LMFDB - Embedded newform 10000.2.a.bl.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.3266578125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 $ x^8 - 3x^7 - 6x^6 + 23x^5 + x^4 - 45x^3 + 25x^2 + 10x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.14505$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31796 q^{3} -0.438060 q^{7} -1.26298 q^{9} +O(q^{10})$	$q+1.31796 q^{3} -0.438060 q^{7} -1.26298 q^{9} +0.0889565 q^{11} +1.98618 q^{13} -3.64217 q^{17} -2.32477 q^{19} -0.577346 q^{21} -1.90531 q^{23} -5.61844 q^{27} -4.96781 q^{29} +3.82913 q^{31} +0.117241 q^{33} +3.20858 q^{37} +2.61770 q^{39} +4.17513 q^{41} -1.93550 q^{43} +0.291012 q^{47} -6.80810 q^{49} -4.80023 q^{51} +8.63482 q^{53} -3.06395 q^{57} +3.70737 q^{59} +6.71522 q^{61} +0.553262 q^{63} +15.1109 q^{67} -2.51112 q^{69} +10.5420 q^{71} +4.60998 q^{73} -0.0389683 q^{77} +13.4735 q^{79} -3.61593 q^{81} -7.36387 q^{83} -6.54738 q^{87} +11.2130 q^{89} -0.870066 q^{91} +5.04664 q^{93} -3.13492 q^{97} -0.112350 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * 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
$3$	1.31796	0.760924	0.380462	−	0.924796i	$-0.375765\pi$
$3$	1.31796	0.760924	0.380462	+	0.924796i	$0.375765\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.438060	−0.165571	−0.0827856	−	0.996567i	$-0.526382\pi$
$7$	−0.438060	−0.165571	−0.0827856	+	0.996567i	$0.526382\pi$
$8$	0	0
$9$	−1.26298	−0.420994
$10$	0	0
$11$	0.0889565	0.0268214	0.0134107	−	0.999910i	$-0.495731\pi$
$11$	0.0889565	0.0268214	0.0134107	+	0.999910i	$0.495731\pi$
$12$	0	0
$13$	1.98618	0.550867	0.275434	−	0.961320i	$-0.411179\pi$
$13$	1.98618	0.550867	0.275434	+	0.961320i	$0.411179\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.64217	−0.883355	−0.441678	−	0.897174i	$-0.645617\pi$
$17$	−3.64217	−0.883355	−0.441678	+	0.897174i	$0.645617\pi$
$18$	0	0
$19$	−2.32477	−0.533339	−0.266669	−	0.963788i	$-0.585923\pi$
$19$	−2.32477	−0.533339	−0.266669	+	0.963788i	$0.585923\pi$
$20$	0	0
$21$	−0.577346	−0.125987
$22$	0	0
$23$	−1.90531	−0.397284	−0.198642	−	0.980072i	$-0.563653\pi$
$23$	−1.90531	−0.397284	−0.198642	+	0.980072i	$0.563653\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.61844	−1.08127
$28$	0	0
$29$	−4.96781	−0.922500	−0.461250	−	0.887270i	$-0.652599\pi$
$29$	−4.96781	−0.922500	−0.461250	+	0.887270i	$0.652599\pi$
$30$	0	0
$31$	3.82913	0.687732	0.343866	−	0.939019i	$-0.388263\pi$
$31$	3.82913	0.687732	0.343866	+	0.939019i	$0.388263\pi$
$32$	0	0
$33$	0.117241	0.0204091
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.20858	0.527487	0.263743	−	0.964593i	$-0.415043\pi$
$37$	3.20858	0.527487	0.263743	+	0.964593i	$0.415043\pi$
$38$	0	0
$39$	2.61770	0.419168
$40$	0	0
$41$	4.17513	0.652046	0.326023	−	0.945362i	$-0.394291\pi$
$41$	4.17513	0.652046	0.326023	+	0.945362i	$0.394291\pi$
$42$	0	0
$43$	−1.93550	−0.295161	−0.147581	−	0.989050i	$-0.547149\pi$
$43$	−1.93550	−0.295161	−0.147581	+	0.989050i	$0.547149\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.291012	0.0424484	0.0212242	−	0.999775i	$-0.493244\pi$
$47$	0.291012	0.0424484	0.0212242	+	0.999775i	$0.493244\pi$
$48$	0	0
$49$	−6.80810	−0.972586
$50$	0	0
$51$	−4.80023	−0.672167
$52$	0	0
$53$	8.63482	1.18608	0.593042	−	0.805172i	$-0.297927\pi$
$53$	8.63482	1.18608	0.593042	+	0.805172i	$0.297927\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.06395	−0.405831
$58$	0	0
$59$	3.70737	0.482658	0.241329	−	0.970443i	$-0.422417\pi$
$59$	3.70737	0.482658	0.241329	+	0.970443i	$0.422417\pi$
$60$	0	0
$61$	6.71522	0.859796	0.429898	−	0.902877i	$-0.358550\pi$
$61$	6.71522	0.859796	0.429898	+	0.902877i	$0.358550\pi$
$62$	0	0
$63$	0.553262	0.0697045
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.1109	1.84609	0.923047	−	0.384688i	$-0.125691\pi$
$67$	15.1109	1.84609	0.923047	+	0.384688i	$0.125691\pi$
$68$	0	0
$69$	−2.51112	−0.302303
$70$	0	0
$71$	10.5420	1.25110	0.625552	−	0.780183i	$-0.284874\pi$
$71$	10.5420	1.25110	0.625552	+	0.780183i	$0.284874\pi$
$72$	0	0
$73$	4.60998	0.539558	0.269779	−	0.962922i	$-0.413049\pi$
$73$	4.60998	0.539558	0.269779	+	0.962922i	$0.413049\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.0389683	−0.00444085
$78$	0	0
$79$	13.4735	1.51589	0.757945	−	0.652318i	$-0.226204\pi$
$79$	13.4735	1.51589	0.757945	+	0.652318i	$0.226204\pi$
$80$	0	0
$81$	−3.61593	−0.401770
$82$	0	0
$83$	−7.36387	−0.808290	−0.404145	−	0.914695i	$-0.632431\pi$
