LMFDB - Embedded newform 441.6.a.u.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{39}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 39 $ x^2 - 39
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.24500$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} -28.0000 q^{4} -74.9400 q^{5} -120.000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} -28.0000 q^{4} -74.9400 q^{5} -120.000 q^{8} -149.880 q^{10} +284.000 q^{11} +524.580 q^{13} +656.000 q^{16} +149.880 q^{17} +2173.26 q^{19} +2098.32 q^{20} +568.000 q^{22} -1496.00 q^{23} +2491.00 q^{25} +1049.16 q^{26} +4366.00 q^{29} -6444.84 q^{31} +5152.00 q^{32} +299.760 q^{34} -12630.0 q^{37} +4346.52 q^{38} +8992.80 q^{40} +9442.44 q^{41} -1356.00 q^{43} -7952.00 q^{44} -2992.00 q^{46} -10042.0 q^{47} +4982.00 q^{50} -14688.2 q^{52} -14150.0 q^{53} -21283.0 q^{55} +8732.00 q^{58} +37395.0 q^{59} -35596.5 q^{61} -12889.7 q^{62} -10688.0 q^{64} -39312.0 q^{65} -3644.00 q^{67} -4196.64 q^{68} -35632.0 q^{71} +40767.3 q^{73} -25260.0 q^{74} -60851.3 q^{76} -54616.0 q^{79} -49160.6 q^{80} +18884.9 q^{82} +524.580 q^{83} -11232.0 q^{85} -2712.00 q^{86} -34080.0 q^{88} +20383.7 q^{89} +41888.0 q^{92} -20083.9 q^{94} -162864. q^{95} +183603. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 56 q^{4} - 240 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 - 56 * q^4 - 240 * q^8	$ 2 q + 4 q^{2} - 56 q^{4} - 240 q^{8} + 568 q^{11} + 1312 q^{16} + 1136 q^{22} - 2992 q^{23} + 4982 q^{25} + 8732 q^{29} + 10304 q^{32} - 25260 q^{37} - 2712 q^{43} - 15904 q^{44} - 5984 q^{46} + 9964 q^{50} - 28300 q^{53} + 17464 q^{58} - 21376 q^{64} - 78624 q^{65} - 7288 q^{67} - 71264 q^{71} - 50520 q^{74} - 109232 q^{79} - 22464 q^{85} - 5424 q^{86} - 68160 q^{88} + 83776 q^{92} - 325728 q^{95}+O(q^{100}) $ 2 * q + 4 * q^2 - 56 * q^4 - 240 * q^8 + 568 * q^11 + 1312 * q^16 + 1136 * q^22 - 2992 * q^23 + 4982 * q^25 + 8732 * q^29 + 10304 * q^32 - 25260 * q^37 - 2712 * q^43 - 15904 * q^44 - 5984 * q^46 + 9964 * q^50 - 28300 * q^53 + 17464 * q^58 - 21376 * q^64 - 78624 * q^65 - 7288 * q^67 - 71264 * q^71 - 50520 * q^74 - 109232 * q^79 - 22464 * q^85 - 5424 * q^86 - 68160 * q^88 + 83776 * q^92 - 325728 * q^95

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 $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−28.0000	−0.875000
$5$	−74.9400	−1.34057	−0.670284	−	0.742105i	$-0.733828\pi$
$5$	−74.9400	−1.34057	−0.670284	+	0.742105i	$0.733828\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−120.000	−0.662913
$9$	0	0
$10$	−149.880	−0.473962
$11$	284.000	0.707680	0.353840	−	0.935306i	$-0.384876\pi$
$11$	284.000	0.707680	0.353840	+	0.935306i	$0.384876\pi$
$12$	0	0
$13$	524.580	0.860901	0.430450	−	0.902614i	$-0.358355\pi$
$13$	524.580	0.860901	0.430450	+	0.902614i	$0.358355\pi$
$14$	0	0
$15$	0	0
$16$	656.000	0.640625
$17$	149.880	0.125783	0.0628914	−	0.998020i	$-0.479968\pi$
$17$	149.880	0.125783	0.0628914	+	0.998020i	$0.479968\pi$
$18$	0	0
$19$	2173.26	1.38111	0.690554	−	0.723281i	$-0.257367\pi$
$19$	2173.26	1.38111	0.690554	+	0.723281i	$0.257367\pi$
$20$	2098.32	1.17300
$21$	0	0
$22$	568.000	0.250202
$23$	−1496.00	−0.589674	−0.294837	−	0.955548i	$-0.595265\pi$
$23$	−1496.00	−0.589674	−0.294837	+	0.955548i	$0.595265\pi$
$24$	0	0
$25$	2491.00	0.797120
$26$	1049.16	0.304374
$27$	0	0
$28$	0	0
$29$	4366.00	0.964026	0.482013	−	0.876164i	$-0.339906\pi$
$29$	4366.00	0.964026	0.482013	+	0.876164i	$0.339906\pi$
$30$	0	0
$31$	−6444.84	−1.20450	−0.602251	−	0.798307i	$-0.705729\pi$
$31$	−6444.84	−1.20450	−0.602251	+	0.798307i	$0.705729\pi$
$32$	5152.00	0.889408
$33$	0	0
$34$	299.760	0.0444709
$35$	0	0
$36$	0	0
$37$	−12630.0	−1.51670	−0.758349	−	0.651849i	$-0.773994\pi$
$37$	−12630.0	−1.51670	−0.758349	+	0.651849i	$0.773994\pi$
$38$	4346.52	0.488295
$39$	0	0
$40$	8992.80	0.888679
$41$	9442.44	0.877252	0.438626	−	0.898670i	$-0.355465\pi$
$41$	9442.44	0.877252	0.438626	+	0.898670i	$0.355465\pi$
$42$	0	0
$43$	−1356.00	−0.111838	−0.0559189	−	0.998435i	$-0.517809\pi$
$43$	−1356.00	−0.111838	−0.0559189	+	0.998435i	$0.517809\pi$
$44$	−7952.00	−0.619220
$45$	0	0
$46$	−2992.00	−0.208481
$47$	−10042.0	−0.663092	−0.331546	−	0.943439i	$-0.607570\pi$
$47$	−10042.0	−0.663092	−0.331546	+	0.943439i	$0.607570\pi$
$48$	0	0
$49$	0	0
$50$	4982.00	0.281824
$51$	0	0
$52$	−14688.2	−0.753288
$53$	−14150.0	−0.691937	−0.345969	−	0.938246i	$-0.612450\pi$
$53$	−14150.0	−0.691937	−0.345969	+	0.938246i	$0.612450\pi$
$54$	0	0
$55$	−21283.0	−0.948692
$56$	0	0
$57$	0	0
$58$	8732.00	0.340835
$59$	37395.0	1.39857	0.699285	−	0.714843i	$-0.253502\pi$
$59$	37395.0	1.39857	0.699285	+	0.714843i	$0.253502\pi$
$60$	0	0
$61$	−35596.5	−1.22485	−0.612425	−	0.790529i	$-0.709806\pi$
$61$	−35596.5	−1.22485	−0.612425	+	0.790529i	$0.709806\pi$
$62$	−12889.7	−0.425856
$63$	0	0
$64$	−10688.0	−0.326172
$65$	−39312.0	−1.15410
$66$	0	0
$67$	−3644.00	−0.0991725	−0.0495863	−	0.998770i	$-0.515790\pi$
$67$	−3644.00	−0.0991725	−0.0495863	+	0.998770i	$0.515790\pi$
$68$	−4196.64	−0.110060
