LMFDB - Embedded newform 210.6.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [210,6,Mod(1,210)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(210, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("210.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$210 = 2 \cdot 3 \cdot 5 \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	210.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:	$33.6806021607$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	210.1

$q$ -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} +9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} +36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -100.000 q^{10} -668.000 q^{11} +144.000 q^{12} -366.000 q^{13} -196.000 q^{14} -225.000 q^{15} +256.000 q^{16} -34.0000 q^{17} +324.000 q^{18} -2016.00 q^{19} -400.000 q^{20} -441.000 q^{21} -2672.00 q^{22} +1908.00 q^{23} +576.000 q^{24} +625.000 q^{25} -1464.00 q^{26} +729.000 q^{27} -784.000 q^{28} -5754.00 q^{29} -900.000 q^{30} -52.0000 q^{31} +1024.00 q^{32} -6012.00 q^{33} -136.000 q^{34} +1225.00 q^{35} +1296.00 q^{36} -10594.0 q^{37} -8064.00 q^{38} -3294.00 q^{39} -1600.00 q^{40} -418.000 q^{41} -1764.00 q^{42} +6676.00 q^{43} -10688.0 q^{44} -2025.00 q^{45} +7632.00 q^{46} +1472.00 q^{47} +2304.00 q^{48} +2401.00 q^{49} +2500.00 q^{50} -306.000 q^{51} -5856.00 q^{52} +19834.0 q^{53} +2916.00 q^{54} +16700.0 q^{55} -3136.00 q^{56} -18144.0 q^{57} -23016.0 q^{58} +10492.0 q^{59} -3600.00 q^{60} -38810.0 q^{61} -208.000 q^{62} -3969.00 q^{63} +4096.00 q^{64} +9150.00 q^{65} -24048.0 q^{66} -61428.0 q^{67} -544.000 q^{68} +17172.0 q^{69} +4900.00 q^{70} -7936.00 q^{71} +5184.00 q^{72} +21134.0 q^{73} -42376.0 q^{74} +5625.00 q^{75} -32256.0 q^{76} +32732.0 q^{77} -13176.0 q^{78} +87088.0 q^{79} -6400.00 q^{80} +6561.00 q^{81} -1672.00 q^{82} -103252. q^{83} -7056.00 q^{84} +850.000 q^{85} +26704.0 q^{86} -51786.0 q^{87} -42752.0 q^{88} -49490.0 q^{89} -8100.00 q^{90} +17934.0 q^{91} +30528.0 q^{92} -468.000 q^{93} +5888.00 q^{94} +50400.0 q^{95} +9216.00 q^{96} +125630. q^{97} +9604.00 q^{98} -54108.0 q^{99} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	36.0000	0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−100.000	−0.316228
$11$	−668.000	−1.66454	−0.832271	−	0.554369i	$-0.812960\pi$
$11$	−668.000	−1.66454	−0.832271	+	0.554369i	$0.812960\pi$
$12$	144.000	0.288675
$13$	−366.000	−0.600652	−0.300326	−	0.953837i	$-0.597095\pi$
$13$	−366.000	−0.600652	−0.300326	+	0.953837i	$0.597095\pi$
$14$	−196.000	−0.267261
$15$	−225.000	−0.258199
$16$	256.000	0.250000
$17$	−34.0000	−0.0285336	−0.0142668	−	0.999898i	$-0.504541\pi$
$17$	−34.0000	−0.0285336	−0.0142668	+	0.999898i	$0.504541\pi$
$18$	324.000	0.235702
$19$	−2016.00	−1.28117	−0.640585	−	0.767888i	$-0.721308\pi$
$19$	−2016.00	−1.28117	−0.640585	+	0.767888i	$0.721308\pi$
$20$	−400.000	−0.223607
$21$	−441.000	−0.218218
$22$	−2672.00	−1.17701
$23$	1908.00	0.752071	0.376035	−	0.926605i	$-0.377287\pi$
$23$	1908.00	0.752071	0.376035	+	0.926605i	$0.377287\pi$
$24$	576.000	0.204124
$25$	625.000	0.200000
$26$	−1464.00	−0.424725
$27$	729.000	0.192450
$28$	−784.000	−0.188982
$29$	−5754.00	−1.27050	−0.635250	−	0.772306i	$-0.719103\pi$
$29$	−5754.00	−1.27050	−0.635250	+	0.772306i	$0.719103\pi$
$30$	−900.000	−0.182574
$31$	−52.0000	−0.00971850	−0.00485925	−	0.999988i	$-0.501547\pi$
$31$	−52.0000	−0.00971850	−0.00485925	+	0.999988i	$0.501547\pi$
$32$	1024.00	0.176777
$33$	−6012.00	−0.961024
$34$	−136.000	−0.0201763
$35$	1225.00	0.169031
$36$	1296.00	0.166667
$37$	−10594.0	−1.27220	−0.636100	−	0.771606i	$-0.719454\pi$
$37$	−10594.0	−1.27220	−0.636100	+	0.771606i	$0.719454\pi$
$38$	−8064.00	−0.905924
$39$	−3294.00	−0.346786
$40$	−1600.00	−0.158114
$41$	−418.000	−0.0388344	−0.0194172	−	0.999811i	$-0.506181\pi$
$41$	−418.000	−0.0388344	−0.0194172	+	0.999811i	$0.506181\pi$
$42$	−1764.00	−0.154303
$43$	6676.00	0.550611	0.275306	−	0.961357i	$-0.411221\pi$
$43$	6676.00	0.550611	0.275306	+	0.961357i	$0.411221\pi$
$44$	−10688.0	−0.832271
$45$	−2025.00	−0.149071
$46$	7632.00	0.531794
$47$	1472.00	0.0971993	0.0485997	−	0.998818i	$-0.484524\pi$
$47$	1472.00	0.0971993	0.0485997	+	0.998818i	$0.484524\pi$
$48$	2304.00	0.144338
$49$	2401.00	0.142857
$50$	2500.00	0.141421
$51$	−306.000	−0.0164739
$52$	−5856.00	−0.300326
$53$	19834.0	0.969886	0.484943	−	0.874546i	$-0.338840\pi$
$53$	19834.0	0.969886	0.484943	+	0.874546i	$0.338840\pi$
$54$	2916.00	0.136083
$55$	16700.0	0.744406
$56$	−3136.00	−0.133631
$57$	−18144.0	−0.739683
$58$	−23016.0	−0.898380
$59$	10492.0	0.392399	0.196200	−	0.980564i	$-0.437140\pi$
$59$	10492.0	0.392399	0.196200	+	0.980564i	$0.437140\pi$
$60$	−3600.00	−0.129099
$61$	−38810.0	−1.33542	−0.667712	−	0.744420i	$-0.732726\pi$
$61$	−38810.0	−1.33542	−0.667712	+	0.744420i	$0.732726\pi$
$62$	−208.000	−0.00687202
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	9150.00	0.268620
$66$	−24048.0	−0.679546
$67$	−61428.0	−1.67178	−0.835890	−	0.548896i	$-0.815048\pi$
$67$	−61428.0	−1.67178	−0.835890	+	0.548896i	$0.815048\pi$
$68$	−544.000	−0.0142668
$69$	17172.0	0.434208
$70$	4900.00	0.119523
$71$	−7936.00	−0.186834	−0.0934170	−	0.995627i	$-0.529779\pi$
$71$	−7936.00	−0.186834	−0.0934170	+	0.995627i	$0.529779\pi$
$72$	5184.00	0.117851
$73$	21134.0	0.464167	0.232084	−	0.972696i	$-0.425446\pi$
$73$	21134.0	0.464167	0.232084	+	0.972696i	$0.425446\pi$
$74$	−42376.0	−0.899582
$75$	5625.00	0.115470
$76$	−32256.0	−0.640585
$77$	32732.0	0.629138
$78$	−13176.0	−0.245215