$83$	−7.36387	−0.808290	−0.404145	+	0.914695i	$0.632431\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.54738	−0.701953
$88$	0	0
$89$	11.2130	1.18858	0.594288	−	0.804252i	$-0.297434\pi$
$89$	11.2130	1.18858	0.594288	+	0.804252i	$0.297434\pi$
$90$	0	0
$91$	−0.870066	−0.0912077
$92$	0	0
$93$	5.04664	0.523312
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.13492	−0.318303	−0.159152	−	0.987254i	$-0.550876\pi$
$97$	−3.13492	−0.318303	−0.159152	+	0.987254i	$0.550876\pi$
$98$	0	0
$99$	−0.112350	−0.0112916
$100$	0	0
$101$	13.5780	1.35106	0.675528	−	0.737334i	$-0.263916\pi$
$101$	13.5780	1.35106	0.675528	+	0.737334i	$0.263916\pi$
$102$	0	0
$103$	5.38399	0.530500	0.265250	−	0.964180i	$-0.414545\pi$
$103$	5.38399	0.530500	0.265250	+	0.964180i	$0.414545\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.62623	0.543908	0.271954	−	0.962310i	$-0.412330\pi$
$107$	5.62623	0.543908	0.271954	+	0.962310i	$0.412330\pi$
$108$	0	0
$109$	−9.60965	−0.920438	−0.460219	−	0.887805i	$-0.652229\pi$
$109$	−9.60965	−0.920438	−0.460219	+	0.887805i	$0.652229\pi$
$110$	0	0
$111$	4.22878	0.401378
$112$	0	0
$113$	9.06681	0.852934	0.426467	−	0.904503i	$-0.359758\pi$
$113$	9.06681	0.852934	0.426467	+	0.904503i	$0.359758\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.50851	−0.231912
$118$	0	0
$119$	1.59549	0.146258
$120$	0	0
$121$	−10.9921	−0.999281
$122$	0	0
$123$	5.50266	0.496158
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.5654	−1.29247	−0.646236	−	0.763137i	$-0.723658\pi$
$127$	−14.5654	−1.29247	−0.646236	+	0.763137i	$0.723658\pi$
$128$	0	0
$129$	−2.55092	−0.224596
$130$	0	0
$131$	13.7201	1.19873	0.599366	−	0.800475i	$-0.295419\pi$
$131$	13.7201	1.19873	0.599366	+	0.800475i	$0.295419\pi$
$132$	0	0
$133$	1.01839	0.0883056
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.7786	1.86067	0.930334	−	0.366712i	$-0.119517\pi$
$137$	21.7786	1.86067	0.930334	+	0.366712i	$0.119517\pi$
$138$	0	0
$139$	17.8147	1.51102	0.755512	−	0.655135i	$-0.227388\pi$
$139$	17.8147	1.51102	0.755512	+	0.655135i	$0.227388\pi$
$140$	0	0
$141$	0.383542	0.0323000
$142$	0	0
$143$	0.176684	0.0147750
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.97281	−0.740065
$148$	0	0
$149$	−19.9748	−1.63640	−0.818200	−	0.574934i	$-0.805028\pi$
$149$	−19.9748	−1.63640	−0.818200	+	0.574934i	$0.805028\pi$
$150$	0	0
$151$	−6.69940	−0.545190	−0.272595	−	0.962129i	$-0.587882\pi$
$151$	−6.69940	−0.545190	−0.272595	+	0.962129i	$0.587882\pi$
$152$	0	0
$153$	4.59999	0.371887
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.99645	−0.797803	−0.398902	−	0.916994i	$-0.630609\pi$
$157$	−9.99645	−0.797803	−0.398902	+	0.916994i	$0.630609\pi$
$158$	0	0
$159$	11.3803	0.902520
$160$	0	0
$161$	0.834639	0.0657788
$162$	0	0
$163$	14.6749	1.14943	0.574714	−	0.818354i	$-0.305113\pi$
$163$	14.6749	1.14943	0.574714	+	0.818354i	$0.305113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.9915	1.16008	0.580039	−	0.814589i	$-0.303037\pi$
$167$	14.9915	1.16008	0.580039	+	0.814589i	$0.303037\pi$
$168$	0	0
$169$	−9.05509	−0.696545
$170$	0	0
$171$	2.93614	0.224532
$172$	0	0
$173$	3.00097	0.228160	0.114080	−	0.993472i	$-0.463608\pi$
$173$	3.00097	0.228160	0.114080	+	0.993472i	$0.463608\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.88616	0.367266
$178$	0	0
$179$	1.90204	0.142165	0.0710824	−	0.997470i	$-0.477355\pi$
$179$	1.90204	0.142165	0.0710824	+	0.997470i	$0.477355\pi$
$180$	0	0
$181$	−16.3412	−1.21463	−0.607316	−	0.794460i	$-0.707754\pi$
$181$	−16.3412	−1.21463	−0.607316	+	0.794460i	$0.707754\pi$
$182$	0	0
$183$	8.85039	0.654240
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.323995	−0.0236928
$188$	0	0
$189$	2.46122	0.179027
$190$	0	0
$191$	−3.06991	−0.222131	−0.111065	−	0.993813i	$-0.535426\pi$
$191$	−3.06991	−0.222131	−0.111065	+	0.993813i	$0.535426\pi$
$192$	0	0
$193$	0.721839	0.0519591	0.0259795	−	0.999662i	$-0.491730\pi$
$193$	0.721839	0.0519591	0.0259795	+	0.999662i	$0.491730\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.17717	0.368858	0.184429	−	0.982846i	$-0.440956\pi$
$197$	5.17717	0.368858	0.184429	+	0.982846i	$0.440956\pi$
$198$	0	0
$199$	−12.6796	−0.898836	−0.449418	−	0.893322i	$-0.648369\pi$
$199$	−12.6796	−0.898836	−0.449418	+	0.893322i	$0.648369\pi$
$200$	0	0
$201$	19.9156	1.40474
$202$	0	0
$203$	2.17620	0.152739
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.40637	0.167254
$208$	0	0
$209$	−0.206803	−0.0143049
$210$	0	0
