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Properties

Label 390.2.t.a
Level $390$
Weight $2$
Character orbit 390.t
Analytic conductor $3.114$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{7} q^{3} - q^{4} + ( - \beta_{8} + \beta_{7} + \beta_{3} + \beta_1 - 1) q^{5} + \beta_{2} q^{6} + (\beta_{7} - \beta_{5} + \beta_{2}) q^{7} + \beta_{4} q^{8} - \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{7} q^{3} - q^{4} + ( - \beta_{8} + \beta_{7} + \beta_{3} + \beta_1 - 1) q^{5} + \beta_{2} q^{6} + (\beta_{7} - \beta_{5} + \beta_{2}) q^{7} + \beta_{4} q^{8} - \beta_{4} q^{9} - \beta_{11} q^{10} + ( - \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{4} + 1) q^{11} + \beta_{7} q^{12} + ( - \beta_{11} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{13} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{14} - \beta_{10} q^{15} + q^{16} + ( - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{4} + 1) q^{17} - q^{18} + ( - \beta_{11} - \beta_{9} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{19}+ \cdots + (\beta_{11} - \beta_{9} + \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 4 q^{5} + 4 q^{11} + 20 q^{13} - 4 q^{15} + 12 q^{16} + 8 q^{17} - 12 q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{21} - 4 q^{22} - 4 q^{25} - 4 q^{26} + 4 q^{30} + 12 q^{31} - 8 q^{34} + 12 q^{35} - 16 q^{37} + 4 q^{38} - 12 q^{39} + 8 q^{41} + 8 q^{42} + 16 q^{43} - 4 q^{44} + 32 q^{47} - 20 q^{49} + 8 q^{50} - 20 q^{52} - 16 q^{53} - 12 q^{55} - 40 q^{58} + 20 q^{59} + 4 q^{60} + 16 q^{61} + 12 q^{62} - 12 q^{64} - 32 q^{65} - 16 q^{66} - 8 q^{68} - 32 q^{69} - 20 q^{70} - 32 q^{71} + 12 q^{72} + 4 q^{76} + 16 q^{77} - 4 q^{80} - 12 q^{81} + 8 q^{82} + 32 q^{83} + 8 q^{84} + 12 q^{85} + 16 q^{86} + 20 q^{87} + 4 q^{88} + 16 q^{89} + 4 q^{90} - 28 q^{91} - 8 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6x^{10} - 24x^{9} + 18x^{8} + 40x^{7} - 82x^{6} + 12x^{5} + 228x^{4} - 284x^{3} + 124x^{2} - 16x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47412266 \nu^{11} + 206226196 \nu^{10} + 498918647 \nu^{9} + 275321548 \nu^{8} - 2696897549 \nu^{7} + 1486501171 \nu^{6} + \cdots - 957747072 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 478873536 \nu^{11} - 47412266 \nu^{10} - 3079467412 \nu^{9} + 10994046217 \nu^{8} - 8895045196 \nu^{7} - 16458043891 \nu^{6} + \cdots + 16498970487 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 562969191 \nu^{11} - 625242846 \nu^{10} - 3845754997 \nu^{9} + 9502475052 \nu^{8} + 1876460724 \nu^{7} - 24232626521 \nu^{6} + \cdots - 278134328 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65564 \nu^{11} + 54701 \nu^{10} - 348926 \nu^{9} + 1928638 \nu^{8} - 2217584 \nu^{7} - 2535511 \nu^{6} + 7789630 \nu^{5} - 3089126 \nu^{4} + \cdots + 1239927 ) / 1476181 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 642662513 \nu^{11} + 232175953 \nu^{10} + 3952388821 \nu^{9} - 13970332336 \nu^{8} + 6804053268 \nu^{7} + 28228005853 \nu^{6} + \cdots - 4015649046 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1158219438 \nu^{11} + 651192603 \nu^{10} + 7299225171 \nu^{9} - 23748128936 \nu^{8} + 7624490093 \nu^{7} + 50974131203 \nu^{6} + \cdots - 2779767646 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2007824523 \nu^{11} + 642662513 \nu^{10} + 12279123091 \nu^{9} - 44235399731 \nu^{8} + 22170509078 \nu^{7} + 87117034188 \nu^{6} + \cdots - 2642262891 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2024040388 \nu^{11} - 481621272 \nu^{10} + 11262741596 \nu^{9} - 52099333786 \nu^{8} + 42254553243 \nu^{7} + 89411069453 \nu^{6} + \cdots + 4694601129 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2281630388 \nu^{11} + 1661658328 \nu^{10} + 14577758021 \nu^{9} - 44298602736 \nu^{8} + 6809317318 \nu^{7} + 102670257953 \nu^{6} + \cdots + 27875512029 ) / 13913005925 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 511017007 \nu^{11} - 120344967 \nu^{10} - 3187847414 \nu^{9} + 11403406359 \nu^{8} - 7135610307 \nu^{7} - 20604017287 \nu^{6} + \cdots + 1016773599 ) / 2782601185 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 771058311 \nu^{11} + 121340051 \nu^{10} + 4638436737 \nu^{9} - 17846443417 \nu^{8} + 10922555446 \nu^{7} + 32171693006 \nu^{6} + \cdots - 7445776057 ) / 2782601185 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 3\beta_{5} - \beta_{4} - 3\beta_{3} - 4\beta_{2} - 4\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{8} + 4\beta_{7} - 5\beta_{6} + 5\beta_{5} - \beta_{4} - 2\beta_{3} + \beta_{2} + 3\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 13 \beta_{7} + 13 \beta_{6} + 5 \beta_{5} + 17 \beta_{4} + 20 \beta_{3} + 5 \beta_{2} + 15 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{11} + 17 \beta_{9} + 13 \beta_{8} - 25 \beta_{7} + 27 \beta_{6} - 72 \beta_{5} - 25 \beta_{4} - 44 \beta_{3} - 28 \beta_{2} - 72 \beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 27 \beta_{11} + 59 \beta_{10} - 27 \beta_{9} - 59 \beta_{8} + 234 \beta_{7} - 218 \beta_{6} + 130 \beta_{5} - 100 \beta_{4} - 130 \beta_{3} + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 218 \beta_{11} - 157 \beta_{10} - 157 \beta_{9} - 301 \beta_{7} + 301 \beta_{6} + 519 \beta_{5} + 564 \beta_{4} + 867 \beta_{3} + 346 \beta_{2} + 867 \beta _1 + 301 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 301 \beta_{11} - 301 \beta_{10} + 710 \beta_{9} + 710 \beta_{8} - 1925 \beta_{7} + 1889 \beta_{6} - 2961 \beta_{5} - 301 \beta_{4} - 710 \beta_{3} - 914 \beta_{2} - 2298 \beta _1 - 2036 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1889 \beta_{11} + 2660 \beta_{10} - 1889 \beta_{8} + 8736 \beta_{7} - 8557 \beta_{6} + 1674 \beta_{5} - 6076 \beta_{4} - 8557 \beta_{3} - 1735 \beta_{2} - 4334 \beta _1 + 1735 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8557 \beta_{11} - 3563 \beta_{10} - 8557 \beta_{9} - 3563 \beta_{8} + 31302 \beta_{5} + 19663 \beta_{4} + 31302 \beta_{3} + 15794 \beta_{2} + 39260 \beta _1 + 19663 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22745 \beta_{10} + 22745 \beta_{9} + 32134 \beta_{8} - 105363 \beta_{7} + 103116 \beta_{6} - 103116 \beta_{5} + 20910 \beta_{4} + 20017 \beta_{3} - 20910 \beta_{2} - 52151 \beta _1 - 73229 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.0572576 − 0.138232i
0.563963 − 1.36153i
−1.32833 + 3.20687i
−1.46953 − 0.608701i
1.52752 + 0.632721i
0.649118 + 0.268874i
0.0572576 + 0.138232i
0.563963 + 1.36153i
−1.32833 − 3.20687i
−1.46953 + 0.608701i
1.52752 − 0.632721i
0.649118 − 0.268874i
− 1.00000i −0.707107 + 0.707107i −1.00000 −2.20289 + 0.383740i 0.707107 + 0.707107i 1.52873 1.00000i − 1.00000i 0.383740 + 2.20289i
307.2 − 1.00000i −0.707107 + 0.707107i −1.00000 1.04698 − 1.97581i 0.707107 + 0.707107i 2.54214 1.00000i − 1.00000i −1.97581 − 1.04698i
307.3 − 1.00000i −0.707107 + 0.707107i −1.00000 1.57013 + 1.59207i 0.707107 + 0.707107i −1.24244 1.00000i − 1.00000i 1.59207 − 1.57013i
307.4 − 1.00000i 0.707107 − 0.707107i −1.00000 −2.23563 − 0.0440169i −0.707107 − 0.707107i −4.35328 1.00000i − 1.00000i −0.0440169 + 2.23563i
307.5 − 1.00000i 0.707107 − 0.707107i −1.00000 −0.623976 − 2.14724i −0.707107 − 0.707107i 1.64083 1.00000i − 1.00000i −2.14724 + 0.623976i
307.6 − 1.00000i 0.707107 − 0.707107i −1.00000 0.445397 + 2.19126i −0.707107 − 0.707107i −0.115977 1.00000i − 1.00000i 2.19126 − 0.445397i
