LMFDB - Embedded newform 8048.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.31567$ of defining polynomial
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)

$f(q)$	$=$	$q-0.315672 q^{3} +2.25024 q^{5} +3.20647 q^{7} -2.90035 q^{9} +O(q^{10})$	$q-0.315672 q^{3} +2.25024 q^{5} +3.20647 q^{7} -2.90035 q^{9} +0.218022 q^{11} -4.17856 q^{13} -0.710339 q^{15} -4.68939 q^{17} -3.43926 q^{19} -1.01219 q^{21} +3.99816 q^{23} +0.0635992 q^{25} +1.86257 q^{27} -0.712153 q^{29} -1.04933 q^{31} -0.0688235 q^{33} +7.21534 q^{35} +1.70998 q^{37} +1.31905 q^{39} +3.18460 q^{41} +6.35890 q^{43} -6.52650 q^{45} +3.87861 q^{47} +3.28144 q^{49} +1.48031 q^{51} -10.8877 q^{53} +0.490604 q^{55} +1.08568 q^{57} +2.63793 q^{59} +11.1544 q^{61} -9.29988 q^{63} -9.40279 q^{65} -6.09170 q^{67} -1.26210 q^{69} -9.40515 q^{71} -2.78532 q^{73} -0.0200765 q^{75} +0.699082 q^{77} -5.16993 q^{79} +8.11309 q^{81} -1.83793 q^{83} -10.5523 q^{85} +0.224807 q^{87} -8.08786 q^{89} -13.3984 q^{91} +0.331245 q^{93} -7.73917 q^{95} -12.9058 q^{97} -0.632342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	$ 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) $ 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.315672	−0.182253	−0.0911266	−	0.995839i	$-0.529047\pi$
$3$	−0.315672	−0.182253	−0.0911266	+	0.995839i	$0.529047\pi$
$4$	0	0
$5$	2.25024	1.00634	0.503170	−	0.864188i	$-0.332167\pi$
$5$	2.25024	1.00634	0.503170	+	0.864188i	$0.332167\pi$
$6$	0	0
$7$	3.20647	1.21193	0.605966	−	0.795491i	$-0.292787\pi$
$7$	3.20647	1.21193	0.605966	+	0.795491i	$0.292787\pi$
$8$	0	0
$9$	−2.90035	−0.966784
$10$	0	0
$11$	0.218022	0.0657362	0.0328681	−	0.999460i	$-0.489536\pi$
$11$	0.218022	0.0657362	0.0328681	+	0.999460i	$0.489536\pi$
$12$	0	0
$13$	−4.17856	−1.15892	−0.579462	−	0.814999i	$-0.696737\pi$
$13$	−4.17856	−1.15892	−0.579462	+	0.814999i	$0.696737\pi$
$14$	0	0
$15$	−0.710339	−0.183409
$16$	0	0
$17$	−4.68939	−1.13734	−0.568672	−	0.822564i	$-0.692543\pi$
$17$	−4.68939	−1.13734	−0.568672	+	0.822564i	$0.692543\pi$
$18$	0	0
$19$	−3.43926	−0.789020	−0.394510	−	0.918892i	$-0.629086\pi$
$19$	−3.43926	−0.789020	−0.394510	+	0.918892i	$0.629086\pi$
$20$	0	0
$21$	−1.01219	−0.220878
$22$	0	0
$23$	3.99816	0.833673	0.416837	−	0.908981i	$-0.363139\pi$
$23$	3.99816	0.833673	0.416837	+	0.908981i	$0.363139\pi$
$24$	0	0
$25$	0.0635992	0.0127198
$26$	0	0
$27$	1.86257	0.358453
$28$	0	0
$29$	−0.712153	−0.132243	−0.0661217	−	0.997812i	$-0.521063\pi$
$29$	−0.712153	−0.132243	−0.0661217	+	0.997812i	$0.521063\pi$
$30$	0	0
$31$	−1.04933	−0.188466	−0.0942329	−	0.995550i	$-0.530040\pi$
$31$	−1.04933	−0.188466	−0.0942329	+	0.995550i	$0.530040\pi$
$32$	0	0
$33$	−0.0688235	−0.0119806
$34$	0	0
$35$	7.21534	1.21961
$36$	0	0
$37$	1.70998	0.281119	0.140560	−	0.990072i	$-0.455110\pi$
$37$	1.70998	0.281119	0.140560	+	0.990072i	$0.455110\pi$
$38$	0	0
$39$	1.31905	0.211218
$40$	0	0
$41$	3.18460	0.497351	0.248675	−	0.968587i	$-0.420005\pi$
$41$	3.18460	0.497351	0.248675	+	0.968587i	$0.420005\pi$
$42$	0	0
$43$	6.35890	0.969723	0.484861	−	0.874591i	$-0.338870\pi$
$43$	6.35890	0.969723	0.484861	+	0.874591i	$0.338870\pi$
$44$	0	0
$45$	−6.52650	−0.972913
$46$	0	0
$47$	3.87861	0.565754	0.282877	−	0.959156i	$-0.408711\pi$
$47$	3.87861	0.565754	0.282877	+	0.959156i	$0.408711\pi$
$48$	0	0
$49$	3.28144	0.468777
$50$	0	0
$51$	1.48031	0.207285
$52$	0	0
$53$	−10.8877	−1.49555	−0.747773	−	0.663954i	$-0.768877\pi$
$53$	−10.8877	−1.49555	−0.747773	+	0.663954i	$0.768877\pi$
$54$	0	0
$55$	0.490604	0.0661530
$56$	0	0
$57$	1.08568	0.143801
$58$	0	0
$59$	2.63793	0.343429	0.171715	−	0.985147i	$-0.445069\pi$
$59$	2.63793	0.343429	0.171715	+	0.985147i	$0.445069\pi$
$60$	0	0
$61$	11.1544	1.42817	0.714084	−	0.700060i	$-0.246843\pi$
$61$	11.1544	1.42817	0.714084	+	0.700060i	$0.246843\pi$
$62$	0	0
$63$	−9.29988	−1.17168
$64$	0	0
$65$	−9.40279	−1.16627
$66$	0	0
$67$	−6.09170	−0.744219	−0.372110	−	0.928189i	$-0.621365\pi$
$67$	−6.09170	−0.744219	−0.372110	+	0.928189i	$0.621365\pi$
$68$	0	0
$69$	−1.26210	−0.151940
$70$	0	0
$71$	−9.40515	−1.11619	−0.558093	−	0.829778i	$-0.688467\pi$
$71$	−9.40515	−1.11619	−0.558093	+	0.829778i	$0.688467\pi$
$72$	0	0
$73$	−2.78532	−0.325997	−0.162998	−	0.986626i	$-0.552117\pi$
$73$	−2.78532	−0.325997	−0.162998	+	0.986626i	$0.552117\pi$
$74$	0	0
$75$	−0.0200765	−0.00231823
$76$	0	0
$77$	0.699082	0.0796678
$78$	0	0
$79$	−5.16993	−0.581662	−0.290831	−	0.956774i	$-0.593932\pi$
$79$	−5.16993	−0.581662	−0.290831	+	0.956774i	$0.593932\pi$
$80$	0	0
$81$	8.11309	0.901455
$82$	0	0
$83$	−1.83793	−0.201739	−0.100870	−	0.994900i	$-0.532162\pi$
