LMFDB - Embedded newform 147.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	147.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$23.5764215125$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +11.0000 q^{5} +18.0000 q^{6} +120.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +11.0000 q^{5} +18.0000 q^{6} +120.000 q^{8} +81.0000 q^{9} -22.0000 q^{10} +269.000 q^{11} +252.000 q^{12} -308.000 q^{13} -99.0000 q^{15} +656.000 q^{16} +1896.00 q^{17} -162.000 q^{18} -164.000 q^{19} -308.000 q^{20} -538.000 q^{22} -3264.00 q^{23} -1080.00 q^{24} -3004.00 q^{25} +616.000 q^{26} -729.000 q^{27} +2417.00 q^{29} +198.000 q^{30} +2841.00 q^{31} -5152.00 q^{32} -2421.00 q^{33} -3792.00 q^{34} -2268.00 q^{36} -11328.0 q^{37} +328.000 q^{38} +2772.00 q^{39} +1320.00 q^{40} -16856.0 q^{41} -7894.00 q^{43} -7532.00 q^{44} +891.000 q^{45} +6528.00 q^{46} +21102.0 q^{47} -5904.00 q^{48} +6008.00 q^{50} -17064.0 q^{51} +8624.00 q^{52} -29691.0 q^{53} +1458.00 q^{54} +2959.00 q^{55} +1476.00 q^{57} -4834.00 q^{58} -8163.00 q^{59} +2772.00 q^{60} +15166.0 q^{61} -5682.00 q^{62} -10688.0 q^{64} -3388.00 q^{65} +4842.00 q^{66} -32078.0 q^{67} -53088.0 q^{68} +29376.0 q^{69} -38274.0 q^{71} +9720.00 q^{72} +34866.0 q^{73} +22656.0 q^{74} +27036.0 q^{75} +4592.00 q^{76} -5544.00 q^{78} +13529.0 q^{79} +7216.00 q^{80} +6561.00 q^{81} +33712.0 q^{82} -68103.0 q^{83} +20856.0 q^{85} +15788.0 q^{86} -21753.0 q^{87} +32280.0 q^{88} -114922. q^{89} -1782.00 q^{90} +91392.0 q^{92} -25569.0 q^{93} -42204.0 q^{94} -1804.00 q^{95} +46368.0 q^{96} +154959. q^{97} +21789.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−2.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−28.0000	−0.875000
$5$	11.0000	0.196774	0.0983870	−	0.995148i	$-0.468632\pi$
$5$	11.0000	0.196774	0.0983870	+	0.995148i	$0.468632\pi$
$6$	18.0000	0.204124
$7$	0	0
$7$	0	0
$8$	120.000	0.662913
$9$	81.0000	0.333333
$10$	−22.0000	−0.0695701
$11$	269.000	0.670302	0.335151	−	0.942164i	$-0.391213\pi$
$11$	269.000	0.670302	0.335151	+	0.942164i	$0.391213\pi$
$12$	252.000	0.505181
$13$	−308.000	−0.505466	−0.252733	−	0.967536i	$-0.581329\pi$
$13$	−308.000	−0.505466	−0.252733	+	0.967536i	$0.581329\pi$
$14$	0	0
$15$	−99.0000	−0.113608
$16$	656.000	0.640625
$17$	1896.00	1.59117	0.795584	−	0.605843i	$-0.207164\pi$
$17$	1896.00	1.59117	0.795584	+	0.605843i	$0.207164\pi$
$18$	−162.000	−0.117851
$19$	−164.000	−0.104222	−0.0521111	−	0.998641i	$-0.516595\pi$
$19$	−164.000	−0.104222	−0.0521111	+	0.998641i	$0.516595\pi$
$20$	−308.000	−0.172177
$21$	0	0
$22$	−538.000	−0.236988
$23$	−3264.00	−1.28656	−0.643281	−	0.765630i	$-0.722427\pi$
$23$	−3264.00	−1.28656	−0.643281	+	0.765630i	$0.722427\pi$
$24$	−1080.00	−0.382733
$25$	−3004.00	−0.961280
$26$	616.000	0.178709
$27$	−729.000	−0.192450
$28$	0	0
$29$	2417.00	0.533681	0.266840	−	0.963741i	$-0.414020\pi$
$29$	2417.00	0.533681	0.266840	+	0.963741i	$0.414020\pi$
$30$	198.000	0.0401663
$31$	2841.00	0.530966	0.265483	−	0.964115i	$-0.414469\pi$
$31$	2841.00	0.530966	0.265483	+	0.964115i	$0.414469\pi$
$32$	−5152.00	−0.889408
$33$	−2421.00	−0.386999
$34$	−3792.00	−0.562563
$35$	0	0
$36$	−2268.00	−0.291667
$37$	−11328.0	−1.36034	−0.680172	−	0.733052i	$-0.738095\pi$
$37$	−11328.0	−1.36034	−0.680172	+	0.733052i	$0.738095\pi$
$38$	328.000	0.0368481
$39$	2772.00	0.291831
$40$	1320.00	0.130444
$41$	−16856.0	−1.56601	−0.783006	−	0.622015i	$-0.786314\pi$
$41$	−16856.0	−1.56601	−0.783006	+	0.622015i	$0.786314\pi$
$42$	0	0
$43$	−7894.00	−0.651067	−0.325534	−	0.945530i	$-0.605544\pi$
$43$	−7894.00	−0.651067	−0.325534	+	0.945530i	$0.605544\pi$
$44$	−7532.00	−0.586514
$45$	891.000	0.0655913
$46$	6528.00	0.454868
$47$	21102.0	1.39341	0.696705	−	0.717358i	$-0.254649\pi$
$47$	21102.0	1.39341	0.696705	+	0.717358i	$0.254649\pi$
$48$	−5904.00	−0.369865
$49$	0	0
$50$	6008.00	0.339864
$51$	−17064.0	−0.918661
$52$	8624.00	0.442283
$53$	−29691.0	−1.45189	−0.725947	−	0.687750i	$-0.758598\pi$
$53$	−29691.0	−1.45189	−0.725947	+	0.687750i	$0.758598\pi$
$54$	1458.00	0.0680414
$55$	2959.00	0.131898
$56$	0	0
$57$	1476.00	0.0601727
$58$	−4834.00	−0.188685
$59$	−8163.00	−0.305295	−0.152648	−	0.988281i	$-0.548780\pi$
$59$	−8163.00	−0.305295	−0.152648	+	0.988281i	$0.548780\pi$
$60$	2772.00	0.0994066
$61$	15166.0	0.521851	0.260925	−	0.965359i	$-0.415972\pi$
$61$	15166.0	0.521851	0.260925	+	0.965359i	$0.415972\pi$
$62$	−5682.00	−0.187725
$63$	0	0
$64$	−10688.0	−0.326172
$65$	−3388.00	−0.0994626
$66$	4842.00	0.136825
$67$	−32078.0	−0.873012	−0.436506	−	0.899701i	$-0.643784\pi$
$67$	−32078.0	−0.873012	−0.436506	+	0.899701i	$0.643784\pi$
$68$	−53088.0	−1.39227
$69$	29376.0	0.742797
$70$	0	0
$71$	−38274.0	−0.901069	−0.450534	−	0.892759i	$-0.648767\pi$
$71$	−38274.0	−0.901069	−0.450534	+	0.892759i	$0.648767\pi$
$72$	9720.00	0.220971
$73$	34866.0	0.765764	0.382882	−	0.923797i	$-0.374932\pi$
$73$	34866.0	0.765764	0.382882	+	0.923797i	$0.374932\pi$
$74$	22656.0	0.480954
$75$	27036.0	0.554995
$76$	4592.00	0.0911943
$77$	0	0
$78$	−5544.00	−0.103178
$79$	13529.0	0.243892	0.121946	−	0.992537i	$-0.461086\pi$
$79$	13529.0	0.243892	0.121946	+	0.992537i	$0.461086\pi$
$80$	7216.00	0.126058
$81$	6561.00	0.111111
$82$	33712.0	0.553669
