40346	0.308444	0.154222	−	0.988036i	$-0.450713\pi$
$431$	6.40346	0.308444	0.154222	+	0.988036i	$0.450713\pi$
$432$	0	0
$433$	15.8009	0.759343	0.379671	−	0.925121i	$-0.376037\pi$
$433$	15.8009	0.759343	0.379671	+	0.925121i	$0.376037\pi$
$434$	0	0
$435$	47.5740	2.28100
$436$	0	0
$437$	−12.6721	−0.606188
$438$	0	0
$439$	−14.4817	−0.691172	−0.345586	−	0.938387i	$-0.612320\pi$
$439$	−14.4817	−0.691172	−0.345586	+	0.938387i	$0.612320\pi$
$440$	0	0
$441$	44.5918	2.12342
$442$	0	0
$443$	−37.3325	−1.77372	−0.886859	−	0.462040i	$-0.847118\pi$
$443$	−37.3325	−1.77372	−0.886859	+	0.462040i	$0.847118\pi$
$444$	0	0
$445$	54.7700	2.59635
$446$	0	0
$447$	16.0189	0.757669
$448$	0	0
$449$	−12.0856	−0.570354	−0.285177	−	0.958475i	$-0.592052\pi$
$449$	−12.0856	−0.570354	−0.285177	+	0.958475i	$0.592052\pi$
$450$	0	0
$451$	12.1558	0.572396
$452$	0	0
$453$	−29.1146	−1.36792
$454$	0	0
$455$	−13.5071	−0.633223
$456$	0	0
$457$	−26.7002	−1.24898	−0.624491	−	0.781032i	$-0.714694\pi$
$457$	−26.7002	−1.24898	−0.624491	+	0.781032i	$0.714694\pi$
$458$	0	0
$459$	−64.1845	−2.99588
$460$	0	0
$461$	6.59638	0.307224	0.153612	−	0.988131i	$-0.450909\pi$
$461$	6.59638	0.307224	0.153612	+	0.988131i	$0.450909\pi$
$462$	0	0
$463$	−4.89916	−0.227683	−0.113842	−	0.993499i	$-0.536316\pi$
$463$	−4.89916	−0.227683	−0.113842	+	0.993499i	$0.536316\pi$
$464$	0	0
$465$	−101.240	−4.69487
$466$	0	0
$467$	−9.57820	−0.443226	−0.221613	−	0.975135i	$-0.571132\pi$
$467$	−9.57820	−0.443226	−0.221613	+	0.975135i	$0.571132\pi$
$468$	0	0
$469$	−29.1185	−1.34457
$470$	0	0
$471$	40.1098	1.84816
$472$	0	0
$473$	−20.2760	−0.932292
$474$	0	0
$475$	−26.4999	−1.21590
$476$	0	0
$477$	15.2503	0.698262
$478$	0	0
$479$	5.73059	0.261837	0.130919	−	0.991393i	$-0.458207\pi$
$479$	5.73059	0.261837	0.130919	+	0.991393i	$0.458207\pi$
$480$	0	0
$481$	3.55230	0.161971
$482$	0	0
$483$	63.8146	2.90366
$484$	0	0
$485$	4.89690	0.222357
$486$	0	0
$487$	−35.5845	−1.61249	−0.806243	−	0.591584i	$-0.798503\pi$
$487$	−35.5845	−1.61249	−0.806243	+	0.591584i	$0.798503\pi$
$488$	0	0
$489$	−33.6241	−1.52053
$490$	0	0
$491$	37.5371	1.69403	0.847013	−	0.531572i	$-0.178399\pi$
$491$	37.5371	1.69403	0.847013	+	0.531572i	$0.178399\pi$
$492$	0	0
$493$	32.0966	1.44556
$494$	0	0
$495$	−55.7065	−2.50382
$496$	0	0
$497$	−28.5335	−1.27990
$498$	0	0
$499$	30.4728	1.36415	0.682074	−	0.731283i	$-0.261078\pi$
$499$	30.4728	1.36415	0.682074	+	0.731283i	$0.261078\pi$
$500$	0	0
$501$	5.50286	0.245850
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−56.7917	−2.52720
$506$	0	0
$507$	36.1368	1.60489
