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Properties

Label 50.4.d.b
Level $50$
Weight $4$
Character orbit 50.d
Analytic conductor $2.950$
Analytic rank $0$
Dimension $16$
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] = [50,4,Mod(11,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.d (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: \(2.95009550029\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 135 x^{14} + 7296 x^{12} + 200295 x^{10} + 2912021 x^{8} + 20937420 x^{6} + 57578496 x^{4} + \cdots + 952576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{6} q^{2} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{3}+ \cdots + (\beta_{15} - 2 \beta_{14} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{6} q^{2} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \cdots + 1) q^{3}+ \cdots + (78 \beta_{15} - 48 \beta_{14} + \cdots + 252) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{2} + 7 q^{3} - 16 q^{4} + 15 q^{5} + 14 q^{6} - 54 q^{7} - 32 q^{8} - 83 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{2} + 7 q^{3} - 16 q^{4} + 15 q^{5} + 14 q^{6} - 54 q^{7} - 32 q^{8} - 83 q^{9} - 50 q^{10} + 152 q^{11} - 12 q^{12} + 77 q^{13} + 52 q^{14} - 70 q^{15} - 64 q^{16} + 76 q^{17} + 304 q^{18} - 20 q^{19} + 20 q^{20} + 257 q^{21} - 226 q^{22} - 383 q^{23} - 64 q^{24} - 215 q^{25} - 716 q^{26} + 415 q^{27} + 4 q^{28} + 105 q^{29} + 140 q^{30} + 177 q^{31} + 512 q^{32} + 64 q^{33} + 342 q^{34} - 210 q^{35} - 332 q^{36} + 446 q^{37} + 90 q^{38} - 146 q^{39} - 40 q^{40} + 462 q^{41} + 514 q^{42} - 1058 q^{43} - 452 q^{44} - 865 q^{45} + 1124 q^{46} - 1114 q^{47} - 48 q^{48} + 1698 q^{49} - 550 q^{50} - 3608 q^{51} + 308 q^{52} + 567 q^{53} + 390 q^{54} - 210 q^{55} + 8 q^{56} - 1520 q^{57} + 210 q^{58} + 1275 q^{59} + 1260 q^{60} + 252 q^{61} - 176 q^{62} + 5402 q^{63} - 256 q^{64} + 2645 q^{65} + 998 q^{66} + 3296 q^{67} - 1976 q^{68} - 5491 q^{69} - 1750 q^{70} + 3712 q^{71} + 56 q^{72} - 1688 q^{73} - 1148 q^{74} - 6315 q^{75} - 200 q^{76} - 5453 q^{77} - 1152 q^{78} + 1750 q^{79} + 160 q^{80} + 4106 q^{81} - 1576 q^{82} - 2153 q^{83} - 612 q^{84} + 5860 q^{85} + 1074 q^{86} - 1255 q^{87} + 1216 q^{88} + 5660 q^{89} + 820 q^{90} - 493 q^{91} + 2248 q^{92} - 256 q^{93} - 2228 q^{94} - 1420 q^{95} + 224 q^{96} - 429 q^{97} - 674 q^{98} - 1786 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 135 x^{14} + 7296 x^{12} + 200295 x^{10} + 2912021 x^{8} + 20937420 x^{6} + 57578496 x^{4} + \cdots + 952576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2606260817 \nu^{14} - 299720681931 \nu^{12} - 13455196227932 \nu^{10} + \cdots + 704732477177408 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 630304983 \nu^{15} - 533360406 \nu^{14} + 73305733293 \nu^{13} - 13349395378 \nu^{12} + \cdots + 18\!\cdots\!04 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 630304983 \nu^{15} + 721297714 \nu^{14} + 73305733293 \nu^{13} + 106532776630 \nu^{12} + \cdots + 24\!\cdots\!72 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 630304983 \nu^{15} + 721297714 \nu^{14} - 73305733293 \nu^{13} + 106532776630 \nu^{12} + \cdots + 24\!\cdots\!72 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 236761278263 \nu^{15} - 76897207926 \nu^{14} - 32027842535037 \nu^{13} + \cdots + 45\!\cdots\!56 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 236761278263 \nu^{15} + 76897207926 \nu^{14} - 32027842535037 \nu^{13} + \cdots - 45\!\cdots\!56 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 382949064251 \nu^{15} + 934959346766 \nu^{14} + 52942143810537 \nu^{13} + \cdots + 94\!\cdots\!00 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 388969833455 \nu^{15} + 76897207926 \nu^{14} + 52269860904677 \nu^{13} + \cdots - 17\!\cdots\!96 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 798661771895 \nu^{15} + 520614588058 \nu^{14} - 110795388652797 \nu^{13} + \cdots + 15\!