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Properties

Label 690.2.e.b
Level $690$
Weight $2$
Character orbit 690.e
Analytic conductor $5.510$
Analytic rank $0$
Dimension $16$
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] = [690,2,Mod(551,690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(690, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("690.551");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.e (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: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 12 x^{13} + 15 x^{12} - 4 x^{11} + 45 x^{10} - 66 x^{9} - 32 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_{2} q^{3} - q^{4} + q^{5} + \beta_1 q^{6} + ( - \beta_{9} + \beta_{7}) q^{7} - \beta_{9} q^{8} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + \beta_{2} q^{3} - q^{4} + q^{5} + \beta_1 q^{6} + ( - \beta_{9} + \beta_{7}) q^{7} - \beta_{9} q^{8} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{9}+ \cdots + (\beta_{14} + \beta_{12} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{5} + 2 q^{6} + 2 q^{9} + 12 q^{11} + 12 q^{14} + 16 q^{16} + 8 q^{18} - 16 q^{20} + 4 q^{21} - 4 q^{23} - 2 q^{24} + 16 q^{25} + 24 q^{27} + 2 q^{30} + 4 q^{31} + 28 q^{33} - 2 q^{36} + 16 q^{38} - 8 q^{39} - 12 q^{44} + 2 q^{45} - 4 q^{46} - 4 q^{49} + 2 q^{51} + 8 q^{53} - 26 q^{54} + 12 q^{55} - 12 q^{56} - 28 q^{57} - 8 q^{58} - 16 q^{64} - 10 q^{66} - 22 q^{69} + 12 q^{70} - 8 q^{72} - 16 q^{73} + 24 q^{74} - 12 q^{78} + 16 q^{80} + 22 q^{81} - 16 q^{82} + 40 q^{83} - 4 q^{84} + 40 q^{86} + 20 q^{87} - 80 q^{89} + 8 q^{90} + 4 q^{92} - 4 q^{93} - 24 q^{94} + 2 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 12 x^{13} + 15 x^{12} - 4 x^{11} + 45 x^{10} - 66 x^{9} - 32 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{15} - 7 \nu^{14} + 252 \nu^{13} - 240 \nu^{12} - 69 \nu^{11} - 1271 \nu^{10} + 1266 \nu^{9} + \cdots - 303993 ) / 23328 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 3 \nu^{13} - 12 \nu^{12} + 15 \nu^{11} - 4 \nu^{10} + 45 \nu^{9} + \cdots - 4374 ) / 2187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{15} - 143 \nu^{14} + 444 \nu^{13} - 264 \nu^{12} + 465 \nu^{11} - 1987 \nu^{10} + \cdots - 356481 ) / 69984 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4 \nu^{15} - 209 \nu^{14} + 396 \nu^{13} - 264 \nu^{12} + 1068 \nu^{11} - 2167 \nu^{10} + \cdots - 352107 ) / 34992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9 \nu^{15} + 32 \nu^{14} + 8 \nu^{13} + 60 \nu^{12} - 159 \nu^{11} - 294 \nu^{10} + 52 \nu^{9} + \cdots - 29160 ) / 11664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71 \nu^{15} - 151 \nu^{14} + 564 \nu^{13} - 816 \nu^{12} + 1281 \nu^{11} - 2147 \nu^{10} + \cdots - 811377 ) / 69984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55 \nu^{15} - 350 \nu^{14} + 384 \nu^{13} - 480 \nu^{12} + 1761 \nu^{11} - 1660 \nu^{10} + \cdots - 284310 ) / 34992 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 139 \nu^{15} + 263 \nu^{14} - 396 \nu^{13} + 912 \nu^{12} - 1365 \nu^{11} + 763 \nu^{10} + \cdots + 356481 ) / 69984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 163 \nu^{15} + 91 \nu^{14} - 300 \nu^{13} - 768 \nu^{12} - 291 \nu^{11} + 3443 \nu^{10} + \cdots + 207765 ) / 69984 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43 \nu^{15} - 62 \nu^{14} - 282 \nu^{12} + 357 \nu^{11} + 674 \nu^{10} + 624 \nu^{9} - 2892 \nu^{8} + \cdots + 34992 ) / 17496 