$83$	−68103.0	−1.08510	−0.542552	−	0.840023i	$-0.682542\pi$
$83$	−68103.0	−1.08510	−0.542552	+	0.840023i	$0.682542\pi$
$84$	0	0
$85$	20856.0	0.313100
$86$	15788.0	0.230187
$87$	−21753.0	−0.308121
$88$	32280.0	0.444352
$89$	−114922.	−1.53790	−0.768950	−	0.639309i	$-0.779221\pi$
$89$	−114922.	−1.53790	−0.768950	+	0.639309i	$0.779221\pi$
$90$	−1782.00	−0.0231900
$91$	0	0
$92$	91392.0	1.12574
$93$	−25569.0	−0.306554
$94$	−42204.0	−0.492645
$95$	−1804.00	−0.0205082
$96$	46368.0	0.513500
$97$	154959.	1.67220	0.836099	−	0.548579i	$-0.184831\pi$
$97$	154959.	1.67220	0.836099	+	0.548579i	$0.184831\pi$
$98$	0	0
$99$	21789.0	0.223434
$100$	84112.0	0.841120
$101$	−107570.	−1.04927	−0.524636	−	0.851327i	$-0.675798\pi$
$101$	−107570.	−1.04927	−0.524636	+	0.851327i	$0.675798\pi$
$102$	34128.0	0.324796
$103$	8936.00	0.0829947	0.0414973	−	0.999139i	$-0.486787\pi$
$103$	8936.00	0.0829947	0.0414973	+	0.999139i	$0.486787\pi$
$104$	−36960.0	−0.335080
$105$	0	0
$106$	59382.0	0.513322
$107$	193667.	1.63530	0.817648	−	0.575719i	$-0.195278\pi$
$107$	193667.	1.63530	0.817648	+	0.575719i	$0.195278\pi$
$108$	20412.0	0.168394
$109$	205110.	1.65356	0.826781	−	0.562524i	$-0.190169\pi$
$109$	205110.	1.65356	0.826781	+	0.562524i	$0.190169\pi$
$110$	−5918.00	−0.0466330
$111$	101952.	0.785395
$112$	0	0
$113$	46664.0	0.343784	0.171892	−	0.985116i	$-0.445012\pi$
$113$	46664.0	0.343784	0.171892	+	0.985116i	$0.445012\pi$
$114$	−2952.00	−0.0212742
$115$	−35904.0	−0.253162
$116$	−67676.0	−0.466971
$117$	−24948.0	−0.168489
$118$	16326.0	0.107938
$119$	0	0
$120$	−11880.0	−0.0753119
$121$	−88690.0	−0.550695
$122$	−30332.0	−0.184502
$123$	151704.	0.904137
$124$	−79548.0	−0.464596
$125$	−67419.0	−0.385929
$126$	0	0
$127$	−304365.	−1.67450	−0.837250	−	0.546820i	$-0.815838\pi$
$127$	−304365.	−1.67450	−0.837250	+	0.546820i	$0.815838\pi$
$128$	186240.	1.00473
$129$	71046.0	0.375894
$130$	6776.00	0.0351654
$131$	−13303.0	−0.0677285	−0.0338642	−	0.999426i	$-0.510781\pi$
$131$	−13303.0	−0.0677285	−0.0338642	+	0.999426i	$0.510781\pi$
$132$	67788.0	0.338624
$133$	0	0
$134$	64156.0	0.308656
$135$	−8019.00	−0.0378692
$136$	227520.	1.05481
$137$	−398262.	−1.81287	−0.906437	−	0.422342i	$-0.861208\pi$
$137$	−398262.	−1.81287	−0.906437	+	0.422342i	$0.861208\pi$
$138$	−58752.0	−0.262618
$139$	230286.	1.01095	0.505476	−	0.862841i	$-0.331317\pi$
$139$	230286.	1.01095	0.505476	+	0.862841i	$0.331317\pi$
$140$	0	0
$141$	−189918.	−0.804486
$142$	76548.0	0.318576
$143$	−82852.0	−0.338815
$144$	53136.0	0.213542
$145$	26587.0	0.105015
$146$	−69732.0	−0.270738
$147$	0	0
$148$	317184.	1.19030
$149$	−97134.0	−0.358431	−0.179216	−	0.983810i	$-0.557356\pi$
$149$	−97134.0	−0.358431	−0.179216	+	0.983810i	$0.557356\pi$
$150$	−54072.0	−0.196220
$151$	−29047.0	−0.103671	−0.0518357	−	0.998656i	$-0.516507\pi$
$151$	−29047.0	−0.103671	−0.0518357	+	0.998656i	$0.516507\pi$
$152$	−19680.0	−0.0690901
$153$	153576.	0.530389
$154$	0	0
$155$	31251.0	0.104480
$156$	−77616.0	−0.255352
$157$	−576500.	−1.86660	−0.933298	−	0.359104i	$-0.883082\pi$
$157$	−576500.	−1.86660	−0.933298	+	0.359104i	$0.883082\pi$
$158$	−27058.0	−0.0862289
$159$	267219.	0.838252
$160$	−56672.0	−0.175012
$161$	0	0
$162$	−13122.0	−0.0392837
$163$	−265232.	−0.781910	−0.390955	−	0.920410i	$-0.627855\pi$
$163$	−265232.	−0.781910	−0.390955	+	0.920410i	$0.627855\pi$
$164$	471968.	1.37026
$165$	−26631.0	−0.0761514
$166$	136206.	0.383642
$167$	−363790.	−1.00939	−0.504696	−	0.863297i	$-0.668395\pi$
$167$	−363790.	−1.00939	−0.504696	+	0.863297i	$0.668395\pi$
$168$	0	0
$169$	−276429.	−0.744504
$170$	−41712.0	−0.110698
$171$	−13284.0	−0.0347407
$172$	221032.	0.569684
$173$	−164846.	−0.418758	−0.209379	−	0.977835i	$-0.567144\pi$
$173$	−164846.	−0.418758	−0.209379	+	0.977835i	$0.567144\pi$
$174$	43506.0	0.108937
$175$	0	0
$176$	176464.	0.429412
$177$	73467.0	0.176262
$178$	229844.	0.543730
$179$	30628.0	0.0714473	0.0357237	−	0.999362i	$-0.488626\pi$
$179$	30628.0	0.0714473	0.0357237	+	0.999362i	$0.488626\pi$
$180$	−24948.0	−0.0573924
$181$	−651392.	−1.47790	−0.738952	−	0.673759i	$-0.764679\pi$
$181$	−651392.	−1.47790	−0.738952	+	0.673759i	$0.764679\pi$
$182$	0	0
$183$	−136494.	−0.301291
$184$	−391680.	−0.852878
$185$	−124608.	−0.267680
$186$	51138.0	0.108383
$187$	510024.	1.06656
$188$	−590856.	−1.21923
$189$	0	0
$190$	3608.00	0.00725074
$191$	−757360.	−1.50217	−0.751085	−	0.660206i	$-0.770469\pi$
$191$	−757360.	−1.50217	−0.751085	+	0.660206i	$0.770469\pi$
$192$	96192.0	0.188315
$193$	−160339.	−0.309846	−0.154923	−	0.987927i	$-0.549513\pi$
$193$	−160339.	−0.309846	−0.154923	+	0.987927i	$0.549513\pi$
$194$	−309918.	−0.591211
$195$	30492.0	0.0574248
$196$	0	0
$197$	−61738.0	−0.113341	−0.0566705	−	0.998393i	$-0.518048\pi$
$197$	−61738.0	−0.113341	−0.0566705	+	0.998393i	$0.518048\pi$
$198$	−43578.0	−0.0789959
$199$	370908.	0.663947	0.331974	−	0.943289i	$-0.392286\pi$
$199$	370908.	0.663947	0.331974	+	0.943289i	$0.392286\pi$
$200$	−360480.	−0.637245
$201$	288702.	0.504034
$202$	215140.	0.370973
$203$	0	0
$204$	477792.	0.803829
$205$	−185416.	−0.308150
$206$	−17872.0	−0.0293430
$207$	−264384.	−0.428854
$208$	−202048.	−0.323814
$209$	−44116.0	−0.0698603
$210$	0	0
$211$	217450.	0.336243	0.168122	−	0.985766i	$-0.446230\pi$
