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Properties

Label 54.7.d.a.35.1
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 35.1
Root \(-8.88570i\) of defining polynomial
Character \(\chi\) \(=\) 54.35
Dual form 54.7.d.a.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.89898 - 2.82843i) q^{2} +(16.0000 + 27.7128i) q^{4} +(-202.253 + 116.771i) q^{5} +(95.5752 - 165.541i) q^{7} -181.019i q^{8} +1321.11 q^{10} +(673.077 + 388.601i) q^{11} +(45.5802 + 78.9472i) q^{13} +(-936.442 + 540.655i) q^{14} +(-512.000 + 886.810i) q^{16} -7047.39i q^{17} +2731.10 q^{19} +(-6472.09 - 3736.66i) q^{20} +(-2198.26 - 3807.50i) q^{22} +(17228.9 - 9947.14i) q^{23} +(19458.3 - 33702.7i) q^{25} -515.681i q^{26} +6116.81 q^{28} +(-27104.3 - 15648.7i) q^{29} +(6174.50 + 10694.6i) q^{31} +(5016.55 - 2896.31i) q^{32} +(-19933.0 + 34525.0i) q^{34} +44641.5i q^{35} -27972.0 q^{37} +(-13379.6 - 7724.70i) q^{38} +(21137.7 + 36611.7i) q^{40} +(37428.2 - 21609.2i) q^{41} +(19256.1 - 33352.5i) q^{43} +24870.5i q^{44} -112539. q^{46} +(143771. + 83006.2i) q^{47} +(40555.3 + 70243.8i) q^{49} +(-190651. + 110073. i) q^{50} +(-1458.57 + 2526.31i) q^{52} +54741.5i q^{53} -181509. q^{55} +(-29966.1 - 17301.0i) q^{56} +(88522.3 + 153325. i) q^{58} +(14102.1 - 8141.84i) q^{59} +(29443.7 - 50998.0i) q^{61} -69856.5i q^{62} -32768.0 q^{64} +(-18437.4 - 10644.9i) q^{65} +(-147998. - 256341. i) q^{67} +(195303. - 112758. i) q^{68} +(126265. - 218698. i) q^{70} -157251. i q^{71} +80297.0 q^{73} +(137034. + 79116.8i) q^{74} +(43697.5 + 75686.3i) q^{76} +(128659. - 74281.3i) q^{77} +(188424. - 326360. i) q^{79} -239146. i q^{80} -244480. q^{82} +(-733992. - 423771. i) q^{83} +(822929. + 1.42535e6i) q^{85} +(-188670. + 108929. i) q^{86} +(70344.3 - 121840. i) q^{88} -1128.91i q^{89} +17425.3 q^{91} +(551326. + 318308. i) q^{92} +(-469554. - 813291. i) q^{94} +(-552371. + 318912. i) q^{95} +(675152. - 1.16940e6i) q^{97} -458831. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 192 q^{4} - 432 q^{5} + 240 q^{7} - 378 q^{11} + 1680 q^{13} + 4752 q^{14} - 6144 q^{16} - 2820 q^{19} - 13824 q^{20} - 3600 q^{22} + 76248 q^{23} + 8094 q^{25} + 15360 q^{28} - 97092 q^{29} + 21480 q^{31}+ \cdots - 38874 q^{97}+O(q^{100}) \) 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
\(5\) −202.253 + 116.771i −1.61802 + 0.934165i −0.630591 + 0.776116i \(0.717187\pi\)
−0.987431 + 0.158050i \(0.949479\pi\)
\(6\) 0 0
\(7\) 95.5752 − 165.541i 0.278645 − 0.482627i −0.692403 − 0.721511i \(-0.743448\pi\)
0.971048 + 0.238884i \(0.0767814\pi\)
\(8\) − 181.019i − 0.353553i
