LMFDB - Embedded newform 912.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,6,Mod(1,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("912.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	912.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:	$146.270043669$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	912.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +21.0000 q^{5} +143.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +21.0000 q^{5} +143.000 q^{7} +81.0000 q^{9} +205.000 q^{11} -78.0000 q^{13} +189.000 q^{15} -2125.00 q^{17} -361.000 q^{19} +1287.00 q^{21} -20.0000 q^{23} -2684.00 q^{25} +729.000 q^{27} -4866.00 q^{29} +1098.00 q^{31} +1845.00 q^{33} +3003.00 q^{35} -15128.0 q^{37} -702.000 q^{39} -9400.00 q^{41} -20073.0 q^{43} +1701.00 q^{45} -14105.0 q^{47} +3642.00 q^{49} -19125.0 q^{51} +26386.0 q^{53} +4305.00 q^{55} -3249.00 q^{57} +13216.0 q^{59} -2293.00 q^{61} +11583.0 q^{63} -1638.00 q^{65} -35976.0 q^{67} -180.000 q^{69} -10180.0 q^{71} +33109.0 q^{73} -24156.0 q^{75} +29315.0 q^{77} +53888.0 q^{79} +6561.00 q^{81} -75196.0 q^{83} -44625.0 q^{85} -43794.0 q^{87} +20618.0 q^{89} -11154.0 q^{91} +9882.00 q^{93} -7581.00 q^{95} -84130.0 q^{97} +16605.0 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	21.0000	0.375659	0.187830	−	0.982202i	$-0.439855\pi$
$5$	21.0000	0.375659	0.187830	+	0.982202i	$0.439855\pi$
$6$	0	0
$7$	143.000	1.10304	0.551520	−	0.834162i	$-0.314048\pi$
$7$	143.000	1.10304	0.551520	+	0.834162i	$0.314048\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	205.000	0.510825	0.255413	−	0.966832i	$-0.417789\pi$
$11$	205.000	0.510825	0.255413	+	0.966832i	$0.417789\pi$
$12$	0	0
$13$	−78.0000	−0.128008	−0.0640039	−	0.997950i	$-0.520387\pi$
$13$	−78.0000	−0.128008	−0.0640039	+	0.997950i	$0.520387\pi$
$14$	0	0
$15$	189.000	0.216887
$16$	0	0
$17$	−2125.00	−1.78335	−0.891675	−	0.452676i	$-0.850469\pi$
$17$	−2125.00	−1.78335	−0.891675	+	0.452676i	$0.850469\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	1287.00	0.636840
$22$	0	0
$23$	−20.0000	−0.00788334	−0.00394167	−	0.999992i	$-0.501255\pi$
$23$	−20.0000	−0.00788334	−0.00394167	+	0.999992i	$0.501255\pi$
$24$	0	0
$25$	−2684.00	−0.858880
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−4866.00	−1.07443	−0.537214	−	0.843446i	$-0.680523\pi$
$29$	−4866.00	−1.07443	−0.537214	+	0.843446i	$0.680523\pi$
$30$	0	0
$31$	1098.00	0.205210	0.102605	−	0.994722i	$-0.467282\pi$
$31$	1098.00	0.205210	0.102605	+	0.994722i	$0.467282\pi$
$32$	0	0
$33$	1845.00	0.294925
$34$	0	0
$35$	3003.00	0.414367
$36$	0	0
$37$	−15128.0	−1.81667	−0.908337	−	0.418238i	$-0.862648\pi$
$37$	−15128.0	−1.81667	−0.908337	+	0.418238i	$0.862648\pi$
$38$	0	0
$39$	−702.000	−0.0739053
$40$	0	0
$41$	−9400.00	−0.873310	−0.436655	−	0.899629i	$-0.643837\pi$
$41$	−9400.00	−0.873310	−0.436655	+	0.899629i	$0.643837\pi$
$42$	0	0
$43$	−20073.0	−1.65555	−0.827773	−	0.561063i	$-0.810392\pi$
$43$	−20073.0	−1.65555	−0.827773	+	0.561063i	$0.810392\pi$
$44$	0	0
$45$	1701.00	0.125220
$46$	0	0
$47$	−14105.0	−0.931383	−0.465692	−	0.884947i	$-0.654194\pi$
$47$	−14105.0	−0.931383	−0.465692	+	0.884947i	$0.654194\pi$
$48$	0	0
$49$	3642.00	0.216695
$50$	0	0
$51$	−19125.0	−1.02962
$52$	0	0
$53$	26386.0	1.29028	0.645140	−	0.764064i	$-0.276799\pi$
$53$	26386.0	1.29028	0.645140	+	0.764064i	$0.276799\pi$
$54$	0	0
$55$	4305.00	0.191896
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	13216.0	0.494277	0.247138	−	0.968980i	$-0.420510\pi$
$59$	13216.0	0.494277	0.247138	+	0.968980i	$0.420510\pi$
$60$	0	0
$61$	−2293.00	−0.0789004	−0.0394502	−	0.999222i	$-0.512561\pi$
$61$	−2293.00	−0.0789004	−0.0394502	+	0.999222i	$0.512561\pi$
$62$	0	0
$63$	11583.0	0.367680
$64$	0	0
$65$	−1638.00	−0.0480873
$66$	0	0
$67$	−35976.0	−0.979097	−0.489549	−	0.871976i	$-0.662838\pi$
$67$	−35976.0	−0.979097	−0.489549	+	0.871976i	$0.662838\pi$
$68$	0	0
$69$	−180.000	−0.00455145
$70$	0	0
$71$	−10180.0	−0.239664	−0.119832	−	0.992794i	$-0.538236\pi$
$71$	−10180.0	−0.239664	−0.119832	+	0.992794i	$0.538236\pi$
$72$	0	0
$73$	33109.0	0.727175	0.363587	−	0.931560i	$-0.381552\pi$
$73$	33109.0	0.727175	0.363587	+	0.931560i	$0.381552\pi$
$74$	0	0
$75$	−24156.0	−0.495875
$76$	0	0
$77$	29315.0	0.563460
$78$	0	0
$79$	53888.0	0.971459	0.485729	−	0.874109i	$-0.338554\pi$
$79$	53888.0	0.971459	0.485729	+	0.874109i	$0.338554\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−75196.0	−1.19812	−0.599059	−	0.800705i	$-0.704458\pi$
