LMFDB - Embedded newform 441.6.a.l.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{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ 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)

$f(q)$	$=$	$q-8.27492 q^{2} +36.4743 q^{4} +28.7492 q^{5} -37.0241 q^{8} +O(q^{10})$	$q-8.27492 q^{2} +36.4743 q^{4} +28.7492 q^{5} -37.0241 q^{8} -237.897 q^{10} +270.090 q^{11} -300.640 q^{13} -860.805 q^{16} +613.106 q^{17} +1700.95 q^{19} +1048.60 q^{20} -2234.97 q^{22} -3188.15 q^{23} -2298.49 q^{25} +2487.77 q^{26} -4299.28 q^{29} -2028.46 q^{31} +8307.86 q^{32} -5073.40 q^{34} +5154.46 q^{37} -14075.2 q^{38} -1064.41 q^{40} -7146.21 q^{41} -19584.3 q^{43} +9851.32 q^{44} +26381.7 q^{46} +19998.4 q^{47} +19019.8 q^{50} -10965.6 q^{52} -3948.82 q^{53} +7764.86 q^{55} +35576.2 q^{58} -29707.6 q^{59} +50519.3 q^{61} +16785.3 q^{62} -41201.1 q^{64} -8643.14 q^{65} +5053.56 q^{67} +22362.6 q^{68} -32853.3 q^{71} +11115.0 q^{73} -42652.7 q^{74} +62040.8 q^{76} +81889.4 q^{79} -24747.4 q^{80} +59134.3 q^{82} +118234. q^{83} +17626.3 q^{85} +162058. q^{86} -9999.83 q^{88} -41695.4 q^{89} -116286. q^{92} -165485. q^{94} +48900.9 q^{95} -43682.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8}+O(q^{10}) $ 2 * q - 9 * q^2 + 5 * q^4 - 18 * q^5 + 9 * q^8	$ 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8} - 204 q^{10} - 396 q^{11} + 350 q^{13} + 113 q^{16} + 1800 q^{17} + 3266 q^{19} + 2520 q^{20} - 1752 q^{22} - 2088 q^{23} - 3238 q^{25} + 2016 q^{26} - 6696 q^{29} + 20 q^{31} + 6129 q^{32} - 5934 q^{34} + 6232 q^{37} - 15210 q^{38} - 3216 q^{40} - 6048 q^{41} - 3020 q^{43} + 30816 q^{44} + 25584 q^{46} + 11700 q^{47} + 19701 q^{50} - 31444 q^{52} - 9468 q^{53} + 38904 q^{55} + 37314 q^{58} - 43938 q^{59} + 64754 q^{61} + 15300 q^{62} - 70783 q^{64} - 39060 q^{65} + 24784 q^{67} - 14994 q^{68} - 97416 q^{71} - 17452 q^{73} - 43434 q^{74} + 12782 q^{76} + 51256 q^{79} - 70272 q^{80} + 58338 q^{82} + 117558 q^{83} - 37860 q^{85} + 150048 q^{86} - 40656 q^{88} + 84276 q^{89} - 150912 q^{92} - 159468 q^{94} - 24264 q^{95} - 20776 q^{97}+O(q^{100}) $ 2 * q - 9 * q^2 + 5 * q^4 - 18 * q^5 + 9 * q^8 - 204 * q^10 - 396 * q^11 + 350 * q^13 + 113 * q^16 + 1800 * q^17 + 3266 * q^19 + 2520 * q^20 - 1752 * q^22 - 2088 * q^23 - 3238 * q^25 + 2016 * q^26 - 6696 * q^29 + 20 * q^31 + 6129 * q^32 - 5934 * q^34 + 6232 * q^37 - 15210 * q^38 - 3216 * q^40 - 6048 * q^41 - 3020 * q^43 + 30816 * q^44 + 25584 * q^46 + 11700 * q^47 + 19701 * q^50 - 31444 * q^52 - 9468 * q^53 + 38904 * q^55 + 37314 * q^58 - 43938 * q^59 + 64754 * q^61 + 15300 * q^62 - 70783 * q^64 - 39060 * q^65 + 24784 * q^67 - 14994 * q^68 - 97416 * q^71 - 17452 * q^73 - 43434 * q^74 + 12782 * q^76 + 51256 * q^79 - 70272 * q^80 + 58338 * q^82 + 117558 * q^83 - 37860 * q^85 + 150048 * q^86 - 40656 * q^88 + 84276 * q^89 - 150912 * q^92 - 159468 * q^94 - 24264 * q^95 - 20776 * 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 $

$2$	−8.27492	−1.46281	−0.731406	−	0.681942i	$-0.761136\pi$
$2$	−8.27492	−1.46281	−0.731406	+	0.681942i	$0.761136\pi$
$3$	0	0
$3$	0	0
$4$	36.4743	1.13982
$5$	28.7492	0.514281	0.257140	−	0.966374i	$-0.417220\pi$
$5$	28.7492	0.514281	0.257140	+	0.966374i	$0.417220\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−37.0241	−0.204531
$9$	0	0
$10$	−237.897	−0.752296
$11$	270.090	0.673018	0.336509	−	0.941680i	$-0.390754\pi$
$11$	270.090	0.673018	0.336509	+	0.941680i	$0.390754\pi$
$12$	0	0
$13$	−300.640	−0.493387	−0.246694	−	0.969094i	$-0.579344\pi$
$13$	−300.640	−0.493387	−0.246694	+	0.969094i	$0.579344\pi$
$14$	0	0
$15$	0	0
$16$	−860.805	−0.840630
$17$	613.106	0.514533	0.257267	−	0.966340i	$-0.417178\pi$
$17$	613.106	0.514533	0.257267	+	0.966340i	$0.417178\pi$
$18$	0	0
$19$	1700.95	1.08095	0.540477	−	0.841359i	$-0.318244\pi$
$19$	1700.95	1.08095	0.540477	+	0.841359i	$0.318244\pi$
$20$	1048.60	0.586188
$21$	0	0
$22$	−2234.97	−0.984498
$23$	−3188.15	−1.25667	−0.628333	−	0.777945i	$-0.716262\pi$
$23$	−3188.15	−1.25667	−0.628333	+	0.777945i	$0.716262\pi$
$24$	0	0
$25$	−2298.49	−0.735515
$26$	2487.77	0.721733
$27$	0	0
$28$	0	0
$29$	−4299.28	−0.949294	−0.474647	−	0.880176i	$-0.657424\pi$
$29$	−4299.28	−0.949294	−0.474647	+	0.880176i	$0.657424\pi$
$30$	0	0
$31$	−2028.46	−0.379106	−0.189553	−	0.981870i	$-0.560704\pi$
$31$	−2028.46	−0.379106	−0.189553	+	0.981870i	$0.560704\pi$
$32$	8307.86	1.43421
$33$	0	0
$34$	−5073.40	−0.752666
$35$	0	0
$36$	0	0
$37$	5154.46	0.618983	0.309491	−	0.950902i	$-0.399841\pi$
$37$	5154.46	0.618983	0.309491	+	0.950902i	$0.399841\pi$
$38$	−14075.2	−1.58123
$39$	0	0
$40$	−1064.41	−0.105186
$41$	−7146.21	−0.663921	−0.331960	−	0.943293i	$-0.607710\pi$
$41$	−7146.21	−0.663921	−0.331960	+	0.943293i	$0.607710\pi$
$42$	0	0
$43$	−19584.3	−1.61524	−0.807620	−	0.589703i	$-0.799245\pi$
$43$	−19584.3	−1.61524	−0.807620	+	0.589703i	$0.799245\pi$
$44$	9851.32	0.767119
$45$	0	0
$46$	26381.7	1.83827
$47$	19998.4	1.32054	0.660268	−	0.751030i	$-0.270443\pi$
$47$	19998.4	1.32054	0.660268	+	0.751030i	$0.270443\pi$
$48$	0	0
$49$	0	0
$50$	19019.8	1.07592
$51$	0	0
$52$	−10965.6	−0.562373
$53$	−3948.82	−0.193098	−0.0965489	−	0.995328i	$-0.530780\pi$
$53$	−3948.82	−0.193098	−0.0965489	+	0.995328i	$0.530780\pi$
$54$	0	0
$55$	7764.86	0.346120
