LMFDB - Embedded newform 864.6.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,6,Mod(1,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("864.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	864.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:	$138.571620318$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 286x^{4} + 1126x^{3} + 1345x^{2} - 3193x - 384 $ x^6 - x^5 - 286x^4 + 1126x^3 + 1345x^2 - 3193x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.82128$ of defining polynomial
Character	$\chi$	$=$	864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-43.4968 q^{5} +238.200 q^{7} +O(q^{10})$	$q-43.4968 q^{5} +238.200 q^{7} -628.981 q^{11} -966.503 q^{13} -758.898 q^{17} +2229.53 q^{19} -1535.42 q^{23} -1233.03 q^{25} -1721.68 q^{29} +5893.78 q^{31} -10360.9 q^{35} -2878.30 q^{37} -1744.06 q^{41} +15025.3 q^{43} +27880.9 q^{47} +39932.3 q^{49} -36718.5 q^{53} +27358.7 q^{55} +11248.2 q^{59} -21070.7 q^{61} +42039.8 q^{65} -27761.0 q^{67} -8983.56 q^{71} -2793.24 q^{73} -149823. q^{77} -21609.0 q^{79} +48373.7 q^{83} +33009.7 q^{85} +136005. q^{89} -230221. q^{91} -96977.5 q^{95} -130323. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 462 q^{11} - 384 q^{13} - 1380 q^{23} + 120 q^{25} - 5598 q^{35} - 396 q^{37} + 15036 q^{47} + 34620 q^{49} + 10824 q^{59} + 5520 q^{61} + 9144 q^{71} + 61734 q^{73} - 1806 q^{83} + 160392 q^{85} - 16476 q^{95} + 125226 q^{97}+O(q^{100}) $ 6 * q - 462 * q^11 - 384 * q^13 - 1380 * q^23 + 120 * q^25 - 5598 * q^35 - 396 * q^37 + 15036 * q^47 + 34620 * q^49 + 10824 * q^59 + 5520 * q^61 + 9144 * q^71 + 61734 * q^73 - 1806 * q^83 + 160392 * q^85 - 16476 * q^95 + 125226 * 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
$5$	−43.4968	−0.778095	−0.389047	−	0.921218i	$-0.627196\pi$
$5$	−43.4968	−0.778095	−0.389047	+	0.921218i	$0.627196\pi$
$6$	0	0
$7$	238.200	1.83737	0.918686	−	0.394990i	$-0.129252\pi$
$7$	238.200	1.83737	0.918686	+	0.394990i	$0.129252\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−628.981	−1.56731	−0.783656	−	0.621195i	$-0.786648\pi$
$11$	−628.981	−1.56731	−0.783656	+	0.621195i	$0.786648\pi$
$12$	0	0
$13$	−966.503	−1.58615	−0.793076	−	0.609122i	$-0.791522\pi$
$13$	−966.503	−1.58615	−0.793076	+	0.609122i	$0.791522\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−758.898	−0.636885	−0.318443	−	0.947942i	$-0.603160\pi$
$17$	−758.898	−0.636885	−0.318443	+	0.947942i	$0.603160\pi$
$18$	0	0
$19$	2229.53	1.41687	0.708434	−	0.705777i	$-0.249402\pi$
$19$	2229.53	1.41687	0.708434	+	0.705777i	$0.249402\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1535.42	−0.605212	−0.302606	−	0.953116i	$-0.597857\pi$
$23$	−1535.42	−0.605212	−0.302606	+	0.953116i	$0.597857\pi$
$24$	0	0
$25$	−1233.03	−0.394568
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1721.68	−0.380153	−0.190077	−	0.981769i	$-0.560874\pi$
$29$	−1721.68	−0.380153	−0.190077	+	0.981769i	$0.560874\pi$
$30$	0	0
$31$	5893.78	1.10151	0.550757	−	0.834666i	$-0.314339\pi$
$31$	5893.78	1.10151	0.550757	+	0.834666i	$0.314339\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10360.9	−1.42965
$36$	0	0
$37$	−2878.30	−0.345646	−0.172823	−	0.984953i	$-0.555289\pi$
$37$	−2878.30	−0.345646	−0.172823	+	0.984953i	$0.555289\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1744.06	−0.162032	−0.0810162	−	0.996713i	$-0.525817\pi$
$41$	−1744.06	−0.162032	−0.0810162	+	0.996713i	$0.525817\pi$
$42$	0	0
$43$	15025.3	1.23923	0.619613	−	0.784907i	$-0.287289\pi$
$43$	15025.3	1.23923	0.619613	+	0.784907i	$0.287289\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	27880.9	1.84103	0.920517	−	0.390702i	$-0.127768\pi$
$47$	27880.9	1.84103	0.920517	+	0.390702i	$0.127768\pi$
$48$	0	0
$49$	39932.3	2.37593
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−36718.5	−1.79554	−0.897771	−	0.440462i	$-0.854815\pi$
$53$	−36718.5	−1.79554	−0.897771	+	0.440462i	$0.854815\pi$
$54$	0	0
$55$	27358.7	1.21952
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11248.2	0.420681	0.210340	−	0.977628i	$-0.432543\pi$
$59$	11248.2	0.420681	0.210340	+	0.977628i	$0.432543\pi$
$60$	0	0
$61$	−21070.7	−0.725027	−0.362513	−	0.931979i	$-0.618081\pi$
$61$	−21070.7	−0.725027	−0.362513	+	0.931979i	$0.618081\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	42039.8	1.23418
$66$	0	0
$67$	−27761.0	−0.755523	−0.377761	−	0.925903i	$-0.623306\pi$
$67$	−27761.0	−0.755523	−0.377761	+	0.925903i	$0.623306\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8983.56	−0.211496	−0.105748	−	0.994393i	$-0.533724\pi$
$71$	−8983.56	−0.211496	−0.105748	+	0.994393i	$0.533724\pi$
$72$	0	0
