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Properties

Label 351.2.h.a.289.9
Level $351$
Weight $2$
Character 351.289
Analytic conductor $2.803$
Analytic rank $0$
Dimension $24$
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] = [351,2,Mod(289,351)] 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("351.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.h (of order \(3\), 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: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.9
Character \(\chi\) \(=\) 351.289
Dual form 351.2.h.a.334.9

$q$-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)
 
\(f(q)\) \(=\) \(q+1.20426 q^{2} -0.549753 q^{4} +(-1.89177 - 3.27665i) q^{5} +(-0.150228 - 0.260203i) q^{7} -3.07057 q^{8} +(-2.27819 - 3.94594i) q^{10} -1.28462 q^{11} +(0.423704 - 3.58057i) q^{13} +(-0.180914 - 0.313352i) q^{14} -2.59827 q^{16} +(2.63848 - 4.56998i) q^{17} +(-0.829906 + 1.43744i) q^{19} +(1.04001 + 1.80135i) q^{20} -1.54702 q^{22} +(-1.67399 + 2.89944i) q^{23} +(-4.65760 + 8.06721i) q^{25} +(0.510251 - 4.31194i) q^{26} +(0.0825883 + 0.143047i) q^{28} +9.63240 q^{29} +(-3.29700 - 5.71056i) q^{31} +3.01215 q^{32} +(3.17742 - 5.50346i) q^{34} +(-0.568394 + 0.984488i) q^{35} +(1.97702 + 3.42430i) q^{37} +(-0.999424 + 1.73105i) q^{38} +(5.80882 + 10.0612i) q^{40} +(1.43368 - 2.48321i) q^{41} +(0.0104220 + 0.0180514i) q^{43} +0.706223 q^{44} +(-2.01592 + 3.49168i) q^{46} +(-0.954216 + 1.65275i) q^{47} +(3.45486 - 5.98400i) q^{49} +(-5.60898 + 9.71503i) q^{50} +(-0.232933 + 1.96843i) q^{52} +7.15248 q^{53} +(2.43021 + 4.20924i) q^{55} +(0.461286 + 0.798970i) q^{56} +11.5999 q^{58} -8.73826 q^{59} +(-2.56839 - 4.44858i) q^{61} +(-3.97045 - 6.87702i) q^{62} +8.82395 q^{64} +(-12.5338 + 5.38529i) q^{65} +(-4.15570 + 7.19788i) q^{67} +(-1.45051 + 2.51236i) q^{68} +(-0.684496 + 1.18558i) q^{70} +(-4.64070 + 8.03792i) q^{71} -4.69504 q^{73} +(2.38085 + 4.12376i) q^{74} +(0.456243 - 0.790236i) q^{76} +(0.192986 + 0.334261i) q^{77} +(6.65302 - 11.5234i) q^{79} +(4.91533 + 8.51360i) q^{80} +(1.72653 - 2.99044i) q^{82} +(6.45522 - 11.1808i) q^{83} -19.9656 q^{85} +(0.0125508 + 0.0217386i) q^{86} +3.94451 q^{88} +(-3.19907 - 5.54096i) q^{89} +(-0.995325 + 0.427653i) q^{91} +(0.920281 - 1.59397i) q^{92} +(-1.14913 + 1.99035i) q^{94} +6.27997 q^{95} +(1.88195 + 3.25964i) q^{97} +(4.16056 - 7.20630i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 18 q^{4} + 2 q^{5} + 3 q^{7} + 18 q^{8} - 6 q^{11} - 2 q^{14} + 6 q^{16} - 6 q^{17} - 3 q^{19} + 11 q^{20} - 18 q^{22} - 17 q^{23} - 6 q^{25} + 12 q^{26} + 24 q^{29} - 6 q^{31} + 38 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.20426 0.851542 0.425771 − 0.904831i \(-0.360003\pi\)
