Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(257,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 9, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.cc (of order \(18\), degree \(6\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + 162 x^{7} + 270 x^{6} + 2916 x^{5} - 4374 x^{4} - 729 x^{3} + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{3} + (\beta_{15} - \beta_{14} - \beta_{2}) q^{5} + ( - \beta_{16} + \beta_{15} - \beta_{6} - \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{17} - \beta_{12} - \beta_{10} - 2 \beta_{8} - \beta_{4} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{3} + (\beta_{15} - \beta_{14} - \beta_{2}) q^{5} + ( - \beta_{16} + \beta_{15} - \beta_{6} - \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{17} - \beta_{12} - \beta_{10} - 2 \beta_{8} - \beta_{4} - \beta_1) q^{9} + (\beta_{17} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{4}) q^{11} + (\beta_{13} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{13} - \beta_{11} + 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{2} - 1) q^{15} + ( - \beta_{17} - \beta_{16} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{16} + 2 \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{5} + \beta_{4} + \cdots - 1) q^{19}+ \cdots + (\beta_{17} - \beta_{16} + 3 \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} - 2 \beta_{9} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{3} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{3} - 3 q^{9} - 12 q^{13} - 18 q^{15} + 6 q^{17} + 6 q^{19} - 18 q^{25} + 6 q^{27} - 6 q^{29} - 24 q^{33} + 24 q^{35} - 6 q^{39} + 3 q^{41} + 6 q^{43} - 54 q^{45} - 30 q^{47} + 21 q^{49} - 42 q^{51} - 60 q^{53} - 30 q^{55} + 12 q^{57} - 3 q^{59} + 54 q^{61} + 18 q^{63} + 24 q^{65} + 15 q^{67} + 30 q^{69} - 36 q^{71} - 42 q^{73} + 6 q^{79} - 3 q^{81} - 36 q^{83} - 60 q^{89} + 18 q^{91} - 66 q^{93} - 6 q^{95} + 9 q^{97} + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + 162 x^{7} + 270 x^{6} + 2916 x^{5} - 4374 x^{4} - 729 x^{3} + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{17} + \nu^{14} + 18 \nu^{13} - 36 \nu^{12} - 10 \nu^{11} - 18 \nu^{10} - 90 \nu^{9} + 567 \nu^{8} - 270 \nu^{7} - 162 \nu^{6} - 270 \nu^{5} - 2916 \nu^{4} + 4374 \nu^{3} + 729 \nu^{2} ) / 6561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{17} + 21 \nu^{16} - 81 \nu^{15} + 97 \nu^{14} - 57 \nu^{13} - 198 \nu^{12} + 1568 \nu^{11} - 3075 \nu^{10} + 2313 \nu^{9} - 117 \nu^{8} - 9261 \nu^{7} + 29268 \nu^{6} - 31374 \nu^{5} + \cdots - 65610 ) / 4374 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{17} + 99 \nu^{16} - 153 \nu^{15} - 22 \nu^{14} + 207 \nu^{13} - 1485 \nu^{12} + 4675 \nu^{11} - 2637 \nu^{10} - 4167 \nu^{9} + 10098 \nu^{8} - 38475 \nu^{7} + 51273 \nu^{6} + \cdots + 334611 ) / 13122 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} - 27 \nu^{15} + 50 \nu^{14} - 47 \nu^{13} + 9 \nu^{12} + 409 \nu^{11} - 1366 \nu^{10} + 1629 \nu^{9} - 822 \nu^{8} - 1809 \nu^{7} + 11025 \nu^{6} - 19359 \nu^{5} + 12420 \nu^{4} + \cdots - 72171 ) / 1458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16 \nu^{17} + 81 \nu^{16} + 36 \nu^{15} - 335 \nu^{14} + 477 \nu^{13} - 1422 \nu^{12} + 1406 \nu^{11} + 6165 \nu^{10} - 12420 \nu^{9} + 12609 \nu^{8} - 20493 \nu^{7} - 25272 \nu^{6} + \cdots + 603612 ) / 13122 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{17} - 3 \nu^{16} - 6 \nu^{15} + 92 \nu^{14} + 48 \nu^{13} - 255 \nu^{12} - 137 \nu^{11} - 1425 \nu^{10} + 3108 \nu^{9} + 1350 \nu^{8} - 504 \nu^{7} + 3645 \nu^{6} - 36045 \nu^{5} + \cdots - 225261 ) / 4374 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{17} - 105 \nu^{16} + 90 \nu^{15} + 275 \nu^{14} - 318 \nu^{13} + 1296 \nu^{12} - 4181 \nu^{11} - 2247 \nu^{10} + 12492 \nu^{9} - 11367 \nu^{8} + 36126 \nu^{7} - 25110 \nu^{6} + \cdots - 787320 ) / 13122 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{17} + 147 \nu^{16} - 216 \nu^{15} - 43 \nu^{14} + 375 \nu^{13} - 2259 \nu^{12} + 6451 \nu^{11} - 3183 \nu^{10} - 6660 \nu^{9} + 15525 \nu^{8} - 52947 \nu^{7} + 64233 \nu^{6} + \cdots + 452709 ) / 13122 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{17} - 5 \nu^{16} + 99 \nu^{15} - 136 \nu^{14} + 284 \nu^{13} - 405 \nu^{12} - 1655 \nu^{11} + 4369 \nu^{10} - 4167 \nu^{9} + 5472 \nu^{8} + 5508 \nu^{7} - 41229 \nu^{6} + \cdots + 63423 ) / 4374 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4 \nu^{17} + 17 \nu^{16} - 10 \nu^{15} - 23 \nu^{14} + 82 \nu^{13} - 332 \nu^{12} + 536 \nu^{11} + 197 \nu^{10} - 1234 \nu^{9} + 2619 \nu^{8} - 5760 \nu^{7} + 2430 \nu^{6} + 9666 \nu^{5} + \cdots + 56862 ) / 1458 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23 \nu^{17} + 10 \nu^{16} + 6 \nu^{15} + 86 \nu^{14} + 161 \nu^{13} - 717 \nu^{12} - 41 \nu^{11} - 908 \nu^{10} + 2814 \nu^{9} + 4716 \nu^{8} - 4941 \nu^{7} - 1377 \nu^{6} - 32373 \nu^{5} + \cdots - 247131 ) / 4374 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9 \nu^{17} - 4 \nu^{16} + 39 \nu^{15} - 21 \nu^{14} + 94 \nu^{13} - 174 \nu^{12} - 717 \nu^{11} + 1220 \nu^{10} - 393 \nu^{9} + 1797 \nu^{8} + 2520 \nu^{7} - 14598 \nu^{6} + 6345 \nu^{5} + \cdots - 34992 ) / 1458 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 76 \nu^{17} - 108 \nu^{16} + 459 \nu^{15} - 194 \nu^{14} + 747 \nu^{13} - 522 \nu^{12} - 9724 \nu^{11} + 13104 \nu^{10} - 1521 \nu^{9} + 9774 \nu^{8} + 50193 \nu^{7} - 167832 \nu^{6} + \cdots - 551124 ) / 13122 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11 \nu^{17} - 17 \nu^{16} - 3 \nu^{15} + 13 \nu^{14} - 145 \nu^{13} + 525 \nu^{12} - 283 \nu^{11} - 413 \nu^{10} + 807 \nu^{9} - 4125 \nu^{8} + 5787 \nu^{7} + 2817 \nu^{6} - 5103 \nu^{5} + \cdots + 10935 ) / 1458 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 103 \nu^{17} - 99 \nu^{16} - 27 \nu^{15} - 157 \nu^{14} - 1026 \nu^{13} + 4140 \nu^{12} - 1265 \nu^{11} + 621 \nu^{10} - 3555 \nu^{9} - 30429 \nu^{8} + 39960 \nu^{7} + 12150 \nu^{6} + \cdots + 669222 ) / 13122 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17 \nu^{17} + 10 \nu^{16} + 27 \nu^{15} - 10 \nu^{14} + 188 \nu^{13} - 576 \nu^{12} - 269 \nu^{11} + 874 \nu^{10} - 351 \nu^{9} + 4680 \nu^{8} - 3078 \nu^{7} - 10746 \nu^{6} + 1161 \nu^{5} + \cdots - 78732 ) / 1458 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} + \beta_{15} + \beta_{9} - \beta_{8} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - 4\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 3 \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} - 2 \beta_{16} + \beta_{15} + 3 \beta_{13} + 3 \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 6 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{16} + 10 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 2 \beta_{7} + 2 \beta_{6} + 5 \beta_{3} + \beta_{2} - 9 \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} + 15 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 9 \beta_{9} + 18 \beta_{8} - 12 \beta_{7} + 7 \beta_{5} + 36 \beta_{4} + 15 \beta_{2} - 3 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18 \beta_{17} - \beta_{16} + 15 \beta_{15} - 9 \beta_{14} + 21 \beta_{12} + 24 \beta_{11} + 21 \beta_{10} - 11 \beta_{9} + 5 \beta_{8} + 36 \beta_{7} + 18 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} - 14 \beta_{3} - 18 \beta_{2} + 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 27 \beta_{17} - 27 \beta_{16} + 11 \beta_{15} + \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - 30 \beta_{11} + 6 \beta_{10} - 27 \beta_{9} - 36 \beta_{8} + 74 \beta_{7} + 76 \beta_{6} - 7 \beta_{5} + 13 \beta_{3} + 44 \beta_{2} + 27 \beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 33 \beta_{17} - 54 \beta_{16} + 63 \beta_{15} + 24 \beta_{14} - 89 \beta_{13} + \beta_{12} + 2 \beta_{11} - 10 \beta_{10} - 27 \beta_{9} + 18 \beta_{8} + 9 \beta_{7} + 66 \beta_{6} - 76 \beta_{5} - 18 \beta_{4} + 63 \beta_{3} + 9 \beta_{2} - 67 \beta _1 + 140 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15 \beta_{17} + 100 \beta_{16} - 103 \beta_{15} + 126 \beta_{14} - 129 \beta_{13} + 102 \beta_{12} - 30 \beta_{11} - 63 \beta_{10} - \beta_{9} - 2 \beta_{8} - 18 \beta_{7} + 18 \beta_{6} - 9 \beta_{5} + 68 \beta_{4} - 51 \beta_{3} + 63 \beta_{2} + \cdots + 120 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 90 \beta_{17} - 81 \beta_{16} + 136 \beta_{15} - 154 \beta_{14} + 33 \beta_{13} + 249 \beta_{12} + 129 \beta_{11} + 111 \beta_{10} + 36 \beta_{9} - 342 \beta_{8} + 148 \beta_{7} + 2 \beta_{6} - 34 \beta_{5} - 612 \beta_{4} + 17 \beta_{3} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 62 \beta_{17} - 171 \beta_{16} - 189 \beta_{15} + 153 \beta_{14} - 19 \beta_{13} - 100 \beta_{12} - 362 \beta_{11} - 101 \beta_{10} - 99 \beta_{9} - 648 \beta_{8} - 96 \beta_{7} - 6 \beta_{6} - 29 \beta_{5} - 486 \beta_{4} + 432 \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 372 \beta_{17} - 106 \beta_{16} + 206 \beta_{15} + 243 \beta_{14} - 426 \beta_{13} + 9 \beta_{12} + 450 \beta_{11} - 39 \beta_{10} - 80 \beta_{9} + 380 \beta_{8} - 1026 \beta_{7} - 756 \beta_{6} - 525 \beta_{5} - 29 \beta_{4} + 710 \beta_{3} + \cdots + 354 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 351 \beta_{17} + 765 \beta_{16} - 802 \beta_{15} + 916 \beta_{14} + 46 \beta_{13} + 327 \beta_{12} + 402 \beta_{11} - 624 \beta_{10} + 351 \beta_{9} - 378 \beta_{8} - 376 \beta_{7} - 974 \beta_{6} + 522 \beta_{5} + 270 \beta_{4} + \cdots - 2208 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 878 \beta_{17} - 630 \beta_{16} + 927 \beta_{15} - 1554 \beta_{14} + 1726 \beta_{13} + 430 \beta_{12} + 2318 \beta_{11} + 998 \beta_{10} - 423 \beta_{9} - 558 \beta_{8} + 390 \beta_{7} - 351 \beta_{6} + 326 \beta_{5} + \cdots - 4210 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 918 \beta_{17} + 406 \beta_{16} - 3786 \beta_{15} + 1656 \beta_{14} - 378 \beta_{13} - 3126 \beta_{12} - 1536 \beta_{11} - 1074 \beta_{10} - 3040 \beta_{9} + 1750 \beta_{8} + 504 \beta_{7} + 738 \beta_{6} - 192 \beta_{5} + \cdots + 54 \) Copy content Toggle raw display

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{4} - \beta_{8}\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−1.72388 + 0.168030i
0.0786547 + 1.73026i
1.47158 0.913487i
−1.72388 0.168030i
0.0786547 1.73026i
1.47158 + 0.913487i
−0.442647 + 1.67453i
−1.73189 0.0237018i
1.40849 1.00804i
−0.442647 1.67453i
−1.73189 + 0.0237018i
1.40849 + 1.00804i
1.69944 + 0.334495i
−0.396613 1.68603i
−0.363139 + 1.69356i
1.69944 0.334495i
−0.396613 + 1.68603i
−0.363139 1.69356i
0 −1.67739 + 0.431705i 0 1.14133 3.13578i 0 1.07356 1.85947i 0 2.62726 1.44827i 0
257.2 0 −0.517874 1.65282i 0 −0.258510 + 0.710252i 0 −0.777943 + 1.34744i 0 −2.46361 + 1.71190i 0
257.3 0 1.69526 + 0.355087i 0 −0.882820 + 2.42553i 0 1.58376 2.74316i 0 2.74783 + 1.20393i 0
401.1 0 −1.67739 0.431705i 0 1.14133 + 3.13578i 0 1.07356 + 1.85947i 0 2.62726 + 1.44827i 0
401.2 0 −0.517874 + 1.65282i 0 −0.258510 0.710252i 0 −0.777943 1.34744i 0 −2.46361 1.71190i 0
401.3 0 1.69526 0.355087i 0 −0.882820 2.42553i 0 1.58376 + 2.74316i 0 2.74783 1.20393i 0
497.1 0 −1.57223 + 0.726702i 0 −1.96615 0.346685i 0 −0.910931 1.57778i 0 1.94381 2.28508i 0
497.2 0 0.324081 + 1.70146i 0 2.22841 + 0.392929i 0 1.16829 + 2.02354i 0 −2.78994 + 1.10282i 0
497.3 0 0.748148 1.56214i 0 −0.262261 0.0462437i 0 −0.604656 1.04730i 0 −1.88055 2.33742i 0
545.1 0 −1.57223 0.726702i 0 −1.96615 + 0.346685i 0 −0.910931 + 1.57778i 0 1.94381 + 2.28508i 0
545.2 0 0.324081 1.70146i 0 2.22841 0.392929i 0 1.16829 2.02354i 0 −2.78994 1.10282i 0
545.3 0 0.748148 + 1.56214i 0 −0.262261 + 0.0462437i 0 −0.604656 + 1.04730i 0 −1.88055 + 2.33742i 0
641.1 0 −1.08684 + 1.34862i 0 0.343148 0.408948i 0 0.716507 + 1.24103i 0 −0.637553 2.93147i 0
641.2 0 −0.779936 1.54651i 0 1.86241 2.21954i 0 −0.562083 0.973556i 0 −1.78340 + 2.41236i 0
641.3 0 1.36678 + 1.06392i 0 −2.20556 + 2.62849i 0 −1.68651 2.92113i 0 0.736160 + 2.90828i 0
737.1 0 −1.08684 1.34862i 0 0.343148 + 0.408948i 0 0.716507 1.24103i 0 −0.637553 + 2.93147i 0
737.2 0 −0.779936 + 1.54651i 0 1.86241 + 2.21954i 0 −0.562083 + 0.973556i 0 −1.78340 2.41236i 0
737.3 0 1.36678 1.06392i 0 −2.20556 2.62849i 0 −1.68651 + 2.92113i 0 0.736160 2.90828i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j 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.cc.c 18
3.b odd 2 1 912.2.cc.d 18
4.b odd 2 1 114.2.l.b yes 18
12.b even 2 1 114.2.l.a 18
19.f odd 18 1 912.2.cc.d 18
