LMFDB - Embedded newform 650.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,6,Mod(1,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$650 = 2 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	650.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,16,32,0,-64,252] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$104.249482878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 14$ x^2 - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} -3.22497 q^{3} +16.0000 q^{4} +12.8999 q^{6} +200.833 q^{7} -64.0000 q^{8} -232.600 q^{9} -530.607 q^{11} -51.5996 q^{12} +169.000 q^{13} -803.333 q^{14} +256.000 q^{16} +15.1196 q^{17} +930.398 q^{18} -392.592 q^{19} -647.681 q^{21} +2122.43 q^{22} -2633.57 q^{23} +206.398 q^{24} -676.000 q^{26} +1533.80 q^{27} +3213.33 q^{28} -7135.52 q^{29} +6828.51 q^{31} -1024.00 q^{32} +1711.19 q^{33} -60.4783 q^{34} -3721.59 q^{36} +13272.7 q^{37} +1570.37 q^{38} -545.020 q^{39} +3210.06 q^{41} +2590.73 q^{42} +11083.4 q^{43} -8489.71 q^{44} +10534.3 q^{46} -9258.69 q^{47} -825.593 q^{48} +23527.0 q^{49} -48.7602 q^{51} +2704.00 q^{52} -3520.99 q^{53} -6135.18 q^{54} -12853.3 q^{56} +1266.10 q^{57} +28542.1 q^{58} +40334.1 q^{59} -44243.3 q^{61} -27314.0 q^{62} -46713.7 q^{63} +4096.00 q^{64} -6844.77 q^{66} -7071.36 q^{67} +241.913 q^{68} +8493.18 q^{69} -36271.3 q^{71} +14886.4 q^{72} +41064.8 q^{73} -53090.8 q^{74} -6281.47 q^{76} -106563. q^{77} +2180.08 q^{78} -19473.4 q^{79} +51575.2 q^{81} -12840.3 q^{82} -67566.0 q^{83} -10362.9 q^{84} -44333.6 q^{86} +23011.9 q^{87} +33958.9 q^{88} +33319.0 q^{89} +33940.8 q^{91} -42137.1 q^{92} -22021.7 q^{93} +37034.8 q^{94} +3302.37 q^{96} +2206.46 q^{97} -94107.8 q^{98} +123419. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{2} + 16 q^{3} + 32 q^{4} - 64 q^{6} + 252 q^{7} - 128 q^{8} - 106 q^{9} - 36 q^{11} + 256 q^{12} + 338 q^{13} - 1008 q^{14} + 512 q^{16} + 2380 q^{17} + 424 q^{18} - 1092 q^{19} + 336 q^{21}+ \cdots + 186036 q^{99}+O(q^{100})$ 2 * q - 8 * q^2 + 16 * q^3 + 32 * q^4 - 64 * q^6 + 252 * q^7 - 128 * q^8 - 106 * q^9 - 36 * q^11 + 256 * q^12 + 338 * q^13 - 1008 * q^14 + 512 * q^16 + 2380 * q^17 + 424 * q^18 - 1092 * q^19 + 336 * q^21 + 144 * q^22 + 1176 * q^23 - 1024 * q^24 - 1352 * q^26 - 704 * q^27 + 4032 * q^28 - 4872 * q^29 - 2260 * q^31 - 2048 * q^32 + 11220 * q^33 - 9520 * q^34 - 1696 * q^36 + 17760 * q^37 + 4368 * q^38 + 2704 * q^39 - 9220 * q^41 - 1344 * q^42 + 23656 * q^43 - 576 * q^44 - 4704 * q^46 - 12860 * q^47 + 4096 * q^48 + 9338 * q^49 + 45416 * q^51 + 5408 * q^52 - 11068 * q^53 + 2816 * q^54 - 16128 * q^56 - 12180 * q^57 + 19488 * q^58 + 90404 * q^59 - 69000 * q^61 + 9040 * q^62 - 40236 * q^63 + 8192 * q^64 - 44880 * q^66 - 50796 * q^67 + 38080 * q^68 + 81732 * q^69 - 28668 * q^71 + 6784 * q^72 + 62688 * q^73 - 71040 * q^74 - 17472 * q^76 - 81256 * q^77 - 10816 * q^78 + 84064 * q^79 - 22210 * q^81 + 36880 * q^82 + 12828 * q^83 + 5376 * q^84 - 94624 * q^86 + 66528 * q^87 + 2304 * q^88 + 92620 * q^89 + 42588 * q^91 + 18816 * q^92 - 196748 * q^93 + 51440 * q^94 - 16384 * q^96 + 75684 * q^97 - 37352 * q^98 + 186036 * 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
$3$	−3.22497	−0.206882	−0.103441	−	0.994636i	$-0.532985\pi$
$3$	−3.22497	−0.206882	−0.103441	+	0.994636i	$0.532985\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	12.8999	0.146288
$7$	200.833	1.54914	0.774569	−	0.632489i	$-0.217967\pi$
$7$	200.833	1.54914	0.774569	+	0.632489i	$0.217967\pi$
$8$	−64.0000	−0.353553
$9$	−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	$0.729906\pi$
$12$	−51.5996	−0.103441
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−803.333	−1.09541
$15$	0	0
$16$	256.000	0.250000
$17$	15.1196	0.0126887	0.00634435	−	0.999980i	$-0.497981\pi$
$17$	15.1196	0.0126887	0.00634435	+	0.999980i	$0.497981\pi$
$18$	930.398	0.676842
$19$	−392.592	−0.249493	−0.124746	−	0.992189i	$-0.539812\pi$
$19$	−392.592	−0.249493	−0.124746	+	0.992189i	$0.539812\pi$
$20$	0	0
$21$	−647.681	−0.320489
$22$	2122.43	0.934924
$23$	−2633.57	−1.03807	−0.519033	−	0.854754i	$-0.673708\pi$
$23$	−2633.57	−1.03807	−0.519033	+	0.854754i	$0.673708\pi$
$24$	206.398	0.0731439
$25$	0	0
$26$	−676.000	−0.196116
$27$	1533.80	0.404910
$28$	3213.33	0.774569
$29$	−7135.52	−1.57554	−0.787772	−	0.615966i	$-0.788766\pi$
$29$	−7135.52	−1.57554	−0.787772	+	0.615966i	$0.788766\pi$
$30$	0	0
$31$	6828.51	1.27621	0.638104	−	0.769950i	$-0.279719\pi$
$31$	6828.51	1.27621	0.638104	+	0.769950i	$0.279719\pi$
$32$	−1024.00	−0.176777
$33$	1711.19	0.273536
$34$	−60.4783	−0.00897227
$35$	0	0
$36$	−3721.59	−0.478600
$37$	13272.7	1.59388	0.796939	−	0.604060i	$-0.206451\pi$
$37$	13272.7	1.59388	0.796939	+	0.604060i	$0.206451\pi$
$38$	1570.37	0.176418
$39$	−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	$0.452357\pi$
$42$	2590.73	0.226620
$43$	11083.4	0.914118	0.457059	−	0.889436i	$-0.348903\pi$
