LMFDB - Embedded newform 114.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,6,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	114.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:	$18.2837554587$
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	$\chi$	$=$	114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +21.0000 q^{5} -36.0000 q^{6} -143.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +21.0000 q^{5} -36.0000 q^{6} -143.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +84.0000 q^{10} -205.000 q^{11} -144.000 q^{12} -78.0000 q^{13} -572.000 q^{14} -189.000 q^{15} +256.000 q^{16} -2125.00 q^{17} +324.000 q^{18} +361.000 q^{19} +336.000 q^{20} +1287.00 q^{21} -820.000 q^{22} +20.0000 q^{23} -576.000 q^{24} -2684.00 q^{25} -312.000 q^{26} -729.000 q^{27} -2288.00 q^{28} -4866.00 q^{29} -756.000 q^{30} -1098.00 q^{31} +1024.00 q^{32} +1845.00 q^{33} -8500.00 q^{34} -3003.00 q^{35} +1296.00 q^{36} -15128.0 q^{37} +1444.00 q^{38} +702.000 q^{39} +1344.00 q^{40} -9400.00 q^{41} +5148.00 q^{42} +20073.0 q^{43} -3280.00 q^{44} +1701.00 q^{45} +80.0000 q^{46} +14105.0 q^{47} -2304.00 q^{48} +3642.00 q^{49} -10736.0 q^{50} +19125.0 q^{51} -1248.00 q^{52} +26386.0 q^{53} -2916.00 q^{54} -4305.00 q^{55} -9152.00 q^{56} -3249.00 q^{57} -19464.0 q^{58} -13216.0 q^{59} -3024.00 q^{60} -2293.00 q^{61} -4392.00 q^{62} -11583.0 q^{63} +4096.00 q^{64} -1638.00 q^{65} +7380.00 q^{66} +35976.0 q^{67} -34000.0 q^{68} -180.000 q^{69} -12012.0 q^{70} +10180.0 q^{71} +5184.00 q^{72} +33109.0 q^{73} -60512.0 q^{74} +24156.0 q^{75} +5776.00 q^{76} +29315.0 q^{77} +2808.00 q^{78} -53888.0 q^{79} +5376.00 q^{80} +6561.00 q^{81} -37600.0 q^{82} +75196.0 q^{83} +20592.0 q^{84} -44625.0 q^{85} +80292.0 q^{86} +43794.0 q^{87} -13120.0 q^{88} +20618.0 q^{89} +6804.00 q^{90} +11154.0 q^{91} +320.000 q^{92} +9882.00 q^{93} +56420.0 q^{94} +7581.00 q^{95} -9216.00 q^{96} -84130.0 q^{97} +14568.0 q^{98} -16605.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	21.0000	0.375659	0.187830	−	0.982202i	$-0.439855\pi$
$5$	21.0000	0.375659	0.187830	+	0.982202i	$0.439855\pi$
$6$	−36.0000	−0.408248
$7$	−143.000	−1.10304	−0.551520	−	0.834162i	$-0.685952\pi$
$7$	−143.000	−1.10304	−0.551520	+	0.834162i	$0.685952\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	84.0000	0.265631
$11$	−205.000	−0.510825	−0.255413	−	0.966832i	$-0.582211\pi$
$11$	−205.000	−0.510825	−0.255413	+	0.966832i	$0.582211\pi$
$12$	−144.000	−0.288675
$13$	−78.0000	−0.128008	−0.0640039	−	0.997950i	$-0.520387\pi$
$13$	−78.0000	−0.128008	−0.0640039	+	0.997950i	$0.520387\pi$
$14$	−572.000	−0.779966
$15$	−189.000	−0.216887
$16$	256.000	0.250000
$17$	−2125.00	−1.78335	−0.891675	−	0.452676i	$-0.850469\pi$
$17$	−2125.00	−1.78335	−0.891675	+	0.452676i	$0.850469\pi$
$18$	324.000	0.235702
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	336.000	0.187830
$21$	1287.00	0.636840
$22$	−820.000	−0.361208
$23$	20.0000	0.00788334	0.00394167	−	0.999992i	$-0.498745\pi$
$23$	20.0000	0.00788334	0.00394167	+	0.999992i	$0.498745\pi$
$24$	−576.000	−0.204124
$25$	−2684.00	−0.858880
$26$	−312.000	−0.0905151
$27$	−729.000	−0.192450
$28$	−2288.00	−0.551520
$29$	−4866.00	−1.07443	−0.537214	−	0.843446i	$-0.680523\pi$
$29$	−4866.00	−1.07443	−0.537214	+	0.843446i	$0.680523\pi$
$30$	−756.000	−0.153362
$31$	−1098.00	−0.205210	−0.102605	−	0.994722i	$-0.532718\pi$
$31$	−1098.00	−0.205210	−0.102605	+	0.994722i	$0.532718\pi$
$32$	1024.00	0.176777
$33$	1845.00	0.294925
$34$	−8500.00	−1.26102
$35$	−3003.00	−0.414367
$36$	1296.00	0.166667
$37$	−15128.0	−1.81667	−0.908337	−	0.418238i	$-0.862648\pi$
$37$	−15128.0	−1.81667	−0.908337	+	0.418238i	$0.862648\pi$
$38$	1444.00	0.162221
$39$	702.000	0.0739053
$40$	1344.00	0.132816
$41$	−9400.00	−0.873310	−0.436655	−	0.899629i	$-0.643837\pi$
$41$	−9400.00	−0.873310	−0.436655	+	0.899629i	$0.643837\pi$
$42$	5148.00	0.450314
$43$	20073.0	1.65555	0.827773	−	0.561063i	$-0.189608\pi$
$43$	20073.0	1.65555	0.827773	+	0.561063i	$0.189608\pi$
$44$	−3280.00	−0.255413
$45$	1701.00	0.125220
$46$	80.0000	0.00557437
$47$	14105.0	0.931383	0.465692	−	0.884947i	$-0.345806\pi$
$47$	14105.0	0.931383	0.465692	+	0.884947i	$0.345806\pi$
$48$	−2304.00	−0.144338
$49$	3642.00	0.216695
$50$	−10736.0	−0.607320
$51$	19125.0	1.02962
$52$	−1248.00	−0.0640039
$53$	26386.0	1.29028	0.645140	−	0.764064i	$-0.276799\pi$
$53$	26386.0	1.29028	0.645140	+	0.764064i	$0.276799\pi$
$54$	−2916.00	−0.136083
$55$	−4305.00	−0.191896
$56$	−9152.00	−0.389983
$57$	−3249.00	−0.132453
$58$	−19464.0	−0.759735
$59$	−13216.0	−0.494277	−0.247138	−	0.968980i	$-0.579490\pi$
$59$	−13216.0	−0.494277	−0.247138	+	0.968980i	$0.579490\pi$
$60$	−3024.00	−0.108444
$61$	−2293.00	−0.0789004	−0.0394502	−	0.999222i	$-0.512561\pi$
$61$	−2293.00	−0.0789004	−0.0394502	+	0.999222i	$0.512561\pi$
$62$	−4392.00	−0.145105
$63$	−11583.0	−0.367680
$64$	4096.00	0.125000
$65$	−1638.00	−0.0480873
$66$	7380.00	0.208543
$67$	35976.0	0.979097	0.489549	−	0.871976i	$-0.337162\pi$
$67$	35976.0	0.979097	0.489549	+	0.871976i	$0.337162\pi$
$68$	−34000.0	−0.891675
$69$	−180.000	−0.00455145
$70$	−12012.0	−0.293002
$71$	10180.0	0.239664	0.119832	−	0.992794i	$-0.461764\pi$
$71$	10180.0	0.239664	0.119832	+	0.992794i	$0.461764\pi$
$72$	5184.00	0.117851
$73$	33109.0	0.727175	0.363587	−	0.931560i	$-0.381552\pi$
$73$	33109.0	0.727175	0.363587	+	0.931560i	$0.381552\pi$
$74$	−60512.0	−1.28458
$75$	24156.0	0.495875
$76$	5776.00	0.114708
