LMFDB - Embedded newform 441.6.a.ba.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$6$
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} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 $ x^6 - 2x^5 - 59x^4 + 122x^3 + 941x^2 - 1856*x - 2338
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-5.61145$ 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)

$f(q)$	$=$	$q+11.1881 q^{2} +93.1729 q^{4} +62.3150 q^{5} +684.407 q^{8} +O(q^{10})$	$q+11.1881 q^{2} +93.1729 q^{4} +62.3150 q^{5} +684.407 q^{8} +697.185 q^{10} +173.495 q^{11} +348.244 q^{13} +4675.66 q^{16} -999.206 q^{17} -2042.17 q^{19} +5806.07 q^{20} +1941.07 q^{22} +1949.67 q^{23} +758.159 q^{25} +3896.18 q^{26} -738.129 q^{29} -2459.99 q^{31} +30410.6 q^{32} -11179.2 q^{34} +8786.52 q^{37} -22848.0 q^{38} +42648.8 q^{40} -17617.4 q^{41} -10317.1 q^{43} +16165.0 q^{44} +21813.1 q^{46} +75.0582 q^{47} +8482.34 q^{50} +32446.9 q^{52} -27705.5 q^{53} +10811.3 q^{55} -8258.24 q^{58} +3457.57 q^{59} +15956.4 q^{61} -27522.5 q^{62} +190615. q^{64} +21700.8 q^{65} +14413.2 q^{67} -93099.0 q^{68} +30337.6 q^{71} +85880.9 q^{73} +98304.2 q^{74} -190275. q^{76} +7613.25 q^{79} +291364. q^{80} -197104. q^{82} -63725.4 q^{83} -62265.5 q^{85} -115428. q^{86} +118741. q^{88} -46391.3 q^{89} +181657. q^{92} +839.757 q^{94} -127258. q^{95} +21461.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 150 q^{4} - 100 q^{5} + 114 q^{8}+O(q^{10}) $ 6 * q - 2 * q^2 + 150 * q^4 - 100 * q^5 + 114 * q^8	$ 6 q - 2 q^{2} + 150 q^{4} - 100 q^{5} + 114 q^{8} + 864 q^{10} - 604 q^{11} + 1352 q^{13} + 4578 q^{16} - 3028 q^{17} + 1728 q^{19} - 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} - 14172 q^{26} + 5320 q^{29} + 3976 q^{31} + 37326 q^{32} - 16336 q^{34} + 22680 q^{37} - 52744 q^{38} + 100600 q^{40} - 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} - 51552 q^{47} + 40622 q^{50} + 119296 q^{52} - 80884 q^{53} + 11656 q^{55} - 70464 q^{58} - 8872 q^{59} + 50896 q^{61} - 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} - 37348 q^{68} + 110852 q^{71} + 64232 q^{73} + 27464 q^{74} - 194864 q^{76} + 111696 q^{79} + 308940 q^{80} - 189640 q^{82} - 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} + 35012 q^{89} + 449260 q^{92} - 121016 q^{94} + 119080 q^{95} + 70952 q^{97}+O(q^{100}) $ 6 * q - 2 * q^2 + 150 * q^4 - 100 * q^5 + 114 * q^8 + 864 * q^10 - 604 * q^11 + 1352 * q^13 + 4578 * q^16 - 3028 * q^17 + 1728 * q^19 - 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 - 14172 * q^26 + 5320 * q^29 + 3976 * q^31 + 37326 * q^32 - 16336 * q^34 + 22680 * q^37 - 52744 * q^38 + 100600 * q^40 - 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 - 51552 * q^47 + 40622 * q^50 + 119296 * q^52 - 80884 * q^53 + 11656 * q^55 - 70464 * q^58 - 8872 * q^59 + 50896 * q^61 - 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 - 37348 * q^68 + 110852 * q^71 + 64232 * q^73 + 27464 * q^74 - 194864 * q^76 + 111696 * q^79 + 308940 * q^80 - 189640 * q^82 - 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 + 35012 * q^89 + 449260 * q^92 - 121016 * q^94 + 119080 * q^95 + 70952 * 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 $

$2$	11.1881	1.97779	0.988895	−	0.148615i	$-0.0474815\pi$
$2$	11.1881	1.97779	0.988895	+	0.148615i	$0.0474815\pi$
$3$	0	0
$3$	0	0
$4$	93.1729	2.91165
$5$	62.3150	1.11472	0.557362	−	0.830269i	$-0.311813\pi$
$5$	62.3150	1.11472	0.557362	+	0.830269i	$0.311813\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	684.407	3.78085
$9$	0	0
$10$	697.185	2.20469
$11$	173.495	0.432320	0.216160	−	0.976358i	$-0.430647\pi$
$11$	173.495	0.432320	0.216160	+	0.976358i	$0.430647\pi$
$12$	0	0
$13$	348.244	0.571512	0.285756	−	0.958302i	$-0.407755\pi$
$13$	348.244	0.571512	0.285756	+	0.958302i	$0.407755\pi$
$14$	0	0
$15$	0	0
$16$	4675.66	4.56608
$17$	−999.206	−0.838557	−0.419279	−	0.907858i	$-0.637717\pi$
$17$	−999.206	−0.838557	−0.419279	+	0.907858i	$0.637717\pi$
$18$	0	0
$19$	−2042.17	−1.29780	−0.648901	−	0.760873i	$-0.724771\pi$
$19$	−2042.17	−1.29780	−0.648901	+	0.760873i	$0.724771\pi$
$20$	5806.07	3.24569
$21$	0	0
$22$	1941.07	0.855038
$23$	1949.67	0.768496	0.384248	−	0.923230i	$-0.374461\pi$
$23$	1949.67	0.768496	0.384248	+	0.923230i	$0.374461\pi$
$24$	0	0
$25$	758.159	0.242611
$26$	3896.18	1.13033
$27$	0	0
$28$	0	0
$29$	−738.129	−0.162981	−0.0814906	−	0.996674i	$-0.525968\pi$
$29$	−738.129	−0.162981	−0.0814906	+	0.996674i	$0.525968\pi$
$30$	0	0
$31$	−2459.99	−0.459758	−0.229879	−	0.973219i	$-0.573833\pi$
$31$	−2459.99	−0.459758	−0.229879	+	0.973219i	$0.573833\pi$
$32$	30410.6	5.24989
$33$	0	0
$34$	−11179.2	−1.65849
$35$	0	0
$36$	0	0
$37$	8786.52	1.05515	0.527573	−	0.849510i	$-0.323102\pi$
$37$	8786.52	1.05515	0.527573	+	0.849510i	$0.323102\pi$
$38$	−22848.0	−2.56678
$39$	0	0
$40$	42648.8	4.21461
$41$	−17617.4	−1.63675	−0.818374	−	0.574686i	$-0.805124\pi$
$41$	−17617.4	−1.63675	−0.818374	+	0.574686i	$0.805124\pi$
$42$	0	0
$43$	−10317.1	−0.850914	−0.425457	−	0.904979i	$-0.639887\pi$
$43$	−10317.1	−0.850914	−0.425457	+	0.904979i	$0.639887\pi$
$44$	16165.0	1.25877
$45$	0	0
$46$	21813.1	1.51992
$47$	75.0582	0.00495626	0.00247813	−	0.999997i	$-0.499211\pi$
