LMFDB - Embedded newform 1110.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,6,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, 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("1110.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1110.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:	$178.026039992$
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	$\chi$	$=$	1110.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -36.0000 q^{6} +33.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -36.0000 q^{6} +33.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +100.000 q^{10} +39.0000 q^{11} -144.000 q^{12} -314.000 q^{13} +132.000 q^{14} -225.000 q^{15} +256.000 q^{16} -1949.00 q^{17} +324.000 q^{18} +922.000 q^{19} +400.000 q^{20} -297.000 q^{21} +156.000 q^{22} +3844.00 q^{23} -576.000 q^{24} +625.000 q^{25} -1256.00 q^{26} -729.000 q^{27} +528.000 q^{28} -8313.00 q^{29} -900.000 q^{30} +5985.00 q^{31} +1024.00 q^{32} -351.000 q^{33} -7796.00 q^{34} +825.000 q^{35} +1296.00 q^{36} -1369.00 q^{37} +3688.00 q^{38} +2826.00 q^{39} +1600.00 q^{40} -13607.0 q^{41} -1188.00 q^{42} +847.000 q^{43} +624.000 q^{44} +2025.00 q^{45} +15376.0 q^{46} +2904.00 q^{47} -2304.00 q^{48} -15718.0 q^{49} +2500.00 q^{50} +17541.0 q^{51} -5024.00 q^{52} -33851.0 q^{53} -2916.00 q^{54} +975.000 q^{55} +2112.00 q^{56} -8298.00 q^{57} -33252.0 q^{58} -2186.00 q^{59} -3600.00 q^{60} +19893.0 q^{61} +23940.0 q^{62} +2673.00 q^{63} +4096.00 q^{64} -7850.00 q^{65} -1404.00 q^{66} -18596.0 q^{67} -31184.0 q^{68} -34596.0 q^{69} +3300.00 q^{70} -17740.0 q^{71} +5184.00 q^{72} -44536.0 q^{73} -5476.00 q^{74} -5625.00 q^{75} +14752.0 q^{76} +1287.00 q^{77} +11304.0 q^{78} +79732.0 q^{79} +6400.00 q^{80} +6561.00 q^{81} -54428.0 q^{82} +36254.0 q^{83} -4752.00 q^{84} -48725.0 q^{85} +3388.00 q^{86} +74817.0 q^{87} +2496.00 q^{88} -57970.0 q^{89} +8100.00 q^{90} -10362.0 q^{91} +61504.0 q^{92} -53865.0 q^{93} +11616.0 q^{94} +23050.0 q^{95} -9216.00 q^{96} +85531.0 q^{97} -62872.0 q^{98} +3159.00 q^{99} +O(q^{100})\)

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
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−36.0000	−0.408248
$7$	33.0000	0.254548	0.127274	−	0.991868i	$-0.459377\pi$
$7$	33.0000	0.254548	0.127274	+	0.991868i	$0.459377\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	100.000	0.316228
$11$	39.0000	0.0971813	0.0485907	−	0.998819i	$-0.484527\pi$
$11$	39.0000	0.0971813	0.0485907	+	0.998819i	$0.484527\pi$
$12$	−144.000	−0.288675
$13$	−314.000	−0.515313	−0.257657	−	0.966237i	$-0.582950\pi$
$13$	−314.000	−0.515313	−0.257657	+	0.966237i	$0.582950\pi$
$14$	132.000	0.179992
$15$	−225.000	−0.258199
$16$	256.000	0.250000
$17$	−1949.00	−1.63565	−0.817823	−	0.575469i	$-0.804819\pi$
$17$	−1949.00	−1.63565	−0.817823	+	0.575469i	$0.804819\pi$
$18$	324.000	0.235702
$19$	922.000	0.585932	0.292966	−	0.956123i	$-0.405358\pi$
$19$	922.000	0.585932	0.292966	+	0.956123i	$0.405358\pi$
$20$	400.000	0.223607
$21$	−297.000	−0.146963
$22$	156.000	0.0687176
$23$	3844.00	1.51518	0.757589	−	0.652732i	$-0.226377\pi$
$23$	3844.00	1.51518	0.757589	+	0.652732i	$0.226377\pi$
$24$	−576.000	−0.204124
$25$	625.000	0.200000
$26$	−1256.00	−0.364381
$27$	−729.000	−0.192450
$28$	528.000	0.127274
$29$	−8313.00	−1.83554	−0.917768	−	0.397118i	$-0.870011\pi$
$29$	−8313.00	−1.83554	−0.917768	+	0.397118i	$0.870011\pi$
$30$	−900.000	−0.182574
$31$	5985.00	1.11856	0.559281	−	0.828978i	$-0.311077\pi$
$31$	5985.00	1.11856	0.559281	+	0.828978i	$0.311077\pi$
$32$	1024.00	0.176777
$33$	−351.000	−0.0561077
$34$	−7796.00	−1.15658
$35$	825.000	0.113837
$36$	1296.00	0.166667
$37$	−1369.00	−0.164399
$37$	−1369.00	−0.164399
$38$	3688.00	0.414316
$39$	2826.00	0.297516
$40$	1600.00	0.158114
$41$	−13607.0	−1.26416	−0.632081	−	0.774902i	$-0.717799\pi$
$41$	−13607.0	−1.26416	−0.632081	+	0.774902i	$0.717799\pi$
$42$	−1188.00	−0.103919
$43$	847.000	0.0698574	0.0349287	−	0.999390i	$-0.488880\pi$
$43$	847.000	0.0698574	0.0349287	+	0.999390i	$0.488880\pi$
$44$	624.000	0.0485907
$45$	2025.00	0.149071
$46$	15376.0	1.07139
$47$	2904.00	0.191757	0.0958787	−	0.995393i	$-0.469434\pi$
$47$	2904.00	0.191757	0.0958787	+	0.995393i	$0.469434\pi$
$48$	−2304.00	−0.144338
$49$	−15718.0	−0.935206
$50$	2500.00	0.141421
$51$	17541.0	0.944341
$52$	−5024.00	−0.257657
$53$	−33851.0	−1.65532	−0.827660	−	0.561230i	$-0.810328\pi$
$53$	−33851.0	−1.65532	−0.827660	+	0.561230i	$0.810328\pi$
$54$	−2916.00	−0.136083
$55$	975.000	0.0434608
$56$	2112.00	0.0899961
$57$	−8298.00	−0.338288
$58$	−33252.0	−1.29792
$59$	−2186.00	−0.0817561	−0.0408780	−	0.999164i	$-0.513016\pi$
$59$	−2186.00	−0.0817561	−0.0408780	+	0.999164i	$0.513016\pi$
$60$	−3600.00	−0.129099
$61$	19893.0	0.684504	0.342252	−	0.939608i	$-0.388810\pi$
$61$	19893.0	0.684504	0.342252	+	0.939608i	$0.388810\pi$
$62$	23940.0	0.790943
$63$	2673.00	0.0848492
$64$	4096.00	0.125000
$65$	−7850.00	−0.230455
$66$	−1404.00	−0.0396741
$67$	−18596.0	−0.506096	−0.253048	−	0.967454i	$-0.581433\pi$
$67$	−18596.0	−0.506096	−0.253048	+	0.967454i	$0.581433\pi$
$68$	−31184.0	−0.817823
$69$	−34596.0	−0.874789
$70$	3300.00	0.0804950
$71$	−17740.0	−0.417645	−0.208823	−	0.977954i	$-0.566963\pi$
$71$	−17740.0	−0.417645	−0.208823	+	0.977954i	$0.566963\pi$
$72$	5184.00	0.117851
$73$	−44536.0	−0.978147	−0.489073	−	0.872243i	$-0.662665\pi$
$73$	−44536.0	−0.978147	−0.489073	+	0.872243i	$0.662665\pi$
$74$	−5476.00	−0.116248
