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Properties

Label 1150.3.d.b
Level $1150$
Weight $3$
Character orbit 1150.d
Analytic conductor $31.335$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1150.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3352304014\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{4} q^{3} + 2 q^{4} - \beta_{9} q^{6} + (\beta_{10} + \beta_{8} - \beta_{3}) q^{7} + 2 \beta_{5} q^{8} + ( - \beta_{14} + \beta_{12} - \beta_{9} + \beta_{5} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{4} q^{3} + 2 q^{4} - \beta_{9} q^{6} + (\beta_{10} + \beta_{8} - \beta_{3}) q^{7} + 2 \beta_{5} q^{8} + ( - \beta_{14} + \beta_{12} - \beta_{9} + \beta_{5} + 5) q^{9} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2}) q^{11} + 2 \beta_{4} q^{12} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{9} + 3 \beta_{5} + \beta_{4} - 1) q^{13} + ( - \beta_{11} + \beta_{7} - \beta_1) q^{14} + 4 q^{16} + ( - \beta_{11} - 3 \beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_1) q^{17} + (\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + 4 \beta_{5} + \cdots + 3) q^{18}+ \cdots + ( - 4 \beta_{10} - 8 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} + 16 \beta_{3} + \cdots + 30 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} - 24 q^{13} + 64 q^{16} + 32 q^{18} - 4 q^{23} - 16 q^{24} + 96 q^{26} + 96 q^{27} - 108 q^{29} - 116 q^{31} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} - 48 q^{52} + 224 q^{54} - 160 q^{58} + 204 q^{59} - 64 q^{62} + 128 q^{64} - 268 q^{69} + 236 q^{71} + 64 q^{72} + 112 q^{73} + 936 q^{77} + 432 q^{78} - 136 q^{81} + 64 q^{82} + 152 q^{87} - 8 q^{92} - 856 q^{93} - 216 q^{94} - 32 q^{96} - 256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 249266 \nu^{15} + 19450521 \nu^{13} + 540300528 \nu^{11} + 7075107694 \nu^{9} + 46294879486 \nu^{7} + 143284317912 \nu^{5} + \cdots + 76734676377 \nu ) / 736991990 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1452833849 \nu^{15} + 114494825161 \nu^{13} + 3236490394640 \nu^{11} + 43627262833926 \nu^{9} + 300170143661102 \nu^{7} + \cdots + 713709368150605 \nu ) / 3401955025840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5957154299 \nu^{15} - 459480090555 \nu^{13} - 12507983573336 \nu^{11} - 158460209288026 \nu^{9} + \cdots - 543467028934999 \nu ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + \cdots + 38708131230591 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 328071 \nu^{14} + 25254583 \nu^{12} + 684448696 \nu^{10} + 8586649314 \nu^{8} + 51832164194 \nu^{6} + 133890163508 \nu^{4} + 92129883455 \nu^{2} + \cdots + 885659459 ) / 505359680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + \cdots + 8733597167783 \nu ) / 114351429440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 229620557 \nu^{15} - 17699955559 \nu^{13} - 480902133464 \nu^{11} - 6059544792898 \nu^{9} - 36877647447520 \nu^{7} + \cdots - 4166619657821 \nu ) / 121498393780 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3600464285 \nu^{15} - 277707100247 \nu^{13} - 7555326565548 \nu^{11} - 95479887197410 \nu^{9} - 585118020325518 \nu^{7} + \cdots - 191041726435967 \nu ) / 1700977512920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14365389855 \nu^{14} + 1105876709199 \nu^{12} + 29972322851736 \nu^{10} + 375976473404450 \nu^{8} + \cdots - 44834433467221 ) / 6803910051680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5871432519 \nu^{15} + 451130209015 \nu^{13} + 12184818665176 \nu^{11} + 151934673265506 \nu^{9} + 905732893319778 \nu^{7} + \cdots - 230413141490301 \nu ) / 1943974300480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43531256921 \nu^{15} - 3343956125401 \nu^{13} - 90283361017128 \nu^{11} + \cdots + 17\!