LMFDB - Embedded newform 336.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +4.00000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +4.00000 q^{5} +49.0000 q^{7} +81.0000 q^{9} -370.000 q^{11} +122.000 q^{13} +36.0000 q^{15} -1428.00 q^{17} -1724.00 q^{19} +441.000 q^{21} +2670.00 q^{23} -3109.00 q^{25} +729.000 q^{27} +4302.00 q^{29} -3104.00 q^{31} -3330.00 q^{33} +196.000 q^{35} -14318.0 q^{37} +1098.00 q^{39} -12272.0 q^{41} +21652.0 q^{43} +324.000 q^{45} +2644.00 q^{47} +2401.00 q^{49} -12852.0 q^{51} -24342.0 q^{53} -1480.00 q^{55} -15516.0 q^{57} +14088.0 q^{59} -24474.0 q^{61} +3969.00 q^{63} +488.000 q^{65} -7208.00 q^{67} +24030.0 q^{69} -54302.0 q^{71} -48962.0 q^{73} -27981.0 q^{75} -18130.0 q^{77} +33332.0 q^{79} +6561.00 q^{81} +4004.00 q^{83} -5712.00 q^{85} +38718.0 q^{87} +64752.0 q^{89} +5978.00 q^{91} -27936.0 q^{93} -6896.00 q^{95} +7038.00 q^{97} -29970.0 q^{99} +O(q^{100})$

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
$5$	4.00000	0.0715542	0.0357771	−	0.999360i	$-0.488609\pi$
$5$	4.00000	0.0715542	0.0357771	+	0.999360i	$0.488609\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−370.000	−0.921977	−0.460988	−	0.887406i	$-0.652505\pi$
$11$	−370.000	−0.921977	−0.460988	+	0.887406i	$0.652505\pi$
$12$	0	0
$13$	122.000	0.200217	0.100109	−	0.994977i	$-0.468081\pi$
$13$	122.000	0.200217	0.100109	+	0.994977i	$0.468081\pi$
$14$	0	0
$15$	36.0000	0.0413118
$16$	0	0
$17$	−1428.00	−1.19841	−0.599206	−	0.800595i	$-0.704517\pi$
$17$	−1428.00	−1.19841	−0.599206	+	0.800595i	$0.704517\pi$
$18$	0	0
$19$	−1724.00	−1.09560	−0.547802	−	0.836608i	$-0.684535\pi$
$19$	−1724.00	−1.09560	−0.547802	+	0.836608i	$0.684535\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	2670.00	1.05243	0.526213	−	0.850353i	$-0.323611\pi$
$23$	2670.00	1.05243	0.526213	+	0.850353i	$0.323611\pi$
$24$	0	0
$25$	−3109.00	−0.994880
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	4302.00	0.949895	0.474947	−	0.880014i	$-0.342467\pi$
$29$	4302.00	0.949895	0.474947	+	0.880014i	$0.342467\pi$
$30$	0	0
$31$	−3104.00	−0.580120	−0.290060	−	0.957009i	$-0.593675\pi$
$31$	−3104.00	−0.580120	−0.290060	+	0.957009i	$0.593675\pi$
$32$	0	0
$33$	−3330.00	−0.532304
$34$	0	0
$35$	196.000	0.0270449
$36$	0	0
$37$	−14318.0	−1.71940	−0.859702	−	0.510796i	$-0.829351\pi$
$37$	−14318.0	−1.71940	−0.859702	+	0.510796i	$0.829351\pi$
$38$	0	0
$39$	1098.00	0.115595
$40$	0	0
$41$	−12272.0	−1.14013	−0.570067	−	0.821598i	$-0.693083\pi$
$41$	−12272.0	−1.14013	−0.570067	+	0.821598i	$0.693083\pi$
$42$	0	0
$43$	21652.0	1.78578	0.892888	−	0.450279i	$-0.148676\pi$
$43$	21652.0	1.78578	0.892888	+	0.450279i	$0.148676\pi$
$44$	0	0
$45$	324.000	0.0238514
$46$	0	0
$47$	2644.00	0.174589	0.0872945	−	0.996183i	$-0.472178\pi$
$47$	2644.00	0.174589	0.0872945	+	0.996183i	$0.472178\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−12852.0	−0.691903
$52$	0	0
$53$	−24342.0	−1.19033	−0.595164	−	0.803604i	$-0.702913\pi$
$53$	−24342.0	−1.19033	−0.595164	+	0.803604i	$0.702913\pi$
$54$	0	0
$55$	−1480.00	−0.0659713
$56$	0	0
$57$	−15516.0	−0.632547
$58$	0	0
$59$	14088.0	0.526889	0.263445	−	0.964675i	$-0.415141\pi$
$59$	14088.0	0.526889	0.263445	+	0.964675i	$0.415141\pi$
$60$	0	0
$61$	−24474.0	−0.842132	−0.421066	−	0.907030i	$-0.638344\pi$
$61$	−24474.0	−0.842132	−0.421066	+	0.907030i	$0.638344\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	488.000	0.0143264
$66$	0	0
$67$	−7208.00	−0.196168	−0.0980839	−	0.995178i	$-0.531271\pi$
$67$	−7208.00	−0.196168	−0.0980839	+	0.995178i	$0.531271\pi$
$68$	0	0
$69$	24030.0	0.607619
$70$	0	0
$71$	−54302.0	−1.27841	−0.639205	−	0.769037i	$-0.720736\pi$
$71$	−54302.0	−1.27841	−0.639205	+	0.769037i	$0.720736\pi$
$72$	0	0
$73$	−48962.0	−1.07536	−0.537678	−	0.843150i	$-0.680698\pi$
$73$	−48962.0	−1.07536	−0.537678	+	0.843150i	$0.680698\pi$
$74$	0	0
$75$	−27981.0	−0.574394
$76$	0	0
$77$	−18130.0	−0.348474
$78$	0	0
$79$	33332.0	0.600888	0.300444	−	0.953799i	$-0.402865\pi$
$79$	33332.0	0.600888	0.300444	+	0.953799i	$0.402865\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	4004.00	0.0637968	0.0318984	−	0.999491i	$-0.489845\pi$
$83$	4004.00	0.0637968	0.0318984	+	0.999491i	$0.489845\pi$
$84$	0	0
$85$	−5712.00	−0.0857513
$86$	0	0
$87$	38718.0	0.548422
$88$	0	0
$89$	64752.0	0.866519	0.433260	−	0.901269i	$-0.357363\pi$
$89$	64752.0	0.866519	0.433260	+	0.901269i	$0.357363\pi$
$90$	0	0
