LMFDB - Embedded newform 441.6.a.bc.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 187x^{4} + 9570x^{2} - 135576 $ x^6 - 187x^4 + 9570x^2 - 135576
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{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			1.1
Root			$-10.6223$ 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)

$f(q)$	$=$	$q-10.6223 q^{2} +80.8340 q^{4} +67.4751 q^{5} -518.731 q^{8} +O(q^{10})$	$q-10.6223 q^{2} +80.8340 q^{4} +67.4751 q^{5} -518.731 q^{8} -716.743 q^{10} +522.647 q^{11} +76.6331 q^{13} +2923.45 q^{16} +1269.60 q^{17} -1892.72 q^{19} +5454.28 q^{20} -5551.73 q^{22} -1150.04 q^{23} +1427.89 q^{25} -814.022 q^{26} -3850.03 q^{29} -10413.7 q^{31} -14454.4 q^{32} -13486.1 q^{34} -5602.27 q^{37} +20105.2 q^{38} -35001.4 q^{40} -14232.7 q^{41} -14827.9 q^{43} +42247.6 q^{44} +12216.2 q^{46} -11549.8 q^{47} -15167.5 q^{50} +6194.56 q^{52} -4677.72 q^{53} +35265.6 q^{55} +40896.3 q^{58} +29102.0 q^{59} +11842.9 q^{61} +110618. q^{62} +59989.5 q^{64} +5170.82 q^{65} -36212.6 q^{67} +102627. q^{68} -13477.6 q^{71} +2608.45 q^{73} +59509.2 q^{74} -152996. q^{76} -78320.3 q^{79} +197260. q^{80} +151185. q^{82} +16746.0 q^{83} +85666.5 q^{85} +157507. q^{86} -271113. q^{88} +36768.5 q^{89} -92962.7 q^{92} +122685. q^{94} -127712. q^{95} +36133.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 182 q^{4}+O(q^{10}) $ 6 * q + 182 * q^4	$ 6 q + 182 q^{4} - 686 q^{10} + 154 q^{13} + 1898 q^{16} - 9422 q^{19} - 9146 q^{22} + 7526 q^{25} - 23422 q^{31} - 27804 q^{34} + 18182 q^{37} - 69258 q^{40} - 43686 q^{43} - 25332 q^{46} - 34272 q^{52} - 48160 q^{55} + 89782 q^{58} + 16156 q^{61} + 190290 q^{64} - 144650 q^{67} + 100058 q^{73} - 342720 q^{76} - 101994 q^{79} - 75712 q^{82} + 301176 q^{85} - 752310 q^{88} + 120456 q^{94} - 433048 q^{97}+O(q^{100}) $ 6 * q + 182 * q^4 - 686 * q^10 + 154 * q^13 + 1898 * q^16 - 9422 * q^19 - 9146 * q^22 + 7526 * q^25 - 23422 * q^31 - 27804 * q^34 + 18182 * q^37 - 69258 * q^40 - 43686 * q^43 - 25332 * q^46 - 34272 * q^52 - 48160 * q^55 + 89782 * q^58 + 16156 * q^61 + 190290 * q^64 - 144650 * q^67 + 100058 * q^73 - 342720 * q^76 - 101994 * q^79 - 75712 * q^82 + 301176 * q^85 - 752310 * q^88 + 120456 * q^94 - 433048 * 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 $

$2$	−10.6223	−1.87778	−0.938891	−	0.344216i	$-0.888145\pi$
$2$	−10.6223	−1.87778	−0.938891	+	0.344216i	$0.888145\pi$
$3$	0	0
$3$	0	0
$4$	80.8340	2.52606
$5$	67.4751	1.20703	0.603516	−	0.797351i	$-0.293766\pi$
$5$	67.4751	1.20703	0.603516	+	0.797351i	$0.293766\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−518.731	−2.86561
$9$	0	0
$10$	−716.743	−2.26654
$11$	522.647	1.30235	0.651173	−	0.758929i	$-0.274277\pi$
$11$	522.647	1.30235	0.651173	+	0.758929i	$0.274277\pi$
$12$	0	0
$13$	76.6331	0.125764	0.0628822	−	0.998021i	$-0.479971\pi$
$13$	76.6331	0.125764	0.0628822	+	0.998021i	$0.479971\pi$
$14$	0	0
$15$	0	0
$16$	2923.45	2.85493
$17$	1269.60	1.06548	0.532740	−	0.846279i	$-0.321162\pi$
$17$	1269.60	1.06548	0.532740	+	0.846279i	$0.321162\pi$
$18$	0	0
$19$	−1892.72	−1.20283	−0.601414	−	0.798938i	$-0.705396\pi$
$19$	−1892.72	−1.20283	−0.601414	+	0.798938i	$0.705396\pi$
$20$	5454.28	3.04904
$21$	0	0
$22$	−5551.73	−2.44552
$23$	−1150.04	−0.453310	−0.226655	−	0.973975i	$-0.572779\pi$
$23$	−1150.04	−0.453310	−0.226655	+	0.973975i	$0.572779\pi$
$24$	0	0
$25$	1427.89	0.456925
$26$	−814.022	−0.236158
$27$	0	0
$28$	0	0
$29$	−3850.03	−0.850099	−0.425049	−	0.905170i	$-0.639743\pi$
$29$	−3850.03	−0.850099	−0.425049	+	0.905170i	$0.639743\pi$
$30$	0	0
$31$	−10413.7	−1.94626	−0.973132	−	0.230248i	$-0.926046\pi$
$31$	−10413.7	−1.94626	−0.973132	+	0.230248i	$0.926046\pi$
$32$	−14454.4	−2.49532
$33$	0	0
$34$	−13486.1	−2.00074
$35$	0	0
$36$	0	0
$37$	−5602.27	−0.672760	−0.336380	−	0.941726i	$-0.609203\pi$
$37$	−5602.27	−0.672760	−0.336380	+	0.941726i	$0.609203\pi$
$38$	20105.2	2.25865
$39$	0	0
$40$	−35001.4	−3.45888
$41$	−14232.7	−1.32229	−0.661147	−	0.750256i	$-0.729930\pi$
$41$	−14232.7	−1.32229	−0.661147	+	0.750256i	$0.729930\pi$
$42$	0	0
$43$	−14827.9	−1.22295	−0.611475	−	0.791264i	$-0.709424\pi$
$43$	−14827.9	−1.22295	−0.611475	+	0.791264i	$0.709424\pi$
$44$	42247.6	3.28981
$45$	0	0
$46$	12216.2	0.851216
$47$	−11549.8	−0.762655	−0.381328	−	0.924440i	$-0.624533\pi$
$47$	−11549.8	−0.762655	−0.381328	+	0.924440i	$0.624533\pi$
$48$	0	0
$49$	0	0
$50$	−15167.5	−0.858004
$51$	0	0
$52$	6194.56	0.317689
$53$	−4677.72	−0.228741	−0.114371	−	0.993438i	$-0.536485\pi$
$53$	−4677.72	−0.228741	−0.114371	+	0.993438i	$0.536485\pi$
$54$	0	0
$55$	35265.6	1.57197
$56$	0	0
$57$	0	0
$58$	40896.3	1.59630
$59$	29102.0	1.08841	0.544205	−	0.838952i	$-0.316831\pi$
$59$	29102.0	1.08841	0.544205	+	0.838952i	$0.316831\pi$
$60$	0	0
$61$	11842.9	0.407505	0.203753	−	0.979022i	$-0.434686\pi$
$61$	11842.9	0.407505	0.203753	+	0.979022i	$0.434686\pi$
$62$	110618.	3.65466
$63$	0	0
$64$	59989.5	1.83073
$65$	5170.82	0.151802
$66$	0	0
