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Properties

Label 370.2.o.b
Level $370$
Weight $2$
Character orbit 370.o
Analytic conductor $2.954$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [370,2,Mod(71,370)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.o (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content 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: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 18 x^{16} - 25 x^{15} + 132 x^{14} - 135 x^{13} + 666 x^{12} - 297 x^{11} + 1845 x^{10} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{8} q^{3} + (\beta_{11} - \beta_{10}) q^{4} - \beta_{10} q^{5} - \beta_{5} q^{6} + (\beta_{17} - \beta_{15} - \beta_{14} + \cdots - 1) q^{7} + \beta_{2} q^{8} + (\beta_{17} - \beta_{15} - \beta_{10} + \cdots - 1) q^{9}+ \cdots + ( - \beta_{17} + \beta_{14} - 2 \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 6 q^{6} - 9 q^{7} + 9 q^{8} - 12 q^{9} - 9 q^{10} + 15 q^{11} + 3 q^{13} - 6 q^{14} + 6 q^{17} - 6 q^{18} - 3 q^{19} - 12 q^{21} - 12 q^{22} + 15 q^{23} + 6 q^{26} + 12 q^{27} + 9 q^{28} - 12 q^{29}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 18 x^{16} - 25 x^{15} + 132 x^{14} - 135 x^{13} + 666 x^{12} - 297 x^{11} + 1845 x^{10} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 467011630737 \nu^{17} + 7985628307079 \nu^{16} - 20288175144489 \nu^{15} + \cdots + 10\!\cdots\!73 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1286483210027 \nu^{17} + 14299357227244 \nu^{16} - 49682295365806 \nu^{15} + \cdots + 734083447796847 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4104600398475 \nu^{17} - 27243735217429 \nu^{16} + 106486954036400 \nu^{15} + \cdots - 15\!\cdots\!05 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9386663199290 \nu^{17} - 28694384497755 \nu^{16} + 169908419865960 \nu^{15} + \cdots - 4203104676633 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14929934022004 \nu^{17} + 32604146863850 \nu^{16} - 236340748141006 \nu^{15} + \cdots - 36941403586275 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18337730018439 \nu^{17} - 31292997456654 \nu^{16} - 118353124680056 \nu^{15} + \cdots - 57\!\cdots\!63 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26105354366996 \nu^{17} - 63238568869762 \nu^{16} + 423969596505896 \nu^{15} + \cdots + 345236246921304 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31986364509206 \nu^{17} - 91923387999016 \nu^{16} + 560235379575988 \nu^{15} + \cdots - 522277892930232 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38359582991256 \nu^{17} + 141184103340764 \nu^{16} - 753711062712370 \nu^{15} + \cdots + 10\!\cdots\!19 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43205244541727 \nu^{17} - 102223896048158 \nu^{16} + 700156475654080 \nu^{15} + \cdots - 207168800466078 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 58030876992248 \nu^{17} + 142106266467538 \nu^{16} - 952632397861448 \nu^{15} + \cdots - 10\!\cdots\!71 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 59935273415844 \nu^{17} - 162040208301516 \nu^{16} + \cdots - 54728443433070 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1086451530723 \nu^{17} + 3619950724209 \nu^{16} - 20271176257636 \nu^{15} + \cdots + 7416666919458 ) / 12981828186153 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 90945361447588 \nu^{17} - 266358564289231 \nu^{16} + \cdots + 627260144849775 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 123305861608837 \nu^{17} - 339448293361062 \nu^{16} + \cdots - 43\!\cdots\!27 ) / 947673457589169 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 153621363022881 \nu^{17} + 442400867850384 \nu^{16} + \cdots + 45\!