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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 15.4
Root \(2.10961 - 3.65395i\) of defining polynomial
Character \(\chi\) \(=\) 112.15
Dual form 112.5.d.b.15.5

$q$-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)
 
\(f(q)\) \(=\) \(q-0.390999i q^{3} -33.2658 q^{5} -18.5203i q^{7} +80.8471 q^{9} +118.953i q^{11} +261.968 q^{13} +13.0069i q^{15} +169.384 q^{17} +672.608i q^{19} -7.24141 q^{21} -759.021i q^{23} +481.614 q^{25} -63.2821i q^{27} +595.855 q^{29} +1548.77i q^{31} +46.5104 q^{33} +616.092i q^{35} +2246.28 q^{37} -102.429i q^{39} +186.373 q^{41} -629.154i q^{43} -2689.45 q^{45} -427.254i q^{47} -343.000 q^{49} -66.2289i q^{51} -2936.86 q^{53} -3957.06i q^{55} +262.989 q^{57} +1263.20i q^{59} -825.491 q^{61} -1497.31i q^{63} -8714.58 q^{65} +851.695i q^{67} -296.777 q^{69} -7424.34i q^{71} +608.347 q^{73} -188.311i q^{75} +2203.04 q^{77} +5914.96i q^{79} +6523.87 q^{81} +6261.33i q^{83} -5634.69 q^{85} -232.979i q^{87} -5976.60 q^{89} -4851.72i q^{91} +605.568 q^{93} -22374.9i q^{95} +436.938 q^{97} +9616.99i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{5} + 80 q^{9} - 236 q^{13} + 24 q^{17} - 196 q^{21} + 624 q^{25} + 3864 q^{29} - 4160 q^{33} - 2408 q^{37} - 1464 q^{41} - 116 q^{45} - 2744 q^{49} + 6960 q^{53} + 23480 q^{57} - 10836 q^{61}+ \cdots + 19576 q^{97}+O(q^{100}) \) 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
\(3\) − 0.390999i − 0.0434444i −0.999764 − 0.0217222i \(-0.993085\pi\)
0.999764 − 0.0217222i \(-0.00691493\pi\)
\(4\) 0 0
\(5\) −33.2658 −1.33063 −0.665316 − 0.746562i \(-0.731703\pi\)
−0.665316 + 0.746562i \(0.731703\pi\)
\(6\) 0 0
\(7\) − 18.5203i − 0.377964i
\(8\) 0 0
\(9\) 80.8471 0.998113
\(10\) 0 0
\(11\) 118.953i 0.983081i 0.870855 + 0.491540i \(0.163566\pi\)
−0.870855 + 0.491540i \(0.836434\pi\)
\(12\) 0 0
\(13\) 261.968 1.55011 0.775053 − 0.631896i \(-0.217723\pi\)
0.775053 + 0.631896i \(0.217723\pi\)
\(14\) 0 0
\(15\) 13.0069i 0.0578085i
\(16\) 0 0
\(17\) 169.384 0.586103 0.293051 − 0.956097i \(-0.405329\pi\)
0.293051 + 0.956097i \(0.405329\pi\)
\(18\) 0 0
\(19\) 672.608i 1.86318i 0.363509 + 0.931590i \(0.381578\pi\)
−0.363509 + 0.931590i \(0.618422\pi\)
\(20\) 0 0
\(21\) −7.24141 −0.0164204
\(22\) 0 0
\(23\) − 759.021i − 1.43482i −0.696650 − 0.717411i \(-0.745327\pi\)
0.696650 − 0.717411i \(-0.254673\pi\)
\(24\) 0 0
\(25\) 481.614 0.770583
\(26\) 0 0
\(27\) − 63.2821i − 0.0868067i
\(28\) 0 0
\(29\) 595.855 0.708508 0.354254 − 0.935149i \(-0.384735\pi\)
0.354254 + 0.935149i \(0.384735\pi\)
\(30\) 0 0
\(31\) 1548.77i 1.61162i 0.592172 + 0.805812i \(0.298271\pi\)
−0.592172 + 0.805812i \(0.701729\pi\)
\(32\) 0 0
\(33\) 46.5104 0.0427093
\(34\) 0 0
\(35\) 616.092i 0.502932i
\(36\) 0 0
\(37\) 2246.28 1.64081 0.820407 − 0.571779i \(-0.193747\pi\)