$56$	0	0
$57$	0	0
$58$	35576.2	1.38864
$59$	−29707.6	−1.11106	−0.555530	−	0.831497i	$-0.687484\pi$
$59$	−29707.6	−1.11106	−0.555530	+	0.831497i	$0.687484\pi$
$60$	0	0
$61$	50519.3	1.73833	0.869165	−	0.494522i	$-0.164657\pi$
$61$	50519.3	1.73833	0.869165	+	0.494522i	$0.164657\pi$
$62$	16785.3	0.554562
$63$	0	0
$64$	−41201.1	−1.25736
$65$	−8643.14	−0.253740
$66$	0	0
$67$	5053.56	0.137534	0.0687671	−	0.997633i	$-0.478093\pi$
$67$	5053.56	0.137534	0.0687671	+	0.997633i	$0.478093\pi$
$68$	22362.6	0.586476
$69$	0	0
$70$	0	0
$71$	−32853.3	−0.773453	−0.386726	−	0.922195i	$-0.626394\pi$
$71$	−32853.3	−0.773453	−0.386726	+	0.922195i	$0.626394\pi$
$72$	0	0
$73$	11115.0	0.244119	0.122059	−	0.992523i	$-0.461050\pi$
$73$	11115.0	0.244119	0.122059	+	0.992523i	$0.461050\pi$
$74$	−42652.7	−0.905456
$75$	0	0
$76$	62040.8	1.23209
$77$	0	0
$78$	0	0
$79$	81889.4	1.47625	0.738125	−	0.674664i	$-0.235712\pi$
$79$	81889.4	1.47625	0.738125	+	0.674664i	$0.235712\pi$
$80$	−24747.4	−0.432320
$81$	0	0
$82$	59134.3	0.971191
$83$	118234.	1.88385	0.941926	−	0.335819i	$-0.109013\pi$
$83$	118234.	1.88385	0.941926	+	0.335819i	$0.109013\pi$
$84$	0	0
$85$	17626.3	0.264615
$86$	162058.	2.36279
$87$	0	0
$88$	−9999.83	−0.137653
$89$	−41695.4	−0.557972	−0.278986	−	0.960295i	$-0.589998\pi$
$89$	−41695.4	−0.557972	−0.278986	+	0.960295i	$0.589998\pi$
$90$	0	0
$91$	0	0
$92$	−116286.	−1.43237
$93$	0	0
$94$	−165485.	−1.93170
$95$	48900.9	0.555914
$96$	0	0
$97$	−43682.8	−0.471391	−0.235695	−	0.971827i	$-0.575737\pi$
$97$	−43682.8	−0.471391	−0.235695	+	0.971827i	$0.575737\pi$
$98$	0	0
$99$	0	0
$100$	−83835.5	−0.838355
$101$	25648.1	0.250179	0.125090	−	0.992145i	$-0.460078\pi$
$101$	25648.1	0.250179	0.125090	+	0.992145i	$0.460078\pi$
$102$	0	0
$103$	14320.0	0.133000	0.0664999	−	0.997786i	$-0.478817\pi$
$103$	14320.0	0.133000	0.0664999	+	0.997786i	$0.478817\pi$
$104$	11130.9	0.100913
$105$	0	0
$106$	32676.1	0.282466
$107$	−17201.8	−0.145249	−0.0726247	−	0.997359i	$-0.523138\pi$
$107$	−17201.8	−0.145249	−0.0726247	+	0.997359i	$0.523138\pi$
$108$	0	0
$109$	−86017.6	−0.693459	−0.346730	−	0.937965i	$-0.612708\pi$
$109$	−86017.6	−0.693459	−0.346730	+	0.937965i	$0.612708\pi$
$110$	−64253.5	−0.506309
$111$	0	0
$112$	0	0
$113$	−137568.	−1.01349	−0.506745	−	0.862096i	$-0.669152\pi$
$113$	−137568.	−1.01349	−0.506745	+	0.862096i	$0.669152\pi$
$114$	0	0
$115$	−91656.8	−0.646279
$116$	−156813.	−1.08202
$117$	0	0
$118$	245828.	1.62527
$119$	0	0
$120$	0	0
$121$	−88102.5	−0.547047
$122$	−418043.	−2.54285
$123$	0	0
$124$	−73986.4	−0.432113
$125$	−155921.	−0.892542
$126$	0	0
$127$	−70567.1	−0.388233	−0.194117	−	0.980978i	$-0.562184\pi$
$127$	−70567.1	−0.388233	−0.194117	+	0.980978i	$0.562184\pi$
$128$	75084.2	0.405064
$129$	0	0
$130$	71521.3	0.371173
$131$	−173712.	−0.884408	−0.442204	−	0.896914i	$-0.645803\pi$
$131$	−173712.	−0.884408	−0.442204	+	0.896914i	$0.645803\pi$
$132$	0	0
$133$	0	0
$134$	−41817.8	−0.201187
$135$	0	0
$136$	−22699.7	−0.105238
$137$	1989.94	0.00905813	0.00452907	−	0.999990i	$-0.498558\pi$
$137$	1989.94	0.00905813	0.00452907	+	0.999990i	$0.498558\pi$
$138$	0	0
$139$	−366409.	−1.60853	−0.804264	−	0.594272i	$-0.797440\pi$
$139$	−366409.	−1.60853	−0.804264	+	0.594272i	$0.797440\pi$
$140$	0	0
$141$	0	0
$142$	271859.	1.13142
$143$	−81199.7	−0.332058
$144$	0	0
$145$	−123601.	−0.488204
$146$	−91975.4	−0.357100
$147$	0	0
$148$	188005.	0.705529
$149$	−140719.	−0.519261	−0.259631	−	0.965708i	$-0.583601\pi$
$149$	−140719.	−0.519261	−0.259631	+	0.965708i	$0.583601\pi$
$150$	0	0
$151$	50064.6	0.178685	0.0893425	−	0.996001i	$-0.471523\pi$
$151$	50064.6	0.178685	0.0893425	+	0.996001i	$0.471523\pi$
$152$	−62976.1	−0.221089
$153$	0	0
$154$	0	0
$155$	−58316.4	−0.194967
$156$	0	0
$157$	89794.6	0.290738	0.145369	−	0.989378i	$-0.453563\pi$
$157$	89794.6	0.290738	0.145369	+	0.989378i	$0.453563\pi$
$158$	−677628.	−2.15948
$159$	0	0
$160$	238844.	0.737589
$161$	0	0
$162$	0	0
$163$	−481230.	−1.41868	−0.709339	−	0.704867i	$-0.751006\pi$
$163$	−481230.	−1.41868	−0.709339	+	0.704867i	$0.751006\pi$
$164$	−260653.	−0.756750
$165$	0	0
$166$	−978376.	−2.75572
$167$	−86572.7	−0.240209	−0.120105	−	0.992761i	$-0.538323\pi$
$167$	−86572.7	−0.240209	−0.120105	+	0.992761i	$0.538323\pi$
$168$	0	0
$169$	−280909.	−0.756569
$170$	−145856.	−0.387082
$171$	0	0
$172$	−714323.	−1.84108
$173$	−58137.4	−0.147686	−0.0738432	−	0.997270i	$-0.523526\pi$
$173$	−58137.4	−0.147686	−0.0738432	+	0.997270i	$0.523526\pi$
$174$	0	0
$175$	0	0
$176$	−232495.	−0.565759
$177$	0	0
$178$	345026.	0.816209
$179$	209380.	0.488431	0.244215	−	0.969721i	$-0.421470\pi$
$179$	209380.	0.488431	0.244215	+	0.969721i	$0.421470\pi$
$180$	0	0
$181$	−278996.	−0.632996	−0.316498	−	0.948593i	$-0.602507\pi$
$181$	−278996.	−0.632996	−0.316498	+	0.948593i	$0.602507\pi$
$182$	0	0
$183$	0	0
$184$	118038.	0.257027
$185$	148186.	0.318331
$186$	0	0
$187$	165594.	0.346290
$188$	729426.	1.50517
$189$	0	0
$190$	−404651.	−0.813198
$191$	445132.	0.882888	0.441444	−	0.897289i	$-0.354466\pi$
$191$	445132.	0.882888	0.441444	+	0.897289i	$0.354466\pi$
$192$	0	0
$193$	−726811.	−1.40452	−0.702260	−	0.711920i	$-0.747826\pi$
$193$	−726811.	−1.40452	−0.702260	+	0.711920i	$0.747826\pi$
$194$	361471.	0.689556
$195$	0	0
$196$	0	0
