Properties

Label 15.12.a.b
Level $15$
Weight $12$
Character orbit 15.a
Self dual yes
Analytic conductor $11.525$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.5251477084\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1609}) \)
Defining polynomial: \( x^{2} - x - 402 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{1609}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 11) q^{2} + 243 q^{3} + (22 \beta - 318) q^{4} - 3125 q^{5} + ( - 243 \beta - 2673) q^{6} + (1276 \beta - 5432) q^{7} + (2124 \beta - 9372) q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 11) q^{2} + 243 q^{3} + (22 \beta - 318) q^{4} - 3125 q^{5} + ( - 243 \beta - 2673) q^{6} + (1276 \beta - 5432) q^{7} + (2124 \beta - 9372) q^{8} + 59049 q^{9} + (3125 \beta + 34375) q^{10} + ( - 10384 \beta - 180896) q^{11} + (5346 \beta - 77274) q^{12} + (1916 \beta - 1066866) q^{13} + ( - 8604 \beta - 1993332) q^{14} - 759375 q^{15} + ( - 59048 \beta - 2663160) q^{16} + ( - 67540 \beta - 3902294) q^{17} + ( - 59049 \beta - 649539) q^{18} + (208364 \beta - 7781112) q^{19} + ( - 68750 \beta + 993750) q^{20} + (310068 \beta - 1319976) q^{21} + (295120 \beta + 18697712) q^{22} + ( - 692028 \beta - 18725124) q^{23} + (516132 \beta - 2277396) q^{24} + 9765625 q^{25} + (1045790 \beta + 8652682) q^{26} + 14348907 q^{27} + ( - 525272 \beta + 46895224) q^{28} + ( - 3089792 \beta - 35160334) q^{29} + (759375 \beta + 8353125) q^{30} + ( - 1112588 \beta + 149292436) q^{31} + ( - 1037264 \beta + 143496848) q^{32} + ( - 2523312 \beta - 43957728) q^{33} + (4645234 \beta + 151597094) q^{34} + ( - 3987500 \beta + 16975000) q^{35} + (1299078 \beta - 18777582) q^{36} + (9027036 \beta + 118000478) q^{37} + (5489108 \beta - 249665444) q^{38} + (465588 \beta - 259248438) q^{39} + ( - 6637500 \beta + 29287500) q^{40} + (22327912 \beta - 232471294) q^{41} + ( - 2090772 \beta - 484379676) q^{42} + ( - 41066656 \beta - 121104300) q^{43} + ( - 677600 \beta - 310047904) q^{44} - 184528125 q^{45} + (26337432 \beta + 1319449416) q^{46} + (19806332 \beta - 2187898460) q^{47} + ( - 14348664 \beta - 647147880) q^{48} + ( - 13862464 \beta + 671915065) q^{49} + ( - 9765625 \beta - 107421875) q^{50} + ( - 16412220 \beta - 948257442) q^{51} + ( - 24080340 \beta + 407085956) q^{52} + ( - 29631928 \beta - 1094770694) q^{53} + ( - 14348907 \beta - 157837977) q^{54} + (32450000 \beta + 565300000) q^{55} + ( - 23496240 \beta + 4411659120) q^{56} + (50632452 \beta - 1890810216) q^{57} + (69148046 \beta + 5358239002) q^{58} + (146298896 \beta - 2740192928) q^{59} + ( - 16706250 \beta + 241481250) q^{60} + ( - 14026936 \beta + 7278951990) q^{61} + ( - 137053968 \beta + 147937296) q^{62} + (75346524 \beta - 320754168) q^{63} + ( - 11156640 \beta + 5544644128) q^{64} + ( - 5987500 \beta + 3333956250) q^{65} + (71714160 \beta + 4543544016) q^{66} + (50570080 \beta - 7959194444) q^{67} + ( - 64372748 \beta - 1149851428) q^{68} + ( - 168162804 \beta - 4550205132) q^{69} + (26887500 \beta + 6229162500) q^{70} + ( - 295730240 \beta + 560280512) q^{71} + (125420076 \beta - 553407228) q^{72} + (71528792 \beta - 12260787174) q^{73} + ( - 217297874 \beta - 15822506182) q^{74} + 2373046875 q^{75} + ( - 237444216 \beta + 9850062488) q^{76} + ( - 174417408 \beta - 20336597184) q^{77} + (254126970 \beta + 2102601726) q^{78} + (90478060 \beta - 39621527780) q^{79} + (184525000 \beta + 8322375000) q^{80} + 3486784401 q^{81} + ( - 13135738 \beta - 33368426174) q^{82} + (1383234048 \beta + 4622613348) q^{83} + ( - 127641096 \beta + 11395539432) q^{84} + (211062500 \beta + 12194668750) q^{85} + (572837516 \beta + 67408396804) q^{86} + ( - 750819456 \beta - 8543961162) q^{87} + ( - 286904256 \beta - 33792128832) q^{88} + ( - 312314856 \beta + 11058660618) q^{89} + (184528125 \beta + 2029809375) q^{90} + ( - 1371728728 \beta + 9728925056) q^{91} + ( - 191887824 \beta - 18541817712) q^{92} + ( - 270358884 \beta + 36278061948) q^{93} + (1970028808 \beta - 7801505128) q^{94} + ( - 651137500 \beta + 24315975000) q^{95} + ( - 252055152 \beta + 34869734064) q^{96} + (127385280 \beta - 80181836734) q^{97} + ( - 519427961 \beta + 14913638861) q^{98} + ( - 613164816 \beta - 10681727904) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 22 q^{2} + 486 q^{3} - 636 q^{4} - 6250 q^{5} - 5346 q^{6} - 10864 q^{7} - 18744 q^{8} + 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 22 q^{2} + 486 q^{3} - 