LMFDB - Embedded newform 768.6.a.be.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,6,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.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:	$123.174773616$
Analytic rank:	$1$
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} - 2x^{5} - 167x^{4} + 62x^{3} + 7709x^{2} + 6040x - 68866 $ x^6 - 2x^5 - 167x^4 + 62x^3 + 7709x^2 + 6040*x - 68866
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$10.8525$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +46.1613 q^{5} +59.9278 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +46.1613 q^{5} +59.9278 q^{7} +81.0000 q^{9} -400.065 q^{11} +351.898 q^{13} -415.451 q^{15} +1662.58 q^{17} -564.314 q^{19} -539.350 q^{21} -3831.57 q^{23} -994.139 q^{25} -729.000 q^{27} -5944.12 q^{29} +2678.95 q^{31} +3600.59 q^{33} +2766.34 q^{35} +771.803 q^{37} -3167.08 q^{39} +5119.70 q^{41} +9518.90 q^{43} +3739.06 q^{45} -8138.39 q^{47} -13215.7 q^{49} -14963.2 q^{51} +11051.9 q^{53} -18467.5 q^{55} +5078.83 q^{57} -32143.3 q^{59} -35454.8 q^{61} +4854.15 q^{63} +16244.0 q^{65} +11362.8 q^{67} +34484.1 q^{69} -10394.7 q^{71} +40098.7 q^{73} +8947.25 q^{75} -23975.0 q^{77} -25754.8 q^{79} +6561.00 q^{81} +2239.75 q^{83} +76746.5 q^{85} +53497.1 q^{87} +15149.4 q^{89} +21088.5 q^{91} -24110.6 q^{93} -26049.5 q^{95} +158426. q^{97} -32405.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} + 486 q^{9}+O(q^{10}) $ 6 * q - 54 * q^3 + 486 * q^9	$ 6 q - 54 q^{3} + 486 q^{9} + 40 q^{11} - 2444 q^{17} - 728 q^{19} + 9330 q^{25} - 4374 q^{27} - 360 q^{33} - 16432 q^{35} + 12548 q^{41} - 24072 q^{43} + 67782 q^{49} + 21996 q^{51} + 6552 q^{57} + 64856 q^{59} + 127136 q^{65} + 21544 q^{67} + 93516 q^{73} - 83970 q^{75} + 39366 q^{81} - 303016 q^{83} + 287772 q^{89} - 262048 q^{91} + 46932 q^{97} + 3240 q^{99}+O(q^{100}) $ 6 * q - 54 * q^3 + 486 * q^9 + 40 * q^11 - 2444 * q^17 - 728 * q^19 + 9330 * q^25 - 4374 * q^27 - 360 * q^33 - 16432 * q^35 + 12548 * q^41 - 24072 * q^43 + 67782 * q^49 + 21996 * q^51 + 6552 * q^57 + 64856 * q^59 + 127136 * q^65 + 21544 * q^67 + 93516 * q^73 - 83970 * q^75 + 39366 * q^81 - 303016 * q^83 + 287772 * q^89 - 262048 * q^91 + 46932 * q^97 + 3240 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	46.1613	0.825758	0.412879	−	0.910786i	$-0.364523\pi$
$5$	46.1613	0.825758	0.412879	+	0.910786i	$0.364523\pi$
$6$	0	0
$7$	59.9278	0.462257	0.231128	−	0.972923i	$-0.425758\pi$
$7$	59.9278	0.462257	0.231128	+	0.972923i	$0.425758\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−400.065	−0.996894	−0.498447	−	0.866920i	$-0.666096\pi$
$11$	−400.065	−0.996894	−0.498447	+	0.866920i	$0.666096\pi$
$12$	0	0
$13$	351.898	0.577508	0.288754	−	0.957403i	$-0.406759\pi$
$13$	351.898	0.577508	0.288754	+	0.957403i	$0.406759\pi$
$14$	0	0
$15$	−415.451	−0.476751
$16$	0	0
$17$	1662.58	1.39527	0.697636	−	0.716452i	$-0.254235\pi$
$17$	1662.58	1.39527	0.697636	+	0.716452i	$0.254235\pi$
$18$	0	0
$19$	−564.314	−0.358622	−0.179311	−	0.983792i	$-0.557387\pi$
$19$	−564.314	−0.358622	−0.179311	+	0.983792i	$0.557387\pi$
$20$	0	0
$21$	−539.350	−0.266884
$22$	0	0
$23$	−3831.57	−1.51028	−0.755139	−	0.655565i	$-0.772430\pi$
$23$	−3831.57	−1.51028	−0.755139	+	0.655565i	$0.772430\pi$
$24$	0	0
$25$	−994.139	−0.318124
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5944.12	−1.31248	−0.656240	−	0.754552i	$-0.727854\pi$
$29$	−5944.12	−1.31248	−0.656240	+	0.754552i	$0.727854\pi$
$30$	0	0
$31$	2678.95	0.500680	0.250340	−	0.968158i	$-0.419458\pi$
$31$	2678.95	0.500680	0.250340	+	0.968158i	$0.419458\pi$
$32$	0	0
$33$	3600.59	0.575557
$34$	0	0
$35$	2766.34	0.381712
$36$	0	0
$37$	771.803	0.0926834	0.0463417	−	0.998926i	$-0.485244\pi$
$37$	771.803	0.0926834	0.0463417	+	0.998926i	$0.485244\pi$
$38$	0	0
$39$	−3167.08	−0.333425
$40$	0	0
$41$	5119.70	0.475647	0.237824	−	0.971308i	$-0.423566\pi$
$41$	5119.70	0.475647	0.237824	+	0.971308i	$0.423566\pi$
$42$	0	0
$43$	9518.90	0.785083	0.392541	−	0.919734i	$-0.371596\pi$
$43$	9518.90	0.785083	0.392541	+	0.919734i	$0.371596\pi$
$44$	0	0
$45$	3739.06	0.275253
$46$	0	0
$47$	−8138.39	−0.537395	−0.268698	−	0.963225i	$-0.586593\pi$
$47$	−8138.39	−0.537395	−0.268698	+	0.963225i	$0.586593\pi$
$48$	0	0
$49$	−13215.7	−0.786319
$50$	0	0
$51$	−14963.2	−0.805561
$52$	0	0
$53$	11051.9	0.540441	0.270220	−	0.962799i	$-0.412903\pi$
$53$	11051.9	0.540441	0.270220	+	0.962799i	$0.412903\pi$
$54$	0	0
$55$	−18467.5	−0.823193
$56$	0	0
$57$	5078.83	0.207051
$58$	0	0
$59$	−32143.3	−1.20215	−0.601077	−	0.799191i	$-0.705261\pi$
$59$	−32143.3	−1.20215	−0.601077	+	0.799191i	$0.705261\pi$
$60$	0	0
$61$	−35454.8	−1.21997	−0.609986	−	0.792412i	$-0.708825\pi$
$61$	−35454.8	−1.21997	−0.609986	+	0.792412i	$0.708825\pi$
$62$	0	0
$63$	4854.15	0.154086
$64$	0	0
$65$	16244.0	0.476882
$66$	0	0
