LMFDB - Embedded newform 1050.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,6,Mod(1,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1050.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:	$168.403010804$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -668.000 q^{11} -144.000 q^{12} +366.000 q^{13} -196.000 q^{14} +256.000 q^{16} +34.0000 q^{17} -324.000 q^{18} -2016.00 q^{19} -441.000 q^{21} +2672.00 q^{22} -1908.00 q^{23} +576.000 q^{24} -1464.00 q^{26} -729.000 q^{27} +784.000 q^{28} -5754.00 q^{29} -52.0000 q^{31} -1024.00 q^{32} +6012.00 q^{33} -136.000 q^{34} +1296.00 q^{36} +10594.0 q^{37} +8064.00 q^{38} -3294.00 q^{39} -418.000 q^{41} +1764.00 q^{42} -6676.00 q^{43} -10688.0 q^{44} +7632.00 q^{46} -1472.00 q^{47} -2304.00 q^{48} +2401.00 q^{49} -306.000 q^{51} +5856.00 q^{52} -19834.0 q^{53} +2916.00 q^{54} -3136.00 q^{56} +18144.0 q^{57} +23016.0 q^{58} +10492.0 q^{59} -38810.0 q^{61} +208.000 q^{62} +3969.00 q^{63} +4096.00 q^{64} -24048.0 q^{66} +61428.0 q^{67} +544.000 q^{68} +17172.0 q^{69} -7936.00 q^{71} -5184.00 q^{72} -21134.0 q^{73} -42376.0 q^{74} -32256.0 q^{76} -32732.0 q^{77} +13176.0 q^{78} +87088.0 q^{79} +6561.00 q^{81} +1672.00 q^{82} +103252. q^{83} -7056.00 q^{84} +26704.0 q^{86} +51786.0 q^{87} +42752.0 q^{88} -49490.0 q^{89} +17934.0 q^{91} -30528.0 q^{92} +468.000 q^{93} +5888.00 q^{94} +9216.00 q^{96} -125630. q^{97} -9604.00 q^{98} -54108.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	0	0
$11$	−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	$-0.402905\pi$
$13$	366.000	0.600652	0.300326	+	0.953837i	$0.402905\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	34.0000	0.0285336	0.0142668	−	0.999898i	$-0.495459\pi$
$17$	34.0000	0.0285336	0.0142668	+	0.999898i	$0.495459\pi$
$18$	−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	$0.721308\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	2672.00	1.17701
$23$	−1908.00	−0.752071	−0.376035	−	0.926605i	$-0.622713\pi$
$23$	−1908.00	−0.752071	−0.376035	+	0.926605i	$0.622713\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−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	$0.719103\pi$
$30$	0	0
$31$	−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
$35$	0	0
$36$	1296.00	0.166667
$37$	10594.0	1.27220	0.636100	−	0.771606i	$-0.280546\pi$
$37$	10594.0	1.27220	0.636100	+	0.771606i	$0.280546\pi$
$38$	8064.00	0.905924
$39$	−3294.00	−0.346786
$40$	0	0
$41$	−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	$-0.588779\pi$
$43$	−6676.00	−0.550611	−0.275306	+	0.961357i	$0.588779\pi$
$44$	−10688.0	−0.832271
$45$	0	0
$46$	7632.00	0.531794
$47$	−1472.00	−0.0971993	−0.0485997	−	0.998818i	$-0.515476\pi$
$47$	−1472.00	−0.0971993	−0.0485997	+	0.998818i	$0.515476\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	0	0
$51$	−306.000	−0.0164739
$52$	5856.00	0.300326
$53$	−19834.0	−0.969886	−0.484943	−	0.874546i	$-0.661160\pi$
$53$	−19834.0	−0.969886	−0.484943	+	0.874546i	$0.661160\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	−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	$0.437140\pi$
$60$	0	0
$61$	−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
$65$	0	0
$66$	−24048.0	−0.679546
$67$	61428.0	1.67178	0.835890	−	0.548896i	$-0.184952\pi$
$67$	61428.0	1.67178	0.835890	+	0.548896i	$0.184952\pi$
$68$	544.000	0.0142668
$69$	17172.0	0.434208
$70$	0	0
$71$	−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	$-0.574554\pi$
$73$	−21134.0	−0.464167	−0.232084	+	0.972696i	$0.574554\pi$
$74$	−42376.0	−0.899582
$75$	0	0
$76$	−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	$0.212671\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	1672.00	0.0274601
$83$	103252.	1.64514	0.822571	−	0.568663i	$-0.192539\pi$
$83$	103252.	1.64514	0.822571	+	0.568663i	$0.192539\pi$
$84$	−7056.00	−0.109109
$85$	0	0
