LMFDB - Embedded newform 784.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+26.0000 q^{3} -16.0000 q^{5} +433.000 q^{9} +O(q^{10})$

$q+26.0000 q^{3} -16.0000 q^{5} +433.000 q^{9} -8.00000 q^{11} -684.000 q^{13} -416.000 q^{15} +2218.00 q^{17} -2698.00 q^{19} -3344.00 q^{23} -2869.00 q^{25} +4940.00 q^{27} -3254.00 q^{29} +4788.00 q^{31} -208.000 q^{33} -11470.0 q^{37} -17784.0 q^{39} -13350.0 q^{41} +928.000 q^{43} -6928.00 q^{45} +1212.00 q^{47} +57668.0 q^{51} +13110.0 q^{53} +128.000 q^{55} -70148.0 q^{57} +34702.0 q^{59} +1032.00 q^{61} +10944.0 q^{65} -10108.0 q^{67} -86944.0 q^{69} -62720.0 q^{71} +18926.0 q^{73} -74594.0 q^{75} -11400.0 q^{79} +23221.0 q^{81} +88958.0 q^{83} -35488.0 q^{85} -84604.0 q^{87} -19722.0 q^{89} +124488. q^{93} +43168.0 q^{95} -17062.0 q^{97} -3464.00 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	26.0000	1.66790	0.833950	−	0.551839i	$-0.186074\pi$
$3$	26.0000	1.66790	0.833950	+	0.551839i	$0.186074\pi$
$4$	0	0
$5$	−16.0000	−0.286217	−0.143108	−	0.989707i	$-0.545710\pi$
$5$	−16.0000	−0.286217	−0.143108	+	0.989707i	$0.545710\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	433.000	1.78189
$10$	0	0
$11$	−8.00000	−0.0199346	−0.00996732	−	0.999950i	$-0.503173\pi$
$11$	−8.00000	−0.0199346	−0.00996732	+	0.999950i	$0.503173\pi$
$12$	0	0
$13$	−684.000	−1.12253	−0.561265	−	0.827636i	$-0.689685\pi$
$13$	−684.000	−1.12253	−0.561265	+	0.827636i	$0.689685\pi$
$14$	0	0
$15$	−416.000	−0.477381
$16$	0	0
$17$	2218.00	1.86140	0.930699	−	0.365786i	$-0.119200\pi$
$17$	2218.00	1.86140	0.930699	+	0.365786i	$0.119200\pi$
$18$	0	0
$19$	−2698.00	−1.71458	−0.857290	−	0.514833i	$-0.827854\pi$
$19$	−2698.00	−1.71458	−0.857290	+	0.514833i	$0.827854\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3344.00	−1.31809	−0.659047	−	0.752101i	$-0.729040\pi$
$23$	−3344.00	−1.31809	−0.659047	+	0.752101i	$0.729040\pi$
$24$	0	0
$25$	−2869.00	−0.918080
$26$	0	0
$27$	4940.00	1.30412
$28$	0	0
$29$	−3254.00	−0.718493	−0.359247	−	0.933243i	$-0.616966\pi$
$29$	−3254.00	−0.718493	−0.359247	+	0.933243i	$0.616966\pi$
$30$	0	0
$31$	4788.00	0.894849	0.447425	−	0.894322i	$-0.352341\pi$
$31$	4788.00	0.894849	0.447425	+	0.894322i	$0.352341\pi$
$32$	0	0
$33$	−208.000	−0.0332490
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11470.0	−1.37740	−0.688698	−	0.725048i	$-0.741818\pi$
$37$	−11470.0	−1.37740	−0.688698	+	0.725048i	$0.741818\pi$
$38$	0	0
$39$	−17784.0	−1.87227
$40$	0	0
$41$	−13350.0	−1.24029	−0.620143	−	0.784489i	$-0.712925\pi$
$41$	−13350.0	−1.24029	−0.620143	+	0.784489i	$0.712925\pi$
$42$	0	0
$43$	928.000	0.0765380	0.0382690	−	0.999267i	$-0.487816\pi$
$43$	928.000	0.0765380	0.0382690	+	0.999267i	$0.487816\pi$
$44$	0	0
$45$	−6928.00	−0.510008
$46$	0	0
$47$	1212.00	0.0800310	0.0400155	−	0.999199i	$-0.487259\pi$
$47$	1212.00	0.0800310	0.0400155	+	0.999199i	$0.487259\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	57668.0	3.10463
$52$	0	0
$53$	13110.0	0.641081	0.320541	−	0.947235i	$-0.396135\pi$
$53$	13110.0	0.641081	0.320541	+	0.947235i	$0.396135\pi$
$54$	0	0
$55$	128.000	0.00570563
$56$	0	0
$57$	−70148.0	−2.85975
$58$	0	0
$59$	34702.0	1.29785	0.648925	−	0.760852i	$-0.275219\pi$
$59$	34702.0	1.29785	0.648925	+	0.760852i	$0.275219\pi$
$60$	0	0
$61$	1032.00	0.0355104	0.0177552	−	0.999842i	$-0.494348\pi$
$61$	1032.00	0.0355104	0.0177552	+	0.999842i	$0.494348\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10944.0	0.321287
$66$	0	0
$67$	−10108.0	−0.275092	−0.137546	−	0.990495i	$-0.543922\pi$
$67$	−10108.0	−0.275092	−0.137546	+	0.990495i	$0.543922\pi$
$68$	0	0
$69$	−86944.0	−2.19845
$70$	0	0
$71$	−62720.0	−1.47659	−0.738295	−	0.674477i	$-0.764369\pi$
$71$	−62720.0	−1.47659	−0.738295	+	0.674477i	$0.764369\pi$
$72$	0	0
$73$	18926.0	0.415673	0.207836	−	0.978164i	$-0.433358\pi$
$73$	18926.0	0.415673	0.207836	+	0.978164i	$0.433358\pi$
$74$	0	0
$75$	−74594.0	−1.53127
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11400.0	−0.205512	−0.102756	−	0.994707i	$-0.532766\pi$
$79$	−11400.0	−0.205512	−0.102756	+	0.994707i	$0.532766\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	0	0
$83$	88958.0	1.41739	0.708696	−	0.705514i	$-0.249284\pi$
$83$	88958.0	1.41739	0.708696	+	0.705514i	$0.249284\pi$
$84$	0	0
$85$	−35488.0	−0.532763
$86$	0	0
$87$	−84604.0	−1.19838
$88$	0	0
$89$	−19722.0	−0.263922	−0.131961	−	0.991255i	$-0.542127\pi$
$89$	−19722.0	−0.263922	−0.131961	+	0.991255i	$0.542127\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	124488.	1.49252
