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Properties

Label 525.3.o.n
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: \( x^{12} - x^{11} + 19 x^{10} + 2 x^{9} + 259 x^{8} - x^{7} + 1351 x^{6} + 488 x^{5} + 5137 x^{4} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{4}+ \cdots - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{4}+ \cdots + (3 \beta_{11} + 3 \beta_{10} + \cdots + 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} - 18 q^{3} - 13 q^{4} + 7 q^{7} + 46 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} - 18 q^{3} - 13 q^{4} + 7 q^{7} + 46 q^{8} + 18 q^{9} - 4 q^{11} + 39 q^{12} + 8 q^{14} - 49 q^{16} + 18 q^{17} + 3 q^{18} - 45 q^{19} - 18 q^{21} - 16 q^{22} - 58 q^{23} - 69 q^{24} + 147 q^{26} - 17 q^{28} + 44 q^{29} - 15 q^{31} - 153 q^{32} + 12 q^{33} - 78 q^{36} + 14 q^{37} + 159 q^{38} + 3 q^{39} - 27 q^{42} - 106 q^{43} + 114 q^{44} - 274 q^{46} - 36 q^{47} + 213 q^{49} - 18 q^{51} + 93 q^{52} - 112 q^{53} - 9 q^{54} + 253 q^{56} + 90 q^{57} - 130 q^{58} - 306 q^{59} + 276 q^{61} + 33 q^{63} + 202 q^{64} + 24 q^{66} - 141 q^{67} + 300 q^{68} - 220 q^{71} + 69 q^{72} + 240 q^{73} - 185 q^{74} - 364 q^{77} - 294 q^{78} - 42 q^{79} - 54 q^{81} + 258 q^{82} + 174 q^{84} - 362 q^{86} - 66 q^{87} - 104 q^{88} + 192 q^{89} + 135 q^{91} + 900 q^{92} + 15 q^{93} - 708 q^{94} + 459 q^{96} - 31 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 19 x^{10} + 2 x^{9} + 259 x^{8} - x^{7} + 1351 x^{6} + 488 x^{5} + 5137 x^{4} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1044027009 \nu^{11} + 5036953105 \nu^{10} - 23577292347 \nu^{9} + 72733992866 \nu^{8} + \cdots + 120312429372 ) / 3358090568244 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2088532087 \nu^{11} - 13805477343 \nu^{10} + 54276478049 \nu^{9} - 219400927878 \nu^{8} + \cdots - 9896536791504 ) / 3358090568244 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1135617979 \nu^{11} - 13240469727 \nu^{10} + 33320853599 \nu^{9} - 224721216950 \nu^{8} + \cdots - 436107232764 ) / 1679045284122 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 935194794 \nu^{11} + 674307559 \nu^{10} - 17442355593 \nu^{9} - 5984239336 \nu^{8} + \cdots + 4855805540586 ) / 839522642061 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 998231524 \nu^{11} - 935194794 \nu^{10} + 18705511721 \nu^{9} + 3259619176 \nu^{8} + \cdots + 37584972324 ) / 839522642061 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33048346701 \nu^{11} + 36218120541 \nu^{10} - 639852860917 \nu^{9} - 356500772 \nu^{8} + \cdots - 4077506477832 ) / 3358090568244 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17910003551 \nu^{11} - 2438935121 \nu^{10} + 318331355323 \nu^{9} + 338723971188 \nu^{8} + \cdots + 11571159407322 ) / 1679045284122 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 37915712279 \nu^{11} + 37906702085 \nu^{10} - 729553809367 \nu^{9} - 57941519778 \nu^{8} + \cdots + 2665838870364 ) / 3358090568244 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39105630635 \nu^{11} + 41869715049 \nu^{10} - 732016721231 \nu^{9} - 42145826696 \nu^{8} + \cdots - 2335570588392 ) / 3358090568244 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 42269939967 \nu^{11} + 52256818043 \nu^{10} - 807549571401 \nu^{9} + 99914641478 \nu^{8} + \cdots - 1106152448364 ) / 3358090568244 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} + 6\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 11\beta_{6} + \beta_{5} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 14 \beta_{5} - 14 \beta_{4} + \beta_{3} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 18 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 139 \beta_{6} + \cdots - 139 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 28 \beta_{11} + 2 \beta_{10} + 40 \beta_{9} - 2 \beta_{8} - 38 \beta_{7} - 323 \beta_{6} - 155 \beta_{5} + \cdots + 756 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96 \beta_{10} + 58 \beta_{9} + 58 \beta_{8} - 96 \beta_{7} - 445 \beta_{5} + 445 \beta_{4} - 271 \beta_{3} + \cdots + 1347 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 561 \beta_{11} + 580 \beta_{10} - 58 \beta_{9} + 638 \beta_{8} - 58 \beta_{7} + 5245 \beta_{6} + \cdots + 5245 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3916 \beta_{11} - 580 \beta_{10} - 1760 \beta_{9} + 580 \beta_{8} + 1180 \beta_{7} + 26633 \beta_{6} + \cdots - 23028 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9592 \beta_{10} - 8412 \beta_{9} - 8412 \beta_{8} + 9592 \beta_{7} + 39561 \beta_{5} - 39561 \beta_{4} + \cdots - 135534 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 56385 \beta_{11} - 20916 \beta_{10} + 8412 \beta_{9} - 29328 \beta_{8} + 8412 \beta_{7} + \cdots - 385495 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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