$86$	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	$0.607433\pi$
$90$	0	0
$91$	17934.0	0.227025
$92$	−30528.0	−0.376035
$93$	468.000	0.00561098
$94$	5888.00	0.0687303
$95$	0	0
$96$	9216.00	0.102062
$97$	−125630.	−1.35570	−0.677851	−	0.735200i	$-0.737088\pi$
$97$	−125630.	−1.35570	−0.677851	+	0.735200i	$0.737088\pi$
$98$	−9604.00	−0.101015
$99$	−54108.0	−0.554847
$100$	0	0
$101$	−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	$-0.489521\pi$
$103$	7088.00	0.0658310	0.0329155	+	0.999458i	$0.489521\pi$
$104$	−23424.0	−0.212362
$105$	0	0
$106$	79336.0	0.685813
$107$	−126048.	−1.06433	−0.532165	−	0.846641i	$-0.678621\pi$
$107$	−126048.	−1.06433	−0.532165	+	0.846641i	$0.678621\pi$
$108$	−11664.0	−0.0962250
$109$	−151554.	−1.22180	−0.610901	−	0.791707i	$-0.709193\pi$
$109$	−151554.	−1.22180	−0.610901	+	0.791707i	$0.709193\pi$
$110$	0	0
$111$	−95346.0	−0.734505
$112$	12544.0	0.0944911
$113$	−83174.0	−0.612762	−0.306381	−	0.951909i	$-0.599118\pi$
$113$	−83174.0	−0.612762	−0.306381	+	0.951909i	$0.599118\pi$
$114$	−72576.0	−0.523035
$115$	0	0
$116$	−92064.0	−0.635250
$117$	29646.0	0.200217
$118$	−41968.0	−0.277468
$119$	1666.00	0.0107847
$120$	0	0
$121$	285173.	1.77070
$122$	155240.	0.944287
$123$	3762.00	0.0224211
$124$	−832.000	−0.00485925
$125$	0	0
$126$	−15876.0	−0.0890871
$127$	−38656.0	−0.212671	−0.106335	−	0.994330i	$-0.533912\pi$
$127$	−38656.0	−0.212671	−0.106335	+	0.994330i	$0.533912\pi$
$128$	−16384.0	−0.0883883
$129$	60084.0	0.317896
$130$	0	0
$131$	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
$135$	0	0
$136$	−2176.00	−0.0100882
$137$	−384366.	−1.74962	−0.874810	−	0.484467i	$-0.839014\pi$
$137$	−384366.	−1.74962	−0.874810	+	0.484467i	$0.839014\pi$
$138$	−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	$0.499821\pi$
$140$	0	0
$141$	13248.0	0.0561180
$142$	31744.0	0.132112
$143$	−244488.	−0.999810
$144$	20736.0	0.0833333
$145$	0	0
$146$	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	$0.375167\pi$
$150$	0	0
$151$	−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
$155$	0	0
$156$	−52704.0	−0.173393
$157$	−71602.0	−0.231833	−0.115917	−	0.993259i	$-0.536981\pi$
$157$	−71602.0	−0.231833	−0.115917	+	0.993259i	$0.536981\pi$
$158$	−348352.	−1.11013
$159$	178506.	0.559964
$160$	0	0
$161$	−93492.0	−0.284256
$162$	−26244.0	−0.0785674
$163$	555196.	1.63673	0.818366	−	0.574698i	$-0.194881\pi$
$163$	555196.	1.63673	0.818366	+	0.574698i	$0.194881\pi$
$164$	−6688.00	−0.0194172
$165$	0	0
$166$	−413008.	−1.16329
$167$	−541152.	−1.50151	−0.750755	−	0.660581i	$-0.770310\pi$
$167$	−541152.	−1.50151	−0.750755	+	0.660581i	$0.770310\pi$
$168$	28224.0	0.0771517
$169$	−237337.	−0.639218
$170$	0	0
$171$	−163296.	−0.427056
$172$	−106816.	−0.275306
$173$	81358.0	0.206674	0.103337	−	0.994646i	$-0.467048\pi$
$173$	81358.0	0.206674	0.103337	+	0.994646i	$0.467048\pi$
$174$	−207144.	−0.518680
$175$	0	0
$176$	−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	$0.172798\pi$
$180$	0	0
$181$	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
$185$	0	0
$186$	−1872.00	−0.00396756
$187$	−22712.0	−0.0474954
$188$	−23552.0	−0.0485997
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	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	$-0.552154\pi$
$193$	−168818.	−0.326231	−0.163116	+	0.986607i	$0.552154\pi$
$194$	502520.	0.958626
$195$	0	0
$196$	38416.0	0.0714286
$197$	−296258.	−0.543882	−0.271941	−	0.962314i	$-0.587666\pi$
$197$	−296258.	−0.543882	−0.271941	+	0.962314i	$0.587666\pi$
$198$	216432.	0.392336
$199$	199676.	0.357432	0.178716	−	0.983901i	$-0.442806\pi$
$199$	199676.	0.357432	0.178716	+	0.983901i	$0.442806\pi$
$200$	0	0
$201$	−552852.	−0.965203
$202$	382200.	0.659041
$203$	−281946.	−0.480204
$204$	−4896.00	−0.00823694
$205$	0	0
$206$	−28352.0	−0.0465496
$207$	−154548.	−0.250690
$208$	93696.0	0.150163
$209$	1.34669e6	2.13256
$210$	0	0
