LMFDB - Embedded newform 768.6.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,6,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.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:	$123.174773616$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	768.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +80.0000 q^{5} +36.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +80.0000 q^{5} +36.0000 q^{7} +81.0000 q^{9} -36.0000 q^{11} +404.000 q^{13} +720.000 q^{15} -2.00000 q^{17} +2916.00 q^{19} +324.000 q^{21} -2808.00 q^{23} +3275.00 q^{25} +729.000 q^{27} -4408.00 q^{29} +5796.00 q^{31} -324.000 q^{33} +2880.00 q^{35} +2316.00 q^{37} +3636.00 q^{39} -6874.00 q^{41} +17244.0 q^{43} +6480.00 q^{45} -5688.00 q^{47} -15511.0 q^{49} -18.0000 q^{51} +32072.0 q^{53} -2880.00 q^{55} +26244.0 q^{57} +43308.0 q^{59} +18012.0 q^{61} +2916.00 q^{63} +32320.0 q^{65} -11628.0 q^{67} -25272.0 q^{69} -8712.00 q^{71} -15846.0 q^{73} +29475.0 q^{75} -1296.00 q^{77} +40356.0 q^{79} +6561.00 q^{81} -66204.0 q^{83} -160.000 q^{85} -39672.0 q^{87} -66086.0 q^{89} +14544.0 q^{91} +52164.0 q^{93} +233280. q^{95} -119682. q^{97} -2916.00 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	80.0000	1.43108	0.715542	−	0.698570i	$-0.246180\pi$
$5$	80.0000	1.43108	0.715542	+	0.698570i	$0.246180\pi$
$6$	0	0
$7$	36.0000	0.277688	0.138844	−	0.990314i	$-0.455661\pi$
$7$	36.0000	0.277688	0.138844	+	0.990314i	$0.455661\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−36.0000	−0.0897059	−0.0448529	−	0.998994i	$-0.514282\pi$
$11$	−36.0000	−0.0897059	−0.0448529	+	0.998994i	$0.514282\pi$
$12$	0	0
$13$	404.000	0.663014	0.331507	−	0.943453i	$-0.392443\pi$
$13$	404.000	0.663014	0.331507	+	0.943453i	$0.392443\pi$
$14$	0	0
$15$	720.000	0.826236
$16$	0	0
$17$	−2.00000	−0.00167845	−0.000839224	−	1.00000i	$-0.500267\pi$
$17$	−2.00000	−0.00167845	−0.000839224		1.00000i	$0.500267\pi$
$18$	0	0
$19$	2916.00	1.85312	0.926560	−	0.376147i	$-0.122751\pi$
$19$	2916.00	1.85312	0.926560	+	0.376147i	$0.122751\pi$
$20$	0	0
$21$	324.000	0.160323
$22$	0	0
$23$	−2808.00	−1.10682	−0.553411	−	0.832909i	$-0.686674\pi$
$23$	−2808.00	−1.10682	−0.553411	+	0.832909i	$0.686674\pi$
$24$	0	0
$25$	3275.00	1.04800
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−4408.00	−0.973300	−0.486650	−	0.873597i	$-0.661781\pi$
$29$	−4408.00	−0.973300	−0.486650	+	0.873597i	$0.661781\pi$
$30$	0	0
$31$	5796.00	1.08324	0.541619	−	0.840624i	$-0.317811\pi$
$31$	5796.00	1.08324	0.541619	+	0.840624i	$0.317811\pi$
$32$	0	0
$33$	−324.000	−0.0517917
$34$	0	0
$35$	2880.00	0.397395
$36$	0	0
$37$	2316.00	0.278121	0.139061	−	0.990284i	$-0.455592\pi$
$37$	2316.00	0.278121	0.139061	+	0.990284i	$0.455592\pi$
$38$	0	0
$39$	3636.00	0.382792
$40$	0	0
$41$	−6874.00	−0.638631	−0.319315	−	0.947648i	$-0.603453\pi$
$41$	−6874.00	−0.638631	−0.319315	+	0.947648i	$0.603453\pi$
$42$	0	0
$43$	17244.0	1.42222	0.711110	−	0.703081i	$-0.248193\pi$
$43$	17244.0	1.42222	0.711110	+	0.703081i	$0.248193\pi$
$44$	0	0
$45$	6480.00	0.477028
$46$	0	0
$47$	−5688.00	−0.375591	−0.187795	−	0.982208i	$-0.560134\pi$
$47$	−5688.00	−0.375591	−0.187795	+	0.982208i	$0.560134\pi$
$48$	0	0
$49$	−15511.0	−0.922889
$50$	0	0
$51$	−18.0000	−0.000969052 0
$52$	0	0
$53$	32072.0	1.56833	0.784163	−	0.620555i	$-0.213093\pi$
$53$	32072.0	1.56833	0.784163	+	0.620555i	$0.213093\pi$
$54$	0	0
$55$	−2880.00	−0.128377
$56$	0	0
$57$	26244.0	1.06990
$58$	0	0
$59$	43308.0	1.61971	0.809857	−	0.586628i	$-0.199545\pi$
$59$	43308.0	1.61971	0.809857	+	0.586628i	$0.199545\pi$
$60$	0	0
$61$	18012.0	0.619780	0.309890	−	0.950772i	$-0.399708\pi$
$61$	18012.0	0.619780	0.309890	+	0.950772i	$0.399708\pi$
$62$	0	0
$63$	2916.00	0.0925627
$64$	0	0
$65$	32320.0	0.948829
$66$	0	0
$67$	−11628.0	−0.316459	−0.158230	−	0.987402i	$-0.550579\pi$
$67$	−11628.0	−0.316459	−0.158230	+	0.987402i	$0.550579\pi$
$68$	0	0
$69$	−25272.0	−0.639024
$70$	0	0
$71$	−8712.00	−0.205103	−0.102551	−	0.994728i	$-0.532701\pi$
$71$	−8712.00	−0.205103	−0.102551	+	0.994728i	$0.532701\pi$
$72$	0	0
$73$	−15846.0	−0.348027	−0.174013	−	0.984743i	$-0.555674\pi$
$73$	−15846.0	−0.348027	−0.174013	+	0.984743i	$0.555674\pi$
$74$	0	0
$75$	29475.0	0.605063
$76$	0	0
$77$	−1296.00	−0.0249103
$78$	0	0
$79$	40356.0	0.727512	0.363756	−	0.931494i	$-0.381494\pi$
$79$	40356.0	0.727512	0.363756	+	0.931494i	$0.381494\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−66204.0	−1.05485	−0.527423	−	0.849603i	$-0.676842\pi$
