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Properties

Label 1450.2.e.j
Level $1450$
Weight $2$
Character orbit 1450.e
Analytic conductor $11.578$
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] = [1450,2,Mod(307,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.e (of order \(4\), degree \(2\), 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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 36 x^{18} + 534 x^{16} + 4248 x^{14} + 19701 x^{12} + 54104 x^{10} + 85176 x^{8} + 70068 x^{6} + \cdots + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_{14} - \beta_{13} + \beta_{11} + \cdots + 1) q^{7}+ \cdots + (\beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_{14} - \beta_{13} + \beta_{11} + \cdots + 1) q^{7}+ \cdots + (\beta_{19} + 2 \beta_{17} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} + 20 q^{4} + 8 q^{7} + 20 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} + 20 q^{4} + 8 q^{7} + 20 q^{8} - 12 q^{9} - 8 q^{11} + 4 q^{13} + 8 q^{14} + 20 q^{16} - 16 q^{17} - 12 q^{18} - 4 q^{19} + 4 q^{21} - 8 q^{22} - 12 q^{23} + 4 q^{26} + 8 q^{28} + 8 q^{29} + 28 q^{31} + 20 q^{32} - 16 q^{34} - 12 q^{36} - 4 q^{38} - 24 q^{39} + 24 q^{41} + 4 q^{42} - 8 q^{44} - 12 q^{46} + 4 q^{52} + 8 q^{56} + 8 q^{57} + 8 q^{58} + 24 q^{61} + 28 q^{62} - 20 q^{63} + 20 q^{64} - 12 q^{67} - 16 q^{68} - 28 q^{69} - 12 q^{72} + 16 q^{73} - 4 q^{76} - 16 q^{77} - 24 q^{78} + 32 q^{79} - 12 q^{81} + 24 q^{82} + 20 q^{83} + 4 q^{84} - 44 q^{87} - 8 q^{88} + 4 q^{89} - 12 q^{92} - 12 q^{93} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 36 x^{18} + 534 x^{16} + 4248 x^{14} + 19701 x^{12} + 54104 x^{10} + 85176 x^{8} + 70068 x^{6} + \cdots + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{18} - 17725 \nu^{16} - 544407 \nu^{14} - 6695597 \nu^{12} - 42240706 \nu^{10} + \cdots - 2363980 ) / 464120 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19 \nu^{18} - 675 \nu^{16} - 9891 \nu^{14} - 77861 \nu^{12} - 357398 \nu^{10} - 965498 \nu^{8} + \cdots - 2280 ) / 1640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{18} + 675 \nu^{16} + 9891 \nu^{14} + 77861 \nu^{12} + 357398 \nu^{10} + 965498 \nu^{8} + \cdots + 7200 ) / 1640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1517 \nu^{19} - 15061 \nu^{18} - 57482 \nu^{17} - 537770 \nu^{16} - 903148 \nu^{15} + \cdots - 5969640 ) / 464120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1517 \nu^{19} + 15061 \nu^{18} - 57482 \nu^{17} + 537770 \nu^{16} - 903148 \nu^{15} + \cdots + 5969640 ) / 464120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20225 \nu^{19} + 56399 \nu^{18} - 712387 \nu^{17} + 2004965 \nu^{16} - 10288255 \nu^{15} + \cdots + 19616460 ) / 928240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 72 \nu^{19} - 2573 \nu^{17} - 37773 \nu^{15} - 295965 \nu^{13} - 1340611 \nu^{11} + \cdots - 15692 \nu ) / 1640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19655 \nu^{19} + 28400 \nu^{18} + 702797 \nu^{17} + 1008095 \nu^{16} + 10333670 \nu^{15} + \cdots + 10637940 ) / 464120 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51703 \nu^{19} + 48695 \nu^{18} + 1853489 \nu^{17} + 1773975 \nu^{16} + 27293017 \nu^{15} + \cdots + 19328300 ) / 928240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51703 \nu^{19} + 48695 \nu^{18} - 1853489 \nu^{17} + 1773975 \nu^{16} - 27293017 \nu^{15} + \cdots + 19328300 ) / 928240 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 92417 \nu^{19} - 11155 \nu^{18} - 3293697 \nu^{17} - 393275 \nu^{16} - 48161163 \nu^{15} + \cdots - 5734620 ) / 928240 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 92417 \nu^{19} - 11155 \nu^{18} + 3293697 \nu^{17} - 393275 \nu^{16} + 48161163 \nu^{15} + \cdots - 5734620 ) / 928240 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 197 \nu^{19} + 7044 \nu^{17} + 103428 \nu^{15} + 810034 \nu^{13} + 3664435 \nu^{11} + \cdots + 38236 \nu ) / 1640 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 120177 \nu^{19} + 24701 \nu^{18} + 4285121 \nu^{17} + 878075 \nu^{16} + 62741153 \nu^{15} + \cdots + 3596340 ) / 928240 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 61343 \nu^{19} - 2883 \nu^{18} - 2193794 \nu^{17} - 110795 \nu^{16} - 32214487 \nu^{15} + \cdots - 2212280 ) / 464120 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 61343 \nu^{19} - 2883 \nu^{18} + 2193794 \nu^{17} - 110795 \nu^{16} + 32214487 \nu^{15} + \cdots - 2212280 ) / 464120 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 120177 \nu^{19} + 24701 \nu^{18} - 4285121 \nu^{17} + 878075 \nu^{16} - 62741153 \nu^{15} + \cdots + 3596340 ) / 928240 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 38202 \nu^{19} + 1365782 \nu^{17} + 20049873 \nu^{15} + 156988371 \nu^{13} + 710070547 \nu^{11} + \cdots + 14292548 \nu ) / 232060 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} - 2\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 2\beta_{8} - \beta_{6} - \beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{18} - \beta_{17} - \beta_{16} - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{19} + 2 \beta_{17} - 2 \beta_{16} + 25 \beta_{14} - 12 \beta_{13} + 13 \beta_{12} + \cdots + 44 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{18} + 17 \beta_{17} + 17 \beta_{16} + 12 \beta_{15} + \beta_{14} + 18 \beta_{13} + 19 \beta_{12} + \cdots - 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 29 \beta_{19} + \beta_{18} - 41 \beta_{17} + 41 \beta_{16} - \beta_{15} - 266 \beta_{14} + \cdots - 364 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 124 \beta_{18} - 211 \beta_{17} - 211 \beta_{16} - 124 \beta_{15} - 25 \beta_{14} - 229 \beta_{13} + \cdots + 1120 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 25 \beta_{19} - 28 \beta_{18} + 567 \beta_{17} - 567 \beta_{16} + 28 \beta_{15} + 2732 \beta_{14} + \cdots + 3266 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1254 \beta_{18} + 2347 \beta_{17} + 2347 \beta_{16} + 1254 \beta_{15} + 381 \beta_{14} + 2590 \beta_{13} + \cdots - 10066 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2379 \beta_{19} + 455 \beta_{18} - 6741 \beta_{17} + 6741 \beta_{16} - 455 \beta_{15} + \cdots - 30854 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 12674 \beta_{18} - 24935 \beta_{17} - 24935 \beta_{16} - 12674 \beta_{15} - 4775 \beta_{14} + \cdots + 94862 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 37631 \beta_{19} - 5944 \beta_{18} + 74625 \beta_{17} - 74625 \beta_{16} + 5944 \beta_{15} + \cdots + 300766 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 128392 \beta_{18} + 259413 \beta_{17} + 259413 \beta_{16} + 128392 \beta_{15} + 54465 \beta_{14} + \cdots - 921738 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 466017 \beta_{19} + 69731 \beta_{18} - 796241 \beta_{17} + 796241 \beta_{16} - 69731 \beta_{15} + \cdots - 2987058 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1303350 \beta_{18} - 2671653 \beta_{17} - 2671653 \beta_{16} - 1303350 \beta_{15} - 591293 \beta_{14} + \cdots + 9129398 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 5252693 \beta_{19} - 771730 \beta_{18} + 8328079 \beta_{17} - 8328079 \beta_{16} + \cdots + 29992222 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 13250104 \beta_{18} + 27376547 \beta_{17} + 27376547 \beta_{16} + 13250104 \beta_{15} + 6247719 \beta_{14} + \cdots - 91486378 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 56588715 \beta_{19} + 8261799 \beta_{18} - 86145587 \beta_{17} + 86145587 \beta_{16} + \cdots - 303067302 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-\beta_{8}\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
− 2.46836i
− 1.80930i
− 1.41267i
− 0.542524i
0.200136i
0.525625i
1.50359i
2.38141i
2.42934i
3.19276i
− 3.19276i
− 2.42934i
− 2.38141i
− 1.50359i
− 0.525625i
− 0.200136i
0.542524i
1.41267i
1.80930i
2.46836i
1.00000 − 2.46836i 1.00000 0 − 2.46836i 1.24550 + 1.24550i 1.00000 −3.09281 0
307.2 1.00000 − 1.80930i 1.00000 0 − 1.80930i −1.92568 − 1.92568i 1.00000 −0.273573 0
