↑

Properties

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

Related objects

Downloads

Learn more

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: \( x^{16} + 32x^{14} + 412x^{12} + 2770x^{10} + 10580x^{8} + 23236x^{6} + 27985x^{4} + 16144x^{2} + 3136 \) 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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_{11} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_{12} q^{6} + \beta_{4} q^{7} - \beta_{9} q^{8} + (\beta_{9} - 2 \beta_{4} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_{11} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_{12} q^{6} + \beta_{4} q^{7} - \beta_{9} q^{8} + (\beta_{9} - 2 \beta_{4} + \cdots - \beta_{2}) q^{9}+ \cdots + (\beta_{13} + \beta_{11} + 3 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 4 q^{11} + 4 q^{13} + 16 q^{16} - 4 q^{17} - 16 q^{18} - 4 q^{22} + 16 q^{29} - 16 q^{31} + 8 q^{33} - 4 q^{34} - 16 q^{35} - 8 q^{38} + 20 q^{39} - 16 q^{41} + 4 q^{44} + 24 q^{45} - 8 q^{47} - 16 q^{50} - 28 q^{51} - 4 q^{52} + 12 q^{55} - 20 q^{57} - 16 q^{58} + 16 q^{61} + 16 q^{62} + 24 q^{63} - 16 q^{64} - 4 q^{65} - 20 q^{67} + 4 q^{68} - 76 q^{69} + 16 q^{72} - 16 q^{73} - 20 q^{78} + 48 q^{79} + 32 q^{81} - 16 q^{82} - 52 q^{86} + 4 q^{88} + 4 q^{89} + 24 q^{90} + 4 q^{91} + 12 q^{95} + 12 q^{97} + 16 q^{98} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 412x^{12} + 2770x^{10} + 10580x^{8} + 23236x^{6} + 27985x^{4} + 16144x^{2} + 3136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11 \nu^{15} + 1416 \nu^{13} + 30292 \nu^{11} + 250214 \nu^{9} + 908780 \nu^{7} + 1436972 \nu^{5} + \cdots + 79640 \nu ) / 56896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 958 \nu^{15} + 32585 \nu^{14} - 27632 \nu^{13} + 920472 \nu^{12} - 308008 \nu^{11} + \cdots + 27391560 ) / 113792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 958 \nu^{15} - 32585 \nu^{14} - 27632 \nu^{13} - 920472 \nu^{12} - 308008 \nu^{11} + \cdots - 27391560 ) / 113792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4655 \nu^{14} - 131496 \nu^{12} - 1424676 \nu^{10} - 7553102 \nu^{8} - 20946492 \nu^{6} + \cdots - 3880568 ) / 8128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{14} - 648\nu^{12} - 6996\nu^{10} - 36926\nu^{8} - 101916\nu^{6} - 143980\nu^{4} - 92071\nu^{2} - 18872 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2125 \nu^{14} + 59816 \nu^{12} + 644956 \nu^{10} + 3397730 \nu^{8} + 9352148 \nu^{6} + 13159588 \nu^{4} + \cdots + 1705576 ) / 2032 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41183 \nu^{15} + 16184 \nu^{14} - 1158312 \nu^{13} + 459200 \nu^{12} - 12474404 \nu^{11} + \cdots + 14304192 ) / 227584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 337 \nu^{15} - 9496 \nu^{13} - 102556 \nu^{11} - 541714 \nu^{9} - 1497604 \nu^{7} + \cdots - 284552 \nu ) / 1792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41183 \nu^{15} + 114156 \nu^{14} - 1158312 \nu^{13} + 3222688 \nu^{12} - 12474404 \nu^{11} + \cdots + 94124128 ) / 227584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4655 \nu^{15} - 432 \nu^{14} + 131496 \nu^{13} - 12384 \nu^{12} + 1424676 \nu^{11} - 136704 \nu^{10} + \cdots - 429184 ) / 16256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4655 \nu^{15} - 432 \nu^{14} - 131496 \nu^{13} - 12384 \nu^{12} - 1424676 \nu^{11} + \cdots - 429184 ) / 16256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17509 \nu^{15} - 492696 \nu^{13} - 5309260 \nu^{11} - 27938970 \nu^{9} - 76745780 \nu^{7} + \cdots - 13940392 \nu ) / 56896 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 23825 \nu^{15} - 23814 \nu^{14} + 673416 \nu^{13} - 672000 \nu^{12} + 7302956 \nu^{11} + \cdots - 20159664 ) / 56896 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23825 \nu^{15} + 23814 \nu^{14} + 673416 \nu^{13} + 672000 \nu^{12} + 7302956 \nu^{11} + \cdots + 20159664 ) / 56896 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{13} - 2\beta_{12} + 2\beta_{11} - \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{10} + 3\beta_{8} - \beta_{6} - 11\beta_{5} + 10\beta_{4} - 10\beta_{3} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{15} + 11 \beta_{14} - 14 \beta_{13} + 21 \beta_{12} - 21 \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} - \beta_{14} - 5 \beta_{12} - 5 \beta_{11} + 38 \beta_{10} - 38 \beta_{8} + \beta_{7} + \cdots - 190 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 104 \beta_{15} - 104 \beta_{14} + 142 \beta_{13} - 194 \beta_{12} + 194 \beta_{11} - 44 \beta_{10} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 22 \beta_{15} + 22 \beta_{14} + 90 \beta_{12} + 90 \beta_{11} - 380 \beta_{10} + 380 \beta_{8} + \cdots + 1576 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 943 \beta_{15} + 943 \beta_{14} - 1317 \beta_{13} + 1746 \beta_{12} - 1746 \beta_{11} + 498 \beta_{10} + \cdots + 498 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 338 \beta_{15} - 338 \beta_{14} - 1166 \beta_{12} - 1166 \beta_{11} + 3561 \beta_{10} - 3561 \beta_{8} + \cdots - 13587 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8433 \beta_{15} - 8433 \beta_{14} + 11870 \beta_{13} - 15627 \beta_{12} + 15627 \beta_{11} + \cdots - 5189 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4401 \beta_{15} + 4401 \beta_{14} + 13371 \beta_{12} + 13371 \beta_{11} - 32686 \beta_{10} + \cdots + 119254 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 75154 \beta_{15} + 75154 \beta_{14} - 106088 \beta_{13} + 139910 \beta_{12} - 139910 \beta_{11} + \cdots + 52210 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 52072 \beta_{15} - 52072 \beta_{14} - 144384 \beta_{12} - 144384 \beta_{11} + 298208 \beta_{10} + \cdots - 1056668 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 670269 \beta_{15} - 670269 \beta_{14} + 947201 \beta_{13} - 1255466 \beta_{12} + 1255466 \beta_{11} + \cdots - 515940 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(71\) \(171\) \(477\)
\(\chi(n)\) \(\beta_{9}\) \(1\) \(1\)

