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Properties

Label 833.2.e.a.18.1
Level $833$
Weight $2$
Character 833.18
Analytic conductor $6.652$
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] = [833,2,Mod(18,833)] 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("833.18"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(833, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.65153848837\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 18.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 833.18
Dual form 833.2.e.a.324.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +3.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +(1.00000 - 1.73205i) q^{10} +2.00000 q^{13} +(0.500000 + 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{17} +(-1.50000 + 2.59808i) q^{18} +(-2.00000 - 3.46410i) q^{19} -2.00000 q^{20} +(-2.00000 - 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +(1.00000 + 1.73205i) q^{26} +6.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(2.50000 - 4.33013i) q^{32} +1.00000 q^{34} +3.00000 q^{36} +(1.00000 + 1.73205i) q^{37} +(2.00000 - 3.46410i) q^{38} +(-3.00000 - 5.19615i) q^{40} +6.00000 q^{41} +4.00000 q^{43} +(3.00000 - 5.19615i) q^{45} +(2.00000 - 3.46410i) q^{46} +1.00000 q^{50} +(1.00000 - 1.73205i) q^{52} +(-3.00000 + 5.19615i) q^{53} +(3.00000 + 5.19615i) q^{58} +(-6.00000 + 10.3923i) q^{59} +(-5.00000 - 8.66025i) q^{61} +4.00000 q^{62} +7.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} +(-2.00000 + 3.46410i) q^{67} +(-0.500000 - 0.866025i) q^{68} -4.00000 q^{71} +(4.50000 + 7.79423i) q^{72} +(-3.00000 + 5.19615i) q^{73} +(-1.00000 + 1.73205i) q^{74} -4.00000 q^{76} +(-6.00000 - 10.3923i) q^{79} +(1.00000 - 1.73205i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(3.00000 + 5.19615i) q^{82} +4.00000 q^{83} -2.00000 q^{85} +(2.00000 + 3.46410i) q^{86} +(5.00000 + 8.66025i) q^{89} +6.00000 q^{90} -4.00000 q^{92} +(-4.00000 + 6.92820i) q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} - 2 q^{5} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 4 q^{13} + q^{16} + q^{17} - 3 q^{18} - 4 q^{19} - 4 q^{20} - 4 q^{23} + q^{25} + 2 q^{26} + 12 q^{29} + 4 q^{31} + 5 q^{32} + 2 q^{34}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(785\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i 0.986869 − 0.161521i \(-0.0516399\pi\)
−0.633316 + 0.773893i \(0.718307\pi\)
\(3\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −1.00000 − 1.73205i −0.447214 − 0.774597i 0.550990 − 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 1.00000 − 1.73205i 0.316228 − 0.547723i
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) 0.500000 − 0.866025i 0.121268 − 0.210042i
\(18\) −1.50000 + 2.59808i −0.353553 + 0.612372i
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 − 3.46410i −0.417029 − 0.722315i 0.578610 − 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 1.00000 + 1.73205i 0.196116 + 0.339683i
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 2.50000 − 4.33013i 0.441942 − 0.765466i
\(33\) 0 0
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 − 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 2.00000 − 3.46410i 0.324443 − 0.561951i
\(39\) 0 0
\(40\) −3.00000 − 5.19615i −0.474342 − 0.821584i
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 3.00000 − 5.19615i 0.447214 − 0.774597i
\(46\) 2.00000 − 3.46410i 0.294884 − 0.510754i
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 − 1.73205i 0.138675 − 0.240192i
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 3.00000 + 5.19615i 0.393919 + 0.682288i
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) −5.00000 − 8.66025i −0.640184 − 1.10883i −0.985391 − 0.170305i \(-0.945525\pi\)
