Properties

Label 9.9.b.a
Level $9$
Weight $9$
Character orbit 9.b
Analytic conductor $3.666$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,9,Mod(8,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.8");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 9.b (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: \(3.66640749055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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 \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8} - 4194 q^{10} - 5036 \beta q^{11} - 26272 q^{13} + 1652 \beta q^{14} + 52036 q^{16} + 3579 \beta q^{17} + 46640 q^{19} + 55454 \beta q^{20} + 90648 q^{22} - 77332 \beta q^{23} - 586577 q^{25} - 26272 \beta q^{26} + 393176 q^{28} - 144953 \beta q^{29} - 196444 q^{31} + 178500 \beta q^{32} - 64422 q^{34} + 384916 \beta q^{35} + 2819414 q^{37} + 46640 \beta q^{38} - 2071836 q^{40} - 166507 \beta q^{41} - 2213464 q^{43} - 1198568 \beta q^{44} + 1391976 q^{46} + 384892 \beta q^{47} - 3035697 q^{49} - 586577 \beta q^{50} - 6252736 q^{52} + 1222983 \beta q^{53} + 21120984 q^{55} + 816088 \beta q^{56} + 2609154 q^{58} + 2768264 \beta q^{59} - 17405302 q^{61} - 196444 \beta q^{62} + 10108216 q^{64} - 6121376 \beta q^{65} - 14322664 q^{67} + 851802 \beta q^{68} - 6928488 q^{70} - 3687708 \beta q^{71} - 8906992 q^{73} + 2819414 \beta q^{74} + 11100320 q^{76} - 8319472 \beta q^{77} + 32758844 q^{79} + 12124388 \beta q^{80} + 2997126 q^{82} + 20055628 \beta q^{83} - 15010326 q^{85} - 2213464 \beta q^{86} + 44780112 q^{88} - 13891371 \beta q^{89} - 43401344 q^{91} - 18405016 \beta q^{92} - 6928056 q^{94} + 10867120 \beta q^{95} - 24451744 q^{97} - 3035697 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 476 q^{4} + 3304 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 476 q^{4} + 3304 q^{7} - 8388 q^{10} - 52544 q^{13} + 104072 q^{16} + 93280 q^{19} + 181296 q^{22} - 1173154 q^{25} + 786352 q^{28} - 392888 q^{31} - 128844 q^{34} + 5638828 q^{37} - 4143672 q^{40} - 4426928 q^{43} + 2783952 q^{46} - 6071394 q^{49} - 12505472 q^{52} + 42241968 q^{55} + 5218308 q^{58} - 34810604 q^{61} + 20216432 q^{64} - 28645328 q^{67} - 13856976 q^{70} - 17813984 q^{73} + 22200640 q^{76} + 65517688 q^{79} + 5994252 q^{82} - 30020652 q^{85} + 89560224 q^{88} - 86802688 q^{91} - 13856112 q^{94} - 48903488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.41421i
1.41421i
4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
8.2 4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.9.b.a 2
3.b odd 2 1 inner 9.9.b.a 2
4.b odd 2 1 144.9.e.a 2
5.b even 2 1 225.9.c.a 2
5.c odd 4 2 225.9.d.a 4
9.c even 3 2 81.9.d.b 4
9.d odd 6 2 81.9.d.b 4
12.b even 2 1 144.9.e.a 2
15.d odd 2 1 225.9.c.a 2
15.e even 4 2 225.9.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.b.a 2 1.a even 1 1 trivial
9.9.b.a 2 3.b odd 2 1 inner
81.9.d.b 4 9.c even 3 2
81.9.d.b 4 9.d odd 6 2
144.9.e.a 2 4.b odd 2 1
144.9.e.a 2 12.b even 2 1
225.9.c.a 2 5.b even 2 1
225.9.c.a 2 15.d odd 2 1
225.9.d.a 4 5.c odd 4 2
225.9.d.a 4 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 18 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 977202 \) Copy content Toggle raw display
$7$ \( (T - 1652)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 456503328 \) Copy content Toggle raw display
$13$ \( (T + 26272)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 230566338 \) Copy content Toggle raw display
$19$ \( (T - 46640)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 107644288032 \) Copy content Toggle raw display
$29$ \( T^{2} + 378204699762 \) Copy content Toggle raw display
$31$ \( (T + 196444)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2819414)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 499042458882 \) Copy content Toggle raw display
$43$ \( (T + 2213464)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2666553329952 \) Copy content Toggle raw display
$53$ \( T^{2} + 26922373529202 \) Copy content Toggle raw display
$59$ \( T^{2} + 137939140326528 \) Copy content Toggle raw display
$61$ \( (T + 17405302)^{2} \) Copy content Toggle raw display
$67$ \( (T + 14322664)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 244785425278752 \) Copy content Toggle raw display
$73$ \( (T + 8906992)^{2} \) Copy content Toggle raw display
$79$ \( (T - 32758844)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 72\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + 34\!\cdots\!38 \) Copy content Toggle raw display
$97$ \( (T + 24451744)^{2} \) Copy content Toggle raw display
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