LMFDB - Embedded newform 350.6.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(1,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-4.00000 q^{2} -8.00000 q^{3} +16.0000 q^{4} +32.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} -179.000 q^{9} -340.000 q^{11} -128.000 q^{12} +294.000 q^{13} -196.000 q^{14} +256.000 q^{16} -1226.00 q^{17} +716.000 q^{18} +2432.00 q^{19} -392.000 q^{21} +1360.00 q^{22} -2000.00 q^{23} +512.000 q^{24} -1176.00 q^{26} +3376.00 q^{27} +784.000 q^{28} -6746.00 q^{29} +8856.00 q^{31} -1024.00 q^{32} +2720.00 q^{33} +4904.00 q^{34} -2864.00 q^{36} -9182.00 q^{37} -9728.00 q^{38} -2352.00 q^{39} -14574.0 q^{41} +1568.00 q^{42} -8108.00 q^{43} -5440.00 q^{44} +8000.00 q^{46} +312.000 q^{47} -2048.00 q^{48} +2401.00 q^{49} +9808.00 q^{51} +4704.00 q^{52} +14634.0 q^{53} -13504.0 q^{54} -3136.00 q^{56} -19456.0 q^{57} +26984.0 q^{58} -27656.0 q^{59} +34338.0 q^{61} -35424.0 q^{62} -8771.00 q^{63} +4096.00 q^{64} -10880.0 q^{66} -12316.0 q^{67} -19616.0 q^{68} +16000.0 q^{69} +36920.0 q^{71} +11456.0 q^{72} +61718.0 q^{73} +36728.0 q^{74} +38912.0 q^{76} -16660.0 q^{77} +9408.00 q^{78} -64752.0 q^{79} +16489.0 q^{81} +58296.0 q^{82} +77056.0 q^{83} -6272.00 q^{84} +32432.0 q^{86} +53968.0 q^{87} +21760.0 q^{88} -8166.00 q^{89} +14406.0 q^{91} -32000.0 q^{92} -70848.0 q^{93} -1248.00 q^{94} +8192.00 q^{96} -20650.0 q^{97} -9604.00 q^{98} +60860.0 q^{99} +O(q^{100})\)

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
$3$	−8.00000	−0.513200	−0.256600	−	0.966518i	$-0.582602\pi$
$3$	−8.00000	−0.513200	−0.256600	+	0.966518i	$0.582602\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	32.0000	0.362887
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	−179.000	−0.736626
$10$	0	0
$11$	−340.000	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$11$	−340.000	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$12$	−128.000	−0.256600
$13$	294.000	0.482491	0.241245	−	0.970464i	$-0.422444\pi$
$13$	294.000	0.482491	0.241245	+	0.970464i	$0.422444\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−1226.00	−1.02889	−0.514444	−	0.857524i	$-0.672002\pi$
$17$	−1226.00	−1.02889	−0.514444	+	0.857524i	$0.672002\pi$
$18$	716.000	0.520873
$19$	2432.00	1.54554	0.772769	−	0.634688i	$-0.218871\pi$
$19$	2432.00	1.54554	0.772769	+	0.634688i	$0.218871\pi$
$20$	0	0
$21$	−392.000	−0.193971
$22$	1360.00	0.599076
$23$	−2000.00	−0.788334	−0.394167	−	0.919039i	$-0.628967\pi$
$23$	−2000.00	−0.788334	−0.394167	+	0.919039i	$0.628967\pi$
$24$	512.000	0.181444
$25$	0	0
$26$	−1176.00	−0.341172
$27$	3376.00	0.891237
$28$	784.000	0.188982
$29$	−6746.00	−1.48954	−0.744769	−	0.667323i	$-0.767440\pi$
$29$	−6746.00	−1.48954	−0.744769	+	0.667323i	$0.767440\pi$
$30$	0	0
$31$	8856.00	1.65513	0.827567	−	0.561366i	$-0.189724\pi$
$31$	8856.00	1.65513	0.827567	+	0.561366i	$0.189724\pi$
$32$	−1024.00	−0.176777
$33$	2720.00	0.434795
$34$	4904.00	0.727534
$35$	0	0
$36$	−2864.00	−0.368313
$37$	−9182.00	−1.10264	−0.551319	−	0.834295i	$-0.685875\pi$
$37$	−9182.00	−1.10264	−0.551319	+	0.834295i	$0.685875\pi$
$38$	−9728.00	−1.09286
$39$	−2352.00	−0.247614
$40$	0	0
$41$	−14574.0	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$41$	−14574.0	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$42$	1568.00	0.137159
$43$	−8108.00	−0.668717	−0.334359	−	0.942446i	$-0.608520\pi$
$43$	−8108.00	−0.668717	−0.334359	+	0.942446i	$0.608520\pi$
$44$	−5440.00	−0.423611
$45$	0	0
$46$	8000.00	0.557437
$47$	312.000	0.0206020	0.0103010	−	0.999947i	$-0.496721\pi$
$47$	312.000	0.0206020	0.0103010	+	0.999947i	$0.496721\pi$
$48$	−2048.00	−0.128300
$49$	2401.00	0.142857
$50$	0	0
$51$	9808.00	0.528026
$52$	4704.00	0.241245
$53$	14634.0	0.715605	0.357803	−	0.933797i	$-0.383526\pi$
$53$	14634.0	0.715605	0.357803	+	0.933797i	$0.383526\pi$
$54$	−13504.0	−0.630199
$55$	0	0
$56$	−3136.00	−0.133631
$57$	−19456.0	−0.793170
$58$	26984.0	1.05326
$59$	−27656.0	−1.03433	−0.517165	−	0.855886i	$-0.673013\pi$
$59$	−27656.0	−1.03433	−0.517165	+	0.855886i	$0.673013\pi$
$60$	0	0
$61$	34338.0	1.18155	0.590773	−	0.806838i	$-0.298823\pi$
$61$	34338.0	1.18155	0.590773	+	0.806838i	$0.298823\pi$
$62$	−35424.0	−1.17036
$63$	−8771.00	−0.278418
$64$	4096.00	0.125000
$65$	0	0
$66$	−10880.0	−0.307446
$67$	−12316.0	−0.335184	−0.167592	−	0.985856i	$-0.553599\pi$
$67$	−12316.0	−0.335184	−0.167592	+	0.985856i	$0.553599\pi$
$68$	−19616.0	−0.514444
$69$	16000.0	0.404573
$70$	0	0
$71$	36920.0	0.869192	0.434596	−	0.900625i	$-0.356891\pi$
$71$	36920.0	0.869192	0.434596	+	0.900625i	$0.356891\pi$
$72$	11456.0	0.260436
$73$	61718.0	1.35552	0.677758	−	0.735285i	$-0.262952\pi$
$73$	61718.0	1.35552	0.677758	+	0.735285i	$0.262952\pi$
$74$	36728.0	0.779683
$75$	0	0
$76$	38912.0	0.772769
$77$	−16660.0	−0.320220
$78$	9408.00	0.175090
$79$	−64752.0	−1.16731	−0.583654	−	0.812002i	$-0.698378\pi$
$79$	−64752.0	−1.16731	−0.583654	+	0.812002i	$0.698378\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	58296.0	0.957424
$83$	77056.0	1.22775	0.613877	−	0.789402i	$-0.289609\pi$
$83$	77056.0	1.22775	0.613877	+	0.789402i	$0.289609\pi$
$84$	−6272.00	−0.0969857