$73$	−2793.24	−0.0613481	−0.0306740	−	0.999529i	$-0.509765\pi$
$73$	−2793.24	−0.0613481	−0.0306740	+	0.999529i	$0.509765\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−149823.	−2.87974
$78$	0	0
$79$	−21609.0	−0.389554	−0.194777	−	0.980848i	$-0.562398\pi$
$79$	−21609.0	−0.389554	−0.194777	+	0.980848i	$0.562398\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	48373.7	0.770750	0.385375	−	0.922760i	$-0.374072\pi$
$83$	48373.7	0.770750	0.385375	+	0.922760i	$0.374072\pi$
$84$	0	0
$85$	33009.7	0.495557
$86$	0	0
$87$	0	0
$88$	0	0
$89$	136005.	1.82004	0.910018	−	0.414569i	$-0.136068\pi$
$89$	136005.	1.82004	0.910018	+	0.414569i	$0.136068\pi$
$90$	0	0
$91$	−230221.	−2.91435
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−96977.5	−1.10246
$96$	0	0
$97$	−130323.	−1.40634	−0.703172	−	0.711019i	$-0.748234\pi$
$97$	−130323.	−1.40634	−0.703172	+	0.711019i	$0.748234\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	141630.	1.38151	0.690753	−	0.723091i	$-0.257279\pi$
$101$	141630.	1.38151	0.690753	+	0.723091i	$0.257279\pi$
$102$	0	0
$103$	−67778.9	−0.629508	−0.314754	−	0.949173i	$-0.601922\pi$
$103$	−67778.9	−0.629508	−0.314754	+	0.949173i	$0.601922\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−138518.	−1.16962	−0.584811	−	0.811170i	$-0.698831\pi$
$107$	−138518.	−1.16962	−0.584811	+	0.811170i	$0.698831\pi$
$108$	0	0
$109$	116731.	0.941066	0.470533	−	0.882382i	$-0.344062\pi$
$109$	116731.	0.941066	0.470533	+	0.882382i	$0.344062\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−168277.	−1.23974	−0.619869	−	0.784705i	$-0.712814\pi$
$113$	−168277.	−1.23974	−0.619869	+	0.784705i	$0.712814\pi$
$114$	0	0
$115$	66785.9	0.470912
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−180770.	−1.17019
$120$	0	0
$121$	234566.	1.45647
$122$	0	0
$123$	0	0
$124$	0	0
$125$	189560.	1.08511
$126$	0	0
$127$	114407.	0.629424	0.314712	−	0.949187i	$-0.398092\pi$
$127$	114407.	0.629424	0.314712	+	0.949187i	$0.398092\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	243693.	1.24070	0.620348	−	0.784327i	$-0.286991\pi$
$131$	243693.	1.24070	0.620348	+	0.784327i	$0.286991\pi$
$132$	0	0
$133$	531075.	2.60331
$134$	0	0
$135$	0	0
$136$	0	0
$137$	245544.	1.11771	0.558854	−	0.829266i	$-0.311241\pi$
$137$	245544.	1.11771	0.558854	+	0.829266i	$0.311241\pi$
$138$	0	0
$139$	274623.	1.20559	0.602795	−	0.797896i	$-0.294054\pi$
$139$	274623.	1.20559	0.602795	+	0.797896i	$0.294054\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	607912.	2.48600
$144$	0	0
$145$	74887.8	0.295795
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−187023.	−0.690126	−0.345063	−	0.938580i	$-0.612142\pi$
$149$	−187023.	−0.690126	−0.345063	+	0.938580i	$0.612142\pi$
$150$	0	0
$151$	504788.	1.80164	0.900818	−	0.434198i	$-0.142968\pi$
$151$	504788.	1.80164	0.900818	+	0.434198i	$0.142968\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−256361.	−0.857082
$156$	0	0
$157$	−307900.	−0.996920	−0.498460	−	0.866913i	$-0.666101\pi$
$157$	−307900.	−0.996920	−0.498460	+	0.866913i	$0.666101\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−365737.	−1.11200
$162$	0	0
$163$	−269060.	−0.793194	−0.396597	−	0.917993i	$-0.629809\pi$
$163$	−269060.	−0.793194	−0.396597	+	0.917993i	$0.629809\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	136016.	0.377396	0.188698	−	0.982035i	$-0.439573\pi$
$167$	136016.	0.377396	0.188698	+	0.982035i	$0.439573\pi$
$168$	0	0
$169$	562836.	1.51588
$170$	0	0
$171$	0	0
$172$	0	0
$173$	654874.	1.66358	0.831788	−	0.555094i	$-0.187318\pi$
$173$	654874.	1.66358	0.831788	+	0.555094i	$0.187318\pi$
$174$	0	0
$175$	−293707.	−0.724968
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−306491.	−0.714965	−0.357482	−	0.933920i	$-0.616365\pi$
$179$	−306491.	−0.714965	−0.357482	+	0.933920i	$0.616365\pi$
$180$	0	0
$181$	330963.	0.750901	0.375450	−	0.926842i	$-0.377488\pi$
$181$	330963.	0.750901	0.375450	+	0.926842i	$0.377488\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	125197.	0.268945
$186$	0	0
$187$	477333.	0.998199
$188$	0	0
$189$	0	0
$190$	0	0
$191$	303420.	0.601811	0.300906	−	0.953654i	$-0.402711\pi$
$191$	303420.	0.601811	0.300906	+	0.953654i	$0.402711\pi$
$192$	0	0
$193$	−43319.2	−0.0837119	−0.0418559	−	0.999124i	$-0.513327\pi$
$193$	−43319.2	−0.0837119	−0.0418559	+	0.999124i	$0.513327\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−160510.	−0.294671	−0.147335	−	0.989087i	$-0.547070\pi$
$197$	−160510.	−0.294671	−0.147335	+	0.989087i	$0.547070\pi$
$198$	0	0
$199$	446301.	0.798905	0.399452	−	0.916754i	$-0.369200\pi$
