Properties

Label 1620.4.i.x
Level $1620$
Weight $4$
Character orbit 1620.i
Analytic conductor $95.583$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 5 \beta_{6} q^{5} + (\beta_{8} - 2 \beta_{6} - \beta_{2} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 \beta_{6} q^{5} + (\beta_{8} - 2 \beta_{6} - \beta_{2} - 2) q^{7} + ( - \beta_{10} + \beta_{9} + \beta_{7}) q^{11} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{7} - 14 \beta_{6} - \beta_{5} + \beta_{4}) q^{13} + (\beta_{5} + 5 \beta_{2} + 3 \beta_1 + 2) q^{17} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_1 - 19) q^{19} + ( - \beta_{11} - \beta_{10} - 7 \beta_{8} + 2 \beta_{7} + 5 \beta_{6} + 2 \beta_{5} - \beta_{4}) q^{23} + ( - 25 \beta_{6} - 25) q^{25} + ( - 4 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - 3 \beta_{7} - 28 \beta_{6} + \cdots - 28) q^{29}+ \cdots + (19 \beta_{11} + 4 \beta_{10} - 16 \beta_{9} + 27 \beta_{8} + 13 \beta_{7} + \cdots + 198) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 30 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 30 q^{5} - 12 q^{7} + 84 q^{13} + 24 q^{17} - 228 q^{19} - 30 q^{23} - 150 q^{25} - 168 q^{29} + 324 q^{31} - 120 q^{35} - 984 q^{37} - 312 q^{41} + 156 q^{43} - 462 q^{47} + 588 q^{49} + 2028 q^{53} - 1008 q^{59} - 36 q^{61} - 420 q^{65} - 144 q^{67} + 2424 q^{71} - 1800 q^{73} - 672 q^{77} + 936 q^{79} - 288 q^{83} + 60 q^{85} - 240 q^{89} + 4572 q^{91} - 570 q^{95} + 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1291 \nu^{10} - 383792 \nu^{8} - 35790499 \nu^{6} - 1330336354 \nu^{4} - 18065389388 \nu^{2} - 45652500136 ) / 1027615680 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5102693 \nu^{10} + 1398025756 \nu^{8} + 112247897357 \nu^{6} + 3452478375842 \nu^{4} + 38122728094324 \nu^{2} + \cdots + 83013902159528 ) / 1103659240320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44857 \nu^{10} + 12522644 \nu^{8} + 1045133377 \nu^{6} + 34081870402 \nu^{4} + 393573716612 \nu^{2} + 423568473640 ) / 8489686464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 119695 \nu^{10} + 32147810 \nu^{8} + 2466092659 \nu^{6} + 69043953676 \nu^{4} + 581554032092 \nu^{2} - 136972654160 ) / 18394320672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10810771 \nu^{10} - 2947200752 \nu^{8} - 234983924419 \nu^{6} - 7244547031114 \nu^{4} - 80864075604908 \nu^{2} + \cdots - 153225963267016 ) / 1103659240320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23357 \nu^{11} - 6338239 \nu^{9} - 500676503 \nu^{7} - 15279946733 \nu^{5} - 175329177676 \nu^{3} - 573920985812 \nu - 233206555680 ) / 466413111360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3422391862 \nu^{11} + 155253482331 \nu^{10} + 678724169054 \nu^{9} + 42324749999472 \nu^{8} + 8571410900578 \nu^{7} + \cdots + 22\!\cdots\!76 ) / 31\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1657489291 \nu^{11} + 24426591391 \nu^{10} + 498195552242 \nu^{9} + 6692349293972 \nu^{8} + 47646796509679 \nu^{7} + \cdots + 39\!\cdots\!36 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 867417539 \nu^{11} + 3220956885 \nu^{10} - 242242649518 \nu^{9} + 899188452420 \nu^{8} - 20179270104191 \nu^{7} + \cdots + 30\!\cdots\!00 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4899606623 \nu^{11} - 17189398950 \nu^{10} + 1275959537596 \nu^{9} - 4616746994100 \nu^{8} + 90300640751687 \nu^{7} + \cdots + 19\!\cdots\!00 ) / 52\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1866498795 \nu^{11} - 1106223043 \nu^{10} - 516891193560 \nu^{9} - 328861002416 \nu^{8} - 42548567539275 \nu^{7} + \cdots - 39\!\cdots\!28 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{10} + 2\beta_{9} - 4\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{5} + 3\beta_{4} - 21\beta_{3} - 26\beta _1 - 918 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 408 \beta_{11} - 164 \beta_{10} - 314 \beta_{9} + 300 \beta_{8} + 304 \beta_{7} - 5616 \beta_{6} + 152 \beta_{5} - 82 \beta_{4} + 157 \beta_{3} - 150 \beta_{2} - 204 \beta _1 - 2808 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1398\beta_{5} - 1131\beta_{4} + 3507\beta_{3} + 1980\beta_{2} + 5278\beta _1 + 96246 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 79160 \beta_{11} + 20624 \beta_{10} + 51938 \beta_{9} - 49668 \beta_{8} - 33208 \beta_{7} + 1092240 \beta_{6} - 16604 \beta_{5} + 10312 \beta_{4} - 25969 \beta_{3} + 24834 \beta_{2} + \cdots + 546120 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -88698\beta_{5} + 68253\beta_{4} - 190725\beta_{3} - 147780\beta_{2} - 300898\beta _1 - 4542450 