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Properties

Label 297.2.f.b.163.1
Level $297$
Weight $2$
Character 297.163
Analytic conductor $2.372$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [297,2,Mod(82,297)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("297.82"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 9x^{14} + 51x^{12} - 249x^{10} + 1476x^{8} - 2875x^{6} + 2335x^{4} + 125x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 163.1
Root \(-1.18970 + 0.386556i\) of defining polynomial
Character \(\chi\) \(=\) 297.163
Dual form 297.2.f.b.82.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.760284 + 2.33991i) q^{2} +(-3.27913 - 2.38243i) q^{4} +(-0.305860 - 0.941339i) q^{5} +(-3.49672 - 2.54052i) q^{7} +(4.08684 - 2.96926i) q^{8} +2.43519 q^{10} +(2.79120 + 1.79142i) q^{11} +(1.11803 - 3.44095i) q^{13} +(8.60310 - 6.25052i) q^{14} +(1.33563 + 4.11065i) q^{16} +(-0.816208 - 2.51203i) q^{17} +(3.09629 - 2.24958i) q^{19} +(-1.23972 + 3.81546i) q^{20} +(-6.31388 + 5.16917i) q^{22} -3.45011 q^{23} +(3.25252 - 2.36309i) q^{25} +(7.20151 + 5.23221i) q^{26} +(5.41361 + 16.6614i) q^{28} +(-8.43168 - 6.12597i) q^{29} +(-0.521582 + 1.60526i) q^{31} -0.530785 q^{32} +6.49848 q^{34} +(-1.32198 + 4.06865i) q^{35} +(-2.61803 - 1.90211i) q^{37} +(2.90977 + 8.95537i) q^{38} +(-4.04508 - 2.93893i) q^{40} +(-9.63335 + 6.99904i) q^{41} -2.12307 q^{43} +(-4.88477 - 12.5241i) q^{44} +(2.62307 - 8.07297i) q^{46} +(3.47861 - 2.52736i) q^{47} +(3.60973 + 11.1096i) q^{49} +(3.05659 + 9.40723i) q^{50} +(-11.8640 + 8.61970i) q^{52} +(2.93021 - 9.01826i) q^{53} +(0.832623 - 3.17539i) q^{55} -21.8340 q^{56} +(20.7447 - 15.0719i) q^{58} +(5.23607 + 3.80423i) q^{59} +(-1.48656 - 4.57516i) q^{61} +(-3.35963 - 2.44091i) q^{62} +(-2.26771 + 6.97930i) q^{64} -3.58107 q^{65} +0.854102 q^{67} +(-3.30828 + 10.1818i) q^{68} +(-8.51520 - 6.18665i) q^{70} +(1.16751 + 3.59324i) q^{71} +(-9.22952 - 6.70564i) q^{73} +(6.44123 - 4.67983i) q^{74} -15.5126 q^{76} +(-5.20891 - 13.3552i) q^{77} +(1.89841 - 5.84272i) q^{79} +(3.46100 - 2.51456i) q^{80} +(-9.05306 - 27.8625i) q^{82} +(-1.03831 - 3.19558i) q^{83} +(-2.11503 + 1.53666i) q^{85} +(1.61413 - 4.96779i) q^{86} +(16.7264 - 0.966542i) q^{88} +13.2605 q^{89} +(-12.6513 + 9.19169i) q^{91} +(11.3134 + 8.21964i) q^{92} +(3.26907 + 10.0612i) q^{94} +(-3.06465 - 2.22660i) q^{95} +(-0.114759 + 0.353191i) q^{97} -28.7399 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{4} + 12 q^{10} - 10 q^{16} - 2 q^{19} - 36 q^{22} + 32 q^{25} + 42 q^{28} - 26 q^{31} - 48 q^{34} - 24 q^{37} - 20 q^{40} + 24 q^{43} - 16 q^{46} + 24 q^{49} - 40 q^{52} - 16 q^{55} + 106 q^{58}+ \cdots + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.760284 + 2.33991i −0.537602 + 1.65457i 0.200357 + 0.979723i \(0.435790\pi\)
−0.737959 + 0.674846i \(0.764210\pi\)
\(3\) 0 0
\(4\) −3.27913 − 2.38243i −1.63956 − 1.19121i
\(5\) −0.305860 − 0.941339i −0.136785 − 0.420980i 0.859079 − 0.511843i \(-0.171037\pi\)
