LMFDB - Embedded newform 882.6.a.bm.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{505}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 126 $ x^2 - x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$11.7361$ 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)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -87.1527 q^{5} +64.0000 q^{8} -348.611 q^{10} +478.069 q^{11} -436.153 q^{13} +256.000 q^{16} -2259.05 q^{17} +2664.12 q^{19} -1394.44 q^{20} +1912.28 q^{22} +2568.61 q^{23} +4470.60 q^{25} -1744.61 q^{26} +708.010 q^{29} +5509.14 q^{31} +1024.00 q^{32} -9036.22 q^{34} -915.630 q^{37} +10656.5 q^{38} -5577.77 q^{40} -8866.08 q^{41} -7915.62 q^{43} +7649.10 q^{44} +10274.4 q^{46} -19572.5 q^{47} +17882.4 q^{50} -6978.44 q^{52} -8247.87 q^{53} -41665.0 q^{55} +2832.04 q^{58} +32190.6 q^{59} +28420.1 q^{61} +22036.6 q^{62} +4096.00 q^{64} +38011.9 q^{65} -9682.41 q^{67} -36144.9 q^{68} -28568.4 q^{71} -3112.88 q^{73} -3662.52 q^{74} +42626.0 q^{76} -6790.07 q^{79} -22311.1 q^{80} -35464.3 q^{82} -14482.3 q^{83} +196883. q^{85} -31662.5 q^{86} +30596.4 q^{88} -27464.6 q^{89} +41097.8 q^{92} -78290.0 q^{94} -232186. q^{95} -91212.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 17 q^{5} + 128 q^{8} - 68 q^{10} - 145 q^{11} - 715 q^{13} + 512 q^{16} - 1372 q^{17} + 1081 q^{19} - 272 q^{20} - 580 q^{22} + 4508 q^{23} + 6267 q^{25} - 2860 q^{26} - 7865 q^{29}+ \cdots - 46671 q^{97}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 17 * q^5 + 128 * q^8 - 68 * q^10 - 145 * q^11 - 715 * q^13 + 512 * q^16 - 1372 * q^17 + 1081 * q^19 - 272 * q^20 - 580 * q^22 + 4508 * q^23 + 6267 * q^25 - 2860 * q^26 - 7865 * q^29 + 8816 * q^31 + 2048 * q^32 - 5488 * q^34 - 14573 * q^37 + 4324 * q^38 - 1088 * q^40 - 7350 * q^41 - 5921 * q^43 - 2320 * q^44 + 18032 * q^46 - 44808 * q^47 + 25068 * q^50 - 11440 * q^52 - 9417 * q^53 - 85375 * q^55 - 31460 * q^58 + 5077 * q^59 + 42368 * q^61 + 35264 * q^62 + 8192 * q^64 + 18450 * q^65 + 30501 * q^67 - 21952 * q^68 - 91744 * q^71 - 85665 * q^73 - 58292 * q^74 + 17296 * q^76 + 94646 * q^79 - 4352 * q^80 - 29400 * q^82 - 33841 * q^83 + 259112 * q^85 - 23684 * q^86 - 9280 * q^88 - 27558 * q^89 + 72128 * q^92 - 179232 * q^94 - 343246 * q^95 - 46671 * q^97

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
$5$	−87.1527	−1.55904	−0.779518	−	0.626380i	$-0.784536\pi$
$5$	−87.1527	−1.55904	−0.779518	+	0.626380i	$0.784536\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	−348.611	−1.10240
$11$	478.069	1.19127	0.595633	−	0.803257i	$-0.296901\pi$
$11$	478.069	1.19127	0.595633	+	0.803257i	$0.296901\pi$
$12$	0	0
$13$	−436.153	−0.715781	−0.357891	−	0.933764i	$-0.616504\pi$
$13$	−436.153	−0.715781	−0.357891	+	0.933764i	$0.616504\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−2259.05	−1.89585	−0.947926	−	0.318491i	$-0.896824\pi$
$17$	−2259.05	−1.89585	−0.947926	+	0.318491i	$0.896824\pi$
$18$	0	0
$19$	2664.12	1.69305	0.846526	−	0.532347i	$-0.178690\pi$
$19$	2664.12	1.69305	0.846526	+	0.532347i	$0.178690\pi$
$20$	−1394.44	−0.779518
$21$	0	0
$22$	1912.28	0.842353
$23$	2568.61	1.01246	0.506231	−	0.862398i	$-0.331038\pi$
$23$	2568.61	1.01246	0.506231	+	0.862398i	$0.331038\pi$
$24$	0	0
$25$	4470.60	1.43059
$26$	−1744.61	−0.506134
$27$	0	0
$28$	0	0
$29$	708.010	0.156331	0.0781654	−	0.996940i	$-0.475094\pi$
$29$	708.010	0.156331	0.0781654	+	0.996940i	$0.475094\pi$
$30$	0	0
$31$	5509.14	1.02963	0.514813	−	0.857303i	$-0.327861\pi$
$31$	5509.14	1.02963	0.514813	+	0.857303i	$0.327861\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−9036.22	−1.34057
$35$	0	0
$36$	0	0
$37$	−915.630	−0.109955	−0.0549776	−	0.998488i	$-0.517509\pi$
$37$	−915.630	−0.109955	−0.0549776	+	0.998488i	$0.517509\pi$
$38$	10656.5	1.19717
$39$	0	0
$40$	−5577.77	−0.551202
$41$	−8866.08	−0.823706	−0.411853	−	0.911250i	$-0.635118\pi$
$41$	−8866.08	−0.823706	−0.411853	+	0.911250i	$0.635118\pi$
$42$	0	0
$43$	−7915.62	−0.652851	−0.326425	−	0.945223i	$-0.605844\pi$
$43$	−7915.62	−0.652851	−0.326425	+	0.945223i	$0.605844\pi$
$44$	7649.10	0.595633
$45$	0	0
$46$	10274.4	0.715919
$47$	−19572.5	−1.29241	−0.646207	−	0.763162i	$-0.723646\pi$
$47$	−19572.5	−1.29241	−0.646207	+	0.763162i	$0.723646\pi$
$48$	0	0
$49$	0	0
$50$	17882.4	1.01158
$51$	0	0
$52$	−6978.44	−0.357891
$53$	−8247.87	−0.403322	−0.201661	−	0.979455i	$-0.564634\pi$
$53$	−8247.87	−0.403322	−0.201661	+	0.979455i	$0.564634\pi$
$54$	0	0
$55$	−41665.0	−1.85723
$56$	0	0
$57$	0	0
$58$	2832.04	0.110543
$59$	32190.6	1.20392	0.601961	−	0.798525i	$-0.294386\pi$
$59$	32190.6	1.20392	0.601961	+	0.798525i	$0.294386\pi$
$60$	0	0
$61$	28420.1	0.977913	0.488957	−	0.872308i	$-0.337378\pi$
$61$	28420.1	0.977913	0.488957	+	0.872308i	$0.337378\pi$
$62$	22036.6	0.728055
$63$	0	0
$64$	4096.00	0.125000
$65$	38011.9	1.11593
$66$	0	0
$67$	−9682.41	−0.263510	−0.131755	−	0.991282i	$-0.542061\pi$
$67$	−9682.41	−0.263510	−0.131755	+	0.991282i	$0.542061\pi$
$68$	−36144.9	−0.947926
$69$	0	0
$70$	0	0
