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Properties

Label 930.2.i.b.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,-3,-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.b.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} +(-1.50000 - 2.59808i) 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} +(1.50000 + 2.59808i) q^{14} +1.00000 q^{15} +1.00000 q^{16} +(-1.00000 - 1.73205i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 - 1.73205i) q^{19} +(0.500000 - 0.866025i) q^{20} +(1.50000 - 2.59808i) q^{21} +(-0.500000 + 0.866025i) q^{22} -4.00000 q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(-1.50000 - 2.59808i) q^{28} -7.00000 q^{29} -1.00000 q^{30} +(3.50000 - 4.33013i) q^{31} -1.00000 q^{32} +1.00000 q^{33} +(1.00000 + 1.73205i) q^{34} -3.00000 q^{35} +(-0.500000 + 0.866025i) q^{36} +(-2.00000 - 3.46410i) q^{37} +(1.00000 + 1.73205i) q^{38} +(-0.500000 + 0.866025i) q^{40} +(-1.50000 + 2.59808i) q^{42} +(-4.00000 - 6.92820i) q^{43} +(0.500000 - 0.866025i) q^{44} +(0.500000 + 0.866025i) q^{45} +4.00000 q^{46} -6.00000 q^{47} +(0.500000 + 0.866025i) q^{48} +(-1.00000 + 1.73205i) q^{49} +(0.500000 + 0.866025i) q^{50} +(1.00000 - 1.73205i) q^{51} +(-2.50000 + 4.33013i) q^{53} +1.00000 q^{54} +(-0.500000 - 0.866025i) q^{55} +(1.50000 + 2.59808i) q^{56} +(1.00000 - 1.73205i) q^{57} +7.00000 q^{58} +(3.50000 + 6.06218i) q^{59} +1.00000 q^{60} +12.0000 q^{61} +(-3.50000 + 4.33013i) q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +(4.00000 - 6.92820i) q^{67} +(-1.00000 - 1.73205i) q^{68} +(-2.00000 - 3.46410i) q^{69} +3.00000 q^{70} +(0.500000 - 0.866025i) q^{72} +(3.00000 - 5.19615i) q^{73} +(2.00000 + 3.46410i) q^{74} +(0.500000 - 0.866025i) q^{75} +(-1.00000 - 1.73205i) q^{76} -3.00000 q^{77} +(-4.00000 - 6.92820i) q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(4.50000 - 7.79423i) q^{83} +(1.50000 - 2.59808i) q^{84} -2.00000 q^{85} +(4.00000 + 6.92820i) q^{86} +(-3.50000 - 6.06218i) q^{87} +(-0.500000 + 0.866025i) q^{88} +6.00000 q^{89} +(-0.500000 - 0.866025i) q^{90} -4.00000 q^{92} +(5.50000 + 0.866025i) q^{93} +6.00000 q^{94} -2.00000 q^{95} +(-0.500000 - 0.866025i) q^{96} -5.00000 q^{97} +(1.00000 - 1.73205i) q^{98} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 3 q^{7} - 2 q^{8} - q^{9} - q^{10} + q^{11} + q^{12} + 3 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} + q^{18} - 2 q^{19} + q^{20} + 3 q^{21} - q^{22}+ \cdots + q^{99}+O(q^{100}) \) 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
\(7\) −1.50000 − 2.59808i −0.566947 − 0.981981i −0.996866 − 0.0791130i \(-0.974791\pi\)
0.429919 − 0.902867i \(-0.358542\pi\)
\(8\) −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
\(13\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 1.50000 + 2.59808i 0.400892 + 0.694365i
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 − 1.73205i −0.242536 − 0.420084i 0.718900 − 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0.500000 − 0.866025i 0.117851 − 0.204124i
\(19\) −1.00000 − 1.73205i −0.229416 − 0.397360i 0.728219 − 0.685344i \(-0.240348\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0.500000 − 0.866025i 0.111803 − 0.193649i
\(21\) 1.50000 − 2.59808i 0.327327 − 0.566947i
\(22\) −0.500000 + 0.866025i −0.106600 + 0.184637i
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −0.500000 − 0.866025i −0.102062 − 0.176777i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.50000 − 2.59808i −0.283473 − 0.490990i