0.425771 + 0.904831i \(0.360003\pi\)
\(3\) 0 0
\(4\) −0.549753 −0.274876
\(5\) −1.89177 − 3.27665i −0.846026 − 1.46536i −0.884727 − 0.466110i \(-0.845655\pi\)
0.0387007 − 0.999251i \(-0.487678\pi\)
\(6\) 0 0
\(7\) −0.150228 − 0.260203i −0.0567809 − 0.0983473i 0.836238 − 0.548367i \(-0.184750\pi\)
−0.893019 + 0.450020i \(0.851417\pi\)
\(8\) −3.07057 −1.08561
\(9\) 0 0
\(10\) −2.27819 − 3.94594i −0.720427 − 1.24782i
\(11\) −1.28462 −0.387327 −0.193664 − 0.981068i \(-0.562037\pi\)
−0.193664 + 0.981068i \(0.562037\pi\)
\(12\) 0 0
\(13\) 0.423704 − 3.58057i 0.117514 − 0.993071i
\(14\) −0.180914 − 0.313352i −0.0483513 − 0.0837469i
\(15\) 0 0
\(16\) −2.59827 −0.649567
\(17\) 2.63848 − 4.56998i 0.639926 − 1.10838i −0.345523 − 0.938410i \(-0.612298\pi\)
0.985449 − 0.169974i \(-0.0543683\pi\)
\(18\) 0 0
\(19\) −0.829906 + 1.43744i −0.190393 + 0.329771i −0.945381 − 0.325968i \(-0.894310\pi\)
0.754987 + 0.655739i \(0.227643\pi\)
\(20\) 1.04001 + 1.80135i 0.232553 + 0.402793i
\(21\) 0 0
\(22\) −1.54702 −0.329825
\(23\) −1.67399 + 2.89944i −0.349051 + 0.604574i −0.986081 − 0.166265i \(-0.946829\pi\)
0.637030 + 0.770839i \(0.280163\pi\)
\(24\) 0 0
\(25\) −4.65760 + 8.06721i −0.931521 + 1.61344i
\(26\) 0.510251 − 4.31194i 0.100068 − 0.845642i
\(27\) 0 0
\(28\) 0.0825883 + 0.143047i 0.0156077 + 0.0270334i
\(29\) 9.63240 1.78869 0.894346 − 0.447376i \(-0.147641\pi\)
0.894346 + 0.447376i \(0.147641\pi\)
\(30\) 0 0
\(31\) −3.29700 − 5.71056i −0.592158 − 1.02565i −0.993941 − 0.109913i \(-0.964943\pi\)
0.401783 − 0.915735i \(-0.368391\pi\)
\(32\) 3.01215 0.532478
\(33\) 0 0
\(34\) 3.17742 − 5.50346i 0.544924 − 0.943836i
\(35\) −0.568394 + 0.984488i −0.0960762 + 0.166409i
\(36\) 0 0
\(37\) 1.97702 + 3.42430i 0.325020 + 0.562952i 0.981517 − 0.191378i \(-0.0612955\pi\)
−0.656496 + 0.754329i \(0.727962\pi\)
\(38\) −0.999424 + 1.73105i −0.162128 + 0.280814i
\(39\) 0 0
\(40\) 5.80882 + 10.0612i 0.918455 + 1.59081i
\(41\) 1.43368 − 2.48321i 0.223904 − 0.387813i −0.732086 − 0.681212i \(-0.761453\pi\)
0.955990 + 0.293399i \(0.0947866\pi\)
\(42\) 0 0
\(43\) 0.0104220 + 0.0180514i 0.00158933 + 0.00275281i 0.866819 − 0.498623i \(-0.166161\pi\)
−0.865230 + 0.501376i \(0.832827\pi\)
\(44\) 0.706223 0.106467
\(45\) 0 0
\(46\) −2.01592 + 3.49168i −0.297232 + 0.514821i
\(47\) −0.954216 + 1.65275i −0.139187 + 0.241079i −0.927189 − 0.374594i \(-0.877782\pi\)
0.788002 + 0.615672i \(0.211116\pi\)
\(48\) 0 0
\(49\) 3.45486 − 5.98400i 0.493552 − 0.854857i
\(50\) −5.60898 + 9.71503i −0.793229 + 1.37391i
\(51\) 0 0
\(52\) −0.232933 + 1.96843i −0.0323019 + 0.272972i
\(53\) 7.15248 0.982469 0.491235 − 0.871027i \(-0.336546\pi\)
0.491235 + 0.871027i \(0.336546\pi\)
\(54\) 0 0
\(55\) 2.43021 + 4.20924i 0.327689 + 0.567574i