$67$	−36212.6	−0.985536	−0.492768	−	0.870161i	$-0.664015\pi$
$67$	−36212.6	−0.985536	−0.492768	+	0.870161i	$0.664015\pi$
$68$	102627.	2.69147
$69$	0	0
$70$	0	0
$71$	−13477.6	−0.317298	−0.158649	−	0.987335i	$-0.550714\pi$
$71$	−13477.6	−0.317298	−0.158649	+	0.987335i	$0.550714\pi$
$72$	0	0
$73$	2608.45	0.0572896	0.0286448	−	0.999590i	$-0.490881\pi$
$73$	2608.45	0.0572896	0.0286448	+	0.999590i	$0.490881\pi$
$74$	59509.2	1.26330
$75$	0	0
$76$	−152996.	−3.03842
$77$	0	0
$78$	0	0
$79$	−78320.3	−1.41191	−0.705955	−	0.708257i	$-0.749482\pi$
$79$	−78320.3	−1.41191	−0.705955	+	0.708257i	$0.749482\pi$
$80$	197260.	3.44599
$81$	0	0
$82$	151185.	2.48298
$83$	16746.0	0.266819	0.133410	−	0.991061i	$-0.457407\pi$
$83$	16746.0	0.266819	0.133410	+	0.991061i	$0.457407\pi$
$84$	0	0
$85$	85666.5	1.28607
$86$	157507.	2.29643
$87$	0	0
$88$	−271113.	−3.73202
$89$	36768.5	0.492040	0.246020	−	0.969265i	$-0.420877\pi$
$89$	36768.5	0.492040	0.246020	+	0.969265i	$0.420877\pi$
$90$	0	0
$91$	0	0
$92$	−92962.7	−1.14509
$93$	0	0
$94$	122685.	1.43210
$95$	−127712.	−1.45185
$96$	0	0
$97$	36133.4	0.389924	0.194962	−	0.980811i	$-0.437542\pi$
$97$	36133.4	0.389924	0.194962	+	0.980811i	$0.437542\pi$
$98$	0	0
$99$	0	0
$100$	115422.	1.15422
$101$	−98675.2	−0.962509	−0.481254	−	0.876581i	$-0.659819\pi$
$101$	−98675.2	−0.962509	−0.481254	+	0.876581i	$0.659819\pi$
$102$	0	0
$103$	7273.50	0.0675539	0.0337769	−	0.999429i	$-0.489246\pi$
$103$	7273.50	0.0675539	0.0337769	+	0.999429i	$0.489246\pi$
$104$	−39751.9	−0.360392
$105$	0	0
$106$	49688.3	0.429526
$107$	−30953.5	−0.261367	−0.130683	−	0.991424i	$-0.541717\pi$
$107$	−30953.5	−0.261367	−0.130683	+	0.991424i	$0.541717\pi$
$108$	0	0
$109$	113831.	0.917689	0.458844	−	0.888517i	$-0.348264\pi$
$109$	113831.	0.917689	0.458844	+	0.888517i	$0.348264\pi$
$110$	−374603.	−2.95182
$111$	0	0
$112$	0	0
$113$	−894.559	−0.00659041	−0.00329521	−	0.999995i	$-0.501049\pi$
$113$	−894.559	−0.00659041	−0.00329521	+	0.999995i	$0.501049\pi$
$114$	0	0
$115$	−77599.3	−0.547159
$116$	−311213.	−2.14740
$117$	0	0
$118$	−309131.	−2.04379
$119$	0	0
$120$	0	0
$121$	112109.	0.696106
$122$	−125799.	−0.765206
$123$	0	0
$124$	−841783.	−4.91638
$125$	−114513.	−0.655509
$126$	0	0
$127$	72325.9	0.397910	0.198955	−	0.980009i	$-0.436245\pi$
$127$	72325.9	0.397910	0.198955	+	0.980009i	$0.436245\pi$
$128$	−174687.	−0.942399
$129$	0	0
$130$	−54926.2	−0.285050
$131$	−70168.1	−0.357241	−0.178620	−	0.983918i	$-0.557163\pi$
$131$	−70168.1	−0.357241	−0.178620	+	0.983918i	$0.557163\pi$
$132$	0	0
$133$	0	0
$134$	384662.	1.85062
$135$	0	0
$136$	−658582.	−3.05325
$137$	372009.	1.69337	0.846685	−	0.532095i	$-0.178595\pi$
$137$	372009.	1.69337	0.846685	+	0.532095i	$0.178595\pi$
$138$	0	0
$139$	5566.26	0.0244358	0.0122179	−	0.999925i	$-0.496111\pi$
$139$	5566.26	0.0244358	0.0122179	+	0.999925i	$0.496111\pi$
$140$	0	0
$141$	0	0
$142$	143164.	0.595816
$143$	40052.0	0.163789
$144$	0	0
$145$	−259781.	−1.02610
$146$	−27707.8	−0.107577
$147$	0	0
$148$	−452854.	−1.69943
$149$	−333914.	−1.23217	−0.616083	−	0.787681i	$-0.711281\pi$
$149$	−333914.	−1.23217	−0.616083	+	0.787681i	$0.711281\pi$
$150$	0	0
$151$	75157.3	0.268243	0.134122	−	0.990965i	$-0.457179\pi$
$151$	75157.3	0.268243	0.134122	+	0.990965i	$0.457179\pi$
$152$	981815.	3.44684
$153$	0	0
$154$	0	0
$155$	−702667.	−2.34920
$156$	0	0
$157$	475585.	1.53985	0.769926	−	0.638134i	$-0.220293\pi$
$157$	475585.	1.53985	0.769926	+	0.638134i	$0.220293\pi$
$158$	831945.	2.65126
$159$	0	0
$160$	−975314.	−3.01193
$161$	0	0
$162$	0	0
$163$	239223.	0.705234	0.352617	−	0.935768i	$-0.385292\pi$
$163$	239223.	0.705234	0.352617	+	0.935768i	$0.385292\pi$
$164$	−1.15049e6	−3.34020
$165$	0	0
$166$	−177882.	−0.501028
$167$	−111308.	−0.308842	−0.154421	−	0.988005i	$-0.549351\pi$
$167$	−111308.	−0.308842	−0.154421	+	0.988005i	$0.549351\pi$
$168$	0	0
$169$	−365420.	−0.984183
$170$	−909978.	−2.41495
$171$	0	0
$172$	−1.19860e6	−3.08925
$173$	−698994.	−1.77565	−0.887826	−	0.460179i	$-0.847785\pi$
$173$	−698994.	−1.77565	−0.887826	+	0.460179i	$0.847785\pi$
$174$	0	0
$175$	0	0
$176$	1.52793e6	3.71810
$177$	0	0
$178$	−390567.	−0.923944
$179$	103909.	0.242393	0.121196	−	0.992629i	$-0.461327\pi$
$179$	103909.	0.242393	0.121196	+	0.992629i	$0.461327\pi$
$180$	0	0
$181$	−415260.	−0.942159	−0.471079	−	0.882091i	$-0.656135\pi$
$181$	−415260.	−0.942159	−0.471079	+	0.882091i	$0.656135\pi$
$182$	0	0
$183$	0	0
$184$	596563.	1.29901
$185$	−378014.	−0.812042
$186$	0	0
$187$	663553.	1.38762
$188$	−933613.	−1.92651
$189$	0	0
$190$	1.35660e6	2.72626
$191$	570116.	1.13078	0.565392	−	0.824823i	$-0.308725\pi$
$191$	570116.	1.13078	0.565392	+	0.824823i	$0.308725\pi$
$192$	0	0
$193$	297709.	0.575305	0.287653	−	0.957735i	$-0.407125\pi$
$193$	297709.	0.575305	0.287653	+	0.957735i	$0.407125\pi$
$194$	−383821.	−0.732192
$195$	0	0
$196$	0	0
$197$	410678.	0.753938	0.376969	−	0.926226i	$-0.376966\pi$
$197$	410678.	0.753938	0.376969	+	0.926226i	$0.376966\pi$
$198$	0	0
$199$	−699345.	−1.25187	−0.625934	−	0.779876i	$-0.715282\pi$