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 94 \nu^{15} + 137 \nu^{14} + 180 \nu^{13} + 336 \nu^{12} - 690 \nu^{11} - 2117 \nu^{10} + \cdots - 242757 ) / 34992 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31 \nu^{15} - 114 \nu^{14} + 56 \nu^{13} - 120 \nu^{12} + 657 \nu^{11} + 32 \nu^{10} - 728 \nu^{9} + \cdots + 39366 ) / 11664 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 247 \nu^{15} + 383 \nu^{14} + 84 \nu^{13} + 1128 \nu^{12} - 1833 \nu^{11} - 4349 \nu^{10} + \cdots - 378351 ) / 69984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 361 \nu^{15} + 269 \nu^{14} - 708 \nu^{13} + 2280 \nu^{12} - 1383 \nu^{11} - 1247 \nu^{10} + \cdots + 789507 ) / 69984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - 2\beta_{11} + \beta_{10} + 2\beta_{9} - \beta_{6} - \beta_{4} + \beta_{3} + 2\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - \beta_{14} + 2 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} + 6 \beta_{14} + \beta_{13} + \beta_{12} + 3 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{15} - 7 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 8 \beta_{11} - \beta_{10} - 13 \beta_{9} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{15} + \beta_{14} - \beta_{13} + 8 \beta_{12} - 10 \beta_{11} + 7 \beta_{10} + \beta_{9} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20 \beta_{15} - \beta_{14} - 19 \beta_{13} + 27 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} - 20 \beta_{9} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14 \beta_{15} - 9 \beta_{14} - 11 \beta_{13} + 5 \beta_{12} + 26 \beta_{11} - 27 \beta_{10} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 48 \beta_{15} + 41 \beta_{14} + 15 \beta_{13} - 3 \beta_{12} - 32 \beta_{11} + 15 \beta_{10} + \cdots + 320 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 86 \beta_{15} - 3 \beta_{14} - 9 \beta_{13} - 25 \beta_{12} - 106 \beta_{11} - 116 \beta_{10} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 136 \beta_{15} + 56 \beta_{14} + 197 \beta_{13} + 46 \beta_{12} + 116 \beta_{11} + 149 \beta_{10} + \cdots - 183 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 46 \beta_{15} + 339 \beta_{14} + 230 \beta_{13} + 64 \beta_{12} + 96 \beta_{11} + 106 \beta_{10} + \cdots + 431 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 86 \beta_{15} + 88 \beta_{14} + 523 \beta_{13} - 309 \beta_{12} - 986 \beta_{11} + 603 \beta_{10} + \cdots - 272 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1
−0.251873 + 1.71364i
−1.14349 + 1.30094i
1.53677 + 0.798958i
−1.62589 + 0.597052i
1.73162 + 0.0386882i
1.37342 − 1.05533i
−0.462710 − 1.66910i
−0.157845 − 1.72484i
−0.251873 − 1.71364i
−1.14349 − 1.30094i
1.53677 − 0.798958i
−1.62589 − 0.597052i
1.73162 − 0.0386882i
1.37342 + 1.05533i
−0.462710 + 1.66910i
−0.157845 + 1.72484i
− 1.00000i −1.71364 − 0.251873i −1.00000 1.00000 −0.251873 + 1.71364i 0.194278i 1.00000i 2.87312 + 0.863238i − 1.00000i
551.2 − 1.00000i −1.30094 − 1.14349i −1.00000 1.00000 −1.14349 + 1.30094i − 1.02070i 1.00000i 0.384868 + 2.97521i − 1.00000i
551.3 − 1.00000i −0.798958 + 1.53677i −1.00000 1.00000 1.53677 + 0.798958i 0.145234i 1.00000i −1.72333 − 2.45563i − 1.00000i
551.4 − 1.00000i −0.597052 − 1.62589i −1.00000 1.00000 −1.62589 + 0.597052i 3.80421i 1.00000i −2.28706 + 1.94149i − 1.00000i
551.5 − 1.00000i −0.0386882 + 1.73162i −1.00000 1.00000 1.73162 + 0.0386882i 2.83902i 1.00000i −2.99701 − 0.133986i − 1.00000i
