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Properties

Label 4100.2.d.g
Level $4100$
Weight $2$
Character orbit 4100.d
Analytic conductor $32.739$
Analytic rank $0$
Dimension $14$
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] = [4100,2,Mod(1149,4100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4100.1149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4100.d (of order \(2\), degree \(1\), not 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: \(32.7386648287\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 27x^{12} + 261x^{10} + 1141x^{8} + 2289x^{6} + 1896x^{4} + 549x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{7} q^{7} + (\beta_{3} + \beta_{2}) q^{9} + ( - \beta_{11} + \beta_{8} + \cdots - \beta_{2}) q^{11}+ \cdots + ( - \beta_{11} + \beta_{5} + 2 \beta_{4} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 27x^{12} + 261x^{10} + 1141x^{8} + 2289x^{6} + 1896x^{4} + 549x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{12} + 317\nu^{10} + 3015\nu^{8} + 9926\nu^{6} + 3665\nu^{4} - 12474\nu^{2} + 1194 ) / 10701 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{12} - 317\nu^{10} - 3015\nu^{8} - 9926\nu^{6} - 3665\nu^{4} + 23175\nu^{2} + 30909 ) / 10701 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -76\nu^{12} - 1974\nu^{10} - 18561\nu^{8} - 84469\nu^{6} - 205077\nu^{4} - 238413\nu^{2} - 55269 ) / 10701 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -299\nu^{12} - 7860\nu^{10} - 72225\nu^{8} - 284399\nu^{6} - 438054\nu^{4} - 117510\nu^{2} + 43749 ) / 10701 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -398\nu^{13} - 10713\nu^{11} - 102927\nu^{9} - 445073\nu^{7} - 881244\nu^{5} - 743613\nu^{3} - 255924\nu ) / 32103 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 541 \nu^{13} + 16023 \nu^{11} + 177792 \nu^{9} + 945079 \nu^{7} + 2488857 \nu^{5} + \cdots + 1088289 \nu ) / 32103 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 858\nu^{12} + 22348\nu^{10} + 203067\nu^{8} + 795630\nu^{6} + 1282252\nu^{4} + 607209\nu^{2} + 21792 ) / 10701 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -387\nu^{13} - 10396\nu^{11} - 99912\nu^{9} - 435147\nu^{7} - 877579\nu^{5} - 756087\nu^{3} - 244029\nu ) / 10701 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2234 \nu^{13} + 61029 \nu^{11} + 600321 \nu^{9} + 2688422 \nu^{7} + 5565507 \nu^{5} + \cdots + 1244169 \nu ) / 32103 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2142\nu^{12} + 56324\nu^{10} + 519654\nu^{8} + 2083974\nu^{6} + 3473993\nu^{4} + 1705125\nu^{2} + 59667 ) / 10701 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -332\nu^{13} - 8811\nu^{11} - 82459\nu^{9} - 337957\nu^{7} - 585784\nu^{5} - 322644\nu^{3} - 34740\nu ) / 3567 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8915 \nu^{13} + 235404 \nu^{11} + 2187345 \nu^{9} + 8885411 \nu^{7} + 15261117 \nu^{5} + \cdots + 846729 \nu ) / 32103 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - 2\beta_{10} + 2\beta_{7} - \beta_{6} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 6\beta_{5} + 4\beta_{4} - 9\beta_{3} - 13\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{13} + 7\beta_{12} + 27\beta_{10} - 8\beta_{9} - 31\beta_{7} + 13\beta_{6} + 73\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{11} + \beta_{8} - 98\beta_{5} - 71\beta_{4} + 85\beta_{3} + 163\beta_{2} - 137 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{13} - 39\beta_{12} - 322\beta_{10} + 150\beta_{9} + 393\beta_{7} - 186\beta_{6} - 736\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 228\beta_{11} - 23\beta_{8} + 1282\beta_{5} + 960\beta_{4} - 869\beta_{3} - 1993\beta_{2} + 1225 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -755\beta_{13} + 185\beta_{12} + 3752\beta_{10} - 2107\beta_{9} - 4712\beta_{7} + 2501\beta_{6} + 7930\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2812\beta_{11} + 315\beta_{8} - 15675\beta_{5} - 11923\beta_{4} + 9390\beta_{3} + 23941\beta_{2} - 12252 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9426\beta_{13} - 589\beta_{12} - 43566\beta_{10} + 26789\beta_{9} + 55489\beta_{7} - 31699\beta_{6} - 88798\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 33551\beta_{11} - 3676\beta_{8} + 186773\beta_{5} + 143207\beta_{4} - 104986\beta_{3} - 284320\beta_{2} + 131435 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 113332 \beta_{13} - 2007 \beta_{12} + 506401 \beta_{10} - 326217 \beta_{9} - 649608 \beta_{7} + \cdots + 1015120 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times\).