$69$	0	0
$70$	0	0
$71$	−35632.0	−0.838869	−0.419435	−	0.907786i	$-0.637772\pi$
$71$	−35632.0	−0.838869	−0.419435	+	0.907786i	$0.637772\pi$
$72$	0	0
$73$	40767.3	0.895376	0.447688	−	0.894190i	$-0.352248\pi$
$73$	40767.3	0.895376	0.447688	+	0.894190i	$0.352248\pi$
$74$	−25260.0	−0.536234
$75$	0	0
$76$	−60851.3	−1.20847
$77$	0	0
$78$	0	0
$79$	−54616.0	−0.984583	−0.492291	−	0.870431i	$-0.663841\pi$
$79$	−54616.0	−0.984583	−0.492291	+	0.870431i	$0.663841\pi$
$80$	−49160.6	−0.858801
$81$	0	0
$82$	18884.9	0.310155
$83$	524.580	0.00835827	0.00417913	−	0.999991i	$-0.498670\pi$
$83$	524.580	0.00835827	0.00417913	+	0.999991i	$0.498670\pi$
$84$	0	0
$85$	−11232.0	−0.168620
$86$	−2712.00	−0.0395406
$87$	0	0
$88$	−34080.0	−0.469130
$89$	20383.7	0.272777	0.136388	−	0.990655i	$-0.456450\pi$
$89$	20383.7	0.272777	0.136388	+	0.990655i	$0.456450\pi$
$90$	0	0
$91$	0	0
$92$	41888.0	0.515965
$93$	0	0
$94$	−20083.9	−0.234438
$95$	−162864.	−1.85147
$96$	0	0
$97$	183603.	1.98130	0.990650	−	0.136427i	$-0.0435619\pi$
$97$	183603.	1.98130	0.990650	+	0.136427i	$0.0435619\pi$
$98$	0	0
$99$	0	0
$100$	−69748.0	−0.697480
$101$	−75914.2	−0.740491	−0.370245	−	0.928934i	$-0.620726\pi$
$101$	−75914.2	−0.740491	−0.370245	+	0.928934i	$0.620726\pi$
$102$	0	0
$103$	−10941.2	−0.101619	−0.0508093	−	0.998708i	$-0.516180\pi$
$103$	−10941.2	−0.101619	−0.0508093	+	0.998708i	$0.516180\pi$
$104$	−62949.6	−0.570702
$105$	0	0
$106$	−28300.0	−0.244637
$107$	−218188.	−1.84235	−0.921173	−	0.389152i	$-0.872768\pi$
$107$	−218188.	−1.84235	−0.921173	+	0.389152i	$0.872768\pi$
$108$	0	0
$109$	−96030.0	−0.774178	−0.387089	−	0.922042i	$-0.626519\pi$
$109$	−96030.0	−0.774178	−0.387089	+	0.922042i	$0.626519\pi$
$110$	−42565.9	−0.335413
$111$	0	0
$112$	0	0
$113$	137422.	1.01242	0.506209	−	0.862411i	$-0.331046\pi$
$113$	137422.	1.01242	0.506209	+	0.862411i	$0.331046\pi$
$114$	0	0
$115$	112110.	0.790498
$116$	−122248.	−0.843523
$117$	0	0
$118$	74790.1	0.494469
$119$	0	0
$120$	0	0
$121$	−80395.0	−0.499190
$122$	−71193.0	−0.433050
$123$	0	0
$124$	180455.	1.05394
$125$	47511.9	0.271974
$126$	0	0
$127$	170368.	0.937300	0.468650	−	0.883384i	$-0.344741\pi$
$127$	170368.	0.937300	0.468650	+	0.883384i	$0.344741\pi$
$128$	−186240.	−1.00473
$129$	0	0
$130$	−78624.0	−0.408034
$131$	348246.	1.77300	0.886498	−	0.462732i	$-0.153131\pi$
$131$	348246.	1.77300	0.886498	+	0.462732i	$0.153131\pi$
$132$	0	0
$133$	0	0
$134$	−7288.00	−0.0350628
$135$	0	0
$136$	−17985.6	−0.0833830
$137$	75562.0	0.343955	0.171978	−	0.985101i	$-0.444984\pi$
$137$	75562.0	0.343955	0.171978	+	0.985101i	$0.444984\pi$
$138$	0	0
$139$	97047.3	0.426036	0.213018	−	0.977048i	$-0.431671\pi$
$139$	97047.3	0.426036	0.213018	+	0.977048i	$0.431671\pi$
$140$	0	0
$141$	0	0
$142$	−71264.0	−0.296585
$143$	148981.	0.609242
$144$	0	0
$145$	−327188.	−1.29234
$146$	81534.7	0.316563
$147$	0	0
$148$	353640.	1.32711
$149$	−361030.	−1.33223	−0.666113	−	0.745851i	$-0.732043\pi$
$149$	−361030.	−1.33223	−0.666113	+	0.745851i	$0.732043\pi$
$150$	0	0
$151$	32280.0	0.115210	0.0576051	−	0.998339i	$-0.481654\pi$
$151$	32280.0	0.115210	0.0576051	+	0.998339i	$0.481654\pi$
$152$	−260791.	−0.915554
$153$	0	0
$154$	0	0
$155$	482976.	1.61472
$156$	0	0
$157$	−132869.	−0.430203	−0.215101	−	0.976592i	$-0.569008\pi$
$157$	−132869.	−0.430203	−0.215101	+	0.976592i	$0.569008\pi$
$158$	−109232.	−0.348103
$159$	0	0
$160$	−386091.	−1.19231
$161$	0	0
$162$	0	0
$163$	−61364.0	−0.180903	−0.0904513	−	0.995901i	$-0.528831\pi$
$163$	−61364.0	−0.180903	−0.0904513	+	0.995901i	$0.528831\pi$
$164$	−264388.	−0.767596
$165$	0	0
$166$	1049.16	0.00295509
$167$	−380845.	−1.05671	−0.528356	−	0.849023i	$-0.677192\pi$
$167$	−380845.	−1.05671	−0.528356	+	0.849023i	$0.677192\pi$
$168$	0	0
$169$	−96109.0	−0.258849
$170$	−22464.0	−0.0596163
$171$	0	0
$172$	37968.0	0.0978581
$173$	−517311.	−1.31412	−0.657062	−	0.753837i	$-0.728201\pi$
$173$	−517311.	−1.31412	−0.657062	+	0.753837i	$0.728201\pi$
$174$	0	0
$175$	0	0
$176$	186304.	0.453357
$177$	0	0
$178$	40767.3	0.0964412
$179$	−610564.	−1.42429	−0.712145	−	0.702032i	$-0.752276\pi$
$179$	−610564.	−1.42429	−0.712145	+	0.702032i	$0.752276\pi$
$180$	0	0
$181$	−433828.	−0.984285	−0.492142	−	0.870515i	$-0.663786\pi$
$181$	−433828.	−0.984285	−0.492142	+	0.870515i	$0.663786\pi$
$182$	0	0
$183$	0	0
$184$	179520.	0.390902
$185$	946492.	2.03324
$186$	0	0
$187$	42565.9	0.0890139
$188$	281175.	0.580205
$189$	0	0
$190$	−325728.	−0.654593
$191$	−341192.	−0.676730	−0.338365	−	0.941015i	$-0.609874\pi$
$191$	−341192.	−0.676730	−0.338365	+	0.941015i	$0.609874\pi$
$192$	0	0
$193$	−616158.	−1.19069	−0.595345	−	0.803470i	$-0.702985\pi$
$193$	−616158.	−1.19069	−0.595345	+	0.803470i	$0.702985\pi$
$194$	367206.	0.700495
$195$	0	0
$196$	0	0
$197$	−231478.	−0.424956	−0.212478	−	0.977166i	$-0.568153\pi$
$197$	−231478.	−0.424956	−0.212478	+	0.977166i	$0.568153\pi$
$198$	0	0
$199$	−405126.	−0.725199	−0.362599	−	0.931945i	$-0.618111\pi$
$199$	−405126.	−0.725199	−0.362599	+	0.931945i	$0.618111\pi$
$200$	−298920.	−0.528421
