Properties

Label 342.2.f.g
Level $342$
Weight $2$
Character orbit 342.f
Analytic conductor $2.731$
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] = [342,2,Mod(7,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.f (of order \(3\), degree \(2\), 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} - 324 x^{6} - 1701 x^{5} + 243 x^{4} - 4374 x^{3} + 8748 x^{2} + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{9} q^{3} + q^{4} + ( - \beta_{16} + \beta_{4}) q^{5} + \beta_{9} q^{6} + ( - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} - q^{8} + (\beta_{13} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{9} q^{3} + q^{4} + ( - \beta_{16} + \beta_{4}) q^{5} + \beta_{9} q^{6} + ( - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - \beta_1 + 1) q^{7} - q^{8} + (\beta_{13} + \beta_{4}) q^{9} + (\beta_{16} - \beta_{4}) q^{10} + ( - \beta_{14} + \beta_{3}) q^{11} - \beta_{9} q^{12} + ( - \beta_{12} + \beta_{6} - \beta_{4} - 1) q^{13} + (\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_1 - 1) q^{14} + (\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} + \beta_1) q^{15} + q^{16} + ( - \beta_{14} - \beta_{5}) q^{17} + ( - \beta_{13} - \beta_{4}) q^{18} + ( - \beta_{15} + 2 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{2} - 2 \beta_1 + 1) q^{19} + ( - \beta_{16} + \beta_{4}) q^{20} + (\beta_{17} - \beta_{16} + \beta_{13} + \beta_{11} - 2 \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{21} + (\beta_{14} - \beta_{3}) q^{22} + ( - \beta_{11} + \beta_{10} - \beta_{7} - \beta_{6} + \beta_{3} - \beta_1) q^{23} + \beta_{9} q^{24} + ( - 2 \beta_{15} - 2 \beta_{14} + \beta_{12} - \beta_{11} - 2 \beta_{10} - 3 \beta_{9} + \cdots - 4 \beta_1) q^{25}+ \cdots + (\beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{2} + 18 q^{4} + 5 q^{7} - 18 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{2} + 18 q^{4} + 5 q^{7} - 18 q^{8} + 4 q^{9} + q^{11} - 2 q^{13} - 5 q^{14} + 18 q^{16} - 5 q^{17} - 4 q^{18} + 9 q^{19} - 4 q^{21} - q^{22} + 4 q^{23} - 9 q^{25} + 2 q^{26} - 18 q^{27} + 5 q^{28} - 9 q^{29} + 4 q^{31} - 18 q^{32} + 16 q^{33} + 5 q^{34} + 6 q^{35} + 4 q^{36} + 20 q^{37} - 9 q^{38} + 4 q^{39} + q^{41} + 4 q^{42} - 14 q^{43} + q^{44} + 30 q^{45} - 4 q^{46} + 19 q^{47} + 6 q^{49} + 9 q^{50} + 16 q^{51} - 2 q^{52} - 10 q^{53} + 18 q^{54} + 6 q^{55} - 5 q^{56} - 36 q^{57} + 9 q^{58} - 5 q^{59} + 18 q^{61} - 4 q^{62} - 15 q^{63} + 18 q^{64} - 45 q^{65} - 16 q^{66} - 44 q^{67} - 5 q^{68} - 26 q^{69} - 6 q^{70} + 11 q^{71} - 4 q^{72} + 44 q^{73} - 20 q^{74} + 9 q^{76} - 2 q^{77} - 4 q^{78} - 4 q^{79} - 32 q^{81} - q^{82} - 7 q^{83} - 4 q^{84} + 14 q^{86} + 3 q^{87} - q^{88} + q^{89} - 30 q^{90} - 25 q^{91} + 4 q^{92} + 10 q^{93} - 19 q^{94} - 24 q^{95} - 6 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 4 x^{16} - 6 x^{15} + x^{14} - 21 x^{13} - 12 x^{12} + 9 x^{10} + 135 x^{9} + 27 x^{8} - 324 x^{6} - 1701 x^{5} + 243 x^{4} - 4374 x^{3} + 8748 x^{2} + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 138 \nu^{17} - 3248 \nu^{16} + 2796 \nu^{15} - 5540 \nu^{14} + 15462 \nu^{13} + 11971 \nu^{12} + 18537 \nu^{11} + 24135 \nu^{10} - 73143 \nu^{9} - 39708 \nu^{8} + \cdots - 19081575 ) / 2574099 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 44 \nu^{17} - 351 \nu^{16} + 397 \nu^{15} - 774 \nu^{14} + 2455 \nu^{13} + 282 \nu^{12} + 2460 \nu^{11} + 2058 \nu^{10} - 11835 \nu^{9} - 3222 \nu^{8} - 