↑

Properties

Label 275.4.b.f
Level $275$
Weight $4$
Character orbit 275.b
Analytic conductor $16.226$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,4,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not 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: \(16.2255252516\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 80x^{8} + 2296x^{6} + 27417x^{4} + 110472x^{2} + 21904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{8} + \beta_1) q^{3} + (\beta_{2} - 8) q^{4} + ( - \beta_{7} + \beta_{5} + \beta_{2} - 8) q^{6} + (\beta_{8} - \beta_{6} + \cdots - 2 \beta_1) q^{7}+ \cdots + (\beta_{5} + 2 \beta_{2} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{8} + \beta_1) q^{3} + (\beta_{2} - 8) q^{4} + ( - \beta_{7} + \beta_{5} + \beta_{2} - 8) q^{6} + (\beta_{8} - \beta_{6} + \cdots - 2 \beta_1) q^{7}+ \cdots + ( - 11 \beta_{5} - 22 \beta_{2} + 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41} + 880 q^{44} + 2012 q^{46} - 454 q^{49} - 2256 q^{51} + 2410 q^{54} - 1670 q^{56} - 2348 q^{59} + 1280 q^{61} + 1898 q^{64} + 924 q^{66} + 3292 q^{69} - 1346 q^{71} + 1122 q^{74} + 2496 q^{76} - 208 q^{79} - 2438 q^{81} - 8706 q^{84} + 4994 q^{86} - 5642 q^{89} + 960 q^{91} + 3574 q^{94} - 3638 q^{96} + 682 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 80x^{8} + 2296x^{6} + 27417x^{4} + 110472x^{2} + 21904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 524\nu^{7} - 24200\nu^{5} - 339993\nu^{3} - 1222692\nu ) / 105376 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{8} - 148\nu^{6} - 1400\nu^{4} + 10997\nu^{2} + 38596 ) / 2848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -113\nu^{8} - 6524\nu^{6} - 114440\nu^{4} - 563593\nu^{2} - 125908 ) / 2848 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -55\nu^{9} - 2476\nu^{7} - 13800\nu^{5} + 478817\nu^{3} + 3590956\nu ) / 52688 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 125\nu^{8} + 7116\nu^{6} + 122888\nu^{4} + 593653\nu^{2} + 131012 ) / 2848 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -791\nu^{9} - 45668\nu^{7} - 806776\nu^{5} - 4124575\nu^{3} - 1747148\nu ) / 210752 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 791\nu^{9} + 45668\nu^{7} + 806776\nu^{5} + 4335327\nu^{3} + 6594444\nu ) / 210752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{5} + 4\beta_{4} - 26\beta_{2} + 360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -35\beta_{9} - 37\beta_{8} + 7\beta_{6} + 21\beta_{3} + 555\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -65\beta_{7} - 68\beta_{5} - 147\beta_{4} + 660\beta_{2} - 8584 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1022\beta_{9} + 1122\beta_{8} - 348\beta_{6} - 1270\beta_{3} - 13695\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2740\beta_{7} + 2888\beta_{5} + 4436\beta_{4} - 16761\beta_{2} + 209692 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -28521\beta_{9} - 32521\beta_{8} + 12952\beta_{6} + 51904\beta_{3} + 342327\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 5.19955i
− 4.78071i
− 4.72095i
− 2.75920i
− 0.457079i
0.457079i
2.75920i
4.72095i
4.78071i
5.19955i
− 5.19955i − 7.06563i −19.0353 0 −36.7381 32.4859i 57.3785i −22.9231 0
199.2 − 4.78071i 5.99252i −14.8552 0 28.6485 − 11.9641i 32.7728i −8.91034 0
199.3 − 4.72095i − 8.29546i −14.2874 0 −39.1625 − 5.48389i 29.6825i −41.8147 0
199.4 − 2.75920i 2.30980i 0.386826 0 6.37319 13.2555i − 23.1409i 21.6648 0
199.5 − 0.457079i − 2.45289i 7.79108 0 −1.12116 23.1894i − 7.21777i 20.9833 0
199.6 0.457079i 2.45289i 7.79108 0 −1.12116 − 23.1894i 7.21777i 20.9833 0
199.7 2.75920i − 2.30980i 0.386826 0 6.37319 − 13.2555i 23.1409i 21.6648 0
199.8 4.72095i 8.29546i −14.2874 0 −39.1625 5.48389i − 29.6825i −41.8147 0
199.9 4.78071i − 5.99252i −14.8552 0 28.6485 11.9641i − 32.7728i −8.91034 0
199.10 5.19955i 7.06563i −19.0353 0 −36.7381 − 32.4859i − 57.3785i −22.9231 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.4.b.f 10
5.b even 2 1 inner 275.4.b.f 10
5.c odd 4 1 275.4.a.g ✓ 5
5.c odd 4 1 275.4.a.h yes 5
15.e even 4 1 2475.4.a.bh 5
15.e even 4 1 2475.4.a.bl 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.4.a.g ✓ 5 5.c odd 4 1
275.4.a.h yes 5 5.c odd 4 1
275.4.b.f 10 1.a even 1 1 trivial
275.4.b.f 10 5.b even 2 1 inner
2475.4.a.bh 5 15.e even 4 1
2475.4.a.bl 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 80T_{2}^{8} + 2296T_{2}^{6} + 27417T_{2}^{4} + 110472T_{2}^{2} + 21904 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 80 T^{8} + \cdots + 21904 \) Copy content Toggle raw display
$3$ \( T^{10} + 166 T^{8} + \cdots + 3960100 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 429234625600 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{5} + 23 T^{4} + \cdots + 2305690119)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 1076856572089 \) Copy content Toggle raw display
$29$ \( (T^{5} - 183 T^{4} + \cdots + 16865962259)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + T^{4} + \cdots + 94367463777)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{5} - 94 T^{4} + \cdots + 29753829648)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 13620640524856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 2812375273130)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{5} + 673 T^{4} + \cdots + 38185247888)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + 104 T^{4} + \cdots + 718883827912)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 30\!\cdots\!09 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 28452224444275)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
show more
show less