LMFDB - Embedded newform 882.6.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{9601}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2400 $ x^2 - x - 2400
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$49.4923$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -22.4923 q^{5} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} -22.4923 q^{5} -64.0000 q^{8} +89.9694 q^{10} -340.462 q^{11} -728.416 q^{13} +256.000 q^{16} -809.847 q^{17} +1026.20 q^{19} -359.878 q^{20} +1361.85 q^{22} -1422.03 q^{23} -2619.09 q^{25} +2913.66 q^{26} -5218.03 q^{29} -7037.74 q^{31} -1024.00 q^{32} +3239.39 q^{34} +12792.1 q^{37} -4104.81 q^{38} +1439.51 q^{40} -1173.51 q^{41} +3664.17 q^{43} -5447.39 q^{44} +5688.12 q^{46} -9313.19 q^{47} +10476.4 q^{50} -11654.7 q^{52} -35642.8 q^{53} +7657.78 q^{55} +20872.1 q^{58} +30376.2 q^{59} +32186.2 q^{61} +28151.0 q^{62} +4096.00 q^{64} +16383.8 q^{65} +21351.1 q^{67} -12957.6 q^{68} -61153.7 q^{71} +41267.8 q^{73} -51168.4 q^{74} +16419.2 q^{76} -35000.5 q^{79} -5758.04 q^{80} +4694.02 q^{82} -86193.0 q^{83} +18215.4 q^{85} -14656.7 q^{86} +21789.6 q^{88} -77992.6 q^{89} -22752.5 q^{92} +37252.8 q^{94} -23081.7 q^{95} -161765. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 32 * q^4 + 53 * q^5 - 128 * q^8	$ 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8} - 212 q^{10} - 191 q^{11} - 379 q^{13} + 512 q^{16} + 340 q^{17} - 1769 q^{19} + 848 q^{20} + 764 q^{22} - 3236 q^{23} - 45 q^{25} + 1516 q^{26} - 4459 q^{29} + 1994 q^{31} - 2048 q^{32} - 1360 q^{34} + 20587 q^{37} + 7076 q^{38} - 3392 q^{40} - 8814 q^{41} + 15853 q^{43} - 3056 q^{44} + 12944 q^{46} - 33912 q^{47} + 180 q^{50} - 6064 q^{52} - 49239 q^{53} + 18941 q^{55} + 17836 q^{58} + 56735 q^{59} + 67508 q^{61} - 7976 q^{62} + 8192 q^{64} + 42762 q^{65} + 75723 q^{67} + 5440 q^{68} + 8992 q^{71} - 3201 q^{73} - 82348 q^{74} - 28304 q^{76} + 26612 q^{79} + 13568 q^{80} + 35256 q^{82} + 949 q^{83} + 105020 q^{85} - 63412 q^{86} + 12224 q^{88} - 176562 q^{89} - 51776 q^{92} + 135648 q^{94} - 234098 q^{95} - 129423 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 53 * q^5 - 128 * q^8 - 212 * q^10 - 191 * q^11 - 379 * q^13 + 512 * q^16 + 340 * q^17 - 1769 * q^19 + 848 * q^20 + 764 * q^22 - 3236 * q^23 - 45 * q^25 + 1516 * q^26 - 4459 * q^29 + 1994 * q^31 - 2048 * q^32 - 1360 * q^34 + 20587 * q^37 + 7076 * q^38 - 3392 * q^40 - 8814 * q^41 + 15853 * q^43 - 3056 * q^44 + 12944 * q^46 - 33912 * q^47 + 180 * q^50 - 6064 * q^52 - 49239 * q^53 + 18941 * q^55 + 17836 * q^58 + 56735 * q^59 + 67508 * q^61 - 7976 * q^62 + 8192 * q^64 + 42762 * q^65 + 75723 * q^67 + 5440 * q^68 + 8992 * q^71 - 3201 * q^73 - 82348 * q^74 - 28304 * q^76 + 26612 * q^79 + 13568 * q^80 + 35256 * q^82 + 949 * q^83 + 105020 * q^85 - 63412 * q^86 + 12224 * q^88 - 176562 * q^89 - 51776 * q^92 + 135648 * q^94 - 234098 * q^95 - 129423 * q^97

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−22.4923	−0.402355	−0.201178	−	0.979555i	$-0.564477\pi$
$5$	−22.4923	−0.402355	−0.201178	+	0.979555i	$0.564477\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	89.9694	0.284508
$11$	−340.462	−0.848373	−0.424186	−	0.905575i	$-0.639440\pi$
$11$	−340.462	−0.848373	−0.424186	+	0.905575i	$0.639440\pi$
$12$	0	0
$13$	−728.416	−1.19542	−0.597711	−	0.801712i	$-0.703923\pi$
$13$	−728.416	−1.19542	−0.597711	+	0.801712i	$0.703923\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−809.847	−0.679643	−0.339821	−	0.940490i	$-0.610367\pi$
$17$	−809.847	−0.679643	−0.339821	+	0.940490i	$0.610367\pi$
$18$	0	0
$19$	1026.20	0.652152	0.326076	−	0.945344i	$-0.394274\pi$
$19$	1026.20	0.652152	0.326076	+	0.945344i	$0.394274\pi$
$20$	−359.878	−0.201178
$21$	0	0
$22$	1361.85	0.599890
$23$	−1422.03	−0.560518	−0.280259	−	0.959924i	$-0.590420\pi$
$23$	−1422.03	−0.560518	−0.280259	+	0.959924i	$0.590420\pi$
$24$	0	0
$25$	−2619.09	−0.838110
$26$	2913.66	0.845290
$27$	0	0
$28$	0	0
$29$	−5218.03	−1.15216	−0.576079	−	0.817394i	$-0.695418\pi$
$29$	−5218.03	−1.15216	−0.576079	+	0.817394i	$0.695418\pi$
$30$	0	0
$31$	−7037.74	−1.31531	−0.657657	−	0.753318i	$-0.728452\pi$
$31$	−7037.74	−1.31531	−0.657657	+	0.753318i	$0.728452\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	3239.39	0.480580
$35$	0	0
$36$	0	0
$37$	12792.1	1.53616	0.768082	−	0.640351i	$-0.221211\pi$
$37$	12792.1	1.53616	0.768082	+	0.640351i	$0.221211\pi$
$38$	−4104.81	−0.461141
$39$	0	0
$40$	1439.51	0.142254
$41$	−1173.51	−0.109025	−0.0545124	−	0.998513i	$-0.517360\pi$
$41$	−1173.51	−0.109025	−0.0545124	+	0.998513i	$0.517360\pi$
$42$	0	0
$43$	3664.17	0.302207	0.151103	−	0.988518i	$-0.451717\pi$
$43$	3664.17	0.302207	0.151103	+	0.988518i	$0.451717\pi$
$44$	−5447.39	−0.424186
$45$	0	0
$46$	5688.12	0.396346
$47$	−9313.19	−0.614970	−0.307485	−	0.951553i	$-0.599487\pi$
$47$	−9313.19	−0.614970	−0.307485	+	0.951553i	$0.599487\pi$
$48$	0	0
$49$	0	0
$50$	10476.4	0.592633
$51$	0	0
$52$	−11654.7	−0.597711
$53$	−35642.8	−1.74294	−0.871469	−	0.490451i	$-0.836832\pi$
$53$	−35642.8	−1.74294	−0.871469	+	0.490451i	$0.836832\pi$
$54$	0	0
$55$	7657.78	0.341347
$56$	0	0
$57$	0	0
$58$	20872.1	0.814698
$59$	30376.2	1.13607	0.568033	−	0.823006i	$-0.307705\pi$
