LMFDB - Embedded newform 441.6.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [441,6,Mod(1,441)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,-9,0,5,-18,0,0,9,0,-204,-396,0,350] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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			1.2
Root			$-3.27492$ 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)

$f(q)$	$=$	$q-0.725083 q^{2} -31.4743 q^{4} -46.7492 q^{5} +46.0241 q^{8} +33.8970 q^{10} -666.090 q^{11} +650.640 q^{13} +973.805 q^{16} +1186.89 q^{17} +1565.05 q^{19} +1471.40 q^{20} +482.970 q^{22} +1100.15 q^{23} -939.515 q^{25} -471.768 q^{26} -2396.72 q^{29} +2048.46 q^{31} -2178.86 q^{32} -860.596 q^{34} +1077.54 q^{37} -1134.79 q^{38} -2151.59 q^{40} +1098.21 q^{41} +16564.3 q^{43} +20964.7 q^{44} -797.702 q^{46} -8298.39 q^{47} +681.226 q^{50} -20478.4 q^{52} -5519.18 q^{53} +31139.1 q^{55} +1737.82 q^{58} -14230.4 q^{59} +14234.7 q^{61} -1485.30 q^{62} -29581.9 q^{64} -30416.9 q^{65} +19730.4 q^{67} -37356.6 q^{68} -64562.7 q^{71} -28567.0 q^{73} -781.309 q^{74} -49258.8 q^{76} -30633.4 q^{79} -45524.6 q^{80} -796.293 q^{82} -675.946 q^{83} -55486.3 q^{85} -12010.5 q^{86} -30656.2 q^{88} +125971. q^{89} -34626.5 q^{92} +6017.02 q^{94} -73164.9 q^{95} +22906.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	−0.725083	−0.128178	−0.0640889	−	0.997944i	$-0.520414\pi$
$2$	−0.725083	−0.128178	−0.0640889	+	0.997944i	$0.520414\pi$
$3$	0	0
$3$	0	0
$4$	−31.4743	−0.983570
$5$	−46.7492	−0.836275	−0.418137	−	0.908384i	$-0.637317\pi$
$5$	−46.7492	−0.836275	−0.418137	+	0.908384i	$0.637317\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	46.0241	0.254250
$9$	0	0
$10$	33.8970	0.107192
$11$	−666.090	−1.65978	−0.829891	−	0.557926i	$-0.811597\pi$
$11$	−666.090	−1.65978	−0.829891	+	0.557926i	$0.811597\pi$
$12$	0	0
$13$	650.640	1.06778	0.533890	−	0.845554i	$-0.320729\pi$
$13$	650.640	1.06778	0.533890	+	0.845554i	$0.320729\pi$
$14$	0	0
$15$	0	0
$16$	973.805	0.950981
$17$	1186.89	0.996069	0.498035	−	0.867157i	$-0.334055\pi$
$17$	1186.89	0.996069	0.498035	+	0.867157i	$0.334055\pi$
$18$	0	0
$19$	1565.05	0.994591	0.497296	−	0.867581i	$-0.334326\pi$
$19$	1565.05	0.994591	0.497296	+	0.867581i	$0.334326\pi$
$20$	1471.40	0.822535
$21$	0	0
$22$	482.970	0.212747
$23$	1100.15	0.433644	0.216822	−	0.976211i	$-0.430431\pi$
$23$	1100.15	0.433644	0.216822	+	0.976211i	$0.430431\pi$
$24$	0	0
$25$	−939.515	−0.300645
$26$	−471.768	−0.136866
$27$	0	0
$28$	0	0
$29$	−2396.72	−0.529203	−0.264602	−	0.964358i	$-0.585240\pi$
$29$	−2396.72	−0.529203	−0.264602	+	0.964358i	$0.585240\pi$
$30$	0	0
$31$	2048.46	0.382844	0.191422	−	0.981508i	$-0.438690\pi$
$31$	2048.46	0.382844	0.191422	+	0.981508i	$0.438690\pi$
$32$	−2178.86	−0.376144
$33$	0	0
$34$	−860.596	−0.127674
$35$	0	0
$36$	0	0
$37$	1077.54	0.129399	0.0646995	−	0.997905i	$-0.479391\pi$
$37$	1077.54	0.129399	0.0646995	+	0.997905i	$0.479391\pi$
$38$	−1134.79	−0.127484
$39$	0	0
$40$	−2151.59	−0.212622
$41$	1098.21	0.102029	0.0510147	−	0.998698i	$-0.483754\pi$
$41$	1098.21	0.102029	0.0510147	+	0.998698i	$0.483754\pi$
$42$	0	0
$43$	16564.3	1.36616	0.683081	−	0.730343i	$-0.260640\pi$
$43$	16564.3	1.36616	0.683081	+	0.730343i	$0.260640\pi$
$44$	20964.7	1.63251
$45$	0	0
$46$	−797.702	−0.0555835
$47$	−8298.39	−0.547960	−0.273980	−	0.961735i	$-0.588340\pi$
$47$	−8298.39	−0.547960	−0.273980	+	0.961735i	$0.588340\pi$
$48$	0	0
$49$	0	0
$50$	681.226	0.0385360
$51$	0	0
$52$	−20478.4	−1.05024
$53$	−5519.18	−0.269889	−0.134944	−	0.990853i	$-0.543086\pi$
$53$	−5519.18	−0.269889	−0.134944	+	0.990853i	$0.543086\pi$
$54$	0	0
$55$	31139.1	1.38803
$56$	0	0
$57$	0	0
$58$	1737.82	0.0678321
$59$	−14230.4	−0.532216	−0.266108	−	0.963943i	$-0.585738\pi$
$59$	−14230.4	−0.532216	−0.266108	+	0.963943i	$0.585738\pi$
$60$	0	0
$61$	14234.7	0.489807	0.244904	−	0.969547i	$-0.421244\pi$
$61$	14234.7	0.489807	0.244904	+	0.969547i	$0.421244\pi$
$62$	−1485.30	−0.0490721
$63$	0	0
$64$	−29581.9	−0.902768
$65$	−30416.9	−0.892958
$66$	0	0
$67$	19730.4	0.536970	0.268485	−	0.963284i	$-0.413477\pi$
$67$	19730.4	0.536970	0.268485	+	0.963284i	$0.413477\pi$
$68$	−37356.6	−0.979704
$69$	0	0
$70$	0	0
$71$	−64562.7	−1.51997	−0.759986	−	0.649940i	$-0.774794\pi$
$71$	−64562.7	−1.51997	−0.759986	+	0.649940i	$0.774794\pi$
$72$	0	0
$73$	−28567.0	−0.627418	−0.313709	−	0.949519i	$-0.601572\pi$
$73$	−28567.0	−0.627418	−0.313709	+	0.949519i	$0.601572\pi$
$74$	−781.309	−0.0165861
$75$	0	0
$76$	−49258.8	−0.978251
$77$	0	0
$78$	0	0
$79$	−30633.4	−0.552239	−0.276119	−	0.961123i	$-0.589049\pi$
$79$	−30633.4	−0.552239	−0.276119	+	0.961123i	$0.589049\pi$
$80$	−45524.6	−0.795282
$81$	0	0
$82$	−796.293	−0.0130779
$83$	−675.946	−0.0107700	−0.00538501	−	0.999986i	$-0.501714\pi$
$83$	−675.946	−0.0107700	−0.00538501	+	0.999986i	$0.501714\pi$
$84$	0	0
$85$	−55486.3	−0.832987
$86$	−12010.5	−0.175111