$94$	0	0
$95$	43168.0	0.490742
$96$	0	0
$97$	−17062.0	−0.184120	−0.0920599	−	0.995753i	$-0.529345\pi$
$97$	−17062.0	−0.184120	−0.0920599	+	0.995753i	$0.529345\pi$
$98$	0	0
$99$	−3464.00	−0.0355214
$100$	0	0
$101$	−45904.0	−0.447762	−0.223881	−	0.974617i	$-0.571873\pi$
$101$	−45904.0	−0.447762	−0.223881	+	0.974617i	$0.571873\pi$
$102$	0	0
$103$	−136012.	−1.26324	−0.631618	−	0.775280i	$-0.717609\pi$
$103$	−136012.	−1.26324	−0.631618	+	0.775280i	$0.717609\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	69156.0	0.583943	0.291971	−	0.956427i	$-0.405689\pi$
$107$	69156.0	0.583943	0.291971	+	0.956427i	$0.405689\pi$
$108$	0	0
$109$	−146414.	−1.18037	−0.590183	−	0.807270i	$-0.700944\pi$
$109$	−146414.	−1.18037	−0.590183	+	0.807270i	$0.700944\pi$
$110$	0	0
$111$	−298220.	−2.29736
$112$	0	0
$113$	−80186.0	−0.590748	−0.295374	−	0.955382i	$-0.595444\pi$
$113$	−80186.0	−0.590748	−0.295374	+	0.955382i	$0.595444\pi$
$114$	0	0
$115$	53504.0	0.377261
$116$	0	0
$117$	−296172.	−2.00023
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−160987.	−0.999603
$122$	0	0
$123$	−347100.	−2.06867
$124$	0	0
$125$	95904.0	0.548987
$126$	0	0
$127$	−274800.	−1.51185	−0.755923	−	0.654661i	$-0.772811\pi$
$127$	−274800.	−1.51185	−0.755923	+	0.654661i	$0.772811\pi$
$128$	0	0
$129$	24128.0	0.127658
$130$	0	0
$131$	180742.	0.920197	0.460099	−	0.887868i	$-0.347814\pi$
$131$	180742.	0.920197	0.460099	+	0.887868i	$0.347814\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−79040.0	−0.373261
$136$	0	0
$137$	−209678.	−0.954446	−0.477223	−	0.878782i	$-0.658357\pi$
$137$	−209678.	−0.954446	−0.477223	+	0.878782i	$0.658357\pi$
$138$	0	0
$139$	17242.0	0.0756921	0.0378461	−	0.999284i	$-0.487950\pi$
$139$	17242.0	0.0756921	0.0378461	+	0.999284i	$0.487950\pi$
$140$	0	0
$141$	31512.0	0.133484
$142$	0	0
$143$	5472.00	0.0223772
$144$	0	0
$145$	52064.0	0.205645
$146$	0	0
$147$	0	0
$148$	0	0
$149$	59358.0	0.219035	0.109518	−	0.993985i	$-0.465069\pi$
$149$	59358.0	0.219035	0.109518	+	0.993985i	$0.465069\pi$
$150$	0	0
$151$	336344.	1.20044	0.600221	−	0.799834i	$-0.295079\pi$
$151$	336344.	1.20044	0.600221	+	0.799834i	$0.295079\pi$
$152$	0	0
$153$	960394.	3.31681
$154$	0	0
$155$	−76608.0	−0.256121
$156$	0	0
$157$	−464588.	−1.50425	−0.752123	−	0.659023i	$-0.770970\pi$
$157$	−464588.	−1.50425	−0.752123	+	0.659023i	$0.770970\pi$
$158$	0	0
$159$	340860.	1.06926
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−314792.	−0.928014	−0.464007	−	0.885831i	$-0.653589\pi$
$163$	−314792.	−0.928014	−0.464007	+	0.885831i	$0.653589\pi$
$164$	0	0
$165$	3328.00	0.00951642
$166$	0	0
$167$	285724.	0.792785	0.396393	−	0.918081i	$-0.370262\pi$
$167$	285724.	0.792785	0.396393	+	0.918081i	$0.370262\pi$
$168$	0	0
$169$	96563.0	0.260072
$170$	0	0
$171$	−1.16823e6	−3.05520
$172$	0	0
$173$	709148.	1.80145	0.900724	−	0.434392i	$-0.143037\pi$
$173$	709148.	1.80145	0.900724	+	0.434392i	$0.143037\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	902252.	2.16468
$178$	0	0
$179$	617148.	1.43965	0.719825	−	0.694156i	$-0.244222\pi$
$179$	617148.	1.43965	0.719825	+	0.694156i	$0.244222\pi$
$180$	0	0
$181$	−237828.	−0.539593	−0.269797	−	0.962917i	$-0.586956\pi$
$181$	−237828.	−0.539593	−0.269797	+	0.962917i	$0.586956\pi$
$182$	0	0
$183$	26832.0	0.0592278
$184$	0	0
$185$	183520.	0.394234
$186$	0	0
$187$	−17744.0	−0.0371063
$188$	0	0
$189$	0	0
$190$	0	0
$191$	133512.	0.264812	0.132406	−	0.991196i	$-0.457730\pi$
$191$	133512.	0.264812	0.132406	+	0.991196i	$0.457730\pi$
$192$	0	0
$193$	270446.	0.522622	0.261311	−	0.965255i	$-0.415845\pi$
$193$	270446.	0.522622	0.261311	+	0.965255i	$0.415845\pi$
$194$	0	0
$195$	284544.	0.535874
$196$	0	0
$197$	875102.	1.60655	0.803273	−	0.595611i	$-0.203090\pi$
$197$	875102.	1.60655	0.803273	+	0.595611i	$0.203090\pi$
$198$	0	0
$199$	−347620.	−0.622260	−0.311130	−	0.950367i	$-0.600708\pi$
$199$	−347620.	−0.622260	−0.311130	+	0.950367i	$0.600708\pi$
$200$	0	0
$201$	−262808.	−0.458826
$202$	0	0
$203$	0	0
$204$	0	0
$205$	213600.	0.354990
$206$	0	0
$207$	−1.44795e6	−2.34870
$208$	0	0
$209$	21584.0	0.0341795
$210$	0	0
$211$	425380.	0.657765	0.328883	−	0.944371i	$-0.393328\pi$
$211$	425380.	0.657765	0.328883	+	0.944371i	$0.393328\pi$
$212$	0	0
$213$	−1.63072e6	−2.46281
$214$	0	0
$215$	−14848.0	−0.0219064
$216$	0	0
$217$	0	0
$218$	0	0
$219$	492076.	0.693301
$220$	0	0
$221$	−1.51711e6	−2.08947
$222$	0	0
$223$	481592.	0.648511	0.324255	−	0.945970i	$-0.394886\pi$
$223$	481592.	0.648511	0.324255	+	0.945970i	$0.394886\pi$