$59$	30376.2	1.13607	0.568033	+	0.823006i	$0.307705\pi$
$60$	0	0
$61$	32186.2	1.10751	0.553753	−	0.832681i	$-0.313195\pi$
$61$	32186.2	1.10751	0.553753	+	0.832681i	$0.313195\pi$
$62$	28151.0	0.930067
$63$	0	0
$64$	4096.00	0.125000
$65$	16383.8	0.480984
$66$	0	0
$67$	21351.1	0.581076	0.290538	−	0.956863i	$-0.406166\pi$
$67$	21351.1	0.581076	0.290538	+	0.956863i	$0.406166\pi$
$68$	−12957.6	−0.339821
$69$	0	0
$70$	0	0
$71$	−61153.7	−1.43972	−0.719859	−	0.694121i	$-0.755793\pi$
$71$	−61153.7	−1.43972	−0.719859	+	0.694121i	$0.755793\pi$
$72$	0	0
$73$	41267.8	0.906367	0.453184	−	0.891417i	$-0.350288\pi$
$73$	41267.8	0.906367	0.453184	+	0.891417i	$0.350288\pi$
$74$	−51168.4	−1.08623
$75$	0	0
$76$	16419.2	0.326076
$77$	0	0
$78$	0	0
$79$	−35000.5	−0.630966	−0.315483	−	0.948931i	$-0.602167\pi$
$79$	−35000.5	−0.630966	−0.315483	+	0.948931i	$0.602167\pi$
$80$	−5758.04	−0.100589
$81$	0	0
$82$	4694.02	0.0770922
$83$	−86193.0	−1.37334	−0.686668	−	0.726972i	$-0.740927\pi$
$83$	−86193.0	−1.37334	−0.686668	+	0.726972i	$0.740927\pi$
$84$	0	0
$85$	18215.4	0.273458
$86$	−14656.7	−0.213692
$87$	0	0
$88$	21789.6	0.299945
$89$	−77992.6	−1.04371	−0.521853	−	0.853035i	$-0.674759\pi$
$89$	−77992.6	−1.04371	−0.521853	+	0.853035i	$0.674759\pi$
$90$	0	0
$91$	0	0
$92$	−22752.5	−0.280259
$93$	0	0
$94$	37252.8	0.434850
$95$	−23081.7	−0.262397
$96$	0	0
$97$	−161765.	−1.74565	−0.872823	−	0.488037i	$-0.837713\pi$
$97$	−161765.	−1.74565	−0.872823	+	0.488037i	$0.837713\pi$
$98$	0	0
$99$	0	0
$100$	−41905.5	−0.419055
$101$	65854.4	0.642364	0.321182	−	0.947017i	$-0.395920\pi$
$101$	65854.4	0.642364	0.321182	+	0.947017i	$0.395920\pi$
$102$	0	0
$103$	130198.	1.20924	0.604619	−	0.796515i	$-0.293326\pi$
$103$	130198.	1.20924	0.604619	+	0.796515i	$0.293326\pi$
$104$	46618.6	0.422645
$105$	0	0
$106$	142571.	1.23244
$107$	−2590.95	−0.0218776	−0.0109388	−	0.999940i	$-0.503482\pi$
$107$	−2590.95	−0.0218776	−0.0109388	+	0.999940i	$0.503482\pi$
$108$	0	0
$109$	110654.	0.892072	0.446036	−	0.895015i	$-0.352835\pi$
$109$	110654.	0.892072	0.446036	+	0.895015i	$0.352835\pi$
$110$	−30631.1	−0.241369
$111$	0	0
$112$	0	0
$113$	193910.	1.42858	0.714291	−	0.699849i	$-0.246749\pi$
$113$	193910.	1.42858	0.714291	+	0.699849i	$0.246749\pi$
$114$	0	0
$115$	31984.8	0.225527
$116$	−83488.5	−0.576079
$117$	0	0
$118$	−121505.	−0.803319
$119$	0	0
$120$	0	0
$121$	−45136.8	−0.280264
$122$	−128745.	−0.783124
$123$	0	0
$124$	−112604.	−0.657657
$125$	129198.	0.739573
$126$	0	0
$127$	27429.2	0.150905	0.0754526	−	0.997149i	$-0.475960\pi$
$127$	27429.2	0.150905	0.0754526	+	0.997149i	$0.475960\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−65535.1	−0.340107
$131$	−150376.	−0.765597	−0.382798	−	0.923832i	$-0.625040\pi$
$131$	−150376.	−0.765597	−0.382798	+	0.923832i	$0.625040\pi$
$132$	0	0
$133$	0	0
$134$	−85404.3	−0.410883
$135$	0	0
$136$	51830.2	0.240290
$137$	−96102.0	−0.437453	−0.218726	−	0.975786i	$-0.570190\pi$
$137$	−96102.0	−0.437453	−0.218726	+	0.975786i	$0.570190\pi$
$138$	0	0
$139$	100854.	0.442749	0.221375	−	0.975189i	$-0.428946\pi$
$139$	100854.	0.442749	0.221375	+	0.975189i	$0.428946\pi$
$140$	0	0
$141$	0	0
$142$	244615.	1.01803
$143$	247998.	1.01416
$144$	0	0
$145$	117366.	0.463577
$146$	−165071.	−0.640898
$147$	0	0
$148$	204674.	0.768082
$149$	360944.	1.33191	0.665954	−	0.745993i	$-0.268025\pi$
$149$	360944.	1.33191	0.665954	+	0.745993i	$0.268025\pi$
$150$	0	0
$151$	−434056.	−1.54918	−0.774592	−	0.632461i	$-0.782045\pi$
$151$	−434056.	−1.54918	−0.774592	+	0.632461i	$0.782045\pi$
$152$	−65676.9	−0.230570
$153$	0	0
$154$	0	0
$155$	158295.	0.529223
$156$	0	0
$157$	−511596.	−1.65645	−0.828225	−	0.560396i	$-0.810649\pi$
$157$	−511596.	−1.65645	−0.828225	+	0.560396i	$0.810649\pi$
$158$	140002.	0.446160
$159$	0	0
$160$	23032.2	0.0711270
$161$	0	0
$162$	0	0
$163$	−251269.	−0.740748	−0.370374	−	0.928883i	$-0.620770\pi$
$163$	−251269.	−0.740748	−0.370374	+	0.928883i	$0.620770\pi$
$164$	−18776.1	−0.0545124
$165$	0	0
$166$	344772.	0.971095
$167$	−419277.	−1.16335	−0.581674	−	0.813422i	$-0.697602\pi$
$167$	−419277.	−1.16335	−0.581674	+	0.813422i	$0.697602\pi$
$168$	0	0
$169$	159297.	0.429032
$170$	−72861.4	−0.193364
$171$	0	0
$172$	58626.7	0.151103
$173$	461376.	1.17203	0.586017	−	0.810299i	$-0.300695\pi$
$173$	461376.	1.17203	0.586017	+	0.810299i	$0.300695\pi$
$174$	0	0
$175$	0	0
$176$	−87158.2	−0.212093
$177$	0	0
$178$	311970.	0.738012
$179$	−741706.	−1.73021	−0.865105	−	0.501590i	$-0.832749\pi$
$179$	−741706.	−1.73021	−0.865105	+	0.501590i	$0.832749\pi$
$180$	0	0
$181$	301371.	0.683762	0.341881	−	0.939743i	$-0.388936\pi$
$181$	301371.	0.683762	0.341881	+	0.939743i	$0.388936\pi$
$182$	0	0
$183$	0	0
$184$	91010.0	0.198173
$185$	−287725.	−0.618084
$186$	0	0
$187$	275722.	0.576590
$188$	−149011.	−0.307485
$189$	0	0
$190$	92326.7	0.185542
$191$	−263183.	−0.522004	−0.261002	−	0.965338i	$-0.584053\pi$
$191$	−263183.	−0.522004	−0.261002	+	0.965338i	$0.584053\pi$
$192$	0	0
$193$	−831127.	−1.60611	−0.803053	−	0.595908i	$-0.796792\pi$
$193$	−831127.	−1.60611	−0.803053	+	0.595908i	$0.796792\pi$
$194$	647061.	1.23436
$195$	0	0
$196$	0	0
$197$	1.05421e6	1.93536	0.967681	−	0.252178i	$-0.0811470\pi$
