LMFDB - Embedded newform 114.6.a.g.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2441})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 610$ x^2 - x - 610
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$25.2032$ of defining polynomial
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)

$f(q)$	$=$	$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -76.6097 q^{5} -36.0000 q^{6} -126.610 q^{7} +64.0000 q^{8} +81.0000 q^{9} -306.439 q^{10} +744.268 q^{11} -144.000 q^{12} +844.439 q^{13} -506.439 q^{14} +689.487 q^{15} +256.000 q^{16} +1306.17 q^{17} +324.000 q^{18} -361.000 q^{19} -1225.76 q^{20} +1139.49 q^{21} +2977.07 q^{22} -1823.41 q^{23} -576.000 q^{24} +2744.05 q^{25} +3377.76 q^{26} -729.000 q^{27} -2025.76 q^{28} -5539.75 q^{29} +2757.95 q^{30} +10200.7 q^{31} +1024.00 q^{32} -6698.41 q^{33} +5224.68 q^{34} +9699.53 q^{35} +1296.00 q^{36} +3328.10 q^{37} -1444.00 q^{38} -7599.95 q^{39} -4903.02 q^{40} +6050.53 q^{41} +4557.95 q^{42} +1922.46 q^{43} +11908.3 q^{44} -6205.39 q^{45} -7293.66 q^{46} +10370.8 q^{47} -2304.00 q^{48} -776.980 q^{49} +10976.2 q^{50} -11755.5 q^{51} +13511.0 q^{52} +3107.66 q^{53} -2916.00 q^{54} -57018.2 q^{55} -8103.02 q^{56} +3249.00 q^{57} -22159.0 q^{58} -3194.44 q^{59} +11031.8 q^{60} +17475.6 q^{61} +40802.9 q^{62} -10255.4 q^{63} +4096.00 q^{64} -64692.2 q^{65} -26793.6 q^{66} +13374.0 q^{67} +20898.7 q^{68} +16410.7 q^{69} +38798.1 q^{70} +6280.96 q^{71} +5184.00 q^{72} +78709.0 q^{73} +13312.4 q^{74} -24696.4 q^{75} -5776.00 q^{76} -94231.6 q^{77} -30399.8 q^{78} +41603.4 q^{79} -19612.1 q^{80} +6561.00 q^{81} +24202.1 q^{82} -9042.74 q^{83} +18231.8 q^{84} -100065. q^{85} +7689.85 q^{86} +49857.8 q^{87} +47633.2 q^{88} +3935.36 q^{89} -24821.5 q^{90} -106914. q^{91} -29174.6 q^{92} -91806.6 q^{93} +41483.4 q^{94} +27656.1 q^{95} -9216.00 q^{96} +144902. q^{97} -3107.92 q^{98} +60285.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} - 5 q^{5} - 72 q^{6} - 105 q^{7} + 128 q^{8} + 162 q^{9} - 20 q^{10} + 451 q^{11} - 288 q^{12} + 1096 q^{13} - 420 q^{14} + 45 q^{15} + 512 q^{16} + 3057 q^{17} + 648 q^{18}+ \cdots + 36531 q^{99}+O(q^{100})$ 2 * q + 8 * q^2 - 18 * q^3 + 32 * q^4 - 5 * q^5 - 72 * q^6 - 105 * q^7 + 128 * q^8 + 162 * q^9 - 20 * q^10 + 451 * q^11 - 288 * q^12 + 1096 * q^13 - 420 * q^14 + 45 * q^15 + 512 * q^16 + 3057 * q^17 + 648 * q^18 - 722 * q^19 - 80 * q^20 + 945 * q^21 + 1804 * q^22 - 386 * q^23 - 1152 * q^24 + 4747 * q^25 + 4384 * q^26 - 1458 * q^27 - 1680 * q^28 + 3446 * q^29 + 180 * q^30 + 15362 * q^31 + 2048 * q^32 - 4059 * q^33 + 12228 * q^34 + 11247 * q^35 + 2592 * q^36 + 5174 * q^37 - 2888 * q^38 - 9864 * q^39 - 320 * q^40 - 2128 * q^41 + 3780 * q^42 - 157 * q^43 + 7216 * q^44 - 405 * q^45 - 1544 * q^46 - 1343 * q^47 - 4608 * q^48 - 17117 * q^49 + 18988 * q^50 - 27513 * q^51 + 17536 * q^52 + 5326 * q^53 - 5832 * q^54 - 78019 * q^55 - 6720 * q^56 + 6498 * q^57 + 13784 * q^58 - 5796 * q^59 + 720 * q^60 + 1009 * q^61 + 61448 * q^62 - 8505 * q^63 + 8192 * q^64 - 46678 * q^65 - 16236 * q^66 + 45720 * q^67 + 48912 * q^68 + 3474 * q^69 + 44988 * q^70 - 50876 * q^71 + 10368 * q^72 + 95907 * q^73 + 20696 * q^74 - 42723 * q^75 - 11552 * q^76 - 100569 * q^77 - 39456 * q^78 - 5132 * q^79 - 1280 * q^80 + 13122 * q^81 - 8512 * q^82 - 61662 * q^83 + 15120 * q^84 + 25311 * q^85 - 628 * q^86 - 31014 * q^87 + 28864 * q^88 - 726 * q^89 - 1620 * q^90 - 101478 * q^91 - 6176 * q^92 - 138258 * q^93 - 5372 * q^94 + 1805 * q^95 - 18432 * q^96 + 75776 * q^97 - 68468 * q^98 + 36531 * q^99

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
$5$	−76.6097	−1.37044	−0.685218	−	0.728338i	$-0.740293\pi$
$5$	−76.6097	−1.37044	−0.685218	+	0.728338i	$0.740293\pi$
$6$	−36.0000	−0.408248
$7$	−126.610	−0.976612	−0.488306	−	0.872673i	$-0.662385\pi$
$7$	−126.610	−0.976612	−0.488306	+	0.872673i	$0.662385\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−306.439	−0.969045
$11$	744.268	1.85459	0.927294	−	0.374333i	$-0.122128\pi$
$11$	744.268	1.85459	0.927294	+	0.374333i	$0.122128\pi$
$12$	−144.000	−0.288675
$13$	844.439	1.38583	0.692915	−	0.721019i	$-0.256326\pi$
$13$	844.439	1.38583	0.692915	+	0.721019i	$0.256326\pi$
$14$	−506.439	−0.690569
$15$	689.487	0.791222
$16$	256.000	0.250000
$17$	1306.17	1.09617	0.548085	−	0.836423i	$-0.315357\pi$
$17$	1306.17	1.09617	0.548085	+	0.836423i	$0.315357\pi$
$18$	324.000	0.235702
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	−1225.76	−0.685218
$21$	1139.49	0.563847
$22$	2977.07	1.31139
$23$	−1823.41	−0.718730	−0.359365	−	0.933197i	$-0.617007\pi$
$23$	−1823.41	−0.718730	−0.359365	+	0.933197i	$0.617007\pi$
$24$	−576.000	−0.204124
$25$	2744.05	0.878096
$26$	3377.76	0.979930
$27$	−729.000	−0.192450
$28$	−2025.76	−0.488306
$29$	−5539.75	−1.22319	−0.611597	−	0.791169i	$-0.709473\pi$
$29$	−5539.75	−1.22319	−0.611597	+	0.791169i	$0.709473\pi$
$30$	2757.95	0.559478
$31$	10200.7	1.90646	0.953229	−	0.302250i	$-0.0977379\pi$
$31$	10200.7	1.90646	0.953229	+	0.302250i	$0.0977379\pi$
$32$	1024.00	0.176777
$33$	−6698.41	−1.07075
$34$	5224.68	0.775109
$35$	9699.53	1.33838
$36$	1296.00	0.166667
$37$	3328.10	0.399661	0.199830	−	0.979830i	$-0.435961\pi$
$37$	3328.10	0.399661	0.199830	+	0.979830i	$0.435961\pi$
$38$	−1444.00	−0.162221
$39$	−7599.95	−0.800109
$40$	−4903.02	−0.484522
$41$	6050.53	0.562126	0.281063	−	0.959689i	$-0.409313\pi$
$41$	6050.53	0.562126	0.281063	+	0.959689i	$0.409313\pi$
$42$	4557.95	0.398700
$43$	1922.46	0.158557	0.0792787	−	0.996852i	$-0.474738\pi$
$43$	1922.46	0.158557	0.0792787	+	0.996852i	$0.474738\pi$
$44$	11908.3	0.927294
$45$	−6205.39	−0.456812
$46$	−7293.66	−0.508219
$47$	10370.8	0.684809	0.342405	−	0.939553i	$-0.388759\pi$
