LMFDB - Embedded newform 624.6.a.m.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:	$2$
Coefficient field:	$\Q(\sqrt{209}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 52 $ x^2 - x - 52
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.72842$ of defining polynomial
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)

$f(q)$	$=$	$q+9.00000 q^{3} -28.9137 q^{5} -71.6547 q^{7} +81.0000 q^{9} +272.050 q^{11} +169.000 q^{13} -260.223 q^{15} +71.9280 q^{17} +93.2373 q^{19} -644.892 q^{21} -3826.37 q^{23} -2289.00 q^{25} +729.000 q^{27} +5613.12 q^{29} +2364.01 q^{31} +2448.45 q^{33} +2071.80 q^{35} +4336.98 q^{37} +1521.00 q^{39} +1938.57 q^{41} +9140.53 q^{43} -2342.01 q^{45} -5464.05 q^{47} -11672.6 q^{49} +647.352 q^{51} -33206.9 q^{53} -7865.97 q^{55} +839.135 q^{57} +27394.4 q^{59} -28746.5 q^{61} -5804.03 q^{63} -4886.41 q^{65} +25822.3 q^{67} -34437.4 q^{69} -32750.7 q^{71} -27763.9 q^{73} -20601.0 q^{75} -19493.7 q^{77} +32269.8 q^{79} +6561.00 q^{81} -51837.1 q^{83} -2079.70 q^{85} +50518.1 q^{87} -70891.9 q^{89} -12109.6 q^{91} +21276.1 q^{93} -2695.83 q^{95} -126615. q^{97} +22036.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 88 q^{7} + 162 q^{9} - 92 q^{11} + 338 q^{13} - 1244 q^{17} - 1664 q^{19} + 792 q^{21} - 5224 q^{23} - 4578 q^{25} + 1458 q^{27} + 3940 q^{29} - 4640 q^{31} - 828 q^{33} + 6688 q^{35} - 1388 q^{37}+ \cdots - 7452 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 88 * q^7 + 162 * q^9 - 92 * q^11 + 338 * q^13 - 1244 * q^17 - 1664 * q^19 + 792 * q^21 - 5224 * q^23 - 4578 * q^25 + 1458 * q^27 + 3940 * q^29 - 4640 * q^31 - 828 * q^33 + 6688 * q^35 - 1388 * q^37 + 3042 * q^39 - 12488 * q^41 - 6816 * q^43 + 21860 * q^47 - 2990 * q^49 - 11196 * q^51 - 13444 * q^53 - 18392 * q^55 - 14976 * q^57 - 12580 * q^59 - 41764 * q^61 + 7128 * q^63 + 54536 * q^67 - 47016 * q^69 + 51252 * q^71 - 100980 * q^73 - 41202 * q^75 - 77616 * q^77 + 119360 * q^79 + 13122 * q^81 - 30060 * q^83 - 40128 * q^85 + 35460 * q^87 - 29888 * q^89 + 14872 * q^91 - 41760 * q^93 - 53504 * q^95 - 244788 * q^97 - 7452 * q^99

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
$5$	−28.9137	−0.517223	−0.258612	−	0.965981i	$-0.583265\pi$
$5$	−28.9137	−0.517223	−0.258612	+	0.965981i	$0.583265\pi$
$6$	0	0
$7$	−71.6547	−0.552713	−0.276356	−	0.961055i	$-0.589127\pi$
$7$	−71.6547	−0.552713	−0.276356	+	0.961055i	$0.589127\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	272.050	0.677903	0.338951	−	0.940804i	$-0.389928\pi$
$11$	272.050	0.677903	0.338951	+	0.940804i	$0.389928\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−260.223	−0.298619
$16$	0	0
$17$	71.9280	0.0603636	0.0301818	−	0.999544i	$-0.490391\pi$
$17$	71.9280	0.0603636	0.0301818	+	0.999544i	$0.490391\pi$
$18$	0	0
$19$	93.2373	0.0592523	0.0296262	−	0.999561i	$-0.490568\pi$
$19$	93.2373	0.0592523	0.0296262	+	0.999561i	$0.490568\pi$
$20$	0	0
$21$	−644.892	−0.319109
$22$	0	0
$23$	−3826.37	−1.50823	−0.754115	−	0.656742i	$-0.771934\pi$
$23$	−3826.37	−1.50823	−0.754115	+	0.656742i	$0.771934\pi$
$24$	0	0
$25$	−2289.00	−0.732480
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	5613.12	1.23939	0.619697	−	0.784841i	$-0.287255\pi$
$29$	5613.12	1.23939	0.619697	+	0.784841i	$0.287255\pi$
$30$	0	0
$31$	2364.01	0.441820	0.220910	−	0.975294i	$-0.429097\pi$
$31$	2364.01	0.441820	0.220910	+	0.975294i	$0.429097\pi$
$32$	0	0
$33$	2448.45	0.391387
$34$	0	0
$35$	2071.80	0.285876
$36$	0	0
$37$	4336.98	0.520814	0.260407	−	0.965499i	$-0.416143\pi$
$37$	4336.98	0.520814	0.260407	+	0.965499i	$0.416143\pi$
$38$	0	0
$39$	1521.00	0.160128
$40$	0	0
$41$	1938.57	0.180103	0.0900516	−	0.995937i	$-0.471297\pi$
$41$	1938.57	0.180103	0.0900516	+	0.995937i	$0.471297\pi$
$42$	0	0
$43$	9140.53	0.753877	0.376938	−	0.926238i	$-0.376977\pi$
$43$	9140.53	0.753877	0.376938	+	0.926238i	$0.376977\pi$
$44$	0	0
$45$	−2342.01	−0.172408
$46$	0	0
$47$	−5464.05	−0.360803	−0.180401	−	0.983593i	$-0.557740\pi$
$47$	−5464.05	−0.360803	−0.180401	+	0.983593i	$0.557740\pi$
$48$	0	0
$49$	−11672.6	−0.694509
$50$	0	0
$51$	647.352	0.0348510
$52$	0	0
$53$	−33206.9	−1.62382	−0.811912	−	0.583780i	$-0.801573\pi$
$53$	−33206.9	−1.62382	−0.811912	+	0.583780i	$0.801573\pi$
$54$	0	0
$55$	−7865.97	−0.350627
$56$	0	0
$57$	839.135	0.0342094
$58$	0	0
$59$	27394.4	1.02455	0.512274	−	0.858822i	$-0.328803\pi$
$59$	27394.4	1.02455	0.512274	+	0.858822i	$0.328803\pi$
$60$	0	0
$61$	−28746.5	−0.989147	−0.494573	−	0.869136i	$-0.664676\pi$
$61$	−28746.5	−0.989147	−0.494573	+	0.869136i	$0.664676\pi$
$62$	0	0
$63$	−5804.03	−0.184238
$64$	0	0
$65$	−4886.41	−0.143452
$66$	0	0
$67$	25822.3	0.702762	0.351381	−	0.936233i	$-0.385712\pi$
$67$	25822.3	0.702762	0.351381	+	0.936233i	$0.385712\pi$
$68$	0	0
$69$	−34437.4	−0.870777
$70$	0	0
