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Properties

Label 370.2.o.a
Level $370$
Weight $2$
Character orbit 370.o
Analytic conductor $2.954$
Analytic rank $0$
Dimension $18$
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] = [370,2,Mod(71,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.o (of order \(9\), degree \(6\), 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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 21 x^{16} - 20 x^{15} + 180 x^{14} - 126 x^{13} + 1002 x^{12} - 270 x^{11} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{17} + \beta_{8} - \beta_{2}) q^{3} + \beta_{6} q^{4} + \beta_{5} q^{5} + ( - \beta_{16} + \beta_{13} + \cdots + \beta_1) q^{6}+ \cdots + ( - \beta_{17} - \beta_{16} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{17} + \beta_{8} - \beta_{2}) q^{3} + \beta_{6} q^{4} + \beta_{5} q^{5} + ( - \beta_{16} + \beta_{13} + \cdots + \beta_1) q^{6}+ \cdots + (3 \beta_{17} - 3 \beta_{16} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{6} - 3 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{6} - 3 q^{7} - 9 q^{8} - 12 q^{9} - 9 q^{10} + 3 q^{11} + 9 q^{13} - 6 q^{17} + 6 q^{18} + 9 q^{19} - 6 q^{21} + 21 q^{23} + 6 q^{26} + 12 q^{27} - 3 q^{28} + 6 q^{29} - 30 q^{31} - 45 q^{33} - 15 q^{34} + 6 q^{35} + 24 q^{36} + 6 q^{37} - 12 q^{38} + 24 q^{39} + 15 q^{41} - 6 q^{42} - 12 q^{43} - 12 q^{45} + 24 q^{47} + 3 q^{48} + 33 q^{49} - 42 q^{51} - 12 q^{53} + 27 q^{54} + 6 q^{56} + 51 q^{57} - 15 q^{58} - 15 q^{59} + 3 q^{60} + 72 q^{61} - 57 q^{62} - 30 q^{63} - 9 q^{64} + 9 q^{65} - 3 q^{66} + 18 q^{67} + 24 q^{69} - 3 q^{70} + 6 q^{72} - 66 q^{73} - 24 q^{74} - 6 q^{75} + 9 q^{76} - 66 q^{77} + 6 q^{78} - 12 q^{79} + 18 q^{80} + 66 q^{81} + 45 q^{82} + 9 q^{83} + 42 q^{84} + 12 q^{86} + 48 q^{87} + 3 q^{88} + 6 q^{90} - 78 q^{91} + 18 q^{92} + 24 q^{94} - 3 q^{97} - 48 q^{98} - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 21 x^{16} - 20 x^{15} + 180 x^{14} - 126 x^{13} + 1002 x^{12} - 270 x^{11} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 818926776113450 \nu^{17} + \cdots - 75\!\cdots\!70 ) / 43\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 499294071546270 \nu^{17} + \cdots - 10\!\cdots\!11 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!30 \nu^{17} + \cdots - 11\!\cdots\!81 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!23 \nu^{17} + \cdots - 14\!\cdots\!22 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37\!\cdots\!93 \nu^{17} + \cdots + 83\!\cdots\!72 ) / 43\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42\!\cdots\!03 \nu^{17} + \cdots - 20\!\cdots\!82 ) / 43\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1267556030104 \nu^{17} + 4354177580338 \nu^{16} - 28169998789216 \nu^{15} + \cdots + 286512494647932 ) / 12\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16\!\cdots\!79 \nu^{17} + \cdots + 30\!\cdots\!19 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 50\!\cdots\!65 \nu^{17} + \cdots + 11\!\cdots\!69 ) / 43\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!06 \nu^{17} + \cdots + 46\!\cdots\!74 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!52 \nu^{17} + \cdots - 19\!\cdots\!74 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 56\!\cdots\!48 \nu^{17} + \cdots + 17\!\cdots\!86 ) / 43\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 31834721627548 \nu^{17} - 84096160611708 \nu^{16} + 629341555955466 \nu^{15} + \cdots + 87\!\cdots\!77 ) / 11\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 56\!\cdots\!54 \nu^{17} + \cdots + 98\!\cdots\!78 ) / 14\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 23\!\cdots\!29 \nu^{17} + \cdots - 35\!\cdots\!51 ) / 25\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 44\!\cdots\!75 \nu^{17} + \cdots + 57\!\cdots\!15 ) / 43\!