$79$	87088.0	1.56997	0.784984	−	0.619517i	$-0.212671\pi$
$79$	87088.0	1.56997	0.784984	+	0.619517i	$0.212671\pi$
$80$	−6400.00	−0.111803
$81$	6561.00	0.111111
$82$	−1672.00	−0.0274601
$83$	−103252.	−1.64514	−0.822571	−	0.568663i	$-0.807461\pi$
$83$	−103252.	−1.64514	−0.822571	+	0.568663i	$0.807461\pi$
$84$	−7056.00	−0.109109
$85$	850.000	0.0127606
$86$	26704.0	0.389341
$87$	−51786.0	−0.733524
$88$	−42752.0	−0.588504
$89$	−49490.0	−0.662281	−0.331141	−	0.943581i	$-0.607433\pi$
$89$	−49490.0	−0.662281	−0.331141	+	0.943581i	$0.607433\pi$
$90$	−8100.00	−0.105409
$91$	17934.0	0.227025
$92$	30528.0	0.376035
$93$	−468.000	−0.00561098
$94$	5888.00	0.0687303
$95$	50400.0	0.572956
$96$	9216.00	0.102062
$97$	125630.	1.35570	0.677851	−	0.735200i	$-0.262912\pi$
$97$	125630.	1.35570	0.677851	+	0.735200i	$0.262912\pi$
$98$	9604.00	0.101015
$99$	−54108.0	−0.554847
$100$	10000.0	0.100000
$101$	−95550.0	−0.932024	−0.466012	−	0.884778i	$-0.654310\pi$
$101$	−95550.0	−0.932024	−0.466012	+	0.884778i	$0.654310\pi$
$102$	−1224.00	−0.0116488
$103$	−7088.00	−0.0658310	−0.0329155	−	0.999458i	$-0.510479\pi$
$103$	−7088.00	−0.0658310	−0.0329155	+	0.999458i	$0.510479\pi$
$104$	−23424.0	−0.212362
$105$	11025.0	0.0975900
$106$	79336.0	0.685813
$107$	126048.	1.06433	0.532165	−	0.846641i	$-0.321379\pi$
$107$	126048.	1.06433	0.532165	+	0.846641i	$0.321379\pi$
$108$	11664.0	0.0962250
$109$	−151554.	−1.22180	−0.610901	−	0.791707i	$-0.709193\pi$
$109$	−151554.	−1.22180	−0.610901	+	0.791707i	$0.709193\pi$
$110$	66800.0	0.526374
$111$	−95346.0	−0.734505
$112$	−12544.0	−0.0944911
$113$	83174.0	0.612762	0.306381	−	0.951909i	$-0.400882\pi$
$113$	83174.0	0.612762	0.306381	+	0.951909i	$0.400882\pi$
$114$	−72576.0	−0.523035
$115$	−47700.0	−0.336336
$116$	−92064.0	−0.635250
$117$	−29646.0	−0.200217
$118$	41968.0	0.277468
$119$	1666.00	0.0107847
$120$	−14400.0	−0.0912871
$121$	285173.	1.77070
$122$	−155240.	−0.944287
$123$	−3762.00	−0.0224211
$124$	−832.000	−0.00485925
$125$	−15625.0	−0.0894427
$126$	−15876.0	−0.0890871
$127$	38656.0	0.212671	0.106335	−	0.994330i	$-0.466088\pi$
$127$	38656.0	0.212671	0.106335	+	0.994330i	$0.466088\pi$
$128$	16384.0	0.0883883
$129$	60084.0	0.317896
$130$	36600.0	0.189943
$131$	108788.	0.553864	0.276932	−	0.960890i	$-0.410682\pi$
$131$	108788.	0.553864	0.276932	+	0.960890i	$0.410682\pi$
$132$	−96192.0	−0.480512
$133$	98784.0	0.484236
$134$	−245712.	−1.18213
$135$	−18225.0	−0.0860663
$136$	−2176.00	−0.0100882
$137$	384366.	1.74962	0.874810	−	0.484467i	$-0.160986\pi$
$137$	384366.	1.74962	0.874810	+	0.484467i	$0.160986\pi$
$138$	68688.0	0.307032
$139$	256.000	0.00112384	0.000561918	−	1.00000i	$-0.499821\pi$
$139$	256.000	0.00112384	0.000561918		1.00000i	$0.499821\pi$
$140$	19600.0	0.0845154
$141$	13248.0	0.0561180
$142$	−31744.0	−0.132112
$143$	244488.	0.999810
$144$	20736.0	0.0833333
$145$	143850.	0.568185
$146$	84536.0	0.328216
$147$	21609.0	0.0824786
$148$	−169504.	−0.636100
$149$	207150.	0.764398	0.382199	−	0.924080i	$-0.375167\pi$
$149$	207150.	0.764398	0.382199	+	0.924080i	$0.375167\pi$
$150$	22500.0	0.0816497
$151$	−276808.	−0.987953	−0.493976	−	0.869475i	$-0.664457\pi$
$151$	−276808.	−0.987953	−0.493976	+	0.869475i	$0.664457\pi$
$152$	−129024.	−0.452962
$153$	−2754.00	−0.00951120
$154$	130928.	0.444868
$155$	1300.00	0.00434624
$156$	−52704.0	−0.173393
$157$	71602.0	0.231833	0.115917	−	0.993259i	$-0.463019\pi$
$157$	71602.0	0.231833	0.115917	+	0.993259i	$0.463019\pi$
$158$	348352.	1.11013
$159$	178506.	0.559964
$160$	−25600.0	−0.0790569
$161$	−93492.0	−0.284256
$162$	26244.0	0.0785674
$163$	−555196.	−1.63673	−0.818366	−	0.574698i	$-0.805119\pi$
$163$	−555196.	−1.63673	−0.818366	+	0.574698i	$0.805119\pi$
$164$	−6688.00	−0.0194172
$165$	150300.	0.429783
$166$	−413008.	−1.16329
$167$	541152.	1.50151	0.750755	−	0.660581i	$-0.229690\pi$
$167$	541152.	1.50151	0.750755	+	0.660581i	$0.229690\pi$
$168$	−28224.0	−0.0771517
$169$	−237337.	−0.639218
$170$	3400.00	0.00902312
$171$	−163296.	−0.427056
$172$	106816.	0.275306
$173$	−81358.0	−0.206674	−0.103337	−	0.994646i	$-0.532952\pi$
$173$	−81358.0	−0.206674	−0.103337	+	0.994646i	$0.532952\pi$
$174$	−207144.	−0.518680
$175$	−30625.0	−0.0755929
$176$	−171008.	−0.416135
$177$	94428.0	0.226552
$178$	−197960.	−0.468304
$179$	734100.	1.71247	0.856234	−	0.516588i	$-0.172798\pi$
$179$	734100.	1.71247	0.856234	+	0.516588i	$0.172798\pi$
$180$	−32400.0	−0.0745356
$181$	128542.	0.291641	0.145821	−	0.989311i	$-0.453418\pi$
$181$	128542.	0.291641	0.145821	+	0.989311i	$0.453418\pi$
$182$	71736.0	0.160531
$183$	−349290.	−0.771007
$184$	122112.	0.265897
$185$	264850.	0.568945
$186$	−1872.00	−0.00396756
$187$	22712.0	0.0474954
$188$	23552.0	0.0485997
$189$	−35721.0	−0.0727393
$190$	201600.	0.405141
$191$	383712.	0.761065	0.380533	−	0.924767i	$-0.375741\pi$
$191$	383712.	0.761065	0.380533	+	0.924767i	$0.375741\pi$
$192$	36864.0	0.0721688
$193$	168818.	0.326231	0.163116	−	0.986607i	$-0.447846\pi$
$193$	168818.	0.326231	0.163116	+	0.986607i	$0.447846\pi$
$194$	502520.	0.958626
$195$	82350.0	0.155088
$196$	38416.0	0.0714286
$197$	296258.	0.543882	0.271941	−	0.962314i	$-0.412334\pi$
$197$	296258.	0.543882	0.271941	+	0.962314i	$0.412334\pi$
$198$	−216432.	−0.392336
$199$	199676.	0.357432	0.178716	−	0.983901i	$-0.442806\pi$
$199$	199676.	0.357432	0.178716	+	0.983901i	$0.442806\pi$
$200$	40000.0	0.0707107
$201$	−552852.	−0.965203
$202$	−382200.	−0.659041
$203$	281946.	0.480204
$204$	−4896.00	−0.00823694
$205$	10450.0	0.0173673
$206$	−28352.0	−0.0465496