$71$	−32750.7	−0.771036	−0.385518	−	0.922700i	$-0.625977\pi$
$71$	−32750.7	−0.771036	−0.385518	+	0.922700i	$0.625977\pi$
$72$	0	0
$73$	−27763.9	−0.609779	−0.304890	−	0.952388i	$-0.598620\pi$
$73$	−27763.9	−0.609779	−0.304890	+	0.952388i	$0.598620\pi$
$74$	0	0
$75$	−20601.0	−0.422898
$76$	0	0
$77$	−19493.7	−0.374685
$78$	0	0
$79$	32269.8	0.581740	0.290870	−	0.956763i	$-0.406055\pi$
$79$	32269.8	0.581740	0.290870	+	0.956763i	$0.406055\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−51837.1	−0.825934	−0.412967	−	0.910746i	$-0.635508\pi$
$83$	−51837.1	−0.825934	−0.412967	+	0.910746i	$0.635508\pi$
$84$	0	0
$85$	−2079.70	−0.0312215
$86$	0	0
$87$	50518.1	0.715565
$88$	0	0
$89$	−70891.9	−0.948685	−0.474342	−	0.880340i	$-0.657314\pi$
$89$	−70891.9	−0.948685	−0.474342	+	0.880340i	$0.657314\pi$
$90$	0	0
$91$	−12109.6	−0.153295
$92$	0	0
$93$	21276.1	0.255085
$94$	0	0
$95$	−2695.83	−0.0306467
$96$	0	0
$97$	−126615.	−1.36633	−0.683167	−	0.730262i	$-0.739398\pi$
$97$	−126615.	−1.36633	−0.683167	+	0.730262i	$0.739398\pi$
$98$	0	0
$99$	22036.1	0.225968
$100$	0	0
$101$	−188761.	−1.84124	−0.920619	−	0.390463i	$-0.872315\pi$
$101$	−188761.	−1.84124	−0.920619	+	0.390463i	$0.872315\pi$
$102$	0	0
$103$	−127344.	−1.18273	−0.591367	−	0.806403i	$-0.701412\pi$
$103$	−127344.	−1.18273	−0.591367	+	0.806403i	$0.701412\pi$
$104$	0	0
$105$	18646.2	0.165050
$106$	0	0
$107$	141201.	1.19228	0.596140	−	0.802881i	$-0.296700\pi$
$107$	141201.	1.19228	0.596140	+	0.802881i	$0.296700\pi$
$108$	0	0
$109$	−136367.	−1.09937	−0.549684	−	0.835372i	$-0.685252\pi$
$109$	−136367.	−1.09937	−0.549684	+	0.835372i	$0.685252\pi$
$110$	0	0
$111$	39032.8	0.300692
$112$	0	0
$113$	108128.	0.796603	0.398301	−	0.917255i	$-0.369600\pi$
$113$	108128.	0.796603	0.398301	+	0.917255i	$0.369600\pi$
$114$	0	0
$115$	110634.	0.780092
$116$	0	0
$117$	13689.0	0.0924500
$118$	0	0
$119$	−5153.97	−0.0333637
$120$	0	0
$121$	−87039.6	−0.540448
$122$	0	0
$123$	17447.1	0.103983
$124$	0	0
$125$	156539.	0.896079
$126$	0	0
$127$	53050.0	0.291861	0.145930	−	0.989295i	$-0.453382\pi$
$127$	53050.0	0.291861	0.145930	+	0.989295i	$0.453382\pi$
$128$	0	0
$129$	82264.8	0.435251
$130$	0	0
$131$	32842.9	0.167210	0.0836051	−	0.996499i	$-0.473357\pi$
$131$	32842.9	0.167210	0.0836051	+	0.996499i	$0.473357\pi$
$132$	0	0
$133$	−6680.88	−0.0327495
$134$	0	0
$135$	−21078.1	−0.0995397
$136$	0	0
$137$	−55848.9	−0.254222	−0.127111	−	0.991888i	$-0.540570\pi$
$137$	−55848.9	−0.254222	−0.127111	+	0.991888i	$0.540570\pi$
$138$	0	0
$139$	−30745.8	−0.134974	−0.0674869	−	0.997720i	$-0.521498\pi$
$139$	−30745.8	−0.134974	−0.0674869	+	0.997720i	$0.521498\pi$
$140$	0	0
$141$	−49176.4	−0.208310
$142$	0	0
$143$	45976.5	0.188016
$144$	0	0
$145$	−162296.	−0.641044
$146$	0	0
$147$	−105053.	−0.400975
$148$	0	0
$149$	−273442.	−1.00902	−0.504510	−	0.863406i	$-0.668327\pi$
$149$	−273442.	−1.00902	−0.504510	+	0.863406i	$0.668327\pi$
$150$	0	0
$151$	−200195.	−0.714515	−0.357258	−	0.934006i	$-0.616288\pi$
$151$	−200195.	−0.714515	−0.357258	+	0.934006i	$0.616288\pi$
$152$	0	0
$153$	5826.16	0.0201212
$154$	0	0
$155$	−68352.3	−0.228520
$156$	0	0
$157$	−82074.0	−0.265740	−0.132870	−	0.991133i	$-0.542419\pi$
$157$	−82074.0	−0.265740	−0.132870	+	0.991133i	$0.542419\pi$
$158$	0	0
$159$	−298862.	−0.937515
$160$	0	0
$161$	274178.	0.833618
$162$	0	0
$163$	−38858.0	−0.114554	−0.0572771	−	0.998358i	$-0.518242\pi$
$163$	−38858.0	−0.114554	−0.0572771	+	0.998358i	$0.518242\pi$
$164$	0	0
$165$	−70793.7	−0.202435
$166$	0	0
$167$	−374616.	−1.03943	−0.519716	−	0.854339i	$-0.673962\pi$
$167$	−374616.	−1.03943	−0.519716	+	0.854339i	$0.673962\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	7552.22	0.0197508
$172$	0	0
$173$	−15493.7	−0.0393586	−0.0196793	−	0.999806i	$-0.506265\pi$
$173$	−15493.7	−0.0393586	−0.0196793	+	0.999806i	$0.506265\pi$
$174$	0	0
$175$	164018.	0.404851
$176$	0	0
$177$	246550.	0.591523
$178$	0	0
$179$	97528.5	0.227509	0.113755	−	0.993509i	$-0.463712\pi$
$179$	97528.5	0.227509	0.113755	+	0.993509i	$0.463712\pi$
$180$	0	0
$181$	−461470.	−1.04700	−0.523501	−	0.852025i	$-0.675374\pi$
$181$	−461470.	−1.04700	−0.523501	+	0.852025i	$0.675374\pi$
$182$	0	0
$183$	−258719.	−0.571084
$184$	0	0
$185$	−125398.	−0.269377
$186$	0	0
$187$	19568.0	0.0409207
$188$	0	0
$189$	−52236.2	−0.106370
$190$	0	0
$191$	−78150.2	−0.155005	−0.0775026	−	0.996992i	$-0.524695\pi$
$191$	−78150.2	−0.155005	−0.0775026	+	0.996992i	$0.524695\pi$
$192$	0	0
$193$	−103100.	−0.199236	−0.0996179	−	0.995026i	$-0.531762\pi$
$193$	−103100.	−0.199236	−0.0996179	+	0.995026i	$0.531762\pi$
$194$	0	0
$195$	−43977.7	−0.0828220
$196$	0	0
$197$	578483.	1.06200	0.531000	−	0.847372i	$-0.321816\pi$
$197$	578483.	1.06200	0.531000	+	0.847372i	$0.321816\pi$
$198$	0	0
