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Properties

Label 525.3.o.m
Level $525$
Weight $3$
Character orbit 525.o
Analytic conductor $14.305$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 456 x^{8} - 1050 x^{7} + 1999 x^{6} - 2844 x^{5} + 2949 x^{4} + \cdots + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + ( - \beta_{5} - 2) q^{3} + (\beta_{11} + 4 \beta_{5}) q^{4} + ( - 2 \beta_{8} - \beta_{2}) q^{6} + ( - \beta_{8} - \beta_{7} - 2) q^{7} + (\beta_{4} + \beta_{3} + 4 \beta_{2} - 5) q^{8} + (3 \beta_{5} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + ( - \beta_{5} - 2) q^{3} + (\beta_{11} + 4 \beta_{5}) q^{4} + ( - 2 \beta_{8} - \beta_{2}) q^{6} + ( - \beta_{8} - \beta_{7} - 2) q^{7} + (\beta_{4} + \beta_{3} + 4 \beta_{2} - 5) q^{8} + (3 \beta_{5} + 3) q^{9} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{2}) q^{11}+ \cdots + (3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 18 q^{3} - 22 q^{4} - 22 q^{7} - 40 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 18 q^{3} - 22 q^{4} - 22 q^{7} - 40 q^{8} + 18 q^{9} + 20 q^{11} + 66 q^{12} + 32 q^{14} - 82 q^{16} + 78 q^{17} + 6 q^{18} - 6 q^{19} + 36 q^{21} - 56 q^{22} - 2 q^{23} + 60 q^{24} + 36 q^{26} + 128 q^{28} - 100 q^{29} + 108 q^{31} + 108 q^{32} - 60 q^{33} - 132 q^{36} + 34 q^{37} - 126 q^{38} - 42 q^{39} - 114 q^{42} + 124 q^{43} + 234 q^{44} + 278 q^{46} - 96 q^{47} - 60 q^{49} - 78 q^{51} + 444 q^{52} + 76 q^{53} - 18 q^{54} + 112 q^{56} + 12 q^{57} + 52 q^{58} - 270 q^{59} - 60 q^{61} - 42 q^{63} + 700 q^{64} + 84 q^{66} + 18 q^{67} - 108 q^{68} - 628 q^{71} - 60 q^{72} - 234 q^{73} + 244 q^{74} + 196 q^{77} - 72 q^{78} + 108 q^{79} - 54 q^{81} - 480 q^{82} - 192 q^{84} + 130 q^{86} + 150 q^{87} + 668 q^{88} - 186 q^{89} + 444 q^{91} - 456 q^{92} - 108 q^{93} + 30 q^{94} - 324 q^{96} - 416 q^{98} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 456 x^{8} - 1050 x^{7} + 1999 x^{6} - 2844 x^{5} + 2949 x^{4} + \cdots + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 32 \nu^{8} - 98 \nu^{7} + 296 \nu^{6} - 566 \nu^{5} + 877 \nu^{4} - 915 \nu^{3} + \cdots - 120 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{10} + 15 \nu^{9} - 101 \nu^{8} + 314 \nu^{7} - 1020 \nu^{6} + 2024 \nu^{5} - 3629 \nu^{4} + \cdots - 342 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40 \nu^{11} + 143 \nu^{10} - 1051 \nu^{9} + 2250 \nu^{8} - 7462 \nu^{7} + 7504 \nu^{6} + \cdots - 5604 ) / 168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40 \nu^{11} - 255 \nu^{10} + 1611 \nu^{9} - 6002 \nu^{8} + 19110 \nu^{7} - 45192 \nu^{6} + 83950 \nu^{5} + \cdots - 6660 ) / 168 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 206 \nu^{11} - 1133 \nu^{10} + 7471 \nu^{9} - 25122 \nu^{8} + 81494 \nu^{7} - 175924 \nu^{6} + \cdots - 15420 ) / 168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 