$47$	75.0582	0.00495626	0.00247813	+	0.999997i	$0.499211\pi$
$48$	0	0
$49$	0	0
$50$	8482.34	0.479833
$51$	0	0
$52$	32446.9	1.66405
$53$	−27705.5	−1.35481	−0.677403	−	0.735612i	$-0.736895\pi$
$53$	−27705.5	−1.35481	−0.677403	+	0.735612i	$0.736895\pi$
$54$	0	0
$55$	10811.3	0.481917
$56$	0	0
$57$	0	0
$58$	−8258.24	−0.322342
$59$	3457.57	0.129313	0.0646564	−	0.997908i	$-0.479405\pi$
$59$	3457.57	0.129313	0.0646564	+	0.997908i	$0.479405\pi$
$60$	0	0
$61$	15956.4	0.549048	0.274524	−	0.961580i	$-0.411480\pi$
$61$	15956.4	0.549048	0.274524	+	0.961580i	$0.411480\pi$
$62$	−27522.5	−0.909304
$63$	0	0
$64$	190615.	5.81711
$65$	21700.8	0.637079
$66$	0	0
$67$	14413.2	0.392259	0.196129	−	0.980578i	$-0.437163\pi$
$67$	14413.2	0.392259	0.196129	+	0.980578i	$0.437163\pi$
$68$	−93099.0	−2.44159
$69$	0	0
$70$	0	0
$71$	30337.6	0.714225	0.357112	−	0.934061i	$-0.383761\pi$
$71$	30337.6	0.714225	0.357112	+	0.934061i	$0.383761\pi$
$72$	0	0
$73$	85880.9	1.88621	0.943104	−	0.332498i	$-0.107892\pi$
$73$	85880.9	1.88621	0.943104	+	0.332498i	$0.107892\pi$
$74$	98304.2	2.08686
$75$	0	0
$76$	−190275.	−3.77875
$77$	0	0
$78$	0	0
$79$	7613.25	0.137247	0.0686234	−	0.997643i	$-0.478139\pi$
$79$	7613.25	0.137247	0.0686234	+	0.997643i	$0.478139\pi$
$80$	291364.	5.08992
$81$	0	0
$82$	−197104.	−3.23714
$83$	−63725.4	−1.01535	−0.507677	−	0.861548i	$-0.669496\pi$
$83$	−63725.4	−1.01535	−0.507677	+	0.861548i	$0.669496\pi$
$84$	0	0
$85$	−62265.5	−0.934761
$86$	−115428.	−1.68293
$87$	0	0
$88$	118741.	1.63454
$89$	−46391.3	−0.620815	−0.310407	−	0.950604i	$-0.600465\pi$
$89$	−46391.3	−0.620815	−0.310407	+	0.950604i	$0.600465\pi$
$90$	0	0
$91$	0	0
$92$	181657.	2.23759
$93$	0	0
$94$	839.757	0.00980244
$95$	−127258.	−1.44669
$96$	0	0
$97$	21461.2	0.231592	0.115796	−	0.993273i	$-0.463058\pi$
$97$	21461.2	0.231592	0.115796	+	0.993273i	$0.463058\pi$
$98$	0	0
$99$	0	0
$100$	70639.9	0.706399
$101$	−145933.	−1.42347	−0.711737	−	0.702446i	$-0.752091\pi$
$101$	−145933.	−1.42347	−0.711737	+	0.702446i	$0.752091\pi$
$102$	0	0
$103$	165629.	1.53830	0.769152	−	0.639066i	$-0.220679\pi$
$103$	165629.	1.53830	0.769152	+	0.639066i	$0.220679\pi$
$104$	238341.	2.16080
$105$	0	0
$106$	−309972.	−2.67952
$107$	−215048.	−1.81584	−0.907918	−	0.419147i	$-0.862329\pi$
$107$	−215048.	−1.81584	−0.907918	+	0.419147i	$0.862329\pi$
$108$	0	0
$109$	−40463.9	−0.326213	−0.163107	−	0.986608i	$-0.552151\pi$
$109$	−40463.9	−0.326213	−0.163107	+	0.986608i	$0.552151\pi$
$110$	120958.	0.953131
$111$	0	0
$112$	0	0
$113$	15002.6	0.110528	0.0552638	−	0.998472i	$-0.482400\pi$
$113$	15002.6	0.110528	0.0552638	+	0.998472i	$0.482400\pi$
$114$	0	0
$115$	121494.	0.856661
$116$	−68773.7	−0.474545
$117$	0	0
$118$	38683.6	0.255753
$119$	0	0
$120$	0	0
$121$	−130951.	−0.813100
$122$	178521.	1.08590
$123$	0	0
$124$	−229204.	−1.33866
$125$	−147490.	−0.844280
$126$	0	0
$127$	118326.	0.650986	0.325493	−	0.945544i	$-0.394470\pi$
$127$	118326.	0.650986	0.325493	+	0.945544i	$0.394470\pi$
$128$	1.15947e6	6.25513
$129$	0	0
$130$	242791.	1.26001
$131$	219465.	1.11735	0.558673	−	0.829388i	$-0.311311\pi$
$131$	219465.	1.11735	0.558673	+	0.829388i	$0.311311\pi$
$132$	0	0
$133$	0	0
$134$	161256.	0.775805
$135$	0	0
$136$	−683864.	−3.17046
$137$	143983.	0.655405	0.327702	−	0.944781i	$-0.393726\pi$
$137$	143983.	0.655405	0.327702	+	0.944781i	$0.393726\pi$
$138$	0	0
$139$	244306.	1.07250	0.536249	−	0.844060i	$-0.319841\pi$
$139$	244306.	1.07250	0.536249	+	0.844060i	$0.319841\pi$
$140$	0	0
$141$	0	0
$142$	339419.	1.41259
$143$	60418.6	0.247076
$144$	0	0
$145$	−45996.5	−0.181679
$146$	960842.	3.73052
$147$	0	0
$148$	818666.	3.07222
$149$	279769.	1.03237	0.516184	−	0.856478i	$-0.327352\pi$
$149$	279769.	1.03237	0.516184	+	0.856478i	$0.327352\pi$
$150$	0	0
$151$	−200417.	−0.715307	−0.357653	−	0.933854i	$-0.616423\pi$
$151$	−200417.	−0.715307	−0.357653	+	0.933854i	$0.616423\pi$
$152$	−1.39768e6	−4.90679
$153$	0	0
$154$	0	0
$155$	−153294.	−0.512503
$156$	0	0
$157$	470002.	1.52178	0.760888	−	0.648883i	$-0.224764\pi$
$157$	470002.	1.52178	0.760888	+	0.648883i	$0.224764\pi$
$158$	85177.6	0.271446
$159$	0	0
$160$	1.89504e6	5.85218
$161$	0	0
$162$	0	0
$163$	−587005.	−1.73051	−0.865253	−	0.501335i	$-0.832842\pi$
$163$	−587005.	−1.73051	−0.865253	+	0.501335i	$0.832842\pi$
$164$	−1.64146e6	−4.76564
$165$	0	0
$166$	−712964.	−2.00816
$167$	−172819.	−0.479513	−0.239757	−	0.970833i	$-0.577068\pi$
$167$	−172819.	−0.479513	−0.239757	+	0.970833i	$0.577068\pi$
$168$	0	0
$169$	−250019.	−0.673374
$170$	−696631.	−1.84876
$171$	0	0
$172$	−961273.	−2.47757
$173$	5090.06	0.0129303	0.00646514	−	0.999979i	$-0.497942\pi$
$173$	5090.06	0.0129303	0.00646514	+	0.999979i	$0.497942\pi$
$174$	0	0
$175$	0	0
$176$	811204.	1.97400
$177$	0	0
$178$	−519030.	−1.22784
$179$	−242278.	−0.565174	−0.282587	−	0.959242i	$-0.591193\pi$
$179$	−242278.	−0.565174	−0.282587	+	0.959242i	$0.591193\pi$
$180$	0	0
$181$	−492254.	−1.11684	−0.558422	−	0.829557i	$-0.688593\pi$
$181$	−492254.	−1.11684	−0.558422	+	0.829557i	$0.688593\pi$
$182$	0	0
$183$	0	0
$184$	1.33437e6	2.90557
$185$	547532.	1.17620
$186$	0	0
$187$	−173357.	−0.362525
$188$	6993.40	0.0144309
$189$	0	0
$190$	−1.42377e6	−2.86125