$71$	−28568.4	−0.672574	−0.336287	−	0.941760i	$-0.609171\pi$
$71$	−28568.4	−0.672574	−0.336287	+	0.941760i	$0.609171\pi$
$72$	0	0
$73$	−3112.88	−0.0683683	−0.0341842	−	0.999416i	$-0.510883\pi$
$73$	−3112.88	−0.0683683	−0.0341842	+	0.999416i	$0.510883\pi$
$74$	−3662.52	−0.0777500
$75$	0	0
$76$	42626.0	0.846526
$77$	0	0
$78$	0	0
$79$	−6790.07	−0.122407	−0.0612035	−	0.998125i	$-0.519494\pi$
$79$	−6790.07	−0.122407	−0.0612035	+	0.998125i	$0.519494\pi$
$80$	−22311.1	−0.389759
$81$	0	0
$82$	−35464.3	−0.582448
$83$	−14482.3	−0.230750	−0.115375	−	0.993322i	$-0.536807\pi$
$83$	−14482.3	−0.230750	−0.115375	+	0.993322i	$0.536807\pi$
$84$	0	0
$85$	196883.	2.95570
$86$	−31662.5	−0.461635
$87$	0	0
$88$	30596.4	0.421176
$89$	−27464.6	−0.367534	−0.183767	−	0.982970i	$-0.558829\pi$
$89$	−27464.6	−0.367534	−0.183767	+	0.982970i	$0.558829\pi$
$90$	0	0
$91$	0	0
$92$	41097.8	0.506231
$93$	0	0
$94$	−78290.0	−0.913875
$95$	−232186.	−2.63953
$96$	0	0
$97$	−91212.8	−0.984298	−0.492149	−	0.870511i	$-0.663788\pi$
$97$	−91212.8	−0.984298	−0.492149	+	0.870511i	$0.663788\pi$
$98$	0	0
$99$	0	0
$100$	71529.5	0.715295
$101$	−107712.	−1.05066	−0.525329	−	0.850899i	$-0.676058\pi$
$101$	−107712.	−1.05066	−0.525329	+	0.850899i	$0.676058\pi$
$102$	0	0
$103$	−50001.3	−0.464396	−0.232198	−	0.972669i	$-0.574592\pi$
$103$	−50001.3	−0.464396	−0.232198	+	0.972669i	$0.574592\pi$
$104$	−27913.8	−0.253067
$105$	0	0
$106$	−32991.5	−0.285192
$107$	−172686.	−1.45813	−0.729066	−	0.684443i	$-0.760046\pi$
$107$	−172686.	−1.45813	−0.729066	+	0.684443i	$0.760046\pi$
$108$	0	0
$109$	149700.	1.20686	0.603429	−	0.797417i	$-0.293801\pi$
$109$	149700.	1.20686	0.603429	+	0.797417i	$0.293801\pi$
$110$	−166660.	−1.31326
$111$	0	0
$112$	0	0
$113$	−143442.	−1.05677	−0.528384	−	0.849005i	$-0.677202\pi$
$113$	−143442.	−1.05677	−0.528384	+	0.849005i	$0.677202\pi$
$114$	0	0
$115$	−223861.	−1.57846
$116$	11328.2	0.0781654
$117$	0	0
$118$	128762.	0.851302
$119$	0	0
$120$	0	0
$121$	67499.0	0.419116
$122$	113680.	0.691489
$123$	0	0
$124$	88146.2	0.514813
$125$	−117272.	−0.671306
$126$	0	0
$127$	−337908.	−1.85904	−0.929520	−	0.368771i	$-0.879779\pi$
$127$	−337908.	−1.85904	−0.929520	+	0.368771i	$0.879779\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	152048.	0.789080
$131$	18676.4	0.0950855	0.0475428	−	0.998869i	$-0.484861\pi$
$131$	18676.4	0.0950855	0.0475428	+	0.998869i	$0.484861\pi$
$132$	0	0
$133$	0	0
$134$	−38729.6	−0.186329
$135$	0	0
$136$	−144579.	−0.670285
$137$	−102639.	−0.467208	−0.233604	−	0.972332i	$-0.575052\pi$
$137$	−102639.	−0.467208	−0.233604	+	0.972332i	$0.575052\pi$
$138$	0	0
$139$	−202436.	−0.888692	−0.444346	−	0.895855i	$-0.646564\pi$
$139$	−202436.	−0.888692	−0.444346	+	0.895855i	$0.646564\pi$
$140$	0	0
$141$	0	0
$142$	−114274.	−0.475582
$143$	−208511.	−0.852686
$144$	0	0
$145$	−61705.0	−0.243725
$146$	−12451.5	−0.0483437
$147$	0	0
$148$	−14650.1	−0.0549776
$149$	93617.7	0.345456	0.172728	−	0.984970i	$-0.444742\pi$
$149$	93617.7	0.345456	0.172728	+	0.984970i	$0.444742\pi$
$150$	0	0
$151$	−189178.	−0.675194	−0.337597	−	0.941291i	$-0.609614\pi$
$151$	−189178.	−0.675194	−0.337597	+	0.941291i	$0.609614\pi$
$152$	170504.	0.598584
$153$	0	0
$154$	0	0
$155$	−480136.	−1.60522
$156$	0	0
$157$	526526.	1.70479	0.852395	−	0.522898i	$-0.175149\pi$
$157$	526526.	1.70479	0.852395	+	0.522898i	$0.175149\pi$
$158$	−27160.3	−0.0865549
$159$	0	0
$160$	−89244.4	−0.275601
$161$	0	0
$162$	0	0
$163$	195092.	0.575137	0.287569	−	0.957760i	$-0.407153\pi$
$163$	195092.	0.575137	0.287569	+	0.957760i	$0.407153\pi$
$164$	−141857.	−0.411853
$165$	0	0
$166$	−57929.1	−0.163165
$167$	−240814.	−0.668175	−0.334087	−	0.942542i	$-0.608428\pi$
$167$	−240814.	−0.668175	−0.334087	+	0.942542i	$0.608428\pi$
$168$	0	0
$169$	−181064.	−0.487657
$170$	787531.	2.09000
$171$	0	0
$172$	−126650.	−0.326425
$173$	−537269.	−1.36482	−0.682412	−	0.730967i	$-0.739069\pi$
$173$	−537269.	−1.36482	−0.682412	+	0.730967i	$0.739069\pi$
$174$	0	0
$175$	0	0
$176$	122386.	0.297817
$177$	0	0
$178$	−109858.	−0.259886
$179$	185893.	0.433642	0.216821	−	0.976211i	$-0.430431\pi$
$179$	185893.	0.433642	0.216821	+	0.976211i	$0.430431\pi$
$180$	0	0
$181$	1575.22	0.00357392	0.00178696	−	0.999998i	$-0.499431\pi$
$181$	1575.22	0.00357392	0.00178696	+	0.999998i	$0.499431\pi$
$182$	0	0
$183$	0	0
$184$	164391.	0.357959
$185$	79799.6	0.171424
$186$	0	0
$187$	−1.07998e6	−2.25846
$188$	−313160.	−0.646207
$189$	0	0
$190$	−928742.	−1.86643
$191$	−197325.	−0.391379	−0.195690	−	0.980666i	$-0.562695\pi$
$191$	−197325.	−0.391379	−0.195690	+	0.980666i	$0.562695\pi$
$192$	0	0
$193$	−457717.	−0.884513	−0.442256	−	0.896889i	$-0.645822\pi$
$193$	−457717.	−0.884513	−0.442256	+	0.896889i	$0.645822\pi$
$194$	−364851.	−0.696004
$195$	0	0
$196$	0	0
$197$	280464.	0.514887	0.257443	−	0.966293i	$-0.417120\pi$
$197$	280464.	0.514887	0.257443	+	0.966293i	$0.417120\pi$
$198$	0	0
$199$	121027.	0.216646	0.108323	−	0.994116i	$-0.465452\pi$
$199$	121027.	0.216646	0.108323	+	0.994116i	$0.465452\pi$
