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Properties

Label 912.2.cc.c.401.1
Level $912$
Weight $2$
Character 912.401
Analytic conductor $7.282$
Analytic rank $0$
Dimension $18$
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] = [912,2,Mod(257,912)] 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("912.257"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 9, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.cc (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,-3,0,0,0,0,0,-3,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 401.1
Root \(-1.72388 - 0.168030i\) of defining polynomial
Character \(\chi\) \(=\) 912.401
Dual form 912.2.cc.c.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.67739 - 0.431705i) q^{3} +(1.14133 + 3.13578i) q^{5} +(1.07356 + 1.85947i) q^{7} +(2.62726 + 1.44827i) q^{9} +(5.41799 + 3.12808i) q^{11} +(-2.56208 - 3.05336i) q^{13} +(-0.560722 - 5.75264i) q^{15} +(-0.403611 - 0.0711674i) q^{17} +(-4.34640 + 0.329887i) q^{19} +(-0.998042 - 3.58251i) q^{21} +(0.280411 - 0.770422i) q^{23} +(-4.70025 + 3.94398i) q^{25} +(-3.78171 - 3.56352i) q^{27} +(0.805141 + 4.56618i) q^{29} +(2.02597 - 1.16970i) q^{31} +(-7.73767 - 7.58597i) q^{33} +(-4.60559 + 5.48872i) q^{35} +6.01346i q^{37} +(2.97944 + 6.22774i) q^{39} +(0.926617 + 0.777524i) q^{41} +(-5.87377 + 2.13788i) q^{43} +(-1.54289 + 9.89147i) q^{45} +(7.59919 - 1.33994i) q^{47} +(1.19492 - 2.06967i) q^{49} +(0.646288 + 0.293616i) q^{51} +(-0.220516 - 0.0802612i) q^{53} +(-3.62525 + 20.5598i) q^{55} +(7.43301 + 1.32301i) q^{57} +(-0.930375 + 5.27642i) q^{59} +(7.30705 + 2.65955i) q^{61} +(0.127515 + 6.44012i) q^{63} +(6.65050 - 11.5190i) q^{65} +(-3.48689 + 0.614832i) q^{67} +(-0.802953 + 1.17124i) q^{69} +(-4.19799 + 1.52794i) q^{71} +(-4.33185 - 3.63485i) q^{73} +(9.58679 - 4.58646i) q^{75} +13.4328i q^{77} +(8.05412 - 9.59853i) q^{79} +(4.80501 + 7.60999i) q^{81} +(-8.01579 + 4.62792i) q^{83} +(-0.237488 - 1.34686i) q^{85} +(0.620710 - 8.00685i) q^{87} +(-5.61888 + 4.71480i) q^{89} +(2.92708 - 8.04207i) q^{91} +(-3.90331 + 1.08741i) q^{93} +(-5.99513 - 13.2528i) q^{95} +(-16.0734 - 2.83418i) q^{97} +(9.70416 + 16.0650i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{3} - 3 q^{9} - 12 q^{13} - 18 q^{15} + 6 q^{17} + 6 q^{19} - 18 q^{25} + 6 q^{27} - 6 q^{29} - 24 q^{33} + 24 q^{35} - 6 q^{39} + 3 q^{41} + 6 q^{43} - 54 q^{45} - 30 q^{47} + 21 q^{49} - 42 q^{51}+ \cdots + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.67739 − 0.431705i −0.968440 − 0.249245i
\(4\) 0 0
