LMFDB - Embedded newform 882.6.a.bk.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -4.50253 q^{5} +64.0000 q^{8} +O(q^{10})$	$q+4.00000 q^{2} +16.0000 q^{4} -4.50253 q^{5} +64.0000 q^{8} -18.0101 q^{10} +116.191 q^{11} -85.4773 q^{13} +256.000 q^{16} -33.2662 q^{17} +635.377 q^{19} -72.0404 q^{20} +464.764 q^{22} -2727.43 q^{23} -3104.73 q^{25} -341.909 q^{26} -5860.48 q^{29} +279.136 q^{31} +1024.00 q^{32} -133.065 q^{34} +3038.49 q^{37} +2541.51 q^{38} -288.162 q^{40} -819.415 q^{41} +11100.2 q^{43} +1859.05 q^{44} -10909.7 q^{46} +7407.40 q^{47} -12418.9 q^{50} -1367.64 q^{52} +13698.8 q^{53} -523.153 q^{55} -23441.9 q^{58} -22375.7 q^{59} -12692.1 q^{61} +1116.55 q^{62} +4096.00 q^{64} +384.864 q^{65} -52303.3 q^{67} -532.259 q^{68} -60230.2 q^{71} +76958.4 q^{73} +12154.0 q^{74} +10166.0 q^{76} -33596.0 q^{79} -1152.65 q^{80} -3277.66 q^{82} -60574.9 q^{83} +149.782 q^{85} +44400.9 q^{86} +7436.22 q^{88} -92190.1 q^{89} -43638.8 q^{92} +29629.6 q^{94} -2860.80 q^{95} +152287. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8} - 432 q^{10} - 124 q^{11} + 720 q^{13} + 512 q^{16} + 1260 q^{17} + 360 q^{19} - 1728 q^{20} - 496 q^{22} - 6524 q^{23} + 4482 q^{25} + 2880 q^{26} - 7088 q^{29} + 5904 q^{31} + 2048 q^{32} + 5040 q^{34} - 6040 q^{37} + 1440 q^{38} - 6912 q^{40} + 17388 q^{41} - 608 q^{43} - 1984 q^{44} - 26096 q^{46} + 30456 q^{47} + 17928 q^{50} + 11520 q^{52} - 3964 q^{53} + 24336 q^{55} - 28352 q^{58} - 40752 q^{59} - 1368 q^{61} + 23616 q^{62} + 8192 q^{64} - 82980 q^{65} - 16224 q^{67} + 20160 q^{68} + 3204 q^{71} + 23976 q^{73} - 24160 q^{74} + 5760 q^{76} - 82160 q^{79} - 27648 q^{80} + 69552 q^{82} - 173736 q^{83} - 133700 q^{85} - 2432 q^{86} - 7936 q^{88} - 200556 q^{89} - 104384 q^{92} + 121824 q^{94} + 25640 q^{95} + 251928 q^{97}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8 - 432 * q^10 - 124 * q^11 + 720 * q^13 + 512 * q^16 + 1260 * q^17 + 360 * q^19 - 1728 * q^20 - 496 * q^22 - 6524 * q^23 + 4482 * q^25 + 2880 * q^26 - 7088 * q^29 + 5904 * q^31 + 2048 * q^32 + 5040 * q^34 - 6040 * q^37 + 1440 * q^38 - 6912 * q^40 + 17388 * q^41 - 608 * q^43 - 1984 * q^44 - 26096 * q^46 + 30456 * q^47 + 17928 * q^50 + 11520 * q^52 - 3964 * q^53 + 24336 * q^55 - 28352 * q^58 - 40752 * q^59 - 1368 * q^61 + 23616 * q^62 + 8192 * q^64 - 82980 * q^65 - 16224 * q^67 + 20160 * q^68 + 3204 * q^71 + 23976 * q^73 - 24160 * q^74 + 5760 * q^76 - 82160 * q^79 - 27648 * q^80 + 69552 * q^82 - 173736 * q^83 - 133700 * q^85 - 2432 * q^86 - 7936 * q^88 - 200556 * q^89 - 104384 * q^92 + 121824 * q^94 + 25640 * q^95 + 251928 * 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
$5$	−4.50253	−0.0805436	−0.0402718	−	0.999189i	$-0.512822\pi$
$5$	−4.50253	−0.0805436	−0.0402718	+	0.999189i	$0.512822\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	−18.0101	−0.0569529
$11$	116.191	0.289528	0.144764	−	0.989466i	$-0.453758\pi$
$11$	116.191	0.289528	0.144764	+	0.989466i	$0.453758\pi$
$12$	0	0
$13$	−85.4773	−0.140279	−0.0701394	−	0.997537i	$-0.522344\pi$
$13$	−85.4773	−0.140279	−0.0701394	+	0.997537i	$0.522344\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−33.2662	−0.0279177	−0.0139589	−	0.999903i	$-0.504443\pi$
$17$	−33.2662	−0.0279177	−0.0139589	+	0.999903i	$0.504443\pi$
$18$	0	0
$19$	635.377	0.403782	0.201891	−	0.979408i	$-0.435291\pi$
$19$	635.377	0.403782	0.201891	+	0.979408i	$0.435291\pi$
$20$	−72.0404	−0.0402718
$21$	0	0
$22$	464.764	0.204727
$23$	−2727.43	−1.07506	−0.537531	−	0.843244i	$-0.680643\pi$
$23$	−2727.43	−1.07506	−0.537531	+	0.843244i	$0.680643\pi$
$24$	0	0
$25$	−3104.73	−0.993513
$26$	−341.909	−0.0991921
$27$	0	0
$28$	0	0
$29$	−5860.48	−1.29401	−0.647006	−	0.762485i	$-0.723979\pi$
$29$	−5860.48	−1.29401	−0.647006	+	0.762485i	$0.723979\pi$
$30$	0	0
$31$	279.136	0.0521690	0.0260845	−	0.999660i	$-0.491696\pi$
$31$	279.136	0.0521690	0.0260845	+	0.999660i	$0.491696\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−133.065	−0.0197408
$35$	0	0
$36$	0	0
$37$	3038.49	0.364883	0.182442	−	0.983217i	$-0.441600\pi$
$37$	3038.49	0.364883	0.182442	+	0.983217i	$0.441600\pi$
$38$	2541.51	0.285517
$39$	0	0
$40$	−288.162	−0.0284765
$41$	−819.415	−0.0761279	−0.0380640	−	0.999275i	$-0.512119\pi$
$41$	−819.415	−0.0761279	−0.0380640	+	0.999275i	$0.512119\pi$
$42$	0	0
$43$	11100.2	0.915504	0.457752	−	0.889080i	$-0.348655\pi$
$43$	11100.2	0.915504	0.457752	+	0.889080i	$0.348655\pi$
$44$	1859.05	0.144764
$45$	0	0
$46$	−10909.7	−0.760184
$47$	7407.40	0.489126	0.244563	−	0.969633i	$-0.421355\pi$
$47$	7407.40	0.489126	0.244563	+	0.969633i	$0.421355\pi$
$48$	0	0
$49$	0	0
$50$	−12418.9	−0.702520
$51$	0	0
$52$	−1367.64	−0.0701394
$53$	13698.8	0.669874	0.334937	−	0.942241i	$-0.391285\pi$
$53$	13698.8	0.669874	0.334937	+	0.942241i	$0.391285\pi$
$54$	0	0
$55$	−523.153	−0.0233196
$56$	0	0
$57$	0	0
$58$	−23441.9	−0.915005
$59$	−22375.7	−0.836848	−0.418424	−	0.908252i	$-0.637417\pi$
$59$	−22375.7	−0.836848	−0.418424	+	0.908252i	$0.637417\pi$
$60$	0	0
$61$	−12692.1	−0.436725	−0.218363	−	0.975868i	$-0.570072\pi$
$61$	−12692.1	−0.436725	−0.218363	+	0.975868i	$0.570072\pi$