$191$	203729.	0.404083	0.202041	−	0.979377i	$-0.435242\pi$
$191$	203729.	0.404083	0.202041	+	0.979377i	$0.435242\pi$
$192$	0	0
$193$	−61060.4	−0.117996	−0.0589979	−	0.998258i	$-0.518791\pi$
$193$	−61060.4	−0.117996	−0.0589979	+	0.998258i	$0.518791\pi$
$194$	240109.	0.458041
$195$	0	0
$196$	0	0
$197$	−457733.	−0.840325	−0.420162	−	0.907449i	$-0.638027\pi$
$197$	−457733.	−0.840325	−0.420162	+	0.907449i	$0.638027\pi$
$198$	0	0
$199$	−270547.	−0.484295	−0.242148	−	0.970239i	$-0.577852\pi$
$199$	−270547.	−0.484295	−0.242148	+	0.970239i	$0.577852\pi$
$200$	518889.	0.917276
$201$	0	0
$202$	−1.63271e6	−2.81533
$203$	0	0
$204$	0	0
$205$	−1.09783e6	−1.82452
$206$	1.85306e6	3.04244
$207$	0	0
$208$	1.62827e6	2.60957
$209$	−354306.	−0.561065
$210$	0	0
$211$	933127.	1.44289	0.721447	−	0.692469i	$-0.243477\pi$
$211$	933127.	1.44289	0.721447	+	0.692469i	$0.243477\pi$
$212$	−2.58141e6	−3.94473
$213$	0	0
$214$	−2.40598e6	−3.59134
$215$	−642909.	−0.948535
$216$	0	0
$217$	0	0
$218$	−452713.	−0.645182
$219$	0	0
$220$	1.00732e6	1.40318
$221$	−347968.	−0.479246
$222$	0	0
$223$	−197037.	−0.265329	−0.132664	−	0.991161i	$-0.542353\pi$
$223$	−197037.	−0.265329	−0.132664	+	0.991161i	$0.542353\pi$
$224$	0	0
$225$	0	0
$226$	167850.	0.218600
$227$	1.19839e6	1.54359	0.771795	−	0.635872i	$-0.219359\pi$
$227$	1.19839e6	1.54359	0.771795	+	0.635872i	$0.219359\pi$
$228$	0	0
$229$	−244069.	−0.307555	−0.153778	−	0.988105i	$-0.549144\pi$
$229$	−244069.	−0.307555	−0.153778	+	0.988105i	$0.549144\pi$
$230$	1.35928e6	1.69430
$231$	0	0
$232$	−505181.	−0.616207
$233$	152260.	0.183737	0.0918686	−	0.995771i	$-0.470716\pi$
$233$	152260.	0.183737	0.0918686	+	0.995771i	$0.470716\pi$
$234$	0	0
$235$	4677.25	0.00552486
$236$	322152.	0.376514
$237$	0	0
$238$	0	0
$239$	−1.51074e6	−1.71078	−0.855390	−	0.517984i	$-0.826683\pi$
$239$	−1.51074e6	−1.71078	−0.855390	+	0.517984i	$0.826683\pi$
$240$	0	0
$241$	232939.	0.258345	0.129172	−	0.991622i	$-0.458768\pi$
$241$	232939.	0.258345	0.129172	+	0.991622i	$0.458768\pi$
$242$	−1.46508e6	−1.60814
$243$	0	0
$244$	1.48670e6	1.59864
$245$	0	0
$246$	0	0
$247$	−711175.	−0.741709
$248$	−1.68363e6	−1.73828
$249$	0	0
$250$	−1.65013e6	−1.66981
$251$	−762727.	−0.764161	−0.382081	−	0.924129i	$-0.624792\pi$
$251$	−762727.	−0.764161	−0.382081	+	0.924129i	$0.624792\pi$
$252$	0	0
$253$	338258.	0.332236
$254$	1.32384e6	1.28751
$255$	0	0
$256$	6.87260e6	6.55422
$257$	−1.38112e6	−1.30436	−0.652181	−	0.758064i	$-0.726146\pi$
$257$	−1.38112e6	−1.30436	−0.652181	+	0.758064i	$0.726146\pi$
$258$	0	0
$259$	0	0
$260$	2.02193e6	1.85495
$261$	0	0
$262$	2.45539e6	2.20988
$263$	633610.	0.564850	0.282425	−	0.959289i	$-0.408861\pi$
$263$	633610.	0.564850	0.282425	+	0.959289i	$0.408861\pi$
$264$	0	0
$265$	−1.72647e6	−1.51023
$266$	0	0
$267$	0	0
$268$	1.34292e6	1.14212
$269$	20533.7	0.0173016	0.00865081	−	0.999963i	$-0.497246\pi$
$269$	20533.7	0.0173016	0.00865081	+	0.999963i	$0.497246\pi$
$270$	0	0
$271$	−2.20939e6	−1.82746	−0.913732	−	0.406318i	$-0.866813\pi$
$271$	−2.20939e6	−1.82746	−0.913732	+	0.406318i	$0.866813\pi$
$272$	−4.67195e6	−3.82892
$273$	0	0
$274$	1.61089e6	1.29625
$275$	131537.	0.104885
$276$	0	0
$277$	−679840.	−0.532362	−0.266181	−	0.963923i	$-0.585762\pi$
$277$	−679840.	−0.532362	−0.266181	+	0.963923i	$0.585762\pi$
$278$	2.73331e6	2.12118
$279$	0	0
$280$	0	0
$281$	831855.	0.628466	0.314233	−	0.949346i	$-0.398253\pi$
$281$	831855.	0.628466	0.314233	+	0.949346i	$0.398253\pi$
$282$	0	0
$283$	2.31071e6	1.71506	0.857531	−	0.514433i	$-0.171997\pi$
$283$	2.31071e6	1.71506	0.857531	+	0.514433i	$0.171997\pi$
$284$	2.82664e6	2.07958
$285$	0	0
$286$	675968.	0.488665
$287$	0	0
$288$	0	0
$289$	−421444.	−0.296821
$290$	−514612.	−0.359323
$291$	0	0
$292$	8.00178e6	5.49198
$293$	−1.54768e6	−1.05321	−0.526603	−	0.850111i	$-0.676535\pi$
$293$	−1.54768e6	−1.05321	−0.526603	+	0.850111i	$0.676535\pi$
$294$	0	0
$295$	215459.	0.144148
$296$	6.01356e6	3.98935
$297$	0	0
$298$	3.13008e6	2.04181
$299$	678962.	0.439205
$300$	0	0
$301$	0	0
$302$	−2.24228e6	−1.41473
$303$	0	0
$304$	−9.54851e6	−5.92586
$305$	994323.	0.612037
$306$	0	0
$307$	149665.	0.0906303	0.0453151	−	0.998973i	$-0.485571\pi$
$307$	149665.	0.0906303	0.0453151	+	0.998973i	$0.485571\pi$
$308$	0	0
$309$	0	0
$310$	−1.71507e6	−1.01362
$311$	146487.	0.0858812	0.0429406	−	0.999078i	$-0.486327\pi$
$311$	146487.	0.0858812	0.0429406	+	0.999078i	$0.486327\pi$
$312$	0	0
$313$	318335.	0.183664	0.0918319	−	0.995775i	$-0.470728\pi$
$313$	318335.	0.183664	0.0918319	+	0.995775i	$0.470728\pi$
$314$	5.25842e6	3.00975
$315$	0	0
$316$	709349.	0.399615
$317$	−2.16592e6	−1.21058	−0.605291	−	0.796004i	$-0.706943\pi$
$317$	−2.16592e6	−1.21058	−0.605291	+	0.796004i	$0.706943\pi$
$318$	0	0
$319$	−128062.	−0.0704599
$320$	1.18782e7	6.48447
$321$	0	0
$322$	0	0
$323$	2.04055e6	1.08828
$324$	0	0
$325$	264024.	0.138655
$326$	−6.56746e6	−3.42258
$327$	0	0
$328$	−1.20575e7	−6.18830
$329$	0	0
$330$	0	0
$331$	−2.54345e6	−1.27601	−0.638005	−	0.770033i	$-0.720240\pi$
$331$	−2.54345e6	−1.27601	−0.638005	+	0.770033i	$0.720240\pi$
$332$	−5.93748e6	−2.95636
$333$	0	0
$334$	−1.93351e6	−0.948377
$335$	898156.	0.437260
$336$	0	0
$337$	−2.57641e6	−1.23578	−0.617888	−	0.786266i	$-0.712012\pi$
