Properties

Label 18.7.d.a
Level $18$
Weight $7$
Character orbit 18.d
Analytic conductor $4.141$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.14097350516\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{3} + ( - 32 \beta_1 + 32) q^{4} + (\beta_{10} + \beta_{8} + \beta_{5} + 24 \beta_1 + 24) q^{5} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 13 \beta_1 - 5) q^{6} + (4 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - 7 \beta_{5} - 3 \beta_{4} + \cdots + 10) q^{7}+ \cdots + (2 \beta_{10} - \beta_{9} - 5 \beta_{8} - 4 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} + \cdots + 181) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{3} + ( - 32 \beta_1 + 32) q^{4} + (\beta_{10} + \beta_{8} + \beta_{5} + 24 \beta_1 + 24) q^{5} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 13 \beta_1 - 5) q^{6} + (4 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - 7 \beta_{5} - 3 \beta_{4} + \cdots + 10) q^{7}+ \cdots + (2637 \beta_{11} + 750 \beta_{10} - 1626 \beta_{9} + 7314 \beta_{8} + \cdots + 190140) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9} + 378 q^{11} + 384 q^{12} + 1680 q^{13} - 4752 q^{14} - 10872 q^{15} - 6144 q^{16} - 2976 q^{18} - 2820 q^{19} + 13824 q^{20} + 24876 q^{21} - 3600 q^{22} - 76248 q^{23} - 6144 q^{24} + 8094 q^{25} + 127008 q^{27} + 15360 q^{28} + 97092 q^{29} + 34272 q^{30} + 21480 q^{31} - 246258 q^{33} - 27360 q^{34} + 38208 q^{36} - 25536 q^{37} + 97632 q^{38} + 42204 q^{39} - 410562 q^{41} - 222144 q^{42} + 71430 q^{43} + 13716 q^{45} - 135072 q^{46} + 347652 q^{47} + 55296 q^{48} - 135954 q^{49} + 311040 q^{50} + 336402 q^{51} - 53760 q^{52} - 173520 q^{54} + 580392 q^{55} - 152064 q^{56} - 522282 q^{57} + 159264 q^{58} + 369738 q^{59} - 170496 q^{60} + 135744 q^{61} - 103800 q^{63} - 393216 q^{64} - 753840 q^{65} + 909216 q^{66} - 289938 q^{67} + 744768 q^{68} + 2059272 q^{69} + 155952 q^{70} - 374784 q^{72} - 977700 q^{73} - 2197152 q^{74} - 2115342 q^{75} - 45120 q^{76} - 159192 q^{77} - 631488 q^{78} - 764796 q^{79} - 1428282 q^{81} + 1073088 q^{82} + 396900 q^{83} + 1441536 q^{84} + 1619568 q^{85} + 3264624 q^{86} + 3072636 q^{87} + 115200 q^{88} - 1987200 q^{90} + 355584 q^{91} - 2439936 q^{92} - 2526576 q^{93} - 736848 q^{94} - 2089260 q^{95} - 49152 q^{96} - 38874 q^{97} + 4398804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 119 \nu^{11} - 59366 \nu^{9} - 10447223 \nu^{7} - 794976432 \nu^{5} - 25420007664 \nu^{3} - 246402608256 \nu + 25705589760 ) / 51411179520 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79977 \nu^{11} - 234158 \nu^{10} - 22976562 \nu^{9} - 160687484 \nu^{8} - 2207867961 \nu^{7} - 34293527246 \nu^{6} + \cdots - 10\!\cdots\!40 ) / 10520012609280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79977 \nu^{11} - 234158 \nu^{10} + 22976562 \nu^{9} - 160687484 \nu^{8} + 2207867961 \nu^{7} - 34293527246 \nu^{6} + \cdots - 10\!\cdots\!40 ) / 10520012609280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2018785 \nu^{11} + 50427040 \nu^{10} + 441850954 \nu^{9} + 13150349632 \nu^{8} + 16441271137 \nu^{7} + 1021898930848 \nu^{6} + \cdots + 62\!\cdots\!40 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2658601 \nu^{11} + 49792016 \nu^{10} + 625663450 \nu^{9} + 13943243552 \nu^{8} + 34104214825 \nu^{7} + 1231527538448 \nu^{6} + \cdots - 76\!\cdots\!20 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3789367 \nu^{11} + 91688548 \nu^{10} + 1383048298 \nu^{9} + 27588769336 \nu^{8} + 185839870831 \nu^{7} + 2837672924164 \nu^{6} + \cdots + 13\!\cdots\!40 ) / 21040025218560 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18455885 \nu^{11} + 113412560 \nu^{10} + 6341669138 \nu^{9} + 38860753760 \nu^{8} + 795126641837 \nu^{7} + 4853986176080 \nu^{6} + \cdots + 60\!\cdots\!80 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3697973 \nu^{11} - 10394400 \nu^{10} - 1223942794 \nu^{9} - 4111867872 \nu^{8} - 146365636165 \nu^{7} - 598349805408 \nu^{6} + \cdots - 95\!