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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,5,Mod(15,448)] 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("448.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.3097434616\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 239.22
Character \(\chi\) \(=\) 448.239
Dual form 448.5.k.a.15.22

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.79635 + 1.79635i) q^{3} +(-4.04343 + 4.04343i) q^{5} -18.5203 q^{7} +74.5463i q^{9} +(-48.6761 - 48.6761i) q^{11} +(-191.638 - 191.638i) q^{13} -14.5268i q^{15} +125.405 q^{17} +(389.439 - 389.439i) q^{19} +(33.2688 - 33.2688i) q^{21} -292.883 q^{23} +592.301i q^{25} +(-279.415 - 279.415i) q^{27} +(1125.64 + 1125.64i) q^{29} -1005.46i q^{31} +174.878 q^{33} +(74.8853 - 74.8853i) q^{35} +(-1332.30 + 1332.30i) q^{37} +688.498 q^{39} +109.259i q^{41} +(-316.699 - 316.699i) q^{43} +(-301.422 - 301.422i) q^{45} -1386.71i q^{47} +343.000 q^{49} +(-225.271 + 225.271i) q^{51} +(3627.01 - 3627.01i) q^{53} +393.636 q^{55} +1399.14i q^{57} +(4697.25 + 4697.25i) q^{59} +(2287.68 + 2287.68i) q^{61} -1380.62i q^{63} +1549.75 q^{65} +(4285.76 - 4285.76i) q^{67} +(526.119 - 526.119i) q^{69} +93.6034 q^{71} -3096.04i q^{73} +(-1063.98 - 1063.98i) q^{75} +(901.494 + 901.494i) q^{77} +7817.52i q^{79} -5034.40 q^{81} +(3301.94 - 3301.94i) q^{83} +(-507.067 + 507.067i) q^{85} -4044.09 q^{87} +13746.1i q^{89} +(3549.19 + 3549.19i) q^{91} +(1806.15 + 1806.15i) q^{93} +3149.34i q^{95} +14974.1 q^{97} +(3628.62 - 3628.62i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.79635 + 1.79635i −0.199594 + 0.199594i −0.799826 − 0.600232i \(-0.795075\pi\)
0.600232 + 0.799826i \(0.295075\pi\)
\(4\) 0 0
\(5\) −4.04343 + 4.04343i −0.161737 + 0.161737i −0.783336 − 0.621599i \(-0.786483\pi\)
0.621599 + 0.783336i \(0.286483\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 0 0
\(9\) 74.5463i 0.920324i
\(10\) 0 0
\(11\) −48.6761 − 48.6761i −0.402282 − 0.402282i 0.476755 − 0.879036i \(-0.341813\pi\)
−0.879036 + 0.476755i \(0.841813\pi\)
\(12\) 0 0
\(13\) −191.638 − 191.638i −1.13396 − 1.13396i −0.989513 − 0.144442i \(-0.953861\pi\)
−0.144442 − 0.989513i \(-0.546139\pi\)
\(14\) 0 0
\(15\) − 14.5268i − 0.0645635i
\(16\) 0 0
\(17\) 125.405 0.433928 0.216964 − 0.976180i \(-0.430385\pi\)
0.216964 + 0.976180i \(0.430385\pi\)
\(18\) 0 0
\(19\) 389.439 − 389.439i 1.07878 − 1.07878i 0.0821596 − 0.996619i \(-0.473818\pi\)
0.996619 − 0.0821596i \(-0.0261817\pi\)
\(20\) 0 0
\(21\) 33.2688 − 33.2688i 0.0754394 − 0.0754394i
\(22\) 0 0
\(23\) −292.883 −0.553654 −0.276827 − 0.960920i \(-0.589283\pi\)
−0.276827 + 0.960920i \(0.589283\pi\)
\(24\) 0 0
\(25\) 592.301i 0.947682i
\(26\) 0 0
\(27\) −279.415 − 279.415i −0.383285 − 0.383285i
\(28\) 0 0
\(29\) 1125.64 + 1125.64i 1.33846 + 1.33846i 0.897555 + 0.440903i \(0.145342\pi\)
0.440903 + 0.897555i \(0.354658\pi\)
\(30\) 0 0
\(31\) − 1005.46i − 1.04626i −0.852253 − 0.523130i \(-0.824764\pi\)
0.852253 − 0.523130i \(-0.175236\pi\)
\(32\) 0 0
\(33\) 174.878 0.160586