$224$	0	0
$225$	−1.24228e6	−1.63592
$226$	0	0
$227$	−6042.00	−0.00778245	−0.00389122	−	0.999992i	$-0.501239\pi$
$227$	−6042.00	−0.00778245	−0.00389122	+	0.999992i	$0.501239\pi$
$228$	0	0
$229$	−1804.00	−0.00227325	−0.00113663	−	0.999999i	$-0.500362\pi$
$229$	−1804.00	−0.00227325	−0.00113663	+	0.999999i	$0.500362\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.61153e6	−1.94468	−0.972339	−	0.233576i	$-0.924957\pi$
$233$	−1.61153e6	−1.94468	−0.972339	+	0.233576i	$0.924957\pi$
$234$	0	0
$235$	−19392.0	−0.0229062
$236$	0	0
$237$	−296400.	−0.342774
$238$	0	0
$239$	987096.	1.11780	0.558901	−	0.829235i	$-0.311223\pi$
$239$	987096.	1.11780	0.558901	+	0.829235i	$0.311223\pi$
$240$	0	0
$241$	−893510.	−0.990962	−0.495481	−	0.868619i	$-0.665008\pi$
$241$	−893510.	−0.990962	−0.495481	+	0.868619i	$0.665008\pi$
$242$	0	0
$243$	−596674.	−0.648219
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.84543e6	1.92467
$248$	0	0
$249$	2.31291e6	2.36407
$250$	0	0
$251$	365946.	0.366634	0.183317	−	0.983054i	$-0.441317\pi$
$251$	365946.	0.366634	0.183317	+	0.983054i	$0.441317\pi$
$252$	0	0
$253$	26752.0	0.0262757
$254$	0	0
$255$	−922688.	−0.888596
$256$	0	0
$257$	−1.40459e6	−1.32653	−0.663266	−	0.748383i	$-0.730830\pi$
$257$	−1.40459e6	−1.32653	−0.663266	+	0.748383i	$0.730830\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.40898e6	−1.28028
$262$	0	0
$263$	−1.09968e6	−0.980341	−0.490170	−	0.871627i	$-0.663065\pi$
$263$	−1.09968e6	−0.980341	−0.490170	+	0.871627i	$0.663065\pi$
$264$	0	0
$265$	−209760.	−0.183488
$266$	0	0
$267$	−512772.	−0.440196
$268$	0	0
$269$	−814948.	−0.686672	−0.343336	−	0.939213i	$-0.611557\pi$
$269$	−814948.	−0.686672	−0.343336	+	0.939213i	$0.611557\pi$
$270$	0	0
$271$	−1.69906e6	−1.40535	−0.702675	−	0.711511i	$-0.748011\pi$
$271$	−1.69906e6	−1.40535	−0.702675	+	0.711511i	$0.748011\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22952.0	0.0183016
$276$	0	0
$277$	−1.36508e6	−1.06895	−0.534477	−	0.845183i	$-0.679492\pi$
$277$	−1.36508e6	−1.06895	−0.534477	+	0.845183i	$0.679492\pi$
$278$	0	0
$279$	2.07320e6	1.59453
$280$	0	0
$281$	−715846.	−0.540821	−0.270411	−	0.962745i	$-0.587159\pi$
$281$	−715846.	−0.540821	−0.270411	+	0.962745i	$0.587159\pi$
$282$	0	0
$283$	217726.	0.161601	0.0808005	−	0.996730i	$-0.474252\pi$
$283$	217726.	0.161601	0.0808005	+	0.996730i	$0.474252\pi$
$284$	0	0
$285$	1.12237e6	0.818508
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.49967e6	2.46480
$290$	0	0
$291$	−443612.	−0.307094
$292$	0	0
$293$	−1.50708e6	−1.02557	−0.512787	−	0.858516i	$-0.671387\pi$
$293$	−1.50708e6	−1.02557	−0.512787	+	0.858516i	$0.671387\pi$
$294$	0	0
$295$	−555232.	−0.371466
$296$	0	0
$297$	−39520.0	−0.0259972
$298$	0	0
$299$	2.28730e6	1.47960
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.19350e6	−0.746822
$304$	0	0
$305$	−16512.0	−0.0101637
$306$	0	0
$307$	12502.0	0.00757066	0.00378533	−	0.999993i	$-0.498795\pi$
$307$	12502.0	0.00757066	0.00378533	+	0.999993i	$0.498795\pi$
$308$	0	0
$309$	−3.53631e6	−2.10695
$310$	0	0
$311$	−647432.	−0.379571	−0.189786	−	0.981826i	$-0.560779\pi$
$311$	−647432.	−0.379571	−0.189786	+	0.981826i	$0.560779\pi$
$312$	0	0
$313$	935978.	0.540014	0.270007	−	0.962858i	$-0.412974\pi$
$313$	935978.	0.540014	0.270007	+	0.962858i	$0.412974\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	705942.	0.394567	0.197284	−	0.980346i	$-0.436788\pi$
$317$	705942.	0.394567	0.197284	+	0.980346i	$0.436788\pi$
$318$	0	0
$319$	26032.0	0.0143229
$320$	0	0
$321$	1.79806e6	0.973959
$322$	0	0
$323$	−5.98416e6	−3.19152
$324$	0	0
$325$	1.96240e6	1.03057
$326$	0	0
$327$	−3.80676e6	−1.96873
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.14304e6	0.573445	0.286722	−	0.958014i	$-0.407434\pi$
$331$	1.14304e6	0.573445	0.286722	+	0.958014i	$0.407434\pi$
$332$	0	0
$333$	−4.96651e6	−2.45437
$334$	0	0
$335$	161728.	0.0787360
$336$	0	0
$337$	−2.36402e6	−1.13390	−0.566952	−	0.823751i	$-0.691877\pi$
$337$	−2.36402e6	−1.13390	−0.566952	+	0.823751i	$0.691877\pi$
$338$	0	0
$339$	−2.08484e6	−0.985309
$340$	0	0
$341$	−38304.0	−0.0178385
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.39110e6	0.629234
$346$	0	0
$347$	−726240.	−0.323785	−0.161892	−	0.986808i	$-0.551760\pi$
$347$	−726240.	−0.323785	−0.161892	+	0.986808i	$0.551760\pi$
$348$	0	0
$349$	−136180.	−0.0598480	−0.0299240	−	0.999552i	$-0.509527\pi$
$349$	−136180.	−0.0598480	−0.0299240	+	0.999552i	$0.509527\pi$
$350$	0	0
$351$	−3.37896e6	−1.46391
