LMFDB - Embedded newform 624.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,6,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, 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("624.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.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:	$100.079503563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} +24.0000 q^{5} +238.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +24.0000 q^{5} +238.000 q^{7} +81.0000 q^{9} -24.0000 q^{11} +169.000 q^{13} -216.000 q^{15} -2262.00 q^{17} -1154.00 q^{19} -2142.00 q^{21} +3744.00 q^{23} -2549.00 q^{25} -729.000 q^{27} -6294.00 q^{29} -7010.00 q^{31} +216.000 q^{33} +5712.00 q^{35} -5182.00 q^{37} -1521.00 q^{39} -9252.00 q^{41} +23044.0 q^{43} +1944.00 q^{45} -27480.0 q^{47} +39837.0 q^{49} +20358.0 q^{51} -5130.00 q^{53} -576.000 q^{55} +10386.0 q^{57} -10164.0 q^{59} +37490.0 q^{61} +19278.0 q^{63} +4056.00 q^{65} -26342.0 q^{67} -33696.0 q^{69} +28668.0 q^{71} -26818.0 q^{73} +22941.0 q^{75} -5712.00 q^{77} -26168.0 q^{79} +6561.00 q^{81} +13308.0 q^{83} -54288.0 q^{85} +56646.0 q^{87} -48264.0 q^{89} +40222.0 q^{91} +63090.0 q^{93} -27696.0 q^{95} +73094.0 q^{97} -1944.00 q^{99} +O(q^{100})$

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
$5$	24.0000	0.429325	0.214663	−	0.976688i	$-0.431135\pi$
$5$	24.0000	0.429325	0.214663	+	0.976688i	$0.431135\pi$
$6$	0	0
$7$	238.000	1.83583	0.917914	−	0.396780i	$-0.129872\pi$
$7$	238.000	1.83583	0.917914	+	0.396780i	$0.129872\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−24.0000	−0.0598039	−0.0299020	−	0.999553i	$-0.509520\pi$
$11$	−24.0000	−0.0598039	−0.0299020	+	0.999553i	$0.509520\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−216.000	−0.247871
$16$	0	0
$17$	−2262.00	−1.89832	−0.949162	−	0.314788i	$-0.898066\pi$
$17$	−2262.00	−1.89832	−0.949162	+	0.314788i	$0.898066\pi$
$18$	0	0
$19$	−1154.00	−0.733368	−0.366684	−	0.930346i	$-0.619507\pi$
$19$	−1154.00	−0.733368	−0.366684	+	0.930346i	$0.619507\pi$
$20$	0	0
$21$	−2142.00	−1.05992
$22$	0	0
$23$	3744.00	1.47576	0.737881	−	0.674931i	$-0.235827\pi$
$23$	3744.00	1.47576	0.737881	+	0.674931i	$0.235827\pi$
$24$	0	0
$25$	−2549.00	−0.815680
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−6294.00	−1.38973	−0.694867	−	0.719138i	$-0.744537\pi$
$29$	−6294.00	−1.38973	−0.694867	+	0.719138i	$0.744537\pi$
$30$	0	0
$31$	−7010.00	−1.31013	−0.655064	−	0.755573i	$-0.727358\pi$
$31$	−7010.00	−1.31013	−0.655064	+	0.755573i	$0.727358\pi$
$32$	0	0
$33$	216.000	0.0345278
$34$	0	0
$35$	5712.00	0.788167
$36$	0	0
$37$	−5182.00	−0.622290	−0.311145	−	0.950362i	$-0.600713\pi$
$37$	−5182.00	−0.622290	−0.311145	+	0.950362i	$0.600713\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	−9252.00	−0.859560	−0.429780	−	0.902934i	$-0.641409\pi$
$41$	−9252.00	−0.859560	−0.429780	+	0.902934i	$0.641409\pi$
$42$	0	0
$43$	23044.0	1.90058	0.950291	−	0.311362i	$-0.100785\pi$
$43$	23044.0	1.90058	0.950291	+	0.311362i	$0.100785\pi$
$44$	0	0
$45$	1944.00	0.143108
$46$	0	0
$47$	−27480.0	−1.81456	−0.907282	−	0.420524i	$-0.861846\pi$
$47$	−27480.0	−1.81456	−0.907282	+	0.420524i	$0.861846\pi$
$48$	0	0
$49$	39837.0	2.37026
$50$	0	0
$51$	20358.0	1.09600
$52$	0	0
$53$	−5130.00	−0.250858	−0.125429	−	0.992103i	$-0.540031\pi$
$53$	−5130.00	−0.250858	−0.125429	+	0.992103i	$0.540031\pi$
$54$	0	0
$55$	−576.000	−0.0256753
$56$	0	0
$57$	10386.0	0.423410
$58$	0	0
$59$	−10164.0	−0.380132	−0.190066	−	0.981771i	$-0.560870\pi$
$59$	−10164.0	−0.380132	−0.190066	+	0.981771i	$0.560870\pi$
$60$	0	0
$61$	37490.0	1.29000	0.645002	−	0.764181i	$-0.276857\pi$
$61$	37490.0	1.29000	0.645002	+	0.764181i	$0.276857\pi$
$62$	0	0
$63$	19278.0	0.611942
$64$	0	0
$65$	4056.00	0.119073
$66$	0	0
$67$	−26342.0	−0.716905	−0.358453	−	0.933548i	$-0.616696\pi$
$67$	−26342.0	−0.716905	−0.358453	+	0.933548i	$0.616696\pi$
$68$	0	0
$69$	−33696.0	−0.852031
$70$	0	0
$71$	28668.0	0.674919	0.337459	−	0.941340i	$-0.390432\pi$
$71$	28668.0	0.674919	0.337459	+	0.941340i	$0.390432\pi$
$72$	0	0
$73$	−26818.0	−0.589005	−0.294503	−	0.955651i	$-0.595154\pi$
$73$	−26818.0	−0.589005	−0.294503	+	0.955651i	$0.595154\pi$
$74$	0	0
$75$	22941.0	0.470933
$76$	0	0
$77$	−5712.00	−0.109790
$78$	0	0
$79$	−26168.0	−0.471740	−0.235870	−	0.971785i	$-0.575794\pi$
$79$	−26168.0	−0.471740	−0.235870	+	0.971785i	$0.575794\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	13308.0	0.212040	0.106020	−	0.994364i	$-0.466189\pi$