551.6 − 1.00000i 1.05533 + 1.37342i −1.00000 1.00000 1.37342 − 1.05533i − 1.91110i 1.00000i −0.772562 + 2.89882i − 1.00000i
551.7 − 1.00000i 1.66910 − 0.462710i −1.00000 1.00000 −0.462710 − 1.66910i − 2.82123i 1.00000i 2.57180 − 1.54462i − 1.00000i
551.8 − 1.00000i 1.72484 − 0.157845i −1.00000 1.00000 −0.157845 − 1.72484i 4.77029i 1.00000i 2.95017 − 0.544517i − 1.00000i
551.9 1.00000i −1.71364 + 0.251873i −1.00000 1.00000 −0.251873 − 1.71364i − 0.194278i − 1.00000i 2.87312 − 0.863238i 1.00000i
551.10 1.00000i −1.30094 + 1.14349i −1.00000 1.00000 −1.14349 − 1.30094i 1.02070i − 1.00000i 0.384868 − 2.97521i 1.00000i
551.11 1.00000i −0.798958 − 1.53677i −1.00000 1.00000 1.53677 − 0.798958i − 0.145234i − 1.00000i −1.72333 + 2.45563i 1.00000i
551.12 1.00000i −0.597052 + 1.62589i −1.00000 1.00000 −1.62589 − 0.597052i − 3.80421i − 1.00000i −2.28706 − 1.94149i 1.00000i
551.13 1.00000i −0.0386882 − 1.73162i −1.00000 1.00000 1.73162 − 0.0386882i − 2.83902i − 1.00000i −2.99701 + 0.133986i 1.00000i
551.14 1.00000i 1.05533 − 1.37342i −1.00000 1.00000 1.37342 + 1.05533i 1.91110i − 1.00000i −0.772562 − 2.89882i 1.00000i
551.15 1.00000i 1.66910 + 0.462710i −1.00000 1.00000 −0.462710 + 1.66910i 2.82123i − 1.00000i 2.57180 + 1.54462i 1.00000i
551.16 1.00000i 1.72484 + 0.157845i −1.00000 1.00000 −0.157845 + 1.72484i − 4.77029i − 1.00000i 2.95017 + 0.544517i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.e.b yes 16
3.b odd 2 1 690.2.e.a ✓ 16
23.b odd 2 1 690.2.e.a ✓ 16
69.c even 2 1 inner 690.2.e.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.e.a ✓ 16 3.b odd 2 1
690.2.e.a ✓ 16 23.b odd 2 1
690.2.e.b yes 16 1.a even 1 1 trivial
690.2.e.b yes 16 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} - 6T_{11}^{7} - 29T_{11}^{6} + 202T_{11}^{5} + 93T_{11}^{4} - 1704T_{11}^{3} + 1560T_{11}^{2} + 920T_{11} + 68 \) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 58 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} - 29 T^{6} + \cdots + 68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 79 T^{6} + \cdots - 43184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 91 T^{6} + \cdots + 90592)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 122 T^{14} + \cdots + 3625216 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2517630976 \) Copy content Toggle raw display
$31$ \( (T^{8} - 2 T^{7} + \cdots + 90304)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 484704256 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 256000000 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 27688960000 \) Copy content Toggle raw display
$47$ \( T^{16} + 360 T^{14} + \cdots + 12845056 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots + 756224)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32943702016 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 5217752377600 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2824120582144 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 4296168198400 \) Copy content Toggle raw display
$73$ \( (T^{8} + 8 T^{7} + \cdots + 51200)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 20404373094400 \) Copy content Toggle raw display
$83$ \( (T^{8} - 20 T^{7} + \cdots - 51542912)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 40 T^{7} + \cdots + 278751040)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 487900948710400 \) Copy content Toggle raw display
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