LMFDB - Embedded newform 882.6.a.z.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:	$0$
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:	$ 1 $
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)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -46.9289 q^{5} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} -46.9289 q^{5} -64.0000 q^{8} +187.716 q^{10} +87.4558 q^{11} +754.566 q^{13} +256.000 q^{16} +1449.04 q^{17} +2540.24 q^{19} -750.863 q^{20} -349.823 q^{22} +912.248 q^{23} -922.675 q^{25} -3018.26 q^{26} -173.217 q^{29} +4531.13 q^{31} -1024.00 q^{32} -5796.16 q^{34} +6829.96 q^{37} -10161.0 q^{38} +3003.45 q^{40} -13069.3 q^{41} -12277.7 q^{43} +1399.29 q^{44} -3648.99 q^{46} -13492.8 q^{47} +3690.70 q^{50} +12073.0 q^{52} +9677.70 q^{53} -4104.21 q^{55} +692.868 q^{58} -30369.2 q^{59} +732.473 q^{61} -18124.5 q^{62} +4096.00 q^{64} -35411.0 q^{65} +46399.9 q^{67} +23184.6 q^{68} +3295.39 q^{71} +10238.2 q^{73} -27319.8 q^{74} +40643.8 q^{76} +98584.0 q^{79} -12013.8 q^{80} +52277.1 q^{82} -87792.7 q^{83} -68001.9 q^{85} +49110.8 q^{86} -5597.17 q^{88} +69099.8 q^{89} +14596.0 q^{92} +53971.2 q^{94} -119211. q^{95} -42181.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 612 q^{17} + 2088 q^{19} - 1728 q^{20} - 496 q^{22} - 772 q^{23} - 318 q^{25} - 2880 q^{26} + 4592 q^{29} + 9792 q^{31} - 2048 q^{32} + 2448 q^{34} - 5992 q^{37} - 8352 q^{38} + 6912 q^{40} - 20196 q^{41} - 1136 q^{43} + 1984 q^{44} + 3088 q^{46} - 36936 q^{47} + 1272 q^{50} + 11520 q^{52} + 16708 q^{53} - 6336 q^{55} - 18368 q^{58} - 74592 q^{59} - 18648 q^{61} - 39168 q^{62} + 8192 q^{64} - 33300 q^{65} + 67344 q^{67} - 9792 q^{68} - 76548 q^{71} + 47304 q^{73} + 23968 q^{74} + 33408 q^{76} + 140656 q^{79} - 27648 q^{80} + 80784 q^{82} - 94104 q^{83} + 57868 q^{85} + 4544 q^{86} - 7936 q^{88} + 17604 q^{89} - 12352 q^{92} + 147744 q^{94} - 91592 q^{95} + 85176 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 - 612 * q^17 + 2088 * q^19 - 1728 * q^20 - 496 * q^22 - 772 * q^23 - 318 * q^25 - 2880 * q^26 + 4592 * q^29 + 9792 * q^31 - 2048 * q^32 + 2448 * q^34 - 5992 * q^37 - 8352 * q^38 + 6912 * q^40 - 20196 * q^41 - 1136 * q^43 + 1984 * q^44 + 3088 * q^46 - 36936 * q^47 + 1272 * q^50 + 11520 * q^52 + 16708 * q^53 - 6336 * q^55 - 18368 * q^58 - 74592 * q^59 - 18648 * q^61 - 39168 * q^62 + 8192 * q^64 - 33300 * q^65 + 67344 * q^67 - 9792 * q^68 - 76548 * q^71 + 47304 * q^73 + 23968 * q^74 + 33408 * q^76 + 140656 * q^79 - 27648 * q^80 + 80784 * q^82 - 94104 * q^83 + 57868 * q^85 + 4544 * q^86 - 7936 * q^88 + 17604 * q^89 - 12352 * q^92 + 147744 * q^94 - 91592 * q^95 + 85176 * 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
$5$	−46.9289	−0.839490	−0.419745	−	0.907642i	$-0.637881\pi$
$5$	−46.9289	−0.839490	−0.419745	+	0.907642i	$0.637881\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	187.716	0.593609
$11$	87.4558	0.217925	0.108963	−	0.994046i	$-0.465247\pi$
$11$	87.4558	0.217925	0.108963	+	0.994046i	$0.465247\pi$
$12$	0	0
$13$	754.566	1.23834	0.619168	−	0.785258i	$-0.287470\pi$
$13$	754.566	1.23834	0.619168	+	0.785258i	$0.287470\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	1449.04	1.21607	0.608034	−	0.793911i	$-0.291958\pi$
$17$	1449.04	1.21607	0.608034	+	0.793911i	$0.291958\pi$
$18$	0	0
$19$	2540.24	1.61432	0.807161	−	0.590331i	$-0.201003\pi$
$19$	2540.24	1.61432	0.807161	+	0.590331i	$0.201003\pi$
$20$	−750.863	−0.419745
$21$	0	0
$22$	−349.823	−0.154096
$23$	912.248	0.359578	0.179789	−	0.983705i	$-0.442458\pi$
$23$	912.248	0.359578	0.179789	+	0.983705i	$0.442458\pi$
$24$	0	0
$25$	−922.675	−0.295256
$26$	−3018.26	−0.875636
$27$	0	0
$28$	0	0
$29$	−173.217	−0.0382468	−0.0191234	−	0.999817i	$-0.506088\pi$
$29$	−173.217	−0.0382468	−0.0191234	+	0.999817i	$0.506088\pi$
$30$	0	0
$31$	4531.13	0.846842	0.423421	−	0.905933i	$-0.360829\pi$
$31$	4531.13	0.846842	0.423421	+	0.905933i	$0.360829\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−5796.16	−0.859890
$35$	0	0
$36$	0	0
$37$	6829.96	0.820188	0.410094	−	0.912043i	$-0.365496\pi$
$37$	6829.96	0.820188	0.410094	+	0.912043i	$0.365496\pi$
$38$	−10161.0	−1.14150
$39$	0	0
$40$	3003.45	0.296805
$41$	−13069.3	−1.21420	−0.607102	−	0.794624i	$-0.707668\pi$
$41$	−13069.3	−1.21420	−0.607102	+	0.794624i	$0.707668\pi$
$42$	0	0
$43$	−12277.7	−1.01262	−0.506309	−	0.862352i	$-0.668990\pi$
$43$	−12277.7	−1.01262	−0.506309	+	0.862352i	$0.668990\pi$
$44$	1399.29	0.108963
$45$	0	0
$46$	−3648.99	−0.254260
$47$	−13492.8	−0.890958	−0.445479	−	0.895292i	$-0.646967\pi$
$47$	−13492.8	−0.890958	−0.445479	+	0.895292i	$0.646967\pi$
$48$	0	0
$49$	0	0
$50$	3690.70	0.208778
$51$	0	0
$52$	12073.0	0.619168
$53$	9677.70	0.473241	0.236621	−	0.971602i	$-0.423960\pi$
$53$	9677.70	0.473241	0.236621	+	0.971602i	$0.423960\pi$
$54$	0	0
$55$	−4104.21	−0.182946
$56$	0	0
$57$	0	0
$58$	692.868	0.0270446
$59$	−30369.2	−1.13580	−0.567902	−	0.823096i	$-0.692245\pi$
$59$	−30369.2	−1.13580	−0.567902	+	0.823096i	$0.692245\pi$
$60$	0	0
$61$	732.473	0.0252038	0.0126019	−	0.999921i	$-0.495989\pi$