$83$	13308.0	0.212040	0.106020	+	0.994364i	$0.466189\pi$
$84$	0	0
$85$	−54288.0	−0.814998
$86$	0	0
$87$	56646.0	0.802363
$88$	0	0
$89$	−48264.0	−0.645875	−0.322937	−	0.946420i	$-0.604670\pi$
$89$	−48264.0	−0.645875	−0.322937	+	0.946420i	$0.604670\pi$
$90$	0	0
$91$	40222.0	0.509167
$92$	0	0
$93$	63090.0	0.756403
$94$	0	0
$95$	−27696.0	−0.314853
$96$	0	0
$97$	73094.0	0.788774	0.394387	−	0.918945i	$-0.370957\pi$
$97$	73094.0	0.788774	0.394387	+	0.918945i	$0.370957\pi$
$98$	0	0
$99$	−1944.00	−0.0199346
$100$	0	0
$101$	51954.0	0.506775	0.253388	−	0.967365i	$-0.418455\pi$
$101$	51954.0	0.506775	0.253388	+	0.967365i	$0.418455\pi$
$102$	0	0
$103$	−109088.	−1.01317	−0.506587	−	0.862189i	$-0.669093\pi$
$103$	−109088.	−1.01317	−0.506587	+	0.862189i	$0.669093\pi$
$104$	0	0
$105$	−51408.0	−0.455048
$106$	0	0
$107$	−40692.0	−0.343597	−0.171799	−	0.985132i	$-0.554958\pi$
$107$	−40692.0	−0.343597	−0.171799	+	0.985132i	$0.554958\pi$
$108$	0	0
$109$	3422.00	0.0275876	0.0137938	−	0.999905i	$-0.495609\pi$
$109$	3422.00	0.0275876	0.0137938	+	0.999905i	$0.495609\pi$
$110$	0	0
$111$	46638.0	0.359280
$112$	0	0
$113$	−12570.0	−0.0926060	−0.0463030	−	0.998927i	$-0.514744\pi$
$113$	−12570.0	−0.0926060	−0.0463030	+	0.998927i	$0.514744\pi$
$114$	0	0
$115$	89856.0	0.633581
$116$	0	0
$117$	13689.0	0.0924500
$118$	0	0
$119$	−538356.	−3.48499
$120$	0	0
$121$	−160475.	−0.996423
$122$	0	0
$123$	83268.0	0.496267
$124$	0	0
$125$	−136176.	−0.779517
$126$	0	0
$127$	−289964.	−1.59527	−0.797636	−	0.603139i	$-0.793916\pi$
$127$	−289964.	−1.59527	−0.797636	+	0.603139i	$0.793916\pi$
$128$	0	0
$129$	−207396.	−1.09730
$130$	0	0
$131$	101460.	0.516555	0.258278	−	0.966071i	$-0.416845\pi$
$131$	101460.	0.516555	0.258278	+	0.966071i	$0.416845\pi$
$132$	0	0
$133$	−274652.	−1.34634
$134$	0	0
$135$	−17496.0	−0.0826236
$136$	0	0
$137$	−95076.0	−0.432782	−0.216391	−	0.976307i	$-0.569429\pi$
$137$	−95076.0	−0.432782	−0.216391	+	0.976307i	$0.569429\pi$
$138$	0	0
$139$	−133712.	−0.586994	−0.293497	−	0.955960i	$-0.594819\pi$
$139$	−133712.	−0.586994	−0.293497	+	0.955960i	$0.594819\pi$
$140$	0	0
$141$	247320.	1.04764
$142$	0	0
$143$	−4056.00	−0.0165866
$144$	0	0
$145$	−151056.	−0.596648
$146$	0	0
$147$	−358533.	−1.36847
$148$	0	0
$149$	4980.00	0.0183765	0.00918827	−	0.999958i	$-0.497075\pi$
$149$	4980.00	0.0183765	0.00918827	+	0.999958i	$0.497075\pi$
$150$	0	0
$151$	111370.	0.397490	0.198745	−	0.980051i	$-0.436313\pi$
$151$	111370.	0.397490	0.198745	+	0.980051i	$0.436313\pi$
$152$	0	0
$153$	−183222.	−0.632775
$154$	0	0
$155$	−168240.	−0.562471
$156$	0	0
$157$	224882.	0.728124	0.364062	−	0.931375i	$-0.381390\pi$
$157$	224882.	0.728124	0.364062	+	0.931375i	$0.381390\pi$
$158$	0	0
$159$	46170.0	0.144833
$160$	0	0
$161$	891072.	2.70924
$162$	0	0
$163$	645586.	1.90320	0.951601	−	0.307335i	$-0.0994371\pi$
$163$	645586.	1.90320	0.951601	+	0.307335i	$0.0994371\pi$
$164$	0	0
$165$	5184.00	0.0148236
$166$	0	0
$167$	−605376.	−1.67971	−0.839854	−	0.542812i	$-0.817360\pi$
$167$	−605376.	−1.67971	−0.839854	+	0.542812i	$0.817360\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−93474.0	−0.244456
$172$	0	0
$173$	−557478.	−1.41616	−0.708080	−	0.706132i	$-0.750439\pi$
$173$	−557478.	−1.41616	−0.708080	+	0.706132i	$0.750439\pi$
$174$	0	0
$175$	−606662.	−1.49745
$176$	0	0
$177$	91476.0	0.219469
$178$	0	0
$179$	405060.	0.944902	0.472451	−	0.881357i	$-0.343369\pi$
$179$	405060.	0.944902	0.472451	+	0.881357i	$0.343369\pi$
$180$	0	0
$181$	−469870.	−1.06606	−0.533030	−	0.846097i	$-0.678947\pi$
$181$	−469870.	−1.06606	−0.533030	+	0.846097i	$0.678947\pi$
$182$	0	0
$183$	−337410.	−0.744784
$184$	0	0
$185$	−124368.	−0.267165
$186$	0	0
$187$	54288.0	0.113527
$188$	0	0
$189$	−173502.	−0.353305
$190$	0	0
$191$	650160.	1.28955	0.644773	−	0.764374i	$-0.276952\pi$
$191$	650160.	1.28955	0.644773	+	0.764374i	$0.276952\pi$
$192$	0	0
$193$	−231454.	−0.447272	−0.223636	−	0.974673i	$-0.571793\pi$
$193$	−231454.	−0.447272	−0.223636	+	0.974673i	$0.571793\pi$
$194$	0	0
$195$	−36504.0	−0.0687470
$196$	0	0
$197$	−459168.	−0.842958	−0.421479	−	0.906838i	$-0.638489\pi$
$197$	−459168.	−0.842958	−0.421479	+	0.906838i	$0.638489\pi$
$198$	0	0
$199$	128860.	0.230667	0.115333	−	0.993327i	$-0.463206\pi$
$199$	128860.	0.230667	0.115333	+	0.993327i	$0.463206\pi$
$200$	0	0
$201$	237078.	0.413905
$202$	0	0
$203$	−1.49797e6	−2.55131
$204$	0	0
$205$	−222048.	−0.369030
$206$	0	0
$207$	303264.	0.491921