$67$	11362.8	0.309242	0.154621	−	0.987974i	$-0.450584\pi$
$67$	11362.8	0.309242	0.154621	+	0.987974i	$0.450584\pi$
$68$	0	0
$69$	34484.1	0.871959
$70$	0	0
$71$	−10394.7	−0.244719	−0.122360	−	0.992486i	$-0.539046\pi$
$71$	−10394.7	−0.244719	−0.122360	+	0.992486i	$0.539046\pi$
$72$	0	0
$73$	40098.7	0.880689	0.440345	−	0.897829i	$-0.354856\pi$
$73$	40098.7	0.880689	0.440345	+	0.897829i	$0.354856\pi$
$74$	0	0
$75$	8947.25	0.183669
$76$	0	0
$77$	−23975.0	−0.460821
$78$	0	0
$79$	−25754.8	−0.464291	−0.232146	−	0.972681i	$-0.574575\pi$
$79$	−25754.8	−0.464291	−0.232146	+	0.972681i	$0.574575\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	2239.75	0.0356866	0.0178433	−	0.999841i	$-0.494320\pi$
$83$	2239.75	0.0356866	0.0178433	+	0.999841i	$0.494320\pi$
$84$	0	0
$85$	76746.5	1.15216
$86$	0	0
$87$	53497.1	0.757761
$88$	0	0
$89$	15149.4	0.202732	0.101366	−	0.994849i	$-0.467679\pi$
$89$	15149.4	0.202732	0.101366	+	0.994849i	$0.467679\pi$
$90$	0	0
$91$	21088.5	0.266957
$92$	0	0
$93$	−24110.6	−0.289068
$94$	0	0
$95$	−26049.5	−0.296135
$96$	0	0
$97$	158426.	1.70961	0.854804	−	0.518951i	$-0.173677\pi$
$97$	158426.	1.70961	0.854804	+	0.518951i	$0.173677\pi$
$98$	0	0
$99$	−32405.3	−0.332298
$100$	0	0
$101$	154639.	1.50840	0.754198	−	0.656647i	$-0.228026\pi$
$101$	154639.	1.50840	0.754198	+	0.656647i	$0.228026\pi$
$102$	0	0
$103$	89671.0	0.832835	0.416417	−	0.909174i	$-0.363286\pi$
$103$	89671.0	0.832835	0.416417	+	0.909174i	$0.363286\pi$
$104$	0	0
$105$	−24897.1	−0.220382
$106$	0	0
$107$	−113338.	−0.957006	−0.478503	−	0.878086i	$-0.658820\pi$
$107$	−113338.	−0.957006	−0.478503	+	0.878086i	$0.658820\pi$
$108$	0	0
$109$	−221403.	−1.78491	−0.892457	−	0.451133i	$-0.851020\pi$
$109$	−221403.	−1.78491	−0.892457	+	0.451133i	$0.851020\pi$
$110$	0	0
$111$	−6946.23	−0.0535108
$112$	0	0
$113$	−6039.39	−0.0444936	−0.0222468	−	0.999753i	$-0.507082\pi$
$113$	−6039.39	−0.0444936	−0.0222468	+	0.999753i	$0.507082\pi$
$114$	0	0
$115$	−176870.	−1.24712
$116$	0	0
$117$	28503.7	0.192503
$118$	0	0
$119$	99634.5	0.644974
$120$	0	0
$121$	−998.858	−0.00620212
$122$	0	0
$123$	−46077.3	−0.274615
$124$	0	0
$125$	−190145.	−1.08845
$126$	0	0
$127$	328162.	1.80542	0.902711	−	0.430247i	$-0.141573\pi$
$127$	328162.	1.80542	0.902711	+	0.430247i	$0.141573\pi$
$128$	0	0
$129$	−85670.1	−0.453268
$130$	0	0
$131$	−176176.	−0.896952	−0.448476	−	0.893795i	$-0.648033\pi$
$131$	−176176.	−0.896952	−0.448476	+	0.893795i	$0.648033\pi$
$132$	0	0
$133$	−33818.1	−0.165776
$134$	0	0
$135$	−33651.6	−0.158917
$136$	0	0
$137$	−50245.9	−0.228718	−0.114359	−	0.993440i	$-0.536481\pi$
$137$	−50245.9	−0.228718	−0.114359	+	0.993440i	$0.536481\pi$
$138$	0	0
$139$	−364925.	−1.60201	−0.801007	−	0.598655i	$-0.795702\pi$
$139$	−364925.	−1.60201	−0.801007	+	0.598655i	$0.795702\pi$
$140$	0	0
$141$	73245.5	0.310265
$142$	0	0
$143$	−140782.	−0.575715
$144$	0	0
$145$	−274388.	−1.08379
$146$	0	0
$147$	118941.	0.453981
$148$	0	0
$149$	502438.	1.85403	0.927016	−	0.375023i	$-0.122365\pi$
$149$	502438.	1.85403	0.927016	+	0.375023i	$0.122365\pi$
$150$	0	0
$151$	−225684.	−0.805485	−0.402743	−	0.915313i	$-0.631943\pi$
$151$	−225684.	−0.805485	−0.402743	+	0.915313i	$0.631943\pi$
$152$	0	0
$153$	134669.	0.465091
$154$	0	0
$155$	123664.	0.413441
$156$	0	0
$157$	−464228.	−1.50308	−0.751539	−	0.659688i	$-0.770688\pi$
$157$	−464228.	−1.50308	−0.751539	+	0.659688i	$0.770688\pi$
$158$	0	0
$159$	−99467.3	−0.312024
$160$	0	0
$161$	−229617.	−0.698136
$162$	0	0
$163$	−143142.	−0.421987	−0.210994	−	0.977487i	$-0.567670\pi$
$163$	−143142.	−0.421987	−0.210994	+	0.977487i	$0.567670\pi$
$164$	0	0
$165$	166208.	0.475271
$166$	0	0
$167$	−170711.	−0.473663	−0.236831	−	0.971551i	$-0.576109\pi$
$167$	−170711.	−0.473663	−0.236831	+	0.971551i	$0.576109\pi$
$168$	0	0
$169$	−247461.	−0.666484
$170$	0	0
$171$	−45709.5	−0.119541
$172$	0	0
$173$	38779.6	0.0985118	0.0492559	−	0.998786i	$-0.484315\pi$
$173$	38779.6	0.0985118	0.0492559	+	0.998786i	$0.484315\pi$
$174$	0	0
$175$	−59576.6	−0.147055
$176$	0	0
$177$	289289.	0.694064
$178$	0	0
$179$	−399489.	−0.931907	−0.465954	−	0.884809i	$-0.654289\pi$
$179$	−399489.	−0.931907	−0.465954	+	0.884809i	$0.654289\pi$
$180$	0	0
$181$	−168069.	−0.381322	−0.190661	−	0.981656i	$-0.561063\pi$
$181$	−168069.	−0.381322	−0.190661	+	0.981656i	$0.561063\pi$
$182$	0	0
$183$	319093.	0.704351
$184$	0	0
$185$	35627.4	0.0765340
$186$	0	0
$187$	−665138.	−1.39094
$188$	0	0
$189$	−43687.4	−0.0889614
$190$	0	0
$191$	230781.	0.457738	0.228869	−	0.973457i	$-0.426497\pi$
$191$	230781.	0.457738	0.228869	+	0.973457i	$0.426497\pi$
$192$	0	0
$193$	7236.58	0.0139843	0.00699214	−	0.999976i	$-0.497774\pi$
$193$	7236.58	0.0139843	0.00699214	+	0.999976i	$0.497774\pi$
$194$	0	0
$195$	−146196.	−0.275328
$196$	0	0
$197$	−582795.	−1.06992	−0.534959	−	0.844878i	$-0.679673\pi$
