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Properties

Label 825.2.n.n
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(301,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.n (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 9 x^{14} - 15 x^{13} + 44 x^{12} - 61 x^{11} + 208 x^{10} - 281 x^{9} + 851 x^{8} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{4} - \beta_{2} + \beta_1) q^{2} + \beta_{11} q^{3} + (\beta_{13} - \beta_{8}) q^{4} + \beta_{6} q^{6} + (\beta_{12} - \beta_{8} + \cdots + \beta_{5}) q^{7}+ \cdots + (\beta_{11} + \beta_{9} + \beta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{4} - \beta_{2} + \beta_1) q^{2} + \beta_{11} q^{3} + (\beta_{13} - \beta_{8}) q^{4} + \beta_{6} q^{6} + (\beta_{12} - \beta_{8} + \cdots + \beta_{5}) q^{7}+ \cdots + ( - \beta_{13} + \beta_{12} + \cdots - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 4 q^{3} - 6 q^{4} - 2 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 4 q^{3} - 6 q^{4} - 2 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9} + 3 q^{11} - 14 q^{12} - 11 q^{13} + 4 q^{14} + 16 q^{16} - 17 q^{17} - 3 q^{18} + 8 q^{19} - 16 q^{21} - 23 q^{22} - 14 q^{23} - q^{24} + 33 q^{26} + 4 q^{27} + 7 q^{28} + 4 q^{29} - 5 q^{31} + 24 q^{32} + 12 q^{33} - 16 q^{34} - 6 q^{36} + 2 q^{37} - 3 q^{38} + 11 q^{39} + 2 q^{41} - 4 q^{42} + 22 q^{43} - 40 q^{46} - 20 q^{47} + 9 q^{48} - 26 q^{49} - 18 q^{51} - 43 q^{52} + 40 q^{53} - 2 q^{54} - 8 q^{56} + 2 q^{57} + 26 q^{58} + 6 q^{59} - 2 q^{61} + 26 q^{62} - 4 q^{63} + 4 q^{64} - 7 q^{66} - 12 q^{67} - 56 q^{68} + 4 q^{69} + 11 q^{71} + q^{72} - 21 q^{73} + 66 q^{74} - 10 q^{76} + 23 q^{77} + 2 q^{78} - 7 q^{79} - 4 q^{81} + 20 q^{82} + 13 q^{83} + 18 q^{84} + 37 q^{86} + 56 q^{87} + 26 q^{88} + 42 q^{89} + 2 q^{91} + 92 q^{92} + 5 q^{93} + 88 q^{94} + 11 q^{96} - 27 q^{97} - 54 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 9 x^{14} - 15 x^{13} + 44 x^{12} - 61 x^{11} + 208 x^{10} - 281 x^{9} + 851 x^{8} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 67\!\cdots\!19 \nu^{15} + \cdots + 82\!\cdots\!50 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39\!\cdots\!75 \nu^{15} + \cdots - 15\!\cdots\!13 ) / 52\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36\!\cdots\!42 \nu^{15} + \cdots + 61\!\cdots\!25 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!96 \nu^{15} + \cdots + 62\!\cdots\!30 ) / 52\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 89\!\cdots\!27 \nu^{15} + \cdots - 35\!\cdots\!50 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!84 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!26 \nu^{15} + \cdots + 77\!\cdots\!85 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!23 \nu^{15} + \cdots + 17\!\cdots\!75 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!81 \nu^{15} + \cdots - 44\!\cdots\!35 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!57 \nu^{15} + \cdots + 38\!\cdots\!80 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 38\!\cdots\!79 \nu^{15} + \cdots + 19\!\cdots\!50 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42\!\cdots\!13 \nu^{15} + \cdots + 22\!\cdots\!05 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 72\!\cdots\!21 \nu^{15} + \cdots - 11\!\cdots\!90 ) / 26\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 73\!\cdots\!70 \nu^{15} + \cdots - 14\!\cdots\!90 ) / 26\!