LMFDB - Embedded newform 8048.2.a.t.1.10

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-0.0212378 q^{3} -0.474891 q^{5} +0.949153 q^{7} -2.99955 q^{9} +O(q^{10})$	$q-0.0212378 q^{3} -0.474891 q^{5} +0.949153 q^{7} -2.99955 q^{9} -5.65202 q^{11} -4.21210 q^{13} +0.0100856 q^{15} +1.35546 q^{17} -5.79165 q^{19} -0.0201579 q^{21} +7.38842 q^{23} -4.77448 q^{25} +0.127417 q^{27} -2.48569 q^{29} +5.09893 q^{31} +0.120036 q^{33} -0.450744 q^{35} +8.76237 q^{37} +0.0894556 q^{39} -0.109364 q^{41} -10.9886 q^{43} +1.42446 q^{45} +9.85603 q^{47} -6.09911 q^{49} -0.0287870 q^{51} -11.4138 q^{53} +2.68409 q^{55} +0.123002 q^{57} -9.75542 q^{59} -13.1203 q^{61} -2.84703 q^{63} +2.00029 q^{65} +6.25867 q^{67} -0.156914 q^{69} +12.2336 q^{71} -15.1143 q^{73} +0.101399 q^{75} -5.36463 q^{77} +11.4869 q^{79} +8.99594 q^{81} +1.29775 q^{83} -0.643696 q^{85} +0.0527905 q^{87} -0.406030 q^{89} -3.99793 q^{91} -0.108290 q^{93} +2.75040 q^{95} +10.1621 q^{97} +16.9535 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * 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
$3$	−0.0212378	−0.0122616	−0.00613081	−	0.999981i	$-0.501952\pi$
$3$	−0.0212378	−0.0122616	−0.00613081	+	0.999981i	$0.501952\pi$
$4$	0	0
$5$	−0.474891	−0.212378	−0.106189	−	0.994346i	$-0.533865\pi$
$5$	−0.474891	−0.212378	−0.106189	+	0.994346i	$0.533865\pi$
$6$	0	0
$7$	0.949153	0.358746	0.179373	−	0.983781i	$-0.442593\pi$
$7$	0.949153	0.358746	0.179373	+	0.983781i	$0.442593\pi$
$8$	0	0
$9$	−2.99955	−0.999850
$10$	0	0
$11$	−5.65202	−1.70415	−0.852074	−	0.523422i	$-0.824655\pi$
$11$	−5.65202	−1.70415	−0.852074	+	0.523422i	$0.824655\pi$
$12$	0	0
$13$	−4.21210	−1.16823	−0.584113	−	0.811672i	$-0.698558\pi$
$13$	−4.21210	−1.16823	−0.584113	+	0.811672i	$0.698558\pi$
$14$	0	0
$15$	0.0100856	0.00260410
$16$	0	0
$17$	1.35546	0.328748	0.164374	−	0.986398i	$-0.447440\pi$
$17$	1.35546	0.328748	0.164374	+	0.986398i	$0.447440\pi$
$18$	0	0
$19$	−5.79165	−1.32870	−0.664348	−	0.747424i	$-0.731291\pi$
$19$	−5.79165	−1.32870	−0.664348	+	0.747424i	$0.731291\pi$
$20$	0	0
$21$	−0.0201579	−0.00439881
$22$	0	0
$23$	7.38842	1.54059	0.770296	−	0.637686i	$-0.220108\pi$
$23$	7.38842	1.54059	0.770296	+	0.637686i	$0.220108\pi$
$24$	0	0
$25$	−4.77448	−0.954896
$26$	0	0
$27$	0.127417	0.0245214
$28$	0	0
$29$	−2.48569	−0.461581	−0.230790	−	0.973003i	$-0.574131\pi$
$29$	−2.48569	−0.461581	−0.230790	+	0.973003i	$0.574131\pi$
$30$	0	0
$31$	5.09893	0.915796	0.457898	−	0.889005i	$-0.348603\pi$
$31$	5.09893	0.915796	0.457898	+	0.889005i	$0.348603\pi$
$32$	0	0
$33$	0.120036	0.0208956
$34$	0	0
$35$	−0.450744	−0.0761897
$36$	0	0
$37$	8.76237	1.44052	0.720262	−	0.693702i	$-0.244022\pi$
$37$	8.76237	1.44052	0.720262	+	0.693702i	$0.244022\pi$
$38$	0	0
$39$	0.0894556	0.0143244
$40$	0	0
$41$	−0.109364	−0.0170798	−0.00853992	−	0.999964i	$-0.502718\pi$
$41$	−0.109364	−0.0170798	−0.00853992	+	0.999964i	$0.502718\pi$
$42$	0	0
$43$	−10.9886	−1.67575	−0.837876	−	0.545860i	$-0.816203\pi$
$43$	−10.9886	−1.67575	−0.837876	+	0.545860i	$0.816203\pi$
$44$	0	0
$45$	1.42446	0.212346
$46$	0	0
$47$	9.85603	1.43765	0.718825	−	0.695191i	$-0.244680\pi$
$47$	9.85603	1.43765	0.718825	+	0.695191i	$0.244680\pi$
$48$	0	0
$49$	−6.09911	−0.871301
$50$	0	0
$51$	−0.0287870	−0.00403098
$52$	0	0
$53$	−11.4138	−1.56780	−0.783900	−	0.620887i	$-0.786773\pi$
$53$	−11.4138	−1.56780	−0.783900	+	0.620887i	$0.786773\pi$
$54$	0	0
$55$	2.68409	0.361923
$56$	0	0
$57$	0.123002	0.0162920
$58$	0	0
$59$	−9.75542	−1.27005	−0.635024	−	0.772493i	$-0.719010\pi$
$59$	−9.75542	−1.27005	−0.635024	+	0.772493i	$0.719010\pi$
$60$	0	0
$61$	−13.1203	−1.67988	−0.839939	−	0.542681i	$-0.817409\pi$
$61$	−13.1203	−1.67988	−0.839939	+	0.542681i	$0.817409\pi$
$62$	0	0
$63$	−2.84703	−0.358692
$64$	0	0
$65$	2.00029	0.248105
$66$	0	0
$67$	6.25867	0.764619	0.382309	−	0.924034i	$-0.375129\pi$
$67$	6.25867	0.764619	0.382309	+	0.924034i	$0.375129\pi$
$68$	0	0
$69$	−0.156914	−0.0188902
$70$	0	0
$71$	12.2336	1.45187	0.725933	−	0.687766i	$-0.241408\pi$
$71$	12.2336	1.45187	0.725933	+	0.687766i	$0.241408\pi$
$72$	0	0
$73$	−15.1143	−1.76899	−0.884495	−	0.466550i	$-0.845497\pi$
$73$	−15.1143	−1.76899	−0.884495	+	0.466550i	$0.845497\pi$
$74$	0	0
$75$	0.101399	0.0117086
$76$	0	0
$77$	−5.36463	−0.611356
$78$	0	0
$79$	11.4869	1.29237	0.646187	−	0.763179i	$-0.276363\pi$
$79$	11.4869	1.29237	0.646187	+	0.763179i	$0.276363\pi$
$80$	0	0
$81$	8.99594	0.999549
$82$	0	0
$83$	1.29775	0.142447	0.0712234	−	0.997460i	$-0.477310\pi$
