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Properties

Label 105.3.e.a
Level $105$
Weight $3$
Character orbit 105.e
Analytic conductor $2.861$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,3,Mod(34,105)]
 
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([0, 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("105.34");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.e (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: \( x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + \cdots + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{2} q^{3} + (\beta_{4} - 2) q^{4} + (\beta_{12} - \beta_{8}) q^{5} - \beta_1 q^{6} + (\beta_{14} - \beta_{3}) q^{7} + (\beta_{5} + 3 \beta_{3}) q^{8} + 3 q^{9} + (\beta_{14} - \beta_{13} + \cdots - 2 \beta_1) q^{10}+ \cdots + (3 \beta_{15} + 3 \beta_{4} - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 48 q^{9} - 56 q^{11} - 84 q^{14} + 24 q^{15} + 112 q^{16} - 12 q^{21} + 16 q^{25} - 32 q^{29} - 72 q^{30} - 4 q^{35} - 96 q^{36} + 72 q^{39} + 568 q^{44} - 96 q^{46} - 152 q^{49} - 96 q^{50}+ \cdots - 168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + \cdots + 1849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 59\!\cdots\!83 \nu^{15} + \cdots + 16\!\cdots\!31 ) / 97\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!36 \nu^{15} + \cdots + 12\!\cdots\!37 ) / 69\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 50\!\cdots\!03 \nu^{15} + \cdots + 16\!\cdots\!07 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 92\!\cdots\!48 \nu^{15} + \cdots + 15\!\cdots\!65 ) / 22\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 60\!\cdots\!14 \nu^{15} + \cdots - 19\!\cdots\!04 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!56 \nu^{15} + \cdots + 16\!\cdots\!46 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!71 \nu^{15} + \cdots - 14\!\cdots\!46 ) / 97\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41\!\cdots\!66 \nu^{15} + \cdots + 37\!\cdots\!92 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36\!\cdots\!93 \nu^{15} + \cdots - 94\!\cdots\!38 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62\!\cdots\!57 \nu^{15} + \cdots - 27\!\cdots\!78 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55\!\cdots\!33 \nu^{15} + \cdots + 48\!\cdots\!43 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!74 \nu^{15} + \cdots + 39\!\cdots\!02 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 53\!\cdots\!37 \nu^{15} + \cdots - 94\!\cdots\!27 ) / 97\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!53 \nu^{15} + \cdots - 10\!\cdots\!98 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18\!\cdots\!24 \nu^{15} + \cdots - 15\!\cdots\!95 ) / 22\!\cdots\!29 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} + \beta_{12} - \beta_{10} - 2\beta_{8} - 2\beta_{6} + 2\beta_{4} - 4\beta_{3} + 4\beta_{2} + 2\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 9 \beta_{14} - 3 \beta_{13} + 9 \beta_{12} + 3 \beta_{11} - 9 \beta_{10} + 3 \beta_{9} + \cdots - 82 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{15} + 24 \beta_{14} - 8 \beta_{13} - 24 \beta_{12} + 15 \beta_{11} + 24 \beta_{10} + \cdots + 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22 \beta_{15} + 220 \beta_{14} + 22 \beta_{13} - 220 \beta_{12} + 16 \beta_{11} + 220 \beta_{10} + \cdots + 1194 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 304 \beta_{15} - 90 \beta_{14} + 330 \beta_{13} + 45 \beta_{12} - 460 \beta_{11} - 90 \beta_{10} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 159 \beta_{15} - 13847 \beta_{14} + 779 \beta_{13} + 12915 \beta_{12} - 4135 \beta_{11} - 13705 \beta_{10} + \cdots - 64130 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 9154 \beta_{15} - 17468 \beta_{14} - 8288 \beta_{13} + 16744 \beta_{12} + 7602 \beta_{11} + \cdots - 78049 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 53655 \beta_{15} + 305481 \beta_{14} - 68391 \beta_{13} - 276777 \beta_{12} + 168135 \beta_{11} + \cdots + 1419200 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 202407 \beta_{15} + 937566 \beta_{14} + 142329 \beta_{13} - 855228 \beta_{12} + 18051 \beta_{11} + \cdots + 4084397 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2710959 \beta_{15} - 3619903 \beta_{14} + 2725383 \beta_{13} + 3310087 \beta_{12} - 4805661 \beta_{11} + \cdots - 19194622 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 2221755 \beta_{15} - 32096940 \beta_{14} - 399360 \beta_{13} + 29102580 \beta_{12} - 7233894 \beta_{11} + \cdots - 143703808 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 91678709 \beta_{15} - 72135045 \beta_{14} - 80600813 \beta_{13} + 62581077 \beta_{12} + \cdots - 222089510 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 61804684 \beta_{15} + 827687867 \beta_{14} - 97795045 \beta_{13} - 755034803 \beta_{12} + \cdots + 3854381958 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1161278471 \beta_{15} + 3307866177 \beta_{14} + 897806217 \beta_{13} - 2960162271 \beta_{12} + \cdots + 13705362729 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