$61$	732.473	0.0252038	0.0126019	+	0.999921i	$0.495989\pi$
$62$	−18124.5	−0.598808
$63$	0	0
$64$	4096.00	0.125000
$65$	−35411.0	−1.03957
$66$	0	0
$67$	46399.9	1.26279	0.631394	−	0.775462i	$-0.282483\pi$
$67$	46399.9	1.26279	0.631394	+	0.775462i	$0.282483\pi$
$68$	23184.6	0.608034
$69$	0	0
$70$	0	0
$71$	3295.39	0.0775821	0.0387910	−	0.999247i	$-0.487649\pi$
$71$	3295.39	0.0775821	0.0387910	+	0.999247i	$0.487649\pi$
$72$	0	0
$73$	10238.2	0.224862	0.112431	−	0.993660i	$-0.464136\pi$
$73$	10238.2	0.224862	0.112431	+	0.993660i	$0.464136\pi$
$74$	−27319.8	−0.579961
$75$	0	0
$76$	40643.8	0.807161
$77$	0	0
$78$	0	0
$79$	98584.0	1.77721	0.888605	−	0.458674i	$-0.151675\pi$
$79$	98584.0	1.77721	0.888605	+	0.458674i	$0.151675\pi$
$80$	−12013.8	−0.209873
$81$	0	0
$82$	52277.1	0.858571
$83$	−87792.7	−1.39882	−0.699412	−	0.714719i	$-0.746555\pi$
$83$	−87792.7	−1.39882	−0.699412	+	0.714719i	$0.746555\pi$
$84$	0	0
$85$	−68001.9	−1.02088
$86$	49110.8	0.716029
$87$	0	0
$88$	−5597.17	−0.0770481
$89$	69099.8	0.924702	0.462351	−	0.886697i	$-0.347006\pi$
$89$	69099.8	0.924702	0.462351	+	0.886697i	$0.347006\pi$
$90$	0	0
$91$	0	0
$92$	14596.0	0.179789
$93$	0	0
$94$	53971.2	0.630003
$95$	−119211.	−1.35521
$96$	0	0
$97$	−42181.4	−0.455189	−0.227594	−	0.973756i	$-0.573086\pi$
$97$	−42181.4	−0.455189	−0.227594	+	0.973756i	$0.573086\pi$
$98$	0	0
$99$	0	0
$100$	−14762.8	−0.147628
$101$	−178474.	−1.74089	−0.870445	−	0.492266i	$-0.836169\pi$
$101$	−178474.	−1.74089	−0.870445	+	0.492266i	$0.836169\pi$
$102$	0	0
$103$	35226.2	0.327169	0.163585	−	0.986529i	$-0.447694\pi$
$103$	35226.2	0.327169	0.163585	+	0.986529i	$0.447694\pi$
$104$	−48292.2	−0.437818
$105$	0	0
$106$	−38710.8	−0.334632
$107$	−89960.9	−0.759616	−0.379808	−	0.925065i	$-0.624010\pi$
$107$	−89960.9	−0.759616	−0.379808	+	0.925065i	$0.624010\pi$
$108$	0	0
$109$	185002.	1.49146	0.745729	−	0.666249i	$-0.232101\pi$
$109$	185002.	1.49146	0.745729	+	0.666249i	$0.232101\pi$
$110$	16416.8	0.129362
$111$	0	0
$112$	0	0
$113$	228275.	1.68175	0.840877	−	0.541227i	$-0.182040\pi$
$113$	228275.	1.68175	0.840877	+	0.541227i	$0.182040\pi$
$114$	0	0
$115$	−42810.8	−0.301862
$116$	−2771.47	−0.0191234
$117$	0	0
$118$	121477.	0.803134
$119$	0	0
$120$	0	0
$121$	−153402.	−0.952509
$122$	−2929.89	−0.0178218
$123$	0	0
$124$	72498.1	0.423421
$125$	189953.	1.08735
$126$	0	0
$127$	−343515.	−1.88989	−0.944944	−	0.327232i	$-0.893884\pi$
$127$	−343515.	−1.88989	−0.944944	+	0.327232i	$0.893884\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	141644.	0.735088
$131$	361272.	1.83931	0.919657	−	0.392722i	$-0.128467\pi$
$131$	361272.	1.83931	0.919657	+	0.392722i	$0.128467\pi$
$132$	0	0
$133$	0	0
$134$	−185600.	−0.892926
$135$	0	0
$136$	−92738.5	−0.429945
$137$	304627.	1.38665	0.693325	−	0.720625i	$-0.256145\pi$
$137$	304627.	1.38665	0.693325	+	0.720625i	$0.256145\pi$
$138$	0	0
$139$	129737.	0.569543	0.284772	−	0.958595i	$-0.408082\pi$
$139$	129737.	0.569543	0.284772	+	0.958595i	$0.408082\pi$
$140$	0	0
$141$	0	0
$142$	−13181.6	−0.0548588
$143$	65991.2	0.269864
$144$	0	0
$145$	8128.88	0.0321078
$146$	−40952.7	−0.159001
$147$	0	0
$148$	109279.	0.410094
$149$	−373908.	−1.37975	−0.689873	−	0.723931i	$-0.742333\pi$
$149$	−373908.	−1.37975	−0.689873	+	0.723931i	$0.742333\pi$
$150$	0	0
$151$	−86639.8	−0.309225	−0.154613	−	0.987975i	$-0.549413\pi$
$151$	−86639.8	−0.309225	−0.154613	+	0.987975i	$0.549413\pi$
$152$	−162575.	−0.570749
$153$	0	0
$154$	0	0
$155$	−212641.	−0.710916
$156$	0	0
$157$	−462173.	−1.49643	−0.748214	−	0.663458i	$-0.769088\pi$
$157$	−462173.	−1.49643	−0.748214	+	0.663458i	$0.769088\pi$
$158$	−394336.	−1.25668
$159$	0	0
$160$	48055.2	0.148402
$161$	0	0
$162$	0	0
$163$	609032.	1.79544	0.897721	−	0.440565i	$-0.145222\pi$
$163$	609032.	1.79544	0.897721	+	0.440565i	$0.145222\pi$
$164$	−209108.	−0.607102
$165$	0	0
$166$	351171.	0.989118
$167$	127786.	0.354562	0.177281	−	0.984160i	$-0.443270\pi$
$167$	127786.	0.354562	0.177281	+	0.984160i	$0.443270\pi$
$168$	0	0
$169$	198076.	0.533477
$170$	272007.	0.721869
$171$	0	0
$172$	−196443.	−0.506309
$173$	142539.	0.362092	0.181046	−	0.983475i	$-0.442052\pi$
$173$	142539.	0.362092	0.181046	+	0.983475i	$0.442052\pi$
$174$	0	0
$175$	0	0
$176$	22388.7	0.0544813
$177$	0	0
$178$	−276399.	−0.653863
$179$	296634.	0.691973	0.345986	−	0.938240i	$-0.387544\pi$
$179$	296634.	0.691973	0.345986	+	0.938240i	$0.387544\pi$
$180$	0	0
$181$	−277654.	−0.629952	−0.314976	−	0.949100i	$-0.601996\pi$
$181$	−277654.	−0.629952	−0.314976	+	0.949100i	$0.601996\pi$
$182$	0	0
$183$	0	0
$184$	−58383.9	−0.127130
$185$	−320523.	−0.688540
$186$	0	0
$187$	126727.	0.265012
$188$	−215885.	−0.445479
$189$	0	0
$190$	476843.	0.958277
$191$	242584.	0.481149	0.240574	−	0.970631i	$-0.422664\pi$
$191$	242584.	0.481149	0.240574	+	0.970631i	$0.422664\pi$
$192$	0	0
$193$	−581525.	−1.12376	−0.561882	−	0.827218i	$-0.689922\pi$
$193$	−581525.	−1.12376	−0.561882	+	0.827218i	$0.689922\pi$
$194$	168726.	0.321867
$195$	0	0
$196$	0	0
$197$	−748916.	−1.37489	−0.687444	−	0.726237i	$-0.741267\pi$
$197$	−748916.	−1.37489	−0.687444	+	0.726237i	$0.741267\pi$