$83$	−66204.0	−1.05485	−0.527423	+	0.849603i	$0.676842\pi$
$84$	0	0
$85$	−160.000	−0.00240200
$86$	0	0
$87$	−39672.0	−0.561935
$88$	0	0
$89$	−66086.0	−0.884371	−0.442185	−	0.896924i	$-0.645797\pi$
$89$	−66086.0	−0.884371	−0.442185	+	0.896924i	$0.645797\pi$
$90$	0	0
$91$	14544.0	0.184111
$92$	0	0
$93$	52164.0	0.625408
$94$	0	0
$95$	233280.	2.65197
$96$	0	0
$97$	−119682.	−1.29152	−0.645758	−	0.763543i	$-0.723458\pi$
$97$	−119682.	−1.29152	−0.645758	+	0.763543i	$0.723458\pi$
$98$	0	0
$99$	−2916.00	−0.0299020
$100$	0	0
$101$	32392.0	0.315962	0.157981	−	0.987442i	$-0.449502\pi$
$101$	32392.0	0.315962	0.157981	+	0.987442i	$0.449502\pi$
$102$	0	0
$103$	−109476.	−1.01678	−0.508389	−	0.861128i	$-0.669759\pi$
$103$	−109476.	−1.01678	−0.508389	+	0.861128i	$0.669759\pi$
$104$	0	0
$105$	25920.0	0.229436
$106$	0	0
$107$	−106596.	−0.900081	−0.450040	−	0.893008i	$-0.648590\pi$
$107$	−106596.	−0.900081	−0.450040	+	0.893008i	$0.648590\pi$
$108$	0	0
$109$	77524.0	0.624985	0.312493	−	0.949920i	$-0.398836\pi$
$109$	77524.0	0.624985	0.312493	+	0.949920i	$0.398836\pi$
$110$	0	0
$111$	20844.0	0.160573
$112$	0	0
$113$	179506.	1.32246	0.661230	−	0.750183i	$-0.270035\pi$
$113$	179506.	1.32246	0.661230	+	0.750183i	$0.270035\pi$
$114$	0	0
$115$	−224640.	−1.58395
$116$	0	0
$117$	32724.0	0.221005
$118$	0	0
$119$	−72.0000	−0.000466085 0
$120$	0	0
$121$	−159755.	−0.991953
$122$	0	0
$123$	−61866.0	−0.368714
$124$	0	0
$125$	12000.0	0.0686920
$126$	0	0
$127$	339804.	1.86947	0.934736	−	0.355342i	$-0.115635\pi$
$127$	339804.	1.86947	0.934736	+	0.355342i	$0.115635\pi$
$128$	0	0
$129$	155196.	0.821119
$130$	0	0
$131$	−334188.	−1.70142	−0.850712	−	0.525632i	$-0.823829\pi$
$131$	−334188.	−1.70142	−0.850712	+	0.525632i	$0.823829\pi$
$132$	0	0
$133$	104976.	0.514589
$134$	0	0
$135$	58320.0	0.275412
$136$	0	0
$137$	393766.	1.79241	0.896204	−	0.443643i	$-0.146314\pi$
$137$	393766.	1.79241	0.896204	+	0.443643i	$0.146314\pi$
$138$	0	0
$139$	244620.	1.07388	0.536939	−	0.843621i	$-0.319581\pi$
$139$	244620.	1.07388	0.536939	+	0.843621i	$0.319581\pi$
$140$	0	0
$141$	−51192.0	−0.216847
$142$	0	0
$143$	−14544.0	−0.0594763
$144$	0	0
$145$	−352640.	−1.39287
$146$	0	0
$147$	−139599.	−0.532830
$148$	0	0
$149$	−368240.	−1.35883	−0.679415	−	0.733754i	$-0.737766\pi$
$149$	−368240.	−1.35883	−0.679415	+	0.733754i	$0.737766\pi$
$150$	0	0
$151$	380052.	1.35644	0.678220	−	0.734859i	$-0.262752\pi$
$151$	380052.	1.35644	0.678220	+	0.734859i	$0.262752\pi$
$152$	0	0
$153$	−162.000	−0.000559482 0
$154$	0	0
$155$	463680.	1.55020
$156$	0	0
$157$	−513204.	−1.66166	−0.830828	−	0.556530i	$-0.812132\pi$
$157$	−513204.	−1.66166	−0.830828	+	0.556530i	$0.812132\pi$
$158$	0	0
$159$	288648.	0.905473
$160$	0	0
$161$	−101088.	−0.307351
$162$	0	0
$163$	−31500.0	−0.0928628	−0.0464314	−	0.998921i	$-0.514785\pi$
$163$	−31500.0	−0.0928628	−0.0464314	+	0.998921i	$0.514785\pi$
$164$	0	0
$165$	−25920.0	−0.0741182
$166$	0	0
$167$	−106560.	−0.295667	−0.147834	−	0.989012i	$-0.547230\pi$
$167$	−106560.	−0.295667	−0.147834	+	0.989012i	$0.547230\pi$
$168$	0	0
$169$	−208077.	−0.560412
$170$	0	0
$171$	236196.	0.617707
$172$	0	0
$173$	249680.	0.634262	0.317131	−	0.948382i	$-0.397281\pi$
$173$	249680.	0.634262	0.317131	+	0.948382i	$0.397281\pi$
$174$	0	0
$175$	117900.	0.291017
$176$	0	0
$177$	389772.	0.935142
$178$	0	0
$179$	362916.	0.846591	0.423295	−	0.905992i	$-0.360873\pi$
$179$	362916.	0.846591	0.423295	+	0.905992i	$0.360873\pi$
$180$	0	0
$181$	506644.	1.14949	0.574747	−	0.818331i	$-0.305101\pi$
$181$	506644.	1.14949	0.574747	+	0.818331i	$0.305101\pi$
$182$	0	0
$183$	162108.	0.357830
$184$	0	0
$185$	185280.	0.398015
$186$	0	0
$187$	72.0000	0.000150567 0
$188$	0	0
$189$	26244.0	0.0534411
$190$	0	0
$191$	806256.	1.59915	0.799576	−	0.600565i	$-0.205058\pi$
$191$	806256.	1.59915	0.799576	+	0.600565i	$0.205058\pi$
$192$	0	0
$193$	−438418.	−0.847218	−0.423609	−	0.905845i	$-0.639237\pi$
$193$	−438418.	−0.847218	−0.423609	+	0.905845i	$0.639237\pi$
$194$	0	0
$195$	290880.	0.547807
$196$	0	0
$197$	−334664.	−0.614389	−0.307195	−	0.951647i	$-0.599390\pi$
$197$	−334664.	−0.614389	−0.307195	+	0.951647i	$0.599390\pi$
$198$	0	0
$199$	63324.0	0.113354	0.0566768	−	0.998393i	$-0.481950\pi$
$199$	63324.0	0.113354	0.0566768	+	0.998393i	$0.481950\pi$
$200$	0	0
$201$	−104652.	−0.182708
$202$	0	0
$203$	−158688.	−0.270274
$204$	0	0
$205$	−549920.	−0.913934
$206$	0	0
$207$	−227448.	−0.368940
$208$	0	0
$209$	−104976.	−0.166236
$210$	0	0
$211$	662436.	1.02433	0.512163	−	0.858889i	$-0.328845\pi$
