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Properties

Label 930.2.i.e.811.1
Level $930$
Weight $2$
Character 930.811
Analytic conductor $7.426$
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] = [930,2,Mod(211,930)] 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("930.211"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-1,2,-1,-1,0,2,-1,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.42608738798\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 811.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 930.811
Dual form 930.2.i.e.211.1

$q$-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+1.00000 q^{2} +(-0.500000 + 0.866025i) q^{3} +1.00000 q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{10} +(0.500000 + 0.866025i) q^{11} +(-0.500000 + 0.866025i) q^{12} +(3.00000 + 5.19615i) q^{13} +1.00000 q^{15} +1.00000 q^{16} +(0.500000 - 0.866025i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-2.00000 + 3.46410i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(0.500000 + 0.866025i) q^{22} +7.00000 q^{23} +(-0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(3.00000 + 5.19615i) q^{26} +1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} +(2.00000 - 5.19615i) q^{31} +1.00000 q^{32} -1.00000 q^{33} +(0.500000 - 0.866025i) q^{34} +(-0.500000 - 0.866025i) q^{36} +(-3.50000 + 6.06218i) q^{37} +(-2.00000 + 3.46410i) q^{38} -6.00000 q^{39} +(-0.500000 - 0.866025i) q^{40} +(1.00000 + 1.73205i) q^{41} +(3.50000 - 6.06218i) q^{43} +(0.500000 + 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{45} +7.00000 q^{46} +3.00000 q^{47} +(-0.500000 + 0.866025i) q^{48} +(3.50000 + 6.06218i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(0.500000 + 0.866025i) q^{51} +(3.00000 + 5.19615i) q^{52} +(2.00000 + 3.46410i) q^{53} +1.00000 q^{54} +(0.500000 - 0.866025i) q^{55} +(-2.00000 - 3.46410i) q^{57} +6.00000 q^{58} +(-6.00000 + 10.3923i) q^{59} +1.00000 q^{60} -14.0000 q^{61} +(2.00000 - 5.19615i) q^{62} +1.00000 q^{64} +(3.00000 - 5.19615i) q^{65} -1.00000 q^{66} +(-2.50000 - 4.33013i) q^{67} +(0.500000 - 0.866025i) q^{68} +(-3.50000 + 6.06218i) q^{69} +(-3.00000 - 5.19615i) q^{71} +(-0.500000 - 0.866025i) q^{72} +(-1.00000 - 1.73205i) q^{73} +(-3.50000 + 6.06218i) q^{74} +(-0.500000 - 0.866025i) q^{75} +(-2.00000 + 3.46410i) q^{76} -6.00000 q^{78} +(-1.50000 + 2.59808i) q^{79} +(-0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(1.00000 + 1.73205i) q^{82} +(-8.00000 - 13.8564i) q^{83} -1.00000 q^{85} +(3.50000 - 6.06218i) q^{86} +(-3.00000 + 5.19615i) q^{87} +(0.500000 + 0.866025i) q^{88} +4.00000 q^{89} +(-0.500000 + 0.866025i) q^{90} +7.00000 q^{92} +(3.50000 + 4.33013i) q^{93} +3.00000 q^{94} +4.00000 q^{95} +(-0.500000 + 0.866025i) q^{96} -6.00000 q^{97} +(3.50000 + 6.06218i) q^{98} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + 2 q^{8} - q^{9} - q^{10} + q^{11} - q^{12} + 6 q^{13} + 2 q^{15} + 2 q^{16} + q^{17} - q^{18} - 4 q^{19} - q^{20} + q^{22} + 14 q^{23} - q^{24}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 1.00000 0.500000
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i
\(6\) −0.500000 + 0.866025i −0.204124 + 0.353553i
\(7\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −0.500000 − 0.866025i −0.158114 − 0.273861i
\(11\) 0.500000 + 0.866025i 0.150756 + 0.261116i 0.931505 − 0.363727i \(-0.118496\pi\)
−0.780750 + 0.624844i \(0.785163\pi\)
\(12\) −0.500000 + 0.866025i −0.144338 + 0.250000i
\(13\) 3.00000 + 5.19615i 0.832050 + 1.44115i 0.896410 + 0.443227i \(0.146166\pi\)
−0.0643593 + 0.997927i \(0.520500\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0.500000 − 0.866025i 0.121268 − 0.210042i −0.799000 − 0.601331i \(-0.794637\pi\)
0.920268 + 0.391289i \(0.127971\pi\)
\(18\) −0.500000 − 0.866025i −0.117851 − 0.204124i
\(19\) −2.00000 + 3.46410i −0.458831 + 0.794719i −0.998899 − 0.0469020i \(-0.985065\pi\)
0.540068 + 0.841621i \(0.318398\pi\)
