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Properties

Label 775.2.o.f
Level $775$
Weight $2$
Character orbit 775.o
Analytic conductor $6.188$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(149,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.o (of order \(6\), degree \(2\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 36x^{12} - 77x^{10} + 119x^{8} - 77x^{6} + 36x^{4} - 7x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_1) q^{2} + (\beta_{14} - \beta_{11} + \cdots - 2 \beta_{7}) q^{3}+ \cdots + (3 \beta_{10} + \beta_{4} + 3 \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_1) q^{2} + (\beta_{14} - \beta_{11} + \cdots - 2 \beta_{7}) q^{3}+ \cdots + (7 \beta_{12} - 6 \beta_{10} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 4 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 4 q^{6} + 10 q^{9} + 8 q^{11} + 16 q^{14} - 12 q^{16} - 6 q^{19} - 6 q^{21} - 20 q^{24} - 16 q^{26} - 76 q^{29} - 10 q^{31} + 32 q^{34} + 46 q^{36} + 28 q^{39} - 8 q^{41} + 6 q^{44} + 4 q^{46} - 22 q^{49} - 32 q^{51} - 64 q^{54} + 30 q^{56} + 36 q^{59} - 20 q^{61} + 12 q^{64} + 20 q^{66} + 12 q^{69} - 24 q^{71} - 28 q^{74} - 22 q^{76} - 28 q^{79} - 32 q^{81} - 18 q^{84} + 14 q^{86} + 4 q^{89} + 8 q^{91} - 64 q^{94} - 62 q^{96} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{14} + 36x^{12} - 77x^{10} + 119x^{8} - 77x^{6} + 36x^{4} - 7x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 271\nu^{14} - 2016\nu^{12} + 10290\nu^{10} - 23177\nu^{8} + 31752\nu^{6} - 17640\nu^{4} - 534\nu^{2} - 546 ) / 3325 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -55\nu^{14} + 307\nu^{12} - 1470\nu^{10} + 1715\nu^{8} - 2009\nu^{6} - 1470\nu^{4} - 510\nu^{2} + 97 ) / 665 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{14} + 104\nu^{12} - 553\nu^{10} + 1421\nu^{8} - 2303\nu^{6} + 1974\nu^{4} - 713\nu^{2} + 139 ) / 133 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 457 \nu^{14} + 3732 \nu^{12} - 19880 \nu^{10} + 52409 \nu^{8} - 85379 \nu^{6} + 79205 \nu^{4} + \cdots + 5167 ) / 3325 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 457 \nu^{15} - 3732 \nu^{13} + 19880 \nu^{11} - 52409 \nu^{9} + 85379 \nu^{7} - 79205 \nu^{5} + \cdots - 5167 \nu ) / 3325 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -97\nu^{15} + 624\nu^{13} - 3185\nu^{11} + 5999\nu^{9} - 9828\nu^{7} + 5460\nu^{5} - 4962\nu^{3} + 169\nu ) / 665 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 546 \nu^{15} - 3551 \nu^{13} + 17640 \nu^{11} - 31752 \nu^{9} + 41797 \nu^{7} - 10290 \nu^{5} + \cdots - 4356 \nu ) / 3325 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 702 \nu^{14} + 4552 \nu^{12} - 22680 \nu^{10} + 40824 \nu^{8} - 55069 \nu^{6} + 13230 \nu^{4} + \cdots - 1538 ) / 3325 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 756 \nu^{14} + 5136 \nu^{12} - 26215 \nu^{10} + 53172 \nu^{8} - 80892 \nu^{6} + 44940 \nu^{4} + \cdots + 1391 ) / 3325 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 756 \nu^{15} + 5136 \nu^{13} - 26215 \nu^{11} + 53172 \nu^{9} - 80892 \nu^{7} + 44940 \nu^{5} + \cdots + 4716 \nu ) / 3325 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\nu^{14} - 84\nu^{12} + 420\nu^{10} - 756\nu^{8} + 1043\nu^{6} - 245\nu^{4} + 48\nu^{2} + 56 ) / 35 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -13\nu^{15} + 84\nu^{13} - 420\nu^{11} + 756\nu^{9} - 1043\nu^{7} + 245\nu^{5} - 48\nu^{3} - 21\nu ) / 35 