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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [56,6,Mod(9,56)] 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("56.9"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 56.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.98149390953\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 119 x^{8} - 521 x^{7} - 898 x^{6} + 27806 x^{5} + 657990 x^{4} + 3648839 x^{3} + \cdots + 92895579 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 9.3
Root \(11.5439 + 0.371202i\) of defining polynomial
Character \(\chi\) \(=\) 56.9
Dual form 56.6.i.a.25.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.11280 - 5.39153i) q^{3} +(-4.70275 + 8.14541i) q^{5} +(105.969 + 74.6834i) q^{7} +(102.121 - 176.879i) q^{9} +(-382.558 - 662.609i) q^{11} +732.172 q^{13} +58.5549 q^{15} +(-241.290 - 417.927i) q^{17} +(1308.54 - 2266.45i) q^{19} +(72.7976 - 803.808i) q^{21} +(267.753 - 463.762i) q^{23} +(1518.27 + 2629.72i) q^{25} -2784.35 q^{27} +3943.87 q^{29} +(-1831.72 - 3172.64i) q^{31} +(-2381.65 + 4125.14i) q^{33} +(-1106.67 + 511.942i) q^{35} +(-6319.51 + 10945.7i) q^{37} +(-2279.11 - 3947.53i) q^{39} +4814.47 q^{41} -4938.11 q^{43} +(960.499 + 1663.63i) q^{45} +(-8702.49 + 15073.2i) q^{47} +(5651.79 + 15828.2i) q^{49} +(-1502.18 + 2601.85i) q^{51} +(2329.22 + 4034.32i) q^{53} +7196.30 q^{55} -16292.8 q^{57} +(-16250.4 - 28146.6i) q^{59} +(21748.1 - 37668.8i) q^{61} +(24031.5 - 11116.9i) q^{63} +(-3443.23 + 5963.84i) q^{65} +(16041.2 + 27784.1i) q^{67} -3333.85 q^{69} +15820.0 q^{71} +(-18257.7 - 31623.3i) q^{73} +(9452.13 - 16371.6i) q^{75} +(8946.70 - 98786.6i) q^{77} +(-12736.1 + 22059.5i) q^{79} +(-16148.3 - 27969.6i) q^{81} +69443.5 q^{83} +4538.91 q^{85} +(-12276.5 - 21263.5i) q^{87} +(-54165.4 + 93817.2i) q^{89} +(77587.5 + 54681.1i) q^{91} +(-11403.6 + 19751.6i) q^{93} +(12307.4 + 21317.1i) q^{95} +94569.1 q^{97} -156269. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 13 q^{3} - 31 q^{5} - 92 q^{7} - 230 q^{9} + 351 q^{11} - 108 q^{13} + 1214 q^{15} - 111 q^{17} - 1035 q^{19} - 1365 q^{21} - 3639 q^{23} - 1540 q^{25} + 7214 q^{27} - 1468 q^{29} - 7677 q^{31} + 7439 q^{33}+ \cdots - 600308 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) −3.11280 − 5.39153i −0.199686 − 0.345867i 0.748740 − 0.662863i \(-0.230659\pi\)
−0.948427 + 0.316997i \(0.897326\pi\)
\(4\) 0 0
\(5\) −4.70275 + 8.14541i −0.0841254 + 0.145709i −0.905018 − 0.425373i \(-0.860143\pi\)
