↑

Properties

Label 2960.2.p.k
Level $2960$
Weight $2$
Character orbit 2960.p
Analytic conductor $23.636$
Analytic rank $0$
Dimension $20$
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] = [2960,2,Mod(961,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2960.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.p (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: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 38 x^{18} + 605 x^{16} + 5300 x^{14} + 28080 x^{12} + 92796 x^{10} + 189756 x^{8} + 229444 x^{6} + \cdots + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 1480)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{16} q^{3} - \beta_{11} q^{5} + \beta_{8} q^{7} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{16} q^{3} - \beta_{11} q^{5} + \beta_{8} q^{7} + ( - \beta_{2} + 1) q^{9} + \beta_{12} q^{11} + (\beta_{9} - \beta_{7} + \beta_1) q^{13} + \beta_1 q^{15} + (\beta_{17} + \beta_{9} - \beta_{5}) q^{17} + \beta_{10} q^{19} + (\beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{21}+ \cdots + ( - \beta_{16} + \beta_{14} - \beta_{13} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} - 6 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} - 6 q^{7} + 16 q^{9} + 6 q^{11} - 20 q^{21} - 20 q^{25} + 16 q^{27} - 16 q^{33} + 18 q^{37} - 14 q^{41} - 28 q^{47} + 26 q^{49} + 26 q^{53} - 50 q^{63} + 28 q^{67} - 40 q^{71} - 12 q^{73} - 4 q^{75} + 58 q^{77} - 36 q^{81} - 16 q^{83} - 2 q^{85} + 4 q^{95} - 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 38 x^{18} + 605 x^{16} + 5300 x^{14} + 28080 x^{12} + 92796 x^{10} + 189756 x^{8} + 229444 x^{6} + \cdots + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 701 \nu^{19} - 79176 \nu^{17} - 2016521 \nu^{15} - 22396790 \nu^{13} - 127730156 \nu^{11} + \cdots + 101955264 \nu ) / 4057728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1283 \nu^{19} + 36136 \nu^{17} + 365479 \nu^{15} + 1442698 \nu^{13} - 398412 \nu^{11} + \cdots + 31794752 \nu ) / 2028864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2701 \nu^{19} - 105428 \nu^{17} - 1727577 \nu^{15} - 15525170 \nu^{13} - 83204468 \nu^{11} + \cdots - 43512416 \nu ) / 4057728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1579 \nu^{18} + 58112 \nu^{16} + 913791 \nu^{14} + 8110866 \nu^{12} + 44315332 \nu^{10} + \cdots + 4949088 ) / 1014432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9379 \nu^{19} - 240804 \nu^{17} - 1951031 \nu^{15} - 1663590 \nu^{13} + 60886052 \nu^{11} + \cdots + 92687136 \nu ) / 4057728 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9117 \nu^{18} + 320168 \nu^{16} + 4624377 \nu^{14} + 35948518 \nu^{12} + 163783212 \nu^{10} + \cdots + 332736 ) / 2028864 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14697 \nu^{19} + 549380 \nu^{17} + 8565765 \nu^{15} + 73029066 \nu^{13} + 372887908 \nu^{11} + \cdots + 189082208 \nu ) / 4057728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7597 \nu^{19} - 242732 \nu^{17} - 3065337 \nu^{15} - 19681898 \nu^{13} - 67768676 \nu^{11} + \cdots - 16648928 \nu ) / 2028864 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15329 \nu^{19} - 518620 \nu^{17} - 7127293 \nu^{15} - 52003570 \nu^{13} - 219538644 \nu^{11} + \cdots + 7067872 \nu ) / 4057728 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1583 \nu^{18} - 56981 \nu^{16} - 837807 \nu^{14} - 6540815 \nu^{12} - 29374658 \nu^{10} + \cdots - 1540728 ) / 253608 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16165 \nu^{18} - 546416 \nu^{16} - 7435281 \nu^{14} - 52803054 \nu^{12} - 210631132 \nu^{10} + \cdots + 1281792 ) / 2028864 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11407 \nu^{18} + 375376 \nu^{16} + 4976587 \nu^{14} + 34723986 \nu^{12} + 138989284 \nu^{10} + \cdots + 10070496 ) / 1014432 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 27367 \nu^{18} - 913712 \nu^{16} - 12385691 \nu^{14} - 89349898 \nu^{12} - 375539572 \nu^{10} + \cdots - 41129664 ) / 2028864 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10647 \nu^{18} + 357792 \nu^{16} + 4873355 \nu^{14} + 35149946 \nu^{12} + 146226292 \nu^{10} + \cdots + 5886336 ) / 676288 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10749 \nu^{19} - 376503 \nu^{17} - 5408441 \nu^{15} - 41698265 \nu^{13} - 188458022 \nu^{11} + \cdots - 27813256 \nu ) / 1014432 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 4685 \nu^{18} - 146771 \nu^{16} - 1805971 \nu^{14} - 11203603 \nu^{12} - 36666794 \nu^{10} + \cdots + 3485232 ) / 253608 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 24735 \nu^{19} - 812550 \nu^{17} - 10731667 \nu^{15} - 74281592 \nu^{13} - 292459128 \nu^{11} + \cdots - 15560592 \nu ) / 2028864 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{5} - \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + 2\beta_{13} - \beta_{12} - \beta_{8} - \beta_{6} - 9\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{17} - 20\beta_{11} + 13\beta_{10} + \beta_{9} + \beta_{7} + 11\beta_{5} + 17\beta_{4} - \beta_{3} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{18} - 23 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 32 \beta_{13} + 12 \beta_{12} + 17 \beta_{8} + \cdots - 189 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{19} - 52 \beta_{17} + 282 \beta_{11} - 143 \beta_{10} - 16 \beta_{9} - 17 \beta_{7} + \cdots - 229 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 22 \beta_{18} + 331 \beta_{16} + 34 \beta_{15} - 55 \beta_{14} + 395 \beta_{13} - 116 \beta_{12} + \cdots + 1631 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 111 \beta_{19} + 662 \beta_{17} - 3450 \beta_{11} + 1517 \beta_{10} + 190 \beta_{9} + 215 \beta_{7} + \cdots + 1947 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 326 \beta_{18} - 4067 \beta_{16} - 440 \beta_{15} + 719 \beta_{14} - 4491 \beta_{13} + 1086 \beta_{12} + \cdots - 15389 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1663 \beta_{19} - 7604 \beta_{17} + 39542 \beta_{11} - 16005 \beta_{10} - 2056 \beta_{9} + \cdots - 18239 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4132 \beta_{18} + 46655 \beta_{16} + 5202 \beta_{15} - 8339 \beta_{14} + 49389 \beta_{13} + \cdots + 153333 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 21227 \beta_{19} + 83806 \beta_{17} - 438614 \beta_{11} + 169199 \beta_{10} + 21604 \beta_{9} + \cdots + 181187 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 48520 \beta_{18} - 517547 \beta_{16} - 58990 \beta_{15} + 92023 \beta_{14} - 534985 \beta_{13} + \cdots - 1575869 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 250187 \beta_{19} - 907306 \beta_{17} + 4781670 \beta_{11} - 1793995 \beta_{10} + \cdots - 1860019 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 546868 \beta_{18} + 5641643 \beta_{16} + 653590 \beta_{15} - 994147 \beta_{14} + 5752525 \beta_{13} + \cdots + 16473529 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2824651 \beta_{19} + 9744654 \beta_{17} - 51637138 \beta_{11} + 19065491 \beta_{10} + \cdots + 19436011 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 6023420 \beta_{18} - 60918435 \beta_{16} - 7139830 \beta_{15} + 10646019 \beta_{14} + \cdots - 173800605 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 31131431 \beta_{19} - 104280174 \beta_{17} + 554692134 \beta_{11} - 202926923 \beta_{10} + \cdots - 205027259 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