$508$	0	0
$509$	−7.45227	−0.330316	−0.165158	−	0.986267i	$-0.552813\pi$
$509$	−7.45227	−0.330316	−0.165158	+	0.986267i	$0.552813\pi$
$510$	0	0
$511$	15.4218	0.682221
$512$	0	0
$513$	17.7876	0.785341
$514$	0	0
$515$	−38.4390	−1.69383
$516$	0	0
$517$	15.8935	0.698997
$518$	0	0
$519$	−52.5491	−2.30665
$520$	0	0
$521$	19.1645	0.839612	0.419806	−	0.907614i	$-0.362098\pi$
$521$	19.1645	0.839612	0.419806	+	0.907614i	$0.362098\pi$
$522$	0	0
$523$	−12.5657	−0.549462	−0.274731	−	0.961521i	$-0.588589\pi$
$523$	−12.5657	−0.549462	−0.274731	+	0.961521i	$0.588589\pi$
$524$	0	0
$525$	133.449	5.82420
$526$	0	0
$527$	−68.3029	−2.97532
$528$	0	0
$529$	8.59303	0.373610
$530$	0	0
$531$	−15.4184	−0.669101
$532$	0	0
$533$	−4.34256	−0.188097
$534$	0	0
$535$	−49.9187	−2.15817
$536$	0	0
$537$	8.32873	0.359411
$538$	0	0
$539$	18.8223	0.810734
$540$	0	0
$541$	−22.0678	−0.948767	−0.474384	−	0.880318i	$-0.657329\pi$
$541$	−22.0678	−0.948767	−0.474384	+	0.880318i	$0.657329\pi$
$542$	0	0
$543$	35.1050	1.50650
$544$	0	0
$545$	22.8621	0.979305
$546$	0	0
$547$	13.3312	0.570001	0.285001	−	0.958527i	$-0.408006\pi$
$547$	13.3312	0.570001	0.285001	+	0.958527i	$0.408006\pi$
$548$	0	0
$549$	−20.8020	−0.887806
$550$	0	0
$551$	−8.89500	−0.378940
$552$	0	0
$553$	38.7843	1.64928
$554$	0	0
$555$	−50.0258	−2.12348
$556$	0	0
$557$	30.5612	1.29492	0.647460	−	0.762100i	$-0.275831\pi$
$557$	30.5612	1.29492	0.647460	+	0.762100i	$0.275831\pi$
$558$	0	0
$559$	7.24342	0.306364
$560$	0	0
$561$	−57.4398	−2.42511
$562$	0	0
$563$	6.59443	0.277922	0.138961	−	0.990298i	$-0.455624\pi$
$563$	6.59443	0.277922	0.138961	+	0.990298i	$0.455624\pi$
$564$	0	0
$565$	−16.7696	−0.705500
$566$	0	0
$567$	−23.9241	−1.00472
$568$	0	0
$569$	45.2137	1.89546	0.947729	−	0.319076i	$-0.103373\pi$
$569$	45.2137	1.89546	0.947729	+	0.319076i	$0.103373\pi$
$570$	0	0
$571$	16.5907	0.694301	0.347151	−	0.937809i	$-0.387149\pi$
$571$	16.5907	0.694301	0.347151	+	0.937809i	$0.387149\pi$
$572$	0	0
$573$	−0.418686	−0.0174909
$574$	0	0
$575$	66.0673	2.75520
$576$	0	0
$577$	−37.0375	−1.54189	−0.770945	−	0.636901i	$-0.780216\pi$
$577$	−37.0375	−1.54189	−0.770945	+	0.636901i	$0.780216\pi$
$578$	0	0
$579$	−22.0087	−0.914648
$580$	0	0
$581$	32.1258	1.33280
$582$	0	0
$583$	6.43718	0.266601
$584$	0	0
$585$	19.9006	0.822790
$586$	0	0
$587$	−1.60910	−0.0664146	−0.0332073	−	0.999448i	$-0.510572\pi$
$587$	−1.60910	−0.0664146	−0.0332073	+	0.999448i	$0.510572\pi$
$588$	0	0
$589$	18.9289	0.779953
$590$	0	0
$591$	−2.58159	−0.106192
$592$	0	0
$593$	−13.7079	−0.562917	−0.281458	−	0.959573i	$-0.590818\pi$