$337$	−2.57641e6	−1.23578	−0.617888	+	0.786266i	$0.712012\pi$
$338$	−2.79723e6	−1.33179
$339$	0	0
$340$	−5.80146e6	−2.72170
$341$	−426796.	−0.198762
$342$	0	0
$343$	0	0
$344$	−7.06108e6	−3.21718
$345$	0	0
$346$	56948.0	0.0255734
$347$	1.99035e6	0.887374	0.443687	−	0.896182i	$-0.353670\pi$
$347$	1.99035e6	0.887374	0.443687	+	0.896182i	$0.353670\pi$
$348$	0	0
$349$	1.37419e6	0.603926	0.301963	−	0.953320i	$-0.402358\pi$
$349$	1.37419e6	0.603926	0.301963	+	0.953320i	$0.402358\pi$
$350$	0	0
$351$	0	0
$352$	5.27609e6	2.26963
$353$	2.93667e6	1.25435	0.627174	−	0.778879i	$-0.284211\pi$
$353$	2.93667e6	1.25435	0.627174	+	0.778879i	$0.284211\pi$
$354$	0	0
$355$	1.89049e6	0.796164
$356$	−4.32242e6	−1.80760
$357$	0	0
$358$	−2.71063e6	−1.11779
$359$	−1.38575e6	−0.567476	−0.283738	−	0.958902i	$-0.591575\pi$
$359$	−1.38575e6	−0.567476	−0.283738	+	0.958902i	$0.591575\pi$
$360$	0	0
$361$	1.69436e6	0.684288
$362$	−5.50737e6	−2.20888
$363$	0	0
$364$	0	0
$365$	5.35167e6	2.10260
$366$	0	0
$367$	3.64088e6	1.41105	0.705523	−	0.708687i	$-0.250712\pi$
$367$	3.64088e6	1.41105	0.705523	+	0.708687i	$0.250712\pi$
$368$	9.11600e6	3.50901
$369$	0	0
$370$	6.12583e6	2.32627
$371$	0	0
$372$	0	0
$373$	4.14938e6	1.54423	0.772114	−	0.635484i	$-0.219200\pi$
$373$	4.14938e6	1.54423	0.772114	+	0.635484i	$0.219200\pi$
$374$	−1.93953e6	−0.716998
$375$	0	0
$376$	51370.4	0.0187389
$377$	−257049.	−0.0931457
$378$	0	0
$379$	−1.26316e6	−0.451711	−0.225856	−	0.974161i	$-0.572518\pi$
$379$	−1.26316e6	−0.451711	−0.225856	+	0.974161i	$0.572518\pi$
$380$	−1.18570e7	−4.21226
$381$	0	0
$382$	2.27934e6	0.799191
$383$	2.28188e6	0.794869	0.397435	−	0.917631i	$-0.369901\pi$
$383$	2.28188e6	0.794869	0.397435	+	0.917631i	$0.369901\pi$
$384$	0	0
$385$	0	0
$386$	−683148.	−0.233371
$387$	0	0
$388$	1.99960e6	0.674317
$389$	−3.25869e6	−1.09186	−0.545932	−	0.837829i	$-0.683824\pi$
$389$	−3.25869e6	−1.09186	−0.545932	+	0.837829i	$0.683824\pi$
$390$	0	0
$391$	−1.94812e6	−0.644428
$392$	0	0
$393$	0	0
$394$	−5.12115e6	−1.66199
$395$	474420.	0.152992
$396$	0	0
$397$	−4.49485e6	−1.43133	−0.715663	−	0.698445i	$-0.753876\pi$
$397$	−4.49485e6	−1.43133	−0.715663	+	0.698445i	$0.753876\pi$
$398$	−3.02690e6	−0.957835
$399$	0	0
$400$	3.54490e6	1.10778
$401$	−552039.	−0.171439	−0.0857194	−	0.996319i	$-0.527319\pi$
$401$	−552039.	−0.171439	−0.0857194	+	0.996319i	$0.527319\pi$
$402$	0	0
$403$	−856677.	−0.262757
$404$	−1.35970e7	−4.14467
$405$	0	0
$406$	0	0
$407$	1.52442e6	0.456160
$408$	0	0
$409$	6.05239e6	1.78903	0.894517	−	0.447035i	$-0.147520\pi$
$409$	6.05239e6	1.78903	0.894517	+	0.447035i	$0.147520\pi$
$410$	−1.22826e7	−3.60852
$411$	0	0
$412$	1.54321e7	4.47901
$413$	0	0
$414$	0	0
$415$	−3.97105e6	−1.13184
$416$	1.05903e7	3.00038
$417$	0	0
$418$	−3.96400e6	−1.10967
$419$	1.48630e6	0.413592	0.206796	−	0.978384i	$-0.433696\pi$
$419$	1.48630e6	0.413592	0.206796	+	0.978384i	$0.433696\pi$
$420$	0	0
$421$	−1.40434e6	−0.386159	−0.193079	−	0.981183i	$-0.561847\pi$
$421$	−1.40434e6	−0.386159	−0.193079	+	0.981183i	$0.561847\pi$
$422$	1.04399e7	2.85374
$423$	0	0
$424$	−1.89619e7	−5.12232
$425$	−757557.	−0.203443
$426$	0	0
$427$	0	0
$428$	−2.00367e7	−5.28709
$429$	0	0
$430$	−7.19291e6	−1.87600
$431$	2.70173e6	0.700567	0.350283	−	0.936644i	$-0.386085\pi$
$431$	2.70173e6	0.700567	0.350283	+	0.936644i	$0.386085\pi$
$432$	0	0
$433$	−4.76987e6	−1.22261	−0.611304	−	0.791396i	$-0.709355\pi$
$433$	−4.76987e6	−1.22261	−0.611304	+	0.791396i	$0.709355\pi$
$434$	0	0
$435$	0	0
$436$	−3.77014e6	−0.949821
$437$	−3.98156e6	−0.997355
$438$	0	0
$439$	6327.49	0.00156700	0.000783502	−	1.00000i	$-0.499751\pi$
$439$	6327.49	0.00156700	0.000783502		1.00000i	$0.499751\pi$
$440$	7.39935e6	1.82206
$441$	0	0
$442$	−3.89309e6	−0.947848
$443$	−1.41589e6	−0.342783	−0.171392	−	0.985203i	$-0.554826\pi$
$443$	−1.41589e6	−0.342783	−0.171392	+	0.985203i	$0.554826\pi$
$444$	0	0
$445$	−2.89088e6	−0.692037
$446$	−2.20446e6	−0.524765
$447$	0	0
$448$	0	0
$449$	1.88150e6	0.440442	0.220221	−	0.975450i	$-0.429322\pi$
$449$	1.88150e6	0.440442	0.220221	+	0.975450i	$0.429322\pi$
$450$	0	0
$451$	−3.05652e6	−0.707598
$452$	1.39784e6	0.321818
$453$	0	0
$454$	1.34076e7	3.05290
$455$	0	0
$456$	0	0
$457$	7.44422e6	1.66736	0.833678	−	0.552250i	$-0.186231\pi$
$457$	7.44422e6	1.66736	0.833678	+	0.552250i	$0.186231\pi$
$458$	−2.73066e6	−0.608280
$459$	0	0
$460$	1.13199e7	2.49430
$461$	−7.95698e6	−1.74380	−0.871899	−	0.489686i	$-0.837112\pi$
$461$	−7.95698e6	−1.74380	−0.871899	+	0.489686i	$0.837112\pi$
$462$	0	0
$463$	6.64462e6	1.44051	0.720257	−	0.693707i	$-0.244024\pi$
$463$	6.64462e6	1.44051	0.720257	+	0.693707i	$0.244024\pi$
$464$	−3.45124e6	−0.744184
$465$	0	0
$466$	1.70350e6	0.363394
$467$	2.97533e6	0.631311	0.315655	−	0.948874i	$-0.397776\pi$
$467$	2.97533e6	0.631311	0.315655	+	0.948874i	$0.397776\pi$
$468$	0	0
$469$	0	0
$470$	52329.5	0.0109270
$471$	0	0
$472$	2.36639e6	0.488912
$473$	−1.78996e6	−0.367867
$474$	0	0
$475$	−1.54829e6	−0.314861
$476$	0	0
$477$	0	0
$478$	−1.69022e7	−3.38356
$479$	3.40858e6	0.678789	0.339395	−	0.940644i	$-0.389778\pi$
$479$	3.40858e6	0.678789	0.339395	+	0.940644i	$0.389778\pi$
$480$	0	0
$481$	3.05985e6	0.603029
$482$	2.60614e6	0.510951
