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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(703,1408)] 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("1408.703"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-8,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2429366046\)
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: \( x^{12} + 12 x^{10} - 34 x^{9} + 83 x^{8} - 206 x^{7} + 242 x^{6} - 514 x^{5} + 873 x^{4} - 416 x^{3} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{10} q^{5} + (\beta_{3} - 1) q^{7} + ( - \beta_{3} + \beta_{2} + 2) q^{9} + ( - \beta_{10} + \beta_{5}) q^{11} + ( - \beta_{9} + \beta_{5} - \beta_{4}) q^{13} + (\beta_{11} - \beta_{8}) q^{15}+ \cdots + ( - 3 \beta_{10} + 3 \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{7} + 20 q^{9} + 4 q^{25} - 8 q^{33} + 8 q^{39} + 12 q^{49} + 56 q^{55} - 112 q^{63} + 16 q^{79} + 28 q^{81} + 96 q^{87} + 16 q^{89} - 8 q^{95} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 12 x^{10} - 34 x^{9} + 83 x^{8} - 206 x^{7} + 242 x^{6} - 514 x^{5} + 873 x^{4} - 416 x^{3} + \cdots + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8575848931 \nu^{11} - 4697650972 \nu^{10} - 109306513636 \nu^{9} + 221787474622 \nu^{8} + \cdots + 4659147562800 ) / 2509283774300 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17520127 \nu^{11} - 45987071 \nu^{10} - 263486715 \nu^{9} - 21100649 \nu^{8} + \cdots - 4216232860 ) / 4326351335 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22911241 \nu^{11} + 5126253 \nu^{10} + 294389585 \nu^{9} - 684558558 \nu^{8} + \cdots - 3881917025 ) / 4326351335 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 82479545761 \nu^{11} + 5996335922 \nu^{10} + 950368799376 \nu^{9} - 2815139409922 \nu^{8} + \cdots - 60121467193200 ) / 5018567548600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 96609280349 \nu^{11} - 74256884378 \nu^{10} - 1186758286104 \nu^{9} + \cdots + 67651990103600 ) / 5018567548600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 219074457 \nu^{11} + 220895118 \nu^{10} + 2806414938 \nu^{9} - 4463949938 \nu^{8} + \cdots - 78464513680 ) / 8652702670 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20134203421 \nu^{11} - 14823368817 \nu^{10} - 248373346311 \nu^{9} + 506700085442 \nu^{8} + \cdots + 9995718535600 ) / 627320943575 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 614570087 \nu^{11} + 441318112 \nu^{10} + 7605516544 \nu^{9} - 15507502182 \nu^{8} + \cdots - 307876011760 ) / 17305405340 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28982544 \nu^{11} - 21967723 \nu^{10} - 364895044 \nu^{9} + 714360048 \nu^{8} + \cdots + 15252088400 ) / 653119150 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43622514396 \nu^{11} - 36829428887 \nu^{10} - 555405490416 \nu^{9} + 1013631465887 \nu^{8} + \cdots + 21289687605600 ) / 627320943575 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2624266183 \nu^{11} + 2152596352 \nu^{10} + 33278284612 \nu^{9} - 61651606142 \nu^{8} + \cdots - 1267138239920 ) / 17305405340 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - 2\beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 2\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - 8 \beta_{8} - 5 \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots + 39 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{11} - 27 \beta_{10} + 32 \beta_{9} + 34 \beta_{8} + 18 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 34 \beta_{11} + 112 \beta_{10} - 52 \beta_{9} - 164 \beta_{8} - 89 \beta_{7} + 33 \beta_{6} - 29 \beta_{5} + \cdots - 385 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 51 \beta_{11} + 187 \beta_{10} - 285 \beta_{9} - 235 \beta_{8} - 119 \beta_{7} + 62 \beta_{6} + \cdots + 840 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 437 \beta_{11} - 1360 \beta_{10} + 1139 \beta_{9} + 1899 \beta_{8} + 1088 \beta_{7} - 325 \beta_{6} + \cdots + 175 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 604 \beta_{11} + 1785 \beta_{10} - 14 \beta_{9} - 2588 \beta_{8} - 1498 \beta_{7} + 392 \beta_{6} + \cdots - 12083 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8396 \beta_{11} + 25784 \beta_{10} - 30690 \beta_{9} - 35910 \beta_{8} - 20485 \beta_{7} + 6105 \beta_{6} + \cdots + 68913 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 20967 \beta_{11} - 65638 \beta_{10} + 50307 \beta_{9} + 90097 \beta_{8} + 51046 \beta_{7} + \cdots + 56544 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 17804 \beta_{11} + 59098 \beta_{10} + 108118 \beta_{9} - 78006 \beta_{8} - 42793 \beta_{7} + \cdots - 1281907 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1408\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(639\) \(1025\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.56843 + 0.205322i
