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Properties

Label 441.2.bg.a.395.18
Level $441$
Weight $2$
Character 441.395
Analytic conductor $3.521$
Analytic rank $0$
Dimension $216$
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] = [441,2,Mod(17,441)] 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("441.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 25])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bg (of order \(42\), degree \(12\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 395.18
Character \(\chi\) \(=\) 441.395
Dual form 441.2.bg.a.278.18

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.45194 + 0.962314i) q^{2} +(3.61984 + 3.35872i) q^{4} +(-1.47762 - 1.00742i) q^{5} +(2.09999 - 1.60937i) q^{7} +(3.35777 + 6.97247i) q^{8} +(-2.65357 - 3.89208i) q^{10} +(0.538606 + 3.57342i) q^{11} +(-2.76752 + 2.20702i) q^{13} +(6.69775 - 1.92522i) q^{14} +(0.785279 + 10.4788i) q^{16} +(-2.32943 - 0.718533i) q^{17} +(6.52328 - 3.76622i) q^{19} +(-1.96509 - 8.60964i) q^{20} +(-2.11812 + 9.28010i) q^{22} +(-1.91813 - 6.21843i) q^{23} +(-0.658247 - 1.67719i) q^{25} +(-8.90963 + 2.74826i) q^{26} +(13.0070 + 1.22762i) q^{28} +(-4.35647 + 0.994336i) q^{29} +(-5.85986 - 3.38319i) q^{31} +(-3.59633 + 11.6590i) q^{32} +(-5.02015 - 4.00344i) q^{34} +(-4.72430 + 0.262454i) q^{35} +(-4.18776 + 3.88567i) q^{37} +(19.6190 - 2.95708i) q^{38} +(2.06274 - 13.6854i) q^{40} +(-3.53525 + 1.70249i) q^{41} +(3.05468 + 1.47106i) q^{43} +(-10.0524 + 14.7442i) q^{44} +(1.28095 - 17.0930i) q^{46} +(-1.81628 + 4.62779i) q^{47} +(1.81988 - 6.75929i) q^{49} -4.74579i q^{50} +(-17.4308 - 1.30625i) q^{52} +(3.20504 - 3.45421i) q^{53} +(2.80409 - 5.82276i) q^{55} +(18.2725 + 9.23822i) q^{56} +(-11.6387 - 1.75424i) q^{58} +(-9.48163 + 6.46446i) q^{59} +(4.54073 + 4.89374i) q^{61} +(-11.1123 - 13.9344i) q^{62} +(-6.93408 + 8.69506i) q^{64} +(6.31275 - 0.473075i) q^{65} +(6.90681 - 11.9629i) q^{67} +(-6.01880 - 10.4249i) q^{68} +(-11.8362 - 3.90274i) q^{70} +(-1.21008 - 0.276193i) q^{71} +(8.19697 - 3.21707i) q^{73} +(-14.0074 + 5.49749i) q^{74} +(36.2629 + 8.27678i) q^{76} +(6.88200 + 6.63731i) q^{77} +(5.24053 + 9.07686i) q^{79} +(9.39628 - 16.2748i) q^{80} +(-10.3066 + 0.772369i) q^{82} +(8.72515 - 10.9410i) q^{83} +(2.71814 + 3.40844i) q^{85} +(6.07427 + 6.54650i) q^{86} +(-23.1070 + 15.7541i) q^{88} +(7.03212 + 1.05992i) q^{89} +(-2.25984 + 9.08866i) q^{91} +(13.9427 - 28.9522i) q^{92} +(-8.90679 + 9.59923i) q^{94} +(-13.4331 - 1.00667i) q^{95} -1.36179i q^{97} +(10.9668 - 14.8221i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q - 16 q^{4} + 2 q^{7} + 12 q^{10} + 12 q^{16} - 6 q^{19} + 44 q^{22} + 26 q^{25} + 84 q^{28} - 6 q^{31} - 112 q^{34} + 60 q^{37} - 304 q^{40} + 20 q^{43} - 20 q^{46} - 86 q^{49} - 168 q^{52} - 84 q^{55}+ \cdots + 52 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{42}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.45194 + 0.962314i 1.73378 + 0.680459i 0.999999 + 0.00107413i \(0.000341906\pi\)