$593$	−13.7079	−0.562917	−0.281458	+	0.959573i	$0.590818\pi$
$594$	0	0
$595$	128.332	5.26111
$596$	0	0
$597$	−4.24818	−0.173866
$598$	0	0
$599$	28.9663	1.18353	0.591766	−	0.806110i	$-0.298431\pi$
$599$	28.9663	1.18353	0.591766	+	0.806110i	$0.298431\pi$
$600$	0	0
$601$	−15.9319	−0.649875	−0.324937	−	0.945736i	$-0.605343\pi$
$601$	−15.9319	−0.649875	−0.324937	+	0.945736i	$0.605343\pi$
$602$	0	0
$603$	42.9017	1.74709
$604$	0	0
$605$	21.5112	0.874553
$606$	0	0
$607$	24.8703	1.00946	0.504728	−	0.863278i	$-0.331593\pi$
$607$	24.8703	1.00946	0.504728	+	0.863278i	$0.331593\pi$
$608$	0	0
$609$	44.7938	1.81514
$610$	0	0
$611$	−5.67782	−0.229700
$612$	0	0
$613$	7.08473	0.286150	0.143075	−	0.989712i	$-0.454301\pi$
$613$	7.08473	0.286150	0.143075	+	0.989712i	$0.454301\pi$
$614$	0	0
$615$	61.1548	2.46600
$616$	0	0
$617$	−2.21783	−0.0892863	−0.0446432	−	0.999003i	$-0.514215\pi$
$617$	−2.21783	−0.0892863	−0.0446432	+	0.999003i	$0.514215\pi$
$618$	0	0
$619$	31.4755	1.26511	0.632553	−	0.774517i	$-0.282007\pi$
$619$	31.4755	1.26511	0.632553	+	0.774517i	$0.282007\pi$
$620$	0	0
$621$	−44.3466	−1.77957
$622$	0	0
$623$	51.5692	2.06607
$624$	0	0
$625$	54.3893	2.17557
$626$	0	0
$627$	15.9184	0.635720
$628$	0	0
$629$	−33.7507	−1.34573
$630$	0	0
$631$	−20.4575	−0.814400	−0.407200	−	0.913339i	$-0.633495\pi$
$631$	−20.4575	−0.814400	−0.407200	+	0.913339i	$0.633495\pi$
$632$	0	0
$633$	−14.2313	−0.565644
$634$	0	0
$635$	−1.42880	−0.0567003
$636$	0	0
$637$	−6.72410	−0.266418
$638$	0	0
$639$	42.0397	1.66306
$640$	0	0
$641$	−35.3720	−1.39711	−0.698554	−	0.715557i	$-0.746173\pi$
$641$	−35.3720	−1.39711	−0.698554	+	0.715557i	$0.746173\pi$
$642$	0	0
$643$	−14.6195	−0.576538	−0.288269	−	0.957549i	$-0.593080\pi$
$643$	−14.6195	−0.576538	−0.288269	+	0.957549i	$0.593080\pi$
$644$	0	0
$645$	−102.007	−4.01651
$646$	0	0
$647$	30.7538	1.20906	0.604529	−	0.796583i	$-0.293361\pi$
$647$	30.7538	1.20906	0.604529	+	0.796583i	$0.293361\pi$
$648$	0	0
$649$	−6.50814	−0.255467
$650$	0	0
$651$	−95.3230	−3.73600
$652$	0	0
$653$	25.7913	1.00929	0.504645	−	0.863327i	$-0.331623\pi$
$653$	25.7913	1.00929	0.504645	+	0.863327i	$0.331623\pi$
$654$	0	0
$655$	12.8778	0.503177
$656$	0	0
$657$	−22.7217	−0.886457
$658$	0	0
$659$	8.95894	0.348991	0.174495	−	0.984658i	$-0.444171\pi$
$659$	8.95894	0.348991	0.174495	+	0.984658i	$0.444171\pi$
$660$	0	0
$661$	−35.2409	−1.37071	−0.685357	−	0.728208i	$-0.740354\pi$
$661$	−35.2409	−1.37071	−0.685357	+	0.728208i	$0.740354\pi$
$662$	0	0
$663$	20.5198	0.796924
$664$	0	0
$665$	−35.5650	−1.37915