$75$	−5625.00	−0.115470
$76$	14752.0	0.292966
$77$	1287.00	0.0247373
$78$	11304.0	0.210376
$79$	79732.0	1.43736	0.718679	−	0.695342i	$-0.244747\pi$
$79$	79732.0	1.43736	0.718679	+	0.695342i	$0.244747\pi$
$80$	6400.00	0.111803
$81$	6561.00	0.111111
$82$	−54428.0	−0.893898
$83$	36254.0	0.577645	0.288822	−	0.957383i	$-0.406736\pi$
$83$	36254.0	0.577645	0.288822	+	0.957383i	$0.406736\pi$
$84$	−4752.00	−0.0734815
$85$	−48725.0	−0.731483
$86$	3388.00	0.0493966
$87$	74817.0	1.05975
$88$	2496.00	0.0343588
$89$	−57970.0	−0.775762	−0.387881	−	0.921710i	$-0.626793\pi$
$89$	−57970.0	−0.775762	−0.387881	+	0.921710i	$0.626793\pi$
$90$	8100.00	0.105409
$91$	−10362.0	−0.131172
$92$	61504.0	0.757589
$93$	−53865.0	−0.645802
$94$	11616.0	0.135593
$95$	23050.0	0.262037
$96$	−9216.00	−0.102062
$97$	85531.0	0.922984	0.461492	−	0.887144i	$-0.347314\pi$
$97$	85531.0	0.922984	0.461492	+	0.887144i	$0.347314\pi$
$98$	−62872.0	−0.661290
$99$	3159.00	0.0323938
$100$	10000.0	0.100000
$101$	102124.	0.996149	0.498075	−	0.867134i	$-0.334041\pi$
$101$	102124.	0.996149	0.498075	+	0.867134i	$0.334041\pi$
$102$	70164.0	0.667750
$103$	107864.	1.00181	0.500903	−	0.865504i	$-0.333001\pi$
$103$	107864.	1.00181	0.500903	+	0.865504i	$0.333001\pi$
$104$	−20096.0	−0.182191
$105$	−7425.00	−0.0657239
$106$	−135404.	−1.17049
$107$	110946.	0.936811	0.468406	−	0.883513i	$-0.344829\pi$
$107$	110946.	0.936811	0.468406	+	0.883513i	$0.344829\pi$
$108$	−11664.0	−0.0962250
$109$	−50987.0	−0.411049	−0.205524	−	0.978652i	$-0.565890\pi$
$109$	−50987.0	−0.411049	−0.205524	+	0.978652i	$0.565890\pi$
$110$	3900.00	0.0307314
$111$	12321.0	0.0949158
$112$	8448.00	0.0636369
$113$	−15919.0	−0.117279	−0.0586394	−	0.998279i	$-0.518676\pi$
$113$	−15919.0	−0.117279	−0.0586394	+	0.998279i	$0.518676\pi$
$114$	−33192.0	−0.239206
$115$	96100.0	0.677608
$116$	−133008.	−0.917768
$117$	−25434.0	−0.171771
$118$	−8744.00	−0.0578103
$119$	−64317.0	−0.416350
$120$	−14400.0	−0.0912871
$121$	−159530.	−0.990556
$122$	79572.0	0.484017
$123$	122463.	0.729864
$124$	95760.0	0.559281
$125$	15625.0	0.0894427
$126$	10692.0	0.0599974
$127$	−217920.	−1.19891	−0.599457	−	0.800407i	$-0.704617\pi$
$127$	−217920.	−1.19891	−0.599457	+	0.800407i	$0.704617\pi$
$128$	16384.0	0.0883883
$129$	−7623.00	−0.0403322
$130$	−31400.0	−0.162956
$131$	−229662.	−1.16926	−0.584630	−	0.811300i	$-0.698760\pi$
$131$	−229662.	−1.16926	−0.584630	+	0.811300i	$0.698760\pi$
$132$	−5616.00	−0.0280538
$133$	30426.0	0.149147
$134$	−74384.0	−0.357864
$135$	−18225.0	−0.0860663
$136$	−124736.	−0.578288
$137$	112422.	0.511741	0.255870	−	0.966711i	$-0.417638\pi$
$137$	112422.	0.511741	0.255870	+	0.966711i	$0.417638\pi$
$138$	−138384.	−0.618569
$139$	151281.	0.664121	0.332061	−	0.943258i	$-0.392256\pi$
$139$	151281.	0.664121	0.332061	+	0.943258i	$0.392256\pi$
$140$	13200.0	0.0569186
$141$	−26136.0	−0.110711
$142$	−70960.0	−0.295320
$143$	−12246.0	−0.0500788
$144$	20736.0	0.0833333
$145$	−207825.	−0.820876
$146$	−178144.	−0.691654
$147$	141462.	0.539941
$148$	−21904.0	−0.0821995
$149$	−488610.	−1.80300	−0.901502	−	0.432775i	$-0.857534\pi$
$149$	−488610.	−1.80300	−0.901502	+	0.432775i	$0.857534\pi$
$150$	−22500.0	−0.0816497
$151$	−268862.	−0.959593	−0.479796	−	0.877380i	$-0.659290\pi$
$151$	−268862.	−0.959593	−0.479796	+	0.877380i	$0.659290\pi$
$152$	59008.0	0.207158
$153$	−157869.	−0.545216
$154$	5148.00	0.0174919
$155$	149625.	0.500236
$156$	45216.0	0.148758
$157$	−222005.	−0.718809	−0.359405	−	0.933182i	$-0.617020\pi$
$157$	−222005.	−0.718809	−0.359405	+	0.933182i	$0.617020\pi$
$158$	318928.	1.01637
$159$	304659.	0.955699
$160$	25600.0	0.0790569
$161$	126852.	0.385685
$162$	26244.0	0.0785674
$163$	24951.0	0.0735561	0.0367781	−	0.999323i	$-0.488291\pi$
$163$	24951.0	0.0735561	0.0367781	+	0.999323i	$0.488291\pi$
$164$	−217712.	−0.632081
$165$	−8775.00	−0.0250921
$166$	145016.	0.408456
$167$	−333728.	−0.925980	−0.462990	−	0.886364i	$-0.653223\pi$
$167$	−333728.	−0.925980	−0.462990	+	0.886364i	$0.653223\pi$
$168$	−19008.0	−0.0519593
$169$	−272697.	−0.734452
$170$	−194900.	−0.517237
$171$	74682.0	0.195311
$172$	13552.0	0.0349287
$173$	210781.	0.535447	0.267723	−	0.963496i	$-0.413729\pi$
$173$	210781.	0.535447	0.267723	+	0.963496i	$0.413729\pi$
$174$	299268.	0.749354
$175$	20625.0	0.0509095
$176$	9984.00	0.0242953
$177$	19674.0	0.0472019
$178$	−231880.	−0.548546
$179$	−90788.0	−0.211785	−0.105893	−	0.994378i	$-0.533770\pi$
$179$	−90788.0	−0.211785	−0.105893	+	0.994378i	$0.533770\pi$
$180$	32400.0	0.0745356
$181$	−238526.	−0.541177	−0.270588	−	0.962695i	$-0.587218\pi$
$181$	−238526.	−0.541177	−0.270588	+	0.962695i	$0.587218\pi$
$182$	−41448.0	−0.0927524
$183$	−179037.	−0.395198
$184$	246016.	0.535696
$185$	−34225.0	−0.0735215
$186$	−215460.	−0.456651
$187$	−76011.0	−0.158954
$188$	46464.0	0.0958787
$189$	−24057.0	−0.0489877
$190$	92200.0	0.185288
$191$	−684575.	−1.35781	−0.678903	−	0.734228i	$-0.737544\pi$
$191$	−684575.	−1.35781	−0.678903	+	0.734228i	$0.737544\pi$
$192$	−36864.0	−0.0721688
$193$	108706.	0.210068	0.105034	−	0.994469i	$-0.466505\pi$
$193$	108706.	0.210068	0.105034	+	0.994469i	$0.466505\pi$
$194$	342124.	0.652648
$195$	70650.0	0.133053
$196$	−251488.	−0.467603
$197$	75198.0	0.138051	0.0690257	−	0.997615i	$-0.478011\pi$
$197$	75198.0	0.138051	0.0690257	+	0.997615i	$0.478011\pi$
$198$	12636.0	0.0229059
$199$	−280268.	−0.501696	−0.250848	−	0.968026i	$-0.580709\pi$
$199$	−280268.	−0.501696	−0.250848	+	0.968026i	$0.580709\pi$
