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Properties

Label 370.2.h.e
Level $370$
Weight $2$
Character orbit 370.h
Analytic conductor $2.954$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(117,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.h (of order \(4\), degree \(2\), 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} + 4 x^{17} + 103 x^{16} - 394 x^{15} + 760 x^{14} + 278 x^{13} + 2009 x^{12} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{4} q^{3} + q^{4} + \beta_{13} q^{5} + \beta_{4} q^{6} - \beta_{17} q^{7} - q^{8} + ( - \beta_{18} - \beta_{10}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{4} q^{3} + q^{4} + \beta_{13} q^{5} + \beta_{4} q^{6} - \beta_{17} q^{7} - q^{8} + ( - \beta_{18} - \beta_{10}) q^{9} - \beta_{13} q^{10} - \beta_{19} q^{11} - \beta_{4} q^{12} + (\beta_{12} - \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{13}+ \cdots + ( - \beta_{17} + \beta_{15} + \beta_{14} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} - 4 q^{5} - 4 q^{6} - 2 q^{7} - 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 4 q^{3} + 20 q^{4} - 4 q^{5} - 4 q^{6} - 2 q^{7} - 20 q^{8} + 4 q^{10} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 20 q^{16} + 6 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{24} + 10 q^{25} - 20 q^{27} - 2 q^{28} + 18 q^{29} + 4 q^{30} + 12 q^{31} - 20 q^{32} + 4 q^{33} - 12 q^{35} - 32 q^{37} - 6 q^{38} + 6 q^{39} + 4 q^{40} + 16 q^{43} + 22 q^{45} + 4 q^{46} - 22 q^{47} + 4 q^{48} - 10 q^{50} + 8 q^{51} - 4 q^{53} + 20 q^{54} + 16 q^{55} + 2 q^{56} + 24 q^{57} - 18 q^{58} - 10 q^{59} - 4 q^{60} + 10 q^{61} - 12 q^{62} - 2 q^{63} + 20 q^{64} + 20 q^{65} - 4 q^{66} + 8 q^{67} - 34 q^{69} + 12 q^{70} + 16 q^{71} - 6 q^{73} + 32 q^{74} - 26 q^{75} + 6 q^{76} - 4 q^{77} - 6 q^{78} + 12 q^{79} - 4 q^{80} - 28 q^{81} + 6 q^{83} + 10 q^{85} - 16 q^{86} - 44 q^{89} - 22 q^{90} - 40 q^{91} - 4 q^{92} - 40 q^{93} + 22 q^{94} + 50 q^{95} - 4 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 8 x^{18} + 4 x^{17} + 103 x^{16} - 394 x^{15} + 760 x^{14} + 278 x^{13} + 2009 x^{12} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58\!\cdots\!94 \nu^{19} + \cdots + 27\!\cdots\!56 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!87 \nu^{19} + \cdots + 38\!\cdots\!52 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\!\cdots\!71 \nu^{19} + \cdots - 27\!\cdots\!56 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!01 \nu^{19} + \cdots - 40\!\cdots\!60 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18\!\cdots\!82 \nu^{19} + \cdots - 91\!\cdots\!64 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!49 \nu^{19} + \cdots + 59\!\cdots\!44 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 74\!\cdots\!19 \nu^{19} + \cdots - 15\!\cdots\!84 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23\!\cdots\!89 \nu^{19} + \cdots + 34\!\cdots\!36 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!51 \nu^{19} + \cdots - 30\!\cdots\!56 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!77 \nu^{19} + \cdots - 42\!\cdots\!88 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24\!\cdots\!73 \nu^{19} + \cdots - 52\!\cdots\!88 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 33\!\cdots\!27 \nu^{19} + \cdots + 16\!\cdots\!12 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 54\!\cdots\!63 \nu^{19} + \cdots + 15\!\cdots\!16 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 31\!\cdots\!11 \nu^{19} + \cdots + 52\!\cdots\!96 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 80\!\cdots\!83 \nu^{19} + \cdots - 31\!\cdots\!28 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 15\!\cdots\!09 \nu^{19} + \cdots + 79\!\cdots\!84 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{19} + \cdots + 30\!\cdots\!56 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 45\!\cdots\!13 \nu^{19} + \cdots + 13\!