$87$	0	0
$88$	−30656.2	−0.421999
$89$	125971.	1.68576	0.842882	−	0.538098i	$-0.180857\pi$
$89$	125971.	1.68576	0.842882	+	0.538098i	$0.180857\pi$
$90$	0	0
$91$	0	0
$92$	−34626.5	−0.426520
$93$	0	0
$94$	6017.02	0.0702363
$95$	−73164.9	−0.831751
$96$	0	0
$97$	22906.8	0.247192	0.123596	−	0.992333i	$-0.460557\pi$
$97$	22906.8	0.247192	0.123596	+	0.992333i	$0.460557\pi$
$98$	0	0
$99$	0	0
$100$	29570.5	0.295705
$101$	−181474.	−1.77015	−0.885077	−	0.465444i	$-0.845895\pi$
$101$	−181474.	−1.77015	−0.885077	+	0.465444i	$0.845895\pi$
$102$	0	0
$103$	−64772.0	−0.601581	−0.300791	−	0.953690i	$-0.597251\pi$
$103$	−64772.0	−0.601581	−0.300791	+	0.953690i	$0.597251\pi$
$104$	29945.1	0.271483
$105$	0	0
$106$	4001.86	0.0345938
$107$	148170.	1.25112	0.625562	−	0.780175i	$-0.284870\pi$
$107$	148170.	1.25112	0.625562	+	0.780175i	$0.284870\pi$
$108$	0	0
$109$	−111294.	−0.897237	−0.448618	−	0.893723i	$-0.648084\pi$
$109$	−111294.	−0.897237	−0.448618	+	0.893723i	$0.648084\pi$
$110$	−22578.5	−0.177915
$111$	0	0
$112$	0	0
$113$	43175.5	0.318084	0.159042	−	0.987272i	$-0.449160\pi$
$113$	43175.5	0.318084	0.159042	+	0.987272i	$0.449160\pi$
$114$	0	0
$115$	−51431.2	−0.362646
$116$	75435.0	0.520509
$117$	0	0
$118$	10318.2	0.0682182
$119$	0	0
$120$	0	0
$121$	282625.	1.75488
$122$	−10321.4	−0.0627824
$123$	0	0
$124$	−64473.6	−0.376554
$125$	190013.	1.08770
$126$	0	0
$127$	−131449.	−0.723182	−0.361591	−	0.932337i	$-0.617766\pi$
$127$	−131449.	−0.723182	−0.361591	+	0.932337i	$0.617766\pi$
$128$	91172.8	0.491859
$129$	0	0
$130$	22054.7	0.114457
$131$	−349458.	−1.77916	−0.889582	−	0.456775i	$-0.849005\pi$
$131$	−349458.	−1.77916	−0.889582	+	0.456775i	$0.849005\pi$
$132$	0	0
$133$	0	0
$134$	−14306.2	−0.0688276
$135$	0	0
$136$	54625.7	0.253250
$137$	−386434.	−1.75903	−0.879516	−	0.475869i	$-0.842134\pi$
$137$	−386434.	−1.75903	−0.879516	+	0.475869i	$0.842134\pi$
$138$	0	0
$139$	−17289.3	−0.0758997	−0.0379498	−	0.999280i	$-0.512083\pi$
$139$	−17289.3	−0.0758997	−0.0379498	+	0.999280i	$0.512083\pi$
$140$	0	0
$141$	0	0
$142$	46813.3	0.194827
$143$	−433384.	−1.77228
$144$	0	0
$145$	112045.	0.442559
$146$	20713.4	0.0804210
$147$	0	0
$148$	−33914.9	−0.127273
$149$	112171.	0.413917	0.206959	−	0.978350i	$-0.433643\pi$
$149$	112171.	0.413917	0.206959	+	0.978350i	$0.433643\pi$
$150$	0	0
$151$	30495.4	0.108841	0.0544205	−	0.998518i	$-0.482669\pi$
$151$	30495.4	0.108841	0.0544205	+	0.998518i	$0.482669\pi$
$152$	72030.1	0.252874
$153$	0	0
$154$	0	0
$155$	−95763.6	−0.320163
$156$	0	0
$157$	−523509.	−1.69502	−0.847510	−	0.530780i	$-0.821899\pi$
$157$	−523509.	−1.69502	−0.847510	+	0.530780i	$0.821899\pi$
$158$	22211.7	0.0707847
$159$	0	0
$160$	101860.	0.314560
$161$	0	0
$162$	0	0
$163$	−439646.	−1.29609	−0.648043	−	0.761604i	$-0.724412\pi$
$163$	−439646.	−1.29609	−0.648043	+	0.761604i	$0.724412\pi$
$164$	−34565.3	−0.100353
$165$	0	0
$166$	490.117	0.00138048
$167$	279353.	0.775107	0.387554	−	0.921847i	$-0.373320\pi$
$167$	279353.	0.775107	0.387554	+	0.921847i	$0.373320\pi$
$168$	0	0
$169$	52038.8	0.140156
$170$	40232.2	0.106770
$171$	0	0
$172$	−521349.	−1.34372
$173$	99699.4	0.253266	0.126633	−	0.991950i	$-0.459583\pi$
$173$	99699.4	0.253266	0.126633	+	0.991950i	$0.459583\pi$
$174$	0	0
$175$	0	0
$176$	−648641.	−1.57842
$177$	0	0
$178$	−91339.7	−0.216077
$179$	−329980.	−0.769760	−0.384880	−	0.922967i	$-0.625757\pi$
$179$	−329980.	−0.769760	−0.384880	+	0.922967i	$0.625757\pi$
$180$	0	0
$181$	505810.	1.14760	0.573800	−	0.818995i	$-0.305469\pi$
$181$	505810.	1.14760	0.573800	+	0.818995i	$0.305469\pi$
$182$	0	0
$183$	0	0
$184$	50633.5	0.110254
$185$	−50374.3	−0.108213
$186$	0	0
$187$	−790578.	−1.65326
$188$	261186.	0.538958
$189$	0	0
$190$	53050.6	0.106612
$191$	63835.6	0.126613	0.0633067	−	0.997994i	$-0.479835\pi$
$191$	63835.6	0.126613	0.0633067	+	0.997994i	$0.479835\pi$
$192$	0	0
$193$	469355.	0.907001	0.453501	−	0.891256i	$-0.350175\pi$
$193$	469355.	0.907001	0.453501	+	0.891256i	$0.350175\pi$
$194$	−16609.3	−0.0316845
$195$	0	0
$196$	0	0
$197$	−268021.	−0.492043	−0.246021	−	0.969264i	$-0.579123\pi$
$197$	−268021.	−0.492043	−0.246021	+	0.969264i	$0.579123\pi$
$198$	0	0
$199$	605167.	1.08328	0.541642	−	0.840609i	$-0.317803\pi$
$199$	605167.	1.08328	0.541642	+	0.840609i	$0.317803\pi$
$200$	−43240.3	−0.0764388
$201$	0	0
$202$	131584.	0.226894
$203$	0	0
$204$	0	0
$205$	−51340.4	−0.0853246
$206$	46965.1	0.0771094
$207$	0	0
$208$	633596.	1.01544
$209$	−1.04246e6	−1.65080
$210$	0	0
$211$	335389.	0.518612	0.259306	−	0.965795i	$-0.416506\pi$
$211$	335389.	0.518612	0.259306	+	0.965795i	$0.416506\pi$
$212$	173712.	0.265455
$213$	0	0
$214$	−107435.	−0.160366
$215$	−774367.	−1.14249
$216$	0	0
$217$	0	0
$218$	80697.7	0.115006
$219$	0	0
$220$	−980081.	−1.36523
$221$	772240.	1.06358
$222$	0	0
$223$	−1.02526e6	−1.38061	−0.690305	−	0.723518i	$-0.742524\pi$
