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Properties

Label 525.3.e.b
Level $525$
Weight $3$
Character orbit 525.e
Analytic conductor $14.305$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [525,3,Mod(349,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.349"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 25 x^{18} + 409 x^{16} - 3886 x^{14} + 26830 x^{12} - 118498 x^{10} + 377533 x^{8} + \cdots + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{10} q^{3} + (\beta_1 - 1) q^{4} + \beta_{3} q^{6} + ( - \beta_{19} - \beta_{17}) q^{7} + (\beta_{17} - \beta_{15} - \beta_{12}) q^{8} + 3 q^{9} + (\beta_{8} + \beta_{6} - 2 \beta_1 + 3) q^{11}+ \cdots + (3 \beta_{8} + 3 \beta_{6} - 6 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} + 60 q^{9} + 64 q^{11} + 32 q^{14} - 108 q^{16} - 30 q^{21} + 80 q^{29} - 60 q^{36} - 60 q^{39} - 612 q^{44} - 452 q^{46} - 14 q^{49} - 96 q^{51} + 296 q^{56} + 336 q^{64} + 376 q^{71} + 28 q^{74}+ \cdots + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 25 x^{18} + 409 x^{16} - 3886 x^{14} + 26830 x^{12} - 118498 x^{10} + 377533 x^{8} + \cdots + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3617939313 \nu^{18} - 84846996190 \nu^{16} + 1351620557815 \nu^{14} - 12018402452941 \nu^{12} + \cdots + 79\!\cdots\!40 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 81122925909 \nu^{18} - 2027467651114 \nu^{16} + 32297748660589 \nu^{14} + \cdots - 56\!\cdots\!88 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1358236543359 \nu^{18} - 32510113143594 \nu^{16} + 520831042391223 \nu^{14} + \cdots - 19\!\cdots\!68 ) / 88\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 156900681985 \nu^{19} - 3911663231686 \nu^{17} + 63917837943295 \nu^{15} + \cdots - 170291618704944 \nu ) / 48\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3432176577821 \nu^{18} + 85212955129946 \nu^{16} + \cdots + 54\!\cdots\!92 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2522994378905 \nu^{18} + 63821474417814 \nu^{16} + \cdots + 19\!\cdots\!16 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2898462399021 \nu^{18} + 73162334590182 \nu^{16} + \cdots + 89\!\cdots\!84 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3061394534611 \nu^{18} - 76733613425810 \nu^{16} + \cdots - 10\!\cdots\!40 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38405224301426 \nu^{18} + 964843179278276 \nu^{16} + \cdots + 64\!\cdots\!72 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1003409 \nu^{19} - 25111712 \nu^{17} + 410994140 \nu^{15} - 3908912321 \nu^{13} + \cdots - 65604353256 \nu ) / 6556964040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16480463415563 \nu^{18} + 411694606518438 \nu^{16} + \cdots + 27\!\cdots\!36 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30519923280476 \nu^{19} + 761722331142833 \nu^{17} + \cdots + 33\!\cdots\!08 \nu ) / 79\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 386170330990279 \nu^{19} + \cdots + 25\!\cdots\!56 \nu ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 599567832223921 \nu^{19} + \cdots + 39\!\cdots\!64 \nu ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 116974916 \nu^{19} + 2917960490 \nu^{17} - 47688006353 \nu^{15} + 452087598962 \nu^{13} + \cdots + 127160845056 \nu ) / 144107115768 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 351083198989003 \nu^{19} + \cdots + 13\!\cdots\!28 \nu ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 8263483285676 \nu^{19} + 206211274631939 \nu^{17} + \cdots + 89\!\cdots\!72 \nu ) / 88\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 8280119671121 \nu^{19} + 207527167722518 \nu^{17} + \cdots + 55\!\cdots\!24 \nu ) / 80\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 261833944163494 \nu^{19} + \cdots - 50\!\cdots\!96 \nu ) / 13\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{19} + \beta_{18} - 2\beta_{16} + 3\beta_{14} - \beta_{13} + 3\beta_{10} + 5\beta_{4} ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 6\beta_{9} - 6\beta_{8} + 6\beta_{7} - 5\beta_{3} - 5\beta _1 + 25 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{15} + \beta_{12} + 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{11} - 51 \beta_{9} - 56 \beta_{8} + 51 \beta_{7} - 5 \beta_{6} - 15 \beta_{5} - 65 \beta_{3} + \cdots - 190 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 92 \beta_{19} - 36 \beta_{18} - 90 \beta_{17} + 82 \beta_{16} + 70 \beta_{15} - 218 \beta_{14} + \cdots + 365 \beta_{4} ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{8} + \beta_{7} - 14\beta_{6} + 34\beta_{2} + 116\beta _1 - 329 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 622 \beta_{19} - 211 \beta_{18} + 1190 \beta_{17} + 822 \beta_{16} - 830 \beta_{15} + \cdots - 3545 \beta_{4} ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 901 \beta_{11} + 3966 \beta_{9} + 3956 \beta_{8} - 4526 \beta_{7} - 830 \beta_{6} + 2790 \beta_{5} + \cdots - 15345 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 278\beta_{19} + 2831\beta_{17} + 278\beta_{16} - 1883\beta_{15} - 1449\beta_{12} - 7124\beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10866 \beta_{11} + 37131 \beta_{9} + 54066 \beta_{8} - 48991 \beta_{7} + 9415 \beta_{6} + 31425 \beta_{5} + \cdots + 149790 ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 