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Properties

Label 3312.2.i.g
Level $3312$
Weight $2$
Character orbit 3312.i
Analytic conductor $26.446$
Analytic rank $0$
Dimension $16$
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] = [3312,2,Mod(2575,3312)] 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("3312.2575"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3312.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,16] 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: \(26.4464531494\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 10x^{14} + 71x^{12} - 250x^{10} + 640x^{8} - 560x^{6} + 371x^{4} - 20x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 1104)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{5} - \beta_{8} q^{7} - \beta_{6} q^{11} + ( - \beta_1 + 1) q^{13} - \beta_{14} q^{17} + ( - \beta_{8} + \beta_{4}) q^{19} + ( - \beta_{11} - \beta_{4}) q^{23} + (\beta_{3} - 2) q^{25}+ \cdots + ( - \beta_{15} + 3 \beta_{14} + \cdots - 3 \beta_{12}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{13} - 32 q^{25} - 16 q^{29} + 32 q^{41} - 32 q^{49} + 32 q^{73} + 16 q^{77} + 48 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 10x^{14} + 71x^{12} - 250x^{10} + 640x^{8} - 560x^{6} + 371x^{4} - 20x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1798 \nu^{14} + 16617 \nu^{12} - 115320 \nu^{10} + 362080 \nu^{8} - 876240 \nu^{6} + \cdots - 1176618 ) / 240645 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\nu^{14} - 269\nu^{12} + 1860\nu^{10} - 5840\nu^{8} + 13700\nu^{6} - 3690\nu^{4} + 199\nu^{2} + 3161 ) / 915 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10237 \nu^{14} + 96421 \nu^{12} - 656580 \nu^{10} + 2061520 \nu^{8} - 4673230 \nu^{6} + \cdots - 441844 ) / 240645 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19403 \nu^{15} - 194588 \nu^{13} + 1382770 \nu^{11} - 4889305 \nu^{9} + 12557950 \nu^{7} + \cdots - 790198 \nu ) / 240645 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -105\nu^{14} + 1051\nu^{12} - 7460\nu^{10} + 26296\nu^{8} - 67220\nu^{6} + 58814\nu^{4} - 36815\nu^{2} + 1073 ) / 789 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 68504 \nu^{15} - 695729 \nu^{13} + 4969040 \nu^{11} - 17864125 \nu^{9} + 46360220 \nu^{7} + \cdots - 3075389 \nu ) / 240645 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53214 \nu^{14} + 528544 \nu^{12} - 3744960 \nu^{10} + 13072860 \nu^{8} - 33332800 \nu^{6} + \cdots + 558314 ) / 240645 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 190739 \nu^{15} + 1914171 \nu^{13} - 13611090 \nu^{11} + 48174990 \nu^{9} - 123831290 \nu^{7} + \cdots + 7816741 \nu ) / 481290 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61441 \nu^{14} - 615704 \nu^{12} + 4372200 \nu^{10} - 15404520 \nu^{8} + 39356010 \nu^{6} + \cdots - 634704 ) / 240645 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 176331 \nu^{15} + 96403 \nu^{14} - 1774803 \nu^{13} - 953311 \nu^{12} + 12631670 \nu^{11} + \cdots - 1013631 ) / 481290 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 176331 \nu^{15} - 96403 \nu^{14} - 1774803 \nu^{13} + 953311 \nu^{12} + 12631670 \nu^{11} + \cdots + 1013631 ) / 481290 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 361523 \nu^{15} - 3598119 \nu^{13} + 25494210 \nu^{11} - 89133920 \nu^{9} + 226916550 \nu^{7} + \cdots - 524349 \nu ) / 481290 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 386767 \nu^{15} + 3850053 \nu^{13} - 27279270 \nu^{11} + 95395870 \nu^{9} - 242804850 \nu^{7} + \cdots + 561063 \nu ) / 481290 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 599623 \nu^{15} - 5964229 \nu^{13} + 42259110 \nu^{11} - 147687310 \nu^{9} + 376136050 \nu^{7} + \cdots - 869159 \nu ) / 481290 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 243765 \nu^{15} + 2424417 \nu^{13} - 17178030 \nu^{11} + 60031660 \nu^{9} - 152896650 \nu^{7} + \cdots + 353307 \nu ) / 160430 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 2\beta_{8} - 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{9} - 3\beta_{7} + \beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} - 3\beta_{14} + \beta_{13} + 2\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{11} - 5\beta_{10} - 5\beta_{9} - 13\beta_{7} - 3\beta_{5} - \beta_{2} - 5\beta _1 - 21 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 19 \beta_{15} - 29 \beta_{14} + 5 \beta_{13} + 15 \beta_{12} - 15 \beta_{11} - 15 \beta_{10} + \cdots + 52 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{3} - 7\beta_{2} - 24\beta _1 - 95 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 93 \beta_{15} + 137 \beta_{14} - 13 \beta_{13} - 53 \beta_{12} - 55 \beta_{11} - 55 \beta_{10} + \cdots + 270 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 105 \beta_{11} + 105 \beta_{10} + 135 \beta_{9} + 272 \beta_{7} + 123 \beta_{5} - 10 \beta_{3} + \cdots - 439 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 