$197$	1.05421e6	1.93536	0.967681	+	0.252178i	$0.0811470\pi$
$198$	0	0
$199$	698568.	1.25048	0.625239	−	0.780434i	$-0.285002\pi$
$199$	698568.	1.25048	0.625239	+	0.780434i	$0.285002\pi$
$200$	167622.	0.296317
$201$	0	0
$202$	−263418.	−0.454220
$203$	0	0
$204$	0	0
$205$	26394.9	0.0438667
$206$	−520792.	−0.855060
$207$	0	0
$208$	−186474.	−0.298855
$209$	−349382.	−0.553268
$210$	0	0
$211$	−99693.6	−0.154156	−0.0770781	−	0.997025i	$-0.524559\pi$
$211$	−99693.6	−0.154156	−0.0770781	+	0.997025i	$0.524559\pi$
$212$	−570284.	−0.871469
$213$	0	0
$214$	10363.8	0.0154698
$215$	−82415.7	−0.121594
$216$	0	0
$217$	0	0
$218$	−442615.	−0.630790
$219$	0	0
$220$	122525.	0.170674
$221$	589905.	0.812459
$222$	0	0
$223$	526194.	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$223$	526194.	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$224$	0	0
$225$	0	0
$226$	−775642.	−1.01016
$227$	981605.	1.26436	0.632182	−	0.774820i	$-0.282160\pi$
$227$	981605.	1.26436	0.632182	+	0.774820i	$0.282160\pi$
$228$	0	0
$229$	−85081.0	−0.107212	−0.0536061	−	0.998562i	$-0.517072\pi$
$229$	−85081.0	−0.107212	−0.0536061	+	0.998562i	$0.517072\pi$
$230$	−127939.	−0.159472
$231$	0	0
$232$	333954.	0.407349
$233$	−113123.	−0.136509	−0.0682544	−	0.997668i	$-0.521743\pi$
$233$	−113123.	−0.136509	−0.0682544	+	0.997668i	$0.521743\pi$
$234$	0	0
$235$	209476.	0.247436
$236$	486019.	0.568033
$237$	0	0
$238$	0	0
$239$	895100.	1.01362	0.506812	−	0.862057i	$-0.330824\pi$
$239$	895100.	1.01362	0.506812	+	0.862057i	$0.330824\pi$
$240$	0	0
$241$	527715.	0.585270	0.292635	−	0.956224i	$-0.405468\pi$
$241$	527715.	0.585270	0.292635	+	0.956224i	$0.405468\pi$
$242$	180547.	0.198177
$243$	0	0
$244$	514980.	0.553753
$245$	0	0
$246$	0	0
$247$	−747501.	−0.779596
$248$	450416.	0.465034
$249$	0	0
$250$	−516793.	−0.522957
$251$	1.19293e6	1.19517	0.597584	−	0.801806i	$-0.296127\pi$
$251$	1.19293e6	1.19517	0.597584	+	0.801806i	$0.296127\pi$
$252$	0	0
$253$	484147.	0.475528
$254$	−109717.	−0.106706
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.19492e6	1.12851	0.564257	−	0.825599i	$-0.309163\pi$
$257$	1.19492e6	1.12851	0.564257	+	0.825599i	$0.309163\pi$
$258$	0	0
$259$	0	0
$260$	262140.	0.240492
$261$	0	0
$262$	601504.	0.541359
$263$	2.12230e6	1.89199	0.945993	−	0.324186i	$-0.105090\pi$
$263$	2.12230e6	1.89199	0.945993	+	0.324186i	$0.105090\pi$
$264$	0	0
$265$	801690.	0.701280
$266$	0	0
$267$	0	0
$268$	341617.	0.290538
$269$	−154977.	−0.130583	−0.0652913	−	0.997866i	$-0.520798\pi$
$269$	−154977.	−0.130583	−0.0652913	+	0.997866i	$0.520798\pi$
$270$	0	0
$271$	1.95888e6	1.62026	0.810129	−	0.586252i	$-0.199397\pi$
$271$	1.95888e6	1.62026	0.810129	+	0.586252i	$0.199397\pi$
$272$	−207321.	−0.169911
$273$	0	0
$274$	384408.	0.309326
$275$	891701.	0.711030
$276$	0	0
$277$	−1.74953e6	−1.37000	−0.685001	−	0.728542i	$-0.740198\pi$
$277$	−1.74953e6	−1.37000	−0.685001	+	0.728542i	$0.740198\pi$
$278$	−403418.	−0.313071
$279$	0	0
$280$	0	0
$281$	−1.40665e6	−1.06272	−0.531361	−	0.847145i	$-0.678319\pi$
$281$	−1.40665e6	−1.06272	−0.531361	+	0.847145i	$0.678319\pi$
$282$	0	0
$283$	1.95707e6	1.45258	0.726291	−	0.687388i	$-0.241243\pi$
$283$	1.95707e6	1.45258	0.726291	+	0.687388i	$0.241243\pi$
$284$	−978460.	−0.719859
$285$	0	0
$286$	−991991.	−0.717121
$287$	0	0
$288$	0	0
$289$	−764005.	−0.538086
$290$	−469463.	−0.327798
$291$	0	0
$292$	660285.	0.453184
$293$	1.26998e6	0.864229	0.432114	−	0.901819i	$-0.357768\pi$
$293$	1.26998e6	0.864229	0.432114	+	0.901819i	$0.357768\pi$
$294$	0	0
$295$	−683232.	−0.457102
$296$	−818695.	−0.543116
$297$	0	0
$298$	−1.44378e6	−0.941802
$299$	1.03583e6	0.670055
$300$	0	0
$301$	0	0
$302$	1.73622e6	1.09544
$303$	0	0
$304$	262708.	0.163038
$305$	−723944.	−0.445611
$306$	0	0
$307$	−41854.4	−0.0253451	−0.0126726	−	0.999920i	$-0.504034\pi$
$307$	−41854.4	−0.0253451	−0.0126726	+	0.999920i	$0.504034\pi$
$308$	0	0
$309$	0	0
$310$	−633182.	−0.374217
$311$	2.29189e6	1.34367	0.671837	−	0.740699i	$-0.265506\pi$
$311$	2.29189e6	1.34367	0.671837	+	0.740699i	$0.265506\pi$
$312$	0	0
$313$	2.27150e6	1.31054	0.655271	−	0.755394i	$-0.272554\pi$
$313$	2.27150e6	1.31054	0.655271	+	0.755394i	$0.272554\pi$
$314$	2.04639e6	1.17129
$315$	0	0
$316$	−560007.	−0.315483
$317$	1.98474e6	1.10932	0.554659	−	0.832078i	$-0.312849\pi$
$317$	1.98474e6	1.10932	0.554659	+	0.832078i	$0.312849\pi$
$318$	0	0
$319$	1.77654e6	0.977459
$320$	−92128.7	−0.0502944
$321$	0	0
$322$	0	0
$323$	−831066.	−0.443230
$324$	0	0
$325$	1.90779e6	1.00189
$326$	1.00508e6	0.523788
$327$	0	0
$328$	75104.3	0.0385461
$329$	0	0
$330$	0	0
$331$	638461.	0.320305	0.160153	−	0.987092i	$-0.448801\pi$
$331$	638461.	0.320305	0.160153	+	0.987092i	$0.448801\pi$
$332$	−1.37909e6	−0.686668
$333$	0	0
$334$	1.67711e6	0.822611
$335$	−480236.	−0.233799
$336$	0	0
$337$	2.72026e6	1.30478	0.652388	−	0.757886i	$-0.273767\pi$
$337$	2.72026e6	1.30478	0.652388	+	0.757886i	$0.273767\pi$
$338$	−637186.	−0.303371
$339$	0	0
$340$	291446.	0.136729
$341$	2.39608e6	1.11588
$342$	0	0
$343$	0	0
$344$	−234507.	−0.106846
$345$	0	0
$346$	−1.84551e6	−0.828753
$347$	−2.62076e6	−1.16843	−0.584217	−	0.811597i	$-0.698598\pi$