$200$	286118.	0.505790
$201$	0	0
$202$	−430849.	−0.742928
$203$	0	0
$204$	0	0
$205$	772703.	1.28419
$206$	−200005.	−0.328378
$207$	0	0
$208$	−111655.	−0.178945
$209$	1.27363e6	2.01688
$210$	0	0
$211$	−270026.	−0.417541	−0.208770	−	0.977965i	$-0.566946\pi$
$211$	−270026.	−0.417541	−0.208770	+	0.977965i	$0.566946\pi$
$212$	−131966.	−0.201661
$213$	0	0
$214$	−690743.	−1.03106
$215$	689868.	1.01782
$216$	0	0
$217$	0	0
$218$	598801.	0.853377
$219$	0	0
$220$	−666640.	−0.928613
$221$	985293.	1.35701
$222$	0	0
$223$	83646.2	0.112638	0.0563189	−	0.998413i	$-0.482064\pi$
$223$	83646.2	0.112638	0.0563189	+	0.998413i	$0.482064\pi$
$224$	0	0
$225$	0	0
$226$	−573767.	−0.747248
$227$	−104452.	−0.134540	−0.0672700	−	0.997735i	$-0.521429\pi$
$227$	−104452.	−0.134540	−0.0672700	+	0.997735i	$0.521429\pi$
$228$	0	0
$229$	−1.56292e6	−1.96946	−0.984732	−	0.174078i	$-0.944306\pi$
$229$	−1.56292e6	−1.96946	−0.984732	+	0.174078i	$0.944306\pi$
$230$	−895446.	−1.11614
$231$	0	0
$232$	45312.7	0.0552713
$233$	898082.	1.08374	0.541871	−	0.840462i	$-0.317716\pi$
$233$	898082.	1.08374	0.541871	+	0.840462i	$0.317716\pi$
$234$	0	0
$235$	1.70580e6	2.01492
$236$	515049.	0.601961
$237$	0	0
$238$	0	0
$239$	884007.	1.00106	0.500531	−	0.865719i	$-0.333138\pi$
$239$	884007.	1.00106	0.500531	+	0.865719i	$0.333138\pi$
$240$	0	0
$241$	−460426.	−0.510643	−0.255322	−	0.966856i	$-0.582181\pi$
$241$	−460426.	−0.510643	−0.255322	+	0.966856i	$0.582181\pi$
$242$	269996.	0.296360
$243$	0	0
$244$	454721.	0.488957
$245$	0	0
$246$	0	0
$247$	−1.16196e6	−1.21185
$248$	352585.	0.364028
$249$	0	0
$250$	−469089.	−0.474685
$251$	−1.89967e6	−1.90324	−0.951621	−	0.307276i	$-0.900583\pi$
$251$	−1.89967e6	−1.90324	−0.951621	+	0.307276i	$0.900583\pi$
$252$	0	0
$253$	1.22797e6	1.20611
$254$	−1.35163e6	−1.31454
$255$	0	0
$256$	65536.0	0.0625000
$257$	−750477.	−0.708769	−0.354384	−	0.935100i	$-0.615310\pi$
$257$	−750477.	−0.708769	−0.354384	+	0.935100i	$0.615310\pi$
$258$	0	0
$259$	0	0
$260$	608190.	0.557964
$261$	0	0
$262$	74705.5	0.0672356
$263$	1.40293e6	1.25068	0.625340	−	0.780353i	$-0.284960\pi$
$263$	1.40293e6	1.25068	0.625340	+	0.780353i	$0.284960\pi$
$264$	0	0
$265$	718824.	0.628794
$266$	0	0
$267$	0	0
$268$	−154919.	−0.131755
$269$	−813515.	−0.685465	−0.342732	−	0.939433i	$-0.611352\pi$
$269$	−813515.	−0.685465	−0.342732	+	0.939433i	$0.611352\pi$
$270$	0	0
$271$	−153342.	−0.126835	−0.0634174	−	0.997987i	$-0.520200\pi$
$271$	−153342.	−0.126835	−0.0634174	+	0.997987i	$0.520200\pi$
$272$	−578318.	−0.473963
$273$	0	0
$274$	−410555.	−0.330366
$275$	2.13725e6	1.70421
$276$	0	0
$277$	−483914.	−0.378939	−0.189469	−	0.981887i	$-0.560677\pi$
$277$	−483914.	−0.378939	−0.189469	+	0.981887i	$0.560677\pi$
$278$	−809745.	−0.628400
$279$	0	0
$280$	0	0
$281$	−845997.	−0.639151	−0.319575	−	0.947561i	$-0.603540\pi$
$281$	−845997.	−0.639151	−0.319575	+	0.947561i	$0.603540\pi$
$282$	0	0
$283$	−1.51529e6	−1.12468	−0.562340	−	0.826906i	$-0.690099\pi$
$283$	−1.51529e6	−1.12468	−0.562340	+	0.826906i	$0.690099\pi$
$284$	−457094.	−0.336287
$285$	0	0
$286$	−834044.	−0.602940
$287$	0	0
$288$	0	0
$289$	3.68347e6	2.59425
$290$	−246820.	−0.172340
$291$	0	0
$292$	−49806.0	−0.0341842
$293$	2.17685e6	1.48135	0.740676	−	0.671862i	$-0.234505\pi$
$293$	2.17685e6	1.48135	0.740676	+	0.671862i	$0.234505\pi$
$294$	0	0
$295$	−2.80550e6	−1.87696
$296$	−58600.3	−0.0388750
$297$	0	0
$298$	374471.	0.244274
$299$	−1.12031e6	−0.724701
$300$	0	0
$301$	0	0
$302$	−756712.	−0.477434
$303$	0	0
$304$	682016.	0.423263
$305$	−2.47688e6	−1.52460
$306$	0	0
$307$	2.52205e6	1.52724	0.763621	−	0.645665i	$-0.223420\pi$
$307$	2.52205e6	1.52724	0.763621	+	0.645665i	$0.223420\pi$
$308$	0	0
$309$	0	0
$310$	−1.92055e6	−1.13506
$311$	−303748.	−0.178079	−0.0890394	−	0.996028i	$-0.528380\pi$
$311$	−303748.	−0.178079	−0.0890394	+	0.996028i	$0.528380\pi$
$312$	0	0
$313$	−1.72483e6	−0.995142	−0.497571	−	0.867423i	$-0.665775\pi$
$313$	−1.72483e6	−0.995142	−0.497571	+	0.867423i	$0.665775\pi$
$314$	2.10611e6	1.20547
$315$	0	0
$316$	−108641.	−0.0612035
$317$	−3.44337e6	−1.92458	−0.962289	−	0.272030i	$-0.912305\pi$
$317$	−3.44337e6	−1.92458	−0.962289	+	0.272030i	$0.912305\pi$
$318$	0	0
$319$	338478.	0.186232
$320$	−356978.	−0.194879
$321$	0	0
$322$	0	0
$323$	−6.01840e6	−3.20978
$324$	0	0
$325$	−1.94986e6	−1.02399
$326$	780370.	0.406683
$327$	0	0
$328$	−567429.	−0.291224
$329$	0	0
$330$	0	0
$331$	1.36359e6	0.684090	0.342045	−	0.939684i	$-0.388881\pi$
$331$	1.36359e6	0.684090	0.342045	+	0.939684i	$0.388881\pi$
$332$	−231716.	−0.115375
$333$	0	0
$334$	−963255.	−0.472471
$335$	843848.	0.410821
$336$	0	0
$337$	−3.63708e6	−1.74453	−0.872265	−	0.489034i	$-0.837349\pi$
$337$	−3.63708e6	−1.74453	−0.872265	+	0.489034i	$0.837349\pi$
$338$	−724255.	−0.344826
$339$	0	0
$340$	3.15012e6	1.47785
$341$	2.63375e6	1.22656
$342$	0	0
$343$	0	0
$344$	−506600.	−0.230818
$345$	0	0
$346$	−2.14908e6	−0.965077
$347$	1.76415e6	0.786523	0.393262	−	0.919427i	$-0.371347\pi$
