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Properties

Label 525.3.c.b
Level $525$
Weight $3$
Character orbit 525.c
Analytic conductor $14.305$
Analytic rank $0$
Dimension $16$
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(176,525)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.176");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.c (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{5} + \beta_{4} - 2) q^{4} + ( - \beta_{15} + \beta_{13} + \beta_{8} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{15} + \beta_{13} - \beta_{7} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{5} + \beta_{4} - 2) q^{4} + ( - \beta_{15} + \beta_{13} + \beta_{8} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} - 28 q^{4} - 28 q^{6} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} - 28 q^{4} - 28 q^{6} + 22 q^{9} - 12 q^{12} + 92 q^{16} + 52 q^{18} - 16 q^{19} - 14 q^{21} - 16 q^{22} + 128 q^{24} + 148 q^{27} - 112 q^{28} - 72 q^{31} + 4 q^{33} - 176 q^{34} - 76 q^{36} + 40 q^{37} + 90 q^{39} - 280 q^{43} + 72 q^{46} + 172 q^{48} + 112 q^{49} + 38 q^{51} + 88 q^{52} + 208 q^{54} + 36 q^{57} + 24 q^{58} - 56 q^{61} + 56 q^{63} - 44 q^{64} - 260 q^{66} + 120 q^{67} + 60 q^{69} - 376 q^{72} + 208 q^{73} + 144 q^{76} + 228 q^{78} - 204 q^{79} + 458 q^{81} + 384 q^{82} - 84 q^{84} + 324 q^{87} - 168 q^{88} - 28 q^{91} - 108 q^{93} + 984 q^{94} + 40 q^{96} - 728 q^{97} - 166 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22 \nu^{15} - 87 \nu^{14} + 749 \nu^{13} - 2851 \nu^{12} + 6822 \nu^{11} - 26756 \nu^{10} + \cdots + 884592 ) / 117168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 355 \nu^{15} + 362 \nu^{14} + 14860 \nu^{13} + 13434 \nu^{12} + 235239 \nu^{11} + 173056 \nu^{10} + \cdots + 480768 ) / 468672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + \cdots + 780032 ) / 78112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 187 \nu^{14} - 7587 \nu^{12} - 114130 \nu^{10} - 791734 \nu^{8} - 2620631 \nu^{6} - 3968375 \nu^{4} + \cdots - 311360 ) / 78112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 107 \nu^{15} + 5751 \nu^{13} + 120390 \nu^{11} + 1231638 \nu^{9} + 6324279 \nu^{7} + \cdots + 3596080 \nu ) / 78112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 527 \nu^{15} - 612 \nu^{14} - 25154 \nu^{13} - 30156 \nu^{12} - 468321 \nu^{11} - 579448 \nu^{10} + \cdots - 443808 ) / 468672 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 767 \nu^{15} + 362 \nu^{14} - 30662 \nu^{13} + 13434 \nu^{12} - 449541 \nu^{11} + 173056 \nu^{10} + \cdots + 480768 ) / 468672 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 156 \nu^{15} + 268 \nu^{14} + 7974 \nu^{13} + 9568 \nu^{12} + 158663 \nu^{11} + 113284 \nu^{10} + \cdots - 527040 ) / 117168 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 187 \nu^{15} + 7587 \nu^{13} + 114130 \nu^{11} + 791734 \nu^{9} + 2620631 \nu^{7} + \cdots + 233248 \nu ) / 78112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 997 \nu^{15} - 454 \nu^{14} - 44484 \nu^{13} - 17010 \nu^{12} - 767181 \nu^{11} - 222000 \nu^{10} + \cdots + 357696 ) / 468672 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 422 \nu^{15} + 833 \nu^{14} - 19693 \nu^{13} + 32909 \nu^{12} - 358759 \nu^{11} + 472448 \nu^{10} + \cdots - 704304 ) / 234336 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 321 \nu^{15} - 14812 \nu^{13} - 265971 \nu^{11} - 2352364 \nu^{9} - 10668555 \nu^{7} + \cdots - 3269960 \nu ) / 117168 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1823 \nu^{15} + 166 \nu^{14} - 86142 \nu^{13} + 4542 \nu^{12} - 1587409 \nu^{11} + \cdots + 3499872 ) / 468672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 511 \nu^{15} - 87 \nu^{14} - 22834 \nu^{13} - 2851 \nu^{12} - 393902 \nu^{11} - 26756 \nu^{10} + \cdots + 884592 ) / 117168 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - \beta_{8} + \beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 2 \beta_{13} + 4 \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 16 \beta_{10} - 2 \beta_{9} + \cdots + 117 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22 \beta_{15} + 42 \beta_{13} - 2 \beta_{12} - 88 \beta_{11} - 20 \beta_{10} + 20 \beta_{9} - 38 \beta_{7} + \cdots - 692 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 34 \beta_{15} - 48 \beta_{14} + 60 \beta_{13} - 48 \beta_{12} - 225 \beta_{10} + 40 \beta_{9} - 259 \beta_{8} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 343 \beta_{15} - 12 \beta_{14} - 692 \beta_{13} + 54 \beta_{12} + 1468 \beta_{11} + 313 \beta_{10} + \cdots + 8626 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 380 \beta_{15} + 854 \beta_{14} - 1178 \beta_{13} + 854 \beta_{12} + 3116 \beta_{10} - 590 \beta_{9} + \cdots - 264 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4724 \beta_{15} + 480 \beta_{14} + 10524 \beta_{13} - 1116 \beta_{12} - 22320 \beta_{11} - 