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Properties

Label 1700.2.o.e
Level $1700$
Weight $2$
Character orbit 1700.o
Analytic conductor $13.575$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1700,2,Mod(701,1700)] 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("1700.701"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1700.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5745683436\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 2 x^{9} + 55 x^{8} - 106 x^{7} + 104 x^{6} + 102 x^{5} + 187 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{11} q^{7} + (\beta_{11} - \beta_{10} - \beta_{9}) q^{9} + (\beta_{10} + \beta_{8} - \beta_{2} - \beta_1) q^{11} + (\beta_{8} + \beta_{7}) q^{13} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{17}+ \cdots + ( - 3 \beta_{9} - 2 \beta_{7} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 4 q^{13} + 2 q^{17} - 8 q^{21} - 12 q^{23} - 2 q^{27} - 14 q^{29} - 14 q^{31} + 12 q^{33} - 2 q^{37} - 10 q^{41} + 4 q^{47} + 12 q^{51} - 30 q^{57} - 8 q^{61} - 38 q^{63} + 20 q^{67} - 8 q^{69}+ \cdots + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 2 x^{9} + 55 x^{8} - 106 x^{7} + 104 x^{6} + 102 x^{5} + 187 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8558105 \nu^{11} - 17417129 \nu^{10} + 23187359 \nu^{9} + 13392690 \nu^{8} + 471766478 \nu^{7} + \cdots - 18382859 ) / 1422458607 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50849063 \nu^{11} + 83315267 \nu^{10} - 73490513 \nu^{9} - 121046715 \nu^{8} + \cdots + 370798178 ) / 1422458607 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 101698126 \nu^{11} - 166630534 \nu^{10} + 146981026 \nu^{9} + 242093430 \nu^{8} + \cdots - 741596356 ) / 474152869 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 370798178 \nu^{11} - 690747293 \nu^{10} + 658281089 \nu^{9} + 815086869 \nu^{8} + \cdots - 2295129914 ) / 1422458607 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 700053115 \nu^{11} + 1449014104 \nu^{10} - 1400967004 \nu^{9} - 1452787167 \nu^{8} + \cdots + 3815040409 ) / 1422458607 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 258819166 \nu^{11} - 410732933 \nu^{10} + 324432374 \nu^{9} + 681681934 \nu^{8} + \cdots + 85670748 ) / 474152869 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 284762816 \nu^{11} - 448778941 \nu^{10} + 345082244 \nu^{9} + 781223542 \nu^{8} + \cdots + 1513080934 ) / 474152869 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1051660429 \nu^{11} + 2154169921 \nu^{10} - 2186636125 \nu^{9} - 2029830345 \nu^{8} + \cdots + 13351914763 ) / 1422458607 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1863597631 \nu^{11} - 3610642399 \nu^{10} + 3453024304 \nu^{9} + 4018023786 \nu^{8} + \cdots - 11062670224 ) / 1422458607 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 641443912 \nu^{11} - 1176238119 \nu^{10} + 1080410482 \nu^{9} + 1477818016 \nu^{8} + \cdots - 1532048401 ) / 474152869 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{11} + 7\beta_{9} - 8\beta_{8} + \beta_{7} + \beta_{3} + \beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{7} - \beta_{6} + \beta_{5} + 9\beta_{2} - 39\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49\beta_{11} - 59\beta_{10} - 49\beta_{9} - 11\beta_{6} - 120\beta_{5} + \beta_{4} - 12\beta_{3} + \beta_{2} + 12\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{10} + 11\beta_{9} + \beta_{8} - 12\beta_{7} - 12\beta_{6} + 15\beta_{5} + 70\beta_{4} + 264\beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 346 \beta_{11} - 346 \beta_{9} + 428 \beta_{8} - 94 \beta_{7} + 13 \beta_{4} - 107 \beta_{3} + \cdots + 829 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 94 \beta_{11} + 13 \beta_{10} + 13 \beta_{8} - 107 \beta_{7} + 107 \beta_{6} - 145 \beta_{5} + \cdots - 145 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2458 \beta_{11} + 3087 \beta_{10} + 2458 \beta_{9} + 736 \beta_{6} + 5821 \beta_{5} + \cdots - 855 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 121 \beta_{10} - 734 \beta_{9} - 121 \beta_{8} + 856 \beta_{7} + 856 \beta_{6} - 1200 \beta_{5} + \cdots + 1200 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(851\) \(1601\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
