LMFDB - Embedded newform 722.6.a.q.1.7

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			1.7
Root			$-3.66933$ 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)

$f(q)$	$=$	$q-4.00000 q^{2} -2.13724 q^{3} +16.0000 q^{4} -10.7945 q^{5} +8.54896 q^{6} -195.108 q^{7} -64.0000 q^{8} -238.432 q^{9} +43.1779 q^{10} -17.0907 q^{11} -34.1959 q^{12} +263.186 q^{13} +780.432 q^{14} +23.0704 q^{15} +256.000 q^{16} -577.725 q^{17} +953.729 q^{18} -172.712 q^{20} +416.993 q^{21} +68.3627 q^{22} -565.508 q^{23} +136.783 q^{24} -3008.48 q^{25} -1052.74 q^{26} +1028.94 q^{27} -3121.73 q^{28} -6712.34 q^{29} -92.2816 q^{30} -9280.96 q^{31} -1024.00 q^{32} +36.5269 q^{33} +2310.90 q^{34} +2106.09 q^{35} -3814.92 q^{36} -1713.65 q^{37} -562.492 q^{39} +690.846 q^{40} -1219.00 q^{41} -1667.97 q^{42} -15472.5 q^{43} -273.451 q^{44} +2573.75 q^{45} +2262.03 q^{46} +1492.69 q^{47} -547.134 q^{48} +21260.1 q^{49} +12033.9 q^{50} +1234.74 q^{51} +4210.97 q^{52} -25666.2 q^{53} -4115.75 q^{54} +184.485 q^{55} +12486.9 q^{56} +26849.4 q^{58} -2971.17 q^{59} +369.126 q^{60} +11469.3 q^{61} +37123.8 q^{62} +46520.0 q^{63} +4096.00 q^{64} -2840.95 q^{65} -146.108 q^{66} +4196.56 q^{67} -9243.60 q^{68} +1208.63 q^{69} -8424.35 q^{70} -67410.0 q^{71} +15259.7 q^{72} -45625.5 q^{73} +6854.61 q^{74} +6429.85 q^{75} +3334.53 q^{77} +2249.97 q^{78} -33434.2 q^{79} -2763.39 q^{80} +55739.9 q^{81} +4876.00 q^{82} +66875.6 q^{83} +6671.88 q^{84} +6236.23 q^{85} +61890.1 q^{86} +14345.9 q^{87} +1093.80 q^{88} +86080.7 q^{89} -10295.0 q^{90} -51349.6 q^{91} -9048.13 q^{92} +19835.6 q^{93} -5970.77 q^{94} +2188.53 q^{96} -130701. q^{97} -85040.4 q^{98} +4074.97 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$15 q - 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} - 960 q^{8} + 2127 q^{9} - 432 q^{10} + 126 q^{11} + 114 q^{13} - 336 q^{14} + 3840 q^{16} + 4119 q^{17} - 8508 q^{18} + 1728 q^{20} - 3408 q^{21} - 504 q^{22}+ \cdots - 149895 q^{99}+O(q^{100})$ 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
$3$	−2.13724	−0.137104	−0.0685520	−	0.997648i	$-0.521838\pi$
$3$	−2.13724	−0.137104	−0.0685520	+	0.997648i	$0.521838\pi$
$4$	16.0000	0.500000
$5$	−10.7945	−0.193097	−0.0965487	−	0.995328i	$-0.530780\pi$
$5$	−10.7945	−0.193097	−0.0965487	+	0.995328i	$0.530780\pi$
$6$	8.54896	0.0969472
$7$	−195.108	−1.50498	−0.752488	−	0.658606i	$-0.771147\pi$
$7$	−195.108	−1.50498	−0.752488	+	0.658606i	$0.771147\pi$
$8$	−64.0000	−0.353553
$9$	−238.432	−0.981202
$10$	43.1779	0.136541
$11$	−17.0907	−0.0425870	−0.0212935	−	0.999773i	$-0.506778\pi$
$11$	−17.0907	−0.0425870	−0.0212935	+	0.999773i	$0.506778\pi$
$12$	−34.1959	−0.0685520
$13$	263.186	0.431921	0.215960	−	0.976402i	$-0.430712\pi$
$13$	263.186	0.431921	0.215960	+	0.976402i	$0.430712\pi$
$14$	780.432	1.06418
$15$	23.0704	0.0264744
$16$	256.000	0.250000
$17$	−577.725	−0.484840	−0.242420	−	0.970171i	$-0.577941\pi$
$17$	−577.725	−0.484840	−0.242420	+	0.970171i	$0.577941\pi$
$18$	953.729	0.693815
$19$	0	0
$19$	0	0
$20$	−172.712	−0.0965487
$21$	416.993	0.206338
$22$	68.3627	0.0301136
$23$	−565.508	−0.222905	−0.111452	−	0.993770i	$-0.535550\pi$
$23$	−565.508	−0.222905	−0.111452	+	0.993770i	$0.535550\pi$
$24$	136.783	0.0484736
$25$	−3008.48	−0.962713
$26$	−1052.74	−0.305414
$27$	1028.94	0.271631
$28$	−3121.73	−0.752488
$29$	−6712.34	−1.48210	−0.741052	−	0.671447i	$-0.765673\pi$
$29$	−6712.34	−1.48210	−0.741052	+	0.671447i	$0.765673\pi$
$30$	−92.2816	−0.0187203
$31$	−9280.96	−1.73456	−0.867279	−	0.497823i	$-0.834133\pi$
$31$	−9280.96	−1.73456	−0.867279	+	0.497823i	$0.834133\pi$
$32$	−1024.00	−0.176777
$33$	36.5269	0.00583886
$34$	2310.90	0.342834
$35$	2106.09	0.290607
$36$	−3814.92	−0.490601
$37$	−1713.65	−0.205787	−0.102894	−	0.994692i	$-0.532810\pi$
$37$	−1713.65	−0.205787	−0.102894	+	0.994692i	$0.532810\pi$
$38$	0	0
$39$	−562.492	−0.0592181
$40$	690.846	0.0682703
$41$	−1219.00	−0.113252	−0.0566258	−	0.998395i	$-0.518034\pi$
$41$	−1219.00	−0.113252	−0.0566258	+	0.998395i	$0.518034\pi$
$42$	−1667.97	−0.145903
$43$	−15472.5	−1.27612	−0.638058	−	0.769988i	$-0.720262\pi$
$43$	−15472.5	−1.27612	−0.638058	+	0.769988i	$0.720262\pi$
$44$	−273.451	−0.0212935
$45$	2573.75	0.189468
$46$	2262.03	0.157617
$47$	1492.69	0.0985657	0.0492829	−	0.998785i	$-0.484306\pi$
$47$	1492.69	0.0985657	0.0492829	+	0.998785i	$0.484306\pi$
$48$	−547.134	−0.0342760
$49$	21260.1	1.26495
$50$	12033.9	0.680741
$51$	1234.74	0.0664736
$52$	4210.97	0.215960
$53$	−25666.2	−1.25508	−0.627541	−	0.778583i	$-0.715939\pi$
$53$	−25666.2	−1.25508	−0.627541	+	0.778583i	$0.715939\pi$
$54$	−4115.75	−0.192072
$55$	184.485	0.00822345
$56$	12486.9	0.532090
$57$	0	0
$58$	26849.4	1.04801
$59$	−2971.17	−0.111121	−0.0555607	−	0.998455i	$-0.517695\pi$
$59$	−2971.17	−0.111121	−0.0555607	+	0.998455i	$0.517695\pi$
$60$	369.126	0.0132372
$61$	11469.3	0.394651	0.197326	−	0.980338i	$-0.436774\pi$
$61$	11469.3	0.394651	0.197326	+	0.980338i	$0.436774\pi$
$62$	37123.8	1.22652
$63$	46520.0	1.47669
$64$	4096.00	0.125000
$65$	−2840.95	−0.0834028
$66$	−146.108	−0.00412870
$67$	4196.56	0.114211	0.0571053	−	0.998368i	$-0.481813\pi$
$67$	4196.56	0.114211	0.0571053	+	0.998368i	$0.481813\pi$
$68$	−9243.60	−0.242420
$69$	1208.63	0.0305611
$70$	−8424.35	−0.205490
$71$	−67410.0	−1.58701	−0.793503	−	0.608567i	$-0.791745\pi$
$71$	−67410.0	−1.58701	−0.793503	+	0.608567i	$0.791745\pi$
$72$	15259.7	0.346907
$73$	−45625.5	−1.00208	−0.501038	−	0.865425i	$-0.667048\pi$
