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Properties

Label 700.2.n.c
Level $700$
Weight $2$
Character orbit 700.n
Analytic conductor $5.590$
Analytic rank $0$
Dimension $20$
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] = [700,2,Mod(141,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.141");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.n (of order \(5\), degree \(4\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + 28 x^{18} - 114 x^{17} + 447 x^{16} - 1130 x^{15} + 3141 x^{14} - 5918 x^{13} + \cdots + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} + \beta_{2}) q^{3} + (\beta_{17} - \beta_{15} + \cdots - \beta_{4}) q^{5}+ \cdots + (\beta_{18} - \beta_{15} - 2 \beta_{13} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} + \beta_{2}) q^{3} + (\beta_{17} - \beta_{15} + \cdots - \beta_{4}) q^{5}+ \cdots + ( - 2 \beta_{19} + 2 \beta_{18} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{5} + 20 q^{7} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{5} + 20 q^{7} - 21 q^{9} + 6 q^{11} + 8 q^{13} - 21 q^{15} + 18 q^{17} - 4 q^{19} - q^{23} + 4 q^{25} + 21 q^{27} + 5 q^{29} - 3 q^{31} + 15 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{39} - 36 q^{41} - 46 q^{43} - 28 q^{45} + 4 q^{47} + 20 q^{49} - 14 q^{51} + 14 q^{53} - 38 q^{55} - 10 q^{57} + 9 q^{59} + 66 q^{61} - 21 q^{63} + 31 q^{65} + 23 q^{67} + 11 q^{69} - 44 q^{71} + 10 q^{73} + 44 q^{75} + 6 q^{77} + 10 q^{79} + 26 q^{81} + 32 q^{83} + 18 q^{85} + 26 q^{87} + 13 q^{89} + 8 q^{91} - 60 q^{93} - 17 q^{95} + 52 q^{97} - 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 5 x^{19} + 28 x^{18} - 114 x^{17} + 447 x^{16} - 1130 x^{15} + 3141 x^{14} - 5918 x^{13} + \cdots + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18\!\cdots\!92 \nu^{19} + \cdots - 25\!\cdots\!82 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\!\cdots\!81 \nu^{19} + \cdots - 11\!\cdots\!76 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!25 \nu^{19} + \cdots - 12\!\cdots\!81 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 33\!\cdots\!00 \nu^{19} + \cdots - 61\!\cdots\!05 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 40\!\cdots\!47 \nu^{19} + \cdots - 26\!\cdots\!23 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 68\!\cdots\!41 \nu^{19} + \cdots + 31\!\cdots\!97 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 74\!\cdots\!70 \nu^{19} + \cdots - 12\!\cdots\!62 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53\!\cdots\!67 \nu^{19} + \cdots + 44\!\cdots\!84 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!24 \nu^{19} + \cdots + 17\!\cdots\!82 ) / 22\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 61\!\cdots\!27 \nu^{19} + \cdots + 62\!\cdots\!77 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 62\!\cdots\!10 \nu^{19} + \cdots - 13\!\cdots\!74 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 91\!\cdots\!33 \nu^{19} + \cdots - 44\!\cdots\!18 ) / 10\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 68\!\cdots\!08 \nu^{19} + \cdots + 15\!\cdots\!82 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!80 \nu^{19} + \cdots + 11\!\cdots\!84 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!29 \nu^{19} + \cdots - 17\!\cdots\!48 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17\!\cdots\!15 \nu^{19} + \cdots + 19\!\cdots\!81 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 17\!\cdots\!47 \nu^{19} + \cdots - 19\!\cdots\!03 ) / 70\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 19\!