$83$	1.29775	0.142447	0.0712234	+	0.997460i	$0.477310\pi$
$84$	0	0
$85$	−0.643696	−0.0698187
$86$	0	0
$87$	0.0527905	0.00565973
$88$	0	0
$89$	−0.406030	−0.0430390	−0.0215195	−	0.999768i	$-0.506850\pi$
$89$	−0.406030	−0.0430390	−0.0215195	+	0.999768i	$0.506850\pi$
$90$	0	0
$91$	−3.99793	−0.419097
$92$	0	0
$93$	−0.108290	−0.0112291
$94$	0	0
$95$	2.75040	0.282185
$96$	0	0
$97$	10.1621	1.03180	0.515902	−	0.856648i	$-0.327457\pi$
$97$	10.1621	1.03180	0.515902	+	0.856648i	$0.327457\pi$
$98$	0	0
$99$	16.9535	1.70389
$100$	0	0
$101$	4.93069	0.490622	0.245311	−	0.969444i	$-0.421110\pi$
$101$	4.93069	0.490622	0.245311	+	0.969444i	$0.421110\pi$
$102$	0	0
$103$	−8.34434	−0.822192	−0.411096	−	0.911592i	$-0.634854\pi$
$103$	−8.34434	−0.822192	−0.411096	+	0.911592i	$0.634854\pi$
$104$	0	0
$105$	0.00957280	0.000934209 0
$106$	0	0
$107$	10.4441	1.00967	0.504835	−	0.863216i	$-0.331553\pi$
$107$	10.4441	1.00967	0.504835	+	0.863216i	$0.331553\pi$
$108$	0	0
$109$	14.9796	1.43478	0.717392	−	0.696670i	$-0.245336\pi$
$109$	14.9796	1.43478	0.717392	+	0.696670i	$0.245336\pi$
$110$	0	0
$111$	−0.186093	−0.0176632
$112$	0	0
$113$	−13.0070	−1.22360	−0.611798	−	0.791014i	$-0.709554\pi$
$113$	−13.0070	−1.22360	−0.611798	+	0.791014i	$0.709554\pi$
$114$	0	0
$115$	−3.50869	−0.327187
$116$	0	0
$117$	12.6344	1.16805
$118$	0	0
$119$	1.28654	0.117937
$120$	0	0
$121$	20.9453	1.90412
$122$	0	0
$123$	0.00232265	0.000209427 0
$124$	0	0
$125$	4.64181	0.415176
$126$	0	0
$127$	6.22792	0.552638	0.276319	−	0.961066i	$-0.410885\pi$
$127$	6.22792	0.552638	0.276319	+	0.961066i	$0.410885\pi$
$128$	0	0
$129$	0.233374	0.0205475
$130$	0	0
$131$	3.38570	0.295810	0.147905	−	0.989002i	$-0.452747\pi$
$131$	3.38570	0.295810	0.147905	+	0.989002i	$0.452747\pi$
$132$	0	0
$133$	−5.49716	−0.476664
$134$	0	0
$135$	−0.0605092	−0.00520780
$136$	0	0
$137$	5.12066	0.437487	0.218744	−	0.975782i	$-0.429804\pi$
$137$	5.12066	0.437487	0.218744	+	0.975782i	$0.429804\pi$
$138$	0	0
$139$	1.51296	0.128327	0.0641636	−	0.997939i	$-0.479562\pi$
$139$	1.51296	0.128327	0.0641636	+	0.997939i	$0.479562\pi$
$140$	0	0
$141$	−0.209320	−0.0176279
$142$	0	0
$143$	23.8069	1.99083
$144$	0	0
$145$	1.18043	0.0980294
$146$	0	0
$147$	0.129531	0.0106836
$148$	0	0
$149$	8.25917	0.676618	0.338309	−	0.941035i	$-0.390145\pi$
$149$	8.25917	0.676618	0.338309	+	0.941035i	$0.390145\pi$
$150$	0	0
$151$	12.4174	1.01051	0.505256	−	0.862970i	$-0.331398\pi$
$151$	12.4174	1.01051	0.505256	+	0.862970i	$0.331398\pi$
$152$	0	0
$153$	−4.06577	−0.328698
$154$	0	0
$155$	−2.42144	−0.194495
$156$	0	0
$157$	−1.22079	−0.0974297	−0.0487148	−	0.998813i	$-0.515513\pi$
$157$	−1.22079	−0.0974297	−0.0487148	+	0.998813i	$0.515513\pi$
$158$	0	0
$159$	0.242403	0.0192238
$160$	0	0
$161$	7.01274	0.552681
$162$	0	0
$163$	16.7898	1.31508	0.657539	−	0.753421i	$-0.271598\pi$
$163$	16.7898	1.31508	0.657539	+	0.753421i	$0.271598\pi$
$164$	0	0
$165$	−0.0570041	−0.00443776
$166$	0	0
$167$	22.2041	1.71821	0.859104	−	0.511802i	$-0.171022\pi$
$167$	22.2041	1.71821	0.859104	+	0.511802i	$0.171022\pi$
$168$	0	0
$169$	4.74181	0.364754
$170$	0	0
$171$	17.3723	1.32850
$172$	0	0
$173$	−3.55746	−0.270468	−0.135234	−	0.990814i	$-0.543179\pi$
$173$	−3.55746	−0.270468	−0.135234	+	0.990814i	$0.543179\pi$
$174$	0	0
$175$	−4.53171	−0.342565
$176$	0	0
$177$	0.207183	0.0155728
$178$	0	0
$179$	−5.00109	−0.373799	−0.186899	−	0.982379i	$-0.559844\pi$
$179$	−5.00109	−0.373799	−0.186899	+	0.982379i	$0.559844\pi$
$180$	0	0
$181$	−7.46665	−0.554992	−0.277496	−	0.960727i	$-0.589505\pi$
$181$	−7.46665	−0.554992	−0.277496	+	0.960727i	$0.589505\pi$
$182$	0	0
$183$	0.278645	0.0205980
$184$	0	0
$185$	−4.16117	−0.305935
$186$	0	0
$187$	−7.66109	−0.560234
$188$	0	0
$189$	0.120938	0.00879696
$190$	0	0
$191$	−26.7231	−1.93362	−0.966808	−	0.255503i	$-0.917759\pi$
$191$	−26.7231	−1.93362	−0.966808	+	0.255503i	$0.917759\pi$
$192$	0	0
$193$	3.10540	0.223532	0.111766	−	0.993735i	$-0.464349\pi$
$193$	3.10540	0.223532	0.111766	+	0.993735i	$0.464349\pi$
$194$	0	0
$195$	−0.0424817	−0.00304217
$196$	0	0
$197$	0.866640	0.0617455	0.0308728	−	0.999523i	$-0.490171\pi$
$197$	0.866640	0.0617455	0.0308728	+	0.999523i	$0.490171\pi$
$198$	0	0
$199$	−25.3795	−1.79911	−0.899553	−	0.436812i	$-0.856107\pi$
$199$	−25.3795	−1.79911	−0.899553	+	0.436812i	$0.856107\pi$
$200$	0	0
$201$	−0.132920	−0.00937547
$202$	0	0
$203$	−2.35930	−0.165590
$204$	0	0
$205$	0.0519361	0.00362738
$206$	0	0
$207$	−22.1619	−1.54036
$208$	0	0
$209$	32.7345	2.26429
$210$	0	0
