Properties

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

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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.

\(f(q)\) \(=\) \( q -i q^{2} -i q^{3} - q^{4} - q^{6} + 2 i q^{7} + i q^{8} - q^{9} +O(q^{10})\) \( q -i q^{2} -i q^{3} - q^{4} - q^{6} + 2 i q^{7} + i q^{8} - q^{9} + i q^{12} -2 i q^{13} + 2 q^{14} + q^{16} + i q^{18} -2 q^{19} + 2 q^{21} + i q^{23} + q^{24} -2 q^{26} + i q^{27} -2 i q^{28} + 6 q^{29} -4 q^{31} -i q^{32} + q^{36} -10 i q^{37} + 2 i q^{38} -2 q^{39} -6 q^{41} -2 i q^{42} -2 i q^{43} + q^{46} -i q^{48} + 3 q^{49} + 2 i q^{52} -12 i q^{53} + q^{54} -2 q^{56} + 2 i q^{57} -6 i q^{58} -12 q^{59} -10 q^{61} + 4 i q^{62} -2 i q^{63} - q^{64} + 14 i q^{67} + q^{69} -i q^{72} -2 i q^{73} -10 q^{74} + 2 q^{76} + 2 i q^{78} + 10 q^{79} + q^{81} + 6 i q^{82} -2 q^{84} -2 q^{86} -6 i q^{87} -12 q^{89} + 4 q^{91} -i q^{92} + 4 i q^{93} - q^{96} -10 i q^{97} -3 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 4q^{14} + 2q^{16} - 4q^{19} + 4q^{21} + 2q^{24} - 4q^{26} + 12q^{29} - 8q^{31} + 2q^{36} - 4q^{39} - 12q^{41} + 2q^{46} + 6q^{49} + 2q^{54} - 4q^{56} - 24q^{59} - 20q^{61} - 2q^{64} + 2q^{69} - 20q^{74} + 4q^{76} + 20q^{79} + 2q^{81} - 4q^{84} - 4q^{86} - 24q^{89} + 8q^{91} - 2q^{96} + O(q^{100}) \)

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 2.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 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.g 2
5.b even 2 1 inner 3450.2.d.g 2
5.c odd 4 1 138.2.a.b 1
5.c odd 4 1 3450.2.a.o 1
15.e even 4 1 414.2.a.c 1
20.e even 4 1 1104.2.a.b 1
35.f even 4 1 6762.2.a.g 1
40.i odd 4 1 4416.2.a.i 1
40.k even 4 1 4416.2.a.t 1
60.l odd 4 1 3312.2.a.h 1
115.e even 4 1 3174.2.a.d 1
345.l odd 4 1 9522.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.b 1 5.c odd 4 1
414.2.a.c 1 15.e even 4 1
1104.2.a.b 1 20.e even 4 1
3174.2.a.d 1 115.e even 4 1
3312.2.a.h 1 60.l odd 4 1
3450.2.a.o 1 5.c odd 4 1
3450.2.d.g 2 1.a even 1 1 trivial
3450.2.d.g 2 5.b even 2 1 inner
4416.2.a.i 1 40.i odd 4 1
4416.2.a.t 1 40.k even 4 1
6762.2.a.g 1 35.f even 4 1
9522.2.a.k 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 \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 196 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 12 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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