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Properties

Label 320.11.b.d
Level $320$
Weight $11$
Character orbit 320.b
Analytic conductor $203.314$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,11,Mod(191,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.191");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 320.b (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: \(203.314320856\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 199481 x^{18} + 16413464051 x^{16} + 725560177607766 x^{14} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{190}\cdot 3^{4}\cdot 5^{29} \)
Twist minimal: no (minimal twist has level 20)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{11} + 3 \beta_1) q^{7} + (\beta_{3} - 4 \beta_{2} - 20743) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + ( - \beta_{11} + 3 \beta_1) q^{7} + (\beta_{3} - 4 \beta_{2} - 20743) q^{9} + (\beta_{14} - 54 \beta_1) q^{11} + (\beta_{4} - 2 \beta_{3} + \cdots + 13942) q^{13}+ \cdots + (1108 \beta_{19} + \cdots - 4837475 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 414868 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 414868 q^{9} + 278864 q^{13} - 1921656 q^{17} - 4157512 q^{21} + 39062500 q^{25} + 66014888 q^{29} + 85980560 q^{33} + 153620656 q^{37} + 477406160 q^{41} + 140125000 q^{45} + 333772012 q^{49} + 1669491824 q^{53} + 3973032960 q^{57} + 4283166080 q^{61} + 290125000 q^{65} + 5321669928 q^{69} + 2474287656 q^{73} - 410885040 q^{77} + 9939722652 q^{81} - 4799500000 q^{85} + 3011851592 q^{89} + 11861394640 q^{93} - 39984502056 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 199481 x^{18} + 16413464051 x^{16} + 725560177607766 x^{14} + \cdots + 21\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!36 \nu^{18} + \cdots + 84\!\cdots\!25 ) / 23\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 59\!\cdots\!44 \nu^{18} + \cdots + 18\!\cdots\!20 ) / 23\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!26 \nu^{18} + \cdots + 30\!\cdots\!75 ) / 18\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27\!\cdots\!02 \nu^{18} + \cdots + 70\!\cdots\!25 ) / 80\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!10 \nu^{18} + \cdots + 21\!\cdots\!75 ) / 80\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37\!\cdots\!92 \nu^{18} + \cdots - 19\!\cdots\!25 ) / 62\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!42 \nu^{18} + \cdots + 48\!\cdots\!50 ) / 28\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!78 \nu^{18} + \cdots + 20\!\cdots\!75 ) / 56\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!66 \nu^{18} + \cdots + 11\!\cdots\!25 ) / 56\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 44\!\cdots\!93 \nu^{19} + \cdots + 42\!\cdots\!89 \nu ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42\!\cdots\!09 \nu^{19} + \cdots - 56\!\cdots\!25 \nu ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36\!\cdots\!71 \nu^{19} + \cdots + 33\!\cdots\!85 \nu ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 58\!\cdots\!55 \nu^{19} + \cdots + 51\!\cdots\!25 \nu ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!87 \nu^{19} + \cdots + 76\!\cdots\!50 \nu ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!75 \nu^{19} + \cdots + 53\!\cdots\!00 \nu ) / 42\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15\!\cdots\!89 \nu^{19} + \cdots + 11\!\cdots\!75 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 35\!\cdots\!05 \nu^{19} + \cdots - 22\!\cdots\!75 \nu ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 71\!\cdots\!13 \nu^{19} + \cdots - 52\!\cdots\!25 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 4\beta_{2} - 79792 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{19} - 5 \beta_{18} - 3 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 26 \beta_{14} + \cdots - 139787 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 78 \beta_{10} + 197 \beta_{9} - 77 \beta_{8} - 438 \beta_{7} + 283 \beta_{6} + 73 \beta_{5} + \cdots + 5572550999 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 538050 \beta_{19} + 554982 \beta_{18} + 337507 \beta_{17} + 357420 \beta_{16} + \cdots + 11728432332 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 8965464 \beta_{10} - 27535979 \beta_{9} + 12193388 \beta_{8} + 69829242 \beta_{7} + \cdots - 467258795879598 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 59128703830 \beta_{19} - 55532351933 \beta_{18} - 33360478993 \beta_{17} - 37377502494 \beta_{16} + \cdots - 10\!