−0.995863 + 0.0908638i \(0.971037\pi\)
\(6\) 0 0
\(7\) −3.49672 − 2.54052i −1.32164 − 0.960226i −0.999910 − 0.0133937i \(-0.995737\pi\)
−0.321727 − 0.946832i \(-0.604263\pi\)
\(8\) 4.08684 − 2.96926i 1.44492 − 1.04979i
\(9\) 0 0
\(10\) 2.43519 0.770075
\(11\) 2.79120 + 1.79142i 0.841578 + 0.540135i
\(12\) 0 0
\(13\) 1.11803 − 3.44095i 0.310087 − 0.954349i −0.667643 − 0.744482i \(-0.732697\pi\)
0.977730 − 0.209868i \(-0.0673033\pi\)
\(14\) 8.60310 − 6.25052i 2.29927 − 1.67052i
\(15\) 0 0
\(16\) 1.33563 + 4.11065i 0.333908 + 1.02766i
\(17\) −0.816208 − 2.51203i −0.197959 − 0.609257i −0.999929 − 0.0118921i \(-0.996215\pi\)
0.801970 − 0.597365i \(-0.203785\pi\)
\(18\) 0 0
\(19\) 3.09629 − 2.24958i 0.710337 − 0.516090i −0.172945 − 0.984931i \(-0.555328\pi\)
0.883282 + 0.468841i \(0.155328\pi\)
\(20\) −1.23972 + 3.81546i −0.277209 + 0.853163i
\(21\) 0 0
\(22\) −6.31388 + 5.16917i −1.34612 + 1.10207i
\(23\) −3.45011 −0.719398 −0.359699 − 0.933068i \(-0.617121\pi\)
−0.359699 + 0.933068i \(0.617121\pi\)
\(24\) 0 0
\(25\) 3.25252 − 2.36309i 0.650503 − 0.472618i
\(26\) 7.20151 + 5.23221i 1.41233 + 1.02612i
\(27\) 0 0
\(28\) 5.41361 + 16.6614i 1.02308 + 3.14870i
\(29\) −8.43168 − 6.12597i −1.56572 − 1.13756i −0.931112 − 0.364733i \(-0.881160\pi\)
−0.634611 − 0.772832i \(-0.718840\pi\)
\(30\) 0 0
\(31\) −0.521582 + 1.60526i −0.0936788 + 0.288314i −0.986907 − 0.161290i \(-0.948435\pi\)
0.893228 + 0.449603i \(0.148435\pi\)
\(32\) −0.530785 −0.0938305
\(33\) 0 0
\(34\) 6.49848 1.11448
\(35\) −1.32198 + 4.06865i −0.223456 + 0.687727i
\(36\) 0 0
\(37\) −2.61803 − 1.90211i −0.430402 − 0.312705i 0.351408 − 0.936223i \(-0.385703\pi\)
−0.781810 + 0.623517i \(0.785703\pi\)
\(38\) 2.90977 + 8.95537i 0.472028 + 1.45275i
\(39\) 0 0
\(40\) −4.04508 − 2.93893i −0.639584 − 0.464685i
\(41\) −9.63335 + 6.99904i −1.50448 + 1.09307i −0.535921 + 0.844268i \(0.680035\pi\)
−0.968555 + 0.248798i \(0.919965\pi\)
\(42\) 0 0
\(43\) −2.12307 −0.323764 −0.161882 − 0.986810i \(-0.551756\pi\)
−0.161882 + 0.986810i \(0.551756\pi\)
\(44\) −4.88477 − 12.5241i −0.736406 − 1.88809i
\(45\) 0 0
\(46\) 2.62307 − 8.07297i 0.386750 − 1.19029i
\(47\) 3.47861 − 2.52736i 0.507407 − 0.368653i −0.304432 − 0.952534i \(-0.598467\pi\)
0.811839 + 0.583881i \(0.198467\pi\)
\(48\) 0 0
\(49\) 3.60973 + 11.1096i 0.515675 + 1.58709i
\(50\) 3.05659 + 9.40723i 0.432267 + 1.33038i
\(51\) 0 0
\(52\) −11.8640 + 8.61970i −1.64524 + 1.19534i
\(53\) 2.93021 − 9.01826i 0.402496 − 1.23875i −0.520473 − 0.853878i \(-0.674244\pi\)
0.922968 − 0.384876i \(-0.125756\pi\)
\(54\) 0 0
\(55\) 0.832623 − 3.17539i 0.112271 − 0.428170i
\(56\) −21.8340 −2.91770
\(57\) 0 0
\(58\) 20.7447 − 15.0719i 2.72391 − 1.97904i
\(59\) 5.23607 + 3.80423i 0.681679 + 0.495269i 0.873914 − 0.486080i \(-0.161574\pi\)
−0.192235 + 0.981349i \(0.561574\pi\)
\(60\) 0 0
\(61\) −1.48656 − 4.57516i −0.190334 − 0.585789i 0.809665 − 0.586892i \(-0.199649\pi\)
