LMFDB - Embedded newform 650.6.b.c.599.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	650.b (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$104.249482878$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 49$ x^4 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.3
Root			$-1.87083 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.c.599.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -3.22497i q^{3} -16.0000 q^{4} +12.8999 q^{6} -200.833i q^{7} -64.0000i q^{8} +232.600 q^{9} -530.607 q^{11} +51.5996i q^{12} +169.000i q^{13} +803.333 q^{14} +256.000 q^{16} -15.1196i q^{17} +930.398i q^{18} +392.592 q^{19} -647.681 q^{21} -2122.43i q^{22} -2633.57i q^{23} -206.398 q^{24} -676.000 q^{26} -1533.80i q^{27} +3213.33i q^{28} +7135.52 q^{29} +6828.51 q^{31} +1024.00i q^{32} +1711.19i q^{33} +60.4783 q^{34} -3721.59 q^{36} -13272.7i q^{37} +1570.37i q^{38} +545.020 q^{39} +3210.06 q^{41} -2590.73i q^{42} +11083.4i q^{43} +8489.71 q^{44} +10534.3 q^{46} +9258.69i q^{47} -825.593i q^{48} -23527.0 q^{49} -48.7602 q^{51} -2704.00i q^{52} -3520.99i q^{53} +6135.18 q^{54} -12853.3 q^{56} -1266.10i q^{57} +28542.1i q^{58} -40334.1 q^{59} -44243.3 q^{61} +27314.0i q^{62} -46713.7i q^{63} -4096.00 q^{64} -6844.77 q^{66} +7071.36i q^{67} +241.913i q^{68} -8493.18 q^{69} -36271.3 q^{71} -14886.4i q^{72} +41064.8i q^{73} +53090.8 q^{74} -6281.47 q^{76} +106563. i q^{77} +2180.08i q^{78} +19473.4 q^{79} +51575.2 q^{81} +12840.3i q^{82} -67566.0i q^{83} +10362.9 q^{84} -44333.6 q^{86} -23011.9i q^{87} +33958.9i q^{88} -33319.0 q^{89} +33940.8 q^{91} +42137.1i q^{92} -22021.7i q^{93} -37034.8 q^{94} +3302.37 q^{96} -2206.46i q^{97} -94107.8i q^{98} -123419. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 64 q^{4} - 128 q^{6} + 212 q^{9} - 72 q^{11} + 2016 q^{14} + 1024 q^{16} + 2184 q^{19} + 672 q^{21} + 2048 q^{24} - 2704 q^{26} + 9744 q^{29} - 4520 q^{31} + 19040 q^{34} - 3392 q^{36} - 5408 q^{39}+ \cdots - 372072 q^{99}+O(q^{100})$ 4 * q - 64 * q^4 - 128 * q^6 + 212 * q^9 - 72 * q^11 + 2016 * q^14 + 1024 * q^16 + 2184 * q^19 + 672 * q^21 + 2048 * q^24 - 2704 * q^26 + 9744 * q^29 - 4520 * q^31 + 19040 * q^34 - 3392 * q^36 - 5408 * q^39 - 18440 * q^41 + 1152 * q^44 - 9408 * q^46 - 18676 * q^49 + 90832 * q^51 - 5632 * q^54 - 32256 * q^56 - 180808 * q^59 - 138000 * q^61 - 16384 * q^64 - 89760 * q^66 - 163464 * q^69 - 57336 * q^71 + 142080 * q^74 - 34944 * q^76 - 168128 * q^79 - 44420 * q^81 - 10752 * q^84 - 189248 * q^86 - 185240 * q^89 + 85176 * q^91 - 102880 * q^94 - 32768 * q^96 - 372072 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$ .

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 3.22497i	− 0.206882i	−0.994636	−	0.103441i	$-0.967015\pi$
$3$	− 3.22497i	− 0.206882i	0.994636	−	0.103441i	$-0.0329853\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	12.8999	0.146288
$7$	− 200.833i	− 1.54914i	−0.632489	−	0.774569i	$-0.717967\pi$
$7$	− 200.833i	− 1.54914i	0.632489	−	0.774569i	$-0.282033\pi$
$8$	− 64.0000i	− 0.353553i
$9$	232.600	0.957200
$10$	0	0
$11$	−530.607	−1.32218	−0.661091	−	0.750306i	$-0.729906\pi$
$11$	−530.607	−1.32218	−0.661091	+	0.750306i	$0.729906\pi$
$12$	51.5996i	0.103441i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	803.333	1.09541
$15$	0	0
$16$	256.000	0.250000
$17$	− 15.1196i	− 0.0126887i	−0.999980	−	0.00634435i	$-0.997981\pi$
$17$	− 15.1196i	− 0.0126887i	0.999980	−	0.00634435i	$-0.00201948\pi$
$18$	930.398i	0.676842i
$19$	392.592	0.249493	0.124746	−	0.992189i	$-0.460188\pi$
$19$	392.592	0.249493	0.124746	+	0.992189i	$0.460188\pi$
$20$	0	0
$21$	−647.681	−0.320489
$22$	− 2122.43i	− 0.934924i
$23$	− 2633.57i	− 1.03807i	−0.854754	−	0.519033i	$-0.826292\pi$
$23$	− 2633.57i	− 1.03807i	0.854754	−	0.519033i	$-0.173708\pi$
$24$	−206.398	−0.0731439
$25$	0	0
$26$	−676.000	−0.196116
$27$	− 1533.80i	− 0.404910i
$28$	3213.33i	0.774569i
$29$	7135.52	1.57554	0.787772	−	0.615966i	$-0.211234\pi$
$29$	7135.52	1.57554	0.787772	+	0.615966i	$0.211234\pi$
$30$	0	0
$31$	6828.51	1.27621	0.638104	−	0.769950i	$-0.279719\pi$
$31$	6828.51	1.27621	0.638104	+	0.769950i	$0.279719\pi$
$32$	1024.00i	0.176777i
$33$	1711.19i	0.273536i
$34$	60.4783	0.00897227
$35$	0	0
$36$	−3721.59	−0.478600
$37$	− 13272.7i	− 1.59388i	−0.604060	−	0.796939i	$-0.706451\pi$
$37$	− 13272.7i	− 1.59388i	0.604060	−	0.796939i	$-0.293549\pi$
$38$	1570.37i	0.176418i
$39$	545.020	0.0573788
$40$	0	0
$41$	3210.06	0.298232	0.149116	−	0.988820i	$-0.452357\pi$
$41$	3210.06	0.298232	0.149116	+	0.988820i	$0.452357\pi$
$42$	− 2590.73i	− 0.226620i
$43$	11083.4i	0.914118i	0.889436	+	0.457059i	$0.151097\pi$
$43$	11083.4i	0.914118i	−0.889436	+	0.457059i	$0.848903\pi$
$44$	8489.71	0.661091
$45$	0	0
$46$	10534.3	0.734023
$47$	9258.69i	0.611371i	0.952132	+	0.305686i	$0.0988857\pi$
$47$	9258.69i	0.611371i	−0.952132	+	0.305686i	$0.901114\pi$
$48$	− 825.593i	− 0.0517205i
$49$	−23527.0	−1.39983
$50$	0	0
$51$	−48.7602	−0.00262507
$52$	− 2704.00i	− 0.138675i
$53$	− 3520.99i	− 0.172177i	−0.996287	−	0.0860885i	$-0.972563\pi$
$53$	− 3520.99i	− 0.172177i	0.996287	−	0.0860885i	$-0.0274368\pi$
$54$	6135.18	0.286314
$55$	0	0
$56$	−12853.3	−0.547703
$57$	− 1266.10i	− 0.0516155i
$58$	28542.1i	1.11408i
$59$	−40334.1	−1.50849	−0.754245	−	0.656593i	$-0.771997\pi$
$59$	−40334.1	−1.50849	−0.754245	+	0.656593i	$0.771997\pi$
$60$	0	0
$61$	−44243.3	−1.52238	−0.761189	−	0.648530i	$-0.775384\pi$
$61$	−44243.3	−1.52238	−0.761189	+	0.648530i	$0.775384\pi$
