LMFDB - Embedded newform 1008.4.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1008.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.71780 q^{5} +7.00000 q^{7} +O(q^{10})$	$q+8.71780 q^{5} +7.00000 q^{7} +43.5890 q^{11} +82.0000 q^{13} -78.4602 q^{17} +20.0000 q^{19} +130.767 q^{23} -49.0000 q^{25} +244.098 q^{29} -156.000 q^{31} +61.0246 q^{35} +186.000 q^{37} -165.638 q^{41} -164.000 q^{43} -470.761 q^{47} +49.0000 q^{49} -156.920 q^{53} +380.000 q^{55} -156.920 q^{59} +790.000 q^{61} +714.859 q^{65} +44.0000 q^{67} -444.608 q^{71} +126.000 q^{73} +305.123 q^{77} +712.000 q^{79} +1464.59 q^{83} -684.000 q^{85} -1455.87 q^{89} +574.000 q^{91} +174.356 q^{95} +798.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{7}+O(q^{10}) $ 2 * q + 14 * q^7	$ 2 q + 14 q^{7} + 164 q^{13} + 40 q^{19} - 98 q^{25} - 312 q^{31} + 372 q^{37} - 328 q^{43} + 98 q^{49} + 760 q^{55} + 1580 q^{61} + 88 q^{67} + 252 q^{73} + 1424 q^{79} - 1368 q^{85} + 1148 q^{91} + 1596 q^{97}+O(q^{100}) $ 2 * q + 14 * q^7 + 164 * q^13 + 40 * q^19 - 98 * q^25 - 312 * q^31 + 372 * q^37 - 328 * q^43 + 98 * q^49 + 760 * q^55 + 1580 * q^61 + 88 * q^67 + 252 * q^73 + 1424 * q^79 - 1368 * q^85 + 1148 * q^91 + 1596 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	8.71780	0.779744	0.389872	−	0.920869i	$-0.372519\pi$
$5$	8.71780	0.779744	0.389872	+	0.920869i	$0.372519\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	43.5890	1.19478	0.597390	−	0.801951i	$-0.296205\pi$
$11$	43.5890	1.19478	0.597390	+	0.801951i	$0.296205\pi$
$12$	0	0
$13$	82.0000	1.74944	0.874720	−	0.484629i	$-0.161046\pi$
$13$	82.0000	1.74944	0.874720	+	0.484629i	$0.161046\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−78.4602	−1.11938	−0.559688	−	0.828703i	$-0.689079\pi$
$17$	−78.4602	−1.11938	−0.559688	+	0.828703i	$0.689079\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	130.767	1.18551	0.592756	−	0.805382i	$-0.298040\pi$
$23$	130.767	1.18551	0.592756	+	0.805382i	$0.298040\pi$
$24$	0	0
$25$	−49.0000	−0.392000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	244.098	1.56303	0.781516	−	0.623885i	$-0.214447\pi$
$29$	244.098	1.56303	0.781516	+	0.623885i	$0.214447\pi$
$30$	0	0
$31$	−156.000	−0.903820	−0.451910	−	0.892063i	$-0.649257\pi$
$31$	−156.000	−0.903820	−0.451910	+	0.892063i	$0.649257\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	61.0246	0.294715
$36$	0	0
$37$	186.000	0.826438	0.413219	−	0.910632i	$-0.364404\pi$
$37$	186.000	0.826438	0.413219	+	0.910632i	$0.364404\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−165.638	−0.630935	−0.315467	−	0.948936i	$-0.602161\pi$
$41$	−165.638	−0.630935	−0.315467	+	0.948936i	$0.602161\pi$
$42$	0	0
$43$	−164.000	−0.581622	−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000	−0.581622	−0.290811	+	0.956780i	$0.593925\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−470.761	−1.46101	−0.730506	−	0.682906i	$-0.760716\pi$
$47$	−470.761	−1.46101	−0.730506	+	0.682906i	$0.760716\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−156.920	−0.406692	−0.203346	−	0.979107i	$-0.565182\pi$
$53$	−156.920	−0.406692	−0.203346	+	0.979107i	$0.565182\pi$
$54$	0	0
$55$	380.000	0.931622
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−156.920	−0.346259	−0.173130	−	0.984899i	$-0.555388\pi$
$59$	−156.920	−0.346259	−0.173130	+	0.984899i	$0.555388\pi$
$60$	0	0
$61$	790.000	1.65818	0.829091	−	0.559113i	$-0.188858\pi$
$61$	790.000	1.65818	0.829091	+	0.559113i	$0.188858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	714.859	1.36411
$66$	0	0
$67$	44.0000	0.0802307	0.0401153	−	0.999195i	$-0.487227\pi$
$67$	44.0000	0.0802307	0.0401153	+	0.999195i	$0.487227\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−444.608	−0.743172	−0.371586	−	0.928398i	$-0.621186\pi$
$71$	−444.608	−0.743172	−0.371586	+	0.928398i	$0.621186\pi$
$72$	0	0
$73$	126.000	0.202016	0.101008	−	0.994886i	$-0.467793\pi$
$73$	126.000	0.202016	0.101008	+	0.994886i	$0.467793\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	305.123	0.451584
$78$	0	0
$79$	712.000	1.01400	0.507002	−	0.861945i	$-0.330754\pi$
$79$	712.000	1.01400	0.507002	+	0.861945i	$0.330754\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1464.59	1.93686	0.968432	−	0.249280i	$-0.0801938\pi$
$83$	1464.59	1.93686	0.968432	+	0.249280i	$0.0801938\pi$
$84$	0	0
$85$	−684.000	−0.872826
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1455.87	−1.73396	−0.866978	−	0.498346i	$-0.833941\pi$
$89$	−1455.87	−1.73396	−0.866978	+	0.498346i	$0.833941\pi$
$90$	0	0