$211$	217450.	0.336243	0.168122	+	0.985766i	$0.446230\pi$
$212$	831348.	1.27041
$213$	344466.	0.520232
$214$	−387334.	−0.578164
$215$	−86834.0	−0.128113
$216$	−87480.0	−0.127578
$217$	0	0
$218$	−410220.	−0.584623
$219$	−313794.	−0.442114
$220$	−82852.0	−0.115411
$221$	−583968.	−0.804282
$222$	−203904.	−0.277679
$223$	589771.	0.794184	0.397092	−	0.917779i	$-0.370019\pi$
$223$	589771.	0.794184	0.397092	+	0.917779i	$0.370019\pi$
$224$	0	0
$225$	−243324.	−0.320427
$226$	−93328.0	−0.121546
$227$	−387045.	−0.498536	−0.249268	−	0.968434i	$-0.580190\pi$
$227$	−387045.	−0.498536	−0.249268	+	0.968434i	$0.580190\pi$
$228$	−41328.0	−0.0526511
$229$	−232732.	−0.293270	−0.146635	−	0.989191i	$-0.546844\pi$
$229$	−232732.	−0.293270	−0.146635	+	0.989191i	$0.546844\pi$
$230$	71808.0	0.0895062
$231$	0	0
$232$	290040.	0.353784
$233$	42096.0	0.0507985	0.0253993	−	0.999677i	$-0.491914\pi$
$233$	42096.0	0.0507985	0.0253993	+	0.999677i	$0.491914\pi$
$234$	49896.0	0.0595698
$235$	232122.	0.274187
$236$	228564.	0.267133
$237$	−121761.	−0.140811
$238$	0	0
$239$	−313416.	−0.354917	−0.177458	−	0.984128i	$-0.556788\pi$
$239$	−313416.	−0.354917	−0.177458	+	0.984128i	$0.556788\pi$
$240$	−64944.0	−0.0727798
$241$	857807.	0.951365	0.475682	−	0.879617i	$-0.342201\pi$
$241$	857807.	0.951365	0.475682	+	0.879617i	$0.342201\pi$
$242$	177380.	0.194700
$243$	−59049.0	−0.0641500
$244$	−424648.	−0.456620
$245$	0	0
$246$	−303408.	−0.319661
$247$	50512.0	0.0526808
$248$	340920.	0.351984
$249$	612927.	0.626485
$250$	134838.	0.136446
$251$	454517.	0.455371	0.227686	−	0.973735i	$-0.426884\pi$
$251$	454517.	0.455371	0.227686	+	0.973735i	$0.426884\pi$
$252$	0	0
$253$	−878016.	−0.862385
$254$	608730.	0.592026
$255$	−187704.	−0.180769
$256$	−30464.0	−0.0290527
$257$	878182.	0.829376	0.414688	−	0.909964i	$-0.363891\pi$
$257$	878182.	0.829376	0.414688	+	0.909964i	$0.363891\pi$
$258$	−142092.	−0.132899
$259$	0	0
$260$	94864.0	0.0870298
$261$	195777.	0.177894
$262$	26606.0	0.0239456
$263$	1.96093e6	1.74813	0.874065	−	0.485809i	$-0.161475\pi$
$263$	1.96093e6	1.74813	0.874065	+	0.485809i	$0.161475\pi$
$264$	−290520.	−0.256547
$265$	−326601.	−0.285695
$266$	0	0
$267$	1.03430e6	0.887907
$268$	898184.	0.763886
$269$	−1.05380e6	−0.887923	−0.443962	−	0.896046i	$-0.646427\pi$
$269$	−1.05380e6	−0.887923	−0.443962	+	0.896046i	$0.646427\pi$
$270$	16038.0	0.0133888
$271$	−105059.	−0.0868981	−0.0434490	−	0.999056i	$-0.513835\pi$
$271$	−105059.	−0.0868981	−0.0434490	+	0.999056i	$0.513835\pi$
$272$	1.24378e6	1.01934
$273$	0	0
$274$	796524.	0.640948
$275$	−808076.	−0.644348
$276$	−822528.	−0.649947
$277$	−427592.	−0.334834	−0.167417	−	0.985886i	$-0.553543\pi$
$277$	−427592.	−0.334834	−0.167417	+	0.985886i	$0.553543\pi$
$278$	−460572.	−0.357426
$279$	230121.	0.176989
$280$	0	0
$281$	638878.	0.482672	0.241336	−	0.970442i	$-0.422414\pi$
$281$	638878.	0.482672	0.241336	+	0.970442i	$0.422414\pi$
$282$	379836.	0.284429
$283$	−2.45142e6	−1.81950	−0.909750	−	0.415157i	$-0.863727\pi$
$283$	−2.45142e6	−1.81950	−0.909750	+	0.415157i	$0.863727\pi$
$284$	1.07167e6	0.788435
$285$	16236.0	0.0118404
$286$	165704.	0.119789
$287$	0	0
$288$	−417312.	−0.296469
$289$	2.17496e6	1.53182
$290$	−53174.0	−0.0371282
$291$	−1.39463e6	−0.965443
$292$	−976248.	−0.670044
$293$	1.71617e6	1.16786	0.583930	−	0.811804i	$-0.301514\pi$
$293$	1.71617e6	1.16786	0.583930	+	0.811804i	$0.301514\pi$
$294$	0	0
$295$	−89793.0	−0.0600741
$296$	−1.35936e6	−0.901790
$297$	−196101.	−0.129000
$298$	194268.	0.126725
$299$	1.00531e6	0.650314
$300$	−757008.	−0.485621
$301$	0	0
$302$	58094.0	0.0366534
$303$	968130.	0.605797
$304$	−107584.	−0.0667673
$305$	166826.	0.102687
$306$	−307152.	−0.187521
$307$	1.80897e6	1.09543	0.547715	−	0.836665i	$-0.315498\pi$
$307$	1.80897e6	1.09543	0.547715	+	0.836665i	$0.315498\pi$
$308$	0	0
$309$	−80424.0	−0.0479170
$310$	−62502.0	−0.0369394
$311$	1.52146e6	0.891987	0.445993	−	0.895036i	$-0.352850\pi$
$311$	1.52146e6	0.891987	0.445993	+	0.895036i	$0.352850\pi$
$312$	332640.	0.193459
$313$	−1.34840e6	−0.777961	−0.388980	−	0.921246i	$-0.627173\pi$
$313$	−1.34840e6	−0.777961	−0.388980	+	0.921246i	$0.627173\pi$
$314$	1.15300e6	0.659941
$315$	0	0
$316$	−378812.	−0.213406
$317$	−49695.0	−0.0277757	−0.0138878	−	0.999904i	$-0.504421\pi$
$317$	−49695.0	−0.0277757	−0.0138878	+	0.999904i	$0.504421\pi$
$318$	−534438.	−0.296367
$319$	650173.	0.357727
$320$	−117568.	−0.0641821
$321$	−1.74300e6	−0.944138
$322$	0	0
$323$	−310944.	−0.165835
$324$	−183708.	−0.0972222
$325$	925232.	0.485895
$326$	530464.	0.276447
$327$	−1.84599e6	−0.954685
$328$	−2.02272e6	−1.03813
$329$	0	0
$330$	53262.0	0.0269236
$331$	1.58784e6	0.796591	0.398296	−	0.917257i	$-0.369602\pi$
$331$	1.58784e6	0.796591	0.398296	+	0.917257i	$0.369602\pi$
$332$	1.90688e6	0.949465
$333$	−917568.	−0.453448
$334$	727580.	0.356874
$335$	−352858.	−0.171786
$336$	0	0
$337$	214825.	0.103041	0.0515205	−	0.998672i	$-0.483593\pi$
$337$	214825.	0.103041	0.0515205	+	0.998672i	$0.483593\pi$
$338$	552858.	0.263222
$339$	−419976.	−0.198484
$340$	−583968.	−0.273963
$341$	764229.	0.355908
$342$	26568.0	0.0122827
$343$	0	0
$344$	−947280.	−0.431601
$345$	323136.	0.146163
$346$	329692.	0.148053
$347$	2.58860e6	1.15409	0.577046	−	0.816711i	$-0.304205\pi$