$73$	−45625.5	−1.00208	−0.501038	+	0.865425i	$0.667048\pi$
$74$	6854.61	0.145514
$75$	6429.85	0.131992
$76$	0	0
$77$	3334.53	0.0640925
$78$	2249.97	0.0418735
$79$	−33434.2	−0.602730	−0.301365	−	0.953509i	$-0.597442\pi$
$79$	−33434.2	−0.602730	−0.301365	+	0.953509i	$0.597442\pi$
$80$	−2763.39	−0.0482744
$81$	55739.9	0.943961
$82$	4876.00	0.0800809
$83$	66875.6	1.06555	0.532773	−	0.846258i	$-0.321150\pi$
$83$	66875.6	1.06555	0.532773	+	0.846258i	$0.321150\pi$
$84$	6671.88	0.103169
$85$	6236.23	0.0936214
$86$	61890.1	0.902351
$87$	14345.9	0.203203
$88$	1093.80	0.0150568
$89$	86080.7	1.15194	0.575972	−	0.817470i	$-0.304624\pi$
$89$	86080.7	1.15194	0.575972	+	0.817470i	$0.304624\pi$
$90$	−10295.0	−0.133974
$91$	−51349.6	−0.650031
$92$	−9048.13	−0.111452
$93$	19835.6	0.237815
$94$	−5970.77	−0.0696965
$95$	0	0
$96$	2188.53	0.0242368
$97$	−130701.	−1.41042	−0.705211	−	0.708998i	$-0.749148\pi$
$97$	−130701.	−1.41042	−0.705211	+	0.708998i	$0.749148\pi$
$98$	−85040.4	−0.894458
$99$	4074.97	0.0417865
$100$	−48135.7	−0.481357
$101$	−18239.4	−0.177913	−0.0889564	−	0.996036i	$-0.528353\pi$
$101$	−18239.4	−0.177913	−0.0889564	+	0.996036i	$0.528353\pi$
$102$	−4938.95	−0.0470039
$103$	−26800.0	−0.248910	−0.124455	−	0.992225i	$-0.539718\pi$
$103$	−26800.0	−0.248910	−0.124455	+	0.992225i	$0.539718\pi$
$104$	−16843.9	−0.152707
$105$	−4501.22	−0.0398434
$106$	102665.	0.887478
$107$	−168247.	−1.42065	−0.710327	−	0.703872i	$-0.751453\pi$
$107$	−168247.	−1.42065	−0.710327	+	0.703872i	$0.751453\pi$
$108$	16463.0	0.135815
$109$	16094.3	0.129750	0.0648748	−	0.997893i	$-0.479335\pi$
$109$	16094.3	0.129750	0.0648748	+	0.997893i	$0.479335\pi$
$110$	−737.939	−0.00581486
$111$	3662.49	0.0282143
$112$	−49947.6	−0.376244
$113$	103218.	0.760427	0.380213	−	0.924899i	$-0.375851\pi$
$113$	103218.	0.760427	0.380213	+	0.924899i	$0.375851\pi$
$114$	0	0
$115$	6104.36	0.0430423
$116$	−107397.	−0.741052
$117$	−62752.0	−0.423802
$118$	11884.7	0.0785747
$119$	112719.	0.729673
$120$	−1476.51	−0.00936013
$121$	−160759.	−0.998186
$122$	−45877.3	−0.279061
$123$	2605.30	0.0155272
$124$	−148495.	−0.867279
$125$	66207.7	0.378995
$126$	−186080.	−1.04418
$127$	213253.	1.17324	0.586620	−	0.809863i	$-0.300458\pi$
$127$	213253.	1.17324	0.586620	+	0.809863i	$0.300458\pi$
$128$	−16384.0	−0.0883883
$129$	33068.5	0.174961
$130$	11363.8	0.0589747
$131$	−322898.	−1.64394	−0.821971	−	0.569529i	$-0.807126\pi$
$131$	−322898.	−1.64394	−0.821971	+	0.569529i	$0.807126\pi$
$132$	584.430	0.00291943
$133$	0	0
$134$	−16786.2	−0.0807591
$135$	−11106.8	−0.0524512
$136$	36974.4	0.171417
$137$	−297608.	−1.35470	−0.677349	−	0.735662i	$-0.736871\pi$
$137$	−297608.	−1.35470	−0.677349	+	0.735662i	$0.736871\pi$
$138$	−4834.51	−0.0216100
$139$	−42527.0	−0.186693	−0.0933464	−	0.995634i	$-0.529756\pi$
$139$	−42527.0	−0.186693	−0.0933464	+	0.995634i	$0.529756\pi$
$140$	33697.4	0.145304
$141$	−3190.25	−0.0135138
$142$	269640.	1.12218
$143$	−4498.02	−0.0183942
$144$	−61038.6	−0.245301
$145$	72456.2	0.286191
$146$	182502.	0.708574
$147$	−45437.9	−0.173430
$148$	−27418.4	−0.102894
$149$	397969.	1.46853	0.734266	−	0.678862i	$-0.237526\pi$
$149$	397969.	1.46853	0.734266	+	0.678862i	$0.237526\pi$
$150$	−25719.4	−0.0933324
$151$	312906.	1.11679	0.558394	−	0.829576i	$-0.311418\pi$
$151$	312906.	1.11679	0.558394	+	0.829576i	$0.311418\pi$
$152$	0	0
$153$	137748.	0.475726
$154$	−13338.1	−0.0453202
$155$	100183.	0.334939
$156$	−8999.87	−0.0296091
$157$	451929.	1.46326	0.731629	−	0.681703i	$-0.238760\pi$
$157$	451929.	1.46326	0.731629	+	0.681703i	$0.238760\pi$
$158$	133737.	0.426195
$159$	54854.9	0.172077
$160$	11053.5	0.0341351
$161$	110335.	0.335466
$162$	−222960.	−0.667481
$163$	440059.	1.29730	0.648652	−	0.761085i	$-0.275333\pi$
$163$	440059.	1.29730	0.648652	+	0.761085i	$0.275333\pi$
$164$	−19504.0	−0.0566258
$165$	−394.289	−0.00112747
$166$	−267502.	−0.753455
$167$	−121109.	−0.336034	−0.168017	−	0.985784i	$-0.553736\pi$
$167$	−121109.	−0.336034	−0.168017	+	0.985784i	$0.553736\pi$
$168$	−26687.5	−0.0729517
$169$	−302026.	−0.813444
$170$	−24944.9	−0.0662003
$171$	0	0
$172$	−247561.	−0.638058
$173$	−562940.	−1.43003	−0.715017	−	0.699107i	$-0.753581\pi$
$173$	−562940.	−1.43003	−0.715017	+	0.699107i	$0.753581\pi$
$174$	−57383.5	−0.143686
$175$	586978.	1.44886
$176$	−4375.21	−0.0106468
$177$	6350.11	0.0152352
$178$	−344323.	−0.814547
$179$	560560.	1.30764	0.653822	−	0.756648i	$-0.273164\pi$
$179$	560560.	1.30764	0.653822	+	0.756648i	$0.273164\pi$
$180$	41180.0	0.0947338
$181$	467614.	1.06094	0.530471	−	0.847703i	$-0.322015\pi$
$181$	467614.	1.06094	0.530471	+	0.847703i	$0.322015\pi$
$182$	205399.	0.459641
$183$	−24512.7	−0.0541083
$184$	36192.5	0.0788087
$185$	18498.0	0.0397370
$186$	−79342.6	−0.168160
$187$	9873.70	0.0206479
$188$	23883.1	0.0492829
$189$	−200754.	−0.408798
$190$	0	0
$191$	50390.1	0.0999452	0.0499726	−	0.998751i	$-0.484087\pi$
$191$	50390.1	0.0999452	0.0499726	+	0.998751i	$0.484087\pi$
$192$	−8754.14	−0.0171380
$193$	−817525.	−1.57982	−0.789910	−	0.613223i	$-0.789873\pi$
$193$	−817525.	−1.57982	−0.789910	+	0.613223i	$0.789873\pi$
$194$	522803.	0.997319
$195$	6071.80	0.0114349
$196$	340162.	0.632477
$197$	455871.	0.836905	0.418453	−	0.908239i	$-0.362573\pi$
$197$	455871.	0.836905	0.418453	+	0.908239i	$0.362573\pi$
$198$	−16299.9	−0.0295475
$199$	547135.	0.979403	0.489702	−	0.871890i	$-0.337106\pi$
$199$	547135.	0.979403	0.489702	+	0.871890i	$0.337106\pi$
$200$	192543.	0.340371
