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Properties

Label 825.2.bx.g
Level $825$
Weight $2$
Character orbit 825.bx
Analytic conductor $6.588$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(49,825)]
 
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, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bx (of order \(10\), degree \(4\), not 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 \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11x^{14} + 80x^{12} - 529x^{10} + 3359x^{8} - 10729x^{6} + 15420x^{4} + 1089x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + \beta_{11} q^{3} + (\beta_{8} - \beta_{7} + \cdots + 2 \beta_{2}) q^{4}+ \cdots + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + \beta_{11} q^{3} + (\beta_{8} - \beta_{7} + \cdots + 2 \beta_{2}) q^{4}+ \cdots + ( - \beta_{8} + \beta_{6} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 4 q^{6} + 4 q^{9} - 6 q^{11} - 48 q^{14} + 8 q^{16} + 4 q^{19} + 32 q^{21} + 32 q^{24} + 28 q^{26} - 28 q^{29} - 10 q^{31} + 140 q^{34} - 4 q^{36} + 20 q^{39} + 2 q^{41} - 94 q^{44} + 84 q^{46} + 30 q^{49} - 16 q^{51} + 4 q^{54} - 48 q^{56} - 26 q^{59} - 6 q^{61} - 38 q^{64} + 74 q^{66} + 6 q^{69} + 18 q^{71} + 34 q^{74} - 92 q^{76} + 20 q^{79} - 4 q^{81} + 36 q^{84} + 40 q^{86} - 8 q^{89} - 86 q^{91} - 114 q^{94} + 12 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 80x^{12} - 529x^{10} + 3359x^{8} - 10729x^{6} + 15420x^{4} + 1089x^{2} + 14641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 14121491458 \nu^{14} - 48922516060 \nu^{12} + 141324576560 \nu^{10} + \cdots + 548537214078931 ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19118240560 \nu^{14} - 221686154525 \nu^{12} + 1769258652890 \nu^{10} + \cdots - 235615624474021 ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24420924323 \nu^{14} + 503982751690 \nu^{12} - 4354382702517 \nu^{10} + \cdots - 422815961881852 ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27458425690 \nu^{14} - 351220616736 \nu^{12} + 2497123846655 \nu^{10} + \cdots + 402771359701876 ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 303353 \nu^{14} + 2230690 \nu^{12} - 15867287 \nu^{10} + 114133707 \nu^{8} + \cdots + 3699802843 ) / 4256875381 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5445979874 \nu^{14} - 55652526464 \nu^{12} + 346658148128 \nu^{10} - 2375497505251 \nu^{8} + \cdots + 32711100255251 ) / 66402999068219 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 80056172618 \nu^{14} - 1068158135055 \nu^{12} + 7584297134503 \nu^{10} + \cdots - 10\!\cdots\!97 ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 192081312287 \nu^{15} + 1869181273960 \nu^{13} - 14608340767679 \nu^{11} + \cdots + 71\!\cdots\!36 \nu ) / 80\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19118240560 \nu^{15} + 221686154525 \nu^{13} - 1769258652890 \nu^{11} + \cdots + 235615624474021 \nu ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 270339671531 \nu^{15} - 3632699951595 \nu^{13} + 28361129424624 \nu^{11} + \cdots + 14\!\cdots\!19 \nu ) / 80\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27458425690 \nu^{15} - 351220616736 \nu^{13} + 2497123846655 \nu^{11} + \cdots + 402771359701876 \nu ) / 730432989750409 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 335759720654 \nu^{15} + 4312905177790 \nu^{13} - 31599184667959 \nu^{11} + \cdots + 42\!\cdots\!96 \nu ) / 80\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2895103593 \nu^{15} - 28802548035 \nu^{13} + 186585436186 \nu^{11} + \cdots + 55346076458801 \nu ) / 66402999068219 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 60698157708 \nu^{15} - 621829287321 \nu^{13} + 4407707076105 \nu^{11} + \cdots - 14740040443623 \nu ) / 730432989750409 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + 4\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - 3\beta_{11} - 5\beta_{10} + 3\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 7\beta_{7} + \beta_{6} - 23\beta_{5} - 7\beta_{4} - 3\beta_{3} + 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{15} + 13\beta_{14} - 12\beta_{13} - 28\beta_{12} + 13\beta_{11} + 12\beta_{9} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55\beta_{8} - 55\beta_{7} + 10\beta_{6} - 55\beta_{5} - 33\beta_{3} + 110\beta_{2} + 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 165\beta_{15} - 175\beta_{14} - 175\beta_{13} - 165\beta_{12} + 188\beta_{10} + 95\beta_{9} + 188\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -363\beta_{7} + 363\beta_{6} + 832\beta_{5} + 291\beta_{4} + 832\beta_{3} + 637\beta_{2} - 637 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1000\beta_{15} - 1161\beta_{14} + 195\beta_{12} - 510\beta_{11} - 195\beta_{10} + 