Properties

Label 380.3.e.a
Level $380$
Weight $3$
Character orbit 380.e
Analytic conductor $10.354$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7} + (\beta_{7} - \beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7} + (\beta_{7} - \beta_{6} - 1) q^{9} + ( - \beta_{11} - \beta_{8} + \beta_{7} - 2) q^{11} + ( - \beta_{4} - \beta_1) q^{13} + \beta_{5} q^{15} + (\beta_{11} - \beta_{8} + 2 \beta_{7} - \beta_{6}) q^{17} + (\beta_{10} + \beta_{8} - \beta_{2} - \beta_1 + 2) q^{19} + (\beta_{9} + \beta_{5} + \beta_{3} - 2 \beta_1) q^{21} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + 1) q^{23} + 5 q^{25} + (\beta_{9} - 2 \beta_{5} + 3 \beta_1) q^{27} + (\beta_{9} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{29} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{31} + (2 \beta_{9} + \beta_{4} - 6 \beta_1) q^{33} + (\beta_{10} - \beta_{8} - \beta_{6} + 4) q^{35} + (\beta_{9} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{37} + ( - 4 \beta_{8} + \beta_{7} + \beta_{6} + 12) q^{39} + (2 \beta_{9} + \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_1) q^{41} + (2 \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 15) q^{43} + (\beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - 2 \beta_{6} - 3) q^{45} + (5 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 7) q^{47} + (\beta_{11} + 2 \beta_{10} - 5 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} + 1) q^{49} + (\beta_{9} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 7 \beta_1) q^{51} + (2 \beta_{9} - 2 \beta_{5} + 4 \beta_{3} + \beta_1) q^{53} + ( - \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{55} + (\beta_{11} + 4 \beta_{10} + \beta_{9} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 13) q^{57}+ \cdots + (\beta_{11} - 4 \beta_{10} + \beta_{8} - 9 \beta_{7} + 6 \beta_{6} + 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59\nu^{11} + 5868\nu^{9} + 184075\nu^{7} + 2290126\nu^{5} + 10898236\nu^{3} + 13285720\nu ) / 407880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -239\nu^{11} - 15993\nu^{9} - 373210\nu^{7} - 3571816\nu^{5} - 13082536\nu^{3} - 14157280\nu ) / 407880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -401\nu^{11} - 20007\nu^{9} - 344590\nu^{7} - 2594164\nu^{5} - 8665084\nu^{3} - 10740520\nu ) / 203940 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1502992\nu^{3} - 417160\nu ) / 37080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1502992\nu^{2} - 417160 ) / 37080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1465912\nu^{2} - 46360 ) / 37080 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 301\nu^{10} + 14382\nu^{8} + 228185\nu^{6} + 1462904\nu^{4} + 3463424\nu^{2} + 1849460 ) / 67980 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1484452\nu^{3} - 139060\nu ) / 18540 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -361\nu^{10} - 17757\nu^{8} - 291230\nu^{6} - 1890134\nu^{4} - 4123544\nu^{2} - 1392200 ) / 67980 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 545\nu^{10} + 28107\nu^{8} + 495898\nu^{6} + 3578728\nu^{4} + 9148096\nu^{2} + 3524896 ) / 81576 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{5} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} - 4\beta_{10} + \beta_{8} - 22\beta_{7} + 37\beta_{6} + 150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -29\beta_{9} + 62\beta_{5} - 5\beta_{4} - 6\beta_{3} - 8\beta_{2} + 269\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -93\beta_{11} + 158\beta_{10} - 15\beta_{8} + 470\beta_{7} - 971\beta_{6} - 2674 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 721\beta_{9} - 1584\beta_{5} + 173\beta_{4} + 236\beta_{3} + 316\beta_{2} - 5403\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2413\beta_{11} - 4526\beta_{10} + 91\beta_{8} - 10310\beta_{7} + 23619\beta_{6} + 53398 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -17249\beta_{9} + 38364\beta_{5} - 4617\beta_{4} - 6848\beta_{3} - 9052\beta_{2} + 116231\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -59373\beta_{11} + 115918\beta_{10} + 2389\beta_{8} + 231734\beta_{7} - 560747\beta_{6} - 1144366 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 407025\beta_{9} - 910788\beta_{5} + 113529\beta_{4} + 177680\beta_{3} + 231836\beta_{2} - 2599023\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
