Properties

Label 207.2.c.a
Level $207$
Weight $2$
Character orbit 207.c
Analytic conductor $1.653$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,2,Mod(206,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.206");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 207.c (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: \(1.65290332184\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3057647616.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{5} + 10x^{4} + 12x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{5} q^{4} - \beta_{7} q^{5} + \beta_{6} q^{7} + \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{5} q^{4} - \beta_{7} q^{5} + \beta_{6} q^{7} + \beta_{4} q^{8} + ( - \beta_{6} - \beta_{3}) q^{10} - \beta_{2} q^{11} + ( - \beta_{5} - 1) q^{13} - \beta_{7} q^{14} + ( - 2 \beta_{5} - 1) q^{16} + (\beta_{7} + \beta_{2}) q^{17} + ( - \beta_{6} + \beta_{3}) q^{19} + (\beta_{7} + \beta_{2}) q^{20} - \beta_{3} q^{22} + (\beta_{7} + 2 \beta_{4} + \beta_1) q^{23} + (3 \beta_{5} + 4) q^{25} + (\beta_{4} - 3 \beta_1) q^{26} + (\beta_{6} - \beta_{3}) q^{28} + ( - 3 \beta_{4} + \beta_1) q^{29} + (2 \beta_{5} - 4) q^{31} + (4 \beta_{4} - 5 \beta_1) q^{32} + (\beta_{6} + 2 \beta_{3}) q^{34} + ( - 3 \beta_{4} - 3 \beta_1) q^{35} + ( - \beta_{7} - \beta_{2}) q^{38} - \beta_{6} q^{40} + ( - 3 \beta_{4} + 7 \beta_1) q^{41} + ( - \beta_{6} + 2 \beta_{3}) q^{43} + (2 \beta_{7} - \beta_{2}) q^{44} + (\beta_{6} - \beta_{5} + \beta_{3} - 4) q^{46} + ( - 3 \beta_{4} - 5 \beta_1) q^{47} + (3 \beta_{5} - 2) q^{49} + ( - 3 \beta_{4} + 10 \beta_1) q^{50} + (\beta_{5} + 3) q^{52} + ( - \beta_{7} - \beta_{2}) q^{53} + 6 \beta_{5} q^{55} + ( - \beta_{7} + \beta_{2}) q^{56} + ( - \beta_{5} + 1) q^{58} + (\beta_{4} - \beta_1) q^{59} - 2 \beta_{3} q^{61} - 2 \beta_{4} q^{62} + (\beta_{5} + 4) q^{64} + (2 \beta_{7} + \beta_{2}) q^{65} + (\beta_{6} + \beta_{3}) q^{67} - 3 \beta_{7} q^{68} + (3 \beta_{5} + 9) q^{70} + ( - 6 \beta_{4} + 4 \beta_1) q^{71} + ( - 4 \beta_{5} + 2) q^{73} - 3 \beta_{6} q^{76} + (12 \beta_{4} - 6 \beta_1) q^{77} + (3 \beta_{6} + \beta_{3}) q^{79} + (3 \beta_{7} + 2 \beta_{2}) q^{80} + ( - 7 \beta_{5} - 11) q^{82} + (2 \beta_{7} + \beta_{2}) q^{83} + ( - 9 \beta_{5} - 9) q^{85} + ( - 3 \beta_{7} - 2 \beta_{2}) q^{86} + (2 \beta_{6} - \beta_{3}) q^{88} + (\beta_{7} - 2 \beta_{2}) q^{89} - \beta_{3} q^{91} + ( - \beta_{7} + 5 \beta_{4} - \beta_{2} - 4 \beta_1) q^{92} + (5 \beta_{5} + 13) q^{94} + (9 \beta_{4} - 9 \beta_1) q^{95} + ( - 4 \beta_{6} - \beta_{3}) q^{97} + ( - 3 \beta_{4} + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} - 8 q^{16} + 32 q^{25} - 32 q^{31} - 32 q^{46} - 16 q^{49} + 24 q^{52} + 8 q^{58} + 32 q^{64} + 72 q^{70} + 16 q^{73} - 88 q^{82} - 72 q^{85} + 104 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{5} + 10x^{4} + 12x^{2} + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -81\nu^{7} + 725\nu^{6} - 1250\nu^{5} - 2413\nu^{4} + 3778\nu^{3} + 6007\nu^{2} + 2166\nu + 2419 ) / 3445 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 244\nu^{7} - 185\nu^{6} - 3720\nu^{5} + 4802\nu^{4} + 7418\nu^{3} - 1338\nu^{2} + 1386\nu + 14574 ) / 3445 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -24\nu^{7} + 205\nu^{6} - 390\nu^{5} - 342\nu^{4} + 452\nu^{3} + 1878\nu^{2} - 6\nu + 756 ) / 265 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -449\nu^{7} + 2190\nu^{6} - 1400\nu^{5} - 4742\nu^{4} - 68\nu^{3} + 5653\nu^{2} - 5346\nu + 2351 ) / 3445 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -46\nu^{7} + 150\nu^{6} + 180\nu^{5} - 523\nu^{4} - 812\nu^{3} + 22\nu^{2} - 144\nu - 141 ) / 265 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 78\nu^{7} - 335\nu^{6} + 75\nu^{5} + 714\nu^{4} + 386\nu^{3} - 141\nu^{2} + 1212\nu - 72 ) / 265 