Properties

Label 370.2.q.d
Level $370$
Weight $2$
Character orbit 370.q
Analytic conductor $2.954$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.q (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2x^{11} - 4x^{10} + 6x^{8} + 44x^{7} + 56x^{6} + 32x^{5} + 92x^{4} - 16x^{3} + 36x^{2} - 24x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - 1) q^{2} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{3} + \beta_{7} q^{4} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_1) q^{5} + (\beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{11} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{7} + q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - 1) q^{2} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 1) q^{3} + \beta_{7} q^{4} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_1) q^{5} + (\beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{11} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{7} + q^{8} + ( - \beta_{11} - \beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{9} + (\beta_{10} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_1) q^{10} + ( - 2 \beta_{8} - 2 \beta_{6} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{12} + ( - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 2) q^{13}+ \cdots + ( - \beta_{11} - 2 \beta_{10} + \beta_{9} - 8 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 6 q^{4} + 2 q^{5} + 6 q^{6} - 4 q^{7} + 12 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} - 6 q^{4} + 2 q^{5} + 6 q^{6} - 4 q^{7} + 12 q^{8} + 18 q^{9} - 4 q^{10} - 6 q^{12} + 8 q^{14} - 10 q^{15} - 6 q^{16} - 12 q^{17} - 18 q^{18} - 8 q^{19} + 2 q^{20} + 54 q^{21} - 6 q^{22} - 4 q^{23} - 14 q^{25} + 18 q^{27} - 4 q^{28} + 4 q^{29} - 4 q^{30} - 2 q^{31} - 6 q^{32} + 20 q^{33} + 12 q^{34} + 6 q^{35} + 14 q^{37} + 16 q^{38} - 30 q^{39} + 2 q^{40} + 12 q^{41} - 54 q^{42} + 44 q^{43} + 6 q^{44} - 60 q^{45} + 2 q^{46} - 28 q^{47} + 6 q^{48} + 24 q^{49} + 10 q^{50} - 8 q^{51} + 8 q^{53} - 18 q^{54} - 4 q^{56} - 28 q^{57} - 14 q^{58} + 14 q^{60} + 24 q^{61} - 14 q^{62} + 8 q^{63} + 12 q^{64} - 10 q^{65} + 8 q^{66} + 18 q^{67} - 40 q^{69} - 30 q^{70} + 18 q^{72} - 28 q^{73} + 38 q^{74} - 30 q^{75} - 8 q^{76} + 10 q^{77} + 48 q^{78} - 56 q^{79} - 4 q^{80} + 4 q^{81} + 14 q^{83} - 28 q^{85} - 22 q^{86} + 18 q^{87} - 2 q^{89} + 18 q^{90} + 38 q^{91} + 2 q^{92} - 16 q^{94} - 28 q^{95} - 6 q^{96} - 24 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} - 4x^{10} + 6x^{8} + 44x^{7} + 56x^{6} + 32x^{5} + 92x^{4} - 16x^{3} + 36x^{2} - 24x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 48023 \nu^{11} - 173427 \nu^{10} + 812758 \nu^{9} + 795518 \nu^{8} - 312576 \nu^{7} - 3181332 \nu^{6} - 14801278 \nu^{5} - 14727942 \nu^{4} + \cdots - 16072116 ) / 8800836 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2660983 \nu^{11} - 7621528 \nu^{10} - 6209960 \nu^{9} + 9898720 \nu^{8} + 18275158 \nu^{7} + 95878200 \nu^{6} + 40728518 \nu^{5} - 40308836 \nu^{4} + \cdots - 77578192 ) / 132012540 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1499 \nu^{11} - 1684 \nu^{10} - 8260 \nu^{9} - 6690 \nu^{8} + 8734 \nu^{7} + 77220 \nu^{6} + 146974 \nu^{5} + 127492 \nu^{4} + 163380 \nu^{3} + 66376 \nu^{2} + 66060 \nu - 22456 ) / 40260 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5362957 \nu^{11} + 9940372 \nu^{10} + 23030390 \nu^{9} + 1380680 \nu^{8} - 29137702 \nu^{7} - 231395340 \nu^{6} - 335137622 \nu^{5} + \cdots + 32870848 ) / 132012540 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8217712 \nu^{11} + 11072467 \nu^{10} + 42811220 \nu^{9} + 23030390 \nu^{8} - 47925592 \nu^{7} - 390717030 \nu^{6} - 691587212 \nu^{5} + \cdots + 35972008 ) / 132012540 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5614 \nu^{11} + 9729 \nu^{10} + 24140 \nu^{9} + 8260 \nu^{8} - 26994 \nu^{7} - 255750 \nu^{6} - 391604 \nu^{5} - 326622 \nu^{4} - 643980 \nu^{3} - 73556 \nu^{2} + \cdots + 68676 ) / 40260 