$85$	0	0
$86$	32432.0	0.472855
$87$	53968.0	0.764431
$88$	21760.0	0.299538
$89$	−8166.00	−0.109278	−0.0546392	−	0.998506i	$-0.517401\pi$
$89$	−8166.00	−0.109278	−0.0546392	+	0.998506i	$0.517401\pi$
$90$	0	0
$91$	14406.0	0.182364
$92$	−32000.0	−0.394167
$93$	−70848.0	−0.849416
$94$	−1248.00	−0.0145678
$95$	0	0
$96$	8192.00	0.0907218
$97$	−20650.0	−0.222839	−0.111419	−	0.993773i	$-0.535540\pi$
$97$	−20650.0	−0.222839	−0.111419	+	0.993773i	$0.535540\pi$
$98$	−9604.00	−0.101015
$99$	60860.0	0.624085
$100$	0	0
$101$	186250.	1.81674	0.908370	−	0.418167i	$-0.137327\pi$
$101$	186250.	1.81674	0.908370	+	0.418167i	$0.137327\pi$
$102$	−39232.0	−0.373371
$103$	60064.0	0.557855	0.278927	−	0.960312i	$-0.410021\pi$
$103$	60064.0	0.557855	0.278927	+	0.960312i	$0.410021\pi$
$104$	−18816.0	−0.170586
$105$	0	0
$106$	−58536.0	−0.506009
$107$	−47892.0	−0.404393	−0.202196	−	0.979345i	$-0.564808\pi$
$107$	−47892.0	−0.404393	−0.202196	+	0.979345i	$0.564808\pi$
$108$	54016.0	0.445618
$109$	22102.0	0.178183	0.0890913	−	0.996023i	$-0.471604\pi$
$109$	22102.0	0.178183	0.0890913	+	0.996023i	$0.471604\pi$
$110$	0	0
$111$	73456.0	0.565874
$112$	12544.0	0.0944911
$113$	245054.	1.80537	0.902684	−	0.430304i	$-0.141594\pi$
$113$	245054.	1.80537	0.902684	+	0.430304i	$0.141594\pi$
$114$	77824.0	0.560856
$115$	0	0
$116$	−107936.	−0.744769
$117$	−52626.0	−0.355415
$118$	110624.	0.731382
$119$	−60074.0	−0.388883
$120$	0	0
$121$	−45451.0	−0.282215
$122$	−137352.	−0.835479
$123$	116592.	0.694874
$124$	141696.	0.827567
$125$	0	0
$126$	35084.0	0.196871
$127$	96696.0	0.531985	0.265992	−	0.963975i	$-0.414300\pi$
$127$	96696.0	0.531985	0.265992	+	0.963975i	$0.414300\pi$
$128$	−16384.0	−0.0883883
$129$	64864.0	0.343186
$130$	0	0
$131$	134368.	0.684097	0.342048	−	0.939682i	$-0.388879\pi$
$131$	134368.	0.684097	0.342048	+	0.939682i	$0.388879\pi$
$132$	43520.0	0.217397
$133$	119168.	0.584158
$134$	49264.0	0.237011
$135$	0	0
$136$	78464.0	0.363767
$137$	294662.	1.34129	0.670645	−	0.741778i	$-0.266017\pi$
$137$	294662.	1.34129	0.670645	+	0.741778i	$0.266017\pi$
$138$	−64000.0	−0.286077
$139$	314944.	1.38260	0.691300	−	0.722568i	$-0.257038\pi$
$139$	314944.	1.38260	0.691300	+	0.722568i	$0.257038\pi$
$140$	0	0
$141$	−2496.00	−0.0105730
$142$	−147680.	−0.614612
$143$	−99960.0	−0.408777
$144$	−45824.0	−0.184156
$145$	0	0
$146$	−246872.	−0.958495
$147$	−19208.0	−0.0733143
$148$	−146912.	−0.551319
$149$	113622.	0.419273	0.209636	−	0.977779i	$-0.432772\pi$
$149$	113622.	0.419273	0.209636	+	0.977779i	$0.432772\pi$
$150$	0	0
$151$	408208.	1.45693	0.728466	−	0.685082i	$-0.240234\pi$
$151$	408208.	1.45693	0.728466	+	0.685082i	$0.240234\pi$
$152$	−155648.	−0.546430
$153$	219454.	0.757905
$154$	66640.0	0.226430
$155$	0	0
$156$	−37632.0	−0.123807
$157$	−293546.	−0.950445	−0.475223	−	0.879866i	$-0.657632\pi$
$157$	−293546.	−0.950445	−0.475223	+	0.879866i	$0.657632\pi$
$158$	259008.	0.825411
$159$	−117072.	−0.367249
$160$	0	0
$161$	−98000.0	−0.297962
$162$	−65956.0	−0.197454
$163$	317116.	0.934866	0.467433	−	0.884029i	$-0.345179\pi$
$163$	317116.	0.934866	0.467433	+	0.884029i	$0.345179\pi$
$164$	−233184.	−0.677001
$165$	0	0
$166$	−308224.	−0.868153
$167$	−141568.	−0.392802	−0.196401	−	0.980524i	$-0.562925\pi$
$167$	−141568.	−0.392802	−0.196401	+	0.980524i	$0.562925\pi$
$168$	25088.0	0.0685793
$169$	−284857.	−0.767203
$170$	0	0
$171$	−435328.	−1.13848
$172$	−129728.	−0.334359
$173$	71222.0	0.180925	0.0904626	−	0.995900i	$-0.471165\pi$
$173$	71222.0	0.180925	0.0904626	+	0.995900i	$0.471165\pi$
$174$	−215872.	−0.540534
$175$	0	0
$176$	−87040.0	−0.211805
$177$	221248.	0.530819
$178$	32664.0	0.0772715
$179$	485628.	1.13285	0.566423	−	0.824114i	$-0.308327\pi$
$179$	485628.	1.13285	0.566423	+	0.824114i	$0.308327\pi$
$180$	0	0
$181$	657090.	1.49083	0.745416	−	0.666600i	$-0.232251\pi$
$181$	657090.	1.49083	0.745416	+	0.666600i	$0.232251\pi$
$182$	−57624.0	−0.128951
$183$	−274704.	−0.606369
$184$	128000.	0.278718
$185$	0	0
$186$	283392.	0.600628
$187$	416840.	0.871697
$188$	4992.00	0.0103010
$189$	165424.	0.336856
$190$	0	0
$191$	68304.0	0.135476	0.0677381	−	0.997703i	$-0.478422\pi$
$191$	68304.0	0.135476	0.0677381	+	0.997703i	$0.478422\pi$
$192$	−32768.0	−0.0641500
$193$	−352754.	−0.681677	−0.340839	−	0.940122i	$-0.610711\pi$
$193$	−352754.	−0.681677	−0.340839	+	0.940122i	$0.610711\pi$
$194$	82600.0	0.157571
$195$	0	0
$196$	38416.0	0.0714286
$197$	−196982.	−0.361627	−0.180814	−	0.983517i	$-0.557873\pi$
$197$	−196982.	−0.361627	−0.180814	+	0.983517i	$0.557873\pi$
$198$	−243440.	−0.441295
$199$	−1.10392e6	−1.97608	−0.988041	−	0.154192i	$-0.950723\pi$
$199$	−1.10392e6	−1.97608	−0.988041	+	0.154192i	$0.950723\pi$
$200$	0	0
$201$	98528.0	0.172016
$202$	−745000.	−1.28463
$203$	−330554.	−0.562992
$204$	156928.	0.264013
$205$	0	0
$206$	−240256.	−0.394463
$207$	358000.	0.580707
$208$	75264.0	0.120623
$209$	−826880.	−1.30941
$210$	0	0
$211$	−103444.	−0.159955	−0.0799777	−	0.996797i	$-0.525485\pi$
$211$	−103444.	−0.159955	−0.0799777	+	0.996797i	$0.525485\pi$
$212$	234144.	0.357803
$213$	−295360.	−0.446070
$214$	191568.	0.285949
$215$	0	0
$216$	−216064.	−0.315100
