LMFDB - Embedded newform 126.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,6,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("126.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	126.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:	$20.2083612964$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	126.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} -26.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} +O(q^{10})$

\(q+4.00000 q^{2} +16.0000 q^{4} -26.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} -104.000 q^{10} -664.000 q^{11} +318.000 q^{13} -196.000 q^{14} +256.000 q^{16} -1582.00 q^{17} +236.000 q^{19} -416.000 q^{20} -2656.00 q^{22} -2212.00 q^{23} -2449.00 q^{25} +1272.00 q^{26} -784.000 q^{28} +4954.00 q^{29} -7128.00 q^{31} +1024.00 q^{32} -6328.00 q^{34} +1274.00 q^{35} +4358.00 q^{37} +944.000 q^{38} -1664.00 q^{40} -10542.0 q^{41} -8452.00 q^{43} -10624.0 q^{44} -8848.00 q^{46} -5352.00 q^{47} +2401.00 q^{49} -9796.00 q^{50} +5088.00 q^{52} +33354.0 q^{53} +17264.0 q^{55} -3136.00 q^{56} +19816.0 q^{58} +15436.0 q^{59} -36762.0 q^{61} -28512.0 q^{62} +4096.00 q^{64} -8268.00 q^{65} +40972.0 q^{67} -25312.0 q^{68} +5096.00 q^{70} +9092.00 q^{71} -73454.0 q^{73} +17432.0 q^{74} +3776.00 q^{76} +32536.0 q^{77} +89400.0 q^{79} -6656.00 q^{80} -42168.0 q^{82} +6428.00 q^{83} +41132.0 q^{85} -33808.0 q^{86} -42496.0 q^{88} +122658. q^{89} -15582.0 q^{91} -35392.0 q^{92} -21408.0 q^{94} -6136.00 q^{95} +21370.0 q^{97} +9604.00 q^{98} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−26.0000	−0.465102	−0.232551	−	0.972584i	$-0.574707\pi$
$5$	−26.0000	−0.465102	−0.232551	+	0.972584i	$0.574707\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	0	0
$10$	−104.000	−0.328877
$11$	−664.000	−1.65457	−0.827287	−	0.561779i	$-0.810117\pi$
$11$	−664.000	−1.65457	−0.827287	+	0.561779i	$0.810117\pi$
$12$	0	0
$13$	318.000	0.521878	0.260939	−	0.965355i	$-0.415968\pi$
$13$	318.000	0.521878	0.260939	+	0.965355i	$0.415968\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1582.00	−1.32765	−0.663826	−	0.747887i	$-0.731068\pi$
$17$	−1582.00	−1.32765	−0.663826	+	0.747887i	$0.731068\pi$
$18$	0	0
$19$	236.000	0.149978	0.0749891	−	0.997184i	$-0.476108\pi$
$19$	236.000	0.149978	0.0749891	+	0.997184i	$0.476108\pi$
$20$	−416.000	−0.232551
$21$	0	0
$22$	−2656.00	−1.16996
$23$	−2212.00	−0.871898	−0.435949	−	0.899971i	$-0.643587\pi$
$23$	−2212.00	−0.871898	−0.435949	+	0.899971i	$0.643587\pi$
$24$	0	0
$25$	−2449.00	−0.783680
$26$	1272.00	0.369023
$27$	0	0
$28$	−784.000	−0.188982
$29$	4954.00	1.09386	0.546929	−	0.837179i	$-0.315797\pi$
$29$	4954.00	1.09386	0.546929	+	0.837179i	$0.315797\pi$
$30$	0	0
$31$	−7128.00	−1.33218	−0.666091	−	0.745871i	$-0.732034\pi$
$31$	−7128.00	−1.33218	−0.666091	+	0.745871i	$0.732034\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−6328.00	−0.938792
$35$	1274.00	0.175792
$36$	0	0
$37$	4358.00	0.523339	0.261669	−	0.965158i	$-0.415727\pi$
$37$	4358.00	0.523339	0.261669	+	0.965158i	$0.415727\pi$
$38$	944.000	0.106051
$39$	0	0
$40$	−1664.00	−0.164438
$41$	−10542.0	−0.979407	−0.489704	−	0.871889i	$-0.662895\pi$
$41$	−10542.0	−0.979407	−0.489704	+	0.871889i	$0.662895\pi$
$42$	0	0
$43$	−8452.00	−0.697089	−0.348545	−	0.937292i	$-0.613324\pi$
$43$	−8452.00	−0.697089	−0.348545	+	0.937292i	$0.613324\pi$
$44$	−10624.0	−0.827287
$45$	0	0
$46$	−8848.00	−0.616525
$47$	−5352.00	−0.353404	−0.176702	−	0.984264i	$-0.556543\pi$
$47$	−5352.00	−0.353404	−0.176702	+	0.984264i	$0.556543\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−9796.00	−0.554145
$51$	0	0
$52$	5088.00	0.260939
$53$	33354.0	1.63102	0.815508	−	0.578746i	$-0.196458\pi$
$53$	33354.0	1.63102	0.815508	+	0.578746i	$0.196458\pi$
$54$	0	0
$55$	17264.0	0.769546
$56$	−3136.00	−0.133631
$57$	0	0
$58$	19816.0	0.773475
$59$	15436.0	0.577304	0.288652	−	0.957434i	$-0.406793\pi$
$59$	15436.0	0.577304	0.288652	+	0.957434i	$0.406793\pi$
$60$	0	0
$61$	−36762.0	−1.26495	−0.632477	−	0.774579i	$-0.717962\pi$
$61$	−36762.0	−1.26495	−0.632477	+	0.774579i	$0.717962\pi$
$62$	−28512.0	−0.941995
$63$	0	0
$64$	4096.00	0.125000
$65$	−8268.00	−0.242726
$66$	0	0
$67$	40972.0	1.11506	0.557532	−	0.830155i	$-0.311748\pi$
$67$	40972.0	1.11506	0.557532	+	0.830155i	$0.311748\pi$
$68$	−25312.0	−0.663826
$69$	0	0
$70$	5096.00	0.124304
$71$	9092.00	0.214049	0.107025	−	0.994256i	$-0.465868\pi$
$71$	9092.00	0.214049	0.107025	+	0.994256i	$0.465868\pi$
$72$	0	0
$73$	−73454.0	−1.61327	−0.806637	−	0.591047i	$-0.798715\pi$
$73$	−73454.0	−1.61327	−0.806637	+	0.591047i	$0.798715\pi$
$74$	17432.0	0.370056
$75$	0	0
$76$	3776.00	0.0749891
$77$	32536.0	0.625370
$78$	0	0
$79$	89400.0	1.61165	0.805823	−	0.592156i	$-0.201723\pi$
$79$	89400.0	1.61165	0.805823	+	0.592156i	$0.201723\pi$
$80$	−6656.00	−0.116276
$81$	0	0
$82$	−42168.0	−0.692546