$347$	2.58860e6	1.15409	0.577046	+	0.816711i	$0.304205\pi$
$348$	609084.	0.269606
$349$	−24878.0	−0.0109333	−0.00546666	−	0.999985i	$-0.501740\pi$
$349$	−24878.0	−0.0109333	−0.00546666	+	0.999985i	$0.501740\pi$
$350$	0	0
$351$	224532.	0.0972771
$352$	−1.38589e6	−0.596172
$353$	−1.73601e6	−0.741506	−0.370753	−	0.928731i	$-0.620900\pi$
$353$	−1.73601e6	−0.741506	−0.370753	+	0.928731i	$0.620900\pi$
$354$	−146934.	−0.0623181
$355$	−421014.	−0.177307
$356$	3.21782e6	1.34566
$357$	0	0
$358$	−61256.0	−0.0252604
$359$	862426.	0.353172	0.176586	−	0.984285i	$-0.443495\pi$
$359$	862426.	0.353172	0.176586	+	0.984285i	$0.443495\pi$
$360$	106920.	0.0434813
$361$	−2.44920e6	−0.989138
$362$	1.30278e6	0.522518
$363$	798210.	0.317944
$364$	0	0
$365$	383526.	0.150682
$366$	272988.	0.106522
$367$	3.11542e6	1.20740	0.603700	−	0.797211i	$-0.293692\pi$
$367$	3.11542e6	1.20740	0.603700	+	0.797211i	$0.293692\pi$
$368$	−2.14118e6	−0.824203
$369$	−1.36534e6	−0.522004
$370$	249216.	0.0946393
$371$	0	0
$372$	715932.	0.268234
$373$	−1.79694e6	−0.668748	−0.334374	−	0.942440i	$-0.608525\pi$
$373$	−1.79694e6	−0.668748	−0.334374	+	0.942440i	$0.608525\pi$
$374$	−1.02005e6	−0.377087
$375$	606771.	0.222816
$376$	2.53224e6	0.923709
$377$	−744436.	−0.269758
$378$	0	0
$379$	3.45466e6	1.23540	0.617699	−	0.786415i	$-0.288065\pi$
$379$	3.45466e6	1.23540	0.617699	+	0.786415i	$0.288065\pi$
$380$	50512.0	0.0179447
$381$	2.73928e6	0.966774
$382$	1.51472e6	0.531097
$383$	2.15504e6	0.750685	0.375343	−	0.926886i	$-0.377525\pi$
$383$	2.15504e6	0.750685	0.375343	+	0.926886i	$0.377525\pi$
$384$	−1.67616e6	−0.580079
$385$	0	0
$386$	320678.	0.109547
$387$	−639414.	−0.217022
$388$	−4.33885e6	−1.46317
$389$	−462774.	−0.155058	−0.0775291	−	0.996990i	$-0.524703\pi$
$389$	−462774.	−0.155058	−0.0775291	+	0.996990i	$0.524703\pi$
$390$	−60984.0	−0.0203027
$391$	−6.18854e6	−2.04714
$392$	0	0
$393$	119727.	0.0391031
$394$	123476.	0.0400721
$395$	148819.	0.0479916
$396$	−610092.	−0.195505
$397$	−4.06621e6	−1.29483	−0.647416	−	0.762136i	$-0.724151\pi$
$397$	−4.06621e6	−1.29483	−0.647416	+	0.762136i	$0.724151\pi$
$398$	−741816.	−0.234741
$399$	0	0
$400$	−1.97062e6	−0.615820
$401$	−5.06863e6	−1.57409	−0.787045	−	0.616895i	$-0.788390\pi$
$401$	−5.06863e6	−1.57409	−0.787045	+	0.616895i	$0.788390\pi$
$402$	−577404.	−0.178203
$403$	−875028.	−0.268386
$404$	3.01196e6	0.918112
$405$	72171.0	0.0218638
$406$	0	0
$407$	−3.04723e6	−0.911842
$408$	−2.04768e6	−0.608992
$409$	2.87734e6	0.850515	0.425258	−	0.905072i	$-0.360183\pi$
$409$	2.87734e6	0.850515	0.425258	+	0.905072i	$0.360183\pi$
$410$	370832.	0.108948
$411$	3.58436e6	1.04666
$412$	−250208.	−0.0726203
$413$	0	0
$414$	528768.	0.151623
$415$	−749133.	−0.213520
$416$	1.58682e6	0.449566
$417$	−2.07257e6	−0.583673
$418$	88232.0	0.0246993
$419$	−3.41342e6	−0.949850	−0.474925	−	0.880026i	$-0.657525\pi$
$419$	−3.41342e6	−0.949850	−0.474925	+	0.880026i	$0.657525\pi$
$420$	0	0
$421$	−1.30737e6	−0.359496	−0.179748	−	0.983713i	$-0.557528\pi$
$421$	−1.30737e6	−0.359496	−0.179748	+	0.983713i	$0.557528\pi$
$422$	−434900.	−0.118880
$423$	1.70926e6	0.464470
$424$	−3.56292e6	−0.962479
$425$	−5.69558e6	−1.52956
$426$	−688932.	−0.183930
$427$	0	0
$428$	−5.42268e6	−1.43088
$429$	745668.	0.195615
$430$	173668.	0.0452948
$431$	1.93547e6	0.501872	0.250936	−	0.968004i	$-0.419262\pi$
$431$	1.93547e6	0.501872	0.250936	+	0.968004i	$0.419262\pi$
$432$	−478224.	−0.123288
$433$	516670.	0.132432	0.0662161	−	0.997805i	$-0.478907\pi$
$433$	516670.	0.132432	0.0662161	+	0.997805i	$0.478907\pi$
$434$	0	0
$435$	−239283.	−0.0606302
$436$	−5.74308e6	−1.44687
$437$	535296.	0.134088
$438$	627588.	0.156311
$439$	2.91530e6	0.721975	0.360987	−	0.932571i	$-0.382440\pi$
$439$	2.91530e6	0.721975	0.360987	+	0.932571i	$0.382440\pi$
$440$	355080.	0.0874369
$441$	0	0
$442$	1.16794e6	0.284357
$443$	−1.78379e6	−0.431852	−0.215926	−	0.976410i	$-0.569277\pi$
$443$	−1.78379e6	−0.431852	−0.215926	+	0.976410i	$0.569277\pi$
$444$	−2.85466e6	−0.687221
$445$	−1.26414e6	−0.302619
$446$	−1.17954e6	−0.280787
$447$	874206.	0.206940
$448$	0	0
$449$	4.00158e6	0.936733	0.468366	−	0.883534i	$-0.344843\pi$
$449$	4.00158e6	0.936733	0.468366	+	0.883534i	$0.344843\pi$
$450$	486648.	0.113288
$451$	−4.53426e6	−1.04970
$452$	−1.30659e6	−0.300811
$453$	261423.	0.0598547
$454$	774090.	0.176259
$455$	0	0
$456$	177120.	0.0398892
$457$	−1.16766e6	−0.261534	−0.130767	−	0.991413i	$-0.541744\pi$
$457$	−1.16766e6	−0.261534	−0.130767	+	0.991413i	$0.541744\pi$
$458$	465464.	0.103687
$459$	−1.38218e6	−0.306220
$460$	1.00531e6	0.221517
$461$	3.61358e6	0.791928	0.395964	−	0.918266i	$-0.370411\pi$
$461$	3.61358e6	0.791928	0.395964	+	0.918266i	$0.370411\pi$
$462$	0	0
$463$	−1.80111e6	−0.390471	−0.195235	−	0.980756i	$-0.562547\pi$
$463$	−1.80111e6	−0.390471	−0.195235	+	0.980756i	$0.562547\pi$
$464$	1.58555e6	0.341889
$465$	−281259.	−0.0603218
$466$	−84192.0	−0.0179600
$467$	−2.36975e6	−0.502817	−0.251409	−	0.967881i	$-0.580894\pi$
$467$	−2.36975e6	−0.502817	−0.251409	+	0.967881i	$0.580894\pi$
$468$	698544.	0.147428
$469$	0	0
$470$	−464244.	−0.0969397
$471$	5.18850e6	1.07768
$472$	−979560.	−0.202384
$473$	−2.12349e6	−0.436412
$474$	243522.	0.0497843
$475$	492656.	0.100187
$476$	0	0