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1538 \nu^{15} - 11468 \nu^{13} + 59920 \nu^{11} - 141106 \nu^{9} + 223846 \nu^{7} - 173495 \nu^{5} + \cdots - 13358 \nu ) / 3325 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1937 \nu^{15} + 12532 \nu^{13} - 62580 \nu^{11} + 112644 \nu^{9} - 154154 \nu^{7} + \cdots - 6858 \nu ) / 3325 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{13} - 3\beta_{11} + \beta_{7} + \beta_{6} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{10} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{14} - 12\beta_{11} - 4\beta_{8} + 4\beta_{7} + 6\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{12} - 23\beta_{9} + 16\beta_{3} - 16\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -23\beta_{15} + 29\beta_{13} - 16\beta_{8} - 51\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -29\beta_{12} + 51\beta_{10} - 103\beta_{9} + 29\beta_{5} - 103\beta_{4} - 67\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -103\beta_{15} + 103\beta_{14} + 132\beta_{13} + 221\beta_{11} - 67\beta_{7} - 132\beta_{6} - 221\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 221\beta_{10} + 132\beta_{5} - 456\beta_{4} - 288\beta_{3} + 221 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 456\beta_{14} + 965\beta_{11} + 288\beta_{8} - 288\beta_{7} - 588\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 588\beta_{12} + 2009\beta_{9} - 1253\beta_{3} + 1253\beta_{2} + 377 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2009\beta_{15} - 2597\beta_{13} + 1253\beta_{8} + 4227\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2597\beta_{12} - 4227\beta_{10} + 8833\beta_{9} - 2597\beta_{5} + 8833\beta_{4} + 5480\beta_{2} - 2597 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 8833 \beta_{15} - 8833 \beta_{14} - 11430 \beta_{13} - 18540 \beta_{11} + 5480 \beta_{7} + \cdots + 18540 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(652\)
\(\chi(n)\) \(-1 - \beta_{10}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−1.81458 − 1.04765i
−1.17405 − 0.677837i
−0.638815 − 0.368820i
−0.413319 − 0.238630i
0.413319 + 0.238630i
0.638815 + 0.368820i
1.17405 + 0.677837i
1.81458 + 1.04765i
1.81458 − 1.04765i
1.17405 − 0.677837i
0.638815 − 0.368820i
0.413319 − 0.238630i
−0.413319 + 0.238630i
−0.638815 + 0.368820i
−1.17405 + 0.677837i
−1.81458 + 1.04765i
− 2.09529i −0.121914 + 0.0703870i −2.39026 0 0.147481 + 0.255445i −0.693107 + 0.400166i 0.817703i −1.49009 + 2.58091i 0
149.2 − 1.35567i −0.762443 + 0.440197i 0.162147 0 0.596764 + 1.03362i −3.07370 + 1.77460i − 2.93117i −1.11245 + 1.92683i 0
149.3 − 0.737640i 2.57531 − 1.48685i 1.45589 0 −1.09676 − 1.89965i −1.67244 + 0.965584i − 2.54920i 2.92147 − 5.06014i 0
149.4 − 0.477260i 2.34981 − 1.35666i 1.77222 0 −0.647481 − 1.12147i −0.157874 + 0.0911485i − 1.80033i 2.18107 − 3.77773i 0
149.5 0.477260i −2.34981 + 1.35666i 1.77222 0 −0.647481 − 1.12147i 0.157874 − 0.0911485i 1.80033i 2.18107 − 3.77773i 0
149.6 0.737640i −2.57531 + 1.48685i 1.45589 0 −1.09676 − 1.89965i 1.67244 − 0.965584i 2.54920i 2.92147 − 5.06014i 0
149.7 1.35567i 0.762443 − 0.440197i 0.162147 0 0.596764 + 1.03362i 3.07370 − 1.77460i 2.93117i −1.11245 + 1.92683i 0
149.8 2.09529i 0.121914 − 0.0703870i −2.39026 0 0.147481 + 0.255445i 0.693107 − 0.400166i − 0.817703i −1.49009 + 2.58091i 0
749.1 − 2.09529i 0.121914 + 0.0703870i −2.39026 0 0.147481 − 0.255445i 0.693107 + 0.400166i 0.817703i −1.49009 − 2.58091i 0
