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Properties

Label 525.3.o.q
Level $525$
Weight $3$
Character orbit 525.o
Analytic conductor $14.305$
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] = [525,3,Mod(376,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.376");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.o (of order \(6\), degree \(2\), 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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24 x^{14} + 405 x^{12} - 30 x^{11} + 3324 x^{10} - 1302 x^{9} + 19731 x^{8} - 8442 x^{7} + \cdots + 22500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{4} + 2) q^{3} + ( - \beta_{9} + 2 \beta_{4}) q^{4} + ( - \beta_{3} - 2 \beta_1) q^{6} + (\beta_{8} + \beta_{2}) q^{7} + ( - 2 \beta_{14} + \beta_{13} + \cdots + 1) q^{8}+ \cdots + (3 \beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{4} + 2) q^{3} + ( - \beta_{9} + 2 \beta_{4}) q^{4} + ( - \beta_{3} - 2 \beta_1) q^{6} + (\beta_{8} + \beta_{2}) q^{7} + ( - 2 \beta_{14} + \beta_{13} + \cdots + 1) q^{8}+ \cdots + ( - 6 \beta_{14} + 3 \beta_{12} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 24 q^{3} - 16 q^{4} + 6 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{3} - 16 q^{4} + 6 q^{7} + 24 q^{9} - 14 q^{11} - 48 q^{12} - 30 q^{14} - 20 q^{16} + 6 q^{17} + 30 q^{19} + 6 q^{21} - 36 q^{22} - 18 q^{23} - 48 q^{26} - 168 q^{28} + 44 q^{29} + 42 q^{31} + 150 q^{32} - 42 q^{33} - 96 q^{36} + 96 q^{37} + 204 q^{38} - 18 q^{39} - 78 q^{42} - 160 q^{44} - 30 q^{46} - 138 q^{47} - 178 q^{49} + 6 q^{51} + 126 q^{52} + 150 q^{53} - 234 q^{56} + 60 q^{57} - 90 q^{58} + 402 q^{59} + 168 q^{61} + 200 q^{64} - 54 q^{66} + 174 q^{67} - 234 q^{68} + 172 q^{71} + 336 q^{73} - 450 q^{74} - 372 q^{77} - 96 q^{78} + 10 q^{79} - 72 q^{81} - 690 q^{82} - 390 q^{84} + 72 q^{86} + 66 q^{87} + 492 q^{88} - 12 q^{89} - 112 q^{91} + 204 q^{92} + 42 q^{93} + 462 q^{94} + 450 q^{96} - 198 q^{98} - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24 x^{14} + 405 x^{12} - 30 x^{11} + 3324 x^{10} - 1302 x^{9} + 19731 x^{8} - 8442 x^{7} + \cdots + 22500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29\!\cdots\!00 \nu^{15} + \cdots - 23\!\cdots\!44 ) / 42\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 66\!\cdots\!91 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 42\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!33 \nu^{15} + \cdots - 89\!\cdots\!00 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!21 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!41 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54\!\cdots\!87 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 57\!\cdots\!81 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 96\!\cdots\!98 \nu^{15} + \cdots - 49\!\cdots\!00 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!11 \nu^{15} + \cdots - 16\!\cdots\!60 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 74\!\cdots\!84 \nu^{15} + \cdots + 72\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 77\!\cdots\!38 \nu^{15} + \cdots - 62\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 52\!\cdots\!47 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 31\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!33 \nu^{15} + \cdots - 40\!\cdots\!80 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!97 \nu^{15} + \cdots - 26\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 6\beta_{4} - \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 10\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 14 \beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + 18 \beta_{14} - 16 \beta_{13} - 36 \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46 \beta_{14} - 25 \beta_{13} - 23 \beta_{12} + 40 \beta_{11} + 21 \beta_{10} + 21 \beta_{9} + \cdots + 620 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 46 \beta_{15} + 266 \beta_{14} - 13 \beta_{13} + 266 \beta_{12} + 266 \beta_{11} + 34 \beta_{10} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 411 \beta_{14} + 600 \beta_{13} + 822 \beta_{12} - 237 \beta_{11} - 552 \beta_{10} + 2127 \beta_{9} + \cdots - 7323 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 381 \beta_{15} - 7452 \beta_{14} + 3393 \beta_{13} + 3726 \beta_{12} - 2949 \beta_{11} - 4059 \beta_{10} + \cdots - 6075 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 60 \beta_{15} - 6723 \beta_{14} - 1521 \beta_{13} - 6723 \beta_{12} - 6723 \beta_{11} + 2301 \beta_{10} + \cdots - 1521 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5601 \beta_{15} + 51360 \beta_{14} - 46050 \beta_{13} - 102720 \beta_{12} - 10041 \beta_{11} + \cdots + 116580 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1272 \beta_{15} + 210774 \beta_{14} - 116187 \beta_{13} - 105387 \beta_{12} + 144048 \beta_{11} + \cdots + 1225524 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 156186 \beta_{15} + 705690 \beta_{14} + 2601 \beta_{13} + 705690 \beta_{12} + 705690 \beta_{11} + \cdots + 2601 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 33654 \beta_{15} - 1611567 \beta_{14} + 1949592 \beta_{13} + 3223134 \beta_{12} - 475173 \beta_{11} + \cdots - 16066347 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1062501 \beta_{15} - 19424424 \beta_{14} + 8887821 \beta_{13} + 9712212 \beta_{12} - 8209695 \beta_{11} + \cdots - 31751193 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
