Properties

Label 50.8.a.j
Level $50$
Weight $8$
Character orbit 50.a
Self dual yes
Analytic conductor $15.619$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,8,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

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: yes
Analytic conductor: \(15.6192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 10\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + (\beta + 28) q^{3} + 64 q^{4} + (8 \beta + 224) q^{6} + ( - 7 \beta + 204) q^{7} + 512 q^{8} + (56 \beta + 1697) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + (\beta + 28) q^{3} + 64 q^{4} + (8 \beta + 224) q^{6} + ( - 7 \beta + 204) q^{7} + 512 q^{8} + (56 \beta + 1697) q^{9} + ( - 16 \beta + 4452) q^{11} + (64 \beta + 1792) q^{12} + ( - 78 \beta - 1752) q^{13} + ( - 56 \beta + 1632) q^{14} + 4096 q^{16} + ( - 168 \beta + 21824) q^{17} + (448 \beta + 13576) q^{18} + ( - 624 \beta - 15900) q^{19} + (8 \beta - 15988) q^{21} + ( - 128 \beta + 35616) q^{22} + ( - 849 \beta + 35828) q^{23} + (512 \beta + 14336) q^{24} + ( - 624 \beta - 14016) q^{26} + (1078 \beta + 159880) q^{27} + ( - 448 \beta + 13056) q^{28} + (2592 \beta - 42390) q^{29} + (1248 \beta - 98528) q^{31} + 32768 q^{32} + (4004 \beta + 75056) q^{33} + ( - 1344 \beta + 174592) q^{34} + (3584 \beta + 108608) q^{36} + ( - 4438 \beta + 97704) q^{37} + ( - 4992 \beta - 127200) q^{38} + ( - 3936 \beta - 290856) q^{39} + ( - 7768 \beta + 58122) q^{41} + (64 \beta - 127904) q^{42} + (653 \beta + 137868) q^{43} + ( - 1024 \beta + 284928) q^{44} + ( - 6792 \beta + 286624) q^{46} + (16413 \beta - 485636) q^{47} + (4096 \beta + 114688) q^{48} + ( - 2856 \beta - 630027) q^{49} + (17120 \beta + 90272) q^{51} + ( - 4992 \beta - 112128) q^{52} + ( - 12738 \beta - 1003752) q^{53} + (8624 \beta + 1279040) q^{54} + ( - 3584 \beta + 104448) q^{56} + ( - 33372 \beta - 2379600) q^{57} + (20736 \beta - 339120) q^{58} + (11824 \beta - 523380) q^{59} + ( - 1464 \beta - 1312858) q^{61} + (9984 \beta - 788224) q^{62} + ( - 455 \beta - 869012) q^{63} + 262144 q^{64} + (32032 \beta + 600448) q^{66} + (31759 \beta + 839364) q^{67} + ( - 10752 \beta + 1396736) q^{68} + (12056 \beta - 1628716) q^{69} + ( - 53184 \beta - 1958088) q^{71} + (28672 \beta + 868864) q^{72} + ( - 21268 \beta + 1303248) q^{73} + ( - 35504 \beta + 781632) q^{74} + ( - 39936 \beta - 1017600) q^{76} + ( - 34428 \beta + 1255408) q^{77} + ( - 31488 \beta - 2326848) q^{78} + ( - 49824 \beta - 1931760) q^{79} + (67592 \beta + 4107101) q^{81} + ( - 62144 \beta + 464976) q^{82} + (118665 \beta + 2591708) q^{83} + (512 \beta - 1023232) q^{84} + (5224 \beta + 1102944) q^{86} + (30186 \beta + 6848280) q^{87} + ( - 8192 \beta + 2279424) q^{88} + ( - 34720 \beta + 8367510) q^{89} + ( - 3648 \beta + 1335192) q^{91} + ( - 54336 \beta + 2292992) q^{92} + ( - 63584 \beta + 1110016) q^{93} + (131304 \beta - 3885088) q^{94} + (32768 \beta + 917504) q^{96} + (114880 \beta + 6302304) q^{97} + ( - 22848 \beta - 5040216) q^{98} + (222160 \beta + 4777444) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} + 56 q^{3} + 128 q^{4} + 448 q^{6} + 408 q^{7} + 1024 q^{8} + 3394 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} + 56 q^{3} + 128 q^{4} + 448 q^{6} + 408 q^{7} + 1024 q^{8} + 3394 q^{9} + 8904 q^{11} + 3584 q^{12} - 3504 q^{13} + 3264 q^{14} + 8192 q^{16} + 43648 q^{17} + 27152 q^{18} - 31800 q^{19} - 31976 q^{21} + 71232 q^{22} + 71656 q^{23} + 28672 q^{24} - 28032 q^{26} + 319760 q^{27} + 26112 q^{28} - 84780 q^{29} - 197056 q^{31} + 65536 q^{32} + 150112 q^{33} + 349184 q^{34} + 217216 q^{36} + 