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Properties

Label 135.5.h.a.44.19
Level $135$
Weight $5$
Character 135.44
Analytic conductor $13.955$
Analytic rank $0$
Dimension $44$
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] = [135,5,Mod(44,135)] 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("135.44"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 135.h (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: \(13.9549450163\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 44.19
Character \(\chi\) \(=\) 135.44
Dual form 135.5.h.a.89.19

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.87361 + 4.97724i) q^{2} +(-8.51525 + 14.7488i) q^{4} +(-7.19048 - 23.9436i) q^{5} +(41.4151 - 23.9110i) q^{7} -5.92246 q^{8} +(98.5104 - 104.593i) q^{10} +(73.2455 - 42.2883i) q^{11} +(193.485 + 111.709i) q^{13} +(238.021 + 137.422i) q^{14} +(119.225 + 206.504i) q^{16} -434.998 q^{17} +378.461 q^{19} +(414.369 + 97.8346i) q^{20} +(420.958 + 243.040i) q^{22} +(326.825 - 566.078i) q^{23} +(-521.594 + 344.332i) q^{25} +1284.03i q^{26} +814.433i q^{28} +(430.201 - 248.377i) q^{29} +(151.498 - 262.402i) q^{31} +(-732.592 + 1268.89i) q^{32} +(-1250.01 - 2165.09i) q^{34} +(-870.311 - 819.695i) q^{35} -55.6914i q^{37} +(1087.55 + 1883.69i) q^{38} +(42.5853 + 141.805i) q^{40} +(-411.405 - 237.525i) q^{41} +(707.073 - 408.229i) q^{43} +1440.38i q^{44} +3756.67 q^{46} +(435.232 + 753.844i) q^{47} +(-57.0270 + 98.7737i) q^{49} +(-3212.68 - 1606.62i) q^{50} +(-3295.14 + 1902.45i) q^{52} -2339.28 q^{53} +(-1539.21 - 1449.69i) q^{55} +(-245.279 + 141.612i) q^{56} +(2472.46 + 1427.47i) q^{58} +(1019.68 + 588.710i) q^{59} +(3456.52 + 5986.86i) q^{61} +1741.38 q^{62} -4605.53 q^{64} +(1283.46 - 5435.97i) q^{65} +(-3356.29 - 1937.75i) q^{67} +(3704.11 - 6415.71i) q^{68} +(1578.89 - 6687.22i) q^{70} +5822.30i q^{71} -6443.82i q^{73} +(277.189 - 160.035i) q^{74} +(-3222.69 + 5581.85i) q^{76} +(2022.31 - 3502.75i) q^{77} +(-2446.72 - 4237.84i) q^{79} +(4087.17 - 4339.54i) q^{80} -2730.21i q^{82} +(-3257.36 - 5641.91i) q^{83} +(3127.84 + 10415.4i) q^{85} +(4063.70 + 2346.18i) q^{86} +(-433.793 + 250.451i) q^{88} -13898.7i q^{89} +10684.3 q^{91} +(5566.00 + 9640.59i) q^{92} +(-2501.37 + 4332.50i) q^{94} +(-2721.31 - 9061.72i) q^{95} +(-11681.7 + 6744.42i) q^{97} -655.493 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 162 q^{4} - 6 q^{5} + 28 q^{10} - 228 q^{11} - 282 q^{14} - 1058 q^{16} - 8 q^{19} + 2196 q^{20} - 148 q^{25} - 2370 q^{29} - 1112 q^{31} - 436 q^{34} - 850 q^{40} - 1830 q^{41} - 5668 q^{46} + 5396 q^{49}+ \cdots - 58746 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.87361 + 4.97724i 0.718402 + 1.24431i 0.961633 + 0.274340i \(0.0884595\pi\)
−0.243231 + 0.969968i \(0.578207\pi\)
\(3\) 0 0
