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Properties

Label 930.2.i.d.211.1
Level $930$
Weight $2$
Character 930.211
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,-5,2,-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 211.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 930.211
Dual form 930.2.i.d.811.1

$q$-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} +(-2.50000 - 4.33013i) q^{7} +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} +(-2.00000 + 3.46410i) q^{13} +(-2.50000 - 4.33013i) q^{14} +1.00000 q^{15} +1.00000 q^{16} +(-2.00000 - 3.46410i) q^{17} +(-0.500000 + 0.866025i) q^{18} +(-2.00000 - 3.46410i) q^{19} +(-0.500000 + 0.866025i) q^{20} +(-2.50000 + 4.33013i) q^{21} +(0.500000 - 0.866025i) q^{22} +2.00000 q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{26} +1.00000 q^{27} +(-2.50000 - 4.33013i) q^{28} -9.00000 q^{29} +1.00000 q^{30} +(-5.50000 + 0.866025i) q^{31} +1.00000 q^{32} -1.00000 q^{33} +(-2.00000 - 3.46410i) q^{34} +5.00000 q^{35} +(-0.500000 + 0.866025i) q^{36} +(-1.00000 - 1.73205i) q^{37} +(-2.00000 - 3.46410i) q^{38} +4.00000 q^{39} +(-0.500000 + 0.866025i) q^{40} +(1.00000 - 1.73205i) q^{41} +(-2.50000 + 4.33013i) q^{42} +(-4.00000 - 6.92820i) q^{43} +(0.500000 - 0.866025i) q^{44} +(-0.500000 - 0.866025i) q^{45} +2.00000 q^{46} +8.00000 q^{47} +(-0.500000 - 0.866025i) q^{48} +(-9.00000 + 15.5885i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-2.00000 + 3.46410i) q^{51} +(-2.00000 + 3.46410i) q^{52} +(4.50000 - 7.79423i) q^{53} +1.00000 q^{54} +(0.500000 + 0.866025i) q^{55} +(-2.50000 - 4.33013i) q^{56} +(-2.00000 + 3.46410i) q^{57} -9.00000 q^{58} +(1.50000 + 2.59808i) q^{59} +1.00000 q^{60} +6.00000 q^{61} +(-5.50000 + 0.866025i) q^{62} +5.00000 q^{63} +1.00000 q^{64} +(-2.00000 - 3.46410i) q^{65} -1.00000 q^{66} +(5.00000 - 8.66025i) q^{67} +(-2.00000 - 3.46410i) q^{68} +(-1.00000 - 1.73205i) q^{69} +5.00000 q^{70} +(2.00000 - 3.46410i) q^{71} +(-0.500000 + 0.866025i) q^{72} +(-1.00000 + 1.73205i) q^{73} +(-1.00000 - 1.73205i) q^{74} +(-0.500000 + 0.866025i) q^{75} +(-2.00000 - 3.46410i) q^{76} -5.00000 q^{77} +4.00000 q^{78} +(6.00000 + 10.3923i) q^{79} +(-0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(1.00000 - 1.73205i) q^{82} +(4.50000 - 7.79423i) q^{83} +(-2.50000 + 4.33013i) q^{84} +4.00000 q^{85} +(-4.00000 - 6.92820i) q^{86} +(4.50000 + 7.79423i) q^{87} +(0.500000 - 0.866025i) q^{88} -6.00000 q^{89} +(-0.500000 - 0.866025i) q^{90} +20.0000 q^{91} +2.00000 q^{92} +(3.50000 + 4.33013i) q^{93} +8.00000 q^{94} +4.00000 q^{95} +(-0.500000 - 0.866025i) q^{96} -1.00000 q^{97} +(-9.00000 + 15.5885i) q^{98} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} + 2 q^{8} - q^{9} - q^{10} + q^{11} - q^{12} - 4 q^{13} - 5 q^{14} + 2 q^{15} + 2 q^{16} - 4 q^{17} - q^{18} - 4 q^{19} - q^{20} - 5 q^{21}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(7\) −2.50000 − 4.33013i −0.944911 − 1.63663i −0.755929 − 0.654654i \(-0.772814\pi\)
−0.188982 − 0.981981i \(-0.560519\pi\)
\(8\) 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 −0.780750 − 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) −0.500000 − 0.866025i −0.144338 − 0.250000i
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) −2.50000 − 4.33013i −0.668153 − 1.15728i
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 − 3.46410i −0.485071 − 0.840168i 0.514782 − 0.857321i \(-0.327873\pi\)
−0.999853 + 0.0171533i \(0.994540\pi\)
\(18\) −0.500000 + 0.866025i −0.117851 + 0.204124i
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) −0.500000 + 0.866025i −0.111803 + 0.193649i
\(21\) −2.50000 + 4.33013i −0.545545 + 0.944911i
\(22\) 0.500000 − 0.866025i 0.106600 − 0.184637i
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −0.500000 − 0.866025i −0.102062 − 0.176777i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) −2.00000 + 3.46410i −0.392232 + 0.679366i