\(56\) 0.461286 + 0.798970i 0.0616419 + 0.106767i
\(57\) 0 0
\(58\) 11.5999 1.52315
\(59\) −8.73826 −1.13762 −0.568812 − 0.822468i \(-0.692597\pi\)
−0.568812 + 0.822468i \(0.692597\pi\)
\(60\) 0 0
\(61\) −2.56839 − 4.44858i −0.328849 − 0.569583i 0.653435 − 0.756983i \(-0.273327\pi\)
−0.982284 + 0.187400i \(0.939994\pi\)
\(62\) −3.97045 − 6.87702i −0.504247 − 0.873382i
\(63\) 0 0
\(64\) 8.82395 1.10299
\(65\) −12.5338 + 5.38529i −1.55463 + 0.667963i
\(66\) 0 0
\(67\) −4.15570 + 7.19788i −0.507700 + 0.879361i 0.492261 + 0.870448i \(0.336171\pi\)
−0.999960 + 0.00891361i \(0.997163\pi\)
\(68\) −1.45051 + 2.51236i −0.175901 + 0.304669i
\(69\) 0 0
\(70\) −0.684496 + 1.18558i −0.0818129 + 0.141704i
\(71\) −4.64070 + 8.03792i −0.550749 + 0.953926i 0.447471 + 0.894298i \(0.352325\pi\)
−0.998221 + 0.0596277i \(0.981009\pi\)
\(72\) 0 0
\(73\) −4.69504 −0.549512 −0.274756 − 0.961514i \(-0.588597\pi\)
−0.274756 + 0.961514i \(0.588597\pi\)
\(74\) 2.38085 + 4.12376i 0.276769 + 0.479377i
\(75\) 0 0
\(76\) 0.456243 − 0.790236i 0.0523346 − 0.0906463i
\(77\) 0.192986 + 0.334261i 0.0219928 + 0.0380926i
\(78\) 0 0
\(79\) 6.65302 − 11.5234i 0.748523 − 1.29648i −0.200007 − 0.979794i \(-0.564097\pi\)
0.948530 − 0.316686i \(-0.102570\pi\)
\(80\) 4.91533 + 8.51360i 0.549550 + 0.951849i
\(81\) 0 0
\(82\) 1.72653 − 2.99044i 0.190663 − 0.330239i
\(83\) 6.45522 − 11.1808i 0.708552 − 1.22725i −0.256842 − 0.966453i \(-0.582682\pi\)
0.965394 − 0.260795i \(-0.0839848\pi\)
\(84\) 0 0
\(85\) −19.9656 −2.16558
\(86\) 0.0125508 + 0.0217386i 0.00135339 + 0.00234413i
\(87\) 0 0
\(88\) 3.94451 0.420486
\(89\) −3.19907 − 5.54096i −0.339101 − 0.587340i 0.645163 − 0.764045i \(-0.276790\pi\)
−0.984264 + 0.176705i \(0.943456\pi\)
\(90\) 0 0
\(91\) −0.995325 + 0.427653i −0.104338 + 0.0448302i
\(92\) 0.920281 − 1.59397i 0.0959459 − 0.166183i
\(93\) 0 0
\(94\) −1.14913 + 1.99035i −0.118523 + 0.205289i
\(95\) 6.27997 0.644311
\(96\) 0 0
\(97\) 1.88195 + 3.25964i 0.191083 + 0.330966i 0.945610 − 0.325304i \(-0.105467\pi\)
−0.754526 + 0.656270i \(0.772133\pi\)
\(98\) 4.16056 − 7.20630i 0.420280 − 0.727946i
\(99\) 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 351.2.h.a.289.9 24
3.2 odd 2 117.2.h.a.16.4 yes 24
9.4 even 3 351.2.f.a.172.4 24
9.5 odd 6 117.2.f.a.94.9 yes 24
13.9 even 3 351.2.f.a.100.4 24
39.35 odd 6 117.2.f.a.61.9 ✓ 24
117.22 even 3 inner 351.2.h.a.334.9 24
117.113 odd 6 117.2.h.a.22.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.9 ✓ 24 39.35 odd 6
117.2.f.a.94.9 yes 24 9.5 odd 6
117.2.h.a.16.4 yes 24 3.2 odd 2
117.2.h.a.22.4 yes 24 117.113 odd 6
351.2.f.a.100.4 24 13.9 even 3
351.2.f.a.172.4 24 9.4 even 3
351.2.h.a.289.9 24 1.1 even 1 trivial
351.2.h.a.334.9 24 117.22 even 3 inner