$197$	364897.	0.669892	0.334946	−	0.942237i	$-0.391282\pi$
$197$	364897.	0.669892	0.334946	+	0.942237i	$0.391282\pi$
$198$	0	0
$199$	−289307.	−0.517877	−0.258938	−	0.965894i	$-0.583373\pi$
$199$	−289307.	−0.517877	−0.258938	+	0.965894i	$0.583373\pi$
$200$	85099.3	0.150436
$201$	0	0
$202$	−212236.	−0.365965
$203$	0	0
$204$	0	0
$205$	−205448.	−0.341442
$206$	−118497.	−0.194554
$207$	0	0
$208$	258792.	0.414756
$209$	459409.	0.727501
$210$	0	0
$211$	750147.	1.15995	0.579976	−	0.814633i	$-0.303062\pi$
$211$	750147.	1.15995	0.579976	+	0.814633i	$0.303062\pi$
$212$	−144030.	−0.220097
$213$	0	0
$214$	142343.	0.212473
$215$	−563033.	−0.830687
$216$	0	0
$217$	0	0
$218$	711788.	1.01440
$219$	0	0
$220$	283217.	0.394515
$221$	−184324.	−0.253864
$222$	0	0
$223$	−534398.	−0.719619	−0.359810	−	0.933026i	$-0.617158\pi$
$223$	−534398.	−0.719619	−0.359810	+	0.933026i	$0.617158\pi$
$224$	0	0
$225$	0	0
$226$	1.13836e6	1.48255
$227$	−410624.	−0.528907	−0.264453	−	0.964398i	$-0.585192\pi$
$227$	−410624.	−0.528907	−0.264453	+	0.964398i	$0.585192\pi$
$228$	0	0
$229$	−1.03036e6	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$229$	−1.03036e6	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$230$	758452.	0.945385
$231$	0	0
$232$	159177.	0.194160
$233$	119211.	0.143856	0.0719278	−	0.997410i	$-0.477085\pi$
$233$	119211.	0.143856	0.0719278	+	0.997410i	$0.477085\pi$
$234$	0	0
$235$	574937.	0.679127
$236$	−1.08356e6	−1.26641
$237$	0	0
$238$	0	0
$239$	254090.	0.287735	0.143868	−	0.989597i	$-0.454046\pi$
$239$	254090.	0.287735	0.143868	+	0.989597i	$0.454046\pi$
$240$	0	0
$241$	−1.41251e6	−1.56656	−0.783282	−	0.621667i	$-0.786456\pi$
$241$	−1.41251e6	−1.56656	−0.783282	+	0.621667i	$0.786456\pi$
$242$	729041.	0.800228
$243$	0	0
$244$	1.84265e6	1.98138
$245$	0	0
$246$	0	0
$247$	−511372.	−0.533329
$248$	75101.7	0.0775391
$249$	0	0
$250$	1.29023e6	1.30562
$251$	−1.67542e6	−1.67857	−0.839286	−	0.543690i	$-0.817027\pi$
$251$	−1.67542e6	−1.67857	−0.839286	+	0.543690i	$0.817027\pi$
$252$	0	0
$253$	−861087.	−0.845758
$254$	583937.	0.567913
$255$	0	0
$256$	697120.	0.664825
$257$	726996.	0.686593	0.343296	−	0.939227i	$-0.388456\pi$
$257$	726996.	0.686593	0.343296	+	0.939227i	$0.388456\pi$
$258$	0	0
$259$	0	0
$260$	−315252.	−0.289217
$261$	0	0
$262$	1.43746e6	1.29372
$263$	225880.	0.201367	0.100684	−	0.994918i	$-0.467897\pi$
$263$	225880.	0.201367	0.100684	+	0.994918i	$0.467897\pi$
$264$	0	0
$265$	−113525.	−0.0993065
$266$	0	0
$267$	0	0
$268$	184325.	0.156764
$269$	1.80527e6	1.52111	0.760557	−	0.649272i	$-0.224926\pi$
$269$	1.80527e6	1.52111	0.760557	+	0.649272i	$0.224926\pi$
$270$	0	0
$271$	1.71380e6	1.41754	0.708771	−	0.705439i	$-0.249250\pi$
$271$	1.71380e6	1.41754	0.708771	+	0.705439i	$0.249250\pi$
$272$	−527765.	−0.432532
$273$	0	0
$274$	−16466.6	−0.0132504
$275$	−620797.	−0.495015
$276$	0	0
$277$	2.23055e6	1.74668	0.873338	−	0.487115i	$-0.161951\pi$
$277$	2.23055e6	1.74668	0.873338	+	0.487115i	$0.161951\pi$
$278$	3.03200e6	2.35298
$279$	0	0
$280$	0	0
$281$	−1.67140e6	−1.26274	−0.631371	−	0.775481i	$-0.717507\pi$
$281$	−1.67140e6	−1.26274	−0.631371	+	0.775481i	$0.717507\pi$
$282$	0	0
$283$	396152.	0.294033	0.147016	−	0.989134i	$-0.453033\pi$
$283$	396152.	0.294033	0.147016	+	0.989134i	$0.453033\pi$
$284$	−1.19830e6	−0.881597
$285$	0	0
$286$	671920.	0.485739
$287$	0	0
$288$	0	0
$289$	−1.04396e6	−0.735256
$290$	1.02279e6	0.714150
$291$	0	0
$292$	405410.	0.278251
$293$	−929465.	−0.632505	−0.316252	−	0.948675i	$-0.602425\pi$
$293$	−929465.	−0.632505	−0.316252	+	0.948675i	$0.602425\pi$
$294$	0	0
$295$	−854068.	−0.571397
$296$	−190839.	−0.126601
$297$	0	0
$298$	1.16443e6	0.759582
$299$	958485.	0.620022
$300$	0	0
$301$	0	0
$302$	−414280.	−0.261383
$303$	0	0
$304$	−1.46418e6	−0.908682
$305$	1.45239e6	0.893990
$306$	0	0
$307$	−1.83295e6	−1.10995	−0.554976	−	0.831866i	$-0.687273\pi$
$307$	−1.83295e6	−1.10995	−0.554976	+	0.831866i	$0.687273\pi$
$308$	0	0
$309$	0	0
$310$	482563.	0.285200
$311$	−2.29685e6	−1.34658	−0.673289	−	0.739379i	$-0.735119\pi$
$311$	−2.29685e6	−1.34658	−0.673289	+	0.739379i	$0.735119\pi$
$312$	0	0
$313$	3.42470e6	1.97589	0.987943	−	0.154817i	$-0.0494787\pi$
$313$	3.42470e6	1.97589	0.987943	+	0.154817i	$0.0494787\pi$
$314$	−743043.	−0.425295
$315$	0	0
$316$	2.98685e6	1.68266
$317$	−2.94305e6	−1.64494	−0.822470	−	0.568808i	$-0.807405\pi$
$317$	−2.94305e6	−1.64494	−0.822470	+	0.568808i	$0.807405\pi$
$318$	0	0
$319$	−1.16119e6	−0.638891
$320$	−1.18450e6	−0.646635
$321$	0	0
$322$	0	0
$323$	1.04286e6	0.556187
$324$	0	0
$325$	691016.	0.362894
$326$	3.98214e6	2.07526
$327$	0	0
$328$	264582.	0.135792
$329$	0	0
$330$	0	0
$331$	966164.	0.484709	0.242354	−	0.970188i	$-0.422080\pi$
$331$	966164.	0.484709	0.242354	+	0.970188i	$0.422080\pi$
$332$	4.31250e6	2.14725
$333$	0	0
$334$	716382.	0.351381
$335$	145286.	0.0707312
$336$	0	0
$337$	136417.	0.0654327	0.0327163	−	0.999465i	$-0.489584\pi$
$337$	136417.	0.0654327	0.0327163	+	0.999465i	$0.489584\pi$
$338$	2.32450e6	1.10672
$339$	0	0
$340$	642906.	0.301613
$341$	−547865.	−0.255145
$342$	0	0
$343$	0	0
$344$	725091.	0.330367
$345$	0	0
$346$	481082.	0.216038
$347$	−355408.	−0.158454	−0.0792270	−	0.996857i	$-0.525245\pi$
$347$	−355408.	−0.158454	−0.0792270	+	0.996857i	$0.525245\pi$
