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Properties

Label 1190.2.p.c
Level $1190$
Weight $2$
Character orbit 1190.p
Analytic conductor $9.502$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1190,2,Mod(421,1190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1190, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1190.421");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1190 = 2 \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1190.p (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.50219784053\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 320x^{12} + 1922x^{10} + 6524x^{8} + 12572x^{6} + 13137x^{4} + 6588x^{2} + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{7} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_{11} q^{6} + \beta_{4} q^{7} - \beta_{5} q^{8} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{7} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_{11} q^{6} + \beta_{4} q^{7} - \beta_{5} q^{8} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_{3}) q^{9}+ \cdots + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 16 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 16 q^{4} - 4 q^{6} + 8 q^{11} + 4 q^{12} + 20 q^{13} + 16 q^{16} - 4 q^{17} + 8 q^{18} - 8 q^{22} + 4 q^{24} + 8 q^{27} + 24 q^{29} + 8 q^{31} - 32 q^{33} - 16 q^{35} - 4 q^{37} + 40 q^{38} - 36 q^{39} - 12 q^{41} - 8 q^{44} + 8 q^{45} + 64 q^{47} - 4 q^{48} + 16 q^{50} - 24 q^{51} - 20 q^{52} - 8 q^{54} + 4 q^{55} - 12 q^{57} + 24 q^{58} - 16 q^{61} + 8 q^{62} + 8 q^{63} - 16 q^{64} - 12 q^{65} - 44 q^{67} + 4 q^{68} + 4 q^{69} + 28 q^{71} - 8 q^{72} - 36 q^{73} - 4 q^{74} + 4 q^{75} - 36 q^{78} - 20 q^{79} + 72 q^{81} + 12 q^{82} - 4 q^{85} - 36 q^{86} + 8 q^{88} + 44 q^{89} - 8 q^{90} + 12 q^{91} - 4 q^{96} + 16 q^{97} - 16 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 320x^{12} + 1922x^{10} + 6524x^{8} + 12572x^{6} + 13137x^{4} + 6588x^{2} + 1156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27 \nu^{15} + 119 \nu^{14} + 654 \nu^{13} + 2550 \nu^{12} + 6294 \nu^{11} + 21216 \nu^{10} + \cdots + 15470 ) / 1496 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27 \nu^{15} - 119 \nu^{14} + 654 \nu^{13} - 2550 \nu^{12} + 6294 \nu^{11} - 21216 \nu^{10} + \cdots - 15470 ) / 1496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{15} - 162\nu^{13} - 1492\nu^{11} - 6994\nu^{9} - 17720\nu^{7} - 23676\nu^{5} - 14915\nu^{3} - 3242\nu ) / 136 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -15\nu^{15} - 301\nu^{13} - 2250\nu^{11} - 7614\nu^{9} - 10752\nu^{7} - 1665\nu^{5} + 8203\nu^{3} + 4353\nu ) / 187 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7 \nu^{15} - 6 \nu^{14} - 150 \nu^{13} - 138 \nu^{12} - 1248 \nu^{11} - 1252 \nu^{10} - 5124 \nu^{9} + \cdots - 1836 ) / 88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{14} + 46\nu^{12} + 421\nu^{10} + 1970\nu^{8} + 5046\nu^{6} + 6965\nu^{4} + 4638\nu^{2} + 1074 ) / 22 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{15} - 17 \nu^{14} - 28 \nu^{13} - 374 \nu^{12} - 320 \nu^{11} - 3230 \nu^{10} - 1922 \nu^{9} + \cdots - 3366 ) / 136 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} + 17 \nu^{14} - 28 \nu^{13} + 374 \nu^{12} - 320 \nu^{11} + 3230 \nu^{10} - 1922 \nu^{9} + \cdots + 3366 ) / 136 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7 \nu^{15} - 6 \nu^{14} + 150 \nu^{13} - 138 \nu^{12} + 1248 \nu^{11} - 1252 \nu^{10} + 5124 \nu^{9} + \cdots - 1836 ) / 88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3 \nu^{15} + 153 \nu^{14} + 135 \nu^{13} + 3332 \nu^{12} + 1946 \nu^{11} + 