$62$	1116.55	0.0368890
$63$	0	0
$64$	4096.00	0.125000
$65$	384.864	0.0112986
$66$	0	0
$67$	−52303.3	−1.42345	−0.711725	−	0.702458i	$-0.752086\pi$
$67$	−52303.3	−1.42345	−0.711725	+	0.702458i	$0.752086\pi$
$68$	−532.259	−0.0139589
$69$	0	0
$70$	0	0
$71$	−60230.2	−1.41798	−0.708988	−	0.705221i	$-0.750848\pi$
$71$	−60230.2	−1.41798	−0.708988	+	0.705221i	$0.750848\pi$
$72$	0	0
$73$	76958.4	1.69024	0.845121	−	0.534575i	$-0.179528\pi$
$73$	76958.4	1.69024	0.845121	+	0.534575i	$0.179528\pi$
$74$	12154.0	0.258011
$75$	0	0
$76$	10166.0	0.201891
$77$	0	0
$78$	0	0
$79$	−33596.0	−0.605647	−0.302824	−	0.953047i	$-0.597929\pi$
$79$	−33596.0	−0.605647	−0.302824	+	0.953047i	$0.597929\pi$
$80$	−1152.65	−0.0201359
$81$	0	0
$82$	−3277.66	−0.0538306
$83$	−60574.9	−0.965157	−0.482578	−	0.875853i	$-0.660300\pi$
$83$	−60574.9	−0.965157	−0.482578	+	0.875853i	$0.660300\pi$
$84$	0	0
$85$	149.782	0.00224860
$86$	44400.9	0.647359
$87$	0	0
$88$	7436.22	0.102364
$89$	−92190.1	−1.23370	−0.616850	−	0.787081i	$-0.711591\pi$
$89$	−92190.1	−1.23370	−0.616850	+	0.787081i	$0.711591\pi$
$90$	0	0
$91$	0	0
$92$	−43638.8	−0.537531
$93$	0	0
$94$	29629.6	0.345865
$95$	−2860.80	−0.0325221
$96$	0	0
$97$	152287.	1.64336	0.821680	−	0.569949i	$-0.193037\pi$
$97$	152287.	1.64336	0.821680	+	0.569949i	$0.193037\pi$
$98$	0	0
$99$	0	0
$100$	−49675.6	−0.496756
$101$	−107644.	−1.04999	−0.524997	−	0.851104i	$-0.675934\pi$
$101$	−107644.	−1.04999	−0.524997	+	0.851104i	$0.675934\pi$
$102$	0	0
$103$	120639.	1.12046	0.560229	−	0.828338i	$-0.310713\pi$
$103$	120639.	1.12046	0.560229	+	0.828338i	$0.310713\pi$
$104$	−5470.55	−0.0495961
$105$	0	0
$106$	54795.2	0.473672
$107$	−86114.3	−0.727136	−0.363568	−	0.931568i	$-0.618442\pi$
$107$	−86114.3	−0.727136	−0.363568	+	0.931568i	$0.618442\pi$
$108$	0	0
$109$	−66139.2	−0.533203	−0.266602	−	0.963807i	$-0.585901\pi$
$109$	−66139.2	−0.533203	−0.266602	+	0.963807i	$0.585901\pi$
$110$	−2092.61	−0.0164895
$111$	0	0
$112$	0	0
$113$	−131608.	−0.969588	−0.484794	−	0.874628i	$-0.661105\pi$
$113$	−131608.	−0.969588	−0.484794	+	0.874628i	$0.661105\pi$
$114$	0	0
$115$	12280.3	0.0865894
$116$	−93767.7	−0.647006
$117$	0	0
$118$	−89502.8	−0.591741
$119$	0	0
$120$	0	0
$121$	−147551.	−0.916174
$122$	−50768.3	−0.308812
$123$	0	0
$124$	4466.18	0.0260845
$125$	28049.5	0.160565
$126$	0	0
$127$	14437.3	0.0794284	0.0397142	−	0.999211i	$-0.487355\pi$
$127$	14437.3	0.0794284	0.0397142	+	0.999211i	$0.487355\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	1539.45	0.00798929
$131$	−319216.	−1.62520	−0.812600	−	0.582821i	$-0.801949\pi$
$131$	−319216.	−1.62520	−0.812600	+	0.582821i	$0.801949\pi$
$132$	0	0
$133$	0	0
$134$	−209213.	−1.00653
$135$	0	0
$136$	−2129.03	−0.00987041
$137$	50534.0	0.230029	0.115014	−	0.993364i	$-0.463309\pi$
$137$	50534.0	0.230029	0.115014	+	0.993364i	$0.463309\pi$
$138$	0	0
$139$	223787.	0.982422	0.491211	−	0.871041i	$-0.336554\pi$
$139$	223787.	0.982422	0.491211	+	0.871041i	$0.336554\pi$
$140$	0	0
$141$	0	0
$142$	−240921.	−1.00266
$143$	−9931.68	−0.0406146
$144$	0	0
$145$	26387.0	0.104224
$146$	307834.	1.19518
$147$	0	0
$148$	48615.9	0.182442
$149$	−259521.	−0.957651	−0.478825	−	0.877910i	$-0.658937\pi$
$149$	−259521.	−0.957651	−0.478825	+	0.877910i	$0.658937\pi$
$150$	0	0
$151$	−110893.	−0.395786	−0.197893	−	0.980224i	$-0.563410\pi$
$151$	−110893.	−0.395786	−0.197893	+	0.980224i	$0.563410\pi$
$152$	40664.1	0.142759
$153$	0	0
$154$	0	0
$155$	−1256.82	−0.00420188
$156$	0	0
$157$	−405385.	−1.31256	−0.656280	−	0.754518i	$-0.727871\pi$
$157$	−405385.	−1.31256	−0.656280	+	0.754518i	$0.727871\pi$
$158$	−134384.	−0.428257
$159$	0	0
$160$	−4610.59	−0.0142382
$161$	0	0
$162$	0	0
$163$	38618.7	0.113849	0.0569244	−	0.998378i	$-0.481871\pi$
$163$	38618.7	0.113849	0.0569244	+	0.998378i	$0.481871\pi$
$164$	−13110.6	−0.0380640
$165$	0	0
$166$	−242300.	−0.682469
$167$	207255.	0.575060	0.287530	−	0.957772i	$-0.407166\pi$
$167$	207255.	0.575060	0.287530	+	0.957772i	$0.407166\pi$
$168$	0	0
$169$	−363987.	−0.980322
$170$	599.127	0.00159000
$171$	0	0
$172$	177603.	0.457752
$173$	128257.	0.325812	0.162906	−	0.986642i	$-0.447913\pi$
$173$	128257.	0.325812	0.162906	+	0.986642i	$0.447913\pi$
$174$	0	0
$175$	0	0
$176$	29744.9	0.0723820
$177$	0	0
$178$	−368760.	−0.872357
$179$	98599.5	0.230007	0.115004	−	0.993365i	$-0.463312\pi$
$179$	98599.5	0.230007	0.115004	+	0.993365i	$0.463312\pi$
$180$	0	0
$181$	599225.	1.35954	0.679772	−	0.733423i	$-0.262079\pi$
$181$	599225.	1.35954	0.679772	+	0.733423i	$0.262079\pi$
$182$	0	0
$183$	0	0
$184$	−174555.	−0.380092
$185$	−13680.9	−0.0293890
$186$	0	0
$187$	−3865.23	−0.00808297
$188$	118518.	0.244563
$189$	0	0
$190$	−11443.2	−0.0229966
$191$	−829216.	−1.64469	−0.822346	−	0.568988i	$-0.807335\pi$
$191$	−829216.	−1.64469	−0.822346	+	0.568988i	$0.807335\pi$
$192$	0	0
$193$	−416385.	−0.804641	−0.402320	−	0.915499i	$-0.631796\pi$
$193$	−416385.	−0.804641	−0.402320	+	0.915499i	$0.631796\pi$
$194$	609147.	1.16203
$195$	0	0
$196$	0	0
$197$	592061.	1.08693	0.543464	−	0.839433i	$-0.317113\pi$
$197$	592061.	1.08693	0.543464	+	0.839433i	$0.317113\pi$