$483$	0	0
$484$	−1.22010e7	−2.36747
$485$	1.33735e6	0.258162
$486$	0	0
$487$	5.91307e6	1.12977	0.564886	−	0.825169i	$-0.308920\pi$
$487$	5.91307e6	1.12977	0.564886	+	0.825169i	$0.308920\pi$
$488$	1.09207e7	2.07587
$489$	0	0
$490$	0	0
$491$	−549890.	−0.102937	−0.0514686	−	0.998675i	$-0.516390\pi$
$491$	−549890.	−0.102937	−0.0514686	+	0.998675i	$0.516390\pi$
$492$	0	0
$493$	737543.	0.136669
$494$	−7.95667e6	−1.46695
$495$	0	0
$496$	−1.15021e7	−2.09929
$497$	0	0
$498$	0	0
$499$	5.60945e6	1.00848	0.504241	−	0.863563i	$-0.331772\pi$
$499$	5.60945e6	1.00848	0.504241	+	0.863563i	$0.331772\pi$
$500$	−1.37420e7	−2.45825
$501$	0	0
$502$	−8.53345e6	−1.51135
$503$	−1.43167e6	−0.252304	−0.126152	−	0.992011i	$-0.540263\pi$
$503$	−1.43167e6	−0.252304	−0.126152	+	0.992011i	$0.540263\pi$
$504$	0	0
$505$	−9.09381e6	−1.58678
$506$	3.78445e6	0.657093
$507$	0	0
$508$	1.10248e7	1.89545
$509$	5.39838e6	0.923568	0.461784	−	0.886992i	$-0.347210\pi$
$509$	5.39838e6	0.923568	0.461784	+	0.886992i	$0.347210\pi$
$510$	0	0
$511$	0	0
$512$	3.97880e7	6.70775
$513$	0	0
$514$	−1.54520e7	−2.57975
$515$	1.03211e7	1.71478
$516$	0	0
$517$	13022.2	0.00214269
$518$	0	0
$519$	0	0
$520$	1.48522e7	2.40870
$521$	1.56889e6	0.253220	0.126610	−	0.991953i	$-0.459590\pi$
$521$	1.56889e6	0.253220	0.126610	+	0.991953i	$0.459590\pi$
$522$	0	0
$523$	4.63319e6	0.740671	0.370336	−	0.928898i	$-0.379243\pi$
$523$	4.63319e6	0.740671	0.370336	+	0.928898i	$0.379243\pi$
$524$	2.04482e7	3.25333
$525$	0	0
$526$	7.08888e6	1.11715
$527$	2.45804e6	0.385533
$528$	0	0
$529$	−2.63513e6	−0.409414
$530$	−1.93159e7	−2.98693
$531$	0	0
$532$	0	0
$533$	−6.13515e6	−0.935421
$534$	0	0
$535$	−1.34007e7	−2.02416
$536$	9.86448e6	1.48307
$537$	0	0
$538$	229733.	0.0342190
$539$	0	0
$540$	0	0
$541$	1.03758e7	1.52415	0.762074	−	0.647490i	$-0.224181\pi$
$541$	1.03758e7	1.52415	0.762074	+	0.647490i	$0.224181\pi$
$542$	−2.47188e7	−3.61434
$543$	0	0
$544$	−3.03865e7	−4.40234
$545$	−2.52151e6	−0.363638
$546$	0	0
$547$	1.07555e7	1.53695	0.768476	−	0.639879i	$-0.221015\pi$
$547$	1.07555e7	1.53695	0.768476	+	0.639879i	$0.221015\pi$
$548$	1.34153e7	1.90831
$549$	0	0
$550$	1.47164e6	0.207441
$551$	1.50739e6	0.211517
$552$	0	0
$553$	0	0
$554$	−7.60610e6	−1.05290
$555$	0	0
$556$	2.27627e7	3.12274
$557$	3.37312e6	0.460674	0.230337	−	0.973111i	$-0.426017\pi$
$557$	3.37312e6	0.460674	0.230337	+	0.973111i	$0.426017\pi$
$558$	0	0
$559$	−3.59286e6	−0.486308
$560$	0	0
$561$	0	0
$562$	9.30685e6	1.24297
$563$	9.76402e6	1.29825	0.649124	−	0.760683i	$-0.275136\pi$
$563$	9.76402e6	1.29825	0.649124	+	0.760683i	$0.275136\pi$
$564$	0	0
$565$	934888.	0.123208
$566$	2.58524e7	3.39203
$567$	0	0
$568$	2.07633e7	2.70038
$569$	9.03945e6	1.17047	0.585236	−	0.810863i	$-0.301002\pi$
$569$	9.03945e6	1.17047	0.585236	+	0.810863i	$0.301002\pi$
$570$	0	0
$571$	−1.06993e7	−1.37329	−0.686646	−	0.726991i	$-0.740918\pi$
$571$	−1.06993e7	−1.37329	−0.686646	+	0.726991i	$0.740918\pi$
$572$	5.62938e6	0.719400
$573$	0	0
$574$	0	0
$575$	1.47816e6	0.186445
$576$	0	0
$577$	−5.00063e6	−0.625295	−0.312648	−	0.949869i	$-0.601216\pi$
$577$	−5.00063e6	−0.625295	−0.312648	+	0.949869i	$0.601216\pi$
$578$	−4.71515e6	−0.587051
$579$	0	0
$580$	−4.28563e6	−0.528987
$581$	0	0
$582$	0	0
$583$	−4.80677e6	−0.585709
$584$	5.87775e7	7.13147
$585$	0	0
$586$	−1.73156e7	−2.08302
$587$	7.54203e6	0.903427	0.451713	−	0.892163i	$-0.350813\pi$
$587$	7.54203e6	0.903427	0.451713	+	0.892163i	$0.350813\pi$
$588$	0	0
$589$	5.02372e6	0.596674
$590$	2.41057e6	0.285095
$591$	0	0
$592$	4.10828e7	4.81788
$593$	5.99664e6	0.700279	0.350139	−	0.936698i	$-0.386134\pi$
$593$	5.99664e6	0.700279	0.350139	+	0.936698i	$0.386134\pi$
$594$	0	0
$595$	0	0
$596$	2.60669e7	3.00590
$597$	0	0
$598$	7.59627e6	0.868655
$599$	7.54150e6	0.858797	0.429399	−	0.903115i	$-0.358726\pi$
$599$	7.54150e6	0.858797	0.429399	+	0.903115i	$0.358726\pi$
$600$	0	0
$601$	−2.34440e6	−0.264756	−0.132378	−	0.991199i	$-0.542261\pi$
$601$	−2.34440e6	−0.264756	−0.132378	+	0.991199i	$0.542261\pi$
$602$	0	0
$603$	0	0
$604$	−1.86734e7	−2.08273
$605$	−8.16018e6	−0.906382
$606$	0	0
$607$	1.41940e7	1.56363	0.781816	−	0.623509i	$-0.214294\pi$
$607$	1.41940e7	1.56363	0.781816	+	0.623509i	$0.214294\pi$
$608$	−6.21037e7	−6.81332
$609$	0	0
$610$	1.11246e7	1.21048
$611$	26138.6	0.00283256
$612$	0	0
$613$	1.53799e7	1.65311	0.826555	−	0.562856i	$-0.190297\pi$
$613$	1.53799e7	1.65311	0.826555	+	0.562856i	$0.190297\pi$
$614$	1.67446e6	0.179248
$615$	0	0
$616$	0	0
$617$	3.26335e6	0.345105	0.172552	−	0.985000i	$-0.444799\pi$
$617$	3.26335e6	0.345105	0.172552	+	0.985000i	$0.444799\pi$
$618$	0	0
$619$	−1.14762e7	−1.20385	−0.601924	−	0.798553i	$-0.705599\pi$
$619$	−1.14762e7	−1.20385	−0.601924	+	0.798553i	$0.705599\pi$
$620$	−1.42829e7	−1.49223
$621$	0	0
$622$	1.63891e6	0.169855
$623$	0	0
$624$	0	0
$625$	−1.15601e7	−1.18375
$626$	3.56155e6	0.363248
$627$	0	0
$628$	4.37915e7	4.43089
$629$	−8.77954e6	−0.884800
$630$	0	0
$631$	9.11249e6	0.911095	0.455548	−	0.890211i	$-0.349444\pi$
$631$	9.11249e6	0.911095	0.455548	+	0.890211i	$0.349444\pi$
$632$	5.21056e6	0.518910
$633$	0	0
$634$	−2.42325e7	−2.39428
$635$	7.37350e6	0.725670
$636$	0	0
$637$	0	0
$638$	−1.43276e6	−0.139355
