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Properties

Label 15.8.a.c
Level 15
Weight 8
Character orbit 15.a
Self dual Yes
Analytic conductor 4.686
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 15.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.68577538226\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{601}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{601})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 4 - \beta ) q^{2} \) \( + 27 q^{3} \) \( + ( 38 - 7 \beta ) q^{4} \) \( + 125 q^{5} \) \( + ( 108 - 27 \beta ) q^{6} \) \( + ( 624 + 56 \beta ) q^{7} \) \( + ( 690 + 69 \beta ) q^{8} \) \( + 729 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 4 - \beta ) q^{2} \) \( + 27 q^{3} \) \( + ( 38 - 7 \beta ) q^{4} \) \( + 125 q^{5} \) \( + ( 108 - 27 \beta ) q^{6} \) \( + ( 624 + 56 \beta ) q^{7} \) \( + ( 690 + 69 \beta ) q^{8} \) \( + 729 q^{9} \) \( + ( 500 - 125 \beta ) q^{10} \) \( + ( 1492 + 464 \beta ) q^{11} \) \( + ( 1026 - 189 \beta ) q^{12} \) \( + ( -4082 - 824 \beta ) q^{13} \) \( + ( -5904 - 456 \beta ) q^{14} \) \( + 3375 q^{15} \) \( + ( -12454 + 413 \beta ) q^{16} \) \( + ( -3446 + 1400 \beta ) q^{17} \) \( + ( 2916 - 729 \beta ) q^{18} \) \( + ( -25820 + 2056 \beta ) q^{19} \) \( + ( 4750 - 875 \beta ) q^{20} \) \( + ( 16848 + 1512 \beta ) q^{21} \) \( + ( -63632 - 100 \beta ) q^{22} \) \( + ( 48528 - 5208 \beta ) q^{23} \) \( + ( 18630 + 1863 \beta ) q^{24} \) \( + 15625 q^{25} \) \( + ( 107272 + 1610 \beta ) q^{26} \) \( + 19683 q^{27} \) \( + ( -35088 - 2632 \beta ) q^{28} \) \( + ( 95030 - 8288 \beta ) q^{29} \) \( + ( 13500 - 3375 \beta ) q^{30} \) \( + ( 151032 + 2168 \beta ) q^{31} \) \( + ( -200086 + 4861 \beta ) q^{32} \) \( + ( 40284 + 12528 \beta ) q^{33} \) \( + ( -223784 + 7646 \beta ) q^{34} \) \( + ( 78000 + 7000 \beta ) q^{35} \) \( + ( 27702 - 5103 \beta ) q^{36} \) \( + ( -243946 - 14424 \beta ) q^{37} \) \( + ( -411680 + 31988 \beta ) q^{38} \) \( + ( -110214 - 22248 \beta ) q^{39} \) \( + ( 86250 + 8625 \beta ) q^{40} \) \( + ( 326282 - 21392 \beta ) q^{41} \) \( + ( -159408 - 12312 \beta ) q^{42} \) \( + ( 180388 - 7136 \beta ) q^{43} \) \( + ( -430504 + 3940 \beta ) q^{44} \) \( + 91125 q^{45} \) \( + ( 975312 - 64152 \beta ) q^{46} \) \( + ( -236696 + 5912 \beta ) q^{47} \) \( + ( -336258 + 11151 \beta ) q^{48} \) \( + ( 36233 + 73024 \beta ) q^{49} \) \( + ( 62500 - 15625 \beta ) q^{50} \) \( + ( -93042 + 37800 \beta ) q^{51} \) \( + ( 710084 + 3030 \beta ) q^{52} \) \( + ( -290642 + 13232 \beta ) q^{53} \) \( + ( 78732 - 19683 \beta ) q^{54} \) \( + ( 186500 + 58000 \beta ) q^{55} \) \( + ( 1010160 + 85560 \beta ) q^{56} \) \( + ( -697140 + 55512 \beta ) q^{57} \) \( + ( 1623320 - 119894 \beta ) q^{58} \) \( + ( 218500 - 149776 \beta ) q^{59} \) \( + ( 128250 - 23625 \beta ) q^{60} \) \( + ( -1257818 + 1456 \beta ) q^{61} \) \( + ( 278928 - 144528 \beta ) q^{62} \) \( + ( 454896 + 40824 \beta ) q^{63} \) \( + ( 64618 + 161805 \beta ) q^{64} \) \( + ( -510250 - 103000 \beta ) q^{65} \) \( + ( -1718064 - 2700 \beta ) q^{66} \) \( + ( -2601876 + 129920 \beta ) q^{67} \) \( + ( -1600948 + 67522 \beta ) q^{68} \) \( + ( 1310256 - 140616 \beta ) q^{69} \) \( + ( -738000 - 57000 \beta ) q^{70} \) \( + ( -1912648 + 76480 \beta ) q^{71} \) \( + ( 503010 + 50301 \beta ) q^{72} \) \( + ( -544502 - 388208 \beta ) q^{73} \) \( + ( 1187816 + 200674 \beta ) q^{74} \) \( + 421875 q^{75} \) \( + ( -3139960 + 244476 \beta ) q^{76} \) \( + ( 4828608 + 399072 \beta ) q^{77} \) \( + ( 2896344 + 43470 \beta ) q^{78} \) \( + ( -2273640 - 80440 \beta ) q^{79} \) \( + ( -1556750 + 51625 \beta ) q^{80} \) \( + 531441 q^{81} \) \( + ( 4513928 - 390458 \beta ) q^{82} \) \( + ( -2990532 - 91872 \beta ) q^{83} \) \( + ( -947376 - 71064 \beta ) q^{84} \) \( + ( -430750 + 175000 \beta ) q^{85} \) \( + ( 1791952 - 201796 \beta ) q^{86} \) \( + ( 2565810 - 223776 \beta ) q^{87} \) \( + ( 5831880 + 455124 \beta ) q^{88} \) \( + ( 8247930 + 20496 \beta ) q^{89} \) \( + ( 364500 - 91125 \beta ) q^{90} \) \( + ( -9468768 - 788912 \beta ) q^{91} \) \( + ( 7312464 - 501144 \beta ) q^{92} \) \( + ( 4077864 + 58536 \beta ) q^{93} \) \( + ( -1833584 + 254432 \beta ) q^{94} \) \( + ( -3227500 + 257000 \beta ) q^{95} \) \( + ( -5402322 + 131247 \beta ) q^{96} \) \( + ( 1147394 + 428640 \beta ) q^{97} \) \( + ( -10808668 + 182839 \beta ) q^{98} \) \( + ( 1087668 + 338256 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 69q^{4} \) \(\mathstrut +\mathstrut 250q^{5} \) \(\mathstrut +\mathstrut 189q^{6} \) \(\mathstrut +\mathstrut 1304q^{7} \) \(\mathstrut +\mathstrut 1449q^{8} \) \(\mathstrut +\mathstrut 1458q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 69q^{4} \) \(\mathstrut +\mathstrut 250q^{5} \) \(\mathstrut +\mathstrut 189q^{6} \) \(\mathstrut +\mathstrut 1304q^{7} \) \(\mathstrut +\mathstrut 1449q^{8} \) \(\mathstrut +\mathstrut 1458q^{9} \) \(\mathstrut +\mathstrut 875q^{10} \) \(\mathstrut +\mathstrut 3448q^{11} \) \(\mathstrut +\mathstrut 1863q^{12} \) \(\mathstrut -\mathstrut 8988q^{13} \) \(\mathstrut -\mathstrut 12264q^{14} \) \(\mathstrut +\mathstrut 6750q^{15} \) \(\mathstrut -\mathstrut 24495q^{16} \) \(\mathstrut -\mathstrut 5492q^{17} \) \(\mathstrut +\mathstrut 5103q^{18} \) \(\mathstrut -\mathstrut 49584q^{19} \) \(\mathstrut +\mathstrut 8625q^{20} \) \(\mathstrut +\mathstrut 35208q^{21} \) \(\mathstrut -\mathstrut 127364q^{22} \) \(\mathstrut +\mathstrut 91848q^{23} \) \(\mathstrut +\mathstrut 39123q^{24} \) \(\mathstrut +\mathstrut 31250q^{25} \) \(\mathstrut +\mathstrut 216154q^{26} \) \(\mathstrut +\mathstrut 39366q^{27} \) \(\mathstrut -\mathstrut 72808q^{28} \) \(\mathstrut +\mathstrut 181772q^{29} \) \(\mathstrut +\mathstrut 23625q^{30} \) \(\mathstrut +\mathstrut 304232q^{31} \) \(\mathstrut -\mathstrut 395311q^{32} \) \(\mathstrut +\mathstrut 93096q^{33} \) \(\mathstrut -\mathstrut 439922q^{34} \) \(\mathstrut +\mathstrut 163000q^{35} \) \(\mathstrut +\mathstrut 50301q^{36} \) \(\mathstrut -\mathstrut 502316q^{37} \) \(\mathstrut -\mathstrut 