\(27\) 1.00000 0.192450
\(28\) −2.50000 − 4.33013i −0.472456 − 0.818317i
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 1.00000 0.182574
\(31\) −5.50000 + 0.866025i −0.987829 + 0.155543i
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.00000 − 3.46410i −0.342997 − 0.594089i
\(35\) 5.00000 0.845154
\(36\) −0.500000 + 0.866025i −0.0833333 + 0.144338i
\(37\) −1.00000 − 1.73205i −0.164399 − 0.284747i 0.772043 − 0.635571i \(-0.219235\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(38\) −2.00000 − 3.46410i −0.324443 − 0.561951i
\(39\) 4.00000 0.640513
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) 1.00000 − 1.73205i 0.156174 − 0.270501i −0.777312 − 0.629115i \(-0.783417\pi\)
0.933486 + 0.358614i \(0.116751\pi\)
\(42\) −2.50000 + 4.33013i −0.385758 + 0.668153i
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 0.500000 − 0.866025i 0.0753778 − 0.130558i
\(45\) −0.500000 − 0.866025i −0.0745356 − 0.129099i
\(46\) 2.00000 0.294884
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −0.500000 − 0.866025i −0.0721688 − 0.125000i
\(49\) −9.00000 + 15.5885i −1.28571 + 2.22692i
\(50\) −0.500000 − 0.866025i −0.0707107 − 0.122474i
\(51\) −2.00000 + 3.46410i −0.280056 + 0.485071i
\(52\) −2.00000 + 3.46410i −0.277350 + 0.480384i
\(53\) 4.50000 − 7.79423i 0.618123 − 1.07062i −0.371706 − 0.928351i \(-0.621227\pi\)
0.989828 − 0.142269i \(-0.0454398\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.500000 + 0.866025i 0.0674200 + 0.116775i
\(56\) −2.50000 − 4.33013i −0.334077 − 0.578638i
\(57\) −2.00000 + 3.46410i −0.264906 + 0.458831i
\(58\) −9.00000 −1.18176
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 − 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −5.50000 + 0.866025i −0.698501 + 0.109985i
\(63\) 5.00000 0.629941
\(64\) 1.00000 0.125000
\(65\) −2.00000 − 3.46410i −0.248069 − 0.429669i
\(66\) −1.00000 −0.123091
\(67\) 5.00000 − 8.66025i 0.610847 − 1.05802i −0.380251 − 0.924883i \(-0.624162\pi\)
0.991098 − 0.133135i \(-0.0425044\pi\)
\(68\) −2.00000 − 3.46410i −0.242536 − 0.420084i
\(69\) −1.00000 − 1.73205i −0.120386 − 0.208514i
\(70\) 5.00000 0.597614
\(71\) 2.00000 − 3.46410i 0.237356 − 0.411113i −0.722599 − 0.691268i \(-0.757052\pi\)
0.959955 + 0.280155i \(0.0903858\pi\)
\(72\) −0.500000 + 0.866025i −0.0589256 + 0.102062i
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 − 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) −1.00000 − 1.73205i −0.116248 − 0.201347i
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) −2.00000 − 3.46410i −0.229416 − 0.397360i
\(77\) −5.00000 −0.569803
\(78\) 4.00000 0.452911
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) −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
\(83\) 4.50000 − 7.79423i 0.493939 − 0.855528i −0.506036 − 0.862512i \(-0.668890\pi\)
0.999976 + 0.00698436i \(0.00222321\pi\)
\(84\) −2.50000 + 4.33013i −0.272772 + 0.472456i
\(85\) 4.00000 0.433861
\(86\) −4.00000 − 6.92820i −0.431331 − 0.747087i
\(87\) 4.50000 + 7.79423i 0.482451 + 0.835629i
\(88\) 0.500000 − 0.866025i 0.0533002 − 0.0923186i
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −0.500000 − 0.866025i −0.0527046 − 0.0912871i
\(91\) 20.0000 2.09657
\(92\) 2.00000 0.208514
\(93\) 3.50000 + 4.33013i 0.362933 + 0.449013i
\(94\) 8.00000 0.825137
\(95\) 4.00000 0.410391
\(96\) −0.500000 − 0.866025i −0.0510310 − 0.0883883i
\(97\) −1.00000 −0.101535 −0.0507673 − 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) −9.00000 + 15.5885i −0.909137 + 1.57467i
\(99\) 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.d.211.1 ✓ 2
31.5 even 3 inner 930.2.i.d.811.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.i.d.211.1 ✓ 2 1.1 even 1 trivial
930.2.i.d.811.1 yes 2 31.5 even 3 inner