$207$	154548.	0.250690
$208$	−93696.0	−0.150163
$209$	1.34669e6	2.13256
$210$	44100.0	0.0690066
$211$	596620.	0.922554	0.461277	−	0.887256i	$-0.347391\pi$
$211$	596620.	0.922554	0.461277	+	0.887256i	$0.347391\pi$
$212$	317344.	0.484943
$213$	−71424.0	−0.107869
$214$	504192.	0.752595
$215$	−166900.	−0.246241
$216$	46656.0	0.0680414
$217$	2548.00	0.00367325
$218$	−606216.	−0.863945
$219$	190206.	0.267987
$220$	267200.	0.372203
$221$	12444.0	0.0171388
$222$	−381384.	−0.519374
$223$	−647432.	−0.871830	−0.435915	−	0.899988i	$-0.643575\pi$
$223$	−647432.	−0.871830	−0.435915	+	0.899988i	$0.643575\pi$
$224$	−50176.0	−0.0668153
$225$	50625.0	0.0666667
$226$	332696.	0.433288
$227$	−809932.	−1.04324	−0.521620	−	0.853178i	$-0.674672\pi$
$227$	−809932.	−1.04324	−0.521620	+	0.853178i	$0.674672\pi$
$228$	−290304.	−0.369842
$229$	−1.06347e6	−1.34009	−0.670046	−	0.742319i	$-0.733726\pi$
$229$	−1.06347e6	−1.34009	−0.670046	+	0.742319i	$0.733726\pi$
$230$	−190800.	−0.237826
$231$	294588.	0.363233
$232$	−368256.	−0.449190
$233$	−1.15017e6	−1.38794	−0.693972	−	0.720002i	$-0.744141\pi$
$233$	−1.15017e6	−1.38794	−0.693972	+	0.720002i	$0.744141\pi$
$234$	−118584.	−0.141575
$235$	−36800.0	−0.0434689
$236$	167872.	0.196200
$237$	783792.	0.906421
$238$	6664.00	0.00762593
$239$	1.05722e6	1.19721	0.598603	−	0.801046i	$-0.295723\pi$
$239$	1.05722e6	1.19721	0.598603	+	0.801046i	$0.295723\pi$
$240$	−57600.0	−0.0645497
$241$	−1.68597e6	−1.86985	−0.934924	−	0.354849i	$-0.884532\pi$
$241$	−1.68597e6	−1.86985	−0.934924	+	0.354849i	$0.884532\pi$
$242$	1.14069e6	1.25207
$243$	59049.0	0.0641500
$244$	−620960.	−0.667712
$245$	−60025.0	−0.0638877
$246$	−15048.0	−0.0158541
$247$	737856.	0.769537
$248$	−3328.00	−0.00343601
$249$	−929268.	−0.949823
$250$	−62500.0	−0.0632456
$251$	−1.16730e6	−1.16949	−0.584747	−	0.811216i	$-0.698806\pi$
$251$	−1.16730e6	−1.16949	−0.584747	+	0.811216i	$0.698806\pi$
$252$	−63504.0	−0.0629941
$253$	−1.27454e6	−1.25185
$254$	154624.	0.150381
$255$	7650.00	0.00736734
$256$	65536.0	0.0625000
$257$	−277570.	−0.262144	−0.131072	−	0.991373i	$-0.541842\pi$
$257$	−277570.	−0.262144	−0.131072	+	0.991373i	$0.541842\pi$
$258$	240336.	0.224786
$259$	519106.	0.480847
$260$	146400.	0.134310
$261$	−466074.	−0.423500
$262$	435152.	0.391641
$263$	124124.	0.110654	0.0553269	−	0.998468i	$-0.482380\pi$
$263$	124124.	0.110654	0.0553269	+	0.998468i	$0.482380\pi$
$264$	−384768.	−0.339773
$265$	−495850.	−0.433746
$266$	395136.	0.342407
$267$	−445410.	−0.382368
$268$	−982848.	−0.835890
$269$	138018.	0.116293	0.0581467	−	0.998308i	$-0.481481\pi$
$269$	138018.	0.116293	0.0581467	+	0.998308i	$0.481481\pi$
$270$	−72900.0	−0.0608581
$271$	−1.32678e6	−1.09743	−0.548714	−	0.836010i	$-0.684882\pi$
$271$	−1.32678e6	−1.09743	−0.548714	+	0.836010i	$0.684882\pi$
$272$	−8704.00	−0.00713340
$273$	161406.	0.131073
$274$	1.53746e6	1.23717
$275$	−417500.	−0.332908
$276$	274752.	0.217104
$277$	−1.04114e6	−0.815284	−0.407642	−	0.913142i	$-0.633649\pi$
$277$	−1.04114e6	−0.815284	−0.407642	+	0.913142i	$0.633649\pi$
$278$	1024.00	0.000794672 0
$279$	−4212.00	−0.00323950
$280$	78400.0	0.0597614
$281$	280490.	0.211910	0.105955	−	0.994371i	$-0.466210\pi$
$281$	280490.	0.211910	0.105955	+	0.994371i	$0.466210\pi$
$282$	52992.0	0.0396815
$283$	−1.95382e6	−1.45017	−0.725084	−	0.688660i	$-0.758199\pi$
$283$	−1.95382e6	−1.45017	−0.725084	+	0.688660i	$0.758199\pi$
$284$	−126976.	−0.0934170
$285$	453600.	0.330796
$286$	977952.	0.706972
$287$	20482.0	0.0146780
$288$	82944.0	0.0589256
$289$	−1.41870e6	−0.999186
$290$	575400.	0.401768
$291$	1.13067e6	0.782715
$292$	338144.	0.232084
$293$	−775910.	−0.528010	−0.264005	−	0.964521i	$-0.585044\pi$
$293$	−775910.	−0.528010	−0.264005	+	0.964521i	$0.585044\pi$
$294$	86436.0	0.0583212
$295$	−262300.	−0.175486
$296$	−678016.	−0.449791
$297$	−486972.	−0.320341
$298$	828600.	0.540511
$299$	−698328.	−0.451733
$300$	90000.0	0.0577350
$301$	−327124.	−0.208112
$302$	−1.10723e6	−0.698588
$303$	−859950.	−0.538105
$304$	−516096.	−0.320292
$305$	970250.	0.597220
$306$	−11016.0	−0.00672543
$307$	−1.69219e6	−1.02471	−0.512357	−	0.858773i	$-0.671228\pi$
$307$	−1.69219e6	−1.02471	−0.512357	+	0.858773i	$0.671228\pi$
$308$	523712.	0.314569
$309$	−63792.0	−0.0380076
$310$	5200.00	0.00307326
$311$	−904416.	−0.530234	−0.265117	−	0.964216i	$-0.585411\pi$
$311$	−904416.	−0.530234	−0.265117	+	0.964216i	$0.585411\pi$
$312$	−210816.	−0.122608
$313$	−3.09618e6	−1.78634	−0.893172	−	0.449715i	$-0.851525\pi$
$313$	−3.09618e6	−1.78634	−0.893172	+	0.449715i	$0.851525\pi$
$314$	286408.	0.163931
$315$	99225.0	0.0563436
$316$	1.39341e6	0.784984
$317$	1.40432e6	0.784908	0.392454	−	0.919772i	$-0.371626\pi$
$317$	1.40432e6	0.784908	0.392454	+	0.919772i	$0.371626\pi$
$318$	714024.	0.395954
$319$	3.84367e6	2.11480
$320$	−102400.	−0.0559017
$321$	1.13443e6	0.614492
$322$	−373968.	−0.200999
$323$	68544.0	0.0365564
$324$	104976.	0.0555556
$325$	−228750.	−0.120130
$326$	−2.22078e6	−1.15734
$327$	−1.36399e6	−0.705408
$328$	−26752.0	−0.0137300
$329$	−72128.0	−0.0367379
$330$	601200.	0.303902
$331$	794932.	0.398804	0.199402	−	0.979918i	$-0.436100\pi$
$331$	794932.	0.398804	0.199402	+	0.979918i	$0.436100\pi$
$332$	−1.65203e6	−0.822571
$333$	−858114.	−0.424067
$334$	2.16461e6	1.06173
$335$	1.53570e6	0.747643
$336$	−112896.	−0.0545545
$337$	−1.78206e6	−0.854768	−0.427384	−	0.904070i	$-0.640565\pi$
$337$	−1.78206e6	−0.854768	−0.427384	+	0.904070i	$0.640565\pi$
$338$	−949348.	−0.451995
$339$	748566.	0.353778
$340$	13600.0	0.00638031
$341$	34736.0	0.0161768
