LMFDB - Embedded newform 384.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.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:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} -20.0000 q^{5} -122.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} -20.0000 q^{5} -122.000 q^{7} +81.0000 q^{9} +724.000 q^{11} -914.000 q^{13} +180.000 q^{15} +1006.00 q^{17} -2920.00 q^{19} +1098.00 q^{21} -3124.00 q^{23} -2725.00 q^{25} -729.000 q^{27} +6744.00 q^{29} -5010.00 q^{31} -6516.00 q^{33} +2440.00 q^{35} -5278.00 q^{37} +8226.00 q^{39} +5238.00 q^{41} +16752.0 q^{43} -1620.00 q^{45} -1108.00 q^{47} -1923.00 q^{49} -9054.00 q^{51} +22008.0 q^{53} -14480.0 q^{55} +26280.0 q^{57} -23716.0 q^{59} +45202.0 q^{61} -9882.00 q^{63} +18280.0 q^{65} +22756.0 q^{67} +28116.0 q^{69} -53436.0 q^{71} +4790.00 q^{73} +24525.0 q^{75} -88328.0 q^{77} +1886.00 q^{79} +6561.00 q^{81} +11268.0 q^{83} -20120.0 q^{85} -60696.0 q^{87} +73522.0 q^{89} +111508. q^{91} +45090.0 q^{93} +58400.0 q^{95} +114154. q^{97} +58644.0 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−20.0000	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$5$	−20.0000	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$6$	0	0
$7$	−122.000	−0.941054	−0.470527	−	0.882385i	$-0.655936\pi$
$7$	−122.000	−0.941054	−0.470527	+	0.882385i	$0.655936\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	724.000	1.80408	0.902042	−	0.431648i	$-0.142068\pi$
$11$	724.000	1.80408	0.902042	+	0.431648i	$0.142068\pi$
$12$	0	0
$13$	−914.000	−1.49999	−0.749994	−	0.661445i	$-0.769944\pi$
$13$	−914.000	−1.49999	−0.749994	+	0.661445i	$0.769944\pi$
$14$	0	0
$15$	180.000	0.206559
$16$	0	0
$17$	1006.00	0.844259	0.422129	−	0.906536i	$-0.361283\pi$
$17$	1006.00	0.844259	0.422129	+	0.906536i	$0.361283\pi$
$18$	0	0
$19$	−2920.00	−1.85566	−0.927831	−	0.373001i	$-0.878329\pi$
$19$	−2920.00	−1.85566	−0.927831	+	0.373001i	$0.878329\pi$
$20$	0	0
$21$	1098.00	0.543318
$22$	0	0
$23$	−3124.00	−1.23138	−0.615689	−	0.787989i	$-0.711122\pi$
$23$	−3124.00	−1.23138	−0.615689	+	0.787989i	$0.711122\pi$
$24$	0	0
$25$	−2725.00	−0.872000
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	6744.00	1.48910	0.744548	−	0.667569i	$-0.232665\pi$
$29$	6744.00	1.48910	0.744548	+	0.667569i	$0.232665\pi$
$30$	0	0
$31$	−5010.00	−0.936340	−0.468170	−	0.883638i	$-0.655086\pi$
$31$	−5010.00	−0.936340	−0.468170	+	0.883638i	$0.655086\pi$
$32$	0	0
$33$	−6516.00	−1.04159
$34$	0	0
$35$	2440.00	0.336682
$36$	0	0
$37$	−5278.00	−0.633819	−0.316909	−	0.948456i	$-0.602645\pi$
$37$	−5278.00	−0.633819	−0.316909	+	0.948456i	$0.602645\pi$
$38$	0	0
$39$	8226.00	0.866019
$40$	0	0
$41$	5238.00	0.486638	0.243319	−	0.969946i	$-0.421764\pi$
$41$	5238.00	0.486638	0.243319	+	0.969946i	$0.421764\pi$
$42$	0	0
$43$	16752.0	1.38164	0.690821	−	0.723026i	$-0.257249\pi$
$43$	16752.0	1.38164	0.690821	+	0.723026i	$0.257249\pi$
$44$	0	0
$45$	−1620.00	−0.119257
$46$	0	0
$47$	−1108.00	−0.0731636	−0.0365818	−	0.999331i	$-0.511647\pi$
$47$	−1108.00	−0.0731636	−0.0365818	+	0.999331i	$0.511647\pi$
$48$	0	0
$49$	−1923.00	−0.114417
$50$	0	0
$51$	−9054.00	−0.487433
$52$	0	0
$53$	22008.0	1.07619	0.538097	−	0.842883i	$-0.319143\pi$
$53$	22008.0	1.07619	0.538097	+	0.842883i	$0.319143\pi$
$54$	0	0
$55$	−14480.0	−0.645449
$56$	0	0
$57$	26280.0	1.07137
$58$	0	0
$59$	−23716.0	−0.886975	−0.443488	−	0.896281i	$-0.646259\pi$
$59$	−23716.0	−0.886975	−0.443488	+	0.896281i	$0.646259\pi$
$60$	0	0
$61$	45202.0	1.55537	0.777684	−	0.628656i	$-0.216394\pi$
$61$	45202.0	1.55537	0.777684	+	0.628656i	$0.216394\pi$
$62$	0	0
$63$	−9882.00	−0.313685
$64$	0	0
$65$	18280.0	0.536652
$66$	0	0
$67$	22756.0	0.619311	0.309656	−	0.950849i	$-0.399786\pi$
$67$	22756.0	0.619311	0.309656	+	0.950849i	$0.399786\pi$
$68$	0	0
$69$	28116.0	0.710936
$70$	0	0
$71$	−53436.0	−1.25802	−0.629011	−	0.777397i	$-0.716540\pi$
$71$	−53436.0	−1.25802	−0.629011	+	0.777397i	$0.716540\pi$
$72$	0	0
$73$	4790.00	0.105203	0.0526015	−	0.998616i	$-0.483249\pi$
$73$	4790.00	0.105203	0.0526015	+	0.998616i	$0.483249\pi$
$74$	0	0
$75$	24525.0	0.503449
$76$	0	0
$77$	−88328.0	−1.69774
$78$	0	0
$79$	1886.00	0.0339996	0.0169998	−	0.999855i	$-0.494589\pi$
$79$	1886.00	0.0339996	0.0169998	+	0.999855i	$0.494589\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	11268.0	0.179536	0.0897680	−	0.995963i	$-0.471387\pi$
$83$	11268.0	0.179536	0.0897680	+	0.995963i	$0.471387\pi$
$84$	0	0
$85$	−20120.0	−0.302051
$86$	0	0
