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Properties

Label 243.7.b.c.242.2
Level $243$
Weight $7$
Character 243.242
Analytic conductor $55.903$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [243,7,Mod(242,243)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("243.242"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 243.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-166,0,0,190] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(55.9031423196\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 242.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 243.242
Dual form 243.7.b.c.242.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+12.1244i q^{2} -83.0000 q^{4} +242.487i q^{5} +95.0000 q^{7} -230.363i q^{8} -2940.00 q^{10} -1842.90i q^{11} -589.000 q^{13} +1151.81i q^{14} -2519.00 q^{16} -2764.35i q^{17} +8315.00 q^{19} -20126.4i q^{20} +22344.0 q^{22} -10233.0i q^{23} -43175.0 q^{25} -7141.25i q^{26} -7885.00 q^{28} -31232.3i q^{29} -35113.0 q^{31} -45284.5i q^{32} +33516.0 q^{34} +23036.3i q^{35} -62245.0 q^{37} +100814. i q^{38} +55860.0 q^{40} -55044.6i q^{41} -11677.0 q^{43} +152961. i q^{44} +124068. q^{46} -140061. i q^{47} -108624. q^{49} -523469. i q^{50} +48887.0 q^{52} +259849. i q^{53} +446880. q^{55} -21884.5i q^{56} +378672. q^{58} +21193.4i q^{59} +218906. q^{61} -425723. i q^{62} +387829. q^{64} -142825. i q^{65} +128186. q^{67} +229441. i q^{68} -279300. q^{70} +143746. i q^{71} -469726. q^{73} -754681. i q^{74} -690145. q^{76} -175076. i q^{77} -225625. q^{79} -610825. i q^{80} +667380. q^{82} -212079. i q^{83} +670320. q^{85} -141576. i q^{86} -424536. q^{88} +706656. i q^{89} -55955.0 q^{91} +849335. i q^{92} +1.69814e6 q^{94} +2.01628e6i q^{95} -1.35408e6 q^{97} -1.31700e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 166 q^{4} + 190 q^{7} - 5880 q^{10} - 1178 q^{13} - 5038 q^{16} + 16630 q^{19} + 44688 q^{22} - 86350 q^{25} - 15770 q^{28} - 70226 q^{31} + 67032 q^{34} - 124490 q^{37} + 111720 q^{40} - 23354 q^{43}+ \cdots - 2708162 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.1244i 1.51554i 0.652519 + 0.757772i \(0.273712\pi\)
−0.652519 + 0.757772i \(0.726288\pi\)
\(3\) 0 0
\(4\) −83.0000 −1.29688
\(5\) 242.487i 1.93990i 0.243311 + 0.969948i \(0.421767\pi\)
−0.243311 + 0.969948i \(0.578233\pi\)
\(6\) 0 0
\(7\) 95.0000 0.276968 0.138484 − 0.990365i \(-0.455777\pi\)
0.138484 + 0.990365i \(0.455777\pi\)
\(8\) − 230.363i − 0.449927i
\(9\) 0 0
\(10\) −2940.00 −2.94000
\(11\) − 1842.90i − 1.38460i −0.721610 − 0.692300i \(-0.756598\pi\)
0.721610 − 0.692300i \(-0.243402\pi\)
\(12\) 0 0
\(13\) −589.000 −0.268093 −0.134046 − 0.990975i \(-0.542797\pi\)
−0.134046 + 0.990975i \(0.542797\pi\)
\(14\) 1151.81i 0.419757i
\(15\) 0 0
\(16\) −2519.00 −0.614990
\(17\) − 2764.35i − 0.562661i −0.959611 − 0.281330i \(-0.909224\pi\)
0.959611 − 0.281330i \(-0.0907757\pi\)
\(18\) 0 0
\(19\) 8315.00 1.21228 0.606138 − 0.795360i \(-0.292718\pi\)
0.606138 + 0.795360i \(0.292718\pi\)
\(20\) − 20126.4i − 2.51580i
\(21\) 0 0
\(22\) 22344.0 2.09842
\(23\) − 10233.0i − 0.841042i −0.907283 − 0.420521i \(-0.861847\pi\)
