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Properties

Label 770.2.e.a
Level $770$
Weight $2$
Character orbit 770.e
Analytic conductor $6.148$
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] = [770,2,Mod(461,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.461");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.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: \(6.14848095564\)
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: \( x^{16} + 30x^{14} + 345x^{12} + 1972x^{10} + 6012x^{8} + 9640x^{6} + 7364x^{4} + 2000x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{10} q^{5} - \beta_{3} q^{6} + \beta_{7} q^{7} - \beta_{10} q^{8} + (\beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{10} q^{5} - \beta_{3} q^{6} + \beta_{7} q^{7} - \beta_{10} q^{8} + (\beta_{4} - \beta_{3} - 1) q^{9} + q^{10} + ( - \beta_{14} - \beta_{6} - \beta_{3}) q^{11} - \beta_1 q^{12} + (\beta_{12} - \beta_{11} + \beta_{7} + \cdots - 1) q^{13}+ \cdots + ( - \beta_{13} + 2 \beta_{11} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 4 q^{6} - 12 q^{9} + 16 q^{10} + 8 q^{13} - 6 q^{14} + 4 q^{15} + 16 q^{16} + 16 q^{17} + 20 q^{19} - 6 q^{21} + 4 q^{22} - 24 q^{23} + 4 q^{24} - 16 q^{25} - 16 q^{33} + 6 q^{35} + 12 q^{36} - 8 q^{37} - 16 q^{40} + 8 q^{41} + 12 q^{42} - 10 q^{49} - 8 q^{52} + 16 q^{53} + 28 q^{54} - 4 q^{55} + 6 q^{56} - 8 q^{58} - 4 q^{60} - 4 q^{61} + 32 q^{62} - 24 q^{63} - 16 q^{64} + 24 q^{66} - 8 q^{67} - 16 q^{68} - 4 q^{71} + 8 q^{73} - 20 q^{76} - 16 q^{77} - 40 q^{78} + 24 q^{81} + 32 q^{83} + 6 q^{84} + 8 q^{86} - 16 q^{87} - 4 q^{88} - 12 q^{90} + 28 q^{91} + 24 q^{92} + 16 q^{93} + 16 q^{94} - 4 q^{96} + 32 q^{98} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30x^{14} + 345x^{12} + 1972x^{10} + 6012x^{8} + 9640x^{6} + 7364x^{4} + 2000x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{14} + 1029 \nu^{12} + 28830 \nu^{10} + 271199 \nu^{8} + 1123589 \nu^{6} + 2112240 \nu^{4} + \cdots + 228148 ) / 43364 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 61 \nu^{14} - 1686 \nu^{12} - 17161 \nu^{10} - 82144 \nu^{8} - 192896 \nu^{6} - 205708 \nu^{4} + \cdots - 1032 ) / 4688 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61 \nu^{14} - 1686 \nu^{12} - 17161 \nu^{10} - 82144 \nu^{8} - 192896 \nu^{6} - 205708 \nu^{4} + \cdots + 17720 ) / 4688 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1631 \nu^{15} + 218 \nu^{14} - 52448 \nu^{13} + 2836 \nu^{12} - 666943 \nu^{11} - 21114 \nu^{10} + \cdots + 512144 ) / 346912 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1169 \nu^{14} + 32613 \nu^{12} + 338719 \nu^{10} + 1696047 \nu^{8} + 4407618 \nu^{6} + 5863342 \nu^{4} + \cdots + 196192 ) / 86728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1631 \nu^{15} - 218 \nu^{14} - 52448 \nu^{13} - 2836 \nu^{12} - 666943 \nu^{11} + 21114 \nu^{10} + \cdots - 512144 ) / 346912 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 645 \nu^{15} - 19033 \nu^{13} - 216213 \nu^{11} - 1245951 \nu^{9} - 4023200 \nu^{7} + \cdots - 2875088 \nu ) / 86728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 437 \nu^{15} - 14139 \nu^{13} - 179667 \nu^{11} - 1162125 \nu^{9} - 4100502 \nu^{7} + \cdots - 2427016 \nu ) / 43364 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 129 \nu^{15} - 3748 \nu^{13} - 41133 \nu^{11} - 220066 \nu^{9} - 611260 \nu^{7} + \cdots - 107624 \nu ) / 9376 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2417 \nu^{15} - 416 \nu^{14} + 72818 \nu^{13} - 9788 \nu^{12} + 839861 \nu^{11} - 73092 \nu^{10} + \cdots - 375776 ) / 173456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2417 \nu^{15} + 416 \nu^{14} + 72818 \nu^{13} + 9788 \nu^{12} + 839861 \nu^{11} + 73092 \nu^{10} + \cdots + 375776 ) / 173456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4753 \nu^{15} - 4116 \nu^{14} + 142792 \nu^{13} - 115608 \nu^{12} + 1637241 \nu^{11} + \cdots + 374880 ) / 346912 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4753 \nu^{15} - 4116 \nu^{14} - 142792 \nu^{13} - 115608 \nu^{12} - 1637241 \nu^{11} + \cdots + 374880 ) / 346912 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 265 \nu^{15} - 7872 \nu^{13} - 89077 \nu^{11} - 495910 \nu^{9} - 1447988 \nu^{7} + \cdots - 311432 \nu ) / 9376 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} + \beta_{14} - \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 6 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{12} + \beta_{11} - 4\beta_{7} + 4\beta_{5} - 12\beta_{4} + 12\beta_{3} + 2\beta_{2} + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 30 \beta_{15} - 9 \beta_{14} + 9 \beta_{13} - 26 \beta_{12} - 26 \beta_{11} + 26 \beta_{10} + \cdots + 54 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{14} + 3 \beta_{13} + 21 \beta_{12} - 21 \beta_{11} + 66 \beta_{7} - 4 \beta_{6} - 66 \beta_{5} + \cdots - 284 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 376 \beta_{15} + 75 \beta_{14} - 75 \beta_{13} + 296 \beta_{12} + 296 \beta_{11} - 308 \beta_{10} + \cdots - 560 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 65 \beta_{14} - 65 \beta_{13} - 295 \beta_{12} + 295 \beta_{11} - 850 \beta_{7} + 84 \beta_{6} + \cdots + 2952 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4460 \beta_{15} - 663 \beta_{14} + 663 \beta_{13} - 3304 \beta_{12} - 3304 \beta_{11} + \cdots + 6088 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 991 \beta_{14} + 991 \beta_{13} + 3651 \beta_{12} - 3651 \beta_{11} + 10146 \beta_{7} - 1212 \beta_{6} + \cdots - 31956 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 51656 \beta_{15} + 6315 \beta_{14} - 6315 \beta_{13} + 36864 \beta_{12} + 36864 \beta_{11} + \cdots - 67428 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 13123 \beta_{14} - 13123 \beta_{13} - 43011 \beta_{12} + 43011 \beta_{11} - 117566 \beta_{7} + \cdots + 352732 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 591232 \beta_{15} - 63931 \beta_{14} + 63931 \beta_{13} - 412508 \beta_{12} - 412508 \beta_{11} + \cdots + 753308 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 161959 \beta_{14} + 161959 \beta_{13} + 495887 \beta_{12} - 495887 \beta_{11} + 1344506 \beta_{7} + \cdots - 3932580 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6724712 \beta_{15} + 675187 \beta_{14} - 675187 \beta_{13} + 4629016 \beta_{12} + 4629016 \beta_{11} + \cdots - 8453756 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
461.1
− 3.35938i
− 2.11620i
− 1.48511i
− 0.741934i
0.0908173i
1.16133i
1.89360i
2.55688i
− 2.55688i
− 1.89360i
− 1.16133i
− 0.0908173i
0.741934i
1.48511i
2.11620i
3.35938i
− 1.00000i − 3.35938i −1.00000 1.00000i −3.35938 −1.41588 − 2.23501i 1.00000i −8.28540 1.00000
461.2 − 1.00000i − 2.11620i −1.00000 1.00000i −2.11620 2.58002 − 0.586068i 1.00000i −1.47832 1.00000
461.3 − 1.00000i − 1.48511i −1.00000 1.00000i −1.48511 0.778797 + 2.52853i 1.00000i 0.794454 1.00000
461.4 − 1.00000i − 0.741934i −1.00000 1.00000i −0.741934 −2.60369 − 0.469884i 1.00000i 2.44953 1.00000