$348$	0	0
$349$	140128.	0.0615830	0.0307915	−	0.999526i	$-0.490197\pi$
$349$	140128.	0.0615830	0.0307915	+	0.999526i	$0.490197\pi$
$350$	0	0
$351$	0	0
$352$	2.24387e6	0.965252
$353$	3.48141e6	1.48703	0.743514	−	0.668721i	$-0.233158\pi$
$353$	3.48141e6	1.48703	0.743514	+	0.668721i	$0.233158\pi$
$354$	0	0
$355$	−944507.	−0.397772
$356$	−1.52081e6	−0.635988
$357$	0	0
$358$	−1.73260e6	−0.714482
$359$	−1.75285e6	−0.717810	−0.358905	−	0.933374i	$-0.616850\pi$
$359$	−1.75285e6	−0.717810	−0.358905	+	0.933374i	$0.616850\pi$
$360$	0	0
$361$	417127.	0.168461
$362$	2.30867e6	0.925955
$363$	0	0
$364$	0	0
$365$	319546.	0.125546
$366$	0	0
$367$	1.76939e6	0.685738	0.342869	−	0.939383i	$-0.388601\pi$
$367$	1.76939e6	0.685738	0.342869	+	0.939383i	$0.388601\pi$
$368$	2.74438e6	1.05639
$369$	0	0
$370$	−1.22623e6	−0.465658
$371$	0	0
$372$	0	0
$373$	−4.16212e6	−1.54897	−0.774485	−	0.632592i	$-0.781991\pi$
$373$	−4.16212e6	−1.54897	−0.774485	+	0.632592i	$0.781991\pi$
$374$	−1.37027e6	−0.506557
$375$	0	0
$376$	−740422.	−0.270091
$377$	1.29253e6	0.468369
$378$	0	0
$379$	618163.	0.221057	0.110529	−	0.993873i	$-0.464746\pi$
$379$	618163.	0.221057	0.110529	+	0.993873i	$0.464746\pi$
$380$	1.78362e6	0.633642
$381$	0	0
$382$	−3.68343e6	−1.29150
$383$	−4.11163e6	−1.43225	−0.716123	−	0.697974i	$-0.754085\pi$
$383$	−4.11163e6	−1.43225	−0.716123	+	0.697974i	$0.754085\pi$
$384$	0	0
$385$	0	0
$386$	6.01430e6	2.05455
$387$	0	0
$388$	−1.59330e6	−0.537301
$389$	−4.62076e6	−1.54824	−0.774122	−	0.633037i	$-0.781808\pi$
$389$	−4.62076e6	−1.54824	−0.774122	+	0.633037i	$0.781808\pi$
$390$	0	0
$391$	−1.95468e6	−0.646596
$392$	0	0
$393$	0	0
$394$	−3.01949e6	−0.979926
$395$	2.35425e6	0.759207
$396$	0	0
$397$	−5.07349e6	−1.61559	−0.807794	−	0.589465i	$-0.799339\pi$
$397$	−5.07349e6	−1.61559	−0.807794	+	0.589465i	$0.799339\pi$
$398$	2.39399e6	0.757557
$399$	0	0
$400$	1.97855e6	0.618296
$401$	1.48056e6	0.459795	0.229898	−	0.973215i	$-0.426161\pi$
$401$	1.48056e6	0.459795	0.229898	+	0.973215i	$0.426161\pi$
$402$	0	0
$403$	609834.	0.187046
$404$	935495.	0.285160
$405$	0	0
$406$	0	0
$407$	1.39217e6	0.416586
$408$	0	0
$409$	4.53379e6	1.34015	0.670075	−	0.742294i	$-0.266262\pi$
$409$	4.53379e6	1.34015	0.670075	+	0.742294i	$0.266262\pi$
$410$	1.70006e6	0.499465
$411$	0	0
$412$	522313.	0.151596
$413$	0	0
$414$	0	0
$415$	3.39913e6	0.968829
$416$	−2.49767e6	−0.707623
$417$	0	0
$418$	−3.80157e6	−1.06420
$419$	111026.	0.0308952	0.0154476	−	0.999881i	$-0.495083\pi$
$419$	111026.	0.0308952	0.0154476	+	0.999881i	$0.495083\pi$
$420$	0	0
$421$	−1.41151e6	−0.388132	−0.194066	−	0.980988i	$-0.562168\pi$
$421$	−1.41151e6	−0.388132	−0.194066	+	0.980988i	$0.562168\pi$
$422$	−6.20740e6	−1.69679
$423$	0	0
$424$	146201.	0.0394945
$425$	−1.40922e6	−0.378447
$426$	0	0
$427$	0	0
$428$	−627422.	−0.165558
$429$	0	0
$430$	4.65905e6	1.21514
$431$	−1.07640e6	−0.279113	−0.139557	−	0.990214i	$-0.544568\pi$
$431$	−1.07640e6	−0.279113	−0.139557	+	0.990214i	$0.544568\pi$
$432$	0	0
$433$	310172.	0.0795029	0.0397515	−	0.999210i	$-0.487343\pi$
$433$	310172.	0.0795029	0.0397515	+	0.999210i	$0.487343\pi$
$434$	0	0
$435$	0	0
$436$	−3.13743e6	−0.790419
$437$	−5.42288e6	−1.35840
$438$	0	0
$439$	−5.67650e6	−1.40579	−0.702893	−	0.711296i	$-0.748109\pi$
$439$	−5.67650e6	−1.40579	−0.702893	+	0.711296i	$0.748109\pi$
$440$	−287487.	−0.0707923
$441$	0	0
$442$	1.52527e6	0.371356
$443$	−4.05966e6	−0.982834	−0.491417	−	0.870924i	$-0.663521\pi$
$443$	−4.05966e6	−0.982834	−0.491417	+	0.870924i	$0.663521\pi$
$444$	0	0
$445$	−1.19871e6	−0.286955
$446$	4.42210e6	1.05267
$447$	0	0
$448$	0	0
$449$	6.96544e6	1.63054	0.815272	−	0.579078i	$-0.196587\pi$
$449$	6.96544e6	1.63054	0.815272	+	0.579078i	$0.196587\pi$
$450$	0	0
$451$	−1.93012e6	−0.446830
$452$	−5.01767e6	−1.15520
$453$	0	0
$454$	3.39788e6	0.773692
$455$	0	0
$456$	0	0
$457$	1.79523e6	0.402096	0.201048	−	0.979581i	$-0.435565\pi$
$457$	1.79523e6	0.402096	0.201048	+	0.979581i	$0.435565\pi$
$458$	8.52616e6	1.89928
$459$	0	0
$460$	−3.34311e6	−0.736642
$461$	−2.11294e6	−0.463058	−0.231529	−	0.972828i	$-0.574373\pi$
$461$	−2.11294e6	−0.463058	−0.231529	+	0.972828i	$0.574373\pi$
$462$	0	0
$463$	1.26223e6	0.273643	0.136822	−	0.990596i	$-0.456311\pi$
$463$	1.26223e6	0.273643	0.136822	+	0.990596i	$0.456311\pi$
$464$	3.70084e6	0.798005
$465$	0	0
$466$	−986462.	−0.210434
$467$	−3.58926e6	−0.761576	−0.380788	−	0.924662i	$-0.624347\pi$
$467$	−3.58926e6	−0.761576	−0.380788	+	0.924662i	$0.624347\pi$
$468$	0	0
$469$	0	0
$470$	−4.75756e6	−0.993435
$471$	0	0
$472$	1.09990e6	0.227246
$473$	−5.28952e6	−1.08708
$474$	0	0
$475$	−3.90960e6	−0.795058
$476$	0	0
$477$	0	0
$478$	−2.10257e6	−0.420903
$479$	−2.41693e6	−0.481311	−0.240655	−	0.970611i	$-0.577362\pi$
$479$	−2.41693e6	−0.481311	−0.240655	+	0.970611i	$0.577362\pi$
$480$	0	0
$481$	−1.54963e6	−0.305398
$482$	1.16884e7	2.29159
$483$	0	0
$484$	−3.21347e6	−0.623536
$485$	−1.25584e6	−0.242427
$486$	0	0
$487$	−5.19403e6	−0.992388	−0.496194	−	0.868212i	$-0.665270\pi$
$487$	−5.19403e6	−0.992388	−0.496194	+	0.868212i	$0.665270\pi$
$488$	−1.87043e6	−0.355543
$489$	0	0
$490$	0	0
$491$	−5.38961e6	−1.00891	−0.504456	−	0.863437i	$-0.668307\pi$
$491$	−5.38961e6	−1.00891	−0.504456	+	0.863437i	$0.668307\pi$
$492$	0	0