$62$	27314.0i	0.902415i
$63$	− 46713.7i	− 1.48284i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−6844.77	−0.193419
$67$	7071.36i	0.192449i	0.995360	+	0.0962246i	$0.0306767\pi$
$67$	7071.36i	0.192449i	−0.995360	+	0.0962246i	$0.969323\pi$
$68$	241.913i	0.00634435i
$69$	−8493.18	−0.214757
$70$	0	0
$71$	−36271.3	−0.853921	−0.426961	−	0.904270i	$-0.640416\pi$
$71$	−36271.3	−0.853921	−0.426961	+	0.904270i	$0.640416\pi$
$72$	− 14886.4i	− 0.338421i
$73$	41064.8i	0.901909i	0.892547	+	0.450955i	$0.148916\pi$
$73$	41064.8i	0.901909i	−0.892547	+	0.450955i	$0.851084\pi$
$74$	53090.8	1.12704
$75$	0	0
$76$	−6281.47	−0.124746
$77$	106563.i	2.04824i
$78$	2180.08i	0.0405729i
$79$	19473.4	0.351053	0.175527	−	0.984475i	$-0.443837\pi$
$79$	19473.4	0.351053	0.175527	+	0.984475i	$0.443837\pi$
$80$	0	0
$81$	51575.2	0.873431
$82$	12840.3i	0.210882i
$83$	− 67566.0i	− 1.07655i	−0.842770	−	0.538274i	$-0.819077\pi$
$83$	− 67566.0i	− 1.07655i	0.842770	−	0.538274i	$-0.180923\pi$
$84$	10362.9	0.160245
$85$	0	0
$86$	−44333.6	−0.646379
$87$	− 23011.9i	− 0.325952i
$88$	33958.9i	0.467462i
$89$	−33319.0	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$89$	−33319.0	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$90$	0	0
$91$	33940.8	0.429654
$92$	42137.1i	0.519033i
$93$	− 22021.7i	− 0.264025i
$94$	−37034.8	−0.432305
$95$	0	0
$96$	3302.37	0.0365719
$97$	− 2206.46i	− 0.0238103i	−0.999929	−	0.0119052i	$-0.996210\pi$
$97$	− 2206.46i	− 0.0238103i	0.999929	−	0.0119052i	$-0.00378962\pi$
$98$	− 94107.8i	− 0.989830i
$99$	−123419.	−1.26559
$100$	0	0
$101$	−93675.8	−0.913743	−0.456871	−	0.889533i	$-0.651030\pi$
$101$	−93675.8	−0.913743	−0.456871	+	0.889533i	$0.651030\pi$
$102$	− 195.041i	− 0.00185620i
$103$	− 116473.i	− 1.08176i	−0.841099	−	0.540881i	$-0.818091\pi$
$103$	− 116473.i	− 1.08176i	0.841099	−	0.540881i	$-0.181909\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	14084.0	0.121747
$107$	− 195303.i	− 1.64911i	−0.565784	−	0.824554i	$-0.691426\pi$
$107$	− 195303.i	− 1.64911i	0.565784	−	0.824554i	$-0.308574\pi$
$108$	24540.7i	0.202455i
$109$	−173562.	−1.39923	−0.699613	−	0.714522i	$-0.746644\pi$
$109$	−173562.	−1.39923	−0.699613	+	0.714522i	$0.746644\pi$
$110$	0	0
$111$	−42804.1	−0.329745
$112$	− 51413.3i	− 0.387285i
$113$	− 3810.27i	− 0.0280711i	−0.999901	−	0.0140356i	$-0.995532\pi$
$113$	− 3810.27i	− 0.0280711i	0.999901	−	0.0140356i	$-0.00446781\pi$
$114$	5064.39	0.0364977
$115$	0	0
$116$	−114168.	−0.787772
$117$	39309.3i	0.265479i
$118$	− 161336.i	− 1.06666i
$119$	−3036.51	−0.0196566
$120$	0	0
$121$	120493.	0.748166
$122$	− 176973.i	− 1.07648i
$123$	− 10352.4i	− 0.0616988i
$124$	−109256.	−0.638104
$125$	0	0
$126$	186855.	1.04852
$127$	− 19926.1i	− 0.109626i	−0.998497	−	0.0548130i	$-0.982544\pi$
$127$	− 19926.1i	− 0.109626i	0.998497	−	0.0548130i	$-0.0174563\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	35743.7	0.189115
$130$	0	0
$131$	266895.	1.35882	0.679409	−	0.733759i	$-0.262236\pi$
$131$	266895.	1.35882	0.679409	+	0.733759i	$0.262236\pi$
$132$	− 27379.1i	− 0.136768i
$133$	− 78845.5i	− 0.386498i
$134$	−28285.4	−0.136082
$135$	0	0
$136$	−967.653	−0.00448614
$137$	9618.96i	0.0437851i	0.999760	+	0.0218926i	$0.00696918\pi$
$137$	9618.96i	0.0437851i	−0.999760	+	0.0218926i	$0.993031\pi$
$138$	− 33972.7i	− 0.151856i
$139$	85185.2	0.373962	0.186981	−	0.982364i	$-0.440130\pi$
$139$	85185.2	0.373962	0.186981	+	0.982364i	$0.440130\pi$
$140$	0	0
$141$	29859.0	0.126482
$142$	− 145085.i	− 0.603813i
$143$	− 89672.6i	− 0.366707i
$144$	59545.5	0.239300
$145$	0	0
$146$	−164259.	−0.637746
$147$	75873.8i	0.289600i
$148$	212363.i	0.796939i
$149$	−222940.	−0.822663	−0.411332	−	0.911486i	$-0.634936\pi$
$149$	−222940.	−0.822663	−0.411332	+	0.911486i	$0.634936\pi$
$150$	0	0
$151$	−363446.	−1.29717	−0.648585	−	0.761142i	$-0.724639\pi$
$151$	−363446.	−1.29717	−0.648585	+	0.761142i	$0.724639\pi$
$152$	− 25125.9i	− 0.0882089i
$153$	− 3516.81i	− 0.0121456i
$154$	−426254.	−1.44833
$155$	0	0
$156$	−8720.32	−0.0286894
$157$	− 323402.i	− 1.04711i	−0.851991	−	0.523556i	$-0.824605\pi$
$157$	− 323402.i	− 1.04711i	0.851991	−	0.523556i	$-0.175395\pi$
$158$	77893.5i	0.248232i
$159$	−11355.1	−0.0356203
$160$	0	0
$161$	−528908.	−1.60811
$162$	206301.i	0.617609i
$163$	361691.i	1.06627i	0.846029	+	0.533137i	$0.178987\pi$
$163$	361691.i	1.06627i	−0.846029	+	0.533137i	$0.821013\pi$
$164$	−51361.0	−0.149116
$165$	0	0
$166$	270264.	0.761234
$167$	− 535913.i	− 1.48697i	−0.668751	−	0.743486i	$-0.733171\pi$
$167$	− 535913.i	− 1.48697i	0.668751	−	0.743486i	$-0.266829\pi$
$168$	41451.6i	0.113310i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	91316.7	0.238814
$172$	− 177335.i	− 0.457059i
$173$	252514.i	0.641462i	0.947170	+	0.320731i	$0.103929\pi$
$173$	252514.i	0.641462i	−0.947170	+	0.320731i	$0.896071\pi$
$174$	92047.4	0.230483
$175$	0	0
$176$	−135835.	−0.330546
$177$	130076.i	0.312079i
$178$	− 133276.i	− 0.315284i
$179$	499988.	1.16634	0.583172	−	0.812349i	$-0.301811\pi$
$179$	499988.	1.16634	0.583172	+	0.812349i	$0.301811\pi$
$180$	0	0
$181$	−607767.	−1.37892	−0.689462	−	0.724322i	$-0.742153\pi$
$181$	−607767.	−1.37892	−0.689462	+	0.724322i	$0.742153\pi$
$182$	135763.i	0.303811i
$183$	142683.i	0.314953i
$184$	−168548.	−0.367012
$185$	0	0
$186$	88087.0	0.186694
$187$	8022.56i	0.0167768i
$188$	− 148139.i	− 0.305686i
$189$	−308037.	−0.627261
$190$	0	0
$191$	51603.7	0.102352	0.0511762	−	0.998690i	$-0.483703\pi$
$191$	51603.7	0.102352	0.0511762	+	0.998690i	$0.483703\pi$