$83$	−75196.0	−1.19812	−0.599059	+	0.800705i	$0.704458\pi$
$84$	0	0
$85$	−44625.0	−0.669932
$86$	0	0
$87$	−43794.0	−0.620321
$88$	0	0
$89$	20618.0	0.275913	0.137956	−	0.990438i	$-0.455947\pi$
$89$	20618.0	0.275913	0.137956	+	0.990438i	$0.455947\pi$
$90$	0	0
$91$	−11154.0	−0.141198
$92$	0	0
$93$	9882.00	0.118478
$94$	0	0
$95$	−7581.00	−0.0861822
$96$	0	0
$97$	−84130.0	−0.907866	−0.453933	−	0.891036i	$-0.649979\pi$
$97$	−84130.0	−0.907866	−0.453933	+	0.891036i	$0.649979\pi$
$98$	0	0
$99$	16605.0	0.170275
$100$	0	0
$101$	163714.	1.59692	0.798459	−	0.602050i	$-0.205649\pi$
$101$	163714.	1.59692	0.798459	+	0.602050i	$0.205649\pi$
$102$	0	0
$103$	139062.	1.29156	0.645781	−	0.763523i	$-0.276532\pi$
$103$	139062.	1.29156	0.645781	+	0.763523i	$0.276532\pi$
$104$	0	0
$105$	27027.0	0.239235
$106$	0	0
$107$	−124690.	−1.05286	−0.526432	−	0.850217i	$-0.676470\pi$
$107$	−124690.	−1.05286	−0.526432	+	0.850217i	$0.676470\pi$
$108$	0	0
$109$	−11836.0	−0.0954198	−0.0477099	−	0.998861i	$-0.515192\pi$
$109$	−11836.0	−0.0954198	−0.0477099	+	0.998861i	$0.515192\pi$
$110$	0	0
$111$	−136152.	−1.04886
$112$	0	0
$113$	57674.0	0.424897	0.212449	−	0.977172i	$-0.431856\pi$
$113$	57674.0	0.424897	0.212449	+	0.977172i	$0.431856\pi$
$114$	0	0
$115$	−420.000	−0.00296145
$116$	0	0
$117$	−6318.00	−0.0426692
$118$	0	0
$119$	−303875.	−1.96711
$120$	0	0
$121$	−119026.	−0.739058
$122$	0	0
$123$	−84600.0	−0.504206
$124$	0	0
$125$	−121989.	−0.698306
$126$	0	0
$127$	−134314.	−0.738945	−0.369472	−	0.929242i	$-0.620462\pi$
$127$	−134314.	−0.738945	−0.369472	+	0.929242i	$0.620462\pi$
$128$	0	0
$129$	−180657.	−0.955830
$130$	0	0
$131$	365955.	1.86316	0.931578	−	0.363540i	$-0.118432\pi$
$131$	365955.	1.86316	0.931578	+	0.363540i	$0.118432\pi$
$132$	0	0
$133$	−51623.0	−0.253055
$134$	0	0
$135$	15309.0	0.0722957
$136$	0	0
$137$	44763.0	0.203759	0.101880	−	0.994797i	$-0.467514\pi$
$137$	44763.0	0.203759	0.101880	+	0.994797i	$0.467514\pi$
$138$	0	0
$139$	422179.	1.85336	0.926680	−	0.375852i	$-0.122650\pi$
$139$	422179.	1.85336	0.926680	+	0.375852i	$0.122650\pi$
$140$	0	0
$141$	−126945.	−0.537734
$142$	0	0
$143$	−15990.0	−0.0653896
$144$	0	0
$145$	−102186.	−0.403619
$146$	0	0
$147$	32778.0	0.125109
$148$	0	0
$149$	−41741.0	−0.154027	−0.0770136	−	0.997030i	$-0.524538\pi$
$149$	−41741.0	−0.154027	−0.0770136	+	0.997030i	$0.524538\pi$
$150$	0	0
$151$	41240.0	0.147189	0.0735947	−	0.997288i	$-0.476553\pi$
$151$	41240.0	0.147189	0.0735947	+	0.997288i	$0.476553\pi$
$152$	0	0
$153$	−172125.	−0.594450
$154$	0	0
$155$	23058.0	0.0770890
$156$	0	0
$157$	−345954.	−1.12013	−0.560066	−	0.828448i	$-0.689224\pi$
$157$	−345954.	−1.12013	−0.560066	+	0.828448i	$0.689224\pi$
$158$	0	0
$159$	237474.	0.744943
$160$	0	0
$161$	−2860.00	−0.00869564
$162$	0	0
$163$	298144.	0.878936	0.439468	−	0.898258i	$-0.355167\pi$
$163$	298144.	0.878936	0.439468	+	0.898258i	$0.355167\pi$
$164$	0	0
$165$	38745.0	0.110791
$166$	0	0
$167$	73290.0	0.203354	0.101677	−	0.994817i	$-0.467579\pi$
$167$	73290.0	0.203354	0.101677	+	0.994817i	$0.467579\pi$
$168$	0	0
$169$	−365209.	−0.983614
$170$	0	0
$171$	−29241.0	−0.0764719
$172$	0	0
$173$	282102.	0.716623	0.358312	−	0.933602i	$-0.383353\pi$
$173$	282102.	0.716623	0.358312	+	0.933602i	$0.383353\pi$
$174$	0	0
$175$	−383812.	−0.947378
$176$	0	0
$177$	118944.	0.285371
$178$	0	0
$179$	−193946.	−0.452427	−0.226213	−	0.974078i	$-0.572635\pi$
$179$	−193946.	−0.452427	−0.226213	+	0.974078i	$0.572635\pi$
$180$	0	0
$181$	−283446.	−0.643093	−0.321547	−	0.946894i	$-0.604203\pi$
$181$	−283446.	−0.643093	−0.321547	+	0.946894i	$0.604203\pi$
$182$	0	0
$183$	−20637.0	−0.0455532
$184$	0	0
$185$	−317688.	−0.682451
$186$	0	0
$187$	−435625.	−0.910980
$188$	0	0
$189$	104247.	0.212280
$190$	0	0
$191$	−50495.0	−0.100153	−0.0500766	−	0.998745i	$-0.515947\pi$
$191$	−50495.0	−0.100153	−0.0500766	+	0.998745i	$0.515947\pi$
$192$	0	0
$193$	231092.	0.446572	0.223286	−	0.974753i	$-0.428322\pi$
$193$	231092.	0.446572	0.223286	+	0.974753i	$0.428322\pi$
$194$	0	0
$195$	−14742.0	−0.0277632
$196$	0	0
$197$	−452482.	−0.830684	−0.415342	−	0.909665i	$-0.636338\pi$
$197$	−452482.	−0.830684	−0.415342	+	0.909665i	$0.636338\pi$
$198$	0	0
$199$	207199.	0.370898	0.185449	−	0.982654i	$-0.440626\pi$
$199$	207199.	0.370898	0.185449	+	0.982654i	$0.440626\pi$
$200$	0	0
$201$	−323784.	−0.565282
$202$	0	0
$203$	−695838.	−1.18514
$204$	0	0
$205$	−197400.	−0.328067
$206$	0	0
$207$	−1620.00	−0.00262778
