Properties

Label 380.3.g.c
Level $380$
Weight $3$
Character orbit 380.g
Analytic conductor $10.354$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.g (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: \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{3} + 1) q^{5} + \beta_{8} q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{3} + 1) q^{5} + \beta_{8} q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 6) q^{9} + ( - \beta_{4} + \beta_{3} - 2) q^{11} + (\beta_{9} + \beta_{5}) q^{13} + (\beta_{9} - \beta_{6} - 2 \beta_{5} + \beta_1) q^{15} + ( - \beta_{8} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{2} - \beta_1 + 6) q^{19} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{21} + 5 \beta_{8} q^{23} + ( - 2 \beta_{10} + \beta_{8} + 2 \beta_{3} + \beta_{2} - 6) q^{25} + ( - \beta_{11} - 6 \beta_{5}) q^{27} + (\beta_{7} + \beta_{6} + \beta_1) q^{29} + (\beta_{9} + 2 \beta_{7}) q^{31} + \beta_{9} q^{33} + (\beta_{10} + 2 \beta_{8} - \beta_{3} + 2 \beta_{2} - 2) q^{35} + ( - \beta_{11} - \beta_{5}) q^{37} + ( - 4 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 21) q^{39} + (\beta_{9} - 2 \beta_{6} - 2 \beta_1) q^{41} + ( - 2 \beta_{10} - 5 \beta_{4} - 5 \beta_{3}) q^{43} + ( - 3 \beta_{10} + 9 \beta_{8} + 5 \beta_{4} + 5 \beta_{3} - \beta_{2} + 18) q^{45} + (6 \beta_{10} - 4 \beta_{8} - 6 \beta_{4} - 6 \beta_{3}) q^{47} + (3 \beta_{4} - 3 \beta_{3} + \beta_{2} + 18) q^{49} + ( - \beta_{9} + \beta_{7} + 3 \beta_{6} - 3 \beta_1) q^{51} + ( - \beta_{9} - 7 \beta_{5}) q^{53} + ( - 2 \beta_{10} + \beta_{8} - 2 \beta_{3} + \beta_{2} + 15) q^{55} + (\beta_{11} + 4 \beta_{10} + \beta_{9} - 7 \beta_{8} + \beta_{5} - 5 \beta_{4} - 5 \beta_{3}) q^{57} + (3 \beta_{9} + 3 \beta_{7} - 3 \beta_{6} + \beta_1) q^{59} + ( - 7 \beta_{4} + 7 \beta_{3} - 50) q^{61} + ( - 2 \beta_{10} + 12 \beta_{8} + 8 \beta_{4} + 8 \beta_{3}) q^{63} + (\beta_{11} - 2 \beta_{9} + \beta_{7} + 4 \beta_{6} - 4 \beta_{5}) q^{65} + ( - 2 \beta_{11} - 5 \beta_{5}) q^{67} + ( - 5 \beta_{7} - 5 \beta_{6} + 5 \beta_1) q^{69} + ( - \beta_{9} - 2 \beta_{7} - 4 \beta_1) q^{71} + (4 \beta_{10} - \beta_{8} + 5 \beta_{4} + 5 \beta_{3}) q^{73} + (\beta_{11} + 3 \beta_{9} + \beta_{7} - \beta_{6} + 11 \beta_{5} + 5 \beta_1) q^{75} + (2 \beta_{10} - \beta_{4} - \beta_{3}) q^{77} + ( - 2 \beta_{7} - 2 \beta_{6} + 6 \beta_1) q^{79} + ( - 12 \beta_{2} + 37) q^{81} + ( - 8 \beta_{10} + 4 \beta_{8} + 4 \beta_{4} + 4 \beta_{3}) q^{83} + (\beta_{10} - 3 \beta_{8} - \beta_{3} - 3 \beta_{2} + 33) q^{85} + ( - 6 \beta_{10} - 7 \beta_{8} + 2 \beta_{4} + 2 \beta_{3}) q^{87} + ( - 3 \beta_{9} - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_1) q^{89} + ( - \beta_{9} - \beta_{7} + \beta_{6} - 5 \beta_1) q^{91} + ( - 2 \beta_{10} - 24 \beta_{8} + 3 \beta_{4} + 3 \beta_{3}) q^{93} + ( - \beta_{11} + \beta_{10} + \beta_{9} - 8 \beta_{8} - \beta_{7} + 2 \beta_{6} - 9 \beta_{5} + \cdots + 11) q^{95}+ \cdots + (4 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 28618\nu^{10} + 524952\nu^{8} + 4391039\nu^{6} + 54167333\nu^{4} + 1825258945\nu^{2} - 558303007 ) / 908842237 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60512 \nu^{10} + 2443821 \nu^{8} + 32340835 \nu^{6} + 100053868 \nu^{4} + 517724371 \nu^{2} + 21947164823 ) / 1817684474 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 194997239 \nu^{11} + 17200092 \nu^{10} - 5048505213 \nu^{9} + 7318528889 \nu^{8} - 28805936576 \nu^{7} + \cdots + 187896630501135 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 194997239 \nu^{11} - 17200092 \nu^{10} - 5048505213 \nu^{9} - 7318528889 \nu^{8} - 