$199$	446301.	0.798905	0.399452	+	0.916754i	$0.369200\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−410105.	−0.698482
$204$	0	0
$205$	75861.1	0.126077
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.40233e6	−2.22068
$210$	0	0
$211$	295670.	0.457195	0.228598	−	0.973521i	$-0.426586\pi$
$211$	295670.	0.457195	0.228598	+	0.973521i	$0.426586\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−653551.	−0.964236
$216$	0	0
$217$	1.40390e6	2.02389
$218$	0	0
$219$	0	0
$220$	0	0
$221$	733478.	1.01020
$222$	0	0
$223$	−591927.	−0.797087	−0.398544	−	0.917149i	$-0.630484\pi$
$223$	−591927.	−0.797087	−0.398544	+	0.917149i	$0.630484\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21754.7	−0.0280213	−0.0140106	−	0.999902i	$-0.504460\pi$
$227$	−21754.7	−0.0280213	−0.0140106	+	0.999902i	$0.504460\pi$
$228$	0	0
$229$	1.20550e6	1.51907	0.759536	−	0.650465i	$-0.225426\pi$
$229$	1.20550e6	1.51907	0.759536	+	0.650465i	$0.225426\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−482208.	−0.581895	−0.290947	−	0.956739i	$-0.593970\pi$
$233$	−482208.	−0.581895	−0.290947	+	0.956739i	$0.593970\pi$
$234$	0	0
$235$	−1.21273e6	−1.43250
$236$	0	0
$237$	0	0
$238$	0	0
$239$	78381.2	0.0887600	0.0443800	−	0.999015i	$-0.485869\pi$
$239$	78381.2	0.0887600	0.0443800	+	0.999015i	$0.485869\pi$
$240$	0	0
$241$	1.60921e6	1.78472	0.892359	−	0.451327i	$-0.149049\pi$
$241$	1.60921e6	1.78472	0.892359	+	0.451327i	$0.149049\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.73693e6	−1.84870
$246$	0	0
$247$	−2.15485e6	−2.24737
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−902802.	−0.904499	−0.452250	−	0.891891i	$-0.649378\pi$
$251$	−902802.	−0.904499	−0.452250	+	0.891891i	$0.649378\pi$
$252$	0	0
$253$	965750.	0.948556
$254$	0	0
$255$	0	0
$256$	0	0
$257$	848294.	0.801150	0.400575	−	0.916264i	$-0.368810\pi$
$257$	848294.	0.801150	0.400575	+	0.916264i	$0.368810\pi$
$258$	0	0
$259$	−685611.	−0.635080
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.07974e6	0.962568	0.481284	−	0.876565i	$-0.340171\pi$
$263$	1.07974e6	0.962568	0.481284	+	0.876565i	$0.340171\pi$
$264$	0	0
$265$	1.59714e6	1.39710
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.05599e6	0.889775	0.444888	−	0.895586i	$-0.353244\pi$
$269$	1.05599e6	0.889775	0.444888	+	0.895586i	$0.353244\pi$
$270$	0	0
$271$	−394981.	−0.326703	−0.163351	−	0.986568i	$-0.552230\pi$
$271$	−394981.	−0.326703	−0.163351	+	0.986568i	$0.552230\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	775550.	0.618412
$276$	0	0
$277$	613031.	0.480046	0.240023	−	0.970767i	$-0.422845\pi$
$277$	613031.	0.480046	0.240023	+	0.970767i	$0.422845\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−64392.2	−0.0486483	−0.0243241	−	0.999704i	$-0.507743\pi$
$281$	−64392.2	−0.0486483	−0.0243241	+	0.999704i	$0.507743\pi$
$282$	0	0
$283$	2.21208e6	1.64186	0.820928	−	0.571032i	$-0.193457\pi$
$283$	2.21208e6	1.64186	0.820928	+	0.571032i	$0.193457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−415435.	−0.297713
$288$	0	0
$289$	−843930.	−0.594377
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.19315e6	−0.811940	−0.405970	−	0.913886i	$-0.633066\pi$
$293$	−1.19315e6	−0.811940	−0.405970	+	0.913886i	$0.633066\pi$
$294$	0	0
$295$	−489260.	−0.327329
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.48399e6	0.959959
$300$	0	0
$301$	3.57902e6	2.27692
$302$	0	0
$303$	0	0
$304$	0	0
$305$	916508.	0.564139
$306$	0	0
$307$	−218811.	−0.132503	−0.0662513	−	0.997803i	$-0.521104\pi$
$307$	−218811.	−0.132503	−0.0662513	+	0.997803i	$0.521104\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.74472e6	1.60915	0.804577	−	0.593848i	$-0.202392\pi$
$311$	2.74472e6	1.60915	0.804577	+	0.593848i	$0.202392\pi$
$312$	0	0
$313$	−1.22904e6	−0.709098	−0.354549	−	0.935038i	$-0.615366\pi$
$313$	−1.22904e6	−0.709098	−0.354549	+	0.935038i	$0.615366\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	317742.	0.177593	0.0887967	−	0.996050i	$-0.471698\pi$
$317$	317742.	0.177593	0.0887967	+	0.996050i	$0.471698\pi$
$318$	0	0
$319$	1.08291e6	0.595819
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.69199e6	−0.902383
$324$	0	0
$325$	1.19172e6	0.625846
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.64123e6	3.38266
$330$	0	0
$331$	2.89037e6	1.45005	0.725026	−	0.688721i	$-0.241828\pi$
$331$	2.89037e6	1.45005	0.725026	+	0.688721i	$0.241828\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.20751e6	0.587869
$336$	0	0
$337$	−65398.3	−0.0313683	−0.0156842	−	0.999877i	$-0.504993\pi$
$337$	−65398.3	−0.0313683	−0.0156842	+	0.999877i	$0.504993\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.70707e6	−1.72642
$342$	0	0
