LMFDB - Embedded newform 24.12.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,12,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	24.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:	$18.4402363334$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+243.000 q^{3} +1870.00 q^{5} -72312.0 q^{7} +59049.0 q^{9} +O(q^{10})$

$q+243.000 q^{3} +1870.00 q^{5} -72312.0 q^{7} +59049.0 q^{9} +147940. q^{11} -1.56286e6 q^{13} +454410. q^{15} -145774. q^{17} +1.09680e6 q^{19} -1.75718e7 q^{21} -6.00143e7 q^{23} -4.53312e7 q^{25} +1.43489e7 q^{27} -1.96270e7 q^{29} -2.39950e8 q^{31} +3.59494e7 q^{33} -1.35223e8 q^{35} +4.88238e8 q^{37} -3.79774e8 q^{39} +4.70660e7 q^{41} +4.28867e8 q^{43} +1.10422e8 q^{45} +4.50903e8 q^{47} +3.25170e9 q^{49} -3.54231e7 q^{51} +4.33669e9 q^{53} +2.76648e8 q^{55} +2.66521e8 q^{57} -8.93756e9 q^{59} +4.67388e9 q^{61} -4.26995e9 q^{63} -2.92254e9 q^{65} +7.49894e9 q^{67} -1.45835e10 q^{69} -2.70321e10 q^{71} +1.16761e10 q^{73} -1.10155e10 q^{75} -1.06978e10 q^{77} +2.47888e9 q^{79} +3.48678e9 q^{81} +4.27456e10 q^{83} -2.72597e8 q^{85} -4.76935e9 q^{87} -9.32708e10 q^{89} +1.13013e11 q^{91} -5.83080e10 q^{93} +2.05101e9 q^{95} +1.18033e11 q^{97} +8.73571e9 q^{99} +O(q^{100})$

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
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	1870.00	0.267613	0.133806	−	0.991008i	$-0.457280\pi$
$5$	1870.00	0.267613	0.133806	+	0.991008i	$0.457280\pi$
$6$	0	0
$7$	−72312.0	−1.62619	−0.813095	−	0.582131i	$-0.802219\pi$
$7$	−72312.0	−1.62619	−0.813095	+	0.582131i	$0.802219\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	147940.	0.276966	0.138483	−	0.990365i	$-0.455777\pi$
$11$	147940.	0.276966	0.138483	+	0.990365i	$0.455777\pi$
$12$	0	0
$13$	−1.56286e6	−1.16743	−0.583715	−	0.811958i	$-0.698402\pi$
$13$	−1.56286e6	−1.16743	−0.583715	+	0.811958i	$0.698402\pi$
$14$	0	0
$15$	454410.	0.154506
$16$	0	0
$17$	−145774.	−0.0249007	−0.0124503	−	0.999922i	$-0.503963\pi$
$17$	−145774.	−0.0249007	−0.0124503	+	0.999922i	$0.503963\pi$
$18$	0	0
$19$	1.09680e6	0.101620	0.0508102	−	0.998708i	$-0.483820\pi$
$19$	1.09680e6	0.101620	0.0508102	+	0.998708i	$0.483820\pi$
$20$	0	0
$21$	−1.75718e7	−0.938881
$22$	0	0
$23$	−6.00143e7	−1.94425	−0.972123	−	0.234470i	$-0.924665\pi$
$23$	−6.00143e7	−1.94425	−0.972123	+	0.234470i	$0.924665\pi$
$24$	0	0
$25$	−4.53312e7	−0.928383
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	−1.96270e7	−0.177690	−0.0888452	−	0.996045i	$-0.528318\pi$
$29$	−1.96270e7	−0.177690	−0.0888452	+	0.996045i	$0.528318\pi$
$30$	0	0
$31$	−2.39950e8	−1.50533	−0.752666	−	0.658403i	$-0.771232\pi$
$31$	−2.39950e8	−1.50533	−0.752666	+	0.658403i	$0.771232\pi$
$32$	0	0
$33$	3.59494e7	0.159906
$34$	0	0
$35$	−1.35223e8	−0.435189
$36$	0	0
$37$	4.88238e8	1.15750	0.578752	−	0.815504i	$-0.303540\pi$
$37$	4.88238e8	1.15750	0.578752	+	0.815504i	$0.303540\pi$
$38$	0	0
$39$	−3.79774e8	−0.674016
$40$	0	0
$41$	4.70660e7	0.0634448	0.0317224	−	0.999497i	$-0.489901\pi$
$41$	4.70660e7	0.0634448	0.0317224	+	0.999497i	$0.489901\pi$
$42$	0	0
$43$	4.28867e8	0.444883	0.222442	−	0.974946i	$-0.428597\pi$
$43$	4.28867e8	0.444883	0.222442	+	0.974946i	$0.428597\pi$
$44$	0	0
$45$	1.10422e8	0.0892042
$46$	0	0
$47$	4.50903e8	0.286778	0.143389	−	0.989666i	$-0.454200\pi$
$47$	4.50903e8	0.286778	0.143389	+	0.989666i	$0.454200\pi$
$48$	0	0
$49$	3.25170e9	1.64449
$50$	0	0
$51$	−3.54231e7	−0.0143764
$52$	0	0
$53$	4.33669e9	1.42443	0.712214	−	0.701962i	$-0.247692\pi$
$53$	4.33669e9	1.42443	0.712214	+	0.701962i	$0.247692\pi$
$54$	0	0
$55$	2.76648e8	0.0741195
$56$	0	0
$57$	2.66521e8	0.0586706
$58$	0	0
$59$	−8.93756e9	−1.62754	−0.813772	−	0.581184i	$-0.802590\pi$
$59$	−8.93756e9	−1.62754	−0.813772	+	0.581184i	$0.802590\pi$
$60$	0	0
$61$	4.67388e9	0.708539	0.354270	−	0.935143i	$-0.384730\pi$
$61$	4.67388e9	0.708539	0.354270	+	0.935143i	$0.384730\pi$
$62$	0	0
$63$	−4.26995e9	−0.542063
$64$	0	0
$65$	−2.92254e9	−0.312419
$66$	0	0
$67$	7.49894e9	0.678560	0.339280	−	0.940685i	$-0.389817\pi$
$67$	7.49894e9	0.678560	0.339280	+	0.940685i	$0.389817\pi$
$68$	0	0
$69$	−1.45835e10	−1.12251
$70$	0	0
$71$	−2.70321e10	−1.77811	−0.889056	−	0.457799i	$-0.848638\pi$
$71$	−2.70321e10	−1.77811	−0.889056	+	0.457799i	$0.848638\pi$
$72$	0	0
$73$	1.16761e10	0.659210	0.329605	−	0.944119i	$-0.393084\pi$
$73$	1.16761e10	0.659210	0.329605	+	0.944119i	$0.393084\pi$
$74$	0	0
$75$	−1.10155e10	−0.536002
$76$	0	0
$77$	−1.06978e10	−0.450399
$78$	0	0
$79$	2.47888e9	0.0906371	0.0453185	−	0.998973i	$-0.485570\pi$
$79$	2.47888e9	0.0906371	0.0453185	+	0.998973i	$0.485570\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	4.27456e10	1.19114	0.595569	−	0.803304i	$-0.296927\pi$
