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Properties

Label 575.3.c.c
Level $575$
Weight $3$
Character orbit 575.c
Analytic conductor $15.668$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,3,Mod(574,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.574");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 575.c (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: \(15.6676152007\)
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: \( x^{12} + 109x^{10} + 4260x^{8} + 71825x^{6} + 495285x^{4} + 936364x^{2} + 512656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 115)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_{8} q^{3} + 3 q^{4} + \beta_{2} q^{6} - \beta_{4} q^{7} + 7 \beta_{9} q^{8} + (\beta_{2} + \beta_1 - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + \beta_{8} q^{3} + 3 q^{4} + \beta_{2} q^{6} - \beta_{4} q^{7} + 7 \beta_{9} q^{8} + (\beta_{2} + \beta_1 - 8) q^{9} + \beta_{6} q^{11} + 3 \beta_{8} q^{12} + (\beta_{11} - \beta_{9} - \beta_{8}) q^{13} - \beta_{10} q^{14} + 5 q^{16} + ( - \beta_{5} + \beta_{3}) q^{17} + ( - \beta_{11} - 8 \beta_{9} - \beta_{8}) q^{18} + ( - \beta_{10} - \beta_{7}) q^{19} + ( - 3 \beta_{10} - 3 \beta_{7} + \beta_{6}) q^{21} - \beta_{5} q^{22} + (\beta_{11} - 2 \beta_{9} + \cdots + \beta_{3}) q^{23}+ \cdots + ( - 17 \beta_{10} + 5 \beta_{7} - 2 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 36 q^{4} - 4 q^{6} - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 36 q^{4} - 4 q^{6} - 96 q^{9} + 60 q^{16} - 28 q^{24} + 20 q^{26} - 144 q^{29} - 12 q^{31} - 288 q^{36} + 296 q^{39} + 284 q^{41} + 20 q^{46} + 608 q^{49} + 112 q^{54} - 472 q^{59} - 156 q^{64} - 312 q^{69} - 436 q^{71} + 708 q^{81} + 224 q^{94} - 132 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 109x^{10} + 4260x^{8} + 71825x^{6} + 495285x^{4} + 936364x^{2} + 512656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -37\nu^{10} - 5040\nu^{8} - 231120\nu^{6} - 3896605\nu^{4} - 8752160\nu^{2} + 130246912 ) / 7852600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -573\nu^{10} - 52584\nu^{8} - 1558784\nu^{6} - 15665549\nu^{4} - 32412224\nu^{2} - 24625936 ) / 15705200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} + 800\nu^{8} + 33440\nu^{6} + 608935\nu^{4} + 4253800\nu^{2} + 4433768 ) / 49700 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -103\nu^{10} - 11024\nu^{8} - 415424\nu^{6} - 6490759\nu^{4} - 37800424\nu^{2} - 37874816 ) / 663600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15619\nu^{10} + 1664288\nu^{8} + 62755328\nu^{6} + 995866483\nu^{4} + 6076925128\nu^{2} + 6158555504 ) / 47115600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19219 \nu^{11} - 1155658 \nu^{9} + 10352872 \nu^{7} + 1559657437 \nu^{5} + \cdots + 74794911656 \nu ) / 2811230800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4824 \nu^{11} + 545685 \nu^{9} + 23256720 \nu^{7} + 470595240 \nu^{5} + 4481731725 \nu^{3} + 13433776056 \nu ) / 702807700 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65111 \nu^{11} + 6873886 \nu^{9} + 254891176 \nu^{7} + 3897515111 \nu^{5} + \cdots + 21705357640 \nu ) / 8433692400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 985\nu^{11} + 104501\nu^{9} + 3917576\nu^{7} + 61522681\nu^{5} + 367972981\nu^{3} + 376132976\nu ) / 59392200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -899\nu^{11} - 89936\nu^{9} - 3042856\nu^{7} - 40195171\nu^{5} - 204087176\nu^{3} - 489383912\nu ) / 39594800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 338201 \nu^{11} - 36750244 \nu^{9} - 1422756784 \nu^{7} - 23329680209 \nu^{5} + \cdots - 151377458888 \nu ) / 1405615400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{9} + 12\beta_{8} + \beta_{7} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} - 8\beta_{4} + \beta_{3} + 12\beta _1 - 222 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{11} - 15\beta_{10} + 123\beta_{9} - 174\beta_{8} - 31\beta_{7} + 15\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 232\beta_{5} + 404\beta_{4} - 83\beta_{3} + 120\beta_{2} - 396\beta _1 + 6894 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -348\beta_{11} + 1200\beta_{10} - 10602\beta_{9} + 11592\beta_{8} + 2855\beta_{7} - 1500\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5241\beta_{5} - 8247\beta_{4} + 2399\beta_{3} - 4170\beta_{2} + 6933\beta _1 - 121185 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17508\beta_{11} - 41300\beta_{10} + 466890\beta_{9} - 416568\beta_{8} - 117473\beta_{7} + 