$347$	−2.62076e6	−1.16843	−0.584217	+	0.811597i	$0.698598\pi$
$348$	0	0
$349$	575592.	0.252960	0.126480	−	0.991969i	$-0.459632\pi$
$349$	575592.	0.252960	0.126480	+	0.991969i	$0.459632\pi$
$350$	0	0
$351$	0	0
$352$	348633.	0.149972
$353$	2.58972e6	1.10615	0.553077	−	0.833130i	$-0.313454\pi$
$353$	2.58972e6	1.10615	0.553077	+	0.833130i	$0.313454\pi$
$354$	0	0
$355$	1.37549e6	0.579278
$356$	−1.24788e6	−0.521853
$357$	0	0
$358$	2.96682e6	1.22344
$359$	−2.86779e6	−1.17439	−0.587193	−	0.809447i	$-0.699767\pi$
$359$	−2.86779e6	−1.17439	−0.587193	+	0.809447i	$0.699767\pi$
$360$	0	0
$361$	−1.42301e6	−0.574698
$362$	−1.20548e6	−0.483493
$363$	0	0
$364$	0	0
$365$	−928210.	−0.364682
$366$	0	0
$367$	−1.95451e6	−0.757482	−0.378741	−	0.925503i	$-0.623643\pi$
$367$	−1.95451e6	−0.757482	−0.378741	+	0.925503i	$0.623643\pi$
$368$	−364040.	−0.140129
$369$	0	0
$370$	1.15090e6	0.437052
$371$	0	0
$372$	0	0
$373$	2.21110e6	0.822878	0.411439	−	0.911437i	$-0.365026\pi$
$373$	2.21110e6	0.822878	0.411439	+	0.911437i	$0.365026\pi$
$374$	−1.10289e6	−0.407711
$375$	0	0
$376$	596044.	0.217425
$377$	3.80090e6	1.37731
$378$	0	0
$379$	3.81232e6	1.36330	0.681649	−	0.731679i	$-0.261263\pi$
$379$	3.81232e6	1.36330	0.681649	+	0.731679i	$0.261263\pi$
$380$	−369307.	−0.131198
$381$	0	0
$382$	1.05273e6	0.369112
$383$	−3.80606e6	−1.32580	−0.662901	−	0.748707i	$-0.730675\pi$
$383$	−3.80606e6	−1.32580	−0.662901	+	0.748707i	$0.730675\pi$
$384$	0	0
$385$	0	0
$386$	3.32451e6	1.13569
$387$	0	0
$388$	−2.58825e6	−0.872823
$389$	−3.07524e6	−1.03040	−0.515200	−	0.857070i	$-0.672282\pi$
$389$	−3.07524e6	−1.03040	−0.515200	+	0.857070i	$0.672282\pi$
$390$	0	0
$391$	1.15163e6	0.380952
$392$	0	0
$393$	0	0
$394$	−4.21685e6	−1.36851
$395$	787242.	0.253873
$396$	0	0
$397$	2.19133e6	0.697800	0.348900	−	0.937160i	$-0.386555\pi$
$397$	2.19133e6	0.697800	0.348900	+	0.937160i	$0.386555\pi$
$398$	−2.79427e6	−0.884221
$399$	0	0
$400$	−670488.	−0.209528
$401$	−1.77668e6	−0.551757	−0.275879	−	0.961192i	$-0.588969\pi$
$401$	−1.77668e6	−0.551757	−0.275879	+	0.961192i	$0.588969\pi$
$402$	0	0
$403$	5.12640e6	1.57235
$404$	1.05367e6	0.321182
$405$	0	0
$406$	0	0
$407$	−4.35522e6	−1.30324
$408$	0	0
$409$	2.36118e6	0.697945	0.348973	−	0.937133i	$-0.386531\pi$
$409$	2.36118e6	0.697945	0.348973	+	0.937133i	$0.386531\pi$
$410$	−105580.	−0.0310185
$411$	0	0
$412$	2.08317e6	0.604619
$413$	0	0
$414$	0	0
$415$	1.93868e6	0.552569
$416$	745898.	0.211323
$417$	0	0
$418$	1.39753e6	0.391219
$419$	3.23493e6	0.900181	0.450090	−	0.892983i	$-0.351392\pi$
$419$	3.23493e6	0.900181	0.450090	+	0.892983i	$0.351392\pi$
$420$	0	0
$421$	−2.85759e6	−0.785769	−0.392884	−	0.919588i	$-0.628523\pi$
$421$	−2.85759e6	−0.785769	−0.392884	+	0.919588i	$0.628523\pi$
$422$	398774.	0.109005
$423$	0	0
$424$	2.28114e6	0.616222
$425$	2.12107e6	0.569615
$426$	0	0
$427$	0	0
$428$	−41455.2	−0.0109388
$429$	0	0
$430$	329663.	0.0859803
$431$	88386.2	0.0229188	0.0114594	−	0.999934i	$-0.496352\pi$
$431$	88386.2	0.0229188	0.0114594	+	0.999934i	$0.496352\pi$
$432$	0	0
$433$	−3.09418e6	−0.793097	−0.396549	−	0.918014i	$-0.629792\pi$
$433$	−3.09418e6	−0.793097	−0.396549	+	0.918014i	$0.629792\pi$
$434$	0	0
$435$	0	0
$436$	1.77046e6	0.446036
$437$	−1.45929e6	−0.365543
$438$	0	0
$439$	485178.	0.120155	0.0600773	−	0.998194i	$-0.480865\pi$
$439$	485178.	0.120155	0.0600773	+	0.998194i	$0.480865\pi$
$440$	−490098.	−0.120684
$441$	0	0
$442$	−2.35962e6	−0.574495
$443$	−1.27520e6	−0.308722	−0.154361	−	0.988015i	$-0.549332\pi$
$443$	−1.27520e6	−0.308722	−0.154361	+	0.988015i	$0.549332\pi$
$444$	0	0
$445$	1.75424e6	0.419941
$446$	−2.10478e6	−0.501036
$447$	0	0
$448$	0	0
$449$	3.79144e6	0.887541	0.443770	−	0.896141i	$-0.353641\pi$
$449$	3.79144e6	0.887541	0.443770	+	0.896141i	$0.353641\pi$
$450$	0	0
$451$	399534.	0.0924937
$452$	3.10257e6	0.714291
$453$	0	0
$454$	−3.92642e6	−0.894040
$455$	0	0
$456$	0	0
$457$	1.70281e6	0.381395	0.190698	−	0.981649i	$-0.438925\pi$
$457$	1.70281e6	0.381395	0.190698	+	0.981649i	$0.438925\pi$
$458$	340324.	0.0758104
$459$	0	0
$460$	511757.	0.112764
$461$	−4.55537e6	−0.998323	−0.499161	−	0.866509i	$-0.666358\pi$
$461$	−4.55537e6	−0.998323	−0.499161	+	0.866509i	$0.666358\pi$
$462$	0	0
$463$	5.82647e6	1.26314	0.631572	−	0.775317i	$-0.282410\pi$
$463$	5.82647e6	1.26314	0.631572	+	0.775317i	$0.282410\pi$
$464$	−1.33582e6	−0.288039
$465$	0	0
$466$	452492.	0.0965263
$467$	3.54261e6	0.751677	0.375839	−	0.926685i	$-0.377355\pi$
$467$	3.54261e6	0.751677	0.375839	+	0.926685i	$0.377355\pi$
$468$	0	0
$469$	0	0
$470$	−837902.	−0.174964
$471$	0	0
$472$	−1.94408e6	−0.401660
$473$	−1.24751e6	−0.256384
$474$	0	0
$475$	−2.68772e6	−0.546575
$476$	0	0
$477$	0	0
$478$	−3.58040e6	−0.716740
$479$	4.23542e6	0.843447	0.421723	−	0.906725i	$-0.361425\pi$
$479$	4.23542e6	0.843447	0.421723	+	0.906725i	$0.361425\pi$
$480$	0	0
$481$	−9.31797e6	−1.83636
$482$	−2.11086e6	−0.413849
$483$	0	0
$484$	−722189.	−0.140132
$485$	3.63848e6	0.702370
$486$	0	0
$487$	−5.65479e6	−1.08042	−0.540212	−	0.841529i	$-0.681656\pi$
$487$	−5.65479e6	−1.08042	−0.540212	+	0.841529i	$0.681656\pi$
$488$	−2.05992e6	−0.391562
$489$	0	0
$490$	0	0
$491$	−8.33183e6	−1.55968	−0.779842	−	0.625976i	$-0.784701\pi$