307.3 1.00000 − 1.41267i 1.00000 0 − 1.41267i 2.52368 + 2.52368i 1.00000 1.00435 0
307.4 1.00000 − 0.542524i 1.00000 0 − 0.542524i 2.83405 + 2.83405i 1.00000 2.70567 0
307.5 1.00000 0.200136i 1.00000 0 0.200136i −1.67827 − 1.67827i 1.00000 2.95995 0
307.6 1.00000 0.525625i 1.00000 0 0.525625i 1.99569 + 1.99569i 1.00000 2.72372 0
307.7 1.00000 1.50359i 1.00000 0 1.50359i −2.85734 − 2.85734i 1.00000 0.739204 0
307.8 1.00000 2.38141i 1.00000 0 2.38141i −0.493332 − 0.493332i 1.00000 −2.67112 0
307.9 1.00000 2.42934i 1.00000 0 2.42934i 0.0920472 + 0.0920472i 1.00000 −2.90168 0
307.10 1.00000 3.19276i 1.00000 0 3.19276i 2.26365 + 2.26365i 1.00000 −7.19370 0
1143.1 1.00000 − 3.19276i 1.00000 0 − 3.19276i 2.26365 − 2.26365i 1.00000 −7.19370 0
1143.2 1.00000 − 2.42934i 1.00000 0 − 2.42934i 0.0920472 − 0.0920472i 1.00000 −2.90168 0
1143.3 1.00000 − 2.38141i 1.00000 0 − 2.38141i −0.493332 + 0.493332i 1.00000 −2.67112 0
1143.4 1.00000 − 1.50359i 1.00000 0 − 1.50359i −2.85734 + 2.85734i 1.00000 0.739204 0
1143.5 1.00000 − 0.525625i 1.00000 0 − 0.525625i 1.99569 − 1.99569i 1.00000 2.72372 0
1143.6 1.00000 − 0.200136i 1.00000 0 − 0.200136i −1.67827 + 1.67827i 1.00000 2.95995 0
1143.7 1.00000 0.542524i 1.00000 0 0.542524i 2.83405 − 2.83405i 1.00000 2.70567 0
1143.8 1.00000 1.41267i 1.00000 0 1.41267i 2.52368 − 2.52368i 1.00000 1.00435 0
1143.9 1.00000 1.80930i 1.00000 0 1.80930i −1.92568 + 1.92568i 1.00000 −0.273573 0
1143.10 1.00000 2.46836i 1.00000 0 2.46836i 1.24550 − 1.24550i 1.00000 −3.09281 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.e.j yes 20
5.b even 2 1 1450.2.e.i ✓ 20
5.c odd 4 1 1450.2.j.i yes 20
5.c odd 4 1 1450.2.j.j yes 20
29.c odd 4 1 1450.2.j.i yes 20
145.e even 4 1 1450.2.e.i ✓ 20
145.f odd 4 1 1450.2.j.j yes 20
145.j even 4 1 inner 1450.2.e.j yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.e.i ✓ 20 5.b even 2 1
1450.2.e.i ✓ 20 145.e even 4 1
1450.2.e.j yes 20 1.a even 1 1 trivial
1450.2.e.j yes 20 145.j even 4 1 inner
1450.2.j.i yes 20 5.c odd 4 1
1450.2.j.i yes 20 29.c odd 4 1
1450.2.j.j yes 20 5.c odd 4 1
1450.2.j.j yes 20 145.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{20} + 36 T_{3}^{18} + 534 T_{3}^{16} + 4248 T_{3}^{14} + 19701 T_{3}^{12} + 54104 T_{3}^{10} + \cdots + 100 \) Copy content Toggle raw display
\( T_{7}^{20} - 8 T_{7}^{19} + 32 T_{7}^{18} - 52 T_{7}^{17} + 380 T_{7}^{16} - 2776 T_{7}^{15} + \cdots + 291600 \) Copy content Toggle raw display
\( T_{11}^{20} + 8 T_{11}^{19} + 32 T_{11}^{18} - 16 T_{11}^{17} + 1550 T_{11}^{16} + 11848 T_{11}^{15} + \cdots + 2560000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 36 T^{18} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 8 T^{19} + \cdots + 291600 \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots + 2560000 \) Copy content Toggle raw display
$13$ \( T^{20} - 4 T^{19} + \cdots + 64000000 \) Copy content Toggle raw display
$17$ \( (T^{10} + 8 T^{9} + \cdots - 960)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + 4 T^{19} + \cdots + 743044 \) Copy content Toggle raw display
$23$ \( T^{20} + 12 T^{19} + \cdots + 18524416 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 420707233300201 \) Copy content Toggle raw display
$31$ \( T^{20} - 28 T^{19} + \cdots + 4260096 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 30512278059264 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 5511577600 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 4842768100 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 86561810899600 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 2387643040000 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 1421741447424 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1549763561025 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} - 8 T^{9} + \cdots + 312545950)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 22399320115264 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
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