0.820893 + 0.571082i \(0.193476\pi\)
\(6\) 0 0
\(7\) 105.969 + 74.6834i 0.817397 + 0.576075i
\(8\) 0 0
\(9\) 102.121 − 176.879i 0.420251 − 0.727896i
\(10\) 0 0
\(11\) −382.558 − 662.609i −0.953269 − 1.65111i −0.738282 − 0.674493i \(-0.764362\pi\)
−0.214987 − 0.976617i \(-0.568971\pi\)
\(12\) 0 0
\(13\) 732.172 1.20159 0.600793 − 0.799405i \(-0.294852\pi\)
0.600793 + 0.799405i \(0.294852\pi\)
\(14\) 0 0
\(15\) 58.5549 0.0671947
\(16\) 0 0
\(17\) −241.290 − 417.927i −0.202496 − 0.350734i 0.746836 − 0.665009i \(-0.231572\pi\)
−0.949332 + 0.314274i \(0.898239\pi\)
\(18\) 0 0
\(19\) 1308.54 − 2266.45i 0.831576 − 1.44033i −0.0652125 − 0.997871i \(-0.520773\pi\)
0.896788 − 0.442460i \(-0.145894\pi\)
\(20\) 0 0
\(21\) 72.7976 − 803.808i 0.0360221 − 0.397745i
\(22\) 0 0
\(23\) 267.753 − 463.762i 0.105539 − 0.182800i −0.808419 − 0.588607i \(-0.799676\pi\)
0.913958 + 0.405808i \(0.133010\pi\)
\(24\) 0 0
\(25\) 1518.27 + 2629.72i 0.485846 + 0.841510i
\(26\) 0 0
\(27\) −2784.35 −0.735046
\(28\) 0 0
\(29\) 3943.87 0.870819 0.435409 − 0.900233i \(-0.356604\pi\)
0.435409 + 0.900233i \(0.356604\pi\)
\(30\) 0 0
\(31\) −1831.72 − 3172.64i −0.342338 − 0.592947i 0.642528 − 0.766262i \(-0.277886\pi\)
−0.984866 + 0.173315i \(0.944552\pi\)
\(32\) 0 0
\(33\) −2381.65 + 4125.14i −0.380709 + 0.659408i
\(34\) 0 0
\(35\) −1106.67 + 511.942i −0.152703 + 0.0706400i
\(36\) 0 0
\(37\) −6319.51 + 10945.7i −0.758890 + 1.31444i 0.184527 + 0.982827i \(0.440925\pi\)
−0.943417 + 0.331609i \(0.892409\pi\)
\(38\) 0 0
\(39\) −2279.11 − 3947.53i −0.239940 − 0.415589i
\(40\) 0 0
\(41\) 4814.47 0.447290 0.223645 − 0.974671i \(-0.428204\pi\)
0.223645 + 0.974671i \(0.428204\pi\)
\(42\) 0 0
\(43\) −4938.11 −0.407277 −0.203638 − 0.979046i \(-0.565277\pi\)
−0.203638 + 0.979046i \(0.565277\pi\)
\(44\) 0 0
\(45\) 960.499 + 1663.63i 0.0707076 + 0.122469i
\(46\) 0 0
\(47\) −8702.49 + 15073.2i −0.574644 + 0.995312i 0.421436 + 0.906858i \(0.361526\pi\)
−0.996080 + 0.0884544i \(0.971807\pi\)
\(48\) 0 0
\(49\) 5651.79 + 15828.2i 0.336276 + 0.941763i
\(50\) 0 0
\(51\) −1502.18 + 2601.85i −0.0808715 + 0.140074i
\(52\) 0 0
\(53\) 2329.22 + 4034.32i 0.113899 + 0.197279i 0.917339 − 0.398107i \(-0.130333\pi\)
−0.803440 + 0.595386i \(0.796999\pi\)
\(54\) 0 0
\(55\) 7196.30 0.320776
\(56\) 0 0
\(57\) −16292.8 −0.664217
\(58\) 0 0
\(59\) −16250.4 − 28146.6i −0.607763 − 1.05268i −0.991608 − 0.129280i \(-0.958734\pi\)
0.383845 − 0.923398i \(-0.374600\pi\)
