Properties

Label 15.6.e.a
Level $15$
Weight $6$
Character orbit 15.e
Analytic conductor $2.406$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.40575729719\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 10768x^{12} + 16341006x^{8} + 4217167600x^{4} + 50906640625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{10}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - 13 \beta_{3}) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 5) q^{6} + (\beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} - 5 \beta_{3} - 5) q^{7} + ( - 2 \beta_{15} + 2 \beta_{14} + \beta_{12} - \beta_{11} - \beta_{5} - \beta_{4} - 7 \beta_{2}) q^{8} + (2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} - 6 \beta_{7} + \cdots + 7 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - 13 \beta_{3}) q^{4} + ( - \beta_{14} + \beta_{9} + \beta_{2}) q^{5} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{8} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 5) q^{6} + (\beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} - 5 \beta_{3} - 5) q^{7} + ( - 2 \beta_{15} + 2 \beta_{14} + \beta_{12} - \beta_{11} - \beta_{5} - \beta_{4} - 7 \beta_{2}) q^{8} + (2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} - 6 \beta_{7} + \cdots + 7 \beta_{2}) q^{9}+ \cdots + ( - 865 \beta_{15} + 364 \beta_{14} - 1043 \beta_{13} + 865 \beta_{12} + \cdots + 2413 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 84 q^{6} - 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 84 q^{6} - 80 q^{7} + 1060 q^{10} + 420 q^{12} - 2120 q^{13} - 3000 q^{15} - 452 q^{16} + 6000 q^{18} + 1416 q^{21} + 3380 q^{22} + 9160 q^{25} - 12960 q^{27} - 10900 q^{28} - 13380 q^{30} - 21568 q^{31} + 5760 q^{33} + 61932 q^{36} + 55720 q^{37} + 21480 q^{40} - 103020 q^{42} - 23360 q^{43} - 21240 q^{45} - 151168 q^{46} + 74820 q^{48} + 138816 q^{51} + 204160 q^{52} + 89120 q^{55} - 163800 q^{57} - 163500 q^{58} - 217380 q^{60} - 175168 q^{61} + 143040 q^{63} + 263400 q^{66} + 140320 q^{67} + 385260 q^{70} - 328680 q^{72} - 240320 q^{73} - 199920 q^{75} - 271536 q^{76} + 429960 q^{78} + 326016 q^{81} + 431120 q^{82} + 2440 q^{85} - 331680 q^{87} - 401700 q^{88} - 348360 q^{90} - 487168 q^{91} - 22320 q^{93} + 311748 q^{96} + 440800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10768x^{12} + 16341006x^{8} + 4217167600x^{4} + 50906640625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -983\nu^{12} - 10530009\nu^{8} - 14985743913\nu^{4} - 1365328353575 ) / 52711868160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -193037\nu^{13} - 2063897891\nu^{9} - 3039554293747\nu^{5} - 800189026191325\nu ) / 175266961632000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65179\nu^{14} + 704329347\nu^{10} + 1093176906949\nu^{6} + 338344643638525\nu^{2} ) / 1156275094100000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17652037 \nu^{14} + 44517950 \nu^{13} - 182440375 \nu^{12} + 186662751291 \nu^{10} + 495889999350 \nu^{9} - 1876904386125 \nu^{8} + \cdots - 16\!\cdots\!75 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17652037 \nu^{14} + 44517950 \nu^{13} + 182440375 \nu^{12} - 186662751291 \nu^{10} + 495889999350 \nu^{9} + \cdots + 16\!\cdots\!75 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 40605037 \nu^{14} + 433934370291 \nu^{10} + 626362399315347 \nu^{6} + 11\!\cdots\!25 \nu^{2} ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2501702961 \nu^{15} - 26901148843423 \nu^{11} + \cdots - 10\!\cdots\!25 \nu^{3} ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2077147966 \nu^{15} - 5030830545 \nu^{14} - 16232545500 \nu^{13} - 120958465000 \nu^{12} - 22414270741638 \nu^{11} + \cdots - 57\!\cdots\!00 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2077147966 \nu^{15} + 1676943515 \nu^{14} + 17331835625 \nu^{12} - 22414270741638 \nu^{11} + 17732961372645 \nu^{10} + \cdots + 16\!\cdots\!25 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2077147966 \nu^{15} + 39891462730 \nu^{14} - 16232545500 \nu^{13} + 51995506875 \nu^{12} + 22414270741638 \nu^{11} + \cdots + 48\!\cdots\!75 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4963927071 \nu^{15} + 3353887030 \nu^{14} - 82856539500 \nu^{13} - 34663671250 \nu^{12} + 52964213316153 \nu^{11} + \cdots - 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4963927071 \nu^{15} - 3353887030 \nu^{14} + 82856539500 \nu^{13} + 34663671250 \nu^{12} + 52964213316153 \nu^{11} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2077147966 \nu^{15} + 1676943515 \nu^{14} + 17331835625 \nu^{12} + 22414270741638 \nu^{11} + 17732961372645 \nu^{10} + \cdots + 16\!