\(4\) −8.51525 + 14.7488i −0.532203 + 0.921802i
\(5\) −7.19048 − 23.9436i −0.287619 − 0.957745i
\(6\) 0 0
\(7\) 41.4151 − 23.9110i 0.845206 − 0.487980i −0.0138245 − 0.999904i \(-0.504401\pi\)
0.859030 + 0.511925i \(0.171067\pi\)
\(8\) −5.92246 −0.0925384
\(9\) 0 0
\(10\) 98.5104 − 104.593i 0.985104 − 1.04593i
\(11\) 73.2455 − 42.2883i 0.605335 − 0.349490i −0.165803 − 0.986159i \(-0.553021\pi\)
0.771137 + 0.636669i \(0.219688\pi\)
\(12\) 0 0
\(13\) 193.485 + 111.709i 1.14488 + 0.660998i 0.947635 − 0.319356i \(-0.103467\pi\)
0.197247 + 0.980354i \(0.436800\pi\)
\(14\) 238.021 + 137.422i 1.21440 + 0.701131i
\(15\) 0 0
\(16\) 119.225 + 206.504i 0.465723 + 0.806656i
\(17\) −434.998 −1.50518 −0.752591 − 0.658488i \(-0.771196\pi\)
−0.752591 + 0.658488i \(0.771196\pi\)
\(18\) 0 0
\(19\) 378.461 1.04837 0.524184 − 0.851605i \(-0.324371\pi\)
0.524184 + 0.851605i \(0.324371\pi\)
\(20\) 414.369 + 97.8346i 1.03592 + 0.244586i
\(21\) 0 0
\(22\) 420.958 + 243.040i 0.869747 + 0.502149i
\(23\) 326.825 − 566.078i 0.617818 − 1.07009i −0.372066 − 0.928206i \(-0.621350\pi\)
0.989883 − 0.141885i \(-0.0453163\pi\)
\(24\) 0 0
\(25\) −521.594 + 344.332i −0.834550 + 0.550932i
\(26\) 1284.03i 1.89945i
\(27\) 0 0
\(28\) 814.433i 1.03882i
\(29\) 430.201 − 248.377i 0.511535 − 0.295335i −0.221929 − 0.975063i \(-0.571235\pi\)
0.733464 + 0.679728i \(0.237902\pi\)
\(30\) 0 0
\(31\) 151.498 − 262.402i 0.157646 − 0.273051i −0.776373 − 0.630273i \(-0.782943\pi\)
0.934019 + 0.357222i \(0.116276\pi\)
\(32\) −732.592 + 1268.89i −0.715422 + 1.23915i
\(33\) 0 0
\(34\) −1250.01 − 2165.09i −1.08133 − 1.87291i
\(35\) −870.311 − 819.695i −0.710458 − 0.669139i
\(36\) 0 0
\(37\) − 55.6914i − 0.0406804i −0.999793 − 0.0203402i \(-0.993525\pi\)
0.999793 − 0.0203402i \(-0.00647493\pi\)
\(38\) 1087.55 + 1883.69i 0.753149 + 1.30449i
\(39\) 0 0
\(40\) 42.5853 + 141.805i 0.0266158 + 0.0886282i
\(41\) −411.405 − 237.525i −0.244738 − 0.141300i 0.372614 − 0.927986i \(-0.378461\pi\)
−0.617353 + 0.786687i \(0.711795\pi\)
\(42\) 0 0
\(43\) 707.073 − 408.229i 0.382409 − 0.220784i −0.296457 − 0.955046i \(-0.595805\pi\)
0.678866 + 0.734262i \(0.262472\pi\)
\(44\) 1440.38i 0.743999i
\(45\) 0 0
\(46\) 3756.67 1.77537
\(47\) 435.232 + 753.844i 0.197027 + 0.341260i 0.947563 − 0.319569i \(-0.103538\pi\)
−0.750536 + 0.660829i \(0.770205\pi\)
\(48\) 0 0
\(49\) −57.0270 + 98.7737i −0.0237514 + 0.0411386i
\(50\) −3212.68 − 1606.62i −1.28507 − 0.642648i
\(51\) 0 0
\(52\) −3295.14 + 1902.45i −1.21862 + 0.703570i
\(53\) −2339.28 −0.832780 −0.416390 − 0.909186i \(-0.636705\pi\)
−0.416390 + 0.909186i \(0.636705\pi\)
\(54\) 0 0
\(55\) −1539.21 − 1449.69i −0.508828 − 0.479236i
\(56\) −245.279 + 141.612i −0.0782140 + 0.0451569i
\(57\) 0 0
\(58\) 2472.46 + 1427.47i 0.734976 + 0.424338i
\(59\) 1019.68 + 588.710i 0.292926 + 0.169121i 0.639261 − 0.768990i \(-0.279240\pi\)
