LMFDB - Embedded newform 8048.2.a.m.1.1

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:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
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			1.1
Root			$2.19869$ 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)

$f(q)$	$=$	$q-2.19869 q^{3} -0.364448 q^{7} +1.83424 q^{9} +O(q^{10})$	$q-2.19869 q^{3} -0.364448 q^{7} +1.83424 q^{9} -1.80131 q^{11} -2.03293 q^{13} +7.66849 q^{17} -4.00000 q^{19} +0.801309 q^{21} +4.00000 q^{23} -5.00000 q^{25} +2.56314 q^{27} -8.06587 q^{29} +3.27110 q^{31} +3.96052 q^{33} -6.79476 q^{37} +4.46980 q^{39} +11.7344 q^{41} +0.105347 q^{43} -0.768374 q^{47} -6.86718 q^{49} -16.8606 q^{51} -6.46325 q^{53} +8.79476 q^{57} +7.23163 q^{59} +2.00000 q^{61} -0.668486 q^{63} -13.5631 q^{67} -8.79476 q^{69} -0.331514 q^{71} +13.4698 q^{73} +10.9935 q^{75} +0.656483 q^{77} -12.4962 q^{79} -11.1383 q^{81} +0.364448 q^{83} +17.7344 q^{87} +12.4633 q^{89} +0.740899 q^{91} -7.19215 q^{93} -11.4907 q^{97} -3.30404 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - q^{7}+O(q^{10}) $ 3 * q - q^3 - q^7	$ 3 q - q^{3} - q^{7} - 11 q^{11} + 5 q^{13} + 12 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{23} - 15 q^{25} + 2 q^{27} - 2 q^{29} + 10 q^{31} - 5 q^{33} + 2 q^{37} + 8 q^{39} + 2 q^{41} - 5 q^{43} - 19 q^{47} - 4 q^{49} - 6 q^{51} + 14 q^{53} + 4 q^{57} + 5 q^{59} + 6 q^{61} + 9 q^{63} - 35 q^{67} - 4 q^{69} - 12 q^{71} + 35 q^{73} + 5 q^{75} - 4 q^{77} + 7 q^{79} - 17 q^{81} + q^{83} + 20 q^{87} + 4 q^{89} - 3 q^{91} + 12 q^{93} - 23 q^{97} + q^{99}+O(q^{100}) $ 3 * q - q^3 - q^7 - 11 * q^11 + 5 * q^13 + 12 * q^17 - 12 * q^19 + 8 * q^21 + 12 * q^23 - 15 * q^25 + 2 * q^27 - 2 * q^29 + 10 * q^31 - 5 * q^33 + 2 * q^37 + 8 * q^39 + 2 * q^41 - 5 * q^43 - 19 * q^47 - 4 * q^49 - 6 * q^51 + 14 * q^53 + 4 * q^57 + 5 * q^59 + 6 * q^61 + 9 * q^63 - 35 * q^67 - 4 * q^69 - 12 * q^71 + 35 * q^73 + 5 * q^75 - 4 * q^77 + 7 * q^79 - 17 * q^81 + q^83 + 20 * q^87 + 4 * q^89 - 3 * q^91 + 12 * q^93 - 23 * q^97 + 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
$3$	−2.19869	−1.26941	−0.634707	−	0.772752i	$-0.718879\pi$
$3$	−2.19869	−1.26941	−0.634707	+	0.772752i	$0.718879\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−0.364448	−0.137748	−0.0688742	−	0.997625i	$-0.521941\pi$
$7$	−0.364448	−0.137748	−0.0688742	+	0.997625i	$0.521941\pi$
$8$	0	0
$9$	1.83424	0.611414
$10$	0	0
$11$	−1.80131	−0.543115	−0.271558	−	0.962422i	$-0.587539\pi$
$11$	−1.80131	−0.543115	−0.271558	+	0.962422i	$0.587539\pi$
$12$	0	0
$13$	−2.03293	−0.563835	−0.281917	−	0.959439i	$-0.590970\pi$
$13$	−2.03293	−0.563835	−0.281917	+	0.959439i	$0.590970\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.66849	1.85988	0.929941	−	0.367710i	$-0.119858\pi$
$17$	7.66849	1.85988	0.929941	+	0.367710i	$0.119858\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0.801309	0.174860
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	2.56314	0.493276
$28$	0	0
$29$	−8.06587	−1.49779	−0.748897	−	0.662686i	$-0.769416\pi$
$29$	−8.06587	−1.49779	−0.748897	+	0.662686i	$0.769416\pi$
$30$	0	0
$31$	3.27110	0.587508	0.293754	−	0.955881i	$-0.405095\pi$
$31$	3.27110	0.587508	0.293754	+	0.955881i	$0.405095\pi$
$32$	0	0
$33$	3.96052	0.689438
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.79476	−1.11705	−0.558526	−	0.829487i	$-0.688633\pi$
$37$	−6.79476	−1.11705	−0.558526	+	0.829487i	$0.688633\pi$
$38$	0	0
$39$	4.46980	0.715740
$40$	0	0
$41$	11.7344	1.83260	0.916299	−	0.400494i	$-0.131162\pi$
$41$	11.7344	1.83260	0.916299	+	0.400494i	$0.131162\pi$
$42$	0	0
$43$	0.105347	0.0160653	0.00803264	−	0.999968i	$-0.497443\pi$
$43$	0.105347	0.0160653	0.00803264	+	0.999968i	$0.497443\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.768374	−0.112079	−0.0560395	−	0.998429i	$-0.517847\pi$
$47$	−0.768374	−0.112079	−0.0560395	+	0.998429i	$0.517847\pi$
$48$	0	0
$49$	−6.86718	−0.981025
$50$	0	0
$51$	−16.8606	−2.36096
$52$	0	0
$53$	−6.46325	−0.887796	−0.443898	−	0.896077i	$-0.646405\pi$
$53$	−6.46325	−0.887796	−0.443898	+	0.896077i	$0.646405\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.79476	1.16490
$58$	0	0
$59$	7.23163	0.941477	0.470739	−	0.882273i	$-0.343987\pi$
$59$	7.23163	0.941477	0.470739	+	0.882273i	$0.343987\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−0.668486	−0.0842214
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.5631	−1.65700	−0.828501	−	0.559988i	$-0.810806\pi$
$67$	−13.5631	−1.65700	−0.828501	+	0.559988i	$0.810806\pi$
$68$	0	0
$69$	−8.79476	−1.05877
$70$	0	0
$71$	−0.331514	−0.0393434	−0.0196717	−	0.999806i	$-0.506262\pi$