$199$	−699345.	−1.25187	−0.625934	+	0.779876i	$0.715282\pi$
$200$	−740690.	−1.30937
$201$	0	0
$202$	1.04816e6	1.80738
$203$	0	0
$204$	0	0
$205$	−960354.	−1.59605
$206$	−77261.5	−0.126851
$207$	0	0
$208$	224033.	0.359048
$209$	−989226.	−1.56650
$210$	0	0
$211$	−58292.9	−0.0901384	−0.0450692	−	0.998984i	$-0.514351\pi$
$211$	−58292.9	−0.0901384	−0.0450692	+	0.998984i	$0.514351\pi$
$212$	−378119.	−0.577815
$213$	0	0
$214$	328799.	0.490790
$215$	−1.00051e6	−1.47614
$216$	0	0
$217$	0	0
$218$	−1.20915e6	−1.72322
$219$	0	0
$220$	2.85066e6	3.97090
$221$	97293.5	0.133999
$222$	0	0
$223$	25758.5	0.0346864	0.0173432	−	0.999850i	$-0.494479\pi$
$223$	25758.5	0.0346864	0.0173432	+	0.999850i	$0.494479\pi$
$224$	0	0
$225$	0	0
$226$	9502.30	0.0123754
$227$	−982245.	−1.26519	−0.632594	−	0.774483i	$-0.718010\pi$
$227$	−982245.	−1.26519	−0.632594	+	0.774483i	$0.718010\pi$
$228$	0	0
$229$	519963.	0.655215	0.327607	−	0.944814i	$-0.393758\pi$
$229$	519963.	0.655215	0.327607	+	0.944814i	$0.393758\pi$
$230$	824286.	1.02744
$231$	0	0
$232$	1.99713e6	2.43605
$233$	1.21941e6	1.47149	0.735747	−	0.677257i	$-0.236831\pi$
$233$	1.21941e6	1.47149	0.735747	+	0.677257i	$0.236831\pi$
$234$	0	0
$235$	−779321.	−0.920548
$236$	2.35243e6	2.74939
$237$	0	0
$238$	0	0
$239$	−909357.	−1.02977	−0.514884	−	0.857260i	$-0.672165\pi$
$239$	−909357.	−1.02977	−0.514884	+	0.857260i	$0.672165\pi$
$240$	0	0
$241$	−641596.	−0.711572	−0.355786	−	0.934567i	$-0.615787\pi$
$241$	−641596.	−0.711572	−0.355786	+	0.934567i	$0.615787\pi$
$242$	−1.19086e6	−1.30714
$243$	0	0
$244$	957308.	1.02938
$245$	0	0
$246$	0	0
$247$	−145045.	−0.151273
$248$	5.40192e6	5.57723
$249$	0	0
$250$	1.21639e6	1.23090
$251$	1.92052e6	1.92413	0.962066	−	0.272816i	$-0.0879550\pi$
$251$	1.92052e6	1.92413	0.962066	+	0.272816i	$0.0879550\pi$
$252$	0	0
$253$	−601067.	−0.590366
$254$	−768270.	−0.747188
$255$	0	0
$256$	−64082.7	−0.0611140
$257$	−986369.	−0.931551	−0.465775	−	0.884903i	$-0.654225\pi$
$257$	−986369.	−0.931551	−0.465775	+	0.884903i	$0.654225\pi$
$258$	0	0
$259$	0	0
$260$	417978.	0.383460
$261$	0	0
$262$	745349.	0.670820
$263$	849019.	0.756881	0.378441	−	0.925626i	$-0.376460\pi$
$263$	849019.	0.756881	0.378441	+	0.925626i	$0.376460\pi$
$264$	0	0
$265$	−315630.	−0.276098
$266$	0	0
$267$	0	0
$268$	−2.92721e6	−2.48952
$269$	−1.73358e6	−1.46071	−0.730355	−	0.683068i	$-0.760645\pi$
$269$	−1.73358e6	−1.46071	−0.730355	+	0.683068i	$0.760645\pi$
$270$	0	0
$271$	−1.05565e6	−0.873167	−0.436583	−	0.899664i	$-0.643812\pi$
$271$	−1.05565e6	−0.873167	−0.436583	+	0.899664i	$0.643812\pi$
$272$	3.71161e6	3.04187
$273$	0	0
$274$	−3.95160e6	−3.17978
$275$	746282.	0.595074
$276$	0	0
$277$	1.62493e6	1.27243	0.636216	−	0.771511i	$-0.280499\pi$
$277$	1.62493e6	1.27243	0.636216	+	0.771511i	$0.280499\pi$
$278$	−59126.7	−0.0458851
$279$	0	0
$280$	0	0
$281$	−1.56484e6	−1.18224	−0.591118	−	0.806585i	$-0.701313\pi$
$281$	−1.56484e6	−1.18224	−0.591118	+	0.806585i	$0.701313\pi$
$282$	0	0
$283$	−694213.	−0.515260	−0.257630	−	0.966244i	$-0.582942\pi$
$283$	−694213.	−0.515260	−0.257630	+	0.966244i	$0.582942\pi$
$284$	−1.08945e6	−0.801515
$285$	0	0
$286$	−425446.	−0.307560
$287$	0	0
$288$	0	0
$289$	192032.	0.135247
$290$	2.75948e6	1.92678
$291$	0	0
$292$	210851.	0.144717
$293$	999262.	0.680002	0.340001	−	0.940425i	$-0.389573\pi$
$293$	999262.	0.680002	0.340001	+	0.940425i	$0.389573\pi$
$294$	0	0
$295$	1.96366e6	1.31374
$296$	2.90607e6	1.92787
$297$	0	0
$298$	3.54695e6	2.31374
$299$	−88131.4	−0.0570102
$300$	0	0
$301$	0	0
$302$	−798346.	−0.503702
$303$	0	0
$304$	−5.53328e6	−3.43399
$305$	799100.	0.491872
$306$	0	0
$307$	800764.	0.484907	0.242453	−	0.970163i	$-0.422048\pi$
$307$	800764.	0.484907	0.242453	+	0.970163i	$0.422048\pi$
$308$	0	0
$309$	0	0
$310$	7.46396e6	4.41129
$311$	−582944.	−0.341764	−0.170882	−	0.985292i	$-0.554662\pi$
$311$	−582944.	−0.341764	−0.170882	+	0.985292i	$0.554662\pi$
$312$	0	0
$313$	−1.99909e6	−1.15338	−0.576689	−	0.816963i	$-0.695656\pi$
$313$	−1.99909e6	−1.15338	−0.576689	+	0.816963i	$0.695656\pi$
$314$	−5.05182e6	−2.89150
$315$	0	0
$316$	−6.33095e6	−3.56657
$317$	1.80208e6	1.00722	0.503610	−	0.863931i	$-0.332005\pi$
$317$	1.80208e6	1.00722	0.503610	+	0.863931i	$0.332005\pi$
$318$	0	0
$319$	−2.01221e6	−1.10712
$320$	4.04780e6	2.20975
$321$	0	0
$322$	0	0
$323$	−2.40301e6	−1.28159
$324$	0	0
$325$	109424.	0.0574649
$326$	−2.54110e6	−1.32428
$327$	0	0
$328$	7.38295e6	3.78918
$329$	0	0
$330$	0	0
$331$	103771.	0.0520603	0.0260301	−	0.999661i	$-0.491713\pi$
$331$	103771.	0.0520603	0.0260301	+	0.999661i	$0.491713\pi$
$332$	1.35365e6	0.674001
$333$	0	0
$334$	1.18235e6	0.579937
$335$	−2.44345e6	−1.18957
$336$	0	0
$337$	961978.	0.461414	0.230707	−	0.973023i	$-0.425896\pi$
$337$	961978.	0.461414	0.230707	+	0.973023i	$0.425896\pi$
$338$	3.88162e6	1.84808
$339$	0	0
$340$	6.92476e6	3.24869
$341$	−5.44270e6	−2.53471
$342$	0	0
$343$	0	0
$344$	7.69169e6	3.50450
$345$	0	0
$346$	7.42494e6	3.33429