$211$	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
$215$	0	0
$216$	46656.0	0.0680414
$217$	−2548.00	−0.00367325
$218$	606216.	0.863945
$219$	190206.	0.267987
$220$	0	0
$221$	12444.0	0.0171388
$222$	381384.	0.519374
$223$	647432.	0.871830	0.435915	−	0.899988i	$-0.356425\pi$
$223$	647432.	0.871830	0.435915	+	0.899988i	$0.356425\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	332696.	0.433288
$227$	809932.	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$227$	809932.	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$228$	290304.	0.369842
$229$	−1.06347e6	−1.34009	−0.670046	−	0.742319i	$-0.733726\pi$
$229$	−1.06347e6	−1.34009	−0.670046	+	0.742319i	$0.733726\pi$
$230$	0	0
$231$	294588.	0.363233
$232$	368256.	0.449190
$233$	1.15017e6	1.38794	0.693972	−	0.720002i	$-0.255859\pi$
$233$	1.15017e6	1.38794	0.693972	+	0.720002i	$0.255859\pi$
$234$	−118584.	−0.141575
$235$	0	0
$236$	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	$0.295723\pi$
$240$	0	0
$241$	−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
$245$	0	0
$246$	−15048.0	−0.0158541
$247$	−737856.	−0.769537
$248$	3328.00	0.00343601
$249$	−929268.	−0.949823
$250$	0	0
$251$	−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
$255$	0	0
$256$	65536.0	0.0625000
$257$	277570.	0.262144	0.131072	−	0.991373i	$-0.458158\pi$
$257$	277570.	0.262144	0.131072	+	0.991373i	$0.458158\pi$
$258$	−240336.	−0.224786
$259$	519106.	0.480847
$260$	0	0
$261$	−466074.	−0.423500
$262$	−435152.	−0.391641
$263$	−124124.	−0.110654	−0.0553269	−	0.998468i	$-0.517620\pi$
$263$	−124124.	−0.110654	−0.0553269	+	0.998468i	$0.517620\pi$
$264$	−384768.	−0.339773
$265$	0	0
$266$	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	$0.481481\pi$
$270$	0	0
$271$	−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
$275$	0	0
$276$	274752.	0.217104
$277$	1.04114e6	0.815284	0.407642	−	0.913142i	$-0.366351\pi$
$277$	1.04114e6	0.815284	0.407642	+	0.913142i	$0.366351\pi$
$278$	−1024.00	−0.000794672 0
$279$	−4212.00	−0.00323950
$280$	0	0
$281$	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	$-0.241801\pi$
$283$	1.95382e6	1.45017	0.725084	+	0.688660i	$0.241801\pi$
$284$	−126976.	−0.0934170
$285$	0	0
$286$	977952.	0.706972
$287$	−20482.0	−0.0146780
$288$	−82944.0	−0.0589256
$289$	−1.41870e6	−0.999186
$290$	0	0
$291$	1.13067e6	0.782715
$292$	−338144.	−0.232084
$293$	775910.	0.528010	0.264005	−	0.964521i	$-0.414956\pi$
$293$	775910.	0.528010	0.264005	+	0.964521i	$0.414956\pi$
$294$	86436.0	0.0583212
$295$	0	0
$296$	−678016.	−0.449791
$297$	486972.	0.320341
$298$	−828600.	−0.540511
$299$	−698328.	−0.451733
$300$	0	0
$301$	−327124.	−0.208112
$302$	1.10723e6	0.698588
$303$	859950.	0.538105
$304$	−516096.	−0.320292
$305$	0	0
$306$	−11016.0	−0.00672543
$307$	1.69219e6	1.02471	0.512357	−	0.858773i	$-0.328772\pi$
$307$	1.69219e6	1.02471	0.512357	+	0.858773i	$0.328772\pi$
$308$	−523712.	−0.314569
$309$	−63792.0	−0.0380076
$310$	0	0
$311$	−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	$-0.148475\pi$
$313$	3.09618e6	1.78634	0.893172	+	0.449715i	$0.148475\pi$
$314$	286408.	0.163931
$315$	0	0
$316$	1.39341e6	0.784984
$317$	−1.40432e6	−0.784908	−0.392454	−	0.919772i	$-0.628374\pi$
$317$	−1.40432e6	−0.784908	−0.392454	+	0.919772i	$0.628374\pi$
$318$	−714024.	−0.395954
$319$	3.84367e6	2.11480
$320$	0	0
$321$	1.13443e6	0.614492
$322$	373968.	0.200999
$323$	−68544.0	−0.0365564
$324$	104976.	0.0555556
$325$	0	0
$326$	−2.22078e6	−1.15734
$327$	1.36399e6	0.705408
$328$	26752.0	0.0137300
$329$	−72128.0	−0.0367379
$330$	0	0
$331$	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
$335$	0	0
$336$	−112896.	−0.0545545
$337$	1.78206e6	0.854768	0.427384	−	0.904070i	$-0.359435\pi$
$337$	1.78206e6	0.854768	0.427384	+	0.904070i	$0.359435\pi$
$338$	949348.	0.451995
$339$	748566.	0.353778
$340$	0	0
$341$	34736.0	0.0161768
$342$	653184.	0.301975