$211$	22.0581	1.51854	0.759271	−	0.650774i	$-0.225556\pi$
$211$	22.0581	1.51854	0.759271	+	0.650774i	$0.225556\pi$
$212$	0	0
$213$	−0.259815	−0.0178022
$214$	0	0
$215$	5.21841	0.355892
$216$	0	0
$217$	4.83967	0.328538
$218$	0	0
$219$	0.320993	0.0216907
$220$	0	0
$221$	−5.70934	−0.384052
$222$	0	0
$223$	15.3564	1.02834	0.514169	−	0.857689i	$-0.328100\pi$
$223$	15.3564	1.02834	0.514169	+	0.857689i	$0.328100\pi$
$224$	0	0
$225$	14.3213	0.954752
$226$	0	0
$227$	0.0398969	0.00264805	0.00132403	−	0.999999i	$-0.499579\pi$
$227$	0.0398969	0.00264805	0.00132403	+	0.999999i	$0.499579\pi$
$228$	0	0
$229$	−2.16561	−0.143108	−0.0715539	−	0.997437i	$-0.522796\pi$
$229$	−2.16561	−0.143108	−0.0715539	+	0.997437i	$0.522796\pi$
$230$	0	0
$231$	0.113933	0.00749622
$232$	0	0
$233$	−8.93213	−0.585163	−0.292582	−	0.956241i	$-0.594514\pi$
$233$	−8.93213	−0.585163	−0.292582	+	0.956241i	$0.594514\pi$
$234$	0	0
$235$	−4.68054	−0.305325
$236$	0	0
$237$	−0.243955	−0.0158466
$238$	0	0
$239$	−0.452001	−0.0292375	−0.0146188	−	0.999893i	$-0.504653\pi$
$239$	−0.452001	−0.0292375	−0.0146188	+	0.999893i	$0.504653\pi$
$240$	0	0
$241$	−22.0101	−1.41779	−0.708896	−	0.705313i	$-0.750806\pi$
$241$	−22.0101	−1.41779	−0.708896	+	0.705313i	$0.750806\pi$
$242$	0	0
$243$	−0.573305	−0.0367775
$244$	0	0
$245$	2.89641	0.185045
$246$	0	0
$247$	24.3950	1.55222
$248$	0	0
$249$	−0.0275613	−0.00174663
$250$	0	0
$251$	−23.9760	−1.51335	−0.756675	−	0.653791i	$-0.773178\pi$
$251$	−23.9760	−1.51335	−0.756675	+	0.653791i	$0.773178\pi$
$252$	0	0
$253$	−41.7595	−2.62540
$254$	0	0
$255$	0.0136707	0.000856090 0
$256$	0	0
$257$	20.8626	1.30137	0.650686	−	0.759347i	$-0.274482\pi$
$257$	20.8626	1.30137	0.650686	+	0.759347i	$0.274482\pi$
$258$	0	0
$259$	8.31683	0.516782
$260$	0	0
$261$	7.45594	0.461511
$262$	0	0
$263$	−24.5642	−1.51469	−0.757346	−	0.653014i	$-0.773504\pi$
$263$	−24.5642	−1.51469	−0.757346	+	0.653014i	$0.773504\pi$
$264$	0	0
$265$	5.42029	0.332966
$266$	0	0
$267$	0.00862316	0.000527729 0
$268$	0	0
$269$	25.7909	1.57250	0.786250	−	0.617909i	$-0.212020\pi$
$269$	25.7909	1.57250	0.786250	+	0.617909i	$0.212020\pi$
$270$	0	0
$271$	22.5342	1.36886	0.684428	−	0.729080i	$-0.260052\pi$
$271$	22.5342	1.36886	0.684428	+	0.729080i	$0.260052\pi$
$272$	0	0
$273$	0.0849071	0.00513881
$274$	0	0
$275$	26.9854	1.62728
$276$	0	0
$277$	15.9778	0.960012	0.480006	−	0.877265i	$-0.340635\pi$
$277$	15.9778	0.960012	0.480006	+	0.877265i	$0.340635\pi$
$278$	0	0
$279$	−15.2945	−0.915658
$280$	0	0
$281$	16.7423	0.998760	0.499380	−	0.866383i	$-0.333561\pi$
$281$	16.7423	0.998760	0.499380	+	0.866383i	$0.333561\pi$
$282$	0	0
$283$	9.65519	0.573941	0.286971	−	0.957939i	$-0.407352\pi$
$283$	9.65519	0.573941	0.286971	+	0.957939i	$0.407352\pi$
$284$	0	0
$285$	−0.0584124	−0.00346005
$286$	0	0
$287$	−0.103803	−0.00612732
$288$	0	0
$289$	−15.1627	−0.891925
$290$	0	0
$291$	−0.215820	−0.0126516
$292$	0	0
$293$	−16.1948	−0.946109	−0.473054	−	0.881033i	$-0.656849\pi$
$293$	−16.1948	−0.946109	−0.473054	+	0.881033i	$0.656849\pi$
$294$	0	0
$295$	4.63276	0.269730
$296$	0	0
$297$	−0.720163	−0.0417881
$298$	0	0
$299$	−31.1208	−1.79976
$300$	0	0
$301$	−10.4299	−0.601170
$302$	0	0
$303$	−0.104717	−0.00601583
$304$	0	0
$305$	6.23069	0.356768
$306$	0	0
$307$	−6.61596	−0.377593	−0.188796	−	0.982016i	$-0.560459\pi$
$307$	−6.61596	−0.377593	−0.188796	+	0.982016i	$0.560459\pi$
$308$	0	0
$309$	0.177215	0.0100814
$310$	0	0
$311$	11.2025	0.635234	0.317617	−	0.948219i	$-0.397117\pi$
$311$	11.2025	0.635234	0.317617	+	0.948219i	$0.397117\pi$
$312$	0	0
$313$	27.4312	1.55050	0.775252	−	0.631652i	$-0.217623\pi$
$313$	27.4312	1.55050	0.775252	+	0.631652i	$0.217623\pi$
$314$	0	0
$315$	1.35203	0.0761782
$316$	0	0
$317$	−11.8545	−0.665817	−0.332909	−	0.942959i	$-0.608030\pi$
$317$	−11.8545	−0.665817	−0.332909	+	0.942959i	$0.608030\pi$
$318$	0	0
$319$	14.0492	0.786602
$320$	0	0
$321$	−0.221809	−0.0123802
$322$	0	0
$323$	−7.85036	−0.436806
$324$	0	0
$325$	20.1106	1.11553
$326$	0	0
$327$	−0.318133	−0.0175928
$328$	0	0
$329$	9.35488	0.515751
$330$	0	0
$331$	6.09504	0.335014	0.167507	−	0.985871i	$-0.446428\pi$
$331$	6.09504	0.335014	0.167507	+	0.985871i	$0.446428\pi$
$332$	0	0
$333$	−26.2831	−1.44031
$334$	0	0
$335$	−2.97219	−0.162388
$336$	0	0
$337$	−28.5402	−1.55468	−0.777342	−	0.629078i	$-0.783433\pi$
$337$	−28.5402	−1.55468	−0.777342	+	0.629078i	$0.783433\pi$
$338$	0	0
$339$	0.276240	0.0150033
$340$	0	0
$341$	−28.8193	−1.56065
$342$	0	0
$343$	−12.4331	−0.671322
$344$	0	0