$201$	−8969.06	−0.0156587
$202$	72957.6	0.125803
$203$	1.30963e6	2.23053
$204$	19755.8	0.0332368
$205$	13158.5	0.0218686
$206$	107200.	0.176006
$207$	134835.	0.218715
$208$	67375.6	0.107980
$209$	0	0
$210$	18004.9	0.0281736
$211$	−31898.8	−0.0493251	−0.0246625	−	0.999696i	$-0.507851\pi$
$211$	−31898.8	−0.0493251	−0.0246625	+	0.999696i	$0.507851\pi$
$212$	−410660.	−0.627541
$213$	144071.	0.217585
$214$	672989.	1.00455
$215$	167018.	0.246415
$216$	−65851.9	−0.0960360
$217$	1.81079e6	2.61047
$218$	−64377.2	−0.0917468
$219$	97512.7	0.137389
$220$	2951.76	0.00411172
$221$	−152049.	−0.209413
$222$	−14649.9	−0.0199505
$223$	−1.22713e6	−1.65245	−0.826224	−	0.563341i	$-0.809516\pi$
$223$	−1.22713e6	−1.65245	−0.826224	+	0.563341i	$0.809516\pi$
$224$	199790.	0.266045
$225$	717318.	0.944617
$226$	−412870.	−0.537703
$227$	−243999.	−0.314285	−0.157143	−	0.987576i	$-0.550228\pi$
$227$	−243999.	−0.314285	−0.157143	+	0.987576i	$0.550228\pi$
$228$	0	0
$229$	−1.29079e6	−1.62654	−0.813272	−	0.581884i	$-0.802316\pi$
$229$	−1.29079e6	−1.62654	−0.813272	+	0.581884i	$0.802316\pi$
$230$	−24417.4	−0.0304355
$231$	−7126.69	−0.00878734
$232$	429590.	0.524003
$233$	1.23246e6	1.48725	0.743624	−	0.668598i	$-0.233105\pi$
$233$	1.23246e6	1.48725	0.743624	+	0.668598i	$0.233105\pi$
$234$	251008.	0.299673
$235$	−16112.8	−0.0190328
$236$	−47538.7	−0.0555607
$237$	71456.9	0.0826368
$238$	−450875.	−0.515957
$239$	−299709.	−0.339395	−0.169698	−	0.985496i	$-0.554279\pi$
$239$	−299709.	−0.339395	−0.169698	+	0.985496i	$0.554279\pi$
$240$	5906.02	0.00661861
$241$	334121.	0.370563	0.185281	−	0.982686i	$-0.440680\pi$
$241$	334121.	0.370563	0.185281	+	0.982686i	$0.440680\pi$
$242$	643036.	0.705824
$243$	−369161.	−0.401052
$244$	183509.	0.197326
$245$	−229492.	−0.244260
$246$	−10421.2	−0.0109794
$247$	0	0
$248$	593981.	0.613259
$249$	−142929.	−0.146091
$250$	−264831.	−0.267990
$251$	1.78722e6	1.79058	0.895290	−	0.445484i	$-0.146968\pi$
$251$	1.78722e6	1.79058	0.895290	+	0.445484i	$0.146968\pi$
$252$	744320.	0.738343
$253$	9664.91	0.00949285
$254$	−853013.	−0.829605
$255$	−13328.3	−0.0128359
$256$	65536.0	0.0625000
$257$	−1.30644e6	−1.23384	−0.616919	−	0.787026i	$-0.711619\pi$
$257$	−1.30644e6	−1.23384	−0.616919	+	0.787026i	$0.711619\pi$
$258$	−132274.	−0.123716
$259$	334347.	0.309705
$260$	−45455.2	−0.0417014
$261$	1.60044e6	1.45424
$262$	1.29159e6	1.16244
$263$	2.21793e6	1.97724	0.988619	−	0.150441i	$-0.0480694\pi$
$263$	2.21793e6	1.97724	0.988619	+	0.150441i	$0.0480694\pi$
$264$	−2337.72	−0.00206435
$265$	277053.	0.242353
$266$	0	0
$267$	−183975.	−0.157936
$268$	67145.0	0.0571053
$269$	−433462.	−0.365233	−0.182616	−	0.983184i	$-0.558457\pi$
$269$	−433462.	−0.365233	−0.182616	+	0.983184i	$0.558457\pi$
$270$	44427.3	0.0370886
$271$	−1.44684e6	−1.19673	−0.598366	−	0.801223i	$-0.704183\pi$
$271$	−1.44684e6	−1.19673	−0.598366	+	0.801223i	$0.704183\pi$
$272$	−147898.	−0.121210
$273$	109747.	0.0891219
$274$	1.19043e6	0.957916
$275$	51416.9	0.0409991
$276$	19338.0	0.0152806
$277$	1.64595e6	1.28889	0.644446	−	0.764650i	$-0.277088\pi$
$277$	1.64595e6	1.28889	0.644446	+	0.764650i	$0.277088\pi$
$278$	170108.	0.132012
$279$	2.21288e6	1.70195
$280$	−134790.	−0.102745
$281$	−925545.	−0.699249	−0.349625	−	0.936890i	$-0.613691\pi$
$281$	−925545.	−0.699249	−0.349625	+	0.936890i	$0.613691\pi$
$282$	12761.0	0.00955567
$283$	−2.12008e6	−1.57357	−0.786786	−	0.617226i	$-0.788257\pi$
$283$	−2.12008e6	−1.57357	−0.786786	+	0.617226i	$0.788257\pi$
$284$	−1.07856e6	−0.793503
$285$	0	0
$286$	17992.1	0.0130067
$287$	237836.	0.170441
$288$	244155.	0.173454
$289$	−1.08609e6	−0.764930
$290$	−289825.	−0.202367
$291$	279339.	0.193375
$292$	−730008.	−0.501038
$293$	943155.	0.641821	0.320910	−	0.947110i	$-0.396011\pi$
$293$	943155.	0.641821	0.320910	+	0.947110i	$0.396011\pi$
$294$	181752.	0.122634
$295$	32072.2	0.0214572
$296$	109674.	0.0727568
$297$	−17585.2	−0.0115680
$298$	−1.59188e6	−1.03841
$299$	−148834.	−0.0962771
$300$	102878.	0.0659960
$301$	3.01881e6	1.92053
$302$	−1.25162e6	−0.789689
$303$	38982.0	0.0243926
$304$	0	0
$305$	−123805.	−0.0762062
$306$	−550993.	−0.336389
$307$	2.09504e6	1.26867	0.634333	−	0.773060i	$-0.281275\pi$
$307$	2.09504e6	1.26867	0.634333	+	0.773060i	$0.281275\pi$
$308$	53352.4	0.0320463
$309$	57278.0	0.0341265
$310$	−400732.	−0.236837
$311$	638862.	0.374547	0.187273	−	0.982308i	$-0.440035\pi$
$311$	638862.	0.374547	0.187273	+	0.982308i	$0.440035\pi$
$312$	35999.5	0.0209368
$313$	1.13152e6	0.652833	0.326417	−	0.945226i	$-0.394159\pi$
$313$	1.13152e6	0.652833	0.326417	+	0.945226i	$0.394159\pi$
$314$	−1.80772e6	−1.03468
$315$	−502159.	−0.285144
$316$	−534947.	−0.301365
$317$	1.17244e6	0.655301	0.327651	−	0.944799i	$-0.393743\pi$
$317$	1.17244e6	0.655301	0.327651	+	0.944799i	$0.393743\pi$
$318$	−219420.	−0.121677
$319$	114718.	0.0631185
$320$	−44214.2	−0.0241372
$321$	359585.	0.194777
$322$	−441340.	−0.237210
$323$	0	0
$324$	891839.	0.471980
$325$	−791789.	−0.415816
$326$	−1.76023e6	−0.917332
$327$	−34397.4	−0.0177892
$328$	78016.0	0.0400405
$329$	−291236.	−0.148339
$330$	1577.15	0.000797241 0
$331$	738377.	0.370432	0.185216	−	0.982698i	$-0.440702\pi$
$331$	738377.	0.370432	0.185216	+	0.982698i	$0.440702\pi$
$332$	1.07001e6	0.532773
$333$	408590.	0.201919
$334$	484434.	0.237612
$335$	−45299.7	−0.0220538
$336$	106750.	0.0515846
$337$	−1.96851e6	−0.944197	−0.472099	−	0.881546i	$-0.656503\pi$
$337$	−1.96851e6	−0.944197	−0.472099	+	0.881546i	$0.656503\pi$