$77$	29315.0	0.563460
$78$	2808.00	0.0522589
$79$	−53888.0	−0.971459	−0.485729	−	0.874109i	$-0.661446\pi$
$79$	−53888.0	−0.971459	−0.485729	+	0.874109i	$0.661446\pi$
$80$	5376.00	0.0939149
$81$	6561.00	0.111111
$82$	−37600.0	−0.617523
$83$	75196.0	1.19812	0.599059	−	0.800705i	$-0.295542\pi$
$83$	75196.0	1.19812	0.599059	+	0.800705i	$0.295542\pi$
$84$	20592.0	0.318420
$85$	−44625.0	−0.669932
$86$	80292.0	1.17065
$87$	43794.0	0.620321
$88$	−13120.0	−0.180604
$89$	20618.0	0.275913	0.137956	−	0.990438i	$-0.455947\pi$
$89$	20618.0	0.275913	0.137956	+	0.990438i	$0.455947\pi$
$90$	6804.00	0.0885438
$91$	11154.0	0.141198
$92$	320.000	0.00394167
$93$	9882.00	0.118478
$94$	56420.0	0.658587
$95$	7581.00	0.0861822
$96$	−9216.00	−0.102062
$97$	−84130.0	−0.907866	−0.453933	−	0.891036i	$-0.649979\pi$
$97$	−84130.0	−0.907866	−0.453933	+	0.891036i	$0.649979\pi$
$98$	14568.0	0.153227
$99$	−16605.0	−0.170275
$100$	−42944.0	−0.429440
$101$	163714.	1.59692	0.798459	−	0.602050i	$-0.205649\pi$
$101$	163714.	1.59692	0.798459	+	0.602050i	$0.205649\pi$
$102$	76500.0	0.728050
$103$	−139062.	−1.29156	−0.645781	−	0.763523i	$-0.723468\pi$
$103$	−139062.	−1.29156	−0.645781	+	0.763523i	$0.723468\pi$
$104$	−4992.00	−0.0452576
$105$	27027.0	0.239235
$106$	105544.	0.912366
$107$	124690.	1.05286	0.526432	−	0.850217i	$-0.323530\pi$
$107$	124690.	1.05286	0.526432	+	0.850217i	$0.323530\pi$
$108$	−11664.0	−0.0962250
$109$	−11836.0	−0.0954198	−0.0477099	−	0.998861i	$-0.515192\pi$
$109$	−11836.0	−0.0954198	−0.0477099	+	0.998861i	$0.515192\pi$
$110$	−17220.0	−0.135691
$111$	136152.	1.04886
$112$	−36608.0	−0.275760
$113$	57674.0	0.424897	0.212449	−	0.977172i	$-0.431856\pi$
$113$	57674.0	0.424897	0.212449	+	0.977172i	$0.431856\pi$
$114$	−12996.0	−0.0936586
$115$	420.000	0.00296145
$116$	−77856.0	−0.537214
$117$	−6318.00	−0.0426692
$118$	−52864.0	−0.349506
$119$	303875.	1.96711
$120$	−12096.0	−0.0766812
$121$	−119026.	−0.739058
$122$	−9172.00	−0.0557910
$123$	84600.0	0.504206
$124$	−17568.0	−0.102605
$125$	−121989.	−0.698306
$126$	−46332.0	−0.259989
$127$	134314.	0.738945	0.369472	−	0.929242i	$-0.379538\pi$
$127$	134314.	0.738945	0.369472	+	0.929242i	$0.379538\pi$
$128$	16384.0	0.0883883
$129$	−180657.	−0.955830
$130$	−6552.00	−0.0340029
$131$	−365955.	−1.86316	−0.931578	−	0.363540i	$-0.881568\pi$
$131$	−365955.	−1.86316	−0.931578	+	0.363540i	$0.881568\pi$
$132$	29520.0	0.147462
$133$	−51623.0	−0.253055
$134$	143904.	0.692326
$135$	−15309.0	−0.0722957
$136$	−136000.	−0.630510
$137$	44763.0	0.203759	0.101880	−	0.994797i	$-0.467514\pi$
$137$	44763.0	0.203759	0.101880	+	0.994797i	$0.467514\pi$
$138$	−720.000	−0.00321836
$139$	−422179.	−1.85336	−0.926680	−	0.375852i	$-0.877350\pi$
$139$	−422179.	−1.85336	−0.926680	+	0.375852i	$0.877350\pi$
$140$	−48048.0	−0.207184
$141$	−126945.	−0.537734
$142$	40720.0	0.169468
$143$	15990.0	0.0653896
$144$	20736.0	0.0833333
$145$	−102186.	−0.403619
$146$	132436.	0.514190
$147$	−32778.0	−0.125109
$148$	−242048.	−0.908337
$149$	−41741.0	−0.154027	−0.0770136	−	0.997030i	$-0.524538\pi$
$149$	−41741.0	−0.154027	−0.0770136	+	0.997030i	$0.524538\pi$
$150$	96624.0	0.350636
$151$	−41240.0	−0.147189	−0.0735947	−	0.997288i	$-0.523447\pi$
$151$	−41240.0	−0.147189	−0.0735947	+	0.997288i	$0.523447\pi$
$152$	23104.0	0.0811107
$153$	−172125.	−0.594450
$154$	117260.	0.398426
$155$	−23058.0	−0.0770890
$156$	11232.0	0.0369527
$157$	−345954.	−1.12013	−0.560066	−	0.828448i	$-0.689224\pi$
$157$	−345954.	−1.12013	−0.560066	+	0.828448i	$0.689224\pi$
$158$	−215552.	−0.686925
$159$	−237474.	−0.744943
$160$	21504.0	0.0664078
$161$	−2860.00	−0.00869564
$162$	26244.0	0.0785674
$163$	−298144.	−0.878936	−0.439468	−	0.898258i	$-0.644833\pi$
$163$	−298144.	−0.878936	−0.439468	+	0.898258i	$0.644833\pi$
$164$	−150400.	−0.436655
$165$	38745.0	0.110791
$166$	300784.	0.847197
$167$	−73290.0	−0.203354	−0.101677	−	0.994817i	$-0.532421\pi$
$167$	−73290.0	−0.203354	−0.101677	+	0.994817i	$0.532421\pi$
$168$	82368.0	0.225157
$169$	−365209.	−0.983614
$170$	−178500.	−0.473714
$171$	29241.0	0.0764719
$172$	321168.	0.827773
$173$	282102.	0.716623	0.358312	−	0.933602i	$-0.383353\pi$
$173$	282102.	0.716623	0.358312	+	0.933602i	$0.383353\pi$
$174$	175176.	0.438633
$175$	383812.	0.947378
$176$	−52480.0	−0.127706
$177$	118944.	0.285371
$178$	82472.0	0.195100
$179$	193946.	0.452427	0.226213	−	0.974078i	$-0.427365\pi$
$179$	193946.	0.452427	0.226213	+	0.974078i	$0.427365\pi$
$180$	27216.0	0.0626099
$181$	−283446.	−0.643093	−0.321547	−	0.946894i	$-0.604203\pi$
$181$	−283446.	−0.643093	−0.321547	+	0.946894i	$0.604203\pi$
$182$	44616.0	0.0998417
$183$	20637.0	0.0455532
$184$	1280.00	0.00278718
$185$	−317688.	−0.682451
$186$	39528.0	0.0837766
$187$	435625.	0.910980
$188$	225680.	0.465692
$189$	104247.	0.212280
$190$	30324.0	0.0609400
$191$	50495.0	0.100153	0.0500766	−	0.998745i	$-0.484053\pi$
$191$	50495.0	0.100153	0.0500766	+	0.998745i	$0.484053\pi$
$192$	−36864.0	−0.0721688
$193$	231092.	0.446572	0.223286	−	0.974753i	$-0.428322\pi$
$193$	231092.	0.446572	0.223286	+	0.974753i	$0.428322\pi$
$194$	−336520.	−0.641958
$195$	14742.0	0.0277632
$196$	58272.0	0.108348
$197$	−452482.	−0.830684	−0.415342	−	0.909665i	$-0.636338\pi$
$197$	−452482.	−0.830684	−0.415342	+	0.909665i	$0.636338\pi$
$198$	−66420.0	−0.120403
$199$	−207199.	−0.370898	−0.185449	−	0.982654i	$-0.559374\pi$
$199$	−207199.	−0.370898	−0.185449	+	0.982654i	$0.559374\pi$
$200$	−171776.	−0.303660
$201$	−323784.	−0.565282
