LMFDB - Embedded newform 624.6.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,6,Mod(1,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [3,0,27,0,54,0,-84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$100.079503563$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.125308.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 55x - 78 $ x^3 - 55*x - 78
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.47673$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} +6.18613 q^{5} -190.614 q^{7} +81.0000 q^{9} -141.200 q^{11} +169.000 q^{13} +55.6751 q^{15} +1854.72 q^{17} +2091.51 q^{19} -1715.52 q^{21} -4682.44 q^{23} -3086.73 q^{25} +729.000 q^{27} -2483.38 q^{29} -4629.17 q^{31} -1270.80 q^{33} -1179.16 q^{35} +1013.68 q^{37} +1521.00 q^{39} +14283.8 q^{41} +625.799 q^{43} +501.076 q^{45} -9057.59 q^{47} +19526.5 q^{49} +16692.5 q^{51} +31642.2 q^{53} -873.483 q^{55} +18823.6 q^{57} +25952.5 q^{59} -4735.49 q^{61} -15439.7 q^{63} +1045.46 q^{65} -19717.3 q^{67} -42142.0 q^{69} +27169.0 q^{71} -13688.2 q^{73} -27780.6 q^{75} +26914.7 q^{77} +75281.6 q^{79} +6561.00 q^{81} +104809. q^{83} +11473.5 q^{85} -22350.4 q^{87} +4771.10 q^{89} -32213.7 q^{91} -41662.5 q^{93} +12938.4 q^{95} +34879.0 q^{97} -11437.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 27 q^{3} + 54 q^{5} - 84 q^{7} + 243 q^{9} - 876 q^{11} + 507 q^{13} + 486 q^{15} + 102 q^{17} + 16 q^{19} - 756 q^{21} - 1363 q^{25} + 2187 q^{27} + 9666 q^{29} + 10196 q^{31} - 7884 q^{33} - 16680 q^{35}+ \cdots - 70956 q^{99}+O(q^{100}) $ 3 * q + 27 * q^3 + 54 * q^5 - 84 * q^7 + 243 * q^9 - 876 * q^11 + 507 * q^13 + 486 * q^15 + 102 * q^17 + 16 * q^19 - 756 * q^21 - 1363 * q^25 + 2187 * q^27 + 9666 * q^29 + 10196 * q^31 - 7884 * q^33 - 16680 * q^35 + 11818 * q^37 + 4563 * q^39 + 35490 * q^41 - 2780 * q^43 + 4374 * q^45 - 25728 * q^47 + 39291 * q^49 + 918 * q^51 + 36786 * q^53 + 6440 * q^55 + 144 * q^57 + 27516 * q^59 + 45994 * q^61 - 6804 * q^63 + 9126 * q^65 + 42536 * q^67 + 54432 * q^71 + 27846 * q^73 - 12267 * q^75 - 78000 * q^77 + 80568 * q^79 + 19683 * q^81 - 24012 * q^83 - 186580 * q^85 + 86994 * q^87 + 117450 * q^89 - 14196 * q^91 + 91764 * q^93 + 64608 * q^95 - 20930 * q^97 - 70956 * 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
$5$	6.18613	0.110661	0.0553304	−	0.998468i	$-0.482379\pi$
$5$	6.18613	0.110661	0.0553304	+	0.998468i	$0.482379\pi$
$6$	0	0
$7$	−190.614	−1.47031	−0.735154	−	0.677900i	$-0.762890\pi$
$7$	−190.614	−1.47031	−0.735154	+	0.677900i	$0.762890\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−141.200	−0.351847	−0.175924	−	0.984404i	$-0.556291\pi$
$11$	−141.200	−0.351847	−0.175924	+	0.984404i	$0.556291\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	55.6751	0.0638900
$16$	0	0
$17$	1854.72	1.55652	0.778262	−	0.627940i	$-0.216102\pi$
$17$	1854.72	1.55652	0.778262	+	0.627940i	$0.216102\pi$
$18$	0	0
$19$	2091.51	1.32916	0.664579	−	0.747218i	$-0.268611\pi$
$19$	2091.51	1.32916	0.664579	+	0.747218i	$0.268611\pi$
$20$	0	0
$21$	−1715.52	−0.848883
$22$	0	0
$23$	−4682.44	−1.84566	−0.922832	−	0.385203i	$-0.874132\pi$
$23$	−4682.44	−1.84566	−0.922832	+	0.385203i	$0.874132\pi$
$24$	0	0
$25$	−3086.73	−0.987754
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−2483.38	−0.548337	−0.274169	−	0.961682i	$-0.588403\pi$
$29$	−2483.38	−0.548337	−0.274169	+	0.961682i	$0.588403\pi$
$30$	0	0
$31$	−4629.17	−0.865164	−0.432582	−	0.901595i	$-0.642397\pi$
$31$	−4629.17	−0.865164	−0.432582	+	0.901595i	$0.642397\pi$
$32$	0	0
$33$	−1270.80	−0.203139
$34$	0	0
$35$	−1179.16	−0.162706
$36$	0	0
$37$	1013.68	0.121730	0.0608650	−	0.998146i	$-0.480614\pi$
$37$	1013.68	0.121730	0.0608650	+	0.998146i	$0.480614\pi$
$38$	0	0
$39$	1521.00	0.160128
$40$	0	0
$41$	14283.8	1.32704	0.663520	−	0.748159i	$-0.269062\pi$
$41$	14283.8	1.32704	0.663520	+	0.748159i	$0.269062\pi$
$42$	0	0
$43$	625.799	0.0516135	0.0258068	−	0.999667i	$-0.491785\pi$
$43$	625.799	0.0516135	0.0258068	+	0.999667i	$0.491785\pi$
$44$	0	0
$45$	501.076	0.0368869
$46$	0	0
$47$	−9057.59	−0.598092	−0.299046	−	0.954239i	$-0.596668\pi$
$47$	−9057.59	−0.598092	−0.299046	+	0.954239i	$0.596668\pi$
$48$	0	0
$49$	19526.5	1.16181
$50$	0	0
$51$	16692.5	0.898659
$52$	0	0
$53$	31642.2	1.54731	0.773654	−	0.633608i	$-0.218427\pi$
$53$	31642.2	1.54731	0.773654	+	0.633608i	$0.218427\pi$
$54$	0	0
$55$	−873.483	−0.0389357
$56$	0	0
$57$	18823.6	0.767390
$58$	0	0
$59$	25952.5	0.970618	0.485309	−	0.874343i	$-0.338707\pi$
$59$	25952.5	0.970618	0.485309	+	0.874343i	$0.338707\pi$
$60$	0	0
$61$	−4735.49	−0.162945	−0.0814724	−	0.996676i	$-0.525962\pi$
$61$	−4735.49	−0.162945	−0.0814724	+	0.996676i	$0.525962\pi$
$62$	0	0
$63$	−15439.7	−0.490103
$64$	0	0
$65$	1045.46	0.0306918
$66$	0	0
$67$	−19717.3	−0.536611	−0.268306	−	0.963334i	$-0.586464\pi$