$211$	−4.85936	−0.334532	−0.167266	−	0.985912i	$-0.553494\pi$
$211$	−4.85936	−0.334532	−0.167266	+	0.985912i	$0.553494\pi$
$212$	0	0
$213$	13.8939	0.951995
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.67739	−0.113869
$218$	0	0
$219$	6.07577	0.410563
$220$	0	0
$221$	−7.23400	−0.486611
$222$	0	0
$223$	3.37347	0.225904	0.112952	−	0.993600i	$-0.463969\pi$
$223$	3.37347	0.225904	0.112952	+	0.993600i	$0.463969\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.46530	−0.628234	−0.314117	−	0.949384i	$-0.601708\pi$
$227$	−9.46530	−0.628234	−0.314117	+	0.949384i	$0.601708\pi$
$228$	0	0
$229$	−7.77910	−0.514057	−0.257029	−	0.966404i	$-0.582743\pi$
$229$	−7.77910	−0.514057	−0.257029	+	0.966404i	$0.582743\pi$
$230$	0	0
$231$	−0.0513587	−0.00337915
$232$	0	0
$233$	−0.583702	−0.0382396	−0.0191198	−	0.999817i	$-0.506086\pi$
$233$	−0.583702	−0.0382396	−0.0191198	+	0.999817i	$0.506086\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	17.7576	1.15348
$238$	0	0
$239$	26.2917	1.70067	0.850336	−	0.526241i	$-0.176399\pi$
$239$	26.2917	1.70067	0.850336	+	0.526241i	$0.176399\pi$
$240$	0	0
$241$	13.6631	0.880119	0.440060	−	0.897969i	$-0.354957\pi$
$241$	13.6631	0.880119	0.440060	+	0.897969i	$0.354957\pi$
$242$	0	0
$243$	12.0897	0.775552
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.61741	−0.293799
$248$	0	0
$249$	−9.70528	−0.615047
$250$	0	0
$251$	−12.7339	−0.803757	−0.401879	−	0.915693i	$-0.631643\pi$
$251$	−12.7339	−0.803757	−0.401879	+	0.915693i	$0.631643\pi$
$252$	0	0
$253$	−0.169489	−0.0106557
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−30.8686	−1.92553	−0.962766	−	0.270337i	$-0.912865\pi$
$257$	−30.8686	−1.92553	−0.962766	+	0.270337i	$0.912865\pi$
$258$	0	0
$259$	−1.40555	−0.0873367
$260$	0	0
$261$	6.27426	0.388367
$262$	0	0
$263$	16.0714	0.991005	0.495503	−	0.868606i	$-0.334984\pi$
$263$	16.0714	0.991005	0.495503	+	0.868606i	$0.334984\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.7783	0.904417
$268$	0	0
$269$	−19.0831	−1.16352	−0.581760	−	0.813361i	$-0.697636\pi$
$269$	−19.0831	−1.16352	−0.581760	+	0.813361i	$0.697636\pi$
$270$	0	0
$271$	29.9073	1.81674	0.908369	−	0.418169i	$-0.137328\pi$
$271$	29.9073	1.81674	0.908369	+	0.418169i	$0.137328\pi$
$272$	0	0
$273$	−1.14671	−0.0694022
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.5744	−1.77696	−0.888478	−	0.458919i	$-0.848237\pi$
$277$	−29.5744	−1.77696	−0.888478	+	0.458919i	$0.848237\pi$
$278$	0	0
$279$	−4.83612	−0.289531
$280$	0	0
$281$	10.1680	0.606571	0.303285	−	0.952900i	$-0.401916\pi$
$281$	10.1680	0.606571	0.303285	+	0.952900i	$0.401916\pi$
$282$	0	0
$283$	−6.39083	−0.379895	−0.189948	−	0.981794i	$-0.560832\pi$
$283$	−6.39083	−0.379895	−0.189948	+	0.981794i	$0.560832\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.82896	−0.107960
$288$	0	0
$289$	−3.73462	−0.219683
$290$	0	0
$291$	−4.13170	−0.242205
$292$	0	0
$293$	−5.73602	−0.335102	−0.167551	−	0.985863i	$-0.553586\pi$
$293$	−5.73602	−0.335102	−0.167551	+	0.985863i	$0.553586\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.499797	−0.0290012
$298$	0	0
$299$	−3.78428	−0.218851
$300$	0	0
$301$	0.847867	0.0488702
$302$	0	0
$303$	17.8952	1.02805
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.7023	1.29569	0.647845	−	0.761772i	$-0.275670\pi$
$307$	22.7023	1.29569	0.647845	+	0.761772i	$0.275670\pi$
$308$	0	0
$309$	7.09588	0.403671
$310$	0	0
$311$	−26.3612	−1.49481	−0.747404	−	0.664370i	$-0.768700\pi$
$311$	−26.3612	−1.49481	−0.747404	+	0.664370i	$0.768700\pi$
$312$	0	0
$313$	−4.48777	−0.253664	−0.126832	−	0.991924i	$-0.540481\pi$
$313$	−4.48777	−0.253664	−0.126832	+	0.991924i	$0.540481\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.7922	0.774645	0.387322	−	0.921944i	$-0.373400\pi$
$317$	13.7922	0.774645	0.387322	+	0.921944i	$0.373400\pi$
$318$	0	0
$319$	−0.441919	−0.0247427
$320$	0	0
$321$	7.41514	0.413873
$322$	0	0
$323$	8.46720	0.471128
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.6651	−0.700384
$328$	0	0
$329$	−0.127481	−0.00702823
$330$	0	0
$331$	22.8538	1.25616	0.628080	−	0.778149i	$-0.283841\pi$
$331$	22.8538	1.25616	0.628080	+	0.778149i	$0.283841\pi$
$332$	0	0
$333$	−4.05238	−0.222069
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.8821	1.62778	0.813891	−	0.581018i	$-0.197345\pi$
$337$	29.8821	1.62778	0.813891	+	0.581018i	$0.197345\pi$
$338$	0	0
$339$	11.9497	0.649018
$340$	0	0
$341$	0.340626	0.0184459
$342$	0	0
$343$	6.04878	0.326604