$202$	654856.	1.12919
$203$	695838.	1.18514
$204$	306000.	0.514809
$205$	−197400.	−0.328067
$206$	−556248.	−0.913273
$207$	1620.00	0.00262778
$208$	−19968.0	−0.0320019
$209$	−74005.0	−0.117191
$210$	108108.	0.169165
$211$	985948.	1.52457	0.762286	−	0.647240i	$-0.224077\pi$
$211$	985948.	1.52457	0.762286	+	0.647240i	$0.224077\pi$
$212$	422176.	0.645140
$213$	−91620.0	−0.138370
$214$	498760.	0.744487
$215$	421533.	0.621921
$216$	−46656.0	−0.0680414
$217$	157014.	0.226354
$218$	−47344.0	−0.0674720
$219$	−297981.	−0.419835
$220$	−68880.0	−0.0959481
$221$	165750.	0.228283
$222$	544608.	0.741654
$223$	177756.	0.239366	0.119683	−	0.992812i	$-0.461812\pi$
$223$	177756.	0.239366	0.119683	+	0.992812i	$0.461812\pi$
$224$	−146432.	−0.194992
$225$	−217404.	−0.286293
$226$	230696.	0.300448
$227$	276382.	0.355996	0.177998	−	0.984031i	$-0.443038\pi$
$227$	276382.	0.355996	0.177998	+	0.984031i	$0.443038\pi$
$228$	−51984.0	−0.0662266
$229$	986125.	1.24263	0.621317	−	0.783559i	$-0.286598\pi$
$229$	986125.	1.24263	0.621317	+	0.783559i	$0.286598\pi$
$230$	1680.00	0.00209406
$231$	−263835.	−0.325314
$232$	−311424.	−0.379867
$233$	−116691.	−0.140815	−0.0704073	−	0.997518i	$-0.522430\pi$
$233$	−116691.	−0.140815	−0.0704073	+	0.997518i	$0.522430\pi$
$234$	−25272.0	−0.0301717
$235$	296205.	0.349883
$236$	−211456.	−0.247138
$237$	484992.	0.560872
$238$	1.21550e6	1.39095
$239$	−1.25870e6	−1.42537	−0.712685	−	0.701484i	$-0.752521\pi$
$239$	−1.25870e6	−1.42537	−0.712685	+	0.701484i	$0.752521\pi$
$240$	−48384.0	−0.0542218
$241$	−143492.	−0.159142	−0.0795710	−	0.996829i	$-0.525355\pi$
$241$	−143492.	−0.159142	−0.0795710	+	0.996829i	$0.525355\pi$
$242$	−476104.	−0.522593
$243$	−59049.0	−0.0641500
$244$	−36688.0	−0.0394502
$245$	76482.0	0.0814037
$246$	338400.	0.356527
$247$	−28158.0	−0.0293670
$248$	−70272.0	−0.0725526
$249$	−676764.	−0.691734
$250$	−487956.	−0.493777
$251$	−884163.	−0.885825	−0.442913	−	0.896565i	$-0.646055\pi$
$251$	−884163.	−0.885825	−0.442913	+	0.896565i	$0.646055\pi$
$252$	−185328.	−0.183840
$253$	−4100.00	−0.00402701
$254$	537256.	0.522513
$255$	401625.	0.386786
$256$	65536.0	0.0625000
$257$	1.04230e6	0.984370	0.492185	−	0.870491i	$-0.336198\pi$
$257$	1.04230e6	0.984370	0.492185	+	0.870491i	$0.336198\pi$
$258$	−722628.	−0.675874
$259$	2.16330e6	2.00386
$260$	−26208.0	−0.0240437
$261$	−394146.	−0.358142
$262$	−1.46382e6	−1.31745
$263$	998121.	0.889803	0.444901	−	0.895580i	$-0.353239\pi$
$263$	998121.	0.889803	0.444901	+	0.895580i	$0.353239\pi$
$264$	118080.	0.104272
$265$	554106.	0.484706
$266$	−206492.	−0.178937
$267$	−185562.	−0.159298
$268$	575616.	0.489549
$269$	1.73550e6	1.46232	0.731162	−	0.682204i	$-0.238978\pi$
$269$	1.73550e6	1.46232	0.731162	+	0.682204i	$0.238978\pi$
$270$	−61236.0	−0.0511208
$271$	−1.01674e6	−0.840982	−0.420491	−	0.907297i	$-0.638142\pi$
$271$	−1.01674e6	−0.840982	−0.420491	+	0.907297i	$0.638142\pi$
$272$	−544000.	−0.445838
$273$	−100386.	−0.0815204
$274$	179052.	0.144080
$275$	550220.	0.438737
$276$	−2880.00	−0.00227573
$277$	−1.37870e6	−1.07962	−0.539810	−	0.841787i	$-0.681504\pi$
$277$	−1.37870e6	−1.07962	−0.539810	+	0.841787i	$0.681504\pi$
$278$	−1.68872e6	−1.31052
$279$	−88938.0	−0.0684033
$280$	−192192.	−0.146501
$281$	−2.09789e6	−1.58495	−0.792476	−	0.609903i	$-0.791208\pi$
$281$	−2.09789e6	−1.58495	−0.792476	+	0.609903i	$0.791208\pi$
$282$	−507780.	−0.380236
$283$	−215963.	−0.160293	−0.0801463	−	0.996783i	$-0.525539\pi$
$283$	−215963.	−0.160293	−0.0801463	+	0.996783i	$0.525539\pi$
$284$	162880.	0.119832
$285$	−68229.0	−0.0497573
$286$	63960.0	0.0462374
$287$	1.34420e6	0.963295
$288$	82944.0	0.0589256
$289$	3.09577e6	2.18034
$290$	−408744.	−0.285402
$291$	757170.	0.524156
$292$	529744.	0.363587
$293$	−2.47872e6	−1.68678	−0.843389	−	0.537304i	$-0.819443\pi$
$293$	−2.47872e6	−1.68678	−0.843389	+	0.537304i	$0.819443\pi$
$294$	−131112.	−0.0884655
$295$	−277536.	−0.185680
$296$	−968192.	−0.642292
$297$	149445.	0.0983083
$298$	−166964.	−0.108914
$299$	−1560.00	−0.00100913
$300$	386496.	0.247937
$301$	−2.87044e6	−1.82613
$302$	−164960.	−0.104079
$303$	−1.47343e6	−0.921981
$304$	92416.0	0.0573539
$305$	−48153.0	−0.0296397
$306$	−688500.	−0.420340
$307$	495420.	0.300004	0.150002	−	0.988686i	$-0.452072\pi$
$307$	495420.	0.300004	0.150002	+	0.988686i	$0.452072\pi$
$308$	469040.	0.281730
$309$	1.25156e6	0.745684
$310$	−92232.0	−0.0545102
$311$	2.15964e6	1.26614	0.633070	−	0.774095i	$-0.281795\pi$
$311$	2.15964e6	1.26614	0.633070	+	0.774095i	$0.281795\pi$
$312$	44928.0	0.0261295
$313$	−1.34757e6	−0.777482	−0.388741	−	0.921347i	$-0.627090\pi$
$313$	−1.34757e6	−0.777482	−0.388741	+	0.921347i	$0.627090\pi$
$314$	−1.38382e6	−0.792053
$315$	−243243.	−0.138122
$316$	−862208.	−0.485729
$317$	−1.01166e6	−0.565440	−0.282720	−	0.959203i	$-0.591237\pi$
$317$	−1.01166e6	−0.565440	−0.282720	+	0.959203i	$0.591237\pi$
$318$	−949896.	−0.526755
$319$	997530.	0.548844
$320$	86016.0	0.0469574
$321$	−1.12221e6	−0.607871
$322$	−11440.0	−0.00614874
$323$	−767125.	−0.409129
$324$	104976.	0.0555556
$325$	209352.	0.109943
$326$	−1.19258e6	−0.621501
$327$	106524.	0.0550907
$328$	−601600.	−0.308762
$329$	−2.01702e6	−1.02735
$330$	154980.	0.0783413
$331$	−1.57062e6	−0.787954	−0.393977	−	0.919120i	$-0.628901\pi$
$331$	−1.57062e6	−0.787954	−0.393977	+	0.919120i	$0.628901\pi$
$332$	1.20314e6	0.599059
$333$	−1.22537e6	−0.605558
$334$	−293160.	−0.143793
$335$	755496.	0.367807
$336$	329472.	0.159210
$337$	2.16228e6	1.03714	0.518570	−	0.855035i	$-0.326464\pi$