\cdots\!30 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + \beta_{10} - 3\beta_{9} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{12} + \beta_{7} - 4\beta_{6} - \beta_{5} - 4\beta_{4} + 4\beta_{2} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 6\beta_{14} - \beta_{12} - 13\beta_{11} - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{11} - \beta_{8} - 6 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 25 \beta_{13} + 8 \beta_{12} - 2 \beta_{10} + 10 \beta_{9} - 52 \beta_{8} - 8 \beta_{7} + 11 \beta_{6} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{15} + 21 \beta_{14} + 12 \beta_{13} + 42 \beta_{12} + 15 \beta_{11} - 12 \beta_{10} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 65 \beta_{15} + 194 \beta_{14} + 23 \beta_{13} + 65 \beta_{12} + 314 \beta_{11} - 129 \beta_{10} + \cdots - 236 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 246 \beta_{15} + 349 \beta_{14} + 88 \beta_{12} + 231 \beta_{11} - 77 \beta_{10} + 140 \beta_{9} + \cdots + 101 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 323 \beta_{15} + 512 \beta_{14} - 189 \beta_{13} + 549 \beta_{11} + 549 \beta_{8} + 114 \beta_{7} + \cdots + 1114 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 771 \beta_{13} - 626 \beta_{12} + 540 \beta_{10} - 1088 \beta_{9} + 674 \beta_{8} + 626 \beta_{7} + \cdots + 1088 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1989 \beta_{15} - 3353 \beta_{14} - 3801 \beta_{13} - 2846 \beta_{12} - 3663 \beta_{11} + 3801 \beta_{10} + \cdots + 3663 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8636 \beta_{15} - 14015 \beta_{14} - 3617 \beta_{13} - 8636 \beta_{12} - 12434 \beta_{11} + 5379 \beta_{10} + \cdots + 4704 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 18225 \beta_{15} - 39793 \beta_{14} - 12253 \beta_{12} - 51642 \beta_{11} + 9237 \beta_{10} + \cdots - 11347 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 27462 \beta_{15} - 51062 \beta_{14} + 23600 \beta_{13} - 52266 \beta_{11} - 52266 \beta_{8} + \cdots - 31651 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
−0.624800 − 1.92294i
−0.211561 − 0.651117i
0.394217 + 1.21327i
0.633127 + 1.94857i
−1.69439 − 1.23104i
−0.152143 − 0.110539i
1.13858 + 0.827228i
2.01697 + 1.46541i
−1.69439 + 1.23104i
−0.152143 + 0.110539i
1.13858 − 0.827228i
2.01697 − 1.46541i
−0.624800 + 1.92294i
−0.211561 + 0.651117i
0.394217 − 1.21327i
0.633127 − 1.94857i
−1.63575 − 1.18844i −0.309017 + 0.951057i 0.645245 + 1.98586i 0 1.63575 − 1.18844i −0.404805 − 1.24586i 0.0550187 − 0.169330i −0.809017 − 0.587785i 0
301.2 −0.553874 − 0.402413i −0.309017 + 0.951057i −0.473194 − 1.45634i 0 0.553874 − 0.402413i 1.30626 + 4.02026i −0.747082 + 2.29928i −0.809017 − 0.587785i 0
301.3 1.03207 + 0.749844i −0.309017 + 0.951057i −0.115127 − 0.354326i 0 −1.03207 + 0.749844i −0.810118 − 2.49329i 0.935302 − 2.87856i −0.809017 − 0.587785i 0
301.4 1.65755 + 1.20428i −0.309017 + 0.951057i 0.679144 + 2.09019i 0 −1.65755 + 1.20428i 1.14473 + 3.52312i −0.125205 + 0.385340i −0.809017 − 0.587785i 0
526.1 −0.647198 + 1.99187i 0.809017 − 0.587785i −1.93065 − 1.40270i 0 0.647198 + 1.99187i −2.09875 − 1.52483i 0.654736 − 0.475694i 0.309017 − 0.951057i 0
526.2 −0.0581136 + 0.178855i 0.809017 − 0.587785i 1.58942 + 1.15478i 0 0.0581136 + 0.178855i −1.33395 − 0.969174i −0.603192 + 0.438245i 0.309017 − 0.951057i 0
526.3 0.434899 − 1.33848i 0.809017 − 0.587785i 0.0156350 + 0.0113595i 0 −0.434899 − 1.33848i 1.75053 + 1.27183i 2.29917 − 1.67044i 0.309017 − 0.951057i 0
526.4 0.770412 − 2.37108i 0.809017 − 0.587785i −3.41047 − 2.47785i 0 −0.770412 − 2.37108i −1.55389 − 1.12897i −4.46875 + 3.24673i 0.309017 − 0.951057i 0
676.1 −0.647198 − 1.99187i 0.809017 + 0.587785i −1.93065 + 1.40270i 0 0.647198 − 1.99187i −2.09875 + 1.52483i 0.654736 + 0.475694i 0.309017 + 0.951057i 0
