Properties

Label 10.7.c.a
Level $10$
Weight $7$
Character orbit 10.c
Analytic conductor $2.301$
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] = [10,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(2.30054083620\)
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.

\(f(q)\) \(=\) \( q + ( - 4 i - 4) q^{2} + (23 i - 23) q^{3} + 32 i q^{4} + (100 i - 75) q^{5} + 184 q^{6} + ( - 247 i - 247) q^{7} + ( - 128 i + 128) q^{8} - 329 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 i - 4) q^{2} + (23 i - 23) q^{3} + 32 i q^{4} + (100 i - 75) q^{5} + 184 q^{6} + ( - 247 i - 247) q^{7} + ( - 128 i + 128) q^{8} - 329 i q^{9} + ( - 100 i + 700) q^{10} + 1402 q^{11} + ( - 736 i - 736) q^{12} + (2703 i - 2703) q^{13} + 1976 i q^{14} + ( - 4025 i - 575) q^{15} - 1024 q^{16} + (2593 i + 2593) q^{17} + (1316 i - 1316) q^{18} + 1720 i q^{19} + ( - 2400 i - 3200) q^{20} + 11362 q^{21} + ( - 5608 i - 5608) q^{22} + ( - 2137 i + 2137) q^{23} + 5888 i q^{24} + ( - 15000 i - 4375) q^{25} + 21624 q^{26} + ( - 9200 i - 9200) q^{27} + ( - 7904 i + 7904) q^{28} + 30520 i q^{29} + (18400 i - 13800) q^{30} - 37838 q^{31} + (4096 i + 4096) q^{32} + (32246 i - 32246) q^{33} - 20744 i q^{34} + ( - 6175 i + 43225) q^{35} + 10528 q^{36} + (37113 i + 37113) q^{37} + ( - 6880 i + 6880) q^{38} - 124338 i q^{39} + (22400 i + 3200) q^{40} - 35438 q^{41} + ( - 45448 i - 45448) q^{42} + ( - 39177 i + 39177) q^{43} + 44864 i q^{44} + (24675 i + 32900) q^{45} - 17096 q^{46} + (95193 i + 95193) q^{47} + ( - 23552 i + 23552) q^{48} + 4369 i q^{49} + (77500 i - 42500) q^{50} - 119278 q^{51} + ( - 86496 i - 86496) q^{52} + ( - 36017 i + 36017) q^{53} + 73600 i q^{54} + (140200 i - 105150) q^{55} - 63232 q^{56} + ( - 39560 i - 39560) q^{57} + ( - 122080 i + 122080) q^{58} - 35960 i q^{59} + ( - 18400 i + 128800) q^{60} + 83322 q^{61} + (151352 i + 151352) q^{62} + (81263 i - 81263) q^{63} - 32768 i q^{64} + ( - 473025 i - 67575) q^{65} + 257968 q^{66} + (60833 i + 60833) q^{67} + (82976 i - 82976) q^{68} + 98302 i q^{69} + ( - 148200 i - 197600) q^{70} - 40318 q^{71} + ( - 42112 i - 42112) q^{72} + (129023 i - 129023) q^{73} - 296904 i q^{74} + (244375 i + 445625) q^{75} - 55040 q^{76} + ( - 346294 i - 346294) q^{77} + (497352 i - 497352) q^{78} + 524640 i q^{79} + ( - 102400 i + 76800) q^{80} + 663041 q^{81} + (141752 i + 141752) q^{82} + (114423 i - 114423) q^{83} + 363584 i q^{84} + (64825 i - 453775) q^{85} - 313416 q^{86} + ( - 701960 i - 701960) q^{87} + ( - 179456 i + 179456) q^{88} + 187280 i q^{89} + ( - 230300 i - 32900) q^{90} + 1335282 q^{91} + (68384 i + 68384) q^{92} + ( - 870274 i + 870274) q^{93} - 761544 i q^{94} + ( - 129000 i - 172000) q^{95} - 188416 q^{96} + (532833 i + 532833) q^{97} + ( - 17476 i + 17476) q^{98} - 461258 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 46 q^{3} - 150 q^{5} + 368 q^{6} - 494 q^{7} + 256 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 46 q^{3} - 150 q^{5} + 368 q^{6} - 494 q^{7} + 256 q^{8} + 1400 q^{10} + 2804 q^{11} - 1472 q^{12} - 5406 q^{13} - 1150 q^{15} - 2048 q^{16} + 5186 q^{17} - 2632 q^{18} - 6400 q^{20} + 22724 q^{21} - 11216 q^{22} + 4274 q^{23} - 8750 q^{25} + 43248 q^{26} - 18400 q^{27} + 15808 q^{28} - 27600 q^{30} - 75676 q^{31} + 8192 q^{32} - 64492 q^{33} + 86450 q^{35} + 21056 q^{36} + 74226 q^{37} + 13760 q^{38} + 6400 q^{40} - 70876 q^{41} - 