\(9\) 0 0
\(10\) 1321.11 1.32111
\(11\) 673.077 + 388.601i 0.505693 + 0.291962i 0.731061 − 0.682312i \(-0.239025\pi\)
−0.225369 + 0.974274i \(0.572359\pi\)
\(12\) 0 0
\(13\) 45.5802 + 78.9472i 0.0207466 + 0.0359341i 0.876212 − 0.481925i \(-0.160062\pi\)
−0.855466 + 0.517859i \(0.826729\pi\)
\(14\) −936.442 + 540.655i −0.341269 + 0.197032i
\(15\) 0 0
\(16\) −512.000 + 886.810i −0.125000 + 0.216506i
\(17\) − 7047.39i − 1.43444i −0.696848 − 0.717219i \(-0.745415\pi\)
0.696848 − 0.717219i \(-0.254585\pi\)
\(18\) 0 0
\(19\) 2731.10 0.398177 0.199088 − 0.979982i \(-0.436202\pi\)
0.199088 + 0.979982i \(0.436202\pi\)
\(20\) −6472.09 − 3736.66i −0.809011 − 0.467083i
\(21\) 0 0
\(22\) −2198.26 − 3807.50i −0.206448 − 0.357579i
\(23\) 17228.9 − 9947.14i 1.41604 − 0.817550i 0.420091 − 0.907482i \(-0.361998\pi\)
0.995948 + 0.0899317i \(0.0286649\pi\)
\(24\) 0 0
\(25\) 19458.3 − 33702.7i 1.24533 − 2.15697i
\(26\) − 515.681i − 0.0293401i
\(27\) 0 0
\(28\) 6116.81 0.278645
\(29\) −27104.3 − 15648.7i −1.11133 − 0.641629i −0.172159 − 0.985069i \(-0.555074\pi\)
−0.939175 + 0.343440i \(0.888408\pi\)
\(30\) 0 0
\(31\) 6174.50 + 10694.6i 0.207261 + 0.358986i 0.950851 − 0.309650i \(-0.100212\pi\)
−0.743590 + 0.668636i \(0.766879\pi\)
\(32\) 5016.55 − 2896.31i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) −19933.0 + 34525.0i −0.507150 + 0.878410i
\(35\) 44641.5i 1.04120i
\(36\) 0 0
\(37\) −27972.0 −0.552228 −0.276114 − 0.961125i \(-0.589047\pi\)
−0.276114 + 0.961125i \(0.589047\pi\)
\(38\) −13379.6 − 7724.70i −0.243833 − 0.140777i
\(39\) 0 0
\(40\) 21137.7 + 36611.7i 0.330277 + 0.572057i
\(41\) 37428.2 − 21609.2i 0.543059 − 0.313535i −0.203259 − 0.979125i \(-0.565153\pi\)
0.746318 + 0.665590i \(0.231820\pi\)
\(42\) 0 0
\(43\) 19256.1 − 33352.5i 0.242193 − 0.419491i −0.719146 − 0.694860i \(-0.755467\pi\)
0.961339 + 0.275369i \(0.0887999\pi\)
\(44\) 24870.5i 0.291962i
\(45\) 0 0
\(46\) −112539. −1.15619
\(47\) 143771. + 83006.2i 1.38477 + 0.799497i 0.992720 − 0.120447i \(-0.0384327\pi\)
0.392050 + 0.919944i \(0.371766\pi\)
\(48\) 0 0
\(49\) 40555.3 + 70243.8i 0.344714 + 0.597062i
\(50\) −190651. + 110073.i −1.52521 + 0.880581i
\(51\) 0 0
\(52\) −1458.57 + 2526.31i −0.0103733 + 0.0179671i
\(53\) 54741.5i 0.367696i 0.982955 + 0.183848i \(0.0588554\pi\)
−0.982955 + 0.183848i \(0.941145\pi\)
\(54\) 0 0
\(55\) −181509. −1.09096
\(56\) −29966.1 − 17301.0i −0.170634 − 0.0985158i
\(57\) 0 0
\(58\) 88522.3 + 153325.i 0.453700 + 0.785832i
\(59\) 14102.1 − 8141.84i 0.0686637 − 0.0396430i −0.465275 − 0.885166i \(-0.654045\pi\)