1161\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2356\beta_{8} + 1900\beta_{7} + 415\beta_{5} - 2356\beta_{4} + 415\beta_{2} - 6161 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1485\beta_{15} + 3344\beta_{14} + 7524\beta_{13} + 7524\beta_{11} - 1485\beta_{10} - 7646\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -2695\beta_{8} - 15170\beta_{6} - 10769\beta_{5} - 15170\beta_{4} - 33928\beta_{3} - 4401 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25950\beta_{13} - 10769\beta_{12} + 48205\beta_{11} + 49098\beta_{10} - 48205\beta_{9} + 10769\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15181 \beta_{8} - 82122 \beta_{7} - 15181 \beta_{6} + 233828 \beta_{5} + 82122 \beta_{4} + \cdots - 15181 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 76100 \beta_{15} - 149063 \beta_{14} + 158027 \beta_{13} + 239850 \beta_{12} - 149063 \beta_{11} + \cdots + 76100 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.49977 + 2.06426i
0.548716 + 0.755243i
−0.548716 − 0.755243i
−1.49977 − 2.06426i
−2.35870 − 0.766388i
−1.77091 − 0.575405i
1.77091 + 0.575405i
2.35870 + 0.766388i
−2.35870 + 0.766388i
−1.77091 + 0.575405i
1.77091 − 0.575405i
2.35870 − 0.766388i
1.49977 − 2.06426i
0.548716 − 0.755243i
−0.548716 + 0.755243i
−1.49977 + 2.06426i
−2.42668 + 0.788477i 0.587785 − 0.809017i 3.64906 − 2.65120i 0 −0.788477 + 2.42668i 2.46575 + 3.39382i −3.76516 + 5.18229i −0.309017 − 0.951057i 0
49.2 −0.887841 + 0.288477i −0.587785 + 0.809017i −0.912991 + 0.663327i 0 0.288477 − 0.887841i −1.19972 − 1.65127i 1.71667 − 2.36279i −0.309017 − 0.951057i 0
49.3 0.887841 − 0.288477i 0.587785 − 0.809017i −0.912991 + 0.663327i 0 0.288477 − 0.887841i 1.19972 + 1.65127i −1.71667 + 2.36279i −0.309017 − 0.951057i 0
49.4 2.42668 − 0.788477i −0.587785 + 0.809017i 3.64906 − 2.65120i 0 −0.788477 + 2.42668i −2.46575 − 3.39382i 3.76516 − 5.18229i −0.309017 − 0.951057i 0
124.1 −1.45776 − 2.00643i 0.951057 − 0.309017i −1.28267 + 3.94765i 0 −2.00643 − 1.45776i 4.17973 + 1.35808i 5.07311 − 1.64835i 0.809017 − 0.587785i 0
124.2 −1.09448 − 1.50643i −0.951057 + 0.309017i −0.453397 + 1.39541i 0 1.50643 + 1.09448i 2.50213 + 0.812990i −0.943499 + 0.306561i 0.809017 − 0.587785i 0
124.3 1.09448 + 1.50643i 0.951057 − 0.309017i −0.453397 + 1.39541i 0 1.50643 + 1.09448i −2.50213 − 0.812990i 0.943499 − 0.306561i 0.809017 − 0.587785i 0
124.4 1.45776 + 2.00643i −0.951057 + 0.309017i −1.28267 + 3.94765i 0 −2.00643 − 1.45776i −4.17973 − 1.35808i −5.07311 + 1.64835i 0.809017 − 0.587785i 0
499.1 −1.45776 + 2.00643i 0.951057 + 0.309017i −1.28267 − 3.94765i 0 −2.00643 + 1.45776i 4.17973 − 1.35808i 5.07311 + 1.64835i 0.809017 + 0.587785i 0
499.2 −1.09448 + 1.50643i −0.951057 − 0.309017i −0.453397 − 1.39541i 0 1.50643 − 1.09448i 2.50213 − 0.812990i −0.943499 − 0.306561i 0.809017 + 0.587785i 0
499.3 1.09448 − 1.50643i 0.951057 + 0.309017i −0.453397 − 1.39541i 0 1.50643 − 1.09448i −2.50213 + 0.812990i 0.943499 + 0.306561i 0.809017 + 0.587785i 0
499.4 1.45776 − 2.00643i −0.951057 − 0.309017i −1.28267 − 3.94765i 0 −2.00643 + 1.45776i −4.17973 + 1.35808i −5.07311 − 1.64835i 0.809017 + 0.587785i 0
724.1 −2.42668 − 0.788477i 0.587785 + 0.809017i 3.64906 + 2.65120i 0 −0.788477 − 2.42668i 2.46575 − 3.39382i −3.76516 − 5.18229i −0.309017 + 0.951057i 0
724.2 −0.887841 − 0.288477i −0.587785 − 0.809017i −0.912991 − 0.663327i 0 0.288477 + 0.887841i −1.19972 + 1.65127i 1.71667 + 2.36279i −0.309017 + 0.951057i 0
724.3 0.887841 + 0.288477i 0.587785 + 0.809017i −0.912991 − 0.663327i 0 0.288477 + 0.887841i 1.19972 − 1.65127i −1.71667 − 2.36279i −0.309017 + 0.951057i 0
724.4 2.42668 + 0.788477i −0.587785 − 0.809017i 3.64906 + 2.65120i 0 −0.788477 − 2.42668i −2.46575 + 3.39382i 3.76516 + 5.18229i −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bx.g 16
5.b even 2 1 inner 825.2.bx.g 16
5.c odd 4 1 165.2.m.b ✓ 8
5.c odd 4 1 825.2.n.i 8
11.c even 5 1 inner 825.2.bx.g 16
15.e even 4 1 495.2.n.b 8
55.j even 10 1 inner 825.2.bx.g 16
55.k odd 20 1 165.2.m.b ✓ 8
55.k odd 20 1 825.2.n.i 8
55.k odd 20 1 1815.2.a.v 4
55.k odd 20 1 9075.2.a.cq 4
55.l even 20 1 1815.2.a.r 4
55.l even 20 1 9075.2.a.dg 4
165.u odd 20 1 5445.2.a.br 4
165.v even 20 1 495.2.n.b 8
165.v even 20 1 5445.2.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.b ✓ 8 5.c odd 4 1
165.2.m.b ✓ 8 55.k odd 20 1
495.2.n.b 8 15.e even 4 1
495.2.n.b 8 165.v even 20 1
825.2.n.i 8 5.c odd 4 1
825.2.n.i 8 55.k odd 20 1
825.2.bx.g 16 1.a even 1 1 trivial
825.2.bx.g 16 5.b even 2 1 inner
825.2.bx.g 16 11.c even 5 1 inner
825.2.bx.g 16 55.j even 10 1 inner
1815.2.a.r 4 55.l even 20 1
1815.2.a.v 4 55.k odd 20 1
5445.2.a.bk 4 165.v even 20 1
5445.2.a.br 4 165.u odd 20 1
9075.2.a.cq 4 55.k odd 20 1
9075.2.a.dg 4 55.l even 20 1