$91$	574.000	0.661226
$92$	0	0
$93$	0	0
$94$	0	0
$95$	174.356	0.188300
$96$	0	0
$97$	798.000	0.835305	0.417653	−	0.908607i	$-0.362853\pi$
$97$	798.000	0.835305	0.417653	+	0.908607i	$0.362853\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	409.737	0.403666	0.201833	−	0.979420i	$-0.435310\pi$
$101$	409.737	0.403666	0.201833	+	0.979420i	$0.435310\pi$
$102$	0	0
$103$	916.000	0.876273	0.438137	−	0.898908i	$-0.355639\pi$
$103$	916.000	0.876273	0.438137	+	0.898908i	$0.355639\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−897.933	−0.811275	−0.405638	−	0.914034i	$-0.632951\pi$
$107$	−897.933	−0.811275	−0.405638	+	0.914034i	$0.632951\pi$
$108$	0	0
$109$	−342.000	−0.300529	−0.150264	−	0.988646i	$-0.548013\pi$
$109$	−342.000	−0.300529	−0.150264	+	0.988646i	$0.548013\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−488.197	−0.406422	−0.203211	−	0.979135i	$-0.565138\pi$
$113$	−488.197	−0.406422	−0.203211	+	0.979135i	$0.565138\pi$
$114$	0	0
$115$	1140.00	0.924396
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−549.221	−0.423084
$120$	0	0
$121$	569.000	0.427498
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1516.90	−1.08540
$126$	0	0
$127$	−456.000	−0.318610	−0.159305	−	0.987229i	$-0.550925\pi$
$127$	−456.000	−0.318610	−0.159305	+	0.987229i	$0.550925\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1499.46	−1.00007	−0.500033	−	0.866007i	$-0.666679\pi$
$131$	−1499.46	−1.00007	−0.500033	+	0.866007i	$0.666679\pi$
$132$	0	0
$133$	140.000	0.0912747
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−889.215	−0.554531	−0.277266	−	0.960793i	$-0.589428\pi$
$137$	−889.215	−0.554531	−0.277266	+	0.960793i	$0.589428\pi$
$138$	0	0
$139$	768.000	0.468640	0.234320	−	0.972160i	$-0.424714\pi$
$139$	768.000	0.468640	0.234320	+	0.972160i	$0.424714\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3574.30	2.09019
$144$	0	0
$145$	2128.00	1.21876
$146$	0	0
$147$	0	0
$148$	0	0
$149$	993.829	0.546427	0.273214	−	0.961953i	$-0.411913\pi$
$149$	993.829	0.546427	0.273214	+	0.961953i	$0.411913\pi$
$150$	0	0
$151$	3496.00	1.88411	0.942054	−	0.335460i	$-0.108892\pi$
$151$	3496.00	1.88411	0.942054	+	0.335460i	$0.108892\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1359.98	−0.704748
$156$	0	0
$157$	−506.000	−0.257218	−0.128609	−	0.991695i	$-0.541051\pi$
$157$	−506.000	−0.257218	−0.128609	+	0.991695i	$0.541051\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	915.369	0.448082
$162$	0	0
$163$	2564.00	1.23207	0.616037	−	0.787717i	$-0.288737\pi$
$163$	2564.00	1.23207	0.616037	+	0.787717i	$0.288737\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	645.117	0.298926	0.149463	−	0.988767i	$-0.452245\pi$
$167$	645.117	0.298926	0.149463	+	0.988767i	$0.452245\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3861.98	1.69723	0.848616	−	0.529009i	$-0.177436\pi$
$173$	3861.98	1.69723	0.848616	+	0.529009i	$0.177436\pi$
$174$	0	0
$175$	−343.000	−0.148162
$176$	0	0
$177$	0	0
$178$	0	0
$179$	270.252	0.112847	0.0564234	−	0.998407i	$-0.482030\pi$
$179$	270.252	0.112847	0.0564234	+	0.998407i	$0.482030\pi$
$180$	0	0
$181$	418.000	0.171656	0.0858279	−	0.996310i	$-0.472646\pi$
$181$	418.000	0.171656	0.0858279	+	0.996310i	$0.472646\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1621.51	0.644410
$186$	0	0
$187$	−3420.00	−1.33741
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1525.61	−0.577956	−0.288978	−	0.957336i	$-0.593315\pi$
$191$	−1525.61	−0.577956	−0.288978	+	0.957336i	$0.593315\pi$
$192$	0	0
$193$	1358.00	0.506482	0.253241	−	0.967403i	$-0.418503\pi$
$193$	1358.00	0.506482	0.253241	+	0.967403i	$0.418503\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3748.65	1.35574	0.677869	−	0.735183i	$-0.262904\pi$
$197$	3748.65	1.35574	0.677869	+	0.735183i	$0.262904\pi$
$198$	0	0
$199$	1056.00	0.376170	0.188085	−	0.982153i	$-0.439772\pi$
$199$	1056.00	0.376170	0.188085	+	0.982153i	$0.439772\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1708.69	0.590771
$204$	0	0
$205$	−1444.00	−0.491967
$206$	0	0
$207$	0	0
$208$	0	0
$209$	871.780	0.288528
$210$	0	0
$211$	3620.00	1.18110	0.590548	−	0.807003i	$-0.298912\pi$
$211$	3620.00	1.18110	0.590548	+	0.807003i	$0.298912\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1429.72	−0.453516
$216$	0	0
$217$	−1092.00	−0.341612
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6433.73	−1.95828
$222$	0	0
$223$	−5368.00	−1.61196	−0.805982	−	0.591940i	$-0.798362\pi$
$223$	−5368.00	−1.61196	−0.805982	+	0.591940i	$0.798362\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1621.51	−0.474112	−0.237056	−	0.971496i	$-0.576182\pi$