$192$	13209.5i	0.0258603i
$193$	946349.i	1.82877i	0.404851	+	0.914383i	$0.367323\pi$
$193$	946349.i	1.82877i	−0.404851	+	0.914383i	$0.632677\pi$
$194$	8825.82	0.0168365
$195$	0	0
$196$	376431.	0.699915
$197$	− 229033.i	− 0.420468i	−0.977651	−	0.210234i	$-0.932577\pi$
$197$	− 229033.i	− 0.420468i	0.977651	−	0.210234i	$-0.0674226\pi$
$198$	− 493676.i	− 0.894909i
$199$	−1.06207e6	−1.90116	−0.950582	−	0.310473i	$-0.899513\pi$
$199$	−1.06207e6	−1.90116	−0.950582	+	0.310473i	$0.899513\pi$
$200$	0	0
$201$	22804.9	0.0398143
$202$	− 374703.i	− 0.646114i
$203$	− 1.43305e6i	− 2.44074i
$204$	780.164	0.00131253
$205$	0	0
$206$	465892.	0.764922
$207$	− 612567.i	− 0.993636i
$208$	43264.0i	0.0693375i
$209$	−208312.	−0.329875
$210$	0	0
$211$	−256699.	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$211$	−256699.	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$212$	56335.8i	0.0860885i
$213$	116974.i	0.176661i
$214$	781211.	1.16610
$215$	0	0
$216$	−98162.9	−0.143157
$217$	− 1.37139e6i	− 1.97702i
$218$	− 694247.i	− 0.989402i
$219$	132433.	0.186589
$220$	0	0
$221$	2555.21	0.00351921
$222$	− 171216.i	− 0.233165i
$223$	− 809910.i	− 1.09062i	−0.838233	−	0.545312i	$-0.816411\pi$
$223$	− 809910.i	− 1.09062i	0.838233	−	0.545312i	$-0.183589\pi$
$224$	205653.	0.273852
$225$	0	0
$226$	15241.1	0.0198493
$227$	− 656318.i	− 0.845375i	−0.906276	−	0.422687i	$-0.861087\pi$
$227$	− 656318.i	− 0.845375i	0.906276	−	0.422687i	$-0.138913\pi$
$228$	20257.6i	0.0258078i
$229$	−1.19398e6	−1.50456	−0.752280	−	0.658844i	$-0.771046\pi$
$229$	−1.19398e6	−1.50456	−0.752280	+	0.658844i	$0.771046\pi$
$230$	0	0
$231$	343664.	0.423745
$232$	− 456673.i	− 0.557039i
$233$	1.06005e6i	1.27920i	0.768709	+	0.639598i	$0.220899\pi$
$233$	1.06005e6i	1.27920i	−0.768709	+	0.639598i	$0.779101\pi$
$234$	−157237.	−0.187722
$235$	0	0
$236$	645346.	0.754245
$237$	− 62801.1i	− 0.0726267i
$238$	− 12146.1i	− 0.0138993i
$239$	−456553.	−0.517006	−0.258503	−	0.966010i	$-0.583229\pi$
$239$	−456553.	−0.517006	−0.258503	+	0.966010i	$0.583229\pi$
$240$	0	0
$241$	1.22007e6	1.35314	0.676569	−	0.736379i	$-0.263466\pi$
$241$	1.22007e6	1.35314	0.676569	+	0.736379i	$0.263466\pi$
$242$	481971.i	0.529033i
$243$	− 539041.i	− 0.585607i
$244$	707892.	0.761189
$245$	0	0
$246$	41409.5	0.0436277
$247$	66348.1i	0.0691968i
$248$	− 437024.i	− 0.451208i
$249$	−217899.	−0.222718
$250$	0	0
$251$	1.44804e6	1.45076	0.725382	−	0.688347i	$-0.241663\pi$
$251$	1.44804e6	1.45076	0.725382	+	0.688347i	$0.241663\pi$
$252$	747419.i	0.741418i
$253$	1.39739e6i	1.37251i
$254$	79704.5	0.0775173
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.21790e6i	− 1.15021i	−0.818079	−	0.575106i	$-0.804961\pi$
$257$	− 1.21790e6i	− 1.15021i	0.818079	−	0.575106i	$-0.195039\pi$
$258$	142975.i	0.133724i
$259$	−2.66560e6	−2.46914
$260$	0	0
$261$	1.65972e6	1.50811
$262$	1.06758e6i	0.960830i
$263$	− 1.51165e6i	− 1.34760i	−0.738915	−	0.673799i	$-0.764661\pi$
$263$	− 1.51165e6i	− 1.34760i	0.738915	−	0.673799i	$-0.235339\pi$
$264$	109516.	0.0967095
$265$	0	0
$266$	315382.	0.273296
$267$	107453.i	0.0922443i
$268$	− 113142.i	− 0.0962246i
$269$	−405154.	−0.341381	−0.170691	−	0.985325i	$-0.554600\pi$
$269$	−405154.	−0.341381	−0.170691	+	0.985325i	$0.554600\pi$
$270$	0	0
$271$	613161.	0.507167	0.253584	−	0.967313i	$-0.418391\pi$
$271$	613161.	0.507167	0.253584	+	0.967313i	$0.418391\pi$
$272$	− 3870.61i	− 0.00317218i
$273$	− 109458.i	− 0.0888877i
$274$	−38475.8	−0.0309608
$275$	0	0
$276$	135891.	0.107379
$277$	− 1.14240e6i	− 0.894579i	−0.894389	−	0.447289i	$-0.852389\pi$
$277$	− 1.14240e6i	− 0.894579i	0.894389	−	0.447289i	$-0.147611\pi$
$278$	340741.i	0.264431i
$279$	1.58831e6	1.22159
$280$	0	0
$281$	−705223.	−0.532795	−0.266398	−	0.963863i	$-0.585833\pi$
$281$	−705223.	−0.532795	−0.266398	+	0.963863i	$0.585833\pi$
$282$	119436.i	0.0894361i
$283$	205148.i	0.152266i	0.997098	+	0.0761328i	$0.0242573\pi$
$283$	205148.i	0.152266i	−0.997098	+	0.0761328i	$0.975743\pi$
$284$	580341.	0.426961
$285$	0	0
$286$	358690.	0.259301
$287$	− 644687.i	− 0.462003i
$288$	238182.i	0.169211i
$289$	1.41963e6	0.999839
$290$	0	0
$291$	−7115.76	−0.00492593
$292$	− 657037.i	− 0.450955i
$293$	2.50408e6i	1.70404i	0.523513	+	0.852018i	$0.324621\pi$
$293$	2.50408e6i	1.70404i	−0.523513	+	0.852018i	$0.675379\pi$
$294$	−303495.	−0.204778
$295$	0	0
$296$	−849453.	−0.563521
$297$	813843.i	0.535364i
$298$	− 891759.i	− 0.581711i
$299$	445073.	0.287908
$300$	0	0
$301$	2.22592e6	1.41610
$302$	− 1.45378e6i	− 0.917238i
$303$	302102.i	0.189037i
$304$	100504.	0.0623731
$305$	0	0
$306$	14067.2	0.00858826
$307$	− 110033.i	− 0.0666312i	−0.999445	−	0.0333156i	$-0.989393\pi$
$307$	− 110033.i	− 0.0666312i	0.999445	−	0.0333156i	$-0.0106067\pi$
$308$	− 1.70502e6i	− 1.02412i
$309$	−375622.	−0.223797
$310$	0	0
$311$	−1.15351e6	−0.676272	−0.338136	−	0.941097i	$-0.609796\pi$
$311$	−1.15351e6	−0.676272	−0.338136	+	0.941097i	$0.609796\pi$
$312$	− 34881.3i	− 0.0202865i
$313$	− 2.00121e6i	− 1.15460i	−0.816531	−	0.577302i	$-0.804106\pi$
$313$	− 2.00121e6i	− 1.15460i	0.816531	−	0.577302i	$-0.195894\pi$
$314$	1.29361e6	0.740421
$315$	0	0
$316$	−311574.	−0.175527
$317$	90375.3i	0.0505128i	0.999681	+	0.0252564i	$0.00804022\pi$
$317$	90375.3i	0.0505128i	−0.999681	+	0.0252564i	$0.991960\pi$
$318$	− 45420.4i	− 0.0251874i
$319$	−3.78616e6	−2.08316
$320$	0	0
$321$	−629846.	−0.341171
$322$	− 2.11563e6i	− 1.13710i
$323$	− 5935.83i	− 0.00316574i
$324$	−825204.	−0.436716