$211$	662436.	1.02433	0.512163	+	0.858889i	$0.328845\pi$
$212$	0	0
$213$	−78408.0	−0.118416
$214$	0	0
$215$	1.37952e6	2.03532
$216$	0	0
$217$	208656.	0.300803
$218$	0	0
$219$	−142614.	−0.200933
$220$	0	0
$221$	−808.000	−0.00111283
$222$	0	0
$223$	1.33038e6	1.79149	0.895743	−	0.444572i	$-0.146644\pi$
$223$	1.33038e6	1.79149	0.895743	+	0.444572i	$0.146644\pi$
$224$	0	0
$225$	265275.	0.349333
$226$	0	0
$227$	717156.	0.923738	0.461869	−	0.886948i	$-0.347179\pi$
$227$	717156.	0.923738	0.461869	+	0.886948i	$0.347179\pi$
$228$	0	0
$229$	−198316.	−0.249902	−0.124951	−	0.992163i	$-0.539877\pi$
$229$	−198316.	−0.249902	−0.124951	+	0.992163i	$0.539877\pi$
$230$	0	0
$231$	−11664.0	−0.0143819
$232$	0	0
$233$	−860662.	−1.03859	−0.519293	−	0.854596i	$-0.673805\pi$
$233$	−860662.	−1.03859	−0.519293	+	0.854596i	$0.673805\pi$
$234$	0	0
$235$	−455040.	−0.537502
$236$	0	0
$237$	363204.	0.420029
$238$	0	0
$239$	443520.	0.502248	0.251124	−	0.967955i	$-0.419200\pi$
$239$	443520.	0.502248	0.251124	+	0.967955i	$0.419200\pi$
$240$	0	0
$241$	−34242.0	−0.0379766	−0.0189883	−	0.999820i	$-0.506045\pi$
$241$	−34242.0	−0.0379766	−0.0189883	+	0.999820i	$0.506045\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−1.24088e6	−1.32073
$246$	0	0
$247$	1.17806e6	1.22865
$248$	0	0
$249$	−595836.	−0.609016
$250$	0	0
$251$	−236052.	−0.236496	−0.118248	−	0.992984i	$-0.537728\pi$
$251$	−236052.	−0.236496	−0.118248	+	0.992984i	$0.537728\pi$
$252$	0	0
$253$	101088.	0.0992884
$254$	0	0
$255$	−1440.00	−0.00138679
$256$	0	0
$257$	929282.	0.877637	0.438818	−	0.898576i	$-0.355397\pi$
$257$	929282.	0.877637	0.438818	+	0.898576i	$0.355397\pi$
$258$	0	0
$259$	83376.0	0.0772310
$260$	0	0
$261$	−357048.	−0.324433
$262$	0	0
$263$	1.69070e6	1.50723	0.753613	−	0.657319i	$-0.228309\pi$
$263$	1.69070e6	1.50723	0.753613	+	0.657319i	$0.228309\pi$
$264$	0	0
$265$	2.56576e6	2.24441
$266$	0	0
$267$	−594774.	−0.510592
$268$	0	0
$269$	−900056.	−0.758383	−0.379192	−	0.925318i	$-0.623798\pi$
$269$	−900056.	−0.758383	−0.379192	+	0.925318i	$0.623798\pi$
$270$	0	0
$271$	−1.58389e6	−1.31009	−0.655047	−	0.755588i	$-0.727351\pi$
$271$	−1.58389e6	−1.31009	−0.655047	+	0.755588i	$0.727351\pi$
$272$	0	0
$273$	130896.	0.106297
$274$	0	0
$275$	−117900.	−0.0940117
$276$	0	0
$277$	1.95466e6	1.53064	0.765318	−	0.643653i	$-0.222582\pi$
$277$	1.95466e6	1.53064	0.765318	+	0.643653i	$0.222582\pi$
$278$	0	0
$279$	469476.	0.361080
$280$	0	0
$281$	−2.16081e6	−1.63249	−0.816244	−	0.577707i	$-0.803947\pi$
$281$	−2.16081e6	−1.63249	−0.816244	+	0.577707i	$0.803947\pi$
$282$	0	0
$283$	−659556.	−0.489537	−0.244769	−	0.969582i	$-0.578712\pi$
$283$	−659556.	−0.489537	−0.244769	+	0.969582i	$0.578712\pi$
$284$	0	0
$285$	2.09952e6	1.53112
$286$	0	0
$287$	−247464.	−0.177340
$288$	0	0
$289$	−1.41985e6	−0.999997
$290$	0	0
$291$	−1.07714e6	−0.745657
$292$	0	0
$293$	802664.	0.546216	0.273108	−	0.961983i	$-0.411948\pi$
$293$	802664.	0.546216	0.273108	+	0.961983i	$0.411948\pi$
$294$	0	0
$295$	3.46464e6	2.31794
$296$	0	0
$297$	−26244.0	−0.0172639
$298$	0	0
$299$	−1.13443e6	−0.733839
$300$	0	0
$301$	620784.	0.394934
$302$	0	0
$303$	291528.	0.182421
$304$	0	0
$305$	1.44096e6	0.886957
$306$	0	0
$307$	178452.	0.108063	0.0540313	−	0.998539i	$-0.482793\pi$
$307$	178452.	0.108063	0.0540313	+	0.998539i	$0.482793\pi$
$308$	0	0
$309$	−985284.	−0.587037
$310$	0	0
$311$	1.25827e6	0.737689	0.368845	−	0.929491i	$-0.379753\pi$
$311$	1.25827e6	0.737689	0.368845	+	0.929491i	$0.379753\pi$
$312$	0	0
$313$	923062.	0.532562	0.266281	−	0.963895i	$-0.414205\pi$
$313$	923062.	0.532562	0.266281	+	0.963895i	$0.414205\pi$
$314$	0	0
$315$	233280.	0.132465
$316$	0	0
$317$	568408.	0.317696	0.158848	−	0.987303i	$-0.449222\pi$
$317$	568408.	0.317696	0.158848	+	0.987303i	$0.449222\pi$
$318$	0	0
$319$	158688.	0.0873107
$320$	0	0
$321$	−959364.	−0.519662
$322$	0	0
$323$	−5832.00	−0.00311036
$324$	0	0
$325$	1.32310e6	0.694839
$326$	0	0
$327$	697716.	0.360836
$328$	0	0
$329$	−204768.	−0.104297
$330$	0	0
$331$	−2.60644e6	−1.30761	−0.653803	−	0.756665i	$-0.726828\pi$
$331$	−2.60644e6	−1.30761	−0.653803	+	0.756665i	$0.726828\pi$
$332$	0	0
$333$	187596.	0.0927071
$334$	0	0
$335$	−930240.	−0.452880
$336$	0	0
$337$	232002.	0.111280	0.0556400	−	0.998451i	$-0.482280\pi$
$337$	232002.	0.111280	0.0556400	+	0.998451i	$0.482280\pi$
$338$	0	0
$339$	1.61555e6	0.763523
$340$	0	0
$341$	−208656.	−0.0971728
$342$	0	0
$343$	−1.16345e6	−0.533964
$344$	0	0
$345$	−2.02176e6	−0.914496
$346$	0	0