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.48599	0.240820	0.120410	−	0.992724i	$-0.461579\pi$
$347$	4.48599	0.240820	0.120410	+	0.992724i	$0.461579\pi$
$348$	0	0
$349$	−24.4969	−1.31129	−0.655646	−	0.755069i	$-0.727603\pi$
$349$	−24.4969	−1.31129	−0.655646	+	0.755069i	$0.727603\pi$
$350$	0	0
$351$	−11.1592	−0.595636
$352$	0	0
$353$	13.7093	0.729675	0.364837	−	0.931071i	$-0.381125\pi$
$353$	13.7093	0.729675	0.364837	+	0.931071i	$0.381125\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.10279	0.111291
$358$	0	0
$359$	22.2217	1.17281	0.586407	−	0.810016i	$-0.300542\pi$
$359$	22.2217	1.17281	0.586407	+	0.810016i	$0.300542\pi$
$360$	0	0
$361$	−13.5954	−0.715550
$362$	0	0
$363$	−14.4871	−0.760377
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.55555	−0.0811992	−0.0405996	−	0.999175i	$-0.512927\pi$
$367$	−1.55555	−0.0811992	−0.0405996	+	0.999175i	$0.512927\pi$
$368$	0	0
$369$	−5.27312	−0.274507
$370$	0	0
$371$	−3.78257	−0.196381
$372$	0	0
$373$	9.24940	0.478916	0.239458	−	0.970907i	$-0.423030\pi$
$373$	9.24940	0.478916	0.239458	+	0.970907i	$0.423030\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.86697	−0.508175
$378$	0	0
$379$	−6.76941	−0.347721	−0.173861	−	0.984770i	$-0.555624\pi$
$379$	−6.76941	−0.347721	−0.173861	+	0.984770i	$0.555624\pi$
$380$	0	0
$381$	−19.1966	−0.983474
$382$	0	0
$383$	33.3334	1.70326	0.851629	−	0.524145i	$-0.175615\pi$
$383$	33.3334	1.70326	0.851629	+	0.524145i	$0.175615\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.44450	0.124261
$388$	0	0
$389$	−0.162454	−0.00823674	−0.00411837	−	0.999992i	$-0.501311\pi$
$389$	−0.162454	−0.00823674	−0.00411837	+	0.999992i	$0.501311\pi$
$390$	0	0
$391$	6.93944	0.350943
$392$	0	0
$393$	18.0826	0.912145
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.64137	0.182755	0.0913775	−	0.995816i	$-0.470873\pi$
$397$	3.64137	0.182755	0.0913775	+	0.995816i	$0.470873\pi$
$398$	0	0
$399$	1.34220	0.0671939
$400$	0	0
$401$	−2.67293	−0.133480	−0.0667399	−	0.997770i	$-0.521260\pi$
$401$	−2.67293	−0.133480	−0.0667399	+	0.997770i	$0.521260\pi$
$402$	0	0
$403$	7.60534	0.378849
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.285424	0.0141479
$408$	0	0
$409$	28.9610	1.43203	0.716015	−	0.698085i	$-0.245964\pi$
$409$	28.9610	1.43203	0.716015	+	0.698085i	$0.245964\pi$
$410$	0	0
$411$	28.7033	1.41583
$412$	0	0
$413$	−1.62405	−0.0799143
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.4791	1.14978
$418$	0	0
$419$	16.1768	0.790290	0.395145	−	0.918619i	$-0.370694\pi$
$419$	16.1768	0.790290	0.395145	+	0.918619i	$0.370694\pi$
$420$	0	0
$421$	−11.2674	−0.549141	−0.274570	−	0.961567i	$-0.588536\pi$
$421$	−11.2674	−0.549141	−0.274570	+	0.961567i	$0.588536\pi$
$422$	0	0
$423$	−0.367542	−0.0178705
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.94167	−0.142357
$428$	0	0
$429$	0.232862	0.0112427
$430$	0	0
$431$	−1.23495	−0.0594852	−0.0297426	−	0.999558i	$-0.509469\pi$
$431$	−1.23495	−0.0594852	−0.0297426	+	0.999558i	$0.509469\pi$
$432$	0	0
$433$	20.7197	0.995723	0.497862	−	0.867256i	$-0.334119\pi$
$433$	20.7197	0.995723	0.497862	+	0.867256i	$0.334119\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.42940	0.211887
$438$	0	0
$439$	14.6704	0.700178	0.350089	−	0.936716i	$-0.386151\pi$
$439$	14.6704	0.700178	0.350089	+	0.936716i	$0.386151\pi$
$440$	0	0
$441$	8.59851	0.409453
$442$	0	0
$443$	24.5559	1.16668	0.583342	−	0.812226i	$-0.301745\pi$
$443$	24.5559	1.16668	0.583342	+	0.812226i	$0.301745\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−26.3260	−1.24518
$448$	0	0
$449$	2.13731	0.100866	0.0504328	−	0.998727i	$-0.483940\pi$
$449$	2.13731	0.100866	0.0504328	+	0.998727i	$0.483940\pi$
$450$	0	0
$451$	0.371405	0.0174888
$452$	0	0
$453$	−8.82954	−0.414848
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.8309	−1.16154	−0.580771	−	0.814067i	$-0.697249\pi$
$457$	−24.8309	−1.16154	−0.580771	+	0.814067i	$0.697249\pi$
$458$	0	0
$459$	20.4633	0.955145
$460$	0	0
$461$	16.2755	0.758028	0.379014	−	0.925391i	$-0.376263\pi$
$461$	16.2755	0.758028	0.379014	+	0.925391i	$0.376263\pi$
$462$	0	0
$463$	25.4358	1.18210	0.591051	−	0.806634i	$-0.298713\pi$
$463$	25.4358	1.18210	0.591051	+	0.806634i	$0.298713\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.2615	0.706218	0.353109	−	0.935582i	$-0.385124\pi$
$467$	15.2615	0.706218	0.353109	+	0.935582i	$0.385124\pi$
$468$	0	0
$469$	−6.61950	−0.305660
$470$	0	0
$471$	−13.1749	−0.607068