$83$	−1.83793	−0.201739	−0.100870	+	0.994900i	$0.532162\pi$
$84$	0	0
$85$	−10.5523	−1.14455
$86$	0	0
$87$	0.224807	0.0241018
$88$	0	0
$89$	−8.08786	−0.857311	−0.428656	−	0.903468i	$-0.641013\pi$
$89$	−8.08786	−0.857311	−0.428656	+	0.903468i	$0.641013\pi$
$90$	0	0
$91$	−13.3984	−1.40454
$92$	0	0
$93$	0.331245	0.0343485
$94$	0	0
$95$	−7.73917	−0.794022
$96$	0	0
$97$	−12.9058	−1.31039	−0.655193	−	0.755462i	$-0.727413\pi$
$97$	−12.9058	−1.31039	−0.655193	+	0.755462i	$0.727413\pi$
$98$	0	0
$99$	−0.632342	−0.0635527
$100$	0	0
$101$	2.17744	0.216663	0.108331	−	0.994115i	$-0.465449\pi$
$101$	2.17744	0.216663	0.108331	+	0.994115i	$0.465449\pi$
$102$	0	0
$103$	−14.7612	−1.45447	−0.727234	−	0.686390i	$-0.759194\pi$
$103$	−14.7612	−1.45447	−0.727234	+	0.686390i	$0.759194\pi$
$104$	0	0
$105$	−2.27768	−0.222279
$106$	0	0
$107$	−10.4459	−1.00984	−0.504920	−	0.863166i	$-0.668478\pi$
$107$	−10.4459	−1.00984	−0.504920	+	0.863166i	$0.668478\pi$
$108$	0	0
$109$	−5.69014	−0.545017	−0.272508	−	0.962153i	$-0.587853\pi$
$109$	−5.69014	−0.545017	−0.272508	+	0.962153i	$0.587853\pi$
$110$	0	0
$111$	−0.539793	−0.0512348
$112$	0	0
$113$	20.9980	1.97533	0.987665	−	0.156584i	$-0.0500481\pi$
$113$	20.9980	1.97533	0.987665	+	0.156584i	$0.0500481\pi$
$114$	0	0
$115$	8.99683	0.838958
$116$	0	0
$117$	12.1193	1.12043
$118$	0	0
$119$	−15.0364	−1.37838
$120$	0	0
$121$	−10.9525	−0.995679
$122$	0	0
$123$	−1.00529	−0.0906438
$124$	0	0
$125$	−11.1081	−0.993539
$126$	0	0
$127$	0.0956906	0.00849117	0.00424558	−	0.999991i	$-0.498649\pi$
$127$	0.0956906	0.00849117	0.00424558	+	0.999991i	$0.498649\pi$
$128$	0	0
$129$	−2.00732	−0.176735
$130$	0	0
$131$	−14.5638	−1.27244	−0.636221	−	0.771507i	$-0.719504\pi$
$131$	−14.5638	−1.27244	−0.636221	+	0.771507i	$0.719504\pi$
$132$	0	0
$133$	−11.0279	−0.956238
$134$	0	0
$135$	4.19125	0.360725
$136$	0	0
$137$	−17.1079	−1.46162	−0.730811	−	0.682579i	$-0.760858\pi$
$137$	−17.1079	−1.46162	−0.730811	+	0.682579i	$0.760858\pi$
$138$	0	0
$139$	17.7270	1.50358	0.751791	−	0.659402i	$-0.229190\pi$
$139$	17.7270	1.50358	0.751791	+	0.659402i	$0.229190\pi$
$140$	0	0
$141$	−1.22437	−0.103110
$142$	0	0
$143$	−0.911020	−0.0761833
$144$	0	0
$145$	−1.60252	−0.133082
$146$	0	0
$147$	−1.03586	−0.0854361
$148$	0	0
$149$	−3.37396	−0.276406	−0.138203	−	0.990404i	$-0.544133\pi$
$149$	−3.37396	−0.276406	−0.138203	+	0.990404i	$0.544133\pi$
$150$	0	0
$151$	−2.76168	−0.224742	−0.112371	−	0.993666i	$-0.535845\pi$
$151$	−2.76168	−0.224742	−0.112371	+	0.993666i	$0.535845\pi$
$152$	0	0
$153$	13.6009	1.09957
$154$	0	0
$155$	−2.36126	−0.189661
$156$	0	0
$157$	−8.93566	−0.713144	−0.356572	−	0.934268i	$-0.616054\pi$
$157$	−8.93566	−0.713144	−0.356572	+	0.934268i	$0.616054\pi$
$158$	0	0
$159$	3.43695	0.272568
$160$	0	0
$161$	12.8200	1.01035
$162$	0	0
$163$	8.75660	0.685870	0.342935	−	0.939359i	$-0.388579\pi$
$163$	8.75660	0.685870	0.342935	+	0.939359i	$0.388579\pi$
$164$	0	0
$165$	−0.154870	−0.0120566
$166$	0	0
$167$	−14.3605	−1.11125	−0.555623	−	0.831434i	$-0.687520\pi$
$167$	−14.3605	−1.11125	−0.555623	+	0.831434i	$0.687520\pi$
$168$	0	0
$169$	4.46038	0.343106
$170$	0	0
$171$	9.97506	0.762812
$172$	0	0
$173$	−11.0498	−0.840098	−0.420049	−	0.907501i	$-0.637987\pi$
$173$	−11.0498	−0.840098	−0.420049	+	0.907501i	$0.637987\pi$
$174$	0	0
$175$	0.203929	0.0154156
$176$	0	0
$177$	−0.832720	−0.0625910
$178$	0	0
$179$	6.55060	0.489615	0.244808	−	0.969572i	$-0.421275\pi$
$179$	6.55060	0.489615	0.244808	+	0.969572i	$0.421275\pi$
$180$	0	0
$181$	−13.9694	−1.03834	−0.519169	−	0.854672i	$-0.673758\pi$
$181$	−13.9694	−1.03834	−0.519169	+	0.854672i	$0.673758\pi$
$182$	0	0
$183$	−3.52111	−0.260288
$184$	0	0
$185$	3.84787	0.282901
$186$	0	0
$187$	−1.02239	−0.0747647
$188$	0	0
$189$	5.97229	0.434420
$190$	0	0
$191$	−2.23996	−0.162078	−0.0810390	−	0.996711i	$-0.525824\pi$
$191$	−2.23996	−0.162078	−0.0810390	+	0.996711i	$0.525824\pi$
$192$	0	0
$193$	22.1924	1.59744	0.798722	−	0.601700i	$-0.205510\pi$
$193$	22.1924	1.59744	0.798722	+	0.601700i	$0.205510\pi$
$194$	0	0
$195$	2.96819	0.212557
$196$	0	0
$197$	−11.0264	−0.785595	−0.392797	−	0.919625i	$-0.628493\pi$
$197$	−11.0264	−0.785595	−0.392797	+	0.919625i	$0.628493\pi$
$198$	0	0
$199$	−14.6196	−1.03636	−0.518179	−	0.855272i	$-0.673390\pi$
$199$	−14.6196	−1.03636	−0.518179	+	0.855272i	$0.673390\pi$
$200$	0	0
$201$	1.92298	0.135636
$202$	0	0
$203$	−2.28350	−0.160270
$204$	0	0
$205$	7.16612	0.500504
$206$	0	0
$207$	−11.5961	−0.805982
$208$	0	0
$209$	−0.749835	−0.0518672
$210$	0	0
$211$	−1.54449	−0.106327	−0.0531635	−	0.998586i	$-0.516930\pi$
$211$	−1.54449	−0.106327	−0.0531635	+	0.998586i	$0.516930\pi$