$87$	−60696.0	−0.859730
$88$	0	0
$89$	73522.0	0.983880	0.491940	−	0.870629i	$-0.336288\pi$
$89$	73522.0	0.983880	0.491940	+	0.870629i	$0.336288\pi$
$90$	0	0
$91$	111508.	1.41157
$92$	0	0
$93$	45090.0	0.540596
$94$	0	0
$95$	58400.0	0.663902
$96$	0	0
$97$	114154.	1.23186	0.615931	−	0.787800i	$-0.288780\pi$
$97$	114154.	1.23186	0.615931	+	0.787800i	$0.288780\pi$
$98$	0	0
$99$	58644.0	0.601361
$100$	0	0
$101$	168024.	1.63896	0.819479	−	0.573109i	$-0.194263\pi$
$101$	168024.	1.63896	0.819479	+	0.573109i	$0.194263\pi$
$102$	0	0
$103$	−38750.0	−0.359897	−0.179949	−	0.983676i	$-0.557593\pi$
$103$	−38750.0	−0.359897	−0.179949	+	0.983676i	$0.557593\pi$
$104$	0	0
$105$	−21960.0	−0.194383
$106$	0	0
$107$	207172.	1.74933	0.874665	−	0.484728i	$-0.161082\pi$
$107$	207172.	1.74933	0.874665	+	0.484728i	$0.161082\pi$
$108$	0	0
$109$	39318.0	0.316975	0.158488	−	0.987361i	$-0.449338\pi$
$109$	39318.0	0.316975	0.158488	+	0.987361i	$0.449338\pi$
$110$	0	0
$111$	47502.0	0.365935
$112$	0	0
$113$	5666.00	0.0417427	0.0208713	−	0.999782i	$-0.493356\pi$
$113$	5666.00	0.0417427	0.0208713	+	0.999782i	$0.493356\pi$
$114$	0	0
$115$	62480.0	0.440551
$116$	0	0
$117$	−74034.0	−0.499996
$118$	0	0
$119$	−122732.	−0.794494
$120$	0	0
$121$	363125.	2.25472
$122$	0	0
$123$	−47142.0	−0.280960
$124$	0	0
$125$	117000.	0.669747
$126$	0	0
$127$	−157894.	−0.868673	−0.434337	−	0.900751i	$-0.643017\pi$
$127$	−157894.	−0.868673	−0.434337	+	0.900751i	$0.643017\pi$
$128$	0	0
$129$	−150768.	−0.797691
$130$	0	0
$131$	94188.0	0.479532	0.239766	−	0.970831i	$-0.422929\pi$
$131$	94188.0	0.479532	0.239766	+	0.970831i	$0.422929\pi$
$132$	0	0
$133$	356240.	1.74628
$134$	0	0
$135$	14580.0	0.0688530
$136$	0	0
$137$	−29234.0	−0.133072	−0.0665360	−	0.997784i	$-0.521195\pi$
$137$	−29234.0	−0.133072	−0.0665360	+	0.997784i	$0.521195\pi$
$138$	0	0
$139$	−123556.	−0.542409	−0.271204	−	0.962522i	$-0.587422\pi$
$139$	−123556.	−0.542409	−0.271204	+	0.962522i	$0.587422\pi$
$140$	0	0
$141$	9972.00	0.0422410
$142$	0	0
$143$	−661736.	−2.70611
$144$	0	0
$145$	−134880.	−0.532755
$146$	0	0
$147$	17307.0	0.0660585
$148$	0	0
$149$	247836.	0.914532	0.457266	−	0.889330i	$-0.348829\pi$
$149$	247836.	0.914532	0.457266	+	0.889330i	$0.348829\pi$
$150$	0	0
$151$	256670.	0.916079	0.458039	−	0.888932i	$-0.348552\pi$
$151$	256670.	0.916079	0.458039	+	0.888932i	$0.348552\pi$
$152$	0	0
$153$	81486.0	0.281420
$154$	0	0
$155$	100200.	0.334995
$156$	0	0
$157$	−591534.	−1.91527	−0.957636	−	0.287981i	$-0.907016\pi$
$157$	−591534.	−1.91527	−0.957636	+	0.287981i	$0.907016\pi$
$158$	0	0
$159$	−198072.	−0.621341
$160$	0	0
$161$	381128.	1.15879
$162$	0	0
$163$	120056.	0.353928	0.176964	−	0.984217i	$-0.443372\pi$
$163$	120056.	0.353928	0.176964	+	0.984217i	$0.443372\pi$
$164$	0	0
$165$	130320.	0.372650
$166$	0	0
$167$	82544.0	0.229031	0.114516	−	0.993421i	$-0.463468\pi$
$167$	82544.0	0.229031	0.114516	+	0.993421i	$0.463468\pi$
$168$	0	0
$169$	464103.	1.24996
$170$	0	0
$171$	−236520.	−0.618554
$172$	0	0
$173$	229404.	0.582755	0.291377	−	0.956608i	$-0.405886\pi$
$173$	229404.	0.582755	0.291377	+	0.956608i	$0.405886\pi$
$174$	0	0
$175$	332450.	0.820599
$176$	0	0
$177$	213444.	0.512095
$178$	0	0
$179$	−175788.	−0.410069	−0.205034	−	0.978755i	$-0.565731\pi$
$179$	−175788.	−0.410069	−0.205034	+	0.978755i	$0.565731\pi$
$180$	0	0
$181$	372934.	0.846127	0.423063	−	0.906100i	$-0.360955\pi$
$181$	372934.	0.846127	0.423063	+	0.906100i	$0.360955\pi$
$182$	0	0
$183$	−406818.	−0.897992
$184$	0	0
$185$	105560.	0.226762
$186$	0	0
$187$	728344.	1.52311
$188$	0	0
$189$	88938.0	0.181106
$190$	0	0
$191$	729016.	1.44595	0.722976	−	0.690874i	$-0.242774\pi$
$191$	729016.	1.44595	0.722976	+	0.690874i	$0.242774\pi$
$192$	0	0
$193$	−852962.	−1.64830	−0.824150	−	0.566371i	$-0.808347\pi$
$193$	−852962.	−1.64830	−0.824150	+	0.566371i	$0.808347\pi$
$194$	0	0
$195$	−164520.	−0.309836
$196$	0	0
$197$	214112.	0.393075	0.196538	−	0.980496i	$-0.437030\pi$
$197$	214112.	0.393075	0.196538	+	0.980496i	$0.437030\pi$
$198$	0	0
$199$	−498902.	−0.893064	−0.446532	−	0.894768i	$-0.647341\pi$
$199$	−498902.	−0.893064	−0.446532	+	0.894768i	$0.647341\pi$
$200$	0	0
$201$	−204804.	−0.357559
$202$	0	0
$203$	−822768.	−1.40132
$204$	0	0
$205$	−104760.	−0.174105
$206$	0	0
$207$	−253044.	−0.410459
$208$	0	0
$209$	−2.11408e6	−3.34777
$210$	0	0
$211$	474932.	0.734388	0.367194	−	0.930144i	$-0.380319\pi$
$211$	474932.	0.734388	0.367194	+	0.930144i	$0.380319\pi$
$212$	0	0
$213$	480924.	0.726319