$200$	40000.0	0.0707107
$201$	167364.	0.292194
$202$	408496.	0.704384
$203$	−274329.	−0.467231
$204$	280656.	0.472171
$205$	−340175.	−0.565350
$206$	431456.	0.708384
$207$	311364.	0.505059
$208$	−80384.0	−0.128828
$209$	35958.0	0.0569416
$210$	−29700.0	−0.0464738
$211$	−25865.0	−0.0399951	−0.0199975	−	0.999800i	$-0.506366\pi$
$211$	−25865.0	−0.0399951	−0.0199975	+	0.999800i	$0.506366\pi$
$212$	−541616.	−0.827660
$213$	159660.	0.241128
$214$	443784.	0.662426
$215$	21175.0	0.0312412
$216$	−46656.0	−0.0680414
$217$	197505.	0.284727
$218$	−203948.	−0.290655
$219$	400824.	0.564733
$220$	15600.0	0.0217304
$221$	611986.	0.842870
$222$	49284.0	0.0671156
$223$	−853541.	−1.14938	−0.574688	−	0.818372i	$-0.694877\pi$
$223$	−853541.	−1.14938	−0.574688	+	0.818372i	$0.694877\pi$
$224$	33792.0	0.0449981
$225$	50625.0	0.0666667
$226$	−63676.0	−0.0829287
$227$	−1.06611e6	−1.37322	−0.686608	−	0.727028i	$-0.740901\pi$
$227$	−1.06611e6	−1.37322	−0.686608	+	0.727028i	$0.740901\pi$
$228$	−132768.	−0.169144
$229$	−262972.	−0.331376	−0.165688	−	0.986178i	$-0.552984\pi$
$229$	−262972.	−0.331376	−0.165688	+	0.986178i	$0.552984\pi$
$230$	384400.	0.479142
$231$	−11583.0	−0.0142821
$232$	−532032.	−0.648960
$233$	−134608.	−0.162436	−0.0812178	−	0.996696i	$-0.525881\pi$
$233$	−134608.	−0.162436	−0.0812178	+	0.996696i	$0.525881\pi$
$234$	−101736.	−0.121460
$235$	72600.0	0.0857565
$236$	−34976.0	−0.0408780
$237$	−717588.	−0.829859
$238$	−257268.	−0.294404
$239$	−1.39927e6	−1.58455	−0.792277	−	0.610161i	$-0.791105\pi$
$239$	−1.39927e6	−1.58455	−0.792277	+	0.610161i	$0.791105\pi$
$240$	−57600.0	−0.0645497
$241$	−933458.	−1.03527	−0.517633	−	0.855603i	$-0.673187\pi$
$241$	−933458.	−1.03527	−0.517633	+	0.855603i	$0.673187\pi$
$242$	−638120.	−0.700429
$243$	−59049.0	−0.0641500
$244$	318288.	0.342252
$245$	−392950.	−0.418237
$246$	489852.	0.516092
$247$	−289508.	−0.301938
$248$	383040.	0.395471
$249$	−326286.	−0.333503
$250$	62500.0	0.0632456
$251$	−123680.	−0.123913	−0.0619563	−	0.998079i	$-0.519734\pi$
$251$	−123680.	−0.123913	−0.0619563	+	0.998079i	$0.519734\pi$
$252$	42768.0	0.0424246
$253$	149916.	0.147247
$254$	−871680.	−0.847760
$255$	438525.	0.422322
$256$	65536.0	0.0625000
$257$	−1.48420e6	−1.40171	−0.700856	−	0.713302i	$-0.747199\pi$
$257$	−1.48420e6	−1.40171	−0.700856	+	0.713302i	$0.747199\pi$
$258$	−30492.0	−0.0285192
$259$	−45177.0	−0.0418474
$260$	−125600.	−0.115228
$261$	−673353.	−0.611845
$262$	−918648.	−0.826791
$263$	1.47930e6	1.31876	0.659380	−	0.751810i	$-0.270819\pi$
$263$	1.47930e6	1.31876	0.659380	+	0.751810i	$0.270819\pi$
$264$	−22464.0	−0.0198371
$265$	−846275.	−0.740281
$266$	121704.	0.105463
$267$	521730.	0.447886
$268$	−297536.	−0.253048
$269$	784136.	0.660710	0.330355	−	0.943857i	$-0.392832\pi$
$269$	784136.	0.660710	0.330355	+	0.943857i	$0.392832\pi$
$270$	−72900.0	−0.0608581
$271$	998872.	0.826203	0.413101	−	0.910685i	$-0.364446\pi$
$271$	998872.	0.826203	0.413101	+	0.910685i	$0.364446\pi$
$272$	−498944.	−0.408912
$273$	93258.0	0.0757320
$274$	449688.	0.361855
$275$	24375.0	0.0194363
$276$	−553536.	−0.437394
$277$	54376.0	0.0425802	0.0212901	−	0.999773i	$-0.493223\pi$
$277$	54376.0	0.0425802	0.0212901	+	0.999773i	$0.493223\pi$
$278$	605124.	0.469605
$279$	484785.	0.372854
$280$	52800.0	0.0402475
$281$	1.65057e6	1.24700	0.623502	−	0.781821i	$-0.285709\pi$
$281$	1.65057e6	1.24700	0.623502	+	0.781821i	$0.285709\pi$
$282$	−104544.	−0.0782846
$283$	234044.	0.173713	0.0868563	−	0.996221i	$-0.472318\pi$
$283$	234044.	0.173713	0.0868563	+	0.996221i	$0.472318\pi$
$284$	−283840.	−0.208823
$285$	−207450.	−0.151287
$286$	−48984.0	−0.0354111
$287$	−449031.	−0.321789
$288$	82944.0	0.0589256
$289$	2.37874e6	1.67534
$290$	−831300.	−0.580447
$291$	−769779.	−0.532885
$292$	−712576.	−0.489073
$293$	953967.	0.649179	0.324589	−	0.945855i	$-0.394774\pi$
$293$	953967.	0.649179	0.324589	+	0.945855i	$0.394774\pi$
$294$	565848.	0.381796
$295$	−54650.0	−0.0365624
$296$	−87616.0	−0.0581238
$297$	−28431.0	−0.0187026
$298$	−1.95444e6	−1.27492
$299$	−1.20702e6	−0.780791
$300$	−90000.0	−0.0577350
$301$	27951.0	0.0177820
$302$	−1.07545e6	−0.678535
$303$	−919116.	−0.575127
$304$	236032.	0.146483
$305$	497325.	0.306119
$306$	−631476.	−0.385526
$307$	255702.	0.154842	0.0774209	−	0.996998i	$-0.475331\pi$
$307$	255702.	0.154842	0.0774209	+	0.996998i	$0.475331\pi$
$308$	20592.0	0.0123686
$309$	−970776.	−0.578393
$310$	598500.	0.353720
$311$	−2.81516e6	−1.65045	−0.825225	−	0.564804i	$-0.808952\pi$
$311$	−2.81516e6	−1.65045	−0.825225	+	0.564804i	$0.808952\pi$
$312$	180864.	0.105188
$313$	−2.65446e6	−1.53150	−0.765748	−	0.643141i	$-0.777631\pi$
$313$	−2.65446e6	−1.53150	−0.765748	+	0.643141i	$0.777631\pi$
$314$	−888020.	−0.508275
$315$	66825.0	0.0379457
$316$	1.27571e6	0.718679
$317$	−401267.	−0.224277	−0.112139	−	0.993693i	$-0.535770\pi$
$317$	−401267.	−0.224277	−0.112139	+	0.993693i	$0.535770\pi$
$318$	1.21864e6	0.675781
$319$	−324207.	−0.178380
$320$	102400.	0.0559017
$321$	−998514.	−0.540868
$322$	507408.	0.272720
$323$	−1.79698e6	−0.958377
$324$	104976.	0.0555556
$325$	−196250.	−0.103063
$326$	99804.0	0.0520120
$327$	458883.	0.237319
$328$	−870848.	−0.446949
$329$	95832.0	0.0488113
$330$	−35100.0	−0.0177428
$331$	−3.11456e6	−1.56252	−0.781261	−	0.624204i	$-0.785423\pi$
$331$	−3.11456e6	−1.56252	−0.781261	+	0.624204i	$0.785423\pi$
$332$	580064.	0.288822
$333$	−110889.	−0.0547997
$334$	−1.33491e6	−0.654767
$335$	−464900.	−0.226333