$212$	0	0
$213$	2.96894	0.203428
$214$	0	0
$215$	14.3091	0.975871
$216$	0	0
$217$	−3.36465	−0.228408
$218$	0	0
$219$	0.879246	0.0594139
$220$	0	0
$221$	19.5949	1.31810
$222$	0	0
$223$	26.6878	1.78715	0.893573	−	0.448918i	$-0.148190\pi$
$223$	26.6878	1.78715	0.893573	+	0.448918i	$0.148190\pi$
$224$	0	0
$225$	−0.184460	−0.0122973
$226$	0	0
$227$	6.41989	0.426103	0.213052	−	0.977041i	$-0.431660\pi$
$227$	6.41989	0.426103	0.213052	+	0.977041i	$0.431660\pi$
$228$	0	0
$229$	−27.8670	−1.84150	−0.920752	−	0.390149i	$-0.872423\pi$
$229$	−27.8670	−1.84150	−0.920752	+	0.390149i	$0.872423\pi$
$230$	0	0
$231$	−0.220680	−0.0145197
$232$	0	0
$233$	0.647350	0.0424093	0.0212047	−	0.999775i	$-0.493250\pi$
$233$	0.647350	0.0424093	0.0212047	+	0.999775i	$0.493250\pi$
$234$	0	0
$235$	8.72782	0.569340
$236$	0	0
$237$	1.63200	0.106010
$238$	0	0
$239$	12.3635	0.799727	0.399864	−	0.916575i	$-0.369057\pi$
$239$	12.3635	0.799727	0.399864	+	0.916575i	$0.369057\pi$
$240$	0	0
$241$	−12.7531	−0.821497	−0.410748	−	0.911749i	$-0.634732\pi$
$241$	−12.7531	−0.821497	−0.410748	+	0.911749i	$0.634732\pi$
$242$	0	0
$243$	−8.14880	−0.522746
$244$	0	0
$245$	7.38404	0.471749
$246$	0	0
$247$	14.3712	0.914415
$248$	0	0
$249$	0.580183	0.0367676
$250$	0	0
$251$	13.4917	0.851588	0.425794	−	0.904820i	$-0.359995\pi$
$251$	13.4917	0.851588	0.425794	+	0.904820i	$0.359995\pi$
$252$	0	0
$253$	0.871688	0.0548025
$254$	0	0
$255$	3.33105	0.208599
$256$	0	0
$257$	−4.56318	−0.284643	−0.142322	−	0.989820i	$-0.545457\pi$
$257$	−4.56318	−0.284643	−0.142322	+	0.989820i	$0.545457\pi$
$258$	0	0
$259$	5.48300	0.340697
$260$	0	0
$261$	2.06549	0.127851
$262$	0	0
$263$	−16.0985	−0.992674	−0.496337	−	0.868130i	$-0.665322\pi$
$263$	−16.0985	−0.992674	−0.496337	+	0.868130i	$0.665322\pi$
$264$	0	0
$265$	−24.5001	−1.50503
$266$	0	0
$267$	2.55311	0.156248
$268$	0	0
$269$	−7.49840	−0.457186	−0.228593	−	0.973522i	$-0.573412\pi$
$269$	−7.49840	−0.457186	−0.228593	+	0.973522i	$0.573412\pi$
$270$	0	0
$271$	−6.47424	−0.393282	−0.196641	−	0.980476i	$-0.563003\pi$
$271$	−6.47424	−0.393282	−0.196641	+	0.980476i	$0.563003\pi$
$272$	0	0
$273$	4.22951	0.255981
$274$	0	0
$275$	0.0138660	0.000836154 0
$276$	0	0
$277$	8.32050	0.499930	0.249965	−	0.968255i	$-0.419581\pi$
$277$	8.32050	0.499930	0.249965	+	0.968255i	$0.419581\pi$
$278$	0	0
$279$	3.04344	0.182206
$280$	0	0
$281$	−26.3798	−1.57369	−0.786844	−	0.617152i	$-0.788286\pi$
$281$	−26.3798	−1.57369	−0.786844	+	0.617152i	$0.788286\pi$
$282$	0	0
$283$	23.4048	1.39127	0.695637	−	0.718394i	$-0.255122\pi$
$283$	23.4048	1.39127	0.695637	+	0.718394i	$0.255122\pi$
$284$	0	0
$285$	2.44304	0.144713
$286$	0	0
$287$	10.2113	0.602755
$288$	0	0
$289$	4.99036	0.293551
$290$	0	0
$291$	4.07400	0.238822
$292$	0	0
$293$	−3.57088	−0.208613	−0.104307	−	0.994545i	$-0.533262\pi$
$293$	−3.57088	−0.208613	−0.104307	+	0.994545i	$0.533262\pi$
$294$	0	0
$295$	5.93598	0.345606
$296$	0	0
$297$	0.406083	0.0235633
$298$	0	0
$299$	−16.7065	−0.966164
$300$	0	0
$301$	20.3896	1.17524
$302$	0	0
$303$	−0.687355	−0.0394875
$304$	0	0
$305$	25.1000	1.43722
$306$	0	0
$307$	2.46828	0.140872	0.0704362	−	0.997516i	$-0.477561\pi$
$307$	2.46828	0.140872	0.0704362	+	0.997516i	$0.477561\pi$
$308$	0	0
$309$	4.65970	0.265081
$310$	0	0
$311$	19.8949	1.12814	0.564069	−	0.825727i	$-0.309235\pi$
$311$	19.8949	1.12814	0.564069	+	0.825727i	$0.309235\pi$
$312$	0	0
$313$	−10.0179	−0.566246	−0.283123	−	0.959084i	$-0.591370\pi$
$313$	−10.0179	−0.566246	−0.283123	+	0.959084i	$0.591370\pi$
$314$	0	0
$315$	−20.9270	−1.17910
$316$	0	0
$317$	−11.2440	−0.631529	−0.315764	−	0.948838i	$-0.602261\pi$
$317$	−11.2440	−0.631529	−0.315764	+	0.948838i	$0.602261\pi$
$318$	0	0
$319$	−0.155265	−0.00869319
$320$	0	0
$321$	3.29747	0.184047
$322$	0	0
$323$	16.1280	0.897387
$324$	0	0
$325$	−0.265753	−0.0147413
$326$	0	0
$327$	1.79622	0.0993311
$328$	0	0
$329$	12.4366	0.685654
$330$	0	0
$331$	30.0728	1.65295	0.826475	−	0.562974i	$-0.190343\pi$
$331$	30.0728	1.65295	0.826475	+	0.562974i	$0.190343\pi$
$332$	0	0
$333$	−4.95954	−0.271781
$334$	0	0
$335$	−13.7078	−0.748937
$336$	0	0
$337$	−17.3224	−0.943613	−0.471807	−	0.881702i	$-0.656398\pi$
$337$	−17.3224	−0.943613	−0.471807	+	0.881702i	$0.656398\pi$
$338$	0	0
$339$	−6.62849	−0.360010
$340$	0	0
$341$	−0.228778	−0.0123890
$342$	0	0
$343$	−11.9234	−0.643806
$344$	0	0
$345$	−2.84004	−0.152903
$346$	0	0
$347$	−21.2412	−1.14029	−0.570143	−	0.821545i	$-0.693112\pi$
$347$	−21.2412	−1.14029	−0.570143	+	0.821545i	$0.693112\pi$
$348$	0	0
$349$	11.6817	0.625308	0.312654	−	0.949867i	$-0.398782\pi$
$349$	11.6817	0.625308	0.312654	+	0.949867i	$0.398782\pi$