$352$	0	0
$353$	1.16907e6	0.499349	0.249674	−	0.968330i	$-0.419676\pi$
$353$	1.16907e6	0.499349	0.249674	+	0.968330i	$0.419676\pi$
$354$	0	0
$355$	1.00352e6	0.422625
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4280.00	0.00175270	0.000876350	−	1.00000i	$-0.499721\pi$
$359$	4280.00	0.00175270	0.000876350		1.00000i	$0.499721\pi$
$360$	0	0
$361$	4.80310e6	1.93979
$362$	0	0
$363$	−4.18566e6	−1.66724
$364$	0	0
$365$	−302816.	−0.118973
$366$	0	0
$367$	2.44796e6	0.948722	0.474361	−	0.880330i	$-0.342679\pi$
$367$	2.44796e6	0.948722	0.474361	+	0.880330i	$0.342679\pi$
$368$	0	0
$369$	−5.78055e6	−2.21006
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−904514.	−0.336623	−0.168311	−	0.985734i	$-0.553831\pi$
$373$	−904514.	−0.336623	−0.168311	+	0.985734i	$0.553831\pi$
$374$	0	0
$375$	2.49350e6	0.915655
$376$	0	0
$377$	2.22574e6	0.806530
$378$	0	0
$379$	4.23034e6	1.51279	0.756393	−	0.654117i	$-0.226960\pi$
$379$	4.23034e6	1.51279	0.756393	+	0.654117i	$0.226960\pi$
$380$	0	0
$381$	−7.14480e6	−2.52161
$382$	0	0
$383$	4.55400e6	1.58634	0.793169	−	0.609002i	$-0.208430\pi$
$383$	4.55400e6	1.58634	0.793169	+	0.609002i	$0.208430\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	401824.	0.136382
$388$	0	0
$389$	−3.98541e6	−1.33536	−0.667680	−	0.744448i	$-0.732713\pi$
$389$	−3.98541e6	−1.33536	−0.667680	+	0.744448i	$0.732713\pi$
$390$	0	0
$391$	−7.41699e6	−2.45350
$392$	0	0
$393$	4.69929e6	1.53480
$394$	0	0
$395$	182400.	0.0588210
$396$	0	0
$397$	−552420.	−0.175911	−0.0879555	−	0.996124i	$-0.528033\pi$
$397$	−552420.	−0.175911	−0.0879555	+	0.996124i	$0.528033\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38190.0	0.0118601	0.00593006	−	0.999982i	$-0.498112\pi$
$401$	38190.0	0.0118601	0.00593006	+	0.999982i	$0.498112\pi$
$402$	0	0
$403$	−3.27499e6	−1.00449
$404$	0	0
$405$	−371536.	−0.112555
$406$	0	0
$407$	91760.0	0.0274579
$408$	0	0
$409$	3.92475e6	1.16012	0.580062	−	0.814573i	$-0.303028\pi$
$409$	3.92475e6	1.16012	0.580062	+	0.814573i	$0.303028\pi$
$410$	0	0
$411$	−5.45163e6	−1.59192
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.42333e6	−0.405681
$416$	0	0
$417$	448292.	0.126247
$418$	0	0
$419$	598386.	0.166512	0.0832562	−	0.996528i	$-0.473468\pi$
$419$	598386.	0.166512	0.0832562	+	0.996528i	$0.473468\pi$
$420$	0	0
$421$	4.61597e6	1.26928	0.634641	−	0.772807i	$-0.281148\pi$
$421$	4.61597e6	1.26928	0.634641	+	0.772807i	$0.281148\pi$
$422$	0	0
$423$	524796.	0.142607
$424$	0	0
$425$	−6.36344e6	−1.70891
$426$	0	0
$427$	0	0
$428$	0	0
$429$	142272.	0.0373230
$430$	0	0
$431$	61560.0	0.0159627	0.00798133	−	0.999968i	$-0.497459\pi$
$431$	61560.0	0.0159627	0.00798133	+	0.999968i	$0.497459\pi$
$432$	0	0
$433$	3.79727e6	0.973310	0.486655	−	0.873594i	$-0.338217\pi$
$433$	3.79727e6	0.973310	0.486655	+	0.873594i	$0.338217\pi$
$434$	0	0
$435$	1.35366e6	0.342995
$436$	0	0
$437$	9.02211e6	2.25998
$438$	0	0
$439$	−2.28852e6	−0.566752	−0.283376	−	0.959009i	$-0.591455\pi$
$439$	−2.28852e6	−0.566752	−0.283376	+	0.959009i	$0.591455\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.75976e6	1.15233	0.576163	−	0.817335i	$-0.304549\pi$
$443$	4.75976e6	1.15233	0.576163	+	0.817335i	$0.304549\pi$
$444$	0	0
$445$	315552.	0.0755389
$446$	0	0
$447$	1.54331e6	0.365329
$448$	0	0
$449$	−4.36715e6	−1.02231	−0.511155	−	0.859489i	$-0.670782\pi$
$449$	−4.36715e6	−1.02231	−0.511155	+	0.859489i	$0.670782\pi$
$450$	0	0
$451$	106800.	0.0247246
$452$	0	0
$453$	8.74494e6	2.00222
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.44994e6	1.22068	0.610339	−	0.792140i	$-0.291033\pi$
$457$	5.44994e6	1.22068	0.610339	+	0.792140i	$0.291033\pi$
$458$	0	0
$459$	1.09569e7	2.42749
$460$	0	0
$461$	−1.66966e6	−0.365911	−0.182956	−	0.983121i	$-0.558566\pi$
$461$	−1.66966e6	−0.365911	−0.182956	+	0.983121i	$0.558566\pi$
$462$	0	0
$463$	−70768.0	−0.0153421	−0.00767104	−	0.999971i	$-0.502442\pi$
$463$	−70768.0	−0.0153421	−0.00767104	+	0.999971i	$0.502442\pi$
$464$	0	0
$465$	−1.99181e6	−0.427184
$466$	0	0
$467$	5.66083e6	1.20112	0.600562	−	0.799578i	$-0.294944\pi$
$467$	5.66083e6	1.20112	0.600562	+	0.799578i	$0.294944\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.20793e7	−2.50893
$472$	0	0
$473$	−7424.00	−0.00152576
$474$	0	0
$475$	7.74056e6	1.57412
$476$	0	0
$477$	5.67663e6	1.14234
$478$	0	0
$479$	−1.44948e6	−0.288652	−0.144326	−	0.989530i	$-0.546101\pi$
$479$	−1.44948e6	−0.288652	−0.144326	+	0.989530i	$0.546101\pi$
$480$	0	0
$481$	7.84548e6	1.54617
$482$	0	0
$483$	0	0
$484$	0	0
$485$	272992.	0.0526982