$345$	0.0745168	0.00401185
$346$	0	0
$347$	−12.0735	−0.648139	−0.324070	−	0.946033i	$-0.605051\pi$
$347$	−12.0735	−0.648139	−0.324070	+	0.946033i	$0.605051\pi$
$348$	0	0
$349$	31.9952	1.71266	0.856332	−	0.516426i	$-0.172738\pi$
$349$	31.9952	1.71266	0.856332	+	0.516426i	$0.172738\pi$
$350$	0	0
$351$	−0.536693	−0.0286466
$352$	0	0
$353$	−16.0944	−0.856620	−0.428310	−	0.903632i	$-0.640891\pi$
$353$	−16.0944	−0.856620	−0.428310	+	0.903632i	$0.640891\pi$
$354$	0	0
$355$	−5.80964	−0.308344
$356$	0	0
$357$	−0.0273232	−0.00144610
$358$	0	0
$359$	18.4423	0.973345	0.486673	−	0.873584i	$-0.338211\pi$
$359$	18.4423	0.973345	0.486673	+	0.873584i	$0.338211\pi$
$360$	0	0
$361$	14.5432	0.765432
$362$	0	0
$363$	−0.444831	−0.0233476
$364$	0	0
$365$	7.17762	0.375694
$366$	0	0
$367$	1.54735	0.0807711	0.0403855	−	0.999184i	$-0.487141\pi$
$367$	1.54735	0.0807711	0.0403855	+	0.999184i	$0.487141\pi$
$368$	0	0
$369$	0.328044	0.0170773
$370$	0	0
$371$	−10.8334	−0.562442
$372$	0	0
$373$	25.8271	1.33728	0.668639	−	0.743587i	$-0.266877\pi$
$373$	25.8271	1.33728	0.668639	+	0.743587i	$0.266877\pi$
$374$	0	0
$375$	−0.0985817	−0.00509074
$376$	0	0
$377$	10.4700	0.539231
$378$	0	0
$379$	−0.923274	−0.0474254	−0.0237127	−	0.999719i	$-0.507549\pi$
$379$	−0.923274	−0.0474254	−0.0237127	+	0.999719i	$0.507549\pi$
$380$	0	0
$381$	−0.132267	−0.00677625
$382$	0	0
$383$	−21.0579	−1.07601	−0.538003	−	0.842943i	$-0.680821\pi$
$383$	−21.0579	−1.07601	−0.538003	+	0.842943i	$0.680821\pi$
$384$	0	0
$385$	2.54761	0.129838
$386$	0	0
$387$	32.9610	1.67550
$388$	0	0
$389$	38.5231	1.95320	0.976600	−	0.215062i	$-0.0689953\pi$
$389$	38.5231	1.95320	0.976600	+	0.215062i	$0.0689953\pi$
$390$	0	0
$391$	10.0147	0.506466
$392$	0	0
$393$	−0.0719047	−0.00362711
$394$	0	0
$395$	−5.45501	−0.274471
$396$	0	0
$397$	20.8948	1.04868	0.524340	−	0.851509i	$-0.324312\pi$
$397$	20.8948	1.04868	0.524340	+	0.851509i	$0.324312\pi$
$398$	0	0
$399$	0.116747	0.00584468
$400$	0	0
$401$	22.7395	1.13556	0.567779	−	0.823181i	$-0.307803\pi$
$401$	22.7395	1.13556	0.567779	+	0.823181i	$0.307803\pi$
$402$	0	0
$403$	−21.4772	−1.06986
$404$	0	0
$405$	−4.27209	−0.212282
$406$	0	0
$407$	−49.5250	−2.45486
$408$	0	0
$409$	−10.2536	−0.507005	−0.253503	−	0.967335i	$-0.581583\pi$
$409$	−10.2536	−0.507005	−0.253503	+	0.967335i	$0.581583\pi$
$410$	0	0
$411$	−0.108751	−0.00536431
$412$	0	0
$413$	−9.25939	−0.455625
$414$	0	0
$415$	−0.616290	−0.0302525
$416$	0	0
$417$	−0.0321318	−0.00157350
$418$	0	0
$419$	−3.50245	−0.171106	−0.0855529	−	0.996334i	$-0.527266\pi$
$419$	−3.50245	−0.171106	−0.0855529	+	0.996334i	$0.527266\pi$
$420$	0	0
$421$	24.9194	1.21450	0.607248	−	0.794513i	$-0.292274\pi$
$421$	24.9194	1.21450	0.607248	+	0.794513i	$0.292274\pi$
$422$	0	0
$423$	−29.5636	−1.43743
$424$	0	0
$425$	−6.47162	−0.313920
$426$	0	0
$427$	−12.4531	−0.602650
$428$	0	0
$429$	−0.505605	−0.0244108
$430$	0	0
$431$	−18.6596	−0.898803	−0.449401	−	0.893330i	$-0.648363\pi$
$431$	−18.6596	−0.898803	−0.449401	+	0.893330i	$0.648363\pi$
$432$	0	0
$433$	−14.8830	−0.715232	−0.357616	−	0.933869i	$-0.616410\pi$
$433$	−14.8830	−0.715232	−0.357616	+	0.933869i	$0.616410\pi$
$434$	0	0
$435$	−0.0250697	−0.00120200
$436$	0	0
$437$	−42.7911	−2.04698
$438$	0	0
$439$	−24.4158	−1.16530	−0.582652	−	0.812722i	$-0.697985\pi$
$439$	−24.4158	−1.16530	−0.582652	+	0.812722i	$0.697985\pi$
$440$	0	0
$441$	18.2946	0.871170
$442$	0	0
$443$	−13.4454	−0.638811	−0.319406	−	0.947618i	$-0.603483\pi$
$443$	−13.4454	−0.638811	−0.319406	+	0.947618i	$0.603483\pi$
$444$	0	0
$445$	0.192820	0.00914053
$446$	0	0
$447$	−0.175406	−0.00829644
$448$	0	0
$449$	8.47229	0.399832	0.199916	−	0.979813i	$-0.435933\pi$
$449$	8.47229	0.399832	0.199916	+	0.979813i	$0.435933\pi$
$450$	0	0
$451$	0.618129	0.0291066
$452$	0	0
$453$	−0.263717	−0.0123905
$454$	0	0
$455$	1.89858	0.0890068
$456$	0	0
$457$	−38.4208	−1.79725	−0.898625	−	0.438719i	$-0.855433\pi$
$457$	−38.4208	−1.79725	−0.898625	+	0.438719i	$0.855433\pi$
$458$	0	0
$459$	0.172709	0.00806136
$460$	0	0
$461$	17.1191	0.797314	0.398657	−	0.917100i	$-0.369476\pi$
$461$	17.1191	0.797314	0.398657	+	0.917100i	$0.369476\pi$
$462$	0	0
$463$	13.2508	0.615817	0.307908	−	0.951416i	$-0.400371\pi$
$463$	13.2508	0.615817	0.307908	+	0.951416i	$0.400371\pi$
$464$	0	0
$465$	0.0514259	0.00238482
$466$	0	0
$467$	−4.26675	−0.197442	−0.0987208	−	0.995115i	$-0.531475\pi$
$467$	−4.26675	−0.197442	−0.0987208	+	0.995115i	$0.531475\pi$
$468$	0	0
$469$	5.94044	0.274304
$470$	0	0
$471$	0.0259268	0.00119465