Hecke kernels

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

\( T_{2}^{16} - 6T_{2}^{14} + 45T_{2}^{12} - 289T_{2}^{10} + 1934T_{2}^{8} - 1819T_{2}^{6} + 19655T_{2}^{4} - 26741T_{2}^{2} + 14641 \) Copy content Toggle raw display
\( T_{13}^{16} - 14 T_{13}^{14} + 1357 T_{13}^{12} - 55097 T_{13}^{10} + 1097230 T_{13}^{8} + \cdots + 96059601 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 6 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 29 T^{14} + \cdots + 96059601 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 14 T^{14} + \cdots + 96059601 \) Copy content Toggle raw display
$17$ \( T^{16} - 38 T^{14} + \cdots + 96059601 \) Copy content Toggle raw display
$19$ \( (T^{8} - 2 T^{7} + T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 79 T^{6} + \cdots + 116281)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 14 T^{7} + \cdots + 383161)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1755097689601 \) Copy content Toggle raw display
$41$ \( (T^{8} - T^{7} + 45 T^{6} + \cdots + 121)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 96 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 151611963712561 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 8570379205441 \) Copy content Toggle raw display
$59$ \( (T^{8} + 13 T^{7} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 3 T^{7} + \cdots + 1234321)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 173 T^{6} + \cdots + 1100401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 9 T^{7} + \cdots + 674041)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 519885601 \) Copy content Toggle raw display
$79$ \( (T^{8} - 10 T^{7} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15214875390625 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} - 55 T^{2} + \cdots + 99)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 630247042161 \) Copy content Toggle raw display
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