$666$	0	0
$667$	22.1763	0.858669
$668$	0	0
$669$	82.8134	3.20175
$670$	0	0
$671$	−8.78056	−0.338970
$672$	0	0
$673$	37.7967	1.45696	0.728478	−	0.685069i	$-0.240228\pi$
$673$	37.7967	1.45696	0.728478	+	0.685069i	$0.240228\pi$
$674$	0	0
$675$	−92.7376	−3.56947
$676$	0	0
$677$	−22.4586	−0.863156	−0.431578	−	0.902076i	$-0.642043\pi$
$677$	−22.4586	−0.863156	−0.431578	+	0.902076i	$0.642043\pi$
$678$	0	0
$679$	4.61072	0.176943
$680$	0	0
$681$	−57.0622	−2.18663
$682$	0	0
$683$	29.6337	1.13390	0.566950	−	0.823752i	$-0.308123\pi$
$683$	29.6337	1.13390	0.566950	+	0.823752i	$0.308123\pi$
$684$	0	0
$685$	28.4225	1.08597
$686$	0	0
$687$	57.8603	2.20751
$688$	0	0
$689$	−2.29962	−0.0876087
$690$	0	0
$691$	13.9124	0.529254	0.264627	−	0.964351i	$-0.414751\pi$
$691$	13.9124	0.529254	0.264627	+	0.964351i	$0.414751\pi$
$692$	0	0
$693$	−52.4510	−1.99245
$694$	0	0
$695$	−56.5891	−2.14655
$696$	0	0
$697$	41.2590	1.56280
$698$	0	0
$699$	−20.1919	−0.763727
$700$	0	0
$701$	45.7800	1.72909	0.864544	−	0.502557i	$-0.167607\pi$
$701$	45.7800	1.72909	0.864544	+	0.502557i	$0.167607\pi$
$702$	0	0
$703$	9.35341	0.352770
$704$	0	0
$705$	79.9588	3.01142
$706$	0	0
$707$	−53.4727	−2.01105
$708$	0	0
$709$	−35.8381	−1.34593	−0.672963	−	0.739676i	$-0.734979\pi$
$709$	−35.8381	−1.34593	−0.672963	+	0.739676i	$0.734979\pi$
$710$	0	0
$711$	−57.1427	−2.14302
$712$	0	0
$713$	−47.1921	−1.76736
$714$	0	0
$715$	8.40011	0.314146
$716$	0	0
$717$	−30.0192	−1.12109
$718$	0	0
$719$	23.2338	0.866475	0.433238	−	0.901280i	$-0.357371\pi$
$719$	23.2338	0.866475	0.433238	+	0.901280i	$0.357371\pi$
$720$	0	0
$721$	−36.1926	−1.34788
$722$	0	0
$723$	−38.8674	−1.44549
$724$	0	0
$725$	46.3750	1.72233
$726$	0	0
$727$	5.11147	0.189574	0.0947870	−	0.995498i	$-0.469783\pi$
$727$	5.11147	0.189574	0.0947870	+	0.995498i	$0.469783\pi$
$728$	0	0
$729$	−34.4786	−1.27698
$730$	0	0
$731$	−68.8204	−2.54541
$732$	0	0
$733$	4.58524	0.169360	0.0846798	−	0.996408i	$-0.473013\pi$
$733$	4.58524	0.169360	0.0846798	+	0.996408i	$0.473013\pi$
$734$	0	0
$735$	94.6931	3.49281
$736$	0	0
$737$	18.1089	0.667050
$738$	0	0
$739$	34.9682	1.28633	0.643163	−	0.765729i	$-0.277622\pi$
$739$	34.9682	1.28633	0.643163	+	0.765729i	$0.277622\pi$
$740$	0	0
$741$	−5.68671	−0.208906
$742$	0	0
$743$	−15.9770	−0.586138	−0.293069	−	0.956091i	$-0.594676\pi$
$743$	−15.9770	−0.586138	−0.293069	+	0.956091i	$0.594676\pi$
$744$	0	0
$745$	22.2576	0.815456
$746$	0	0
$747$	−47.3324	−1.73180
$748$	0	0
$749$	−47.0014	−1.71739
$750$	0	0
$751$	43.3320	1.58121	0.790603	−	0.612329i	$-0.209767\pi$