$493$	−2.63592e6	−0.488443
$494$	4.23156e6	0.780160
$495$	0	0
$496$	1.74610e6	0.318688
$497$	0	0
$498$	0	0
$499$	−3.29606e6	−0.592576	−0.296288	−	0.955099i	$-0.595749\pi$
$499$	−3.29606e6	−0.592576	−0.296288	+	0.955099i	$0.595749\pi$
$500$	−5.68709e6	−1.01734
$501$	0	0
$502$	1.38640e7	2.45544
$503$	−1.06512e7	−1.87706	−0.938528	−	0.345204i	$-0.887810\pi$
$503$	−1.06512e7	−1.87706	−0.938528	+	0.345204i	$0.887810\pi$
$504$	0	0
$505$	737361.	0.128662
$506$	7.12543e6	1.23718
$507$	0	0
$508$	−2.57388e6	−0.442516
$509$	−2.74268e6	−0.469225	−0.234612	−	0.972089i	$-0.575382\pi$
$509$	−2.74268e6	−0.469225	−0.234612	+	0.972089i	$0.575382\pi$
$510$	0	0
$511$	0	0
$512$	−8.17130e6	−1.37758
$513$	0	0
$514$	−6.01583e6	−1.00436
$515$	411689.	0.0683992
$516$	0	0
$517$	5.40136e6	0.888744
$518$	0	0
$519$	0	0
$520$	320004.	0.0518976
$521$	4.97077e6	0.802286	0.401143	−	0.916015i	$-0.368613\pi$
$521$	4.97077e6	0.802286	0.401143	+	0.916015i	$0.368613\pi$
$522$	0	0
$523$	−2.41579e6	−0.386193	−0.193096	−	0.981180i	$-0.561853\pi$
$523$	−2.41579e6	−0.386193	−0.193096	+	0.981180i	$0.561853\pi$
$524$	−6.33603e6	−1.00807
$525$	0	0
$526$	−1.86914e6	−0.294563
$527$	−1.24366e6	−0.195063
$528$	0	0
$529$	3.72798e6	0.579207
$530$	939412.	0.145267
$531$	0	0
$532$	0	0
$533$	2.14843e6	0.327570
$534$	0	0
$535$	−494537.	−0.0746989
$536$	−187103.	−0.0281300
$537$	0	0
$538$	−1.49385e7	−2.22510
$539$	0	0
$540$	0	0
$541$	472165.	0.0693587	0.0346794	−	0.999398i	$-0.488959\pi$
$541$	472165.	0.0693587	0.0346794	+	0.999398i	$0.488959\pi$
$542$	−1.41815e7	−2.07360
$543$	0	0
$544$	5.09360e6	0.737951
$545$	−2.47293e6	−0.356633
$546$	0	0
$547$	7.63716e6	1.09135	0.545675	−	0.837997i	$-0.316273\pi$
$547$	7.63716e6	1.09135	0.545675	+	0.837997i	$0.316273\pi$
$548$	72581.6	0.0103246
$549$	0	0
$550$	5.13705e6	0.724114
$551$	−7.31285e6	−1.02614
$552$	0	0
$553$	0	0
$554$	−1.84576e7	−2.55506
$555$	0	0
$556$	−1.33645e7	−1.83343
$557$	4.48807e6	0.612946	0.306473	−	0.951879i	$-0.400851\pi$
$557$	4.48807e6	0.612946	0.306473	+	0.951879i	$0.400851\pi$
$558$	0	0
$559$	5.88782e6	0.796938
$560$	0	0
$561$	0	0
$562$	1.38307e7	1.84715
$563$	−2.16500e6	−0.287864	−0.143932	−	0.989588i	$-0.545975\pi$
$563$	−2.16500e6	−0.287864	−0.143932	+	0.989588i	$0.545975\pi$
$564$	0	0
$565$	−3.95495e6	−0.521219
$566$	−3.27812e6	−0.430115
$567$	0	0
$568$	1.21637e6	0.158195
$569$	1.13325e7	1.46739	0.733696	−	0.679478i	$-0.237794\pi$
$569$	1.13325e7	1.46739	0.733696	+	0.679478i	$0.237794\pi$
$570$	0	0
$571$	−843773.	−0.108302	−0.0541509	−	0.998533i	$-0.517245\pi$
$571$	−843773.	−0.108302	−0.0541509	+	0.998533i	$0.517245\pi$
$572$	−2.96170e6	−0.378487
$573$	0	0
$574$	0	0
$575$	7.32792e6	0.924296
$576$	0	0
$577$	2.23784e6	0.279827	0.139914	−	0.990164i	$-0.455318\pi$
$577$	2.23784e6	0.279827	0.139914	+	0.990164i	$0.455318\pi$
$578$	8.63866e6	1.07554
$579$	0	0
$580$	−4.50824e6	−0.556464
$581$	0	0
$582$	0	0
$583$	−1.06653e6	−0.129958
$584$	−411521.	−0.0499299
$585$	0	0
$586$	7.69124e6	0.925236
$587$	1.21190e7	1.45168	0.725839	−	0.687864i	$-0.241452\pi$
$587$	1.21190e7	1.45168	0.725839	+	0.687864i	$0.241452\pi$
$588$	0	0
$589$	−3.45030e6	−0.409797
$590$	7.06734e6	0.835846
$591$	0	0
$592$	−4.43698e6	−0.520335
$593$	8.00167e6	0.934424	0.467212	−	0.884145i	$-0.345258\pi$
$593$	8.00167e6	0.934424	0.467212	+	0.884145i	$0.345258\pi$
$594$	0	0
$595$	0	0
$596$	−5.13261e6	−0.591865
$597$	0	0
$598$	−7.93138e6	−0.906976
$599$	−1.45899e7	−1.66144	−0.830719	−	0.556692i	$-0.812070\pi$
$599$	−1.45899e7	−1.66144	−0.830719	+	0.556692i	$0.812070\pi$
$600$	0	0
$601$	8.67178e6	0.979314	0.489657	−	0.871915i	$-0.337122\pi$
$601$	8.67178e6	0.979314	0.489657	+	0.871915i	$0.337122\pi$
$602$	0	0
$603$	0	0
$604$	1.82607e6	0.203669
$605$	−2.53287e6	−0.281336
$606$	0	0
$607$	1.33059e7	1.46580	0.732898	−	0.680339i	$-0.238167\pi$
$607$	1.33059e7	1.46580	0.732898	+	0.680339i	$0.238167\pi$
$608$	1.41312e7	1.55032
$609$	0	0
$610$	−1.20184e7	−1.30774
$611$	−6.01231e6	−0.651536
$612$	0	0
$613$	2.35101e6	0.252699	0.126350	−	0.991986i	$-0.459674\pi$
$613$	2.35101e6	0.252699	0.126350	+	0.991986i	$0.459674\pi$
$614$	1.51675e7	1.62365
$615$	0	0
$616$	0	0
$617$	−9.63523e6	−1.01894	−0.509470	−	0.860488i	$-0.670159\pi$
$617$	−9.63523e6	−1.01894	−0.509470	+	0.860488i	$0.670159\pi$
$618$	0	0
$619$	4.86148e6	0.509967	0.254983	−	0.966945i	$-0.417930\pi$
$619$	4.86148e6	0.509967	0.254983	+	0.966945i	$0.417930\pi$
$620$	−2.12705e6	−0.222228
$621$	0	0
$622$	1.90062e7	1.96979
$623$	0	0
$624$	0	0
$625$	2.70017e6	0.276498
$626$	−2.83391e7	−2.89035
$627$	0	0
$628$	3.27519e6	0.331389
$629$	3.16023e6	0.318487
$630$	0	0
$631$	−6.59770e6	−0.659659	−0.329829	−	0.944041i	$-0.606991\pi$
$631$	−6.59770e6	−0.659659	−0.329829	+	0.944041i	$0.606991\pi$
$632$	−3.03188e6	−0.301939
$633$	0	0
$634$	2.43535e7	2.40624
$635$	−2.02874e6	−0.199661
$636$	0	0
$637$	0	0
$638$	9.60876e6	0.934578
$639$	0	0
$640$	2.15861e6	0.208317
$641$	−1.44525e7	−1.38930	−0.694651	−	0.719347i	$-0.744441\pi$
$641$	−1.44525e7	−1.38930	−0.694651	+	0.719347i	$0.744441\pi$
$642$	0	0
$643$	1.54720e7	1.47577	0.737886	−	0.674926i	$-0.235824\pi$
$643$	1.54720e7	1.47577	0.737886	+	0.674926i	$0.235824\pi$
$644$	0	0
$645$	0	0
$646$	−8.62960e6	−0.813597
$647$	1.66647e7	1.56508	0.782540	−	0.622601i	$-0.213924\pi$