$71$	−0.331514	−0.0393434	−0.0196717	+	0.999806i	$0.506262\pi$
$72$	0	0
$73$	13.4698	1.57652	0.788260	−	0.615342i	$-0.210982\pi$
$73$	13.4698	1.57652	0.788260	+	0.615342i	$0.210982\pi$
$74$	0	0
$75$	10.9935	1.26941
$76$	0	0
$77$	0.656483	0.0748132
$78$	0	0
$79$	−12.4962	−1.40593	−0.702965	−	0.711224i	$-0.748141\pi$
$79$	−12.4962	−1.40593	−0.702965	+	0.711224i	$0.748141\pi$
$80$	0	0
$81$	−11.1383	−1.23759
$82$	0	0
$83$	0.364448	0.0400034	0.0200017	−	0.999800i	$-0.493633\pi$
$83$	0.364448	0.0400034	0.0200017	+	0.999800i	$0.493633\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.7344	1.90132
$88$	0	0
$89$	12.4633	1.32110	0.660551	−	0.750781i	$-0.270323\pi$
$89$	12.4633	1.32110	0.660551	+	0.750781i	$0.270323\pi$
$90$	0	0
$91$	0.740899	0.0776673
$92$	0	0
$93$	−7.19215	−0.745791
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.4907	−1.16671	−0.583353	−	0.812219i	$-0.698260\pi$
$97$	−11.4907	−1.16671	−0.583353	+	0.812219i	$0.698260\pi$
$98$	0	0
$99$	−3.30404	−0.332068
$100$	0	0
$101$	−2.87372	−0.285946	−0.142973	−	0.989727i	$-0.545666\pi$
$101$	−2.87372	−0.285946	−0.142973	+	0.989727i	$0.545666\pi$
$102$	0	0
$103$	−8.25256	−0.813149	−0.406574	−	0.913618i	$-0.633277\pi$
$103$	−8.25256	−0.813149	−0.406574	+	0.913618i	$0.633277\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.79476	−0.463527	−0.231764	−	0.972772i	$-0.574450\pi$
$107$	−4.79476	−0.463527	−0.231764	+	0.972772i	$0.574450\pi$
$108$	0	0
$109$	19.4028	1.85846	0.929228	−	0.369508i	$-0.120474\pi$
$109$	19.4028	1.85846	0.929228	+	0.369508i	$0.120474\pi$
$110$	0	0
$111$	14.9396	1.41800
$112$	0	0
$113$	−9.08680	−0.854814	−0.427407	−	0.904059i	$-0.640573\pi$
$113$	−9.08680	−0.854814	−0.427407	+	0.904059i	$0.640573\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.72890	−0.344737
$118$	0	0
$119$	−2.79476	−0.256196
$120$	0	0
$121$	−7.75529	−0.705026
$122$	0	0
$123$	−25.8002	−2.32633
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.2766	−1.44431	−0.722156	−	0.691731i	$-0.756849\pi$
$127$	−16.2766	−1.44431	−0.722156	+	0.691731i	$0.756849\pi$
$128$	0	0
$129$	−0.231626	−0.0203935
$130$	0	0
$131$	−0.105347	−0.00920422	−0.00460211	−	0.999989i	$-0.501465\pi$
$131$	−0.105347	−0.00920422	−0.00460211	+	0.999989i	$0.501465\pi$
$132$	0	0
$133$	1.45779	0.126407
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.6685	1.33865	0.669325	−	0.742970i	$-0.266584\pi$
$137$	15.6685	1.33865	0.669325	+	0.742970i	$0.266584\pi$
$138$	0	0
$139$	13.5895	1.15265	0.576324	−	0.817221i	$-0.304486\pi$
$139$	13.5895	1.15265	0.576324	+	0.817221i	$0.304486\pi$
$140$	0	0
$141$	1.68942	0.142275
$142$	0	0
$143$	3.66194	0.306227
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.0988	1.24533
$148$	0	0
$149$	2.33151	0.191005	0.0955025	−	0.995429i	$-0.469554\pi$
$149$	2.33151	0.191005	0.0955025	+	0.995429i	$0.469554\pi$
$150$	0	0
$151$	16.6499	1.35495	0.677476	−	0.735544i	$-0.263074\pi$
$151$	16.6499	1.35495	0.677476	+	0.735544i	$0.263074\pi$
$152$	0	0
$153$	14.0659	1.13716
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0659	0.962961	0.481481	−	0.876457i	$-0.340099\pi$
$157$	12.0659	0.962961	0.481481	+	0.876457i	$0.340099\pi$
$158$	0	0
$159$	14.2107	1.12698
$160$	0	0
$161$	−1.45779	−0.114890
$162$	0	0
$163$	−0.397382	−0.0311254	−0.0155627	−	0.999879i	$-0.504954\pi$
$163$	−0.397382	−0.0311254	−0.0155627	+	0.999879i	$0.504954\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.79476	−0.216265	−0.108133	−	0.994136i	$-0.534487\pi$
$167$	−2.79476	−0.216265	−0.108133	+	0.994136i	$0.534487\pi$
$168$	0	0
$169$	−8.86718	−0.682091
$170$	0	0
$171$	−7.33697	−0.561072
$172$	0	0
$173$	4.38538	0.333414	0.166707	−	0.986006i	$-0.446687\pi$
$173$	4.38538	0.333414	0.166707	+	0.986006i	$0.446687\pi$
$174$	0	0
$175$	1.82224	0.137748
$176$	0	0
$177$	−15.9001	−1.19513
$178$	0	0
$179$	16.7289	1.25038	0.625188	−	0.780474i	$-0.285022\pi$
$179$	16.7289	1.25038	0.625188	+	0.780474i	$0.285022\pi$
$180$	0	0
$181$	−13.8002	−1.02576	−0.512881	−	0.858460i	$-0.671422\pi$
$181$	−13.8002	−1.02576	−0.512881	+	0.858460i	$0.671422\pi$
$182$	0	0
$183$	−4.39738	−0.325064
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−13.8133	−1.01013
$188$	0	0
$189$	−0.934131	−0.0679480
$190$	0	0
$191$	−3.73436	−0.270208	−0.135104	−	0.990831i	$-0.543137\pi$
$191$	−3.73436	−0.270208	−0.135104	+	0.990831i	$0.543137\pi$
$192$	0	0
$193$	−7.80022	−0.561472	−0.280736	−	0.959785i	$-0.590579\pi$
$193$	−7.80022	−0.561472	−0.280736	+	0.959785i	$0.590579\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.2526	−0.872959	−0.436479	−	0.899714i	$-0.643775\pi$
$197$	−12.2526	−0.872959	−0.436479	+	0.899714i	$0.643775\pi$