$83$	6428.00	0.102419	0.0512095	−	0.998688i	$-0.483692\pi$
$83$	6428.00	0.102419	0.0512095	+	0.998688i	$0.483692\pi$
$84$	0	0
$85$	41132.0	0.617494
$86$	−33808.0	−0.492916
$87$	0	0
$88$	−42496.0	−0.584980
$89$	122658.	1.64142	0.820712	−	0.571342i	$-0.193577\pi$
$89$	122658.	1.64142	0.820712	+	0.571342i	$0.193577\pi$
$90$	0	0
$91$	−15582.0	−0.197251
$92$	−35392.0	−0.435949
$93$	0	0
$94$	−21408.0	−0.249894
$95$	−6136.00	−0.0697552
$96$	0	0
$97$	21370.0	0.230608	0.115304	−	0.993330i	$-0.463216\pi$
$97$	21370.0	0.230608	0.115304	+	0.993330i	$0.463216\pi$
$98$	9604.00	0.101015
$99$	0	0
$100$	−39184.0	−0.391840
$101$	36814.0	0.359095	0.179548	−	0.983749i	$-0.442537\pi$
$101$	36814.0	0.359095	0.179548	+	0.983749i	$0.442537\pi$
$102$	0	0
$103$	104528.	0.970822	0.485411	−	0.874286i	$-0.338670\pi$
$103$	104528.	0.970822	0.485411	+	0.874286i	$0.338670\pi$
$104$	20352.0	0.184512
$105$	0	0
$106$	133416.	1.15330
$107$	−214440.	−1.81070	−0.905350	−	0.424667i	$-0.860391\pi$
$107$	−214440.	−1.81070	−0.905350	+	0.424667i	$0.860391\pi$
$108$	0	0
$109$	28798.0	0.232165	0.116082	−	0.993240i	$-0.462966\pi$
$109$	28798.0	0.232165	0.116082	+	0.993240i	$0.462966\pi$
$110$	69056.0	0.544151
$111$	0	0
$112$	−12544.0	−0.0944911
$113$	56014.0	0.412668	0.206334	−	0.978482i	$-0.433847\pi$
$113$	56014.0	0.412668	0.206334	+	0.978482i	$0.433847\pi$
$114$	0	0
$115$	57512.0	0.405521
$116$	79264.0	0.546929
$117$	0	0
$118$	61744.0	0.408216
$119$	77518.0	0.501805
$120$	0	0
$121$	279845.	1.73762
$122$	−147048.	−0.894457
$123$	0	0
$124$	−114048.	−0.666091
$125$	144924.	0.829593
$126$	0	0
$127$	185400.	1.02000	0.510000	−	0.860174i	$-0.329645\pi$
$127$	185400.	1.02000	0.510000	+	0.860174i	$0.329645\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−33072.0	−0.171634
$131$	−64532.0	−0.328547	−0.164273	−	0.986415i	$-0.552528\pi$
$131$	−64532.0	−0.328547	−0.164273	+	0.986415i	$0.552528\pi$
$132$	0	0
$133$	−11564.0	−0.0566864
$134$	163888.	0.788470
$135$	0	0
$136$	−101248.	−0.469396
$137$	−152930.	−0.696131	−0.348066	−	0.937470i	$-0.613161\pi$
$137$	−152930.	−0.696131	−0.348066	+	0.937470i	$0.613161\pi$
$138$	0	0
$139$	−343460.	−1.50778	−0.753892	−	0.656998i	$-0.771826\pi$
$139$	−343460.	−1.50778	−0.753892	+	0.656998i	$0.771826\pi$
$140$	20384.0	0.0878960
$141$	0	0
$142$	36368.0	0.151356
$143$	−211152.	−0.863486
$144$	0	0
$145$	−128804.	−0.508756
$146$	−293816.	−1.14076
$147$	0	0
$148$	69728.0	0.261669
$149$	174858.	0.645238	0.322619	−	0.946529i	$-0.395437\pi$
$149$	174858.	0.645238	0.322619	+	0.946529i	$0.395437\pi$
$150$	0	0
$151$	−452552.	−1.61520	−0.807600	−	0.589731i	$-0.799234\pi$
$151$	−452552.	−1.61520	−0.807600	+	0.589731i	$0.799234\pi$
$152$	15104.0	0.0530253
$153$	0	0
$154$	130144.	0.442204
$155$	185328.	0.619601
$156$	0	0
$157$	−499066.	−1.61588	−0.807940	−	0.589265i	$-0.799417\pi$
$157$	−499066.	−1.61588	−0.807940	+	0.589265i	$0.799417\pi$
$158$	357600.	1.13961
$159$	0	0
$160$	−26624.0	−0.0822192
$161$	108388.	0.329546
$162$	0	0
$163$	−475588.	−1.40204	−0.701022	−	0.713139i	$-0.747273\pi$
$163$	−475588.	−1.40204	−0.701022	+	0.713139i	$0.747273\pi$
$164$	−168672.	−0.489704
$165$	0	0
$166$	25712.0	0.0724212
$167$	−120224.	−0.333580	−0.166790	−	0.985992i	$-0.553340\pi$
$167$	−120224.	−0.333580	−0.166790	+	0.985992i	$0.553340\pi$
$168$	0	0
$169$	−270169.	−0.727644
$170$	164528.	0.436634
$171$	0	0
$172$	−135232.	−0.348545
$173$	−508874.	−1.29269	−0.646346	−	0.763045i	$-0.723704\pi$
$173$	−508874.	−1.29269	−0.646346	+	0.763045i	$0.723704\pi$
$174$	0	0
$175$	120001.	0.296203
$176$	−169984.	−0.413644
$177$	0	0
$178$	490632.	1.16066
$179$	−487560.	−1.13735	−0.568677	−	0.822561i	$-0.692544\pi$
$179$	−487560.	−1.13735	−0.568677	+	0.822561i	$0.692544\pi$
$180$	0	0
$181$	−544410.	−1.23518	−0.617589	−	0.786501i	$-0.711891\pi$
$181$	−544410.	−1.23518	−0.617589	+	0.786501i	$0.711891\pi$
$182$	−62328.0	−0.139478
$183$	0	0
$184$	−141568.	−0.308262
$185$	−113308.	−0.243406
$186$	0	0
$187$	1.05045e6	2.19670
$188$	−85632.0	−0.176702
$189$	0	0
$190$	−24544.0	−0.0493243
$191$	−376404.	−0.746570	−0.373285	−	0.927717i	$-0.621769\pi$
$191$	−376404.	−0.746570	−0.373285	+	0.927717i	$0.621769\pi$
$192$	0	0
$193$	844946.	1.63281	0.816405	−	0.577480i	$-0.195964\pi$
$193$	844946.	1.63281	0.816405	+	0.577480i	$0.195964\pi$
$194$	85480.0	0.163065
$195$	0	0
$196$	38416.0	0.0714286
$197$	492794.	0.904690	0.452345	−	0.891843i	$-0.350588\pi$
$197$	492794.	0.904690	0.452345	+	0.891843i	$0.350588\pi$
$198$	0	0
$199$	−914776.	−1.63750	−0.818751	−	0.574148i	$-0.805333\pi$
$199$	−914776.	−1.63750	−0.818751	+	0.574148i	$0.805333\pi$
$200$	−156736.	−0.277073
$201$	0	0
$202$	147256.	0.253919
$203$	−242746.	−0.413440
$204$	0	0
$205$	274092.	0.455524
$206$	418112.	0.686475
$207$	0	0
$208$	81408.0	0.130469
$209$	−156704.	−0.248150
$210$	0	0
$211$	311780.	0.482106	0.241053	−	0.970512i	$-0.422507\pi$
$211$	311780.	0.482106	0.241053	+	0.970512i	$0.422507\pi$
