Properties

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

Related objects

Downloads

Learn more

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.

\(f(q)\) \(=\) \( q + 5 i q^{2} -3 i q^{3} -17 q^{4} + 15 q^{6} + 3 i q^{7} -45 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 5 i q^{2} -3 i q^{3} -17 q^{4} + 15 q^{6} + 3 i q^{7} -45 i q^{8} -9 q^{9} -11 q^{11} + 51 i q^{12} -32 i q^{13} -15 q^{14} + 89 q^{16} + 33 i q^{17} -45 i q^{18} -47 q^{19} + 9 q^{21} -55 i q^{22} -113 i q^{23} -135 q^{24} + 160 q^{26} + 27 i q^{27} -51 i q^{28} + 54 q^{29} + 178 q^{31} + 85 i q^{32} + 33 i q^{33} -165 q^{34} + 153 q^{36} + 19 i q^{37} -235 i q^{38} -96 q^{39} + 139 q^{41} + 45 i q^{42} + 308 i q^{43} + 187 q^{44} + 565 q^{46} + 195 i q^{47} -267 i q^{48} + 334 q^{49} + 99 q^{51} + 544 i q^{52} -152 i q^{53} -135 q^{54} + 135 q^{56} + 141 i q^{57} + 270 i q^{58} + 625 q^{59} + 320 q^{61} + 890 i q^{62} -27 i q^{63} + 287 q^{64} -165 q^{66} + 200 i q^{67} -561 i q^{68} -339 q^{69} -947 q^{71} + 405 i q^{72} + 448 i q^{73} -95 q^{74} + 799 q^{76} -33 i q^{77} -480 i q^{78} + 721 q^{79} + 81 q^{81} + 695 i q^{82} -142 i q^{83} -153 q^{84} -1540 q^{86} -162 i q^{87} + 495 i q^{88} -404 q^{89} + 96 q^{91} + 1921 i q^{92} -534 i q^{93} -975 q^{94} + 255 q^{96} + 79 i q^{97} + 1670 i q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 34q^{4} + 30q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 34q^{4} + 30q^{6} - 18q^{9} - 22q^{11} - 30q^{14} + 178q^{16} - 94q^{19} + 18q^{21} - 270q^{24} + 320q^{26} + 108q^{29} + 356q^{31} - 330q^{34} + 306q^{36} - 192q^{39} + 278q^{41} + 374q^{44} + 1130q^{46} + 668q^{49} + 198q^{51} - 270q^{54} + 270q^{56} + 1250q^{59} + 640q^{61} + 574q^{64} - 330q^{66} - 678q^{69} - 1894q^{71} - 190q^{74} + 1598q^{76} + 1442q^{79} + 162q^{81} - 306q^{84} - 3080q^{86} - 808q^{89} + 192q^{91} - 1950q^{94} + 510q^{96} + 198q^{99} + O(q^{100}) \)

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
5.00000i 3.00000i −17.0000 0 15.0000 3.00000i 45.0000i −9.00000 0
199.2 5.00000i 3.00000i −17.0000 0 15.0000 3.00000i 45.0000i −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.b 2
5.b even 2 1 inner 825.4.c.b 2
5.c odd 4 1 825.4.a.a 1
5.c odd 4 1 825.4.a.j yes 1
15.e even 4 1 2475.4.a.a 1
15.e even 4 1 2475.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.a 1 5.c odd 4 1
825.4.a.j yes 1 5.c odd 4 1
825.4.c.b 2 1.a even 1 1 trivial
825.4.c.b 2 5.b even 2 1 inner
2475.4.a.a 1 15.e even 4 1
2475.4.a.k 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 25 \)
\( T_{7}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 25 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( 11 + T )^{2} \)
$13$ \( 1024 + T^{2} \)
$17$ \( 1089 + T^{2} \)
$19$ \( ( 47 + T )^{2} \)
$23$ \( 12769 + T^{2} \)
$29$ \( ( -54 + T )^{2} \)
$31$ \( ( -178 + T )^{2} \)
$37$ \( 361 + T^{2} \)
$41$ \( ( -139 + T )^{2} \)
$43$ \( 94864 + T^{2} \)
$47$ \( 38025 + T^{2} \)
$53$ \( 23104 + T^{2} \)
$59$ \( ( -625 + T )^{2} \)
$61$ \( ( -320 + T )^{2} \)
$67$ \( 40000 + T^{2} \)
$71$ \( ( 947 + T )^{2} \)
$73$ \( 200704 + T^{2} \)
$79$ \( ( -721 + T )^{2} \)
$83$ \( 20164 + T^{2} \)
$89$ \( ( 404 + T )^{2} \)
$97$ \( 6241 + T^{2} \)
show more
show less