$338$	1.20810e6	0.575192
$339$	−220601.	−0.104258
$340$	99779.8	0.0468107
$341$	158618.	0.0738697
$342$	0	0
$343$	−868834.	−0.398751
$344$	990242.	0.451175
$345$	−13046.5	−0.00590128
$346$	2.25176e6	1.01119
$347$	2.63680e6	1.17559	0.587793	−	0.809012i	$-0.299997\pi$
$347$	2.63680e6	1.17559	0.587793	+	0.809012i	$0.299997\pi$
$348$	229534.	0.101601
$349$	−496602.	−0.218245	−0.109123	−	0.994028i	$-0.534804\pi$
$349$	−496602.	−0.218245	−0.109123	+	0.994028i	$0.534804\pi$
$350$	−2.34791e6	−1.02450
$351$	270802.	0.117323
$352$	17500.9	0.00752840
$353$	−2.17439e6	−0.928753	−0.464377	−	0.885638i	$-0.653722\pi$
$353$	−2.17439e6	−0.928753	−0.464377	+	0.885638i	$0.653722\pi$
$354$	−25400.4	−0.0107729
$355$	727655.	0.306447
$356$	1.37729e6	0.575972
$357$	−240907.	−0.100041
$358$	−2.24224e6	−0.924644
$359$	−257893.	−0.105610	−0.0528048	−	0.998605i	$-0.516816\pi$
$359$	−257893.	−0.105610	−0.0528048	+	0.998605i	$0.516816\pi$
$360$	−164720.	−0.0669869
$361$	0	0
$362$	−1.87046e6	−0.750199
$363$	343581.	0.136855
$364$	−821594.	−0.325015
$365$	492503.	0.193498
$366$	98050.9	0.0382603
$367$	−2.62085e6	−1.01573	−0.507863	−	0.861438i	$-0.669564\pi$
$367$	−2.62085e6	−1.01573	−0.507863	+	0.861438i	$0.669564\pi$
$368$	−144770.	−0.0557262
$369$	290649.	0.111123
$370$	−73991.9	−0.0280983
$371$	5.00768e6	1.88887
$372$	317370.	0.118907
$373$	3.45002e6	1.28396	0.641978	−	0.766723i	$-0.278114\pi$
$373$	3.45002e6	1.28396	0.641978	+	0.766723i	$0.278114\pi$
$374$	−39494.8	−0.0146003
$375$	−141502.	−0.0519617
$376$	−95532.4	−0.0348482
$377$	−1.76659e6	−0.640152
$378$	803015.	0.289064
$379$	3.09133e6	1.10547	0.552736	−	0.833356i	$-0.313584\pi$
$379$	3.09133e6	1.10547	0.552736	+	0.833356i	$0.313584\pi$
$380$	0	0
$381$	−455774.	−0.160856
$382$	−201560.	−0.0706719
$383$	−625110.	−0.217751	−0.108875	−	0.994055i	$-0.534725\pi$
$383$	−625110.	−0.217751	−0.108875	+	0.994055i	$0.534725\pi$
$384$	35016.6	0.0121184
$385$	−35994.5	−0.0123761
$386$	3.27010e6	1.11710
$387$	3.68915e6	1.25213
$388$	−2.09121e6	−0.705211
$389$	−3.13033e6	−1.04886	−0.524428	−	0.851455i	$-0.675721\pi$
$389$	−3.13033e6	−1.04886	−0.524428	+	0.851455i	$0.675721\pi$
$390$	−24287.2	−0.00808567
$391$	326708.	0.108073
$392$	−1.36065e6	−0.447229
$393$	690110.	0.225391
$394$	−1.82348e6	−0.591781
$395$	360905.	0.116386
$396$	65199.5	0.0208933
$397$	4.04982e6	1.28961	0.644807	−	0.764346i	$-0.276938\pi$
$397$	4.04982e6	1.28961	0.644807	+	0.764346i	$0.276938\pi$
$398$	−2.18854e6	−0.692543
$399$	0	0
$400$	−770171.	−0.240678
$401$	−4.13849e6	−1.28523	−0.642615	−	0.766189i	$-0.722151\pi$
$401$	−4.13849e6	−1.28523	−0.642615	+	0.766189i	$0.722151\pi$
$402$	35876.3	0.0110724
$403$	−2.44262e6	−0.749191
$404$	−291830.	−0.0889564
$405$	−601683.	−0.182276
$406$	−5.23852e6	−1.57722
$407$	29287.5	0.00876387
$408$	−79023.2	−0.0235020
$409$	3.28825e6	0.971979	0.485990	−	0.873965i	$-0.338459\pi$
$409$	3.28825e6	0.971979	0.485990	+	0.873965i	$0.338459\pi$
$410$	−52633.8	−0.0154634
$411$	636059.	0.185735
$412$	−428800.	−0.124455
$413$	579699.	0.167235
$414$	−539341.	−0.154655
$415$	−721887.	−0.205754
$416$	−269502.	−0.0763535
$417$	90890.4	0.0255964
$418$	0	0
$419$	3.77322e6	1.04997	0.524985	−	0.851111i	$-0.324071\pi$
$419$	3.77322e6	1.04997	0.524985	+	0.851111i	$0.324071\pi$
$420$	−72019.5	−0.0199217
$421$	392214.	0.107849	0.0539247	−	0.998545i	$-0.482827\pi$
$421$	392214.	0.107849	0.0539247	+	0.998545i	$0.482827\pi$
$422$	127595.	0.0348781
$423$	−355906.	−0.0967129
$424$	1.64264e6	0.443739
$425$	1.73807e6	0.466762
$426$	−576286.	−0.153856
$427$	−2.23776e6	−0.593941
$428$	−2.69195e6	−0.710327
$429$	9613.36	0.00252192
$430$	−668072.	−0.174242
$431$	−5.99845e6	−1.55541	−0.777707	−	0.628627i	$-0.783617\pi$
$431$	−5.99845e6	−1.55541	−0.777707	+	0.628627i	$0.783617\pi$
$432$	263408.	0.0679077
$433$	−2.29152e6	−0.587359	−0.293680	−	0.955904i	$-0.594880\pi$
$433$	−2.29152e6	−0.587359	−0.293680	+	0.955904i	$0.594880\pi$
$434$	−7.24315e6	−1.84588
$435$	−154856.	−0.0392379
$436$	257509.	0.0648748
$437$	0	0
$438$	−390051.	−0.0971484
$439$	−168827.	−0.0418101	−0.0209050	−	0.999781i	$-0.506655\pi$
$439$	−168827.	−0.0418101	−0.0209050	+	0.999781i	$0.506655\pi$
$440$	−11807.0	−0.00290743
$441$	−5.06909e6	−1.24118
$442$	608196.	0.148077
$443$	−3.11020e6	−0.752972	−0.376486	−	0.926422i	$-0.622868\pi$
$443$	−3.11020e6	−0.752972	−0.376486	+	0.926422i	$0.622868\pi$
$444$	58599.8	0.0141071
$445$	−929196.	−0.222437
$446$	4.90851e6	1.16846
$447$	−850555.	−0.201342
$448$	−799162.	−0.188122
$449$	−5.05277e6	−1.18281	−0.591403	−	0.806376i	$-0.701426\pi$
$449$	−5.05277e6	−1.18281	−0.591403	+	0.806376i	$0.701426\pi$
$450$	−2.86927e6	−0.667945
$451$	20833.5	0.00482305
$452$	1.65148e6	0.380213
$453$	−668755.	−0.153116
$454$	975997.	0.222233
$455$	554292.	0.125519
$456$	0	0
$457$	2.57909e6	0.577665	0.288832	−	0.957380i	$-0.406733\pi$
$457$	2.57909e6	0.577665	0.288832	+	0.957380i	$0.406733\pi$
$458$	5.16315e6	1.15014
$459$	−594442.	−0.131698
$460$	97669.8	0.0215212
$461$	−7.85829e6	−1.72217	−0.861085	−	0.508461i	$-0.830214\pi$
$461$	−7.85829e6	−1.72217	−0.861085	+	0.508461i	$0.830214\pi$
$462$	28506.7	0.00621359
$463$	4.86915e6	1.05560	0.527802	−	0.849367i	$-0.323016\pi$
$463$	4.86915e6	1.05560	0.527802	+	0.849367i	$0.323016\pi$
$464$	−1.71836e6	−0.370526
$465$	−214115.	−0.0459214
$466$	−4.92984e6	−1.05164
$467$	1.77798e6	0.377254	0.188627	−	0.982049i	$-0.439596\pi$
$467$	1.77798e6	0.377254	0.188627	+	0.982049i	$0.439596\pi$