$201$	0	0
$202$	−151828.	−0.261803
$203$	0	0
$204$	0	0
$205$	−707616.	−1.17602
$206$	−21882.5	−0.0359276
$207$	0	0
$208$	344124.	0.551515
$209$	617206.	0.977382
$210$	0	0
$211$	776820.	1.20120	0.600599	−	0.799551i	$-0.294929\pi$
$211$	776820.	1.20120	0.600599	+	0.799551i	$0.294929\pi$
$212$	396200.	0.605445
$213$	0	0
$214$	−436376.	−0.651368
$215$	101619.	0.149926
$216$	0	0
$217$	0	0
$218$	−192060.	−0.273713
$219$	0	0
$220$	595923.	0.830105
$221$	78624.0	0.108287
$222$	0	0
$223$	81834.5	0.110198	0.0550990	−	0.998481i	$-0.482453\pi$
$223$	81834.5	0.110198	0.0550990	+	0.998481i	$0.482453\pi$
$224$	0	0
$225$	0	0
$226$	274844.	0.357944
$227$	−753671.	−0.970772	−0.485386	−	0.874300i	$-0.661321\pi$
$227$	−753671.	−0.970772	−0.485386	+	0.874300i	$0.661321\pi$
$228$	0	0
$229$	26303.9	0.0331461	0.0165730	−	0.999863i	$-0.494724\pi$
$229$	26303.9	0.0331461	0.0165730	+	0.999863i	$0.494724\pi$
$230$	224220.	0.279483
$231$	0	0
$232$	−523920.	−0.639065
$233$	−47542.0	−0.0573704	−0.0286852	−	0.999588i	$-0.509132\pi$
$233$	−47542.0	−0.0573704	−0.0286852	+	0.999588i	$0.509132\pi$
$234$	0	0
$235$	752544.	0.888919
$236$	−1.04706e6	−1.22375
$237$	0	0
$238$	0	0
$239$	−1.08899e6	−1.23319	−0.616595	−	0.787281i	$-0.711488\pi$
$239$	−1.08899e6	−1.23319	−0.616595	+	0.787281i	$0.711488\pi$
$240$	0	0
$241$	−1.34937e6	−1.49654	−0.748270	−	0.663395i	$-0.769115\pi$
$241$	−1.34937e6	−1.49654	−0.748270	+	0.663395i	$0.769115\pi$
$242$	−160790.	−0.176490
$243$	0	0
$244$	996702.	1.07174
$245$	0	0
$246$	0	0
$247$	1.14005e6	1.18900
$248$	773381.	0.798480
$249$	0	0
$250$	95023.9	0.0961574
$251$	−630020.	−0.631205	−0.315602	−	0.948892i	$-0.602207\pi$
$251$	−630020.	−0.631205	−0.315602	+	0.948892i	$0.602207\pi$
$252$	0	0
$253$	−424864.	−0.417300
$254$	340736.	0.331386
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	1.93405e6	1.82656	0.913282	−	0.407327i	$-0.133539\pi$
$257$	1.93405e6	1.82656	0.913282	+	0.407327i	$0.133539\pi$
$258$	0	0
$259$	0	0
$260$	1.10074e6	1.00983
$261$	0	0
$262$	696492.	0.626849
$263$	1.11712e6	0.995888	0.497944	−	0.867209i	$-0.334089\pi$
$263$	1.11712e6	0.995888	0.497944	+	0.867209i	$0.334089\pi$
$264$	0	0
$265$	1.06040e6	0.927588
$266$	0	0
$267$	0	0
$268$	102032.	0.0867760
$269$	−1.52061e6	−1.28126	−0.640629	−	0.767851i	$-0.721326\pi$
$269$	−1.52061e6	−1.28126	−0.640629	+	0.767851i	$0.721326\pi$
$270$	0	0
$271$	1.02038e6	0.843995	0.421997	−	0.906597i	$-0.361329\pi$
$271$	1.02038e6	0.843995	0.421997	+	0.906597i	$0.361329\pi$
$272$	98321.2	0.0805796
$273$	0	0
$274$	151124.	0.121607
$275$	707444.	0.564105
$276$	0	0
$277$	1.89642e6	1.48503	0.742516	−	0.669829i	$-0.233633\pi$
$277$	1.89642e6	1.48503	0.742516	+	0.669829i	$0.233633\pi$
$278$	194095.	0.150626
$279$	0	0
$280$	0	0
$281$	−1.31911e6	−0.996587	−0.498293	−	0.867008i	$-0.666040\pi$
$281$	−1.31911e6	−0.996587	−0.498293	+	0.867008i	$0.666040\pi$
$282$	0	0
$283$	478792.	0.355370	0.177685	−	0.984087i	$-0.443139\pi$
$283$	478792.	0.355370	0.177685	+	0.984087i	$0.443139\pi$
$284$	997696.	0.734011
$285$	0	0
$286$	297961.	0.215400
$287$	0	0
$288$	0	0
$289$	−1.39739e6	−0.984179
$290$	−654376.	−0.456912
$291$	0	0
$292$	−1.14149e6	−0.783454
$293$	−1.50187e6	−1.02203	−0.511015	−	0.859572i	$-0.670730\pi$
$293$	−1.50187e6	−1.02203	−0.511015	+	0.859572i	$0.670730\pi$
$294$	0	0
$295$	−2.80238e6	−1.87488
$296$	1.51560e6	1.00544
$297$	0	0
$298$	−722060.	−0.471013
$299$	−784771.	−0.507651
$300$	0	0
$301$	0	0
$302$	64560.0	0.0407330
$303$	0	0
$304$	1.42566e6	0.884772
$305$	2.66760e6	1.64199
$306$	0	0
$307$	−1.19657e6	−0.724588	−0.362294	−	0.932064i	$-0.618006\pi$
$307$	−1.19657e6	−0.724588	−0.362294	+	0.932064i	$0.618006\pi$
$308$	0	0
$309$	0	0
$310$	965952.	0.570889
$311$	1.31475e6	0.770799	0.385400	−	0.922750i	$-0.374064\pi$
$311$	1.31475e6	0.770799	0.385400	+	0.922750i	$0.374064\pi$
$312$	0	0
$313$	−2.65273e6	−1.53049	−0.765247	−	0.643737i	$-0.777383\pi$
$313$	−2.65273e6	−1.53049	−0.765247	+	0.643737i	$0.777383\pi$
$314$	−265737.	−0.152100
$315$	0	0
$316$	1.52925e6	0.861510
$317$	1.59992e6	0.894233	0.447116	−	0.894476i	$-0.352451\pi$
$317$	1.59992e6	0.894233	0.447116	+	0.894476i	$0.352451\pi$
$318$	0	0
$319$	1.23994e6	0.682221
$320$	800958.	0.437255
$321$	0	0
$322$	0	0
$323$	325728.	0.173720
$324$	0	0
$325$	1.30673e6	0.686241
$326$	−122728.	−0.0639587
$327$	0	0
$328$	−1.13309e6	−0.581542
$329$	0	0
$330$	0	0
$331$	−111708.	−0.0560421	−0.0280210	−	0.999607i	$-0.508921\pi$
$331$	−111708.	−0.0560421	−0.0280210	+	0.999607i	$0.508921\pi$
$332$	−14688.2	−0.00731349
$333$	0	0
$334$	−761690.	−0.373604
$335$	273081.	0.132947
$336$	0	0
$337$	1.59301e6	0.764087	0.382043	−	0.924144i	$-0.375220\pi$
$337$	1.59301e6	0.764087	0.382043	+	0.924144i	$0.375220\pi$
$338$	−192218.	−0.0915171
$339$	0	0
$340$	314496.	0.147543
$341$	−1.83033e6	−0.852402
$342$	0	0
$343$	0	0
$344$	162720.	0.0741387
$345$	0	0
$346$	−1.03462e6	−0.464613
$347$	3.33676e6	1.48765	0.743827	−	0.668372i	$-0.233009\pi$
$347$	3.33676e6	1.48765	0.743827	+	0.668372i	$0.233009\pi$
$348$	0	0
