LMFDB - Embedded newform 722.6.a.r.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,6,Mod(1,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$722 = 2 \cdot 19^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	722.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:	$115.797117905$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} + \cdots - 34\!\cdots\!72$ x^15 - 2871x^13 - 4674x^12 + 3170019x^11 + 9081402x^10 - 1680307373x^9 - 6060225486x^8 + 437045334939x^7 + 1625551725606x^6 - 51993847360191x^5 - 166593470925336x^4 + 2435236025483701x^3 + 5743342864926480x^2 - 19853455953893436*x - 34308015556724472
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{6}\cdot 3\cdot 19^{6}$
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-10.7498$ of defining polynomial
Character	$\chi$	$=$	722.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +10.4025 q^{3} +16.0000 q^{4} +95.5276 q^{5} +41.6100 q^{6} +232.219 q^{7} +64.0000 q^{8} -134.788 q^{9} +382.111 q^{10} +345.880 q^{11} +166.440 q^{12} +506.327 q^{13} +928.874 q^{14} +993.725 q^{15} +256.000 q^{16} -797.089 q^{17} -539.153 q^{18} +1528.44 q^{20} +2415.65 q^{21} +1383.52 q^{22} -1190.67 q^{23} +665.760 q^{24} +6000.53 q^{25} +2025.31 q^{26} -3929.94 q^{27} +3715.50 q^{28} -425.364 q^{29} +3974.90 q^{30} +6647.85 q^{31} +1024.00 q^{32} +3598.01 q^{33} -3188.36 q^{34} +22183.3 q^{35} -2156.61 q^{36} -7850.99 q^{37} +5267.06 q^{39} +6113.77 q^{40} -16080.1 q^{41} +9662.61 q^{42} +12922.4 q^{43} +5534.07 q^{44} -12876.0 q^{45} -4762.67 q^{46} -21190.2 q^{47} +2663.04 q^{48} +37118.5 q^{49} +24002.1 q^{50} -8291.71 q^{51} +8101.23 q^{52} -29091.8 q^{53} -15719.8 q^{54} +33041.1 q^{55} +14862.0 q^{56} -1701.45 q^{58} +9346.25 q^{59} +15899.6 q^{60} -8320.98 q^{61} +26591.4 q^{62} -31300.3 q^{63} +4096.00 q^{64} +48368.2 q^{65} +14392.0 q^{66} -14347.1 q^{67} -12753.4 q^{68} -12385.9 q^{69} +88733.1 q^{70} -31200.2 q^{71} -8626.44 q^{72} +1992.28 q^{73} -31404.0 q^{74} +62420.4 q^{75} +80319.7 q^{77} +21068.2 q^{78} -9623.27 q^{79} +24455.1 q^{80} -8127.64 q^{81} -64320.3 q^{82} -34179.5 q^{83} +38650.4 q^{84} -76144.0 q^{85} +51689.4 q^{86} -4424.84 q^{87} +22136.3 q^{88} -111322. q^{89} -51504.0 q^{90} +117578. q^{91} -19050.7 q^{92} +69154.2 q^{93} -84760.6 q^{94} +10652.2 q^{96} +96994.8 q^{97} +148474. q^{98} -46620.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$15 q + 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} + 960 q^{8} + 2127 q^{9} + 432 q^{10} + 126 q^{11} - 114 q^{13} + 336 q^{14} + 3840 q^{16} + 4119 q^{17} + 8508 q^{18} + 1728 q^{20} + 3408 q^{21} + 504 q^{22}+ \cdots - 149895 q^{99}+O(q^{100})$ 15 * q + 60 * q^2 + 240 * q^4 + 108 * q^5 + 84 * q^7 + 960 * q^8 + 2127 * q^9 + 432 * q^10 + 126 * q^11 - 114 * q^13 + 336 * q^14 + 3840 * q^16 + 4119 * q^17 + 8508 * q^18 + 1728 * q^20 + 3408 * q^21 + 504 * q^22 + 3936 * q^23 + 26895 * q^25 - 456 * q^26 - 13017 * q^27 + 1344 * q^28 + 14658 * q^29 + 6840 * q^31 + 15360 * q^32 - 3945 * q^33 + 16476 * q^34 + 12636 * q^35 + 34032 * q^36 - 4278 * q^37 + 4956 * q^39 + 6912 * q^40 + 5112 * q^41 + 13632 * q^42 + 94191 * q^43 + 2016 * q^44 + 31770 * q^45 + 15744 * q^46 + 702 * q^47 + 63777 * q^49 + 107580 * q^50 - 108 * q^51 - 1824 * q^52 + 47544 * q^53 - 52068 * q^54 + 16848 * q^55 + 5376 * q^56 + 58632 * q^58 - 8832 * q^59 + 119196 * q^61 + 27360 * q^62 - 88068 * q^63 + 61440 * q^64 + 80646 * q^65 - 15780 * q^66 + 64248 * q^67 + 65904 * q^68 + 124224 * q^69 + 50544 * q^70 - 53364 * q^71 + 136128 * q^72 - 4908 * q^73 - 17112 * q^74 - 87480 * q^75 + 121218 * q^77 + 19824 * q^78 - 115500 * q^79 + 27648 * q^80 + 481659 * q^81 + 20448 * q^82 + 201630 * q^83 + 54528 * q^84 - 150282 * q^85 + 376764 * q^86 + 376512 * q^87 + 8064 * q^88 - 101505 * q^89 + 127080 * q^90 + 414918 * q^91 + 62976 * q^92 + 165960 * q^93 + 2808 * q^94 + 297114 * q^97 + 255108 * q^98 - 149895 * 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
$3$	10.4025	0.667320	0.333660	−	0.942693i	$-0.391716\pi$
$3$	10.4025	0.667320	0.333660	+	0.942693i	$0.391716\pi$
$4$	16.0000	0.500000
$5$	95.5276	1.70885	0.854425	−	0.519575i	$-0.173910\pi$
$5$	95.5276	1.70885	0.854425	+	0.519575i	$0.173910\pi$
$6$	41.6100	0.471867
$7$	232.219	1.79123	0.895616	−	0.444828i	$-0.146735\pi$
$7$	232.219	1.79123	0.895616	+	0.444828i	$0.146735\pi$
$8$	64.0000	0.353553
$9$	−134.788	−0.554684
$10$	382.111	1.20834
$11$	345.880	0.861873	0.430936	−	0.902382i	$-0.358183\pi$
$11$	345.880	0.861873	0.430936	+	0.902382i	$0.358183\pi$
$12$	166.440	0.333660
$13$	506.327	0.830946	0.415473	−	0.909606i	$-0.363616\pi$
$13$	506.327	0.830946	0.415473	+	0.909606i	$0.363616\pi$
$14$	928.874	1.26659
$15$	993.725	1.14035
$16$	256.000	0.250000
$17$	−797.089	−0.668936	−0.334468	−	0.942407i	$-0.608557\pi$
$17$	−797.089	−0.668936	−0.334468	+	0.942407i	$0.608557\pi$
$18$	−539.153	−0.392221
$19$	0	0
$19$	0	0
$20$	1528.44	0.854425
$21$	2415.65	1.19533
$22$	1383.52	0.609436
$23$	−1190.67	−0.469322	−0.234661	−	0.972077i	$-0.575398\pi$
$23$	−1190.67	−0.469322	−0.234661	+	0.972077i	$0.575398\pi$
$24$	665.760	0.235933
$25$	6000.53	1.92017
$26$	2025.31	0.587567
$27$	−3929.94	−1.03747
$28$	3715.50	0.895616
$29$	−425.364	−0.0939216	−0.0469608	−	0.998897i	$-0.514954\pi$
$29$	−425.364	−0.0939216	−0.0469608	+	0.998897i	$0.514954\pi$
$30$	3974.90	0.806349
$31$	6647.85	1.24244	0.621222	−	0.783635i	$-0.286637\pi$
$31$	6647.85	1.24244	0.621222	+	0.783635i	$0.286637\pi$
$32$	1024.00	0.176777
$33$	3598.01	0.575145
$34$	−3188.36	−0.473009
$35$	22183.3	3.06095
$36$	−2156.61	−0.277342
$37$	−7850.99	−0.942801	−0.471401	−	0.881919i	$-0.656251\pi$
$37$	−7850.99	−0.942801	−0.471401	+	0.881919i	$0.656251\pi$
$38$	0	0
$39$	5267.06	0.554507
