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Properties

Label 855.2.c.f
Level $855$
Weight $2$
Character orbit 855.c
Analytic conductor $6.827$
Analytic rank $0$
Dimension $12$
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] = [855,2,Mod(514,855)] 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("855.514"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(12\)
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: \( x^{12} - x^{10} + 59x^{8} - 234x^{6} + 291x^{4} + 39x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + (\beta_{2} - 1) q^{4} - \beta_{8} q^{5} - \beta_{6} q^{7} + (\beta_{11} - 2 \beta_{9} + \cdots - \beta_{3}) q^{8} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{10} - \beta_{10} q^{11}+ \cdots + ( - 5 \beta_{11} - 4 \beta_{9} + \cdots + \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} - 12 q^{10} + 40 q^{16} + 12 q^{19} + 18 q^{25} + 24 q^{31} - 56 q^{34} - 8 q^{40} + 80 q^{46} - 40 q^{49} + 6 q^{55} + 44 q^{61} - 72 q^{64} - 20 q^{70} - 16 q^{76} - 48 q^{79} - 22 q^{85}+ \cdots - 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{10} + 59x^{8} - 234x^{6} + 291x^{4} + 39x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -841\nu^{10} + 7010\nu^{8} - 52841\nu^{6} + 570967\nu^{4} - 1398420\nu^{2} + 1117105 ) / 332254 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1155\nu^{10} + 447\nu^{8} + 67039\nu^{6} - 189368\nu^{4} - 51254\nu^{2} - 13963 ) / 166127 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6530\nu^{11} + 98875\nu^{9} - 444461\nu^{7} + 6832572\nu^{5} - 21462795\nu^{3} + 15195061\nu ) / 7309588 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6523\nu^{10} + 2222\nu^{8} - 400761\nu^{6} + 1232441\nu^{4} - 1792662\nu^{2} + 91515 ) / 664508 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5107\nu^{10} - 6222\nu^{8} + 294409\nu^{6} - 1249687\nu^{4} + 1215154\nu^{2} + 1013843 ) / 332254 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1137\nu^{10} - 1120\nu^{8} + 67853\nu^{6} - 263987\nu^{4} + 382172\nu^{2} - 17131 ) / 51116 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -22689\nu^{11} + 11931\nu^{9} - 1338068\nu^{7} + 4718031\nu^{5} - 4952587\nu^{3} + 3214092\nu ) / 3654794 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51908\nu^{11} - 122737\nu^{9} + 3120597\nu^{7} - 16268634\nu^{5} + 31367969\nu^{3} - 7004069\nu ) / 7309588 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -65305\nu^{11} + 13561\nu^{9} - 3906246\nu^{7} + 12045445\nu^{5} - 13377453\nu^{3} - 10982520\nu ) / 7309588 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -77277\nu^{11} + 68906\nu^{9} - 4624287\nu^{7} + 17774147\nu^{5} - 22679898\nu^{3} + 13017905\nu ) / 7309588 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -137805\nu^{11} + 171047\nu^{9} - 7923024\nu^{7} + 34319793\nu^{5} - 33166283\nu^{3} - 27704010\nu ) / 7309588 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 4\beta_{10} + \beta_{9} + 6\beta_{8} - 5\beta_{7} + 4\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 4\beta_{5} - 4\beta_{4} - 7\beta_{2} + 3\beta _1 - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{11} + 12\beta_{10} - 40\beta_{9} - 79\beta_{8} - 17\beta_{7} - 83\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -94\beta_{6} + 11\beta_{5} - 138\beta_{4} + 55\beta_{2} - 73\beta _1 + 204 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -11\beta_{11} - 266\beta_{10} - 65\beta_{9} - 162\beta_{8} + 407\beta_{7} - 86\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 288\beta_{6} - 331\beta_{5} + 394\beta_{4} + 347\beta_{2} - 54\beta _1 + 1263 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 662\beta_{11} - 14\beta_{10} + 2464\beta_{9} + 5541\beta_{8} + 7\beta_{7} + 5551\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4666\beta_{6} + 885\beta_{5} + 6482\beta_{4} - 5513\beta_{2} + 5307\beta _1 - 20336 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -885\beta_{11} + 17302\beta_{10} - 3407\beta_{9} - 5128\beta_{8} - 26623\beta_{7} - 10000\beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