$208$	0	0
$209$	−74005.0	−0.117191
$210$	0	0
$211$	−985948.	−1.52457	−0.762286	−	0.647240i	$-0.775923\pi$
$211$	−985948.	−1.52457	−0.762286	+	0.647240i	$0.775923\pi$
$212$	0	0
$213$	−91620.0	−0.138370
$214$	0	0
$215$	−421533.	−0.621921
$216$	0	0
$217$	157014.	0.226354
$218$	0	0
$219$	297981.	0.419835
$220$	0	0
$221$	165750.	0.228283
$222$	0	0
$223$	−177756.	−0.239366	−0.119683	−	0.992812i	$-0.538188\pi$
$223$	−177756.	−0.239366	−0.119683	+	0.992812i	$0.538188\pi$
$224$	0	0
$225$	−217404.	−0.286293
$226$	0	0
$227$	−276382.	−0.355996	−0.177998	−	0.984031i	$-0.556962\pi$
$227$	−276382.	−0.355996	−0.177998	+	0.984031i	$0.556962\pi$
$228$	0	0
$229$	986125.	1.24263	0.621317	−	0.783559i	$-0.286598\pi$
$229$	986125.	1.24263	0.621317	+	0.783559i	$0.286598\pi$
$230$	0	0
$231$	263835.	0.325314
$232$	0	0
$233$	−116691.	−0.140815	−0.0704073	−	0.997518i	$-0.522430\pi$
$233$	−116691.	−0.140815	−0.0704073	+	0.997518i	$0.522430\pi$
$234$	0	0
$235$	−296205.	−0.349883
$236$	0	0
$237$	484992.	0.560872
$238$	0	0
$239$	1.25870e6	1.42537	0.712685	−	0.701484i	$-0.247479\pi$
$239$	1.25870e6	1.42537	0.712685	+	0.701484i	$0.247479\pi$
$240$	0	0
$241$	−143492.	−0.159142	−0.0795710	−	0.996829i	$-0.525355\pi$
$241$	−143492.	−0.159142	−0.0795710	+	0.996829i	$0.525355\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	76482.0	0.0814037
$246$	0	0
$247$	28158.0	0.0293670
$248$	0	0
$249$	−676764.	−0.691734
$250$	0	0
$251$	884163.	0.885825	0.442913	−	0.896565i	$-0.353945\pi$
$251$	884163.	0.885825	0.442913	+	0.896565i	$0.353945\pi$
$252$	0	0
$253$	−4100.00	−0.00402701
$254$	0	0
$255$	−401625.	−0.386786
$256$	0	0
$257$	1.04230e6	0.984370	0.492185	−	0.870491i	$-0.336198\pi$
$257$	1.04230e6	0.984370	0.492185	+	0.870491i	$0.336198\pi$
$258$	0	0
$259$	−2.16330e6	−2.00386
$260$	0	0
$261$	−394146.	−0.358142
$262$	0	0
$263$	−998121.	−0.889803	−0.444901	−	0.895580i	$-0.646761\pi$
$263$	−998121.	−0.889803	−0.444901	+	0.895580i	$0.646761\pi$
$264$	0	0
$265$	554106.	0.484706
$266$	0	0
$267$	185562.	0.159298
$268$	0	0
$269$	1.73550e6	1.46232	0.731162	−	0.682204i	$-0.238978\pi$
$269$	1.73550e6	1.46232	0.731162	+	0.682204i	$0.238978\pi$
$270$	0	0
$271$	1.01674e6	0.840982	0.420491	−	0.907297i	$-0.361858\pi$
$271$	1.01674e6	0.840982	0.420491	+	0.907297i	$0.361858\pi$
$272$	0	0
$273$	−100386.	−0.0815204
$274$	0	0
$275$	−550220.	−0.438737
$276$	0	0
$277$	−1.37870e6	−1.07962	−0.539810	−	0.841787i	$-0.681504\pi$
$277$	−1.37870e6	−1.07962	−0.539810	+	0.841787i	$0.681504\pi$
$278$	0	0
$279$	88938.0	0.0684033
$280$	0	0
$281$	−2.09789e6	−1.58495	−0.792476	−	0.609903i	$-0.791208\pi$
$281$	−2.09789e6	−1.58495	−0.792476	+	0.609903i	$0.791208\pi$
$282$	0	0
$283$	215963.	0.160293	0.0801463	−	0.996783i	$-0.474461\pi$
$283$	215963.	0.160293	0.0801463	+	0.996783i	$0.474461\pi$
$284$	0	0
$285$	−68229.0	−0.0497573
$286$	0	0
$287$	−1.34420e6	−0.963295
$288$	0	0
$289$	3.09577e6	2.18034
$290$	0	0
$291$	−757170.	−0.524156
$292$	0	0
$293$	−2.47872e6	−1.68678	−0.843389	−	0.537304i	$-0.819443\pi$
$293$	−2.47872e6	−1.68678	−0.843389	+	0.537304i	$0.819443\pi$
$294$	0	0
$295$	277536.	0.185680
$296$	0	0
$297$	149445.	0.0983083
$298$	0	0
$299$	1560.00	0.00100913
$300$	0	0
$301$	−2.87044e6	−1.82613
$302$	0	0
$303$	1.47343e6	0.921981
$304$	0	0
$305$	−48153.0	−0.0296397
$306$	0	0
$307$	−495420.	−0.300004	−0.150002	−	0.988686i	$-0.547928\pi$
$307$	−495420.	−0.300004	−0.150002	+	0.988686i	$0.547928\pi$
$308$	0	0
$309$	1.25156e6	0.745684
$310$	0	0
$311$	−2.15964e6	−1.26614	−0.633070	−	0.774095i	$-0.718205\pi$
$311$	−2.15964e6	−1.26614	−0.633070	+	0.774095i	$0.718205\pi$
$312$	0	0
$313$	−1.34757e6	−0.777482	−0.388741	−	0.921347i	$-0.627090\pi$
$313$	−1.34757e6	−0.777482	−0.388741	+	0.921347i	$0.627090\pi$
$314$	0	0
$315$	243243.	0.138122
$316$	0	0
$317$	−1.01166e6	−0.565440	−0.282720	−	0.959203i	$-0.591237\pi$
$317$	−1.01166e6	−0.565440	−0.282720	+	0.959203i	$0.591237\pi$
$318$	0	0
$319$	−997530.	−0.548844
$320$	0	0
$321$	−1.12221e6	−0.607871
$322$	0	0
$323$	767125.	0.409129
$324$	0	0
$325$	209352.	0.109943
$326$	0	0
$327$	−106524.	−0.0550907
$328$	0	0
$329$	−2.01702e6	−1.02735
$330$	0	0
$331$	1.57062e6	0.787954	0.393977	−	0.919120i	$-0.371099\pi$
$331$	1.57062e6	0.787954	0.393977	+	0.919120i	$0.371099\pi$
$332$	0	0
$333$	−1.22537e6	−0.605558
$334$	0	0
$335$	−755496.	−0.367807
$336$	0	0
$337$	2.16228e6	1.03714	0.518570	−	0.855035i	$-0.326464\pi$
$337$	2.16228e6	1.03714	0.518570	+	0.855035i	$0.326464\pi$