$472$	0	0
$473$	62.1080	2.85573
$474$	0	0
$475$	27.6521	1.26877
$476$	0	0
$477$	34.2361	1.56756
$478$	0	0
$479$	−0.511377	−0.0233654	−0.0116827	−	0.999932i	$-0.503719\pi$
$479$	−0.511377	−0.0233654	−0.0116827	+	0.999932i	$0.503719\pi$
$480$	0	0
$481$	−36.9080	−1.68286
$482$	0	0
$483$	−0.148935	−0.00677677
$484$	0	0
$485$	−4.82588	−0.219132
$486$	0	0
$487$	−7.71323	−0.349520	−0.174760	−	0.984611i	$-0.555915\pi$
$487$	−7.71323	−0.349520	−0.174760	+	0.984611i	$0.555915\pi$
$488$	0	0
$489$	−0.356577	−0.0161250
$490$	0	0
$491$	32.8947	1.48452	0.742258	−	0.670115i	$-0.233755\pi$
$491$	32.8947	1.48452	0.742258	+	0.670115i	$0.233755\pi$
$492$	0	0
$493$	−3.36926	−0.151744
$494$	0	0
$495$	−8.05106	−0.361868
$496$	0	0
$497$	11.6116	0.520851
$498$	0	0
$499$	26.1879	1.17233	0.586166	−	0.810191i	$-0.300637\pi$
$499$	26.1879	1.17233	0.586166	+	0.810191i	$0.300637\pi$
$500$	0	0
$501$	−0.471566	−0.0210680
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−2.34154	−0.104197
$506$	0	0
$507$	−0.100705	−0.00447248
$508$	0	0
$509$	11.5932	0.513860	0.256930	−	0.966430i	$-0.417289\pi$
$509$	11.5932	0.513860	0.256930	+	0.966430i	$0.417289\pi$
$510$	0	0
$511$	−14.3457	−0.634618
$512$	0	0
$513$	−0.737955	−0.0325815
$514$	0	0
$515$	3.96265	0.174615
$516$	0	0
$517$	−55.7064	−2.44997
$518$	0	0
$519$	0.0755524	0.00331638
$520$	0	0
$521$	−15.5774	−0.682457	−0.341228	−	0.939980i	$-0.610843\pi$
$521$	−15.5774	−0.682457	−0.341228	+	0.939980i	$0.610843\pi$
$522$	0	0
$523$	8.85293	0.387112	0.193556	−	0.981089i	$-0.437998\pi$
$523$	8.85293	0.387112	0.193556	+	0.981089i	$0.437998\pi$
$524$	0	0
$525$	0.0962434	0.00420041
$526$	0	0
$527$	6.91141	0.301066
$528$	0	0
$529$	31.5888	1.37342
$530$	0	0
$531$	29.2619	1.26986
$532$	0	0
$533$	0.460654	0.0199531
$534$	0	0
$535$	−4.95981	−0.214431
$536$	0	0
$537$	0.106212	0.00458338
$538$	0	0
$539$	34.4723	1.48483
$540$	0	0
$541$	2.23701	0.0961764	0.0480882	−	0.998843i	$-0.484687\pi$
$541$	2.23701	0.0961764	0.0480882	+	0.998843i	$0.484687\pi$
$542$	0	0
$543$	0.158575	0.00680511
$544$	0	0
$545$	−7.11367	−0.304716
$546$	0	0
$547$	11.3416	0.484933	0.242467	−	0.970160i	$-0.422044\pi$
$547$	11.3416	0.484933	0.242467	+	0.970160i	$0.422044\pi$
$548$	0	0
$549$	39.3549	1.67963
$550$	0	0
$551$	14.3962	0.613300
$552$	0	0
$553$	10.9028	0.463634
$554$	0	0
$555$	0.0883739	0.00375126
$556$	0	0
$557$	−7.86920	−0.333429	−0.166714	−	0.986005i	$-0.553316\pi$
$557$	−7.86920	−0.333429	−0.166714	+	0.986005i	$0.553316\pi$
$558$	0	0
$559$	46.2853	1.95766
$560$	0	0
$561$	0.162704	0.00686939
$562$	0	0
$563$	−23.6359	−0.996136	−0.498068	−	0.867138i	$-0.665957\pi$
$563$	−23.6359	−0.996136	−0.498068	+	0.867138i	$0.665957\pi$
$564$	0	0
$565$	6.17691	0.259864
$566$	0	0
$567$	8.53852	0.358584
$568$	0	0
$569$	0.853432	0.0357777	0.0178889	−	0.999840i	$-0.494305\pi$
$569$	0.853432	0.0357777	0.0178889	+	0.999840i	$0.494305\pi$
$570$	0	0
$571$	19.1184	0.800080	0.400040	−	0.916498i	$-0.368996\pi$
$571$	19.1184	0.800080	0.400040	+	0.916498i	$0.368996\pi$
$572$	0	0
$573$	0.567539	0.0237093
$574$	0	0
$575$	−35.2759	−1.47110
$576$	0	0
$577$	14.4493	0.601531	0.300765	−	0.953698i	$-0.402758\pi$
$577$	14.4493	0.601531	0.300765	+	0.953698i	$0.402758\pi$
$578$	0	0
$579$	−0.0659517	−0.00274086
$580$	0	0
$581$	1.23176	0.0511022
$582$	0	0
$583$	64.5107	2.67176
$584$	0	0
$585$	−5.99996	−0.248068
$586$	0	0
$587$	−26.3329	−1.08687	−0.543437	−	0.839450i	$-0.682877\pi$
$587$	−26.3329	−1.08687	−0.543437	+	0.839450i	$0.682877\pi$
$588$	0	0
$589$	−29.5312	−1.21681
$590$	0	0
$591$	−0.0184055	−0.000757100 0
$592$	0	0
$593$	−22.5482	−0.925942	−0.462971	−	0.886374i	$-0.653217\pi$
$593$	−22.5482	−0.925942	−0.462971	+	0.886374i	$0.653217\pi$
$594$	0	0
$595$	−0.610966	−0.0250472
$596$	0	0
$597$	0.539004	0.0220600
$598$	0	0
$599$	−40.3703	−1.64949	−0.824743	−	0.565508i	$-0.808680\pi$
$599$	−40.3703	−1.64949	−0.824743	+	0.565508i	$0.808680\pi$
$600$	0	0
$601$	5.47781	0.223445	0.111722	−	0.993739i	$-0.464363\pi$
$601$	5.47781	0.223445	0.111722	+	0.993739i	$0.464363\pi$
$602$	0	0
$603$	−18.7732	−0.764504
$604$	0	0
$605$	−9.94673	−0.404392
$606$	0	0
$607$	−22.8596	−0.927841	−0.463921	−	0.885877i	$-0.653558\pi$
$607$	−22.8596	−0.927841	−0.463921	+	0.885877i	$0.653558\pi$
$608$	0	0
$609$	0.0501062	0.00203041
$610$	0	0
$611$	−41.5146	−1.67950
$612$	0	0
$613$	16.7922	0.678230	0.339115	−	0.940745i	$-0.389872\pi$
$613$	16.7922	0.678230	0.339115	+	0.940745i	$0.389872\pi$
$614$	0	0
$615$	−0.00110301	−4.44775e−5 0
$616$	0	0