514.1
1.48591 + 0.374357i
−1.48591 + 0.374357i
−0.451100 + 0.613929i
0.451100 + 0.613929i
1.82718 + 2.17553i
−1.82718 + 2.17553i
1.82718 − 2.17553i
−1.82718 − 2.17553i
−0.451100 − 0.613929i
0.451100 − 0.613929i
1.48591 − 0.374357i
−1.48591 − 0.374357i
− 2.67125i 0 −5.13555 −2.20451 − 0.374357i 0 − 2.04966i 8.37582i 0 −1.00000 + 5.88878i
514.2 − 2.67125i 0 −5.13555 2.20451 − 0.374357i 0 2.04966i 8.37582i 0 −1.00000 − 5.88878i
514.3 − 1.62885i 0 −0.653165 −2.15014 − 0.613929i 0 4.97181i − 2.19380i 0 −1.00000 + 3.50226i
514.4 − 1.62885i 0 −0.653165 2.15014 − 0.613929i 0 − 4.97181i − 2.19380i 0 −1.00000 − 3.50226i
514.5 − 0.459657i 0 1.78872 −0.516770 − 2.17553i 0 − 1.44222i − 1.74151i 0 −1.00000 + 0.237537i
514.6 − 0.459657i 0 1.78872 0.516770 − 2.17553i 0 1.44222i − 1.74151i 0 −1.00000 − 0.237537i
514.7 0.459657i 0 1.78872 −0.516770 + 2.17553i 0 1.44222i 1.74151i 0 −1.00000 − 0.237537i
514.8 0.459657i 0 1.78872 0.516770 + 2.17553i 0 − 1.44222i 1.74151i 0 −1.00000 + 0.237537i
514.9 1.62885i 0 −0.653165 −2.15014 + 0.613929i 0 − 4.97181i 2.19380i 0 −1.00000 − 3.50226i
514.10 1.62885i 0 −0.653165 2.15014 + 0.613929i 0 4.97181i 2.19380i 0 −1.00000 + 3.50226i
514.11 2.67125i 0 −5.13555 −2.20451 + 0.374357i 0 2.04966i − 8.37582i 0 −1.00000 − 5.88878i
514.12 2.67125i 0 −5.13555 2.20451 + 0.374357i 0 − 2.04966i − 8.37582i 0 −1.00000 + 5.88878i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 514.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.c.f ✓ 12
3.b odd 2 1 inner 855.2.c.f ✓ 12
5.b even 2 1 inner 855.2.c.f ✓ 12
5.c odd 4 2 4275.2.a.bx 12
15.d odd 2 1 inner 855.2.c.f ✓ 12
15.e even 4 2 4275.2.a.bx 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.c.f ✓ 12 1.a even 1 1 trivial
855.2.c.f ✓ 12 3.b odd 2 1 inner
855.2.c.f ✓ 12 5.b even 2 1 inner
855.2.c.f ✓ 12 15.d odd 2 1 inner
4275.2.a.bx 12 5.c odd 4 2
4275.2.a.bx 12 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\):

\( T_{2}^{6} + 10T_{2}^{4} + 21T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{6} - 51T_{11}^{4} + 676T_{11}^{2} - 1944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 10 T^{4} + 21 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 9 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 31 T^{4} + \cdots + 216)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 51 T^{4} + \cdots - 1944)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 64 T^{4} + \cdots + 96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 53 T^{4} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + 96 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 96 T^{4} + \cdots - 7776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} - 32 T - 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 116 T^{4} + \cdots + 55296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 292 T^{4} + \cdots - 812544)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 175 T^{4} + \cdots + 190104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 37 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 60 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 144 T^{4} + \cdots - 24576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 11 T^{2} + 28 T + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 112 T^{4} + \cdots + 24576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 144 T^{4} + \cdots - 24576)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 251 T^{4} + \cdots + 46464)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots - 1408)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 96 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 256 T^{4} + \cdots - 161376)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 64 T^{4} + \cdots + 96)^{2} \) Copy content Toggle raw display
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