$472$	0	0
$473$	−0.172176	−0.00791664
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.9056	−0.499334
$478$	0	0
$479$	−29.1223	−1.33063	−0.665316	−	0.746561i	$-0.731703\pi$
$479$	−29.1223	−1.33063	−0.665316	+	0.746561i	$0.731703\pi$
$480$	0	0
$481$	6.37281	0.290575
$482$	0	0
$483$	1.10002	0.0500527
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.65245	−0.120194	−0.0600969	−	0.998193i	$-0.519141\pi$
$487$	−2.65245	−0.120194	−0.0600969	+	0.998193i	$0.519141\pi$
$488$	0	0
$489$	19.3410	0.874628
$490$	0	0
$491$	−15.6924	−0.708186	−0.354093	−	0.935210i	$-0.615210\pi$
$491$	−15.6924	−0.708186	−0.354093	+	0.935210i	$0.615210\pi$
$492$	0	0
$493$	18.0936	0.814895
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.61802	−0.207147
$498$	0	0
$499$	−28.2560	−1.26491	−0.632456	−	0.774596i	$-0.717953\pi$
$499$	−28.2560	−1.26491	−0.632456	+	0.774596i	$0.717953\pi$
$500$	0	0
$501$	19.7582	0.882731
$502$	0	0
$503$	9.89026	0.440985	0.220492	−	0.975389i	$-0.429234\pi$
$503$	9.89026	0.440985	0.220492	+	0.975389i	$0.429234\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−11.9342	−0.530018
$508$	0	0
$509$	3.09540	0.137201	0.0686005	−	0.997644i	$-0.478147\pi$
$509$	3.09540	0.137201	0.0686005	+	0.997644i	$0.478147\pi$
$510$	0	0
$511$	−2.01945	−0.0893353
$512$	0	0
$513$	13.0616	0.576683
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.0258874	0.00113853
$518$	0	0
$519$	3.95516	0.173612
$520$	0	0
$521$	−20.1003	−0.880611	−0.440305	−	0.897848i	$-0.645130\pi$
$521$	−20.1003	−0.880611	−0.440305	+	0.897848i	$0.645130\pi$
$522$	0	0
$523$	30.1796	1.31966	0.659830	−	0.751415i	$-0.270628\pi$
$523$	30.1796	1.31966	0.659830	+	0.751415i	$0.270628\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.9463	−0.607512
$528$	0	0
$529$	−19.3698	−0.842166
$530$	0	0
$531$	−4.68234	−0.203196
$532$	0	0
$533$	8.29256	0.359191
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.50681	0.108177
$538$	0	0
$539$	−0.605625	−0.0260861
$540$	0	0
$541$	−26.0503	−1.11999	−0.559995	−	0.828496i	$-0.689197\pi$
$541$	−26.0503	−1.11999	−0.559995	+	0.828496i	$0.689197\pi$
$542$	0	0
$543$	−21.5371	−0.924243
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.96681	0.383393	0.191697	−	0.981454i	$-0.438601\pi$
$547$	8.96681	0.383393	0.191697	+	0.981454i	$0.438601\pi$
$548$	0	0
$549$	−8.48120	−0.361969
$550$	0	0
$551$	11.5490	0.492005
$552$	0	0
$553$	−5.90222	−0.250988
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.7175	0.454115	0.227057	−	0.973881i	$-0.427090\pi$
$557$	10.7175	0.454115	0.227057	+	0.973881i	$0.427090\pi$
$558$	0	0
$559$	−3.84426	−0.162595
$560$	0	0
$561$	−0.427012	−0.0180285
$562$	0	0
$563$	−10.5279	−0.443699	−0.221850	−	0.975081i	$-0.571209\pi$
$563$	−10.5279	−0.443699	−0.221850	+	0.975081i	$0.571209\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.58400	0.0665216
$568$	0	0
$569$	5.02954	0.210849	0.105425	−	0.994427i	$-0.466380\pi$
$569$	5.02954	0.210849	0.105425	+	0.994427i	$0.466380\pi$
$570$	0	0
$571$	−14.0833	−0.589369	−0.294685	−	0.955595i	$-0.595215\pi$
$571$	−14.0833	−0.589369	−0.294685	+	0.955595i	$0.595215\pi$
$572$	0	0
$573$	−4.04601	−0.169025
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−0.479431	−0.0199590	−0.00997948	−	0.999950i	$-0.503177\pi$
$577$	−0.479431	−0.0199590	−0.00997948	+	0.999950i	$0.503177\pi$
$578$	0	0
$579$	0.951355	0.0395369
$580$	0	0
$581$	3.22582	0.133830
$582$	0	0
$583$	0.768123	0.0318124
$584$	0	0
$585$	0	0
$586$	0	0
$587$	43.2754	1.78617	0.893084	−	0.449891i	$-0.148537\pi$
$587$	43.2754	1.78617	0.893084	+	0.449891i	$0.148537\pi$
$588$	0	0
$589$	−8.90185	−0.366794
$590$	0	0
$591$	6.82330	0.280673
$592$	0	0
$593$	−11.3833	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$593$	−11.3833	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.7113	−0.683946
$598$	0	0
$599$	−12.4626	−0.509206	−0.254603	−	0.967046i	$-0.581945\pi$
$599$	−12.4626	−0.509206	−0.254603	+	0.967046i	$0.581945\pi$
$600$	0	0
$601$	−33.4833	−1.36581	−0.682906	−	0.730507i	$-0.739284\pi$
$601$	−33.4833	−1.36581	−0.682906	+	0.730507i	$0.739284\pi$
$602$	0	0
$603$	−19.0848	−0.777194
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.3124	−0.986809	−0.493405	−	0.869800i	$-0.664248\pi$
$607$	−24.3124	−0.986809	−0.493405	+	0.869800i	$0.664248\pi$
$608$	0	0
$609$	2.86815	0.116223
$610$	0	0
$611$	0.578001	0.0233834
$612$	0	0
$613$	−25.4940	−1.02969	−0.514846	−	0.857282i	$-0.672151\pi$