$350$	0	0
$351$	−7.78288	−0.415420
$352$	0	0
$353$	14.0775	0.749271	0.374636	−	0.927172i	$-0.377768\pi$
$353$	14.0775	0.749271	0.374636	+	0.927172i	$0.377768\pi$
$354$	0	0
$355$	−21.1639	−1.12326
$356$	0	0
$357$	4.74656	0.251215
$358$	0	0
$359$	−7.85706	−0.414680	−0.207340	−	0.978269i	$-0.566481\pi$
$359$	−7.85706	−0.414680	−0.207340	+	0.978269i	$0.566481\pi$
$360$	0	0
$361$	−7.17150	−0.377448
$362$	0	0
$363$	3.45738	0.181466
$364$	0	0
$365$	−6.26764	−0.328063
$366$	0	0
$367$	19.7726	1.03212	0.516062	−	0.856551i	$-0.327398\pi$
$367$	19.7726	1.03212	0.516062	+	0.856551i	$0.327398\pi$
$368$	0	0
$369$	−9.23646	−0.480831
$370$	0	0
$371$	−34.9112	−1.81250
$372$	0	0
$373$	7.49088	0.387863	0.193931	−	0.981015i	$-0.437876\pi$
$373$	7.49088	0.387863	0.193931	+	0.981015i	$0.437876\pi$
$374$	0	0
$375$	3.50652	0.181076
$376$	0	0
$377$	2.97577	0.153260
$378$	0	0
$379$	−27.5064	−1.41291	−0.706455	−	0.707758i	$-0.749707\pi$
$379$	−27.5064	−1.41291	−0.706455	+	0.707758i	$0.749707\pi$
$380$	0	0
$381$	−0.0302068	−0.00154754
$382$	0	0
$383$	−4.92412	−0.251611	−0.125805	−	0.992055i	$-0.540151\pi$
$383$	−4.92412	−0.251611	−0.125805	+	0.992055i	$0.540151\pi$
$384$	0	0
$385$	1.57311	0.0801729
$386$	0	0
$387$	−18.4430	−0.937512
$388$	0	0
$389$	38.9916	1.97696	0.988478	−	0.151368i	$-0.0483678\pi$
$389$	38.9916	1.97696	0.988478	+	0.151368i	$0.0483678\pi$
$390$	0	0
$391$	−18.7489	−0.948173
$392$	0	0
$393$	4.59737	0.231907
$394$	0	0
$395$	−11.6336	−0.585350
$396$	0	0
$397$	−12.6661	−0.635694	−0.317847	−	0.948142i	$-0.602960\pi$
$397$	−12.6661	−0.635694	−0.317847	+	0.948142i	$0.602960\pi$
$398$	0	0
$399$	3.48119	0.174277
$400$	0	0
$401$	26.7959	1.33812	0.669062	−	0.743207i	$-0.266696\pi$
$401$	26.7959	1.33812	0.669062	+	0.743207i	$0.266696\pi$
$402$	0	0
$403$	4.38471	0.218418
$404$	0	0
$405$	18.2564	0.907170
$406$	0	0
$407$	0.372814	0.0184797
$408$	0	0
$409$	35.3472	1.74781	0.873903	−	0.486101i	$-0.161581\pi$
$409$	35.3472	1.74781	0.873903	+	0.486101i	$0.161581\pi$
$410$	0	0
$411$	5.40047	0.266385
$412$	0	0
$413$	8.45844	0.416212
$414$	0	0
$415$	−4.13580	−0.203018
$416$	0	0
$417$	−5.59590	−0.274033
$418$	0	0
$419$	2.49358	0.121819	0.0609096	−	0.998143i	$-0.480600\pi$
$419$	2.49358	0.121819	0.0609096	+	0.998143i	$0.480600\pi$
$420$	0	0
$421$	17.2671	0.841547	0.420773	−	0.907166i	$-0.361759\pi$
$421$	17.2671	0.841547	0.420773	+	0.907166i	$0.361759\pi$
$422$	0	0
$423$	−11.2493	−0.546961
$424$	0	0
$425$	−0.298241	−0.0144668
$426$	0	0
$427$	35.7661	1.73084
$428$	0	0
$429$	0.287583	0.0138847
$430$	0	0
$431$	−31.0699	−1.49659	−0.748293	−	0.663368i	$-0.769126\pi$
$431$	−31.0699	−1.49659	−0.748293	+	0.663368i	$0.769126\pi$
$432$	0	0
$433$	16.4828	0.792112	0.396056	−	0.918226i	$-0.370379\pi$
$433$	16.4828	0.792112	0.396056	+	0.918226i	$0.370379\pi$
$434$	0	0
$435$	0.505870	0.0242546
$436$	0	0
$437$	−13.7507	−0.657785
$438$	0	0
$439$	−33.0499	−1.57739	−0.788693	−	0.614787i	$-0.789242\pi$
$439$	−33.0499	−1.57739	−0.788693	+	0.614787i	$0.789242\pi$
$440$	0	0
$441$	−9.51733	−0.453206
$442$	0	0
$443$	24.8330	1.17985	0.589925	−	0.807458i	$-0.299157\pi$
$443$	24.8330	1.17985	0.589925	+	0.807458i	$0.299157\pi$
$444$	0	0
$445$	−18.1997	−0.862747
$446$	0	0
$447$	1.06506	0.0503758
$448$	0	0
$449$	−18.6217	−0.878811	−0.439405	−	0.898289i	$-0.644811\pi$
$449$	−18.6217	−0.878811	−0.439405	+	0.898289i	$0.644811\pi$
$450$	0	0
$451$	0.694314	0.0326940
$452$	0	0
$453$	0.871784	0.0409600
$454$	0	0
$455$	−30.1497	−1.41344
$456$	0	0
$457$	−34.2762	−1.60337	−0.801686	−	0.597745i	$-0.796063\pi$
$457$	−34.2762	−1.60337	−0.801686	+	0.597745i	$0.796063\pi$
$458$	0	0
$459$	−8.73433	−0.407684
$460$	0	0
$461$	11.2220	0.522659	0.261329	−	0.965250i	$-0.415839\pi$
$461$	11.2220	0.522659	0.261329	+	0.965250i	$0.415839\pi$
$462$	0	0
$463$	26.9770	1.25373	0.626863	−	0.779129i	$-0.284338\pi$
$463$	26.9770	1.25373	0.626863	+	0.779129i	$0.284338\pi$
$464$	0	0
$465$	0.745382	0.0345663
$466$	0	0
$467$	−11.5336	−0.533711	−0.266856	−	0.963737i	$-0.585985\pi$
$467$	−11.5336	−0.533711	−0.266856	+	0.963737i	$0.585985\pi$
$468$	0	0
$469$	−19.5328	−0.901942
$470$	0	0
$471$	2.82074	0.129973
$472$	0	0
$473$	1.38638	0.0637459
$474$	0	0
$475$	−0.218734	−0.0100362
$476$	0	0
$477$	31.5783	1.44587
$478$	0	0
$479$	28.8893	1.31998	0.659992	−	0.751272i	$-0.270559\pi$
$479$	28.8893	1.31998	0.659992	+	0.751272i	$0.270559\pi$
$480$	0	0
$481$	−7.14526	−0.325796
$482$	0	0
$483$	−4.04690	−0.184140
$484$	0	0
$485$	−29.0412	−1.31869
$486$	0	0
$487$	−1.63083	−0.0739001	−0.0369501	−	0.999317i	$-0.511764\pi$
$487$	−1.63083	−0.0739001	−0.0369501	+	0.999317i	$0.511764\pi$
$488$	0	0