$617$	−15.8055	−0.636305	−0.318153	−	0.948040i	$-0.603062\pi$
$617$	−15.8055	−0.636305	−0.318153	+	0.948040i	$0.603062\pi$
$618$	0	0
$619$	10.5871	0.425530	0.212765	−	0.977103i	$-0.431753\pi$
$619$	10.5871	0.425530	0.212765	+	0.977103i	$0.431753\pi$
$620$	0	0
$621$	0.941410	0.0377775
$622$	0	0
$623$	−0.385384	−0.0154401
$624$	0	0
$625$	21.6680	0.866722
$626$	0	0
$627$	−0.695208	−0.0277639
$628$	0	0
$629$	11.8770	0.473569
$630$	0	0
$631$	32.8530	1.30786	0.653929	−	0.756556i	$-0.273119\pi$
$631$	32.8530	1.30786	0.653929	+	0.756556i	$0.273119\pi$
$632$	0	0
$633$	−0.468465	−0.0186198
$634$	0	0
$635$	−2.95758	−0.117368
$636$	0	0
$637$	25.6901	1.01788
$638$	0	0
$639$	−36.6954	−1.45165
$640$	0	0
$641$	−10.1090	−0.399281	−0.199640	−	0.979869i	$-0.563977\pi$
$641$	−10.1090	−0.399281	−0.199640	+	0.979869i	$0.563977\pi$
$642$	0	0
$643$	44.3707	1.74981	0.874905	−	0.484295i	$-0.160924\pi$
$643$	44.3707	1.74981	0.874905	+	0.484295i	$0.160924\pi$
$644$	0	0
$645$	−0.110827	−0.00436382
$646$	0	0
$647$	25.4109	0.999006	0.499503	−	0.866312i	$-0.333516\pi$
$647$	25.4109	0.999006	0.499503	+	0.866312i	$0.333516\pi$
$648$	0	0
$649$	55.1378	2.16435
$650$	0	0
$651$	−0.102784	−0.00402841
$652$	0	0
$653$	7.41205	0.290056	0.145028	−	0.989428i	$-0.453673\pi$
$653$	7.41205	0.290056	0.145028	+	0.989428i	$0.453673\pi$
$654$	0	0
$655$	−1.60784	−0.0628235
$656$	0	0
$657$	45.3359	1.76872
$658$	0	0
$659$	22.7929	0.887885	0.443942	−	0.896055i	$-0.353580\pi$
$659$	22.7929	0.887885	0.443942	+	0.896055i	$0.353580\pi$
$660$	0	0
$661$	−5.04383	−0.196182	−0.0980911	−	0.995177i	$-0.531274\pi$
$661$	−5.04383	−0.196182	−0.0980911	+	0.995177i	$0.531274\pi$
$662$	0	0
$663$	0.121254	0.00470910
$664$	0	0
$665$	2.61055	0.101233
$666$	0	0
$667$	−18.3653	−0.711108
$668$	0	0
$669$	−0.326135	−0.0126091
$670$	0	0
$671$	74.1560	2.86276
$672$	0	0
$673$	15.4165	0.594262	0.297131	−	0.954837i	$-0.403970\pi$
$673$	15.4165	0.594262	0.297131	+	0.954837i	$0.403970\pi$
$674$	0	0
$675$	−0.608350	−0.0234154
$676$	0	0
$677$	12.1239	0.465961	0.232981	−	0.972481i	$-0.425152\pi$
$677$	12.1239	0.465961	0.232981	+	0.972481i	$0.425152\pi$
$678$	0	0
$679$	9.64537	0.370155
$680$	0	0
$681$	−0.000847322 0	−3.24694e−5 0
$682$	0	0
$683$	−34.1413	−1.30638	−0.653191	−	0.757193i	$-0.726570\pi$
$683$	−34.1413	−1.30638	−0.653191	+	0.757193i	$0.726570\pi$
$684$	0	0
$685$	−2.43175	−0.0929126
$686$	0	0
$687$	0.0459928	0.00175473
$688$	0	0
$689$	48.0759	1.83155
$690$	0	0
$691$	−28.6315	−1.08919	−0.544597	−	0.838698i	$-0.683318\pi$
$691$	−28.6315	−1.08919	−0.544597	+	0.838698i	$0.683318\pi$
$692$	0	0
$693$	16.0915	0.611264
$694$	0	0
$695$	−0.718489	−0.0272538
$696$	0	0
$697$	−0.148239	−0.00561496
$698$	0	0
$699$	0.189698	0.00717506
$700$	0	0
$701$	−30.5446	−1.15365	−0.576826	−	0.816867i	$-0.695709\pi$
$701$	−30.5446	−1.15365	−0.576826	+	0.816867i	$0.695709\pi$
$702$	0	0
$703$	−50.7485	−1.91402
$704$	0	0
$705$	0.0994041	0.00374378
$706$	0	0
$707$	4.67998	0.176009
$708$	0	0
$709$	−28.7595	−1.08009	−0.540043	−	0.841637i	$-0.681592\pi$
$709$	−28.7595	−1.08009	−0.540043	+	0.841637i	$0.681592\pi$
$710$	0	0
$711$	−34.4554	−1.29218
$712$	0	0
$713$	37.6731	1.41087
$714$	0	0
$715$	−11.3057	−0.422808
$716$	0	0
$717$	0.00959949	0.000358500 0
$718$	0	0
$719$	45.6495	1.70244	0.851220	−	0.524809i	$-0.175863\pi$
$719$	45.6495	1.70244	0.851220	+	0.524809i	$0.175863\pi$
$720$	0	0
$721$	−7.92006	−0.294958
$722$	0	0
$723$	0.467444	0.0173844
$724$	0	0
$725$	11.8679	0.440762
$726$	0	0
$727$	−27.2057	−1.00901	−0.504503	−	0.863410i	$-0.668324\pi$
$727$	−27.2057	−1.00901	−0.504503	+	0.863410i	$0.668324\pi$
$728$	0	0
$729$	−26.9756	−0.999098
$730$	0	0
$731$	−14.8947	−0.550900
$732$	0	0
$733$	7.19554	0.265773	0.132887	−	0.991131i	$-0.457575\pi$
$733$	7.19554	0.265773	0.132887	+	0.991131i	$0.457575\pi$
$734$	0	0
$735$	−0.0615133	−0.00226895
$736$	0	0
$737$	−35.3741	−1.30302
$738$	0	0
$739$	8.77115	0.322652	0.161326	−	0.986901i	$-0.448423\pi$
$739$	8.77115	0.322652	0.161326	+	0.986901i	$0.448423\pi$
$740$	0	0
$741$	−0.518096	−0.0190327
$742$	0	0
$743$	−26.0345	−0.955114	−0.477557	−	0.878601i	$-0.658478\pi$
$743$	−26.0345	−0.955114	−0.477557	+	0.878601i	$0.658478\pi$
$744$	0	0
$745$	−3.92221	−0.143699
$746$	0	0
$747$	−3.89267	−0.142425
$748$	0	0
$749$	9.91305	0.362215
$750$	0	0
$751$	44.8577	1.63688	0.818441	−	0.574591i	$-0.194839\pi$
$751$	44.8577	1.63688	0.818441	+	0.574591i	$0.194839\pi$
$752$	0	0
$753$	0.509196	0.0185561
$754$	0	0
$755$	−5.89690	−0.214610
$756$	0	0