$486$	0	0
$487$	−4.07504e6	−0.778591	−0.389296	−	0.921113i	$-0.627282\pi$
$487$	−4.07504e6	−0.778591	−0.389296	+	0.921113i	$0.627282\pi$
$488$	0	0
$489$	−8.18459e6	−1.54784
$490$	0	0
$491$	−986100.	−0.184594	−0.0922969	−	0.995732i	$-0.529421\pi$
$491$	−986100.	−0.184594	−0.0922969	+	0.995732i	$0.529421\pi$
$492$	0	0
$493$	−7.21737e6	−1.33740
$494$	0	0
$495$	55424.0	0.0101668
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.98342e6	−1.07572	−0.537859	−	0.843035i	$-0.680767\pi$
$499$	−5.98342e6	−1.07572	−0.537859	+	0.843035i	$0.680767\pi$
$500$	0	0
$501$	7.42882e6	1.32229
$502$	0	0
$503$	−3.49373e6	−0.615700	−0.307850	−	0.951435i	$-0.599609\pi$
$503$	−3.49373e6	−0.615700	−0.307850	+	0.951435i	$0.599609\pi$
$504$	0	0
$505$	734464.	0.128157
$506$	0	0
$507$	2.51064e6	0.433775
$508$	0	0
$509$	−2.15711e6	−0.369043	−0.184522	−	0.982828i	$-0.559074\pi$
$509$	−2.15711e6	−0.369043	−0.184522	+	0.982828i	$0.559074\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.33281e7	−2.23602
$514$	0	0
$515$	2.17619e6	0.361559
$516$	0	0
$517$	−9696.00	−0.00159539
$518$	0	0
$519$	1.84378e7	3.00464
$520$	0	0
$521$	6.65817e6	1.07463	0.537317	−	0.843380i	$-0.319438\pi$
$521$	6.65817e6	1.07463	0.537317	+	0.843380i	$0.319438\pi$
$522$	0	0
$523$	5.95223e6	0.951537	0.475768	−	0.879571i	$-0.342170\pi$
$523$	5.95223e6	0.951537	0.475768	+	0.879571i	$0.342170\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.06198e7	1.66567
$528$	0	0
$529$	4.74599e6	0.737374
$530$	0	0
$531$	1.50260e7	2.31263
$532$	0	0
$533$	9.13140e6	1.39226
$534$	0	0
$535$	−1.10650e6	−0.167134
$536$	0	0
$537$	1.60458e7	2.40119
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.39681e6	−0.939659	−0.469830	−	0.882757i	$-0.655685\pi$
$541$	−6.39681e6	−0.939659	−0.469830	+	0.882757i	$0.655685\pi$
$542$	0	0
$543$	−6.18353e6	−0.899988
$544$	0	0
$545$	2.34262e6	0.337840
$546$	0	0
$547$	5.51851e6	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$547$	5.51851e6	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$548$	0	0
$549$	446856.	0.0632757
$550$	0	0
$551$	8.77929e6	1.23191
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.77152e6	0.657543
$556$	0	0
$557$	2.02159e6	0.276093	0.138046	−	0.990426i	$-0.455918\pi$
$557$	2.02159e6	0.276093	0.138046	+	0.990426i	$0.455918\pi$
$558$	0	0
$559$	−634752.	−0.0859161
$560$	0	0
$561$	−461344.	−0.0618896
$562$	0	0
$563$	8.14678e6	1.08322	0.541608	−	0.840631i	$-0.317816\pi$
$563$	8.14678e6	1.08322	0.541608	+	0.840631i	$0.317816\pi$
$564$	0	0
$565$	1.28298e6	0.169082
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.19824e7	−1.55154	−0.775772	−	0.631013i	$-0.782639\pi$
$569$	−1.19824e7	−1.55154	−0.775772	+	0.631013i	$0.782639\pi$
$570$	0	0
$571$	−1.39582e6	−0.179159	−0.0895793	−	0.995980i	$-0.528552\pi$
$571$	−1.39582e6	−0.179159	−0.0895793	+	0.995980i	$0.528552\pi$
$572$	0	0
$573$	3.47131e6	0.441679
$574$	0	0
$575$	9.59394e6	1.21012
$576$	0	0
$577$	−1.96784e6	−0.246065	−0.123033	−	0.992403i	$-0.539262\pi$
$577$	−1.96784e6	−0.246065	−0.123033	+	0.992403i	$0.539262\pi$
$578$	0	0
$579$	7.03160e6	0.871681
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−104880.	−0.0127797
$584$	0	0
$585$	4.73875e6	0.572498
$586$	0	0
$587$	−3.18897e6	−0.381993	−0.190997	−	0.981591i	$-0.561172\pi$
$587$	−3.18897e6	−0.381993	−0.190997	+	0.981591i	$0.561172\pi$
$588$	0	0
$589$	−1.29180e7	−1.53429
$590$	0	0
$591$	2.27527e7	2.67956
$592$	0	0
$593$	−1.67500e6	−0.195604	−0.0978022	−	0.995206i	$-0.531181\pi$
$593$	−1.67500e6	−0.195604	−0.0978022	+	0.995206i	$0.531181\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.03812e6	−1.03787
$598$	0	0
$599$	1.00635e7	1.14599	0.572994	−	0.819559i	$-0.305782\pi$
$599$	1.00635e7	1.14599	0.572994	+	0.819559i	$0.305782\pi$
$600$	0	0
$601$	−1.72798e6	−0.195143	−0.0975713	−	0.995229i	$-0.531107\pi$
$601$	−1.72798e6	−0.195143	−0.0975713	+	0.995229i	$0.531107\pi$
$602$	0	0
$603$	−4.37676e6	−0.490185
$604$	0	0
$605$	2.57579e6	0.286103
$606$	0	0
$607$	−1.69523e7	−1.86748	−0.933740	−	0.357953i	$-0.883475\pi$
$607$	−1.69523e7	−1.86748	−0.933740	+	0.357953i	$0.883475\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−829008.	−0.0898371
$612$	0	0
$613$	1.01942e7	1.09572	0.547861	−	0.836569i	$-0.315442\pi$
$613$	1.01942e7	1.09572	0.547861	+	0.836569i	$0.315442\pi$
$614$	0	0
$615$	5.55360e6	0.592089
$616$	0	0
$617$	1.57452e7	1.66508	0.832540	−	0.553965i	$-0.186886\pi$
$617$	1.57452e7	1.66508	0.832540	+	0.553965i	$0.186886\pi$
$618$	0	0