$347$	2.43348e6	1.08494	0.542468	−	0.840077i	$-0.317490\pi$
$347$	2.43348e6	1.08494	0.542468	+	0.840077i	$0.317490\pi$
$348$	0	0
$349$	−3.42269e6	−1.50419	−0.752096	−	0.659053i	$-0.770957\pi$
$349$	−3.42269e6	−1.50419	−0.752096	+	0.659053i	$0.770957\pi$
$350$	0	0
$351$	0	0
$352$	−7.55456e6	−3.24977
$353$	3.85497e6	1.64658	0.823292	−	0.567618i	$-0.192135\pi$
$353$	3.85497e6	1.64658	0.823292	+	0.567618i	$0.192135\pi$
$354$	0	0
$355$	−909404.	−0.382989
$356$	2.97214e6	1.24292
$357$	0	0
$358$	−1.10375e6	−0.455161
$359$	−2.48369e6	−1.01709	−0.508547	−	0.861034i	$-0.669817\pi$
$359$	−2.48369e6	−1.01709	−0.508547	+	0.861034i	$0.669817\pi$
$360$	0	0
$361$	1.10631e6	0.446795
$362$	4.41104e6	1.76917
$363$	0	0
$364$	0	0
$365$	176005.	0.0691503
$366$	0	0
$367$	−4.07224e6	−1.57822	−0.789111	−	0.614250i	$-0.789458\pi$
$367$	−4.07224e6	−1.57822	−0.789111	+	0.614250i	$0.789458\pi$
$368$	−3.36209e6	−1.29417
$369$	0	0
$370$	4.01539e6	1.52484
$371$	0	0
$372$	0	0
$373$	−4.69272e6	−1.74644	−0.873218	−	0.487329i	$-0.837971\pi$
$373$	−4.69272e6	−1.74644	−0.873218	+	0.487329i	$0.837971\pi$
$374$	−7.04848e6	−2.60565
$375$	0	0
$376$	5.99121e6	2.18547
$377$	−295040.	−0.106912
$378$	0	0
$379$	5.04599e6	1.80446	0.902232	−	0.431251i	$-0.141928\pi$
$379$	5.04599e6	1.80446	0.902232	+	0.431251i	$0.141928\pi$
$380$	−1.03235e7	−3.66747
$381$	0	0
$382$	−6.05596e6	−2.12336
$383$	1.65284e6	0.575749	0.287874	−	0.957668i	$-0.407051\pi$
$383$	1.65284e6	0.575749	0.287874	+	0.957668i	$0.407051\pi$
$384$	0	0
$385$	0	0
$386$	−3.16236e6	−1.08030
$387$	0	0
$388$	2.92081e6	0.984972
$389$	2.91455e6	0.976556	0.488278	−	0.872688i	$-0.337625\pi$
$389$	2.91455e6	0.976556	0.488278	+	0.872688i	$0.337625\pi$
$390$	0	0
$391$	−1.46010e6	−0.482992
$392$	0	0
$393$	0	0
$394$	−4.36235e6	−1.41573
$395$	−5.28467e6	−1.70422
$396$	0	0
$397$	−474943.	−0.151239	−0.0756197	−	0.997137i	$-0.524094\pi$
$397$	−474943.	−0.151239	−0.0756197	+	0.997137i	$0.524094\pi$
$398$	7.42867e6	2.35073
$399$	0	0
$400$	4.17436e6	1.30449
$401$	48881.8	0.0151805	0.00759025	−	0.999971i	$-0.497584\pi$
$401$	48881.8	0.0151805	0.00759025	+	0.999971i	$0.497584\pi$
$402$	0	0
$403$	−798035.	−0.244771
$404$	−7.97631e6	−2.43136
$405$	0	0
$406$	0	0
$407$	−2.92801e6	−0.876166
$408$	0	0
$409$	3.73894e6	1.10520	0.552599	−	0.833447i	$-0.313636\pi$
$409$	3.73894e6	1.10520	0.552599	+	0.833447i	$0.313636\pi$
$410$	1.02012e7	2.99703
$411$	0	0
$412$	587946.	0.170645
$413$	0	0
$414$	0	0
$415$	1.12994e6	0.322059
$416$	−1.10769e6	−0.313822
$417$	0	0
$418$	1.05079e7	2.94154
$419$	−681425.	−0.189619	−0.0948097	−	0.995495i	$-0.530224\pi$
$419$	−681425.	−0.189619	−0.0948097	+	0.995495i	$0.530224\pi$
$420$	0	0
$421$	1.54803e6	0.425670	0.212835	−	0.977088i	$-0.431730\pi$
$421$	1.54803e6	0.425670	0.212835	+	0.977088i	$0.431730\pi$
$422$	619207.	0.169260
$423$	0	0
$424$	2.42648e6	0.655483
$425$	1.81285e6	0.486844
$426$	0	0
$427$	0	0
$428$	−2.50210e6	−0.660229
$429$	0	0
$430$	1.06278e7	2.77187
$431$	−6.37710e6	−1.65360	−0.826799	−	0.562497i	$-0.809841\pi$
$431$	−6.37710e6	−1.65360	−0.826799	+	0.562497i	$0.809841\pi$
$432$	0	0
$433$	1.32669e6	0.340056	0.170028	−	0.985439i	$-0.445614\pi$
$433$	1.32669e6	0.340056	0.170028	+	0.985439i	$0.445614\pi$
$434$	0	0
$435$	0	0
$436$	9.20144e6	2.31814
$437$	2.17672e6	0.545253
$438$	0	0
$439$	3.85328e6	0.954267	0.477133	−	0.878831i	$-0.341676\pi$
$439$	3.85328e6	0.954267	0.477133	+	0.878831i	$0.341676\pi$
$440$	−1.82934e7	−4.50466
$441$	0	0
$442$	−1.03348e6	−0.251622
$443$	−1.29104e6	−0.312559	−0.156279	−	0.987713i	$-0.549950\pi$
$443$	−1.29104e6	−0.312559	−0.156279	+	0.987713i	$0.549950\pi$
$444$	0	0
$445$	2.48096e6	0.593908
$446$	−273616.	−0.0651334
$447$	0	0
$448$	0	0
$449$	7.49006e6	1.75335	0.876677	−	0.481080i	$-0.159755\pi$
$449$	7.49006e6	1.75335	0.876677	+	0.481080i	$0.159755\pi$
$450$	0	0
$451$	−7.43868e6	−1.72208
$452$	−72310.7	−0.0166478
$453$	0	0
$454$	1.04337e7	2.37575
$455$	0	0
$456$	0	0
$457$	499323.	0.111839	0.0559193	−	0.998435i	$-0.482191\pi$
$457$	499323.	0.111839	0.0559193	+	0.998435i	$0.482191\pi$
$458$	−5.52322e6	−1.23035
$459$	0	0
$460$	−6.27266e6	−1.38216
$461$	−5.34926e6	−1.17231	−0.586153	−	0.810200i	$-0.699358\pi$
$461$	−5.34926e6	−1.17231	−0.586153	+	0.810200i	$0.699358\pi$
$462$	0	0
$463$	−6.52516e6	−1.41462	−0.707308	−	0.706906i	$-0.750091\pi$
$463$	−6.52516e6	−1.41462	−0.707308	+	0.706906i	$0.750091\pi$
$464$	−1.12554e7	−2.42697
$465$	0	0
$466$	−1.29529e7	−2.76314
$467$	−3.20648e6	−0.680357	−0.340179	−	0.940361i	$-0.610488\pi$
$467$	−3.20648e6	−0.680357	−0.340179	+	0.940361i	$0.610488\pi$
$468$	0	0
$469$	0	0
$470$	8.27821e6	1.72859
$471$	0	0
$472$	−1.50961e7	−3.11896
$473$	−7.74976e6	−1.59270
$474$	0	0
$475$	−2.70260e6	−0.549602
$476$	0	0
$477$	0	0
$478$	9.65949e6	1.93368
$479$	−8.86811e6	−1.76601	−0.883003	−	0.469367i	$-0.844482\pi$
$479$	−8.86811e6	−1.76601	−0.883003	+	0.469367i	$0.844482\pi$
$480$	0	0
$481$	−429319.	−0.0846092
$482$	6.81525e6	1.33618
$483$	0	0
$484$	9.06218e6	1.75841