$212$	533664.	0.815508
$213$	0	0
$214$	−857760.	−1.28036
$215$	219752.	0.324218
$216$	0	0
$217$	349272.	0.503517
$218$	115192.	0.164165
$219$	0	0
$220$	276224.	0.384773
$221$	−503076.	−0.692872
$222$	0	0
$223$	−1.28776e6	−1.73409	−0.867047	−	0.498226i	$-0.833985\pi$
$223$	−1.28776e6	−1.73409	−0.867047	+	0.498226i	$0.833985\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	224056.	0.291800
$227$	−1.28905e6	−1.66037	−0.830187	−	0.557485i	$-0.811766\pi$
$227$	−1.28905e6	−1.66037	−0.830187	+	0.557485i	$0.811766\pi$
$228$	0	0
$229$	678214.	0.854630	0.427315	−	0.904103i	$-0.359460\pi$
$229$	678214.	0.854630	0.427315	+	0.904103i	$0.359460\pi$
$230$	230048.	0.286747
$231$	0	0
$232$	317056.	0.386737
$233$	1.11731e6	1.34829	0.674146	−	0.738598i	$-0.264512\pi$
$233$	1.11731e6	1.34829	0.674146	+	0.738598i	$0.264512\pi$
$234$	0	0
$235$	139152.	0.164369
$236$	246976.	0.288652
$237$	0	0
$238$	310072.	0.354830
$239$	1.26196e6	1.42906	0.714528	−	0.699606i	$-0.246641\pi$
$239$	1.26196e6	1.42906	0.714528	+	0.699606i	$0.246641\pi$
$240$	0	0
$241$	948218.	1.05164	0.525818	−	0.850597i	$-0.323759\pi$
$241$	948218.	1.05164	0.525818	+	0.850597i	$0.323759\pi$
$242$	1.11938e6	1.22868
$243$	0	0
$244$	−588192.	−0.632477
$245$	−62426.0	−0.0664432
$246$	0	0
$247$	75048.0	0.0782703
$248$	−456192.	−0.470997
$249$	0	0
$250$	579696.	0.586611
$251$	486396.	0.487310	0.243655	−	0.969862i	$-0.421653\pi$
$251$	486396.	0.487310	0.243655	+	0.969862i	$0.421653\pi$
$252$	0	0
$253$	1.46877e6	1.44262
$254$	741600.	0.721249
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.03910e6	0.981349	0.490675	−	0.871343i	$-0.336750\pi$
$257$	1.03910e6	0.981349	0.490675	+	0.871343i	$0.336750\pi$
$258$	0	0
$259$	−213542.	−0.197803
$260$	−132288.	−0.121363
$261$	0	0
$262$	−258128.	−0.232317
$263$	−1.35104e6	−1.20443	−0.602213	−	0.798335i	$-0.705714\pi$
$263$	−1.35104e6	−1.20443	−0.602213	+	0.798335i	$0.705714\pi$
$264$	0	0
$265$	−867204.	−0.758589
$266$	−46256.0	−0.0400833
$267$	0	0
$268$	655552.	0.557532
$269$	1.11811e6	0.942115	0.471057	−	0.882103i	$-0.343872\pi$
$269$	1.11811e6	0.942115	0.471057	+	0.882103i	$0.343872\pi$
$270$	0	0
$271$	−190104.	−0.157242	−0.0786209	−	0.996905i	$-0.525052\pi$
$271$	−190104.	−0.157242	−0.0786209	+	0.996905i	$0.525052\pi$
$272$	−404992.	−0.331913
$273$	0	0
$274$	−611720.	−0.492239
$275$	1.62614e6	1.29666
$276$	0	0
$277$	−200506.	−0.157010	−0.0785051	−	0.996914i	$-0.525015\pi$
$277$	−200506.	−0.157010	−0.0785051	+	0.996914i	$0.525015\pi$
$278$	−1.37384e6	−1.06616
$279$	0	0
$280$	81536.0	0.0621519
$281$	−1.09237e6	−0.825285	−0.412643	−	0.910893i	$-0.635394\pi$
$281$	−1.09237e6	−0.825285	−0.412643	+	0.910893i	$0.635394\pi$
$282$	0	0
$283$	1.81258e6	1.34534	0.672669	−	0.739944i	$-0.265148\pi$
$283$	1.81258e6	1.34534	0.672669	+	0.739944i	$0.265148\pi$
$284$	145472.	0.107025
$285$	0	0
$286$	−844608.	−0.610577
$287$	516558.	0.370181
$288$	0	0
$289$	1.08287e6	0.762659
$290$	−515216.	−0.359745
$291$	0	0
$292$	−1.17526e6	−0.806637
$293$	−2.10031e6	−1.42927	−0.714634	−	0.699499i	$-0.753407\pi$
$293$	−2.10031e6	−1.42927	−0.714634	+	0.699499i	$0.753407\pi$
$294$	0	0
$295$	−401336.	−0.268505
$296$	278912.	0.185028
$297$	0	0
$298$	699432.	0.456252
$299$	−703416.	−0.455024
$300$	0	0
$301$	414148.	0.263475
$302$	−1.81021e6	−1.14212
$303$	0	0
$304$	60416.0	0.0374945
$305$	955812.	0.588333
$306$	0	0
$307$	−1.64104e6	−0.993743	−0.496872	−	0.867824i	$-0.665518\pi$
$307$	−1.64104e6	−0.993743	−0.496872	+	0.867824i	$0.665518\pi$
$308$	520576.	0.312685
$309$	0	0
$310$	741312.	0.438124
$311$	945232.	0.554163	0.277081	−	0.960846i	$-0.410633\pi$
$311$	945232.	0.554163	0.277081	+	0.960846i	$0.410633\pi$
$312$	0	0
$313$	415354.	0.239639	0.119820	−	0.992796i	$-0.461768\pi$
$313$	415354.	0.239639	0.119820	+	0.992796i	$0.461768\pi$
$314$	−1.99626e6	−1.14260
$315$	0	0
$316$	1.43040e6	0.805823
$317$	−1.18481e6	−0.662220	−0.331110	−	0.943592i	$-0.607423\pi$
$317$	−1.18481e6	−0.662220	−0.331110	+	0.943592i	$0.607423\pi$
$318$	0	0
$319$	−3.28946e6	−1.80987
$320$	−106496.	−0.0581378
$321$	0	0
$322$	433552.	0.233024
$323$	−373352.	−0.199119
$324$	0	0
$325$	−778782.	−0.408985
$326$	−1.90235e6	−0.991395
$327$	0	0
$328$	−674688.	−0.346273
$329$	262248.	0.133574
$330$	0	0
$331$	1.37155e6	0.688083	0.344042	−	0.938954i	$-0.388204\pi$
$331$	1.37155e6	0.688083	0.344042	+	0.938954i	$0.388204\pi$
$332$	102848.	0.0512095
$333$	0	0
$334$	−480896.	−0.235877
$335$	−1.06527e6	−0.518619
$336$	0	0
$337$	963522.	0.462154	0.231077	−	0.972935i	$-0.425775\pi$
$337$	963522.	0.462154	0.231077	+	0.972935i	$0.425775\pi$
$338$	−1.08068e6	−0.514522
$339$	0	0
$340$	658112.	0.308747
$341$	4.73299e6	2.20419
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−540928.	−0.246458
$345$	0	0
$346$	−2.03550e6	−0.914071
$347$	−2.57731e6	−1.14906	−0.574531	−	0.818483i	$-0.694815\pi$
