Properties

Label 10.12.a.d
Level $10$
Weight $12$
Character orbit 10.a
Self dual yes
Analytic conductor $7.683$
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] = [10,12,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(7.68343180560\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1969}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 492 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
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 \(\beta = 10\sqrt{1969}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 32 q^{2} + ( - \beta + 302) q^{3} + 1024 q^{4} + 3125 q^{5} + ( - 32 \beta + 9664) q^{6} + (177 \beta + 7046) q^{7} + 32768 q^{8} + ( - 604 \beta + 110957) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 q^{2} + ( - \beta + 302) q^{3} + 1024 q^{4} + 3125 q^{5} + ( - 32 \beta + 9664) q^{6} + (177 \beta + 7046) q^{7} + 32768 q^{8} + ( - 604 \beta + 110957) q^{9} + 100000 q^{10} + (1254 \beta + 210792) q^{11} + ( - 1024 \beta + 309248) q^{12} + ( - 1452 \beta + 865262) q^{13} + (5664 \beta + 225472) q^{14} + ( - 3125 \beta + 943750) q^{15} + 1048576 q^{16} + (5028 \beta - 3161814) q^{17} + ( - 19328 \beta + 3550624) q^{18} + ( - 7404 \beta - 14448700) q^{19} + 3200000 q^{20} + (46408 \beta - 32723408) q^{21} + (40128 \beta + 6745344) q^{22} + ( - 39081 \beta - 22618038) q^{23} + ( - 32768 \beta + 9895936) q^{24} + 9765625 q^{25} + ( - 46464 \beta + 27688384) q^{26} + ( - 116218 \beta + 98938220) q^{27} + (181248 \beta + 7215104) q^{28} + (107592 \beta + 29113110) q^{29} + ( - 100000 \beta + 30200000) q^{30} + ( - 113442 \beta + 20706692) q^{31} + 33554432 q^{32} + (167916 \beta - 183253416) q^{33} + (160896 \beta - 101178048) q^{34} + (553125 \beta + 22018750) q^{35} + ( - 618496 \beta + 113619968) q^{36} + ( - 303192 \beta + 188627726) q^{37} + ( - 236928 \beta - 462358400) q^{38} + ( - 1303766 \beta + 547207924) q^{39} + 102400000 q^{40} + (1168812 \beta - 392635518) q^{41} + (1485056 \beta - 1047149056) q^{42} + (986667 \beta - 726493618) q^{43} + (1284096 \beta + 215851008) q^{44} + ( - 1887500 \beta + 346740625) q^{45} + ( - 1250592 \beta - 723777216) q^{46} + ( - 2086863 \beta - 644063874) q^{47} + ( - 1048576 \beta + 316669952) q^{48} + (2494284 \beta + 4240999473) q^{49} + 312500000 q^{50} + (4680270 \beta - 1944881028) q^{51} + ( - 1486848 \beta + 886028288) q^{52} + ( - 9183012 \beta - 15245418) q^{53} + ( - 3718976 \beta + 3166023040) q^{54} + (3918750 \beta + 658725000) q^{55} + (5799936 \beta + 230883328) q^{56} + (12212692 \beta - 2905659800) q^{57} + (3442944 \beta + 931619520) q^{58} + ( - 1261416 \beta + 4338551220) q^{59} + ( - 3200000 \beta + 966400000) q^{60} + ( - 22983984 \beta + 557749382) q^{61} + ( - 3630144 \beta + 662614144) q^{62} + (15383605 \beta - 20268382178) q^{63} + 1073741824 q^{64} + ( - 4537500 \beta + 2703943750) q^{65} + (5373312 \beta - 5864109312) q^{66} + ( - 22637439 \beta - 6336884854) q^{67} + (5148672 \beta - 3237697536) q^{68} + (10815576 \beta + 864401424) q^{69} + (17700000 \beta + 704600000) q^{70} + (7768566 \beta + 6899916492) q^{71} + ( - 19791872 \beta + 3635838976) q^{72} + (29730708 \beta - 8921039758) q^{73} + ( - 9702144 \beta + 6036087232) q^{74} + ( - 9765625 \beta + 2949218750) q^{75} + ( - 7581696 \beta - 14795468800) q^{76} + (46145868 \beta + 45188770632) q^{77} + ( - 41720512 \beta + 17510653568) q^{78} + ( - 30074724 \beta - 6318465160) q^{79} + 3276800000 q^{80} + ( - 27039268 \beta + 33106966961) q^{81} + (37401984 \beta - 12564336576) q^{82} + ( - 77912265 \beta + 20993244462) q^{83} + (47521792 \beta - 33508769792) q^{84} + (15712500 \beta - 9880668750) q^{85} + (31573344 \beta - 23247795776) q^{86} + (3379674 \beta - 12392705580) q^{87} + (41091072 \beta + 6907232256) q^{88} + (35835480 \beta + 6604370010) q^{89} + ( - 60400000 \beta + 11095700000) q^{90} + (142920582 \beta - 44507451548) q^{91} + ( - 40018944 \beta - 23160870912) q^{92} + ( - 54966176 \beta + 28590150784) q^{93} + ( - 