\cdots\!90 \nu^{19} + \cdots - 16\!\cdots\!40 ) / 70\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} - \beta_{17} + \beta_{15} + \beta_{14} + 6\beta_{13} + \beta_{11} + 2\beta_{7} + \beta_{6} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{18} + 2 \beta_{15} + 14 \beta_{13} + 9 \beta_{11} - \beta_{10} + 13 \beta_{8} + \beta_{7} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14 \beta_{19} - 10 \beta_{18} + 10 \beta_{17} + 3 \beta_{16} + 2 \beta_{15} + 3 \beta_{14} - \beta_{13} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{19} + 7 \beta_{18} + 25 \beta_{17} - \beta_{16} + 3 \beta_{14} - \beta_{13} + 32 \beta_{12} + \cdots - 156 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 33 \beta_{19} + \beta_{18} + 89 \beta_{17} - 4 \beta_{16} + 34 \beta_{15} - 130 \beta_{14} + 83 \beta_{13} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 380 \beta_{19} - 90 \beta_{18} + 418 \beta_{17} + 219 \beta_{16} - 147 \beta_{15} - 147 \beta_{14} + \cdots + 361 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 101 \beta_{19} + 1448 \beta_{18} + 205 \beta_{17} - 1448 \beta_{15} + 588 \beta_{14} - 5243 \beta_{13} + \cdots - 5799 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7707 \beta_{19} + 4223 \beta_{18} - 4138 \beta_{17} - 2804 \beta_{16} - 1295 \beta_{15} - 2719 \beta_{14} + \cdots - 2804 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 8337 \beta_{19} - 8338 \beta_{18} - 8213 \beta_{17} + 1895 \beta_{16} + 1771 \beta_{15} + \cdots + 67117 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9757 \beta_{19} - 399 \beta_{18} - 25342 \beta_{17} + 7691 \beta_{16} - 22922 \beta_{15} + 63247 \beta_{14} + \cdots - 36907 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 191020 \beta_{19} + 85283 \beta_{18} - 188513 \beta_{17} - 120869 \beta_{16} + 40236 \beta_{15} + \cdots - 295065 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 24244 \beta_{19} - 635149 \beta_{18} - 90954 \beta_{17} + 635149 \beta_{15} - 408725 \beta_{14} + \cdots + 2562107 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3593135 \beta_{19} - 1822620 \beta_{18} + 1844457 \beta_{17} + 1465997 \beta_{16} + 291144 \beta_{15} + \cdots + 1465997 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1831260 \beta_{19} + 5316983 \beta_{18} + 1940281 \beta_{17} - 1632066 \beta_{16} - 1741087 \beta_{15} + \cdots - 31774778 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 12159706 \beta_{19} - 514261 \beta_{18} + 4994623 \beta_{17} - 6049286 \beta_{16} + 13009828 \beta_{15} + \cdots + 28780362 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 94359953 \beta_{19} - 56946331 \beta_{18} + 81013822 \beta_{17} + 61295702 \beta_{16} - 8613077 \beta_{15} + \cdots + 189309611 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 44552977 \beta_{19} + 282249207 \beta_{18} + 36844528 \beta_{17} - 282249207 \beta_{15} + \cdots - 1234041720 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1660293504 \beta_{19} + 800913218 \beta_{18} - 836483602 \beta_{17} - 737217717 \beta_{16} + \cdots - 737217717 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
141.1
0.615213 − 1.89343i
0.547964 − 1.68646i
0.287623 − 0.885211i
−0.798039 + 2.45611i
−1.07981 + 3.32332i
2.57121 + 1.86809i
2.07610 + 1.50837i
0.163622 + 0.118878i
−0.460734 − 0.334743i
−1.42314 − 1.03397i
2.57121 − 1.86809i
2.07610 − 1.50837i
0.163622 − 0.118878i
−0.460734 + 0.334743i
−1.42314 + 1.03397i
0.615213 + 1.89343i
0.547964 + 1.68646i
0.287623 + 0.885211i
−0.798039 − 2.45611i
−1.07981 − 3.32332i