28805936576 \nu^{7} + \cdots - 187896630501135 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9729001 \nu^{11} - 195485288 \nu^{9} - 813261775 \nu^{7} - 3883239581 \nu^{5} - 354532580162 \nu^{3} + 2065201213063 \nu ) / 1506860428946 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 744934843 \nu^{11} + 1291775986 \nu^{10} + 9572417455 \nu^{9} + 18377506607 \nu^{8} - 122950074222 \nu^{7} + \cdots - 104748473351889 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 744934843 \nu^{11} + 13046909770 \nu^{10} - 9572417455 \nu^{9} + 267705610063 \nu^{8} + 122950074222 \nu^{7} + \cdots - 679853386340457 ) / 57260696299948 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 651724712 \nu^{11} + 15674641213 \nu^{9} + 65363306369 \nu^{7} - 565819505556 \nu^{5} + 17994962875673 \nu^{3} + \cdots + 25942480991957 \nu ) / 28630348149974 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39207097 \nu^{11} + 503811445 \nu^{9} - 6471056538 \nu^{7} - 52572021371 \nu^{5} + 1532495976047 \nu^{3} + \cdots - 10974976155720 \nu ) / 1506860428946 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1021426750 \nu^{11} + 23103082157 \nu^{9} + 96267253819 \nu^{7} - 418256401478 \nu^{5} + 31467200921829 \nu^{3} + \cdots + 61986227495459 \nu ) / 28630348149974 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 108734495 \nu^{11} + 992670198 \nu^{9} - 24848763895 \nu^{7} - 240525327873 \nu^{5} + 4527798347604 \nu^{3} + \cdots - 34021640801649 \nu ) / 1506860428946 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{8} + 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{4} + 3\beta_{3} - \beta_{2} + \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} - 7\beta_{10} - 6\beta_{9} + 5\beta_{8} - 6\beta_{5} - 16\beta_{4} - 16\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{9} + \beta_{7} - 21\beta_{6} + 37\beta_{4} - 37\beta_{3} + 9\beta_{2} - \beta _1 + 107 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{11} + 117\beta_{10} + 174\beta_{9} - 193\beta_{8} + 46\beta_{5} + 38\beta_{4} + 38\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -205\beta_{9} - 67\beta_{7} + 343\beta_{6} - 152\beta_{4} + 152\beta_{3} + 38\beta_{2} + 196\beta _1 - 1504 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 844\beta_{11} + 61\beta_{10} - 3572\beta_{9} + 3\beta_{8} - 4914\beta_{5} + 284\beta_{4} + 284\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3663 \beta_{9} + 1169 \beta_{7} - 6157 \beta_{6} - 4487 \beta_{4} + 4487 \beta_{3} - 339 \beta_{2} - 3775 \beta _1 - 25057 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 15750 \beta_{11} - 42099 \beta_{10} + 57402 \beta_{9} + 57429 \beta_{8} + 51230 \beta_{5} - 24724 \beta_{4} - 24724 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 56558 \beta_{9} - 13056 \beta_{7} + 100060 \beta_{6} + 226937 \beta_{4} - 226937 \beta_{3} + 47133 \beta_{2} + 40801 \beta _1 + 973351 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 193788 \beta_{11} + 1261457 \beta_{10} - 705598 \beta_{9} - 1471617 \beta_{8} - 613302 \beta_{5} + 1040926 \beta_{4} + 1040926 \beta_{3} ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
2.80718 + 1.46321i
2.80718 1.46321i
1.63363 + 1.34185i
1.63363 1.34185i
0.975196 4.19028i
0.975196 + 4.19028i
−0.975196 4.19028i
−0.975196 + 4.19028i
−1.63363 + 1.34185i
−1.63363 1.34185i
−2.80718 + 1.46321i
−2.80718 1.46321i
0 −5.61436 0 1.48938 4.77302i 0 7.03410i 0 22.5210 0
189.2 0 −5.61436 0 1.48938 + 4.77302i 0 7.03410i 0 22.5210 0
189.3 0 −3.26725 0 4.26535 2.60898i 0 6.30368i 0 1.67495 0
189.4 0 −3.26725 0 4.26535 + 2.60898i 0 6.30368i 0 1.67495 0