$347$	−3.21973e6	−1.43548	−0.717738	−	0.696313i	$-0.754823\pi$
$347$	−3.21973e6	−1.43548	−0.717738	+	0.696313i	$0.754823\pi$
$348$	0	0
$349$	−4.40341e6	−1.93520	−0.967600	−	0.252489i	$-0.918751\pi$
$349$	−4.40341e6	−1.93520	−0.967600	+	0.252489i	$0.918751\pi$
$350$	0	0
$351$	294516.	0.127597
$352$	0	0
$353$	−556766.	−0.237813	−0.118907	−	0.992905i	$-0.537939\pi$
$353$	−556766.	−0.237813	−0.118907	+	0.992905i	$0.537939\pi$
$354$	0	0
$355$	−696960.	−0.293520
$356$	0	0
$357$	−648.000	−0.000269094 0
$358$	0	0
$359$	−1.86314e6	−0.762975	−0.381487	−	0.924374i	$-0.624588\pi$
$359$	−1.86314e6	−0.762975	−0.381487	+	0.924374i	$0.624588\pi$
$360$	0	0
$361$	6.02696e6	2.43405
$362$	0	0
$363$	−1.43780e6	−0.572704
$364$	0	0
$365$	−1.26768e6	−0.498055
$366$	0	0
$367$	4.19936e6	1.62749	0.813745	−	0.581222i	$-0.197425\pi$
$367$	4.19936e6	1.62749	0.813745	+	0.581222i	$0.197425\pi$
$368$	0	0
$369$	−556794.	−0.212877
$370$	0	0
$371$	1.15459e6	0.435506
$372$	0	0
$373$	3.74152e6	1.39244	0.696218	−	0.717830i	$-0.254864\pi$
$373$	3.74152e6	1.39244	0.696218	+	0.717830i	$0.254864\pi$
$374$	0	0
$375$	108000.	0.0396593
$376$	0	0
$377$	−1.78083e6	−0.645312
$378$	0	0
$379$	2.47100e6	0.883640	0.441820	−	0.897104i	$-0.354333\pi$
$379$	2.47100e6	0.883640	0.441820	+	0.897104i	$0.354333\pi$
$380$	0	0
$381$	3.05824e6	1.07934
$382$	0	0
$383$	−472176.	−0.164478	−0.0822388	−	0.996613i	$-0.526207\pi$
$383$	−472176.	−0.164478	−0.0822388	+	0.996613i	$0.526207\pi$
$384$	0	0
$385$	−103680.	−0.0356487
$386$	0	0
$387$	1.39676e6	0.474073
$388$	0	0
$389$	−3.39157e6	−1.13639	−0.568194	−	0.822895i	$-0.692358\pi$
$389$	−3.39157e6	−1.13639	−0.568194	+	0.822895i	$0.692358\pi$
$390$	0	0
$391$	5616.00	0.00185774
$392$	0	0
$393$	−3.00769e6	−0.982318
$394$	0	0
$395$	3.22848e6	1.04113
$396$	0	0
$397$	3.94950e6	1.25767	0.628834	−	0.777540i	$-0.283533\pi$
$397$	3.94950e6	1.25767	0.628834	+	0.777540i	$0.283533\pi$
$398$	0	0
$399$	944784.	0.297098
$400$	0	0
$401$	−200354.	−0.0622210	−0.0311105	−	0.999516i	$-0.509904\pi$
$401$	−200354.	−0.0622210	−0.0311105	+	0.999516i	$0.509904\pi$
$402$	0	0
$403$	2.34158e6	0.718203
$404$	0	0
$405$	524880.	0.159009
$406$	0	0
$407$	−83376.0	−0.0249491
$408$	0	0
$409$	3.29357e6	0.973552	0.486776	−	0.873527i	$-0.338173\pi$
$409$	3.29357e6	0.973552	0.486776	+	0.873527i	$0.338173\pi$
$410$	0	0
$411$	3.54389e6	1.03485
$412$	0	0
$413$	1.55909e6	0.449775
$414$	0	0
$415$	−5.29632e6	−1.50957
$416$	0	0
$417$	2.20158e6	0.620004
$418$	0	0
$419$	970740.	0.270127	0.135063	−	0.990837i	$-0.456876\pi$
$419$	970740.	0.270127	0.135063	+	0.990837i	$0.456876\pi$
$420$	0	0
$421$	5.40490e6	1.48622	0.743109	−	0.669171i	$-0.233350\pi$
$421$	5.40490e6	1.48622	0.743109	+	0.669171i	$0.233350\pi$
$422$	0	0
$423$	−460728.	−0.125197
$424$	0	0
$425$	−6550.00	−0.00175901
$426$	0	0
$427$	648432.	0.172106
$428$	0	0
$429$	−130896.	−0.0343386
$430$	0	0
$431$	5.46646e6	1.41747	0.708733	−	0.705477i	$-0.249267\pi$
$431$	5.46646e6	1.41747	0.708733	+	0.705477i	$0.249267\pi$
$432$	0	0
$433$	−2.99423e6	−0.767476	−0.383738	−	0.923442i	$-0.625363\pi$
$433$	−2.99423e6	−0.767476	−0.383738	+	0.923442i	$0.625363\pi$
$434$	0	0
$435$	−3.17376e6	−0.804176
$436$	0	0
$437$	−8.18813e6	−2.05107
$438$	0	0
$439$	−2.04469e6	−0.506368	−0.253184	−	0.967418i	$-0.581478\pi$
$439$	−2.04469e6	−0.506368	−0.253184	+	0.967418i	$0.581478\pi$
$440$	0	0
$441$	−1.25639e6	−0.307630
$442$	0	0
$443$	7.49344e6	1.81414	0.907072	−	0.420976i	$-0.138312\pi$
$443$	7.49344e6	1.81414	0.907072	+	0.420976i	$0.138312\pi$
$444$	0	0
$445$	−5.28688e6	−1.26561
$446$	0	0
$447$	−3.31416e6	−0.784521
$448$	0	0
$449$	−1.92095e6	−0.449676	−0.224838	−	0.974396i	$-0.572185\pi$
$449$	−1.92095e6	−0.449676	−0.224838	+	0.974396i	$0.572185\pi$
$450$	0	0
$451$	247464.	0.0572889
$452$	0	0
$453$	3.42047e6	0.783141
$454$	0	0
$455$	1.16352e6	0.263479
$456$	0	0
$457$	−6.20241e6	−1.38922	−0.694608	−	0.719388i	$-0.744422\pi$
$457$	−6.20241e6	−1.38922	−0.694608	+	0.719388i	$0.744422\pi$
$458$	0	0
$459$	−1458.00	−0.000323017 0
$460$	0	0
$461$	−4.10840e6	−0.900369	−0.450184	−	0.892936i	$-0.648642\pi$
$461$	−4.10840e6	−0.900369	−0.450184	+	0.892936i	$0.648642\pi$
$462$	0	0
$463$	−5940.00	−0.00128776	−0.000643879	−	1.00000i	$-0.500205\pi$
$463$	−5940.00	−0.00128776	−0.000643879		1.00000i	$0.500205\pi$
$464$	0	0
$465$	4.17312e6	0.895011
$466$	0	0
$467$	−4.12715e6	−0.875705	−0.437853	−	0.899047i	$-0.644261\pi$
$467$	−4.12715e6	−0.875705	−0.437853	+	0.899047i	$0.644261\pi$
$468$	0	0
