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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,2,Mod(289,912)] 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("912.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bo (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 769.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 912.769
Dual form 912.2.bo.f.625.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.766044 - 0.642788i) q^{3} +(3.20574 + 1.16679i) q^{5} +(-1.43969 + 2.49362i) q^{7} +(0.173648 - 0.984808i) q^{9} +(-0.173648 - 0.300767i) q^{11} +(-1.26604 - 1.06234i) q^{13} +(3.20574 - 1.16679i) q^{15} +(1.20574 + 6.83807i) q^{17} +(2.82635 + 3.31839i) q^{19} +(0.500000 + 2.83564i) q^{21} +(6.39053 - 2.32596i) q^{23} +(5.08512 + 4.26692i) q^{25} +(-0.500000 - 0.866025i) q^{27} +(1.10354 - 6.25849i) q^{29} +(0.798133 - 1.38241i) q^{31} +(-0.326352 - 0.118782i) q^{33} +(-7.52481 + 6.31407i) q^{35} -11.2121 q^{37} -1.65270 q^{39} +(-2.67365 + 2.24346i) q^{41} +(2.14543 + 0.780873i) q^{43} +(1.70574 - 2.95442i) q^{45} +(-0.971782 + 5.51125i) q^{47} +(-0.645430 - 1.11792i) q^{49} +(5.31908 + 4.46324i) q^{51} +(-1.86097 + 0.677337i) q^{53} +(-0.205737 - 1.16679i) q^{55} +(4.29813 + 0.725293i) q^{57} +(-0.0773815 - 0.438852i) q^{59} +(11.7763 - 4.28623i) q^{61} +(2.20574 + 1.85083i) q^{63} +(-2.81908 - 4.88279i) q^{65} +(0.187319 - 1.06234i) q^{67} +(3.40033 - 5.88954i) q^{69} +(-15.6211 - 5.68561i) q^{71} +(9.51367 - 7.98292i) q^{73} +6.63816 q^{75} +1.00000 q^{77} +(8.36824 - 7.02179i) q^{79} +(-0.939693 - 0.342020i) q^{81} +(5.85844 - 10.1471i) q^{83} +(-4.11334 + 23.3279i) q^{85} +(-3.17752 - 5.50362i) q^{87} +(1.37346 + 1.15247i) q^{89} +(4.47178 - 1.62760i) q^{91} +(-0.277189 - 1.57202i) q^{93} +(5.18866 + 13.9357i) q^{95} +(0.634285 + 3.59721i) q^{97} +(-0.326352 + 0.118782i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{5} - 3 q^{7} - 3 q^{13} + 9 q^{15} - 3 q^{17} + 18 q^{19} + 3 q^{21} + 21 q^{23} + 9 q^{25} - 3 q^{27} - 3 q^{29} - 9 q^{31} - 3 q^{33} - 18 q^{35} - 18 q^{37} - 12 q^{39} - 15 q^{41} - 3 q^{43}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.766044 − 0.642788i 0.442276 − 0.371114i
\(4\) 0 0
\(5\) 3.20574 + 1.16679i 1.43365 + 0.521806i 0.937975 − 0.346703i \(-0.112699\pi\)
0.495674 + 0.868509i \(0.334921\pi\)
\(6\) 0 0
\(7\) −1.43969 + 2.49362i −0.544153 + 0.942500i 0.454507 + 0.890743i \(0.349815\pi\)
−0.998660 + 0.0517569i \(0.983518\pi\)
\(8\) 0 0