$47$	10370.8	0.684809	0.342405	+	0.939553i	$0.388759\pi$
$48$	−2304.00	−0.144338
$49$	−776.980	−0.0462295
$50$	10976.2	0.620907
$51$	−11755.5	−0.632874
$52$	13511.0	0.692915
$53$	3107.66	0.151965	0.0759825	−	0.997109i	$-0.475791\pi$
$53$	3107.66	0.151965	0.0759825	+	0.997109i	$0.475791\pi$
$54$	−2916.00	−0.136083
$55$	−57018.2	−2.54160
$56$	−8103.02	−0.345284
$57$	3249.00	0.132453
$58$	−22159.0	−0.864929
$59$	−3194.44	−0.119472	−0.0597358	−	0.998214i	$-0.519026\pi$
$59$	−3194.44	−0.119472	−0.0597358	+	0.998214i	$0.519026\pi$
$60$	11031.8	0.395611
$61$	17475.6	0.601323	0.300662	−	0.953731i	$-0.402792\pi$
$61$	17475.6	0.601323	0.300662	+	0.953731i	$0.402792\pi$
$62$	40802.9	1.34807
$63$	−10255.4	−0.325537
$64$	4096.00	0.125000
$65$	−64692.2	−1.89919
$66$	−26793.6	−0.757133
$67$	13374.0	0.363976	0.181988	−	0.983301i	$-0.441747\pi$
$67$	13374.0	0.363976	0.181988	+	0.983301i	$0.441747\pi$
$68$	20898.7	0.548085
$69$	16410.7	0.414959
$70$	38798.1	0.946380
$71$	6280.96	0.147870	0.0739350	−	0.997263i	$-0.476444\pi$
$71$	6280.96	0.147870	0.0739350	+	0.997263i	$0.476444\pi$
$72$	5184.00	0.117851
$73$	78709.0	1.72869	0.864346	−	0.502898i	$-0.167733\pi$
$73$	78709.0	1.72869	0.864346	+	0.502898i	$0.167733\pi$
$74$	13312.4	0.282603
$75$	−24696.4	−0.506969
$76$	−5776.00	−0.114708
$77$	−94231.6	−1.81121
$78$	−30399.8	−0.565763
$79$	41603.4	0.750000	0.375000	−	0.927025i	$-0.377643\pi$
$79$	41603.4	0.750000	0.375000	+	0.927025i	$0.377643\pi$
$80$	−19612.1	−0.342609
$81$	6561.00	0.111111
$82$	24202.1	0.397483
$83$	−9042.74	−0.144080	−0.0720402	−	0.997402i	$-0.522951\pi$
$83$	−9042.74	−0.144080	−0.0720402	+	0.997402i	$0.522951\pi$
$84$	18231.8	0.281924
$85$	−100065.	−1.50223
$86$	7689.85	0.112117
$87$	49857.8	0.706211
$88$	47633.2	0.655696
$89$	3935.36	0.0526635	0.0263318	−	0.999653i	$-0.491617\pi$
$89$	3935.36	0.0526635	0.0263318	+	0.999653i	$0.491617\pi$
$90$	−24821.5	−0.323015
$91$	−106914.	−1.35342
$92$	−29174.6	−0.359365
$93$	−91806.6	−1.10069
$94$	41483.4	0.484233
$95$	27656.1	0.314400
$96$	−9216.00	−0.102062
$97$	144902.	1.56367	0.781837	−	0.623483i	$-0.214283\pi$
$97$	144902.	1.56367	0.781837	+	0.623483i	$0.214283\pi$
$98$	−3107.92	−0.0326892
$99$	60285.7	0.618196
$100$	43904.8	0.439048
$101$	−86084.3	−0.839692	−0.419846	−	0.907595i	$-0.637916\pi$
$101$	−86084.3	−0.839692	−0.419846	+	0.907595i	$0.637916\pi$
$102$	−47022.2	−0.447509
$103$	−194549.	−1.80691	−0.903454	−	0.428686i	$-0.858977\pi$
$103$	−194549.	−1.80691	−0.903454	+	0.428686i	$0.858977\pi$
$104$	54044.1	0.489965
$105$	−87295.8	−0.772716
$106$	12430.6	0.107455
$107$	−214997.	−1.81540	−0.907702	−	0.419615i	$-0.862165\pi$
$107$	−214997.	−1.81540	−0.907702	+	0.419615i	$0.862165\pi$
$108$	−11664.0	−0.0962250
$109$	143607.	1.15774	0.578870	−	0.815420i	$-0.303494\pi$
$109$	143607.	1.15774	0.578870	+	0.815420i	$0.303494\pi$
$110$	−228073.	−1.79718
$111$	−29952.9	−0.230744
$112$	−32412.1	−0.244153
$113$	−229554.	−1.69117	−0.845586	−	0.533838i	$-0.820749\pi$
$113$	−229554.	−1.69117	−0.845586	+	0.533838i	$0.820749\pi$
$114$	12996.0	0.0936586
$115$	139691.	0.984973
$116$	−88636.0	−0.611597
$117$	68399.5	0.461943
$118$	−12777.8	−0.0844792
$119$	−165374.	−1.07053
$120$	44127.2	0.279739
$121$	392884.	2.43950
$122$	69902.5	0.425200
$123$	−54454.8	−0.324544
$124$	163212.	0.953229
$125$	29184.6	0.167062
$126$	−41021.5	−0.230190
$127$	−220694.	−1.21418	−0.607088	−	0.794635i	$-0.707662\pi$
$127$	−220694.	−1.21418	−0.607088	+	0.794635i	$0.707662\pi$
$128$	16384.0	0.0883883
$129$	−17302.2	−0.0915432
$130$	−258769.	−1.34293
$131$	271930.	1.38445	0.692227	−	0.721680i	$-0.256630\pi$
$131$	271930.	1.38445	0.692227	+	0.721680i	$0.256630\pi$
$132$	−107175.	−0.535374
$133$	45706.1	0.224050
$134$	53495.8	0.257370
$135$	55848.5	0.263741
$136$	83594.9	0.387554
$137$	302764.	1.37817	0.689085	−	0.724680i	$-0.258013\pi$
$137$	302764.	1.37817	0.689085	+	0.724680i	$0.258013\pi$
$138$	65642.9	0.293420
$139$	405848.	1.78167	0.890833	−	0.454330i	$-0.150121\pi$
$139$	405848.	1.78167	0.890833	+	0.454330i	$0.150121\pi$
$140$	155193.	0.669192
$141$	−93337.6	−0.395375
$142$	25123.8	0.104560
$143$	628489.	2.57014
$144$	20736.0	0.0833333
$145$	424399.	1.67631
$146$	314836.	1.22237
$147$	6992.82	0.0266906
$148$	53249.6	0.199830
$149$	−18679.2	−0.0689274	−0.0344637	−	0.999406i	$-0.510972\pi$
$149$	−18679.2	−0.0689274	−0.0344637	+	0.999406i	$0.510972\pi$
$150$	−98785.7	−0.358481
$151$	−31638.8	−0.112922	−0.0564609	−	0.998405i	$-0.517982\pi$
$151$	−31638.8	−0.112922	−0.0564609	+	0.998405i	$0.517982\pi$
$152$	−23104.0	−0.0811107
$153$	105800.	0.365390
$154$	−376926.	−1.28072
$155$	−781475.	−2.61268
$156$	−121599.	−0.400055
$157$	−7817.53	−0.0253116	−0.0126558	−	0.999920i	$-0.504029\pi$
$157$	−7817.53	−0.0253116	−0.0126558	+	0.999920i	$0.504029\pi$
$158$	166414.	0.530330
$159$	−27968.9	−0.0877370
$160$	−78448.3	−0.242261
$161$	230862.	0.701920
$162$	26244.0	0.0785674
$163$	−245394.	−0.723428	−0.361714	−	0.932289i	$-0.617808\pi$
$163$	−245394.	−0.723428	−0.361714	+	0.932289i	$0.617808\pi$
$164$	96808.5	0.281063
$165$	513163.	1.46739
$166$	−36171.0	−0.101880
$167$	−687602.	−1.90786	−0.953929	−	0.300034i	$-0.903002\pi$
$167$	−687602.	−1.90786	−0.953929	+	0.300034i	$0.903002\pi$
$168$	72927.2	0.199350
$169$	341784.	0.920524
$170$	−400262.	−1.06224
$171$	−29241.0	−0.0764719
$172$	30759.4	0.0792787
$173$	475094.	1.20688	0.603441	−	0.797408i	$-0.293796\pi$
$173$	475094.	1.20688	0.603441	+	0.797408i	$0.293796\pi$
$174$	199431.	0.499367
$175$	−347423.	−0.857558
$176$	190533.	0.463647
$177$	28749.9	0.0689769
$178$	15741.5	0.0372387
$179$	−151716.	−0.353914	−0.176957	−	0.984219i	$-0.556625\pi$