\cdots\!48 ) / 27\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{18} + 4\beta_{10} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{17} + 2\beta_{15} - 2\beta_{11} + 2\beta_{10} - \beta_{7} - \beta_{5} + 7\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{18} + \beta_{17} - 2 \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} - \beta_{9} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{19} - 15 \beta_{18} - 16 \beta_{16} - 2 \beta_{15} - 12 \beta_{14} - 27 \beta_{13} - 2 \beta_{12} + \cdots - 29 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{19} - 113 \beta_{18} - 15 \beta_{17} - 47 \beta_{16} - 12 \beta_{15} - 15 \beta_{14} + \cdots - 39 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 143 \beta_{19} - 26 \beta_{18} - 124 \beta_{17} - 37 \beta_{16} - 312 \beta_{15} - 33 \beta_{13} + \cdots + 363 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 367 \beta_{18} - 179 \beta_{17} + 367 \beta_{16} - 303 \beta_{15} + 179 \beta_{14} + 303 \beta_{13} + \cdots + 2878 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1540 \beta_{19} + 2345 \beta_{18} + 2411 \beta_{16} + 420 \beta_{15} + 1278 \beta_{14} + 3492 \beta_{13} + \cdots + 4428 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4690 \beta_{19} + 12450 \beta_{18} + 2064 \beta_{17} + 7228 \beta_{16} + 4345 \beta_{15} + \cdots + 7975 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 16755 \beta_{19} + 6769 \beta_{18} + 13508 \beta_{17} + 6658 \beta_{16} + 38934 \beta_{15} + \cdots - 53553 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 50135 \beta_{18} + 23917 \beta_{17} - 50135 \beta_{16} + 53046 \beta_{15} - 23917 \beta_{14} + \cdots - 342236 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 185137 \beta_{19} - 342404 \beta_{18} - 331289 \beta_{16} - 56474 \beta_{15} - 146608 \beta_{14} + \cdots - 643707 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 684808 \beta_{19} - 1490121 \beta_{18} - 279555 \beta_{17} - 1006273 \beta_{16} - 788457 \beta_{15} + \cdots - 1290752 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2074136 \beta_{19} - 1212487 \beta_{18} - 1625416 \beta_{17} - 983149 \beta_{16} - 4912634 \beta_{15} + \cdots + 7694530 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6811496 \beta_{18} - 3284842 \beta_{17} + 6811496 \beta_{16} - 8090351 \beta_{15} + 3284842 \beta_{14} + \cdots + 43593678 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 23489413 \beta_{19} + 48801367 \beta_{18} + 45162738 \beta_{16} + 7223690 \beta_{15} + \cdots + 91527319 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 97602734 \beta_{19} + 188655987 \beta_{18} + 38674947 \beta_{17} + 137877993 \beta_{16} + \cdots + 191691961 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 268170457 \beta_{19} + 189375606 \beta_{18} + 208351524 \beta_{17} + 137889655 \beta_{16} + \cdots - 1084260885 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\beta_{10}\) \(-\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
−2.09082 + 2.09082i
−1.28900 + 1.28900i
−1.23675 + 1.23675i
−0.794932 + 0.794932i
0.0477388 − 0.0477388i
0.536506 − 0.536506i
1.28931 − 1.28931i
1.29397 − 1.29397i
1.82785 − 1.82785i
2.41612 − 2.41612i
−2.09082 − 2.09082i
−1.28900 − 1.28900i
−1.23675 − 1.23675i
−0.794932 − 0.794932i
0.0477388 + 0.0477388i
0.536506 + 0.536506i
1.28931 + 1.28931i
1.29397 + 1.29397i
1.82785 + 1.82785i
2.41612 + 2.41612i
−1.00000 −2.09082 − 2.09082i 1.00000 −1.81614 + 1.30447i 2.09082 + 2.09082i −0.643605 − 0.643605i −1.00000 5.74304i 1.81614 − 1.30447i
117.2 −1.00000 −1.28900 − 1.28900i 1.00000 1.63786 − 1.52231i 1.28900 + 1.28900i −3.01566 − 3.01566i −1.00000 0.323040i −1.63786 + 1.52231i
117.3 −1.00000 −1.23675 − 1.23675i 1.00000 −1.26849 − 1.84145i 1.23675 + 1.23675i 1.28708 + 1.28708i −1.00000 0.0591090i 1.26849 + 1.84145i
117.4 −1.00000 −0.794932 − 0.794932i 1.00000 2.22550 + 0.217179i 0.794932 + 0.794932i 2.93770 + 2.93770i −1.00000 − 1.73617i −2.22550 − 0.217179i