$91$	5978.00	0.0756750
$92$	0	0
$93$	−27936.0	−0.334932
$94$	0	0
$95$	−6896.00	−0.0783950
$96$	0	0
$97$	7038.00	0.0759486	0.0379743	−	0.999279i	$-0.487909\pi$
$97$	7038.00	0.0759486	0.0379743	+	0.999279i	$0.487909\pi$
$98$	0	0
$99$	−29970.0	−0.307326
$100$	0	0
$101$	24784.0	0.241751	0.120875	−	0.992668i	$-0.461430\pi$
$101$	24784.0	0.241751	0.120875	+	0.992668i	$0.461430\pi$
$102$	0	0
$103$	−108080.	−1.00381	−0.501906	−	0.864922i	$-0.667368\pi$
$103$	−108080.	−1.00381	−0.501906	+	0.864922i	$0.667368\pi$
$104$	0	0
$105$	1764.00	0.0156144
$106$	0	0
$107$	−108834.	−0.918978	−0.459489	−	0.888183i	$-0.651967\pi$
$107$	−108834.	−0.918978	−0.459489	+	0.888183i	$0.651967\pi$
$108$	0	0
$109$	−202642.	−1.63367	−0.816833	−	0.576874i	$-0.804272\pi$
$109$	−202642.	−1.63367	−0.816833	+	0.576874i	$0.804272\pi$
$110$	0	0
$111$	−128862.	−0.992699
$112$	0	0
$113$	−50198.0	−0.369820	−0.184910	−	0.982755i	$-0.559199\pi$
$113$	−50198.0	−0.369820	−0.184910	+	0.982755i	$0.559199\pi$
$114$	0	0
$115$	10680.0	0.0753055
$116$	0	0
$117$	9882.00	0.0667391
$118$	0	0
$119$	−69972.0	−0.452957
$120$	0	0
$121$	−24151.0	−0.149959
$122$	0	0
$123$	−110448.	−0.658256
$124$	0	0
$125$	−24936.0	−0.142742
$126$	0	0
$127$	−3052.00	−0.0167909	−0.00839547	−	0.999965i	$-0.502672\pi$
$127$	−3052.00	−0.0167909	−0.00839547	+	0.999965i	$0.502672\pi$
$128$	0	0
$129$	194868.	1.03102
$130$	0	0
$131$	−218788.	−1.11390	−0.556949	−	0.830547i	$-0.688028\pi$
$131$	−218788.	−1.11390	−0.556949	+	0.830547i	$0.688028\pi$
$132$	0	0
$133$	−84476.0	−0.414099
$134$	0	0
$135$	2916.00	0.0137706
$136$	0	0
$137$	203070.	0.924367	0.462183	−	0.886784i	$-0.347066\pi$
$137$	203070.	0.924367	0.462183	+	0.886784i	$0.347066\pi$
$138$	0	0
$139$	−49540.0	−0.217480	−0.108740	−	0.994070i	$-0.534682\pi$
$139$	−49540.0	−0.217480	−0.108740	+	0.994070i	$0.534682\pi$
$140$	0	0
$141$	23796.0	0.100799
$142$	0	0
$143$	−45140.0	−0.184596
$144$	0	0
$145$	17208.0	0.0679689
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	93354.0	0.344483	0.172241	−	0.985055i	$-0.444899\pi$
$149$	93354.0	0.344483	0.172241	+	0.985055i	$0.444899\pi$
$150$	0	0
$151$	−70872.0	−0.252949	−0.126474	−	0.991970i	$-0.540366\pi$
$151$	−70872.0	−0.252949	−0.126474	+	0.991970i	$0.540366\pi$
$152$	0	0
$153$	−115668.	−0.399470
$154$	0	0
$155$	−12416.0	−0.0415100
$156$	0	0
$157$	−374898.	−1.21385	−0.606924	−	0.794760i	$-0.707597\pi$
$157$	−374898.	−1.21385	−0.606924	+	0.794760i	$0.707597\pi$
$158$	0	0
$159$	−219078.	−0.687236
$160$	0	0
$161$	130830.	0.397780
$162$	0	0
$163$	125184.	0.369045	0.184523	−	0.982828i	$-0.440926\pi$
$163$	125184.	0.369045	0.184523	+	0.982828i	$0.440926\pi$
$164$	0	0
$165$	−13320.0	−0.0380885
$166$	0	0
$167$	−444068.	−1.23214	−0.616068	−	0.787693i	$-0.711275\pi$
$167$	−444068.	−1.23214	−0.616068	+	0.787693i	$0.711275\pi$
$168$	0	0
$169$	−356409.	−0.959913
$170$	0	0
$171$	−139644.	−0.365201
$172$	0	0
$173$	544368.	1.38286	0.691429	−	0.722445i	$-0.256982\pi$
$173$	544368.	1.38286	0.691429	+	0.722445i	$0.256982\pi$
$174$	0	0
$175$	−152341.	−0.376029
$176$	0	0
$177$	126792.	0.304200
$178$	0	0
$179$	616130.	1.43727	0.718637	−	0.695385i	$-0.244766\pi$
$179$	616130.	1.43727	0.718637	+	0.695385i	$0.244766\pi$
$180$	0	0
$181$	98266.0	0.222950	0.111475	−	0.993767i	$-0.464443\pi$
$181$	98266.0	0.222950	0.111475	+	0.993767i	$0.464443\pi$
$182$	0	0
$183$	−220266.	−0.486205
$184$	0	0
$185$	−57272.0	−0.123031
$186$	0	0
$187$	528360.	1.10491
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	515138.	1.02174	0.510870	−	0.859658i	$-0.329323\pi$
$191$	515138.	1.02174	0.510870	+	0.859658i	$0.329323\pi$
$192$	0	0
$193$	−449930.	−0.869464	−0.434732	−	0.900560i	$-0.643157\pi$
$193$	−449930.	−0.869464	−0.434732	+	0.900560i	$0.643157\pi$
$194$	0	0
$195$	4392.00	0.00827134
$196$	0	0
$197$	−404934.	−0.743393	−0.371697	−	0.928354i	$-0.621224\pi$
$197$	−404934.	−0.743393	−0.371697	+	0.928354i	$0.621224\pi$
$198$	0	0
$199$	342576.	0.613231	0.306616	−	0.951833i	$-0.400803\pi$
$199$	342576.	0.613231	0.306616	+	0.951833i	$0.400803\pi$
$200$	0	0
$201$	−64872.0	−0.113258
$202$	0	0
$203$	210798.	0.359026
$204$	0	0
$205$	−49088.0	−0.0815813
$206$	0	0
$207$	216270.	0.350809
$208$	0	0
$209$	637880.	1.01012
$210$	0	0
$211$	812820.	1.25686	0.628432	−	0.777865i	$-0.283697\pi$
$211$	812820.	1.25686	0.628432	+	0.777865i	$0.283697\pi$
$212$	0	0
$213$	−488718.	−0.738090
$214$	0	0
$215$	86608.0	0.127780
$216$	0	0