$67$	−19717.3	−0.536611	−0.268306	+	0.963334i	$0.586464\pi$
$68$	0	0
$69$	−42142.0	−1.06559
$70$	0	0
$71$	27169.0	0.639629	0.319815	−	0.947480i	$-0.396379\pi$
$71$	27169.0	0.639629	0.319815	+	0.947480i	$0.396379\pi$
$72$	0	0
$73$	−13688.2	−0.300634	−0.150317	−	0.988638i	$-0.548029\pi$
$73$	−13688.2	−0.300634	−0.150317	+	0.988638i	$0.548029\pi$
$74$	0	0
$75$	−27780.6	−0.570280
$76$	0	0
$77$	26914.7	0.517324
$78$	0	0
$79$	75281.6	1.35713	0.678564	−	0.734541i	$-0.262603\pi$
$79$	75281.6	1.35713	0.678564	+	0.734541i	$0.262603\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	104809.	1.66995	0.834974	−	0.550289i	$-0.185483\pi$
$83$	104809.	1.66995	0.834974	+	0.550289i	$0.185483\pi$
$84$	0	0
$85$	11473.5	0.172246
$86$	0	0
$87$	−22350.4	−0.316583
$88$	0	0
$89$	4771.10	0.0638475	0.0319237	−	0.999490i	$-0.489837\pi$
$89$	4771.10	0.0638475	0.0319237	+	0.999490i	$0.489837\pi$
$90$	0	0
$91$	−32213.7	−0.407790
$92$	0	0
$93$	−41662.5	−0.499503
$94$	0	0
$95$	12938.4	0.147086
$96$	0	0
$97$	34879.0	0.376387	0.188194	−	0.982132i	$-0.439737\pi$
$97$	34879.0	0.376387	0.188194	+	0.982132i	$0.439737\pi$
$98$	0	0
$99$	−11437.2	−0.117282
$100$	0	0
$101$	123059.	1.20036	0.600179	−	0.799865i	$-0.295096\pi$
$101$	123059.	1.20036	0.600179	+	0.799865i	$0.295096\pi$
$102$	0	0
$103$	70651.4	0.656187	0.328094	−	0.944645i	$-0.393594\pi$
$103$	70651.4	0.656187	0.328094	+	0.944645i	$0.393594\pi$
$104$	0	0
$105$	−10612.4	−0.0939381
$106$	0	0
$107$	51075.5	0.431274	0.215637	−	0.976474i	$-0.430817\pi$
$107$	51075.5	0.431274	0.215637	+	0.976474i	$0.430817\pi$
$108$	0	0
$109$	172935.	1.39418	0.697088	−	0.716985i	$-0.254479\pi$
$109$	172935.	1.39418	0.697088	+	0.716985i	$0.254479\pi$
$110$	0	0
$111$	9123.15	0.0702809
$112$	0	0
$113$	15198.7	0.111972	0.0559860	−	0.998432i	$-0.482170\pi$
$113$	15198.7	0.111972	0.0559860	+	0.998432i	$0.482170\pi$
$114$	0	0
$115$	−28966.2	−0.204243
$116$	0	0
$117$	13689.0	0.0924500
$118$	0	0
$119$	−353534.	−2.28857
$120$	0	0
$121$	−141113.	−0.876204
$122$	0	0
$123$	128554.	0.766167
$124$	0	0
$125$	−38426.6	−0.219966
$126$	0	0
$127$	−82441.7	−0.453563	−0.226781	−	0.973946i	$-0.572820\pi$
$127$	−82441.7	−0.453563	−0.226781	+	0.973946i	$0.572820\pi$
$128$	0	0
$129$	5632.19	0.0297991
$130$	0	0
$131$	301955.	1.53732	0.768658	−	0.639660i	$-0.220925\pi$
$131$	301955.	1.53732	0.768658	+	0.639660i	$0.220925\pi$
$132$	0	0
$133$	−398671.	−1.95427
$134$	0	0
$135$	4509.69	0.0212967
$136$	0	0
$137$	−387921.	−1.76580	−0.882900	−	0.469562i	$-0.844412\pi$
$137$	−387921.	−1.76580	−0.882900	+	0.469562i	$0.844412\pi$
$138$	0	0
$139$	215864.	0.947640	0.473820	−	0.880622i	$-0.342875\pi$
$139$	215864.	0.947640	0.473820	+	0.880622i	$0.342875\pi$
$140$	0	0
$141$	−81518.3	−0.345309
$142$	0	0
$143$	−23862.9	−0.0975849
$144$	0	0
$145$	−15362.5	−0.0606794
$146$	0	0
$147$	175739.	0.670770
$148$	0	0
$149$	306915.	1.13254	0.566268	−	0.824221i	$-0.308387\pi$
$149$	306915.	1.13254	0.566268	+	0.824221i	$0.308387\pi$
$150$	0	0
$151$	−188241.	−0.671851	−0.335925	−	0.941889i	$-0.609049\pi$
$151$	−188241.	−0.671851	−0.335925	+	0.941889i	$0.609049\pi$
$152$	0	0
$153$	150232.	0.518841
$154$	0	0
$155$	−28636.6	−0.0957398
$156$	0	0
$157$	5384.05	0.0174325	0.00871626	−	0.999962i	$-0.497225\pi$
$157$	5384.05	0.0174325	0.00871626	+	0.999962i	$0.497225\pi$
$158$	0	0
$159$	284780.	0.893339
$160$	0	0
$161$	892536.	2.71370
$162$	0	0
$163$	650379.	1.91733	0.958666	−	0.284533i	$-0.0918385\pi$
$163$	650379.	1.91733	0.958666	+	0.284533i	$0.0918385\pi$
$164$	0	0
$165$	−7861.35	−0.0224795
$166$	0	0
$167$	−235868.	−0.654452	−0.327226	−	0.944946i	$-0.606114\pi$
$167$	−235868.	−0.654452	−0.327226	+	0.944946i	$0.606114\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	169413.	0.443053
$172$	0	0
$173$	332282.	0.844096	0.422048	−	0.906574i	$-0.361311\pi$
$173$	332282.	0.844096	0.422048	+	0.906574i	$0.361311\pi$
$174$	0	0
$175$	588373.	1.45230
$176$	0	0
$177$	233572.	0.560387
$178$	0	0
$179$	−612970.	−1.42990	−0.714952	−	0.699174i	$-0.753551\pi$
$179$	−612970.	−1.42990	−0.714952	+	0.699174i	$0.753551\pi$
$180$	0	0
$181$	138097.	0.313319	0.156659	−	0.987653i	$-0.449927\pi$
$181$	138097.	0.313319	0.156659	+	0.987653i	$0.449927\pi$
$182$	0	0
$183$	−42619.4	−0.0940762
$184$	0	0
$185$	6270.77	0.0134707
$186$	0	0
$187$	−261887.	−0.547658
$188$	0	0
$189$	−138957.	−0.282961
$190$	0	0
$191$	−94267.9	−0.186974	−0.0934868	−	0.995621i	$-0.529801\pi$
$191$	−94267.9	−0.186974	−0.0934868	+	0.995621i	$0.529801\pi$
$192$	0	0
$193$	1.03276e6	1.99574	0.997870	−	0.0652281i	$-0.0207775\pi$
$193$	1.03276e6	1.99574	0.997870	+	0.0652281i	$0.0207775\pi$
$194$	0	0
$195$	9409.10	0.0177199
$196$	0	0
$197$	−157331.	−0.288834	−0.144417	−	0.989517i	$-0.546131\pi$
$197$	−157331.	−0.288834	−0.144417	+	0.989517i	$0.546131\pi$
$198$	0	0
