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Properties

Label 525.6.d.p
Level $525$
Weight $6$
Character orbit 525.d
Analytic conductor $84.202$
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] = [525,6,Mod(274,525)]
 
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, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), 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: \(84.2015054018\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 322x^{10} + 38621x^{8} + 2116025x^{6} + 52188850x^{4} + 497085625x^{2} + 1556302500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{2} - 9 \beta_{6} q^{3} + (\beta_{3} - \beta_{2} - 22) q^{4} + (9 \beta_{2} + 9) q^{6} - 49 \beta_{6} q^{7} + (3 \beta_{9} + \beta_{8} + \cdots - 18 \beta_1) q^{8}+ \cdots - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{2} - 9 \beta_{6} q^{3} + (\beta_{3} - \beta_{2} - 22) q^{4} + (9 \beta_{2} + 9) q^{6} - 49 \beta_{6} q^{7} + (3 \beta_{9} + \beta_{8} + \cdots - 18 \beta_1) q^{8}+ \cdots + (81 \beta_{7} + 243 \beta_{3} + \cdots - 486) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 264 q^{4} + 72 q^{6} - 972 q^{9} + 392 q^{14} + 3568 q^{16} - 520 q^{19} - 5292 q^{21} - 5454 q^{24} - 6534 q^{26} + 4244 q^{29} + 4524 q^{31} - 24778 q^{34} + 21384 q^{36} + 3816 q^{39} + 16248 q^{41} - 62322 q^{44} + 58720 q^{46} - 28812 q^{49} + 67320 q^{51} - 5832 q^{54} - 29694 q^{56} - 51316 q^{59} + 82376 q^{61} - 131894 q^{64} + 124218 q^{66} + 33336 q^{69} + 35160 q^{71} - 7942 q^{74} + 507092 q^{76} + 19472 q^{79} + 78732 q^{81} + 116424 q^{84} + 287632 q^{86} + 660720 q^{89} + 20776 q^{91} + 747360 q^{94} + 491220 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 322x^{10} + 38621x^{8} + 2116025x^{6} + 52188850x^{4} + 497085625x^{2} + 1556302500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 169\nu^{10} + 19533\nu^{8} - 1468596\nu^{6} - 248876635\nu^{4} - 8152256175\nu^{2} - 49099075500 ) / 1674596000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -169\nu^{10} - 19533\nu^{8} + 1468596\nu^{6} + 248876635\nu^{4} + 9826852175\nu^{2} + 137852663500 ) / 1674596000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -169\nu^{10} - 19533\nu^{8} + 1468596\nu^{6} + 39552135\nu^{4} - 14873438825\nu^{2} - 264887674500 ) / 627973500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21027 \nu^{10} + 6393839 \nu^{8} + 726909732 \nu^{6} + 37380055895 \nu^{4} + \cdots + 4357797749500 ) / 1674596000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 124459 \nu^{11} - 40742503 \nu^{9} - 4883788724 \nu^{7} - 257564744255 \nu^{5} + \cdots - 29706129191500 \nu ) / 6606281220000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8979 \nu^{10} - 2796613 \nu^{8} - 315156484 \nu^{6} - 15378447175 \nu^{4} + \cdots - 1522141818300 ) / 251189400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5102819 \nu^{11} + 1670442623 \nu^{9} + 200235337684 \nu^{7} + 10560154514455 \nu^{5} + \cdots + 17\!\cdots\!00 \nu ) / 6606281220000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 129697 \nu^{11} - 39935899 \nu^{9} - 4518699842 \nu^{7} - 226234162865 \nu^{5} + \cdots - 24773844205750 \nu ) / 117969307500 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17895307 \nu^{11} + 5552587119 \nu^{9} + 620640351252 \nu^{7} + 29348488220415 \nu^{5} + \cdots + 13\!\cdots\!00 \nu ) / 6606281220000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1632063 \nu^{11} + 507302331 \nu^{9} + 57277374388 \nu^{7} + 2818558436995 \nu^{5} + \cdots + 295496067546700 \nu ) / 264251248800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 53 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 41\beta_{6} - 82\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{4} - 110\beta_{3} - 118\beta_{2} + 4330 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{11} - 12\beta_{10} + 17\beta_{9} - 128\beta_{8} - 5015\beta_{6} + 7522\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{7} + 71\beta_{5} + 495\beta_{4} + 11224\beta_{3} + 12211\beta_{2} - 396465 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1866\beta_{11} + 1956\beta_{10} - 4676\beta_{9} + 14194\beta_{8} + 524340\beta_{6} - 721315\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -5508\beta_{7} - 15872\beta_{5} - 61872\beta_{4} - 1137273\beta_{3} - 1278409\beta_{2} + 37965565 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 271788\beta_{11} - 241980\beta_{10} + 820740\beta_{9} - 1508505\beta_{8} - 55436865\beta_{6} + 70818418\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 845172\beta_{7} + 2451468\beta_{5} + 7034739\beta_{4} + 115229002\beta_{3} + 138246142\beta_{2} - 3722857790 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 34374825 \beta_{11} + 27293784 \beta_{10} - 120369009 \beta_{9} + 157437436 \beta_{8} + \cdots - 7051047106 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 9.88653i
− 10.2315i
− 8.14446i
− 6.12018i
− 2.98787i
− 2.61865i
2.61865i
2.98787i
6.12018i
8.14446i
10.2315i
9.88653i
− 10.8865i 9.00000i −86.5165 0 97.9787 49.0000i 593.495i −81.0000 0
274.2 − 9.23147i − 9.00000i −53.2201 0 −83.0833 − 49.0000i 195.893i −81.0000 0
274.3 − 7.14446i − 9.00000i −19.0433 0 −64.3001 − 49.0000i − 92.5687i −81.0000 0
274.4 − 7.12018i 9.00000i −18.6970 0 64.0817 49.0000i − 94.7196i −81.0000 0
274.5 − 3.98787i 9.00000i 16.0969 0 35.8908 49.0000i − 191.804i −81.0000 0
274.6 − 1.61865i − 9.00000i 29.3800 0 −14.5678 − 49.0000i − 99.3526i −81.0000 0
274.7 1.61865i 9.00000i 29.3800 0 −14.5678 49.0000i 99.3526i −81.0000 0
274.8 3.98787i − 9.00000i 16.0969 0 35.8908 − 49.0000i 191.804i −81.0000 0
274.9 7.12018i − 9.00000i −18.6970 0 64.0817 − 49.0000i 94.7196i −81.0000 0
274.10 7.14446i 9.00000i −19.0433 0 −64.3001 49.0000i 92.5687i −81.0000 0
274.11 9.23147i 9.00000i −53.2201 0 −83.0833 49.0000i − 195.893i −81.0000 0
274.12 10.8865i − 9.00000i −86.5165 0 97.9787 − 49.0000i − 593.495i −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.6.d.p 12
5.b even 2 1 inner 525.6.d.p 12
5.c odd 4 1 525.6.a.r ✓ 6
5.c odd 4 1 525.6.a.s yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.6.a.r ✓ 6 5.c odd 4 1
525.6.a.s yes 6 5.c odd 4 1
525.6.d.p 12 1.a even 1 1 trivial
525.6.d.p 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 324T_{2}^{10} + 39116T_{2}^{8} + 2186489T_{2}^{6} + 56328096T_{2}^{4} + 548905600T_{2}^{2} + 1089000000 \) acting on \(S_{6}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 1089000000 \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 846458085475044)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 40\!\cdots\!91)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 47\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 27\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
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