195408 q^{37} - 254400 q^{38} - 581712 q^{39} + 116244 q^{41} - 255808 q^{42} + 275736 q^{43} + 569856 q^{44} + 573248 q^{46} - 971272 q^{47} + 229376 q^{48} - 1260054 q^{49} + 180544 q^{51} - 224256 q^{52} - 2007504 q^{53} + 2558080 q^{54} + 208896 q^{56} - 4759200 q^{57} - 678240 q^{58} - 1046760 q^{59} - 2625716 q^{61} - 1576448 q^{62} - 1738024 q^{63} + 524288 q^{64} + 1200896 q^{66} + 1678728 q^{67} + 2793472 q^{68} - 3257432 q^{69} - 3916176 q^{71} + 1737728 q^{72} + 2606496 q^{73} + 1563264 q^{74} - 2035200 q^{76} + 2510816 q^{77} - 4653696 q^{78} - 3863520 q^{79} + 8214202 q^{81} + 929952 q^{82} + 5183416 q^{83} - 2046464 q^{84} + 2205888 q^{86} + 13696560 q^{87} + 4558848 q^{88} + 16735020 q^{89} + 2670384 q^{91} + 4585984 q^{92} + 2220032 q^{93} - 7770176 q^{94} + 1835008 q^{96} + 12604608 q^{97} - 10080432 q^{98} + 9554888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.56776
5.56776
8.00000 −27.6776 64.0000 0 −221.421 593.744 512.000 −1420.95 0
1.2 8.00000 83.6776 64.0000 0 669.421 −185.744 512.000 4814.95 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.8.a.j 2
3.b odd 2 1 450.8.a.bd 2
4.b odd 2 1 400.8.a.t 2
5.b even 2 1 50.8.a.i 2
5.c odd 4 2 10.8.b.a 4
15.d odd 2 1 450.8.a.bi 2
15.e even 4 2 90.8.c.c 4
20.d odd 2 1 400.8.a.bf 2
20.e even 4 2 80.8.c.d 4
40.i odd 4 2 320.8.c.f 4
40.k even 4 2 320.8.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.8.b.a 4 5.c odd 4 2
50.8.a.i 2 5.b even 2 1
50.8.a.j 2 1.a even 1 1 trivial
80.8.c.d 4 20.e even 4 2
90.8.c.c 4 15.e even 4 2
320.8.c.f 4 40.i odd 4 2
320.8.c.g 4 40.k even 4 2
400.8.a.t 2 4.b odd 2 1
400.8.a.bf 2 20.d odd 2 1
450.8.a.bd 2 3.b odd 2 1
450.8.a.bi 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 56T_{3} - 2316 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(50))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 56T - 2316 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 408T - 110284 \) Copy content Toggle raw display
$11$ \( T^{2} - 8904 T + 19026704 \) Copy content Toggle raw display
$13$ \( T^{2} + 3504 T - 15790896 \) Copy content Toggle raw display
$17$ \( T^{2} - 43648 T + 388792576 \) Copy content Toggle raw display
$19$ \( T^{2} + 31800 T - 954255600 \) Copy content Toggle raw display
$23$ \( T^{2} - 71656 T - 950837516 \) Copy content Toggle raw display
$29$ \( T^{2} + 84780 T - 19030326300 \) Copy content Toggle raw display
$31$ \( T^{2} + 197056 T + 4879504384 \) Copy content Toggle raw display
$37$ \( T^{2} - 195408 T - 51511044784 \) Copy content Toggle raw display
$41$ \( T^{2} - 116244 T - 183681487516 \) Copy content Toggle raw display
$43$ \( T^{2} - 275736 T + 17685717524 \) Copy content Toggle raw display
$47$ \( T^{2} + 971272 T - 599256039404 \) Copy content Toggle raw display
$53$ \( T^{2} + 2007504 T + 504522481104 \) Copy content Toggle raw display
$59$ \( T^{2} + 1046760 T - 159475001200 \) Copy content Toggle raw display
$61$ \( T^{2} + 2625716 T + 1716951910564 \) Copy content Toggle raw display
$67$ \( T^{2} - 1678728 T - 2422233726604 \) Copy content Toggle raw display
$71$ \( T^{2} + 3916176 T - 4934358737856 \) Copy content Toggle raw display
$73$ \( T^{2} - 2606496 T + 296239095104 \) Copy content Toggle raw display
$79$ \( T^{2} + 3863520 T - 3963839328000 \) Copy content Toggle raw display
$83$ \( T^{2} - 5183416 T - 36935334540236 \) Copy content Toggle raw display
$89$ \( T^{2} - 16735020 T + 66278240560100 \) Copy content Toggle raw display
$97$ \( T^{2} - 12604608 T - 1192948931584 \) Copy content Toggle raw display
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