$197$	−582795.	−1.06992	−0.534959	+	0.844878i	$0.679673\pi$
$198$	0	0
$199$	−1.07918e6	−1.93180	−0.965899	−	0.258920i	$-0.916633\pi$
$199$	−1.07918e6	−1.93180	−0.965899	+	0.258920i	$0.916633\pi$
$200$	0	0
$201$	−102265.	−0.178541
$202$	0	0
$203$	−356218.	−0.606703
$204$	0	0
$205$	236332.	0.392769
$206$	0	0
$207$	−310357.	−0.503426
$208$	0	0
$209$	225763.	0.357508
$210$	0	0
$211$	−860927.	−1.33125	−0.665626	−	0.746286i	$-0.731835\pi$
$211$	−860927.	−1.33125	−0.665626	+	0.746286i	$0.731835\pi$
$212$	0	0
$213$	93552.7	0.141289
$214$	0	0
$215$	439404.	0.648288
$216$	0	0
$217$	160544.	0.231443
$218$	0	0
$219$	−360888.	−0.508466
$220$	0	0
$221$	585057.	0.805781
$222$	0	0
$223$	442484.	0.595848	0.297924	−	0.954590i	$-0.403706\pi$
$223$	442484.	0.595848	0.297924	+	0.954590i	$0.403706\pi$
$224$	0	0
$225$	−80525.2	−0.106041
$226$	0	0
$227$	−1.25053e6	−1.61076	−0.805378	−	0.592761i	$-0.798038\pi$
$227$	−1.25053e6	−1.61076	−0.805378	+	0.592761i	$0.798038\pi$
$228$	0	0
$229$	508471.	0.640734	0.320367	−	0.947293i	$-0.396194\pi$
$229$	508471.	0.640734	0.320367	+	0.947293i	$0.396194\pi$
$230$	0	0
$231$	215775.	0.266055
$232$	0	0
$233$	−1.28856e6	−1.55495	−0.777474	−	0.628915i	$-0.783499\pi$
$233$	−1.28856e6	−1.55495	−0.777474	+	0.628915i	$0.783499\pi$
$234$	0	0
$235$	−375678.	−0.443758
$236$	0	0
$237$	231793.	0.268059
$238$	0	0
$239$	−1.42040e6	−1.60848	−0.804240	−	0.594304i	$-0.797428\pi$
$239$	−1.42040e6	−1.60848	−0.804240	+	0.594304i	$0.797428\pi$
$240$	0	0
$241$	−196969.	−0.218452	−0.109226	−	0.994017i	$-0.534837\pi$
$241$	−196969.	−0.218452	−0.109226	+	0.994017i	$0.534837\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−610051.	−0.649309
$246$	0	0
$247$	−198581.	−0.207107
$248$	0	0
$249$	−20157.8	−0.0206037
$250$	0	0
$251$	−1.87291e6	−1.87643	−0.938215	−	0.346052i	$-0.887522\pi$
$251$	−1.87291e6	−1.87643	−0.938215	+	0.346052i	$0.887522\pi$
$252$	0	0
$253$	1.53288e6	1.50559
$254$	0	0
$255$	−690719.	−0.665198
$256$	0	0
$257$	−854792.	−0.807287	−0.403643	−	0.914916i	$-0.632256\pi$
$257$	−854792.	−0.807287	−0.403643	+	0.914916i	$0.632256\pi$
$258$	0	0
$259$	46252.4	0.0428435
$260$	0	0
$261$	−481474.	−0.437493
$262$	0	0
$263$	1.94301e6	1.73215	0.866076	−	0.499913i	$-0.166635\pi$
$263$	1.94301e6	1.73215	0.866076	+	0.499913i	$0.166635\pi$
$264$	0	0
$265$	510170.	0.446273
$266$	0	0
$267$	−136345.	−0.117047
$268$	0	0
$269$	223635.	0.188434	0.0942168	−	0.995552i	$-0.469965\pi$
$269$	223635.	0.188434	0.0942168	+	0.995552i	$0.469965\pi$
$270$	0	0
$271$	1.73900e6	1.43839	0.719196	−	0.694807i	$-0.244510\pi$
$271$	1.73900e6	1.43839	0.719196	+	0.694807i	$0.244510\pi$
$272$	0	0
$273$	−189796.	−0.154128
$274$	0	0
$275$	397720.	0.317136
$276$	0	0
$277$	−2.35793e6	−1.84642	−0.923211	−	0.384294i	$-0.874445\pi$
$277$	−2.35793e6	−1.84642	−0.923211	+	0.384294i	$0.874445\pi$
$278$	0	0
$279$	216995.	0.166893
$280$	0	0
$281$	923073.	0.697382	0.348691	−	0.937238i	$-0.386626\pi$
$281$	923073.	0.697382	0.348691	+	0.937238i	$0.386626\pi$
$282$	0	0
$283$	848669.	0.629901	0.314951	−	0.949108i	$-0.398012\pi$
$283$	848669.	0.629901	0.314951	+	0.949108i	$0.398012\pi$
$284$	0	0
$285$	234445.	0.170974
$286$	0	0
$287$	306813.	0.219871
$288$	0	0
$289$	1.34430e6	0.946785
$290$	0	0
$291$	−1.42583e6	−0.987043
$292$	0	0
$293$	−1.09123e6	−0.742585	−0.371293	−	0.928516i	$-0.621085\pi$
$293$	−1.09123e6	−0.742585	−0.371293	+	0.928516i	$0.621085\pi$
$294$	0	0
$295$	−1.48377e6	−0.992687
$296$	0	0
$297$	291648.	0.191852
$298$	0	0
$299$	−1.34832e6	−0.872197
$300$	0	0
$301$	570447.	0.362910
$302$	0	0
$303$	−1.39175e6	−0.870873
$304$	0	0
$305$	−1.63664e6	−1.00740
$306$	0	0
$307$	−1.55616e6	−0.942340	−0.471170	−	0.882042i	$-0.656168\pi$
$307$	−1.55616e6	−0.942340	−0.471170	+	0.882042i	$0.656168\pi$
$308$	0	0
$309$	−807039.	−0.480837
$310$	0	0
$311$	−457231.	−0.268062	−0.134031	−	0.990977i	$-0.542792\pi$
$311$	−457231.	−0.268062	−0.134031	+	0.990977i	$0.542792\pi$
$312$	0	0
$313$	1.73593e6	1.00154	0.500772	−	0.865579i	$-0.333049\pi$
$313$	1.73593e6	1.00154	0.500772	+	0.865579i	$0.333049\pi$
$314$	0	0
$315$	224074.	0.127237
$316$	0	0
$317$	450224.	0.251640	0.125820	−	0.992053i	$-0.459844\pi$
$317$	450224.	0.251640	0.125820	+	0.992053i	$0.459844\pi$
$318$	0	0
$319$	2.37804e6	1.30840
$320$	0	0
$321$	1.02004e6	0.552528
$322$	0	0
$323$	−938215.	−0.500376
$324$	0	0
$325$	−349835.	−0.183719
$326$	0	0
$327$	1.99263e6	1.03052
$328$	0	0
$329$	−487716.	−0.248415
$330$	0	0
$331$	−2.79853e6	−1.40397	−0.701987	−	0.712189i	$-0.747704\pi$
$331$	−2.79853e6	−1.40397	−0.701987	+	0.712189i	$0.747704\pi$
$332$	0	0
$333$	62516.0	0.0308945
$334$	0	0
$335$	524522.	0.255359
$336$	0	0
$337$	1.88545e6	0.904359	0.452180	−	0.891927i	$-0.350647\pi$
$337$	1.88545e6	0.904359	0.452180	+	0.891927i	$0.350647\pi$
$338$	0	0
$339$	54354.5	0.0256884
$340$	0	0
$341$	−1.07176e6	−0.499125
$342$	0	0