$217$	433944.	0.625582
$218$	−88408.0	−0.125994
$219$	−493744.	−0.695651
$220$	0	0
$221$	−360444.	−0.496429
$222$	−293824.	−0.400133
$223$	−307328.	−0.413847	−0.206924	−	0.978357i	$-0.566345\pi$
$223$	−307328.	−0.413847	−0.206924	+	0.978357i	$0.566345\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	−980216.	−1.27659
$227$	891792.	1.14868	0.574340	−	0.818617i	$-0.305259\pi$
$227$	891792.	1.14868	0.574340	+	0.818617i	$0.305259\pi$
$228$	−311296.	−0.396585
$229$	276706.	0.348682	0.174341	−	0.984685i	$-0.444220\pi$
$229$	276706.	0.348682	0.174341	+	0.984685i	$0.444220\pi$
$230$	0	0
$231$	133280.	0.164337
$232$	431744.	0.526631
$233$	−1.47943e6	−1.78528	−0.892639	−	0.450772i	$-0.851149\pi$
$233$	−1.47943e6	−1.78528	−0.892639	+	0.450772i	$0.851149\pi$
$234$	210504.	0.251316
$235$	0	0
$236$	−442496.	−0.517165
$237$	518016.	0.599063
$238$	240296.	0.274982
$239$	1.00034e6	1.13280	0.566402	−	0.824129i	$-0.308335\pi$
$239$	1.00034e6	1.13280	0.566402	+	0.824129i	$0.308335\pi$
$240$	0	0
$241$	1.35833e6	1.50648	0.753239	−	0.657747i	$-0.228490\pi$
$241$	1.35833e6	1.50648	0.753239	+	0.657747i	$0.228490\pi$
$242$	181804.	0.199556
$243$	−952280.	−1.03454
$244$	549408.	0.590773
$245$	0	0
$246$	−466368.	−0.491350
$247$	715008.	0.745708
$248$	−566784.	−0.585179
$249$	−616448.	−0.630083
$250$	0	0
$251$	−177408.	−0.177742	−0.0888708	−	0.996043i	$-0.528326\pi$
$251$	−177408.	−0.177742	−0.0888708	+	0.996043i	$0.528326\pi$
$252$	−140336.	−0.139209
$253$	680000.	0.667894
$254$	−386784.	−0.376170
$255$	0	0
$256$	65536.0	0.0625000
$257$	−326658.	−0.308504	−0.154252	−	0.988032i	$-0.549297\pi$
$257$	−326658.	−0.308504	−0.154252	+	0.988032i	$0.549297\pi$
$258$	−259456.	−0.242669
$259$	−449918.	−0.416758
$260$	0	0
$261$	1.20753e6	1.09723
$262$	−537472.	−0.483730
$263$	34920.0	0.0311304	0.0155652	−	0.999879i	$-0.495045\pi$
$263$	34920.0	0.0311304	0.0155652	+	0.999879i	$0.495045\pi$
$264$	−174080.	−0.153723
$265$	0	0
$266$	−476672.	−0.413062
$267$	65328.0	0.0560817
$268$	−197056.	−0.167592
$269$	716458.	0.603685	0.301842	−	0.953358i	$-0.402398\pi$
$269$	716458.	0.603685	0.301842	+	0.953358i	$0.402398\pi$
$270$	0	0
$271$	−953376.	−0.788571	−0.394286	−	0.918988i	$-0.629008\pi$
$271$	−953376.	−0.788571	−0.394286	+	0.918988i	$0.629008\pi$
$272$	−313856.	−0.257222
$273$	−115248.	−0.0935894
$274$	−1.17865e6	−0.948435
$275$	0	0
$276$	256000.	0.202287
$277$	1.84729e6	1.44656	0.723279	−	0.690556i	$-0.242634\pi$
$277$	1.84729e6	1.44656	0.723279	+	0.690556i	$0.242634\pi$
$278$	−1.25978e6	−0.977645
$279$	−1.58522e6	−1.21921
$280$	0	0
$281$	−1.99601e6	−1.50798	−0.753991	−	0.656885i	$-0.771874\pi$
$281$	−1.99601e6	−1.50798	−0.753991	+	0.656885i	$0.771874\pi$
$282$	9984.00	0.00747622
$283$	−234088.	−0.173745	−0.0868726	−	0.996219i	$-0.527687\pi$
$283$	−234088.	−0.173745	−0.0868726	+	0.996219i	$0.527687\pi$
$284$	590720.	0.434596
$285$	0	0
$286$	399840.	0.289049
$287$	−714126.	−0.511764
$288$	183296.	0.130218
$289$	83219.0	0.0586108
$290$	0	0
$291$	165200.	0.114361
$292$	987488.	0.677758
$293$	2.50081e6	1.70181	0.850905	−	0.525320i	$-0.176054\pi$
$293$	2.50081e6	1.70181	0.850905	+	0.525320i	$0.176054\pi$
$294$	76832.0	0.0518411
$295$	0	0
$296$	587648.	0.389841
$297$	−1.14784e6	−0.755075
$298$	−454488.	−0.296471
$299$	−588000.	−0.380364
$300$	0	0
$301$	−397292.	−0.252751
$302$	−1.63283e6	−1.03021
$303$	−1.49000e6	−0.932352
$304$	622592.	0.386384
$305$	0	0
$306$	−877816.	−0.535920
$307$	−2.34203e6	−1.41823	−0.709115	−	0.705092i	$-0.750905\pi$
$307$	−2.34203e6	−1.41823	−0.709115	+	0.705092i	$0.750905\pi$
$308$	−266560.	−0.160110
$309$	−480512.	−0.286291
$310$	0	0
$311$	−163064.	−0.0955998	−0.0477999	−	0.998857i	$-0.515221\pi$
$311$	−163064.	−0.0955998	−0.0477999	+	0.998857i	$0.515221\pi$
$312$	150528.	0.0875449
$313$	−1.73965e6	−1.00369	−0.501847	−	0.864957i	$-0.667346\pi$
$313$	−1.73965e6	−1.00369	−0.501847	+	0.864957i	$0.667346\pi$
$314$	1.17418e6	0.672066
$315$	0	0
$316$	−1.03603e6	−0.583654
$317$	1.79771e6	1.00478	0.502392	−	0.864640i	$-0.332454\pi$
$317$	1.79771e6	1.00478	0.502392	+	0.864640i	$0.332454\pi$
$318$	468288.	0.259684
$319$	2.29364e6	1.26197
$320$	0	0
$321$	383136.	0.207535
$322$	392000.	0.210691
$323$	−2.98163e6	−1.59019
$324$	263824.	0.139621
$325$	0	0
$326$	−1.26846e6	−0.661050
$327$	−176816.	−0.0914434
$328$	932736.	0.478712
$329$	15288.0	0.00778683
$330$	0	0
$331$	−2.47541e6	−1.24187	−0.620937	−	0.783861i	$-0.713248\pi$
$331$	−2.47541e6	−1.24187	−0.620937	+	0.783861i	$0.713248\pi$
$332$	1.23290e6	0.613877
$333$	1.64358e6	0.812231
$334$	566272.	0.277753
$335$	0	0
$336$	−100352.	−0.0484929
$337$	−89154.0	−0.0427628	−0.0213814	−	0.999771i	$-0.506806\pi$
$337$	−89154.0	−0.0427628	−0.0213814	+	0.999771i	$0.506806\pi$
$338$	1.13943e6	0.542494
$339$	−1.96043e6	−0.926515
$340$	0	0
$341$	−3.01104e6	−1.40227
$342$	1.74131e6	0.805029
$343$	117649.	0.0539949
$344$	518912.	0.236427
$345$	0	0
$346$	−284888.	−0.127933
$347$	−938556.	−0.418443	−0.209222	−	0.977868i	$-0.567093\pi$
$347$	−938556.	−0.418443	−0.209222	+	0.977868i	$0.567093\pi$
$348$	863488.	0.382215
$349$	3.34268e6	1.46903	0.734516	−	0.678591i	$-0.237409\pi$