$349$	−1.60259e6	−0.704303	−0.352151	−	0.935943i	$-0.614550\pi$
$349$	−1.60259e6	−0.704303	−0.352151	+	0.935943i	$0.614550\pi$
$350$	0	0
$351$	0	0
$352$	1.46317e6	0.629416
$353$	−1.80965e6	−0.772962	−0.386481	−	0.922297i	$-0.626309\pi$
$353$	−1.80965e6	−0.772962	−0.386481	+	0.922297i	$0.626309\pi$
$354$	0	0
$355$	2.67026e6	1.12456
$356$	−570743.	−0.238680
$357$	0	0
$358$	−1.22113e6	−0.503563
$359$	−920792.	−0.377073	−0.188536	−	0.982066i	$-0.560374\pi$
$359$	−920792.	−0.377073	−0.188536	+	0.982066i	$0.560374\pi$
$360$	0	0
$361$	2.24696e6	0.907458
$362$	−867655.	−0.347997
$363$	0	0
$364$	0	0
$365$	−3.05510e6	−1.20031
$366$	0	0
$367$	1.34802e6	0.522434	0.261217	−	0.965280i	$-0.415876\pi$
$367$	1.34802e6	0.522434	0.261217	+	0.965280i	$0.415876\pi$
$368$	−981376.	−0.377760
$369$	0	0
$370$	1.89298e6	0.718857
$371$	0	0
$372$	0	0
$373$	3.13773e6	1.16773	0.583867	−	0.811849i	$-0.301539\pi$
$373$	3.13773e6	1.16773	0.583867	+	0.811849i	$0.301539\pi$
$374$	85131.8	0.0314712
$375$	0	0
$376$	1.20503e6	0.439572
$377$	2.29032e6	0.829931
$378$	0	0
$379$	−1.83188e6	−0.655088	−0.327544	−	0.944836i	$-0.606221\pi$
$379$	−1.83188e6	−0.655088	−0.327544	+	0.944836i	$0.606221\pi$
$380$	4.56019e6	1.62003
$381$	0	0
$382$	−682384.	−0.239260
$383$	27727.8	0.00965869	0.00482935	−	0.999988i	$-0.498463\pi$
$383$	27727.8	0.00965869	0.00482935	+	0.999988i	$0.498463\pi$
$384$	0	0
$385$	0	0
$386$	−1.23232e6	−0.420973
$387$	0	0
$388$	−5.14088e6	−1.73364
$389$	548342.	0.183729	0.0918645	−	0.995772i	$-0.470717\pi$
$389$	548342.	0.183729	0.0918645	+	0.995772i	$0.470717\pi$
$390$	0	0
$391$	−224220.	−0.0741708
$392$	0	0
$393$	0	0
$394$	−462956.	−0.150245
$395$	4.09292e6	1.31990
$396$	0	0
$397$	−403852.	−0.128601	−0.0643007	−	0.997931i	$-0.520482\pi$
$397$	−403852.	−0.128601	−0.0643007	+	0.997931i	$0.520482\pi$
$398$	−810251.	−0.256396
$399$	0	0
$400$	1.63410e6	0.510655
$401$	4.16427e6	1.29324	0.646619	−	0.762813i	$-0.276182\pi$
$401$	4.16427e6	1.29324	0.646619	+	0.762813i	$0.276182\pi$
$402$	0	0
$403$	−3.38083e6	−1.03696
$404$	2.12560e6	0.647929
$405$	0	0
$406$	0	0
$407$	−3.58692e6	−1.07334
$408$	0	0
$409$	−2.54121e6	−0.751161	−0.375581	−	0.926790i	$-0.622557\pi$
$409$	−2.54121e6	−0.751161	−0.375581	+	0.926790i	$0.622557\pi$
$410$	−1.41523e6	−0.415784
$411$	0	0
$412$	306355.	0.0889163
$413$	0	0
$414$	0	0
$415$	−39312.0	−0.0112048
$416$	2.70264e6	0.765692
$417$	0	0
$418$	1.23441e6	0.345557
$419$	2.30133e6	0.640389	0.320195	−	0.947352i	$-0.396252\pi$
$419$	2.30133e6	0.640389	0.320195	+	0.947352i	$0.396252\pi$
$420$	0	0
$421$	−2.79991e6	−0.769909	−0.384955	−	0.922936i	$-0.625783\pi$
$421$	−2.79991e6	−0.769909	−0.384955	+	0.922936i	$0.625783\pi$
$422$	1.55364e6	0.424687
$423$	0	0
$424$	1.69800e6	0.458694
$425$	373351.	0.100264
$426$	0	0
$427$	0	0
$428$	6.10926e6	1.61205
$429$	0	0
$430$	203237.	0.0530069
$431$	−954320.	−0.247458	−0.123729	−	0.992316i	$-0.539485\pi$
$431$	−954320.	−0.247458	−0.123729	+	0.992316i	$0.539485\pi$
$432$	0	0
$433$	519334.	0.133115	0.0665575	−	0.997783i	$-0.478798\pi$
$433$	519334.	0.133115	0.0665575	+	0.997783i	$0.478798\pi$
$434$	0	0
$435$	0	0
$436$	2.68884e6	0.677406
$437$	−3.25120e6	−0.814403
$438$	0	0
$439$	5.98081e6	1.48115	0.740574	−	0.671974i	$-0.234554\pi$
$439$	5.98081e6	1.48115	0.740574	+	0.671974i	$0.234554\pi$
$440$	2.55395e6	0.628900
$441$	0	0
$442$	157248.	0.0382851
$443$	−3.74820e6	−0.907432	−0.453716	−	0.891146i	$-0.649902\pi$
$443$	−3.74820e6	−0.907432	−0.453716	+	0.891146i	$0.649902\pi$
$444$	0	0
$445$	−1.52755e6	−0.365676
$446$	163669.	0.0389609
$447$	0	0
$448$	0	0
$449$	−99458.0	−0.0232822	−0.0116411	−	0.999932i	$-0.503706\pi$
$449$	−99458.0	−0.0232822	−0.0116411	+	0.999932i	$0.503706\pi$
$450$	0	0
$451$	2.68165e6	0.620813
$452$	−3.84782e6	−0.885866
$453$	0	0
$454$	−1.50734e6	−0.343220
$455$	0	0
$456$	0	0
$457$	−161814.	−0.0362431	−0.0181216	−	0.999836i	$-0.505769\pi$
$457$	−161814.	−0.0362431	−0.0181216	+	0.999836i	$0.505769\pi$
$458$	52607.9	0.0117189
$459$	0	0
$460$	−3.13909e6	−0.691685
$461$	4.49198e6	0.984431	0.492215	−	0.870473i	$-0.336187\pi$
$461$	4.49198e6	0.984431	0.492215	+	0.870473i	$0.336187\pi$
$462$	0	0
$463$	3.59382e6	0.779118	0.389559	−	0.921001i	$-0.372627\pi$
$463$	3.59382e6	0.779118	0.389559	+	0.921001i	$0.372627\pi$
$464$	2.86410e6	0.617579
$465$	0	0
$466$	−95084.0	−0.0202835
$467$	−2.05223e6	−0.435446	−0.217723	−	0.976011i	$-0.569863\pi$
$467$	−2.05223e6	−0.435446	−0.217723	+	0.976011i	$0.569863\pi$
$468$	0	0
$469$	0	0
$470$	1.50509e6	0.314280
$471$	0	0
$472$	−4.48741e6	−0.927129
$473$	−385104.	−0.0791453
$474$	0	0
$475$	5.41359e6	1.10091
$476$	0	0
$477$	0	0
$478$	−2.17798e6	−0.435998
$479$	1.99985e6	0.398252	0.199126	−	0.979974i	$-0.436190\pi$
$479$	1.99985e6	0.398252	0.199126	+	0.979974i	$0.436190\pi$
$480$	0	0
$481$	−6.62544e6	−1.30573
$482$	−2.69874e6	−0.529107
$483$	0	0
$484$	2.25106e6	0.436791
$485$	−1.37592e7	−2.65607
$486$	0	0
$487$	2.17126e6	0.414848	0.207424	−	0.978251i	$-0.433492\pi$
$487$	2.17126e6	0.414848	0.207424	+	0.978251i	$0.433492\pi$