0.907283 − 0.420521i \(-0.138153\pi\)
\(24\) 0 0
\(25\) −43175.0 −2.76320
\(26\) − 7141.25i − 0.406307i
\(27\) 0 0
\(28\) −7885.00 −0.359193
\(29\) − 31232.3i − 1.28059i −0.768129 − 0.640296i \(-0.778812\pi\)
0.768129 − 0.640296i \(-0.221188\pi\)
\(30\) 0 0
\(31\) −35113.0 −1.17864 −0.589322 − 0.807898i \(-0.700605\pi\)
−0.589322 + 0.807898i \(0.700605\pi\)
\(32\) − 45284.5i − 1.38197i
\(33\) 0 0
\(34\) 33516.0 0.852738
\(35\) 23036.3i 0.537289i
\(36\) 0 0
\(37\) −62245.0 −1.22885 −0.614426 − 0.788975i \(-0.710612\pi\)
−0.614426 + 0.788975i \(0.710612\pi\)
\(38\) 100814.i 1.83726i
\(39\) 0 0
\(40\) 55860.0 0.872812
\(41\) − 55044.6i − 0.798662i −0.916807 − 0.399331i \(-0.869242\pi\)
0.916807 − 0.399331i \(-0.130758\pi\)
\(42\) 0 0
\(43\) −11677.0 −0.146868 −0.0734338 − 0.997300i \(-0.523396\pi\)
−0.0734338 + 0.997300i \(0.523396\pi\)
\(44\) 152961.i 1.79565i
\(45\) 0 0
\(46\) 124068. 1.27464
\(47\) − 140061.i − 1.34903i −0.738260 − 0.674516i \(-0.764352\pi\)
0.738260 − 0.674516i \(-0.235648\pi\)
\(48\) 0 0
\(49\) −108624. −0.923289
\(50\) − 523469.i − 4.18775i
\(51\) 0 0
\(52\) 48887.0 0.347683
\(53\) 259849.i 1.74540i 0.488261 + 0.872698i \(0.337631\pi\)
−0.488261 + 0.872698i \(0.662369\pi\)
\(54\) 0 0
\(55\) 446880. 2.68598
\(56\) − 21884.5i − 0.124615i
\(57\) 0 0
\(58\) 378672. 1.94079
\(59\) 21193.4i 0.103192i 0.998668 + 0.0515958i \(0.0164307\pi\)
−0.998668 + 0.0515958i \(0.983569\pi\)
\(60\) 0 0
\(61\) 218906. 0.964424 0.482212 − 0.876054i \(-0.339833\pi\)
0.482212 + 0.876054i \(0.339833\pi\)
\(62\) − 425723.i − 1.78629i
\(63\) 0 0
\(64\) 387829. 1.47945
\(65\) − 142825.i − 0.520072i
\(66\) 0 0
\(67\) 128186. 0.426203 0.213101 − 0.977030i \(-0.431644\pi\)
0.213101 + 0.977030i \(0.431644\pi\)
\(68\) 229441.i 0.729701i
\(69\) 0 0
\(70\) −279300. −0.814286
\(71\) 143746.i 0.401626i 0.979630 + 0.200813i \(0.0643584\pi\)
−0.979630 + 0.200813i \(0.935642\pi\)
\(72\) 0 0
\(73\) −469726. −1.20747 −0.603735 − 0.797185i \(-0.706321\pi\)
−0.603735 + 0.797185i \(0.706321\pi\)
\(74\) − 754681.i − 1.86238i
\(75\) 0 0
\(76\) −690145. −1.57217
\(77\) − 175076.i − 0.383490i
\(78\) 0 0
\(79\) −225625. −0.457621 −0.228810 − 0.973471i \(-0.573484\pi\)
−0.228810 + 0.973471i \(0.573484\pi\)
\(80\) − 610825.i − 1.19302i
\(81\) 0 0
\(82\) 667380. 1.21041
\(83\) − 212079.i − 0.370906i −0.982653 − 0.185453i \(-0.940625\pi\)
0.982653 − 0.185453i \(-0.0593752\pi\)
\(84\) 0 0
\(85\) 670320. 1.09150
\(86\) − 141576.i − 0.222584i
\(87\) 0 0
\(88\) −424536. −0.622969
\(89\) 706656.i 1.00239i 0.865334 + 0.501196i \(0.167107\pi\)
−0.865334 + 0.501196i \(0.832893\pi\)
\(90\) 0 0
\(91\) −55955.0 −0.0742531
\(92\) 849335.i 1.09073i
\(93\) 0 0
\(94\) 1.69814e6 2.04452
\(95\) 2.01628e6i 2.35169i
\(96\) 0 0
\(97\) −1.35408e6 −1.48364 −0.741822 − 0.670597i \(-0.766038\pi\)
−0.741822 + 0.670597i \(0.766038\pi\)
\(98\) − 1.31700e6i − 1.39929i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 243.7.b.c.242.2 yes 2
3.2 odd 2 inner 243.7.b.c.242.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.7.b.c.242.1 ✓ 2 3.2 odd 2 inner
243.7.b.c.242.2 yes 2 1.1 even 1 trivial