$208$	0	0
$209$	27696.0	0.0438583
$210$	0	0
$211$	675460.	1.04446	0.522232	−	0.852803i	$-0.325099\pi$
$211$	675460.	1.04446	0.522232	+	0.852803i	$0.325099\pi$
$212$	0	0
$213$	−258012.	−0.389665
$214$	0	0
$215$	553056.	0.815968
$216$	0	0
$217$	−1.66838e6	−2.40517
$218$	0	0
$219$	241362.	0.340062
$220$	0	0
$221$	−382278.	−0.526500
$222$	0	0
$223$	−172058.	−0.231693	−0.115846	−	0.993267i	$-0.536958\pi$
$223$	−172058.	−0.231693	−0.115846	+	0.993267i	$0.536958\pi$
$224$	0	0
$225$	−206469.	−0.271893
$226$	0	0
$227$	−157536.	−0.202915	−0.101458	−	0.994840i	$-0.532351\pi$
$227$	−157536.	−0.202915	−0.101458	+	0.994840i	$0.532351\pi$
$228$	0	0
$229$	−505666.	−0.637199	−0.318599	−	0.947889i	$-0.603212\pi$
$229$	−505666.	−0.637199	−0.318599	+	0.947889i	$0.603212\pi$
$230$	0	0
$231$	51408.0	0.0633871
$232$	0	0
$233$	881334.	1.06353	0.531766	−	0.846891i	$-0.321529\pi$
$233$	881334.	1.06353	0.531766	+	0.846891i	$0.321529\pi$
$234$	0	0
$235$	−659520.	−0.779037
$236$	0	0
$237$	235512.	0.272359
$238$	0	0
$239$	722964.	0.818695	0.409347	−	0.912379i	$-0.365756\pi$
$239$	722964.	0.818695	0.409347	+	0.912379i	$0.365756\pi$
$240$	0	0
$241$	−921658.	−1.02218	−0.511090	−	0.859527i	$-0.670758\pi$
$241$	−921658.	−1.02218	−0.511090	+	0.859527i	$0.670758\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	956088.	1.01761
$246$	0	0
$247$	−195026.	−0.203400
$248$	0	0
$249$	−119772.	−0.122421
$250$	0	0
$251$	−958236.	−0.960037	−0.480019	−	0.877258i	$-0.659370\pi$
$251$	−958236.	−0.960037	−0.480019	+	0.877258i	$0.659370\pi$
$252$	0	0
$253$	−89856.0	−0.0882563
$254$	0	0
$255$	488592.	0.470539
$256$	0	0
$257$	−842058.	−0.795260	−0.397630	−	0.917546i	$-0.630167\pi$
$257$	−842058.	−0.795260	−0.397630	+	0.917546i	$0.630167\pi$
$258$	0	0
$259$	−1.23332e6	−1.14242
$260$	0	0
$261$	−509814.	−0.463245
$262$	0	0
$263$	640488.	0.570981	0.285490	−	0.958382i	$-0.407844\pi$
$263$	640488.	0.570981	0.285490	+	0.958382i	$0.407844\pi$
$264$	0	0
$265$	−123120.	−0.107700
$266$	0	0
$267$	434376.	0.372896
$268$	0	0
$269$	−833214.	−0.702063	−0.351031	−	0.936364i	$-0.614169\pi$
$269$	−833214.	−0.702063	−0.351031	+	0.936364i	$0.614169\pi$
$270$	0	0
$271$	106774.	0.0883166	0.0441583	−	0.999025i	$-0.485939\pi$
$271$	106774.	0.0883166	0.0441583	+	0.999025i	$0.485939\pi$
$272$	0	0
$273$	−361998.	−0.293968
$274$	0	0
$275$	61176.0	0.0487808
$276$	0	0
$277$	2.23223e6	1.74799	0.873996	−	0.485933i	$-0.161520\pi$
$277$	2.23223e6	1.74799	0.873996	+	0.485933i	$0.161520\pi$
$278$	0	0
$279$	−567810.	−0.436709
$280$	0	0
$281$	−8196.00	−0.00619207	−0.00309604	−	0.999995i	$-0.500986\pi$
$281$	−8196.00	−0.00619207	−0.00309604	+	0.999995i	$0.500986\pi$
$282$	0	0
$283$	376624.	0.279539	0.139769	−	0.990184i	$-0.455364\pi$
$283$	376624.	0.279539	0.139769	+	0.990184i	$0.455364\pi$
$284$	0	0
$285$	249264.	0.181781
$286$	0	0
$287$	−2.20198e6	−1.57800
$288$	0	0
$289$	3.69679e6	2.60363
$290$	0	0
$291$	−657846.	−0.455399
$292$	0	0
$293$	−2.02564e6	−1.37845	−0.689227	−	0.724545i	$-0.742050\pi$
$293$	−2.02564e6	−1.37845	−0.689227	+	0.724545i	$0.742050\pi$
$294$	0	0
$295$	−243936.	−0.163200
$296$	0	0
$297$	17496.0	0.0115093
$298$	0	0
$299$	632736.	0.409303
$300$	0	0
$301$	5.48447e6	3.48914
$302$	0	0
$303$	−467586.	−0.292587
$304$	0	0
$305$	899760.	0.553831
$306$	0	0
$307$	−1.38168e6	−0.836685	−0.418343	−	0.908289i	$-0.637389\pi$
$307$	−1.38168e6	−0.836685	−0.418343	+	0.908289i	$0.637389\pi$
$308$	0	0
$309$	981792.	0.584956
$310$	0	0
$311$	−2.60066e6	−1.52470	−0.762348	−	0.647167i	$-0.775954\pi$
$311$	−2.60066e6	−1.52470	−0.762348	+	0.647167i	$0.775954\pi$
$312$	0	0
$313$	1.45467e6	0.839271	0.419636	−	0.907693i	$-0.362158\pi$
$313$	1.45467e6	0.839271	0.419636	+	0.907693i	$0.362158\pi$
$314$	0	0
$315$	462672.	0.262722
$316$	0	0
$317$	−1.16304e6	−0.650050	−0.325025	−	0.945705i	$-0.605373\pi$
$317$	−1.16304e6	−0.650050	−0.325025	+	0.945705i	$0.605373\pi$
$318$	0	0
$319$	151056.	0.0831115
$320$	0	0
$321$	366228.	0.198376
$322$	0	0
$323$	2.61035e6	1.39217
$324$	0	0
$325$	−430781.	−0.226229
$326$	0	0
$327$	−30798.0	−0.0159277
$328$	0	0
$329$	−6.54024e6	−3.33122
$330$	0	0
$331$	−687110.	−0.344712	−0.172356	−	0.985035i	$-0.555138\pi$
$331$	−687110.	−0.344712	−0.172356	+	0.985035i	$0.555138\pi$
$332$	0	0
$333$	−419742.	−0.207430
$334$	0	0
$335$	−632208.	−0.307785
$336$	0	0
$337$	−2.53949e6	−1.21807	−0.609033	−	0.793145i	$-0.708442\pi$
$337$	−2.53949e6	−1.21807	−0.609033	+	0.793145i	$0.708442\pi$