\cdots\!83 \nu ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 29269066035 \nu^{15} + 21002456926 \nu^{14} - 2263755902763 \nu^{13} + 1617708410646 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 82437933075262 ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29269066035 \nu^{15} - 28278344419 \nu^{14} - 2263755902763 \nu^{13} - 2174316072571 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 28012979344985 ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 29269066035 \nu^{15} + 40620036785 \nu^{14} - 2263755902763 \nu^{13} + 3124787270393 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 6504254102973 ) / 13607820103360 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 29269066035 \nu^{15} + 68821388859 \nu^{14} - 2263755902763 \nu^{13} + 5299632441843 \nu^{12} - 61890479395272 \nu^{11} + \cdots + 78181554830047 ) / 13607820103360 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{6} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} + \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{11} + 2\beta_{10} - 3\beta_{8} + 5\beta_{7} - 17\beta_{6} + 8\beta_{3} + 7\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{15} - 47 \beta_{14} - 14 \beta_{13} - 43 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 79 \beta_{9} + 45 \beta_{8} - 45 \beta_{6} - 179 \beta_{5} + 117 \beta_{4} - 45 \beta_{3} - 45 \beta_{2} - 45 \beta _1 + 400 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 698 \beta_{11} - 56 \beta_{10} + 3 \beta_{8} - 386 \beta_{7} + 1219 \beta_{6} - 449 \beta_{3} - 626 \beta_{2} - 878 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 359 \beta_{15} + 898 \beta_{14} + 325 \beta_{13} + 872 \beta_{12} - 868 \beta_{11} - 868 \beta_{10} - 1235 \beta_{9} - 868 \beta_{8} + 868 \beta_{6} + 3226 \beta_{5} - 2129 \beta_{4} + 868 \beta_{3} + 868 \beta_{2} + \cdots - 5998 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 27820 \beta_{11} + 104 \beta_{10} + 1733 \beta_{8} + 13932 \beta_{7} - 43859 \beta_{6} + 15295 \beta_{3} + 23596 \beta_{2} + 32356 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 28822 \beta_{15} - 66115 \beta_{14} - 25026 \beta_{13} - 66439 \beta_{12} + 64379 \beta_{11} + 64379 \beta_{10} + 84671 \beta_{9} + 64379 \beta_{8} - 64379 \beta_{6} - 232271 \beta_{5} + 154427 \beta_{4} + \cdots + 406476 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 520505 \beta_{11} + 12974 \beta_{10} - 40318 \beta_{8} - 250339 \beta_{7} + 793004 \beta_{6} - 272833 \beta_{3} - 430885 \beta_{2} - 589491 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1081278 \beta_{15} + 2410481 \beta_{14} + 923762 \beta_{13} + 2458153 \beta_{12} - 2355559 \beta_{11} - 2355559 \beta_{10} - 3010165 \beta_{9} - 2355559 \beta_{8} + \cdots - 14401944 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 38167834 \beta_{11} - 1359064 \beta_{10} + 3109921 \beta_{8} + 18059642 \beta_{7} - 57484087 \beta_{6} + 19690853 \beta_{3} + 31286770 \beta_{2} + 42843970 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 19844966 \beta_{15} - 43809734 \beta_{14} - 16848718 \beta_{13} - 44933550 \beta_{12} + 42873518 \beta_{11} + 42873518 \beta_{10} + 54218408 \beta_{9} + 42873518 \beta_{8} + \cdots + 259136963 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1389854096 \beta_{11} + 54975824 \beta_{10} - 114833875 \beta_{8} - 653397248 \beta_{7} + 2085143085 \beta_{6} - 712996205 \beta_{3} - 1134920000 \beta_{2} - 1555514512 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1446079476 \beta_{15} + 3181503425 \beta_{14} + 1224795972 \beta_{13} + 3270129841 \beta_{12} - 3115174881 \beta_{11} - 3115174881 \beta_{10} + \cdots - 18751391662 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 25246432839 \beta_{11} - 1035217254 \beta_{10} + 2094391749 \beta_{8} + 11839142477 \beta_{7} - 37826840705 \beta_{6} + 12924576256 \beta_{3} + \cdots + 28227614047 \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1
1.00527i
− 1.00527i
− 2.26343i
2.26343i
− 3.68124i
3.68124i
− 6.02373i
6.02373i
− 3.47734i
3.47734i
− 1.01877i
1.01877i
− 2.98291i
2.98291i
0.0431371i