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
− 3.05678i
− 1.76245i
1.51013i
1.04163i
− 0.607476i
− 1.94428i
1.94991i
2.86932i
3.05678i
1.76245i
− 1.51013i
− 1.04163i
0.607476i
1.94428i
− 1.94991i
− 2.86932i
1.00000i −2.16147 − 2.16147i −1.00000 −0.707107 − 0.707107i 2.16147 − 2.16147i 0.707107 − 0.707107i − 1.00000i 6.34393i 0.707107 − 0.707107i
421.2 1.00000i −1.24624 − 1.24624i −1.00000 −0.707107 − 0.707107i 1.24624 − 1.24624i 0.707107 − 0.707107i − 1.00000i 0.106213i 0.707107 − 0.707107i
421.3 1.00000i −1.06782 − 1.06782i −1.00000 0.707107 + 0.707107i 1.06782 − 1.06782i −0.707107 + 0.707107i − 1.00000i − 0.719517i −0.707107 + 0.707107i
421.4 1.00000i −0.736542 − 0.736542i −1.00000 0.707107 + 0.707107i 0.736542 − 0.736542i −0.707107 + 0.707107i − 1.00000i − 1.91501i −0.707107 + 0.707107i
421.5 1.00000i 0.429551 + 0.429551i −1.00000 0.707107 + 0.707107i −0.429551 + 0.429551i −0.707107 + 0.707107i − 1.00000i − 2.63097i −0.707107 + 0.707107i
421.6 1.00000i 1.37481 + 1.37481i −1.00000 0.707107 + 0.707107i −1.37481 + 1.37481i −0.707107 + 0.707107i − 1.00000i 0.780219i −0.707107 + 0.707107i
421.7 1.00000i 1.37880 + 1.37880i −1.00000 −0.707107 − 0.707107i −1.37880 + 1.37880i 0.707107 − 0.707107i − 1.00000i 0.802158i 0.707107 − 0.707107i
421.8 1.00000i 2.02891 + 2.02891i −1.00000 −0.707107 − 0.707107i −2.02891 + 2.02891i 0.707107 − 0.707107i − 1.00000i 5.23298i 0.707107 − 0.707107i
701.1 − 1.00000i −2.16147 + 2.16147i −1.00000 −0.707107 + 0.707107i 2.16147 + 2.16147i 0.707107 + 0.707107i 1.00000i − 6.34393i 0.707107 + 0.707107i
701.2 − 1.00000i −1.24624 + 1.24624i −1.00000 −0.707107 + 0.707107i 1.24624 + 1.24624i 0.707107 + 0.707107i 1.00000i − 0.106213i 0.707107 + 0.707107i
701.3 − 1.00000i −1.06782 + 1.06782i −1.00000 0.707107 − 0.707107i 1.06782 + 1.06782i −0.707107 − 0.707107i 1.00000i 0.719517i −0.707107 − 0.707107i
701.4 − 1.00000i −0.736542 + 0.736542i −1.00000 0.707107 − 0.707107i 0.736542 + 0.736542i −0.707107 − 0.707107i 1.00000i 1.91501i −0.707107 − 0.707107i
701.5 − 1.00000i 0.429551 − 0.429551i −1.00000 0.707107 − 0.707107i −0.429551 − 0.429551i −0.707107 − 0.707107i 1.00000i 2.63097i −0.707107 − 0.707107i
701.6 − 1.00000i 1.37481 − 1.37481i −1.00000 0.707107 − 0.707107i −1.37481 − 1.37481i −0.707107 − 0.707107i 1.00000i − 0.780219i −0.707107 − 0.707107i
701.7 − 1.00000i 1.37880 − 1.37880i −1.00000 −0.707107 + 0.707107i −1.37880 − 1.37880i 0.707107 + 0.707107i 1.00000i − 0.802158i 0.707107 + 0.707107i
701.8 − 1.00000i 2.02891 − 2.02891i −1.00000 −0.707107 + 0.707107i −2.02891 − 2.02891i 0.707107 + 0.707107i 1.00000i − 5.23298i 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.d ✓ 16
17.c even 4 1 inner 1190.2.p.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1190.2.p.d ✓ 16 1.a even 1 1 trivial
1190.2.p.d ✓ 16 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 100 T_{3}^{12} - 12 T_{3}^{11} + 152 T_{3}^{9} + 1812 T_{3}^{8} + 296 T_{3}^{7} + \cdots + 3136 \) 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
$3$ \( T^{16} + 100 T^{12} + \cdots + 3136 \) 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
$11$ \( T^{16} + \cdots + 177635584 \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + \cdots + 4352)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 365497924 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 259081216 \) Copy content Toggle raw display
$29$ \( T^{16} - 16 T^{15} + \cdots + 141376 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 5958604864 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 2644427776 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 53909409856 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 48217133056 \) Copy content Toggle raw display
$47$ \( (T^{8} + 4 T^{7} + \cdots - 851758)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1065630736 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 3976311364 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1083199744 \) Copy content Toggle raw display
$67$ \( (T^{8} + 10 T^{7} + \cdots + 876472)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 59993644096 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1776867672064 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 19092606976 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 12904960000 \) Copy content Toggle raw display
$89$ \( (T^{8} - 2 T^{7} + \cdots + 8481776)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
show more
show less