$214$	0	0
$215$	−335040.	−0.494311
$216$	0	0
$217$	611220.	0.881147
$218$	0	0
$219$	−43110.0	−0.0607390
$220$	0	0
$221$	−919484.	−1.26638
$222$	0	0
$223$	−728118.	−0.980482	−0.490241	−	0.871587i	$-0.663091\pi$
$223$	−728118.	−0.980482	−0.490241	+	0.871587i	$0.663091\pi$
$224$	0	0
$225$	−220725.	−0.290667
$226$	0	0
$227$	355044.	0.457317	0.228659	−	0.973507i	$-0.426566\pi$
$227$	355044.	0.457317	0.228659	+	0.973507i	$0.426566\pi$
$228$	0	0
$229$	−987490.	−1.24435	−0.622177	−	0.782877i	$-0.713752\pi$
$229$	−987490.	−1.24435	−0.622177	+	0.782877i	$0.713752\pi$
$230$	0	0
$231$	794952.	0.980192
$232$	0	0
$233$	993210.	1.19854	0.599268	−	0.800548i	$-0.295458\pi$
$233$	993210.	1.19854	0.599268	+	0.800548i	$0.295458\pi$
$234$	0	0
$235$	22160.0	0.0261758
$236$	0	0
$237$	−16974.0	−0.0196297
$238$	0	0
$239$	765984.	0.867411	0.433706	−	0.901055i	$-0.357206\pi$
$239$	765984.	0.867411	0.433706	+	0.901055i	$0.357206\pi$
$240$	0	0
$241$	1.66425e6	1.84577	0.922884	−	0.385079i	$-0.125826\pi$
$241$	1.66425e6	1.84577	0.922884	+	0.385079i	$0.125826\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	38460.0	0.0409349
$246$	0	0
$247$	2.66888e6	2.78347
$248$	0	0
$249$	−101412.	−0.103655
$250$	0	0
$251$	−1.02534e6	−1.02727	−0.513634	−	0.858010i	$-0.671701\pi$
$251$	−1.02534e6	−1.02727	−0.513634	+	0.858010i	$0.671701\pi$
$252$	0	0
$253$	−2.26178e6	−2.22151
$254$	0	0
$255$	181080.	0.174389
$256$	0	0
$257$	−862254.	−0.814334	−0.407167	−	0.913354i	$-0.633483\pi$
$257$	−862254.	−0.814334	−0.407167	+	0.913354i	$0.633483\pi$
$258$	0	0
$259$	643916.	0.596458
$260$	0	0
$261$	546264.	0.496365
$262$	0	0
$263$	−233832.	−0.208456	−0.104228	−	0.994553i	$-0.533237\pi$
$263$	−233832.	−0.208456	−0.104228	+	0.994553i	$0.533237\pi$
$264$	0	0
$265$	−440160.	−0.385031
$266$	0	0
$267$	−661698.	−0.568044
$268$	0	0
$269$	−461848.	−0.389151	−0.194576	−	0.980888i	$-0.562333\pi$
$269$	−461848.	−0.389151	−0.194576	+	0.980888i	$0.562333\pi$
$270$	0	0
$271$	1.45220e6	1.20117	0.600584	−	0.799562i	$-0.294935\pi$
$271$	1.45220e6	1.20117	0.600584	+	0.799562i	$0.294935\pi$
$272$	0	0
$273$	−1.00357e6	−0.814971
$274$	0	0
$275$	−1.97290e6	−1.57316
$276$	0	0
$277$	−1.53748e6	−1.20396	−0.601978	−	0.798513i	$-0.705621\pi$
$277$	−1.53748e6	−1.20396	−0.601978	+	0.798513i	$0.705621\pi$
$278$	0	0
$279$	−405810.	−0.312113
$280$	0	0
$281$	193266.	0.146012	0.0730062	−	0.997331i	$-0.476741\pi$
$281$	193266.	0.146012	0.0730062	+	0.997331i	$0.476741\pi$
$282$	0	0
$283$	−1.58718e6	−1.17804	−0.589020	−	0.808118i	$-0.700486\pi$
$283$	−1.58718e6	−1.17804	−0.589020	+	0.808118i	$0.700486\pi$
$284$	0	0
$285$	−525600.	−0.383304
$286$	0	0
$287$	−639036.	−0.457953
$288$	0	0
$289$	−407821.	−0.287227
$290$	0	0
$291$	−1.02739e6	−0.711215
$292$	0	0
$293$	2.55274e6	1.73715	0.868577	−	0.495555i	$-0.165035\pi$
$293$	2.55274e6	1.73715	0.868577	+	0.495555i	$0.165035\pi$
$294$	0	0
$295$	474320.	0.317334
$296$	0	0
$297$	−527796.	−0.347196
$298$	0	0
$299$	2.85534e6	1.84705
$300$	0	0
$301$	−2.04374e6	−1.30020
$302$	0	0
$303$	−1.51222e6	−0.946253
$304$	0	0
$305$	−904040.	−0.556465
$306$	0	0
$307$	−461868.	−0.279687	−0.139843	−	0.990174i	$-0.544660\pi$
$307$	−461868.	−0.279687	−0.139843	+	0.990174i	$0.544660\pi$
$308$	0	0
$309$	348750.	0.207787
$310$	0	0
$311$	−1.13480e6	−0.665301	−0.332651	−	0.943050i	$-0.607943\pi$
$311$	−1.13480e6	−0.665301	−0.332651	+	0.943050i	$0.607943\pi$
$312$	0	0
$313$	−720986.	−0.415974	−0.207987	−	0.978132i	$-0.566691\pi$
$313$	−720986.	−0.415974	−0.207987	+	0.978132i	$0.566691\pi$
$314$	0	0
$315$	197640.	0.112227
$316$	0	0
$317$	−651792.	−0.364301	−0.182151	−	0.983271i	$-0.558306\pi$
$317$	−651792.	−0.364301	−0.182151	+	0.983271i	$0.558306\pi$
$318$	0	0
$319$	4.88266e6	2.68645
$320$	0	0
$321$	−1.86455e6	−1.00998
$322$	0	0
$323$	−2.93752e6	−1.56666
$324$	0	0
$325$	2.49065e6	1.30799
$326$	0	0
$327$	−353862.	−0.183006
$328$	0	0
$329$	135176.	0.0688509
$330$	0	0
$331$	2.10895e6	1.05802	0.529012	−	0.848614i	$-0.322562\pi$
$331$	2.10895e6	1.05802	0.529012	+	0.848614i	$0.322562\pi$
$332$	0	0
$333$	−427518.	−0.211273
$334$	0	0
$335$	−455120.	−0.221572
$336$	0	0
$337$	−1.45008e6	−0.695533	−0.347767	−	0.937581i	$-0.613060\pi$
$337$	−1.45008e6	−0.695533	−0.347767	+	0.937581i	$0.613060\pi$
$338$	0	0
$339$	−50994.0	−0.0241002
$340$	0	0
$341$	−3.62724e6	−1.68924
$342$	0	0
$343$	2.28506e6	1.04873
$344$	0	0
$345$	−562320.	−0.254352
$346$	0	0
$347$	−1.70980e6	−0.762291	−0.381145	−	0.924515i	$-0.624470\pi$