$343$	−1.79919e6	−0.825738
$344$	0	0
$345$	1.59183e6	0.720027
$346$	0	0
$347$	841512.	0.375178	0.187589	−	0.982248i	$-0.439933\pi$
$347$	841512.	0.375178	0.187589	+	0.982248i	$0.439933\pi$
$348$	0	0
$349$	3.05425e6	1.34227	0.671137	−	0.741333i	$-0.265806\pi$
$349$	3.05425e6	1.34227	0.671137	+	0.741333i	$0.265806\pi$
$350$	0	0
$351$	−256533.	−0.111142
$352$	0	0
$353$	3.20066e6	1.36711	0.683555	−	0.729899i	$-0.260433\pi$
$353$	3.20066e6	1.36711	0.683555	+	0.729899i	$0.260433\pi$
$354$	0	0
$355$	−479834.	−0.202079
$356$	0	0
$357$	−896710.	−0.372376
$358$	0	0
$359$	1.95304e6	0.799788	0.399894	−	0.916561i	$-0.369047\pi$
$359$	1.95304e6	0.799788	0.399894	+	0.916561i	$0.369047\pi$
$360$	0	0
$361$	−2.15765e6	−0.871390
$362$	0	0
$363$	8989.72	0.00358080
$364$	0	0
$365$	1.85100e6	0.727236
$366$	0	0
$367$	3.31321e6	1.28406	0.642028	−	0.766682i	$-0.278093\pi$
$367$	3.31321e6	1.28406	0.642028	+	0.766682i	$0.278093\pi$
$368$	0	0
$369$	414696.	0.158549
$370$	0	0
$371$	662317.	0.249822
$372$	0	0
$373$	3.65917e6	1.36179	0.680895	−	0.732381i	$-0.261591\pi$
$373$	3.65917e6	1.36179	0.680895	+	0.732381i	$0.261591\pi$
$374$	0	0
$375$	1.71130e6	0.628418
$376$	0	0
$377$	−2.09172e6	−0.757968
$378$	0	0
$379$	1.76629e6	0.631631	0.315816	−	0.948821i	$-0.397722\pi$
$379$	1.76629e6	0.631631	0.315816	+	0.948821i	$0.397722\pi$
$380$	0	0
$381$	−2.95346e6	−1.04236
$382$	0	0
$383$	−3.69691e6	−1.28778	−0.643890	−	0.765118i	$-0.722680\pi$
$383$	−3.69691e6	−1.28778	−0.643890	+	0.765118i	$0.722680\pi$
$384$	0	0
$385$	−1.10672e6	−0.380526
$386$	0	0
$387$	771031.	0.261694
$388$	0	0
$389$	−669156.	−0.224209	−0.112105	−	0.993696i	$-0.535759\pi$
$389$	−669156.	−0.224209	−0.112105	+	0.993696i	$0.535759\pi$
$390$	0	0
$391$	−6.37027e6	−2.10725
$392$	0	0
$393$	1.58559e6	0.517855
$394$	0	0
$395$	−1.18887e6	−0.383392
$396$	0	0
$397$	−2.19806e6	−0.699945	−0.349973	−	0.936760i	$-0.613809\pi$
$397$	−2.19806e6	−0.699945	−0.349973	+	0.936760i	$0.613809\pi$
$398$	0	0
$399$	304363.	0.0957105
$400$	0	0
$401$	−4.54750e6	−1.41225	−0.706125	−	0.708087i	$-0.749558\pi$
$401$	−4.54750e6	−1.41225	−0.706125	+	0.708087i	$0.749558\pi$
$402$	0	0
$403$	942717.	0.289147
$404$	0	0
$405$	302864.	0.0917508
$406$	0	0
$407$	−308771.	−0.0923956
$408$	0	0
$409$	1.46461e6	0.432926	0.216463	−	0.976291i	$-0.430548\pi$
$409$	1.46461e6	0.432926	0.216463	+	0.976291i	$0.430548\pi$
$410$	0	0
$411$	452213.	0.132050
$412$	0	0
$413$	−1.92628e6	−0.555704
$414$	0	0
$415$	103390.	0.0294685
$416$	0	0
$417$	3.28432e6	0.924923
$418$	0	0
$419$	868371.	0.241641	0.120820	−	0.992674i	$-0.461447\pi$
$419$	868371.	0.241641	0.120820	+	0.992674i	$0.461447\pi$
$420$	0	0
$421$	1.31760e6	0.362308	0.181154	−	0.983455i	$-0.442017\pi$
$421$	1.31760e6	0.362308	0.181154	+	0.983455i	$0.442017\pi$
$422$	0	0
$423$	−659210.	−0.179132
$424$	0	0
$425$	−1.65283e6	−0.443870
$426$	0	0
$427$	−2.12473e6	−0.563941
$428$	0	0
$429$	1.26704e6	0.332389
$430$	0	0
$431$	−3.26185e6	−0.845805	−0.422902	−	0.906175i	$-0.638989\pi$
$431$	−3.26185e6	−0.845805	−0.422902	+	0.906175i	$0.638989\pi$
$432$	0	0
$433$	−278010.	−0.0712591	−0.0356295	−	0.999365i	$-0.511344\pi$
$433$	−278010.	−0.0712591	−0.0356295	+	0.999365i	$0.511344\pi$
$434$	0	0
$435$	2.46949e6	0.625727
$436$	0	0
$437$	2.16221e6	0.541619
$438$	0	0
$439$	−2.77446e6	−0.687096	−0.343548	−	0.939135i	$-0.611629\pi$
$439$	−2.77446e6	−0.687096	−0.343548	+	0.939135i	$0.611629\pi$
$440$	0	0
$441$	−1.07047e6	−0.262106
$442$	0	0
$443$	5.97809e6	1.44728	0.723641	−	0.690176i	$-0.242467\pi$
$443$	5.97809e6	1.44728	0.723641	+	0.690176i	$0.242467\pi$
$444$	0	0
$445$	699317.	0.167407
$446$	0	0
$447$	−4.52194e6	−1.07043
$448$	0	0
$449$	−5.68566e6	−1.33096	−0.665480	−	0.746416i	$-0.731773\pi$
$449$	−5.68566e6	−1.33096	−0.665480	+	0.746416i	$0.731773\pi$
$450$	0	0
$451$	−2.04821e6	−0.474170
$452$	0	0
$453$	2.03115e6	0.465047
$454$	0	0
$455$	973470.	0.220442
$456$	0	0
$457$	2.65598e6	0.594886	0.297443	−	0.954740i	$-0.403866\pi$
$457$	2.65598e6	0.594886	0.297443	+	0.954740i	$0.403866\pi$
$458$	0	0
$459$	−1.21202e6	−0.268520
$460$	0	0
$461$	−3.88629e6	−0.851692	−0.425846	−	0.904796i	$-0.640024\pi$
$461$	−3.88629e6	−0.851692	−0.425846	+	0.904796i	$0.640024\pi$
$462$	0	0
$463$	3.78966e6	0.821576	0.410788	−	0.911731i	$-0.365254\pi$
$463$	3.78966e6	0.821576	0.410788	+	0.911731i	$0.365254\pi$
$464$	0	0
$465$	−1.11297e6	−0.238700
$466$	0	0
$467$	−2.31109e6	−0.490371	−0.245185	−	0.969476i	$-0.578849\pi$
$467$	−2.31109e6	−0.490371	−0.245185	+	0.969476i	$0.578849\pi$
$468$	0	0
$469$	680948.	0.142949
$470$	0	0
$471$	4.17805e6	0.867803
$472$	0	0
$473$	−3.80818e6	−0.782645
$474$	0	0
$475$	561007.	0.114086
$476$	0	0
$477$	895206.	0.180147
$478$	0	0
$479$	−4.62388e6	−0.920804	−0.460402	−	0.887710i	$-0.652295\pi$
$479$	−4.62388e6	−0.920804	−0.460402	+	0.887710i	$0.652295\pi$
$480$	0	0
$481$	271596.	0.0535254