$342$	−653184.	−0.301975
$343$	−117649.	−0.0539949
$344$	427264.	0.194671
$345$	−429300.	−0.194184
$346$	−325432.	−0.146140
$347$	−1.54578e6	−0.689164	−0.344582	−	0.938756i	$-0.611979\pi$
$347$	−1.54578e6	−0.689164	−0.344582	+	0.938756i	$0.611979\pi$
$348$	−828576.	−0.366762
$349$	1.90197e6	0.835874	0.417937	−	0.908476i	$-0.362753\pi$
$349$	1.90197e6	0.835874	0.417937	+	0.908476i	$0.362753\pi$
$350$	−122500.	−0.0534522
$351$	−266814.	−0.115595
$352$	−684032.	−0.294252
$353$	−1.74813e6	−0.746684	−0.373342	−	0.927694i	$-0.621788\pi$
$353$	−1.74813e6	−0.746684	−0.373342	+	0.927694i	$0.621788\pi$
$354$	377712.	0.160196
$355$	198400.	0.0835547
$356$	−791840.	−0.331141
$357$	14994.0	0.00622654
$358$	2.93640e6	1.21090
$359$	462536.	0.189413	0.0947064	−	0.995505i	$-0.469809\pi$
$359$	462536.	0.189413	0.0947064	+	0.995505i	$0.469809\pi$
$360$	−129600.	−0.0527046
$361$	1.58816e6	0.641395
$362$	514168.	0.206221
$363$	2.56656e6	1.02231
$364$	286944.	0.113513
$365$	−528350.	−0.207582
$366$	−1.39716e6	−0.545184
$367$	−383768.	−0.148732	−0.0743659	−	0.997231i	$-0.523693\pi$
$367$	−383768.	−0.148732	−0.0743659	+	0.997231i	$0.523693\pi$
$368$	488448.	0.188018
$369$	−33858.0	−0.0129448
$370$	1.05940e6	0.402305
$371$	−971866.	−0.366582
$372$	−7488.00	−0.00280549
$373$	−3.40835e6	−1.26844	−0.634222	−	0.773151i	$-0.718680\pi$
$373$	−3.40835e6	−1.26844	−0.634222	+	0.773151i	$0.718680\pi$
$374$	90848.0	0.0335843
$375$	−140625.	−0.0516398
$376$	94208.0	0.0343651
$377$	2.10596e6	0.763128
$378$	−142884.	−0.0514344
$379$	4.02918e6	1.44085	0.720425	−	0.693533i	$-0.243947\pi$
$379$	4.02918e6	1.44085	0.720425	+	0.693533i	$0.243947\pi$
$380$	806400.	0.286478
$381$	347904.	0.122785
$382$	1.53485e6	0.538154
$383$	−4.25208e6	−1.48117	−0.740584	−	0.671963i	$-0.765451\pi$
$383$	−4.25208e6	−1.48117	−0.740584	+	0.671963i	$0.765451\pi$
$384$	147456.	0.0510310
$385$	−818300.	−0.281359
$386$	675272.	0.230680
$387$	540756.	0.183537
$388$	2.01008e6	0.677851
$389$	4.60602e6	1.54331	0.771653	−	0.636044i	$-0.219430\pi$
$389$	4.60602e6	1.54331	0.771653	+	0.636044i	$0.219430\pi$
$390$	329400.	0.109663
$391$	−64872.0	−0.0214593
$392$	153664.	0.0505076
$393$	979092.	0.319773
$394$	1.18503e6	0.384583
$395$	−2.17720e6	−0.702111
$396$	−865728.	−0.277424
$397$	−4.02580e6	−1.28196	−0.640982	−	0.767556i	$-0.721473\pi$
$397$	−4.02580e6	−1.28196	−0.640982	+	0.767556i	$0.721473\pi$
$398$	798704.	0.252742
$399$	889056.	0.279574
$400$	160000.	0.0500000
$401$	−3.37261e6	−1.04738	−0.523692	−	0.851908i	$-0.675446\pi$
$401$	−3.37261e6	−1.04738	−0.523692	+	0.851908i	$0.675446\pi$
$402$	−2.21141e6	−0.682502
$403$	19032.0	0.00583743
$404$	−1.52880e6	−0.466012
$405$	−164025.	−0.0496904
$406$	1.12778e6	0.339556
$407$	7.07679e6	2.11763
$408$	−19584.0	−0.00582440
$409$	735666.	0.217457	0.108728	−	0.994072i	$-0.465322\pi$
$409$	735666.	0.217457	0.108728	+	0.994072i	$0.465322\pi$
$410$	41800.0	0.0122805
$411$	3.45929e6	1.01014
$412$	−113408.	−0.0329155
$413$	−514108.	−0.148313
$414$	618192.	0.177265
$415$	2.58130e6	0.735730
$416$	−374784.	−0.106181
$417$	2304.00	0.000648847 0
$418$	5.38675e6	1.50795
$419$	4.23222e6	1.17770	0.588848	−	0.808244i	$-0.299582\pi$
$419$	4.23222e6	1.17770	0.588848	+	0.808244i	$0.299582\pi$
$420$	176400.	0.0487950
$421$	5.51413e6	1.51625	0.758127	−	0.652107i	$-0.226115\pi$
$421$	5.51413e6	1.51625	0.758127	+	0.652107i	$0.226115\pi$
$422$	2.38648e6	0.652344
$423$	119232.	0.0323998
$424$	1.26938e6	0.342906
$425$	−21250.0	−0.00570672
$426$	−285696.	−0.0762746
$427$	1.90169e6	0.504743
$428$	2.01677e6	0.532165
$429$	2.20039e6	0.577241
$430$	−667600.	−0.174119
$431$	4.12711e6	1.07017	0.535085	−	0.844798i	$-0.320279\pi$
$431$	4.12711e6	1.07017	0.535085	+	0.844798i	$0.320279\pi$
$432$	186624.	0.0481125
$433$	3.11505e6	0.798444	0.399222	−	0.916854i	$-0.369280\pi$
$433$	3.11505e6	0.798444	0.399222	+	0.916854i	$0.369280\pi$
$434$	10192.0	0.00259738
$435$	1.29465e6	0.328042
$436$	−2.42486e6	−0.610901
$437$	−3.84653e6	−0.963530
$438$	760824.	0.189496
$439$	4.23065e6	1.04772	0.523861	−	0.851804i	$-0.324491\pi$
$439$	4.23065e6	1.04772	0.523861	+	0.851804i	$0.324491\pi$
$440$	1.06880e6	0.263187
$441$	194481.	0.0476190
$442$	49776.0	0.0121189
$443$	5.15771e6	1.24867	0.624335	−	0.781157i	$-0.285370\pi$
$443$	5.15771e6	1.24867	0.624335	+	0.781157i	$0.285370\pi$
$444$	−1.52554e6	−0.367253
$445$	1.23725e6	0.296181
$446$	−2.58973e6	−0.616477
$447$	1.86435e6	0.441325
$448$	−200704.	−0.0472456
$449$	−4.88653e6	−1.14389	−0.571945	−	0.820292i	$-0.693811\pi$
$449$	−4.88653e6	−1.14389	−0.571945	+	0.820292i	$0.693811\pi$
$450$	202500.	0.0471405
$451$	279224.	0.0646415
$452$	1.33078e6	0.306381
$453$	−2.49127e6	−0.570395
$454$	−3.23973e6	−0.737682
$455$	−448350.	−0.101529
$456$	−1.16122e6	−0.261518
$457$	−8.14264e6	−1.82379	−0.911895	−	0.410425i	$-0.865381\pi$
$457$	−8.14264e6	−1.82379	−0.911895	+	0.410425i	$0.865381\pi$
$458$	−4.25386e6	−0.947589
$459$	−24786.0	−0.00549129
$460$	−763200.	−0.168168
$461$	−3.95476e6	−0.866698	−0.433349	−	0.901226i	$-0.642668\pi$
$461$	−3.95476e6	−0.866698	−0.433349	+	0.901226i	$0.642668\pi$
$462$	1.17835e6	0.256844
$463$	5.91742e6	1.28286	0.641431	−	0.767180i	$-0.278341\pi$
$463$	5.91742e6	1.28286	0.641431	+	0.767180i	$0.278341\pi$
$464$	−1.47302e6	−0.317625
$465$	11700.0	0.00250931
$466$	−4.60068e6	−0.981425
$467$	−2.20904e6	−0.468717	−0.234358	−	0.972150i	$-0.575299\pi$
$467$	−2.20904e6	−0.468717	−0.234358	+	0.972150i	$0.575299\pi$
$468$	−474336.	−0.100109
$469$	3.00997e6	0.631874