$217$	−152096.	−0.219265
$218$	0	0
$219$	−440658.	−0.620857
$220$	0	0
$221$	−174216.	−0.239943
$222$	0	0
$223$	655320.	0.882452	0.441226	−	0.897396i	$-0.354544\pi$
$223$	655320.	0.882452	0.441226	+	0.897396i	$0.354544\pi$
$224$	0	0
$225$	−251829.	−0.331627
$226$	0	0
$227$	1.19258e6	1.53611	0.768053	−	0.640386i	$-0.221226\pi$
$227$	1.19258e6	1.53611	0.768053	+	0.640386i	$0.221226\pi$
$228$	0	0
$229$	374506.	0.471922	0.235961	−	0.971763i	$-0.424176\pi$
$229$	374506.	0.471922	0.235961	+	0.971763i	$0.424176\pi$
$230$	0	0
$231$	−163170.	−0.201192
$232$	0	0
$233$	1.47249e6	1.77690	0.888452	−	0.458970i	$-0.151782\pi$
$233$	1.47249e6	1.77690	0.888452	+	0.458970i	$0.151782\pi$
$234$	0	0
$235$	10576.0	0.0124926
$236$	0	0
$237$	299988.	0.346923
$238$	0	0
$239$	1.28908e6	1.45977	0.729885	−	0.683570i	$-0.239574\pi$
$239$	1.28908e6	1.45977	0.729885	+	0.683570i	$0.239574\pi$
$240$	0	0
$241$	174478.	0.193508	0.0967538	−	0.995308i	$-0.469154\pi$
$241$	174478.	0.193508	0.0967538	+	0.995308i	$0.469154\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	9604.00	0.0102220
$246$	0	0
$247$	−210328.	−0.219359
$248$	0	0
$249$	36036.0	0.0368331
$250$	0	0
$251$	875928.	0.877575	0.438787	−	0.898591i	$-0.355408\pi$
$251$	875928.	0.877575	0.438787	+	0.898591i	$0.355408\pi$
$252$	0	0
$253$	−987900.	−0.970313
$254$	0	0
$255$	−51408.0	−0.0495086
$256$	0	0
$257$	−674408.	−0.636927	−0.318464	−	0.947935i	$-0.603167\pi$
$257$	−674408.	−0.636927	−0.318464	+	0.947935i	$0.603167\pi$
$258$	0	0
$259$	−701582.	−0.649874
$260$	0	0
$261$	348462.	0.316632
$262$	0	0
$263$	−1.47155e6	−1.31186	−0.655929	−	0.754823i	$-0.727723\pi$
$263$	−1.47155e6	−1.31186	−0.655929	+	0.754823i	$0.727723\pi$
$264$	0	0
$265$	−97368.0	−0.0851729
$266$	0	0
$267$	582768.	0.500285
$268$	0	0
$269$	2.12577e6	1.79117	0.895583	−	0.444894i	$-0.146759\pi$
$269$	2.12577e6	1.79117	0.895583	+	0.444894i	$0.146759\pi$
$270$	0	0
$271$	−1.10940e6	−0.917624	−0.458812	−	0.888533i	$-0.651725\pi$
$271$	−1.10940e6	−0.917624	−0.458812	+	0.888533i	$0.651725\pi$
$272$	0	0
$273$	53802.0	0.0436910
$274$	0	0
$275$	1.15033e6	0.917256
$276$	0	0
$277$	−229094.	−0.179397	−0.0896983	−	0.995969i	$-0.528590\pi$
$277$	−229094.	−0.179397	−0.0896983	+	0.995969i	$0.528590\pi$
$278$	0	0
$279$	−251424.	−0.193373
$280$	0	0
$281$	−625962.	−0.472914	−0.236457	−	0.971642i	$-0.575986\pi$
$281$	−625962.	−0.472914	−0.236457	+	0.971642i	$0.575986\pi$
$282$	0	0
$283$	2.04374e6	1.51691	0.758455	−	0.651726i	$-0.225955\pi$
$283$	2.04374e6	1.51691	0.758455	+	0.651726i	$0.225955\pi$
$284$	0	0
$285$	−62064.0	−0.0452614
$286$	0	0
$287$	−601328.	−0.430930
$288$	0	0
$289$	619327.	0.436190
$290$	0	0
$291$	63342.0	0.0438490
$292$	0	0
$293$	1.96838e6	1.33949	0.669747	−	0.742589i	$-0.266403\pi$
$293$	1.96838e6	1.33949	0.669747	+	0.742589i	$0.266403\pi$
$294$	0	0
$295$	56352.0	0.0377011
$296$	0	0
$297$	−269730.	−0.177435
$298$	0	0
$299$	325740.	0.210714
$300$	0	0
$301$	1.06095e6	0.674960
$302$	0	0
$303$	223056.	0.139575
$304$	0	0
$305$	−97896.0	−0.0602581
$306$	0	0
$307$	−3.04359e6	−1.84306	−0.921531	−	0.388305i	$-0.873061\pi$
$307$	−3.04359e6	−1.84306	−0.921531	+	0.388305i	$0.873061\pi$
$308$	0	0
$309$	−972720.	−0.579551
$310$	0	0
$311$	325724.	0.190963	0.0954814	−	0.995431i	$-0.469561\pi$
$311$	325724.	0.190963	0.0954814	+	0.995431i	$0.469561\pi$
$312$	0	0
$313$	2.28769e6	1.31989	0.659943	−	0.751316i	$-0.270580\pi$
$313$	2.28769e6	1.31989	0.659943	+	0.751316i	$0.270580\pi$
$314$	0	0
$315$	15876.0	0.00901498
$316$	0	0
$317$	902562.	0.504463	0.252231	−	0.967667i	$-0.418836\pi$
$317$	902562.	0.504463	0.252231	+	0.967667i	$0.418836\pi$
$318$	0	0
$319$	−1.59174e6	−0.875781
$320$	0	0
$321$	−979506.	−0.530572
$322$	0	0
$323$	2.46187e6	1.31298
$324$	0	0
$325$	−379298.	−0.199192
$326$	0	0
$327$	−1.82378e6	−0.943197
$328$	0	0
$329$	129556.	0.0659884
$330$	0	0
$331$	−1.56194e6	−0.783600	−0.391800	−	0.920050i	$-0.628147\pi$
$331$	−1.56194e6	−0.783600	−0.391800	+	0.920050i	$0.628147\pi$
$332$	0	0
$333$	−1.15976e6	−0.573135
$334$	0	0
$335$	−28832.0	−0.0140366
$336$	0	0
$337$	−572230.	−0.274471	−0.137235	−	0.990538i	$-0.543822\pi$
$337$	−572230.	−0.274471	−0.137235	+	0.990538i	$0.543822\pi$
$338$	0	0
$339$	−451782.	−0.213516
$340$	0	0
$341$	1.14848e6	0.534857
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	96120.0	0.0434776
$346$	0	0
$347$	1.88039e6	0.838350	0.419175	−	0.907906i	$-0.362319\pi$