$227$	−1621.51	−0.474112	−0.237056	+	0.971496i	$0.576182\pi$
$228$	0	0
$229$	2186.00	0.630808	0.315404	−	0.948958i	$-0.397860\pi$
$229$	2186.00	0.630808	0.315404	+	0.948958i	$0.397860\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4132.24	−1.16185	−0.580927	−	0.813956i	$-0.697310\pi$
$233$	−4132.24	−1.16185	−0.580927	+	0.813956i	$0.697310\pi$
$234$	0	0
$235$	−4104.00	−1.13921
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4838.38	1.30949	0.654746	−	0.755849i	$-0.272776\pi$
$239$	4838.38	1.30949	0.654746	+	0.755849i	$0.272776\pi$
$240$	0	0
$241$	1286.00	0.343728	0.171864	−	0.985121i	$-0.445021\pi$
$241$	1286.00	0.343728	0.171864	+	0.985121i	$0.445021\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	427.172	0.111392
$246$	0	0
$247$	1640.00	0.422472
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1795.87	0.451610	0.225805	−	0.974173i	$-0.427499\pi$
$251$	1795.87	0.451610	0.225805	+	0.974173i	$0.427499\pi$
$252$	0	0
$253$	5700.00	1.41643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1944.07	−0.471859	−0.235929	−	0.971770i	$-0.575813\pi$
$257$	−1944.07	−0.471859	−0.235929	+	0.971770i	$0.575813\pi$
$258$	0	0
$259$	1302.00	0.312364
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5344.01	−1.25295	−0.626475	−	0.779442i	$-0.715503\pi$
$263$	−5344.01	−1.25295	−0.626475	+	0.779442i	$0.715503\pi$
$264$	0	0
$265$	−1368.00	−0.317115
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4001.47	−0.906966	−0.453483	−	0.891265i	$-0.649819\pi$
$269$	−4001.47	−0.906966	−0.453483	+	0.891265i	$0.649819\pi$
$270$	0	0
$271$	−2788.00	−0.624941	−0.312470	−	0.949928i	$-0.601156\pi$
$271$	−2788.00	−0.624941	−0.312470	+	0.949928i	$0.601156\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2135.86	−0.468354
$276$	0	0
$277$	−4562.00	−0.989545	−0.494773	−	0.869022i	$-0.664749\pi$
$277$	−4562.00	−0.989545	−0.494773	+	0.869022i	$0.664749\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1551.77	−0.329433	−0.164717	−	0.986341i	$-0.552671\pi$
$281$	−1551.77	−0.329433	−0.164717	+	0.986341i	$0.552671\pi$
$282$	0	0
$283$	6788.00	1.42581	0.712906	−	0.701260i	$-0.247379\pi$
$283$	6788.00	1.42581	0.712906	+	0.701260i	$0.247379\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1159.47	−0.238471
$288$	0	0
$289$	1243.00	0.253002
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1142.03	0.227707	0.113854	−	0.993498i	$-0.463681\pi$
$293$	1142.03	0.227707	0.113854	+	0.993498i	$0.463681\pi$
$294$	0	0
$295$	−1368.00	−0.269993
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10722.9	2.07398
$300$	0	0
$301$	−1148.00	−0.219833
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6887.06	1.29296
$306$	0	0
$307$	−532.000	−0.0989018	−0.0494509	−	0.998777i	$-0.515747\pi$
$307$	−532.000	−0.0989018	−0.0494509	+	0.998777i	$0.515747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6538.35	−1.19214	−0.596070	−	0.802932i	$-0.703272\pi$
$311$	−6538.35	−1.19214	−0.596070	+	0.802932i	$0.703272\pi$
$312$	0	0
$313$	4994.00	0.901845	0.450923	−	0.892563i	$-0.351095\pi$
$313$	4994.00	0.901845	0.450923	+	0.892563i	$0.351095\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−470.761	−0.0834088	−0.0417044	−	0.999130i	$-0.513279\pi$
$317$	−470.761	−0.0834088	−0.0417044	+	0.999130i	$0.513279\pi$
$318$	0	0
$319$	10640.0	1.86748
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1569.20	−0.270318
$324$	0	0
$325$	−4018.00	−0.685780
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3295.33	−0.552211
$330$	0	0
$331$	−2588.00	−0.429756	−0.214878	−	0.976641i	$-0.568935\pi$
$331$	−2588.00	−0.429756	−0.214878	+	0.976641i	$0.568935\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	383.583	0.0625594
$336$	0	0
$337$	238.000	0.0384709	0.0192354	−	0.999815i	$-0.493877\pi$
$337$	238.000	0.0384709	0.0192354	+	0.999815i	$0.493877\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6799.88	−1.07987
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7052.70	−1.09109	−0.545546	−	0.838081i	$-0.683678\pi$
$347$	−7052.70	−1.09109	−0.545546	+	0.838081i	$0.683678\pi$
$348$	0	0
$349$	−10850.0	−1.66415	−0.832073	−	0.554666i	$-0.812846\pi$
$349$	−10850.0	−1.66415	−0.832073	+	0.554666i	$0.812846\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5291.70	0.797872	0.398936	−	0.916979i	$-0.369379\pi$
$353$	5291.70	0.797872	0.398936	+	0.916979i	$0.369379\pi$
$354$	0	0
$355$	−3876.00	−0.579484
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4820.94	0.708745	0.354373	−	0.935104i	$-0.384694\pi$