$470$	−147200.	−0.0307371
$471$	644418.	0.133849
$472$	671488.	0.138734
$473$	−4.45957e6	−0.916516
$474$	3.13517e6	0.640936
$475$	−1.26000e6	−0.256234
$476$	26656.0	0.00539234
$477$	1.60655e6	0.323295
$478$	4.22886e6	0.846553
$479$	−3.92527e6	−0.781684	−0.390842	−	0.920458i	$-0.627816\pi$
$479$	−3.92527e6	−0.781684	−0.390842	+	0.920458i	$0.627816\pi$
$480$	−230400.	−0.0456435
$481$	3.87740e6	0.764150
$482$	−6.74386e6	−1.32218
$483$	−841428.	−0.164115
$484$	4.56277e6	0.885350
$485$	−3.14075e6	−0.606288
$486$	236196.	0.0453609
$487$	−236992.	−0.0452805	−0.0226403	−	0.999744i	$-0.507207\pi$
$487$	−236992.	−0.0452805	−0.0226403	+	0.999744i	$0.507207\pi$
$488$	−2.48384e6	−0.472144
$489$	−4.99676e6	−0.944967
$490$	−240100.	−0.0451754
$491$	−4.25346e6	−0.796230	−0.398115	−	0.917335i	$-0.630336\pi$
$491$	−4.25346e6	−0.796230	−0.398115	+	0.917335i	$0.630336\pi$
$492$	−60192.0	−0.0112105
$493$	195636.	0.0362520
$494$	2.95142e6	0.544144
$495$	1.35270e6	0.248135
$496$	−13312.0	−0.00242962
$497$	388864.	0.0706166
$498$	−3.71707e6	−0.671626
$499$	1.39319e6	0.250472	0.125236	−	0.992127i	$-0.460031\pi$
$499$	1.39319e6	0.250472	0.125236	+	0.992127i	$0.460031\pi$
$500$	−250000.	−0.0447214
$501$	4.87037e6	0.866897
$502$	−4.66920e6	−0.826957
$503$	−2.67065e6	−0.470649	−0.235324	−	0.971917i	$-0.575615\pi$
$503$	−2.67065e6	−0.470649	−0.235324	+	0.971917i	$0.575615\pi$
$504$	−254016.	−0.0445435
$505$	2.38875e6	0.416814
$506$	−5.09818e6	−0.885194
$507$	−2.13603e6	−0.369052
$508$	618496.	0.106335
$509$	8.23813e6	1.40940	0.704700	−	0.709506i	$-0.251082\pi$
$509$	8.23813e6	1.40940	0.704700	+	0.709506i	$0.251082\pi$
$510$	30600.0	0.00520950
$511$	−1.03557e6	−0.175439
$512$	262144.	0.0441942
$513$	−1.46966e6	−0.246561
$514$	−1.11028e6	−0.185364
$515$	177200.	0.0294405
$516$	961344.	0.158948
$517$	−983296.	−0.161792
$518$	2.07642e6	0.340010
$519$	−732222.	−0.119323
$520$	585600.	0.0949714
$521$	−1.14135e7	−1.84216	−0.921078	−	0.389379i	$-0.872690\pi$
$521$	−1.14135e7	−1.84216	−0.921078	+	0.389379i	$0.872690\pi$
$522$	−1.86430e6	−0.299460
$523$	−4.15665e6	−0.664492	−0.332246	−	0.943193i	$-0.607806\pi$
$523$	−4.15665e6	−0.664492	−0.332246	+	0.943193i	$0.607806\pi$
$524$	1.74061e6	0.276932
$525$	−275625.	−0.0436436
$526$	496496.	0.0782441
$527$	1768.00	0.000277304 0
$528$	−1.53907e6	−0.240256
$529$	−2.79588e6	−0.434389
$530$	−1.98340e6	−0.306705
$531$	849852.	0.130800
$532$	1.58054e6	0.242118
$533$	152988.	0.0233260
$534$	−1.78164e6	−0.270375
$535$	−3.15120e6	−0.475983
$536$	−3.93139e6	−0.591064
$537$	6.60690e6	0.988694
$538$	552072.	0.0822318
$539$	−1.60387e6	−0.237792
$540$	−291600.	−0.0430331
$541$	−1.06683e7	−1.56711	−0.783557	−	0.621320i	$-0.786597\pi$
$541$	−1.06683e7	−1.56711	−0.783557	+	0.621320i	$0.786597\pi$
$542$	−5.30712e6	−0.775998
$543$	1.15688e6	0.168379
$544$	−34816.0	−0.00504408
$545$	3.78885e6	0.546407
$546$	645624.	0.0926826
$547$	1.13950e7	1.62835	0.814173	−	0.580622i	$-0.197191\pi$
$547$	1.13950e7	1.62835	0.814173	+	0.580622i	$0.197191\pi$
$548$	6.14986e6	0.874810
$549$	−3.14361e6	−0.445141
$550$	−1.67000e6	−0.235402
$551$	1.16001e7	1.62773
$552$	1.09901e6	0.153516
$553$	−4.26731e6	−0.593392
$554$	−4.16455e6	−0.576493
$555$	2.38365e6	0.328481
$556$	4096.00	0.000561918 0
$557$	2.41328e6	0.329587	0.164794	−	0.986328i	$-0.447304\pi$
$557$	2.41328e6	0.329587	0.164794	+	0.986328i	$0.447304\pi$
$558$	−16848.0	−0.00229067
$559$	−2.44342e6	−0.330726
$560$	313600.	0.0422577
$561$	204408.	0.0274215
$562$	1.12196e6	0.149843
$563$	−1.24262e7	−1.65222	−0.826109	−	0.563511i	$-0.809450\pi$
$563$	−1.24262e7	−1.65222	−0.826109	+	0.563511i	$0.809450\pi$
$564$	211968.	0.0280590
$565$	−2.07935e6	−0.274035
$566$	−7.81528e6	−1.02542
$567$	−321489.	−0.0419961
$568$	−507904.	−0.0660558
$569$	−9.41797e6	−1.21948	−0.609742	−	0.792600i	$-0.708727\pi$
$569$	−9.41797e6	−1.21948	−0.609742	+	0.792600i	$0.708727\pi$
$570$	1.81440e6	0.233908
$571$	−2.59936e6	−0.333638	−0.166819	−	0.985988i	$-0.553350\pi$
$571$	−2.59936e6	−0.333638	−0.166819	+	0.985988i	$0.553350\pi$
$572$	3.91181e6	0.499905
$573$	3.45341e6	0.439401
$574$	81928.0	0.0103789
$575$	1.19250e6	0.150414
$576$	331776.	0.0416667
$577$	−1.04806e7	−1.31053	−0.655267	−	0.755398i	$-0.727444\pi$
$577$	−1.04806e7	−1.31053	−0.655267	+	0.755398i	$0.727444\pi$
$578$	−5.67480e6	−0.706531
$579$	1.51936e6	0.188350
$580$	2.30160e6	0.284093
$581$	5.05935e6	0.621805
$582$	4.52268e6	0.553463
$583$	−1.32491e7	−1.61442
$584$	1.35258e6	0.164108
$585$	741150.	0.0895399
$586$	−3.10364e6	−0.373360
$587$	1.50698e7	1.80515	0.902574	−	0.430536i	$-0.141675\pi$
$587$	1.50698e7	1.80515	0.902574	+	0.430536i	$0.141675\pi$
$588$	345744.	0.0412393
$589$	104832.	0.0124510
$590$	−1.04920e6	−0.124088
$591$	2.66632e6	0.314010
$592$	−2.71206e6	−0.318050
$593$	8.35160e6	0.975288	0.487644	−	0.873043i	$-0.337856\pi$
$593$	8.35160e6	0.975288	0.487644	+	0.873043i	$0.337856\pi$
$594$	−1.94789e6	−0.226515
$595$	−41650.0	−0.00482306
$596$	3.31440e6	0.382199
$597$	1.79708e6	0.206363
$598$	−2.79331e6	−0.319423
$599$	4.78980e6	0.545444	0.272722	−	0.962093i	$-0.412076\pi$
$599$	4.78980e6	0.545444	0.272722	+	0.962093i	$0.412076\pi$
$600$	360000.	0.0408248
$601$	1.54853e7	1.74877	0.874386	−	0.485230i	$-0.161264\pi$
$601$	1.54853e7	1.74877	0.874386	+	0.485230i	$0.161264\pi$
$602$	−1.30850e6	−0.147157
$603$	−4.97567e6	−0.557260
$604$	−4.42893e6	−0.493976
$605$	−7.12932e6	−0.791881
$606$	−3.43980e6	−0.380497
$607$	−8.86825e6	−0.976936	−0.488468	−	0.872582i	$-0.662444\pi$
$607$	−8.86825e6	−0.976936	−0.488468	+	0.872582i	$0.662444\pi$