$491$	−8.33183e6	−1.55968	−0.779842	+	0.625976i	$0.784701\pi$
$492$	0	0
$493$	4.22581e6	0.783055
$494$	2.99001e6	0.551258
$495$	0	0
$496$	−1.80166e6	−0.328828
$497$	0	0
$498$	0	0
$499$	−7.23769e6	−1.30121	−0.650607	−	0.759415i	$-0.725485\pi$
$499$	−7.23769e6	−1.30121	−0.650607	+	0.759415i	$0.725485\pi$
$500$	2.06717e6	0.369787
$501$	0	0
$502$	−4.77170e6	−0.845112
$503$	6.16761e6	1.08692	0.543459	−	0.839436i	$-0.317114\pi$
$503$	6.16761e6	1.08692	0.543459	+	0.839436i	$0.317114\pi$
$504$	0	0
$505$	−1.48122e6	−0.258459
$506$	−1.93659e6	−0.336249
$507$	0	0
$508$	438868.	0.0754526
$509$	−1.83488e6	−0.313916	−0.156958	−	0.987605i	$-0.550169\pi$
$509$	−1.83488e6	−0.313916	−0.156958	+	0.987605i	$0.550169\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−4.77969e6	−0.797980
$515$	−2.92846e6	−0.486543
$516$	0	0
$517$	3.17079e6	0.521724
$518$	0	0
$519$	0	0
$520$	−1.04856e6	−0.170054
$521$	−5.31040e6	−0.857103	−0.428551	−	0.903517i	$-0.640976\pi$
$521$	−5.31040e6	−0.857103	−0.428551	+	0.903517i	$0.640976\pi$
$522$	0	0
$523$	−2.98835e6	−0.477725	−0.238862	−	0.971053i	$-0.576774\pi$
$523$	−2.98835e6	−0.477725	−0.238862	+	0.971053i	$0.576774\pi$
$524$	−2.40601e6	−0.382798
$525$	0	0
$526$	−8.48921e6	−1.33784
$527$	5.69950e6	0.893943
$528$	0	0
$529$	−4.41417e6	−0.685820
$530$	−3.20676e6	−0.495880
$531$	0	0
$532$	0	0
$533$	854800.	0.130331
$534$	0	0
$535$	58276.6	0.00880257
$536$	−1.36647e6	−0.205441
$537$	0	0
$538$	619907.	0.0923359
$539$	0	0
$540$	0	0
$541$	2.16832e6	0.318516	0.159258	−	0.987237i	$-0.449090\pi$
$541$	2.16832e6	0.318516	0.159258	+	0.987237i	$0.449090\pi$
$542$	−7.83551e6	−1.14570
$543$	0	0
$544$	829283.	0.120145
$545$	−2.48886e6	−0.358930
$546$	0	0
$547$	−9.13272e6	−1.30506	−0.652532	−	0.757761i	$-0.726293\pi$
$547$	−9.13272e6	−1.30506	−0.652532	+	0.757761i	$0.726293\pi$
$548$	−1.53763e6	−0.218726
$549$	0	0
$550$	−3.56681e6	−0.502774
$551$	−5.35475e6	−0.751381
$552$	0	0
$553$	0	0
$554$	6.99810e6	0.968737
$555$	0	0
$556$	1.61367e6	0.221375
$557$	9.22028e6	1.25923	0.629617	−	0.776906i	$-0.283212\pi$
$557$	9.22028e6	1.25923	0.629617	+	0.776906i	$0.283212\pi$
$558$	0	0
$559$	−2.66904e6	−0.361264
$560$	0	0
$561$	0	0
$562$	5.62660e6	0.751458
$563$	−8.49277e6	−1.12922	−0.564610	−	0.825358i	$-0.690973\pi$
$563$	−8.49277e6	−1.12922	−0.564610	+	0.825358i	$0.690973\pi$
$564$	0	0
$565$	−4.36150e6	−0.574797
$566$	−7.82828e6	−1.02713
$567$	0	0
$568$	3.91384e6	0.509017
$569$	4.70912e6	0.609760	0.304880	−	0.952391i	$-0.401384\pi$
$569$	4.70912e6	0.609760	0.304880	+	0.952391i	$0.401384\pi$
$570$	0	0
$571$	−1.72807e6	−0.221805	−0.110902	−	0.993831i	$-0.535374\pi$
$571$	−1.72807e6	−0.221805	−0.110902	+	0.993831i	$0.535374\pi$
$572$	3.96796e6	0.507081
$573$	0	0
$574$	0	0
$575$	3.72443e6	0.469776
$576$	0	0
$577$	2.38499e6	0.298227	0.149113	−	0.988820i	$-0.452358\pi$
$577$	2.38499e6	0.298227	0.149113	+	0.988820i	$0.452358\pi$
$578$	3.05602e6	0.380484
$579$	0	0
$580$	1.87785e6	0.231788
$581$	0	0
$582$	0	0
$583$	1.21350e7	1.47866
$584$	−2.64114e6	−0.320449
$585$	0	0
$586$	−5.07993e6	−0.611102
$587$	3.44904e6	0.413145	0.206573	−	0.978431i	$-0.433769\pi$
$587$	3.44904e6	0.413145	0.206573	+	0.978431i	$0.433769\pi$
$588$	0	0
$589$	−7.22214e6	−0.857784
$590$	2.73293e6	0.323220
$591$	0	0
$592$	3.27478e6	0.384041
$593$	−1.13468e6	−0.132507	−0.0662533	−	0.997803i	$-0.521105\pi$
$593$	−1.13468e6	−0.132507	−0.0662533	+	0.997803i	$0.521105\pi$
$594$	0	0
$595$	0	0
$596$	5.77511e6	0.665954
$597$	0	0
$598$	−4.14332e6	−0.473800
$599$	−7.61865e6	−0.867583	−0.433792	−	0.901013i	$-0.642825\pi$
$599$	−7.61865e6	−0.867583	−0.433792	+	0.901013i	$0.642825\pi$
$600$	0	0
$601$	9.78414e6	1.10493	0.552467	−	0.833535i	$-0.313686\pi$
$601$	9.78414e6	1.10493	0.552467	+	0.833535i	$0.313686\pi$
$602$	0	0
$603$	0	0
$604$	−6.94489e6	−0.774592
$605$	1.01523e6	0.112766
$606$	0	0
$607$	1.85335e6	0.204167	0.102083	−	0.994776i	$-0.467449\pi$
$607$	1.85335e6	0.204167	0.102083	+	0.994776i	$0.467449\pi$
$608$	−1.05083e6	−0.115285
$609$	0	0
$610$	2.89578e6	0.315094
$611$	6.78388e6	0.735148
$612$	0	0
$613$	−7.12097e6	−0.765399	−0.382699	−	0.923873i	$-0.625005\pi$
$613$	−7.12097e6	−0.765399	−0.382699	+	0.923873i	$0.625005\pi$
$614$	167417.	0.0179217
$615$	0	0
$616$	0	0
$617$	2.75212e6	0.291041	0.145521	−	0.989355i	$-0.453514\pi$
$617$	2.75212e6	0.291041	0.145521	+	0.989355i	$0.453514\pi$
$618$	0	0
$619$	−1.30332e7	−1.36718	−0.683588	−	0.729868i	$-0.739581\pi$
$619$	−1.30332e7	−1.36718	−0.683588	+	0.729868i	$0.739581\pi$
$620$	2.53273e6	0.264612
$621$	0	0
$622$	−9.16758e6	−0.950120
$623$	0	0
$624$	0	0
$625$	5.27870e6	0.540539
$626$	−9.08598e6	−0.926693
$627$	0	0
$628$	−8.18554e6	−0.828225
$629$	−1.03597e7	−1.04404
$630$	0	0
$631$	−4.40820e6	−0.440745	−0.220373	−	0.975416i	$-0.570727\pi$
$631$	−4.40820e6	−0.440745	−0.220373	+	0.975416i	$0.570727\pi$
$632$	2.24003e6	0.223080
$633$	0	0
$634$	−7.93897e6	−0.784406
$635$	−616948.	−0.0607175
$636$	0	0
$637$	0	0
$638$	−7.10616e6	−0.691168
$639$	0	0
$640$	368515.	0.0355635
$641$	7.98390e6	0.767485	0.383742	−	0.923440i	$-0.374635\pi$
$641$	7.98390e6	0.767485	0.383742	+	0.923440i	$0.374635\pi$
$642$	0	0
$643$	1.52102e6	0.145080	0.0725398	−	0.997366i	$-0.476890\pi$