$639$	0	0
$640$	7.22526e7	6.97275
$641$	1.34121e7	1.28930	0.644649	−	0.764479i	$-0.277004\pi$
$641$	1.34121e7	1.28930	0.644649	+	0.764479i	$0.277004\pi$
$642$	0	0
$643$	5.00517e6	0.477410	0.238705	−	0.971092i	$-0.423277\pi$
$643$	5.00517e6	0.477410	0.238705	+	0.971092i	$0.423277\pi$
$644$	0	0
$645$	0	0
$646$	2.28298e7	2.15239
$647$	5.83524e6	0.548022	0.274011	−	0.961727i	$-0.411650\pi$
$647$	5.83524e6	0.548022	0.274011	+	0.961727i	$0.411650\pi$
$648$	0	0
$649$	599871.	0.0559044
$650$	2.95392e6	0.274231
$651$	0	0
$652$	−5.46930e7	−5.03864
$653$	−1.51369e7	−1.38916	−0.694581	−	0.719415i	$-0.744410\pi$
$653$	−1.51369e7	−1.38916	−0.694581	+	0.719415i	$0.744410\pi$
$654$	0	0
$655$	1.36760e7	1.24553
$656$	−8.23729e7	−7.47352
$657$	0	0
$658$	0	0
$659$	−1.26786e7	−1.13725	−0.568625	−	0.822597i	$-0.692525\pi$
$659$	−1.26786e7	−1.13725	−0.568625	+	0.822597i	$0.692525\pi$
$660$	0	0
$661$	1.40024e6	0.124652	0.0623261	−	0.998056i	$-0.480148\pi$
$661$	1.40024e6	0.124652	0.0623261	+	0.998056i	$0.480148\pi$
$662$	−2.84563e7	−2.52368
$663$	0	0
$664$	−4.36141e7	−3.83890
$665$	0	0
$666$	0	0
$667$	−1.43911e6	−0.125250
$668$	−1.61021e7	−1.39618
$669$	0	0
$670$	1.00486e7	0.864809
$671$	2.76835e6	0.237364
$672$	0	0
$673$	−6.01514e6	−0.511927	−0.255963	−	0.966686i	$-0.582393\pi$
$673$	−6.01514e6	−0.511927	−0.255963	+	0.966686i	$0.582393\pi$
$674$	−2.88250e7	−2.44411
$675$	0	0
$676$	−2.32950e7	−1.96063
$677$	6.60009e6	0.553450	0.276725	−	0.960949i	$-0.410751\pi$
$677$	6.60009e6	0.553450	0.276725	+	0.960949i	$0.410751\pi$
$678$	0	0
$679$	0	0
$680$	−4.26150e7	−3.53419
$681$	0	0
$682$	−4.77502e6	−0.393110
$683$	−213734.	−0.0175316	−0.00876581	−	0.999962i	$-0.502790\pi$
$683$	−213734.	−0.0175316	−0.00876581	+	0.999962i	$0.502790\pi$
$684$	0	0
$685$	8.97230e6	0.730596
$686$	0	0
$687$	0	0
$688$	−4.82392e7	−3.88534
$689$	−9.64830e6	−0.774288
$690$	0	0
$691$	−1.43510e6	−0.114337	−0.0571684	−	0.998365i	$-0.518207\pi$
$691$	−1.43510e6	−0.114337	−0.0571684	+	0.998365i	$0.518207\pi$
$692$	474256.	0.0376485
$693$	0	0
$694$	2.22682e7	1.75504
$695$	1.52239e7	1.19554
$696$	0	0
$697$	1.76034e7	1.37251
$698$	1.53745e7	1.19444
$699$	0	0
$700$	0	0
$701$	1.27397e7	0.979186	0.489593	−	0.871951i	$-0.337145\pi$
$701$	1.27397e7	0.979186	0.489593	+	0.871951i	$0.337145\pi$
$702$	0	0
$703$	−1.79436e7	−1.36937
$704$	3.30707e7	2.51485
$705$	0	0
$706$	3.28557e7	2.48084
$707$	0	0
$708$	0	0
$709$	7.79504e6	0.582375	0.291187	−	0.956666i	$-0.405950\pi$
$709$	7.79504e6	0.582375	0.291187	+	0.956666i	$0.405950\pi$
$710$	2.11509e7	1.57465
$711$	0	0
$712$	−3.17506e7	−2.34721
$713$	−4.79617e6	−0.353322
$714$	0	0
$715$	3.76498e6	0.275422
$716$	−2.25738e7	−1.64559
$717$	0	0
$718$	−1.55038e7	−1.12235
$719$	−4.86960e6	−0.351294	−0.175647	−	0.984453i	$-0.556202\pi$
$719$	−4.86960e6	−0.351294	−0.175647	+	0.984453i	$0.556202\pi$
$720$	0	0
$721$	0	0
$722$	1.89567e7	1.35338
$723$	0	0
$724$	−4.58647e7	−3.25186
$725$	−559619.	−0.0395410
$726$	0	0
$727$	−1.29961e7	−0.911966	−0.455983	−	0.889989i	$-0.650712\pi$
$727$	−1.29961e7	−0.911966	−0.455983	+	0.889989i	$0.650712\pi$
$728$	0	0
$729$	0	0
$730$	5.98749e7	4.15851
$731$	1.03089e7	0.713540
$732$	0	0
$733$	−4.80604e6	−0.330391	−0.165195	−	0.986261i	$-0.552825\pi$
$733$	−4.80604e6	−0.330391	−0.165195	+	0.986261i	$0.552825\pi$
$734$	4.07344e7	2.79075
$735$	0	0
$736$	5.92907e7	4.03452
$737$	2.50061e6	0.169581
$738$	0	0
$739$	1.85725e7	1.25101	0.625503	−	0.780222i	$-0.284894\pi$
$739$	1.85725e7	1.25101	0.625503	+	0.780222i	$0.284894\pi$
$740$	5.10152e7	3.42468
$741$	0	0
$742$	0	0
$743$	−1.17402e7	−0.780192	−0.390096	−	0.920774i	$-0.627558\pi$
$743$	−1.17402e7	−0.780192	−0.390096	+	0.920774i	$0.627558\pi$
$744$	0	0
$745$	1.74338e7	1.15081
$746$	4.64236e7	3.05416
$747$	0	0
$748$	−1.61522e7	−1.05555
$749$	0	0
$750$	0	0
$751$	1.50501e7	0.973734	0.486867	−	0.873476i	$-0.338140\pi$
$751$	1.50501e7	0.973734	0.486867	+	0.873476i	$0.338140\pi$
$752$	350947.	0.0226307
$753$	0	0
$754$	−2.87589e6	−0.184223
$755$	−1.24890e7	−0.797370
$756$	0	0
$757$	2.04785e6	0.129885	0.0649423	−	0.997889i	$-0.479314\pi$
$757$	2.04785e6	0.129885	0.0649423	+	0.997889i	$0.479314\pi$
$758$	−1.41323e7	−0.893390
$759$	0	0
$760$	−8.70962e7	−5.46972
$761$	−1.65151e7	−1.03376	−0.516881	−	0.856057i	$-0.672907\pi$
$761$	−1.65151e7	−1.03376	−0.516881	+	0.856057i	$0.672907\pi$
$762$	0	0
$763$	0	0
$764$	1.89821e7	1.17655
$765$	0	0
$766$	2.55298e7	1.57208
$767$	1.20408e6	0.0739038
$768$	0	0
$769$	2.10148e7	1.28147	0.640736	−	0.767761i	$-0.278629\pi$
$769$	2.10148e7	1.28147	0.640736	+	0.767761i	$0.278629\pi$
$770$	0	0
$771$	0	0
$772$	−5.68917e6	−0.343563
$773$	−2.79653e7	−1.68333	−0.841667	−	0.539996i	$-0.818426\pi$
$773$	−2.79653e7	−1.68333	−0.841667	+	0.539996i	$0.818426\pi$
$774$	0	0
$775$	−1.86506e6	−0.111542
$776$	1.46882e7	0.875616
$777$	0	0
$778$	−3.64584e7	−2.15948
$779$	3.59777e7	2.12417
$780$	0	0
$781$	5.26341e6	0.308773
$782$	−2.17957e7	−1.27454
$783$	0	0
$784$	0	0
$785$	2.92882e7	1.69636
$786$	0	0
$787$	−2.67284e6	−0.153828	−0.0769141	−	0.997038i	$-0.524507\pi$
$787$	−2.67284e6	−0.153828	−0.0769141	+	0.997038i	$0.524507\pi$
$788$	−4.26484e7	−2.44673
$789$	0	0
$790$	5.30784e6	0.302587
