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Properties

Label 54.7.d.a.35.6
Level $54$
Weight $7$
Character 54.35
Analytic conductor $12.423$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [54,7,Mod(17,54)] 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("54.17"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(54, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 54.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.4229205155\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{18} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 35.6
Root \(-7.20150i\) of defining polynomial
Character \(\chi\) \(=\) 54.35
Dual form 54.7.d.a.17.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.89898 + 2.82843i) q^{2} +(16.0000 + 27.7128i) q^{4} +(39.5602 - 22.8401i) q^{5} +(-245.097 + 424.521i) q^{7} +181.019i q^{8} +258.406 q^{10} +(873.336 + 504.221i) q^{11} +(466.801 + 808.523i) q^{13} +(-2401.45 + 1386.48i) q^{14} +(-512.000 + 886.810i) q^{16} +8090.59i q^{17} -7727.36 q^{19} +(1265.93 + 730.884i) q^{20} +(2852.30 + 4940.34i) q^{22} +(11848.0 - 6840.45i) q^{23} +(-6769.16 + 11724.5i) q^{25} +5281.25i q^{26} -15686.2 q^{28} +(1964.70 + 1134.32i) q^{29} +(-17062.6 - 29553.2i) q^{31} +(-5016.55 + 2896.31i) q^{32} +(-22883.6 + 39635.6i) q^{34} +22392.2i q^{35} +92058.0 q^{37} +(-37856.2 - 21856.3i) q^{38} +(4134.50 + 7161.17i) q^{40} +(31021.6 - 17910.3i) q^{41} +(34570.9 - 59878.5i) q^{43} +32270.1i q^{44} +77390.9 q^{46} +(-13211.1 - 7627.46i) q^{47} +(-61321.0 - 106211. i) q^{49} +(-66323.9 + 38292.1i) q^{50} +(-14937.6 + 25872.7i) q^{52} -236591. i q^{53} +46065.9 q^{55} +(-76846.5 - 44367.4i) q^{56} +(6416.70 + 11114.0i) q^{58} +(221890. - 128108. i) q^{59} +(-19919.6 + 34501.8i) q^{61} -193041. i q^{62} -32768.0 q^{64} +(36933.5 + 21323.6i) q^{65} +(-160204. - 277482. i) q^{67} +(-224213. + 129449. i) q^{68} +(-63334.7 + 109699. i) q^{70} +404593. i q^{71} +393719. q^{73} +(450990. + 260379. i) q^{74} +(-123638. - 214147. i) q^{76} +(-428105. + 247167. i) q^{77} +(-449184. + 778010. i) q^{79} +46776.5i q^{80} +202632. q^{82} +(-154916. - 89441.0i) q^{83} +(184790. + 320066. i) q^{85} +(338724. - 195562. i) q^{86} +(-91273.8 + 158091. i) q^{88} +826458. i q^{89} -457647. q^{91} +(379136. + 218894. i) q^{92} +(-43147.4 - 74733.5i) q^{94} +(-305696. + 176494. i) q^{95} +(-317981. + 550760. i) q^{97} -693768. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 192 q^{4} - 432 q^{5} + 240 q^{7} - 378 q^{11} + 1680 q^{13} + 4752 q^{14} - 6144 q^{16} - 2820 q^{19} - 13824 q^{20} - 3600 q^{22} + 76248 q^{23} + 8094 q^{25} + 15360 q^{28} - 97092 q^{29} + 21480 q^{31}+ \cdots - 38874 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.89898 + 2.82843i 0.612372 + 0.353553i