$647$	1.66647e7	1.56508	0.782540	+	0.622601i	$0.213924\pi$
$648$	0	0
$649$	−8.02371e6	−0.747762
$650$	−5.71810e6	−0.530845
$651$	0	0
$652$	−1.75525e7	−1.61704
$653$	1.33451e7	1.22472	0.612361	−	0.790578i	$-0.290220\pi$
$653$	1.33451e7	1.22472	0.612361	+	0.790578i	$0.290220\pi$
$654$	0	0
$655$	−4.99409e6	−0.454834
$656$	6.15149e6	0.558111
$657$	0	0
$658$	0	0
$659$	4.00667e6	0.359393	0.179697	−	0.983722i	$-0.442488\pi$
$659$	4.00667e6	0.359393	0.179697	+	0.983722i	$0.442488\pi$
$660$	0	0
$661$	−1.08005e7	−0.961478	−0.480739	−	0.876864i	$-0.659632\pi$
$661$	−1.08005e7	−0.961478	−0.480739	+	0.876864i	$0.659632\pi$
$662$	−7.99493e6	−0.709038
$663$	0	0
$664$	−4.37750e6	−0.385307
$665$	0	0
$666$	0	0
$667$	1.37068e7	1.19294
$668$	−3.15767e6	−0.273795
$669$	0	0
$670$	−1.20223e6	−0.103466
$671$	1.36447e7	1.16993
$672$	0	0
$673$	1.09119e7	0.928676	0.464338	−	0.885658i	$-0.346292\pi$
$673$	1.09119e7	0.928676	0.464338	+	0.885658i	$0.346292\pi$
$674$	−1.12884e6	−0.0957158
$675$	0	0
$676$	−1.02459e7	−0.862353
$677$	−1.35765e7	−1.13846	−0.569229	−	0.822179i	$-0.692758\pi$
$677$	−1.35765e7	−1.13846	−0.569229	+	0.822179i	$0.692758\pi$
$678$	0	0
$679$	0	0
$680$	−652598.	−0.0541219
$681$	0	0
$682$	4.53354e6	0.373230
$683$	1.26726e7	1.03948	0.519738	−	0.854326i	$-0.326030\pi$
$683$	1.26726e7	1.03948	0.519738	+	0.854326i	$0.326030\pi$
$684$	0	0
$685$	57209.2	0.00465843
$686$	0	0
$687$	0	0
$688$	1.68583e7	1.35782
$689$	1.18717e6	0.0952720
$690$	0	0
$691$	−7.11964e6	−0.567235	−0.283617	−	0.958938i	$-0.591535\pi$
$691$	−7.11964e6	−0.567235	−0.283617	+	0.958938i	$0.591535\pi$
$692$	−2.12052e6	−0.168336
$693$	0	0
$694$	2.94097e6	0.231789
$695$	−1.05339e7	−0.827235
$696$	0	0
$697$	−4.38139e6	−0.341609
$698$	−1.15955e6	−0.0900844
$699$	0	0
$700$	0	0
$701$	1.00155e7	0.769803	0.384902	−	0.922958i	$-0.374235\pi$
$701$	1.00155e7	0.769803	0.384902	+	0.922958i	$0.374235\pi$
$702$	0	0
$703$	8.76746e6	0.669092
$704$	−1.11280e7	−0.846224
$705$	0	0
$706$	−2.88084e7	−2.17524
$707$	0	0
$708$	0	0
$709$	−8.84454e6	−0.660784	−0.330392	−	0.943844i	$-0.607181\pi$
$709$	−8.84454e6	−0.660784	−0.330392	+	0.943844i	$0.607181\pi$
$710$	7.81571e6	0.581866
$711$	0	0
$712$	1.54373e6	0.114123
$713$	6.46703e6	0.476410
$714$	0	0
$715$	−2.33442e6	−0.170771
$716$	7.63698e6	0.556723
$717$	0	0
$718$	1.45047e7	1.05002
$719$	6.58086e6	0.474745	0.237373	−	0.971419i	$-0.423714\pi$
$719$	6.58086e6	0.474745	0.237373	+	0.971419i	$0.423714\pi$
$720$	0	0
$721$	0	0
$722$	−3.45169e6	−0.246427
$723$	0	0
$724$	−1.01762e7	−0.721502
$725$	9.88183e6	0.698220
$726$	0	0
$727$	−1.88401e7	−1.32205	−0.661023	−	0.750365i	$-0.729878\pi$
$727$	−1.88401e7	−1.32205	−0.661023	+	0.750365i	$0.729878\pi$
$728$	0	0
$729$	0	0
$730$	−2.64422e6	−0.183650
$731$	−1.20073e7	−0.831095
$732$	0	0
$733$	2.78330e6	0.191337	0.0956687	−	0.995413i	$-0.469501\pi$
$733$	2.78330e6	0.191337	0.0956687	+	0.995413i	$0.469501\pi$
$734$	−1.46416e7	−1.00311
$735$	0	0
$736$	−2.64867e7	−1.80233
$737$	1.36491e6	0.0925629
$738$	0	0
$739$	−2.48970e7	−1.67701	−0.838505	−	0.544894i	$-0.816570\pi$
$739$	−2.48970e7	−1.67701	−0.838505	+	0.544894i	$0.816570\pi$
$740$	5.40499e6	0.362840
$741$	0	0
$742$	0	0
$743$	3.86085e6	0.256573	0.128286	−	0.991737i	$-0.459052\pi$
$743$	3.86085e6	0.256573	0.128286	+	0.991737i	$0.459052\pi$
$744$	0	0
$745$	−4.04554e6	−0.267046
$746$	3.44412e7	2.26585
$747$	0	0
$748$	6.03991e6	0.394708
$749$	0	0
$750$	0	0
$751$	6.72737e6	0.435257	0.217628	−	0.976032i	$-0.430168\pi$
$751$	6.72737e6	0.435257	0.217628	+	0.976032i	$0.430168\pi$
$752$	−1.72147e7	−1.11008
$753$	0	0
$754$	−1.06956e7	−0.685136
$755$	1.43932e6	0.0918943
$756$	0	0
$757$	2.17782e7	1.38128	0.690642	−	0.723197i	$-0.257328\pi$
$757$	2.17782e7	1.38128	0.690642	+	0.723197i	$0.257328\pi$
$758$	−5.11525e6	−0.323366
$759$	0	0
$760$	−1.81051e6	−0.113702
$761$	−2.57074e7	−1.60915	−0.804575	−	0.593851i	$-0.797607\pi$
$761$	−2.57074e7	−1.60915	−0.804575	+	0.593851i	$0.797607\pi$
$762$	0	0
$763$	0	0
$764$	1.62359e7	1.00633
$765$	0	0
$766$	3.40234e7	2.09511
$767$	8.93127e6	0.548182
$768$	0	0
$769$	1.34375e7	0.819413	0.409706	−	0.912217i	$-0.365631\pi$
$769$	1.34375e7	0.819413	0.409706	+	0.912217i	$0.365631\pi$
$770$	0	0
$771$	0	0
$772$	−2.65099e7	−1.60090
$773$	3.05572e7	1.83935	0.919674	−	0.392682i	$-0.128453\pi$
$773$	3.05572e7	1.83935	0.919674	+	0.392682i	$0.128453\pi$
$774$	0	0
$775$	4.66237e6	0.278839
$776$	1.61731e6	0.0964140
$777$	0	0
$778$	3.82364e7	2.26479
$779$	−1.21553e7	−0.717667
$780$	0	0
$781$	−8.87335e6	−0.520547
$782$	1.61748e7	0.945849
$783$	0	0
$784$	0	0
$785$	2.58152e6	0.149521
$786$	0	0
$787$	2.07672e6	0.119520	0.0597602	−	0.998213i	$-0.480966\pi$
$787$	2.07672e6	0.119520	0.0597602	+	0.998213i	$0.480966\pi$
$788$	1.33093e7	0.763556
$789$	0	0
$790$	−1.94812e7	−1.11058
$791$	0	0
$792$	0	0
$793$	−1.51881e7	−0.857670
$794$	4.19827e7	2.36330
$795$	0	0
$796$	−1.05523e7	−0.590286
$797$	−5.98563e6	−0.333783	−0.166892	−	0.985975i	$-0.553373\pi$
$797$	−5.98563e6	−0.333783	−0.166892	+	0.985975i	$0.553373\pi$
$798$	0	0
$799$	1.22611e7	0.679460
$800$	−1.90955e7	−1.05489
$801$	0	0
$802$	−1.22515e7	−0.672594
$803$	3.00204e6	0.164296
$804$	0	0
$805$	0	0
$806$	−5.04633e6	−0.273614
$807$	0	0
$808$	−949597.	−0.0511695