376.1
−1.70407 − 2.95153i
−1.46615 − 2.53945i
−0.990928 − 1.71634i
−0.247129 − 0.428040i
0.549841 + 0.952353i
0.765175 + 1.32532i
1.20633 + 2.08943i
1.88692 + 3.26825i
−1.70407 + 2.95153i
−1.46615 + 2.53945i
−0.990928 + 1.71634i
−0.247129 + 0.428040i
0.549841 − 0.952353i
0.765175 − 1.32532i
1.20633 − 2.08943i
1.88692 − 3.26825i
−1.70407 + 2.95153i 1.50000 − 0.866025i −3.80769 − 6.59511i 0 5.90306i 4.74314 + 5.14807i 12.3217 1.50000 − 2.59808i 0
376.2 −1.46615 + 2.53945i 1.50000 − 0.866025i −2.29920 − 3.98234i 0 5.07890i 0.976973 − 6.93149i 1.75471 1.50000 − 2.59808i 0
376.3 −0.990928 + 1.71634i 1.50000 − 0.866025i 0.0361245 + 0.0625695i 0 3.43267i −6.67372 + 2.11222i −8.07061 1.50000 − 2.59808i 0
376.4 −0.247129 + 0.428040i 1.50000 − 0.866025i 1.87785 + 3.25254i 0 0.856079i 3.87787 + 5.82770i −3.83332 1.50000 − 2.59808i 0
376.5 0.549841 − 0.952353i 1.50000 − 0.866025i 1.39535 + 2.41681i 0 − 1.90471i −3.45923 − 6.08554i 7.46761 1.50000 − 2.59808i 0
376.6 0.765175 − 1.32532i 1.50000 − 0.866025i 0.829016 + 1.43590i 0 − 2.65064i −2.94719 + 6.34933i 8.65876 1.50000 − 2.59808i 0
376.7 1.20633 − 2.08943i 1.50000 − 0.866025i −0.910488 − 1.57701i 0 − 4.17887i −0.420230 − 6.98737i 5.25727 1.50000 − 2.59808i 0
376.8 1.88692 − 3.26825i 1.50000 − 0.866025i −5.12097 − 8.86977i 0 − 6.53650i 6.90238 − 1.16497i −23.5561 1.50000 − 2.59808i 0
451.1 −1.70407 − 2.95153i 1.50000 + 0.866025i −3.80769 + 6.59511i 0 − 5.90306i 4.74314 − 5.14807i 12.3217 1.50000 + 2.59808i 0
451.2 −1.46615 − 2.53945i 1.50000 + 0.866025i −2.29920 + 3.98234i 0 − 5.07890i 0.976973 + 6.93149i 1.75471 1.50000 + 2.59808i 0
451.3 −0.990928 − 1.71634i 1.50000 + 0.866025i 0.0361245 − 0.0625695i 0 − 3.43267i −6.67372 − 2.11222i −8.07061 1.50000 + 2.59808i 0
451.4 −0.247129 − 0.428040i 1.50000 + 0.866025i 1.87785 − 3.25254i 0 − 0.856079i 3.87787 − 5.82770i −3.83332 1.50000 + 2.59808i 0
451.5 0.549841 + 0.952353i 1.50000 + 0.866025i 1.39535 − 2.41681i 0 1.90471i −3.45923 + 6.08554i 7.46761 1.50000 + 2.59808i 0
451.6 0.765175 + 1.32532i 1.50000 + 0.866025i 0.829016 − 1.43590i 0 2.65064i −2.94719 − 6.34933i 8.65876 1.50000 + 2.59808i 0
451.7 1.20633 + 2.08943i 1.50000 + 0.866025i −0.910488 + 1.57701i 0 4.17887i −0.420230 + 6.98737i 5.25727 1.50000 + 2.59808i 0
451.8 1.88692 + 3.26825i 1.50000 + 0.866025i −5.12097 + 8.86977i 0 6.53650i 6.90238 + 1.16497i −23.5561 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 376.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.o.q 16
5.b even 2 1 525.3.o.p 16
5.c odd 4 2 105.3.r.a ✓ 32
7.d odd 6 1 inner 525.3.o.q 16
15.e even 4 2 315.3.bi.e 32
35.i odd 6 1 525.3.o.p 16
35.k even 12 2 105.3.r.a ✓ 32
35.k even 12 2 735.3.e.a 32
35.l odd 12 2 735.3.e.a 32
105.w odd 12 2 315.3.bi.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.r.a ✓ 32 5.c odd 4 2
105.3.r.a ✓ 32 35.k even 12 2
315.3.bi.e 32 15.e even 4 2
315.3.bi.e 32 105.w odd 12 2
525.3.o.p 16 5.b even 2 1
525.3.o.p 16 35.i odd 6 1
525.3.o.q 16 1.a even 1 1 trivial
525.3.o.q 16 7.d odd 6 1 inner
735.3.e.a 32 35.k even 12 2
735.3.e.a 32 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{16} + 24 T_{2}^{14} + 405 T_{2}^{12} - 30 T_{2}^{11} + 3324 T_{2}^{10} - 1302 T_{2}^{9} + \cdots + 22500 \) Copy content Toggle raw display
\( T_{11}^{16} + 14 T_{11}^{15} + 714 T_{11}^{14} + 2788 T_{11}^{13} + 252509 T_{11}^{12} + \cdots + 423613372696900 \) Copy content Toggle raw display
\( T_{13}^{16} + 1418 T_{13}^{14} + 755863 T_{13}^{12} + 186856544 T_{13}^{10} + 21218695183 T_{13}^{8} + \cdots + 121335512257600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 24 T^{14} + \cdots + 22500 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 423613372696900 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 121335512257600 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} - 22 T^{7} + \cdots + 943225000)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{8} - 6258 T^{6} + \cdots - 460968510576)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 27057106624256)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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