$198$	0	0
$199$	−1.06608e6	−1.90835	−0.954173	−	0.299255i	$-0.903262\pi$
$199$	−1.06608e6	−1.90835	−0.954173	+	0.299255i	$0.903262\pi$
$200$	−198703.	−0.351260
$201$	0	0
$202$	−430577.	−0.742458
$203$	0	0
$204$	0	0
$205$	3689.44	0.00613162
$206$	482557.	0.792283
$207$	0	0
$208$	−21882.2	−0.0350697
$209$	73825.0	0.116906
$210$	0	0
$211$	−846029.	−1.30822	−0.654108	−	0.756401i	$-0.726956\pi$
$211$	−846029.	−1.30822	−0.654108	+	0.756401i	$0.726956\pi$
$212$	219181.	0.334937
$213$	0	0
$214$	−344457.	−0.514163
$215$	−49979.0	−0.0737380
$216$	0	0
$217$	0	0
$218$	−264557.	−0.377032
$219$	0	0
$220$	−8370.44	−0.0116598
$221$	2843.50	0.00391627
$222$	0	0
$223$	112299.	0.151222	0.0756109	−	0.997137i	$-0.475909\pi$
$223$	112299.	0.151222	0.0756109	+	0.997137i	$0.475909\pi$
$224$	0	0
$225$	0	0
$226$	−526433.	−0.685602
$227$	178477.	0.229889	0.114944	−	0.993372i	$-0.463331\pi$
$227$	178477.	0.229889	0.114944	+	0.993372i	$0.463331\pi$
$228$	0	0
$229$	459753.	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$229$	459753.	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$230$	49121.2	0.0612280
$231$	0	0
$232$	−375071.	−0.457502
$233$	−748667.	−0.903440	−0.451720	−	0.892160i	$-0.649189\pi$
$233$	−748667.	−0.903440	−0.451720	+	0.892160i	$0.649189\pi$
$234$	0	0
$235$	−33352.0	−0.0393960
$236$	−358011.	−0.418424
$237$	0	0
$238$	0	0
$239$	814752.	0.922637	0.461319	−	0.887235i	$-0.347377\pi$
$239$	814752.	0.922637	0.461319	+	0.887235i	$0.347377\pi$
$240$	0	0
$241$	−1.04763e6	−1.16189	−0.580947	−	0.813942i	$-0.697318\pi$
$241$	−1.04763e6	−1.16189	−0.580947	+	0.813942i	$0.697318\pi$
$242$	−590203.	−0.647833
$243$	0	0
$244$	−203073.	−0.218363
$245$	0	0
$246$	0	0
$247$	−54310.3	−0.0566421
$248$	17864.7	0.0184445
$249$	0	0
$250$	112198.	0.113536
$251$	1.09401e6	1.09607	0.548034	−	0.836456i	$-0.315377\pi$
$251$	1.09401e6	1.09607	0.548034	+	0.836456i	$0.315377\pi$
$252$	0	0
$253$	−316902.	−0.311261
$254$	57749.1	0.0561644
$255$	0	0
$256$	65536.0	0.0625000
$257$	778391.	0.735131	0.367566	−	0.929998i	$-0.380191\pi$
$257$	778391.	0.735131	0.367566	+	0.929998i	$0.380191\pi$
$258$	0	0
$259$	0	0
$260$	6157.82	0.00564928
$261$	0	0
$262$	−1.27687e6	−1.14919
$263$	−1.10976e6	−0.989329	−0.494664	−	0.869084i	$-0.664709\pi$
$263$	−1.10976e6	−0.989329	−0.494664	+	0.869084i	$0.664709\pi$
$264$	0	0
$265$	−61679.2	−0.0539540
$266$	0	0
$267$	0	0
$268$	−836854.	−0.711725
$269$	−1.33640e6	−1.12605	−0.563024	−	0.826440i	$-0.690362\pi$
$269$	−1.33640e6	−1.12605	−0.563024	+	0.826440i	$0.690362\pi$
$270$	0	0
$271$	1.79201e6	1.48224	0.741118	−	0.671375i	$-0.234296\pi$
$271$	1.79201e6	1.48224	0.741118	+	0.671375i	$0.234296\pi$
$272$	−8516.14	−0.00697944
$273$	0	0
$274$	202136.	0.162655
$275$	−360741.	−0.287650
$276$	0	0
$277$	57577.6	0.0450873	0.0225436	−	0.999746i	$-0.492824\pi$
$277$	57577.6	0.0450873	0.0225436	+	0.999746i	$0.492824\pi$
$278$	895148.	0.694677
$279$	0	0
$280$	0	0
$281$	550257.	0.415719	0.207860	−	0.978159i	$-0.433350\pi$
$281$	550257.	0.415719	0.207860	+	0.978159i	$0.433350\pi$
$282$	0	0
$283$	−1.17788e6	−0.874249	−0.437125	−	0.899401i	$-0.644003\pi$
$283$	−1.17788e6	−0.874249	−0.437125	+	0.899401i	$0.644003\pi$
$284$	−963684.	−0.708988
$285$	0	0
$286$	−39726.7	−0.0287189
$287$	0	0
$288$	0	0
$289$	−1.41875e6	−0.999221
$290$	105548.	0.0736978
$291$	0	0
$292$	1.23133e6	0.845121
$293$	−300762.	−0.204670	−0.102335	−	0.994750i	$-0.532631\pi$
$293$	−300762.	−0.204670	−0.102335	+	0.994750i	$0.532631\pi$
$294$	0	0
$295$	100747.	0.0674028
$296$	194463.	0.129006
$297$	0	0
$298$	−1.03808e6	−0.677161
$299$	233133.	0.150809
$300$	0	0
$301$	0	0
$302$	−443571.	−0.279863
$303$	0	0
$304$	162656.	0.100946
$305$	57146.4	0.0351754
$306$	0	0
$307$	146787.	0.0888874	0.0444437	−	0.999012i	$-0.485848\pi$
$307$	146787.	0.0888874	0.0444437	+	0.999012i	$0.485848\pi$
$308$	0	0
$309$	0	0
$310$	−5027.27	−0.00297118
$311$	−1.70311e6	−0.998484	−0.499242	−	0.866462i	$-0.666388\pi$
$311$	−1.70311e6	−0.998484	−0.499242	+	0.866462i	$0.666388\pi$
$312$	0	0
$313$	1.11695e6	0.644424	0.322212	−	0.946668i	$-0.395574\pi$
$313$	1.11695e6	0.644424	0.322212	+	0.946668i	$0.395574\pi$
$314$	−1.62154e6	−0.928120
$315$	0	0
$316$	−537536.	−0.302824
$317$	962894.	0.538184	0.269092	−	0.963115i	$-0.413276\pi$
$317$	962894.	0.538184	0.269092	+	0.963115i	$0.413276\pi$
$318$	0	0
$319$	−680935.	−0.374653
$320$	−18442.3	−0.0100680
$321$	0	0
$322$	0	0
$323$	−21136.5	−0.0112727
$324$	0	0
$325$	265384.	0.139369
$326$	154475.	0.0805032
$327$	0	0
$328$	−52442.5	−0.0269153
$329$	0	0
$330$	0	0
$331$	−2.25995e6	−1.13378	−0.566891	−	0.823793i	$-0.691854\pi$
$331$	−2.25995e6	−1.13378	−0.566891	+	0.823793i	$0.691854\pi$
$332$	−969199.	−0.482578
$333$	0	0
$334$	829019.	0.406629
$335$	235497.	0.114650
$336$	0	0
$337$	1.33338e6	0.639557	0.319778	−	0.947492i	$-0.396392\pi$
$337$	1.33338e6	0.639557	0.319778	+	0.947492i	$0.396392\pi$
$338$	−1.45595e6	−0.693192
$339$	0	0
$340$	2396.51	0.00112430
$341$	32433.1	0.0151044
$342$	0	0
$343$	0	0
$344$	710414.	0.323680
$345$	0	0
$346$	513029.	0.230384
$347$	1.43486e6	0.639715	0.319857	−	0.947466i	$-0.396365\pi$