$619$	−332690.	−0.0348990	−0.0174495	−	0.999848i	$-0.505555\pi$
$619$	−332690.	−0.0348990	−0.0174495	+	0.999848i	$0.505555\pi$
$620$	0	0
$621$	−1.65194e7	−1.71895
$622$	0	0
$623$	0	0
$624$	0	0
$625$	7.43116e6	0.760951
$626$	0	0
$627$	561184.	0.0570081
$628$	0	0
$629$	−2.54405e7	−2.56388
$630$	0	0
$631$	−3.59720e6	−0.359659	−0.179830	−	0.983698i	$-0.557555\pi$
$631$	−3.59720e6	−0.359659	−0.179830	+	0.983698i	$0.557555\pi$
$632$	0	0
$633$	1.10599e7	1.09709
$634$	0	0
$635$	4.39680e6	0.432715
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.71578e7	−2.63113
$640$	0	0
$641$	−1.46389e7	−1.40723	−0.703614	−	0.710583i	$-0.748431\pi$
$641$	−1.46389e7	−1.40723	−0.703614	+	0.710583i	$0.748431\pi$
$642$	0	0
$643$	−1.38386e7	−1.31997	−0.659987	−	0.751277i	$-0.729438\pi$
$643$	−1.38386e7	−1.31997	−0.659987	+	0.751277i	$0.729438\pi$
$644$	0	0
$645$	−386048.	−0.0365378
$646$	0	0
$647$	−1.40358e7	−1.31819	−0.659093	−	0.752061i	$-0.729060\pi$
$647$	−1.40358e7	−1.31819	−0.659093	+	0.752061i	$0.729060\pi$
$648$	0	0
$649$	−277616.	−0.0258722
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.61063e7	1.47813	0.739064	−	0.673635i	$-0.235268\pi$
$653$	1.61063e7	1.47813	0.739064	+	0.673635i	$0.235268\pi$
$654$	0	0
$655$	−2.89187e6	−0.263376
$656$	0	0
$657$	8.19496e6	0.740685
$658$	0	0
$659$	−4.80075e6	−0.430622	−0.215311	−	0.976546i	$-0.569076\pi$
$659$	−4.80075e6	−0.430622	−0.215311	+	0.976546i	$0.569076\pi$
$660$	0	0
$661$	1.76565e7	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$661$	1.76565e7	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$662$	0	0
$663$	−3.94449e7	−3.48504
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.08814e7	0.947042
$668$	0	0
$669$	1.25214e7	1.08165
$670$	0	0
$671$	−8256.00	−0.000707886 0
$672$	0	0
$673$	6.59225e6	0.561043	0.280521	−	0.959848i	$-0.409493\pi$
$673$	6.59225e6	0.561043	0.280521	+	0.959848i	$0.409493\pi$
$674$	0	0
$675$	−1.41729e7	−1.19729
$676$	0	0
$677$	−9.77178e6	−0.819411	−0.409706	−	0.912218i	$-0.634369\pi$
$677$	−9.77178e6	−0.819411	−0.409706	+	0.912218i	$0.634369\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−157092.	−0.0129803
$682$	0	0
$683$	1.88663e7	1.54752	0.773758	−	0.633481i	$-0.218374\pi$
$683$	1.88663e7	1.54752	0.773758	+	0.633481i	$0.218374\pi$
$684$	0	0
$685$	3.35485e6	0.273178
$686$	0	0
$687$	−46904.0	−0.00379156
$688$	0	0
$689$	−8.96724e6	−0.719632
$690$	0	0
$691$	8.67018e6	0.690769	0.345385	−	0.938461i	$-0.387748\pi$
$691$	8.67018e6	0.690769	0.345385	+	0.938461i	$0.387748\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−275872.	−0.0216643
$696$	0	0
$697$	−2.96103e7	−2.30866
$698$	0	0
$699$	−4.18997e7	−3.24353
$700$	0	0
$701$	7.93482e6	0.609877	0.304938	−	0.952372i	$-0.401364\pi$
$701$	7.93482e6	0.609877	0.304938	+	0.952372i	$0.401364\pi$
$702$	0	0
$703$	3.09461e7	2.36166
$704$	0	0
$705$	−504192.	−0.0382053
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.62600e7	1.96191	0.980956	−	0.194228i	$-0.0622202\pi$
$709$	2.62600e7	1.96191	0.980956	+	0.194228i	$0.0622202\pi$
$710$	0	0
$711$	−4.93620e6	−0.366200
$712$	0	0
$713$	−1.60111e7	−1.17950
$714$	0	0
$715$	−87552.0	−0.00640473
$716$	0	0
$717$	2.56645e7	1.86438
$718$	0	0
$719$	2.20763e7	1.59259	0.796295	−	0.604909i	$-0.206790\pi$
$719$	2.20763e7	1.59259	0.796295	+	0.604909i	$0.206790\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.32313e7	−1.65283
$724$	0	0
$725$	9.33573e6	0.659634
$726$	0	0
$727$	8.49245e6	0.595933	0.297966	−	0.954576i	$-0.403692\pi$
$727$	8.49245e6	0.595933	0.297966	+	0.954576i	$0.403692\pi$
$728$	0	0
$729$	−2.11562e7	−1.47441
$730$	0	0
$731$	2.05830e6	0.142468
$732$	0	0
$733$	1.90713e7	1.31105	0.655526	−	0.755172i	$-0.272447\pi$
$733$	1.90713e7	1.31105	0.655526	+	0.755172i	$0.272447\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	80864.0	0.00548386
$738$	0	0
$739$	1.46832e7	0.989032	0.494516	−	0.869169i	$-0.335345\pi$
$739$	1.46832e7	0.989032	0.494516	+	0.869169i	$0.335345\pi$
$740$	0	0
$741$	4.79812e7	3.21015
$742$	0	0
$743$	−1.64265e7	−1.09162	−0.545812	−	0.837908i	$-0.683779\pi$
$743$	−1.64265e7	−1.09162	−0.545812	+	0.837908i	$0.683779\pi$
$744$	0	0
$745$	−949728.	−0.0626915
$746$	0	0
$747$	3.85188e7	2.52564
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.44357e7	1.58097	0.790486	−	0.612479i	$-0.209828\pi$
$751$	2.44357e7	1.58097	0.790486	+	0.612479i	$0.209828\pi$
$752$	0	0
$753$	9.51460e6	0.611509
$754$	0	0
$755$	−5.38150e6	−0.343587
$756$	0	0
$757$	−295566.	−0.0187463	−0.00937313	−	0.999956i	$-0.502984\pi$