$179$	−151716.	−0.353914	−0.176957	+	0.984219i	$0.556625\pi$
$180$	−99286.2	−0.228406
$181$	138508.	0.314251	0.157126	−	0.987579i	$-0.449777\pi$
$181$	138508.	0.314251	0.157126	+	0.987579i	$0.449777\pi$
$182$	−427657.	−0.957011
$183$	−157281.	−0.347174
$184$	−116698.	−0.254109
$185$	−254965.	−0.547710
$186$	−367226.	−0.778308
$187$	972141.	2.03294
$188$	165934.	0.342405
$189$	92298.5	0.187949
$190$	110624.	0.222314
$191$	−308867.	−0.612616	−0.306308	−	0.951933i	$-0.599094\pi$
$191$	−308867.	−0.612616	−0.306308	+	0.951933i	$0.599094\pi$
$192$	−36864.0	−0.0721688
$193$	508873.	0.983367	0.491684	−	0.870774i	$-0.336382\pi$
$193$	508873.	0.983367	0.491684	+	0.870774i	$0.336382\pi$
$194$	579610.	1.10568
$195$	582230.	1.09650
$196$	−12431.7	−0.0231148
$197$	−811685.	−1.49012	−0.745061	−	0.666996i	$-0.767580\pi$
$197$	−811685.	−1.49012	−0.745061	+	0.666996i	$0.767580\pi$
$198$	241143.	0.437131
$199$	−707620.	−1.26668	−0.633341	−	0.773873i	$-0.718317\pi$
$199$	−707620.	−1.26668	−0.633341	+	0.773873i	$0.718317\pi$
$200$	175619.	0.310454
$201$	−120366.	−0.210142
$202$	−344337.	−0.593752
$203$	701386.	1.19459
$204$	−188089.	−0.316437
$205$	−463530.	−0.770358
$206$	−778196.	−1.27768
$207$	−147697.	−0.239577
$208$	216176.	0.346457
$209$	−268681.	−0.425472
$210$	−349183.	−0.546393
$211$	756138.	1.16922	0.584608	−	0.811316i	$-0.301248\pi$
$211$	756138.	1.16922	0.584608	+	0.811316i	$0.301248\pi$
$212$	49722.5	0.0759825
$213$	−56528.6	−0.0853728
$214$	−859989.	−1.28368
$215$	−147279.	−0.217293
$216$	−46656.0	−0.0680414
$217$	−1.29151e6	−1.86187
$218$	574430.	0.818645
$219$	−708381.	−0.998060
$220$	−912291.	−1.27080
$221$	1.10298e6	1.51910
$222$	−119811.	−0.163161
$223$	−420229.	−0.565880	−0.282940	−	0.959138i	$-0.591310\pi$
$223$	−420229.	−0.565880	−0.282940	+	0.959138i	$0.591310\pi$
$224$	−129648.	−0.172642
$225$	222268.	0.292699
$226$	−918214.	−1.19584
$227$	918724.	1.18337	0.591684	−	0.806170i	$-0.298463\pi$
$227$	918724.	1.18337	0.591684	+	0.806170i	$0.298463\pi$
$228$	51984.0	0.0662266
$229$	306139.	0.385772	0.192886	−	0.981221i	$-0.438215\pi$
$229$	306139.	0.385772	0.192886	+	0.981221i	$0.438215\pi$
$230$	558765.	0.696481
$231$	848084.	1.04570
$232$	−354544.	−0.432464
$233$	−394089.	−0.475559	−0.237779	−	0.971319i	$-0.576420\pi$
$233$	−394089.	−0.475559	−0.237779	+	0.971319i	$0.576420\pi$
$234$	273598.	0.326643
$235$	−794508.	−0.938487
$236$	−51111.0	−0.0597358
$237$	−374431.	−0.433012
$238$	−661496.	−0.756980
$239$	−1.44062e6	−1.63138	−0.815691	−	0.578488i	$-0.803643\pi$
$239$	−1.44062e6	−1.63138	−0.815691	+	0.578488i	$0.803643\pi$
$240$	176509.	0.197805
$241$	−253673.	−0.281340	−0.140670	−	0.990057i	$-0.544926\pi$
$241$	−253673.	−0.281340	−0.140670	+	0.990057i	$0.544926\pi$
$242$	1.57154e6	1.72499
$243$	−59049.0	−0.0641500
$244$	279610.	0.300662
$245$	59524.2	0.0633546
$246$	−217819.	−0.229487
$247$	−304842.	−0.317931
$248$	652847.	0.674034
$249$	81384.7	0.0831849
$250$	116738.	0.118131
$251$	1.07742e6	1.07945	0.539724	−	0.841842i	$-0.318528\pi$
$251$	1.07742e6	1.07945	0.539724	+	0.841842i	$0.318528\pi$
$252$	−164086.	−0.162769
$253$	−1.35711e6	−1.33295
$254$	−882777.	−0.858552
$255$	900588.	0.867313
$256$	65536.0	0.0625000
$257$	−1.65751e6	−1.56539	−0.782695	−	0.622405i	$-0.786156\pi$
$257$	−1.65751e6	−1.56539	−0.782695	+	0.622405i	$0.786156\pi$
$258$	−69208.6	−0.0647308
$259$	−421369.	−0.390314
$260$	−1.03508e6	−0.949596
$261$	−448720.	−0.407731
$262$	1.08772e6	0.978957
$263$	−475083.	−0.423526	−0.211763	−	0.977321i	$-0.567921\pi$
$263$	−475083.	−0.423526	−0.211763	+	0.977321i	$0.567921\pi$
$264$	−428698.	−0.378566
$265$	−238077.	−0.208258
$266$	182824.	0.158427
$267$	−35418.3	−0.0304053
$268$	213983.	0.181988
$269$	1.14894e6	0.968090	0.484045	−	0.875043i	$-0.339167\pi$
$269$	1.14894e6	0.968090	0.484045	+	0.875043i	$0.339167\pi$
$270$	223394.	0.186493
$271$	−844947.	−0.698886	−0.349443	−	0.936958i	$-0.613629\pi$
$271$	−844947.	−0.698886	−0.349443	+	0.936958i	$0.613629\pi$
$272$	334380.	0.274042
$273$	962227.	0.781396
$274$	1.21106e6	0.974514
$275$	2.04231e6	1.62851
$276$	262572.	0.207479
$277$	1.25965e6	0.986391	0.493195	−	0.869919i	$-0.335829\pi$
$277$	1.25965e6	0.986391	0.493195	+	0.869919i	$0.335829\pi$
$278$	1.62339e6	1.25983
$279$	826259.	0.635486
$280$	620770.	0.473190
$281$	227822.	0.172119	0.0860595	−	0.996290i	$-0.472572\pi$
$281$	227822.	0.172119	0.0860595	+	0.996290i	$0.472572\pi$
$282$	−373351.	−0.279572
$283$	−2.47321e6	−1.83567	−0.917834	−	0.396965i	$-0.870064\pi$
$283$	−2.47321e6	−1.83567	−0.917834	+	0.396965i	$0.870064\pi$
$284$	100495.	0.0739350
$285$	−248905.	−0.181519
$286$	2.51396e6	1.81737
$287$	−766056.	−0.548979
$288$	82944.0	0.0589256
$289$	286225.	0.201587
$290$	1.69760e6	1.18533
$291$	−1.30412e6	−0.902788
$292$	1.25934e6	0.864346
$293$	−159716.	−0.108687	−0.0543436	−	0.998522i	$-0.517307\pi$
$293$	−159716.	−0.108687	−0.0543436	+	0.998522i	$0.517307\pi$
$294$	27971.3	0.0188731
$295$	244725.	0.163728
$296$	212998.	0.141301
$297$	−542571.	−0.356916
$298$	−74716.7	−0.0487390
$299$	−1.53976e6	−0.996037
$300$	−395143.	−0.253484
$301$	−243402.	−0.154849
$302$	−126555.	−0.0798477
$303$	774758.	0.484797
$304$	−92416.0	−0.0573539
$305$	−1.33880e6	−0.824075
$306$	423199.	0.258370
$307$	722474.	0.437498	0.218749	−	0.975781i	$-0.429802\pi$
$307$	722474.	0.437498	0.218749	+	0.975781i	$0.429802\pi$
$308$	−1.50770e6	−0.905607
$309$	1.75094e6	1.04322
$310$	−3.12590e6	−1.84744
$311$	568481.	0.333284	0.166642	−	0.986017i	$-0.446708\pi$
$311$	568481.	0.333284	0.166642	+	0.986017i	$0.446708\pi$
$312$	−486397.	−0.282881
$313$	−39902.0	−0.0230215	−0.0115108	−	0.999934i	$-0.503664\pi$
$313$	−39902.0	−0.0230215	−0.0115108	+	0.999934i	$0.503664\pi$