$347$	−1.70980e6	−0.762291	−0.381145	+	0.924515i	$0.624470\pi$
$348$	0	0
$349$	1.75191e6	0.769923	0.384961	−	0.922933i	$-0.374215\pi$
$349$	1.75191e6	0.769923	0.384961	+	0.922933i	$0.374215\pi$
$350$	0	0
$351$	666306.	0.288673
$352$	0	0
$353$	−2.13401e6	−0.911508	−0.455754	−	0.890106i	$-0.650630\pi$
$353$	−2.13401e6	−0.911508	−0.455754	+	0.890106i	$0.650630\pi$
$354$	0	0
$355$	1.06872e6	0.450083
$356$	0	0
$357$	1.10459e6	0.458701
$358$	0	0
$359$	387220.	0.158570	0.0792851	−	0.996852i	$-0.474736\pi$
$359$	387220.	0.158570	0.0792851	+	0.996852i	$0.474736\pi$
$360$	0	0
$361$	6.05030e6	2.44348
$362$	0	0
$363$	−3.26812e6	−1.30176
$364$	0	0
$365$	−95800.0	−0.0376386
$366$	0	0
$367$	−1.30194e6	−0.504574	−0.252287	−	0.967652i	$-0.581183\pi$
$367$	−1.30194e6	−0.504574	−0.252287	+	0.967652i	$0.581183\pi$
$368$	0	0
$369$	424278.	0.162213
$370$	0	0
$371$	−2.68498e6	−1.01276
$372$	0	0
$373$	−2.77593e6	−1.03309	−0.516544	−	0.856261i	$-0.672782\pi$
$373$	−2.77593e6	−1.03309	−0.516544	+	0.856261i	$0.672782\pi$
$374$	0	0
$375$	−1.05300e6	−0.386679
$376$	0	0
$377$	−6.16402e6	−2.23363
$378$	0	0
$379$	2.08101e6	0.744176	0.372088	−	0.928197i	$-0.378642\pi$
$379$	2.08101e6	0.744176	0.372088	+	0.928197i	$0.378642\pi$
$380$	0	0
$381$	1.42105e6	0.501529
$382$	0	0
$383$	936408.	0.326188	0.163094	−	0.986611i	$-0.447853\pi$
$383$	936408.	0.326188	0.163094	+	0.986611i	$0.447853\pi$
$384$	0	0
$385$	1.76656e6	0.607402
$386$	0	0
$387$	1.35691e6	0.460547
$388$	0	0
$389$	923420.	0.309404	0.154702	−	0.987961i	$-0.450558\pi$
$389$	923420.	0.309404	0.154702	+	0.987961i	$0.450558\pi$
$390$	0	0
$391$	−3.14274e6	−1.03960
$392$	0	0
$393$	−847692.	−0.276858
$394$	0	0
$395$	−37720.0	−0.0121641
$396$	0	0
$397$	−4.76055e6	−1.51594	−0.757968	−	0.652292i	$-0.773808\pi$
$397$	−4.76055e6	−1.51594	−0.757968	+	0.652292i	$0.773808\pi$
$398$	0	0
$399$	−3.20616e6	−1.00821
$400$	0	0
$401$	1.99791e6	0.620462	0.310231	−	0.950661i	$-0.399594\pi$
$401$	1.99791e6	0.620462	0.310231	+	0.950661i	$0.399594\pi$
$402$	0	0
$403$	4.57914e6	1.40450
$404$	0	0
$405$	−131220.	−0.0397523
$406$	0	0
$407$	−3.82127e6	−1.14346
$408$	0	0
$409$	2.35352e6	0.695681	0.347840	−	0.937554i	$-0.386915\pi$
$409$	2.35352e6	0.695681	0.347840	+	0.937554i	$0.386915\pi$
$410$	0	0
$411$	263106.	0.0768292
$412$	0	0
$413$	2.89335e6	0.834692
$414$	0	0
$415$	−225360.	−0.0642328
$416$	0	0
$417$	1.11200e6	0.313160
$418$	0	0
$419$	−1.15232e6	−0.320654	−0.160327	−	0.987064i	$-0.551255\pi$
$419$	−1.15232e6	−0.320654	−0.160327	+	0.987064i	$0.551255\pi$
$420$	0	0
$421$	−2.34116e6	−0.643763	−0.321882	−	0.946780i	$-0.604315\pi$
$421$	−2.34116e6	−0.643763	−0.321882	+	0.946780i	$0.604315\pi$
$422$	0	0
$423$	−89748.0	−0.0243879
$424$	0	0
$425$	−2.74135e6	−0.736194
$426$	0	0
$427$	−5.51464e6	−1.46369
$428$	0	0
$429$	5.95562e6	1.56237
$430$	0	0
$431$	1.65631e6	0.429485	0.214742	−	0.976671i	$-0.431109\pi$
$431$	1.65631e6	0.429485	0.214742	+	0.976671i	$0.431109\pi$
$432$	0	0
$433$	1.80847e6	0.463544	0.231772	−	0.972770i	$-0.425548\pi$
$433$	1.80847e6	0.463544	0.231772	+	0.972770i	$0.425548\pi$
$434$	0	0
$435$	1.21392e6	0.307586
$436$	0	0
$437$	9.12208e6	2.28502
$438$	0	0
$439$	3.34811e6	0.829161	0.414581	−	0.910013i	$-0.363928\pi$
$439$	3.34811e6	0.829161	0.414581	+	0.910013i	$0.363928\pi$
$440$	0	0
$441$	−155763.	−0.0381389
$442$	0	0
$443$	1.59409e6	0.385926	0.192963	−	0.981206i	$-0.438190\pi$
$443$	1.59409e6	0.385926	0.192963	+	0.981206i	$0.438190\pi$
$444$	0	0
$445$	−1.47044e6	−0.352004
$446$	0	0
$447$	−2.23052e6	−0.528005
$448$	0	0
$449$	2.55174e6	0.597339	0.298670	−	0.954357i	$-0.403457\pi$
$449$	2.55174e6	0.597339	0.298670	+	0.954357i	$0.403457\pi$
$450$	0	0
$451$	3.79231e6	0.877936
$452$	0	0
$453$	−2.31003e6	−0.528898
$454$	0	0
$455$	−2.23016e6	−0.505019
$456$	0	0
$457$	996882.	0.223282	0.111641	−	0.993749i	$-0.464389\pi$
$457$	996882.	0.223282	0.111641	+	0.993749i	$0.464389\pi$
$458$	0	0
$459$	−733374.	−0.162478
$460$	0	0
$461$	1.89230e6	0.414703	0.207352	−	0.978266i	$-0.433516\pi$
$461$	1.89230e6	0.414703	0.207352	+	0.978266i	$0.433516\pi$
$462$	0	0
$463$	−3.30062e6	−0.715555	−0.357778	−	0.933807i	$-0.616465\pi$
$463$	−3.30062e6	−0.715555	−0.357778	+	0.933807i	$0.616465\pi$
$464$	0	0
$465$	−901800.	−0.193410
$466$	0	0
$467$	3.50482e6	0.743658	0.371829	−	0.928301i	$-0.378731\pi$
$467$	3.50482e6	0.743658	0.371829	+	0.928301i	$0.378731\pi$
$468$	0	0
$469$	−2.77623e6	−0.582806
$470$	0	0
$471$	5.32381e6	1.10578
$472$	0	0