$643$	1.52102e6	0.145080	0.0725398	+	0.997366i	$0.476890\pi$
$644$	0	0
$645$	0	0
$646$	3.32426e6	0.313411
$647$	1.25249e7	1.17629	0.588146	−	0.808755i	$-0.299858\pi$
$647$	1.25249e7	1.17629	0.588146	+	0.808755i	$0.299858\pi$
$648$	0	0
$649$	−1.03419e7	−0.963806
$650$	−7.63116e6	−0.708447
$651$	0	0
$652$	−4.02031e6	−0.370374
$653$	1.70740e7	1.56694	0.783471	−	0.621428i	$-0.213447\pi$
$653$	1.70740e7	1.56694	0.783471	+	0.621428i	$0.213447\pi$
$654$	0	0
$655$	3.38231e6	0.308042
$656$	−300417.	−0.0272562
$657$	0	0
$658$	0	0
$659$	2.12585e6	0.190686	0.0953432	−	0.995444i	$-0.469605\pi$
$659$	2.12585e6	0.190686	0.0953432	+	0.995444i	$0.469605\pi$
$660$	0	0
$661$	−2.60011e6	−0.231466	−0.115733	−	0.993280i	$-0.536922\pi$
$661$	−2.60011e6	−0.231466	−0.115733	+	0.993280i	$0.536922\pi$
$662$	−2.55384e6	−0.226490
$663$	0	0
$664$	5.51635e6	0.485547
$665$	0	0
$666$	0	0
$667$	7.42020e6	0.645805
$668$	−6.70843e6	−0.581674
$669$	0	0
$670$	1.92094e6	0.165321
$671$	−1.09582e7	−0.939577
$672$	0	0
$673$	−1.44746e7	−1.23188	−0.615942	−	0.787792i	$-0.711224\pi$
$673$	−1.44746e7	−1.23188	−0.615942	+	0.787792i	$0.711224\pi$
$674$	−1.08810e7	−0.922615
$675$	0	0
$676$	2.54875e6	0.214516
$677$	3.28845e6	0.275753	0.137876	−	0.990449i	$-0.455972\pi$
$677$	3.28845e6	0.275753	0.137876	+	0.990449i	$0.455972\pi$
$678$	0	0
$679$	0	0
$680$	−1.16578e6	−0.0966819
$681$	0	0
$682$	−9.58433e6	−0.789043
$683$	−8.81887e6	−0.723371	−0.361685	−	0.932300i	$-0.617799\pi$
$683$	−8.81887e6	−0.723371	−0.361685	+	0.932300i	$0.617799\pi$
$684$	0	0
$685$	2.16156e6	0.176011
$686$	0	0
$687$	0	0
$688$	938026.	0.0755517
$689$	2.59628e7	2.08354
$690$	0	0
$691$	−8.59617e6	−0.684873	−0.342436	−	0.939541i	$-0.611252\pi$
$691$	−8.59617e6	−0.684873	−0.342436	+	0.939541i	$0.611252\pi$
$692$	7.38202e6	0.586017
$693$	0	0
$694$	1.04831e7	0.826208
$695$	−2.26845e6	−0.178143
$696$	0	0
$697$	950360.	0.0740979
$698$	−2.30237e6	−0.178870
$699$	0	0
$700$	0	0
$701$	−2.06437e7	−1.58669	−0.793347	−	0.608770i	$-0.791663\pi$
$701$	−2.06437e7	−1.58669	−0.793347	+	0.608770i	$0.791663\pi$
$702$	0	0
$703$	1.31273e7	1.00181
$704$	−1.39453e6	−0.106047
$705$	0	0
$706$	−1.03589e7	−0.782169
$707$	0	0
$708$	0	0
$709$	−426882.	−0.0318928	−0.0159464	−	0.999873i	$-0.505076\pi$
$709$	−426882.	−0.0318928	−0.0159464	+	0.999873i	$0.505076\pi$
$710$	−5.50196e6	−0.409611
$711$	0	0
$712$	4.99153e6	0.369006
$713$	1.00079e7	0.737257
$714$	0	0
$715$	−5.57805e6	−0.408054
$716$	−1.18673e7	−0.865105
$717$	0	0
$718$	1.14712e7	0.830417
$719$	−3.80724e6	−0.274655	−0.137328	−	0.990526i	$-0.543851\pi$
$719$	−3.80724e6	−0.274655	−0.137328	+	0.990526i	$0.543851\pi$
$720$	0	0
$721$	0	0
$722$	5.69204e6	0.406373
$723$	0	0
$724$	4.82193e6	0.341881
$725$	1.36665e7	0.965635
$726$	0	0
$727$	−2.22044e7	−1.55813	−0.779065	−	0.626943i	$-0.784306\pi$
$727$	−2.22044e7	−1.55813	−0.779065	+	0.626943i	$0.784306\pi$
$728$	0	0
$729$	0	0
$730$	3.71284e6	0.257869
$731$	−2.96741e6	−0.205393
$732$	0	0
$733$	−2.90572e7	−1.99753	−0.998766	−	0.0496729i	$-0.984182\pi$
$733$	−2.90572e7	−1.99753	−0.998766	+	0.0496729i	$0.984182\pi$
$734$	7.81804e6	0.535621
$735$	0	0
$736$	1.45616e6	0.0990865
$737$	−7.26923e6	−0.492969
$738$	0	0
$739$	2.13523e7	1.43824	0.719122	−	0.694883i	$-0.244544\pi$
$739$	2.13523e7	1.43824	0.719122	+	0.694883i	$0.244544\pi$
$740$	−4.60359e6	−0.309042
$741$	0	0
$742$	0	0
$743$	−3.91874e6	−0.260420	−0.130210	−	0.991486i	$-0.541565\pi$
$743$	−3.91874e6	−0.260420	−0.130210	+	0.991486i	$0.541565\pi$
$744$	0	0
$745$	−8.11848e6	−0.535900
$746$	−8.84438e6	−0.581863
$747$	0	0
$748$	4.41155e6	0.288295
$749$	0	0
$750$	0	0
$751$	5.66912e6	0.366789	0.183394	−	0.983039i	$-0.441291\pi$
$751$	5.66912e6	0.366789	0.183394	+	0.983039i	$0.441291\pi$
$752$	−2.38418e6	−0.153743
$753$	0	0
$754$	−1.52036e7	−0.973908
$755$	9.76293e6	0.623323
$756$	0	0
$757$	−1.91706e7	−1.21590	−0.607949	−	0.793976i	$-0.708007\pi$
$757$	−1.91706e7	−1.21590	−0.607949	+	0.793976i	$0.708007\pi$
$758$	−1.52493e7	−0.963998
$759$	0	0
$760$	1.47723e6	0.0927712
$761$	−1.46336e7	−0.915987	−0.457993	−	0.888956i	$-0.651432\pi$
$761$	−1.46336e7	−0.915987	−0.457993	+	0.888956i	$0.651432\pi$
$762$	0	0
$763$	0	0
$764$	−4.21092e6	−0.261002
$765$	0	0
$766$	1.52242e7	0.937483
$767$	−2.21265e7	−1.35808
$768$	0	0
$769$	1.57337e7	0.959432	0.479716	−	0.877424i	$-0.340740\pi$
$769$	1.57337e7	0.959432	0.479716	+	0.877424i	$0.340740\pi$
$770$	0	0
$771$	0	0
$772$	−1.32980e7	−0.803053
$773$	−4.04904e6	−0.243727	−0.121864	−	0.992547i	$-0.538887\pi$
$773$	−4.04904e6	−0.243727	−0.121864	+	0.992547i	$0.538887\pi$
$774$	0	0
$775$	1.84325e7	1.10238
$776$	1.03530e7	0.617179
$777$	0	0
$778$	1.23010e7	0.728602
$779$	−1.20425e6	−0.0711007
$780$	0	0
$781$	2.08205e7	1.22142
$782$	−4.60651e6	−0.269374
$783$	0	0
$784$	0	0
$785$	1.15070e7	0.666481
$786$	0	0
$787$	−2.16111e7	−1.24377	−0.621886	−	0.783108i	$-0.713633\pi$
$787$	−2.16111e7	−1.24377	−0.621886	+	0.783108i	$0.713633\pi$
$788$	1.68674e7	0.967681
$789$	0	0
$790$	−3.14897e6	−0.179515
$791$	0	0
$792$	0	0
$793$	−2.34450e7	−1.32394
$794$	−8.76530e6	−0.493419
$795$	0	0
$796$	1.11771e7	0.625239
$797$	−1.38162e7	−0.770445	−0.385222	−	0.922824i	$-0.625875\pi$