65240\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 440764\beta_{5} + 632168\beta_{4} - 239081\beta_{3} + 458520\beta_{2} - 511212\beta _1 + 8994606 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 419403 \beta_{11} + 669015 \beta_{10} - 10308219 \beta_{9} + 7768686 \beta_{8} + \cdots - 1357335 \beta_{6} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 18050120 \beta_{5} - 23736340 \beta_{4} + 11100555 \beta_{3} - 22999800 \beta_{2} + 19340460 \beta _1 - 342527358 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 38851980 \beta_{11} - 41439200 \beta_{10} + 903627354 \beta_{9} - 590217288 \beta_{8} + \cdots + 111013100 \beta_{6} ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
574.1
− 5.95286i
− 3.71679i
1.22183i
− 1.01424i
6.34907i
4.11300i
− 6.34907i
− 4.11300i
− 1.22183i
1.01424i
5.95286i
3.71679i
− 1.00000i − 5.33482i 3.00000 0 −5.33482 −13.3268 − 7.00000i −19.4603 0
574.2 − 1.00000i − 5.33482i 3.00000 0 −5.33482 13.3268 − 7.00000i −19.4603 0
574.3 − 1.00000i − 0.396209i 3.00000 0 −0.396209 −8.16531 − 7.00000i 8.84302 0
574.4 − 1.00000i − 0.396209i 3.00000 0 −0.396209 8.16531 − 7.00000i 8.84302 0
574.5 − 1.00000i 4.73103i 3.00000 0 4.73103 −7.39757 − 7.00000i −13.3827 0
574.6 − 1.00000i 4.73103i 3.00000 0 4.73103 7.39757 − 7.00000i −13.3827 0
574.7 1.00000i − 4.73103i 3.00000 0 4.73103 −7.39757 7.00000i −13.3827 0
574.8 1.00000i − 4.73103i 3.00000 0 4.73103 7.39757 7.00000i −13.3827 0
574.9 1.00000i 0.396209i 3.00000 0 −0.396209 −8.16531 7.00000i 8.84302 0
574.10 1.00000i 0.396209i 3.00000 0 −0.396209 8.16531 7.00000i 8.84302 0
574.11 1.00000i 5.33482i 3.00000 0 −5.33482 −13.3268 7.00000i −19.4603 0
574.12 1.00000i 5.33482i 3.00000 0 −5.33482 13.3268 7.00000i −19.4603 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 574.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.b odd 2 1 inner
115.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.3.c.c 12
5.b even 2 1 inner 575.3.c.c 12
5.c odd 4 1 115.3.d.a ✓ 6
5.c odd 4 1 575.3.d.e 6
15.e even 4 1 1035.3.g.a 6
20.e even 4 1 1840.3.k.a 6
23.b odd 2 1 inner 575.3.c.c 12
115.c odd 2 1 inner 575.3.c.c 12
115.e even 4 1 115.3.d.a ✓ 6
115.e even 4 1 575.3.d.e 6
345.l odd 4 1 1035.3.g.a 6
460.k odd 4 1 1840.3.k.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.d.a ✓ 6 5.c odd 4 1
115.3.d.a ✓ 6 115.e even 4 1
575.3.c.c 12 1.a even 1 1 trivial
575.3.c.c 12 5.b even 2 1 inner
575.3.c.c 12 23.b odd 2 1 inner
575.3.c.c 12 115.c odd 2 1 inner
575.3.d.e 6 5.c odd 4 1
575.3.d.e 6 115.e even 4 1
1035.3.g.a 6 15.e even 4 1
1035.3.g.a 6 345.l odd 4 1
1840.3.k.a 6 20.e even 4 1
1840.3.k.a 6 460.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} + 51 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 299 T^{4} + \cdots - 648000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 731 T^{4} + \cdots + 8632980)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 539 T^{4} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 971 T^{4} + \cdots - 524880)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 779 T^{4} + \cdots + 714420)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{3} + 36 T^{2} + \cdots - 8872)^{4} \) Copy content Toggle raw display
$31$ \( (T^{3} + 3 T^{2} + \cdots - 30050)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 1856 T^{4} + \cdots - 123405120)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 71 T^{2} + \cdots + 8462)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} - 5936 T^{4} + \cdots - 2411208000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 3168 T^{4} + \cdots + 268173376)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 1620)^{6} \) Copy content Toggle raw display
$59$ \( (T^{3} + 118 T^{2} + \cdots - 118680)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} + 18819 T^{4} + \cdots + 649116180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 5876 T^{4} + \cdots - 5202247680)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 109 T^{2} + \cdots - 33718)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 1824 T^{4} + \cdots + 118336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 20844 T^{4} + \cdots + 70054917120)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 23924 T^{4} + \cdots - 29786849280)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 23436 T^{4} + \cdots + 39983258880)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 35411 T^{4} + \cdots - 25687244880)^{2} \) Copy content Toggle raw display
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