$43$	11083.4	0.914118	0.457059	+	0.889436i	$0.348903\pi$
$44$	−8489.71	−0.661091
$45$	0	0
$46$	10534.3	0.734023
$47$	−9258.69	−0.611371	−0.305686	−	0.952132i	$-0.598886\pi$
$47$	−9258.69	−0.611371	−0.305686	+	0.952132i	$0.598886\pi$
$48$	−825.593	−0.0517205
$49$	23527.0	1.39983
$50$	0	0
$51$	−48.7602	−0.00262507
$52$	2704.00	0.138675
$53$	−3520.99	−0.172177	−0.0860885	−	0.996287i	$-0.527437\pi$
$53$	−3520.99	−0.172177	−0.0860885	+	0.996287i	$0.527437\pi$
$54$	−6135.18	−0.286314
$55$	0	0
$56$	−12853.3	−0.547703
$57$	1266.10	0.0516155
$58$	28542.1	1.11408
$59$	40334.1	1.50849	0.754245	−	0.656593i	$-0.228003\pi$
$59$	40334.1	1.50849	0.754245	+	0.656593i	$0.228003\pi$
$60$	0	0
$61$	−44243.3	−1.52238	−0.761189	−	0.648530i	$-0.775384\pi$
$61$	−44243.3	−1.52238	−0.761189	+	0.648530i	$0.775384\pi$
$62$	−27314.0	−0.902415
$63$	−46713.7	−1.48284
$64$	4096.00	0.125000
$65$	0	0
$66$	−6844.77	−0.193419
$67$	−7071.36	−0.192449	−0.0962246	−	0.995360i	$-0.530677\pi$
$67$	−7071.36	−0.192449	−0.0962246	+	0.995360i	$0.530677\pi$
$68$	241.913	0.00634435
$69$	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	$0.640416\pi$
$72$	14886.4	0.338421
$73$	41064.8	0.901909	0.450955	−	0.892547i	$-0.351084\pi$
$73$	41064.8	0.901909	0.450955	+	0.892547i	$0.351084\pi$
$74$	−53090.8	−1.12704
$75$	0	0
$76$	−6281.47	−0.124746
$77$	−106563.	−2.04824
$78$	2180.08	0.0405729
$79$	−19473.4	−0.351053	−0.175527	−	0.984475i	$-0.556163\pi$
$79$	−19473.4	−0.351053	−0.175527	+	0.984475i	$0.556163\pi$
$80$	0	0
$81$	51575.2	0.873431
$82$	−12840.3	−0.210882
$83$	−67566.0	−1.07655	−0.538274	−	0.842770i	$-0.680923\pi$
$83$	−67566.0	−1.07655	−0.538274	+	0.842770i	$0.680923\pi$
$84$	−10362.9	−0.160245
$85$	0	0
$86$	−44333.6	−0.646379
$87$	23011.9	0.325952
$88$	33958.9	0.467462
$89$	33319.0	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$89$	33319.0	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$90$	0	0
$91$	33940.8	0.429654
$92$	−42137.1	−0.519033
$93$	−22021.7	−0.264025
$94$	37034.8	0.432305
$95$	0	0
$96$	3302.37	0.0365719
$97$	2206.46	0.0238103	0.0119052	−	0.999929i	$-0.496210\pi$
$97$	2206.46	0.0238103	0.0119052	+	0.999929i	$0.496210\pi$
$98$	−94107.8	−0.989830
$99$	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	$0.651030\pi$
$102$	195.041	0.00185620
$103$	−116473.	−1.08176	−0.540881	−	0.841099i	$-0.681909\pi$
$103$	−116473.	−1.08176	−0.540881	+	0.841099i	$0.681909\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	14084.0	0.121747
$107$	195303.	1.64911	0.824554	−	0.565784i	$-0.191426\pi$
$107$	195303.	1.64911	0.824554	+	0.565784i	$0.191426\pi$
$108$	24540.7	0.202455
$109$	173562.	1.39923	0.699613	−	0.714522i	$-0.253356\pi$
$109$	173562.	1.39923	0.699613	+	0.714522i	$0.253356\pi$
$110$	0	0
$111$	−42804.1	−0.329745
$112$	51413.3	0.387285
$113$	−3810.27	−0.0280711	−0.0140356	−	0.999901i	$-0.504468\pi$
$113$	−3810.27	−0.0280711	−0.0140356	+	0.999901i	$0.504468\pi$
$114$	−5064.39	−0.0364977
$115$	0	0
$116$	−114168.	−0.787772
$117$	−39309.3	−0.265479
$118$	−161336.	−1.06666
$119$	3036.51	0.0196566
$120$	0	0
$121$	120493.	0.748166
$122$	176973.	1.07648
$123$	−10352.4	−0.0616988
$124$	109256.	0.638104
$125$	0	0
$126$	186855.	1.04852
$127$	19926.1	0.109626	0.0548130	−	0.998497i	$-0.482544\pi$
$127$	19926.1	0.109626	0.0548130	+	0.998497i	$0.482544\pi$
$128$	−16384.0	−0.0883883
$129$	−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	$0.262236\pi$
$132$	27379.1	0.136768
$133$	−78845.5	−0.386498
$134$	28285.4	0.136082
$135$	0	0
$136$	−967.653	−0.00448614
$137$	−9618.96	−0.0437851	−0.0218926	−	0.999760i	$-0.506969\pi$
$137$	−9618.96	−0.0437851	−0.0218926	+	0.999760i	$0.506969\pi$
$138$	−33972.7	−0.151856
$139$	−85185.2	−0.373962	−0.186981	−	0.982364i	$-0.559870\pi$
$139$	−85185.2	−0.373962	−0.186981	+	0.982364i	$0.559870\pi$
$140$	0	0
$141$	29859.0	0.126482
$142$	145085.	0.603813
$143$	−89672.6	−0.366707
$144$	−59545.5	−0.239300
$145$	0	0
$146$	−164259.	−0.637746
$147$	−75873.8	−0.289600
$148$	212363.	0.796939
$149$	222940.	0.822663	0.411332	−	0.911486i	$-0.365064\pi$
$149$	222940.	0.822663	0.411332	+	0.911486i	$0.365064\pi$
$150$	0	0
$151$	−363446.	−1.29717	−0.648585	−	0.761142i	$-0.724639\pi$
$151$	−363446.	−1.29717	−0.648585	+	0.761142i	$0.724639\pi$
$152$	25125.9	0.0882089
$153$	−3516.81	−0.0121456
$154$	426254.	1.44833
$155$	0	0
$156$	−8720.32	−0.0286894
$157$	323402.	1.04711	0.523556	−	0.851991i	$-0.324605\pi$
$157$	323402.	1.04711	0.523556	+	0.851991i	$0.324605\pi$
$158$	77893.5	0.248232
$159$	11355.1	0.0356203
$160$	0	0
$161$	−528908.	−1.60811
$162$	−206301.	−0.617609
$163$	361691.	1.06627	0.533137	−	0.846029i	$-0.321013\pi$
$163$	361691.	1.06627	0.533137	+	0.846029i	$0.321013\pi$
$164$	51361.0	0.149116
$165$	0	0
$166$	270264.	0.761234
$167$	535913.	1.48697	0.743486	−	0.668751i	$-0.233171\pi$
$167$	535913.	1.48697	0.743486	+	0.668751i	$0.233171\pi$
$168$	41451.6	0.113310
$169$	28561.0	0.0769231
$170$	0	0
$171$	91316.7	0.238814