1.89391 + 1.89391i
1.44435 + 1.44435i
0.254476 + 0.254476i
0.116232 + 0.116232i
−0.816234 − 0.816234i
−1.89274 − 1.89274i
1.89391 − 1.89391i
1.44435 − 1.44435i
0.254476 − 0.254476i
0.116232 − 0.116232i
−0.816234 + 0.816234i
−1.89274 + 1.89274i
0 −1.89391 + 1.89391i 0 0 0 −0.735996 − 0.735996i 0 − 4.17377i 0
701.2 0 −1.44435 + 1.44435i 0 0 0 −0.653825 − 0.653825i 0 − 1.17232i 0
701.3 0 −0.254476 + 0.254476i 0 0 0 0.964825 + 0.964825i 0 2.87048i 0
701.4 0 −0.116232 + 0.116232i 0 0 0 3.30173 + 3.30173i 0 2.97298i 0
701.5 0 0.816234 − 0.816234i 0 0 0 −1.61257 − 1.61257i 0 1.66752i 0
701.6 0 1.89274 − 1.89274i 0 0 0 −1.26417 − 1.26417i 0 − 4.16490i 0
1101.1 0 −1.89391 − 1.89391i 0 0 0 −0.735996 + 0.735996i 0 4.17377i 0
1101.2 0 −1.44435 − 1.44435i 0 0 0 −0.653825 + 0.653825i 0 1.17232i 0
1101.3 0 −0.254476 − 0.254476i 0 0 0 0.964825 − 0.964825i 0 − 2.87048i 0
1101.4 0 −0.116232 − 0.116232i 0 0 0 3.30173 − 3.30173i 0 − 2.97298i 0
1101.5 0 0.816234 + 0.816234i 0 0 0 −1.61257 + 1.61257i 0 − 1.66752i 0
1101.6 0 1.89274 + 1.89274i 0 0 0 −1.26417 + 1.26417i 0 4.16490i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.2.o.e ✓ 12
5.b even 2 1 1700.2.o.g yes 12
5.c odd 4 1 1700.2.m.d 12
5.c odd 4 1 1700.2.m.e 12
17.c even 4 1 inner 1700.2.o.e ✓ 12
85.f odd 4 1 1700.2.m.e 12
85.i odd 4 1 1700.2.m.d 12
85.j even 4 1 1700.2.o.g yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1700.2.m.d 12 5.c odd 4 1
1700.2.m.d 12 85.i odd 4 1
1700.2.m.e 12 5.c odd 4 1
1700.2.m.e 12 85.f odd 4 1
1700.2.o.e ✓ 12 1.a even 1 1 trivial
1700.2.o.e ✓ 12 17.c even 4 1 inner
1700.2.o.g yes 12 5.b even 2 1
1700.2.o.g yes 12 85.j even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 2 T_{3}^{11} + 2 T_{3}^{10} - 2 T_{3}^{9} + 55 T_{3}^{8} + 106 T_{3}^{7} + 104 T_{3}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(1700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 40 T^{9} + \cdots + 625 \) Copy content Toggle raw display
$11$ \( T^{12} - 56 T^{9} + \cdots + 75076 \) Copy content Toggle raw display
$13$ \( (T^{6} - 2 T^{5} + \cdots - 2917)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 2 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 164 T^{10} + \cdots + 6250000 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 1304164 \) Copy content Toggle raw display
$29$ \( T^{12} + 14 T^{11} + \cdots + 8880400 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 834689881 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 548496400 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 111091600 \) Copy content Toggle raw display
$43$ \( T^{12} + 248 T^{10} + \cdots + 45319824 \) Copy content Toggle raw display
$47$ \( (T^{6} - 2 T^{5} + \cdots - 40376)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 198 T^{10} + \cdots + 28561 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 15141794704 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots + 295936 \) Copy content Toggle raw display
$67$ \( (T^{6} - 10 T^{5} + \cdots - 544)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 190081369 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 33931113616 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 7617972961 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 167516666944 \) Copy content Toggle raw display
$89$ \( (T^{6} - 20 T^{5} + \cdots - 179476)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 318592513600 \) Copy content Toggle raw display
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