$613$	−25.4940	−1.02969	−0.514846	+	0.857282i	$0.672151\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.71662	−0.109367	−0.0546835	−	0.998504i	$-0.517415\pi$
$617$	−2.71662	−0.109367	−0.0546835	+	0.998504i	$0.517415\pi$
$618$	0	0
$619$	−16.3563	−0.657414	−0.328707	−	0.944432i	$-0.606613\pi$
$619$	−16.3563	−0.657414	−0.328707	+	0.944432i	$0.606613\pi$
$620$	0	0
$621$	10.7048	0.429571
$622$	0	0
$623$	−4.91198	−0.196794
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.272559	−0.0108849
$628$	0	0
$629$	−11.6862	−0.465958
$630$	0	0
$631$	−39.4650	−1.57108	−0.785538	−	0.618813i	$-0.787614\pi$
$631$	−39.4650	−1.57108	−0.785538	+	0.618813i	$0.787614\pi$
$632$	0	0
$633$	−6.40444	−0.254554
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−13.5221	−0.535766
$638$	0	0
$639$	−13.3143	−0.526707
$640$	0	0
$641$	−3.29535	−0.130159	−0.0650793	−	0.997880i	$-0.520730\pi$
$641$	−3.29535	−0.130159	−0.0650793	+	0.997880i	$0.520730\pi$
$642$	0	0
$643$	34.9425	1.37800	0.688999	−	0.724762i	$-0.258050\pi$
$643$	34.9425	1.37800	0.688999	+	0.724762i	$0.258050\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.87789	−0.270398	−0.135199	−	0.990818i	$-0.543167\pi$
$647$	−6.87789	−0.270398	−0.135199	+	0.990818i	$0.543167\pi$
$648$	0	0
$649$	0.329794	0.0129456
$650$	0	0
$651$	−2.21073	−0.0866454
$652$	0	0
$653$	34.3278	1.34335	0.671675	−	0.740846i	$-0.265575\pi$
$653$	34.3278	1.34335	0.671675	+	0.740846i	$0.265575\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.82233	−0.227151
$658$	0	0
$659$	30.7582	1.19817	0.599084	−	0.800686i	$-0.295531\pi$
$659$	30.7582	1.19817	0.599084	+	0.800686i	$0.295531\pi$
$660$	0	0
$661$	0.882629	0.0343303	0.0171651	−	0.999853i	$-0.494536\pi$
$661$	0.882629	0.0343303	0.0171651	+	0.999853i	$0.494536\pi$
$662$	0	0
$663$	−9.53412	−0.370275
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.46521	0.366494
$668$	0	0
$669$	4.44610	0.171896
$670$	0	0
$671$	0.597363	0.0230609
$672$	0	0
$673$	33.9553	1.30888	0.654441	−	0.756113i	$-0.272904\pi$
$673$	33.9553	1.30888	0.654441	+	0.756113i	$0.272904\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20.7475	−0.797392	−0.398696	−	0.917083i	$-0.630537\pi$
$677$	−20.7475	−0.797392	−0.398696	+	0.917083i	$0.630537\pi$
$678$	0	0
$679$	1.37328	0.0527018
$680$	0	0
$681$	−12.4749	−0.478039
$682$	0	0
$683$	20.1497	0.771009	0.385504	−	0.922706i	$-0.374027\pi$
$683$	20.1497	0.771009	0.385504	+	0.922706i	$0.374027\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.2525	−0.391159
$688$	0	0
$689$	17.1503	0.653374
$690$	0	0
$691$	−1.76445	−0.0671230	−0.0335615	−	0.999437i	$-0.510685\pi$
$691$	−1.76445	−0.0671230	−0.0335615	+	0.999437i	$0.510685\pi$
$692$	0	0
$693$	0.0492163	0.00186957
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.2065	−0.575988
$698$	0	0
$699$	−0.769296	−0.0290974
$700$	0	0
$701$	−20.1362	−0.760533	−0.380267	−	0.924877i	$-0.624168\pi$
$701$	−20.1362	−0.760533	−0.380267	+	0.924877i	$0.624168\pi$
$702$	0	0
$703$	−7.45920	−0.281329
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.94796	−0.223696
$708$	0	0
$709$	40.3254	1.51445	0.757226	−	0.653153i	$-0.226554\pi$
$709$	40.3254	1.51445	0.757226	+	0.653153i	$0.226554\pi$
$710$	0	0
$711$	−17.0168	−0.638180
$712$	0	0
$713$	−7.29567	−0.273225
$714$	0	0
$715$	0	0
$716$	0	0
$717$	34.6515	1.29408
$718$	0	0
$719$	−28.7614	−1.07262	−0.536309	−	0.844021i	$-0.680182\pi$
$719$	−28.7614	−1.07262	−0.536309	+	0.844021i	$0.680182\pi$
$720$	0	0
$721$	−2.35851	−0.0878356
$722$	0	0
$723$	18.0075	0.669704
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−30.5035	−1.13131	−0.565655	−	0.824642i	$-0.691377\pi$
$727$	−30.5035	−1.13131	−0.565655	+	0.824642i	$0.691377\pi$
$728$	0	0
$729$	26.7815	0.991907
$730$	0	0
$731$	7.04943	0.260732
$732$	0	0
$733$	50.8858	1.87951	0.939756	−	0.341847i	$-0.111052\pi$
$733$	50.8858	1.87951	0.939756	+	0.341847i	$0.111052\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.34422	0.0495148
$738$	0	0
$739$	1.75006	0.0643769	0.0321884	−	0.999482i	$-0.489752\pi$
$739$	1.75006	0.0643769	0.0321884	+	0.999482i	$0.489752\pi$
$740$	0	0
$741$	−6.08556	−0.223559
$742$	0	0
$743$	−31.0643	−1.13964	−0.569820	−	0.821770i	$-0.692987\pi$
$743$	−31.0643	−1.13964	−0.569820	+	0.821770i	$0.692987\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.30043	0.340285
$748$	0	0
$749$	−2.46463	−0.0900555
$750$	0	0
$751$	8.74819	0.319226	0.159613	−	0.987180i	$-0.448975\pi$