$325$	0	0
$326$	−1.44676e6	−0.753970
$327$	559732.i	0.289475i
$328$	− 205444.i	− 0.105441i
$329$	1.85945e6	0.947099
$330$	0	0
$331$	−3.83957e6	−1.92625	−0.963126	−	0.269051i	$-0.913290\pi$
$331$	−3.83957e6	−1.92625	−0.963126	+	0.269051i	$0.913290\pi$
$332$	1.08106e6i	0.538274i
$333$	− 3.08723e6i	− 1.52566i
$334$	2.14365e6	1.05145
$335$	0	0
$336$	−165806.	−0.0801223
$337$	2.29930e6i	1.10286i	0.834221	+	0.551430i	$0.185917\pi$
$337$	2.29930e6i	1.10286i	−0.834221	+	0.551430i	$0.814083\pi$
$338$	− 114244.i	− 0.0543928i
$339$	−12288.0	−0.00580742
$340$	0	0
$341$	−3.62325e6	−1.68738
$342$	365267.i	0.168867i
$343$	1.34959e6i	0.619393i
$344$	709338.	0.323190
$345$	0	0
$346$	−1.01006e6	−0.453582
$347$	− 1.82848e6i	− 0.815204i	−0.913160	−	0.407602i	$-0.866365\pi$
$347$	− 1.82848e6i	− 0.815204i	0.913160	−	0.407602i	$-0.133635\pi$
$348$	368190.i	0.162976i
$349$	−2.54054e6	−1.11651	−0.558254	−	0.829670i	$-0.688528\pi$
$349$	−2.54054e6	−1.11651	−0.558254	+	0.829670i	$0.688528\pi$
$350$	0	0
$351$	259211.	0.112302
$352$	− 543342.i	− 0.233731i
$353$	3.67598e6i	1.57013i	0.619412	+	0.785066i	$0.287371\pi$
$353$	3.67598e6i	1.57013i	−0.619412	+	0.785066i	$0.712629\pi$
$354$	−520305.	−0.220674
$355$	0	0
$356$	533103.	0.222939
$357$	9792.67i	0.00406659i
$358$	1.99995e6i	0.824730i
$359$	920537.	0.376969	0.188484	−	0.982076i	$-0.439643\pi$
$359$	920537.	0.376969	0.188484	+	0.982076i	$0.439643\pi$
$360$	0	0
$361$	−2.32197e6	−0.937753
$362$	− 2.43107e6i	− 0.975047i
$363$	− 388586.i	− 0.154782i
$364$	−543053.	−0.214827
$365$	0	0
$366$	−570733.	−0.222705
$367$	2.28316e6i	0.884854i	0.896804	+	0.442427i	$0.145882\pi$
$367$	2.28316e6i	0.884854i	−0.896804	+	0.442427i	$0.854118\pi$
$368$	− 674193.i	− 0.259516i
$369$	746659.	0.285467
$370$	0	0
$371$	−707131.	−0.266726
$372$	352348.i	0.132012i
$373$	− 2.84732e6i	− 1.05965i	−0.848106	−	0.529827i	$-0.822257\pi$
$373$	− 2.84732e6i	− 1.05965i	0.848106	−	0.529827i	$-0.177743\pi$
$374$	−32090.2	−0.0118630
$375$	0	0
$376$	592556.	0.216152
$377$	1.20590e6i	0.436977i
$378$	− 1.23215e6i	− 0.443541i
$379$	−3.75104e6	−1.34139	−0.670693	−	0.741735i	$-0.734003\pi$
$379$	−3.75104e6	−1.34139	−0.670693	+	0.741735i	$0.734003\pi$
$380$	0	0
$381$	−64261.2	−0.0226797
$382$	206415.i	0.0723740i
$383$	1.81532e6i	0.632349i	0.948701	+	0.316174i	$0.102398\pi$
$383$	1.81532e6i	0.632349i	−0.948701	+	0.316174i	$0.897602\pi$
$384$	−52837.9	−0.0182860
$385$	0	0
$386$	−3.78540e6	−1.29313
$387$	2.57800e6i	0.874994i
$388$	35303.3i	0.0119052i
$389$	4.37154e6	1.46474	0.732371	−	0.680906i	$-0.238414\pi$
$389$	4.37154e6	1.46474	0.732371	+	0.680906i	$0.238414\pi$
$390$	0	0
$391$	−39818.4	−0.0131717
$392$	1.50573e6i	0.494915i
$393$	− 860728.i	− 0.281115i
$394$	916133.	0.297316
$395$	0	0
$396$	1.97470e6	0.632796
$397$	− 2.28486e6i	− 0.727584i	−0.931480	−	0.363792i	$-0.881482\pi$
$397$	− 2.28486e6i	− 0.727584i	0.931480	−	0.363792i	$-0.118518\pi$
$398$	− 4.24827e6i	− 1.34433i
$399$	−254275.	−0.0799596
$400$	0	0
$401$	5.53357e6	1.71848	0.859240	−	0.511573i	$-0.170937\pi$
$401$	5.53357e6	1.71848	0.859240	+	0.511573i	$0.170937\pi$
$402$	91219.8i	0.0281529i
$403$	1.15402e6i	0.353956i
$404$	1.49881e6	0.456871
$405$	0	0
$406$	5.73220e6	1.72586
$407$	7.04259e6i	2.10740i
$408$	3120.65i	0 0.000928101i
$409$	2.28293e6	0.674815	0.337408	−	0.941359i	$-0.390450\pi$
$409$	2.28293e6	0.674815	0.337408	+	0.941359i	$0.390450\pi$
$410$	0	0
$411$	31020.9	0.00905836
$412$	1.86357e6i	0.540881i
$413$	8.10043e6i	2.33686i
$414$	2.45027e6	0.702607
$415$	0	0
$416$	−173056.	−0.0490290
$417$	− 274720.i	− 0.0773660i
$418$	− 833248.i	− 0.233257i
$419$	−2.75540e6	−0.766741	−0.383371	−	0.923595i	$-0.625237\pi$
$419$	−2.75540e6	−0.766741	−0.383371	+	0.923595i	$0.625237\pi$
$420$	0	0
$421$	−6.19470e6	−1.70339	−0.851697	−	0.524035i	$-0.824426\pi$
$421$	−6.19470e6	−1.70339	−0.851697	+	0.524035i	$0.824426\pi$
$422$	− 1.02680e6i	− 0.280675i
$423$	2.15357e6i	0.585204i
$424$	−225343.	−0.0608737
$425$	0	0
$426$	−467896.	−0.124918
$427$	8.88552e6i	2.35838i
$428$	3.12484e6i	0.824554i
$429$	−289192.	−0.0758652
$430$	0	0
$431$	5.52072e6	1.43154	0.715768	−	0.698338i	$-0.246077\pi$
$431$	5.52072e6	1.43154	0.715768	+	0.698338i	$0.246077\pi$
$432$	− 392652.i	− 0.101227i
$433$	4.76773e6i	1.22206i	0.791608	+	0.611030i	$0.209244\pi$
$433$	4.76773e6i	1.22206i	−0.791608	+	0.611030i	$0.790756\pi$
$434$	5.48556e6	1.39797
$435$	0	0
$436$	2.77699e6	0.699613
$437$	− 1.03392e6i	− 0.258990i
$438$	529732.i	0.131938i
$439$	−275097.	−0.0681279	−0.0340639	−	0.999420i	$-0.510845\pi$
$439$	−275097.	−0.0681279	−0.0340639	+	0.999420i	$0.510845\pi$
$440$	0	0
$441$	−5.47236e6	−1.33992
$442$	10220.8i	0.00248846i
$443$	5.89806e6i	1.42791i	0.700194	+	0.713953i	$0.253097\pi$
$443$	5.89806e6i	1.42791i	−0.700194	+	0.713953i	$0.746903\pi$
$444$	684866.	0.164872
$445$	0	0
$446$	3.23964e6	0.771187
$447$	718975.i	0.170194i
$448$	822613.i	0.193642i
$449$	−2.69527e6	−0.630938	−0.315469	−	0.948936i	$-0.602162\pi$
$449$	−2.69527e6	−0.630938	−0.315469	+	0.948936i	$0.602162\pi$
$450$	0	0
$451$	−1.70328e6	−0.394317
$452$	60964.4i	0.0140356i
$453$	1.17210e6i	0.268361i
$454$	2.62527e6	0.597770
$455$	0	0
$456$	−81030.3	−0.0182488
$457$	5.00128e6i	1.12019i	0.828429	+	0.560094i	$0.189235\pi$
$457$	5.00128e6i	1.12019i	−0.828429	+	0.560094i	$0.810765\pi$
$458$	− 4.77593e6i	− 1.06388i
$459$	−23190.3	−0.00513778
$460$	0	0
$461$	1.31041e6	0.287179	0.143590	−	0.989637i	$-0.454135\pi$