$338$	0	0
$339$	113130.	0.0534661
$340$	0	0
$341$	168240.	0.0783508
$342$	0	0
$343$	5.48114e6	2.51557
$344$	0	0
$345$	−808704.	−0.365798
$346$	0	0
$347$	1.39844e6	0.623478	0.311739	−	0.950168i	$-0.399088\pi$
$347$	1.39844e6	0.623478	0.311739	+	0.950168i	$0.399088\pi$
$348$	0	0
$349$	3.45684e6	1.51920	0.759602	−	0.650388i	$-0.225394\pi$
$349$	3.45684e6	1.51920	0.759602	+	0.650388i	$0.225394\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	1.48348e6	0.633642	0.316821	−	0.948485i	$-0.397385\pi$
$353$	1.48348e6	0.633642	0.316821	+	0.948485i	$0.397385\pi$
$354$	0	0
$355$	688032.	0.289760
$356$	0	0
$357$	4.84520e6	2.01206
$358$	0	0
$359$	−876096.	−0.358770	−0.179385	−	0.983779i	$-0.557411\pi$
$359$	−876096.	−0.358770	−0.179385	+	0.983779i	$0.557411\pi$
$360$	0	0
$361$	−1.14438e6	−0.462172
$362$	0	0
$363$	1.44428e6	0.575285
$364$	0	0
$365$	−643632.	−0.252875
$366$	0	0
$367$	2.49344e6	0.966347	0.483173	−	0.875525i	$-0.339484\pi$
$367$	2.49344e6	0.966347	0.483173	+	0.875525i	$0.339484\pi$
$368$	0	0
$369$	−749412.	−0.286520
$370$	0	0
$371$	−1.22094e6	−0.460532
$372$	0	0
$373$	−4.91696e6	−1.82989	−0.914945	−	0.403580i	$-0.867766\pi$
$373$	−4.91696e6	−1.82989	−0.914945	+	0.403580i	$0.867766\pi$
$374$	0	0
$375$	1.22558e6	0.450054
$376$	0	0
$377$	−1.06369e6	−0.385443
$378$	0	0
$379$	−781562.	−0.279489	−0.139745	−	0.990188i	$-0.544628\pi$
$379$	−781562.	−0.279489	−0.139745	+	0.990188i	$0.544628\pi$
$380$	0	0
$381$	2.60968e6	0.921031
$382$	0	0
$383$	2.45299e6	0.854475	0.427237	−	0.904140i	$-0.359487\pi$
$383$	2.45299e6	0.854475	0.427237	+	0.904140i	$0.359487\pi$
$384$	0	0
$385$	−137088.	−0.0471354
$386$	0	0
$387$	1.86656e6	0.633528
$388$	0	0
$389$	−605922.	−0.203022	−0.101511	−	0.994834i	$-0.532368\pi$
$389$	−605922.	−0.203022	−0.101511	+	0.994834i	$0.532368\pi$
$390$	0	0
$391$	−8.46893e6	−2.80147
$392$	0	0
$393$	−913140.	−0.298233
$394$	0	0
$395$	−628032.	−0.202530
$396$	0	0
$397$	1.09159e6	0.347604	0.173802	−	0.984781i	$-0.444395\pi$
$397$	1.09159e6	0.347604	0.173802	+	0.984781i	$0.444395\pi$
$398$	0	0
$399$	2.47187e6	0.777308
$400$	0	0
$401$	−2.65176e6	−0.823518	−0.411759	−	0.911293i	$-0.635086\pi$
$401$	−2.65176e6	−0.823518	−0.411759	+	0.911293i	$0.635086\pi$
$402$	0	0
$403$	−1.18469e6	−0.363364
$404$	0	0
$405$	157464.	0.0477028
$406$	0	0
$407$	124368.	0.0372154
$408$	0	0
$409$	−863866.	−0.255351	−0.127676	−	0.991816i	$-0.540752\pi$
$409$	−863866.	−0.255351	−0.127676	+	0.991816i	$0.540752\pi$
$410$	0	0
$411$	855684.	0.249867
$412$	0	0
$413$	−2.41903e6	−0.697857
$414$	0	0
$415$	319392.	0.0910340
$416$	0	0
$417$	1.20341e6	0.338901
$418$	0	0
$419$	−4.38066e6	−1.21900	−0.609501	−	0.792785i	$-0.708630\pi$
$419$	−4.38066e6	−1.21900	−0.609501	+	0.792785i	$0.708630\pi$
$420$	0	0
$421$	3.75006e6	1.03118	0.515588	−	0.856836i	$-0.327573\pi$
$421$	3.75006e6	1.03118	0.515588	+	0.856836i	$0.327573\pi$
$422$	0	0
$423$	−2.22588e6	−0.604854
$424$	0	0
$425$	5.76584e6	1.54842
$426$	0	0
$427$	8.92262e6	2.36822
$428$	0	0
$429$	36504.0	0.00957629
$430$	0	0
$431$	−1.20864e6	−0.313403	−0.156702	−	0.987646i	$-0.550086\pi$
$431$	−1.20864e6	−0.313403	−0.156702	+	0.987646i	$0.550086\pi$
$432$	0	0
$433$	−1.23368e6	−0.316216	−0.158108	−	0.987422i	$-0.550539\pi$
$433$	−1.23368e6	−0.316216	−0.158108	+	0.987422i	$0.550539\pi$
$434$	0	0
$435$	1.35950e6	0.344475
$436$	0	0
$437$	−4.32058e6	−1.08228
$438$	0	0
$439$	5.75831e6	1.42605	0.713024	−	0.701140i	$-0.247325\pi$
$439$	5.75831e6	1.42605	0.713024	+	0.701140i	$0.247325\pi$
$440$	0	0
$441$	3.22680e6	0.790087
$442$	0	0
$443$	−2.04316e6	−0.494643	−0.247322	−	0.968933i	$-0.579550\pi$
$443$	−2.04316e6	−0.494643	−0.247322	+	0.968933i	$0.579550\pi$
$444$	0	0
$445$	−1.15834e6	−0.277290
$446$	0	0
$447$	−44820.0	−0.0106097
$448$	0	0
$449$	452184.	0.105852	0.0529260	−	0.998598i	$-0.483145\pi$
$449$	452184.	0.105852	0.0529260	+	0.998598i	$0.483145\pi$
$450$	0	0
$451$	222048.	0.0514050
$452$	0	0
$453$	−1.00233e6	−0.229491
$454$	0	0
$455$	965328.	0.218598
$456$	0	0
$457$	−1.79300e6	−0.401597	−0.200798	−	0.979633i	$-0.564354\pi$
$457$	−1.79300e6	−0.401597	−0.200798	+	0.979633i	$0.564354\pi$
$458$	0	0
$459$	1.64900e6	0.365333
$460$	0	0
$461$	4.17074e6	0.914032	0.457016	−	0.889459i	$-0.348918\pi$
$461$	4.17074e6	0.914032	0.457016	+	0.889459i	$0.348918\pi$
$462$	0	0
$463$	−1.31215e6	−0.284467	−0.142234	−	0.989833i	$-0.545428\pi$
$463$	−1.31215e6	−0.284467	−0.142234	+	0.989833i	$0.545428\pi$
$464$	0	0
$465$	1.51416e6	0.324743