\(5\) 1.14133 + 3.13578i 0.510418 + 1.40236i 0.880802 + 0.473484i \(0.157004\pi\)
−0.370384 + 0.928879i \(0.620774\pi\)
\(6\) 0 0
\(7\) 1.07356 + 1.85947i 0.405769 + 0.702812i 0.994411 − 0.105582i \(-0.0336704\pi\)
−0.588642 + 0.808394i \(0.700337\pi\)
\(8\) 0 0
\(9\) 2.62726 + 1.44827i 0.875754 + 0.482758i
\(10\) 0 0
\(11\) 5.41799 + 3.12808i 1.63359 + 0.943151i 0.982975 + 0.183740i \(0.0588203\pi\)
0.650611 + 0.759412i \(0.274513\pi\)
\(12\) 0 0
\(13\) −2.56208 − 3.05336i −0.710592 − 0.846850i 0.283089 − 0.959094i \(-0.408641\pi\)
−0.993681 + 0.112243i \(0.964196\pi\)
\(14\) 0 0
\(15\) −0.560722 − 5.75264i −0.144778 − 1.48532i
\(16\) 0 0
\(17\) −0.403611 − 0.0711674i −0.0978899 − 0.0172606i 0.124489 − 0.992221i \(-0.460271\pi\)
−0.222379 + 0.974960i \(0.571382\pi\)
\(18\) 0 0
\(19\) −4.34640 + 0.329887i −0.997132 + 0.0756812i
\(20\) 0 0
\(21\) −0.998042 − 3.58251i −0.217791 − 0.781768i
\(22\) 0 0
\(23\) 0.280411 − 0.770422i 0.0584697 − 0.160644i −0.907019 − 0.421090i \(-0.861647\pi\)
0.965488 + 0.260446i \(0.0838697\pi\)
\(24\) 0 0
\(25\) −4.70025 + 3.94398i −0.940051 + 0.788796i
\(26\) 0 0
\(27\) −3.78171 − 3.56352i −0.727790 − 0.685800i
\(28\) 0 0
\(29\) 0.805141 + 4.56618i 0.149511 + 0.847919i 0.963634 + 0.267226i \(0.0861071\pi\)
−0.814123 + 0.580693i \(0.802782\pi\)
\(30\) 0 0
\(31\) 2.02597 − 1.16970i 0.363875 − 0.210084i −0.306904 − 0.951740i \(-0.599293\pi\)
0.670779 + 0.741657i \(0.265960\pi\)
\(32\) 0 0
\(33\) −7.73767 − 7.58597i −1.34695 − 1.32055i
\(34\) 0 0
\(35\) −4.60559 + 5.48872i −0.778486 + 0.927764i
\(36\) 0 0
\(37\) 6.01346i 0.988607i 0.869289 + 0.494303i \(0.164577\pi\)
−0.869289 + 0.494303i \(0.835423\pi\)
\(38\) 0 0
\(39\) 2.97944 + 6.22774i 0.477093 + 0.997236i
\(40\) 0 0
\(41\) 0.926617 + 0.777524i 0.144713 + 0.121429i 0.712270 − 0.701905i \(-0.247667\pi\)
−0.567557 + 0.823334i \(0.692111\pi\)
\(42\) 0 0
\(43\) −5.87377 + 2.13788i −0.895741 + 0.326023i −0.748545 − 0.663084i \(-0.769247\pi\)
−0.147196 + 0.989107i \(0.547025\pi\)
\(44\) 0 0
\(45\) −1.54289 + 9.89147i −0.230001 + 1.47453i
\(46\) 0 0
\(47\) 7.59919 − 1.33994i 1.10846 − 0.195451i 0.410689 − 0.911776i \(-0.365288\pi\)
0.697767 + 0.716325i \(0.254177\pi\)
\(48\) 0 0
\(49\) 1.19492 − 2.06967i 0.170703 − 0.295666i
\(50\) 0 0
\(51\) 0.646288 + 0.293616i 0.0904984 + 0.0411145i
\(52\) 0 0
\(53\) −0.220516 − 0.0802612i −0.0302902 − 0.0110247i 0.326831 − 0.945083i \(-0.394019\pi\)
−0.357121 + 0.934058i \(0.616242\pi\)
\(54\) 0 0
\(55\) −3.62525 + 20.5598i −0.488828 + 2.77228i
\(56\) 0 0
\(57\) 7.43301 + 1.32301i 0.984526 + 0.175237i
\(58\) 0 0