$199$	−510479.	−0.913788	−0.456894	−	0.889521i	$-0.651038\pi$
$199$	−510479.	−0.913788	−0.456894	+	0.889521i	$0.651038\pi$
$200$	0	0
$201$	−177455.	−0.309813
$202$	0	0
$203$	473365.	0.806225
$204$	0	0
$205$	88361.4	0.146851
$206$	0	0
$207$	−379278.	−0.615221
$208$	0	0
$209$	−295322.	−0.467660
$210$	0	0
$211$	442016.	0.683490	0.341745	−	0.939793i	$-0.388982\pi$
$211$	442016.	0.683490	0.341745	+	0.939793i	$0.388982\pi$
$212$	0	0
$213$	244521.	0.369290
$214$	0	0
$215$	3871.27	0.00571160
$216$	0	0
$217$	882382.	1.27206
$218$	0	0
$219$	−123194.	−0.173571
$220$	0	0
$221$	313447.	0.431702
$222$	0	0
$223$	972760.	1.30992	0.654958	−	0.755665i	$-0.272686\pi$
$223$	972760.	1.30992	0.654958	+	0.755665i	$0.272686\pi$
$224$	0	0
$225$	−250025.	−0.329251
$226$	0	0
$227$	−257972.	−0.332283	−0.166141	−	0.986102i	$-0.553131\pi$
$227$	−257972.	−0.332283	−0.166141	+	0.986102i	$0.553131\pi$
$228$	0	0
$229$	133806.	0.168612	0.0843060	−	0.996440i	$-0.473133\pi$
$229$	133806.	0.168612	0.0843060	+	0.996440i	$0.473133\pi$
$230$	0	0
$231$	242232.	0.298677
$232$	0	0
$233$	−1.62143e6	−1.95663	−0.978317	−	0.207112i	$-0.933594\pi$
$233$	−1.62143e6	−1.95663	−0.978317	+	0.207112i	$0.933594\pi$
$234$	0	0
$235$	−56031.4	−0.0661853
$236$	0	0
$237$	677534.	0.783538
$238$	0	0
$239$	811498.	0.918952	0.459476	−	0.888190i	$-0.348037\pi$
$239$	811498.	0.918952	0.459476	+	0.888190i	$0.348037\pi$
$240$	0	0
$241$	252994.	0.280587	0.140294	−	0.990110i	$-0.455195\pi$
$241$	252994.	0.280587	0.140294	+	0.990110i	$0.455195\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	120793.	0.128567
$246$	0	0
$247$	353466.	0.368642
$248$	0	0
$249$	943280.	0.964145
$250$	0	0
$251$	−959107.	−0.960910	−0.480455	−	0.877019i	$-0.659529\pi$
$251$	−959107.	−0.960910	−0.480455	+	0.877019i	$0.659529\pi$
$252$	0	0
$253$	661162.	0.649392
$254$	0	0
$255$	103262.	0.0994464
$256$	0	0
$257$	−424311.	−0.400730	−0.200365	−	0.979721i	$-0.564213\pi$
$257$	−424311.	−0.400730	−0.200365	+	0.979721i	$0.564213\pi$
$258$	0	0
$259$	−193222.	−0.178981
$260$	0	0
$261$	−201153.	−0.182779
$262$	0	0
$263$	1.63733e6	1.45964	0.729822	−	0.683638i	$-0.239603\pi$
$263$	1.63733e6	1.45964	0.729822	+	0.683638i	$0.239603\pi$
$264$	0	0
$265$	195743.	0.171226
$266$	0	0
$267$	42939.9	0.0368624
$268$	0	0
$269$	−1.32752e6	−1.11857	−0.559283	−	0.828977i	$-0.688923\pi$
$269$	−1.32752e6	−1.11857	−0.559283	+	0.828977i	$0.688923\pi$
$270$	0	0
$271$	−105128.	−0.0869547	−0.0434774	−	0.999054i	$-0.513844\pi$
$271$	−105128.	−0.0869547	−0.0434774	+	0.999054i	$0.513844\pi$
$272$	0	0
$273$	−289923.	−0.235438
$274$	0	0
$275$	435848.	0.347539
$276$	0	0
$277$	−1.28791e6	−1.00852	−0.504260	−	0.863552i	$-0.668235\pi$
$277$	−1.28791e6	−1.00852	−0.504260	+	0.863552i	$0.668235\pi$
$278$	0	0
$279$	−374963.	−0.288388
$280$	0	0
$281$	−297907.	−0.225068	−0.112534	−	0.993648i	$-0.535897\pi$
$281$	−297907.	−0.225068	−0.112534	+	0.993648i	$0.535897\pi$
$282$	0	0
$283$	925425.	0.686871	0.343435	−	0.939176i	$-0.388409\pi$
$283$	925425.	0.686871	0.343435	+	0.939176i	$0.388409\pi$
$284$	0	0
$285$	116445.	0.0849199
$286$	0	0
$287$	−2.72268e6	−1.95116
$288$	0	0
$289$	2.02012e6	1.42277
$290$	0	0
$291$	313911.	0.217307
$292$	0	0
$293$	2.14512e6	1.45976	0.729880	−	0.683575i	$-0.239576\pi$
$293$	2.14512e6	1.45976	0.729880	+	0.683575i	$0.239576\pi$
$294$	0	0
$295$	160545.	0.107409
$296$	0	0
$297$	−102935.	−0.0677130
$298$	0	0
$299$	−791332.	−0.511895
$300$	0	0
$301$	−119286.	−0.0758879
$302$	0	0
$303$	1.10753e6	0.693028
$304$	0	0
$305$	−29294.3	−0.0180316
$306$	0	0
$307$	−2.53446e6	−1.53475	−0.767377	−	0.641196i	$-0.778438\pi$
$307$	−2.53446e6	−1.53475	−0.767377	+	0.641196i	$0.778438\pi$
$308$	0	0
$309$	635863.	0.378850
$310$	0	0
$311$	1.40549e6	0.824001	0.412000	−	0.911184i	$-0.364830\pi$
$311$	1.40549e6	0.824001	0.412000	+	0.911184i	$0.364830\pi$
$312$	0	0
$313$	−3.05827e6	−1.76447	−0.882236	−	0.470808i	$-0.843962\pi$
$313$	−3.05827e6	−1.76447	−0.882236	+	0.470808i	$0.843962\pi$
$314$	0	0
$315$	−95511.9	−0.0542352
$316$	0	0
$317$	543899.	0.303997	0.151999	−	0.988381i	$-0.451429\pi$
$317$	543899.	0.303997	0.151999	+	0.988381i	$0.451429\pi$
$318$	0	0
$319$	350654.	0.192931
$320$	0	0
$321$	459679.	0.248996
$322$	0	0
$323$	3.87917e6	2.06887
$324$	0	0
$325$	−521658.	−0.273954
$326$	0	0
$327$	1.55642e6	0.804928
$328$	0	0
$329$	1.72650e6	0.879380
$330$	0	0
$331$	3.20780e6	1.60930	0.804650	−	0.593749i	$-0.202353\pi$
$331$	3.20780e6	1.60930	0.804650	+	0.593749i	$0.202353\pi$
$332$	0	0
$333$	82108.3	0.0405767
$334$	0	0
$335$	−121974.	−0.0593818
$336$	0	0
$337$	1.89691e6	0.909855	0.454928	−	0.890528i	$-0.349665\pi$
$337$	1.89691e6	0.909855	0.454928	+	0.890528i	$0.349665\pi$
$338$	0	0
$339$	136788.	0.0646471
$340$	0	0
$341$	653640.	0.304406
$342$	0	0
$343$	−518376.	−0.237908
$344$	0	0