$349$	3.34268e6	1.46903	0.734516	+	0.678591i	$0.237409\pi$
$350$	0	0
$351$	992544.	0.430013
$352$	348160.	0.149769
$353$	3.76606e6	1.60861	0.804305	−	0.594217i	$-0.202538\pi$
$353$	3.76606e6	1.60861	0.804305	+	0.594217i	$0.202538\pi$
$354$	−884992.	−0.375345
$355$	0	0
$356$	−130656.	−0.0546392
$357$	480592.	0.199575
$358$	−1.94251e6	−0.801044
$359$	−1.53934e6	−0.630376	−0.315188	−	0.949029i	$-0.602068\pi$
$359$	−1.53934e6	−0.630376	−0.315188	+	0.949029i	$0.602068\pi$
$360$	0	0
$361$	3.43852e6	1.38869
$362$	−2.62836e6	−1.05418
$363$	363608.	0.144833
$364$	230496.	0.0911822
$365$	0	0
$366$	1.09882e6	0.428768
$367$	859312.	0.333032	0.166516	−	0.986039i	$-0.446748\pi$
$367$	859312.	0.333032	0.166516	+	0.986039i	$0.446748\pi$
$368$	−512000.	−0.197084
$369$	2.60875e6	0.997392
$370$	0	0
$371$	717066.	0.270473
$372$	−1.13357e6	−0.424708
$373$	976586.	0.363445	0.181722	−	0.983350i	$-0.441833\pi$
$373$	976586.	0.363445	0.181722	+	0.983350i	$0.441833\pi$
$374$	−1.66736e6	−0.616383
$375$	0	0
$376$	−19968.0	−0.00728392
$377$	−1.98332e6	−0.718688
$378$	−661696.	−0.238193
$379$	106444.	0.0380648	0.0190324	−	0.999819i	$-0.493941\pi$
$379$	106444.	0.0380648	0.0190324	+	0.999819i	$0.493941\pi$
$380$	0	0
$381$	−773568.	−0.273015
$382$	−273216.	−0.0957961
$383$	2.00634e6	0.698889	0.349445	−	0.936957i	$-0.386370\pi$
$383$	2.00634e6	0.698889	0.349445	+	0.936957i	$0.386370\pi$
$384$	131072.	0.0453609
$385$	0	0
$386$	1.41102e6	0.482018
$387$	1.45133e6	0.492594
$388$	−330400.	−0.111419
$389$	−684002.	−0.229184	−0.114592	−	0.993413i	$-0.536556\pi$
$389$	−684002.	−0.229184	−0.114592	+	0.993413i	$0.536556\pi$
$390$	0	0
$391$	2.45200e6	0.811108
$392$	−153664.	−0.0505076
$393$	−1.07494e6	−0.351079
$394$	787928.	0.255709
$395$	0	0
$396$	973760.	0.312043
$397$	222870.	0.0709701	0.0354850	−	0.999370i	$-0.488702\pi$
$397$	222870.	0.0709701	0.0354850	+	0.999370i	$0.488702\pi$
$398$	4.41568e6	1.39730
$399$	−953344.	−0.299790
$400$	0	0
$401$	1.90072e6	0.590279	0.295140	−	0.955454i	$-0.404634\pi$
$401$	1.90072e6	0.590279	0.295140	+	0.955454i	$0.404634\pi$
$402$	−394112.	−0.121634
$403$	2.60366e6	0.798587
$404$	2.98000e6	0.908370
$405$	0	0
$406$	1.32222e6	0.398096
$407$	3.12188e6	0.934179
$408$	−627712.	−0.186685
$409$	1.77715e6	0.525311	0.262656	−	0.964890i	$-0.415402\pi$
$409$	1.77715e6	0.525311	0.262656	+	0.964890i	$0.415402\pi$
$410$	0	0
$411$	−2.35730e6	−0.688350
$412$	961024.	0.278927
$413$	−1.35514e6	−0.390940
$414$	−1.43200e6	−0.410622
$415$	0	0
$416$	−301056.	−0.0852931
$417$	−2.51955e6	−0.709550
$418$	3.30752e6	0.925895
$419$	28056.0	0.00780712	0.00390356	−	0.999992i	$-0.498757\pi$
$419$	28056.0	0.00780712	0.00390356	+	0.999992i	$0.498757\pi$
$420$	0	0
$421$	−2.70897e6	−0.744902	−0.372451	−	0.928052i	$-0.621482\pi$
$421$	−2.70897e6	−0.744902	−0.372451	+	0.928052i	$0.621482\pi$
$422$	413776.	0.113106
$423$	−55848.0	−0.0151760
$424$	−936576.	−0.253005
$425$	0	0
$426$	1.18144e6	0.315419
$427$	1.68256e6	0.446582
$428$	−766272.	−0.202196
$429$	799680.	0.209784
$430$	0	0
$431$	5.53898e6	1.43627	0.718136	−	0.695902i	$-0.244995\pi$
$431$	5.53898e6	1.43627	0.718136	+	0.695902i	$0.244995\pi$
$432$	864256.	0.222809
$433$	868294.	0.222560	0.111280	−	0.993789i	$-0.464505\pi$
$433$	868294.	0.222560	0.111280	+	0.993789i	$0.464505\pi$
$434$	−1.73578e6	−0.442353
$435$	0	0
$436$	353632.	0.0890913
$437$	−4.86400e6	−1.21840
$438$	1.97498e6	0.491900
$439$	−1.13767e6	−0.281745	−0.140872	−	0.990028i	$-0.544991\pi$
$439$	−1.13767e6	−0.281745	−0.140872	+	0.990028i	$0.544991\pi$
$440$	0	0
$441$	−429779.	−0.105232
$442$	1.44178e6	0.351028
$443$	−1.75399e6	−0.424636	−0.212318	−	0.977201i	$-0.568101\pi$
$443$	−1.75399e6	−0.424636	−0.212318	+	0.977201i	$0.568101\pi$
$444$	1.17530e6	0.282937
$445$	0	0
$446$	1.22931e6	0.292634
$447$	−908976.	−0.215171
$448$	200704.	0.0472456
$449$	2.41674e6	0.565736	0.282868	−	0.959159i	$-0.408714\pi$
$449$	2.41674e6	0.565736	0.282868	+	0.959159i	$0.408714\pi$
$450$	0	0
$451$	4.95516e6	1.14714
$452$	3.92086e6	0.902684
$453$	−3.26566e6	−0.747698
$454$	−3.56717e6	−0.812239
$455$	0	0
$456$	1.24518e6	0.280428
$457$	127430.	0.0285418	0.0142709	−	0.999898i	$-0.495457\pi$
$457$	127430.	0.0285418	0.0142709	+	0.999898i	$0.495457\pi$
$458$	−1.10682e6	−0.246556
$459$	−4.13898e6	−0.916983
$460$	0	0
$461$	−128198.	−0.0280950	−0.0140475	−	0.999901i	$-0.504472\pi$
$461$	−128198.	−0.0280950	−0.0140475	+	0.999901i	$0.504472\pi$
$462$	−533120.	−0.116204
$463$	4.01653e6	0.870760	0.435380	−	0.900247i	$-0.356614\pi$
$463$	4.01653e6	0.870760	0.435380	+	0.900247i	$0.356614\pi$
$464$	−1.72698e6	−0.372384
$465$	0	0
$466$	5.91774e6	1.26238
$467$	−8.67246e6	−1.84014	−0.920069	−	0.391757i	$-0.871867\pi$
$467$	−8.67246e6	−1.84014	−0.920069	+	0.391757i	$0.871867\pi$
$468$	−842016.	−0.177707
$469$	−603484.	−0.126687
$470$	0	0
$471$	2.34837e6	0.487769
$472$	1.76998e6	0.365691
$473$	2.75672e6	0.566552
$474$	−2.07206e6	−0.423601
$475$	0	0
$476$	−961184.	−0.194442
$477$	−2.61949e6	−0.527133
$478$	−4.00138e6	−0.801013
$479$	8.28946e6	1.65077	0.825387	−	0.564567i	$-0.190957\pi$
$479$	8.28946e6	1.65077	0.825387	+	0.564567i	$0.190957\pi$
$480$	0	0