$172$	177335.	0.457059
$173$	252514.	0.641462	0.320731	−	0.947170i	$-0.396071\pi$
$173$	252514.	0.641462	0.320731	+	0.947170i	$0.396071\pi$
$174$	−92047.4	−0.230483
$175$	0	0
$176$	−135835.	−0.330546
$177$	−130076.	−0.312079
$178$	−133276.	−0.315284
$179$	−499988.	−1.16634	−0.583172	−	0.812349i	$-0.698189\pi$
$179$	−499988.	−1.16634	−0.583172	+	0.812349i	$0.698189\pi$
$180$	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.	−0.303811
$183$	142683.	0.314953
$184$	168548.	0.367012
$185$	0	0
$186$	88087.0	0.186694
$187$	−8022.56	−0.0167768
$188$	−148139.	−0.305686
$189$	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	$0.483703\pi$
$192$	−13209.5	−0.0258603
$193$	946349.	1.82877	0.914383	−	0.404851i	$-0.132677\pi$
$193$	946349.	1.82877	0.914383	+	0.404851i	$0.132677\pi$
$194$	−8825.82	−0.0168365
$195$	0	0
$196$	376431.	0.699915
$197$	229033.	0.420468	0.210234	−	0.977651i	$-0.432577\pi$
$197$	229033.	0.420468	0.210234	+	0.977651i	$0.432577\pi$
$198$	−493676.	−0.894909
$199$	1.06207e6	1.90116	0.950582	−	0.310473i	$-0.100487\pi$
$199$	1.06207e6	1.90116	0.950582	+	0.310473i	$0.100487\pi$
$200$	0	0
$201$	22804.9	0.0398143
$202$	374703.	0.646114
$203$	−1.43305e6	−2.44074
$204$	−780.164	−0.00131253
$205$	0	0
$206$	465892.	0.764922
$207$	612567.	0.993636
$208$	43264.0	0.0693375
$209$	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	$0.563596\pi$
$212$	−56335.8	−0.0860885
$213$	116974.	0.176661
$214$	−781211.	−1.16610
$215$	0	0
$216$	−98162.9	−0.143157
$217$	1.37139e6	1.97702
$218$	−694247.	−0.989402
$219$	−132433.	−0.186589
$220$	0	0
$221$	2555.21	0.00351921
$222$	171216.	0.233165
$223$	−809910.	−1.09062	−0.545312	−	0.838233i	$-0.683589\pi$
$223$	−809910.	−1.09062	−0.545312	+	0.838233i	$0.683589\pi$
$224$	−205653.	−0.273852
$225$	0	0
$226$	15241.1	0.0198493
$227$	656318.	0.845375	0.422687	−	0.906276i	$-0.361087\pi$
$227$	656318.	0.845375	0.422687	+	0.906276i	$0.361087\pi$
$228$	20257.6	0.0258078
$229$	1.19398e6	1.50456	0.752280	−	0.658844i	$-0.228954\pi$
$229$	1.19398e6	1.50456	0.752280	+	0.658844i	$0.228954\pi$
$230$	0	0
$231$	343664.	0.423745
$232$	456673.	0.557039
$233$	1.06005e6	1.27920	0.639598	−	0.768709i	$-0.279101\pi$
$233$	1.06005e6	1.27920	0.639598	+	0.768709i	$0.279101\pi$
$234$	157237.	0.187722
$235$	0	0
$236$	645346.	0.754245
$237$	62801.1	0.0726267
$238$	−12146.1	−0.0138993
$239$	456553.	0.517006	0.258503	−	0.966010i	$-0.416771\pi$
$239$	456553.	0.517006	0.258503	+	0.966010i	$0.416771\pi$
$240$	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.	−0.529033
$243$	−539041.	−0.585607
$244$	−707892.	−0.761189
$245$	0	0
$246$	41409.5	0.0436277
$247$	−66348.1	−0.0691968
$248$	−437024.	−0.451208
$249$	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.	−0.741418
$253$	1.39739e6	1.37251
$254$	−79704.5	−0.0775173
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.21790e6	1.15021	0.575106	−	0.818079i	$-0.304961\pi$
$257$	1.21790e6	1.15021	0.575106	+	0.818079i	$0.304961\pi$
$258$	142975.	0.133724
$259$	2.66560e6	2.46914
$260$	0	0
$261$	1.65972e6	1.50811
$262$	−1.06758e6	−0.960830
$263$	−1.51165e6	−1.34760	−0.673799	−	0.738915i	$-0.735339\pi$
$263$	−1.51165e6	−1.34760	−0.673799	+	0.738915i	$0.735339\pi$
$264$	−109516.	−0.0967095
$265$	0	0
$266$	315382.	0.273296
$267$	−107453.	−0.0922443
$268$	−113142.	−0.0962246
$269$	405154.	0.341381	0.170691	−	0.985325i	$-0.445400\pi$
$269$	405154.	0.341381	0.170691	+	0.985325i	$0.445400\pi$
$270$	0	0
$271$	613161.	0.507167	0.253584	−	0.967313i	$-0.418391\pi$
$271$	613161.	0.507167	0.253584	+	0.967313i	$0.418391\pi$
$272$	3870.61	0.00317218
$273$	−109458.	−0.0888877
$274$	38475.8	0.0309608
$275$	0	0
$276$	135891.	0.107379
$277$	1.14240e6	0.894579	0.447289	−	0.894389i	$-0.352389\pi$
$277$	1.14240e6	0.894579	0.447289	+	0.894389i	$0.352389\pi$
$278$	340741.	0.264431
$279$	−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.	−0.0894361
$283$	205148.	0.152266	0.0761328	−	0.997098i	$-0.475743\pi$
$283$	205148.	0.152266	0.0761328	+	0.997098i	$0.475743\pi$
$284$	−580341.	−0.426961
$285$	0	0
$286$	358690.	0.259301
$287$	644687.	0.462003
$288$	238182.	0.169211
$289$	−1.41963e6	−0.999839
$290$	0	0
$291$	−7115.76	−0.00492593
$292$	657037.	0.450955
$293$	2.50408e6	1.70404	0.852018	−	0.523513i	$-0.175379\pi$
$293$	2.50408e6	1.70404	0.852018	+	0.523513i	$0.175379\pi$
$294$	303495.	0.204778
$295$	0	0
$296$	−849453.	−0.563521
$297$	−813843.	−0.535364
$298$	−891759.	−0.581711
$299$	−445073.	−0.287908
$300$	0	0
$301$	2.22592e6	1.41610
$302$	1.45378e6	0.917238
$303$	302102.	0.189037
$304$	−100504.	−0.0623731
$305$	0	0
$306$	14067.2	0.00858826
$307$	110033.	0.0666312	0.0333156	−	0.999445i	$-0.489393\pi$
$307$	110033.	0.0666312	0.0333156	+	0.999445i	$0.489393\pi$
$308$	−1.70502e6	−1.02412
$309$	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	$0.609796\pi$
$312$	34881.3	0.0202865
$313$	−2.00121e6	−1.15460	−0.577302	−	0.816531i	$-0.695894\pi$