$199$	414351.	0.741713	0.370857	−	0.928690i	$-0.379064\pi$
$199$	414351.	0.741713	0.370857	+	0.928690i	$0.379064\pi$
$200$	0	0
$201$	232401.	0.405740
$202$	0	0
$203$	−402206.	−0.685029
$204$	0	0
$205$	−56051.1	−0.0931535
$206$	0	0
$207$	−309936.	−0.502744
$208$	0	0
$209$	25365.2	0.0401673
$210$	0	0
$211$	−941613.	−1.45602	−0.728009	−	0.685568i	$-0.759554\pi$
$211$	−941613.	−1.45602	−0.728009	+	0.685568i	$0.759554\pi$
$212$	0	0
$213$	−294756.	−0.445158
$214$	0	0
$215$	−264286.	−0.389923
$216$	0	0
$217$	−169393.	−0.244200
$218$	0	0
$219$	−249875.	−0.352056
$220$	0	0
$221$	12155.8	0.0167419
$222$	0	0
$223$	369966.	0.498195	0.249097	−	0.968478i	$-0.419866\pi$
$223$	369966.	0.498195	0.249097	+	0.968478i	$0.419866\pi$
$224$	0	0
$225$	−185409.	−0.244160
$226$	0	0
$227$	−362430.	−0.466831	−0.233416	−	0.972377i	$-0.574990\pi$
$227$	−362430.	−0.466831	−0.233416	+	0.972377i	$0.574990\pi$
$228$	0	0
$229$	−939778.	−1.18423	−0.592115	−	0.805853i	$-0.701707\pi$
$229$	−939778.	−1.18423	−0.592115	+	0.805853i	$0.701707\pi$
$230$	0	0
$231$	−175443.	−0.216325
$232$	0	0
$233$	256173.	0.309132	0.154566	−	0.987982i	$-0.450602\pi$
$233$	256173.	0.309132	0.154566	+	0.987982i	$0.450602\pi$
$234$	0	0
$235$	157986.	0.186616
$236$	0	0
$237$	290429.	0.335868
$238$	0	0
$239$	−1.05916e6	−1.19941	−0.599705	−	0.800221i	$-0.704715\pi$
$239$	−1.05916e6	−1.19941	−0.599705	+	0.800221i	$0.704715\pi$
$240$	0	0
$241$	−753699.	−0.835902	−0.417951	−	0.908469i	$-0.637252\pi$
$241$	−753699.	−0.835902	−0.417951	+	0.908469i	$0.637252\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	337498.	0.359216
$246$	0	0
$247$	15757.1	0.0164336
$248$	0	0
$249$	−466534.	−0.476853
$250$	0	0
$251$	94598.6	0.0947765	0.0473882	−	0.998877i	$-0.484910\pi$
$251$	94598.6	0.0947765	0.0473882	+	0.998877i	$0.484910\pi$
$252$	0	0
$253$	−1.04097e6	−1.02243
$254$	0	0
$255$	−18717.3	−0.0180257
$256$	0	0
$257$	−26297.2	−0.0248357	−0.0124178	−	0.999923i	$-0.503953\pi$
$257$	−26297.2	−0.0248357	−0.0124178	+	0.999923i	$0.503953\pi$
$258$	0	0
$259$	−310765.	−0.287861
$260$	0	0
$261$	454663.	0.413131
$262$	0	0
$263$	−1.05604e6	−0.941433	−0.470717	−	0.882284i	$-0.656005\pi$
$263$	−1.05604e6	−0.941433	−0.470717	+	0.882284i	$0.656005\pi$
$264$	0	0
$265$	960134.	0.839880
$266$	0	0
$267$	−638027.	−0.547723
$268$	0	0
$269$	2.02103e6	1.70291	0.851455	−	0.524428i	$-0.175721\pi$
$269$	2.02103e6	1.70291	0.851455	+	0.524428i	$0.175721\pi$
$270$	0	0
$271$	−2.19354e6	−1.81436	−0.907178	−	0.420748i	$-0.861768\pi$
$271$	−2.19354e6	−1.81436	−0.907178	+	0.420748i	$0.861768\pi$
$272$	0	0
$273$	−108987.	−0.0885048
$274$	0	0
$275$	−622723.	−0.496550
$276$	0	0
$277$	1.35211e6	1.05880	0.529399	−	0.848373i	$-0.322418\pi$
$277$	1.35211e6	1.05880	0.529399	+	0.848373i	$0.322418\pi$
$278$	0	0
$279$	191485.	0.147273
$280$	0	0
$281$	111733.	0.0844145	0.0422073	−	0.999109i	$-0.486561\pi$
$281$	111733.	0.0844145	0.0422073	+	0.999109i	$0.486561\pi$
$282$	0	0
$283$	1.94906e6	1.44663	0.723316	−	0.690517i	$-0.242617\pi$
$283$	1.94906e6	1.44663	0.723316	+	0.690517i	$0.242617\pi$
$284$	0	0
$285$	−24262.5	−0.0176939
$286$	0	0
$287$	−138907.	−0.0995452
$288$	0	0
$289$	−1.41468e6	−0.996356
$290$	0	0
$291$	−1.13954e6	−0.788854
$292$	0	0
$293$	−1.05039e6	−0.714798	−0.357399	−	0.933952i	$-0.616336\pi$
$293$	−1.05039e6	−0.714798	−0.357399	+	0.933952i	$0.616336\pi$
$294$	0	0
$295$	−792073.	−0.529920
$296$	0	0
$297$	198325.	0.130462
$298$	0	0
$299$	−646657.	−0.418308
$300$	0	0
$301$	−654962.	−0.416677
$302$	0	0
$303$	−1.69885e6	−1.06304
$304$	0	0
$305$	831167.	0.511610
$306$	0	0
$307$	567029.	0.343367	0.171684	−	0.985152i	$-0.445079\pi$
$307$	567029.	0.343367	0.171684	+	0.985152i	$0.445079\pi$
$308$	0	0
$309$	−1.14610e6	−0.682852
$310$	0	0
$311$	655377.	0.384229	0.192115	−	0.981372i	$-0.438465\pi$
$311$	655377.	0.384229	0.192115	+	0.981372i	$0.438465\pi$
$312$	0	0
$313$	1.86421e6	1.07556	0.537778	−	0.843086i	$-0.319264\pi$
$313$	1.86421e6	1.07556	0.537778	+	0.843086i	$0.319264\pi$
$314$	0	0
$315$	167816.	0.0952919
$316$	0	0
$317$	−1.49838e6	−0.837478	−0.418739	−	0.908107i	$-0.637528\pi$
$317$	−1.49838e6	−0.837478	−0.418739	+	0.908107i	$0.637528\pi$
$318$	0	0
$319$	1.52705e6	0.840189
$320$	0	0
$321$	1.27081e6	0.688363
$322$	0	0
$323$	6706.37	0.00357669
$324$	0	0
$325$	−386841.	−0.203153
$326$	0	0
$327$	−1.22730e6	−0.634721
$328$	0	0
$329$	391524.	0.199420
$330$	0	0
$331$	622230.	0.312163	0.156081	−	0.987744i	$-0.450114\pi$
$331$	622230.	0.312163	0.156081	+	0.987744i	$0.450114\pi$
$332$	0	0
$333$	351295.	0.173605
$334$	0	0
$335$	−746618.	−0.363485
$336$	0	0
$337$	−1.68061e6	−0.806106	−0.403053	−	0.915177i	$-0.632051\pi$
$337$	−1.68061e6	−0.806106	−0.403053	+	0.915177i	$0.632051\pi$
$338$	0	0
$339$	973152.	0.459919
$340$	0	0
$341$	643131.	0.299511
$342$	0	0
