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Properties

Label 408.2.ba.b
Level $408$
Weight $2$
Character orbit 408.ba
Analytic conductor $3.258$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [408,2,Mod(25,408)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(408, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("408.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 408.ba (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.25789640247\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
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: \( x^{16} - 8 x^{15} + 40 x^{14} - 112 x^{13} + 166 x^{12} - 328 x^{11} + 728 x^{10} - 368 x^{9} + \cdots + 12689 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{15} - \beta_{12} + \cdots + \beta_1) q^{5} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{2}) q^{7} + \beta_{6} q^{9} + (\beta_{14} - 2 \beta_{11} + \beta_{10} + \cdots + 2) q^{11}+ \cdots + (\beta_{14} + \beta_{13} + \beta_{10} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{11} - 8 q^{15} - 16 q^{17} + 8 q^{19} + 8 q^{23} + 24 q^{25} + 8 q^{31} + 24 q^{33} + 16 q^{35} - 8 q^{37} + 24 q^{39} - 8 q^{41} + 8 q^{43} + 8 q^{45} - 40 q^{49} - 8 q^{51} - 24 q^{53} + 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 40 x^{14} - 112 x^{13} + 166 x^{12} - 328 x^{11} + 728 x^{10} - 368 x^{9} + \cdots + 12689 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!76 \nu^{15} + \cdots - 24\!\cdots\!47 ) / 20\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 42\!\cdots\!98 \nu^{15} + \cdots + 66\!\cdots\!17 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 64\!\cdots\!89 \nu^{15} + \cdots + 29\!\cdots\!76 ) / 20\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!71 \nu^{15} + \cdots - 36\!\cdots\!24 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 58\!\cdots\!06 \nu^{15} + \cdots - 54\!\cdots\!81 ) / 17\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59\!\cdots\!80 \nu^{15} + \cdots + 54\!\cdots\!77 ) / 17\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 69\!\cdots\!49 \nu^{15} + \cdots + 82\!\cdots\!14 ) / 20\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14\!\cdots\!65 \nu^{15} + \cdots + 17\!\cdots\!65 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!38 \nu^{15} + \cdots + 18\!\cdots\!19 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2142808205348 \nu^{15} - 14344019676505 \nu^{14} + 66884037638778 \nu^{13} + \cdots - 20\!\cdots\!28 ) / 45692994084548 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19\!\cdots\!05 \nu^{15} + \cdots + 23\!\cdots\!54 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37\!\cdots\!46 \nu^{15} + \cdots + 39\!\cdots\!69 ) / 40\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!35 \nu^{15} + \cdots - 18\!\cdots\!35 ) / 20\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!59 \nu^{15} + \cdots + 25\!\cdots\!73 ) / 20\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2 \beta_{14} + 2 \beta_{13} + \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + 7 \beta_{14} + 6 \beta_{13} - 11 \beta_{12} - 12 \beta_{11} + 6 \beta_{10} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{15} - 13 \beta_{14} - 17 \beta_{13} - 26 \beta_{12} - 38 \beta_{11} + 10 \beta_{10} + \cdots - 19 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{15} - 165 \beta_{14} - 154 \beta_{13} + 83 \beta_{12} + 56 \beta_{11} - 117 \beta_{10} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 115 \beta_{15} - 363 \beta_{14} - 234 \beta_{13} + 712 \beta_{12} + 845 \beta_{11} - 620 \beta_{10} + \cdots + 136 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 575 \beta_{15} + 1671 \beta_{14} + 2019 \beta_{13} + 1076 \beta_{12} + 2137 \beta_{11} - 80 \beta_{10} + \cdots - 104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 412 \beta_{15} + 13222 \beta_{14} + 11884 \beta_{13} - 9654 \beta_{12} - 9326 \beta_{11} + \cdots - 2961 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11326 \beta_{15} + 15652 \beta_{14} + 3770 \beta_{13} - 56442 \beta_{12} - 75740 \beta_{11} + \cdots - 8356 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32000 \beta_{15} - 199820 \beta_{14} - 215088 \beta_{13} - 16506 \beta_{12} - 92270 \beta_{11} + \cdots + 27928 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 101259 \beta_{15} - 1016695 \beta_{14} - 846562 \beta_{13} + 1026113 \beta_{12} + 1112240 \beta_{11} + \cdots + 259440 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 987989 \beta_{15} + 188967 \beta_{14} + 1083973 \beta_{13} + 3996068 \beta_{12} + 5741484 \beta_{11} + \cdots + 392320 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1626641 \beta_{15} + 20159001 \beta_{14} + 20114724 \beta_{13} - 5335889 \beta_{12} - 739652 \beta_{11} + \cdots - 3543986 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 13087183 \beta_{15} + 68702061 \beta_{14} + 51355564 \beta_{13} - 95740282 \beta_{12} + \cdots - 20252752 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 78075357 \beta_{15} - 139932103 \beta_{14} - 201467159 \beta_{13} - 241208818 \beta_{12} + \cdots - 4804008 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/408\mathbb{Z}\right)^\times\).