$83$	4.27456e10	1.19114	0.595569	+	0.803304i	$0.296927\pi$
$84$	0	0
$85$	−2.72597e8	−0.00666373
$86$	0	0
$87$	−4.76935e9	−0.102590
$88$	0	0
$89$	−9.32708e10	−1.77052	−0.885259	−	0.465098i	$-0.846019\pi$
$89$	−9.32708e10	−1.77052	−0.885259	+	0.465098i	$0.846019\pi$
$90$	0	0
$91$	1.13013e11	1.89846
$92$	0	0
$93$	−5.83080e10	−0.869104
$94$	0	0
$95$	2.05101e9	0.0271949
$96$	0	0
$97$	1.18033e11	1.39559	0.697795	−	0.716297i	$-0.254164\pi$
$97$	1.18033e11	1.39559	0.697795	+	0.716297i	$0.254164\pi$
$98$	0	0
$99$	8.73571e9	0.0923219
$100$	0	0
$101$	1.69409e11	1.60387	0.801933	−	0.597413i	$-0.203805\pi$
$101$	1.69409e11	1.60387	0.801933	+	0.597413i	$0.203805\pi$
$102$	0	0
$103$	1.99274e11	1.69374	0.846868	−	0.531803i	$-0.178485\pi$
$103$	1.99274e11	1.69374	0.846868	+	0.531803i	$0.178485\pi$
$104$	0	0
$105$	−3.28593e10	−0.251256
$106$	0	0
$107$	−1.51791e11	−1.04625	−0.523123	−	0.852257i	$-0.675233\pi$
$107$	−1.51791e11	−1.04625	−0.523123	+	0.852257i	$0.675233\pi$
$108$	0	0
$109$	−2.38323e11	−1.48361	−0.741805	−	0.670616i	$-0.766030\pi$
$109$	−2.38323e11	−1.48361	−0.741805	+	0.670616i	$0.766030\pi$
$110$	0	0
$111$	1.18642e11	0.668285
$112$	0	0
$113$	8.31507e10	0.424556	0.212278	−	0.977209i	$-0.431912\pi$
$113$	8.31507e10	0.424556	0.212278	+	0.977209i	$0.431912\pi$
$114$	0	0
$115$	−1.12227e11	−0.520305
$116$	0	0
$117$	−9.22852e10	−0.389144
$118$	0	0
$119$	1.05412e10	0.0404932
$120$	0	0
$121$	−2.63425e11	−0.923290
$122$	0	0
$123$	1.14370e10	0.0366299
$124$	0	0
$125$	−1.76078e11	−0.516060
$126$	0	0
$127$	−3.66342e11	−0.983935	−0.491968	−	0.870613i	$-0.663722\pi$
$127$	−3.66342e11	−0.983935	−0.491968	+	0.870613i	$0.663722\pi$
$128$	0	0
$129$	1.04215e11	0.256853
$130$	0	0
$131$	−4.82671e11	−1.09310	−0.546549	−	0.837427i	$-0.684059\pi$
$131$	−4.82671e11	−1.09310	−0.546549	+	0.837427i	$0.684059\pi$
$132$	0	0
$133$	−7.93115e10	−0.165254
$134$	0	0
$135$	2.68325e10	0.0515021
$136$	0	0
$137$	8.27104e11	1.46419	0.732094	−	0.681203i	$-0.238543\pi$
$137$	8.27104e11	1.46419	0.732094	+	0.681203i	$0.238543\pi$
$138$	0	0
$139$	−1.50200e11	−0.245521	−0.122760	−	0.992436i	$-0.539175\pi$
$139$	−1.50200e11	−0.245521	−0.122760	+	0.992436i	$0.539175\pi$
$140$	0	0
$141$	1.09569e11	0.165571
$142$	0	0
$143$	−2.31209e11	−0.323338
$144$	0	0
$145$	−3.67024e10	−0.0475522
$146$	0	0
$147$	7.90163e11	0.949448
$148$	0	0
$149$	1.02471e12	1.14308	0.571538	−	0.820576i	$-0.306347\pi$
$149$	1.02471e12	1.14308	0.571538	+	0.820576i	$0.306347\pi$
$150$	0	0
$151$	−9.74174e11	−1.00987	−0.504933	−	0.863159i	$-0.668483\pi$
$151$	−9.74174e11	−1.00987	−0.504933	+	0.863159i	$0.668483\pi$
$152$	0	0
$153$	−8.60781e9	−0.00830022
$154$	0	0
$155$	−4.48707e11	−0.402846
$156$	0	0
$157$	−1.18243e12	−0.989302	−0.494651	−	0.869092i	$-0.664704\pi$
$157$	−1.18243e12	−0.989302	−0.494651	+	0.869092i	$0.664704\pi$
$158$	0	0
$159$	1.05381e12	0.822394
$160$	0	0
$161$	4.33975e12	3.16171
$162$	0	0
$163$	3.27873e11	0.223190	0.111595	−	0.993754i	$-0.464404\pi$
$163$	3.27873e11	0.223190	0.111595	+	0.993754i	$0.464404\pi$
$164$	0	0
$165$	6.72254e10	0.0427929
$166$	0	0
$167$	−9.04008e11	−0.538557	−0.269278	−	0.963062i	$-0.586785\pi$
$167$	−9.04008e11	−0.538557	−0.269278	+	0.963062i	$0.586785\pi$
$168$	0	0
$169$	6.50365e11	0.362894
$170$	0	0
$171$	6.47647e10	0.0338735
$172$	0	0
$173$	−2.95463e12	−1.44960	−0.724802	−	0.688958i	$-0.758069\pi$
$173$	−2.95463e12	−1.44960	−0.724802	+	0.688958i	$0.758069\pi$
$174$	0	0
$175$	3.27799e12	1.50973
$176$	0	0
$177$	−2.17183e12	−0.939663
$178$	0	0
$179$	−3.91792e11	−0.159354	−0.0796771	−	0.996821i	$-0.525389\pi$
$179$	−3.91792e11	−0.159354	−0.0796771	+	0.996821i	$0.525389\pi$
$180$	0	0
$181$	−2.93601e12	−1.12338	−0.561688	−	0.827349i	$-0.689848\pi$
$181$	−2.93601e12	−1.12338	−0.561688	+	0.827349i	$0.689848\pi$
$182$	0	0
$183$	1.13575e12	0.409075
$184$	0	0
$185$	9.13005e11	0.309762
$186$	0	0
$187$	−2.15658e10	−0.00689663
$188$	0	0
$189$	−1.03760e12	−0.312960
$190$	0	0
$191$	−3.22908e12	−0.919168	−0.459584	−	0.888134i	$-0.652002\pi$
$191$	−3.22908e12	−0.919168	−0.459584	+	0.888134i	$0.652002\pi$
$192$	0	0
$193$	−3.47369e12	−0.933739	−0.466869	−	0.884326i	$-0.654618\pi$
$193$	−3.47369e12	−0.933739	−0.466869	+	0.884326i	$0.654618\pi$
$194$	0	0
$195$	−7.10178e11	−0.180375
$196$	0	0
$197$	−5.03894e12	−1.20997	−0.604986	−	0.796236i	$-0.706821\pi$
$197$	−5.03894e12	−1.20997	−0.604986	+	0.796236i	$0.706821\pi$
$198$	0	0
$199$	−9.41356e11	−0.213827	−0.106913	−	0.994268i	$-0.534097\pi$
$199$	−9.41356e11	−0.213827	−0.106913	+	0.994268i	$0.534097\pi$
$200$	0	0
$201$	1.82224e12	0.391767
$202$	0	0