$485$	2.43811e6	0.470650
$486$	0	0
$487$	5.84722e6	1.11719	0.558595	−	0.829441i	$-0.311341\pi$
$487$	5.84722e6	1.11719	0.558595	+	0.829441i	$0.311341\pi$
$488$	−6.14327e6	−1.16775
$489$	0	0
$490$	0	0
$491$	−114241.	−0.0213854	−0.0106927	−	0.999943i	$-0.503404\pi$
$491$	−114241.	−0.0213854	−0.0106927	+	0.999943i	$0.503404\pi$
$492$	0	0
$493$	−4.88801e6	−0.905763
$494$	1.54072e6	0.284057
$495$	0	0
$496$	−3.04439e7	−5.55644
$497$	0	0
$498$	0	0
$499$	−9.34482e6	−1.68004	−0.840020	−	0.542556i	$-0.817457\pi$
$499$	−9.34482e6	−1.68004	−0.840020	+	0.542556i	$0.817457\pi$
$500$	−9.25652e6	−1.65586
$501$	0	0
$502$	−2.04004e7	−3.61310
$503$	−892118.	−0.157218	−0.0786091	−	0.996906i	$-0.525048\pi$
$503$	−892118.	−0.157218	−0.0786091	+	0.996906i	$0.525048\pi$
$504$	0	0
$505$	−6.65812e6	−1.16178
$506$	6.38473e6	1.10858
$507$	0	0
$508$	5.84639e6	1.00515
$509$	6.84023e6	1.17024	0.585122	−	0.810946i	$-0.301047\pi$
$509$	6.84023e6	1.17024	0.585122	+	0.810946i	$0.301047\pi$
$510$	0	0
$511$	0	0
$512$	6.27068e6	1.05716
$513$	0	0
$514$	1.04775e7	1.74925
$515$	490780.	0.0815396
$516$	0	0
$517$	−6.03644e6	−0.993241
$518$	0	0
$519$	0	0
$520$	−2.68227e6	−0.435004
$521$	8.91887e6	1.43951	0.719756	−	0.694227i	$-0.244254\pi$
$521$	8.91887e6	1.43951	0.719756	+	0.694227i	$0.244254\pi$
$522$	0	0
$523$	611823.	0.0978073	0.0489037	−	0.998803i	$-0.484427\pi$
$523$	611823.	0.0978073	0.0489037	+	0.998803i	$0.484427\pi$
$524$	−5.67196e6	−0.902413
$525$	0	0
$526$	−9.01856e6	−1.42126
$527$	−1.32213e7	−2.07370
$528$	0	0
$529$	−5.11374e6	−0.794510
$530$	3.35272e6	0.518451
$531$	0	0
$532$	0	0
$533$	−1.09070e6	−0.166298
$534$	0	0
$535$	−2.08859e6	−0.315478
$536$	1.87846e7	2.82416
$537$	0	0
$538$	1.84147e7	2.74289
$539$	0	0
$540$	0	0
$541$	−244343.	−0.0358928	−0.0179464	−	0.999839i	$-0.505713\pi$
$541$	−244343.	−0.0358928	−0.0179464	+	0.999839i	$0.505713\pi$
$542$	1.12135e7	1.63962
$543$	0	0
$544$	−1.83514e7	−2.65871
$545$	7.68078e6	1.10768
$546$	0	0
$547$	−7.55551e6	−1.07968	−0.539840	−	0.841767i	$-0.681515\pi$
$547$	−7.55551e6	−1.07968	−0.539840	+	0.841767i	$0.681515\pi$
$548$	3.00709e7	4.27756
$549$	0	0
$550$	−7.92725e6	−1.11742
$551$	7.28705e6	1.02252
$552$	0	0
$553$	0	0
$554$	−1.72605e7	−2.38935
$555$	0	0
$556$	449943.	0.0617264
$557$	2.57276e6	0.351368	0.175684	−	0.984447i	$-0.443786\pi$
$557$	2.57276e6	0.351368	0.175684	+	0.984447i	$0.443786\pi$
$558$	0	0
$559$	−1.13631e6	−0.153804
$560$	0	0
$561$	0	0
$562$	1.66223e7	2.21998
$563$	1.10693e7	1.47181	0.735903	−	0.677087i	$-0.236758\pi$
$563$	1.10693e7	1.47181	0.735903	+	0.677087i	$0.236758\pi$
$564$	0	0
$565$	−60360.4	−0.00795484
$566$	7.37416e6	0.967546
$567$	0	0
$568$	6.99126e6	0.909253
$569$	3.11210e6	0.402970	0.201485	−	0.979492i	$-0.435423\pi$
$569$	3.11210e6	0.402970	0.201485	+	0.979492i	$0.435423\pi$
$570$	0	0
$571$	5.73724e6	0.736398	0.368199	−	0.929747i	$-0.379975\pi$
$571$	5.73724e6	0.736398	0.368199	+	0.929747i	$0.379975\pi$
$572$	3.23756e6	0.413741
$573$	0	0
$574$	0	0
$575$	−1.64214e6	−0.207128
$576$	0	0
$577$	−1.11306e7	−1.39180	−0.695901	−	0.718138i	$-0.744995\pi$
$577$	−1.11306e7	−1.39180	−0.695901	+	0.718138i	$0.744995\pi$
$578$	−2.03982e6	−0.253964
$579$	0	0
$580$	−2.09992e7	−2.59198
$581$	0	0
$582$	0	0
$583$	−2.44479e6	−0.297900
$584$	−1.35308e6	−0.164170
$585$	0	0
$586$	−1.06145e7	−1.27690
$587$	7.71211e6	0.923800	0.461900	−	0.886932i	$-0.347168\pi$
$587$	7.71211e6	0.923800	0.461900	+	0.886932i	$0.347168\pi$
$588$	0	0
$589$	1.97103e7	2.34102
$590$	−2.08586e7	−2.46692
$591$	0	0
$592$	−1.63779e7	−1.92068
$593$	8.33301e6	0.973117	0.486558	−	0.873648i	$-0.338252\pi$
$593$	8.33301e6	0.973117	0.486558	+	0.873648i	$0.338252\pi$
$594$	0	0
$595$	0	0
$596$	−2.69916e7	−3.11253
$597$	0	0
$598$	936161.	0.107053
$599$	5.33581e6	0.607621	0.303811	−	0.952732i	$-0.401741\pi$
$599$	5.33581e6	0.607621	0.303811	+	0.952732i	$0.401741\pi$
$600$	0	0
$601$	−7.53972e6	−0.851470	−0.425735	−	0.904848i	$-0.639984\pi$
$601$	−7.53972e6	−0.851470	−0.425735	+	0.904848i	$0.639984\pi$
$602$	0	0
$603$	0	0
$604$	6.07526e6	0.677599
$605$	7.56454e6	0.840222
$606$	0	0
$607$	−9.68565e6	−1.06698	−0.533491	−	0.845806i	$-0.679120\pi$
$607$	−9.68565e6	−1.06698	−0.533491	+	0.845806i	$0.679120\pi$
$608$	2.73582e7	3.00144
$609$	0	0
$610$	−8.48831e6	−0.923627
$611$	−885093.	−0.0959149
$612$	0	0
$613$	1.36761e7	1.46998	0.734989	−	0.678079i	$-0.237187\pi$
$613$	1.36761e7	1.46998	0.734989	+	0.678079i	$0.237187\pi$
$614$	−8.50598e6	−0.910549
$615$	0	0
$616$	0	0
$617$	1.53796e7	1.62642	0.813211	−	0.581969i	$-0.197717\pi$
$617$	1.53796e7	1.62642	0.813211	+	0.581969i	$0.197717\pi$
$618$	0	0
$619$	5.41608e6	0.568144	0.284072	−	0.958803i	$-0.408315\pi$
$619$	5.41608e6	0.568144	0.284072	+	0.958803i	$0.408315\pi$
$620$	−5.67994e7	−5.93423
$621$	0	0
$622$	6.19223e6	0.641757
$623$	0	0
$624$	0	0
$625$	−1.21889e7	−1.24814
$626$	2.12350e7	2.16579
$627$	0	0
$628$	3.84434e7	3.88976
$629$	−7.11266e6	−0.716812
$630$	0	0
$631$	1.44178e7	1.44154	0.720770	−	0.693174i	$-0.243788\pi$