\(n\) \(103\) \(137\) \(205\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−0.449188 − 2.16886i
1.12476 + 0.186724i
1.17854 − 0.119294i
−1.26833 + 1.51565i
−0.449188 + 2.16886i
1.12476 − 0.186724i
1.17854 + 0.119294i
−1.26833 − 1.51565i
0.836451 − 0.822218i
1.79454 + 3.99440i
2.30326 − 0.259235i
−1.52004 + 0.501266i
0.836451 + 0.822218i
1.79454 − 3.99440i
2.30326 + 0.259235i
−1.52004 − 0.501266i
0 −0.382683 + 0.923880i 0 −2.08178 − 0.862301i 0 −0.829992 + 0.343794i 0 −0.707107 − 0.707107i 0
25.2 0 −0.382683 + 0.923880i 0 3.60511 + 1.49329i 0 2.07829 − 0.860858i 0 −0.707107 − 0.707107i 0
25.3 0 0.382683 − 0.923880i 0 −3.44232 − 1.42586i 0 −2.17765 + 0.902013i 0 −0.707107 − 0.707107i 0
25.4 0 0.382683 − 0.923880i 0 0.504774 + 0.209084i 0 2.34356 − 0.970735i 0 −0.707107 − 0.707107i 0
49.1 0 −0.382683 − 0.923880i 0 −2.08178 + 0.862301i 0 −0.829992 − 0.343794i 0 −0.707107 + 0.707107i 0
49.2 0 −0.382683 − 0.923880i 0 3.60511 − 1.49329i 0 2.07829 + 0.860858i 0 −0.707107 + 0.707107i 0
49.3 0 0.382683 + 0.923880i 0 −3.44232 + 1.42586i 0 −2.17765 − 0.902013i 0 −0.707107 + 0.707107i 0
49.4 0 0.382683 + 0.923880i 0 0.504774 − 0.209084i 0 2.34356 + 0.970735i 0 −0.707107 + 0.707107i 0
121.1 0 −0.923880 + 0.382683i 0 −0.564745 − 1.36341i 0 −0.640192 + 1.54556i 0 0.707107 − 0.707107i 0
121.2 0 −0.923880 + 0.382683i 0 1.43036 + 3.45320i 0 −1.37348 + 3.31587i 0 0.707107 − 0.707107i 0
121.3 0 0.923880 − 0.382683i 0 0.116792 + 0.281961i 0 1.76284 − 4.25587i 0 0.707107 − 0.707107i 0
121.4 0 0.923880 − 0.382683i 0 0.431802 + 1.04246i 0 −1.16338 + 2.80866i 0 0.707107 − 0.707107i 0
145.1 0 −0.923880 − 0.382683i 0 −0.564745 + 1.36341i 0 −0.640192 − 1.54556i 0 0.707107 + 0.707107i 0
145.2 0 −0.923880 − 0.382683i 0 1.43036 − 3.45320i 0 −1.37348 − 3.31587i 0 0.707107 + 0.707107i 0
145.3 0 0.923880 + 0.382683i 0 0.116792 − 0.281961i 0 1.76284 + 4.25587i 0 0.707107 + 0.707107i 0
145.4 0 0.923880 + 0.382683i 0 0.431802 − 1.04246i 0 −1.16338 − 2.80866i 0 0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 408.2.ba.b ✓ 16
3.b odd 2 1 1224.2.bq.d 16
4.b odd 2 1 816.2.bq.e 16
17.d even 8 1 inner 408.2.ba.b ✓ 16
17.e odd 16 1 6936.2.a.bm 8
17.e odd 16 1 6936.2.a.bn 8
51.g odd 8 1 1224.2.bq.d 16
68.g odd 8 1 816.2.bq.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.ba.b ✓ 16 1.a even 1 1 trivial
408.2.ba.b ✓ 16 17.d even 8 1 inner
816.2.bq.e 16 4.b odd 2 1
816.2.bq.e 16 68.g odd 8 1
1224.2.bq.d 16 3.b odd 2 1
1224.2.bq.d 16 51.g odd 8 1
6936.2.a.bm 8 17.e odd 16 1
6936.2.a.bn 8 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 12 T_{5}^{14} + 32 T_{5}^{13} + 72 T_{5}^{12} - 632 T_{5}^{11} + 1072 T_{5}^{10} + \cdots + 1156 \) acting on \(S_{2}^{\mathrm{new}}(408, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 12 T^{14} + \cdots + 1156 \) Copy content Toggle raw display
$7$ \( T^{16} + 20 T^{14} + \cdots + 1032256 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + \cdots + 1597696 \) Copy content Toggle raw display
$13$ \( T^{16} + 100 T^{14} + \cdots + 1056784 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} - 8 T^{15} + \cdots + 17774656 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 4404180496 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 3272755264 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 653722624 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 11601013264 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 94865232004 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 173242418176 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 2959941320704 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 755703353344 \) Copy content Toggle raw display
$59$ \( T^{16} + 32 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 8706382864 \) Copy content Toggle raw display
$67$ \( (T^{8} - 300 T^{6} + \cdots - 8795168)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 55249132216576 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 685478020096 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 90254579776 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 761388875776 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 459694848064 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 59206992273664 \) Copy content Toggle raw display
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