\cdots\!80 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 221584638216 \beta_{10} + 777054137908 \beta_{9} - 357045896842 \beta_{8} - 2063129997396 \beta_{7} + \cdots + 10\!\cdots\!56 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 15\!\cdots\!32 \beta_{19} + \cdots + 25\!\cdots\!56 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10\!\cdots\!46 \beta_{10} + \cdots - 50\!\cdots\!62 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 77\!\cdots\!31 \beta_{19} + \cdots - 12\!\cdots\!04 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 10\!\cdots\!24 \beta_{10} + \cdots + 49\!\cdots\!90 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 77\!\cdots\!52 \beta_{19} + \cdots + 12\!\cdots\!84 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 50\!\cdots\!10 \beta_{10} + \cdots - 24\!\cdots\!36 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 38\!\cdots\!04 \beta_{19} + \cdots - 60\!\cdots\!92 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 30\!\cdots\!32 \beta_{10} + \cdots + 14\!\cdots\!86 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 23\!\cdots\!90 \beta_{19} + \cdots + 37\!\cdots\!13 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 59\!\cdots\!64 \beta_{10} + \cdots - 29\!\cdots\!44 ) / 8 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 47\!\cdots\!13 \beta_{19} + \cdots - 73\!\cdots\!74 \beta_1 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
− 224.354i
− 221.211i
− 165.390i
− 160.986i
− 137.813i
− 104.388i
− 103.213i
− 68.5823i
− 40.2810i
− 8.56570i
8.56570i
40.2810i
68.5823i
103.213i
104.388i
137.813i
160.986i
165.390i
221.211i
224.354i
0 − 448.707i 0 1397.54 0 − 19334.7i 0 −142289. 0
191.2 0 − 442.423i 0 −1397.54 0 − 2455.96i 0 −136689. 0
191.3 0 − 330.781i 0 −1397.54 0 29449.0i 0 −50366.8 0
191.4 0 − 321.971i 0 1397.54 0 9880.78i 0 −44616.5 0
191.5 0 − 275.626i 0 −1397.54 0 − 19327.7i 0 −16920.9 0
191.6 0 − 208.777i 0 1397.54 0 − 17557.3i 0 15461.2 0
191.7 0 − 206.425i 0 −1397.54 0 1527.38i 0 16437.6 0
191.8 0 − 137.165i 0 1397.54 0 24910.8i 0 40234.9 0
191.9 0 − 80.5620i 0 1397.54 0 − 345.112i 0 52558.8 0
191.10 0 − 17.1314i 0 −1397.54 0 2883.30i 0 58755.5 0
191.11 0 17.1314i 0 −1397.54 0 − 2883.30i 0 58755.5 0
191.12 0 80.5620i 0 1397.54 0 345.112i 0 52558.8 0
191.13 0 137.165i 0 1397.54 0 − 24910.8i 0 40234.9 0
191.14 0 206.425i 0 −1397.54 0 − 1527.38i 0 16437.6 0
191.15 0 208.777i 0 1397.54 0 17557.3i 0 15461.2 0
191.16 0 275.626i 0 −1397.54 0 19327.7i 0 −16920.9 0
191.17 0 321.971i 0 1397.54 0 − 9880.78i 0 −44616.5 0
191.18 0 330.781i 0 −1397.54 0 − 29449.0i 0 −50366.8 0
191.19 0 442.423i 0 −1397.54 0 2455.96i 0 −136689. 0
191.20 0 448.707i 0 1397.54 0 19334.7i 0 −142289. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.11.b.d 20
4.b odd 2 1 inner 320.11.b.d 20
8.b even 2 1 20.11.b.a ✓ 20
8.d odd 2 1 20.11.b.a ✓ 20
24.f even 2 1 180.11.c.a 20
24.h odd 2 1 180.11.c.a 20
40.e odd 2 1 100.11.b.e 20
40.f even 2 1 100.11.b.e 20
40.i odd 4 2 100.11.d.c 40
40.k even 4 2 100.11.d.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.b.a ✓ 20 8.b even 2 1
20.11.b.a ✓ 20 8.d odd 2 1
100.11.b.e 20 40.e odd 2 1
100.11.b.e 20 40.f even 2 1
100.11.d.c 40 40.i odd 4 2
100.11.d.c 40 40.k even 4 2
180.11.c.a 20 24.f even 2 1
180.11.c.a 20 24.h odd 2 1
320.11.b.d 20 1.a even 1 1 trivial
320.11.b.d 20 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 797924 T_{3}^{18} + 262615424816 T_{3}^{16} + \cdots + 22\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} - 1953125)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
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