\(29\) −7.00000 −1.29987 −0.649934 − 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) −1.00000 −0.182574
\(31\) 3.50000 − 4.33013i 0.628619 − 0.777714i
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 1.00000 + 1.73205i 0.171499 + 0.297044i
\(35\) −3.00000 −0.507093
\(36\) −0.500000 + 0.866025i −0.0833333 + 0.144338i
\(37\) −2.00000 − 3.46410i −0.328798 − 0.569495i 0.653476 − 0.756948i \(-0.273310\pi\)
−0.982274 + 0.187453i \(0.939977\pi\)
\(38\) 1.00000 + 1.73205i 0.162221 + 0.280976i
\(39\) 0 0
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) −1.50000 + 2.59808i −0.231455 + 0.400892i
\(43\) −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
\(46\) 4.00000 0.589768
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0.500000 + 0.866025i 0.0721688 + 0.125000i
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0.500000 + 0.866025i 0.0707107 + 0.122474i
\(51\) 1.00000 − 1.73205i 0.140028 − 0.242536i
\(52\) 0 0
\(53\) −2.50000 + 4.33013i −0.343401 + 0.594789i −0.985062 − 0.172200i \(-0.944912\pi\)
0.641661 + 0.766989i \(0.278246\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.500000 − 0.866025i −0.0674200 − 0.116775i
\(56\) 1.50000 + 2.59808i 0.200446 + 0.347183i
\(57\) 1.00000 − 1.73205i 0.132453 − 0.229416i
\(58\) 7.00000 0.919145
\(59\) 3.50000 + 6.06218i 0.455661 + 0.789228i 0.998726 − 0.0504625i \(-0.0160695\pi\)
−0.543065 + 0.839691i \(0.682736\pi\)
\(60\) 1.00000 0.129099
\(61\) 12.0000 1.53644 0.768221 − 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −3.50000 + 4.33013i −0.444500 + 0.549927i
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 4.00000 − 6.92820i 0.488678 − 0.846415i −0.511237 − 0.859440i \(-0.670813\pi\)
0.999915 + 0.0130248i \(0.00414604\pi\)
\(68\) −1.00000 − 1.73205i −0.121268 − 0.210042i
\(69\) −2.00000 − 3.46410i −0.240772 − 0.417029i
\(70\) 3.00000 0.358569
\(71\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) 0.500000 − 0.866025i 0.0589256 − 0.102062i
\(73\) 3.00000 − 5.19615i 0.351123 − 0.608164i −0.635323 − 0.772246i \(-0.719133\pi\)
0.986447 + 0.164083i \(0.0524664\pi\)
\(74\) 2.00000 + 3.46410i 0.232495 + 0.402694i
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) −1.00000 − 1.73205i −0.114708 − 0.198680i
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −4.00000 − 6.92820i −0.450035 − 0.779484i 0.548352 − 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 0.500000 − 0.866025i 0.0559017 − 0.0968246i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 4.50000 − 7.79423i 0.493939 − 0.855528i −0.506036 − 0.862512i \(-0.668890\pi\)
0.999976 + 0.00698436i \(0.00222321\pi\)
\(84\) 1.50000 − 2.59808i 0.163663 − 0.283473i
\(85\) −2.00000 −0.216930
\(86\) 4.00000 + 6.92820i 0.431331 + 0.747087i
\(87\) −3.50000 − 6.06218i −0.375239 − 0.649934i
\(88\) −0.500000 + 0.866025i −0.0533002 + 0.0923186i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −0.500000 − 0.866025i −0.0527046 − 0.0912871i
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 5.50000 + 0.866025i 0.570323 + 0.0898027i
\(94\) 6.00000 0.618853
\(95\) −2.00000 −0.205196
\(96\) −0.500000 − 0.866025i −0.0510310 − 0.0883883i
\(97\) −5.00000 −0.507673 −0.253837 − 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 1.00000 − 1.73205i 0.101015 − 0.174964i
\(99\) 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.b.211.1 ✓ 2
31.5 even 3 inner 930.2.i.b.811.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.i.b.211.1 ✓ 2 1.1 even 1 trivial
930.2.i.b.811.1 yes 2 31.5 even 3 inner