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Properties

Label 930.2.d.d.559.1
Level $930$
Weight $2$
Character 930.559
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(559,930)] 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("930.559"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,4,-2,0,0,-2,2,0] 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: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 559.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 930.559
Dual form 930.2.d.d.559.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(1.00000 - 2.00000i) q^{10} +1.00000i q^{12} +2.00000i q^{13} +2.00000 q^{14} +(1.00000 - 2.00000i) q^{15} +1.00000 q^{16} +1.00000i q^{18} +4.00000 q^{19} +(-2.00000 - 1.00000i) q^{20} +2.00000 q^{21} +2.00000i q^{23} +1.00000 q^{24} +(3.00000 + 4.00000i) q^{25} +2.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +10.0000 q^{29} +(-2.00000 - 1.00000i) q^{30} -1.00000 q^{31} -1.00000i q^{32} +(-2.00000 + 4.00000i) q^{35} +1.00000 q^{36} +2.00000i q^{37} -4.00000i q^{38} +2.00000 q^{39} +(-1.00000 + 2.00000i) q^{40} +2.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +(-2.00000 - 1.00000i) q^{45} +2.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -2.00000i q^{52} +2.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} -4.00000i q^{57} -10.0000i q^{58} +6.00000 q^{59} +(-1.00000 + 2.00000i) q^{60} -8.00000 q^{61} +1.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +(-2.00000 + 4.00000i) q^{65} -8.00000i q^{67} +2.00000 q^{69} +(4.00000 + 2.00000i) q^{70} -8.00000 q^{71} -1.00000i q^{72} -10.0000i q^{73} +2.00000 q^{74} +(4.00000 - 3.00000i) q^{75} -4.00000 q^{76} -2.00000i q^{78} -12.0000 q^{79} +(2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -2.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -10.0000i q^{87} +14.0000 q^{89} +(-1.00000 + 2.00000i) q^{90} -4.00000 q^{91} -2.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} +(8.00000 + 4.00000i) q^{95} -1.00000 q^{96} +4.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{9} + 2 q^{10} + 4 q^{14} + 2 q^{15} + 2 q^{16} + 8 q^{19} - 4 q^{20} + 4 q^{21} + 2 q^{24} + 6 q^{25} + 4 q^{26} + 20 q^{29} - 4 q^{30} - 2 q^{31} - 4 q^{35} + 2 q^{36}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) −1.00000 −0.408248
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 1.00000 − 2.00000i 0.316228 − 0.632456i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 2.00000 0.534522
\(15\) 1.00000 − 2.00000i 0.258199 − 0.516398i
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 − 1.00000i −0.447214 − 0.223607i
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 1.00000 0.204124
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) −2.00000 − 1.00000i −0.365148 − 0.182574i
\(31\) −1.00000 −0.179605
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 + 4.00000i −0.338062 + 0.676123i
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 2.00000 0.320256
\(40\) −1.00000 + 2.00000i −0.158114 + 0.316228i
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) −2.00000 − 1.00000i −0.298142 − 0.149071i
\(46\) 2.00000 0.294884
\(47\) − 4.00000i − 0.583460i −0.956501 − 0.291730i \(-0.905769\pi\)
0.956501 − 0.291730i \(-0.0942309\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 4.00000 − 3.00000i 0.565685 − 0.424264i
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) − 4.00000i − 0.529813i
\(58\) − 10.0000i − 1.31306i
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 + 2.00000i −0.129099 + 0.258199i
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 1.00000i 0.127000i
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) −2.00000 + 4.00000i −0.248069 + 0.496139i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 4.00000 + 2.00000i 0.478091 + 0.239046i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) 2.00000 0.232495
\(75\) 4.00000 − 3.00000i 0.461880 − 0.346410i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 2.00000 + 1.00000i 0.223607 + 0.111803i
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 10.0000i − 1.07211i
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −1.00000 + 2.00000i −0.105409 + 0.210819i
\(91\) −4.00000 −0.419314
\(92\) − 2.00000i − 0.208514i
\(93\) 1.00000i 0.103695i
\(94\) −4.00000 −0.412568
\(95\) 8.00000 + 4.00000i 0.820783 + 0.410391i
\(96\) −1.00000 −0.102062
\(97\) 4.00000i 0.406138i 0.979164 + 0.203069i \(0.0650917\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) 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 930.2.d.d.559.1 ✓ 2
3.2 odd 2 2790.2.d.d.559.2 2
5.2 odd 4 4650.2.a.z.1.1 1
5.3 odd 4 4650.2.a.u.1.1 1
5.4 even 2 inner 930.2.d.d.559.2 yes 2
15.14 odd 2 2790.2.d.d.559.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.d.559.1 ✓ 2 1.1 even 1 trivial
930.2.d.d.559.2 yes 2 5.4 even 2 inner
2790.2.d.d.559.1 2 15.14 odd 2
2790.2.d.d.559.2 2 3.2 odd 2
4650.2.a.u.1.1 1 5.3 odd 4
4650.2.a.z.1.1 1 5.2 odd 4