$631$	1.44178e7	1.44154	0.720770	+	0.693174i	$0.243788\pi$
$632$	4.06272e7	4.04598
$633$	0	0
$634$	−1.91422e7	−1.89134
$635$	4.88020e6	0.480290
$636$	0	0
$637$	0	0
$638$	2.13743e7	2.07893
$639$	0	0
$640$	−1.17870e7	−1.13751
$641$	6.13817e6	0.590057	0.295029	−	0.955488i	$-0.404671\pi$
$641$	6.13817e6	0.590057	0.295029	+	0.955488i	$0.404671\pi$
$642$	0	0
$643$	−402177.	−0.0383610	−0.0191805	−	0.999816i	$-0.506106\pi$
$643$	−402177.	−0.0383610	−0.0191805	+	0.999816i	$0.506106\pi$
$644$	0	0
$645$	0	0
$646$	2.55255e7	2.40654
$647$	1.32816e7	1.24736	0.623678	−	0.781682i	$-0.285638\pi$
$647$	1.32816e7	1.24736	0.623678	+	0.781682i	$0.285638\pi$
$648$	0	0
$649$	1.52100e7	1.41749
$650$	−1.16233e6	−0.107906
$651$	0	0
$652$	1.93373e7	1.78147
$653$	1.12731e7	1.03457	0.517287	−	0.855812i	$-0.326942\pi$
$653$	1.12731e7	1.03457	0.517287	+	0.855812i	$0.326942\pi$
$654$	0	0
$655$	−4.73460e6	−0.431201
$656$	−4.16086e7	−3.77505
$657$	0	0
$658$	0	0
$659$	9.29624e6	0.833861	0.416930	−	0.908938i	$-0.363106\pi$
$659$	9.29624e6	0.833861	0.416930	+	0.908938i	$0.363106\pi$
$660$	0	0
$661$	−628772.	−0.0559744	−0.0279872	−	0.999608i	$-0.508910\pi$
$661$	−628772.	−0.0559744	−0.0279872	+	0.999608i	$0.508910\pi$
$662$	−1.10229e6	−0.0977578
$663$	0	0
$664$	−8.68669e6	−0.764599
$665$	0	0
$666$	0	0
$667$	4.42771e6	0.385358
$668$	−8.99749e6	−0.780153
$669$	0	0
$670$	2.59551e7	2.23376
$671$	6.18965e6	0.530713
$672$	0	0
$673$	5.15635e6	0.438838	0.219419	−	0.975631i	$-0.429584\pi$
$673$	5.15635e6	0.438838	0.219419	+	0.975631i	$0.429584\pi$
$674$	−1.02185e7	−0.866434
$675$	0	0
$676$	−2.95384e7	−2.48611
$677$	−6.56297e6	−0.550337	−0.275169	−	0.961396i	$-0.588734\pi$
$677$	−6.56297e6	−0.550337	−0.275169	+	0.961396i	$0.588734\pi$
$678$	0	0
$679$	0	0
$680$	−4.44379e7	−3.68537
$681$	0	0
$682$	5.78141e7	4.75963
$683$	−2.01533e7	−1.65308	−0.826540	−	0.562879i	$-0.809694\pi$
$683$	−2.01533e7	−1.65308	−0.826540	+	0.562879i	$0.809694\pi$
$684$	0	0
$685$	2.51013e7	2.04395
$686$	0	0
$687$	0	0
$688$	−4.33486e7	−3.49143
$689$	−358468.	−0.0287675
$690$	0	0
$691$	−1.01196e7	−0.806246	−0.403123	−	0.915146i	$-0.632075\pi$
$691$	−1.01196e7	−0.806246	−0.403123	+	0.915146i	$0.632075\pi$
$692$	−5.65024e7	−4.48541
$693$	0	0
$694$	−2.58492e7	−2.03727
$695$	375584.	0.0294948
$696$	0	0
$697$	−1.80699e7	−1.40888
$698$	3.63569e7	2.82455
$699$	0	0
$700$	0	0
$701$	−3.27446e6	−0.251678	−0.125839	−	0.992051i	$-0.540162\pi$
$701$	−3.27446e6	−0.251678	−0.125839	+	0.992051i	$0.540162\pi$
$702$	0	0
$703$	1.06036e7	0.809214
$704$	3.13533e7	2.38425
$705$	0	0
$706$	−4.09488e7	−3.09193
$707$	0	0
$708$	0	0
$709$	5.99446e6	0.447852	0.223926	−	0.974606i	$-0.428113\pi$
$709$	5.99446e6	0.447852	0.223926	+	0.974606i	$0.428113\pi$
$710$	9.65999e6	0.719169
$711$	0	0
$712$	−1.90729e7	−1.41000
$713$	1.19762e7	0.882260
$714$	0	0
$715$	2.70251e6	0.197698
$716$	8.39936e6	0.612299
$717$	0	0
$718$	2.63826e7	1.90988
$719$	2.47737e7	1.78718	0.893590	−	0.448885i	$-0.148179\pi$
$719$	2.47737e7	1.78718	0.893590	+	0.448885i	$0.148179\pi$
$720$	0	0
$721$	0	0
$722$	−1.17516e7	−0.838983
$723$	0	0
$724$	−3.35672e7	−2.37995
$725$	−5.49742e6	−0.388431
$726$	0	0
$727$	−1.16194e7	−0.815357	−0.407678	−	0.913126i	$-0.633662\pi$
$727$	−1.16194e7	−0.815357	−0.407678	+	0.913126i	$0.633662\pi$
$728$	0	0
$729$	0	0
$730$	−1.86959e6	−0.129849
$731$	−1.88255e7	−1.30303
$732$	0	0
$733$	−1.27434e7	−0.876041	−0.438020	−	0.898965i	$-0.644320\pi$
$733$	−1.27434e7	−0.876041	−0.438020	+	0.898965i	$0.644320\pi$
$734$	4.32567e7	2.96356
$735$	0	0
$736$	1.66232e7	1.13115
$737$	−1.89264e7	−1.28351
$738$	0	0
$739$	1.20918e7	0.814477	0.407238	−	0.913322i	$-0.366492\pi$
$739$	1.20918e7	0.814477	0.407238	+	0.913322i	$0.366492\pi$
$740$	−3.05564e7	−2.05127
$741$	0	0
$742$	0	0
$743$	9.79136e6	0.650685	0.325343	−	0.945596i	$-0.394520\pi$
$743$	9.79136e6	0.650685	0.325343	+	0.945596i	$0.394520\pi$
$744$	0	0
$745$	−2.25309e7	−1.48726
$746$	4.98477e7	3.27943
$747$	0	0
$748$	5.36376e7	3.50522
$749$	0	0
$750$	0	0
$751$	−1.75303e7	−1.13420	−0.567098	−	0.823650i	$-0.691934\pi$
$751$	−1.75303e7	−1.13420	−0.567098	+	0.823650i	$0.691934\pi$
$752$	−3.37651e7	−2.17732
$753$	0	0
$754$	3.13401e6	0.200758
$755$	5.07125e6	0.323778
$756$	0	0
$757$	6.01558e6	0.381538	0.190769	−	0.981635i	$-0.438902\pi$
$757$	6.01558e6	0.381538	0.190769	+	0.981635i	$0.438902\pi$
$758$	−5.36002e7	−3.38839
$759$	0	0
$760$	6.62481e7	4.16044
$761$	1.27680e7	0.799214	0.399607	−	0.916687i	$-0.369147\pi$
$761$	1.27680e7	0.799214	0.399607	+	0.916687i	$0.369147\pi$
$762$	0	0
$763$	0	0
$764$	4.60847e7	2.85643
$765$	0	0
$766$	−1.75570e7	−1.08113
$767$	2.23017e6	0.136883
$768$	0	0
$769$	−6.32718e6	−0.385829	−0.192914	−	0.981216i	$-0.561794\pi$
$769$	−6.32718e6	−0.385829	−0.192914	+	0.981216i	$0.561794\pi$
$770$	0	0
$771$	0	0
$772$	2.40650e7	1.45326
$773$	−1.62017e7	−0.975240	−0.487620	−	0.873056i	$-0.662135\pi$
$773$	−1.62017e7	−0.975240	−0.487620	+	0.873056i	$0.662135\pi$
$774$	0	0
$775$	−1.48696e7	−0.889296