$314$	−31270.1	−0.0178980
$315$	785662.	0.446128
$316$	665654.	0.375000
$317$	2.42559e6	1.35572	0.677859	−	0.735192i	$-0.262908\pi$
$317$	2.42559e6	1.35572	0.677859	+	0.735192i	$0.262908\pi$
$318$	−111876.	−0.0620395
$319$	−4.12306e6	−2.26852
$320$	−313793.	−0.171305
$321$	1.93497e6	1.04812
$322$	923448.	0.496332
$323$	−471528.	−0.251479
$324$	104976.	0.0555556
$325$	2.31718e6	1.21689
$326$	−981577.	−0.511541
$327$	−1.29247e6	−0.668421
$328$	387234.	0.198742
$329$	−1.31305e6	−0.668793
$330$	2.05265e6	1.03760
$331$	−87388.5	−0.0438414	−0.0219207	−	0.999760i	$-0.506978\pi$
$331$	−87388.5	−0.0438414	−0.0219207	+	0.999760i	$0.506978\pi$
$332$	−144684.	−0.0720402
$333$	269576.	0.133220
$334$	−2.75041e6	−1.34906
$335$	−1.02457e6	−0.498806
$336$	291709.	0.140962
$337$	2.25546e6	1.08183	0.540917	−	0.841076i	$-0.318077\pi$
$337$	2.25546e6	1.08183	0.540917	+	0.841076i	$0.318077\pi$
$338$	1.36714e6	0.650909
$339$	2.06598e6	0.976399
$340$	−1.60105e6	−0.751115
$341$	7.59208e6	3.53569
$342$	−116964.	−0.0540738
$343$	2.22630e6	1.02176
$344$	123038.	0.0560585
$345$	−1.25722e6	−0.568675
$346$	1.90038e6	0.853394
$347$	−2.30600e6	−1.02810	−0.514050	−	0.857760i	$-0.671855\pi$
$347$	−2.30600e6	−1.02810	−0.514050	+	0.857760i	$0.671855\pi$
$348$	797724.	0.353106
$349$	3.57261e6	1.57008	0.785041	−	0.619443i	$-0.212642\pi$
$349$	3.57261e6	1.57008	0.785041	+	0.619443i	$0.212642\pi$
$350$	−1.38969e6	−0.606385
$351$	−615596.	−0.266703
$352$	762130.	0.327848
$353$	−1.11293e6	−0.475371	−0.237685	−	0.971342i	$-0.576389\pi$
$353$	−1.11293e6	−0.475371	−0.237685	+	0.971342i	$0.576389\pi$
$354$	115000.	0.0487741
$355$	−481182.	−0.202646
$356$	62965.8	0.0263318
$357$	1.48837e6	0.618072
$358$	−606863.	−0.250255
$359$	−3.44175e6	−1.40943	−0.704714	−	0.709492i	$-0.748925\pi$
$359$	−3.44175e6	−1.40943	−0.704714	+	0.709492i	$0.748925\pi$
$360$	−397145.	−0.161507
$361$	130321.	0.0526316
$362$	554030.	0.222209
$363$	−3.53595e6	−1.40845
$364$	−1.71063e6	−0.676709
$365$	−6.02988e6	−2.36906
$366$	−629122.	−0.245489
$367$	−1.46107e6	−0.566248	−0.283124	−	0.959083i	$-0.591371\pi$
$367$	−1.46107e6	−0.566248	−0.283124	+	0.959083i	$0.591371\pi$
$368$	−466794.	−0.179682
$369$	490093.	0.187375
$370$	−1.01986e6	−0.387289
$371$	−393460.	−0.148411
$372$	−1.46891e6	−0.550347
$373$	−2.62654e6	−0.977488	−0.488744	−	0.872427i	$-0.662545\pi$
$373$	−2.62654e6	−0.977488	−0.488744	+	0.872427i	$0.662545\pi$
$374$	3.88856e6	1.43751
$375$	−262661.	−0.0964535
$376$	663734.	0.242117
$377$	−4.67798e6	−1.69514
$378$	369194.	0.132900
$379$	302415.	0.108145	0.0540724	−	0.998537i	$-0.482780\pi$
$379$	302415.	0.108145	0.0540724	+	0.998537i	$0.482780\pi$
$380$	442498.	0.157200
$381$	1.98625e6	0.701005
$382$	−1.23547e6	−0.433185
$383$	−161905.	−0.0563981	−0.0281991	−	0.999602i	$-0.508977\pi$
$383$	−161905.	−0.0563981	−0.0281991	+	0.999602i	$0.508977\pi$
$384$	−147456.	−0.0510310
$385$	7.21905e6	2.48215
$386$	2.03549e6	0.695346
$387$	155719.	0.0528525
$388$	2.31844e6	0.781837
$389$	3.05129e6	1.02237	0.511187	−	0.859470i	$-0.329206\pi$
$389$	3.05129e6	1.02237	0.511187	+	0.859470i	$0.329206\pi$
$390$	2.32892e6	0.775342
$391$	−2.38169e6	−0.787850
$392$	−49726.7	−0.0163446
$393$	−2.44737e6	−0.799315
$394$	−3.24674e6	−1.05368
$395$	−3.18722e6	−1.02783
$396$	964571.	0.309098
$397$	2.59147e6	0.825219	0.412610	−	0.910908i	$-0.364617\pi$
$397$	2.59147e6	0.825219	0.412610	+	0.910908i	$0.364617\pi$
$398$	−2.83048e6	−0.895679
$399$	−411355.	−0.129355
$400$	702476.	0.219524
$401$	3.91055e6	1.21444	0.607221	−	0.794533i	$-0.292284\pi$
$401$	3.91055e6	1.21444	0.607221	+	0.794533i	$0.292284\pi$
$402$	−481462.	−0.148593
$403$	8.61389e6	2.64202
$404$	−1.37735e6	−0.419846
$405$	−502636.	−0.152271
$406$	2.80555e6	0.844700
$407$	2.47700e6	0.741207
$408$	−752354.	−0.223755
$409$	−3.71122e6	−1.09700	−0.548502	−	0.836149i	$-0.684802\pi$
$409$	−3.71122e6	−1.09700	−0.548502	+	0.836149i	$0.684802\pi$
$410$	−1.85412e6	−0.544726
$411$	−2.72488e6	−0.795687
$412$	−3.11278e6	−0.903454
$413$	404447.	0.116677
$414$	−590786.	−0.169406
$415$	692762.	0.197453
$416$	864705.	0.244982
$417$	−3.65263e6	−1.02865
$418$	−1.07472e6	−0.300854
$419$	−2.87932e6	−0.801225	−0.400613	−	0.916248i	$-0.631203\pi$
$419$	−2.87932e6	−0.801225	−0.400613	+	0.916248i	$0.631203\pi$
$420$	−1.39673e6	−0.386358
$421$	−6.67740e6	−1.83612	−0.918062	−	0.396436i	$-0.870247\pi$
$421$	−6.67740e6	−1.83612	−0.918062	+	0.396436i	$0.870247\pi$
$422$	3.02455e6	0.826761
$423$	840039.	0.228270
$424$	198890.	0.0537277
$425$	3.58420e6	0.962541
$426$	−226115.	−0.0603677
$427$	−2.21258e6	−0.587260
$428$	−3.43996e6	−0.907702
$429$	−5.65640e6	−1.48387
$430$	−589117.	−0.153649
$431$	1.04017e6	0.269720	0.134860	−	0.990865i	$-0.456942\pi$
$431$	1.04017e6	0.269720	0.134860	+	0.990865i	$0.456942\pi$
$432$	−186624.	−0.0481125
$433$	−924840.	−0.237054	−0.118527	−	0.992951i	$-0.537817\pi$
$433$	−924840.	−0.237054	−0.118527	+	0.992951i	$0.537817\pi$
$434$	−5.16605e6	−1.31654
$435$	−3.81959e6	−0.967818
$436$	2.29772e6	0.578870
$437$	658252.	0.164888
$438$	−2.83353e6	−0.705735
$439$	689018.	0.170636	0.0853178	−	0.996354i	$-0.472809\pi$
$439$	689018.	0.170636	0.0853178	+	0.996354i	$0.472809\pi$
$440$	−3.64916e6	−0.898590
$441$	−62935.4	−0.0154098
$442$	4.41193e6	1.07417
$443$	721377.	0.174644	0.0873219	−	0.996180i	$-0.472169\pi$
$443$	721377.	0.174644	0.0873219	+	0.996180i	$0.472169\pi$
$444$	−479246.	−0.115372
$445$	−301487.	−0.0721720
$446$	−1.68092e6	−0.400138
$447$	168113.	0.0397953
$448$	−518593.	−0.122076
$449$	6.19115e6	1.44929	0.724645	−	0.689123i	$-0.242004\pi$
$449$	6.19115e6	1.44929	0.724645	+	0.689123i	$0.242004\pi$
$450$	889072.	0.206969
$451$	4.50322e6	1.04251