\(9\) 0.173648 − 0.984808i 0.0578827 − 0.328269i
\(10\) 0 0
\(11\) −0.173648 − 0.300767i −0.0523569 − 0.0906848i 0.838659 − 0.544657i \(-0.183340\pi\)
−0.891016 + 0.453972i \(0.850007\pi\)
\(12\) 0 0
\(13\) −1.26604 − 1.06234i −0.351138 − 0.294639i 0.450109 − 0.892974i \(-0.351385\pi\)
−0.801247 + 0.598334i \(0.795830\pi\)
\(14\) 0 0
\(15\) 3.20574 − 1.16679i 0.827718 − 0.301265i
\(16\) 0 0
\(17\) 1.20574 + 6.83807i 0.292434 + 1.65848i 0.677452 + 0.735567i \(0.263084\pi\)
−0.385017 + 0.922909i \(0.625805\pi\)
\(18\) 0 0
\(19\) 2.82635 + 3.31839i 0.648410 + 0.761292i
\(20\) 0 0
\(21\) 0.500000 + 2.83564i 0.109109 + 0.618788i
\(22\) 0 0
\(23\) 6.39053 − 2.32596i 1.33252 − 0.484997i 0.425069 − 0.905161i \(-0.360250\pi\)
0.907448 + 0.420164i \(0.138027\pi\)
\(24\) 0 0
\(25\) 5.08512 + 4.26692i 1.01702 + 0.853385i
\(26\) 0 0
\(27\) −0.500000 − 0.866025i −0.0962250 − 0.166667i
\(28\) 0 0
\(29\) 1.10354 − 6.25849i 0.204922 − 1.16217i −0.692640 − 0.721284i \(-0.743552\pi\)
0.897562 − 0.440888i \(-0.145337\pi\)
\(30\) 0 0
\(31\) 0.798133 − 1.38241i 0.143349 − 0.248288i −0.785407 − 0.618980i \(-0.787546\pi\)
0.928756 + 0.370692i \(0.120880\pi\)
\(32\) 0 0
\(33\) −0.326352 − 0.118782i −0.0568106 − 0.0206774i
\(34\) 0 0
\(35\) −7.52481 + 6.31407i −1.27193 + 1.06727i
\(36\) 0 0
\(37\) −11.2121 −1.84326 −0.921632 − 0.388066i \(-0.873143\pi\)
−0.921632 + 0.388066i \(0.873143\pi\)
\(38\) 0 0
\(39\) −1.65270 −0.264644
\(40\) 0 0
\(41\) −2.67365 + 2.24346i −0.417554 + 0.350369i −0.827232 − 0.561861i \(-0.810086\pi\)
0.409678 + 0.912230i \(0.365641\pi\)
\(42\) 0 0
\(43\) 2.14543 + 0.780873i 0.327175 + 0.119082i 0.500385 − 0.865803i \(-0.333192\pi\)
−0.173210 + 0.984885i \(0.555414\pi\)
\(44\) 0 0
\(45\) 1.70574 − 2.95442i 0.254276 − 0.440419i
\(46\) 0 0
\(47\) −0.971782 + 5.51125i −0.141749 + 0.803898i 0.828171 + 0.560475i \(0.189381\pi\)
−0.969920 + 0.243423i \(0.921730\pi\)
\(48\) 0 0
\(49\) −0.645430 − 1.11792i −0.0922042 − 0.159702i
\(50\) 0 0
\(51\) 5.31908 + 4.46324i 0.744820 + 0.624978i
\(52\) 0 0
\(53\) −1.86097 + 0.677337i −0.255623 + 0.0930393i −0.466653 − 0.884440i \(-0.654540\pi\)
0.211030 + 0.977480i \(0.432318\pi\)
\(54\) 0 0
\(55\) −0.205737 − 1.16679i −0.0277416 − 0.157330i
\(56\) 0 0
\(57\) 4.29813 + 0.725293i 0.569302 + 0.0960674i
\(58\) 0 0
\(59\) −0.0773815 − 0.438852i −0.0100742 − 0.0571337i 0.979356 − 0.202142i \(-0.0647902\pi\)
−0.989430 + 0.145008i \(0.953679\pi\)
\(60\) 0 0