$466$	0	0
$467$	2.77297e6	0.588374	0.294187	−	0.955748i	$-0.404951\pi$
$467$	2.77297e6	0.588374	0.294187	+	0.955748i	$0.404951\pi$
$468$	0	0
$469$	−6.26940e6	−1.31611
$470$	0	0
$471$	−2.02394e6	−0.420383
$472$	0	0
$473$	−553056.	−0.113662
$474$	0	0
$475$	2.94155e6	0.598193
$476$	0	0
$477$	−415530.	−0.0836193
$478$	0	0
$479$	−7.07737e6	−1.40940	−0.704698	−	0.709507i	$-0.748918\pi$
$479$	−7.07737e6	−1.40940	−0.704698	+	0.709507i	$0.748918\pi$
$480$	0	0
$481$	−875758.	−0.172592
$482$	0	0
$483$	−8.01965e6	−1.56418
$484$	0	0
$485$	1.75426e6	0.338640
$486$	0	0
$487$	−1.60503e6	−0.306662	−0.153331	−	0.988175i	$-0.549000\pi$
$487$	−1.60503e6	−0.306662	−0.153331	+	0.988175i	$0.549000\pi$
$488$	0	0
$489$	−5.81027e6	−1.09881
$490$	0	0
$491$	5.13335e6	0.960942	0.480471	−	0.877011i	$-0.340466\pi$
$491$	5.13335e6	0.960942	0.480471	+	0.877011i	$0.340466\pi$
$492$	0	0
$493$	1.42370e7	2.63817
$494$	0	0
$495$	−46656.0	−0.00855844
$496$	0	0
$497$	6.82298e6	1.23903
$498$	0	0
$499$	9.61136e6	1.72796	0.863980	−	0.503527i	$-0.167964\pi$
$499$	9.61136e6	1.72796	0.863980	+	0.503527i	$0.167964\pi$
$500$	0	0
$501$	5.44838e6	0.969780
$502$	0	0
$503$	−2.16924e6	−0.382285	−0.191143	−	0.981562i	$-0.561219\pi$
$503$	−2.16924e6	−0.382285	−0.191143	+	0.981562i	$0.561219\pi$
$504$	0	0
$505$	1.24690e6	0.217571
$506$	0	0
$507$	−257049.	−0.0444116
$508$	0	0
$509$	8.20505e6	1.40374	0.701870	−	0.712305i	$-0.252349\pi$
$509$	8.20505e6	1.40374	0.701870	+	0.712305i	$0.252349\pi$
$510$	0	0
$511$	−6.38268e6	−1.08131
$512$	0	0
$513$	841266.	0.141137
$514$	0	0
$515$	−2.61811e6	−0.434981
$516$	0	0
$517$	659520.	0.108518
$518$	0	0
$519$	5.01730e6	0.817621
$520$	0	0
$521$	−8.54743e6	−1.37956	−0.689781	−	0.724018i	$-0.742293\pi$
$521$	−8.54743e6	−1.37956	−0.689781	+	0.724018i	$0.742293\pi$
$522$	0	0
$523$	1.04792e7	1.67523	0.837613	−	0.546265i	$-0.183951\pi$
$523$	1.04792e7	1.67523	0.837613	+	0.546265i	$0.183951\pi$
$524$	0	0
$525$	5.45996e6	0.864552
$526$	0	0
$527$	1.58566e7	2.48705
$528$	0	0
$529$	7.58119e6	1.17787
$530$	0	0
$531$	−823284.	−0.126711
$532$	0	0
$533$	−1.56359e6	−0.238399
$534$	0	0
$535$	−976608.	−0.147515
$536$	0	0
$537$	−3.64554e6	−0.545539
$538$	0	0
$539$	−956088.	−0.141751
$540$	0	0
$541$	3.85434e6	0.566183	0.283092	−	0.959093i	$-0.408640\pi$
$541$	3.85434e6	0.566183	0.283092	+	0.959093i	$0.408640\pi$
$542$	0	0
$543$	4.22883e6	0.615490
$544$	0	0
$545$	82128.0	0.0118440
$546$	0	0
$547$	9.92056e6	1.41765	0.708823	−	0.705386i	$-0.249226\pi$
$547$	9.92056e6	1.41765	0.708823	+	0.705386i	$0.249226\pi$
$548$	0	0
$549$	3.03669e6	0.430001
$550$	0	0
$551$	7.26328e6	1.01919
$552$	0	0
$553$	−6.22798e6	−0.866033
$554$	0	0
$555$	1.11931e6	0.154248
$556$	0	0
$557$	3.51577e6	0.480156	0.240078	−	0.970754i	$-0.422827\pi$
$557$	3.51577e6	0.480156	0.240078	+	0.970754i	$0.422827\pi$
$558$	0	0
$559$	3.89444e6	0.527127
$560$	0	0
$561$	−488592.	−0.0655449
$562$	0	0
$563$	−8.86364e6	−1.17853	−0.589266	−	0.807939i	$-0.700583\pi$
$563$	−8.86364e6	−1.17853	−0.589266	+	0.807939i	$0.700583\pi$
$564$	0	0
$565$	−301680.	−0.0397581
$566$	0	0
$567$	1.56152e6	0.203981
$568$	0	0
$569$	1.07245e6	0.138866	0.0694328	−	0.997587i	$-0.477881\pi$
$569$	1.07245e6	0.138866	0.0694328	+	0.997587i	$0.477881\pi$
$570$	0	0
$571$	−9.74492e6	−1.25080	−0.625400	−	0.780304i	$-0.715064\pi$
$571$	−9.74492e6	−1.25080	−0.625400	+	0.780304i	$0.715064\pi$
$572$	0	0
$573$	−5.85144e6	−0.744520
$574$	0	0
$575$	−9.54346e6	−1.20375
$576$	0	0
$577$	−8.89632e6	−1.11243	−0.556213	−	0.831040i	$-0.687746\pi$
$577$	−8.89632e6	−1.11243	−0.556213	+	0.831040i	$0.687746\pi$
$578$	0	0
$579$	2.08309e6	0.258232
$580$	0	0
$581$	3.16730e6	0.389269
$582$	0	0
$583$	123120.	0.0150023
$584$	0	0
$585$	328536.	0.0396911
$586$	0	0
$587$	−2796.00	−0.000334921 0	−0.000167460	−	1.00000i	$-0.500053\pi$
$587$	−2796.00	−0.000334921 0	−0.000167460		1.00000i	$0.500053\pi$
$588$	0	0
$589$	8.08954e6	0.960806
$590$	0	0
$591$	4.13251e6	0.486682
$592$	0	0
$593$	5.54911e6	0.648018	0.324009	−	0.946054i	$-0.394969\pi$
$593$	5.54911e6	0.648018	0.324009	+	0.946054i	$0.394969\pi$
$594$	0	0
$595$	−1.29205e7	−1.49620
$596$	0	0
$597$	−1.15974e6	−0.133176
$598$	0	0
$599$	1.41044e7	1.60616	0.803079	−	0.595873i	$-0.203194\pi$
$599$	1.41044e7	1.60616	0.803079	+	0.595873i	$0.203194\pi$
$600$	0	0
$601$	−7.39572e6	−0.835207	−0.417604	−	0.908629i	$-0.637130\pi$
$601$	−7.39572e6	−0.835207	−0.417604	+	0.908629i	$0.637130\pi$
$602$	0	0
