Properties

Label 912.2.ci.f
Level $912$
Weight $2$
Character orbit 912.ci
Analytic conductor $7.282$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.ci (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \( x^{18} - 6 x^{17} - 3 x^{16} + 100 x^{15} - 171 x^{14} - 471 x^{13} + 1537 x^{12} + 321 x^{11} - 3936 x^{10} - 1317 x^{9} + 4941 x^{8} + 21078 x^{7} - 14829 x^{6} + \cdots + 1367631 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{16} + \beta_{12} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{16} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{2}) q^{7} + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{16} + \beta_{12} - \beta_{5} - \beta_1) q^{5} + ( - \beta_{16} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{2}) q^{7} + \beta_{6} q^{9} + (\beta_{17} + \beta_{14} - \beta_{10} - \beta_{9} - \beta_{6} + \beta_{5}) q^{11} + (\beta_{15} - \beta_{13} + \beta_{12} + \beta_{6} - \beta_{2}) q^{13} + (\beta_{17} - \beta_{7}) q^{15} + (\beta_{17} + \beta_{14} - \beta_{13} + 2 \beta_{11} + \beta_{9} - \beta_{3}) q^{17} + ( - \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} - \beta_1 + 1) q^{19} + ( - \beta_{17} + \beta_{15} - \beta_{14} + \beta_{9} - \beta_{2} + 1) q^{21} + ( - \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} - 2 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 1) q^{23}+ \cdots + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{13} + 12 q^{17} + 3 q^{19} + 15 q^{21} + 6 q^{23} + 24 q^{25} - 9 q^{27} + 12 q^{29} + 12 q^{31} + 6 q^{33} - 36 q^{35} + 12 q^{41} + 21 q^{43} - 6 q^{45} - 24 q^{47} - 3 q^{49} - 6 q^{51} + 6 q^{53} + 12 q^{55} + 54 q^{59} - 24 q^{61} + 12 q^{63} + 36 q^{65} - 21 q^{67} + 15 q^{73} + 18 q^{75} - 60 q^{79} + 54 q^{85} + 18 q^{87} + 36 q^{89} + 24 q^{91} + 24 q^{93} - 6 q^{95} - 6 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} - 3 x^{16} + 100 x^{15} - 171 x^{14} - 471 x^{13} + 1537 x^{12} + 321 x^{11} - 3936 x^{10} - 1317 x^{9} + 4941 x^{8} + 21078 x^{7} - 14829 x^{6} + \cdots + 1367631 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97\!\cdots\!17 \nu^{17} + \cdots - 23\!\cdots\!81 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!28 \nu^{17} + \cdots - 24\!\cdots\!68 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!57 \nu^{17} + \cdots + 21\!\cdots\!99 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22\!\cdots\!40 \nu^{17} + \cdots + 18\!\cdots\!98 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!58 \nu^{17} + \cdots + 16\!\cdots\!11 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!50 \nu^{17} + \cdots + 38\!\cdots\!37 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!87 \nu^{17} + \cdots - 68\!\cdots\!41 ) / 32\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!92 \nu^{17} + \cdots + 68\!\cdots\!91 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 57\!\cdots\!91 \nu^{17} + \cdots + 19\!\cdots\!65 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66\!\cdots\!55 \nu^{17} + \cdots + 58\!\cdots\!23 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 68\!\cdots\!21 \nu^{17} + \cdots - 50\!\cdots\!22 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!65 \nu^{17} + \cdots - 13\!