$751$	8.74819	0.319226	0.159613	+	0.987180i	$0.448975\pi$
$752$	0	0
$753$	−16.7828	−0.611599
$754$	0	0
$755$	0	0
$756$	0	0
$757$	40.9463	1.48822	0.744109	−	0.668058i	$-0.232874\pi$
$757$	40.9463	1.48822	0.744109	+	0.668058i	$0.232874\pi$
$758$	0	0
$759$	−0.223380	−0.00810819
$760$	0	0
$761$	−35.0146	−1.26928	−0.634639	−	0.772809i	$-0.718851\pi$
$761$	−35.0146	−1.26928	−0.634639	+	0.772809i	$0.718851\pi$
$762$	0	0
$763$	4.20961	0.152398
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.36350	0.265880
$768$	0	0
$769$	43.4607	1.56723	0.783616	−	0.621246i	$-0.213373\pi$
$769$	43.4607	1.56723	0.783616	+	0.621246i	$0.213373\pi$
$770$	0	0
$771$	−40.6836	−1.46518
$772$	0	0
$773$	12.0184	0.432273	0.216136	−	0.976363i	$-0.430654\pi$
$773$	12.0184	0.432273	0.216136	+	0.976363i	$0.430654\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.85246	−0.0664566
$778$	0	0
$779$	−9.70622	−0.347761
$780$	0	0
$781$	0.937778	0.0335563
$782$	0	0
$783$	27.9114	0.997471
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.0694	1.14315	0.571575	−	0.820550i	$-0.306333\pi$
$787$	32.0694	1.14315	0.571575	+	0.820550i	$0.306333\pi$
$788$	0	0
$789$	21.1815	0.754080
$790$	0	0
$791$	−3.97181	−0.141221
$792$	0	0
$793$	13.3376	0.473633
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.8035	0.630633	0.315316	−	0.948987i	$-0.397889\pi$
$797$	17.8035	0.630633	0.315316	+	0.948987i	$0.397889\pi$
$798$	0	0
$799$	−1.05991	−0.0374970
$800$	0	0
$801$	−14.1618	−0.500384
$802$	0	0
$803$	0.410088	0.0144717
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25.1508	−0.885350
$808$	0	0
$809$	12.3429	0.433953	0.216976	−	0.976177i	$-0.430381\pi$
$809$	12.3429	0.433953	0.216976	+	0.976177i	$0.430381\pi$
$810$	0	0
$811$	20.2636	0.711552	0.355776	−	0.934571i	$-0.384217\pi$
$811$	20.2636	0.711552	0.355776	+	0.934571i	$0.384217\pi$
$812$	0	0
$813$	39.4166	1.38240
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.49960	0.157421
$818$	0	0
$819$	1.09888	0.0383979
$820$	0	0
$821$	35.8848	1.25239	0.626195	−	0.779667i	$-0.284612\pi$
$821$	35.8848	1.25239	0.626195	+	0.779667i	$0.284612\pi$
$822$	0	0
$823$	−43.8418	−1.52823	−0.764115	−	0.645080i	$-0.776824\pi$
$823$	−43.8418	−1.52823	−0.764115	+	0.645080i	$0.776824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.76097	0.0612351	0.0306175	−	0.999531i	$-0.490253\pi$
$827$	1.76097	0.0612351	0.0306175	+	0.999531i	$0.490253\pi$
$828$	0	0
$829$	−49.9114	−1.73349	−0.866747	−	0.498748i	$-0.833793\pi$
$829$	−49.9114	−1.73349	−0.866747	+	0.498748i	$0.833793\pi$
$830$	0	0
$831$	−38.9779	−1.35213
$832$	0	0
$833$	24.7963	0.859139
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−21.5137	−0.743623
$838$	0	0
$839$	−47.4064	−1.63665	−0.818326	−	0.574754i	$-0.805098\pi$
$839$	−47.4064	−1.63665	−0.818326	+	0.574754i	$0.805098\pi$
$840$	0	0
$841$	−4.32082	−0.148994
$842$	0	0
$843$	13.4010	0.461555
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.81520	0.165452
$848$	0	0
$849$	−8.42286	−0.289072
$850$	0	0
$851$	−6.11332	−0.209562
$852$	0	0
$853$	−17.5673	−0.601495	−0.300747	−	0.953704i	$-0.597236\pi$
$853$	−17.5673	−0.601495	−0.300747	+	0.953704i	$0.597236\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.2296	1.51086	0.755428	−	0.655232i	$-0.227429\pi$
$857$	44.2296	1.51086	0.755428	+	0.655232i	$0.227429\pi$
$858$	0	0
$859$	−21.4146	−0.730658	−0.365329	−	0.930879i	$-0.619043\pi$
$859$	−21.4146	−0.730658	−0.365329	+	0.930879i	$0.619043\pi$
$860$	0	0
$861$	−2.41050	−0.0821495
$862$	0	0
$863$	−37.1695	−1.26526	−0.632632	−	0.774452i	$-0.718026\pi$
$863$	−37.1695	−1.26526	−0.632632	+	0.774452i	$0.718026\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.92208	−0.167162
$868$	0	0
$869$	1.19856	0.0406583
$870$	0	0
$871$	30.0130	1.01695
$872$	0	0
$873$	3.95935	0.134004
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.4486	−0.656732	−0.328366	−	0.944551i	$-0.606498\pi$
$877$	−19.4486	−0.656732	−0.328366	+	0.944551i	$0.606498\pi$
$878$	0	0
$879$	−7.55984	−0.254987
$880$	0	0
$881$	48.7106	1.64110	0.820551	−	0.571573i	$-0.193667\pi$
$881$	48.7106	1.64110	0.820551	+	0.571573i	$0.193667\pi$
$882$	0	0
$883$	−51.4967	−1.73300	−0.866500	−	0.499177i	$-0.833636\pi$
$883$	−51.4967	−1.73300	−0.866500	+	0.499177i	$0.833636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.7017	0.896557	0.448278	−	0.893894i	$-0.352037\pi$
$887$	26.7017	0.896557	0.448278	+	0.893894i	$0.352037\pi$
$888$	0	0
$889$	6.38053	0.213996
$890$	0	0