$347$	1.88039e6	0.838350	0.419175	+	0.907906i	$0.362319\pi$
$348$	0	0
$349$	1.43343e6	0.629960	0.314980	−	0.949098i	$-0.398002\pi$
$349$	1.43343e6	0.629960	0.314980	+	0.949098i	$0.398002\pi$
$350$	0	0
$351$	88938.0	0.0385318
$352$	0	0
$353$	−3.09802e6	−1.32327	−0.661633	−	0.749827i	$-0.730136\pi$
$353$	−3.09802e6	−1.32327	−0.661633	+	0.749827i	$0.730136\pi$
$354$	0	0
$355$	−217208.	−0.0914755
$356$	0	0
$357$	−629748.	−0.261515
$358$	0	0
$359$	−663918.	−0.271881	−0.135940	−	0.990717i	$-0.543406\pi$
$359$	−663918.	−0.271881	−0.135940	+	0.990717i	$0.543406\pi$
$360$	0	0
$361$	496077.	0.200346
$362$	0	0
$363$	−217359.	−0.0865787
$364$	0	0
$365$	−195848.	−0.0769462
$366$	0	0
$367$	−4.33787e6	−1.68117	−0.840585	−	0.541680i	$-0.817789\pi$
$367$	−4.33787e6	−1.68117	−0.840585	+	0.541680i	$0.817789\pi$
$368$	0	0
$369$	−994032.	−0.380044
$370$	0	0
$371$	−1.19276e6	−0.449902
$372$	0	0
$373$	4.49537e6	1.67299	0.836494	−	0.547976i	$-0.184601\pi$
$373$	4.49537e6	1.67299	0.836494	+	0.547976i	$0.184601\pi$
$374$	0	0
$375$	−224424.	−0.0824121
$376$	0	0
$377$	524844.	0.190185
$378$	0	0
$379$	1.44910e6	0.518203	0.259102	−	0.965850i	$-0.416574\pi$
$379$	1.44910e6	0.518203	0.259102	+	0.965850i	$0.416574\pi$
$380$	0	0
$381$	−27468.0	−0.00969426
$382$	0	0
$383$	−3.62465e6	−1.26261	−0.631305	−	0.775535i	$-0.717480\pi$
$383$	−3.62465e6	−1.26261	−0.631305	+	0.775535i	$0.717480\pi$
$384$	0	0
$385$	−72520.0	−0.0249348
$386$	0	0
$387$	1.75381e6	0.595259
$388$	0	0
$389$	−585274.	−0.196103	−0.0980517	−	0.995181i	$-0.531261\pi$
$389$	−585274.	−0.196103	−0.0980517	+	0.995181i	$0.531261\pi$
$390$	0	0
$391$	−3.81276e6	−1.26124
$392$	0	0
$393$	−1.96909e6	−0.643109
$394$	0	0
$395$	133328.	0.0429961
$396$	0	0
$397$	−4.36239e6	−1.38915	−0.694573	−	0.719422i	$-0.744407\pi$
$397$	−4.36239e6	−1.38915	−0.694573	+	0.719422i	$0.744407\pi$
$398$	0	0
$399$	−760284.	−0.239080
$400$	0	0
$401$	4.16168e6	1.29243	0.646216	−	0.763155i	$-0.276351\pi$
$401$	4.16168e6	1.29243	0.646216	+	0.763155i	$0.276351\pi$
$402$	0	0
$403$	−378688.	−0.116150
$404$	0	0
$405$	26244.0	0.00795046
$406$	0	0
$407$	5.29766e6	1.58525
$408$	0	0
$409$	−4.54131e6	−1.34237	−0.671185	−	0.741290i	$-0.734215\pi$
$409$	−4.54131e6	−1.34237	−0.671185	+	0.741290i	$0.734215\pi$
$410$	0	0
$411$	1.82763e6	0.533683
$412$	0	0
$413$	690312.	0.199145
$414$	0	0
$415$	16016.0	0.00456493
$416$	0	0
$417$	−445860.	−0.125562
$418$	0	0
$419$	1.93357e6	0.538052	0.269026	−	0.963133i	$-0.413298\pi$
$419$	1.93357e6	0.538052	0.269026	+	0.963133i	$0.413298\pi$
$420$	0	0
$421$	−2.16777e6	−0.596084	−0.298042	−	0.954553i	$-0.596333\pi$
$421$	−2.16777e6	−0.596084	−0.298042	+	0.954553i	$0.596333\pi$
$422$	0	0
$423$	214164.	0.0581963
$424$	0	0
$425$	4.43965e6	1.19228
$426$	0	0
$427$	−1.19923e6	−0.318296
$428$	0	0
$429$	−406260.	−0.106576
$430$	0	0
$431$	3.36421e6	0.872348	0.436174	−	0.899862i	$-0.356333\pi$
$431$	3.36421e6	0.872348	0.436174	+	0.899862i	$0.356333\pi$
$432$	0	0
$433$	6.09510e6	1.56229	0.781144	−	0.624351i	$-0.214637\pi$
$433$	6.09510e6	1.56229	0.781144	+	0.624351i	$0.214637\pi$
$434$	0	0
$435$	154872.	0.0392419
$436$	0	0
$437$	−4.60308e6	−1.15304
$438$	0	0
$439$	−7.65252e6	−1.89515	−0.947574	−	0.319536i	$-0.896473\pi$
$439$	−7.65252e6	−1.89515	−0.947574	+	0.319536i	$0.896473\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	762570.	0.184616	0.0923082	−	0.995730i	$-0.470575\pi$
$443$	762570.	0.184616	0.0923082	+	0.995730i	$0.470575\pi$
$444$	0	0
$445$	259008.	0.0620031
$446$	0	0
$447$	840186.	0.198887
$448$	0	0
$449$	−4.44504e6	−1.04054	−0.520271	−	0.854001i	$-0.674169\pi$
$449$	−4.44504e6	−1.04054	−0.520271	+	0.854001i	$0.674169\pi$
$450$	0	0
$451$	4.54064e6	1.05118
$452$	0	0
$453$	−637848.	−0.146040
$454$	0	0
$455$	23912.0	0.00541486
$456$	0	0
$457$	−6.00527e6	−1.34506	−0.672531	−	0.740069i	$-0.734793\pi$
$457$	−6.00527e6	−1.34506	−0.672531	+	0.740069i	$0.734793\pi$
$458$	0	0
$459$	−1.04101e6	−0.230634
$460$	0	0
$461$	−2.86783e6	−0.628494	−0.314247	−	0.949341i	$-0.601752\pi$
$461$	−2.86783e6	−0.628494	−0.314247	+	0.949341i	$0.601752\pi$
$462$	0	0
$463$	−4.45140e6	−0.965037	−0.482518	−	0.875886i	$-0.660278\pi$
$463$	−4.45140e6	−0.965037	−0.482518	+	0.875886i	$0.660278\pi$
$464$	0	0
$465$	−111744.	−0.0239658
$466$	0	0
$467$	2.34522e6	0.497613	0.248807	−	0.968553i	$-0.419962\pi$
$467$	2.34522e6	0.497613	0.248807	+	0.968553i	$0.419962\pi$
$468$	0	0
$469$	−353192.	−0.0741445