$347$	−2.57731e6	−1.14906	−0.574531	+	0.818483i	$0.694815\pi$
$348$	0	0
$349$	−3.06751e6	−1.34810	−0.674051	−	0.738684i	$-0.735447\pi$
$349$	−3.06751e6	−1.34810	−0.674051	+	0.738684i	$0.735447\pi$
$350$	480004.	0.209447
$351$	0	0
$352$	−679936.	−0.292490
$353$	3.10144e6	1.32473	0.662364	−	0.749182i	$-0.269553\pi$
$353$	3.10144e6	1.32473	0.662364	+	0.749182i	$0.269553\pi$
$354$	0	0
$355$	−236392.	−0.0995547
$356$	1.96253e6	0.820712
$357$	0	0
$358$	−1.95024e6	−0.804230
$359$	327508.	0.134118	0.0670588	−	0.997749i	$-0.478638\pi$
$359$	327508.	0.134118	0.0670588	+	0.997749i	$0.478638\pi$
$360$	0	0
$361$	−2.42040e6	−0.977507
$362$	−2.17764e6	−0.873403
$363$	0	0
$364$	−249312.	−0.0986256
$365$	1.90980e6	0.750337
$366$	0	0
$367$	−2.86739e6	−1.11128	−0.555638	−	0.831424i	$-0.687526\pi$
$367$	−2.86739e6	−1.11128	−0.555638	+	0.831424i	$0.687526\pi$
$368$	−566272.	−0.217974
$369$	0	0
$370$	−453232.	−0.172114
$371$	−1.63435e6	−0.616466
$372$	0	0
$373$	3.58029e6	1.33244	0.666218	−	0.745757i	$-0.267912\pi$
$373$	3.58029e6	1.33244	0.666218	+	0.745757i	$0.267912\pi$
$374$	4.20179e6	1.55330
$375$	0	0
$376$	−342528.	−0.124947
$377$	1.57537e6	0.570860
$378$	0	0
$379$	1.64235e6	0.587310	0.293655	−	0.955912i	$-0.405128\pi$
$379$	1.64235e6	0.587310	0.293655	+	0.955912i	$0.405128\pi$
$380$	−98176.0	−0.0348776
$381$	0	0
$382$	−1.50562e6	−0.527905
$383$	2.05698e6	0.716527	0.358263	−	0.933621i	$-0.383369\pi$
$383$	2.05698e6	0.716527	0.358263	+	0.933621i	$0.383369\pi$
$384$	0	0
$385$	−845936.	−0.290861
$386$	3.37978e6	1.15457
$387$	0	0
$388$	341920.	0.115304
$389$	−616142.	−0.206446	−0.103223	−	0.994658i	$-0.532916\pi$
$389$	−616142.	−0.206446	−0.103223	+	0.994658i	$0.532916\pi$
$390$	0	0
$391$	3.49938e6	1.15758
$392$	153664.	0.0505076
$393$	0	0
$394$	1.97118e6	0.639713
$395$	−2.32440e6	−0.749580
$396$	0	0
$397$	2.19212e6	0.698052	0.349026	−	0.937113i	$-0.386513\pi$
$397$	2.19212e6	0.698052	0.349026	+	0.937113i	$0.386513\pi$
$398$	−3.65910e6	−1.15789
$399$	0	0
$400$	−626944.	−0.195920
$401$	−3.28454e6	−1.02003	−0.510015	−	0.860165i	$-0.670360\pi$
$401$	−3.28454e6	−1.02003	−0.510015	+	0.860165i	$0.670360\pi$
$402$	0	0
$403$	−2.26670e6	−0.695236
$404$	589024.	0.179548
$405$	0	0
$406$	−970984.	−0.292346
$407$	−2.89371e6	−0.865903
$408$	0	0
$409$	−3.61219e6	−1.06773	−0.533866	−	0.845569i	$-0.679261\pi$
$409$	−3.61219e6	−1.06773	−0.533866	+	0.845569i	$0.679261\pi$
$410$	1.09637e6	0.322104
$411$	0	0
$412$	1.67245e6	0.485411
$413$	−756364.	−0.218200
$414$	0	0
$415$	−167128.	−0.0476353
$416$	325632.	0.0922558
$417$	0	0
$418$	−626816.	−0.175469
$419$	−5.41489e6	−1.50680	−0.753398	−	0.657564i	$-0.771587\pi$
$419$	−5.41489e6	−1.50680	−0.753398	+	0.657564i	$0.771587\pi$
$420$	0	0
$421$	3.60629e6	0.991644	0.495822	−	0.868424i	$-0.334867\pi$
$421$	3.60629e6	0.991644	0.495822	+	0.868424i	$0.334867\pi$
$422$	1.24712e6	0.340900
$423$	0	0
$424$	2.13466e6	0.576651
$425$	3.87432e6	1.04045
$426$	0	0
$427$	1.80134e6	0.478107
$428$	−3.43104e6	−0.905350
$429$	0	0
$430$	879008.	0.229257
$431$	2.78214e6	0.721416	0.360708	−	0.932679i	$-0.382535\pi$
$431$	2.78214e6	0.721416	0.360708	+	0.932679i	$0.382535\pi$
$432$	0	0
$433$	6.27619e6	1.60871	0.804353	−	0.594152i	$-0.202512\pi$
$433$	6.27619e6	1.60871	0.804353	+	0.594152i	$0.202512\pi$
$434$	1.39709e6	0.356041
$435$	0	0
$436$	460768.	0.116082
$437$	−522032.	−0.130766
$438$	0	0
$439$	641592.	0.158890	0.0794452	−	0.996839i	$-0.474685\pi$
$439$	641592.	0.158890	0.0794452	+	0.996839i	$0.474685\pi$
$440$	1.10490e6	0.272076
$441$	0	0
$442$	−2.01230e6	−0.489934
$443$	−6.05546e6	−1.46601	−0.733006	−	0.680222i	$-0.761883\pi$
$443$	−6.05546e6	−1.46601	−0.733006	+	0.680222i	$0.761883\pi$
$444$	0	0
$445$	−3.18911e6	−0.763430
$446$	−5.15104e6	−1.22619
$447$	0	0
$448$	−200704.	−0.0472456
$449$	5.16681e6	1.20950	0.604752	−	0.796414i	$-0.293272\pi$
$449$	5.16681e6	1.20950	0.604752	+	0.796414i	$0.293272\pi$
$450$	0	0
$451$	6.99989e6	1.62050
$452$	896224.	0.206334
$453$	0	0
$454$	−5.15621e6	−1.17406
$455$	405132.	0.0917420
$456$	0	0
$457$	−227798.	−0.0510222	−0.0255111	−	0.999675i	$-0.508121\pi$
$457$	−227798.	−0.0510222	−0.0255111	+	0.999675i	$0.508121\pi$
$458$	2.71286e6	0.604315
$459$	0	0
$460$	920192.	0.202761
$461$	−585146.	−0.128237	−0.0641183	−	0.997942i	$-0.520423\pi$
$461$	−585146.	−0.128237	−0.0641183	+	0.997942i	$0.520423\pi$
$462$	0	0
$463$	−3.41454e6	−0.740251	−0.370126	−	0.928982i	$-0.620685\pi$
$463$	−3.41454e6	−0.740251	−0.370126	+	0.928982i	$0.620685\pi$
$464$	1.26822e6	0.273465
$465$	0	0
$466$	4.46924e6	0.953386
$467$	−716300.	−0.151986	−0.0759929	−	0.997108i	$-0.524213\pi$
$467$	−716300.	−0.151986	−0.0759929	+	0.997108i	$0.524213\pi$
$468$	0	0
$469$	−2.00763e6	−0.421455
$470$	556608.	0.116226
$471$	0	0
$472$	987904.	0.204108
$473$	5.61213e6	1.15339
$474$	0	0
$475$	−577964.	−0.117535
$476$	1.24029e6	0.250903
$477$	0	0
$478$	5.04782e6	1.01050