$452$	−3.67286e6	−0.845586
$453$	284749.	0.0651954
$454$	3.67489e6	0.836768
$455$	8.19066e6	1.85477
$456$	207936.	0.0468293
$457$	−3.93674e6	−0.881753	−0.440876	−	0.897568i	$-0.645332\pi$
$457$	−3.93674e6	−0.881753	−0.440876	+	0.897568i	$0.645332\pi$
$458$	1.22456e6	0.272782
$459$	−952199.	−0.210958
$460$	2.23506e6	0.492487
$461$	1.63169e6	0.357589	0.178794	−	0.983886i	$-0.442780\pi$
$461$	1.63169e6	0.357589	0.178794	+	0.983886i	$0.442780\pi$
$462$	3.39234e6	0.739425
$463$	7.30809e6	1.58435	0.792176	−	0.610293i	$-0.208948\pi$
$463$	7.30809e6	1.58435	0.792176	+	0.610293i	$0.208948\pi$
$464$	−1.41818e6	−0.305798
$465$	7.03328e6	1.50843
$466$	−1.57636e6	−0.336271
$467$	−567857.	−0.120489	−0.0602444	−	0.998184i	$-0.519188\pi$
$467$	−567857.	−0.120489	−0.0602444	+	0.998184i	$0.519188\pi$
$468$	1.09439e6	0.230972
$469$	−1.69327e6	−0.355463
$470$	−3.17803e6	−0.663611
$471$	70357.7	0.0146137
$472$	−204444.	−0.0422396
$473$	1.43083e6	0.294059
$474$	−1.49772e6	−0.306186
$475$	−990602.	−0.201449
$476$	−2.64598e6	−0.535266
$477$	251720.	0.0506550
$478$	−5.76249e6	−1.15356
$479$	−704030.	−0.140201	−0.0701007	−	0.997540i	$-0.522332\pi$
$479$	−704030.	−0.140201	−0.0701007	+	0.997540i	$0.522332\pi$
$480$	706035.	0.139870
$481$	2.81037e6	0.553862
$482$	−1.01469e6	−0.198938
$483$	−2.07776e6	−0.405254
$484$	6.28614e6	1.21975
$485$	−1.11009e7	−2.14292
$486$	−236196.	−0.0453609
$487$	1.05439e6	0.201455	0.100727	−	0.994914i	$-0.467883\pi$
$487$	1.05439e6	0.201455	0.100727	+	0.994914i	$0.467883\pi$
$488$	1.11844e6	0.212600
$489$	2.20855e6	0.417671
$490$	238097.	0.0447985
$491$	4.58542e6	0.858371	0.429185	−	0.903216i	$-0.358801\pi$
$491$	4.58542e6	0.858371	0.429185	+	0.903216i	$0.358801\pi$
$492$	−871277.	−0.162272
$493$	−7.23586e6	−1.34083
$494$	−1.21937e6	−0.224811
$495$	−4.61847e6	−0.847199
$496$	2.61139e6	0.476614
$497$	−795230.	−0.144412
$498$	325539.	0.0588206
$499$	−4.71445e6	−0.847577	−0.423789	−	0.905761i	$-0.639300\pi$
$499$	−4.71445e6	−0.847577	−0.423789	+	0.905761i	$0.639300\pi$
$500$	466953.	0.0835311
$501$	6.18842e6	1.10150
$502$	4.30969e6	0.763285
$503$	−8.80792e6	−1.55222	−0.776110	−	0.630597i	$-0.782810\pi$
$503$	−8.80792e6	−1.55222	−0.776110	+	0.630597i	$0.782810\pi$
$504$	−656345.	−0.115095
$505$	6.59489e6	1.15075
$506$	−5.42843e6	−0.942537
$507$	−3.07606e6	−0.531465
$508$	−3.53111e6	−0.607088
$509$	−2.92609e6	−0.500602	−0.250301	−	0.968168i	$-0.580530\pi$
$509$	−2.92609e6	−0.500602	−0.250301	+	0.968168i	$0.580530\pi$
$510$	3.60235e6	0.613283
$511$	−9.96533e6	−1.68826
$512$	262144.	0.0441942
$513$	263169.	0.0441511
$514$	−6.63003e6	−1.10690
$515$	1.49043e7	2.47625
$516$	−276835.	−0.0457716
$517$	7.71869e6	1.27004
$518$	−1.68548e6	−0.275993
$519$	−4.27585e6	−0.696793
$520$	−4.14030e6	−0.671465
$521$	−8.26613e6	−1.33416	−0.667080	−	0.744986i	$-0.732456\pi$
$521$	−8.26613e6	−1.33416	−0.667080	+	0.744986i	$0.732456\pi$
$522$	−1.79488e6	−0.288310
$523$	1.64703e6	0.263298	0.131649	−	0.991296i	$-0.457973\pi$
$523$	1.64703e6	0.263298	0.131649	+	0.991296i	$0.457973\pi$
$524$	4.35088e6	0.692227
$525$	3.12681e6	0.495112
$526$	−1.90033e6	−0.299478
$527$	1.33239e7	2.08980
$528$	−1.71479e6	−0.267687
$529$	−3.11151e6	−0.483428
$530$	−952307.	−0.147261
$531$	−258750.	−0.0398239
$532$	731298.	0.112025
$533$	5.10930e6	0.779011
$534$	−141673.	−0.0214998
$535$	1.64709e7	2.48790
$536$	855933.	0.128685
$537$	1.36544e6	0.204333
$538$	4.59575e6	0.684543
$539$	−578281.	−0.0857368
$540$	893576.	0.131870
$541$	−9.24181e6	−1.35758	−0.678788	−	0.734335i	$-0.737494\pi$
$541$	−9.24181e6	−1.35758	−0.678788	+	0.734335i	$0.737494\pi$
$542$	−3.37979e6	−0.494187
$543$	−1.24657e6	−0.181433
$544$	1.33752e6	0.193777
$545$	−1.10017e7	−1.58661
$546$	3.84891e6	0.552530
$547$	−6.58733e6	−0.941329	−0.470664	−	0.882312i	$-0.655986\pi$
$547$	−6.58733e6	−0.941329	−0.470664	+	0.882312i	$0.655986\pi$
$548$	4.84423e6	0.689085
$549$	1.41553e6	0.200441
$550$	8.16923e6	1.15153
$551$	1.99985e6	0.280620
$552$	1.05029e6	0.146710
$553$	−5.26739e6	−0.732458
$554$	5.03858e6	0.697484
$555$	2.29468e6	0.316220
$556$	6.49357e6	0.890833
$557$	7.38856e6	1.00907	0.504536	−	0.863391i	$-0.331664\pi$
$557$	7.38856e6	1.00907	0.504536	+	0.863391i	$0.331664\pi$
$558$	3.30504e6	0.449356
$559$	1.62340e6	0.219734
$560$	2.48308e6	0.334596
$561$	−8.74927e6	−1.17372
$562$	911286.	0.121707
$563$	1.25576e7	1.66969	0.834843	−	0.550488i	$-0.185558\pi$
$563$	1.25576e7	1.66969	0.834843	+	0.550488i	$0.185558\pi$
$564$	−1.49340e6	−0.197687
$565$	1.75860e7	2.31764
$566$	−9.89282e6	−1.29801
$567$	−830686.	−0.108512
$568$	401981.	0.0522799
$569$	1.02834e7	1.33155	0.665775	−	0.746152i	$-0.268101\pi$
$569$	1.02834e7	1.33155	0.665775	+	0.746152i	$0.268101\pi$
$570$	−995620.	−0.128353
$571$	−1.81737e6	−0.233267	−0.116633	−	0.993175i	$-0.537210\pi$
$571$	−1.81737e6	−0.233267	−0.116633	+	0.993175i	$0.537210\pi$
$572$	1.00558e7	1.28507
$573$	2.77980e6	0.353694
$574$	−3.06422e6	−0.388187
$575$	−5.00354e6	−0.631113
$576$	331776.	0.0416667
$577$	−9.08710e6	−1.13628	−0.568140	−	0.822932i	$-0.692337\pi$
$577$	−9.08710e6	−1.13628	−0.568140	+	0.822932i	$0.692337\pi$
$578$	1.14490e6	0.142544
$579$	−4.57985e6	−0.567747
$580$	6.79038e6	0.838155
$581$	1.14490e6	0.140711
$582$	−5.21649e6	−0.638367
$583$	2.31293e6	0.281833
$584$	5.03738e6	0.611185
$585$	−5.24007e6	−0.633064
$586$	−638863.	−0.0768535
$587$	1.14406e7	1.37042	0.685211	−	0.728344i	$-0.259710\pi$
$587$	1.14406e7	1.37042	0.685211	+	0.728344i	$0.259710\pi$
$588$	111885.	0.0133453
$589$	−3.68246e6	−0.437371
$590$	978900.	0.115773
$591$	7.30516e6	0.860322
$592$	851993.	0.0999152
$593$	−1.10567e7	−1.29119	−0.645596	−	0.763679i	$-0.723391\pi$
$593$	−1.10567e7	−1.29119	−0.645596	+	0.763679i	$0.723391\pi$