28373 \nu^{10} + \cdots + 37842 ) / 748 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{14} - 22\nu^{12} - 190\nu^{10} - 822\nu^{8} - 1892\nu^{6} - 2266\nu^{4} - 1261\nu^{2} - 238 ) / 4 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 57 \nu^{15} - 425 \nu^{14} - 1256 \nu^{13} - 9214 \nu^{12} - 10794 \nu^{11} - 77962 \nu^{10} + \cdots - 92650 ) / 1496 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21 \nu^{15} + 425 \nu^{14} + 384 \nu^{13} + 9214 \nu^{12} + 2402 \nu^{11} + 77962 \nu^{10} + \cdots + 92650 ) / 1496 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{11} - \beta_{7} + \beta_{6} + 2\beta_{5} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - \beta_{10} + \beta_{9} - 8\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{15} + 8 \beta_{14} - \beta_{12} - 8 \beta_{11} + \beta_{10} + \beta_{9} + 8 \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + 10 \beta_{13} + \beta_{12} + 10 \beta_{10} - 10 \beta_{9} - \beta_{8} + 2 \beta_{4} + \cdots - 132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 65 \beta_{15} - 56 \beta_{14} + 9 \beta_{12} + 52 \beta_{11} - 11 \beta_{10} - 11 \beta_{9} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{15} - 74 \beta_{13} - 12 \beta_{12} - 76 \beta_{10} + 76 \beta_{9} + 14 \beta_{8} + \cdots + 816 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 441 \beta_{15} + 377 \beta_{14} - 64 \beta_{12} - 323 \beta_{11} + 88 \beta_{10} + 88 \beta_{9} + \cdots - 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 100 \beta_{15} + 493 \beta_{13} + 100 \beta_{12} + 4 \beta_{11} + 529 \beta_{10} - 529 \beta_{9} + \cdots - 5108 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2921 \beta_{15} - 2488 \beta_{14} + 433 \beta_{12} + 1992 \beta_{11} - 629 \beta_{10} - 629 \beta_{9} + \cdots + 433 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 725 \beta_{15} - 3134 \beta_{13} - 725 \beta_{12} - 82 \beta_{11} - 3550 \beta_{10} + 3550 \beta_{9} + \cdots + 32168 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 19133 \beta_{15} + 16226 \beta_{14} - 2907 \beta_{12} - 12314 \beta_{11} + 4275 \beta_{10} + 4275 \beta_{9} + \cdots - 2907 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4922 \beta_{15} + 19448 \beta_{13} + 4922 \beta_{12} + 1044 \beta_{11} + 23408 \beta_{10} + \cdots - 203230 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 124591 \beta_{15} - 105061 \beta_{14} + 19530 \beta_{12} + 76417 \beta_{11} - 28330 \beta_{10} + \cdots + 19530 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1190\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(171\) \(477\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1
− 2.53809i
2.52860i
− 1.96133i
1.30507i
0.599080i
− 0.938035i
− 1.48143i
2.48612i
2.53809i
− 2.52860i
1.96133i
− 1.30507i
− 0.599080i
0.938035i
1.48143i
− 2.48612i
− 1.00000i −1.79470 − 1.79470i −1.00000 −0.707107 − 0.707107i −1.79470 + 1.79470i 0.707107 − 0.707107i 1.00000i 3.44191i −0.707107 + 0.707107i
421.2 − 1.00000i −1.78799 − 1.78799i −1.00000 0.707107 + 0.707107i −1.78799 + 1.78799i −0.707107 + 0.707107i 1.00000i 3.39383i 0.707107 − 0.707107i
421.3 − 1.00000i −1.38687 − 1.38687i −1.00000 −0.707107 − 0.707107i −1.38687 + 1.38687i 0.707107 − 0.707107i 1.00000i 0.846801i −0.707107 + 0.707107i
421.4 − 1.00000i −0.922825 − 0.922825i −1.00000 0.707107 + 0.707107i −0.922825 + 0.922825i −0.707107 + 0.707107i 1.00000i − 1.29679i 0.707107 − 0.707107i
421.5 − 1.00000i 0.423614 + 0.423614i −1.00000 −0.707107 − 0.707107i 0.423614 − 0.423614i 0.707107 − 0.707107i 1.00000i − 2.64110i −0.707107 + 0.707107i