57.j even 18 1 inner 912.2.cc.c 18
76.k even 18 1 114.2.l.a 18
228.u odd 18 1 114.2.l.b yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.l.a 18 12.b even 2 1
114.2.l.a 18 76.k even 18 1
114.2.l.b yes 18 4.b odd 2 1
114.2.l.b yes 18 228.u odd 18 1
912.2.cc.c 18 1.a even 1 1 trivial
912.2.cc.c 18 57.j even 18 1 inner
912.2.cc.d 18 3.b odd 2 1
912.2.cc.d 18 19.f odd 18 1

Hecke kernels

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

\( T_{5}^{18} + 9 T_{5}^{16} + 108 T_{5}^{14} - 162 T_{5}^{13} + 46 T_{5}^{12} + 324 T_{5}^{11} - 1458 T_{5}^{10} - 1458 T_{5}^{9} - 30258 T_{5}^{8} + 5202 T_{5}^{7} + 141193 T_{5}^{6} + 60552 T_{5}^{5} + 70344 T_{5}^{4} + \cdots + 1728 \) Copy content Toggle raw display
\( T_{7}^{18} + 21 T_{7}^{16} + 4 T_{7}^{15} + 297 T_{7}^{14} + 90 T_{7}^{13} + 2222 T_{7}^{12} + 1494 T_{7}^{11} + 11985 T_{7}^{10} + 9622 T_{7}^{9} + 43740 T_{7}^{8} + 41064 T_{7}^{7} + 117161 T_{7}^{6} + 100062 T_{7}^{5} + \cdots + 87616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 3 T^{17} + 6 T^{16} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} + 9 T^{16} + 108 T^{14} + \cdots + 1728 \) Copy content Toggle raw display
$7$ \( T^{18} + 21 T^{16} + 4 T^{15} + \cdots + 87616 \) Copy content Toggle raw display
$11$ \( T^{18} - 51 T^{16} + 1884 T^{14} + \cdots + 35769627 \) Copy content Toggle raw display
$13$ \( T^{18} + 12 T^{17} + 27 T^{16} + \cdots + 2365632 \) Copy content Toggle raw display
$17$ \( T^{18} - 6 T^{17} + 21 T^{16} + \cdots + 3878307 \) Copy content Toggle raw display
$19$ \( T^{18} - 6 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} - 105 T^{16} + \cdots + 123187392 \) Copy content Toggle raw display
$29$ \( T^{18} + 6 T^{17} + \cdots + 52719833664 \) Copy content Toggle raw display
$31$ \( T^{18} - 165 T^{16} + \cdots + 6231379854528 \) Copy content Toggle raw display
$37$ \( T^{18} + 342 T^{16} + \cdots + 7868768303808 \) Copy content Toggle raw display
$41$ \( T^{18} - 3 T^{17} + \cdots + 1846709769969 \) Copy content Toggle raw display
$43$ \( T^{18} - 6 T^{17} + \cdots + 390621250009 \) Copy content Toggle raw display
$47$ \( T^{18} + 30 T^{17} + \cdots + 3499077312 \) Copy content Toggle raw display
$53$ \( T^{18} + 60 T^{17} + \cdots + 3426463296 \) Copy content Toggle raw display
$59$ \( T^{18} + 3 T^{17} + \cdots + 38983402581561 \) Copy content Toggle raw display
$61$ \( T^{18} - 54 T^{17} + \cdots + 65033160256 \) Copy content Toggle raw display
$67$ \( T^{18} - 15 T^{17} + \cdots + 56\!\cdots\!23 \) Copy content Toggle raw display
$71$ \( T^{18} + 36 T^{17} + \cdots + 404099233344 \) Copy content Toggle raw display
$73$ \( T^{18} + 42 T^{17} + \cdots + 192753487369 \) Copy content Toggle raw display
$79$ \( T^{18} - 6 T^{17} + \cdots + 21259626441408 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 176145902499843 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 381874169643201 \) Copy content Toggle raw display
$97$ \( T^{18} - 9 T^{17} + \cdots + 17447631785307 \) Copy content Toggle raw display
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