90896 q^{42} + 78354 q^{43} + 65800 q^{45} - 34192 q^{46} + 190386 q^{47} + 47104 q^{48} - 85000 q^{50} - 238556 q^{51} - 172992 q^{52} + 72034 q^{53} - 210300 q^{55} - 126464 q^{56} - 79120 q^{57} + 244160 q^{58} + 257600 q^{60} + 166644 q^{61} + 302704 q^{62} - 162526 q^{63} - 135150 q^{65} + 515936 q^{66} + 121666 q^{67} - 165952 q^{68} - 395200 q^{70} - 80636 q^{71} - 84224 q^{72} - 258046 q^{73} + 891250 q^{75} - 110080 q^{76} - 692588 q^{77} - 994704 q^{78} + 153600 q^{80} + 1326082 q^{81} + 283504 q^{82} - 228846 q^{83} - 907550 q^{85} - 626832 q^{86} - 1403920 q^{87} + 358912 q^{88} - 65800 q^{90} + 2670564 q^{91} + 136768 q^{92} + 1740548 q^{93} - 344000 q^{95} - 376832 q^{96} + 1065666 q^{97} + 34952 q^{98}+O(q^{100}) \) 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
−4.00000 + 4.00000i −23.0000 23.0000i 32.0000i −75.0000 100.000i 184.000 −247.000 + 247.000i 128.000 + 128.000i 329.000i 700.000 + 100.000i
7.1 −4.00000 4.00000i −23.0000 + 23.0000i 32.0000i −75.0000 + 100.000i 184.000 −247.000 247.000i 128.000 128.000i 329.000i 700.000 100.000i
\(n\): 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 10.7.c.a 2
3.b odd 2 1 90.7.g.a 2
4.b odd 2 1 80.7.p.a 2
5.b even 2 1 50.7.c.c 2
5.c odd 4 1 inner 10.7.c.a 2
5.c odd 4 1 50.7.c.c 2
15.d odd 2 1 450.7.g.b 2
15.e even 4 1 90.7.g.a 2
15.e even 4 1 450.7.g.b 2
20.e even 4 1 80.7.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.7.c.a 2 1.a even 1 1 trivial
10.7.c.a 2 5.c odd 4 1 inner
50.7.c.c 2 5.b even 2 1
50.7.c.c 2 5.c odd 4 1
80.7.p.a 2 4.b odd 2 1
80.7.p.a 2 20.e even 4 1
90.7.g.a 2 3.b odd 2 1
90.7.g.a 2 15.e even 4 1
450.7.g.b 2 15.d odd 2 1
450.7.g.b 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 46T_{3} + 1058 \) acting on \(S_{7}^{\mathrm{new}}(10, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$3$ \( T^{2} + 46T + 1058 \) Copy content Toggle raw display
$5$ \( T^{2} + 150T + 15625 \) Copy content Toggle raw display
$7$ \( T^{2} + 494T + 122018 \) Copy content Toggle raw display
$11$ \( (T - 1402)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5406 T + 14612418 \) Copy content Toggle raw display
$17$ \( T^{2} - 5186 T + 13447298 \) Copy content Toggle raw display
$19$ \( T^{2} + 2958400 \) Copy content Toggle raw display
$23$ \( T^{2} - 4274 T + 9133538 \) Copy content Toggle raw display
$29$ \( T^{2} + 931470400 \) Copy content Toggle raw display
$31$ \( (T + 37838)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 2754749538 \) Copy content Toggle raw display
$41$ \( (T + 35438)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 3069674658 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 18123414498 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 2594448578 \) Copy content Toggle raw display
$59$ \( T^{2} + 1293121600 \) Copy content Toggle raw display
$61$ \( (T - 83322)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 7401307778 \) Copy content Toggle raw display
$71$ \( (T + 40318)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 33293869058 \) Copy content Toggle raw display
$79$ \( T^{2} + 275247129600 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 26185245858 \) Copy content Toggle raw display
$89$ \( T^{2} + 35073798400 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 567822011778 \) Copy content Toggle raw display
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