Properties

Label 10.5.c.b
Level $10$
Weight $5$
Character orbit 10.c
Analytic conductor $1.034$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,5,Mod(3,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 10.c (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: \(1.03369963084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 i + 2) q^{2} + ( - i + 1) q^{3} + 8 i q^{4} + ( - 20 i - 15) q^{5} + 4 q^{6} + ( - 19 i - 19) q^{7} + (16 i - 16) q^{8} + 79 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 i + 2) q^{2} + ( - i + 1) q^{3} + 8 i q^{4} + ( - 20 i - 15) q^{5} + 4 q^{6} + ( - 19 i - 19) q^{7} + (16 i - 16) q^{8} + 79 i q^{9} + ( - 70 i + 10) q^{10} + 202 q^{11} + (8 i + 8) q^{12} + (99 i - 99) q^{13} - 76 i q^{14} + ( - 5 i - 35) q^{15} - 64 q^{16} + ( - 239 i - 239) q^{17} + (158 i - 158) q^{18} + 40 i q^{19} + ( - 120 i + 160) q^{20} - 38 q^{21} + (404 i + 404) q^{22} + ( - 541 i + 541) q^{23} + 32 i q^{24} + (600 i - 175) q^{25} - 396 q^{26} + (160 i + 160) q^{27} + ( - 152 i + 152) q^{28} - 200 i q^{29} + ( - 80 i - 60) q^{30} - 758 q^{31} + ( - 128 i - 128) q^{32} + ( - 202 i + 202) q^{33} - 956 i q^{34} + (665 i - 95) q^{35} - 632 q^{36} + (141 i + 141) q^{37} + (80 i - 80) q^{38} + 198 i q^{39} + (80 i + 560) q^{40} + 1042 q^{41} + ( - 76 i - 76) q^{42} + (759 i - 759) q^{43} + 1616 i q^{44} + ( - 1185 i + 1580) q^{45} + 2164 q^{46} + ( - 459 i - 459) q^{47} + (64 i - 64) q^{48} - 1679 i q^{49} + (850 i - 1550) q^{50} - 478 q^{51} + ( - 792 i - 792) q^{52} + (1819 i - 1819) q^{53} + 640 i q^{54} + ( - 4040 i - 3030) q^{55} + 608 q^{56} + (40 i + 40) q^{57} + ( - 400 i + 400) q^{58} + 4600 i q^{59} + ( - 280 i + 40) q^{60} + 2082 q^{61} + ( - 1516 i - 1516) q^{62} + ( - 1501 i + 1501) q^{63} - 512 i q^{64} + (495 i + 3465) q^{65} + 808 q^{66} + (5081 i + 5081) q^{67} + ( - 1912 i + 1912) q^{68} - 1082 i q^{69} + (1140 i - 1520) q^{70} - 3478 q^{71} + ( - 1264 i - 1264) q^{72} + (3479 i - 3479) q^{73} + 564 i q^{74} + (775 i + 425) q^{75} - 320 q^{76} + ( - 3838 i - 3838) q^{77} + (396 i - 396) q^{78} - 7680 i q^{79} + (1280 i + 960) q^{80} - 6079 q^{81} + (2084 i + 2084) q^{82} + ( - 6081 i + 6081) q^{83} - 304 i q^{84} + (8365 i - 1195) q^{85} - 3036 q^{86} + ( - 200 i - 200) q^{87} + (3232 i - 3232) q^{88} - 5680 i q^{89} + (790 i + 5530) q^{90} + 3762 q^{91} + (4328 i + 4328) q^{92} + (758 i - 758) q^{93} - 1836 i q^{94} + ( - 600 i + 800) q^{95} - 256 q^{96} + (561 i + 561) q^{97} + ( - 3358 i + 3358) q^{98} + 15958 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{3} - 30 q^{5} + 8 q^{6} - 38 q^{7} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 2 q^{3} - 30 q^{5} + 8 q^{6} - 38 q^{7} - 32 q^{8} + 20 q^{10} + 404 q^{11} + 16 q^{12} - 198 q^{13} - 70 q^{15} - 128 q^{16} - 478 q^{17} - 316 q^{18} + 320 q^{20} - 76 q^{21} + 808 q^{22} + 1082 q^{23} - 350 q^{25} - 792 q^{26} + 320 q^{27} + 304 q^{28} - 120 q^{30} - 1516 q^{31} - 256 q^{32} + 404 q^{33} - 190 q^{35} - 1264 q^{36} + 282 q^{37} - 160 q^{38} + 1120 q^{40} + 2084 q^{41} - 152 q^{42} - 1518 q^{43} + 3160 q^{45} + 4328 q^{46} - 918 q^{47} - 128 q^{48} - 3100 q^{50} - 956 q^{51} - 1584 q^{52} - 3638 q^{53} - 6060 q^{55} + 1216 q^{56} + 80 q^{57} + 800 q^{58} + 80 q^{60} + 4164 q^{61} - 3032 q^{62} + 3002 q^{63} + 6930 q^{65} + 1616 q^{66} + 10162 q^{67} + 3824 q^{68} - 3040 q^{70} - 6956 q^{71} - 2528 q^{72} - 6958 q^{73} + 850 q^{75} - 640 q^{76} - 7676 q^{77} - 792 q^{78} + 1920 q^{80} - 12158 q^{81} + 4168 q^{82} + 12162 q^{83} - 2390 q^{85} - 6072 q^{86} - 400 q^{87} - 6464 q^{88} + 11060 q^{90} + 7524 q^{91} + 8656 q^{92} - 1516 q^{93} + 1600 q^{95} - 512 q^{96} + 1122 q^{97} + 6716 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.00000i