$469$	−418608.	−0.0878770
$470$	0	0
$471$	−4.61884e6	−0.959357
$472$	0	0
$473$	−620784.	−0.127581
$474$	0	0
$475$	9.54990e6	1.94207
$476$	0	0
$477$	2.59783e6	0.522775
$478$	0	0
$479$	−4.34297e6	−0.864864	−0.432432	−	0.901667i	$-0.642344\pi$
$479$	−4.34297e6	−0.864864	−0.432432	+	0.901667i	$0.642344\pi$
$480$	0	0
$481$	935664.	0.184398
$482$	0	0
$483$	−909792.	−0.177449
$484$	0	0
$485$	−9.57456e6	−1.84827
$486$	0	0
$487$	−9.15822e6	−1.74980	−0.874901	−	0.484303i	$-0.839073\pi$
$487$	−9.15822e6	−1.74980	−0.874901	+	0.484303i	$0.839073\pi$
$488$	0	0
$489$	−283500.	−0.0536143
$490$	0	0
$491$	−3.33205e6	−0.623746	−0.311873	−	0.950124i	$-0.600956\pi$
$491$	−3.33205e6	−0.623746	−0.311873	+	0.950124i	$0.600956\pi$
$492$	0	0
$493$	8816.00	0.00163363
$494$	0	0
$495$	−233280.	−0.0427922
$496$	0	0
$497$	−313632.	−0.0569547
$498$	0	0
$499$	−4.36028e6	−0.783905	−0.391952	−	0.919985i	$-0.628200\pi$
$499$	−4.36028e6	−0.783905	−0.391952	+	0.919985i	$0.628200\pi$
$500$	0	0
$501$	−959040.	−0.170703
$502$	0	0
$503$	−7.82474e6	−1.37896	−0.689478	−	0.724307i	$-0.742160\pi$
$503$	−7.82474e6	−1.37896	−0.689478	+	0.724307i	$0.742160\pi$
$504$	0	0
$505$	2.59136e6	0.452167
$506$	0	0
$507$	−1.87269e6	−0.323554
$508$	0	0
$509$	2.18598e6	0.373982	0.186991	−	0.982362i	$-0.440126\pi$
$509$	2.18598e6	0.373982	0.186991	+	0.982362i	$0.440126\pi$
$510$	0	0
$511$	−570456.	−0.0966429
$512$	0	0
$513$	2.12576e6	0.356633
$514$	0	0
$515$	−8.75808e6	−1.45509
$516$	0	0
$517$	204768.	0.0336927
$518$	0	0
$519$	2.24712e6	0.366191
$520$	0	0
$521$	4.42385e6	0.714012	0.357006	−	0.934102i	$-0.383798\pi$
$521$	4.42385e6	0.714012	0.357006	+	0.934102i	$0.383798\pi$
$522$	0	0
$523$	−2.86866e6	−0.458590	−0.229295	−	0.973357i	$-0.573642\pi$
$523$	−2.86866e6	−0.458590	−0.229295	+	0.973357i	$0.573642\pi$
$524$	0	0
$525$	1.06110e6	0.168019
$526$	0	0
$527$	−11592.0	−0.00181816
$528$	0	0
$529$	1.44852e6	0.225053
$530$	0	0
$531$	3.50795e6	0.539904
$532$	0	0
$533$	−2.77710e6	−0.423421
$534$	0	0
$535$	−8.52768e6	−1.28809
$536$	0	0
$537$	3.26624e6	0.488779
$538$	0	0
$539$	558396.	0.0827886
$540$	0	0
$541$	−4.40059e6	−0.646424	−0.323212	−	0.946327i	$-0.604763\pi$
$541$	−4.40059e6	−0.646424	−0.323212	+	0.946327i	$0.604763\pi$
$542$	0	0
$543$	4.55980e6	0.663660
$544$	0	0
$545$	6.20192e6	0.894406
$546$	0	0
$547$	−7.05280e6	−1.00784	−0.503922	−	0.863749i	$-0.668110\pi$
$547$	−7.05280e6	−1.00784	−0.503922	+	0.863749i	$0.668110\pi$
$548$	0	0
$549$	1.45897e6	0.206593
$550$	0	0
$551$	−1.28537e7	−1.80364
$552$	0	0
$553$	1.45282e6	0.202022
$554$	0	0
$555$	1.66752e6	0.229794
$556$	0	0
$557$	−1.08665e7	−1.48406	−0.742028	−	0.670369i	$-0.766136\pi$
$557$	−1.08665e7	−1.48406	−0.742028	+	0.670369i	$0.766136\pi$
$558$	0	0
$559$	6.96658e6	0.942953
$560$	0	0
$561$	648.000	8.69296e−5 0
$562$	0	0
$563$	4.50709e6	0.599274	0.299637	−	0.954053i	$-0.403134\pi$
$563$	4.50709e6	0.599274	0.299637	+	0.954053i	$0.403134\pi$
$564$	0	0
$565$	1.43605e7	1.89255
$566$	0	0
$567$	236196.	0.0308542
$568$	0	0
$569$	−2.57873e6	−0.333907	−0.166953	−	0.985965i	$-0.553393\pi$
$569$	−2.57873e6	−0.333907	−0.166953	+	0.985965i	$0.553393\pi$
$570$	0	0
$571$	−6.02212e6	−0.772963	−0.386482	−	0.922297i	$-0.626310\pi$
$571$	−6.02212e6	−0.772963	−0.386482	+	0.922297i	$0.626310\pi$
$572$	0	0
$573$	7.25630e6	0.923270
$574$	0	0
$575$	−9.19620e6	−1.15995
$576$	0	0
$577$	−1.51475e7	−1.89409	−0.947047	−	0.321095i	$-0.895949\pi$
$577$	−1.51475e7	−1.89409	−0.947047	+	0.321095i	$0.895949\pi$
$578$	0	0
$579$	−3.94576e6	−0.489141
$580$	0	0
$581$	−2.38334e6	−0.292918
$582$	0	0
$583$	−1.15459e6	−0.140688
$584$	0	0
$585$	2.61792e6	0.316276
$586$	0	0
$587$	−1.30090e7	−1.55829	−0.779145	−	0.626843i	$-0.784347\pi$
$587$	−1.30090e7	−1.55829	−0.779145	+	0.626843i	$0.784347\pi$
$588$	0	0
$589$	1.69011e7	2.00737
$590$	0	0
$591$	−3.01198e6	−0.354718
$592$	0	0
$593$	900818.	0.105196	0.0525981	−	0.998616i	$-0.483250\pi$
$593$	900818.	0.105196	0.0525981	+	0.998616i	$0.483250\pi$
$594$	0	0
$595$	−5760.00	−0.000667007 0
$596$	0	0
$597$	569916.	0.0654448
$598$	0	0
$599$	−1.39649e7	−1.59027	−0.795135	−	0.606432i	$-0.792600\pi$
$599$	−1.39649e7	−1.59027	−0.795135	+	0.606432i	$0.792600\pi$
$600$	0	0
$601$	−1.28411e7	−1.45017	−0.725083	−	0.688662i	$-0.758198\pi$
$601$	−1.28411e7	−1.45017	−0.725083	+	0.688662i	$0.758198\pi$
$602$	0	0
$603$	−941868.	−0.105486
$604$	0	0
$605$	−1.27804e7	−1.41957
$606$	0	0
$607$	−7.62023e6	−0.839453	−0.419726	−	0.907651i	$-0.637874\pi$
$607$	−7.62023e6	−0.839453	−0.419726	+	0.907651i	$0.637874\pi$