$223$	−1.02526e6	−1.38061	−0.690305	+	0.723518i	$0.742524\pi$
$224$	0	0
$225$	0	0
$226$	−31305.8	−0.0407713
$227$	−504226.	−0.649473	−0.324736	−	0.945805i	$-0.605276\pi$
$227$	−504226.	−0.649473	−0.324736	+	0.945805i	$0.605276\pi$
$228$	0	0
$229$	1.11939e6	1.41057	0.705283	−	0.708925i	$-0.250820\pi$
$229$	1.11939e6	1.41057	0.705283	+	0.708925i	$0.250820\pi$
$230$	37291.9	0.0464831
$231$	0	0
$232$	−110307.	−0.134550
$233$	−770703.	−0.930031	−0.465015	−	0.885303i	$-0.653951\pi$
$233$	−770703.	−0.930031	−0.465015	+	0.885303i	$0.653951\pi$
$234$	0	0
$235$	387943.	0.458245
$236$	447892.	0.523472
$237$	0	0
$238$	0	0
$239$	171646.	0.194374	0.0971871	−	0.995266i	$-0.469015\pi$
$239$	171646.	0.194374	0.0971871	+	0.995266i	$0.469015\pi$
$240$	0	0
$241$	383779.	0.425637	0.212818	−	0.977092i	$-0.431736\pi$
$241$	383779.	0.425637	0.212818	+	0.977092i	$0.431736\pi$
$242$	−204926.	−0.224936
$243$	0	0
$244$	−448028.	−0.481760
$245$	0	0
$246$	0	0
$247$	1.01828e6	1.06201
$248$	94278.3	0.0973380
$249$	0	0
$250$	−137775.	−0.139418
$251$	−1.57046e6	−1.57342	−0.786708	−	0.617325i	$-0.788216\pi$
$251$	−1.57046e6	−1.57342	−0.786708	+	0.617325i	$0.788216\pi$
$252$	0	0
$253$	−732801.	−0.719755
$254$	95311.4	0.0926959
$255$	0	0
$256$	880513.	0.839723
$257$	790656.	0.746715	0.373357	−	0.927688i	$-0.378207\pi$
$257$	790656.	0.746715	0.373357	+	0.927688i	$0.378207\pi$
$258$	0	0
$259$	0	0
$260$	957348.	0.878287
$261$	0	0
$262$	253386.	0.228049
$263$	−464416.	−0.414017	−0.207008	−	0.978339i	$-0.566373\pi$
$263$	−464416.	−0.414017	−0.207008	+	0.978339i	$0.566373\pi$
$264$	0	0
$265$	258017.	0.225701
$266$	0	0
$267$	0	0
$268$	−621001.	−0.528147
$269$	1.99959e6	1.68484	0.842422	−	0.538818i	$-0.181129\pi$
$269$	1.99959e6	1.68484	0.842422	+	0.538818i	$0.181129\pi$
$270$	0	0
$271$	−1.61296e6	−1.33414	−0.667070	−	0.744995i	$-0.732452\pi$
$271$	−1.61296e6	−1.33414	−0.667070	+	0.744995i	$0.732452\pi$
$272$	1.15580e6	0.947243
$273$	0	0
$274$	280197.	0.225469
$275$	625801.	0.499005
$276$	0	0
$277$	−2.08119e6	−1.62972	−0.814860	−	0.579658i	$-0.803186\pi$
$277$	−2.08119e6	−1.62972	−0.814860	+	0.579658i	$0.803186\pi$
$278$	12536.2	0.00972865
$279$	0	0
$280$	0	0
$281$	982035.	0.741927	0.370964	−	0.928647i	$-0.379027\pi$
$281$	982035.	0.741927	0.370964	+	0.928647i	$0.379027\pi$
$282$	0	0
$283$	1.39622e6	1.03630	0.518152	−	0.855289i	$-0.326620\pi$
$283$	1.39622e6	1.03630	0.518152	+	0.855289i	$0.326620\pi$
$284$	2.03206e6	1.49500
$285$	0	0
$286$	314240.	0.227167
$287$	0	0
$288$	0	0
$289$	−11140.3	−0.00784609
$290$	−81241.7	−0.0567262
$291$	0	0
$292$	899124.	0.617110
$293$	2.56205e6	1.74348	0.871742	−	0.489965i	$-0.162990\pi$
$293$	2.56205e6	1.74348	0.871742	+	0.489965i	$0.162990\pi$
$294$	0	0
$295$	665260.	0.445078
$296$	49593.0	0.0328996
$297$	0	0
$298$	−81333.0	−0.0530550
$299$	715803.	0.463037
$300$	0	0
$301$	0	0
$302$	−22111.7	−0.0139510
$303$	0	0
$304$	1.52405e6	0.945838
$305$	−665463.	−0.409613
$306$	0	0
$307$	884855.	0.535829	0.267915	−	0.963443i	$-0.413666\pi$
$307$	884855.	0.535829	0.267915	+	0.963443i	$0.413666\pi$
$308$	0	0
$309$	0	0
$310$	69436.5	0.0410378
$311$	1.80120e6	1.05599	0.527997	−	0.849246i	$-0.322943\pi$
$311$	1.80120e6	1.05599	0.527997	+	0.849246i	$0.322943\pi$
$312$	0	0
$313$	−950366.	−0.548315	−0.274158	−	0.961685i	$-0.588399\pi$
$313$	−950366.	−0.548315	−0.274158	+	0.961685i	$0.588399\pi$
$314$	379587.	0.217264
$315$	0	0
$316$	964162.	0.543166
$317$	−3.04277e6	−1.70068	−0.850338	−	0.526237i	$-0.823602\pi$
$317$	−3.04277e6	−1.70068	−0.850338	+	0.526237i	$0.823602\pi$
$318$	0	0
$319$	1.59643e6	0.878362
$320$	1.38293e6	0.754962
$321$	0	0
$322$	0	0
$323$	1.85755e6	0.990682
$324$	0	0
$325$	−611286.	−0.321023
$326$	318780.	0.166129
$327$	0	0
$328$	50544.1	0.0259409
$329$	0	0
$330$	0	0
$331$	−2.19616e6	−1.10178	−0.550889	−	0.834579i	$-0.685711\pi$
$331$	−2.19616e6	−1.10178	−0.550889	+	0.834579i	$0.685711\pi$
$332$	21274.9	0.0105931
$333$	0	0
$334$	−202554.	−0.0993515
$335$	−922382.	−0.449054
$336$	0	0
$337$	−2.41491e6	−1.15832	−0.579158	−	0.815216i	$-0.696618\pi$
$337$	−2.41491e6	−1.15832	−0.579158	+	0.815216i	$0.696618\pi$
$338$	−37732.5	−0.0179648
$339$	0	0
$340$	1.74639e6	0.819302
$341$	−1.36446e6	−0.635438
$342$	0	0
$343$	0	0
$344$	762357.	0.347346
$345$	0	0
$346$	−72290.3	−0.0324631
$347$	1.08833e6	0.485219	0.242609	−	0.970124i	$-0.421997\pi$
$347$	1.08833e6	0.485219	0.242609	+	0.970124i	$0.421997\pi$
$348$	0	0
$349$	−2.79267e6	−1.22731	−0.613657	−	0.789573i	$-0.710302\pi$
$349$	−2.79267e6	−1.22731	−0.613657	+	0.789573i	$0.710302\pi$
$350$	0	0
$351$	0	0
$352$	1.45132e6	0.624317
$353$	2.53134e6	1.08122	0.540610	−	0.841273i	$-0.318193\pi$
$353$	2.53134e6	1.08122	0.540610	+	0.841273i	$0.318193\pi$
$354$	0	0
$355$	3.01825e6	1.27111
$356$	−3.96485e6	−1.65807
$357$	0	0
$358$	239263.	0.0986661
$359$	−1.09028e6	−0.446480	−0.223240	−	0.974763i	$-0.571663\pi$