$347$	1.43486e6	0.639715	0.319857	+	0.947466i	$0.396365\pi$
$348$	0	0
$349$	3.34711e6	1.47098	0.735489	−	0.677536i	$-0.236952\pi$
$349$	3.34711e6	1.47098	0.735489	+	0.677536i	$0.236952\pi$
$350$	0	0
$351$	0	0
$352$	118979.	0.0511818
$353$	−4.00418e6	−1.71032	−0.855160	−	0.518365i	$-0.826541\pi$
$353$	−4.00418e6	−1.71032	−0.855160	+	0.518365i	$0.826541\pi$
$354$	0	0
$355$	271188.	0.114209
$356$	−1.47504e6	−0.616850
$357$	0	0
$358$	394398.	0.162640
$359$	595986.	0.244062	0.122031	−	0.992526i	$-0.461059\pi$
$359$	595986.	0.244062	0.122031	+	0.992526i	$0.461059\pi$
$360$	0	0
$361$	−2.07240e6	−0.836960
$362$	2.39690e6	0.961343
$363$	0	0
$364$	0	0
$365$	−346507.	−0.136138
$366$	0	0
$367$	−3.30390e6	−1.28045	−0.640224	−	0.768188i	$-0.721158\pi$
$367$	−3.30390e6	−1.28045	−0.640224	+	0.768188i	$0.721158\pi$
$368$	−698221.	−0.268766
$369$	0	0
$370$	−54723.5	−0.0207812
$371$	0	0
$372$	0	0
$373$	−2.83063e6	−1.05344	−0.526722	−	0.850038i	$-0.676579\pi$
$373$	−2.83063e6	−1.05344	−0.526722	+	0.850038i	$0.676579\pi$
$374$	−15460.9	−0.00571552
$375$	0	0
$376$	474074.	0.172932
$377$	500938.	0.181523
$378$	0	0
$379$	2.45025e6	0.876218	0.438109	−	0.898922i	$-0.355648\pi$
$379$	2.45025e6	0.876218	0.438109	+	0.898922i	$0.355648\pi$
$380$	−45772.8	−0.0162610
$381$	0	0
$382$	−3.31687e6	−1.16297
$383$	3.38405e6	1.17880	0.589399	−	0.807842i	$-0.299364\pi$
$383$	3.38405e6	1.17880	0.589399	+	0.807842i	$0.299364\pi$
$384$	0	0
$385$	0	0
$386$	−1.66554e6	−0.568967
$387$	0	0
$388$	2.43659e6	0.821680
$389$	4.14961e6	1.39038	0.695189	−	0.718827i	$-0.255321\pi$
$389$	4.14961e6	1.39038	0.695189	+	0.718827i	$0.255321\pi$
$390$	0	0
$391$	90731.0	0.0300133
$392$	0	0
$393$	0	0
$394$	2.36824e6	0.768574
$395$	151267.	0.0487810
$396$	0	0
$397$	−1.64666e6	−0.524358	−0.262179	−	0.965019i	$-0.584441\pi$
$397$	−1.64666e6	−0.524358	−0.262179	+	0.965019i	$0.584441\pi$
$398$	−4.26432e6	−1.34940
$399$	0	0
$400$	−794810.	−0.248378
$401$	4.38664e6	1.36229	0.681147	−	0.732147i	$-0.261481\pi$
$401$	4.38664e6	1.36229	0.681147	+	0.732147i	$0.261481\pi$
$402$	0	0
$403$	−23859.8	−0.00731820
$404$	−1.72231e6	−0.524997
$405$	0	0
$406$	0	0
$407$	353045.	0.105644
$408$	0	0
$409$	354592.	0.104814	0.0524072	−	0.998626i	$-0.483311\pi$
$409$	354592.	0.104814	0.0524072	+	0.998626i	$0.483311\pi$
$410$	14757.7	0.00433571
$411$	0	0
$412$	1.93023e6	0.560229
$413$	0	0
$414$	0	0
$415$	272740.	0.0777372
$416$	−87528.7	−0.0247980
$417$	0	0
$418$	295300.	0.0826652
$419$	1.79802e6	0.500335	0.250167	−	0.968203i	$-0.419514\pi$
$419$	1.79802e6	0.500335	0.250167	+	0.968203i	$0.419514\pi$
$420$	0	0
$421$	6.10881e6	1.67978	0.839888	−	0.542760i	$-0.182621\pi$
$421$	6.10881e6	1.67978	0.839888	+	0.542760i	$0.182621\pi$
$422$	−3.38412e6	−0.925048
$423$	0	0
$424$	876723.	0.236836
$425$	103282.	0.0277366
$426$	0	0
$427$	0	0
$428$	−1.37783e6	−0.363568
$429$	0	0
$430$	−199916.	−0.0521407
$431$	1.91176e6	0.495724	0.247862	−	0.968795i	$-0.420272\pi$
$431$	1.91176e6	0.495724	0.247862	+	0.968795i	$0.420272\pi$
$432$	0	0
$433$	385530.	0.0988186	0.0494093	−	0.998779i	$-0.484266\pi$
$433$	385530.	0.0988186	0.0494093	+	0.998779i	$0.484266\pi$
$434$	0	0
$435$	0	0
$436$	−1.05823e6	−0.266602
$437$	−1.73294e6	−0.434091
$438$	0	0
$439$	−1.83382e6	−0.454146	−0.227073	−	0.973878i	$-0.572916\pi$
$439$	−1.83382e6	−0.454146	−0.227073	+	0.973878i	$0.572916\pi$
$440$	−33481.8	−0.00824473
$441$	0	0
$442$	11374.0	0.00276922
$443$	1.93392e6	0.468198	0.234099	−	0.972213i	$-0.424786\pi$
$443$	1.93392e6	0.468198	0.234099	+	0.972213i	$0.424786\pi$
$444$	0	0
$445$	415088.	0.0993666
$446$	449197.	0.106930
$447$	0	0
$448$	0	0
$449$	−6.76354e6	−1.58328	−0.791641	−	0.610987i	$-0.790773\pi$
$449$	−6.76354e6	−1.58328	−0.791641	+	0.610987i	$0.790773\pi$
$450$	0	0
$451$	−95208.5	−0.0220412
$452$	−2.10573e6	−0.484794
$453$	0	0
$454$	713907.	0.162556
$455$	0	0
$456$	0	0
$457$	−2.39492e6	−0.536414	−0.268207	−	0.963361i	$-0.586431\pi$
$457$	−2.39492e6	−0.536414	−0.268207	+	0.963361i	$0.586431\pi$
$458$	1.83901e6	0.409658
$459$	0	0
$460$	196485.	0.0432947
$461$	6.42949e6	1.40904	0.704522	−	0.709682i	$-0.251162\pi$
$461$	6.42949e6	1.40904	0.704522	+	0.709682i	$0.251162\pi$
$462$	0	0
$463$	1.36524e6	0.295976	0.147988	−	0.988989i	$-0.452720\pi$
$463$	1.36524e6	0.295976	0.147988	+	0.988989i	$0.452720\pi$
$464$	−1.50028e6	−0.323503
$465$	0	0
$466$	−2.99467e6	−0.638828
$467$	−4.36362e6	−0.925880	−0.462940	−	0.886390i	$-0.653205\pi$
$467$	−4.36362e6	−0.925880	−0.462940	+	0.886390i	$0.653205\pi$
$468$	0	0
$469$	0	0
$470$	−133408.	−0.0278572
$471$	0	0
$472$	−1.43204e6	−0.295870
$473$	1.28974e6	0.265064
$474$	0	0
$475$	−1.97267e6	−0.401163
$476$	0	0
$477$	0	0
$478$	3.25901e6	0.652403
$479$	1.39250e6	0.277303	0.138652	−	0.990341i	$-0.455723\pi$
$479$	1.39250e6	0.277303	0.138652	+	0.990341i	$0.455723\pi$
$480$	0	0
$481$	−259722.	−0.0511854
$482$	−4.19053e6	−0.821583
$483$	0	0
$484$	−2.36081e6	−0.458087
$485$	−685675.	−0.132362
$486$	0	0
$487$	9.82919e6	1.87800	0.939000	−	0.343918i	$-0.111754\pi$
$487$	9.82919e6	1.87800	0.939000	+	0.343918i	$0.111754\pi$
$488$	−812294.	−0.154406
$489$	0	0