$203$	1.41926e12	0.288958
$204$	0	0
$205$	8.80134e10	0.0169786
$206$	0	0
$207$	−3.54378e12	−0.648082
$208$	0	0
$209$	1.62260e11	0.0281454
$210$	0	0
$211$	1.02571e13	1.68838	0.844191	−	0.536043i	$-0.180081\pi$
$211$	1.02571e13	1.68838	0.844191	+	0.536043i	$0.180081\pi$
$212$	0	0
$213$	−6.56880e12	−1.02659
$214$	0	0
$215$	8.01981e11	0.119056
$216$	0	0
$217$	1.73513e13	2.44796
$218$	0	0
$219$	2.83730e12	0.380595
$220$	0	0
$221$	2.27824e11	0.0290698
$222$	0	0
$223$	6.03153e12	0.732404	0.366202	−	0.930535i	$-0.380658\pi$
$223$	6.03153e12	0.732404	0.366202	+	0.930535i	$0.380658\pi$
$224$	0	0
$225$	−2.67676e12	−0.309461
$226$	0	0
$227$	−6.01399e12	−0.662248	−0.331124	−	0.943587i	$-0.607428\pi$
$227$	−6.01399e12	−0.662248	−0.331124	+	0.943587i	$0.607428\pi$
$228$	0	0
$229$	−7.32258e12	−0.768368	−0.384184	−	0.923257i	$-0.625517\pi$
$229$	−7.32258e12	−0.768368	−0.384184	+	0.923257i	$0.625517\pi$
$230$	0	0
$231$	−2.59957e12	−0.260038
$232$	0	0
$233$	9.47000e12	0.903425	0.451713	−	0.892163i	$-0.350813\pi$
$233$	9.47000e12	0.903425	0.451713	+	0.892163i	$0.350813\pi$
$234$	0	0
$235$	8.43189e11	0.0767453
$236$	0	0
$237$	6.02367e11	0.0523293
$238$	0	0
$239$	−2.05626e12	−0.170565	−0.0852823	−	0.996357i	$-0.527179\pi$
$239$	−2.05626e12	−0.170565	−0.0852823	+	0.996357i	$0.527179\pi$
$240$	0	0
$241$	1.79367e13	1.42118	0.710590	−	0.703607i	$-0.248428\pi$
$241$	1.79367e13	1.42118	0.710590	+	0.703607i	$0.248428\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	6.08068e12	0.440087
$246$	0	0
$247$	−1.71414e12	−0.118635
$248$	0	0
$249$	1.03872e13	0.687704
$250$	0	0
$251$	−3.63087e12	−0.230041	−0.115021	−	0.993363i	$-0.536693\pi$
$251$	−3.63087e12	−0.230041	−0.115021	+	0.993363i	$0.536693\pi$
$252$	0	0
$253$	−8.87851e12	−0.538489
$254$	0	0
$255$	−6.62412e10	−0.00384731
$256$	0	0
$257$	−1.81673e13	−1.01078	−0.505391	−	0.862891i	$-0.668652\pi$
$257$	−1.81673e13	−1.01078	−0.505391	+	0.862891i	$0.668652\pi$
$258$	0	0
$259$	−3.53055e13	−1.88232
$260$	0	0
$261$	−1.15895e12	−0.0592301
$262$	0	0
$263$	2.56780e13	1.25836	0.629178	−	0.777261i	$-0.283391\pi$
$263$	2.56780e13	1.25836	0.629178	+	0.777261i	$0.283391\pi$
$264$	0	0
$265$	8.10960e12	0.381195
$266$	0	0
$267$	−2.26648e13	−1.02221
$268$	0	0
$269$	−2.67233e13	−1.15679	−0.578393	−	0.815759i	$-0.696320\pi$
$269$	−2.67233e13	−1.15679	−0.578393	+	0.815759i	$0.696320\pi$
$270$	0	0
$271$	−1.70265e13	−0.707610	−0.353805	−	0.935319i	$-0.615112\pi$
$271$	−1.70265e13	−0.707610	−0.353805	+	0.935319i	$0.615112\pi$
$272$	0	0
$273$	2.74623e13	1.09608
$274$	0	0
$275$	−6.70630e12	−0.257130
$276$	0	0
$277$	2.96076e12	0.109085	0.0545424	−	0.998511i	$-0.482630\pi$
$277$	2.96076e12	0.109085	0.0545424	+	0.998511i	$0.482630\pi$
$278$	0	0
$279$	−1.41688e13	−0.501777
$280$	0	0
$281$	5.72445e12	0.194917	0.0974583	−	0.995240i	$-0.468929\pi$
$281$	5.72445e12	0.194917	0.0974583	+	0.995240i	$0.468929\pi$
$282$	0	0
$283$	−2.76343e13	−0.904948	−0.452474	−	0.891778i	$-0.649459\pi$
$283$	−2.76343e13	−0.904948	−0.452474	+	0.891778i	$0.649459\pi$
$284$	0	0
$285$	4.98395e11	0.0157010
$286$	0	0
$287$	−3.40344e12	−0.103173
$288$	0	0
$289$	−3.42506e13	−0.999380
$290$	0	0
$291$	2.86820e13	0.805745
$292$	0	0
$293$	1.02668e12	0.0277756	0.0138878	−	0.999904i	$-0.495579\pi$
$293$	1.02668e12	0.0277756	0.0138878	+	0.999904i	$0.495579\pi$
$294$	0	0
$295$	−1.67132e13	−0.435551
$296$	0	0
$297$	2.12278e12	0.0533021
$298$	0	0
$299$	9.37938e13	2.26977
$300$	0	0
$301$	−3.10122e13	−0.723464
$302$	0	0
$303$	4.11663e13	0.925993
$304$	0	0
$305$	8.74016e12	0.189614
$306$	0	0
$307$	5.67757e13	1.18823	0.594116	−	0.804379i	$-0.297502\pi$
$307$	5.67757e13	1.18823	0.594116	+	0.804379i	$0.297502\pi$
$308$	0	0
$309$	4.84236e13	0.977879
$310$	0	0
$311$	5.93553e13	1.15685	0.578425	−	0.815735i	$-0.303668\pi$
$311$	5.93553e13	1.15685	0.578425	+	0.815735i	$0.303668\pi$
$312$	0	0
$313$	−2.96894e13	−0.558608	−0.279304	−	0.960203i	$-0.590104\pi$
$313$	−2.96894e13	−0.558608	−0.279304	+	0.960203i	$0.590104\pi$
$314$	0	0
$315$	−7.98481e12	−0.145063
$316$	0	0
$317$	−3.95905e13	−0.694648	−0.347324	−	0.937745i	$-0.612910\pi$
$317$	−3.95905e13	−0.694648	−0.347324	+	0.937745i	$0.612910\pi$
$318$	0	0
$319$	−2.90361e12	−0.0492141
$320$	0	0
$321$	−3.68851e13	−0.604050
$322$	0	0
$323$	−1.59884e11	−0.00253042
$324$	0	0
$325$	7.08463e13	1.08382
$326$	0	0
$327$	−5.79124e13	−0.856563
$328$	0	0
$329$	−3.26057e13	−0.466355
$330$	0	0
$331$	3.37083e13	0.466318	0.233159	−	0.972439i	$-0.425094\pi$
$331$	3.37083e13	0.466318	0.233159	+	0.972439i	$0.425094\pi$
$332$	0	0
$333$	2.88300e13	0.385834
$334$	0	0
$335$	1.40230e13	0.181591
$336$	0	0