$608$	0	0
$609$	−1.42819e6	−0.156043
$610$	0	0
$611$	−2.29795e6	−0.249022
$612$	0	0
$613$	−1.50444e7	−1.61705	−0.808527	−	0.588459i	$-0.799735\pi$
$613$	−1.50444e7	−1.61705	−0.808527	+	0.588459i	$0.799735\pi$
$614$	0	0
$615$	−4.94928e6	−0.527660
$616$	0	0
$617$	1.25329e6	0.132537	0.0662687	−	0.997802i	$-0.478891\pi$
$617$	1.25329e6	0.132537	0.0662687	+	0.997802i	$0.478891\pi$
$618$	0	0
$619$	−7.56551e6	−0.793618	−0.396809	−	0.917901i	$-0.629882\pi$
$619$	−7.56551e6	−0.793618	−0.396809	+	0.917901i	$0.629882\pi$
$620$	0	0
$621$	−2.04703e6	−0.213008
$622$	0	0
$623$	−2.37910e6	−0.245579
$624$	0	0
$625$	−9.27438e6	−0.949696
$626$	0	0
$627$	−944784.	−0.0959762
$628$	0	0
$629$	−4632.00	−0.000466812 0
$630$	0	0
$631$	8.00600e6	0.800465	0.400233	−	0.916414i	$-0.368929\pi$
$631$	8.00600e6	0.800465	0.400233	+	0.916414i	$0.368929\pi$
$632$	0	0
$633$	5.96192e6	0.591394
$634$	0	0
$635$	2.71843e7	2.67537
$636$	0	0
$637$	−6.26644e6	−0.611889
$638$	0	0
$639$	−705672.	−0.0683677
$640$	0	0
$641$	5.20862e6	0.500700	0.250350	−	0.968155i	$-0.419454\pi$
$641$	5.20862e6	0.500700	0.250350	+	0.968155i	$0.419454\pi$
$642$	0	0
$643$	−5.08288e6	−0.484822	−0.242411	−	0.970174i	$-0.577938\pi$
$643$	−5.08288e6	−0.484822	−0.242411	+	0.970174i	$0.577938\pi$
$644$	0	0
$645$	1.24157e7	1.17509
$646$	0	0
$647$	1.02670e7	0.964233	0.482117	−	0.876107i	$-0.339868\pi$
$647$	1.02670e7	0.964233	0.482117	+	0.876107i	$0.339868\pi$
$648$	0	0
$649$	−1.55909e6	−0.145298
$650$	0	0
$651$	1.87790e6	0.173668
$652$	0	0
$653$	−2.06434e7	−1.89452	−0.947259	−	0.320469i	$-0.896160\pi$
$653$	−2.06434e7	−1.89452	−0.947259	+	0.320469i	$0.896160\pi$
$654$	0	0
$655$	−2.67350e7	−2.43488
$656$	0	0
$657$	−1.28353e6	−0.116009
$658$	0	0
$659$	5.87552e6	0.527027	0.263514	−	0.964656i	$-0.415119\pi$
$659$	5.87552e6	0.527027	0.263514	+	0.964656i	$0.415119\pi$
$660$	0	0
$661$	5.07688e6	0.451953	0.225976	−	0.974133i	$-0.427443\pi$
$661$	5.07688e6	0.451953	0.225976	+	0.974133i	$0.427443\pi$
$662$	0	0
$663$	−7272.00	−0.000642495 0
$664$	0	0
$665$	8.39808e6	0.736421
$666$	0	0
$667$	1.23777e7	1.07727
$668$	0	0
$669$	1.19734e7	1.03432
$670$	0	0
$671$	−648432.	−0.0555979
$672$	0	0
$673$	6.12989e6	0.521693	0.260846	−	0.965380i	$-0.415998\pi$
$673$	6.12989e6	0.521693	0.260846	+	0.965380i	$0.415998\pi$
$674$	0	0
$675$	2.38748e6	0.201688
$676$	0	0
$677$	3.14086e6	0.263377	0.131688	−	0.991291i	$-0.457960\pi$
$677$	3.14086e6	0.263377	0.131688	+	0.991291i	$0.457960\pi$
$678$	0	0
$679$	−4.30855e6	−0.358638
$680$	0	0
$681$	6.45440e6	0.533321
$682$	0	0
$683$	−1.57417e7	−1.29122	−0.645609	−	0.763668i	$-0.723396\pi$
$683$	−1.57417e7	−1.29122	−0.645609	+	0.763668i	$0.723396\pi$
$684$	0	0
$685$	3.15013e7	2.56509
$686$	0	0
$687$	−1.78484e6	−0.144281
$688$	0	0
$689$	1.29571e7	1.03982
$690$	0	0
$691$	1.05145e7	0.837708	0.418854	−	0.908054i	$-0.362432\pi$
$691$	1.05145e7	0.837708	0.418854	+	0.908054i	$0.362432\pi$
$692$	0	0
$693$	−104976.	−0.00830342
$694$	0	0
$695$	1.95696e7	1.53681
$696$	0	0
$697$	13748.0	0.00107191
$698$	0	0
$699$	−7.74596e6	−0.599628
$700$	0	0
$701$	−5.86967e6	−0.451148	−0.225574	−	0.974226i	$-0.572426\pi$
$701$	−5.86967e6	−0.451148	−0.225574	+	0.974226i	$0.572426\pi$
$702$	0	0
$703$	6.75346e6	0.515392
$704$	0	0
$705$	−4.09536e6	−0.310327
$706$	0	0
$707$	1.16611e6	0.0877388
$708$	0	0
$709$	1.27697e7	0.954037	0.477018	−	0.878893i	$-0.341718\pi$
$709$	1.27697e7	0.954037	0.477018	+	0.878893i	$0.341718\pi$
$710$	0	0
$711$	3.26884e6	0.242504
$712$	0	0
$713$	−1.62752e7	−1.19895
$714$	0	0
$715$	−1.16352e6	−0.0851155
$716$	0	0
$717$	3.99168e6	0.289973
$718$	0	0
$719$	−2.43243e7	−1.75476	−0.877380	−	0.479797i	$-0.840710\pi$
$719$	−2.43243e7	−1.75476	−0.877380	+	0.479797i	$0.840710\pi$
$720$	0	0
$721$	−3.94114e6	−0.282347
$722$	0	0
$723$	−308178.	−0.0219258
$724$	0	0
$725$	−1.44362e7	−1.02002
$726$	0	0
$727$	−1.65775e7	−1.16328	−0.581639	−	0.813447i	$-0.697589\pi$
$727$	−1.65775e7	−1.16328	−0.581639	+	0.813447i	$0.697589\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−34488.0	−0.00238712
$732$	0	0
$733$	1.50948e7	1.03769	0.518844	−	0.854869i	$-0.326363\pi$
$733$	1.50948e7	1.03769	0.518844	+	0.854869i	$0.326363\pi$
$734$	0	0
$735$	−1.11679e7	−0.762525
$736$	0	0
$737$	418608.	0.0283883
$738$	0	0
$739$	1.81265e7	1.22096	0.610481	−	0.792031i	$-0.290976\pi$
$739$	1.81265e7	1.22096	0.610481	+	0.792031i	$0.290976\pi$
$740$	0	0
$741$	1.06026e7	0.709359
$742$	0	0
$743$	−5.97038e6	−0.396762	−0.198381	−	0.980125i	$-0.563568\pi$
