LMFDB - Embedded newform 441.6.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.358541904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 111x^{2} + 756 $ x^4 - 111*x^2 + 756
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.69991$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69991 q^{2} -24.7105 q^{4} +87.9366 q^{5} +153.113 q^{8} +O(q^{10})$	$q-2.69991 q^{2} -24.7105 q^{4} +87.9366 q^{5} +153.113 q^{8} -237.421 q^{10} -41.6592 q^{11} -989.368 q^{13} +377.343 q^{16} +2121.25 q^{17} -928.632 q^{19} -2172.95 q^{20} +112.476 q^{22} +4392.21 q^{23} +4607.84 q^{25} +2671.21 q^{26} +3893.93 q^{29} -1134.10 q^{31} -5918.42 q^{32} -5727.20 q^{34} -10056.5 q^{37} +2507.23 q^{38} +13464.3 q^{40} +131.039 q^{41} -3534.94 q^{43} +1029.42 q^{44} -11858.6 q^{46} -16072.6 q^{47} -12440.8 q^{50} +24447.7 q^{52} -2959.15 q^{53} -3663.37 q^{55} -10513.3 q^{58} -18324.9 q^{59} -14896.5 q^{61} +3061.98 q^{62} +3904.23 q^{64} -87001.6 q^{65} +55504.2 q^{67} -52417.2 q^{68} +60717.8 q^{71} +49613.7 q^{73} +27151.7 q^{74} +22946.9 q^{76} +40997.4 q^{79} +33182.2 q^{80} -353.794 q^{82} +34040.1 q^{83} +186536. q^{85} +9544.03 q^{86} -6378.58 q^{88} +5098.59 q^{89} -108534. q^{92} +43394.5 q^{94} -81660.7 q^{95} -98414.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 94 q^{4}+O(q^{10}) $ 4 * q + 94 * q^4	$ 4 q + 94 q^{4} - 564 q^{10} - 872 q^{13} + 4402 q^{16} - 6800 q^{19} + 10092 q^{22} + 3004 q^{25} + 4720 q^{31} + 32244 q^{34} + 6056 q^{37} + 23388 q^{40} + 10544 q^{43} - 47820 q^{46} + 128260 q^{52} - 11568 q^{55} - 106848 q^{58} - 13304 q^{61} + 5782 q^{64} + 203504 q^{67} - 17528 q^{73} - 308552 q^{76} + 34400 q^{79} + 182556 q^{82} + 354288 q^{85} + 379068 q^{88} + 561576 q^{94} - 436856 q^{97}+O(q^{100}) $ 4 * q + 94 * q^4 - 564 * q^10 - 872 * q^13 + 4402 * q^16 - 6800 * q^19 + 10092 * q^22 + 3004 * q^25 + 4720 * q^31 + 32244 * q^34 + 6056 * q^37 + 23388 * q^40 + 10544 * q^43 - 47820 * q^46 + 128260 * q^52 - 11568 * q^55 - 106848 * q^58 - 13304 * q^61 + 5782 * q^64 + 203504 * q^67 - 17528 * q^73 - 308552 * q^76 + 34400 * q^79 + 182556 * q^82 + 354288 * q^85 + 379068 * q^88 + 561576 * q^94 - 436856 * 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 $

$2$	−2.69991	−0.477281	−0.238641	−	0.971108i	$-0.576702\pi$
$2$	−2.69991	−0.477281	−0.238641	+	0.971108i	$0.576702\pi$
$3$	0	0
$3$	0	0
$4$	−24.7105	−0.772202
$5$	87.9366	1.57306	0.786528	−	0.617554i	$-0.211876\pi$
$5$	87.9366	1.57306	0.786528	+	0.617554i	$0.211876\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	153.113	0.845839
$9$	0	0
$10$	−237.421	−0.750791
$11$	−41.6592	−0.103808	−0.0519038	−	0.998652i	$-0.516529\pi$
$11$	−41.6592	−0.103808	−0.0519038	+	0.998652i	$0.516529\pi$
$12$	0	0
$13$	−989.368	−1.62368	−0.811838	−	0.583883i	$-0.801533\pi$
$13$	−989.368	−1.62368	−0.811838	+	0.583883i	$0.801533\pi$
$14$	0	0
$15$	0	0
$16$	377.343	0.368499
$17$	2121.25	1.78021	0.890103	−	0.455760i	$-0.150632\pi$
$17$	2121.25	1.78021	0.890103	+	0.455760i	$0.150632\pi$
$18$	0	0
$19$	−928.632	−0.590146	−0.295073	−	0.955475i	$-0.595344\pi$
$19$	−928.632	−0.590146	−0.295073	+	0.955475i	$0.595344\pi$
$20$	−2172.95	−1.21472
$21$	0	0
$22$	112.476	0.0495455
$23$	4392.21	1.73126	0.865632	−	0.500680i	$-0.166917\pi$
$23$	4392.21	1.73126	0.865632	+	0.500680i	$0.166917\pi$
$24$	0	0
$25$	4607.84	1.47451
$26$	2671.21	0.774950
$27$	0	0
$28$	0	0
$29$	3893.93	0.859793	0.429896	−	0.902878i	$-0.358550\pi$
$29$	3893.93	0.859793	0.429896	+	0.902878i	$0.358550\pi$
$30$	0	0
$31$	−1134.10	−0.211957	−0.105979	−	0.994368i	$-0.533798\pi$
$31$	−1134.10	−0.211957	−0.105979	+	0.994368i	$0.533798\pi$
$32$	−5918.42	−1.02172
$33$	0	0
$34$	−5727.20	−0.849659
$35$	0	0
$36$	0	0
$37$	−10056.5	−1.20766	−0.603828	−	0.797115i	$-0.706359\pi$
$37$	−10056.5	−1.20766	−0.603828	+	0.797115i	$0.706359\pi$
$38$	2507.23	0.281666
$39$	0	0
$40$	13464.3	1.33055
$41$	131.039	0.0121742	0.00608711	−	0.999981i	$-0.498062\pi$
$41$	131.039	0.0121742	0.00608711	+	0.999981i	$0.498062\pi$
$42$	0	0
$43$	−3534.94	−0.291549	−0.145774	−	0.989318i	$-0.546567\pi$
$43$	−3534.94	−0.291549	−0.145774	+	0.989318i	$0.546567\pi$
$44$	1029.42	0.0801605
$45$	0	0
$46$	−11858.6	−0.826301
$47$	−16072.6	−1.06131	−0.530653	−	0.847589i	$-0.678053\pi$
$47$	−16072.6	−1.06131	−0.530653	+	0.847589i	$0.678053\pi$
$48$	0	0
$49$	0	0
$50$	−12440.8	−0.703755
$51$	0	0
$52$	24447.7	1.25381
$53$	−2959.15	−0.144703	−0.0723515	−	0.997379i	$-0.523050\pi$
$53$	−2959.15	−0.144703	−0.0723515	+	0.997379i	$0.523050\pi$
$54$	0	0
$55$	−3663.37	−0.163295
$56$	0	0
$57$	0	0
$58$	−10513.3	−0.410363
$59$	−18324.9	−0.685347	−0.342674	−	0.939455i	$-0.611333\pi$
$59$	−18324.9	−0.685347	−0.342674	+	0.939455i	$0.611333\pi$
$60$	0	0
$61$	−14896.5	−0.512578	−0.256289	−	0.966600i	$-0.582500\pi$
$61$	−14896.5	−0.512578	−0.256289	+	0.966600i	$0.582500\pi$
$62$	3061.98	0.101163
$63$	0	0
$64$	3904.23	0.119148
$65$	−87001.6	−2.55413
$66$	0	0
$67$	55504.2	1.51056	0.755282	−	0.655400i	$-0.227500\pi$
