Properties

Label 825.4.c.c
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 3 i q^{3} -8 q^{4} -12 q^{6} -21 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 4 i q^{2} + 3 i q^{3} -8 q^{4} -12 q^{6} -21 i q^{7} -9 q^{9} + 11 q^{11} -24 i q^{12} -68 i q^{13} + 84 q^{14} -64 q^{16} -21 i q^{17} -36 i q^{18} -125 q^{19} + 63 q^{21} + 44 i q^{22} + 137 i q^{23} + 272 q^{26} -27 i q^{27} + 168 i q^{28} + 150 q^{29} + 292 q^{31} -256 i q^{32} + 33 i q^{33} + 84 q^{34} + 72 q^{36} + 349 i q^{37} -500 i q^{38} + 204 q^{39} + 497 q^{41} + 252 i q^{42} -208 i q^{43} -88 q^{44} -548 q^{46} + 369 i q^{47} -192 i q^{48} -98 q^{49} + 63 q^{51} + 544 i q^{52} + 542 i q^{53} + 108 q^{54} -375 i q^{57} + 600 i q^{58} -235 q^{59} + 482 q^{61} + 1168 i q^{62} + 189 i q^{63} + 512 q^{64} -132 q^{66} + 734 i q^{67} + 168 i q^{68} -411 q^{69} + 587 q^{71} -518 i q^{73} -1396 q^{74} + 1000 q^{76} -231 i q^{77} + 816 i q^{78} + 1045 q^{79} + 81 q^{81} + 1988 i q^{82} -608 i q^{83} -504 q^{84} + 832 q^{86} + 450 i q^{87} + 770 q^{89} -1428 q^{91} -1096 i q^{92} + 876 i q^{93} -1476 q^{94} + 768 q^{96} -1541 i q^{97} -392 i q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{4} - 24q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 16q^{4} - 24q^{6} - 18q^{9} + 22q^{11} + 168q^{14} - 128q^{16} - 250q^{19} + 126q^{21} + 544q^{26} + 300q^{29} + 584q^{31} + 168q^{34} + 144q^{36} + 408q^{39} + 994q^{41} - 176q^{44} - 1096q^{46} - 196q^{49} + 126q^{51} + 216q^{54} - 470q^{59} + 964q^{61} + 1024q^{64} - 264q^{66} - 822q^{69} + 1174q^{71} - 2792q^{74} + 2000q^{76} + 2090q^{79} + 162q^{81} - 1008q^{84} + 1664q^{86} + 1540q^{89} - 2856q^{91} - 2952q^{94} + 1536q^{96} - 198q^{99} + O(q^{100}) \)

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
4.00000i 3.00000i −8.00000 0 −12.0000 21.0000i 0 −9.00000 0
199.2 4.00000i 3.00000i −8.00000 0 −12.0000 21.0000i 0 −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.c 2
5.b even 2 1 inner 825.4.c.c 2
5.c odd 4 1 825.4.a.b 1
5.c odd 4 1 825.4.a.h yes 1
15.e even 4 1 2475.4.a.c 1
15.e even 4 1 2475.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.b 1 5.c odd 4 1
825.4.a.h yes 1 5.c odd 4 1
825.4.c.c 2 1.a even 1 1 trivial
825.4.c.c 2 5.b even 2 1 inner
2475.4.a.c 1 15.e even 4 1
2475.4.a.j 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 16 \)
\( T_{7}^{2} + 441 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 441 + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 4624 + T^{2} \)
$17$ \( 441 + T^{2} \)
$19$ \( ( 125 + T )^{2} \)
$23$ \( 18769 + T^{2} \)
$29$ \( ( -150 + T )^{2} \)
$31$ \( ( -292 + T )^{2} \)
$37$ \( 121801 + T^{2} \)
$41$ \( ( -497 + T )^{2} \)
$43$ \( 43264 + T^{2} \)
$47$ \( 136161 + T^{2} \)
$53$ \( 293764 + T^{2} \)
$59$ \( ( 235 + T )^{2} \)
$61$ \( ( -482 + T )^{2} \)
$67$ \( 538756 + T^{2} \)
$71$ \( ( -587 + T )^{2} \)
$73$ \( 268324 + T^{2} \)
$79$ \( ( -1045 + T )^{2} \)
$83$ \( 369664 + T^{2} \)
$89$ \( ( -770 + T )^{2} \)
$97$ \( 2374681 + T^{2} \)
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