$337$	2.16228e6	1.03714	0.518570	+	0.855035i	$0.326464\pi$
$338$	−1.46084e6	−0.695520
$339$	−519066.	−0.245315
$340$	−714000.	−0.334966
$341$	225090.	0.104826
$342$	116964.	0.0540738
$343$	1.88259e6	0.864016
$344$	1.28467e6	0.585324
$345$	−3780.00	−0.00170980
$346$	1.12841e6	0.506729
$347$	4.16873e6	1.85858	0.929288	−	0.369356i	$-0.120422\pi$
$347$	4.16873e6	1.85858	0.929288	+	0.369356i	$0.120422\pi$
$348$	700704.	0.310160
$349$	722845.	0.317674	0.158837	−	0.987305i	$-0.449226\pi$
$349$	722845.	0.317674	0.158837	+	0.987305i	$0.449226\pi$
$350$	1.53525e6	0.669898
$351$	56862.0	0.0246351
$352$	−209920.	−0.0903020
$353$	1.45102e6	0.619780	0.309890	−	0.950772i	$-0.399708\pi$
$353$	1.45102e6	0.619780	0.309890	+	0.950772i	$0.399708\pi$
$354$	475776.	0.201788
$355$	213780.	0.0900319
$356$	329888.	0.137956
$357$	−2.73488e6	−1.13571
$358$	775784.	0.319914
$359$	517179.	0.211790	0.105895	−	0.994377i	$-0.466229\pi$
$359$	517179.	0.211790	0.105895	+	0.994377i	$0.466229\pi$
$360$	108864.	0.0442719
$361$	130321.	0.0526316
$362$	−1.13378e6	−0.454736
$363$	1.07123e6	0.426695
$364$	178464.	0.0705988
$365$	695289.	0.273170
$366$	82548.0	0.0322110
$367$	−3.59392e6	−1.39285	−0.696423	−	0.717631i	$-0.745226\pi$
$367$	−3.59392e6	−1.39285	−0.696423	+	0.717631i	$0.745226\pi$
$368$	5120.00	0.00197084
$369$	−761400.	−0.291103
$370$	−1.27075e6	−0.482566
$371$	−3.77320e6	−1.42323
$372$	158112.	0.0592390
$373$	−1.77578e6	−0.660870	−0.330435	−	0.943829i	$-0.607195\pi$
$373$	−1.77578e6	−0.660870	−0.330435	+	0.943829i	$0.607195\pi$
$374$	1.74250e6	0.644160
$375$	1.09790e6	0.403167
$376$	902720.	0.329294
$377$	379548.	0.137535
$378$	416988.	0.150105
$379$	2.48464e6	0.888516	0.444258	−	0.895899i	$-0.353467\pi$
$379$	2.48464e6	0.888516	0.444258	+	0.895899i	$0.353467\pi$
$380$	121296.	0.0430911
$381$	−1.20883e6	−0.426630
$382$	201980.	0.0708190
$383$	1.70987e6	0.595614	0.297807	−	0.954626i	$-0.403745\pi$
$383$	1.70987e6	0.595614	0.297807	+	0.954626i	$0.403745\pi$
$384$	−147456.	−0.0510310
$385$	615615.	0.211669
$386$	924368.	0.315774
$387$	1.62591e6	0.551849
$388$	−1.34608e6	−0.453933
$389$	−244715.	−0.0819949	−0.0409974	−	0.999159i	$-0.513054\pi$
$389$	−244715.	−0.0819949	−0.0409974	+	0.999159i	$0.513054\pi$
$390$	58968.0	0.0196316
$391$	−42500.0	−0.0140588
$392$	233088.	0.0766134
$393$	3.29359e6	1.07569
$394$	−1.80993e6	−0.587382
$395$	−1.13165e6	−0.364938
$396$	−265680.	−0.0851375
$397$	−4.11195e6	−1.30940	−0.654698	−	0.755890i	$-0.727204\pi$
$397$	−4.11195e6	−1.30940	−0.654698	+	0.755890i	$0.727204\pi$
$398$	−828796.	−0.262265
$399$	464607.	0.146101
$400$	−687104.	−0.214720
$401$	−1.87955e6	−0.583704	−0.291852	−	0.956464i	$-0.594271\pi$
$401$	−1.87955e6	−0.583704	−0.291852	+	0.956464i	$0.594271\pi$
$402$	−1.29514e6	−0.399715
$403$	85644.0	0.0262684
$404$	2.61942e6	0.798459
$405$	137781.	0.0417399
$406$	2.78335e6	0.838017
$407$	3.10124e6	0.928003
$408$	1.22400e6	0.364025
$409$	−2.68258e6	−0.792948	−0.396474	−	0.918046i	$-0.629766\pi$
$409$	−2.68258e6	−0.792948	−0.396474	+	0.918046i	$0.629766\pi$
$410$	−789600.	−0.231978
$411$	−402867.	−0.117641
$412$	−2.22499e6	−0.645781
$413$	1.88989e6	0.545206
$414$	6480.00	0.00185812
$415$	1.57912e6	0.450084
$416$	−79872.0	−0.0226288
$417$	3.79961e6	1.07004
$418$	−296020.	−0.0828668
$419$	−5.70599e6	−1.58780	−0.793900	−	0.608048i	$-0.791953\pi$
$419$	−5.70599e6	−1.58780	−0.793900	+	0.608048i	$0.791953\pi$
$420$	432432.	0.119617
$421$	−4.24578e6	−1.16749	−0.583744	−	0.811938i	$-0.698413\pi$
$421$	−4.24578e6	−1.16749	−0.583744	+	0.811938i	$0.698413\pi$
$422$	3.94379e6	1.07804
$423$	1.14250e6	0.310461
$424$	1.68870e6	0.456183
$425$	5.70350e6	1.53168
$426$	−366480.	−0.0978422
$427$	327899.	0.0870303
$428$	1.99504e6	0.526432
$429$	−143910.	−0.0377527
$430$	1.68613e6	0.439765
$431$	−1.28341e6	−0.332793	−0.166396	−	0.986059i	$-0.553213\pi$
$431$	−1.28341e6	−0.332793	−0.166396	+	0.986059i	$0.553213\pi$
$432$	−186624.	−0.0481125
$433$	−6.39182e6	−1.63834	−0.819171	−	0.573549i	$-0.805566\pi$
$433$	−6.39182e6	−1.63834	−0.819171	+	0.573549i	$0.805566\pi$
$434$	628056.	0.160057
$435$	919674.	0.233029
$436$	−189376.	−0.0477099
$437$	7220.00	0.00180856
$438$	−1.19192e6	−0.296868
$439$	1.27464e6	0.315664	0.157832	−	0.987466i	$-0.449549\pi$
$439$	1.27464e6	0.315664	0.157832	+	0.987466i	$0.449549\pi$
$440$	−275520.	−0.0678456
$441$	295002.	0.0722318
$442$	663000.	0.161420
$443$	2.35546e6	0.570253	0.285126	−	0.958490i	$-0.407964\pi$
$443$	2.35546e6	0.570253	0.285126	+	0.958490i	$0.407964\pi$
$444$	2.17843e6	0.524429
$445$	432978.	0.103649
$446$	711024.	0.169257
$447$	375669.	0.0889276
$448$	−585728.	−0.137880
$449$	7.99412e6	1.87135	0.935675	−	0.352863i	$-0.114792\pi$
$449$	7.99412e6	1.87135	0.935675	+	0.352863i	$0.114792\pi$
$450$	−869616.	−0.202440
$451$	1.92700e6	0.446108
$452$	922784.	0.212449
$453$	371160.	0.0849798
$454$	1.10553e6	0.251727
$455$	234234.	0.0530422
$456$	−207936.	−0.0468293
$457$	−3.03844e6	−0.680550	−0.340275	−	0.940326i	$-0.610520\pi$
$457$	−3.03844e6	−0.680550	−0.340275	+	0.940326i	$0.610520\pi$
$458$	3.94450e6	0.878675
$459$	1.54912e6	0.343206
$460$	6720.00	0.00148073
$461$	−5.98744e6	−1.31217	−0.656084	−	0.754688i	$-0.727788\pi$
$461$	−5.98744e6	−1.31217	−0.656084	+	0.754688i	$0.727788\pi$
$462$	−1.05534e6	−0.230032
$463$	8.05385e6	1.74603	0.873014	−	0.487695i	$-0.162162\pi$
$463$	8.05385e6	1.74603	0.873014	+	0.487695i	$0.162162\pi$
$464$	−1.24570e6	−0.268607
$465$	207522.	0.0445074