$470$	0	0
$471$	−3.37408e6	−0.700815
$472$	0	0
$473$	−8.01124e6	−1.64644
$474$	0	0
$475$	5.35992e6	1.08999
$476$	0	0
$477$	−1.97170e6	−0.396776
$478$	0	0
$479$	−1.16154e6	−0.231311	−0.115655	−	0.993289i	$-0.536897\pi$
$479$	−1.16154e6	−0.231311	−0.115655	+	0.993289i	$0.536897\pi$
$480$	0	0
$481$	−1.74680e6	−0.344254
$482$	0	0
$483$	1.17747e6	0.229658
$484$	0	0
$485$	28152.0	0.00543444
$486$	0	0
$487$	17776.0	0.00339634	0.00169817	−	0.999999i	$-0.499459\pi$
$487$	17776.0	0.00339634	0.00169817	+	0.999999i	$0.499459\pi$
$488$	0	0
$489$	1.12666e6	0.213068
$490$	0	0
$491$	2.09827e6	0.392787	0.196393	−	0.980525i	$-0.437077\pi$
$491$	2.09827e6	0.392787	0.196393	+	0.980525i	$0.437077\pi$
$492$	0	0
$493$	−6.14326e6	−1.13836
$494$	0	0
$495$	−119880.	−0.0219904
$496$	0	0
$497$	−2.66080e6	−0.483193
$498$	0	0
$499$	8.44714e6	1.51865	0.759326	−	0.650710i	$-0.225529\pi$
$499$	8.44714e6	1.51865	0.759326	+	0.650710i	$0.225529\pi$
$500$	0	0
$501$	−3.99661e6	−0.711373
$502$	0	0
$503$	−7.42743e6	−1.30894	−0.654468	−	0.756089i	$-0.727108\pi$
$503$	−7.42743e6	−1.30894	−0.654468	+	0.756089i	$0.727108\pi$
$504$	0	0
$505$	99136.0	0.0172983
$506$	0	0
$507$	−3.20768e6	−0.554206
$508$	0	0
$509$	2.06342e6	0.353015	0.176508	−	0.984299i	$-0.443520\pi$
$509$	2.06342e6	0.353015	0.176508	+	0.984299i	$0.443520\pi$
$510$	0	0
$511$	−2.39914e6	−0.406446
$512$	0	0
$513$	−1.25680e6	−0.210849
$514$	0	0
$515$	−432320.	−0.0718269
$516$	0	0
$517$	−978280.	−0.160967
$518$	0	0
$519$	4.89931e6	0.798393
$520$	0	0
$521$	233724.	0.0377232	0.0188616	−	0.999822i	$-0.493996\pi$
$521$	233724.	0.0377232	0.0188616	+	0.999822i	$0.493996\pi$
$522$	0	0
$523$	−1.15859e7	−1.85214	−0.926071	−	0.377350i	$-0.876835\pi$
$523$	−1.15859e7	−1.85214	−0.926071	+	0.377350i	$0.876835\pi$
$524$	0	0
$525$	−1.37107e6	−0.217101
$526$	0	0
$527$	4.43251e6	0.695222
$528$	0	0
$529$	692557.	0.107601
$530$	0	0
$531$	1.14113e6	0.175630
$532$	0	0
$533$	−1.49718e6	−0.228274
$534$	0	0
$535$	−435336.	−0.0657567
$536$	0	0
$537$	5.54517e6	0.829811
$538$	0	0
$539$	−888370.	−0.131711
$540$	0	0
$541$	8.75141e6	1.28554	0.642769	−	0.766060i	$-0.277786\pi$
$541$	8.75141e6	1.28554	0.642769	+	0.766060i	$0.277786\pi$
$542$	0	0
$543$	884394.	0.128720
$544$	0	0
$545$	−810568.	−0.116896
$546$	0	0
$547$	1.29193e7	1.84617	0.923084	−	0.384598i	$-0.125660\pi$
$547$	1.29193e7	1.84617	0.923084	+	0.384598i	$0.125660\pi$
$548$	0	0
$549$	−1.98239e6	−0.280711
$550$	0	0
$551$	−7.41665e6	−1.04071
$552$	0	0
$553$	1.63327e6	0.227114
$554$	0	0
$555$	−515448.	−0.0710317
$556$	0	0
$557$	5.28030e6	0.721141	0.360571	−	0.932732i	$-0.382582\pi$
$557$	5.28030e6	0.721141	0.360571	+	0.932732i	$0.382582\pi$
$558$	0	0
$559$	2.64154e6	0.357543
$560$	0	0
$561$	4.75524e6	0.637919
$562$	0	0
$563$	−8.48097e6	−1.12765	−0.563825	−	0.825894i	$-0.690671\pi$
$563$	−8.48097e6	−1.12765	−0.563825	+	0.825894i	$0.690671\pi$
$564$	0	0
$565$	−200792.	−0.0264622
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	7.01798e6	0.908723	0.454362	−	0.890817i	$-0.349867\pi$
$569$	7.01798e6	0.908723	0.454362	+	0.890817i	$0.349867\pi$
$570$	0	0
$571$	−4.79490e6	−0.615446	−0.307723	−	0.951476i	$-0.599567\pi$
$571$	−4.79490e6	−0.615446	−0.307723	+	0.951476i	$0.599567\pi$
$572$	0	0
$573$	4.63624e6	0.589902
$574$	0	0
$575$	−8.30103e6	−1.04704
$576$	0	0
$577$	−6.39709e6	−0.799914	−0.399957	−	0.916534i	$-0.630975\pi$
$577$	−6.39709e6	−0.799914	−0.399957	+	0.916534i	$0.630975\pi$
$578$	0	0
$579$	−4.04937e6	−0.501985
$580$	0	0
$581$	196196.	0.0241129
$582$	0	0
$583$	9.00654e6	1.09745
$584$	0	0
$585$	39528.0	0.00477546
$586$	0	0
$587$	5.49917e6	0.658721	0.329361	−	0.944204i	$-0.393167\pi$
$587$	5.49917e6	0.658721	0.329361	+	0.944204i	$0.393167\pi$
$588$	0	0
$589$	5.35130e6	0.635581
$590$	0	0
$591$	−3.64441e6	−0.429198
$592$	0	0
$593$	−1.34543e7	−1.57118	−0.785588	−	0.618750i	$-0.787639\pi$
$593$	−1.34543e7	−1.57118	−0.785588	+	0.618750i	$0.787639\pi$
$594$	0	0
$595$	−279888.	−0.0324110
$596$	0	0
$597$	3.08318e6	0.354049
$598$	0	0
$599$	−4.57493e6	−0.520976	−0.260488	−	0.965477i	$-0.583883\pi$
$599$	−4.57493e6	−0.520976	−0.260488	+	0.965477i	$0.583883\pi$
$600$	0	0
$601$	1.07837e7	1.21782	0.608910	−	0.793240i	$-0.291607\pi$
$601$	1.07837e7	1.21782	0.608910	+	0.793240i	$0.291607\pi$
$602$	0	0
$603$	−583848.	−0.0653893
$604$	0	0
$605$	−96604.0	−0.0107302
$606$	0	0
$607$	1.28728e7	1.41808	0.709041	−	0.705167i	$-0.249128\pi$