0.533939 + 0.845523i \(0.320711\pi\)
\(60\) 0 0
\(61\) 29443.7 − 50998.0i 0.129719 − 0.224680i −0.793849 − 0.608115i \(-0.791926\pi\)
0.923568 + 0.383436i \(0.125259\pi\)
\(62\) − 69856.5i − 0.293111i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) −18437.4 − 10644.9i −0.0671368 − 0.0387614i
\(66\) 0 0
\(67\) −147998. − 256341.i −0.492076 − 0.852301i 0.507882 − 0.861427i \(-0.330429\pi\)
−0.999958 + 0.00912565i \(0.997095\pi\)
\(68\) 195303. − 112758.i 0.621130 − 0.358609i
\(69\) 0 0
\(70\) 126265. − 218698.i 0.368120 − 0.637603i
\(71\) − 157251.i − 0.439358i −0.975572 − 0.219679i \(-0.929499\pi\)
0.975572 − 0.219679i \(-0.0705010\pi\)
\(72\) 0 0
\(73\) 80297.0 0.206410 0.103205 − 0.994660i \(-0.467090\pi\)
0.103205 + 0.994660i \(0.467090\pi\)
\(74\) 137034. + 79116.8i 0.338169 + 0.195242i
\(75\) 0 0
\(76\) 43697.5 + 75686.3i 0.0995442 + 0.172416i
\(77\) 128659. − 74281.3i 0.281817 − 0.162707i
\(78\) 0 0
\(79\) 188424. − 326360.i 0.382169 − 0.661936i −0.609203 − 0.793014i \(-0.708511\pi\)
0.991372 + 0.131078i \(0.0418438\pi\)
\(80\) − 239146.i − 0.467083i
\(81\) 0 0
\(82\) −244480. −0.443406
\(83\) −733992. − 423771.i −1.28368 − 0.741134i −0.306162 − 0.951980i \(-0.599045\pi\)
−0.977519 + 0.210846i \(0.932378\pi\)
\(84\) 0 0
\(85\) 822929. + 1.42535e6i 1.34000 + 2.32095i
\(86\) −188670. + 108929.i −0.296625 + 0.171256i
\(87\) 0 0
\(88\) 70344.3 − 121840.i 0.103224 − 0.178789i
\(89\) − 1128.91i − 0.00160136i −1.00000 0.000800679i \(-0.999745\pi\)
1.00000 0.000800679i \(-0.000254864\pi\)
\(90\) 0 0
\(91\) 17425.3 0.0231237
\(92\) 551326. + 318308.i 0.708019 + 0.408775i
\(93\) 0 0
\(94\) −469554. − 813291.i −0.565330 − 0.979180i
\(95\) −552371. + 318912.i −0.644259 + 0.371963i
\(96\) 0 0
\(97\) 675152. − 1.16940e6i 0.739753 − 1.28129i −0.212854 − 0.977084i \(-0.568276\pi\)
0.952607 − 0.304205i \(-0.0983908\pi\)
\(98\) − 458831.i − 0.487499i
\(99\) 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.35.1 12
3.2 odd 2 18.7.d.a.11.5 yes 12
4.3 odd 2 432.7.q.b.305.1 12
9.2 odd 6 162.7.b.c.161.6 12
9.4 even 3 18.7.d.a.5.5 ✓ 12
9.5 odd 6 inner 54.7.d.a.17.1 12
9.7 even 3 162.7.b.c.161.7 12
12.11 even 2 144.7.q.c.65.4 12
36.23 even 6 432.7.q.b.17.1 12
36.31 odd 6 144.7.q.c.113.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.5 ✓ 12 9.4 even 3
18.7.d.a.11.5 yes 12 3.2 odd 2
54.7.d.a.17.1 12 9.5 odd 6 inner
54.7.d.a.35.1 12 1.1 even 1 trivial
144.7.q.c.65.4 12 12.11 even 2
144.7.q.c.113.4 12 36.31 odd 6
162.7.b.c.161.6 12 9.2 odd 6
162.7.b.c.161.7 12 9.7 even 3
432.7.q.b.17.1 12 36.23 even 6
432.7.q.b.305.1 12 4.3 odd 2