$198$	0	0
$199$	−725693.	−1.29903	−0.649516	−	0.760348i	$-0.725029\pi$
$199$	−725693.	−1.29903	−0.649516	+	0.760348i	$0.725029\pi$
$200$	59051.2	0.104389
$201$	0	0
$202$	713896.	1.23100
$203$	0	0
$204$	0	0
$205$	613327.	1.01931
$206$	−140905.	−0.231343
$207$	0	0
$208$	193169.	0.309584
$209$	222159.	0.351801
$210$	0	0
$211$	814718.	1.25980	0.629899	−	0.776677i	$-0.283096\pi$
$211$	814718.	1.25980	0.629899	+	0.776677i	$0.283096\pi$
$212$	154843.	0.236621
$213$	0	0
$214$	359843.	0.537130
$215$	576179.	0.850083
$216$	0	0
$217$	0	0
$218$	−740010.	−1.05462
$219$	0	0
$220$	−65667.4	−0.0914730
$221$	1.09339e6	1.50590
$222$	0	0
$223$	−335874.	−0.452288	−0.226144	−	0.974094i	$-0.572612\pi$
$223$	−335874.	−0.452288	−0.226144	+	0.974094i	$0.572612\pi$
$224$	0	0
$225$	0	0
$226$	−913100.	−1.18918
$227$	64721.2	0.0833647	0.0416823	−	0.999131i	$-0.486728\pi$
$227$	64721.2	0.0833647	0.0416823	+	0.999131i	$0.486728\pi$
$228$	0	0
$229$	750789.	0.946083	0.473041	−	0.881040i	$-0.343156\pi$
$229$	750789.	0.946083	0.473041	+	0.881040i	$0.343156\pi$
$230$	171243.	0.213449
$231$	0	0
$232$	11085.9	0.0135223
$233$	1.38718e6	1.67395	0.836974	−	0.547242i	$-0.184322\pi$
$233$	1.38718e6	1.67395	0.836974	+	0.547242i	$0.184322\pi$
$234$	0	0
$235$	633203.	0.747951
$236$	−485907.	−0.567902
$237$	0	0
$238$	0	0
$239$	160907.	0.182213	0.0911064	−	0.995841i	$-0.470960\pi$
$239$	160907.	0.182213	0.0911064	+	0.995841i	$0.470960\pi$
$240$	0	0
$241$	−1.28482e6	−1.42495	−0.712473	−	0.701699i	$-0.752425\pi$
$241$	−1.28482e6	−1.42495	−0.712473	+	0.701699i	$0.752425\pi$
$242$	613610.	0.673525
$243$	0	0
$244$	11719.6	0.0126019
$245$	0	0
$246$	0	0
$247$	1.91678e6	1.99907
$248$	−289993.	−0.299404
$249$	0	0
$250$	−759812.	−0.768876
$251$	481311.	0.482216	0.241108	−	0.970498i	$-0.422489\pi$
$251$	481311.	0.482216	0.241108	+	0.970498i	$0.422489\pi$
$252$	0	0
$253$	79781.4	0.0783611
$254$	1.37406e6	1.33635
$255$	0	0
$256$	65536.0	0.0625000
$257$	−33088.7	−0.0312497	−0.0156249	−	0.999878i	$-0.504974\pi$
$257$	−33088.7	−0.0312497	−0.0156249	+	0.999878i	$0.504974\pi$
$258$	0	0
$259$	0	0
$260$	−566575.	−0.519786
$261$	0	0
$262$	−1.44509e6	−1.30059
$263$	−446200.	−0.397778	−0.198889	−	0.980022i	$-0.563733\pi$
$263$	−446200.	−0.397778	−0.198889	+	0.980022i	$0.563733\pi$
$264$	0	0
$265$	−454164.	−0.397281
$266$	0	0
$267$	0	0
$268$	742399.	0.631394
$269$	1.94854e6	1.64183	0.820915	−	0.571051i	$-0.193464\pi$
$269$	1.94854e6	1.64183	0.820915	+	0.571051i	$0.193464\pi$
$270$	0	0
$271$	−96873.0	−0.0801271	−0.0400636	−	0.999197i	$-0.512756\pi$
$271$	−96873.0	−0.0801271	−0.0400636	+	0.999197i	$0.512756\pi$
$272$	370954.	0.304017
$273$	0	0
$274$	−1.21851e6	−0.980510
$275$	−80693.3	−0.0643437
$276$	0	0
$277$	1.63679e6	1.28172	0.640859	−	0.767658i	$-0.278578\pi$
$277$	1.63679e6	1.28172	0.640859	+	0.767658i	$0.278578\pi$
$278$	−518948.	−0.402728
$279$	0	0
$280$	0	0
$281$	1.79558e6	1.35656	0.678281	−	0.734803i	$-0.262725\pi$
$281$	1.79558e6	1.35656	0.678281	+	0.734803i	$0.262725\pi$
$282$	0	0
$283$	−1.28048e6	−0.950397	−0.475199	−	0.879879i	$-0.657624\pi$
$283$	−1.28048e6	−0.950397	−0.475199	+	0.879879i	$0.657624\pi$
$284$	52726.3	0.0387910
$285$	0	0
$286$	−263965.	−0.190823
$287$	0	0
$288$	0	0
$289$	679857.	0.478821
$290$	−32515.5	−0.0227037
$291$	0	0
$292$	163811.	0.112431
$293$	330814.	0.225120	0.112560	−	0.993645i	$-0.464095\pi$
$293$	330814.	0.225120	0.112560	+	0.993645i	$0.464095\pi$
$294$	0	0
$295$	1.42519e6	0.953496
$296$	−437117.	−0.289980
$297$	0	0
$298$	1.49563e6	0.975627
$299$	688351.	0.445279
$300$	0	0
$301$	0	0
$302$	346559.	0.218655
$303$	0	0
$304$	650301.	0.403581
$305$	−34374.2	−0.0211584
$306$	0	0
$307$	554088.	0.335531	0.167766	−	0.985827i	$-0.446345\pi$
$307$	554088.	0.335531	0.167766	+	0.985827i	$0.446345\pi$
$308$	0	0
$309$	0	0
$310$	850565.	0.502693
$311$	−1.75926e6	−1.03140	−0.515701	−	0.856769i	$-0.672469\pi$
$311$	−1.75926e6	−1.03140	−0.515701	+	0.856769i	$0.672469\pi$
$312$	0	0
$313$	2.33819e6	1.34902	0.674512	−	0.738264i	$-0.264354\pi$
$313$	2.33819e6	1.34902	0.674512	+	0.738264i	$0.264354\pi$
$314$	1.84869e6	1.05813
$315$	0	0
$316$	1.57734e6	0.888605
$317$	967392.	0.540697	0.270349	−	0.962762i	$-0.412861\pi$
$317$	967392.	0.540697	0.270349	+	0.962762i	$0.412861\pi$
$318$	0	0
$319$	−15148.8	−0.00833494
$320$	−192221.	−0.104936
$321$	0	0
$322$	0	0
$323$	3.68090e6	1.96313
$324$	0	0
$325$	−696219.	−0.365626
$326$	−2.43613e6	−1.26957
$327$	0	0
$328$	836433.	0.429286
$329$	0	0
$330$	0	0
$331$	−327808.	−0.164456	−0.0822281	−	0.996614i	$-0.526204\pi$
$331$	−327808.	−0.164456	−0.0822281	+	0.996614i	$0.526204\pi$
$332$	−1.40468e6	−0.699412
$333$	0	0
$334$	−511144.	−0.250713
$335$	−2.17750e6	−1.06010
$336$	0	0
$337$	1.34052e6	0.642983	0.321491	−	0.946913i	$-0.395816\pi$
$337$	1.34052e6	0.642983	0.321491	+	0.946913i	$0.395816\pi$
$338$	−792305.	−0.377225
$339$	0	0
$340$	−1.08803e6	−0.510439
$341$	396274.	0.184548
$342$	0	0
$343$	0	0
$344$	785772.	0.358014
$345$	0	0
$346$	−570157.	−0.256038
$347$	3.53374e6	1.57547	0.787736	−	0.616013i	$-0.211253\pi$