$338$	0	0
$339$	519066.	0.245315
$340$	0	0
$341$	225090.	0.104826
$342$	0	0
$343$	−1.88259e6	−0.864016
$344$	0	0
$345$	−3780.00	−0.00170980
$346$	0	0
$347$	−4.16873e6	−1.85858	−0.929288	−	0.369356i	$-0.879578\pi$
$347$	−4.16873e6	−1.85858	−0.929288	+	0.369356i	$0.879578\pi$
$348$	0	0
$349$	722845.	0.317674	0.158837	−	0.987305i	$-0.449226\pi$
$349$	722845.	0.317674	0.158837	+	0.987305i	$0.449226\pi$
$350$	0	0
$351$	−56862.0	−0.0246351
$352$	0	0
$353$	1.45102e6	0.619780	0.309890	−	0.950772i	$-0.399708\pi$
$353$	1.45102e6	0.619780	0.309890	+	0.950772i	$0.399708\pi$
$354$	0	0
$355$	−213780.	−0.0900319
$356$	0	0
$357$	−2.73488e6	−1.13571
$358$	0	0
$359$	−517179.	−0.211790	−0.105895	−	0.994377i	$-0.533771\pi$
$359$	−517179.	−0.211790	−0.105895	+	0.994377i	$0.533771\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−1.07123e6	−0.426695
$364$	0	0
$365$	695289.	0.273170
$366$	0	0
$367$	3.59392e6	1.39285	0.696423	−	0.717631i	$-0.254774\pi$
$367$	3.59392e6	1.39285	0.696423	+	0.717631i	$0.254774\pi$
$368$	0	0
$369$	−761400.	−0.291103
$370$	0	0
$371$	3.77320e6	1.42323
$372$	0	0
$373$	−1.77578e6	−0.660870	−0.330435	−	0.943829i	$-0.607195\pi$
$373$	−1.77578e6	−0.660870	−0.330435	+	0.943829i	$0.607195\pi$
$374$	0	0
$375$	−1.09790e6	−0.403167
$376$	0	0
$377$	379548.	0.137535
$378$	0	0
$379$	−2.48464e6	−0.888516	−0.444258	−	0.895899i	$-0.646533\pi$
$379$	−2.48464e6	−0.888516	−0.444258	+	0.895899i	$0.646533\pi$
$380$	0	0
$381$	−1.20883e6	−0.426630
$382$	0	0
$383$	−1.70987e6	−0.595614	−0.297807	−	0.954626i	$-0.596255\pi$
$383$	−1.70987e6	−0.595614	−0.297807	+	0.954626i	$0.596255\pi$
$384$	0	0
$385$	615615.	0.211669
$386$	0	0
$387$	−1.62591e6	−0.551849
$388$	0	0
$389$	−244715.	−0.0819949	−0.0409974	−	0.999159i	$-0.513054\pi$
$389$	−244715.	−0.0819949	−0.0409974	+	0.999159i	$0.513054\pi$
$390$	0	0
$391$	42500.0	0.0140588
$392$	0	0
$393$	3.29359e6	1.07569
$394$	0	0
$395$	1.13165e6	0.364938
$396$	0	0
$397$	−4.11195e6	−1.30940	−0.654698	−	0.755890i	$-0.727204\pi$
$397$	−4.11195e6	−1.30940	−0.654698	+	0.755890i	$0.727204\pi$
$398$	0	0
$399$	−464607.	−0.146101
$400$	0	0
$401$	−1.87955e6	−0.583704	−0.291852	−	0.956464i	$-0.594271\pi$
$401$	−1.87955e6	−0.583704	−0.291852	+	0.956464i	$0.594271\pi$
$402$	0	0
$403$	−85644.0	−0.0262684
$404$	0	0
$405$	137781.	0.0417399
$406$	0	0
$407$	−3.10124e6	−0.928003
$408$	0	0
$409$	−2.68258e6	−0.792948	−0.396474	−	0.918046i	$-0.629766\pi$
$409$	−2.68258e6	−0.792948	−0.396474	+	0.918046i	$0.629766\pi$
$410$	0	0
$411$	402867.	0.117641
$412$	0	0
$413$	1.88989e6	0.545206
$414$	0	0
$415$	−1.57912e6	−0.450084
$416$	0	0
$417$	3.79961e6	1.07004
$418$	0	0
$419$	5.70599e6	1.58780	0.793900	−	0.608048i	$-0.208047\pi$
$419$	5.70599e6	1.58780	0.793900	+	0.608048i	$0.208047\pi$
$420$	0	0
$421$	−4.24578e6	−1.16749	−0.583744	−	0.811938i	$-0.698413\pi$
$421$	−4.24578e6	−1.16749	−0.583744	+	0.811938i	$0.698413\pi$
$422$	0	0
$423$	−1.14250e6	−0.310461
$424$	0	0
$425$	5.70350e6	1.53168
$426$	0	0
$427$	−327899.	−0.0870303
$428$	0	0
$429$	−143910.	−0.0377527
$430$	0	0
$431$	1.28341e6	0.332793	0.166396	−	0.986059i	$-0.446787\pi$
$431$	1.28341e6	0.332793	0.166396	+	0.986059i	$0.446787\pi$
$432$	0	0
$433$	−6.39182e6	−1.63834	−0.819171	−	0.573549i	$-0.805566\pi$
$433$	−6.39182e6	−1.63834	−0.819171	+	0.573549i	$0.805566\pi$
$434$	0	0
$435$	−919674.	−0.233029
$436$	0	0
$437$	7220.00	0.00180856
$438$	0	0
$439$	−1.27464e6	−0.315664	−0.157832	−	0.987466i	$-0.550451\pi$
$439$	−1.27464e6	−0.315664	−0.157832	+	0.987466i	$0.550451\pi$
$440$	0	0
$441$	295002.	0.0722318
$442$	0	0
$443$	−2.35546e6	−0.570253	−0.285126	−	0.958490i	$-0.592036\pi$
$443$	−2.35546e6	−0.570253	−0.285126	+	0.958490i	$0.592036\pi$
$444$	0	0
$445$	432978.	0.103649
$446$	0	0
$447$	−375669.	−0.0889276
$448$	0	0
$449$	7.99412e6	1.87135	0.935675	−	0.352863i	$-0.114792\pi$
$449$	7.99412e6	1.87135	0.935675	+	0.352863i	$0.114792\pi$
$450$	0	0
$451$	−1.92700e6	−0.446108
$452$	0	0
$453$	371160.	0.0849798
$454$	0	0
$455$	−234234.	−0.0530422
$456$	0	0
$457$	−3.03844e6	−0.680550	−0.340275	−	0.940326i	$-0.610520\pi$
$457$	−3.03844e6	−0.680550	−0.340275	+	0.940326i	$0.610520\pi$
$458$	0	0
$459$	−1.54912e6	−0.343206
$460$	0	0
$461$	−5.98744e6	−1.31217	−0.656084	−	0.754688i	$-0.727788\pi$
$461$	−5.98744e6	−1.31217	−0.656084	+	0.754688i	$0.727788\pi$
$462$	0	0
$463$	−8.05385e6	−1.74603	−0.873014	−	0.487695i	$-0.837838\pi$
$463$	−8.05385e6	−1.74603	−0.873014	+	0.487695i	$0.837838\pi$