$337$	3.99669e13	0.500882	0.250441	−	0.968132i	$-0.419424\pi$
$337$	3.99669e13	0.500882	0.250441	+	0.968132i	$0.419424\pi$
$338$	0	0
$339$	2.02056e13	0.245117
$340$	0	0
$341$	−3.54983e13	−0.416925
$342$	0	0
$343$	−9.21524e13	−1.04807
$344$	0	0
$345$	−2.72711e13	−0.300398
$346$	0	0
$347$	−2.94245e13	−0.313976	−0.156988	−	0.987600i	$-0.550178\pi$
$347$	−2.94245e13	−0.313976	−0.156988	+	0.987600i	$0.550178\pi$
$348$	0	0
$349$	−9.44210e12	−0.0976177	−0.0488089	−	0.998808i	$-0.515543\pi$
$349$	−9.44210e12	−0.0976177	−0.0488089	+	0.998808i	$0.515543\pi$
$350$	0	0
$351$	−2.24253e13	−0.224672
$352$	0	0
$353$	−6.80249e13	−0.660552	−0.330276	−	0.943884i	$-0.607142\pi$
$353$	−6.80249e13	−0.660552	−0.330276	+	0.943884i	$0.607142\pi$
$354$	0	0
$355$	−5.05500e13	−0.475845
$356$	0	0
$357$	2.56151e12	0.0233788
$358$	0	0
$359$	8.11738e13	0.718449	0.359225	−	0.933251i	$-0.383041\pi$
$359$	8.11738e13	0.718449	0.359225	+	0.933251i	$0.383041\pi$
$360$	0	0
$361$	−1.15287e14	−0.989673
$362$	0	0
$363$	−6.40124e13	−0.533062
$364$	0	0
$365$	2.18344e13	0.176413
$366$	0	0
$367$	9.16664e13	0.718699	0.359349	−	0.933203i	$-0.382999\pi$
$367$	9.16664e13	0.718699	0.359349	+	0.933203i	$0.382999\pi$
$368$	0	0
$369$	2.77920e12	0.0211483
$370$	0	0
$371$	−3.13594e14	−2.31639
$372$	0	0
$373$	2.65457e14	1.90369	0.951844	−	0.306584i	$-0.0991860\pi$
$373$	2.65457e14	1.90369	0.951844	+	0.306584i	$0.0991860\pi$
$374$	0	0
$375$	−4.27870e13	−0.297947
$376$	0	0
$377$	3.06741e13	0.207441
$378$	0	0
$379$	7.93925e13	0.521511	0.260756	−	0.965405i	$-0.416028\pi$
$379$	7.93925e13	0.521511	0.260756	+	0.965405i	$0.416028\pi$
$380$	0	0
$381$	−8.90212e13	−0.568075
$382$	0	0
$383$	−2.11948e14	−1.31412	−0.657062	−	0.753837i	$-0.728201\pi$
$383$	−2.11948e14	−1.31412	−0.657062	+	0.753837i	$0.728201\pi$
$384$	0	0
$385$	−2.00050e13	−0.120532
$386$	0	0
$387$	2.53242e13	0.148294
$388$	0	0
$389$	1.56709e14	0.892014	0.446007	−	0.895029i	$-0.352846\pi$
$389$	1.56709e14	0.892014	0.446007	+	0.895029i	$0.352846\pi$
$390$	0	0
$391$	8.74852e12	0.0484130
$392$	0	0
$393$	−1.17289e14	−0.631101
$394$	0	0
$395$	4.63550e12	0.0242556
$396$	0	0
$397$	1.82571e14	0.929144	0.464572	−	0.885535i	$-0.346208\pi$
$397$	1.82571e14	0.929144	0.464572	+	0.885535i	$0.346208\pi$
$398$	0	0
$399$	−1.92727e13	−0.0954095
$400$	0	0
$401$	−5.02944e13	−0.242228	−0.121114	−	0.992639i	$-0.538647\pi$
$401$	−5.02944e13	−0.242228	−0.121114	+	0.992639i	$0.538647\pi$
$402$	0	0
$403$	3.75009e14	1.75737
$404$	0	0
$405$	6.52029e12	0.0297347
$406$	0	0
$407$	7.22299e13	0.320589
$408$	0	0
$409$	1.00388e14	0.433712	0.216856	−	0.976204i	$-0.430420\pi$
$409$	1.00388e14	0.433712	0.216856	+	0.976204i	$0.430420\pi$
$410$	0	0
$411$	2.00986e14	0.845350
$412$	0	0
$413$	6.46293e14	2.64669
$414$	0	0
$415$	7.99343e13	0.318763
$416$	0	0
$417$	−3.64986e13	−0.141752
$418$	0	0
$419$	−2.25997e14	−0.854921	−0.427460	−	0.904034i	$-0.640592\pi$
$419$	−2.25997e14	−0.854921	−0.427460	+	0.904034i	$0.640592\pi$
$420$	0	0
$421$	−2.27720e14	−0.839171	−0.419585	−	0.907716i	$-0.637824\pi$
$421$	−2.27720e14	−0.839171	−0.419585	+	0.907716i	$0.637824\pi$
$422$	0	0
$423$	2.66254e13	0.0955925
$424$	0	0
$425$	6.60811e12	0.0231174
$426$	0	0
$427$	−3.37978e14	−1.15222
$428$	0	0
$429$	−5.61838e13	−0.186679
$430$	0	0
$431$	−9.26042e12	−0.0299920	−0.0149960	−	0.999888i	$-0.504774\pi$
$431$	−9.26042e12	−0.0299920	−0.0149960	+	0.999888i	$0.504774\pi$
$432$	0	0
$433$	−1.51510e14	−0.478363	−0.239181	−	0.970975i	$-0.576879\pi$
$433$	−1.51510e14	−0.478363	−0.239181	+	0.970975i	$0.576879\pi$
$434$	0	0
$435$	−8.91868e12	−0.0274543
$436$	0	0
$437$	−6.58234e13	−0.197575
$438$	0	0
$439$	−1.22038e14	−0.357223	−0.178612	−	0.983920i	$-0.557161\pi$
$439$	−1.22038e14	−0.357223	−0.178612	+	0.983920i	$0.557161\pi$
$440$	0	0
$441$	1.92010e14	0.548164
$442$	0	0
$443$	2.64225e14	0.735788	0.367894	−	0.929868i	$-0.380079\pi$
$443$	2.64225e14	0.735788	0.367894	+	0.929868i	$0.380079\pi$
$444$	0	0
$445$	−1.74416e14	−0.473813
$446$	0	0
$447$	2.49004e14	0.659955
$448$	0	0
$449$	−7.66733e14	−1.98285	−0.991423	−	0.130690i	$-0.958281\pi$
$449$	−7.66733e14	−1.98285	−0.991423	+	0.130690i	$0.958281\pi$
$450$	0	0
$451$	6.96295e12	0.0175720
$452$	0	0
$453$	−2.36724e14	−0.583046
$454$	0	0
$455$	2.11335e14	0.508053
$456$	0	0
$457$	−4.72988e14	−1.10997	−0.554985	−	0.831860i	$-0.687276\pi$
$457$	−4.72988e14	−1.10997	−0.554985	+	0.831860i	$0.687276\pi$
$458$	0	0
$459$	−2.09170e12	−0.00479214
$460$	0	0
$461$	2.25139e14	0.503611	0.251805	−	0.967778i	$-0.418976\pi$
$461$	2.25139e14	0.503611	0.251805	+	0.967778i	$0.418976\pi$
$462$	0	0
$463$	3.78465e14	0.826665	0.413333	−	0.910580i	$-0.364365\pi$
