Properties

Label 18.7.b.a
Level $18$
Weight $7$
Character orbit 18.b
Analytic conductor $4.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 18.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.14097350516\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 \beta q^{2} - 32 q^{4} + 123 \beta q^{5} - 484 q^{7} - 128 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta q^{2} - 32 q^{4} + 123 \beta q^{5} - 484 q^{7} - 128 \beta q^{8} - 984 q^{10} + 948 \beta q^{11} + 3368 q^{13} - 1936 \beta q^{14} + 1024 q^{16} + 9 \beta q^{17} + 5744 q^{19} - 3936 \beta q^{20} - 7584 q^{22} - 2388 \beta q^{23} - 14633 q^{25} + 13472 \beta q^{26} + 15488 q^{28} + 20757 \beta q^{29} - 39796 q^{31} + 4096 \beta q^{32} - 72 q^{34} - 59532 \beta q^{35} + 52526 q^{37} + 22976 \beta q^{38} + 31488 q^{40} - 26193 \beta q^{41} + 3800 q^{43} - 30336 \beta q^{44} + 19104 q^{46} + 54300 \beta q^{47} + 116607 q^{49} - 58532 \beta q^{50} - 107776 q^{52} + 168813 \beta q^{53} - 233208 q^{55} + 61952 \beta q^{56} - 166056 q^{58} - 176664 \beta q^{59} + 13250 q^{61} - 159184 \beta q^{62} - 32768 q^{64} + 414264 \beta q^{65} + 168968 q^{67} - 288 \beta q^{68} + 476256 q^{70} - 375804 \beta q^{71} + 236144 q^{73} + 210104 \beta q^{74} - 183808 q^{76} - 458832 \beta q^{77} - 35116 q^{79} + 125952 \beta q^{80} + 209544 q^{82} - 7764 \beta q^{83} - 2214 q^{85} + 15200 \beta q^{86} + 242688 q^{88} - 91449 \beta q^{89} - 1630112 q^{91} + 76416 \beta q^{92} - 434400 q^{94} + 706512 \beta q^{95} - 321424 q^{97} + 466428 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 968 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43} + 38208 q^{46} + 233214 q^{49} - 215552 q^{52} - 466416 q^{55} - 332112 q^{58} + 26500 q^{61} - 65536 q^{64} + 337936 q^{67} + 952512 q^{70} + 472288 q^{73} - 367616 q^{76} - 70232 q^{79} + 419088 q^{82} - 4428 q^{85} + 485376 q^{88} - 3260224 q^{91} - 868800 q^{94} - 642848 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
5.65685i 0 −32.0000 173.948i 0 −484.000 181.019i 0 −984.000
17.2 5.65685i 0 −32.0000 173.948i 0 −484.000 181.019i 0 −984.000
\(n\): 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 18.7.b.a 2
3.b odd 2 1 inner 18.7.b.a 2
4.b odd 2 1 144.7.e.d 2
5.b even 2 1 450.7.d.a 2
5.c odd 4 2 450.7.b.a 4
8.b even 2 1 576.7.e.b 2
8.d odd 2 1 576.7.e.k 2
9.c even 3 2 162.7.d.d 4
9.d odd 6 2 162.7.d.d 4
12.b even 2 1 144.7.e.d 2
15.d odd 2 1 450.7.d.a 2
15.e even 4 2 450.7.b.a 4
24.f even 2 1 576.7.e.k 2
24.h odd 2 1 576.7.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.b.a 2 1.a even 1 1 trivial
18.7.b.a 2 3.b odd 2 1 inner
144.7.e.d 2 4.b odd 2 1
144.7.e.d 2 12.b even 2 1
162.7.d.d 4 9.c even 3 2
162.7.d.d 4 9.d odd 6 2
450.7.b.a 4 5.c odd 4 2
450.7.b.a 4 15.e even 4 2
450.7.d.a 2 5.b even 2 1
450.7.d.a 2 15.d odd 2 1
576.7.e.b 2 8.b even 2 1
576.7.e.b 2 24.h odd 2 1
576.7.e.k 2 8.d odd 2 1
576.7.e.k 2 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 32 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 30258 \) Copy content Toggle raw display
$7$ \( (T + 484)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 1797408 \) Copy content Toggle raw display
$13$ \( (T - 3368)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 162 \) Copy content Toggle raw display
$19$ \( (T - 5744)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11405088 \) Copy content Toggle raw display
$29$ \( T^{2} + 861706098 \) Copy content Toggle raw display
$31$ \( (T + 39796)^{2} \) Copy content Toggle raw display
$37$ \( (T - 52526)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1372146498 \) Copy content Toggle raw display
$43$ \( (T - 3800)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5896980000 \) Copy content Toggle raw display
$53$ \( T^{2} + 56995657938 \) Copy content Toggle raw display
$59$ \( T^{2} + 62420337792 \) Copy content Toggle raw display
$61$ \( (T - 13250)^{2} \) Copy content Toggle raw display
$67$ \( (T - 168968)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 282457292832 \) Copy content Toggle raw display
$73$ \( (T - 236144)^{2} \) Copy content Toggle raw display
$79$ \( (T + 35116)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 120559392 \) Copy content Toggle raw display
$89$ \( T^{2} + 16725839202 \) Copy content Toggle raw display
$97$ \( (T + 321424)^{2} \) Copy content Toggle raw display
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