1.94247 + 3.36446i
1.26136 + 2.18474i
0.213109 + 0.369116i
−0.200737 − 0.347687i
−1.12234 − 1.94396i
−1.59386 − 2.76065i
1.94247 − 3.36446i
1.26136 − 2.18474i
0.213109 − 0.369116i
−0.200737 + 0.347687i
−1.12234 + 1.94396i
−1.59386 + 2.76065i
−1.94247 + 3.36446i −1.50000 + 0.866025i −5.54641 − 9.60667i 0 − 6.72893i 5.91645 + 3.74107i 27.5553 1.50000 − 2.59808i 0
376.2 −1.26136 + 2.18474i −1.50000 + 0.866025i −1.18206 − 2.04738i 0 − 4.36948i −3.66055 − 5.96660i −4.12688 1.50000 − 2.59808i 0
376.3 −0.213109 + 0.369116i −1.50000 + 0.866025i 1.90917 + 3.30678i 0 − 0.738232i −4.56839 + 5.30375i −3.33232 1.50000 − 2.59808i 0
376.4 0.200737 − 0.347687i −1.50000 + 0.866025i 1.91941 + 3.32451i 0 0.695374i 6.75245 − 1.84511i 3.14709 1.50000 − 2.59808i 0
376.5 1.12234 − 1.94396i −1.50000 + 0.866025i −0.519314 − 0.899478i 0 3.88791i −6.98682 − 0.429379i 6.64736 1.50000 − 2.59808i 0
376.6 1.59386 − 2.76065i −1.50000 + 0.866025i −3.08079 − 5.33609i 0 5.52130i 6.04686 + 3.52639i −6.89053 1.50000 − 2.59808i 0
451.1 −1.94247 − 3.36446i −1.50000 − 0.866025i −5.54641 + 9.60667i 0 6.72893i 5.91645 − 3.74107i 27.5553 1.50000 + 2.59808i 0
451.2 −1.26136 − 2.18474i −1.50000 − 0.866025i −1.18206 + 2.04738i 0 4.36948i −3.66055 + 5.96660i −4.12688 1.50000 + 2.59808i 0
451.3 −0.213109 − 0.369116i −1.50000 − 0.866025i 1.90917 − 3.30678i 0 0.738232i −4.56839 − 5.30375i −3.33232 1.50000 + 2.59808i 0
451.4 0.200737 + 0.347687i −1.50000 − 0.866025i 1.91941 − 3.32451i 0 − 0.695374i 6.75245 + 1.84511i 3.14709 1.50000 + 2.59808i 0
451.5 1.12234 + 1.94396i −1.50000 − 0.866025i −0.519314 + 0.899478i 0 − 3.88791i −6.98682 + 0.429379i 6.64736 1.50000 + 2.59808i 0
451.6 1.59386 + 2.76065i −1.50000 − 0.866025i −3.08079 + 5.33609i 0 − 5.52130i 6.04686 − 3.52639i −6.89053 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.n ✓ 12
5.b even 2 1 525.3.o.o yes 12
5.c odd 4 2 525.3.s.i 24
7.d odd 6 1 inner 525.3.o.n ✓ 12
35.i odd 6 1 525.3.o.o yes 12
35.k even 12 2 525.3.s.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.3.o.n ✓ 12 1.a even 1 1 trivial
525.3.o.n ✓ 12 7.d odd 6 1 inner
525.3.o.o yes 12 5.b even 2 1
525.3.o.o yes 12 35.i odd 6 1
525.3.s.i 24 5.c odd 4 2
525.3.s.i 24 35.k even 12 2

Hecke kernels

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

\( T_{2}^{12} + T_{2}^{11} + 19 T_{2}^{10} - 2 T_{2}^{9} + 259 T_{2}^{8} + T_{2}^{7} + 1351 T_{2}^{6} + \cdots + 144 \) Copy content Toggle raw display
\( T_{11}^{12} + 4 T_{11}^{11} + 568 T_{11}^{10} + 1384 T_{11}^{9} + 224680 T_{11}^{8} + \cdots + 10697609318400 \) Copy content Toggle raw display
\( T_{13}^{12} + 921 T_{13}^{10} + 290211 T_{13}^{8} + 38720907 T_{13}^{6} + 2416609224 T_{13}^{4} + \cdots + 765779007744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 144 \) 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
$11$ \( T^{12} + \cdots + 10697609318400 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 765779007744 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{11} + \cdots + 5645376 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 53642850960384 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 135355540377600 \) Copy content Toggle raw display
$29$ \( (T^{6} - 22 T^{5} + \cdots + 548632320)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{6} + 53 T^{5} + \cdots + 3475739104)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 43\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + 110 T^{5} + \cdots - 7273534512)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 53\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 56\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 29\!\cdots\!89 \) Copy content Toggle raw display
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