$479$	−5.24092e6	−1.04368	−0.521842	−	0.853042i	$-0.674755\pi$
$479$	−5.24092e6	−1.04368	−0.521842	+	0.853042i	$0.674755\pi$
$480$	0	0
$481$	1.38584e6	0.273119
$482$	3.79287e6	0.743619
$483$	0	0
$484$	4.47752e6	0.868809
$485$	−555620.	−0.107256
$486$	0	0
$487$	1.11702e6	0.213421	0.106710	−	0.994290i	$-0.465968\pi$
$487$	1.11702e6	0.213421	0.106710	+	0.994290i	$0.465968\pi$
$488$	−2.35277e6	−0.447229
$489$	0	0
$490$	−249704.	−0.0469824
$491$	−1.34458e6	−0.251699	−0.125850	−	0.992049i	$-0.540166\pi$
$491$	−1.34458e6	−0.251699	−0.125850	+	0.992049i	$0.540166\pi$
$492$	0	0
$493$	−7.83723e6	−1.45226
$494$	300192.	0.0553454
$495$	0	0
$496$	−1.82477e6	−0.333045
$497$	−445508.	−0.0809030
$498$	0	0
$499$	−6.54648e6	−1.17695	−0.588473	−	0.808517i	$-0.700271\pi$
$499$	−6.54648e6	−1.17695	−0.588473	+	0.808517i	$0.700271\pi$
$500$	2.31878e6	0.414797
$501$	0	0
$502$	1.94558e6	0.344580
$503$	8.22050e6	1.44870	0.724350	−	0.689432i	$-0.242140\pi$
$503$	8.22050e6	1.44870	0.724350	+	0.689432i	$0.242140\pi$
$504$	0	0
$505$	−957164.	−0.167016
$506$	5.87507e6	1.02009
$507$	0	0
$508$	2.96640e6	0.510000
$509$	5.11045e6	0.874308	0.437154	−	0.899387i	$-0.355987\pi$
$509$	5.11045e6	0.874308	0.437154	+	0.899387i	$0.355987\pi$
$510$	0	0
$511$	3.59925e6	0.609760
$512$	262144.	0.0441942
$513$	0	0
$514$	4.15639e6	0.693919
$515$	−2.71773e6	−0.451531
$516$	0	0
$517$	3.55373e6	0.584733
$518$	−854168.	−0.139868
$519$	0	0
$520$	−529152.	−0.0858168
$521$	−9.69999e6	−1.56559	−0.782793	−	0.622282i	$-0.786206\pi$
$521$	−9.69999e6	−1.56559	−0.782793	+	0.622282i	$0.786206\pi$
$522$	0	0
$523$	−3.17295e6	−0.507234	−0.253617	−	0.967305i	$-0.581620\pi$
$523$	−3.17295e6	−0.507234	−0.253617	+	0.967305i	$0.581620\pi$
$524$	−1.03251e6	−0.164273
$525$	0	0
$526$	−5.40418e6	−0.851658
$527$	1.12765e7	1.76867
$528$	0	0
$529$	−1.54340e6	−0.239794
$530$	−3.46882e6	−0.536403
$531$	0	0
$532$	−185024.	−0.0283432
$533$	−3.35236e6	−0.511131
$534$	0	0
$535$	5.57544e6	0.842160
$536$	2.62221e6	0.394235
$537$	0	0
$538$	4.47244e6	0.666176
$539$	−1.59426e6	−0.236368
$540$	0	0
$541$	−6.62575e6	−0.973289	−0.486644	−	0.873600i	$-0.661779\pi$
$541$	−6.62575e6	−0.973289	−0.486644	+	0.873600i	$0.661779\pi$
$542$	−760416.	−0.111187
$543$	0	0
$544$	−1.61997e6	−0.234698
$545$	−748748.	−0.107980
$546$	0	0
$547$	3.84707e6	0.549745	0.274873	−	0.961481i	$-0.411364\pi$
$547$	3.84707e6	0.549745	0.274873	+	0.961481i	$0.411364\pi$
$548$	−2.44688e6	−0.348066
$549$	0	0
$550$	6.50454e6	0.916875
$551$	1.16914e6	0.164055
$552$	0	0
$553$	−4.38060e6	−0.609145
$554$	−802024.	−0.111023
$555$	0	0
$556$	−5.49536e6	−0.753892
$557$	−5.00176e6	−0.683101	−0.341550	−	0.939863i	$-0.610952\pi$
$557$	−5.00176e6	−0.683101	−0.341550	+	0.939863i	$0.610952\pi$
$558$	0	0
$559$	−2.68774e6	−0.363795
$560$	326144.	0.0439480
$561$	0	0
$562$	−4.36948e6	−0.583565
$563$	−2.27772e6	−0.302852	−0.151426	−	0.988469i	$-0.548386\pi$
$563$	−2.27772e6	−0.302852	−0.151426	+	0.988469i	$0.548386\pi$
$564$	0	0
$565$	−1.45636e6	−0.191933
$566$	7.25032e6	0.951297
$567$	0	0
$568$	581888.	0.0756778
$569$	−8.86979e6	−1.14850	−0.574252	−	0.818678i	$-0.694707\pi$
$569$	−8.86979e6	−1.14850	−0.574252	+	0.818678i	$0.694707\pi$
$570$	0	0
$571$	1.40102e7	1.79826	0.899132	−	0.437678i	$-0.144199\pi$
$571$	1.40102e7	1.79826	0.899132	+	0.437678i	$0.144199\pi$
$572$	−3.37843e6	−0.431743
$573$	0	0
$574$	2.06623e6	0.261758
$575$	5.41719e6	0.683289
$576$	0	0
$577$	8.75327e6	1.09454	0.547269	−	0.836957i	$-0.315668\pi$
$577$	8.75327e6	1.09454	0.547269	+	0.836957i	$0.315668\pi$
$578$	4.33147e6	0.539281
$579$	0	0
$580$	−2.06086e6	−0.254378
$581$	−314972.	−0.0387108
$582$	0	0
$583$	−2.21471e7	−2.69864
$584$	−4.70106e6	−0.570379
$585$	0	0
$586$	−8.40122e6	−1.01064
$587$	1.06117e7	1.27113	0.635564	−	0.772048i	$-0.280768\pi$
$587$	1.06117e7	1.27113	0.635564	+	0.772048i	$0.280768\pi$
$588$	0	0
$589$	−1.68221e6	−0.199798
$590$	−1.60534e6	−0.189862
$591$	0	0
$592$	1.11565e6	0.130835
$593$	−1.88552e6	−0.220188	−0.110094	−	0.993921i	$-0.535115\pi$
$593$	−1.88552e6	−0.220188	−0.110094	+	0.993921i	$0.535115\pi$
$594$	0	0
$595$	−2.01547e6	−0.233391
$596$	2.79773e6	0.322619
$597$	0	0
$598$	−2.81366e6	−0.321751
$599$	−1.27256e7	−1.44915	−0.724573	−	0.689198i	$-0.757963\pi$
$599$	−1.27256e7	−1.44915	−0.724573	+	0.689198i	$0.757963\pi$
$600$	0	0
$601$	7.18846e6	0.811801	0.405900	−	0.913917i	$-0.366958\pi$
$601$	7.18846e6	0.811801	0.405900	+	0.913917i	$0.366958\pi$
$602$	1.65659e6	0.186305
$603$	0	0
$604$	−7.24083e6	−0.807600
$605$	−7.27597e6	−0.808170
$606$	0	0
$607$	1.08494e7	1.19519	0.597593	−	0.801800i	$-0.296124\pi$
$607$	1.08494e7	1.19519	0.597593	+	0.801800i	$0.296124\pi$
$608$	241664.	0.0265126
$609$	0	0
$610$	3.82325e6	0.416014
$611$	−1.70194e6	−0.184434
$612$	0	0
$613$	−4.90511e6	−0.527227	−0.263614	−	0.964628i	$-0.584914\pi$
$613$	−4.90511e6	−0.527227	−0.263614	+	0.964628i	$0.584914\pi$