791372q^{38} \) \(\mathstrut -\mathstrut 242676q^{39} \) \(\mathstrut +\mathstrut 181125q^{40} \) \(\mathstrut +\mathstrut 631172q^{41} \) \(\mathstrut -\mathstrut 331128q^{42} \) \(\mathstrut +\mathstrut 353640q^{43} \) \(\mathstrut -\mathstrut 857068q^{44} \) \(\mathstrut +\mathstrut 182250q^{45} \) \(\mathstrut +\mathstrut 1886472q^{46} \) \(\mathstrut -\mathstrut 467480q^{47} \) \(\mathstrut -\mathstrut 661365q^{48} \) \(\mathstrut +\mathstrut 145490q^{49} \) \(\mathstrut +\mathstrut 109375q^{50} \) \(\mathstrut -\mathstrut 148284q^{51} \) \(\mathstrut +\mathstrut 1423198q^{52} \) \(\mathstrut -\mathstrut 568052q^{53} \) \(\mathstrut +\mathstrut 137781q^{54} \) \(\mathstrut +\mathstrut 431000q^{55} \) \(\mathstrut +\mathstrut 2105880q^{56} \) \(\mathstrut -\mathstrut 1338768q^{57} \) \(\mathstrut +\mathstrut 3126746q^{58} \) \(\mathstrut +\mathstrut 287224q^{59} \) \(\mathstrut +\mathstrut 232875q^{60} \) \(\mathstrut -\mathstrut 2514180q^{61} \) \(\mathstrut +\mathstrut 413328q^{62} \) \(\mathstrut +\mathstrut 950616q^{63} \) \(\mathstrut +\mathstrut 291041q^{64} \) \(\mathstrut -\mathstrut 1123500q^{65} \) \(\mathstrut -\mathstrut 3438828q^{66} \) \(\mathstrut -\mathstrut 5073832q^{67} \) \(\mathstrut -\mathstrut 3134374q^{68} \) \(\mathstrut +\mathstrut 2479896q^{69} \) \(\mathstrut -\mathstrut 1533000q^{70} \) \(\mathstrut -\mathstrut 3748816q^{71} \) \(\mathstrut +\mathstrut 1056321q^{72} \) \(\mathstrut -\mathstrut 1477212q^{73} \) \(\mathstrut +\mathstrut 2576306q^{74} \) \(\mathstrut +\mathstrut 843750q^{75} \) \(\mathstrut -\mathstrut 6035444q^{76} \) \(\mathstrut +\mathstrut 10056288q^{77} \) \(\mathstrut +\mathstrut 5836158q^{78} \) \(\mathstrut -\mathstrut 4627720q^{79} \) \(\mathstrut -\mathstrut 3061875q^{80} \) \(\mathstrut +\mathstrut 1062882q^{81} \) \(\mathstrut +\mathstrut 8637398q^{82} \) \(\mathstrut -\mathstrut 6072936q^{83} \) \(\mathstrut -\mathstrut 1965816q^{84} \) \(\mathstrut -\mathstrut 686500q^{85} \) \(\mathstrut +\mathstrut 3382108q^{86} \) \(\mathstrut +\mathstrut 4907844q^{87} \) \(\mathstrut +\mathstrut 12118884q^{88} \) \(\mathstrut +\mathstrut 16516356q^{89} \) \(\mathstrut +\mathstrut 637875q^{90} \) \(\mathstrut -\mathstrut 19726448q^{91} \) \(\mathstrut +\mathstrut 14123784q^{92} \) \(\mathstrut +\mathstrut 8214264q^{93} \) \(\mathstrut -\mathstrut 3412736q^{94} \) \(\mathstrut -\mathstrut 6198000q^{95} \) \(\mathstrut -\mathstrut 10673397q^{96} \) \(\mathstrut +\mathstrut 2723428q^{97} \) \(\mathstrut -\mathstrut 21434497q^{98} \) \(\mathstrut +\mathstrut 2513592q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Display 100 coefficients 10 coefficients

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
12.7577
−11.7577
−8.75765 27.0000 −51.3036 125.000 −236.457 1338.43 1570.28 729.000 −1094.71
1.2 15.7577 27.0000 120.304 125.000 425.457 −34.4284 −121.278 729.000 1969.71
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 7 T_{2} \) \(\mathstrut -\mathstrut 138 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).