$751$	43.3320	1.58121	0.790603	+	0.612329i	$0.209767\pi$
$752$	0	0
$753$	76.8239	2.79962
$754$	0	0
$755$	−40.4534	−1.47225
$756$	0	0
$757$	−13.4721	−0.489651	−0.244825	−	0.969567i	$-0.578731\pi$
$757$	−13.4721	−0.489651	−0.244825	+	0.969567i	$0.578731\pi$
$758$	0	0
$759$	−39.6865	−1.44053
$760$	0	0
$761$	37.7313	1.36776	0.683879	−	0.729595i	$-0.260292\pi$
$761$	37.7313	1.36776	0.683879	+	0.729595i	$0.260292\pi$
$762$	0	0
$763$	21.5260	0.779294
$764$	0	0
$765$	−189.078	−6.83612
$766$	0	0
$767$	2.32497	0.0839499
$768$	0	0
$769$	12.8783	0.464404	0.232202	−	0.972668i	$-0.425407\pi$
$769$	12.8783	0.464404	0.232202	+	0.972668i	$0.425407\pi$
$770$	0	0
$771$	−20.1972	−0.727384
$772$	0	0
$773$	−38.6106	−1.38873	−0.694363	−	0.719625i	$-0.744314\pi$
$773$	−38.6106	−1.38873	−0.694363	+	0.719625i	$0.744314\pi$
$774$	0	0
$775$	−98.6881	−3.54498
$776$	0	0
$777$	−47.1022	−1.68978
$778$	0	0
$779$	−11.4342	−0.409673
$780$	0	0
$781$	17.7451	0.634968
$782$	0	0
$783$	−31.1285	−1.11244
$784$	0	0
$785$	55.7308	1.98912
$786$	0	0
$787$	1.10497	0.0393879	0.0196940	−	0.999806i	$-0.493731\pi$
$787$	1.10497	0.0393879	0.0196940	+	0.999806i	$0.493731\pi$
$788$	0	0
$789$	−35.9421	−1.27957
$790$	0	0
$791$	−15.7895	−0.561411
$792$	0	0
$793$	3.13677	0.111390
$794$	0	0
$795$	32.3848	1.14857
$796$	0	0
$797$	−39.2261	−1.38946	−0.694731	−	0.719270i	$-0.744476\pi$
$797$	−39.2261	−1.38946	−0.694731	+	0.719270i	$0.744476\pi$
$798$	0	0
$799$	53.9455	1.90845
$800$	0	0
$801$	−75.9792	−2.68459
$802$	0	0
$803$	−9.59087	−0.338454
$804$	0	0
$805$	88.6677	3.12513
$806$	0	0
$807$	59.8497	2.10681
$808$	0	0
$809$	−24.5083	−0.861665	−0.430833	−	0.902432i	$-0.641780\pi$
$809$	−24.5083	−0.861665	−0.430833	+	0.902432i	$0.641780\pi$
$810$	0	0
$811$	−18.2197	−0.639779	−0.319890	−	0.947455i	$-0.603646\pi$
$811$	−18.2197	−0.639779	−0.319890	+	0.947455i	$0.603646\pi$
$812$	0	0
$813$	86.6586	3.03925
$814$	0	0
$815$	−46.7193	−1.63650
$816$	0	0
$817$	19.0723	0.667257
$818$	0	0
$819$	18.7376	0.654746
$820$	0	0
$821$	9.83542	0.343258	0.171629	−	0.985162i	$-0.445097\pi$
$821$	9.83542	0.343258	0.171629	+	0.985162i	$0.445097\pi$
$822$	0	0
$823$	−34.9866	−1.21956	−0.609778	−	0.792572i	$-0.708741\pi$
$823$	−34.9866	−1.21956	−0.609778	+	0.792572i	$0.708741\pi$
$824$	0	0
$825$	−82.9924	−2.88942
$826$	0	0
$827$	−39.1096	−1.35998	−0.679988	−	0.733224i	$-0.738015\pi$
$827$	−39.1096	−1.35998	−0.679988	+	0.733224i	$0.738015\pi$
$828$	0	0
$829$	26.4357	0.918149	0.459074	−	0.888398i	$-0.348181\pi$
$829$	26.4357	0.918149	0.459074	+	0.888398i	$0.348181\pi$
$830$	0	0