$603$	−2.13370e6	−0.238968
$604$	0	0
$605$	−3.85140e6	−0.427790
$606$	0	0
$607$	−2.44689e6	−0.269552	−0.134776	−	0.990876i	$-0.543032\pi$
$607$	−2.44689e6	−0.269552	−0.134776	+	0.990876i	$0.543032\pi$
$608$	0	0
$609$	1.34817e7	1.47300
$610$	0	0
$611$	−4.64412e6	−0.503269
$612$	0	0
$613$	−3.70159e6	−0.397866	−0.198933	−	0.980013i	$-0.563748\pi$
$613$	−3.70159e6	−0.397866	−0.198933	+	0.980013i	$0.563748\pi$
$614$	0	0
$615$	1.99843e6	0.213060
$616$	0	0
$617$	−1.55469e7	−1.64411	−0.822054	−	0.569410i	$-0.807172\pi$
$617$	−1.55469e7	−1.64411	−0.822054	+	0.569410i	$0.807172\pi$
$618$	0	0
$619$	1.66849e7	1.75023	0.875117	−	0.483911i	$-0.160784\pi$
$619$	1.66849e7	1.75023	0.875117	+	0.483911i	$0.160784\pi$
$620$	0	0
$621$	−2.72938e6	−0.284010
$622$	0	0
$623$	−1.14868e7	−1.18571
$624$	0	0
$625$	4.69740e6	0.481014
$626$	0	0
$627$	−249264.	−0.0253216
$628$	0	0
$629$	1.17217e7	1.18131
$630$	0	0
$631$	−1.14295e7	−1.14275	−0.571377	−	0.820688i	$-0.693590\pi$
$631$	−1.14295e7	−1.14275	−0.571377	+	0.820688i	$0.693590\pi$
$632$	0	0
$633$	−6.07914e6	−0.603022
$634$	0	0
$635$	−6.95914e6	−0.684890
$636$	0	0
$637$	6.73245e6	0.657393
$638$	0	0
$639$	2.32211e6	0.224973
$640$	0	0
$641$	−7.16326e6	−0.688598	−0.344299	−	0.938860i	$-0.611883\pi$
$641$	−7.16326e6	−0.688598	−0.344299	+	0.938860i	$0.611883\pi$
$642$	0	0
$643$	−1.12894e7	−1.07682	−0.538409	−	0.842684i	$-0.680974\pi$
$643$	−1.12894e7	−1.07682	−0.538409	+	0.842684i	$0.680974\pi$
$644$	0	0
$645$	−4.97750e6	−0.471099
$646$	0	0
$647$	−9.36372e6	−0.879403	−0.439701	−	0.898144i	$-0.644916\pi$
$647$	−9.36372e6	−0.879403	−0.439701	+	0.898144i	$0.644916\pi$
$648$	0	0
$649$	243936.	0.0227334
$650$	0	0
$651$	1.50154e7	1.38863
$652$	0	0
$653$	−4.00520e6	−0.367571	−0.183785	−	0.982966i	$-0.558835\pi$
$653$	−4.00520e6	−0.367571	−0.183785	+	0.982966i	$0.558835\pi$
$654$	0	0
$655$	2.43504e6	0.221770
$656$	0	0
$657$	−2.17226e6	−0.196335
$658$	0	0
$659$	−1.36299e7	−1.22258	−0.611290	−	0.791406i	$-0.709349\pi$
$659$	−1.36299e7	−1.22258	−0.611290	+	0.791406i	$0.709349\pi$
$660$	0	0
$661$	1.85919e7	1.65509	0.827544	−	0.561401i	$-0.189737\pi$
$661$	1.85919e7	1.65509	0.827544	+	0.561401i	$0.189737\pi$
$662$	0	0
$663$	3.44050e6	0.303975
$664$	0	0
$665$	−6.59165e6	−0.578016
$666$	0	0
$667$	−2.35647e7	−2.05092
$668$	0	0
$669$	1.54852e6	0.133768
$670$	0	0
$671$	−899760.	−0.0771472
$672$	0	0
$673$	6.14745e6	0.523187	0.261594	−	0.965178i	$-0.415752\pi$
$673$	6.14745e6	0.523187	0.261594	+	0.965178i	$0.415752\pi$
$674$	0	0
$675$	1.85822e6	0.156978
$676$	0	0
$677$	974550.	0.0817208	0.0408604	−	0.999165i	$-0.486990\pi$
$677$	974550.	0.0817208	0.0408604	+	0.999165i	$0.486990\pi$
$678$	0	0
$679$	1.73964e7	1.44805
$680$	0	0
$681$	1.41782e6	0.117153
$682$	0	0
$683$	−1.38486e7	−1.13593	−0.567967	−	0.823051i	$-0.692270\pi$
$683$	−1.38486e7	−1.13593	−0.567967	+	0.823051i	$0.692270\pi$
$684$	0	0
$685$	−2.28182e6	−0.185804
$686$	0	0
$687$	4.55099e6	0.367887
$688$	0	0
$689$	−866970.	−0.0695754
$690$	0	0
$691$	6.17139e6	0.491686	0.245843	−	0.969310i	$-0.420935\pi$
$691$	6.17139e6	0.491686	0.245843	+	0.969310i	$0.420935\pi$
$692$	0	0
$693$	−462672.	−0.0365965
$694$	0	0
$695$	−3.20909e6	−0.252011
$696$	0	0
$697$	2.09280e7	1.63172
$698$	0	0
$699$	−7.93201e6	−0.614031
$700$	0	0
$701$	−2.18574e7	−1.67998	−0.839989	−	0.542603i	$-0.817439\pi$
$701$	−2.18574e7	−1.67998	−0.839989	+	0.542603i	$0.817439\pi$
$702$	0	0
$703$	5.98003e6	0.456368
$704$	0	0
$705$	5.93568e6	0.449777
$706$	0	0
$707$	1.23651e7	0.930352
$708$	0	0
$709$	7.27281e6	0.543358	0.271679	−	0.962388i	$-0.412421\pi$
$709$	7.27281e6	0.543358	0.271679	+	0.962388i	$0.412421\pi$
$710$	0	0
$711$	−2.11961e6	−0.157247
$712$	0	0
$713$	−2.62454e7	−1.93344
$714$	0	0
$715$	−97344.0	−0.00712105
$716$	0	0
$717$	−6.50668e6	−0.472674
$718$	0	0
$719$	1.08042e7	0.779420	0.389710	−	0.920938i	$-0.372575\pi$
$719$	1.08042e7	0.779420	0.389710	+	0.920938i	$0.372575\pi$
$720$	0	0
$721$	−2.59629e7	−1.86001
$722$	0	0
$723$	8.29492e6	0.590156
$724$	0	0
$725$	1.60434e7	1.13358
$726$	0	0
$727$	−5.28192e6	−0.370643	−0.185321	−	0.982678i	$-0.559333\pi$
$727$	−5.28192e6	−0.370643	−0.185321	+	0.982678i	$0.559333\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.21255e7	−3.60792
$732$	0	0
$733$	−1.76844e7	−1.21571	−0.607854	−	0.794049i	$-0.707970\pi$
$733$	−1.76844e7	−1.21571	−0.607854	+	0.794049i	$0.707970\pi$
$734$	0	0
$735$	−8.60479e6	−0.587519
$736$	0	0
$737$	632208.	0.0428737
$738$	0	0