\cdots\!89 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} + \beta_{15} + 3\beta_{14} + \beta_{12} + \beta_{10} - \beta_{6} - \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{17} + 3 \beta_{16} + \beta_{15} - 3 \beta_{13} - 2 \beta_{12} + 3 \beta_{11} - \beta_{10} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{17} - 12 \beta_{15} - 19 \beta_{14} - 14 \beta_{13} - 25 \beta_{12} + 8 \beta_{11} + \cdots - 31 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34 \beta_{17} - 48 \beta_{16} - 29 \beta_{15} - 37 \beta_{14} - 50 \beta_{12} - 36 \beta_{11} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 208 \beta_{17} - 188 \beta_{16} + 42 \beta_{15} + 188 \beta_{13} + 136 \beta_{12} - 204 \beta_{11} + \cdots + 183 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 256 \beta_{17} + 541 \beta_{15} + 542 \beta_{14} + 682 \beta_{13} + 1162 \beta_{12} - 180 \beta_{11} + \cdots + 1621 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1921 \beta_{17} + 2585 \beta_{16} + 1557 \beta_{15} + 2201 \beta_{14} + 2718 \beta_{12} + 2097 \beta_{11} + \cdots - 2201 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 10769 \beta_{17} + 9577 \beta_{16} - 2560 \beta_{15} - 9577 \beta_{13} - 6057 \beta_{12} + \cdots - 7606 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14002 \beta_{17} - 31733 \beta_{15} - 29136 \beta_{14} - 36009 \beta_{13} - 61561 \beta_{12} + \cdots - 82993 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 99580 \beta_{17} - 134422 \beta_{16} - 78532 \beta_{15} - 106227 \beta_{14} - 145315 \beta_{12} + \cdots + 106227 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 572097 \beta_{17} - 504309 \beta_{16} + 152081 \beta_{15} + 504309 \beta_{13} + 317054 \beta_{12} + \cdots + 400331 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 742603 \beta_{17} + 1673031 \beta_{15} + 1486432 \beta_{14} + 1887683 \beta_{13} + 3225607 \beta_{12} + \cdots + 4351972 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5252245 \beta_{17} + 7077729 \beta_{16} + 4130201 \beta_{15} + 5578129 \beta_{14} + 7667879 \beta_{12} + \cdots - 5578129 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 30133351 \beta_{17} + 26518247 \beta_{16} - 8144025 \beta_{15} - 26518247 \beta_{13} + \cdots - 20841006 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 39219205 \beta_{17} - 88590124 \beta_{15} - 78134348 \beta_{14} - 99413071 \beta_{13} + \cdots - 228888829 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 276527785 \beta_{17} - 372601478 \beta_{16} - 217043820 \beta_{15} - 292555769 \beta_{14} + \cdots + 292555769 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
−0.978711 − 1.69518i
0.571872 + 0.990512i
0.140794 + 0.243862i
−1.05539 − 1.82798i
1.87438 + 3.24652i
0.620701 + 1.07509i
−0.231777 − 0.401450i
−0.649209 − 1.12446i
1.20734 + 2.09117i
−1.05539 + 1.82798i
1.87438 − 3.24652i
0.620701 − 1.07509i
−0.231777 + 0.401450i
−0.649209 + 1.12446i
1.20734 − 2.09117i
−0.978711 + 1.69518i
0.571872 − 0.990512i
0.140794 − 0.243862i
0.766044 + 0.642788i −1.90337 + 1.59711i 0.173648 + 0.984808i −0.939693 − 0.342020i −2.48467 −4.17420 − 1.51928i −0.500000 + 0.866025i 0.551085 − 3.12536i −0.500000 − 0.866025i
71.2 0.766044 + 0.642788i −0.646156 + 0.542189i 0.173648 + 0.984808i −0.939693 − 0.342020i −0.843496 0.282141 + 0.102691i −0.500000 + 0.866025i −0.397396 + 2.25375i −0.500000 − 0.866025i
71.3 0.766044 + 0.642788i 1.78348 − 1.49651i 0.173648 + 0.984808i −0.939693 − 0.342020i 2.32816 2.45236 + 0.892588i −0.500000 + 0.866025i 0.420289 − 2.38358i −0.500000 − 0.866025i
81.1 −0.939693 + 0.342020i −1.84510 − 0.671562i 0.766044 − 0.642788i 0.173648 + 0.984808i 1.96352 0.0255713 + 0.145022i −0.500000 + 0.866025i 0.655269 + 0.549836i −0.500000 − 0.866025i
81.2 −0.939693 + 0.342020i 0.528795 + 0.192466i 0.766044 − 0.642788i 0.173648 + 0.984808i −0.562732 −0.553248 − 3.13762i −0.500000 + 0.866025i −2.05555 − 1.72481i −0.500000 − 0.866025i
81.3 −0.939693 + 0.342020i 2.25600 + 0.821116i 0.766044 − 0.642788i 0.173648 + 0.984808i −2.40078 0.201324 + 1.14177i −0.500000 + 0.866025i 2.11716 + 1.77651i −0.500000 − 0.866025i
181.1 0.173648 − 0.984808i −0.568073 − 3.22170i −0.939693 − 0.342020i 0.766044 − 0.642788i −3.27140 2.86114 − 2.40078i −0.500000 + 0.866025i −7.23759 + 2.63427i −0.500000 − 0.866025i