$359$	4820.94	0.708745	0.354373	+	0.935104i	$0.384694\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1098.44	0.157521
$366$	0	0
$367$	−11712.0	−1.66583	−0.832917	−	0.553397i	$-0.813331\pi$
$367$	−11712.0	−1.66583	−0.832917	+	0.553397i	$0.813331\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1098.44	−0.153715
$372$	0	0
$373$	−10450.0	−1.45062	−0.725309	−	0.688423i	$-0.758303\pi$
$373$	−10450.0	−1.45062	−0.725309	+	0.688423i	$0.758303\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20016.1	2.73443
$378$	0	0
$379$	756.000	0.102462	0.0512310	−	0.998687i	$-0.483686\pi$
$379$	756.000	0.102462	0.0512310	+	0.998687i	$0.483686\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6381.43	−0.851373	−0.425686	−	0.904871i	$-0.639967\pi$
$383$	−6381.43	−0.851373	−0.425686	+	0.904871i	$0.639967\pi$
$384$	0	0
$385$	2660.00	0.352120
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−418.454	−0.0545411	−0.0272705	−	0.999628i	$-0.508682\pi$
$389$	−418.454	−0.0545411	−0.0272705	+	0.999628i	$0.508682\pi$
$390$	0	0
$391$	−10260.0	−1.32703
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6207.07	0.790663
$396$	0	0
$397$	−5802.00	−0.733486	−0.366743	−	0.930322i	$-0.619527\pi$
$397$	−5802.00	−0.733486	−0.366743	+	0.930322i	$0.619527\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4132.24	−0.514599	−0.257299	−	0.966332i	$-0.582833\pi$
$401$	−4132.24	−0.514599	−0.257299	+	0.966332i	$0.582833\pi$
$402$	0	0
$403$	−12792.0	−1.58118
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8107.55	0.987411
$408$	0	0
$409$	−1330.00	−0.160793	−0.0803964	−	0.996763i	$-0.525619\pi$
$409$	−1330.00	−0.160793	−0.0803964	+	0.996763i	$0.525619\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1098.44	−0.130874
$414$	0	0
$415$	12768.0	1.51026
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10409.1	−1.21364	−0.606820	−	0.794839i	$-0.707555\pi$
$419$	−10409.1	−1.21364	−0.606820	+	0.794839i	$0.707555\pi$
$420$	0	0
$421$	−12274.0	−1.42090	−0.710449	−	0.703749i	$-0.751508\pi$
$421$	−12274.0	−1.42090	−0.710449	+	0.703749i	$0.751508\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3844.55	0.438795
$426$	0	0
$427$	5530.00	0.626734
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4681.46	0.523197	0.261598	−	0.965177i	$-0.415750\pi$
$431$	4681.46	0.523197	0.261598	+	0.965177i	$0.415750\pi$
$432$	0	0
$433$	5770.00	0.640389	0.320195	−	0.947352i	$-0.396252\pi$
$433$	5770.00	0.640389	0.320195	+	0.947352i	$0.396252\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2615.34	0.286290
$438$	0	0
$439$	1872.00	0.203521	0.101760	−	0.994809i	$-0.467552\pi$
$439$	1872.00	0.203521	0.101760	+	0.994809i	$0.467552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11115.2	1.19210	0.596048	−	0.802949i	$-0.296737\pi$
$443$	11115.2	1.19210	0.596048	+	0.802949i	$0.296737\pi$
$444$	0	0
$445$	−12692.0	−1.35204
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7636.79	−0.802678	−0.401339	−	0.915930i	$-0.631455\pi$
$449$	−7636.79	−0.802678	−0.401339	+	0.915930i	$0.631455\pi$
$450$	0	0
$451$	−7220.00	−0.753828
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5004.02	0.515587
$456$	0	0
$457$	15142.0	1.54992	0.774959	−	0.632011i	$-0.217770\pi$
$457$	15142.0	1.54992	0.774959	+	0.632011i	$0.217770\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13190.0	−1.33258	−0.666292	−	0.745691i	$-0.732119\pi$
$461$	−13190.0	−1.33258	−0.666292	+	0.745691i	$0.732119\pi$
$462$	0	0
$463$	−9328.00	−0.936304	−0.468152	−	0.883648i	$-0.655080\pi$
$463$	−9328.00	−0.936304	−0.468152	+	0.883648i	$0.655080\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3399.94	−0.336896	−0.168448	−	0.985711i	$-0.553876\pi$
$467$	−3399.94	−0.336896	−0.168448	+	0.985711i	$0.553876\pi$
$468$	0	0
$469$	308.000	0.0303243
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7148.59	−0.694911
$474$	0	0
$475$	−980.000	−0.0946642
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3574.30	−0.340947	−0.170474	−	0.985362i	$-0.554530\pi$
$479$	−3574.30	−0.340947	−0.170474	+	0.985362i	$0.554530\pi$
$480$	0	0
$481$	15252.0	1.44580
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6956.80	0.651324
$486$	0	0
$487$	−8968.00	−0.834454	−0.417227	−	0.908802i	$-0.636998\pi$
$487$	−8968.00	−0.834454	−0.417227	+	0.908802i	$0.636998\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5169.65	−0.475159	−0.237580	−	0.971368i	$-0.576354\pi$
$491$	−5169.65	−0.475159	−0.237580	+	0.971368i	$0.576354\pi$
$492$	0	0