$791$	0	0
$792$	0	0
$793$	5.55673e6	0.313788
$794$	−5.02887e7	−2.83086
$795$	0	0
$796$	−2.52077e7	−1.41010
$797$	−8.17692e6	−0.455978	−0.227989	−	0.973664i	$-0.573215\pi$
$797$	−8.17692e6	−0.455978	−0.227989	+	0.973664i	$0.573215\pi$
$798$	0	0
$799$	−74998.7	−0.00415611
$800$	2.30561e7	1.27368
$801$	0	0
$802$	−6.17625e6	−0.339070
$803$	1.48999e7	0.815445
$804$	0	0
$805$	0	0
$806$	−9.58457e6	−0.519679
$807$	0	0
$808$	−9.98775e7	−5.38195
$809$	−2.37801e7	−1.27745	−0.638723	−	0.769436i	$-0.720537\pi$
$809$	−2.37801e7	−1.27745	−0.638723	+	0.769436i	$0.720537\pi$
$810$	0	0
$811$	2.42489e7	1.29461	0.647306	−	0.762231i	$-0.275896\pi$
$811$	2.42489e7	1.29461	0.647306	+	0.762231i	$0.275896\pi$
$812$	0	0
$813$	0	0
$814$	1.70553e7	0.902189
$815$	−3.65792e7	−1.92904
$816$	0	0
$817$	2.10692e7	1.10432
$818$	6.77145e7	3.53833
$819$	0	0
$820$	−1.02288e8	−5.31238
$821$	3.60109e6	0.186456	0.0932280	−	0.995645i	$-0.470281\pi$
$821$	3.60109e6	0.186456	0.0932280	+	0.995645i	$0.470281\pi$
$822$	0	0
$823$	−1.68438e7	−0.866844	−0.433422	−	0.901191i	$-0.642694\pi$
$823$	−1.68438e7	−0.866844	−0.433422	+	0.901191i	$0.642694\pi$
$824$	1.13357e8	5.81610
$825$	0	0
$826$	0	0
$827$	1.81504e7	0.922829	0.461415	−	0.887185i	$-0.347342\pi$
$827$	1.81504e7	0.922829	0.461415	+	0.887185i	$0.347342\pi$
$828$	0	0
$829$	1.07241e7	0.541969	0.270985	−	0.962584i	$-0.412651\pi$
$829$	1.07241e7	0.541969	0.270985	+	0.962584i	$0.412651\pi$
$830$	−4.44284e7	−2.23854
$831$	0	0
$832$	6.63806e7	3.32455
$833$	0	0
$834$	0	0
$835$	−1.07692e7	−0.534525
$836$	−3.30118e7	−1.63363
$837$	0	0
$838$	1.66288e7	0.817997
$839$	−1.18159e7	−0.579510	−0.289755	−	0.957101i	$-0.593574\pi$
$839$	−1.18159e7	−0.579510	−0.289755	+	0.957101i	$0.593574\pi$
$840$	0	0
$841$	−1.99663e7	−0.973437
$842$	−1.57118e7	−0.763741
$843$	0	0
$844$	8.69422e7	4.20121
$845$	−1.55799e7	−0.750626
$846$	0	0
$847$	0	0
$848$	−1.29542e8	−6.18615
$849$	0	0
$850$	−8.47560e6	−0.402368
$851$	1.71308e7	0.810875
$852$	0	0
$853$	6.26477e6	0.294803	0.147402	−	0.989077i	$-0.452909\pi$
$853$	6.26477e6	0.294803	0.147402	+	0.989077i	$0.452909\pi$
$854$	0	0
$855$	0	0
$856$	−1.47181e8	−6.86541
$857$	2.51137e7	1.16804	0.584020	−	0.811739i	$-0.301479\pi$
$857$	2.51137e7	1.16804	0.584020	+	0.811739i	$0.301479\pi$
$858$	0	0
$859$	4.14654e6	0.191736	0.0958679	−	0.995394i	$-0.469437\pi$
$859$	4.14654e6	0.191736	0.0958679	+	0.995394i	$0.469437\pi$
$860$	−5.99017e7	−2.76180
$861$	0	0
$862$	3.02272e7	1.38557
$863$	1.38124e7	0.631308	0.315654	−	0.948874i	$-0.397776\pi$
$863$	1.38124e7	0.631308	0.315654	+	0.948874i	$0.397776\pi$
$864$	0	0
$865$	317187.	0.0144137
$866$	−5.33657e7	−2.41806
$867$	0	0
$868$	0	0
$869$	1.32086e6	0.0593345
$870$	0	0
$871$	5.01930e6	0.224181
$872$	−2.76938e7	−1.23336
$873$	0	0
$874$	−4.45460e7	−1.97256
$875$	0	0
$876$	0	0
$877$	9.15841e6	0.402088	0.201044	−	0.979582i	$-0.435567\pi$
$877$	9.15841e6	0.402088	0.201044	+	0.979582i	$0.435567\pi$
$878$	70792.4	0.00309920
$879$	0	0
$880$	5.05502e7	2.20047
$881$	−2.45789e7	−1.06690	−0.533449	−	0.845833i	$-0.679104\pi$
$881$	−2.45789e7	−1.06690	−0.533449	+	0.845833i	$0.679104\pi$
$882$	0	0
$883$	−1.10185e7	−0.475575	−0.237787	−	0.971317i	$-0.576422\pi$
$883$	−1.10185e7	−0.475575	−0.237787	+	0.971317i	$0.576422\pi$
$884$	−3.24212e7	−1.39540
$885$	0	0
$886$	−1.58411e7	−0.677954
$887$	−2.86983e7	−1.22475	−0.612374	−	0.790569i	$-0.709785\pi$
$887$	−2.86983e7	−1.22475	−0.612374	+	0.790569i	$0.709785\pi$
$888$	0	0
$889$	0	0
$890$	−3.23433e7	−1.36870
$891$	0	0
$892$	−1.83585e7	−0.772546
$893$	−153282.	−0.00643224
$894$	0	0
$895$	−1.50976e7	−0.630013
$896$	0	0
$897$	0	0
$898$	2.10504e7	0.871103
$899$	1.81579e6	0.0749318
$900$	0	0
$901$	2.76835e7	1.13608
$902$	−3.41966e7	−1.39948
$903$	0	0
$904$	1.02679e7	0.417889
$905$	−3.06748e7	−1.24497
$906$	0	0
$907$	−1.55637e7	−0.628195	−0.314098	−	0.949391i	$-0.601702\pi$
$907$	−1.55637e7	−0.628195	−0.314098	+	0.949391i	$0.601702\pi$
$908$	1.11657e8	4.49440
$909$	0	0
$910$	0	0
$911$	−2.36784e7	−0.945273	−0.472637	−	0.881257i	$-0.656698\pi$
$911$	−2.36784e7	−0.945273	−0.472637	+	0.881257i	$0.656698\pi$
$912$	0	0
$913$	−1.10560e7	−0.438957
$914$	8.32864e7	3.29768
$915$	0	0
$916$	−2.27406e7	−0.895495
$917$	0	0
$918$	0	0
$919$	1.31869e7	0.515054	0.257527	−	0.966271i	$-0.417092\pi$
$919$	1.31869e7	0.515054	0.257527	+	0.966271i	$0.417092\pi$
$920$	8.31512e7	3.23891
$921$	0	0
$922$	−8.90233e7	−3.44887
$923$	1.05649e7	0.408188
$924$	0	0
$925$	6.66158e6	0.255990
$926$	7.43405e7	2.84903
$927$	0	0
$928$	−2.24470e7	−0.855633
$929$	−2.36801e7	−0.900211	−0.450105	−	0.892975i	$-0.648614\pi$
$929$	−2.36801e7	−0.900211	−0.450105	+	0.892975i	$0.648614\pi$
$930$	0	0
$931$	0	0
$932$	1.41865e7	0.534979
$933$	0	0
$934$	3.32882e7	1.24860
$935$	−1.08028e7	−0.404115
$936$	0	0
$937$	9.81602e6	0.365247	0.182623	−	0.983183i	$-0.441541\pi$
$937$	9.81602e6	0.365247	0.182623	+	0.983183i	$0.441541\pi$
$938$	0	0
$939$	0	0
$940$	435794.	0.0160865
$941$	3.32172e7	1.22289	0.611446	−	0.791286i	$-0.290588\pi$
$941$	3.32172e7	1.22289	0.611446	+	0.791286i	$0.290588\pi$
$942$	0	0
$943$	−3.43481e7	−1.25783
$944$	1.61664e7	0.590452
$945$	0	0
$946$	−2.00262e7	−0.727563
$947$	−7.39103e6	−0.267812	−0.133906	−	0.990994i	$-0.542752\pi$