\cdots\!54 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 78\!\cdots\!46 \nu^{17} + \cdots - 83\!\cdots\!67 ) / 64\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!44 \nu^{17} + \cdots - 17\!\cdots\!25 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 13\!\cdots\!64 \nu^{17} + \cdots - 78\!\cdots\!81 ) / 87\!\cdots\!53 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 31\!\cdots\!36 \nu^{17} + \cdots - 21\!\cdots\!33 ) / 17\!\cdots\!37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 5 \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{17} - 4 \beta_{16} + \beta_{15} + 3 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} - 4 \beta_{11} - 11 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 14 \beta_{3} - 9 \beta_{2} + 9 \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{17} - 8 \beta_{16} - 7 \beta_{15} + 8 \beta_{14} - 14 \beta_{13} - 14 \beta_{12} - 6 \beta_{11} + 10 \beta_{10} - 17 \beta_{9} - 4 \beta_{8} + 15 \beta_{7} + 26 \beta_{6} + 10 \beta_{5} - 18 \beta_{4} + 23 \beta_{3} - 4 \beta_{2} + 21 \beta _1 - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{17} - 20 \beta_{16} - 7 \beta_{15} + 27 \beta_{14} - 3 \beta_{13} - 50 \beta_{12} - 13 \beta_{11} - 19 \beta_{10} - 13 \beta_{9} - 84 \beta_{8} + 15 \beta_{7} - 46 \beta_{6} + 19 \beta_{5} - 44 \beta_{4} + 50 \beta_{3} - 55 \beta_{2} + 7 \beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{17} + 59 \beta_{16} - 20 \beta_{15} + 16 \beta_{14} + \beta_{13} - 17 \beta_{12} - 32 \beta_{11} + 192 \beta_{10} + 24 \beta_{9} - 38 \beta_{8} + 36 \beta_{7} + 242 \beta_{6} + 49 \beta_{5} - 190 \beta_{4} - \beta_{3} - 16 \beta_{2} - 27 \beta _1 - 292 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 121 \beta_{17} + 135 \beta_{16} - 144 \beta_{15} + 78 \beta_{14} + 249 \beta_{13} + 44 \beta_{12} + 119 \beta_{11} - 27 \beta_{10} + 186 \beta_{9} - 503 \beta_{8} - 103 \beta_{7} - 332 \beta_{6} - 61 \beta_{5} - 379 \beta_{4} + \cdots - 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 208 \beta_{17} + 952 \beta_{16} - 496 \beta_{15} - 327 \beta_{14} + 693 \beta_{13} + 688 \beta_{12} + 137 \beta_{11} + 1478 \beta_{10} + 836 \beta_{9} + 306 \beta_{8} - 420 \beta_{7} + 1388 \beta_{6} - 296 \beta_{5} + \cdots - 1434 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1427 \beta_{17} + 1967 \beta_{16} - 1439 \beta_{15} - 416 \beta_{14} + 2290 \beta_{13} + 1723 \beta_{12} + 1723 \beta_{11} - 240 \beta_{10} + 2304 \beta_{9} - 1922 \beta_{8} - 2532 \beta_{7} - 2302 \beta_{6} + \cdots - 298 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1510 \beta_{17} + 7056 \beta_{16} - 4053 \beta_{15} - 3543 \beta_{14} + 4563 \beta_{13} + 6119 \beta_{12} + 2858 \beta_{11} + 7194 \beta_{10} + 6876 \beta_{9} + 9454 \beta_{8} - 5500 \beta_{7} + 10396 \beta_{6} + \cdots - 1030 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6211 \beta_{17} + 7531 \beta_{16} - 6772 \beta_{15} - 2826 \beta_{14} + 9171 \beta_{13} + 10096 \beta_{12} + 14735 \beta_{11} - 7096 \beta_{10} + 11228 \beta_{9} - 738 \beta_{8} - 19284 \beta_{7} + \cdots + 9582 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 