$468$	−1.00403e6	−0.211901
$469$	−818782.	−0.171884
$470$	64451.4	0.0134582
$471$	−965881.	−0.200619
$472$	190155.	0.0392873
$473$	264436.	0.0543460
$474$	−285828.	−0.0584330
$475$	0	0
$476$	1.80350e6	0.364837
$477$	6.11966e6	1.23149
$478$	1.19884e6	0.239989
$479$	−9.95603e6	−1.98266	−0.991328	−	0.131408i	$-0.958050\pi$
$479$	−9.95603e6	−1.98266	−0.991328	+	0.131408i	$0.958050\pi$
$480$	−23624.1	−0.00468007
$481$	−451009.	−0.0888838
$482$	−1.33649e6	−0.262027
$483$	−235813.	−0.0459938
$484$	−2.57214e6	−0.499093
$485$	1.41085e6	0.272349
$486$	1.47665e6	0.283586
$487$	758898.	0.144998	0.0724988	−	0.997368i	$-0.476903\pi$
$487$	758898.	0.144998	0.0724988	+	0.997368i	$0.476903\pi$
$488$	−734037.	−0.139530
$489$	−940512.	−0.177866
$490$	917966.	0.172718
$491$	1.85836e6	0.347877	0.173938	−	0.984757i	$-0.444351\pi$
$491$	1.85836e6	0.347877	0.173938	+	0.984757i	$0.444351\pi$
$492$	41684.7	0.00776362
$493$	3.87788e6	0.718584
$494$	0	0
$495$	−43987.1	−0.00806887
$496$	−2.37593e6	−0.433639
$497$	1.31522e7	2.38841
$498$	571717.	0.103302
$499$	−7.70818e6	−1.38580	−0.692900	−	0.721034i	$-0.743667\pi$
$499$	−7.70818e6	−1.38580	−0.692900	+	0.721034i	$0.743667\pi$
$500$	1.05932e6	0.189497
$501$	258838.	0.0460717
$502$	−7.14888e6	−1.26613
$503$	−6.63172e6	−1.16871	−0.584354	−	0.811499i	$-0.698652\pi$
$503$	−6.63172e6	−1.16871	−0.584354	+	0.811499i	$0.698652\pi$
$504$	−2.97728e6	−0.522088
$505$	196885.	0.0343545
$506$	−38659.6	−0.00671246
$507$	645503.	0.111527
$508$	3.41205e6	0.586620
$509$	−1.10429e7	−1.88924	−0.944620	−	0.328166i	$-0.893569\pi$
$509$	−1.10429e7	−1.88924	−0.944620	+	0.328166i	$0.893569\pi$
$510$	53313.3	0.00907634
$511$	8.90190e6	1.50810
$512$	−262144.	−0.0441942
$513$	0	0
$514$	5.22578e6	0.872456
$515$	289292.	0.0480638
$516$	529097.	0.0874804
$517$	−25511.1	−0.00419762
$518$	−1.33739e6	−0.218994
$519$	1.20314e6	0.196064
$520$	181821.	0.0294873
$521$	−5.31342e6	−0.857590	−0.428795	−	0.903402i	$-0.641062\pi$
$521$	−5.31342e6	−0.857590	−0.428795	+	0.903402i	$0.641062\pi$
$522$	−6.40175e6	−1.02831
$523$	−186790.	−0.0298607	−0.0149303	−	0.999889i	$-0.504753\pi$
$523$	−186790.	−0.0298607	−0.0149303	+	0.999889i	$0.504753\pi$
$524$	−5.16636e6	−0.821971
$525$	−1.25451e6	−0.198645
$526$	−8.87173e6	−1.39812
$527$	5.36184e6	0.840983
$528$	9350.88	0.00145971
$529$	−6.11654e6	−0.950314
$530$	−1.10821e6	−0.171370
$531$	708423.	0.109033
$532$	0	0
$533$	−320823.	−0.0489157
$534$	735901.	0.111678
$535$	1.81614e6	0.274325
$536$	−268580.	−0.0403796
$537$	−1.19805e6	−0.179283
$538$	1.73385e6	0.258259
$539$	−363349.	−0.0538707
$540$	−177709.	−0.0262256
$541$	−486912.	−0.0715249	−0.0357624	−	0.999360i	$-0.511386\pi$
$541$	−486912.	−0.0715249	−0.0357624	+	0.999360i	$0.511386\pi$
$542$	5.78735e6	0.846217
$543$	−999405.	−0.145459
$544$	591590.	0.0857085
$545$	−173730.	−0.0250543
$546$	−438986.	−0.0630187
$547$	5.31296e6	0.759221	0.379610	−	0.925146i	$-0.376058\pi$
$547$	5.31296e6	0.759221	0.379610	+	0.925146i	$0.376058\pi$
$548$	−4.76172e6	−0.677349
$549$	−2.73466e6	−0.387233
$550$	−205668.	−0.0289908
$551$	0	0
$552$	−77352.1	−0.0108050
$553$	6.52328e6	0.907095
$554$	−6.58379e6	−0.911384
$555$	−39534.6	−0.00544810
$556$	−680432.	−0.0933464
$557$	−3.60173e6	−0.491897	−0.245948	−	0.969283i	$-0.579099\pi$
$557$	−3.60173e6	−0.491897	−0.245948	+	0.969283i	$0.579099\pi$
$558$	−8.85152e6	−1.20346
$559$	−4.07215e6	−0.551181
$560$	539158.	0.0726518
$561$	−21102.5	−0.00283091
$562$	3.70218e6	0.494444
$563$	1.99766e6	0.265614	0.132807	−	0.991142i	$-0.457601\pi$
$563$	1.99766e6	0.265614	0.132807	+	0.991142i	$0.457601\pi$
$564$	−51043.9	−0.00675688
$565$	−1.11418e6	−0.146836
$566$	8.48033e6	1.11268
$567$	−1.08753e7	−1.42064
$568$	4.31424e6	0.561091
$569$	5.80414e6	0.751549	0.375775	−	0.926711i	$-0.377377\pi$
$569$	5.80414e6	0.751549	0.375775	+	0.926711i	$0.377377\pi$
$570$	0	0
$571$	2.27354e6	0.291818	0.145909	−	0.989298i	$-0.453389\pi$
$571$	2.27354e6	0.291818	0.145909	+	0.989298i	$0.453389\pi$
$572$	−71968.4	−0.00919712
$573$	−107696.	−0.0137029
$574$	−951346.	−0.120520
$575$	1.70132e6	0.214593
$576$	−976618.	−0.122650
$577$	1.48551e7	1.85753	0.928766	−	0.370667i	$-0.120871\pi$
$577$	1.48551e7	1.85753	0.928766	+	0.370667i	$0.120871\pi$
$578$	4.34436e6	0.540887
$579$	1.74725e6	0.216600
$580$	1.15930e6	0.143095
$581$	−1.30480e7	−1.60362
$582$	−1.11736e6	−0.136736
$583$	438653.	0.0534503
$584$	2.92003e6	0.354287
$585$	677375.	0.0818350
$586$	−3.77262e6	−0.453836
$587$	9.06829e6	1.08625	0.543125	−	0.839652i	$-0.317241\pi$
$587$	9.06829e6	1.08625	0.543125	+	0.839652i	$0.317241\pi$
$588$	−727007.	−0.0867152
$589$	0	0
$590$	−128289.	−0.0151726
$591$	−974306.	−0.114743
$592$	−438695.	−0.0514468
$593$	1.15630e6	0.135031	0.0675157	−	0.997718i	$-0.478493\pi$
$593$	1.15630e6	0.135031	0.0675157	+	0.997718i	$0.478493\pi$
$594$	70340.9	0.00817978
$595$	−1.21674e6	−0.140898
$596$	6.36750e6	0.734266
$597$	−1.16936e6	−0.134280
$598$	595335.	0.0680782
$599$	7.49253e6	0.853221	0.426610	−	0.904436i	$-0.359708\pi$
$599$	7.49253e6	0.853221	0.426610	+	0.904436i	$0.359708\pi$
$600$	−411510.	−0.0466662
$601$	−2.46573e6	−0.278458	−0.139229	−	0.990260i	$-0.544462\pi$
$601$	−2.46573e6	−0.278458	−0.139229	+	0.990260i	$0.544462\pi$
$602$	−1.20753e7	−1.35802
$603$	−1.00060e6	−0.112064
$604$	5.00649e6	0.558394
$605$	1.73531e6	0.192747
$606$	−155928.	−0.0172481
$607$	−1.30412e7	−1.43663	−0.718316	−	0.695717i	$-0.755087\pi$