\(3\) 0 0
\(4\) 16.0000 + 27.7128i 0.250000 + 0.433013i
\(5\) 39.5602 − 22.8401i 0.316482 − 0.182721i −0.333341 − 0.942806i \(-0.608176\pi\)
0.649823 + 0.760085i \(0.274843\pi\)
\(6\) 0 0
\(7\) −245.097 + 424.521i −0.714570 + 1.23767i 0.248556 + 0.968618i \(0.420044\pi\)
−0.963125 + 0.269053i \(0.913289\pi\)
\(8\) 181.019i 0.353553i
\(9\) 0 0
\(10\) 258.406 0.258406
\(11\) 873.336 + 504.221i 0.656151 + 0.378829i 0.790809 − 0.612063i \(-0.209660\pi\)
−0.134658 + 0.990892i \(0.542994\pi\)
\(12\) 0 0
\(13\) 466.801 + 808.523i 0.212472 + 0.368012i 0.952488 − 0.304577i \(-0.0985152\pi\)
−0.740016 + 0.672590i \(0.765182\pi\)
\(14\) −2401.45 + 1386.48i −0.875166 + 0.505277i
\(15\) 0 0
\(16\) −512.000 + 886.810i −0.125000 + 0.216506i
\(17\) 8090.59i 1.64677i 0.567482 + 0.823386i \(0.307918\pi\)
−0.567482 + 0.823386i \(0.692082\pi\)
\(18\) 0 0
\(19\) −7727.36 −1.12660 −0.563301 − 0.826252i \(-0.690469\pi\)
−0.563301 + 0.826252i \(0.690469\pi\)
\(20\) 1265.93 + 730.884i 0.158241 + 0.0913604i
\(21\) 0 0
\(22\) 2852.30 + 4940.34i 0.267872 + 0.463969i
\(23\) 11848.0 − 6840.45i 0.973782 − 0.562213i 0.0733950 − 0.997303i \(-0.476617\pi\)
0.900387 + 0.435090i \(0.143283\pi\)
\(24\) 0 0
\(25\) −6769.16 + 11724.5i −0.433226 + 0.750370i
\(26\) 5281.25i 0.300481i
\(27\) 0 0
\(28\) −15686.2 −0.714570
\(29\) 1964.70 + 1134.32i 0.0805570 + 0.0465096i 0.539737 − 0.841833i \(-0.318524\pi\)
−0.459180 + 0.888343i \(0.651857\pi\)
\(30\) 0 0
\(31\) −17062.6 − 29553.2i −0.572742 − 0.992019i −0.996283 − 0.0861417i \(-0.972546\pi\)
0.423541 − 0.905877i \(-0.360787\pi\)
\(32\) −5016.55 + 2896.31i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −22883.6 + 39635.6i −0.582222 + 1.00844i
\(35\) 22392.2i 0.522267i
\(36\) 0 0
\(37\) 92058.0 1.81743 0.908713 − 0.417422i \(-0.137066\pi\)
0.908713 + 0.417422i \(0.137066\pi\)
\(38\) −37856.2 − 21856.3i −0.689900 − 0.398314i
\(39\) 0 0
\(40\) 4134.50 + 7161.17i 0.0646016 + 0.111893i
\(41\) 31021.6 − 17910.3i 0.450103 − 0.259867i −0.257771 − 0.966206i \(-0.582988\pi\)
0.707874 + 0.706339i \(0.249655\pi\)
\(42\) 0 0
\(43\) 34570.9 − 59878.5i 0.434816 − 0.753123i −0.562465 − 0.826821i \(-0.690147\pi\)
0.997281 + 0.0736985i \(0.0234803\pi\)
\(44\) 32270.1i 0.378829i
\(45\) 0 0
\(46\) 77390.9 0.795090
\(47\) −13211.1 − 7627.46i −0.127247 − 0.0734659i 0.435025 − 0.900418i \(-0.356739\pi\)
−0.562272 + 0.826952i \(0.690073\pi\)
\(48\) 0 0
\(49\) −61321.0 − 106211.i −0.521220 − 0.902779i
\(50\) −66323.9 + 38292.1i −0.530592 + 0.306337i
\(51\) 0 0
\(52\) −14937.6 + 25872.7i −0.106236 + 0.184006i
\(53\) − 236591.i − 1.58917i −0.607153 − 0.794585i \(-0.707688\pi\)
0.607153 − 0.794585i \(-0.292312\pi\)
\(54\) 0 0
\(55\) 46065.9 0.276880
\(56\) −76846.5 − 44367.4i −0.437583 − 0.252639i