$345$	−260696.	−0.117920
$346$	0	0
$347$	1.45684e6	0.649512	0.324756	−	0.945798i	$-0.394718\pi$
$347$	1.45684e6	0.649512	0.324756	+	0.945798i	$0.394718\pi$
$348$	0	0
$349$	−3.62193e6	−1.59176	−0.795878	−	0.605457i	$-0.792990\pi$
$349$	−3.62193e6	−1.59176	−0.795878	+	0.605457i	$0.792990\pi$
$350$	0	0
$351$	123201.	0.0533761
$352$	0	0
$353$	−1.74392e6	−0.744888	−0.372444	−	0.928055i	$-0.621480\pi$
$353$	−1.74392e6	−0.744888	−0.372444	+	0.928055i	$0.621480\pi$
$354$	0	0
$355$	168071.	0.0707819
$356$	0	0
$357$	−3.18181e6	−1.32131
$358$	0	0
$359$	2.52775e6	1.03514	0.517569	−	0.855641i	$-0.326837\pi$
$359$	2.52775e6	1.03514	0.517569	+	0.855641i	$0.326837\pi$
$360$	0	0
$361$	1.89833e6	0.766661
$362$	0	0
$363$	−1.27002e6	−0.505876
$364$	0	0
$365$	−84676.8	−0.0332684
$366$	0	0
$367$	−1.53298e6	−0.594116	−0.297058	−	0.954859i	$-0.596006\pi$
$367$	−1.53298e6	−0.594116	−0.297058	+	0.954859i	$0.596006\pi$
$368$	0	0
$369$	1.15699e6	0.442347
$370$	0	0
$371$	−6.03143e6	−2.27502
$372$	0	0
$373$	−331711.	−0.123449	−0.0617246	−	0.998093i	$-0.519660\pi$
$373$	−331711.	−0.123449	−0.0617246	+	0.998093i	$0.519660\pi$
$374$	0	0
$375$	−345839.	−0.126998
$376$	0	0
$377$	−419691.	−0.152081
$378$	0	0
$379$	3.12182e6	1.11638	0.558188	−	0.829715i	$-0.311497\pi$
$379$	3.12182e6	1.11638	0.558188	+	0.829715i	$0.311497\pi$
$380$	0	0
$381$	−741975.	−0.261865
$382$	0	0
$383$	−2.43441e6	−0.848003	−0.424001	−	0.905662i	$-0.639375\pi$
$383$	−2.43441e6	−0.848003	−0.424001	+	0.905662i	$0.639375\pi$
$384$	0	0
$385$	166498.	0.0572475
$386$	0	0
$387$	50689.7	0.0172045
$388$	0	0
$389$	−5.21717e6	−1.74808	−0.874039	−	0.485857i	$-0.838508\pi$
$389$	−5.21717e6	−1.74808	−0.874039	+	0.485857i	$0.838508\pi$
$390$	0	0
$391$	−8.68461e6	−2.87282
$392$	0	0
$393$	2.71759e6	0.887570
$394$	0	0
$395$	465701.	0.150181
$396$	0	0
$397$	4.19247e6	1.33504	0.667519	−	0.744593i	$-0.267356\pi$
$397$	4.19247e6	1.33504	0.667519	+	0.744593i	$0.267356\pi$
$398$	0	0
$399$	−3.58804e6	−1.12830
$400$	0	0
$401$	730988.	0.227012	0.113506	−	0.993537i	$-0.463792\pi$
$401$	730988.	0.227012	0.113506	+	0.993537i	$0.463792\pi$
$402$	0	0
$403$	−782329.	−0.239953
$404$	0	0
$405$	40587.2	0.0122956
$406$	0	0
$407$	−143132.	−0.0428304
$408$	0	0
$409$	5.26806e6	1.55719	0.778596	−	0.627525i	$-0.215932\pi$
$409$	5.26806e6	1.55719	0.778596	+	0.627525i	$0.215932\pi$
$410$	0	0
$411$	−3.49129e6	−1.01948
$412$	0	0
$413$	−4.94689e6	−1.42711
$414$	0	0
$415$	648361.	0.184798
$416$	0	0
$417$	1.94278e6	0.547120
$418$	0	0
$419$	−2.12251e6	−0.590630	−0.295315	−	0.955400i	$-0.595425\pi$
$419$	−2.12251e6	−0.590630	−0.295315	+	0.955400i	$0.595425\pi$
$420$	0	0
$421$	−2.85481e6	−0.785005	−0.392502	−	0.919751i	$-0.628391\pi$
$421$	−2.85481e6	−0.785005	−0.392502	+	0.919751i	$0.628391\pi$
$422$	0	0
$423$	−733665.	−0.199364
$424$	0	0
$425$	−5.72502e6	−1.53746
$426$	0	0
$427$	902648.	0.239579
$428$	0	0
$429$	−214766.	−0.0563406
$430$	0	0
$431$	−5.25703e6	−1.36316	−0.681581	−	0.731743i	$-0.738707\pi$
$431$	−5.25703e6	−1.36316	−0.681581	+	0.731743i	$0.738707\pi$
$432$	0	0
$433$	−3.05346e6	−0.782659	−0.391329	−	0.920251i	$-0.627985\pi$
$433$	−3.05346e6	−0.782659	−0.391329	+	0.920251i	$0.627985\pi$
$434$	0	0
$435$	−138262.	−0.0350333
$436$	0	0
$437$	−9.79338e6	−2.45318
$438$	0	0
$439$	2.76642e6	0.685105	0.342552	−	0.939499i	$-0.388709\pi$
$439$	2.76642e6	0.685105	0.342552	+	0.939499i	$0.388709\pi$
$440$	0	0
$441$	1.58165e6	0.387269
$442$	0	0
$443$	−2.38712e6	−0.577915	−0.288958	−	0.957342i	$-0.593309\pi$
$443$	−2.38712e6	−0.577915	−0.288958	+	0.957342i	$0.593309\pi$
$444$	0	0
$445$	29514.7	0.00706541
$446$	0	0
$447$	2.76223e6	0.653870
$448$	0	0
$449$	88547.7	0.0207282	0.0103641	−	0.999946i	$-0.496701\pi$
$449$	88547.7	0.0207282	0.0103641	+	0.999946i	$0.496701\pi$
$450$	0	0
$451$	−2.01688e6	−0.466915
$452$	0	0
$453$	−1.69417e6	−0.387893
$454$	0	0
$455$	−199278.	−0.0451264
$456$	0	0
$457$	−5.82616e6	−1.30494	−0.652472	−	0.757812i	$-0.726268\pi$
$457$	−5.82616e6	−1.30494	−0.652472	+	0.757812i	$0.726268\pi$
$458$	0	0
$459$	1.35209e6	0.299553
$460$	0	0
$461$	4.32111e6	0.946984	0.473492	−	0.880798i	$-0.342993\pi$
$461$	4.32111e6	0.946984	0.473492	+	0.880798i	$0.342993\pi$
$462$	0	0
$463$	−5.57763e6	−1.20920	−0.604599	−	0.796530i	$-0.706667\pi$
$463$	−5.57763e6	−1.20920	−0.604599	+	0.796530i	$0.706667\pi$
$464$	0	0
$465$	−257730.	−0.0552754
$466$	0	0
$467$	−1.95457e6	−0.414724	−0.207362	−	0.978264i	$-0.566488\pi$
$467$	−1.95457e6	−0.414724	−0.207362	+	0.978264i	$0.566488\pi$
$468$	0	0
$469$	3.75838e6	0.788984
$470$	0	0
$471$	48456.5	0.0100647
$472$	0	0
$473$	−88363.0	−0.0181601
$474$	0	0
$475$	−6.45594e6	−1.31288
$476$	0	0
$477$	2.56302e6	0.515769
$478$	0	0
$479$	−3.03700e6	−0.604793	−0.302396	−	0.953182i	$-0.597787\pi$
$479$	−3.03700e6	−0.604793	−0.302396	+	0.953182i	$0.597787\pi$