− 0.0431371i
−1.41421 −4.30716 2.00000 0 6.09125 − 1.47532i −2.82843 9.55167 0
551.2 −1.41421 −4.30716 2.00000 0 6.09125 1.47532i −2.82843 9.55167 0
551.3 −1.41421 −1.43837 2.00000 0 2.03417 − 10.1866i −2.82843 −6.93108 0
551.4 −1.41421 −1.43837 2.00000 0 2.03417 10.1866i −2.82843 −6.93108 0
551.5 −1.41421 3.36596 2.00000 0 −4.76019 − 1.16919i −2.82843 2.32968 0
551.6 −1.41421 3.36596 2.00000 0 −4.76019 1.16919i −2.82843 2.32968 0
551.7 −1.41421 3.79379 2.00000 0 −5.36524 − 7.10180i −2.82843 5.39287 0
551.8 −1.41421 3.79379 2.00000 0 −5.36524 7.10180i −2.82843 5.39287 0
551.9 1.41421 −4.76369 2.00000 0 −6.73687 − 7.05858i 2.82843 13.6927 0
551.10 1.41421 −4.76369 2.00000 0 −6.73687 7.05858i 2.82843 13.6927 0
551.11 1.41421 −2.34854 2.00000 0 −3.32134 − 7.61815i 2.82843 −3.48436 0
551.12 1.41421 −2.34854 2.00000 0 −3.32134 7.61815i 2.82843 −3.48436 0
551.13 1.41421 0.278523 2.00000 0 0.393890 − 8.51262i 2.82843 −8.92243 0
551.14 1.41421 0.278523 2.00000 0 0.393890 8.51262i 2.82843 −8.92243 0
551.15 1.41421 5.41949 2.00000 0 7.66432 − 8.24199i 2.82843 20.3709 0
551.16 1.41421 5.41949 2.00000 0 7.66432 8.24199i 2.82843 20.3709 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1150.3.d.b 16
5.b even 2 1 230.3.d.a ✓ 16
5.c odd 4 2 1150.3.c.c 32
15.d odd 2 1 2070.3.c.a 16
20.d odd 2 1 1840.3.k.d 16
23.b odd 2 1 inner 1150.3.d.b 16
115.c odd 2 1 230.3.d.a ✓ 16
115.e even 4 2 1150.3.c.c 32
345.h even 2 1 2070.3.c.a 16
460.g even 2 1 1840.3.k.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.d.a ✓ 16 5.b even 2 1
230.3.d.a ✓ 16 115.c odd 2 1
1150.3.c.c 32 5.c odd 4 2
1150.3.c.c 32 115.e even 4 2
1150.3.d.b 16 1.a even 1 1 trivial
1150.3.d.b 16 23.b odd 2 1 inner
1840.3.k.d 16 20.d odd 2 1
1840.3.k.d 16 460.g even 2 1
2070.3.c.a 16 15.d odd 2 1
2070.3.c.a 16 345.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 52T_{3}^{6} - 16T_{3}^{5} + 829T_{3}^{4} + 456T_{3}^{3} - 4114T_{3}^{2} - 3704T_{3} + 1336 \) acting on \(S_{3}^{\mathrm{new}}(1150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - 52 T^{6} - 16 T^{5} + 829 T^{4} + \cdots + 1336)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 406 T^{14} + \cdots + 221645107264 \) Copy content Toggle raw display
$11$ \( T^{16} + 1016 T^{14} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} - 898 T^{6} + \cdots - 343464224)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 1858 T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + 4184 T^{14} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + 4 T^{15} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 54 T^{7} - 2309 T^{6} + \cdots - 61767459836)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 58 T^{7} + \cdots + 229759835104)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 14482 T^{14} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{8} + 78 T^{7} + \cdots - 212194449184)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 13412 T^{14} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{8} - 64 T^{7} - 5764 T^{6} + \cdots + 13232824136)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 21250 T^{14} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{8} - 102 T^{7} + \cdots - 42922529206784)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 28128 T^{14} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{16} + 52678 T^{14} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{8} - 118 T^{7} + \cdots - 24390990617024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 56 T^{7} + \cdots + 1317400530416)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 82216 T^{14} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{16} + 69862 T^{14} + \cdots + 39\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + 67928 T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + 83856 T^{14} + \cdots + 37\!\cdots\!04 \) Copy content Toggle raw display
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