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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(15,112)] 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("112.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.207528535809.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 265x^{4} - 66x^{3} + 1344x^{2} + 1080x + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 15.2
Root \(-1.60961 - 2.78792i\) of defining polynomial
Character \(\chi\) \(=\) 112.15
Dual form 112.5.d.b.15.7

$q$-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)
 
\(f(q)\) \(=\) \(q-6.40550i q^{3} +38.0135 q^{5} +18.5203i q^{7} +39.9695 q^{9} +70.7571i q^{11} -59.7612 q^{13} -243.496i q^{15} +359.030 q^{17} +24.0021i q^{19} +118.632 q^{21} -985.841i q^{23} +820.029 q^{25} -774.871i q^{27} +1112.52 q^{29} +656.478i q^{31} +453.235 q^{33} +704.021i q^{35} -1922.60 q^{37} +382.800i q^{39} -304.913 q^{41} -1214.41i q^{43} +1519.38 q^{45} +3315.46i q^{47} -343.000 q^{49} -2299.77i q^{51} +57.6259 q^{53} +2689.73i q^{55} +153.746 q^{57} -4473.09i q^{59} -4637.64 q^{61} +740.246i q^{63} -2271.73 q^{65} +2605.64i q^{67} -6314.81 q^{69} +4949.48i q^{71} -7893.99 q^{73} -5252.70i q^{75} -1310.44 q^{77} +9566.11i q^{79} -1725.90 q^{81} +5924.74i q^{83} +13648.0 q^{85} -7126.26i q^{87} -4470.34 q^{89} -1106.79i q^{91} +4205.07 q^{93} +912.406i q^{95} +15831.0 q^{97} +2828.13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{5} + 80 q^{9} - 236 q^{13} + 24 q^{17} - 196 q^{21} + 624 q^{25} + 3864 q^{29} - 4160 q^{33} - 2408 q^{37} - 1464 q^{41} - 116 q^{45} - 2744 q^{49} + 6960 q^{53} + 23480 q^{57} - 10836 q^{61}+ \cdots + 19576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 6.40550i − 0.711722i −0.934539 − 0.355861i \(-0.884188\pi\)
0.934539 − 0.355861i \(-0.115812\pi\)
\(4\) 0 0
\(5\) 38.0135 1.52054 0.760271 − 0.649606i \(-0.225066\pi\)
0.760271 + 0.649606i \(0.225066\pi\)
\(6\) 0 0
\(7\) 18.5203i 0.377964i
\(8\) 0 0
\(9\) 39.9695 0.493451
\(10\) 0 0
\(11\) 70.7571i 0.584770i 0.956301 + 0.292385i \(0.0944488\pi\)
−0.956301 + 0.292385i \(0.905551\pi\)
\(12\) 0 0
\(13\) −59.7612 −0.353616 −0.176808 − 0.984245i \(-0.556577\pi\)
−0.176808 + 0.984245i \(0.556577\pi\)
\(14\) 0 0
\(15\) − 243.496i − 1.08220i
\(16\) 0 0
\(17\) 359.030 1.24232 0.621159 − 0.783685i \(-0.286662\pi\)
0.621159 + 0.783685i \(0.286662\pi\)
\(18\) 0 0
\(19\) 24.0021i 0.0664879i 0.999447 + 0.0332439i \(0.0105838\pi\)
−0.999447 + 0.0332439i \(0.989416\pi\)
\(20\) 0 0
\(21\) 118.632 0.269006
\(22\) 0 0
\(23\) − 985.841i − 1.86359i −0.362980 − 0.931797i \(-0.618241\pi\)
0.362980 − 0.931797i \(-0.381759\pi\)
\(24\) 0 0
\(25\) 820.029 1.31205
\(26\) 0 0
\(27\) − 774.871i − 1.06292i
\(28\) 0 0
\(29\) 1112.52 1.32286 0.661428 − 0.750009i \(-0.269951\pi\)
0.661428 + 0.750009i \(0.269951\pi\)
\(30\) 0 0
\(31\) 656.478i 0.683120i 0.939860 + 0.341560i \(0.110955\pi\)
−0.939860 + 0.341560i \(0.889045\pi\)