$198$	0	0
$199$	−26.7278	−1.89468	−0.947342	−	0.320223i	$-0.896242\pi$
$199$	−26.7278	−1.89468	−0.947342	+	0.320223i	$0.896242\pi$
$200$	0	0
$201$	29.8212	2.10342
$202$	0	0
$203$	2.93959	0.206319
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.33697	0.509955
$208$	0	0
$209$	7.20524	0.498397
$210$	0	0
$211$	−2.46325	−0.169577	−0.0847886	−	0.996399i	$-0.527021\pi$
$211$	−2.46325	−0.169577	−0.0847886	+	0.996399i	$0.527021\pi$
$212$	0	0
$213$	0.728896	0.0499431
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.19215	−0.0809282
$218$	0	0
$219$	−29.6159	−2.00126
$220$	0	0
$221$	−15.5895	−1.04867
$222$	0	0
$223$	9.40393	0.629733	0.314867	−	0.949136i	$-0.398040\pi$
$223$	9.40393	0.629733	0.314867	+	0.949136i	$0.398040\pi$
$224$	0	0
$225$	−9.17122	−0.611414
$226$	0	0
$227$	14.1317	0.937956	0.468978	−	0.883210i	$-0.344622\pi$
$227$	14.1317	0.937956	0.468978	+	0.883210i	$0.344622\pi$
$228$	0	0
$229$	17.5082	1.15697	0.578487	−	0.815692i	$-0.303643\pi$
$229$	17.5082	1.15697	0.578487	+	0.815692i	$0.303643\pi$
$230$	0	0
$231$	−1.44340	−0.0949690
$232$	0	0
$233$	−21.9989	−1.44120	−0.720598	−	0.693353i	$-0.756133\pi$
$233$	−21.9989	−1.44120	−0.720598	+	0.693353i	$0.756133\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	27.4753	1.78471
$238$	0	0
$239$	7.06041	0.456700	0.228350	−	0.973579i	$-0.426667\pi$
$239$	7.06041	0.456700	0.228350	+	0.973579i	$0.426667\pi$
$240$	0	0
$241$	5.12628	0.330213	0.165106	−	0.986276i	$-0.447203\pi$
$241$	5.12628	0.330213	0.165106	+	0.986276i	$0.447203\pi$
$242$	0	0
$243$	16.8002	1.07773
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.13174	0.517410
$248$	0	0
$249$	−0.801309	−0.0507809
$250$	0	0
$251$	7.33697	0.463106	0.231553	−	0.972822i	$-0.425619\pi$
$251$	7.33697	0.463106	0.231553	+	0.972822i	$0.425619\pi$
$252$	0	0
$253$	−7.20524	−0.452989
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.9594	1.68168	0.840842	−	0.541281i	$-0.182060\pi$
$257$	26.9594	1.68168	0.840842	+	0.541281i	$0.182060\pi$
$258$	0	0
$259$	2.47634	0.153872
$260$	0	0
$261$	−14.7948	−0.915773
$262$	0	0
$263$	15.0384	0.927307	0.463654	−	0.886017i	$-0.346538\pi$
$263$	15.0384	0.927307	0.463654	+	0.886017i	$0.346538\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−27.4028	−1.67703
$268$	0	0
$269$	4.25256	0.259283	0.129641	−	0.991561i	$-0.458617\pi$
$269$	4.25256	0.259283	0.129641	+	0.991561i	$0.458617\pi$
$270$	0	0
$271$	−0.316041	−0.0191981	−0.00959907	−	0.999954i	$-0.503056\pi$
$271$	−0.316041	−0.0191981	−0.00959907	+	0.999954i	$0.503056\pi$
$272$	0	0
$273$	−1.62901	−0.0985921
$274$	0	0
$275$	9.00654	0.543115
$276$	0	0
$277$	11.3370	0.681173	0.340586	−	0.940213i	$-0.389374\pi$
$277$	11.3370	0.681173	0.340586	+	0.940213i	$0.389374\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	21.4687	1.28072	0.640358	−	0.768077i	$-0.278786\pi$
$281$	21.4687	1.28072	0.640358	+	0.768077i	$0.278786\pi$
$282$	0	0
$283$	−0.105347	−0.00626223	−0.00313112	−	0.999995i	$-0.500997\pi$
$283$	−0.105347	−0.00626223	−0.00313112	+	0.999995i	$0.500997\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.27656	−0.252438
$288$	0	0
$289$	41.8057	2.45916
$290$	0	0
$291$	25.2646	1.48103
$292$	0	0
$293$	−2.83424	−0.165578	−0.0827891	−	0.996567i	$-0.526383\pi$
$293$	−2.83424	−0.165578	−0.0827891	+	0.996567i	$0.526383\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.61701	−0.267906
$298$	0	0
$299$	−8.13174	−0.470271
$300$	0	0
$301$	−0.0383935	−0.00221297
$302$	0	0
$303$	6.31843	0.362984
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.9815	0.855037	0.427518	−	0.904007i	$-0.359388\pi$
$307$	14.9815	0.855037	0.427518	+	0.904007i	$0.359388\pi$
$308$	0	0
$309$	18.1448	1.03222
$310$	0	0
$311$	−11.3919	−0.645977	−0.322988	−	0.946403i	$-0.604687\pi$
$311$	−11.3919	−0.645977	−0.322988	+	0.946403i	$0.604687\pi$
$312$	0	0
$313$	−16.2107	−0.916283	−0.458141	−	0.888879i	$-0.651485\pi$
$313$	−16.2107	−0.916283	−0.458141	+	0.888879i	$0.651485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.7080	−0.882247	−0.441124	−	0.897446i	$-0.645420\pi$
$317$	−15.7080	−0.882247	−0.441124	+	0.897446i	$0.645420\pi$
$318$	0	0
$319$	14.5291	0.813475
$320$	0	0
$321$	10.5422	0.588409
$322$	0	0
$323$	−30.6739	−1.70674
$324$	0	0
$325$	10.1647	0.563835
$326$	0	0
$327$	−42.6609	−2.35915
$328$	0	0
$329$	0.280033	0.0154387
$330$	0	0
$331$	−13.3370	−0.733066	−0.366533	−	0.930405i	$-0.619455\pi$
$331$	−13.3370	−0.733066	−0.366533	+	0.930405i	$0.619455\pi$
$332$	0	0
$333$	−12.4633	−0.682982
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.8846	1.46450	0.732250	−	0.681036i	$-0.238470\pi$
