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Properties

Label 2070.2.e.b
Level $2070$
Weight $2$
Character orbit 2070.e
Analytic conductor $16.529$
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] = [2070,2,Mod(1241,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2070.1241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.e (of order \(2\), degree \(1\), 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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 392x^{12} + 2324x^{10} + 6930x^{8} + 9856x^{6} + 5740x^{4} + 1108x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - q^{4} + q^{5} - \beta_{5} q^{7} - \beta_{8} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - q^{4} + q^{5} - \beta_{5} q^{7} - \beta_{8} q^{8} + \beta_{8} q^{10} + (\beta_{3} - 1) q^{11} + (\beta_{7} + \beta_{2}) q^{13} + \beta_1 q^{14} + q^{16} + (\beta_{2} - \beta_1) q^{17} + (\beta_{15} + \beta_{14} + \cdots - \beta_{5}) q^{19}+ \cdots + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{5} - 24 q^{11} + 16 q^{16} - 16 q^{20} - 4 q^{23} + 16 q^{25} - 8 q^{31} - 8 q^{38} + 24 q^{44} - 4 q^{46} - 16 q^{49} + 8 q^{53} - 24 q^{55} + 16 q^{58} - 16 q^{64} + 32 q^{73} + 16 q^{80} + 32 q^{82} - 56 q^{83} - 8 q^{86} + 40 q^{89} + 4 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 392x^{12} + 2324x^{10} + 6930x^{8} + 9856x^{6} + 5740x^{4} + 1108x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 31\nu^{12} + 353\nu^{10} + 1747\nu^{8} + 3031\nu^{6} - 1479\nu^{4} - 4661\nu^{2} - 71 ) / 1568 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{14} - 411\nu^{12} - 4925\nu^{10} - 27999\nu^{8} - 76603\nu^{6} - 88749\nu^{4} - 24879\nu^{2} + 2195 ) / 1568 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{14} + 153\nu^{12} + 1751\nu^{10} + 9303\nu^{8} + 22823\nu^{6} + 21583\nu^{4} + 3137\nu^{2} - 275 ) / 392 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{15} - 91\nu^{13} - 1017\nu^{11} - 5039\nu^{9} - 9481\nu^{7} + 2899\nu^{5} + 22933\nu^{3} + 9247\nu ) / 392 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13 \nu^{15} + 411 \nu^{13} + 4925 \nu^{11} + 27999 \nu^{9} + 76603 \nu^{7} + 88749 \nu^{5} + \cdots - 2195 \nu ) / 1568 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{14} + 593\nu^{12} + 7015\nu^{10} + 39505\nu^{8} + 108221\nu^{6} + 129879\nu^{4} + 45877\nu^{2} - 557 ) / 784 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55 \nu^{14} + 1729 \nu^{12} + 20591 \nu^{10} + 116373 \nu^{8} + 317281 \nu^{6} + 371943 \nu^{4} + \cdots - 1057 ) / 1568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8\nu^{15} + 255\nu^{13} + 3104\nu^{11} + 18204\nu^{9} + 53214\nu^{7} + 72735\nu^{5} + 38816\nu^{3} + 6544\nu ) / 196 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 69 \nu^{15} + 2195 \nu^{13} + 26653 \nu^{11} + 155823 \nu^{9} + 453411 \nu^{7} + 613797 \nu^{5} + \cdots + 39117 \nu ) / 1568 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 71 \nu^{15} - 2273 \nu^{13} - 27863 \nu^{11} - 165357 \nu^{9} - 493777 \nu^{7} - 702807 \nu^{5} + \cdots - 74007 \nu ) / 1568 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41 \nu^{15} + 1303 \nu^{13} + 15789 \nu^{11} + 91911 \nu^{9} + 265007 \nu^{7} + 351273 \nu^{5} + \cdots + 20029 \nu ) / 784 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 85 \nu^{15} + 12 \nu^{14} - 2747 \nu^{13} + 388 \nu^{12} - 34149 \nu^{11} + 4796 \nu^{10} + \cdots + 7804 ) / 1568 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 85 \nu^{15} + 25 \nu^{14} + 2747 \nu^{13} + 799 \nu^{12} + 34149 \nu^{11} + 9721 \nu^{10} + \cdots + 5609 ) / 1568 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 169 \nu^{15} + 18 \nu^{14} - 5399 \nu^{13} + 574 \nu^{12} - 65985 \nu^{11} + 6970 \nu^{10} + \cdots - 3766 ) / 1568 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 169 \nu^{15} - 18 \nu^{14} - 5399 \nu^{13} - 574 \nu^{12} - 65985 \nu^{11} - 6970 \nu^{10} + \cdots + 3766 ) / 1568 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{9} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + \beta_{12} - 2\beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + 2\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - \beta_{14} - 8\beta_{11} + 7\beta_{9} - 5\beta_{8} + 11\beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + \beta_{14} - 9 \beta_{13} - 9 \beta_{12} + 22 \beta_{7} - 12 \beta_{6} - 8 \beta_{3} + \cdots + 52 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{15} + 14 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 69 \beta_{11} - 14 \beta_{10} + \cdots - 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19 \beta_{15} - 19 \beta_{14} + 86 \beta_{13} + 86 \beta_{12} - 228 \beta_{7} + 133 \beta_{6} + \cdots - 461 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 173 \beta_{15} - 173 \beta_{14} + 42 \beta_{13} - 42 \beta_{12} - 642 \beta_{11} + \cdots + 42 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 126 \beta_{15} + 126 \beta_{14} - 429 \beta_{13} - 429 \beta_{12} + 1192 \beta_{7} - 729 \beta_{6} + \cdots + 2212 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2038 \beta_{15} + 2038 \beta_{14} - 628 \beta_{13} + 628 \beta_{12} + 6291 \beta_{11} + \cdots - 628 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2998 \beta_{15} - 2998 \beta_{14} + 8833 \beta_{13} + 8833 \beta_{12} - 25326 \beta_{7} + 16039 \beta_{6} + \cdots - 44493 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23423 \beta_{15} - 23423 \beta_{14} + 8184 \beta_{13} - 8184 \beta_{12} - 63958 \beta_{11} + \cdots + 8184 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34197 \beta_{15} + 34197 \beta_{14} - 92959 \beta_{13} - 92959 \beta_{12} + 272502 \beta_{7} + \cdots + 461354 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 265372 \beta_{15} + 265372 \beta_{14} - 99474 \beta_{13} + 99474 \beta_{12} + 667523 \beta_{11} + \cdots - 99474 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 382989 \beta_{15} - 382989 \beta_{14} + 993290 \beta_{13} + 993290 \beta_{12} - 2958048 \beta_{7} + \cdots - 4882959 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2979759 \beta_{15} - 2979759 \beta_{14} + 1163166 \beta_{13} - 1163166 \beta_{12} + \cdots + 1163166 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1241.1
− 3.32099i
1.90023i
1.48712i
− 0.0301128i
0.630652i
− 0.795887i
− 2.59147i
2.72045i
− 2.72045i
2.59147i
0.795887i
− 0.630652i
0.0301128i
− 1.48712i
− 1.90023i
3.32099i
− 1.00000i 0 −1.00000 1.00000 0 − 4.69659i 1.00000i 0 − 1.00000i
1241.2 − 1.00000i 0 −1.00000 1.00000 0 − 2.68734i 1.00000i 0 − 1.00000i
1241.3 − 1.00000i 0 −1.00000 1.00000 0 − 2.10311i 1.00000i 0 − 1.00000i
1241.4 − 1.00000i 0 −1.00000 1.00000 0 − 0.0425859i 1.00000i 0 − 1.00000i
1241.5 − 1.00000i 0 −1.00000 1.00000 0 0.891877i 1.00000i 0 − 1.00000i
1241.6 − 1.00000i 0 −1.00000 1.00000 0 1.12555i 1.00000i 0 − 1.00000i
1241.7 − 1.00000i 0 −1.00000 1.00000 0 3.66489i 1.00000i 0 − 1.00000i
1241.8 − 1.00000i 0 −1.00000 1.00000 0 3.84729i 1.00000i 0 − 1.00000i
1241.9 1.00000i 0 −1.00000 1.00000 0 − 3.84729i − 1.00000i 0 1.00000i
1241.10 1.00000i 0 −1.00000 1.00000 0 − 3.66489i − 1.00000i 0 1.00000i
1241.11 1.00000i 0 −1.00000 1.00000 0 − 1.12555i − 1.00000i 0 1.00000i
1241.12 1.00000i 0 −1.00000 1.00000 0 − 0.891877i − 1.00000i 0 1.00000i
1241.13 1.00000i 0 −1.00000 1.00000 0 0.0425859i − 1.00000i 0 1.00000i
1241.14 1.00000i 0 −1.00000 1.00000 0 2.10311i − 1.00000i 0 1.00000i
1241.15 1.00000i 0 −1.00000 1.00000 0 2.68734i − 1.00000i 0 1.00000i
1241.16 1.00000i 0 −1.00000 1.00000 0 4.69659i − 1.00000i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2070.2.e.b yes 16
3.b odd 2 1 2070.2.e.a ✓ 16
23.b odd 2 1 2070.2.e.a ✓ 16
69.c even 2 1 inner 2070.2.e.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2070.2.e.a ✓ 16 3.b odd 2 1
2070.2.e.a ✓ 16 23.b odd 2 1
2070.2.e.b yes 16 1.a even 1 1 trivial
2070.2.e.b yes 16 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 12T_{11}^{7} + 22T_{11}^{6} - 200T_{11}^{5} - 732T_{11}^{4} + 192T_{11}^{3} + 2568T_{11}^{2} + 896T_{11} - 1792 \) acting on \(S_{2}^{\mathrm{new}}(2070, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 64 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{8} + 12 T^{7} + \cdots - 1792)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 46 T^{6} + \cdots - 1568)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 40 T^{6} + \cdots - 128)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 4228120576 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 45474709504 \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 28672)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 286015744 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 4787974164736 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 1584676864 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1670008274944 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots + 1035776)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 716888201453824 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 8469889024 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 203119673344 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 20025696600064 \) Copy content Toggle raw display
$73$ \( (T^{8} - 16 T^{7} + \cdots + 4600064)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 825757696 \) Copy content Toggle raw display
$83$ \( (T^{8} + 28 T^{7} + \cdots - 17706752)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 20 T^{7} + \cdots - 180512)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 71572141441024 \) Copy content Toggle raw display
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