4560 \beta_{10} + \cdots - 112226 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2828 \beta_{15} - 13672 \beta_{14} + 19768 \beta_{13} - 13672 \beta_{12} - 43313 \beta_{10} + \cdots + 5984 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 61357 \beta_{15} - 12128 \beta_{14} - 154710 \beta_{13} + 20572 \beta_{12} + 326308 \beta_{11} + \cdots + 1494128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1122 \beta_{15} + 208722 \beta_{14} - 309042 \beta_{13} + 208722 \beta_{12} + 604952 \beta_{10} + \cdots - 115392 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 771362 \beta_{15} + 249912 \beta_{14} + 2237598 \beta_{13} - 354134 \beta_{12} - 4683640 \beta_{11} + \cdots - 20142648 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 596674 \beta_{15} - 3109496 \beta_{14} + 4662788 \beta_{13} - 3109496 \beta_{12} - 8474417 \beta_{10} + \cdots + 2035176 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1
− 3.73696i
− 3.57278i
− 2.62785i
− 2.60953i
− 2.02253i
− 1.02879i
− 0.601965i
− 0.209282i
0.209282i
0.601965i
1.02879i
2.02253i
2.60953i
2.62785i
3.57278i
3.73696i
− 3.73696i −1.47033 − 2.61498i −9.96484 0 −9.77207 + 5.49456i 2.64575 22.2903i −4.67625 + 7.68978i 0
176.2 − 3.57278i 2.99481 + 0.176471i −8.76473 0 0.630492 − 10.6998i 2.64575 17.0233i 8.93772 + 1.05699i 0
176.3 − 2.62785i 0.217001 − 2.99214i −2.90558 0 −7.86289 − 0.570245i −2.64575 − 2.87597i −8.90582 − 1.29859i 0
176.4 − 2.60953i −2.93192 + 0.635503i −2.80964 0 1.65836 + 7.65092i 2.64575 − 3.10627i 8.19227 − 3.72648i 0
176.5 − 2.02253i −2.91626 − 0.703870i −0.0906451 0 −1.42360 + 5.89823i −2.64575 − 7.90680i 8.00914 + 4.10533i 0
176.6 − 1.02879i 2.94860 + 0.552947i 2.94159 0 0.568867 − 3.03349i −2.64575 − 7.14144i 8.38850 + 3.26084i 0
176.7 − 0.601965i −0.926467 + 2.85336i 3.63764 0 1.71762 + 0.557701i −2.64575 − 4.59759i −7.28332 − 5.28709i 0
176.8 − 0.209282i −1.91543 + 2.30892i 3.95620 0 0.483215 + 0.400865i 2.64575 − 1.66509i −1.66223 − 8.84517i 0
176.9 0.209282i −1.91543 − 2.30892i 3.95620 0 0.483215 − 0.400865i 2.64575 1.66509i −1.66223 + 8.84517i 0
176.10 0.601965i −0.926467 − 2.85336i 3.63764 0 1.71762 − 0.557701i −2.64575 4.59759i −7.28332 + 5.28709i 0
176.11 1.02879i 2.94860 − 0.552947i 2.94159 0 0.568867 + 3.03349i −2.64575 7.14144i 8.38850 − 3.26084i 0
176.12 2.02253i −2.91626 + 0.703870i −0.0906451 0 −1.42360 − 5.89823i −2.64575 7.90680i 8.00914 − 4.10533i 0
176.13 2.60953i −2.93192 − 0.635503i −2.80964 0 1.65836 − 7.65092i 2.64575 3.10627i 8.19227 + 3.72648i 0
176.14 2.62785i 0.217001 + 2.99214i −2.90558 0 −7.86289 + 0.570245i −2.64575 2.87597i −8.90582 + 1.29859i 0
176.15 3.57278i 2.99481 − 0.176471i −8.76473 0 0.630492 + 10.6998i 2.64575 − 17.0233i 8.93772 − 1.05699i 0
176.16 3.73696i −1.47033 + 2.61498i −9.96484 0 −9.77207 − 5.49456i 2.64575 − 22.2903i −4.67625 − 7.68978i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.c.b 16
3.b odd 2 1 inner 525.3.c.b 16
5.b even 2 1 105.3.c.a ✓ 16
5.c odd 4 2 525.3.f.b 32
15.d odd 2 1 105.3.c.a ✓ 16
15.e even 4 2 525.3.f.b 32
20.d odd 2 1 1680.3.l.a 16
60.h even 2 1 1680.3.l.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.c.a ✓ 16 5.b even 2 1
105.3.c.a ✓ 16 15.d odd 2 1
525.3.c.b 16 1.a even 1 1 trivial
525.3.c.b 16 3.b odd 2 1 inner
525.3.f.b 32 5.c odd 4 2
525.3.f.b 32 15.e even 4 2
1680.3.l.a 16 20.d odd 2 1
1680.3.l.a 16 60.h even 2 1

Hecke kernels

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

\( T_{2}^{16} + 46T_{2}^{14} + 823T_{2}^{12} + 7252T_{2}^{10} + 32831T_{2}^{8} + 71486T_{2}^{6} + 60809T_{2}^{4} + 15680T_{2}^{2} + 576 \) Copy content Toggle raw display
\( T_{13}^{8} - 607 T_{13}^{6} - 1452 T_{13}^{5} + 112572 T_{13}^{4} + 325952 T_{13}^{3} - 7726480 T_{13}^{2} + \cdots + 187600704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 46 T^{14} + \cdots + 576 \) Copy content Toggle raw display
$3$ \( T^{16} + 8 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{8} - 607 T^{6} + \cdots + 187600704)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{7} + \cdots + 18126732544)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 53\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} + 36 T^{7} + \cdots - 949999104)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 20 T^{7} + \cdots - 722645680896)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 59\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{8} + 140 T^{7} + \cdots + 104284349696)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 58902532217856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 49453479094784)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 1772600244224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 110155963598144)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 99213562086336)^{2} \) Copy content Toggle raw display
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