Properties

Label 3800.2.d.f
Level $3800$
Weight $2$
Character orbit 3800.d
Analytic conductor $30.343$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.3431527681\)
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 152)
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.

\(f(q)\) \(=\) \( q + i q^{3} - 3 i q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 3 i q^{7} + 2 q^{9} + 2 q^{11} + i q^{13} + 5 i q^{17} - q^{19} + 3 q^{21} - i q^{23} + 5 i q^{27} + 3 q^{29} + 4 q^{31} + 2 i q^{33} - 2 i q^{37} - q^{39} - 8 q^{41} - 8 i q^{43} + 8 i q^{47} - 2 q^{49} - 5 q^{51} + 9 i q^{53} - i q^{57} - q^{59} + 14 q^{61} - 6 i q^{63} - 13 i q^{67} + q^{69} + 10 q^{71} + 9 i q^{73} - 6 i q^{77} + 10 q^{79} + q^{81} + 10 i q^{83} + 3 i q^{87} + 12 q^{89} + 3 q^{91} + 4 i q^{93} - 14 i q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} + 4 q^{11} - 2 q^{19} + 6 q^{21} + 6 q^{29} + 8 q^{31} - 2 q^{39} - 16 q^{41} - 4 q^{49} - 10 q^{51} - 2 q^{59} + 28 q^{61} + 2 q^{69} + 20 q^{71} + 20 q^{79} + 2 q^{81} + 24 q^{89} + 6 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
1.00000i
1.00000i
0 1.00000i 0 0 0 3.00000i 0 2.00000 0
3649.2 0 1.00000i 0 0 0 3.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.2.d.f 2
5.b even 2 1 inner 3800.2.d.f 2
5.c odd 4 1 152.2.a.b 1
5.c odd 4 1 3800.2.a.d 1
15.e even 4 1 1368.2.a.g 1
20.e even 4 1 304.2.a.b 1
20.e even 4 1 7600.2.a.o 1
35.f even 4 1 7448.2.a.g 1
40.i odd 4 1 1216.2.a.f 1
40.k even 4 1 1216.2.a.l 1
60.l odd 4 1 2736.2.a.k 1
95.g even 4 1 2888.2.a.b 1
380.j odd 4 1 5776.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.a.b 1 5.c odd 4 1
304.2.a.b 1 20.e even 4 1
1216.2.a.f 1 40.i odd 4 1
1216.2.a.l 1 40.k even 4 1
1368.2.a.g 1 15.e even 4 1
2736.2.a.k 1 60.l odd 4 1
2888.2.a.b 1 95.g even 4 1
3800.2.a.d 1 5.c odd 4 1
3800.2.d.f 2 1.a even 1 1 trivial
3800.2.d.f 2 5.b even 2 1 inner
5776.2.a.l 1 380.j odd 4 1
7448.2.a.g 1 35.f even 4 1
7600.2.a.o 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11} - 2 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 25 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 81 \) Copy content Toggle raw display
$59$ \( (T + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 169 \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 81 \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 100 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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