$466$	−466764.	−0.0995709
$467$	−8.29332e6	−1.75969	−0.879845	−	0.475261i	$-0.842354\pi$
$467$	−8.29332e6	−1.75969	−0.879845	+	0.475261i	$0.842354\pi$
$468$	−101088.	−0.0213346
$469$	−5.14457e6	−1.07998
$470$	1.18482e6	0.247405
$471$	3.11359e6	0.646709
$472$	−845824.	−0.174753
$473$	−4.11496e6	−0.845694
$474$	1.93997e6	0.396596
$475$	−968924.	−0.197041
$476$	4.86200e6	0.983553
$477$	2.13727e6	0.430093
$478$	−5.03480e6	−1.00789
$479$	−4.58472e6	−0.913007	−0.456503	−	0.889722i	$-0.650898\pi$
$479$	−4.58472e6	−0.913007	−0.456503	+	0.889722i	$0.650898\pi$
$480$	−193536.	−0.0383406
$481$	1.17998e6	0.232548
$482$	−573968.	−0.112530
$483$	25740.0	0.00502043
$484$	−1.90442e6	−0.369529
$485$	−1.76673e6	−0.341048
$486$	−236196.	−0.0453609
$487$	55304.0	0.0105666	0.00528329	−	0.999986i	$-0.498318\pi$
$487$	55304.0	0.0105666	0.00528329	+	0.999986i	$0.498318\pi$
$488$	−146752.	−0.0278955
$489$	2.68330e6	0.507454
$490$	305928.	0.0575611
$491$	−6.88932e6	−1.28965	−0.644826	−	0.764329i	$-0.723070\pi$
$491$	−6.88932e6	−1.28965	−0.644826	+	0.764329i	$0.723070\pi$
$492$	1.35360e6	0.252103
$493$	1.03402e7	1.91608
$494$	−112632.	−0.0207656
$495$	−348705.	−0.0639654
$496$	−281088.	−0.0513025
$497$	−1.45574e6	−0.264358
$498$	−2.70706e6	−0.489130
$499$	5.61676e6	1.00980	0.504899	−	0.863179i	$-0.331530\pi$
$499$	5.61676e6	1.00980	0.504899	+	0.863179i	$0.331530\pi$
$500$	−1.95182e6	−0.349153
$501$	659610.	0.117407
$502$	−3.53665e6	−0.626373
$503$	−5.16131e6	−0.909578	−0.454789	−	0.890599i	$-0.650285\pi$
$503$	−5.16131e6	−0.909578	−0.454789	+	0.890599i	$0.650285\pi$
$504$	−741312.	−0.129994
$505$	3.43799e6	0.599897
$506$	−16400.0	−0.00284753
$507$	3.28688e6	0.567890
$508$	2.14902e6	0.369472
$509$	−1.81839e6	−0.311094	−0.155547	−	0.987828i	$-0.549714\pi$
$509$	−1.81839e6	−0.311094	−0.155547	+	0.987828i	$0.549714\pi$
$510$	1.60650e6	0.273499
$511$	−4.73459e6	−0.802102
$512$	262144.	0.0441942
$513$	−263169.	−0.0441511
$514$	4.16918e6	0.696055
$515$	−2.92030e6	−0.485188
$516$	−2.89051e6	−0.477915
$517$	−2.89152e6	−0.475774
$518$	8.65322e6	1.41695
$519$	−2.53892e6	−0.413743
$520$	−104832.	−0.0170014
$521$	1.10957e7	1.79085	0.895424	−	0.445215i	$-0.146873\pi$
$521$	1.10957e7	1.79085	0.895424	+	0.445215i	$0.146873\pi$
$522$	−1.57658e6	−0.253245
$523$	−7.51297e6	−1.20104	−0.600520	−	0.799610i	$-0.705040\pi$
$523$	−7.51297e6	−1.20104	−0.600520	+	0.799610i	$0.705040\pi$
$524$	−5.85528e6	−0.931578
$525$	−3.45431e6	−0.546969
$526$	3.99248e6	0.629186
$527$	2.33325e6	0.365961
$528$	472320.	0.0737312
$529$	−6.43594e6	−0.999938
$530$	2.21642e6	0.342739
$531$	−1.07050e6	−0.164759
$532$	−825968.	−0.126527
$533$	733200.	0.111790
$534$	−742248.	−0.112641
$535$	2.61849e6	0.395518
$536$	2.30246e6	0.346163
$537$	−1.74551e6	−0.261209
$538$	6.94199e6	1.03402
$539$	−746610.	−0.110693
$540$	−244944.	−0.0361478
$541$	3.15622e6	0.463632	0.231816	−	0.972760i	$-0.425533\pi$
$541$	3.15622e6	0.463632	0.231816	+	0.972760i	$0.425533\pi$
$542$	−4.06696e6	−0.594664
$543$	2.55101e6	0.371290
$544$	−2.17600e6	−0.315255
$545$	−248556.	−0.0358454
$546$	−401544.	−0.0576437
$547$	−1.17063e7	−1.67283	−0.836417	−	0.548094i	$-0.815353\pi$
$547$	−1.17063e7	−1.67283	−0.836417	+	0.548094i	$0.815353\pi$
$548$	716208.	0.101880
$549$	−185733.	−0.0263001
$550$	2.20088e6	0.310234
$551$	−1.75663e6	−0.246491
$552$	−11520.0	−0.00160918
$553$	7.70598e6	1.07156
$554$	−5.51481e6	−0.763407
$555$	2.85919e6	0.394013
$556$	−6.75486e6	−0.926680
$557$	5.56111e6	0.759493	0.379746	−	0.925091i	$-0.376011\pi$
$557$	5.56111e6	0.759493	0.379746	+	0.925091i	$0.376011\pi$
$558$	−355752.	−0.0483684
$559$	−1.56569e6	−0.211923
$560$	−768768.	−0.103592
$561$	−3.92062e6	−0.525954
$562$	−8.39154e6	−1.12073
$563$	−1.17669e7	−1.56456	−0.782280	−	0.622928i	$-0.785943\pi$
$563$	−1.17669e7	−1.56456	−0.782280	+	0.622928i	$0.785943\pi$
$564$	−2.03112e6	−0.268867
$565$	1.21115e6	0.159617
$566$	−863852.	−0.113344
$567$	−938223.	−0.122560
$568$	651520.	0.0847338
$569$	6.49649e6	0.841198	0.420599	−	0.907247i	$-0.361820\pi$
$569$	6.49649e6	0.841198	0.420599	+	0.907247i	$0.361820\pi$
$570$	−272916.	−0.0351837
$571$	−7.15432e6	−0.918286	−0.459143	−	0.888362i	$-0.651843\pi$
$571$	−7.15432e6	−0.918286	−0.459143	+	0.888362i	$0.651843\pi$
$572$	255840.	0.0326948
$573$	−454455.	−0.0578235
$574$	5.37680e6	0.681152
$575$	−53680.0	−0.00677085
$576$	331776.	0.0416667
$577$	−8.50781e6	−1.06385	−0.531923	−	0.846793i	$-0.678530\pi$
$577$	−8.50781e6	−1.06385	−0.531923	+	0.846793i	$0.678530\pi$
$578$	1.23831e7	1.54173
$579$	−2.07983e6	−0.257829
$580$	−1.63498e6	−0.201809
$581$	−1.07530e7	−1.32157
$582$	3.02868e6	0.370635
$583$	−5.40913e6	−0.659107
$584$	2.11898e6	0.257095
$585$	−132678.	−0.0160291
$586$	−9.91486e6	−1.19273
$587$	−854707.	−0.102382	−0.0511908	−	0.998689i	$-0.516302\pi$
$587$	−854707.	−0.102382	−0.0511908	+	0.998689i	$0.516302\pi$
$588$	−524448.	−0.0625546
$589$	−396378.	−0.0470784
$590$	−1.11014e6	−0.131295
$591$	4.07234e6	0.479596
$592$	−3.87277e6	−0.454169
$593$	−5.55213e6	−0.648370	−0.324185	−	0.945994i	$-0.605090\pi$
$593$	−5.55213e6	−0.648370	−0.324185	+	0.945994i	$0.605090\pi$
$594$	597780.	0.0695145
$595$	6.38138e6	0.738962
$596$	−667856.	−0.0770136
$597$	1.86479e6	0.214138
$598$	−6240.00	−0.000713562 0
$599$	−1.30669e6	−0.148801	−0.0744003	−	0.997228i	$-0.523704\pi$
$599$	−1.30669e6	−0.148801	−0.0744003	+	0.997228i	$0.523704\pi$
$600$	1.54598e6	0.175318
$601$	−7.94858e6	−0.897643	−0.448821	−	0.893621i	$-0.648156\pi$