$337$	26.8846	1.46450	0.732250	+	0.681036i	$0.238470\pi$
$338$	0	0
$339$	19.9791	1.08511
$340$	0	0
$341$	−5.89227	−0.319084
$342$	0	0
$343$	5.05387	0.272883
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.58953	−0.407427	−0.203714	−	0.979031i	$-0.565301\pi$
$347$	−7.58953	−0.407427	−0.203714	+	0.979031i	$0.565301\pi$
$348$	0	0
$349$	−13.8552	−0.741650	−0.370825	−	0.928703i	$-0.620925\pi$
$349$	−13.8552	−0.741650	−0.370825	+	0.928703i	$0.620925\pi$
$350$	0	0
$351$	−5.21069	−0.278126
$352$	0	0
$353$	−9.66849	−0.514602	−0.257301	−	0.966331i	$-0.582833\pi$
$353$	−9.66849	−0.514602	−0.257301	+	0.966331i	$0.582833\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.14483	0.325219
$358$	0	0
$359$	11.8552	0.625692	0.312846	−	0.949804i	$-0.398718\pi$
$359$	11.8552	0.625692	0.312846	+	0.949804i	$0.398718\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	17.0515	0.894971
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.34352	0.226730	0.113365	−	0.993553i	$-0.463837\pi$
$367$	4.34352	0.226730	0.113365	+	0.993553i	$0.463837\pi$
$368$	0	0
$369$	21.5237	1.12048
$370$	0	0
$371$	2.35552	0.122292
$372$	0	0
$373$	20.4094	1.05676	0.528379	−	0.849009i	$-0.322800\pi$
$373$	20.4094	1.05676	0.528379	+	0.849009i	$0.322800\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.3974	0.844508
$378$	0	0
$379$	−30.9725	−1.59095	−0.795476	−	0.605985i	$-0.792779\pi$
$379$	−30.9725	−1.59095	−0.795476	+	0.605985i	$0.792779\pi$
$380$	0	0
$381$	35.7871	1.83343
$382$	0	0
$383$	38.1163	1.94765	0.973825	−	0.227299i	$-0.0729894\pi$
$383$	38.1163	1.94765	0.973825	+	0.227299i	$0.0729894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.193232	0.00982254
$388$	0	0
$389$	−13.3040	−0.674542	−0.337271	−	0.941408i	$-0.609504\pi$
$389$	−13.3040	−0.674542	−0.337271	+	0.941408i	$0.609504\pi$
$390$	0	0
$391$	30.6739	1.55125
$392$	0	0
$393$	0.231626	0.0116840
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.9714	1.60460	0.802300	−	0.596921i	$-0.203609\pi$
$397$	31.9714	1.60460	0.802300	+	0.596921i	$0.203609\pi$
$398$	0	0
$399$	−3.20524	−0.160462
$400$	0	0
$401$	−10.9594	−0.547288	−0.273644	−	0.961831i	$-0.588229\pi$
$401$	−10.9594	−0.547288	−0.273644	+	0.961831i	$0.588229\pi$
$402$	0	0
$403$	−6.64994	−0.331257
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.2395	0.606688
$408$	0	0
$409$	4.84972	0.239803	0.119902	−	0.992786i	$-0.461742\pi$
$409$	4.84972	0.239803	0.119902	+	0.992786i	$0.461742\pi$
$410$	0	0
$411$	−34.4502	−1.69930
$412$	0	0
$413$	−2.63555	−0.129687
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−29.8792	−1.46319
$418$	0	0
$419$	13.3788	0.653599	0.326799	−	0.945094i	$-0.394030\pi$
$419$	13.3788	0.653599	0.326799	+	0.945094i	$0.394030\pi$
$420$	0	0
$421$	12.3710	0.602925	0.301463	−	0.953478i	$-0.402525\pi$
$421$	12.3710	0.602925	0.301463	+	0.953478i	$0.402525\pi$
$422$	0	0
$423$	−1.40939	−0.0685267
$424$	0	0
$425$	−38.3424	−1.85988
$426$	0	0
$427$	−0.728896	−0.0352738
$428$	0	0
$429$	−8.05148	−0.388729
$430$	0	0
$431$	38.9924	1.87820	0.939098	−	0.343649i	$-0.111663\pi$
$431$	38.9924	1.87820	0.939098	+	0.343649i	$0.111663\pi$
$432$	0	0
$433$	3.20762	0.154148	0.0770742	−	0.997025i	$-0.475442\pi$
$433$	3.20762	0.154148	0.0770742	+	0.997025i	$0.475442\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	4.27656	0.204109	0.102055	−	0.994779i	$-0.467458\pi$
$439$	4.27656	0.204109	0.102055	+	0.994779i	$0.467458\pi$
$440$	0	0
$441$	−12.5961	−0.599813
$442$	0	0
$443$	38.0198	1.80638	0.903189	−	0.429244i	$-0.141220\pi$
$443$	38.0198	1.80638	0.903189	+	0.429244i	$0.141220\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.12628	−0.242465
$448$	0	0
$449$	1.19215	0.0562609	0.0281305	−	0.999604i	$-0.491045\pi$
$449$	1.19215	0.0562609	0.0281305	+	0.999604i	$0.491045\pi$
$450$	0	0
$451$	−21.1372	−0.995312
$452$	0	0
$453$	−36.6081	−1.72000
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.12082	0.192764	0.0963819	−	0.995344i	$-0.469273\pi$
$457$	4.12082	0.192764	0.0963819	+	0.995344i	$0.469273\pi$
$458$	0	0
$459$	19.6554	0.917435
$460$	0	0
$461$	−14.7289	−0.685993	−0.342997	−	0.939337i	$-0.611442\pi$
$461$	−14.7289	−0.685993	−0.342997	+	0.939337i	$0.611442\pi$
$462$	0	0
$463$	−32.9265	−1.53022	−0.765112	−	0.643897i	$-0.777317\pi$
$463$	−32.9265	−1.53022	−0.765112	+	0.643897i	$0.777317\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.19215	0.332813	0.166406	−	0.986057i	$-0.446784\pi$
$467$	7.19215	0.332813	0.166406	+	0.986057i	$0.446784\pi$
$468$	0	0
$469$	4.94306	0.228249
$470$	0	0
$471$	−26.5291	−1.22240
$472$	0	0
$473$	−0.189763	−0.00872529
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	−11.8552	−0.542811