0.345207 − 0.938527i \(-0.387809\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −2.00000 − 3.46410i −0.248069 − 0.429669i
\(66\) 0 0
\(67\) −2.00000 + 3.46410i −0.244339 + 0.423207i −0.961946 − 0.273241i \(-0.911904\pi\)
0.717607 + 0.696449i \(0.245238\pi\)
\(68\) −0.500000 − 0.866025i −0.0606339 − 0.105021i
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 4.50000 + 7.79423i 0.530330 + 0.918559i
\(73\) −3.00000 + 5.19615i −0.351123 + 0.608164i −0.986447 − 0.164083i \(-0.947534\pi\)
0.635323 + 0.772246i \(0.280867\pi\)
\(74\) −1.00000 + 1.73205i −0.116248 + 0.201347i
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 − 10.3923i −0.675053 − 1.16923i −0.976453 − 0.215728i \(-0.930788\pi\)
0.301401 − 0.953498i \(-0.402546\pi\)
\(80\) 1.00000 − 1.73205i 0.111803 − 0.193649i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 3.00000 + 5.19615i 0.331295 + 0.573819i
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 2.00000 + 3.46410i 0.215666 + 0.373544i
\(87\) 0 0
\(88\) 0 0
\(89\) 5.00000 + 8.66025i 0.529999 + 0.917985i 0.999388 + 0.0349934i \(0.0111410\pi\)
−0.469389 + 0.882992i \(0.655526\pi\)
\(90\) 6.00000 0.632456
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 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 833.2.e.a.18.1 2
7.2 even 3 inner 833.2.e.a.324.1 2
7.3 odd 6 17.2.a.a.1.1 ✓ 1
7.4 even 3 833.2.a.a.1.1 1
7.5 odd 6 833.2.e.b.324.1 2
7.6 odd 2 833.2.e.b.18.1 2
21.11 odd 6 7497.2.a.l.1.1 1
21.17 even 6 153.2.a.c.1.1 1
28.3 even 6 272.2.a.b.1.1 1
35.3 even 12 425.2.b.b.324.2 2
35.17 even 12 425.2.b.b.324.1 2
35.24 odd 6 425.2.a.d.1.1 1
56.3 even 6 1088.2.a.h.1.1 1
56.45 odd 6 1088.2.a.i.1.1 1
77.10 even 6 2057.2.a.e.1.1 1
84.59 odd 6 2448.2.a.o.1.1 1
91.38 odd 6 2873.2.a.c.1.1 1
105.59 even 6 3825.2.a.d.1.1 1
119.3 even 48 289.2.d.d.179.2 8
119.10 even 48 289.2.d.d.134.1 8
119.24 even 48 289.2.d.d.134.2 8
119.31 even 48 289.2.d.d.179.1 8
119.38 odd 12 289.2.b.a.288.2 2
119.45 even 48 289.2.d.d.155.1 8
119.59 odd 24 289.2.c.a.251.1 4
119.66 odd 24 289.2.c.a.38.1 4
119.73 even 48 289.2.d.d.110.2 8
119.80 even 48 289.2.d.d.110.1 8
119.87 odd 24 289.2.c.a.38.2 4
119.94 odd 24 289.2.c.a.251.2 4
119.101 odd 6 289.2.a.a.1.1 1
119.108 even 48 289.2.d.d.155.2 8
119.115 odd 12 289.2.b.a.288.1 2
133.94 even 6 6137.2.a.b.1.1 1
140.59 even 6 6800.2.a.n.1.1 1
161.45 even 6 8993.2.a.a.1.1 1
168.59 odd 6 9792.2.a.i.1.1 1
168.101 even 6 9792.2.a.n.1.1 1
357.101 even 6 2601.2.a.g.1.1 1
476.339 even 6 4624.2.a.d.1.1 1
595.339 odd 6 7225.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 ✓ 1 7.3 odd 6
153.2.a.c.1.1 1 21.17 even 6
272.2.a.b.1.1 1 28.3 even 6
289.2.a.a.1.1 1 119.101 odd 6
289.2.b.a.288.1 2 119.115 odd 12
289.2.b.a.288.2 2 119.38 odd 12
289.2.c.a.38.1 4 119.66 odd 24
289.2.c.a.38.2 4 119.87 odd 24
289.2.c.a.251.1 4 119.59 odd 24
289.2.c.a.251.2 4 119.94 odd 24
289.2.d.d.110.1 8 119.80 even 48
289.2.d.d.110.2 8 119.73 even 48
289.2.d.d.134.1 8 119.10 even 48
289.2.d.d.134.2 8 119.24 even 48
289.2.d.d.155.1 8 119.45 even 48
289.2.d.d.155.2 8 119.108 even 48
289.2.d.d.179.1 8 119.31 even 48
289.2.d.d.179.2 8 119.3 even 48
425.2.a.d.1.1 1 35.24 odd 6
425.2.b.b.324.1 2 35.17 even 12
425.2.b.b.324.2 2 35.3 even 12
833.2.a.a.1.1 1 7.4 even 3
833.2.e.a.18.1 2 1.1 even 1 trivial
833.2.e.a.324.1 2 7.2 even 3 inner
833.2.e.b.18.1 2 7.6 odd 2
833.2.e.b.324.1 2 7.5 odd 6
1088.2.a.h.1.1 1 56.3 even 6
1088.2.a.i.1.1 1 56.45 odd 6
2057.2.a.e.1.1 1 77.10 even 6
2448.2.a.o.1.1 1 84.59 odd 6
2601.2.a.g.1.1 1 357.101 even 6
2873.2.a.c.1.1 1 91.38 odd 6
3825.2.a.d.1.1 1 105.59 even 6
4624.2.a.d.1.1 1 476.339 even 6
6137.2.a.b.1.1 1 133.94 even 6
6800.2.a.n.1.1 1 140.59 even 6
7225.2.a.g.1.1 1 595.339 odd 6
7497.2.a.l.1.1 1 21.11 odd 6
8993.2.a.a.1.1 1 161.45 even 6
9792.2.a.i.1.1 1 168.59 odd 6
9792.2.a.n.1.1 1 168.101 even 6