$739$	4.86907e6	0.327971	0.163985	−	0.986463i	$-0.447565\pi$
$739$	4.86907e6	0.327971	0.163985	+	0.986463i	$0.447565\pi$
$740$	0	0
$741$	1.75523e6	0.117433
$742$	0	0
$743$	−2.73229e6	−0.181575	−0.0907873	−	0.995870i	$-0.528938\pi$
$743$	−2.73229e6	−0.181575	−0.0907873	+	0.995870i	$0.528938\pi$
$744$	0	0
$745$	119520.	0.00788951
$746$	0	0
$747$	1.07795e6	0.0706800
$748$	0	0
$749$	−9.68470e6	−0.630785
$750$	0	0
$751$	1.83506e6	0.118727	0.0593635	−	0.998236i	$-0.481093\pi$
$751$	1.83506e6	0.118727	0.0593635	+	0.998236i	$0.481093\pi$
$752$	0	0
$753$	8.62412e6	0.554278
$754$	0	0
$755$	2.67288e6	0.170652
$756$	0	0
$757$	−2.61341e7	−1.65756	−0.828778	−	0.559577i	$-0.810963\pi$
$757$	−2.61341e7	−1.65756	−0.828778	+	0.559577i	$0.810963\pi$
$758$	0	0
$759$	808704.	0.0509548
$760$	0	0
$761$	−2.81858e7	−1.76429	−0.882143	−	0.470981i	$-0.843900\pi$
$761$	−2.81858e7	−1.76429	−0.882143	+	0.470981i	$0.843900\pi$
$762$	0	0
$763$	814436.	0.0506461
$764$	0	0
$765$	−4.39733e6	−0.271666
$766$	0	0
$767$	−1.71772e6	−0.105430
$768$	0	0
$769$	1.14050e7	0.695470	0.347735	−	0.937593i	$-0.386951\pi$
$769$	1.14050e7	0.695470	0.347735	+	0.937593i	$0.386951\pi$
$770$	0	0
$771$	7.57852e6	0.459144
$772$	0	0
$773$	2.22191e6	0.133745	0.0668725	−	0.997762i	$-0.478698\pi$
$773$	2.22191e6	0.133745	0.0668725	+	0.997762i	$0.478698\pi$
$774$	0	0
$775$	1.78685e7	1.06865
$776$	0	0
$777$	1.10998e7	0.659575
$778$	0	0
$779$	1.06768e7	0.630373
$780$	0	0
$781$	−688032.	−0.0403628
$782$	0	0
$783$	4.58833e6	0.267454
$784$	0	0
$785$	5.39717e6	0.312602
$786$	0	0
$787$	3.98114e6	0.229124	0.114562	−	0.993416i	$-0.463454\pi$
$787$	3.98114e6	0.229124	0.114562	+	0.993416i	$0.463454\pi$
$788$	0	0
$789$	−5.76439e6	−0.329656
$790$	0	0
$791$	−2.99166e6	−0.170009
$792$	0	0
$793$	6.33581e6	0.357783
$794$	0	0
$795$	1.10808e6	0.0621804
$796$	0	0
$797$	2.20126e7	1.22751	0.613755	−	0.789496i	$-0.289658\pi$
$797$	2.20126e7	1.22751	0.613755	+	0.789496i	$0.289658\pi$
$798$	0	0
$799$	6.21598e7	3.44463
$800$	0	0
$801$	−3.90938e6	−0.215292
$802$	0	0
$803$	643632.	0.0352248
$804$	0	0
$805$	2.13857e7	1.16315
$806$	0	0
$807$	7.49893e6	0.405336
$808$	0	0
$809$	−2.68804e7	−1.44399	−0.721995	−	0.691898i	$-0.756775\pi$
$809$	−2.68804e7	−1.44399	−0.721995	+	0.691898i	$0.756775\pi$
$810$	0	0
$811$	−8.80164e6	−0.469907	−0.234953	−	0.972007i	$-0.575494\pi$
$811$	−8.80164e6	−0.469907	−0.234953	+	0.972007i	$0.575494\pi$
$812$	0	0
$813$	−960966.	−0.0509896
$814$	0	0
$815$	1.54941e7	0.817093
$816$	0	0
$817$	−2.65928e7	−1.39383
$818$	0	0
$819$	3.25798e6	0.169722
$820$	0	0
$821$	1.69316e7	0.876677	0.438338	−	0.898810i	$-0.355567\pi$
$821$	1.69316e7	0.876677	0.438338	+	0.898810i	$0.355567\pi$
$822$	0	0
$823$	−1.05941e7	−0.545210	−0.272605	−	0.962126i	$-0.587885\pi$
$823$	−1.05941e7	−0.545210	−0.272605	+	0.962126i	$0.587885\pi$
$824$	0	0
$825$	−550584.	−0.0281636
$826$	0	0
$827$	2.54154e7	1.29221	0.646104	−	0.763249i	$-0.276397\pi$
$827$	2.54154e7	1.29221	0.646104	+	0.763249i	$0.276397\pi$
$828$	0	0
$829$	1.92970e7	0.975222	0.487611	−	0.873061i	$-0.337868\pi$
$829$	1.92970e7	0.975222	0.487611	+	0.873061i	$0.337868\pi$
$830$	0	0
$831$	−2.00901e7	−1.00920
$832$	0	0
$833$	−9.01113e7	−4.49953
$834$	0	0
$835$	−1.45290e7	−0.721141
$836$	0	0
$837$	5.11029e6	0.252134
$838$	0	0
$839$	−2.67637e7	−1.31262	−0.656312	−	0.754489i	$-0.727885\pi$
$839$	−2.67637e7	−1.31262	−0.656312	+	0.754489i	$0.727885\pi$
$840$	0	0
$841$	1.91033e7	0.931361
$842$	0	0
$843$	73764.0	0.00357500
$844$	0	0
$845$	685464.	0.0330250
$846$	0	0
$847$	−3.81930e7	−1.82926
$848$	0	0
$849$	−3.38962e6	−0.161392
$850$	0	0
$851$	−1.94014e7	−0.918352
$852$	0	0
$853$	9.51435e6	0.447720	0.223860	−	0.974621i	$-0.428134\pi$
$853$	9.51435e6	0.447720	0.223860	+	0.974621i	$0.428134\pi$
$854$	0	0
$855$	−2.24338e6	−0.104951
$856$	0	0
$857$	2.73757e7	1.27325	0.636623	−	0.771175i	$-0.280331\pi$
$857$	2.73757e7	1.27325	0.636623	+	0.771175i	$0.280331\pi$
$858$	0	0
$859$	−1.05377e7	−0.487261	−0.243631	−	0.969868i	$-0.578338\pi$
$859$	−1.05377e7	−0.487261	−0.243631	+	0.969868i	$0.578338\pi$
$860$	0	0
$861$	1.98178e7	0.911060
$862$	0	0
$863$	−3.82773e7	−1.74950	−0.874750	−	0.484574i	$-0.838975\pi$
$863$	−3.82773e7	−1.74950	−0.874750	+	0.484574i	$0.838975\pi$
$864$	0	0
$865$	−1.33795e7	−0.607993
$866$	0	0
$867$	−3.32711e7	−1.50321
$868$	0	0
$869$	628032.	0.0282119
$870$	0	0
$871$	−4.45180e6	−0.198834
$872$	0	0
$873$	5.92061e6	0.262925
$874$	0	0
$875$	−3.24099e7	−1.43106