$359$	−1.09028e6	−0.446480	−0.223240	+	0.974763i	$0.571663\pi$
$360$	0	0
$361$	−26712.8	−0.0107883
$362$	−366754.	−0.147097
$363$	0	0
$364$	0	0
$365$	1.33548e6	0.524694
$366$	0	0
$367$	−188070.	−0.0728879	−0.0364439	−	0.999336i	$-0.511603\pi$
$367$	−188070.	−0.0728879	−0.0364439	+	0.999336i	$0.511603\pi$
$368$	1.07133e6	0.412387
$369$	0	0
$370$	36525.6	0.0138705
$371$	0	0
$372$	0	0
$373$	−1.79371e6	−0.667545	−0.333772	−	0.942654i	$-0.608322\pi$
$373$	−1.79371e6	−0.667545	−0.333772	+	0.942654i	$0.608322\pi$
$374$	573234.	0.211911
$375$	0	0
$376$	−381926.	−0.139319
$377$	−1.55940e6	−0.565073
$378$	0	0
$379$	3.58806e6	1.28310	0.641551	−	0.767080i	$-0.278291\pi$
$379$	3.58806e6	1.28310	0.641551	+	0.767080i	$0.278291\pi$
$380$	2.30281e6	0.818086
$381$	0	0
$382$	−46286.1	−0.0162290
$383$	−3.42457e6	−1.19291	−0.596457	−	0.802645i	$-0.703425\pi$
$383$	−3.42457e6	−1.19291	−0.596457	+	0.802645i	$0.703425\pi$
$384$	0	0
$385$	0	0
$386$	−340321.	−0.116257
$387$	0	0
$388$	−720974.	−0.243131
$389$	−8625.27	−0.00289001	−0.00144500	−	0.999999i	$-0.500460\pi$
$389$	−8625.27	−0.00289001	−0.00144500	+	0.999999i	$0.500460\pi$
$390$	0	0
$391$	1.30576e6	0.431940
$392$	0	0
$393$	0	0
$394$	194337.	0.0630690
$395$	1.43208e6	0.461823
$396$	0	0
$397$	−1.25709e6	−0.400306	−0.200153	−	0.979765i	$-0.564144\pi$
$397$	−1.25709e6	−0.400306	−0.200153	+	0.979765i	$0.564144\pi$
$398$	−438796.	−0.138853
$399$	0	0
$400$	−914904.	−0.285908
$401$	−1.42670e6	−0.443070	−0.221535	−	0.975152i	$-0.571107\pi$
$401$	−1.42670e6	−0.443070	−0.221535	+	0.975152i	$0.571107\pi$
$402$	0	0
$403$	1.33281e6	0.408794
$404$	5.71176e6	1.74107
$405$	0	0
$406$	0	0
$407$	−717741.	−0.214774
$408$	0	0
$409$	3.06529e6	0.906073	0.453036	−	0.891492i	$-0.350341\pi$
$409$	3.06529e6	0.906073	0.453036	+	0.891492i	$0.350341\pi$
$410$	37226.0	0.0109367
$411$	0	0
$412$	2.03865e6	0.591698
$413$	0	0
$414$	0	0
$415$	31599.9	0.00900670
$416$	−1.41765e6	−0.401640
$417$	0	0
$418$	755873.	0.211596
$419$	−248240.	−0.0690776	−0.0345388	−	0.999403i	$-0.510996\pi$
$419$	−248240.	−0.0690776	−0.0345388	+	0.999403i	$0.510996\pi$
$420$	0	0
$421$	5.96280e6	1.63963	0.819814	−	0.572630i	$-0.194077\pi$
$421$	5.96280e6	1.63963	0.819814	+	0.572630i	$0.194077\pi$
$422$	−243185.	−0.0664746
$423$	0	0
$424$	−254015.	−0.0686192
$425$	−1.11510e6	−0.299463
$426$	0	0
$427$	0	0
$428$	−4.66353e6	−1.23057
$429$	0	0
$430$	561481.	0.146441
$431$	4.93538e6	1.27976	0.639879	−	0.768476i	$-0.278985\pi$
$431$	4.93538e6	1.27976	0.639879	+	0.768476i	$0.278985\pi$
$432$	0	0
$433$	−4.15513e6	−1.06504	−0.532519	−	0.846418i	$-0.678755\pi$
$433$	−4.15513e6	−1.06504	−0.532519	+	0.846418i	$0.678755\pi$
$434$	0	0
$435$	0	0
$436$	3.50291e6	0.882496
$437$	1.72180e6	0.431299
$438$	0	0
$439$	227955.	0.0564531	0.0282265	−	0.999602i	$-0.491014\pi$
$439$	227955.	0.0564531	0.0282265	+	0.999602i	$0.491014\pi$
$440$	1.43315e6	0.352907
$441$	0	0
$442$	−559938.	−0.136328
$443$	1.98462e6	0.480472	0.240236	−	0.970715i	$-0.422775\pi$
$443$	1.98462e6	0.480472	0.240236	+	0.970715i	$0.422775\pi$
$444$	0	0
$445$	−5.88906e6	−1.40976
$446$	743397.	0.176963
$447$	0	0
$448$	0	0
$449$	−2.61077e6	−0.611157	−0.305579	−	0.952167i	$-0.598850\pi$
$449$	−2.61077e6	−0.611157	−0.305579	+	0.952167i	$0.598850\pi$
$450$	0	0
$451$	−731506.	−0.169347
$452$	−1.35892e6	−0.312858
$453$	0	0
$454$	365606.	0.0832480
$455$	0	0
$456$	0	0
$457$	−4.09917e6	−0.918132	−0.459066	−	0.888402i	$-0.651816\pi$
$457$	−4.09917e6	−0.918132	−0.459066	+	0.888402i	$0.651816\pi$
$458$	−811652.	−0.180803
$459$	0	0
$460$	1.61876e6	0.356687
$461$	−2.62378e6	−0.575009	−0.287505	−	0.957779i	$-0.592826\pi$
$461$	−2.62378e6	−0.575009	−0.287505	+	0.957779i	$0.592826\pi$
$462$	0	0
$463$	−4.28563e6	−0.929100	−0.464550	−	0.885547i	$-0.653784\pi$
$463$	−4.28563e6	−0.929100	−0.464550	+	0.885547i	$0.653784\pi$
$464$	−2.33394e6	−0.503262
$465$	0	0
$466$	558824.	0.119209
$467$	990118.	0.210085	0.105042	−	0.994468i	$-0.466502\pi$
$467$	990118.	0.210085	0.105042	+	0.994468i	$0.466502\pi$
$468$	0	0
$469$	0	0
$470$	−281291.	−0.0587369
$471$	0	0
$472$	−654942.	−0.135316
$473$	−1.10333e7	−2.26753
$474$	0	0
$475$	−1.47039e6	−0.299019
$476$	0	0
$477$	0	0
$478$	−124457.	−0.0249144
$479$	2.57976e6	0.513737	0.256868	−	0.966446i	$-0.417309\pi$
$479$	2.57976e6	0.513737	0.256868	+	0.966446i	$0.417309\pi$
$480$	0	0
$481$	701093.	0.138170
$482$	−278272.	−0.0545571
$483$	0	0
$484$	−8.89540e6	−1.72604
$485$	−1.07087e6	−0.206720
$486$	0	0
$487$	3.21474e6	0.614219	0.307109	−	0.951674i	$-0.400638\pi$
$487$	3.21474e6	0.614219	0.307109	+	0.951674i	$0.400638\pi$
$488$	655141.	0.124533
$489$	0	0
$490$	0	0
$491$	7.86108e6	1.47156	0.735781	−	0.677220i	$-0.236815\pi$
$491$	7.86108e6	1.47156	0.735781	+	0.677220i	$0.236815\pi$
$492$	0	0
$493$	−2.84465e6	−0.527123
$494$	−738341.	−0.136125
$495$	0	0
$496$	1.99480e6	0.364078
$497$	0	0
