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Properties

Label 15.9.c.a
Level $15$
Weight $9$
Character orbit 15.c
Analytic conductor $6.111$
Analytic rank $0$
Dimension $10$
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] = [15,9,Mod(11,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.11");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 15.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: \(6.11067915092\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{10}\cdot 5^{10} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} - 2 \beta_1 - 11) q^{3} + ( - \beta_{3} - 79) q^{4} + ( - \beta_{4} - 3 \beta_1) q^{5} + (\beta_{8} - \beta_{4} - 2 \beta_{3} + \cdots - 529) q^{6}+ \cdots + ( - \beta_{9} + 2 \beta_{8} + \cdots + 385) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} - 2 \beta_1 - 11) q^{3} + ( - \beta_{3} - 79) q^{4} + ( - \beta_{4} - 3 \beta_1) q^{5} + (\beta_{8} - \beta_{4} - 2 \beta_{3} + \cdots - 529) q^{6}+ \cdots + ( - 25421 \beta_{9} + 15604 \beta_{8} + \cdots - 3323878) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 112 q^{3} - 786 q^{4} - 5282 q^{6} + 7156 q^{7} + 3922 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 112 q^{3} - 786 q^{4} - 5282 q^{6} + 7156 q^{7} + 3922 q^{9} - 8750 q^{10} - 3812 q^{12} - 55464 q^{13} - 21250 q^{15} + 280386 q^{16} - 419800 q^{18} - 231516 q^{19} + 289572 q^{21} + 1129940 q^{22} + 1136334 q^{24} - 781250 q^{25} - 335512 q^{27} - 3340724 q^{28} - 965000 q^{30} + 881620 q^{31} + 1266460 q^{33} - 1111276 q^{34} - 668662 q^{36} + 4672616 q^{37} + 1826792 q^{39} + 2913750 q^{40} - 5392860 q^{42} + 7731336 q^{43} - 2142500 q^{45} - 25424604 q^{46} + 22413388 q^{48} + 9354214 q^{49} - 27732692 q^{51} + 21064016 q^{52} - 7979798 q^{54} - 4377500 q^{55} - 2856304 q^{57} - 4351100 q^{58} + 23016250 q^{60} + 22417020 q^{61} + 8830596 q^{63} - 22935002 q^{64} - 27419800 q^{66} - 46646024 q^{67} + 33562632 q^{69} - 62992500 q^{70} + 54175560 q^{72} - 129964884 q^{73} + 8750000 q^{75} + 198922436 q^{76} + 60388360 q^{78} + 162310924 q^{79} - 93575390 q^{81} + 202877560 q^{82} - 197346768 q^{84} - 110682500 q^{85} - 168322540 q^{87} - 484775700 q^{88} + 171878750 q^{90} + 444288464 q^{91} + 463412376 q^{93} - 92050036 q^{94} - 360807406 q^{96} - 258825724 q^{97} - 33965200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 422181448349 \nu^{9} + 6737848279449 \nu^{8} + 139312194537578 \nu^{7} + \cdots + 41\!\cdots\!04 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2454900232379 \nu^{9} + 8124069454611 \nu^{8} + \cdots - 14\!\cdots\!12 ) / 52\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6879197511740 \nu^{9} + 17194261561041 \nu^{8} + \cdots - 24\!\cdots\!12 ) / 26\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5233252857042 \nu^{9} + 39209403295667 \nu^{8} + \cdots - 12\!\cdots\!68 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 136514483776663 \nu^{9} - 765773291245803 \nu^{8} + \cdots + 15\!\cdots\!32 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13995282752553 \nu^{9} + 72894475678928 \nu^{8} + \cdots - 11\!\cdots\!92 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 157357802456293 \nu^{9} + \cdots + 20\!\cdots\!48 ) / 87\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21633109424708 \nu^{9} + 115213115150653 \nu^{8} + \cdots - 16\!\cdots\!44 ) / 87\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 143077927645465 \nu^{9} - 910044828994923 \nu^{8} + \cdots + 11\!