$343$	117649.	0.0539949
$344$	427264.	0.194671
$345$	0	0
$346$	−325432.	−0.146140
$347$	1.54578e6	0.689164	0.344582	−	0.938756i	$-0.388021\pi$
$347$	1.54578e6	0.689164	0.344582	+	0.938756i	$0.388021\pi$
$348$	828576.	0.366762
$349$	1.90197e6	0.835874	0.417937	−	0.908476i	$-0.362753\pi$
$349$	1.90197e6	0.835874	0.417937	+	0.908476i	$0.362753\pi$
$350$	0	0
$351$	−266814.	−0.115595
$352$	684032.	0.294252
$353$	1.74813e6	0.746684	0.373342	−	0.927694i	$-0.378212\pi$
$353$	1.74813e6	0.746684	0.373342	+	0.927694i	$0.378212\pi$
$354$	377712.	0.160196
$355$	0	0
$356$	−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	$0.469809\pi$
$360$	0	0
$361$	1.58816e6	0.641395
$362$	−514168.	−0.206221
$363$	−2.56656e6	−1.02231
$364$	286944.	0.113513
$365$	0	0
$366$	−1.39716e6	−0.545184
$367$	383768.	0.148732	0.0743659	−	0.997231i	$-0.476307\pi$
$367$	383768.	0.148732	0.0743659	+	0.997231i	$0.476307\pi$
$368$	−488448.	−0.188018
$369$	−33858.0	−0.0129448
$370$	0	0
$371$	−971866.	−0.366582
$372$	7488.00	0.00280549
$373$	3.40835e6	1.26844	0.634222	−	0.773151i	$-0.281320\pi$
$373$	3.40835e6	1.26844	0.634222	+	0.773151i	$0.281320\pi$
$374$	90848.0	0.0335843
$375$	0	0
$376$	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	$0.243947\pi$
$380$	0	0
$381$	347904.	0.122785
$382$	−1.53485e6	−0.538154
$383$	4.25208e6	1.48117	0.740584	−	0.671963i	$-0.234549\pi$
$383$	4.25208e6	1.48117	0.740584	+	0.671963i	$0.234549\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	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	$0.219430\pi$
$390$	0	0
$391$	−64872.0	−0.0214593
$392$	−153664.	−0.0505076
$393$	−979092.	−0.319773
$394$	1.18503e6	0.384583
$395$	0	0
$396$	−865728.	−0.277424
$397$	4.02580e6	1.28196	0.640982	−	0.767556i	$-0.278527\pi$
$397$	4.02580e6	1.28196	0.640982	+	0.767556i	$0.278527\pi$
$398$	−798704.	−0.252742
$399$	889056.	0.279574
$400$	0	0
$401$	−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
$405$	0	0
$406$	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	$0.465322\pi$
$410$	0	0
$411$	3.45929e6	1.01014
$412$	113408.	0.0329155
$413$	514108.	0.148313
$414$	618192.	0.177265
$415$	0	0
$416$	−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	$0.299582\pi$
$420$	0	0
$421$	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
$425$	0	0
$426$	−285696.	−0.0762746
$427$	−1.90169e6	−0.504743
$428$	−2.01677e6	−0.532165
$429$	2.20039e6	0.577241
$430$	0	0
$431$	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	$-0.630720\pi$
$433$	−3.11505e6	−0.798444	−0.399222	+	0.916854i	$0.630720\pi$
$434$	10192.0	0.00259738
$435$	0	0
$436$	−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	$0.324491\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	−49776.0	−0.0121189
$443$	−5.15771e6	−1.24867	−0.624335	−	0.781157i	$-0.714630\pi$
$443$	−5.15771e6	−1.24867	−0.624335	+	0.781157i	$0.714630\pi$
$444$	−1.52554e6	−0.367253
$445$	0	0
$446$	−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	$0.693811\pi$
$450$	0	0
$451$	279224.	0.0646415
$452$	−1.33078e6	−0.306381
$453$	2.49127e6	0.570395
$454$	−3.23973e6	−0.737682
$455$	0	0
$456$	−1.16122e6	−0.261518
$457$	8.14264e6	1.82379	0.911895	−	0.410425i	$-0.134619\pi$
$457$	8.14264e6	1.82379	0.911895	+	0.410425i	$0.134619\pi$
$458$	4.25386e6	0.947589
$459$	−24786.0	−0.00549129
$460$	0	0
$461$	−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	$-0.721659\pi$
$463$	−5.91742e6	−1.28286	−0.641431	+	0.767180i	$0.721659\pi$
$464$	−1.47302e6	−0.317625
$465$	0	0
$466$	−4.60068e6	−0.981425
$467$	2.20904e6	0.468717	0.234358	−	0.972150i	$-0.424701\pi$
$467$	2.20904e6	0.468717	0.234358	+	0.972150i	$0.424701\pi$
$468$	474336.	0.100109
$469$	3.00997e6	0.631874
$470$	0	0
$471$	644418.	0.133849
$472$	−671488.	−0.138734
$473$	4.45957e6	0.916516
$474$	3.13517e6	0.640936
$475$	0	0
$476$	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	$0.627816\pi$