$480$	0	0
$481$	171312.	0.0337619
$482$	0	0
$483$	8.03283e6	1.56675
$484$	0	0
$485$	215766.	0.0416513
$486$	0	0
$487$	8.37573e6	1.60030	0.800148	−	0.599803i	$-0.204754\pi$
$487$	8.37573e6	1.60030	0.800148	+	0.599803i	$0.204754\pi$
$488$	0	0
$489$	5.85341e6	1.10697
$490$	0	0
$491$	1.40343e6	0.262716	0.131358	−	0.991335i	$-0.458066\pi$
$491$	1.40343e6	0.262716	0.131358	+	0.991335i	$0.458066\pi$
$492$	0	0
$493$	−4.60596e6	−0.853500
$494$	0	0
$495$	−70752.1	−0.0129786
$496$	0	0
$497$	−5.17878e6	−0.940452
$498$	0	0
$499$	2.14531e6	0.385691	0.192845	−	0.981229i	$-0.438228\pi$
$499$	2.14531e6	0.385691	0.192845	+	0.981229i	$0.438228\pi$
$500$	0	0
$501$	−2.12281e6	−0.377848
$502$	0	0
$503$	−5.55969e6	−0.979785	−0.489892	−	0.871783i	$-0.662964\pi$
$503$	−5.55969e6	−0.979785	−0.489892	+	0.871783i	$0.662964\pi$
$504$	0	0
$505$	761261.	0.132833
$506$	0	0
$507$	257049.	0.0444116
$508$	0	0
$509$	−1.12359e6	−0.192227	−0.0961136	−	0.995370i	$-0.530641\pi$
$509$	−1.12359e6	−0.192227	−0.0961136	+	0.995370i	$0.530641\pi$
$510$	0	0
$511$	2.60915e6	0.442025
$512$	0	0
$513$	1.52471e6	0.255797
$514$	0	0
$515$	437059.	0.0726142
$516$	0	0
$517$	1.27893e6	0.210437
$518$	0	0
$519$	2.99054e6	0.487339
$520$	0	0
$521$	−3.76416e6	−0.607538	−0.303769	−	0.952746i	$-0.598245\pi$
$521$	−3.76416e6	−0.607538	−0.303769	+	0.952746i	$0.598245\pi$
$522$	0	0
$523$	−591216.	−0.0945131	−0.0472565	−	0.998883i	$-0.515048\pi$
$523$	−591216.	−0.0945131	−0.0472565	+	0.998883i	$0.515048\pi$
$524$	0	0
$525$	5.29536e6	0.838488
$526$	0	0
$527$	−8.58580e6	−1.34665
$528$	0	0
$529$	1.54889e7	2.40648
$530$	0	0
$531$	2.10215e6	0.323539
$532$	0	0
$533$	2.41396e6	0.368055
$534$	0	0
$535$	315959.	0.0477251
$536$	0	0
$537$	−5.51673e6	−0.825555
$538$	0	0
$539$	−2.75715e6	−0.408779
$540$	0	0
$541$	−5.87874e6	−0.863557	−0.431778	−	0.901980i	$-0.642114\pi$
$541$	−5.87874e6	−0.863557	−0.431778	+	0.901980i	$0.642114\pi$
$542$	0	0
$543$	1.24287e6	0.180895
$544$	0	0
$545$	1.06980e6	0.154281
$546$	0	0
$547$	9.43385e6	1.34810	0.674048	−	0.738688i	$-0.264554\pi$
$547$	9.43385e6	1.34810	0.674048	+	0.738688i	$0.264554\pi$
$548$	0	0
$549$	−383575.	−0.0543149
$550$	0	0
$551$	−5.19401e6	−0.728826
$552$	0	0
$553$	−1.43497e7	−1.99540
$554$	0	0
$555$	56437.0	0.00777734
$556$	0	0
$557$	9.68718e6	1.32300	0.661499	−	0.749946i	$-0.269921\pi$
$557$	9.68718e6	1.32300	0.661499	+	0.749946i	$0.269921\pi$
$558$	0	0
$559$	105760.	0.0143150
$560$	0	0
$561$	−2.35698e6	−0.316191
$562$	0	0
$563$	−1.16995e7	−1.55560	−0.777800	−	0.628512i	$-0.783664\pi$
$563$	−1.16995e7	−1.55560	−0.777800	+	0.628512i	$0.783664\pi$
$564$	0	0
$565$	94020.9	0.0123909
$566$	0	0
$567$	−1.25062e6	−0.163368
$568$	0	0
$569$	1.47102e6	0.190475	0.0952373	−	0.995455i	$-0.469639\pi$
$569$	1.47102e6	0.190475	0.0952373	+	0.995455i	$0.469639\pi$
$570$	0	0
$571$	5.53198e6	0.710052	0.355026	−	0.934856i	$-0.384472\pi$
$571$	5.53198e6	0.710052	0.355026	+	0.934856i	$0.384472\pi$
$572$	0	0
$573$	−848411.	−0.107949
$574$	0	0
$575$	1.44534e7	1.82306
$576$	0	0
$577$	1.46652e7	1.83378	0.916891	−	0.399138i	$-0.130691\pi$
$577$	1.46652e7	1.83378	0.916891	+	0.399138i	$0.130691\pi$
$578$	0	0
$579$	9.29480e6	1.15224
$580$	0	0
$581$	−1.99780e7	−2.45534
$582$	0	0
$583$	−4.46789e6	−0.544416
$584$	0	0
$585$	84681.9	0.0102306
$586$	0	0
$587$	−4.66463e6	−0.558756	−0.279378	−	0.960181i	$-0.590128\pi$
$587$	−4.66463e6	−0.558756	−0.279378	+	0.960181i	$0.590128\pi$
$588$	0	0
$589$	−9.68196e6	−1.14994
$590$	0	0
$591$	−1.41598e6	−0.166759
$592$	0	0
$593$	9.26934e6	1.08246	0.541230	−	0.840875i	$-0.317959\pi$
$593$	9.26934e6	1.08246	0.541230	+	0.840875i	$0.317959\pi$
$594$	0	0
$595$	−2.18701e6	−0.253255
$596$	0	0
$597$	−4.59431e6	−0.527576
$598$	0	0
$599$	−4.96013e6	−0.564840	−0.282420	−	0.959291i	$-0.591137\pi$
$599$	−4.96013e6	−0.564840	−0.282420	+	0.959291i	$0.591137\pi$
$600$	0	0
$601$	−7.53100e6	−0.850485	−0.425243	−	0.905079i	$-0.639811\pi$
$601$	−7.53100e6	−0.850485	−0.425243	+	0.905079i	$0.639811\pi$
$602$	0	0
$603$	−1.59710e6	−0.178870
$604$	0	0
$605$	−872946.	−0.0969614
$606$	0	0
$607$	1.70472e7	1.87794	0.938970	−	0.343999i	$-0.111782\pi$
$607$	1.70472e7	1.87794	0.938970	+	0.343999i	$0.111782\pi$
$608$	0	0
$609$	4.26029e6	0.465474
$610$	0	0
$611$	−1.53073e6	−0.165881
$612$	0	0
$613$	−5.65370e6	−0.607689	−0.303845	−	0.952722i	$-0.598270\pi$
$613$	−5.65370e6	−0.607689	−0.303845	+	0.952722i	$0.598270\pi$
$614$	0	0
$615$	795252.	0.0847846
$616$	0	0
$617$	−5.06541e6	−0.535676	−0.267838	−	0.963464i	$-0.586309\pi$
$617$	−5.06541e6	−0.535676	−0.267838	+	0.963464i	$0.586309\pi$
$618$	0	0
$619$	−8.67695e6	−0.910208	−0.455104	−	0.890438i	$-0.650398\pi$
$619$	−8.67695e6	−0.910208	−0.455104	+	0.890438i	$0.650398\pi$
$620$	0	0
$621$	−3.41350e6	−0.355198
$622$	0	0
$623$	−909437.	−0.0938755
$624$	0	0
