Properties

Label 10.5.c.a
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.

\(f(q)\) \(=\) \( q + ( - 2 i - 2) q^{2} + ( - 9 i + 9) q^{3} + 8 i q^{4} + (20 i - 15) q^{5} - 36 q^{6} + (29 i + 29) q^{7} + ( - 16 i + 16) q^{8} - 81 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 i - 2) q^{2} + ( - 9 i + 9) q^{3} + 8 i q^{4} + (20 i - 15) q^{5} - 36 q^{6} + (29 i + 29) q^{7} + ( - 16 i + 16) q^{8} - 81 i q^{9} + ( - 10 i + 70) q^{10} - 118 q^{11} + (72 i + 72) q^{12} + ( - 69 i + 69) q^{13} - 116 i q^{14} + (315 i + 45) q^{15} - 64 q^{16} + ( - 271 i - 271) q^{17} + (162 i - 162) q^{18} + 280 i q^{19} + ( - 120 i - 160) q^{20} + 522 q^{21} + (236 i + 236) q^{22} + ( - 269 i + 269) q^{23} - 288 i q^{24} + ( - 600 i - 175) q^{25} - 276 q^{26} + (232 i - 232) q^{28} + 680 i q^{29} + ( - 720 i + 540) q^{30} + 202 q^{31} + (128 i + 128) q^{32} + (1062 i - 1062) q^{33} + 1084 i q^{34} + (145 i - 1015) q^{35} + 648 q^{36} + ( - 651 i - 651) q^{37} + ( - 560 i + 560) q^{38} - 1242 i q^{39} + (560 i + 80) q^{40} + 1682 q^{41} + ( - 1044 i - 1044) q^{42} + ( - 1089 i + 1089) q^{43} - 944 i q^{44} + (1215 i + 1620) q^{45} - 1076 q^{46} + (1269 i + 1269) q^{47} + (576 i - 576) q^{48} - 719 i q^{49} + (1550 i - 850) q^{50} - 4878 q^{51} + (552 i + 552) q^{52} + (611 i - 611) q^{53} + ( - 2360 i + 1770) q^{55} + 928 q^{56} + (2520 i + 2520) q^{57} + ( - 1360 i + 1360) q^{58} + 1160 i q^{59} + (360 i - 2520) q^{60} - 5598 q^{61} + ( - 404 i - 404) q^{62} + ( - 2349 i + 2349) q^{63} - 512 i q^{64} + (2415 i + 345) q^{65} + 4248 q^{66} + ( - 751 i - 751) q^{67} + ( - 2168 i + 2168) q^{68} - 4842 i q^{69} + (1740 i + 2320) q^{70} + 6442 q^{71} + ( - 1296 i - 1296) q^{72} + (2951 i - 2951) q^{73} + 2604 i q^{74} + ( - 3825 i - 6975) q^{75} - 2240 q^{76} + ( - 3422 i - 3422) q^{77} + (2484 i - 2484) q^{78} + 10560 i q^{79} + ( - 1280 i + 960) q^{80} + 6561 q^{81} + ( - 3364 i - 3364) q^{82} + (6231 i - 6231) q^{83} + 4176 i q^{84} + ( - 1355 i + 9485) q^{85} - 4356 q^{86} + (6120 i + 6120) q^{87} + (1888 i - 1888) q^{88} - 14480 i q^{89} + ( - 5670 i - 810) q^{90} + 4002 q^{91} + (2152 i + 2152) q^{92} + ( - 1818 i + 1818) q^{93} - 5076 i q^{94} + ( - 4200 i - 5600) q^{95} + 2304 q^{96} + ( - 7311 i - 7311) q^{97} + (1438 i - 1438) q^{98} + 9558 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 18 q^{3} - 30 q^{5} - 72 q^{6} + 58 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 18 q^{3} - 30 q^{5} - 72 q^{6} + 58 q^{7} + 32 q^{8} + 140 q^{10} - 236 q^{11} + 144 q^{12} + 138 q^{13} + 90 q^{15} - 128 q^{16} - 542 q^{17} - 324 q^{18} - 320 q^{20} + 1044 q^{21} + 472 q^{22} + 538 q^{23} - 350 q^{25} - 552 q^{26} - 464 q^{28} + 1080 q^{30} + 404 q^{31} + 256 q^{32} - 2124 q^{33} - 2030 q^{35} + 1296 q^{36} - 1302 q^{37} + 1120 q^{38} + 160 q^{40} + 3364 q^{41} - 2088 q^{42} + 2178 q^{43} + 3240 q^{45} - 2152 q^{46} + 2538 q^{47} - 1152 q^{48} - 1700 q^{50} - 9756 q^{51} + 1104 q^{52} - 1222 q^{53} + 3540 q^{55} + 1856 q^{56} + 5040 q^{57} + 2720 q^{58} - 5040 q^{60} - 11196 q^{61} - 808 q^{62} + 4698 q^{63} + 690 q^{65} + 8496 q^{66} - 1502 q^{67} + 4336 q^{68} + 4640 q^{70} + 12884 q^{71} - 2592 q^{72} - 5902 