$347$	3.53374e6	1.57547	0.787736	+	0.616013i	$0.211253\pi$
$348$	0	0
$349$	3.98108e6	1.74959	0.874796	−	0.484491i	$-0.160995\pi$
$349$	3.98108e6	1.74959	0.874796	+	0.484491i	$0.160995\pi$
$350$	0	0
$351$	0	0
$352$	−89554.8	−0.0385241
$353$	44279.1	0.0189131	0.00945654	−	0.999955i	$-0.496990\pi$
$353$	44279.1	0.0189131	0.00945654	+	0.999955i	$0.496990\pi$
$354$	0	0
$355$	−154649.	−0.0651294
$356$	1.10560e6	0.462351
$357$	0	0
$358$	−1.18654e6	−0.489299
$359$	1.75114e6	0.717108	0.358554	−	0.933509i	$-0.383270\pi$
$359$	1.75114e6	0.717108	0.358554	+	0.933509i	$0.383270\pi$
$360$	0	0
$361$	3.97671e6	1.60604
$362$	1.11062e6	0.445443
$363$	0	0
$364$	0	0
$365$	−480467.	−0.188769
$366$	0	0
$367$	3.97114e6	1.53904	0.769521	−	0.638621i	$-0.220495\pi$
$367$	3.97114e6	1.53904	0.769521	+	0.638621i	$0.220495\pi$
$368$	233536.	0.0898945
$369$	0	0
$370$	1.28209e6	0.486871
$371$	0	0
$372$	0	0
$373$	2.23960e6	0.833486	0.416743	−	0.909024i	$-0.363172\pi$
$373$	2.23960e6	0.833486	0.416743	+	0.909024i	$0.363172\pi$
$374$	−506908.	−0.187392
$375$	0	0
$376$	863539.	0.315001
$377$	−130703.	−0.0473624
$378$	0	0
$379$	−4.25421e6	−1.52132	−0.760661	−	0.649149i	$-0.775125\pi$
$379$	−4.25421e6	−1.52132	−0.760661	+	0.649149i	$0.775125\pi$
$380$	−1.90737e6	−0.677604
$381$	0	0
$382$	−970338.	−0.340224
$383$	−1.61883e6	−0.563904	−0.281952	−	0.959429i	$-0.590982\pi$
$383$	−1.61883e6	−0.563904	−0.281952	+	0.959429i	$0.590982\pi$
$384$	0	0
$385$	0	0
$386$	2.32610e6	0.794621
$387$	0	0
$388$	−674902.	−0.227594
$389$	1.35315e6	0.453391	0.226696	−	0.973966i	$-0.427208\pi$
$389$	1.35315e6	0.453391	0.226696	+	0.973966i	$0.427208\pi$
$390$	0	0
$391$	1.32188e6	0.437271
$392$	0	0
$393$	0	0
$394$	2.99566e6	0.972193
$395$	−4.62644e6	−1.49195
$396$	0	0
$397$	−428056.	−0.136309	−0.0681545	−	0.997675i	$-0.521711\pi$
$397$	−428056.	−0.136309	−0.0681545	+	0.997675i	$0.521711\pi$
$398$	2.90277e6	0.918555
$399$	0	0
$400$	−236205.	−0.0738140
$401$	118228.	0.0367164	0.0183582	−	0.999831i	$-0.494156\pi$
$401$	118228.	0.0367164	0.0183582	+	0.999831i	$0.494156\pi$
$402$	0	0
$403$	3.41904e6	1.04868
$404$	−2.85558e6	−0.870445
$405$	0	0
$406$	0	0
$407$	597320.	0.178740
$408$	0	0
$409$	1.20974e6	0.357589	0.178795	−	0.983886i	$-0.442780\pi$
$409$	1.20974e6	0.357589	0.178795	+	0.983886i	$0.442780\pi$
$410$	−2.45331e6	−0.720762
$411$	0	0
$412$	563619.	0.163585
$413$	0	0
$414$	0	0
$415$	4.12002e6	1.17430
$416$	−772675.	−0.218909
$417$	0	0
$418$	−888635.	−0.248761
$419$	−4.12593e6	−1.14812	−0.574059	−	0.818814i	$-0.694632\pi$
$419$	−4.12593e6	−1.14812	−0.574059	+	0.818814i	$0.694632\pi$
$420$	0	0
$421$	3.12374e6	0.858953	0.429477	−	0.903078i	$-0.358698\pi$
$421$	3.12374e6	0.858953	0.429477	+	0.903078i	$0.358698\pi$
$422$	−3.25887e6	−0.890812
$423$	0	0
$424$	−619373.	−0.167316
$425$	−1.33699e6	−0.359051
$426$	0	0
$427$	0	0
$428$	−1.43937e6	−0.379808
$429$	0	0
$430$	−2.30472e6	−0.601099
$431$	−1.11459e6	−0.289015	−0.144507	−	0.989504i	$-0.546160\pi$
$431$	−1.11459e6	−0.289015	−0.144507	+	0.989504i	$0.546160\pi$
$432$	0	0
$433$	266138.	0.0682161	0.0341080	−	0.999418i	$-0.489141\pi$
$433$	266138.	0.0682161	0.0341080	+	0.999418i	$0.489141\pi$
$434$	0	0
$435$	0	0
$436$	2.96004e6	0.745729
$437$	2.31733e6	0.580475
$438$	0	0
$439$	2.46047e6	0.609336	0.304668	−	0.952459i	$-0.401455\pi$
$439$	2.46047e6	0.609336	0.304668	+	0.952459i	$0.401455\pi$
$440$	262669.	0.0646812
$441$	0	0
$442$	−4.37358e6	−1.06483
$443$	1.21081e6	0.293134	0.146567	−	0.989201i	$-0.453178\pi$
$443$	1.21081e6	0.293134	0.146567	+	0.989201i	$0.453178\pi$
$444$	0	0
$445$	−3.24278e6	−0.776279
$446$	1.34350e6	0.319816
$447$	0	0
$448$	0	0
$449$	3.66337e6	0.857561	0.428781	−	0.903409i	$-0.358943\pi$
$449$	3.66337e6	0.857561	0.428781	+	0.903409i	$0.358943\pi$
$450$	0	0
$451$	−1.14298e6	−0.264605
$452$	3.65240e6	0.840877
$453$	0	0
$454$	−258885.	−0.0589477
$455$	0	0
$456$	0	0
$457$	−8.73868e6	−1.95729	−0.978645	−	0.205556i	$-0.934100\pi$
$457$	−8.73868e6	−1.95729	−0.978645	+	0.205556i	$0.934100\pi$
$458$	−3.00316e6	−0.668982
$459$	0	0
$460$	−684973.	−0.150931
$461$	−1.23476e6	−0.270602	−0.135301	−	0.990805i	$-0.543200\pi$
$461$	−1.23476e6	−0.270602	−0.135301	+	0.990805i	$0.543200\pi$
$462$	0	0
$463$	−4.07764e6	−0.884008	−0.442004	−	0.897013i	$-0.645732\pi$
$463$	−4.07764e6	−0.884008	−0.442004	+	0.897013i	$0.645732\pi$
$464$	−44343.5	−0.00956170
$465$	0	0
$466$	−5.54871e6	−1.18366
$467$	−5.25021e6	−1.11400	−0.556999	−	0.830513i	$-0.688047\pi$
$467$	−5.25021e6	−1.11400	−0.556999	+	0.830513i	$0.688047\pi$
$468$	0	0
$469$	0	0
$470$	−2.53281e6	−0.528881
$471$	0	0
$472$	1.94363e6	0.401567
$473$	−1.07376e6	−0.220675
$474$	0	0
$475$	−2.34381e6	−0.476639
$476$	0	0
$477$	0	0
$478$	−643626.	−0.128844
$479$	−1.68707e6	−0.335965	−0.167982	−	0.985790i	$-0.553725\pi$
$479$	−1.68707e6	−0.335965	−0.167982	+	0.985790i	$0.553725\pi$
$480$	0	0
$481$	5.15365e6	1.01567
$482$	5.13927e6	1.00759
$483$	0	0
$484$	−2.45444e6	−0.476254
$485$	1.97953e6	0.382126
$486$	0	0
$487$	4.39520e6	0.839762	0.419881	−	0.907579i	$-0.362072\pi$
$487$	4.39520e6	0.839762	0.419881	+	0.907579i	$0.362072\pi$