$594$	−2.17029e6	−0.252378
$595$	1.26692e7	1.46710
$596$	−298867.	−0.0344637
$597$	6.36858e6	0.731319
$598$	−6.15905e6	−0.704305
$599$	6.40039e6	0.728852	0.364426	−	0.931232i	$-0.381265\pi$
$599$	6.40039e6	0.728852	0.364426	+	0.931232i	$0.381265\pi$
$600$	−1.58057e6	−0.179241
$601$	1.52981e7	1.72764	0.863818	−	0.503804i	$-0.168067\pi$
$601$	1.52981e7	1.72764	0.863818	+	0.503804i	$0.168067\pi$
$602$	−973610.	−0.109495
$603$	1.08329e6	0.121325
$604$	−506221.	−0.0564609
$605$	−3.00987e7	−3.34318
$606$	3.09903e6	0.342803
$607$	241645.	0.0266199	0.0133100	−	0.999911i	$-0.495763\pi$
$607$	241645.	0.0266199	0.0133100	+	0.999911i	$0.495763\pi$
$608$	−369664.	−0.0405554
$609$	−6.31248e6	−0.689694
$610$	−5.35521e6	−0.582709
$611$	8.75755e6	0.949029
$612$	1.69280e6	0.182695
$613$	−883791.	−0.0949945	−0.0474973	−	0.998871i	$-0.515125\pi$
$613$	−883791.	−0.0949945	−0.0474973	+	0.998871i	$0.515125\pi$
$614$	2.88989e6	0.309358
$615$	4.17177e6	0.444767
$616$	−6.03082e6	−0.640361
$617$	−1.03688e7	−1.09652	−0.548260	−	0.836308i	$-0.684710\pi$
$617$	−1.03688e7	−1.09652	−0.548260	+	0.836308i	$0.684710\pi$
$618$	7.00376e6	0.737667
$619$	−1.27847e7	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$619$	−1.27847e7	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$620$	−1.25036e7	−1.30634
$621$	1.32927e6	0.138320
$622$	2.27392e6	0.235667
$623$	−498255.	−0.0514318
$624$	−1.94559e6	−0.200027
$625$	−1.08110e7	−1.10704
$626$	−159608.	−0.0162787
$627$	2.41813e6	0.245646
$628$	−125080.	−0.0126558
$629$	4.34706e6	0.438096
$630$	3.14265e6	0.315460
$631$	−1.69339e7	−1.69311	−0.846554	−	0.532304i	$-0.821326\pi$
$631$	−1.69339e7	−1.69311	−0.846554	+	0.532304i	$0.821326\pi$
$632$	2.66262e6	0.265165
$633$	−6.80524e6	−0.675048
$634$	9.70237e6	0.958638
$635$	1.69073e7	1.66395
$636$	−447503.	−0.0438685
$637$	−656112.	−0.0640663
$638$	−1.64922e7	−1.60409
$639$	508758.	0.0492900
$640$	−1.25517e6	−0.121131
$641$	4.09564e6	0.393711	0.196855	−	0.980433i	$-0.436927\pi$
$641$	4.09564e6	0.393711	0.196855	+	0.980433i	$0.436927\pi$
$642$	7.73990e6	0.741136
$643$	−6.06799e6	−0.578785	−0.289393	−	0.957211i	$-0.593453\pi$
$643$	−6.06799e6	−0.578785	−0.289393	+	0.957211i	$0.593453\pi$
$644$	3.69379e6	0.350960
$645$	1.32551e6	0.125454
$646$	−1.88611e6	−0.177822
$647$	−2.00355e7	−1.88165	−0.940825	−	0.338894i	$-0.889947\pi$
$647$	−2.00355e7	−1.88165	−0.940825	+	0.338894i	$0.889947\pi$
$648$	419904.	0.0392837
$649$	−2.37752e6	−0.221571
$650$	9.26873e6	0.860472
$651$	1.16236e7	1.07495
$652$	−3.92631e6	−0.361714
$653$	−441926.	−0.0405571	−0.0202785	−	0.999794i	$-0.506455\pi$
$653$	−441926.	−0.0405571	−0.0202785	+	0.999794i	$0.506455\pi$
$654$	−5.16987e6	−0.472645
$655$	−2.08325e7	−1.89731
$656$	1.54894e6	0.140532
$657$	6.37543e6	0.576230
$658$	−5.25220e6	−0.472908
$659$	−2.53352e6	−0.227253	−0.113627	−	0.993524i	$-0.536247\pi$
$659$	−2.53352e6	−0.227253	−0.113627	+	0.993524i	$0.536247\pi$
$660$	8.21062e6	0.733696
$661$	−6.52457e6	−0.580829	−0.290414	−	0.956901i	$-0.593793\pi$
$661$	−6.52457e6	−0.580829	−0.290414	+	0.956901i	$0.593793\pi$
$662$	−349554.	−0.0310005
$663$	−9.92683e6	−0.877055
$664$	−578736.	−0.0509401
$665$	−3.50153e6	−0.307046
$666$	1.07830e6	0.0942010
$667$	1.01013e7	0.879146
$668$	−1.10016e7	−0.953929
$669$	3.78206e6	0.326711
$670$	−4.09830e6	−0.352709
$671$	1.30065e7	1.11521
$672$	1.16684e6	0.0996750
$673$	2.45884e6	0.209263	0.104631	−	0.994511i	$-0.466634\pi$
$673$	2.45884e6	0.209263	0.104631	+	0.994511i	$0.466634\pi$
$674$	9.02184e6	0.764972
$675$	−2.00041e6	−0.168990
$676$	5.46854e6	0.460262
$677$	−2.06428e7	−1.73100	−0.865500	−	0.500909i	$-0.832999\pi$
$677$	−2.06428e7	−1.73100	−0.865500	+	0.500909i	$0.832999\pi$
$678$	8.26393e6	0.690418
$679$	−1.83461e7	−1.52710
$680$	−6.40418e6	−0.531119
$681$	−8.26851e6	−0.683218
$682$	3.03683e7	2.50011
$683$	−1.33015e7	−1.09106	−0.545532	−	0.838090i	$-0.683672\pi$
$683$	−1.33015e7	−1.09106	−0.545532	+	0.838090i	$0.683672\pi$
$684$	−467856.	−0.0382360
$685$	−2.31947e7	−1.88870
$686$	8.90521e6	0.722493
$687$	−2.75525e6	−0.222725
$688$	492150.	0.0396394
$689$	2.62423e6	0.210598
$690$	−5.02888e6	−0.402114
$691$	3.88585e6	0.309592	0.154796	−	0.987946i	$-0.450528\pi$
$691$	3.88585e6	0.309592	0.154796	+	0.987946i	$0.450528\pi$
$692$	7.60151e6	0.603441
$693$	−7.63276e6	−0.603738
$694$	−9.22400e6	−0.726977
$695$	−3.10919e7	−2.44166
$696$	3.19090e6	0.249683
$697$	7.90303e6	0.616186
$698$	1.42905e7	1.11022
$699$	3.54680e6	0.274564
$700$	−5.55877e6	−0.428779
$701$	−6.00468e6	−0.461524	−0.230762	−	0.973010i	$-0.574122\pi$
$701$	−6.00468e6	−0.461524	−0.230762	+	0.973010i	$0.574122\pi$
$702$	−2.46238e6	−0.188588
$703$	−1.20144e6	−0.0916885
$704$	3.04852e6	0.231824
$705$	7.15057e6	0.541836
$706$	−4.45174e6	−0.336138
$707$	1.08991e7	0.820054
$708$	459999.	0.0344885
$709$	1.88042e7	1.40488	0.702442	−	0.711741i	$-0.252093\pi$
$709$	1.88042e7	1.40488	0.702442	+	0.711741i	$0.252093\pi$
$710$	−1.92473e6	−0.143293
$711$	3.36987e6	0.250000
$712$	251863.	0.0186194
$713$	−1.86002e7	−1.37023
$714$	5.95346e6	0.437043
$715$	−4.81484e7	−3.52222
$716$	−2.42745e6	−0.176957
$717$	1.29656e7	0.941879
$718$	−1.37670e7	−0.996616
$719$	2.03339e7	1.46689	0.733446	−	0.679748i	$-0.237911\pi$
$719$	2.03339e7	1.46689	0.733446	+	0.679748i	$0.237911\pi$
$720$	−1.58858e6	−0.114203
$721$	2.46318e7	1.76465
$722$	521284.	0.0372161
$723$	2.28306e6	0.162432
$724$	2.21612e6	0.157126
$725$	−1.52013e7	−1.07408
$726$	−1.41438e7	−0.995922
$727$	2.53227e7	1.77694	0.888471	−	0.458933i	$-0.151768\pi$
$727$	2.53227e7	1.77694	0.888471	+	0.458933i	$0.151768\pi$
$728$	−6.84251e6	−0.478505
$729$	531441.	0.0370370
$730$	−2.41195e7	−1.67518