$343$	5.50845e6	2.52810
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.54353e6	−1.57984	−0.789919	−	0.613211i	$-0.789878\pi$
$347$	−3.54353e6	−1.57984	−0.789919	+	0.613211i	$0.789878\pi$
$348$	0	0
$349$	−1.13278e6	−0.497833	−0.248916	−	0.968525i	$-0.580074\pi$
$349$	−1.13278e6	−0.497833	−0.248916	+	0.968525i	$0.580074\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.33228e6	−0.569063	−0.284531	−	0.958667i	$-0.591838\pi$
$353$	−1.33228e6	−0.569063	−0.284531	+	0.958667i	$0.591838\pi$
$354$	0	0
$355$	390756.	0.164564
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17438.3	−0.00714115	−0.00357057	−	0.999994i	$-0.501137\pi$
$359$	−17438.3	−0.00714115	−0.00357057	+	0.999994i	$0.501137\pi$
$360$	0	0
$361$	2.49471e6	1.00752
$362$	0	0
$363$	0	0
$364$	0	0
$365$	121497.	0.0477346
$366$	0	0
$367$	−1.92930e6	−0.747713	−0.373857	−	0.927486i	$-0.621965\pi$
$367$	−1.92930e6	−0.747713	−0.373857	+	0.927486i	$0.621965\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.74636e6	−3.29908
$372$	0	0
$373$	2.25187e6	0.838053	0.419026	−	0.907974i	$-0.362372\pi$
$373$	2.25187e6	0.838053	0.419026	+	0.907974i	$0.362372\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.66401e6	0.602981
$378$	0	0
$379$	937118.	0.335117	0.167558	−	0.985862i	$-0.446412\pi$
$379$	937118.	0.335117	0.167558	+	0.985862i	$0.446412\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.13598e6	1.78907	0.894533	−	0.447001i	$-0.147508\pi$
$383$	5.13598e6	1.78907	0.894533	+	0.447001i	$0.147508\pi$
$384$	0	0
$385$	6.51684e6	2.24071
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.90747e6	0.639123	0.319562	−	0.947566i	$-0.396464\pi$
$389$	1.90747e6	0.639123	0.319562	+	0.947566i	$0.396464\pi$
$390$	0	0
$391$	1.16523e6	0.385451
$392$	0	0
$393$	0	0
$394$	0	0
$395$	939925.	0.303110
$396$	0	0
$397$	1.88120e6	0.599045	0.299522	−	0.954089i	$-0.403173\pi$
$397$	1.88120e6	0.599045	0.299522	+	0.954089i	$0.403173\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.92637e6	−1.21935	−0.609677	−	0.792650i	$-0.708701\pi$
$401$	−3.92637e6	−1.21935	−0.609677	+	0.792650i	$0.708701\pi$
$402$	0	0
$403$	−5.69636e6	−1.74717
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.81039e6	0.541735
$408$	0	0
$409$	−1.29230e6	−0.381991	−0.190996	−	0.981591i	$-0.561172\pi$
$409$	−1.29230e6	−0.381991	−0.190996	+	0.981591i	$0.561172\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.67932e6	0.772946
$414$	0	0
$415$	−2.10410e6	−0.599717
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.51195e6	−0.420729	−0.210365	−	0.977623i	$-0.567465\pi$
$419$	−1.51195e6	−0.420729	−0.210365	+	0.977623i	$0.567465\pi$
$420$	0	0
$421$	728989.	0.200454	0.100227	−	0.994965i	$-0.468043\pi$
$421$	728989.	0.200454	0.100227	+	0.994965i	$0.468043\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	935741.	0.251295
$426$	0	0
$427$	−5.01904e6	−1.33214
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.55233e6	−0.921127	−0.460564	−	0.887627i	$-0.652353\pi$
$431$	−3.55233e6	−0.921127	−0.460564	+	0.887627i	$0.652353\pi$
$432$	0	0
$433$	912422.	0.233871	0.116935	−	0.993140i	$-0.462693\pi$
$433$	912422.	0.233871	0.116935	+	0.993140i	$0.462693\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.42327e6	−0.857506
$438$	0	0
$439$	−704468.	−0.174462	−0.0872308	−	0.996188i	$-0.527802\pi$
$439$	−704468.	−0.174462	−0.0872308	+	0.996188i	$0.527802\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.89511e6	0.458801	0.229401	−	0.973332i	$-0.426323\pi$
$443$	1.89511e6	0.458801	0.229401	+	0.973332i	$0.426323\pi$
$444$	0	0
$445$	−5.91579e6	−1.41616
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.18292e6	−1.21327	−0.606637	−	0.794979i	$-0.707482\pi$
$449$	−5.18292e6	−1.21327	−0.606637	+	0.794979i	$0.707482\pi$
$450$	0	0
$451$	1.09698e6	0.253955
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.00139e7	2.26764
$456$	0	0
$457$	−3.14278e6	−0.703920	−0.351960	−	0.936015i	$-0.614485\pi$
$457$	−3.14278e6	−0.703920	−0.351960	+	0.936015i	$0.614485\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.24377e6	1.80665	0.903324	−	0.428958i	$-0.141119\pi$
$461$	8.24377e6	1.80665	0.903324	+	0.428958i	$0.141119\pi$
$462$	0	0
$463$	1.70691e6	0.370049	0.185025	−	0.982734i	$-0.440764\pi$
$463$	1.70691e6	0.370049	0.185025	+	0.982734i	$0.440764\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.29616e6	−1.12375	−0.561874	−	0.827223i	$-0.689919\pi$
$467$	−5.29616e6	−1.12375	−0.561874	+	0.827223i	$0.689919\pi$
$468$	0	0
$469$	−6.61267e6	−1.38818
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.45060e6	−1.94226
$474$	0	0
$475$	−2.74907e6	−0.559051