$313$	−2.00121e6	−1.15460	−0.577302	+	0.816531i	$0.695894\pi$
$314$	−1.29361e6	−0.740421
$315$	0	0
$316$	−311574.	−0.175527
$317$	−90375.3	−0.0505128	−0.0252564	−	0.999681i	$-0.508040\pi$
$317$	−90375.3	−0.0505128	−0.0252564	+	0.999681i	$0.508040\pi$
$318$	−45420.4	−0.0251874
$319$	3.78616e6	2.08316
$320$	0	0
$321$	−629846.	−0.341171
$322$	2.11563e6	1.13710
$323$	−5935.83	−0.00316574
$324$	825204.	0.436716
$325$	0	0
$326$	−1.44676e6	−0.753970
$327$	−559732.	−0.289475
$328$	−205444.	−0.105441
$329$	−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	$0.913290\pi$
$332$	−1.08106e6	−0.538274
$333$	−3.08723e6	−1.52566
$334$	−2.14365e6	−1.05145
$335$	0	0
$336$	−165806.	−0.0801223
$337$	−2.29930e6	−1.10286	−0.551430	−	0.834221i	$-0.685917\pi$
$337$	−2.29930e6	−1.10286	−0.551430	+	0.834221i	$0.685917\pi$
$338$	−114244.	−0.0543928
$339$	12288.0	0.00580742
$340$	0	0
$341$	−3.62325e6	−1.68738
$342$	−365267.	−0.168867
$343$	1.34959e6	0.619393
$344$	−709338.	−0.323190
$345$	0	0
$346$	−1.01006e6	−0.453582
$347$	1.82848e6	0.815204	0.407602	−	0.913160i	$-0.366365\pi$
$347$	1.82848e6	0.815204	0.407602	+	0.913160i	$0.366365\pi$
$348$	368190.	0.162976
$349$	2.54054e6	1.11651	0.558254	−	0.829670i	$-0.311472\pi$
$349$	2.54054e6	1.11651	0.558254	+	0.829670i	$0.311472\pi$
$350$	0	0
$351$	259211.	0.112302
$352$	543342.	0.233731
$353$	3.67598e6	1.57013	0.785066	−	0.619412i	$-0.212629\pi$
$353$	3.67598e6	1.57013	0.785066	+	0.619412i	$0.212629\pi$
$354$	520305.	0.220674
$355$	0	0
$356$	533103.	0.222939
$357$	−9792.67	−0.00406659
$358$	1.99995e6	0.824730
$359$	−920537.	−0.376969	−0.188484	−	0.982076i	$-0.560357\pi$
$359$	−920537.	−0.376969	−0.188484	+	0.982076i	$0.560357\pi$
$360$	0	0
$361$	−2.32197e6	−0.937753
$362$	2.43107e6	0.975047
$363$	−388586.	−0.154782
$364$	543053.	0.214827
$365$	0	0
$366$	−570733.	−0.222705
$367$	−2.28316e6	−0.884854	−0.442427	−	0.896804i	$-0.645882\pi$
$367$	−2.28316e6	−0.884854	−0.442427	+	0.896804i	$0.645882\pi$
$368$	−674193.	−0.259516
$369$	−746659.	−0.285467
$370$	0	0
$371$	−707131.	−0.266726
$372$	−352348.	−0.132012
$373$	−2.84732e6	−1.05965	−0.529827	−	0.848106i	$-0.677743\pi$
$373$	−2.84732e6	−1.05965	−0.529827	+	0.848106i	$0.677743\pi$
$374$	32090.2	0.0118630
$375$	0	0
$376$	592556.	0.216152
$377$	−1.20590e6	−0.436977
$378$	−1.23215e6	−0.443541
$379$	3.75104e6	1.34139	0.670693	−	0.741735i	$-0.265997\pi$
$379$	3.75104e6	1.34139	0.670693	+	0.741735i	$0.265997\pi$
$380$	0	0
$381$	−64261.2	−0.0226797
$382$	−206415.	−0.0723740
$383$	1.81532e6	0.632349	0.316174	−	0.948701i	$-0.397602\pi$
$383$	1.81532e6	0.632349	0.316174	+	0.948701i	$0.397602\pi$
$384$	52837.9	0.0182860
$385$	0	0
$386$	−3.78540e6	−1.29313
$387$	−2.57800e6	−0.874994
$388$	35303.3	0.0119052
$389$	−4.37154e6	−1.46474	−0.732371	−	0.680906i	$-0.761586\pi$
$389$	−4.37154e6	−1.46474	−0.732371	+	0.680906i	$0.761586\pi$
$390$	0	0
$391$	−39818.4	−0.0131717
$392$	−1.50573e6	−0.494915
$393$	−860728.	−0.281115
$394$	−916133.	−0.297316
$395$	0	0
$396$	1.97470e6	0.632796
$397$	2.28486e6	0.727584	0.363792	−	0.931480i	$-0.381482\pi$
$397$	2.28486e6	0.727584	0.363792	+	0.931480i	$0.381482\pi$
$398$	−4.24827e6	−1.34433
$399$	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	$0.170937\pi$
$402$	−91219.8	−0.0281529
$403$	1.15402e6	0.353956
$404$	−1.49881e6	−0.456871
$405$	0	0
$406$	5.73220e6	1.72586
$407$	−7.04259e6	−2.10740
$408$	3120.65	0.000928101 0
$409$	−2.28293e6	−0.674815	−0.337408	−	0.941359i	$-0.609550\pi$
$409$	−2.28293e6	−0.674815	−0.337408	+	0.941359i	$0.609550\pi$
$410$	0	0
$411$	31020.9	0.00905836
$412$	−1.86357e6	−0.540881
$413$	8.10043e6	2.33686
$414$	−2.45027e6	−0.702607
$415$	0	0
$416$	−173056.	−0.0490290
$417$	274720.	0.0773660
$418$	−833248.	−0.233257
$419$	2.75540e6	0.766741	0.383371	−	0.923595i	$-0.374763\pi$
$419$	2.75540e6	0.766741	0.383371	+	0.923595i	$0.374763\pi$
$420$	0	0
$421$	−6.19470e6	−1.70339	−0.851697	−	0.524035i	$-0.824426\pi$
$421$	−6.19470e6	−1.70339	−0.851697	+	0.524035i	$0.824426\pi$
$422$	1.02680e6	0.280675
$423$	2.15357e6	0.585204
$424$	225343.	0.0608737
$425$	0	0
$426$	−467896.	−0.124918
$427$	−8.88552e6	−2.35838
$428$	3.12484e6	0.824554
$429$	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.	0.101227
$433$	4.76773e6	1.22206	0.611030	−	0.791608i	$-0.290756\pi$
$433$	4.76773e6	1.22206	0.611030	+	0.791608i	$0.290756\pi$
$434$	−5.48556e6	−1.39797
$435$	0	0
$436$	2.77699e6	0.699613
$437$	1.03392e6	0.258990
$438$	529732.	0.131938
$439$	275097.	0.0681279	0.0340639	−	0.999420i	$-0.489155\pi$
$439$	275097.	0.0681279	0.0340639	+	0.999420i	$0.489155\pi$
$440$	0	0
$441$	−5.47236e6	−1.33992
$442$	−10220.8	−0.00248846
$443$	5.89806e6	1.42791	0.713953	−	0.700194i	$-0.246903\pi$
$443$	5.89806e6	1.42791	0.713953	+	0.700194i	$0.246903\pi$
$444$	−684866.	−0.164872
$445$	0	0
$446$	3.23964e6	0.771187
$447$	−718975.	−0.170194
$448$	822613.	0.193642
$449$	2.69527e6	0.630938	0.315469	−	0.948936i	$-0.397838\pi$