$731$	2.51106e6	0.173806
$732$	−2.51649e6	−0.173587
$733$	−2.71535e6	−0.186666	−0.0933331	−	0.995635i	$-0.529752\pi$
$733$	−2.71535e6	−0.186666	−0.0933331	+	0.995635i	$0.529752\pi$
$734$	−5.84429e6	−0.400398
$735$	−535718.	−0.0365778
$736$	−1.86718e6	−0.127055
$737$	9.95381e6	0.675026
$738$	1.96037e6	0.132494
$739$	−2.19660e7	−1.47958	−0.739792	−	0.672835i	$-0.765076\pi$
$739$	−2.19660e7	−1.47958	−0.739792	+	0.672835i	$0.765076\pi$
$740$	−4.07943e6	−0.273855
$741$	2.74358e6	0.183558
$742$	−1.57384e6	−0.104942
$743$	−3.22060e6	−0.214025	−0.107013	−	0.994258i	$-0.534129\pi$
$743$	−3.22060e6	−0.214025	−0.107013	+	0.994258i	$0.534129\pi$
$744$	−5.87562e6	−0.389154
$745$	1.43101e6	0.0944606
$746$	−1.05062e7	−0.691189
$747$	−732462.	−0.0480268
$748$	1.55543e7	1.01647
$749$	2.72207e7	1.77295
$750$	−1.05064e6	−0.0682029
$751$	1.03346e7	0.668643	0.334322	−	0.942459i	$-0.391493\pi$
$751$	1.03346e7	0.668643	0.334322	+	0.942459i	$0.391493\pi$
$752$	2.65494e6	0.171202
$753$	−9.69681e6	−0.623220
$754$	−1.87119e7	−1.19864
$755$	2.42384e6	0.154752
$756$	1.47678e6	0.0939745
$757$	−189369.	−0.0120107	−0.00600536	−	0.999982i	$-0.501912\pi$
$757$	−189369.	−0.0120107	−0.00600536	+	0.999982i	$0.501912\pi$
$758$	1.20966e6	0.0764699
$759$	1.22140e7	0.769578
$760$	1.76999e6	0.111157
$761$	2.13628e7	1.33720	0.668601	−	0.743621i	$-0.266893\pi$
$761$	2.13628e7	1.33720	0.668601	+	0.743621i	$0.266893\pi$
$762$	7.94499e6	0.495685
$763$	−1.81821e7	−1.13066
$764$	−4.94187e6	−0.306308
$765$	−8.10530e6	−0.500743
$766$	−647622.	−0.0398795
$767$	−2.69751e6	−0.165567
$768$	−589824.	−0.0360844
$769$	4.12263e6	0.251396	0.125698	−	0.992069i	$-0.459883\pi$
$769$	4.12263e6	0.251396	0.125698	+	0.992069i	$0.459883\pi$
$770$	2.88762e7	1.75515
$771$	1.49176e7	0.903778
$772$	8.14196e6	0.491684
$773$	2.63212e6	0.158437	0.0792185	−	0.996857i	$-0.474758\pi$
$773$	2.63212e6	0.158437	0.0792185	+	0.996857i	$0.474758\pi$
$774$	622878.	0.0373724
$775$	2.79913e7	1.67405
$776$	9.27376e6	0.552842
$777$	3.79232e6	0.225348
$778$	1.22052e7	0.722927
$779$	−2.18424e6	−0.128961
$780$	9.31568e6	0.548249
$781$	4.67472e6	0.274238
$782$	−9.52676e6	−0.557094
$783$	4.03848e6	0.235404
$784$	−198907.	−0.0115574
$785$	598898.	0.0346880
$786$	−9.78947e6	−0.565201
$787$	−9.05359e6	−0.521055	−0.260528	−	0.965466i	$-0.583897\pi$
$787$	−9.05359e6	−0.521055	−0.260528	+	0.965466i	$0.583897\pi$
$788$	−1.29870e7	−0.745061
$789$	4.27575e6	0.244523
$790$	−1.27489e7	−0.726783
$791$	2.90637e7	1.65162
$792$	3.85829e6	0.218565
$793$	1.47571e7	0.833332
$794$	1.03659e7	0.583518
$795$	2.14269e6	0.120238
$796$	−1.13219e7	−0.633341
$797$	−1.36889e7	−0.763349	−0.381675	−	0.924297i	$-0.624653\pi$
$797$	−1.36889e7	−0.763349	−0.381675	+	0.924297i	$0.624653\pi$
$798$	−1.64542e6	−0.0914681
$799$	1.35461e7	0.750667
$800$	2.80991e6	0.155227
$801$	318764.	0.0175545
$802$	1.56422e7	0.858740
$803$	5.85806e7	3.20601
$804$	−1.92585e6	−0.105071
$805$	−1.76863e7	−0.961937
$806$	3.44556e7	1.86819
$807$	−1.03404e7	−0.558927
$808$	−5.50939e6	−0.296876
$809$	−2.60118e7	−1.39733	−0.698666	−	0.715448i	$-0.746222\pi$
$809$	−2.60118e7	−1.39733	−0.698666	+	0.715448i	$0.746222\pi$
$810$	−2.01055e6	−0.107672
$811$	2.60995e7	1.39341	0.696706	−	0.717356i	$-0.254648\pi$
$811$	2.60995e7	1.39341	0.696706	+	0.717356i	$0.254648\pi$
$812$	1.12222e7	0.597293
$813$	7.60453e6	0.403502
$814$	9.90799e6	0.524112
$815$	1.87996e7	0.991412
$816$	−3.00942e6	−0.158218
$817$	−694009.	−0.0363756
$818$	−1.48449e7	−0.775699
$819$	−8.66005e6	−0.451139
$820$	−7.41647e6	−0.385179
$821$	1.26552e7	0.655255	0.327627	−	0.944807i	$-0.393751\pi$
$821$	1.26552e7	0.655255	0.327627	+	0.944807i	$0.393751\pi$
$822$	−1.08995e7	−0.562636
$823$	8.61575e6	0.443398	0.221699	−	0.975115i	$-0.428840\pi$
$823$	8.61575e6	0.443398	0.221699	+	0.975115i	$0.428840\pi$
$824$	−1.24511e7	−0.638838
$825$	−1.83808e7	−0.940218
$826$	1.61779e6	0.0825033
$827$	2.37393e7	1.20699	0.603495	−	0.797367i	$-0.293774\pi$
$827$	2.37393e7	1.20699	0.603495	+	0.797367i	$0.293774\pi$
$828$	−2.36314e6	−0.119788
$829$	−1.01922e7	−0.515087	−0.257544	−	0.966267i	$-0.582913\pi$
$829$	−1.01922e7	−0.515087	−0.257544	+	0.966267i	$0.582913\pi$
$830$	2.77105e6	0.139620
$831$	−1.13368e7	−0.569493
$832$	3.45882e6	0.173229
$833$	−1.01487e6	−0.0506754
$834$	−1.46105e7	−0.727362
$835$	5.26770e7	2.61460
$836$	−4.29889e6	−0.212736
$837$	−7.43633e6	−0.366898
$838$	−1.15173e7	−0.566552
$839$	1.57496e7	0.772438	0.386219	−	0.922407i	$-0.373781\pi$
$839$	1.57496e7	0.772438	0.386219	+	0.922407i	$0.373781\pi$
$840$	−5.58693e6	−0.273197
$841$	1.01777e7	0.496204
$842$	−2.67096e7	−1.29834
$843$	−2.05039e6	−0.0993730
$844$	1.20982e7	0.584608
$845$	−2.61840e7	−1.26152
$846$	3.36015e6	0.161411
$847$	−4.97429e7	−2.38244
$848$	795561.	0.0379913
$849$	2.22588e7	1.05982
$850$	1.43368e7	0.680620
$851$	−6.06850e6	−0.287248
$852$	−904458.	−0.0426864
$853$	−3.51443e7	−1.65380	−0.826899	−	0.562350i	$-0.809897\pi$
$853$	−3.51443e7	−1.65380	−0.826899	+	0.562350i	$0.809897\pi$
$854$	−8.85034e6	−0.415255
$855$	2.24014e6	0.104800
$856$	−1.37598e7	−0.641842
$857$	−7.76419e6	−0.361114	−0.180557	−	0.983565i	$-0.557790\pi$
$857$	−7.76419e6	−0.361114	−0.180557	+	0.983565i	$0.557790\pi$
$858$	−2.26256e7	−1.04926
$859$	3.50287e7	1.61973	0.809863	−	0.586619i	$-0.199541\pi$
$859$	3.50287e7	1.61973	0.809863	+	0.586619i	$0.199541\pi$
$860$	−2.35647e6	−0.108646
$861$	6.89451e6	0.316953
$862$	4.16069e6	0.190721
$863$	−1.31015e7	−0.598818	−0.299409	−	0.954125i	$-0.596790\pi$
$863$	−1.31015e7	−0.598818	−0.299409	+	0.954125i	$0.596790\pi$
$864$	−746496.	−0.0340207
$865$	−3.63968e7	−1.65395
$866$	−3.69936e6	−0.167622
$867$	−2.57603e6	−0.116387
$868$	−2.06642e7	−0.930934