$493$	−19152.0	−1.74962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3112.25	−0.280893
$498$	0	0
$499$	−3940.00	−0.353464	−0.176732	−	0.984259i	$-0.556553\pi$
$499$	−3940.00	−0.353464	−0.176732	+	0.984259i	$0.556553\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10252.1	−0.908787	−0.454394	−	0.890801i	$-0.650144\pi$
$503$	−10252.1	−0.908787	−0.454394	+	0.890801i	$0.650144\pi$
$504$	0	0
$505$	3572.00	0.314756
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9772.65	0.851012	0.425506	−	0.904956i	$-0.360096\pi$
$509$	9772.65	0.851012	0.425506	+	0.904956i	$0.360096\pi$
$510$	0	0
$511$	882.000	0.0763550
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7985.50	0.683269
$516$	0	0
$517$	−20520.0	−1.74559
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7401.41	−0.622383	−0.311192	−	0.950347i	$-0.600728\pi$
$521$	−7401.41	−0.622383	−0.311192	+	0.950347i	$0.600728\pi$
$522$	0	0
$523$	2768.00	0.231427	0.115713	−	0.993283i	$-0.463085\pi$
$523$	2768.00	0.231427	0.115713	+	0.993283i	$0.463085\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12239.8	1.01171
$528$	0	0
$529$	4933.00	0.405441
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−13582.3	−1.10378
$534$	0	0
$535$	−7828.00	−0.632587
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2135.86	0.170683
$540$	0	0
$541$	16310.0	1.29616	0.648079	−	0.761573i	$-0.275573\pi$
$541$	16310.0	1.29616	0.648079	+	0.761573i	$0.275573\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2981.49	−0.234336
$546$	0	0
$547$	−11140.0	−0.870771	−0.435386	−	0.900244i	$-0.643388\pi$
$547$	−11140.0	−0.870771	−0.435386	+	0.900244i	$0.643388\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4881.97	0.377457
$552$	0	0
$553$	4984.00	0.383257
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22788.3	1.73352	0.866761	−	0.498723i	$-0.166197\pi$
$557$	22788.3	1.73352	0.866761	+	0.498723i	$0.166197\pi$
$558$	0	0
$559$	−13448.0	−1.01751
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11524.9	0.862732	0.431366	−	0.902177i	$-0.358032\pi$
$563$	11524.9	0.862732	0.431366	+	0.902177i	$0.358032\pi$
$564$	0	0
$565$	−4256.00	−0.316905
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1691.25	−0.124606	−0.0623032	−	0.998057i	$-0.519845\pi$
$569$	−1691.25	−0.124606	−0.0623032	+	0.998057i	$0.519845\pi$
$570$	0	0
$571$	−11228.0	−0.822902	−0.411451	−	0.911432i	$-0.634978\pi$
$571$	−11228.0	−0.822902	−0.411451	+	0.911432i	$0.634978\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6407.58	−0.464721
$576$	0	0
$577$	2050.00	0.147907	0.0739537	−	0.997262i	$-0.476438\pi$
$577$	2050.00	0.147907	0.0739537	+	0.997262i	$0.476438\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10252.1	0.732065
$582$	0	0
$583$	−6840.00	−0.485907
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18394.6	−1.29340	−0.646699	−	0.762745i	$-0.723851\pi$
$587$	−18394.6	−1.29340	−0.646699	+	0.762745i	$0.723851\pi$
$588$	0	0
$589$	−3120.00	−0.218264
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12632.1	−0.874769	−0.437384	−	0.899275i	$-0.644095\pi$
$593$	−12632.1	−0.874769	−0.437384	+	0.899275i	$0.644095\pi$
$594$	0	0
$595$	−4788.00	−0.329897
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9598.30	0.654717	0.327359	−	0.944900i	$-0.393841\pi$
$599$	9598.30	0.654717	0.327359	+	0.944900i	$0.393841\pi$
$600$	0	0
$601$	−10758.0	−0.730163	−0.365082	−	0.930976i	$-0.618959\pi$
$601$	−10758.0	−0.730163	−0.365082	+	0.930976i	$0.618959\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4960.43	0.333339
$606$	0	0
$607$	21352.0	1.42776	0.713881	−	0.700268i	$-0.246936\pi$
$607$	21352.0	1.42776	0.713881	+	0.700268i	$0.246936\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−38602.4	−2.55595
$612$	0	0
$613$	−5714.00	−0.376487	−0.188243	−	0.982122i	$-0.560279\pi$
$613$	−5714.00	−0.376487	−0.188243	+	0.982122i	$0.560279\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6747.58	0.440271	0.220135	−	0.975469i	$-0.429350\pi$
$617$	6747.58	0.440271	0.220135	+	0.975469i	$0.429350\pi$
$618$	0	0
$619$	1880.00	0.122074	0.0610368	−	0.998136i	$-0.480559\pi$
$619$	1880.00	0.122074	0.0610368	+	0.998136i	$0.480559\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10191.1	−0.655374
$624$	0	0
$625$	−7099.00	−0.454336
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14593.6	−0.925095
$630$	0	0
$631$	28888.0	1.82252	0.911262	−	0.411826i	$-0.135109\pi$
$631$	28888.0	1.82252	0.911262	+	0.411826i	$0.135109\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3975.32	−0.248434
$636$	0	0
$637$	4018.00	0.249920