755 \beta_{17} + 22457 \beta_{16} - 12245 \beta_{15} - 14276 \beta_{14} + 5827 \beta_{13} + 22486 \beta_{12} + 14605 \beta_{11} + 27444 \beta_{10} + 22155 \beta_{9} + 96934 \beta_{8} - 21933 \beta_{7} + \cdots + 37664 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11569 \beta_{17} - 47250 \beta_{16} + 33870 \beta_{15} + 4347 \beta_{14} - 13107 \beta_{13} + 8345 \beta_{12} + 60185 \beta_{11} - 71460 \beta_{10} - 22863 \beta_{9} + 21253 \beta_{8} - 60457 \beta_{7} + \cdots + 97034 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 25748 \beta_{17} - 116042 \beta_{16} + 136559 \beta_{15} - 31065 \beta_{14} - 93582 \beta_{13} - 23678 \beta_{12} - 91438 \beta_{11} + 50984 \beta_{10} - 95947 \beta_{9} + 676734 \beta_{8} + \cdots + 373716 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 47696 \beta_{17} - 956557 \beta_{16} + 854071 \beta_{15} + 63757 \beta_{14} - 322373 \beta_{13} - 223460 \beta_{12} - 188606 \beta_{11} - 531198 \beta_{10} - 759699 \beta_{9} - 250835 \beta_{8} + \cdots + 372131 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 150682 \beta_{17} - 1835490 \beta_{16} + 2049249 \beta_{15} - 317313 \beta_{14} - 557694 \beta_{13} - 635779 \beta_{12} - 2201278 \beta_{11} - 377322 \beta_{10} - 1456380 \beta_{9} + \cdots + 2261204 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.12178 + 1.48644i
−0.489413 1.79677i
−1.33811 + 1.29514i
1.12178 1.48644i
−0.489413 + 1.79677i
−1.33811 1.29514i
2.46004 + 0.909722i
−2.61262 0.231530i
2.26592 1.32098i
2.46004 0.909722i
−2.61262 + 0.231530i
2.26592 + 1.32098i
1.98709 0.839158i
1.76137 + 1.24510i
−2.15607 0.0639189i
1.98709 + 0.839158i
1.76137 1.24510i
−2.15607 + 0.0639189i
0 −0.939693 + 0.342020i 0 −0.189533 + 1.07490i 0 1.46125 0.843655i 0 0.766044 0.642788i 0
79.2 0 −0.939693 + 0.342020i 0 −0.0537040 + 0.304570i 0 −4.22544 + 2.43956i 0 0.766044 0.642788i 0
79.3 0 −0.939693 + 0.342020i 0 0.590533 3.34908i 0 1.12990 0.652348i 0 0.766044 0.642788i 0
127.1 0 −0.939693 0.342020i 0 −0.189533 1.07490i 0 1.46125 + 0.843655i 0 0.766044 + 0.642788i 0
127.2 0 −0.939693 0.342020i 0 −0.0537040 0.304570i 0 −4.22544 2.43956i 0 0.766044 + 0.642788i 0
127.3 0 −0.939693 0.342020i 0 0.590533 + 3.34908i 0 1.12990 + 0.652348i 0 0.766044 + 0.642788i 0
223.1 0 0.173648 0.984808i 0 −0.718121 0.602575i 0 0.983288 0.567702i 0 −0.939693 0.342020i 0
223.2 0 0.173648 0.984808i 0 −0.345737 0.290108i 0 −0.993418 + 0.573550i 0 −0.939693 0.342020i 0
223.3 0 0.173648 0.984808i 0 2.59595 + 2.17826i 0 −2.88040 + 1.66300i 0 −0.939693 0.342020i 0
319.1 0 0.173648 + 0.984808i 0 −0.718121 + 0.602575i 0 0.983288 + 0.567702i 0 −0.939693 + 0.342020i 0
319.2 0 0.173648 + 0.984808i 0 −0.345737 + 0.290108i 0 −0.993418 0.573550i 0 −0.939693 + 0.342020i 0
319.3 0 0.173648 + 0.984808i 0 2.59595 2.17826i 0 −2.88040 1.66300i 0 −0.939693 + 0.342020i 0
751.1 0 0.766044 0.642788i 0 −3.87454 1.41022i 0 3.15920 + 1.82397i 0 0.173648 0.984808i 0
751.2 0 0.766044 0.642788i 0 −1.03170 0.375509i 0 −0.450834 0.260289i 0 0.173648 0.984808i 0
751.3 0 0.766044 0.642788i 0 3.02685 + 1.10168i 0 1.81645 + 1.04873i 0 0.173648 0.984808i 0
