Properties

Label 15.10.a.d
Level $15$
Weight $10$
Character orbit 15.a
Self dual yes
Analytic conductor $7.726$
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] = [15,10,Mod(1,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.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: \(7.72553754246\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{241})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 15) q^{2} - 81 q^{3} + ( - 31 \beta + 255) q^{4} + 625 q^{5} + (81 \beta - 1215) q^{6} + ( - 224 \beta + 6944) q^{7} + ( - 239 \beta + 12947) q^{8} + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 15) q^{2} - 81 q^{3} + ( - 31 \beta + 255) q^{4} + 625 q^{5} + (81 \beta - 1215) q^{6} + ( - 224 \beta + 6944) q^{7} + ( - 239 \beta + 12947) q^{8} + 6561 q^{9} + ( - 625 \beta + 9375) q^{10} + (2368 \beta - 9572) q^{11} + (2511 \beta - 20655) q^{12} + (5344 \beta + 14814) q^{13} + ( - 10528 \beta + 225568) q^{14} - 50625 q^{15} + ( - 899 \beta + 193183) q^{16} + ( - 7520 \beta - 82238) q^{17} + ( - 6561 \beta + 98415) q^{18} + (5728 \beta - 45084) q^{19} + ( - 19375 \beta + 159375) q^{20} + (18144 \beta - 562464) q^{21} + (47460 \beta - 1427036) q^{22} + ( - 26272 \beta - 380768) q^{23} + (19359 \beta - 1048707) q^{24} + 390625 q^{25} + (70690 \beta - 2674238) q^{26} - 531441 q^{27} + ( - 279328 \beta + 5534368) q^{28} + (168576 \beta - 1254818) q^{29} + (50625 \beta - 759375) q^{30} + (152736 \beta + 5467584) q^{31} + ( - 85199 \beta - 3243861) q^{32} + ( - 191808 \beta + 775332) q^{33} + ( - 38082 \beta + 2842270) q^{34} + ( - 140000 \beta + 4340000) q^{35} + ( - 203391 \beta + 1673055) q^{36} + (198496 \beta + 11083414) q^{37} + (136732 \beta - 3780836) q^{38} + ( - 432864 \beta - 1199934) q^{39} + ( - 149375 \beta + 8091875) q^{40} + (492096 \beta + 13276234) q^{41} + (852768 \beta - 18271008) q^{42} + (702336 \beta - 3244412) q^{43} + (973980 \beta - 42227996) q^{44} + 4100625 q^{45} + ( - 39584 \beta + 8527904) q^{46} + ( - 1970528 \beta - 16775384) q^{47} + (72819 \beta - 15647823) q^{48} + ( - 3161088 \beta + 35060921) q^{49} + ( - 390625 \beta + 5859375) q^{50} + (609120 \beta + 6661278) q^{51} + (1069150 \beta - 86012318) q^{52} + (177728 \beta + 1654422) q^{53} + (531441 \beta - 7971615) q^{54} + (1480000 \beta - 5982500) q^{55} + ( - 4613280 \beta + 118920480) q^{56} + ( - 463968 \beta + 3651804) q^{57} + (3952034 \beta - 110190462) q^{58} + (4348352 \beta - 15573156) q^{59} + (1569375 \beta - 12909375) q^{60} + ( - 950208 \beta + 170273566) q^{61} + ( - 3023808 \beta - 769152) q^{62} + ( - 1469664 \beta + 45559584) q^{63} + (2340965 \beta - 101389753) q^{64} + (3340000 \beta + 9258750) q^{65} + ( - 3844260 \beta + 115589916) q^{66} + ( - 1026560 \beta - 144611188) q^{67} + (398658 \beta + 105380350) q^{68} + (2128032 \beta + 30842208) q^{69} + ( - 6580000 \beta + 140980000) q^{70} + (4545280 \beta + 107415672) q^{71} + ( - 1568079 \beta + 84945267) q^{72} + ( - 1168192 \beta - 116915638) q^{73} + ( - 7907478 \beta + 58666378) q^{74} - 31640625 q^{75} + (3035812 \beta - 107738276) q^{76} + (19117952 \beta - 353962112) q^{77} + ( - 5725890 \beta + 216613278) q^{78} + ( - 19049120 \beta - 21902080) q^{79} + ( - 561875 \beta + 120739375) q^{80} + 43046721 q^{81} + ( - 5402698 \beta - 67572522) q^{82} + ( - 7260288 \beta - 189671220) q^{83} + (22625568 \beta - 448283808) q^{84} + ( - 4700000 \beta - 51398750) q^{85} + (14481788 \beta - 429332292) q^{86} + ( - 13654656 \beta + 101640258) q^{87} + (33512156 \beta - 430674668) q^{88} + ( - 9049152 \beta - 218344134) q^{89} + ( - 4100625 \beta + 61509375) q^{90} + (34987456 \beta - 545935936) q^{91} + (4290016 \beta + 344326304) q^{92} + ( - 12371616 \beta - 442874304) q^{93} + ( - 14753064 \beta + 816395416) q^{94} + (3580000 \beta - 28177500) q^{95} + (6901119 \beta + 262752741) q^{96} + (13450880 \beta + 892554882) q^{97} + ( - 85638329 \beta + 2239223511) q^{98} + (15536448 \beta - 62801892) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 31 q^{2} - 162 q^{3} + 541 q^{4} + 1250 q^{5} - 2511 q^{6} + 14112 q^{7} + 26133 q^{8} + 13122 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 31 q^{2} - 162 q^{3} + 541 q^{4} + 1250 q^{5} - 2511 q^{6} + 14112 q^{7} + 26133 q^{8} + 13122 q^{9} + 19375 q^{10} - 21512 q^{11} - 43821 q^{12} + 24284 