$488$	4.27158e6	0.811968
$489$	0	0
$490$	0	0
$491$	−3.04555e6	−0.570114	−0.285057	−	0.958511i	$-0.592013\pi$
$491$	−3.04555e6	−0.570114	−0.285057	+	0.958511i	$0.592013\pi$
$492$	0	0
$493$	654376.	0.121258
$494$	2.28010e6	0.420374
$495$	0	0
$496$	−4.22781e6	−0.771635
$497$	0	0
$498$	0	0
$499$	−7.99225e6	−1.43687	−0.718436	−	0.695594i	$-0.755141\pi$
$499$	−7.99225e6	−1.43687	−0.718436	+	0.695594i	$0.755141\pi$
$500$	−1.33033e6	−0.237977
$501$	0	0
$502$	−1.26004e6	−0.223165
$503$	−9.47811e6	−1.67033	−0.835164	−	0.550001i	$-0.814627\pi$
$503$	−9.47811e6	−1.67033	−0.835164	+	0.550001i	$0.814627\pi$
$504$	0	0
$505$	5.68901e6	0.992677
$506$	−849728.	−0.147538
$507$	0	0
$508$	−4.77030e6	−0.820138
$509$	−1.01003e7	−1.72799	−0.863995	−	0.503500i	$-0.832045\pi$
$509$	−1.01003e7	−1.72799	−0.863995	+	0.503500i	$0.832045\pi$
$510$	0	0
$511$	0	0
$512$	5.89875e6	0.994455
$513$	0	0
$514$	3.86810e6	0.645788
$515$	819936.	0.136227
$516$	0	0
$517$	−2.85192e6	−0.469257
$518$	0	0
$519$	0	0
$520$	4.71744e6	0.765064
$521$	−5.11975e6	−0.826332	−0.413166	−	0.910656i	$-0.635577\pi$
$521$	−5.11975e6	−0.826332	−0.413166	+	0.910656i	$0.635577\pi$
$522$	0	0
$523$	3.51476e6	0.561877	0.280939	−	0.959726i	$-0.409354\pi$
$523$	3.51476e6	0.561877	0.280939	+	0.959726i	$0.409354\pi$
$524$	−9.75089e6	−1.55137
$525$	0	0
$526$	2.23424e6	0.352100
$527$	−965952.	−0.151506
$528$	0	0
$529$	−4.19833e6	−0.652285
$530$	2.12080e6	0.327952
$531$	0	0
$532$	0	0
$533$	4.95331e6	0.755227
$534$	0	0
$535$	1.63510e7	2.46979
$536$	437280.	0.0657427
$537$	0	0
$538$	−3.04121e6	−0.452993
$539$	0	0
$540$	0	0
$541$	−3.47547e6	−0.510530	−0.255265	−	0.966871i	$-0.582163\pi$
$541$	−3.47547e6	−0.510530	−0.255265	+	0.966871i	$0.582163\pi$
$542$	2.04077e6	0.298397
$543$	0	0
$544$	772182.	0.111872
$545$	7.19649e6	1.03784
$546$	0	0
$547$	−7.85765e6	−1.12286	−0.561429	−	0.827525i	$-0.689748\pi$
$547$	−7.85765e6	−1.12286	−0.561429	+	0.827525i	$0.689748\pi$
$548$	−2.11574e6	−0.300961
$549$	0	0
$550$	1.41489e6	0.199441
$551$	9.48845e6	1.33142
$552$	0	0
$553$	0	0
$554$	3.79284e6	0.525038
$555$	0	0
$556$	−2.71732e6	−0.372782
$557$	−9.06537e6	−1.23808	−0.619039	−	0.785361i	$-0.712478\pi$
$557$	−9.06537e6	−1.23808	−0.619039	+	0.785361i	$0.712478\pi$
$558$	0	0
$559$	−711330.	−0.0962813
$560$	0	0
$561$	0	0
$562$	−2.63822e6	−0.352347
$563$	−5.80957e6	−0.772455	−0.386227	−	0.922404i	$-0.626222\pi$
$563$	−5.80957e6	−0.772455	−0.386227	+	0.922404i	$0.626222\pi$
$564$	0	0
$565$	−1.02984e7	−1.35722
$566$	957583.	0.125642
$567$	0	0
$568$	4.27584e6	0.556097
$569$	−1.05559e7	−1.36684	−0.683418	−	0.730027i	$-0.739507\pi$
$569$	−1.05559e7	−1.36684	−0.683418	+	0.730027i	$0.739507\pi$
$570$	0	0
$571$	7.99584e6	1.02630	0.513150	−	0.858299i	$-0.328479\pi$
$571$	7.99584e6	1.02630	0.513150	+	0.858299i	$0.328479\pi$
$572$	−4.17146e6	−0.533087
$573$	0	0
$574$	0	0
$575$	−3.72654e6	−0.470041
$576$	0	0
$577$	−2.60941e6	−0.326289	−0.163145	−	0.986602i	$-0.552164\pi$
$577$	−2.60941e6	−0.326289	−0.163145	+	0.986602i	$0.552164\pi$
$578$	−2.79479e6	−0.347960
$579$	0	0
$580$	9.16126e6	1.13080
$581$	0	0
$582$	0	0
$583$	−4.01860e6	−0.489670
$584$	−4.89208e6	−0.593556
$585$	0	0
$586$	−3.00374e6	−0.361342
$587$	−4.41749e6	−0.529151	−0.264576	−	0.964365i	$-0.585232\pi$
$587$	−4.41749e6	−0.529151	−0.264576	+	0.964365i	$0.585232\pi$
$588$	0	0
$589$	−1.40063e7	−1.66355
$590$	−5.60477e6	−0.662869
$591$	0	0
$592$	−8.28528e6	−0.971634
$593$	5.54106e6	0.647077	0.323539	−	0.946215i	$-0.395127\pi$
$593$	5.54106e6	0.647077	0.323539	+	0.946215i	$0.395127\pi$
$594$	0	0
$595$	0	0
$596$	1.01088e7	1.16570
$597$	0	0
$598$	−1.56954e6	−0.179482
$599$	−5.08611e6	−0.579187	−0.289594	−	0.957150i	$-0.593520\pi$
$599$	−5.08611e6	−0.579187	−0.289594	+	0.957150i	$0.593520\pi$
$600$	0	0
$601$	2.41307e6	0.272511	0.136255	−	0.990674i	$-0.456493\pi$
$601$	2.41307e6	0.272511	0.136255	+	0.990674i	$0.456493\pi$
$602$	0	0
$603$	0	0
$604$	−903840.	−0.100809
$605$	6.02480e6	0.669197
$606$	0	0
$607$	7.62170e6	0.839614	0.419807	−	0.907613i	$-0.362098\pi$
$607$	7.62170e6	0.839614	0.419807	+	0.907613i	$0.362098\pi$
$608$	1.11966e7	1.22837
$609$	0	0
$610$	5.33520e6	0.580532
$611$	−5.26781e6	−0.570856
$612$	0	0
$613$	3.60126e6	0.387082	0.193541	−	0.981092i	$-0.438003\pi$
$613$	3.60126e6	0.387082	0.193541	+	0.981092i	$0.438003\pi$
$614$	−2.39313e6	−0.256180
$615$	0	0
$616$	0	0
$617$	−7.22901e6	−0.764480	−0.382240	−	0.924063i	$-0.624847\pi$
$617$	−7.22901e6	−0.764480	−0.382240	+	0.924063i	$0.624847\pi$
$618$	0	0
$619$	−1.25832e7	−1.31998	−0.659988	−	0.751276i	$-0.729439\pi$
$619$	−1.25832e7	−1.31998	−0.659988	+	0.751276i	$0.729439\pi$
$620$	−1.35233e7	−1.41288
$621$	0	0
$622$	2.62949e6	0.272519
$623$	0	0
$624$	0	0
$625$	−1.13449e7	−1.16172
$626$	−5.30545e6	−0.541111
$627$	0	0
$628$	3.72032e6	0.376427
$629$	−1.89298e6	−0.190774
$630$	0	0
$631$	5.98350e6	0.598249	0.299125	−	0.954214i	$-0.403305\pi$
$631$	5.98350e6	0.598249	0.299125	+	0.954214i	$0.403305\pi$
$632$	6.55392e6	0.652692
$633$	0	0