\(n\) \(1477\) \(2051\) \(3901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1149.1
3.41812i
2.47795i
2.20421i
1.68828i
0.894124i
0.806545i
0.131983i
− 0.131983i
− 0.806545i
− 0.894124i
− 1.68828i
− 2.20421i
− 2.47795i
− 3.41812i
0 − 3.41812i 0 0 0 − 2.44400i 0 −8.68355 0
1149.2 0 − 2.47795i 0 0 0 0.154217i 0 −3.14025 0
1149.3 0 − 2.20421i 0 0 0 0.688745i 0 −1.85852 0
1149.4 0 − 1.68828i 0 0 0 4.03576i 0 0.149714 0
1149.5 0 − 0.894124i 0 0 0 − 2.24085i 0 2.20054 0
1149.6 0 − 0.806545i 0 0 0 0.0998195i 0 2.34949 0
1149.7 0 − 0.131983i 0 0 0 4.26732i 0 2.98258 0
1149.8 0 0.131983i 0 0 0 − 4.26732i 0 2.98258 0
1149.9 0 0.806545i 0 0 0 − 0.0998195i 0 2.34949 0
1149.10 0 0.894124i 0 0 0 2.24085i 0 2.20054 0
1149.11 0 1.68828i 0 0 0 − 4.03576i 0 0.149714 0
1149.12 0 2.20421i 0 0 0 − 0.688745i 0 −1.85852 0
1149.13 0 2.47795i 0 0 0 − 0.154217i 0 −3.14025 0
1149.14 0 3.41812i 0 0 0 2.44400i 0 −8.68355 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1149.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4100.2.d.g 14
5.b even 2 1 inner 4100.2.d.g 14
5.c odd 4 1 4100.2.a.g ✓ 7
5.c odd 4 1 4100.2.a.j yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4100.2.a.g ✓ 7 5.c odd 4 1
4100.2.a.j yes 7 5.c odd 4 1
4100.2.d.g 14 1.a even 1 1 trivial
4100.2.d.g 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4100, [\chi])\):

\( T_{3}^{14} + 27T_{3}^{12} + 261T_{3}^{10} + 1141T_{3}^{8} + 2289T_{3}^{6} + 1896T_{3}^{4} + 549T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{14} + 46T_{7}^{12} + 729T_{7}^{10} + 4655T_{7}^{8} + 11090T_{7}^{6} + 4590T_{7}^{4} + 145T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 27 T^{12} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 46 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{7} + T^{6} - 42 T^{5} + \cdots + 432)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 87 T^{12} + \cdots + 2614689 \) Copy content Toggle raw display
$17$ \( T^{14} + 133 T^{12} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( (T^{7} - T^{6} - 61 T^{5} + \cdots - 1269)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 154927809 \) Copy content Toggle raw display
$29$ \( (T^{7} - 5 T^{6} + \cdots + 2673)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 4 T^{6} + \cdots - 7857)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 269977761 \) Copy content Toggle raw display
$41$ \( (T - 1)^{14} \) Copy content Toggle raw display
$43$ \( T^{14} + 358 T^{12} + \cdots + 6091024 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 172423161 \) Copy content Toggle raw display
$53$ \( T^{14} + 400 T^{12} + \cdots + 83777409 \) Copy content Toggle raw display
$59$ \( (T^{7} - 16 T^{6} + \cdots + 270621)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 10 T^{6} + \cdots - 5747)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 2251329196249 \) Copy content Toggle raw display
$71$ \( (T^{7} - 27 T^{6} + \cdots - 55479)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 430297952841 \) Copy content Toggle raw display
$79$ \( (T^{7} - T^{6} + \cdots + 2704419)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 88 T^{12} + \cdots + 11664 \) Copy content Toggle raw display
$89$ \( (T^{7} - 16 T^{6} + \cdots - 979827)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 27891338049 \) Copy content Toggle raw display
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