$40$	6113.77	0.604170
$41$	−16080.1	−1.49392	−0.746962	−	0.664867i	$-0.768488\pi$
$41$	−16080.1	−1.49392	−0.746962	+	0.664867i	$0.768488\pi$
$42$	9662.61	0.845223
$43$	12922.4	1.06579	0.532894	−	0.846182i	$-0.321105\pi$
$43$	12922.4	1.06579	0.532894	+	0.846182i	$0.321105\pi$
$44$	5534.07	0.430936
$45$	−12876.0	−0.947871
$46$	−4762.67	−0.331861
$47$	−21190.2	−1.39923	−0.699616	−	0.714519i	$-0.746645\pi$
$47$	−21190.2	−1.39923	−0.699616	+	0.714519i	$0.746645\pi$
$48$	2663.04	0.166830
$49$	37118.5	2.20851
$50$	24002.1	1.35776
$51$	−8291.71	−0.446395
$52$	8101.23	0.415473
$53$	−29091.8	−1.42259	−0.711296	−	0.702893i	$-0.751891\pi$
$53$	−29091.8	−1.42259	−0.711296	+	0.702893i	$0.751891\pi$
$54$	−15719.8	−0.733603
$55$	33041.1	1.47281
$56$	14862.0	0.633296
$57$	0	0
$58$	−1701.45	−0.0664126
$59$	9346.25	0.349549	0.174774	−	0.984609i	$-0.444080\pi$
$59$	9346.25	0.349549	0.174774	+	0.984609i	$0.444080\pi$
$60$	15899.6	0.570175
$61$	−8320.98	−0.286319	−0.143159	−	0.989700i	$-0.545726\pi$
$61$	−8320.98	−0.286319	−0.143159	+	0.989700i	$0.545726\pi$
$62$	26591.4	0.878541
$63$	−31300.3	−0.993567
$64$	4096.00	0.125000
$65$	48368.2	1.41996
$66$	14392.0	0.406689
$67$	−14347.1	−0.390459	−0.195230	−	0.980758i	$-0.562545\pi$
$67$	−14347.1	−0.390459	−0.195230	+	0.980758i	$0.562545\pi$
$68$	−12753.4	−0.334468
$69$	−12385.9	−0.313188
$70$	88733.1	2.16442
$71$	−31200.2	−0.734532	−0.367266	−	0.930116i	$-0.619706\pi$
$71$	−31200.2	−0.734532	−0.367266	+	0.930116i	$0.619706\pi$
$72$	−8626.44	−0.196110
$73$	1992.28	0.0437566	0.0218783	−	0.999761i	$-0.493035\pi$
$73$	1992.28	0.0437566	0.0218783	+	0.999761i	$0.493035\pi$
$74$	−31404.0	−0.666661
$75$	62420.4	1.28137
$76$	0	0
$77$	80319.7	1.54381
$78$	21068.2	0.392096
$79$	−9623.27	−0.173482	−0.0867411	−	0.996231i	$-0.527645\pi$
$79$	−9623.27	−0.173482	−0.0867411	+	0.996231i	$0.527645\pi$
$80$	24455.1	0.427213
$81$	−8127.64	−0.137642
$82$	−64320.3	−1.05636
$83$	−34179.5	−0.544590	−0.272295	−	0.962214i	$-0.587783\pi$
$83$	−34179.5	−0.544590	−0.272295	+	0.962214i	$0.587783\pi$
$84$	38650.4	0.597663
$85$	−76144.0	−1.14311
$86$	51689.4	0.753625
$87$	−4424.84	−0.0626758
$88$	22136.3	0.304718
$89$	−111322.	−1.48973	−0.744863	−	0.667218i	$-0.767485\pi$
$89$	−111322.	−1.48973	−0.744863	+	0.667218i	$0.767485\pi$
$90$	−51504.0	−0.670246
$91$	117578.	1.48842
$92$	−19050.7	−0.234661
$93$	69154.2	0.829108
$94$	−84760.6	−0.989406
$95$	0	0
$96$	10652.2	0.117967
$97$	96994.8	1.04669	0.523346	−	0.852120i	$-0.324684\pi$
$97$	96994.8	1.04669	0.523346	+	0.852120i	$0.324684\pi$
$98$	148474.	1.56165
$99$	−46620.5	−0.478067
$100$	96008.4	0.960084
$101$	−170094.	−1.65915	−0.829575	−	0.558396i	$-0.811417\pi$
$101$	−170094.	−1.65915	−0.829575	+	0.558396i	$0.811417\pi$
$102$	−33166.9	−0.315649
$103$	−72570.9	−0.674015	−0.337008	−	0.941502i	$-0.609415\pi$
$103$	−72570.9	−0.674015	−0.337008	+	0.941502i	$0.609415\pi$
$104$	32404.9	0.293784
$105$	230761.	2.04263
$106$	−116367.	−1.00592
$107$	−68021.4	−0.574363	−0.287181	−	0.957876i	$-0.592718\pi$
$107$	−68021.4	−0.574363	−0.287181	+	0.957876i	$0.592718\pi$
$108$	−62879.0	−0.518736
$109$	−19382.6	−0.156259	−0.0781296	−	0.996943i	$-0.524895\pi$
$109$	−19382.6	−0.156259	−0.0781296	+	0.996943i	$0.524895\pi$
$110$	132164.	1.04144
$111$	−81669.9	−0.629150
$112$	59448.0	0.447808
$113$	−128260.	−0.944917	−0.472458	−	0.881353i	$-0.656633\pi$
$113$	−128260.	−0.944917	−0.472458	+	0.881353i	$0.656633\pi$
$114$	0	0
$115$	−113742.	−0.802001
$116$	−6805.82	−0.0469608
$117$	−68246.9	−0.460912
$118$	37385.0	0.247168
$119$	−185099.	−1.19822
$120$	63598.4	0.403175
$121$	−41418.3	−0.257175
$122$	−33283.9	−0.202458
$123$	−167273.	−0.996926
$124$	106366.	0.621222
$125$	274692.	1.57243
$126$	−125201.	−0.702558
$127$	94926.0	0.522247	0.261123	−	0.965305i	$-0.415907\pi$
$127$	94926.0	0.522247	0.261123	+	0.965305i	$0.415907\pi$
$128$	16384.0	0.0883883
$129$	134425.	0.711221
$130$	193473.	1.00406
$131$	265435.	1.35139	0.675695	−	0.737182i	$-0.263844\pi$
$131$	265435.	1.35139	0.675695	+	0.737182i	$0.263844\pi$
$132$	57568.2	0.287573
$133$	0	0
$134$	−57388.2	−0.276096
$135$	−375418.	−1.77288
$136$	−51013.7	−0.236505
$137$	−69622.2	−0.316918	−0.158459	−	0.987366i	$-0.550652\pi$
$137$	−69622.2	−0.316918	−0.158459	+	0.987366i	$0.550652\pi$
$138$	−49543.6	−0.221457
$139$	369479.	1.62201	0.811004	−	0.585041i	$-0.198922\pi$
$139$	369479.	1.62201	0.811004	+	0.585041i	$0.198922\pi$
$140$	354933.	1.53047
$141$	−220430.	−0.933735
$142$	−124801.	−0.519393
$143$	175128.	0.716170
$144$	−34505.8	−0.138671
$145$	−40634.0	−0.160498
$146$	7969.13	0.0309406
$147$	386124.	1.47378
$148$	−125616.	−0.471401
$149$	10845.5	0.0400206	0.0200103	−	0.999800i	$-0.493630\pi$
$149$	10845.5	0.0400206	0.0200103	+	0.999800i	$0.493630\pi$
$150$	249682.	0.906064
$151$	33593.4	0.119898	0.0599489	−	0.998201i	$-0.480906\pi$
$151$	33593.4	0.119898	0.0599489	+	0.998201i	$0.480906\pi$
$152$	0	0
$153$	107438.	0.371048
$154$	321279.	1.09164
$155$	635053.	2.12315
$156$	84273.0	0.277253
$157$	571612.	1.85077	0.925385	−	0.379029i	$-0.123742\pi$
$157$	571612.	1.85077	0.925385	+	0.379029i	$0.123742\pi$
$158$	−38493.1	−0.122670
$159$	−302627.	−0.949324
$160$	97820.3	0.302085
$161$	−276495.	−0.840665
$162$	−32510.5	−0.0973277
$163$	441057.	1.30025	0.650123	−	0.759829i	$-0.274717\pi$
$163$	441057.	1.30025	0.650123	+	0.759829i	$0.274717\pi$
$164$	−257281.	−0.746962
$165$	343709.	0.982837
$166$	−136718.	−0.385084
$167$	−32306.8	−0.0896401	−0.0448200	−	0.998995i	$-0.514271\pi$
$167$	−32306.8	−0.0896401	−0.0448200	+	0.998995i	$0.514271\pi$
$168$	154602.	0.422611
$169$	−114926.	−0.309529
$170$	−304576.	−0.808302
$171$	0	0
$172$	206758.	0.532894
$173$	−140107.	−0.355915	−0.177957	−	0.984038i	$-0.556949\pi$