$608$	−2.06438e6	−0.226481
$609$	2.53751e6	0.277246
$610$	3.88100e6	0.422298
$611$	−538752.	−0.0583829
$612$	−44064.0	−0.00475560
$613$	1.16930e7	1.25683	0.628414	−	0.777879i	$-0.283705\pi$
$613$	1.16930e7	1.25683	0.628414	+	0.777879i	$0.283705\pi$
$614$	−6.76875e6	−0.724582
$615$	94050.0	0.0100270
$616$	2.09485e6	0.222434
$617$	−1.14604e7	−1.21196	−0.605978	−	0.795482i	$-0.707218\pi$
$617$	−1.14604e7	−1.21196	−0.605978	+	0.795482i	$0.707218\pi$
$618$	−255168.	−0.0268754
$619$	237488.	0.0249124	0.0124562	−	0.999922i	$-0.496035\pi$
$619$	237488.	0.0249124	0.0124562	+	0.999922i	$0.496035\pi$
$620$	20800.0	0.00217312
$621$	1.39093e6	0.144736
$622$	−3.61766e6	−0.374932
$623$	2.42501e6	0.250319
$624$	−843264.	−0.0866966
$625$	390625.	0.0400000
$626$	−1.23847e7	−1.26314
$627$	1.21202e7	1.23123
$628$	1.14563e6	0.115917
$629$	360196.	0.0363005
$630$	396900.	0.0398410
$631$	−14344.0	−0.00143416	−0.000717079	−	1.00000i	$-0.500228\pi$
$631$	−14344.0	−0.00143416	−0.000717079		1.00000i	$0.500228\pi$
$632$	5.57363e6	0.555067
$633$	5.36958e6	0.532637
$634$	5.61729e6	0.555013
$635$	−966400.	−0.0951092
$636$	2.85610e6	0.279982
$637$	−878766.	−0.0858074
$638$	1.53747e7	1.49539
$639$	−642816.	−0.0622780
$640$	−409600.	−0.0395285
$641$	−8.98925e6	−0.864128	−0.432064	−	0.901843i	$-0.642215\pi$
$641$	−8.98925e6	−0.864128	−0.432064	+	0.901843i	$0.642215\pi$
$642$	4.53773e6	0.434511
$643$	1.28790e7	1.22844	0.614222	−	0.789133i	$-0.289470\pi$
$643$	1.28790e7	1.22844	0.614222	+	0.789133i	$0.289470\pi$
$644$	−1.49587e6	−0.142128
$645$	−1.50210e6	−0.142167
$646$	274176.	0.0258493
$647$	2.06233e7	1.93685	0.968427	−	0.249299i	$-0.0802002\pi$
$647$	2.06233e7	1.93685	0.968427	+	0.249299i	$0.0802002\pi$
$648$	419904.	0.0392837
$649$	−7.00866e6	−0.653165
$650$	−915000.	−0.0849450
$651$	22932.0	0.00212075
$652$	−8.88314e6	−0.818366
$653$	−6.76833e6	−0.621153	−0.310577	−	0.950548i	$-0.600522\pi$
$653$	−6.76833e6	−0.621153	−0.310577	+	0.950548i	$0.600522\pi$
$654$	−5.45594e6	−0.498799
$655$	−2.71970e6	−0.247695
$656$	−107008.	−0.00970860
$657$	1.71185e6	0.154722
$658$	−288512.	−0.0259776
$659$	−6.58008e6	−0.590225	−0.295112	−	0.955463i	$-0.595357\pi$
$659$	−6.58008e6	−0.590225	−0.295112	+	0.955463i	$0.595357\pi$
$660$	2.40480e6	0.214891
$661$	1.30970e7	1.16592	0.582961	−	0.812500i	$-0.301894\pi$
$661$	1.30970e7	1.16592	0.582961	+	0.812500i	$0.301894\pi$
$662$	3.17973e6	0.281997
$663$	111996.	0.00989507
$664$	−6.60813e6	−0.581645
$665$	−2.46960e6	−0.216557
$666$	−3.43246e6	−0.299861
$667$	−1.09786e7	−0.955506
$668$	8.65843e6	0.750755
$669$	−5.82689e6	−0.503352
$670$	6.14280e6	0.528664
$671$	2.59251e7	2.22287
$672$	−451584.	−0.0385758
$673$	−1.13071e7	−0.962304	−0.481152	−	0.876637i	$-0.659781\pi$
$673$	−1.13071e7	−0.962304	−0.481152	+	0.876637i	$0.659781\pi$
$674$	−7.12825e6	−0.604412
$675$	455625.	0.0384900
$676$	−3.79739e6	−0.319609
$677$	−3.55455e6	−0.298066	−0.149033	−	0.988832i	$-0.547616\pi$
$677$	−3.55455e6	−0.298066	−0.149033	+	0.988832i	$0.547616\pi$
$678$	2.99426e6	0.250159
$679$	−6.15587e6	−0.512407
$680$	54400.0	0.00451156
$681$	−7.28939e6	−0.602314
$682$	138944.	0.0114388
$683$	−1.46799e7	−1.20412	−0.602062	−	0.798449i	$-0.705654\pi$
$683$	−1.46799e7	−1.20412	−0.602062	+	0.798449i	$0.705654\pi$
$684$	−2.61274e6	−0.213528
$685$	−9.60915e6	−0.782454
$686$	−470596.	−0.0381802
$687$	−9.57119e6	−0.773703
$688$	1.70906e6	0.137653
$689$	−7.25924e6	−0.582564
$690$	−1.71720e6	−0.137309
$691$	1.94548e7	1.55000	0.775002	−	0.631959i	$-0.217749\pi$
$691$	1.94548e7	1.55000	0.775002	+	0.631959i	$0.217749\pi$
$692$	−1.30173e6	−0.103337
$693$	2.65129e6	0.209713
$694$	−6.18310e6	−0.487313
$695$	−6400.00	−0.000502595 0
$696$	−3.31430e6	−0.259340
$697$	14212.0	0.00110809
$698$	7.60790e6	0.591052
$699$	−1.03515e7	−0.801330
$700$	−490000.	−0.0377964
$701$	1.04704e7	0.804764	0.402382	−	0.915472i	$-0.368182\pi$
$701$	1.04704e7	0.804764	0.402382	+	0.915472i	$0.368182\pi$
$702$	−1.06726e6	−0.0817383
$703$	2.13575e7	1.62990
$704$	−2.73613e6	−0.208068
$705$	−331200.	−0.0250968
$706$	−6.99252e6	−0.527985
$707$	4.68195e6	0.352272
$708$	1.51085e6	0.113276
$709$	−1.45289e7	−1.08547	−0.542734	−	0.839904i	$-0.682611\pi$
$709$	−1.45289e7	−1.08547	−0.542734	+	0.839904i	$0.682611\pi$
$710$	793600.	0.0590821
$711$	7.05413e6	0.523322
$712$	−3.16736e6	−0.234152
$713$	−99216.0	−0.00730900
$714$	59976.0	0.00440283
$715$	−6.11220e6	−0.447129
$716$	1.17456e7	0.856234
$717$	9.51494e6	0.691207
$718$	1.85014e6	0.133935
$719$	−1.75805e7	−1.26826	−0.634131	−	0.773225i	$-0.718642\pi$
$719$	−1.75805e7	−1.26826	−0.634131	+	0.773225i	$0.718642\pi$
$720$	−518400.	−0.0372678
$721$	347312.	0.0248818
$722$	6.35263e6	0.453535
$723$	−1.51737e7	−1.07956
$724$	2.05667e6	0.145821
$725$	−3.59625e6	−0.254100
$726$	1.02662e7	0.722885
$727$	1.63370e7	1.14640	0.573201	−	0.819415i	$-0.305702\pi$
$727$	1.63370e7	1.14640	0.573201	+	0.819415i	$0.305702\pi$
$728$	1.14778e6	0.0802655
$729$	531441.	0.0370370
$730$	−2.11340e6	−0.146783
$731$	−226984.	−0.0157109
$732$	−5.58864e6	−0.385504
$733$	−1.80106e7	−1.23813	−0.619067	−	0.785338i	$-0.712489\pi$
$733$	−1.80106e7	−1.23813	−0.619067	+	0.785338i	$0.712489\pi$
$734$	−1.53507e6	−0.105169
$735$	−540225.	−0.0368856
$736$	1.95379e6	0.132949
$737$	4.10339e7	2.78275
$738$	−135432.	−0.00915336
$739$	2.32873e7	1.56858	0.784292	−	0.620392i	$-0.213027\pi$
$739$	2.32873e7	1.56858	0.784292	+	0.620392i	$0.213027\pi$
$740$	4.23760e6	0.284473
$741$	6.64070e6	0.444292
$742$	−3.88746e6	−0.259213