\cdots\!44 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 671993916829 \nu^{15} - 919553009500 \nu^{14} + 90194832923047 \nu^{13} + \cdots - 55\!\cdots\!88 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1845294772439 \nu^{15} + 649280159806 \nu^{14} - 244901174066797 \nu^{13} + \cdots + 52\!\cdots\!08 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2305026603065 \nu^{15} - 363600972846 \nu^{14} - 307837187881651 \nu^{13} + \cdots - 10\!\cdots\!16 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 631399667451 \nu^{15} + 238722614524 \nu^{14} - 86031799577357 \nu^{13} + \cdots + 21\!\cdots\!04 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1419571357221 \nu^{15} + 191253753108 \nu^{14} + 193074455424391 \nu^{13} + \cdots + 16\!\cdots\!36 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 951579173651 \nu^{15} - 94481789129 \nu^{14} + 127968394858977 \nu^{13} + \cdots + 15\!\cdots\!68 ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} - 2\beta_{5} + \beta_{4} + 3\beta_{3} - 4\beta_{2} + 2\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{13} + \beta_{12} - 4 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots - 90 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{15} - 7 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} - 5 \beta_{11} - 13 \beta_{10} - 3 \beta_{9} + \cdots + 30 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 26 \beta_{15} - 28 \beta_{14} + 50 \beta_{13} - 32 \beta_{12} + 28 \beta_{11} + 122 \beta_{10} + \cdots + 2458 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 472 \beta_{15} + 334 \beta_{14} + 230 \beta_{13} + 321 \beta_{12} + 216 \beta_{11} + 564 \beta_{10} + \cdots - 1564 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 158 \beta_{15} + 1263 \beta_{14} - 1532 \beta_{13} + 838 \beta_{12} - 1263 \beta_{11} - 3961 \beta_{10} + \cdots - 73558 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17362 \beta_{15} - 12886 \beta_{14} - 7146 \beta_{13} - 12522 \beta_{12} - 8546 \beta_{11} + \cdots + 59454 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8418 \beta_{15} - 47196 \beta_{14} + 48810 \beta_{13} - 22419 \beta_{12} + 47196 \beta_{11} + \cdots + 2290606 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 590334 \beta_{15} + 463593 \beta_{14} + 205890 \beta_{13} + 439422 \beta_{12} + 320067 \beta_{11} + \cdots - 2008578 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 570266 \beta_{15} + 1667676 \beta_{14} - 1559302 \beta_{13} + 630440 \beta_{12} - 1667676 \beta_{11} + \cdots - 72791046 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19425824 \beta_{15} - 16156190 \beta_{14} - 5666226 \beta_{13} - 14750163 \beta_{12} - 11598052 \beta_{11} + \cdots + 64290828 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 25656542 \beta_{15} - 57512849 \beta_{14} + 49765900 \beta_{13} - 18453886 \beta_{12} + \cdots + 2339020874 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 629892446 \beta_{15} + 553726202 \beta_{14} + 149623630 \beta_{13} + 485440158 \beta_{12} + \cdots - 2002447682 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1018604030 \beta_{15} + 1957736304 \beta_{14} - 1589371942 \beta_{13} + 553720217 \beta_{12} + \cdots - 75637271474 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 20293660586 \beta_{15} - 18798456239 \beta_{14} - 3758233926 \beta_{13} - 15842375454 \beta_{12} + \cdots + 61459129302 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1 + \beta_{5} - \beta_{6} + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
− 5.43776i
− 0.276776i
0.493198i
4.04577i
5.76335i
4.19209i
− 2.36461i
− 5.68871i
− 5.76335i
− 4.19209i
2.36461i
5.68871i
5.43776i
0.276776i
− 0.493198i
− 4.04577i