q^{13} + 461664 q^{14} - 101250 q^{15} + 387265 q^{16} - 156956 q^{17} + 203391 q^{18} - 95896 q^{19} + 338125 q^{20} - 1143072 q^{21} - 2901532 q^{22} - 735264 q^{23} - 2116773 q^{24} + 781250 q^{25} - 5419166 q^{26} - 1062882 q^{27} + 11348064 q^{28} - 2678212 q^{29} - 1569375 q^{30} + 10782432 q^{31} - 6402523 q^{32} + 1742472 q^{33} + 5722622 q^{34} + 8820000 q^{35} + 3549501 q^{36} + 21968332 q^{37} - 7698404 q^{38} - 1967004 q^{39} + 16333125 q^{40} + 26060372 q^{41} - 37394784 q^{42} - 7191160 q^{43} - 85429972 q^{44} + 8201250 q^{45} + 17095392 q^{46} - 31580240 q^{47} - 31368465 q^{48} + 73282930 q^{49} + 12109375 q^{50} + 12713436 q^{51} - 173093786 q^{52} + 3131116 q^{53} - 16474671 q^{54} - 13445000 q^{55} + 242454240 q^{56} + 7767576 q^{57} - 224332958 q^{58} - 35494664 q^{59} - 27388125 q^{60} + 341497340 q^{61} + 1485504 q^{62} + 92588832 q^{63} - 205120471 q^{64} + 15177500 q^{65} + 235024092 q^{66} - 288195816 q^{67} + 210362042 q^{68} + 59556384 q^{69} + 288540000 q^{70} + 210286064 q^{71} + 171458613 q^{72} - 232663084 q^{73} + 125240234 q^{74} - 63281250 q^{75} - 218512364 q^{76} - 727042176 q^{77} + 438952446 q^{78} - 24755040 q^{79} + 242040625 q^{80} + 86093442 q^{81} - 129742346 q^{82} - 372082152 q^{83} - 919193184 q^{84} - 98097500 q^{85} - 873146372 q^{86} + 216935172 q^{87} - 894861492 q^{88} - 427639116 q^{89} + 127119375 q^{90} - 1126859328 q^{91} + 684362592 q^{92} - 873376992 q^{93} + 1647543896 q^{94} - 59935000 q^{95} + 518604363 q^{96} + 1771658884 q^{97} + 4564085351 q^{98} - 141140232 q^{99}+O(q^{100}) \) 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
8.26209
−7.26209
−7.78626 −81.0000 −451.374 625.000 630.687 1839.88 7501.08 6561.00 −4866.41
1.2 38.7863 −81.0000 992.374 625.000 −3141.69 12272.1 18631.9 6561.00 24241.4
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)

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 15.10.a.d 2
3.b odd 2 1 45.10.a.d 2
4.b odd 2 1 240.10.a.r 2
5.b even 2 1 75.10.a.f 2
5.c odd 4 2 75.10.b.f 4
15.d odd 2 1 225.10.a.k 2
15.e even 4 2 225.10.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.d 2 1.a even 1 1 trivial
45.10.a.d 2 3.b odd 2 1
75.10.a.f 2 5.b even 2 1
75.10.b.f 4 5.c odd 4 2
225.10.a.k 2 15.d odd 2 1
225.10.b.i 4 15.e even 4 2
240.10.a.r 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 31T_{2} - 302 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(15))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 31T - 302 \) Copy content Toggle raw display
$3$ \( (T + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T - 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 14112 T + 22579200 \) Copy content Toggle raw display
$11$ \( T^{2} + 21512 T - 2924934128 \) Copy content Toggle raw display
$13$ \( T^{2} - 24284 T - 15338329532 \) Copy content Toggle raw display
$17$ \( T^{2} + 156956 T - 24505657916 \) Copy content Toggle raw display
$19$ \( T^{2} + 95896 T - 15492203120 \) Copy content Toggle raw display
$23$ \( T^{2} + 735264 T - 239117414400 \) Copy content Toggle raw display
$29$ \( T^{2} + 2678212 T - 13616383922300 \) Copy content Toggle raw display
$31$ \( T^{2} - 10782432 T + 16415447040000 \) Copy content Toggle raw display
$37$ \( T^{2} - 21968332 T + 99286893737380 \) Copy content Toggle raw display
$41$ \( T^{2} - 26060372 T + 38475315093220 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 254550637865456 \) Copy content Toggle raw display
$47$ \( T^{2} + 31580240 T - 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} - 3131116 T - 14677210114460 \) Copy content Toggle raw display
$59$ \( T^{2} + 35494664 T - 99\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{2} - 341497340 T + 28\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{2} + 288195816 T + 20\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 147594805309376 \) Copy content Toggle raw display
$73$ \( T^{2} + 232663084 T + 12\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{2} + 24755040 T - 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 372082152 T + 60\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{2} + 427639116 T + 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} - 1771658884 T + 68\!\cdots\!64 \) Copy content Toggle raw display
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