$776$	−1.87435e7	−1.11737
$777$	0	0
$778$	−3.09593e7	−1.83376
$779$	2.69386e7	1.59049
$780$	0	0
$781$	−7.04404e6	−0.413232
$782$	1.55096e7	0.906954
$783$	0	0
$784$	0	0
$785$	3.20901e7	1.85865
$786$	0	0
$787$	−1.96547e7	−1.13118	−0.565588	−	0.824688i	$-0.691351\pi$
$787$	−1.96547e7	−1.13118	−0.565588	+	0.824688i	$0.691351\pi$
$788$	3.31967e7	1.90449
$789$	0	0
$790$	5.61356e7	3.20015
$791$	0	0
$792$	0	0
$793$	907557.	0.0512497
$794$	5.04500e6	0.283995
$795$	0	0
$796$	−5.65308e7	−3.16230
$797$	6.39709e6	0.356728	0.178364	−	0.983965i	$-0.442920\pi$
$797$	6.39709e6	0.356728	0.178364	+	0.983965i	$0.442920\pi$
$798$	0	0
$799$	−1.46636e7	−0.812593
$800$	−2.06393e7	−1.14017
$801$	0	0
$802$	−519238.	−0.0285056
$803$	1.36330e6	0.0746108
$804$	0	0
$805$	0	0
$806$	8.47700e6	0.459626
$807$	0	0
$808$	5.11859e7	2.75817
$809$	−2.38705e7	−1.28230	−0.641152	−	0.767414i	$-0.721543\pi$
$809$	−2.38705e7	−1.28230	−0.641152	+	0.767414i	$0.721543\pi$
$810$	0	0
$811$	−3.15356e7	−1.68364	−0.841819	−	0.539759i	$-0.818515\pi$
$811$	−3.15356e7	−1.68364	−0.841819	+	0.539759i	$0.818515\pi$
$812$	0	0
$813$	0	0
$814$	3.11023e7	1.64525
$815$	1.61416e7	0.851240
$816$	0	0
$817$	2.80651e7	1.47100
$818$	−3.97163e7	−2.07532
$819$	0	0
$820$	−7.76292e7	−4.03172
$821$	−2.34305e7	−1.21317	−0.606587	−	0.795017i	$-0.707462\pi$
$821$	−2.34305e7	−1.21317	−0.606587	+	0.795017i	$0.707462\pi$
$822$	0	0
$823$	3.71839e7	1.91362	0.956810	−	0.290712i	$-0.0938923\pi$
$823$	3.71839e7	1.91362	0.956810	+	0.290712i	$0.0938923\pi$
$824$	−3.77299e6	−0.193583
$825$	0	0
$826$	0	0
$827$	−494429.	−0.0251386	−0.0125693	−	0.999921i	$-0.504001\pi$
$827$	−494429.	−0.0251386	−0.0125693	+	0.999921i	$0.504001\pi$
$828$	0	0
$829$	−1.33424e7	−0.674293	−0.337147	−	0.941452i	$-0.609462\pi$
$829$	−1.33424e7	−0.674293	−0.337147	+	0.941452i	$0.609462\pi$
$830$	−1.20026e7	−0.604756
$831$	0	0
$832$	4.59718e6	0.230241
$833$	0	0
$834$	0	0
$835$	−7.51053e6	−0.372782
$836$	−7.99631e7	−3.95707
$837$	0	0
$838$	7.23832e6	0.356064
$839$	−1.74957e7	−0.858075	−0.429038	−	0.903287i	$-0.641147\pi$
$839$	−1.74957e7	−0.858075	−0.429038	+	0.903287i	$0.641147\pi$
$840$	0	0
$841$	−5.68840e6	−0.277332
$842$	−1.64437e7	−0.799316
$843$	0	0
$844$	−4.71205e6	−0.227695
$845$	−2.46568e7	−1.18794
$846$	0	0
$847$	0	0
$848$	−1.36751e7	−0.653040
$849$	0	0
$850$	−1.92567e7	−0.914186
$851$	6.44286e6	0.304968
$852$	0	0
$853$	1.83938e7	0.865565	0.432782	−	0.901498i	$-0.357532\pi$
$853$	1.83938e7	0.865565	0.432782	+	0.901498i	$0.357532\pi$
$854$	0	0
$855$	0	0
$856$	1.60565e7	0.748975
$857$	3.65151e6	0.169832	0.0849161	−	0.996388i	$-0.472938\pi$
$857$	3.65151e6	0.169832	0.0849161	+	0.996388i	$0.472938\pi$
$858$	0	0
$859$	−5.12785e6	−0.237111	−0.118556	−	0.992947i	$-0.537826\pi$
$859$	−5.12785e6	−0.237111	−0.118556	+	0.992947i	$0.537826\pi$
$860$	−8.08756e7	−3.72882
$861$	0	0
$862$	6.77397e7	3.10510
$863$	1.96489e7	0.898074	0.449037	−	0.893513i	$-0.351767\pi$
$863$	1.96489e7	0.898074	0.449037	+	0.893513i	$0.351767\pi$
$864$	0	0
$865$	−4.71647e7	−2.14327
$866$	−1.40926e7	−0.638551
$867$	0	0
$868$	0	0
$869$	−4.09339e7	−1.83880
$870$	0	0
$871$	−2.77508e6	−0.123945
$872$	−5.90478e7	−2.62974
$873$	0	0
$874$	−2.31218e7	−1.02387
$875$	0	0
$876$	0	0
$877$	−3.07559e7	−1.35030	−0.675150	−	0.737681i	$-0.735921\pi$
$877$	−3.07559e7	−1.35030	−0.675150	+	0.737681i	$0.735921\pi$
$878$	−4.09309e7	−1.79190
$879$	0	0
$880$	1.03097e8	4.48787
$881$	−3.34518e7	−1.45204	−0.726022	−	0.687671i	$-0.758633\pi$
$881$	−3.34518e7	−1.45204	−0.726022	+	0.687671i	$0.758633\pi$
$882$	0	0
$883$	−2.87952e7	−1.24285	−0.621425	−	0.783474i	$-0.713446\pi$
$883$	−2.87952e7	−1.24285	−0.621425	+	0.783474i	$0.713446\pi$
$884$	7.86462e6	0.338491
$885$	0	0
$886$	1.37139e7	0.586917
$887$	−1.68365e7	−0.718527	−0.359264	−	0.933236i	$-0.616972\pi$
$887$	−1.68365e7	−0.718527	−0.359264	+	0.933236i	$0.616972\pi$
$888$	0	0
$889$	0	0
$890$	−2.63536e7	−1.11523
$891$	0	0
$892$	2.08217e6	0.0876200
$893$	2.18605e7	0.917343
$894$	0	0
$895$	7.01126e6	0.292576
$896$	0	0
$897$	0	0
$898$	−7.95619e7	−3.29241
$899$	4.00932e7	1.65452
$900$	0	0
$901$	−5.93884e6	−0.243719
$902$	7.90161e7	3.23370
$903$	0	0
$904$	464035.	0.0188856
$905$	−2.80197e7	−1.13722
$906$	0	0
$907$	9.75606e6	0.393782	0.196891	−	0.980425i	$-0.436915\pi$
$907$	9.75606e6	0.393782	0.196891	+	0.980425i	$0.436915\pi$
$908$	−7.93988e7	−3.19594
$909$	0	0
$910$	0	0
$911$	3.67934e7	1.46884	0.734420	−	0.678696i	$-0.237454\pi$
$911$	3.67934e7	1.46884	0.734420	+	0.678696i	$0.237454\pi$
$912$	0	0
$913$	8.75226e6	0.347491
$914$	−5.30398e6	−0.210008
$915$	0	0
$916$	4.20307e7	1.65511
$917$	0	0
$918$	0	0
$919$	1.72178e7	0.672495	0.336248	−	0.941774i	$-0.390842\pi$
$919$	1.72178e7	0.672495	0.336248	+	0.941774i	$0.390842\pi$
$920$	4.02532e7	1.56794
$921$	0	0
$922$	5.68216e7	2.20133
$923$	−1.03283e6	−0.0399048
$924$	0	0
$925$	−7.99943e6	−0.307400
$926$	6.93124e7	2.65634
$927$	0	0
$928$	5.56500e7	2.12127
$929$	4.90559e7	1.86489	0.932443	−	0.361318i	$-0.117673\pi$