$343$	2.04070e6	0.936576
$344$	0	0
$345$	995710.	0.450386
$346$	0	0
$347$	2.17232e6	0.968502	0.484251	−	0.874929i	$-0.339092\pi$
$347$	2.17232e6	0.968502	0.484251	+	0.874929i	$0.339092\pi$
$348$	0	0
$349$	−2.90123e6	−1.27503	−0.637513	−	0.770440i	$-0.720037\pi$
$349$	−2.90123e6	−1.27503	−0.637513	+	0.770440i	$0.720037\pi$
$350$	0	0
$351$	123201.	0.0533761
$352$	0	0
$353$	355946.	0.152036	0.0760182	−	0.997106i	$-0.475779\pi$
$353$	355946.	0.152036	0.0760182	+	0.997106i	$0.475779\pi$
$354$	0	0
$355$	946942.	0.398798
$356$	0	0
$357$	−46385.8	−0.0192626
$358$	0	0
$359$	1.56214e6	0.639710	0.319855	−	0.947467i	$-0.396366\pi$
$359$	1.56214e6	0.639710	0.319855	+	0.947467i	$0.396366\pi$
$360$	0	0
$361$	−2.46741e6	−0.996489
$362$	0	0
$363$	−783357.	−0.312028
$364$	0	0
$365$	802755.	0.315392
$366$	0	0
$367$	2.22068e6	0.860640	0.430320	−	0.902676i	$-0.358401\pi$
$367$	2.22068e6	0.860640	0.430320	+	0.902676i	$0.358401\pi$
$368$	0	0
$369$	157024.	0.0600344
$370$	0	0
$371$	2.37943e6	0.897508
$372$	0	0
$373$	−2.20109e6	−0.819155	−0.409577	−	0.912275i	$-0.634324\pi$
$373$	−2.20109e6	−0.819155	−0.409577	+	0.912275i	$0.634324\pi$
$374$	0	0
$375$	1.40885e6	0.517352
$376$	0	0
$377$	948618.	0.343746
$378$	0	0
$379$	−71429.7	−0.0255435	−0.0127718	−	0.999918i	$-0.504065\pi$
$379$	−71429.7	−0.0255435	−0.0127718	+	0.999918i	$0.504065\pi$
$380$	0	0
$381$	477450.	0.168506
$382$	0	0
$383$	2.57371e6	0.896525	0.448262	−	0.893902i	$-0.352043\pi$
$383$	2.57371e6	0.896525	0.448262	+	0.893902i	$0.352043\pi$
$384$	0	0
$385$	563633.	0.193796
$386$	0	0
$387$	740383.	0.251292
$388$	0	0
$389$	2.26737e6	0.759710	0.379855	−	0.925046i	$-0.375974\pi$
$389$	2.26737e6	0.759710	0.379855	+	0.925046i	$0.375974\pi$
$390$	0	0
$391$	−275223.	−0.0910423
$392$	0	0
$393$	295586.	0.0965389
$394$	0	0
$395$	−933040.	−0.300890
$396$	0	0
$397$	611054.	0.194582	0.0972911	−	0.995256i	$-0.468982\pi$
$397$	611054.	0.194582	0.0972911	+	0.995256i	$0.468982\pi$
$398$	0	0
$399$	−60128.0	−0.0189079
$400$	0	0
$401$	3.38562e6	1.05142	0.525711	−	0.850663i	$-0.323799\pi$
$401$	3.38562e6	1.05142	0.525711	+	0.850663i	$0.323799\pi$
$402$	0	0
$403$	399518.	0.122539
$404$	0	0
$405$	−189703.	−0.0574693
$406$	0	0
$407$	1.17988e6	0.353062
$408$	0	0
$409$	−2.20943e6	−0.653087	−0.326544	−	0.945182i	$-0.605884\pi$
$409$	−2.20943e6	−0.653087	−0.326544	+	0.945182i	$0.605884\pi$
$410$	0	0
$411$	−502640.	−0.146775
$412$	0	0
$413$	−1.96294e6	−0.566280
$414$	0	0
$415$	1.49880e6	0.427192
$416$	0	0
$417$	−276713.	−0.0779272
$418$	0	0
$419$	4.01767e6	1.11799	0.558997	−	0.829170i	$-0.311186\pi$
$419$	4.01767e6	1.11799	0.558997	+	0.829170i	$0.311186\pi$
$420$	0	0
$421$	−2.54149e6	−0.698849	−0.349424	−	0.936965i	$-0.613623\pi$
$421$	−2.54149e6	−0.698849	−0.349424	+	0.936965i	$0.613623\pi$
$422$	0	0
$423$	−442588.	−0.120268
$424$	0	0
$425$	−164643.	−0.0442152
$426$	0	0
$427$	2.05982e6	0.546714
$428$	0	0
$429$	413789.	0.108551
$430$	0	0
$431$	5.33002e6	1.38209	0.691044	−	0.722813i	$-0.257151\pi$
$431$	5.33002e6	1.38209	0.691044	+	0.722813i	$0.257151\pi$
$432$	0	0
$433$	4.82254e6	1.23611	0.618053	−	0.786136i	$-0.287922\pi$
$433$	4.82254e6	1.23611	0.618053	+	0.786136i	$0.287922\pi$
$434$	0	0
$435$	−1.46066e6	−0.370107
$436$	0	0
$437$	−356761.	−0.0893662
$438$	0	0
$439$	1.88724e6	0.467375	0.233688	−	0.972312i	$-0.424921\pi$
$439$	1.88724e6	0.467375	0.233688	+	0.972312i	$0.424921\pi$
$440$	0	0
$441$	−945481.	−0.231503
$442$	0	0
$443$	6.73150e6	1.62968	0.814840	−	0.579686i	$-0.196825\pi$
$443$	6.73150e6	1.62968	0.814840	+	0.579686i	$0.196825\pi$
$444$	0	0
$445$	2.04975e6	0.490682
$446$	0	0
$447$	−2.46098e6	−0.582558
$448$	0	0
$449$	999884.	0.234063	0.117032	−	0.993128i	$-0.462662\pi$
$449$	999884.	0.234063	0.117032	+	0.993128i	$0.462662\pi$
$450$	0	0
$451$	527388.	0.122092
$452$	0	0
$453$	−1.80176e6	−0.412526
$454$	0	0
$455$	350134.	0.0792877
$456$	0	0
$457$	−2.70696e6	−0.606305	−0.303153	−	0.952942i	$-0.598039\pi$
$457$	−2.70696e6	−0.606305	−0.303153	+	0.952942i	$0.598039\pi$
$458$	0	0
$459$	52435.5	0.0116170
$460$	0	0
$461$	6.23493e6	1.36640	0.683202	−	0.730229i	$-0.260587\pi$
$461$	6.23493e6	1.36640	0.683202	+	0.730229i	$0.260587\pi$
$462$	0	0
$463$	7.11687e6	1.54290	0.771448	−	0.636293i	$-0.219533\pi$
$463$	7.11687e6	1.54290	0.771448	+	0.636293i	$0.219533\pi$
$464$	0	0
$465$	−615171.	−0.131936
$466$	0	0
$467$	−4.15699e6	−0.882037	−0.441018	−	0.897498i	$-0.645383\pi$
$467$	−4.15699e6	−0.882037	−0.441018	+	0.897498i	$0.645383\pi$
$468$	0	0
$469$	−1.85029e6	−0.388425
$470$	0	0
$471$	−738666.	−0.153425
$472$	0	0
$473$	2.48668e6	0.511055
$474$	0	0
$475$	−213420.	−0.0434012
$476$	0	0
$477$	−2.68976e6	−0.541275
$478$	0	0
$479$	−4.27365e6	−0.851060	−0.425530	−	0.904944i	$-0.639912\pi$
$479$	−4.27365e6	−0.851060	−0.425530	+	0.904944i	$0.639912\pi$