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25996.5	1.60187	0.800935	−	0.598751i	$-0.204336\pi$
$641$	25996.5	1.60187	0.800935	+	0.598751i	$0.204336\pi$
$642$	0	0
$643$	−24788.0	−1.52029	−0.760143	−	0.649756i	$-0.774871\pi$
$643$	−24788.0	−1.52029	−0.760143	+	0.649756i	$0.774871\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28472.3	−1.73008	−0.865041	−	0.501702i	$-0.832708\pi$
$647$	−28472.3	−1.73008	−0.865041	+	0.501702i	$0.832708\pi$
$648$	0	0
$649$	−6840.00	−0.413703
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3243.02	0.194348	0.0971740	−	0.995267i	$-0.469020\pi$
$653$	3243.02	0.194348	0.0971740	+	0.995267i	$0.469020\pi$
$654$	0	0
$655$	−13072.0	−0.779794
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1176.90	−0.0695685	−0.0347842	−	0.999395i	$-0.511074\pi$
$659$	−1176.90	−0.0695685	−0.0347842	+	0.999395i	$0.511074\pi$
$660$	0	0
$661$	11590.0	0.681995	0.340998	−	0.940064i	$-0.389235\pi$
$661$	11590.0	0.681995	0.340998	+	0.940064i	$0.389235\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1220.49	0.0711709
$666$	0	0
$667$	31920.0	1.85299
$668$	0	0
$669$	0	0
$670$	0	0
$671$	34435.3	1.98116
$672$	0	0
$673$	23062.0	1.32091	0.660457	−	0.750864i	$-0.270363\pi$
$673$	23062.0	1.32091	0.660457	+	0.750864i	$0.270363\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22884.2	−1.29913	−0.649566	−	0.760305i	$-0.725049\pi$
$677$	−22884.2	−1.29913	−0.649566	+	0.760305i	$0.725049\pi$
$678$	0	0
$679$	5586.00	0.315716
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24715.0	1.38461	0.692307	−	0.721603i	$-0.256594\pi$
$683$	24715.0	1.38461	0.692307	+	0.721603i	$0.256594\pi$
$684$	0	0
$685$	−7752.00	−0.432392
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12867.5	−0.711483
$690$	0	0
$691$	10600.0	0.583564	0.291782	−	0.956485i	$-0.405752\pi$
$691$	10600.0	0.583564	0.291782	+	0.956485i	$0.405752\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6695.27	0.365419
$696$	0	0
$697$	12996.0	0.706253
$698$	0	0
$699$	0	0
$700$	0	0
$701$	12449.0	0.670746	0.335373	−	0.942085i	$-0.391138\pi$
$701$	12449.0	0.670746	0.335373	+	0.942085i	$0.391138\pi$
$702$	0	0
$703$	3720.00	0.199577
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2868.16	0.152572
$708$	0	0
$709$	−13710.0	−0.726220	−0.363110	−	0.931746i	$-0.618285\pi$
$709$	−13710.0	−0.726220	−0.363110	+	0.931746i	$0.618285\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20399.6	−1.07149
$714$	0	0
$715$	31160.0	1.62982
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2510.73	0.130228	0.0651142	−	0.997878i	$-0.479259\pi$
$719$	2510.73	0.130228	0.0651142	+	0.997878i	$0.479259\pi$
$720$	0	0
$721$	6412.00	0.331200
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−11960.8	−0.612708
$726$	0	0
$727$	−620.000	−0.0316293	−0.0158147	−	0.999875i	$-0.505034\pi$
$727$	−620.000	−0.0316293	−0.0158147	+	0.999875i	$0.505034\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12867.5	0.651054
$732$	0	0
$733$	−20214.0	−1.01858	−0.509291	−	0.860594i	$-0.670092\pi$
$733$	−20214.0	−1.01858	−0.509291	+	0.860594i	$0.670092\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1917.92	0.0958580
$738$	0	0
$739$	12324.0	0.613458	0.306729	−	0.951797i	$-0.400765\pi$
$739$	12324.0	0.613458	0.306729	+	0.951797i	$0.400765\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29736.4	−1.46827	−0.734134	−	0.679005i	$-0.762412\pi$
$743$	−29736.4	−1.46827	−0.734134	+	0.679005i	$0.762412\pi$
$744$	0	0
$745$	8664.00	0.426073
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6285.53	−0.306633
$750$	0	0
$751$	−19336.0	−0.939522	−0.469761	−	0.882794i	$-0.655660\pi$
$751$	−19336.0	−0.939522	−0.469761	+	0.882794i	$0.655660\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	30477.4	1.46912
$756$	0	0
$757$	15986.0	0.767531	0.383766	−	0.923431i	$-0.374627\pi$
$757$	15986.0	0.767531	0.383766	+	0.923431i	$0.374627\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	37007.1	1.76282	0.881409	−	0.472354i	$-0.156596\pi$
$761$	37007.1	1.76282	0.881409	+	0.472354i	$0.156596\pi$
$762$	0	0
$763$	−2394.00	−0.113589
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12867.5	−0.605759
$768$	0	0
$769$	−36070.0	−1.69144	−0.845720	−	0.533627i	$-0.820829\pi$
$769$	−36070.0	−1.69144	−0.845720	+	0.533627i	$0.820829\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−531.786	−0.0247439	−0.0123719	−	0.999923i	$-0.503938\pi$
$773$	−531.786	−0.0247439	−0.0123719	+	0.999923i	$0.503938\pi$
$774$	0	0
$775$	7644.00	0.354298
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3312.76	−0.152365