$473$	1.21284e7	2.49260
$474$	0	0
$475$	7.95700e6	1.61814
$476$	0	0
$477$	1.78265e6	0.358732
$478$	0	0
$479$	3.32248e6	0.661642	0.330821	−	0.943694i	$-0.392674\pi$
$479$	3.32248e6	0.661642	0.330821	+	0.943694i	$0.392674\pi$
$480$	0	0
$481$	4.82409e6	0.950721
$482$	0	0
$483$	−3.43015e6	−0.669030
$484$	0	0
$485$	−2.28308e6	−0.440724
$486$	0	0
$487$	6.61506e6	1.26390	0.631948	−	0.775011i	$-0.282256\pi$
$487$	6.61506e6	1.26390	0.631948	+	0.775011i	$0.282256\pi$
$488$	0	0
$489$	−1.08050e6	−0.204340
$490$	0	0
$491$	1.03231e7	1.93244	0.966218	−	0.257727i	$-0.0829734\pi$
$491$	1.03231e7	1.93244	0.966218	+	0.257727i	$0.0829734\pi$
$492$	0	0
$493$	6.78446e6	1.25718
$494$	0	0
$495$	−1.17288e6	−0.215150
$496$	0	0
$497$	6.51919e6	1.18387
$498$	0	0
$499$	8.63184e6	1.55186	0.775929	−	0.630821i	$-0.217282\pi$
$499$	8.63184e6	1.55186	0.775929	+	0.630821i	$0.217282\pi$
$500$	0	0
$501$	−742896.	−0.132231
$502$	0	0
$503$	−4.66876e6	−0.822777	−0.411388	−	0.911460i	$-0.634956\pi$
$503$	−4.66876e6	−0.822777	−0.411388	+	0.911460i	$0.634956\pi$
$504$	0	0
$505$	−3.36048e6	−0.586372
$506$	0	0
$507$	−4.17693e6	−0.721667
$508$	0	0
$509$	4.27885e6	0.732036	0.366018	−	0.930608i	$-0.380721\pi$
$509$	4.27885e6	0.732036	0.366018	+	0.930608i	$0.380721\pi$
$510$	0	0
$511$	−584380.	−0.0990018
$512$	0	0
$513$	2.12868e6	0.357122
$514$	0	0
$515$	775000.	0.128761
$516$	0	0
$517$	−802192.	−0.131993
$518$	0	0
$519$	−2.06464e6	−0.336454
$520$	0	0
$521$	−9.86240e6	−1.59180	−0.795900	−	0.605428i	$-0.793002\pi$
$521$	−9.86240e6	−1.59180	−0.795900	+	0.605428i	$0.793002\pi$
$522$	0	0
$523$	3.30272e6	0.527980	0.263990	−	0.964525i	$-0.414961\pi$
$523$	3.30272e6	0.527980	0.263990	+	0.964525i	$0.414961\pi$
$524$	0	0
$525$	−2.99205e6	−0.473773
$526$	0	0
$527$	−5.04006e6	−0.790513
$528$	0	0
$529$	3.32303e6	0.516292
$530$	0	0
$531$	−1.92100e6	−0.295658
$532$	0	0
$533$	−4.78753e6	−0.729951
$534$	0	0
$535$	−4.14344e6	−0.625859
$536$	0	0
$537$	1.58209e6	0.236753
$538$	0	0
$539$	−1.39225e6	−0.206417
$540$	0	0
$541$	1.91152e6	0.280792	0.140396	−	0.990095i	$-0.455162\pi$
$541$	1.91152e6	0.280792	0.140396	+	0.990095i	$0.455162\pi$
$542$	0	0
$543$	−3.35641e6	−0.488512
$544$	0	0
$545$	−786360.	−0.113404
$546$	0	0
$547$	−4.91005e6	−0.701645	−0.350822	−	0.936442i	$-0.614098\pi$
$547$	−4.91005e6	−0.701645	−0.350822	+	0.936442i	$0.614098\pi$
$548$	0	0
$549$	3.66136e6	0.518456
$550$	0	0
$551$	−1.96925e7	−2.76326
$552$	0	0
$553$	−230092.	−0.0319955
$554$	0	0
$555$	−950040.	−0.130921
$556$	0	0
$557$	1.24364e7	1.69847	0.849234	−	0.528017i	$-0.177064\pi$
$557$	1.24364e7	1.69847	0.849234	+	0.528017i	$0.177064\pi$
$558$	0	0
$559$	−1.53113e7	−2.07245
$560$	0	0
$561$	−6.55510e6	−0.879371
$562$	0	0
$563$	−8.30351e6	−1.10405	−0.552027	−	0.833826i	$-0.686146\pi$
$563$	−8.30351e6	−1.10405	−0.552027	+	0.833826i	$0.686146\pi$
$564$	0	0
$565$	−113320.	−0.0149343
$566$	0	0
$567$	−800442.	−0.104562
$568$	0	0
$569$	−2.46481e6	−0.319156	−0.159578	−	0.987185i	$-0.551013\pi$
$569$	−2.46481e6	−0.319156	−0.159578	+	0.987185i	$0.551013\pi$
$570$	0	0
$571$	−457540.	−0.0587271	−0.0293636	−	0.999569i	$-0.509348\pi$
$571$	−457540.	−0.0587271	−0.0293636	+	0.999569i	$0.509348\pi$
$572$	0	0
$573$	−6.56114e6	−0.834820
$574$	0	0
$575$	8.51290e6	1.07376
$576$	0	0
$577$	1.36331e7	1.70473	0.852364	−	0.522950i	$-0.175168\pi$
$577$	1.36331e7	1.70473	0.852364	+	0.522950i	$0.175168\pi$
$578$	0	0
$579$	7.67666e6	0.951647
$580$	0	0
$581$	−1.37470e6	−0.168953
$582$	0	0
$583$	1.59338e7	1.94155
$584$	0	0
$585$	1.48068e6	0.178884
$586$	0	0
$587$	−570820.	−0.0683760	−0.0341880	−	0.999415i	$-0.510885\pi$
$587$	−570820.	−0.0683760	−0.0341880	+	0.999415i	$0.510885\pi$
$588$	0	0
$589$	1.46292e7	1.73753
$590$	0	0
$591$	−1.92701e6	−0.226942
$592$	0	0
$593$	9.51962e6	1.11169	0.555844	−	0.831287i	$-0.312395\pi$
$593$	9.51962e6	1.11169	0.555844	+	0.831287i	$0.312395\pi$
$594$	0	0
$595$	2.45464e6	0.284247
$596$	0	0
$597$	4.49012e6	0.515611
$598$	0	0
$599$	1.58794e7	1.80829	0.904145	−	0.427225i	$-0.140509\pi$
$599$	1.58794e7	1.80829	0.904145	+	0.427225i	$0.140509\pi$
$600$	0	0
$601$	−6.15280e6	−0.694843	−0.347422	−	0.937709i	$-0.612943\pi$
$601$	−6.15280e6	−0.694843	−0.347422	+	0.937709i	$0.612943\pi$
$602$	0	0
$603$	1.84324e6	0.206437
$604$	0	0
$605$	−7.26250e6	−0.806673
$606$	0	0
$607$	5.81252e6	0.640314	0.320157	−	0.947365i	$-0.396264\pi$
$607$	5.81252e6	0.640314	0.320157	+	0.947365i	$0.396264\pi$
$608$	0	0
$609$	7.40491e6	0.809052
$610$	0	0
