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Properties

Label 15.7.c.a
Level $15$
Weight $7$
Character orbit 15.c
Analytic conductor $3.451$
Analytic rank $0$
Dimension $8$
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] = [15,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.11");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(3.45081125430\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 143x^{6} - 134x^{5} + 7489x^{4} + 24994x^{3} - 39153x^{2} - 3258x + 1105326 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{4}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{5} + 2) q^{3} + ( - \beta_{5} - \beta_{3} - 41) q^{4} + (\beta_{6} - \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 30) q^{6}+ \cdots + ( - 5 \beta_{7} + 6 \beta_{6} + \cdots - 199) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{5} + 2) q^{3} + ( - \beta_{5} - \beta_{3} - 41) q^{4} + (\beta_{6} - \beta_{2}) q^{5} + (2 \beta_{7} - \beta_{5} + \beta_{4} + \cdots + 30) q^{6}+ \cdots + ( - 8965 \beta_{7} - 4902 \beta_{6} + \cdots - 106685) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{3} - 322 q^{4} + 238 q^{6} + 160 q^{7} - 1580 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{3} - 322 q^{4} + 238 q^{6} + 160 q^{7} - 1580 q^{9} - 750 q^{10} + 6220 q^{12} - 1400 q^{13} + 2000 q^{15} + 674 q^{16} - 14920 q^{18} + 31000 q^{19} - 3252 q^{21} - 40620 q^{22} - 33714 q^{24} - 25000 q^{25} + 71180 q^{27} + 95500 q^{28} + 1000 q^{30} - 86504 q^{31} - 55040 q^{33} - 30684 q^{34} + 67946 q^{36} + 27640 q^{37} + 88472 q^{39} + 147750 q^{40} + 1140 q^{42} + 85960 q^{43} - 159500 q^{45} - 420684 q^{46} - 456020 q^{48} - 165888 q^{49} + 518608 q^{51} + 100720 q^{52} + 366538 q^{54} + 201000 q^{55} - 294760 q^{57} + 1227780 q^{58} - 635750 q^{60} - 325304 q^{61} - 1091640 q^{63} - 712426 q^{64} + 1969640 q^{66} - 1151480 q^{67} + 154068 q^{69} + 895500 q^{70} + 1935720 q^{72} + 1759840 q^{73} - 62500 q^{75} - 3314348 q^{76} - 4499000 q^{78} + 955960 q^{79} + 1505248 q^{81} - 2395080 q^{82} - 800208 q^{84} + 1014000 q^{85} + 2525000 q^{87} + 6660780 q^{88} - 1569250 q^{90} - 2730544 q^{91} - 3639840 q^{93} - 657444 q^{94} + 6013634 q^{96} - 3158960 q^{97} - 897280 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 143x^{6} - 134x^{5} + 7489x^{4} + 24994x^{3} - 39153x^{2} - 3258x + 1105326 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 669 \nu^{7} - 641803 \nu^{6} + 6364872 \nu^{5} + 48086450 \nu^{4} - 413418861 \nu^{3} + \cdots - 6070703346 ) / 299842920 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11307 \nu^{7} + 135457 \nu^{6} + 1179786 \nu^{5} - 13279034 \nu^{4} - 69304611 \nu^{3} + \cdots - 4779973890 ) / 449764380 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28303 \nu^{7} + 405069 \nu^{6} + 1016888 \nu^{5} - 20508114 \nu^{4} - 87123457 \nu^{3} + \cdots + 17120998494 ) / 299842920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 58179 \nu^{7} + 809020 \nu^{6} + 6213666 \nu^{5} - 120336326 \nu^{4} - 259451085 \nu^{3} + \cdots - 42075445356 ) / 449764380 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59581 \nu^{7} + 300237 \nu^{6} + 7807412 \nu^{5} - 18559830 \nu^{4} - 376464691 \nu^{3} + \cdots - 6520335066 ) / 299842920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 130422 \nu^{7} + 659717 \nu^{6} + 15411636 \nu^{5} - 26117074 \nu^{4} - 783750216 \nu^{3} + \cdots - 20686104810 ) / 449764380 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 94649 \nu^{7} - 