$876$	0	0
$877$	−3.91309e7	−1.71799	−0.858995	−	0.511983i	$-0.828911\pi$
$877$	−3.91309e7	−1.71799	−0.858995	+	0.511983i	$0.828911\pi$
$878$	0	0
$879$	1.82307e7	0.795851
$880$	0	0
$881$	−1.74541e7	−0.757633	−0.378816	−	0.925472i	$-0.623669\pi$
$881$	−1.74541e7	−0.757633	−0.378816	+	0.925472i	$0.623669\pi$
$882$	0	0
$883$	7.42293e6	0.320386	0.160193	−	0.987086i	$-0.448788\pi$
$883$	7.42293e6	0.320386	0.160193	+	0.987086i	$0.448788\pi$
$884$	0	0
$885$	2.19542e6	0.0942237
$886$	0	0
$887$	−1.98525e7	−0.847239	−0.423619	−	0.905840i	$-0.639241\pi$
$887$	−1.98525e7	−0.847239	−0.423619	+	0.905840i	$0.639241\pi$
$888$	0	0
$889$	−6.90114e7	−2.92864
$890$	0	0
$891$	−157464.	−0.00664488
$892$	0	0
$893$	3.17119e7	1.33074
$894$	0	0
$895$	9.72144e6	0.405670
$896$	0	0
$897$	−5.69462e6	−0.236311
$898$	0	0
$899$	4.41209e7	1.82073
$900$	0	0
$901$	1.16041e7	0.476209
$902$	0	0
$903$	−4.93602e7	−2.01446
$904$	0	0
$905$	−1.12769e7	−0.457686
$906$	0	0
$907$	1.80284e7	0.727678	0.363839	−	0.931462i	$-0.381466\pi$
$907$	1.80284e7	0.727678	0.363839	+	0.931462i	$0.381466\pi$
$908$	0	0
$909$	4.20827e6	0.168925
$910$	0	0
$911$	1.16246e7	0.464070	0.232035	−	0.972707i	$-0.425462\pi$
$911$	1.16246e7	0.464070	0.232035	+	0.972707i	$0.425462\pi$
$912$	0	0
$913$	−319392.	−0.0126808
$914$	0	0
$915$	−8.09784e6	−0.319754
$916$	0	0
$917$	2.41475e7	0.948306
$918$	0	0
$919$	−2.19382e7	−0.856866	−0.428433	−	0.903574i	$-0.640934\pi$
$919$	−2.19382e7	−0.856866	−0.428433	+	0.903574i	$0.640934\pi$
$920$	0	0
$921$	1.24351e7	0.483061
$922$	0	0
$923$	4.84489e6	0.187189
$924$	0	0
$925$	1.32089e7	0.507590
$926$	0	0
$927$	−8.83613e6	−0.337725
$928$	0	0
$929$	−2.92599e7	−1.11233	−0.556165	−	0.831072i	$-0.687728\pi$
$929$	−2.92599e7	−1.11233	−0.556165	+	0.831072i	$0.687728\pi$
$930$	0	0
$931$	−4.59719e7	−1.73827
$932$	0	0
$933$	2.34060e7	0.880284
$934$	0	0
$935$	1.30291e6	0.0487401
$936$	0	0
$937$	2.31218e7	0.860343	0.430172	−	0.902747i	$-0.358453\pi$
$937$	2.31218e7	0.860343	0.430172	+	0.902747i	$0.358453\pi$
$938$	0	0
$939$	−1.30920e7	−0.484554
$940$	0	0
$941$	−1.57256e7	−0.578940	−0.289470	−	0.957187i	$-0.593479\pi$
$941$	−1.57256e7	−0.578940	−0.289470	+	0.957187i	$0.593479\pi$
$942$	0	0
$943$	−3.46395e7	−1.26851
$944$	0	0
$945$	−4.16405e6	−0.151683
$946$	0	0
$947$	−1.57405e7	−0.570352	−0.285176	−	0.958475i	$-0.592052\pi$
$947$	−1.57405e7	−0.570352	−0.285176	+	0.958475i	$0.592052\pi$
$948$	0	0
$949$	−4.53224e6	−0.163361
$950$	0	0
$951$	1.04674e7	0.375306
$952$	0	0
$953$	312414.	0.0111429	0.00557145	−	0.999984i	$-0.498227\pi$
$953$	312414.	0.0111429	0.00557145	+	0.999984i	$0.498227\pi$
$954$	0	0
$955$	1.56038e7	0.553634
$956$	0	0
$957$	−1.35950e6	−0.0479845
$958$	0	0
$959$	−2.26281e7	−0.794514
$960$	0	0
$961$	2.05109e7	0.716436
$962$	0	0
$963$	−3.29605e6	−0.114532
$964$	0	0
$965$	−5.55490e6	−0.192025
$966$	0	0
$967$	−3.14651e7	−1.08209	−0.541045	−	0.840994i	$-0.681971\pi$
$967$	−3.14651e7	−1.08209	−0.541045	+	0.840994i	$0.681971\pi$
$968$	0	0
$969$	−2.34931e7	−0.803769
$970$	0	0
$971$	5.61172e7	1.91006	0.955032	−	0.296504i	$-0.0958209\pi$
$971$	5.61172e7	1.91006	0.955032	+	0.296504i	$0.0958209\pi$
$972$	0	0
$973$	−3.18235e7	−1.07762
$974$	0	0
$975$	3.87703e6	0.130613
$976$	0	0
$977$	4.01018e7	1.34409	0.672044	−	0.740511i	$-0.265416\pi$
$977$	4.01018e7	1.34409	0.672044	+	0.740511i	$0.265416\pi$
$978$	0	0
$979$	1.15834e6	0.0386258
$980$	0	0
$981$	277182.	0.00919586
$982$	0	0
$983$	3.82807e7	1.26356	0.631781	−	0.775147i	$-0.282324\pi$
$983$	3.82807e7	1.26356	0.631781	+	0.775147i	$0.282324\pi$
$984$	0	0
$985$	−1.10200e7	−0.361903
$986$	0	0
$987$	5.88622e7	1.92328
$988$	0	0
$989$	8.62767e7	2.80481
$990$	0	0
$991$	1.87594e6	0.0606785	0.0303392	−	0.999540i	$-0.490341\pi$
$991$	1.87594e6	0.0606785	0.0303392	+	0.999540i	$0.490341\pi$
$992$	0	0
$993$	6.18399e6	0.199020
$994$	0	0
$995$	3.09264e6	0.0990311
$996$	0	0
$997$	1.34521e7	0.428601	0.214301	−	0.976768i	$-0.431253\pi$
$997$	1.34521e7	0.428601	0.214301	+	0.976768i	$0.431253\pi$
$998$	0	0
$999$	3.77768e6	0.119760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.c.1.1		1
4.3	odd	2		78.6.a.b.1.1	✓	1
12.11	even	2		234.6.a.e.1.1		1
52.51	odd	2		1014.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.a.b.1.1	✓	1	4.3	odd	2
234.6.a.e.1.1		1	12.11	even	2
624.6.a.c.1.1		1	1.1	even	1	trivial
1014.6.a.g.1.1		1	52.51	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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