0.820407 + 0.571779i \(0.193747\pi\)
\(38\) 0 0
\(39\) − 102.429i − 0.0673434i
\(40\) 0 0
\(41\) 186.373 0.110870 0.0554350 − 0.998462i \(-0.482345\pi\)
0.0554350 + 0.998462i \(0.482345\pi\)
\(42\) 0 0
\(43\) − 629.154i − 0.340267i −0.985421 − 0.170133i \(-0.945580\pi\)
0.985421 − 0.170133i \(-0.0544199\pi\)
\(44\) 0 0
\(45\) −2689.45 −1.32812
\(46\) 0 0
\(47\) − 427.254i − 0.193415i −0.995313 − 0.0967076i \(-0.969169\pi\)
0.995313 − 0.0967076i \(-0.0308312\pi\)
\(48\) 0 0
\(49\) −343.000 −0.142857
\(50\) 0 0
\(51\) − 66.2289i − 0.0254629i
\(52\) 0 0
\(53\) −2936.86 −1.04552 −0.522759 − 0.852480i \(-0.675097\pi\)
−0.522759 + 0.852480i \(0.675097\pi\)
\(54\) 0 0
\(55\) − 3957.06i − 1.30812i
\(56\) 0 0
\(57\) 262.989 0.0809447
\(58\) 0 0
\(59\) 1263.20i 0.362885i 0.983402 + 0.181442i \(0.0580766\pi\)
−0.983402 + 0.181442i \(0.941923\pi\)
\(60\) 0 0
\(61\) −825.491 −0.221846 −0.110923 − 0.993829i \(-0.535381\pi\)
−0.110923 + 0.993829i \(0.535381\pi\)
\(62\) 0 0
\(63\) − 1497.31i − 0.377251i
\(64\) 0 0
\(65\) −8714.58 −2.06262
\(66\) 0 0
\(67\) 851.695i 0.189729i 0.995490 + 0.0948647i \(0.0302419\pi\)
−0.995490 + 0.0948647i \(0.969758\pi\)
\(68\) 0 0
\(69\) −296.777 −0.0623350
\(70\) 0 0
\(71\) − 7424.34i − 1.47279i −0.676551 − 0.736396i \(-0.736526\pi\)
0.676551 − 0.736396i \(-0.263474\pi\)
\(72\) 0 0
\(73\) 608.347 0.114158 0.0570789 − 0.998370i \(-0.481821\pi\)
0.0570789 + 0.998370i \(0.481821\pi\)
\(74\) 0 0
\(75\) − 188.311i − 0.0334775i
\(76\) 0 0
\(77\) 2203.04 0.371570
\(78\) 0 0
\(79\) 5914.96i 0.947759i 0.880590 + 0.473879i \(0.157147\pi\)
−0.880590 + 0.473879i \(0.842853\pi\)
\(80\) 0 0
\(81\) 6523.87 0.994341
\(82\) 0 0
\(83\) 6261.33i 0.908888i 0.890775 + 0.454444i \(0.150162\pi\)
−0.890775 + 0.454444i \(0.849838\pi\)
\(84\) 0 0
\(85\) −5634.69 −0.779887
\(86\) 0 0
\(87\) − 232.979i − 0.0307807i
\(88\) 0 0
\(89\) −5976.60 −0.754526 −0.377263 − 0.926106i \(-0.623135\pi\)
−0.377263 + 0.926106i \(0.623135\pi\)
\(90\) 0 0
\(91\) − 4851.72i − 0.585885i
\(92\) 0 0
\(93\) 605.568 0.0700159
\(94\) 0 0
\(95\) − 22374.9i − 2.47921i
\(96\) 0 0
\(97\) 436.938 0.0464383 0.0232192 − 0.999730i \(-0.492608\pi\)
0.0232192 + 0.999730i \(0.492608\pi\)
\(98\) 0 0
\(99\) 9616.99i 0.981225i
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.15.4 ✓ 8
3.2 odd 2 1008.5.m.e.127.7 8
4.3 odd 2 inner 112.5.d.b.15.5 yes 8
7.6 odd 2 784.5.d.j.687.5 8
8.3 odd 2 448.5.d.d.127.4 8
8.5 even 2 448.5.d.d.127.5 8
12.11 even 2 1008.5.m.e.127.8 8
28.27 even 2 784.5.d.j.687.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.4 ✓ 8 1.1 even 1 trivial
112.5.d.b.15.5 yes 8 4.3 odd 2 inner
448.5.d.d.127.4 8 8.3 odd 2
448.5.d.d.127.5 8 8.5 even 2
784.5.d.j.687.4 8 28.27 even 2
784.5.d.j.687.5 8 7.6 odd 2
1008.5.m.e.127.7 8 3.2 odd 2
1008.5.m.e.127.8 8 12.11 even 2