Properties

Label 525.3.h.b
Level $525$
Weight $3$
Character orbit 525.h
Analytic conductor $14.305$
Analytic rank $0$
Dimension $10$
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(76,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.76");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.h (of order \(2\), degree \(1\), 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 13x^{8} + 2x^{7} + 118x^{6} + 8x^{5} + 403x^{4} + 299x^{3} + 931x^{2} + 186x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{7} q^{3} + ( - \beta_{3} + \beta_1 + 1) q^{4} - \beta_{2} q^{6} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} - 1) q^{7} + (\beta_{4} + \beta_{3} - 3) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{7} q^{3} + ( - \beta_{3} + \beta_1 + 1) q^{4} - \beta_{2} q^{6} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} - 1) q^{7} + (\beta_{4} + \beta_{3} - 3) q^{8} - 3 q^{9} + (\beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 3) q^{11} + (\beta_{9} - \beta_{7} + \beta_{2}) q^{12} + ( - \beta_{9} + 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{13} + ( - \beta_{8} - \beta_{7} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{14} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 6) q^{16} + (3 \beta_{8} - 3 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{2}) q^{17} + 3 \beta_1 q^{18} + (2 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} + \beta_{5}) q^{19} + ( - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2) q^{21} + (\beta_{6} - \beta_{5} + 4 \beta_{4} + \beta_{3} - 9 \beta_1 - 6) q^{22} + (2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} + 4 \beta_1 - 5) q^{23} + ( - \beta_{9} - \beta_{8} + 4 \beta_{7} + \beta_{6} + \beta_{5}) q^{24} + (2 \beta_{9} + \beta_{8} - 9 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{2}) q^{26} + 3 \beta_{7} q^{27} + ( - \beta_{9} + 2 \beta_{8} - 6 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 5 \beta_{3} + \cdots - 9) q^{28}+ \cdots + ( - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} - 6 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 10 q^{4} - 9 q^{7} - 28 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} + 10 q^{4} - 9 q^{7} - 28 q^{8} - 30 q^{9} + 32 q^{11} - 16 q^{14} - 54 q^{16} + 6 q^{18} - 15 q^{21} - 74 q^{22} - 32 q^{23} - 86 q^{28} - 40 q^{29} + 66 q^{32} - 30 q^{36} + 216 q^{37} + 30 q^{39} - 30 q^{42} + 6 q^{43} + 306 q^{44} - 226 q^{46} + 7 q^{49} - 48 q^{51} - 32 q^{53} + 148 q^{56} - 150 q^{57} + 130 q^{58} + 27 q^{63} - 168 q^{64} - 26 q^{67} + 188 q^{71} + 84 q^{72} - 14 q^{74} - 446 q^{77} - 312 q^{78} + 176 q^{79} + 90 q^{81} - 192 q^{84} - 626 q^{86} - 28 q^{88} + 255 q^{91} - 354 q^{92} + 234 q^{93} - 68 q^{98} - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 13x^{8} + 2x^{7} + 118x^{6} + 8x^{5} + 403x^{4} + 299x^{3} + 931x^{2} + 186x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1889 \nu^{9} + 8166 \nu^{8} - 23137 \nu^{7} + 138911 \nu^{6} - 137461 \nu^{5} + 817961 \nu^{4} - 4016082 \nu^{3} + 2483825 \nu^{2} + 498126 \nu - 2557980 ) / 13246392 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1889 \nu^{9} - 8166 \nu^{8} + 23137 \nu^{7} - 138911 \nu^{6} + 137461 \nu^{5} - 817961 \nu^{4} + 4016082 \nu^{3} - 2483825 \nu^{2} + 25994658 \nu + 2557980 ) / 13246392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4083 \nu^{9} + 27930 \nu^{8} - 79135 \nu^{7} + 249637 \nu^{6} - 470155 \nu^{5} + 2797655 \nu^{4} - 2967548 \nu^{3} + 8495375 \nu^{2} + 1703730 \nu + 31870992 ) / 6623196 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11639 \nu^{9} + 4734 \nu^{8} - 13413 \nu^{7} + 306007 \nu^{6} - 79689 \nu^{5} + 474189 \nu^{4} - 6473584 \nu^{3} + 1439925 \nu^{2} + 288774 \nu - 22233324 ) / 6623196 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12709 \nu^{9} - 48512 \nu^{8} - 46527 \nu^{7} - 1097617 \nu^{6} - 471225 \nu^{5} - 9458727 \nu^{4} + 1615680 \nu^{3} - 40328629 \nu^{2} + \cdots - 45901620 ) / 6623196 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41651 \nu^{9} + 155956 \nu^{8} - 625853 \nu^{7} + 1259669 \nu^{6} - 3913103 \nu^{5} + 11022151 \nu^{4} - 10888160 \nu^{3} + 21863721 \nu^{2} + \cdots + 46713888 ) / 6623196 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71055 \nu^{9} - 69166 \nu^{8} + 931881 \nu^{7} + 118973 \nu^{6} + 8523401 \nu^{5} + 430979 \nu^{4} + 29453126 \nu^{3} + 17229363 \nu^{2} + 68636030 \nu + 7091160 ) / 6623196 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 80735 \nu^{9} + 116728 \nu^{8} - 1066640 \nu^{7} + 162904 \nu^{6} - 9324028 \nu^{5} + 2125416 \nu^{4} - 32315787 \nu^{3} - 22632176 \nu^{2} + \cdots - 7779900 ) / 3311598 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 361247 \nu^{9} - 365594 \nu^{8} + 4715403 \nu^{7} + 484139 \nu^{6} + 42949699 \nu^{5} + 175201 \nu^{4} + 146217096 \nu^{3} + 93381657 \nu^{2} + \cdots + 34142808 ) / 6623196 