$757$	−44.3441	−1.61171	−0.805857	−	0.592111i	$-0.798295\pi$
$757$	−44.3441	−1.61171	−0.805857	+	0.592111i	$0.798295\pi$
$758$	0	0
$759$	0.886878	0.0321916
$760$	0	0
$761$	20.0122	0.725440	0.362720	−	0.931898i	$-0.381848\pi$
$761$	20.0122	0.725440	0.362720	+	0.931898i	$0.381848\pi$
$762$	0	0
$763$	14.2179	0.514723
$764$	0	0
$765$	1.93080	0.0698082
$766$	0	0
$767$	41.0908	1.48370
$768$	0	0
$769$	26.4510	0.953849	0.476924	−	0.878944i	$-0.341752\pi$
$769$	26.4510	0.953849	0.476924	+	0.878944i	$0.341752\pi$
$770$	0	0
$771$	−0.443074	−0.0159569
$772$	0	0
$773$	−22.1328	−0.796061	−0.398031	−	0.917372i	$-0.630306\pi$
$773$	−22.1328	−0.796061	−0.398031	+	0.917372i	$0.630306\pi$
$774$	0	0
$775$	−24.3448	−0.874489
$776$	0	0
$777$	−0.176631	−0.00633659
$778$	0	0
$779$	0.633400	0.0226939
$780$	0	0
$781$	−69.1447	−2.47419
$782$	0	0
$783$	−0.316719	−0.0113186
$784$	0	0
$785$	0.579742	0.0206919
$786$	0	0
$787$	−23.3516	−0.832393	−0.416197	−	0.909275i	$-0.636637\pi$
$787$	−23.3516	−0.832393	−0.416197	+	0.909275i	$0.636637\pi$
$788$	0	0
$789$	0.521688	0.0185726
$790$	0	0
$791$	−12.3456	−0.438960
$792$	0	0
$793$	55.2639	1.96248
$794$	0	0
$795$	−0.115115	−0.00408270
$796$	0	0
$797$	−23.3851	−0.828343	−0.414172	−	0.910199i	$-0.635929\pi$
$797$	−23.3851	−0.828343	−0.414172	+	0.910199i	$0.635929\pi$
$798$	0	0
$799$	13.3595	0.472624
$800$	0	0
$801$	1.21791	0.0430326
$802$	0	0
$803$	85.4260	3.01462
$804$	0	0
$805$	−3.33029	−0.117377
$806$	0	0
$807$	−0.547741	−0.0192814
$808$	0	0
$809$	−48.1723	−1.69365	−0.846825	−	0.531872i	$-0.821489\pi$
$809$	−48.1723	−1.69365	−0.846825	+	0.531872i	$0.821489\pi$
$810$	0	0
$811$	52.2988	1.83646	0.918229	−	0.396049i	$-0.129619\pi$
$811$	52.2988	1.83646	0.918229	+	0.396049i	$0.129619\pi$
$812$	0	0
$813$	−0.478576	−0.0167844
$814$	0	0
$815$	−7.97331	−0.279293
$816$	0	0
$817$	63.6424	2.22657
$818$	0	0
$819$	11.9920	0.419034
$820$	0	0
$821$	11.1196	0.388076	0.194038	−	0.980994i	$-0.437841\pi$
$821$	11.1196	0.388076	0.194038	+	0.980994i	$0.437841\pi$
$822$	0	0
$823$	36.3903	1.26848	0.634242	−	0.773134i	$-0.281312\pi$
$823$	36.3903	1.26848	0.634242	+	0.773134i	$0.281312\pi$
$824$	0	0
$825$	−0.573110	−0.0199531
$826$	0	0
$827$	1.53095	0.0532363	0.0266181	−	0.999646i	$-0.491526\pi$
$827$	1.53095	0.0532363	0.0266181	+	0.999646i	$0.491526\pi$
$828$	0	0
$829$	22.1291	0.768576	0.384288	−	0.923213i	$-0.374447\pi$
$829$	22.1291	0.768576	0.384288	+	0.923213i	$0.374447\pi$
$830$	0	0
$831$	−0.339332	−0.0117713
$832$	0	0
$833$	−8.26711	−0.286438
$834$	0	0
$835$	−10.5445	−0.364909
$836$	0	0
$837$	0.649691	0.0224566
$838$	0	0
$839$	−33.4376	−1.15440	−0.577198	−	0.816604i	$-0.695854\pi$
$839$	−33.4376	−1.15440	−0.577198	+	0.816604i	$0.695854\pi$
$840$	0	0
$841$	−22.8214	−0.786943
$842$	0	0
$843$	−0.355568	−0.0122464
$844$	0	0
$845$	−2.25184	−0.0774656
$846$	0	0
$847$	19.8803	0.683095
$848$	0	0
$849$	−0.205055	−0.00703746
$850$	0	0
$851$	64.7400	2.21926
$852$	0	0
$853$	33.2308	1.13780	0.568900	−	0.822406i	$-0.307369\pi$
$853$	33.2308	1.13780	0.568900	+	0.822406i	$0.307369\pi$
$854$	0	0
$855$	−8.24996	−0.282143
$856$	0	0
$857$	−46.6091	−1.59214	−0.796068	−	0.605207i	$-0.793090\pi$
$857$	−46.6091	−1.59214	−0.796068	+	0.605207i	$0.793090\pi$
$858$	0	0
$859$	24.7875	0.845739	0.422869	−	0.906191i	$-0.361023\pi$
$859$	24.7875	0.845739	0.422869	+	0.906191i	$0.361023\pi$
$860$	0	0
$861$	0.00220455	7.51310e−5 0
$862$	0	0
$863$	−5.95065	−0.202562	−0.101281	−	0.994858i	$-0.532294\pi$
$863$	−5.95065	−0.202562	−0.101281	+	0.994858i	$0.532294\pi$
$864$	0	0
$865$	1.68940	0.0574414
$866$	0	0
$867$	0.322022	0.0109365
$868$	0	0
$869$	−64.9240	−2.20239
$870$	0	0
$871$	−26.3622	−0.893248
$872$	0	0
$873$	−30.4817	−1.03165
$874$	0	0
$875$	4.40579	0.148943
$876$	0	0
$877$	18.4487	0.622968	0.311484	−	0.950251i	$-0.399174\pi$
$877$	18.4487	0.622968	0.311484	+	0.950251i	$0.399174\pi$
$878$	0	0
$879$	0.343941	0.0116008
$880$	0	0
$881$	50.8873	1.71444	0.857218	−	0.514954i	$-0.172191\pi$
$881$	50.8873	1.71444	0.857218	+	0.514954i	$0.172191\pi$
$882$	0	0
$883$	34.4156	1.15818	0.579088	−	0.815265i	$-0.303409\pi$
$883$	34.4156	1.15818	0.579088	+	0.815265i	$0.303409\pi$
$884$	0	0
$885$	−0.0983894	−0.00330733
$886$	0	0
$887$	−14.1642	−0.475588	−0.237794	−	0.971316i	$-0.576424\pi$
$887$	−14.1642	−0.475588	−0.237794	+	0.971316i	$0.576424\pi$
$888$	0	0
$889$	5.91125	0.198257
$890$	0	0
$891$	−50.8452	−1.70338
$892$	0	0
$893$	−57.0827	−1.91020
$894$	0	0
$895$	2.37497	0.0793865
$896$	0	0
$897$	0.660936	0.0220680