$477$	−2.40497e6	−0.483965
$478$	626832.	0.125482
$479$	518146.	0.103184	0.0515921	−	0.998668i	$-0.483570\pi$
$479$	518146.	0.103184	0.0515921	+	0.998668i	$0.483570\pi$
$480$	510048.	0.101043
$481$	3.48902e6	0.687609
$482$	−1.71561e6	−0.336358
$483$	0	0
$484$	2.48332e6	0.481858
$485$	1.70455e6	0.329045
$486$	118098.	0.0226805
$487$	2.82613e6	0.539970	0.269985	−	0.962865i	$-0.412981\pi$
$487$	2.82613e6	0.539970	0.269985	+	0.962865i	$0.412981\pi$
$488$	1.81992e6	0.345942
$489$	2.38709e6	0.451436
$490$	0	0
$491$	9.34747e6	1.74981	0.874904	−	0.484296i	$-0.160924\pi$
$491$	9.34747e6	1.74981	0.874904	+	0.484296i	$0.160924\pi$
$492$	−4.24771e6	−0.791120
$493$	4.58263e6	0.849176
$494$	−101024.	−0.0186255
$495$	239679.	0.0439660
$496$	1.86370e6	0.340150
$497$	0	0
$498$	−1.22585e6	−0.221496
$499$	8.17185e6	1.46916	0.734580	−	0.678522i	$-0.237379\pi$
$499$	8.17185e6	1.46916	0.734580	+	0.678522i	$0.237379\pi$
$500$	1.88773e6	0.337688
$501$	3.27411e6	0.582772
$502$	−909034.	−0.160998
$503$	−7.37713e6	−1.30007	−0.650036	−	0.759903i	$-0.725246\pi$
$503$	−7.37713e6	−1.30007	−0.650036	+	0.759903i	$0.725246\pi$
$504$	0	0
$505$	−1.18327e6	−0.206469
$506$	1.75603e6	0.304899
$507$	2.48786e6	0.429839
$508$	8.52222e6	1.46519
$509$	−326315.	−0.0558268	−0.0279134	−	0.999610i	$-0.508886\pi$
$509$	−326315.	−0.0558268	−0.0279134	+	0.999610i	$0.508886\pi$
$510$	375408.	0.0639114
$511$	0	0
$512$	−5.89875e6	−0.994455
$513$	119556.	0.0200576
$514$	−1.75636e6	−0.293229
$515$	98296.0	0.0163312
$516$	−1.98929e6	−0.328907
$517$	5.67644e6	0.934006
$518$	0	0
$519$	1.48361e6	0.241770
$520$	−406560.	−0.0659350
$521$	−2.16703e6	−0.349760	−0.174880	−	0.984590i	$-0.555954\pi$
$521$	−2.16703e6	−0.349760	−0.174880	+	0.984590i	$0.555954\pi$
$522$	−391554.	−0.0628949
$523$	−723404.	−0.115645	−0.0578225	−	0.998327i	$-0.518416\pi$
$523$	−723404.	−0.115645	−0.0578225	+	0.998327i	$0.518416\pi$
$524$	372484.	0.0592624
$525$	0	0
$526$	−3.92187e6	−0.618057
$527$	5.38654e6	0.844857
$528$	−1.58818e6	−0.247921
$529$	4.21735e6	0.655241
$530$	653202.	0.101008
$531$	−661203.	−0.101765
$532$	0	0
$533$	5.19165e6	0.791566
$534$	−2.06860e6	−0.313923
$535$	2.13034e6	0.321784
$536$	−3.84936e6	−0.578731
$537$	−275652.	−0.0412501
$538$	2.10759e6	0.313928
$539$	0	0
$540$	224532.	0.0331355
$541$	5.99964e6	0.881317	0.440659	−	0.897675i	$-0.354745\pi$
$541$	5.99964e6	0.881317	0.440659	+	0.897675i	$0.354745\pi$
$542$	210118.	0.0307231
$543$	5.86253e6	0.853268
$544$	−9.76819e6	−1.41520
$545$	2.25621e6	0.325378
$546$	0	0
$547$	7.01570e6	1.00254	0.501271	−	0.865290i	$-0.332866\pi$
$547$	7.01570e6	1.00254	0.501271	+	0.865290i	$0.332866\pi$
$548$	1.11513e7	1.58626
$549$	1.22845e6	0.173950
$550$	1.61615e6	0.227811
$551$	−396388.	−0.0556213
$552$	3.52512e6	0.492409
$553$	0	0
$554$	855184.	0.118382
$555$	1.12147e6	0.154545
$556$	−6.44801e6	−0.884583
$557$	−8.91872e6	−1.21805	−0.609025	−	0.793151i	$-0.708439\pi$
$557$	−8.91872e6	−1.21805	−0.609025	+	0.793151i	$0.708439\pi$
$558$	−460242.	−0.0625750
$559$	2.43135e6	0.329093
$560$	0	0
$561$	−4.59022e6	−0.615781
$562$	−1.27776e6	−0.170650
$563$	−1.33482e7	−1.77481	−0.887407	−	0.460987i	$-0.847495\pi$
$563$	−1.33482e7	−1.77481	−0.887407	+	0.460987i	$0.847495\pi$
$564$	5.31770e6	0.703925
$565$	513304.	0.0676478
$566$	4.90284e6	0.643290
$567$	0	0
$568$	−4.59288e6	−0.597330
$569$	1.10215e6	0.142712	0.0713558	−	0.997451i	$-0.477267\pi$
$569$	1.10215e6	0.142712	0.0713558	+	0.997451i	$0.477267\pi$
$570$	−32472.0	−0.00418622
$571$	1.89348e6	0.243036	0.121518	−	0.992589i	$-0.461224\pi$
$571$	1.89348e6	0.243036	0.121518	+	0.992589i	$0.461224\pi$
$572$	2.31986e6	0.296463
$573$	6.81624e6	0.867278
$574$	0	0
$575$	9.80506e6	1.23675
$576$	−865728.	−0.108724
$577$	2.82951e6	0.353811	0.176906	−	0.984228i	$-0.443391\pi$
$577$	2.82951e6	0.353811	0.176906	+	0.984228i	$0.443391\pi$
$578$	−4.34992e6	−0.541579
$579$	1.44305e6	0.178890
$580$	−744436.	−0.0918877
$581$	0	0
$582$	2.78926e6	0.341336
$583$	−7.98688e6	−0.973208
$584$	4.18392e6	0.507635
$585$	−274428.	−0.0331542
$586$	−3.43234e6	−0.412901
$587$	−1.06799e7	−1.27930	−0.639649	−	0.768667i	$-0.720920\pi$
$587$	−1.06799e7	−1.27930	−0.639649	+	0.768667i	$0.720920\pi$
$588$	0	0
$589$	−465924.	−0.0553384
$590$	179586.	0.0212394
$591$	555642.	0.0654374
$592$	−7.43117e6	−0.871471
$593$	−1.46997e7	−1.71661	−0.858304	−	0.513141i	$-0.828482\pi$
$593$	−1.46997e7	−1.71661	−0.858304	+	0.513141i	$0.828482\pi$
$594$	392202.	0.0456083
$595$	0	0
$596$	2.71975e6	0.313627
$597$	−3.33817e6	−0.383330
$598$	−2.01062e6	−0.229921
$599$	8.49163e6	0.966994	0.483497	−	0.875346i	$-0.339366\pi$
$599$	8.49163e6	0.966994	0.483497	+	0.875346i	$0.339366\pi$
$600$	3.24432e6	0.367913
$601$	8.62947e6	0.974536	0.487268	−	0.873253i	$-0.337994\pi$
$601$	8.62947e6	0.974536	0.487268	+	0.873253i	$0.337994\pi$
$602$	0	0
$603$	−2.59832e6	−0.291004
$604$	813316.	0.0907125
$605$	−975590.	−0.108362
$606$	−1.93626e6	−0.214182
$607$	1.05807e7	1.16559	0.582793	−	0.812621i	$-0.301960\pi$
$607$	1.05807e7	1.16559	0.582793	+	0.812621i	$0.301960\pi$
$608$	844928.	0.0926959
$609$	0	0
$610$	−333652.	−0.0363052
$611$	−6.49942e6	−0.704322
$612$	−4.30013e6	−0.464091
$613$	3.84784e6	0.413586	0.206793	−	0.978385i	$-0.433697\pi$