351 \nu^{11} - 1948 \nu^{10} + 12827 \nu^{9} - 43437 \nu^{8} + 140994 \nu^{7} - 306530 \nu^{6} + \cdots - 28842 ) / 84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 890 \nu^{11} - 4923 \nu^{10} + 32385 \nu^{9} - 109342 \nu^{8} + 354018 \nu^{7} - 766836 \nu^{6} + \cdots - 64332 ) / 168 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 471 \nu^{11} - 2580 \nu^{10} + 17016 \nu^{9} - 57026 \nu^{8} + 184814 \nu^{7} - 397488 \nu^{6} + \cdots - 31896 ) / 84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1004 \nu^{11} - 5529 \nu^{10} + 36423 \nu^{9} - 122590 \nu^{8} + 397194 \nu^{7} - 858060 \nu^{6} + \cdots - 72540 ) / 168 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1576 \nu^{11} + 8619 \nu^{10} - 56871 \nu^{9} + 190374 \nu^{8} - 617134 \nu^{7} + 1326024 \nu^{6} + \cdots + 107340 ) / 168 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 121 \nu^{11} - 665 \nu^{10} + 4384 \nu^{9} - 14732 \nu^{8} + 47760 \nu^{7} - 103018 \nu^{6} + \cdots - 8844 ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - 20 \beta_{8} + 7 \beta_{7} - 3 \beta_{6} + 11 \beta_{5} + \cdots + 27 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 9 \beta_{10} + 11 \beta_{9} - 23 \beta_{8} - 7 \beta_{7} - \beta_{6} + 6 \beta_{5} + \cdots - 131 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{11} + 45 \beta_{10} + 29 \beta_{9} + 115 \beta_{8} - 70 \beta_{7} + 12 \beta_{6} + \cdots - 227 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{11} + 129 \beta_{10} - 55 \beta_{9} + 283 \beta_{8} - 7 \beta_{7} + 19 \beta_{6} + \cdots + 823 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 43 \beta_{11} - 261 \beta_{10} - 283 \beta_{9} - 653 \beta_{8} + 560 \beta_{7} - 22 \beta_{6} + \cdots + 2305 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 143 \beta_{11} - 1385 \beta_{10} + 117 \beta_{9} - 2953 \beta_{8} + 595 \beta_{7} - 115 \beta_{6} + \cdots - 4859 ) / 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 33 \beta_{11} + 131 \beta_{10} + 325 \beta_{9} + 363 \beta_{8} - 570 \beta_{7} - 34 \beta_{6} + \cdots - 3131 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1851 \beta_{11} + 12977 \beta_{10} + 1297 \beta_{9} + 27159 \beta_{8} - 8841 \beta_{7} + \cdots + 23857 ) / 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 181 \beta_{11} + 4545 \beta_{10} - 16447 \beta_{9} + 4531 \beta_{8} + 24248 \beta_{7} + 2934 \beta_{6} + \cdots + 198931 ) / 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 18905 \beta_{11} - 109049 \beta_{10} - 24753 \beta_{9} - 227251 \beta_{8} + 98455 \beta_{7} + \cdots - 51503 ) / 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16553 \beta_{11} - 144371 \beta_{10} + 109143 \beta_{9} - 255223 \beta_{8} - 103460 \beta_{7} + \cdots - 1729109 ) / 42 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(1 + \beta_{5}\)