$601$	−7.94858e6	−0.897643	−0.448821	+	0.893621i	$0.648156\pi$
$602$	−1.14818e7	−1.29127
$603$	2.91406e6	0.326366
$604$	−659840.	−0.0735947
$605$	−2.49955e6	−0.277634
$606$	−5.89370e6	−0.651939
$607$	2.83746e6	0.312578	0.156289	−	0.987711i	$-0.450047\pi$
$607$	2.83746e6	0.312578	0.156289	+	0.987711i	$0.450047\pi$
$608$	369664.	0.0405554
$609$	−6.26254e6	−0.684238
$610$	−192612.	−0.0209584
$611$	−1.10019e6	−0.119224
$612$	−2.75400e6	−0.297225
$613$	1.49674e7	1.60877	0.804387	−	0.594106i	$-0.202494\pi$
$613$	1.49674e7	1.60877	0.804387	+	0.594106i	$0.202494\pi$
$614$	1.98168e6	0.212135
$615$	1.77660e6	0.189410
$616$	1.87616e6	0.199213
$617$	−3.59751e6	−0.380443	−0.190221	−	0.981741i	$-0.560921\pi$
$617$	−3.59751e6	−0.380443	−0.190221	+	0.981741i	$0.560921\pi$
$618$	5.00623e6	0.527278
$619$	6.74758e6	0.707818	0.353909	−	0.935280i	$-0.384852\pi$
$619$	6.74758e6	0.707818	0.353909	+	0.935280i	$0.384852\pi$
$620$	−368928.	−0.0385445
$621$	−14580.0	−0.00151715
$622$	8.63858e6	0.895295
$623$	−2.94837e6	−0.304342
$624$	179712.	0.0184763
$625$	5.82573e6	0.596555
$626$	−5.39028e6	−0.549763
$627$	666045.	0.0676604
$628$	−5.53526e6	−0.560066
$629$	3.21470e7	3.23977
$630$	−972972.	−0.0976673
$631$	1.36044e7	1.36021	0.680103	−	0.733117i	$-0.261935\pi$
$631$	1.36044e7	1.36021	0.680103	+	0.733117i	$0.261935\pi$
$632$	−3.44883e6	−0.343463
$633$	−8.87353e6	−0.880212
$634$	−4.04664e6	−0.399826
$635$	2.82059e6	0.277592
$636$	−3.79958e6	−0.372472
$637$	−284076.	−0.0277387
$638$	3.99012e6	0.388092
$639$	824580.	0.0798878
$640$	344064.	0.0332039
$641$	1.91221e7	1.83819	0.919097	−	0.394030i	$-0.128920\pi$
$641$	1.91221e7	1.83819	0.919097	+	0.394030i	$0.128920\pi$
$642$	−4.48884e6	−0.429830
$643$	8.38738e6	0.800016	0.400008	−	0.916512i	$-0.369007\pi$
$643$	8.38738e6	0.800016	0.400008	+	0.916512i	$0.369007\pi$
$644$	−45760.0	−0.00434782
$645$	−3.79380e6	−0.359066
$646$	−3.06850e6	−0.289298
$647$	7.90221e6	0.742143	0.371072	−	0.928604i	$-0.378990\pi$
$647$	7.90221e6	0.742143	0.371072	+	0.928604i	$0.378990\pi$
$648$	419904.	0.0392837
$649$	2.70928e6	0.252489
$650$	837408.	0.0777416
$651$	−1.41313e6	−0.130686
$652$	−4.77030e6	−0.439468
$653$	−279213.	−0.0256243	−0.0128122	−	0.999918i	$-0.504078\pi$
$653$	−279213.	−0.0256243	−0.0128122	+	0.999918i	$0.504078\pi$
$654$	426096.	0.0389550
$655$	−7.68506e6	−0.699912
$656$	−2.40640e6	−0.218327
$657$	2.68183e6	0.242392
$658$	−8.06806e6	−0.726448
$659$	−6.82671e6	−0.612348	−0.306174	−	0.951976i	$-0.599049\pi$
$659$	−6.82671e6	−0.612348	−0.306174	+	0.951976i	$0.599049\pi$
$660$	619920.	0.0553957
$661$	1.85059e7	1.64742	0.823712	−	0.567008i	$-0.191899\pi$
$661$	1.85059e7	1.64742	0.823712	+	0.567008i	$0.191899\pi$
$662$	−6.28248e6	−0.557168
$663$	−1.49175e6	−0.131799
$664$	4.81254e6	0.423599
$665$	−1.08408e6	−0.0950623
$666$	−4.90147e6	−0.428194
$667$	−97320.0	−0.00847008
$668$	−1.17264e6	−0.101677
$669$	−1.59980e6	−0.138198
$670$	3.02198e6	0.260079
$671$	470065.	0.0403043
$672$	1.31789e6	0.112578
$673$	1.25059e7	1.06433	0.532166	−	0.846640i	$-0.321378\pi$
$673$	1.25059e7	1.06433	0.532166	+	0.846640i	$0.321378\pi$
$674$	8.64913e6	0.733369
$675$	1.95664e6	0.165292
$676$	−5.84334e6	−0.491807
$677$	5.47029e6	0.458710	0.229355	−	0.973343i	$-0.426338\pi$
$677$	5.47029e6	0.458710	0.229355	+	0.973343i	$0.426338\pi$
$678$	−2.07626e6	−0.173464
$679$	1.20306e7	1.00141
$680$	−2.85600e6	−0.236857
$681$	−2.48744e6	−0.205534
$682$	900360.	0.0741234
$683$	1.02415e7	0.840061	0.420031	−	0.907510i	$-0.362019\pi$
$683$	1.02415e7	0.840061	0.420031	+	0.907510i	$0.362019\pi$
$684$	467856.	0.0382360
$685$	940023.	0.0765442
$686$	7.53038e6	0.610951
$687$	−8.87512e6	−0.717435
$688$	5.13869e6	0.413886
$689$	−2.05811e6	−0.165166
$690$	−15120.0	−0.00120901
$691$	1.84945e6	0.147349	0.0736744	−	0.997282i	$-0.476527\pi$
$691$	1.84945e6	0.147349	0.0736744	+	0.997282i	$0.476527\pi$
$692$	4.51363e6	0.358312
$693$	2.37452e6	0.187820
$694$	1.66749e7	1.31421
$695$	−8.86576e6	−0.696232
$696$	2.80282e6	0.219317
$697$	1.99750e7	1.55742
$698$	2.89138e6	0.224629
$699$	1.05022e6	0.0812993
$700$	6.14099e6	0.473689
$701$	1.65813e7	1.27445	0.637226	−	0.770677i	$-0.280082\pi$
$701$	1.65813e7	1.27445	0.637226	+	0.770677i	$0.280082\pi$
$702$	227448.	0.0174196
$703$	−5.46121e6	−0.416774
$704$	−839680.	−0.0638531
$705$	−2.66584e6	−0.202005
$706$	5.80409e6	0.438250
$707$	−2.34111e7	−1.76146
$708$	1.90310e6	0.142685
$709$	1.10501e7	0.825562	0.412781	−	0.910830i	$-0.364558\pi$
$709$	1.10501e7	0.825562	0.412781	+	0.910830i	$0.364558\pi$
$710$	855120.	0.0636621
$711$	−4.36493e6	−0.323820
$712$	1.31955e6	0.0975498
$713$	−21960.0	−0.00161774
$714$	−1.09395e7	−0.803067
$715$	335790.	0.0245642
$716$	3.10314e6	0.226213
$717$	1.13283e7	0.822938
$718$	2.06872e6	0.149758
$719$	1.02255e7	0.737669	0.368835	−	0.929495i	$-0.379757\pi$
$719$	1.02255e7	0.737669	0.368835	+	0.929495i	$0.379757\pi$
$720$	435456.	0.0313050
$721$	1.98859e7	1.42464
$722$	521284.	0.0372161
$723$	1.29143e6	0.0918807
$724$	−4.53514e6	−0.321547
$725$	1.30603e7	0.922804
$726$	4.28494e6	0.301719
$727$	2.61985e7	1.83840	0.919202	−	0.393787i	$-0.128835\pi$
$727$	2.61985e7	1.83840	0.919202	+	0.393787i	$0.128835\pi$
$728$	713856.	0.0499209
$729$	531441.	0.0370370
$730$	2.78116e6	0.193160
$731$	−4.26551e7	−2.95242
$732$	330192.	0.0227766
$733$	−2.56029e7	−1.76007	−0.880033	−	0.474913i	$-0.842480\pi$
$733$	−2.56029e7	−1.76007	−0.880033	+	0.474913i	$0.842480\pi$
$734$	−1.43757e7	−0.984891