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1327\nu^{7} + 5030\nu^{6} + 1425\nu^{5} - 11291\nu^{4} - 17894\nu^{3} - 771\nu^{2} - 3048\nu - 732 ) / 3445 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{7} - 3\beta_{5} - 3\beta_{4} - \beta_{2} + 3\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta_{5} - 3\beta_{4} + \beta_{3} - 2\beta_{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 4\beta_{6} - 6\beta_{5} - 10\beta_{4} + \beta_{3} - 3\beta_{2} + 4\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{7} - 20\beta_{6} - 21\beta_{5} - 54\beta_{4} + 8\beta_{3} - 10\beta_{2} + 12\beta _1 + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{7} - 70\beta_{6} - 36\beta_{5} - 186\beta_{4} + 25\beta_{3} - 20\beta_{2} + 51\beta _1 + 69 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{7} - 74\beta_{6} - 15\beta_{5} - 197\beta_{4} + 27\beta_{3} - 7\beta_{2} + 53\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -37\beta_{7} - 665\beta_{6} + 96\beta_{5} - 1773\beta_{4} + 245\beta_{3} + 47\beta_{2} + 471\beta _1 - 156 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
206.1
0.430200 0.707107i
−1.16225 0.707107i
2.90421 + 0.707107i
−0.172164 + 0.707107i
2.90421 0.707107i
−0.172164 0.707107i
0.430200 + 0.707107i
−1.16225 + 0.707107i
1.93185i 0 −1.73205 −3.76778 0 1.95035i 0.517638i 0 7.27879i
206.2 1.93185i 0 −1.73205 3.76778 0 1.95035i 0.517638i 0 7.27879i
206.3 0.517638i 0 1.73205 −1.95035 0 3.76778i 1.93185i 0 1.00957i
206.4 0.517638i 0 1.73205 1.95035 0 3.76778i 1.93185i 0 1.00957i
206.5 0.517638i 0 1.73205 −1.95035 0 3.76778i 1.93185i 0 1.00957i
206.6 0.517638i 0 1.73205 1.95035 0 3.76778i 1.93185i 0 1.00957i
206.7 1.93185i 0 −1.73205 −3.76778 0 1.95035i 0.517638i 0 7.27879i
206.8 1.93185i 0 −1.73205 3.76778 0 1.95035i 0.517638i 0 7.27879i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 206.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
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 207.2.c.a 8
3.b odd 2 1 inner 207.2.c.a 8
4.b odd 2 1 3312.2.m.b 8
12.b even 2 1 3312.2.m.b 8
23.b odd 2 1 inner 207.2.c.a 8
69.c even 2 1 inner 207.2.c.a 8
92.b even 2 1 3312.2.m.b 8
276.h odd 2 1 3312.2.m.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.c.a 8 1.a even 1 1 trivial
207.2.c.a 8 3.b odd 2 1 inner
207.2.c.a 8 23.b odd 2 1 inner
207.2.c.a 8 69.c even 2 1 inner
3312.2.m.b 8 4.b odd 2 1
3312.2.m.b 8 12.b even 2 1
3312.2.m.b 8 92.b even 2 1
3312.2.m.b 8 276.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(207, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 18 T^{2} + 54)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 18 T^{2} + 54)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 36 T^{2} + 216)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 54 T^{2} + 486)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 54 T^{2} + 486)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 20 T^{6} + 726 T^{4} + \cdots + 279841 \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 148 T^{2} + 676)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 162 T^{2} + 6534)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 196 T^{2} + 8836)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 54 T^{2} + 486)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 144 T^{2} + 3456)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 54 T^{2} + 54)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 44)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 198 T^{2} + 9126)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 108 T^{2} + 216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 162 T^{2} + 486)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 324 T^{2} + 26136)^{2} \) Copy content Toggle raw display
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