$743$	1.61253e7	1.07161	0.535805	−	0.844342i	$-0.320008\pi$
$743$	1.61253e7	1.07161	0.535805	+	0.844342i	$0.320008\pi$
$744$	−29952.0	−0.00198378
$745$	−5.17875e6	−0.341849
$746$	−1.36334e7	−0.896926
$747$	−8.36341e6	−0.548380
$748$	363392.	0.0237477
$749$	−6.17635e6	−0.402279
$750$	−562500.	−0.0365148
$751$	−4.61740e6	−0.298743	−0.149371	−	0.988781i	$-0.547725\pi$
$751$	−4.61740e6	−0.298743	−0.149371	+	0.988781i	$0.547725\pi$
$752$	376832.	0.0242998
$753$	−1.05057e7	−0.675208
$754$	8.42386e6	0.539613
$755$	6.92020e6	0.441826
$756$	−571536.	−0.0363696
$757$	−7.09630e6	−0.450083	−0.225041	−	0.974349i	$-0.572252\pi$
$757$	−7.09630e6	−0.450083	−0.225041	+	0.974349i	$0.572252\pi$
$758$	1.61167e7	1.01883
$759$	−1.14709e7	−0.722758
$760$	3.22560e6	0.202571
$761$	2.59097e6	0.162182	0.0810908	−	0.996707i	$-0.474160\pi$
$761$	2.59097e6	0.162182	0.0810908	+	0.996707i	$0.474160\pi$
$762$	1.39162e6	0.0868224
$763$	7.42615e6	0.461798
$764$	6.13939e6	0.380533
$765$	68850.0	0.00425354
$766$	−1.70083e7	−1.04734
$767$	−3.84007e6	−0.235695
$768$	589824.	0.0360844
$769$	−2.18633e7	−1.33321	−0.666606	−	0.745410i	$-0.732254\pi$
$769$	−2.18633e7	−1.33321	−0.666606	+	0.745410i	$0.732254\pi$
$770$	−3.27320e6	−0.198951
$771$	−2.49813e6	−0.151349
$772$	2.70109e6	0.163116
$773$	−1.37603e7	−0.828284	−0.414142	−	0.910212i	$-0.635918\pi$
$773$	−1.37603e7	−0.828284	−0.414142	+	0.910212i	$0.635918\pi$
$774$	2.16302e6	0.129780
$775$	−32500.0	−0.00194370
$776$	8.04032e6	0.479313
$777$	4.67195e6	0.277617
$778$	1.84241e7	1.09128
$779$	842688.	0.0497534
$780$	1.31760e6	0.0775438
$781$	5.30125e6	0.310993
$782$	−259488.	−0.0151740
$783$	−4.19467e6	−0.244508
$784$	614656.	0.0357143
$785$	−1.79005e6	−0.103679
$786$	3.91637e6	0.226114
$787$	−1.85194e7	−1.06584	−0.532918	−	0.846167i	$-0.678905\pi$
$787$	−1.85194e7	−1.06584	−0.532918	+	0.846167i	$0.678905\pi$
$788$	4.74013e6	0.271941
$789$	1.11712e6	0.0638860
$790$	−8.70880e6	−0.496467
$791$	−4.07553e6	−0.231602
$792$	−3.46291e6	−0.196168
$793$	1.42045e7	0.802124
$794$	−1.61032e7	−0.906485
$795$	−4.46265e6	−0.250423
$796$	3.19482e6	0.178716
$797$	2.01727e7	1.12491	0.562457	−	0.826827i	$-0.309856\pi$
$797$	2.01727e7	1.12491	0.562457	+	0.826827i	$0.309856\pi$
$798$	3.55622e6	0.197689
$799$	−50048.0	−0.00277345
$800$	640000.	0.0353553
$801$	−4.00869e6	−0.220760
$802$	−1.34905e7	−0.740612
$803$	−1.41175e7	−0.772626
$804$	−8.84563e6	−0.482602
$805$	2.33730e6	0.127123
$806$	76128.0	0.00412769
$807$	1.24216e6	0.0671420
$808$	−6.11520e6	−0.329520
$809$	2.31331e7	1.24269	0.621344	−	0.783538i	$-0.286587\pi$
$809$	2.31331e7	1.24269	0.621344	+	0.783538i	$0.286587\pi$
$810$	−656100.	−0.0351364
$811$	−4.43094e6	−0.236562	−0.118281	−	0.992980i	$-0.537738\pi$
$811$	−4.43094e6	−0.236562	−0.118281	+	0.992980i	$0.537738\pi$
$812$	4.51114e6	0.240102
$813$	−1.19410e7	−0.633600
$814$	2.83072e7	1.49739
$815$	1.38799e7	0.731968
$816$	−78336.0	−0.00411847
$817$	−1.34588e7	−0.705426
$818$	2.94266e6	0.153765
$819$	1.45265e6	0.0756750
$820$	167200.	0.00868364
$821$	3.68300e7	1.90697	0.953485	−	0.301439i	$-0.0974669\pi$
$821$	3.68300e7	1.90697	0.953485	+	0.301439i	$0.0974669\pi$
$822$	1.38372e7	0.714279
$823$	2.16829e7	1.11588	0.557940	−	0.829881i	$-0.311592\pi$
$823$	2.16829e7	1.11588	0.557940	+	0.829881i	$0.311592\pi$
$824$	−453632.	−0.0232748
$825$	−3.75750e6	−0.192205
$826$	−2.05643e6	−0.104873
$827$	1.10996e7	0.564344	0.282172	−	0.959364i	$-0.408945\pi$
$827$	1.10996e7	0.564344	0.282172	+	0.959364i	$0.408945\pi$
$828$	2.47277e6	0.125345
$829$	9.57052e6	0.483670	0.241835	−	0.970317i	$-0.422251\pi$
$829$	9.57052e6	0.483670	0.241835	+	0.970317i	$0.422251\pi$
$830$	1.03252e7	0.520239
$831$	−9.37024e6	−0.470704
$832$	−1.49914e6	−0.0750815
$833$	−81634.0	−0.00407623
$834$	9216.00	0.000458804 0
$835$	−1.35288e7	−0.671495
$836$	2.15470e7	1.06628
$837$	−37908.0	−0.00187033
$838$	1.69289e7	0.832757
$839$	2.40996e7	1.18196	0.590982	−	0.806685i	$-0.298740\pi$
$839$	2.40996e7	1.18196	0.590982	+	0.806685i	$0.298740\pi$
$840$	705600.	0.0345033
$841$	1.25974e7	0.614172
$842$	2.20565e7	1.07215
$843$	2.52441e6	0.122346
$844$	9.54592e6	0.461277
$845$	5.93342e6	0.285867
$846$	476928.	0.0229101
$847$	−1.39735e7	−0.669262
$848$	5.07750e6	0.242471
$849$	−1.75844e7	−0.837255
$850$	−85000.0	−0.00403526
$851$	−2.02134e7	−0.956785
$852$	−1.14278e6	−0.0539343
$853$	−1.62218e7	−0.763357	−0.381678	−	0.924295i	$-0.624654\pi$
$853$	−1.62218e7	−0.763357	−0.381678	+	0.924295i	$0.624654\pi$
$854$	7.60676e6	0.356907
$855$	4.08240e6	0.190985
$856$	8.06707e6	0.376298
$857$	1.82957e7	0.850936	0.425468	−	0.904974i	$-0.360110\pi$
$857$	1.82957e7	0.850936	0.425468	+	0.904974i	$0.360110\pi$
$858$	8.80157e6	0.408171
$859$	3.49800e6	0.161747	0.0808736	−	0.996724i	$-0.474229\pi$
$859$	3.49800e6	0.161747	0.0808736	+	0.996724i	$0.474229\pi$
$860$	−2.67040e6	−0.123120
$861$	184338.	0.00847436
$862$	1.65084e7	0.756725
$863$	−3.70241e7	−1.69222	−0.846110	−	0.533008i	$-0.821062\pi$
$863$	−3.70241e7	−1.69222	−0.846110	+	0.533008i	$0.821062\pi$
$864$	746496.	0.0340207
$865$	2.03395e6	0.0924272
$866$	1.24602e7	0.564586
$867$	−1.27683e7	−0.576880
$868$	40768.0	0.00183662
$869$	−5.81748e7	−2.61328
$870$	5.17860e6	0.231961
$871$	2.24826e7	1.00416
$872$	−9.69946e6	−0.431973
$873$	1.01760e7	0.451900
$874$	−1.53861e7	−0.681319
$875$	765625.	0.0338062
$876$	3.04330e6	0.133994
$877$	186038.	0.00816775	0.00408388	−	0.999992i	$-0.498700\pi$
$877$	186038.	0.00816775	0.00408388	+	0.999992i	$0.498700\pi$
$878$	1.69226e7	0.740851