\(34\) 0 0
\(35\) 74.8853 − 74.8853i 0.0611309 − 0.0611309i
\(36\) 0 0
\(37\) −1332.30 + 1332.30i −0.973194 + 0.973194i −0.999650 − 0.0264561i \(-0.991578\pi\)
0.0264561 + 0.999650i \(0.491578\pi\)
\(38\) 0 0
\(39\) 688.498 0.452661
\(40\) 0 0
\(41\) 109.259i 0.0649967i 0.999472 + 0.0324983i \(0.0103464\pi\)
−0.999472 + 0.0324983i \(0.989654\pi\)
\(42\) 0 0
\(43\) −316.699 − 316.699i −0.171281 − 0.171281i 0.616261 − 0.787542i \(-0.288647\pi\)
−0.787542 + 0.616261i \(0.788647\pi\)
\(44\) 0 0
\(45\) −301.422 − 301.422i −0.148851 − 0.148851i
\(46\) 0 0
\(47\) − 1386.71i − 0.627755i −0.949464 − 0.313877i \(-0.898372\pi\)
0.949464 − 0.313877i \(-0.101628\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) −225.271 + 225.271i −0.0866095 + 0.0866095i
\(52\) 0 0
\(53\) 3627.01 − 3627.01i 1.29121 − 1.29121i 0.357171 − 0.934039i \(-0.383741\pi\)
0.934039 − 0.357171i \(-0.116259\pi\)
\(54\) 0 0
\(55\) 393.636 0.130128
\(56\) 0 0
\(57\) 1399.14i 0.430636i
\(58\) 0 0
\(59\) 4697.25 + 4697.25i 1.34940 + 1.34940i 0.886316 + 0.463080i \(0.153256\pi\)
0.463080 + 0.886316i \(0.346744\pi\)
\(60\) 0 0
\(61\) 2287.68 + 2287.68i 0.614801 + 0.614801i 0.944193 − 0.329392i \(-0.106844\pi\)
−0.329392 + 0.944193i \(0.606844\pi\)
\(62\) 0 0
\(63\) − 1380.62i − 0.347850i
\(64\) 0 0
\(65\) 1549.75 0.366805
\(66\) 0 0
\(67\) 4285.76 − 4285.76i 0.954725 − 0.954725i −0.0442932 − 0.999019i \(-0.514104\pi\)
0.999019 + 0.0442932i \(0.0141036\pi\)
\(68\) 0 0
\(69\) 526.119 − 526.119i 0.110506 − 0.110506i
\(70\) 0 0
\(71\) 93.6034 0.0185684 0.00928421 − 0.999957i \(-0.497045\pi\)
0.00928421 + 0.999957i \(0.497045\pi\)
\(72\) 0 0
\(73\) − 3096.04i − 0.580980i −0.956878 − 0.290490i \(-0.906182\pi\)
0.956878 − 0.290490i \(-0.0938184\pi\)
\(74\) 0 0
\(75\) −1063.98 − 1063.98i −0.189152 − 0.189152i
\(76\) 0 0
\(77\) 901.494 + 901.494i 0.152048 + 0.152048i
\(78\) 0 0
\(79\) 7817.52i 1.25261i 0.779579 + 0.626304i \(0.215433\pi\)
−0.779579 + 0.626304i \(0.784567\pi\)
\(80\) 0 0
\(81\) −5034.40 −0.767322
\(82\) 0 0
\(83\) 3301.94 − 3301.94i 0.479306 − 0.479306i −0.425603 − 0.904910i \(-0.639938\pi\)
0.904910 + 0.425603i \(0.139938\pi\)
\(84\) 0 0
\(85\) −507.067 + 507.067i −0.0701823 + 0.0701823i
\(86\) 0 0
\(87\) −4044.09 −0.534296
\(88\) 0 0
\(89\) 13746.1i 1.73540i 0.497090 + 0.867699i \(0.334402\pi\)
−0.497090 + 0.867699i \(0.665598\pi\)
\(90\) 0 0
\(91\) 3549.19 + 3549.19i 0.428595 + 0.428595i
\(92\) 0 0
\(93\) 1806.15 + 1806.15i 0.208827 + 0.208827i
\(94\) 0 0
\(95\) 3149.34i 0.348957i
\(96\) 0 0
\(97\) 14974.1 1.59146 0.795731 − 0.605651i \(-0.207087\pi\)
0.795731 + 0.605651i \(0.207087\pi\)
\(98\) 0 0
\(99\) 3628.62 − 3628.62i 0.370230 − 0.370230i
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 448.5.k.a.239.22 96
4.3 odd 2 112.5.k.a.99.2 yes 96
16.5 even 4 112.5.k.a.43.2 ✓ 96
16.11 odd 4 inner 448.5.k.a.15.22 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.2 ✓ 96 16.5 even 4
112.5.k.a.99.2 yes 96 4.3 odd 2
448.5.k.a.15.22 96 16.11 odd 4 inner
448.5.k.a.239.22 96 1.1 even 1 trivial