−1.61803 + 1.17557i −3.00525 + 9.24922i 1.23607 − 3.80423i 10.4354 + 4.01272i −6.01051 − 18.4984i −23.3628 2.47214 + 7.60845i −54.6730 − 39.7223i −21.6021 + 5.77486i
11.2 −1.61803 + 1.17557i 0.0282983 − 0.0870932i 1.23607 − 3.80423i 4.90949 − 10.0447i 0.0565966 + 0.174186i 10.7092 2.47214 + 7.60845i 21.8367 + 15.8653i 3.86458 + 22.0242i
11.3 −1.61803 + 1.17557i 0.480877 − 1.47999i 1.23607 − 3.80423i 1.22328 + 11.1132i 0.961755 + 2.95998i 19.9781 2.47214 + 7.60845i 19.8843 + 14.4468i −15.0437 − 16.5435i
11.4 −1.61803 + 1.17557i 2.56903 − 7.90665i 1.23607 − 3.80423i −11.1411 + 0.935417i 5.13805 + 15.8133i −26.4146 2.47214 + 7.60845i −34.0718 − 24.7546i 16.9271 − 14.6107i
21.1 0.618034 − 1.90211i −4.17225 − 3.03132i −3.23607 − 2.35114i −10.1806 + 4.62126i −8.34450 + 6.06264i −3.36421 −6.47214 + 4.70228i −0.124662 − 0.383671i 2.49822 + 22.2207i
21.2 0.618034 − 1.90211i −2.67790 − 1.94561i −3.23607 − 2.35114i 7.17512 − 8.57424i −5.35580 + 3.89121i −25.0474 −6.47214 + 4.70228i −4.95771 − 15.2583i −11.8747 − 18.9471i
21.3 0.618034 − 1.90211i 3.55790 + 2.58496i −3.23607 − 2.35114i −0.591780 − 11.1647i 7.11579 − 5.16993i 32.4633 −6.47214 + 4.70228i −2.36687 − 7.28447i −21.6022 − 5.77451i
21.4 0.618034 − 1.90211i 6.71931 + 4.88186i −3.23607 − 2.35114i 5.67017 + 9.63582i 13.4386 − 9.76372i −11.9615 −6.47214 + 4.70228i 12.9730 + 39.9269i 21.8328 − 4.83005i
31.1 0.618034 + 1.90211i −4.17225 + 3.03132i −3.23607 + 2.35114i −10.1806 − 4.62126i −8.34450 − 6.06264i −3.36421 −6.47214 − 4.70228i −0.124662 + 0.383671i 2.49822 − 22.2207i
31.2 0.618034 + 1.90211i −2.67790 + 1.94561i −3.23607 + 2.35114i 7.17512 + 8.57424i −5.35580 − 3.89121i −25.0474 −6.47214 − 4.70228i −4.95771 + 15.2583i −11.8747 + 18.9471i
31.3 0.618034 + 1.90211i 3.55790 − 2.58496i −3.23607 + 2.35114i −0.591780 + 11.1647i 7.11579 + 5.16993i 32.4633 −6.47214 − 4.70228i −2.36687 + 7.28447i −21.6022 + 5.77451i
31.4 0.618034 + 1.90211i 6.71931 − 4.88186i −3.23607 + 2.35114i 5.67017 − 9.63582i 13.4386 + 9.76372i −11.9615 −6.47214 − 4.70228i 12.9730 − 39.9269i 21.8328 + 4.83005i
41.1 −1.61803 − 1.17557i −3.00525 − 9.24922i 1.23607 + 3.80423i 10.4354 − 4.01272i −6.01051 + 18.4984i −23.3628 2.47214 − 7.60845i −54.6730 + 39.7223i −21.6021 − 5.77486i
41.2 −1.61803 − 1.17557i 0.0282983 + 0.0870932i 1.23607 + 3.80423i 4.90949 + 10.0447i 0.0565966 − 0.174186i 10.7092 2.47214 − 7.60845i 21.8367 − 15.8653i 3.86458 − 22.0242i
41.3 −1.61803 − 1.17557i 0.480877 + 1.47999i 1.23607 + 3.80423i 1.22328 − 11.1132i 0.961755 − 2.95998i 19.9781 2.47214 − 7.60845i 19.8843 − 14.4468i −15.0437 + 16.5435i
41.4 −1.61803 − 1.17557i 2.56903 + 7.90665i 1.23607 + 3.80423i −11.1411 − 0.935417i 5.13805 − 15.8133i −26.4146 2.47214 − 7.60845i −34.0718 + 24.7546i 16.9271 + 14.6107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
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 50.4.d.b ✓ 16
5.b even 2 1 250.4.d.b 16
5.c odd 4 2 250.4.e.c 32
25.d even 5 1 inner 50.4.d.b ✓ 16
25.d even 5 1 1250.4.a.j 8
25.e even 10 1 250.4.d.b 16
25.e even 10 1 1250.4.a.g 8
25.f odd 20 2 250.4.e.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.d.b ✓ 16 1.a even 1 1 trivial
50.4.d.b ✓ 16 25.d even 5 1 inner
250.4.d.b 16 5.b even 2 1
250.4.d.b 16 25.e even 10 1
250.4.e.c 32 5.c odd 4 2
250.4.e.c 32 25.f odd 20 2
1250.4.a.g 8 25.e even 10 1
1250.4.a.j 8 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 7 T_{3}^{15} + 120 T_{3}^{14} - 810 T_{3}^{13} + 7880 T_{3}^{12} - 15911 T_{3}^{11} + \cdots + 51609856 \) acting on \(S_{4}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{15} + \cdots + 51609856 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 27 T^{7} + \cdots - 4320209664)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 96\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 78\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
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