$625$	9.40833e6	0.963413
$626$	0	0
$627$	−2.65790e6	−0.270004
$628$	0	0
$629$	1.88010e6	0.189476
$630$	0	0
$631$	1.39822e7	1.39798	0.698992	−	0.715129i	$-0.253632\pi$
$631$	1.39822e7	1.39798	0.698992	+	0.715129i	$0.253632\pi$
$632$	0	0
$633$	3.97815e6	0.394613
$634$	0	0
$635$	−509995.	−0.0501916
$636$	0	0
$637$	3.29998e6	0.322228
$638$	0	0
$639$	2.20069e6	0.213210
$640$	0	0
$641$	1.51645e6	0.145775	0.0728876	−	0.997340i	$-0.476779\pi$
$641$	1.51645e6	0.145775	0.0728876	+	0.997340i	$0.476779\pi$
$642$	0	0
$643$	−1.42916e7	−1.36318	−0.681589	−	0.731735i	$-0.738711\pi$
$643$	−1.42916e7	−1.36318	−0.681589	+	0.731735i	$0.738711\pi$
$644$	0	0
$645$	34841.4	0.00329759
$646$	0	0
$647$	−1.98547e6	−0.186468	−0.0932338	−	0.995644i	$-0.529720\pi$
$647$	−1.98547e6	−0.186468	−0.0932338	+	0.995644i	$0.529720\pi$
$648$	0	0
$649$	−3.66450e6	−0.341509
$650$	0	0
$651$	7.94144e6	0.734423
$652$	0	0
$653$	−1.55847e7	−1.43027	−0.715133	−	0.698989i	$-0.753634\pi$
$653$	−1.55847e7	−1.43027	−0.715133	+	0.698989i	$0.753634\pi$
$654$	0	0
$655$	1.86793e6	0.170121
$656$	0	0
$657$	−1.10874e6	−0.100211
$658$	0	0
$659$	75439.4	0.00676682	0.00338341	−	0.999994i	$-0.498923\pi$
$659$	75439.4	0.00676682	0.00338341	+	0.999994i	$0.498923\pi$
$660$	0	0
$661$	3.11557e6	0.277353	0.138677	−	0.990338i	$-0.455715\pi$
$661$	3.11557e6	0.277353	0.138677	+	0.990338i	$0.455715\pi$
$662$	0	0
$663$	2.82103e6	0.249243
$664$	0	0
$665$	−2.46623e6	−0.216261
$666$	0	0
$667$	1.16283e7	1.01205
$668$	0	0
$669$	8.75484e6	0.756280
$670$	0	0
$671$	668653.	0.0573316
$672$	0	0
$673$	1.40632e7	1.19687	0.598435	−	0.801172i	$-0.295790\pi$
$673$	1.40632e7	1.19687	0.598435	+	0.801172i	$0.295790\pi$
$674$	0	0
$675$	−2.25023e6	−0.190093
$676$	0	0
$677$	−6.04060e6	−0.506533	−0.253267	−	0.967396i	$-0.581505\pi$
$677$	−6.04060e6	−0.506533	−0.253267	+	0.967396i	$0.581505\pi$
$678$	0	0
$679$	−6.64842e6	−0.553406
$680$	0	0
$681$	−2.32175e6	−0.191843
$682$	0	0
$683$	−1.32539e7	−1.08716	−0.543578	−	0.839359i	$-0.682931\pi$
$683$	−1.32539e7	−1.08716	−0.543578	+	0.839359i	$0.682931\pi$
$684$	0	0
$685$	−2.39973e6	−0.195405
$686$	0	0
$687$	1.20426e6	0.0973482
$688$	0	0
$689$	5.34753e6	0.429146
$690$	0	0
$691$	−459579.	−0.0366155	−0.0183077	−	0.999832i	$-0.505828\pi$
$691$	−459579.	−0.0366155	−0.0183077	+	0.999832i	$0.505828\pi$
$692$	0	0
$693$	2.18009e6	0.172441
$694$	0	0
$695$	1.33536e6	0.104867
$696$	0	0
$697$	2.64924e7	2.06557
$698$	0	0
$699$	−1.45929e7	−1.12966
$700$	0	0
$701$	−1.53638e7	−1.18087	−0.590437	−	0.807083i	$-0.701045\pi$
$701$	−1.53638e7	−1.18087	−0.590437	+	0.807083i	$0.701045\pi$
$702$	0	0
$703$	2.12013e6	0.161799
$704$	0	0
$705$	−504282.	−0.0382121
$706$	0	0
$707$	−2.34568e7	−1.76490
$708$	0	0
$709$	7.38149e6	0.551478	0.275739	−	0.961232i	$-0.411077\pi$
$709$	7.38149e6	0.551478	0.275739	+	0.961232i	$0.411077\pi$
$710$	0	0
$711$	6.09781e6	0.452376
$712$	0	0
$713$	2.16758e7	1.59680
$714$	0	0
$715$	−147619.	−0.0107988
$716$	0	0
$717$	7.30349e6	0.530557
$718$	0	0
$719$	2.47769e7	1.78741	0.893707	−	0.448651i	$-0.148096\pi$
$719$	2.47769e7	1.78741	0.893707	+	0.448651i	$0.148096\pi$
$720$	0	0
$721$	−1.34671e7	−0.964798
$722$	0	0
$723$	2.27695e6	0.161997
$724$	0	0
$725$	7.66552e6	0.541622
$726$	0	0
$727$	−1.24758e7	−0.875452	−0.437726	−	0.899108i	$-0.644216\pi$
$727$	−1.24758e7	−0.875452	−0.437726	+	0.899108i	$0.644216\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.16068e6	0.0803377
$732$	0	0
$733$	2.67161e6	0.183659	0.0918296	−	0.995775i	$-0.470728\pi$
$733$	2.67161e6	0.183659	0.0918296	+	0.995775i	$0.470728\pi$
$734$	0	0
$735$	1.08714e6	0.0742280
$736$	0	0
$737$	2.78409e6	0.188805
$738$	0	0
$739$	1.53282e7	1.03248	0.516239	−	0.856445i	$-0.327332\pi$
$739$	1.53282e7	1.03248	0.516239	+	0.856445i	$0.327332\pi$
$740$	0	0
$741$	3.18119e6	0.212836
$742$	0	0
$743$	9.13987e6	0.607391	0.303695	−	0.952769i	$-0.401780\pi$
$743$	9.13987e6	0.607391	0.303695	+	0.952769i	$0.401780\pi$
$744$	0	0
$745$	1.89861e6	0.125327
$746$	0	0
$747$	8.48952e6	0.556649
$748$	0	0
$749$	−9.73567e6	−0.634105
$750$	0	0
$751$	1.95912e6	0.126754	0.0633769	−	0.997990i	$-0.479813\pi$
$751$	1.95912e6	0.126754	0.0633769	+	0.997990i	$0.479813\pi$
$752$	0	0
$753$	−8.63196e6	−0.554782
$754$	0	0
$755$	−1.16449e6	−0.0743475
$756$	0	0
$757$	−1.81437e7	−1.15076	−0.575382	−	0.817885i	$-0.695147\pi$
$757$	−1.81437e7	−1.15076	−0.575382	+	0.817885i	$0.695147\pi$
$758$	0	0
$759$	5.95046e6	0.374926
$760$	0	0
$761$	1.52710e7	0.955883	0.477941	−	0.878392i	$-0.341383\pi$
$761$	1.52710e7	0.955883	0.477941	+	0.878392i	$0.341383\pi$
$762$	0	0
$763$	−3.29638e7	−2.04987
$764$	0	0
$765$	929355.	0.0574154
$766$	0	0
$767$	4.38596e6	0.269201
$768$	0	0
$769$	1.54324e7	0.941062	0.470531	−	0.882384i	$-0.344062\pi$
$769$	1.54324e7	0.941062	0.470531	+	0.882384i	$0.344062\pi$