$743$	−5.97038e6	−0.396762	−0.198381	+	0.980125i	$0.563568\pi$
$744$	0	0
$745$	−2.94592e7	−1.94460
$746$	0	0
$747$	−5.36252e6	−0.351615
$748$	0	0
$749$	−3.83746e6	−0.249942
$750$	0	0
$751$	−4.93672e6	−0.319402	−0.159701	−	0.987165i	$-0.551053\pi$
$751$	−4.93672e6	−0.319402	−0.159701	+	0.987165i	$0.551053\pi$
$752$	0	0
$753$	−2.12447e6	−0.136541
$754$	0	0
$755$	3.04042e7	1.94118
$756$	0	0
$757$	−1.06262e7	−0.673965	−0.336982	−	0.941511i	$-0.609406\pi$
$757$	−1.06262e7	−0.673965	−0.336982	+	0.941511i	$0.609406\pi$
$758$	0	0
$759$	909792.	0.0573242
$760$	0	0
$761$	3.05189e7	1.91032	0.955162	−	0.296085i	$-0.0956813\pi$
$761$	3.05189e7	1.91032	0.955162	+	0.296085i	$0.0956813\pi$
$762$	0	0
$763$	2.79086e6	0.173551
$764$	0	0
$765$	−12960.0	−0.000800666 0
$766$	0	0
$767$	1.74964e7	1.07389
$768$	0	0
$769$	2.25735e7	1.37652	0.688260	−	0.725464i	$-0.258375\pi$
$769$	2.25735e7	1.37652	0.688260	+	0.725464i	$0.258375\pi$
$770$	0	0
$771$	8.36354e6	0.506704
$772$	0	0
$773$	−8.22743e6	−0.495240	−0.247620	−	0.968857i	$-0.579648\pi$
$773$	−8.22743e6	−0.495240	−0.247620	+	0.968857i	$0.579648\pi$
$774$	0	0
$775$	1.89819e7	1.13523
$776$	0	0
$777$	750384.	0.0445893
$778$	0	0
$779$	−2.00446e7	−1.18346
$780$	0	0
$781$	313632.	0.0183989
$782$	0	0
$783$	−3.21343e6	−0.187312
$784$	0	0
$785$	−4.10563e7	−2.37797
$786$	0	0
$787$	2.64585e7	1.52275	0.761373	−	0.648314i	$-0.224526\pi$
$787$	2.64585e7	1.52275	0.761373	+	0.648314i	$0.224526\pi$
$788$	0	0
$789$	1.52163e7	0.870197
$790$	0	0
$791$	6.46222e6	0.367232
$792$	0	0
$793$	7.27685e6	0.410923
$794$	0	0
$795$	2.30918e7	1.29581
$796$	0	0
$797$	−599296.	−0.0334192	−0.0167096	−	0.999860i	$-0.505319\pi$
$797$	−599296.	−0.0334192	−0.0167096	+	0.999860i	$0.505319\pi$
$798$	0	0
$799$	11376.0	0.000630409 0
$800$	0	0
$801$	−5.35297e6	−0.294790
$802$	0	0
$803$	570456.	0.0312200
$804$	0	0
$805$	−8.08704e6	−0.439845
$806$	0	0
$807$	−8.10050e6	−0.437853
$808$	0	0
$809$	−5.97875e6	−0.321173	−0.160586	−	0.987022i	$-0.551339\pi$
$809$	−5.97875e6	−0.321173	−0.160586	+	0.987022i	$0.551339\pi$
$810$	0	0
$811$	−3.22959e7	−1.72423	−0.862116	−	0.506711i	$-0.830861\pi$
$811$	−3.22959e7	−1.72423	−0.862116	+	0.506711i	$0.830861\pi$
$812$	0	0
$813$	−1.42550e7	−0.756383
$814$	0	0
$815$	−2.52000e6	−0.132894
$816$	0	0
$817$	5.02835e7	2.63554
$818$	0	0
$819$	1.17806e6	0.0613704
$820$	0	0
$821$	−1.94847e7	−1.00887	−0.504437	−	0.863449i	$-0.668300\pi$
$821$	−1.94847e7	−1.00887	−0.504437	+	0.863449i	$0.668300\pi$
$822$	0	0
$823$	2.52634e7	1.30015	0.650073	−	0.759872i	$-0.274738\pi$
$823$	2.52634e7	1.30015	0.650073	+	0.759872i	$0.274738\pi$
$824$	0	0
$825$	−1.06110e6	−0.0542777
$826$	0	0
$827$	6.51164e6	0.331075	0.165538	−	0.986203i	$-0.447064\pi$
$827$	6.51164e6	0.331075	0.165538	+	0.986203i	$0.447064\pi$
$828$	0	0
$829$	1.29742e7	0.655682	0.327841	−	0.944733i	$-0.393679\pi$
$829$	1.29742e7	0.655682	0.327841	+	0.944733i	$0.393679\pi$
$830$	0	0
$831$	1.75919e7	0.883713
$832$	0	0
$833$	31022.0	0.00154902
$834$	0	0
$835$	−8.52480e6	−0.423124
$836$	0	0
$837$	4.22528e6	0.208469
$838$	0	0
$839$	−700056.	−0.0343343	−0.0171671	−	0.999853i	$-0.505465\pi$
$839$	−700056.	−0.0343343	−0.0171671	+	0.999853i	$0.505465\pi$
$840$	0	0
$841$	−1.08068e6	−0.0526877
$842$	0	0
$843$	−1.94473e7	−0.942517
$844$	0	0
$845$	−1.66462e7	−0.801996
$846$	0	0
$847$	−5.75118e6	−0.275454
$848$	0	0
$849$	−5.93600e6	−0.282634
$850$	0	0
$851$	−6.50333e6	−0.307831
$852$	0	0
$853$	2.98508e6	0.140470	0.0702351	−	0.997530i	$-0.477625\pi$
$853$	2.98508e6	0.140470	0.0702351	+	0.997530i	$0.477625\pi$
$854$	0	0
$855$	1.88957e7	0.883990
$856$	0	0
$857$	−1.82849e7	−0.850436	−0.425218	−	0.905091i	$-0.639803\pi$
$857$	−1.82849e7	−0.850436	−0.425218	+	0.905091i	$0.639803\pi$
$858$	0	0
$859$	9.64825e6	0.446134	0.223067	−	0.974803i	$-0.428393\pi$
$859$	9.64825e6	0.446134	0.223067	+	0.974803i	$0.428393\pi$
$860$	0	0
$861$	−2.22718e6	−0.102387
$862$	0	0
$863$	1.02411e7	0.468081	0.234041	−	0.972227i	$-0.424805\pi$
$863$	1.02411e7	0.468081	0.234041	+	0.972227i	$0.424805\pi$
$864$	0	0
$865$	1.99744e7	0.907681
$866$	0	0
$867$	−1.27787e7	−0.577349
$868$	0	0
$869$	−1.45282e6	−0.0652621
$870$	0	0
$871$	−4.69771e6	−0.209817
$872$	0	0
$873$	−9.69424e6	−0.430505
$874$	0	0
$875$	432000.	0.0190750
$876$	0	0
$877$	1.42253e7	0.624545	0.312272	−	0.949993i	$-0.398910\pi$
$877$	1.42253e7	0.624545	0.312272	+	0.949993i	$0.398910\pi$
$878$	0	0
$879$	7.22398e6	0.315358
$880$	0	0
$881$	−3.91991e7	−1.70152	−0.850758	−	0.525558i	$-0.823856\pi$