$173$	−140107.	−0.355915	−0.177957	+	0.984038i	$0.556949\pi$
$174$	−17699.4	−0.0443185
$175$	1.39343e6	3.43947
$176$	88545.2	0.215468
$177$	97224.3	0.233261
$178$	−445288.	−1.05339
$179$	205087.	0.478416	0.239208	−	0.970968i	$-0.423112\pi$
$179$	205087.	0.478416	0.239208	+	0.970968i	$0.423112\pi$
$180$	−206016.	−0.473936
$181$	−596816.	−1.35408	−0.677039	−	0.735947i	$-0.736737\pi$
$181$	−596816.	−1.35408	−0.677039	+	0.735947i	$0.736737\pi$
$182$	470314.	1.05247
$183$	−86558.9	−0.191066
$184$	−76202.7	−0.165930
$185$	−749986.	−1.61111
$186$	276617.	0.586268
$187$	−275697.	−0.576538
$188$	−339042.	−0.699616
$189$	−912605.	−1.85835
$190$	0	0
$191$	374224.	0.742246	0.371123	−	0.928584i	$-0.378973\pi$
$191$	374224.	0.742246	0.371123	+	0.928584i	$0.378973\pi$
$192$	42608.6	0.0834150
$193$	185090.	0.357676	0.178838	−	0.983879i	$-0.442766\pi$
$193$	185090.	0.357676	0.178838	+	0.983879i	$0.442766\pi$
$194$	387979.	0.740123
$195$	503150.	0.947569
$196$	593895.	1.10426
$197$	149519.	0.274493	0.137247	−	0.990537i	$-0.456175\pi$
$197$	149519.	0.274493	0.137247	+	0.990537i	$0.456175\pi$
$198$	−186482.	−0.338044
$199$	−305923.	−0.547619	−0.273810	−	0.961784i	$-0.588284\pi$
$199$	−305923.	−0.547619	−0.273810	+	0.961784i	$0.588284\pi$
$200$	384034.	0.678882
$201$	−149245.	−0.260561
$202$	−680376.	−1.17320
$203$	−98777.3	−0.168235
$204$	−132667.	−0.223197
$205$	−1.53609e6	−2.55289
$206$	−290284.	−0.476601
$207$	160488.	0.260325
$208$	129620.	0.207736
$209$	0	0
$210$	923046.	1.44436
$211$	809738.	1.25210	0.626049	−	0.779784i	$-0.284671\pi$
$211$	809738.	1.25210	0.626049	+	0.779784i	$0.284671\pi$
$212$	−465468.	−0.711296
$213$	−324559.	−0.490168
$214$	−272086.	−0.406136
$215$	1.23444e6	1.82127
$216$	−251516.	−0.366802
$217$	1.54375e6	2.22551
$218$	−77530.4	−0.110492
$219$	20724.7	0.0291997
$220$	528657.	0.736406
$221$	−403588.	−0.555849
$222$	−326679.	−0.444876
$223$	634589.	0.854536	0.427268	−	0.904125i	$-0.359476\pi$
$223$	634589.	0.854536	0.427268	+	0.904125i	$0.359476\pi$
$224$	237792.	0.316648
$225$	−808800.	−1.06509
$226$	−513038.	−0.668157
$227$	−93859.7	−0.120897	−0.0604484	−	0.998171i	$-0.519253\pi$
$227$	−93859.7	−0.120897	−0.0604484	+	0.998171i	$0.519253\pi$
$228$	0	0
$229$	176185.	0.222014	0.111007	−	0.993820i	$-0.464592\pi$
$229$	176185.	0.222014	0.111007	+	0.993820i	$0.464592\pi$
$230$	−454967.	−0.567100
$231$	835525.	1.03022
$232$	−27223.3	−0.0332063
$233$	−226160.	−0.272915	−0.136457	−	0.990646i	$-0.543572\pi$
$233$	−226160.	−0.272915	−0.136457	+	0.990646i	$0.543572\pi$
$234$	−272987.	−0.325914
$235$	−2.02425e6	−2.39108
$236$	149540.	0.174774
$237$	−100106.	−0.115768
$238$	−740395.	−0.847269
$239$	−343996.	−0.389546	−0.194773	−	0.980848i	$-0.562397\pi$
$239$	−343996.	−0.389546	−0.194773	+	0.980848i	$0.562397\pi$
$240$	254394.	0.285088
$241$	1.58686e6	1.75993	0.879966	−	0.475036i	$-0.157565\pi$
$241$	1.58686e6	1.75993	0.879966	+	0.475036i	$0.157565\pi$
$242$	−165673.	−0.181850
$243$	870427.	0.945620
$244$	−133136.	−0.143159
$245$	3.54584e6	3.77402
$246$	−669092.	−0.704933
$247$	0	0
$248$	425462.	0.439270
$249$	−355552.	−0.363416
$250$	1.09877e6	1.11188
$251$	745968.	0.747370	0.373685	−	0.927556i	$-0.378094\pi$
$251$	745968.	0.747370	0.373685	+	0.927556i	$0.378094\pi$
$252$	−500805.	−0.496784
$253$	−411828.	−0.404496
$254$	379704.	0.369284
$255$	−792088.	−0.762821
$256$	65536.0	0.0625000
$257$	1.92630e6	1.81925	0.909623	−	0.415436i	$-0.136371\pi$
$257$	1.92630e6	1.81925	0.909623	+	0.415436i	$0.136371\pi$
$258$	537699.	0.502910
$259$	−1.82315e6	−1.68878
$260$	773891.	0.709981
$261$	57334.0	0.0520968
$262$	1.06174e6	0.955577
$263$	−964619.	−0.859937	−0.429968	−	0.902844i	$-0.641475\pi$
$263$	−964619.	−0.859937	−0.429968	+	0.902844i	$0.641475\pi$
$264$	230273.	0.203345
$265$	−2.77907e6	−2.43100
$266$	0	0
$267$	−1.15803e6	−0.994124
$268$	−229553.	−0.195230
$269$	−1.61062e6	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$269$	−1.61062e6	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$270$	−1.50167e6	−1.25362
$271$	1.13650e6	0.940038	0.470019	−	0.882656i	$-0.344247\pi$
$271$	1.13650e6	0.940038	0.470019	+	0.882656i	$0.344247\pi$
$272$	−204055.	−0.167234
$273$	1.22311e6	0.993250
$274$	−278489.	−0.224095
$275$	2.07546e6	1.65494
$276$	−198175.	−0.156594
$277$	648448.	0.507780	0.253890	−	0.967233i	$-0.418290\pi$
$277$	648448.	0.507780	0.253890	+	0.967233i	$0.418290\pi$
$278$	1.47792e6	1.14693
$279$	−896051.	−0.689163
$280$	1.41973e6	1.08221
$281$	−501822.	−0.379126	−0.189563	−	0.981869i	$-0.560707\pi$
$281$	−501822.	−0.379126	−0.189563	+	0.981869i	$0.560707\pi$
$282$	−881722.	−0.660251
$283$	1.36912e6	1.01619	0.508096	−	0.861301i	$-0.330350\pi$
$283$	1.36912e6	1.01619	0.508096	+	0.861301i	$0.330350\pi$
$284$	−499203.	−0.367266
$285$	0	0
$286$	700513.	0.506408
$287$	−3.73409e6	−2.67596
$288$	−138023.	−0.0980552
$289$	−784506.	−0.552525
$290$	−162536.	−0.113489
$291$	1.00899e6	0.698479
$292$	31876.5	0.0218783
$293$	−684715.	−0.465952	−0.232976	−	0.972483i	$-0.574846\pi$
$293$	−684715.	−0.465952	−0.232976	+	0.972483i	$0.574846\pi$
$294$	1.54450e6	1.04212
$295$	892825.	0.597326
$296$	−502463.	−0.333331
$297$	−1.35929e6	−0.894169
$298$	43382.0	0.0282988
$299$	−602867.	−0.389981
$300$	998727.	0.640684
$301$	3.00081e6	1.90907
$302$	134373.	0.0847805
$303$	−1.76940e6	−1.10718
$304$	0	0
$305$	−794883.	−0.489276
$306$	429753.	0.262370
$307$	−1.43283e6	−0.867656	−0.433828	−	0.900996i	$-0.642837\pi$
$307$	−1.43283e6	−0.867656	−0.433828	+	0.900996i	$0.642837\pi$
$308$	1.28511e6	0.771907
$309$	−754919.	−0.449784
$310$	2.54021e6	1.50129
$311$	−286576.	−0.168012	−0.0840058	−	0.996465i	$-0.526771\pi$
$311$	−286576.	−0.168012	−0.0840058	+	0.996465i	$0.526771\pi$