$481$	−2.69951e6	−0.532013
$482$	−5.43332e6	−1.06524
$483$	784000.	0.152914
$484$	−727216.	−0.141107
$485$	0	0
$486$	3.80912e6	0.731533
$487$	8.91770e6	1.70385	0.851923	−	0.523667i	$-0.175437\pi$
$487$	8.91770e6	1.70385	0.851923	+	0.523667i	$0.175437\pi$
$488$	−2.19763e6	−0.417739
$489$	−2.53693e6	−0.479773
$490$	0	0
$491$	−5.71537e6	−1.06989	−0.534947	−	0.844886i	$-0.679668\pi$
$491$	−5.71537e6	−1.06989	−0.534947	+	0.844886i	$0.679668\pi$
$492$	1.86547e6	0.347437
$493$	8.27060e6	1.53257
$494$	−2.86003e6	−0.527295
$495$	0	0
$496$	2.26714e6	0.413784
$497$	1.80908e6	0.328524
$498$	2.46579e6	0.445536
$499$	125116.	0.0224937	0.0112469	−	0.999937i	$-0.496420\pi$
$499$	125116.	0.0224937	0.0112469	+	0.999937i	$0.496420\pi$
$500$	0	0
$501$	1.13254e6	0.201586
$502$	709632.	0.125682
$503$	2.77116e6	0.488362	0.244181	−	0.969730i	$-0.421481\pi$
$503$	2.77116e6	0.488362	0.244181	+	0.969730i	$0.421481\pi$
$504$	561344.	0.0984357
$505$	0	0
$506$	−2.72000e6	−0.472272
$507$	2.27886e6	0.393729
$508$	1.54714e6	0.265992
$509$	−138534.	−0.0237007	−0.0118504	−	0.999930i	$-0.503772\pi$
$509$	−138534.	−0.0237007	−0.0118504	+	0.999930i	$0.503772\pi$
$510$	0	0
$511$	3.02418e6	0.512337
$512$	−262144.	−0.0441942
$513$	8.21043e6	1.37744
$514$	1.30663e6	0.218145
$515$	0	0
$516$	1.03782e6	0.171593
$517$	−106080.	−0.0174545
$518$	1.79967e6	0.294692
$519$	−569776.	−0.0928508
$520$	0	0
$521$	−1.80281e6	−0.290976	−0.145488	−	0.989360i	$-0.546475\pi$
$521$	−1.80281e6	−0.290976	−0.145488	+	0.989360i	$0.546475\pi$
$522$	−4.83014e6	−0.775860
$523$	−9.77247e6	−1.56225	−0.781124	−	0.624375i	$-0.785354\pi$
$523$	−9.77247e6	−1.56225	−0.781124	+	0.624375i	$0.785354\pi$
$524$	2.14989e6	0.342048
$525$	0	0
$526$	−139680.	−0.0220125
$527$	−1.08575e7	−1.70295
$528$	696320.	0.108699
$529$	−2.43634e6	−0.378529
$530$	0	0
$531$	4.95042e6	0.761914
$532$	1.90669e6	0.292079
$533$	−4.28476e6	−0.653293
$534$	−261312.	−0.0396558
$535$	0	0
$536$	788224.	0.118505
$537$	−3.88502e6	−0.581377
$538$	−2.86583e6	−0.426869
$539$	−816340.	−0.121032
$540$	0	0
$541$	2.45504e6	0.360633	0.180316	−	0.983609i	$-0.442288\pi$
$541$	2.45504e6	0.360633	0.180316	+	0.983609i	$0.442288\pi$
$542$	3.81350e6	0.557604
$543$	−5.25672e6	−0.765095
$544$	1.25542e6	0.181883
$545$	0	0
$546$	460992.	0.0661777
$547$	−1.32081e7	−1.88744	−0.943721	−	0.330743i	$-0.892701\pi$
$547$	−1.32081e7	−1.88744	−0.943721	+	0.330743i	$0.892701\pi$
$548$	4.71459e6	0.670645
$549$	−6.14650e6	−0.870356
$550$	0	0
$551$	−1.64063e7	−2.30214
$552$	−1.02400e6	−0.143038
$553$	−3.17285e6	−0.441201
$554$	−7.38916e6	−1.02287
$555$	0	0
$556$	5.03910e6	0.691300
$557$	−7.83293e6	−1.06976	−0.534880	−	0.844928i	$-0.679643\pi$
$557$	−7.83293e6	−1.06976	−0.534880	+	0.844928i	$0.679643\pi$
$558$	6.34090e6	0.862115
$559$	−2.38375e6	−0.322650
$560$	0	0
$561$	−3.33472e6	−0.447355
$562$	7.98402e6	1.06630
$563$	−3.57908e6	−0.475883	−0.237942	−	0.971279i	$-0.576473\pi$
$563$	−3.57908e6	−0.475883	−0.237942	+	0.971279i	$0.576473\pi$
$564$	−39936.0	−0.00528648
$565$	0	0
$566$	936352.	0.122856
$567$	807961.	0.105544
$568$	−2.36288e6	−0.307306
$569$	−3.39581e6	−0.439707	−0.219853	−	0.975533i	$-0.570558\pi$
$569$	−3.39581e6	−0.439707	−0.219853	+	0.975533i	$0.570558\pi$
$570$	0	0
$571$	−1.47695e6	−0.189572	−0.0947862	−	0.995498i	$-0.530217\pi$
$571$	−1.47695e6	−0.189572	−0.0947862	+	0.995498i	$0.530217\pi$
$572$	−1.59936e6	−0.204388
$573$	−546432.	−0.0695264
$574$	2.85650e6	0.361872
$575$	0	0
$576$	−733184.	−0.0920782
$577$	1.49961e7	1.87516	0.937580	−	0.347771i	$-0.113061\pi$
$577$	1.49961e7	1.87516	0.937580	+	0.347771i	$0.113061\pi$
$578$	−332876.	−0.0414441
$579$	2.82203e6	0.349837
$580$	0	0
$581$	3.77574e6	0.464047
$582$	−660800.	−0.0808654
$583$	−4.97556e6	−0.606276
$584$	−3.94995e6	−0.479247
$585$	0	0
$586$	−1.00032e7	−1.20336
$587$	3.29291e6	0.394444	0.197222	−	0.980359i	$-0.436808\pi$
$587$	3.29291e6	0.394444	0.197222	+	0.980359i	$0.436808\pi$
$588$	−307328.	−0.0366572
$589$	2.15378e7	2.55807
$590$	0	0
$591$	1.57586e6	0.185587
$592$	−2.35059e6	−0.275660
$593$	1.17908e7	1.37692	0.688459	−	0.725275i	$-0.258287\pi$
$593$	1.17908e7	1.37692	0.688459	+	0.725275i	$0.258287\pi$
$594$	4.59136e6	0.533919
$595$	0	0
$596$	1.81795e6	0.209636
$597$	8.83136e6	1.01413
$598$	2.35200e6	0.268958
$599$	−1.52642e6	−0.173823	−0.0869117	−	0.996216i	$-0.527700\pi$
$599$	−1.52642e6	−0.173823	−0.0869117	+	0.996216i	$0.527700\pi$
$600$	0	0
$601$	−1.00142e7	−1.13092	−0.565458	−	0.824777i	$-0.691301\pi$
$601$	−1.00142e7	−1.13092	−0.565458	+	0.824777i	$0.691301\pi$
$602$	1.58917e6	0.178722
$603$	2.20456e6	0.246905
$604$	6.53133e6	0.728466
$605$	0	0
$606$	5.96000e6	0.659272
$607$	−1.20660e7	−1.32920	−0.664599	−	0.747200i	$-0.731398\pi$
$607$	−1.20660e7	−1.32920	−0.664599	+	0.747200i	$0.731398\pi$
$608$	−2.49037e6	−0.273215
$609$	2.64443e6	0.288928
$610$	0	0
$611$	91728.0	0.00994029
$612$	3.51126e6	0.378953
$613$	−5.81950e6	−0.625511	−0.312755	−	0.949834i	$-0.601252\pi$
$613$	−5.81950e6	−0.625511	−0.312755	+	0.949834i	$0.601252\pi$
$614$	9.36813e6	1.00284
$615$	0	0
$616$	1.06624e6	0.113215
$617$	4.16589e6	0.440550	0.220275	−	0.975438i	$-0.429305\pi$