$614$	−6.56418e6	−0.702683
$615$	0	0
$616$	2.08230e6	0.221102
$617$	−2.58445e6	−0.273310	−0.136655	−	0.990619i	$-0.543635\pi$
$617$	−2.58445e6	−0.273310	−0.136655	+	0.990619i	$0.543635\pi$
$618$	0	0
$619$	−4.99336e6	−0.523801	−0.261901	−	0.965095i	$-0.584349\pi$
$619$	−4.99336e6	−0.523801	−0.261901	+	0.965095i	$0.584349\pi$
$620$	2.96525e6	0.309800
$621$	0	0
$622$	3.78093e6	0.391852
$623$	−6.01024e6	−0.620400
$624$	0	0
$625$	3.88510e6	0.397834
$626$	1.66142e6	0.169450
$627$	0	0
$628$	−7.98506e6	−0.807940
$629$	−6.89436e6	−0.694812
$630$	0	0
$631$	−1.18219e7	−1.18199	−0.590997	−	0.806674i	$-0.701265\pi$
$631$	−1.18219e7	−1.18199	−0.590997	+	0.806674i	$0.701265\pi$
$632$	5.72160e6	0.569803
$633$	0	0
$634$	−4.73926e6	−0.468260
$635$	−4.82040e6	−0.474404
$636$	0	0
$637$	763518.	0.0745540
$638$	−1.31578e7	−1.27977
$639$	0	0
$640$	−425984.	−0.0411096
$641$	5.47007e6	0.525833	0.262916	−	0.964819i	$-0.415316\pi$
$641$	5.47007e6	0.525833	0.262916	+	0.964819i	$0.415316\pi$
$642$	0	0
$643$	9.64934e6	0.920386	0.460193	−	0.887819i	$-0.347780\pi$
$643$	9.64934e6	0.920386	0.460193	+	0.887819i	$0.347780\pi$
$644$	1.73421e6	0.164773
$645$	0	0
$646$	−1.49341e6	−0.140798
$647$	−292368.	−0.0274580	−0.0137290	−	0.999906i	$-0.504370\pi$
$647$	−292368.	−0.0274580	−0.0137290	+	0.999906i	$0.504370\pi$
$648$	0	0
$649$	−1.02495e7	−0.955193
$650$	−3.11513e6	−0.289196
$651$	0	0
$652$	−7.60941e6	−0.701022
$653$	−6.94081e6	−0.636982	−0.318491	−	0.947926i	$-0.603176\pi$
$653$	−6.94081e6	−0.636982	−0.318491	+	0.947926i	$0.603176\pi$
$654$	0	0
$655$	1.67783e6	0.152808
$656$	−2.69875e6	−0.244852
$657$	0	0
$658$	1.04899e6	0.0944512
$659$	1.32912e7	1.19221	0.596104	−	0.802908i	$-0.296715\pi$
$659$	1.32912e7	1.19221	0.596104	+	0.802908i	$0.296715\pi$
$660$	0	0
$661$	2.05219e6	0.182690	0.0913448	−	0.995819i	$-0.470883\pi$
$661$	2.05219e6	0.182690	0.0913448	+	0.995819i	$0.470883\pi$
$662$	5.48619e6	0.486548
$663$	0	0
$664$	411392.	0.0362106
$665$	300664.	0.0263650
$666$	0	0
$667$	−1.09582e7	−0.953732
$668$	−1.92358e6	−0.166790
$669$	0	0
$670$	−4.26109e6	−0.366719
$671$	2.44100e7	2.09296
$672$	0	0
$673$	−1.57039e7	−1.33650	−0.668252	−	0.743935i	$-0.732957\pi$
$673$	−1.57039e7	−1.33650	−0.668252	+	0.743935i	$0.732957\pi$
$674$	3.85409e6	0.326792
$675$	0	0
$676$	−4.32270e6	−0.363822
$677$	969534.	0.0813002	0.0406501	−	0.999173i	$-0.487057\pi$
$677$	969534.	0.0813002	0.0406501	+	0.999173i	$0.487057\pi$
$678$	0	0
$679$	−1.04713e6	−0.0871618
$680$	2.63245e6	0.218317
$681$	0	0
$682$	1.89320e7	1.55860
$683$	1.49908e7	1.22962	0.614812	−	0.788673i	$-0.289232\pi$
$683$	1.49908e7	1.22962	0.614812	+	0.788673i	$0.289232\pi$
$684$	0	0
$685$	3.97618e6	0.323772
$686$	−470596.	−0.0381802
$687$	0	0
$688$	−2.16371e6	−0.174272
$689$	1.06066e7	0.851191
$690$	0	0
$691$	−7.16038e6	−0.570481	−0.285240	−	0.958456i	$-0.592073\pi$
$691$	−7.16038e6	−0.570481	−0.285240	+	0.958456i	$0.592073\pi$
$692$	−8.14198e6	−0.646346
$693$	0	0
$694$	−1.03092e7	−0.812509
$695$	8.92996e6	0.701274
$696$	0	0
$697$	1.66774e7	1.30031
$698$	−1.22701e7	−0.953253
$699$	0	0
$700$	1.92002e6	0.148102
$701$	91834.0	0.00705844	0.00352922	−	0.999994i	$-0.498877\pi$
$701$	91834.0	0.00705844	0.00352922	+	0.999994i	$0.498877\pi$
$702$	0	0
$703$	1.02849e6	0.0784894
$704$	−2.71974e6	−0.206822
$705$	0	0
$706$	1.24058e7	0.936725
$707$	−1.80389e6	−0.135725
$708$	0	0
$709$	2.20981e7	1.65097	0.825487	−	0.564422i	$-0.190901\pi$
$709$	2.20981e7	1.65097	0.825487	+	0.564422i	$0.190901\pi$
$710$	−945568.	−0.0703958
$711$	0	0
$712$	7.85011e6	0.580331
$713$	1.57671e7	1.16153
$714$	0	0
$715$	5.48995e6	0.401609
$716$	−7.80096e6	−0.568677
$717$	0	0
$718$	1.31003e6	0.0948355
$719$	−1.58388e7	−1.14262	−0.571308	−	0.820736i	$-0.693564\pi$
$719$	−1.58388e7	−1.14262	−0.571308	+	0.820736i	$0.693564\pi$
$720$	0	0
$721$	−5.12187e6	−0.366936
$722$	−9.68161e6	−0.691202
$723$	0	0
$724$	−8.71056e6	−0.617589
$725$	−1.21323e7	−0.857235
$726$	0	0
$727$	6.31418e6	0.443078	0.221539	−	0.975151i	$-0.428892\pi$
$727$	6.31418e6	0.443078	0.221539	+	0.975151i	$0.428892\pi$
$728$	−997248.	−0.0697388
$729$	0	0
$730$	7.63922e6	0.530569
$731$	1.33711e7	0.925492
$732$	0	0
$733$	6.93003e6	0.476404	0.238202	−	0.971216i	$-0.423442\pi$
$733$	6.93003e6	0.476404	0.238202	+	0.971216i	$0.423442\pi$
$734$	−1.14696e7	−0.785791
$735$	0	0
$736$	−2.26509e6	−0.154131
$737$	−2.72054e7	−1.84496
$738$	0	0
$739$	1.42331e7	0.958714	0.479357	−	0.877620i	$-0.340870\pi$
$739$	1.42331e7	0.958714	0.479357	+	0.877620i	$0.340870\pi$
$740$	−1.81293e6	−0.121703
$741$	0	0
$742$	−6.53738e6	−0.435907
$743$	5.94460e6	0.395048	0.197524	−	0.980298i	$-0.436710\pi$
$743$	5.94460e6	0.395048	0.197524	+	0.980298i	$0.436710\pi$
$744$	0	0
$745$	−4.54631e6	−0.300102
$746$	1.43212e7	0.942175
$747$	0	0
$748$	1.68072e7	1.09835
$749$	1.05076e7	0.684380
$750$	0	0
$751$	−682752.	−0.0441736	−0.0220868	−	0.999756i	$-0.507031\pi$