$881$	−3.91991e7	−1.70152	−0.850758	+	0.525558i	$0.823856\pi$
$882$	0	0
$883$	−2.17875e7	−0.940383	−0.470191	−	0.882564i	$-0.655815\pi$
$883$	−2.17875e7	−0.940383	−0.470191	+	0.882564i	$0.655815\pi$
$884$	0	0
$885$	3.11818e7	1.33827
$886$	0	0
$887$	2.95406e7	1.26070	0.630348	−	0.776313i	$-0.282912\pi$
$887$	2.95406e7	1.26070	0.630348	+	0.776313i	$0.282912\pi$
$888$	0	0
$889$	1.22329e7	0.519131
$890$	0	0
$891$	−236196.	−0.00996732
$892$	0	0
$893$	−1.65862e7	−0.696015
$894$	0	0
$895$	2.90333e7	1.21154
$896$	0	0
$897$	−1.02099e7	−0.423682
$898$	0	0
$899$	−2.55488e7	−1.05432
$900$	0	0
$901$	−64144.0	−0.00263235
$902$	0	0
$903$	5.58706e6	0.228015
$904$	0	0
$905$	4.05315e7	1.64502
$906$	0	0
$907$	−4.26701e7	−1.72229	−0.861144	−	0.508362i	$-0.830251\pi$
$907$	−4.26701e7	−1.72229	−0.861144	+	0.508362i	$0.830251\pi$
$908$	0	0
$909$	2.62375e6	0.105321
$910$	0	0
$911$	−2.80109e7	−1.11823	−0.559115	−	0.829090i	$-0.688859\pi$
$911$	−2.80109e7	−1.11823	−0.559115	+	0.829090i	$0.688859\pi$
$912$	0	0
$913$	2.38334e6	0.0946258
$914$	0	0
$915$	1.29686e7	0.512085
$916$	0	0
$917$	−1.20308e7	−0.472465
$918$	0	0
$919$	2.94685e7	1.15098	0.575492	−	0.817808i	$-0.304811\pi$
$919$	2.94685e7	1.15098	0.575492	+	0.817808i	$0.304811\pi$
$920$	0	0
$921$	1.60607e6	0.0623900
$922$	0	0
$923$	−3.51965e6	−0.135986
$924$	0	0
$925$	7.58490e6	0.291471
$926$	0	0
$927$	−8.86756e6	−0.338926
$928$	0	0
$929$	−2.29283e7	−0.871631	−0.435816	−	0.900036i	$-0.643540\pi$
$929$	−2.29283e7	−0.871631	−0.435816	+	0.900036i	$0.643540\pi$
$930$	0	0
$931$	−4.52301e7	−1.71022
$932$	0	0
$933$	1.13244e7	0.425905
$934$	0	0
$935$	5760.00	0.000215473 0
$936$	0	0
$937$	2.94030e6	0.109406	0.0547032	−	0.998503i	$-0.482579\pi$
$937$	2.94030e6	0.109406	0.0547032	+	0.998503i	$0.482579\pi$
$938$	0	0
$939$	8.30756e6	0.307475
$940$	0	0
$941$	−3.11062e7	−1.14518	−0.572589	−	0.819843i	$-0.694061\pi$
$941$	−3.11062e7	−1.14518	−0.572589	+	0.819843i	$0.694061\pi$
$942$	0	0
$943$	1.93022e7	0.706850
$944$	0	0
$945$	2.09952e6	0.0764787
$946$	0	0
$947$	−2.15022e7	−0.779126	−0.389563	−	0.921000i	$-0.627374\pi$
$947$	−2.15022e7	−0.779126	−0.389563	+	0.921000i	$0.627374\pi$
$948$	0	0
$949$	−6.40178e6	−0.230747
$950$	0	0
$951$	5.11567e6	0.183422
$952$	0	0
$953$	−7.04724e6	−0.251355	−0.125677	−	0.992071i	$-0.540110\pi$
$953$	−7.04724e6	−0.251355	−0.125677	+	0.992071i	$0.540110\pi$
$954$	0	0
$955$	6.45005e7	2.28852
$956$	0	0
$957$	1.42819e6	0.0504088
$958$	0	0
$959$	1.41756e7	0.497730
$960$	0	0
$961$	4.96446e6	0.173406
$962$	0	0
$963$	−8.63428e6	−0.300027
$964$	0	0
$965$	−3.50734e7	−1.21244
$966$	0	0
$967$	1.33230e7	0.458179	0.229090	−	0.973405i	$-0.426425\pi$
$967$	1.33230e7	0.458179	0.229090	+	0.973405i	$0.426425\pi$
$968$	0	0
$969$	−52488.0	−0.00179577
$970$	0	0
$971$	−1.72053e7	−0.585618	−0.292809	−	0.956171i	$-0.594590\pi$
$971$	−1.72053e7	−0.585618	−0.292809	+	0.956171i	$0.594590\pi$
$972$	0	0
$973$	8.80632e6	0.298203
$974$	0	0
$975$	1.19079e7	0.401166
$976$	0	0
$977$	3.43520e7	1.15137	0.575686	−	0.817671i	$-0.304735\pi$
$977$	3.43520e7	1.15137	0.575686	+	0.817671i	$0.304735\pi$
$978$	0	0
$979$	2.37910e6	0.0793332
$980$	0	0
$981$	6.27944e6	0.208328
$982$	0	0
$983$	−1.83486e7	−0.605647	−0.302824	−	0.953047i	$-0.597929\pi$
$983$	−1.83486e7	−0.605647	−0.302824	+	0.953047i	$0.597929\pi$
$984$	0	0
$985$	−2.67731e7	−0.879242
$986$	0	0
$987$	−1.84291e6	−0.0602160
$988$	0	0
$989$	−4.84212e7	−1.57414
$990$	0	0
$991$	2.27810e7	0.736866	0.368433	−	0.929654i	$-0.379894\pi$
$991$	2.27810e7	0.736866	0.368433	+	0.929654i	$0.379894\pi$
$992$	0	0
$993$	−2.34579e7	−0.754947
$994$	0	0
$995$	5.06592e6	0.162219
$996$	0	0
$997$	−2.50268e7	−0.797382	−0.398691	−	0.917085i	$-0.630535\pi$
$997$	−2.50268e7	−0.797382	−0.398691	+	0.917085i	$0.630535\pi$
$998$	0	0
$999$	1.68836e6	0.0535245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.k.1.1		1
4.3	odd	2		768.6.a.e.1.1		1
8.3	odd	2		768.6.a.h.1.1		1
8.5	even	2		768.6.a.b.1.1		1
16.3	odd	4		384.6.d.d.193.1	yes	2
16.5	even	4		384.6.d.c.193.1	✓	2
16.11	odd	4		384.6.d.d.193.2	yes	2
16.13	even	4		384.6.d.c.193.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.c.193.1	✓	2	16.5	even	4
384.6.d.c.193.2	yes	2	16.13	even	4
384.6.d.d.193.1	yes	2	16.3	odd	4
384.6.d.d.193.2	yes	2	16.11	odd	4
768.6.a.b.1.1		1	8.5	even	2
768.6.a.e.1.1		1	4.3	odd	2
768.6.a.h.1.1		1	8.3	odd	2
768.6.a.k.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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