$482$	0	0
$483$	2.06656e6	0.403069
$484$	0	0
$485$	7.31313e6	1.41172
$486$	0	0
$487$	7.55618e6	1.44371	0.721855	−	0.692044i	$-0.243290\pi$
$487$	7.55618e6	1.44371	0.721855	+	0.692044i	$0.243290\pi$
$488$	0	0
$489$	1.28828e6	0.243634
$490$	0	0
$491$	4.40919e6	0.825382	0.412691	−	0.910871i	$-0.364589\pi$
$491$	4.40919e6	0.825382	0.412691	+	0.910871i	$0.364589\pi$
$492$	0	0
$493$	−9.88255e6	−1.83127
$494$	0	0
$495$	−1.49587e6	−0.274398
$496$	0	0
$497$	−622934.	−0.113123
$498$	0	0
$499$	8.45546e6	1.52015	0.760074	−	0.649837i	$-0.225163\pi$
$499$	8.45546e6	1.52015	0.760074	+	0.649837i	$0.225163\pi$
$500$	0	0
$501$	1.53639e6	0.273469
$502$	0	0
$503$	−1.17007e6	−0.206201	−0.103100	−	0.994671i	$-0.532876\pi$
$503$	−1.17007e6	−0.206201	−0.103100	+	0.994671i	$0.532876\pi$
$504$	0	0
$505$	7.13833e6	1.24557
$506$	0	0
$507$	2.22715e6	0.384795
$508$	0	0
$509$	−9.48209e6	−1.62222	−0.811110	−	0.584893i	$-0.801136\pi$
$509$	−9.48209e6	−1.62222	−0.811110	+	0.584893i	$0.801136\pi$
$510$	0	0
$511$	2.40302e6	0.407105
$512$	0	0
$513$	411385.	0.0690169
$514$	0	0
$515$	4.13932e6	0.687720
$516$	0	0
$517$	3.25589e6	0.535726
$518$	0	0
$519$	−349016.	−0.0568758
$520$	0	0
$521$	−1.08606e7	−1.75290	−0.876451	−	0.481491i	$-0.840095\pi$
$521$	−1.08606e7	−1.75290	−0.876451	+	0.481491i	$0.840095\pi$
$522$	0	0
$523$	7.05571e6	1.12794	0.563971	−	0.825795i	$-0.309273\pi$
$523$	7.05571e6	1.12794	0.563971	+	0.825795i	$0.309273\pi$
$524$	0	0
$525$	536189.	0.0849023
$526$	0	0
$527$	4.45396e6	0.698585
$528$	0	0
$529$	8.24455e6	1.28094
$530$	0	0
$531$	−2.60360e6	−0.400718
$532$	0	0
$533$	1.80161e6	0.274690
$534$	0	0
$535$	−5.23181e6	−0.790255
$536$	0	0
$537$	3.59541e6	0.538037
$538$	0	0
$539$	5.28712e6	0.783876
$540$	0	0
$541$	−2.11215e6	−0.310264	−0.155132	−	0.987894i	$-0.549580\pi$
$541$	−2.11215e6	−0.310264	−0.155132	+	0.987894i	$0.549580\pi$
$542$	0	0
$543$	1.51262e6	0.220157
$544$	0	0
$545$	−1.02202e7	−1.47391
$546$	0	0
$547$	9.37947e6	1.34032	0.670162	−	0.742215i	$-0.266225\pi$
$547$	9.37947e6	1.34032	0.670162	+	0.742215i	$0.266225\pi$
$548$	0	0
$549$	−2.87184e6	−0.406657
$550$	0	0
$551$	3.35435e6	0.470685
$552$	0	0
$553$	−1.54343e6	−0.214622
$554$	0	0
$555$	−320646.	−0.0441869
$556$	0	0
$557$	3.61461e6	0.493655	0.246827	−	0.969059i	$-0.420612\pi$
$557$	3.61461e6	0.493655	0.246827	+	0.969059i	$0.420612\pi$
$558$	0	0
$559$	3.34968e6	0.453392
$560$	0	0
$561$	5.98625e6	0.803059
$562$	0	0
$563$	1.04180e7	1.38520	0.692601	−	0.721321i	$-0.256465\pi$
$563$	1.04180e7	1.38520	0.692601	+	0.721321i	$0.256465\pi$
$564$	0	0
$565$	−278786.	−0.0367409
$566$	0	0
$567$	393186.	0.0513619
$568$	0	0
$569$	−7.80034e6	−1.01003	−0.505014	−	0.863111i	$-0.668513\pi$
$569$	−7.80034e6	−1.01003	−0.505014	+	0.863111i	$0.668513\pi$
$570$	0	0
$571$	1.13367e7	1.45511	0.727555	−	0.686049i	$-0.240657\pi$
$571$	1.13367e7	1.45511	0.727555	+	0.686049i	$0.240657\pi$
$572$	0	0
$573$	−2.07703e6	−0.264275
$574$	0	0
$575$	3.80911e6	0.480456
$576$	0	0
$577$	3.41197e6	0.426644	0.213322	−	0.976982i	$-0.431572\pi$
$577$	3.41197e6	0.426644	0.213322	+	0.976982i	$0.431572\pi$
$578$	0	0
$579$	−65129.2	−0.00807383
$580$	0	0
$581$	134224.	0.0164964
$582$	0	0
$583$	−4.42149e6	−0.538762
$584$	0	0
$585$	1.31577e6	0.158961
$586$	0	0
$587$	−9.10881e6	−1.09111	−0.545553	−	0.838077i	$-0.683680\pi$
$587$	−9.10881e6	−1.09111	−0.545553	+	0.838077i	$0.683680\pi$
$588$	0	0
$589$	−1.51177e6	−0.179555
$590$	0	0
$591$	5.24516e6	0.617717
$592$	0	0
$593$	−5.87052e6	−0.685551	−0.342776	−	0.939417i	$-0.611367\pi$
$593$	−5.87052e6	−0.685551	−0.342776	+	0.939417i	$0.611367\pi$
$594$	0	0
$595$	4.59925e6	0.532592
$596$	0	0
$597$	9.71263e6	1.11532
$598$	0	0
$599$	9.18159e6	1.04556	0.522782	−	0.852466i	$-0.324894\pi$
$599$	9.18159e6	1.04556	0.522782	+	0.852466i	$0.324894\pi$
$600$	0	0
$601$	−1.01852e7	−1.15022	−0.575112	−	0.818075i	$-0.695041\pi$
$601$	−1.01852e7	−1.15022	−0.575112	+	0.818075i	$0.695041\pi$
$602$	0	0
$603$	920388.	0.103081
$604$	0	0
$605$	−46108.5	−0.00512145
$606$	0	0
$607$	−4.74894e6	−0.523148	−0.261574	−	0.965183i	$-0.584242\pi$
$607$	−4.74894e6	−0.523148	−0.261574	+	0.965183i	$0.584242\pi$
$608$	0	0
$609$	3.20596e6	0.350280
$610$	0	0
$611$	−2.86388e6	−0.310350
$612$	0	0
$613$	1.16623e6	0.125352	0.0626760	−	0.998034i	$-0.480037\pi$
$613$	1.16623e6	0.125352	0.0626760	+	0.998034i	$0.480037\pi$
$614$	0	0
$615$	−2.12699e6	−0.226766
$616$	0	0
$617$	1.29443e6	0.136888	0.0684440	−	0.997655i	$-0.478197\pi$
$617$	1.29443e6	0.136888	0.0684440	+	0.997655i	$0.478197\pi$
$618$	0	0
$619$	−4.23949e6	−0.444721	−0.222360	−	0.974965i	$-0.571376\pi$
$619$	−4.23949e6	−0.444721	−0.222360	+	0.974965i	$0.571376\pi$
$620$	0	0
$621$	2.79321e6	0.290653
$622$	0	0
$623$	907873.	0.0937141
$624$	0	0
$625$	−5.67063e6	−0.580672
$626$	0	0
$627$	−2.03186e6	−0.206408
$628$	0	0
$629$	1.28318e6	0.129319