$449$	2.69527e6	0.630938	0.315469	+	0.948936i	$0.397838\pi$
$450$	0	0
$451$	−1.70328e6	−0.394317
$452$	−60964.4	−0.0140356
$453$	1.17210e6	0.268361
$454$	−2.62527e6	−0.597770
$455$	0	0
$456$	−81030.3	−0.0182488
$457$	−5.00128e6	−1.12019	−0.560094	−	0.828429i	$-0.689235\pi$
$457$	−5.00128e6	−1.12019	−0.560094	+	0.828429i	$0.689235\pi$
$458$	−4.77593e6	−1.06388
$459$	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	$0.454135\pi$
$462$	−1.37466e6	−0.299633
$463$	−4.81314e6	−1.04346	−0.521731	−	0.853110i	$-0.674713\pi$
$463$	−4.81314e6	−1.04346	−0.521731	+	0.853110i	$0.674713\pi$
$464$	−1.82669e6	−0.393886
$465$	0	0
$466$	−4.24021e6	−0.904529
$467$	8.29674e6	1.76042	0.880208	−	0.474589i	$-0.157403\pi$
$467$	8.29674e6	1.76042	0.880208	+	0.474589i	$0.157403\pi$
$468$	−628949.	−0.132740
$469$	−1.42016e6	−0.298130
$470$	0	0
$471$	−1.04296e6	−0.216629
$472$	−2.58138e6	−0.533332
$473$	−5.88094e6	−1.20863
$474$	−251204.	−0.0513548
$475$	0	0
$476$	48584.2	0.00982828
$477$	818980.	0.164808
$478$	−1.82621e6	−0.365579
$479$	−1.37501e6	−0.273821	−0.136910	−	0.990583i	$-0.543717\pi$
$479$	−1.37501e6	−0.273821	−0.136910	+	0.990583i	$0.543717\pi$
$480$	0	0
$481$	2.24309e6	0.442062
$482$	−4.88028e6	−0.956813
$483$	1.70571e6	0.332689
$484$	1.92789e6	0.374083
$485$	0	0
$486$	2.15616e6	0.414087
$487$	3.68612e6	0.704283	0.352142	−	0.935947i	$-0.385454\pi$
$487$	3.68612e6	0.704283	0.352142	+	0.935947i	$0.385454\pi$
$488$	2.83157e6	0.538242
$489$	−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.	−0.0308494
$493$	−107886.	−0.0199916
$494$	265392.	0.0489295
$495$	0	0
$496$	1.74810e6	0.319052
$497$	−7.28449e6	−1.32284
$498$	−871595.	−0.157486
$499$	8.78314e6	1.57906	0.789529	−	0.613713i	$-0.210325\pi$
$499$	8.78314e6	1.57906	0.789529	+	0.613713i	$0.210325\pi$
$500$	0	0
$501$	−1.72830e6	−0.307628
$502$	−5.79216e6	−1.02584
$503$	8.16254e6	1.43849	0.719243	−	0.694759i	$-0.244489\pi$
$503$	8.16254e6	1.43849	0.719243	+	0.694759i	$0.244489\pi$
$504$	2.98968e6	0.524261
$505$	0	0
$506$	−5.58956e6	−0.970512
$507$	−92108.4	−0.0159140
$508$	318818.	0.0548130
$509$	1.76886e6	0.302621	0.151310	−	0.988486i	$-0.451651\pi$
$509$	1.76886e6	0.302621	0.151310	+	0.988486i	$0.451651\pi$
$510$	0	0
$511$	8.24718e6	1.39718
$512$	−262144.	−0.0441942
$513$	−602156.	−0.101022
$514$	−4.87159e6	−0.813322
$515$	0	0
$516$	−571899.	−0.0945573
$517$	4.91273e6	0.808344
$518$	−1.06624e7	−1.74594
$519$	−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	$0.529059\pi$
$522$	−6.63888e6	−1.06640
$523$	−3.65785e6	−0.584753	−0.292376	−	0.956303i	$-0.594446\pi$
$523$	−3.65785e6	−0.584753	−0.292376	+	0.956303i	$0.594446\pi$
$524$	4.27031e6	0.679409
$525$	0	0
$526$	6.04658e6	0.952896
$527$	103244.	0.0161934
$528$	438065.	0.0683839
$529$	499332.	0.0775801
$530$	0	0
$531$	−9.38169e6	−1.44393
$532$	−1.26153e6	−0.193249
$533$	542501.	0.0827146
$534$	429811.	0.0652265
$535$	0	0
$536$	452567.	0.0680411
$537$	1.61245e6	0.241296
$538$	−1.62062e6	−0.241393
$539$	−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	$0.602928\pi$
$542$	−2.45264e6	−0.358621
$543$	1.96003e6	0.285275
$544$	−15482.5	−0.00224307
$545$	0	0
$546$	437833.	0.0628531
$547$	−7.59075e6	−1.08472	−0.542359	−	0.840147i	$-0.682469\pi$
$547$	−7.59075e6	−1.08472	−0.542359	+	0.840147i	$0.682469\pi$
$548$	−153903.	−0.0218926
$549$	1.02910e7	1.45722
$550$	0	0
$551$	2.80135e6	0.393087
$552$	−543564.	−0.0759281
$553$	−3.91090e6	−0.543830
$554$	−4.56960e6	−0.632563
$555$	0	0
$556$	−1.36296e6	−0.186981
$557$	−1.04397e7	−1.42577	−0.712886	−	0.701280i	$-0.752612\pi$
$557$	−1.04397e7	−1.42577	−0.712886	+	0.701280i	$0.752612\pi$
$558$	6.35323e6	0.863792
$559$	1.87310e6	0.253531
$560$	0	0
$561$	25872.5	0.00347082
$562$	2.82089e6	0.376743
$563$	3.38176e6	0.449646	0.224823	−	0.974400i	$-0.427820\pi$
$563$	3.38176e6	0.449646	0.224823	+	0.974400i	$0.427820\pi$
$564$	477744.	0.0632409
$565$	0	0
$566$	−820593.	−0.107668
$567$	1.03580e7	1.35307
$568$	2.32137e6	0.301907
$569$	−1.09186e7	−1.41380	−0.706900	−	0.707313i	$-0.749907\pi$
$569$	−1.09186e7	−1.41380	−0.706900	+	0.707313i	$0.749907\pi$
$570$	0	0
$571$	1.15547e7	1.48310	0.741549	−	0.670899i	$-0.234092\pi$
$571$	1.15547e7	1.48310	0.741549	+	0.670899i	$0.234092\pi$
$572$	−1.43476e6	−0.183354
$573$	−166421.	−0.0211749
$574$	−2.57875e6	−0.326685
$575$	0	0
$576$	−952728.	−0.119650
$577$	−1.71415e6	−0.214343	−0.107171	−	0.994241i	$-0.534179\pi$
$577$	−1.71415e6	−0.214343	−0.107171	+	0.994241i	$0.534179\pi$
$578$	5.67851e6	0.706993
$579$	−3.05195e6	−0.378339
$580$	0	0
$581$	−1.35695e7	−1.66772
$582$	28463.0	0.00348316
$583$	1.86826e6	0.227649
$584$	−2.62815e6	−0.318873
$585$	0	0
$586$	−1.00163e7	−1.20494
$587$	2.85910e6	0.342479	0.171239	−	0.985229i	$-0.445223\pi$
$587$	2.85910e6	0.342479	0.171239	+	0.985229i	$0.445223\pi$
$588$	−1.21398e6	−0.144800
$589$	−2.68082e6	−0.318404
$590$	0	0
$591$	−738626.	−0.0869873
$592$	3.39781e6	0.398470