$751$	−682752.	−0.0441736	−0.0220868	+	0.999756i	$0.507031\pi$
$752$	−1.37011e6	−0.0883510
$753$	0	0
$754$	6.30149e6	0.403659
$755$	1.17664e7	0.751233
$756$	0	0
$757$	1.46333e7	0.928116	0.464058	−	0.885805i	$-0.346393\pi$
$757$	1.46333e7	0.928116	0.464058	+	0.885805i	$0.346393\pi$
$758$	6.56939e6	0.415291
$759$	0	0
$760$	−392704.	−0.0246622
$761$	1.16367e7	0.728399	0.364200	−	0.931321i	$-0.381343\pi$
$761$	1.16367e7	0.728399	0.364200	+	0.931321i	$0.381343\pi$
$762$	0	0
$763$	−1.41110e6	−0.0877500
$764$	−6.02246e6	−0.373285
$765$	0	0
$766$	8.22790e6	0.506661
$767$	4.90865e6	0.301282
$768$	0	0
$769$	1.91472e7	1.16759	0.583793	−	0.811902i	$-0.301568\pi$
$769$	1.91472e7	1.16759	0.583793	+	0.811902i	$0.301568\pi$
$770$	−3.38374e6	−0.205670
$771$	0	0
$772$	1.35191e7	0.816405
$773$	5.39261e6	0.324601	0.162301	−	0.986741i	$-0.448109\pi$
$773$	5.39261e6	0.324601	0.162301	+	0.986741i	$0.448109\pi$
$774$	0	0
$775$	1.74565e7	1.04400
$776$	1.36768e6	0.0815324
$777$	0	0
$778$	−2.46457e6	−0.145979
$779$	−2.48791e6	−0.146890
$780$	0	0
$781$	−6.03709e6	−0.354160
$782$	1.39975e7	0.818530
$783$	0	0
$784$	614656.	0.0357143
$785$	1.29757e7	0.751549
$786$	0	0
$787$	3.04348e6	0.175159	0.0875796	−	0.996158i	$-0.472087\pi$
$787$	3.04348e6	0.175159	0.0875796	+	0.996158i	$0.472087\pi$
$788$	7.88470e6	0.452345
$789$	0	0
$790$	−9.29760e6	−0.530033
$791$	−2.74469e6	−0.155974
$792$	0	0
$793$	−1.16903e7	−0.660151
$794$	8.76847e6	0.493597
$795$	0	0
$796$	−1.46364e7	−0.818751
$797$	−2.29652e7	−1.28063	−0.640316	−	0.768111i	$-0.721197\pi$
$797$	−2.29652e7	−1.28063	−0.640316	+	0.768111i	$0.721197\pi$
$798$	0	0
$799$	8.46686e6	0.469197
$800$	−2.50778e6	−0.138536
$801$	0	0
$802$	−1.31382e7	−0.721271
$803$	4.87735e7	2.66928
$804$	0	0
$805$	−2.81809e6	−0.153273
$806$	−9.06682e6	−0.491606
$807$	0	0
$808$	2.35610e6	0.126959
$809$	−1.90787e7	−1.02489	−0.512445	−	0.858720i	$-0.671260\pi$
$809$	−1.90787e7	−1.02489	−0.512445	+	0.858720i	$0.671260\pi$
$810$	0	0
$811$	1.09414e7	0.584147	0.292074	−	0.956396i	$-0.405655\pi$
$811$	1.09414e7	0.584147	0.292074	+	0.956396i	$0.405655\pi$
$812$	−3.88394e6	−0.206720
$813$	0	0
$814$	−1.15748e7	−0.612286
$815$	1.23653e7	0.652094
$816$	0	0
$817$	−1.99467e6	−0.104548
$818$	−1.44488e7	−0.755001
$819$	0	0
$820$	4.38547e6	0.227762
$821$	−2.12594e7	−1.10076	−0.550380	−	0.834914i	$-0.685517\pi$
$821$	−2.12594e7	−1.10076	−0.550380	+	0.834914i	$0.685517\pi$
$822$	0	0
$823$	−1.42256e7	−0.732103	−0.366052	−	0.930595i	$-0.619291\pi$
$823$	−1.42256e7	−0.732103	−0.366052	+	0.930595i	$0.619291\pi$
$824$	6.68979e6	0.343237
$825$	0	0
$826$	−3.02546e6	−0.154291
$827$	−2.76103e6	−0.140381	−0.0701904	−	0.997534i	$-0.522361\pi$
$827$	−2.76103e6	−0.140381	−0.0701904	+	0.997534i	$0.522361\pi$
$828$	0	0
$829$	−3.82147e7	−1.93127	−0.965637	−	0.259895i	$-0.916312\pi$
$829$	−3.82147e7	−1.93127	−0.965637	+	0.259895i	$0.916312\pi$
$830$	−668512.	−0.0336832
$831$	0	0
$832$	1.30253e6	0.0652347
$833$	−3.79838e6	−0.189665
$834$	0	0
$835$	3.12582e6	0.155149
$836$	−2.50726e6	−0.124075
$837$	0	0
$838$	−2.16596e7	−1.06547
$839$	−1.06044e7	−0.520094	−0.260047	−	0.965596i	$-0.583738\pi$
$839$	−1.06044e7	−0.520094	−0.260047	+	0.965596i	$0.583738\pi$
$840$	0	0
$841$	4.03097e6	0.196526
$842$	1.44252e7	0.701198
$843$	0	0
$844$	4.98848e6	0.241053
$845$	7.02439e6	0.338429
$846$	0	0
$847$	−1.37124e7	−0.656758
$848$	8.53862e6	0.407754
$849$	0	0
$850$	1.54973e7	0.735712
$851$	−9.63990e6	−0.456298
$852$	0	0
$853$	−4.07009e7	−1.91527	−0.957637	−	0.287977i	$-0.907017\pi$
$853$	−4.07009e7	−1.91527	−0.957637	+	0.287977i	$0.907017\pi$
$854$	7.20535e6	0.338073
$855$	0	0
$856$	−1.37242e7	−0.640179
$857$	3.10120e7	1.44237	0.721187	−	0.692741i	$-0.243597\pi$
$857$	3.10120e7	1.44237	0.721187	+	0.692741i	$0.243597\pi$
$858$	0	0
$859$	1.09104e7	0.504495	0.252247	−	0.967663i	$-0.418830\pi$
$859$	1.09104e7	0.504495	0.252247	+	0.967663i	$0.418830\pi$
$860$	3.51603e6	0.162109
$861$	0	0
$862$	1.11286e7	0.510118
$863$	−1.04089e7	−0.475751	−0.237875	−	0.971296i	$-0.576451\pi$
$863$	−1.04089e7	−0.475751	−0.237875	+	0.971296i	$0.576451\pi$
$864$	0	0
$865$	1.32307e7	0.601234
$866$	2.51048e7	1.13753
$867$	0	0
$868$	5.58835e6	0.251759
$869$	−5.93616e7	−2.66659
$870$	0	0
$871$	1.30291e7	0.581928
$872$	1.84307e6	0.0820826
$873$	0	0
$874$	−2.08813e6	−0.0924652
$875$	−7.10128e6	−0.313557
$876$	0	0
$877$	1.64064e7	0.720299	0.360150	−	0.932895i	$-0.382726\pi$
$877$	1.64064e7	0.720299	0.360150	+	0.932895i	$0.382726\pi$
$878$	2.56637e6	0.112352
$879$	0	0
$880$	4.41958e6	0.192387
$881$	−1.48577e7	−0.644927	−0.322464	−	0.946582i	$-0.604511\pi$
$881$	−1.48577e7	−0.644927	−0.322464	+	0.946582i	$0.604511\pi$
$882$	0	0
$883$	−2.72018e7	−1.17407	−0.587037	−	0.809560i	$-0.699706\pi$
$883$	−2.72018e7	−1.17407	−0.587037	+	0.809560i	$0.699706\pi$
$884$	−8.04922e6	−0.346436
$885$	0	0