$831$	−22.1219	−0.767400
$832$	0	0
$833$	63.8862	2.21353
$834$	0	0
$835$	7.64600	0.264601
$836$	0	0
$837$	66.2427	2.28968
$838$	0	0
$839$	7.64496	0.263933	0.131967	−	0.991254i	$-0.457871\pi$
$839$	7.64496	0.263933	0.131967	+	0.991254i	$0.457871\pi$
$840$	0	0
$841$	−13.4337	−0.463230
$842$	0	0
$843$	−3.69847	−0.127382
$844$	0	0
$845$	50.2105	1.72729
$846$	0	0
$847$	20.2540	0.695937
$848$	0	0
$849$	−39.5124	−1.35606
$850$	0	0
$851$	−23.3192	−0.799370
$852$	0	0
$853$	41.9888	1.43767	0.718835	−	0.695181i	$-0.244676\pi$
$853$	41.9888	1.43767	0.718835	+	0.695181i	$0.244676\pi$
$854$	0	0
$855$	52.3995	1.79203
$856$	0	0
$857$	13.2187	0.451542	0.225771	−	0.974180i	$-0.427510\pi$
$857$	13.2187	0.451542	0.225771	+	0.974180i	$0.427510\pi$
$858$	0	0
$859$	−2.18221	−0.0744560	−0.0372280	−	0.999307i	$-0.511853\pi$
$859$	−2.18221	−0.0744560	−0.0372280	+	0.999307i	$0.511853\pi$
$860$	0	0
$861$	57.5808	1.96235
$862$	0	0
$863$	32.0245	1.09013	0.545063	−	0.838395i	$-0.316506\pi$
$863$	32.0245	1.09013	0.545063	+	0.838395i	$0.316506\pi$
$864$	0	0
$865$	−73.0148	−2.48258
$866$	0	0
$867$	−144.881	−4.92041
$868$	0	0
$869$	−24.1201	−0.818217
$870$	0	0
$871$	−6.46924	−0.219202
$872$	0	0
$873$	−6.79319	−0.229915
$874$	0	0
$875$	106.547	3.60194
$876$	0	0
$877$	18.3986	0.621276	0.310638	−	0.950528i	$-0.399457\pi$
$877$	18.3986	0.621276	0.310638	+	0.950528i	$0.399457\pi$
$878$	0	0
$879$	43.5071	1.46746
$880$	0	0
$881$	26.5283	0.893761	0.446880	−	0.894594i	$-0.352535\pi$
$881$	26.5283	0.893761	0.446880	+	0.894594i	$0.352535\pi$
$882$	0	0
$883$	−49.9159	−1.67980	−0.839901	−	0.542739i	$-0.817387\pi$
$883$	−49.9159	−1.67980	−0.839901	+	0.542739i	$0.817387\pi$
$884$	0	0
$885$	−32.7418	−1.10060
$886$	0	0
$887$	−49.7602	−1.67078	−0.835392	−	0.549655i	$-0.814759\pi$
$887$	−49.7602	−1.67078	−0.835392	+	0.549655i	$0.814759\pi$
$888$	0	0
$889$	−1.34530	−0.0451199
$890$	0	0
$891$	14.8785	0.498448
$892$	0	0
$893$	−14.9500	−0.500283
$894$	0	0
$895$	11.5724	0.386824
$896$	0	0
$897$	14.1776	0.473377
$898$	0	0
$899$	−33.1258	−1.10481
$900$	0	0
$901$	21.8489	0.727893
$902$	0	0
$903$	−96.0452	−3.19618
$904$	0	0
$905$	48.7770	1.62140
$906$	0	0
$907$	−14.6671	−0.487013	−0.243506	−	0.969899i	$-0.578298\pi$
$907$	−14.6671	−0.487013	−0.243506	+	0.969899i	$0.578298\pi$
$908$	0	0
$909$	78.7838	2.61309
$910$	0	0
$911$	17.4614	0.578522	0.289261	−	0.957250i	$-0.406590\pi$
$911$	17.4614	0.578522	0.289261	+	0.957250i	$0.406590\pi$
$912$	0	0
$913$	−19.9791	−0.661212
$914$	0	0
$915$	−44.1741	−1.46035
$916$	0	0
$917$	12.1252	0.400409
$918$	0	0