$498$	0	0
$499$	−1.35382e6	−0.243395	−0.121697	−	0.992567i	$-0.538834\pi$
$499$	−1.35382e6	−0.243395	−0.121697	+	0.992567i	$0.538834\pi$
$500$	−5.98051e6	−1.06983
$501$	0	0
$502$	1.13872e6	0.201677
$503$	3.85775e6	0.679851	0.339926	−	0.940452i	$-0.389598\pi$
$503$	3.85775e6	0.679851	0.339926	+	0.940452i	$0.389598\pi$
$504$	0	0
$505$	8.48376e6	1.48034
$506$	531341.	0.0922565
$507$	0	0
$508$	4.13726e6	0.711301
$509$	−1.06060e7	−1.81451	−0.907253	−	0.420585i	$-0.861825\pi$
$509$	−1.06060e7	−1.81451	−0.907253	+	0.420585i	$0.861825\pi$
$510$	0	0
$511$	0	0
$512$	−3.55598e6	−0.599493
$513$	0	0
$514$	−573291.	−0.0957122
$515$	3.02804e6	0.503087
$516$	0	0
$517$	5.52747e6	0.909495
$518$	0	0
$519$	0	0
$520$	−1.39991e6	−0.227034
$521$	−9.17989e6	−1.48164	−0.740821	−	0.671703i	$-0.765563\pi$
$521$	−9.17989e6	−1.48164	−0.740821	+	0.671703i	$0.765563\pi$
$522$	0	0
$523$	−9.05585e6	−1.44769	−0.723844	−	0.689964i	$-0.757627\pi$
$523$	−9.05585e6	−1.44769	−0.723844	+	0.689964i	$0.757627\pi$
$524$	1.09989e7	1.74993
$525$	0	0
$526$	336740.	0.0530677
$527$	2.43130e6	0.381339
$528$	0	0
$529$	−5.22601e6	−0.811953
$530$	−187084.	−0.0289299
$531$	0	0
$532$	0	0
$533$	714539.	0.108945
$534$	0	0
$535$	−6.92681e6	−1.04628
$536$	908075.	0.136524
$537$	0	0
$538$	−1.44987e6	−0.215960
$539$	0	0
$540$	0	0
$541$	1.21783e7	1.78894	0.894468	−	0.447132i	$-0.147555\pi$
$541$	1.21783e7	1.78894	0.894468	+	0.447132i	$0.147555\pi$
$542$	1.16953e6	0.171007
$543$	0	0
$544$	−2.58608e6	−0.374666
$545$	5.20292e6	0.750336
$546$	0	0
$547$	−9.00451e6	−1.28674	−0.643372	−	0.765554i	$-0.722465\pi$
$547$	−9.00451e6	−1.28674	−0.643372	+	0.765554i	$0.722465\pi$
$548$	1.21627e7	1.73013
$549$	0	0
$550$	−453758.	−0.0639613
$551$	−3.75099e6	−0.526341
$552$	0	0
$553$	0	0
$554$	1.50904e6	0.208894
$555$	0	0
$556$	544167.	0.0746527
$557$	−1.54461e6	−0.210950	−0.105475	−	0.994422i	$-0.533636\pi$
$557$	−1.54461e6	−0.210950	−0.105475	+	0.994422i	$0.533636\pi$
$558$	0	0
$559$	1.07774e7	1.45876
$560$	0	0
$561$	0	0
$562$	−712057.	−0.0950986
$563$	−1.18748e7	−1.57890	−0.789449	−	0.613816i	$-0.789634\pi$
$563$	−1.18748e7	−1.57890	−0.789449	+	0.613816i	$0.789634\pi$
$564$	0	0
$565$	−2.01842e6	−0.266005
$566$	−1.01237e6	−0.132831
$567$	0	0
$568$	−2.97144e6	−0.386452
$569$	1.54308e6	0.199806	0.0999031	−	0.994997i	$-0.468147\pi$
$569$	1.54308e6	0.199806	0.0999031	+	0.994997i	$0.468147\pi$
$570$	0	0
$571$	−7.01812e6	−0.900804	−0.450402	−	0.892826i	$-0.648719\pi$
$571$	−7.01812e6	−0.900804	−0.450402	+	0.892826i	$0.648719\pi$
$572$	1.36404e7	1.74317
$573$	0	0
$574$	0	0
$575$	−1.03361e6	−0.130373
$576$	0	0
$577$	5.37543e6	0.672162	0.336081	−	0.941833i	$-0.390898\pi$
$577$	5.37543e6	0.672162	0.336081	+	0.941833i	$0.390898\pi$
$578$	8077.66	0.00100569
$579$	0	0
$580$	−3.52652e6	−0.435288
$581$	0	0
$582$	0	0
$583$	3.67627e6	0.447957
$584$	−1.31477e6	−0.159521
$585$	0	0
$586$	−1.85770e6	−0.223476
$587$	−2.06682e6	−0.247575	−0.123788	−	0.992309i	$-0.539504\pi$
$587$	−2.06682e6	−0.247575	−0.123788	+	0.992309i	$0.539504\pi$
$588$	0	0
$589$	3.20594e6	0.380774
$590$	−482369.	−0.0570492
$591$	0	0
$592$	1.04932e6	0.123056
$593$	−5.46947e6	−0.638717	−0.319358	−	0.947634i	$-0.603467\pi$
$593$	−5.46947e6	−0.638717	−0.319358	+	0.947634i	$0.603467\pi$
$594$	0	0
$595$	0	0
$596$	−3.53049e6	−0.407117
$597$	0	0
$598$	−519016.	−0.0593510
$599$	−1.09943e7	−1.25199	−0.625996	−	0.779826i	$-0.715307\pi$
$599$	−1.09943e7	−1.25199	−0.625996	+	0.779826i	$0.715307\pi$
$600$	0	0
$601$	−1.58788e7	−1.79322	−0.896608	−	0.442826i	$-0.853976\pi$
$601$	−1.58788e7	−1.79322	−0.896608	+	0.442826i	$0.853976\pi$
$602$	0	0
$603$	0	0
$604$	−959820.	−0.107053
$605$	−1.32125e7	−1.46756
$606$	0	0
$607$	−5.33262e6	−0.587447	−0.293724	−	0.955890i	$-0.594895\pi$
$607$	−5.33262e6	−0.587447	−0.293724	+	0.955890i	$0.594895\pi$
$608$	−3.41003e6	−0.374110
$609$	0	0
$610$	482516.	0.0525033
$611$	−5.39926e6	−0.585102
$612$	0	0
$613$	−8.91838e6	−0.958594	−0.479297	−	0.877653i	$-0.659108\pi$
$613$	−8.91838e6	−0.958594	−0.479297	+	0.877653i	$0.659108\pi$
$614$	−641593.	−0.0686814
$615$	0	0
$616$	0	0
$617$	−9.63586e6	−1.01901	−0.509504	−	0.860468i	$-0.670171\pi$
$617$	−9.63586e6	−1.01901	−0.509504	+	0.860468i	$0.670171\pi$
$618$	0	0
$619$	−1.21747e7	−1.27712	−0.638560	−	0.769572i	$-0.720470\pi$
$619$	−1.21747e7	−1.27712	−0.638560	+	0.769572i	$0.720470\pi$
$620$	3.01409e6	0.314903
$621$	0	0
$622$	−1.30602e6	−0.135355
$623$	0	0
$624$	0	0
$625$	−5.94695e6	−0.608968
$626$	689094.	0.0702818
$627$	0	0
$628$	1.64770e7	1.66717
$629$	1.27893e6	0.128890
$630$	0	0
$631$	−1.30854e7	−1.30832	−0.654161	−	0.756356i	$-0.726978\pi$
$631$	−1.30854e7	−1.30832	−0.654161	+	0.756356i	$0.726978\pi$
$632$	−1.40987e6	−0.140407
$633$	0	0
$634$	2.20626e6	0.217989
$635$	6.14513e6	0.604779
$636$	0	0
$637$	0	0
$638$	−1.15754e6	−0.112586
$639$	0	0
$640$	−4.26226e6	−0.411329