\(61\) 11.7763 − 4.28623i 1.50780 − 0.548795i 0.549735 − 0.835339i \(-0.314729\pi\)
0.958067 + 0.286544i \(0.0925064\pi\)
\(62\) 0 0
\(63\) 2.20574 + 1.85083i 0.277897 + 0.233183i
\(64\) 0 0
\(65\) −2.81908 − 4.88279i −0.349664 − 0.605635i
\(66\) 0 0
\(67\) 0.187319 − 1.06234i 0.0228846 − 0.129785i −0.971225 − 0.238163i \(-0.923455\pi\)
0.994110 + 0.108378i \(0.0345657\pi\)
\(68\) 0 0
\(69\) 3.40033 − 5.88954i 0.409352 − 0.709018i
\(70\) 0 0
\(71\) −15.6211 − 5.68561i −1.85388 − 0.674758i −0.983095 − 0.183096i \(-0.941388\pi\)
−0.870786 − 0.491662i \(-0.836390\pi\)
\(72\) 0 0
\(73\) 9.51367 − 7.98292i 1.11349 − 0.934330i 0.115233 − 0.993338i \(-0.463238\pi\)
0.998257 + 0.0590086i \(0.0187939\pi\)
\(74\) 0 0
\(75\) 6.63816 0.766508
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 8.36824 − 7.02179i 0.941501 − 0.790013i −0.0363452 − 0.999339i \(-0.511572\pi\)
0.977846 + 0.209326i \(0.0671271\pi\)
\(80\) 0 0
\(81\) −0.939693 − 0.342020i −0.104410 − 0.0380022i
\(82\) 0 0
\(83\) 5.85844 − 10.1471i 0.643047 − 1.11379i −0.341701 − 0.939809i \(-0.611003\pi\)
0.984749 − 0.173982i \(-0.0556635\pi\)
\(84\) 0 0
\(85\) −4.11334 + 23.3279i −0.446154 + 2.53027i
\(86\) 0 0
\(87\) −3.17752 − 5.50362i −0.340666 − 0.590050i
\(88\) 0 0
\(89\) 1.37346 + 1.15247i 0.145586 + 0.122161i 0.712672 − 0.701497i \(-0.247485\pi\)
−0.567086 + 0.823659i \(0.691929\pi\)
\(90\) 0 0
\(91\) 4.47178 − 1.62760i 0.468770 − 0.170618i
\(92\) 0 0
\(93\) −0.277189 − 1.57202i −0.0287431 − 0.163010i
\(94\) 0 0
\(95\) 5.18866 + 13.9357i 0.532346 + 1.42977i
\(96\) 0 0
\(97\) 0.634285 + 3.59721i 0.0644019 + 0.365241i 0.999928 + 0.0119843i \(0.00381481\pi\)
−0.935526 + 0.353257i \(0.885074\pi\)
\(98\) 0 0
\(99\) −0.326352 + 0.118782i −0.0327996 + 0.0119381i
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 912.2.bo.f.769.1 6
4.3 odd 2 114.2.i.d.85.1 yes 6
12.11 even 2 342.2.u.a.199.1 6
19.17 even 9 inner 912.2.bo.f.625.1 6
76.51 even 18 2166.2.a.u.1.1 3
76.55 odd 18 114.2.i.d.55.1 ✓ 6
76.63 odd 18 2166.2.a.o.1.1 3
228.131 even 18 342.2.u.a.55.1 6
228.203 odd 18 6498.2.a.bn.1.3 3
228.215 even 18 6498.2.a.bs.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.55.1 ✓ 6 76.55 odd 18
114.2.i.d.85.1 yes 6 4.3 odd 2
342.2.u.a.55.1 6 228.131 even 18
342.2.u.a.199.1 6 12.11 even 2
912.2.bo.f.625.1 6 19.17 even 9 inner
912.2.bo.f.769.1 6 1.1 even 1 trivial
2166.2.a.o.1.1 3 76.63 odd 18
2166.2.a.u.1.1 3 76.51 even 18
6498.2.a.bn.1.3 3 228.203 odd 18
6498.2.a.bs.1.3 3 228.215 even 18