$607$	1.28728e7	1.41808	0.709041	+	0.705167i	$0.249128\pi$
$608$	0	0
$609$	1.89718e6	0.207284
$610$	0	0
$611$	322568.	0.0349557
$612$	0	0
$613$	−1.13805e7	−1.22323	−0.611616	−	0.791155i	$-0.709480\pi$
$613$	−1.13805e7	−1.22323	−0.611616	+	0.791155i	$0.709480\pi$
$614$	0	0
$615$	−441792.	−0.0471010
$616$	0	0
$617$	1.15644e7	1.22295	0.611477	−	0.791263i	$-0.290576\pi$
$617$	1.15644e7	1.22295	0.611477	+	0.791263i	$0.290576\pi$
$618$	0	0
$619$	−641356.	−0.0672779	−0.0336390	−	0.999434i	$-0.510710\pi$
$619$	−641356.	−0.0672779	−0.0336390	+	0.999434i	$0.510710\pi$
$620$	0	0
$621$	1.94643e6	0.202540
$622$	0	0
$623$	3.17285e6	0.327513
$624$	0	0
$625$	9.61588e6	0.984666
$626$	0	0
$627$	5.74092e6	0.583193
$628$	0	0
$629$	2.04461e7	2.06055
$630$	0	0
$631$	−1.18685e7	−1.18665	−0.593323	−	0.804964i	$-0.702184\pi$
$631$	−1.18685e7	−1.18665	−0.593323	+	0.804964i	$0.702184\pi$
$632$	0	0
$633$	7.31538e6	0.725651
$634$	0	0
$635$	−12208.0	−0.00120146
$636$	0	0
$637$	292922.	0.0286025
$638$	0	0
$639$	−4.39846e6	−0.426136
$640$	0	0
$641$	2.80700e6	0.269834	0.134917	−	0.990857i	$-0.456923\pi$
$641$	2.80700e6	0.269834	0.134917	+	0.990857i	$0.456923\pi$
$642$	0	0
$643$	3.23109e6	0.308192	0.154096	−	0.988056i	$-0.450753\pi$
$643$	3.23109e6	0.308192	0.154096	+	0.988056i	$0.450753\pi$
$644$	0	0
$645$	779472.	0.0737736
$646$	0	0
$647$	−1.13192e7	−1.06305	−0.531527	−	0.847041i	$-0.678382\pi$
$647$	−1.13192e7	−1.06305	−0.531527	+	0.847041i	$0.678382\pi$
$648$	0	0
$649$	−5.21256e6	−0.485780
$650$	0	0
$651$	−1.36886e6	−0.126592
$652$	0	0
$653$	4.54002e6	0.416653	0.208327	−	0.978059i	$-0.433198\pi$
$653$	4.54002e6	0.416653	0.208327	+	0.978059i	$0.433198\pi$
$654$	0	0
$655$	−875152.	−0.0797040
$656$	0	0
$657$	−3.96592e6	−0.358452
$658$	0	0
$659$	1.24941e7	1.12070	0.560352	−	0.828254i	$-0.310666\pi$
$659$	1.24941e7	1.12070	0.560352	+	0.828254i	$0.310666\pi$
$660$	0	0
$661$	−1.61074e7	−1.43391	−0.716954	−	0.697121i	$-0.754464\pi$
$661$	−1.61074e7	−1.43391	−0.716954	+	0.697121i	$0.754464\pi$
$662$	0	0
$663$	−1.56794e6	−0.138531
$664$	0	0
$665$	−337904.	−0.0296305
$666$	0	0
$667$	1.14863e7	0.999694
$668$	0	0
$669$	5.89788e6	0.509484
$670$	0	0
$671$	9.05538e6	0.776427
$672$	0	0
$673$	1.42956e7	1.21665	0.608326	−	0.793688i	$-0.291841\pi$
$673$	1.42956e7	1.21665	0.608326	+	0.793688i	$0.291841\pi$
$674$	0	0
$675$	−2.26646e6	−0.191465
$676$	0	0
$677$	2.53658e6	0.212705	0.106352	−	0.994328i	$-0.466083\pi$
$677$	2.53658e6	0.212705	0.106352	+	0.994328i	$0.466083\pi$
$678$	0	0
$679$	344862.	0.0287059
$680$	0	0
$681$	1.07332e7	0.886872
$682$	0	0
$683$	−6.58174e6	−0.539870	−0.269935	−	0.962879i	$-0.587002\pi$
$683$	−6.58174e6	−0.539870	−0.269935	+	0.962879i	$0.587002\pi$
$684$	0	0
$685$	812280.	0.0661423
$686$	0	0
$687$	3.37055e6	0.272464
$688$	0	0
$689$	−2.96972e6	−0.238324
$690$	0	0
$691$	8.68272e6	0.691768	0.345884	−	0.938277i	$-0.387579\pi$
$691$	8.68272e6	0.691768	0.345884	+	0.938277i	$0.387579\pi$
$692$	0	0
$693$	−1.46853e6	−0.116158
$694$	0	0
$695$	−198160.	−0.0155616
$696$	0	0
$697$	1.75244e7	1.36635
$698$	0	0
$699$	1.32524e7	1.02590
$700$	0	0
$701$	4.22777e6	0.324950	0.162475	−	0.986713i	$-0.448052\pi$
$701$	4.22777e6	0.324950	0.162475	+	0.986713i	$0.448052\pi$
$702$	0	0
$703$	2.46842e7	1.88378
$704$	0	0
$705$	95184.0	0.00721259
$706$	0	0
$707$	1.21442e6	0.0913732
$708$	0	0
$709$	−9.66791e6	−0.722299	−0.361150	−	0.932508i	$-0.617616\pi$
$709$	−9.66791e6	−0.722299	−0.361150	+	0.932508i	$0.617616\pi$
$710$	0	0
$711$	2.69989e6	0.200296
$712$	0	0
$713$	−8.28768e6	−0.610533
$714$	0	0
$715$	−180560.	−0.0132086
$716$	0	0
$717$	1.16017e7	0.842799
$718$	0	0
$719$	−2.23997e7	−1.61592	−0.807962	−	0.589235i	$-0.799429\pi$
$719$	−2.23997e7	−1.61592	−0.807962	+	0.589235i	$0.799429\pi$
$720$	0	0
$721$	−5.29592e6	−0.379405
$722$	0	0
$723$	1.57030e6	0.111722
$724$	0	0
$725$	−1.33749e7	−0.945031
$726$	0	0
$727$	1.06700e7	0.748735	0.374368	−	0.927280i	$-0.377860\pi$
$727$	1.06700e7	0.748735	0.374368	+	0.927280i	$0.377860\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−3.09191e7	−2.14009
$732$	0	0
$733$	−1.30456e7	−0.896818	−0.448409	−	0.893828i	$-0.648009\pi$
$733$	−1.30456e7	−0.896818	−0.448409	+	0.893828i	$0.648009\pi$
$734$	0	0
$735$	86436.0	0.00590169
$736$	0	0
$737$	2.66696e6	0.180862
$738$	0	0
$739$	2.77150e6	0.186683	0.0933414	−	0.995634i	$-0.470245\pi$
$739$	2.77150e6	0.186683	0.0933414	+	0.995634i	$0.470245\pi$