1.36603 − 5.19914i
−0.366025 + 1.39311i
1.36603 − 3.14303i
−0.366025 + 0.842173i
1.36603 − 2.63709i
−0.366025 + 0.706606i
1.36603 − 0.959486i
−0.366025 + 0.257094i
1.36603 + 0.959486i
−0.366025 − 0.257094i
1.36603 + 2.63709i
−0.366025 − 0.706606i
1.36603 + 3.14303i
−0.366025 − 0.842173i
1.36603 + 5.19914i
−0.366025 − 1.39311i
− 3.80604i −1.73205 −10.4859 −3.71318 + 3.34847i 6.59225i 5.08005 − 4.81592i 24.6856i 3.00000 12.7444 + 14.1325i
34.2 − 3.80604i 1.73205 −10.4859 3.71318 − 3.34847i − 6.59225i −5.08005 − 4.81592i 24.6856i 3.00000 −12.7444 − 14.1325i
34.3 − 2.30086i −1.73205 −1.29396 −3.65761 − 3.40909i 3.98521i −6.39480 + 2.84720i − 6.22623i 3.00000 −7.84383 + 8.41565i
34.4 − 2.30086i 1.73205 −1.29396 3.65761 + 3.40909i − 3.98521i 6.39480 + 2.84720i − 6.22623i 3.00000 7.84383 − 8.41565i
34.5 − 1.93048i −1.73205 0.273228 4.88618 + 1.06077i 3.34370i −0.433408 − 6.98657i − 8.24940i 3.00000 2.04780 − 9.43270i
34.6 − 1.93048i 1.73205 0.273228 −4.88618 − 1.06077i − 3.34370i 0.433408 − 6.98657i − 8.24940i 3.00000 −2.04780 + 9.43270i
34.7 − 0.702393i −1.73205 3.50664 −0.979490 + 4.90312i 1.21658i 3.48021 + 6.07356i − 5.27261i 3.00000 3.44392 + 0.687987i
34.8 − 0.702393i 1.73205 3.50664 0.979490 − 4.90312i − 1.21658i −3.48021 + 6.07356i − 5.27261i 3.00000 −3.44392 − 0.687987i
34.9 0.702393i −1.73205 3.50664 −0.979490 − 4.90312i − 1.21658i 3.48021 − 6.07356i 5.27261i 3.00000 3.44392 − 0.687987i
34.10 0.702393i 1.73205 3.50664 0.979490 + 4.90312i 1.21658i −3.48021 − 6.07356i 5.27261i 3.00000 −3.44392 + 0.687987i
34.11 1.93048i −1.73205 0.273228 4.88618 − 1.06077i − 3.34370i −0.433408 + 6.98657i 8.24940i 3.00000 2.04780 + 9.43270i
34.12 1.93048i 1.73205 0.273228 −4.88618 + 1.06077i 3.34370i 0.433408 + 6.98657i 8.24940i 3.00000 −2.04780 − 9.43270i
34.13 2.30086i −1.73205 −1.29396 −3.65761 + 3.40909i − 3.98521i −6.39480 − 2.84720i 6.22623i 3.00000 −7.84383 − 8.41565i
34.14 2.30086i 1.73205 −1.29396 3.65761 − 3.40909i 3.98521i 6.39480 − 2.84720i 6.22623i 3.00000 7.84383 + 8.41565i
34.15 3.80604i −1.73205 −10.4859 −3.71318 − 3.34847i − 6.59225i 5.08005 + 4.81592i − 24.6856i 3.00000 12.7444 − 14.1325i
34.16 3.80604i 1.73205 −10.4859 3.71318 + 3.34847i 6.59225i −5.08005 + 4.81592i − 24.6856i 3.00000 −12.7444 + 14.1325i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c 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.e.a ✓ 16
3.b odd 2 1 315.3.e.e 16
4.b odd 2 1 1680.3.bd.c 16
5.b even 2 1 inner 105.3.e.a ✓ 16
5.c odd 4 2 525.3.h.e 16
7.b odd 2 1 inner 105.3.e.a ✓ 16
15.d odd 2 1 315.3.e.e 16
20.d odd 2 1 1680.3.bd.c 16
21.c even 2 1 315.3.e.e 16
28.d even 2 1 1680.3.bd.c 16
35.c odd 2 1 inner 105.3.e.a ✓ 16
35.f even 4 2 525.3.h.e 16
105.g even 2 1 315.3.e.e 16
140.c even 2 1 1680.3.bd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.e.a ✓ 16 1.a even 1 1 trivial
105.3.e.a ✓ 16 5.b even 2 1 inner
105.3.e.a ✓ 16 7.b odd 2 1 inner
105.3.e.a ✓ 16 35.c odd 2 1 inner
315.3.e.e 16 3.b odd 2 1
315.3.e.e 16 15.d odd 2 1
315.3.e.e 16 21.c even 2 1
315.3.e.e 16 105.g even 2 1
525.3.h.e 16 5.c odd 4 2
525.3.h.e 16 35.f even 4 2
1680.3.bd.c 16 4.b odd 2 1
1680.3.bd.c 16 20.d odd 2 1
1680.3.bd.c 16 28.d even 2 1
1680.3.bd.c 16 140.c even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 24 T^{6} + \cdots + 141)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} + 14 T^{3} + \cdots + 3280)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 556 T^{6} + \cdots + 60715264)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1468 T^{6} + \cdots + 4251040000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 396 T^{6} + \cdots + 324864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2904 T^{6} + \cdots + 382942464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} + \cdots - 26384)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3840 T^{6} + \cdots + 1892910336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 3497148290304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 26090305209600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 2508 T^{6} + \cdots + 86666496)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1152376486144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 55452408710400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 2149592204544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 9868871097600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 674847686079744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots - 3270032)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 36785098365184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 116 T^{3} + \cdots + 28622512)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 162581970541824)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 338426355591424)^{2} \) Copy content Toggle raw display
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