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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(481,1860)] 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("1860.481"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.z (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,4,0,-16] 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: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 15 x^{13} + 101 x^{12} - 273 x^{11} + 1747 x^{10} - 2144 x^{9} + \cdots + 961 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 841.1
Root \(2.73953 + 1.99039i\) of defining polynomial
Character \(\chi\) \(=\) 1860.841
Dual form 1860.2.z.b.721.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.309017 + 0.951057i) q^{3} -1.00000 q^{5} +(-2.73953 + 1.99039i) q^{7} +(-0.809017 - 0.587785i) q^{9} +(3.84117 - 2.79078i) q^{11} +(0.355424 - 1.09388i) q^{13} +(0.309017 - 0.951057i) q^{15} +(2.48359 + 1.80443i) q^{17} +(-1.73199 - 5.33053i) q^{19} +(-1.04641 - 3.22051i) q^{21} +(0.993407 + 0.721753i) q^{23} +1.00000 q^{25} +(0.809017 - 0.587785i) q^{27} +(-1.98250 - 6.10149i) q^{29} +(5.21442 + 1.95188i) q^{31} +(1.46720 + 4.51557i) q^{33} +(2.73953 - 1.99039i) q^{35} -1.21578 q^{37} +(0.930513 + 0.676057i) q^{39} +(1.57999 + 4.86270i) q^{41} +(1.52098 + 4.68109i) q^{43} +(0.809017 + 0.587785i) q^{45} +(-2.06016 + 6.34053i) q^{47} +(1.38027 - 4.24804i) q^{49} +(-2.48359 + 1.80443i) q^{51} +(4.83458 + 3.51253i) q^{53} +(-3.84117 + 2.79078i) q^{55} +5.60485 q^{57} +(0.950833 - 2.92636i) q^{59} -5.08010 q^{61} +3.38625 q^{63} +(-0.355424 + 1.09388i) q^{65} +12.2497 q^{67} +(-0.993407 + 0.721753i) q^{69} +(12.9317 + 9.39541i) q^{71} +(9.80610 - 7.12455i) q^{73} +(-0.309017 + 0.951057i) q^{75} +(-4.96829 + 15.2908i) q^{77} +(8.77049 + 6.37213i) q^{79} +(0.309017 + 0.951057i) q^{81} +(0.606376 + 1.86623i) q^{83} +(-2.48359 - 1.80443i) q^{85} +6.41549 q^{87} +(-7.27230 + 5.28364i) q^{89} +(1.20355 + 3.70416i) q^{91} +(-3.46769 + 4.35604i) q^{93} +(1.73199 + 5.33053i) q^{95} +(10.0874 - 7.32895i) q^{97} -4.74795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 16 q^{5} - 3 q^{7} - 4 q^{9} + 2 q^{11} - 18 q^{13} - 4 q^{15} + 20 q^{17} - q^{19} - 2 q^{21} - 12 q^{23} + 16 q^{25} + 4 q^{27} - 7 q^{29} - 3 q^{31} + 3 q^{33} + 3 q^{35} - 44 q^{37}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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
\(3\) −0.309017 + 0.951057i −0.178411 + 0.549093i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.73953 + 1.99039i −1.03545 + 0.752295i −0.969391 − 0.245521i \(-0.921041\pi\)
−0.0660538 + 0.997816i \(0.521041\pi\)
\(8\) 0 0
\(9\) −0.809017 − 0.587785i −0.269672 − 0.195928i
\(10\) 0 0