$67$	55504.2	1.51056	0.755282	+	0.655400i	$0.227500\pi$
$68$	−52417.2	−1.37468
$69$	0	0
$70$	0	0
$71$	60717.8	1.42945	0.714727	−	0.699404i	$-0.246551\pi$
$71$	60717.8	1.42945	0.714727	+	0.699404i	$0.246551\pi$
$72$	0	0
$73$	49613.7	1.08967	0.544835	−	0.838543i	$-0.316592\pi$
$73$	49613.7	1.08967	0.544835	+	0.838543i	$0.316592\pi$
$74$	27151.7	0.576392
$75$	0	0
$76$	22946.9	0.455713
$77$	0	0
$78$	0	0
$79$	40997.4	0.739076	0.369538	−	0.929216i	$-0.379516\pi$
$79$	40997.4	0.739076	0.369538	+	0.929216i	$0.379516\pi$
$80$	33182.2	0.579670
$81$	0	0
$82$	−353.794	−0.00581053
$83$	34040.1	0.542370	0.271185	−	0.962527i	$-0.412584\pi$
$83$	34040.1	0.542370	0.271185	+	0.962527i	$0.412584\pi$
$84$	0	0
$85$	186536.	2.80036
$86$	9544.03	0.139151
$87$	0	0
$88$	−6378.58	−0.0878046
$89$	5098.59	0.0682299	0.0341150	−	0.999418i	$-0.489139\pi$
$89$	5098.59	0.0682299	0.0341150	+	0.999418i	$0.489139\pi$
$90$	0	0
$91$	0	0
$92$	−108534.	−1.33689
$93$	0	0
$94$	43394.5	0.506542
$95$	−81660.7	−0.928334
$96$	0	0
$97$	−98414.9	−1.06202	−0.531008	−	0.847367i	$-0.678187\pi$
$97$	−98414.9	−1.06202	−0.531008	+	0.847367i	$0.678187\pi$
$98$	0	0
$99$	0	0
$100$	−113862.	−1.13862
$101$	2468.76	0.0240811	0.0120405	−	0.999928i	$-0.496167\pi$
$101$	2468.76	0.0240811	0.0120405	+	0.999928i	$0.496167\pi$
$102$	0	0
$103$	67609.4	0.627934	0.313967	−	0.949434i	$-0.398342\pi$
$103$	67609.4	0.627934	0.313967	+	0.949434i	$0.398342\pi$
$104$	−151485.	−1.37337
$105$	0	0
$106$	7989.45	0.0690641
$107$	169653.	1.43253	0.716264	−	0.697829i	$-0.245851\pi$
$107$	169653.	1.43253	0.716264	+	0.697829i	$0.245851\pi$
$108$	0	0
$109$	88433.1	0.712933	0.356466	−	0.934308i	$-0.383981\pi$
$109$	88433.1	0.712933	0.356466	+	0.934308i	$0.383981\pi$
$110$	9890.77	0.0779378
$111$	0	0
$112$	0	0
$113$	195325.	1.43901	0.719503	−	0.694489i	$-0.244370\pi$
$113$	195325.	1.43901	0.719503	+	0.694489i	$0.244370\pi$
$114$	0	0
$115$	386236.	2.72338
$116$	−96221.0	−0.663934
$117$	0	0
$118$	49475.5	0.327104
$119$	0	0
$120$	0	0
$121$	−159316.	−0.989224
$122$	40219.3	0.244644
$123$	0	0
$124$	28024.2	0.163674
$125$	130396.	0.746428
$126$	0	0
$127$	−61155.8	−0.336456	−0.168228	−	0.985748i	$-0.553804\pi$
$127$	−61155.8	−0.336456	−0.168228	+	0.985748i	$0.553804\pi$
$128$	178848.	0.964850
$129$	0	0
$130$	234897.	1.21904
$131$	288044.	1.46649	0.733247	−	0.679962i	$-0.238004\pi$
$131$	288044.	1.46649	0.733247	+	0.679962i	$0.238004\pi$
$132$	0	0
$133$	0	0
$134$	−149856.	−0.720964
$135$	0	0
$136$	324792.	1.50577
$137$	273332.	1.24420	0.622099	−	0.782938i	$-0.286280\pi$
$137$	273332.	1.24420	0.622099	+	0.782938i	$0.286280\pi$
$138$	0	0
$139$	−288985.	−1.26864	−0.634320	−	0.773071i	$-0.718720\pi$
$139$	−288985.	−1.26864	−0.634320	+	0.773071i	$0.718720\pi$
$140$	0	0
$141$	0	0
$142$	−163933.	−0.682252
$143$	41216.3	0.168550
$144$	0	0
$145$	342419.	1.35250
$146$	−133953.	−0.520079
$147$	0	0
$148$	248501.	0.932555
$149$	−191595.	−0.706999	−0.353499	−	0.935435i	$-0.615008\pi$
$149$	−191595.	−0.706999	−0.353499	+	0.935435i	$0.615008\pi$
$150$	0	0
$151$	175529.	0.626478	0.313239	−	0.949674i	$-0.398586\pi$
$151$	175529.	0.626478	0.313239	+	0.949674i	$0.398586\pi$
$152$	−142186.	−0.499169
$153$	0	0
$154$	0	0
$155$	−99729.1	−0.333421
$156$	0	0
$157$	67098.4	0.217252	0.108626	−	0.994083i	$-0.465355\pi$
$157$	67098.4	0.217252	0.108626	+	0.994083i	$0.465355\pi$
$158$	−110689.	−0.352747
$159$	0	0
$160$	−520445.	−1.60722
$161$	0	0
$162$	0	0
$163$	−549231.	−1.61915	−0.809573	−	0.587019i	$-0.800301\pi$
$163$	−549231.	−1.61915	−0.809573	+	0.587019i	$0.800301\pi$
$164$	−3238.04	−0.00940097
$165$	0	0
$166$	−91905.3	−0.258863
$167$	244929.	0.679592	0.339796	−	0.940499i	$-0.389642\pi$
$167$	244929.	0.679592	0.339796	+	0.940499i	$0.389642\pi$
$168$	0	0
$169$	607555.	1.63632
$170$	−503630.	−1.33656
$171$	0	0
$172$	87350.1	0.225135
$173$	−44848.6	−0.113929	−0.0569644	−	0.998376i	$-0.518142\pi$
$173$	−44848.6	−0.113929	−0.0569644	+	0.998376i	$0.518142\pi$
$174$	0	0
$175$	0	0
$176$	−15719.8	−0.0382530
$177$	0	0
$178$	−13765.7	−0.0325649
$179$	442204.	1.03155	0.515775	−	0.856724i	$-0.327504\pi$
$179$	442204.	1.03155	0.515775	+	0.856724i	$0.327504\pi$
$180$	0	0
$181$	215860.	0.489752	0.244876	−	0.969554i	$-0.421253\pi$
$181$	215860.	0.489752	0.244876	+	0.969554i	$0.421253\pi$
$182$	0	0
$183$	0	0
$184$	672506.	1.46437
$185$	−884335.	−1.89971
$186$	0	0
$187$	−88369.7	−0.184799
$188$	397161.	0.819543
$189$	0	0
$190$	220477.	0.443077
$191$	187532.	0.371957	0.185978	−	0.982554i	$-0.440455\pi$
$191$	187532.	0.371957	0.185978	+	0.982554i	$0.440455\pi$
$192$	0	0
$193$	735933.	1.42215	0.711075	−	0.703117i	$-0.248209\pi$
$193$	735933.	1.42215	0.711075	+	0.703117i	$0.248209\pi$
$194$	265711.	0.506881
$195$	0	0
$196$	0	0
$197$	−195426.	−0.358770	−0.179385	−	0.983779i	$-0.557411\pi$
$197$	−195426.	−0.358770	−0.179385	+	0.983779i	$0.557411\pi$
$198$	0	0
$199$	966468.	1.73003	0.865017	−	0.501742i	$-0.167307\pi$
