Properties

Label 74.2.h.a
Level $74$
Weight $2$
Character orbit 74.h
Analytic conductor $0.591$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{36}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{36} q^{2} + ( - \zeta_{36}^{6} - \zeta_{36}^{4} + 1) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} - \zeta_{36}^{8} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} - \zeta_{36}^{2} - \zeta_{36}) q^{5} + ( - \zeta_{36}^{7} - \zeta_{36}^{5} + \zeta_{36}) q^{6} + (2 \zeta_{36}^{11} + \zeta_{36}^{10} - \zeta_{36}^{5} - 1) q^{7} + \zeta_{36}^{3} q^{8} + ( - \zeta_{36}^{10} + \zeta_{36}^{8} - \zeta_{36}^{6} + \zeta_{36}^{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36} q^{2} + ( - \zeta_{36}^{6} - \zeta_{36}^{4} + 1) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{9} - \zeta_{36}^{8} + \zeta_{36}^{7} + \zeta_{36}^{5} + \zeta_{36}^{3} - \zeta_{36}^{2} - \zeta_{36}) q^{5} + ( - \zeta_{36}^{7} - \zeta_{36}^{5} + \zeta_{36}) q^{6} + (2 \zeta_{36}^{11} + \zeta_{36}^{10} - \zeta_{36}^{5} - 1) q^{7} + \zeta_{36}^{3} q^{8} + ( - \zeta_{36}^{10} + \zeta_{36}^{8} - \zeta_{36}^{6} + \zeta_{36}^{4}) q^{9} + ( - \zeta_{36}^{10} - \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{6} + \zeta_{36}^{4} - \zeta_{36}^{3} - \zeta_{36}^{2}) q^{10} + ( - \zeta_{36}^{10} + 4 \zeta_{36}^{9} - \zeta_{36}^{8} + \zeta_{36}^{6} - 2 \zeta_{36}^{3} + \zeta_{36}^{2} - 1) q^{11} + ( - \zeta_{36}^{8} - \zeta_{36}^{6} + \zeta_{36}^{2}) q^{12} + ( - 2 \zeta_{36}^{11} + \zeta_{36}^{10} + 2 \zeta_{36}^{8} + \zeta_{36}^{6} - \zeta_{36}^{4} + \zeta_{36}^{3} - \zeta_{36}^{2} - \zeta_{36}) q^{13} + (\zeta_{36}^{11} + \zeta_{36}^{6} - \zeta_{36} - 2) q^{14} + ( - 2 \zeta_{36}^{11} - 2 \zeta_{36}^{9} + \zeta_{36}^{8} + \zeta_{36}^{7} + 2 \zeta_{36}^{6} + 2 \zeta_{36}^{5} - 2 \zeta_{36}^{2} + \cdots - 1) q^{15}+ \cdots + (2 \zeta_{36}^{11} + 2 \zeta_{36}^{10} - 2 \zeta_{36}^{9} - \zeta_{36}^{8} + 4 \zeta_{36}^{7} - \zeta_{36}^{6} - 4 \zeta_{36}^{5} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 12 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} - 12 q^{7} - 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} + 6 q^{13} - 18 q^{14} - 18 q^{19} - 6 q^{21} - 18 q^{25} + 12 q^{26} - 6 q^{27} - 6 q^{28} + 18 q^{29} + 24 q^{30} - 6 q^{33} + 12 q^{34} + 18 q^{35} + 12 q^{36} + 30 q^{37} - 24 q^{38} + 30 q^{39} + 12 q^{40} + 24 q^{41} + 6 q^{44} - 18 q^{45} + 30 q^{46} + 6 q^{47} + 12 q^{49} - 36 q^{50} - 12 q^{52} - 12 q^{53} - 18 q^{55} - 36 q^{57} + 6 q^{58} - 36 q^{61} - 6 q^{63} + 6 q^{64} + 36 q^{65} - 30 q^{67} - 18 q^{69} - 12 q^{70} + 12 q^{71} - 48 q^{74} - 36 q^{75} - 18 q^{76} + 12 q^{77} - 6 q^{78} + 6 q^{79} + 24 q^{81} - 48 q^{83} + 12 q^{84} + 18 q^{85} - 36 q^{86} + 36 q^{87} - 36 q^{88} - 18 q^{89} + 6 q^{90} - 6 q^{91} + 18 q^{92} - 12 q^{93} + 36 q^{97} + 36 q^{98} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(\zeta_{36}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.642788 + 0.766044i
0.642788 0.766044i
−0.342020 + 0.939693i
0.342020 0.939693i
−0.642788 0.766044i
0.642788 + 0.766044i
−0.984808 0.173648i
0.984808 + 0.173648i
−0.984808 + 0.173648i
0.984808 0.173648i
−0.342020 0.939693i
0.342020 + 0.939693i
−0.642788 + 0.766044i 1.43969 1.20805i −0.173648 0.984808i 0.273629 0.751790i 1.87939i −0.138449 0.0503913i 0.866025 + 0.500000i 0.0923963 0.524005i 0.400019 + 0.692853i
3.2 0.642788 0.766044i 1.43969 1.20805i −0.173648 0.984808i −1.45842 + 4.00698i 1.87939i −3.39364 1.23518i −0.866025 0.500000i 0.0923963 0.524005i 2.13207 + 3.69285i
21.1 −0.342020 + 0.939693i 0.326352 0.118782i −0.766044 0.642788i 2.57176 + 0.453471i 0.347296i −0.361075 + 2.04776i 0.866025 0.500000i −2.20574 + 1.85083i −1.30572 + 2.26157i
21.2 0.342020 0.939693i 0.326352 0.118782i −0.766044 0.642788i 0.839712 + 0.148064i 0.347296i 0.240460 1.36372i −0.866025 + 0.500000i −2.20574 + 1.85083i 0.426333 0.738430i
