Properties

Label 3450.2.d.f
Level $3450$
Weight $2$
Character orbit 3450.d
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} -i q^{8} - q^{9} -i q^{12} + 2 i q^{13} + q^{16} + 2 i q^{17} -i q^{18} + 8 q^{19} + i q^{23} + q^{24} -2 q^{26} -i q^{27} + 2 q^{29} -8 q^{31} + i q^{32} -2 q^{34} + q^{36} + 2 i q^{37} + 8 i q^{38} -2 q^{39} + 10 q^{41} -8 i q^{43} - q^{46} + 8 i q^{47} + i q^{48} + 7 q^{49} -2 q^{51} -2 i q^{52} -2 i q^{53} + q^{54} + 8 i q^{57} + 2 i q^{58} + 4 q^{59} + 2 q^{61} -8 i q^{62} - q^{64} + 8 i q^{67} -2 i q^{68} - q^{69} + i q^{72} + 6 i q^{73} -2 q^{74} -8 q^{76} -2 i q^{78} -8 q^{79} + q^{81} + 10 i q^{82} + 16 i q^{83} + 8 q^{86} + 2 i q^{87} -18 q^{89} -i q^{92} -8 i q^{93} -8 q^{94} - q^{96} + 10 i q^{97} + 7 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 2q^{16} + 16q^{19} + 2q^{24} - 4q^{26} + 4q^{29} - 16q^{31} - 4q^{34} + 2q^{36} - 4q^{39} + 20q^{41} - 2q^{46} + 14q^{49} - 4q^{51} + 2q^{54} + 8q^{59} + 4q^{61} - 2q^{64} - 2q^{69} - 4q^{74} - 16q^{76} - 16q^{79} + 2q^{81} + 16q^{86} - 36q^{89} - 16q^{94} - 2q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2899.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i −1.00000 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 3450.2.d.f 2
5.b even 2 1 inner 3450.2.d.f 2
5.c odd 4 1 138.2.a.c 1
5.c odd 4 1 3450.2.a.k 1
15.e even 4 1 414.2.a.a 1
20.e even 4 1 1104.2.a.g 1
35.f even 4 1 6762.2.a.bg 1
40.i odd 4 1 4416.2.a.s 1
40.k even 4 1 4416.2.a.c 1
60.l odd 4 1 3312.2.a.d 1
115.e even 4 1 3174.2.a.e 1
345.l odd 4 1 9522.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.c 1 5.c odd 4 1
414.2.a.a 1 15.e even 4 1
1104.2.a.g 1 20.e even 4 1
3174.2.a.e 1 115.e even 4 1
3312.2.a.d 1 60.l odd 4 1
3450.2.a.k 1 5.c odd 4 1
3450.2.d.f 2 1.a even 1 1 trivial
3450.2.d.f 2 5.b even 2 1 inner
4416.2.a.c 1 40.k even 4 1
4416.2.a.s 1 40.i odd 4 1
6762.2.a.bg 1 35.f even 4 1
9522.2.a.d 1 345.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7} \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( 64 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 256 + T^{2} \)
$89$ \( ( 18 + T )^{2} \)
$97$ \( 100 + T^{2} \)
show more
show less