$480$	0	0
$481$	3.87740e6	0.764150
$482$	6.74386e6	1.32218
$483$	841428.	0.164115
$484$	4.56277e6	0.885350
$485$	0	0
$486$	236196.	0.0453609
$487$	236992.	0.0452805	0.0226403	−	0.999744i	$-0.492793\pi$
$487$	236992.	0.0452805	0.0226403	+	0.999744i	$0.492793\pi$
$488$	2.48384e6	0.472144
$489$	−4.99676e6	−0.944967
$490$	0	0
$491$	−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
$495$	0	0
$496$	−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	$0.460031\pi$
$500$	0	0
$501$	4.87037e6	0.866897
$502$	4.66920e6	0.826957
$503$	2.67065e6	0.470649	0.235324	−	0.971917i	$-0.424385\pi$
$503$	2.67065e6	0.470649	0.235324	+	0.971917i	$0.424385\pi$
$504$	−254016.	−0.0445435
$505$	0	0
$506$	−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	$0.251082\pi$
$510$	0	0
$511$	−1.03557e6	−0.175439
$512$	−262144.	−0.0441942
$513$	1.46966e6	0.246561
$514$	−1.11028e6	−0.185364
$515$	0	0
$516$	961344.	0.158948
$517$	983296.	0.161792
$518$	−2.07642e6	−0.340010
$519$	−732222.	−0.119323
$520$	0	0
$521$	−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	$-0.392194\pi$
$523$	4.15665e6	0.664492	0.332246	+	0.943193i	$0.392194\pi$
$524$	1.74061e6	0.276932
$525$	0	0
$526$	496496.	0.0782441
$527$	−1768.00	−0.000277304 0
$528$	1.53907e6	0.240256
$529$	−2.79588e6	−0.434389
$530$	0	0
$531$	849852.	0.130800
$532$	−1.58054e6	−0.242118
$533$	−152988.	−0.0233260
$534$	−1.78164e6	−0.270375
$535$	0	0
$536$	−3.93139e6	−0.591064
$537$	−6.60690e6	−0.988694
$538$	−552072.	−0.0822318
$539$	−1.60387e6	−0.237792
$540$	0	0
$541$	−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
$545$	0	0
$546$	645624.	0.0926826
$547$	−1.13950e7	−1.62835	−0.814173	−	0.580622i	$-0.802809\pi$
$547$	−1.13950e7	−1.62835	−0.814173	+	0.580622i	$0.802809\pi$
$548$	−6.14986e6	−0.874810
$549$	−3.14361e6	−0.445141
$550$	0	0
$551$	1.16001e7	1.62773
$552$	−1.09901e6	−0.153516
$553$	4.26731e6	0.593392
$554$	−4.16455e6	−0.576493
$555$	0	0
$556$	4096.00	0.000561918 0
$557$	−2.41328e6	−0.329587	−0.164794	−	0.986328i	$-0.552696\pi$
$557$	−2.41328e6	−0.329587	−0.164794	+	0.986328i	$0.552696\pi$
$558$	16848.0	0.00229067
$559$	−2.44342e6	−0.330726
$560$	0	0
$561$	204408.	0.0274215
$562$	−1.12196e6	−0.149843
$563$	1.24262e7	1.65222	0.826109	−	0.563511i	$-0.190550\pi$
$563$	1.24262e7	1.65222	0.826109	+	0.563511i	$0.190550\pi$
$564$	211968.	0.0280590
$565$	0	0
$566$	−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	$0.708727\pi$
$570$	0	0
$571$	−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
$575$	0	0
$576$	331776.	0.0416667
$577$	1.04806e7	1.31053	0.655267	−	0.755398i	$-0.272556\pi$
$577$	1.04806e7	1.31053	0.655267	+	0.755398i	$0.272556\pi$
$578$	5.67480e6	0.706531
$579$	1.51936e6	0.188350
$580$	0	0
$581$	5.05935e6	0.621805
$582$	−4.52268e6	−0.553463
$583$	1.32491e7	1.61442
$584$	1.35258e6	0.164108
$585$	0	0
$586$	−3.10364e6	−0.373360
$587$	−1.50698e7	−1.80515	−0.902574	−	0.430536i	$-0.858325\pi$
$587$	−1.50698e7	−1.80515	−0.902574	+	0.430536i	$0.858325\pi$
$588$	−345744.	−0.0412393
$589$	104832.	0.0124510
$590$	0	0
$591$	2.66632e6	0.314010
$592$	2.71206e6	0.318050
$593$	−8.35160e6	−0.975288	−0.487644	−	0.873043i	$-0.662144\pi$
$593$	−8.35160e6	−0.975288	−0.487644	+	0.873043i	$0.662144\pi$
$594$	−1.94789e6	−0.226515
$595$	0	0
$596$	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	$0.412076\pi$
$600$	0	0
$601$	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
$605$	0	0
$606$	−3.43980e6	−0.380497
$607$	8.86825e6	0.976936	0.488468	−	0.872582i	$-0.337556\pi$
$607$	8.86825e6	0.976936	0.488468	+	0.872582i	$0.337556\pi$
$608$	2.06438e6	0.226481
$609$	2.53751e6	0.277246
$610$	0	0
$611$	−538752.	−0.0583829
$612$	44064.0	0.00475560
$613$	−1.16930e7	−1.25683	−0.628414	−	0.777879i	$-0.716295\pi$
$613$	−1.16930e7	−1.25683	−0.628414	+	0.777879i	$0.716295\pi$
$614$	−6.76875e6	−0.724582
$615$	0	0