$199$	966468.	1.73003	0.865017	+	0.501742i	$0.167307\pi$
$200$	705521.	1.24720
$201$	0	0
$202$	−6665.44	−0.0114934
$203$	0	0
$204$	0	0
$205$	11523.1	0.0191508
$206$	−182539.	−0.299701
$207$	0	0
$208$	−373331.	−0.598323
$209$	38686.1	0.0612617
$210$	0	0
$211$	−819549.	−1.26727	−0.633634	−	0.773633i	$-0.718438\pi$
$211$	−819549.	−1.26727	−0.633634	+	0.773633i	$0.718438\pi$
$212$	73122.0	0.111740
$213$	0	0
$214$	−458049.	−0.683719
$215$	−310851.	−0.458623
$216$	0	0
$217$	0	0
$218$	−238762.	−0.340270
$219$	0	0
$220$	90523.6	0.126097
$221$	−2.09870e6	−2.89048
$222$	0	0
$223$	953789.	1.28437	0.642185	−	0.766550i	$-0.278028\pi$
$223$	953789.	1.28437	0.642185	+	0.766550i	$0.278028\pi$
$224$	0	0
$225$	0	0
$226$	−527361.	−0.686811
$227$	1.47007e6	1.89354	0.946770	−	0.321912i	$-0.104326\pi$
$227$	1.47007e6	1.89354	0.946770	+	0.321912i	$0.104326\pi$
$228$	0	0
$229$	621804.	0.783547	0.391773	−	0.920062i	$-0.371862\pi$
$229$	621804.	0.783547	0.391773	+	0.920062i	$0.371862\pi$
$230$	−1.04280e6	−1.29982
$231$	0	0
$232$	596213.	0.727246
$233$	1.13371e6	1.36808	0.684042	−	0.729443i	$-0.260221\pi$
$233$	1.13371e6	1.36808	0.684042	+	0.729443i	$0.260221\pi$
$234$	0	0
$235$	−1.41337e6	−1.66950
$236$	452816.	0.529227
$237$	0	0
$238$	0	0
$239$	66697.1	0.0755287	0.0377644	−	0.999287i	$-0.487976\pi$
$239$	66697.1	0.0755287	0.0377644	+	0.999287i	$0.487976\pi$
$240$	0	0
$241$	−662390.	−0.734634	−0.367317	−	0.930096i	$-0.619724\pi$
$241$	−662390.	−0.734634	−0.367317	+	0.930096i	$0.619724\pi$
$242$	430138.	0.472138
$243$	0	0
$244$	368100.	0.395814
$245$	0	0
$246$	0	0
$247$	918759.	0.958207
$248$	−173646.	−0.179282
$249$	0	0
$250$	−352057.	−0.356256
$251$	−256918.	−0.257401	−0.128700	−	0.991684i	$-0.541081\pi$
$251$	−256918.	−0.257401	−0.128700	+	0.991684i	$0.541081\pi$
$252$	0	0
$253$	−182976.	−0.179719
$254$	165115.	0.160584
$255$	0	0
$256$	−607810.	−0.579653
$257$	167500.	0.158191	0.0790957	−	0.996867i	$-0.474797\pi$
$257$	167500.	0.158191	0.0790957	+	0.996867i	$0.474797\pi$
$258$	0	0
$259$	0	0
$260$	2.14985e6	1.97231
$261$	0	0
$262$	−777693.	−0.699931
$263$	1.30152e6	1.16028	0.580138	−	0.814518i	$-0.302999\pi$
$263$	1.30152e6	1.16028	0.580138	+	0.814518i	$0.302999\pi$
$264$	0	0
$265$	−260218.	−0.227626
$266$	0	0
$267$	0	0
$268$	−1.37154e6	−1.16646
$269$	1.67615e6	1.41232	0.706158	−	0.708054i	$-0.250427\pi$
$269$	1.67615e6	1.41232	0.706158	+	0.708054i	$0.250427\pi$
$270$	0	0
$271$	704119.	0.582402	0.291201	−	0.956662i	$-0.405945\pi$
$271$	704119.	0.582402	0.291201	+	0.956662i	$0.405945\pi$
$272$	800440.	0.656004
$273$	0	0
$274$	−737973.	−0.593833
$275$	−191959.	−0.153065
$276$	0	0
$277$	1.19116e6	0.932760	0.466380	−	0.884585i	$-0.345558\pi$
$277$	1.19116e6	0.932760	0.466380	+	0.884585i	$0.345558\pi$
$278$	780234.	0.605498
$279$	0	0
$280$	0	0
$281$	−473761.	−0.357926	−0.178963	−	0.983856i	$-0.557274\pi$
$281$	−473761.	−0.357926	−0.178963	+	0.983856i	$0.557274\pi$
$282$	0	0
$283$	−2.35457e6	−1.74761	−0.873805	−	0.486276i	$-0.838355\pi$
$283$	−2.35457e6	−1.74761	−0.873805	+	0.486276i	$0.838355\pi$
$284$	−1.50037e6	−1.10383
$285$	0	0
$286$	−111280.	−0.0804458
$287$	0	0
$288$	0	0
$289$	3.07986e6	2.16913
$290$	−924502.	−0.645524
$291$	0	0
$292$	−1.22598e6	−0.841445
$293$	−1.49266e6	−1.01576	−0.507880	−	0.861428i	$-0.669571\pi$
$293$	−1.49266e6	−1.01576	−0.507880	+	0.861428i	$0.669571\pi$
$294$	0	0
$295$	−1.61143e6	−1.07809
$296$	−1.53979e6	−1.02148
$297$	0	0
$298$	517290.	0.337437
$299$	−4.34551e6	−2.81101
$300$	0	0
$301$	0	0
$302$	−473912.	−0.299006
$303$	0	0
$304$	−350413.	−0.217468
$305$	−1.30995e6	−0.806315
$306$	0	0
$307$	−588357.	−0.356283	−0.178141	−	0.984005i	$-0.557008\pi$
$307$	−588357.	−0.356283	−0.178141	+	0.984005i	$0.557008\pi$
$308$	0	0
$309$	0	0
$310$	269260.	0.159136
$311$	−1.37662e6	−0.807075	−0.403537	−	0.914963i	$-0.632219\pi$
$311$	−1.37662e6	−0.807075	−0.403537	+	0.914963i	$0.632219\pi$
$312$	0	0
$313$	−2.20183e6	−1.27035	−0.635175	−	0.772368i	$-0.719072\pi$
$313$	−2.20183e6	−1.27035	−0.635175	+	0.772368i	$0.719072\pi$
$314$	−181160.	−0.103690
$315$	0	0
$316$	−1.01307e6	−0.570716
$317$	−1.30372e6	−0.728677	−0.364338	−	0.931267i	$-0.618705\pi$
$317$	−1.30372e6	−0.728677	−0.364338	+	0.931267i	$0.618705\pi$
$318$	0	0
$319$	−162218.	−0.0892530
$320$	343325.	0.187426
$321$	0	0
$322$	0	0
$323$	−1.96986e6	−1.05058
$324$	0	0
$325$	−4.55885e6	−2.39412
$326$	1.48288e6	0.772788
$327$	0	0
$328$	20063.8	0.0102974
$329$	0	0
$330$	0	0
$331$	−498571.	−0.250125	−0.125062	−	0.992149i	$-0.539913\pi$
$331$	−498571.	−0.250125	−0.125062	+	0.992149i	$0.539913\pi$
$332$	−841147.	−0.418819
$333$	0	0
$334$	−661285.	−0.324357
$335$	4.88085e6	2.37620
$336$	0	0
$337$	−1.97199e6	−0.945864	−0.472932	−	0.881099i	$-0.656804\pi$
$337$	−1.97199e6	−0.945864	−0.472932	+	0.881099i	$0.656804\pi$
$338$	−1.64035e6	−0.780987
$339$	0	0
$340$	−4.60939e6	−2.16245
$341$	47245.8	0.0220028
$342$	0	0
$343$	0	0
$344$	−541246.	−0.246603
$345$	0	0
$346$	121087.	0.0543761