q^{73} - 13950 q^{75} - 4480 q^{76} - 6844 q^{77} - 4968 q^{78} + 1920 q^{80} + 13122 q^{81} - 6728 q^{82} - 12462 q^{83} + 18970 q^{85} - 8712 q^{86} + 12240 q^{87} - 3776 q^{88} - 1620 q^{90} + 8004 q^{91} + 4304 q^{92} + 3636 q^{93} - 11200 q^{95} + 4608 q^{96} - 14622 q^{97} - 2876 q^{98}+O(q^{100}) \) 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 9.00000 + 9.00000i 8.00000i −15.0000 20.0000i −36.0000 29.0000 29.0000i 16.0000 + 16.0000i 81.0000i 70.0000 + 10.0000i
7.1 −2.00000 2.00000i 9.00000 9.00000i 8.00000i −15.0000 + 20.0000i −36.0000 29.0000 + 29.0000i 16.0000 16.0000i 81.0000i 70.0000 10.0000i
\(n\): 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.a 2
3.b odd 2 1 90.5.g.b 2
4.b odd 2 1 80.5.p.b 2
5.b even 2 1 50.5.c.b 2
5.c odd 4 1 inner 10.5.c.a 2
5.c odd 4 1 50.5.c.b 2
8.b even 2 1 320.5.p.b 2
8.d odd 2 1 320.5.p.i 2
15.d odd 2 1 450.5.g.a 2
15.e even 4 1 90.5.g.b 2
15.e even 4 1 450.5.g.a 2
20.d odd 2 1 400.5.p.c 2
20.e even 4 1 80.5.p.b 2
20.e even 4 1 400.5.p.c 2
40.i odd 4 1 320.5.p.b 2
40.k even 4 1 320.5.p.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.5.c.a 2 1.a even 1 1 trivial
10.5.c.a 2 5.c odd 4 1 inner
50.5.c.b 2 5.b even 2 1
50.5.c.b 2 5.c odd 4 1
80.5.p.b 2 4.b odd 2 1
80.5.p.b 2 20.e even 4 1
90.5.g.b 2 3.b odd 2 1
90.5.g.b 2 15.e even 4 1
320.5.p.b 2 8.b even 2 1
320.5.p.b 2 40.i odd 4 1
320.5.p.i 2 8.d odd 2 1
320.5.p.i 2 40.k even 4 1
400.5.p.c 2 20.d odd 2 1
400.5.p.c 2 20.e even 4 1
450.5.g.a 2 15.d odd 2 1
450.5.g.a 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 18T_{3} + 162 \) 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
$3$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$5$ \( T^{2} + 30T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} - 58T + 1682 \) Copy content Toggle raw display
$11$ \( (T + 118)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 138T + 9522 \) Copy content Toggle raw display
$17$ \( T^{2} + 542T + 146882 \) Copy content Toggle raw display
$19$ \( T^{2} + 78400 \) Copy content Toggle raw display
$23$ \( T^{2} - 538T + 144722 \) Copy content Toggle raw display
$29$ \( T^{2} + 462400 \) Copy content Toggle raw display
$31$ \( (T - 202)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1302 T + 847602 \) Copy content Toggle raw display
$41$ \( (T - 1682)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 2178 T + 2371842 \) Copy content Toggle raw display
$47$ \( T^{2} - 2538 T + 3220722 \) Copy content Toggle raw display
$53$ \( T^{2} + 1222 T + 746642 \) Copy content Toggle raw display
$59$ \( T^{2} + 1345600 \) Copy content Toggle raw display
$61$ \( (T + 5598)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1502 T + 1128002 \) Copy content Toggle raw display
$71$ \( (T - 6442)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5902 T + 17416802 \) Copy content Toggle raw display
$79$ \( T^{2} + 111513600 \) Copy content Toggle raw display
$83$ \( T^{2} + 12462 T + 77650722 \) Copy content Toggle raw display
$89$ \( T^{2} + 209670400 \) Copy content Toggle raw display
$97$ \( T^{2} + 14622 T + 106901442 \) Copy content Toggle raw display
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