$780$	0	0
$781$	−19380.0	−0.887927
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4411.21	−0.200564
$786$	0	0
$787$	1136.00	0.0514537	0.0257268	−	0.999669i	$-0.491810\pi$
$787$	1136.00	0.0514537	0.0257268	+	0.999669i	$0.491810\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3417.38	−0.153613
$792$	0	0
$793$	64780.0	2.90089
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18054.6	0.802416	0.401208	−	0.915987i	$-0.368591\pi$
$797$	18054.6	0.802416	0.401208	+	0.915987i	$0.368591\pi$
$798$	0	0
$799$	36936.0	1.63542
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5492.21	0.241365
$804$	0	0
$805$	7980.00	0.349389
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38707.0	−1.68216	−0.841079	−	0.540912i	$-0.818079\pi$
$809$	−38707.0	−1.68216	−0.841079	+	0.540912i	$0.818079\pi$
$810$	0	0
$811$	−17936.0	−0.776595	−0.388297	−	0.921534i	$-0.626937\pi$
$811$	−17936.0	−0.776595	−0.388297	+	0.921534i	$0.626937\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22352.4	0.960701
$816$	0	0
$817$	−3280.00	−0.140456
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24915.5	1.05914	0.529571	−	0.848266i	$-0.322353\pi$
$821$	24915.5	1.05914	0.529571	+	0.848266i	$0.322353\pi$
$822$	0	0
$823$	−23424.0	−0.992113	−0.496057	−	0.868290i	$-0.665219\pi$
$823$	−23424.0	−0.992113	−0.496057	+	0.868290i	$0.665219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26650.3	1.12058	0.560291	−	0.828296i	$-0.310689\pi$
$827$	26650.3	1.12058	0.560291	+	0.828296i	$0.310689\pi$
$828$	0	0
$829$	−26254.0	−1.09993	−0.549963	−	0.835189i	$-0.685358\pi$
$829$	−26254.0	−1.09993	−0.549963	+	0.835189i	$0.685358\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3844.55	−0.159911
$834$	0	0
$835$	5624.00	0.233086
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6189.64	0.254696	0.127348	−	0.991858i	$-0.459353\pi$
$839$	6189.64	0.254696	0.127348	+	0.991858i	$0.459353\pi$
$840$	0	0
$841$	35195.0	1.44307
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39465.5	1.60669
$846$	0	0
$847$	3983.00	0.161579
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24322.7	0.979753
$852$	0	0
$853$	−45322.0	−1.81922	−0.909611	−	0.415462i	$-0.863620\pi$
$853$	−45322.0	−1.81922	−0.909611	+	0.415462i	$0.863620\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8691.64	0.346442	0.173221	−	0.984883i	$-0.444582\pi$
$857$	8691.64	0.346442	0.173221	+	0.984883i	$0.444582\pi$
$858$	0	0
$859$	43252.0	1.71797	0.858987	−	0.511998i	$-0.171094\pi$
$859$	43252.0	1.71797	0.858987	+	0.511998i	$0.171094\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29318.0	1.15642	0.578212	−	0.815886i	$-0.303750\pi$
$863$	29318.0	1.15642	0.578212	+	0.815886i	$0.303750\pi$
$864$	0	0
$865$	33668.0	1.32341
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31035.4	1.21151
$870$	0	0
$871$	3608.00	0.140359
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10618.3	−0.410244
$876$	0	0
$877$	47110.0	1.81390	0.906951	−	0.421237i	$-0.138404\pi$
$877$	47110.0	1.81390	0.906951	+	0.421237i	$0.138404\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42133.1	−1.61124	−0.805619	−	0.592434i	$-0.798167\pi$
$881$	−42133.1	−1.61124	−0.805619	+	0.592434i	$0.798167\pi$
$882$	0	0
$883$	22732.0	0.866356	0.433178	−	0.901308i	$-0.357392\pi$
$883$	22732.0	0.866356	0.433178	+	0.901308i	$0.357392\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21463.2	−0.812474	−0.406237	−	0.913768i	$-0.633159\pi$
$887$	−21463.2	−0.812474	−0.406237	+	0.913768i	$0.633159\pi$
$888$	0	0
$889$	−3192.00	−0.120423
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9415.22	−0.352820
$894$	0	0
$895$	2356.00	0.0879915
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38079.3	−1.41270
$900$	0	0
$901$	12312.0	0.455241
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3644.04	0.133847
$906$	0	0
$907$	6916.00	0.253189	0.126594	−	0.991955i	$-0.459595\pi$
$907$	6916.00	0.253189	0.126594	+	0.991955i	$0.459595\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38210.1	−1.38963	−0.694817	−	0.719186i	$-0.744515\pi$
$911$	−38210.1	−1.38963	−0.694817	+	0.719186i	$0.744515\pi$
$912$	0	0
$913$	63840.0	2.31412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10496.2	−0.377989
$918$	0	0
$919$	47632.0	1.70972	0.854861	−	0.518857i	$-0.173642\pi$
$919$	47632.0	1.70972	0.854861	+	0.518857i	$0.173642\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−36457.8	−1.30013
$924$	0	0
$925$	−9114.00	−0.323964
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3304.05	−0.116687	−0.0583435	−	0.998297i	$-0.518582\pi$