$347$	1.76415e6	0.786523	0.393262	+	0.919427i	$0.371347\pi$
$348$	0	0
$349$	−3.15882e6	−1.38823	−0.694114	−	0.719865i	$-0.744204\pi$
$349$	−3.15882e6	−1.38823	−0.694114	+	0.719865i	$0.744204\pi$
$350$	0	0
$351$	0	0
$352$	489543.	0.210588
$353$	1.77170e6	0.756754	0.378377	−	0.925652i	$-0.376482\pi$
$353$	1.77170e6	0.756754	0.378377	+	0.925652i	$0.376482\pi$
$354$	0	0
$355$	2.48981e6	1.04857
$356$	−439433.	−0.183767
$357$	0	0
$358$	743573.	0.306631
$359$	1.48525e6	0.608226	0.304113	−	0.952636i	$-0.401640\pi$
$359$	1.48525e6	0.608226	0.304113	+	0.952636i	$0.401640\pi$
$360$	0	0
$361$	4.62145e6	1.86643
$362$	6300.87	0.00252714
$363$	0	0
$364$	0	0
$365$	271296.	0.106589
$366$	0	0
$367$	530304.	0.205523	0.102761	−	0.994706i	$-0.467232\pi$
$367$	530304.	0.205523	0.102761	+	0.994706i	$0.467232\pi$
$368$	657564.	0.253115
$369$	0	0
$370$	319199.	0.121215
$371$	0	0
$372$	0	0
$373$	−3.44714e6	−1.28288	−0.641442	−	0.767172i	$-0.721663\pi$
$373$	−3.44714e6	−1.28288	−0.641442	+	0.767172i	$0.721663\pi$
$374$	−4.31994e6	−1.59698
$375$	0	0
$376$	−1.25264e6	−0.456937
$377$	−308801.	−0.111899
$378$	0	0
$379$	−1.59447e6	−0.570189	−0.285094	−	0.958499i	$-0.592025\pi$
$379$	−1.59447e6	−0.570189	−0.285094	+	0.958499i	$0.592025\pi$
$380$	−3.71497e6	−1.31976
$381$	0	0
$382$	−789298.	−0.276747
$383$	2.18196e6	0.760064	0.380032	−	0.924973i	$-0.375913\pi$
$383$	2.18196e6	0.760064	0.380032	+	0.924973i	$0.375913\pi$
$384$	0	0
$385$	0	0
$386$	−1.83087e6	−0.625445
$387$	0	0
$388$	−1.45940e6	−0.492149
$389$	−2.11491e6	−0.708628	−0.354314	−	0.935127i	$-0.615286\pi$
$389$	−2.11491e6	−0.708628	−0.354314	+	0.935127i	$0.615286\pi$
$390$	0	0
$391$	−5.80263e6	−1.91948
$392$	0	0
$393$	0	0
$394$	1.12186e6	0.364080
$395$	591773.	0.190837
$396$	0	0
$397$	−491562.	−0.156532	−0.0782658	−	0.996933i	$-0.524938\pi$
$397$	−491562.	−0.156532	−0.0782658	+	0.996933i	$0.524938\pi$
$398$	484109.	0.153192
$399$	0	0
$400$	1.14447e6	0.357648
$401$	2.72121e6	0.845085	0.422542	−	0.906343i	$-0.361138\pi$
$401$	2.72121e6	0.845085	0.422542	+	0.906343i	$0.361138\pi$
$402$	0	0
$403$	−2.40283e6	−0.736987
$404$	−1.72340e6	−0.525329
$405$	0	0
$406$	0	0
$407$	−437734.	−0.130986
$408$	0	0
$409$	3.19908e6	0.945620	0.472810	−	0.881164i	$-0.343240\pi$
$409$	3.19908e6	0.945620	0.472810	+	0.881164i	$0.343240\pi$
$410$	3.09081e6	0.908057
$411$	0	0
$412$	−800021.	−0.232198
$413$	0	0
$414$	0	0
$415$	1.26217e6	0.359747
$416$	−446620.	−0.126533
$417$	0	0
$418$	5.09454e6	1.42615
$419$	4.89012e6	1.36077	0.680385	−	0.732855i	$-0.261812\pi$
$419$	4.89012e6	1.36077	0.680385	+	0.732855i	$0.261812\pi$
$420$	0	0
$421$	4.98359e6	1.37037	0.685184	−	0.728370i	$-0.259722\pi$
$421$	4.98359e6	1.37037	0.685184	+	0.728370i	$0.259722\pi$
$422$	−1.08010e6	−0.295246
$423$	0	0
$424$	−527864.	−0.142596
$425$	−1.00993e7	−2.71219
$426$	0	0
$427$	0	0
$428$	−2.76297e6	−0.729066
$429$	0	0
$430$	2.75947e6	0.719705
$431$	−1.74879e6	−0.453466	−0.226733	−	0.973957i	$-0.572804\pi$
$431$	−1.74879e6	−0.453466	−0.226733	+	0.973957i	$0.572804\pi$
$432$	0	0
$433$	5.36175e6	1.37432	0.687159	−	0.726507i	$-0.258858\pi$
$433$	5.36175e6	1.37432	0.687159	+	0.726507i	$0.258858\pi$
$434$	0	0
$435$	0	0
$436$	2.39520e6	0.603429
$437$	6.84310e6	1.71415
$438$	0	0
$439$	6.17448e6	1.52911	0.764555	−	0.644558i	$-0.222959\pi$
$439$	6.17448e6	1.52911	0.764555	+	0.644558i	$0.222959\pi$
$440$	−2.66656e6	−0.656629
$441$	0	0
$442$	3.94117e6	0.959554
$443$	3.58277e6	0.867380	0.433690	−	0.901062i	$-0.357211\pi$
$443$	3.58277e6	0.867380	0.433690	+	0.901062i	$0.357211\pi$
$444$	0	0
$445$	2.39361e6	0.572999
$446$	334585.	0.0796469
$447$	0	0
$448$	0	0
$449$	4.93126e6	1.15436	0.577182	−	0.816616i	$-0.304152\pi$
$449$	4.93126e6	1.15436	0.577182	+	0.816616i	$0.304152\pi$
$450$	0	0
$451$	−4.23860e6	−0.981253
$452$	−2.29507e6	−0.528384
$453$	0	0
$454$	−417807.	−0.0951341
$455$	0	0
$456$	0	0
$457$	−4.27229e6	−0.956908	−0.478454	−	0.878113i	$-0.658803\pi$
$457$	−4.27229e6	−0.956908	−0.478454	+	0.878113i	$0.658803\pi$
$458$	−6.25168e6	−1.39262
$459$	0	0
$460$	−3.58178e6	−0.789232
$461$	4.13355e6	0.905882	0.452941	−	0.891541i	$-0.350375\pi$
$461$	4.13355e6	0.905882	0.452941	+	0.891541i	$0.350375\pi$
$462$	0	0
$463$	−7.66515e6	−1.66176	−0.830880	−	0.556452i	$-0.812162\pi$
$463$	−7.66515e6	−1.66176	−0.830880	+	0.556452i	$0.812162\pi$
$464$	181251.	0.0390827
$465$	0	0
$466$	3.59233e6	0.766321
$467$	624673.	0.132544	0.0662720	−	0.997802i	$-0.478889\pi$
$467$	624673.	0.132544	0.0662720	+	0.997802i	$0.478889\pi$
$468$	0	0
$469$	0	0
$470$	6.82319e6	1.42476
$471$	0	0
$472$	2.06020e6	0.425651
$473$	−3.78421e6	−0.777719
$474$	0	0
$475$	1.19102e7	2.42206
$476$	0	0
$477$	0	0
$478$	3.53603e6	0.707858
$479$	7.30068e6	1.45387	0.726933	−	0.686708i	$-0.240945\pi$
$479$	7.30068e6	1.45387	0.726933	+	0.686708i	$0.240945\pi$
$480$	0	0
$481$	399354.	0.0787038
$482$	−1.84171e6	−0.361079
$483$	0	0
$484$	1.07998e6	0.209558
$485$	7.94944e6	1.53455
$486$	0	0