$463$	3.78465e14	0.826665	0.413333	+	0.910580i	$0.364365\pi$
$464$	0	0
$465$	−1.09036e14	−0.232583
$466$	0	0
$467$	−6.93495e14	−1.44478	−0.722388	−	0.691488i	$-0.756955\pi$
$467$	−6.93495e14	−1.44478	−0.722388	+	0.691488i	$0.756955\pi$
$468$	0	0
$469$	−5.42263e14	−1.10347
$470$	0	0
$471$	−2.87331e14	−0.571174
$472$	0	0
$473$	6.34466e13	0.123217
$474$	0	0
$475$	−4.97191e13	−0.0943427
$476$	0	0
$477$	2.56077e14	0.474810
$478$	0	0
$479$	7.29474e14	1.32180	0.660898	−	0.750475i	$-0.270175\pi$
$479$	7.29474e14	1.32180	0.660898	+	0.750475i	$0.270175\pi$
$480$	0	0
$481$	−7.63047e14	−1.35130
$482$	0	0
$483$	1.05456e15	1.82542
$484$	0	0
$485$	2.20721e14	0.373478
$486$	0	0
$487$	−7.20456e14	−1.19179	−0.595893	−	0.803064i	$-0.703202\pi$
$487$	−7.20456e14	−1.19179	−0.595893	+	0.803064i	$0.703202\pi$
$488$	0	0
$489$	7.96732e13	0.128859
$490$	0	0
$491$	−7.56442e14	−1.19627	−0.598133	−	0.801397i	$-0.704090\pi$
$491$	−7.56442e14	−1.19627	−0.598133	+	0.801397i	$0.704090\pi$
$492$	0	0
$493$	2.86110e12	0.00442461
$494$	0	0
$495$	1.63358e13	0.0247065
$496$	0	0
$497$	1.95475e15	2.89155
$498$	0	0
$499$	2.94967e14	0.426796	0.213398	−	0.976965i	$-0.431547\pi$
$499$	2.94967e14	0.426796	0.213398	+	0.976965i	$0.431547\pi$
$500$	0	0
$501$	−2.19674e14	−0.310936
$502$	0	0
$503$	2.91878e14	0.404182	0.202091	−	0.979367i	$-0.435226\pi$
$503$	2.91878e14	0.404182	0.202091	+	0.979367i	$0.435226\pi$
$504$	0	0
$505$	3.16794e14	0.429215
$506$	0	0
$507$	1.58039e14	0.209517
$508$	0	0
$509$	−1.01712e15	−1.31954	−0.659772	−	0.751466i	$-0.729347\pi$
$509$	−1.01712e15	−1.31954	−0.659772	+	0.751466i	$0.729347\pi$
$510$	0	0
$511$	−8.44325e14	−1.07200
$512$	0	0
$513$	1.57378e13	0.0195569
$514$	0	0
$515$	3.72642e14	0.453265
$516$	0	0
$517$	6.67066e13	0.0794275
$518$	0	0
$519$	−7.17974e14	−0.836929
$520$	0	0
$521$	9.62156e13	0.109809	0.0549045	−	0.998492i	$-0.482515\pi$
$521$	9.62156e13	0.109809	0.0549045	+	0.998492i	$0.482515\pi$
$522$	0	0
$523$	1.70097e15	1.90080	0.950401	−	0.311027i	$-0.100673\pi$
$523$	1.70097e15	1.90080	0.950401	+	0.311027i	$0.100673\pi$
$524$	0	0
$525$	7.96552e14	0.871642
$526$	0	0
$527$	3.49785e13	0.0374838
$528$	0	0
$529$	2.64890e15	2.78010
$530$	0	0
$531$	−5.27754e14	−0.542515
$532$	0	0
$533$	−7.35575e13	−0.0740674
$534$	0	0
$535$	−2.83848e14	−0.279989
$536$	0	0
$537$	−9.52054e13	−0.0920032
$538$	0	0
$539$	4.81056e14	0.455468
$540$	0	0
$541$	6.48649e14	0.601762	0.300881	−	0.953662i	$-0.402719\pi$
$541$	6.48649e14	0.601762	0.300881	+	0.953662i	$0.402719\pi$
$542$	0	0
$543$	−7.13450e14	−0.648581
$544$	0	0
$545$	−4.45664e14	−0.397033
$546$	0	0
$547$	−1.74938e15	−1.52740	−0.763701	−	0.645570i	$-0.776620\pi$
$547$	−1.74938e15	−1.52740	−0.763701	+	0.645570i	$0.776620\pi$
$548$	0	0
$549$	2.75988e14	0.236180
$550$	0	0
$551$	−2.15268e13	−0.0180570
$552$	0	0
$553$	−1.79253e14	−0.147393
$554$	0	0
$555$	2.21860e14	0.178841
$556$	0	0
$557$	−1.66657e15	−1.31710	−0.658552	−	0.752535i	$-0.728831\pi$
$557$	−1.66657e15	−1.31710	−0.658552	+	0.752535i	$0.728831\pi$
$558$	0	0
$559$	−6.70258e14	−0.519370
$560$	0	0
$561$	−5.24049e12	−0.00398177
$562$	0	0
$563$	−1.81221e15	−1.35025	−0.675124	−	0.737705i	$-0.735910\pi$
$563$	−1.81221e15	−1.35025	−0.675124	+	0.737705i	$0.735910\pi$
$564$	0	0
$565$	1.55492e14	0.113616
$566$	0	0
$567$	−2.52136e14	−0.180688
$568$	0	0
$569$	−5.11949e14	−0.359840	−0.179920	−	0.983681i	$-0.557584\pi$
$569$	−5.11949e14	−0.359840	−0.179920	+	0.983681i	$0.557584\pi$
$570$	0	0
$571$	8.13548e14	0.560899	0.280449	−	0.959869i	$-0.409516\pi$
$571$	8.13548e14	0.560899	0.280449	+	0.959869i	$0.409516\pi$
$572$	0	0
$573$	−7.84666e14	−0.530682
$574$	0	0
$575$	2.72052e15	1.80501
$576$	0	0
$577$	−2.87800e14	−0.187337	−0.0936686	−	0.995603i	$-0.529859\pi$
$577$	−2.87800e14	−0.187337	−0.0936686	+	0.995603i	$0.529859\pi$
$578$	0	0
$579$	−8.44106e14	−0.539094
$580$	0	0
$581$	−3.09102e15	−1.93702
$582$	0	0
$583$	6.41569e14	0.394518
$584$	0	0
$585$	−1.72573e14	−0.104140
$586$	0	0
$587$	−1.73295e15	−1.02631	−0.513153	−	0.858297i	$-0.671523\pi$
$587$	−1.73295e15	−1.02631	−0.513153	+	0.858297i	$0.671523\pi$
$588$	0	0
$589$	−2.63177e14	−0.152972
$590$	0	0
$591$	−1.22446e15	−0.698577
$592$	0	0
$593$	2.47295e15	1.38489	0.692443	−	0.721472i	$-0.256534\pi$
$593$	2.47295e15	1.38489	0.692443	+	0.721472i	$0.256534\pi$
$594$	0	0
$595$	1.97121e13	0.0108365
$596$	0	0
$597$	−2.28749e14	−0.123453
$598$	0	0
$599$	−5.62022e14	−0.297787	−0.148893	−	0.988853i	$-0.547571\pi$
$599$	−5.62022e14	−0.297787	−0.148893	+	0.988853i	$0.547571\pi$
$600$	0	0
$601$	1.52329e15	0.792452	0.396226	−	0.918153i	$-0.370320\pi$
$601$	1.52329e15	0.792452	0.396226	+	0.918153i	$0.370320\pi$
$602$	0	0