$480$	0	0
$481$	732949.	0.144448
$482$	0	0
$483$	2.46760e6	0.481290
$484$	0	0
$485$	3.66092e6	0.706700
$486$	0	0
$487$	−7.00352e6	−1.33812	−0.669059	−	0.743210i	$-0.733303\pi$
$487$	−7.00352e6	−1.33812	−0.669059	+	0.743210i	$0.733303\pi$
$488$	0	0
$489$	−349722.	−0.0661379
$490$	0	0
$491$	5.13989e6	0.962167	0.481083	−	0.876675i	$-0.340243\pi$
$491$	5.13989e6	0.962167	0.481083	+	0.876675i	$0.340243\pi$
$492$	0	0
$493$	403740.	0.0748143
$494$	0	0
$495$	−637144.	−0.116876
$496$	0	0
$497$	2.34674e6	0.426161
$498$	0	0
$499$	−9.42361e6	−1.69421	−0.847103	−	0.531429i	$-0.821655\pi$
$499$	−9.42361e6	−1.69421	−0.847103	+	0.531429i	$0.821655\pi$
$500$	0	0
$501$	−3.37155e6	−0.600116
$502$	0	0
$503$	3.77027e6	0.664434	0.332217	−	0.943203i	$-0.392203\pi$
$503$	3.77027e6	0.664434	0.332217	+	0.943203i	$0.392203\pi$
$504$	0	0
$505$	5.45778e6	0.952331
$506$	0	0
$507$	257049.	0.0444116
$508$	0	0
$509$	−1.03784e6	−0.177557	−0.0887783	−	0.996051i	$-0.528296\pi$
$509$	−1.03784e6	−0.177557	−0.0887783	+	0.996051i	$0.528296\pi$
$510$	0	0
$511$	1.98941e6	0.337033
$512$	0	0
$513$	67970.0	0.0114031
$514$	0	0
$515$	3.68200e6	0.611738
$516$	0	0
$517$	−1.48650e6	−0.244589
$518$	0	0
$519$	−139443.	−0.0227237
$520$	0	0
$521$	1.82521e6	0.294590	0.147295	−	0.989093i	$-0.452943\pi$
$521$	1.82521e6	0.294590	0.147295	+	0.989093i	$0.452943\pi$
$522$	0	0
$523$	3.56769e6	0.570339	0.285170	−	0.958477i	$-0.407950\pi$
$523$	3.56769e6	0.570339	0.285170	+	0.958477i	$0.407950\pi$
$524$	0	0
$525$	1.47616e6	0.233741
$526$	0	0
$527$	170039.	0.0266699
$528$	0	0
$529$	8.20479e6	1.27476
$530$	0	0
$531$	2.21895e6	0.341516
$532$	0	0
$533$	327618.	0.0499516
$534$	0	0
$535$	−4.08264e6	−0.616675
$536$	0	0
$537$	877756.	0.131352
$538$	0	0
$539$	−3.17554e6	−0.470810
$540$	0	0
$541$	−2.85109e6	−0.418811	−0.209405	−	0.977829i	$-0.567153\pi$
$541$	−2.85109e6	−0.418811	−0.209405	+	0.977829i	$0.567153\pi$
$542$	0	0
$543$	−4.15323e6	−0.604486
$544$	0	0
$545$	3.94287e6	0.568619
$546$	0	0
$547$	−5.09692e6	−0.728349	−0.364174	−	0.931331i	$-0.618649\pi$
$547$	−5.09692e6	−0.728349	−0.364174	+	0.931331i	$0.618649\pi$
$548$	0	0
$549$	−2.32847e6	−0.329716
$550$	0	0
$551$	523352.	0.0734370
$552$	0	0
$553$	−2.31228e6	−0.321535
$554$	0	0
$555$	−1.12858e6	−0.155525
$556$	0	0
$557$	−2.31522e6	−0.316194	−0.158097	−	0.987424i	$-0.550536\pi$
$557$	−2.31522e6	−0.316194	−0.158097	+	0.987424i	$0.550536\pi$
$558$	0	0
$559$	1.54475e6	0.209088
$560$	0	0
$561$	176112.	0.0236256
$562$	0	0
$563$	304153.	0.0404409	0.0202205	−	0.999796i	$-0.493563\pi$
$563$	304153.	0.0404409	0.0202205	+	0.999796i	$0.493563\pi$
$564$	0	0
$565$	−3.12638e6	−0.412022
$566$	0	0
$567$	−470126.	−0.0614125
$568$	0	0
$569$	−3.22725e6	−0.417881	−0.208940	−	0.977928i	$-0.567001\pi$
$569$	−3.22725e6	−0.417881	−0.208940	+	0.977928i	$0.567001\pi$
$570$	0	0
$571$	−4.88990e6	−0.627639	−0.313819	−	0.949483i	$-0.601609\pi$
$571$	−4.88990e6	−0.627639	−0.313819	+	0.949483i	$0.601609\pi$
$572$	0	0
$573$	−703351.	−0.0894923
$574$	0	0
$575$	8.75857e6	1.10475
$576$	0	0
$577$	3.58483e6	0.448259	0.224130	−	0.974559i	$-0.428046\pi$
$577$	3.58483e6	0.448259	0.224130	+	0.974559i	$0.428046\pi$
$578$	0	0
$579$	−927904.	−0.115029
$580$	0	0
$581$	3.71437e6	0.456504
$582$	0	0
$583$	−9.03395e6	−1.10079
$584$	0	0
$585$	−395799.	−0.0478173
$586$	0	0
$587$	3.46214e6	0.414715	0.207358	−	0.978265i	$-0.433514\pi$
$587$	3.46214e6	0.414715	0.207358	+	0.978265i	$0.433514\pi$
$588$	0	0
$589$	220414.	0.0261789
$590$	0	0
$591$	5.20634e6	0.613146
$592$	0	0
$593$	1.56077e7	1.82265	0.911324	−	0.411689i	$-0.135061\pi$
$593$	1.56077e7	1.82265	0.911324	+	0.411689i	$0.135061\pi$
$594$	0	0
$595$	149020.	0.0172565
$596$	0	0
$597$	3.72916e6	0.428228
$598$	0	0
$599$	4.38283e6	0.499100	0.249550	−	0.968362i	$-0.419717\pi$
$599$	4.38283e6	0.499100	0.249550	+	0.968362i	$0.419717\pi$
$600$	0	0
$601$	3.77116e6	0.425882	0.212941	−	0.977065i	$-0.431696\pi$
$601$	3.77116e6	0.425882	0.212941	+	0.977065i	$0.431696\pi$
$602$	0	0
$603$	2.09161e6	0.234254
$604$	0	0
$605$	2.51663e6	0.279532
$606$	0	0
$607$	−3.16199e6	−0.348329	−0.174164	−	0.984717i	$-0.555722\pi$
$607$	−3.16199e6	−0.348329	−0.174164	+	0.984717i	$0.555722\pi$
$608$	0	0
$609$	−3.61986e6	−0.395502
$610$	0	0
$611$	−923424.	−0.100069
$612$	0	0
$613$	−1.35843e7	−1.46011	−0.730057	−	0.683386i	$-0.760507\pi$
$613$	−1.35843e7	−1.46011	−0.730057	+	0.683386i	$0.760507\pi$
$614$	0	0
$615$	−504460.	−0.0537822
$616$	0	0
$617$	1.11980e7	1.18421	0.592104	−	0.805861i	$-0.298297\pi$
$617$	1.11980e7	1.18421	0.592104	+	0.805861i	$0.298297\pi$
$618$	0	0
$619$	−1.13447e7	−1.19006	−0.595028	−	0.803705i	$-0.702859\pi$
$619$	−1.13447e7	−1.19006	−0.595028	+	0.803705i	$0.702859\pi$
$620$	0	0
$621$	−2.78943e6	−0.290259
$622$	0	0
$623$	5.07974e6	0.524350