$461$	1.31041e6	0.287179	0.143590	+	0.989637i	$0.454135\pi$
$462$	1.37466e6i	0.299633i
$463$	− 4.81314e6i	− 1.04346i	−0.853110	−	0.521731i	$-0.825287\pi$
$463$	− 4.81314e6i	− 1.04346i	0.853110	−	0.521731i	$-0.174713\pi$
$464$	1.82669e6	0.393886
$465$	0	0
$466$	−4.24021e6	−0.904529
$467$	− 8.29674e6i	− 1.76042i	−0.474589	−	0.880208i	$-0.657403\pi$
$467$	− 8.29674e6i	− 1.76042i	0.474589	−	0.880208i	$-0.342597\pi$
$468$	− 628949.i	− 0.132740i
$469$	1.42016e6	0.298130
$470$	0	0
$471$	−1.04296e6	−0.216629
$472$	2.58138e6i	0.533332i
$473$	− 5.88094e6i	− 1.20863i
$474$	251204.	0.0513548
$475$	0	0
$476$	48584.2	0.00982828
$477$	− 818980.i	− 0.164808i
$478$	− 1.82621e6i	− 0.365579i
$479$	1.37501e6	0.273821	0.136910	−	0.990583i	$-0.456283\pi$
$479$	1.37501e6	0.273821	0.136910	+	0.990583i	$0.456283\pi$
$480$	0	0
$481$	2.24309e6	0.442062
$482$	4.88028e6i	0.956813i
$483$	1.70571e6i	0.332689i
$484$	−1.92789e6	−0.374083
$485$	0	0
$486$	2.15616e6	0.414087
$487$	− 3.68612e6i	− 0.704283i	−0.935947	−	0.352142i	$-0.885454\pi$
$487$	− 3.68612e6i	− 0.704283i	0.935947	−	0.352142i	$-0.114546\pi$
$488$	2.83157e6i	0.538242i
$489$	1.16644e6	0.220593
$490$	0	0
$491$	−4.43125e6	−0.829512	−0.414756	−	0.909933i	$-0.636133\pi$
$491$	−4.43125e6	−0.829512	−0.414756	+	0.909933i	$0.636133\pi$
$492$	165638.i	0.0308494i
$493$	− 107886.i	− 0.0199916i
$494$	−265392.	−0.0489295
$495$	0	0
$496$	1.74810e6	0.319052
$497$	7.28449e6i	1.32284i
$498$	− 871595.i	− 0.157486i
$499$	−8.78314e6	−1.57906	−0.789529	−	0.613713i	$-0.789675\pi$
$499$	−8.78314e6	−1.57906	−0.789529	+	0.613713i	$0.789675\pi$
$500$	0	0
$501$	−1.72830e6	−0.307628
$502$	5.79216e6i	1.02584i
$503$	8.16254e6i	1.43849i	0.694759	+	0.719243i	$0.255511\pi$
$503$	8.16254e6i	1.43849i	−0.694759	+	0.719243i	$0.744489\pi$
$504$	−2.98968e6	−0.524261
$505$	0	0
$506$	−5.58956e6	−0.970512
$507$	92108.4i	0.0159140i
$508$	318818.i	0.0548130i
$509$	−1.76886e6	−0.302621	−0.151310	−	0.988486i	$-0.548349\pi$
$509$	−1.76886e6	−0.302621	−0.151310	+	0.988486i	$0.548349\pi$
$510$	0	0
$511$	8.24718e6	1.39718
$512$	262144.i	0.0441942i
$513$	− 602156.i	− 0.101022i
$514$	4.87159e6	0.813322
$515$	0	0
$516$	−571899.	−0.0945573
$517$	− 4.91273e6i	− 0.808344i
$518$	− 1.06624e7i	− 1.74594i
$519$	814352.	0.132707
$520$	0	0
$521$	−1.12965e6	−0.182327	−0.0911634	−	0.995836i	$-0.529059\pi$
$521$	−1.12965e6	−0.182327	−0.0911634	+	0.995836i	$0.529059\pi$
$522$	6.63888e6i	1.06640i
$523$	− 3.65785e6i	− 0.584753i	−0.956303	−	0.292376i	$-0.905554\pi$
$523$	− 3.65785e6i	− 0.584753i	0.956303	−	0.292376i	$-0.0944460\pi$
$524$	−4.27031e6	−0.679409
$525$	0	0
$526$	6.04658e6	0.952896
$527$	− 103244.i	− 0.0161934i
$528$	438065.i	0.0683839i
$529$	−499332.	−0.0775801
$530$	0	0
$531$	−9.38169e6	−1.44393
$532$	1.26153e6i	0.193249i
$533$	542501.i	0.0827146i
$534$	−429811.	−0.0652265
$535$	0	0
$536$	452567.	0.0680411
$537$	− 1.61245e6i	− 0.241296i
$538$	− 1.62062e6i	− 0.241393i
$539$	1.24836e7	1.85083
$540$	0	0
$541$	−4.32623e6	−0.635502	−0.317751	−	0.948174i	$-0.602928\pi$
$541$	−4.32623e6	−0.635502	−0.317751	+	0.948174i	$0.602928\pi$
$542$	2.45264e6i	0.358621i
$543$	1.96003e6i	0.285275i
$544$	15482.5	0.00224307
$545$	0	0
$546$	437833.	0.0628531
$547$	7.59075e6i	1.08472i	0.840147	+	0.542359i	$0.182469\pi$
$547$	7.59075e6i	1.08472i	−0.840147	+	0.542359i	$0.817531\pi$
$548$	− 153903.i	− 0.0218926i
$549$	−1.02910e7	−1.45722
$550$	0	0
$551$	2.80135e6	0.393087
$552$	543564.i	0.0759281i
$553$	− 3.91090e6i	− 0.543830i
$554$	4.56960e6	0.632563
$555$	0	0
$556$	−1.36296e6	−0.186981
$557$	1.04397e7i	1.42577i	0.701280	+	0.712886i	$0.252612\pi$
$557$	1.04397e7i	1.42577i	−0.701280	+	0.712886i	$0.747388\pi$
$558$	6.35323e6i	0.863792i
$559$	−1.87310e6	−0.253531
$560$	0	0
$561$	25872.5	0.00347082
$562$	− 2.82089e6i	− 0.376743i
$563$	3.38176e6i	0.449646i	0.974400	+	0.224823i	$0.0721805\pi$
$563$	3.38176e6i	0.449646i	−0.974400	+	0.224823i	$0.927820\pi$
$564$	−477744.	−0.0632409
$565$	0	0
$566$	−820593.	−0.107668
$567$	− 1.03580e7i	− 1.35307i
$568$	2.32137e6i	0.301907i
$569$	1.09186e7	1.41380	0.706900	−	0.707313i	$-0.250093\pi$
$569$	1.09186e7	1.41380	0.706900	+	0.707313i	$0.250093\pi$
$570$	0	0
$571$	1.15547e7	1.48310	0.741549	−	0.670899i	$-0.234092\pi$
$571$	1.15547e7	1.48310	0.741549	+	0.670899i	$0.234092\pi$
$572$	1.43476e6i	0.183354i
$573$	− 166421.i	− 0.0211749i
$574$	2.57875e6	0.326685
$575$	0	0
$576$	−952728.	−0.119650
$577$	1.71415e6i	0.214343i	0.994241	+	0.107171i	$0.0341793\pi$
$577$	1.71415e6i	0.214343i	−0.994241	+	0.107171i	$0.965821\pi$
$578$	5.67851e6i	0.706993i
$579$	3.05195e6	0.378339
$580$	0	0
$581$	−1.35695e7	−1.66772
$582$	− 28463.0i	− 0.00348316i
$583$	1.86826e6i	0.227649i
$584$	2.62815e6	0.318873
$585$	0	0
$586$	−1.00163e7	−1.20494
$587$	− 2.85910e6i	− 0.342479i	−0.985229	−	0.171239i	$-0.945223\pi$
$587$	− 2.85910e6i	− 0.342479i	0.985229	−	0.171239i	$-0.0547771\pi$
$588$	− 1.21398e6i	− 0.144800i
$589$	2.68082e6	0.318404
$590$	0	0
$591$	−738626.	−0.0869873
$592$	− 3.39781e6i	− 0.398470i
$593$	− 4.96085e6i	− 0.579321i	−0.957129	−	0.289661i	$-0.906458\pi$
$593$	− 4.96085e6i	− 0.579321i	0.957129	−	0.289661i	$-0.0935424\pi$
$594$	−3.25537e6	−0.378560
$595$	0	0
$596$	3.56704e6	0.411332
$597$	3.42514e6i	0.393317i
$598$	1.78029e6i	0.203581i
$599$	1.24276e7	1.41520	0.707601	−	0.706612i	$-0.249777\pi$
$599$	1.24276e7	1.41520	0.707601	+	0.706612i	$0.249777\pi$
$600$	0	0
$601$	3.58410e6	0.404757	0.202378	−	0.979307i	$-0.435133\pi$
$601$	3.58410e6	0.404757	0.202378	+	0.979307i	$0.435133\pi$
