↑

Properties

Label 605.2.j.h
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,2,Mod(9,605)] 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("605.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,6,-2,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{3} + (\beta_{11} - \beta_{7} + \cdots + \beta_{3}) q^{4} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{2}) q^{5} + (2 \beta_{11} + 2 \beta_{8} - \beta_{7} + \cdots + 1) q^{6}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{4} - 2 q^{5} + 12 q^{6} + 2 q^{9} + 8 q^{14} + 24 q^{15} + 6 q^{16} + 6 q^{19} + 12 q^{20} + 8 q^{21} - 4 q^{24} + 24 q^{25} - 50 q^{26} + 22 q^{29} - 4 q^{30} - 22 q^{31} - 16 q^{34} - 8 q^{35}+ \cdots + 94 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{14} + 25x^{12} - 57x^{10} + 194x^{8} - 303x^{6} + 235x^{4} - 33x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1829973 \nu^{15} - 24476532 \nu^{13} + 152757677 \nu^{11} - 557238290 \nu^{9} + \cdots - 7498263355 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 608673 \nu^{14} + 4779625 \nu^{12} - 9344761 \nu^{10} - 2560494 \nu^{8} - 9430964 \nu^{6} + \cdots + 31775447 ) / 609644662 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2744937 \nu^{14} - 22433610 \nu^{12} + 99837091 \nu^{10} - 276654262 \nu^{8} + 814713604 \nu^{6} + \cdots - 771926881 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2744937 \nu^{15} - 22433610 \nu^{13} + 99837091 \nu^{11} - 276654262 \nu^{9} + \cdots - 771926881 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2924455 \nu^{14} + 43736988 \nu^{12} - 193036323 \nu^{10} + 488836470 \nu^{8} + \cdots - 236512683 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1351905 \nu^{14} + 5558819 \nu^{12} - 7678521 \nu^{10} - 13223046 \nu^{8} - 42401020 \nu^{6} + \cdots - 629543387 ) / 609644662 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6960689 \nu^{14} + 30587176 \nu^{12} - 61040185 \nu^{10} + 38446538 \nu^{8} + \cdots - 1738523017 ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3962283 \nu^{14} + 31992860 \nu^{12} - 118526613 \nu^{10} + 271533274 \nu^{8} + \cdots - 383811549 ) / 1219289324 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8234811 \nu^{15} - 67300830 \nu^{13} + 299511273 \nu^{11} - 829962786 \nu^{9} + \cdots - 2315780643 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 770947 \nu^{14} - 4704418 \nu^{12} + 13758165 \nu^{10} - 28071690 \nu^{8} + 126275508 \nu^{6} + \cdots - 243303327 ) / 221688968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10669503 \nu^{15} - 86419330 \nu^{13} + 336890317 \nu^{11} - 819720810 \nu^{9} + \cdots - 4303783 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8815096 \nu^{15} + 58503253 \nu^{13} - 198965251 \nu^{11} + 429083674 \nu^{9} + \cdots + 352179707 \nu ) / 1219289324 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19639683 \nu^{15} + 162786416 \nu^{13} - 643887411 \nu^{11} + 1627587846 \nu^{9} + \cdots + 4755604557 \nu ) / 2438578648 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1496498 \nu^{15} + 9707543 \nu^{13} - 31032777 \nu^{11} + 60570826 \nu^{9} + \cdots - 172220447 \nu ) / 110844484 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{9} - 4\beta_{8} + \beta_{7} - 4\beta_{6} + 4\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 5\beta_{14} - \beta_{13} - 10\beta_{12} - \beta_{10} + 15\beta_{5} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{11} - 22\beta_{8} + 21\beta_{7} + \beta_{4} - 7\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21\beta_{15} - 21\beta_{14} - 36\beta_{13} - 36\beta_{12} + 29\beta_{5} - 28\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -85\beta_{11} - 78\beta_{9} - 87\beta_{8} + 85\beta_{7} + 49\beta_{6} - 7\beta_{4} - 49\beta_{3} - 87 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85\beta_{15} - 36\beta_{14} - 