$929$	−3304.05	−0.116687	−0.0583435	+	0.998297i	$0.518582\pi$
$930$	0	0
$931$	980.000	0.0344986
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29814.9	−1.04283
$936$	0	0
$937$	21858.0	0.762081	0.381040	−	0.924558i	$-0.375566\pi$
$937$	21858.0	0.762081	0.381040	+	0.924558i	$0.375566\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−50380.2	−1.74532	−0.872660	−	0.488328i	$-0.837607\pi$
$941$	−50380.2	−1.74532	−0.872660	+	0.488328i	$0.837607\pi$
$942$	0	0
$943$	−21660.0	−0.747982
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31061.5	−1.06585	−0.532927	−	0.846161i	$-0.678908\pi$
$947$	−31061.5	−1.06585	−0.532927	+	0.846161i	$0.678908\pi$
$948$	0	0
$949$	10332.0	0.353415
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22770.9	0.773999	0.387000	−	0.922080i	$-0.373511\pi$
$953$	22770.9	0.773999	0.387000	+	0.922080i	$0.373511\pi$
$954$	0	0
$955$	−13300.0	−0.450657
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6224.51	−0.209593
$960$	0	0
$961$	−5455.00	−0.183109
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11838.8	0.394926
$966$	0	0
$967$	36416.0	1.21102	0.605512	−	0.795836i	$-0.292968\pi$
$967$	36416.0	1.21102	0.605512	+	0.795836i	$0.292968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	18621.2	0.615431	0.307715	−	0.951478i	$-0.400436\pi$
$971$	18621.2	0.615431	0.307715	+	0.951478i	$0.400436\pi$
$972$	0	0
$973$	5376.00	0.177129
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5806.05	−0.190125	−0.0950625	−	0.995471i	$-0.530305\pi$
$977$	−5806.05	−0.190125	−0.0950625	+	0.995471i	$0.530305\pi$
$978$	0	0
$979$	−63460.0	−2.07170
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3243.02	0.105225	0.0526126	−	0.998615i	$-0.483245\pi$
$983$	3243.02	0.105225	0.0526126	+	0.998615i	$0.483245\pi$
$984$	0	0
$985$	32680.0	1.05713
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21445.8	−0.689521
$990$	0	0
$991$	−49448.0	−1.58503	−0.792516	−	0.609851i	$-0.791229\pi$
$991$	−49448.0	−1.58503	−0.792516	+	0.609851i	$0.791229\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9205.99	0.293316
$996$	0	0
$997$	16294.0	0.517589	0.258794	−	0.965932i	$-0.416675\pi$
$997$	16294.0	0.517589	0.258794	+	0.965932i	$0.416675\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.a.be.1.2		2
3.2	odd	2	inner	1008.4.a.be.1.1		2
4.3	odd	2		63.4.a.d.1.2	yes	2
12.11	even	2		63.4.a.d.1.1	✓	2
20.19	odd	2		1575.4.a.t.1.1		2
28.3	even	6		441.4.e.s.226.1		4
28.11	odd	6		441.4.e.r.226.1		4
28.19	even	6		441.4.e.s.361.1		4
28.23	odd	6		441.4.e.r.361.1		4
28.27	even	2		441.4.a.q.1.2		2
60.59	even	2		1575.4.a.t.1.2		2
84.11	even	6		441.4.e.r.226.2		4
84.23	even	6		441.4.e.r.361.2		4
84.47	odd	6		441.4.e.s.361.2		4
84.59	odd	6		441.4.e.s.226.2		4
84.83	odd	2		441.4.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.a.d.1.1	✓	2	12.11	even	2
63.4.a.d.1.2	yes	2	4.3	odd	2
441.4.a.q.1.1		2	84.83	odd	2
441.4.a.q.1.2		2	28.27	even	2
441.4.e.r.226.1		4	28.11	odd	6
441.4.e.r.226.2		4	84.11	even	6
441.4.e.r.361.1		4	28.23	odd	6
441.4.e.r.361.2		4	84.23	even	6
441.4.e.s.226.1		4	28.3	even	6
441.4.e.s.226.2		4	84.59	odd	6
441.4.e.s.361.1		4	28.19	even	6
441.4.e.s.361.2		4	84.47	odd	6
1008.4.a.be.1.1		2	3.2	odd	2	inner
1008.4.a.be.1.2		2	1.1	even	1	trivial
1575.4.a.t.1.1		2	20.19	odd	2
1575.4.a.t.1.2		2	60.59	even	2

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q+8.71780 q^{5} +7.00000 q^{7} +O(q^{10})\)	\(q+8.71780 q^{5} +7.00000 q^{7} +43.5890 q^{11} +82.0000 q^{13} -78.4602 q^{17} +20.0000 q^{19} +130.767 q^{23} -49.0000 q^{25} +244.098 q^{29} -156.000 q^{31} +61.0246 q^{35} +186.000 q^{37} -165.638 q^{41} -164.000 q^{43} -470.761 q^{47} +49.0000 q^{49} -156.920 q^{53} +380.000 q^{55} -156.920 q^{59} +790.000 q^{61} +714.859 q^{65} +44.0000 q^{67} -444.608 q^{71} +126.000 q^{73} +305.123 q^{77} +712.000 q^{79} +1464.59 q^{83} -684.000 q^{85} -1455.87 q^{89} +574.000 q^{91} +174.356 q^{95} +798.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{7}+O(q^{10}) \) 2 * q + 14 * q^7	\( 2 q + 14 q^{7} + 164 q^{13} + 40 q^{19} - 98 q^{25} - 312 q^{31} + 372 q^{37} - 328 q^{43} + 98 q^{49} + 760 q^{55} + 1580 q^{61} + 88 q^{67} + 252 q^{73} + 1424 q^{79} - 1368 q^{85} + 1148 q^{91} + 1596 q^{97}+O(q^{100}) \) 2 * q + 14 * q^7 + 164 * q^13 + 40 * q^19 - 98 * q^25 - 312 * q^31 + 372 * q^37 - 328 * q^43 + 98 * q^49 + 760 * q^55 + 1580 * q^61 + 88 * q^67 + 252 * q^73 + 1424 * q^79 - 1368 * q^85 + 1148 * q^91 + 1596 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more