Properties

Label 3328.2.b.j
Level $3328$
Weight $2$
Character orbit 3328.b
Analytic conductor $26.574$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5742137927\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} - 3 i q^{5} - q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 3 i q^{5} - q^{7} + 2 q^{9} - 6 i q^{11} - i q^{13} + 3 q^{15} - 3 q^{17} + 2 i q^{19} - i q^{21} - 4 q^{25} + 5 i q^{27} - 6 i q^{29} + 4 q^{31} + 6 q^{33} + 3 i q^{35} - 7 i q^{37} + q^{39} + i q^{43} - 6 i q^{45} - 3 q^{47} - 6 q^{49} - 3 i q^{51} - 18 q^{55} - 2 q^{57} + 6 i q^{59} - 8 i q^{61} - 2 q^{63} - 3 q^{65} + 14 i q^{67} - 3 q^{71} - 2 q^{73} - 4 i q^{75} + 6 i q^{77} - 8 q^{79} + q^{81} + 12 i q^{83} + 9 i q^{85} + 6 q^{87} + 6 q^{89} + i q^{91} + 4 i q^{93} + 6 q^{95} - 10 q^{97} - 12 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} + 4 q^{9} + 6 q^{15} - 6 q^{17} - 8 q^{25} + 8 q^{31} + 12 q^{33} + 2 q^{39} - 6 q^{47} - 12 q^{49} - 36 q^{55} - 4 q^{57} - 4 q^{63} - 6 q^{65} - 6 q^{71} - 4 q^{73} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 12 q^{89} + 12 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3328\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1665.1
1.00000i
1.00000i
0 1.00000i 0 3.00000i 0 −1.00000 0 2.00000 0
1665.2 0 1.00000i 0 3.00000i 0 −1.00000 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.b.j 2
4.b odd 2 1 3328.2.b.m 2
8.b even 2 1 inner 3328.2.b.j 2
8.d odd 2 1 3328.2.b.m 2
16.e even 4 1 208.2.a.a 1
16.e even 4 1 832.2.a.i 1
16.f odd 4 1 26.2.a.a 1
16.f odd 4 1 832.2.a.d 1
48.i odd 4 1 1872.2.a.q 1
48.i odd 4 1 7488.2.a.h 1
48.k even 4 1 234.2.a.e 1
48.k even 4 1 7488.2.a.g 1
80.j even 4 1 650.2.b.d 2
80.k odd 4 1 650.2.a.j 1
80.q even 4 1 5200.2.a.x 1
80.s even 4 1 650.2.b.d 2
112.j even 4 1 1274.2.a.d 1
112.u odd 12 2 1274.2.f.p 2
112.v even 12 2 1274.2.f.r 2
144.u even 12 2 2106.2.e.b 2
144.v odd 12 2 2106.2.e.ba 2
176.i even 4 1 3146.2.a.n 1
208.l even 4 1 338.2.b.c 2
208.m odd 4 1 2704.2.f.d 2
208.o odd 4 1 338.2.a.f 1
208.p even 4 1 2704.2.a.f 1
208.r odd 4 1 2704.2.f.d 2
208.s even 4 1 338.2.b.c 2
208.bf even 12 2 338.2.e.a 4
208.bg odd 12 2 338.2.c.d 2
208.bi odd 12 2 338.2.c.a 2
208.bk even 12 2 338.2.e.a 4
240.t even 4 1 5850.2.a.p 1
240.z odd 4 1 5850.2.e.a 2
240.bd odd 4 1 5850.2.e.a 2
272.k odd 4 1 7514.2.a.c 1
304.m even 4 1 9386.2.a.j 1
624.s odd 4 1 3042.2.b.a 2
624.v even 4 1 3042.2.a.a 1
624.bo odd 4 1 3042.2.b.a 2
1040.cb odd 4 1 8450.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 16.f odd 4 1
208.2.a.a 1 16.e even 4 1
234.2.a.e 1 48.k even 4 1
338.2.a.f 1 208.o odd 4 1
338.2.b.c 2 208.l even 4 1
338.2.b.c 2 208.s even 4 1
338.2.c.a 2 208.bi odd 12 2
338.2.c.d 2 208.bg odd 12 2
338.2.e.a 4 208.bf even 12 2
338.2.e.a 4 208.bk even 12 2
650.2.a.j 1 80.k odd 4 1
650.2.b.d 2 80.j even 4 1
650.2.b.d 2 80.s even 4 1
832.2.a.d 1 16.f odd 4 1
832.2.a.i 1 16.e even 4 1
1274.2.a.d 1 112.j even 4 1
1274.2.f.p 2 112.u odd 12 2
1274.2.f.r 2 112.v even 12 2
1872.2.a.q 1 48.i odd 4 1
2106.2.e.b 2 144.u even 12 2
2106.2.e.ba 2 144.v odd 12 2
2704.2.a.f 1 208.p even 4 1
2704.2.f.d 2 208.m odd 4 1
2704.2.f.d 2 208.r odd 4 1
3042.2.a.a 1 624.v even 4 1
3042.2.b.a 2 624.s odd 4 1
3042.2.b.a 2 624.bo odd 4 1
3146.2.a.n 1 176.i even 4 1
3328.2.b.j 2 1.a even 1 1 trivial
3328.2.b.j 2 8.b even 2 1 inner
3328.2.b.m 2 4.b odd 2 1
3328.2.b.m 2 8.d odd 2 1
5200.2.a.x 1 80.q even 4 1
5850.2.a.p 1 240.t even 4 1
5850.2.e.a 2 240.z odd 4 1
5850.2.e.a 2 240.bd odd 4 1
7488.2.a.g 1 48.k even 4 1
7488.2.a.h 1 48.i odd 4 1
7514.2.a.c 1 272.k odd 4 1
8450.2.a.c 1 1040.cb odd 4 1
9386.2.a.j 1 304.m even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3328, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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