$593$	−4.96085e6	−0.579321	−0.289661	−	0.957129i	$-0.593542\pi$
$593$	−4.96085e6	−0.579321	−0.289661	+	0.957129i	$0.593542\pi$
$594$	3.25537e6	0.378560
$595$	0	0
$596$	3.56704e6	0.411332
$597$	−3.42514e6	−0.393317
$598$	1.78029e6	0.203581
$599$	−1.24276e7	−1.41520	−0.707601	−	0.706612i	$-0.750223\pi$
$599$	−1.24276e7	−1.41520	−0.707601	+	0.706612i	$0.750223\pi$
$600$	0	0
$601$	3.58410e6	0.404757	0.202378	−	0.979307i	$-0.435133\pi$
$601$	3.58410e6	0.404757	0.202378	+	0.979307i	$0.435133\pi$
$602$	−8.90366e6	−1.00133
$603$	1.64480e6	0.184212
$604$	−5.81513e6	−0.648585
$605$	0	0
$606$	−1.20841e6	−0.133669
$607$	3.78467e6	0.416923	0.208462	−	0.978031i	$-0.433154\pi$
$607$	3.78467e6	0.416923	0.208462	+	0.978031i	$0.433154\pi$
$608$	402014.	0.0441045
$609$	4.62154e6	0.504945
$610$	0	0
$611$	−1.56472e6	−0.169564
$612$	−56268.9	−0.00607281
$613$	−1.54450e7	−1.66011	−0.830053	−	0.557685i	$-0.811690\pi$
$613$	−1.54450e7	−1.66011	−0.830053	+	0.557685i	$0.811690\pi$
$614$	−440133.	−0.0471154
$615$	0	0
$616$	6.82006e6	0.724163
$617$	−6.78866e6	−0.717912	−0.358956	−	0.933355i	$-0.616867\pi$
$617$	−6.78866e6	−0.717912	−0.358956	+	0.933355i	$0.616867\pi$
$618$	−1.50249e6	−0.158249
$619$	−1.31549e7	−1.37994	−0.689970	−	0.723838i	$-0.742376\pi$
$619$	−1.31549e7	−1.37994	−0.689970	+	0.723838i	$0.742376\pi$
$620$	0	0
$621$	−4.03935e6	−0.420323
$622$	4.61405e6	0.478197
$623$	6.69155e6	0.690728
$624$	−139525.	−0.0143447
$625$	0	0
$626$	8.00486e6	0.816428
$627$	−671801.	−0.0682451
$628$	5.17443e6	0.523556
$629$	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	$0.343173\pi$
$632$	1.24630e6	0.124116
$633$	827849.	0.0821186
$634$	361501.	0.0357179
$635$	0	0
$636$	181681.	0.0178102
$637$	3.97606e6	0.388243
$638$	−1.51446e7	−1.47301
$639$	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	$0.795300\pi$
$642$	2.51938e6	0.241244
$643$	1.31070e7	1.25019	0.625096	−	0.780548i	$-0.285060\pi$
$643$	1.31070e7	1.25019	0.625096	+	0.780548i	$0.285060\pi$
$644$	−8.46252e6	−0.804054
$645$	0	0
$646$	23743.3	0.00223851
$647$	1.08998e7	1.02366	0.511832	−	0.859086i	$-0.328967\pi$
$647$	1.08998e7	1.02366	0.511832	+	0.859086i	$0.328967\pi$
$648$	−3.30082e6	−0.308805
$649$	−2.14016e7	−1.99450
$650$	0	0
$651$	−4.42270e6	−0.409011
$652$	5.78706e6	0.533137
$653$	1.18879e7	1.09099	0.545497	−	0.838113i	$-0.316341\pi$
$653$	1.18879e7	1.09099	0.545497	+	0.838113i	$0.316341\pi$
$654$	2.23893e6	0.204689
$655$	0	0
$656$	821776.	0.0745580
$657$	−9.55166e6	−0.863307
$658$	7.43781e6	0.669700
$659$	5.65451e6	0.507203	0.253601	−	0.967309i	$-0.418385\pi$
$659$	5.65451e6	0.507203	0.253601	+	0.967309i	$0.418385\pi$
$660$	0	0
$661$	8.05716e6	0.717263	0.358631	−	0.933479i	$-0.383243\pi$
$661$	8.05716e6	0.717263	0.358631	+	0.933479i	$0.383243\pi$
$662$	1.53583e7	1.36207
$663$	−8240.48	−0.000728062 0
$664$	4.32423e6	0.380617
$665$	0	0
$666$	1.23489e7	1.07880
$667$	1.87919e7	1.63552
$668$	8.57461e6	0.743486
$669$	2.61194e6	0.225630
$670$	0	0
$671$	2.34758e7	2.01286
$672$	663226.	0.0566550
$673$	8.86976e6	0.754874	0.377437	−	0.926035i	$-0.376806\pi$
$673$	8.86976e6	0.754874	0.377437	+	0.926035i	$0.376806\pi$
$674$	9.19718e6	0.779839
$675$	0	0
$676$	456976.	0.0384615
$677$	1.86550e7	1.56431	0.782156	−	0.623083i	$-0.214120\pi$
$677$	1.86550e7	1.56431	0.782156	+	0.623083i	$0.214120\pi$
$678$	−49152.1	−0.00410646
$679$	443129.	0.0368855
$680$	0	0
$681$	−2.11661e6	−0.174893
$682$	1.44930e7	1.19316
$683$	−1.79463e7	−1.47205	−0.736027	−	0.676952i	$-0.763300\pi$
$683$	−1.79463e7	−1.47205	−0.736027	+	0.676952i	$0.763300\pi$
$684$	1.46107e6	0.119407
$685$	0	0
$686$	−5.39836e6	−0.437977
$687$	−3.85056e6	−0.311266
$688$	2.83735e6	0.228530
$689$	−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	$0.312778\pi$
$692$	4.04023e6	0.320731
$693$	2.47866e7	1.96058
$694$	−7.31391e6	−0.576436
$695$	0	0
$696$	−1.47276e6	−0.115241
$697$	48534.8	0.00378418
$698$	−1.01621e7	−0.789490
$699$	−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	$0.156651\pi$
$702$	−1.03685e6	−0.0794093
$703$	−5.21076e6	−0.397661
$704$	−2.17337e6	−0.165273
$705$	0	0
$706$	−1.47039e7	−1.11025
$707$	−1.88132e7	−1.41551
$708$	−2.08122e6	−0.156040
$709$	2.11865e7	1.58286	0.791430	−	0.611260i	$-0.209337\pi$
$709$	2.11865e7	1.58286	0.791430	+	0.611260i	$0.209337\pi$
$710$	0	0
$711$	4.52950e6	0.336028
$712$	−2.13241e6	−0.157642
$713$	−1.79833e7	−1.32479
$714$	39170.7	0.00287551
$715$	0	0
$716$	−7.99980e6	−0.583172
$717$	−1.47237e6	−0.106959
$718$	3.68215e6	0.266557
$719$	2.03595e7	1.46874	0.734371	−	0.678749i	$-0.237477\pi$
$719$	2.03595e7	1.46874	0.734371	+	0.678749i	$0.237477\pi$
$720$	0	0
$721$	−2.33916e7	−1.67580
$722$	9.28788e6	0.663092
$723$	−3.93469e6	−0.279940
$724$	−9.72426e6	−0.689462
$725$	0	0
$726$	1.55434e6	0.109447
$727$	−3.11392e6	−0.218510	−0.109255	−	0.994014i	$-0.534847\pi$
$727$	−3.11392e6	−0.218510	−0.109255	+	0.994014i	$0.534847\pi$
$728$	−2.17221e6	−0.151906