$602$	8.90366e6i	1.00133i
$603$	1.64480e6i	0.184212i
$604$	5.81513e6	0.648585
$605$	0	0
$606$	−1.20841e6	−0.133669
$607$	− 3.78467e6i	− 0.416923i	−0.978031	−	0.208462i	$-0.933154\pi$
$607$	− 3.78467e6i	− 0.416923i	0.978031	−	0.208462i	$-0.0668456\pi$
$608$	402014.i	0.0441045i
$609$	−4.62154e6	−0.504945
$610$	0	0
$611$	−1.56472e6	−0.169564
$612$	56268.9i	0.00607281i
$613$	− 1.54450e7i	− 1.66011i	−0.557685	−	0.830053i	$-0.688310\pi$
$613$	− 1.54450e7i	− 1.66011i	0.557685	−	0.830053i	$-0.311690\pi$
$614$	440133.	0.0471154
$615$	0	0
$616$	6.82006e6	0.724163
$617$	6.78866e6i	0.717912i	0.933355	+	0.358956i	$0.116867\pi$
$617$	6.78866e6i	0.717912i	−0.933355	+	0.358956i	$0.883133\pi$
$618$	− 1.50249e6i	− 0.158249i
$619$	1.31549e7	1.37994	0.689970	−	0.723838i	$-0.257624\pi$
$619$	1.31549e7	1.37994	0.689970	+	0.723838i	$0.257624\pi$
$620$	0	0
$621$	−4.03935e6	−0.420323
$622$	− 4.61405e6i	− 0.478197i
$623$	6.69155e6i	0.690728i
$624$	139525.	0.0143447
$625$	0	0
$626$	8.00486e6	0.816428
$627$	671801.i	0.0682451i
$628$	5.17443e6i	0.523556i
$629$	−200678.	−0.0202243
$630$	0	0
$631$	9.46150e6	0.945990	0.472995	−	0.881065i	$-0.343173\pi$
$631$	9.46150e6	0.945990	0.472995	+	0.881065i	$0.343173\pi$
$632$	− 1.24630e6i	− 0.124116i
$633$	827849.i	0.0821186i
$634$	−361501.	−0.0357179
$635$	0	0
$636$	181681.	0.0178102
$637$	− 3.97606e6i	− 0.388243i
$638$	− 1.51446e7i	− 1.47301i
$639$	−8.43670e6	−0.817373
$640$	0	0
$641$	−1.66495e7	−1.60050	−0.800250	−	0.599666i	$-0.795300\pi$
$641$	−1.66495e7	−1.60050	−0.800250	+	0.599666i	$0.795300\pi$
$642$	− 2.51938e6i	− 0.241244i
$643$	1.31070e7i	1.25019i	0.780548	+	0.625096i	$0.214940\pi$
$643$	1.31070e7i	1.25019i	−0.780548	+	0.625096i	$0.785060\pi$
$644$	8.46252e6	0.804054
$645$	0	0
$646$	23743.3	0.00223851
$647$	− 1.08998e7i	− 1.02366i	−0.859086	−	0.511832i	$-0.828967\pi$
$647$	− 1.08998e7i	− 1.02366i	0.859086	−	0.511832i	$-0.171033\pi$
$648$	− 3.30082e6i	− 0.308805i
$649$	2.14016e7	1.99450
$650$	0	0
$651$	−4.42270e6	−0.409011
$652$	− 5.78706e6i	− 0.533137i
$653$	1.18879e7i	1.09099i	0.838113	+	0.545497i	$0.183659\pi$
$653$	1.18879e7i	1.09099i	−0.838113	+	0.545497i	$0.816341\pi$
$654$	−2.23893e6	−0.204689
$655$	0	0
$656$	821776.	0.0745580
$657$	9.55166e6i	0.863307i
$658$	7.43781e6i	0.669700i
$659$	−5.65451e6	−0.507203	−0.253601	−	0.967309i	$-0.581615\pi$
$659$	−5.65451e6	−0.507203	−0.253601	+	0.967309i	$0.581615\pi$
$660$	0	0
$661$	8.05716e6	0.717263	0.358631	−	0.933479i	$-0.383243\pi$
$661$	8.05716e6	0.717263	0.358631	+	0.933479i	$0.383243\pi$
$662$	− 1.53583e7i	− 1.36207i
$663$	− 8240.48i	0 0.000728062i
$664$	−4.32423e6	−0.380617
$665$	0	0
$666$	1.23489e7	1.07880
$667$	− 1.87919e7i	− 1.63552i
$668$	8.57461e6i	0.743486i
$669$	−2.61194e6	−0.225630
$670$	0	0
$671$	2.34758e7	2.01286
$672$	− 663226.i	− 0.0566550i
$673$	8.86976e6i	0.754874i	0.926035	+	0.377437i	$0.123194\pi$
$673$	8.86976e6i	0.754874i	−0.926035	+	0.377437i	$0.876806\pi$
$674$	−9.19718e6	−0.779839
$675$	0	0
$676$	456976.	0.0384615
$677$	− 1.86550e7i	− 1.56431i	−0.623083	−	0.782156i	$-0.714120\pi$
$677$	− 1.86550e7i	− 1.56431i	0.623083	−	0.782156i	$-0.285880\pi$
$678$	− 49152.1i	− 0.00410646i
$679$	−443129.	−0.0368855
$680$	0	0
$681$	−2.11661e6	−0.174893
$682$	− 1.44930e7i	− 1.19316i
$683$	− 1.79463e7i	− 1.47205i	−0.676952	−	0.736027i	$-0.736700\pi$
$683$	− 1.79463e7i	− 1.47205i	0.676952	−	0.736027i	$-0.263300\pi$
$684$	−1.46107e6	−0.119407
$685$	0	0
$686$	−5.39836e6	−0.437977
$687$	3.85056e6i	0.311266i
$688$	2.83735e6i	0.228530i
$689$	595047.	0.0477533
$690$	0	0
$691$	1.39282e7	1.10969	0.554844	−	0.831954i	$-0.312778\pi$
$691$	1.39282e7	1.10969	0.554844	+	0.831954i	$0.312778\pi$
$692$	− 4.04023e6i	− 0.320731i
$693$	2.47866e7i	1.96058i
$694$	7.31391e6	0.576436
$695$	0	0
$696$	−1.47276e6	−0.115241
$697$	− 48534.8i	− 0.00378418i
$698$	− 1.01621e7i	− 0.789490i
$699$	3.41864e6	0.264643
$700$	0	0
$701$	2.29331e7	1.76265	0.881326	−	0.472508i	$-0.156651\pi$
$701$	2.29331e7	1.76265	0.881326	+	0.472508i	$0.156651\pi$
$702$	1.03685e6i	0.0794093i
$703$	− 5.21076e6i	− 0.397661i
$704$	2.17337e6	0.165273
$705$	0	0
$706$	−1.47039e7	−1.11025
$707$	1.88132e7i	1.41551i
$708$	− 2.08122e6i	− 0.156040i
$709$	−2.11865e7	−1.58286	−0.791430	−	0.611260i	$-0.790663\pi$
$709$	−2.11865e7	−1.58286	−0.791430	+	0.611260i	$0.790663\pi$
$710$	0	0
$711$	4.52950e6	0.336028
$712$	2.13241e6i	0.157642i
$713$	− 1.79833e7i	− 1.32479i
$714$	−39170.7	−0.00287551
$715$	0	0
$716$	−7.99980e6	−0.583172
$717$	1.47237e6i	0.106959i
$718$	3.68215e6i	0.266557i
$719$	−2.03595e7	−1.46874	−0.734371	−	0.678749i	$-0.762523\pi$
$719$	−2.03595e7	−1.46874	−0.734371	+	0.678749i	$0.762523\pi$
$720$	0	0
$721$	−2.33916e7	−1.67580
$722$	− 9.28788e6i	− 0.663092i
$723$	− 3.93469e6i	− 0.279940i
$724$	9.72426e6	0.689462
$725$	0	0
$726$	1.55434e6	0.109447
$727$	3.11392e6i	0.218510i	0.994014	+	0.109255i	$0.0348465\pi$
$727$	3.11392e6i	0.218510i	−0.994014	+	0.109255i	$0.965153\pi$
$728$	− 2.17221e6i	− 0.151906i
$729$	1.07944e7	0.752280
$730$	0	0
$731$	167577.	0.0115990
$732$	− 2.28293e6i	− 0.157476i
$733$	− 2.20488e7i	− 1.51574i	−0.652405	−	0.757871i	$-0.726240\pi$
$733$	− 2.20488e7i	− 1.51574i	0.652405	−	0.757871i	$-0.273760\pi$
$734$	−9.13265e6	−0.625686
$735$	0	0
$736$	2.69677e6	0.183506
$737$	− 3.75211e6i	− 0.254453i
$738$	2.98664e6i	0.201856i
$739$	−2.29831e6	−0.154809	−0.0774046	−	0.997000i	$-0.524663\pi$
$739$	−2.29831e6	−0.154809	−0.0774046	+	0.997000i	$0.524663\pi$