−0.346335 + 0.938111i \(0.612574\pi\)
\(60\) 0 0
\(61\) 3456.52 + 5986.86i 0.928921 + 1.60894i 0.785130 + 0.619331i \(0.212596\pi\)
0.143792 + 0.989608i \(0.454070\pi\)
\(62\) 1741.38 0.453013
\(63\) 0 0
\(64\) −4605.53 −1.12440
\(65\) 1283.46 − 5435.97i 0.303777 − 1.28662i
\(66\) 0 0
\(67\) −3356.29 − 1937.75i −0.747669 − 0.431667i 0.0771820 − 0.997017i \(-0.475408\pi\)
−0.824851 + 0.565350i \(0.808741\pi\)
\(68\) 3704.11 − 6415.71i 0.801063 − 1.38748i
\(69\) 0 0
\(70\) 1578.89 − 6687.22i 0.322222 − 1.36474i
\(71\) 5822.30i 1.15499i 0.816394 + 0.577495i \(0.195970\pi\)
−0.816394 + 0.577495i \(0.804030\pi\)
\(72\) 0 0
\(73\) − 6443.82i − 1.20920i −0.796530 − 0.604599i \(-0.793333\pi\)
0.796530 − 0.604599i \(-0.206667\pi\)
\(74\) 277.189 − 160.035i 0.0506189 − 0.0292249i
\(75\) 0 0
\(76\) −3222.69 + 5581.85i −0.557944 + 0.966388i
\(77\) 2022.31 − 3502.75i 0.341088 − 0.590782i
\(78\) 0 0
\(79\) −2446.72 − 4237.84i −0.392039 − 0.679032i 0.600679 − 0.799490i \(-0.294897\pi\)
−0.992718 + 0.120459i \(0.961564\pi\)
\(80\) 4087.17 − 4339.54i 0.638620 − 0.678054i
\(81\) 0 0
\(82\) − 2730.21i − 0.406040i
\(83\) −3257.36 − 5641.91i −0.472835 − 0.818974i 0.526682 − 0.850063i \(-0.323436\pi\)
−0.999517 + 0.0310882i \(0.990103\pi\)
\(84\) 0 0
\(85\) 3127.84 + 10415.4i 0.432920 + 1.44158i
\(86\) 4063.70 + 2346.18i 0.549446 + 0.317223i
\(87\) 0 0
\(88\) −433.793 + 250.451i −0.0560167 + 0.0323413i
\(89\) − 13898.7i − 1.75467i −0.479881 − 0.877333i \(-0.659320\pi\)
0.479881 − 0.877333i \(-0.340680\pi\)
\(90\) 0 0
\(91\) 10684.3 1.29021
\(92\) 5566.00 + 9640.59i 0.657609 + 1.13901i
\(93\) 0 0
\(94\) −2501.37 + 4332.50i −0.283089 + 0.490324i
\(95\) −2721.31 − 9061.72i −0.301531 − 1.00407i
\(96\) 0 0
\(97\) −11681.7 + 6744.42i −1.24154 + 0.716805i −0.969408 − 0.245454i \(-0.921063\pi\)
−0.272134 + 0.962259i \(0.587729\pi\)
\(98\) −655.493 −0.0682521
\(99\) 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 135.5.h.a.44.19 44
3.2 odd 2 45.5.h.a.14.4 ✓ 44
5.4 even 2 inner 135.5.h.a.44.4 44
9.2 odd 6 inner 135.5.h.a.89.4 44
9.4 even 3 405.5.d.a.404.8 44
9.5 odd 6 405.5.d.a.404.38 44
9.7 even 3 45.5.h.a.29.19 yes 44
15.14 odd 2 45.5.h.a.14.19 yes 44
45.4 even 6 405.5.d.a.404.37 44
45.14 odd 6 405.5.d.a.404.7 44
45.29 odd 6 inner 135.5.h.a.89.19 44
45.34 even 6 45.5.h.a.29.4 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.5.h.a.14.4 ✓ 44 3.2 odd 2
45.5.h.a.14.19 yes 44 15.14 odd 2
45.5.h.a.29.4 yes 44 45.34 even 6
45.5.h.a.29.19 yes 44 9.7 even 3
135.5.h.a.44.4 44 5.4 even 2 inner
135.5.h.a.44.19 44 1.1 even 1 trivial
135.5.h.a.89.4 44 9.2 odd 6 inner
135.5.h.a.89.19 44 45.29 odd 6 inner
405.5.d.a.404.7 44 45.14 odd 6
405.5.d.a.404.8 44 9.4 even 3
405.5.d.a.404.37 44 45.4 even 6
405.5.d.a.404.38 44 9.5 odd 6