$487$	−8.45722e6	−1.61587	−0.807933	−	0.589274i	$-0.799414\pi$
$487$	−8.45722e6	−1.61587	−0.807933	+	0.589274i	$0.799414\pi$
$488$	1.81888e6	0.345744
$489$	0	0
$490$	0	0
$491$	2.09490e6	0.392156	0.196078	−	0.980588i	$-0.437179\pi$
$491$	2.09490e6	0.392156	0.196078	+	0.980588i	$0.437179\pi$
$492$	0	0
$493$	−1.59943e6	−0.296380
$494$	−4.64786e6	−0.856911
$495$	0	0
$496$	1.41034e6	0.257406
$497$	0	0
$498$	0	0
$499$	−5.92467e6	−1.06515	−0.532577	−	0.846381i	$-0.678776\pi$
$499$	−5.92467e6	−1.06515	−0.532577	+	0.846381i	$0.678776\pi$
$500$	−1.87636e6	−0.335653
$501$	0	0
$502$	−7.59868e6	−1.34579
$503$	−4.80370e6	−0.846557	−0.423278	−	0.906000i	$-0.639121\pi$
$503$	−4.80370e6	−0.846557	−0.423278	+	0.906000i	$0.639121\pi$
$504$	0	0
$505$	9.38741e6	1.63801
$506$	4.91189e6	0.852850
$507$	0	0
$508$	−5.40652e6	−0.929520
$509$	5.20283e6	0.890112	0.445056	−	0.895503i	$-0.353184\pi$
$509$	5.20283e6	0.890112	0.445056	+	0.895503i	$0.353184\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.00191e6	−0.501175
$515$	4.35775e6	0.724010
$516$	0	0
$517$	−9.35701e6	−1.53961
$518$	0	0
$519$	0	0
$520$	2.43276e6	0.394540
$521$	6.81153e6	1.09939	0.549693	−	0.835367i	$-0.314745\pi$
$521$	6.81153e6	1.09939	0.549693	+	0.835367i	$0.314745\pi$
$522$	0	0
$523$	−1.07231e7	−1.71421	−0.857106	−	0.515139i	$-0.827740\pi$
$523$	−1.07231e7	−1.71421	−0.857106	+	0.515139i	$0.827740\pi$
$524$	298822.	0.0475428
$525$	0	0
$526$	5.61171e6	0.884364
$527$	−1.24454e7	−1.95202
$528$	0	0
$529$	161419.	0.0250793
$530$	2.87530e6	0.444624
$531$	0	0
$532$	0	0
$533$	3.86696e6	0.589593
$534$	0	0
$535$	1.50500e7	2.27328
$536$	−619674.	−0.0931647
$537$	0	0
$538$	−3.25406e6	−0.484697
$539$	0	0
$540$	0	0
$541$	1.27890e7	1.87864	0.939320	−	0.343043i	$-0.111458\pi$
$541$	1.27890e7	1.87864	0.939320	+	0.343043i	$0.111458\pi$
$542$	−613368.	−0.0896857
$543$	0	0
$544$	−2.31327e6	−0.335142
$545$	−1.30468e7	−1.88153
$546$	0	0
$547$	−1.14426e7	−1.63514	−0.817570	−	0.575829i	$-0.804679\pi$
$547$	−1.14426e7	−1.63514	−0.817570	+	0.575829i	$0.804679\pi$
$548$	−1.64222e6	−0.233604
$549$	0	0
$550$	8.54901e6	1.20506
$551$	1.88623e6	0.264676
$552$	0	0
$553$	0	0
$554$	−1.93566e6	−0.267950
$555$	0	0
$556$	−3.23898e6	−0.444346
$557$	2.90688e6	0.396999	0.198499	−	0.980101i	$-0.436393\pi$
$557$	2.90688e6	0.396999	0.198499	+	0.980101i	$0.436393\pi$
$558$	0	0
$559$	3.45242e6	0.467298
$560$	0	0
$561$	0	0
$562$	−3.38399e6	−0.451948
$563$	−3.42695e6	−0.455656	−0.227828	−	0.973701i	$-0.573162\pi$
$563$	−3.42695e6	−0.455656	−0.227828	+	0.973701i	$0.573162\pi$
$564$	0	0
$565$	1.25013e7	1.64754
$566$	−6.06115e6	−0.795268
$567$	0	0
$568$	−1.82838e6	−0.237791
$569$	1.66523e6	0.215622	0.107811	−	0.994171i	$-0.465616\pi$
$569$	1.66523e6	0.215622	0.107811	+	0.994171i	$0.465616\pi$
$570$	0	0
$571$	−1.49751e7	−1.92212	−0.961060	−	0.276341i	$-0.910878\pi$
$571$	−1.49751e7	−1.92212	−0.961060	+	0.276341i	$0.910878\pi$
$572$	−3.33618e6	−0.426343
$573$	0	0
$574$	0	0
$575$	1.14832e7	1.44842
$576$	0	0
$577$	1.12185e7	1.40280	0.701398	−	0.712770i	$-0.252559\pi$
$577$	1.12185e7	1.40280	0.701398	+	0.712770i	$0.252559\pi$
$578$	1.47339e7	1.83441
$579$	0	0
$580$	−987280.	−0.121863
$581$	0	0
$582$	0	0
$583$	−3.94305e6	−0.480464
$584$	−199224.	−0.0241719
$585$	0	0
$586$	8.70738e6	1.04747
$587$	−1.13200e6	−0.135597	−0.0677984	−	0.997699i	$-0.521597\pi$
$587$	−1.13200e6	−0.135597	−0.0677984	+	0.997699i	$0.521597\pi$
$588$	0	0
$589$	1.46770e7	1.74321
$590$	−1.12220e7	−1.32721
$591$	0	0
$592$	−234401.	−0.0274888
$593$	−1.17278e7	−1.36956	−0.684779	−	0.728750i	$-0.740101\pi$
$593$	−1.17278e7	−1.36956	−0.684779	+	0.728750i	$0.740101\pi$
$594$	0	0
$595$	0	0
$596$	1.49788e6	0.172728
$597$	0	0
$598$	−4.48123e6	−0.512441
$599$	−2.04830e6	−0.233253	−0.116626	−	0.993176i	$-0.537208\pi$
$599$	−2.04830e6	−0.233253	−0.116626	+	0.993176i	$0.537208\pi$
$600$	0	0
$601$	−5.75746e6	−0.650196	−0.325098	−	0.945680i	$-0.605397\pi$
$601$	−5.75746e6	−0.650196	−0.325098	+	0.945680i	$0.605397\pi$
$602$	0	0
$603$	0	0
$604$	−3.02685e6	−0.337597
$605$	−5.88272e6	−0.653416
$606$	0	0
$607$	−4.97797e6	−0.548378	−0.274189	−	0.961676i	$-0.588409\pi$
$607$	−4.97797e6	−0.548378	−0.274189	+	0.961676i	$0.588409\pi$
$608$	2.72806e6	0.299292
$609$	0	0
$610$	−9.90754e6	−1.07806
$611$	8.53660e6	0.925086
$612$	0	0
$613$	−4.56305e6	−0.490461	−0.245230	−	0.969465i	$-0.578864\pi$
$613$	−4.56305e6	−0.490461	−0.245230	+	0.969465i	$0.578864\pi$
$614$	1.00882e7	1.07992
$615$	0	0
$616$	0	0
$617$	−1.41767e7	−1.49921	−0.749603	−	0.661888i	$-0.769756\pi$
$617$	−1.41767e7	−1.49921	−0.749603	+	0.661888i	$0.769756\pi$
$618$	0	0
$619$	−7.54794e6	−0.791775	−0.395888	−	0.918299i	$-0.629563\pi$
$619$	−7.54794e6	−0.791775	−0.395888	+	0.918299i	$0.629563\pi$
$620$	−7.68218e6	−0.802611
$621$	0	0
$622$	−1.21499e6	−0.125921
$623$	0	0
$624$	0	0
$625$	−3.75001e6	−0.384001
$626$	−6.89932e6	−0.703672
$627$	0	0
$628$	8.42442e6	0.852395
$629$	2.06846e6	0.208459
$630$	0	0
$631$	−87076.0	−0.00870612	−0.00435306	−	0.999991i	$-0.501386\pi$