$740$	0	0
$741$	213971.	0.0143156
$742$	− 2.82852e6i	− 0.188604i
$743$	− 711970.i	− 0.0473140i	−0.999720	−	0.0236570i	$-0.992469\pi$
$743$	− 711970.i	− 0.0473140i	0.999720	−	0.0236570i	$-0.00753096\pi$
$744$	−1.40939e6	−0.0933468
$745$	0	0
$746$	1.13893e7	0.749288
$747$	− 1.57158e7i	− 1.03047i
$748$	− 128361.i	− 0.00838839i
$749$	−3.92233e7	−2.55470
$750$	0	0
$751$	−3.24479e6	−0.209936	−0.104968	−	0.994476i	$-0.533474\pi$
$751$	−3.24479e6	−0.209936	−0.104968	+	0.994476i	$0.533474\pi$
$752$	2.37023e6i	0.152843i
$753$	− 4.66989e6i	− 0.300137i
$754$	−4.82361e6	−0.308990
$755$	0	0
$756$	4.92859e6	0.313631
$757$	2.38887e7i	1.51514i	0.652756	+	0.757569i	$0.273613\pi$
$757$	2.38887e7i	1.51514i	−0.652756	+	0.757569i	$0.726387\pi$
$758$	− 1.50042e7i	− 0.948504i
$759$	4.50654e6	0.283948
$760$	0	0
$761$	2340.75	0.000146519 0	7.32593e−5	−	1.00000i	$-0.499977\pi$
$761$	2340.75	0.000146519 0	7.32593e−5		1.00000i	$0.499977\pi$
$762$	− 257045.i	− 0.0160369i
$763$	3.48569e7i	2.16759i
$764$	−825660.	−0.0511762
$765$	0	0
$766$	−7.26128e6	−0.447138
$767$	− 6.81646e6i	− 0.418380i
$768$	− 211352.i	− 0.0129301i
$769$	365454.	0.0222852	0.0111426	−	0.999938i	$-0.496453\pi$
$769$	365454.	0.0222852	0.0111426	+	0.999938i	$0.496453\pi$
$770$	0	0
$771$	−3.92768e6	−0.237958
$772$	− 1.51416e7i	− 0.914383i
$773$	− 2.76260e7i	− 1.66291i	−0.555589	−	0.831457i	$-0.687507\pi$
$773$	− 2.76260e7i	− 1.66291i	0.555589	−	0.831457i	$-0.312493\pi$
$774$	−1.03120e7	−0.618714
$775$	0	0
$776$	−141213.	−0.00841823
$777$	8.59648e6i	0.510820i
$778$	1.74862e7i	1.03573i
$779$	1.26025e6	0.0744066
$780$	0	0
$781$	1.92458e7	1.12904
$782$	− 159274.i	− 0.00931381i
$783$	− 1.09444e7i	− 0.637953i
$784$	−6.02290e6	−0.349958
$785$	0	0
$786$	3.44291e6	0.198779
$787$	9.92420e6i	0.571161i	0.958355	+	0.285580i	$0.0921864\pi$
$787$	9.92420e6i	0.571161i	−0.958355	+	0.285580i	$0.907814\pi$
$788$	3.66453e6i	0.210234i
$789$	−4.87501e6	−0.278794
$790$	0	0
$791$	−765229.	−0.0434861
$792$	7.89881e6i	0.447455i
$793$	− 7.47711e6i	− 0.422232i
$794$	9.13944e6	0.514480
$795$	0	0
$796$	1.69931e7	0.950582
$797$	− 1.49499e6i	− 0.0833668i	−0.999131	−	0.0416834i	$-0.986728\pi$
$797$	− 1.49499e6i	− 0.0833668i	0.999131	−	0.0416834i	$-0.0132721\pi$
$798$	− 1.01710e6i	− 0.0565400i
$799$	139988.	0.00775751
$800$	0	0
$801$	−7.74998e6	−0.426795
$802$	2.21343e7i	1.21515i
$803$	− 2.17893e7i	− 1.19249i
$804$	−364879.	−0.0199071
$805$	0	0
$806$	−4.61607e6	−0.250285
$807$	1.30661e6i	0.0706257i
$808$	5.99525e6i	0.323057i
$809$	2.85707e6	0.153479	0.0767396	−	0.997051i	$-0.475549\pi$
$809$	2.85707e6	0.153479	0.0767396	+	0.997051i	$0.475549\pi$
$810$	0	0
$811$	−9.42188e6	−0.503020	−0.251510	−	0.967855i	$-0.580927\pi$
$811$	−9.42188e6	−0.503020	−0.251510	+	0.967855i	$0.580927\pi$
$812$	2.29288e7i	1.22037i
$813$	− 1.97743e6i	− 0.104924i
$814$	−2.81704e7	−1.49016
$815$	0	0
$816$	−12482.6	−0.000656267 0
$817$	4.35126e6i	0.228066i
$818$	9.13173e6i	0.477167i
$819$	7.89462e6	0.411265
$820$	0	0
$821$	1.54084e7	0.797809	0.398904	−	0.916992i	$-0.369390\pi$
$821$	1.54084e7	0.797809	0.398904	+	0.916992i	$0.369390\pi$
$822$	124084.i	0.00640523i
$823$	− 4.13964e6i	− 0.213041i	−0.994311	−	0.106520i	$-0.966029\pi$
$823$	− 4.13964e6i	− 0.213041i	0.994311	−	0.106520i	$-0.0339710\pi$
$824$	−7.45427e6	−0.382461
$825$	0	0
$826$	−3.24017e7	−1.65241
$827$	− 1.01580e7i	− 0.516470i	−0.966082	−	0.258235i	$-0.916859\pi$
$827$	− 1.01580e7i	− 0.516470i	0.966082	−	0.258235i	$-0.0831409\pi$
$828$	9.80106e6i	0.496818i
$829$	−3.45331e7	−1.74521	−0.872607	−	0.488423i	$-0.837572\pi$
$829$	−3.45331e7	−1.74521	−0.872607	+	0.488423i	$0.837572\pi$
$830$	0	0
$831$	−3.68421e6	−0.185072
$832$	− 692224.i	− 0.0346688i
$833$	355718.i	0.0177620i
$834$	1.09888e6	0.0547060
$835$	0	0
$836$	3.33299e6	0.164937
$837$	− 1.04735e7i	− 0.516749i
$838$	− 1.10216e7i	− 0.542168i
$839$	2.33327e7	1.14435	0.572177	−	0.820130i	$-0.306099\pi$
$839$	2.33327e7	1.14435	0.572177	+	0.820130i	$0.306099\pi$
$840$	0	0
$841$	3.04045e7	1.48234
$842$	− 2.47788e7i	− 1.20448i
$843$	2.27432e6i	0.110226i
$844$	4.10719e6	0.198467
$845$	0	0
$846$	−8.61427e6	−0.413802
$847$	− 2.41990e7i	− 1.15901i
$848$	− 901373.i	− 0.0430442i
$849$	661597.	0.0315010
$850$	0	0
$851$	−3.49546e7	−1.65455
$852$	− 1.87158e6i	− 0.0883305i
$853$	1.80270e7i	0.848304i	0.905591	+	0.424152i	$0.139428\pi$
$853$	1.80270e7i	0.848304i	−0.905591	+	0.424152i	$0.860572\pi$
$854$	−3.55421e7	−1.66762
$855$	0	0
$856$	−1.24994e7	−0.583048
$857$	1.82132e7i	0.847100i	0.905873	+	0.423550i	$0.139216\pi$
$857$	1.82132e7i	0.847100i	−0.905873	+	0.423550i	$0.860784\pi$
$858$	− 1.15677e6i	− 0.0536448i
$859$	−728581.	−0.0336895	−0.0168448	−	0.999858i	$-0.505362\pi$
$859$	−728581.	−0.0336895	−0.0168448	+	0.999858i	$0.505362\pi$
$860$	0	0
$861$	−2.07910e6	−0.0955800
$862$	2.20829e7i	1.01225i
$863$	− 4.10201e7i	− 1.87486i	−0.348168	−	0.937432i	$-0.613196\pi$
$863$	− 4.10201e7i	− 1.87486i	0.348168	−	0.937432i	$-0.386804\pi$
$864$	1.57061e6	0.0715786
$865$	0	0
$866$	−1.90709e7	−0.864127
$867$	− 4.57826e6i	− 0.206849i
$868$	2.19422e7i	0.988511i
$869$	−1.03327e7	−0.464157
$870$	0	0
$871$	−1.19506e6	−0.0533758
$872$	1.11079e7i	0.494701i
$873$	− 513220.i	− 0.0227913i
$874$	4.13567e6	0.183133
$875$	0	0
$876$	−2.11893e6	−0.0932944
$877$	− 9.34651e6i	− 0.410346i	−0.978726	−	0.205173i	$-0.934224\pi$
$877$	− 9.34651e6i	− 0.410346i	0.978726	−	0.205173i	$-0.0657758\pi$
$878$	− 1.10039e6i	− 0.0481737i
$879$	8.07558e6	0.352534