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13312824 \beta_{11} - 3150776 \beta_{10} - 8464034 \beta_{9} + 7729524 \beta_{8} + 4620712 \beta_{7} - 185682672 \beta_{6} + 2310356 \beta_{5} - 1575388 \beta_{4} + \cdots - 92841336 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 45823038 \beta_{5} - 33969423 \beta_{4} + 93192375 \beta_{3} + 78683724 \beta_{2} + 148284326 \beta _1 + 2128577670 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2181126936 \beta_{11} + 505095320 \beta_{10} + 1375227746 \beta_{9} - 1225681140 \beta_{8} - 713179240 \beta_{7} + 30601018512 \beta_{6} + \cdots + 15300509256 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7602088494 \beta_{5} + 5545453335 \beta_{4} - 15162026055 \beta_{3} - 13140890748 \beta_{2} - 24146228950 \beta _1 - 341966456118 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 355230206072 \beta_{11} - 81835116872 \beta_{10} - 223422680258 \beta_{9} + 197156937876 \beta_{8} + 114022803976 \beta_{7} - 4996477881648 \beta_{6} + \cdots - 2498238940824 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
1.37492i
5.63924i
4.35846i
12.7486i
6.01416i
7.39017i
1.37492i
5.63924i
4.35846i
12.7486i
6.01416i
7.39017i
0 0 0 2.50000 + 4.33013i 0 −11.2133 + 19.4221i 0 0 0
541.2 0 0 0 2.50000 + 4.33013i 0 −8.28565 + 14.3512i 0 0 0
541.3 0 0 0 2.50000 + 4.33013i 0 −3.10707 + 5.38160i 0 0 0
541.4 0 0 0 2.50000 + 4.33013i 0 −0.292449 + 0.506536i 0 0 0
541.5 0 0 0 2.50000 + 4.33013i 0 5.24065 9.07707i 0 0 0
541.6 0 0 0 2.50000 + 4.33013i 0 11.6578 20.1920i 0 0 0
1081.1 0 0 0 2.50000 4.33013i 0 −11.2133 19.4221i 0 0 0
1081.2 0 0 0 2.50000 4.33013i 0 −8.28565 14.3512i 0 0 0
1081.3 0 0 0 2.50000 4.33013i 0 −3.10707 5.38160i 0 0 0
1081.4 0 0 0 2.50000 4.33013i 0 −0.292449 0.506536i 0 0 0
1081.5 0 0 0 2.50000 4.33013i 0 5.24065 + 9.07707i 0 0 0
1081.6 0 0 0 2.50000 4.33013i 0 11.6578 + 20.1920i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1081.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.4.i.x 12
3.b odd 2 1 1620.4.i.w 12
9.c even 3 1 1620.4.a.i 6
9.c even 3 1 inner 1620.4.i.x 12
9.d odd 6 1 1620.4.a.j yes 6
9.d odd 6 1 1620.4.i.w 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.4.a.i 6 9.c even 3 1
1620.4.a.j yes 6 9.d odd 6 1
1620.4.i.w 12 3.b odd 2 1
1620.4.i.w 12 9.d odd 6 1
1620.4.i.x 12 1.a even 1 1 trivial
1620.4.i.x 12 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{12} + 12 T_{7}^{11} + 807 T_{7}^{10} + 7612 T_{7}^{9} + 465309 T_{7}^{8} + 4143240 T_{7}^{7} + 98836980 T_{7}^{6} + 281425968 T_{7}^{5} + 9518654844 T_{7}^{4} + 46178285440 T_{7}^{3} + \cdots + 108966010000 \) Copy content Toggle raw display
\( T_{11}^{12} + 5826 T_{11}^{10} + 120960 T_{11}^{9} + 27242307 T_{11}^{8} + 421485120 T_{11}^{7} + 39574013994 T_{11}^{6} + 400272788160 T_{11}^{5} + 39988229062761 T_{11}^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{11} + \cdots + 108966010000 \) Copy content Toggle raw display
$11$ \( T^{12} + 5826 T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} - 84 T^{11} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} - 12 T^{5} - 14628 T^{4} + \cdots + 20589584400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 114 T^{5} + \cdots - 171382733975)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 30 T^{11} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + 168 T^{11} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{12} - 324 T^{11} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{6} + 492 T^{5} + \cdots + 147577423801600)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 312 T^{11} + \cdots + 30\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{12} - 156 T^{11} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + 462 T^{11} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} - 1014 T^{5} + \cdots + 24472129608900)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 1008 T^{11} + \cdots + 35\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{12} + 36 T^{11} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + 144 T^{11} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} - 1212 T^{5} + \cdots - 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 900 T^{5} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 936 T^{11} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{12} + 288 T^{11} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + 120 T^{5} + \cdots + 157896529778448)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 1188 T^{11} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
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