$624$	0	0
$625$	2.62702e6	0.269007
$626$	0	0
$627$	228287.	0.0231906
$628$	0	0
$629$	311950.	0.0314382
$630$	0	0
$631$	−6.93729e6	−0.693611	−0.346806	−	0.937937i	$-0.612734\pi$
$631$	−6.93729e6	−0.693611	−0.346806	+	0.937937i	$0.612734\pi$
$632$	0	0
$633$	−8.47452e6	−0.840632
$634$	0	0
$635$	−1.53387e6	−0.150957
$636$	0	0
$637$	−1.97267e6	−0.192622
$638$	0	0
$639$	−2.65281e6	−0.257012
$640$	0	0
$641$	1.12159e7	1.07818	0.539089	−	0.842249i	$-0.318768\pi$
$641$	1.12159e7	1.07818	0.539089	+	0.842249i	$0.318768\pi$
$642$	0	0
$643$	5.50189e6	0.524788	0.262394	−	0.964961i	$-0.415488\pi$
$643$	5.50189e6	0.524788	0.262394	+	0.964961i	$0.415488\pi$
$644$	0	0
$645$	−2.37858e6	−0.225122
$646$	0	0
$647$	1.22447e7	1.14998	0.574988	−	0.818162i	$-0.305007\pi$
$647$	1.22447e7	1.14998	0.574988	+	0.818162i	$0.305007\pi$
$648$	0	0
$649$	7.45266e6	0.694544
$650$	0	0
$651$	−1.52453e6	−0.140989
$652$	0	0
$653$	−9.57856e6	−0.879058	−0.439529	−	0.898228i	$-0.644855\pi$
$653$	−9.57856e6	−0.879058	−0.439529	+	0.898228i	$0.644855\pi$
$654$	0	0
$655$	−949608.	−0.0864850
$656$	0	0
$657$	−2.24887e6	−0.203260
$658$	0	0
$659$	−1.61762e7	−1.45098	−0.725490	−	0.688232i	$-0.758387\pi$
$659$	−1.61762e7	−1.45098	−0.725490	+	0.688232i	$0.758387\pi$
$660$	0	0
$661$	6.90601e6	0.614785	0.307393	−	0.951583i	$-0.400544\pi$
$661$	6.90601e6	0.614785	0.307393	+	0.951583i	$0.400544\pi$
$662$	0	0
$663$	109402.	0.00966592
$664$	0	0
$665$	193169.	0.0169388
$666$	0	0
$667$	−2.14779e7	−1.86929
$668$	0	0
$669$	3.32969e6	0.287633
$670$	0	0
$671$	−7.82050e6	−0.670545
$672$	0	0
$673$	1.94177e7	1.65257	0.826287	−	0.563250i	$-0.190449\pi$
$673$	1.94177e7	1.65257	0.826287	+	0.563250i	$0.190449\pi$
$674$	0	0
$675$	−1.66868e6	−0.140966
$676$	0	0
$677$	767961.	0.0643973	0.0321986	−	0.999481i	$-0.489749\pi$
$677$	767961.	0.0643973	0.0321986	+	0.999481i	$0.489749\pi$
$678$	0	0
$679$	9.07258e6	0.755190
$680$	0	0
$681$	−3.26187e6	−0.269525
$682$	0	0
$683$	2.18810e6	0.179480	0.0897399	−	0.995965i	$-0.471396\pi$
$683$	2.18810e6	0.179480	0.0897399	+	0.995965i	$0.471396\pi$
$684$	0	0
$685$	1.61480e6	0.131490
$686$	0	0
$687$	−8.45800e6	−0.683716
$688$	0	0
$689$	−5.61197e6	−0.450368
$690$	0	0
$691$	−223392.	−0.0177980	−0.00889902	−	0.999960i	$-0.502833\pi$
$691$	−223392.	−0.0177980	−0.00889902	+	0.999960i	$0.502833\pi$
$692$	0	0
$693$	−1.57899e6	−0.124895
$694$	0	0
$695$	888975.	0.0698116
$696$	0	0
$697$	139437.	0.0108717
$698$	0	0
$699$	2.30556e6	0.178477
$700$	0	0
$701$	−1.07733e7	−0.828041	−0.414021	−	0.910268i	$-0.635876\pi$
$701$	−1.07733e7	−0.828041	−0.414021	+	0.910268i	$0.635876\pi$
$702$	0	0
$703$	404368.	0.0308595
$704$	0	0
$705$	1.42187e6	0.107743
$706$	0	0
$707$	1.35256e7	1.01767
$708$	0	0
$709$	5.30994e6	0.396711	0.198355	−	0.980130i	$-0.436440\pi$
$709$	5.30994e6	0.396711	0.198355	+	0.980130i	$0.436440\pi$
$710$	0	0
$711$	2.61386e6	0.193913
$712$	0	0
$713$	−9.04560e6	−0.666367
$714$	0	0
$715$	−1.32935e6	−0.0972465
$716$	0	0
$717$	−9.53246e6	−0.692480
$718$	0	0
$719$	3.79956e6	0.274101	0.137050	−	0.990564i	$-0.456238\pi$
$719$	3.79956e6	0.274101	0.137050	+	0.990564i	$0.456238\pi$
$720$	0	0
$721$	9.12483e6	0.653712
$722$	0	0
$723$	−6.78330e6	−0.482609
$724$	0	0
$725$	−1.28484e7	−0.907831
$726$	0	0
$727$	−6.78114e6	−0.475846	−0.237923	−	0.971284i	$-0.576467\pi$
$727$	−6.78114e6	−0.475846	−0.237923	+	0.971284i	$0.576467\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	657460.	0.0455067
$732$	0	0
$733$	−1.62233e7	−1.11527	−0.557633	−	0.830088i	$-0.688290\pi$
$733$	−1.62233e7	−1.11527	−0.557633	+	0.830088i	$0.688290\pi$
$734$	0	0
$735$	3.03748e6	0.207394
$736$	0	0
$737$	7.02497e6	0.476404
$738$	0	0
$739$	−2.78852e7	−1.87829	−0.939143	−	0.343525i	$-0.888379\pi$
$739$	−2.78852e7	−1.87829	−0.939143	+	0.343525i	$0.888379\pi$
$740$	0	0
$741$	141814.	0.00948797
$742$	0	0
$743$	−1.27984e6	−0.0850516	−0.0425258	−	0.999095i	$-0.513540\pi$
$743$	−1.27984e6	−0.0850516	−0.0425258	+	0.999095i	$0.513540\pi$
$744$	0	0
$745$	7.90621e6	0.521889
$746$	0	0
$747$	−4.19880e6	−0.275311
$748$	0	0
$749$	−1.01177e7	−0.658988
$750$	0	0
$751$	−1.56219e7	−1.01073	−0.505364	−	0.862906i	$-0.668642\pi$
$751$	−1.56219e7	−1.01073	−0.505364	+	0.862906i	$0.668642\pi$
$752$	0	0
$753$	851388.	0.0547192
$754$	0	0
$755$	5.78838e6	0.369564
$756$	0	0
$757$	1.09700e7	0.695771	0.347885	−	0.937537i	$-0.386900\pi$
$757$	1.09700e7	0.695771	0.347885	+	0.937537i	$0.386900\pi$
$758$	0	0
$759$	−9.36870e6	−0.590303
$760$	0	0
$761$	9.61776e6	0.602022	0.301011	−	0.953621i	$-0.402676\pi$
$761$	9.61776e6	0.602022	0.301011	+	0.953621i	$0.402676\pi$
$762$	0	0
$763$	9.77134e6	0.607635
$764$	0	0
$765$	−168456.	−0.0104072
$766$	0	0
$767$	4.62966e6	0.284158
$768$	0	0
$769$	−8.40921e6	−0.512790	−0.256395	−	0.966572i	$-0.582535\pi$
