Properties

Label 68.2.e.a
Level $68$
Weight $2$
Character orbit 68.e
Analytic conductor $0.543$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.542982733745\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} + 4 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + ( - \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{3} - \beta_{2} + 4 \beta_1) q^{9} + ( - \beta_{3} - \beta_1 + 1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{15} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{17} + (\beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + ( - \beta_{3} + \beta_{2} + \beta_1 - 8) q^{21} + ( - \beta_{3} + 3 \beta_1 - 3) q^{23} - 3 \beta_1 q^{25} + (2 \beta_{3} - 8 \beta_1 + 8) q^{27} + (2 \beta_{2} + 3 \beta_1 + 1) q^{29} + ( - \beta_{2} + 2 \beta_1 + 3) q^{31} + ( - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{33} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{35} + (3 \beta_1 + 3) q^{37} + ( - 6 \beta_1 - 6) q^{39} + ( - \beta_1 + 1) q^{41} + (\beta_{3} + \beta_{2} + 7 \beta_1) q^{43} + (2 \beta_{3} - 5 \beta_1 + 5) q^{45} - 4 q^{47} + (\beta_{3} + \beta_{2}) q^{49} + ( - 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 9) q^{51} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{53} + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{55} + (2 \beta_{3} + 4 \beta_1 - 4) q^{57} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{59} + ( - 4 \beta_{3} + \beta_1 - 1) q^{61} + ( - 5 \beta_{2} + 6 \beta_1 + 11) q^{63} + (2 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{67} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 12) q^{69} + ( - 3 \beta_{2} - 4 \beta_1 - 1) q^{71} + ( - 7 \beta_1 - 7) q^{73} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{75} + (\beta_{3} + \beta_{2} + 5 \beta_1) q^{77} + ( - \beta_{3} - \beta_1 + 1) q^{79} + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 13) q^{81} + (3 \beta_{3} + 3 \beta_{2} - 7 \beta_1) q^{83} + (2 \beta_{2} + 3 \beta_1 - 3) q^{85} + (\beta_{3} + \beta_{2} + 11 \beta_1) q^{87} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{89} + ( - 6 \beta_1 + 6) q^{91} + (3 \beta_{3} + 3 \beta_{2} - 9 \beta_1) q^{93} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{95} + (2 \beta_{2} - 3 \beta_1 - 5) q^{97} + (\beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{11} - 4 q^{13} - 28 q^{21} - 10 q^{23} + 28 q^{27} + 8 q^{29} + 10 q^{31} + 20 q^{33} - 4 q^{35} + 12 q^{37} - 24 q^{39} + 4 q^{41} + 16 q^{45} - 16 q^{47} - 30 q^{51} - 12 q^{55} - 20 q^{57} + 4 q^{61} + 34 q^{63} + 4 q^{65} + 8 q^{67} + 36 q^{69} - 10 q^{71} - 28 q^{73} - 6 q^{75} + 6 q^{79} - 32 q^{81} - 8 q^{85} - 12 q^{89} + 24 q^{91} + 12 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 3\nu^{2} + 7\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} - 2\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/68\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
2.30278i
1.30278i
2.30278i
1.30278i
0 −2.30278 2.30278i 0 −1.00000 1.00000i 0 2.30278 2.30278i 0 7.60555i 0
13.2 0 1.30278 + 1.30278i 0 −1.00000 1.00000i 0 −1.30278 + 1.30278i 0 0.394449i 0
21.1 0 −2.30278 + 2.30278i 0 −1.00000 + 1.00000i 0 2.30278 + 2.30278i 0 7.60555i 0
21.2 0 1.30278 1.30278i 0 −1.00000 + 1.00000i 0 −1.30278 1.30278i 0 0.394449i 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
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.2.e.a 4
3.b odd 2 1 612.2.k.e 4
4.b odd 2 1 272.2.o.g 4
5.b even 2 1 1700.2.o.c 4
5.c odd 4 1 1700.2.m.a 4
5.c odd 4 1 1700.2.m.b 4
8.b even 2 1 1088.2.o.t 4
8.d odd 2 1 1088.2.o.s 4
12.b even 2 1 2448.2.be.u 4
17.b even 2 1 1156.2.e.c 4
17.c even 4 1 inner 68.2.e.a 4
17.c even 4 1 1156.2.e.c 4
17.d even 8 2 1156.2.a.h 4
17.d even 8 2 1156.2.b.a 4
17.e odd 16 8 1156.2.h.e 16
51.f odd 4 1 612.2.k.e 4
68.f odd 4 1 272.2.o.g 4
68.g odd 8 2 4624.2.a.bq 4
85.f odd 4 1 1700.2.m.b 4
85.i odd 4 1 1700.2.m.a 4
85.j even 4 1 1700.2.o.c 4
136.i even 4 1 1088.2.o.t 4
136.j odd 4 1 1088.2.o.s 4
204.l even 4 1 2448.2.be.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.e.a 4 1.a even 1 1 trivial
68.2.e.a 4 17.c even 4 1 inner
272.2.o.g 4 4.b odd 2 1
272.2.o.g 4 68.f odd 4 1
612.2.k.e 4 3.b odd 2 1
612.2.k.e 4 51.f odd 4 1
1088.2.o.s 4 8.d odd 2 1
1088.2.o.s 4 136.j odd 4 1
1088.2.o.t 4 8.b even 2 1
1088.2.o.t 4 136.i even 4 1
1156.2.a.h 4 17.d even 8 2
1156.2.b.a 4 17.d even 8 2
1156.2.e.c 4 17.b even 2 1
1156.2.e.c 4 17.c even 4 1
1156.2.h.e 16 17.e odd 16 8
1700.2.m.a 4 5.c odd 4 1
1700.2.m.a 4 85.i odd 4 1
1700.2.m.b 4 5.c odd 4 1
1700.2.m.b 4 85.f odd 4 1
1700.2.o.c 4 5.b even 2 1
1700.2.o.c 4 85.j even 4 1
2448.2.be.u 4 12.b even 2 1
2448.2.be.u 4 204.l even 4 1
4624.2.a.bq 4 68.g odd 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(68, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} - 12 T + 36 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 2 T^{2} + 12 T + 36 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 18 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 18T^{2} + 289 \) Copy content Toggle raw display
$19$ \( T^{4} + 44T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + 50 T^{2} + 60 T + 36 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} + 32 T^{2} + 144 T + 324 \) Copy content Toggle raw display
$31$ \( T^{4} - 10 T^{3} + 50 T^{2} - 60 T + 36 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 124T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( (T + 4)^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$59$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 10404 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 48)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + 18 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$83$ \( T^{4} + 332T^{2} + 4624 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 108)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 36 \) Copy content Toggle raw display
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