749.2 − 1.35567i 0.762443 + 0.440197i 0.162147 0 0.596764 − 1.03362i 3.07370 + 1.77460i − 2.93117i −1.11245 − 1.92683i 0
749.3 − 0.737640i −2.57531 − 1.48685i 1.45589 0 −1.09676 + 1.89965i 1.67244 + 0.965584i − 2.54920i 2.92147 + 5.06014i 0
749.4 − 0.477260i −2.34981 − 1.35666i 1.77222 0 −0.647481 + 1.12147i 0.157874 + 0.0911485i − 1.80033i 2.18107 + 3.77773i 0
749.5 0.477260i 2.34981 + 1.35666i 1.77222 0 −0.647481 + 1.12147i −0.157874 − 0.0911485i 1.80033i 2.18107 + 3.77773i 0
749.6 0.737640i 2.57531 + 1.48685i 1.45589 0 −1.09676 + 1.89965i −1.67244 − 0.965584i 2.54920i 2.92147 + 5.06014i 0
749.7 1.35567i −0.762443 − 0.440197i 0.162147 0 0.596764 − 1.03362i −3.07370 − 1.77460i 2.93117i −1.11245 − 1.92683i 0
749.8 2.09529i −0.121914 − 0.0703870i −2.39026 0 0.147481 − 0.255445i −0.693107 − 0.400166i − 0.817703i −1.49009 − 2.58091i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.o.f 16
5.b even 2 1 inner 775.2.o.f 16
5.c odd 4 1 155.2.e.d ✓ 8
5.c odd 4 1 775.2.e.f 8
31.c even 3 1 inner 775.2.o.f 16
155.j even 6 1 inner 775.2.o.f 16
155.o odd 12 1 155.2.e.d ✓ 8
155.o odd 12 1 775.2.e.f 8
155.o odd 12 1 4805.2.a.o 4
155.p even 12 1 4805.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.e.d ✓ 8 5.c odd 4 1
155.2.e.d ✓ 8 155.o odd 12 1
775.2.e.f 8 5.c odd 4 1
775.2.e.f 8 155.o odd 12 1
775.2.o.f 16 1.a even 1 1 trivial
775.2.o.f 16 5.b even 2 1 inner
775.2.o.f 16 31.c even 3 1 inner
775.2.o.f 16 155.j even 6 1 inner
4805.2.a.m 4 155.p even 12 1
4805.2.a.o 4 155.o odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\):

\( T_{2}^{8} + 7T_{2}^{6} + 13T_{2}^{4} + 7T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{16} - 17T_{7}^{14} + 231T_{7}^{12} - 922T_{7}^{10} + 2819T_{7}^{8} - 1822T_{7}^{6} + 966T_{7}^{4} - 32T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 7 T^{6} + 13 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 17 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 17 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots + 6241)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 21 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} - 78 T^{14} + \cdots + 38950081 \) Copy content Toggle raw display
$19$ \( (T^{8} + 3 T^{7} + \cdots + 841)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 119 T^{6} + \cdots + 44521)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19 T^{3} + \cdots - 1421)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{7} + \cdots + 923521)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 298248146641 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 101 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( (T^{8} + 192 T^{6} + \cdots + 73441)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 18395872699681 \) Copy content Toggle raw display
$59$ \( (T^{4} - 9 T^{3} + \cdots + 1681)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 5 T^{3} + \cdots + 589)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} - 81 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$71$ \( (T^{8} + 12 T^{7} + \cdots + 183358681)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 185189072896 \) Copy content Toggle raw display
$79$ \( (T^{8} + 14 T^{7} + \cdots + 11566801)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} - T^{3} - 38 T^{2} + \cdots + 121)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 143 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
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