$490$	0	0
$491$	7.67255e6	1.43627	0.718135	−	0.695904i	$-0.244996\pi$
$491$	7.67255e6	1.43627	0.718135	+	0.695904i	$0.244996\pi$
$492$	0	0
$493$	194956.	0.0361259
$494$	−217241.	−0.0400520
$495$	0	0
$496$	71458.9	0.0130422
$497$	0	0
$498$	0	0
$499$	3.39383e6	0.610152	0.305076	−	0.952328i	$-0.401318\pi$
$499$	3.39383e6	0.610152	0.305076	+	0.952328i	$0.401318\pi$
$500$	448792.	0.0802824
$501$	0	0
$502$	4.37604e6	0.775037
$503$	8.36268e6	1.47376	0.736878	−	0.676026i	$-0.236299\pi$
$503$	8.36268e6	1.47376	0.736878	+	0.676026i	$0.236299\pi$
$504$	0	0
$505$	484671.	0.0845704
$506$	−1.26761e6	−0.220094
$507$	0	0
$508$	230996.	0.0397142
$509$	−3.87324e6	−0.662643	−0.331322	−	0.943518i	$-0.607494\pi$
$509$	−3.87324e6	−0.662643	−0.331322	+	0.943518i	$0.607494\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	3.11356e6	0.519816
$515$	−543181.	−0.0902457
$516$	0	0
$517$	860672.	0.141616
$518$	0	0
$519$	0	0
$520$	24631.3	0.00399465
$521$	−731064.	−0.117994	−0.0589971	−	0.998258i	$-0.518790\pi$
$521$	−731064.	−0.117994	−0.0589971	+	0.998258i	$0.518790\pi$
$522$	0	0
$523$	−7.10854e6	−1.13639	−0.568193	−	0.822895i	$-0.692357\pi$
$523$	−7.10854e6	−1.13639	−0.568193	+	0.822895i	$0.692357\pi$
$524$	−5.10746e6	−0.812600
$525$	0	0
$526$	−4.43905e6	−0.699561
$527$	−9285.80	−0.00145644
$528$	0	0
$529$	1.00252e6	0.155759
$530$	−246717.	−0.0381513
$531$	0	0
$532$	0	0
$533$	70041.3	0.0106791
$534$	0	0
$535$	387732.	0.0585662
$536$	−3.34741e6	−0.503266
$537$	0	0
$538$	−5.34562e6	−0.796237
$539$	0	0
$540$	0	0
$541$	3.79514e6	0.557487	0.278743	−	0.960366i	$-0.410082\pi$
$541$	3.79514e6	0.557487	0.278743	+	0.960366i	$0.410082\pi$
$542$	7.16804e6	1.04810
$543$	0	0
$544$	−34064.5	−0.00493521
$545$	297793.	0.0429461
$546$	0	0
$547$	−1.34686e7	−1.92466	−0.962332	−	0.271879i	$-0.912355\pi$
$547$	−1.34686e7	−1.92466	−0.962332	+	0.271879i	$0.912355\pi$
$548$	808543.	0.115014
$549$	0	0
$550$	−1.44296e6	−0.203399
$551$	−3.72361e6	−0.522499
$552$	0	0
$553$	0	0
$554$	230310.	0.0318815
$555$	0	0
$556$	3.58059e6	0.491211
$557$	−4.89487e6	−0.668503	−0.334251	−	0.942484i	$-0.608483\pi$
$557$	−4.89487e6	−0.668503	−0.334251	+	0.942484i	$0.608483\pi$
$558$	0	0
$559$	−948816.	−0.128426
$560$	0	0
$561$	0	0
$562$	2.20103e6	0.293958
$563$	6.25606e6	0.831821	0.415910	−	0.909406i	$-0.363463\pi$
$563$	6.25606e6	0.831821	0.415910	+	0.909406i	$0.363463\pi$
$564$	0	0
$565$	592569.	0.0780941
$566$	−4.71152e6	−0.618188
$567$	0	0
$568$	−3.85474e6	−0.501330
$569$	−1.42825e6	−0.184937	−0.0924685	−	0.995716i	$-0.529476\pi$
$569$	−1.42825e6	−0.184937	−0.0924685	+	0.995716i	$0.529476\pi$
$570$	0	0
$571$	9.05147e6	1.16179	0.580897	−	0.813977i	$-0.302702\pi$
$571$	9.05147e6	1.16179	0.580897	+	0.813977i	$0.302702\pi$
$572$	−158907.	−0.0203073
$573$	0	0
$574$	0	0
$575$	8.46792e6	1.06809
$576$	0	0
$577$	8.01919e6	1.00275	0.501373	−	0.865231i	$-0.332828\pi$
$577$	8.01919e6	1.00275	0.501373	+	0.865231i	$0.332828\pi$
$578$	−5.67500e6	−0.706556
$579$	0	0
$580$	422191.	0.0521122
$581$	0	0
$582$	0	0
$583$	1.59168e6	0.193947
$584$	4.92534e6	0.597591
$585$	0	0
$586$	−1.20305e6	−0.144723
$587$	8.28470e6	0.992388	0.496194	−	0.868212i	$-0.334730\pi$
$587$	8.28470e6	0.992388	0.496194	+	0.868212i	$0.334730\pi$
$588$	0	0
$589$	177357.	0.0210649
$590$	402989.	0.0476610
$591$	0	0
$592$	777854.	0.0912208
$593$	−5.91443e6	−0.690679	−0.345340	−	0.938478i	$-0.612236\pi$
$593$	−5.91443e6	−0.690679	−0.345340	+	0.938478i	$0.612236\pi$
$594$	0	0
$595$	0	0
$596$	−4.15234e6	−0.478825
$597$	0	0
$598$	932532.	0.106638
$599$	−3.32897e6	−0.379091	−0.189545	−	0.981872i	$-0.560701\pi$
$599$	−3.32897e6	−0.379091	−0.189545	+	0.981872i	$0.560701\pi$
$600$	0	0
$601$	−4.87081e6	−0.550066	−0.275033	−	0.961435i	$-0.588689\pi$
$601$	−4.87081e6	−0.550066	−0.275033	+	0.961435i	$0.588689\pi$
$602$	0	0
$603$	0	0
$604$	−1.77428e6	−0.197893
$605$	664351.	0.0737919
$606$	0	0
$607$	−1.51701e7	−1.67115	−0.835577	−	0.549374i	$-0.814866\pi$
$607$	−1.51701e7	−1.67115	−0.835577	+	0.549374i	$0.814866\pi$
$608$	650626.	0.0713793
$609$	0	0
$610$	228586.	0.0248728
$611$	−633164.	−0.0686141
$612$	0	0
$613$	5.18156e6	0.556941	0.278471	−	0.960445i	$-0.410173\pi$
$613$	5.18156e6	0.556941	0.278471	+	0.960445i	$0.410173\pi$
$614$	587146.	0.0628529
$615$	0	0
$616$	0	0
$617$	−1.55854e6	−0.164818	−0.0824092	−	0.996599i	$-0.526261\pi$
$617$	−1.55854e6	−0.164818	−0.0824092	+	0.996599i	$0.526261\pi$
$618$	0	0
$619$	−1.40751e6	−0.147647	−0.0738236	−	0.997271i	$-0.523520\pi$
$619$	−1.40751e6	−0.147647	−0.0738236	+	0.997271i	$0.523520\pi$
$620$	−20109.1	−0.00210094
$621$	0	0
$622$	−6.81243e6	−0.706035
$623$	0	0
$624$	0	0
$625$	9.57598e6	0.980580
$626$	4.46779e6	0.455677
$627$	0	0
$628$	−6.48617e6	−0.656280
$629$	−101079.	−0.0101867
$630$	0	0
$631$	−1.09643e7	−1.09624	−0.548120	−	0.836399i	$-0.684656\pi$
$631$	−1.09643e7	−1.09624	−0.548120	+	0.836399i	$0.684656\pi$
$632$	−2.15014e6	−0.214129
$633$	0	0
$634$	3.85158e6	0.380553
$635$	−65004.2	−0.00639745
$636$	0	0
$637$	0	0
$638$	−2.72374e6	−0.264919
$639$	0	0
$640$	−73769.4	−0.00711912
$641$	1.42122e7	1.36621	0.683103	−	0.730322i	$-0.260630\pi$