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19394548 \nu^{11} + 36128113 \nu^{10} + 85199720 \nu^{9} + 6209960 \nu^{8} - 126266008 \nu^{7} - 871635270 \nu^{6} - 1181972888 \nu^{5} + \cdots + 277303252 ) / 132012540 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12537719 \nu^{11} - 22791104 \nu^{10} - 56749885 \nu^{9} - 5276635 \nu^{8} + 85217429 \nu^{7} + 561759990 \nu^{6} + 785090434 \nu^{5} + \cdots - 249596696 ) / 66006270 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15731201 \nu^{11} + 24513101 \nu^{10} + 73188085 \nu^{9} + 35555635 \nu^{8} - 80700356 \nu^{7} - 738220065 \nu^{6} - 1202248126 \nu^{5} + \cdots + 164311814 ) / 66006270 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16622963 \nu^{11} + 31633798 \nu^{10} + 68509300 \nu^{9} + 8136630 \nu^{8} - 92738998 \nu^{7} - 739730970 \nu^{6} - 1013258878 \nu^{5} + \cdots + 249278272 ) / 44004180 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{7} - 2\beta_{6} - \beta_{3} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{8} - 2\beta_{7} - \beta_{6} - 3\beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{11} + 7 \beta_{10} - \beta_{9} - 8 \beta_{8} - 16 \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 7 \beta_{2} + 7 \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{11} + 18 \beta_{10} - 9 \beta_{9} - 24 \beta_{8} - 34 \beta_{7} - 21 \beta_{6} - 16 \beta_{5} + 16 \beta_{4} + 8 \beta_{3} + 13 \beta_{2} + 9 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 54 \beta_{11} + 54 \beta_{10} - 42 \beta_{9} - 112 \beta_{8} - 124 \beta_{7} - 70 \beta_{6} - 34 \beta_{5} + 34 \beta_{4} + 36 \beta_{3} + 42 \beta_{2} + 2 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 152 \beta_{11} + 124 \beta_{10} - 124 \beta_{9} - 310 \beta_{8} - 310 \beta_{7} - 148 \beta_{6} - 112 \beta_{5} + 124 \beta_{4} + 188 \beta_{3} + 76 \beta_{2} - 62 \beta _1 - 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 436 \beta_{11} + 326 \beta_{10} - 436 \beta_{9} - 996 \beta_{8} - 874 \beta_{7} - 388 \beta_{6} - 224 \beta_{5} + 310 \beta_{4} + 636 \beta_{3} + 110 \beta_{2} - 334 \beta _1 - 438 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1080 \beta_{11} + 636 \beta_{10} - 1272 \beta_{9} - 2662 \beta_{8} - 1940 \beta_{7} - 722 \beta_{6} - 498 \beta_{5} + 874 \beta_{4} + 2076 \beta_{3} - 1510 \beta _1 - 1582 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2644 \beta_{11} + 942 \beta_{10} - 3586 \beta_{9} - 7104 \beta_{8} - 4084 \beta_{7} - 942 \beta_{6} - 722 \beta_{5} + 1940 \beta_{4} + 6248 \beta_{3} - 942 \beta_{2} - 5306 \beta _1 - 5524 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5306 \beta_{11} - 9130 \beta_{9} - 16376 \beta_{8} - 6028 \beta_{7} + 722 \beta_{6} + 4084 \beta_{4} + 17716 \beta_{3} - 5306 \beta_{2} - 17298 \beta _1 - 17098 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\beta_{8}\) \(-\beta_{6} - \beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.270517 + 0.0724848i
2.78638 + 0.746609i
−1.69087 0.453068i
0.270517 0.0724848i
2.78638 0.746609i
−1.69087 + 0.453068i
−0.208144 0.776804i
−0.431120 1.60896i
0.273238 + 1.01974i
−0.208144 + 0.776804i
−0.431120 + 1.60896i
0.273238 1.01974i
−0.500000 + 0.866025i −0.176529 + 0.658815i −0.500000 0.866025i 0.242559 + 2.22287i −0.482286 0.482286i 1.09930 4.10264i 1.00000 2.19520 + 1.26740i −2.04634 0.901374i
97.2 −0.500000 + 0.866025i 0.434319 1.62090i −0.500000 0.866025i 1.34690 + 1.78490i 1.18658 + 1.18658i −0.632846 + 2.36181i 1.00000 0.159392 + 0.0920251i −2.21922 + 0.274000i
97.3 −0.500000 + 0.866025i 0.608236 2.26997i −0.500000 0.866025i 0.642592 2.14175i 1.66173 + 1.66173i 0.265598 0.991227i 1.00000 −2.18472 1.26135i 1.53351 + 1.62737i
103.1 −0.500000 0.866025i −0.176529 0.658815i −0.500000 + 0.866025i 0.242559 2.22287i −0.482286 + 0.482286i 1.09930 + 4.10264i 1.00000 2.19520 1.26740i −2.04634 + 0.901374i
103.2 −0.500000 0.866025i 0.434319 + 1.62090i −0.500000 + 0.866025i 1.34690 1.78490i 1.18658 1.18658i −0.632846 2.36181i 1.00000 0.159392 0.0920251i −2.21922 0.274000i
