Properties

Label 825.4.c.d
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} + 7 i q^{7} + 21 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{2} -3 i q^{3} - q^{4} + 9 q^{6} + 7 i q^{7} + 21 i q^{8} -9 q^{9} + 11 q^{11} + 3 i q^{12} -16 i q^{13} -21 q^{14} -71 q^{16} -21 i q^{17} -27 i q^{18} -125 q^{19} + 21 q^{21} + 33 i q^{22} -81 i q^{23} + 63 q^{24} + 48 q^{26} + 27 i q^{27} -7 i q^{28} -186 q^{29} -58 q^{31} -45 i q^{32} -33 i q^{33} + 63 q^{34} + 9 q^{36} + 253 i q^{37} -375 i q^{38} -48 q^{39} + 63 q^{41} + 63 i q^{42} -100 i q^{43} -11 q^{44} + 243 q^{46} + 219 i q^{47} + 213 i q^{48} + 294 q^{49} -63 q^{51} + 16 i q^{52} -192 i q^{53} -81 q^{54} -147 q^{56} + 375 i q^{57} -558 i q^{58} -249 q^{59} -64 q^{61} -174 i q^{62} -63 i q^{63} -433 q^{64} + 99 q^{66} -272 i q^{67} + 21 i q^{68} -243 q^{69} -645 q^{71} -189 i q^{72} -112 i q^{73} -759 q^{74} + 125 q^{76} + 77 i q^{77} -144 i q^{78} -509 q^{79} + 81 q^{81} + 189 i q^{82} -1254 i q^{83} -21 q^{84} + 300 q^{86} + 558 i q^{87} + 231 i q^{88} -756 q^{89} + 112 q^{91} + 81 i q^{92} + 174 i q^{93} -657 q^{94} -135 q^{96} -839 i q^{97} + 882 i q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 18q^{6} - 18q^{9} + 22q^{11} - 42q^{14} - 142q^{16} - 250q^{19} + 42q^{21} + 126q^{24} + 96q^{26} - 372q^{29} - 116q^{31} + 126q^{34} + 18q^{36} - 96q^{39} + 126q^{41} - 22q^{44} + 486q^{46} + 588q^{49} - 126q^{51} - 162q^{54} - 294q^{56} - 498q^{59} - 128q^{61} - 866q^{64} + 198q^{66} - 486q^{69} - 1290q^{71} - 1518q^{74} + 250q^{76} - 1018q^{79} + 162q^{81} - 42q^{84} + 600q^{86} - 1512q^{89} + 224q^{91} - 1314q^{94} - 270q^{96} - 198q^{99} + O(q^{100}) \)

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
3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −9.00000 0
199.2 3.00000i 3.00000i −1.00000 0 9.00000 7.00000i 21.0000i −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.d 2
5.b even 2 1 inner 825.4.c.d 2
5.c odd 4 1 825.4.a.c 1
5.c odd 4 1 825.4.a.g yes 1
15.e even 4 1 2475.4.a.d 1
15.e even 4 1 2475.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.c 1 5.c odd 4 1
825.4.a.g yes 1 5.c odd 4 1
825.4.c.d 2 1.a even 1 1 trivial
825.4.c.d 2 5.b even 2 1 inner
2475.4.a.d 1 15.e even 4 1
2475.4.a.i 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 9 \)
\( T_{7}^{2} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 256 + T^{2} \)
$17$ \( 441 + T^{2} \)
$19$ \( ( 125 + T )^{2} \)
$23$ \( 6561 + T^{2} \)
$29$ \( ( 186 + T )^{2} \)
$31$ \( ( 58 + T )^{2} \)
$37$ \( 64009 + T^{2} \)
$41$ \( ( -63 + T )^{2} \)
$43$ \( 10000 + T^{2} \)
$47$ \( 47961 + T^{2} \)
$53$ \( 36864 + T^{2} \)
$59$ \( ( 249 + T )^{2} \)
$61$ \( ( 64 + T )^{2} \)
$67$ \( 73984 + T^{2} \)
$71$ \( ( 645 + T )^{2} \)
$73$ \( 12544 + T^{2} \)
$79$ \( ( 509 + T )^{2} \)
$83$ \( 1572516 + T^{2} \)
$89$ \( ( 756 + T )^{2} \)
$97$ \( 703921 + T^{2} \)
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