$769$	−8.40921e6	−0.512790	−0.256395	+	0.966572i	$0.582535\pi$
$770$	0	0
$771$	−236674.	−0.0143389
$772$	0	0
$773$	−1.03787e7	−0.624732	−0.312366	−	0.949962i	$-0.601122\pi$
$773$	−1.03787e7	−0.624732	−0.312366	+	0.949962i	$0.601122\pi$
$774$	0	0
$775$	−5.41123e6	−0.323625
$776$	0	0
$777$	−2.79688e6	−0.166196
$778$	0	0
$779$	180747.	0.0106715
$780$	0	0
$781$	−8.90984e6	−0.522687
$782$	0	0
$783$	4.09197e6	0.238522
$784$	0	0
$785$	2.37306e6	0.137447
$786$	0	0
$787$	2.65654e7	1.52890	0.764451	−	0.644682i	$-0.223010\pi$
$787$	2.65654e7	1.52890	0.764451	+	0.644682i	$0.223010\pi$
$788$	0	0
$789$	−9.50433e6	−0.543537
$790$	0	0
$791$	−7.74787e6	−0.440292
$792$	0	0
$793$	−4.85816e6	−0.274340
$794$	0	0
$795$	8.64120e6	0.484905
$796$	0	0
$797$	−1.92434e7	−1.07309	−0.536545	−	0.843872i	$-0.680271\pi$
$797$	−1.92434e7	−1.07309	−0.536545	+	0.843872i	$0.680271\pi$
$798$	0	0
$799$	−393018.	−0.0217794
$800$	0	0
$801$	−5.74225e6	−0.316228
$802$	0	0
$803$	−7.55317e6	−0.413371
$804$	0	0
$805$	−7.92748e6	−0.431167
$806$	0	0
$807$	1.81892e7	0.983175
$808$	0	0
$809$	1.43425e7	0.770467	0.385233	−	0.922819i	$-0.374121\pi$
$809$	1.43425e7	0.770467	0.385233	+	0.922819i	$0.374121\pi$
$810$	0	0
$811$	−2.56772e7	−1.37087	−0.685434	−	0.728135i	$-0.740388\pi$
$811$	−2.56772e7	−1.37087	−0.685434	+	0.728135i	$0.740388\pi$
$812$	0	0
$813$	−1.97419e7	−1.04752
$814$	0	0
$815$	1.12353e6	0.0592501
$816$	0	0
$817$	852238.	0.0446690
$818$	0	0
$819$	−980881.	−0.0510983
$820$	0	0
$821$	−2.36193e7	−1.22295	−0.611476	−	0.791263i	$-0.709424\pi$
$821$	−2.36193e7	−1.22295	−0.611476	+	0.791263i	$0.709424\pi$
$822$	0	0
$823$	−1.69943e7	−0.874591	−0.437295	−	0.899318i	$-0.644064\pi$
$823$	−1.69943e7	−0.874591	−0.437295	+	0.899318i	$0.644064\pi$
$824$	0	0
$825$	−5.60451e6	−0.286683
$826$	0	0
$827$	9.50945e6	0.483495	0.241747	−	0.970339i	$-0.422279\pi$
$827$	9.50945e6	0.483495	0.241747	+	0.970339i	$0.422279\pi$
$828$	0	0
$829$	−3.50009e7	−1.76886	−0.884428	−	0.466676i	$-0.845451\pi$
$829$	−3.50009e7	−1.76886	−0.884428	+	0.466676i	$0.845451\pi$
$830$	0	0
$831$	1.21690e7	0.611297
$832$	0	0
$833$	−839587.	−0.0419231
$834$	0	0
$835$	1.08315e7	0.537618
$836$	0	0
$837$	1.72337e6	0.0850284
$838$	0	0
$839$	−2.59134e6	−0.127092	−0.0635462	−	0.997979i	$-0.520241\pi$
$839$	−2.59134e6	−0.127092	−0.0635462	+	0.997979i	$0.520241\pi$
$840$	0	0
$841$	1.09960e7	0.536098
$842$	0	0
$843$	1.00560e6	0.0487368
$844$	0	0
$845$	−825803.	−0.0397864
$846$	0	0
$847$	6.23679e6	0.298712
$848$	0	0
$849$	1.75415e7	0.835214
$850$	0	0
$851$	−1.65949e7	−0.785508
$852$	0	0
$853$	3.31365e7	1.55932	0.779658	−	0.626206i	$-0.215393\pi$
$853$	3.31365e7	1.55932	0.779658	+	0.626206i	$0.215393\pi$
$854$	0	0
$855$	−218362.	−0.0102156
$856$	0	0
$857$	2.10642e6	0.0979697	0.0489849	−	0.998800i	$-0.484401\pi$
$857$	2.10642e6	0.0979697	0.0489849	+	0.998800i	$0.484401\pi$
$858$	0	0
$859$	−1.55967e7	−0.721191	−0.360595	−	0.932722i	$-0.617426\pi$
$859$	−1.55967e7	−0.721191	−0.360595	+	0.932722i	$0.617426\pi$
$860$	0	0
$861$	−1.25017e6	−0.0574725
$862$	0	0
$863$	2.74535e7	1.25479	0.627395	−	0.778701i	$-0.284121\pi$
$863$	2.74535e7	1.25479	0.627395	+	0.778701i	$0.284121\pi$
$864$	0	0
$865$	447980.	0.0203572
$866$	0	0
$867$	−1.27322e7	−0.575247
$868$	0	0
$869$	8.77902e6	0.394363
$870$	0	0
$871$	4.36397e6	0.194911
$872$	0	0
$873$	−1.02558e7	−0.455445
$874$	0	0
$875$	−1.12167e7	−0.495274
$876$	0	0
$877$	891772.	0.0391521	0.0195760	−	0.999808i	$-0.493768\pi$
$877$	891772.	0.0391521	0.0195760	+	0.999808i	$0.493768\pi$
$878$	0	0
$879$	−9.45355e6	−0.412689
$880$	0	0
$881$	−3.19746e7	−1.38792	−0.693960	−	0.720013i	$-0.744136\pi$
$881$	−3.19746e7	−1.38792	−0.693960	+	0.720013i	$0.744136\pi$
$882$	0	0
$883$	2.95484e7	1.27536	0.637679	−	0.770302i	$-0.279894\pi$
$883$	2.95484e7	1.27536	0.637679	+	0.770302i	$0.279894\pi$
$884$	0	0
$885$	−7.12866e6	−0.305949
$886$	0	0
$887$	5.15586e6	0.220035	0.110018	−	0.993930i	$-0.464909\pi$
$887$	5.15586e6	0.220035	0.110018	+	0.993930i	$0.464909\pi$
$888$	0	0
$889$	−3.80128e6	−0.161315
$890$	0	0
$891$	1.78492e6	0.0753225
$892$	0	0
$893$	−509453.	−0.0213784
$894$	0	0
$895$	−2.81991e6	−0.117673
$896$	0	0
$897$	−5.81991e6	−0.241510
$898$	0	0
$899$	1.32695e7	0.547590
$900$	0	0
$901$	−2.38851e6	−0.0980199
$902$	0	0
$903$	−5.89465e6	−0.240569
$904$	0	0
$905$	1.33428e7	0.541533
$906$	0	0
$907$	1.85404e7	0.748342	0.374171	−	0.927360i	$-0.377927\pi$
$907$	1.85404e7	0.748342	0.374171	+	0.927360i	$0.377927\pi$
$908$	0	0
$909$	−1.52897e7	−0.613746
$910$	0	0
$911$	3.33448e7	1.33117	0.665583	−	0.746324i	$-0.268183\pi$
$911$	3.33448e7	1.33117	0.665583	+	0.746324i	$0.268183\pi$
$912$	0	0
$913$	−1.41023e7	−0.559903
$914$	0	0
$915$	7.48050e6	0.295378
$916$	0	0
$917$	−2.35334e6	−0.0924192
$918$	0	0
