Properties

Label 5.10.b.a
Level $5$
Weight $10$
Character orbit 5.b
Analytic conductor $2.575$
Analytic rank $0$
Dimension $4$
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] = [5,10,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), 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: \(2.57517918082\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.49740556.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 45x^{2} + 304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 342) q^{4} + (\beta_{3} + 7 \beta_{2} + 19 \beta_1 + 285) q^{5} + (\beta_{3} + 702) q^{6} + ( - 49 \beta_{2} + 133 \beta_1) q^{7} + (8 \beta_{2} - 684 \beta_1) q^{8} + (18 \beta_{3} + 2907) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{3} - 342) q^{4} + (\beta_{3} + 7 \beta_{2} + 19 \beta_1 + 285) q^{5} + (\beta_{3} + 702) q^{6} + ( - 49 \beta_{2} + 133 \beta_1) q^{7} + (8 \beta_{2} - 684 \beta_1) q^{8} + (18 \beta_{3} + 2907) q^{9} + ( - 19 \beta_{3} - 8 \beta_{2} + 1139 \beta_1 - 17290) q^{10} + ( - 20 \beta_{3} + 27492) q^{11} + (504 \beta_{2} + 1044 \beta_1) q^{12} + ( - 110 \beta_{2} - 1918 \beta_1) q^{13} + ( - 133 \beta_{3} - 106134) q^{14} + (152 \beta_{3} - 561 \beta_{2} - 987 \beta_1 - 99180) q^{15} + (172 \beta_{3} + 407816) q^{16} + ( - 1632 \beta_{2} - 7448 \beta_1) q^{17} + ( - 144 \beta_{2} + 18279 \beta_1) q^{18} + (476 \beta_{3} - 159220) q^{19} + ( - 627 \beta_{3} + 3736 \beta_{2} - 23788 \beta_1 - 825570) q^{20} + ( - 798 \beta_{3} + 880992) q^{21} + (160 \beta_{2} + 10412 \beta_1) q^{22} + (5199 \beta_{2} + 17005 \beta_1) q^{23} + ( - 532 \beta_{3} - 608760) q^{24} + (1140 \beta_{3} - 8270 \beta_{2} + 45410 \beta_1 - 334475) q^{25} + (1918 \beta_{3} + 1654692) q^{26} + (7362 \beta_{2} - 35226 \beta_1) q^{27} + ( - 24024 \beta_{2} - 151620 \beta_1) q^{28} + ( - 1976 \beta_{3} - 882930) q^{29} + (987 \beta_{3} - 1216 \beta_{2} + 30628 \beta_1 + 928170) q^{30} + ( - 760 \beta_{3} - 2646928) q^{31} + (2720 \beta_{2} + 204496 \beta_1) q^{32} + (44412 \beta_{2} - 13452 \beta_1) q^{33} + (7448 \beta_{3} + 6608656) q^{34} + ( - 9044 \beta_{3} + 26817 \beta_{2} + 144039 \beta_1 + 3407460) q^{35} + ( - 9063 \beta_{3} - 14099994) q^{36} + ( - 51378 \beta_{2} - 335370 \beta_1) q^{37} + ( - 3808 \beta_{2} + 247284 \beta_1) q^{38} + ( - 4008 \beta_{3} + 421704) q^{39} + (14060 \beta_{3} + 920 \beta_{2} - 777860 \beta_1 + 10894600) q^{40} + (24890 \beta_{3} - 4197138) q^{41} + (6384 \beta_{2} + 199500 \beta_1) q^{42} + ( - 132643 \beta_{2} + 719131 \beta_1) q^{43} + ( - 20652 \beta_{3} + 5159736) q^{44} + (8037 \beta_{3} - 89991 \beta_{2} + 366453 \beta_1 + 13934295) q^{45} + ( - 17005 \beta_{3} - 15312518) q^{46} + (214259 \beta_{2} - 1012111 \beta_1) q^{47} + (262304 \beta_{2} - 528560 \beta_1) q^{48} + (27930 \beta_{3} - 11730257) q^{49} + ( - 45410 \beta_{3} - 9120 \beta_{2} + 639085 \beta_1 - 37523100) q^{50} + ( - 38456 \beta_{3} + 21004272) q^{51} + ( - 71664 \beta_{2} + 2310648 \beta_1) q^{52} + ( - 259794 \beta_{2} - 1349666 \beta_1) q^{53} + (35226 \beta_{3} + 28963980) q^{54} + (21792 \beta_{3} + 315044 \beta_{2} + 176548 \beta_1 - 6726780) q^{55} + (83524 \beta_{3} + 78794520) q^{56} + ( - 561916 \beta_{2} - 174932 \beta_1) q^{57} + (15808 \beta_{2} - 2570434 \beta_1) q^{58} + ( - 52972 \beta_{3} - 115207260) q^{59} + (47196 \beta_{3} - 295128 \beta_{2} + 1265724 \beta_1 - 76751640) q^{60} + ( - 43150 \beta_{3} + 90122642) q^{61} + (6080 \beta_{2} - 3295968 \beta_1) q^{62} + (591633 \beta_{2} + 2297043 \beta_1) q^{63} + ( - 116432 \beta_{3} + 33748768) q^{64} + (21812 \beta_{3} + 77934 \beta_{2} - 2201322 \beta_1 + 45973920) q^{65} + (13452 \beta_{3} + 4737384) q^{66} + (1448953 \beta_{2} + 6669647 \beta_1) q^{67} + ( - 895168 \beta_{2} + 9155872 \beta_1) q^{68} + (115786 \beta_{3} - 71631216) q^{69} + ( - 144039 \beta_{3} + 72352 \beta_{2} + \cdots - 127085490) q^{70}+ \cdots + (436716 \beta_{3} - 182196756) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1368 q^{4} + 1140 q^{5} + 2808 q^{6} + 11628 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 