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Properties

Label 575.4.b.j
Level $575$
Weight $4$
Character orbit 575.b
Analytic conductor $33.926$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,4,Mod(24,575)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (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: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{8} + \beta_{6} + \beta_1) q^{3} + (\beta_{5} + \beta_{4} - 4) q^{4} + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots - 5) q^{6} + ( - \beta_{12} - \beta_{8}) q^{7} + (2 \beta_{10} + 2 \beta_{8} + \cdots - 3 \beta_1) q^{8}+ \cdots + (23 \beta_{9} - 25 \beta_{7} + \cdots + 288) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2761 \nu^{12} + 193316 \nu^{10} + 4988175 \nu^{8} + 57142094 \nu^{6} + 264366298 \nu^{4} + \cdots - 132164352 ) / 21505248 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2899 \nu^{12} + 208820 \nu^{10} + 5584101 \nu^{8} + 67767626 \nu^{6} + 365072926 \nu^{4} + \cdots + 573138432 ) / 21505248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 673 \nu^{12} + 46391 \nu^{10} + 1172553 \nu^{8} + 13181345 \nu^{6} + 62463766 \nu^{4} + \cdots + 29332800 ) / 3584208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 673 \nu^{12} - 46391 \nu^{10} - 1172553 \nu^{8} - 13181345 \nu^{6} - 62463766 \nu^{4} + \cdots + 13677696 ) / 3584208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16975 \nu^{13} - 1118189 \nu^{11} - 26046213 \nu^{9} - 244991279 \nu^{7} + \cdots + 3227461056 \nu ) / 1548377856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7196 \nu^{12} + 487045 \nu^{10} + 12004194 \nu^{8} + 129952549 \nu^{6} + 577259876 \nu^{4} + \cdots + 189432288 ) / 21505248 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26717 \nu^{13} - 2037367 \nu^{11} - 58609191 \nu^{9} - 779396941 \nu^{7} + \cdots + 1827097920 \nu ) / 1548377856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27685 \nu^{12} + 1912367 \nu^{10} + 48919983 \nu^{8} + 569920133 \nu^{6} + 2981644672 \nu^{4} + \cdots + 3235561344 ) / 43010496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1077 \nu^{13} - 65807 \nu^{11} - 1326331 \nu^{9} - 8251101 \nu^{7} + 27581252 \nu^{5} + \cdots + 537403648 \nu ) / 28673664 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7253 \nu^{13} + 551887 \nu^{11} + 16165695 \nu^{9} + 229537141 \nu^{7} + 1623583616 \nu^{5} + \cdots + 4412473344 \nu ) / 129031488 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20576 \nu^{13} + 1347439 \nu^{11} + 31418445 \nu^{9} + 302528533 \nu^{7} + 890890415 \nu^{5} + \cdots - 6944411376 \nu ) / 193547232 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 183923 \nu^{13} + 12861853 \nu^{11} + 334482897 \nu^{9} + 3987741031 \nu^{7} + \cdots + 32486214528 \nu ) / 774188928 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} + 2\beta_{8} - 10\beta_{6} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 23\beta_{5} - 33\beta_{4} + 4\beta_{3} + 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{13} - 8\beta_{12} - 14\beta_{11} - 62\beta_{10} - 62\beta_{8} + 334\beta_{6} + 395\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -160\beta_{7} + 505\beta_{5} + 921\beta_{4} - 148\beta_{3} - 36\beta_{2} - 4332 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -344\beta_{13} + 296\beta_{12} + 636\beta_{11} + 1578\beta_{10} + 1698\beta_{8} - 9402\beta_{6} - 8675\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{9} + 4844\beta_{7} - 11243\beta_{5} - 24481\beta_{4} + 4212\beta_{3} + 1832\beta_{2} + 92368 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11080 \beta_{13} - 8232 \beta_{12} - 