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 5\beta_{7} + \beta_{3} + \beta_{2} - \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} - 8\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9 \beta_{9} - 3 \beta_{8} + 39 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 11 \beta_{3} - 13 \beta_{2} - 13 \beta _1 - 38 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 16 \beta_{9} - 15 \beta_{8} + 56 \beta_{7} + 13 \beta_{6} + 11 \beta_{5} - 13 \beta_{4} - 18 \beta_{3} - 73 \beta_{2} + 73 \beta _1 + 43 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{6} - 13\beta_{5} - 20\beta_{4} - 114\beta_{3} + 152\beta _1 + 329 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 198 \beta_{9} + 180 \beta_{8} - 672 \beta_{7} - 100 \beta_{6} - 140 \beta_{5} - 140 \beta_{4} - 238 \beta_{3} + 709 \beta_{2} + 709 \beta _1 + 532 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 889 \beta_{9} + 558 \beta_{8} - 3347 \beta_{7} - 278 \beta_{6} + 2 \beta_{5} + 278 \beta_{4} + 1169 \beta_{3} + 1717 \beta_{2} - 1717 \beta _1 - 3069 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -278\beta_{6} + 278\beta_{5} + 1449\beta_{4} + 2831\beta_{3} - 7124\beta _1 - 6247 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.63138 2.82562i
1.63138 + 2.82562i
1.08647 1.88183i
1.08647 + 1.88183i
−0.101707 + 0.176161i
−0.101707 0.176161i
−0.776277 + 1.34455i
−0.776277 1.34455i
−1.33987 + 2.32071i
−1.33987 2.32071i
−3.26275 1.73205i 6.64554 0 5.65125i −5.10886 4.78534i −8.63174 −3.00000 0
76.2 −3.26275 1.73205i 6.64554 0 5.65125i −5.10886 + 4.78534i −8.63174 −3.00000 0
76.3 −2.17295 1.73205i 0.721702 0 3.76366i 6.11016 + 3.41555i 7.12357 −3.00000 0
76.4 −2.17295 1.73205i 0.721702 0 3.76366i 6.11016 3.41555i 7.12357 −3.00000 0
76.5 0.203414 1.73205i −3.95862 0 0.352323i −3.03444 + 6.30811i −1.61889 −3.00000 0
76.6 0.203414 1.73205i −3.95862 0 0.352323i −3.03444 6.30811i −1.61889 −3.00000 0
76.7 1.55255 1.73205i −1.58958 0 2.68910i 3.69430 5.94577i −8.67812 −3.00000 0
76.8 1.55255 1.73205i −1.58958 0 2.68910i 3.69430 + 5.94577i −8.67812 −3.00000 0
76.9 2.67973 1.73205i 3.18096 0 4.64143i −6.16116 3.32267i −2.19482 −3.00000 0
76.10 2.67973 1.73205i 3.18096 0 4.64143i −6.16116 + 3.32267i −2.19482 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.h.b 10
5.b even 2 1 525.3.h.c yes 10
5.c odd 4 2 525.3.e.b 20
7.b odd 2 1 inner 525.3.h.b 10
35.c odd 2 1 525.3.h.c yes 10
35.f even 4 2 525.3.e.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.e.b 20 5.c odd 4 2
525.3.e.b 20 35.f even 4 2
525.3.h.b 10 1.a even 1 1 trivial
525.3.h.b 10 7.b odd 2 1 inner
525.3.h.c yes 10 5.b even 2 1
525.3.h.c yes 10 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + T_{2}^{4} - 12T_{2}^{3} - 5T_{2}^{2} + 31T_{2} - 6 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + T^{4} - 12 T^{3} - 5 T^{2} + 31 T - 6)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 9 T^{9} + 37 T^{8} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( (T^{5} - 16 T^{4} - 207 T^{3} + 4226 T^{2} + \cdots + 8400)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 1269 T^{8} + \cdots + 11663315712 \) Copy content Toggle raw display
$17$ \( T^{10} + 2298 T^{8} + \cdots + 874800000000 \) Copy content Toggle raw display
$19$ \( T^{10} + 2163 T^{8} + \cdots + 128517193728 \) Copy content Toggle raw display
$23$ \( (T^{5} + 16 T^{4} - 1542 T^{3} + \cdots + 3662082)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 20 T^{4} - 1698 T^{3} + \cdots + 3075678)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 3333 T^{8} + \cdots + 338397758208 \) Copy content Toggle raw display
$37$ \( (T^{5} - 108 T^{4} + 3761 T^{3} + \cdots + 46312)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 6654 T^{8} + \cdots + 1097349120000 \) Copy content Toggle raw display
$43$ \( (T^{5} - 3 T^{4} - 4810 T^{3} + \cdots - 46365899)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 12324 T^{8} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( (T^{5} + 16 T^{4} - 5463 T^{3} + \cdots + 22722000)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 24738 T^{8} + \cdots + 13\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{10} + 17013 T^{8} + \cdots + 18172516320000 \) Copy content Toggle raw display
$67$ \( (T^{5} + 13 T^{4} - 11477 T^{3} + \cdots - 304035028)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 94 T^{4} - 7779 T^{3} + \cdots - 260099904)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 34008 T^{8} + \cdots + 16\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( (T^{5} - 88 T^{4} - 12131 T^{3} + \cdots + 272274352)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 25938 T^{8} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + 80832 T^{8} + \cdots + 52\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{10} + 48771 T^{8} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
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