$947$	−7.39103e6	−0.267812	−0.133906	+	0.990994i	$0.542752\pi$
$948$	0	0
$949$	2.99075e7	1.07799
$950$	−1.73224e7	−0.622728
$951$	0	0
$952$	0	0
$953$	1.81495e7	0.647341	0.323671	−	0.946170i	$-0.395083\pi$
$953$	1.81495e7	0.647341	0.323671	+	0.946170i	$0.395083\pi$
$954$	0	0
$955$	1.26954e7	0.450441
$956$	−1.40760e8	−4.98120
$957$	0	0
$958$	3.81354e7	1.34250
$959$	0	0
$960$	0	0
$961$	−2.25776e7	−0.788623
$962$	3.42339e7	1.19266
$963$	0	0
$964$	2.17036e7	0.752210
$965$	−3.80498e6	−0.131533
$966$	0	0
$967$	3.12321e7	1.07408	0.537038	−	0.843558i	$-0.319543\pi$
$967$	3.12321e7	1.07408	0.537038	+	0.843558i	$0.319543\pi$
$968$	−8.96235e7	−3.07421
$969$	0	0
$970$	1.49624e7	0.510590
$971$	1.36711e7	0.465323	0.232662	−	0.972558i	$-0.425257\pi$
$971$	1.36711e7	0.465323	0.232662	+	0.972558i	$0.425257\pi$
$972$	0	0
$973$	0	0
$974$	6.61558e7	2.23445
$975$	0	0
$976$	7.46068e7	2.50700
$977$	−4.99027e7	−1.67258	−0.836291	−	0.548285i	$-0.815281\pi$
$977$	−4.99027e7	−1.67258	−0.836291	+	0.548285i	$0.815281\pi$
$978$	0	0
$979$	−8.04866e6	−0.268390
$980$	0	0
$981$	0	0
$982$	−6.15221e6	−0.203588
$983$	2.21662e7	0.731656	0.365828	−	0.930682i	$-0.380786\pi$
$983$	2.21662e7	0.731656	0.365828	+	0.930682i	$0.380786\pi$
$984$	0	0
$985$	−2.85237e7	−0.936730
$986$	8.25169e6	0.270303
$987$	0	0
$988$	−6.62622e7	−2.15960
$989$	−2.01149e7	−0.653924
$990$	0	0
$991$	−4.18915e7	−1.35501	−0.677504	−	0.735519i	$-0.736938\pi$
$991$	−4.18915e7	−1.35501	−0.677504	+	0.735519i	$0.736938\pi$
$992$	−7.48098e7	−2.41368
$993$	0	0
$994$	0	0
$995$	−1.68592e7	−0.539856
$996$	0	0
$997$	4.23041e7	1.34786	0.673930	−	0.738795i	$-0.264605\pi$
$997$	4.23041e7	1.34786	0.673930	+	0.738795i	$0.264605\pi$
$998$	6.27589e7	1.99457
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.ba.1.6		6
3.2	odd	2		147.6.a.o.1.1	yes	6
7.6	odd	2		441.6.a.bb.1.6		6
21.2	odd	6		147.6.e.p.67.6		12
21.5	even	6		147.6.e.q.67.6		12
21.11	odd	6		147.6.e.p.79.6		12
21.17	even	6		147.6.e.q.79.6		12
21.20	even	2		147.6.a.n.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.n.1.1	✓	6	21.20	even	2
147.6.a.o.1.1	yes	6	3.2	odd	2
147.6.e.p.67.6		12	21.2	odd	6
147.6.e.p.79.6		12	21.11	odd	6
147.6.e.q.67.6		12	21.5	even	6
147.6.e.q.79.6		12	21.17	even	6
441.6.a.ba.1.6		6	1.1	even	1	trivial
441.6.a.bb.1.6		6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+11.1881 q^{2} +93.1729 q^{4} +62.3150 q^{5} +684.407 q^{8} +O(q^{10})\)	\(q+11.1881 q^{2} +93.1729 q^{4} +62.3150 q^{5} +684.407 q^{8} +697.185 q^{10} +173.495 q^{11} +348.244 q^{13} +4675.66 q^{16} -999.206 q^{17} -2042.17 q^{19} +5806.07 q^{20} +1941.07 q^{22} +1949.67 q^{23} +758.159 q^{25} +3896.18 q^{26} -738.129 q^{29} -2459.99 q^{31} +30410.6 q^{32} -11179.2 q^{34} +8786.52 q^{37} -22848.0 q^{38} +42648.8 q^{40} -17617.4 q^{41} -10317.1 q^{43} +16165.0 q^{44} +21813.1 q^{46} +75.0582 q^{47} +8482.34 q^{50} +32446.9 q^{52} -27705.5 q^{53} +10811.3 q^{55} -8258.24 q^{58} +3457.57 q^{59} +15956.4 q^{61} -27522.5 q^{62} +190615. q^{64} +21700.8 q^{65} +14413.2 q^{67} -93099.0 q^{68} +30337.6 q^{71} +85880.9 q^{73} +98304.2 q^{74} -190275. q^{76} +7613.25 q^{79} +291364. q^{80} -197104. q^{82} -63725.4 q^{83} -62265.5 q^{85} -115428. q^{86} +118741. q^{88} -46391.3 q^{89} +181657. q^{92} +839.757 q^{94} -127258. q^{95} +21461.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 150 q^{4} - 100 q^{5} + 114 q^{8}+O(q^{10}) \) 6 * q - 2 * q^2 + 150 * q^4 - 100 * q^5 + 114 * q^8	\( 6 q - 2 q^{2} + 150 q^{4} - 100 q^{5} + 114 q^{8} + 864 q^{10} - 604 q^{11} + 1352 q^{13} + 4578 q^{16} - 3028 q^{17} + 1728 q^{19} - 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} - 14172 q^{26} + 5320 q^{29} + 3976 q^{31} + 37326 q^{32} - 16336 q^{34} + 22680 q^{37} - 52744 q^{38} + 100600 q^{40} - 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} - 51552 q^{47} + 40622 q^{50} + 119296 q^{52} - 80884 q^{53} + 11656 q^{55} - 70464 q^{58} - 8872 q^{59} + 50896 q^{61} - 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} - 37348 q^{68} + 110852 q^{71} + 64232 q^{73} + 27464 q^{74} - 194864 q^{76} + 111696 q^{79} + 308940 q^{80} - 189640 q^{82} - 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} + 35012 q^{89} + 449260 q^{92} - 121016 q^{94} + 119080 q^{95} + 70952 q^{97}+O(q^{100}) \) 6 * q - 2 * q^2 + 150 * q^4 - 100 * q^5 + 114 * q^8 + 864 * q^10 - 604 * q^11 + 1352 * q^13 + 4578 * q^16 - 3028 * q^17 + 1728 * q^19 - 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 - 14172 * q^26 + 5320 * q^29 + 3976 * q^31 + 37326 * q^32 - 16336 * q^34 + 22680 * q^37 - 52744 * q^38 + 100600 * q^40 - 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 - 51552 * q^47 + 40622 * q^50 + 119296 * q^52 - 80884 * q^53 + 11656 * q^55 - 70464 * q^58 - 8872 * q^59 + 50896 * q^61 - 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 - 37348 * q^68 + 110852 * q^71 + 64232 * q^73 + 27464 * q^74 - 194864 * q^76 + 111696 * q^79 + 308940 * q^80 - 189640 * q^82 - 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 + 35012 * q^89 + 449260 * q^92 - 121016 * q^94 + 119080 * q^95 + 70952 * q^97

Introduction

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