4.83157i
3.53755i
3.42504i
3.14288i
2.03369i
0.630185i
0.630185i
2.03369i
3.14288i
3.42504i
3.53755i
4.83157i
0 4.83157i 0 2.23607 0 −3.72856 0 −14.3441 0
341.2 0 3.53755i 0 −2.23607 0 0.468611 0 −3.51429 0
341.3 0 3.42504i 0 2.23607 0 9.18599 0 −2.73093 0
341.4 0 3.14288i 0 −2.23607 0 −12.2192 0 −0.877687 0
341.5 0 2.03369i 0 −2.23607 0 4.27843 0 4.86411 0
341.6 0 0.630185i 0 2.23607 0 −3.98530 0 8.60287 0
341.7 0 0.630185i 0 2.23607 0 −3.98530 0 8.60287 0
341.8 0 2.03369i 0 −2.23607 0 4.27843 0 4.86411 0
341.9 0 3.14288i 0 −2.23607 0 −12.2192 0 −0.877687 0
341.10 0 3.42504i 0 2.23607 0 9.18599 0 −2.73093 0
341.11 0 3.53755i 0 −2.23607 0 0.468611 0 −3.51429 0
341.12 0 4.83157i 0 2.23607 0 −3.72856 0 −14.3441 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.e.a 12
3.b odd 2 1 3420.3.o.a 12
4.b odd 2 1 1520.3.h.b 12
5.b even 2 1 1900.3.e.f 12
5.c odd 4 2 1900.3.g.c 24
19.b odd 2 1 inner 380.3.e.a 12
57.d even 2 1 3420.3.o.a 12
76.d even 2 1 1520.3.h.b 12
95.d odd 2 1 1900.3.e.f 12
95.g even 4 2 1900.3.g.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.e.a 12 1.a even 1 1 trivial
380.3.e.a 12 19.b odd 2 1 inner
1520.3.h.b 12 4.b odd 2 1
1520.3.h.b 12 76.d even 2 1
1900.3.e.f 12 5.b even 2 1
1900.3.e.f 12 95.d odd 2 1
1900.3.g.c 24 5.c odd 4 2
1900.3.g.c 24 95.g even 4 2
3420.3.o.a 12 3.b odd 2 1
3420.3.o.a 12 57.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 62 T^{10} + 1445 T^{8} + \cdots + 55600 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{5} - 123 T^{4} - 448 T^{3} + \cdots - 3344)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 16 T^{5} - 164 T^{4} + \cdots - 43264)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 1174 T^{10} + \cdots + 18127879600 \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} - 703 T^{4} + \cdots + 818576)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 24 T^{11} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} - 1379 T^{4} + \cdots - 752400)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 6442 T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 119374584217600 \) Copy content Toggle raw display
$37$ \( T^{12} + 9204 T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + 7120 T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + 88 T^{5} + 460 T^{4} + \cdots - 2451136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 36 T^{5} - 7500 T^{4} + \cdots - 9167899456)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 28390 T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 492787956121600 \) Copy content Toggle raw display
$61$ \( (T^{6} - 76 T^{5} - 2952 T^{4} + \cdots - 1301500864)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 19710 T^{10} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + 52160 T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + 74 T^{5} - 7103 T^{4} + \cdots + 5637612816)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 49000 T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + 104 T^{5} - 12064 T^{4} + \cdots - 2227337984)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 42776 T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + 55476 T^{10} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
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