Properties

Label 825.4.c.e
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
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 165)
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.

\(f(q)\) \(=\) \( q + i q^{2} -3 i q^{3} + 7 q^{4} + 3 q^{6} + 36 i q^{7} + 15 i q^{8} -9 q^{9} +O(q^{10})\) \( q + i q^{2} -3 i q^{3} + 7 q^{4} + 3 q^{6} + 36 i q^{7} + 15 i q^{8} -9 q^{9} + 11 q^{11} -21 i q^{12} -2 i q^{13} -36 q^{14} + 41 q^{16} + 66 i q^{17} -9 i q^{18} -140 q^{19} + 108 q^{21} + 11 i q^{22} + 68 i q^{23} + 45 q^{24} + 2 q^{26} + 27 i q^{27} + 252 i q^{28} -150 q^{29} -128 q^{31} + 161 i q^{32} -33 i q^{33} -66 q^{34} -63 q^{36} -314 i q^{37} -140 i q^{38} -6 q^{39} -118 q^{41} + 108 i q^{42} -172 i q^{43} + 77 q^{44} -68 q^{46} -324 i q^{47} -123 i q^{48} -953 q^{49} + 198 q^{51} -14 i q^{52} -82 i q^{53} -27 q^{54} -540 q^{56} + 420 i q^{57} -150 i q^{58} + 740 q^{59} + 122 q^{61} -128 i q^{62} -324 i q^{63} + 167 q^{64} + 33 q^{66} -124 i q^{67} + 462 i q^{68} + 204 q^{69} -988 q^{71} -135 i q^{72} -2 i q^{73} + 314 q^{74} -980 q^{76} + 396 i q^{77} -6 i q^{78} -1100 q^{79} + 81 q^{81} -118 i q^{82} + 868 i q^{83} + 756 q^{84} + 172 q^{86} + 450 i q^{87} + 165 i q^{88} + 470 q^{89} + 72 q^{91} + 476 i q^{92} + 384 i q^{93} + 324 q^{94} + 483 q^{96} + 1186 i q^{97} -953 i q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{4} + 6q^{6} - 18q^{9} + O(q^{10}) \) \( 2q + 14q^{4} + 6q^{6} - 18q^{9} + 22q^{11} - 72q^{14} + 82q^{16} - 280q^{19} + 216q^{21} + 90q^{24} + 4q^{26} - 300q^{29} - 256q^{31} - 132q^{34} - 126q^{36} - 12q^{39} - 236q^{41} + 154q^{44} - 136q^{46} - 1906q^{49} + 396q^{51} - 54q^{54} - 1080q^{56} + 1480q^{59} + 244q^{61} + 334q^{64} + 66q^{66} + 408q^{69} - 1976q^{71} + 628q^{74} - 1960q^{76} - 2200q^{79} + 162q^{81} + 1512q^{84} + 344q^{86} + 940q^{89} + 144q^{91} + 648q^{94} + 966q^{96} - 198q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 3.00000i 7.00000 0 3.00000 36.0000i 15.0000i −9.00000 0
199.2 1.00000i 3.00000i 7.00000 0 3.00000 36.0000i 15.0000i −9.00000 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 825.4.c.e 2
5.b even 2 1 inner 825.4.c.e 2
5.c odd 4 1 165.4.a.b 1
5.c odd 4 1 825.4.a.d 1
15.e even 4 1 495.4.a.b 1
15.e even 4 1 2475.4.a.g 1
55.e even 4 1 1815.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.b 1 5.c odd 4 1
495.4.a.b 1 15.e even 4 1
825.4.a.d 1 5.c odd 4 1
825.4.c.e 2 1.a even 1 1 trivial
825.4.c.e 2 5.b even 2 1 inner
1815.4.a.c 1 55.e even 4 1
2475.4.a.g 1 15.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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7}^{2} + 1296 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1296 + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4356 + T^{2} \)
$19$ \( ( 140 + T )^{2} \)
$23$ \( 4624 + T^{2} \)
$29$ \( ( 150 + T )^{2} \)
$31$ \( ( 128 + T )^{2} \)
$37$ \( 98596 + T^{2} \)
$41$ \( ( 118 + T )^{2} \)
$43$ \( 29584 + T^{2} \)
$47$ \( 104976 + T^{2} \)
$53$ \( 6724 + T^{2} \)
$59$ \( ( -740 + T )^{2} \)
$61$ \( ( -122 + T )^{2} \)
$67$ \( 15376 + T^{2} \)
$71$ \( ( 988 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( 1100 + T )^{2} \)
$83$ \( 753424 + T^{2} \)
$89$ \( ( -470 + T )^{2} \)
$97$ \( 1406596 + T^{2} \)
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