136\beta_{13} - 45\beta_{12} + 36\beta_{10} + 36\beta_{5} - 121\beta_{2} - 85\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -342\beta_{11} - 396\beta_{9} - 230\beta_{8} + 166\beta_{7} + 166\beta_{6} - 166\beta_{4} - 529 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 166\beta_{15} - 220\beta_{13} + 342\beta_{10} - 220\beta_{5} - 342\beta_{2} - 475\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -728\beta_{11} - 1653\beta_{9} + 728\beta_{6} - 1170\beta_{4} + 651\beta_{3} - 1653 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 728\beta_{14} + 1002\beta_{12} + 1379\beta_{10} - 2823\beta_{5} - 1002\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -4644\beta_{9} + 3748\beta_{8} - 3109\beta_{7} + 2472\beta_{6} - 3748\beta_{4} + 3109\beta_{3} - 3109 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3109 \beta_{15} + 5581 \beta_{14} + 4385 \beta_{13} + 8392 \beta_{12} + 3109 \beta_{10} + \cdots - 1276 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{4} - \beta_{8} - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.92464 + 0.625353i
−1.17360 + 0.381325i
1.17360 − 0.381325i
1.92464 − 0.625353i
−0.972539 + 1.33858i
−0.471815 + 0.649397i
0.471815 − 0.649397i
0.972539 − 1.33858i
−1.92464 − 0.625353i
−1.17360 − 0.381325i
1.17360 + 0.381325i
1.92464 + 0.625353i
−0.972539 − 1.33858i
−0.471815 − 0.649397i
0.471815 + 0.649397i
0.972539 + 1.33858i
−1.92464 − 0.625353i −1.54035 − 2.12011i 1.69513 + 1.23158i −2.19919 + 0.404418i 1.63880 + 5.04369i 0.567697 − 0.781367i −0.113351 − 0.156015i −1.19513 + 3.67823i 4.48555 + 0.596911i
9.2 −1.17360 − 0.381325i 0.213943 + 0.294468i −0.386111 − 0.280526i 2.13387 + 0.668267i −0.138796 − 0.427169i −1.51977 + 2.09178i 1.79681 + 2.47310i 0.886111 − 2.72717i −2.24948 − 1.59798i
9.3 1.17360 + 0.381325i −0.213943 − 0.294468i −0.386111 − 0.280526i −1.33354 + 1.79490i −0.138796 − 0.427169i 1.51977 − 2.09178i −1.79681 − 2.47310i 0.886111 − 2.72717i −2.24948 + 1.59798i
9.4 1.92464 + 0.625353i 1.54035 + 2.12011i 1.69513 + 1.23158i 2.01689 − 0.965471i 1.63880 + 5.04369i −0.567697 + 0.781367i 0.113351 + 0.156015i −1.19513 + 3.67823i 4.48555 − 0.596911i
124.1 −0.972539 − 1.33858i −1.87813 + 0.610243i −0.227943 + 0.701538i −1.51945 − 1.64051i 2.64342 + 1.92056i −2.13329 − 0.693148i −1.98645 + 0.645437i 0.727943 − 0.528882i −0.718246 + 3.62937i
124.2 −0.471815 − 0.649397i 1.67457 − 0.544099i 0.418926 − 1.28932i 2.07703 − 0.828231i −1.14342 − 0.830744i 0.563124 + 0.182970i −2.56176 + 0.832367i 0.0810736 − 0.0589034i −1.51782 − 0.958043i
124.3 0.471815 + 0.649397i −1.67457 + 0.544099i 0.418926 − 1.28932i −0.145858 + 2.23131i −1.14342 − 0.830744i −0.563124 − 0.182970i 2.56176 − 0.832367i 0.0810736 − 0.0589034i −1.51782 + 0.958043i
124.4 0.972539 + 1.33858i 1.87813 − 0.610243i −0.227943 + 0.701538i −2.02976 − 0.938132i 2.64342 + 1.92056i 2.13329 + 0.693148i 1.98645 − 0.645437i 0.727943 − 0.528882i −0.718246 − 3.62937i
269.1 −1.92464 + 0.625353i −1.54035 + 2.12011i 1.69513 − 1.23158i −2.19919 − 0.404418i 1.63880 − 5.04369i 0.567697 + 0.781367i −0.113351 + 0.156015i −1.19513 − 3.67823i 4.48555 − 0.596911i
269.2 −1.17360 + 0.381325i 0.213943 − 0.294468i −0.386111 + 0.280526i 2.13387 − 0.668267i −0.138796 + 0.427169i −1.51977 − 2.09178i 1.79681 − 2.47310i 0.886111 + 2.72717i −2.24948 + 1.59798i
269.3 1.17360 − 0.381325i −0.213943 + 0.294468i −0.386111 + 0.280526i −1.33354 − 1.79490i −0.138796 + 0.427169i 1.51977 + 2.09178i −1.79681 + 2.47310i 0.886111 + 2.72717i −2.24948 − 1.59798i
269.4 1.92464 − 0.625353i 1.54035 − 2.12011i 1.69513 − 1.23158i 2.01689 + 0.965471i 1.63880 − 5.04369i −0.567697 − 0.781367i 0.113351 − 0.156015i −1.19513 − 3.67823i 4.48555 + 0.596911i
444.1 −0.972539 + 1.33858i −1.87813 − 0.610243i −0.227943 − 0.701538i −1.51945 + 1.64051i 2.64342 − 1.92056i −2.13329 + 0.693148i −1.98645 − 0.645437i 0.727943 + 0.528882i −0.718246 − 3.62937i