$757$	−295566.	−0.0187463	−0.00937313	+	0.999956i	$0.502984\pi$
$758$	0	0
$759$	695552.	0.0438253
$760$	0	0
$761$	473842.	0.0296601	0.0148300	−	0.999890i	$-0.495279\pi$
$761$	473842.	0.0296601	0.0148300	+	0.999890i	$0.495279\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.53663e7	−0.949327
$766$	0	0
$767$	−2.37362e7	−1.45687
$768$	0	0
$769$	−2.33241e7	−1.42229	−0.711145	−	0.703045i	$-0.751823\pi$
$769$	−2.33241e7	−1.42229	−0.711145	+	0.703045i	$0.751823\pi$
$770$	0	0
$771$	−3.65194e7	−2.21253
$772$	0	0
$773$	−1.55583e7	−0.936511	−0.468255	−	0.883593i	$-0.655117\pi$
$773$	−1.55583e7	−0.936511	−0.468255	+	0.883593i	$0.655117\pi$
$774$	0	0
$775$	−1.37368e7	−0.821543
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.60183e7	2.12657
$780$	0	0
$781$	501760.	0.0294353
$782$	0	0
$783$	−1.60748e7	−0.937001
$784$	0	0
$785$	7.43341e6	0.430540
$786$	0	0
$787$	6.66843e6	0.383784	0.191892	−	0.981416i	$-0.438538\pi$
$787$	6.66843e6	0.383784	0.191892	+	0.981416i	$0.438538\pi$
$788$	0	0
$789$	−2.85917e7	−1.63511
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−705888.	−0.0398614
$794$	0	0
$795$	−5.45376e6	−0.306040
$796$	0	0
$797$	1.22461e7	0.682892	0.341446	−	0.939901i	$-0.389083\pi$
$797$	1.22461e7	0.682892	0.341446	+	0.939901i	$0.389083\pi$
$798$	0	0
$799$	2.68822e6	0.148969
$800$	0	0
$801$	−8.53963e6	−0.470281
$802$	0	0
$803$	−151408.	−0.00828629
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.11886e7	−1.14530
$808$	0	0
$809$	−2.91495e7	−1.56588	−0.782941	−	0.622095i	$-0.786282\pi$
$809$	−2.91495e7	−1.56588	−0.782941	+	0.622095i	$0.786282\pi$
$810$	0	0
$811$	7.58849e6	0.405138	0.202569	−	0.979268i	$-0.435071\pi$
$811$	7.58849e6	0.405138	0.202569	+	0.979268i	$0.435071\pi$
$812$	0	0
$813$	−4.41755e7	−2.34398
$814$	0	0
$815$	5.03667e6	0.265613
$816$	0	0
$817$	−2.50374e6	−0.131230
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.98849e6	−0.310070	−0.155035	−	0.987909i	$-0.549549\pi$
$821$	−5.98849e6	−0.310070	−0.155035	+	0.987909i	$0.549549\pi$
$822$	0	0
$823$	−817960.	−0.0420952	−0.0210476	−	0.999778i	$-0.506700\pi$
$823$	−817960.	−0.0420952	−0.0210476	+	0.999778i	$0.506700\pi$
$824$	0	0
$825$	596752.	0.0305252
$826$	0	0
$827$	−2.51963e6	−0.128107	−0.0640535	−	0.997946i	$-0.520403\pi$
$827$	−2.51963e6	−0.128107	−0.0640535	+	0.997946i	$0.520403\pi$
$828$	0	0
$829$	1.61006e7	0.813684	0.406842	−	0.913499i	$-0.366630\pi$
$829$	1.61006e7	0.813684	0.406842	+	0.913499i	$0.366630\pi$
$830$	0	0
$831$	−3.54921e7	−1.78291
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.57158e6	−0.226908
$836$	0	0
$837$	2.36527e7	1.16699
$838$	0	0
$839$	−2.58167e7	−1.26618	−0.633091	−	0.774077i	$-0.718214\pi$
$839$	−2.58167e7	−1.26618	−0.633091	+	0.774077i	$0.718214\pi$
$840$	0	0
$841$	−9.92263e6	−0.483768
$842$	0	0
$843$	−1.86120e7	−0.902036
$844$	0	0
$845$	−1.54501e6	−0.0744370
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.66088e6	0.269535
$850$	0	0
$851$	3.83557e7	1.81554
$852$	0	0
$853$	1.54270e7	0.725954	0.362977	−	0.931798i	$-0.381760\pi$
$853$	1.54270e7	0.725954	0.362977	+	0.931798i	$0.381760\pi$
$854$	0	0
$855$	1.86917e7	0.874449
$856$	0	0
$857$	−3.60517e6	−0.167677	−0.0838384	−	0.996479i	$-0.526718\pi$
$857$	−3.60517e6	−0.167677	−0.0838384	+	0.996479i	$0.526718\pi$
$858$	0	0
$859$	4.06995e6	0.188194	0.0940970	−	0.995563i	$-0.470004\pi$
$859$	4.06995e6	0.188194	0.0940970	+	0.995563i	$0.470004\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.25111e6	−0.331419	−0.165710	−	0.986175i	$-0.552991\pi$
$863$	−7.25111e6	−0.331419	−0.165710	+	0.986175i	$0.552991\pi$
$864$	0	0
$865$	−1.13464e7	−0.515604
$866$	0	0
$867$	9.09913e7	4.11105
$868$	0	0
$869$	91200.0	0.00409681
$870$	0	0
$871$	6.91387e6	0.308799
$872$	0	0
$873$	−7.38785e6	−0.328082
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.37414e6	0.104233	0.0521167	−	0.998641i	$-0.483403\pi$
$877$	2.37414e6	0.104233	0.0521167	+	0.998641i	$0.483403\pi$
$878$	0	0
$879$	−3.91841e7	−1.71056
$880$	0	0
$881$	−3.03558e7	−1.31766	−0.658828	−	0.752293i	$-0.728948\pi$
$881$	−3.03558e7	−1.31766	−0.658828	+	0.752293i	$0.728948\pi$
$882$	0	0
$883$	−1.53338e6	−0.0661832	−0.0330916	−	0.999452i	$-0.510535\pi$
$883$	−1.53338e6	−0.0661832	−0.0330916	+	0.999452i	$0.510535\pi$
$884$	0	0
$885$	−1.44360e7	−0.619569
$886$	0	0
$887$	−2.92379e7	−1.24778	−0.623888	−	0.781514i	$-0.714448\pi$
$887$	−2.92379e7	−1.24778	−0.623888	+	0.781514i	$0.714448\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−185768.	−0.00783929