$770$	0	0
$771$	−3.81880e6	−0.231361
$772$	0	0
$773$	342144.	0.0205949	0.0102975	−	0.999947i	$-0.496722\pi$
$773$	342144.	0.0205949	0.0102975	+	0.999947i	$0.496722\pi$
$774$	0	0
$775$	1.42890e7	0.854570
$776$	0	0
$777$	−1.73900e6	−0.103335
$778$	0	0
$779$	2.98747e7	1.76385
$780$	0	0
$781$	−3.83628e6	−0.225052
$782$	0	0
$783$	−1.81038e6	−0.105528
$784$	0	0
$785$	33306.4	0.00192910
$786$	0	0
$787$	2.07885e7	1.19643	0.598213	−	0.801337i	$-0.295878\pi$
$787$	2.07885e7	1.19643	0.598213	+	0.801337i	$0.295878\pi$
$788$	0	0
$789$	1.47360e7	0.842725
$790$	0	0
$791$	−2.89707e6	−0.164633
$792$	0	0
$793$	−800298.	−0.0451927
$794$	0	0
$795$	1.76168e6	0.0988576
$796$	0	0
$797$	−2.31380e7	−1.29027	−0.645133	−	0.764070i	$-0.723198\pi$
$797$	−2.31380e7	−1.29027	−0.645133	+	0.764070i	$0.723198\pi$
$798$	0	0
$799$	−1.67993e7	−0.930944
$800$	0	0
$801$	386459.	0.0212825
$802$	0	0
$803$	1.93278e6	0.105777
$804$	0	0
$805$	5.52134e6	0.300300
$806$	0	0
$807$	−1.19477e7	−0.645804
$808$	0	0
$809$	1.89827e7	1.01973	0.509866	−	0.860254i	$-0.329695\pi$
$809$	1.89827e7	1.01973	0.509866	+	0.860254i	$0.329695\pi$
$810$	0	0
$811$	1.05961e7	0.565712	0.282856	−	0.959162i	$-0.408718\pi$
$811$	1.05961e7	0.565712	0.282856	+	0.959162i	$0.408718\pi$
$812$	0	0
$813$	−946148.	−0.0502033
$814$	0	0
$815$	4.02333e6	0.212174
$816$	0	0
$817$	1.30887e6	0.0686025
$818$	0	0
$819$	−2.60931e6	−0.135930
$820$	0	0
$821$	1.09767e7	0.568346	0.284173	−	0.958773i	$-0.408281\pi$
$821$	1.09767e7	0.568346	0.284173	+	0.958773i	$0.408281\pi$
$822$	0	0
$823$	1.13830e7	0.585810	0.292905	−	0.956142i	$-0.405378\pi$
$823$	1.13830e7	0.585810	0.292905	+	0.956142i	$0.405378\pi$
$824$	0	0
$825$	3.92263e6	0.200651
$826$	0	0
$827$	−9.41841e6	−0.478866	−0.239433	−	0.970913i	$-0.576961\pi$
$827$	−9.41841e6	−0.478866	−0.239433	+	0.970913i	$0.576961\pi$
$828$	0	0
$829$	1.04972e7	0.530502	0.265251	−	0.964179i	$-0.414545\pi$
$829$	1.04972e7	0.530502	0.265251	+	0.964179i	$0.414545\pi$
$830$	0	0
$831$	−1.15911e7	−0.582269
$832$	0	0
$833$	3.62162e7	1.80838
$834$	0	0
$835$	−1.45911e6	−0.0724222
$836$	0	0
$837$	−3.37466e6	−0.166501
$838$	0	0
$839$	6.67378e6	0.327316	0.163658	−	0.986517i	$-0.447671\pi$
$839$	6.67378e6	0.327316	0.163658	+	0.986517i	$0.447671\pi$
$840$	0	0
$841$	−1.43440e7	−0.699327
$842$	0	0
$843$	−2.68116e6	−0.129943
$844$	0	0
$845$	176682.	0.00851237
$846$	0	0
$847$	2.68981e7	1.28829
$848$	0	0
$849$	8.32882e6	0.396565
$850$	0	0
$851$	−4.74651e6	−0.224673
$852$	0	0
$853$	3.58582e6	0.168739	0.0843695	−	0.996435i	$-0.473112\pi$
$853$	3.58582e6	0.168739	0.0843695	+	0.996435i	$0.473112\pi$
$854$	0	0
$855$	1.04801e6	0.0490286
$856$	0	0
$857$	1.55304e7	0.722321	0.361160	−	0.932504i	$-0.382381\pi$
$857$	1.55304e7	0.722321	0.361160	+	0.932504i	$0.382381\pi$
$858$	0	0
$859$	−3.10835e7	−1.43730	−0.718649	−	0.695373i	$-0.755239\pi$
$859$	−3.10835e7	−1.43730	−0.718649	+	0.695373i	$0.755239\pi$
$860$	0	0
$861$	−2.45042e7	−1.12650
$862$	0	0
$863$	−3.59926e7	−1.64508	−0.822539	−	0.568708i	$-0.807443\pi$
$863$	−3.59926e7	−1.64508	−0.822539	+	0.568708i	$0.807443\pi$
$864$	0	0
$865$	2.05554e6	0.0934083
$866$	0	0
$867$	1.81811e7	0.821434
$868$	0	0
$869$	−1.06298e7	−0.477502
$870$	0	0
$871$	−3.33222e6	−0.148829
$872$	0	0
$873$	2.82520e6	0.125462
$874$	0	0
$875$	7.32462e6	0.323419
$876$	0	0
$877$	−1.74870e7	−0.767746	−0.383873	−	0.923386i	$-0.625410\pi$
$877$	−1.74870e7	−0.767746	−0.383873	+	0.923386i	$0.625410\pi$
$878$	0	0
$879$	1.93060e7	0.842793
$880$	0	0
$881$	1.52380e7	0.661437	0.330719	−	0.943729i	$-0.392709\pi$
$881$	1.52380e7	0.661437	0.330719	+	0.943729i	$0.392709\pi$
$882$	0	0
$883$	3.22630e7	1.39252	0.696262	−	0.717787i	$-0.254845\pi$
$883$	3.22630e7	1.39252	0.696262	+	0.717787i	$0.254845\pi$
$884$	0	0
$885$	1.44491e6	0.0620128
$886$	0	0
$887$	−2.82559e7	−1.20587	−0.602934	−	0.797791i	$-0.706002\pi$
$887$	−2.82559e7	−1.20587	−0.602934	+	0.797791i	$0.706002\pi$
$888$	0	0
$889$	1.57145e7	0.666877
$890$	0	0
$891$	−926416.	−0.0390941
$892$	0	0
$893$	−1.89441e7	−0.794959
$894$	0	0
$895$	−3.79191e6	−0.158234
$896$	0	0
$897$	−7.12199e6	−0.295543
$898$	0	0
$899$	1.14960e7	0.474402
$900$	0	0
$901$	5.86873e7	2.40842
$902$	0	0
$903$	−1.07357e6	−0.0438139
$904$	0	0
$905$	854283.	0.0346721
$906$	0	0
$907$	1.87244e7	0.755771	0.377885	−	0.925852i	$-0.376651\pi$
$907$	1.87244e7	0.755771	0.377885	+	0.925852i	$0.376651\pi$
$908$	0	0
$909$	9.96781e6	0.400120
$910$	0	0
$911$	1.24649e7	0.497616	0.248808	−	0.968553i	$-0.419961\pi$
$911$	1.24649e7	0.497616	0.248808	+	0.968553i	$0.419961\pi$
$912$	0	0
$913$	−1.47991e7	−0.587567
$914$	0	0
$915$	−263649.	−0.0104105
$916$	0	0
$917$	−5.75566e7	−2.26033
$918$	0	0
$919$	3.05164e7	1.19191	0.595956	−	0.803017i	$-0.296773\pi$
$919$	3.05164e7	1.19191	0.595956	+	0.803017i	$0.296773\pi$
$920$	0	0
$921$	−2.28101e7	−0.886091