103.3 −0.500000 0.866025i 0.608236 + 2.26997i −0.500000 + 0.866025i 0.642592 + 2.14175i 1.66173 1.66173i 0.265598 + 0.991227i 1.00000 −2.18472 + 1.26135i 1.53351 1.62737i
267.1 −0.500000 0.866025i −2.78144 + 0.745286i −0.500000 + 0.866025i −2.02520 + 0.947920i 2.03616 + 2.03616i −4.90683 + 1.31478i 1.00000 4.58291 2.64594i 1.83352 + 1.27992i
267.2 −0.500000 0.866025i −1.08428 + 0.290531i −0.500000 + 0.866025i 1.94546 + 1.10235i 0.793745 + 0.793745i −1.13508 + 0.304145i 1.00000 −1.50683 + 0.869970i −0.0180717 2.23599i
267.3 −0.500000 0.866025i 2.99969 0.803766i −0.500000 + 0.866025i −1.15231 1.91629i −2.19593 2.19593i 3.30986 0.886874i 1.00000 5.75405 3.32210i −1.08340 + 1.95608i
273.1 −0.500000 + 0.866025i −2.78144 0.745286i −0.500000 0.866025i −2.02520 0.947920i 2.03616 2.03616i −4.90683 1.31478i 1.00000 4.58291 + 2.64594i 1.83352 1.27992i
273.2 −0.500000 + 0.866025i −1.08428 0.290531i −0.500000 0.866025i 1.94546 1.10235i 0.793745 0.793745i −1.13508 0.304145i 1.00000 −1.50683 0.869970i −0.0180717 + 2.23599i
273.3 −0.500000 + 0.866025i 2.99969 + 0.803766i −0.500000 0.866025i −1.15231 + 1.91629i −2.19593 + 2.19593i 3.30986 + 0.886874i 1.00000 5.75405 + 3.32210i −1.08340 1.95608i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 273.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.d 12
5.c odd 4 1 370.2.r.d yes 12
37.g odd 12 1 370.2.r.d yes 12
185.p even 12 1 inner 370.2.q.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.d 12 1.a even 1 1 trivial
370.2.q.d 12 185.p even 12 1 inner
370.2.r.d yes 12 5.c odd 4 1
370.2.r.d yes 12 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 9 T_{3}^{10} + 6 T_{3}^{9} - 10 T_{3}^{8} - 96 T_{3}^{7} + 351 T_{3}^{6} + 522 T_{3}^{5} + 886 T_{3}^{4} + 2340 T_{3}^{3} + 2511 T_{3}^{2} + 1458 T_{3} + 729 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 9 T^{10} + 6 T^{9} - 10 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + 9 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} - 4 T^{10} + \cdots + 47524 \) Copy content Toggle raw display
$11$ \( T^{12} + 36 T^{10} + 428 T^{8} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{12} + 39 T^{10} + 92 T^{9} + \cdots + 97969 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + 38 T^{10} + \cdots + 47524 \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + 56 T^{10} + \cdots + 30976 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} - 26 T^{4} + 2 T^{3} + \cdots - 74)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 238144 \) Copy content Toggle raw display
$31$ \( T^{12} + 2 T^{11} + 2 T^{10} - 170 T^{9} + \cdots + 529 \) Copy content Toggle raw display
$37$ \( T^{12} - 14 T^{11} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} - 23 T^{10} + \cdots + 1390041 \) Copy content Toggle raw display
$43$ \( (T^{6} - 22 T^{5} - T^{4} + 2744 T^{3} + \cdots + 366553)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 28 T^{11} + \cdots + 16333351204 \) Copy content Toggle raw display
$53$ \( T^{12} - 8 T^{11} + \cdots + 24894212841 \) Copy content Toggle raw display
$59$ \( T^{12} + 18 T^{10} - 140 T^{9} + \cdots + 1747684 \) Copy content Toggle raw display
$61$ \( T^{12} - 24 T^{11} + \cdots + 898081024 \) Copy content Toggle raw display
$67$ \( T^{12} - 18 T^{11} + \cdots + 47924215056 \) Copy content Toggle raw display
$71$ \( T^{12} + 304 T^{10} + \cdots + 165408143616 \) Copy content Toggle raw display
$73$ \( T^{12} + 28 T^{11} + \cdots + 573985764 \) Copy content Toggle raw display
$79$ \( T^{12} + 56 T^{11} + \cdots + 4859205264 \) Copy content Toggle raw display
$83$ \( T^{12} - 14 T^{11} + \cdots + 6754209856 \) Copy content Toggle raw display
$89$ \( T^{12} + 2 T^{11} + \cdots + 2304384016 \) Copy content Toggle raw display
$97$ \( T^{12} + 640 T^{10} + \cdots + 234403159104 \) Copy content Toggle raw display
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