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.97109e6	1.18909	0.594545	−	0.804063i	$-0.297332\pi$
$479$	5.97109e6	1.18909	0.594545	+	0.804063i	$0.297332\pi$
$480$	0	0
$481$	2.78188e6	0.548247
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.66864e6	1.09427
$486$	0	0
$487$	−3.86439e6	−0.738344	−0.369172	−	0.929361i	$-0.620359\pi$
$487$	−3.86439e6	−0.738344	−0.369172	+	0.929361i	$0.620359\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.72054e6	0.883665	0.441832	−	0.897098i	$-0.354329\pi$
$491$	4.72054e6	0.883665	0.441832	+	0.897098i	$0.354329\pi$
$492$	0	0
$493$	1.30658e6	0.242114
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.13988e6	−0.388597
$498$	0	0
$499$	−5.76698e6	−1.03680	−0.518402	−	0.855137i	$-0.673473\pi$
$499$	−5.76698e6	−1.03680	−0.518402	+	0.855137i	$0.673473\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.56779e6	0.276292	0.138146	−	0.990412i	$-0.455886\pi$
$503$	1.56779e6	0.276292	0.138146	+	0.990412i	$0.455886\pi$
$504$	0	0
$505$	−6.16047e6	−1.07494
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.02710e7	−1.75719	−0.878594	−	0.477570i	$-0.841518\pi$
$509$	−1.02710e7	−1.75719	−0.878594	+	0.477570i	$0.841518\pi$
$510$	0	0
$511$	−665350.	−0.112719
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.94817e6	0.489817
$516$	0	0
$517$	−1.75365e7	−2.88548
$518$	0	0
$519$	0	0
$520$	0	0
$521$	329958.	0.0532554	0.0266277	−	0.999645i	$-0.491523\pi$
$521$	329958.	0.0532554	0.0266277	+	0.999645i	$0.491523\pi$
$522$	0	0
$523$	9.54496e6	1.52588	0.762939	−	0.646470i	$-0.223756\pi$
$523$	9.54496e6	1.52588	0.762939	+	0.646470i	$0.223756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.47278e6	−0.701538
$528$	0	0
$529$	−4.07883e6	−0.633718
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.68564e6	0.257008
$534$	0	0
$535$	6.02508e6	0.910077
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.51166e7	−3.72383
$540$	0	0
$541$	1.26583e7	1.85944	0.929721	−	0.368266i	$-0.120048\pi$
$541$	1.26583e7	1.85944	0.929721	+	0.368266i	$0.120048\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.07743e6	−0.732239
$546$	0	0
$547$	−852516.	−0.121824	−0.0609122	−	0.998143i	$-0.519401\pi$
$547$	−852516.	−0.121824	−0.0609122	+	0.998143i	$0.519401\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.83855e6	−0.538627
$552$	0	0
$553$	−5.14728e6	−0.715755
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.73795e6	−0.510500	−0.255250	−	0.966875i	$-0.582158\pi$
$557$	−3.73795e6	−0.510500	−0.255250	+	0.966875i	$0.582158\pi$
$558$	0	0
$559$	−1.45220e7	−1.96560
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.40018e6	0.718021	0.359011	−	0.933333i	$-0.383114\pi$
$563$	5.40018e6	0.718021	0.359011	+	0.933333i	$0.383114\pi$
$564$	0	0
$565$	7.31954e6	0.964634
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.31015e6	0.946555	0.473277	−	0.880913i	$-0.343071\pi$
$569$	7.31015e6	0.946555	0.473277	+	0.880913i	$0.343071\pi$
$570$	0	0
$571$	−1.20437e6	−0.154585	−0.0772926	−	0.997008i	$-0.524628\pi$
$571$	−1.20437e6	−0.154585	−0.0772926	+	0.997008i	$0.524628\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.89321e6	0.238797
$576$	0	0
$577$	−1.18554e7	−1.48244	−0.741219	−	0.671263i	$-0.765752\pi$
$577$	−1.18554e7	−1.48244	−0.741219	+	0.671263i	$0.765752\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.15226e7	1.41615
$582$	0	0
$583$	2.30953e7	2.81418
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−758999.	−0.0909172	−0.0454586	−	0.998966i	$-0.514475\pi$
$587$	−758999.	−0.0909172	−0.0454586	+	0.998966i	$0.514475\pi$
$588$	0	0
$589$	1.31404e7	1.56070
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.25788e7	1.46893	0.734466	−	0.678645i	$-0.237433\pi$
$593$	1.25788e7	1.46893	0.734466	+	0.678645i	$0.237433\pi$
$594$	0	0
$595$	7.86291e6	0.910523
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.50146e7	1.70980	0.854902	−	0.518790i	$-0.173617\pi$
$599$	1.50146e7	1.70980	0.854902	+	0.518790i	$0.173617\pi$
$600$	0	0
$601$	−9.13086e6	−1.03116	−0.515579	−	0.856842i	$-0.672423\pi$
$601$	−9.13086e6	−1.03116	−0.515579	+	0.856842i	$0.672423\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.02029e7	−1.13327
$606$	0	0
$607$	1.70428e7	1.87745	0.938726	−	0.344664i	$-0.112007\pi$
$607$	1.70428e7	1.87745	0.938726	+	0.344664i	$0.112007\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.69470e7	−2.92016
$612$	0	0
$613$	−1.33457e7	−1.43447	−0.717234	−	0.696833i	$-0.754592\pi$
$613$	−1.33457e7	−1.43447	−0.717234	+	0.696833i	$0.754592\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.54154e6	0.374524	0.187262	−	0.982310i	$-0.440039\pi$