$634$	3.19984e6	0.316159
$635$	−1.27674e7	−1.25651
$636$	0	0
$637$	0	0
$638$	2.47989e6	0.241202
$639$	0	0
$640$	1.39568e7	1.34690
$641$	7.74000e6	0.744040	0.372020	−	0.928225i	$-0.378665\pi$
$641$	7.74000e6	0.744040	0.372020	+	0.928225i	$0.378665\pi$
$642$	0	0
$643$	6.81377e6	0.649920	0.324960	−	0.945728i	$-0.394649\pi$
$643$	6.81377e6	0.649920	0.324960	+	0.945728i	$0.394649\pi$
$644$	0	0
$645$	0	0
$646$	651456.	0.0614191
$647$	1.07121e7	1.00603	0.503017	−	0.864276i	$-0.332223\pi$
$647$	1.07121e7	1.00603	0.503017	+	0.864276i	$0.332223\pi$
$648$	0	0
$649$	1.06202e7	0.989739
$650$	2.61346e6	0.242623
$651$	0	0
$652$	1.71819e6	0.158290
$653$	−1.34167e7	−1.23129	−0.615647	−	0.788022i	$-0.711105\pi$
$653$	−1.34167e7	−1.23129	−0.615647	+	0.788022i	$0.711105\pi$
$654$	0	0
$655$	−2.60976e7	−2.37682
$656$	6.19424e6	0.561990
$657$	0	0
$658$	0	0
$659$	−1.38574e7	−1.24299	−0.621494	−	0.783419i	$-0.713474\pi$
$659$	−1.38574e7	−1.24299	−0.621494	+	0.783419i	$0.713474\pi$
$660$	0	0
$661$	−1.20734e7	−1.07479	−0.537396	−	0.843330i	$-0.680592\pi$
$661$	−1.20734e7	−1.07479	−0.537396	+	0.843330i	$0.680592\pi$
$662$	−223416.	−0.0198139
$663$	0	0
$664$	−62949.6	−0.00554080
$665$	0	0
$666$	0	0
$667$	−6.53154e6	−0.568461
$668$	1.06637e7	0.924624
$669$	0	0
$670$	546163.	0.0470040
$671$	−1.01094e7	−0.866801
$672$	0	0
$673$	−5.32490e6	−0.453183	−0.226592	−	0.973990i	$-0.572758\pi$
$673$	−5.32490e6	−0.453183	−0.226592	+	0.973990i	$0.572758\pi$
$674$	3.18601e6	0.270145
$675$	0	0
$676$	2.69105e6	0.226493
$677$	2.34518e7	1.96655	0.983274	−	0.182135i	$-0.0583008\pi$
$677$	2.34518e7	1.96655	0.983274	+	0.182135i	$0.0583008\pi$
$678$	0	0
$679$	0	0
$680$	1.34784e6	0.111781
$681$	0	0
$682$	−3.66067e6	−0.301370
$683$	−1.82270e7	−1.49508	−0.747538	−	0.664219i	$-0.768764\pi$
$683$	−1.82270e7	−1.49508	−0.747538	+	0.664219i	$0.768764\pi$
$684$	0	0
$685$	−5.66261e6	−0.461095
$686$	0	0
$687$	0	0
$688$	−889536.	−0.0716461
$689$	−7.42280e6	−0.595690
$690$	0	0
$691$	7.60858e6	0.606190	0.303095	−	0.952960i	$-0.401980\pi$
$691$	7.60858e6	0.606190	0.303095	+	0.952960i	$0.401980\pi$
$692$	1.44847e7	1.14986
$693$	0	0
$694$	6.67353e6	0.525965
$695$	−7.27272e6	−0.571130
$696$	0	0
$697$	1.41523e6	0.110343
$698$	−3.20518e6	−0.249009
$699$	0	0
$700$	0	0
$701$	−314162.	−0.0241467	−0.0120734	−	0.999927i	$-0.503843\pi$
$701$	−314162.	−0.0241467	−0.0120734	+	0.999927i	$0.503843\pi$
$702$	0	0
$703$	−2.74483e7	−2.09472
$704$	−3.03539e6	−0.230825
$705$	0	0
$706$	−3.61930e6	−0.273283
$707$	0	0
$708$	0	0
$709$	2.48285e7	1.85496	0.927482	−	0.373869i	$-0.121969\pi$
$709$	2.48285e7	1.85496	0.927482	+	0.373869i	$0.121969\pi$
$710$	5.34052e6	0.397592
$711$	0	0
$712$	−2.44604e6	−0.180827
$713$	9.64148e6	0.710264
$714$	0	0
$715$	−1.11646e7	−0.816730
$716$	1.70958e7	1.24625
$717$	0	0
$718$	−1.84158e6	−0.133315
$719$	−1.71932e7	−1.24032	−0.620160	−	0.784475i	$-0.712932\pi$
$719$	−1.71932e7	−1.24032	−0.620160	+	0.784475i	$0.712932\pi$
$720$	0	0
$721$	0	0
$722$	4.49391e6	0.320835
$723$	0	0
$724$	1.21472e7	0.861249
$725$	1.08757e7	0.768444
$726$	0	0
$727$	−2.12927e7	−1.49415	−0.747076	−	0.664739i	$-0.768543\pi$
$727$	−2.12927e7	−1.49415	−0.747076	+	0.664739i	$0.768543\pi$
$728$	0	0
$729$	0	0
$730$	−6.11021e6	−0.424374
$731$	−203237.	−0.0140673
$732$	0	0
$733$	−1.96024e7	−1.34757	−0.673783	−	0.738930i	$-0.735332\pi$
$733$	−1.96024e7	−1.34757	−0.673783	+	0.738930i	$0.735332\pi$
$734$	2.69604e6	0.184708
$735$	0	0
$736$	−7.70739e6	−0.524461
$737$	−1.03490e6	−0.0701824
$738$	0	0
$739$	1.44181e7	0.971173	0.485587	−	0.874189i	$-0.338606\pi$
$739$	1.44181e7	0.971173	0.485587	+	0.874189i	$0.338606\pi$
$740$	−2.65018e7	−1.77908
$741$	0	0
$742$	0	0
$743$	−5.57521e6	−0.370501	−0.185250	−	0.982691i	$-0.559310\pi$
$743$	−5.57521e6	−0.370501	−0.185250	+	0.982691i	$0.559310\pi$
$744$	0	0
$745$	2.70556e7	1.78594
$746$	6.27547e6	0.412856
$747$	0	0
$748$	−1.19185e6	−0.0778872
$749$	0	0
$750$	0	0
$751$	508800.	0.0329190	0.0164595	−	0.999865i	$-0.494761\pi$
$751$	508800.	0.0329190	0.0164595	+	0.999865i	$0.494761\pi$
$752$	−6.58752e6	−0.424793
$753$	0	0
$754$	4.58063e6	0.293425
$755$	−2.41906e6	−0.154447
$756$	0	0
$757$	1.10466e7	0.700631	0.350316	−	0.936632i	$-0.386074\pi$
$757$	1.10466e7	0.700631	0.350316	+	0.936632i	$0.386074\pi$
$758$	−3.66377e6	−0.231609
$759$	0	0
$760$	1.95437e7	1.22736
$761$	6.77173e6	0.423875	0.211937	−	0.977283i	$-0.432023\pi$
$761$	6.77173e6	0.423875	0.211937	+	0.977283i	$0.432023\pi$
$762$	0	0
$763$	0	0
$764$	9.55338e6	0.592139
$765$	0	0
$766$	55455.6	0.00341486
$767$	1.96167e7	1.20403
$768$	0	0
$769$	−2.48053e7	−1.51261	−0.756307	−	0.654216i	$-0.772999\pi$
$769$	−2.48053e7	−1.51261	−0.756307	+	0.654216i	$0.772999\pi$
$770$	0	0
$771$	0	0
$772$	1.72524e7	1.04185
$773$	1.62952e7	0.980867	0.490434	−	0.871479i	$-0.336838\pi$
$773$	1.62952e7	0.980867	0.490434	+	0.871479i	$0.336838\pi$
$774$	0	0
$775$	−1.60541e7	−0.960133
$776$	−2.20324e7	−1.31343
$777$	0	0
$778$	1.09668e6	0.0649580
$779$	2.05209e7	1.21158
$780$	0	0
$781$	−1.01195e7	−0.593651