343.1 1.00000i −0.707107 − 0.707107i −1.00000 −2.20289 − 0.383740i 0.707107 − 0.707107i 1.52873 − 1.00000i 1.00000i 0.383740 − 2.20289i
343.2 1.00000i −0.707107 − 0.707107i −1.00000 1.04698 + 1.97581i 0.707107 − 0.707107i 2.54214 − 1.00000i 1.00000i −1.97581 + 1.04698i
343.3 1.00000i −0.707107 − 0.707107i −1.00000 1.57013 − 1.59207i 0.707107 − 0.707107i −1.24244 − 1.00000i 1.00000i 1.59207 + 1.57013i
343.4 1.00000i 0.707107 + 0.707107i −1.00000 −2.23563 + 0.0440169i −0.707107 + 0.707107i −4.35328 − 1.00000i 1.00000i −0.0440169 − 2.23563i
343.5 1.00000i 0.707107 + 0.707107i −1.00000 −0.623976 + 2.14724i −0.707107 + 0.707107i 1.64083 − 1.00000i 1.00000i −2.14724 − 0.623976i
343.6 1.00000i 0.707107 + 0.707107i −1.00000 0.445397 − 2.19126i −0.707107 + 0.707107i −0.115977 − 1.00000i 1.00000i 2.19126 + 0.445397i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.t.a yes 12
3.b odd 2 1 1170.2.w.f 12
5.b even 2 1 1950.2.t.b 12
5.c odd 4 1 390.2.j.a ✓ 12
5.c odd 4 1 1950.2.j.c 12
13.d odd 4 1 390.2.j.a ✓ 12
15.e even 4 1 1170.2.m.f 12
39.f even 4 1 1170.2.m.f 12
65.f even 4 1 inner 390.2.t.a yes 12
65.g odd 4 1 1950.2.j.c 12
65.k even 4 1 1950.2.t.b 12
195.u odd 4 1 1170.2.w.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.a ✓ 12 5.c odd 4 1
390.2.j.a ✓ 12 13.d odd 4 1
390.2.t.a yes 12 1.a even 1 1 trivial
390.2.t.a yes 12 65.f even 4 1 inner
1170.2.m.f 12 15.e even 4 1
1170.2.m.f 12 39.f even 4 1
1170.2.w.f 12 3.b odd 2 1
1170.2.w.f 12 195.u odd 4 1
1950.2.j.c 12 5.c odd 4 1
1950.2.j.c 12 65.g odd 4 1
1950.2.t.b 12 5.b even 2 1
1950.2.t.b 12 65.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 16T_{7}^{4} + 20T_{7}^{3} + 24T_{7}^{2} - 32T_{7} - 4 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + 10 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 16 T^{4} + 20 T^{3} + 24 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 141376 \) Copy content Toggle raw display
$13$ \( T^{12} - 20 T^{11} + 188 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 446224 \) Copy content Toggle raw display
$19$ \( T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 23309584 \) Copy content Toggle raw display
$23$ \( T^{12} + 104 T^{9} + 5768 T^{8} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( T^{12} + 332 T^{10} + \cdots + 932203024 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$37$ \( (T^{6} + 8 T^{5} - 118 T^{4} - 840 T^{3} + \cdots + 25144)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 8 T^{11} + \cdots + 3237154816 \) Copy content Toggle raw display
$43$ \( T^{12} - 16 T^{11} + \cdots + 289272064 \) Copy content Toggle raw display
$47$ \( (T^{6} - 16 T^{5} - 24 T^{4} + 800 T^{3} + \cdots - 9248)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 16 T^{11} + 128 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$59$ \( T^{12} - 20 T^{11} + 200 T^{10} + \cdots + 94789696 \) Copy content Toggle raw display
$61$ \( (T^{6} - 8 T^{5} - 160 T^{4} + 480 T^{3} + \cdots - 33056)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 428 T^{10} + \cdots + 15038607424 \) Copy content Toggle raw display
$71$ \( T^{12} + 32 T^{11} + \cdots + 209295270144 \) Copy content Toggle raw display
$73$ \( T^{12} + 536 T^{10} + \cdots + 43930384 \) Copy content Toggle raw display
$79$ \( T^{12} + 368 T^{10} + \cdots + 485409024 \) Copy content Toggle raw display
$83$ \( (T^{6} - 16 T^{5} - 32 T^{4} + 520 T^{3} + \cdots - 1424)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 16 T^{11} + \cdots + 24551129344 \) Copy content Toggle raw display
$97$ \( T^{12} + 840 T^{10} + \cdots + 2393122368784 \) Copy content Toggle raw display
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