$641$	441107.	0.0424032	0.0212016	−	0.999775i	$-0.493251\pi$
$641$	441107.	0.0424032	0.0212016	+	0.999775i	$0.493251\pi$
$642$	0	0
$643$	4.18888e6	0.399550	0.199775	−	0.979842i	$-0.435979\pi$
$643$	4.18888e6	0.399550	0.199775	+	0.979842i	$0.435979\pi$
$644$	0	0
$645$	0	0
$646$	−1.34688e6	−0.126983
$647$	1.87822e7	1.76395	0.881973	−	0.471300i	$-0.156215\pi$
$647$	1.87822e7	1.76395	0.881973	+	0.471300i	$0.156215\pi$
$648$	0	0
$649$	9.47874e6	0.883362
$650$	443233.	0.0411480
$651$	0	0
$652$	1.38375e7	1.27479
$653$	1.51733e6	0.139251	0.0696254	−	0.997573i	$-0.477820\pi$
$653$	1.51733e6	0.139251	0.0696254	+	0.997573i	$0.477820\pi$
$654$	0	0
$655$	1.63368e7	1.48787
$656$	1.06944e6	0.0970281
$657$	0	0
$658$	0	0
$659$	−1.84809e7	−1.65772	−0.828859	−	0.559458i	$-0.811009\pi$
$659$	−1.84809e7	−1.65772	−0.828859	+	0.559458i	$0.811009\pi$
$660$	0	0
$661$	1.03952e7	0.925403	0.462702	−	0.886514i	$-0.346880\pi$
$661$	1.03952e7	0.925403	0.462702	+	0.886514i	$0.346880\pi$
$662$	1.59240e6	0.141223
$663$	0	0
$664$	−31109.8	−0.00273828
$665$	0	0
$666$	0	0
$667$	−2.63676e6	−0.229486
$668$	−8.79242e6	−0.762372
$669$	0	0
$670$	668803.	0.0575587
$671$	−9.48162e6	−0.812973
$672$	0	0
$673$	−1.10398e7	−0.939556	−0.469778	−	0.882785i	$-0.655666\pi$
$673$	−1.10398e7	−0.939556	−0.469778	+	0.882785i	$0.655666\pi$
$674$	1.75101e6	0.148470
$675$	0	0
$676$	−1.63788e6	−0.137853
$677$	−8.23485e6	−0.690532	−0.345266	−	0.938505i	$-0.612211\pi$
$677$	−8.23485e6	−0.690532	−0.345266	+	0.938505i	$0.612211\pi$
$678$	0	0
$679$	0	0
$680$	−2.55371e6	−0.211787
$681$	0	0
$682$	989343.	0.0814490
$683$	−1.98437e7	−1.62768	−0.813842	−	0.581086i	$-0.802628\pi$
$683$	−1.98437e7	−1.62768	−0.813842	+	0.581086i	$0.802628\pi$
$684$	0	0
$685$	1.80655e7	1.47103
$686$	0	0
$687$	0	0
$688$	1.61304e7	1.29919
$689$	−3.59100e6	−0.288182
$690$	0	0
$691$	2.37019e6	0.188837	0.0944185	−	0.995533i	$-0.469901\pi$
$691$	2.37019e6	0.188837	0.0944185	+	0.995533i	$0.469901\pi$
$692$	−3.13796e6	−0.249105
$693$	0	0
$694$	−789131.	−0.0621943
$695$	808259.	0.0634730
$696$	0	0
$697$	1.30346e6	0.101628
$698$	2.02491e6	0.157314
$699$	0	0
$700$	0	0
$701$	1.56833e7	1.20543	0.602714	−	0.797957i	$-0.294086\pi$
$701$	1.56833e7	1.20543	0.602714	+	0.797957i	$0.294086\pi$
$702$	0	0
$703$	1.68641e6	0.128699
$704$	1.97042e7	1.49840
$705$	0	0
$706$	−1.83543e6	−0.138588
$707$	0	0
$708$	0	0
$709$	3.68544e6	0.275343	0.137671	−	0.990478i	$-0.456038\pi$
$709$	3.68544e6	0.275343	0.137671	+	0.990478i	$0.456038\pi$
$710$	−2.18848e6	−0.162928
$711$	0	0
$712$	5.79772e6	0.428605
$713$	2.25361e6	0.166018
$714$	0	0
$715$	2.02604e7	1.48212
$716$	1.03859e7	0.757113
$717$	0	0
$718$	790544.	0.0572289
$719$	−1.56427e7	−1.12847	−0.564233	−	0.825615i	$-0.690828\pi$
$719$	−1.56427e7	−1.12847	−0.564233	+	0.825615i	$0.690828\pi$
$720$	0	0
$721$	0	0
$722$	19369.0	0.00138282
$723$	0	0
$724$	−1.59200e7	−1.12875
$725$	2.25175e6	0.159102
$726$	0	0
$727$	−1.85908e7	−1.30456	−0.652279	−	0.757979i	$-0.726187\pi$
$727$	−1.85908e7	−1.30456	−0.652279	+	0.757979i	$0.726187\pi$
$728$	0	0
$729$	0	0
$730$	−968335.	−0.0672541
$731$	1.96601e7	1.36079
$732$	0	0
$733$	−2.49466e7	−1.71495	−0.857476	−	0.514524i	$-0.827969\pi$
$733$	−2.49466e7	−1.71495	−0.857476	+	0.514524i	$0.827969\pi$
$734$	136367.	0.00934260
$735$	0	0
$736$	−2.39708e6	−0.163113
$737$	−1.31422e7	−0.891253
$738$	0	0
$739$	−2.42944e7	−1.63642	−0.818211	−	0.574918i	$-0.805034\pi$
$739$	−2.42944e7	−1.63642	−0.818211	+	0.574918i	$0.805034\pi$
$740$	1.58549e6	0.106435
$741$	0	0
$742$	0	0
$743$	−4.16541e6	−0.276812	−0.138406	−	0.990376i	$-0.544198\pi$
$743$	−4.16541e6	−0.276812	−0.138406	+	0.990376i	$0.544198\pi$
$744$	0	0
$745$	−5.24388e6	−0.346148
$746$	1.30059e6	0.0855644
$747$	0	0
$748$	2.48828e7	1.62610
$749$	0	0
$750$	0	0
$751$	−2.36434e7	−1.52972	−0.764858	−	0.644199i	$-0.777191\pi$
$751$	−2.36434e7	−1.52972	−0.764858	+	0.644199i	$0.777191\pi$
$752$	−8.08101e6	−0.521100
$753$	0	0
$754$	1.13070e6	0.0724298
$755$	−1.42564e6	−0.0910209
$756$	0	0
$757$	1.50108e7	0.952062	0.476031	−	0.879429i	$-0.342075\pi$
$757$	1.50108e7	0.952062	0.476031	+	0.879429i	$0.342075\pi$
$758$	−2.60164e6	−0.164465
$759$	0	0
$760$	−3.36735e6	−0.211472
$761$	2.92191e6	0.182897	0.0914483	−	0.995810i	$-0.470850\pi$
$761$	2.92191e6	0.182897	0.0914483	+	0.995810i	$0.470850\pi$
$762$	0	0
$763$	0	0
$764$	−2.00918e6	−0.124533
$765$	0	0
$766$	2.48310e6	0.152905
$767$	−9.25887e6	−0.568290
$768$	0	0
$769$	−1.42847e7	−0.871073	−0.435536	−	0.900171i	$-0.643441\pi$
$769$	−1.42847e7	−0.871073	−0.435536	+	0.900171i	$0.643441\pi$
$770$	0	0
$771$	0	0
$772$	−1.47726e7	−0.892100
$773$	−1.09012e7	−0.656186	−0.328093	−	0.944645i	$-0.606406\pi$
$773$	−1.09012e7	−0.656186	−0.328093	+	0.944645i	$0.606406\pi$
$774$	0	0
$775$	−1.92455e6	−0.115100
$776$	1.05426e6	0.0628485
$777$	0	0
$778$	6254.04	0.000370434 0
$779$	1.71875e6	0.101478