$347$	736785.	0.328486	0.164243	−	0.986420i	$-0.447482\pi$
$347$	736785.	0.328486	0.164243	+	0.986420i	$0.447482\pi$
$348$	0	0
$349$	−821566.	−0.361060	−0.180530	−	0.983570i	$-0.557781\pi$
$349$	−821566.	−0.361060	−0.180530	+	0.983570i	$0.557781\pi$
$350$	0	0
$351$	0	0
$352$	246557.	0.106062
$353$	−1.61612e6	−0.690299	−0.345149	−	0.938548i	$-0.612172\pi$
$353$	−1.61612e6	−0.690299	−0.345149	+	0.938548i	$0.612172\pi$
$354$	0	0
$355$	5.33931e6	2.24861
$356$	−125989.	−0.0526873
$357$	0	0
$358$	−1.19391e6	−0.492340
$359$	−385547.	−0.157885	−0.0789425	−	0.996879i	$-0.525154\pi$
$359$	−385547.	−0.157885	−0.0789425	+	0.996879i	$0.525154\pi$
$360$	0	0
$361$	−1.61374e6	−0.651727
$362$	−582803.	−0.233749
$363$	0	0
$364$	0	0
$365$	4.36286e6	1.71411
$366$	0	0
$367$	−3.21688e6	−1.24672	−0.623362	−	0.781934i	$-0.714234\pi$
$367$	−3.21688e6	−1.24672	−0.623362	+	0.781934i	$0.714234\pi$
$368$	1.65737e6	0.637969
$369$	0	0
$370$	2.38763e6	0.906697
$371$	0	0
$372$	0	0
$373$	1.41089e6	0.525074	0.262537	−	0.964922i	$-0.415441\pi$
$373$	1.41089e6	0.525074	0.262537	+	0.964922i	$0.415441\pi$
$374$	238590.	0.0882011
$375$	0	0
$376$	−2.46092e6	−0.897695
$377$	−3.85253e6	−1.39602
$378$	0	0
$379$	−297560.	−0.106409	−0.0532043	−	0.998584i	$-0.516943\pi$
$379$	−297560.	−0.106409	−0.0532043	+	0.998584i	$0.516943\pi$
$380$	2.01788e6	0.716862
$381$	0	0
$382$	−506320.	−0.177528
$383$	2.08516e6	0.726345	0.363173	−	0.931722i	$-0.381694\pi$
$383$	2.08516e6	0.726345	0.363173	+	0.931722i	$0.381694\pi$
$384$	0	0
$385$	0	0
$386$	−1.98695e6	−0.678765
$387$	0	0
$388$	2.43188e6	0.820092
$389$	−3.60863e6	−1.20912	−0.604559	−	0.796560i	$-0.706651\pi$
$389$	−3.60863e6	−1.20912	−0.604559	+	0.796560i	$0.706651\pi$
$390$	0	0
$391$	9.31699e6	3.08201
$392$	0	0
$393$	0	0
$394$	527632.	0.171234
$395$	3.60517e6	1.16261
$396$	0	0
$397$	−3.04068e6	−0.968265	−0.484132	−	0.874995i	$-0.660865\pi$
$397$	−3.04068e6	−0.968265	−0.484132	+	0.874995i	$0.660865\pi$
$398$	−2.60938e6	−0.825713
$399$	0	0
$400$	1.73873e6	0.543355
$401$	3.51735e6	1.09233	0.546167	−	0.837677i	$-0.316087\pi$
$401$	3.51735e6	1.09233	0.546167	+	0.837677i	$0.316087\pi$
$402$	0	0
$403$	1.12204e6	0.344150
$404$	−61004.3	−0.0185955
$405$	0	0
$406$	0	0
$407$	418946.	0.125364
$408$	0	0
$409$	−1.88540e6	−0.557307	−0.278654	−	0.960392i	$-0.589888\pi$
$409$	−1.88540e6	−0.557307	−0.278654	+	0.960392i	$0.589888\pi$
$410$	−31111.4	−0.00914030
$411$	0	0
$412$	−1.67066e6	−0.484892
$413$	0	0
$414$	0	0
$415$	2.99337e6	0.853179
$416$	5.85549e6	1.65894
$417$	0	0
$418$	−104449.	−0.0292391
$419$	2.23298e6	0.621369	0.310685	−	0.950513i	$-0.399442\pi$
$419$	2.23298e6	0.621369	0.310685	+	0.950513i	$0.399442\pi$
$420$	0	0
$421$	−4.33567e6	−1.19221	−0.596103	−	0.802908i	$-0.703285\pi$
$421$	−4.33567e6	−1.19221	−0.596103	+	0.802908i	$0.703285\pi$
$422$	2.21271e6	0.604844
$423$	0	0
$424$	−453085.	−0.122395
$425$	9.77439e6	2.62493
$426$	0	0
$427$	0	0
$428$	−4.19222e6	−1.10620
$429$	0	0
$430$	839269.	0.218892
$431$	6.40103e6	1.65980	0.829901	−	0.557910i	$-0.188397\pi$
$431$	6.40103e6	1.65980	0.829901	+	0.557910i	$0.188397\pi$
$432$	0	0
$433$	−2.41837e6	−0.619872	−0.309936	−	0.950757i	$-0.600308\pi$
$433$	−2.41837e6	−0.619872	−0.309936	+	0.950757i	$0.600308\pi$
$434$	0	0
$435$	0	0
$436$	−2.18522e6	−0.550528
$437$	−4.07875e6	−1.02170
$438$	0	0
$439$	2.01835e6	0.499845	0.249923	−	0.968266i	$-0.419595\pi$
$439$	2.01835e6	0.499845	0.249923	+	0.968266i	$0.419595\pi$
$440$	−560910.	−0.138122
$441$	0	0
$442$	5.66630e6	1.37957
$443$	4.51656e6	1.09345	0.546725	−	0.837312i	$-0.315874\pi$
$443$	4.51656e6	1.09345	0.546725	+	0.837312i	$0.315874\pi$
$444$	0	0
$445$	448352.	0.107330
$446$	−2.57515e6	−0.613006
$447$	0	0
$448$	0	0
$449$	3.63815e6	0.851658	0.425829	−	0.904804i	$-0.359983\pi$
$449$	3.63815e6	0.851658	0.425829	+	0.904804i	$0.359983\pi$
$450$	0	0
$451$	−5458.99	−0.00126378
$452$	−4.82658e6	−1.11120
$453$	0	0
$454$	−3.96907e6	−0.903751
$455$	0	0
$456$	0	0
$457$	3.89454e6	0.872299	0.436150	−	0.899874i	$-0.356342\pi$
$457$	3.89454e6	0.872299	0.436150	+	0.899874i	$0.356342\pi$
$458$	−1.67882e6	−0.373972
$459$	0	0
$460$	−9.54407e6	−2.10300
$461$	959953.	0.210377	0.105188	−	0.994452i	$-0.466455\pi$
$461$	959953.	0.210377	0.105188	+	0.994452i	$0.466455\pi$
$462$	0	0
$463$	514391.	0.111517	0.0557584	−	0.998444i	$-0.482242\pi$
$463$	514391.	0.111517	0.0557584	+	0.998444i	$0.482242\pi$
$464$	1.46935e6	0.316833
$465$	0	0
$466$	−3.06092e6	−0.652961
$467$	1.64057e6	0.348098	0.174049	−	0.984737i	$-0.444315\pi$
$467$	1.64057e6	0.348098	0.174049	+	0.984737i	$0.444315\pi$
$468$	0	0
$469$	0	0
$470$	3.81596e6	0.796819
$471$	0	0
$472$	−2.80578e6	−0.579694
$473$	147263.	0.0302650
$474$	0	0
$475$	−4.27899e6	−0.870176
$476$	0	0
$477$	0	0
$478$	−180076.	−0.0360485
$479$	−3.82165e6	−0.761048	−0.380524	−	0.924771i	$-0.624256\pi$
$479$	−3.82165e6	−0.761048	−0.380524	+	0.924771i	$0.624256\pi$
$480$	0	0
$481$	9.94959e6	1.96084
$482$	1.78839e6	0.350627
$483$	0	0
$484$	3.93676e6	0.763881