$603$	4.42805e14	0.226187
$604$	0	0
$605$	−4.92606e14	−0.247084
$606$	0	0
$607$	1.26403e15	0.622616	0.311308	−	0.950309i	$-0.399233\pi$
$607$	1.26403e15	0.622616	0.311308	+	0.950309i	$0.399233\pi$
$608$	0	0
$609$	3.44881e14	0.166830
$610$	0	0
$611$	−7.04698e14	−0.334793
$612$	0	0
$613$	1.55065e15	0.723572	0.361786	−	0.932261i	$-0.382167\pi$
$613$	1.55065e15	0.723572	0.361786	+	0.932261i	$0.382167\pi$
$614$	0	0
$615$	2.13873e13	0.00980262
$616$	0	0
$617$	−2.16506e15	−0.974767	−0.487384	−	0.873188i	$-0.662049\pi$
$617$	−2.16506e15	−0.974767	−0.487384	+	0.873188i	$0.662049\pi$
$618$	0	0
$619$	2.65766e15	1.17544	0.587720	−	0.809064i	$-0.300026\pi$
$619$	2.65766e15	1.17544	0.587720	+	0.809064i	$0.300026\pi$
$620$	0	0
$621$	−8.61139e14	−0.374170
$622$	0	0
$623$	6.74460e15	2.87920
$624$	0	0
$625$	1.88417e15	0.790279
$626$	0	0
$627$	3.94292e13	0.0162497
$628$	0	0
$629$	−7.11724e13	−0.0288226
$630$	0	0
$631$	−1.85186e15	−0.736964	−0.368482	−	0.929635i	$-0.620122\pi$
$631$	−1.85186e15	−0.736964	−0.368482	+	0.929635i	$0.620122\pi$
$632$	0	0
$633$	2.49247e15	0.974788
$634$	0	0
$635$	−6.85060e14	−0.263314
$636$	0	0
$637$	−5.08194e15	−1.91983
$638$	0	0
$639$	−1.59622e15	−0.592704
$640$	0	0
$641$	−1.80163e15	−0.657578	−0.328789	−	0.944403i	$-0.606641\pi$
$641$	−1.80163e15	−0.657578	−0.328789	+	0.944403i	$0.606641\pi$
$642$	0	0
$643$	1.26341e15	0.453299	0.226650	−	0.973976i	$-0.427223\pi$
$643$	1.26341e15	0.453299	0.226650	+	0.973976i	$0.427223\pi$
$644$	0	0
$645$	1.94881e14	0.0687372
$646$	0	0
$647$	2.16146e15	0.749505	0.374753	−	0.927125i	$-0.377728\pi$
$647$	2.16146e15	0.749505	0.374753	+	0.927125i	$0.377728\pi$
$648$	0	0
$649$	−1.32222e15	−0.450774
$650$	0	0
$651$	4.21637e15	1.41333
$652$	0	0
$653$	2.01054e15	0.662659	0.331329	−	0.943515i	$-0.392503\pi$
$653$	2.01054e15	0.662659	0.331329	+	0.943515i	$0.392503\pi$
$654$	0	0
$655$	−9.02595e14	−0.292527
$656$	0	0
$657$	6.89464e14	0.219737
$658$	0	0
$659$	2.58466e15	0.810090	0.405045	−	0.914297i	$-0.367256\pi$
$659$	2.58466e15	0.810090	0.405045	+	0.914297i	$0.367256\pi$
$660$	0	0
$661$	−2.17422e15	−0.670187	−0.335093	−	0.942185i	$-0.608768\pi$
$661$	−2.17422e15	−0.670187	−0.335093	+	0.942185i	$0.608768\pi$
$662$	0	0
$663$	5.53612e13	0.0167835
$664$	0	0
$665$	−1.48313e14	−0.0442241
$666$	0	0
$667$	1.17790e15	0.345474
$668$	0	0
$669$	1.46566e15	0.422853
$670$	0	0
$671$	6.91454e14	0.196241
$672$	0	0
$673$	2.26389e15	0.632080	0.316040	−	0.948746i	$-0.397647\pi$
$673$	2.26389e15	0.632080	0.316040	+	0.948746i	$0.397647\pi$
$674$	0	0
$675$	−6.50454e14	−0.178667
$676$	0	0
$677$	3.88590e15	1.05016	0.525079	−	0.851054i	$-0.324036\pi$
$677$	3.88590e15	1.05016	0.525079	+	0.851054i	$0.324036\pi$
$678$	0	0
$679$	−8.53519e15	−2.26950
$680$	0	0
$681$	−1.46140e15	−0.382349
$682$	0	0
$683$	−3.59484e15	−0.925478	−0.462739	−	0.886494i	$-0.653133\pi$
$683$	−3.59484e15	−0.925478	−0.462739	+	0.886494i	$0.653133\pi$
$684$	0	0
$685$	1.54668e15	0.391835
$686$	0	0
$687$	−1.77939e15	−0.443617
$688$	0	0
$689$	−6.77762e15	−1.66292
$690$	0	0
$691$	−3.27801e15	−0.791556	−0.395778	−	0.918346i	$-0.629525\pi$
$691$	−3.27801e15	−0.791556	−0.395778	+	0.918346i	$0.629525\pi$
$692$	0	0
$693$	−6.31697e14	−0.150133
$694$	0	0
$695$	−2.80874e14	−0.0657045
$696$	0	0
$697$	−6.86100e12	−0.00157982
$698$	0	0
$699$	2.30121e15	0.521593
$700$	0	0
$701$	2.00750e15	0.447927	0.223964	−	0.974598i	$-0.428100\pi$
$701$	2.00750e15	0.447927	0.223964	+	0.974598i	$0.428100\pi$
$702$	0	0
$703$	5.35498e14	0.117626
$704$	0	0
$705$	2.04895e14	0.0443089
$706$	0	0
$707$	−1.22503e16	−2.60819
$708$	0	0
$709$	−5.04111e15	−1.05675	−0.528374	−	0.849012i	$-0.677198\pi$
$709$	−5.04111e15	−1.05675	−0.528374	+	0.849012i	$0.677198\pi$
$710$	0	0
$711$	1.46375e14	0.0302124
$712$	0	0
$713$	1.44005e16	2.92674
$714$	0	0
$715$	−4.32361e14	−0.0865294
$716$	0	0
$717$	−4.99671e14	−0.0984755
$718$	0	0
$719$	1.69264e15	0.328516	0.164258	−	0.986417i	$-0.447477\pi$
$719$	1.69264e15	0.328516	0.164258	+	0.986417i	$0.447477\pi$
$720$	0	0
$721$	−1.44099e16	−2.75434
$722$	0	0
$723$	4.35862e15	0.820518
$724$	0	0
$725$	8.89714e14	0.164965
$726$	0	0
$727$	−1.15879e15	−0.211625	−0.105812	−	0.994386i	$-0.533744\pi$
$727$	−1.15879e15	−0.211625	−0.105812	+	0.994386i	$0.533744\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−6.25177e13	−0.0110779
$732$	0	0
$733$	8.71945e15	1.52201	0.761005	−	0.648747i	$-0.224706\pi$
$733$	8.71945e15	1.52201	0.761005	+	0.648747i	$0.224706\pi$
$734$	0	0
$735$	1.47760e15	0.254084
$736$	0	0
$737$	1.10939e15	0.187938
$738$	0	0
$739$	−3.58724e15	−0.598710	−0.299355	−	0.954142i	$-0.596771\pi$
$739$	−3.58724e15	−0.598710	−0.299355	+	0.954142i	$0.596771\pi$