181.2 0.173648 − 0.984808i −0.0317483 − 0.180053i −0.939693 − 0.342020i 0.766044 − 0.642788i −0.182831 1.13470 − 0.952128i −0.500000 + 0.866025i 2.78767 − 1.01463i −0.500000 − 0.866025i
181.3 0.173648 − 0.984808i 0.426173 + 2.41695i −0.939693 − 0.342020i 0.766044 − 0.642788i 2.45423 −3.72980 + 3.12967i −0.500000 + 0.866025i −2.84094 + 1.03402i −0.500000 − 0.866025i
201.1 −0.939693 − 0.342020i −1.84510 + 0.671562i 0.766044 + 0.642788i 0.173648 − 0.984808i 1.96352 0.0255713 − 0.145022i −0.500000 − 0.866025i 0.655269 − 0.549836i −0.500000 + 0.866025i
201.2 −0.939693 − 0.342020i 0.528795 − 0.192466i 0.766044 + 0.642788i 0.173648 − 0.984808i −0.562732 −0.553248 + 3.13762i −0.500000 − 0.866025i −2.05555 + 1.72481i −0.500000 + 0.866025i
201.3 −0.939693 − 0.342020i 2.25600 − 0.821116i 0.766044 + 0.642788i 0.173648 − 0.984808i −2.40078 0.201324 − 1.14177i −0.500000 − 0.866025i 2.11716 − 1.77651i −0.500000 + 0.866025i
231.1 0.173648 + 0.984808i −0.568073 + 3.22170i −0.939693 + 0.342020i 0.766044 + 0.642788i −3.27140 2.86114 + 2.40078i −0.500000 − 0.866025i −7.23759 − 2.63427i −0.500000 + 0.866025i
231.2 0.173648 + 0.984808i −0.0317483 + 0.180053i −0.939693 + 0.342020i 0.766044 + 0.642788i −0.182831 1.13470 + 0.952128i −0.500000 − 0.866025i 2.78767 + 1.01463i −0.500000 + 0.866025i
231.3 0.173648 + 0.984808i 0.426173 − 2.41695i −0.939693 + 0.342020i 0.766044 + 0.642788i 2.45423 −3.72980 − 3.12967i −0.500000 − 0.866025i −2.84094 − 1.03402i −0.500000 + 0.866025i
271.1 0.766044 − 0.642788i −1.90337 − 1.59711i 0.173648 − 0.984808i −0.939693 + 0.342020i −2.48467 −4.17420 + 1.51928i −0.500000 − 0.866025i 0.551085 + 3.12536i −0.500000 + 0.866025i
271.2 0.766044 − 0.642788i −0.646156 − 0.542189i 0.173648 − 0.984808i −0.939693 + 0.342020i −0.843496 0.282141 − 0.102691i −0.500000 − 0.866025i −0.397396 − 2.25375i −0.500000 + 0.866025i
271.3 0.766044 − 0.642788i 1.78348 + 1.49651i 0.173648 − 0.984808i −0.939693 + 0.342020i 2.32816 2.45236 − 0.892588i −0.500000 − 0.866025i 0.420289 + 2.38358i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.o.a ✓ 18
37.f even 9 1 inner 370.2.o.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.o.a ✓ 18 1.a even 1 1 trivial
370.2.o.a ✓ 18 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 6 T_{3}^{16} - 10 T_{3}^{15} - 12 T_{3}^{14} + 21 T_{3}^{13} + 254 T_{3}^{12} - 240 T_{3}^{11} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} + 6 T^{16} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 3 T^{17} + \cdots + 2601 \) Copy content Toggle raw display
$11$ \( T^{18} - 3 T^{17} + \cdots + 651249 \) Copy content Toggle raw display
$13$ \( T^{18} - 9 T^{17} + \cdots + 289 \) Copy content Toggle raw display
$17$ \( T^{18} + 6 T^{17} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 122556706561 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 153909489969 \) Copy content Toggle raw display
$29$ \( T^{18} - 6 T^{17} + \cdots + 8485569 \) Copy content Toggle raw display
$31$ \( (T^{9} + 15 T^{8} + \cdots + 3187)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 129961739795077 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 5887606220721 \) Copy content Toggle raw display
$43$ \( (T^{9} + 6 T^{8} + \cdots + 1578531)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 197880489 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 53445579489 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 165096801 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 58\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 6156844727401 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 458961767632449 \) Copy content Toggle raw display
$73$ \( (T^{9} + 33 T^{8} + \cdots + 31500201)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 68548580271649 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 21019490361 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 50074347200329 \) Copy content Toggle raw display
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