$485$	−8.65426e6	−1.67061
$486$	0	0
$487$	6.42923e6	1.22839	0.614196	−	0.789154i	$-0.289480\pi$
$487$	6.42923e6	1.22839	0.614196	+	0.789154i	$0.289480\pi$
$488$	−2.28085e6	−0.433559
$489$	0	0
$490$	0	0
$491$	−7.38701e6	−1.38282	−0.691409	−	0.722464i	$-0.743010\pi$
$491$	−7.38701e6	−1.38282	−0.691409	+	0.722464i	$0.743010\pi$
$492$	0	0
$493$	8.26002e6	1.53061
$494$	−2.48057e6	−0.457334
$495$	0	0
$496$	−427946.	−0.0781060
$497$	0	0
$498$	0	0
$499$	−3.90542e6	−0.702129	−0.351064	−	0.936351i	$-0.614180\pi$
$499$	−3.90542e6	−0.702129	−0.351064	+	0.936351i	$0.614180\pi$
$500$	−3.22214e6	−0.576394
$501$	0	0
$502$	693655.	0.122853
$503$	4.00168e6	0.705216	0.352608	−	0.935771i	$-0.385295\pi$
$503$	4.00168e6	0.705216	0.352608	+	0.935771i	$0.385295\pi$
$504$	0	0
$505$	217094.	0.0378809
$506$	494019.	0.0857763
$507$	0	0
$508$	1.51119e6	0.259812
$509$	−6.62556e6	−1.13352	−0.566759	−	0.823884i	$-0.691803\pi$
$509$	−6.62556e6	−1.13352	−0.566759	+	0.823884i	$0.691803\pi$
$510$	0	0
$511$	0	0
$512$	−4.08211e6	−0.688192
$513$	0	0
$514$	−452236.	−0.0755018
$515$	5.94534e6	0.987776
$516$	0	0
$517$	669571.	0.110172
$518$	0	0
$519$	0	0
$520$	−1.33211e7	−2.16039
$521$	−1.88364e6	−0.304021	−0.152010	−	0.988379i	$-0.548575\pi$
$521$	−1.88364e6	−0.304021	−0.152010	+	0.988379i	$0.548575\pi$
$522$	0	0
$523$	9.13549e6	1.46042	0.730210	−	0.683223i	$-0.239422\pi$
$523$	9.13549e6	1.46042	0.730210	+	0.683223i	$0.239422\pi$
$524$	−7.11770e6	−1.13243
$525$	0	0
$526$	−3.51399e6	−0.553778
$527$	−2.40572e6	−0.377327
$528$	0	0
$529$	1.28552e7	1.99728
$530$	702565.	0.108642
$531$	0	0
$532$	0	0
$533$	−129646.	−0.0197670
$534$	0	0
$535$	1.49187e7	2.25345
$536$	8.49843e6	1.27769
$537$	0	0
$538$	−4.52545e6	−0.674072
$539$	0	0
$540$	0	0
$541$	−4.95445e6	−0.727783	−0.363891	−	0.931441i	$-0.618552\pi$
$541$	−4.95445e6	−0.727783	−0.363891	+	0.931441i	$0.618552\pi$
$542$	−1.90106e6	−0.277970
$543$	0	0
$544$	−1.25545e7	−1.81887
$545$	7.77650e6	1.12148
$546$	0	0
$547$	1.48240e6	0.211835	0.105917	−	0.994375i	$-0.466222\pi$
$547$	1.48240e6	0.211835	0.105917	+	0.994375i	$0.466222\pi$
$548$	−6.75417e6	−0.960773
$549$	0	0
$550$	518272.	0.0730552
$551$	−3.61603e6	−0.507404
$552$	0	0
$553$	0	0
$554$	−3.21602e6	−0.445189
$555$	0	0
$556$	7.14095e6	0.979646
$557$	−5.50541e6	−0.751885	−0.375942	−	0.926643i	$-0.622681\pi$
$557$	−5.50541e6	−0.751885	−0.375942	+	0.926643i	$0.622681\pi$
$558$	0	0
$559$	3.49736e6	0.473380
$560$	0	0
$561$	0	0
$562$	1.27911e6	0.170831
$563$	1.06459e7	1.41551	0.707754	−	0.706459i	$-0.249708\pi$
$563$	1.06459e7	1.41551	0.707754	+	0.706459i	$0.249708\pi$
$564$	0	0
$565$	1.71762e7	2.26364
$566$	6.35712e6	0.834102
$567$	0	0
$568$	9.29670e6	1.20909
$569$	−1.25123e7	−1.62016	−0.810080	−	0.586319i	$-0.800576\pi$
$569$	−1.25123e7	−1.62016	−0.810080	+	0.586319i	$0.800576\pi$
$570$	0	0
$571$	−4.03238e6	−0.517573	−0.258786	−	0.965935i	$-0.583323\pi$
$571$	−4.03238e6	−0.517573	−0.258786	+	0.965935i	$0.583323\pi$
$572$	−1.01847e6	−0.130155
$573$	0	0
$574$	0	0
$575$	2.02386e7	2.55276
$576$	0	0
$577$	3.15333e6	0.394303	0.197151	−	0.980373i	$-0.436831\pi$
$577$	3.15333e6	0.394303	0.197151	+	0.980373i	$0.436831\pi$
$578$	−8.31534e6	−1.03529
$579$	0	0
$580$	−8.46134e6	−1.04441
$581$	0	0
$582$	0	0
$583$	123276.	0.0150213
$584$	7.59652e6	0.921685
$585$	0	0
$586$	4.03005e6	0.484804
$587$	−1.50298e7	−1.80036	−0.900179	−	0.435520i	$-0.856564\pi$
$587$	−1.50298e7	−1.80036	−0.900179	+	0.435520i	$0.856564\pi$
$588$	0	0
$589$	1.05316e6	0.125086
$590$	4.35071e6	0.514553
$591$	0	0
$592$	−3.79475e6	−0.445020
$593$	2.57952e6	0.301233	0.150616	−	0.988592i	$-0.451874\pi$
$593$	2.57952e6	0.301233	0.150616	+	0.988592i	$0.451874\pi$
$594$	0	0
$595$	0	0
$596$	4.73440e6	0.545946
$597$	0	0
$598$	1.17325e7	1.34164
$599$	1.27808e6	0.145543	0.0727713	−	0.997349i	$-0.476816\pi$
$599$	1.27808e6	0.145543	0.0727713	+	0.997349i	$0.476816\pi$
$600$	0	0
$601$	−6.74349e6	−0.761550	−0.380775	−	0.924668i	$-0.624343\pi$
$601$	−6.74349e6	−0.761550	−0.380775	+	0.924668i	$0.624343\pi$
$602$	0	0
$603$	0	0
$604$	−4.33740e6	−0.483768
$605$	−1.40097e7	−1.55611
$606$	0	0
$607$	−1.02861e7	−1.13313	−0.566566	−	0.824017i	$-0.691728\pi$
$607$	−1.02861e7	−1.13313	−0.566566	+	0.824017i	$0.691728\pi$
$608$	5.49603e6	0.602963
$609$	0	0
$610$	3.53674e6	0.384839
$611$	1.59017e7	1.72322
$612$	0	0
$613$	3.94586e6	0.424122	0.212061	−	0.977256i	$-0.431983\pi$
$613$	3.94586e6	0.424122	0.212061	+	0.977256i	$0.431983\pi$
$614$	1.58851e6	0.170047
$615$	0	0
$616$	0	0
$617$	−1.19763e7	−1.26652	−0.633259	−	0.773940i	$-0.718283\pi$
$617$	−1.19763e7	−1.26652	−0.633259	+	0.773940i	$0.718283\pi$
$618$	0	0
$619$	−1.11019e7	−1.16458	−0.582291	−	0.812980i	$-0.697844\pi$
$619$	−1.11019e7	−1.16458	−0.582291	+	0.812980i	$0.697844\pi$
$620$	2.46435e6	0.257468
$621$	0	0
$622$	3.71676e6	0.385202
$623$	0	0
$624$	0	0
$625$	−2.93295e6	−0.300334
$626$	5.94475e6	0.606315
$627$	0	0
$628$	−1.65803e6	−0.167762
$629$	−2.13324e7	−2.14988
$630$	0	0