$464$	0	0
$465$	207522.	0.0445074
$466$	0	0
$467$	8.29332e6	1.75969	0.879845	−	0.475261i	$-0.157646\pi$
$467$	8.29332e6	1.75969	0.879845	+	0.475261i	$0.157646\pi$
$468$	0	0
$469$	−5.14457e6	−1.07998
$470$	0	0
$471$	−3.11359e6	−0.646709
$472$	0	0
$473$	−4.11496e6	−0.845694
$474$	0	0
$475$	968924.	0.197041
$476$	0	0
$477$	2.13727e6	0.430093
$478$	0	0
$479$	4.58472e6	0.913007	0.456503	−	0.889722i	$-0.349102\pi$
$479$	4.58472e6	0.913007	0.456503	+	0.889722i	$0.349102\pi$
$480$	0	0
$481$	1.17998e6	0.232548
$482$	0	0
$483$	−25740.0	−0.00502043
$484$	0	0
$485$	−1.76673e6	−0.341048
$486$	0	0
$487$	−55304.0	−0.0105666	−0.00528329	−	0.999986i	$-0.501682\pi$
$487$	−55304.0	−0.0105666	−0.00528329	+	0.999986i	$0.501682\pi$
$488$	0	0
$489$	2.68330e6	0.507454
$490$	0	0
$491$	6.88932e6	1.28965	0.644826	−	0.764329i	$-0.276930\pi$
$491$	6.88932e6	1.28965	0.644826	+	0.764329i	$0.276930\pi$
$492$	0	0
$493$	1.03402e7	1.91608
$494$	0	0
$495$	348705.	0.0639654
$496$	0	0
$497$	−1.45574e6	−0.264358
$498$	0	0
$499$	−5.61676e6	−1.00980	−0.504899	−	0.863179i	$-0.668470\pi$
$499$	−5.61676e6	−1.00980	−0.504899	+	0.863179i	$0.668470\pi$
$500$	0	0
$501$	659610.	0.117407
$502$	0	0
$503$	5.16131e6	0.909578	0.454789	−	0.890599i	$-0.349715\pi$
$503$	5.16131e6	0.909578	0.454789	+	0.890599i	$0.349715\pi$
$504$	0	0
$505$	3.43799e6	0.599897
$506$	0	0
$507$	−3.28688e6	−0.567890
$508$	0	0
$509$	−1.81839e6	−0.311094	−0.155547	−	0.987828i	$-0.549714\pi$
$509$	−1.81839e6	−0.311094	−0.155547	+	0.987828i	$0.549714\pi$
$510$	0	0
$511$	4.73459e6	0.802102
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	2.92030e6	0.485188
$516$	0	0
$517$	−2.89152e6	−0.475774
$518$	0	0
$519$	2.53892e6	0.413743
$520$	0	0
$521$	1.10957e7	1.79085	0.895424	−	0.445215i	$-0.146873\pi$
$521$	1.10957e7	1.79085	0.895424	+	0.445215i	$0.146873\pi$
$522$	0	0
$523$	7.51297e6	1.20104	0.600520	−	0.799610i	$-0.294960\pi$
$523$	7.51297e6	1.20104	0.600520	+	0.799610i	$0.294960\pi$
$524$	0	0
$525$	−3.45431e6	−0.546969
$526$	0	0
$527$	−2.33325e6	−0.365961
$528$	0	0
$529$	−6.43594e6	−0.999938
$530$	0	0
$531$	1.07050e6	0.164759
$532$	0	0
$533$	733200.	0.111790
$534$	0	0
$535$	−2.61849e6	−0.395518
$536$	0	0
$537$	−1.74551e6	−0.261209
$538$	0	0
$539$	746610.	0.110693
$540$	0	0
$541$	3.15622e6	0.463632	0.231816	−	0.972760i	$-0.425533\pi$
$541$	3.15622e6	0.463632	0.231816	+	0.972760i	$0.425533\pi$
$542$	0	0
$543$	−2.55101e6	−0.371290
$544$	0	0
$545$	−248556.	−0.0358454
$546$	0	0
$547$	1.17063e7	1.67283	0.836417	−	0.548094i	$-0.184647\pi$
$547$	1.17063e7	1.67283	0.836417	+	0.548094i	$0.184647\pi$
$548$	0	0
$549$	−185733.	−0.0263001
$550$	0	0
$551$	1.75663e6	0.246491
$552$	0	0
$553$	7.70598e6	1.07156
$554$	0	0
$555$	−2.85919e6	−0.394013
$556$	0	0
$557$	5.56111e6	0.759493	0.379746	−	0.925091i	$-0.376011\pi$
$557$	5.56111e6	0.759493	0.379746	+	0.925091i	$0.376011\pi$
$558$	0	0
$559$	1.56569e6	0.211923
$560$	0	0
$561$	−3.92062e6	−0.525954
$562$	0	0
$563$	1.17669e7	1.56456	0.782280	−	0.622928i	$-0.214057\pi$
$563$	1.17669e7	1.56456	0.782280	+	0.622928i	$0.214057\pi$
$564$	0	0
$565$	1.21115e6	0.159617
$566$	0	0
$567$	938223.	0.122560
$568$	0	0
$569$	6.49649e6	0.841198	0.420599	−	0.907247i	$-0.361820\pi$
$569$	6.49649e6	0.841198	0.420599	+	0.907247i	$0.361820\pi$
$570$	0	0
$571$	7.15432e6	0.918286	0.459143	−	0.888362i	$-0.348157\pi$
$571$	7.15432e6	0.918286	0.459143	+	0.888362i	$0.348157\pi$
$572$	0	0
$573$	−454455.	−0.0578235
$574$	0	0
$575$	53680.0	0.00677085
$576$	0	0
$577$	−8.50781e6	−1.06385	−0.531923	−	0.846793i	$-0.678530\pi$
$577$	−8.50781e6	−1.06385	−0.531923	+	0.846793i	$0.678530\pi$
$578$	0	0
$579$	2.07983e6	0.257829
$580$	0	0
$581$	−1.07530e7	−1.32157
$582$	0	0
$583$	5.40913e6	0.659107
$584$	0	0
$585$	−132678.	−0.0160291
$586$	0	0
$587$	854707.	0.102382	0.0511908	−	0.998689i	$-0.483698\pi$
$587$	854707.	0.102382	0.0511908	+	0.998689i	$0.483698\pi$
$588$	0	0
$589$	−396378.	−0.0470784
$590$	0	0
$591$	−4.07234e6	−0.479596
$592$	0	0
$593$	−5.55213e6	−0.648370	−0.324185	−	0.945994i	$-0.605090\pi$
$593$	−5.55213e6	−0.648370	−0.324185	+	0.945994i	$0.605090\pi$
$594$	0	0
$595$	−6.38138e6	−0.738962
$596$	0	0
$597$	1.86479e6	0.214138
$598$	0	0
$599$	1.30669e6	0.148801	0.0744003	−	0.997228i	$-0.476296\pi$
$599$	1.30669e6	0.148801	0.0744003	+	0.997228i	$0.476296\pi$
$600$	0	0
$601$	−7.94858e6	−0.897643	−0.448821	−	0.893621i	$-0.648156\pi$
$601$	−7.94858e6	−0.897643	−0.448821	+	0.893621i	$0.648156\pi$