1.00000i
2.00000 2.00000i 1.00000 + 1.00000i 8.00000i −15.0000 + 20.0000i 4.00000 −19.0000 + 19.0000i −16.0000 16.0000i 79.0000i 10.0000 + 70.0000i
7.1 2.00000 + 2.00000i 1.00000 1.00000i 8.00000i −15.0000 20.0000i 4.00000 −19.0000 19.0000i −16.0000 + 16.0000i 79.0000i 10.0000 70.0000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.5.c.b 2
3.b odd 2 1 90.5.g.a 2
4.b odd 2 1 80.5.p.c 2
5.b even 2 1 50.5.c.a 2
5.c odd 4 1 inner 10.5.c.b 2
5.c odd 4 1 50.5.c.a 2
8.b even 2 1 320.5.p.d 2
8.d odd 2 1 320.5.p.g 2
15.d odd 2 1 450.5.g.b 2
15.e even 4 1 90.5.g.a 2
15.e even 4 1 450.5.g.b 2
20.d odd 2 1 400.5.p.b 2
20.e even 4 1 80.5.p.c 2
20.e even 4 1 400.5.p.b 2
40.i odd 4 1 320.5.p.d 2
40.k even 4 1 320.5.p.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.5.c.b 2 1.a even 1 1 trivial
10.5.c.b 2 5.c odd 4 1 inner
50.5.c.a 2 5.b even 2 1
50.5.c.a 2 5.c odd 4 1
80.5.p.c 2 4.b odd 2 1
80.5.p.c 2 20.e even 4 1
90.5.g.a 2 3.b odd 2 1
90.5.g.a 2 15.e even 4 1
320.5.p.d 2 8.b even 2 1
320.5.p.d 2 40.i odd 4 1
320.5.p.g 2 8.d odd 2 1
320.5.p.g 2 40.k even 4 1
400.5.p.b 2 20.d odd 2 1
400.5.p.b 2 20.e even 4 1
450.5.g.b 2 15.d odd 2 1
450.5.g.b 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 2 \) acting on \(S_{5}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 30T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} + 38T + 722 \) Copy content Toggle raw display
$11$ \( (T - 202)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 198T + 19602 \) Copy content Toggle raw display
$17$ \( T^{2} + 478T + 114242 \) Copy content Toggle raw display
$19$ \( T^{2} + 1600 \) Copy content Toggle raw display
$23$ \( T^{2} - 1082 T + 585362 \) Copy content Toggle raw display
$29$ \( T^{2} + 40000 \) Copy content Toggle raw display
$31$ \( (T + 758)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 282T + 39762 \) Copy content Toggle raw display
$41$ \( (T - 1042)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1518 T + 1152162 \) Copy content Toggle raw display
$47$ \( T^{2} + 918T + 421362 \) Copy content Toggle raw display
$53$ \( T^{2} + 3638 T + 6617522 \) Copy content Toggle raw display
$59$ \( T^{2} + 21160000 \) Copy content Toggle raw display
$61$ \( (T - 2082)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 10162 T + 51633122 \) Copy content Toggle raw display
$71$ \( (T + 3478)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6958 T + 24206882 \) Copy content Toggle raw display
$79$ \( T^{2} + 58982400 \) Copy content Toggle raw display
$83$ \( T^{2} - 12162 T + 73957122 \) Copy content Toggle raw display
$89$ \( T^{2} + 32262400 \) Copy content Toggle raw display
$97$ \( T^{2} - 1122 T + 629442 \) Copy content Toggle raw display
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