$880$	0	0
$881$	2.76158e7	1.19872	0.599361	−	0.800479i	$-0.295422\pi$
$881$	2.76158e7	1.19872	0.599361	+	0.800479i	$0.295422\pi$
$882$	− 2.18894e7i	− 0.947465i
$883$	1.71751e7i	0.741305i	0.928772	+	0.370652i	$0.120866\pi$
$883$	1.71751e7i	0.741305i	−0.928772	+	0.370652i	$0.879134\pi$
$884$	−40883.3	−0.00175961
$885$	0	0
$886$	−2.35922e7	−1.00968
$887$	5.13121e6i	0.218983i	0.993988	+	0.109492i	$0.0349222\pi$
$887$	5.13121e6i	0.218983i	−0.993988	+	0.109492i	$0.965078\pi$
$888$	2.73946e6i	0.116582i
$889$	−4.00183e6	−0.169826
$890$	0	0
$891$	−2.73662e7	−1.15484
$892$	1.29586e7i	0.545312i
$893$	3.63489e6i	0.152533i
$894$	−2.87590e6	−0.120346
$895$	0	0
$896$	−3.29045e6	−0.136926
$897$	− 1.43535e6i	− 0.0595629i
$898$	− 1.07811e7i	− 0.446140i
$899$	4.87249e7	2.01072
$900$	0	0
$901$	−53235.9	−0.00218470
$902$	− 6.81313e6i	− 0.278824i
$903$	− 7.17852e6i	− 0.292965i
$904$	−243858.	−0.00992465
$905$	0	0
$906$	−4.68841e6	−0.189760
$907$	− 4.53644e7i	− 1.83104i	−0.402275	−	0.915519i	$-0.631780\pi$
$907$	− 4.53644e7i	− 1.83104i	0.402275	−	0.915519i	$-0.368220\pi$
$908$	1.05011e7i	0.422687i
$909$	−2.17889e7	−0.874634
$910$	0	0
$911$	3.56713e7	1.42404	0.712021	−	0.702158i	$-0.247780\pi$
$911$	3.56713e7	1.42404	0.712021	+	0.702158i	$0.247780\pi$
$912$	− 324121.i	− 0.0129039i
$913$	3.58510e7i	1.42339i
$914$	−2.00051e7	−0.792093
$915$	0	0
$916$	1.91037e7	0.752280
$917$	− 5.36013e7i	− 2.10500i
$918$	− 92761.4i	− 0.00363296i
$919$	−1.32697e6	−0.0518289	−0.0259144	−	0.999664i	$-0.508250\pi$
$919$	−1.32697e6	−0.0518289	−0.0259144	+	0.999664i	$0.508250\pi$
$920$	0	0
$921$	−354854.	−0.0137848
$922$	5.24162e6i	0.203066i
$923$	− 6.12986e6i	− 0.236835i
$924$	−5.49863e6	−0.211872
$925$	0	0
$926$	1.92526e7	0.737838
$927$	− 2.70916e7i	− 1.03546i
$928$	7.30677e6i	0.278520i
$929$	−1.87784e7	−0.713869	−0.356935	−	0.934129i	$-0.616178\pi$
$929$	−1.87784e7	−0.713869	−0.356935	+	0.934129i	$0.616178\pi$
$930$	0	0
$931$	−9.23649e6	−0.349247
$932$	− 1.69608e7i	− 0.639598i
$933$	3.72005e6i	0.139909i
$934$	3.31870e7	1.24480
$935$	0	0
$936$	2.51580e6	0.0938612
$937$	− 2.71600e7i	− 1.01060i	−0.862942	−	0.505302i	$-0.831381\pi$
$937$	− 2.71600e7i	− 1.01060i	0.862942	−	0.505302i	$-0.168619\pi$
$938$	5.68066e6i	0.210810i
$939$	−6.45386e6	−0.238867
$940$	0	0
$941$	−3.93036e7	−1.44696	−0.723482	−	0.690343i	$-0.757460\pi$
$941$	−3.93036e7	−1.44696	−0.723482	+	0.690343i	$0.757460\pi$
$942$	− 4.17185e6i	− 0.153180i
$943$	− 8.45392e6i	− 0.309584i
$944$	−1.03255e7	−0.377122
$945$	0	0
$946$	2.35237e7	0.854631
$947$	− 1.05666e7i	− 0.382877i	−0.981505	−	0.191439i	$-0.938685\pi$
$947$	− 1.05666e7i	− 0.382877i	0.981505	−	0.191439i	$-0.0613153\pi$
$948$	1.00482e6i	0.0363133i
$949$	−6.93996e6	−0.250145
$950$	0	0
$951$	291458.	0.0104502
$952$	194337.i	0.00694965i
$953$	− 1.99531e7i	− 0.711667i	−0.934549	−	0.355834i	$-0.884197\pi$
$953$	− 1.99531e7i	− 0.711667i	0.934549	−	0.355834i	$-0.115803\pi$
$954$	3.27592e6	0.116537
$955$	0	0
$956$	7.30484e6	0.258503
$957$	1.22103e7i	0.430968i
$958$	5.50004e6i	0.193621i
$959$	1.93181e6	0.0678293
$960$	0	0
$961$	1.79993e7	0.628706
$962$	8.97235e6i	0.312585i
$963$	− 4.54273e7i	− 1.57853i
$964$	−1.95211e7	−0.676569
$965$	0	0
$966$	−6.82285e6	−0.235246
$967$	− 2.42835e6i	− 0.0835113i	−0.999128	−	0.0417556i	$-0.986705\pi$
$967$	− 2.42835e6i	− 0.0835113i	0.999128	−	0.0417556i	$-0.0132951\pi$
$968$	− 7.71154e6i	− 0.264517i
$969$	−19142.9	−0.000654934 0
$970$	0	0
$971$	1.43525e7	0.488517	0.244259	−	0.969710i	$-0.421455\pi$
$971$	1.43525e7	0.488517	0.244259	+	0.969710i	$0.421455\pi$
$972$	8.62466e6i	0.292803i
$973$	− 1.71080e7i	− 0.579319i
$974$	1.47445e7	0.498004
$975$	0	0
$976$	−1.13263e7	−0.380595
$977$	2.56112e7i	0.858409i	0.903207	+	0.429205i	$0.141206\pi$
$977$	2.56112e7i	0.858409i	−0.903207	+	0.429205i	$0.858794\pi$
$978$	4.66578e6i	0.155983i
$979$	1.76793e7	0.589533
$980$	0	0
$981$	−4.03704e7	−1.33934
$982$	− 1.77250e7i	− 0.586554i
$983$	2.23389e6i	0.0737356i	0.999320	+	0.0368678i	$0.0117381\pi$
$983$	2.23389e6i	0.0737356i	−0.999320	+	0.0368678i	$0.988262\pi$
$984$	−662551.	−0.0218138
$985$	0	0
$986$	431544.	0.0141362
$987$	− 5.99668e6i	− 0.195938i
$988$	− 1.06157e6i	− 0.0345984i
$989$	2.91889e7	0.948914
$990$	0	0
$991$	−5.77025e6	−0.186643	−0.0933213	−	0.995636i	$-0.529748\pi$
$991$	−5.77025e6	−0.186643	−0.0933213	+	0.995636i	$0.529748\pi$
$992$	6.99239e6i	0.225604i
$993$	1.23825e7i	0.398507i
$994$	−2.91379e7	−0.935391
$995$	0	0
$996$	3.48638e6	0.111359
$997$	1.12116e7i	0.357216i	0.983920	+	0.178608i	$0.0571594\pi$
$997$	1.12116e7i	0.357216i	−0.983920	+	0.178608i	$0.942841\pi$
$998$	− 3.51325e7i	− 1.11656i
$999$	−2.03576e7	−0.645377

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

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	650.6.b.c.599.3		4
5.2	odd	4		650.6.a.c.1.1		2
5.3	odd	4		130.6.a.e.1.2	✓	2
5.4	even	2	inner	650.6.b.c.599.2		4
20.3	even	4		1040.6.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.e.1.2	✓	2	5.3	odd	4
650.6.a.c.1.1		2	5.2	odd	4
650.6.b.c.599.2		4	5.4	even	2	inner
650.6.b.c.599.3		4	1.1	even	1	trivial
1040.6.a.h.1.1		2	20.3	even	4

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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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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	650.6.b.c.599.3

Level	$650$
Weight	$6$
Character	650.599
Analytic conductor	$104.249$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$