$602$	0	0
$603$	−2.91406e6	−0.326366
$604$	0	0
$605$	−2.49955e6	−0.277634
$606$	0	0
$607$	−2.83746e6	−0.312578	−0.156289	−	0.987711i	$-0.549953\pi$
$607$	−2.83746e6	−0.312578	−0.156289	+	0.987711i	$0.549953\pi$
$608$	0	0
$609$	−6.26254e6	−0.684238
$610$	0	0
$611$	1.10019e6	0.119224
$612$	0	0
$613$	1.49674e7	1.60877	0.804387	−	0.594106i	$-0.202494\pi$
$613$	1.49674e7	1.60877	0.804387	+	0.594106i	$0.202494\pi$
$614$	0	0
$615$	−1.77660e6	−0.189410
$616$	0	0
$617$	−3.59751e6	−0.380443	−0.190221	−	0.981741i	$-0.560921\pi$
$617$	−3.59751e6	−0.380443	−0.190221	+	0.981741i	$0.560921\pi$
$618$	0	0
$619$	−6.74758e6	−0.707818	−0.353909	−	0.935280i	$-0.615148\pi$
$619$	−6.74758e6	−0.707818	−0.353909	+	0.935280i	$0.615148\pi$
$620$	0	0
$621$	−14580.0	−0.00151715
$622$	0	0
$623$	2.94837e6	0.304342
$624$	0	0
$625$	5.82573e6	0.596555
$626$	0	0
$627$	−666045.	−0.0676604
$628$	0	0
$629$	3.21470e7	3.23977
$630$	0	0
$631$	−1.36044e7	−1.36021	−0.680103	−	0.733117i	$-0.738065\pi$
$631$	−1.36044e7	−1.36021	−0.680103	+	0.733117i	$0.738065\pi$
$632$	0	0
$633$	−8.87353e6	−0.880212
$634$	0	0
$635$	−2.82059e6	−0.277592
$636$	0	0
$637$	−284076.	−0.0277387
$638$	0	0
$639$	−824580.	−0.0798878
$640$	0	0
$641$	1.91221e7	1.83819	0.919097	−	0.394030i	$-0.128920\pi$
$641$	1.91221e7	1.83819	0.919097	+	0.394030i	$0.128920\pi$
$642$	0	0
$643$	−8.38738e6	−0.800016	−0.400008	−	0.916512i	$-0.630993\pi$
$643$	−8.38738e6	−0.800016	−0.400008	+	0.916512i	$0.630993\pi$
$644$	0	0
$645$	−3.79380e6	−0.359066
$646$	0	0
$647$	−7.90221e6	−0.742143	−0.371072	−	0.928604i	$-0.621010\pi$
$647$	−7.90221e6	−0.742143	−0.371072	+	0.928604i	$0.621010\pi$
$648$	0	0
$649$	2.70928e6	0.252489
$650$	0	0
$651$	1.41313e6	0.130686
$652$	0	0
$653$	−279213.	−0.0256243	−0.0128122	−	0.999918i	$-0.504078\pi$
$653$	−279213.	−0.0256243	−0.0128122	+	0.999918i	$0.504078\pi$
$654$	0	0
$655$	7.68506e6	0.699912
$656$	0	0
$657$	2.68183e6	0.242392
$658$	0	0
$659$	6.82671e6	0.612348	0.306174	−	0.951976i	$-0.400951\pi$
$659$	6.82671e6	0.612348	0.306174	+	0.951976i	$0.400951\pi$
$660$	0	0
$661$	1.85059e7	1.64742	0.823712	−	0.567008i	$-0.191899\pi$
$661$	1.85059e7	1.64742	0.823712	+	0.567008i	$0.191899\pi$
$662$	0	0
$663$	1.49175e6	0.131799
$664$	0	0
$665$	−1.08408e6	−0.0950623
$666$	0	0
$667$	97320.0	0.00847008
$668$	0	0
$669$	−1.59980e6	−0.138198
$670$	0	0
$671$	−470065.	−0.0403043
$672$	0	0
$673$	1.25059e7	1.06433	0.532166	−	0.846640i	$-0.321378\pi$
$673$	1.25059e7	1.06433	0.532166	+	0.846640i	$0.321378\pi$
$674$	0	0
$675$	−1.95664e6	−0.165292
$676$	0	0
$677$	5.47029e6	0.458710	0.229355	−	0.973343i	$-0.426338\pi$
$677$	5.47029e6	0.458710	0.229355	+	0.973343i	$0.426338\pi$
$678$	0	0
$679$	−1.20306e7	−1.00141
$680$	0	0
$681$	−2.48744e6	−0.205534
$682$	0	0
$683$	−1.02415e7	−0.840061	−0.420031	−	0.907510i	$-0.637981\pi$
$683$	−1.02415e7	−0.840061	−0.420031	+	0.907510i	$0.637981\pi$
$684$	0	0
$685$	940023.	0.0765442
$686$	0	0
$687$	8.87512e6	0.717435
$688$	0	0
$689$	−2.05811e6	−0.165166
$690$	0	0
$691$	−1.84945e6	−0.147349	−0.0736744	−	0.997282i	$-0.523473\pi$
$691$	−1.84945e6	−0.147349	−0.0736744	+	0.997282i	$0.523473\pi$
$692$	0	0
$693$	2.37452e6	0.187820
$694$	0	0
$695$	8.86576e6	0.696232
$696$	0	0
$697$	1.99750e7	1.55742
$698$	0	0
$699$	−1.05022e6	−0.0812993
$700$	0	0
$701$	1.65813e7	1.27445	0.637226	−	0.770677i	$-0.280082\pi$
$701$	1.65813e7	1.27445	0.637226	+	0.770677i	$0.280082\pi$
$702$	0	0
$703$	5.46121e6	0.416774
$704$	0	0
$705$	−2.66584e6	−0.202005
$706$	0	0
$707$	2.34111e7	1.76146
$708$	0	0
$709$	1.10501e7	0.825562	0.412781	−	0.910830i	$-0.364558\pi$
$709$	1.10501e7	0.825562	0.412781	+	0.910830i	$0.364558\pi$
$710$	0	0
$711$	4.36493e6	0.323820
$712$	0	0
$713$	−21960.0	−0.00161774
$714$	0	0
$715$	−335790.	−0.0245642
$716$	0	0
$717$	1.13283e7	0.822938
$718$	0	0
$719$	−1.02255e7	−0.737669	−0.368835	−	0.929495i	$-0.620243\pi$
$719$	−1.02255e7	−0.737669	−0.368835	+	0.929495i	$0.620243\pi$
$720$	0	0
$721$	1.98859e7	1.42464
$722$	0	0
$723$	−1.29143e6	−0.0918807
$724$	0	0
$725$	1.30603e7	0.922804
$726$	0	0
$727$	−2.61985e7	−1.83840	−0.919202	−	0.393787i	$-0.871165\pi$
$727$	−2.61985e7	−1.83840	−0.919202	+	0.393787i	$0.871165\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	4.26551e7	2.95242
$732$	0	0
$733$	−2.56029e7	−1.76007	−0.880033	−	0.474913i	$-0.842480\pi$
$733$	−2.56029e7	−1.76007	−0.880033	+	0.474913i	$0.842480\pi$
$734$	0	0
$735$	688338.	0.0469984
$736$	0	0