$809$	−1.96864e7	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$809$	−1.96864e7	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$810$	0	0
$811$	−8.50101e6	−0.453856	−0.226928	−	0.973912i	$-0.572868\pi$
$811$	−8.50101e6	−0.453856	−0.226928	+	0.973912i	$0.572868\pi$
$812$	0	0
$813$	0	0
$814$	−1.15201e7	−0.609387
$815$	−1.38350e7	−0.729599
$816$	0	0
$817$	−3.33119e7	−1.74600
$818$	−3.75168e7	−1.96039
$819$	0	0
$820$	−7.49355e6	−0.389182
$821$	1.36199e6	0.0705204	0.0352602	−	0.999378i	$-0.488774\pi$
$821$	1.36199e6	0.0705204	0.0352602	+	0.999378i	$0.488774\pi$
$822$	0	0
$823$	−1.35934e6	−0.0699566	−0.0349783	−	0.999388i	$-0.511136\pi$
$823$	−1.35934e6	−0.0699566	−0.0349783	+	0.999388i	$0.511136\pi$
$824$	−530186.	−0.0272026
$825$	0	0
$826$	0	0
$827$	−1.00727e7	−0.512132	−0.256066	−	0.966659i	$-0.582426\pi$
$827$	−1.00727e7	−0.512132	−0.256066	+	0.966659i	$0.582426\pi$
$828$	0	0
$829$	5.63984e6	0.285023	0.142512	−	0.989793i	$-0.454482\pi$
$829$	5.63984e6	0.285023	0.142512	+	0.989793i	$0.454482\pi$
$830$	−2.81275e7	−1.41722
$831$	0	0
$832$	1.23867e7	0.620364
$833$	0	0
$834$	0	0
$835$	−2.48889e6	−0.123535
$836$	1.67566e7	0.829220
$837$	0	0
$838$	−918733.	−0.0451938
$839$	−1.16351e7	−0.570642	−0.285321	−	0.958432i	$-0.592100\pi$
$839$	−1.16351e7	−0.570642	−0.285321	+	0.958432i	$0.592100\pi$
$840$	0	0
$841$	−2.02735e6	−0.0988413
$842$	1.16801e7	0.567764
$843$	0	0
$844$	2.73610e7	1.32214
$845$	−8.07590e6	−0.389089
$846$	0	0
$847$	0	0
$848$	3.39916e6	0.162324
$849$	0	0
$850$	1.16611e7	0.553597
$851$	−1.64332e7	−0.777854
$852$	0	0
$853$	−2.85205e7	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$853$	−2.85205e7	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$854$	0	0
$855$	0	0
$856$	636880.	0.0297080
$857$	−9.95725e6	−0.463113	−0.231557	−	0.972821i	$-0.574382\pi$
$857$	−9.95725e6	−0.463113	−0.231557	+	0.972821i	$0.574382\pi$
$858$	0	0
$859$	1.49322e7	0.690463	0.345232	−	0.938517i	$-0.387800\pi$
$859$	1.49322e7	0.690463	0.345232	+	0.938517i	$0.387800\pi$
$860$	−2.05362e7	−0.946834
$861$	0	0
$862$	8.90711e6	0.408290
$863$	−3.84933e7	−1.75937	−0.879687	−	0.475553i	$-0.842248\pi$
$863$	−3.84933e7	−1.75937	−0.879687	+	0.475553i	$0.842248\pi$
$864$	0	0
$865$	−1.67140e6	−0.0759523
$866$	−2.56665e6	−0.116298
$867$	0	0
$868$	0	0
$869$	2.21175e7	0.993542
$870$	0	0
$871$	−1.51930e6	−0.0678576
$872$	3.18472e6	0.141834
$873$	0	0
$874$	4.48739e7	1.98708
$875$	0	0
$876$	0	0
$877$	9.40311e6	0.412831	0.206416	−	0.978464i	$-0.433820\pi$
$877$	9.40311e6	0.412831	0.206416	+	0.978464i	$0.433820\pi$
$878$	4.69726e7	2.05640
$879$	0	0
$880$	−6.68403e6	−0.290959
$881$	1.10395e6	0.0479194	0.0239597	−	0.999713i	$-0.492373\pi$
$881$	1.10395e6	0.0479194	0.0239597	+	0.999713i	$0.492373\pi$
$882$	0	0
$883$	8.06579e6	0.348133	0.174067	−	0.984734i	$-0.444309\pi$
$883$	8.06579e6	0.348133	0.174067	+	0.984734i	$0.444309\pi$
$884$	−6.72308e6	−0.289359
$885$	0	0
$886$	3.35933e7	1.43770
$887$	−1.49902e7	−0.639732	−0.319866	−	0.947463i	$-0.603638\pi$
$887$	−1.49902e7	−0.639732	−0.319866	+	0.947463i	$0.603638\pi$
$888$	0	0
$889$	0	0
$890$	9.91920e6	0.419761
$891$	0	0
$892$	−1.94918e7	−0.820237
$893$	3.40162e7	1.42744
$894$	0	0
$895$	6.01950e6	0.251190
$896$	0	0
$897$	0	0
$898$	−5.76384e7	−2.38518
$899$	8.72090e6	0.359883
$900$	0	0
$901$	−2.42104e6	−0.0993553
$902$	1.59716e7	0.653629
$903$	0	0
$904$	5.09331e6	0.207290
$905$	−8.02089e6	−0.325538
$906$	0	0
$907$	4.12622e6	0.166546	0.0832730	−	0.996527i	$-0.473463\pi$
$907$	4.12622e6	0.166546	0.0832730	+	0.996527i	$0.473463\pi$
$908$	−1.49772e7	−0.602859
$909$	0	0
$910$	0	0
$911$	−4.04272e7	−1.61391	−0.806953	−	0.590616i	$-0.798885\pi$
$911$	−4.04272e7	−1.61391	−0.806953	+	0.590616i	$0.798885\pi$
$912$	0	0
$913$	3.19338e7	1.26787
$914$	−1.48554e7	−0.588191
$915$	0	0
$916$	−3.75817e7	−1.47992
$917$	0	0
$918$	0	0
$919$	2.18546e7	0.853600	0.426800	−	0.904346i	$-0.359641\pi$
$919$	2.18546e7	0.853600	0.426800	+	0.904346i	$0.359641\pi$
$920$	3.39351e6	0.132184
$921$	0	0
$922$	1.74844e7	0.677366
$923$	9.87702e6	0.381612
$924$	0	0
$925$	−1.18474e7	−0.455271
$926$	−1.04448e7	−0.400289
$927$	0	0
$928$	−3.57178e7	−1.36149
$929$	1.06843e7	0.406169	0.203085	−	0.979161i	$-0.434903\pi$
$929$	1.06843e7	0.406169	0.203085	+	0.979161i	$0.434903\pi$
$930$	0	0
$931$	0	0
$932$	4.34813e6	0.163970
$933$	0	0
$934$	2.97009e7	1.11404
$935$	4.76068e6	0.178090
$936$	0	0
$937$	3.99105e7	1.48504	0.742521	−	0.669823i	$-0.233630\pi$
$937$	3.99105e7	1.48504	0.742521	+	0.669823i	$0.233630\pi$
$938$	0	0
$939$	0	0
$940$	2.09704e7	0.774082
$941$	1.32350e6	0.0487248	0.0243624	−	0.999703i	$-0.492244\pi$
$941$	1.32350e6	0.0487248	0.0243624	+	0.999703i	$0.492244\pi$
$942$	0	0
$943$	2.27832e7	0.834326
$944$	2.55724e7	0.933990
$945$	0	0
$946$	4.37703e7	1.59020
$947$	2.76322e7	1.00124	0.500622	−	0.865666i	$-0.333105\pi$
$947$	2.76322e7	1.00124	0.500622	+	0.865666i	$0.333105\pi$
$948$	0	0
$949$	−3.34160e6	−0.120445
$950$	3.23517e7	1.16302
$951$	0	0
$952$	0	0
$953$	3.07901e7	1.09819	0.549096	−	0.835759i	$-0.314972\pi$
$953$	3.07901e7	1.09819	0.549096	+	0.835759i	$0.314972\pi$
$954$	0	0
$955$	1.27972e7	0.454053
$956$	9.26775e6	0.327966
$957$	0	0
$958$	1.99999e7	0.704068
$959$	0	0
$960$	0	0
$961$	−2.45145e7	−0.856278