$489$	−2.76421	−0.125002
$490$	0	0
$491$	6.45592	0.291352	0.145676	−	0.989332i	$-0.453464\pi$
$491$	6.45592	0.291352	0.145676	+	0.989332i	$0.453464\pi$
$492$	0	0
$493$	3.33956	0.150406
$494$	0	0
$495$	−1.42292	−0.0639556
$496$	0	0
$497$	−30.1573	−1.35274
$498$	0	0
$499$	−29.7187	−1.33039	−0.665196	−	0.746669i	$-0.731652\pi$
$499$	−29.7187	−1.33039	−0.665196	+	0.746669i	$0.731652\pi$
$500$	0	0
$501$	4.53320	0.202528
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	4.89976	0.218037
$506$	0	0
$507$	−1.40802	−0.0625323
$508$	0	0
$509$	7.29140	0.323186	0.161593	−	0.986858i	$-0.448337\pi$
$509$	7.29140	0.323186	0.161593	+	0.986858i	$0.448337\pi$
$510$	0	0
$511$	−8.93103	−0.395086
$512$	0	0
$513$	−6.40587	−0.282826
$514$	0	0
$515$	−33.2164	−1.46369
$516$	0	0
$517$	0.845624	0.0371905
$518$	0	0
$519$	3.48810	0.153110
$520$	0	0
$521$	1.73107	0.0758394	0.0379197	−	0.999281i	$-0.487927\pi$
$521$	1.73107	0.0758394	0.0379197	+	0.999281i	$0.487927\pi$
$522$	0	0
$523$	−15.8653	−0.693742	−0.346871	−	0.937913i	$-0.612756\pi$
$523$	−15.8653	−0.693742	−0.346871	+	0.937913i	$0.612756\pi$
$524$	0	0
$525$	−0.0643745	−0.00280953
$526$	0	0
$527$	4.92073	0.214350
$528$	0	0
$529$	−7.01475	−0.304989
$530$	0	0
$531$	−7.65092	−0.332022
$532$	0	0
$533$	−13.3070	−0.576392
$534$	0	0
$535$	−23.5058	−1.01624
$536$	0	0
$537$	−2.06784	−0.0892339
$538$	0	0
$539$	0.715427	0.0308156
$540$	0	0
$541$	0.384069	0.0165124	0.00825621	−	0.999966i	$-0.497372\pi$
$541$	0.384069	0.0165124	0.00825621	+	0.999966i	$0.497372\pi$
$542$	0	0
$543$	4.40975	0.189240
$544$	0	0
$545$	−12.8042	−0.548472
$546$	0	0
$547$	−38.3367	−1.63916	−0.819579	−	0.572967i	$-0.805792\pi$
$547$	−38.3367	−1.63916	−0.819579	+	0.572967i	$0.805792\pi$
$548$	0	0
$549$	−32.3515	−1.38073
$550$	0	0
$551$	2.44928	0.104343
$552$	0	0
$553$	−16.5772	−0.704935
$554$	0	0
$555$	−1.21467	−0.0515597
$556$	0	0
$557$	25.2935	1.07172	0.535860	−	0.844307i	$-0.319987\pi$
$557$	25.2935	1.07172	0.535860	+	0.844307i	$0.319987\pi$
$558$	0	0
$559$	−26.5710	−1.12384
$560$	0	0
$561$	0.322740	0.0136261
$562$	0	0
$563$	−32.5764	−1.37293	−0.686465	−	0.727163i	$-0.740839\pi$
$563$	−32.5764	−1.37293	−0.686465	+	0.727163i	$0.740839\pi$
$564$	0	0
$565$	47.2507	1.98785
$566$	0	0
$567$	26.0144	1.09250
$568$	0	0
$569$	−36.9548	−1.54923	−0.774614	−	0.632435i	$-0.782056\pi$
$569$	−36.9548	−1.54923	−0.774614	+	0.632435i	$0.782056\pi$
$570$	0	0
$571$	32.4941	1.35984	0.679919	−	0.733288i	$-0.262015\pi$
$571$	32.4941	1.35984	0.679919	+	0.733288i	$0.262015\pi$
$572$	0	0
$573$	0.707093	0.0295393
$574$	0	0
$575$	0.254279	0.0106042
$576$	0	0
$577$	−3.08889	−0.128592	−0.0642960	−	0.997931i	$-0.520480\pi$
$577$	−3.08889	−0.128592	−0.0642960	+	0.997931i	$0.520480\pi$
$578$	0	0
$579$	−7.00552	−0.291139
$580$	0	0
$581$	−5.89327	−0.244494
$582$	0	0
$583$	−2.37377	−0.0983116
$584$	0	0
$585$	27.2714	1.12753
$586$	0	0
$587$	39.5581	1.63274	0.816369	−	0.577530i	$-0.195983\pi$
$587$	39.5581	1.63274	0.816369	+	0.577530i	$0.195983\pi$
$588$	0	0
$589$	3.60893	0.148703
$590$	0	0
$591$	3.48071	0.143177
$592$	0	0
$593$	37.0809	1.52273	0.761366	−	0.648323i	$-0.224529\pi$
$593$	37.0809	1.52273	0.761366	+	0.648323i	$0.224529\pi$
$594$	0	0
$595$	−33.8355	−1.38712
$596$	0	0
$597$	4.61501	0.188880
$598$	0	0
$599$	7.21058	0.294616	0.147308	−	0.989091i	$-0.452939\pi$
$599$	7.21058	0.294616	0.147308	+	0.989091i	$0.452939\pi$
$600$	0	0
$601$	−15.6443	−0.638145	−0.319072	−	0.947730i	$-0.603371\pi$
$601$	−15.6443	−0.638145	−0.319072	+	0.947730i	$0.603371\pi$
$602$	0	0
$603$	17.6681	0.719499
$604$	0	0
$605$	−24.6457	−1.00199
$606$	0	0
$607$	5.03996	0.204566	0.102283	−	0.994755i	$-0.467385\pi$
$607$	5.03996	0.204566	0.102283	+	0.994755i	$0.467385\pi$
$608$	0	0
$609$	0.720835	0.0292097
$610$	0	0
$611$	−16.2070	−0.655666
$612$	0	0
$613$	−41.4393	−1.67372	−0.836858	−	0.547420i	$-0.815610\pi$
$613$	−41.4393	−1.67372	−0.836858	+	0.547420i	$0.815610\pi$
$614$	0	0
$615$	−2.26214	−0.0912184
$616$	0	0
$617$	17.1685	0.691178	0.345589	−	0.938386i	$-0.387679\pi$
$617$	17.1685	0.691178	0.345589	+	0.938386i	$0.387679\pi$
$618$	0	0
$619$	16.9228	0.680186	0.340093	−	0.940392i	$-0.389541\pi$
$619$	16.9228	0.680186	0.340093	+	0.940392i	$0.389541\pi$
$620$	0	0
$621$	7.44686	0.298832
$622$	0	0
$623$	−25.9335	−1.03900
$624$	0	0
$625$	−25.3140	−1.01256
$626$	0	0
$627$	0.236702	0.00945296
$628$	0	0
$629$	−8.01876	−0.319729
$630$	0	0
$631$	−15.9640	−0.635518	−0.317759	−	0.948171i	$-0.602930\pi$
$631$	−15.9640	−0.635518	−0.317759	+	0.948171i	$0.602930\pi$
$632$	0	0
$633$	0.487551	0.0193784
$634$	0	0
$635$	0.215327	0.00854500
$636$	0	0
$637$	−13.7117	−0.543277