$740$	0	0
$741$	−1.89295e6	−0.126647
$742$	0	0
$743$	1.37895e7	0.916381	0.458191	−	0.888854i	$-0.348498\pi$
$743$	1.37895e7	0.916381	0.458191	+	0.888854i	$0.348498\pi$
$744$	0	0
$745$	373416.	0.0246492
$746$	0	0
$747$	324324.	0.0212656
$748$	0	0
$749$	−5.33287e6	−0.347341
$750$	0	0
$751$	1.84892e7	1.19624	0.598118	−	0.801408i	$-0.295915\pi$
$751$	1.84892e7	1.19624	0.598118	+	0.801408i	$0.295915\pi$
$752$	0	0
$753$	7.88335e6	0.506668
$754$	0	0
$755$	−283488.	−0.0180995
$756$	0	0
$757$	2.05745e7	1.30494	0.652469	−	0.757816i	$-0.273733\pi$
$757$	2.05745e7	1.30494	0.652469	+	0.757816i	$0.273733\pi$
$758$	0	0
$759$	−8.89110e6	−0.560210
$760$	0	0
$761$	−1.75512e7	−1.09862	−0.549309	−	0.835620i	$-0.685109\pi$
$761$	−1.75512e7	−1.09862	−0.549309	+	0.835620i	$0.685109\pi$
$762$	0	0
$763$	−9.92946e6	−0.617468
$764$	0	0
$765$	−462672.	−0.0285838
$766$	0	0
$767$	1.71874e6	0.105492
$768$	0	0
$769$	1.47910e6	0.0901947	0.0450974	−	0.998983i	$-0.485640\pi$
$769$	1.47910e6	0.0901947	0.0450974	+	0.998983i	$0.485640\pi$
$770$	0	0
$771$	−6.06967e6	−0.367730
$772$	0	0
$773$	9.71423e6	0.584736	0.292368	−	0.956306i	$-0.405557\pi$
$773$	9.71423e6	0.584736	0.292368	+	0.956306i	$0.405557\pi$
$774$	0	0
$775$	9.65034e6	0.577149
$776$	0	0
$777$	−6.31424e6	−0.375205
$778$	0	0
$779$	2.11569e7	1.24913
$780$	0	0
$781$	2.00917e7	1.17866
$782$	0	0
$783$	3.13616e6	0.182807
$784$	0	0
$785$	−1.49959e6	−0.0868558
$786$	0	0
$787$	−1.45994e7	−0.840231	−0.420115	−	0.907471i	$-0.638010\pi$
$787$	−1.45994e7	−0.840231	−0.420115	+	0.907471i	$0.638010\pi$
$788$	0	0
$789$	−1.32440e7	−0.757402
$790$	0	0
$791$	−2.45970e6	−0.139779
$792$	0	0
$793$	−2.98583e6	−0.168609
$794$	0	0
$795$	−876312.	−0.0491746
$796$	0	0
$797$	−2.46095e7	−1.37233	−0.686163	−	0.727448i	$-0.740706\pi$
$797$	−2.46095e7	−1.37233	−0.686163	+	0.727448i	$0.740706\pi$
$798$	0	0
$799$	−3.77563e6	−0.209229
$800$	0	0
$801$	5.24491e6	0.288840
$802$	0	0
$803$	1.81159e7	0.991453
$804$	0	0
$805$	523320.	0.0284628
$806$	0	0
$807$	1.91319e7	1.03413
$808$	0	0
$809$	−2.00982e7	−1.07966	−0.539828	−	0.841776i	$-0.681511\pi$
$809$	−2.00982e7	−1.07966	−0.539828	+	0.841776i	$0.681511\pi$
$810$	0	0
$811$	1.12008e7	0.597993	0.298996	−	0.954254i	$-0.403348\pi$
$811$	1.12008e7	0.597993	0.298996	+	0.954254i	$0.403348\pi$
$812$	0	0
$813$	−9.98460e6	−0.529791
$814$	0	0
$815$	500736.	0.0264067
$816$	0	0
$817$	−3.73280e7	−1.95650
$818$	0	0
$819$	484218.	0.0252250
$820$	0	0
$821$	2.38971e7	1.23733	0.618667	−	0.785653i	$-0.287673\pi$
$821$	2.38971e7	1.23733	0.618667	+	0.785653i	$0.287673\pi$
$822$	0	0
$823$	−9.17972e6	−0.472422	−0.236211	−	0.971702i	$-0.575906\pi$
$823$	−9.17972e6	−0.472422	−0.236211	+	0.971702i	$0.575906\pi$
$824$	0	0
$825$	1.03530e7	0.529578
$826$	0	0
$827$	−2.82018e7	−1.43388	−0.716939	−	0.697135i	$-0.754458\pi$
$827$	−2.82018e7	−1.43388	−0.716939	+	0.697135i	$0.754458\pi$
$828$	0	0
$829$	−1.81930e7	−0.919429	−0.459714	−	0.888067i	$-0.652048\pi$
$829$	−1.81930e7	−0.919429	−0.459714	+	0.888067i	$0.652048\pi$
$830$	0	0
$831$	−2.06185e6	−0.103575
$832$	0	0
$833$	−3.42863e6	−0.171202
$834$	0	0
$835$	−1.77627e6	−0.0881644
$836$	0	0
$837$	−2.26282e6	−0.111644
$838$	0	0
$839$	2.97349e6	0.145835	0.0729175	−	0.997338i	$-0.476769\pi$
$839$	2.97349e6	0.145835	0.0729175	+	0.997338i	$0.476769\pi$
$840$	0	0
$841$	−2.00394e6	−0.0977003
$842$	0	0
$843$	−5.63366e6	−0.273037
$844$	0	0
$845$	−1.42564e6	−0.0686858
$846$	0	0
$847$	−1.18340e6	−0.0566791
$848$	0	0
$849$	1.83937e7	0.875788
$850$	0	0
$851$	−3.82291e7	−1.80955
$852$	0	0
$853$	−1.43801e7	−0.676690	−0.338345	−	0.941022i	$-0.609867\pi$
$853$	−1.43801e7	−0.676690	−0.338345	+	0.941022i	$0.609867\pi$
$854$	0	0
$855$	−558576.	−0.0261317
$856$	0	0
$857$	−2.74223e7	−1.27542	−0.637708	−	0.770278i	$-0.720117\pi$
$857$	−2.74223e7	−1.27542	−0.637708	+	0.770278i	$0.720117\pi$
$858$	0	0
$859$	2.79786e7	1.29373	0.646864	−	0.762606i	$-0.276080\pi$
$859$	2.79786e7	1.29373	0.646864	+	0.762606i	$0.276080\pi$
$860$	0	0
$861$	−5.41195e6	−0.248798
$862$	0	0
$863$	5.67312e6	0.259295	0.129648	−	0.991560i	$-0.458615\pi$
$863$	5.67312e6	0.259295	0.129648	+	0.991560i	$0.458615\pi$
$864$	0	0
$865$	2.17747e6	0.0989492
$866$	0	0
$867$	5.57394e6	0.251834
$868$	0	0
$869$	−1.23328e7	−0.554005
$870$	0	0
$871$	−879376.	−0.0392762
$872$	0	0
$873$	570078.	0.0253162
$874$	0	0
$875$	−1.22186e6	−0.0539514
$876$	0	0
$877$	−1.09211e7	−0.479478	−0.239739	−	0.970837i	$-0.577062\pi$