$312$	337092.	0.196048
$313$	326263.	0.188238	0.0941189	−	0.995561i	$-0.469997\pi$
$313$	326263.	0.188238	0.0941189	+	0.995561i	$0.469997\pi$
$314$	2.28645e6	1.30869
$315$	−2.99004e6	−1.69786
$316$	−153972.	−0.0867411
$317$	3.04382e6	1.70126	0.850631	−	0.525764i	$-0.176220\pi$
$317$	3.04382e6	1.70126	0.850631	+	0.525764i	$0.176220\pi$
$318$	−1.21051e6	−0.671274
$319$	−147125.	−0.0809485
$320$	391281.	0.213606
$321$	−707592.	−0.383284
$322$	−1.10598e6	−0.594440
$323$	0	0
$324$	−130042.	−0.0688211
$325$	3.03823e6	1.59556
$326$	1.76423e6	0.919412
$327$	−201627.	−0.104275
$328$	−1.02912e6	−0.528182
$329$	−4.92075e6	−2.50635
$330$	1.37484e6	0.694971
$331$	−2.26302e6	−1.13532	−0.567660	−	0.823263i	$-0.692151\pi$
$331$	−2.26302e6	−1.13532	−0.567660	+	0.823263i	$0.692151\pi$
$332$	−546871.	−0.272295
$333$	1.05822e6	0.522956
$334$	−129227.	−0.0633851
$335$	−1.37054e6	−0.667237
$336$	618407.	0.298831
$337$	2.34013e6	1.12245	0.561223	−	0.827665i	$-0.310331\pi$
$337$	2.34013e6	1.12245	0.561223	+	0.827665i	$0.310331\pi$
$338$	−459704.	−0.218870
$339$	−1.33422e6	−0.630562
$340$	−1.21830e6	−0.571556
$341$	2.29936e6	1.07083
$342$	0	0
$343$	4.71670e6	2.16472
$344$	827031.	0.376813
$345$	−1.18320e6	−0.535191
$346$	−560430.	−0.251670
$347$	−1.61523e6	−0.720128	−0.360064	−	0.932928i	$-0.617245\pi$
$347$	−1.61523e6	−0.720128	−0.360064	+	0.932928i	$0.617245\pi$
$348$	−70797.5	−0.0313379
$349$	49878.2	0.0219203	0.0109602	−	0.999940i	$-0.496511\pi$
$349$	49878.2	0.0219203	0.0109602	+	0.999940i	$0.496511\pi$
$350$	5.57374e6	2.43207
$351$	−1.98983e6	−0.862083
$352$	354181.	0.152359
$353$	3.46683e6	1.48080	0.740399	−	0.672167i	$-0.234636\pi$
$353$	3.46683e6	1.48080	0.740399	+	0.672167i	$0.234636\pi$
$354$	388897.	0.164940
$355$	−2.98048e6	−1.25521
$356$	−1.78115e6	−0.744863
$357$	−1.92549e6	−0.799596
$358$	820348.	0.338291
$359$	2.97986e6	1.22028	0.610140	−	0.792294i	$-0.291113\pi$
$359$	2.97986e6	1.22028	0.610140	+	0.792294i	$0.291113\pi$
$360$	−824063.	−0.335123
$361$	0	0
$362$	−2.38726e6	−0.957478
$363$	−430853.	−0.171618
$364$	1.88126e6	0.744208
$365$	190318.	0.0747735
$366$	−346236.	−0.135104
$367$	2.23837e6	0.867495	0.433748	−	0.901034i	$-0.357191\pi$
$367$	2.23837e6	0.867495	0.433748	+	0.901034i	$0.357191\pi$
$368$	−304811.	−0.117331
$369$	2.16740e6	0.828655
$370$	−2.99995e6	−1.13922
$371$	−6.75565e6	−2.54819
$372$	1.10647e6	0.414554
$373$	1.11334e6	0.414338	0.207169	−	0.978305i	$-0.433575\pi$
$373$	1.11334e6	0.414338	0.207169	+	0.978305i	$0.433575\pi$
$374$	−1.10279e6	−0.407674
$375$	2.85748e6	1.04931
$376$	−1.35617e6	−0.494703
$377$	−215373.	−0.0780437
$378$	−3.65042e6	−1.31405
$379$	3.21599e6	1.15005	0.575026	−	0.818135i	$-0.304992\pi$
$379$	3.21599e6	1.15005	0.575026	+	0.818135i	$0.304992\pi$
$380$	0	0
$381$	987467.	0.348506
$382$	1.49690e6	0.524847
$383$	491800.	0.171313	0.0856567	−	0.996325i	$-0.472701\pi$
$383$	491800.	0.171313	0.0856567	+	0.996325i	$0.472701\pi$
$384$	170434.	0.0589833
$385$	7.67275e6	2.63815
$386$	740360.	0.252915
$387$	−1.74178e6	−0.591175
$388$	1.55192e6	0.523346
$389$	−2.10114e6	−0.704015	−0.352007	−	0.935997i	$-0.614501\pi$
$389$	−2.10114e6	−0.704015	−0.352007	+	0.935997i	$0.614501\pi$
$390$	2.01260e6	0.670032
$391$	949068.	0.313946
$392$	2.37558e6	0.780827
$393$	2.76119e6	0.901809
$394$	598077.	0.194096
$395$	−919288.	−0.296455
$396$	−745928.	−0.239033
$397$	−5.27092e6	−1.67846	−0.839229	−	0.543778i	$-0.816993\pi$
$397$	−5.27092e6	−1.67846	−0.839229	+	0.543778i	$0.816993\pi$
$398$	−1.22369e6	−0.387225
$399$	0	0
$400$	1.53614e6	0.480042
$401$	−4.12441e6	−1.28086	−0.640430	−	0.768017i	$-0.721244\pi$
$401$	−4.12441e6	−1.28086	−0.640430	+	0.768017i	$0.721244\pi$
$402$	−596981.	−0.184245
$403$	3.36598e6	1.03240
$404$	−2.72150e6	−0.829575
$405$	−776414.	−0.235210
$406$	−395109.	−0.118960
$407$	−2.71550e6	−0.812575
$408$	−530670.	−0.157824
$409$	−4.71220e6	−1.39289	−0.696443	−	0.717613i	$-0.745235\pi$
$409$	−4.71220e6	−1.39289	−0.696443	+	0.717613i	$0.745235\pi$
$410$	−6.14437e6	−1.80517
$411$	−724244.	−0.211485
$412$	−1.16113e6	−0.337008
$413$	2.17037e6	0.626122
$414$	641951.	0.184078
$415$	−3.26508e6	−0.930623
$416$	518479.	0.146892
$417$	3.84350e6	1.08240
$418$	0	0
$419$	−3.47084e6	−0.965827	−0.482914	−	0.875668i	$-0.660422\pi$
$419$	−3.47084e6	−0.965827	−0.482914	+	0.875668i	$0.660422\pi$
$420$	3.69218e6	1.02132
$421$	−1.19223e6	−0.327835	−0.163918	−	0.986474i	$-0.552413\pi$
$421$	−1.19223e6	−0.327835	−0.163918	+	0.986474i	$0.552413\pi$
$422$	3.23895e6	0.885367
$423$	2.85618e6	0.776131
$424$	−1.86187e6	−0.502962
$425$	−4.78295e6	−1.28447
$426$	−1.29824e6	−0.346601
$427$	−1.93229e6	−0.512863
$428$	−1.08834e6	−0.287181
$429$	1.82177e6	0.477914
$430$	4.93777e6	1.28783
$431$	−679294.	−0.176143	−0.0880714	−	0.996114i	$-0.528070\pi$
$431$	−679294.	−0.176143	−0.0880714	+	0.996114i	$0.528070\pi$
$432$	−1.00606e6	−0.259368
$433$	−7.14219e6	−1.83068	−0.915339	−	0.402685i	$-0.868077\pi$
$433$	−7.14219e6	−1.83068	−0.915339	+	0.402685i	$0.868077\pi$
$434$	6.17501e6	1.57367
$435$	−422695.	−0.107103
$436$	−310121.	−0.0781296
$437$	0	0
$438$	82898.8	0.0206473
$439$	1.61627e6	0.400271	0.200135	−	0.979768i	$-0.435862\pi$
$439$	1.61627e6	0.400271	0.200135	+	0.979768i	$0.435862\pi$
$440$	2.11463e6	0.520718
$441$	−5.00313e6	−1.22503
$442$	−1.61435e6	−0.393045
$443$	−3.15786e6	−0.764511	−0.382255	−	0.924057i	$-0.624853\pi$
$443$	−3.15786e6	−0.764511	−0.382255	+	0.924057i	$0.624853\pi$
$444$	−1.30672e6	−0.314575
$445$	−1.06343e7	−2.54572
$446$	2.53836e6	0.604249
$447$	112820.	0.0267066
$448$	951167.	0.223904
$449$	1.47951e6	0.346338	0.173169	−	0.984892i	$-0.444599\pi$
$449$	1.47951e6	0.346338	0.173169	+	0.984892i	$0.444599\pi$