\(60\) 0 0
\(61\) 21748.1 − 37668.8i 0.748335 − 1.29615i −0.200285 − 0.979738i \(-0.564187\pi\)
0.948620 − 0.316417i \(-0.102480\pi\)
\(62\) 0 0
\(63\) 24031.5 − 11116.9i 0.762834 − 0.352884i
\(64\) 0 0
\(65\) −3443.23 + 5963.84i −0.101084 + 0.175083i
\(66\) 0 0
\(67\) 16041.2 + 27784.1i 0.436565 + 0.756154i 0.997422 − 0.0717595i \(-0.0228614\pi\)
−0.560857 + 0.827913i \(0.689528\pi\)
\(68\) 0 0
\(69\) −3333.85 −0.0842991
\(70\) 0 0
\(71\) 15820.0 0.372443 0.186221 − 0.982508i \(-0.440376\pi\)
0.186221 + 0.982508i \(0.440376\pi\)
\(72\) 0 0
\(73\) −18257.7 − 31623.3i −0.400996 − 0.694545i 0.592851 − 0.805312i \(-0.298002\pi\)
−0.993846 + 0.110768i \(0.964669\pi\)
\(74\) 0 0
\(75\) 9452.13 − 16371.6i 0.194033 − 0.336076i
\(76\) 0 0
\(77\) 8946.70 − 98786.6i 0.171963 − 1.89877i
\(78\) 0 0
\(79\) −12736.1 + 22059.5i −0.229598 + 0.397675i −0.957689 − 0.287805i \(-0.907074\pi\)
0.728091 + 0.685480i \(0.240408\pi\)
\(80\) 0 0
\(81\) −16148.3 − 27969.6i −0.273472 − 0.473668i
\(82\) 0 0
\(83\) 69443.5 1.10646 0.553231 − 0.833028i \(-0.313395\pi\)
0.553231 + 0.833028i \(0.313395\pi\)
\(84\) 0 0
\(85\) 4538.91 0.0681404
\(86\) 0 0
\(87\) −12276.5 − 21263.5i −0.173890 − 0.301187i
\(88\) 0 0
\(89\) −54165.4 + 93817.2i −0.724848 + 1.25547i 0.234188 + 0.972191i \(0.424757\pi\)
−0.959037 + 0.283283i \(0.908577\pi\)
\(90\) 0 0
\(91\) 77587.5 + 54681.1i 0.982173 + 0.692203i
\(92\) 0 0
\(93\) −11403.6 + 19751.6i −0.136720 + 0.236807i
\(94\) 0 0
\(95\) 12307.4 + 21317.1i 0.139913 + 0.242337i
\(96\) 0 0
\(97\) 94569.1 1.02052 0.510258 − 0.860021i \(-0.329550\pi\)
0.510258 + 0.860021i \(0.329550\pi\)
\(98\) 0 0
\(99\) −156269. −1.60245
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 56.6.i.a.9.3 ✓ 10
3.2 odd 2 504.6.s.d.289.3 10
4.3 odd 2 112.6.i.g.65.3 10
7.2 even 3 392.6.a.l.1.3 5
7.3 odd 6 392.6.i.p.361.3 10
7.4 even 3 inner 56.6.i.a.25.3 yes 10
7.5 odd 6 392.6.a.i.1.3 5
7.6 odd 2 392.6.i.p.177.3 10
21.11 odd 6 504.6.s.d.361.3 10
28.11 odd 6 112.6.i.g.81.3 10
28.19 even 6 784.6.a.bm.1.3 5
28.23 odd 6 784.6.a.bj.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.i.a.9.3 ✓ 10 1.1 even 1 trivial
56.6.i.a.25.3 yes 10 7.4 even 3 inner
112.6.i.g.65.3 10 4.3 odd 2
112.6.i.g.81.3 10 28.11 odd 6
392.6.a.i.1.3 5 7.5 odd 6
392.6.a.l.1.3 5 7.2 even 3
392.6.i.p.177.3 10 7.6 odd 2
392.6.i.p.361.3 10 7.3 odd 6
504.6.s.d.289.3 10 3.2 odd 2
504.6.s.d.361.3 10 21.11 odd 6
784.6.a.bj.1.3 5 28.23 odd 6
784.6.a.bm.1.3 5 28.19 even 6