$898$	0	0
$899$	−12.6744	−0.422714
$900$	0	0
$901$	−15.4709	−0.515411
$902$	0	0
$903$	0.221508	0.00737132
$904$	0	0
$905$	3.54585	0.117868
$906$	0	0
$907$	−1.56622	−0.0520056	−0.0260028	−	0.999662i	$-0.508278\pi$
$907$	−1.56622	−0.0520056	−0.0260028	+	0.999662i	$0.508278\pi$
$908$	0	0
$909$	−14.7899	−0.490549
$910$	0	0
$911$	−18.0858	−0.599211	−0.299605	−	0.954063i	$-0.596855\pi$
$911$	−18.0858	−0.599211	−0.299605	+	0.954063i	$0.596855\pi$
$912$	0	0
$913$	−7.33491	−0.242750
$914$	0	0
$915$	−0.132326	−0.00437456
$916$	0	0
$917$	3.21355	0.106121
$918$	0	0
$919$	−45.3956	−1.49746	−0.748732	−	0.662873i	$-0.769337\pi$
$919$	−45.3956	−1.49746	−0.748732	+	0.662873i	$0.769337\pi$
$920$	0	0
$921$	0.140508	0.00462990
$922$	0	0
$923$	−51.5293	−1.69611
$924$	0	0
$925$	−41.8357	−1.37555
$926$	0	0
$927$	25.0293	0.822069
$928$	0	0
$929$	−40.1721	−1.31800	−0.659001	−	0.752142i	$-0.729021\pi$
$929$	−40.1721	−1.31800	−0.659001	+	0.752142i	$0.729021\pi$
$930$	0	0
$931$	35.3239	1.15769
$932$	0	0
$933$	−0.237915	−0.00778900
$934$	0	0
$935$	3.63818	0.118981
$936$	0	0
$937$	19.9535	0.651853	0.325926	−	0.945395i	$-0.394324\pi$
$937$	19.9535	0.651853	0.325926	+	0.945395i	$0.394324\pi$
$938$	0	0
$939$	−0.582578	−0.0190117
$940$	0	0
$941$	−15.9493	−0.519934	−0.259967	−	0.965618i	$-0.583712\pi$
$941$	−15.9493	−0.519934	−0.259967	+	0.965618i	$0.583712\pi$
$942$	0	0
$943$	−0.808030	−0.0263131
$944$	0	0
$945$	−0.0574325	−0.00186828
$946$	0	0
$947$	−17.9923	−0.584670	−0.292335	−	0.956316i	$-0.594432\pi$
$947$	−17.9923	−0.584670	−0.292335	+	0.956316i	$0.594432\pi$
$948$	0	0
$949$	63.6628	2.06658
$950$	0	0
$951$	0.251764	0.00816400
$952$	0	0
$953$	−36.3190	−1.17649	−0.588244	−	0.808683i	$-0.700181\pi$
$953$	−36.3190	−1.17649	−0.588244	+	0.808683i	$0.700181\pi$
$954$	0	0
$955$	12.6906	0.410657
$956$	0	0
$957$	−0.298373	−0.00964502
$958$	0	0
$959$	4.86029	0.156947
$960$	0	0
$961$	−5.00087	−0.161318
$962$	0	0
$963$	−31.3276	−1.00952
$964$	0	0
$965$	−1.47473	−0.0474731
$966$	0	0
$967$	52.0694	1.67444	0.837219	−	0.546867i	$-0.184180\pi$
$967$	52.0694	1.67444	0.837219	+	0.546867i	$0.184180\pi$
$968$	0	0
$969$	0.166724	0.00535595
$970$	0	0
$971$	−2.24833	−0.0721523	−0.0360762	−	0.999349i	$-0.511486\pi$
$971$	−2.24833	−0.0721523	−0.0360762	+	0.999349i	$0.511486\pi$
$972$	0	0
$973$	1.43603	0.0460369
$974$	0	0
$975$	−0.427104	−0.0136783
$976$	0	0
$977$	−34.3199	−1.09799	−0.548996	−	0.835825i	$-0.684990\pi$
$977$	−34.3199	−1.09799	−0.548996	+	0.835825i	$0.684990\pi$
$978$	0	0
$979$	2.29489	0.0733449
$980$	0	0
$981$	−44.9320	−1.43457
$982$	0	0
$983$	32.6691	1.04198	0.520992	−	0.853562i	$-0.325562\pi$
$983$	32.6691	1.04198	0.520992	+	0.853562i	$0.325562\pi$
$984$	0	0
$985$	−0.411559	−0.0131134
$986$	0	0
$987$	−0.198677	−0.00632395
$988$	0	0
$989$	−81.1887	−2.58165
$990$	0	0
$991$	12.1978	0.387477	0.193738	−	0.981053i	$-0.437939\pi$
$991$	12.1978	0.387477	0.193738	+	0.981053i	$0.437939\pi$
$992$	0	0
$993$	−0.129445	−0.00410781
$994$	0	0
$995$	12.0525	0.382090
$996$	0	0
$997$	−12.2031	−0.386478	−0.193239	−	0.981152i	$-0.561899\pi$
$997$	−12.2031	−0.386478	−0.193239	+	0.981152i	$0.561899\pi$
$998$	0	0
$999$	1.11647	0.0353237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.10		21
4.3	odd	2		2012.2.a.a.1.12	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.12	✓	21	4.3	odd	2
8048.2.a.t.1.10		21	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)

\(f(q)\)	\(=\)	\(q-0.0212378 q^{3} -0.474891 q^{5} +0.949153 q^{7} -2.99955 q^{9} +O(q^{10})\)	\(q-0.0212378 q^{3} -0.474891 q^{5} +0.949153 q^{7} -2.99955 q^{9} -5.65202 q^{11} -4.21210 q^{13} +0.0100856 q^{15} +1.35546 q^{17} -5.79165 q^{19} -0.0201579 q^{21} +7.38842 q^{23} -4.77448 q^{25} +0.127417 q^{27} -2.48569 q^{29} +5.09893 q^{31} +0.120036 q^{33} -0.450744 q^{35} +8.76237 q^{37} +0.0894556 q^{39} -0.109364 q^{41} -10.9886 q^{43} +1.42446 q^{45} +9.85603 q^{47} -6.09911 q^{49} -0.0287870 q^{51} -11.4138 q^{53} +2.68409 q^{55} +0.123002 q^{57} -9.75542 q^{59} -13.1203 q^{61} -2.84703 q^{63} +2.00029 q^{65} +6.25867 q^{67} -0.156914 q^{69} +12.2336 q^{71} -15.1143 q^{73} +0.101399 q^{75} -5.36463 q^{77} +11.4869 q^{79} +8.99594 q^{81} +1.29775 q^{83} -0.643696 q^{85} +0.0527905 q^{87} -0.406030 q^{89} -3.99793 q^{91} -0.108290 q^{93} +2.75040 q^{95} +10.1621 q^{97} +16.9535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

Introduction

L-functions

Modular forms

Varieties

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