$617$	3.54154e6	0.374524	0.187262	+	0.982310i	$0.440039\pi$
$618$	0	0
$619$	7.40154e6	0.776417	0.388209	−	0.921571i	$-0.373094\pi$
$619$	7.40154e6	0.776417	0.388209	+	0.921571i	$0.373094\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.23964e7	3.34408
$624$	0	0
$625$	−4.39207e6	−0.449748
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.18434e6	0.220137
$630$	0	0
$631$	−4.09670e6	−0.409601	−0.204801	−	0.978804i	$-0.565655\pi$
$631$	−4.09670e6	−0.409601	−0.204801	+	0.978804i	$0.565655\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.97634e6	−0.489752
$636$	0	0
$637$	−3.85947e7	−3.76859
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.04505e6	0.677234	0.338617	−	0.940924i	$-0.390041\pi$
$641$	7.04505e6	0.677234	0.338617	+	0.940924i	$0.390041\pi$
$642$	0	0
$643$	9.48089e6	0.904319	0.452159	−	0.891937i	$-0.350654\pi$
$643$	9.48089e6	0.904319	0.452159	+	0.891937i	$0.350654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.50542e6	−0.798794	−0.399397	−	0.916778i	$-0.630781\pi$
$647$	−8.50542e6	−0.798794	−0.399397	+	0.916778i	$0.630781\pi$
$648$	0	0
$649$	−7.07489e6	−0.659338
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.14712e7	1.05275	0.526374	−	0.850253i	$-0.323551\pi$
$653$	1.14712e7	1.05275	0.526374	+	0.850253i	$0.323551\pi$
$654$	0	0
$655$	−1.05999e7	−0.965379
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.16294e7	−1.04314	−0.521571	−	0.853208i	$-0.674654\pi$
$659$	−1.16294e7	−1.04314	−0.521571	+	0.853208i	$0.674654\pi$
$660$	0	0
$661$	6.90433e6	0.614636	0.307318	−	0.951607i	$-0.400568\pi$
$661$	6.90433e6	0.614636	0.307318	+	0.951607i	$0.400568\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.31001e7	−2.02562
$666$	0	0
$667$	2.64351e6	0.230073
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.32531e7	1.13634
$672$	0	0
$673$	−3.35548e6	−0.285573	−0.142787	−	0.989753i	$-0.545606\pi$
$673$	−3.35548e6	−0.285573	−0.142787	+	0.989753i	$0.545606\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.27857e6	−0.526488	−0.263244	−	0.964729i	$-0.584792\pi$
$677$	−6.27857e6	−0.526488	−0.263244	+	0.964729i	$0.584792\pi$
$678$	0	0
$679$	−3.10430e7	−2.58398
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.03583e6	0.495091	0.247546	−	0.968876i	$-0.420376\pi$
$683$	6.03583e6	0.495091	0.247546	+	0.968876i	$0.420376\pi$
$684$	0	0
$685$	−1.06804e7	−0.869683
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.54886e7	2.84800
$690$	0	0
$691$	−2.02558e6	−0.161382	−0.0806909	−	0.996739i	$-0.525713\pi$
$691$	−2.02558e6	−0.161382	−0.0806909	+	0.996739i	$0.525713\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.19452e7	−0.938063
$696$	0	0
$697$	1.32356e6	0.103196
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.24146e7	−0.954199	−0.477100	−	0.878849i	$-0.658312\pi$
$701$	−1.24146e7	−0.954199	−0.477100	+	0.878849i	$0.658312\pi$
$702$	0	0
$703$	−6.41725e6	−0.489735
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.37364e7	2.53834
$708$	0	0
$709$	−1.81440e7	−1.35555	−0.677777	−	0.735267i	$-0.737057\pi$
$709$	−1.81440e7	−1.35555	−0.677777	+	0.735267i	$0.737057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.04943e6	−0.666649
$714$	0	0
$715$	−2.64422e7	−1.93434
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.43642e7	1.03623	0.518117	−	0.855309i	$-0.326633\pi$
$719$	1.43642e7	1.03623	0.518117	+	0.855309i	$0.326633\pi$
$720$	0	0
$721$	−1.61449e7	−1.15664
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.12288e6	0.149996
$726$	0	0
$727$	−7.61647e6	−0.534463	−0.267232	−	0.963632i	$-0.586109\pi$
$727$	−7.61647e6	−0.534463	−0.267232	+	0.963632i	$0.586109\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.14026e7	−0.789246
$732$	0	0
$733$	−2.52045e7	−1.73268	−0.866338	−	0.499457i	$-0.833533\pi$
$733$	−2.52045e7	−1.73268	−0.866338	+	0.499457i	$0.833533\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.74611e7	1.18414
$738$	0	0
$739$	−2.76133e7	−1.85998	−0.929989	−	0.367587i	$-0.880184\pi$
$739$	−2.76133e7	−1.85998	−0.929989	+	0.367587i	$0.880184\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.37947e7	0.916728	0.458364	−	0.888764i	$-0.348436\pi$
$743$	1.37947e7	0.916728	0.458364	+	0.888764i	$0.348436\pi$
$744$	0	0
$745$	8.13489e6	0.536984
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.29949e7	−2.14903
$750$	0	0
$751$	4.01785e6	0.259952	0.129976	−	0.991517i	$-0.458510\pi$
$751$	4.01785e6	0.259952	0.129976	+	0.991517i	$0.458510\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.19567e7	−1.40184
$756$	0	0
$757$	1.91655e7	1.21557	0.607787	−	0.794100i	$-0.292058\pi$