895.1 0 0.766044 + 0.642788i 0 −3.87454 + 1.41022i 0 3.15920 1.82397i 0 0.173648 + 0.984808i 0
895.2 0 0.766044 + 0.642788i 0 −1.03170 + 0.375509i 0 −0.450834 + 0.260289i 0 0.173648 + 0.984808i 0
895.3 0 0.766044 + 0.642788i 0 3.02685 1.10168i 0 1.81645 1.04873i 0 0.173648 + 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 895.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ci.f yes 18
4.b odd 2 1 912.2.ci.e 18
19.f odd 18 1 912.2.ci.e 18
76.k even 18 1 inner 912.2.ci.f yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.e 18 4.b odd 2 1
912.2.ci.e 18 19.f odd 18 1
912.2.ci.f yes 18 1.a even 1 1 trivial
912.2.ci.f yes 18 76.k even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{18} - 12 T_{5}^{16} + 38 T_{5}^{15} + 48 T_{5}^{14} - 486 T_{5}^{13} + 2238 T_{5}^{12} - 306 T_{5}^{11} - 10332 T_{5}^{10} + 332 T_{5}^{9} + 61155 T_{5}^{8} + 159078 T_{5}^{7} + 239065 T_{5}^{6} + 237444 T_{5}^{5} + \cdots + 576 \) Copy content Toggle raw display
\( T_{7}^{18} - 30 T_{7}^{16} + 675 T_{7}^{14} - 855 T_{7}^{13} - 5477 T_{7}^{12} + 10431 T_{7}^{11} + 33117 T_{7}^{10} - 120555 T_{7}^{9} + 85032 T_{7}^{8} + 130302 T_{7}^{7} - 169952 T_{7}^{6} - 139707 T_{7}^{5} + \cdots + 34347 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} - 12 T^{16} + 38 T^{15} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{18} - 30 T^{16} + 675 T^{14} + \cdots + 34347 \) Copy content Toggle raw display
$11$ \( T^{18} - 33 T^{16} + 813 T^{14} + \cdots + 1728 \) Copy content Toggle raw display
$13$ \( T^{18} - 3 T^{17} - 18 T^{16} + \cdots + 162867 \) Copy content Toggle raw display
$17$ \( T^{18} - 12 T^{17} + \cdots + 268697664 \) Copy content Toggle raw display
$19$ \( T^{18} - 3 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} - 6 T^{17} + \cdots + 9032809152 \) Copy content Toggle raw display
$29$ \( T^{18} - 12 T^{17} + \cdots + 122712760512 \) Copy content Toggle raw display
$31$ \( T^{18} - 12 T^{17} + \cdots + 544949574849 \) Copy content Toggle raw display
$37$ \( T^{18} + 282 T^{16} + \cdots + 13733220843 \) Copy content Toggle raw display
$41$ \( T^{18} - 12 T^{17} + \cdots + 108865728 \) Copy content Toggle raw display
$43$ \( T^{18} - 21 T^{17} + \cdots + 11432149083 \) Copy content Toggle raw display
$47$ \( T^{18} + 24 T^{17} + \cdots + 7316338368 \) Copy content Toggle raw display
$53$ \( T^{18} - 6 T^{17} + \cdots + 3614768832 \) Copy content Toggle raw display
$59$ \( T^{18} - 54 T^{17} + \cdots + 7831445534784 \) Copy content Toggle raw display
$61$ \( T^{18} + 24 T^{17} + \cdots + 573075721 \) Copy content Toggle raw display
$67$ \( T^{18} + 21 T^{17} + \cdots + 7453904896 \) Copy content Toggle raw display
$71$ \( T^{18} + 378 T^{16} + \cdots + 4653477324864 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 855648672669729 \) Copy content Toggle raw display
$79$ \( T^{18} + 60 T^{17} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{18} - 381 T^{16} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{18} - 36 T^{17} + \cdots + 123685470912 \) Copy content Toggle raw display
$97$ \( T^{18} + 6 T^{17} + \cdots + 56816030267643 \) Copy content Toggle raw display
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