$630$	0	0
$631$	−1.24132e6	−0.124111	−0.0620553	−	0.998073i	$-0.519766\pi$
$631$	−1.24132e6	−0.124111	−0.0620553	+	0.998073i	$0.519766\pi$
$632$	0	0
$633$	7.74834e6	0.768599
$634$	0	0
$635$	1.51484e7	1.49084
$636$	0	0
$637$	−4.65056e6	−0.454105
$638$	0	0
$639$	−841974.	−0.0815730
$640$	0	0
$641$	−1.09152e7	−1.04927	−0.524635	−	0.851327i	$-0.675798\pi$
$641$	−1.09152e7	−1.04927	−0.524635	+	0.851327i	$0.675798\pi$
$642$	0	0
$643$	−1.48054e7	−1.41219	−0.706095	−	0.708118i	$-0.749545\pi$
$643$	−1.48054e7	−1.41219	−0.706095	+	0.708118i	$0.749545\pi$
$644$	0	0
$645$	−3.95464e6	−0.374289
$646$	0	0
$647$	−7.50071e6	−0.704436	−0.352218	−	0.935918i	$-0.614572\pi$
$647$	−7.50071e6	−0.704436	−0.352218	+	0.935918i	$0.614572\pi$
$648$	0	0
$649$	1.28594e7	1.19842
$650$	0	0
$651$	−1.44489e6	−0.133624
$652$	0	0
$653$	1.84442e7	1.69269	0.846343	−	0.532638i	$-0.178799\pi$
$653$	1.84442e7	1.69269	0.846343	+	0.532638i	$0.178799\pi$
$654$	0	0
$655$	−8.13252e6	−0.740665
$656$	0	0
$657$	3.24799e6	0.293563
$658$	0	0
$659$	1.50202e7	1.34729	0.673645	−	0.739055i	$-0.264728\pi$
$659$	1.50202e7	1.34729	0.673645	+	0.739055i	$0.264728\pi$
$660$	0	0
$661$	−1.62211e7	−1.44403	−0.722017	−	0.691876i	$-0.756785\pi$
$661$	−1.62211e7	−1.44403	−0.722017	+	0.691876i	$0.756785\pi$
$662$	0	0
$663$	−5.26551e6	−0.465218
$664$	0	0
$665$	−1.56109e6	−0.136890
$666$	0	0
$667$	2.27753e7	1.98221
$668$	0	0
$669$	−3.98236e6	−0.344013
$670$	0	0
$671$	1.41842e7	1.21618
$672$	0	0
$673$	8.38719e6	0.713804	0.356902	−	0.934142i	$-0.383833\pi$
$673$	8.38719e6	0.713804	0.356902	+	0.934142i	$0.383833\pi$
$674$	0	0
$675$	724727.	0.0612231
$676$	0	0
$677$	−5.22589e6	−0.438216	−0.219108	−	0.975701i	$-0.570315\pi$
$677$	−5.22589e6	−0.438216	−0.219108	+	0.975701i	$0.570315\pi$
$678$	0	0
$679$	9.49411e6	0.790278
$680$	0	0
$681$	1.12548e7	0.929970
$682$	0	0
$683$	−1.61671e7	−1.32611	−0.663056	−	0.748570i	$-0.730741\pi$
$683$	−1.61671e7	−1.32611	−0.663056	+	0.748570i	$0.730741\pi$
$684$	0	0
$685$	−2.31942e6	−0.188865
$686$	0	0
$687$	−4.57624e6	−0.369928
$688$	0	0
$689$	3.88915e6	0.312109
$690$	0	0
$691$	2.38372e7	1.89915	0.949577	−	0.313533i	$-0.101513\pi$
$691$	2.38372e7	1.89915	0.949577	+	0.313533i	$0.101513\pi$
$692$	0	0
$693$	−1.94198e6	−0.153607
$694$	0	0
$695$	−1.68454e7	−1.32287
$696$	0	0
$697$	8.51189e6	0.663658
$698$	0	0
$699$	1.15971e7	0.897750
$700$	0	0
$701$	9.30715e6	0.715355	0.357678	−	0.933845i	$-0.383569\pi$
$701$	9.30715e6	0.715355	0.357678	+	0.933845i	$0.383569\pi$
$702$	0	0
$703$	−435539.	−0.0332383
$704$	0	0
$705$	3.38110e6	0.256204
$706$	0	0
$707$	9.26717e6	0.697266
$708$	0	0
$709$	−5.17752e6	−0.386818	−0.193409	−	0.981118i	$-0.561954\pi$
$709$	−5.17752e6	−0.386818	−0.193409	+	0.981118i	$0.561954\pi$
$710$	0	0
$711$	−2.08614e6	−0.154764
$712$	0	0
$713$	−1.02646e7	−0.756166
$714$	0	0
$715$	−6.49868e6	−0.475401
$716$	0	0
$717$	1.27836e7	0.928657
$718$	0	0
$719$	−3.77696e6	−0.272471	−0.136236	−	0.990676i	$-0.543500\pi$
$719$	−3.77696e6	−0.272471	−0.136236	+	0.990676i	$0.543500\pi$
$720$	0	0
$721$	5.37378e6	0.384983
$722$	0	0
$723$	1.77272e6	0.126123
$724$	0	0
$725$	5.90928e6	0.417532
$726$	0	0
$727$	2.50517e7	1.75793	0.878966	−	0.476885i	$-0.158234\pi$
$727$	2.50517e7	1.75793	0.878966	+	0.476885i	$0.158234\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.58259e7	1.09540
$732$	0	0
$733$	2.58312e7	1.77576	0.887881	−	0.460074i	$-0.152177\pi$
$733$	2.58312e7	1.77576	0.887881	+	0.460074i	$0.152177\pi$
$734$	0	0
$735$	5.49046e6	0.374879
$736$	0	0
$737$	−4.54587e6	−0.308282
$738$	0	0
$739$	−1.40532e7	−0.946593	−0.473297	−	0.880903i	$-0.656936\pi$
$739$	−1.40532e7	−0.946593	−0.473297	+	0.880903i	$0.656936\pi$
$740$	0	0
$741$	1.78723e6	0.119573
$742$	0	0
$743$	3.60964e6	0.239878	0.119939	−	0.992781i	$-0.461730\pi$
$743$	3.60964e6	0.239878	0.119939	+	0.992781i	$0.461730\pi$
$744$	0	0
$745$	2.31932e7	1.53098
$746$	0	0
$747$	181420.	0.0118955
$748$	0	0
$749$	−6.79207e6	−0.442382
$750$	0	0
$751$	−2.91748e7	−1.88759	−0.943795	−	0.330531i	$-0.892772\pi$
$751$	−2.91748e7	−1.88759	−0.943795	+	0.330531i	$0.892772\pi$
$752$	0	0
$753$	1.68562e7	1.08336
$754$	0	0
$755$	−1.04178e7	−0.665136
$756$	0	0
$757$	−1.12305e7	−0.712294	−0.356147	−	0.934430i	$-0.615910\pi$
$757$	−1.12305e7	−0.712294	−0.356147	+	0.934430i	$0.615910\pi$
$758$	0	0
$759$	−1.37959e7	−0.869251
$760$	0	0
$761$	−1.69614e7	−1.06170	−0.530849	−	0.847466i	$-0.678127\pi$
$761$	−1.69614e7	−1.06170	−0.530849	+	0.847466i	$0.678127\pi$
$762$	0	0
$763$	−1.32682e7	−0.825089
$764$	0	0
$765$	6.21647e6	0.384052
$766$	0	0
$767$	−1.13111e7	−0.694254
$768$	0	0
$769$	2.61093e7	1.59214	0.796068	−	0.605208i	$-0.206910\pi$
$769$	2.61093e7	1.59214	0.796068	+	0.605208i	$0.206910\pi$
$770$	0	0
$771$	7.69313e6	0.466087
$772$	0	0
$773$	−2.28744e7	−1.37690	−0.688448	−	0.725286i	$-0.741708\pi$