$638$	0	0
$639$	27.2782	1.07911
$640$	0	0
$641$	34.1774	1.34993	0.674964	−	0.737851i	$-0.264159\pi$
$641$	34.1774	1.34993	0.674964	+	0.737851i	$0.264159\pi$
$642$	0	0
$643$	1.27460	0.0502653	0.0251326	−	0.999684i	$-0.491999\pi$
$643$	1.27460	0.0502653	0.0251326	+	0.999684i	$0.491999\pi$
$644$	0	0
$645$	−4.51697	−0.177856
$646$	0	0
$647$	−16.4300	−0.645930	−0.322965	−	0.946411i	$-0.604680\pi$
$647$	−16.4300	−0.645930	−0.322965	+	0.946411i	$0.604680\pi$
$648$	0	0
$649$	0.575128	0.0225757
$650$	0	0
$651$	1.06213	0.0416280
$652$	0	0
$653$	46.5637	1.82218	0.911088	−	0.412211i	$-0.135243\pi$
$653$	46.5637	1.82218	0.911088	+	0.412211i	$0.135243\pi$
$654$	0	0
$655$	−32.7720	−1.28051
$656$	0	0
$657$	8.07840	0.315168
$658$	0	0
$659$	21.2263	0.826861	0.413431	−	0.910536i	$-0.364330\pi$
$659$	21.2263	0.826861	0.413431	+	0.910536i	$0.364330\pi$
$660$	0	0
$661$	31.5049	1.22540	0.612700	−	0.790316i	$-0.290084\pi$
$661$	31.5049	1.22540	0.612700	+	0.790316i	$0.290084\pi$
$662$	0	0
$663$	−6.18556	−0.240227
$664$	0	0
$665$	−24.8154	−0.962300
$666$	0	0
$667$	−2.84730	−0.110248
$668$	0	0
$669$	−8.42458	−0.325713
$670$	0	0
$671$	2.43190	0.0938824
$672$	0	0
$673$	−46.0920	−1.77672	−0.888359	−	0.459150i	$-0.848154\pi$
$673$	−46.0920	−1.77672	−0.888359	+	0.459150i	$0.848154\pi$
$674$	0	0
$675$	0.118458	0.00455946
$676$	0	0
$677$	5.22605	0.200853	0.100427	−	0.994944i	$-0.467979\pi$
$677$	5.22605	0.200853	0.100427	+	0.994944i	$0.467979\pi$
$678$	0	0
$679$	−41.3820	−1.58810
$680$	0	0
$681$	−2.02658	−0.0776587
$682$	0	0
$683$	26.7163	1.02227	0.511135	−	0.859500i	$-0.329225\pi$
$683$	26.7163	1.02227	0.511135	+	0.859500i	$0.329225\pi$
$684$	0	0
$685$	−38.4969	−1.47089
$686$	0	0
$687$	8.79683	0.335620
$688$	0	0
$689$	45.4951	1.73323
$690$	0	0
$691$	20.7575	0.789653	0.394826	−	0.918756i	$-0.370805\pi$
$691$	20.7575	0.789653	0.394826	+	0.918756i	$0.370805\pi$
$692$	0	0
$693$	−2.02758	−0.0770215
$694$	0	0
$695$	39.8900	1.51311
$696$	0	0
$697$	−14.9338	−0.565659
$698$	0	0
$699$	−0.204350	−0.00772924
$700$	0	0
$701$	−48.5428	−1.83344	−0.916719	−	0.399533i	$-0.869172\pi$
$701$	−48.5428	−1.83344	−0.916719	+	0.399533i	$0.869172\pi$
$702$	0	0
$703$	−5.88106	−0.221809
$704$	0	0
$705$	−2.75513	−0.103764
$706$	0	0
$707$	6.98188	0.262581
$708$	0	0
$709$	−21.9935	−0.825981	−0.412991	−	0.910735i	$-0.635516\pi$
$709$	−21.9935	−0.825981	−0.412991	+	0.910735i	$0.635516\pi$
$710$	0	0
$711$	14.9946	0.562342
$712$	0	0
$713$	−4.19540	−0.157119
$714$	0	0
$715$	−2.05002	−0.0766663
$716$	0	0
$717$	−3.90280	−0.145753
$718$	0	0
$719$	−9.78228	−0.364817	−0.182409	−	0.983223i	$-0.558389\pi$
$719$	−9.78228	−0.364817	−0.182409	+	0.983223i	$0.558389\pi$
$720$	0	0
$721$	−47.3314	−1.76271
$722$	0	0
$723$	4.02578	0.149720
$724$	0	0
$725$	−0.0452923	−0.00168211
$726$	0	0
$727$	41.0223	1.52143	0.760716	−	0.649085i	$-0.224848\pi$
$727$	41.0223	1.52143	0.760716	+	0.649085i	$0.224848\pi$
$728$	0	0
$729$	−21.7669	−0.806183
$730$	0	0
$731$	−29.8193	−1.10291
$732$	0	0
$733$	19.7580	0.729780	0.364890	−	0.931051i	$-0.381107\pi$
$733$	19.7580	0.729780	0.364890	+	0.931051i	$0.381107\pi$
$734$	0	0
$735$	−2.33093	−0.0859777
$736$	0	0
$737$	−1.32813	−0.0489222
$738$	0	0
$739$	−8.08268	−0.297326	−0.148663	−	0.988888i	$-0.547497\pi$
$739$	−8.08268	−0.297326	−0.148663	+	0.988888i	$0.547497\pi$
$740$	0	0
$741$	−4.53657	−0.166655
$742$	0	0
$743$	−10.5995	−0.388859	−0.194429	−	0.980917i	$-0.562286\pi$
$743$	−10.5995	−0.388859	−0.194429	+	0.980917i	$0.562286\pi$
$744$	0	0
$745$	−7.59224	−0.278158
$746$	0	0
$747$	5.33065	0.195038
$748$	0	0
$749$	−33.4944	−1.22386
$750$	0	0
$751$	−14.0003	−0.510877	−0.255438	−	0.966825i	$-0.582220\pi$
$751$	−14.0003	−0.510877	−0.255438	+	0.966825i	$0.582220\pi$
$752$	0	0
$753$	−4.25895	−0.155205
$754$	0	0
$755$	−6.21445	−0.226167
$756$	0	0
$757$	−28.8413	−1.04826	−0.524128	−	0.851640i	$-0.675609\pi$
$757$	−28.8413	−1.04826	−0.524128	+	0.851640i	$0.675609\pi$
$758$	0	0
$759$	−0.275167	−0.00998794
$760$	0	0
$761$	−14.2005	−0.514767	−0.257383	−	0.966309i	$-0.582860\pi$
$761$	−14.2005	−0.514767	−0.257383	+	0.966309i	$0.582860\pi$
$762$	0	0
$763$	−18.2453	−0.660523
$764$	0	0
$765$	30.6053	1.10654
$766$	0	0
$767$	−11.0228	−0.398008
$768$	0	0
$769$	19.1077	0.689043	0.344521	−	0.938778i	$-0.388041\pi$
$769$	19.1077	0.689043	0.344521	+	0.938778i	$0.388041\pi$
$770$	0	0
$771$	1.44047	0.0518771
$772$	0	0
$773$	−29.5204	−1.06178	−0.530888	−	0.847442i	$-0.678141\pi$
$773$	−29.5204	−1.06178	−0.530888	+	0.847442i	$0.678141\pi$
$774$	0	0
$775$	−0.0667367	−0.00239725
$776$	0	0
$777$	−1.73083	−0.0620931
$778$	0	0
$779$	−10.9527	−0.392420