$780$	0	0
$781$	4.30045e7	2.52282
$782$	−946787.	−0.0553650
$783$	0	0
$784$	0	0
$785$	2.44736e7	1.41750
$786$	0	0
$787$	2.56449e7	1.47593	0.737963	−	0.674841i	$-0.235788\pi$
$787$	2.56449e7	1.47593	0.737963	+	0.674841i	$0.235788\pi$
$788$	8.43576e6	0.483959
$789$	0	0
$790$	−1.03838e6	−0.0591955
$791$	0	0
$792$	0	0
$793$	9.26169e6	0.523007
$794$	911498.	0.0513103
$795$	0	0
$796$	−1.90472e7	−1.06549
$797$	7.73086e6	0.431104	0.215552	−	0.976492i	$-0.430845\pi$
$797$	7.73086e6	0.431104	0.215552	+	0.976492i	$0.430845\pi$
$798$	0	0
$799$	−9.84931e6	−0.545807
$800$	2.04707e6	0.113086
$801$	0	0
$802$	1.03448e6	0.0567917
$803$	1.90282e7	1.04138
$804$	0	0
$805$	0	0
$806$	−966395.	−0.0523983
$807$	0	0
$808$	−8.35218e6	−0.450061
$809$	1.87811e7	1.00890	0.504452	−	0.863440i	$-0.331695\pi$
$809$	1.87811e7	1.00890	0.504452	+	0.863440i	$0.331695\pi$
$810$	0	0
$811$	9.00729e6	0.480886	0.240443	−	0.970663i	$-0.422707\pi$
$811$	9.00729e6	0.480886	0.240443	+	0.970663i	$0.422707\pi$
$812$	0	0
$813$	0	0
$814$	520422.	0.0275293
$815$	2.05531e7	1.08388
$816$	0	0
$817$	2.59240e7	1.35877
$818$	−2.22259e6	−0.116138
$819$	0	0
$820$	1.61590e6	0.0839228
$821$	9.27965e6	0.480478	0.240239	−	0.970714i	$-0.422774\pi$
$821$	9.27965e6	0.480478	0.240239	+	0.970714i	$0.422774\pi$
$822$	0	0
$823$	−1.08308e7	−0.557393	−0.278697	−	0.960379i	$-0.589902\pi$
$823$	−1.08308e7	−0.557393	−0.278697	+	0.960379i	$0.589902\pi$
$824$	−2.98107e6	−0.152952
$825$	0	0
$826$	0	0
$827$	2.05230e7	1.04346	0.521731	−	0.853110i	$-0.325286\pi$
$827$	2.05230e7	1.04346	0.521731	+	0.853110i	$0.325286\pi$
$828$	0	0
$829$	1.42216e7	0.718724	0.359362	−	0.933198i	$-0.382994\pi$
$829$	1.42216e7	0.718724	0.359362	+	0.933198i	$0.382994\pi$
$830$	−22912.6	−0.00115446
$831$	0	0
$832$	−1.92472e7	−0.963958
$833$	0	0
$834$	0	0
$835$	−1.30595e7	−0.648202
$836$	3.28108e7	1.62368
$837$	0	0
$838$	179995.	0.00885421
$839$	−9.29934e6	−0.456087	−0.228043	−	0.973651i	$-0.573233\pi$
$839$	−9.29934e6	−0.456087	−0.228043	+	0.973651i	$0.573233\pi$
$840$	0	0
$841$	−1.47669e7	−0.719944
$842$	−4.32353e6	−0.210164
$843$	0	0
$844$	−1.05561e7	−0.510092
$845$	−2.43277e6	−0.117209
$846$	0	0
$847$	0	0
$848$	−5.37461e6	−0.256659
$849$	0	0
$850$	808543.	0.0383845
$851$	1.18546e6	0.0561131
$852$	0	0
$853$	3.07436e6	0.144671	0.0723357	−	0.997380i	$-0.476955\pi$
$853$	3.07436e6	0.144671	0.0723357	+	0.997380i	$0.476955\pi$
$854$	0	0
$855$	0	0
$856$	6.81938e6	0.318098
$857$	3.45835e7	1.60848	0.804242	−	0.594302i	$-0.202572\pi$
$857$	3.45835e7	1.60848	0.804242	+	0.594302i	$0.202572\pi$
$858$	0	0
$859$	−1.63022e7	−0.753814	−0.376907	−	0.926251i	$-0.623012\pi$
$859$	−1.63022e7	−0.753814	−0.376907	+	0.926251i	$0.623012\pi$
$860$	2.43726e7	1.12372
$861$	0	0
$862$	−3.57856e6	−0.164036
$863$	−2.56962e7	−1.17447	−0.587235	−	0.809416i	$-0.699784\pi$
$863$	−2.56962e7	−1.17447	−0.587235	+	0.809416i	$0.699784\pi$
$864$	0	0
$865$	−4.66087e6	−0.211800
$866$	3.01281e6	0.136514
$867$	0	0
$868$	0	0
$869$	2.04046e7	0.916596
$870$	0	0
$871$	1.28374e7	0.573366
$872$	−5.12222e6	−0.228122
$873$	0	0
$874$	−1.24844e6	−0.0552829
$875$	0	0
$876$	0	0
$877$	3.30060e7	1.44908	0.724542	−	0.689230i	$-0.242051\pi$
$877$	3.30060e7	1.44908	0.724542	+	0.689230i	$0.242051\pi$
$878$	−165286.	−0.00723603
$879$	0	0
$880$	3.03235e7	1.31999
$881$	2.26705e6	0.0984060	0.0492030	−	0.998789i	$-0.484332\pi$
$881$	2.26705e6	0.0984060	0.0492030	+	0.998789i	$0.484332\pi$
$882$	0	0
$883$	1.97779e7	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$883$	1.97779e7	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$884$	−2.43057e7	−1.04611
$885$	0	0
$886$	−1.43901e6	−0.0615858
$887$	−36468.3	−0.00155635	−0.000778173	−	1.00000i	$-0.500248\pi$
$887$	−36468.3	−0.00155635	−0.000778173		1.00000i	$0.500248\pi$
$888$	0	0
$889$	0	0
$890$	4.27005e6	0.180700
$891$	0	0
$892$	3.22692e7	1.35793
$893$	−1.29874e7	−0.544997
$894$	0	0
$895$	1.54263e7	0.643730
$896$	0	0
$897$	0	0
$898$	1.89303e6	0.0783367
$899$	−4.90958e6	−0.202602
$900$	0	0
$901$	−6.55068e6	−0.268828
$902$	530403.	0.0217065
$903$	0	0
$904$	1.98711e6	0.0808727
$905$	−2.36462e7	−0.959709
$906$	0	0
$907$	1.62298e7	0.655079	0.327540	−	0.944837i	$-0.393781\pi$
$907$	1.62298e7	0.655079	0.327540	+	0.944837i	$0.393781\pi$
$908$	1.58702e7	0.638802
$909$	0	0
$910$	0	0
$911$	2.61699e7	1.04474	0.522368	−	0.852720i	$-0.325049\pi$
$911$	2.61699e7	1.04474	0.522368	+	0.852720i	$0.325049\pi$
$912$	0	0
$913$	450241.	0.0178759
$914$	2.97224e6	0.117684
$915$	0	0
$916$	−3.52320e7	−1.38739
$917$	0	0
$918$	0	0
$919$	4.05973e6	0.158565	0.0792826	−	0.996852i	$-0.474737\pi$
$919$	4.05973e6	0.158565	0.0792826	+	0.996852i	$0.474737\pi$
$920$	−2.36708e6	−0.0922025
$921$	0	0
$922$	1.90246e6	0.0737034
$923$	−4.20070e7	−1.62300
$924$	0	0
$925$	−1.01237e6	−0.0389031
$926$	3.10744e6	0.119090
$927$	0	0
$928$	5.22212e6	0.199057