$631$	−87076.0	−0.00870612	−0.00435306	+	0.999991i	$0.501386\pi$
$632$	−434564.	−0.0432774
$633$	0	0
$634$	−1.37735e7	−1.36088
$635$	2.94496e7	2.89831
$636$	0	0
$637$	0	0
$638$	1.35391e6	0.131686
$639$	0	0
$640$	−1.42791e6	−0.137801
$641$	−1.35335e7	−1.30096	−0.650480	−	0.759524i	$-0.725432\pi$
$641$	−1.35335e7	−1.30096	−0.650480	+	0.759524i	$0.725432\pi$
$642$	0	0
$643$	−595140.	−0.0567664	−0.0283832	−	0.999597i	$-0.509036\pi$
$643$	−595140.	−0.0567664	−0.0283832	+	0.999597i	$0.509036\pi$
$644$	0	0
$645$	0	0
$646$	−2.40736e7	−2.26965
$647$	−3.90809e6	−0.367032	−0.183516	−	0.983017i	$-0.558748\pi$
$647$	−3.90809e6	−0.367032	−0.183516	+	0.983017i	$0.558748\pi$
$648$	0	0
$649$	1.53893e7	1.43419
$650$	−7.79945e6	−0.724070
$651$	0	0
$652$	3.12148e6	0.287569
$653$	5.18455e6	0.475804	0.237902	−	0.971289i	$-0.423540\pi$
$653$	5.18455e6	0.475804	0.237902	+	0.971289i	$0.423540\pi$
$654$	0	0
$655$	−1.62770e6	−0.148242
$656$	−2.26972e6	−0.205926
$657$	0	0
$658$	0	0
$659$	−2.73626e6	−0.245439	−0.122719	−	0.992441i	$-0.539162\pi$
$659$	−2.73626e6	−0.245439	−0.122719	+	0.992441i	$0.539162\pi$
$660$	0	0
$661$	5.99119e6	0.533347	0.266673	−	0.963787i	$-0.414076\pi$
$661$	5.99119e6	0.533347	0.266673	+	0.963787i	$0.414076\pi$
$662$	5.45435e6	0.483724
$663$	0	0
$664$	−926865.	−0.0815824
$665$	0	0
$666$	0	0
$667$	1.81860e6	0.158279
$668$	−3.85302e6	−0.334087
$669$	0	0
$670$	3.37539e6	0.290494
$671$	1.35867e7	1.16495
$672$	0	0
$673$	−6.70138e6	−0.570330	−0.285165	−	0.958478i	$-0.592048\pi$
$673$	−6.70138e6	−0.570330	−0.285165	+	0.958478i	$0.592048\pi$
$674$	−1.45483e7	−1.23357
$675$	0	0
$676$	−2.89702e6	−0.243829
$677$	7.32665e6	0.614375	0.307188	−	0.951649i	$-0.400612\pi$
$677$	7.32665e6	0.614375	0.307188	+	0.951649i	$0.400612\pi$
$678$	0	0
$679$	0	0
$680$	1.26005e7	1.04500
$681$	0	0
$682$	1.05350e7	0.867308
$683$	−191311.	−0.0156923	−0.00784617	−	0.999969i	$-0.502498\pi$
$683$	−191311.	−0.0156923	−0.00784617	+	0.999969i	$0.502498\pi$
$684$	0	0
$685$	8.94524e6	0.728393
$686$	0	0
$687$	0	0
$688$	−2.02640e6	−0.163213
$689$	3.59733e6	0.288690
$690$	0	0
$691$	−1.02923e7	−0.820010	−0.410005	−	0.912083i	$-0.634473\pi$
$691$	−1.02923e7	−0.820010	−0.410005	+	0.912083i	$0.634473\pi$
$692$	−8.59631e6	−0.682412
$693$	0	0
$694$	7.05660e6	0.556156
$695$	1.76429e7	1.38550
$696$	0	0
$697$	2.00290e7	1.56162
$698$	−1.26353e7	−0.981625
$699$	0	0
$700$	0	0
$701$	−7.48444e6	−0.575260	−0.287630	−	0.957742i	$-0.592867\pi$
$701$	−7.48444e6	−0.575260	−0.287630	+	0.957742i	$0.592867\pi$
$702$	0	0
$703$	−2.43935e6	−0.186160
$704$	1.95817e6	0.148908
$705$	0	0
$706$	7.08682e6	0.535106
$707$	0	0
$708$	0	0
$709$	1.66785e7	1.24607	0.623035	−	0.782194i	$-0.285899\pi$
$709$	1.66785e7	1.24607	0.623035	+	0.782194i	$0.285899\pi$
$710$	9.95926e6	0.741449
$711$	0	0
$712$	−1.75773e6	−0.129943
$713$	1.41508e7	1.04246
$714$	0	0
$715$	1.81723e7	1.32937
$716$	2.97429e6	0.216821
$717$	0	0
$718$	5.94102e6	0.430081
$719$	1.39388e7	1.00555	0.502775	−	0.864417i	$-0.332312\pi$
$719$	1.39388e7	1.00555	0.502775	+	0.864417i	$0.332312\pi$
$720$	0	0
$721$	0	0
$722$	1.84858e7	1.31976
$723$	0	0
$724$	25203.5	0.00178696
$725$	3.16523e6	0.223645
$726$	0	0
$727$	1.05038e7	0.737075	0.368537	−	0.929613i	$-0.379859\pi$
$727$	1.05038e7	0.737075	0.368537	+	0.929613i	$0.379859\pi$
$728$	0	0
$729$	0	0
$730$	1.08518e6	0.0753695
$731$	1.78818e7	1.23771
$732$	0	0
$733$	1.93063e7	1.32721	0.663603	−	0.748085i	$-0.269027\pi$
$733$	1.93063e7	1.32721	0.663603	+	0.748085i	$0.269027\pi$
$734$	2.12122e6	0.145327
$735$	0	0
$736$	2.63026e6	0.178980
$737$	−4.62886e6	−0.313910
$738$	0	0
$739$	−1.09473e7	−0.737390	−0.368695	−	0.929550i	$-0.620195\pi$
$739$	−1.09473e7	−0.737390	−0.368695	+	0.929550i	$0.620195\pi$
$740$	1.27679e6	0.0857120
$741$	0	0
$742$	0	0
$743$	1.37335e7	0.912661	0.456331	−	0.889810i	$-0.349163\pi$
$743$	1.37335e7	0.912661	0.456331	+	0.889810i	$0.349163\pi$
$744$	0	0
$745$	−8.15904e6	−0.538578
$746$	−1.37886e7	−0.907136
$747$	0	0
$748$	−1.72797e7	−1.12923
$749$	0	0
$750$	0	0
$751$	−6.91285e6	−0.447257	−0.223629	−	0.974674i	$-0.571790\pi$
$751$	−6.91285e6	−0.447257	−0.223629	+	0.974674i	$0.571790\pi$
$752$	−5.01056e6	−0.323104
$753$	0	0
$754$	−1.23520e6	−0.0791243
$755$	1.64874e7	1.05265
$756$	0	0
$757$	6.68202e6	0.423807	0.211904	−	0.977291i	$-0.432034\pi$
$757$	6.68202e6	0.423807	0.211904	+	0.977291i	$0.432034\pi$
$758$	−6.37789e6	−0.403184
$759$	0	0
$760$	−1.48599e7	−0.933214
$761$	−2.34859e7	−1.47010	−0.735049	−	0.678014i	$-0.762841\pi$
$761$	−2.34859e7	−1.47010	−0.735049	+	0.678014i	$0.762841\pi$
$762$	0	0
$763$	0	0
$764$	−3.15719e6	−0.195690
$765$	0	0
$766$	8.72784e6	0.537446
$767$	−1.40400e7	−0.861745
$768$	0	0
$769$	1.24562e7	0.759571	0.379785	−	0.925075i	$-0.375998\pi$
$769$	1.24562e7	0.759571	0.379785	+	0.925075i	$0.375998\pi$
$770$	0	0
$771$	0	0
$772$	−7.32348e6	−0.442256
$773$	373957.	0.0225099	0.0112549	−	0.999937i	$-0.496417\pi$
$773$	373957.	0.0225099	0.0112549	+	0.999937i	$0.496417\pi$