$737$	−7.37508e6	−0.500147
$738$	0	0
$739$	2.10847e7	1.42022	0.710112	−	0.704089i	$-0.248644\pi$
$739$	2.10847e7	1.42022	0.710112	+	0.704089i	$0.248644\pi$
$740$	0	0
$741$	253422.	0.0169550
$742$	0	0
$743$	−8.71807e6	−0.579360	−0.289680	−	0.957124i	$-0.593549\pi$
$743$	−8.71807e6	−0.579360	−0.289680	+	0.957124i	$0.593549\pi$
$744$	0	0
$745$	−876561.	−0.0578617
$746$	0	0
$747$	−6.09088e6	−0.399373
$748$	0	0
$749$	−1.78307e7	−1.16135
$750$	0	0
$751$	1.26501e7	0.818452	0.409226	−	0.912433i	$-0.365799\pi$
$751$	1.26501e7	0.818452	0.409226	+	0.912433i	$0.365799\pi$
$752$	0	0
$753$	7.95747e6	0.511431
$754$	0	0
$755$	866040.	0.0552931
$756$	0	0
$757$	−5.69653e6	−0.361302	−0.180651	−	0.983547i	$-0.557821\pi$
$757$	−5.69653e6	−0.361302	−0.180651	+	0.983547i	$0.557821\pi$
$758$	0	0
$759$	−36900.0	−0.00232499
$760$	0	0
$761$	−2.51658e7	−1.57525	−0.787625	−	0.616155i	$-0.788690\pi$
$761$	−2.51658e7	−1.57525	−0.787625	+	0.616155i	$0.788690\pi$
$762$	0	0
$763$	−1.69255e6	−0.105252
$764$	0	0
$765$	−3.61462e6	−0.223311
$766$	0	0
$767$	−1.03085e6	−0.0632712
$768$	0	0
$769$	−2.35673e7	−1.43713	−0.718563	−	0.695462i	$-0.755200\pi$
$769$	−2.35673e7	−1.43713	−0.718563	+	0.695462i	$0.755200\pi$
$770$	0	0
$771$	9.38066e6	0.568326
$772$	0	0
$773$	1.61071e7	0.969548	0.484774	−	0.874639i	$-0.338902\pi$
$773$	1.61071e7	0.969548	0.484774	+	0.874639i	$0.338902\pi$
$774$	0	0
$775$	−2.94703e6	−0.176251
$776$	0	0
$777$	−1.94697e7	−1.15693
$778$	0	0
$779$	3.39340e6	0.200351
$780$	0	0
$781$	−2.08690e6	−0.122426
$782$	0	0
$783$	−3.54731e6	−0.206774
$784$	0	0
$785$	−7.26503e6	−0.420788
$786$	0	0
$787$	8.09622e6	0.465956	0.232978	−	0.972482i	$-0.425153\pi$
$787$	8.09622e6	0.465956	0.232978	+	0.972482i	$0.425153\pi$
$788$	0	0
$789$	−8.98309e6	−0.513728
$790$	0	0
$791$	8.24738e6	0.468678
$792$	0	0
$793$	178854.	0.0100999
$794$	0	0
$795$	4.98695e6	0.279845
$796$	0	0
$797$	2.23729e7	1.24760	0.623800	−	0.781584i	$-0.285588\pi$
$797$	2.23729e7	1.24760	0.623800	+	0.781584i	$0.285588\pi$
$798$	0	0
$799$	2.99731e7	1.66098
$800$	0	0
$801$	1.67006e6	0.0919709
$802$	0	0
$803$	6.78735e6	0.371459
$804$	0	0
$805$	−60060.0	−0.00326660
$806$	0	0
$807$	1.56195e7	0.844273
$808$	0	0
$809$	915669.	0.0491889	0.0245945	−	0.999698i	$-0.492171\pi$
$809$	915669.	0.0491889	0.0245945	+	0.999698i	$0.492171\pi$
$810$	0	0
$811$	3.00496e7	1.60430	0.802151	−	0.597121i	$-0.203689\pi$
$811$	3.00496e7	1.60430	0.802151	+	0.597121i	$0.203689\pi$
$812$	0	0
$813$	9.15066e6	0.485541
$814$	0	0
$815$	6.26102e6	0.330180
$816$	0	0
$817$	7.24635e6	0.379808
$818$	0	0
$819$	−903474.	−0.0470658
$820$	0	0
$821$	3.25959e6	0.168774	0.0843868	−	0.996433i	$-0.473107\pi$
$821$	3.25959e6	0.168774	0.0843868	+	0.996433i	$0.473107\pi$
$822$	0	0
$823$	−2.63514e7	−1.35614	−0.678068	−	0.734999i	$-0.737183\pi$
$823$	−2.63514e7	−1.35614	−0.678068	+	0.734999i	$0.737183\pi$
$824$	0	0
$825$	−4.95198e6	−0.253305
$826$	0	0
$827$	−3.88895e6	−0.197728	−0.0988640	−	0.995101i	$-0.531521\pi$
$827$	−3.88895e6	−0.197728	−0.0988640	+	0.995101i	$0.531521\pi$
$828$	0	0
$829$	−2.49909e7	−1.26298	−0.631488	−	0.775385i	$-0.717556\pi$
$829$	−2.49909e7	−1.26298	−0.631488	+	0.775385i	$0.717556\pi$
$830$	0	0
$831$	−1.24083e7	−0.623319
$832$	0	0
$833$	−7.73925e6	−0.386444
$834$	0	0
$835$	1.53909e6	0.0763920
$836$	0	0
$837$	800442.	0.0394926
$838$	0	0
$839$	2.77733e6	0.136214	0.0681072	−	0.997678i	$-0.478304\pi$
$839$	2.77733e6	0.136214	0.0681072	+	0.997678i	$0.478304\pi$
$840$	0	0
$841$	3.16681e6	0.154394
$842$	0	0
$843$	−1.88810e7	−0.915072
$844$	0	0
$845$	−7.66939e6	−0.369504
$846$	0	0
$847$	−1.70207e7	−0.815210
$848$	0	0
$849$	1.94367e6	0.0925449
$850$	0	0
$851$	302560.	0.0143215
$852$	0	0
$853$	1.63379e6	0.0768820	0.0384410	−	0.999261i	$-0.487761\pi$
$853$	1.63379e6	0.0768820	0.0384410	+	0.999261i	$0.487761\pi$
$854$	0	0
$855$	−614061.	−0.0287274
$856$	0	0
$857$	2.45037e7	1.13967	0.569836	−	0.821758i	$-0.307007\pi$
$857$	2.45037e7	1.13967	0.569836	+	0.821758i	$0.307007\pi$
$858$	0	0
$859$	−1.45576e7	−0.673143	−0.336572	−	0.941658i	$-0.609267\pi$
$859$	−1.45576e7	−0.673143	−0.336572	+	0.941658i	$0.609267\pi$
$860$	0	0
$861$	−1.20978e7	−0.556158
$862$	0	0
$863$	−1.76706e7	−0.807651	−0.403826	−	0.914836i	$-0.632320\pi$
$863$	−1.76706e7	−0.807651	−0.403826	+	0.914836i	$0.632320\pi$
$864$	0	0
$865$	5.92414e6	0.269206
$866$	0	0
$867$	2.78619e7	1.25882
$868$	0	0
$869$	1.10470e7	0.496245
$870$	0	0
$871$	2.80613e6	0.125332
$872$	0	0
