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Properties

Label 240.11.l.b
Level $240$
Weight $11$
Character orbit 240.l
Analytic conductor $152.486$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,11,Mod(161,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.161");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 240.l (of order \(2\), degree \(1\), not 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: \(152.485740642\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 11554 x^{12} + 52224391 x^{10} + 115670558124 x^{8} + 127683454012911 x^{6} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{20}\cdot 5^{21} \)
Twist minimal: no (minimal twist has level 15)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 3) q^{3} + \beta_{5} q^{5} + ( - \beta_{3} + 2 \beta_{2} + 3610) q^{7} + ( - \beta_{7} - 10 \beta_{5} + \cdots + 8310) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 3) q^{3} + \beta_{5} q^{5} + ( - \beta_{3} + 2 \beta_{2} + 3610) q^{7} + ( - \beta_{7} - 10 \beta_{5} + \cdots + 8310) q^{9}+ \cdots + ( - 10218 \beta_{13} + \cdots - 2591405620) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 44 q^{3} + 50548 q^{7} + 116362 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 44 q^{3} + 50548 q^{7} + 116362 q^{9} + 699408 q^{13} + 343750 q^{15} - 3814644 q^{19} - 2191008 q^{21} - 27343750 q^{25} - 13322636 q^{27} - 105444308 q^{31} - 187570700 q^{33} - 152902928 q^{37} + 262995952 q^{39} + 82568592 q^{43} + 284500000 q^{45} + 1339929050 q^{49} + 519773324 q^{51} + 414437500 q^{55} + 2459677832 q^{57} - 2372907732 q^{61} - 253855908 q^{63} + 7807415008 q^{67} - 1032380604 q^{69} + 10465834068 q^{73} + 85937500 q^{75} + 8333919076 q^{79} - 4284635426 q^{81} + 4711812500 q^{85} + 11735627260 q^{87} - 4013221984 q^{91} - 9561672552 q^{93} + 31262487532 q^{97} - 36258312560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 11554 x^{12} + 52224391 x^{10} + 115670558124 x^{8} + 127683454012911 x^{6} + \cdots + 62\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 87\!\cdots\!87 \nu^{13} + \cdots - 24\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!63 \nu^{13} + \cdots + 24\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!63 \nu^{13} + \cdots - 13\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!63 \nu^{13} + \cdots - 19\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47553301111585 \nu^{13} + \cdots - 21\!\cdots\!00 \nu ) / 39\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!33 \nu^{13} + \cdots - 68\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 63\!\cdots\!43 \nu^{13} + \cdots + 51\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!57 \nu^{13} + \cdots + 16\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!73 \nu^{13} + \cdots - 18\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43\!\cdots\!71 \nu^{13} + \cdots - 15\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!75 \nu^{13} + \cdots + 28\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!71 \nu^{13} + \cdots + 92\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 91\!\cdots\!43 \nu^{13} + \cdots + 19\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 32\beta_{5} - 625\beta_{2} - 625\beta_1 ) / 20000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 8 \beta_{13} + 8 \beta_{12} + 32 \beta_{11} + 8 \beta_{10} - 16 \beta_{9} + 8 \beta_{8} + \cdots - 33011769 ) / 20000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2140 \beta_{13} + 2260 \beta_{12} + 3860 \beta_{10} - 3220 \beta_{9} + 12800 \beta_{8} + \cdots + 96080 ) / 20000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 62876 \beta_{13} + 66724 \beta_{12} + 1696 \beta_{11} - 65876 \beta_{10} + 124252 \beta_{9} + \cdots + 82993027893 ) / 20000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1726480 \beta_{13} - 1550320 \beta_{12} - 3013520 \beta_{10} + 3789040 \beta_{9} - 9293600 \beta_{8} + \cdots - 61878560 ) / 4000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 247393864 \beta_{13} - 353220536 \beta_{12} - 137037344 \beta_{11} + 284701864 \beta_{10} + \cdots - 226336546920177 ) / 20000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29793592060 \beta_{13} + 23947723540 \beta_{12} + 51315739940 \beta_{10} - 77274407380 \beta_{9} + \cdots + 797129544320 ) / 20000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 863304579932 \beta_{13} + 1356567975268 \beta_{12} + 570575896672 \beta_{11} - 1071280026932 \beta_{10} + \cdots + 64\!\cdots\!01 ) / 20000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 98405968479920 \beta_{13} - 71989873927280 \beta_{12} - 166942423502080 \beta_{10} + \cdots - 18\!\cdots\!40 ) / 20000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 572776032605288 \beta_{13} - 938068862063512 \beta_{12} - 376481166132448 \beta_{11} + \cdots - 36\!\cdots\!09 ) / 4000 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 31\!\cdots\!80 \beta_{13} + \cdots + 39\!\cdots\!60 ) / 20000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 92\!\cdots\!48 \beta_{13} + \cdots + 54\!\cdots\!89 ) / 20000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 10\!\cdots\!40 \beta_{13} + \cdots - 74\!\cdots\!80 ) / 20000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
− 42.9372i
42.9372i
49.8576i
− 49.8576i
− 2.70449i
2.70449i
− 55.5349i
55.5349i
− 29.7613i
29.7613i
54.9539i
− 54.9539i
− 15.0833i
15.0833i
0 −230.652 − 76.4761i 0 − 1397.54i 0 −19744.7 0 47351.8 + 35278.8i 0
161.2 0 −230.652 + 76.4761i 0 1397.54i 0 −19744.7 0 47351.8 − 35278.8i 0
161.3 0 −196.615 − 142.800i 0 − 1397.54i 0 32323.0 0 18265.6 + 56152.9i 0
161.4 0 −196.615 + 142.800i 0 1397.54i 0 32323.0 0 18265.6 − 56152.9i 0
161.5 0 −182.813 − 160.089i 0 1397.54i 0 5515.83 0 7791.87 + 58532.7i 0
161.6 0 −182.813 + 160.089i 0 − 1397.54i 0 5515.83 0 7791.87 − 58532.7i 0
161.7 0 60.6159 − 235.318i 0 1397.54i 0 −22792.7 0 −51700.4 − 28528.1i 0
161.8 0 60.6159 + 235.318i 0 − 1397.54i 0 −22792.7 0 −51700.4 + 28528.1i 0
161.9 0 80.1400 − 229.405i 0 − 1397.54i 0 24115.7 0 −46204.2 − 36769.0i 0
161.10 0 80.1400 + 229.405i 0 1397.54i 0 24115.7 0 −46204.2 + 36769.0i 0
161.11 0 210.660 − 121.125i 0 1397.54i 0 8585.72 0 29706.2 − 51032.6i 0
161.12 0 210.660 + 121.125i 0 − 1397.54i 0 8585.72 0 29706.2 + 51032.6i 0
161.13 0 236.663 − 55.1313i 0 1397.54i 0 −2728.90 0 52970.1 − 26095.1i 0
161.14 0 236.663 + 55.1313i 0 − 1397.54i 0 −2728.90 0 52970.1 + 26095.1i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.14
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 240.11.l.b 14
3.b odd 2 1 inner 240.11.l.b 14
4.b odd 2 1 15.11.c.a ✓ 14
12.b even 2 1 15.11.c.a ✓ 14
20.d odd 2 1 75.11.c.g 14
20.e even 4 2 75.11.d.d 28
60.h even 2 1 75.11.c.g 14
60.l odd 4 2 75.11.d.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.c.a ✓ 14 4.b odd 2 1
15.11.c.a ✓ 14 12.b even 2 1
75.11.c.g 14 20.d odd 2 1
75.11.c.g 14 60.h even 2 1
75.11.d.d 28 20.e even 4 2
75.11.d.d 28 60.l odd 4 2
240.11.l.b 14 1.a even 1 1 trivial
240.11.l.b 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} - 25274 T_{7}^{6} - 1004258096 T_{7}^{5} + 21084027831504 T_{7}^{4} + \cdots + 45\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1953125)^{7} \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 71\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 24\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
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