$613$	3.84784e6	0.413586	0.206793	+	0.978385i	$0.433697\pi$
$614$	−3.61794e6	−0.387293
$615$	1.66874e6	0.177911
$616$	0	0
$617$	−1.51001e7	−1.59686	−0.798428	−	0.602090i	$-0.794335\pi$
$617$	−1.51001e7	−1.59686	−0.798428	+	0.602090i	$0.794335\pi$
$618$	160848.	0.0169412
$619$	9.93102e6	1.04176	0.520879	−	0.853630i	$-0.325604\pi$
$619$	9.93102e6	1.04176	0.520879	+	0.853630i	$0.325604\pi$
$620$	−875028.	−0.0914203
$621$	2.37946e6	0.247599
$622$	−3.04291e6	−0.315365
$623$	0	0
$624$	1.81843e6	0.186954
$625$	8.64589e6	0.885339
$626$	2.69680e6	0.275051
$627$	397044.	0.0403339
$628$	1.61420e7	1.63327
$629$	−2.14779e7	−2.16454
$630$	0	0
$631$	−9.25224e6	−0.925068	−0.462534	−	0.886602i	$-0.653060\pi$
$631$	−9.25224e6	−0.925068	−0.462534	+	0.886602i	$0.653060\pi$
$632$	1.62348e6	0.161679
$633$	−1.95705e6	−0.194130
$634$	99390.0	0.00982018
$635$	−3.34802e6	−0.329498
$636$	−7.48213e6	−0.733470
$637$	0	0
$638$	−1.30035e6	−0.126476
$639$	−3.10019e6	−0.300356
$640$	2.04864e6	0.197704
$641$	5.00428e6	0.481057	0.240529	−	0.970642i	$-0.422679\pi$
$641$	5.00428e6	0.481057	0.240529	+	0.970642i	$0.422679\pi$
$642$	3.48601e6	0.333803
$643$	1.26137e7	1.20314	0.601569	−	0.798821i	$-0.294543\pi$
$643$	1.26137e7	1.20314	0.601569	+	0.798821i	$0.294543\pi$
$644$	0	0
$645$	781506.	0.0739662
$646$	621888.	0.0586315
$647$	−1.25383e7	−1.17755	−0.588774	−	0.808298i	$-0.700389\pi$
$647$	−1.25383e7	−1.17755	−0.588774	+	0.808298i	$0.700389\pi$
$648$	787320.	0.0736570
$649$	−2.19585e6	−0.204640
$650$	−1.85046e6	−0.171790
$651$	0	0
$652$	7.42650e6	0.684171
$653$	−8.66066e6	−0.794819	−0.397409	−	0.917641i	$-0.630091\pi$
$653$	−8.66066e6	−0.794819	−0.397409	+	0.917641i	$0.630091\pi$
$654$	3.69198e6	0.337532
$655$	−146333.	−0.0133272
$656$	−1.10575e7	−1.00323
$657$	2.82415e6	0.255255
$658$	0	0
$659$	7.94177e6	0.712367	0.356183	−	0.934416i	$-0.384078\pi$
$659$	7.94177e6	0.712367	0.356183	+	0.934416i	$0.384078\pi$
$660$	745668.	0.0666324
$661$	−2.11416e6	−0.188206	−0.0941032	−	0.995562i	$-0.529998\pi$
$661$	−2.11416e6	−0.188206	−0.0941032	+	0.995562i	$0.529998\pi$
$662$	−3.17567e6	−0.281638
$663$	5.25571e6	0.464352
$664$	−8.17236e6	−0.719329
$665$	0	0
$666$	1.83514e6	0.160318
$667$	−7.88909e6	−0.686613
$668$	1.01861e7	0.883217
$669$	−5.30794e6	−0.458522
$670$	705716.	0.0607355
$671$	4.07965e6	0.349798
$672$	0	0
$673$	−442307.	−0.0376432	−0.0188216	−	0.999823i	$-0.505991\pi$
$673$	−442307.	−0.0376432	−0.0188216	+	0.999823i	$0.505991\pi$
$674$	−429650.	−0.0364305
$675$	2.18992e6	0.184998
$676$	7.74001e6	0.651441
$677$	1.07561e7	0.901949	0.450975	−	0.892537i	$-0.351077\pi$
$677$	1.07561e7	0.901949	0.450975	+	0.892537i	$0.351077\pi$
$678$	839952.	0.0701746
$679$	0	0
$680$	2.50272e6	0.207558
$681$	3.48340e6	0.287830
$682$	−1.52846e6	−0.125832
$683$	−1.14886e7	−0.942356	−0.471178	−	0.882038i	$-0.656171\pi$
$683$	−1.14886e7	−0.942356	−0.471178	+	0.882038i	$0.656171\pi$
$684$	371952.	0.0303981
$685$	−4.38088e6	−0.356726
$686$	0	0
$687$	2.09459e6	0.169319
$688$	−5.17846e6	−0.417090
$689$	9.14483e6	0.733884
$690$	−646272.	−0.0516764
$691$	1.01388e7	0.807779	0.403890	−	0.914808i	$-0.367658\pi$
$691$	1.01388e7	0.807779	0.403890	+	0.914808i	$0.367658\pi$
$692$	4.61569e6	0.366413
$693$	0	0
$694$	−5.17719e6	−0.408033
$695$	2.53315e6	0.198929
$696$	−2.61036e6	−0.204257
$697$	−3.19590e7	−2.49179
$698$	49756.0	0.00386551
$699$	−378864.	−0.0293285
$700$	0	0
$701$	−1.96839e7	−1.51292	−0.756459	−	0.654041i	$-0.773073\pi$
$701$	−1.96839e7	−1.51292	−0.756459	+	0.654041i	$0.773073\pi$
$702$	−449064.	−0.0343926
$703$	1.85779e6	0.141778
$704$	−2.87507e6	−0.218634
$705$	−2.08910e6	−0.158302
$706$	3.47202e6	0.262162
$707$	0	0
$708$	−2.05708e6	−0.154229
$709$	−2.01717e7	−1.50705	−0.753524	−	0.657420i	$-0.771648\pi$
$709$	−2.01717e7	−1.50705	−0.753524	+	0.657420i	$0.771648\pi$
$710$	842028.	0.0626875
$711$	1.09585e6	0.0812974
$712$	−1.37906e7	−1.01949
$713$	−9.27302e6	−0.683121
$714$	0	0
$715$	−911372.	−0.0666700
$716$	−857584.	−0.0625164
$717$	2.82074e6	0.204911
$718$	−1.72485e6	−0.124865
$719$	4.15735e6	0.299912	0.149956	−	0.988693i	$-0.452087\pi$
$719$	4.15735e6	0.299912	0.149956	+	0.988693i	$0.452087\pi$
$720$	584496.	0.0420194
$721$	0	0
$722$	4.89841e6	0.349713
$723$	−7.72026e6	−0.549271
$724$	1.82390e7	1.29317
$725$	−7.26067e6	−0.513017
$726$	−1.59642e6	−0.112410
$727$	−1.54433e7	−1.08369	−0.541845	−	0.840479i	$-0.682274\pi$
$727$	−1.54433e7	−1.08369	−0.541845	+	0.840479i	$0.682274\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	−767052.	−0.0532743
$731$	−1.49670e7	−1.03596
$732$	3.82183e6	0.263629
$733$	−6.20414e6	−0.426502	−0.213251	−	0.976997i	$-0.568405\pi$
$733$	−6.20414e6	−0.426502	−0.213251	+	0.976997i	$0.568405\pi$
$734$	−6.23084e6	−0.426880
$735$	0	0
$736$	1.68161e7	1.14428
$737$	−8.62898e6	−0.585182
$738$	2.73067e6	0.184556
$739$	2.18984e7	1.47503	0.737517	−	0.675328i	$-0.235998\pi$
$739$	2.18984e7	1.47503	0.737517	+	0.675328i	$0.235998\pi$
$740$	3.48902e6	0.234220
$741$	−454608.	−0.0304153
$742$	0	0
$743$	−2.75483e6	−0.183073	−0.0915363	−	0.995802i	$-0.529178\pi$
$743$	−2.75483e6	−0.183073	−0.0915363	+	0.995802i	$0.529178\pi$
$744$	−3.06828e6	−0.203218
$745$	−1.06847e6	−0.0705299
$746$	3.59389e6	0.236438
$747$	−5.51634e6	−0.361701