$729$	−1.07944e7	−0.752280
$730$	0	0
$731$	167577.	0.0115990
$732$	2.28293e6	0.157476
$733$	−2.20488e7	−1.51574	−0.757871	−	0.652405i	$-0.773760\pi$
$733$	−2.20488e7	−1.51574	−0.757871	+	0.652405i	$0.773760\pi$
$734$	9.13265e6	0.625686
$735$	0	0
$736$	2.69677e6	0.183506
$737$	3.75211e6	0.254453
$738$	2.98664e6	0.201856
$739$	2.29831e6	0.154809	0.0774046	−	0.997000i	$-0.475337\pi$
$739$	2.29831e6	0.154809	0.0774046	+	0.997000i	$0.475337\pi$
$740$	0	0
$741$	213971.	0.0143156
$742$	2.82852e6	0.188604
$743$	−711970.	−0.0473140	−0.0236570	−	0.999720i	$-0.507531\pi$
$743$	−711970.	−0.0473140	−0.0236570	+	0.999720i	$0.507531\pi$
$744$	1.40939e6	0.0933468
$745$	0	0
$746$	1.13893e7	0.749288
$747$	1.57158e7	1.03047
$748$	−128361.	−0.00838839
$749$	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	$0.533474\pi$
$752$	−2.37023e6	−0.152843
$753$	−4.66989e6	−0.300137
$754$	4.82361e6	0.308990
$755$	0	0
$756$	4.92859e6	0.313631
$757$	−2.38887e7	−1.51514	−0.757569	−	0.652756i	$-0.773613\pi$
$757$	−2.38887e7	−1.51514	−0.757569	+	0.652756i	$0.773613\pi$
$758$	−1.50042e7	−0.948504
$759$	−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.	0.0160369
$763$	3.48569e7	2.16759
$764$	825660.	0.0511762
$765$	0	0
$766$	−7.26128e6	−0.447138
$767$	6.81646e6	0.418380
$768$	−211352.	−0.0129301
$769$	−365454.	−0.0222852	−0.0111426	−	0.999938i	$-0.503547\pi$
$769$	−365454.	−0.0222852	−0.0111426	+	0.999938i	$0.503547\pi$
$770$	0	0
$771$	−3.92768e6	−0.237958
$772$	1.51416e7	0.914383
$773$	−2.76260e7	−1.66291	−0.831457	−	0.555589i	$-0.812493\pi$
$773$	−2.76260e7	−1.66291	−0.831457	+	0.555589i	$0.812493\pi$
$774$	1.03120e7	0.618714
$775$	0	0
$776$	−141213.	−0.00841823
$777$	−8.59648e6	−0.510820
$778$	1.74862e7	1.03573
$779$	−1.26025e6	−0.0744066
$780$	0	0
$781$	1.92458e7	1.12904
$782$	159274.	0.00931381
$783$	−1.09444e7	−0.637953
$784$	6.02290e6	0.349958
$785$	0	0
$786$	3.44291e6	0.198779
$787$	−9.92420e6	−0.571161	−0.285580	−	0.958355i	$-0.592186\pi$
$787$	−9.92420e6	−0.571161	−0.285580	+	0.958355i	$0.592186\pi$
$788$	3.66453e6	0.210234
$789$	4.87501e6	0.278794
$790$	0	0
$791$	−765229.	−0.0434861
$792$	−7.89881e6	−0.447455
$793$	−7.47711e6	−0.422232
$794$	−9.13944e6	−0.514480
$795$	0	0
$796$	1.69931e7	0.950582
$797$	1.49499e6	0.0833668	0.0416834	−	0.999131i	$-0.486728\pi$
$797$	1.49499e6	0.0833668	0.0416834	+	0.999131i	$0.486728\pi$
$798$	−1.01710e6	−0.0565400
$799$	−139988.	−0.00775751
$800$	0	0
$801$	−7.74998e6	−0.426795
$802$	−2.21343e7	−1.21515
$803$	−2.17893e7	−1.19249
$804$	364879.	0.0199071
$805$	0	0
$806$	−4.61607e6	−0.250285
$807$	−1.30661e6	−0.0706257
$808$	5.99525e6	0.323057
$809$	−2.85707e6	−0.153479	−0.0767396	−	0.997051i	$-0.524451\pi$
$809$	−2.85707e6	−0.153479	−0.0767396	+	0.997051i	$0.524451\pi$
$810$	0	0
$811$	−9.42188e6	−0.503020	−0.251510	−	0.967855i	$-0.580927\pi$
$811$	−9.42188e6	−0.503020	−0.251510	+	0.967855i	$0.580927\pi$
$812$	−2.29288e7	−1.22037
$813$	−1.97743e6	−0.104924
$814$	2.81704e7	1.49016
$815$	0	0
$816$	−12482.6	−0.000656267 0
$817$	−4.35126e6	−0.228066
$818$	9.13173e6	0.477167
$819$	−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.	−0.00640523
$823$	−4.13964e6	−0.213041	−0.106520	−	0.994311i	$-0.533971\pi$
$823$	−4.13964e6	−0.213041	−0.106520	+	0.994311i	$0.533971\pi$
$824$	7.45427e6	0.382461
$825$	0	0
$826$	−3.24017e7	−1.65241
$827$	1.01580e7	0.516470	0.258235	−	0.966082i	$-0.416859\pi$
$827$	1.01580e7	0.516470	0.258235	+	0.966082i	$0.416859\pi$
$828$	9.80106e6	0.496818
$829$	3.45331e7	1.74521	0.872607	−	0.488423i	$-0.162428\pi$
$829$	3.45331e7	1.74521	0.872607	+	0.488423i	$0.162428\pi$
$830$	0	0
$831$	−3.68421e6	−0.185072
$832$	692224.	0.0346688
$833$	355718.	0.0177620
$834$	−1.09888e6	−0.0547060
$835$	0	0
$836$	3.33299e6	0.164937
$837$	1.04735e7	0.516749
$838$	−1.10216e7	−0.542168
$839$	−2.33327e7	−1.14435	−0.572177	−	0.820130i	$-0.693901\pi$
$839$	−2.33327e7	−1.14435	−0.572177	+	0.820130i	$0.693901\pi$
$840$	0	0
$841$	3.04045e7	1.48234
$842$	2.47788e7	1.20448
$843$	2.27432e6	0.110226
$844$	−4.10719e6	−0.198467
$845$	0	0
$846$	−8.61427e6	−0.413802
$847$	2.41990e7	1.15901
$848$	−901373.	−0.0430442
$849$	−661597.	−0.0315010
$850$	0	0
$851$	−3.49546e7	−1.65455
$852$	1.87158e6	0.0883305
$853$	1.80270e7	0.848304	0.424152	−	0.905591i	$-0.360572\pi$
$853$	1.80270e7	0.848304	0.424152	+	0.905591i	$0.360572\pi$
$854$	3.55421e7	1.66762
$855$	0	0
$856$	−1.24994e7	−0.583048
$857$	−1.82132e7	−0.847100	−0.423550	−	0.905873i	$-0.639216\pi$
$857$	−1.82132e7	−0.847100	−0.423550	+	0.905873i	$0.639216\pi$
$858$	−1.15677e6	−0.0536448
$859$	728581.	0.0336895	0.0168448	−	0.999858i	$-0.494638\pi$
$859$	728581.	0.0336895	0.0168448	+	0.999858i	$0.494638\pi$
$860$	0	0
$861$	−2.07910e6	−0.0955800
$862$	−2.20829e7	−1.01225
$863$	−4.10201e7	−1.87486	−0.937432	−	0.348168i	$-0.886804\pi$
$863$	−4.10201e7	−1.87486	−0.937432	+	0.348168i	$0.886804\pi$
$864$	−1.57061e6	−0.0715786
$865$	0	0