$919$	−14.9447	−0.492979	−0.246489	−	0.969145i	$-0.579277\pi$
$919$	−14.9447	−0.492979	−0.246489	+	0.969145i	$0.579277\pi$
$920$	0	0
$921$	−22.6589	−0.746636
$922$	0	0
$923$	−6.33926	−0.208659
$924$	0	0
$925$	−48.7650	−1.60338
$926$	0	0
$927$	53.3242	1.75140
$928$	0	0
$929$	−32.6470	−1.07111	−0.535557	−	0.844499i	$-0.679898\pi$
$929$	−32.6470	−1.07111	−0.535557	+	0.844499i	$0.679898\pi$
$930$	0	0
$931$	−17.7049	−0.580256
$932$	0	0
$933$	29.7631	0.974399
$934$	0	0
$935$	−79.8102	−2.61007
$936$	0	0
$937$	−19.4445	−0.635224	−0.317612	−	0.948221i	$-0.602881\pi$
$937$	−19.4445	−0.635224	−0.317612	+	0.948221i	$0.602881\pi$
$938$	0	0
$939$	−81.5827	−2.66235
$940$	0	0
$941$	19.1242	0.623430	0.311715	−	0.950176i	$-0.399097\pi$
$941$	19.1242	0.623430	0.311715	+	0.950176i	$0.399097\pi$
$942$	0	0
$943$	28.5068	0.928310
$944$	0	0
$945$	−124.461	−4.04873
$946$	0	0
$947$	11.4645	0.372547	0.186274	−	0.982498i	$-0.440359\pi$
$947$	11.4645	0.372547	0.186274	+	0.982498i	$0.440359\pi$
$948$	0	0
$949$	3.42625	0.111221
$950$	0	0
$951$	17.3610	0.562969
$952$	0	0
$953$	−34.0787	−1.10392	−0.551959	−	0.833871i	$-0.686119\pi$
$953$	−34.0787	−1.10392	−0.551959	+	0.833871i	$0.686119\pi$
$954$	0	0
$955$	−0.581747	−0.0188249
$956$	0	0
$957$	−27.8574	−0.900501
$958$	0	0
$959$	26.7614	0.864172
$960$	0	0
$961$	39.4931	1.27397
$962$	0	0
$963$	69.2493	2.23153
$964$	0	0
$965$	−30.5801	−0.984408
$966$	0	0
$967$	26.0712	0.838392	0.419196	−	0.907896i	$-0.362312\pi$
$967$	26.0712	0.838392	0.419196	+	0.907896i	$0.362312\pi$
$968$	0	0
$969$	54.0299	1.73569
$970$	0	0
$971$	0.547670	0.0175756	0.00878779	−	0.999961i	$-0.497203\pi$
$971$	0.547670	0.0175756	0.00878779	+	0.999961i	$0.497203\pi$
$972$	0	0
$973$	−53.2820	−1.70814
$974$	0	0
$975$	29.6483	0.949504
$976$	0	0
$977$	9.54080	0.305237	0.152619	−	0.988285i	$-0.451229\pi$
$977$	9.54080	0.305237	0.152619	+	0.988285i	$0.451229\pi$
$978$	0	0
$979$	−32.0710	−1.02499
$980$	0	0
$981$	−31.7153	−1.01259
$982$	0	0
$983$	35.6875	1.13825	0.569127	−	0.822249i	$-0.307281\pi$
$983$	35.6875	1.13825	0.569127	+	0.822249i	$0.307281\pi$
$984$	0	0
$985$	−3.58701	−0.114292
$986$	0	0
$987$	75.2859	2.39638
$988$	0	0
$989$	−47.5496	−1.51199
$990$	0	0
$991$	17.6407	0.560376	0.280188	−	0.959945i	$-0.409603\pi$
$991$	17.6407	0.560376	0.280188	+	0.959945i	$0.409603\pi$
$992$	0	0
$993$	−7.54698	−0.239496
$994$	0	0
$995$	−5.90266	−0.187127
$996$	0	0
$997$	−3.25227	−0.103000	−0.0515002	−	0.998673i	$-0.516400\pi$
$997$	−3.25227	−0.103000	−0.0515002	+	0.998673i	$0.516400\pi$
$998$	0	0