117.5 −1.00000 0.0477388 + 0.0477388i 1.00000 0.531109 + 2.17208i −0.0477388 − 0.0477388i −2.77997 − 2.77997i −1.00000 − 2.99544i −0.531109 − 2.17208i
117.6 −1.00000 0.536506 + 0.536506i 1.00000 −2.23241 − 0.127776i −0.536506 − 0.536506i 0.767774 + 0.767774i −1.00000 − 2.42432i 2.23241 + 0.127776i
117.7 −1.00000 1.28931 + 1.28931i 1.00000 1.69364 + 1.45999i −1.28931 − 1.28931i 0.579841 + 0.579841i −1.00000 0.324646i −1.69364 − 1.45999i
117.8 −1.00000 1.29397 + 1.29397i 1.00000 −2.20164 − 0.390851i −1.29397 − 1.29397i −2.67087 − 2.67087i −1.00000 0.348729i 2.20164 + 0.390851i
117.9 −1.00000 1.82785 + 1.82785i 1.00000 −1.23698 + 1.86276i −1.82785 − 1.82785i 3.41332 + 3.41332i −1.00000 3.68208i 1.23698 − 1.86276i
117.10 −1.00000 2.41612 + 2.41612i 1.00000 0.667560 − 2.13410i −2.41612 − 2.41612i −0.875609 − 0.875609i −1.00000 8.67529i −0.667560 + 2.13410i
253.1 −1.00000 −2.09082 + 2.09082i 1.00000 −1.81614 − 1.30447i 2.09082 − 2.09082i −0.643605 + 0.643605i −1.00000 − 5.74304i 1.81614 + 1.30447i
253.2 −1.00000 −1.28900 + 1.28900i 1.00000 1.63786 + 1.52231i 1.28900 − 1.28900i −3.01566 + 3.01566i −1.00000 − 0.323040i −1.63786 − 1.52231i
253.3 −1.00000 −1.23675 + 1.23675i 1.00000 −1.26849 + 1.84145i 1.23675 − 1.23675i 1.28708 − 1.28708i −1.00000 − 0.0591090i 1.26849 − 1.84145i
253.4 −1.00000 −0.794932 + 0.794932i 1.00000 2.22550 − 0.217179i 0.794932 − 0.794932i 2.93770 − 2.93770i −1.00000 1.73617i −2.22550 + 0.217179i
253.5 −1.00000 0.0477388 − 0.0477388i 1.00000 0.531109 − 2.17208i −0.0477388 + 0.0477388i −2.77997 + 2.77997i −1.00000 2.99544i −0.531109 + 2.17208i
253.6 −1.00000 0.536506 − 0.536506i 1.00000 −2.23241 + 0.127776i −0.536506 + 0.536506i 0.767774 − 0.767774i −1.00000 2.42432i 2.23241 − 0.127776i
253.7 −1.00000 1.28931 − 1.28931i 1.00000 1.69364 − 1.45999i −1.28931 + 1.28931i 0.579841 − 0.579841i −1.00000 − 0.324646i −1.69364 + 1.45999i
253.8 −1.00000 1.29397 − 1.29397i 1.00000 −2.20164 + 0.390851i −1.29397 + 1.29397i −2.67087 + 2.67087i −1.00000 − 0.348729i 2.20164 − 0.390851i
253.9 −1.00000 1.82785 − 1.82785i 1.00000 −1.23698 − 1.86276i −1.82785 + 1.82785i 3.41332 − 3.41332i −1.00000 − 3.68208i 1.23698 + 1.86276i
253.10 −1.00000 2.41612 − 2.41612i 1.00000 0.667560 + 2.13410i −2.41612 + 2.41612i −0.875609 + 0.875609i −1.00000 − 8.67529i −0.667560 − 2.13410i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.h.e yes 20
5.c odd 4 1 370.2.g.e ✓ 20
37.d odd 4 1 370.2.g.e ✓ 20
185.k even 4 1 inner 370.2.h.e yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.e ✓ 20 5.c odd 4 1
370.2.g.e ✓ 20 37.d odd 4 1
370.2.h.e yes 20 1.a even 1 1 trivial
370.2.h.e yes 20 185.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 4 T_{3}^{19} + 8 T_{3}^{18} + 4 T_{3}^{17} + 103 T_{3}^{16} - 394 T_{3}^{15} + 760 T_{3}^{14} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{20} + 4 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 2 T^{19} + \cdots + 5382400 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 2130745600 \) Copy content Toggle raw display
$13$ \( (T^{10} - 83 T^{8} + \cdots - 32672)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 6206918656 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 36191257600 \) Copy content Toggle raw display
$23$ \( (T^{10} + 2 T^{9} + \cdots - 23040)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 651474576 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 11580342544 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 3430496665600 \) Copy content Toggle raw display
$43$ \( (T^{10} - 8 T^{9} + \cdots - 1401856)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 7191040000 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 12096484000000 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 778824091056400 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{10} - 8 T^{9} + \cdots - 147456)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 573388694327296 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
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