$773$	−2.28744e7	−1.37690	−0.688448	+	0.725286i	$0.741708\pi$
$774$	0	0
$775$	−2.66325e6	−0.159279
$776$	0	0
$777$	−416272.	−0.0247357
$778$	0	0
$779$	−2.88912e6	−0.170578
$780$	0	0
$781$	4.15857e6	0.243959
$782$	0	0
$783$	4.33327e6	0.252587
$784$	0	0
$785$	−2.14293e7	−1.24118
$786$	0	0
$787$	1.44464e6	0.0831423	0.0415712	−	0.999136i	$-0.486764\pi$
$787$	1.44464e6	0.0831423	0.0415712	+	0.999136i	$0.486764\pi$
$788$	0	0
$789$	−1.74871e7	−1.00006
$790$	0	0
$791$	−361927.	−0.0205674
$792$	0	0
$793$	−1.24765e7	−0.704544
$794$	0	0
$795$	−4.59153e6	−0.257656
$796$	0	0
$797$	5.37136e6	0.299529	0.149764	−	0.988722i	$-0.452148\pi$
$797$	5.37136e6	0.299529	0.149764	+	0.988722i	$0.452148\pi$
$798$	0	0
$799$	−1.35307e7	−0.749813
$800$	0	0
$801$	1.22710e6	0.0675772
$802$	0	0
$803$	−1.60421e7	−0.877954
$804$	0	0
$805$	−1.05994e7	−0.576491
$806$	0	0
$807$	−2.01271e6	−0.108792
$808$	0	0
$809$	−8.86137e6	−0.476025	−0.238012	−	0.971262i	$-0.576496\pi$
$809$	−8.86137e6	−0.476025	−0.238012	+	0.971262i	$0.576496\pi$
$810$	0	0
$811$	−2.91251e7	−1.55495	−0.777474	−	0.628915i	$-0.783499\pi$
$811$	−2.91251e7	−1.55495	−0.777474	+	0.628915i	$0.783499\pi$
$812$	0	0
$813$	−1.56510e7	−0.830456
$814$	0	0
$815$	−6.60763e6	−0.348459
$816$	0	0
$817$	−5.37165e6	−0.281548
$818$	0	0
$819$	1.70817e6	0.0889857
$820$	0	0
$821$	−2.62586e7	−1.35961	−0.679803	−	0.733394i	$-0.737935\pi$
$821$	−2.62586e7	−1.35961	−0.679803	+	0.733394i	$0.737935\pi$
$822$	0	0
$823$	1.88150e7	0.968287	0.484143	−	0.874989i	$-0.339131\pi$
$823$	1.88150e7	0.968287	0.484143	+	0.874989i	$0.339131\pi$
$824$	0	0
$825$	−3.57948e6	−0.183099
$826$	0	0
$827$	1.26962e7	0.645520	0.322760	−	0.946481i	$-0.395389\pi$
$827$	1.26962e7	0.645520	0.322760	+	0.946481i	$0.395389\pi$
$828$	0	0
$829$	2.01838e7	1.02004	0.510018	−	0.860164i	$-0.329639\pi$
$829$	2.01838e7	1.02004	0.510018	+	0.860164i	$0.329639\pi$
$830$	0	0
$831$	2.12213e7	1.06603
$832$	0	0
$833$	−2.19720e7	−1.09713
$834$	0	0
$835$	−7.88021e6	−0.391130
$836$	0	0
$837$	−1.95296e6	−0.0963560
$838$	0	0
$839$	−1.27185e7	−0.623781	−0.311890	−	0.950118i	$-0.600962\pi$
$839$	−1.27185e7	−0.623781	−0.311890	+	0.950118i	$0.600962\pi$
$840$	0	0
$841$	1.48215e7	0.722605
$842$	0	0
$843$	−8.30766e6	−0.402633
$844$	0	0
$845$	−1.14231e7	−0.550354
$846$	0	0
$847$	−59859.3	−0.00286697
$848$	0	0
$849$	−7.63802e6	−0.363674
$850$	0	0
$851$	−2.95721e6	−0.139978
$852$	0	0
$853$	1.95957e7	0.922121	0.461060	−	0.887369i	$-0.347469\pi$
$853$	1.95957e7	0.922121	0.461060	+	0.887369i	$0.347469\pi$
$854$	0	0
$855$	−2.11001e6	−0.0987117
$856$	0	0
$857$	−3.68287e7	−1.71291	−0.856454	−	0.516224i	$-0.827337\pi$
$857$	−3.68287e7	−1.71291	−0.856454	+	0.516224i	$0.827337\pi$
$858$	0	0
$859$	1.70593e7	0.788823	0.394411	−	0.918934i	$-0.370949\pi$
$859$	1.70593e7	0.788823	0.394411	+	0.918934i	$0.370949\pi$
$860$	0	0
$861$	−2.76131e6	−0.126943
$862$	0	0
$863$	3.56411e7	1.62901	0.814506	−	0.580155i	$-0.197008\pi$
$863$	3.56411e7	1.62901	0.814506	+	0.580155i	$0.197008\pi$
$864$	0	0
$865$	1.79012e6	0.0813468
$866$	0	0
$867$	−1.20987e7	−0.546626
$868$	0	0
$869$	1.03036e7	0.462849
$870$	0	0
$871$	3.99855e6	0.178590
$872$	0	0
$873$	1.28325e7	0.569869
$874$	0	0
$875$	−1.13949e7	−0.503144
$876$	0	0
$877$	6.78596e6	0.297929	0.148964	−	0.988843i	$-0.452406\pi$
$877$	6.78596e6	0.297929	0.148964	+	0.988843i	$0.452406\pi$
$878$	0	0
$879$	9.82105e6	0.428732
$880$	0	0
$881$	6.27597e6	0.272421	0.136211	−	0.990680i	$-0.456508\pi$
$881$	6.27597e6	0.272421	0.136211	+	0.990680i	$0.456508\pi$
$882$	0	0
$883$	−2.46975e7	−1.06599	−0.532993	−	0.846120i	$-0.678933\pi$
$883$	−2.46975e7	−1.06599	−0.532993	+	0.846120i	$0.678933\pi$
$884$	0	0
$885$	1.33540e7	0.573128
$886$	0	0
$887$	−1.04054e6	−0.0444070	−0.0222035	−	0.999753i	$-0.507068\pi$
$887$	−1.04054e6	−0.0444070	−0.0222035	+	0.999753i	$0.507068\pi$
$888$	0	0
$889$	1.96660e7	0.834569
$890$	0	0
$891$	−2.62483e6	−0.110766
$892$	0	0
$893$	4.59261e6	0.192722
$894$	0	0
$895$	−1.84409e7	−0.769530
$896$	0	0
$897$	1.21349e7	0.503563
$898$	0	0
$899$	−1.59240e7	−0.657133
$900$	0	0
$901$	1.83746e7	0.754062
$902$	0	0
$903$	−5.13402e6	−0.209526
$904$	0	0
$905$	−7.75829e6	−0.314880
$906$	0	0
$907$	−2.41754e7	−0.975789	−0.487894	−	0.872903i	$-0.662235\pi$
$907$	−2.41754e7	−0.975789	−0.487894	+	0.872903i	$0.662235\pi$
$908$	0	0
$909$	1.25258e7	0.502799
$910$	0	0
$911$	3.64416e7	1.45479	0.727396	−	0.686217i	$-0.240730\pi$
$911$	3.64416e7	1.45479	0.727396	+	0.686217i	$0.240730\pi$
$912$	0	0
$913$	−896047.	−0.0355758
$914$	0	0
$915$	1.47297e7	0.581624
$916$	0	0
$917$	−1.05579e7	−0.414622
$918$	0	0
$919$	3.29796e7	1.28812	0.644060	−	0.764975i	$-0.277249\pi$
$919$	3.29796e7	1.28812	0.644060	+	0.764975i	$0.277249\pi$
$920$	0	0
$921$	1.40054e7	0.544060
$922$	0	0
$923$	−3.65789e6	−0.141327
$924$	0	0
$925$	−767279.	−0.0294849
$926$	0	0