229\beta_{15} + 324\beta_{14} - 12\beta_{13} - 87\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 482 \beta_{11} + 482 \beta_{10} + 695 \beta_{9} + 1266 \beta_{7} + 681 \beta_{5} + 71 \beta_{3} + \cdots + 2050 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2259 \beta_{15} + 3081 \beta_{14} + 39 \beta_{13} - 499 \beta_{12} + 641 \beta_{11} + \cdots - 7100 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 220\beta_{3} + 610\beta_{2} + 1335\beta _1 + 4824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 11149 \beta_{15} - 14729 \beta_{14} - 831 \beta_{13} + 989 \beta_{12} + 1869 \beta_{11} + \cdots - 36038 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 10399 \beta_{11} - 10399 \beta_{10} - 18019 \beta_{9} - 28057 \beta_{7} - 19290 \beta_{5} + 2540 \beta_{3} + \cdots + 45715 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -27523\beta_{15} - 35382\beta_{14} - 3356\beta_{13} - 577\beta_{12} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(2305\) \(2485\) \(2945\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2575.1
1.75074 − 1.01079i
−1.75074 − 1.01079i
0.201575 − 0.116380i
−0.201575 − 0.116380i
1.92597 + 1.11196i
−1.92597 + 1.11196i
0.827585 + 0.477806i
−0.827585 + 0.477806i
0.827585 − 0.477806i
−0.827585 − 0.477806i
1.92597 − 1.11196i
−1.92597 − 1.11196i
0.201575 + 0.116380i
−0.201575 + 0.116380i
1.75074 + 1.01079i
−1.75074 + 1.01079i
0 0 0 − 4.17163i 0 −1.35144 0 0 0
2575.2 0 0 0 − 4.17163i 0 1.35144 0 0 0
2575.3 0 0 0 − 2.97471i 0 −3.14510 0 0 0
2575.4 0 0 0 − 2.97471i 0 3.14510 0 0 0
2575.5 0 0 0 − 1.29885i 0 −0.329171 0 0 0
2575.6 0 0 0 − 1.29885i 0 0.329171 0 0 0
2575.7 0 0 0 − 0.248171i 0 −2.85895 0 0 0
2575.8 0 0 0 − 0.248171i 0 2.85895 0 0 0
2575.9 0 0 0 0.248171i 0 −2.85895 0 0 0
2575.10 0 0 0 0.248171i 0 2.85895 0 0 0
2575.11 0 0 0 1.29885i 0 −0.329171 0 0 0
2575.12 0 0 0 1.29885i 0 0.329171 0 0 0
2575.13 0 0 0 2.97471i 0 −3.14510 0 0 0
2575.14 0 0 0 2.97471i 0 3.14510 0 0 0
2575.15 0 0 0 4.17163i 0 −1.35144 0 0 0
2575.16 0 0 0 4.17163i 0 1.35144 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2575.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3312.2.i.g 16
3.b odd 2 1 1104.2.i.b ✓ 16
4.b odd 2 1 inner 3312.2.i.g 16
12.b even 2 1 1104.2.i.b ✓ 16
23.b odd 2 1 inner 3312.2.i.g 16
24.f even 2 1 4416.2.i.b 16
24.h odd 2 1 4416.2.i.b 16
69.c even 2 1 1104.2.i.b ✓ 16
92.b even 2 1 inner 3312.2.i.g 16
276.h odd 2 1 1104.2.i.b ✓ 16
552.b even 2 1 4416.2.i.b 16
552.h odd 2 1 4416.2.i.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1104.2.i.b ✓ 16 3.b odd 2 1
1104.2.i.b ✓ 16 12.b even 2 1
1104.2.i.b ✓ 16 69.c even 2 1
1104.2.i.b ✓ 16 276.h odd 2 1
3312.2.i.g 16 1.a even 1 1 trivial
3312.2.i.g 16 4.b odd 2 1 inner
3312.2.i.g 16 23.b odd 2 1 inner
3312.2.i.g 16 92.b even 2 1 inner
4416.2.i.b 16 24.f even 2 1
4416.2.i.b 16 24.h odd 2 1
4416.2.i.b 16 552.b even 2 1
4416.2.i.b 16 552.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3312, [\chi])\):

\( T_{5}^{8} + 28T_{5}^{6} + 200T_{5}^{4} + 272T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 20T_{7}^{6} + 116T_{7}^{4} - 160T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 40T_{11}^{6} + 404T_{11}^{4} - 1520T_{11}^{2} + 1936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 28 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 20 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 40 T^{6} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots + 208)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 52 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 76 T^{6} + \cdots + 59536)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots - 176)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 104 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 40 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + \cdots + 1744)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 284 T^{6} + \cdots + 10061584)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 192 T^{6} + \cdots + 692224)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 300 T^{6} + \cdots + 26501904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 120 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 200 T^{6} + \cdots + 3489424)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 364 T^{6} + \cdots + 4682896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 104 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} + \cdots - 1328)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 148 T^{6} + \cdots + 234256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 424 T^{6} + \cdots + 2676496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 404 T^{6} + \cdots + 31494544)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 608 T^{6} + \cdots + 11075584)^{2} \) Copy content Toggle raw display
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