$616$	2.09485e6	0.222434
$617$	1.14604e7	1.21196	0.605978	−	0.795482i	$-0.292782\pi$
$617$	1.14604e7	1.21196	0.605978	+	0.795482i	$0.292782\pi$
$618$	255168.	0.0268754
$619$	237488.	0.0249124	0.0124562	−	0.999922i	$-0.496035\pi$
$619$	237488.	0.0249124	0.0124562	+	0.999922i	$0.496035\pi$
$620$	0	0
$621$	1.39093e6	0.144736
$622$	3.61766e6	0.374932
$623$	−2.42501e6	−0.250319
$624$	−843264.	−0.0866966
$625$	0	0
$626$	−1.23847e7	−1.26314
$627$	−1.21202e7	−1.23123
$628$	−1.14563e6	−0.115917
$629$	360196.	0.0363005
$630$	0	0
$631$	−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
$635$	0	0
$636$	2.85610e6	0.279982
$637$	878766.	0.0858074
$638$	−1.53747e7	−1.49539
$639$	−642816.	−0.0622780
$640$	0	0
$641$	−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	$-0.710530\pi$
$643$	−1.28790e7	−1.22844	−0.614222	+	0.789133i	$0.710530\pi$
$644$	−1.49587e6	−0.142128
$645$	0	0
$646$	274176.	0.0258493
$647$	−2.06233e7	−1.93685	−0.968427	−	0.249299i	$-0.919800\pi$
$647$	−2.06233e7	−1.93685	−0.968427	+	0.249299i	$0.919800\pi$
$648$	−419904.	−0.0392837
$649$	−7.00866e6	−0.653165
$650$	0	0
$651$	22932.0	0.00212075
$652$	8.88314e6	0.818366
$653$	6.76833e6	0.621153	0.310577	−	0.950548i	$-0.399478\pi$
$653$	6.76833e6	0.621153	0.310577	+	0.950548i	$0.399478\pi$
$654$	−5.45594e6	−0.498799
$655$	0	0
$656$	−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	$0.595357\pi$
$660$	0	0
$661$	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
$665$	0	0
$666$	−3.43246e6	−0.299861
$667$	1.09786e7	0.955506
$668$	−8.65843e6	−0.750755
$669$	−5.82689e6	−0.503352
$670$	0	0
$671$	2.59251e7	2.22287
$672$	451584.	0.0385758
$673$	1.13071e7	0.962304	0.481152	−	0.876637i	$-0.340219\pi$
$673$	1.13071e7	0.962304	0.481152	+	0.876637i	$0.340219\pi$
$674$	−7.12825e6	−0.604412
$675$	0	0
$676$	−3.79739e6	−0.319609
$677$	3.55455e6	0.298066	0.149033	−	0.988832i	$-0.452384\pi$
$677$	3.55455e6	0.298066	0.149033	+	0.988832i	$0.452384\pi$
$678$	−2.99426e6	−0.250159
$679$	−6.15587e6	−0.512407
$680$	0	0
$681$	−7.28939e6	−0.602314
$682$	−138944.	−0.0114388
$683$	1.46799e7	1.20412	0.602062	−	0.798449i	$-0.294346\pi$
$683$	1.46799e7	1.20412	0.602062	+	0.798449i	$0.294346\pi$
$684$	−2.61274e6	−0.213528
$685$	0	0
$686$	−470596.	−0.0381802
$687$	9.57119e6	0.773703
$688$	−1.70906e6	−0.137653
$689$	−7.25924e6	−0.582564
$690$	0	0
$691$	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
$695$	0	0
$696$	−3.31430e6	−0.259340
$697$	−14212.0	−0.00110809
$698$	−7.60790e6	−0.591052
$699$	−1.03515e7	−0.801330
$700$	0	0
$701$	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
$705$	0	0
$706$	−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	$0.682611\pi$
$710$	0	0
$711$	7.05413e6	0.523322
$712$	3.16736e6	0.234152
$713$	99216.0	0.00730900
$714$	59976.0	0.00440283
$715$	0	0
$716$	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	$0.718642\pi$
$720$	0	0
$721$	347312.	0.0248818
$722$	−6.35263e6	−0.453535
$723$	1.51737e7	1.07956
$724$	2.05667e6	0.145821
$725$	0	0
$726$	1.02662e7	0.722885
$727$	−1.63370e7	−1.14640	−0.573201	−	0.819415i	$-0.694298\pi$
$727$	−1.63370e7	−1.14640	−0.573201	+	0.819415i	$0.694298\pi$
$728$	−1.14778e6	−0.0802655
$729$	531441.	0.0370370
$730$	0	0
$731$	−226984.	−0.0157109
$732$	5.58864e6	0.385504
$733$	1.80106e7	1.23813	0.619067	−	0.785338i	$-0.287511\pi$
$733$	1.80106e7	1.23813	0.619067	+	0.785338i	$0.287511\pi$
$734$	−1.53507e6	−0.105169
$735$	0	0
$736$	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	$0.213027\pi$
$740$	0	0
$741$	6.64070e6	0.444292
$742$	3.88746e6	0.259213
$743$	−1.61253e7	−1.07161	−0.535805	−	0.844342i	$-0.679992\pi$
$743$	−1.61253e7	−1.07161	−0.535805	+	0.844342i	$0.679992\pi$
$744$	−29952.0	−0.00198378
$745$	0	0
$746$	−1.36334e7	−0.896926
$747$	8.36341e6	0.548380
$748$	−363392.	−0.0237477
$749$	−6.17635e6	−0.402279
$750$	0	0