421.6 − 1.00000i 0.663291 + 0.663291i −1.00000 0.707107 + 0.707107i 0.663291 − 0.663291i −0.707107 + 0.707107i 1.00000i − 2.12009i 0.707107 − 0.707107i
421.7 − 1.00000i 1.04753 + 1.04753i −1.00000 0.707107 + 0.707107i 1.04753 − 1.04753i −0.707107 + 0.707107i 1.00000i − 0.805378i 0.707107 − 0.707107i
421.8 − 1.00000i 1.75796 + 1.75796i −1.00000 −0.707107 − 0.707107i 1.75796 − 1.75796i 0.707107 − 0.707107i 1.00000i 3.18082i −0.707107 + 0.707107i
701.1 1.00000i −1.79470 + 1.79470i −1.00000 −0.707107 + 0.707107i −1.79470 − 1.79470i 0.707107 + 0.707107i − 1.00000i − 3.44191i −0.707107 − 0.707107i
701.2 1.00000i −1.78799 + 1.78799i −1.00000 0.707107 − 0.707107i −1.78799 − 1.78799i −0.707107 − 0.707107i − 1.00000i − 3.39383i 0.707107 + 0.707107i
701.3 1.00000i −1.38687 + 1.38687i −1.00000 −0.707107 + 0.707107i −1.38687 − 1.38687i 0.707107 + 0.707107i − 1.00000i − 0.846801i −0.707107 − 0.707107i
701.4 1.00000i −0.922825 + 0.922825i −1.00000 0.707107 − 0.707107i −0.922825 − 0.922825i −0.707107 − 0.707107i − 1.00000i 1.29679i 0.707107 + 0.707107i
701.5 1.00000i 0.423614 − 0.423614i −1.00000 −0.707107 + 0.707107i 0.423614 + 0.423614i 0.707107 + 0.707107i − 1.00000i 2.64110i −0.707107 − 0.707107i
701.6 1.00000i 0.663291 − 0.663291i −1.00000 0.707107 − 0.707107i 0.663291 + 0.663291i −0.707107 − 0.707107i − 1.00000i 2.12009i 0.707107 + 0.707107i
701.7 1.00000i 1.04753 − 1.04753i −1.00000 0.707107 − 0.707107i 1.04753 + 1.04753i −0.707107 − 0.707107i − 1.00000i 0.805378i 0.707107 + 0.707107i
701.8 1.00000i 1.75796 − 1.75796i −1.00000 −0.707107 + 0.707107i 1.75796 + 1.75796i 0.707107 + 0.707107i − 1.00000i − 3.18082i −0.707107 − 0.707107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1190.2.p.c ✓ 16
17.c even 4 1 inner 1190.2.p.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1190.2.p.c ✓ 16 1.a even 1 1 trivial
1190.2.p.c ✓ 16 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 4 T_{3}^{15} + 8 T_{3}^{14} + 40 T_{3}^{12} + 172 T_{3}^{11} + 368 T_{3}^{10} - 24 T_{3}^{9} + \cdots + 1156 \) acting on \(S_{2}^{\mathrm{new}}(1190, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 1156 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + \cdots + 153664 \) Copy content Toggle raw display
$13$ \( (T^{8} - 10 T^{7} + \cdots + 1552)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 5552038144 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 150209536 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 105596176 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 45242991616 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 252341456896 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 139956795664 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10977400262656 \) Copy content Toggle raw display
$47$ \( (T^{8} - 32 T^{7} + \cdots - 112456)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1099788469264 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 658830915856 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 2004195164416 \) Copy content Toggle raw display
$67$ \( (T^{8} + 22 T^{7} + \cdots + 261832)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 31190385664 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 15640632870976 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 14048223625216 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 636216807424 \) Copy content Toggle raw display
$89$ \( (T^{8} - 22 T^{7} + \cdots + 12682304)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 118656054700096 \) Copy content Toggle raw display
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