$748$	−1.42807e7	−0.933243
$749$	0	0
$750$	−1.21354e6	−0.0787774
$751$	1.29121e7	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$751$	1.29121e7	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$752$	1.38429e7	0.892653
$753$	−4.09065e6	−0.262909
$754$	1.48887e6	0.0953738
$755$	−319517.	−0.0203998
$756$	0	0
$757$	−2.64315e7	−1.67642	−0.838209	−	0.545349i	$-0.816397\pi$
$757$	−2.64315e7	−1.67642	−0.838209	+	0.545349i	$0.816397\pi$
$758$	−6.90931e6	−0.436779
$759$	7.90214e6	0.497898
$760$	−216480.	−0.0135951
$761$	1.22214e7	0.764996	0.382498	−	0.923956i	$-0.375064\pi$
$761$	1.22214e7	0.764996	0.382498	+	0.923956i	$0.375064\pi$
$762$	−5.47857e6	−0.341806
$763$	0	0
$764$	2.12061e7	1.31440
$765$	1.68934e6	0.104367
$766$	−4.31008e6	−0.265407
$767$	2.51420e6	0.154316
$768$	274176.	0.0167736
$769$	6.10654e6	0.372374	0.186187	−	0.982514i	$-0.440387\pi$
$769$	6.10654e6	0.372374	0.186187	+	0.982514i	$0.440387\pi$
$770$	0	0
$771$	−7.90364e6	−0.478841
$772$	4.48949e6	0.271115
$773$	−3.02220e6	−0.181918	−0.0909588	−	0.995855i	$-0.528993\pi$
$773$	−3.02220e6	−0.181918	−0.0909588	+	0.995855i	$0.528993\pi$
$774$	1.27883e6	0.0767290
$775$	−8.53436e6	−0.510407
$776$	1.85951e7	1.10852
$777$	0	0
$778$	925548.	0.0548214
$779$	2.76438e6	0.163213
$780$	−853776.	−0.0502467
$781$	−1.02957e7	−0.603988
$782$	1.23771e7	0.723772
$783$	−1.76199e6	−0.102707
$784$	0	0
$785$	−6.34150e6	−0.367297
$786$	−239454.	−0.0138250
$787$	−2.08285e7	−1.19873	−0.599365	−	0.800476i	$-0.704580\pi$
$787$	−2.08285e7	−1.19873	−0.599365	+	0.800476i	$0.704580\pi$
$788$	1.72866e6	0.0991734
$789$	−1.76484e7	−1.00928
$790$	−297638.	−0.0169676
$791$	0	0
$792$	2.61468e6	0.148117
$793$	−4.67113e6	−0.263778
$794$	8.13242e6	0.457793
$795$	2.93941e6	0.164946
$796$	−1.03854e7	−0.580954
$797$	2.32328e7	1.29556	0.647778	−	0.761829i	$-0.275698\pi$
$797$	2.32328e7	1.29556	0.647778	+	0.761829i	$0.275698\pi$
$798$	0	0
$799$	4.00094e7	2.21715
$800$	1.54766e7	0.854970
$801$	−9.30868e6	−0.512633
$802$	1.01373e7	0.556525
$803$	9.37895e6	0.513293
$804$	−8.08366e6	−0.441030
$805$	0	0
$806$	1.75006e6	0.0948887
$807$	9.48416e6	0.512643
$808$	−1.29084e7	−0.695575
$809$	1.08668e7	0.583753	0.291876	−	0.956456i	$-0.405720\pi$
$809$	1.08668e7	0.583753	0.291876	+	0.956456i	$0.405720\pi$
$810$	−144342.	−0.00773001
$811$	−2.22632e7	−1.18860	−0.594299	−	0.804244i	$-0.702570\pi$
$811$	−2.22632e7	−1.18860	−0.594299	+	0.804244i	$0.702570\pi$
$812$	0	0
$813$	945531.	0.0501706
$814$	6.09446e6	0.322385
$815$	−2.91755e6	−0.153860
$816$	−1.11940e7	−0.588517
$817$	1.29462e6	0.0678556
$818$	−5.75467e6	−0.300703
$819$	0	0
$820$	5.19165e6	0.269631
$821$	−1.23881e7	−0.641426	−0.320713	−	0.947176i	$-0.603922\pi$
$821$	−1.23881e7	−0.641426	−0.320713	+	0.947176i	$0.603922\pi$
$822$	−7.16872e6	−0.370051
$823$	1.69481e6	0.0872210	0.0436105	−	0.999049i	$-0.486114\pi$
$823$	1.69481e6	0.0872210	0.0436105	+	0.999049i	$0.486114\pi$
$824$	1.07232e6	0.0550182
$825$	7.27268e6	0.372014
$826$	0	0
$827$	−378495.	−0.0192440	−0.00962202	−	0.999954i	$-0.503063\pi$
$827$	−378495.	−0.0192440	−0.00962202	+	0.999954i	$0.503063\pi$
$828$	7.40275e6	0.375247
$829$	−1.04287e7	−0.527043	−0.263521	−	0.964654i	$-0.584884\pi$
$829$	−1.04287e7	−0.527043	−0.263521	+	0.964654i	$0.584884\pi$
$830$	1.49827e6	0.0754907
$831$	3.84833e6	0.193317
$832$	3.29190e6	0.164869
$833$	0	0
$834$	4.14515e6	0.206360
$835$	−4.00169e6	−0.198622
$836$	1.23525e6	0.0611278
$837$	−2.07109e6	−0.102185
$838$	6.82685e6	0.335823
$839$	3.04082e7	1.49137	0.745686	−	0.666297i	$-0.232122\pi$
$839$	3.04082e7	1.49137	0.745686	+	0.666297i	$0.232122\pi$
$840$	0	0
$841$	−1.46693e7	−0.715185
$842$	2.61475e6	0.127101
$843$	−5.74990e6	−0.278671
$844$	−6.08860e6	−0.294213
$845$	−3.04072e6	−0.146499
$846$	−3.41852e6	−0.164215
$847$	0	0
$848$	−1.94773e7	−0.930120
$849$	2.20628e7	1.05049
$850$	1.13912e7	0.540780
$851$	3.69746e7	1.75017
$852$	−9.64505e6	−0.455203
$853$	2.80315e7	1.31909	0.659544	−	0.751666i	$-0.270750\pi$
$853$	2.80315e7	1.31909	0.659544	+	0.751666i	$0.270750\pi$
$854$	0	0
$855$	−146124.	−0.00683607
$856$	2.32400e7	1.08406
$857$	1.88030e7	0.874529	0.437264	−	0.899333i	$-0.355947\pi$
$857$	1.88030e7	0.874529	0.437264	+	0.899333i	$0.355947\pi$
$858$	−1.49134e6	−0.0691604
$859$	7.86323e6	0.363595	0.181798	−	0.983336i	$-0.441808\pi$
$859$	7.86323e6	0.363595	0.181798	+	0.983336i	$0.441808\pi$
$860$	2.43135e6	0.112099
$861$	0	0
$862$	−3.87094e6	−0.177438
$863$	−1.12858e7	−0.515827	−0.257913	−	0.966168i	$-0.583035\pi$
$863$	−1.12858e7	−0.515827	−0.257913	+	0.966168i	$0.583035\pi$
$864$	3.75581e6	0.171167
$865$	−1.81331e6	−0.0824007
$866$	−1.03334e6	−0.0468218
$867$	−1.95746e7	−0.884394
$868$	0	0
$869$	3.63930e6	0.163481
$870$	478566.	0.0214360
$871$	9.88002e6	0.441278
$872$	2.46132e7	1.09617
$873$	1.25517e7	0.557399
$874$	−1.07059e6	−0.0474073
$875$	0	0
$876$	8.78623e6	0.386850
$877$	1.34150e7	0.588968	0.294484	−	0.955656i	$-0.404852\pi$
$877$	1.34150e7	0.588968	0.294484	+	0.955656i	$0.404852\pi$
$878$	−5.83060e6	−0.255257
$879$	−1.54455e7	−0.674265
$880$	1.94110e6	0.0844972
$881$	3.18547e7	1.38272	0.691359	−	0.722511i	$-0.257012\pi$
$881$	3.18547e7	1.38272	0.691359	+	0.722511i	$0.257012\pi$
$882$	0	0
$883$	−3.05922e7	−1.32041	−0.660205	−	0.751086i	$-0.729530\pi$