$879$	−6.98319e6	−0.304847
$880$	4.27520e6	0.186101
$881$	3.33731e6	0.144863	0.0724314	−	0.997373i	$-0.476924\pi$
$881$	3.33731e6	0.144863	0.0724314	+	0.997373i	$0.476924\pi$
$882$	777924.	0.0336718
$883$	1.62975e7	0.703426	0.351713	−	0.936108i	$-0.385599\pi$
$883$	1.62975e7	0.703426	0.351713	+	0.936108i	$0.385599\pi$
$884$	199104.	0.00856938
$885$	−2.36070e6	−0.101317
$886$	2.06308e7	0.882943
$887$	−2.32942e7	−0.994120	−0.497060	−	0.867716i	$-0.665587\pi$
$887$	−2.32942e7	−0.994120	−0.497060	+	0.867716i	$0.665587\pi$
$888$	−6.10214e6	−0.259687
$889$	−1.89414e6	−0.0803820
$890$	4.94900e6	0.209432
$891$	−4.38275e6	−0.184949
$892$	−1.03589e7	−0.435915
$893$	−2.96755e6	−0.124529
$894$	7.45740e6	0.312064
$895$	−1.83525e7	−0.765839
$896$	−802816.	−0.0334077
$897$	−6.28495e6	−0.260808
$898$	−1.95461e7	−0.808853
$899$	299208.	0.0123474
$900$	810000.	0.0333333
$901$	−674356.	−0.0276743
$902$	1.11690e6	0.0457084
$903$	−2.94412e6	−0.120153
$904$	5.32314e6	0.216644
$905$	−3.21355e6	−0.130426
$906$	−9.96509e6	−0.403330
$907$	3.13298e7	1.26456	0.632281	−	0.774739i	$-0.282119\pi$
$907$	3.13298e7	1.26456	0.632281	+	0.774739i	$0.282119\pi$
$908$	−1.29589e7	−0.521620
$909$	−7.73955e6	−0.310675
$910$	−1.79340e6	−0.0717916
$911$	−3.63983e7	−1.45307	−0.726533	−	0.687132i	$-0.758869\pi$
$911$	−3.63983e7	−1.45307	−0.726533	+	0.687132i	$0.758869\pi$
$912$	−4.64486e6	−0.184921
$913$	6.89723e7	2.73841
$914$	−3.25706e7	−1.28961
$915$	8.73225e6	0.344805
$916$	−1.70155e7	−0.670046
$917$	−5.33061e6	−0.209341
$918$	−99144.0	−0.00388293
$919$	2.86227e7	1.11795	0.558975	−	0.829185i	$-0.311195\pi$
$919$	2.86227e7	1.11795	0.558975	+	0.829185i	$0.311195\pi$
$920$	−3.05280e6	−0.118913
$921$	−1.52297e7	−0.591619
$922$	−1.58190e7	−0.612848
$923$	2.90458e6	0.112222
$924$	4.71341e6	0.181616
$925$	−6.62125e6	−0.254440
$926$	2.36697e7	0.907121
$927$	−574128.	−0.0219437
$928$	−5.89210e6	−0.224595
$929$	−4.13345e6	−0.157135	−0.0785676	−	0.996909i	$-0.525035\pi$
$929$	−4.13345e6	−0.157135	−0.0785676	+	0.996909i	$0.525035\pi$
$930$	46800.0	0.00177435
$931$	−4.84042e6	−0.183024
$932$	−1.84027e7	−0.693972
$933$	−8.13974e6	−0.306131
$934$	−8.83614e6	−0.331433
$935$	−567800.	−0.0212406
$936$	−1.89734e6	−0.0707875
$937$	−4.16051e6	−0.154809	−0.0774047	−	0.997000i	$-0.524663\pi$
$937$	−4.16051e6	−0.154809	−0.0774047	+	0.997000i	$0.524663\pi$
$938$	1.20399e7	0.446802
$939$	−2.78656e7	−1.03135
$940$	−588800.	−0.0217344
$941$	−5.42587e6	−0.199754	−0.0998770	−	0.995000i	$-0.531845\pi$
$941$	−5.42587e6	−0.199754	−0.0998770	+	0.995000i	$0.531845\pi$
$942$	2.57767e6	0.0946456
$943$	−797544.	−0.0292062
$944$	2.68595e6	0.0980998
$945$	893025.	0.0325300
$946$	−1.78383e7	−0.648075
$947$	−8.32317e6	−0.301588	−0.150794	−	0.988565i	$-0.548183\pi$
$947$	−8.32317e6	−0.301588	−0.150794	+	0.988565i	$0.548183\pi$
$948$	1.25407e7	0.453211
$949$	−7.73504e6	−0.278803
$950$	−5.04000e6	−0.181185
$951$	1.26389e7	0.453167
$952$	106624.	0.00381296
$953$	4.27085e7	1.52329	0.761644	−	0.647995i	$-0.224392\pi$
$953$	4.27085e7	1.52329	0.761644	+	0.647995i	$0.224392\pi$
$954$	6.42622e6	0.228604
$955$	−9.59280e6	−0.340359
$956$	1.69155e7	0.598603
$957$	3.45930e7	1.22098
$958$	−1.57011e7	−0.552734
$959$	−1.88339e7	−0.661294
$960$	−921600.	−0.0322749
$961$	−2.86264e7	−0.999906
$962$	1.55096e7	0.540335
$963$	1.02099e7	0.354777
$964$	−2.69755e7	−0.934924
$965$	−4.22045e6	−0.145895
$966$	−3.36571e6	−0.116047
$967$	−3.75367e7	−1.29089	−0.645445	−	0.763807i	$-0.723328\pi$
$967$	−3.75367e7	−1.29089	−0.645445	+	0.763807i	$0.723328\pi$
$968$	1.82511e7	0.626037
$969$	616896.	0.0211058
$970$	−1.25630e7	−0.428710
$971$	−4.06557e7	−1.38380	−0.691900	−	0.721993i	$-0.743226\pi$
$971$	−4.06557e7	−1.38380	−0.691900	+	0.721993i	$0.743226\pi$
$972$	944784.	0.0320750
$973$	−12544.0	−0.000424770 0
$974$	−947968.	−0.0320182
$975$	−2.05875e6	−0.0693573
$976$	−9.93536e6	−0.333856
$977$	2.37522e7	0.796098	0.398049	−	0.917364i	$-0.369687\pi$
$977$	2.37522e7	0.796098	0.398049	+	0.917364i	$0.369687\pi$
$978$	−1.99871e7	−0.668193
$979$	3.30593e7	1.10239
$980$	−960400.	−0.0319438
$981$	−1.22759e7	−0.407268
$982$	−1.70138e7	−0.563020
$983$	4.51922e7	1.49169	0.745847	−	0.666118i	$-0.232045\pi$
$983$	4.51922e7	1.49169	0.745847	+	0.666118i	$0.232045\pi$
$984$	−240768.	−0.00792704
$985$	−7.40645e6	−0.243231
$986$	782544.	0.0256340
$987$	−649152.	−0.0212106
$988$	1.18057e7	0.384768
$989$	1.27378e7	0.414099
$990$	5.41080e6	0.175458
$991$	−3.84745e7	−1.24448	−0.622241	−	0.782826i	$-0.713777\pi$
$991$	−3.84745e7	−1.24448	−0.622241	+	0.782826i	$0.713777\pi$
$992$	−53248.0	−0.00171800
$993$	7.15439e6	0.230250
$994$	1.55546e6	0.0499335
$995$	−4.99190e6	−0.159848
$996$	−1.48683e7	−0.474911
$997$	1.95602e7	0.623213	0.311606	−	0.950211i	$-0.399133\pi$
$997$	1.95602e7	0.623213	0.311606	+	0.950211i	$0.399133\pi$
$998$	5.57275e6	0.177110
$999$	−7.72303e6	−0.244835

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	210.6.a.j.1.1	✓	1
3.2	odd	2		630.6.a.d.1.1		1
5.2	odd	4		1050.6.g.j.799.2		2
5.3	odd	4		1050.6.g.j.799.1		2
5.4	even	2		1050.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.6.a.j.1.1	✓	1	1.1	even	1	trivial
630.6.a.d.1.1		1	3.2	odd	2
1050.6.a.b.1.1		1	5.4	even	2
1050.6.g.j.799.1		2	5.3	odd	4
1050.6.g.j.799.2		2	5.2	odd	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 $\F_{q}$

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	210.6.a.j.1.1

Level	$210$
Weight	$6$
Character	210.1
Self dual	yes
Analytic conductor	$33.681$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$