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Properties

Label 105.3.c.a
Level $105$
Weight $3$
Character orbit 105.c
Analytic conductor $2.861$
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] = [105,3,Mod(71,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, 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("105.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.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: \(2.86104277578\)
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: \( 2^{10}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{3} + ( - \beta_{8} + \beta_{7} - 2) q^{4} + \beta_{9} q^{5} + ( - \beta_{15} + \beta_{13} - \beta_{9} + \cdots - 3) q^{6}+ \cdots + ( - \beta_{15} + \beta_{13} - \beta_{12} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{3} + ( - \beta_{8} + \beta_{7} - 2) q^{4} + \beta_{9} q^{5} + ( - \beta_{15} + \beta_{13} - \beta_{9} + \cdots - 3) q^{6}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - 15) q^{99}+O(q^{100}) \) 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 \( 16 q + 8 q^{3} - 28 q^{4} - 28 q^{6} + 22 q^{9} - 20 q^{10} + 12 q^{12} + 10 q^{15} + 92 q^{16} - 52 q^{18} - 16 q^{19} - 14 q^{21} + 16 q^{22} + 128 q^{24} - 80 q^{25} - 148 q^{27} + 112 q^{28} + 80 q^{30} - 72 q^{31} - 4 q^{33} - 176 q^{34} - 76 q^{36} - 40 q^{37} + 90 q^{39} - 60 q^{40} + 280 q^{43} + 40 q^{45} + 72 q^{46} - 172 q^{48} + 112 q^{49} + 38 q^{51} - 88 q^{52} + 208 q^{54} + 80 q^{55} - 36 q^{57} - 24 q^{58} - 80 q^{60} - 56 q^{61} - 56 q^{63} - 44 q^{64} - 260 q^{66} - 120 q^{67} + 60 q^{69} + 376 q^{72} - 208 q^{73} - 40 q^{75} + 144 q^{76} - 228 q^{78} - 204 q^{79} + 458 q^{81} - 384 q^{82} - 84 q^{84} + 100 q^{85} - 324 q^{87} + 168 q^{88} - 160 q^{90} - 28 q^{91} + 108 q^{93} + 984 q^{94} + 40 q^{96} + 728 q^{97} - 166 q^{99}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( - 19 \nu^{15} + 346 \nu^{14} - 314 \nu^{13} + 11114 \nu^{12} + 6979 \nu^{11} + 96196 \nu^{10} + \cdots - 1400928 ) / 468672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22 \nu^{15} - 87 \nu^{14} - 749 \nu^{13} - 2851 \nu^{12} - 6822 \nu^{11} - 26756 \nu^{10} + \cdots + 884592 ) / 117168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 355 \nu^{15} - 362 \nu^{14} + 14860 \nu^{13} - 13434 \nu^{12} + 235239 \nu^{11} - 173056 \nu^{10} + \cdots - 480768 ) / 468672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 355 \nu^{15} - 454 \nu^{14} + 14860 \nu^{13} - 17010 \nu^{12} + 235239 \nu^{11} - 222000 \nu^{10} + \cdots + 357696 ) / 468672 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9 \nu^{15} - 536 \nu^{14} + 1136 \nu^{13} - 19136 \nu^{12} + 51355 \nu^{11} - 226568 \nu^{10} + \cdots + 1054080 ) / 234336 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + \cdots + 780032 ) / 78112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 187 \nu^{14} + 7587 \nu^{12} + 114130 \nu^{10} + 791734 \nu^{8} + 2620631 \nu^{6} + 3968375 \nu^{4} + \cdots + 311360 ) / 78112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 321 \nu^{15} - 14812 \nu^{13} - 265971 \nu^{11} - 2352364 \nu^{9} - 10668555 \nu^{7} + \cdots - 3269960 \nu ) / 234336 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 48 \nu^{15} - 1040 \nu^{14} - 4519 \nu^{13} - 43396 \nu^{12} - 130499 \nu^{11} - 680212 \nu^{10} + \cdots - 455520 ) / 234336 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1141 \nu^{15} - 346 \nu^{14} + 45836 \nu^{13} - 11114 \nu^{12} + 677801 \nu^{11} - 96196 \nu^{10} + \cdots + 1400928 ) / 468672 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 695 \nu^{15} + 354 \nu^{14} + 32427 \nu^{13} + 12274 \nu^{12} + 589430 \nu^{11} + 134626 \nu^{10} + \cdots - 225744 ) / 234336 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 695 \nu^{15} + 354 \nu^{14} + 29986 \nu^{13} + 12274 \nu^{12} + 494231 \nu^{11} + 134626 \nu^{10} + \cdots - 225744 ) / 234336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1823 \nu^{15} - 956 \nu^{14} + 86142 \nu^{13} - 40980 \nu^{12} + 1587409 \nu^{11} - 664816 \nu^{10} + \cdots + 1631712 ) / 468672 \) 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