$797$	−1.38162e7	−0.770445	−0.385222	+	0.922824i	$0.625875\pi$
$798$	0	0
$799$	7.54226e6	0.417960
$800$	2.68195e6	0.148158
$801$	0	0
$802$	7.10672e6	0.390151
$803$	−1.40501e7	−0.768937
$804$	0	0
$805$	0	0
$806$	−2.05056e7	−1.11182
$807$	0	0
$808$	−4.21468e6	−0.227110
$809$	3.50756e7	1.88423	0.942116	−	0.335287i	$-0.108833\pi$
$809$	3.50756e7	1.88423	0.942116	+	0.335287i	$0.108833\pi$
$810$	0	0
$811$	6.56287e6	0.350382	0.175191	−	0.984534i	$-0.443946\pi$
$811$	6.56287e6	0.350382	0.175191	+	0.984534i	$0.443946\pi$
$812$	0	0
$813$	0	0
$814$	1.74209e7	0.921530
$815$	5.65164e6	0.298044
$816$	0	0
$817$	3.76017e6	0.197085
$818$	−9.44473e6	−0.493522
$819$	0	0
$820$	422318.	0.0219334
$821$	1.77894e6	0.0921093	0.0460547	−	0.998939i	$-0.485335\pi$
$821$	1.77894e6	0.0921093	0.0460547	+	0.998939i	$0.485335\pi$
$822$	0	0
$823$	1.89586e7	0.975676	0.487838	−	0.872934i	$-0.337786\pi$
$823$	1.89586e7	0.975676	0.487838	+	0.872934i	$0.337786\pi$
$824$	−8.33268e6	−0.427530
$825$	0	0
$826$	0	0
$827$	−2.82110e7	−1.43435	−0.717174	−	0.696894i	$-0.754565\pi$
$827$	−2.82110e7	−1.43435	−0.717174	+	0.696894i	$0.754565\pi$
$828$	0	0
$829$	−2.24021e7	−1.13214	−0.566072	−	0.824356i	$-0.691538\pi$
$829$	−2.24021e7	−1.13214	−0.566072	+	0.824356i	$0.691538\pi$
$830$	−7.75473e6	−0.390725
$831$	0	0
$832$	−2.98359e6	−0.149428
$833$	0	0
$834$	0	0
$835$	9.43051e6	0.468079
$836$	−5.59012e6	−0.276634
$837$	0	0
$838$	−1.29397e7	−0.636524
$839$	1.50303e7	0.737162	0.368581	−	0.929596i	$-0.379844\pi$
$839$	1.50303e7	0.737162	0.368581	+	0.929596i	$0.379844\pi$
$840$	0	0
$841$	6.71672e6	0.327467
$842$	1.14304e7	0.555622
$843$	0	0
$844$	−1.59510e6	−0.0770781
$845$	−3.58295e6	−0.172623
$846$	0	0
$847$	0	0
$848$	−9.12455e6	−0.435734
$849$	0	0
$850$	−8.48426e6	−0.402779
$851$	−1.81908e7	−0.861048
$852$	0	0
$853$	−3.18297e7	−1.49782	−0.748911	−	0.662670i	$-0.769423\pi$
$853$	−3.18297e7	−1.49782	−0.748911	+	0.662670i	$0.769423\pi$
$854$	0	0
$855$	0	0
$856$	165821.	0.00773490
$857$	−2.17823e7	−1.01310	−0.506549	−	0.862211i	$-0.669079\pi$
$857$	−2.17823e7	−1.01310	−0.506549	+	0.862211i	$0.669079\pi$
$858$	0	0
$859$	3.73349e7	1.72636	0.863181	−	0.504895i	$-0.168469\pi$
$859$	3.73349e7	1.72636	0.863181	+	0.504895i	$0.168469\pi$
$860$	−1.31865e6	−0.0607972
$861$	0	0
$862$	−353545.	−0.0162060
$863$	1.54613e7	0.706675	0.353338	−	0.935496i	$-0.385047\pi$
$863$	1.54613e7	0.706675	0.353338	+	0.935496i	$0.385047\pi$
$864$	0	0
$865$	−1.03774e7	−0.471574
$866$	1.23767e7	0.560805
$867$	0	0
$868$	0	0
$869$	1.19163e7	0.535294
$870$	0	0
$871$	−1.55525e7	−0.694630
$872$	−7.08184e6	−0.315395
$873$	0	0
$874$	5.83716e6	0.258478
$875$	0	0
$876$	0	0
$877$	−6.81762e6	−0.299319	−0.149659	−	0.988738i	$-0.547818\pi$
$877$	−6.81762e6	−0.299319	−0.149659	+	0.988738i	$0.547818\pi$
$878$	−1.94071e6	−0.0849621
$879$	0	0
$880$	1.96039e6	0.0853368
$881$	1.20903e7	0.524805	0.262402	−	0.964959i	$-0.415485\pi$
$881$	1.20903e7	0.524805	0.262402	+	0.964959i	$0.415485\pi$
$882$	0	0
$883$	1.63966e7	0.707706	0.353853	−	0.935301i	$-0.384871\pi$
$883$	1.63966e7	0.707706	0.353853	+	0.935301i	$0.384871\pi$
$884$	9.43848e6	0.406230
$885$	0	0
$886$	5.10078e6	0.218299
$887$	2.57006e7	1.09682	0.548409	−	0.836210i	$-0.315234\pi$
$887$	2.57006e7	1.09682	0.548409	+	0.836210i	$0.315234\pi$
$888$	0	0
$889$	0	0
$890$	−7.01695e6	−0.296943
$891$	0	0
$892$	8.41911e6	0.354286
$893$	−9.55721e6	−0.401054
$894$	0	0
$895$	1.66827e7	0.696160
$896$	0	0
$897$	0	0
$898$	−1.51658e7	−0.627586
$899$	3.67232e7	1.51545
$900$	0	0
$901$	2.88652e7	1.18457
$902$	−1.59813e6	−0.0654029
$903$	0	0
$904$	−1.24103e7	−0.505080
$905$	−6.77854e6	−0.275115
$906$	0	0
$907$	−2.34481e7	−0.946434	−0.473217	−	0.880946i	$-0.656907\pi$
$907$	−2.34481e7	−0.946434	−0.473217	+	0.880946i	$0.656907\pi$
$908$	1.57057e7	0.632182
$909$	0	0
$910$	0	0
$911$	3.29134e7	1.31395	0.656973	−	0.753914i	$-0.271837\pi$
$911$	3.29134e7	1.31395	0.656973	+	0.753914i	$0.271837\pi$
$912$	0	0
$913$	2.93454e7	1.16510
$914$	−6.81123e6	−0.269687
$915$	0	0
$916$	−1.36130e6	−0.0536061
$917$	0	0
$918$	0	0
$919$	−2.52264e7	−0.985294	−0.492647	−	0.870229i	$-0.663971\pi$
$919$	−2.52264e7	−0.985294	−0.492647	+	0.870229i	$0.663971\pi$
$920$	−2.04703e6	−0.0797359
$921$	0	0
$922$	1.82215e7	0.705921
$923$	4.45454e7	1.72107
$924$	0	0
$925$	−3.35037e7	−1.28748
$926$	−2.33059e7	−0.893178
$927$	0	0
$928$	5.34327e6	0.203675
$929$	−2.51164e7	−0.954812	−0.477406	−	0.878683i	$-0.658423\pi$
$929$	−2.51164e7	−0.954812	−0.477406	+	0.878683i	$0.658423\pi$
$930$	0	0
$931$	0	0
$932$	−1.80997e6	−0.0682544
$933$	0	0
$934$	−1.41704e7	−0.531516
$935$	−6.20163e6	−0.231994
$936$	0	0
$937$	8.25845e6	0.307291	0.153645	−	0.988126i	$-0.450899\pi$
$937$	8.25845e6	0.307291	0.153645	+	0.988126i	$0.450899\pi$
$938$	0	0
$939$	0	0
$940$	3.35161e6	0.123718
$941$	−1.27088e7	−0.467875	−0.233938	−	0.972252i	$-0.575161\pi$
$941$	−1.27088e7	−0.467875	−0.233938	+	0.972252i	$0.575161\pi$
$942$	0	0
$943$	1.66876e6	0.0611103
$944$	7.77630e6	0.284016
$945$	0	0
$946$	4.99003e6	0.181291
$947$	−1.06278e7	−0.385096	−0.192548	−	0.981288i	$-0.561675\pi$
$947$	−1.06278e7	−0.385096	−0.192548	+	0.981288i	$0.561675\pi$
$948$	0	0