$488$	−46878.2	−0.00891091
$489$	0	0
$490$	0	0
$491$	−2.13582e6	−0.399817	−0.199909	−	0.979815i	$-0.564065\pi$
$491$	−2.13582e6	−0.399817	−0.199909	+	0.979815i	$0.564065\pi$
$492$	0	0
$493$	−250998.	−0.0465107
$494$	−7.66710e6	−1.41356
$495$	0	0
$496$	1.15997e6	0.211711
$497$	0	0
$498$	0	0
$499$	−1.93439e6	−0.347771	−0.173885	−	0.984766i	$-0.555632\pi$
$499$	−1.93439e6	−0.347771	−0.173885	+	0.984766i	$0.555632\pi$
$500$	3.03925e6	0.543677
$501$	0	0
$502$	−1.92524e6	−0.340978
$503$	−9.47334e6	−1.66949	−0.834744	−	0.550639i	$-0.814384\pi$
$503$	−9.47334e6	−1.66949	−0.834744	+	0.550639i	$0.814384\pi$
$504$	0	0
$505$	8.37559e6	1.46146
$506$	−319126.	−0.0554097
$507$	0	0
$508$	−5.49624e6	−0.944944
$509$	1.05589e7	1.80645	0.903225	−	0.429167i	$-0.141193\pi$
$509$	1.05589e7	1.80645	0.903225	+	0.429167i	$0.141193\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	132355.	0.0220969
$515$	−1.65313e6	−0.274655
$516$	0	0
$517$	−1.18002e6	−0.194162
$518$	0	0
$519$	0	0
$520$	2.26630e6	0.367544
$521$	7.53190e6	1.21565	0.607827	−	0.794069i	$-0.292041\pi$
$521$	7.53190e6	1.21565	0.607827	+	0.794069i	$0.292041\pi$
$522$	0	0
$523$	−7.33316e6	−1.17230	−0.586148	−	0.810204i	$-0.699356\pi$
$523$	−7.33316e6	−1.17230	−0.586148	+	0.810204i	$0.699356\pi$
$524$	5.78035e6	0.919657
$525$	0	0
$526$	1.78480e6	0.281271
$527$	6.56579e6	1.02982
$528$	0	0
$529$	−5.60415e6	−0.870704
$530$	1.81666e6	0.280920
$531$	0	0
$532$	0	0
$533$	−9.86162e6	−1.50359
$534$	0	0
$535$	4.22177e6	0.637690
$536$	−2.96960e6	−0.446463
$537$	0	0
$538$	−7.79415e6	−1.16095
$539$	0	0
$540$	0	0
$541$	−1.25979e7	−1.85056	−0.925281	−	0.379282i	$-0.876171\pi$
$541$	−1.25979e7	−1.85056	−0.925281	+	0.379282i	$0.876171\pi$
$542$	387492.	0.0566584
$543$	0	0
$544$	−1.48382e6	−0.214972
$545$	−8.68197e6	−1.25206
$546$	0	0
$547$	−4.21767e6	−0.602704	−0.301352	−	0.953513i	$-0.597438\pi$
$547$	−4.21767e6	−0.602704	−0.301352	+	0.953513i	$0.597438\pi$
$548$	4.87403e6	0.693325
$549$	0	0
$550$	322773.	0.0454979
$551$	−440012.	−0.0617427
$552$	0	0
$553$	0	0
$554$	−6.54715e6	−0.906312
$555$	0	0
$556$	2.07579e6	0.284772
$557$	9.56529e6	1.30635	0.653176	−	0.757206i	$-0.273436\pi$
$557$	9.56529e6	1.30635	0.653176	+	0.757206i	$0.273436\pi$
$558$	0	0
$559$	−9.26432e6	−1.25396
$560$	0	0
$561$	0	0
$562$	−7.18233e6	−0.959234
$563$	9.39186e6	1.24876	0.624382	−	0.781119i	$-0.285351\pi$
$563$	9.39186e6	1.24876	0.624382	+	0.781119i	$0.285351\pi$
$564$	0	0
$565$	−1.07127e7	−1.41182
$566$	5.12190e6	0.672032
$567$	0	0
$568$	−210905.	−0.0274294
$569$	−1.03867e7	−1.34493	−0.672463	−	0.740131i	$-0.734763\pi$
$569$	−1.03867e7	−1.34493	−0.672463	+	0.740131i	$0.734763\pi$
$570$	0	0
$571$	4.42960e6	0.568557	0.284279	−	0.958742i	$-0.408246\pi$
$571$	4.42960e6	0.568557	0.284279	+	0.958742i	$0.408246\pi$
$572$	1.05586e6	0.134932
$573$	0	0
$574$	0	0
$575$	−841709.	−0.106168
$576$	0	0
$577$	−1.20182e7	−1.50280	−0.751398	−	0.659849i	$-0.770620\pi$
$577$	−1.20182e7	−1.50280	−0.751398	+	0.659849i	$0.770620\pi$
$578$	−2.71943e6	−0.338577
$579$	0	0
$580$	130062.	0.0160539
$581$	0	0
$582$	0	0
$583$	846372.	0.103131
$584$	−655244.	−0.0795007
$585$	0	0
$586$	−1.32326e6	−0.159184
$587$	−6.97624e6	−0.835653	−0.417826	−	0.908527i	$-0.637208\pi$
$587$	−6.97624e6	−0.835653	−0.417826	+	0.908527i	$0.637208\pi$
$588$	0	0
$589$	1.15102e7	1.36708
$590$	−5.70077e6	−0.674223
$591$	0	0
$592$	1.74847e6	0.205047
$593$	1.78148e6	0.208038	0.104019	−	0.994575i	$-0.466830\pi$
$593$	1.78148e6	0.208038	0.104019	+	0.994575i	$0.466830\pi$
$594$	0	0
$595$	0	0
$596$	−5.98253e6	−0.689873
$597$	0	0
$598$	−2.75340e6	−0.314860
$599$	−1.60365e6	−0.182617	−0.0913085	−	0.995823i	$-0.529105\pi$
$599$	−1.60365e6	−0.182617	−0.0913085	+	0.995823i	$0.529105\pi$
$600$	0	0
$601$	−1.51165e7	−1.70712	−0.853562	−	0.520991i	$-0.825562\pi$
$601$	−1.51165e7	−1.70712	−0.853562	+	0.520991i	$0.825562\pi$
$602$	0	0
$603$	0	0
$604$	−1.38624e6	−0.154613
$605$	7.19901e6	0.799622
$606$	0	0
$607$	1.05167e7	1.15853	0.579263	−	0.815141i	$-0.303340\pi$
$607$	1.05167e7	1.15853	0.579263	+	0.815141i	$0.303340\pi$
$608$	−2.60120e6	−0.285375
$609$	0	0
$610$	137497.	0.0149612
$611$	−1.01812e7	−1.10331
$612$	0	0
$613$	2.71298e6	0.291605	0.145802	−	0.989314i	$-0.453424\pi$
$613$	2.71298e6	0.291605	0.145802	+	0.989314i	$0.453424\pi$
$614$	−2.21635e6	−0.237256
$615$	0	0
$616$	0	0
$617$	−1.14961e7	−1.21573	−0.607864	−	0.794041i	$-0.707974\pi$
$617$	−1.14961e7	−1.21573	−0.607864	+	0.794041i	$0.707974\pi$
$618$	0	0
$619$	1.09950e7	1.15337	0.576684	−	0.816967i	$-0.304346\pi$
$619$	1.09950e7	1.15337	0.576684	+	0.816967i	$0.304346\pi$
$620$	−3.40226e6	−0.355458
$621$	0	0
$622$	7.03702e6	0.729312
$623$	0	0
$624$	0	0
$625$	−6.03093e6	−0.617568
$626$	−9.35277e6	−0.953904
$627$	0	0
$628$	−7.39477e6	−0.748214
$629$	9.89687e6	0.997405
$630$	0	0
$631$	−4.19503e6	−0.419432	−0.209716	−	0.977762i	$-0.567254\pi$
$631$	−4.19503e6	−0.419432	−0.209716	+	0.977762i	$0.567254\pi$
$632$	−6.30938e6	−0.628338
$633$	0	0
$634$	−3.86957e6	−0.382331
$635$	1.61208e7	1.58654
$636$	0	0
$637$	0	0
$638$	60595.3	0.00589369
$639$	0	0