0.733782 + 0.679385i \(0.237753\pi\)
\(3\) 0 0
\(4\) 3.61984 + 3.35872i 1.80992 + 1.67936i
\(5\) −1.47762 − 1.00742i −0.660812 − 0.450534i 0.185912 − 0.982566i \(-0.440476\pi\)
−0.846724 + 0.532032i \(0.821428\pi\)
\(6\) 0 0
\(7\) 2.09999 − 1.60937i 0.793720 − 0.608283i
\(8\) 3.35777 + 6.97247i 1.18715 + 2.46514i
\(9\) 0 0
\(10\) −2.65357 − 3.89208i −0.839134 − 1.23078i
\(11\) 0.538606 + 3.57342i 0.162396 + 1.07743i 0.910399 + 0.413731i \(0.135774\pi\)
−0.748004 + 0.663695i \(0.768987\pi\)
\(12\) 0 0
\(13\) −2.76752 + 2.20702i −0.767571 + 0.612118i −0.926987 − 0.375094i \(-0.877610\pi\)
0.159415 + 0.987212i \(0.449039\pi\)
\(14\) 6.69775 − 1.92522i 1.79005 − 0.514536i
\(15\) 0 0
\(16\) 0.785279 + 10.4788i 0.196320 + 2.61970i
\(17\) −2.32943 − 0.718533i −0.564969 − 0.174270i −0.000902867 − 1.00000i \(-0.500287\pi\)
−0.564066 + 0.825730i \(0.690764\pi\)
\(18\) 0 0
\(19\) 6.52328 − 3.76622i 1.49654 − 0.864030i 0.496551 − 0.868008i \(-0.334600\pi\)
0.999992 + 0.00397809i \(0.00126627\pi\)
\(20\) −1.96509 − 8.60964i −0.439408 − 1.92517i
\(21\) 0 0
\(22\) −2.11812 + 9.28010i −0.451585 + 1.97852i
\(23\) −1.91813 − 6.21843i −0.399958 − 1.29663i −0.901980 − 0.431778i \(-0.857887\pi\)
0.502022 − 0.864855i \(-0.332590\pi\)
\(24\) 0 0
\(25\) −0.658247 − 1.67719i −0.131649 − 0.335437i
\(26\) −8.90963 + 2.74826i −1.74732 + 0.538977i
\(27\) 0 0
\(28\) 13.0070 + 1.22762i 2.45810 + 0.231999i
\(29\) −4.35647 + 0.994336i −0.808976 + 0.184644i −0.606953 − 0.794737i \(-0.707609\pi\)
−0.202023 + 0.979381i \(0.564751\pi\)
\(30\) 0 0
\(31\) −5.85986 − 3.38319i −1.05246 − 0.607639i −0.129124 − 0.991628i \(-0.541216\pi\)
−0.923337 + 0.383990i \(0.874550\pi\)
\(32\) −3.59633 + 11.6590i −0.635747 + 2.06104i
\(33\) 0 0
\(34\) −5.02015 − 4.00344i −0.860949 − 0.686584i
\(35\) −4.72430 + 0.262454i −0.798552 + 0.0443628i
\(36\) 0 0
\(37\) −4.18776 + 3.88567i −0.688464 + 0.638801i −0.944747 − 0.327801i \(-0.893692\pi\)
0.256283 + 0.966602i \(0.417502\pi\)
\(38\) 19.6190 − 2.95708i 3.18262 − 0.479702i
\(39\) 0 0
\(40\) 2.06274 − 13.6854i 0.326147 − 2.16385i
\(41\) −3.53525 + 1.70249i −0.552114 + 0.265884i −0.689072 − 0.724693i \(-0.741982\pi\)
0.136958 + 0.990577i \(0.456267\pi\)
\(42\) 0 0
\(43\) 3.05468 + 1.47106i 0.465834 + 0.224334i 0.652052 − 0.758175i \(-0.273909\pi\)
−0.186217 + 0.982509i \(0.559623\pi\)
\(44\) −10.0524 + 14.7442i −1.51546 + 2.22278i
\(45\) 0 0
\(46\) 1.28095 − 17.0930i 0.188865 − 2.52023i
\(47\) −1.81628 + 4.62779i −0.264931 + 0.675033i −0.999996 − 0.00296394i \(-0.999057\pi\)
0.735065 + 0.677997i \(0.237152\pi\)
\(48\) 0 0
\(49\) 1.81988 − 6.75929i 0.259983 − 0.965613i
\(50\) − 4.74579i − 0.671157i
\(51\) 0 0
\(52\) −17.4308 − 1.30625i −2.41721 − 0.181145i