$641$	1.42122e7	1.36621	0.683103	+	0.730322i	$0.260630\pi$
$642$	0	0
$643$	−9.13928e6	−0.871735	−0.435868	−	0.900011i	$-0.643558\pi$
$643$	−9.13928e6	−0.871735	−0.435868	+	0.900011i	$0.643558\pi$
$644$	0	0
$645$	0	0
$646$	−84546.2	−0.00797100
$647$	1.89272e7	1.77757	0.888785	−	0.458324i	$-0.151550\pi$
$647$	1.89272e7	1.77757	0.888785	+	0.458324i	$0.151550\pi$
$648$	0	0
$649$	−2.59985e6	−0.242291
$650$	1.06153e6	0.0985487
$651$	0	0
$652$	617899.	0.0569244
$653$	4.29983e6	0.394610	0.197305	−	0.980342i	$-0.436781\pi$
$653$	4.29983e6	0.394610	0.197305	+	0.980342i	$0.436781\pi$
$654$	0	0
$655$	1.43728e6	0.130900
$656$	−209770.	−0.0190320
$657$	0	0
$658$	0	0
$659$	1.15303e7	1.03425	0.517126	−	0.855909i	$-0.327002\pi$
$659$	1.15303e7	1.03425	0.517126	+	0.855909i	$0.327002\pi$
$660$	0	0
$661$	1.47047e7	1.30904	0.654521	−	0.756044i	$-0.272870\pi$
$661$	1.47047e7	1.30904	0.654521	+	0.756044i	$0.272870\pi$
$662$	−9.03981e6	−0.801704
$663$	0	0
$664$	−3.87680e6	−0.341234
$665$	0	0
$666$	0	0
$667$	1.59840e7	1.39114
$668$	3.31608e6	0.287530
$669$	0	0
$670$	941989.	0.0810697
$671$	−1.47471e6	−0.126444
$672$	0	0
$673$	1.42971e7	1.21678	0.608389	−	0.793639i	$-0.291816\pi$
$673$	1.42971e7	1.21678	0.608389	+	0.793639i	$0.291816\pi$
$674$	5.33352e6	0.452235
$675$	0	0
$676$	−5.82379e6	−0.490161
$677$	−9.12025e6	−0.764777	−0.382389	−	0.924002i	$-0.624898\pi$
$677$	−9.12025e6	−0.764777	−0.382389	+	0.924002i	$0.624898\pi$
$678$	0	0
$679$	0	0
$680$	9586.03	0.000794999 0
$681$	0	0
$682$	129732.	0.0106804
$683$	1.45152e7	1.19062	0.595309	−	0.803497i	$-0.297030\pi$
$683$	1.45152e7	1.19062	0.595309	+	0.803497i	$0.297030\pi$
$684$	0	0
$685$	−227530.	−0.0185273
$686$	0	0
$687$	0	0
$688$	2.84166e6	0.228876
$689$	−1.17094e6	−0.0939691
$690$	0	0
$691$	2.08507e6	0.166122	0.0830608	−	0.996544i	$-0.473530\pi$
$691$	2.08507e6	0.166122	0.0830608	+	0.996544i	$0.473530\pi$
$692$	2.05212e6	0.162906
$693$	0	0
$694$	5.73945e6	0.452347
$695$	−1.00761e6	−0.0791278
$696$	0	0
$697$	27258.8	0.00212532
$698$	1.33884e7	1.04014
$699$	0	0
$700$	0	0
$701$	−9.96513e6	−0.765928	−0.382964	−	0.923763i	$-0.625097\pi$
$701$	−9.96513e6	−0.765928	−0.382964	+	0.923763i	$0.625097\pi$
$702$	0	0
$703$	1.93059e6	0.147333
$704$	475918.	0.0361910
$705$	0	0
$706$	−1.60167e7	−1.20938
$707$	0	0
$708$	0	0
$709$	7.09035e6	0.529727	0.264864	−	0.964286i	$-0.414673\pi$
$709$	7.09035e6	0.529727	0.264864	+	0.964286i	$0.414673\pi$
$710$	1.08475e6	0.0807579
$711$	0	0
$712$	−5.90017e6	−0.436179
$713$	−761324.	−0.0560849
$714$	0	0
$715$	44717.6	0.00327125
$716$	1.57759e6	0.115004
$717$	0	0
$718$	2.38394e6	0.172578
$719$	−1.20167e7	−0.866886	−0.433443	−	0.901181i	$-0.642701\pi$
$719$	−1.20167e7	−0.866886	−0.433443	+	0.901181i	$0.642701\pi$
$720$	0	0
$721$	0	0
$722$	−8.28958e6	−0.591820
$723$	0	0
$724$	9.58760e6	0.679772
$725$	1.81952e7	1.28562
$726$	0	0
$727$	1.32577e7	0.930318	0.465159	−	0.885227i	$-0.345997\pi$
$727$	1.32577e7	0.930318	0.465159	+	0.885227i	$0.345997\pi$
$728$	0	0
$729$	0	0
$730$	−1.38603e6	−0.0962642
$731$	−369262.	−0.0255588
$732$	0	0
$733$	−909303.	−0.0625099	−0.0312549	−	0.999511i	$-0.509950\pi$
$733$	−909303.	−0.0625099	−0.0312549	+	0.999511i	$0.509950\pi$
$734$	−1.32156e7	−0.905413
$735$	0	0
$736$	−2.79289e6	−0.190046
$737$	−6.07717e6	−0.412129
$738$	0	0
$739$	−286851.	−0.0193217	−0.00966086	−	0.999953i	$-0.503075\pi$
$739$	−286851.	−0.0193217	−0.00966086	+	0.999953i	$0.503075\pi$
$740$	−218894.	−0.0146945
$741$	0	0
$742$	0	0
$743$	1.35906e6	0.0903163	0.0451582	−	0.998980i	$-0.485621\pi$
$743$	1.35906e6	0.0903163	0.0451582	+	0.998980i	$0.485621\pi$
$744$	0	0
$745$	1.16850e6	0.0771327
$746$	−1.13225e7	−0.744897
$747$	0	0
$748$	−61843.6	−0.00404148
$749$	0	0
$750$	0	0
$751$	1.54404e7	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$751$	1.54404e7	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$752$	1.89629e6	0.122282
$753$	0	0
$754$	2.00375e6	0.128356
$755$	499298.	0.0318781
$756$	0	0
$757$	1.70683e7	1.08256	0.541279	−	0.840843i	$-0.317940\pi$
$757$	1.70683e7	1.08256	0.541279	+	0.840843i	$0.317940\pi$
$758$	9.80100e6	0.619580
$759$	0	0
$760$	−183091.	−0.0114983
$761$	−3.08345e6	−0.193008	−0.0965041	−	0.995333i	$-0.530766\pi$
$761$	−3.08345e6	−0.193008	−0.0965041	+	0.995333i	$0.530766\pi$
$762$	0	0
$763$	0	0
$764$	−1.32675e7	−0.822346
$765$	0	0
$766$	1.35362e7	0.833537
$767$	1.91261e6	0.117392
$768$	0	0
$769$	3.00821e7	1.83439	0.917195	−	0.398439i	$-0.130448\pi$
$769$	3.00821e7	1.83439	0.917195	+	0.398439i	$0.130448\pi$
$770$	0	0
$771$	0	0
$772$	−6.66216e6	−0.402320
$773$	−2.16914e7	−1.30569	−0.652843	−	0.757494i	$-0.726424\pi$
$773$	−2.16914e7	−1.30569	−0.652843	+	0.757494i	$0.726424\pi$
$774$	0	0
$775$	−866642.	−0.0518305
$776$	9.74635e6	0.581016
$777$	0	0
$778$	1.65984e7	0.983146
$779$	−520637.	−0.0307391
$780$	0	0
$781$	−6.99821e6	−0.410544
$782$	362924.	0.0212226
$783$	0	0
$784$	0	0
$785$	1.82526e6	0.105718
$786$	0	0
$787$	1.21019e6	0.0696490	0.0348245	−	0.999393i	$-0.488913\pi$
$787$	1.21019e6	0.0696490	0.0348245	+	0.999393i	$0.488913\pi$
$788$	9.47297e6	0.543464
$789$	0	0
$790$	605067.	0.0344934