$631$	3.99794e6	0.399726	0.199863	−	0.979824i	$-0.435950\pi$
$631$	3.99794e6	0.399726	0.199863	+	0.979824i	$0.435950\pi$
$632$	6.27725e6	0.625139
$633$	0	0
$634$	3.51992e6	0.347784
$635$	−5.37783e6	−0.529264
$636$	0	0
$637$	0	0
$638$	437975.	0.0425988
$639$	0	0
$640$	1.57273e7	1.51776
$641$	−1.75881e7	−1.69072	−0.845362	−	0.534194i	$-0.820615\pi$
$641$	−1.75881e7	−1.69072	−0.845362	+	0.534194i	$0.820615\pi$
$642$	0	0
$643$	−6.83576e6	−0.652018	−0.326009	−	0.945367i	$-0.605704\pi$
$643$	−6.83576e6	−0.652018	−0.326009	+	0.945367i	$0.605704\pi$
$644$	0	0
$645$	0	0
$646$	5.31846e6	0.501423
$647$	1.78136e7	1.67298	0.836490	−	0.547982i	$-0.184604\pi$
$647$	1.78136e7	1.67298	0.836490	+	0.547982i	$0.184604\pi$
$648$	0	0
$649$	763399.	0.0711443
$650$	1.23085e7	1.14267
$651$	0	0
$652$	1.35718e7	1.25031
$653$	1.48677e7	1.36446	0.682231	−	0.731137i	$-0.261010\pi$
$653$	1.48677e7	1.36446	0.682231	+	0.731137i	$0.261010\pi$
$654$	0	0
$655$	2.53296e7	2.30688
$656$	49446.7	0.00448619
$657$	0	0
$658$	0	0
$659$	−1.57120e7	−1.40935	−0.704673	−	0.709532i	$-0.748906\pi$
$659$	−1.57120e7	−1.40935	−0.704673	+	0.709532i	$0.748906\pi$
$660$	0	0
$661$	−188221.	−0.0167557	−0.00837787	−	0.999965i	$-0.502667\pi$
$661$	−188221.	−0.0167557	−0.00837787	+	0.999965i	$0.502667\pi$
$662$	1.34610e6	0.119380
$663$	0	0
$664$	5.21199e6	0.458758
$665$	0	0
$666$	0	0
$667$	1.71030e7	1.48853
$668$	−6.05230e6	−0.524782
$669$	0	0
$670$	−1.31779e7	−1.13412
$671$	620577.	0.0532095
$672$	0	0
$673$	8.65246e6	0.736380	0.368190	−	0.929751i	$-0.379978\pi$
$673$	8.65246e6	0.736380	0.368190	+	0.929751i	$0.379978\pi$
$674$	5.32419e6	0.451444
$675$	0	0
$676$	−1.50130e7	−1.26357
$677$	−2.55362e6	−0.214134	−0.107067	−	0.994252i	$-0.534146\pi$
$677$	−2.55362e6	−0.214134	−0.107067	+	0.994252i	$0.534146\pi$
$678$	0	0
$679$	0	0
$680$	2.85611e7	2.36866
$681$	0	0
$682$	−127560.	−0.0105015
$683$	−3.83987e6	−0.314966	−0.157483	−	0.987522i	$-0.550338\pi$
$683$	−3.83987e6	−0.314966	−0.157483	+	0.987522i	$0.550338\pi$
$684$	0	0
$685$	2.40359e7	1.95720
$686$	0	0
$687$	0	0
$688$	−1.33388e6	−0.107435
$689$	2.92769e6	0.234951
$690$	0	0
$691$	5.32370e6	0.424149	0.212074	−	0.977254i	$-0.431978\pi$
$691$	5.32370e6	0.424149	0.212074	+	0.977254i	$0.431978\pi$
$692$	1.10823e6	0.0879761
$693$	0	0
$694$	−1.98925e6	−0.156780
$695$	−2.54123e7	−1.99564
$696$	0	0
$697$	277967.	0.0216726
$698$	2.21816e6	0.172327
$699$	0	0
$700$	0	0
$701$	−2.44705e6	−0.188082	−0.0940409	−	0.995568i	$-0.529978\pi$
$701$	−2.44705e6	−0.188082	−0.0940409	+	0.995568i	$0.529978\pi$
$702$	0	0
$703$	9.33880e6	0.712694
$704$	−162647.	−0.0123684
$705$	0	0
$706$	4.36339e6	0.329467
$707$	0	0
$708$	0	0
$709$	−1.35056e7	−1.00902	−0.504509	−	0.863407i	$-0.668326\pi$
$709$	−1.35056e7	−1.00902	−0.504509	+	0.863407i	$0.668326\pi$
$710$	−1.44157e7	−1.07322
$711$	0	0
$712$	780662.	0.0577116
$713$	−4.98122e6	−0.366954
$714$	0	0
$715$	3.62442e6	0.265139
$716$	−1.09271e7	−0.796566
$717$	0	0
$718$	1.04094e6	0.0753556
$719$	602676.	0.0434772	0.0217386	−	0.999764i	$-0.493080\pi$
$719$	602676.	0.0434772	0.0217386	+	0.999764i	$0.493080\pi$
$720$	0	0
$721$	0	0
$722$	4.35696e6	0.311057
$723$	0	0
$724$	−5.33401e6	−0.378187
$725$	1.79426e7	1.26777
$726$	0	0
$727$	−2.25663e7	−1.58352	−0.791762	−	0.610830i	$-0.790836\pi$
$727$	−2.25663e7	−1.58352	−0.791762	+	0.610830i	$0.790836\pi$
$728$	0	0
$729$	0	0
$730$	−1.17793e7	−0.818114
$731$	−7.49850e6	−0.519017
$732$	0	0
$733$	1.64126e7	1.12828	0.564140	−	0.825679i	$-0.309208\pi$
$733$	1.64126e7	1.12828	0.564140	+	0.825679i	$0.309208\pi$
$734$	8.68530e6	0.595038
$735$	0	0
$736$	−2.59949e7	−1.76886
$737$	−2.31226e6	−0.156808
$738$	0	0
$739$	−1.85396e7	−1.24879	−0.624394	−	0.781110i	$-0.714654\pi$
$739$	−1.85396e7	−1.24879	−0.624394	+	0.781110i	$0.714654\pi$
$740$	2.18523e7	1.46696
$741$	0	0
$742$	0	0
$743$	−2.35170e7	−1.56282	−0.781412	−	0.624016i	$-0.785500\pi$
$743$	−2.35170e7	−1.56282	−0.781412	+	0.624016i	$0.785500\pi$
$744$	0	0
$745$	−1.68482e7	−1.11215
$746$	−3.80928e6	−0.250608
$747$	0	0
$748$	2.18366e6	0.142702
$749$	0	0
$750$	0	0
$751$	1.53822e7	0.995218	0.497609	−	0.867402i	$-0.334211\pi$
$751$	1.53822e7	0.995218	0.497609	+	0.867402i	$0.334211\pi$
$752$	−6.06487e6	−0.391090
$753$	0	0
$754$	1.04015e7	0.666297
$755$	1.54354e7	0.985486
$756$	0	0
$757$	−1.75980e7	−1.11615	−0.558075	−	0.829790i	$-0.688460\pi$
$757$	−1.75980e7	−1.11615	−0.558075	+	0.829790i	$0.688460\pi$
$758$	803387.	0.0507869
$759$	0	0
$760$	−1.25033e7	−0.785221
$761$	−1.67807e7	−1.05039	−0.525194	−	0.850983i	$-0.676007\pi$
$761$	−1.67807e7	−1.05039	−0.525194	+	0.850983i	$0.676007\pi$
$762$	0	0
$763$	0	0
$764$	−4.63401e6	−0.287226
$765$	0	0
$766$	−5.62976e6	−0.346671
$767$	1.81300e7	1.11278
$768$	0	0
$769$	−261764.	−0.0159622	−0.00798112	−	0.999968i	$-0.502540\pi$
$769$	−261764.	−0.0159622	−0.00798112	+	0.999968i	$0.502540\pi$
$770$	0	0
$771$	0	0
$772$	−1.81853e7	−1.09819
$773$	2.14255e7	1.28968	0.644840	−	0.764317i	$-0.276924\pi$