2.48532i
− 2.48532i
2.24149i
− 2.24149i
2.03573i
− 2.03573i
0.764728i
− 0.764728i
0.271032i
− 0.271032i
− 1.11768i
1.11768i
− 1.38988i
1.38988i
− 1.85576i
1.85576i
− 2.16880i
2.16880i
− 3.26618i
3.26618i
0 −2.48532 0 − 1.00000i 0 −1.56717 0 3.17683 0
961.2 0 −2.48532 0 1.00000i 0 −1.56717 0 3.17683 0
961.3 0 −2.24149 0 − 1.00000i 0 −0.939521 0 2.02428 0
961.4 0 −2.24149 0 1.00000i 0 −0.939521 0 2.02428 0
961.5 0 −2.03573 0 − 1.00000i 0 3.00287 0 1.14418 0
961.6 0 −2.03573 0 1.00000i 0 3.00287 0 1.14418 0
961.7 0 −0.764728 0 − 1.00000i 0 2.05051 0 −2.41519 0
961.8 0 −0.764728 0 1.00000i 0 2.05051 0 −2.41519 0
961.9 0 −0.271032 0 − 1.00000i 0 −3.29463 0 −2.92654 0
961.10 0 −0.271032 0 1.00000i 0 −3.29463 0 −2.92654 0
961.11 0 1.11768 0 − 1.00000i 0 3.46608 0 −1.75080 0
961.12 0 1.11768 0 1.00000i 0 3.46608 0 −1.75080 0
961.13 0 1.38988 0 − 1.00000i 0 −4.92034 0 −1.06823 0
961.14 0 1.38988 0 1.00000i 0 −4.92034 0 −1.06823 0
961.15 0 1.85576 0 − 1.00000i 0 1.08808 0 0.443859 0
961.16 0 1.85576 0 1.00000i 0 1.08808 0 0.443859 0
961.17 0 2.16880 0 − 1.00000i 0 1.92057 0 1.70369 0
961.18 0 2.16880 0 1.00000i 0 1.92057 0 1.70369 0
961.19 0 3.26618 0 − 1.00000i 0 −3.80644 0 7.66793 0
961.20 0 3.26618 0 1.00000i 0 −3.80644 0 7.66793 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.p.k 20
4.b odd 2 1 1480.2.p.d ✓ 20
37.b even 2 1 inner 2960.2.p.k 20
148.b odd 2 1 1480.2.p.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.p.d ✓ 20 4.b odd 2 1
1480.2.p.d ✓ 20 148.b odd 2 1
2960.2.p.k 20 1.a even 1 1 trivial
2960.2.p.k 20 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2960, [\chi])\):

\( T_{3}^{10} - 2 T_{3}^{9} - 17 T_{3}^{8} + 30 T_{3}^{7} + 98 T_{3}^{6} - 158 T_{3}^{5} - 218 T_{3}^{4} + \cdots - 48 \) Copy content Toggle raw display
\( T_{7}^{10} + 3 T_{7}^{9} - 37 T_{7}^{8} - 77 T_{7}^{7} + 512 T_{7}^{6} + 578 T_{7}^{5} - 3062 T_{7}^{4} + \cdots - 4052 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - 2 T^{9} + \cdots - 48)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$7$ \( (T^{10} + 3 T^{9} + \cdots - 4052)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 3 T^{9} + \cdots + 3072)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 75312922624 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 43477254144 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 191988736 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 21743271936 \) Copy content Toggle raw display
$29$ \( T^{20} + 229 T^{18} + \cdots + 4194304 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 2155930624 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 48\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( (T^{10} + 7 T^{9} + \cdots + 107438784)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 458144874496 \) Copy content Toggle raw display
$47$ \( (T^{10} + 14 T^{9} + \cdots + 5580384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} - 13 T^{9} + \cdots - 221184)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + 536 T^{18} + \cdots + 37748736 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{10} - 14 T^{9} + \cdots - 36423808)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 20 T^{9} + \cdots - 107026432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 6 T^{9} + \cdots - 36634112)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 84\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{10} + 8 T^{9} + \cdots - 90018832)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 2021583486976 \) Copy content Toggle raw display
show more
show less