0 −0.924230 − 2.84449i 0 −1.90971 + 1.16318i 0 1.00000 0 −4.80986 + 3.49456i 0
141.2 0 −0.856981 − 2.63752i 0 1.74511 + 1.39807i 0 1.00000 0 −3.79503 + 2.75725i 0
141.3 0 −0.596640 − 1.83627i 0 −0.344964 − 2.20930i 0 1.00000 0 −0.588849 + 0.427824i 0
141.4 0 0.489022 + 1.50505i 0 2.15677 − 0.590193i 0 1.00000 0 0.401006 − 0.291348i 0
141.5 0 0.770795 + 2.37226i 0 −0.647206 + 2.14036i 0 1.00000 0 −2.60646 + 1.89370i 0
281.1 0 −1.76219 + 1.28031i 0 2.06573 − 0.856025i 0 1.00000 0 0.539080 − 1.65912i 0
281.2 0 −1.26708 + 0.920587i 0 0.334526 + 2.21090i 0 1.00000 0 −0.169041 + 0.520256i 0
281.3 0 0.645395 − 0.468907i 0 1.95814 − 1.07968i 0 1.00000 0 −0.730390 + 2.24791i 0
281.4 0 1.26975 − 0.922528i 0 −1.31503 + 1.80850i 0 1.00000 0 −0.165842 + 0.510409i 0
281.5 0 2.23216 − 1.62176i 0 −2.04335 − 0.908134i 0 1.00000 0 1.42538 − 4.38687i 0
421.1 0 −1.76219 − 1.28031i 0 2.06573 + 0.856025i 0 1.00000 0 0.539080 + 1.65912i 0
421.2 0 −1.26708 − 0.920587i 0 0.334526 − 2.21090i 0 1.00000 0 −0.169041 − 0.520256i 0
421.3 0 0.645395 + 0.468907i 0 1.95814 + 1.07968i 0 1.00000 0 −0.730390 − 2.24791i 0
421.4 0 1.26975 + 0.922528i 0 −1.31503 − 1.80850i 0 1.00000 0 −0.165842 − 0.510409i 0
421.5 0 2.23216 + 1.62176i 0 −2.04335 + 0.908134i 0 1.00000 0 1.42538 + 4.38687i 0
561.1 0 −0.924230 + 2.84449i 0 −1.90971 − 1.16318i 0 1.00000 0 −4.80986 − 3.49456i 0
561.2 0 −0.856981 + 2.63752i 0 1.74511 − 1.39807i 0 1.00000 0 −3.79503 − 2.75725i 0
561.3 0 −0.596640 + 1.83627i 0 −0.344964 + 2.20930i 0 1.00000 0 −0.588849 − 0.427824i 0
561.4 0 0.489022 − 1.50505i 0 2.15677 + 0.590193i 0 1.00000 0 0.401006 + 0.291348i 0
561.5 0 0.770795 − 2.37226i 0 −0.647206 − 2.14036i 0 1.00000 0 −2.60646 − 1.89370i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 141.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.n.c ✓ 20
25.d even 5 1 inner 700.2.n.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.n.c ✓ 20 1.a even 1 1 trivial
700.2.n.c ✓ 20 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} + 18 T_{3}^{18} - 17 T_{3}^{17} + 169 T_{3}^{16} - 132 T_{3}^{15} + 1246 T_{3}^{14} + \cdots + 555025 \) acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 18 T^{18} + \cdots + 555025 \) Copy content Toggle raw display
$5$ \( T^{20} - 4 T^{19} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T - 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} - 6 T^{19} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 6404640841 \) Copy content Toggle raw display
$17$ \( T^{20} - 18 T^{19} + \cdots + 1500625 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 3934927441 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 115341065161 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 339142004881 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 1755495152401 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 442050625 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 222194390625 \) Copy content Toggle raw display
$43$ \( (T^{10} + 23 T^{9} + \cdots - 34915759)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 168714121 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 28610031025 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 346302825625 \) Copy content Toggle raw display
$67$ \( T^{20} - 23 T^{19} + \cdots + 62331025 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 155058750625 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 11405000282641 \) Copy content Toggle raw display
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