$607$	−1.30412e7	−1.43663	−0.718316	+	0.695717i	$0.755087\pi$
$608$	0	0
$609$	−2.79900e6	−0.305815
$610$	495222.	0.0538859
$611$	392856.	0.0425726
$612$	2.20397e6	0.237863
$613$	−3.93935e6	−0.423422	−0.211711	−	0.977332i	$-0.567904\pi$
$613$	−3.93935e6	−0.423422	−0.211711	+	0.977332i	$0.567904\pi$
$614$	−8.38017e6	−0.897082
$615$	−28122.8	−0.00299827
$616$	−213410.	−0.0226601
$617$	−7.55827e6	−0.799299	−0.399649	−	0.916668i	$-0.630868\pi$
$617$	−7.55827e6	−0.799299	−0.399649	+	0.916668i	$0.630868\pi$
$618$	−229112.	−0.0241311
$619$	−7.12628e6	−0.747543	−0.373772	−	0.927521i	$-0.621936\pi$
$619$	−7.12628e6	−0.747543	−0.373772	+	0.927521i	$0.621936\pi$
$620$	1.60293e6	0.167469
$621$	−581872.	−0.0605478
$622$	−2.55545e6	−0.264845
$623$	−1.67950e7	−1.73365
$624$	−143998.	−0.0148045
$625$	8.68682e6	0.889530
$626$	−4.52609e6	−0.461623
$627$	0	0
$628$	7.23086e6	0.731629
$629$	990019.	0.0997739
$630$	2.00864e6	0.201628
$631$	−8.65646e6	−0.865499	−0.432750	−	0.901514i	$-0.642457\pi$
$631$	−8.65646e6	−0.865499	−0.432750	+	0.901514i	$0.642457\pi$
$632$	2.13979e6	0.213097
$633$	68175.4	0.00676267
$634$	−4.68975e6	−0.463368
$635$	−2.30196e6	−0.226549
$636$	877679.	0.0860385
$637$	5.59536e6	0.546360
$638$	−458874.	−0.0446315
$639$	1.60727e7	1.55717
$640$	176857.	0.0170676
$641$	−1.37461e7	−1.32141	−0.660703	−	0.750648i	$-0.729742\pi$
$641$	−1.37461e7	−1.32141	−0.660703	+	0.750648i	$0.729742\pi$
$642$	−1.43834e6	−0.137728
$643$	5.32095e6	0.507530	0.253765	−	0.967266i	$-0.418331\pi$
$643$	5.32095e6	0.507530	0.253765	+	0.967266i	$0.418331\pi$
$644$	1.76536e6	0.167733
$645$	−356958.	−0.0337845
$646$	0	0
$647$	−1.91181e7	−1.79549	−0.897747	−	0.440512i	$-0.854797\pi$
$647$	−1.91181e7	−1.79549	−0.897747	+	0.440512i	$0.854797\pi$
$648$	−3.56736e6	−0.333741
$649$	50779.3	0.00473233
$650$	3.16716e6	0.294026
$651$	−3.87009e6	−0.357906
$652$	7.04094e6	0.648652
$653$	−1.02824e7	−0.943648	−0.471824	−	0.881693i	$-0.656404\pi$
$653$	−1.02824e7	−0.943648	−0.471824	+	0.881693i	$0.656404\pi$
$654$	137590.	0.0125789
$655$	3.48551e6	0.317441
$656$	−312064.	−0.0283129
$657$	1.08786e7	0.983239
$658$	1.16495e6	0.104892
$659$	−1.01629e7	−0.911598	−0.455799	−	0.890083i	$-0.650646\pi$
$659$	−1.01629e7	−0.911598	−0.455799	+	0.890083i	$0.650646\pi$
$660$	−6308.62	−0.000563734 0
$661$	1.78429e7	1.58841	0.794203	−	0.607653i	$-0.207889\pi$
$661$	1.78429e7	1.58841	0.794203	+	0.607653i	$0.207889\pi$
$662$	−2.95351e6	−0.261935
$663$	324965.	0.0287113
$664$	−4.28004e6	−0.376728
$665$	0	0
$666$	−1.63436e6	−0.142778
$667$	3.79588e6	0.330368
$668$	−1.93774e6	−0.168017
$669$	2.62267e6	0.226557
$670$	181199.	0.0155944
$671$	−196019.	−0.0168070
$672$	−427000.	−0.0364758
$673$	−9.67106e6	−0.823069	−0.411535	−	0.911394i	$-0.635007\pi$
$673$	−9.67106e6	−0.823069	−0.411535	+	0.911394i	$0.635007\pi$
$674$	7.87404e6	0.667648
$675$	−3.09553e6	−0.261503
$676$	−4.83242e6	−0.406722
$677$	1.11065e7	0.931331	0.465665	−	0.884961i	$-0.345815\pi$
$677$	1.11065e7	0.931331	0.465665	+	0.884961i	$0.345815\pi$
$678$	882403.	0.0737213
$679$	2.55008e7	2.12265
$680$	−399119.	−0.0331002
$681$	521485.	0.0430898
$682$	−634471.	−0.0522337
$683$	−323437.	−0.0265301	−0.0132650	−	0.999912i	$-0.504223\pi$
$683$	−323437.	−0.0265301	−0.0132650	+	0.999912i	$0.504223\pi$
$684$	0	0
$685$	3.21252e6	0.261589
$686$	3.47534e6	0.281959
$687$	2.75872e6	0.223006
$688$	−3.96097e6	−0.319029
$689$	−6.75499e6	−0.542096
$690$	52186.0	0.00417283
$691$	7.15505e6	0.570056	0.285028	−	0.958519i	$-0.407997\pi$
$691$	7.15505e6	0.570056	0.285028	+	0.958519i	$0.407997\pi$
$692$	−9.00704e6	−0.715017
$693$	−795058.	−0.0628877
$694$	−1.05472e7	−0.831264
$695$	459057.	0.0360499
$696$	−918137.	−0.0718430
$697$	704246.	0.0549089
$698$	1.98641e6	0.154323
$699$	−2.63407e6	−0.203908
$700$	9.39165e6	0.724431
$701$	−1.67109e7	−1.28441	−0.642206	−	0.766532i	$-0.721981\pi$
$701$	−1.67109e7	−1.28441	−0.642206	+	0.766532i	$0.721981\pi$
$702$	−1.08321e6	−0.0829599
$703$	0	0
$704$	−70003.4	−0.00532338
$705$	34437.0	0.00260947
$706$	8.69755e6	0.656728
$707$	3.55865e6	0.267755
$708$	101602.	0.00761760
$709$	−4.46568e6	−0.333635	−0.166818	−	0.985988i	$-0.553349\pi$
$709$	−4.46568e6	−0.333635	−0.166818	+	0.985988i	$0.553349\pi$
$710$	−2.91062e6	−0.216691
$711$	7.97179e6	0.591401
$712$	−5.50917e6	−0.407273
$713$	5.24846e6	0.386641
$714$	963628.	0.0707398
$715$	48553.8	0.00355188
$716$	8.96896e6	0.653822
$717$	640551.	0.0465325
$718$	1.03157e6	0.0746772
$719$	8.11538e6	0.585446	0.292723	−	0.956197i	$-0.405439\pi$
$719$	8.11538e6	0.585446	0.292723	+	0.956197i	$0.405439\pi$
$720$	658880.	0.0473669
$721$	5.22889e6	0.374603
$722$	0	0
$723$	−714098.	−0.0508056
$724$	7.48183e6	0.530471
$725$	2.01939e7	1.42684
$726$	−1.37432e6	−0.0967714
$727$	8.96301e6	0.628952	0.314476	−	0.949265i	$-0.398171\pi$
$727$	8.96301e6	0.628952	0.314476	+	0.949265i	$0.398171\pi$
$728$	3.28638e6	0.229821
$729$	−1.27558e7	−0.888975
$730$	−1.97001e6	−0.136824
$731$	8.93887e6	0.618713
$732$	−392204.	−0.0270542
$733$	−9.84603e6	−0.676864	−0.338432	−	0.940991i	$-0.609896\pi$
$733$	−9.84603e6	−0.676864	−0.338432	+	0.940991i	$0.609896\pi$
$734$	1.04834e7	0.718227
$735$	490479.	0.0334890
$736$	579080.	0.0394043
$737$	−71722.1	−0.00486389
$738$	−1.16260e6	−0.0785756
$739$	−1.22360e7	−0.824189	−0.412094	−	0.911141i	$-0.635203\pi$
$739$	−1.22360e7	−0.824189	−0.412094	+	0.911141i	$0.635203\pi$
$740$	295968.	0.0198685
$741$	0	0
$742$	−2.00307e7	−1.33563
$743$	−2.14702e6	−0.142680	−0.0713402	−	0.997452i	$-0.522728\pi$