$640$	768884.	0.0742012
$641$	1.06270e7	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$641$	1.06270e7	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$642$	0	0
$643$	1.54228e6	0.147108	0.0735538	−	0.997291i	$-0.476566\pi$
$643$	1.54228e6	0.147108	0.0735538	+	0.997291i	$0.476566\pi$
$644$	0	0
$645$	0	0
$646$	−1.47236e7	−1.38814
$647$	1.28170e7	1.20372	0.601861	−	0.798601i	$-0.294426\pi$
$647$	1.28170e7	1.20372	0.601861	+	0.798601i	$0.294426\pi$
$648$	0	0
$649$	−2.65596e6	−0.247520
$650$	2.78488e6	0.258537
$651$	0	0
$652$	9.74452e6	0.897721
$653$	2.84549e6	0.261141	0.130570	−	0.991439i	$-0.458319\pi$
$653$	2.84549e6	0.261141	0.130570	+	0.991439i	$0.458319\pi$
$654$	0	0
$655$	−1.69541e7	−1.54409
$656$	−3.34573e6	−0.303551
$657$	0	0
$658$	0	0
$659$	2.04185e7	1.83151	0.915755	−	0.401736i	$-0.131593\pi$
$659$	2.04185e7	1.83151	0.915755	+	0.401736i	$0.131593\pi$
$660$	0	0
$661$	2.18928e7	1.94893	0.974467	−	0.224529i	$-0.0720843\pi$
$661$	2.18928e7	1.94893	0.974467	+	0.224529i	$0.0720843\pi$
$662$	1.31123e6	0.116288
$663$	0	0
$664$	5.61873e6	0.494559
$665$	0	0
$666$	0	0
$667$	−158017.	−0.0137527
$668$	2.04458e6	0.177281
$669$	0	0
$670$	8.71000e6	0.749602
$671$	64059.0	0.00549255
$672$	0	0
$673$	−992084.	−0.0844327	−0.0422164	−	0.999108i	$-0.513442\pi$
$673$	−992084.	−0.0844327	−0.0422164	+	0.999108i	$0.513442\pi$
$674$	−5.36209e6	−0.454657
$675$	0	0
$676$	3.16922e6	0.266738
$677$	7.83497e6	0.657000	0.328500	−	0.944504i	$-0.393457\pi$
$677$	7.83497e6	0.657000	0.328500	+	0.944504i	$0.393457\pi$
$678$	0	0
$679$	0	0
$680$	4.35212e6	0.360935
$681$	0	0
$682$	−1.58510e6	−0.130495
$683$	7.41182e6	0.607957	0.303979	−	0.952679i	$-0.401685\pi$
$683$	7.41182e6	0.607957	0.303979	+	0.952679i	$0.401685\pi$
$684$	0	0
$685$	−1.42958e7	−1.16408
$686$	0	0
$687$	0	0
$688$	−3.14309e6	−0.253154
$689$	7.30246e6	0.586032
$690$	0	0
$691$	1.00517e7	0.800836	0.400418	−	0.916333i	$-0.368865\pi$
$691$	1.00517e7	0.800836	0.400418	+	0.916333i	$0.368865\pi$
$692$	2.28063e6	0.181046
$693$	0	0
$694$	−1.41350e7	−1.11403
$695$	−6.08841e6	−0.478126
$696$	0	0
$697$	−1.89379e7	−1.47655
$698$	−1.59243e7	−1.23715
$699$	0	0
$700$	0	0
$701$	−1.59518e7	−1.22607	−0.613034	−	0.790057i	$-0.710051\pi$
$701$	−1.59518e7	−1.22607	−0.613034	+	0.790057i	$0.710051\pi$
$702$	0	0
$703$	1.73497e7	1.32405
$704$	358219.	0.0272406
$705$	0	0
$706$	−177116.	−0.0133736
$707$	0	0
$708$	0	0
$709$	−3.82623e6	−0.285862	−0.142931	−	0.989733i	$-0.545653\pi$
$709$	−3.82623e6	−0.285862	−0.142931	+	0.989733i	$0.545653\pi$
$710$	618597.	0.0460534
$711$	0	0
$712$	−4.42239e6	−0.326932
$713$	4.13352e6	0.304506
$714$	0	0
$715$	−3.09690e6	−0.226549
$716$	4.74615e6	0.345986
$717$	0	0
$718$	−7.00456e6	−0.507072
$719$	−7.92731e6	−0.571878	−0.285939	−	0.958248i	$-0.592306\pi$
$719$	−7.92731e6	−0.571878	−0.285939	+	0.958248i	$0.592306\pi$
$720$	0	0
$721$	0	0
$722$	−1.59068e7	−1.13564
$723$	0	0
$724$	−4.44246e6	−0.314976
$725$	159823.	0.0112926
$726$	0	0
$727$	−3.24519e6	−0.227722	−0.113861	−	0.993497i	$-0.536322\pi$
$727$	−3.24519e6	−0.227722	−0.113861	+	0.993497i	$0.536322\pi$
$728$	0	0
$729$	0	0
$730$	1.92187e6	0.133480
$731$	−1.77908e7	−1.23141
$732$	0	0
$733$	−5.97679e6	−0.410874	−0.205437	−	0.978670i	$-0.565862\pi$
$733$	−5.97679e6	−0.410874	−0.205437	+	0.978670i	$0.565862\pi$
$734$	−1.58846e7	−1.08827
$735$	0	0
$736$	−934142.	−0.0635650
$737$	4.05794e6	0.275193
$738$	0	0
$739$	−2.60064e6	−0.175174	−0.0875870	−	0.996157i	$-0.527916\pi$
$739$	−2.60064e6	−0.175174	−0.0875870	+	0.996157i	$0.527916\pi$
$740$	−5.12836e6	−0.344270
$741$	0	0
$742$	0	0
$743$	2.29585e7	1.52571	0.762854	−	0.646571i	$-0.223797\pi$
$743$	2.29585e7	1.52571	0.762854	+	0.646571i	$0.223797\pi$
$744$	0	0
$745$	1.75471e7	1.15828
$746$	−8.95839e6	−0.589363
$747$	0	0
$748$	2.02763e6	0.132506
$749$	0	0
$750$	0	0
$751$	4.60318e6	0.297823	0.148911	−	0.988851i	$-0.452423\pi$
$751$	4.60318e6	0.297823	0.148911	+	0.988851i	$0.452423\pi$
$752$	−3.45416e6	−0.222740
$753$	0	0
$754$	522814.	0.0334903
$755$	4.06591e6	0.259592
$756$	0	0
$757$	4.78695e6	0.303612	0.151806	−	0.988410i	$-0.451491\pi$
$757$	4.78695e6	0.303612	0.151806	+	0.988410i	$0.451491\pi$
$758$	1.70169e7	1.07574
$759$	0	0
$760$	7.62948e6	0.479138
$761$	−2.28441e7	−1.42992	−0.714962	−	0.699164i	$-0.753556\pi$
$761$	−2.28441e7	−1.42992	−0.714962	+	0.699164i	$0.753556\pi$
$762$	0	0
$763$	0	0
$764$	3.88135e6	0.240574
$765$	0	0
$766$	6.47533e6	0.398740
$767$	−2.29155e7	−1.40651
$768$	0	0
$769$	−1.02787e7	−0.626793	−0.313397	−	0.949622i	$-0.601467\pi$
$769$	−1.02787e7	−0.626793	−0.313397	+	0.949622i	$0.601467\pi$
$770$	0	0
$771$	0	0
$772$	−9.30439e6	−0.561882
$773$	1.38227e7	0.832038	0.416019	−	0.909356i	$-0.363425\pi$
$773$	1.38227e7	0.832038	0.416019	+	0.909356i	$0.363425\pi$
$774$	0	0
$775$	−4.18076e6	−0.250035
$776$	2.69961e6	0.160933
$777$	0	0
$778$	−5.41262e6	−0.320596
$779$	−3.31990e7	−1.96012
$780$	0	0
$781$	288201.	0.0169071
$782$	−5.28753e6	−0.309198
$783$	0	0
$784$	0	0
$785$	2.16893e7	1.25624
$786$	0	0
$787$	−736248.	−0.0423728	−0.0211864	−	0.999776i	$-0.506744\pi$