$869$	3.09641e7	1.39094
$870$	−1.52784e7	−0.684350
$871$	1.12935e7	0.504409
$872$	9.19088e6	0.409323
$873$	1.17371e7	0.521225
$874$	2.63301e6	0.116593
$875$	−3.69505e6	−0.163155
$876$	−1.13341e7	−0.499030
$877$	−8.96391e6	−0.393549	−0.196774	−	0.980449i	$-0.563047\pi$
$877$	−8.96391e6	−0.393549	−0.196774	+	0.980449i	$0.563047\pi$
$878$	2.75607e6	0.120658
$879$	1.43744e6	0.0627506
$880$	−1.45966e7	−0.635399
$881$	−1.15038e7	−0.499347	−0.249673	−	0.968330i	$-0.580323\pi$
$881$	−1.15038e7	−0.499347	−0.249673	+	0.968330i	$0.580323\pi$
$882$	−251741.	−0.0108964
$883$	5.08324e6	0.219401	0.109701	−	0.993965i	$-0.465011\pi$
$883$	5.08324e6	0.219401	0.109701	+	0.993965i	$0.465011\pi$
$884$	1.76477e7	0.759552
$885$	−2.20253e6	−0.0945285
$886$	2.88551e6	0.123492
$887$	−2.73348e7	−1.16656	−0.583280	−	0.812271i	$-0.698231\pi$
$887$	−2.73348e7	−1.16656	−0.583280	+	0.812271i	$0.698231\pi$
$888$	−1.91698e6	−0.0815804
$889$	2.79420e7	1.18578
$890$	−1.20595e6	−0.0510333
$891$	4.88314e6	0.206065
$892$	−6.72367e6	−0.282940
$893$	−3.74388e6	−0.157106
$894$	672450.	0.0281395
$895$	1.16229e7	0.485017
$896$	−2.07437e6	−0.0863211
$897$	1.38579e7	0.575062
$898$	2.47646e7	1.02480
$899$	−5.65095e7	−2.33197
$900$	3.55629e6	0.146349
$901$	4.05913e6	0.166579
$902$	1.80129e7	0.737168
$903$	2.19062e6	0.0894022
$904$	−1.46914e7	−0.597920
$905$	−1.06110e7	−0.430661
$906$	1.13900e6	0.0461001
$907$	4.01778e7	1.62169	0.810846	−	0.585260i	$-0.199007\pi$
$907$	4.01778e7	1.62169	0.810846	+	0.585260i	$0.199007\pi$
$908$	1.46996e7	0.591684
$909$	−6.97282e6	−0.279897
$910$	3.27627e7	1.31152
$911$	1.93885e7	0.774014	0.387007	−	0.922077i	$-0.373509\pi$
$911$	1.93885e7	0.774014	0.387007	+	0.922077i	$0.373509\pi$
$912$	831744.	0.0331133
$913$	−6.73022e6	−0.267210
$914$	−1.57470e7	−0.623493
$915$	1.20492e7	0.475780
$916$	4.89823e6	0.192886
$917$	−3.44289e7	−1.35207
$918$	−3.80879e6	−0.149170
$919$	−3.38001e7	−1.32017	−0.660085	−	0.751191i	$-0.729480\pi$
$919$	−3.38001e7	−1.32017	−0.660085	+	0.751191i	$0.729480\pi$
$920$	8.94024e6	0.348241
$921$	−6.50226e6	−0.252590
$922$	6.52674e6	0.252854
$923$	5.30389e6	0.204923
$924$	1.35693e7	0.522852
$925$	9.13246e6	0.350940
$926$	2.92324e7	1.12031
$927$	−1.57585e7	−0.602302
$928$	−5.67271e6	−0.216232
$929$	−3.88357e6	−0.147636	−0.0738179	−	0.997272i	$-0.523518\pi$
$929$	−3.88357e6	−0.147636	−0.0738179	+	0.997272i	$0.523518\pi$
$930$	2.81331e7	1.06662
$931$	280490.	0.0106058
$932$	−6.30542e6	−0.237779
$933$	−5.11632e6	−0.192422
$934$	−2.27143e6	−0.0851984
$935$	−7.44755e7	−2.78602
$936$	4.37757e6	0.163322
$937$	2.61871e7	0.974402	0.487201	−	0.873290i	$-0.338018\pi$
$937$	2.61871e7	0.974402	0.487201	+	0.873290i	$0.338018\pi$
$938$	−6.77309e6	−0.251351
$939$	359118.	0.0132915
$940$	−1.27121e7	−0.469244
$941$	−225612.	−0.00830591	−0.00415296	−	0.999991i	$-0.501322\pi$
$941$	−225612.	−0.00830591	−0.00415296	+	0.999991i	$0.501322\pi$
$942$	281431.	0.0103334
$943$	−1.10326e7	−0.404017
$944$	−817776.	−0.0298679
$945$	−7.07096e6	−0.257572
$946$	5.72331e6	0.207931
$947$	3.64108e6	0.131934	0.0659669	−	0.997822i	$-0.478987\pi$
$947$	3.64108e6	0.131934	0.0659669	+	0.997822i	$0.478987\pi$
$948$	−5.99089e6	−0.216506
$949$	6.64650e7	2.39567
$950$	−3.96241e6	−0.142446
$951$	−2.18303e7	−0.782724
$952$	−1.05839e7	−0.378490
$953$	2.56251e7	0.913972	0.456986	−	0.889474i	$-0.348929\pi$
$953$	2.56251e7	0.913972	0.456986	+	0.889474i	$0.348929\pi$
$954$	1.00688e6	0.0358185
$955$	2.36622e7	0.839551
$956$	−2.30500e7	−0.815691
$957$	3.71075e7	1.30973
$958$	−2.81612e6	−0.0991373
$959$	−3.83329e7	−1.34594
$960$	2.82414e6	0.0989027
$961$	7.54257e7	2.63458
$962$	1.12415e7	0.391640
$963$	−1.74148e7	−0.605135
$964$	−4.05877e6	−0.140670
$965$	−3.89846e7	−1.34764
$966$	−8.31103e6	−0.286558
$967$	−1.66891e7	−0.573941	−0.286971	−	0.957939i	$-0.592648\pi$
$967$	−1.66891e7	−0.573941	−0.286971	+	0.957939i	$0.592648\pi$
$968$	2.51446e7	0.862493
$969$	4.24375e6	0.145191
$970$	−4.44037e7	−1.51527
$971$	−3.65408e6	−0.124374	−0.0621870	−	0.998065i	$-0.519808\pi$
$971$	−3.65408e6	−0.124374	−0.0621870	+	0.998065i	$0.519808\pi$
$972$	−944784.	−0.0320750
$973$	−5.13843e7	−1.74000
$974$	4.21755e6	0.142450
$975$	−2.08546e7	−0.702572
$976$	4.47376e6	0.150331
$977$	1.05458e6	0.0353463	0.0176732	−	0.999844i	$-0.494374\pi$
$977$	1.05458e6	0.0353463	0.0176732	+	0.999844i	$0.494374\pi$
$978$	8.83419e6	0.295338
$979$	2.92897e6	0.0976692
$980$	952387.	0.0316773
$981$	1.16322e7	0.385913
$982$	1.83417e7	0.606960
$983$	−3.53385e7	−1.16644	−0.583222	−	0.812313i	$-0.698208\pi$
$983$	−3.53385e7	−1.16644	−0.583222	+	0.812313i	$0.698208\pi$
$984$	−3.48511e6	−0.114744
$985$	6.21829e7	2.04212
$986$	−2.89435e7	−0.948108
$987$	1.18175e7	0.386128
$988$	−4.87748e6	−0.158966
$989$	−3.50544e6	−0.113960
$990$	−1.84739e7	−0.599060
$991$	−1.88975e7	−0.611252	−0.305626	−	0.952152i	$-0.598866\pi$
$991$	−1.88975e7	−0.611252	−0.305626	+	0.952152i	$0.598866\pi$
$992$	1.04455e7	0.337017
$993$	786496.	0.0253118
$994$	−3.18092e6	−0.102114
$995$	5.42106e7	1.73591
$996$	1.30216e6	0.0415924
$997$	−4.84709e6	−0.154434	−0.0772170	−	0.997014i	$-0.524603\pi$
$997$	−4.84709e6	−0.154434	−0.0772170	+	0.997014i	$0.524603\pi$
$998$	−1.88578e7	−0.599328
$999$	−2.42618e6	−0.0769148

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.6.a.g.1.1	✓	2
3.2	odd	2		342.6.a.g.1.2		2
4.3	odd	2		912.6.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.6.a.g.1.1	✓	2	1.1	even	1	trivial
342.6.a.g.1.2		2	3.2	odd	2
912.6.a.j.1.1		2	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	114.6.a.g.1.1

Level	$114$
Weight	$6$
Character	114.1
Self dual	yes
Analytic conductor	$18.284$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$