444.2 −0.471815 + 0.649397i 1.67457 + 0.544099i 0.418926 + 1.28932i 2.07703 + 0.828231i −1.14342 + 0.830744i 0.563124 − 0.182970i −2.56176 − 0.832367i 0.0810736 + 0.0589034i −1.51782 + 0.958043i
444.3 0.471815 − 0.649397i −1.67457 − 0.544099i 0.418926 + 1.28932i −0.145858 − 2.23131i −1.14342 + 0.830744i −0.563124 + 0.182970i 2.56176 + 0.832367i 0.0810736 + 0.0589034i −1.51782 − 0.958043i
444.4 0.972539 − 1.33858i 1.87813 + 0.610243i −0.227943 − 0.701538i −2.02976 + 0.938132i 2.64342 − 1.92056i 2.13329 − 0.693148i 1.98645 + 0.645437i 0.727943 + 0.528882i −0.718246 + 3.62937i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 9.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.h 16
5.b even 2 1 inner 605.2.j.h 16
11.b odd 2 1 605.2.j.g 16
11.c even 5 2 55.2.j.a ✓ 16
11.c even 5 1 605.2.b.g 8
11.c even 5 1 inner 605.2.j.h 16
11.d odd 10 1 605.2.b.f 8
11.d odd 10 2 605.2.j.d 16
11.d odd 10 1 605.2.j.g 16
33.h odd 10 2 495.2.ba.a 16
44.h odd 10 2 880.2.cd.c 16
55.d odd 2 1 605.2.j.g 16
55.h odd 10 1 605.2.b.f 8
55.h odd 10 2 605.2.j.d 16
55.h odd 10 1 605.2.j.g 16
55.j even 10 2 55.2.j.a ✓ 16
55.j even 10 1 605.2.b.g 8
55.j even 10 1 inner 605.2.j.h 16
55.k odd 20 4 275.2.h.d 16
55.k odd 20 2 3025.2.a.bl 8
55.l even 20 2 3025.2.a.bk 8
165.o odd 10 2 495.2.ba.a 16
220.n odd 10 2 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a ✓ 16 11.c even 5 2
55.2.j.a ✓ 16 55.j even 10 2
275.2.h.d 16 55.k odd 20 4
495.2.ba.a 16 33.h odd 10 2
495.2.ba.a 16 165.o odd 10 2
605.2.b.f 8 11.d odd 10 1
605.2.b.f 8 55.h odd 10 1
605.2.b.g 8 11.c even 5 1
605.2.b.g 8 55.j even 10 1
605.2.j.d 16 11.d odd 10 2
605.2.j.d 16 55.h odd 10 2
605.2.j.g 16 11.b odd 2 1
605.2.j.g 16 11.d odd 10 1
605.2.j.g 16 55.d odd 2 1
605.2.j.g 16 55.h odd 10 1
605.2.j.h 16 1.a even 1 1 trivial
605.2.j.h 16 5.b even 2 1 inner
605.2.j.h 16 11.c even 5 1 inner
605.2.j.h 16 55.j even 10 1 inner
880.2.cd.c 16 44.h odd 10 2
880.2.cd.c 16 220.n odd 10 2
3025.2.a.bk 8 55.l even 20 2
3025.2.a.bl 8 55.k odd 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{16} - 7T_{2}^{14} + 25T_{2}^{12} - 57T_{2}^{10} + 194T_{2}^{8} - 303T_{2}^{6} + 235T_{2}^{4} - 33T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{19}^{8} - 3T_{19}^{7} + 28T_{19}^{6} - 141T_{19}^{5} + 505T_{19}^{4} - 771T_{19}^{3} + 1158T_{19}^{2} + 627T_{19} + 121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 7 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 35 T^{14} + \cdots + 47265625 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1675346761 \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 99 T^{6} + \cdots + 26411)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 11 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 11 T^{7} + 49 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 73 T^{14} + \cdots + 15768841 \) Copy content Toggle raw display
$41$ \( (T^{8} - 9 T^{7} + \cdots + 3876961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 173 T^{6} + \cdots + 212531)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 59639012521 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 239836452361 \) Copy content Toggle raw display
$59$ \( (T^{8} + 4 T^{7} - 41 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots + 43681)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 149 T^{6} + \cdots + 18491)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 3 T^{7} + \cdots + 6305121)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1636073786281 \) Copy content Toggle raw display
$79$ \( (T^{8} + 11 T^{7} + \cdots + 101761)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45169425961 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} + \cdots + 1871)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25937424601 \) Copy content Toggle raw display
show more
show less