$478$	0	0
$479$	32.8606	1.50144	0.750720	−	0.660620i	$-0.229707\pi$
$479$	32.8606	1.50144	0.750720	+	0.660620i	$0.229707\pi$
$480$	0	0
$481$	13.8133	0.629833
$482$	0	0
$483$	3.20524	0.145843
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.6739	1.93374	0.966871	−	0.255267i	$-0.0821635\pi$
$487$	42.6739	1.93374	0.966871	+	0.255267i	$0.0821635\pi$
$488$	0	0
$489$	0.873721	0.0395110
$490$	0	0
$491$	−25.7762	−1.16326	−0.581632	−	0.813452i	$-0.697586\pi$
$491$	−25.7762	−1.16326	−0.581632	+	0.813452i	$0.697586\pi$
$492$	0	0
$493$	−61.8530	−2.78572
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.120819	0.00541950
$498$	0	0
$499$	32.8057	1.46858	0.734292	−	0.678834i	$-0.237514\pi$
$499$	32.8057	1.46858	0.734292	+	0.678834i	$0.237514\pi$
$500$	0	0
$501$	6.14483	0.274531
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	0	0
$506$	0	0
$507$	19.4962	0.865856
$508$	0	0
$509$	18.2406	0.808498	0.404249	−	0.914649i	$-0.367533\pi$
$509$	18.2406	0.808498	0.404249	+	0.914649i	$0.367533\pi$
$510$	0	0
$511$	−4.90904	−0.217163
$512$	0	0
$513$	−10.2526	−0.452661
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.38408	0.0608717
$518$	0	0
$519$	−9.64210	−0.423241
$520$	0	0
$521$	−2.63009	−0.115226	−0.0576132	−	0.998339i	$-0.518349\pi$
$521$	−2.63009	−0.115226	−0.0576132	+	0.998339i	$0.518349\pi$
$522$	0	0
$523$	−7.21615	−0.315540	−0.157770	−	0.987476i	$-0.550431\pi$
$523$	−7.21615	−0.315540	−0.157770	+	0.987476i	$0.550431\pi$
$524$	0	0
$525$	−4.00654	−0.174860
$526$	0	0
$527$	25.0844	1.09269
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	13.2646	0.575633
$532$	0	0
$533$	−23.8552	−1.03328
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−36.7817	−1.58725
$538$	0	0
$539$	12.3699	0.532810
$540$	0	0
$541$	19.4687	0.837025	0.418513	−	0.908211i	$-0.362552\pi$
$541$	19.4687	0.837025	0.418513	+	0.908211i	$0.362552\pi$
$542$	0	0
$543$	30.3424	1.30212
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.8266	1.61735	0.808675	−	0.588256i	$-0.200185\pi$
$547$	37.8266	1.61735	0.808675	+	0.588256i	$0.200185\pi$
$548$	0	0
$549$	3.66849	0.156567
$550$	0	0
$551$	32.2635	1.37447
$552$	0	0
$553$	4.55421	0.193665
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.51712	−0.360882	−0.180441	−	0.983586i	$-0.557752\pi$
$557$	−8.51712	−0.360882	−0.180441	+	0.983586i	$0.557752\pi$
$558$	0	0
$559$	−0.214164	−0.00905816
$560$	0	0
$561$	30.3712	1.28227
$562$	0	0
$563$	22.0528	0.929414	0.464707	−	0.885465i	$-0.346160\pi$
$563$	22.0528	0.929414	0.464707	+	0.885465i	$0.346160\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.05933	0.170476
$568$	0	0
$569$	36.4094	1.52636	0.763180	−	0.646185i	$-0.223637\pi$
$569$	36.4094	1.52636	0.763180	+	0.646185i	$0.223637\pi$
$570$	0	0
$571$	−18.3184	−0.766602	−0.383301	−	0.923623i	$-0.625213\pi$
$571$	−18.3184	−0.766602	−0.383301	+	0.923623i	$0.625213\pi$
$572$	0	0
$573$	8.21069	0.343007
$574$	0	0
$575$	−20.0000	−0.834058
$576$	0	0
$577$	12.5950	0.524336	0.262168	−	0.965022i	$-0.415562\pi$
$577$	12.5950	0.524336	0.262168	+	0.965022i	$0.415562\pi$
$578$	0	0
$579$	17.1503	0.712741
$580$	0	0
$581$	−0.132822	−0.00551040
$582$	0	0
$583$	11.6423	0.482175
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.4622	1.09221	0.546105	−	0.837717i	$-0.316110\pi$
$587$	26.4622	1.09221	0.546105	+	0.837717i	$0.316110\pi$
$588$	0	0
$589$	−13.0844	−0.539134
$590$	0	0
$591$	26.9396	1.10815
$592$	0	0
$593$	−4.26564	−0.175169	−0.0875845	−	0.996157i	$-0.527915\pi$
$593$	−4.26564	−0.175169	−0.0875845	+	0.996157i	$0.527915\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	58.7662	2.40514
$598$	0	0
$599$	−14.0198	−0.572835	−0.286418	−	0.958105i	$-0.592465\pi$
$599$	−14.0198	−0.572835	−0.286418	+	0.958105i	$0.592465\pi$
$600$	0	0
$601$	19.4183	0.792090	0.396045	−	0.918231i	$-0.370383\pi$
$601$	19.4183	0.792090	0.396045	+	0.918231i	$0.370383\pi$
$602$	0	0
$603$	−24.8781	−1.01311
$604$	0	0
$605$	0	0
$606$	0	0
$607$	33.8222	1.37280	0.686401	−	0.727223i	$-0.259189\pi$
$607$	33.8222	1.37280	0.686401	+	0.727223i	$0.259189\pi$
$608$	0	0
$609$	−6.46325	−0.261904
$610$	0	0
$611$	1.56205	0.0631940
$612$	0	0
$613$	44.2085	1.78557	0.892783	−	0.450487i	$-0.148750\pi$
$613$	44.2085	1.78557	0.892783	+	0.450487i	$0.148750\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.53129	−0.182423	−0.0912114	−	0.995832i	$-0.529074\pi$
$617$	−4.53129	−0.182423	−0.0912114	+	0.995832i	$0.529074\pi$
$618$	0	0
$619$	−25.1372	−1.01035	−0.505175	−	0.863017i	$-0.668572\pi$
$619$	−25.1372	−1.01035	−0.505175	+	0.863017i	$0.668572\pi$
$620$	0	0
$621$	10.2526	0.411421
$622$	0	0