$617$	4.16589e6	0.440550	0.220275	+	0.975438i	$0.429305\pi$
$618$	1.92205e6	0.202438
$619$	−8.08090e6	−0.847683	−0.423841	−	0.905736i	$-0.639319\pi$
$619$	−8.08090e6	−0.847683	−0.423841	+	0.905736i	$0.639319\pi$
$620$	0	0
$621$	−6.75200e6	−0.702592
$622$	652256.	0.0675993
$623$	−400134.	−0.0413034
$624$	−602112.	−0.0619036
$625$	0	0
$626$	6.95860e6	0.709718
$627$	6.61504e6	0.671991
$628$	−4.69674e6	−0.475223
$629$	1.12571e7	1.13449
$630$	0	0
$631$	−8.40878e6	−0.840735	−0.420368	−	0.907354i	$-0.638099\pi$
$631$	−8.40878e6	−0.840735	−0.420368	+	0.907354i	$0.638099\pi$
$632$	4.14413e6	0.412706
$633$	827552.	0.0820892
$634$	−7.19086e6	−0.710489
$635$	0	0
$636$	−1.87315e6	−0.183624
$637$	705894.	0.0689272
$638$	−9.17456e6	−0.892347
$639$	−6.60868e6	−0.640269
$640$	0	0
$641$	6.29760e6	0.605383	0.302691	−	0.953089i	$-0.402115\pi$
$641$	6.29760e6	0.605383	0.302691	+	0.953089i	$0.402115\pi$
$642$	−1.53254e6	−0.146749
$643$	−4.39762e6	−0.419460	−0.209730	−	0.977759i	$-0.567259\pi$
$643$	−4.39762e6	−0.419460	−0.209730	+	0.977759i	$0.567259\pi$
$644$	−1.56800e6	−0.148981
$645$	0	0
$646$	1.19265e7	1.12443
$647$	−6.55397e6	−0.615522	−0.307761	−	0.951464i	$-0.599580\pi$
$647$	−6.55397e6	−0.615522	−0.307761	+	0.951464i	$0.599580\pi$
$648$	−1.05530e6	−0.0987272
$649$	9.40304e6	0.876308
$650$	0	0
$651$	−3.47155e6	−0.321049
$652$	5.07386e6	0.467433
$653$	−3.79652e6	−0.348420	−0.174210	−	0.984709i	$-0.555737\pi$
$653$	−3.79652e6	−0.348420	−0.174210	+	0.984709i	$0.555737\pi$
$654$	707264.	0.0646602
$655$	0	0
$656$	−3.73094e6	−0.338500
$657$	−1.10475e7	−0.998508
$658$	−61152.0	−0.00550612
$659$	−8.82684e6	−0.791757	−0.395879	−	0.918303i	$-0.629560\pi$
$659$	−8.82684e6	−0.791757	−0.395879	+	0.918303i	$0.629560\pi$
$660$	0	0
$661$	−341270.	−0.0303805	−0.0151902	−	0.999885i	$-0.504835\pi$
$661$	−341270.	−0.0303805	−0.0151902	+	0.999885i	$0.504835\pi$
$662$	9.90165e6	0.878137
$663$	2.88355e6	0.254767
$664$	−4.93158e6	−0.434076
$665$	0	0
$666$	−6.57431e6	−0.574334
$667$	1.34920e7	1.17425
$668$	−2.26509e6	−0.196401
$669$	2.45862e6	0.212386
$670$	0	0
$671$	−1.16749e7	−1.00103
$672$	401408.	0.0342896
$673$	−4.41807e6	−0.376006	−0.188003	−	0.982168i	$-0.560201\pi$
$673$	−4.41807e6	−0.376006	−0.188003	+	0.982168i	$0.560201\pi$
$674$	356616.	0.0302379
$675$	0	0
$676$	−4.55771e6	−0.383601
$677$	−1.63858e7	−1.37403	−0.687014	−	0.726644i	$-0.741079\pi$
$677$	−1.63858e7	−1.37403	−0.687014	+	0.726644i	$0.741079\pi$
$678$	7.84173e6	0.655145
$679$	−1.01185e6	−0.0842251
$680$	0	0
$681$	−7.13434e6	−0.589503
$682$	1.20442e7	0.991552
$683$	1.75399e7	1.43872	0.719360	−	0.694638i	$-0.244435\pi$
$683$	1.75399e7	1.43872	0.719360	+	0.694638i	$0.244435\pi$
$684$	−6.96525e6	−0.569241
$685$	0	0
$686$	−470596.	−0.0381802
$687$	−2.21365e6	−0.178944
$688$	−2.07565e6	−0.167179
$689$	4.30240e6	0.345273
$690$	0	0
$691$	3.14638e6	0.250678	0.125339	−	0.992114i	$-0.459998\pi$
$691$	3.14638e6	0.250678	0.125339	+	0.992114i	$0.459998\pi$
$692$	1.13955e6	0.0904626
$693$	2.98214e6	0.235882
$694$	3.75422e6	0.295884
$695$	0	0
$696$	−3.45395e6	−0.270267
$697$	1.78677e7	1.39312
$698$	−1.33707e7	−1.03876
$699$	1.18355e7	0.916205
$700$	0	0
$701$	−1.90919e7	−1.46742	−0.733709	−	0.679464i	$-0.762212\pi$
$701$	−1.90919e7	−1.46742	−0.733709	+	0.679464i	$0.762212\pi$
$702$	−3.97018e6	−0.304065
$703$	−2.23306e7	−1.70417
$704$	−1.39264e6	−0.105903
$705$	0	0
$706$	−1.50642e7	−1.13746
$707$	9.12625e6	0.686663
$708$	3.53997e6	0.265409
$709$	990974.	0.0740366	0.0370183	−	0.999315i	$-0.488214\pi$
$709$	990974.	0.0740366	0.0370183	+	0.999315i	$0.488214\pi$
$710$	0	0
$711$	1.15906e7	0.859869
$712$	522624.	0.0386358
$713$	−1.77120e7	−1.30480
$714$	−1.92237e6	−0.141121
$715$	0	0
$716$	7.77005e6	0.566423
$717$	−8.00275e6	−0.581355
$718$	6.15738e6	0.445743
$719$	−1.69014e7	−1.21928	−0.609638	−	0.792680i	$-0.708685\pi$
$719$	−1.69014e7	−1.21928	−0.609638	+	0.792680i	$0.708685\pi$
$720$	0	0
$721$	2.94314e6	0.210849
$722$	−1.37541e7	−0.981950
$723$	−1.08666e7	−0.773125
$724$	1.05134e7	0.745416
$725$	0	0
$726$	−1.45443e6	−0.102412
$727$	2.34302e7	1.64414	0.822071	−	0.569384i	$-0.192818\pi$
$727$	2.34302e7	1.64414	0.822071	+	0.569384i	$0.192818\pi$
$728$	−921984.	−0.0644755
$729$	3.61141e6	0.251686
$730$	0	0
$731$	9.94041e6	0.688035
$732$	−4.39526e6	−0.303185
$733$	−975810.	−0.0670819	−0.0335409	−	0.999437i	$-0.510678\pi$
$733$	−975810.	−0.0670819	−0.0335409	+	0.999437i	$0.510678\pi$
$734$	−3.43725e6	−0.235489
$735$	0	0
$736$	2.04800e6	0.139359
$737$	4.18744e6	0.283975
$738$	−1.04350e7	−0.705263
$739$	−6.30208e6	−0.424495	−0.212247	−	0.977216i	$-0.568078\pi$
$739$	−6.30208e6	−0.424495	−0.212247	+	0.977216i	$0.568078\pi$
$740$	0	0
$741$	−5.72006e6	−0.382697
$742$	−2.86826e6	−0.191253
$743$	6.95698e6	0.462326	0.231163	−	0.972915i	$-0.425747\pi$
$743$	6.95698e6	0.462326	0.231163	+	0.972915i	$0.425747\pi$
$744$	4.53427e6	0.300314
$745$	0	0
$746$	−3.90634e6	−0.256994
$747$	−1.37930e7	−0.904395
$748$	6.66944e6	0.435848
$749$	−2.34671e6	−0.152846
$750$	0	0
$751$	2.74535e7	1.77622	0.888112	−	0.459628i	$-0.152017\pi$
$751$	2.74535e7	1.77622	0.888112	+	0.459628i	$0.152017\pi$