$929$	3.69518e6	0.140474	0.0702370	−	0.997530i	$-0.477624\pi$
$929$	3.69518e6	0.140474	0.0702370	+	0.997530i	$0.477624\pi$
$930$	0	0
$931$	0	0
$932$	2.42573e7	0.914751
$933$	0	0
$934$	−717918.	−0.0269282
$935$	3.69589e7	1.38258
$936$	0	0
$937$	1.91384e7	0.712126	0.356063	−	0.934462i	$-0.384119\pi$
$937$	1.91384e7	0.712126	0.356063	+	0.934462i	$0.384119\pi$
$938$	0	0
$939$	0	0
$940$	−1.22102e7	−0.450717
$941$	1.20954e7	0.445294	0.222647	−	0.974899i	$-0.428530\pi$
$941$	1.20954e7	0.445294	0.222647	+	0.974899i	$0.428530\pi$
$942$	0	0
$943$	1.20820e6	0.0442445
$944$	−1.38577e7	−0.506127
$945$	0	0
$946$	8.00007e6	0.290647
$947$	1.95969e7	0.710087	0.355044	−	0.934850i	$-0.384466\pi$
$947$	1.95969e7	0.710087	0.355044	+	0.934850i	$0.384466\pi$
$948$	0	0
$949$	−1.85868e7	−0.669945
$950$	1.06615e6	0.0383275
$951$	0	0
$952$	0	0
$953$	−4.23265e7	−1.50966	−0.754832	−	0.655918i	$-0.772282\pi$
$953$	−4.23265e7	−1.50966	−0.754832	+	0.655918i	$0.772282\pi$
$954$	0	0
$955$	−2.98426e6	−0.105884
$956$	−5.40243e6	−0.191181
$957$	0	0
$958$	−1.87054e6	−0.0658496
$959$	0	0
$960$	0	0
$961$	−2.44330e7	−0.853430
$962$	−508351.	−0.0177103
$963$	0	0
$964$	−1.20792e7	−0.418644
$965$	−2.19419e7	−0.758502
$966$	0	0
$967$	1.39211e6	0.0478749	0.0239375	−	0.999713i	$-0.492380\pi$
$967$	1.39211e6	0.0478749	0.0239375	+	0.999713i	$0.492380\pi$
$968$	1.30075e7	0.446176
$969$	0	0
$970$	776471.	0.0264970
$971$	4.41877e7	1.50402	0.752009	−	0.659153i	$-0.229085\pi$
$971$	4.41877e7	1.50402	0.752009	+	0.659153i	$0.229085\pi$
$972$	0	0
$973$	0	0
$974$	−2.33095e6	−0.0787292
$975$	0	0
$976$	1.38619e7	0.465798
$977$	5.60457e7	1.87848	0.939239	−	0.343264i	$-0.111533\pi$
$977$	5.60457e7	1.87848	0.939239	+	0.343264i	$0.111533\pi$
$978$	0	0
$979$	−8.39082e7	−2.79800
$980$	0	0
$981$	0	0
$982$	−5.69993e6	−0.188621
$983$	−5.00560e7	−1.65224	−0.826118	−	0.563497i	$-0.809456\pi$
$983$	−5.00560e7	−1.65224	−0.826118	+	0.563497i	$0.809456\pi$
$984$	0	0
$985$	1.25298e7	0.411483
$986$	2.06261e6	0.0675654
$987$	0	0
$988$	−3.20497e7	−1.04456
$989$	1.82233e7	0.592428
$990$	0	0
$991$	1.59116e7	0.514670	0.257335	−	0.966322i	$-0.417156\pi$
$991$	1.59116e7	0.514670	0.257335	+	0.966322i	$0.417156\pi$
$992$	−4.46330e6	−0.144005
$993$	0	0
$994$	0	0
$995$	−2.82911e7	−0.905924
$996$	0	0
$997$	−4.25995e7	−1.35727	−0.678635	−	0.734476i	$-0.737428\pi$
$997$	−4.25995e7	−1.35727	−0.678635	+	0.734476i	$0.737428\pi$
$998$	981634.	0.0311978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.l.1.2		2
3.2	odd	2		49.6.a.f.1.1		2
7.6	odd	2		63.6.a.f.1.2		2
12.11	even	2		784.6.a.v.1.2		2
21.2	odd	6		49.6.c.d.18.2		4
21.5	even	6		49.6.c.e.18.2		4
21.11	odd	6		49.6.c.d.30.2		4
21.17	even	6		49.6.c.e.30.2		4
21.20	even	2		7.6.a.b.1.1	✓	2
28.27	even	2		1008.6.a.bq.1.2		2
84.83	odd	2		112.6.a.h.1.1		2
105.62	odd	4		175.6.b.c.99.3		4
105.83	odd	4		175.6.b.c.99.2		4
105.104	even	2		175.6.a.c.1.2		2
168.83	odd	2		448.6.a.u.1.2		2
168.125	even	2		448.6.a.w.1.1		2
231.230	odd	2		847.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.1	✓	2	21.20	even	2
49.6.a.f.1.1		2	3.2	odd	2
49.6.c.d.18.2		4	21.2	odd	6
49.6.c.d.30.2		4	21.11	odd	6
49.6.c.e.18.2		4	21.5	even	6
49.6.c.e.30.2		4	21.17	even	6
63.6.a.f.1.2		2	7.6	odd	2
112.6.a.h.1.1		2	84.83	odd	2
175.6.a.c.1.2		2	105.104	even	2
175.6.b.c.99.2		4	105.83	odd	4
175.6.b.c.99.3		4	105.62	odd	4
441.6.a.l.1.2		2	1.1	even	1	trivial
448.6.a.u.1.2		2	168.83	odd	2
448.6.a.w.1.1		2	168.125	even	2
784.6.a.v.1.2		2	12.11	even	2
847.6.a.c.1.2		2	231.230	odd	2
1008.6.a.bq.1.2		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.725083 q^{2} -31.4743 q^{4} -46.7492 q^{5} +46.0241 q^{8} +33.8970 q^{10} -666.090 q^{11} +650.640 q^{13} +973.805 q^{16} +1186.89 q^{17} +1565.05 q^{19} +1471.40 q^{20} +482.970 q^{22} +1100.15 q^{23} -939.515 q^{25} -471.768 q^{26} -2396.72 q^{29} +2048.46 q^{31} -2178.86 q^{32} -860.596 q^{34} +1077.54 q^{37} -1134.79 q^{38} -2151.59 q^{40} +1098.21 q^{41} +16564.3 q^{43} +20964.7 q^{44} -797.702 q^{46} -8298.39 q^{47} +681.226 q^{50} -20478.4 q^{52} -5519.18 q^{53} +31139.1 q^{55} +1737.82 q^{58} -14230.4 q^{59} +14234.7 q^{61} -1485.30 q^{62} -29581.9 q^{64} -30416.9 q^{65} +19730.4 q^{67} -37356.6 q^{68} -64562.7 q^{71} -28567.0 q^{73} -781.309 q^{74} -49258.8 q^{76} -30633.4 q^{79} -45524.6 q^{80} -796.293 q^{82} -675.946 q^{83} -55486.3 q^{85} -12010.5 q^{86} -30656.2 q^{88} +125971. q^{89} -34626.5 q^{92} +6017.02 q^{94} -73164.9 q^{95} +22906.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 9 q^{2} + 5 q^{4} - 18 q^{5} + 9 q^{8} - 204 q^{10} - 396 q^{11} + 350 q^{13} + 113 q^{16} + 1800 q^{17} + 3266 q^{19} + 2520 q^{20} - 1752 q^{22} - 2088 q^{23} - 3238 q^{25} + 2016 q^{26} - 6696 q^{29}+ \cdots - 20776 q^{97}+O(q^{100}) \) 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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