$450$	−3.23520e6	−0.753130
$451$	−5.56177e6	−1.28757
$452$	−2.05215e6	−0.472458
$453$	349455.	0.0800102
$454$	−375439.	−0.0854869
$455$	1.12320e7	2.54348
$456$	0	0
$457$	−5.30763e6	−1.18880	−0.594402	−	0.804168i	$-0.702611\pi$
$457$	−5.30763e6	−1.18880	−0.594402	+	0.804168i	$0.702611\pi$
$458$	704739.	0.156987
$459$	3.13251e6	0.694002
$460$	−1.81987e6	−0.401000
$461$	−1.54790e6	−0.339227	−0.169613	−	0.985511i	$-0.554252\pi$
$461$	−1.54790e6	−0.339227	−0.169613	+	0.985511i	$0.554252\pi$
$462$	3.34210e6	0.728474
$463$	7.32198e6	1.58736	0.793682	−	0.608333i	$-0.208162\pi$
$463$	7.32198e6	1.58736	0.793682	+	0.608333i	$0.208162\pi$
$464$	−108893.	−0.0234804
$465$	6.60614e6	1.41682
$466$	−904642.	−0.192980
$467$	6.91574e6	1.46739	0.733696	−	0.679478i	$-0.237794\pi$
$467$	6.91574e6	1.46739	0.733696	+	0.679478i	$0.237794\pi$
$468$	−1.09195e6	−0.230456
$469$	−3.33165e6	−0.699403
$470$	−8.09698e6	−1.69075
$471$	5.94619e6	1.23506
$472$	598160.	0.123584
$473$	4.46958e6	0.918573
$474$	−400424.	−0.0818605
$475$	0	0
$476$	−2.96158e6	−0.599110
$477$	3.92122e6	0.789089
$478$	−1.37599e6	−0.275451
$479$	3.22667e6	0.642563	0.321281	−	0.946984i	$-0.395886\pi$
$479$	3.22667e6	0.642563	0.321281	+	0.946984i	$0.395886\pi$
$480$	1.01757e6	0.201587
$481$	−3.97517e6	−0.783416
$482$	6.34744e6	1.24446
$483$	−2.87624e6	−0.560992
$484$	−662693.	−0.128588
$485$	9.26568e6	1.78864
$486$	3.48171e6	0.668655
$487$	−4.64992e6	−0.888430	−0.444215	−	0.895920i	$-0.646517\pi$
$487$	−4.64992e6	−0.888430	−0.444215	+	0.895920i	$0.646517\pi$
$488$	−532543.	−0.101229
$489$	4.58809e6	0.867680
$490$	1.41834e7	2.66863
$491$	8.23773e6	1.54207	0.771034	−	0.636794i	$-0.219740\pi$
$491$	8.23773e6	1.54207	0.771034	+	0.636794i	$0.219740\pi$
$492$	−2.67637e6	−0.498463
$493$	339053.	0.0628275
$494$	0	0
$495$	−4.45354e6	−0.816945
$496$	1.70185e6	0.310611
$497$	−7.24526e6	−1.31572
$498$	−1.42221e6	−0.256974
$499$	−3.72990e6	−0.670572	−0.335286	−	0.942116i	$-0.608833\pi$
$499$	−3.72990e6	−0.670572	−0.335286	+	0.942116i	$0.608833\pi$
$500$	4.39508e6	0.786215
$501$	−336071.	−0.0598186
$502$	2.98387e6	0.528470
$503$	−1.51655e6	−0.267261	−0.133630	−	0.991031i	$-0.542664\pi$
$503$	−1.51655e6	−0.267261	−0.133630	+	0.991031i	$0.542664\pi$
$504$	−2.00322e6	−0.351279
$505$	−1.62487e7	−2.83524
$506$	−1.64731e6	−0.286022
$507$	−1.19552e6	−0.206555
$508$	1.51882e6	0.261123
$509$	1.12361e7	1.92231	0.961154	−	0.276014i	$-0.0890135\pi$
$509$	1.12361e7	1.92231	0.961154	+	0.276014i	$0.0890135\pi$
$510$	−3.16835e6	−0.539396
$511$	462645.	0.0783782
$512$	262144.	0.0441942
$513$	0	0
$514$	7.70520e6	1.28640
$515$	−6.93253e6	−1.15179
$516$	2.15080e6	0.355611
$517$	−7.32924e6	−1.20596
$518$	−7.29258e6	−1.19414
$519$	−1.45747e6	−0.237509
$520$	3.09557e6	0.502032
$521$	4.99957e6	0.806934	0.403467	−	0.914994i	$-0.367805\pi$
$521$	4.99957e6	0.806934	0.403467	+	0.914994i	$0.367805\pi$
$522$	229336.	0.0368380
$523$	−4.90563e6	−0.784225	−0.392113	−	0.919917i	$-0.628256\pi$
$523$	−4.90563e6	−0.784225	−0.392113	+	0.919917i	$0.628256\pi$
$524$	4.24697e6	0.675695
$525$	1.44952e7	2.29523
$526$	−3.85848e6	−0.608067
$527$	−5.29893e6	−0.831116
$528$	921091.	0.143786
$529$	−5.01865e6	−0.779737
$530$	−1.11163e7	−1.71897
$531$	−1.25976e6	−0.193889
$532$	0	0
$533$	−8.14178e6	−1.24137
$534$	−4.63211e6	−0.702952
$535$	−6.49792e6	−0.981500
$536$	−918212.	−0.138048
$537$	2.13342e6	0.319256
$538$	−6.44248e6	−0.959616
$539$	1.28385e7	1.90346
$540$	−6.00668e6	−0.886442
$541$	3.83360e6	0.563137	0.281568	−	0.959541i	$-0.409145\pi$
$541$	3.83360e6	0.563137	0.281568	+	0.959541i	$0.409145\pi$
$542$	4.54599e6	0.664707
$543$	−6.20837e6	−0.903604
$544$	−816219.	−0.118252
$545$	−1.85157e6	−0.267024
$546$	4.89244e6	0.702334
$547$	−5.83247e6	−0.833458	−0.416729	−	0.909031i	$-0.636824\pi$
$547$	−5.83247e6	−0.833458	−0.416729	+	0.909031i	$0.636824\pi$
$548$	−1.11396e6	−0.158459
$549$	1.12157e6	0.158816
$550$	8.30184e6	1.17022
$551$	0	0
$552$	−792698.	−0.110729
$553$	−2.23470e6	−0.310747
$554$	2.59379e6	0.359055
$555$	−7.80173e6	−1.07512
$556$	5.91166e6	0.811004
$557$	1.47427e6	0.201344	0.100672	−	0.994920i	$-0.467901\pi$
$557$	1.47427e6	0.201344	0.100672	+	0.994920i	$0.467901\pi$
$558$	−3.58420e6	−0.487312
$559$	6.54294e6	0.885611
$560$	5.67892e6	0.765237
$561$	−2.86793e6	−0.384735
$562$	−2.00729e6	−0.268083
$563$	−1.71502e6	−0.228033	−0.114016	−	0.993479i	$-0.536372\pi$
$563$	−1.71502e6	−0.228033	−0.114016	+	0.993479i	$0.536372\pi$
$564$	−3.52689e6	−0.466868
$565$	−1.22523e7	−1.61472
$566$	5.47648e6	0.718556
$567$	−1.88739e6	−0.246549
$568$	−1.99681e6	−0.259696
$569$	1.09443e7	1.41712	0.708558	−	0.705652i	$-0.249346\pi$
$569$	1.09443e7	1.41712	0.708558	+	0.705652i	$0.249346\pi$
$570$	0	0
$571$	−7.80064e6	−1.00124	−0.500622	−	0.865666i	$-0.666895\pi$
$571$	−7.80064e6	−1.00124	−0.500622	+	0.865666i	$0.666895\pi$
$572$	2.80205e6	0.358085
$573$	3.89286e6	0.495316
$574$	−1.49364e7	−1.89219
$575$	−7.14463e6	−0.901177
$576$	−552092.	−0.0693355
$577$	−8.00141e6	−1.00052	−0.500262	−	0.865874i	$-0.666763\pi$
$577$	−8.00141e6	−1.00052	−0.500262	+	0.865874i	$0.666763\pi$
$578$	−3.13802e6	−0.390694
$579$	1.92540e6	0.238684
$580$	−650144.	−0.0802489
$581$	−7.93711e6	−0.975488
$582$	4.03595e6	0.493899
$583$	−1.00622e7	−1.22609
$584$	127506.	0.0154703
$585$	−6.51946e6	−0.787630
$586$	−2.73886e6	−0.329478
$587$	−7.68911e6	−0.921045	−0.460523	−	0.887648i	$-0.652338\pi$
$587$	−7.68911e6	−0.921045	−0.460523	+	0.887648i	$0.652338\pi$
$588$	6.17799e6	0.736892
$589$	0	0
$590$	3.57130e6	0.422373
$591$	1.55537e6	0.183175
$592$	−2.00985e6	−0.235700
$593$	3.87367e6	0.452362	0.226181	−	0.974085i	$-0.427376\pi$
$593$	3.87367e6	0.452362	0.226181	+	0.974085i	$0.427376\pi$
$594$	−5.43714e6	−0.632273