$929$	4.90559e7	1.86489	0.932443	+	0.361318i	$0.117673\pi$
$930$	0	0
$931$	0	0
$932$	9.85694e7	3.71708
$933$	0	0
$934$	3.40604e7	1.27756
$935$	4.47733e7	1.67491
$936$	0	0
$937$	−2.80929e7	−1.04532	−0.522658	−	0.852542i	$-0.675060\pi$
$937$	−2.80929e7	−1.04532	−0.522658	+	0.852542i	$0.675060\pi$
$938$	0	0
$939$	0	0
$940$	−6.29956e7	−2.32536
$941$	−2.28001e7	−0.839389	−0.419695	−	0.907665i	$-0.637863\pi$
$941$	−2.28001e7	−0.839389	−0.419695	+	0.907665i	$0.637863\pi$
$942$	0	0
$943$	1.63682e7	0.599409
$944$	8.50780e7	3.10733
$945$	0	0
$946$	8.23205e7	2.99075
$947$	−4.35503e7	−1.57803	−0.789016	−	0.614372i	$-0.789409\pi$
$947$	−4.35503e7	−1.57803	−0.789016	+	0.614372i	$0.789409\pi$
$948$	0	0
$949$	199894.	0.00720499
$950$	2.87079e7	1.03203
$951$	0	0
$952$	0	0
$953$	−4.54239e7	−1.62014	−0.810069	−	0.586335i	$-0.800570\pi$
$953$	−4.54239e7	−1.62014	−0.810069	+	0.586335i	$0.800570\pi$
$954$	0	0
$955$	3.84686e7	1.36489
$956$	−7.35069e7	−2.60126
$957$	0	0
$958$	9.42000e7	3.31617
$959$	0	0
$960$	0	0
$961$	7.98164e7	2.78794
$962$	4.56037e6	0.158878
$963$	0	0
$964$	−5.18628e7	−1.79748
$965$	2.00879e7	0.694411
$966$	0	0
$967$	5.43677e6	0.186971	0.0934855	−	0.995621i	$-0.470199\pi$
$967$	5.43677e6	0.186971	0.0934855	+	0.995621i	$0.470199\pi$
$968$	−5.81542e7	−1.99477
$969$	0	0
$970$	−2.58984e7	−0.883779
$971$	−3.62705e7	−1.23454	−0.617270	−	0.786751i	$-0.711762\pi$
$971$	−3.62705e7	−1.23454	−0.617270	+	0.786751i	$0.711762\pi$
$972$	0	0
$973$	0	0
$974$	−6.21111e7	−2.09784
$975$	0	0
$976$	3.46220e7	1.16340
$977$	5.07312e6	0.170035	0.0850175	−	0.996379i	$-0.472905\pi$
$977$	5.07312e6	0.170035	0.0850175	+	0.996379i	$0.472905\pi$
$978$	0	0
$979$	1.92169e7	0.640807
$980$	0	0
$981$	0	0
$982$	1.21350e6	0.0401571
$983$	915660.	0.0302239	0.0151119	−	0.999886i	$-0.495190\pi$
$983$	915660.	0.0302239	0.0151119	+	0.999886i	$0.495190\pi$
$984$	0	0
$985$	2.77105e7	0.910026
$986$	5.19220e7	1.70082
$987$	0	0
$988$	−1.17246e7	−0.382125
$989$	1.70527e7	0.554375
$990$	0	0
$991$	−2.83986e7	−0.918571	−0.459285	−	0.888289i	$-0.651894\pi$
$991$	−2.83986e7	−0.918571	−0.459285	+	0.888289i	$0.651894\pi$
$992$	1.50524e8	4.85655
$993$	0	0
$994$	0	0
$995$	−4.71884e7	−1.51104
$996$	0	0
$997$	5.44486e7	1.73480	0.867399	−	0.497613i	$-0.165790\pi$
$997$	5.44486e7	1.73480	0.867399	+	0.497613i	$0.165790\pi$
$998$	9.92638e7	3.15475
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bc.1.1		6
3.2	odd	2	inner	441.6.a.bc.1.6		6
7.2	even	3		63.6.e.f.46.6	yes	12
7.4	even	3		63.6.e.f.37.6	yes	12
7.6	odd	2		441.6.a.bd.1.1		6
21.2	odd	6		63.6.e.f.46.1	yes	12
21.11	odd	6		63.6.e.f.37.1	✓	12
21.20	even	2		441.6.a.bd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.e.f.37.1	✓	12	21.11	odd	6
63.6.e.f.37.6	yes	12	7.4	even	3
63.6.e.f.46.1	yes	12	21.2	odd	6
63.6.e.f.46.6	yes	12	7.2	even	3
441.6.a.bc.1.1		6	1.1	even	1	trivial
441.6.a.bc.1.6		6	3.2	odd	2	inner
441.6.a.bd.1.1		6	7.6	odd	2
441.6.a.bd.1.6		6	21.20	even	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)

\(f(q)\)	\(=\)	\(q-10.6223 q^{2} +80.8340 q^{4} +67.4751 q^{5} -518.731 q^{8} +O(q^{10})\)	\(q-10.6223 q^{2} +80.8340 q^{4} +67.4751 q^{5} -518.731 q^{8} -716.743 q^{10} +522.647 q^{11} +76.6331 q^{13} +2923.45 q^{16} +1269.60 q^{17} -1892.72 q^{19} +5454.28 q^{20} -5551.73 q^{22} -1150.04 q^{23} +1427.89 q^{25} -814.022 q^{26} -3850.03 q^{29} -10413.7 q^{31} -14454.4 q^{32} -13486.1 q^{34} -5602.27 q^{37} +20105.2 q^{38} -35001.4 q^{40} -14232.7 q^{41} -14827.9 q^{43} +42247.6 q^{44} +12216.2 q^{46} -11549.8 q^{47} -15167.5 q^{50} +6194.56 q^{52} -4677.72 q^{53} +35265.6 q^{55} +40896.3 q^{58} +29102.0 q^{59} +11842.9 q^{61} +110618. q^{62} +59989.5 q^{64} +5170.82 q^{65} -36212.6 q^{67} +102627. q^{68} -13477.6 q^{71} +2608.45 q^{73} +59509.2 q^{74} -152996. q^{76} -78320.3 q^{79} +197260. q^{80} +151185. q^{82} +16746.0 q^{83} +85666.5 q^{85} +157507. q^{86} -271113. q^{88} +36768.5 q^{89} -92962.7 q^{92} +122685. q^{94} -127712. q^{95} +36133.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 182 q^{4}+O(q^{10}) \) 6 * q + 182 * q^4	\( 6 q + 182 q^{4} - 686 q^{10} + 154 q^{13} + 1898 q^{16} - 9422 q^{19} - 9146 q^{22} + 7526 q^{25} - 23422 q^{31} - 27804 q^{34} + 18182 q^{37} - 69258 q^{40} - 43686 q^{43} - 25332 q^{46} - 34272 q^{52} - 48160 q^{55} + 89782 q^{58} + 16156 q^{61} + 190290 q^{64} - 144650 q^{67} + 100058 q^{73} - 342720 q^{76} - 101994 q^{79} - 75712 q^{82} + 301176 q^{85} - 752310 q^{88} + 120456 q^{94} - 433048 q^{97}+O(q^{100}) \) 6 * q + 182 * q^4 - 686 * q^10 + 154 * q^13 + 1898 * q^16 - 9422 * q^19 - 9146 * q^22 + 7526 * q^25 - 23422 * q^31 - 27804 * q^34 + 18182 * q^37 - 69258 * q^40 - 43686 * q^43 - 25332 * q^46 - 34272 * q^52 - 48160 * q^55 + 89782 * q^58 + 16156 * q^61 + 190290 * q^64 - 144650 * q^67 + 100058 * q^73 - 342720 * q^76 - 101994 * q^79 - 75712 * q^82 + 301176 * q^85 - 752310 * q^88 + 120456 * q^94 - 433048 * q^97

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