$757$	1.91655e7	1.21557	0.607787	+	0.794100i	$0.292058\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.34719e7	−0.843271	−0.421635	−	0.906765i	$-0.638544\pi$
$761$	−1.34719e7	−0.843271	−0.421635	+	0.906765i	$0.638544\pi$
$762$	0	0
$763$	2.78054e7	1.72909
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.08714e7	−0.667264
$768$	0	0
$769$	−2.03586e7	−1.24146	−0.620728	−	0.784026i	$-0.713163\pi$
$769$	−2.03586e7	−1.24146	−0.620728	+	0.784026i	$0.713163\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	606409.	0.0365020	0.0182510	−	0.999833i	$-0.494190\pi$
$773$	606409.	0.0365020	0.0182510	+	0.999833i	$0.494190\pi$
$774$	0	0
$775$	−7.26718e6	−0.434622
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.88843e6	−0.229578
$780$	0	0
$781$	5.65049e6	0.331481
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.33927e7	0.775699
$786$	0	0
$787$	2.70088e7	1.55442	0.777211	−	0.629240i	$-0.216634\pi$
$787$	2.70088e7	1.55442	0.777211	+	0.629240i	$0.216634\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.00837e7	−2.27786
$792$	0	0
$793$	2.03649e7	1.15000
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.67959e7	0.936608	0.468304	−	0.883567i	$-0.344865\pi$
$797$	1.67959e7	0.936608	0.468304	+	0.883567i	$0.344865\pi$
$798$	0	0
$799$	−2.11588e7	−1.17253
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.75689e6	0.0961516
$804$	0	0
$805$	1.59084e7	0.865241
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.60402e7	0.861662	0.430831	−	0.902433i	$-0.358220\pi$
$809$	1.60402e7	0.861662	0.430831	+	0.902433i	$0.358220\pi$
$810$	0	0
$811$	7.00506e6	0.373990	0.186995	−	0.982361i	$-0.440125\pi$
$811$	7.00506e6	0.373990	0.186995	+	0.982361i	$0.440125\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.17032e7	0.617180
$816$	0	0
$817$	3.34993e7	1.75582
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.40734e7	−0.728690	−0.364345	−	0.931264i	$-0.618707\pi$
$821$	−1.40734e7	−0.728690	−0.364345	+	0.931264i	$0.618707\pi$
$822$	0	0
$823$	5.03356e6	0.259045	0.129523	−	0.991576i	$-0.458656\pi$
$823$	5.03356e6	0.259045	0.129523	+	0.991576i	$0.458656\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.14029e7	−1.08820	−0.544101	−	0.839020i	$-0.683129\pi$
$827$	−2.14029e7	−1.08820	−0.544101	+	0.839020i	$0.683129\pi$
$828$	0	0
$829$	−2.39490e7	−1.21032	−0.605162	−	0.796102i	$-0.706892\pi$
$829$	−2.39490e7	−1.21032	−0.605162	+	0.796102i	$0.706892\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.03046e7	−1.51320
$834$	0	0
$835$	−5.91625e6	−0.293650
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.21737e6	0.157796	0.0788981	−	0.996883i	$-0.474860\pi$
$839$	3.21737e6	0.157796	0.0788981	+	0.996883i	$0.474860\pi$
$840$	0	0
$841$	−1.75470e7	−0.855484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.44816e7	−1.17950
$846$	0	0
$847$	5.58736e7	2.67607
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.41940e6	0.209189
$852$	0	0
$853$	−2.11126e7	−0.993502	−0.496751	−	0.867893i	$-0.665474\pi$
$853$	−2.11126e7	−0.993502	−0.496751	+	0.867893i	$0.665474\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.91130e7	−1.81915	−0.909577	−	0.415535i	$-0.863594\pi$
$857$	−3.91130e7	−1.81915	−0.909577	+	0.415535i	$0.863594\pi$
$858$	0	0
$859$	−3.11440e7	−1.44010	−0.720049	−	0.693924i	$-0.755881\pi$
$859$	−3.11440e7	−1.44010	−0.720049	+	0.693924i	$0.755881\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.12386e7	−1.42779	−0.713896	−	0.700251i	$-0.753071\pi$
$863$	−3.12386e7	−1.42779	−0.713896	+	0.700251i	$0.753071\pi$
$864$	0	0
$865$	−2.84849e7	−1.29442
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.35917e7	0.610553
$870$	0	0
$871$	2.68311e7	1.19837
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.51533e7	1.99374
$876$	0	0
$877$	1.06670e7	0.468320	0.234160	−	0.972198i	$-0.424766\pi$
$877$	1.06670e7	0.468320	0.234160	+	0.972198i	$0.424766\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.05809e7	−0.459284	−0.229642	−	0.973275i	$-0.573756\pi$
$881$	−1.05809e7	−0.459284	−0.229642	+	0.973275i	$0.573756\pi$
$882$	0	0
$883$	−2.05413e7	−0.886596	−0.443298	−	0.896374i	$-0.646192\pi$
$883$	−2.05413e7	−0.886596	−0.443298	+	0.896374i	$0.646192\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.31469e7	1.41460	0.707301	−	0.706913i	$-0.249913\pi$
$887$	3.31469e7	1.41460	0.707301	+	0.706913i	$0.249913\pi$
$888$	0	0
$889$	2.72518e7	1.15649
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.21613e7	2.60850
$894$	0	0
$895$	1.33314e7	0.556310
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.01472e7	−0.418744
$900$	0	0
$901$	2.78656e7	1.14355
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.43958e7	−0.584272