$595$	−1.76821e7	−2.04758
$596$	173528.	0.0200103
$597$	−3.18236e6	−0.365438
$598$	−2.41147e6	−0.275758
$599$	−1.25262e7	−1.42644	−0.713220	−	0.700940i	$-0.752764\pi$
$599$	−1.25262e7	−1.42644	−0.713220	+	0.700940i	$0.752764\pi$
$600$	3.99491e6	0.453032
$601$	293637.	0.0331608	0.0165804	−	0.999863i	$-0.494722\pi$
$601$	293637.	0.0331608	0.0165804	+	0.999863i	$0.494722\pi$
$602$	1.20032e7	1.34992
$603$	1.93381e6	0.216581
$604$	537494.	0.0599489
$605$	−3.95659e6	−0.439474
$606$	−7.07761e6	−0.782897
$607$	−4.13932e6	−0.455993	−0.227996	−	0.973662i	$-0.573217\pi$
$607$	−4.13932e6	−0.455993	−0.227996	+	0.973662i	$0.573217\pi$
$608$	0	0
$609$	−1.02753e6	−0.112267
$610$	−3.17953e6	−0.345970
$611$	−1.07291e7	−1.16269
$612$	1.71901e6	0.185524
$613$	5.73897e6	0.616854	0.308427	−	0.951248i	$-0.400197\pi$
$613$	5.73897e6	0.616854	0.308427	+	0.951248i	$0.400197\pi$
$614$	−5.73130e6	−0.613525
$615$	−1.59792e7	−1.70360
$616$	5.14046e6	0.545821
$617$	−1.15672e7	−1.22325	−0.611627	−	0.791146i	$-0.709485\pi$
$617$	−1.15672e7	−1.22325	−0.611627	+	0.791146i	$0.709485\pi$
$618$	−3.01967e6	−0.318045
$619$	−1.84780e7	−1.93833	−0.969167	−	0.246404i	$-0.920751\pi$
$619$	−1.84780e7	−1.93833	−0.969167	+	0.246404i	$0.920751\pi$
$620$	1.01608e7	1.06158
$621$	4.67925e6	0.486908
$622$	−1.14630e6	−0.118802
$623$	−2.58510e7	−2.66844
$624$	1.34837e6	0.138627
$625$	7.48906e6	0.766879
$626$	1.30505e6	0.133104
$627$	0	0
$628$	9.14580e6	0.925385
$629$	6.25794e6	0.630674
$630$	−1.19602e7	−1.20057
$631$	8.95725e6	0.895573	0.447787	−	0.894140i	$-0.352212\pi$
$631$	8.95725e6	0.895573	0.447787	+	0.894140i	$0.352212\pi$
$632$	−615889.	−0.0613352
$633$	8.42329e6	0.835550
$634$	1.21753e7	1.20297
$635$	9.06805e6	0.892441
$636$	−4.84203e6	−0.474662
$637$	1.87941e7	1.83515
$638$	−588498.	−0.0572392
$639$	4.20541e6	0.407433
$640$	1.56512e6	0.151042
$641$	1.46979e7	1.41290	0.706450	−	0.707763i	$-0.250296\pi$
$641$	1.46979e7	1.41290	0.706450	+	0.707763i	$0.250296\pi$
$642$	−2.83037e6	−0.271023
$643$	1.55533e7	1.48352	0.741762	−	0.670664i	$-0.233991\pi$
$643$	1.55533e7	1.48352	0.741762	+	0.670664i	$0.233991\pi$
$644$	−4.42392e6	−0.420332
$645$	1.28413e7	1.21537
$646$	0	0
$647$	−5.91558e6	−0.555567	−0.277784	−	0.960644i	$-0.589600\pi$
$647$	−5.91558e6	−0.555567	−0.277784	+	0.960644i	$0.589600\pi$
$648$	−520169.	−0.0486639
$649$	3.23268e6	0.301266
$650$	1.21529e7	1.12823
$651$	1.60589e7	1.48512
$652$	7.05691e6	0.650123
$653$	1.27377e6	0.116898	0.0584492	−	0.998290i	$-0.481384\pi$
$653$	1.27377e6	0.116898	0.0584492	+	0.998290i	$0.481384\pi$
$654$	−806509.	−0.0737335
$655$	2.53564e7	2.30932
$656$	−4.11650e6	−0.373481
$657$	−268536.	−0.0242711
$658$	−1.96830e7	−1.77226
$659$	−6.98055e6	−0.626147	−0.313073	−	0.949729i	$-0.601359\pi$
$659$	−6.98055e6	−0.626147	−0.313073	+	0.949729i	$0.601359\pi$
$660$	5.49935e6	0.491419
$661$	−1.02987e7	−0.916812	−0.458406	−	0.888743i	$-0.651579\pi$
$661$	−1.02987e7	−0.916812	−0.458406	+	0.888743i	$0.651579\pi$
$662$	−9.05208e6	−0.802792
$663$	−4.19832e6	−0.370930
$664$	−2.18749e6	−0.192542
$665$	0	0
$666$	4.23288e6	0.369786
$667$	506467.	0.0440795
$668$	−516908.	−0.0448200
$669$	6.60131e6	0.570249
$670$	−5.48216e6	−0.471808
$671$	−2.87806e6	−0.246770
$672$	2.47363e6	0.211306
$673$	−5.48362e6	−0.466692	−0.233346	−	0.972394i	$-0.574967\pi$
$673$	−5.48362e6	−0.466692	−0.233346	+	0.972394i	$0.574967\pi$
$674$	9.36052e6	0.793689
$675$	−2.35817e7	−1.99212
$676$	−1.83882e6	−0.154765
$677$	4.99313e6	0.418698	0.209349	−	0.977841i	$-0.432866\pi$
$677$	4.99313e6	0.418698	0.209349	+	0.977841i	$0.432866\pi$
$678$	−5.33687e6	−0.445875
$679$	2.25240e7	1.87487
$680$	−4.87322e6	−0.404151
$681$	−976375.	−0.0806768
$682$	9.19742e6	0.757190
$683$	−6.96239e6	−0.571093	−0.285546	−	0.958365i	$-0.592175\pi$
$683$	−6.96239e6	−0.571093	−0.285546	+	0.958365i	$0.592175\pi$
$684$	0	0
$685$	−6.65084e6	−0.541565
$686$	1.88668e7	1.53069
$687$	1.83276e6	0.148154
$688$	3.30812e6	0.266447
$689$	−1.47299e7	−1.18210
$690$	−4.73279e6	−0.378438
$691$	9.29521e6	0.740566	0.370283	−	0.928919i	$-0.379261\pi$
$691$	9.29521e6	0.740566	0.370283	+	0.928919i	$0.379261\pi$
$692$	−2.24172e6	−0.177957
$693$	−1.08261e7	−0.856329
$694$	−6.46091e6	−0.509208
$695$	3.52955e7	2.77177
$696$	−283190.	−0.0221592
$697$	1.28173e7	0.999340
$698$	199513.	0.0155000
$699$	−2.35263e6	−0.182121
$700$	2.22949e7	1.71973
$701$	−7.70223e6	−0.591999	−0.296000	−	0.955188i	$-0.595653\pi$
$701$	−7.70223e6	−0.591999	−0.296000	+	0.955188i	$0.595653\pi$
$702$	−7.95933e6	−0.609585
$703$	0	0
$704$	1.41672e6	0.107734
$705$	−2.10572e7	−1.59561
$706$	1.38673e7	1.04708
$707$	−3.94990e7	−2.97192
$708$	1.55559e6	0.116630
$709$	513218.	0.0383430	0.0191715	−	0.999816i	$-0.493897\pi$
$709$	513218.	0.0383430	0.0191715	+	0.999816i	$0.493897\pi$
$710$	−1.19219e7	−0.887565
$711$	1.29710e6	0.0962278
$712$	−7.12461e6	−0.526697
$713$	−7.91538e6	−0.583106
$714$	−7.70196e6	−0.565400
$715$	1.67296e7	1.22383
$716$	3.28139e6	0.239208
$717$	−3.57842e6	−0.259952
$718$	1.19194e7	0.862868
$719$	3.80611e6	0.274574	0.137287	−	0.990531i	$-0.456162\pi$
$719$	3.80611e6	0.274574	0.137287	+	0.990531i	$0.456162\pi$
$720$	−3.29625e6	−0.236968
$721$	−1.68523e7	−1.20732
$722$	0	0
$723$	1.65073e7	1.17444
$724$	−9.54905e6	−0.677039
$725$	−2.55241e6	−0.180345
$726$	−1.72341e6	−0.121352
$727$	−2.43735e7	−1.71033	−0.855167	−	0.518352i	$-0.826546\pi$
$727$	−2.43735e7	−1.71033	−0.855167	+	0.518352i	$0.826546\pi$
$728$	7.52502e6	0.526235
$729$	1.10296e7	0.768674
$730$	761272.	0.0528728
$731$	−1.03003e7	−0.712944
$732$	−1.38494e6	−0.0955331
$733$	4.46559e6	0.306986	0.153493	−	0.988150i	$-0.450948\pi$
$733$	4.46559e6	0.306986	0.153493	+	0.988150i	$0.450948\pi$
$734$	8.95349e6	0.613412