$774$	0	0
$775$	2.46291e7	1.47297
$776$	−5.83762e6	−0.348002
$777$	0	0
$778$	−8.45964e6	−0.501075
$779$	−2.36203e7	−1.39458
$780$	0	0
$781$	−1.36577e7	−0.801215
$782$	−2.32105e7	−1.35728
$783$	0	0
$784$	0	0
$785$	−4.58882e7	−2.65783
$786$	0	0
$787$	3.03820e7	1.74856	0.874278	−	0.485426i	$-0.161336\pi$
$787$	3.03820e7	1.74856	0.874278	+	0.485426i	$0.161336\pi$
$788$	4.48743e6	0.257443
$789$	0	0
$790$	2.36709e6	0.134942
$791$	0	0
$792$	0	0
$793$	−1.23955e7	−0.699972
$794$	−1.96625e6	−0.110685
$795$	0	0
$796$	1.93643e6	0.108323
$797$	1.17189e7	0.653492	0.326746	−	0.945112i	$-0.394048\pi$
$797$	1.17189e7	0.653492	0.326746	+	0.945112i	$0.394048\pi$
$798$	0	0
$799$	4.42153e7	2.45023
$800$	4.57789e6	0.252895
$801$	0	0
$802$	1.08848e7	0.597565
$803$	−1.48817e6	−0.0814449
$804$	0	0
$805$	0	0
$806$	−9.61130e6	−0.521128
$807$	0	0
$808$	−6.89358e6	−0.371464
$809$	2.23437e6	0.120028	0.0600141	−	0.998198i	$-0.480885\pi$
$809$	2.23437e6	0.120028	0.0600141	+	0.998198i	$0.480885\pi$
$810$	0	0
$811$	3.29804e6	0.176077	0.0880387	−	0.996117i	$-0.471940\pi$
$811$	3.29804e6	0.176077	0.0880387	+	0.996117i	$0.471940\pi$
$812$	0	0
$813$	0	0
$814$	−1.75094e6	−0.0926210
$815$	−1.70028e7	−0.896659
$816$	0	0
$817$	−2.10882e7	−1.10531
$818$	1.27963e7	0.668654
$819$	0	0
$820$	1.23632e7	0.642093
$821$	−1.71494e6	−0.0887957	−0.0443979	−	0.999014i	$-0.514137\pi$
$821$	−1.71494e6	−0.0887957	−0.0443979	+	0.999014i	$0.514137\pi$
$822$	0	0
$823$	9.73937e6	0.501223	0.250612	−	0.968088i	$-0.419368\pi$
$823$	9.73937e6	0.501223	0.250612	+	0.968088i	$0.419368\pi$
$824$	−3.20009e6	−0.164189
$825$	0	0
$826$	0	0
$827$	2.95256e7	1.50119	0.750593	−	0.660765i	$-0.229768\pi$
$827$	2.95256e7	1.50119	0.750593	+	0.660765i	$0.229768\pi$
$828$	0	0
$829$	1.73499e7	0.876822	0.438411	−	0.898775i	$-0.355541\pi$
$829$	1.73499e7	0.876822	0.438411	+	0.898775i	$0.355541\pi$
$830$	5.04868e6	0.254380
$831$	0	0
$832$	−1.78648e6	−0.0894726
$833$	0	0
$834$	0	0
$835$	2.09876e7	1.04171
$836$	2.03782e7	1.00844
$837$	0	0
$838$	1.95605e7	0.962210
$839$	−2.22091e7	−1.08924	−0.544622	−	0.838681i	$-0.683327\pi$
$839$	−2.22091e7	−1.08924	−0.544622	+	0.838681i	$0.683327\pi$
$840$	0	0
$841$	−2.00099e7	−0.975561
$842$	1.99344e7	0.968996
$843$	0	0
$844$	−4.32041e6	−0.208770
$845$	1.57802e7	0.760275
$846$	0	0
$847$	0	0
$848$	−2.11146e6	−0.100831
$849$	0	0
$850$	−4.03973e7	−1.91781
$851$	−2.35190e6	−0.111325
$852$	0	0
$853$	−1.55354e7	−0.731053	−0.365527	−	0.930801i	$-0.619111\pi$
$853$	−1.55354e7	−0.731053	−0.365527	+	0.930801i	$0.619111\pi$
$854$	0	0
$855$	0	0
$856$	−1.10519e7	−0.515528
$857$	2.87116e7	1.33538	0.667690	−	0.744439i	$-0.267283\pi$
$857$	2.87116e7	1.33538	0.667690	+	0.744439i	$0.267283\pi$
$858$	0	0
$859$	1.15908e7	0.535957	0.267979	−	0.963425i	$-0.413644\pi$
$859$	1.15908e7	0.535957	0.267979	+	0.963425i	$0.413644\pi$
$860$	1.10379e7	0.508909
$861$	0	0
$862$	−6.99516e6	−0.320649
$863$	−7.76503e6	−0.354908	−0.177454	−	0.984129i	$-0.556786\pi$
$863$	−7.76503e6	−0.354908	−0.177454	+	0.984129i	$0.556786\pi$
$864$	0	0
$865$	4.68245e7	2.12781
$866$	2.14470e7	0.971789
$867$	0	0
$868$	0	0
$869$	−3.24612e6	−0.145819
$870$	0	0
$871$	4.22301e6	0.188615
$872$	9.58081e6	0.426689
$873$	0	0
$874$	2.73724e7	1.21209
$875$	0	0
$876$	0	0
$877$	2.50789e7	1.10106	0.550529	−	0.834816i	$-0.314426\pi$
$877$	2.50789e7	1.10106	0.550529	+	0.834816i	$0.314426\pi$
$878$	2.46979e7	1.08124
$879$	0	0
$880$	−1.06662e7	−0.464307
$881$	−6.69476e6	−0.290600	−0.145300	−	0.989388i	$-0.546415\pi$
$881$	−6.69476e6	−0.290600	−0.145300	+	0.989388i	$0.546415\pi$
$882$	0	0
$883$	6.82227e6	0.294460	0.147230	−	0.989102i	$-0.452964\pi$
$883$	6.82227e6	0.294460	0.147230	+	0.989102i	$0.452964\pi$
$884$	1.57647e7	0.678507
$885$	0	0
$886$	1.43311e7	0.613330
$887$	−1.81125e7	−0.772982	−0.386491	−	0.922293i	$-0.626313\pi$
$887$	−1.81125e7	−0.772982	−0.386491	+	0.922293i	$0.626313\pi$
$888$	0	0
$889$	0	0
$890$	9.57445e6	0.405171
$891$	0	0
$892$	1.33834e6	0.0563189
$893$	−5.21436e7	−2.18812
$894$	0	0
$895$	−1.62011e7	−0.676062
$896$	0	0
$897$	0	0
$898$	1.97251e7	0.816258
$899$	3.90053e6	0.160962
$900$	0	0
$901$	1.86324e7	0.764639
$902$	−1.69544e7	−0.693850
$903$	0	0
$904$	−9.18028e6	−0.373624
$905$	−137285.	−0.00557186
$906$	0	0
$907$	1.93366e7	0.780482	0.390241	−	0.920713i	$-0.372392\pi$
$907$	1.93366e7	0.780482	0.390241	+	0.920713i	$0.372392\pi$
$908$	−1.67123e6	−0.0672700
$909$	0	0
$910$	0	0
$911$	4.70161e6	0.187694	0.0938471	−	0.995587i	$-0.470084\pi$
$911$	4.70161e6	0.187694	0.0938471	+	0.995587i	$0.470084\pi$
$912$	0	0
$913$	−6.92352e6	−0.274884
$914$	−1.70892e7	−0.676636
$915$	0	0
$916$	−2.50067e7	−0.984732
$917$	0	0
$918$	0	0
$919$	1.37727e7	0.537935	0.268967	−	0.963149i	$-0.413318\pi$
$919$	1.37727e7	0.537935	0.268967	+	0.963149i	$0.413318\pi$
$920$	−1.43271e7	−0.558071
$921$	0	0
$922$	1.65342e7	0.640555
$923$	1.24602e7	0.481416
$924$	0	0
$925$	−4.09341e6	−0.157301
$926$	−3.06606e7	−1.17504
$927$	0	0
$928$	725003.	0.0276356