\(32\) 0 0
\(33\) 453.235 0.416194
\(34\) 0 0
\(35\) 704.021i 0.574711i
\(36\) 0 0
\(37\) −1922.60 −1.40438 −0.702190 − 0.711990i \(-0.747794\pi\)
−0.702190 + 0.711990i \(0.747794\pi\)
\(38\) 0 0
\(39\) 382.800i 0.251677i
\(40\) 0 0
\(41\) −304.913 −0.181388 −0.0906941 − 0.995879i \(-0.528909\pi\)
−0.0906941 + 0.995879i \(0.528909\pi\)
\(42\) 0 0
\(43\) − 1214.41i − 0.656794i −0.944540 − 0.328397i \(-0.893492\pi\)
0.944540 − 0.328397i \(-0.106508\pi\)
\(44\) 0 0
\(45\) 1519.38 0.750313
\(46\) 0 0
\(47\) 3315.46i 1.50089i 0.660935 + 0.750443i \(0.270160\pi\)
−0.660935 + 0.750443i \(0.729840\pi\)
\(48\) 0 0
\(49\) −343.000 −0.142857
\(50\) 0 0
\(51\) − 2299.77i − 0.884186i
\(52\) 0 0
\(53\) 57.6259 0.0205147 0.0102574 − 0.999947i \(-0.496735\pi\)
0.0102574 + 0.999947i \(0.496735\pi\)
\(54\) 0 0
\(55\) 2689.73i 0.889167i
\(56\) 0 0
\(57\) 153.746 0.0473209
\(58\) 0 0
\(59\) − 4473.09i − 1.28500i −0.766286 − 0.642500i \(-0.777897\pi\)
0.766286 − 0.642500i \(-0.222103\pi\)
\(60\) 0 0
\(61\) −4637.64 −1.24634 −0.623171 − 0.782086i \(-0.714156\pi\)
−0.623171 + 0.782086i \(0.714156\pi\)
\(62\) 0 0
\(63\) 740.246i 0.186507i
\(64\) 0 0
\(65\) −2271.73 −0.537689
\(66\) 0 0
\(67\) 2605.64i 0.580450i 0.956958 + 0.290225i \(0.0937301\pi\)
−0.956958 + 0.290225i \(0.906270\pi\)
\(68\) 0 0
\(69\) −6314.81 −1.32636
\(70\) 0 0
\(71\) 4949.48i 0.981844i 0.871203 + 0.490922i \(0.163340\pi\)
−0.871203 + 0.490922i \(0.836660\pi\)
\(72\) 0 0
\(73\) −7893.99 −1.48133 −0.740663 − 0.671877i \(-0.765488\pi\)
−0.740663 + 0.671877i \(0.765488\pi\)
\(74\) 0 0
\(75\) − 5252.70i − 0.933813i
\(76\) 0 0
\(77\) −1310.44 −0.221022
\(78\) 0 0
\(79\) 9566.11i 1.53278i 0.642373 + 0.766392i \(0.277950\pi\)
−0.642373 + 0.766392i \(0.722050\pi\)
\(80\) 0 0
\(81\) −1725.90 −0.263055
\(82\) 0 0
\(83\) 5924.74i 0.860029i 0.902822 + 0.430015i \(0.141492\pi\)
−0.902822 + 0.430015i \(0.858508\pi\)
\(84\) 0 0
\(85\) 13648.0 1.88900
\(86\) 0 0
\(87\) − 7126.26i − 0.941506i
\(88\) 0 0
\(89\) −4470.34 −0.564366 −0.282183 − 0.959361i \(-0.591059\pi\)
−0.282183 + 0.959361i \(0.591059\pi\)
\(90\) 0 0
\(91\) − 1106.79i − 0.133654i
\(92\) 0 0
\(93\) 4205.07 0.486192
\(94\) 0 0
\(95\) 912.406i 0.101098i
\(96\) 0 0
\(97\) 15831.0 1.68254 0.841270 − 0.540616i \(-0.181809\pi\)
0.841270 + 0.540616i \(0.181809\pi\)
\(98\) 0 0
\(99\) 2828.13i 0.288555i
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 112.5.d.b.15.2 ✓ 8
3.2 odd 2 1008.5.m.e.127.2 8
4.3 odd 2 inner 112.5.d.b.15.7 yes 8
7.6 odd 2 784.5.d.j.687.7 8
8.3 odd 2 448.5.d.d.127.2 8
8.5 even 2 448.5.d.d.127.7 8
12.11 even 2 1008.5.m.e.127.1 8
28.27 even 2 784.5.d.j.687.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.2 ✓ 8 1.1 even 1 trivial
112.5.d.b.15.7 yes 8 4.3 odd 2 inner
448.5.d.d.127.2 8 8.3 odd 2
448.5.d.d.127.7 8 8.5 even 2
784.5.d.j.687.2 8 28.27 even 2
784.5.d.j.687.7 8 7.6 odd 2
1008.5.m.e.127.1 8 12.11 even 2
1008.5.m.e.127.2 8 3.2 odd 2