$743$	−2.14702e6	−0.142680	−0.0713402	+	0.997452i	$0.522728\pi$
$744$	−1.26948e6	−0.0840802
$745$	−4.29586e6	−0.283570
$746$	−1.38001e7	−0.907894
$747$	−1.59453e7	−1.04552
$748$	157979.	0.0103240
$749$	3.28263e7	2.13805
$750$	566007.	0.0367425
$751$	−1.97210e7	−1.27594	−0.637968	−	0.770063i	$-0.720225\pi$
$751$	−1.97210e7	−1.27594	−0.637968	+	0.770063i	$0.720225\pi$
$752$	382130.	0.0246414
$753$	−3.81972e6	−0.245496
$754$	7.06637e6	0.452656
$755$	−3.37765e6	−0.215649
$756$	−3.21206e6	−0.204399
$757$	2.62195e7	1.66297	0.831484	−	0.555549i	$-0.187492\pi$
$757$	2.62195e7	1.66297	0.831484	+	0.555549i	$0.187492\pi$
$758$	−1.23653e7	−0.781687
$759$	−20656.2	−0.00130151
$760$	0	0
$761$	2.89872e7	1.81445	0.907225	−	0.420645i	$-0.138196\pi$
$761$	2.89872e7	1.81445	0.907225	+	0.420645i	$0.138196\pi$
$762$	1.82310e6	0.113742
$763$	−3.14013e6	−0.195270
$764$	806242.	0.0499726
$765$	−1.48692e6	−0.0918616
$766$	2.50044e6	0.153973
$767$	−781970.	−0.0479956
$768$	−140066.	−0.00856900
$769$	−1.35288e7	−0.824982	−0.412491	−	0.910962i	$-0.635341\pi$
$769$	−1.35288e7	−0.824982	−0.412491	+	0.910962i	$0.635341\pi$
$770$	143978.	0.00875122
$771$	2.79219e6	0.169164
$772$	−1.30804e7	−0.789910
$773$	−1.28806e7	−0.775331	−0.387665	−	0.921800i	$-0.626718\pi$
$773$	−1.28806e7	−0.775331	−0.387665	+	0.921800i	$0.626718\pi$
$774$	−1.47566e7	−0.885389
$775$	2.79216e7	1.66988
$776$	8.36485e6	0.498659
$777$	−714580.	−0.0424618
$778$	1.25213e7	0.741653
$779$	0	0
$780$	97148.8	0.00571743
$781$	1.15208e6	0.0675859
$782$	−1.30683e6	−0.0764192
$783$	−6.90657e6	−0.402585
$784$	5.44258e6	0.316239
$785$	−4.87834e6	−0.282551
$786$	−2.76044e6	−0.159376
$787$	−332357.	−0.0191279	−0.00956395	−	0.999954i	$-0.503044\pi$
$787$	−332357.	−0.0191279	−0.00956395	+	0.999954i	$0.503044\pi$
$788$	7.29393e6	0.418453
$789$	−4.74026e6	−0.271087
$790$	−1.44362e6	−0.0822971
$791$	−2.01386e7	−1.14442
$792$	−260798.	−0.0147738
$793$	3.01857e6	0.170458
$794$	−1.61993e7	−0.911894
$795$	−592130.	−0.0332276
$796$	8.75415e6	0.489702
$797$	−3.13377e7	−1.74752	−0.873760	−	0.486358i	$-0.838325\pi$
$797$	−3.13377e7	−1.74752	−0.873760	+	0.486358i	$0.838325\pi$
$798$	0	0
$799$	−862366.	−0.0477886
$800$	3.08068e6	0.170185
$801$	−2.05244e7	−1.13029
$802$	1.65540e7	0.908795
$803$	779771.	0.0426754
$804$	−143505.	−0.00782937
$805$	−1.19101e6	−0.0647777
$806$	9.77047e6	0.529758
$807$	926412.	0.0500749
$808$	1.16732e6	0.0629016
$809$	−6.58593e6	−0.353790	−0.176895	−	0.984230i	$-0.556605\pi$
$809$	−6.58593e6	−0.353790	−0.176895	+	0.984230i	$0.556605\pi$
$810$	2.40673e6	0.128889
$811$	2.66167e7	1.42103	0.710514	−	0.703683i	$-0.248463\pi$
$811$	2.66167e7	1.42103	0.710514	+	0.703683i	$0.248463\pi$
$812$	2.09541e7	1.11527
$813$	3.09224e6	0.164077
$814$	−117150.	−0.00619699
$815$	−4.75020e6	−0.250506
$816$	316093.	0.0166184
$817$	0	0
$818$	−1.31530e7	−0.687293
$819$	1.22434e7	0.637812
$820$	210535.	0.0109343
$821$	3.28657e7	1.70171	0.850854	−	0.525403i	$-0.176085\pi$
$821$	3.28657e7	1.70171	0.850854	+	0.525403i	$0.176085\pi$
$822$	−2.54424e6	−0.131334
$823$	8.94904e6	0.460550	0.230275	−	0.973126i	$-0.426037\pi$
$823$	8.94904e6	0.460550	0.230275	+	0.973126i	$0.426037\pi$
$824$	1.71520e6	0.0880028
$825$	−109890.	−0.00562115
$826$	−2.31880e6	−0.118253
$827$	1.31486e7	0.668523	0.334262	−	0.942480i	$-0.391513\pi$
$827$	1.31486e7	0.668523	0.334262	+	0.942480i	$0.391513\pi$
$828$	2.15736e6	0.109357
$829$	1.07264e6	0.0542088	0.0271044	−	0.999633i	$-0.491371\pi$
$829$	1.07264e6	0.0542088	0.0271044	+	0.999633i	$0.491371\pi$
$830$	2.88755e6	0.145490
$831$	−3.51779e6	−0.176712
$832$	1.07801e6	0.0539901
$833$	−1.22825e7	−0.613301
$834$	−363562.	−0.0180994
$835$	1.30730e6	0.0648873
$836$	0	0
$837$	−9.54952e6	−0.471159
$838$	−1.50929e7	−0.742441
$839$	−3.48927e7	−1.71131	−0.855656	−	0.517544i	$-0.826846\pi$
$839$	−3.48927e7	−1.71131	−0.855656	+	0.517544i	$0.826846\pi$
$840$	288078.	0.0140868
$841$	2.45443e7	1.19663
$842$	−1.56886e6	−0.0762610
$843$	1.97811e6	0.0958699
$844$	−510380.	−0.0246625
$845$	3.26021e6	0.157074
$846$	1.42362e6	0.0683864
$847$	3.13653e7	1.50225
$848$	−6.57055e6	−0.313771
$849$	4.53113e6	0.215743
$850$	−6.95229e6	−0.330051
$851$	969084.	0.0458709
$852$	2.30514e6	0.108792
$853$	1.81891e7	0.855931	0.427966	−	0.903795i	$-0.359230\pi$
$853$	1.81891e7	0.855931	0.427966	+	0.903795i	$0.359230\pi$
$854$	8.95103e6	0.419980
$855$	0	0
$856$	1.07678e7	0.502277
$857$	−3.36435e7	−1.56477	−0.782383	−	0.622797i	$-0.785996\pi$
$857$	−3.36435e7	−1.56477	−0.782383	+	0.622797i	$0.785996\pi$
$858$	−38453.4	−0.00178327
$859$	3.14828e7	1.45576	0.727882	−	0.685702i	$-0.240505\pi$
$859$	3.14828e7	1.45576	0.727882	+	0.685702i	$0.240505\pi$
$860$	2.67229e6	0.123207
$861$	−508314.	−0.0233681
$862$	2.39938e7	1.09984
$863$	4.88137e6	0.223108	0.111554	−	0.993758i	$-0.464417\pi$
$863$	4.88137e6	0.223108	0.111554	+	0.993758i	$0.464417\pi$
$864$	−1.05363e6	−0.0480180
$865$	6.07664e6	0.276136
$866$	9.16607e6	0.415326
$867$	2.32124e6	0.104875
$868$	2.89726e7	1.30523
$869$	571413.	0.0256685
$870$	619425.	0.0277454
$871$	1.10448e6	0.0493300
$872$	−1.03004e6	−0.0458734
$873$	3.11633e7	1.38391
$874$	0	0
$875$	−1.29176e7	−0.570379
$876$	1.56020e6	0.0686943
$877$	5.94390e6	0.260959	0.130480	−	0.991451i	$-0.458348\pi$
$877$	5.94390e6	0.260959	0.130480	+	0.991451i	$0.458348\pi$
$878$	675308.	0.0295642
$879$	−2.01575e6	−0.0879963
$880$	47228.1	0.00205586
$881$	−1.95504e7	−0.848627	−0.424313	−	0.905515i	$-0.639484\pi$