$336$	−76032.0	−0.0367408
$337$	2.82311e6	1.35411	0.677053	−	0.735935i	$-0.263257\pi$
$337$	2.82311e6	1.35411	0.677053	+	0.735935i	$0.263257\pi$
$338$	−1.09079e6	−0.519336
$339$	143271.	0.0677110
$340$	−779600.	−0.365742
$341$	233415.	0.108703
$342$	298728.	0.138105
$343$	−1.07332e6	−0.492602
$344$	54208.0	0.0246983
$345$	−864900.	−0.391217
$346$	843124.	0.378618
$347$	−265980.	−0.118584	−0.0592919	−	0.998241i	$-0.518884\pi$
$347$	−265980.	−0.118584	−0.0592919	+	0.998241i	$0.518884\pi$
$348$	1.19707e6	0.529873
$349$	1.33488e6	0.586648	0.293324	−	0.956013i	$-0.405239\pi$
$349$	1.33488e6	0.586648	0.293324	+	0.956013i	$0.405239\pi$
$350$	82500.0	0.0359985
$351$	228906.	0.0991721
$352$	39936.0	0.0171794
$353$	−2.92493e6	−1.24934	−0.624668	−	0.780891i	$-0.714766\pi$
$353$	−2.92493e6	−1.24934	−0.624668	+	0.780891i	$0.714766\pi$
$354$	78696.0	0.0333768
$355$	−443500.	−0.186777
$356$	−927520.	−0.387881
$357$	578853.	0.240380
$358$	−363152.	−0.149755
$359$	461730.	0.189083	0.0945414	−	0.995521i	$-0.469862\pi$
$359$	461730.	0.189083	0.0945414	+	0.995521i	$0.469862\pi$
$360$	129600.	0.0527046
$361$	−1.62602e6	−0.656684
$362$	−954104.	−0.382670
$363$	1.43577e6	0.571898
$364$	−165792.	−0.0655858
$365$	−1.11340e6	−0.437441
$366$	−716148.	−0.279447
$367$	−224343.	−0.0869456	−0.0434728	−	0.999055i	$-0.513842\pi$
$367$	−224343.	−0.0869456	−0.0434728	+	0.999055i	$0.513842\pi$
$368$	984064.	0.378795
$369$	−1.10217e6	−0.421387
$370$	−136900.	−0.0519875
$371$	−1.11708e6	−0.421357
$372$	−861840.	−0.322901
$373$	−2.42261e6	−0.901596	−0.450798	−	0.892626i	$-0.648861\pi$
$373$	−2.42261e6	−0.901596	−0.450798	+	0.892626i	$0.648861\pi$
$374$	−304044.	−0.112398
$375$	−140625.	−0.0516398
$376$	185856.	0.0677965
$377$	2.61028e6	0.945876
$378$	−96228.0	−0.0346395
$379$	1.39673e6	0.499475	0.249738	−	0.968314i	$-0.419656\pi$
$379$	1.39673e6	0.499475	0.249738	+	0.968314i	$0.419656\pi$
$380$	368800.	0.131018
$381$	1.96128e6	0.692193
$382$	−2.73830e6	−0.960114
$383$	−3.82780e6	−1.33337	−0.666687	−	0.745338i	$-0.732288\pi$
$383$	−3.82780e6	−1.33337	−0.666687	+	0.745338i	$0.732288\pi$
$384$	−147456.	−0.0510310
$385$	32175.0	0.0110628
$386$	434824.	0.148541
$387$	68607.0	0.0232858
$388$	1.36850e6	0.461492
$389$	5.08828e6	1.70489	0.852447	−	0.522814i	$-0.175118\pi$
$389$	5.08828e6	1.70489	0.852447	+	0.522814i	$0.175118\pi$
$390$	282600.	0.0940829
$391$	−7.49196e6	−2.47830
$392$	−1.00595e6	−0.330645
$393$	2.06696e6	0.675072
$394$	300792.	0.0976171
$395$	1.99330e6	0.642806
$396$	50544.0	0.0161969
$397$	−2.26217e6	−0.720360	−0.360180	−	0.932883i	$-0.617285\pi$
$397$	−2.26217e6	−0.720360	−0.360180	+	0.932883i	$0.617285\pi$
$398$	−1.12107e6	−0.354753
$399$	−273834.	−0.0861103
$400$	160000.	0.0500000
$401$	1.18719e6	0.368688	0.184344	−	0.982862i	$-0.440984\pi$
$401$	1.18719e6	0.368688	0.184344	+	0.982862i	$0.440984\pi$
$402$	669456.	0.206613
$403$	−1.87929e6	−0.576410
$404$	1.63398e6	0.498075
$405$	164025.	0.0496904
$406$	−1.09732e6	−0.330382
$407$	−53391.0	−0.0159765
$408$	1.12262e6	0.333875
$409$	−466906.	−0.138013	−0.0690067	−	0.997616i	$-0.521983\pi$
$409$	−466906.	−0.138013	−0.0690067	+	0.997616i	$0.521983\pi$
$410$	−1.36070e6	−0.399763
$411$	−1.01180e6	−0.295454
$412$	1.72582e6	0.500903
$413$	−72138.0	−0.0208108
$414$	1.24546e6	0.357131
$415$	906350.	0.258331
$416$	−321536.	−0.0910954
$417$	−1.36153e6	−0.383431
$418$	143832.	0.0402638
$419$	−1.49434e6	−0.415829	−0.207914	−	0.978147i	$-0.566668\pi$
$419$	−1.49434e6	−0.415829	−0.207914	+	0.978147i	$0.566668\pi$
$420$	−118800.	−0.0328619
$421$	−3.25272e6	−0.894419	−0.447210	−	0.894429i	$-0.647582\pi$
$421$	−3.25272e6	−0.894419	−0.447210	+	0.894429i	$0.647582\pi$
$422$	−103460.	−0.0282808
$423$	235224.	0.0639191
$424$	−2.16646e6	−0.585244
$425$	−1.21812e6	−0.327129
$426$	638640.	0.170503
$427$	656469.	0.174239
$428$	1.77514e6	0.468406
$429$	110214.	0.0289130
$430$	84700.0	0.0220908
$431$	5.68679e6	1.47460	0.737300	−	0.675566i	$-0.236101\pi$
$431$	5.68679e6	1.47460	0.737300	+	0.675566i	$0.236101\pi$
$432$	−186624.	−0.0481125
$433$	6.82025e6	1.74816	0.874078	−	0.485785i	$-0.161466\pi$
$433$	6.82025e6	1.74816	0.874078	+	0.485785i	$0.161466\pi$
$434$	790020.	0.201332
$435$	1.87042e6	0.473933
$436$	−815792.	−0.205524
$437$	3.54417e6	0.887791
$438$	1.60330e6	0.399327
$439$	−2.25054e6	−0.557346	−0.278673	−	0.960386i	$-0.589895\pi$
$439$	−2.25054e6	−0.557346	−0.278673	+	0.960386i	$0.589895\pi$
$440$	62400.0	0.0153657
$441$	−1.27316e6	−0.311735
$442$	2.44794e6	0.595999
$443$	4.16289e6	1.00783	0.503914	−	0.863754i	$-0.331893\pi$
$443$	4.16289e6	1.00783	0.503914	+	0.863754i	$0.331893\pi$
$444$	197136.	0.0474579
$445$	−1.44925e6	−0.346931
$446$	−3.41416e6	−0.812732
$447$	4.39749e6	1.04096
$448$	135168.	0.0318184
$449$	3.84574e6	0.900252	0.450126	−	0.892965i	$-0.351379\pi$
$449$	3.84574e6	0.900252	0.450126	+	0.892965i	$0.351379\pi$
$450$	202500.	0.0471405
$451$	−530673.	−0.122853
$452$	−254704.	−0.0586394
$453$	2.41976e6	0.554021
$454$	−4.26445e6	−0.971010
$455$	−259050.	−0.0586618
$456$	−531072.	−0.119603
$457$	−1.00536e6	−0.225182	−0.112591	−	0.993641i	$-0.535915\pi$
$457$	−1.00536e6	−0.225182	−0.112591	+	0.993641i	$0.535915\pi$
$458$	−1.05189e6	−0.234318
$459$	1.42082e6	0.314780
$460$	1.53760e6	0.338804
$461$	1.70673e6	0.374034	0.187017	−	0.982357i	$-0.440118\pi$
$461$	1.70673e6	0.374034	0.187017	+	0.982357i	$0.440118\pi$
$462$	−46332.0	−0.0100989
$463$	5.72391e6	1.24091	0.620455	−	0.784242i	$-0.286948\pi$