\(59\) −0.930375 + 5.27642i −0.121124 + 0.686931i 0.862410 + 0.506210i \(0.168954\pi\)
−0.983534 + 0.180721i \(0.942157\pi\)
\(60\) 0 0
\(61\) 7.30705 + 2.65955i 0.935572 + 0.340520i 0.764416 − 0.644723i \(-0.223027\pi\)
0.171156 + 0.985244i \(0.445250\pi\)
\(62\) 0 0
\(63\) 0.127515 + 6.44012i 0.0160654 + 0.811379i
\(64\) 0 0
\(65\) 6.65050 − 11.5190i 0.824893 − 1.42876i
\(66\) 0 0
\(67\) −3.48689 + 0.614832i −0.425991 + 0.0751137i −0.382534 − 0.923942i \(-0.624948\pi\)
−0.0434574 + 0.999055i \(0.513837\pi\)
\(68\) 0 0
\(69\) −0.802953 + 1.17124i −0.0966642 + 0.141001i
\(70\) 0 0
\(71\) −4.19799 + 1.52794i −0.498210 + 0.181334i −0.578889 − 0.815407i \(-0.696513\pi\)
0.0806788 + 0.996740i \(0.474291\pi\)
\(72\) 0 0
\(73\) −4.33185 − 3.63485i −0.507005 − 0.425427i 0.353069 − 0.935597i \(-0.385138\pi\)
−0.860074 + 0.510170i \(0.829583\pi\)
\(74\) 0 0
\(75\) 9.58679 − 4.58646i 1.10699 − 0.529599i
\(76\) 0 0
\(77\) 13.4328i 1.53081i
\(78\) 0 0
\(79\) 8.05412 − 9.59853i 0.906159 − 1.07992i −0.0903059 − 0.995914i \(-0.528784\pi\)
0.996465 − 0.0840047i \(-0.0267711\pi\)
\(80\) 0 0
\(81\) 4.80501 + 7.60999i 0.533889 + 0.845554i
\(82\) 0 0
\(83\) −8.01579 + 4.62792i −0.879848 + 0.507980i −0.870608 − 0.491977i \(-0.836274\pi\)
−0.00923947 + 0.999957i \(0.502941\pi\)
\(84\) 0 0
\(85\) −0.237488 − 1.34686i −0.0257591 − 0.146087i
\(86\) 0 0
\(87\) 0.620710 − 8.00685i 0.0665471 − 0.858424i
\(88\) 0 0
\(89\) −5.61888 + 4.71480i −0.595600 + 0.499767i −0.890028 − 0.455906i \(-0.849315\pi\)
0.294428 + 0.955674i \(0.404871\pi\)
\(90\) 0 0
\(91\) 2.92708 − 8.04207i 0.306841 − 0.843038i
\(92\) 0 0
\(93\) −3.90331 + 1.08741i −0.404754 + 0.112759i
\(94\) 0 0
\(95\) −5.99513 − 13.2528i −0.615087 − 1.35971i
\(96\) 0 0
\(97\) −16.0734 − 2.83418i −1.63201 − 0.287767i −0.718786 − 0.695231i \(-0.755302\pi\)
−0.913221 + 0.407464i \(0.866413\pi\)
\(98\) 0 0
\(99\) 9.70416 + 16.0650i 0.975305 + 1.61459i
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 912.2.cc.c.401.1 18
3.2 odd 2 912.2.cc.d.401.3 18
4.3 odd 2 114.2.l.b.59.3 yes 18
12.11 even 2 114.2.l.a.59.1 yes 18
19.10 odd 18 912.2.cc.d.257.3 18
57.29 even 18 inner 912.2.cc.c.257.1 18
76.67 even 18 114.2.l.a.29.1 ✓ 18
228.143 odd 18 114.2.l.b.29.3 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.l.a.29.1 ✓ 18 76.67 even 18
114.2.l.a.59.1 yes 18 12.11 even 2
114.2.l.b.29.3 yes 18 228.143 odd 18
114.2.l.b.59.3 yes 18 4.3 odd 2
912.2.cc.c.257.1 18 57.29 even 18 inner
912.2.cc.c.401.1 18 1.1 even 1 trivial
912.2.cc.d.257.3 18 19.10 odd 18
912.2.cc.d.401.3 18 3.2 odd 2