$740$	0	0
$741$	−4.16535e14	−0.0684938
$742$	0	0
$743$	−9.86226e15	−1.59786	−0.798928	−	0.601426i	$-0.794599\pi$
$743$	−9.86226e15	−1.59786	−0.798928	+	0.601426i	$0.794599\pi$
$744$	0	0
$745$	1.91620e15	0.305902
$746$	0	0
$747$	2.52408e15	0.397046
$748$	0	0
$749$	1.09763e16	1.70139
$750$	0	0
$751$	1.05913e15	0.161782	0.0808912	−	0.996723i	$-0.474223\pi$
$751$	1.05913e15	0.161782	0.0808912	+	0.996723i	$0.474223\pi$
$752$	0	0
$753$	−8.82302e14	−0.132814
$754$	0	0
$755$	−1.82171e15	−0.270253
$756$	0	0
$757$	1.06759e16	1.56091	0.780454	−	0.625213i	$-0.214988\pi$
$757$	1.06759e16	1.56091	0.780454	+	0.625213i	$0.214988\pi$
$758$	0	0
$759$	−2.15748e15	−0.310897
$760$	0	0
$761$	−6.87039e15	−0.975810	−0.487905	−	0.872897i	$-0.662239\pi$
$761$	−6.87039e15	−0.975810	−0.487905	+	0.872897i	$0.662239\pi$
$762$	0	0
$763$	1.72336e16	2.41263
$764$	0	0
$765$	−1.60966e13	−0.00222124
$766$	0	0
$767$	1.39681e16	1.90004
$768$	0	0
$769$	2.46573e15	0.330636	0.165318	−	0.986240i	$-0.447135\pi$
$769$	2.46573e15	0.330636	0.165318	+	0.986240i	$0.447135\pi$
$770$	0	0
$771$	−4.41464e15	−0.583575
$772$	0	0
$773$	3.28785e14	0.0428474	0.0214237	−	0.999770i	$-0.493180\pi$
$773$	3.28785e14	0.0428474	0.0214237	+	0.999770i	$0.493180\pi$
$774$	0	0
$775$	1.08772e16	1.39753
$776$	0	0
$777$	−8.57923e15	−1.08676
$778$	0	0
$779$	5.16218e13	0.00644729
$780$	0	0
$781$	−3.99913e15	−0.492476
$782$	0	0
$783$	−2.81625e14	−0.0341965
$784$	0	0
$785$	−2.21115e15	−0.264750
$786$	0	0
$787$	−1.93735e15	−0.228743	−0.114371	−	0.993438i	$-0.536485\pi$
$787$	−1.93735e15	−0.228743	−0.114371	+	0.993438i	$0.536485\pi$
$788$	0	0
$789$	6.23974e15	0.726513
$790$	0	0
$791$	−6.01280e15	−0.690408
$792$	0	0
$793$	−7.30462e15	−0.827170
$794$	0	0
$795$	1.97063e15	0.220083
$796$	0	0
$797$	−1.35232e16	−1.48957	−0.744783	−	0.667306i	$-0.767447\pi$
$797$	−1.35232e16	−1.48957	−0.744783	+	0.667306i	$0.767447\pi$
$798$	0	0
$799$	−6.57300e13	−0.00714095
$800$	0	0
$801$	−5.50755e15	−0.590173
$802$	0	0
$803$	1.72737e15	0.182578
$804$	0	0
$805$	8.11534e15	0.846114
$806$	0	0
$807$	−6.49376e15	−0.667870
$808$	0	0
$809$	4.13063e15	0.419082	0.209541	−	0.977800i	$-0.432803\pi$
$809$	4.13063e15	0.419082	0.209541	+	0.977800i	$0.432803\pi$
$810$	0	0
$811$	1.00597e16	1.00687	0.503433	−	0.864035i	$-0.332070\pi$
$811$	1.00597e16	1.00687	0.503433	+	0.864035i	$0.332070\pi$
$812$	0	0
$813$	−4.13744e15	−0.408539
$814$	0	0
$815$	6.13123e14	0.0597284
$816$	0	0
$817$	4.70380e14	0.0452092
$818$	0	0
$819$	6.67333e15	0.632821
$820$	0	0
$821$	1.94165e15	0.181670	0.0908350	−	0.995866i	$-0.471046\pi$
$821$	1.94165e15	0.181670	0.0908350	+	0.995866i	$0.471046\pi$
$822$	0	0
$823$	−9.16505e15	−0.846127	−0.423063	−	0.906100i	$-0.639045\pi$
$823$	−9.16505e15	−0.846127	−0.423063	+	0.906100i	$0.639045\pi$
$824$	0	0
$825$	−1.62963e15	−0.148454
$826$	0	0
$827$	−9.59429e15	−0.862447	−0.431224	−	0.902245i	$-0.641918\pi$
$827$	−9.59429e15	−0.862447	−0.431224	+	0.902245i	$0.641918\pi$
$828$	0	0
$829$	−2.44426e15	−0.216819	−0.108410	−	0.994106i	$-0.534576\pi$
$829$	−2.44426e15	−0.216819	−0.108410	+	0.994106i	$0.534576\pi$
$830$	0	0
$831$	7.19464e14	0.0629801
$832$	0	0
$833$	−4.74013e14	−0.0409490
$834$	0	0
$835$	−1.69049e15	−0.144125
$836$	0	0
$837$	−3.44303e15	−0.289701
$838$	0	0
$839$	−3.82370e15	−0.317536	−0.158768	−	0.987316i	$-0.550752\pi$
$839$	−3.82370e15	−0.317536	−0.158768	+	0.987316i	$0.550752\pi$
$840$	0	0
$841$	−1.18153e16	−0.968426
$842$	0	0
$843$	1.39104e15	0.112535
$844$	0	0
$845$	1.21618e15	0.0971151
$846$	0	0
$847$	1.90488e16	1.50144
$848$	0	0
$849$	−6.71514e15	−0.522472
$850$	0	0
$851$	−2.93012e16	−2.25047
$852$	0	0
$853$	−3.18100e14	−0.0241181	−0.0120591	−	0.999927i	$-0.503839\pi$
$853$	−3.18100e14	−0.0241181	−0.0120591	+	0.999927i	$0.503839\pi$
$854$	0	0
$855$	1.21110e14	0.00906497
$856$	0	0
$857$	−1.65616e16	−1.22379	−0.611895	−	0.790939i	$-0.709592\pi$
$857$	−1.65616e16	−1.22379	−0.611895	+	0.790939i	$0.709592\pi$
$858$	0	0
$859$	3.98595e15	0.290783	0.145392	−	0.989374i	$-0.453556\pi$
$859$	3.98595e15	0.290783	0.145392	+	0.989374i	$0.453556\pi$
$860$	0	0
$861$	−8.27035e14	−0.0595671
$862$	0	0
$863$	2.44385e16	1.73786	0.868932	−	0.494932i	$-0.164807\pi$
$863$	2.44385e16	1.73786	0.868932	+	0.494932i	$0.164807\pi$
$864$	0	0
$865$	−5.52515e15	−0.387932
$866$	0	0
$867$	−8.32291e15	−0.576992
$868$	0	0
$869$	3.66725e14	0.0251034
$870$	0	0
$871$	−1.17198e16	−0.792172
$872$	0	0
$873$	6.96972e15	0.465197
$874$	0	0
$875$	1.27326e16	0.839211
$876$	0	0
$877$	1.04477e16	0.680019	0.340009	−	0.940422i	$-0.389570\pi$
$877$	1.04477e16	0.680019	0.340009	+	0.940422i	$0.389570\pi$
$878$	0	0