$886$	−2.42218e7	−1.03663
$887$	−2.71242e7	−1.15757	−0.578785	−	0.815480i	$-0.696473\pi$
$887$	−2.71242e7	−1.15757	−0.578785	+	0.815480i	$0.696473\pi$
$888$	0	0
$889$	−9.08460e6	−0.385524
$890$	−1.27564e7	−0.539827
$891$	0	0
$892$	−2.06042e7	−0.867047
$893$	−1.26307e6	−0.0530029
$894$	0	0
$895$	1.26766e7	0.528986
$896$	−802816.	−0.0334077
$897$	0	0
$898$	2.06673e7	0.855248
$899$	−3.53121e7	−1.45722
$900$	0	0
$901$	−5.27660e7	−2.16542
$902$	2.79996e7	1.14587
$903$	0	0
$904$	3.58490e6	0.145900
$905$	1.41547e7	0.574484
$906$	0	0
$907$	−8.42269e6	−0.339964	−0.169982	−	0.985447i	$-0.554371\pi$
$907$	−8.42269e6	−0.339964	−0.169982	+	0.985447i	$0.554371\pi$
$908$	−2.06248e7	−0.830187
$909$	0	0
$910$	1.62053e6	0.0648714
$911$	−3.08637e7	−1.23212	−0.616060	−	0.787700i	$-0.711272\pi$
$911$	−3.08637e7	−1.23212	−0.616060	+	0.787700i	$0.711272\pi$
$912$	0	0
$913$	−4.26819e6	−0.169460
$914$	−911192.	−0.0360782
$915$	0	0
$916$	1.08514e7	0.427315
$917$	3.16207e6	0.124179
$918$	0	0
$919$	4.93895e6	0.192906	0.0964531	−	0.995338i	$-0.469250\pi$
$919$	4.93895e6	0.192906	0.0964531	+	0.995338i	$0.469250\pi$
$920$	3.68077e6	0.143373
$921$	0	0
$922$	−2.34058e6	−0.0906770
$923$	2.89126e6	0.111707
$924$	0	0
$925$	−1.06727e7	−0.410130
$926$	−1.36581e7	−0.523437
$927$	0	0
$928$	5.07290e6	0.193369
$929$	−5.62575e6	−0.213866	−0.106933	−	0.994266i	$-0.534103\pi$
$929$	−5.62575e6	−0.213866	−0.106933	+	0.994266i	$0.534103\pi$
$930$	0	0
$931$	566636.	0.0214255
$932$	1.78770e7	0.674146
$933$	0	0
$934$	−2.86520e6	−0.107470
$935$	−2.73116e7	−1.02169
$936$	0	0
$937$	2.60073e7	0.967714	0.483857	−	0.875147i	$-0.339236\pi$
$937$	2.60073e7	0.967714	0.483857	+	0.875147i	$0.339236\pi$
$938$	−8.03051e6	−0.298014
$939$	0	0
$940$	2.22643e6	0.0821845
$941$	−3.02160e6	−0.111241	−0.0556203	−	0.998452i	$-0.517714\pi$
$941$	−3.02160e6	−0.111241	−0.0556203	+	0.998452i	$0.517714\pi$
$942$	0	0
$943$	2.33189e7	0.853943
$944$	3.95162e6	0.144326
$945$	0	0
$946$	2.24485e7	0.815567
$947$	3.48282e7	1.26199	0.630995	−	0.775787i	$-0.282647\pi$
$947$	3.48282e7	1.26199	0.630995	+	0.775787i	$0.282647\pi$
$948$	0	0
$949$	−2.33584e7	−0.841932
$950$	−2.31186e6	−0.0831097
$951$	0	0
$952$	4.96115e6	0.177415
$953$	9.39009e6	0.334917	0.167459	−	0.985879i	$-0.446444\pi$
$953$	9.39009e6	0.334917	0.167459	+	0.985879i	$0.446444\pi$
$954$	0	0
$955$	9.78650e6	0.347232
$956$	2.01913e7	0.714528
$957$	0	0
$958$	−2.09637e7	−0.737996
$959$	7.49357e6	0.263113
$960$	0	0
$961$	2.21792e7	0.774708
$962$	5.54338e6	0.193124
$963$	0	0
$964$	1.51715e7	0.525818
$965$	−2.19686e7	−0.759423
$966$	0	0
$967$	1.44768e7	0.497860	0.248930	−	0.968521i	$-0.419921\pi$
$967$	1.44768e7	0.497860	0.248930	+	0.968521i	$0.419921\pi$
$968$	1.79101e7	0.614340
$969$	0	0
$970$	−2.22248e6	−0.0758418
$971$	−9.24976e6	−0.314834	−0.157417	−	0.987532i	$-0.550317\pi$
$971$	−9.24976e6	−0.314834	−0.157417	+	0.987532i	$0.550317\pi$
$972$	0	0
$973$	1.68295e7	0.569889
$974$	4.46806e6	0.150911
$975$	0	0
$976$	−9.41107e6	−0.316238
$977$	4.97780e7	1.66840	0.834202	−	0.551459i	$-0.185929\pi$
$977$	4.97780e7	1.66840	0.834202	+	0.551459i	$0.185929\pi$
$978$	0	0
$979$	−8.14449e7	−2.71586
$980$	−998816.	−0.0332216
$981$	0	0
$982$	−5.37830e6	−0.177978
$983$	8.95601e6	0.295618	0.147809	−	0.989016i	$-0.452778\pi$
$983$	8.95601e6	0.295618	0.147809	+	0.989016i	$0.452778\pi$
$984$	0	0
$985$	−1.28126e7	−0.420773
$986$	−3.13489e7	−1.02690
$987$	0	0
$988$	1.20077e6	0.0391351
$989$	1.86958e7	0.607790
$990$	0	0
$991$	2.62400e7	0.848751	0.424376	−	0.905486i	$-0.360494\pi$
$991$	2.62400e7	0.848751	0.424376	+	0.905486i	$0.360494\pi$
$992$	−7.29907e6	−0.235499
$993$	0	0
$994$	−1.78203e6	−0.0572070
$995$	2.37842e7	0.761606
$996$	0	0
$997$	2.80506e7	0.893727	0.446863	−	0.894602i	$-0.352541\pi$
$997$	2.80506e7	0.893727	0.446863	+	0.894602i	$0.352541\pi$
$998$	−2.61859e7	−0.832226
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.6.a.i.1.1		1
3.2	odd	2		42.6.a.d.1.1	✓	1
4.3	odd	2		1008.6.a.j.1.1		1
7.6	odd	2		882.6.a.s.1.1		1
12.11	even	2		336.6.a.h.1.1		1
15.2	even	4		1050.6.g.i.799.1		2
15.8	even	4		1050.6.g.i.799.2		2
15.14	odd	2		1050.6.a.k.1.1		1
21.2	odd	6		294.6.e.i.67.1		2
21.5	even	6		294.6.e.p.67.1		2
21.11	odd	6		294.6.e.i.79.1		2
21.17	even	6		294.6.e.p.79.1		2
21.20	even	2		294.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.d.1.1	✓	1	3.2	odd	2
126.6.a.i.1.1		1	1.1	even	1	trivial
294.6.a.b.1.1		1	21.20	even	2
294.6.e.i.67.1		2	21.2	odd	6
294.6.e.i.79.1		2	21.11	odd	6
294.6.e.p.67.1		2	21.5	even	6
294.6.e.p.79.1		2	21.17	even	6
336.6.a.h.1.1		1	12.11	even	2
882.6.a.s.1.1		1	7.6	odd	2
1008.6.a.j.1.1		1	4.3	odd	2
1050.6.a.k.1.1		1	15.14	odd	2
1050.6.g.i.799.1		2	15.2	even	4
1050.6.g.i.799.2		2	15.8	even	4

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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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