$791$	0	0
$792$	0	0
$793$	1.08489e6	0.0612633
$794$	−6.58664e6	−0.370777
$795$	0	0
$796$	−1.70573e7	−0.954173
$797$	−2.44457e7	−1.36319	−0.681596	−	0.731728i	$-0.738714\pi$
$797$	−2.44457e7	−1.36319	−0.681596	+	0.731728i	$0.738714\pi$
$798$	0	0
$799$	−246416.	−0.0136553
$800$	−3.17924e6	−0.175630
$801$	0	0
$802$	1.75466e7	0.963288
$803$	8.94186e6	0.489372
$804$	0	0
$805$	0	0
$806$	−95439.3	−0.00517475
$807$	0	0
$808$	−6.88923e6	−0.371229
$809$	1.18645e7	0.637352	0.318676	−	0.947864i	$-0.396762\pi$
$809$	1.18645e7	0.637352	0.318676	+	0.947864i	$0.396762\pi$
$810$	0	0
$811$	−1.63289e7	−0.871778	−0.435889	−	0.900000i	$-0.643566\pi$
$811$	−1.63289e7	−0.871778	−0.435889	+	0.900000i	$0.643566\pi$
$812$	0	0
$813$	0	0
$814$	1.41218e6	0.0747015
$815$	−173882.	−0.00916979
$816$	0	0
$817$	7.05282e6	0.369664
$818$	1.41837e6	0.0741150
$819$	0	0
$820$	59031.0	0.00306581
$821$	2.20202e7	1.14015	0.570077	−	0.821591i	$-0.306913\pi$
$821$	2.20202e7	1.14015	0.570077	+	0.821591i	$0.306913\pi$
$822$	0	0
$823$	9.43220e6	0.485415	0.242708	−	0.970099i	$-0.421964\pi$
$823$	9.43220e6	0.485415	0.242708	+	0.970099i	$0.421964\pi$
$824$	7.72091e6	0.396142
$825$	0	0
$826$	0	0
$827$	1.93147e7	0.982027	0.491014	−	0.871152i	$-0.336626\pi$
$827$	1.93147e7	0.982027	0.491014	+	0.871152i	$0.336626\pi$
$828$	0	0
$829$	3.26260e7	1.64883	0.824417	−	0.565982i	$-0.191503\pi$
$829$	3.26260e7	1.64883	0.824417	+	0.565982i	$0.191503\pi$
$830$	1.09096e6	0.0549685
$831$	0	0
$832$	−350115.	−0.0175349
$833$	0	0
$834$	0	0
$835$	−933170.	−0.0463174
$836$	1.18120e6	0.0584531
$837$	0	0
$838$	7.19210e6	0.353790
$839$	−2.39890e7	−1.17654	−0.588272	−	0.808663i	$-0.700191\pi$
$839$	−2.39890e7	−1.17654	−0.588272	+	0.808663i	$0.700191\pi$
$840$	0	0
$841$	1.38341e7	0.674467
$842$	2.44352e7	1.18778
$843$	0	0
$844$	−1.35365e7	−0.654108
$845$	1.63886e6	0.0789587
$846$	0	0
$847$	0	0
$848$	3.50689e6	0.167468
$849$	0	0
$850$	413129.	0.0196128
$851$	−8.28726e6	−0.392272
$852$	0	0
$853$	1.36214e7	0.640988	0.320494	−	0.947251i	$-0.396151\pi$
$853$	1.36214e7	0.640988	0.320494	+	0.947251i	$0.396151\pi$
$854$	0	0
$855$	0	0
$856$	−5.51132e6	−0.257082
$857$	−3.97793e7	−1.85014	−0.925071	−	0.379793i	$-0.875995\pi$
$857$	−3.97793e7	−1.85014	−0.925071	+	0.379793i	$0.875995\pi$
$858$	0	0
$859$	1.88973e7	0.873807	0.436904	−	0.899508i	$-0.356075\pi$
$859$	1.88973e7	0.873807	0.436904	+	0.899508i	$0.356075\pi$
$860$	−799664.	−0.0368690
$861$	0	0
$862$	7.64704e6	0.350530
$863$	1.81214e7	0.828256	0.414128	−	0.910219i	$-0.364087\pi$
$863$	1.81214e7	0.828256	0.414128	+	0.910219i	$0.364087\pi$
$864$	0	0
$865$	−577482.	−0.0262421
$866$	1.54212e6	0.0698753
$867$	0	0
$868$	0	0
$869$	−3.90355e6	−0.175352
$870$	0	0
$871$	4.47075e6	0.199680
$872$	−4.23291e6	−0.188516
$873$	0	0
$874$	−6.93178e6	−0.306949
$875$	0	0
$876$	0	0
$877$	−2.38248e7	−1.04600	−0.522999	−	0.852333i	$-0.675187\pi$
$877$	−2.38248e7	−1.04600	−0.522999	+	0.852333i	$0.675187\pi$
$878$	−7.33528e6	−0.321129
$879$	0	0
$880$	−133927.	−0.00582991
$881$	6.59299e6	0.286182	0.143091	−	0.989710i	$-0.454296\pi$
$881$	6.59299e6	0.286182	0.143091	+	0.989710i	$0.454296\pi$
$882$	0	0
$883$	1.13391e7	0.489413	0.244707	−	0.969597i	$-0.421308\pi$
$883$	1.13391e7	0.489413	0.244707	+	0.969597i	$0.421308\pi$
$884$	45496.0	0.00195814
$885$	0	0
$886$	7.73568e6	0.331066
$887$	1.16981e7	0.499237	0.249619	−	0.968344i	$-0.419695\pi$
$887$	1.16981e7	0.499237	0.249619	+	0.968344i	$0.419695\pi$
$888$	0	0
$889$	0	0
$890$	1.66035e6	0.0702628
$891$	0	0
$892$	1.79679e6	0.0756109
$893$	4.70649e6	0.197501
$894$	0	0
$895$	−443947.	−0.0185256
$896$	0	0
$897$	0	0
$898$	−2.70542e7	−1.11955
$899$	−1.63587e6	−0.0675073
$900$	0	0
$901$	−455706.	−0.0187014
$902$	−380834.	−0.0155855
$903$	0	0
$904$	−8.42293e6	−0.342801
$905$	−2.69803e6	−0.109503
$906$	0	0
$907$	2.74600e6	0.110837	0.0554183	−	0.998463i	$-0.482351\pi$
$907$	2.74600e6	0.110837	0.0554183	+	0.998463i	$0.482351\pi$
$908$	2.85563e6	0.114944
$909$	0	0
$910$	0	0
$911$	8.70382e6	0.347467	0.173734	−	0.984793i	$-0.444417\pi$
$911$	8.70382e6	0.347467	0.173734	+	0.984793i	$0.444417\pi$
$912$	0	0
$913$	−7.03826e6	−0.279440
$914$	−9.57967e6	−0.379302
$915$	0	0
$916$	7.35606e6	0.289672
$917$	0	0
$918$	0	0
$919$	3.65884e7	1.42907	0.714536	−	0.699598i	$-0.246638\pi$
$919$	3.65884e7	1.42907	0.714536	+	0.699598i	$0.246638\pi$
$920$	785940.	0.0306140
$921$	0	0
$922$	2.57180e7	0.996344
$923$	5.14832e6	0.198912
$924$	0	0
$925$	−9.43369e6	−0.362516
$926$	5.46097e6	0.209287
$927$	0	0
$928$	−6.00113e6	−0.228751
$929$	3.19411e6	0.121426	0.0607128	−	0.998155i	$-0.480663\pi$
$929$	3.19411e6	0.121426	0.0607128	+	0.998155i	$0.480663\pi$
$930$	0	0
$931$	0	0
$932$	−1.19787e7	−0.451720
$933$	0	0
$934$	−1.74545e7	−0.654696
$935$	17403.3	0.000651031 0
$936$	0	0
$937$	1.53039e7	0.569447	0.284724	−	0.958610i	$-0.408098\pi$
$937$	1.53039e7	0.569447	0.284724	+	0.958610i	$0.408098\pi$
$938$	0	0
$939$	0	0
$940$	−533632.	−0.0196980
$941$	−1.23623e7	−0.455121	−0.227560	−	0.973764i	$-0.573075\pi$
$941$	−1.23623e7	−0.455121	−0.227560	+	0.973764i	$0.573075\pi$
$942$	0	0
$943$	2.23489e6	0.0818423