$879$	2.49483e14	0.0160362
$880$	0	0
$881$	−2.55705e15	−0.162320	−0.0811601	−	0.996701i	$-0.525862\pi$
$881$	−2.55705e15	−0.162320	−0.0811601	+	0.996701i	$0.525862\pi$
$882$	0	0
$883$	−1.02015e16	−0.639557	−0.319778	−	0.947492i	$-0.603608\pi$
$883$	−1.02015e16	−0.639557	−0.319778	+	0.947492i	$0.603608\pi$
$884$	0	0
$885$	−4.06132e15	−0.251466
$886$	0	0
$887$	7.27643e15	0.444978	0.222489	−	0.974935i	$-0.428582\pi$
$887$	7.27643e15	0.444978	0.222489	+	0.974935i	$0.428582\pi$
$888$	0	0
$889$	2.64909e16	1.60007
$890$	0	0
$891$	5.15835e14	0.0307740
$892$	0	0
$893$	4.94549e14	0.0291425
$894$	0	0
$895$	−7.32651e14	−0.0426452
$896$	0	0
$897$	2.27919e16	1.31045
$898$	0	0
$899$	4.70950e15	0.267483
$900$	0	0
$901$	−6.32176e14	−0.0354692
$902$	0	0
$903$	−7.53597e15	−0.417692
$904$	0	0
$905$	−5.49033e15	−0.300629
$906$	0	0
$907$	−1.93510e16	−1.04680	−0.523399	−	0.852088i	$-0.675336\pi$
$907$	−1.93510e16	−1.04680	−0.523399	+	0.852088i	$0.675336\pi$
$908$	0	0
$909$	1.00034e16	0.534622
$910$	0	0
$911$	9.32789e15	0.492529	0.246265	−	0.969203i	$-0.420797\pi$
$911$	9.32789e15	0.492529	0.246265	+	0.969203i	$0.420797\pi$
$912$	0	0
$913$	6.32378e15	0.329904
$914$	0	0
$915$	2.12386e15	0.109474
$916$	0	0
$917$	3.49029e16	1.77758
$918$	0	0
$919$	−2.71469e16	−1.36611	−0.683053	−	0.730369i	$-0.739348\pi$
$919$	−2.71469e16	−1.36611	−0.683053	+	0.730369i	$0.739348\pi$
$920$	0	0
$921$	1.37965e16	0.686026
$922$	0	0
$923$	4.22473e16	2.07582
$924$	0	0
$925$	−2.21324e16	−1.07461
$926$	0	0
$927$	1.17669e16	0.564579
$928$	0	0
$929$	2.27709e16	1.07968	0.539838	−	0.841769i	$-0.318485\pi$
$929$	2.27709e16	1.07968	0.539838	+	0.841769i	$0.318485\pi$
$930$	0	0
$931$	3.56645e15	0.167114
$932$	0	0
$933$	1.44233e16	0.667908
$934$	0	0
$935$	−4.03281e13	−0.00184562
$936$	0	0
$937$	4.79249e15	0.216767	0.108384	−	0.994109i	$-0.465433\pi$
$937$	4.79249e15	0.216767	0.108384	+	0.994109i	$0.465433\pi$
$938$	0	0
$939$	−7.21452e15	−0.322513
$940$	0	0
$941$	−1.91053e16	−0.844131	−0.422066	−	0.906565i	$-0.638695\pi$
$941$	−1.91053e16	−0.844131	−0.422066	+	0.906565i	$0.638695\pi$
$942$	0	0
$943$	−2.82463e15	−0.123352
$944$	0	0
$945$	−1.94031e15	−0.0837521
$946$	0	0
$947$	−1.19017e16	−0.507791	−0.253895	−	0.967232i	$-0.581712\pi$
$947$	−1.19017e16	−0.507791	−0.253895	+	0.967232i	$0.581712\pi$
$948$	0	0
$949$	−1.82482e16	−0.769582
$950$	0	0
$951$	−9.62048e15	−0.401055
$952$	0	0
$953$	−2.76011e16	−1.13740	−0.568702	−	0.822543i	$-0.692554\pi$
$953$	−2.76011e16	−1.13740	−0.568702	+	0.822543i	$0.692554\pi$
$954$	0	0
$955$	−6.03838e15	−0.245981
$956$	0	0
$957$	−7.05578e14	−0.0284138
$958$	0	0
$959$	−5.98096e16	−2.38105
$960$	0	0
$961$	3.21678e16	1.26602
$962$	0	0
$963$	−8.96308e15	−0.348749
$964$	0	0
$965$	−6.49579e15	−0.249880
$966$	0	0
$967$	−4.20202e16	−1.59813	−0.799066	−	0.601243i	$-0.794672\pi$
$967$	−4.20202e16	−1.59813	−0.799066	+	0.601243i	$0.794672\pi$
$968$	0	0
$969$	−3.88519e13	−0.00146094
$970$	0	0
$971$	4.99560e16	1.85730	0.928651	−	0.370956i	$-0.120970\pi$
$971$	4.99560e16	1.85730	0.928651	+	0.370956i	$0.120970\pi$
$972$	0	0
$973$	1.08613e16	0.399264
$974$	0	0
$975$	1.72156e16	0.625746
$976$	0	0
$977$	5.50836e15	0.197971	0.0989856	−	0.995089i	$-0.468440\pi$
$977$	5.50836e15	0.197971	0.0989856	+	0.995089i	$0.468440\pi$
$978$	0	0
$979$	−1.37985e16	−0.490373
$980$	0	0
$981$	−1.40727e16	−0.494537
$982$	0	0
$983$	1.93192e14	0.00671342	0.00335671	−	0.999994i	$-0.498932\pi$
$983$	1.93192e14	0.00671342	0.00335671	+	0.999994i	$0.498932\pi$
$984$	0	0
$985$	−9.42282e15	−0.323804
$986$	0	0
$987$	−7.92319e15	−0.269250
$988$	0	0
$989$	−2.57381e16	−0.864963
$990$	0	0
$991$	3.08029e16	1.02373	0.511866	−	0.859065i	$-0.328955\pi$
$991$	3.08029e16	1.02373	0.511866	+	0.859065i	$0.328955\pi$
$992$	0	0
$993$	8.19111e15	0.269229
$994$	0	0
$995$	−1.76034e15	−0.0572227
$996$	0	0
$997$	−4.87843e16	−1.56840	−0.784199	−	0.620509i	$-0.786926\pi$
$997$	−4.87843e16	−1.56840	−0.784199	+	0.620509i	$0.786926\pi$
$998$	0	0
$999$	7.00568e15	0.222762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.12.a.c.1.1	✓	1
3.2	odd	2		72.12.a.a.1.1		1
4.3	odd	2		48.12.a.b.1.1		1
8.3	odd	2		192.12.a.o.1.1		1
8.5	even	2		192.12.a.e.1.1		1
12.11	even	2		144.12.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.12.a.c.1.1	✓	1	1.1	even	1	trivial
48.12.a.b.1.1		1	4.3	odd	2
72.12.a.a.1.1		1	3.2	odd	2
144.12.a.g.1.1		1	12.11	even	2
192.12.a.e.1.1		1	8.5	even	2
192.12.a.o.1.1		1	8.3	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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	24.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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