636 q^{4} - 6250 q^{5} - 5346 q^{6} - 10864 q^{7} - 18744 q^{8} + 118098 q^{9} + 68750 q^{10} - 361792 q^{11} - 154548 q^{12} - 2133732 q^{13} - 3986664 q^{14} - 1518750 q^{15} - 5326320 q^{16} - 7804588 q^{17} - 1299078 q^{18} - 15562224 q^{19} + 1987500 q^{20} - 2639952 q^{21} + 37395424 q^{22} - 37450248 q^{23} - 4554792 q^{24} + 19531250 q^{25} + 17305364 q^{26} + 28697814 q^{27} + 93790448 q^{28} - 70320668 q^{29} + 16706250 q^{30} + 298584872 q^{31} + 286993696 q^{32} - 87915456 q^{33} + 303194188 q^{34} + 33950000 q^{35} - 37555164 q^{36} + 236000956 q^{37} - 499330888 q^{38} - 518496876 q^{39} + 58575000 q^{40} - 464942588 q^{41} - 968759352 q^{42} - 242208600 q^{43} - 620095808 q^{44} - 369056250 q^{45} + 2638898832 q^{46} - 4375796920 q^{47} - 1294295760 q^{48} + 1343830130 q^{49} - 214843750 q^{50} - 1896514884 q^{51} + 814171912 q^{52} - 2189541388 q^{53} - 315675954 q^{54} + 1130600000 q^{55} + 8823318240 q^{56} - 3781620432 q^{57} + 10716478004 q^{58} - 5480385856 q^{59} + 482962500 q^{60} + 14557903980 q^{61} + 295874592 q^{62} - 641508336 q^{63} + 11089288256 q^{64} + 6667912500 q^{65} + 9087088032 q^{66} - 15918388888 q^{67} - 2299702856 q^{68} - 9100410264 q^{69} + 12458325000 q^{70} + 1120561024 q^{71} - 1106814456 q^{72} - 24521574348 q^{73} - 31645012364 q^{74} + 4746093750 q^{75} + 19700124976 q^{76} - 40673194368 q^{77} + 4205203452 q^{78} - 79243055560 q^{79} + 16644750000 q^{80} + 6973568802 q^{81} - 66736852348 q^{82} + 9245226696 q^{83} + 22791078864 q^{84} + 24389337500 q^{85} + 134816793608 q^{86} - 17087922324 q^{87} - 67584257664 q^{88} + 22117321236 q^{89} + 4059618750 q^{90} + 19457850112 q^{91} - 37083635424 q^{92} + 72556123896 q^{93} - 15603010256 q^{94} + 48631950000 q^{95} + 69739468128 q^{96} - 160363673468 q^{97} + 29827277722 q^{98} - 21363455808 q^{99}+O(q^{100}) \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
20.5562
−19.5562
−51.1123 243.000 564.472 −3125.00 −12420.3 45751.3 75826.6 59049.0 159726.
1.2 29.1123 243.000 −1200.47 −3125.00 7074.30 −56615.3 −94570.6 59049.0 −90976.1
\(n\): 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 15.12.a.b 2
3.b odd 2 1 45.12.a.e 2
4.b odd 2 1 240.12.a.j 2
5.b even 2 1 75.12.a.d 2
5.c odd 4 2 75.12.b.c 4
15.d odd 2 1 225.12.a.g 2
15.e even 4 2 225.12.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.12.a.b 2 1.a even 1 1 trivial
45.12.a.e 2 3.b odd 2 1
75.12.a.d 2 5.b even 2 1
75.12.b.c 4 5.c odd 4 2
225.12.a.g 2 15.d odd 2 1
225.12.b.h 4 15.e even 4 2
240.12.a.j 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 22T_{2} - 1488 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 22T - 1488 \) Copy content Toggle raw display
$3$ \( (T - 243)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 10864 T - 2590228560 \) Copy content Toggle raw display
$11$ \( T^{2} + 361792 T - 140771013888 \) Copy content Toggle raw display
$13$ \( T^{2} + 2133732 T + 1132296332852 \) Copy content Toggle raw display
$17$ \( T^{2} + 7804588 T + 7888201038036 \) Copy content Toggle raw display
$19$ \( T^{2} + 15562224 T - 9309926445520 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 419924260414080 \) Copy content Toggle raw display
$29$ \( T^{2} + 70320668 T - 14\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{2} - 298584872 T + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} - 236000956 T - 11\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + 464942588 T - 74\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{2} + 242208600 T - 26\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + 4375796920 T + 41\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{2} + 2189541388 T - 21\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} + 5480385856 T - 26\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{2} - 14557903980 T + 52\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + 15918388888 T + 59\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} - 1120561024 T - 14\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + 24521574348 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + 79243055560 T + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} - 9245226696 T - 30\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{2} - 22117321236 T - 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + 160363673468 T + 64\!\cdots\!56 \) Copy content Toggle raw display
show more
show less