25.1 −0.642788 0.766044i 1.43969 + 1.20805i −0.173648 + 0.984808i 0.273629 + 0.751790i 1.87939i −0.138449 + 0.0503913i 0.866025 0.500000i 0.0923963 + 0.524005i 0.400019 0.692853i
25.2 0.642788 + 0.766044i 1.43969 + 1.20805i −0.173648 + 0.984808i −1.45842 4.00698i 1.87939i −3.39364 + 1.23518i −0.866025 + 0.500000i 0.0923963 + 0.524005i 2.13207 3.69285i
41.1 −0.984808 0.173648i −0.266044 1.50881i 0.939693 + 0.342020i −1.97937 2.35892i 1.53209i 0.153180 0.128533i −0.866025 0.500000i 0.613341 0.223238i 1.53967 + 2.66679i
41.2 0.984808 + 0.173648i −0.266044 1.50881i 0.939693 + 0.342020i −0.247315 0.294739i 1.53209i −2.50048 + 2.09815i 0.866025 + 0.500000i 0.613341 0.223238i −0.192377 0.333207i
65.1 −0.984808 + 0.173648i −0.266044 + 1.50881i 0.939693 0.342020i −1.97937 + 2.35892i 1.53209i 0.153180 + 0.128533i −0.866025 + 0.500000i 0.613341 + 0.223238i 1.53967 2.66679i
65.2 0.984808 0.173648i −0.266044 + 1.50881i 0.939693 0.342020i −0.247315 + 0.294739i 1.53209i −2.50048 2.09815i 0.866025 0.500000i 0.613341 + 0.223238i −0.192377 + 0.333207i
67.1 −0.342020 0.939693i 0.326352 + 0.118782i −0.766044 + 0.642788i 2.57176 0.453471i 0.347296i −0.361075 2.04776i 0.866025 + 0.500000i −2.20574 1.85083i −1.30572 2.26157i
67.2 0.342020 + 0.939693i 0.326352 + 0.118782i −0.766044 + 0.642788i 0.839712 0.148064i 0.347296i 0.240460 + 1.36372i −0.866025 0.500000i −2.20574 1.85083i 0.426333 + 0.738430i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.h.a 12
3.b odd 2 1 666.2.bj.c 12
4.b odd 2 1 592.2.bq.b 12
37.h even 18 1 inner 74.2.h.a 12
37.i odd 36 1 2738.2.a.r 6
37.i odd 36 1 2738.2.a.s 6
111.n odd 18 1 666.2.bj.c 12
148.o odd 18 1 592.2.bq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.h.a 12 1.a even 1 1 trivial
74.2.h.a 12 37.h even 18 1 inner
592.2.bq.b 12 4.b odd 2 1
592.2.bq.b 12 148.o odd 18 1
666.2.bj.c 12 3.b odd 2 1
666.2.bj.c 12 111.n odd 18 1
2738.2.a.r 6 37.i odd 36 1
2738.2.a.s 6 37.i odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{10} - 72 T^{9} + 36 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{11} + 66 T^{10} + 218 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + 63 T^{10} + \cdots + 408321 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + 30 T^{10} + \cdots + 288369 \) Copy content Toggle raw display
$17$ \( T^{12} + 36 T^{10} + 234 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{12} + 18 T^{11} + 135 T^{10} + \cdots + 10439361 \) Copy content Toggle raw display
$23$ \( T^{12} - 63 T^{10} + 3303 T^{8} + \cdots + 431649 \) Copy content Toggle raw display
$29$ \( T^{12} - 18 T^{11} + 126 T^{10} + \cdots + 110889 \) Copy content Toggle raw display
$31$ \( T^{12} + 210 T^{10} + \cdots + 317445489 \) Copy content Toggle raw display
$37$ \( T^{12} - 30 T^{11} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} - 24 T^{11} + 216 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$43$ \( T^{12} + 156 T^{10} + 8910 T^{8} + \cdots + 2277081 \) Copy content Toggle raw display
$47$ \( T^{12} - 6 T^{11} + \cdots + 2027430729 \) Copy content Toggle raw display
$53$ \( T^{12} + 12 T^{11} + \cdots + 45041148441 \) Copy content Toggle raw display
$59$ \( T^{12} + 1404 T^{9} + \cdots + 17477104401 \) Copy content Toggle raw display
$61$ \( T^{12} + 36 T^{11} + 756 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$67$ \( T^{12} + 30 T^{11} + \cdots + 59697637561 \) Copy content Toggle raw display
$71$ \( T^{12} - 12 T^{11} + \cdots + 4131885551616 \) Copy content Toggle raw display
$73$ \( (T^{6} - 366 T^{4} + 322 T^{3} + \cdots + 94609)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 6 T^{11} + 12 T^{10} + \cdots + 47961 \) Copy content Toggle raw display
$83$ \( T^{12} + 48 T^{11} + 1008 T^{10} + \cdots + 110889 \) Copy content Toggle raw display
$89$ \( T^{12} + 18 T^{11} + \cdots + 687331089 \) Copy content Toggle raw display
$97$ \( T^{12} - 36 T^{11} + \cdots + 27455164416 \) Copy content Toggle raw display
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