1368 q^{4} + 1140 q^{5} + 2808 q^{6} + 11628 q^{9} - 69160 q^{10} + 109968 q^{11} - 424536 q^{14} - 396720 q^{15} + 1631264 q^{16} - 636880 q^{19} - 3302280 q^{20} + 3523968 q^{21} - 2435040 q^{24} - 1337900 q^{25} + 6618768 q^{26} - 3531720 q^{29} + 3712680 q^{30} - 10587712 q^{31} + 26434624 q^{34} + 13629840 q^{35} - 56399976 q^{36} + 1686816 q^{39} + 43578400 q^{40} - 16788552 q^{41} + 20638944 q^{44} + 55737180 q^{45} - 61250072 q^{46} - 46921028 q^{49} - 150092400 q^{50} + 84017088 q^{51} + 115855920 q^{54} - 26907120 q^{55} + 315178080 q^{56} - 460829040 q^{59} - 307006560 q^{60} + 360490568 q^{61} + 134995072 q^{64} + 183895680 q^{65} + 18949536 q^{66} - 286524864 q^{69} - 508341960 q^{70} - 47611872 q^{71} + 1176861744 q^{74} + 659239200 q^{75} - 1168489440 q^{76} - 728043520 q^{79} + 965843040 q^{80} - 343387836 q^{81} + 1118898144 q^{84} + 1275419840 q^{85} - 2375904552 q^{86} - 1582700760 q^{89} - 1197088920 q^{90} + 473322528 q^{91} + 3327101704 q^{94} + 1204791600 q^{95} + 399339648 q^{96} - 728787024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 45x^{2} + 304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 37\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} - 7\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 60\nu^{2} + 1350 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 1350 ) / 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -37\beta_{2} - 14\beta_1 ) / 30 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
2.87724i
6.05982i
6.05982i
2.87724i
41.3193i 37.6407i −1195.29 1138.29 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 47033.3i
4.2 0.843944i 179.263i 511.288 −568.288 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 + 479.603i
4.3 0.843944i 179.263i 511.288 −568.288 + 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 479.603i
4.4 41.3193i 37.6407i −1195.29 1138.29 + 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 + 47033.3i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.10.b.a 4
3.b odd 2 1 45.10.b.b 4
4.b odd 2 1 80.10.c.c 4
5.b even 2 1 inner 5.10.b.a 4
5.c odd 4 2 25.10.a.e 4
15.d odd 2 1 45.10.b.b 4
15.e even 4 2 225.10.a.s 4
20.d odd 2 1 80.10.c.c 4
20.e even 4 2 400.10.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 1.a even 1 1 trivial
5.10.b.a 4 5.b even 2 1 inner
25.10.a.e 4 5.c odd 4 2
45.10.b.b 4 3.b odd 2 1
45.10.b.b 4 15.d odd 2 1
80.10.c.c 4 4.b odd 2 1
80.10.c.c 4 20.d odd 2 1
225.10.a.s 4 15.e even 4 2
400.10.a.ba 4 20.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1708 T^{2} + 1216 \) Copy content Toggle raw display
$3$ \( T^{4} + 33552 T^{2} + \cdots + 45529776 \) Copy content Toggle raw display
$5$ \( T^{4} - 1140 T^{3} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( T^{4} + 104167728 T^{2} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{2} - 54984 T + 464570064)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 6804205632 T^{2} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{4} + 188571136768 T^{2} + \cdots + 88\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{2} + 318440 T - 139618977200)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1424819683312 T^{2} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1765860 T - 2063356400700)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5293856 T + 6585677277184)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 288243684243648 T^{2} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{2} + 8394276 T - 433450792618956)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 73\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{2} + 230414520 T + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 180245284 T + 67\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{2} + 23805936 T - 59\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{2} + 364021760 T - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{2} + 791350380 T + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
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