$735$	−688338.	−0.0469984
$736$	20480.0	0.00139359
$737$	−7.37508e6	−0.500147
$738$	−3.04560e6	−0.205841
$739$	−2.10847e7	−1.42022	−0.710112	−	0.704089i	$-0.751356\pi$
$739$	−2.10847e7	−1.42022	−0.710112	+	0.704089i	$0.751356\pi$
$740$	−5.08301e6	−0.341226
$741$	253422.	0.0169550
$742$	−1.50928e7	−1.00637
$743$	8.71807e6	0.579360	0.289680	−	0.957124i	$-0.406451\pi$
$743$	8.71807e6	0.579360	0.289680	+	0.957124i	$0.406451\pi$
$744$	632448.	0.0418883
$745$	−876561.	−0.0578617
$746$	−7.10310e6	−0.467306
$747$	6.09088e6	0.399373
$748$	6.97000e6	0.455490
$749$	−1.78307e7	−1.16135
$750$	4.39160e6	0.285082
$751$	−1.26501e7	−0.818452	−0.409226	−	0.912433i	$-0.634201\pi$
$751$	−1.26501e7	−0.818452	−0.409226	+	0.912433i	$0.634201\pi$
$752$	3.61088e6	0.232846
$753$	7.95747e6	0.511431
$754$	1.51819e6	0.0972520
$755$	−866040.	−0.0552931
$756$	1.66795e6	0.106140
$757$	−5.69653e6	−0.361302	−0.180651	−	0.983547i	$-0.557821\pi$
$757$	−5.69653e6	−0.361302	−0.180651	+	0.983547i	$0.557821\pi$
$758$	9.93855e6	0.628275
$759$	36900.0	0.00232499
$760$	485184.	0.0304700
$761$	−2.51658e7	−1.57525	−0.787625	−	0.616155i	$-0.788690\pi$
$761$	−2.51658e7	−1.57525	−0.787625	+	0.616155i	$0.788690\pi$
$762$	−4.83530e6	−0.301673
$763$	1.69255e6	0.105252
$764$	807920.	0.0500766
$765$	−3.61462e6	−0.223311
$766$	6.83946e6	0.421163
$767$	1.03085e6	0.0632712
$768$	−589824.	−0.0360844
$769$	−2.35673e7	−1.43713	−0.718563	−	0.695462i	$-0.755200\pi$
$769$	−2.35673e7	−1.43713	−0.718563	+	0.695462i	$0.755200\pi$
$770$	2.46246e6	0.149673
$771$	−9.38066e6	−0.568326
$772$	3.69747e6	0.223286
$773$	1.61071e7	0.969548	0.484774	−	0.874639i	$-0.338902\pi$
$773$	1.61071e7	0.969548	0.484774	+	0.874639i	$0.338902\pi$
$774$	6.50365e6	0.390216
$775$	2.94703e6	0.176251
$776$	−5.38432e6	−0.320979
$777$	−1.94697e7	−1.15693
$778$	−978860.	−0.0579791
$779$	−3.39340e6	−0.200351
$780$	235872.	0.0138816
$781$	−2.08690e6	−0.122426
$782$	−170000.	−0.00994104
$783$	3.54731e6	0.206774
$784$	932352.	0.0541739
$785$	−7.26503e6	−0.420788
$786$	1.31744e7	0.760631
$787$	−8.09622e6	−0.465956	−0.232978	−	0.972482i	$-0.574847\pi$
$787$	−8.09622e6	−0.465956	−0.232978	+	0.972482i	$0.574847\pi$
$788$	−7.23971e6	−0.415342
$789$	−8.98309e6	−0.513728
$790$	−4.52659e6	−0.258050
$791$	−8.24738e6	−0.468678
$792$	−1.06272e6	−0.0602013
$793$	178854.	0.0100999
$794$	−1.64478e7	−0.925883
$795$	−4.98695e6	−0.279845
$796$	−3.31518e6	−0.185449
$797$	2.23729e7	1.24760	0.623800	−	0.781584i	$-0.285588\pi$
$797$	2.23729e7	1.24760	0.623800	+	0.781584i	$0.285588\pi$
$798$	1.85843e6	0.103309
$799$	−2.99731e7	−1.66098
$800$	−2.74842e6	−0.151830
$801$	1.67006e6	0.0919709
$802$	−7.51819e6	−0.412741
$803$	−6.78735e6	−0.371459
$804$	−5.18054e6	−0.282641
$805$	−60060.0	−0.00326660
$806$	342576.	0.0185746
$807$	−1.56195e7	−0.844273
$808$	1.04777e7	0.564595
$809$	915669.	0.0491889	0.0245945	−	0.999698i	$-0.492171\pi$
$809$	915669.	0.0491889	0.0245945	+	0.999698i	$0.492171\pi$
$810$	551124.	0.0295146
$811$	−3.00496e7	−1.60430	−0.802151	−	0.597121i	$-0.796311\pi$
$811$	−3.00496e7	−1.60430	−0.802151	+	0.597121i	$0.796311\pi$
$812$	1.11334e7	0.592568
$813$	9.15066e6	0.485541
$814$	1.24050e7	0.656197
$815$	−6.26102e6	−0.330180
$816$	4.89600e6	0.257404
$817$	7.24635e6	0.379808
$818$	−1.07303e7	−0.560699
$819$	903474.	0.0470658
$820$	−3.15840e6	−0.164033
$821$	3.25959e6	0.168774	0.0843868	−	0.996433i	$-0.473107\pi$
$821$	3.25959e6	0.168774	0.0843868	+	0.996433i	$0.473107\pi$
$822$	−1.61147e6	−0.0831844
$823$	2.63514e7	1.35614	0.678068	−	0.734999i	$-0.262817\pi$
$823$	2.63514e7	1.35614	0.678068	+	0.734999i	$0.262817\pi$
$824$	−8.89997e6	−0.456636
$825$	−4.95198e6	−0.253305
$826$	7.55955e6	0.385519
$827$	3.88895e6	0.197728	0.0988640	−	0.995101i	$-0.468479\pi$
$827$	3.88895e6	0.197728	0.0988640	+	0.995101i	$0.468479\pi$
$828$	25920.0	0.00131389
$829$	−2.49909e7	−1.26298	−0.631488	−	0.775385i	$-0.717556\pi$
$829$	−2.49909e7	−1.26298	−0.631488	+	0.775385i	$0.717556\pi$
$830$	6.31646e6	0.318258
$831$	1.24083e7	0.623319
$832$	−319488.	−0.0160010
$833$	−7.73925e6	−0.386444
$834$	1.51984e7	0.756631
$835$	−1.53909e6	−0.0763920
$836$	−1.18408e6	−0.0585956
$837$	800442.	0.0394926
$838$	−2.28240e7	−1.12274
$839$	−2.77733e6	−0.136214	−0.0681072	−	0.997678i	$-0.521696\pi$
$839$	−2.77733e6	−0.136214	−0.0681072	+	0.997678i	$0.521696\pi$
$840$	1.72973e6	0.0845823
$841$	3.16681e6	0.154394
$842$	−1.69831e7	−0.825539
$843$	1.88810e7	0.915072
$844$	1.57752e7	0.762286
$845$	−7.66939e6	−0.369504
$846$	4.57002e6	0.219529
$847$	1.70207e7	0.815210
$848$	6.75482e6	0.322570
$849$	1.94367e6	0.0925449
$850$	2.28140e7	1.08306
$851$	−302560.	−0.0143215
$852$	−1.46592e6	−0.0691849
$853$	1.63379e6	0.0768820	0.0384410	−	0.999261i	$-0.487761\pi$
$853$	1.63379e6	0.0768820	0.0384410	+	0.999261i	$0.487761\pi$
$854$	1.31160e6	0.0615397
$855$	614061.	0.0287274
$856$	7.98016e6	0.372244
$857$	2.45037e7	1.13967	0.569836	−	0.821758i	$-0.307007\pi$
$857$	2.45037e7	1.13967	0.569836	+	0.821758i	$0.307007\pi$
$858$	−575640.	−0.0266952
$859$	1.45576e7	0.673143	0.336572	−	0.941658i	$-0.390733\pi$
$859$	1.45576e7	0.673143	0.336572	+	0.941658i	$0.390733\pi$
$860$	6.74453e6	0.310961
$861$	−1.20978e7	−0.556158
$862$	−5.13366e6	−0.235320
$863$	1.76706e7	0.807651	0.403826	−	0.914836i	$-0.367680\pi$
$863$	1.76706e7	0.807651	0.403826	+	0.914836i	$0.367680\pi$
$864$	−746496.	−0.0340207
$865$	5.92414e6	0.269206
$866$	−2.55673e7	−1.15848
$867$	−2.78619e7	−1.25882
$868$	2.51222e6	0.113177