\cdots\!25 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14854266548 \nu^{15} + 3353887030 \nu^{14} - 33222243375 \nu^{13} + 34663671250 \nu^{12} + 159176103876364 \nu^{11} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14854266548 \nu^{15} + 3353887030 \nu^{14} + 33222243375 \nu^{13} + 34663671250 \nu^{12} + 159176103876364 \nu^{11} + \cdots + 32\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 6\beta_{15} - 6\beta_{14} - 3\beta_{12} + 3\beta_{11} + 5\beta_{5} + 9\beta_{4} + 12\beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + 12 \beta_{6} + 5 \beta_{5} - 9 \beta_{4} + 690 \beta_{3} - 6 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{15} - 27\beta_{14} + 95\beta_{13} + 134\beta_{12} + 134\beta_{11} - 231\beta_{9} + 1068\beta_{7} ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 129 \beta_{15} - 129 \beta_{14} + 2 \beta_{13} - 129 \beta_{12} + 129 \beta_{11} + 264 \beta_{9} - 258 \beta_{8} + 260 \beta_{5} - 522 \beta_{4} - 258 \beta_{2} + 732 \beta _1 - 24045 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 31344 \beta_{15} + 31344 \beta_{14} + 1707 \beta_{12} - 1707 \beta_{11} - 25195 \beta_{5} - 72171 \beta_{4} - 350088 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2535 \beta_{15} + 2535 \beta_{14} + 685 \beta_{13} + 2535 \beta_{12} - 2535 \beta_{11} + 5070 \beta_{10} + 7125 \beta_{9} - 16332 \beta_{6} - 5755 \beta_{5} + 12195 \beta_{4} - 453290 \beta_{3} + 5070 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15171 \beta_{15} - 15171 \beta_{14} - 2388515 \beta_{13} - 2702868 \beta_{12} - 2702868 \beta_{11} + 7195887 \beta_{9} - 34754736 \beta_{7} ) / 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1048344 \beta_{15} + 1048344 \beta_{14} - 419512 \beta_{13} + 1048344 \beta_{12} - 1048344 \beta_{11} - 3355224 \beta_{9} + 2096688 \beta_{8} - 2516200 \beta_{5} + \cdots + 185581065 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 81977282 \beta_{15} - 81977282 \beta_{14} + 2406879 \beta_{12} - 2406879 \beta_{11} + 75890135 \beta_{5} + 232484163 \beta_{4} + 1117919364 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 196307187 \beta_{15} - 196307187 \beta_{14} - 89861071 \beta_{13} - 196307187 \beta_{12} + 196307187 \beta_{11} - 392614374 \beta_{10} - 662197587 \beta_{9} + \cdots - 392614374 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 887812239 \beta_{15} + 887812239 \beta_{14} + 21662418685 \beta_{13} + 22932690162 \beta_{12} + 22932690162 \beta_{11} - 66762880533 \beta_{9} + \cdots + 320332617924 \beta_{7} ) / 90 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1029265377 \beta_{15} - 1029265377 \beta_{14} + 495930106 \beta_{13} - 1029265377 \beta_{12} + 1029265377 \beta_{11} + 3546321072 \beta_{9} - 2058530754 \beta_{8} + \cdots - 181544707525 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2160764841264 \beta_{15} + 2160764841264 \beta_{14} - 91643620533 \beta_{12} + 91643620533 \beta_{11} - 2058195285995 \beta_{5} + \cdots - 30476432312328 \beta_{2} ) / 90 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1754232431151 \beta_{15} + 1754232431151 \beta_{14} + 862455839813 \beta_{13} + 1754232431151 \beta_{12} - 1754232431151 \beta_{11} + \cdots + 3508464862302 \beta_{2} ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2990689930017 \beta_{15} - 2990689930017 \beta_{14} - 65138846432705 \beta_{13} - 68153498840636 \beta_{12} - 68153498840636 \beta_{11} + \cdots - 965018230513872 \beta_{7} ) / 30 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−6.88827 6.88827i
1.33460 + 1.33460i
4.35647 + 4.35647i
2.96509 + 2.96509i
−2.96509 2.96509i
−4.35647 4.35647i
−1.33460 1.33460i
6.88827 + 6.88827i
−6.88827 + 6.88827i
1.33460 1.33460i
4.35647 4.35647i
2.96509 2.96509i
−2.96509 + 2.96509i
−4.35647 + 4.35647i
−1.33460 + 1.33460i
6.88827 6.88827i
−6.77944 + 6.77944i 3.70445 + 15.1419i 59.9217i −52.1700 20.0821i −127.768 77.5396i 47.9711 + 47.9711i 189.294 + 189.294i −215.554 + 112.185i 489.829 217.538i
2.2 −5.80199 + 5.80199i −7.07428 13.8908i 35.3261i 42.8643 35.8839i 121.639 + 39.5494i −140.332 140.332i 19.2981 + 19.2981i −142.909 + 196.535i −40.5000 + 456.896i