$999$	32.7327	1.03562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.u.1.4		26
4.3	odd	2		503.2.a.f.1.6	✓	26
12.11	even	2		4527.2.a.o.1.21		26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.f.1.6	✓	26	4.3	odd	2
4527.2.a.o.1.21		26	12.11	even	2
8048.2.a.u.1.4		26	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)

\(f(q)\)	\(=\)	\(q-2.94588 q^{3} -4.09318 q^{5} -3.85397 q^{7} +5.67824 q^{9} +O(q^{10})\)	\(q-2.94588 q^{3} -4.09318 q^{5} -3.85397 q^{7} +5.67824 q^{9} +2.39680 q^{11} -0.856234 q^{13} +12.0580 q^{15} +8.13515 q^{17} -2.25451 q^{19} +11.3534 q^{21} +5.62077 q^{23} +11.7541 q^{25} -7.88977 q^{27} +3.94542 q^{29} -8.39602 q^{31} -7.06069 q^{33} +15.7750 q^{35} -4.14875 q^{37} +2.52237 q^{39} +5.07170 q^{41} -8.45963 q^{43} -23.2421 q^{45} +6.63115 q^{47} +7.85311 q^{49} -23.9652 q^{51} +2.68574 q^{53} -9.81053 q^{55} +6.64153 q^{57} -2.71535 q^{59} -3.66345 q^{61} -21.8838 q^{63} +3.50472 q^{65} +7.55546 q^{67} -16.5581 q^{69} +7.40365 q^{71} -4.00154 q^{73} -34.6264 q^{75} -9.23719 q^{77} -10.0635 q^{79} +6.20766 q^{81} -8.33576 q^{83} -33.2987 q^{85} -11.6228 q^{87} -13.3808 q^{89} +3.29990 q^{91} +24.7337 q^{93} +9.22813 q^{95} -1.19636 q^{97} +13.6096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9}+O(q^{10}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9	\( 26 q - 4 q^{3} + 9 q^{5} - 11 q^{7} + 42 q^{9} + 17 q^{11} + 14 q^{13} - 18 q^{15} + 17 q^{17} + 22 q^{19} - 16 q^{21} - 27 q^{23} + 93 q^{25} - 31 q^{27} + 13 q^{29} - 26 q^{31} + 6 q^{33} + 22 q^{35} + 55 q^{37} + 15 q^{39} + 24 q^{41} - 20 q^{43} - 8 q^{45} + 25 q^{47} + 65 q^{49} - 7 q^{51} + 30 q^{53} - 25 q^{55} + 9 q^{57} + 26 q^{59} + 15 q^{61} + 19 q^{63} + 20 q^{65} + 20 q^{67} - 27 q^{69} + 35 q^{71} + 38 q^{73} - 2 q^{75} - 6 q^{77} - 21 q^{79} + 70 q^{81} + 48 q^{83} + 6 q^{85} + 9 q^{87} - 5 q^{89} + 24 q^{91} - 8 q^{93} - 43 q^{95} + 142 q^{97} + 50 q^{99}+O(q^{100}) \) 26 * q - 4 * q^3 + 9 * q^5 - 11 * q^7 + 42 * q^9 + 17 * q^11 + 14 * q^13 - 18 * q^15 + 17 * q^17 + 22 * q^19 - 16 * q^21 - 27 * q^23 + 93 * q^25 - 31 * q^27 + 13 * q^29 - 26 * q^31 + 6 * q^33 + 22 * q^35 + 55 * q^37 + 15 * q^39 + 24 * q^41 - 20 * q^43 - 8 * q^45 + 25 * q^47 + 65 * q^49 - 7 * q^51 + 30 * q^53 - 25 * q^55 + 9 * q^57 + 26 * q^59 + 15 * q^61 + 19 * q^63 + 20 * q^65 + 20 * q^67 - 27 * q^69 + 35 * q^71 + 38 * q^73 - 2 * q^75 - 6 * q^77 - 21 * q^79 + 70 * q^81 + 48 * q^83 + 6 * q^85 + 9 * q^87 - 5 * q^89 + 24 * q^91 - 8 * q^93 - 43 * q^95 + 142 * q^97 + 50 * q^99

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