$869$	1.10470e7	0.496245
$870$	3.67870e6	0.164777
$871$	−2.80613e6	−0.125332
$872$	−757504.	−0.0337360
$873$	−6.81453e6	−0.302622
$874$	28880.0	0.00127885
$875$	1.74444e7	0.770259
$876$	−4.76770e6	−0.209917
$877$	−4.46464e7	−1.96014	−0.980070	−	0.198650i	$-0.936344\pi$
$877$	−4.46464e7	−1.96014	−0.980070	+	0.198650i	$0.936344\pi$
$878$	5.09855e6	0.223208
$879$	2.23084e7	0.973861
$880$	−1.10208e6	−0.0479741
$881$	−1.97316e7	−0.856490	−0.428245	−	0.903663i	$-0.640868\pi$
$881$	−1.97316e7	−0.856490	−0.428245	+	0.903663i	$0.640868\pi$
$882$	1.18001e6	0.0510756
$883$	−2.12686e7	−0.917989	−0.458994	−	0.888439i	$-0.651790\pi$
$883$	−2.12686e7	−0.917989	−0.458994	+	0.888439i	$0.651790\pi$
$884$	2.65200e6	0.114141
$885$	2.49782e6	0.107202
$886$	9.42186e6	0.403230
$887$	−3.40623e7	−1.45367	−0.726835	−	0.686813i	$-0.759009\pi$
$887$	−3.40623e7	−1.45367	−0.726835	+	0.686813i	$0.759009\pi$
$888$	8.71373e6	0.370827
$889$	−1.92069e7	−0.815085
$890$	1.73191e6	0.0732910
$891$	−1.34500e6	−0.0567583
$892$	2.84410e6	0.119683
$893$	5.09190e6	0.213674
$894$	1.50268e6	0.0628813
$895$	4.07287e6	0.169958
$896$	−2.34291e6	−0.0974958
$897$	14040.0	0.000582621 0
$898$	3.19765e7	1.32324
$899$	5.34287e6	0.220483
$900$	−3.47846e6	−0.143147
$901$	−5.60702e7	−2.30102
$902$	7.70800e6	0.315446
$903$	2.58340e7	1.05432
$904$	3.69114e6	0.150224
$905$	−5.95237e6	−0.241584
$906$	1.48464e6	0.0600898
$907$	−1.59880e7	−0.645323	−0.322662	−	0.946514i	$-0.604578\pi$
$907$	−1.59880e7	−0.645323	−0.322662	+	0.946514i	$0.604578\pi$
$908$	4.42211e6	0.177998
$909$	1.32608e7	0.532306
$910$	936936.	0.0375065
$911$	2.70045e7	1.07805	0.539026	−	0.842289i	$-0.318792\pi$
$911$	2.70045e7	1.07805	0.539026	+	0.842289i	$0.318792\pi$
$912$	−831744.	−0.0331133
$913$	−1.54152e7	−0.612029
$914$	−1.21537e7	−0.481221
$915$	433377.	0.0171125
$916$	1.57780e7	0.621317
$917$	5.23316e7	2.05514
$918$	6.19650e6	0.242683
$919$	−6.71840e6	−0.262408	−0.131204	−	0.991355i	$-0.541884\pi$
$919$	−6.71840e6	−0.262408	−0.131204	+	0.991355i	$0.541884\pi$
$920$	26880.0	0.00104703
$921$	−4.45878e6	−0.173208
$922$	−2.39498e7	−0.927842
$923$	−794040.	−0.0306788
$924$	−4.22136e6	−0.162657
$925$	4.06036e7	1.56031
$926$	3.22154e7	1.23463
$927$	−1.12640e7	−0.430521
$928$	−4.98278e6	−0.189934
$929$	−1.29929e7	−0.493933	−0.246967	−	0.969024i	$-0.579434\pi$
$929$	−1.29929e7	−0.493933	−0.246967	+	0.969024i	$0.579434\pi$
$930$	830088.	0.0314715
$931$	1.31476e6	0.0497133
$932$	−1.86706e6	−0.0704073
$933$	−1.94368e7	−0.731006
$934$	−3.31733e7	−1.24429
$935$	9.14812e6	0.342218
$936$	−404352.	−0.0150859
$937$	2.87830e6	0.107099	0.0535497	−	0.998565i	$-0.482946\pi$
$937$	2.87830e6	0.107099	0.0535497	+	0.998565i	$0.482946\pi$
$938$	−2.05783e7	−0.763663
$939$	1.21281e7	0.448880
$940$	4.73928e6	0.174941
$941$	−1.11189e7	−0.409345	−0.204672	−	0.978831i	$-0.565613\pi$
$941$	−1.11189e7	−0.409345	−0.204672	+	0.978831i	$0.565613\pi$
$942$	1.24543e7	0.457292
$943$	−188000.	−0.00688460
$944$	−3.38330e6	−0.123569
$945$	2.18919e6	0.0797450
$946$	−1.64599e7	−0.597996
$947$	−1.59967e7	−0.579638	−0.289819	−	0.957081i	$-0.593595\pi$
$947$	−1.59967e7	−0.579638	−0.289819	+	0.957081i	$0.593595\pi$
$948$	7.75987e6	0.280436
$949$	−2.58250e6	−0.0930840
$950$	−3.87570e6	−0.139329
$951$	9.10494e6	0.326457
$952$	1.94480e7	0.695477
$953$	3.80244e7	1.35622	0.678110	−	0.734960i	$-0.262799\pi$
$953$	3.80244e7	1.35622	0.678110	+	0.734960i	$0.262799\pi$
$954$	8.54906e6	0.304122
$955$	1.06040e6	0.0376235
$956$	−2.01392e7	−0.712685
$957$	−8.97777e6	−0.316875
$958$	−1.83389e7	−0.645593
$959$	−6.40111e6	−0.224755
$960$	−774144.	−0.0271109
$961$	−2.74235e7	−0.957889
$962$	4.71994e6	0.164437
$963$	1.00999e7	0.350955
$964$	−2.29587e6	−0.0795710
$965$	4.85293e6	0.167759
$966$	102960.	0.00354998
$967$	2.60950e6	0.0897409	0.0448705	−	0.998993i	$-0.485712\pi$
$967$	2.60950e6	0.0897409	0.0448705	+	0.998993i	$0.485712\pi$
$968$	−7.61766e6	−0.261296
$969$	6.90413e6	0.236211
$970$	−7.06692e6	−0.241158
$971$	1.40618e7	0.478623	0.239312	−	0.970943i	$-0.423078\pi$
$971$	1.40618e7	0.478623	0.239312	+	0.970943i	$0.423078\pi$
$972$	−944784.	−0.0320750
$973$	6.03716e7	2.04433
$974$	221216.	0.00747170
$975$	−1.88417e6	−0.0634758
$976$	−587008.	−0.0197251
$977$	4.66346e7	1.56305	0.781523	−	0.623876i	$-0.214443\pi$
$977$	4.66346e7	1.56305	0.781523	+	0.623876i	$0.214443\pi$
$978$	1.07332e7	0.358824
$979$	−4.22669e6	−0.140943
$980$	1.22371e6	0.0407018
$981$	−958716.	−0.0318066
$982$	−2.75573e7	−0.911922
$983$	−2.96980e6	−0.0980263	−0.0490132	−	0.998798i	$-0.515608\pi$
$983$	−2.96980e6	−0.0980263	−0.0490132	+	0.998798i	$0.515608\pi$
$984$	5.41440e6	0.178264
$985$	−9.50212e6	−0.312054
$986$	4.13610e7	1.35487
$987$	1.81531e7	0.593142
$988$	−450528.	−0.0146835
$989$	401460.	0.0130512
$990$	−1.39482e6	−0.0452304
$991$	3.88690e7	1.25724	0.628621	−	0.777712i	$-0.283620\pi$
$991$	3.88690e7	1.25724	0.628621	+	0.777712i	$0.283620\pi$
$992$	−1.12435e6	−0.0362763
$993$	1.41356e7	0.454926
$994$	−5.82296e6	−0.186930
$995$	−4.35118e6	−0.139331
$996$	−1.08282e7	−0.345867
$997$	−4.92115e7	−1.56794	−0.783969	−	0.620801i	$-0.786808\pi$
$997$	−4.92115e7	−1.56794	−0.783969	+	0.620801i	$0.786808\pi$
$998$	2.24670e7	0.714034
$999$	1.10283e7	0.349619

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.6.a.c.1.1	✓	1	1.1	even	1	trivial
342.6.a.a.1.1		1	3.2	odd	2
912.6.a.f.1.1		1	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:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more