\(53\) 3.20504 − 3.45421i 0.440247 − 0.474473i −0.473479 − 0.880805i \(-0.657002\pi\)
0.913725 + 0.406332i \(0.133192\pi\)
\(54\) 0 0
\(55\) 2.80409 − 5.82276i 0.378104 − 0.785141i
\(56\) 18.2725 + 9.23822i 2.44177 + 1.23451i
\(57\) 0 0
\(58\) −11.6387 − 1.75424i −1.52823 − 0.230344i
\(59\) −9.48163 + 6.46446i −1.23440 + 0.841601i −0.991709 − 0.128502i \(-0.958983\pi\)
−0.242693 + 0.970103i \(0.578031\pi\)
\(60\) 0 0
\(61\) 4.54073 + 4.89374i 0.581381 + 0.626580i 0.953423 − 0.301636i \(-0.0975328\pi\)
−0.372042 + 0.928216i \(0.621342\pi\)
\(62\) −11.1123 − 13.9344i −1.41126 − 1.76967i
\(63\) 0 0
\(64\) −6.93408 + 8.69506i −0.866759 + 1.08688i
\(65\) 6.31275 − 0.473075i 0.783000 − 0.0586778i
\(66\) 0 0
\(67\) 6.90681 − 11.9629i 0.843801 − 1.46151i −0.0428576 − 0.999081i \(-0.513646\pi\)
0.886658 − 0.462425i \(-0.153020\pi\)
\(68\) −6.01880 − 10.4249i −0.729887 − 1.26420i
\(69\) 0 0
\(70\) −11.8362 − 3.90274i −1.41470 − 0.466467i
\(71\) −1.21008 − 0.276193i −0.143610 − 0.0327781i 0.150112 − 0.988669i \(-0.452037\pi\)
−0.293722 + 0.955891i \(0.594894\pi\)
\(72\) 0 0
\(73\) 8.19697 − 3.21707i 0.959383 − 0.376530i 0.166594 − 0.986026i \(-0.446723\pi\)
0.792790 + 0.609495i \(0.208628\pi\)
\(74\) −14.0074 + 5.49749i −1.62832 + 0.639070i
\(75\) 0 0
\(76\) 36.2629 + 8.27678i 4.15964 + 0.949412i
\(77\) 6.88200 + 6.63731i 0.784276 + 0.756392i
\(78\) 0 0
\(79\) 5.24053 + 9.07686i 0.589606 + 1.02123i 0.994284 + 0.106768i \(0.0340502\pi\)
−0.404678 + 0.914459i \(0.632616\pi\)
\(80\) 9.39628 − 16.2748i 1.05054 − 1.81958i
\(81\) 0 0
\(82\) −10.3066 + 0.772369i −1.13817 + 0.0852939i
\(83\) 8.72515 − 10.9410i 0.957710 − 1.20093i −0.0218467 − 0.999761i \(-0.506955\pi\)
0.979557 − 0.201169i \(-0.0644740\pi\)
\(84\) 0 0
\(85\) 2.71814 + 3.40844i 0.294824 + 0.369697i
\(86\) 6.07427 + 6.54650i 0.655005 + 0.705927i
\(87\) 0 0
\(88\) −23.1070 + 15.7541i −2.46322 + 1.67939i
\(89\) 7.03212 + 1.05992i 0.745403 + 0.112351i 0.510752 − 0.859728i \(-0.329367\pi\)
0.234651 + 0.972080i \(0.424605\pi\)
\(90\) 0 0
\(91\) −2.25984 + 9.08866i −0.236896 + 0.952751i
\(92\) 13.9427 − 28.9522i 1.45362 − 3.01848i
\(93\) 0 0
\(94\) −8.90679 + 9.59923i −0.918665 + 0.990085i
\(95\) −13.4331 − 1.00667i −1.37821 − 0.103282i
\(96\) 0 0
\(97\) − 1.36179i − 0.138269i −0.997607 − 0.0691347i \(-0.977976\pi\)
0.997607 − 0.0691347i \(-0.0220238\pi\)
\(98\) 10.9668 − 14.8221i 1.10781 − 1.49725i
\(99\) 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 441.2.bg.a.395.18 yes 216
3.2 odd 2 inner 441.2.bg.a.395.1 yes 216
49.33 odd 42 inner 441.2.bg.a.278.1 ✓ 216
147.131 even 42 inner 441.2.bg.a.278.18 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.bg.a.278.1 ✓ 216 49.33 odd 42 inner
441.2.bg.a.278.18 yes 216 147.131 even 42 inner
441.2.bg.a.395.1 yes 216 3.2 odd 2 inner
441.2.bg.a.395.18 yes 216 1.1 even 1 trivial