$773$	2.14255e7	1.28968	0.644840	+	0.764317i	$0.276924\pi$
$774$	0	0
$775$	−5.22576e6	−0.312533
$776$	−1.50686e7	−0.898295
$777$	0	0
$778$	9.74299e6	0.577090
$779$	−121687.	−0.00718458
$780$	0	0
$781$	−2.52946e6	−0.148388
$782$	−2.51550e7	−1.47098
$783$	0	0
$784$	0	0
$785$	5.90040e6	0.341749
$786$	0	0
$787$	−1.84568e7	−1.06223	−0.531117	−	0.847299i	$-0.678227\pi$
$787$	−1.84568e7	−1.06223	−0.531117	+	0.847299i	$0.678227\pi$
$788$	4.82906e6	0.277043
$789$	0	0
$790$	−9.73365e6	−0.554891
$791$	0	0
$792$	0	0
$793$	1.47381e7	0.832261
$794$	8.20956e6	0.462135
$795$	0	0
$796$	−2.38819e7	−1.33594
$797$	4.94457e6	0.275729	0.137865	−	0.990451i	$-0.455976\pi$
$797$	4.94457e6	0.275729	0.137865	+	0.990451i	$0.455976\pi$
$798$	0	0
$799$	−3.40940e7	−1.88934
$800$	−2.72711e7	−1.50653
$801$	0	0
$802$	−9.49655e6	−0.521350
$803$	−2.06687e6	−0.113116
$804$	0	0
$805$	0	0
$806$	−3.02942e6	−0.164256
$807$	0	0
$808$	378000.	0.0203687
$809$	1.77802e7	0.955138	0.477569	−	0.878594i	$-0.341518\pi$
$809$	1.77802e7	0.955138	0.477569	+	0.878594i	$0.341518\pi$
$810$	0	0
$811$	2.32483e7	1.24119	0.620595	−	0.784131i	$-0.286891\pi$
$811$	2.32483e7	1.24119	0.620595	+	0.784131i	$0.286891\pi$
$812$	0	0
$813$	0	0
$814$	−1.13112e6	−0.0598339
$815$	−4.82975e7	−2.54701
$816$	0	0
$817$	3.28266e6	0.172056
$818$	5.09041e6	0.265992
$819$	0	0
$820$	−284742.	−0.0147883
$821$	1.58161e7	0.818919	0.409459	−	0.912328i	$-0.365717\pi$
$821$	1.58161e7	0.818919	0.409459	+	0.912328i	$0.365717\pi$
$822$	0	0
$823$	1.41435e7	0.727875	0.363938	−	0.931423i	$-0.381432\pi$
$823$	1.41435e7	0.727875	0.363938	+	0.931423i	$0.381432\pi$
$824$	1.03519e7	0.531131
$825$	0	0
$826$	0	0
$827$	−1.64670e6	−0.0837240	−0.0418620	−	0.999123i	$-0.513329\pi$
$827$	−1.64670e6	−0.0837240	−0.0418620	+	0.999123i	$0.513329\pi$
$828$	0	0
$829$	−4.32819e6	−0.218736	−0.109368	−	0.994001i	$-0.534883\pi$
$829$	−4.32819e6	−0.218736	−0.109368	+	0.994001i	$0.534883\pi$
$830$	−8.08183e6	−0.407206
$831$	0	0
$832$	−3.86272e6	−0.193457
$833$	0	0
$834$	0	0
$835$	2.15382e7	1.06904
$836$	−955952.	−0.0473064
$837$	0	0
$838$	−6.02885e6	−0.296568
$839$	9.30618e6	0.456422	0.228211	−	0.973612i	$-0.426712\pi$
$839$	9.30618e6	0.456422	0.228211	+	0.973612i	$0.426712\pi$
$840$	0	0
$841$	−5.34842e6	−0.260757
$842$	1.17059e7	0.569018
$843$	0	0
$844$	2.02514e7	0.978588
$845$	5.34263e7	2.57403
$846$	0	0
$847$	0	0
$848$	−1.11661e6	−0.0533229
$849$	0	0
$850$	−2.63900e7	−1.25283
$851$	−4.41703e7	−2.09077
$852$	0	0
$853$	−2.39111e7	−1.12519	−0.562596	−	0.826732i	$-0.690197\pi$
$853$	−2.39111e7	−1.12519	−0.562596	+	0.826732i	$0.690197\pi$
$854$	0	0
$855$	0	0
$856$	2.59762e7	1.21169
$857$	1.42935e7	0.664794	0.332397	−	0.943140i	$-0.392143\pi$
$857$	1.42935e7	0.664794	0.332397	+	0.943140i	$0.392143\pi$
$858$	0	0
$859$	−1.13017e7	−0.522590	−0.261295	−	0.965259i	$-0.584150\pi$
$859$	−1.13017e7	−0.522590	−0.261295	+	0.965259i	$0.584150\pi$
$860$	7.68126e6	0.354149
$861$	0	0
$862$	−1.72822e7	−0.792193
$863$	5.31215e6	0.242797	0.121398	−	0.992604i	$-0.461262\pi$
$863$	5.31215e6	0.242797	0.121398	+	0.992604i	$0.461262\pi$
$864$	0	0
$865$	−3.94383e6	−0.179217
$866$	6.52937e6	0.295854
$867$	0	0
$868$	0	0
$869$	−1.70792e6	−0.0767217
$870$	0	0
$871$	−5.49141e7	−2.45266
$872$	1.35403e7	0.603027
$873$	0	0
$874$	1.10123e7	0.487638
$875$	0	0
$876$	0	0
$877$	−8.04440e6	−0.353179	−0.176590	−	0.984285i	$-0.556507\pi$
$877$	−8.04440e6	−0.353179	−0.176590	+	0.984285i	$0.556507\pi$
$878$	−5.44937e6	−0.238567
$879$	0	0
$880$	−1.38235e6	−0.0601741
$881$	−1.26000e7	−0.546930	−0.273465	−	0.961882i	$-0.588170\pi$
$881$	−1.26000e7	−0.546930	−0.273465	+	0.961882i	$0.588170\pi$
$882$	0	0
$883$	2.15646e7	0.930766	0.465383	−	0.885109i	$-0.345917\pi$
$883$	2.15646e7	0.930766	0.465383	+	0.885109i	$0.345917\pi$
$884$	5.18599e7	2.23203
$885$	0	0
$886$	−1.21943e7	−0.521883
$887$	2.49881e7	1.06641	0.533205	−	0.845986i	$-0.320987\pi$
$887$	2.49881e7	1.06641	0.533205	+	0.845986i	$0.320987\pi$
$888$	0	0
$889$	0	0
$890$	−1.21051e6	−0.0512264
$891$	0	0
$892$	−2.35686e7	−0.991794
$893$	1.49255e7	0.626326
$894$	0	0
$895$	3.88859e7	1.62269
$896$	0	0
$897$	0	0
$898$	−9.82269e6	−0.406480
$899$	−4.41612e6	−0.182239
$900$	0	0
$901$	−6.27711e6	−0.257601
$902$	14738.8	0.000603178 0
$903$	0	0
$904$	2.99069e7	1.21717
$905$	1.89820e7	0.770407
$906$	0	0
$907$	−3.42314e7	−1.38168	−0.690839	−	0.723008i	$-0.742759\pi$
$907$	−3.42314e7	−1.38168	−0.690839	+	0.723008i	$0.742759\pi$
$908$	−3.63262e7	−1.46220
$909$	0	0
$910$	0	0
$911$	1.32335e7	0.528296	0.264148	−	0.964482i	$-0.414909\pi$
$911$	1.32335e7	0.528296	0.264148	+	0.964482i	$0.414909\pi$
$912$	0	0
$913$	−1.41808e6	−0.0563022
$914$	−1.05149e7	−0.416332
$915$	0	0
$916$	−1.53651e7	−0.605057
$917$	0	0
$918$	0	0
$919$	2.85336e7	1.11447	0.557235	−	0.830355i	$-0.311862\pi$
$919$	2.85336e7	1.11447	0.557235	+	0.830355i	$0.311862\pi$
$920$	5.91378e7	2.30354
$921$	0	0
$922$	−2.59179e6	−0.100409
$923$	−6.00722e7	−2.32097
$924$	0	0