$873$	−6.81453e6	−0.302622
$874$	0	0
$875$	−1.74444e7	−0.770259
$876$	0	0
$877$	−4.46464e7	−1.96014	−0.980070	−	0.198650i	$-0.936344\pi$
$877$	−4.46464e7	−1.96014	−0.980070	+	0.198650i	$0.936344\pi$
$878$	0	0
$879$	−2.23084e7	−0.973861
$880$	0	0
$881$	−1.97316e7	−0.856490	−0.428245	−	0.903663i	$-0.640868\pi$
$881$	−1.97316e7	−0.856490	−0.428245	+	0.903663i	$0.640868\pi$
$882$	0	0
$883$	2.12686e7	0.917989	0.458994	−	0.888439i	$-0.348210\pi$
$883$	2.12686e7	0.917989	0.458994	+	0.888439i	$0.348210\pi$
$884$	0	0
$885$	2.49782e6	0.107202
$886$	0	0
$887$	3.40623e7	1.45367	0.726835	−	0.686813i	$-0.240991\pi$
$887$	3.40623e7	1.45367	0.726835	+	0.686813i	$0.240991\pi$
$888$	0	0
$889$	−1.92069e7	−0.815085
$890$	0	0
$891$	1.34500e6	0.0567583
$892$	0	0
$893$	5.09190e6	0.213674
$894$	0	0
$895$	−4.07287e6	−0.169958
$896$	0	0
$897$	14040.0	0.000582621 0
$898$	0	0
$899$	−5.34287e6	−0.220483
$900$	0	0
$901$	−5.60702e7	−2.30102
$902$	0	0
$903$	−2.58340e7	−1.05432
$904$	0	0
$905$	−5.95237e6	−0.241584
$906$	0	0
$907$	1.59880e7	0.645323	0.322662	−	0.946514i	$-0.395422\pi$
$907$	1.59880e7	0.645323	0.322662	+	0.946514i	$0.395422\pi$
$908$	0	0
$909$	1.32608e7	0.532306
$910$	0	0
$911$	−2.70045e7	−1.07805	−0.539026	−	0.842289i	$-0.681208\pi$
$911$	−2.70045e7	−1.07805	−0.539026	+	0.842289i	$0.681208\pi$
$912$	0	0
$913$	−1.54152e7	−0.612029
$914$	0	0
$915$	−433377.	−0.0171125
$916$	0	0
$917$	5.23316e7	2.05514
$918$	0	0
$919$	6.71840e6	0.262408	0.131204	−	0.991355i	$-0.458116\pi$
$919$	6.71840e6	0.262408	0.131204	+	0.991355i	$0.458116\pi$
$920$	0	0
$921$	−4.45878e6	−0.173208
$922$	0	0
$923$	794040.	0.0306788
$924$	0	0
$925$	4.06036e7	1.56031
$926$	0	0
$927$	1.12640e7	0.430521
$928$	0	0
$929$	−1.29929e7	−0.493933	−0.246967	−	0.969024i	$-0.579434\pi$
$929$	−1.29929e7	−0.493933	−0.246967	+	0.969024i	$0.579434\pi$
$930$	0	0
$931$	−1.31476e6	−0.0497133
$932$	0	0
$933$	−1.94368e7	−0.731006
$934$	0	0
$935$	−9.14812e6	−0.342218
$936$	0	0
$937$	2.87830e6	0.107099	0.0535497	−	0.998565i	$-0.482946\pi$
$937$	2.87830e6	0.107099	0.0535497	+	0.998565i	$0.482946\pi$
$938$	0	0
$939$	−1.21281e7	−0.448880
$940$	0	0
$941$	−1.11189e7	−0.409345	−0.204672	−	0.978831i	$-0.565613\pi$
$941$	−1.11189e7	−0.409345	−0.204672	+	0.978831i	$0.565613\pi$
$942$	0	0
$943$	188000.	0.00688460
$944$	0	0
$945$	2.18919e6	0.0797450
$946$	0	0
$947$	1.59967e7	0.579638	0.289819	−	0.957081i	$-0.406405\pi$
$947$	1.59967e7	0.579638	0.289819	+	0.957081i	$0.406405\pi$
$948$	0	0
$949$	−2.58250e6	−0.0930840
$950$	0	0
$951$	−9.10494e6	−0.326457
$952$	0	0
$953$	3.80244e7	1.35622	0.678110	−	0.734960i	$-0.262799\pi$
$953$	3.80244e7	1.35622	0.678110	+	0.734960i	$0.262799\pi$
$954$	0	0
$955$	−1.06040e6	−0.0376235
$956$	0	0
$957$	−8.97777e6	−0.316875
$958$	0	0
$959$	6.40111e6	0.224755
$960$	0	0
$961$	−2.74235e7	−0.957889
$962$	0	0
$963$	−1.00999e7	−0.350955
$964$	0	0
$965$	4.85293e6	0.167759
$966$	0	0
$967$	−2.60950e6	−0.0897409	−0.0448705	−	0.998993i	$-0.514288\pi$
$967$	−2.60950e6	−0.0897409	−0.0448705	+	0.998993i	$0.514288\pi$
$968$	0	0
$969$	6.90413e6	0.236211
$970$	0	0
$971$	−1.40618e7	−0.478623	−0.239312	−	0.970943i	$-0.576922\pi$
$971$	−1.40618e7	−0.478623	−0.239312	+	0.970943i	$0.576922\pi$
$972$	0	0
$973$	6.03716e7	2.04433
$974$	0	0
$975$	1.88417e6	0.0634758
$976$	0	0
$977$	4.66346e7	1.56305	0.781523	−	0.623876i	$-0.214443\pi$
$977$	4.66346e7	1.56305	0.781523	+	0.623876i	$0.214443\pi$
$978$	0	0
$979$	4.22669e6	0.140943
$980$	0	0
$981$	−958716.	−0.0318066
$982$	0	0
$983$	2.96980e6	0.0980263	0.0490132	−	0.998798i	$-0.484392\pi$
$983$	2.96980e6	0.0980263	0.0490132	+	0.998798i	$0.484392\pi$
$984$	0	0
$985$	−9.50212e6	−0.312054
$986$	0	0
$987$	−1.81531e7	−0.593142
$988$	0	0
$989$	401460.	0.0130512
$990$	0	0
$991$	−3.88690e7	−1.25724	−0.628621	−	0.777712i	$-0.716380\pi$
$991$	−3.88690e7	−1.25724	−0.628621	+	0.777712i	$0.716380\pi$
$992$	0	0
$993$	1.41356e7	0.454926
$994$	0	0
$995$	4.35118e6	0.139331
$996$	0	0
$997$	−4.92115e7	−1.56794	−0.783969	−	0.620801i	$-0.786808\pi$
$997$	−4.92115e7	−1.56794	−0.783969	+	0.620801i	$0.786808\pi$
$998$	0	0
$999$	−1.10283e7	−0.349619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.6.a.f.1.1		1
4.3	odd	2		114.6.a.c.1.1	✓	1
12.11	even	2		342.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.6.a.c.1.1	✓	1	4.3	odd	2
342.6.a.a.1.1		1	12.11	even	2
912.6.a.f.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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