1.56843 − 0.205322i
0.804896 + 0.0616281i
0.804896 − 0.0616281i
0.626231 − 2.14325i
0.626231 + 2.14325i
−0.626231 − 0.890790i
−0.626231 + 0.890790i
−0.804896 − 1.54816i
−0.804896 + 1.54816i
−1.56843 + 3.34218i
−1.56843 − 3.34218i
0 −3.13686 0 − 2.29240i 0 −3.54751 0 6.83991 0
703.2 0 −3.13686 0 2.29240i 0 −3.54751 0 6.83991 0
703.3 0 −1.60979 0 − 2.89511i 0 −1.48654 0 −0.408571 0
703.4 0 −1.60979 0 2.89511i 0 −1.48654 0 −0.408571 0
703.5 0 −1.25246 0 − 0.602705i 0 3.03404 0 −1.43134 0
703.6 0 −1.25246 0 0.602705i 0 3.03404 0 −1.43134 0
703.7 0 1.25246 0 − 0.602705i 0 3.03404 0 −1.43134 0
703.8 0 1.25246 0 0.602705i 0 3.03404 0 −1.43134 0
703.9 0 1.60979 0 − 2.89511i 0 −1.48654 0 −0.408571 0
703.10 0 1.60979 0 2.89511i 0 −1.48654 0 −0.408571 0
703.11 0 3.13686 0 − 2.29240i 0 −3.54751 0 6.83991 0
703.12 0 3.13686 0 2.29240i 0 −3.54751 0 6.83991 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 703.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
44.c even 2 1 inner
88.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1408.2.g.f ✓ 12
4.b odd 2 1 1408.2.g.g yes 12
8.b even 2 1 inner 1408.2.g.f ✓ 12
8.d odd 2 1 1408.2.g.g yes 12
11.b odd 2 1 1408.2.g.g yes 12
16.e even 4 1 2816.2.e.i 6
16.e even 4 1 2816.2.e.j 6
16.f odd 4 1 2816.2.e.g 6
16.f odd 4 1 2816.2.e.h 6
44.c even 2 1 inner 1408.2.g.f ✓ 12
88.b odd 2 1 1408.2.g.g yes 12
88.g even 2 1 inner 1408.2.g.f ✓ 12
176.i even 4 1 2816.2.e.i 6
176.i even 4 1 2816.2.e.j 6
176.l odd 4 1 2816.2.e.g 6
176.l odd 4 1 2816.2.e.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.g.f ✓ 12 1.a even 1 1 trivial
1408.2.g.f ✓ 12 8.b even 2 1 inner
1408.2.g.f ✓ 12 44.c even 2 1 inner
1408.2.g.f ✓ 12 88.g even 2 1 inner
1408.2.g.g yes 12 4.b odd 2 1
1408.2.g.g yes 12 8.d odd 2 1
1408.2.g.g yes 12 11.b odd 2 1
1408.2.g.g yes 12 88.b odd 2 1
2816.2.e.g 6 16.f odd 4 1
2816.2.e.g 6 176.l odd 4 1
2816.2.e.h 6 16.f odd 4 1
2816.2.e.h 6 176.l odd 4 1
2816.2.e.i 6 16.e even 4 1
2816.2.e.i 6 176.i even 4 1
2816.2.e.j 6 16.e even 4 1
2816.2.e.j 6 176.i even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1408, [\chi])\):

\( T_{3}^{6} - 14T_{3}^{4} + 45T_{3}^{2} - 40 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 10T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 14 T^{4} + \cdots - 40)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 14 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 10 T - 16)^{4} \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{6} - 56 T^{4} + \cdots - 160)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 64 T^{4} + \cdots + 640)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 44 T^{4} + \cdots + 400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 90 T^{4} + \cdots + 1000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 104 T^{4} + \cdots - 10240)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 66 T^{4} + \cdots + 1000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 102 T^{4} + \cdots + 35344)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 176 T^{4} + \cdots + 40960)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 68 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 136 T^{4} + \cdots + 40960)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 152 T^{4} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 246 T^{4} + \cdots - 1000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 104 T^{4} + \cdots - 10240)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 14 T^{4} + \cdots - 40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 186 T^{4} + \cdots + 179560)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 240 T^{4} + \cdots + 400000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 4 T^{2} + \cdots - 256)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 388 T^{4} + \cdots + 1290496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} + \cdots + 1114)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} - 12 T^{2} + \cdots + 746)^{4} \) Copy content Toggle raw display
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