$877$	−1.09211e7	−0.479478	−0.239739	+	0.970837i	$0.577062\pi$
$878$	0	0
$879$	1.77155e7	0.773357
$880$	0	0
$881$	−3.72031e7	−1.61488	−0.807438	−	0.589952i	$-0.799146\pi$
$881$	−3.72031e7	−1.61488	−0.807438	+	0.589952i	$0.799146\pi$
$882$	0	0
$883$	−3.57619e7	−1.54355	−0.771773	−	0.635898i	$-0.780630\pi$
$883$	−3.57619e7	−1.54355	−0.771773	+	0.635898i	$0.780630\pi$
$884$	0	0
$885$	507168.	0.0217668
$886$	0	0
$887$	2.25301e7	0.961510	0.480755	−	0.876855i	$-0.340363\pi$
$887$	2.25301e7	0.961510	0.480755	+	0.876855i	$0.340363\pi$
$888$	0	0
$889$	−149548.	−0.00634638
$890$	0	0
$891$	−2.42757e6	−0.102442
$892$	0	0
$893$	−4.55826e6	−0.191280
$894$	0	0
$895$	2.46452e6	0.102843
$896$	0	0
$897$	2.93166e6	0.121656
$898$	0	0
$899$	−1.33534e7	−0.551052
$900$	0	0
$901$	3.47604e7	1.42650
$902$	0	0
$903$	9.54853e6	0.389688
$904$	0	0
$905$	393064.	0.0159530
$906$	0	0
$907$	−3.18871e7	−1.28705	−0.643526	−	0.765424i	$-0.722529\pi$
$907$	−3.18871e7	−1.28705	−0.643526	+	0.765424i	$0.722529\pi$
$908$	0	0
$909$	2.00750e6	0.0805836
$910$	0	0
$911$	−2.21745e7	−0.885233	−0.442617	−	0.896711i	$-0.645950\pi$
$911$	−2.21745e7	−0.885233	−0.442617	+	0.896711i	$0.645950\pi$
$912$	0	0
$913$	−1.48148e6	−0.0588192
$914$	0	0
$915$	−881064.	−0.0347900
$916$	0	0
$917$	−1.07206e7	−0.421014
$918$	0	0
$919$	2.87465e7	1.12279	0.561393	−	0.827549i	$-0.310266\pi$
$919$	2.87465e7	1.12279	0.561393	+	0.827549i	$0.310266\pi$
$920$	0	0
$921$	−2.73923e7	−1.06409
$922$	0	0
$923$	−6.62484e6	−0.255960
$924$	0	0
$925$	4.45147e7	1.71060
$926$	0	0
$927$	−8.75448e6	−0.334604
$928$	0	0
$929$	−2.58792e7	−0.983810	−0.491905	−	0.870649i	$-0.663699\pi$
$929$	−2.58792e7	−0.983810	−0.491905	+	0.870649i	$0.663699\pi$
$930$	0	0
$931$	−4.13932e6	−0.156515
$932$	0	0
$933$	2.93152e6	0.110252
$934$	0	0
$935$	2.11344e6	0.0790607
$936$	0	0
$937$	4.35364e7	1.61996	0.809979	−	0.586459i	$-0.199478\pi$
$937$	4.35364e7	1.61996	0.809979	+	0.586459i	$0.199478\pi$
$938$	0	0
$939$	2.05892e7	0.762036
$940$	0	0
$941$	4.20702e7	1.54882	0.774410	−	0.632684i	$-0.218047\pi$
$941$	4.20702e7	1.54882	0.774410	+	0.632684i	$0.218047\pi$
$942$	0	0
$943$	−3.27662e7	−1.19991
$944$	0	0
$945$	142884.	0.00520480
$946$	0	0
$947$	−505082.	−0.0183015	−0.00915076	−	0.999958i	$-0.502913\pi$
$947$	−505082.	−0.0183015	−0.00915076	+	0.999958i	$0.502913\pi$
$948$	0	0
$949$	−5.97336e6	−0.215305
$950$	0	0
$951$	8.12306e6	0.291252
$952$	0	0
$953$	2.18755e7	0.780234	0.390117	−	0.920765i	$-0.372434\pi$
$953$	2.18755e7	0.780234	0.390117	+	0.920765i	$0.372434\pi$
$954$	0	0
$955$	2.06055e6	0.0731097
$956$	0	0
$957$	−1.43257e7	−0.505632
$958$	0	0
$959$	9.95043e6	0.349378
$960$	0	0
$961$	−1.89943e7	−0.663461
$962$	0	0
$963$	−8.81555e6	−0.306326
$964$	0	0
$965$	−1.79972e6	−0.0622138
$966$	0	0
$967$	−2.79738e7	−0.962024	−0.481012	−	0.876714i	$-0.659731\pi$
$967$	−2.79738e7	−0.962024	−0.481012	+	0.876714i	$0.659731\pi$
$968$	0	0
$969$	2.21568e7	0.758051
$970$	0	0
$971$	−4.37517e7	−1.48918	−0.744590	−	0.667522i	$-0.767355\pi$
$971$	−4.37517e7	−1.48918	−0.744590	+	0.667522i	$0.767355\pi$
$972$	0	0
$973$	−2.42746e6	−0.0821997
$974$	0	0
$975$	−3.41368e6	−0.115004
$976$	0	0
$977$	−4.53190e7	−1.51895	−0.759476	−	0.650535i	$-0.774545\pi$
$977$	−4.53190e7	−1.51895	−0.759476	+	0.650535i	$0.774545\pi$
$978$	0	0
$979$	−2.39582e7	−0.798911
$980$	0	0
$981$	−1.64140e7	−0.544555
$982$	0	0
$983$	−4.15059e7	−1.37002	−0.685009	−	0.728534i	$-0.740202\pi$
$983$	−4.15059e7	−1.37002	−0.685009	+	0.728534i	$0.740202\pi$
$984$	0	0
$985$	−1.61974e6	−0.0531929
$986$	0	0
$987$	1.16600e6	0.0380984
$988$	0	0
$989$	5.78108e7	1.87940
$990$	0	0
$991$	−3.40757e7	−1.10220	−0.551101	−	0.834439i	$-0.685792\pi$
$991$	−3.40757e7	−1.10220	−0.551101	+	0.834439i	$0.685792\pi$
$992$	0	0
$993$	−1.40575e7	−0.452412
$994$	0	0
$995$	1.37030e6	0.0438793
$996$	0	0
$997$	−4.22181e7	−1.34512	−0.672560	−	0.740042i	$-0.734805\pi$
$997$	−4.22181e7	−1.34512	−0.672560	+	0.740042i	$0.734805\pi$
$998$	0	0
$999$	−1.04378e7	−0.330900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.m.1.1		1
3.2	odd	2		1008.6.a.q.1.1		1
4.3	odd	2		168.6.a.b.1.1	✓	1
12.11	even	2		504.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.b.1.1	✓	1	4.3	odd	2
336.6.a.m.1.1		1	1.1	even	1	trivial
504.6.a.d.1.1		1	12.11	even	2
1008.6.a.q.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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