\cdots\!61 ) / 947673457589169 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{5} + 3\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{11} + \beta_{9} + \beta_{8} + 5\beta_{5} + \beta_{4} - \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - 6 \beta_{13} + \beta_{12} - 8 \beta_{10} + \beta_{9} + \beta_{8} - 7 \beta_{7} + \cdots - 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{14} - 9 \beta_{13} + 12 \beta_{12} - 5 \beta_{11} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 39 \beta_{17} + 11 \beta_{16} + 10 \beta_{15} - 10 \beta_{14} + 10 \beta_{13} + 48 \beta_{12} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{17} + 8 \beta_{16} - 8 \beta_{15} - 24 \beta_{14} + 82 \beta_{13} - 53 \beta_{12} + \cdots + 211 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 280 \beta_{17} - 84 \beta_{16} - 93 \beta_{15} - 9 \beta_{14} + 196 \beta_{13} - 426 \beta_{12} + \cdots - 672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 593 \beta_{17} - 218 \beta_{16} - 172 \beta_{15} + 172 \beta_{14} - 172 \beta_{13} - 573 \beta_{12} + \cdots - 1391 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 47 \beta_{17} - 47 \beta_{16} + 47 \beta_{15} + 728 \beta_{14} - 2008 \beta_{13} + 562 \beta_{12} + \cdots - 3820 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4890 \beta_{17} + 1598 \beta_{16} + 1798 \beta_{15} + 200 \beta_{14} - 3292 \beta_{13} + 7077 \beta_{12} + \cdots + 10758 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 14546 \beta_{17} + 5553 \beta_{16} + 5472 \beta_{15} - 5472 \beta_{14} + 5472 \beta_{13} + \cdots + 33757 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 363 \beta_{17} + 363 \beta_{16} - 363 \beta_{15} - 14228 \beta_{14} + 37152 \beta_{13} + \cdots + 79983 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 107792 \beta_{17} - 43811 \beta_{16} - 42037 \beta_{15} + 1774 \beta_{14} + 63981 \beta_{13} + \cdots - 246687 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 287040 \beta_{17} - 110485 \beta_{16} - 115961 \beta_{15} + 115961 \beta_{14} - 115961 \beta_{13} + \cdots - 628343 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 31685 \beta_{17} + 31685 \beta_{16} - 31685 \beta_{15} + 318069 \beta_{14} - 801590 \beta_{13} + \cdots - 1601012 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2133659 \beta_{17} + 949035 \beta_{16} + 850716 \beta_{15} - 98319 \beta_{14} - 1184624 \beta_{13} + \cdots + 4783620 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\beta_{10} + \beta_{11}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
1.28031 − 2.21756i
−0.132544 + 0.229572i
−0.647764 + 1.12196i
−0.707815 + 1.22597i
0.247857 − 0.429301i
0.959958 − 1.66270i
1.39238 − 2.41167i
0.206805 − 0.358197i
−1.09918 + 1.90384i
−0.707815 − 1.22597i
0.247857 + 0.429301i
0.959958 + 1.66270i
1.39238 + 2.41167i
0.206805 + 0.358197i
−1.09918 − 1.90384i
1.28031 + 2.21756i
−0.132544 − 0.229572i
−0.647764 − 1.12196i
−0.766044 − 0.642788i −1.96154 + 1.64593i 0.173648 + 0.984808i 0.939693 + 0.342020i 2.56061 1.15931 + 0.421953i 0.500000 − 0.866025i 0.617622 − 3.50271i −0.500000 − 0.866025i
71.2 −0.766044 − 0.642788i 0.203068 − 0.170395i 0.173648 + 0.984808i 0.939693 + 0.342020i −0.265087 1.72587 + 0.628164i 0.500000 − 0.866025i −0.508742 + 2.88522i −0.500000 − 0.866025i
71.3 −0.766044 − 0.642788i 0.992431 − 0.832749i 0.173648 + 0.984808i 0.939693 + 0.342020i −1.29553 −3.09818 − 1.12765i 0.500000 − 0.866025i −0.229495 + 1.30153i −0.500000 − 0.866025i
81.1 0.939693 − 0.342020i −1.33026 − 0.484174i 0.766044 − 0.642788i −0.173648 − 0.984808i −1.41563 0.162212 + 0.919948i 0.500000 − 0.866025i −0.762973 − 0.640210i −0.500000 − 0.866025i
81.2 0.939693 − 0.342020i 0.465818 + 0.169544i 0.766044 − 0.642788i −0.173648 − 0.984808i 0.495714 −0.827523 − 4.69312i 0.500000 − 0.866025i −2.10989 − 1.77041i −0.500000 − 0.866025i
81.3 0.939693 − 0.342020i 1.80413 + 0.656650i 0.766044 − 0.642788i −0.173648 − 0.984808i 1.91992 0.523752 + 2.97035i 0.500000 − 0.866025i 0.525568 + 0.441004i −0.500000 − 0.866025i
181.1 −0.173648 + 0.984808i −0.483568 − 2.74245i −0.939693 − 0.342020i −0.766044 + 0.642788i 2.78476 −2.74168 + 2.30055i 0.500000 − 0.866025i −4.46812 + 1.62626i −0.500000 − 0.866025i