\cdots\!60 ) / 14026683479040 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2357021 \nu^{11} - 26258072 \nu^{10} + 793738762 \nu^{9} - 9474316112 \nu^{8} + 95300421325 \nu^{7} - 1246411646168 \nu^{6} + \cdots - 16\!\cdots\!60 ) / 7013341739520 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30942853 \nu^{11} + 63612592 \nu^{10} - 10150528594 \nu^{9} + 6352050784 \nu^{8} - 1177709270725 \nu^{7} + \cdots - 76\!\cdots\!60 ) / 84160100874240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 93257311 \nu^{11} - 164458720 \nu^{10} + 31877901190 \nu^{9} - 53050300288 \nu^{8} + 3970946505535 \nu^{7} + \cdots - 64\!\cdots\!80 ) / 84160100874240 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{8} + 3\beta_{7} - \beta_{6} + 3\beta_{5} - 5\beta_{3} + 4\beta_{2} - 10\beta _1 + 2 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - 11 \beta_{7} + 3 \beta_{6} - 7 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 21 \beta_{2} - 2 \beta _1 - 1104 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 36 \beta_{11} + 24 \beta_{10} - 24 \beta_{9} - 154 \beta_{8} - 249 \beta_{7} + 83 \beta_{6} - 285 \beta_{5} - 12 \beta_{4} - 368 \beta_{3} + 463 \beta_{2} + 3602 \beta _1 - 1498 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 394 \beta_{11} + 346 \beta_{10} + 346 \beta_{9} - 394 \beta_{8} + 2191 \beta_{7} - 615 \beta_{6} + 671 \beta_{5} - 940 \beta_{4} + 1450 \beta_{3} + 4581 \beta_{2} + 394 \beta _1 + 98760 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6804 \beta_{11} - 4536 \beta_{10} + 4536 \beta_{9} + 13802 \beta_{8} + 25677 \beta_{7} - 9607 \beta_{6} + 37197 \beta_{5} + 3840 \beta_{4} + 81832 \beta_{3} - 95279 \beta_{2} - 635866 \beta _1 + 278906 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 61598 \beta_{11} - 49382 \beta_{10} - 49382 \beta_{9} + 61598 \beta_{8} - 344663 \beta_{7} + 98271 \beta_{6} - 58987 \beta_{5} + 107876 \beta_{4} - 333770 \beta_{3} - 786309 \beta_{2} + \cdots - 10730004 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1082880 \beta_{11} + 738660 \beta_{10} - 738660 \beta_{9} - 1382062 \beta_{8} - 2994321 \beta_{7} + 1217819 \beta_{6} - 5116125 \beta_{5} - 724008 \beta_{4} - 12377120 \beta_{3} + \cdots - 43323814 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8984866 \beta_{11} + 6747970 \beta_{10} + 6747970 \beta_{9} - 8984866 \beta_{8} + 50477575 \beta_{7} - 14538111 \beta_{6} + 5493695 \beta_{5} - 13283836 \beta_{4} + \cdots + 1299912672 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 161899740 \beta_{11} - 112694880 \beta_{10} + 112694880 \beta_{9} + 152703602 \beta_{8} + 376483773 \beta_{7} - 160290151 \beta_{6} + 709248693 \beta_{5} + \cdots + 6352973042 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1275517622 \beta_{11} - 917170862 \beta_{10} - 917170862 \beta_{9} + 1275517622 \beta_{8} - 7189238831 \beta_{7} + 2087168343 \beta_{6} - 556996747 \beta_{5} + \cdots - 167725156524 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 23463449496 \beta_{11} + 16548163164 \beta_{10} - 16548163164 \beta_{9} - 18294797254 \beta_{8} - 49497749649 \beta_{7} + 21570075563 \beta_{6} + \cdots - 909341860558 ) / 18 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
4.28281i
7.20150i
11.8022i
8.15670i
8.88570i
3.87527i
4.28281i
7.20150i
11.8022i
8.15670i
8.88570i
3.87527i
−4.89898 + 2.82843i −22.5447 14.8572i 16.0000 27.7128i 156.951 + 90.6160i 152.469 + 9.01919i 104.306 + 180.663i 181.019i 287.526 + 669.903i −1025.20
5.2 −4.89898 + 2.82843i −11.1983 + 24.5683i 16.0000 27.7128i −39.5602 22.8401i −14.6295 152.033i −245.097 424.521i 181.019i −478.198 550.243i 258.406
5.3 −4.89898 + 2.82843i 25.6924 + 8.30052i 16.0000 27.7128i −9.39126 5.42205i −149.344 + 32.0051i 322.041 + 557.792i 181.019i 591.203 + 426.521i 61.3435