$735$	3.68856e7	2.51848
$736$	−1.21924e6	−0.0829652
$737$	−4.96236e6	−0.336526
$738$	8.66962e6	0.585948
$739$	−2.04421e6	−0.137694	−0.0688468	−	0.997627i	$-0.521932\pi$
$739$	−2.04421e6	−0.137694	−0.0688468	+	0.997627i	$0.521932\pi$
$740$	−1.19998e7	−0.805553
$741$	0	0
$742$	−2.70226e7	−1.80184
$743$	1.90490e7	1.26590	0.632950	−	0.774193i	$-0.281844\pi$
$743$	1.90490e7	1.26590	0.632950	+	0.774193i	$0.281844\pi$
$744$	4.42587e6	0.293134
$745$	1.03604e6	0.0683892
$746$	4.45335e6	0.292981
$747$	4.60699e6	0.302075
$748$	−4.41115e6	−0.288269
$749$	−1.57958e7	−1.02882
$750$	1.14299e7	0.741978
$751$	1.08826e7	0.704097	0.352048	−	0.935982i	$-0.385485\pi$
$751$	1.08826e7	0.704097	0.352048	+	0.935982i	$0.385485\pi$
$752$	−5.42468e6	−0.349808
$753$	7.75992e6	0.498735
$754$	−861492.	−0.0551852
$755$	3.20909e6	0.204887
$756$	−1.46017e7	−0.929176
$757$	2.24795e6	0.142576	0.0712881	−	0.997456i	$-0.477289\pi$
$757$	2.24795e6	0.142576	0.0712881	+	0.997456i	$0.477289\pi$
$758$	1.28640e7	0.813209
$759$	−4.28403e6	−0.269928
$760$	0	0
$761$	1.37969e7	0.863614	0.431807	−	0.901966i	$-0.357876\pi$
$761$	1.37969e7	0.863614	0.431807	+	0.901966i	$0.357876\pi$
$762$	3.94987e6	0.246431
$763$	−4.50100e6	−0.279896
$764$	5.98758e6	0.371123
$765$	1.02633e7	0.634065
$766$	1.96720e6	0.121137
$767$	4.73226e6	0.290456
$768$	681738.	0.0417075
$769$	3.16040e7	1.92720	0.963599	−	0.267352i	$-0.0861487\pi$
$769$	3.16040e7	1.92720	0.963599	+	0.267352i	$0.0861487\pi$
$770$	3.06910e7	1.86545
$771$	2.00383e7	1.21402
$772$	2.96144e6	0.178838
$773$	−2.13708e7	−1.28639	−0.643195	−	0.765702i	$-0.722392\pi$
$773$	−2.13708e7	−1.28639	−0.643195	+	0.765702i	$0.722392\pi$
$774$	−6.96712e6	−0.418024
$775$	3.98906e7	2.38570
$776$	6.20767e6	0.370062
$777$	−1.89653e7	−1.12695
$778$	−8.40458e6	−0.497814
$779$	0	0
$780$	8.05040e6	0.473785
$781$	−1.07915e7	−0.633074
$782$	3.79627e6	0.221994
$783$	1.67165e6	0.0974410
$784$	9.50233e6	0.552128
$785$	5.46048e7	3.16269
$786$	1.10448e7	0.637676
$787$	−1.91269e7	−1.10080	−0.550399	−	0.834901i	$-0.685525\pi$
$787$	−1.91269e7	−1.10080	−0.550399	+	0.834901i	$0.685525\pi$
$788$	2.39231e6	0.137247
$789$	−1.00344e7	−0.573853
$790$	−3.67715e6	−0.209625
$791$	−2.97842e7	−1.69256
$792$	−2.98371e6	−0.169022
$793$	−4.21313e6	−0.237915
$794$	−2.10837e7	−1.18685
$795$	−2.89092e7	−1.62225
$796$	−4.89476e6	−0.273810
$797$	−8.28296e6	−0.461891	−0.230946	−	0.972967i	$-0.574182\pi$
$797$	−8.28296e6	−0.461891	−0.230946	+	0.972967i	$0.574182\pi$
$798$	0	0
$799$	1.68904e7	0.935996
$800$	6.14454e6	0.339441
$801$	1.50049e7	0.826326
$802$	−1.64977e7	−0.905704
$803$	689090.	0.0377126
$804$	−2.38792e6	−0.130281
$805$	−2.64129e7	−1.43657
$806$	1.34639e7	0.730019
$807$	−1.67545e7	−0.905622
$808$	−1.08860e7	−0.586598
$809$	−8.91658e6	−0.478991	−0.239495	−	0.970898i	$-0.576982\pi$
$809$	−8.91658e6	−0.478991	−0.239495	+	0.970898i	$0.576982\pi$
$810$	−3.10565e6	−0.166319
$811$	−1.52149e7	−0.812302	−0.406151	−	0.913806i	$-0.633129\pi$
$811$	−1.52149e7	−0.812302	−0.406151	+	0.913806i	$0.633129\pi$
$812$	−1.58044e6	−0.0841177
$813$	1.18224e7	0.627306
$814$	−1.08620e7	−0.574577
$815$	4.21331e7	2.22192
$816$	−2.12268e6	−0.111599
$817$	0	0
$818$	−1.88488e7	−0.984918
$819$	−1.58482e7	−0.825600
$820$	−2.45775e7	−1.27645
$821$	2.23907e7	1.15934	0.579669	−	0.814852i	$-0.303182\pi$
$821$	2.23907e7	1.15934	0.579669	+	0.814852i	$0.303182\pi$
$822$	−2.89698e6	−0.149543
$823$	−2.10545e7	−1.08354	−0.541772	−	0.840526i	$-0.682246\pi$
$823$	−2.10545e7	−1.08354	−0.541772	+	0.840526i	$0.682246\pi$
$824$	−4.64454e6	−0.238300
$825$	2.15900e7	1.10438
$826$	8.68149e6	0.442735
$827$	2.23530e7	1.13651	0.568253	−	0.822854i	$-0.307619\pi$
$827$	2.23530e7	1.13651	0.568253	+	0.822854i	$0.307619\pi$
$828$	2.56781e6	0.130163
$829$	−3.04175e7	−1.53723	−0.768613	−	0.639714i	$-0.779053\pi$
$829$	−3.04175e7	−1.53723	−0.768613	+	0.639714i	$0.779053\pi$
$830$	−1.30603e7	−0.658050
$831$	6.74548e6	0.338852
$832$	2.07391e6	0.103868
$833$	−2.95867e7	−1.47735
$834$	1.53740e7	0.765371
$835$	−3.08619e6	−0.153181
$836$	0	0
$837$	−2.61256e7	−1.28900
$838$	−1.38834e7	−0.682943
$839$	−2.41182e7	−1.18288	−0.591439	−	0.806350i	$-0.701440\pi$
$839$	−2.41182e7	−1.18288	−0.591439	+	0.806350i	$0.701440\pi$
$840$	1.47687e7	0.722179
$841$	−2.03302e7	−0.991179
$842$	−4.76893e6	−0.231814
$843$	−5.22020e6	−0.252999
$844$	1.29558e7	0.626049
$845$	−1.09786e7	−0.528939
$846$	1.14247e7	0.548807
$847$	−9.61810e6	−0.460660
$848$	−7.44749e6	−0.355648
$849$	1.42423e7	0.678125
$850$	−1.91318e7	−0.908257
$851$	9.34792e6	0.442477
$852$	−5.19295e6	−0.245084
$853$	1.65084e7	0.776842	0.388421	−	0.921482i	$-0.373021\pi$
$853$	1.65084e7	0.776842	0.388421	+	0.921482i	$0.373021\pi$
$854$	−7.72914e6	−0.362649
$855$	0	0
$856$	−4.35337e6	−0.203068
$857$	−1.13507e7	−0.527923	−0.263962	−	0.964533i	$-0.585029\pi$
$857$	−1.13507e7	−0.527923	−0.263962	+	0.964533i	$0.585029\pi$
$858$	7.28708e6	0.337937
$859$	2.00932e7	0.929108	0.464554	−	0.885545i	$-0.346215\pi$
$859$	2.00932e7	0.929108	0.464554	+	0.885545i	$0.346215\pi$
$860$	1.97511e7	0.910635
$861$	−3.88439e7	−1.78573
$862$	−2.71718e6	−0.124552
$863$	1.97337e7	0.901950	0.450975	−	0.892537i	$-0.351076\pi$
$863$	1.97337e7	0.901950	0.450975	+	0.892537i	$0.351076\pi$
$864$	−4.02426e6	−0.183401
$865$	−1.33841e7	−0.608205
$866$	−2.85688e7	−1.29448
$867$	−8.16082e6	−0.368711
$868$	2.47001e7	1.11275
$869$	−3.32849e6	−0.149520
$870$	−1.69078e6	−0.0757336
$871$	−7.26430e6	−0.324451
$872$	−1.24049e6	−0.0552460
$873$	−1.30737e7	−0.580583
$874$	0	0
$875$	6.37887e7	2.81659
$876$	331595.	0.0145998
$877$	1.84073e7	0.808149	0.404074	−	0.914726i	$-0.367594\pi$
$877$	1.84073e7	0.808149	0.404074	+	0.914726i	$0.367594\pi$
$878$	6.46510e6	0.283034