$883$	−3.05922e7	−1.32041	−0.660205	+	0.751086i	$0.729530\pi$
$884$	1.63511e7	0.703747
$885$	808137.	0.0346838
$886$	3.56758e6	0.152683
$887$	−4.63772e6	−0.197923	−0.0989613	−	0.995091i	$-0.531552\pi$
$887$	−4.63772e6	−0.197923	−0.0989613	+	0.995091i	$0.531552\pi$
$888$	1.22342e7	0.520648
$889$	0	0
$890$	2.52828e6	0.106992
$891$	1.76491e6	0.0744780
$892$	−1.65136e7	−0.694911
$893$	−3.46073e6	−0.145224
$894$	−1.74841e6	−0.0731644
$895$	336908.	0.0140590
$896$	0	0
$897$	−9.04781e6	−0.375459
$898$	−8.00316e6	−0.331185
$899$	6.86670e6	0.283367
$900$	6.81307e6	0.280373
$901$	−5.62941e7	−2.31021
$902$	9.06853e6	0.371125
$903$	0	0
$904$	5.59968e6	0.227899
$905$	−7.16531e6	−0.290813
$906$	−522846.	−0.0211618
$907$	604376.	0.0243943	0.0121972	−	0.999926i	$-0.496117\pi$
$907$	604376.	0.0243943	0.0121972	+	0.999926i	$0.496117\pi$
$908$	1.08373e7	0.436219
$909$	−8.71317e6	−0.349757
$910$	0	0
$911$	−2.44059e7	−0.974315	−0.487157	−	0.873314i	$-0.661966\pi$
$911$	−2.44059e7	−0.974315	−0.487157	+	0.873314i	$0.661966\pi$
$912$	968256.	0.0385481
$913$	−1.83197e7	−0.727347
$914$	2.33533e6	0.0924661
$915$	−1.50143e6	−0.0592862
$916$	6.51650e6	0.256611
$917$	0	0
$918$	2.76437e6	0.108265
$919$	3.67095e7	1.43380	0.716902	−	0.697174i	$-0.245560\pi$
$919$	3.67095e7	1.43380	0.716902	+	0.697174i	$0.245560\pi$
$920$	−4.30848e6	−0.167824
$921$	−1.62807e7	−0.632447
$922$	−7.22716e6	−0.279989
$923$	1.17884e7	0.455460
$924$	0	0
$925$	3.40293e7	1.30767
$926$	3.60222e6	0.138052
$927$	723816.	0.0276649
$928$	−1.24524e7	−0.474660
$929$	−2.29089e7	−0.870892	−0.435446	−	0.900215i	$-0.643409\pi$
$929$	−2.29089e7	−0.870892	−0.435446	+	0.900215i	$0.643409\pi$
$930$	562518.	0.0213270
$931$	0	0
$932$	−1.17869e6	−0.0444487
$933$	−1.36931e7	−0.514989
$934$	4.73950e6	0.177773
$935$	5.61026e6	0.209872
$936$	−2.99376e6	−0.111693
$937$	−5.99611e6	−0.223111	−0.111555	−	0.993758i	$-0.535583\pi$
$937$	−5.99611e6	−0.223111	−0.111555	+	0.993758i	$0.535583\pi$
$938$	0	0
$939$	1.21356e7	0.449156
$940$	−6.49942e6	−0.239914
$941$	−1.16516e7	−0.428954	−0.214477	−	0.976729i	$-0.568805\pi$
$941$	−1.16516e7	−0.428954	−0.214477	+	0.976729i	$0.568805\pi$
$942$	−1.03770e7	−0.381017
$943$	5.50180e7	2.01477
$944$	−5.35493e6	−0.195580
$945$	0	0
$946$	4.24697e6	0.154295
$947$	−1.10926e6	−0.0401939	−0.0200969	−	0.999798i	$-0.506397\pi$
$947$	−1.10926e6	−0.0401939	−0.0200969	+	0.999798i	$0.506397\pi$
$948$	3.40931e6	0.123210
$949$	−1.07387e7	−0.387068
$950$	−985312.	−0.0354213
$951$	447255.	0.0160363
$952$	0	0
$953$	1.05743e7	0.377155	0.188578	−	0.982058i	$-0.439612\pi$
$953$	1.05743e7	0.377155	0.188578	+	0.982058i	$0.439612\pi$
$954$	4.80994e6	0.171107
$955$	−8.33096e6	−0.295588
$956$	8.77565e6	0.310552
$957$	−5.85156e6	−0.206534
$958$	−1.03629e6	−0.0364811
$959$	0	0
$960$	1.05811e6	0.0370556
$961$	−2.05579e7	−0.718075
$962$	−6.97805e6	−0.243106
$963$	1.56870e7	0.545098
$964$	−2.40186e7	−0.832444
$965$	−1.76373e6	−0.0609696
$966$	0	0
$967$	6.32666e6	0.217575	0.108787	−	0.994065i	$-0.465303\pi$
$967$	6.32666e6	0.217575	0.108787	+	0.994065i	$0.465303\pi$
$968$	−1.06428e7	−0.365063
$969$	2.79850e6	0.0957448
$970$	−3.40910e6	−0.116335
$971$	3.92395e7	1.33560	0.667798	−	0.744343i	$-0.267237\pi$
$971$	3.92395e7	1.33560	0.667798	+	0.744343i	$0.267237\pi$
$972$	1.65337e6	0.0561313
$973$	0	0
$974$	−5.65225e6	−0.190908
$975$	−8.32709e6	−0.280531
$976$	9.94890e6	0.334311
$977$	−1.55074e6	−0.0519760	−0.0259880	−	0.999662i	$-0.508273\pi$
$977$	−1.55074e6	−0.0519760	−0.0259880	+	0.999662i	$0.508273\pi$
$978$	−4.77418e6	−0.159607
$979$	−3.09140e7	−1.03086
$980$	0	0
$981$	1.66139e7	0.551187
$982$	−1.86949e7	−0.618651
$983$	−4.87484e7	−1.60908	−0.804538	−	0.593901i	$-0.797587\pi$
$983$	−4.87484e7	−1.60908	−0.804538	+	0.593901i	$0.797587\pi$
$984$	1.82045e7	0.599364
$985$	−679118.	−0.0223026
$986$	−9.16526e6	−0.300229
$987$	0	0
$988$	−1.41434e6	−0.0460957
$989$	2.57660e7	0.837638
$990$	−479358.	−0.0155443
$991$	−1.92552e6	−0.0622820	−0.0311410	−	0.999515i	$-0.509914\pi$
$991$	−1.92552e6	−0.0622820	−0.0311410	+	0.999515i	$0.509914\pi$
$992$	−1.46368e7	−0.472246
$993$	−1.42905e7	−0.459912
$994$	0	0
$995$	4.07999e6	0.130648
$996$	−1.71620e7	−0.548174
$997$	5.42564e7	1.72867	0.864337	−	0.502913i	$-0.167739\pi$
$997$	5.42564e7	1.72867	0.864337	+	0.502913i	$0.167739\pi$
$998$	−1.63437e7	−0.519427
$999$	8.25811e6	0.261798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.6.a.c.1.1		1
3.2	odd	2		441.6.a.g.1.1		1
7.2	even	3		21.6.e.a.4.1	✓	2
7.3	odd	6		147.6.e.g.79.1		2
7.4	even	3		21.6.e.a.16.1	yes	2
7.5	odd	6		147.6.e.g.67.1		2
7.6	odd	2		147.6.a.d.1.1		1
21.2	odd	6		63.6.e.a.46.1		2
21.11	odd	6		63.6.e.a.37.1		2
21.20	even	2		441.6.a.h.1.1		1
28.11	odd	6		336.6.q.b.289.1		2
28.23	odd	6		336.6.q.b.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.a.4.1	✓	2	7.2	even	3
21.6.e.a.16.1	yes	2	7.4	even	3
63.6.e.a.37.1		2	21.11	odd	6
63.6.e.a.46.1		2	21.2	odd	6
147.6.a.c.1.1		1	1.1	even	1	trivial
147.6.a.d.1.1		1	7.6	odd	2
147.6.e.g.67.1		2	7.5	odd	6
147.6.e.g.79.1		2	7.3	odd	6
336.6.q.b.193.1		2	28.23	odd	6
336.6.q.b.289.1		2	28.11	odd	6
441.6.a.g.1.1		1	3.2	odd	2
441.6.a.h.1.1		1	21.20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more