$623$	−4.54221	−0.181980
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−15.8421	−0.632672
$628$	0	0
$629$	−52.1056	−2.07758
$630$	0	0
$631$	28.8212	1.14735	0.573676	−	0.819082i	$-0.305517\pi$
$631$	28.8212	1.14735	0.573676	+	0.819082i	$0.305517\pi$
$632$	0	0
$633$	5.41593	0.215264
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.9605	0.553136
$638$	0	0
$639$	−0.608077	−0.0240551
$640$	0	0
$641$	9.82878	0.388214	0.194107	−	0.980980i	$-0.437819\pi$
$641$	9.82878	0.388214	0.194107	+	0.980980i	$0.437819\pi$
$642$	0	0
$643$	10.1736	0.401208	0.200604	−	0.979672i	$-0.435710\pi$
$643$	10.1736	0.401208	0.200604	+	0.979672i	$0.435710\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.9374	−1.84530	−0.922650	−	0.385638i	$-0.873981\pi$
$647$	−46.9374	−1.84530	−0.922650	+	0.385638i	$0.873981\pi$
$648$	0	0
$649$	−13.0264	−0.511331
$650$	0	0
$651$	2.62116	0.102732
$652$	0	0
$653$	19.2669	0.753974	0.376987	−	0.926219i	$-0.376960\pi$
$653$	19.2669	0.753974	0.376987	+	0.926219i	$0.376960\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24.7069	0.963907
$658$	0	0
$659$	32.1887	1.25389	0.626946	−	0.779062i	$-0.284304\pi$
$659$	32.1887	1.25389	0.626946	+	0.779062i	$0.284304\pi$
$660$	0	0
$661$	10.5202	0.409188	0.204594	−	0.978847i	$-0.434413\pi$
$661$	10.5202	0.409188	0.204594	+	0.978847i	$0.434413\pi$
$662$	0	0
$663$	34.2766	1.33119
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−32.2635	−1.24925
$668$	0	0
$669$	−20.6763	−0.799393
$670$	0	0
$671$	−3.60262	−0.139078
$672$	0	0
$673$	−39.2295	−1.51218	−0.756092	−	0.654465i	$-0.772894\pi$
$673$	−39.2295	−1.51218	−0.756092	+	0.654465i	$0.772894\pi$
$674$	0	0
$675$	−12.8157	−0.493276
$676$	0	0
$677$	−10.0790	−0.387366	−0.193683	−	0.981064i	$-0.562043\pi$
$677$	−10.0790	−0.387366	−0.193683	+	0.981064i	$0.562043\pi$
$678$	0	0
$679$	4.18777	0.160712
$680$	0	0
$681$	−31.0713	−1.19066
$682$	0	0
$683$	16.6609	0.637510	0.318755	−	0.947837i	$-0.396735\pi$
$683$	16.6609	0.637510	0.318755	+	0.947837i	$0.396735\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−38.4951	−1.46868
$688$	0	0
$689$	13.1394	0.500570
$690$	0	0
$691$	25.2251	0.959607	0.479804	−	0.877376i	$-0.340708\pi$
$691$	25.2251	0.959607	0.479804	+	0.877376i	$0.340708\pi$
$692$	0	0
$693$	1.20415	0.0457419
$694$	0	0
$695$	0	0
$696$	0	0
$697$	89.9847	3.40842
$698$	0	0
$699$	48.3688	1.82948
$700$	0	0
$701$	−15.8541	−0.598801	−0.299400	−	0.954128i	$-0.596787\pi$
$701$	−15.8541	−0.598801	−0.299400	+	0.954128i	$0.596787\pi$
$702$	0	0
$703$	27.1791	1.02508
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.04732	0.0393886
$708$	0	0
$709$	−24.3184	−0.913298	−0.456649	−	0.889647i	$-0.650950\pi$
$709$	−24.3184	−0.913298	−0.456649	+	0.889647i	$0.650950\pi$
$710$	0	0
$711$	−22.9210	−0.859606
$712$	0	0
$713$	13.0844	0.490015
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.5237	−0.579742
$718$	0	0
$719$	−7.81223	−0.291347	−0.145673	−	0.989333i	$-0.546535\pi$
$719$	−7.81223	−0.291347	−0.145673	+	0.989333i	$0.546535\pi$
$720$	0	0
$721$	3.00763	0.112010
$722$	0	0
$723$	−11.2711	−0.419177
$724$	0	0
$725$	40.3293	1.49779
$726$	0	0
$727$	−2.06695	−0.0766591	−0.0383295	−	0.999265i	$-0.512204\pi$
$727$	−2.06695	−0.0766591	−0.0383295	+	0.999265i	$0.512204\pi$
$728$	0	0
$729$	−3.52366	−0.130506
$730$	0	0
$731$	0.807853	0.0298795
$732$	0	0
$733$	32.7158	1.20839	0.604193	−	0.796838i	$-0.293496\pi$
$733$	32.7158	1.20839	0.604193	+	0.796838i	$0.293496\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.4314	0.899942
$738$	0	0
$739$	−3.78822	−0.139352	−0.0696760	−	0.997570i	$-0.522197\pi$
$739$	−3.78822	−0.139352	−0.0696760	+	0.997570i	$0.522197\pi$
$740$	0	0
$741$	−17.8792	−0.656808
$742$	0	0
$743$	37.7213	1.38386	0.691930	−	0.721965i	$-0.256761\pi$
$743$	37.7213	1.38386	0.691930	+	0.721965i	$0.256761\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.668486	0.0244586
$748$	0	0
$749$	1.74744	0.0638502
$750$	0	0
$751$	13.5477	0.494361	0.247181	−	0.968969i	$-0.420496\pi$
$751$	13.5477	0.494361	0.247181	+	0.968969i	$0.420496\pi$
$752$	0	0
$753$	−16.1317	−0.587873
$754$	0	0
$755$	0	0
$756$	0	0
$757$	52.9924	1.92604	0.963020	−	0.269429i	$-0.0868350\pi$
$757$	52.9924	1.92604	0.963020	+	0.269429i	$0.0868350\pi$
$758$	0	0
$759$	15.8421	0.575031
$760$	0	0
$761$	22.1066	0.801365	0.400683	−	0.916217i	$-0.368773\pi$
$761$	22.1066	0.801365	0.400683	+	0.916217i	$0.368773\pi$
$762$	0	0
$763$	−7.07133	−0.255999
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.7014	−0.530838
$768$	0	0
$769$	−11.8792	−0.428374	−0.214187	−	0.976793i	$-0.568710\pi$
$769$	−11.8792	−0.428374	−0.214187	+	0.976793i	$0.568710\pi$