$782$	−448441.	−0.0262234
$783$	0	0
$784$	0	0
$785$	9.95717e6	0.576716
$786$	0	0
$787$	−1.44214e7	−0.829984	−0.414992	−	0.909825i	$-0.636216\pi$
$787$	−1.44214e7	−0.829984	−0.414992	+	0.909825i	$0.636216\pi$
$788$	6.48138e6	0.371837
$789$	0	0
$790$	8.18584e6	0.466655
$791$	0	0
$792$	0	0
$793$	−1.86732e7	−1.05447
$794$	−807703.	−0.0454674
$795$	0	0
$796$	1.13435e7	0.634549
$797$	1.10109e6	0.0614014	0.0307007	−	0.999529i	$-0.490226\pi$
$797$	1.10109e6	0.0614014	0.0307007	+	0.999529i	$0.490226\pi$
$798$	0	0
$799$	−1.50509e6	−0.0834055
$800$	1.28336e7	0.708965
$801$	0	0
$802$	8.32855e6	0.457229
$803$	1.15779e7	0.633639
$804$	0	0
$805$	0	0
$806$	−6.76166e6	−0.366620
$807$	0	0
$808$	9.10970e6	0.490881
$809$	2.45146e7	1.31690	0.658450	−	0.752625i	$-0.271213\pi$
$809$	2.45146e7	1.31690	0.658450	+	0.752625i	$0.271213\pi$
$810$	0	0
$811$	3.03580e7	1.62077	0.810383	−	0.585900i	$-0.199259\pi$
$811$	3.03580e7	1.62077	0.810383	+	0.585900i	$0.199259\pi$
$812$	0	0
$813$	0	0
$814$	−7.17384e6	−0.379482
$815$	4.59862e6	0.242512
$816$	0	0
$817$	−2.94694e6	−0.154460
$818$	−5.08243e6	−0.265576
$819$	0	0
$820$	1.98132e7	1.02901
$821$	2.54599e7	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$821$	2.54599e7	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$822$	0	0
$823$	5.95232e6	0.306328	0.153164	−	0.988201i	$-0.451054\pi$
$823$	5.95232e6	0.306328	0.153164	+	0.988201i	$0.451054\pi$
$824$	1.31295e6	0.0673643
$825$	0	0
$826$	0	0
$827$	7.85900e6	0.399580	0.199790	−	0.979839i	$-0.435974\pi$
$827$	7.85900e6	0.399580	0.199790	+	0.979839i	$0.435974\pi$
$828$	0	0
$829$	1.12642e7	0.569262	0.284631	−	0.958637i	$-0.408129\pi$
$829$	1.12642e7	0.569262	0.284631	+	0.958637i	$0.408129\pi$
$830$	−78624.0	−0.00396150
$831$	0	0
$832$	−5.60671e6	−0.280802
$833$	0	0
$834$	0	0
$835$	2.85405e7	1.41659
$836$	−1.72818e7	−0.855209
$837$	0	0
$838$	4.60266e6	0.226412
$839$	7.80470e6	0.382782	0.191391	−	0.981514i	$-0.438700\pi$
$839$	7.80470e6	0.382782	0.191391	+	0.981514i	$0.438700\pi$
$840$	0	0
$841$	−1.44919e6	−0.0706539
$842$	−5.59983e6	−0.272204
$843$	0	0
$844$	−2.17510e7	−1.05105
$845$	7.20241e6	0.347005
$846$	0	0
$847$	0	0
$848$	−9.28240e6	−0.443272
$849$	0	0
$850$	746702.	0.0354487
$851$	1.88945e7	0.894357
$852$	0	0
$853$	1.50581e7	0.708593	0.354296	−	0.935133i	$-0.384720\pi$
$853$	1.50581e7	0.708593	0.354296	+	0.935133i	$0.384720\pi$
$854$	0	0
$855$	0	0
$856$	2.61826e7	1.22132
$857$	−2.28736e7	−1.06386	−0.531928	−	0.846789i	$-0.678532\pi$
$857$	−2.28736e7	−1.06386	−0.531928	+	0.846789i	$0.678532\pi$
$858$	0	0
$859$	8.22863e6	0.380491	0.190246	−	0.981737i	$-0.439072\pi$
$859$	8.22863e6	0.380491	0.190246	+	0.981737i	$0.439072\pi$
$860$	−2.84532e6	−0.131185
$861$	0	0
$862$	−1.90864e6	−0.0874895
$863$	−4.09902e6	−0.187350	−0.0936748	−	0.995603i	$-0.529861\pi$
$863$	−4.09902e6	−0.187350	−0.0936748	+	0.995603i	$0.529861\pi$
$864$	0	0
$865$	3.87672e7	1.76167
$866$	1.03867e6	0.0470633
$867$	0	0
$868$	0	0
$869$	−1.55109e7	−0.696769
$870$	0	0
$871$	−1.91157e6	−0.0853777
$872$	1.15236e7	0.513212
$873$	0	0
$874$	−6.50239e6	−0.287935
$875$	0	0
$876$	0	0
$877$	787778.	0.0345864	0.0172932	−	0.999850i	$-0.494495\pi$
$877$	787778.	0.0345864	0.0172932	+	0.999850i	$0.494495\pi$
$878$	1.19616e7	0.523665
$879$	0	0
$880$	−1.39616e7	−0.607756
$881$	−1.73321e7	−0.752336	−0.376168	−	0.926551i	$-0.622758\pi$
$881$	−1.73321e7	−0.752336	−0.376168	+	0.926551i	$0.622758\pi$
$882$	0	0
$883$	1.23991e7	0.535164	0.267582	−	0.963535i	$-0.413775\pi$
$883$	1.23991e7	0.535164	0.267582	+	0.963535i	$0.413775\pi$
$884$	−2.20147e6	−0.0947507
$885$	0	0
$886$	−7.49641e6	−0.320826
$887$	1.67555e7	0.715071	0.357535	−	0.933900i	$-0.383617\pi$
$887$	1.67555e7	0.715071	0.357535	+	0.933900i	$0.383617\pi$
$888$	0	0
$889$	0	0
$890$	−3.05510e6	−0.129286
$891$	0	0
$892$	−2.29136e6	−0.0964233
$893$	−2.18238e7	−0.915801
$894$	0	0
$895$	4.57557e7	1.90936
$896$	0	0
$897$	0	0
$898$	−198916.	−0.00823150
$899$	−2.81382e7	−1.16117
$900$	0	0
$901$	−2.12080e6	−0.0870338
$902$	5.36330e6	0.219491
$903$	0	0
$904$	−1.64906e7	−0.671145
$905$	3.25110e7	1.31950
$906$	0	0
$907$	−3.96543e7	−1.60056	−0.800280	−	0.599627i	$-0.795316\pi$
$907$	−3.96543e7	−1.60056	−0.800280	+	0.599627i	$0.795316\pi$
$908$	2.11028e7	0.849426
$909$	0	0
$910$	0	0
$911$	−2.99138e7	−1.19419	−0.597097	−	0.802169i	$-0.703679\pi$
$911$	−2.99138e7	−1.19419	−0.597097	+	0.802169i	$0.703679\pi$
$912$	0	0
$913$	148981.	0.00591498
$914$	−323628.	−0.0128139
$915$	0	0
$916$	−736510.	−0.0290028
$917$	0	0
$918$	0	0
$919$	1.98997e7	0.777245	0.388622	−	0.921397i	$-0.372951\pi$
$919$	1.98997e7	0.777245	0.388622	+	0.921397i	$0.372951\pi$
$920$	−1.34532e7	−0.524031
$921$	0	0
$922$	8.98395e6	0.348049
$923$	−1.86918e7	−0.722183
$924$	0	0
$925$	−3.14613e7	−1.20899
$926$	7.18763e6	0.275460
$927$	0	0
$928$	2.24936e7	0.857412
$929$	−1.19663e7	−0.454904	−0.227452	−	0.973789i	$-0.573039\pi$
$929$	−1.19663e7	−0.454904	−0.227452	+	0.973789i	$0.573039\pi$
$930$	0	0
$931$	0	0
$932$	1.33118e6	0.0501991
$933$	0	0