5.4 4.89898 2.82843i −23.6571 13.0130i 16.0000 27.7128i −95.8504 55.3393i −152.702 + 3.16189i −163.169 282.617i 181.019i 390.321 + 615.703i −626.092
5.5 4.89898 2.82843i −14.9408 + 22.4894i 16.0000 27.7128i 202.253 + 116.771i −9.58533 + 152.434i 95.5752 + 165.541i 181.019i −282.543 672.020i 1321.11
5.6 4.89898 2.82843i 25.6485 8.43532i 16.0000 27.7128i 1.59771 + 0.922438i 101.793 113.869i 6.34411 + 10.9883i 181.019i 586.691 432.707i 10.4362
11.1 −4.89898 2.82843i −22.5447 + 14.8572i 16.0000 + 27.7128i 156.951 90.6160i 152.469 9.01919i 104.306 180.663i 181.019i 287.526 669.903i −1025.20
11.2 −4.89898 2.82843i −11.1983 24.5683i 16.0000 + 27.7128i −39.5602 + 22.8401i −14.6295 + 152.033i −245.097 + 424.521i 181.019i −478.198 + 550.243i 258.406
11.3 −4.89898 2.82843i 25.6924 8.30052i 16.0000 + 27.7128i −9.39126 + 5.42205i −149.344 32.0051i 322.041 557.792i 181.019i 591.203 426.521i 61.3435
11.4 4.89898 + 2.82843i −23.6571 + 13.0130i 16.0000 + 27.7128i −95.8504 + 55.3393i −152.702 3.16189i −163.169 + 282.617i 181.019i 390.321 615.703i −626.092
11.5 4.89898 + 2.82843i −14.9408 22.4894i 16.0000 + 27.7128i 202.253 116.771i −9.58533 152.434i 95.5752 165.541i 181.019i −282.543 + 672.020i 1321.11
11.6 4.89898 + 2.82843i 25.6485 + 8.43532i 16.0000 + 27.7128i 1.59771 0.922438i 101.793 + 113.869i 6.34411 10.9883i 181.019i 586.691 + 432.707i 10.4362
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 18.7.d.a 12
3.b odd 2 1 54.7.d.a 12
4.b odd 2 1 144.7.q.c 12
9.c even 3 1 54.7.d.a 12
9.c even 3 1 162.7.b.c 12
9.d odd 6 1 inner 18.7.d.a 12
9.d odd 6 1 162.7.b.c 12
12.b even 2 1 432.7.q.b 12
36.f odd 6 1 432.7.q.b 12
36.h even 6 1 144.7.q.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.7.d.a 12 1.a even 1 1 trivial
18.7.d.a 12 9.d odd 6 1 inner
54.7.d.a 12 3.b odd 2 1
54.7.d.a 12 9.c even 3 1
144.7.q.c 12 4.b odd 2 1
144.7.q.c 12 36.h even 6 1
162.7.b.c 12 9.c even 3 1
162.7.b.c 12 9.d odd 6 1
432.7.q.b 12 12.b even 2 1
432.7.q.b 12 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(18, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 32 T^{2} + 1024)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + 42 T^{11} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} - 432 T^{11} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} - 240 T^{11} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} - 378 T^{11} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$13$ \( T^{12} - 1680 T^{11} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{12} + 215347950 T^{10} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + 1410 T^{5} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 76248 T^{11} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} - 97092 T^{11} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 21480 T^{11} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + 12768 T^{5} + \cdots + 54\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 410562 T^{11} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} - 71430 T^{11} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} - 347652 T^{11} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + 154278905256 T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} - 369738 T^{11} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} - 135744 T^{11} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + 289938 T^{11} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{12} + 745338015792 T^{10} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} + 488850 T^{5} + \cdots - 49\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 764796 T^{11} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} - 396900 T^{11} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{12} + 1318308583464 T^{10} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + 38874 T^{11} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
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