66779616 \beta - 20610043968) q^{94} + ( - 23137500 \beta - 45152187500) q^{95} + ( - 33554432 \beta + 10133438464) q^{96} + ( - 175966980 \beta - 30893731414) q^{97} + (79817088 \beta + 135711983136) q^{98} + (11821710 \beta - 125746362456) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{2} + 604 q^{3} + 2048 q^{4} + 6250 q^{5} + 19328 q^{6} + 14092 q^{7} + 65536 q^{8} + 221914 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{2} + 604 q^{3} + 2048 q^{4} + 6250 q^{5} + 19328 q^{6} + 14092 q^{7} + 65536 q^{8} + 221914 q^{9} + 200000 q^{10} + 421584 q^{11} + 618496 q^{12} + 1730524 q^{13} + 450944 q^{14} + 1887500 q^{15} + 2097152 q^{16} - 6323628 q^{17} + 7101248 q^{18} - 28897400 q^{19} + 6400000 q^{20} - 65446816 q^{21} + 13490688 q^{22} - 45236076 q^{23} + 19791872 q^{24} + 19531250 q^{25} + 55376768 q^{26} + 197876440 q^{27} + 14430208 q^{28} + 58226220 q^{29} + 60400000 q^{30} + 41413384 q^{31} + 67108864 q^{32} - 366506832 q^{33} - 202356096 q^{34} + 44037500 q^{35} + 227239936 q^{36} + 377255452 q^{37} - 924716800 q^{38} + 1094415848 q^{39} + 204800000 q^{40} - 785271036 q^{41} - 2094298112 q^{42} - 1452987236 q^{43} + 431702016 q^{44} + 693481250 q^{45} - 1447554432 q^{46} - 1288127748 q^{47} + 633339904 q^{48} + 8481998946 q^{49} + 625000000 q^{50} - 3889762056 q^{51} + 1772056576 q^{52} - 30490836 q^{53} + 6332046080 q^{54} + 1317450000 q^{55} + 461766656 q^{56} - 5811319600 q^{57} + 1863239040 q^{58} + 8677102440 q^{59} + 1932800000 q^{60} + 1115498764 q^{61} + 1325228288 q^{62} - 40536764356 q^{63} + 2147483648 q^{64} + 5407887500 q^{65} - 11728218624 q^{66} - 12673769708 q^{67} - 6475395072 q^{68} + 1728802848 q^{69} + 1409200000 q^{70} + 13799832984 q^{71} + 7271677952 q^{72} - 17842079516 q^{73} + 12072174464 q^{74} + 5898437500 q^{75} - 29590937600 q^{76} + 90377541264 q^{77} + 35021307136 q^{78} - 12636930320 q^{79} + 6553600000 q^{80} + 66213933922 q^{81} - 25128673152 q^{82} + 41986488924 q^{83} - 67017539584 q^{84} - 19761337500 q^{85} - 46495591552 q^{86} - 24785411160 q^{87} + 13814464512 q^{88} + 13208740020 q^{89} + 22191400000 q^{90} - 89014903096 q^{91} - 46321741824 q^{92} + 57180301568 q^{93} - 41220087936 q^{94} - 90304375000 q^{95} + 20266876928 q^{96} - 61787462828 q^{97} + 271423966272 q^{98} - 251492724912 q^{99}+O(q^{100}) \) 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
22.6867
−21.6867
32.0000 −141.734 1024.00 3125.00 −4535.49 85586.9 32768.0 −157058. 100000.
1.2 32.0000 745.734 1024.00 3125.00 23863.5 −71494.9 32768.0 378972. 100000.
\(n\): 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 10.12.a.d 2
3.b odd 2 1 90.12.a.l 2
4.b odd 2 1 80.12.a.g 2
5.b even 2 1 50.12.a.f 2
5.c odd 4 2 50.12.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.d 2 1.a even 1 1 trivial
50.12.a.f 2 5.b even 2 1
50.12.b.f 4 5.c odd 4 2
80.12.a.g 2 4.b odd 2 1
90.12.a.l 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 604T_{3} - 105696 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 604T - 105696 \) Copy content Toggle raw display
$5$ \( (T - 3125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 6119033984 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 265195133136 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 333553271044 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 5019281400996 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 197971028059600 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 210845436908544 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 21\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 44\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 16\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 60\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 94\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 75\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 51\!\cdots\!04 \) Copy content Toggle raw display
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