$780$	0	0
$781$	−2.05053	−0.0733739
$782$	0	0
$783$	−1.32644	−0.0474030
$784$	0	0
$785$	−20.1074	−0.717665
$786$	0	0
$787$	20.3547	0.725568	0.362784	−	0.931873i	$-0.381826\pi$
$787$	20.3547	0.725568	0.362784	+	0.931873i	$0.381826\pi$
$788$	0	0
$789$	5.08183	0.180918
$790$	0	0
$791$	67.3295	2.39396
$792$	0	0
$793$	−46.6092	−1.65514
$794$	0	0
$795$	7.73398	0.274296
$796$	0	0
$797$	51.2687	1.81603	0.908016	−	0.418935i	$-0.137597\pi$
$797$	51.2687	1.81603	0.908016	+	0.418935i	$0.137597\pi$
$798$	0	0
$799$	−18.1883	−0.643456
$800$	0	0
$801$	23.4576	0.828835
$802$	0	0
$803$	−0.607262	−0.0214298
$804$	0	0
$805$	28.8480	1.01676
$806$	0	0
$807$	2.36703	0.0833235
$808$	0	0
$809$	53.5568	1.88296	0.941478	−	0.337073i	$-0.109437\pi$
$809$	53.5568	1.88296	0.941478	+	0.337073i	$0.109437\pi$
$810$	0	0
$811$	−42.8963	−1.50629	−0.753146	−	0.657854i	$-0.771465\pi$
$811$	−42.8963	−1.50629	−0.753146	+	0.657854i	$0.771465\pi$
$812$	0	0
$813$	2.04373	0.0716769
$814$	0	0
$815$	19.7045	0.690218
$816$	0	0
$817$	−21.8699	−0.765130
$818$	0	0
$819$	38.8601	1.35788
$820$	0	0
$821$	5.52686	0.192889	0.0964444	−	0.995338i	$-0.469253\pi$
$821$	5.52686	0.192889	0.0964444	+	0.995338i	$0.469253\pi$
$822$	0	0
$823$	19.0939	0.665572	0.332786	−	0.943002i	$-0.392011\pi$
$823$	19.0939	0.665572	0.332786	+	0.943002i	$0.392011\pi$
$824$	0	0
$825$	−0.00437712	−0.000152392 0
$826$	0	0
$827$	47.8774	1.66486	0.832430	−	0.554130i	$-0.186949\pi$
$827$	47.8774	1.66486	0.832430	+	0.554130i	$0.186949\pi$
$828$	0	0
$829$	13.1461	0.456583	0.228292	−	0.973593i	$-0.426686\pi$
$829$	13.1461	0.456583	0.228292	+	0.973593i	$0.426686\pi$
$830$	0	0
$831$	−2.62655	−0.0911139
$832$	0	0
$833$	−15.3879	−0.533161
$834$	0	0
$835$	−32.3146	−1.11829
$836$	0	0
$837$	−1.95446	−0.0675561
$838$	0	0
$839$	−33.7580	−1.16545	−0.582727	−	0.812668i	$-0.698014\pi$
$839$	−33.7580	−1.16545	−0.582727	+	0.812668i	$0.698014\pi$
$840$	0	0
$841$	−28.4928	−0.982512
$842$	0	0
$843$	8.32736	0.286810
$844$	0	0
$845$	10.0370	0.345282
$846$	0	0
$847$	−35.1187	−1.20669
$848$	0	0
$849$	−7.38825	−0.253564
$850$	0	0
$851$	6.83677	0.234361
$852$	0	0
$853$	16.1494	0.552946	0.276473	−	0.961022i	$-0.410834\pi$
$853$	16.1494	0.552946	0.276473	+	0.961022i	$0.410834\pi$
$854$	0	0
$855$	22.4463	0.767648
$856$	0	0
$857$	21.1749	0.723322	0.361661	−	0.932310i	$-0.382210\pi$
$857$	21.1749	0.723322	0.361661	+	0.932310i	$0.382210\pi$
$858$	0	0
$859$	21.2136	0.723798	0.361899	−	0.932217i	$-0.382129\pi$
$859$	21.2136	0.723798	0.361899	+	0.932217i	$0.382129\pi$
$860$	0	0
$861$	−3.22342	−0.109854
$862$	0	0
$863$	45.3619	1.54414	0.772068	−	0.635539i	$-0.219222\pi$
$863$	45.3619	1.54414	0.772068	+	0.635539i	$0.219222\pi$
$864$	0	0
$865$	−24.8647	−0.845424
$866$	0	0
$867$	−1.57532	−0.0535005
$868$	0	0
$869$	−1.12716	−0.0382363
$870$	0	0
$871$	25.4545	0.862494
$872$	0	0
$873$	37.4313	1.26686
$874$	0	0
$875$	−35.6178	−1.20410
$876$	0	0
$877$	12.1962	0.411837	0.205918	−	0.978569i	$-0.433982\pi$
$877$	12.1962	0.411837	0.205918	+	0.978569i	$0.433982\pi$
$878$	0	0
$879$	1.12723	0.0380204
$880$	0	0
$881$	8.50736	0.286620	0.143310	−	0.989678i	$-0.454225\pi$
$881$	8.50736	0.286620	0.143310	+	0.989678i	$0.454225\pi$
$882$	0	0
$883$	26.5027	0.891888	0.445944	−	0.895061i	$-0.352868\pi$
$883$	26.5027	0.891888	0.445944	+	0.895061i	$0.352868\pi$
$884$	0	0
$885$	−1.87382	−0.0629879
$886$	0	0
$887$	−33.1863	−1.11429	−0.557144	−	0.830416i	$-0.688103\pi$
$887$	−33.1863	−1.11429	−0.557144	+	0.830416i	$0.688103\pi$
$888$	0	0
$889$	0.306829	0.0102907
$890$	0	0
$891$	1.76884	0.0592582
$892$	0	0
$893$	−13.3395	−0.446391
$894$	0	0
$895$	14.7405	0.492719
$896$	0	0
$897$	5.27378	0.176087
$898$	0	0
$899$	0.747286	0.0249234
$900$	0	0
$901$	51.0568	1.70095
$902$	0	0
$903$	−6.43642	−0.214191
$904$	0	0
$905$	−31.4346	−1.04492
$906$	0	0
$907$	−6.31486	−0.209682	−0.104841	−	0.994489i	$-0.533433\pi$
$907$	−6.31486	−0.209682	−0.104841	+	0.994489i	$0.533433\pi$
$908$	0	0
$909$	−6.31533	−0.209466
$910$	0	0
$911$	−43.0949	−1.42780	−0.713898	−	0.700249i	$-0.753072\pi$
$911$	−43.0949	−1.42780	−0.713898	+	0.700249i	$0.753072\pi$
$912$	0	0
$913$	−0.400710	−0.0132616
$914$	0	0
$915$	−7.92337	−0.261938
$916$	0	0
$917$	−46.6983	−1.54211
$918$	0	0
$919$	19.4088	0.640236	0.320118	−	0.947378i	$-0.396277\pi$
$919$	19.4088	0.640236	0.320118	+	0.947378i	$0.396277\pi$
$920$	0	0
$921$	−0.779167	−0.0256744
$922$	0	0
$923$	39.3000	1.29358
$924$	0	0
$925$	0.108753	0.00357579
$926$	0	0
$927$	42.8128	1.40616
$928$	0	0
$929$	−28.3914	−0.931493	−0.465747	−	0.884918i	$-0.654214\pi$
$929$	−28.3914	−0.931493	−0.465747	+	0.884918i	$0.654214\pi$
$930$	0	0
