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Properties

Label 930.2.i.h.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,-5] 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.h.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} +(-2.50000 + 4.33013i) q^{11} +(0.500000 + 0.866025i) q^{12} +(-3.00000 + 5.19615i) q^{13} +(1.50000 + 2.59808i) q^{14} +1.00000 q^{15} +1.00000 q^{16} +(-4.00000 - 6.92820i) q^{17} +(-0.500000 + 0.866025i) q^{18} +(2.00000 + 3.46410i) q^{19} +(0.500000 - 0.866025i) q^{20} +(-1.50000 + 2.59808i) q^{21} +(-2.50000 + 4.33013i) q^{22} +8.00000 q^{23} +(0.500000 + 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +(-3.00000 + 5.19615i) q^{26} -1.00000 q^{27} +(1.50000 + 2.59808i) q^{28} -1.00000 q^{29} +1.00000 q^{30} +(3.50000 - 4.33013i) q^{31} +1.00000 q^{32} -5.00000 q^{33} +(-4.00000 - 6.92820i) q^{34} +3.00000 q^{35} +(-0.500000 + 0.866025i) q^{36} +(1.00000 + 1.73205i) q^{37} +(2.00000 + 3.46410i) q^{38} -6.00000 q^{39} +(0.500000 - 0.866025i) q^{40} +(3.00000 - 5.19615i) q^{41} +(-1.50000 + 2.59808i) q^{42} +(-4.00000 - 6.92820i) q^{43} +(-2.50000 + 4.33013i) q^{44} +(0.500000 + 0.866025i) q^{45} +8.00000 q^{46} +12.0000 q^{47} +(0.500000 + 0.866025i) q^{48} +(-1.00000 + 1.73205i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(4.00000 - 6.92820i) q^{51} +(-3.00000 + 5.19615i) q^{52} +(-2.50000 + 4.33013i) q^{53} -1.00000 q^{54} +(2.50000 + 4.33013i) q^{55} +(1.50000 + 2.59808i) q^{56} +(-2.00000 + 3.46410i) q^{57} -1.00000 q^{58} +(-5.50000 - 9.52628i) q^{59} +1.00000 q^{60} +12.0000 q^{61} +(3.50000 - 4.33013i) q^{62} -3.00000 q^{63} +1.00000 q^{64} +(3.00000 + 5.19615i) q^{65} -5.00000 q^{66} +(-5.00000 + 8.66025i) q^{67} +(-4.00000 - 6.92820i) q^{68} +(4.00000 + 6.92820i) q^{69} +3.00000 q^{70} +(-0.500000 + 0.866025i) q^{72} +(-3.00000 + 5.19615i) q^{73} +(1.00000 + 1.73205i) q^{74} +(0.500000 - 0.866025i) q^{75} +(2.00000 + 3.46410i) q^{76} -15.0000 q^{77} -6.00000 q^{78} +(-4.00000 - 6.92820i) q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(3.00000 - 5.19615i) q^{82} +(-1.50000 + 2.59808i) q^{83} +(-1.50000 + 2.59808i) q^{84} -8.00000 q^{85} +(-4.00000 - 6.92820i) q^{86} +(-0.500000 - 0.866025i) q^{87} +(-2.50000 + 4.33013i) q^{88} +(0.500000 + 0.866025i) q^{90} -18.0000 q^{91} +8.00000 q^{92} +(5.50000 + 0.866025i) q^{93} +12.0000 q^{94} +4.00000 q^{95} +(0.500000 + 0.866025i) q^{96} +13.0000 q^{97} +(-1.00000 + 1.73205i) q^{98} +(-2.50000 - 4.33013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} + 3 q^{7} + 2 q^{8} - q^{9} + q^{10} - 5 q^{11} + q^{12} - 6 q^{13} + 3 q^{14} + 2 q^{15} + 2 q^{16} - 8 q^{17} - q^{18} + 4 q^{19} + q^{20} - 3 q^{21}+ \cdots - 5 q^{99}+O(q^{100}) \) 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 \(0.0252088\pi\)
−0.429919 + 0.902867i \(0.641458\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0.500000 − 0.866025i 0.158114 − 0.273861i
\(11\) −2.50000 + 4.33013i −0.753778 + 1.30558i 0.192201 + 0.981356i \(0.438437\pi\)
−0.945979 + 0.324227i \(0.894896\pi\)
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) −3.00000 + 5.19615i −0.832050 + 1.44115i 0.0643593 + 0.997927i \(0.479500\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) 1.50000 + 2.59808i 0.400892 + 0.694365i
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −4.00000 − 6.92820i −0.970143 − 1.68034i −0.695113 − 0.718900i \(-0.744646\pi\)
−0.275029 − 0.961436i \(-0.588688\pi\)