$922$	0	0
$923$	4.59157e6	0.177401
$924$	0	0
$925$	−3.12897e6	−0.120239
$926$	0	0
$927$	5.72276e6	0.218729
$928$	0	0
$929$	−1.89467e7	−0.720267	−0.360134	−	0.932901i	$-0.617269\pi$
$929$	−1.89467e7	−0.720267	−0.360134	+	0.932901i	$0.617269\pi$
$930$	0	0
$931$	4.08400e7	1.54423
$932$	0	0
$933$	1.26494e7	0.475737
$934$	0	0
$935$	−1.62007e6	−0.0606043
$936$	0	0
$937$	6.77539e6	0.252107	0.126054	−	0.992023i	$-0.459769\pi$
$937$	6.77539e6	0.252107	0.126054	+	0.992023i	$0.459769\pi$
$938$	0	0
$939$	−2.75244e7	−1.01872
$940$	0	0
$941$	−4.09175e7	−1.50638	−0.753190	−	0.657802i	$-0.771486\pi$
$941$	−4.09175e7	−1.50638	−0.753190	+	0.657802i	$0.771486\pi$
$942$	0	0
$943$	−6.68830e7	−2.44927
$944$	0	0
$945$	−859607.	−0.0313127
$946$	0	0
$947$	3.30315e7	1.19689	0.598445	−	0.801164i	$-0.295786\pi$
$947$	3.30315e7	1.19689	0.598445	+	0.801164i	$0.295786\pi$
$948$	0	0
$949$	−2.31330e6	−0.0833810
$950$	0	0
$951$	4.89509e6	0.175513
$952$	0	0
$953$	7.16060e6	0.255398	0.127699	−	0.991813i	$-0.459241\pi$
$953$	7.16060e6	0.255398	0.127699	+	0.991813i	$0.459241\pi$
$954$	0	0
$955$	−583153.	−0.0206906
$956$	0	0
$957$	3.15588e6	0.111389
$958$	0	0
$959$	7.39429e7	2.59627
$960$	0	0
$961$	−7.19996e6	−0.251491
$962$	0	0
$963$	4.13711e6	0.143758
$964$	0	0
$965$	6.38875e6	0.220850
$966$	0	0
$967$	−5.72729e6	−0.196962	−0.0984812	−	0.995139i	$-0.531398\pi$
$967$	−5.72729e6	−0.196962	−0.0984812	+	0.995139i	$0.531398\pi$
$968$	0	0
$969$	3.49125e7	1.19446
$970$	0	0
$971$	−1.19944e7	−0.408254	−0.204127	−	0.978944i	$-0.565436\pi$
$971$	−1.19944e7	−0.408254	−0.204127	+	0.978944i	$0.565436\pi$
$972$	0	0
$973$	−4.11466e7	−1.39332
$974$	0	0
$975$	−4.69492e6	−0.158167
$976$	0	0
$977$	−2.39435e7	−0.802512	−0.401256	−	0.915966i	$-0.631426\pi$
$977$	−2.39435e7	−0.802512	−0.401256	+	0.915966i	$0.631426\pi$
$978$	0	0
$979$	−673682.	−0.0224646
$980$	0	0
$981$	1.40078e7	0.464726
$982$	0	0
$983$	−5.44669e7	−1.79783	−0.898915	−	0.438123i	$-0.855643\pi$
$983$	−5.44669e7	−1.79783	−0.898915	+	0.438123i	$0.855643\pi$
$984$	0	0
$985$	−973270.	−0.0319626
$986$	0	0
$987$	1.55385e7	0.507710
$988$	0	0
$989$	−2.93027e6	−0.0952613
$990$	0	0
$991$	−1.99942e7	−0.646724	−0.323362	−	0.946275i	$-0.604813\pi$
$991$	−1.99942e7	−0.646724	−0.323362	+	0.946275i	$0.604813\pi$
$992$	0	0
$993$	2.88702e7	0.929130
$994$	0	0
$995$	−3.15789e6	−0.101120
$996$	0	0
$997$	5.61651e6	0.178949	0.0894744	−	0.995989i	$-0.471481\pi$
$997$	5.61651e6	0.178949	0.0894744	+	0.995989i	$0.471481\pi$
$998$	0	0
$999$	738975.	0.0234270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.t.1.2		3
4.3	odd	2		39.6.a.c.1.2	✓	3
12.11	even	2		117.6.a.e.1.2		3
20.19	odd	2		975.6.a.d.1.2		3
52.51	odd	2		507.6.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.6.a.c.1.2	✓	3	4.3	odd	2
117.6.a.e.1.2		3	12.11	even	2
507.6.a.d.1.2		3	52.51	odd	2
624.6.a.t.1.2		3	1.1	even	1	trivial
975.6.a.d.1.2		3	20.19	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +6.18613 q^{5} -190.614 q^{7} +81.0000 q^{9} -141.200 q^{11} +169.000 q^{13} +55.6751 q^{15} +1854.72 q^{17} +2091.51 q^{19} -1715.52 q^{21} -4682.44 q^{23} -3086.73 q^{25} +729.000 q^{27} -2483.38 q^{29} -4629.17 q^{31} -1270.80 q^{33} -1179.16 q^{35} +1013.68 q^{37} +1521.00 q^{39} +14283.8 q^{41} +625.799 q^{43} +501.076 q^{45} -9057.59 q^{47} +19526.5 q^{49} +16692.5 q^{51} +31642.2 q^{53} -873.483 q^{55} +18823.6 q^{57} +25952.5 q^{59} -4735.49 q^{61} -15439.7 q^{63} +1045.46 q^{65} -19717.3 q^{67} -42142.0 q^{69} +27169.0 q^{71} -13688.2 q^{73} -27780.6 q^{75} +26914.7 q^{77} +75281.6 q^{79} +6561.00 q^{81} +104809. q^{83} +11473.5 q^{85} -22350.4 q^{87} +4771.10 q^{89} -32213.7 q^{91} -41662.5 q^{93} +12938.4 q^{95} +34879.0 q^{97} -11437.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 27 q^{3} + 54 q^{5} - 84 q^{7} + 243 q^{9} - 876 q^{11} + 507 q^{13} + 486 q^{15} + 102 q^{17} + 16 q^{19} - 756 q^{21} - 1363 q^{25} + 2187 q^{27} + 9666 q^{29} + 10196 q^{31} - 7884 q^{33} - 16680 q^{35}+ \cdots - 70956 q^{99}+O(q^{100}) \) 3 * q + 27 * q^3 + 54 * q^5 - 84 * q^7 + 243 * q^9 - 876 * q^11 + 507 * q^13 + 486 * q^15 + 102 * q^17 + 16 * q^19 - 756 * q^21 - 1363 * q^25 + 2187 * q^27 + 9666 * q^29 + 10196 * q^31 - 7884 * q^33 - 16680 * q^35 + 11818 * q^37 + 4563 * q^39 + 35490 * q^41 - 2780 * q^43 + 4374 * q^45 - 25728 * q^47 + 39291 * q^49 + 918 * q^51 + 36786 * q^53 + 6440 * q^55 + 144 * q^57 + 27516 * q^59 + 45994 * q^61 - 6804 * q^63 + 9126 * q^65 + 42536 * q^67 + 54432 * q^71 + 27846 * q^73 - 12267 * q^75 - 78000 * q^77 + 80568 * q^79 + 19683 * q^81 - 24012 * q^83 - 186580 * q^85 + 86994 * q^87 + 117450 * q^89 - 14196 * q^91 + 91764 * q^93 + 64608 * q^95 - 20930 * q^97 - 70956 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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