$752$	79872.0	0.00515051
$753$	1.41926e6	0.0912170
$754$	7.93330e6	0.508189
$755$	0	0
$756$	2.64678e6	0.168428
$757$	1.96889e7	1.24877	0.624384	−	0.781118i	$-0.285350\pi$
$757$	1.96889e7	1.24877	0.624384	+	0.781118i	$0.285350\pi$
$758$	−425776.	−0.0269159
$759$	−5.44000e6	−0.342763
$760$	0	0
$761$	−2.82079e7	−1.76567	−0.882835	−	0.469684i	$-0.844368\pi$
$761$	−2.82079e7	−1.76567	−0.882835	+	0.469684i	$0.844368\pi$
$762$	3.09427e6	0.193051
$763$	1.08300e6	0.0673467
$764$	1.09286e6	0.0677381
$765$	0	0
$766$	−8.02538e6	−0.494189
$767$	−8.13086e6	−0.499055
$768$	−524288.	−0.0320750
$769$	−1.38081e6	−0.0842009	−0.0421005	−	0.999113i	$-0.513405\pi$
$769$	−1.38081e6	−0.0842009	−0.0421005	+	0.999113i	$0.513405\pi$
$770$	0	0
$771$	2.61326e6	0.158324
$772$	−5.64406e6	−0.340839
$773$	1.54347e7	0.929074	0.464537	−	0.885554i	$-0.346221\pi$
$773$	1.54347e7	0.929074	0.464537	+	0.885554i	$0.346221\pi$
$774$	−5.80533e6	−0.348317
$775$	0	0
$776$	1.32160e6	0.0787854
$777$	3.59934e6	0.213880
$778$	2.73601e6	0.162057
$779$	−3.54440e7	−2.09266
$780$	0	0
$781$	−1.25528e7	−0.736399
$782$	−9.80800e6	−0.573540
$783$	−2.27745e7	−1.32753
$784$	614656.	0.0357143
$785$	0	0
$786$	4.29978e6	0.248250
$787$	7.10107e6	0.408683	0.204342	−	0.978900i	$-0.434495\pi$
$787$	7.10107e6	0.408683	0.204342	+	0.978900i	$0.434495\pi$
$788$	−3.15171e6	−0.180814
$789$	−279360.	−0.0159761
$790$	0	0
$791$	1.20076e7	0.682365
$792$	−3.89504e6	−0.220647
$793$	1.00954e7	0.570085
$794$	−891480.	−0.0501834
$795$	0	0
$796$	−1.76627e7	−0.988041
$797$	−6.48182e6	−0.361452	−0.180726	−	0.983533i	$-0.557845\pi$
$797$	−6.48182e6	−0.361452	−0.180726	+	0.983533i	$0.557845\pi$
$798$	3.81338e6	0.211984
$799$	−382512.	−0.0211972
$800$	0	0
$801$	1.46171e6	0.0804973
$802$	−7.60289e6	−0.417391
$803$	−2.09841e7	−1.14842
$804$	1.57645e6	0.0860081
$805$	0	0
$806$	−1.04147e7	−0.564686
$807$	−5.73166e6	−0.309811
$808$	−1.19200e7	−0.642315
$809$	1.60578e7	0.862610	0.431305	−	0.902206i	$-0.358053\pi$
$809$	1.60578e7	0.862610	0.431305	+	0.902206i	$0.358053\pi$
$810$	0	0
$811$	4.84775e6	0.258814	0.129407	−	0.991592i	$-0.458693\pi$
$811$	4.84775e6	0.258814	0.129407	+	0.991592i	$0.458693\pi$
$812$	−5.28886e6	−0.281496
$813$	7.62701e6	0.404695
$814$	−1.24875e7	−0.660564
$815$	0	0
$816$	2.51085e6	0.132006
$817$	−1.97187e7	−1.03353
$818$	−7.10862e6	−0.371451
$819$	−2.57867e6	−0.134334
$820$	0	0
$821$	2.17976e7	1.12863	0.564314	−	0.825560i	$-0.309141\pi$
$821$	2.17976e7	1.12863	0.564314	+	0.825560i	$0.309141\pi$
$822$	9.42918e6	0.486737
$823$	−3.20206e7	−1.64790	−0.823948	−	0.566665i	$-0.808233\pi$
$823$	−3.20206e7	−1.64790	−0.823948	+	0.566665i	$0.808233\pi$
$824$	−3.84410e6	−0.197231
$825$	0	0
$826$	5.42058e6	0.276436
$827$	−2.19008e7	−1.11352	−0.556758	−	0.830675i	$-0.687955\pi$
$827$	−2.19008e7	−1.11352	−0.556758	+	0.830675i	$0.687955\pi$
$828$	5.72800e6	0.290354
$829$	−1.45999e7	−0.737844	−0.368922	−	0.929460i	$-0.620273\pi$
$829$	−1.45999e7	−0.737844	−0.368922	+	0.929460i	$0.620273\pi$
$830$	0	0
$831$	−1.47783e7	−0.742374
$832$	1.20422e6	0.0603113
$833$	−2.94363e6	−0.146984
$834$	1.00782e7	0.501728
$835$	0	0
$836$	−1.32301e7	−0.654707
$837$	2.98979e7	1.47512
$838$	−112224.	−0.00552047
$839$	4.60947e6	0.226072	0.113036	−	0.993591i	$-0.463942\pi$
$839$	4.60947e6	0.226072	0.113036	+	0.993591i	$0.463942\pi$
$840$	0	0
$841$	2.49974e7	1.21872
$842$	1.08359e7	0.526725
$843$	1.59680e7	0.773897
$844$	−1.65510e6	−0.0799777
$845$	0	0
$846$	223392.	0.0107310
$847$	−2.22710e6	−0.106667
$848$	3.74630e6	0.178901
$849$	1.87270e6	0.0891661
$850$	0	0
$851$	1.83640e7	0.869247
$852$	−4.72576e6	−0.223035
$853$	1.98437e7	0.933793	0.466897	−	0.884312i	$-0.345372\pi$
$853$	1.98437e7	0.933793	0.466897	+	0.884312i	$0.345372\pi$
$854$	−6.73025e6	−0.315781
$855$	0	0
$856$	3.06509e6	0.142974
$857$	1.22960e6	0.0571888	0.0285944	−	0.999591i	$-0.490897\pi$
$857$	1.22960e6	0.0571888	0.0285944	+	0.999591i	$0.490897\pi$
$858$	−3.19872e6	−0.148340
$859$	3.33041e7	1.53998	0.769989	−	0.638058i	$-0.220262\pi$
$859$	3.33041e7	1.53998	0.769989	+	0.638058i	$0.220262\pi$
$860$	0	0
$861$	5.71301e6	0.262638
$862$	−2.21559e7	−1.01560
$863$	2.36616e7	1.08148	0.540738	−	0.841191i	$-0.318145\pi$
$863$	2.36616e7	1.08148	0.540738	+	0.841191i	$0.318145\pi$
$864$	−3.45702e6	−0.157550
$865$	0	0
$866$	−3.47318e6	−0.157374
$867$	−665752.	−0.0300791
$868$	6.94310e6	0.312791
$869$	2.20157e7	0.988969
$870$	0	0
$871$	−3.62090e6	−0.161723
$872$	−1.41453e6	−0.0629971
$873$	3.69635e6	0.164149
$874$	1.94560e7	0.861539
$875$	0	0
$876$	−7.89990e6	−0.347826
$877$	2.37812e7	1.04408	0.522042	−	0.852920i	$-0.325170\pi$
$877$	2.37812e7	1.04408	0.522042	+	0.852920i	$0.325170\pi$
$878$	4.55069e6	0.199224
$879$	−2.00064e7	−0.873369
$880$	0	0
$881$	−1.41871e7	−0.615818	−0.307909	−	0.951416i	$-0.599629\pi$
$881$	−1.41871e7	−0.615818	−0.307909	+	0.951416i	$0.599629\pi$
$882$	1.71912e6	0.0744104
$883$	−2.09281e7	−0.903293	−0.451647	−	0.892197i	$-0.649163\pi$
$883$	−2.09281e7	−0.903293	−0.451647	+	0.892197i	$0.649163\pi$
$884$	−5.76710e6	−0.248214
$885$	0	0
$886$	7.01595e6	0.300263
$887$	7.98586e6	0.340810	0.170405	−	0.985374i	$-0.445492\pi$