676.2 −0.0581136 − 0.178855i 0.809017 + 0.587785i 1.58942 − 1.15478i 0 0.0581136 − 0.178855i −1.33395 + 0.969174i −0.603192 − 0.438245i 0.309017 + 0.951057i 0
676.3 0.434899 + 1.33848i 0.809017 + 0.587785i 0.0156350 − 0.0113595i 0 −0.434899 + 1.33848i 1.75053 − 1.27183i 2.29917 + 1.67044i 0.309017 + 0.951057i 0
676.4 0.770412 + 2.37108i 0.809017 + 0.587785i −3.41047 + 2.47785i 0 −0.770412 + 2.37108i −1.55389 + 1.12897i −4.46875 − 3.24673i 0.309017 + 0.951057i 0
751.1 −1.63575 + 1.18844i −0.309017 − 0.951057i 0.645245 − 1.98586i 0 1.63575 + 1.18844i −0.404805 + 1.24586i 0.0550187 + 0.169330i −0.809017 + 0.587785i 0
751.2 −0.553874 + 0.402413i −0.309017 − 0.951057i −0.473194 + 1.45634i 0 0.553874 + 0.402413i 1.30626 − 4.02026i −0.747082 − 2.29928i −0.809017 + 0.587785i 0
751.3 1.03207 − 0.749844i −0.309017 − 0.951057i −0.115127 + 0.354326i 0 −1.03207 − 0.749844i −0.810118 + 2.49329i 0.935302 + 2.87856i −0.809017 + 0.587785i 0
751.4 1.65755 − 1.20428i −0.309017 − 0.951057i 0.679144 − 2.09019i 0 −1.65755 − 1.20428i 1.14473 − 3.52312i −0.125205 − 0.385340i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.n yes 16
5.b even 2 1 825.2.n.m ✓ 16
5.c odd 4 2 825.2.bx.j 32
11.c even 5 1 inner 825.2.n.n yes 16
11.c even 5 1 9075.2.a.dv 8
11.d odd 10 1 9075.2.a.dt 8
55.h odd 10 1 9075.2.a.dw 8
55.j even 10 1 825.2.n.m ✓ 16
55.j even 10 1 9075.2.a.du 8
55.k odd 20 2 825.2.bx.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.m ✓ 16 5.b even 2 1
825.2.n.m ✓ 16 55.j even 10 1
825.2.n.n yes 16 1.a even 1 1 trivial
825.2.n.n yes 16 11.c even 5 1 inner
825.2.bx.j 32 5.c odd 4 2
825.2.bx.j 32 55.k odd 20 2
9075.2.a.dt 8 11.d odd 10 1
9075.2.a.du 8 55.j even 10 1
9075.2.a.dv 8 11.c even 5 1
9075.2.a.dw 8 55.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{16} - 2 T_{2}^{15} + 9 T_{2}^{14} - 10 T_{2}^{13} + 34 T_{2}^{12} - 44 T_{2}^{11} + 143 T_{2}^{10} + \cdots + 25 \) Copy content Toggle raw display
\( T_{13}^{16} + 11 T_{13}^{15} + 116 T_{13}^{14} + 798 T_{13}^{13} + 5523 T_{13}^{12} + 28322 T_{13}^{11} + \cdots + 3933296656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{15} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 913936 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 3933296656 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 983700496 \) Copy content Toggle raw display
$19$ \( T^{16} - 8 T^{15} + \cdots + 15840400 \) Copy content Toggle raw display
$23$ \( (T^{8} + 7 T^{7} + \cdots + 3169)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2484922801 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 941814721 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 294263936521 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 5078356897441 \) Copy content Toggle raw display
$43$ \( (T^{8} - 11 T^{7} + \cdots + 5045)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 21461214390625 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 256128016 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 3563493025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 32428806400 \) Copy content Toggle raw display
$67$ \( (T^{8} + 6 T^{7} + \cdots + 45905)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 15356166400 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 989565363361 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 2392184995561 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 248885245456 \) Copy content Toggle raw display
$89$ \( (T^{8} - 21 T^{7} + \cdots - 60824801)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 20\!\cdots\!41 \) Copy content Toggle raw display
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