\(57\) 0 0
\(58\) 6416.70 + 11114.0i 0.0328873 + 0.0569624i
\(59\) 221890. − 128108.i 1.08039 − 0.623766i 0.149392 − 0.988778i \(-0.452268\pi\)
0.931003 + 0.365012i \(0.118935\pi\)
\(60\) 0 0
\(61\) −19919.6 + 34501.8i −0.0877589 + 0.152003i −0.906563 − 0.422069i \(-0.861304\pi\)
0.818805 + 0.574072i \(0.194637\pi\)
\(62\) − 193041.i − 0.809980i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) 36933.5 + 21323.6i 0.134487 + 0.0776462i
\(66\) 0 0
\(67\) −160204. − 277482.i −0.532660 − 0.922593i −0.999273 − 0.0381319i \(-0.987859\pi\)
0.466613 − 0.884461i \(-0.345474\pi\)
\(68\) −224213. + 129449.i −0.713073 + 0.411693i
\(69\) 0 0
\(70\) −63334.7 + 109699.i −0.184649 + 0.319822i
\(71\) 404593.i 1.13043i 0.824944 + 0.565215i \(0.191207\pi\)
−0.824944 + 0.565215i \(0.808793\pi\)
\(72\) 0 0
\(73\) 393719. 1.01209 0.506044 − 0.862508i \(-0.331107\pi\)
0.506044 + 0.862508i \(0.331107\pi\)
\(74\) 450990. + 260379.i 1.11294 + 0.642557i
\(75\) 0 0
\(76\) −123638. − 214147.i −0.281650 − 0.487833i
\(77\) −428105. + 247167.i −0.937731 + 0.541399i
\(78\) 0 0
\(79\) −449184. + 778010.i −0.911052 + 1.57799i −0.0984709 + 0.995140i \(0.531395\pi\)
−0.812581 + 0.582848i \(0.801938\pi\)
\(80\) 46776.5i 0.0913604i
\(81\) 0 0
\(82\) 202632. 0.367508
\(83\) −154916. − 89441.0i −0.270934 − 0.156424i 0.358378 − 0.933576i \(-0.383330\pi\)
−0.629312 + 0.777153i \(0.716663\pi\)
\(84\) 0 0
\(85\) 184790. + 320066.i 0.300900 + 0.521173i
\(86\) 338724. − 195562.i 0.532538 − 0.307461i
\(87\) 0 0
\(88\) −91273.8 + 158091.i −0.133936 + 0.231984i
\(89\) 826458.i 1.17233i 0.810191 + 0.586166i \(0.199364\pi\)
−0.810191 + 0.586166i \(0.800636\pi\)
\(90\) 0 0
\(91\) −457647. −0.607304
\(92\) 379136. + 218894.i 0.486891 + 0.281107i
\(93\) 0 0
\(94\) −43147.4 − 74733.5i −0.0519483 − 0.0899770i
\(95\) −305696. + 176494.i −0.356549 + 0.205854i
\(96\) 0 0
\(97\) −317981. + 550760.i −0.348406 + 0.603458i −0.985967 − 0.166943i \(-0.946610\pi\)
0.637560 + 0.770401i \(0.279944\pi\)
\(98\) − 693768.i − 0.737116i
\(99\) 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 54.7.d.a.35.6 12
3.2 odd 2 18.7.d.a.11.2 yes 12
4.3 odd 2 432.7.q.b.305.5 12
9.2 odd 6 162.7.b.c.161.8 12
9.4 even 3 18.7.d.a.5.2 ✓ 12
9.5 odd 6 inner 54.7.d.a.17.6 12
9.7 even 3 162.7.b.c.161.5 12
12.11 even 2 144.7.q.c.65.3 12
36.23 even 6 432.7.q.b.17.5 12
36.31 odd 6 144.7.q.c.113.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.2 ✓ 12 9.4 even 3
18.7.d.a.11.2 yes 12 3.2 odd 2
54.7.d.a.17.6 12 9.5 odd 6 inner
54.7.d.a.35.6 12 1.1 even 1 trivial
144.7.q.c.65.3 12 12.11 even 2
144.7.q.c.113.3 12 36.31 odd 6
162.7.b.c.161.5 12 9.7 even 3
162.7.b.c.161.8 12 9.2 odd 6
432.7.q.b.17.5 12 36.23 even 6
432.7.q.b.305.5 12 4.3 odd 2