$787$	−736248.	−0.0423728	−0.0211864	+	0.999776i	$0.506744\pi$
$788$	−1.19827e7	−0.687444
$789$	0	0
$790$	1.85058e7	1.05497
$791$	0	0
$792$	0	0
$793$	552699.	0.0312108
$794$	1.71222e6	0.0963850
$795$	0	0
$796$	−1.16111e7	−0.649516
$797$	−3.29571e7	−1.83782	−0.918912	−	0.394464i	$-0.870930\pi$
$797$	−3.29571e7	−1.83782	−0.918912	+	0.394464i	$0.870930\pi$
$798$	0	0
$799$	−1.95516e7	−1.08347
$800$	944820.	0.0521944
$801$	0	0
$802$	−472913.	−0.0259624
$803$	895389.	0.0490030
$804$	0	0
$805$	0	0
$806$	−1.36761e7	−0.741526
$807$	0	0
$808$	1.14223e7	0.615498
$809$	1.09629e7	0.588918	0.294459	−	0.955664i	$-0.404861\pi$
$809$	1.09629e7	0.588918	0.294459	+	0.955664i	$0.404861\pi$
$810$	0	0
$811$	2.77365e7	1.48081	0.740406	−	0.672160i	$-0.234633\pi$
$811$	2.77365e7	1.48081	0.740406	+	0.672160i	$0.234633\pi$
$812$	0	0
$813$	0	0
$814$	−2.38928e6	−0.126388
$815$	−2.85812e7	−1.50726
$816$	0	0
$817$	−3.11882e7	−1.63469
$818$	−4.83897e6	−0.252854
$819$	0	0
$820$	9.81322e6	0.509656
$821$	−1.25487e7	−0.649743	−0.324872	−	0.945758i	$-0.605321\pi$
$821$	−1.25487e7	−0.649743	−0.324872	+	0.945758i	$0.605321\pi$
$822$	0	0
$823$	2.88571e7	1.48509	0.742545	−	0.669796i	$-0.233618\pi$
$823$	2.88571e7	1.48509	0.742545	+	0.669796i	$0.233618\pi$
$824$	−2.25447e6	−0.115672
$825$	0	0
$826$	0	0
$827$	2.83888e7	1.44339	0.721694	−	0.692213i	$-0.243364\pi$
$827$	2.83888e7	1.44339	0.721694	+	0.692213i	$0.243364\pi$
$828$	0	0
$829$	−2.40584e7	−1.21585	−0.607926	−	0.793994i	$-0.707998\pi$
$829$	−2.40584e7	−1.21585	−0.607926	+	0.793994i	$0.707998\pi$
$830$	−1.64801e7	−0.830355
$831$	0	0
$832$	3.09070e6	0.154792
$833$	0	0
$834$	0	0
$835$	−5.99686e6	−0.297651
$836$	3.55454e6	0.175901
$837$	0	0
$838$	1.65037e7	0.811842
$839$	−2.99511e6	−0.146895	−0.0734477	−	0.997299i	$-0.523400\pi$
$839$	−2.99511e6	−0.146895	−0.0734477	+	0.997299i	$0.523400\pi$
$840$	0	0
$841$	−2.04811e7	−0.998537
$842$	−1.24950e7	−0.607372
$843$	0	0
$844$	1.30355e7	0.629899
$845$	−9.29551e6	−0.447849
$846$	0	0
$847$	0	0
$848$	2.47749e6	0.118310
$849$	0	0
$850$	5.34797e6	0.253888
$851$	6.23061e6	0.294922
$852$	0	0
$853$	−1.46234e6	−0.0688137	−0.0344069	−	0.999408i	$-0.510954\pi$
$853$	−1.46234e6	−0.0688137	−0.0344069	+	0.999408i	$0.510954\pi$
$854$	0	0
$855$	0	0
$856$	5.75750e6	0.268565
$857$	1.66909e7	0.776295	0.388148	−	0.921597i	$-0.373115\pi$
$857$	1.66909e7	0.776295	0.388148	+	0.921597i	$0.373115\pi$
$858$	0	0
$859$	−3.01647e7	−1.39481	−0.697406	−	0.716676i	$-0.745663\pi$
$859$	−3.01647e7	−1.39481	−0.697406	+	0.716676i	$0.745663\pi$
$860$	9.21886e6	0.425041
$861$	0	0
$862$	4.45834e6	0.204364
$863$	2.88352e7	1.31794	0.658970	−	0.752169i	$-0.270992\pi$
$863$	2.88352e7	1.31794	0.658970	+	0.752169i	$0.270992\pi$
$864$	0	0
$865$	−6.68921e6	−0.303973
$866$	−1.06455e6	−0.0482361
$867$	0	0
$868$	0	0
$869$	8.62175e6	0.387298
$870$	0	0
$871$	3.50118e7	1.56376
$872$	−1.18402e7	−0.527310
$873$	0	0
$874$	−9.26931e6	−0.410458
$875$	0	0
$876$	0	0
$877$	3.15732e7	1.38618	0.693090	−	0.720851i	$-0.256249\pi$
$877$	3.15732e7	1.38618	0.693090	+	0.720851i	$0.256249\pi$
$878$	−9.84188e6	−0.430865
$879$	0	0
$880$	−1.05068e6	−0.0457365
$881$	1.42247e7	0.617452	0.308726	−	0.951151i	$-0.400097\pi$
$881$	1.42247e7	0.617452	0.308726	+	0.951151i	$0.400097\pi$
$882$	0	0
$883$	−9.42179e6	−0.406660	−0.203330	−	0.979110i	$-0.565176\pi$
$883$	−9.42179e6	−0.406660	−0.203330	+	0.979110i	$0.565176\pi$
$884$	1.74943e7	0.752950
$885$	0	0
$886$	−4.84323e6	−0.207277
$887$	1.50474e7	0.642174	0.321087	−	0.947050i	$-0.395952\pi$
$887$	1.50474e7	0.642174	0.321087	+	0.947050i	$0.395952\pi$
$888$	0	0
$889$	0	0
$890$	1.29711e7	0.548912
$891$	0	0
$892$	−5.37399e6	−0.226144
$893$	−3.42749e7	−1.43829
$894$	0	0
$895$	−1.39207e7	−0.580904
$896$	0	0
$897$	0	0
$898$	−1.46535e7	−0.606388
$899$	−784869.	−0.0323890
$900$	0	0
$901$	1.40234e7	0.575494
$902$	4.57193e6	0.187104
$903$	0	0
$904$	−1.46096e7	−0.594589
$905$	1.30300e7	0.528838
$906$	0	0
$907$	−1.87480e7	−0.756725	−0.378362	−	0.925658i	$-0.623513\pi$
$907$	−1.87480e7	−0.756725	−0.378362	+	0.925658i	$0.623513\pi$
$908$	1.03554e6	0.0416823
$909$	0	0
$910$	0	0
$911$	−1.49384e7	−0.596358	−0.298179	−	0.954510i	$-0.596379\pi$
$911$	−1.49384e7	−0.596358	−0.298179	+	0.954510i	$0.596379\pi$
$912$	0	0
$913$	−7.67798e6	−0.304839
$914$	3.49547e7	1.38401
$915$	0	0
$916$	1.20126e7	0.473041
$917$	0	0
$918$	0	0
$919$	4.21133e7	1.64487	0.822434	−	0.568861i	$-0.192616\pi$
$919$	4.21133e7	1.64487	0.822434	+	0.568861i	$0.192616\pi$
$920$	2.73989e6	0.106724
$921$	0	0
$922$	4.93905e6	0.191345
$923$	2.48659e6	0.0960727
$924$	0	0
$925$	−6.30183e6	−0.242166
$926$	1.63106e7	0.625088
$927$	0	0
$928$	177374.	0.00676114
$929$	−5.87883e6	−0.223487	−0.111743	−	0.993737i	$-0.535643\pi$
$929$	−5.87883e6	−0.223487	−0.111743	+	0.993737i	$0.535643\pi$
$930$	0	0
$931$	0	0
$932$	2.21948e7	0.836974
$933$	0	0
$934$	2.10008e7	0.787715
$935$	−5.94716e6	−0.222475
$936$	0	0
$937$	1.42337e7	0.529625	0.264813	−	0.964300i	$-0.414690\pi$
$937$	1.42337e7	0.529625	0.264813	+	0.964300i	$0.414690\pi$
$938$	0	0
$939$	0	0
$940$	1.01312e7	0.373975