\(20\) −0.500000 − 0.866025i −0.111803 − 0.193649i
\(21\) 0 0
\(22\) 0.500000 + 0.866025i 0.106600 + 0.184637i
\(23\) 7.00000 1.45960 0.729800 − 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) −0.500000 + 0.866025i −0.102062 + 0.176777i
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 3.00000 + 5.19615i 0.588348 + 1.01905i
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) 2.00000 − 5.19615i 0.359211 − 0.933257i
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) 0.500000 − 0.866025i 0.0857493 − 0.148522i
\(35\) 0 0
\(36\) −0.500000 − 0.866025i −0.0833333 − 0.144338i
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) −2.00000 + 3.46410i −0.324443 + 0.561951i
\(39\) −6.00000 −0.960769
\(40\) −0.500000 − 0.866025i −0.0790569 − 0.136931i
\(41\) 1.00000 + 1.73205i 0.156174 + 0.270501i 0.933486 − 0.358614i \(-0.116751\pi\)
−0.777312 + 0.629115i \(0.783417\pi\)
\(42\) 0 0
\(43\) 3.50000 − 6.06218i 0.533745 − 0.924473i −0.465478 − 0.885059i \(-0.654118\pi\)
0.999223 − 0.0394140i \(-0.0125491\pi\)
\(44\) 0.500000 + 0.866025i 0.0753778 + 0.130558i
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) 7.00000 1.03209
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −0.500000 + 0.866025i −0.0721688 + 0.125000i
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) −0.500000 + 0.866025i −0.0707107 + 0.122474i
\(51\) 0.500000 + 0.866025i 0.0700140 + 0.121268i
\(52\) 3.00000 + 5.19615i 0.416025 + 0.720577i
\(53\) 2.00000 + 3.46410i 0.274721 + 0.475831i 0.970065 − 0.242846i \(-0.0780811\pi\)
−0.695344 + 0.718677i \(0.744748\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.500000 − 0.866025i 0.0674200 − 0.116775i
\(56\) 0 0
\(57\) −2.00000 − 3.46410i −0.264906 − 0.458831i
\(58\) 6.00000 0.787839
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 2.00000 − 5.19615i 0.254000 − 0.659912i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 − 5.19615i 0.372104 − 0.644503i
\(66\) −1.00000 −0.123091
\(67\) −2.50000 − 4.33013i −0.305424 − 0.529009i 0.671932 − 0.740613i \(-0.265465\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0.500000 − 0.866025i 0.0606339 − 0.105021i
\(69\) −3.50000 + 6.06218i −0.421350 + 0.729800i
\(70\) 0 0
\(71\) −3.00000 − 5.19615i −0.356034 − 0.616670i 0.631260 − 0.775571i \(-0.282538\pi\)
−0.987294 + 0.158901i \(0.949205\pi\)
\(72\) −0.500000 − 0.866025i −0.0589256 − 0.102062i
\(73\) −1.00000 − 1.73205i −0.117041 − 0.202721i 0.801553 − 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) −3.50000 + 6.06218i −0.406867 + 0.704714i
\(75\) −0.500000 − 0.866025i −0.0577350 − 0.100000i
\(76\) −2.00000 + 3.46410i −0.229416 + 0.397360i
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −1.50000 + 2.59808i −0.168763 + 0.292306i −0.937985 − 0.346675i \(-0.887311\pi\)
0.769222 + 0.638982i \(0.220644\pi\)
\(80\) −0.500000 − 0.866025i −0.0559017 − 0.0968246i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 1.00000 + 1.73205i 0.110432 + 0.191273i
\(83\) −8.00000 − 13.8564i −0.878114 − 1.52094i −0.853408 − 0.521243i \(-0.825468\pi\)
−0.0247060 − 0.999695i \(-0.507865\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 3.50000 − 6.06218i 0.377415 − 0.653701i
\(87\) −3.00000 + 5.19615i −0.321634 + 0.557086i
\(88\) 0.500000 + 0.866025i 0.0533002 + 0.0923186i
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) −0.500000 + 0.866025i −0.0527046 + 0.0912871i
\(91\) 0 0
\(92\) 7.00000 0.729800
\(93\) 3.50000 + 4.33013i 0.362933 + 0.449013i
\(94\) 3.00000 0.309426
\(95\) 4.00000 0.410391
\(96\) −0.500000 + 0.866025i −0.0510310 + 0.0883883i
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.50000 + 6.06218i 0.353553 + 0.612372i
\(99\) 0.500000 − 0.866025i 0.0502519 − 0.0870388i
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 930.2.i.e.811.1 yes 2
31.25 even 3 inner 930.2.i.e.211.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.i.e.211.1 ✓ 2 31.25 even 3 inner
930.2.i.e.811.1 yes 2 1.1 even 1 trivial