$892$	0	0
$893$	−3.26998e6	−0.137220
$894$	0	0
$895$	−9.87437e6	−0.412052
$896$	0	0
$897$	5.94697e7	2.46783
$898$	0	0
$899$	−1.55802e7	−0.642943
$900$	0	0
$901$	2.90780e7	1.19331
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.80525e6	0.154441
$906$	0	0
$907$	−4.48227e7	−1.80917	−0.904587	−	0.426289i	$-0.859821\pi$
$907$	−4.48227e7	−1.80917	−0.904587	+	0.426289i	$0.859821\pi$
$908$	0	0
$909$	−1.98764e7	−0.797864
$910$	0	0
$911$	3.62906e7	1.44877	0.724384	−	0.689397i	$-0.242124\pi$
$911$	3.62906e7	1.44877	0.724384	+	0.689397i	$0.242124\pi$
$912$	0	0
$913$	−711664.	−0.0282552
$914$	0	0
$915$	−429312.	−0.0169520
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−3.25350e7	−1.27076	−0.635378	−	0.772201i	$-0.719156\pi$
$919$	−3.25350e7	−1.27076	−0.635378	+	0.772201i	$0.719156\pi$
$920$	0	0
$921$	325052.	0.0126271
$922$	0	0
$923$	4.29005e7	1.65752
$924$	0	0
$925$	3.29074e7	1.26456
$926$	0	0
$927$	−5.88932e7	−2.25095
$928$	0	0
$929$	4.46676e7	1.69806	0.849030	−	0.528344i	$-0.177187\pi$
$929$	4.46676e7	1.69806	0.849030	+	0.528344i	$0.177187\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.68332e7	−0.633087
$934$	0	0
$935$	283904.	0.0106204
$936$	0	0
$937$	−1.56680e7	−0.582995	−0.291498	−	0.956572i	$-0.594154\pi$
$937$	−1.56680e7	−0.582995	−0.291498	+	0.956572i	$0.594154\pi$
$938$	0	0
$939$	2.43354e7	0.900689
$940$	0	0
$941$	2.01175e7	0.740627	0.370313	−	0.928907i	$-0.379250\pi$
$941$	2.01175e7	0.740627	0.370313	+	0.928907i	$0.379250\pi$
$942$	0	0
$943$	4.46424e7	1.63481
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.84518e6	0.320503	0.160251	−	0.987076i	$-0.448769\pi$
$947$	8.84518e6	0.320503	0.160251	+	0.987076i	$0.448769\pi$
$948$	0	0
$949$	−1.29454e7	−0.466605
$950$	0	0
$951$	1.83545e7	0.658099
$952$	0	0
$953$	−3.14364e7	−1.12124	−0.560622	−	0.828072i	$-0.689438\pi$
$953$	−3.14364e7	−1.12124	−0.560622	+	0.828072i	$0.689438\pi$
$954$	0	0
$955$	−2.13619e6	−0.0757935
$956$	0	0
$957$	676832.	0.0238892
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−5.70421e6	−0.199245
$962$	0	0
$963$	2.99445e7	1.04052
$964$	0	0
$965$	−4.32714e6	−0.149583
$966$	0	0
$967$	9.52158e6	0.327449	0.163724	−	0.986506i	$-0.447649\pi$
$967$	9.52158e6	0.327449	0.163724	+	0.986506i	$0.447649\pi$
$968$	0	0
$969$	−1.55588e8	−5.32313
$970$	0	0
$971$	1.06520e7	0.362564	0.181282	−	0.983431i	$-0.441975\pi$
$971$	1.06520e7	0.362564	0.181282	+	0.983431i	$0.441975\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5.10223e7	1.71889
$976$	0	0
$977$	2.72931e7	0.914779	0.457389	−	0.889266i	$-0.348785\pi$
$977$	2.72931e7	0.914779	0.457389	+	0.889266i	$0.348785\pi$
$978$	0	0
$979$	157776.	0.00526119
$980$	0	0
$981$	−6.33973e7	−2.10328
$982$	0	0
$983$	1.04764e7	0.345802	0.172901	−	0.984939i	$-0.444686\pi$
$983$	1.04764e7	0.345802	0.172901	+	0.984939i	$0.444686\pi$
$984$	0	0
$985$	−1.40016e7	−0.459820
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.10323e6	−0.100884
$990$	0	0
$991$	−1.88230e6	−0.0608843	−0.0304422	−	0.999537i	$-0.509692\pi$
$991$	−1.88230e6	−0.0608843	−0.0304422	+	0.999537i	$0.509692\pi$
$992$	0	0
$993$	2.97190e7	0.956449
$994$	0	0
$995$	5.56192e6	0.178101
$996$	0	0
$997$	−2.71518e7	−0.865090	−0.432545	−	0.901612i	$-0.642384\pi$
$997$	−2.71518e7	−0.865090	−0.432545	+	0.901612i	$0.642384\pi$
$998$	0	0
$999$	−5.66618e7	−1.79629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.m.1.1		1
4.3	odd	2		196.6.a.a.1.1		1
7.6	odd	2		112.6.a.b.1.1		1
21.20	even	2		1008.6.a.l.1.1		1
28.3	even	6		196.6.e.a.177.1		2
28.11	odd	6		196.6.e.i.177.1		2
28.19	even	6		196.6.e.a.165.1		2
28.23	odd	6		196.6.e.i.165.1		2
28.27	even	2		28.6.a.b.1.1	✓	1
56.13	odd	2		448.6.a.o.1.1		1
56.27	even	2		448.6.a.b.1.1		1
84.83	odd	2		252.6.a.a.1.1		1
140.27	odd	4		700.6.e.b.449.1		2
140.83	odd	4		700.6.e.b.449.2		2
140.139	even	2		700.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.b.1.1	✓	1	28.27	even	2
112.6.a.b.1.1		1	7.6	odd	2
196.6.a.a.1.1		1	4.3	odd	2
196.6.e.a.165.1		2	28.19	even	6
196.6.e.a.177.1		2	28.3	even	6
196.6.e.i.165.1		2	28.23	odd	6
196.6.e.i.177.1		2	28.11	odd	6
252.6.a.a.1.1		1	84.83	odd	2
448.6.a.b.1.1		1	56.27	even	2
448.6.a.o.1.1		1	56.13	odd	2
700.6.a.b.1.1		1	140.139	even	2
700.6.e.b.449.1		2	140.27	odd	4
700.6.e.b.449.2		2	140.83	odd	4
784.6.a.m.1.1		1	1.1	even	1	trivial
1008.6.a.l.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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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