$949$	−3.00601e7	−1.08349
$950$	1.07509e7	0.386487
$951$	0	0
$952$	0	0
$953$	−749727.	−0.0267406	−0.0133703	−	0.999911i	$-0.504256\pi$
$953$	−749727.	−0.0267406	−0.0133703	+	0.999911i	$0.504256\pi$
$954$	0	0
$955$	5.91959e6	0.210031
$956$	1.43216e7	0.506812
$957$	0	0
$958$	−1.69417e7	−0.596407
$959$	0	0
$960$	0	0
$961$	2.09007e7	0.730050
$962$	3.72719e7	1.29851
$963$	0	0
$964$	8.44343e6	0.292635
$965$	1.86940e7	0.646225
$966$	0	0
$967$	2.97713e7	1.02384	0.511920	−	0.859033i	$-0.328934\pi$
$967$	2.97713e7	1.02384	0.511920	+	0.859033i	$0.328934\pi$
$968$	2.88876e6	0.0990883
$969$	0	0
$970$	−1.45539e7	−0.496651
$971$	1.51210e7	0.514676	0.257338	−	0.966321i	$-0.417155\pi$
$971$	1.51210e7	0.514676	0.257338	+	0.966321i	$0.417155\pi$
$972$	0	0
$973$	0	0
$974$	2.26192e7	0.763975
$975$	0	0
$976$	8.23968e6	0.276876
$977$	−1.93107e6	−0.0647235	−0.0323617	−	0.999476i	$-0.510303\pi$
$977$	−1.93107e6	−0.0647235	−0.0323617	+	0.999476i	$0.510303\pi$
$978$	0	0
$979$	2.65535e7	0.885452
$980$	0	0
$981$	0	0
$982$	3.33273e7	1.10286
$983$	−4.22343e7	−1.39406	−0.697030	−	0.717042i	$-0.745496\pi$
$983$	−4.22343e7	−1.39406	−0.697030	+	0.717042i	$0.745496\pi$
$984$	0	0
$985$	−2.37117e7	−0.778703
$986$	−1.69032e7	−0.553704
$987$	0	0
$988$	−1.19600e7	−0.389798
$989$	−5.21056e6	−0.169392
$990$	0	0
$991$	5.83480e6	0.188730	0.0943651	−	0.995538i	$-0.469918\pi$
$991$	5.83480e6	0.188730	0.0943651	+	0.995538i	$0.469918\pi$
$992$	7.20665e6	0.232517
$993$	0	0
$994$	0	0
$995$	−1.57124e7	−0.503136
$996$	0	0
$997$	1.23461e7	0.393360	0.196680	−	0.980468i	$-0.436984\pi$
$997$	1.23461e7	0.393360	0.196680	+	0.980468i	$0.436984\pi$
$998$	2.89508e7	0.920097
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bh.1.1		2
3.2	odd	2		294.6.a.r.1.2		2
7.2	even	3		126.6.g.h.109.2		4
7.4	even	3		126.6.g.h.37.2		4
7.6	odd	2		882.6.a.bb.1.2		2
21.2	odd	6		42.6.e.c.25.1	✓	4
21.5	even	6		294.6.e.s.67.2		4
21.11	odd	6		42.6.e.c.37.1	yes	4
21.17	even	6		294.6.e.s.79.2		4
21.20	even	2		294.6.a.w.1.1		2
84.11	even	6		336.6.q.f.289.1		4
84.23	even	6		336.6.q.f.193.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.c.25.1	✓	4	21.2	odd	6
42.6.e.c.37.1	yes	4	21.11	odd	6
126.6.g.h.37.2		4	7.4	even	3
126.6.g.h.109.2		4	7.2	even	3
294.6.a.r.1.2		2	3.2	odd	2
294.6.a.w.1.1		2	21.20	even	2
294.6.e.s.67.2		4	21.5	even	6
294.6.e.s.79.2		4	21.17	even	6
336.6.q.f.193.1		4	84.23	even	6
336.6.q.f.289.1		4	84.11	even	6
882.6.a.bb.1.2		2	7.6	odd	2
882.6.a.bh.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -22.4923 q^{5} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} -22.4923 q^{5} -64.0000 q^{8} +89.9694 q^{10} -340.462 q^{11} -728.416 q^{13} +256.000 q^{16} -809.847 q^{17} +1026.20 q^{19} -359.878 q^{20} +1361.85 q^{22} -1422.03 q^{23} -2619.09 q^{25} +2913.66 q^{26} -5218.03 q^{29} -7037.74 q^{31} -1024.00 q^{32} +3239.39 q^{34} +12792.1 q^{37} -4104.81 q^{38} +1439.51 q^{40} -1173.51 q^{41} +3664.17 q^{43} -5447.39 q^{44} +5688.12 q^{46} -9313.19 q^{47} +10476.4 q^{50} -11654.7 q^{52} -35642.8 q^{53} +7657.78 q^{55} +20872.1 q^{58} +30376.2 q^{59} +32186.2 q^{61} +28151.0 q^{62} +4096.00 q^{64} +16383.8 q^{65} +21351.1 q^{67} -12957.6 q^{68} -61153.7 q^{71} +41267.8 q^{73} -51168.4 q^{74} +16419.2 q^{76} -35000.5 q^{79} -5758.04 q^{80} +4694.02 q^{82} -86193.0 q^{83} +18215.4 q^{85} -14656.7 q^{86} +21789.6 q^{88} -77992.6 q^{89} -22752.5 q^{92} +37252.8 q^{94} -23081.7 q^{95} -161765. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 + 53 * q^5 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} + 53 q^{5} - 128 q^{8} - 212 q^{10} - 191 q^{11} - 379 q^{13} + 512 q^{16} + 340 q^{17} - 1769 q^{19} + 848 q^{20} + 764 q^{22} - 3236 q^{23} - 45 q^{25} + 1516 q^{26} - 4459 q^{29} + 1994 q^{31} - 2048 q^{32} - 1360 q^{34} + 20587 q^{37} + 7076 q^{38} - 3392 q^{40} - 8814 q^{41} + 15853 q^{43} - 3056 q^{44} + 12944 q^{46} - 33912 q^{47} + 180 q^{50} - 6064 q^{52} - 49239 q^{53} + 18941 q^{55} + 17836 q^{58} + 56735 q^{59} + 67508 q^{61} - 7976 q^{62} + 8192 q^{64} + 42762 q^{65} + 75723 q^{67} + 5440 q^{68} + 8992 q^{71} - 3201 q^{73} - 82348 q^{74} - 28304 q^{76} + 26612 q^{79} + 13568 q^{80} + 35256 q^{82} + 949 q^{83} + 105020 q^{85} - 63412 q^{86} + 12224 q^{88} - 176562 q^{89} - 51776 q^{92} + 135648 q^{94} - 234098 q^{95} - 129423 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 53 * q^5 - 128 * q^8 - 212 * q^10 - 191 * q^11 - 379 * q^13 + 512 * q^16 + 340 * q^17 - 1769 * q^19 + 848 * q^20 + 764 * q^22 - 3236 * q^23 - 45 * q^25 + 1516 * q^26 - 4459 * q^29 + 1994 * q^31 - 2048 * q^32 - 1360 * q^34 + 20587 * q^37 + 7076 * q^38 - 3392 * q^40 - 8814 * q^41 + 15853 * q^43 - 3056 * q^44 + 12944 * q^46 - 33912 * q^47 + 180 * q^50 - 6064 * q^52 - 49239 * q^53 + 18941 * q^55 + 17836 * q^58 + 56735 * q^59 + 67508 * q^61 - 7976 * q^62 + 8192 * q^64 + 42762 * q^65 + 75723 * q^67 + 5440 * q^68 + 8992 * q^71 - 3201 * q^73 - 82348 * q^74 - 28304 * q^76 + 26612 * q^79 + 13568 * q^80 + 35256 * q^82 + 949 * q^83 + 105020 * q^85 - 63412 * q^86 + 12224 * q^88 - 176562 * q^89 - 51776 * q^92 + 135648 * q^94 - 234098 * q^95 - 129423 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more