−0.999999 + 0.00110319i \(0.999649\pi\)
\(62\) −3.35963 − 2.44091i −0.426673 − 0.309996i
\(63\) 0 0
\(64\) −2.26771 + 6.97930i −0.283464 + 0.872413i
\(65\) −3.58107 −0.444177
\(66\) 0 0
\(67\) 0.854102 0.104345 0.0521726 − 0.998638i \(-0.483385\pi\)
0.0521726 + 0.998638i \(0.483385\pi\)
\(68\) −3.30828 + 10.1818i −0.401187 + 1.23473i
\(69\) 0 0
\(70\) −8.51520 − 6.18665i −1.01776 − 0.739446i
\(71\) 1.16751 + 3.59324i 0.138558 + 0.426439i 0.996127 − 0.0879314i \(-0.0280256\pi\)
−0.857568 + 0.514371i \(0.828026\pi\)
\(72\) 0 0
\(73\) −9.22952 − 6.70564i −1.08023 − 0.784835i −0.102510 − 0.994732i \(-0.532687\pi\)
−0.977724 + 0.209897i \(0.932687\pi\)
\(74\) 6.44123 − 4.67983i 0.748778 − 0.544019i
\(75\) 0 0
\(76\) −15.5126 −1.77942
\(77\) −5.20891 − 13.3552i −0.593610 − 1.52197i
\(78\) 0 0
\(79\) 1.89841 − 5.84272i 0.213588 − 0.657357i −0.785663 − 0.618655i \(-0.787678\pi\)
0.999251 − 0.0387017i \(-0.0123222\pi\)
\(80\) 3.46100 − 2.51456i 0.386951 − 0.281137i
\(81\) 0 0
\(82\) −9.05306 − 27.8625i −0.999743 − 3.07689i
\(83\) −1.03831 − 3.19558i −0.113969 − 0.350761i 0.877762 − 0.479098i \(-0.159036\pi\)
−0.991731 + 0.128337i \(0.959036\pi\)
\(84\) 0 0
\(85\) −2.11503 + 1.53666i −0.229407 + 0.166674i
\(86\) 1.61413 − 4.96779i 0.174056 − 0.535690i
\(87\) 0 0
\(88\) 16.7264 − 0.966542i 1.78304 − 0.103034i
\(89\) 13.2605 1.40561 0.702806 − 0.711381i \(-0.251930\pi\)
0.702806 + 0.711381i \(0.251930\pi\)
\(90\) 0 0
\(91\) −12.6513 + 9.19169i −1.32621 + 0.963550i
\(92\) 11.3134 + 8.21964i 1.17950 + 0.856957i
\(93\) 0 0
\(94\) 3.26907 + 10.0612i 0.337178 + 1.03773i
\(95\) −3.06465 − 2.22660i −0.314427 − 0.228444i
\(96\) 0 0
\(97\) −0.114759 + 0.353191i −0.0116520 + 0.0358612i −0.956713 − 0.291031i \(-0.906002\pi\)
0.945061 + 0.326893i \(0.106002\pi\)
\(98\) −28.7399 −2.90317
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 297.2.f.b.163.1 yes 16
3.2 odd 2 inner 297.2.f.b.163.4 yes 16
9.2 odd 6 891.2.n.h.757.1 32
9.4 even 3 891.2.n.h.460.1 32
9.5 odd 6 891.2.n.h.460.4 32
9.7 even 3 891.2.n.h.757.4 32
11.4 even 5 3267.2.a.bj.1.1 8
11.5 even 5 inner 297.2.f.b.82.1 ✓ 16
11.7 odd 10 3267.2.a.bi.1.8 8
33.5 odd 10 inner 297.2.f.b.82.4 yes 16
33.26 odd 10 3267.2.a.bj.1.8 8
33.29 even 10 3267.2.a.bi.1.1 8
99.5 odd 30 891.2.n.h.379.1 32
99.16 even 15 891.2.n.h.676.1 32
99.38 odd 30 891.2.n.h.676.4 32
99.49 even 15 891.2.n.h.379.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
297.2.f.b.82.1 ✓ 16 11.5 even 5 inner
297.2.f.b.82.4 yes 16 33.5 odd 10 inner
297.2.f.b.163.1 yes 16 1.1 even 1 trivial
297.2.f.b.163.4 yes 16 3.2 odd 2 inner
891.2.n.h.379.1 32 99.5 odd 30
891.2.n.h.379.4 32 99.49 even 15
891.2.n.h.460.1 32 9.4 even 3
891.2.n.h.460.4 32 9.5 odd 6
891.2.n.h.676.1 32 99.16 even 15
891.2.n.h.676.4 32 99.38 odd 30
891.2.n.h.757.1 32 9.2 odd 6
891.2.n.h.757.4 32 9.7 even 3
3267.2.a.bi.1.1 8 33.29 even 10
3267.2.a.bi.1.8 8 11.7 odd 10
3267.2.a.bj.1.1 8 11.4 even 5
3267.2.a.bj.1.8 8 33.26 odd 10