$770$	0	0
$771$	−59.2755	−2.13475
$772$	0	0
$773$	−3.45779	−0.124368	−0.0621841	−	0.998065i	$-0.519807\pi$
$773$	−3.45779	−0.124368	−0.0621841	+	0.998065i	$0.519807\pi$
$774$	0	0
$775$	−16.3555	−0.587508
$776$	0	0
$777$	−5.44470	−0.195328
$778$	0	0
$779$	−46.9374	−1.68171
$780$	0	0
$781$	0.597158	0.0213680
$782$	0	0
$783$	−20.6739	−0.738827
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.6499	1.23514	0.617568	−	0.786517i	$-0.288118\pi$
$787$	34.6499	1.23514	0.617568	+	0.786517i	$0.288118\pi$
$788$	0	0
$789$	−33.0648	−1.17714
$790$	0	0
$791$	3.31167	0.117749
$792$	0	0
$793$	−4.06587	−0.144383
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.2515	1.00072	0.500359	−	0.865818i	$-0.333201\pi$
$797$	28.2515	1.00072	0.500359	+	0.865818i	$0.333201\pi$
$798$	0	0
$799$	−5.89227	−0.208453
$800$	0	0
$801$	22.8606	0.807741
$802$	0	0
$803$	−24.2633	−0.856232
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.35006	−0.329138
$808$	0	0
$809$	−10.1867	−0.358145	−0.179072	−	0.983836i	$-0.557310\pi$
$809$	−10.1867	−0.358145	−0.179072	+	0.983836i	$0.557310\pi$
$810$	0	0
$811$	36.2635	1.27338	0.636691	−	0.771119i	$-0.280303\pi$
$811$	36.2635	1.27338	0.636691	+	0.771119i	$0.280303\pi$
$812$	0	0
$813$	0.694877	0.0243704
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−0.421388	−0.0147425
$818$	0	0
$819$	1.35899	0.0474869
$820$	0	0
$821$	−38.6499	−1.34889	−0.674446	−	0.738324i	$-0.735618\pi$
$821$	−38.6499	−1.34889	−0.674446	+	0.738324i	$0.735618\pi$
$822$	0	0
$823$	−40.8585	−1.42424	−0.712118	−	0.702060i	$-0.752264\pi$
$823$	−40.8585	−1.42424	−0.712118	+	0.702060i	$0.752264\pi$
$824$	0	0
$825$	−19.8026	−0.689438
$826$	0	0
$827$	−30.2526	−1.05198	−0.525992	−	0.850489i	$-0.676306\pi$
$827$	−30.2526	−1.05198	−0.525992	+	0.850489i	$0.676306\pi$
$828$	0	0
$829$	−20.6739	−0.718036	−0.359018	−	0.933331i	$-0.616888\pi$
$829$	−20.6739	−0.718036	−0.359018	+	0.933331i	$0.616888\pi$
$830$	0	0
$831$	−24.9265	−0.864691
$832$	0	0
$833$	−52.6609	−1.82459
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.38429	0.289804
$838$	0	0
$839$	−41.8881	−1.44614	−0.723069	−	0.690776i	$-0.757269\pi$
$839$	−41.8881	−1.44614	−0.723069	+	0.690776i	$0.757269\pi$
$840$	0	0
$841$	36.0582	1.24339
$842$	0	0
$843$	−47.2031	−1.62576
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.82640	0.0971162
$848$	0	0
$849$	0.231626	0.00794937
$850$	0	0
$851$	−27.1791	−0.931686
$852$	0	0
$853$	−6.03293	−0.206564	−0.103282	−	0.994652i	$-0.532934\pi$
$853$	−6.03293	−0.206564	−0.103282	+	0.994652i	$0.532934\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.7937	1.59844	0.799221	−	0.601037i	$-0.205246\pi$
$857$	46.7937	1.59844	0.799221	+	0.601037i	$0.205246\pi$
$858$	0	0
$859$	−12.0528	−0.411236	−0.205618	−	0.978632i	$-0.565920\pi$
$859$	−12.0528	−0.411236	−0.205618	+	0.978632i	$0.565920\pi$
$860$	0	0
$861$	9.40284	0.320448
$862$	0	0
$863$	38.5051	1.31073	0.655365	−	0.755313i	$-0.272515\pi$
$863$	38.5051	1.31073	0.655365	+	0.755313i	$0.272515\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−91.9178	−3.12169
$868$	0	0
$869$	22.5095	0.763582
$870$	0	0
$871$	27.5730	0.934275
$872$	0	0
$873$	−21.0768	−0.713341
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.3053	1.49608	0.748042	−	0.663651i	$-0.230994\pi$
$877$	44.3053	1.49608	0.748042	+	0.663651i	$0.230994\pi$
$878$	0	0
$879$	6.23163	0.210188
$880$	0	0
$881$	−18.3030	−0.616642	−0.308321	−	0.951282i	$-0.599767\pi$
$881$	−18.3030	−0.616642	−0.308321	+	0.951282i	$0.599767\pi$
$882$	0	0
$883$	−34.3064	−1.15450	−0.577252	−	0.816566i	$-0.695875\pi$
$883$	−34.3064	−1.15450	−0.577252	+	0.816566i	$0.695875\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.0628	0.707219	0.353610	−	0.935393i	$-0.384954\pi$
$887$	21.0628	0.707219	0.353610	+	0.935393i	$0.384954\pi$
$888$	0	0
$889$	5.93196	0.198952
$890$	0	0
$891$	20.0635	0.672152
$892$	0	0
$893$	3.07350	0.102851
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.8792	0.596968
$898$	0	0
$899$	−26.3843	−0.879966
$900$	0	0
$901$	−49.5634	−1.65119
$902$	0	0
$903$	0.0844155	0.00280917
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.53129	−0.150459	−0.0752295	−	0.997166i	$-0.523969\pi$
$907$	−4.53129	−0.150459	−0.0752295	+	0.997166i	$0.523969\pi$
$908$	0	0
$909$	−5.27110	−0.174831
$910$	0	0
$911$	−44.4083	−1.47131	−0.735656	−	0.677355i	$-0.763126\pi$
$911$	−44.4083	−1.47131	−0.735656	+	0.677355i	$0.763126\pi$
$912$	0	0
$913$	−0.656483	−0.0217264
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.0383935	0.00126787
$918$	0	0
$919$	−17.8881	−0.590074	−0.295037	−	0.955486i	$-0.595332\pi$