$611$	1.01271e6	0.109745
$612$	0	0
$613$	−8.28312e6	−0.890313	−0.445156	−	0.895453i	$-0.646852\pi$
$613$	−8.28312e6	−0.890313	−0.445156	+	0.895453i	$0.646852\pi$
$614$	0	0
$615$	942840.	0.100519
$616$	0	0
$617$	−3.87774e6	−0.410078	−0.205039	−	0.978754i	$-0.565732\pi$
$617$	−3.87774e6	−0.410078	−0.205039	+	0.978754i	$0.565732\pi$
$618$	0	0
$619$	−1.22670e7	−1.28680	−0.643400	−	0.765531i	$-0.722477\pi$
$619$	−1.22670e7	−1.28680	−0.643400	+	0.765531i	$0.722477\pi$
$620$	0	0
$621$	2.27740e6	0.236979
$622$	0	0
$623$	−8.96968e6	−0.925885
$624$	0	0
$625$	6.17562e6	0.632384
$626$	0	0
$627$	1.90267e7	1.93284
$628$	0	0
$629$	−5.30967e6	−0.535107
$630$	0	0
$631$	−9.29180e6	−0.929023	−0.464512	−	0.885567i	$-0.653770\pi$
$631$	−9.29180e6	−0.929023	−0.464512	+	0.885567i	$0.653770\pi$
$632$	0	0
$633$	−4.27439e6	−0.423999
$634$	0	0
$635$	3.15788e6	0.310786
$636$	0	0
$637$	1.75762e6	0.171624
$638$	0	0
$639$	−4.32832e6	−0.419341
$640$	0	0
$641$	−5.39961e6	−0.519060	−0.259530	−	0.965735i	$-0.583568\pi$
$641$	−5.39961e6	−0.519060	−0.259530	+	0.965735i	$0.583568\pi$
$642$	0	0
$643$	5.49216e6	0.523861	0.261930	−	0.965087i	$-0.415641\pi$
$643$	5.49216e6	0.523861	0.261930	+	0.965087i	$0.415641\pi$
$644$	0	0
$645$	3.01536e6	0.285391
$646$	0	0
$647$	−1.15586e7	−1.08554	−0.542770	−	0.839882i	$-0.682624\pi$
$647$	−1.15586e7	−1.08554	−0.542770	+	0.839882i	$0.682624\pi$
$648$	0	0
$649$	−1.71704e7	−1.60018
$650$	0	0
$651$	−5.50098e6	−0.508730
$652$	0	0
$653$	−7.22411e6	−0.662981	−0.331491	−	0.943459i	$-0.607552\pi$
$653$	−7.22411e6	−0.662981	−0.331491	+	0.943459i	$0.607552\pi$
$654$	0	0
$655$	−1.88376e6	−0.171562
$656$	0	0
$657$	387990.	0.0350677
$658$	0	0
$659$	−1.57503e7	−1.41278	−0.706391	−	0.707822i	$-0.749678\pi$
$659$	−1.57503e7	−1.41278	−0.706391	+	0.707822i	$0.749678\pi$
$660$	0	0
$661$	5.87009e6	0.522566	0.261283	−	0.965262i	$-0.415854\pi$
$661$	5.87009e6	0.522566	0.261283	+	0.965262i	$0.415854\pi$
$662$	0	0
$663$	8.27536e6	0.731144
$664$	0	0
$665$	−7.12480e6	−0.624768
$666$	0	0
$667$	−2.10683e7	−1.83364
$668$	0	0
$669$	6.55306e6	0.566082
$670$	0	0
$671$	3.27262e7	2.80601
$672$	0	0
$673$	−1.38791e7	−1.18120	−0.590599	−	0.806965i	$-0.701108\pi$
$673$	−1.38791e7	−1.18120	−0.590599	+	0.806965i	$0.701108\pi$
$674$	0	0
$675$	1.98652e6	0.167816
$676$	0	0
$677$	−1.13566e7	−0.952303	−0.476151	−	0.879363i	$-0.657969\pi$
$677$	−1.13566e7	−0.952303	−0.476151	+	0.879363i	$0.657969\pi$
$678$	0	0
$679$	−1.39268e7	−1.15925
$680$	0	0
$681$	−3.19540e6	−0.264032
$682$	0	0
$683$	2.19391e7	1.79957	0.899783	−	0.436338i	$-0.143725\pi$
$683$	2.19391e7	1.79957	0.899783	+	0.436338i	$0.143725\pi$
$684$	0	0
$685$	584680.	0.0476093
$686$	0	0
$687$	8.88741e6	0.718428
$688$	0	0
$689$	−2.01153e7	−1.61428
$690$	0	0
$691$	−1.91337e7	−1.52442	−0.762210	−	0.647330i	$-0.775886\pi$
$691$	−1.91337e7	−1.52442	−0.762210	+	0.647330i	$0.775886\pi$
$692$	0	0
$693$	−7.15457e6	−0.565914
$694$	0	0
$695$	2.47112e6	0.194058
$696$	0	0
$697$	5.26943e6	0.410848
$698$	0	0
$699$	−8.93889e6	−0.691975
$700$	0	0
$701$	2.33384e7	1.79381	0.896905	−	0.442222i	$-0.145810\pi$
$701$	2.33384e7	1.79381	0.896905	+	0.442222i	$0.145810\pi$
$702$	0	0
$703$	1.54118e7	1.17615
$704$	0	0
$705$	−199440.	−0.0151126
$706$	0	0
$707$	−2.04989e7	−1.54235
$708$	0	0
$709$	−3.44683e6	−0.257516	−0.128758	−	0.991676i	$-0.541099\pi$
$709$	−3.44683e6	−0.257516	−0.128758	+	0.991676i	$0.541099\pi$
$710$	0	0
$711$	152766.	0.0113332
$712$	0	0
$713$	1.56512e7	1.15299
$714$	0	0
$715$	1.32347e7	0.968166
$716$	0	0
$717$	−6.89386e6	−0.500800
$718$	0	0
$719$	5.42504e6	0.391364	0.195682	−	0.980667i	$-0.437308\pi$
$719$	5.42504e6	0.391364	0.195682	+	0.980667i	$0.437308\pi$
$720$	0	0
$721$	4.72750e6	0.338683
$722$	0	0
$723$	−1.49783e7	−1.06565
$724$	0	0
$725$	−1.83774e7	−1.29849
$726$	0	0
$727$	9.03488e6	0.633996	0.316998	−	0.948426i	$-0.397325\pi$
$727$	9.03488e6	0.633996	0.316998	+	0.948426i	$0.397325\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.68525e7	1.16646
$732$	0	0
$733$	5.58049e6	0.383630	0.191815	−	0.981431i	$-0.438563\pi$
$733$	5.58049e6	0.383630	0.191815	+	0.981431i	$0.438563\pi$
$734$	0	0
$735$	−346140.	−0.0236338
$736$	0	0
$737$	1.64753e7	1.11729
$738$	0	0
$739$	−10572.0	−0.000712108 0	−0.000356054	−	1.00000i	$-0.500113\pi$
$739$	−10572.0	−0.000712108 0	−0.000356054		1.00000i	$0.500113\pi$
$740$	0	0
$741$	−2.40199e7	−1.60704
$742$	0	0
$743$	−1.51669e7	−1.00792	−0.503959	−	0.863728i	$-0.668124\pi$