189.5 0 −1.95039 0 −2.75473 4.17271i 0 2.18749i 0 −5.19597 0
189.6 0 −1.95039 0 −2.75473 + 4.17271i 0 2.18749i 0 −5.19597 0
189.7 0 1.95039 0 −2.75473 4.17271i 0 2.18749i 0 −5.19597 0
189.8 0 1.95039 0 −2.75473 + 4.17271i 0 2.18749i 0 −5.19597 0
189.9 0 3.26725 0 4.26535 2.60898i 0 6.30368i 0 1.67495 0
189.10 0 3.26725 0 4.26535 + 2.60898i 0 6.30368i 0 1.67495 0
189.11 0 5.61436 0 1.48938 4.77302i 0 7.03410i 0 22.5210 0
189.12 0 5.61436 0 1.48938 + 4.77302i 0 7.03410i 0 22.5210 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.b odd 2 1 inner
95.d 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.g.c 12
3.b odd 2 1 3420.3.h.e 12
5.b even 2 1 inner 380.3.g.c 12
5.c odd 4 2 1900.3.e.e 12
15.d odd 2 1 3420.3.h.e 12
19.b odd 2 1 inner 380.3.g.c 12
57.d even 2 1 3420.3.h.e 12
95.d odd 2 1 inner 380.3.g.c 12
95.g even 4 2 1900.3.e.e 12
285.b even 2 1 3420.3.h.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.c 12 1.a even 1 1 trivial
380.3.g.c 12 5.b even 2 1 inner
380.3.g.c 12 19.b odd 2 1 inner
380.3.g.c 12 95.d odd 2 1 inner
1900.3.e.e 12 5.c odd 4 2
1900.3.e.e 12 95.g even 4 2
3420.3.h.e 12 3.b odd 2 1
3420.3.h.e 12 15.d odd 2 1
3420.3.h.e 12 57.d even 2 1
3420.3.h.e 12 285.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 46T_{3}^{4} + 497T_{3}^{2} - 1280 \) acting on \(S_{3}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 46 T^{4} + 497 T^{2} - 1280)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 6 T^{5} + 37 T^{4} - 160 T^{3} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 94 T^{4} + 2393 T^{2} + \cdots + 9408)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 6 T^{2} - 38 T - 44)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} - 682 T^{4} + 127153 T^{2} + \cdots - 7200000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 314 T^{4} + 10793 T^{2} + \cdots + 12288)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 34 T^{5} + 723 T^{4} + \cdots + 47045881)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 2350 T^{4} + 1495625 T^{2} + \cdots + 147000000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 2306 T^{4} + 1274297 T^{2} + \cdots + 51671040)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 4652 T^{4} + \cdots + 1345781760)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 5312 T^{4} + \cdots - 5202247680)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 7140 T^{4} + 3194532 T^{2} + \cdots + 318504960)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 7540 T^{4} + 12661412 T^{2} + \cdots + 95158272)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 14056 T^{4} + \cdots + 75606592512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 3130 T^{4} + 2302273 T^{2} + \cdots - 136869120)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 18086 T^{4} + \cdots + 7228354560)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 150 T^{2} + 5050 T + 18964)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 22430 T^{4} + \cdots - 301871934720)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 11980 T^{4} + \cdots + 23313776640)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 12714 T^{4} + \cdots + 4401589248)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 26440 T^{4} + \cdots + 4282122240)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 19616 T^{4} + \cdots + 216760320000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 18508 T^{4} + \cdots + 187730165760)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 22008 T^{4} + \cdots - 318730752000)^{2} \) Copy content Toggle raw display
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