$866$	−1.90709e7	−0.864127
$867$	4.57826e6	0.206849
$868$	2.19422e7	0.988511
$869$	1.03327e7	0.464157
$870$	0	0
$871$	−1.19506e6	−0.0533758
$872$	−1.11079e7	−0.494701
$873$	−513220.	−0.0227913
$874$	−4.13567e6	−0.183133
$875$	0	0
$876$	−2.11893e6	−0.0932944
$877$	9.34651e6	0.410346	0.205173	−	0.978726i	$-0.434224\pi$
$877$	9.34651e6	0.410346	0.205173	+	0.978726i	$0.434224\pi$
$878$	−1.10039e6	−0.0481737
$879$	−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	$0.295422\pi$
$882$	2.18894e7	0.947465
$883$	1.71751e7	0.741305	0.370652	−	0.928772i	$-0.379134\pi$
$883$	1.71751e7	0.741305	0.370652	+	0.928772i	$0.379134\pi$
$884$	40883.3	0.00175961
$885$	0	0
$886$	−2.35922e7	−1.00968
$887$	−5.13121e6	−0.218983	−0.109492	−	0.993988i	$-0.534922\pi$
$887$	−5.13121e6	−0.218983	−0.109492	+	0.993988i	$0.534922\pi$
$888$	2.73946e6	0.116582
$889$	4.00183e6	0.169826
$890$	0	0
$891$	−2.73662e7	−1.15484
$892$	−1.29586e7	−0.545312
$893$	3.63489e6	0.152533
$894$	2.87590e6	0.120346
$895$	0	0
$896$	−3.29045e6	−0.136926
$897$	1.43535e6	0.0595629
$898$	−1.07811e7	−0.446140
$899$	−4.87249e7	−2.01072
$900$	0	0
$901$	−53235.9	−0.00218470
$902$	6.81313e6	0.278824
$903$	−7.17852e6	−0.292965
$904$	243858.	0.00992465
$905$	0	0
$906$	−4.68841e6	−0.189760
$907$	4.53644e7	1.83104	0.915519	−	0.402275i	$-0.131780\pi$
$907$	4.53644e7	1.83104	0.915519	+	0.402275i	$0.131780\pi$
$908$	1.05011e7	0.422687
$909$	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.	0.0129039
$913$	3.58510e7	1.42339
$914$	2.00051e7	0.792093
$915$	0	0
$916$	1.91037e7	0.752280
$917$	5.36013e7	2.10500
$918$	−92761.4	−0.00363296
$919$	1.32697e6	0.0518289	0.0259144	−	0.999664i	$-0.491750\pi$
$919$	1.32697e6	0.0518289	0.0259144	+	0.999664i	$0.491750\pi$
$920$	0	0
$921$	−354854.	−0.0137848
$922$	−5.24162e6	−0.203066
$923$	−6.12986e6	−0.236835
$924$	5.49863e6	0.211872
$925$	0	0
$926$	1.92526e7	0.737838
$927$	2.70916e7	1.03546
$928$	7.30677e6	0.278520
$929$	1.87784e7	0.713869	0.356935	−	0.934129i	$-0.383822\pi$
$929$	1.87784e7	0.713869	0.356935	+	0.934129i	$0.383822\pi$
$930$	0	0
$931$	−9.23649e6	−0.349247
$932$	1.69608e7	0.639598
$933$	3.72005e6	0.139909
$934$	−3.31870e7	−1.24480
$935$	0	0
$936$	2.51580e6	0.0938612
$937$	2.71600e7	1.01060	0.505302	−	0.862942i	$-0.331381\pi$
$937$	2.71600e7	1.01060	0.505302	+	0.862942i	$0.331381\pi$
$938$	5.68066e6	0.210810
$939$	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	$0.757460\pi$
$942$	4.17185e6	0.153180
$943$	−8.45392e6	−0.309584
$944$	1.03255e7	0.377122
$945$	0	0
$946$	2.35237e7	0.854631
$947$	1.05666e7	0.382877	0.191439	−	0.981505i	$-0.438685\pi$
$947$	1.05666e7	0.382877	0.191439	+	0.981505i	$0.438685\pi$
$948$	1.00482e6	0.0363133
$949$	6.93996e6	0.250145
$950$	0	0
$951$	291458.	0.0104502
$952$	−194337.	−0.00694965
$953$	−1.99531e7	−0.711667	−0.355834	−	0.934549i	$-0.615803\pi$
$953$	−1.99531e7	−0.711667	−0.355834	+	0.934549i	$0.615803\pi$
$954$	−3.27592e6	−0.116537
$955$	0	0
$956$	7.30484e6	0.258503
$957$	−1.22103e7	−0.430968
$958$	5.50004e6	0.193621
$959$	−1.93181e6	−0.0678293
$960$	0	0
$961$	1.79993e7	0.628706
$962$	−8.97235e6	−0.312585
$963$	−4.54273e7	−1.57853
$964$	1.95211e7	0.676569
$965$	0	0
$966$	−6.82285e6	−0.235246
$967$	2.42835e6	0.0835113	0.0417556	−	0.999128i	$-0.486705\pi$
$967$	2.42835e6	0.0835113	0.0417556	+	0.999128i	$0.486705\pi$
$968$	−7.71154e6	−0.264517
$969$	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	$0.421455\pi$
$972$	−8.62466e6	−0.292803
$973$	−1.71080e7	−0.579319
$974$	−1.47445e7	−0.498004
$975$	0	0
$976$	−1.13263e7	−0.380595
$977$	−2.56112e7	−0.858409	−0.429205	−	0.903207i	$-0.641206\pi$
$977$	−2.56112e7	−0.858409	−0.429205	+	0.903207i	$0.641206\pi$
$978$	4.66578e6	0.155983
$979$	−1.76793e7	−0.589533
$980$	0	0
$981$	−4.03704e7	−1.33934
$982$	1.77250e7	0.586554
$983$	2.23389e6	0.0737356	0.0368678	−	0.999320i	$-0.488262\pi$
$983$	2.23389e6	0.0737356	0.0368678	+	0.999320i	$0.488262\pi$
$984$	662551.	0.0218138
$985$	0	0
$986$	431544.	0.0141362
$987$	5.99668e6	0.195938
$988$	−1.06157e6	−0.0345984
$989$	−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	$0.529748\pi$
$992$	−6.99239e6	−0.225604
$993$	1.23825e7	0.398507
$994$	2.91379e7	0.935391
$995$	0	0
$996$	3.48638e6	0.111359
$997$	−1.12116e7	−0.357216	−0.178608	−	0.983920i	$-0.557159\pi$
$997$	−1.12116e7	−0.357216	−0.178608	+	0.983920i	$0.557159\pi$
$998$	−3.51325e7	−1.11656
$999$	2.03576e7	0.645377

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.e.1.2	✓	2	5.4	even	2
650.6.a.c.1.1		2	1.1	even	1	trivial
650.6.b.c.599.2		4	5.2	odd	4
650.6.b.c.599.3		4	5.3	odd	4
1040.6.a.h.1.1		2	20.19	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 $\F_{q}$
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

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.a.c.1.1

Level	$650$
Weight	$6$
Character	650.1
Self dual	yes
Analytic conductor	$104.249$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$