$751$	−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
$755$	0	0
$756$	−571536.	−0.0363696
$757$	7.09630e6	0.450083	0.225041	−	0.974349i	$-0.427748\pi$
$757$	7.09630e6	0.450083	0.225041	+	0.974349i	$0.427748\pi$
$758$	−1.61167e7	−1.01883
$759$	−1.14709e7	−0.722758
$760$	0	0
$761$	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
$765$	0	0
$766$	−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	$0.732254\pi$
$770$	0	0
$771$	−2.49813e6	−0.151349
$772$	−2.70109e6	−0.163116
$773$	1.37603e7	0.828284	0.414142	−	0.910212i	$-0.364082\pi$
$773$	1.37603e7	0.828284	0.414142	+	0.910212i	$0.364082\pi$
$774$	2.16302e6	0.129780
$775$	0	0
$776$	8.04032e6	0.479313
$777$	−4.67195e6	−0.277617
$778$	−1.84241e7	−1.09128
$779$	842688.	0.0497534
$780$	0	0
$781$	5.30125e6	0.310993
$782$	259488.	0.0151740
$783$	4.19467e6	0.244508
$784$	614656.	0.0357143
$785$	0	0
$786$	3.91637e6	0.226114
$787$	1.85194e7	1.06584	0.532918	−	0.846167i	$-0.321095\pi$
$787$	1.85194e7	1.06584	0.532918	+	0.846167i	$0.321095\pi$
$788$	−4.74013e6	−0.271941
$789$	1.11712e6	0.0638860
$790$	0	0
$791$	−4.07553e6	−0.231602
$792$	3.46291e6	0.196168
$793$	−1.42045e7	−0.802124
$794$	−1.61032e7	−0.906485
$795$	0	0
$796$	3.19482e6	0.178716
$797$	−2.01727e7	−1.12491	−0.562457	−	0.826827i	$-0.690144\pi$
$797$	−2.01727e7	−1.12491	−0.562457	+	0.826827i	$0.690144\pi$
$798$	−3.55622e6	−0.197689
$799$	−50048.0	−0.00277345
$800$	0	0
$801$	−4.00869e6	−0.220760
$802$	1.34905e7	0.740612
$803$	1.41175e7	0.772626
$804$	−8.84563e6	−0.482602
$805$	0	0
$806$	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	$0.286587\pi$
$810$	0	0
$811$	−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
$815$	0	0
$816$	−78336.0	−0.00411847
$817$	1.34588e7	0.705426
$818$	−2.94266e6	−0.153765
$819$	1.45265e6	0.0756750
$820$	0	0
$821$	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	$-0.688408\pi$
$823$	−2.16829e7	−1.11588	−0.557940	+	0.829881i	$0.688408\pi$
$824$	−453632.	−0.0232748
$825$	0	0
$826$	−2.05643e6	−0.104873
$827$	−1.10996e7	−0.564344	−0.282172	−	0.959364i	$-0.591055\pi$
$827$	−1.10996e7	−0.564344	−0.282172	+	0.959364i	$0.591055\pi$
$828$	−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	$0.422251\pi$
$830$	0	0
$831$	−9.37024e6	−0.470704
$832$	1.49914e6	0.0750815
$833$	81634.0	0.00407623
$834$	9216.00	0.000458804 0
$835$	0	0
$836$	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	$0.298740\pi$
$840$	0	0
$841$	1.25974e7	0.614172
$842$	−2.20565e7	−1.07215
$843$	−2.52441e6	−0.122346
$844$	9.54592e6	0.461277
$845$	0	0
$846$	476928.	0.0229101
$847$	1.39735e7	0.669262
$848$	−5.07750e6	−0.242471
$849$	−1.75844e7	−0.837255
$850$	0	0
$851$	−2.02134e7	−0.956785
$852$	1.14278e6	0.0539343
$853$	1.62218e7	0.763357	0.381678	−	0.924295i	$-0.375346\pi$
$853$	1.62218e7	0.763357	0.381678	+	0.924295i	$0.375346\pi$
$854$	7.60676e6	0.356907
$855$	0	0
$856$	8.06707e6	0.376298
$857$	−1.82957e7	−0.850936	−0.425468	−	0.904974i	$-0.639890\pi$
$857$	−1.82957e7	−0.850936	−0.425468	+	0.904974i	$0.639890\pi$
$858$	−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	$0.474229\pi$
$860$	0	0
$861$	184338.	0.00847436
$862$	−1.65084e7	−0.756725
$863$	3.70241e7	1.69222	0.846110	−	0.533008i	$-0.178938\pi$
$863$	3.70241e7	1.69222	0.846110	+	0.533008i	$0.178938\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	1.24602e7	0.564586
$867$	1.27683e7	0.576880
$868$	−40768.0	−0.00183662
$869$	−5.81748e7	−2.61328
$870$	0	0
$871$	2.24826e7	1.00416
$872$	9.69946e6	0.431973
$873$	−1.01760e7	−0.451900
$874$	−1.53861e7	−0.681319
$875$	0	0
$876$	3.04330e6	0.133994
$877$	−186038.	−0.00816775	−0.00408388	−	0.999992i	$-0.501300\pi$
$877$	−186038.	−0.00816775	−0.00408388	+	0.999992i	$0.501300\pi$
$878$	−1.69226e7	−0.740851
$879$	−6.98319e6	−0.304847
$880$	0	0
$881$	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	$-0.614401\pi$
$883$	−1.62975e7	−0.703426	−0.351713	+	0.936108i	$0.614401\pi$