\cdots\!96 ) / 52\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{9} - 9 \beta_{8} + 8 \beta_{7} - 9 \beta_{6} - 5 \beta_{5} - 55 \beta_{4} + 174 \beta_{3} + \cdots + 2822 ) / 6750 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 61 \beta_{9} + 124 \beta_{8} + 87 \beta_{7} - 576 \beta_{6} + 180 \beta_{5} + 55 \beta_{4} + \cdots + 596958 ) / 6750 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21 \beta_{9} + 334 \beta_{8} + 2477 \beta_{7} - 6966 \beta_{6} + 1780 \beta_{5} - 11841 \beta_{4} + \cdots + 8080028 ) / 6750 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20657 \beta_{9} + 50548 \beta_{8} + 47629 \beta_{7} - 148302 \beta_{6} + 90560 \beta_{5} + \cdots + 162283666 ) / 6750 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 313201 \beta_{9} + 1080634 \beta_{8} + 1007967 \beta_{7} - 2524536 \beta_{6} + 1886880 \beta_{5} + \cdots + 2815161348 ) / 6750 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8412225 \beta_{9} + 27927400 \beta_{8} + 20048825 \beta_{7} - 44262990 \beta_{6} + 46838500 \beta_{5} + \cdots + 49105415690 ) / 6750 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 173861933 \beta_{9} + 628042342 \beta_{8} + 398846731 \beta_{7} - 755515368 \beta_{6} + \cdots + 838451682124 ) / 6750 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3726354937 \beta_{9} + 13831361068 \beta_{8} + 7774933389 \beta_{7} - 12398581422 \beta_{6} + \cdots + 13782771805146 ) / 6750 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 76838521617 \beta_{9} + 294645646858 \beta_{8} + 149605572119 \beta_{7} - 197196436872 \beta_{6} + \cdots + 218981655222716 ) / 6750 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
18.9110 − 2.23607i
−0.616448 + 2.23607i
−7.95862 − 2.23607i
−1.94191 + 2.23607i
−6.39402 − 2.23607i
−6.39402 + 2.23607i
−1.94191 − 2.23607i
−7.95862 + 2.23607i
−0.616448 − 2.23607i
18.9110 + 2.23607i
− 29.5009i 41.5815 − 69.5124i −614.301 − 279.508i −2050.68 − 1226.69i 3174.27 10570.2i −3102.96 − 5780.86i −8245.74
11.2 − 23.7888i −80.3707 − 10.0775i −309.906 279.508i −239.732 + 1911.92i −692.753 1282.36i 6357.89 + 1619.88i 6649.17
11.3 − 10.2357i −57.8099 + 56.7364i 151.230 − 279.508i 580.739 + 591.726i 3448.05 − 4168.30i 122.959 − 6559.85i −2860.98
11.4 − 8.27106i 70.4414 − 39.9876i 187.590 279.508i −330.740 − 582.625i 860.291 − 3668.96i 3362.98 − 5633.57i 2311.83
11.5 − 7.97572i −29.8424 − 75.3023i 192.388 − 279.508i −600.590 + 238.014i −3211.86 − 3576.22i −4779.87 + 4494.40i −2229.28
11.6 7.97572i −29.8424 + 75.3023i 192.388 279.508i −600.590 − 238.014i −3211.86 3576.22i −4779.87 − 4494.40i −2229.28
11.7 8.27106i 70.4414 + 39.9876i 187.590 − 279.508i −330.740 + 582.625i 860.291 3668.96i 3362.98 + 5633.57i 2311.83
11.8 10.2357i −57.8099 − 56.7364i 151.230 279.508i 580.739 − 591.726i 3448.05 4168.30i 122.959 + 6559.85i −2860.98
11.9 23.7888i −80.3707 + 10.0775i −309.906 − 279.508i −239.732 − 1911.92i −692.753 − 1282.36i 6357.89 − 1619.88i 6649.17
11.10 29.5009i 41.5815 + 69.5124i −614.301 279.508i −2050.68 + 1226.69i 3174.27 − 10570.2i −3102.96 + 5780.86i −8245.74
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
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 15.9.c.a ✓ 10
3.b odd 2 1 inner 15.9.c.a ✓ 10
4.b odd 2 1 240.9.l.b 10
5.b even 2 1 75.9.c.g 10
5.c odd 4 2 75.9.d.c 20
12.b even 2 1 240.9.l.b 10
15.d odd 2 1 75.9.c.g 10
15.e even 4 2 75.9.d.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.c.a ✓ 10 1.a even 1 1 trivial
15.9.c.a ✓ 10 3.b odd 2 1 inner
75.9.c.g 10 5.b even 2 1
75.9.c.g 10 15.d odd 2 1
75.9.d.c 20 5.c odd 4 2
75.9.d.c 20 15.e even 4 2
240.9.l.b 10 4.b odd 2 1
240.9.l.b 10 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 224550432000 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( (T^{2} + 78125)^{5} \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 23\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
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