Properties

Label 3450.2.d.h
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: yes
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.

\(f(q)\) \(=\) \( q -i q^{2} -i q^{3} - q^{4} - q^{6} + 5 i q^{7} + i q^{8} - q^{9} +O(q^{10})\) \( q -i q^{2} -i q^{3} - q^{4} - q^{6} + 5 i q^{7} + i q^{8} - q^{9} + i q^{12} -2 i q^{13} + 5 q^{14} + q^{16} + 3 i q^{17} + i q^{18} -2 q^{19} + 5 q^{21} -i q^{23} + q^{24} -2 q^{26} + i q^{27} -5 i q^{28} -3 q^{29} + 2 q^{31} -i q^{32} + 3 q^{34} + q^{36} -7 i q^{37} + 2 i q^{38} -2 q^{39} -5 i q^{42} -2 i q^{43} - q^{46} -3 i q^{47} -i q^{48} -18 q^{49} + 3 q^{51} + 2 i q^{52} + 12 i q^{53} + q^{54} -5 q^{56} + 2 i q^{57} + 3 i q^{58} -6 q^{59} + 2 q^{61} -2 i q^{62} -5 i q^{63} - q^{64} + 2 i q^{67} -3 i q^{68} - q^{69} -15 q^{71} -i q^{72} -11 i q^{73} -7 q^{74} + 2 q^{76} + 2 i q^{78} -8 q^{79} + q^{81} + 9 i q^{83} -5 q^{84} -2 q^{86} + 3 i q^{87} -3 q^{89} + 10 q^{91} + i q^{92} -2 i q^{93} -3 q^{94} - q^{96} -10 i q^{97} + 18 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 10q^{14} + 2q^{16} - 4q^{19} + 10q^{21} + 2q^{24} - 4q^{26} - 6q^{29} + 4q^{31} + 6q^{34} + 2q^{36} - 4q^{39} - 2q^{46} - 36q^{49} + 6q^{51} + 2q^{54} - 10q^{56} - 12q^{59} + 4q^{61} - 2q^{64} - 2q^{69} - 30q^{71} - 14q^{74} + 4q^{76} - 16q^{79} + 2q^{81} - 10q^{84} - 4q^{86} - 6q^{89} + 20q^{91} - 6q^{94} - 2q^{96} + O(q^{100}) \)

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 5.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 5.00000i 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.h 2
5.b even 2 1 inner 3450.2.d.h 2
5.c odd 4 1 3450.2.a.m 1
5.c odd 4 1 3450.2.a.n yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.m 1 5.c odd 4 1
3450.2.a.n yes 1 5.c odd 4 1
3450.2.d.h 2 1.a even 1 1 trivial
3450.2.d.h 2 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} + 25 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( ( -2 + T )^{2} \)
$37$ \( 49 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 4 + T^{2} \)
$47$ \( 9 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( ( 6 + T )^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( ( 15 + T )^{2} \)
$73$ \( 121 + T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( ( 3 + T )^{2} \)
$97$ \( 100 + T^{2} \)
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