$934$	−4.10446e6	−0.153953
$935$	−3.18989e6	−0.119329
$936$	0	0
$937$	−6.18165e6	−0.230015	−0.115007	−	0.993365i	$-0.536689\pi$
$937$	−6.18165e6	−0.230015	−0.115007	+	0.993365i	$0.536689\pi$
$938$	0	0
$939$	0	0
$940$	−2.10712e7	−0.777804
$941$	2.99426e7	1.10234	0.551171	−	0.834393i	$-0.314181\pi$
$941$	2.99426e7	1.10234	0.551171	+	0.834393i	$0.314181\pi$
$942$	0	0
$943$	−1.41259e7	−0.517293
$944$	2.45312e7	0.895959
$945$	0	0
$946$	−770208.	−0.0279821
$947$	4.84279e7	1.75477	0.877386	−	0.479785i	$-0.159286\pi$
$947$	4.84279e7	1.75477	0.877386	+	0.479785i	$0.159286\pi$
$948$	0	0
$949$	2.13857e7	0.770830
$950$	1.08272e7	0.389230
$951$	0	0
$952$	0	0
$953$	−2.26780e7	−0.808860	−0.404430	−	0.914569i	$-0.632530\pi$
$953$	−2.26780e7	−0.808860	−0.404430	+	0.914569i	$0.632530\pi$
$954$	0	0
$955$	2.55689e7	0.907202
$956$	3.04918e7	1.07904
$957$	0	0
$958$	3.99970e6	0.140803
$959$	0	0
$960$	0	0
$961$	1.29068e7	0.450827
$962$	−1.32509e7	−0.461644
$963$	0	0
$964$	3.77823e7	1.30947
$965$	4.61749e7	1.59620
$966$	0	0
$967$	−3.60431e6	−0.123953	−0.0619764	−	0.998078i	$-0.519740\pi$
$967$	−3.60431e6	−0.123953	−0.0619764	+	0.998078i	$0.519740\pi$
$968$	9.64740e6	0.330919
$969$	0	0
$970$	−2.75184e7	−0.939061
$971$	−2.90807e7	−0.989821	−0.494910	−	0.868944i	$-0.664799\pi$
$971$	−2.90807e7	−0.989821	−0.494910	+	0.868944i	$0.664799\pi$
$972$	0	0
$973$	0	0
$974$	4.34251e6	0.146671
$975$	0	0
$976$	−2.33513e7	−0.784669
$977$	1.81960e7	0.609873	0.304936	−	0.952373i	$-0.401365\pi$
$977$	1.81960e7	0.609873	0.304936	+	0.952373i	$0.401365\pi$
$978$	0	0
$979$	5.78896e6	0.193039
$980$	0	0
$981$	0	0
$982$	−6.09110e6	−0.201566
$983$	3.32808e6	0.109853	0.0549263	−	0.998490i	$-0.482508\pi$
$983$	3.32808e6	0.109853	0.0549263	+	0.998490i	$0.482508\pi$
$984$	0	0
$985$	1.73470e7	0.569682
$986$	1.30875e6	0.0428711
$987$	0	0
$988$	−3.19213e7	−1.04037
$989$	2.02858e6	0.0659478
$990$	0	0
$991$	5.25420e7	1.69951	0.849753	−	0.527181i	$-0.176751\pi$
$991$	5.25420e7	1.69951	0.849753	+	0.527181i	$0.176751\pi$
$992$	−3.32038e7	−1.07129
$993$	0	0
$994$	0	0
$995$	3.03601e7	0.972177
$996$	0	0
$997$	1.22999e6	0.0391889	0.0195945	−	0.999808i	$-0.493762\pi$
$997$	1.22999e6	0.0391889	0.0195945	+	0.999808i	$0.493762\pi$
$998$	−1.59845e7	−0.508011
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.u.1.1		2
3.2	odd	2		49.6.a.c.1.2	yes	2
7.6	odd	2	inner	441.6.a.u.1.2		2
12.11	even	2		784.6.a.z.1.1		2
21.2	odd	6		49.6.c.g.18.1		4
21.5	even	6		49.6.c.g.18.2		4
21.11	odd	6		49.6.c.g.30.1		4
21.17	even	6		49.6.c.g.30.2		4
21.20	even	2		49.6.a.c.1.1	✓	2
84.83	odd	2		784.6.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.a.c.1.1	✓	2	21.20	even	2
49.6.a.c.1.2	yes	2	3.2	odd	2
49.6.c.g.18.1		4	21.2	odd	6
49.6.c.g.18.2		4	21.5	even	6
49.6.c.g.30.1		4	21.11	odd	6
49.6.c.g.30.2		4	21.17	even	6
441.6.a.u.1.1		2	1.1	even	1	trivial
441.6.a.u.1.2		2	7.6	odd	2	inner
784.6.a.z.1.1		2	12.11	even	2
784.6.a.z.1.2		2	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -28.0000 q^{4} -74.9400 q^{5} -120.000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} -28.0000 q^{4} -74.9400 q^{5} -120.000 q^{8} -149.880 q^{10} +284.000 q^{11} +524.580 q^{13} +656.000 q^{16} +149.880 q^{17} +2173.26 q^{19} +2098.32 q^{20} +568.000 q^{22} -1496.00 q^{23} +2491.00 q^{25} +1049.16 q^{26} +4366.00 q^{29} -6444.84 q^{31} +5152.00 q^{32} +299.760 q^{34} -12630.0 q^{37} +4346.52 q^{38} +8992.80 q^{40} +9442.44 q^{41} -1356.00 q^{43} -7952.00 q^{44} -2992.00 q^{46} -10042.0 q^{47} +4982.00 q^{50} -14688.2 q^{52} -14150.0 q^{53} -21283.0 q^{55} +8732.00 q^{58} +37395.0 q^{59} -35596.5 q^{61} -12889.7 q^{62} -10688.0 q^{64} -39312.0 q^{65} -3644.00 q^{67} -4196.64 q^{68} -35632.0 q^{71} +40767.3 q^{73} -25260.0 q^{74} -60851.3 q^{76} -54616.0 q^{79} -49160.6 q^{80} +18884.9 q^{82} +524.580 q^{83} -11232.0 q^{85} -2712.00 q^{86} -34080.0 q^{88} +20383.7 q^{89} +41888.0 q^{92} -20083.9 q^{94} -162864. q^{95} +183603. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 56 q^{4} - 240 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 - 56 * q^4 - 240 * q^8	\( 2 q + 4 q^{2} - 56 q^{4} - 240 q^{8} + 568 q^{11} + 1312 q^{16} + 1136 q^{22} - 2992 q^{23} + 4982 q^{25} + 8732 q^{29} + 10304 q^{32} - 25260 q^{37} - 2712 q^{43} - 15904 q^{44} - 5984 q^{46} + 9964 q^{50} - 28300 q^{53} + 17464 q^{58} - 21376 q^{64} - 78624 q^{65} - 7288 q^{67} - 71264 q^{71} - 50520 q^{74} - 109232 q^{79} - 22464 q^{85} - 5424 q^{86} - 68160 q^{88} + 83776 q^{92} - 325728 q^{95}+O(q^{100}) \) 2 * q + 4 * q^2 - 56 * q^4 - 240 * q^8 + 568 * q^11 + 1312 * q^16 + 1136 * q^22 - 2992 * q^23 + 4982 * q^25 + 8732 * q^29 + 10304 * q^32 - 25260 * q^37 - 2712 * q^43 - 15904 * q^44 - 5984 * q^46 + 9964 * q^50 - 28300 * q^53 + 17464 * q^58 - 21376 * q^64 - 78624 * q^65 - 7288 * q^67 - 71264 * q^71 - 50520 * q^74 - 109232 * q^79 - 22464 * q^85 - 5424 * q^86 - 68160 * q^88 + 83776 * q^92 - 325728 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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