$931$	−11.2857	−0.369874
$932$	0	0
$933$	−6.28027	−0.205607
$934$	0	0
$935$	−2.30063	−0.0752387
$936$	0	0
$937$	−4.00280	−0.130766	−0.0653829	−	0.997860i	$-0.520827\pi$
$937$	−4.00280	−0.130766	−0.0653829	+	0.997860i	$0.520827\pi$
$938$	0	0
$939$	3.16237	0.103200
$940$	0	0
$941$	13.9675	0.455328	0.227664	−	0.973740i	$-0.426891\pi$
$941$	13.9675	0.455328	0.227664	+	0.973740i	$0.426891\pi$
$942$	0	0
$943$	12.7325	0.414628
$944$	0	0
$945$	13.4391	0.437174
$946$	0	0
$947$	−57.2963	−1.86188	−0.930940	−	0.365171i	$-0.881010\pi$
$947$	−57.2963	−1.86188	−0.930940	+	0.365171i	$0.881010\pi$
$948$	0	0
$949$	11.6386	0.377806
$950$	0	0
$951$	3.54943	0.115098
$952$	0	0
$953$	5.29452	0.171506	0.0857532	−	0.996316i	$-0.472670\pi$
$953$	5.29452	0.171506	0.0857532	+	0.996316i	$0.472670\pi$
$954$	0	0
$955$	−5.04047	−0.163106
$956$	0	0
$957$	0.0490129	0.00158436
$958$	0	0
$959$	−54.8558	−1.77139
$960$	0	0
$961$	−29.8989	−0.964481
$962$	0	0
$963$	30.2967	0.976298
$964$	0	0
$965$	49.9383	1.60757
$966$	0	0
$967$	−44.3615	−1.42657	−0.713284	−	0.700875i	$-0.752793\pi$
$967$	−44.3615	−1.42657	−0.713284	+	0.700875i	$0.752793\pi$
$968$	0	0
$969$	−5.09116	−0.163552
$970$	0	0
$971$	−51.4789	−1.65204	−0.826018	−	0.563644i	$-0.809399\pi$
$971$	−51.4789	−1.65204	−0.826018	+	0.563644i	$0.809399\pi$
$972$	0	0
$973$	56.8410	1.82224
$974$	0	0
$975$	0.0838907	0.00268665
$976$	0	0
$977$	−0.892938	−0.0285676	−0.0142838	−	0.999898i	$-0.504547\pi$
$977$	−0.892938	−0.0285676	−0.0142838	+	0.999898i	$0.504547\pi$
$978$	0	0
$979$	−1.76333	−0.0563564
$980$	0	0
$981$	16.5034	0.526913
$982$	0	0
$983$	−36.2679	−1.15677	−0.578384	−	0.815765i	$-0.696316\pi$
$983$	−36.2679	−1.15677	−0.578384	+	0.815765i	$0.696316\pi$
$984$	0	0
$985$	−24.8120	−0.790576
$986$	0	0
$987$	−3.92590	−0.124963
$988$	0	0
$989$	25.4239	0.808432
$990$	0	0
$991$	−29.3407	−0.932036	−0.466018	−	0.884775i	$-0.654312\pi$
$991$	−29.3407	−0.932036	−0.466018	+	0.884775i	$0.654312\pi$
$992$	0	0
$993$	−9.49313	−0.301255
$994$	0	0
$995$	−32.8977	−1.04293
$996$	0	0
$997$	18.6388	0.590296	0.295148	−	0.955452i	$-0.404631\pi$
$997$	18.6388	0.590296	0.295148	+	0.955452i	$0.404631\pi$
$998$	0	0
$999$	3.18497	0.100768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.p.1.3		10
4.3	odd	2		503.2.a.e.1.7	✓	10
12.11	even	2		4527.2.a.k.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.7	✓	10	4.3	odd	2
4527.2.a.k.1.4		10	12.11	even	2
8048.2.a.p.1.3		10	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)

\(f(q)\)	\(=\)	\(q-0.315672 q^{3} +2.25024 q^{5} +3.20647 q^{7} -2.90035 q^{9} +O(q^{10})\)	\(q-0.315672 q^{3} +2.25024 q^{5} +3.20647 q^{7} -2.90035 q^{9} +0.218022 q^{11} -4.17856 q^{13} -0.710339 q^{15} -4.68939 q^{17} -3.43926 q^{19} -1.01219 q^{21} +3.99816 q^{23} +0.0635992 q^{25} +1.86257 q^{27} -0.712153 q^{29} -1.04933 q^{31} -0.0688235 q^{33} +7.21534 q^{35} +1.70998 q^{37} +1.31905 q^{39} +3.18460 q^{41} +6.35890 q^{43} -6.52650 q^{45} +3.87861 q^{47} +3.28144 q^{49} +1.48031 q^{51} -10.8877 q^{53} +0.490604 q^{55} +1.08568 q^{57} +2.63793 q^{59} +11.1544 q^{61} -9.29988 q^{63} -9.40279 q^{65} -6.09170 q^{67} -1.26210 q^{69} -9.40515 q^{71} -2.78532 q^{73} -0.0200765 q^{75} +0.699082 q^{77} -5.16993 q^{79} +8.11309 q^{81} -1.83793 q^{83} -10.5523 q^{85} +0.224807 q^{87} -8.08786 q^{89} -13.3984 q^{91} +0.331245 q^{93} -7.73917 q^{95} -12.9058 q^{97} -0.632342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9	\( 10 q + 8 q^{3} - q^{5} + 5 q^{7} - 2 q^{9} + 3 q^{11} - 18 q^{13} + 2 q^{15} - 11 q^{17} + q^{21} + 2 q^{23} - 27 q^{25} + 2 q^{27} - 9 q^{29} + 22 q^{31} - 10 q^{33} + 6 q^{35} - 35 q^{37} - 8 q^{39} - 4 q^{41} + 20 q^{43} + 2 q^{45} - 7 q^{47} - 27 q^{49} - 9 q^{51} - 24 q^{53} + 11 q^{55} - 23 q^{57} - 17 q^{59} - 4 q^{61} - 10 q^{63} - 16 q^{65} + 6 q^{67} - 2 q^{69} + q^{71} - 31 q^{73} - 30 q^{75} + 3 q^{77} + 10 q^{79} - 6 q^{81} - 22 q^{83} - 6 q^{85} - 25 q^{87} + q^{89} - 10 q^{91} - 6 q^{93} - 39 q^{95} - 57 q^{97} - 35 q^{99}+O(q^{100}) \) 10 * q + 8 * q^3 - q^5 + 5 * q^7 - 2 * q^9 + 3 * q^11 - 18 * q^13 + 2 * q^15 - 11 * q^17 + q^21 + 2 * q^23 - 27 * q^25 + 2 * q^27 - 9 * q^29 + 22 * q^31 - 10 * q^33 + 6 * q^35 - 35 * q^37 - 8 * q^39 - 4 * q^41 + 20 * q^43 + 2 * q^45 - 7 * q^47 - 27 * q^49 - 9 * q^51 - 24 * q^53 + 11 * q^55 - 23 * q^57 - 17 * q^59 - 4 * q^61 - 10 * q^63 - 16 * q^65 + 6 * q^67 - 2 * q^69 + q^71 - 31 * q^73 - 30 * q^75 + 3 * q^77 + 10 * q^79 - 6 * q^81 - 22 * q^83 - 6 * q^85 - 25 * q^87 + q^89 - 10 * q^91 - 6 * q^93 - 39 * q^95 - 57 * q^97 - 35 * q^99

Introduction

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