$884$	199104.	0.00856938
$885$	0	0
$886$	2.06308e7	0.882943
$887$	2.32942e7	0.994120	0.497060	−	0.867716i	$-0.334413\pi$
$887$	2.32942e7	0.994120	0.497060	+	0.867716i	$0.334413\pi$
$888$	6.10214e6	0.259687
$889$	−1.89414e6	−0.0803820
$890$	0	0
$891$	−4.38275e6	−0.184949
$892$	1.03589e7	0.435915
$893$	2.96755e6	0.124529
$894$	7.45740e6	0.312064
$895$	0	0
$896$	−802816.	−0.0334077
$897$	6.28495e6	0.260808
$898$	1.95461e7	0.808853
$899$	299208.	0.0123474
$900$	0	0
$901$	−674356.	−0.0276743
$902$	−1.11690e6	−0.0457084
$903$	2.94412e6	0.120153
$904$	5.32314e6	0.216644
$905$	0	0
$906$	−9.96509e6	−0.403330
$907$	−3.13298e7	−1.26456	−0.632281	−	0.774739i	$-0.717881\pi$
$907$	−3.13298e7	−1.26456	−0.632281	+	0.774739i	$0.717881\pi$
$908$	1.29589e7	0.521620
$909$	−7.73955e6	−0.310675
$910$	0	0
$911$	−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
$915$	0	0
$916$	−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	$0.311195\pi$
$920$	0	0
$921$	−1.52297e7	−0.591619
$922$	1.58190e7	0.612848
$923$	−2.90458e6	−0.112222
$924$	4.71341e6	0.181616
$925$	0	0
$926$	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	$0.525035\pi$
$930$	0	0
$931$	−4.84042e6	−0.183024
$932$	1.84027e7	0.693972
$933$	8.13974e6	0.306131
$934$	−8.83614e6	−0.331433
$935$	0	0
$936$	−1.89734e6	−0.0707875
$937$	4.16051e6	0.154809	0.0774047	−	0.997000i	$-0.475337\pi$
$937$	4.16051e6	0.154809	0.0774047	+	0.997000i	$0.475337\pi$
$938$	−1.20399e7	−0.446802
$939$	−2.78656e7	−1.03135
$940$	0	0
$941$	−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
$945$	0	0
$946$	−1.78383e7	−0.648075
$947$	8.32317e6	0.301588	0.150794	−	0.988565i	$-0.451817\pi$
$947$	8.32317e6	0.301588	0.150794	+	0.988565i	$0.451817\pi$
$948$	−1.25407e7	−0.453211
$949$	−7.73504e6	−0.278803
$950$	0	0
$951$	1.26389e7	0.453167
$952$	−106624.	−0.00381296
$953$	−4.27085e7	−1.52329	−0.761644	−	0.647995i	$-0.775608\pi$
$953$	−4.27085e7	−1.52329	−0.761644	+	0.647995i	$0.775608\pi$
$954$	6.42622e6	0.228604
$955$	0	0
$956$	1.69155e7	0.598603
$957$	−3.45930e7	−1.22098
$958$	1.57011e7	0.552734
$959$	−1.88339e7	−0.661294
$960$	0	0
$961$	−2.86264e7	−0.999906
$962$	−1.55096e7	−0.540335
$963$	−1.02099e7	−0.354777
$964$	−2.69755e7	−0.934924
$965$	0	0
$966$	−3.36571e6	−0.116047
$967$	3.75367e7	1.29089	0.645445	−	0.763807i	$-0.276672\pi$
$967$	3.75367e7	1.29089	0.645445	+	0.763807i	$0.276672\pi$
$968$	−1.82511e7	−0.626037
$969$	616896.	0.0211058
$970$	0	0
$971$	−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
$975$	0	0
$976$	−9.93536e6	−0.333856
$977$	−2.37522e7	−0.796098	−0.398049	−	0.917364i	$-0.630313\pi$
$977$	−2.37522e7	−0.796098	−0.398049	+	0.917364i	$0.630313\pi$
$978$	1.99871e7	0.668193
$979$	3.30593e7	1.10239
$980$	0	0
$981$	−1.22759e7	−0.407268
$982$	1.70138e7	0.563020
$983$	−4.51922e7	−1.49169	−0.745847	−	0.666118i	$-0.767955\pi$
$983$	−4.51922e7	−1.49169	−0.745847	+	0.666118i	$0.767955\pi$
$984$	−240768.	−0.00792704
$985$	0	0
$986$	782544.	0.0256340
$987$	649152.	0.0212106
$988$	−1.18057e7	−0.384768
$989$	1.27378e7	0.414099
$990$	0	0
$991$	−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
$995$	0	0
$996$	−1.48683e7	−0.474911
$997$	−1.95602e7	−0.623213	−0.311606	−	0.950211i	$-0.600867\pi$
$997$	−1.95602e7	−0.623213	−0.311606	+	0.950211i	$0.600867\pi$
$998$	−5.57275e6	−0.177110
$999$	−7.72303e6	−0.244835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.6.a.b.1.1		1
5.2	odd	4		1050.6.g.j.799.1		2
5.3	odd	4		1050.6.g.j.799.2		2
5.4	even	2		210.6.a.j.1.1	✓	1
15.14	odd	2		630.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.6.a.j.1.1	✓	1	5.4	even	2
630.6.a.d.1.1		1	15.14	odd	2
1050.6.a.b.1.1		1	1.1	even	1	trivial
1050.6.g.j.799.1		2	5.2	odd	4
1050.6.g.j.799.2		2	5.3	odd	4

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:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more