$962$	1.28231e7	0.446740
$963$	0	0
$964$	−5.15202e7	−1.78560
$965$	−2.08952e7	−0.722318
$966$	0	0
$967$	2.92557e6	0.100611	0.0503055	−	0.998734i	$-0.483981\pi$
$967$	2.92557e6	0.100611	0.0503055	+	0.998734i	$0.483981\pi$
$968$	3.26192e6	0.111888
$969$	0	0
$970$	1.03920e7	0.354625
$971$	2.78109e6	0.0946601	0.0473301	−	0.998879i	$-0.484929\pi$
$971$	2.78109e6	0.0946601	0.0473301	+	0.998879i	$0.484929\pi$
$972$	0	0
$973$	0	0
$974$	4.29801e7	1.45168
$975$	0	0
$976$	−4.34872e7	−1.46129
$977$	−7.48673e6	−0.250932	−0.125466	−	0.992098i	$-0.540043\pi$
$977$	−7.48673e6	−0.250932	−0.125466	+	0.992098i	$0.540043\pi$
$978$	0	0
$979$	−1.12615e7	−0.375525
$980$	0	0
$981$	0	0
$982$	4.45985e7	1.47585
$983$	1.79815e7	0.593528	0.296764	−	0.954951i	$-0.404093\pi$
$983$	1.79815e7	0.593528	0.296764	+	0.954951i	$0.404093\pi$
$984$	0	0
$985$	1.04905e7	0.344512
$986$	2.18120e7	0.714501
$987$	0	0
$988$	−1.86519e7	−0.607899
$989$	6.24378e7	2.02982
$990$	0	0
$991$	3.72778e7	1.20578	0.602888	−	0.797826i	$-0.294017\pi$
$991$	3.72778e7	1.20578	0.602888	+	0.797826i	$0.294017\pi$
$992$	−1.68521e7	−0.543720
$993$	0	0
$994$	0	0
$995$	−8.31734e6	−0.266334
$996$	0	0
$997$	−4.87422e7	−1.55298	−0.776492	−	0.630128i	$-0.783003\pi$
$997$	−4.87422e7	−1.55298	−0.776492	+	0.630128i	$0.783003\pi$
$998$	2.72747e7	0.866828
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.l.1.1		2
3.2	odd	2		49.6.a.f.1.2		2
7.6	odd	2		63.6.a.f.1.1		2
12.11	even	2		784.6.a.v.1.1		2
21.2	odd	6		49.6.c.d.18.1		4
21.5	even	6		49.6.c.e.18.1		4
21.11	odd	6		49.6.c.d.30.1		4
21.17	even	6		49.6.c.e.30.1		4
21.20	even	2		7.6.a.b.1.2	✓	2
28.27	even	2		1008.6.a.bq.1.1		2
84.83	odd	2		112.6.a.h.1.2		2
105.62	odd	4		175.6.b.c.99.4		4
105.83	odd	4		175.6.b.c.99.1		4
105.104	even	2		175.6.a.c.1.1		2
168.83	odd	2		448.6.a.u.1.1		2
168.125	even	2		448.6.a.w.1.2		2
231.230	odd	2		847.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.2	✓	2	21.20	even	2
49.6.a.f.1.2		2	3.2	odd	2
49.6.c.d.18.1		4	21.2	odd	6
49.6.c.d.30.1		4	21.11	odd	6
49.6.c.e.18.1		4	21.5	even	6
49.6.c.e.30.1		4	21.17	even	6
63.6.a.f.1.1		2	7.6	odd	2
112.6.a.h.1.2		2	84.83	odd	2
175.6.a.c.1.1		2	105.104	even	2
175.6.b.c.99.1		4	105.83	odd	4
175.6.b.c.99.4		4	105.62	odd	4
441.6.a.l.1.1		2	1.1	even	1	trivial
448.6.a.u.1.1		2	168.83	odd	2
448.6.a.w.1.2		2	168.125	even	2
784.6.a.v.1.1		2	12.11	even	2
847.6.a.c.1.1		2	231.230	odd	2
1008.6.a.bq.1.1		2	28.27	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)

\(f(q)\)	\(=\)	\(q-8.27492 q^{2} +36.4743 q^{4} +28.7492 q^{5} -37.0241 q^{8} +O(q^{10})\)	\(q-8.27492 q^{2} +36.4743 q^{4} +28.7492 q^{5} -37.0241 q^{8} -237.897 q^{10} +270.090 q^{11} -300.640 q^{13} -860.805 q^{16} +613.106 q^{17} +1700.95 q^{19} +1048.60 q^{20} -2234.97 q^{22} -3188.15 q^{23} -2298.49 q^{25} +2487.77 q^{26} -4299.28 q^{29} -2028.46 q^{31} +8307.86 q^{32} -5073.40 q^{34} +5154.46 q^{37} -14075.2 q^{38} -1064.41 q^{40} -7146.21 q^{41} -19584.3 q^{43} +9851.32 q^{44} +26381.7 q^{46} +19998.4 q^{47} +19019.8 q^{50} -10965.6 q^{52} -3948.82 q^{53} +7764.86 q^{55} +35576.2 q^{58} -29707.6 q^{59} +50519.3 q^{61} +16785.3 q^{62} -41201.1 q^{64} -8643.14 q^{65} +5053.56 q^{67} +22362.6 q^{68} -32853.3 q^{71} +11115.0 q^{73} -42652.7 q^{74} +62040.8 q^{76} +81889.4 q^{79} -24747.4 q^{80} +59134.3 q^{82} +118234. q^{83} +17626.3 q^{85} +162058. q^{86} -9999.83 q^{88} -41695.4 q^{89} -116286. q^{92} -165485. q^{94} +48900.9 q^{95} -43682.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8}+O(q^{10}) \) 2 * q - 9 * q^2 + 5 * q^4 - 18 * q^5 + 9 * q^8	\( 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8} - 204 q^{10} - 396 q^{11} + 350 q^{13} + 113 q^{16} + 1800 q^{17} + 3266 q^{19} + 2520 q^{20} - 1752 q^{22} - 2088 q^{23} - 3238 q^{25} + 2016 q^{26} - 6696 q^{29} + 20 q^{31} + 6129 q^{32} - 5934 q^{34} + 6232 q^{37} - 15210 q^{38} - 3216 q^{40} - 6048 q^{41} - 3020 q^{43} + 30816 q^{44} + 25584 q^{46} + 11700 q^{47} + 19701 q^{50} - 31444 q^{52} - 9468 q^{53} + 38904 q^{55} + 37314 q^{58} - 43938 q^{59} + 64754 q^{61} + 15300 q^{62} - 70783 q^{64} - 39060 q^{65} + 24784 q^{67} - 14994 q^{68} - 97416 q^{71} - 17452 q^{73} - 43434 q^{74} + 12782 q^{76} + 51256 q^{79} - 70272 q^{80} + 58338 q^{82} + 117558 q^{83} - 37860 q^{85} + 150048 q^{86} - 40656 q^{88} + 84276 q^{89} - 150912 q^{92} - 159468 q^{94} - 24264 q^{95} - 20776 q^{97}+O(q^{100}) \) 2 * q - 9 * q^2 + 5 * q^4 - 18 * q^5 + 9 * q^8 - 204 * q^10 - 396 * q^11 + 350 * q^13 + 113 * q^16 + 1800 * q^17 + 3266 * q^19 + 2520 * q^20 - 1752 * q^22 - 2088 * q^23 - 3238 * q^25 + 2016 * q^26 - 6696 * q^29 + 20 * q^31 + 6129 * q^32 - 5934 * q^34 + 6232 * q^37 - 15210 * q^38 - 3216 * q^40 - 6048 * q^41 - 3020 * q^43 + 30816 * q^44 + 25584 * q^46 + 11700 * q^47 + 19701 * q^50 - 31444 * q^52 - 9468 * q^53 + 38904 * q^55 + 37314 * q^58 - 43938 * q^59 + 64754 * q^61 + 15300 * q^62 - 70783 * q^64 - 39060 * q^65 + 24784 * q^67 - 14994 * q^68 - 97416 * q^71 - 17452 * q^73 - 43434 * q^74 + 12782 * q^76 + 51256 * q^79 - 70272 * q^80 + 58338 * q^82 + 117558 * q^83 - 37860 * q^85 + 150048 * q^86 - 40656 * q^88 + 84276 * q^89 - 150912 * q^92 - 159468 * q^94 - 24264 * q^95 - 20776 * q^97

Introduction

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