$879$	−7.12275e6	−0.310939
$880$	8.45851e6	0.368203
$881$	1.42310e7	0.617727	0.308864	−	0.951106i	$-0.400051\pi$
$881$	1.42310e7	0.617727	0.308864	+	0.951106i	$0.400051\pi$
$882$	−2.00125e7	−0.866224
$883$	1.06212e7	0.458430	0.229215	−	0.973376i	$-0.426384\pi$
$883$	1.06212e7	0.458430	0.229215	+	0.973376i	$0.426384\pi$
$884$	−6.45740e6	−0.277925
$885$	9.28761e6	0.398608
$886$	−1.26314e7	−0.540591
$887$	4.79046e6	0.204441	0.102220	−	0.994762i	$-0.467405\pi$
$887$	4.79046e6	0.204441	0.102220	+	0.994762i	$0.467405\pi$
$888$	−5.22687e6	−0.222438
$889$	2.20436e7	0.935465
$890$	−4.25373e7	−1.80009
$891$	−2.81118e6	−0.118630
$892$	1.01534e7	0.427268
$893$	0	0
$894$	451281.	0.0188844
$895$	1.95915e7	0.817541
$896$	3.80467e6	0.158324
$897$	−6.27132e6	−0.260242
$898$	5.91802e6	0.244898
$899$	−2.82775e6	−0.116692
$900$	−1.29408e7	−0.532543
$901$	2.31887e7	0.951623
$902$	−2.22471e7	−0.910451
$903$	3.12159e7	1.27396
$904$	−8.20861e6	−0.334078
$905$	−5.70124e7	−2.31392
$906$	1.39782e6	0.0565758
$907$	−3.87079e7	−1.56236	−0.781180	−	0.624306i	$-0.785382\pi$
$907$	−3.87079e7	−1.56236	−0.781180	+	0.624306i	$0.785382\pi$
$908$	−1.50176e6	−0.0604484
$909$	2.29267e7	0.920303
$910$	4.49280e7	1.79851
$911$	2.75114e7	1.09829	0.549146	−	0.835727i	$-0.314953\pi$
$911$	2.75114e7	1.09829	0.549146	+	0.835727i	$0.314953\pi$
$912$	0	0
$913$	−1.18220e7	−0.469368
$914$	−2.12305e7	−0.840612
$915$	−8.26877e6	−0.326504
$916$	2.81896e6	0.111007
$917$	6.16390e7	2.42065
$918$	1.25300e7	0.490734
$919$	−1.32340e7	−0.516896	−0.258448	−	0.966025i	$-0.583211\pi$
$919$	−1.32340e7	−0.516896	−0.258448	+	0.966025i	$0.583211\pi$
$920$	−7.27947e6	−0.283550
$921$	−1.49050e7	−0.579004
$922$	−6.19159e6	−0.239870
$923$	−1.57975e7	−0.610357
$924$	1.33684e7	0.515109
$925$	−4.71101e7	−1.81034
$926$	2.92879e7	1.12244
$927$	9.78170e6	0.373865
$928$	−435572.	−0.0166031
$929$	−1.94057e7	−0.737717	−0.368858	−	0.929486i	$-0.620251\pi$
$929$	−1.94057e7	−0.737717	−0.368858	+	0.929486i	$0.620251\pi$
$930$	2.64245e7	1.00184
$931$	0	0
$932$	−3.61857e6	−0.136457
$933$	−2.98111e6	−0.112118
$934$	2.76629e7	1.03760
$935$	−2.63367e7	−0.985217
$936$	−4.36780e6	−0.162957
$937$	4.70727e7	1.75154	0.875770	−	0.482729i	$-0.160354\pi$
$937$	4.70727e7	1.75154	0.875770	+	0.482729i	$0.160354\pi$
$938$	−1.33266e7	−0.494553
$939$	3.39395e6	0.125615
$940$	−3.23879e7	−1.19554
$941$	5.01568e7	1.84653	0.923264	−	0.384167i	$-0.125511\pi$
$941$	5.01568e7	1.84653	0.923264	+	0.384167i	$0.125511\pi$
$942$	2.37848e7	0.873317
$943$	1.91460e7	0.701131
$944$	2.39264e6	0.0873871
$945$	−8.71790e7	−3.17565
$946$	1.78783e7	0.649529
$947$	1.85612e7	0.672561	0.336281	−	0.941762i	$-0.390831\pi$
$947$	1.85612e7	0.672561	0.336281	+	0.941762i	$0.390831\pi$
$948$	−1.60170e6	−0.0578841
$949$	1.00875e6	0.0363594
$950$	0	0
$951$	3.16633e7	1.13529
$952$	−1.18463e7	−0.423635
$953$	3.60296e7	1.28507	0.642536	−	0.766255i	$-0.277882\pi$
$953$	3.60296e7	1.28507	0.642536	+	0.766255i	$0.277882\pi$
$954$	1.56849e7	0.557970
$955$	3.57487e7	1.26839
$956$	−5.50394e6	−0.194773
$957$	−1.53046e6	−0.0540185
$958$	1.29067e7	0.454361
$959$	−1.61676e7	−0.567673
$960$	4.07030e6	0.142544
$961$	1.55647e7	0.543667
$962$	−1.59007e7	−0.553959
$963$	9.16848e6	0.318590
$964$	2.53898e7	0.879966
$965$	1.76812e7	0.611215
$966$	−1.15050e7	−0.396682
$967$	1.22925e7	0.422740	0.211370	−	0.977406i	$-0.432207\pi$
$967$	1.22925e7	0.422740	0.211370	+	0.977406i	$0.432207\pi$
$968$	−2.65077e6	−0.0909251
$969$	0	0
$970$	3.70627e7	1.26476
$971$	1.73779e7	0.591493	0.295746	−	0.955267i	$-0.404432\pi$
$971$	1.73779e7	0.591493	0.295746	+	0.955267i	$0.404432\pi$
$972$	1.39268e7	0.472810
$973$	8.57999e7	2.90539
$974$	−1.85997e7	−0.628215
$975$	3.16051e7	1.06475
$976$	−2.13017e6	−0.0715797
$977$	−1.54100e7	−0.516496	−0.258248	−	0.966079i	$-0.583145\pi$
$977$	−1.54100e7	−0.516496	−0.258248	+	0.966079i	$0.583145\pi$
$978$	1.83524e7	0.613542
$979$	−3.85040e7	−1.28395
$980$	5.67334e7	1.88701
$981$	2.61254e6	0.0866744
$982$	3.29509e7	1.09041
$983$	5.45907e7	1.80192	0.900959	−	0.433903i	$-0.142864\pi$
$983$	5.45907e7	1.80192	0.900959	+	0.433903i	$0.142864\pi$
$984$	−1.07055e7	−0.352466
$985$	1.42832e7	0.469068
$986$	1.35621e6	0.0444258
$987$	−5.11880e7	−1.67254
$988$	0	0
$989$	−1.53862e7	−0.500197
$990$	−1.78142e7	−0.577667
$991$	4.29839e7	1.39034	0.695171	−	0.718845i	$-0.255329\pi$
$991$	4.29839e7	1.39034	0.695171	+	0.718845i	$0.255329\pi$
$992$	6.80740e6	0.219635
$993$	−2.35410e7	−0.757622
$994$	−2.89810e7	−0.930353
$995$	−2.92241e7	−0.935800
$996$	−5.68883e6	−0.181708
$997$	−5.32456e7	−1.69647	−0.848234	−	0.529622i	$-0.822334\pi$
$997$	−5.32456e7	−1.69647	−0.848234	+	0.529622i	$0.822334\pi$
$998$	−1.49196e7	−0.474166
$999$	3.08539e7	0.978130

Display

a_p

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p

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a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

a_p

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a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.6.a.r.1.10		15
19.6	even	9		38.6.e.b.17.4	yes	30
19.16	even	9		38.6.e.b.9.4	✓	30
19.18	odd	2		722.6.a.q.1.6		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.9.4	✓	30	19.16	even	9
38.6.e.b.17.4	yes	30	19.6	even	9
722.6.a.q.1.6		15	19.18	odd	2
722.6.a.r.1.10		15	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 $\F_{q}$

Introduction

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Groups

Database

Properties

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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	722.6.a.r.1.10

Level	$722$
Weight	$6$
Character	722.1
Self dual	yes
Analytic conductor	$115.797$
Analytic rank	$0$
Dimension	$15$
CM	no
Inner twists	$1$