2.3 −3.17835 + 3.17835i 14.5679 5.54767i 11.7962i 11.9990 + 54.5987i −28.6694 + 63.9343i 83.6967 + 83.6967i −139.200 139.200i 181.447 161.636i −211.671 135.397i
2.4 −0.879873 + 0.879873i −15.5882 0.0935382i 30.4516i −51.8842 + 20.8094i 13.7979 13.6333i −11.3356 11.3356i −54.9495 54.9495i 242.983 + 2.91618i 27.3419 63.9612i
2.5 0.879873 0.879873i 0.0935382 + 15.5882i 30.4516i 51.8842 20.8094i 13.7979 + 13.6333i −11.3356 11.3356i 54.9495 + 54.9495i −242.983 + 2.91618i 27.3419 63.9612i
2.6 3.17835 3.17835i 5.54767 14.5679i 11.7962i −11.9990 54.5987i −28.6694 63.9343i 83.6967 + 83.6967i 139.200 + 139.200i −181.447 161.636i −211.671 135.397i
2.7 5.80199 5.80199i 13.8908 + 7.07428i 35.3261i −42.8643 + 35.8839i 121.639 39.5494i −140.332 140.332i −19.2981 19.2981i 142.909 + 196.535i −40.5000 + 456.896i
2.8 6.77944 6.77944i −15.1419 3.70445i 59.9217i 52.1700 + 20.0821i −127.768 + 77.5396i 47.9711 + 47.9711i −189.294 189.294i 215.554 + 112.185i 489.829 217.538i
8.1 −6.77944 6.77944i 3.70445 15.1419i 59.9217i −52.1700 + 20.0821i −127.768 + 77.5396i 47.9711 47.9711i 189.294 189.294i −215.554 112.185i 489.829 + 217.538i
8.2 −5.80199 5.80199i −7.07428 + 13.8908i 35.3261i 42.8643 + 35.8839i 121.639 39.5494i −140.332 + 140.332i 19.2981 19.2981i −142.909 196.535i −40.5000 456.896i
8.3 −3.17835 3.17835i 14.5679 + 5.54767i 11.7962i 11.9990 54.5987i −28.6694 63.9343i 83.6967 83.6967i −139.200 + 139.200i 181.447 + 161.636i −211.671 + 135.397i
8.4 −0.879873 0.879873i −15.5882 + 0.0935382i 30.4516i −51.8842 20.8094i 13.7979 + 13.6333i −11.3356 + 11.3356i −54.9495 + 54.9495i 242.983 2.91618i 27.3419 + 63.9612i
8.5 0.879873 + 0.879873i 0.0935382 15.5882i 30.4516i 51.8842 + 20.8094i 13.7979 13.6333i −11.3356 + 11.3356i 54.9495 54.9495i −242.983 2.91618i 27.3419 + 63.9612i
8.6 3.17835 + 3.17835i 5.54767 + 14.5679i 11.7962i −11.9990 + 54.5987i −28.6694 + 63.9343i 83.6967 83.6967i 139.200 139.200i −181.447 + 161.636i −211.671 + 135.397i
8.7 5.80199 + 5.80199i 13.8908 7.07428i 35.3261i −42.8643 35.8839i 121.639 + 39.5494i −140.332 + 140.332i −19.2981 + 19.2981i 142.909 196.535i −40.5000 456.896i
8.8 6.77944 + 6.77944i −15.1419 + 3.70445i 59.9217i 52.1700 20.0821i −127.768 77.5396i 47.9711 47.9711i −189.294 + 189.294i 215.554 112.185i 489.829 + 217.538i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.6.e.a 16
3.b odd 2 1 inner 15.6.e.a 16
5.b even 2 1 75.6.e.e 16
5.c odd 4 1 inner 15.6.e.a 16
5.c odd 4 1 75.6.e.e 16
15.d odd 2 1 75.6.e.e 16
15.e even 4 1 inner 15.6.e.a 16
15.e even 4 1 75.6.e.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.e.a 16 1.a even 1 1 trivial
15.6.e.a 16 3.b odd 2 1 inner
15.6.e.a 16 5.c odd 4 1 inner
15.6.e.a 16 15.e even 4 1 inner
75.6.e.e 16 5.b even 2 1
75.6.e.e 16 5.c odd 4 1
75.6.e.e 16 15.d odd 2 1
75.6.e.e 16 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 13393 T^{12} + \cdots + 37480960000 \) Copy content Toggle raw display
$3$ \( T^{16} + 4320 T^{13} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} - 4580 T^{14} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 40 T^{7} + \cdots + 652674756250000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 532700 T^{6} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 1060 T^{7} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 9871879225768 T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9996984 T^{6} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 92370076420168 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} - 69254700 T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 5392 T^{3} + \cdots + 12072937186816)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 27860 T^{7} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 257067200 T^{6} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 11680 T^{7} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} - 525588900 T^{6} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 43792 T^{3} + \cdots - 18\!\cdots\!84)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 70160 T^{7} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 8891090600 T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 120160 T^{7} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 6750333984 T^{6} + \cdots + 33\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} - 17794791000 T^{6} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 220400 T^{7} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
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