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{7} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{11} + \beta_{9} + \beta_{4} - 2\beta_{2} - 11\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 18 \beta_{8} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - 2 \beta_{14} - 22 \beta_{13} + 2 \beta_{12} + 16 \beta_{11} + 2 \beta_{10} - 24 \beta_{9} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22 \beta_{15} + 16 \beta_{13} - 18 \beta_{12} + 20 \beta_{11} - 36 \beta_{10} + 40 \beta_{9} + \cdots - 692 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 34 \beta_{15} + 48 \beta_{14} + 353 \beta_{13} - 46 \beta_{12} - 225 \beta_{11} - 48 \beta_{10} + \cdots - 267 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 343 \beta_{15} - 12 \beta_{14} - 217 \beta_{13} + 271 \beta_{12} - 313 \beta_{11} + 530 \beta_{10} + \cdots + 8626 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 380 \beta_{15} - 854 \beta_{14} - 5158 \beta_{13} + 860 \beta_{12} + 3116 \beta_{11} + 854 \beta_{10} + \cdots + 3708 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4724 \beta_{15} + 480 \beta_{14} + 2808 \beta_{13} - 3924 \beta_{12} + 4560 \beta_{11} - 7368 \beta_{10} + \cdots - 112226 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2828 \beta_{15} + 13672 \beta_{14} + 72729 \beta_{13} - 14956 \beta_{12} - 43313 \beta_{11} + \cdots - 50085 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 61357 \beta_{15} - 12128 \beta_{14} - 35657 \beta_{13} + 56229 \beta_{12} - 64673 \beta_{11} + \cdots + 1494128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1122 \beta_{15} - 208722 \beta_{14} - 1010382 \beta_{13} + 248790 \beta_{12} + 604952 \beta_{11} + \cdots + 668262 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 771362 \beta_{15} + 249912 \beta_{14} + 448648 \beta_{13} - 802782 \beta_{12} + 907004 \beta_{11} + \cdots - 20142648 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 596674 \beta_{15} + 3109496 \beta_{14} + 13949913 \beta_{13} - 4004922 \beta_{12} - 8474417 \beta_{11} + \cdots - 8870671 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
− 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 2.23607i −9.77207 − 5.49456i −2.64575 22.2903i −4.67625 − 7.68978i 8.35609
71.2 − 3.57278i −2.99481 + 0.176471i −8.76473 − 2.23607i 0.630492 + 10.6998i −2.64575 17.0233i 8.93772 − 1.05699i −7.98897
71.3 − 2.62785i −0.217001 − 2.99214i −2.90558 − 2.23607i −7.86289 + 0.570245i 2.64575 − 2.87597i −8.90582 + 1.29859i −5.87604
71.4 − 2.60953i 2.93192 + 0.635503i −2.80964 − 2.23607i 1.65836 − 7.65092i −2.64575 − 3.10627i 8.19227 + 3.72648i −5.83508
71.5 − 2.02253i 2.91626 − 0.703870i −0.0906451 2.23607i −1.42360 − 5.89823i 2.64575 − 7.90680i 8.00914 − 4.10533i 4.52252
71.6 − 1.02879i −2.94860 + 0.552947i 2.94159 − 2.23607i 0.568867 + 3.03349i 2.64575 − 7.14144i 8.38850 − 3.26084i −2.30045
71.7 − 0.601965i 0.926467 + 2.85336i 3.63764 − 2.23607i 1.71762 − 0.557701i 2.64575 − 4.59759i −7.28332 + 5.28709i −1.34603
71.8 − 0.209282i 1.91543 + 2.30892i 3.95620 2.23607i 0.483215 − 0.400865i −2.64575 − 1.66509i −1.66223 + 8.84517i 0.467968
71.9 0.209282i 1.91543 − 2.30892i 3.95620 − 2.23607i 0.483215 + 0.400865i −2.64575 1.66509i −1.66223 − 8.84517i 0.467968
71.10 0.601965i 0.926467 − 2.85336i 3.63764 2.23607i 1.71762 + 0.557701i 2.64575 4.59759i −7.28332 − 5.28709i −1.34603
71.11 1.02879i −2.94860 − 0.552947i 2.94159 2.23607i 0.568867 − 3.03349i 2.64575 7.14144i 8.38850 + 3.26084i −2.30045
71.12 2.02253i 2.91626 + 0.703870i −0.0906451 − 2.23607i −1.42360 + 5.89823i 2.64575 7.90680i 8.00914 + 4.10533i 4.52252
71.13 2.60953i 2.93192 − 0.635503i −2.80964 2.23607i 1.65836 + 7.65092i −2.64575 3.10627i 8.19227 − 3.72648i −5.83508
71.14 2.62785i −0.217001 + 2.99214i −2.90558 2.23607i −7.86289 − 0.570245i 2.64575 2.87597i −8.90582 − 1.29859i −5.87604
71.15 3.57278i −2.99481 − 0.176471i −8.76473 2.23607i 0.630492 − 10.6998i −2.64575 − 17.0233i 8.93772 + 1.05699i −7.98897
71.16 3.73696i 1.47033 + 2.61498i −9.96484 − 2.23607i −9.77207 + 5.49456i −2.64575 − 22.2903i −4.67625 + 7.68978i 8.35609
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.16
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 105.3.c.a ✓ 16
3.b odd 2 1 inner 105.3.c.a ✓ 16
4.b odd 2 1 1680.3.l.a 16
5.b even 2 1 525.3.c.b 16
5.c odd 4 2 525.3.f.b 32
12.b even 2 1 1680.3.l.a 16
15.d odd 2 1 525.3.c.b 16
15.e even 4 2 525.3.f.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.c.a ✓ 16 1.a even 1 1 trivial
105.3.c.a ✓ 16 3.b odd 2 1 inner
525.3.c.b 16 5.b even 2 1
525.3.c.b 16 15.d odd 2 1
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 4.b odd 2 1
1680.3.l.a 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(105, [\chi])\).

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
$5$ \( (T^{2} + 5)^{8} \) 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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