21018 \beta_{11} - 37942 \beta_{10} - 43814 \beta_{8} + \cdots + 197115 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5696 \beta_{9} - 134520 \beta_{7} + 254829 \beta_{5} + 636777 \beta_{4} - 109300 \beta_{3} + \cdots - 2054900 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 319896 \beta_{13} + 207208 \beta_{12} + 614920 \beta_{11} + 896002 \beta_{10} + 1092906 \beta_{8} + \cdots - 4579603 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 225376 \beta_{9} + 3595588 \beta_{7} - 5869375 \beta_{5} - 16347697 \beta_{4} + 2723220 \beta_{3} + \cdots + 47101912 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8741576 \beta_{13} - 4995688 \beta_{12} - 16920198 \beta_{11} - 21081230 \beta_{10} + \cdots + 108029099 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 4.96146i
− 4.39200i
− 3.84004i
− 3.74805i
− 2.80947i
− 1.37904i
− 0.711047i
0.711047i
1.37904i
2.80947i
3.74805i
3.84004i
4.39200i
4.96146i
− 4.96146i − 7.15622i −16.6161 0 −35.5053 3.69237i 42.7482i −24.2114 0
24.2 − 4.39200i − 3.27927i −11.2896 0 −14.4025 − 13.7978i 14.4481i 16.2464 0
24.3 − 3.84004i 4.08871i −6.74592 0 15.7008 27.0878i − 4.81574i 10.2825 0
24.4 − 3.74805i 2.78570i −6.04789 0 10.4409 9.96305i − 7.31660i 19.2399 0
24.5 − 2.80947i − 9.02517i 0.106856 0 −25.3560 − 11.4533i − 22.7760i −54.4537 0
24.6 − 1.37904i 5.53432i 6.09824 0 7.63206 − 8.23397i − 19.4421i −3.62871 0
24.7 − 0.711047i 0.689108i 7.49441 0 0.489988 − 19.0526i − 11.0173i 26.5251 0
24.8 0.711047i − 0.689108i 7.49441 0 0.489988 19.0526i 11.0173i 26.5251 0
24.9 1.37904i − 5.53432i 6.09824 0 7.63206 8.23397i 19.4421i −3.62871 0
24.10 2.80947i 9.02517i 0.106856 0 −25.3560 11.4533i 22.7760i −54.4537 0
24.11 3.74805i − 2.78570i −6.04789 0 10.4409 − 9.96305i 7.31660i 19.2399 0
24.12 3.84004i − 4.08871i −6.74592 0 15.7008 − 27.0878i 4.81574i 10.2825 0
24.13 4.39200i 3.27927i −11.2896 0 −14.4025 13.7978i − 14.4481i 16.2464 0
24.14 4.96146i 7.15622i −16.6161 0 −35.5053 − 3.69237i − 42.7482i −24.2114 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.14
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 575.4.b.j 14
5.b even 2 1 inner 575.4.b.j 14
5.c odd 4 1 575.4.a.l ✓ 7
5.c odd 4 1 575.4.a.m yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.4.a.l ✓ 7 5.c odd 4 1
575.4.a.m yes 7 5.c odd 4 1
575.4.b.j 14 1.a even 1 1 trivial
575.4.b.j 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(575, [\chi])\):

\( T_{2}^{14} + 83T_{2}^{12} + 2715T_{2}^{10} + 44273T_{2}^{8} + 372280T_{2}^{6} + 1482448T_{2}^{4} + 2136384T_{2}^{2} + 746496 \) Copy content Toggle raw display
\( T_{3}^{14} + 199 T_{3}^{12} + 14475 T_{3}^{10} + 490274 T_{3}^{8} + 8194505 T_{3}^{6} + 65475300 T_{3}^{4} + \cdots + 84640000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 83 T^{12} + \cdots + 746496 \) Copy content Toggle raw display
$3$ \( T^{14} + 199 T^{12} + \cdots + 84640000 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 610305007739904 \) Copy content Toggle raw display
$11$ \( (T^{7} + 52 T^{6} + \cdots - 935052376)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} - 10 T^{6} + \cdots - 451105530840)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 323393919122280)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 38841367049280)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 15\!\cdots\!85)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 771684969846528)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 67\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 30\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
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