34533 \nu^{7} + \cdots - 2755620 ) / 234009 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18 \nu^{17} - 85 \nu^{16} + 144 \nu^{15} - 267 \nu^{14} + 411 \nu^{13} - 57 \nu^{12} + 15 \nu^{11} + 802 \nu^{10} - 1272 \nu^{9} - 141 \nu^{8} - 5877 \nu^{7} + 1431 \nu^{6} + 3186 \nu^{5} + \cdots - 475308 ) / 78003 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1832 \nu^{17} - 15738 \nu^{16} + 19523 \nu^{15} - 39879 \nu^{14} + 80051 \nu^{13} - 12477 \nu^{12} + 47199 \nu^{11} + 83412 \nu^{10} - 298899 \nu^{9} + 74061 \nu^{8} + \cdots - 86769225 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1940 \nu^{17} - 13275 \nu^{16} + 17804 \nu^{15} - 34275 \nu^{14} + 69926 \nu^{13} - 6459 \nu^{12} + 37956 \nu^{11} + 93591 \nu^{10} - 230022 \nu^{9} + 103329 \nu^{8} + \cdots - 72689319 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67 \nu^{17} + 515 \nu^{16} - 736 \nu^{15} + 1505 \nu^{14} - 2365 \nu^{13} - 142 \nu^{12} - 1749 \nu^{11} - 3888 \nu^{10} + 9603 \nu^{9} - 6606 \nu^{8} + 33966 \nu^{7} + \cdots + 2790612 ) / 234009 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3248 \nu^{17} + 3348 \nu^{16} - 6368 \nu^{15} + 15600 \nu^{14} + 9073 \nu^{13} + 16881 \nu^{12} + 24135 \nu^{11} - 71901 \nu^{10} - 21078 \nu^{9} - 196371 \nu^{8} + \cdots - 5006043 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 372 \nu^{17} - 736 \nu^{16} + 432 \nu^{15} - 1369 \nu^{14} + 5703 \nu^{13} + 1649 \nu^{12} + 7989 \nu^{11} - 906 \nu^{10} - 26901 \nu^{9} - 32139 \nu^{8} - 45171 \nu^{7} + \cdots - 7103376 ) / 858033 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 323 \nu^{17} - 699 \nu^{16} + 1193 \nu^{15} - 3204 \nu^{14} + 2150 \nu^{13} - 3252 \nu^{12} - 2067 \nu^{11} + 12834 \nu^{10} + 2070 \nu^{9} + 31050 \nu^{8} - 38745 \nu^{7} + \cdots - 2427570 ) / 702027 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4154 \nu^{17} + 3903 \nu^{16} - 6001 \nu^{15} - 3741 \nu^{14} - 47794 \nu^{13} + 3753 \nu^{12} - 47427 \nu^{11} + 47781 \nu^{10} + 213885 \nu^{9} + 240597 \nu^{8} + \cdots + 55296108 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4646 \nu^{17} + 1410 \nu^{16} + 611 \nu^{15} - 8502 \nu^{14} - 29536 \nu^{13} - 3258 \nu^{12} - 36294 \nu^{11} + 64287 \nu^{10} + 123705 \nu^{9} + 261036 \nu^{8} + \cdots + 32896854 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 538 \nu^{17} - 985 \nu^{16} + 2017 \nu^{15} - 3138 \nu^{14} + 1642 \nu^{13} - 2898 \nu^{12} + 741 \nu^{11} + 14731 \nu^{10} - 4089 \nu^{9} + 36678 \nu^{8} - 39735 \nu^{7} + \cdots - 2093688 ) / 858033 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6235 \nu^{17} - 14928 \nu^{16} + 21007 \nu^{15} - 52014 \nu^{14} + 30850 \nu^{13} - 21834 \nu^{12} - 10695 \nu^{11} + 257778 \nu^{10} - 162882 \nu^{9} + 469125 \nu^{8} + \cdots - 31893021 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 8599 \nu^{17} + 9192 \nu^{16} - 9736 \nu^{15} + 48816 \nu^{14} + 45905 \nu^{13} + 27537 \nu^{12} + 87042 \nu^{11} - 246780 \nu^{10} - 256617 \nu^{9} - 571374 \nu^{8} + \cdots - 53741151 ) / 7722297 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3391 \nu^{17} + 477 \nu^{16} - 5620 \nu^{15} + 8307 \nu^{14} + 17675 \nu^{13} + 14505 \nu^{12} + 43011 \nu^{11} - 45996 \nu^{10} - 62559 \nu^{9} - 253791 \nu^{8} + \cdots - 22447368 ) / 2574099 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3391 \nu^{17} + 477 \nu^{16} - 5620 \nu^{15} + 8307 \nu^{14} + 17675 \nu^{13} + 14505 \nu^{12} + 43011 \nu^{11} - 45996 \nu^{10} - 62559 \nu^{9} - 253791 \nu^{8} + \cdots - 22447368 ) / 2574099 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - \beta_{16} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{10} - \beta_{9} - \beta_{8} - 2\beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{16} + 2 \beta_{15} + \beta_{14} - 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} - 2 \beta_{3} + \beta_{2} + 3 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - 2 \beta_{16} - \beta_{14} + \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 7 \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{17} + 2 \beta_{16} - 3 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} - 6 \beta_{11} + 3 \beta_{10} + \beta_{9} - 9 \beta_{8} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{2} + 5 \beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{17} - \beta_{16} + 3 \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 4 \beta_{10} + 8 \beta_{9} + 14 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} + 10 \beta_{5} - \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + 7 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{17} - 4 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} - 9 \beta_{13} - 9 \beta_{11} + 5 \beta_{10} - 23 \beta_{9} + 12 \beta_{8} - 21 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 27 \beta_{4} + 2 \beta_{3} - \beta_{2} - 14 \beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{17} - 3 \beta_{16} - 24 \beta_{15} - 28 \beta_{14} + 27 \beta_{13} - 3 \beta_{11} + 7 \beta_{10} - 31 \beta_{9} + 56 \beta_{8} - 6 \beta_{7} + 16 \beta_{6} - 8 \beta_{5} + 15 \beta_{4} + \beta_{3} - 2 \beta_{2} - 29 \beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 17 \beta_{17} + 17 \beta_{16} + 50 \beta_{15} + 70 \beta_{14} + 18 \beta_{13} - 33 \beta_{12} + 33 \beta_{11} + 61 \beta_{10} + 40 \beta_{9} - 3 \beta_{8} + 57 \beta_{7} - 45 \beta_{5} + 9 \beta_{4} - 32 \beta_{3} + 58 \beta_{2} + 114 \beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 62 \beta_{17} + 7 \beta_{16} - \beta_{14} + 136 \beta_{13} + 87 \beta_{12} + 3 \beta_{11} - 44 \beta_{10} + 41 \beta_{9} + 29 \beta_{8} - 42 \beta_{7} + 4 \beta_{6} - 29 \beta_{5} - 38 \beta_{4} - 8 \beta_{3} - 71 \beta_{2} + 43 \beta _1 + 104 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 11 \beta_{17} - 43 \beta_{16} - 81 \beta_{15} - 66 \beta_{14} - 57 \beta_{13} + 132 \beta_{12} - 78 \beta_{11} - 42 \beta_{10} - 44 \beta_{9} + 279 \beta_{8} - 99 \beta_{7} + 12 \beta_{6} - 75 \beta_{5} + 24 \beta_{4} + 36 \beta_{3} + \cdots - 135 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 133 \beta_{17} - 109 \beta_{16} + 48 \beta_{15} + 50 \beta_{14} + 11 \beta_{13} + 3 \beta_{12} + 135 \beta_{11} + 112 \beta_{10} - 190 \beta_{9} + 482 \beta_{8} + 84 \beta_{7} - 47 \beta_{6} + 190 \beta_{5} + 116 \beta_{4} + 19 \beta_{3} + \cdots + 575 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 94 \beta_{17} + 41 \beta_{16} + 178 \beta_{15} + 2 \beta_{14} + 162 \beta_{13} - 432 \beta_{12} + 99 \beta_{11} + 104 \beta_{10} - 770 \beta_{9} - 456 \beta_{8} + 195 \beta_{7} - 291 \beta_{6} + 336 \beta_{5} - 45 \beta_{4} + 47 \beta_{3} + \cdots + 159 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1215 \beta_{17} - 1416 \beta_{16} + 57 \beta_{15} - 127 \beta_{14} + 540 \beta_{13} - 270 \beta_{12} - 111 \beta_{11} - 164 \beta_{10} + 257 \beta_{9} + 308 \beta_{8} - 357 \beta_{7} + 115 \beta_{6} - 800 \beta_{5} + 555 \beta_{4} + \cdots + 509 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 287 \beta_{17} - 388 \beta_{16} + 518 \beta_{15} - 812 \beta_{14} + 252 \beta_{13} + 777 \beta_{12} + 708 \beta_{11} + 1465 \beta_{10} + 1048 \beta_{9} + 582 \beta_{8} + 138 \beta_{7} - 1017 \beta_{6} - 873 \beta_{5} + \cdots + 1416 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1187 \beta_{17} - 929 \beta_{16} + 2898 \beta_{15} + 539 \beta_{14} + 478 \beta_{13} - 1668 \beta_{12} + 4431 \beta_{11} + 694 \beta_{10} + 842 \beta_{9} - 4957 \beta_{8} + 1524 \beta_{7} + 481 \beta_{6} + 1609 \beta_{5} + \cdots + 3299 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{8}\) \(-1 + \beta_{8}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.672818 + 1.59603i
−0.238928 + 1.71549i
−1.68875 + 0.384872i
0.837220 + 1.51627i
−1.24302 1.20619i
1.73011 + 0.0819856i
−0.614525 1.61937i
1.55117 0.770640i
0.339544 1.69844i
−0.672818 1.59603i
−0.238928 1.71549i
−1.68875 0.384872i
0.837220 1.51627i
−1.24302 + 1.20619i
1.73011 0.0819856i
−0.614525 + 1.61937i
1.55117 + 0.770640i
0.339544 + 1.69844i
−1.00000 −1.71861 + 0.215338i 1.00000 2.13008 3.68941i 1.71861 0.215338i −0.603898 + 1.04598i −1.00000 2.90726 0.740167i −2.13008 + 3.68941i
7.2 −1.00000 −1.60512 + 0.650828i 1.00000 −0.706161 + 1.22311i 1.60512 0.650828i 1.53389 2.65678i −1.00000 2.15285 2.08932i 0.706161 1.22311i
7.3 −1.00000 −1.17768 1.27006i 1.00000 −0.0748074 + 0.129570i 1.17768 + 1.27006i −0.733568 + 1.27058i −1.00000 −0.226122 + 2.99147i 0.0748074 0.129570i
7.4 −1.00000 −0.894515 + 1.48319i 1.00000 −0.365468 + 0.633010i 0.894515 1.48319i −1.79034 + 3.10095i −1.00000 −1.39969 2.65347i 0.365468 0.633010i
7.5 −1.00000 0.423085 1.67958i 1.00000 −1.97181 + 3.41528i −0.423085 + 1.67958i 1.02646 1.77787i −1.00000 −2.64200 1.42121i 1.97181 3.41528i
7.6 −1.00000 0.794053 + 1.53931i 1.00000 −1.46844 + 2.54341i −0.794053 1.53931i 0.360554 0.624499i −1.00000 −1.73896 + 2.44459i 1.46844 2.54341i
7.7 −1.00000 1.09515 1.34188i 1.00000 0.789777 1.36793i −1.09515 + 1.34188i 2.31561 4.01075i −1.00000 −0.601280 2.93913i −0.789777 + 1.36793i
7.8 −1.00000 1.44298 + 0.958029i 1.00000 1.19924 2.07714i −1.44298 0.958029i 0.959469 1.66185i −1.00000 1.16436 + 2.76483i −1.19924 + 2.07714i
7.9 −1.00000 1.64067 0.555168i 1.00000 0.467593 0.809895i −1.64067 + 0.555168i −0.568176 + 0.984110i −1.00000 2.38358 1.82169i −0.467593 + 0.809895i
49.1 −1.00000 −1.71861 0.215338i 1.00000 2.13008 + 3.68941i 1.71861 + 0.215338i −0.603898 1.04598i −1.00000 2.90726 + 0.740167i −2.13008 3.68941i
49.2 −1.00000 −1.60512 0.650828i 1.00000 −0.706161 1.22311i 1.60512 + 0.650828i 1.53389 + 2.65678i −1.00000 2.15285 + 2.08932i 0.706161 + 1.22311i
49.3 −1.00000 −1.17768 + 1.27006i 1.00000 −0.0748074 0.129570i 1.17768 1.27006i −0.733568 1.27058i −1.00000 −0.226122 2.99147i 0.0748074 + 0.129570i
49.4 −1.00000 −0.894515 1.48319i 1.00000 −0.365468 0.633010i 0.894515 + 1.48319i −1.79034 3.10095i −1.00000 −1.39969 + 2.65347i 0.365468 + 0.633010i
49.5 −1.00000 0.423085 + 1.67958i 1.00000 −1.97181 3.41528i −0.423085 1.67958i 1.02646 + 1.77787i −1.00000 −2.64200 + 1.42121i 1.97181 + 3.41528i
49.6 −1.00000 0.794053 1.53931i 1.00000 −1.46844 2.54341i −0.794053 + 1.53931i 0.360554 + 0.624499i −1.00000 −1.73896 2.44459i 1.46844 + 2.54341i