461.5 − 1.00000i 0.0908173i −1.00000 1.00000i 0.0908173 −1.53620 − 2.15409i 1.00000i 2.99175 1.00000
461.6 − 1.00000i 1.16133i −1.00000 1.00000i 1.16133 −0.209794 + 2.63742i 1.00000i 1.65132 1.00000
461.7 − 1.00000i 1.89360i −1.00000 1.00000i 1.89360 −0.237610 − 2.63506i 1.00000i −0.585717 1.00000
461.8 − 1.00000i 2.55688i −1.00000 1.00000i 2.55688 2.64436 − 0.0858442i 1.00000i −3.53762 1.00000
461.9 1.00000i − 2.55688i −1.00000 − 1.00000i 2.55688 2.64436 + 0.0858442i − 1.00000i −3.53762 1.00000
461.10 1.00000i − 1.89360i −1.00000 − 1.00000i 1.89360 −0.237610 + 2.63506i − 1.00000i −0.585717 1.00000
461.11 1.00000i − 1.16133i −1.00000 − 1.00000i 1.16133 −0.209794 − 2.63742i − 1.00000i 1.65132 1.00000
461.12 1.00000i − 0.0908173i −1.00000 − 1.00000i 0.0908173 −1.53620 + 2.15409i − 1.00000i 2.99175 1.00000
461.13 1.00000i 0.741934i −1.00000 − 1.00000i −0.741934 −2.60369 + 0.469884i − 1.00000i 2.44953 1.00000
461.14 1.00000i 1.48511i −1.00000 − 1.00000i −1.48511 0.778797 − 2.52853i − 1.00000i 0.794454 1.00000
461.15 1.00000i 2.11620i −1.00000 − 1.00000i −2.11620 2.58002 + 0.586068i − 1.00000i −1.47832 1.00000
461.16 1.00000i 3.35938i −1.00000 − 1.00000i −3.35938 −1.41588 + 2.23501i − 1.00000i −8.28540 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 461.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.e.a ✓ 16
7.b odd 2 1 770.2.e.b yes 16
11.b odd 2 1 770.2.e.b yes 16
77.b even 2 1 inner 770.2.e.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.e.a ✓ 16 1.a even 1 1 trivial
770.2.e.a ✓ 16 77.b even 2 1 inner
770.2.e.b yes 16 7.b odd 2 1
770.2.e.b yes 16 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{8} - 4T_{13}^{7} - 62T_{13}^{6} + 248T_{13}^{5} + 732T_{13}^{4} - 3024T_{13}^{3} + 792T_{13}^{2} + 3552T_{13} - 1984 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 30 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 5 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{7} + \cdots - 1984)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 8 T^{7} + \cdots - 26688)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 10 T^{7} + \cdots + 3336)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 12 T^{7} + \cdots - 293184)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2226329856 \) Copy content Toggle raw display
$31$ \( T^{16} + 210 T^{14} + \cdots + 1420864 \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 944080)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 4 T^{7} + \cdots + 3760)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 209983897600 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 63262310400 \) Copy content Toggle raw display
$53$ \( (T^{8} - 8 T^{7} + \cdots + 622032)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 436 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$61$ \( (T^{8} + 2 T^{7} + \cdots - 44736)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 4 T^{7} + \cdots + 370432)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 2 T^{7} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{7} + \cdots + 907264)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 112869376 \) Copy content Toggle raw display
$83$ \( (T^{8} - 16 T^{7} + \cdots + 1025536)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 3384187027456 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 96191842369536 \) Copy content Toggle raw display
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