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
0.500000 − 0.396977i
0.500000 − 2.38770i
0.500000 − 2.68684i
0.500000 + 0.792243i
0.500000 + 2.96550i
0.500000 − 0.0182799i
0.500000 + 0.396977i
0.500000 + 2.38770i
0.500000 + 2.68684i
0.500000 − 0.792243i
0.500000 − 2.96550i
0.500000 + 0.0182799i
−1.80299 + 3.12287i −1.50000 + 0.866025i −4.50153 − 7.79688i 0 − 6.24573i −2.45857 − 6.55404i 18.0409 1.50000 − 2.59808i 0
376.2 −1.46731 + 2.54146i −1.50000 + 0.866025i −2.30602 − 3.99415i 0 − 5.08293i 5.41652 + 4.43411i 1.79613 1.50000 − 2.59808i 0
376.3 −0.687692 + 1.19112i −1.50000 + 0.866025i 1.05416 + 1.82586i 0 − 2.38224i −6.56639 + 2.42539i −8.40129 1.50000 − 2.59808i 0
376.4 −0.288563 + 0.499806i −1.50000 + 0.866025i 1.83346 + 3.17565i 0 − 0.999611i 1.64406 − 6.80420i −4.42478 1.50000 − 2.59808i 0
376.5 1.25588 − 2.17524i −1.50000 + 0.866025i −1.15446 − 1.99958i 0 4.35049i −6.75110 − 1.85004i 4.24760 1.50000 − 2.59808i 0
376.6 1.99068 − 3.44796i −1.50000 + 0.866025i −5.92561 − 10.2635i 0 6.89592i −2.28451 + 6.61672i −31.2586 1.50000 − 2.59808i 0
451.1 −1.80299 − 3.12287i −1.50000 − 0.866025i −4.50153 + 7.79688i 0 6.24573i −2.45857 + 6.55404i 18.0409 1.50000 + 2.59808i 0
451.2 −1.46731 − 2.54146i −1.50000 − 0.866025i −2.30602 + 3.99415i 0 5.08293i 5.41652 − 4.43411i 1.79613 1.50000 + 2.59808i 0
451.3 −0.687692 − 1.19112i −1.50000 − 0.866025i 1.05416 − 1.82586i 0 2.38224i −6.56639 − 2.42539i −8.40129 1.50000 + 2.59808i 0
451.4 −0.288563 − 0.499806i −1.50000 − 0.866025i 1.83346 − 3.17565i 0 0.999611i 1.64406 + 6.80420i −4.42478 1.50000 + 2.59808i 0
451.5 1.25588 + 2.17524i −1.50000 − 0.866025i −1.15446 + 1.99958i 0 − 4.35049i −6.75110 + 1.85004i 4.24760 1.50000 + 2.59808i 0
451.6 1.99068 + 3.44796i −1.50000 − 0.866025i −5.92561 + 10.2635i 0 − 6.89592i −2.28451 − 6.61672i −31.2586 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 376.6
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.m 12
5.b even 2 1 105.3.n.b ✓ 12
5.c odd 4 2 525.3.s.j 24
7.d odd 6 1 inner 525.3.o.m 12
15.d odd 2 1 315.3.w.b 12
35.i odd 6 1 105.3.n.b ✓ 12
35.i odd 6 1 735.3.h.b 12
35.j even 6 1 735.3.h.b 12
35.k even 12 2 525.3.s.j 24
105.p even 6 1 315.3.w.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.n.b ✓ 12 5.b even 2 1
105.3.n.b ✓ 12 35.i odd 6 1
315.3.w.b 12 15.d odd 2 1
315.3.w.b 12 105.p even 6 1
525.3.o.m 12 1.a even 1 1 trivial
525.3.o.m 12 7.d odd 6 1 inner
525.3.s.j 24 5.c odd 4 2
525.3.s.j 24 35.k even 12 2
735.3.h.b 12 35.i odd 6 1
735.3.h.b 12 35.j even 6 1

Hecke kernels

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

\( T_{2}^{12} + 2 T_{2}^{11} + 25 T_{2}^{10} + 50 T_{2}^{9} + 451 T_{2}^{8} + 842 T_{2}^{7} + 3598 T_{2}^{6} + \cdots + 7056 \) Copy content Toggle raw display
\( T_{11}^{12} - 20 T_{11}^{11} + 649 T_{11}^{10} - 1808 T_{11}^{9} + 99307 T_{11}^{8} + 399706 T_{11}^{7} + \cdots + 15854839056 \) Copy content Toggle raw display
\( T_{13}^{12} + 978 T_{13}^{10} + 326655 T_{13}^{8} + 49084380 T_{13}^{6} + 3468233295 T_{13}^{4} + \cdots + 732611029329 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 7056 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 15854839056 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 732611029329 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 93022784308224 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 1794623769 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 170135605308816 \) Copy content Toggle raw display
$29$ \( (T^{6} + 50 T^{5} + \cdots + 32868624)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{6} - 62 T^{5} + \cdots - 961914584)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 796594176 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{6} + 314 T^{5} + \cdots - 210165130176)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
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