741269 \nu^{6} - 8733292 \nu^{5} + 46784758 \nu^{4} + 318964727 \nu^{3} + \cdots + 24254854290 ) / 299842920 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -25\beta_{7} - 18\beta_{6} - 25\beta_{5} + 20\beta_{3} + 18\beta_{2} + 20\beta _1 + 105 ) / 450 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -115\beta_{7} + 36\beta_{6} - 265\beta_{5} - 60\beta_{4} - 100\beta_{3} + 924\beta_{2} + 50\beta _1 + 16125 ) / 450 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2095 \beta_{7} - 2124 \beta_{6} - 295 \beta_{5} - 630 \beta_{4} - 70 \beta_{3} + 4104 \beta_{2} + \cdots + 70845 ) / 450 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11485 \beta_{7} - 6768 \beta_{6} - 10735 \beta_{5} - 11040 \beta_{4} - 5620 \beta_{3} + \cdots + 793095 ) / 450 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22655 \beta_{7} - 42048 \beta_{6} + 24145 \beta_{5} - 22050 \beta_{4} - 14690 \beta_{3} + \cdots + 1222185 ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 618745 \beta_{7} - 1576872 \beta_{6} + 1059605 \beta_{5} - 1355280 \beta_{4} - 572200 \beta_{3} + \cdots + 24923175 ) / 450 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1205455 \beta_{7} - 20627856 \beta_{6} + 24824345 \beta_{5} - 13544370 \beta_{4} - 9268330 \beta_{3} + \cdots + 101232105 ) / 450 \) 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
1.97274 + 2.23607i
−4.61416 − 2.23607i
9.98122 + 2.23607i
−6.33980 − 2.23607i
−6.33980 + 2.23607i
9.98122 − 2.23607i
−4.61416 + 2.23607i
1.97274 − 2.23607i
− 14.6035i −17.0405 + 20.9433i −149.262 − 55.9017i 305.845 + 248.851i −197.732 1245.12i −148.242 − 713.768i −816.359
11.2 − 10.9245i −5.04767 − 26.5240i −55.3439 55.9017i −289.760 + 55.1430i 40.1383 − 94.5635i −678.042 + 267.768i 610.696
11.3 − 7.83150i 26.8218 − 3.09730i 2.66768 − 55.9017i −24.2565 − 210.054i −280.401 − 522.108i 709.813 − 166.150i −437.794
11.4 − 4.80231i 5.26642 + 26.4814i 40.9378 55.9017i 127.172 − 25.2910i 517.995 − 503.944i −673.530 + 278.925i 268.457
11.5 4.80231i 5.26642 − 26.4814i 40.9378 − 55.9017i 127.172 + 25.2910i 517.995 503.944i −673.530 − 278.925i 268.457
11.6 7.83150i 26.8218 + 3.09730i 2.66768 55.9017i −24.2565 + 210.054i −280.401 522.108i 709.813 + 166.150i −437.794
11.7 10.9245i −5.04767 + 26.5240i −55.3439 − 55.9017i −289.760 − 55.1430i 40.1383 94.5635i −678.042 − 267.768i 610.696
11.8 14.6035i −17.0405 − 20.9433i −149.262 55.9017i 305.845 − 248.851i −197.732 − 1245.12i −148.242 + 713.768i −816.359
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
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 15.7.c.a ✓ 8
3.b odd 2 1 inner 15.7.c.a ✓ 8
4.b odd 2 1 240.7.l.a 8
5.b even 2 1 75.7.c.c 8
5.c odd 4 2 75.7.d.c 16
12.b even 2 1 240.7.l.a 8
15.d odd 2 1 75.7.c.c 8
15.e even 4 2 75.7.d.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.7.c.a ✓ 8 1.a even 1 1 trivial
15.7.c.a ✓ 8 3.b odd 2 1 inner
75.7.c.c 8 5.b even 2 1
75.7.c.c 8 15.d odd 2 1
75.7.d.c 16 5.c odd 4 2
75.7.d.c 16 15.e even 4 2
240.7.l.a 8 4.b odd 2 1
240.7.l.a 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 417 T^{6} + \cdots + 36000000 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3125)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 80 T^{3} + \cdots + 1152765000)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 38578703705600)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 40848903895136)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
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