$927$	7.26335e6	0.277612
$928$	0	0
$929$	−3.83111e7	−1.45642	−0.728209	−	0.685356i	$-0.759647\pi$
$929$	−3.83111e7	−1.45642	−0.728209	+	0.685356i	$0.759647\pi$
$930$	0	0
$931$	7.45779e6	0.281991
$932$	0	0
$933$	4.11508e6	0.154765
$934$	0	0
$935$	−3.07036e7	−1.14858
$936$	0	0
$937$	−1.48708e7	−0.553332	−0.276666	−	0.960966i	$-0.589230\pi$
$937$	−1.48708e7	−0.553332	−0.276666	+	0.960966i	$0.589230\pi$
$938$	0	0
$939$	−1.56233e7	−0.578242
$940$	0	0
$941$	2.67852e7	0.986099	0.493049	−	0.870001i	$-0.335882\pi$
$941$	2.67852e7	0.986099	0.493049	+	0.870001i	$0.335882\pi$
$942$	0	0
$943$	−1.96165e7	−0.718359
$944$	0	0
$945$	−2.01666e6	−0.0734605
$946$	0	0
$947$	−1.24319e7	−0.450467	−0.225233	−	0.974305i	$-0.572314\pi$
$947$	−1.24319e7	−0.450467	−0.225233	+	0.974305i	$0.572314\pi$
$948$	0	0
$949$	1.41106e7	0.508605
$950$	0	0
$951$	−4.05201e6	−0.145285
$952$	0	0
$953$	3.05330e7	1.08902	0.544511	−	0.838753i	$-0.316715\pi$
$953$	3.05330e7	1.08902	0.544511	+	0.838753i	$0.316715\pi$
$954$	0	0
$955$	1.06532e7	0.377981
$956$	0	0
$957$	−2.14023e7	−0.755407
$958$	0	0
$959$	−3.01113e6	−0.105726
$960$	0	0
$961$	−2.14524e7	−0.749319
$962$	0	0
$963$	−9.18035e6	−0.319002
$964$	0	0
$965$	334050.	0.0115476
$966$	0	0
$967$	3.23903e7	1.11391	0.556954	−	0.830543i	$-0.311970\pi$
$967$	3.23903e7	1.11391	0.556954	+	0.830543i	$0.311970\pi$
$968$	0	0
$969$	8.44394e6	0.288892
$970$	0	0
$971$	−5.00157e7	−1.70239	−0.851193	−	0.524853i	$-0.824120\pi$
$971$	−5.00157e7	−1.70239	−0.851193	+	0.524853i	$0.824120\pi$
$972$	0	0
$973$	−2.18691e7	−0.740542
$974$	0	0
$975$	3.14852e6	0.106070
$976$	0	0
$977$	621156.	0.0208192	0.0104096	−	0.999946i	$-0.496686\pi$
$977$	621156.	0.0208192	0.0104096	+	0.999946i	$0.496686\pi$
$978$	0	0
$979$	−6.06076e6	−0.202102
$980$	0	0
$981$	−1.79336e7	−0.594971
$982$	0	0
$983$	6.73840e6	0.222420	0.111210	−	0.993797i	$-0.464527\pi$
$983$	6.73840e6	0.222420	0.111210	+	0.993797i	$0.464527\pi$
$984$	0	0
$985$	−2.69026e7	−0.883493
$986$	0	0
$987$	4.38944e6	0.143422
$988$	0	0
$989$	−3.64723e7	−1.18569
$990$	0	0
$991$	1.73933e7	0.562597	0.281298	−	0.959620i	$-0.409235\pi$
$991$	1.73933e7	0.562597	0.281298	+	0.959620i	$0.409235\pi$
$992$	0	0
$993$	2.51867e7	0.810585
$994$	0	0
$995$	−4.98163e7	−1.59520
$996$	0	0
$997$	9.67436e6	0.308237	0.154118	−	0.988052i	$-0.450746\pi$
$997$	9.67436e6	0.308237	0.154118	+	0.988052i	$0.450746\pi$
$998$	0	0
$999$	−562644.	−0.0178369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.be.1.5		6
4.3	odd	2		768.6.a.bf.1.5		6
8.3	odd	2	inner	768.6.a.be.1.2		6
8.5	even	2		768.6.a.bf.1.2		6
16.3	odd	4		384.6.d.j.193.8	yes	12
16.5	even	4		384.6.d.j.193.11	yes	12
16.11	odd	4		384.6.d.j.193.5	yes	12
16.13	even	4		384.6.d.j.193.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.j.193.2	✓	12	16.13	even	4
384.6.d.j.193.5	yes	12	16.11	odd	4
384.6.d.j.193.8	yes	12	16.3	odd	4
384.6.d.j.193.11	yes	12	16.5	even	4
768.6.a.be.1.2		6	8.3	odd	2	inner
768.6.a.be.1.5		6	1.1	even	1	trivial
768.6.a.bf.1.2		6	8.5	even	2
768.6.a.bf.1.5		6	4.3	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +46.1613 q^{5} +59.9278 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +46.1613 q^{5} +59.9278 q^{7} +81.0000 q^{9} -400.065 q^{11} +351.898 q^{13} -415.451 q^{15} +1662.58 q^{17} -564.314 q^{19} -539.350 q^{21} -3831.57 q^{23} -994.139 q^{25} -729.000 q^{27} -5944.12 q^{29} +2678.95 q^{31} +3600.59 q^{33} +2766.34 q^{35} +771.803 q^{37} -3167.08 q^{39} +5119.70 q^{41} +9518.90 q^{43} +3739.06 q^{45} -8138.39 q^{47} -13215.7 q^{49} -14963.2 q^{51} +11051.9 q^{53} -18467.5 q^{55} +5078.83 q^{57} -32143.3 q^{59} -35454.8 q^{61} +4854.15 q^{63} +16244.0 q^{65} +11362.8 q^{67} +34484.1 q^{69} -10394.7 q^{71} +40098.7 q^{73} +8947.25 q^{75} -23975.0 q^{77} -25754.8 q^{79} +6561.00 q^{81} +2239.75 q^{83} +76746.5 q^{85} +53497.1 q^{87} +15149.4 q^{89} +21088.5 q^{91} -24110.6 q^{93} -26049.5 q^{95} +158426. q^{97} -32405.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} + 486 q^{9}+O(q^{10}) \) 6 * q - 54 * q^3 + 486 * q^9	\( 6 q - 54 q^{3} + 486 q^{9} + 40 q^{11} - 2444 q^{17} - 728 q^{19} + 9330 q^{25} - 4374 q^{27} - 360 q^{33} - 16432 q^{35} + 12548 q^{41} - 24072 q^{43} + 67782 q^{49} + 21996 q^{51} + 6552 q^{57} + 64856 q^{59} + 127136 q^{65} + 21544 q^{67} + 93516 q^{73} - 83970 q^{75} + 39366 q^{81} - 303016 q^{83} + 287772 q^{89} - 262048 q^{91} + 46932 q^{97} + 3240 q^{99}+O(q^{100}) \) 6 * q - 54 * q^3 + 486 * q^9 + 40 * q^11 - 2444 * q^17 - 728 * q^19 + 9330 * q^25 - 4374 * q^27 - 360 * q^33 - 16432 * q^35 + 12548 * q^41 - 24072 * q^43 + 67782 * q^49 + 21996 * q^51 + 6552 * q^57 + 64856 * q^59 + 127136 * q^65 + 21544 * q^67 + 93516 * q^73 - 83970 * q^75 + 39366 * q^81 - 303016 * q^83 + 287772 * q^89 - 262048 * q^91 + 46932 * q^97 + 3240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more