49.7 −1.00000 1.09515 + 1.34188i 1.00000 0.789777 + 1.36793i −1.09515 1.34188i 2.31561 + 4.01075i −1.00000 −0.601280 + 2.93913i −0.789777 1.36793i
49.8 −1.00000 1.44298 0.958029i 1.00000 1.19924 + 2.07714i −1.44298 + 0.958029i 0.959469 + 1.66185i −1.00000 1.16436 2.76483i −1.19924 2.07714i
49.9 −1.00000 1.64067 + 0.555168i 1.00000 0.467593 + 0.809895i −1.64067 0.555168i −0.568176 0.984110i −1.00000 2.38358 + 1.82169i −0.467593 0.809895i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.f.g 18
3.b odd 2 1 1026.2.f.g 18
9.c even 3 1 342.2.h.g yes 18
9.d odd 6 1 1026.2.h.g 18
19.c even 3 1 342.2.h.g yes 18
57.h odd 6 1 1026.2.h.g 18
171.h even 3 1 inner 342.2.f.g 18
171.j odd 6 1 1026.2.f.g 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.f.g 18 1.a even 1 1 trivial
342.2.f.g 18 171.h even 3 1 inner
342.2.h.g yes 18 9.c even 3 1
342.2.h.g yes 18 19.c even 3 1
1026.2.f.g 18 3.b odd 2 1
1026.2.f.g 18 171.j odd 6 1
1026.2.h.g 18 9.d odd 6 1
1026.2.h.g 18 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 27 T_{5}^{16} + 4 T_{5}^{15} + 537 T_{5}^{14} + 45 T_{5}^{13} + 4414 T_{5}^{12} - 870 T_{5}^{11} + 26208 T_{5}^{10} - 3330 T_{5}^{9} + 64179 T_{5}^{8} + 2376 T_{5}^{7} + 113373 T_{5}^{6} + 8586 T_{5}^{5} + 73386 T_{5}^{4} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 2 T^{16} + 6 T^{15} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} + 27 T^{16} + 4 T^{15} + 537 T^{14} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{18} - 5 T^{17} + 41 T^{16} + \cdots + 84681 \) Copy content Toggle raw display
$11$ \( T^{18} - T^{17} + 45 T^{16} + 114 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{9} + T^{8} - 66 T^{7} - 82 T^{6} + \cdots + 1776)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + 5 T^{17} + 84 T^{16} + \cdots + 130439241 \) Copy content Toggle raw display
$19$ \( T^{18} - 9 T^{17} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( (T^{9} - 2 T^{8} - 89 T^{7} + 146 T^{6} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + 9 T^{17} + \cdots + 640923532929 \) Copy content Toggle raw display
$31$ \( T^{18} - 4 T^{17} + \cdots + 2690808129 \) Copy content Toggle raw display
$37$ \( (T^{9} - 10 T^{8} - 61 T^{7} + 1060 T^{6} + \cdots - 2416)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} - T^{17} + 87 T^{16} + \cdots + 1679616 \) Copy content Toggle raw display
$43$ \( (T^{9} + 7 T^{8} - 195 T^{7} + \cdots - 299856)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} - 19 T^{17} + \cdots + 1579423076001 \) Copy content Toggle raw display
$53$ \( T^{18} + 10 T^{17} + \cdots + 20372138361 \) Copy content Toggle raw display
$59$ \( T^{18} + 5 T^{17} + \cdots + 13854720729249 \) Copy content Toggle raw display
$61$ \( T^{18} - 18 T^{17} + \cdots + 50781270409 \) Copy content Toggle raw display
$67$ \( (T^{9} + 22 T^{8} - 228 T^{7} + \cdots - 18567936)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} - 11 T^{17} + \cdots + 120758121 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 490005727861329 \) Copy content Toggle raw display
$79$ \( (T^{9} + 2 T^{8} - 370 T^{7} + \cdots + 53092528)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + 7 T^{17} + \cdots + 25101031868649 \) Copy content Toggle raw display
$89$ \( T^{18} - T^{17} + \cdots + 28991407990689 \) Copy content Toggle raw display
$97$ \( (T^{9} - 504 T^{7} - 785 T^{6} + \cdots - 21665872)^{2} \) Copy content Toggle raw display
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