$463$	5.72391e6	1.24091	0.620455	+	0.784242i	$0.286948\pi$
$464$	−2.12813e6	−0.458884
$465$	−1.34662e6	−0.288811
$466$	−538432.	−0.114859
$467$	−669435.	−0.142042	−0.0710209	−	0.997475i	$-0.522626\pi$
$467$	−669435.	−0.142042	−0.0710209	+	0.997475i	$0.522626\pi$
$468$	−406944.	−0.0858855
$469$	−613668.	−0.128825
$470$	290400.	0.0606390
$471$	1.99804e6	0.415005
$472$	−139904.	−0.0289051
$473$	33033.0	0.00678883
$474$	−2.87035e6	−0.586799
$475$	576250.	0.117186
$476$	−1.02907e6	−0.208175
$477$	−2.74193e6	−0.551773
$478$	−5.59708e6	−1.12045
$479$	−3.52900e6	−0.702769	−0.351385	−	0.936231i	$-0.614289\pi$
$479$	−3.52900e6	−0.702769	−0.351385	+	0.936231i	$0.614289\pi$
$480$	−230400.	−0.0456435
$481$	429866.	0.0847170
$482$	−3.73383e6	−0.732044
$483$	−1.14167e6	−0.222675
$484$	−2.55248e6	−0.495278
$485$	2.13828e6	0.412771
$486$	−236196.	−0.0453609
$487$	577114.	0.110265	0.0551327	−	0.998479i	$-0.482442\pi$
$487$	577114.	0.110265	0.0551327	+	0.998479i	$0.482442\pi$
$488$	1.27315e6	0.242009
$489$	−224559.	−0.0424677
$490$	−1.57180e6	−0.295738
$491$	318012.	0.0595305	0.0297653	−	0.999557i	$-0.490524\pi$
$491$	318012.	0.0595305	0.0297653	+	0.999557i	$0.490524\pi$
$492$	1.95941e6	0.364932
$493$	1.62020e7	3.00229
$494$	−1.15803e6	−0.213503
$495$	78975.0	0.0144869
$496$	1.53216e6	0.279640
$497$	−585420.	−0.106311
$498$	−1.30514e6	−0.235822
$499$	6.13994e6	1.10386	0.551928	−	0.833892i	$-0.313892\pi$
$499$	6.13994e6	1.10386	0.551928	+	0.833892i	$0.313892\pi$
$500$	250000.	0.0447214
$501$	3.00355e6	0.534615
$502$	−494720.	−0.0876194
$503$	3.15074e6	0.555255	0.277628	−	0.960689i	$-0.410452\pi$
$503$	3.15074e6	0.555255	0.277628	+	0.960689i	$0.410452\pi$
$504$	171072.	0.0299987
$505$	2.55310e6	0.445491
$506$	599664.	0.104119
$507$	2.45427e6	0.424036
$508$	−3.48672e6	−0.599457
$509$	−4.25165e6	−0.727382	−0.363691	−	0.931520i	$-0.618484\pi$
$509$	−4.25165e6	−0.727382	−0.363691	+	0.931520i	$0.618484\pi$
$510$	1.75410e6	0.298627
$511$	−1.46969e6	−0.248985
$512$	262144.	0.0441942
$513$	−672138.	−0.112763
$514$	−5.93679e6	−0.991161
$515$	2.69660e6	0.448021
$516$	−121968.	−0.0201661
$517$	113256.	0.0186352
$518$	−180708.	−0.0295905
$519$	−1.89703e6	−0.309140
$520$	−502400.	−0.0814782
$521$	−601477.	−0.0970789	−0.0485394	−	0.998821i	$-0.515457\pi$
$521$	−601477.	−0.0970789	−0.0485394	+	0.998821i	$0.515457\pi$
$522$	−2.69341e6	−0.432640
$523$	8.90864e6	1.42415	0.712077	−	0.702101i	$-0.247755\pi$
$523$	8.90864e6	1.42415	0.712077	+	0.702101i	$0.247755\pi$
$524$	−3.67459e6	−0.584630
$525$	−185625.	−0.0293926
$526$	5.91719e6	0.932505
$527$	−1.16648e7	−1.82957
$528$	−89856.0	−0.0140269
$529$	8.33999e6	1.29577
$530$	−3.38510e6	−0.523458
$531$	−177066.	−0.0272520
$532$	486816.	0.0745737
$533$	4.27260e6	0.651439
$534$	2.08692e6	0.316703
$535$	2.77365e6	0.418955
$536$	−1.19014e6	−0.178932
$537$	817092.	0.122274
$538$	3.13654e6	0.467192
$539$	−613002.	−0.0908845
$540$	−291600.	−0.0430331
$541$	−1.13061e7	−1.66081	−0.830406	−	0.557158i	$-0.811892\pi$
$541$	−1.13061e7	−1.66081	−0.830406	+	0.557158i	$0.811892\pi$
$542$	3.99549e6	0.584214
$543$	2.14673e6	0.312449
$544$	−1.99578e6	−0.289144
$545$	−1.27468e6	−0.183827
$546$	373032.	0.0535506
$547$	−697065.	−0.0996105	−0.0498052	−	0.998759i	$-0.515860\pi$
$547$	−697065.	−0.0996105	−0.0498052	+	0.998759i	$0.515860\pi$
$548$	1.79875e6	0.255870
$549$	1.61133e6	0.228168
$550$	97500.0	0.0137435
$551$	−7.66459e6	−1.07550
$552$	−2.21414e6	−0.309285
$553$	2.63116e6	0.365876
$554$	217504.	0.0301088
$555$	308025.	0.0424476
$556$	2.42050e6	0.332061
$557$	−3.60869e6	−0.492846	−0.246423	−	0.969162i	$-0.579255\pi$
$557$	−3.60869e6	−0.492846	−0.246423	+	0.969162i	$0.579255\pi$
$558$	1.93914e6	0.263648
$559$	−265958.	−0.0359984
$560$	211200.	0.0284593
$561$	684099.	0.0917723
$562$	6.60228e6	0.881766
$563$	2.19924e6	0.292417	0.146208	−	0.989254i	$-0.453293\pi$
$563$	2.19924e6	0.292417	0.146208	+	0.989254i	$0.453293\pi$
$564$	−418176.	−0.0553556
$565$	−397975.	−0.0524487
$566$	936176.	0.122833
$567$	216513.	0.0282831
$568$	−1.13536e6	−0.147660
$569$	−5.91609e6	−0.766045	−0.383022	−	0.923739i	$-0.625117\pi$
$569$	−5.91609e6	−0.766045	−0.383022	+	0.923739i	$0.625117\pi$
$570$	−829800.	−0.106976
$571$	1.20739e7	1.54974	0.774868	−	0.632123i	$-0.217816\pi$
$571$	1.20739e7	1.54974	0.774868	+	0.632123i	$0.217816\pi$
$572$	−195936.	−0.0250394
$573$	6.16118e6	0.783929
$574$	−1.79612e6	−0.227539
$575$	2.40250e6	0.303036
$576$	331776.	0.0416667
$577$	−1.08768e7	−1.36007	−0.680037	−	0.733178i	$-0.738036\pi$
$577$	−1.08768e7	−1.36007	−0.680037	+	0.733178i	$0.738036\pi$
$578$	9.51498e6	1.18464
$579$	−978354.	−0.121283
$580$	−3.32520e6	−0.410438
$581$	1.19638e6	0.147038
$582$	−3.07912e6	−0.376807
$583$	−1.32019e6	−0.160866
$584$	−2.85030e6	−0.345827
$585$	−635850.	−0.0768184
$586$	3.81587e6	0.459039
$587$	−2.08187e6	−0.249378	−0.124689	−	0.992196i	$-0.539793\pi$
$587$	−2.08187e6	−0.249378	−0.124689	+	0.992196i	$0.539793\pi$
$588$	2.26339e6	0.269971
$589$	5.51817e6	0.655401
$590$	−218600.	−0.0258535
$591$	−676782.	−0.0797040
$592$	−350464.	−0.0410997
$593$	−1.16666e7	−1.36240	−0.681202	−	0.732095i	$-0.738543\pi$
$593$	−1.16666e7	−1.36240	−0.681202	+	0.732095i	$0.738543\pi$
$594$	−113724.	−0.0132247
$595$	−1.60792e6	−0.186197
$596$	−7.81776e6	−0.901502
$597$	2.52241e6	0.289654
$598$	−4.82806e6	−0.552103
$599$	1.29817e6	0.147830	0.0739152	−	0.997265i	$-0.476451\pi$
$599$	1.29817e6	0.147830	0.0739152	+	0.997265i	$0.476451\pi$