\(18\) −0.500000 + 0.866025i −0.117851 + 0.204124i
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0.500000 − 0.866025i 0.111803 − 0.193649i
\(21\) −1.50000 + 2.59808i −0.327327 + 0.566947i
\(22\) −2.50000 + 4.33013i −0.533002 + 0.923186i
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 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
\(28\) 1.50000 + 2.59808i 0.283473 + 0.490990i
\(29\) −1.00000 −0.185695 −0.0928477 − 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.50000 − 4.33013i 0.628619 − 0.777714i
\(32\) 1.00000 0.176777
\(33\) −5.00000 −0.870388
\(34\) −4.00000 − 6.92820i −0.685994 − 1.18818i
\(35\) 3.00000 0.507093
\(36\) −0.500000 + 0.866025i −0.0833333 + 0.144338i
\(37\) 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
\(39\) −6.00000 −0.960769
\(40\) 0.500000 − 0.866025i 0.0790569 − 0.136931i
\(41\) 3.00000 − 5.19615i 0.468521 − 0.811503i −0.530831 − 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) −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 \(-0.375495\pi\)
\(44\) −2.50000 + 4.33013i −0.376889 + 0.652791i
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 8.00000 1.17954
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 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
\(51\) 4.00000 − 6.92820i 0.560112 − 0.970143i
\(52\) −3.00000 + 5.19615i −0.416025 + 0.720577i
\(53\) −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
\(55\) 2.50000 + 4.33013i 0.337100 + 0.583874i
\(56\) 1.50000 + 2.59808i 0.200446 + 0.347183i
\(57\) −2.00000 + 3.46410i −0.264906 + 0.458831i
\(58\) −1.00000 −0.131306
\(59\) −5.50000 − 9.52628i −0.716039 − 1.24022i −0.962557 − 0.271078i \(-0.912620\pi\)
0.246518 − 0.969138i \(-0.420713\pi\)
\(60\) 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
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) −5.00000 −0.615457
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) −4.00000 − 6.92820i −0.485071 − 0.840168i
\(69\) 4.00000 + 6.92820i 0.481543 + 0.834058i
\(70\) 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 −0.986447 − 0.164083i \(-0.947534\pi\)
0.635323 + 0.772246i \(0.280867\pi\)
\(74\) 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
\(77\) −15.0000 −1.70941
\(78\) −6.00000 −0.679366
\(79\) −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
\(82\) 3.00000 − 5.19615i 0.331295 − 0.573819i
\(83\) −1.50000 + 2.59808i −0.164646 + 0.285176i −0.936530 − 0.350588i \(-0.885982\pi\)
0.771883 + 0.635764i \(0.219315\pi\)
\(84\) −1.50000 + 2.59808i −0.163663 + 0.283473i
\(85\) −8.00000 −0.867722
\(86\) −4.00000 − 6.92820i −0.431331 − 0.747087i
\(87\) −0.500000 − 0.866025i −0.0536056 − 0.0928477i
\(88\) −2.50000 + 4.33013i −0.266501 + 0.461593i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0.500000 + 0.866025i 0.0527046 + 0.0912871i
\(91\) −18.0000 −1.88691
\(92\) 8.00000 0.834058
\(93\) 5.50000 + 0.866025i 0.570323 + 0.0898027i
\(94\) 12.0000 1.23771
\(95\) 4.00000 0.410391
\(96\) 0.500000 + 0.866025i 0.0510310 + 0.0883883i
\(97\) 13.0000 1.31995 0.659975 − 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) −1.00000 + 1.73205i −0.101015 + 0.174964i
\(99\) −2.50000 − 4.33013i −0.251259 − 0.435194i
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.h.211.1 ✓ 2
31.5 even 3 inner 930.2.i.h.811.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.i.h.211.1 ✓ 2 1.1 even 1 trivial
930.2.i.h.811.1 yes 2 31.5 even 3 inner