181.2 −0.173648 + 0.984808i −0.0718226 − 0.407326i −0.939693 − 0.342020i −0.766044 + 0.642788i 0.413610 1.40064 − 1.17528i 0.500000 − 0.866025i 2.65832 − 0.967550i −0.500000 − 0.866025i
181.3 −0.173648 + 0.984808i 0.381743 + 2.16497i −0.939693 − 0.342020i −0.766044 + 0.642788i −2.19837 −2.80439 + 2.35316i 0.500000 − 0.866025i −1.72229 + 0.626862i −0.500000 − 0.866025i
201.1 0.939693 + 0.342020i −1.33026 + 0.484174i 0.766044 + 0.642788i −0.173648 + 0.984808i −1.41563 0.162212 − 0.919948i 0.500000 + 0.866025i −0.762973 + 0.640210i −0.500000 + 0.866025i
201.2 0.939693 + 0.342020i 0.465818 − 0.169544i 0.766044 + 0.642788i −0.173648 + 0.984808i 0.495714 −0.827523 + 4.69312i 0.500000 + 0.866025i −2.10989 + 1.77041i −0.500000 + 0.866025i
201.3 0.939693 + 0.342020i 1.80413 − 0.656650i 0.766044 + 0.642788i −0.173648 + 0.984808i 1.91992 0.523752 − 2.97035i 0.500000 + 0.866025i 0.525568 − 0.441004i −0.500000 + 0.866025i
231.1 −0.173648 − 0.984808i −0.483568 + 2.74245i −0.939693 + 0.342020i −0.766044 − 0.642788i 2.78476 −2.74168 − 2.30055i 0.500000 + 0.866025i −4.46812 − 1.62626i −0.500000 + 0.866025i
231.2 −0.173648 − 0.984808i −0.0718226 + 0.407326i −0.939693 + 0.342020i −0.766044 − 0.642788i 0.413610 1.40064 + 1.17528i 0.500000 + 0.866025i 2.65832 + 0.967550i −0.500000 + 0.866025i
231.3 −0.173648 − 0.984808i 0.381743 − 2.16497i −0.939693 + 0.342020i −0.766044 − 0.642788i −2.19837 −2.80439 − 2.35316i 0.500000 + 0.866025i −1.72229 − 0.626862i −0.500000 + 0.866025i
271.1 −0.766044 + 0.642788i −1.96154 − 1.64593i 0.173648 − 0.984808i 0.939693 − 0.342020i 2.56061 1.15931 − 0.421953i 0.500000 + 0.866025i 0.617622 + 3.50271i −0.500000 + 0.866025i
271.2 −0.766044 + 0.642788i 0.203068 + 0.170395i 0.173648 − 0.984808i 0.939693 − 0.342020i −0.265087 1.72587 − 0.628164i 0.500000 + 0.866025i −0.508742 − 2.88522i −0.500000 + 0.866025i
271.3 −0.766044 + 0.642788i 0.992431 + 0.832749i 0.173648 − 0.984808i 0.939693 − 0.342020i −1.29553 −3.09818 + 1.12765i 0.500000 + 0.866025i −0.229495 − 1.30153i −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.b ✓ 18
37.f even 9 1 inner 370.2.o.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.o.b ✓ 18 1.a even 1 1 trivial
370.2.o.b ✓ 18 37.f even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 6 T_{3}^{16} - 10 T_{3}^{15} + 6 T_{3}^{14} - 63 T_{3}^{13} + 168 T_{3}^{12} - 306 T_{3}^{11} + \cdots + 9 \) 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
$3$ \( T^{18} + 6 T^{16} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{18} + 9 T^{17} + \cdots + 5774409 \) Copy content Toggle raw display
$11$ \( T^{18} - 15 T^{17} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} + \cdots + 811801 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 172423161 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 8051293441 \) Copy content Toggle raw display
$23$ \( T^{18} - 15 T^{17} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 375017837769 \) Copy content Toggle raw display
$31$ \( (T^{9} - 27 T^{8} + \cdots - 6497261)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 129961739795077 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 15742469961 \) Copy content Toggle raw display
$43$ \( (T^{9} + 30 T^{8} + \cdots - 31280563)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 4886429409 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 29987502561 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 43\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 821069424267481 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 4747623409 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 81\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( (T^{9} + 3 T^{8} + \cdots - 563149)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 21673019174041 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 244054772361 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 167859825849 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 133252011369 \) Copy content Toggle raw display
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