$743$	−1.51669e7	−1.00792	−0.503959	+	0.863728i	$0.668124\pi$
$744$	0	0
$745$	−4.95672e6	−0.327193
$746$	0	0
$747$	912708.	0.0598453
$748$	0	0
$749$	−2.52750e7	−1.64621
$750$	0	0
$751$	−2.15869e7	−1.39666	−0.698331	−	0.715775i	$-0.746074\pi$
$751$	−2.15869e7	−1.39666	−0.698331	+	0.715775i	$0.746074\pi$
$752$	0	0
$753$	9.22806e6	0.593093
$754$	0	0
$755$	−5.13340e6	−0.327746
$756$	0	0
$757$	1.07700e7	0.683087	0.341543	−	0.939866i	$-0.389050\pi$
$757$	1.07700e7	0.683087	0.341543	+	0.939866i	$0.389050\pi$
$758$	0	0
$759$	2.03560e7	1.28259
$760$	0	0
$761$	1.75578e7	1.09903	0.549514	−	0.835485i	$-0.314813\pi$
$761$	1.75578e7	1.09903	0.549514	+	0.835485i	$0.314813\pi$
$762$	0	0
$763$	−4.79680e6	−0.298291
$764$	0	0
$765$	−1.62972e6	−0.100684
$766$	0	0
$767$	2.16764e7	1.33045
$768$	0	0
$769$	−2.64544e7	−1.61318	−0.806589	−	0.591112i	$-0.798689\pi$
$769$	−2.64544e7	−1.61318	−0.806589	+	0.591112i	$0.798689\pi$
$770$	0	0
$771$	7.76029e6	0.470156
$772$	0	0
$773$	−4.90282e6	−0.295119	−0.147559	−	0.989053i	$-0.547142\pi$
$773$	−4.90282e6	−0.295119	−0.147559	+	0.989053i	$0.547142\pi$
$774$	0	0
$775$	1.36522e7	0.816488
$776$	0	0
$777$	−5.79524e6	−0.344365
$778$	0	0
$779$	−1.52950e7	−0.903035
$780$	0	0
$781$	−3.86877e7	−2.26958
$782$	0	0
$783$	−4.91638e6	−0.286577
$784$	0	0
$785$	1.18307e7	0.685229
$786$	0	0
$787$	3.93416e6	0.226420	0.113210	−	0.993571i	$-0.463887\pi$
$787$	3.93416e6	0.226420	0.113210	+	0.993571i	$0.463887\pi$
$788$	0	0
$789$	2.10449e6	0.120352
$790$	0	0
$791$	−691252.	−0.0392821
$792$	0	0
$793$	−4.13146e7	−2.33303
$794$	0	0
$795$	3.96144e6	0.222298
$796$	0	0
$797$	−3.91796e6	−0.218481	−0.109241	−	0.994015i	$-0.534842\pi$
$797$	−3.91796e6	−0.218481	−0.109241	+	0.994015i	$0.534842\pi$
$798$	0	0
$799$	−1.11465e6	−0.0617690
$800$	0	0
$801$	5.95528e6	0.327960
$802$	0	0
$803$	3.46796e6	0.189795
$804$	0	0
$805$	−7.62256e6	−0.414583
$806$	0	0
$807$	4.15663e6	0.224677
$808$	0	0
$809$	−8.08192e6	−0.434154	−0.217077	−	0.976155i	$-0.569652\pi$
$809$	−8.08192e6	−0.434154	−0.217077	+	0.976155i	$0.569652\pi$
$810$	0	0
$811$	−2.39763e7	−1.28006	−0.640030	−	0.768350i	$-0.721078\pi$
$811$	−2.39763e7	−1.28006	−0.640030	+	0.768350i	$0.721078\pi$
$812$	0	0
$813$	−1.30698e7	−0.693495
$814$	0	0
$815$	−2.40112e6	−0.126625
$816$	0	0
$817$	−4.89158e7	−2.56386
$818$	0	0
$819$	9.03215e6	0.470523
$820$	0	0
$821$	2.04662e7	1.05969	0.529845	−	0.848095i	$-0.322250\pi$
$821$	2.04662e7	1.05969	0.529845	+	0.848095i	$0.322250\pi$
$822$	0	0
$823$	2.21937e7	1.14217	0.571083	−	0.820892i	$-0.306523\pi$
$823$	2.21937e7	1.14217	0.571083	+	0.820892i	$0.306523\pi$
$824$	0	0
$825$	1.77561e7	0.908265
$826$	0	0
$827$	1.56952e7	0.798000	0.399000	−	0.916951i	$-0.369357\pi$
$827$	1.56952e7	0.798000	0.399000	+	0.916951i	$0.369357\pi$
$828$	0	0
$829$	−7.27719e6	−0.367771	−0.183886	−	0.982948i	$-0.558868\pi$
$829$	−7.27719e6	−0.367771	−0.183886	+	0.982948i	$0.558868\pi$
$830$	0	0
$831$	1.38373e7	0.695104
$832$	0	0
$833$	−1.93454e6	−0.0965972
$834$	0	0
$835$	−1.65088e6	−0.0819406
$836$	0	0
$837$	3.65229e6	0.180199
$838$	0	0
$839$	2.14069e7	1.04990	0.524950	−	0.851133i	$-0.324084\pi$
$839$	2.14069e7	1.04990	0.524950	+	0.851133i	$0.324084\pi$
$840$	0	0
$841$	2.49704e7	1.21741
$842$	0	0
$843$	−1.73939e6	−0.0843003
$844$	0	0
$845$	−9.28206e6	−0.447201
$846$	0	0
$847$	−4.43012e7	−2.12181
$848$	0	0
$849$	1.42846e7	0.680142
$850$	0	0
$851$	1.64885e7	0.780471
$852$	0	0
$853$	2.04603e7	0.962810	0.481405	−	0.876498i	$-0.340127\pi$
$853$	2.04603e7	0.962810	0.481405	+	0.876498i	$0.340127\pi$
$854$	0	0
$855$	4.73040e6	0.221301
$856$	0	0
$857$	−3.91290e7	−1.81990	−0.909949	−	0.414721i	$-0.863879\pi$
$857$	−3.91290e7	−1.81990	−0.909949	+	0.414721i	$0.863879\pi$
$858$	0	0
$859$	−2.48593e7	−1.14949	−0.574745	−	0.818333i	$-0.694899\pi$
$859$	−2.48593e7	−1.14949	−0.574745	+	0.818333i	$0.694899\pi$
$860$	0	0
$861$	5.75132e6	0.264399
$862$	0	0
$863$	2.90387e7	1.32724	0.663620	−	0.748070i	$-0.269019\pi$
$863$	2.90387e7	1.32724	0.663620	+	0.748070i	$0.269019\pi$
$864$	0	0
$865$	−4.58808e6	−0.208493
$866$	0	0
$867$	3.67039e6	0.165830
$868$	0	0
$869$	1.36546e6	0.0613382
$870$	0	0
$871$	−2.07990e7	−0.928959
$872$	0	0
$873$	9.24647e6	0.410620
$874$	0	0
$875$	−1.42740e7	−0.630268
$876$	0	0
$877$	3.91274e7	1.71784	0.858919	−	0.512111i	$-0.171136\pi$
$877$	3.91274e7	1.71784	0.858919	+	0.512111i	$0.171136\pi$
$878$	0	0
$879$	−2.29747e7	−1.00295
$880$	0	0
$881$	−8.65214e6	−0.375564	−0.187782	−	0.982211i	$-0.560130\pi$