$887$	7.98586e6	0.340810	0.170405	+	0.985374i	$0.445492\pi$
$888$	−4.70118e6	−0.200067
$889$	4.73810e6	0.201071
$890$	0	0
$891$	−5.60626e6	−0.236581
$892$	−4.91725e6	−0.206924
$893$	758784.	0.0318412
$894$	3.63590e6	0.152149
$895$	0	0
$896$	−802816.	−0.0334077
$897$	4.70400e6	0.195203
$898$	−9.66695e6	−0.400036
$899$	−5.97426e7	−2.46538
$900$	0	0
$901$	−1.79413e7	−0.736278
$902$	−1.98206e7	−0.811150
$903$	3.17834e6	0.129712
$904$	−1.56835e7	−0.638294
$905$	0	0
$906$	1.30627e7	0.528702
$907$	2.31861e7	0.935856	0.467928	−	0.883767i	$-0.345001\pi$
$907$	2.31861e7	0.935856	0.467928	+	0.883767i	$0.345001\pi$
$908$	1.42687e7	0.574340
$909$	−3.33388e7	−1.33826
$910$	0	0
$911$	1.65299e7	0.659895	0.329948	−	0.943999i	$-0.392969\pi$
$911$	1.65299e7	0.659895	0.329948	+	0.943999i	$0.392969\pi$
$912$	−4.98074e6	−0.198293
$913$	−2.61990e7	−1.04018
$914$	−509720.	−0.0201821
$915$	0	0
$916$	4.42730e6	0.174341
$917$	6.58403e6	0.258564
$918$	1.65559e7	0.648405
$919$	1.28087e7	0.500283	0.250142	−	0.968209i	$-0.419523\pi$
$919$	1.28087e7	0.500283	0.250142	+	0.968209i	$0.419523\pi$
$920$	0	0
$921$	1.87363e7	0.727836
$922$	512792.	0.0198662
$923$	1.08545e7	0.419377
$924$	2.13248e6	0.0821684
$925$	0	0
$926$	−1.60661e7	−0.615720
$927$	−1.07515e7	−0.410930
$928$	6.90790e6	0.263315
$929$	2.97319e7	1.13027	0.565136	−	0.824998i	$-0.308824\pi$
$929$	2.97319e7	1.13027	0.565136	+	0.824998i	$0.308824\pi$
$930$	0	0
$931$	5.83923e6	0.220791
$932$	−2.36709e7	−0.892639
$933$	1.30451e6	0.0490619
$934$	3.46899e7	1.30117
$935$	0	0
$936$	3.36806e6	0.125658
$937$	−1.10970e7	−0.412911	−0.206456	−	0.978456i	$-0.566193\pi$
$937$	−1.10970e7	−0.412911	−0.206456	+	0.978456i	$0.566193\pi$
$938$	2.41394e6	0.0895816
$939$	1.39172e7	0.515096
$940$	0	0
$941$	3.74313e7	1.37804	0.689019	−	0.724743i	$-0.258042\pi$
$941$	3.74313e7	1.37804	0.689019	+	0.724743i	$0.258042\pi$
$942$	−9.39347e6	−0.344905
$943$	2.91480e7	1.06741
$944$	−7.07994e6	−0.258583
$945$	0	0
$946$	−1.10269e7	−0.400613
$947$	−1.50907e7	−0.546808	−0.273404	−	0.961899i	$-0.588150\pi$
$947$	−1.50907e7	−0.546808	−0.273404	+	0.961899i	$0.588150\pi$
$948$	8.28826e6	0.299531
$949$	1.81451e7	0.654024
$950$	0	0
$951$	−1.43817e7	−0.515655
$952$	3.84474e6	0.137491
$953$	2.15741e7	0.769484	0.384742	−	0.923024i	$-0.374290\pi$
$953$	2.15741e7	0.769484	0.384742	+	0.923024i	$0.374290\pi$
$954$	1.04779e7	0.372739
$955$	0	0
$956$	1.60055e7	0.566402
$957$	−1.83491e7	−0.647643
$958$	−3.31579e7	−1.16727
$959$	1.44384e7	0.506960
$960$	0	0
$961$	4.97996e7	1.73947
$962$	1.07980e7	0.376190
$963$	8.57267e6	0.297886
$964$	2.17333e7	0.753239
$965$	0	0
$966$	−3.13600e6	−0.108127
$967$	3.29467e7	1.13304	0.566520	−	0.824048i	$-0.308289\pi$
$967$	3.29467e7	1.13304	0.566520	+	0.824048i	$0.308289\pi$
$968$	2.90886e6	0.0997781
$969$	2.38531e7	0.816083
$970$	0	0
$971$	2.24599e7	0.764470	0.382235	−	0.924065i	$-0.375154\pi$
$971$	2.24599e7	0.764470	0.382235	+	0.924065i	$0.375154\pi$
$972$	−1.52365e7	−0.517272
$973$	1.54323e7	0.522573
$974$	−3.56708e7	−1.20480
$975$	0	0
$976$	8.79053e6	0.295386
$977$	5.16236e7	1.73026	0.865132	−	0.501545i	$-0.167235\pi$
$977$	5.16236e7	1.73026	0.865132	+	0.501545i	$0.167235\pi$
$978$	1.01477e7	0.339251
$979$	2.77644e6	0.0925831
$980$	0	0
$981$	−3.95626e6	−0.131254
$982$	2.28615e7	0.756529
$983$	1.10202e7	0.363751	0.181876	−	0.983322i	$-0.441783\pi$
$983$	1.10202e7	0.363751	0.181876	+	0.983322i	$0.441783\pi$
$984$	−7.46189e6	−0.245675
$985$	0	0
$986$	−3.30824e7	−1.08369
$987$	−122304.	−0.00399621
$988$	1.14401e7	0.372854
$989$	1.62160e7	0.527173
$990$	0	0
$991$	3.21029e7	1.03839	0.519194	−	0.854656i	$-0.326232\pi$
$991$	3.21029e7	1.03839	0.519194	+	0.854656i	$0.326232\pi$
$992$	−9.06854e6	−0.292589
$993$	1.98033e7	0.637330
$994$	−7.23632e6	−0.232301
$995$	0	0
$996$	−9.86317e6	−0.315042
$997$	−2.81772e7	−0.897759	−0.448879	−	0.893592i	$-0.648177\pi$
$997$	−2.81772e7	−0.897759	−0.448879	+	0.893592i	$0.648177\pi$
$998$	−500464.	−0.0159055
$999$	−3.09984e7	−0.982711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.b.1.1		1
5.2	odd	4		350.6.c.f.99.1		2
5.3	odd	4		350.6.c.f.99.2		2
5.4	even	2		14.6.a.b.1.1	✓	1
15.14	odd	2		126.6.a.c.1.1		1
20.19	odd	2		112.6.a.d.1.1		1
35.4	even	6		98.6.c.a.79.1		2
35.9	even	6		98.6.c.a.67.1		2
35.19	odd	6		98.6.c.b.67.1		2
35.24	odd	6		98.6.c.b.79.1		2
35.34	odd	2		98.6.a.b.1.1		1
40.19	odd	2		448.6.a.k.1.1		1
40.29	even	2		448.6.a.f.1.1		1
60.59	even	2		1008.6.a.n.1.1		1
105.104	even	2		882.6.a.g.1.1		1
140.139	even	2		784.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.b.1.1	✓	1	5.4	even	2
98.6.a.b.1.1		1	35.34	odd	2
98.6.c.a.67.1		2	35.9	even	6
98.6.c.a.79.1		2	35.4	even	6
98.6.c.b.67.1		2	35.19	odd	6
98.6.c.b.79.1		2	35.24	odd	6
112.6.a.d.1.1		1	20.19	odd	2
126.6.a.c.1.1		1	15.14	odd	2
350.6.a.b.1.1		1	1.1	even	1	trivial
350.6.c.f.99.1		2	5.2	odd	4
350.6.c.f.99.2		2	5.3	odd	4
448.6.a.f.1.1		1	40.29	even	2
448.6.a.k.1.1		1	40.19	odd	2
784.6.a.h.1.1		1	140.139	even	2
882.6.a.g.1.1		1	105.104	even	2
1008.6.a.n.1.1		1	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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