$600$	−360000.	−0.0408248
$601$	9.83205e6	1.11034	0.555172	−	0.831735i	$-0.312652\pi$
$601$	9.83205e6	1.11034	0.555172	+	0.831735i	$0.312652\pi$
$602$	111804.	0.0125738
$603$	−1.50628e6	−0.168699
$604$	−4.30179e6	−0.479796
$605$	−3.98825e6	−0.442990
$606$	−3.67646e6	−0.406676
$607$	−5.97191e6	−0.657872	−0.328936	−	0.944352i	$-0.606690\pi$
$607$	−5.97191e6	−0.657872	−0.328936	+	0.944352i	$0.606690\pi$
$608$	944128.	0.103579
$609$	2.46896e6	0.269756
$610$	1.98930e6	0.216459
$611$	−911856.	−0.0988151
$612$	−2.52590e6	−0.272608
$613$	8.52197e6	0.915986	0.457993	−	0.888956i	$-0.348568\pi$
$613$	8.52197e6	0.915986	0.457993	+	0.888956i	$0.348568\pi$
$614$	1.02281e6	0.109490
$615$	3.06158e6	0.326405
$616$	82368.0	0.00874594
$617$	6.24075e6	0.659970	0.329985	−	0.943986i	$-0.392956\pi$
$617$	6.24075e6	0.659970	0.329985	+	0.943986i	$0.392956\pi$
$618$	−3.88310e6	−0.408985
$619$	1.66424e7	1.74578	0.872891	−	0.487916i	$-0.162243\pi$
$619$	1.66424e7	1.74578	0.872891	+	0.487916i	$0.162243\pi$
$620$	2.39400e6	0.250118
$621$	−2.80228e6	−0.291596
$622$	−1.12607e7	−1.16704
$623$	−1.91301e6	−0.197468
$624$	723456.	0.0743791
$625$	390625.	0.0400000
$626$	−1.06178e7	−1.08293
$627$	−323622.	−0.0328753
$628$	−3.55208e6	−0.359405
$629$	2.66818e6	0.268899
$630$	267300.	0.0268317
$631$	−4.06681e6	−0.406612	−0.203306	−	0.979115i	$-0.565169\pi$
$631$	−4.06681e6	−0.406612	−0.203306	+	0.979115i	$0.565169\pi$
$632$	5.10285e6	0.508183
$633$	232785.	0.0230912
$634$	−1.60507e6	−0.158588
$635$	−5.44800e6	−0.536170
$636$	4.87454e6	0.477850
$637$	4.93545e6	0.481924
$638$	−1.29683e6	−0.126134
$639$	−1.43694e6	−0.139215
$640$	409600.	0.0395285
$641$	1.07901e7	1.03724	0.518619	−	0.855005i	$-0.326446\pi$
$641$	1.07901e7	1.03724	0.518619	+	0.855005i	$0.326446\pi$
$642$	−3.99406e6	−0.382452
$643$	−5.25226e6	−0.500978	−0.250489	−	0.968119i	$-0.580591\pi$
$643$	−5.25226e6	−0.500978	−0.250489	+	0.968119i	$0.580591\pi$
$644$	2.02963e6	0.192842
$645$	−190575.	−0.0180371
$646$	−7.18791e6	−0.677675
$647$	1.04141e7	0.978049	0.489025	−	0.872270i	$-0.337353\pi$
$647$	1.04141e7	0.978049	0.489025	+	0.872270i	$0.337353\pi$
$648$	419904.	0.0392837
$649$	−85254.0	−0.00794517
$650$	−785000.	−0.0728763
$651$	−1.77754e6	−0.164387
$652$	399216.	0.0367781
$653$	848008.	0.0778246	0.0389123	−	0.999243i	$-0.487611\pi$
$653$	848008.	0.0778246	0.0389123	+	0.999243i	$0.487611\pi$
$654$	1.83553e6	0.167810
$655$	−5.74155e6	−0.522909
$656$	−3.48339e6	−0.316041
$657$	−3.60742e6	−0.326049
$658$	383328.	0.0345148
$659$	−2.09114e6	−0.187573	−0.0937865	−	0.995592i	$-0.529897\pi$
$659$	−2.09114e6	−0.187573	−0.0937865	+	0.995592i	$0.529897\pi$
$660$	−140400.	−0.0125461
$661$	−5.74771e6	−0.511671	−0.255836	−	0.966720i	$-0.582351\pi$
$661$	−5.74771e6	−0.511671	−0.255836	+	0.966720i	$0.582351\pi$
$662$	−1.24582e7	−1.10487
$663$	−5.50787e6	−0.486631
$664$	2.32026e6	0.204228
$665$	760650.	0.0667008
$666$	−443556.	−0.0387492
$667$	−3.19552e7	−2.78116
$668$	−5.33965e6	−0.462990
$669$	7.68187e6	0.663593
$670$	−1.85960e6	−0.160041
$671$	775827.	0.0665210
$672$	−304128.	−0.0259796
$673$	−1.11962e7	−0.952866	−0.476433	−	0.879211i	$-0.658070\pi$
$673$	−1.11962e7	−0.952866	−0.476433	+	0.879211i	$0.658070\pi$
$674$	1.12924e7	0.957497
$675$	−455625.	−0.0384900
$676$	−4.36315e6	−0.367226
$677$	1.39483e7	1.16964	0.584818	−	0.811165i	$-0.301166\pi$
$677$	1.39483e7	1.16964	0.584818	+	0.811165i	$0.301166\pi$
$678$	573084.	0.0478789
$679$	2.82252e6	0.234943
$680$	−3.11840e6	−0.258618
$681$	9.59502e6	0.792826
$682$	933660.	0.0768649
$683$	1.28224e7	1.05176	0.525879	−	0.850559i	$-0.323736\pi$
$683$	1.28224e7	1.05176	0.525879	+	0.850559i	$0.323736\pi$
$684$	1.19491e6	0.0976553
$685$	2.81055e6	0.228857
$686$	−4.29330e6	−0.348322
$687$	2.36675e6	0.191320
$688$	216832.	0.0174643
$689$	1.06292e7	0.853008
$690$	−3.45960e6	−0.276632
$691$	−1.33622e7	−1.06459	−0.532295	−	0.846559i	$-0.678670\pi$
$691$	−1.33622e7	−1.06459	−0.532295	+	0.846559i	$0.678670\pi$
$692$	3.37250e6	0.267723
$693$	104247.	0.00824576
$694$	−1.06392e6	−0.0838514
$695$	3.78202e6	0.297004
$696$	4.78829e6	0.374677
$697$	2.65200e7	2.06772
$698$	5.33950e6	0.414823
$699$	1.21147e6	0.0937822
$700$	330000.	0.0254548
$701$	−968218.	−0.0744180	−0.0372090	−	0.999308i	$-0.511847\pi$
$701$	−968218.	−0.0744180	−0.0372090	+	0.999308i	$0.511847\pi$
$702$	915624.	0.0701252
$703$	−1.26222e6	−0.0963266
$704$	159744.	0.0121477
$705$	−653400.	−0.0495115
$706$	−1.16997e7	−0.883414
$707$	3.37009e6	0.253567
$708$	314784.	0.0236010
$709$	−1.81516e7	−1.35613	−0.678063	−	0.735003i	$-0.737181\pi$
$709$	−1.81516e7	−1.35613	−0.678063	+	0.735003i	$0.737181\pi$
$710$	−1.77400e6	−0.132071
$711$	6.45829e6	0.479119
$712$	−3.71008e6	−0.274273
$713$	2.30063e7	1.69482
$714$	2.31541e6	0.169974
$715$	−306150.	−0.0223959
$716$	−1.45261e6	−0.105893
$717$	1.25934e7	0.914843
$718$	1.84692e6	0.133702
$719$	6.74731e6	0.486753	0.243376	−	0.969932i	$-0.421745\pi$
$719$	6.74731e6	0.486753	0.243376	+	0.969932i	$0.421745\pi$
$720$	518400.	0.0372678
$721$	3.55951e6	0.255007
$722$	−6.50406e6	−0.464346
$723$	8.40112e6	0.597711
$724$	−3.81642e6	−0.270588
$725$	−5.19562e6	−0.367107
$726$	5.74308e6	0.404393
$727$	−8.47793e6	−0.594914	−0.297457	−	0.954735i	$-0.596138\pi$
$727$	−8.47793e6	−0.594914	−0.297457	+	0.954735i	$0.596138\pi$
$728$	−663168.	−0.0463762
$729$	531441.	0.0370370
$730$	−4.45360e6	−0.309317
$731$	−1.65080e6	−0.114262
$732$	−2.86459e6	−0.197599