$929$	2.14844e7	0.816740	0.408370	−	0.912817i	$-0.366097\pi$
$929$	2.14844e7	0.816740	0.408370	+	0.912817i	$0.366097\pi$
$930$	0	0
$931$	0	0
$932$	1.43693e7	0.541871
$933$	0	0
$934$	2.49869e6	0.0937228
$935$	9.41235e7	3.52103
$936$	0	0
$937$	−2.27448e7	−0.846318	−0.423159	−	0.906055i	$-0.639079\pi$
$937$	−2.27448e7	−0.846318	−0.423159	+	0.906055i	$0.639079\pi$
$938$	0	0
$939$	0	0
$940$	2.72927e7	1.00746
$941$	1.91857e7	0.706324	0.353162	−	0.935562i	$-0.385106\pi$
$941$	1.91857e7	0.706324	0.353162	+	0.935562i	$0.385106\pi$
$942$	0	0
$943$	−2.27735e7	−0.833971
$944$	8.24079e6	0.300981
$945$	0	0
$946$	−1.51369e7	−0.549930
$947$	1.09546e6	0.0396936	0.0198468	−	0.999803i	$-0.493682\pi$
$947$	1.09546e6	0.0396936	0.0198468	+	0.999803i	$0.493682\pi$
$948$	0	0
$949$	1.35769e6	0.0489367
$950$	4.76409e7	1.71266
$951$	0	0
$952$	0	0
$953$	276787.	0.00987218	0.00493609	−	0.999988i	$-0.498429\pi$
$953$	276787.	0.00987218	0.00493609	+	0.999988i	$0.498429\pi$
$954$	0	0
$955$	1.71974e7	0.610174
$956$	1.41441e7	0.500531
$957$	0	0
$958$	2.92027e7	1.02804
$959$	0	0
$960$	0	0
$961$	1.72145e6	0.0601293
$962$	1.59742e6	0.0556520
$963$	0	0
$964$	−7.36682e6	−0.255322
$965$	3.98913e7	1.37899
$966$	0	0
$967$	−1.39867e7	−0.481005	−0.240502	−	0.970649i	$-0.577312\pi$
$967$	−1.39867e7	−0.481005	−0.240502	+	0.970649i	$0.577312\pi$
$968$	4.31994e6	0.148180
$969$	0	0
$970$	3.17978e7	1.08509
$971$	−2.71982e7	−0.925748	−0.462874	−	0.886424i	$-0.653182\pi$
$971$	−2.71982e7	−0.925748	−0.462874	+	0.886424i	$0.653182\pi$
$972$	0	0
$973$	0	0
$974$	−3.38289e7	−1.14259
$975$	0	0
$976$	7.27553e6	0.244478
$977$	4.63391e7	1.55314	0.776571	−	0.630029i	$-0.216957\pi$
$977$	4.63391e7	1.55314	0.776571	+	0.630029i	$0.216957\pi$
$978$	0	0
$979$	−1.31300e7	−0.437831
$980$	0	0
$981$	0	0
$982$	8.37958e6	0.277296
$983$	−4.50002e6	−0.148536	−0.0742679	−	0.997238i	$-0.523662\pi$
$983$	−4.50002e6	−0.148536	−0.0742679	+	0.997238i	$0.523662\pi$
$984$	0	0
$985$	−2.44432e7	−0.802727
$986$	−6.39774e6	−0.209572
$987$	0	0
$988$	−1.85914e7	−0.605927
$989$	−2.03322e7	−0.660987
$990$	0	0
$991$	−4.16811e7	−1.34820	−0.674101	−	0.738639i	$-0.735469\pi$
$991$	−4.16811e7	−1.34820	−0.674101	+	0.738639i	$0.735469\pi$
$992$	5.64136e6	0.182014
$993$	0	0
$994$	0	0
$995$	−1.05478e7	−0.337758
$996$	0	0
$997$	−3.37807e7	−1.07629	−0.538147	−	0.842851i	$-0.680876\pi$
$997$	−3.37807e7	−1.07629	−0.538147	+	0.842851i	$0.680876\pi$
$998$	−2.36987e7	−0.753178
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.e.d.25.1	✓	4	21.2	odd	6
42.6.e.d.37.1	yes	4	21.11	odd	6
126.6.g.g.37.2		4	7.4	even	3
126.6.g.g.109.2		4	7.2	even	3
294.6.a.o.1.1		2	21.20	even	2
294.6.a.p.1.2		2	3.2	odd	2
294.6.e.y.67.2		4	21.5	even	6
294.6.e.y.79.2		4	21.17	even	6
336.6.q.h.193.1		4	84.23	even	6
336.6.q.h.289.1		4	84.11	even	6
882.6.a.bm.1.1		2	1.1	even	1	trivial
882.6.a.bs.1.2		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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -87.1527 q^{5} +64.0000 q^{8} -348.611 q^{10} +478.069 q^{11} -436.153 q^{13} +256.000 q^{16} -2259.05 q^{17} +2664.12 q^{19} -1394.44 q^{20} +1912.28 q^{22} +2568.61 q^{23} +4470.60 q^{25} -1744.61 q^{26} +708.010 q^{29} +5509.14 q^{31} +1024.00 q^{32} -9036.22 q^{34} -915.630 q^{37} +10656.5 q^{38} -5577.77 q^{40} -8866.08 q^{41} -7915.62 q^{43} +7649.10 q^{44} +10274.4 q^{46} -19572.5 q^{47} +17882.4 q^{50} -6978.44 q^{52} -8247.87 q^{53} -41665.0 q^{55} +2832.04 q^{58} +32190.6 q^{59} +28420.1 q^{61} +22036.6 q^{62} +4096.00 q^{64} +38011.9 q^{65} -9682.41 q^{67} -36144.9 q^{68} -28568.4 q^{71} -3112.88 q^{73} -3662.52 q^{74} +42626.0 q^{76} -6790.07 q^{79} -22311.1 q^{80} -35464.3 q^{82} -14482.3 q^{83} +196883. q^{85} -31662.5 q^{86} +30596.4 q^{88} -27464.6 q^{89} +41097.8 q^{92} -78290.0 q^{94} -232186. q^{95} -91212.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 17 q^{5} + 128 q^{8} - 68 q^{10} - 145 q^{11} - 715 q^{13} + 512 q^{16} - 1372 q^{17} + 1081 q^{19} - 272 q^{20} - 580 q^{22} + 4508 q^{23} + 6267 q^{25} - 2860 q^{26} - 7865 q^{29}+ \cdots - 46671 q^{97}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 17 * q^5 + 128 * q^8 - 68 * q^10 - 145 * q^11 - 715 * q^13 + 512 * q^16 - 1372 * q^17 + 1081 * q^19 - 272 * q^20 - 580 * q^22 + 4508 * q^23 + 6267 * q^25 - 2860 * q^26 - 7865 * q^29 + 8816 * q^31 + 2048 * q^32 - 5488 * q^34 - 14573 * q^37 + 4324 * q^38 - 1088 * q^40 - 7350 * q^41 - 5921 * q^43 - 2320 * q^44 + 18032 * q^46 - 44808 * q^47 + 25068 * q^50 - 11440 * q^52 - 9417 * q^53 - 85375 * q^55 - 31460 * q^58 + 5077 * q^59 + 42368 * q^61 + 35264 * q^62 + 8192 * q^64 + 18450 * q^65 + 30501 * q^67 - 21952 * q^68 - 91744 * q^71 - 85665 * q^73 - 58292 * q^74 + 17296 * q^76 + 94646 * q^79 - 4352 * q^80 - 29400 * q^82 - 33841 * q^83 + 259112 * q^85 - 23684 * q^86 - 9280 * q^88 - 27558 * q^89 + 72128 * q^92 - 179232 * q^94 - 343246 * q^95 - 46671 * q^97

Introduction

L-functions

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