$919$	−17.8881	−0.590074	−0.295037	+	0.955486i	$0.595332\pi$
$920$	0	0
$921$	−32.9396	−1.08540
$922$	0	0
$923$	0.673945	0.0221832
$924$	0	0
$925$	33.9738	1.11705
$926$	0	0
$927$	−15.1372	−0.497171
$928$	0	0
$929$	−10.6630	−0.349843	−0.174921	−	0.984582i	$-0.555967\pi$
$929$	−10.6630	−0.349843	−0.174921	+	0.984582i	$0.555967\pi$
$930$	0	0
$931$	27.4687	0.900251
$932$	0	0
$933$	25.0473	0.820013
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28.0528	0.916444	0.458222	−	0.888838i	$-0.348486\pi$
$937$	28.0528	0.916444	0.458222	+	0.888838i	$0.348486\pi$
$938$	0	0
$939$	35.6423	1.16314
$940$	0	0
$941$	5.17230	0.168612	0.0843061	−	0.996440i	$-0.473133\pi$
$941$	5.17230	0.168612	0.0843061	+	0.996440i	$0.473133\pi$
$942$	0	0
$943$	46.9374	1.52849
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.9210	−0.972303	−0.486152	−	0.873874i	$-0.661600\pi$
$947$	−29.9210	−0.972303	−0.486152	+	0.873874i	$0.661600\pi$
$948$	0	0
$949$	−27.3832	−0.888897
$950$	0	0
$951$	34.5370	1.11994
$952$	0	0
$953$	27.0449	0.876071	0.438036	−	0.898958i	$-0.355674\pi$
$953$	27.0449	0.876071	0.438036	+	0.898958i	$0.355674\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−31.9450	−1.03264
$958$	0	0
$959$	−5.71035	−0.184397
$960$	0	0
$961$	−20.2999	−0.654835
$962$	0	0
$963$	−8.79476	−0.283407
$964$	0	0
$965$	0	0
$966$	0	0
$967$	52.9924	1.70412	0.852060	−	0.523444i	$-0.175353\pi$
$967$	52.9924	1.70412	0.852060	+	0.523444i	$0.175353\pi$
$968$	0	0
$969$	67.4425	2.16657
$970$	0	0
$971$	30.5422	0.980146	0.490073	−	0.871681i	$-0.336970\pi$
$971$	30.5422	0.980146	0.490073	+	0.871681i	$0.336970\pi$
$972$	0	0
$973$	−4.95268	−0.158776
$974$	0	0
$975$	−22.3490	−0.715740
$976$	0	0
$977$	−45.9824	−1.47111	−0.735553	−	0.677467i	$-0.763078\pi$
$977$	−45.9824	−1.47111	−0.735553	+	0.677467i	$0.763078\pi$
$978$	0	0
$979$	−22.4502	−0.717510
$980$	0	0
$981$	35.5895	1.13629
$982$	0	0
$983$	9.94505	0.317198	0.158599	−	0.987343i	$-0.449302\pi$
$983$	9.94505	0.317198	0.158599	+	0.987343i	$0.449302\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.615705	−0.0195981
$988$	0	0
$989$	0.421388	0.0133994
$990$	0	0
$991$	−21.5357	−0.684103	−0.342051	−	0.939681i	$-0.611122\pi$
$991$	−21.5357	−0.684103	−0.342051	+	0.939681i	$0.611122\pi$
$992$	0	0
$993$	29.3239	0.930565
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−53.9847	−1.70971	−0.854857	−	0.518864i	$-0.826355\pi$
$997$	−53.9847	−1.70971	−0.854857	+	0.518864i	$0.826355\pi$
$998$	0	0
$999$	−17.4159	−0.551016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.m.1.1		3
4.3	odd	2		503.2.a.d.1.3	✓	3
12.11	even	2		4527.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.d.1.3	✓	3	4.3	odd	2
4527.2.a.j.1.1		3	12.11	even	2
8048.2.a.m.1.1		3	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)

\(f(q)\)	\(=\)	\(q-2.19869 q^{3} -0.364448 q^{7} +1.83424 q^{9} +O(q^{10})\)	\(q-2.19869 q^{3} -0.364448 q^{7} +1.83424 q^{9} -1.80131 q^{11} -2.03293 q^{13} +7.66849 q^{17} -4.00000 q^{19} +0.801309 q^{21} +4.00000 q^{23} -5.00000 q^{25} +2.56314 q^{27} -8.06587 q^{29} +3.27110 q^{31} +3.96052 q^{33} -6.79476 q^{37} +4.46980 q^{39} +11.7344 q^{41} +0.105347 q^{43} -0.768374 q^{47} -6.86718 q^{49} -16.8606 q^{51} -6.46325 q^{53} +8.79476 q^{57} +7.23163 q^{59} +2.00000 q^{61} -0.668486 q^{63} -13.5631 q^{67} -8.79476 q^{69} -0.331514 q^{71} +13.4698 q^{73} +10.9935 q^{75} +0.656483 q^{77} -12.4962 q^{79} -11.1383 q^{81} +0.364448 q^{83} +17.7344 q^{87} +12.4633 q^{89} +0.740899 q^{91} -7.19215 q^{93} -11.4907 q^{97} -3.30404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - q^{7}+O(q^{10}) \) 3 * q - q^3 - q^7	\( 3 q - q^{3} - q^{7} - 11 q^{11} + 5 q^{13} + 12 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{23} - 15 q^{25} + 2 q^{27} - 2 q^{29} + 10 q^{31} - 5 q^{33} + 2 q^{37} + 8 q^{39} + 2 q^{41} - 5 q^{43} - 19 q^{47} - 4 q^{49} - 6 q^{51} + 14 q^{53} + 4 q^{57} + 5 q^{59} + 6 q^{61} + 9 q^{63} - 35 q^{67} - 4 q^{69} - 12 q^{71} + 35 q^{73} + 5 q^{75} - 4 q^{77} + 7 q^{79} - 17 q^{81} + q^{83} + 20 q^{87} + 4 q^{89} - 3 q^{91} + 12 q^{93} - 23 q^{97} + q^{99}+O(q^{100}) \) 3 * q - q^3 - q^7 - 11 * q^11 + 5 * q^13 + 12 * q^17 - 12 * q^19 + 8 * q^21 + 12 * q^23 - 15 * q^25 + 2 * q^27 - 2 * q^29 + 10 * q^31 - 5 * q^33 + 2 * q^37 + 8 * q^39 + 2 * q^41 - 5 * q^43 - 19 * q^47 - 4 * q^49 - 6 * q^51 + 14 * q^53 + 4 * q^57 + 5 * q^59 + 6 * q^61 + 9 * q^63 - 35 * q^67 - 4 * q^69 - 12 * q^71 + 35 * q^73 + 5 * q^75 - 4 * q^77 + 7 * q^79 - 17 * q^81 + q^83 + 20 * q^87 + 4 * q^89 - 3 * q^91 + 12 * q^93 - 23 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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