$733$	−5.48169e6	−0.376838	−0.188419	−	0.982089i	$-0.560336\pi$
$733$	−5.48169e6	−0.376838	−0.188419	+	0.982089i	$0.560336\pi$
$734$	−897372.	−0.0614798
$735$	3.53655e6	0.241469
$736$	3.93626e6	0.267848
$737$	−725244.	−0.0491830
$738$	−4.40867e6	−0.297966
$739$	−6.20108e6	−0.417692	−0.208846	−	0.977949i	$-0.566971\pi$
$739$	−6.20108e6	−0.417692	−0.208846	+	0.977949i	$0.566971\pi$
$740$	−547600.	−0.0367607
$741$	2.60557e6	0.174324
$742$	−4.46833e6	−0.297945
$743$	2.80639e6	0.186499	0.0932495	−	0.995643i	$-0.470275\pi$
$743$	2.80639e6	0.186499	0.0932495	+	0.995643i	$0.470275\pi$
$744$	−3.44736e6	−0.228325
$745$	−1.22152e7	−0.806328
$746$	−9.69046e6	−0.637525
$747$	2.93657e6	0.192548
$748$	−1.21618e6	−0.0794772
$749$	3.66122e6	0.238463
$750$	−562500.	−0.0365148
$751$	1.53333e7	0.992056	0.496028	−	0.868307i	$-0.334791\pi$
$751$	1.53333e7	0.992056	0.496028	+	0.868307i	$0.334791\pi$
$752$	743424.	0.0479393
$753$	1.11312e6	0.0715409
$754$	1.04411e7	0.668835
$755$	−6.72155e6	−0.429143
$756$	−384912.	−0.0244938
$757$	−7.76521e6	−0.492508	−0.246254	−	0.969205i	$-0.579200\pi$
$757$	−7.76521e6	−0.492508	−0.246254	+	0.969205i	$0.579200\pi$
$758$	5.58691e6	0.353182
$759$	−1.34924e6	−0.0850131
$760$	1.47520e6	0.0926439
$761$	−2.21402e7	−1.38586	−0.692931	−	0.721004i	$-0.743681\pi$
$761$	−2.21402e7	−1.38586	−0.692931	+	0.721004i	$0.743681\pi$
$762$	7.84512e6	0.489454
$763$	−1.68257e6	−0.104631
$764$	−1.09532e7	−0.678903
$765$	−3.94672e6	−0.243828
$766$	−1.53112e7	−0.942838
$767$	686404.	0.0421300
$768$	−589824.	−0.0360844
$769$	−7.73520e6	−0.471689	−0.235845	−	0.971791i	$-0.575786\pi$
$769$	−7.73520e6	−0.471689	−0.235845	+	0.971791i	$0.575786\pi$
$770$	128700.	0.00782261
$771$	1.33578e7	0.809279
$772$	1.73930e6	0.105034
$773$	−4.80564e6	−0.289269	−0.144635	−	0.989485i	$-0.546201\pi$
$773$	−4.80564e6	−0.289269	−0.144635	+	0.989485i	$0.546201\pi$
$774$	274428.	0.0164655
$775$	3.74062e6	0.223712
$776$	5.47398e6	0.326324
$777$	406593.	0.0241606
$778$	2.03531e7	1.20554
$779$	−1.25457e7	−0.740712
$780$	1.13040e6	0.0665266
$781$	−691860.	−0.0405873
$782$	−2.99678e7	−1.75242
$783$	6.06018e6	0.353249
$784$	−4.02381e6	−0.233801
$785$	−5.55012e6	−0.321461
$786$	8.26783e6	0.477348
$787$	6.16670e6	0.354908	0.177454	−	0.984129i	$-0.443214\pi$
$787$	6.16670e6	0.354908	0.177454	+	0.984129i	$0.443214\pi$
$788$	1.20317e6	0.0690257
$789$	−1.33137e7	−0.761387
$790$	7.97320e6	0.454532
$791$	−525327.	−0.0298530
$792$	202176.	0.0114529
$793$	−6.24640e6	−0.352734
$794$	−9.04870e6	−0.509372
$795$	7.61648e6	0.427402
$796$	−4.48429e6	−0.250848
$797$	−1.64535e7	−0.917516	−0.458758	−	0.888561i	$-0.651705\pi$
$797$	−1.64535e7	−0.917516	−0.458758	+	0.888561i	$0.651705\pi$
$798$	−1.09534e6	−0.0608892
$799$	−5.65990e6	−0.313647
$800$	640000.	0.0353553
$801$	−4.69557e6	−0.258587
$802$	4.74876e6	0.260702
$803$	−1.73690e6	−0.0950576
$804$	2.67782e6	0.146097
$805$	3.17130e6	0.172484
$806$	−7.51716e6	−0.407583
$807$	−7.05722e6	−0.381461
$808$	6.53594e6	0.352192
$809$	−1.92853e7	−1.03599	−0.517994	−	0.855384i	$-0.673321\pi$
$809$	−1.92853e7	−1.03599	−0.517994	+	0.855384i	$0.673321\pi$
$810$	656100.	0.0351364
$811$	2.03924e7	1.08872	0.544361	−	0.838851i	$-0.316772\pi$
$811$	2.03924e7	1.08872	0.544361	+	0.838851i	$0.316772\pi$
$812$	−4.38926e6	−0.233615
$813$	−8.98985e6	−0.477008
$814$	−213564.	−0.0112971
$815$	623775.	0.0328953
$816$	4.49050e6	0.236085
$817$	780934.	0.0409316
$818$	−1.86762e6	−0.0975902
$819$	−839322.	−0.0437239
$820$	−5.44280e6	−0.282675
$821$	−1.77357e7	−0.918312	−0.459156	−	0.888356i	$-0.651848\pi$
$821$	−1.77357e7	−0.918312	−0.459156	+	0.888356i	$0.651848\pi$
$822$	−4.04719e6	−0.208917
$823$	2.12567e7	1.09395	0.546974	−	0.837150i	$-0.315780\pi$
$823$	2.12567e7	1.09395	0.546974	+	0.837150i	$0.315780\pi$
$824$	6.90330e6	0.354192
$825$	−219375.	−0.0112215
$826$	−288552.	−0.0147155
$827$	−1.80840e7	−0.919457	−0.459728	−	0.888060i	$-0.652053\pi$
$827$	−1.80840e7	−0.919457	−0.459728	+	0.888060i	$0.652053\pi$
$828$	4.98182e6	0.252530
$829$	−7.22698e6	−0.365233	−0.182617	−	0.983184i	$-0.558457\pi$
$829$	−7.22698e6	−0.365233	−0.182617	+	0.983184i	$0.558457\pi$
$830$	3.62540e6	0.182667
$831$	−489384.	−0.0245837
$832$	−1.28614e6	−0.0644142
$833$	3.06344e7	1.52967
$834$	−5.44612e6	−0.271126
$835$	−8.34320e6	−0.414111
$836$	575328.	0.0284708
$837$	−4.36306e6	−0.215267
$838$	−5.97736e6	−0.294035
$839$	2.07234e7	1.01638	0.508191	−	0.861245i	$-0.330314\pi$
$839$	2.07234e7	1.01638	0.508191	+	0.861245i	$0.330314\pi$
$840$	−475200.	−0.0232369
$841$	4.85948e7	2.36919
$842$	−1.30109e7	−0.632450
$843$	−1.48551e7	−0.719959
$844$	−413840.	−0.0199975
$845$	−6.81742e6	−0.328457
$846$	940896.	0.0451976
$847$	−5.26449e6	−0.252144
$848$	−8.66586e6	−0.413830
$849$	−2.10640e6	−0.100293
$850$	−4.87250e6	−0.231315
$851$	−5.26244e6	−0.249094
$852$	2.55456e6	0.120564
$853$	3.18129e7	1.49703	0.748516	−	0.663117i	$-0.230767\pi$
$853$	3.18129e7	1.49703	0.748516	+	0.663117i	$0.230767\pi$
$854$	2.62588e6	0.123205
$855$	1.86705e6	0.0873455
$856$	7.10054e6	0.331213
$857$	2.10862e7	0.980723	0.490362	−	0.871519i	$-0.336865\pi$
$857$	2.10862e7	0.980723	0.490362	+	0.871519i	$0.336865\pi$
$858$	440856.	0.0204446
$859$	1.17976e7	0.545521	0.272760	−	0.962082i	$-0.412063\pi$
$859$	1.17976e7	0.545521	0.272760	+	0.962082i	$0.412063\pi$
$860$	338800.	0.0156206
$861$	4.04128e6	0.185785
$862$	2.27472e7	1.04270
$863$	2.96415e7	1.35479	0.677397	−	0.735617i	$-0.263108\pi$