$891$	−0.321661	−0.0107760
$892$	0	0
$893$	−0.676535	−0.0226394
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.98753	−0.166529
$898$	0	0
$899$	−19.0224	−0.634433
$900$	0	0
$901$	−31.4495	−1.04773
$902$	0	0
$903$	1.11745	0.0371866
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12.5555	−0.416899	−0.208449	−	0.978033i	$-0.566842\pi$
$907$	−12.5555	−0.416899	−0.208449	+	0.978033i	$0.566842\pi$
$908$	0	0
$909$	−17.1487	−0.568787
$910$	0	0
$911$	−39.1848	−1.29825	−0.649126	−	0.760681i	$-0.724865\pi$
$911$	−39.1848	−1.29825	−0.649126	+	0.760681i	$0.724865\pi$
$912$	0	0
$913$	−0.655064	−0.0216795
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.01024	−0.198476
$918$	0	0
$919$	−3.80982	−0.125674	−0.0628372	−	0.998024i	$-0.520015\pi$
$919$	−3.80982	−0.125674	−0.0628372	+	0.998024i	$0.520015\pi$
$920$	0	0
$921$	29.9208	0.985922
$922$	0	0
$923$	20.9383	0.689192
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6.79988	−0.223337
$928$	0	0
$929$	11.9147	0.390908	0.195454	−	0.980713i	$-0.437382\pi$
$929$	11.9147	0.390908	0.195454	+	0.980713i	$0.437382\pi$
$930$	0	0
$931$	15.8273	0.518718
$932$	0	0
$933$	−34.7430	−1.13744
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.2901	−0.793522	−0.396761	−	0.917922i	$-0.629866\pi$
$937$	−24.2901	−0.793522	−0.396761	+	0.917922i	$0.629866\pi$
$938$	0	0
$939$	−5.91471	−0.193019
$940$	0	0
$941$	−17.7237	−0.577775	−0.288887	−	0.957363i	$-0.593285\pi$
$941$	−17.7237	−0.577775	−0.288887	+	0.957363i	$0.593285\pi$
$942$	0	0
$943$	−7.95491	−0.259047
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.2599	1.43825	0.719127	−	0.694878i	$-0.244542\pi$
$947$	44.2599	1.43825	0.719127	+	0.694878i	$0.244542\pi$
$948$	0	0
$949$	9.15626	0.297225
$950$	0	0
$951$	18.1775	0.589446
$952$	0	0
$953$	11.6717	0.378083	0.189041	−	0.981969i	$-0.439462\pi$
$953$	11.6717	0.378083	0.189041	+	0.981969i	$0.439462\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.582432	−0.0188274
$958$	0	0
$959$	−9.54033	−0.308073
$960$	0	0
$961$	−16.3378	−0.527025
$962$	0	0
$963$	−7.10582	−0.228982
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−42.0853	−1.35337	−0.676686	−	0.736272i	$-0.736584\pi$
$967$	−42.0853	−1.35337	−0.676686	+	0.736272i	$0.736584\pi$
$968$	0	0
$969$	11.1594	0.358493
$970$	0	0
$971$	−52.3346	−1.67950	−0.839749	−	0.542974i	$-0.817298\pi$
$971$	−52.3346	−1.67950	−0.839749	+	0.542974i	$0.817298\pi$
$972$	0	0
$973$	−7.80392	−0.250182
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.62562	0.115994	0.0579970	−	0.998317i	$-0.481529\pi$
$977$	3.62562	0.115994	0.0579970	+	0.998317i	$0.481529\pi$
$978$	0	0
$979$	0.997471	0.0318793
$980$	0	0
$981$	12.1368	0.387499
$982$	0	0
$983$	41.6847	1.32953	0.664767	−	0.747050i	$-0.268531\pi$
$983$	41.6847	1.32953	0.664767	+	0.747050i	$0.268531\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.168014	−0.00534796
$988$	0	0
$989$	3.68773	0.117263
$990$	0	0
$991$	48.2952	1.53415	0.767074	−	0.641558i	$-0.221712\pi$
$991$	48.2952	1.53415	0.767074	+	0.641558i	$0.221712\pi$
$992$	0	0
$993$	30.1204	0.955843
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.41240	−0.234753	−0.117377	−	0.993087i	$-0.537448\pi$
$997$	−7.41240	−0.234753	−0.117377	+	0.993087i	$0.537448\pi$
$998$	0	0
$999$	−18.0272	−0.570355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bl.1.5		8
4.3	odd	2		5000.2.a.k.1.4	✓	8
5.4	even	2		10000.2.a.bg.1.4		8
20.19	odd	2		5000.2.a.n.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.k.1.4	✓	8	4.3	odd	2
5000.2.a.n.1.5	yes	8	20.19	odd	2
10000.2.a.bg.1.4		8	5.4	even	2
10000.2.a.bl.1.5		8	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31796 q^{3} -0.438060 q^{7} -1.26298 q^{9} +O(q^{10})\)	\(q+1.31796 q^{3} -0.438060 q^{7} -1.26298 q^{9} +0.0889565 q^{11} +1.98618 q^{13} -3.64217 q^{17} -2.32477 q^{19} -0.577346 q^{21} -1.90531 q^{23} -5.61844 q^{27} -4.96781 q^{29} +3.82913 q^{31} +0.117241 q^{33} +3.20858 q^{37} +2.61770 q^{39} +4.17513 q^{41} -1.93550 q^{43} +0.291012 q^{47} -6.80810 q^{49} -4.80023 q^{51} +8.63482 q^{53} -3.06395 q^{57} +3.70737 q^{59} +6.71522 q^{61} +0.553262 q^{63} +15.1109 q^{67} -2.51112 q^{69} +10.5420 q^{71} +4.60998 q^{73} -0.0389683 q^{77} +13.4735 q^{79} -3.61593 q^{81} -7.36387 q^{83} -6.54738 q^{87} +11.2130 q^{89} -0.870066 q^{91} +5.04664 q^{93} -3.13492 q^{97} -0.112350 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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