$925$	−4.63388e7	−1.78070
$926$	−1.38881e6	−0.0532249
$927$	0	0
$928$	−2.30459e7	−0.878465
$929$	−3.17038e7	−1.20523	−0.602617	−	0.798030i	$-0.705875\pi$
$929$	−3.17038e7	−1.20523	−0.602617	+	0.798030i	$0.705875\pi$
$930$	0	0
$931$	0	0
$932$	−2.80145e7	−1.05644
$933$	0	0
$934$	−4.42938e6	−0.166141
$935$	−7.77093e6	−0.290699
$936$	0	0
$937$	3.54930e7	1.32067	0.660333	−	0.750973i	$-0.270415\pi$
$937$	3.54930e7	1.32067	0.660333	+	0.750973i	$0.270415\pi$
$938$	0	0
$939$	0	0
$940$	3.49250e7	1.28919
$941$	2.40587e7	0.885724	0.442862	−	0.896590i	$-0.353963\pi$
$941$	2.40587e7	0.885724	0.442862	+	0.896590i	$0.353963\pi$
$942$	0	0
$943$	575551.	0.0210768
$944$	−6.91476e6	−0.252550
$945$	0	0
$946$	−397597.	−0.0144449
$947$	−8.23520e6	−0.298400	−0.149200	−	0.988807i	$-0.547670\pi$
$947$	−8.23520e6	−0.298400	−0.149200	+	0.988807i	$0.547670\pi$
$948$	0	0
$949$	−4.90862e7	−1.76927
$950$	1.15529e7	0.415319
$951$	0	0
$952$	0	0
$953$	2.49957e7	0.891525	0.445763	−	0.895151i	$-0.352933\pi$
$953$	2.49957e7	0.891525	0.445763	+	0.895151i	$0.352933\pi$
$954$	0	0
$955$	1.64909e7	0.585109
$956$	−1.64812e6	−0.0583235
$957$	0	0
$958$	1.03181e7	0.363234
$959$	0	0
$960$	0	0
$961$	−2.73430e7	−0.955074
$962$	−2.68630e7	−0.935873
$963$	0	0
$964$	1.63680e7	0.567286
$965$	6.47154e7	2.23712
$966$	0	0
$967$	−1.67903e7	−0.577420	−0.288710	−	0.957417i	$-0.593226\pi$
$967$	−1.67903e7	−0.577420	−0.288710	+	0.957417i	$0.593226\pi$
$968$	−2.43933e7	−0.836725
$969$	0	0
$970$	2.33657e7	0.797352
$971$	2.25923e7	0.768974	0.384487	−	0.923130i	$-0.374378\pi$
$971$	2.25923e7	0.768974	0.384487	+	0.923130i	$0.374378\pi$
$972$	0	0
$973$	0	0
$974$	−1.73584e7	−0.586289
$975$	0	0
$976$	−5.62109e6	−0.188884
$977$	371534.	0.0124527	0.00622634	−	0.999981i	$-0.498018\pi$
$977$	371534.	0.0124527	0.00622634	+	0.999981i	$0.498018\pi$
$978$	0	0
$979$	−212403.	−0.00708279
$980$	0	0
$981$	0	0
$982$	1.99443e7	0.659993
$983$	−2.30996e7	−0.762466	−0.381233	−	0.924479i	$-0.624500\pi$
$983$	−2.30996e7	−0.762466	−0.381233	+	0.924479i	$0.624500\pi$
$984$	0	0
$985$	−1.71850e7	−0.564365
$986$	−2.23013e7	−0.730531
$987$	0	0
$988$	−2.27030e7	−0.739929
$989$	−1.55262e7	−0.504748
$990$	0	0
$991$	−7.06806e6	−0.228621	−0.114311	−	0.993445i	$-0.536466\pi$
$991$	−7.06806e6	−0.228621	−0.114311	+	0.993445i	$0.536466\pi$
$992$	6.71209e6	0.216560
$993$	0	0
$994$	0	0
$995$	8.49878e7	2.72144
$996$	0	0
$997$	1.11936e7	0.356643	0.178321	−	0.983972i	$-0.442933\pi$
$997$	1.11936e7	0.356643	0.178321	+	0.983972i	$0.442933\pi$
$998$	1.05443e7	0.335113
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.x.1.2		4
3.2	odd	2	inner	441.6.a.x.1.3		4
7.6	odd	2		63.6.a.h.1.2	✓	4
21.20	even	2		63.6.a.h.1.3	yes	4
28.27	even	2		1008.6.a.by.1.1		4
84.83	odd	2		1008.6.a.by.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.a.h.1.2	✓	4	7.6	odd	2
63.6.a.h.1.3	yes	4	21.20	even	2
441.6.a.x.1.2		4	1.1	even	1	trivial
441.6.a.x.1.3		4	3.2	odd	2	inner
1008.6.a.by.1.1		4	28.27	even	2
1008.6.a.by.1.4		4	84.83	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69991 q^{2} -24.7105 q^{4} +87.9366 q^{5} +153.113 q^{8} +O(q^{10})\)	\(q-2.69991 q^{2} -24.7105 q^{4} +87.9366 q^{5} +153.113 q^{8} -237.421 q^{10} -41.6592 q^{11} -989.368 q^{13} +377.343 q^{16} +2121.25 q^{17} -928.632 q^{19} -2172.95 q^{20} +112.476 q^{22} +4392.21 q^{23} +4607.84 q^{25} +2671.21 q^{26} +3893.93 q^{29} -1134.10 q^{31} -5918.42 q^{32} -5727.20 q^{34} -10056.5 q^{37} +2507.23 q^{38} +13464.3 q^{40} +131.039 q^{41} -3534.94 q^{43} +1029.42 q^{44} -11858.6 q^{46} -16072.6 q^{47} -12440.8 q^{50} +24447.7 q^{52} -2959.15 q^{53} -3663.37 q^{55} -10513.3 q^{58} -18324.9 q^{59} -14896.5 q^{61} +3061.98 q^{62} +3904.23 q^{64} -87001.6 q^{65} +55504.2 q^{67} -52417.2 q^{68} +60717.8 q^{71} +49613.7 q^{73} +27151.7 q^{74} +22946.9 q^{76} +40997.4 q^{79} +33182.2 q^{80} -353.794 q^{82} +34040.1 q^{83} +186536. q^{85} +9544.03 q^{86} -6378.58 q^{88} +5098.59 q^{89} -108534. q^{92} +43394.5 q^{94} -81660.7 q^{95} -98414.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 94 q^{4}+O(q^{10}) \) 4 * q + 94 * q^4	\( 4 q + 94 q^{4} - 564 q^{10} - 872 q^{13} + 4402 q^{16} - 6800 q^{19} + 10092 q^{22} + 3004 q^{25} + 4720 q^{31} + 32244 q^{34} + 6056 q^{37} + 23388 q^{40} + 10544 q^{43} - 47820 q^{46} + 128260 q^{52} - 11568 q^{55} - 106848 q^{58} - 13304 q^{61} + 5782 q^{64} + 203504 q^{67} - 17528 q^{73} - 308552 q^{76} + 34400 q^{79} + 182556 q^{82} + 354288 q^{85} + 379068 q^{88} + 561576 q^{94} - 436856 q^{97}+O(q^{100}) \) 4 * q + 94 * q^4 - 564 * q^10 - 872 * q^13 + 4402 * q^16 - 6800 * q^19 + 10092 * q^22 + 3004 * q^25 + 4720 * q^31 + 32244 * q^34 + 6056 * q^37 + 23388 * q^40 + 10544 * q^43 - 47820 * q^46 + 128260 * q^52 - 11568 * q^55 - 106848 * q^58 - 13304 * q^61 + 5782 * q^64 + 203504 * q^67 - 17528 * q^73 - 308552 * q^76 + 34400 * q^79 + 182556 * q^82 + 354288 * q^85 + 379068 * q^88 + 561576 * q^94 - 436856 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more