$881$	−1.95504e7	−0.848627	−0.424313	+	0.905515i	$0.639484\pi$
$882$	2.02764e7	0.877644
$883$	2.19741e7	0.948439	0.474219	−	0.880407i	$-0.342730\pi$
$883$	2.19741e7	0.948439	0.474219	+	0.880407i	$0.342730\pi$
$884$	−2.43278e6	−0.104706
$885$	−68546.1	−0.00294188
$886$	1.24408e7	0.532432
$887$	−2.36382e6	−0.100880	−0.0504401	−	0.998727i	$-0.516062\pi$
$887$	−2.36382e6	−0.100880	−0.0504401	+	0.998727i	$0.516062\pi$
$888$	−234399.	−0.00997525
$889$	−4.16074e7	−1.76570
$890$	3.71679e6	0.157287
$891$	−952633.	−0.0402005
$892$	−1.96341e7	−0.826224
$893$	0	0
$894$	3.40222e6	0.142370
$895$	−6.05095e6	−0.252503
$896$	3.19665e6	0.133022
$897$	318093.	0.0132000
$898$	2.02111e7	0.836370
$899$	6.22969e7	2.57080
$900$	1.14771e7	0.472308
$901$	1.48280e7	0.608515
$902$	−83334.1	−0.00341041
$903$	−6.45193e6	−0.263312
$904$	−6.60592e6	−0.268851
$905$	−5.04765e6	−0.204865
$906$	2.67502e6	0.108270
$907$	−2.62203e7	−1.05833	−0.529163	−	0.848520i	$-0.677494\pi$
$907$	−2.62203e7	−1.05833	−0.529163	+	0.848520i	$0.677494\pi$
$908$	−3.90399e6	−0.157143
$909$	4.34886e6	0.174568
$910$	−2.21717e6	−0.0887555
$911$	1.31649e7	0.525558	0.262779	−	0.964856i	$-0.415361\pi$
$911$	1.31649e7	0.525558	0.262779	+	0.964856i	$0.415361\pi$
$912$	0	0
$913$	−1.14295e6	−0.0453785
$914$	−1.03164e7	−0.408471
$915$	264602.	0.0104482
$916$	−2.06526e7	−0.813272
$917$	6.29999e7	2.47409
$918$	2.37777e6	0.0931243
$919$	778178.	0.0303942	0.0151971	−	0.999885i	$-0.495162\pi$
$919$	778178.	0.0303942	0.0151971	+	0.999885i	$0.495162\pi$
$920$	−390679.	−0.0152178
$921$	−4.47761e6	−0.173939
$922$	3.14332e7	1.21776
$923$	−1.77414e7	−0.685461
$924$	−114027.	−0.00439367
$925$	5.15549e6	0.198114
$926$	−1.94766e7	−0.746425
$927$	6.38998e6	0.244231
$928$	6.87343e6	0.262002
$929$	−3.15227e7	−1.19835	−0.599176	−	0.800618i	$-0.704505\pi$
$929$	−3.15227e7	−1.19835	−0.599176	+	0.800618i	$0.704505\pi$
$930$	856462.	0.0324714
$931$	0	0
$932$	1.97194e7	0.743624
$933$	−1.36540e6	−0.0513519
$934$	−7.11190e6	−0.266759
$935$	−106581.	−0.00398706
$936$	4.01613e6	0.149837
$937$	2.86621e7	1.06650	0.533248	−	0.845959i	$-0.320971\pi$
$937$	2.86621e7	1.06650	0.533248	+	0.845959i	$0.320971\pi$
$938$	3.27513e6	0.121541
$939$	−2.41834e6	−0.0895061
$940$	−257805.	−0.00951640
$941$	−4.04755e7	−1.49011	−0.745054	−	0.667004i	$-0.767577\pi$
$941$	−4.04755e7	−1.49011	−0.745054	+	0.667004i	$0.767577\pi$
$942$	3.86352e6	0.141859
$943$	689354.	0.0252443
$944$	−760620.	−0.0277803
$945$	2.16703e6	0.0789379
$946$	−1.05774e6	−0.0384285
$947$	−1.22165e7	−0.442663	−0.221331	−	0.975199i	$-0.571040\pi$
$947$	−1.22165e7	−0.442663	−0.221331	+	0.975199i	$0.571040\pi$
$948$	1.14331e6	0.0413184
$949$	−1.20080e7	−0.432817
$950$	0	0
$951$	−2.50578e6	−0.0898445
$952$	−7.21399e6	−0.257978
$953$	−3.60105e7	−1.28439	−0.642195	−	0.766541i	$-0.721976\pi$
$953$	−3.60105e7	−1.28439	−0.642195	+	0.766541i	$0.721976\pi$
$954$	−2.44786e7	−0.870795
$955$	−543935.	−0.0192992
$956$	−4.79535e6	−0.169698
$957$	−245181.	−0.00865380
$958$	3.98241e7	1.40195
$959$	5.80656e7	2.03879
$960$	94496.3	0.00330931
$961$	5.75070e7	2.00869
$962$	1.80404e6	0.0628503
$963$	4.01155e7	1.39395
$964$	5.34594e6	0.185281
$965$	8.82475e6	0.305059
$966$	943250.	0.0325225
$967$	5.59094e6	0.192273	0.0961366	−	0.995368i	$-0.469351\pi$
$967$	5.59094e6	0.192273	0.0961366	+	0.995368i	$0.469351\pi$
$968$	1.02886e7	0.352912
$969$	0	0
$970$	−5.64339e6	−0.192580
$971$	1.20028e7	0.408539	0.204270	−	0.978915i	$-0.434518\pi$
$971$	1.20028e7	0.408539	0.204270	+	0.978915i	$0.434518\pi$
$972$	−5.90658e6	−0.200526
$973$	8.29735e6	0.280968
$974$	−3.03559e6	−0.102529
$975$	1.69224e6	0.0570101
$976$	2.93615e6	0.0986628
$977$	−3.21099e7	−1.07622	−0.538111	−	0.842874i	$-0.680862\pi$
$977$	−3.21099e7	−1.07622	−0.538111	+	0.842874i	$0.680862\pi$
$978$	3.76205e6	0.125770
$979$	−1.47118e6	−0.0490578
$980$	−3.67186e6	−0.122130
$981$	−3.83740e6	−0.127311
$982$	−7.43342e6	−0.245986
$983$	−5.11520e6	−0.168842	−0.0844208	−	0.996430i	$-0.526904\pi$
$983$	−5.11520e6	−0.168842	−0.0844208	+	0.996430i	$0.526904\pi$
$984$	−166739.	−0.00548971
$985$	−4.92089e6	−0.161604
$986$	−1.55115e7	−0.508116
$987$	622442.	0.0203379
$988$	0	0
$989$	8.74984e6	0.284452
$990$	175949.	0.00570555
$991$	−4.69664e6	−0.151916	−0.0759580	−	0.997111i	$-0.524201\pi$
$991$	−4.69664e6	−0.151916	−0.0759580	+	0.997111i	$0.524201\pi$
$992$	9.50370e6	0.306629
$993$	−1.57809e6	−0.0507877
$994$	−5.26089e7	−1.68886
$995$	−5.90603e6	−0.189120
$996$	−2.28687e6	−0.0730454
$997$	−6.09683e6	−0.194252	−0.0971262	−	0.995272i	$-0.530965\pi$
$997$	−6.09683e6	−0.194252	−0.0971262	+	0.995272i	$0.530965\pi$
$998$	3.08327e7	0.979908
$999$	−1.76324e6	−0.0558982

Display

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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up to: 50 250 1000 (See only

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.6.a.q.1.7		15
19.14	odd	18		38.6.e.b.25.3	✓	30
19.15	odd	18		38.6.e.b.35.3	yes	30
19.18	odd	2		722.6.a.r.1.9		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.25.3	✓	30	19.14	odd	18
38.6.e.b.35.3	yes	30	19.15	odd	18
722.6.a.q.1.7		15	1.1	even	1	trivial
722.6.a.r.1.9		15	19.18	odd	2

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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

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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.q.1.7

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