79152 \beta_{19} + 5706 \beta_{18} + 160460 \beta_{17} - 45282 \beta_{16} - 104580 \beta_{15} + \cdots - 365225 \beta_{4} ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 14925\beta_{8} - 5991\beta_{7} + 20916\beta_{6} - 59782\beta_{2} - 136576\beta _1 + 301041 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 420942 \beta_{19} + 16141 \beta_{18} - 1773940 \beta_{17} - 809602 \beta_{16} + \cdots + 3792845 \beta_{4} ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1406521 \beta_{11} - 3579486 \beta_{9} - 4197646 \beta_{8} + 4595806 \beta_{7} + 1146280 \beta_{6} + \cdots + 15416345 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 432520 \beta_{19} - 3867105 \beta_{17} - 432520 \beta_{16} + 2489913 \beta_{15} + \cdots + 7943192 \beta_{4} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 15490566 \beta_{11} - 36338091 \beta_{9} - 59998816 \beta_{8} + 56363171 \beta_{7} + \cdots - 159908750 ) / 10 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 87827212 \beta_{19} + 3407844 \beta_{18} - 208974490 \beta_{17} + 40505762 \beta_{16} + \cdots + 418153805 \beta_{4} ) / 10 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( -16850837\beta_{8} + 10020497\beta_{7} - 26871334\beta_{6} + 78130522\beta_{2} + 163688556\beta _1 - 334564009 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 411542942 \beta_{19} + 50524469 \beta_{18} + 2246696550 \beta_{17} + 924134822 \beta_{16} + \cdots - 4418596025 \beta_{4} ) / 10 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
2.82562 − 1.63138i
−2.82562 − 1.63138i
−2.32071 − 1.33987i
2.32071 − 1.33987i
1.88183 − 1.08647i
−1.88183 − 1.08647i
−1.34455 − 0.776277i
1.34455 − 0.776277i
−0.176161 − 0.101707i
0.176161 − 0.101707i
−0.176161 + 0.101707i
0.176161 + 0.101707i
−1.34455 + 0.776277i
1.34455 + 0.776277i
1.88183 + 1.08647i
−1.88183 + 1.08647i
−2.32071 + 1.33987i
2.32071 + 1.33987i
2.82562 + 1.63138i
−2.82562 + 1.63138i
− 3.26275i −1.73205 −6.64554 0 5.65125i 4.78534 − 5.10886i 8.63174i 3.00000 0
349.2 − 3.26275i 1.73205 −6.64554 0 − 5.65125i −4.78534 − 5.10886i 8.63174i 3.00000 0
349.3 − 2.67973i −1.73205 −3.18096 0 4.64143i 3.32267 + 6.16116i − 2.19482i 3.00000 0
349.4 − 2.67973i 1.73205 −3.18096 0 − 4.64143i −3.32267 + 6.16116i − 2.19482i 3.00000 0
349.5 − 2.17295i −1.73205 −0.721702 0 3.76366i −3.41555 + 6.11016i − 7.12357i 3.00000 0
349.6 − 2.17295i 1.73205 −0.721702 0 − 3.76366i 3.41555 + 6.11016i − 7.12357i 3.00000 0
349.7 − 1.55255i −1.73205 1.58958 0 2.68910i 5.94577 − 3.69430i − 8.67812i 3.00000 0
349.8 − 1.55255i 1.73205 1.58958 0 − 2.68910i −5.94577 − 3.69430i − 8.67812i 3.00000 0
349.9 − 0.203414i −1.73205 3.95862 0 0.352323i −6.30811 + 3.03444i − 1.61889i 3.00000 0
349.10 − 0.203414i 1.73205 3.95862 0 − 0.352323i 6.30811 + 3.03444i − 1.61889i 3.00000 0
349.11 0.203414i −1.73205 3.95862 0 − 0.352323i −6.30811 − 3.03444i 1.61889i 3.00000 0
349.12 0.203414i 1.73205 3.95862 0 0.352323i 6.30811 − 3.03444i 1.61889i 3.00000 0
349.13 1.55255i −1.73205 1.58958 0 − 2.68910i 5.94577 + 3.69430i 8.67812i 3.00000 0
349.14 1.55255i 1.73205 1.58958 0 2.68910i −5.94577 + 3.69430i 8.67812i 3.00000 0
349.15 2.17295i −1.73205 −0.721702 0 − 3.76366i −3.41555 − 6.11016i 7.12357i 3.00000 0
349.16 2.17295i 1.73205 −0.721702 0 3.76366i 3.41555 − 6.11016i 7.12357i 3.00000 0
349.17 2.67973i −1.73205 −3.18096 0 − 4.64143i 3.32267 − 6.16116i 2.19482i 3.00000 0
349.18 2.67973i 1.73205 −3.18096 0 4.64143i −3.32267 − 6.16116i 2.19482i 3.00000 0
349.19 3.26275i −1.73205 −6.64554 0 − 5.65125i 4.78534 + 5.10886i − 8.63174i 3.00000 0
349.20 3.26275i 1.73205 −6.64554 0 5.65125i −4.78534 + 5.10886i − 8.63174i 3.00000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 349.20
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 25T_{2}^{8} + 216T_{2}^{6} + 757T_{2}^{4} + 901T_{2}^{2} + 36 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 25 T^{8} + \cdots + 36)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{5} - 16 T^{4} + \cdots + 8400)^{4} \) Copy content Toggle raw display
$13$ \( (T^{10} - 1269 T^{8} + \cdots - 11663315712)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 874800000000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 128517193728)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 13410844574724)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 20 T^{4} + \cdots - 3075678)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 338397758208)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 4142 T^{8} + \cdots + 2144801344)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 1097349120000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 21\!\cdots\!01)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 17\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 516289284000000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 18172516320000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 92\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 94 T^{4} + \cdots - 260099904)^{4} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 88 T^{4} + \cdots - 272274352)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 52\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots - 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
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