$941$	3.24512e7	1.19469	0.597347	−	0.801983i	$-0.296222\pi$
$941$	3.24512e7	1.19469	0.597347	+	0.801983i	$0.296222\pi$
$942$	0	0
$943$	−1.19224e7	−0.436601
$944$	−7.77451e6	−0.283951
$945$	0	0
$946$	4.29502e6	0.156041
$947$	2.60957e7	0.945570	0.472785	−	0.881178i	$-0.343249\pi$
$947$	2.60957e7	0.945570	0.472785	+	0.881178i	$0.343249\pi$
$948$	0	0
$949$	7.72538e6	0.278455
$950$	9.37526e6	0.337034
$951$	0	0
$952$	0	0
$953$	−2.98631e7	−1.06513	−0.532565	−	0.846389i	$-0.678772\pi$
$953$	−2.98631e7	−1.06513	−0.532565	+	0.846389i	$0.678772\pi$
$954$	0	0
$955$	−1.13842e7	−0.403920
$956$	2.57451e6	0.0911064
$957$	0	0
$958$	6.74827e6	0.237563
$959$	0	0
$960$	0	0
$961$	−8.09799e6	−0.282858
$962$	−2.06146e7	−0.718186
$963$	0	0
$964$	−2.05571e7	−0.712473
$965$	2.72903e7	0.943388
$966$	0	0
$967$	6.83038e6	0.234898	0.117449	−	0.993079i	$-0.462528\pi$
$967$	6.83038e6	0.234898	0.117449	+	0.993079i	$0.462528\pi$
$968$	9.81776e6	0.336763
$969$	0	0
$970$	−7.91811e6	−0.270204
$971$	−8.99328e6	−0.306105	−0.153052	−	0.988218i	$-0.548910\pi$
$971$	−8.99328e6	−0.306105	−0.153052	+	0.988218i	$0.548910\pi$
$972$	0	0
$973$	0	0
$974$	−1.75808e7	−0.593801
$975$	0	0
$976$	187513.	0.00630096
$977$	2.94394e7	0.986718	0.493359	−	0.869826i	$-0.335769\pi$
$977$	2.94394e7	0.986718	0.493359	+	0.869826i	$0.335769\pi$
$978$	0	0
$979$	6.04318e6	0.201516
$980$	0	0
$981$	0	0
$982$	8.54329e6	0.282713
$983$	−3.96261e7	−1.30797	−0.653984	−	0.756508i	$-0.726904\pi$
$983$	−3.96261e7	−1.30797	−0.653984	+	0.756508i	$0.726904\pi$
$984$	0	0
$985$	3.51458e7	1.15421
$986$	1.00399e6	0.0328880
$987$	0	0
$988$	3.06684e7	0.999537
$989$	−1.12003e7	−0.364115
$990$	0	0
$991$	5.34988e7	1.73045	0.865226	−	0.501382i	$-0.167175\pi$
$991$	5.34988e7	1.73045	0.865226	+	0.501382i	$0.167175\pi$
$992$	−4.63988e6	−0.149702
$993$	0	0
$994$	0	0
$995$	3.40560e7	1.09053
$996$	0	0
$997$	−3.76239e7	−1.19874	−0.599371	−	0.800471i	$-0.704583\pi$
$997$	−3.76239e7	−1.19874	−0.599371	+	0.800471i	$0.704583\pi$
$998$	7.73756e6	0.245911
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.t.1.1	✓	2	3.2	odd	2
294.6.a.u.1.2	yes	2	21.20	even	2
294.6.e.u.67.1		4	21.5	even	6
294.6.e.u.79.1		4	21.17	even	6
294.6.e.v.67.2		4	21.2	odd	6
294.6.e.v.79.2		4	21.11	odd	6
882.6.a.z.1.2		2	1.1	even	1	trivial
882.6.a.bj.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)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -46.9289 q^{5} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} -46.9289 q^{5} -64.0000 q^{8} +187.716 q^{10} +87.4558 q^{11} +754.566 q^{13} +256.000 q^{16} +1449.04 q^{17} +2540.24 q^{19} -750.863 q^{20} -349.823 q^{22} +912.248 q^{23} -922.675 q^{25} -3018.26 q^{26} -173.217 q^{29} +4531.13 q^{31} -1024.00 q^{32} -5796.16 q^{34} +6829.96 q^{37} -10161.0 q^{38} +3003.45 q^{40} -13069.3 q^{41} -12277.7 q^{43} +1399.29 q^{44} -3648.99 q^{46} -13492.8 q^{47} +3690.70 q^{50} +12073.0 q^{52} +9677.70 q^{53} -4104.21 q^{55} +692.868 q^{58} -30369.2 q^{59} +732.473 q^{61} -18124.5 q^{62} +4096.00 q^{64} -35411.0 q^{65} +46399.9 q^{67} +23184.6 q^{68} +3295.39 q^{71} +10238.2 q^{73} -27319.8 q^{74} +40643.8 q^{76} +98584.0 q^{79} -12013.8 q^{80} +52277.1 q^{82} -87792.7 q^{83} -68001.9 q^{85} +49110.8 q^{86} -5597.17 q^{88} +69099.8 q^{89} +14596.0 q^{92} +53971.2 q^{94} -119211. q^{95} -42181.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 108 q^{5} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 - 108 * q^5 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} - 108 q^{5} - 128 q^{8} + 432 q^{10} + 124 q^{11} + 720 q^{13} + 512 q^{16} - 612 q^{17} + 2088 q^{19} - 1728 q^{20} - 496 q^{22} - 772 q^{23} - 318 q^{25} - 2880 q^{26} + 4592 q^{29} + 9792 q^{31} - 2048 q^{32} + 2448 q^{34} - 5992 q^{37} - 8352 q^{38} + 6912 q^{40} - 20196 q^{41} - 1136 q^{43} + 1984 q^{44} + 3088 q^{46} - 36936 q^{47} + 1272 q^{50} + 11520 q^{52} + 16708 q^{53} - 6336 q^{55} - 18368 q^{58} - 74592 q^{59} - 18648 q^{61} - 39168 q^{62} + 8192 q^{64} - 33300 q^{65} + 67344 q^{67} - 9792 q^{68} - 76548 q^{71} + 47304 q^{73} + 23968 q^{74} + 33408 q^{76} + 140656 q^{79} - 27648 q^{80} + 80784 q^{82} - 94104 q^{83} + 57868 q^{85} + 4544 q^{86} - 7936 q^{88} + 17604 q^{89} - 12352 q^{92} + 147744 q^{94} - 91592 q^{95} + 85176 q^{97}+O(q^{100}) \) 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 - 612 * q^17 + 2088 * q^19 - 1728 * q^20 - 496 * q^22 - 772 * q^23 - 318 * q^25 - 2880 * q^26 + 4592 * q^29 + 9792 * q^31 - 2048 * q^32 + 2448 * q^34 - 5992 * q^37 - 8352 * q^38 + 6912 * q^40 - 20196 * q^41 - 1136 * q^43 + 1984 * q^44 + 3088 * q^46 - 36936 * q^47 + 1272 * q^50 + 11520 * q^52 + 16708 * q^53 - 6336 * q^55 - 18368 * q^58 - 74592 * q^59 - 18648 * q^61 - 39168 * q^62 + 8192 * q^64 - 33300 * q^65 + 67344 * q^67 - 9792 * q^68 - 76548 * q^71 + 47304 * q^73 + 23968 * q^74 + 33408 * q^76 + 140656 * q^79 - 27648 * q^80 + 80784 * q^82 - 94104 * q^83 + 57868 * q^85 + 4544 * q^86 - 7936 * q^88 + 17604 * q^89 - 12352 * q^92 + 147744 * q^94 - 91592 * q^95 + 85176 * q^97

Introduction

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