$944$	−5.72818e6	−0.209212
$945$	0	0
$946$	5.15898e6	0.187429
$947$	−5.83773e6	−0.211528	−0.105764	−	0.994391i	$-0.533729\pi$
$947$	−5.83773e6	−0.211528	−0.105764	+	0.994391i	$0.533729\pi$
$948$	0	0
$949$	−6.57819e6	−0.237105
$950$	−7.89069e6	−0.283665
$951$	0	0
$952$	0	0
$953$	−1.59935e7	−0.570443	−0.285222	−	0.958462i	$-0.592067\pi$
$953$	−1.59935e7	−0.570443	−0.285222	+	0.958462i	$0.592067\pi$
$954$	0	0
$955$	3.73357e6	0.132469
$956$	1.30360e7	0.461319
$957$	0	0
$958$	5.56998e6	0.196083
$959$	0	0
$960$	0	0
$961$	−2.85512e7	−0.997278
$962$	−1.03889e6	−0.0361935
$963$	0	0
$964$	−1.67621e7	−0.580947
$965$	1.87478e6	0.0648087
$966$	0	0
$967$	−4.65279e6	−0.160010	−0.0800050	−	0.996794i	$-0.525494\pi$
$967$	−4.65279e6	−0.160010	−0.0800050	+	0.996794i	$0.525494\pi$
$968$	−9.44324e6	−0.323916
$969$	0	0
$970$	−2.74270e6	−0.0935942
$971$	−2.17271e7	−0.739528	−0.369764	−	0.929126i	$-0.620561\pi$
$971$	−2.17271e7	−0.739528	−0.369764	+	0.929126i	$0.620561\pi$
$972$	0	0
$973$	0	0
$974$	3.93168e7	1.32795
$975$	0	0
$976$	−3.24917e6	−0.109181
$977$	−2.27382e7	−0.762113	−0.381056	−	0.924552i	$-0.624440\pi$
$977$	−2.27382e7	−0.762113	−0.381056	+	0.924552i	$0.624440\pi$
$978$	0	0
$979$	−1.07117e7	−0.357190
$980$	0	0
$981$	0	0
$982$	3.06902e7	1.01560
$983$	4.40870e7	1.45521	0.727607	−	0.685994i	$-0.240632\pi$
$983$	4.40870e7	1.45521	0.727607	+	0.685994i	$0.240632\pi$
$984$	0	0
$985$	−2.66577e6	−0.0875451
$986$	779823.	0.0255449
$987$	0	0
$988$	−868964.	−0.0283211
$989$	−3.02750e7	−0.984224
$990$	0	0
$991$	2.80292e7	0.906623	0.453312	−	0.891352i	$-0.350242\pi$
$991$	2.80292e7	0.906623	0.453312	+	0.891352i	$0.350242\pi$
$992$	285836.	0.00922226
$993$	0	0
$994$	0	0
$995$	4.80005e6	0.153705
$996$	0	0
$997$	−4.46315e7	−1.42201	−0.711006	−	0.703186i	$-0.751760\pi$
$997$	−4.46315e7	−1.42201	−0.711006	+	0.703186i	$0.751760\pi$
$998$	1.35753e7	0.431443
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bk.1.2		2
3.2	odd	2		294.6.a.q.1.1	yes	2
7.6	odd	2		882.6.a.bu.1.1		2
21.2	odd	6		294.6.e.x.67.2		4
21.5	even	6		294.6.e.z.67.1		4
21.11	odd	6		294.6.e.x.79.2		4
21.17	even	6		294.6.e.z.79.1		4
21.20	even	2		294.6.a.n.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.n.1.2	✓	2	21.20	even	2
294.6.a.q.1.1	yes	2	3.2	odd	2
294.6.e.x.67.2		4	21.2	odd	6
294.6.e.x.79.2		4	21.11	odd	6
294.6.e.z.67.1		4	21.5	even	6
294.6.e.z.79.1		4	21.17	even	6
882.6.a.bk.1.2		2	1.1	even	1	trivial
882.6.a.bu.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -4.50253 q^{5} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -4.50253 q^{5} +64.0000 q^{8} -18.0101 q^{10} +116.191 q^{11} -85.4773 q^{13} +256.000 q^{16} -33.2662 q^{17} +635.377 q^{19} -72.0404 q^{20} +464.764 q^{22} -2727.43 q^{23} -3104.73 q^{25} -341.909 q^{26} -5860.48 q^{29} +279.136 q^{31} +1024.00 q^{32} -133.065 q^{34} +3038.49 q^{37} +2541.51 q^{38} -288.162 q^{40} -819.415 q^{41} +11100.2 q^{43} +1859.05 q^{44} -10909.7 q^{46} +7407.40 q^{47} -12418.9 q^{50} -1367.64 q^{52} +13698.8 q^{53} -523.153 q^{55} -23441.9 q^{58} -22375.7 q^{59} -12692.1 q^{61} +1116.55 q^{62} +4096.00 q^{64} +384.864 q^{65} -52303.3 q^{67} -532.259 q^{68} -60230.2 q^{71} +76958.4 q^{73} +12154.0 q^{74} +10166.0 q^{76} -33596.0 q^{79} -1152.65 q^{80} -3277.66 q^{82} -60574.9 q^{83} +149.782 q^{85} +44400.9 q^{86} +7436.22 q^{88} -92190.1 q^{89} -43638.8 q^{92} +29629.6 q^{94} -2860.80 q^{95} +152287. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8} - 432 q^{10} - 124 q^{11} + 720 q^{13} + 512 q^{16} + 1260 q^{17} + 360 q^{19} - 1728 q^{20} - 496 q^{22} - 6524 q^{23} + 4482 q^{25} + 2880 q^{26} - 7088 q^{29} + 5904 q^{31} + 2048 q^{32} + 5040 q^{34} - 6040 q^{37} + 1440 q^{38} - 6912 q^{40} + 17388 q^{41} - 608 q^{43} - 1984 q^{44} - 26096 q^{46} + 30456 q^{47} + 17928 q^{50} + 11520 q^{52} - 3964 q^{53} + 24336 q^{55} - 28352 q^{58} - 40752 q^{59} - 1368 q^{61} + 23616 q^{62} + 8192 q^{64} - 82980 q^{65} - 16224 q^{67} + 20160 q^{68} + 3204 q^{71} + 23976 q^{73} - 24160 q^{74} + 5760 q^{76} - 82160 q^{79} - 27648 q^{80} + 69552 q^{82} - 173736 q^{83} - 133700 q^{85} - 2432 q^{86} - 7936 q^{88} - 200556 q^{89} - 104384 q^{92} + 121824 q^{94} + 25640 q^{95} + 251928 q^{97}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8 - 432 * q^10 - 124 * q^11 + 720 * q^13 + 512 * q^16 + 1260 * q^17 + 360 * q^19 - 1728 * q^20 - 496 * q^22 - 6524 * q^23 + 4482 * q^25 + 2880 * q^26 - 7088 * q^29 + 5904 * q^31 + 2048 * q^32 + 5040 * q^34 - 6040 * q^37 + 1440 * q^38 - 6912 * q^40 + 17388 * q^41 - 608 * q^43 - 1984 * q^44 - 26096 * q^46 + 30456 * q^47 + 17928 * q^50 + 11520 * q^52 - 3964 * q^53 + 24336 * q^55 - 28352 * q^58 - 40752 * q^59 - 1368 * q^61 + 23616 * q^62 + 8192 * q^64 - 82980 * q^65 - 16224 * q^67 + 20160 * q^68 + 3204 * q^71 + 23976 * q^73 - 24160 * q^74 + 5760 * q^76 - 82160 * q^79 - 27648 * q^80 + 69552 * q^82 - 173736 * q^83 - 133700 * q^85 - 2432 * q^86 - 7936 * q^88 - 200556 * q^89 - 104384 * q^92 + 121824 * q^94 + 25640 * q^95 + 251928 * q^97

Introduction

L-functions

Modular forms

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