$919$	1.57476e7	0.615071	0.307535	−	0.951537i	$-0.400496\pi$
$919$	1.57476e7	0.615071	0.307535	+	0.951537i	$0.400496\pi$
$920$	0	0
$921$	5.10326e6	0.198243
$922$	0	0
$923$	−5.53487e6	−0.213847
$924$	0	0
$925$	−9.92734e6	−0.381486
$926$	0	0
$927$	−1.03149e7	−0.394245
$928$	0	0
$929$	1.19196e7	0.453128	0.226564	−	0.973996i	$-0.427251\pi$
$929$	1.19196e7	0.453128	0.226564	+	0.973996i	$0.427251\pi$
$930$	0	0
$931$	−1.08832e6	−0.0411513
$932$	0	0
$933$	5.89840e6	0.221835
$934$	0	0
$935$	−565783.	−0.0211651
$936$	0	0
$937$	−5.03173e7	−1.87227	−0.936134	−	0.351644i	$-0.885623\pi$
$937$	−5.03173e7	−1.87227	−0.936134	+	0.351644i	$0.885623\pi$
$938$	0	0
$939$	1.67779e7	0.620973
$940$	0	0
$941$	−6.57081e6	−0.241905	−0.120953	−	0.992658i	$-0.538595\pi$
$941$	−6.57081e6	−0.241905	−0.120953	+	0.992658i	$0.538595\pi$
$942$	0	0
$943$	−7.41768e6	−0.271637
$944$	0	0
$945$	1.51034e6	0.0550168
$946$	0	0
$947$	−2.79943e7	−1.01437	−0.507183	−	0.861838i	$-0.669313\pi$
$947$	−2.79943e7	−1.01437	−0.507183	+	0.861838i	$0.669313\pi$
$948$	0	0
$949$	−4.69209e6	−0.169122
$950$	0	0
$951$	−1.34854e7	−0.483518
$952$	0	0
$953$	1.67553e7	0.597614	0.298807	−	0.954314i	$-0.403411\pi$
$953$	1.67553e7	0.597614	0.298807	+	0.954314i	$0.403411\pi$
$954$	0	0
$955$	2.25961e6	0.0801723
$956$	0	0
$957$	1.37435e7	0.485083
$958$	0	0
$959$	4.00184e6	0.140512
$960$	0	0
$961$	−2.30406e7	−0.804795
$962$	0	0
$963$	1.14373e7	0.397427
$964$	0	0
$965$	2.98101e6	0.103049
$966$	0	0
$967$	−1.60885e7	−0.553287	−0.276644	−	0.960973i	$-0.589222\pi$
$967$	−1.60885e7	−0.553287	−0.276644	+	0.960973i	$0.589222\pi$
$968$	0	0
$969$	60357.3	0.00206500
$970$	0	0
$971$	4.59929e7	1.56546	0.782731	−	0.622361i	$-0.213826\pi$
$971$	4.59929e7	1.56546	0.782731	+	0.622361i	$0.213826\pi$
$972$	0	0
$973$	2.20308e6	0.0746017
$974$	0	0
$975$	−3.48157e6	−0.117291
$976$	0	0
$977$	6.62641e6	0.222096	0.111048	−	0.993815i	$-0.464579\pi$
$977$	6.62641e6	0.222096	0.111048	+	0.993815i	$0.464579\pi$
$978$	0	0
$979$	−1.92862e7	−0.643116
$980$	0	0
$981$	−1.10457e7	−0.366456
$982$	0	0
$983$	−5.08829e7	−1.67953	−0.839766	−	0.542948i	$-0.817308\pi$
$983$	−5.08829e7	−1.67953	−0.839766	+	0.542948i	$0.817308\pi$
$984$	0	0
$985$	−1.67261e7	−0.549292
$986$	0	0
$987$	3.52372e6	0.115135
$988$	0	0
$989$	−3.49751e7	−1.13702
$990$	0	0
$991$	1.70488e7	0.551454	0.275727	−	0.961236i	$-0.411081\pi$
$991$	1.70488e7	0.551454	0.275727	+	0.961236i	$0.411081\pi$
$992$	0	0
$993$	5.60007e6	0.180227
$994$	0	0
$995$	−1.19804e7	−0.383631
$996$	0	0
$997$	−3.01764e7	−0.961455	−0.480728	−	0.876870i	$-0.659627\pi$
$997$	−3.01764e7	−0.961455	−0.480728	+	0.876870i	$0.659627\pi$
$998$	0	0
$999$	3.16166e6	0.100231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.m.1.1		2
4.3	odd	2		312.6.a.c.1.1	✓	2
12.11	even	2		936.6.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.6.a.c.1.1	✓	2	4.3	odd	2
624.6.a.m.1.1		2	1.1	even	1	trivial
936.6.a.d.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -28.9137 q^{5} -71.6547 q^{7} +81.0000 q^{9} +272.050 q^{11} +169.000 q^{13} -260.223 q^{15} +71.9280 q^{17} +93.2373 q^{19} -644.892 q^{21} -3826.37 q^{23} -2289.00 q^{25} +729.000 q^{27} +5613.12 q^{29} +2364.01 q^{31} +2448.45 q^{33} +2071.80 q^{35} +4336.98 q^{37} +1521.00 q^{39} +1938.57 q^{41} +9140.53 q^{43} -2342.01 q^{45} -5464.05 q^{47} -11672.6 q^{49} +647.352 q^{51} -33206.9 q^{53} -7865.97 q^{55} +839.135 q^{57} +27394.4 q^{59} -28746.5 q^{61} -5804.03 q^{63} -4886.41 q^{65} +25822.3 q^{67} -34437.4 q^{69} -32750.7 q^{71} -27763.9 q^{73} -20601.0 q^{75} -19493.7 q^{77} +32269.8 q^{79} +6561.00 q^{81} -51837.1 q^{83} -2079.70 q^{85} +50518.1 q^{87} -70891.9 q^{89} -12109.6 q^{91} +21276.1 q^{93} -2695.83 q^{95} -126615. q^{97} +22036.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 88 q^{7} + 162 q^{9} - 92 q^{11} + 338 q^{13} - 1244 q^{17} - 1664 q^{19} + 792 q^{21} - 5224 q^{23} - 4578 q^{25} + 1458 q^{27} + 3940 q^{29} - 4640 q^{31} - 828 q^{33} + 6688 q^{35} - 1388 q^{37}+ \cdots - 7452 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 88 * q^7 + 162 * q^9 - 92 * q^11 + 338 * q^13 - 1244 * q^17 - 1664 * q^19 + 792 * q^21 - 5224 * q^23 - 4578 * q^25 + 1458 * q^27 + 3940 * q^29 - 4640 * q^31 - 828 * q^33 + 6688 * q^35 - 1388 * q^37 + 3042 * q^39 - 12488 * q^41 - 6816 * q^43 + 21860 * q^47 - 2990 * q^49 - 11196 * q^51 - 13444 * q^53 - 18392 * q^55 - 14976 * q^57 - 12580 * q^59 - 41764 * q^61 + 7128 * q^63 + 54536 * q^67 - 47016 * q^69 + 51252 * q^71 - 100980 * q^73 - 41202 * q^75 - 77616 * q^77 + 119360 * q^79 + 13122 * q^81 - 30060 * q^83 - 40128 * q^85 + 35460 * q^87 - 29888 * q^89 + 14872 * q^91 - 41760 * q^93 - 53504 * q^95 - 244788 * q^97 - 7452 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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