$863$	2.96415e7	1.35479	0.677397	+	0.735617i	$0.263108\pi$
$864$	−746496.	−0.0340207
$865$	5.26952e6	0.239459
$866$	2.72810e7	1.23613
$867$	−2.14087e7	−0.967258
$868$	3.16008e6	0.142364
$869$	3.10955e6	0.139684
$870$	7.48170e6	0.335121
$871$	5.83914e6	0.260798
$872$	−3.26317e6	−0.145328
$873$	6.92801e6	0.307661
$874$	1.41767e7	0.627763
$875$	515625.	0.0227674
$876$	6.41318e6	0.282367
$877$	2.78687e7	1.22354	0.611769	−	0.791036i	$-0.290458\pi$
$877$	2.78687e7	1.22354	0.611769	+	0.791036i	$0.290458\pi$
$878$	−9.00215e6	−0.394103
$879$	−8.58570e6	−0.374804
$880$	249600.	0.0108652
$881$	6.85755e6	0.297666	0.148833	−	0.988862i	$-0.452448\pi$
$881$	6.85755e6	0.297666	0.148833	+	0.988862i	$0.452448\pi$
$882$	−5.09263e6	−0.220430
$883$	3.20274e7	1.38235	0.691177	−	0.722685i	$-0.257092\pi$
$883$	3.20274e7	1.38235	0.691177	+	0.722685i	$0.257092\pi$
$884$	9.79178e6	0.421435
$885$	491850.	0.0211093
$886$	1.66516e7	0.712641
$887$	1.72295e7	0.735297	0.367649	−	0.929965i	$-0.380163\pi$
$887$	1.72295e7	0.735297	0.367649	+	0.929965i	$0.380163\pi$
$888$	788544.	0.0335578
$889$	−7.19136e6	−0.305180
$890$	−5.79700e6	−0.245317
$891$	255879.	0.0107979
$892$	−1.36567e7	−0.574688
$893$	2.67749e6	0.112357
$894$	1.75900e7	0.736073
$895$	−2.26970e6	−0.0947133
$896$	540672.	0.0224990
$897$	1.08631e7	0.450790
$898$	1.53830e7	0.636574
$899$	−4.97533e7	−2.05316
$900$	810000.	0.0333333
$901$	6.59756e7	2.70752
$902$	−2.12269e6	−0.0868702
$903$	−251559.	−0.0102665
$904$	−1.01882e6	−0.0414643
$905$	−5.96315e6	−0.242022
$906$	9.67903e6	0.391752
$907$	5.48652e6	0.221452	0.110726	−	0.993851i	$-0.464682\pi$
$907$	5.48652e6	0.221452	0.110726	+	0.993851i	$0.464682\pi$
$908$	−1.70578e7	−0.686608
$909$	8.27204e6	0.332050
$910$	−1.03620e6	−0.0414801
$911$	4.96866e6	0.198355	0.0991776	−	0.995070i	$-0.468379\pi$
$911$	4.96866e6	0.198355	0.0991776	+	0.995070i	$0.468379\pi$
$912$	−2.12429e6	−0.0845719
$913$	1.41391e6	0.0561363
$914$	−4.02146e6	−0.159228
$915$	−4.47592e6	−0.176738
$916$	−4.20755e6	−0.165688
$917$	−7.57885e6	−0.297632
$918$	5.68328e6	0.222583
$919$	3.54149e7	1.38324	0.691619	−	0.722262i	$-0.256898\pi$
$919$	3.54149e7	1.38324	0.691619	+	0.722262i	$0.256898\pi$
$920$	6.15040e6	0.239571
$921$	−2.30132e6	−0.0893979
$922$	6.82690e6	0.264482
$923$	5.57036e6	0.215218
$924$	−185328.	−0.00714103
$925$	−855625.	−0.0328798
$926$	2.28957e7	0.877457
$927$	8.73698e6	0.333935
$928$	−8.51251e6	−0.324480
$929$	−2.56797e7	−0.976228	−0.488114	−	0.872780i	$-0.662315\pi$
$929$	−2.56797e7	−0.976228	−0.488114	+	0.872780i	$0.662315\pi$
$930$	−5.38650e6	−0.204220
$931$	−1.44920e7	−0.547966
$932$	−2.15373e6	−0.0812178
$933$	2.53365e7	0.952888
$934$	−2.67774e6	−0.100439
$935$	−1.90028e6	−0.0710865
$936$	−1.62778e6	−0.0607302
$937$	2.44479e7	0.909687	0.454843	−	0.890571i	$-0.349695\pi$
$937$	2.44479e7	0.909687	0.454843	+	0.890571i	$0.349695\pi$
$938$	−2.45467e6	−0.0910933
$939$	2.38902e7	0.884209
$940$	1.16160e6	0.0428782
$941$	−5.02352e7	−1.84941	−0.924707	−	0.380679i	$-0.875690\pi$
$941$	−5.02352e7	−1.84941	−0.924707	+	0.380679i	$0.875690\pi$
$942$	7.99218e6	0.293453
$943$	−5.23053e7	−1.91543
$944$	−559616.	−0.0204390
$945$	−601425.	−0.0219080
$946$	132132.	0.00480043
$947$	2.01510e7	0.730165	0.365083	−	0.930975i	$-0.381041\pi$
$947$	2.01510e7	0.730165	0.365083	+	0.930975i	$0.381041\pi$
$948$	−1.14814e7	−0.414929
$949$	1.39843e7	0.504052
$950$	2.30500e6	0.0828632
$951$	3.61140e6	0.129487
$952$	−4.11629e6	−0.147202
$953$	8.29606e6	0.295896	0.147948	−	0.988995i	$-0.452733\pi$
$953$	8.29606e6	0.295896	0.147948	+	0.988995i	$0.452733\pi$
$954$	−1.09677e7	−0.390163
$955$	−1.71144e7	−0.607229
$956$	−2.23883e7	−0.792277
$957$	2.91786e6	0.102988
$958$	−1.41160e7	−0.496933
$959$	3.70993e6	0.130262
$960$	−921600.	−0.0322749
$961$	7.19107e6	0.251180
$962$	1.71946e6	0.0599039
$963$	8.98663e6	0.312270
$964$	−1.49353e7	−0.517633
$965$	2.71765e6	0.0939453
$966$	−4.56667e6	−0.157455
$967$	−1.08855e7	−0.374355	−0.187178	−	0.982326i	$-0.559934\pi$
$967$	−1.08855e7	−0.374355	−0.187178	+	0.982326i	$0.559934\pi$
$968$	−1.02099e7	−0.350214
$969$	1.61728e7	0.553319
$970$	8.55310e6	0.291873
$971$	−2.54160e7	−0.865087	−0.432544	−	0.901613i	$-0.642384\pi$
$971$	−2.54160e7	−0.865087	−0.432544	+	0.901613i	$0.642384\pi$
$972$	−944784.	−0.0320750
$973$	4.99227e6	0.169050
$974$	2.30846e6	0.0779694
$975$	1.76625e6	0.0595032
$976$	5.09261e6	0.171126
$977$	4.03548e7	1.35257	0.676283	−	0.736642i	$-0.263590\pi$
$977$	4.03548e7	1.35257	0.676283	+	0.736642i	$0.263590\pi$
$978$	−898236.	−0.0300292
$979$	−2.26083e6	−0.0753896
$980$	−6.28720e6	−0.209118
$981$	−4.12995e6	−0.137016
$982$	1.27205e6	0.0420944
$983$	8.20067e6	0.270686	0.135343	−	0.990799i	$-0.456786\pi$
$983$	8.20067e6	0.270686	0.135343	+	0.990799i	$0.456786\pi$
$984$	7.83763e6	0.258046
$985$	1.87995e6	0.0617385
$986$	6.48081e7	2.12294
$987$	−862488.	−0.0281812
$988$	−4.63213e6	−0.150969
$989$	3.25587e6	0.105846
$990$	315900.	0.0102438
$991$	2.13917e7	0.691928	0.345964	−	0.938248i	$-0.387552\pi$
$991$	2.13917e7	0.691928	0.345964	+	0.938248i	$0.387552\pi$
$992$	6.12864e6	0.197736
$993$	2.80310e7	0.902123
$994$	−2.34168e6	−0.0751729
$995$	−7.00670e6	−0.224365
$996$	−5.22058e6	−0.166752
$997$	9.54605e6	0.304149	0.152074	−	0.988369i	$-0.451405\pi$
$997$	9.54605e6	0.304149	0.152074	+	0.988369i	$0.451405\pi$
$998$	2.45597e7	0.780544
$999$	998001.	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.6.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more