$906$	0	0
$907$	1.45989e7	0.589252	0.294626	−	0.955613i	$-0.404805\pi$
$907$	1.45989e7	0.589252	0.294626	+	0.955613i	$0.404805\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.24495e7	0.896211	0.448106	−	0.893981i	$-0.352099\pi$
$911$	2.24495e7	0.896211	0.448106	+	0.893981i	$0.352099\pi$
$912$	0	0
$913$	−3.04261e7	−1.20801
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.80478e7	2.27962
$918$	0	0
$919$	83468.9	0.00326014	0.00163007	−	0.999999i	$-0.499481\pi$
$919$	83468.9	0.00326014	0.00163007	+	0.999999i	$0.499481\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.68264e6	0.335465
$924$	0	0
$925$	3.54902e6	0.136381
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.39387e7	−1.29020	−0.645098	−	0.764100i	$-0.723183\pi$
$929$	−3.39387e7	−1.29020	−0.645098	+	0.764100i	$0.723183\pi$
$930$	0	0
$931$	8.90303e7	3.36638
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.07625e7	−0.776693
$936$	0	0
$937$	−2.39563e7	−0.891397	−0.445698	−	0.895183i	$-0.647045\pi$
$937$	−2.39563e7	−0.891397	−0.445698	+	0.895183i	$0.647045\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.54402e7	−0.936582	−0.468291	−	0.883574i	$-0.655130\pi$
$941$	−2.54402e7	−0.936582	−0.468291	+	0.883574i	$0.655130\pi$
$942$	0	0
$943$	2.67786e6	0.0980639
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.63563e7	−0.592665	−0.296332	−	0.955085i	$-0.595764\pi$
$947$	−1.63563e7	−0.592665	−0.296332	+	0.955085i	$0.595764\pi$
$948$	0	0
$949$	2.69968e6	0.0973074
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.87396e7	−1.38173	−0.690865	−	0.722984i	$-0.742770\pi$
$953$	−3.87396e7	−1.38173	−0.690865	+	0.722984i	$0.742770\pi$
$954$	0	0
$955$	−1.31978e7	−0.468266
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.84887e7	2.05364
$960$	0	0
$961$	6.10749e6	0.213331
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.88425e6	0.0651358
$966$	0	0
$967$	−3.27238e7	−1.12538	−0.562689	−	0.826669i	$-0.690233\pi$
$967$	−3.27238e7	−1.12538	−0.562689	+	0.826669i	$0.690233\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.49297e7	0.508163	0.254082	−	0.967183i	$-0.418227\pi$
$971$	1.49297e7	0.508163	0.254082	+	0.967183i	$0.418227\pi$
$972$	0	0
$973$	6.54151e7	2.21511
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.98709e6	0.0666011	0.0333005	−	0.999445i	$-0.489398\pi$
$977$	1.98709e6	0.0666011	0.0333005	+	0.999445i	$0.489398\pi$
$978$	0	0
$979$	−8.55445e7	−2.85257
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.35652e7	1.10791	0.553956	−	0.832546i	$-0.313118\pi$
$983$	3.35652e7	1.10791	0.553956	+	0.832546i	$0.313118\pi$
$984$	0	0
$985$	6.98168e6	0.229282
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.30701e7	−0.749995
$990$	0	0
$991$	2.45527e7	0.794174	0.397087	−	0.917781i	$-0.370021\pi$
$991$	2.45527e7	0.794174	0.397087	+	0.917781i	$0.370021\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.94127e7	−0.621624
$996$	0	0
$997$	5.76681e6	0.183737	0.0918687	−	0.995771i	$-0.470716\pi$
$997$	5.76681e6	0.183737	0.0918687	+	0.995771i	$0.470716\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.6.a.o.1.3	✓	6
3.2	odd	2		864.6.a.p.1.4	yes	6
4.3	odd	2		864.6.a.p.1.3	yes	6
12.11	even	2	inner	864.6.a.o.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.6.a.o.1.3	✓	6	1.1	even	1	trivial
864.6.a.o.1.4	yes	6	12.11	even	2	inner
864.6.a.p.1.3	yes	6	4.3	odd	2
864.6.a.p.1.4	yes	6	3.2	odd	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	864.a (trivial)

\(f(q)\)	\(=\)	\(q-43.4968 q^{5} +238.200 q^{7} +O(q^{10})\)	\(q-43.4968 q^{5} +238.200 q^{7} -628.981 q^{11} -966.503 q^{13} -758.898 q^{17} +2229.53 q^{19} -1535.42 q^{23} -1233.03 q^{25} -1721.68 q^{29} +5893.78 q^{31} -10360.9 q^{35} -2878.30 q^{37} -1744.06 q^{41} +15025.3 q^{43} +27880.9 q^{47} +39932.3 q^{49} -36718.5 q^{53} +27358.7 q^{55} +11248.2 q^{59} -21070.7 q^{61} +42039.8 q^{65} -27761.0 q^{67} -8983.56 q^{71} -2793.24 q^{73} -149823. q^{77} -21609.0 q^{79} +48373.7 q^{83} +33009.7 q^{85} +136005. q^{89} -230221. q^{91} -96977.5 q^{95} -130323. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 462 q^{11} - 384 q^{13} - 1380 q^{23} + 120 q^{25} - 5598 q^{35} - 396 q^{37} + 15036 q^{47} + 34620 q^{49} + 10824 q^{59} + 5520 q^{61} + 9144 q^{71} + 61734 q^{73} - 1806 q^{83} + 160392 q^{85} - 16476 q^{95} + 125226 q^{97}+O(q^{100}) \) 6 * q - 462 * q^11 - 384 * q^13 - 1380 * q^23 + 120 * q^25 - 5598 * q^35 - 396 * q^37 + 15036 * q^47 + 34620 * q^49 + 10824 * q^59 + 5520 * q^61 + 9144 * q^71 + 61734 * q^73 - 1806 * q^83 + 160392 * q^85 - 16476 * q^95 + 125226 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more