$881$	−8.65214e6	−0.375564	−0.187782	+	0.982211i	$0.560130\pi$
$882$	0	0
$883$	−3.32706e7	−1.43602	−0.718008	−	0.696035i	$-0.754946\pi$
$883$	−3.32706e7	−1.43602	−0.718008	+	0.696035i	$0.754946\pi$
$884$	0	0
$885$	−4.26888e6	−0.183213
$886$	0	0
$887$	−1.18364e7	−0.505138	−0.252569	−	0.967579i	$-0.581276\pi$
$887$	−1.18364e7	−0.505138	−0.252569	+	0.967579i	$0.581276\pi$
$888$	0	0
$889$	1.92631e7	0.817469
$890$	0	0
$891$	4.75016e6	0.200454
$892$	0	0
$893$	3.23536e6	0.135767
$894$	0	0
$895$	3.51576e6	0.146711
$896$	0	0
$897$	−2.56980e7	−1.06640
$898$	0	0
$899$	−3.37874e7	−1.39430
$900$	0	0
$901$	2.21400e7	0.908587
$902$	0	0
$903$	1.83937e7	0.750671
$904$	0	0
$905$	−7.45868e6	−0.302720
$906$	0	0
$907$	−1.41312e7	−0.570377	−0.285189	−	0.958471i	$-0.592056\pi$
$907$	−1.41312e7	−0.570377	−0.285189	+	0.958471i	$0.592056\pi$
$908$	0	0
$909$	1.36099e7	0.546319
$910$	0	0
$911$	3.78316e7	1.51028	0.755142	−	0.655561i	$-0.227568\pi$
$911$	3.78316e7	1.51028	0.755142	+	0.655561i	$0.227568\pi$
$912$	0	0
$913$	8.15803e6	0.323898
$914$	0	0
$915$	8.13636e6	0.321275
$916$	0	0
$917$	−1.14909e7	−0.451265
$918$	0	0
$919$	−2.08170e7	−0.813074	−0.406537	−	0.913634i	$-0.633264\pi$
$919$	−2.08170e7	−0.813074	−0.406537	+	0.913634i	$0.633264\pi$
$920$	0	0
$921$	4.15681e6	0.161477
$922$	0	0
$923$	4.88405e7	1.88702
$924$	0	0
$925$	1.43826e7	0.552690
$926$	0	0
$927$	−3.13875e6	−0.119966
$928$	0	0
$929$	6.29002e6	0.239118	0.119559	−	0.992827i	$-0.461852\pi$
$929$	6.29002e6	0.239118	0.119559	+	0.992827i	$0.461852\pi$
$930$	0	0
$931$	5.61516e6	0.212319
$932$	0	0
$933$	1.02132e7	0.384112
$934$	0	0
$935$	−1.45669e7	−0.544926
$936$	0	0
$937$	2.27077e7	0.844937	0.422468	−	0.906378i	$-0.361164\pi$
$937$	2.27077e7	0.844937	0.422468	+	0.906378i	$0.361164\pi$
$938$	0	0
$939$	6.48887e6	0.240163
$940$	0	0
$941$	−4.28149e7	−1.57623	−0.788117	−	0.615525i	$-0.788944\pi$
$941$	−4.28149e7	−1.57623	−0.788117	+	0.615525i	$0.788944\pi$
$942$	0	0
$943$	−1.63635e7	−0.599235
$944$	0	0
$945$	−1.77876e6	−0.0647945
$946$	0	0
$947$	1.30949e7	0.474491	0.237246	−	0.971450i	$-0.423755\pi$
$947$	1.30949e7	0.474491	0.237246	+	0.971450i	$0.423755\pi$
$948$	0	0
$949$	−4.37806e6	−0.157803
$950$	0	0
$951$	5.86613e6	0.210330
$952$	0	0
$953$	2.06761e7	0.737458	0.368729	−	0.929537i	$-0.379793\pi$
$953$	2.06761e7	0.737458	0.368729	+	0.929537i	$0.379793\pi$
$954$	0	0
$955$	−1.45803e7	−0.517319
$956$	0	0
$957$	−4.39439e7	−1.55102
$958$	0	0
$959$	3.56655e6	0.125228
$960$	0	0
$961$	−3.52905e6	−0.123268
$962$	0	0
$963$	1.67809e7	0.583110
$964$	0	0
$965$	1.70592e7	0.589714
$966$	0	0
$967$	−5.52496e7	−1.90004	−0.950021	−	0.312187i	$-0.898938\pi$
$967$	−5.52496e7	−1.90004	−0.950021	+	0.312187i	$0.898938\pi$
$968$	0	0
$969$	2.64377e7	0.904511
$970$	0	0
$971$	−1.22186e7	−0.415887	−0.207943	−	0.978141i	$-0.566677\pi$
$971$	−1.22186e7	−0.415887	−0.207943	+	0.978141i	$0.566677\pi$
$972$	0	0
$973$	1.50738e7	0.510436
$974$	0	0
$975$	−2.24158e7	−0.755168
$976$	0	0
$977$	−6.95889e6	−0.233240	−0.116620	−	0.993177i	$-0.537206\pi$
$977$	−6.95889e6	−0.233240	−0.116620	+	0.993177i	$0.537206\pi$
$978$	0	0
$979$	5.32299e7	1.77500
$980$	0	0
$981$	3.18476e6	0.105658
$982$	0	0
$983$	539272.	0.0178002	0.00890008	−	0.999960i	$-0.497167\pi$
$983$	539272.	0.0178002	0.00890008	+	0.999960i	$0.497167\pi$
$984$	0	0
$985$	−4.28224e6	−0.140631
$986$	0	0
$987$	−1.21658e6	−0.0397511
$988$	0	0
$989$	−5.23332e7	−1.70132
$990$	0	0
$991$	2.14279e7	0.693100	0.346550	−	0.938031i	$-0.387353\pi$
$991$	2.14279e7	0.693100	0.346550	+	0.938031i	$0.387353\pi$
$992$	0	0
$993$	−1.89805e7	−0.610851
$994$	0	0
$995$	9.97804e6	0.319512
$996$	0	0
$997$	4.93966e7	1.57383	0.786917	−	0.617059i	$-0.211676\pi$
$997$	4.93966e7	1.57383	0.786917	+	0.617059i	$0.211676\pi$
$998$	0	0
$999$	3.84766e6	0.121978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.a.a.1.1	✓	1
4.3	odd	2		384.6.a.c.1.1	yes	1
8.3	odd	2		384.6.a.b.1.1	yes	1
8.5	even	2		384.6.a.d.1.1	yes	1
16.3	odd	4		768.6.d.e.385.2		2
16.5	even	4		768.6.d.n.385.2		2
16.11	odd	4		768.6.d.e.385.1		2
16.13	even	4		768.6.d.n.385.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.a.a.1.1	✓	1	1.1	even	1	trivial
384.6.a.b.1.1	yes	1	8.3	odd	2
384.6.a.c.1.1	yes	1	4.3	odd	2
384.6.a.d.1.1	yes	1	8.5	even	2
768.6.d.e.385.1		2	16.11	odd	4
768.6.d.e.385.2		2	16.3	odd	4
768.6.d.n.385.1		2	16.13	even	4
768.6.d.n.385.2		2	16.5	even	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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