\(11\) 3.84117 − 2.79078i 1.15816 − 0.841451i 0.168613 − 0.985682i \(-0.446071\pi\)
0.989544 + 0.144232i \(0.0460711\pi\)
\(12\) 0 0
\(13\) 0.355424 − 1.09388i 0.0985770 − 0.303389i −0.889592 − 0.456755i \(-0.849012\pi\)
0.988169 + 0.153367i \(0.0490115\pi\)
\(14\) 0 0
\(15\) 0.309017 − 0.951057i 0.0797878 − 0.245562i
\(16\) 0 0
\(17\) 2.48359 + 1.80443i 0.602359 + 0.437639i 0.846715 − 0.532046i \(-0.178577\pi\)
−0.244357 + 0.969685i \(0.578577\pi\)
\(18\) 0 0
\(19\) −1.73199 − 5.33053i −0.397347 − 1.22291i −0.927119 − 0.374767i \(-0.877723\pi\)
0.529772 − 0.848140i \(-0.322277\pi\)
\(20\) 0 0
\(21\) −1.04641 − 3.22051i −0.228345 − 0.702773i
\(22\) 0 0
\(23\) 0.993407 + 0.721753i 0.207140 + 0.150496i 0.686519 − 0.727112i \(-0.259138\pi\)
−0.479379 + 0.877608i \(0.659138\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.809017 − 0.587785i 0.155695 − 0.113119i
\(28\) 0 0
\(29\) −1.98250 − 6.10149i −0.368140 − 1.13302i −0.947991 − 0.318296i \(-0.896889\pi\)
0.579851 − 0.814722i \(-0.303111\pi\)
\(30\) 0 0
\(31\) 5.21442 + 1.95188i 0.936537 + 0.350568i
\(32\) 0 0
\(33\) 1.46720 + 4.51557i 0.255406 + 0.786060i
\(34\) 0 0
\(35\) 2.73953 − 1.99039i 0.463065 − 0.336436i
\(36\) 0 0
\(37\) −1.21578 −0.199873 −0.0999367 − 0.994994i \(-0.531864\pi\)
−0.0999367 + 0.994994i \(0.531864\pi\)
\(38\) 0 0
\(39\) 0.930513 + 0.676057i 0.149001 + 0.108256i
\(40\) 0 0
\(41\) 1.57999 + 4.86270i 0.246753 + 0.759427i 0.995343 + 0.0963942i \(0.0307309\pi\)
−0.748591 + 0.663033i \(0.769269\pi\)
\(42\) 0 0
\(43\) 1.52098 + 4.68109i 0.231947 + 0.713860i 0.997512 + 0.0705009i \(0.0224598\pi\)
−0.765565 + 0.643359i \(0.777540\pi\)
\(44\) 0 0
\(45\) 0.809017 + 0.587785i 0.120601 + 0.0876219i
\(46\) 0 0
\(47\) −2.06016 + 6.34053i −0.300505 + 0.924861i 0.680811 + 0.732459i \(0.261628\pi\)
−0.981316 + 0.192401i \(0.938372\pi\)
\(48\) 0 0
\(49\) 1.38027 − 4.24804i 0.197182 − 0.606863i
\(50\) 0 0
\(51\) −2.48359 + 1.80443i −0.347772 + 0.252671i
\(52\) 0 0
\(53\) 4.83458 + 3.51253i 0.664081 + 0.482483i 0.868039 − 0.496497i \(-0.165381\pi\)
−0.203958 + 0.978980i \(0.565381\pi\)
\(54\) 0 0
\(55\) −3.84117 + 2.79078i −0.517944 + 0.376308i
\(56\) 0 0
\(57\) 5.60485 0.742380
\(58\) 0 0
\(59\) 0.950833 − 2.92636i 0.123788 − 0.380980i −0.869890 − 0.493245i \(-0.835811\pi\)
0.993678 + 0.112265i \(0.0358106\pi\)
\(60\) 0 0
\(61\) −5.08010 −0.650440 −0.325220 − 0.945638i \(-0.605438\pi\)
−0.325220 + 0.945638i \(0.605438\pi\)
\(62\) 0 0
\(63\) 3.38625 0.426627
\(64\) 0 0
\(65\) −0.355424 + 1.09388i −0.0440850 + 0.135680i
\(66\) 0 0
\(67\) 12.2497 1.49654 0.748268 − 0.663397i \(-0.230886\pi\)
0.748268 + 0.663397i \(0.230886\pi\)
\(68\) 0 0
\(69\) −0.993407 + 0.721753i −0.119592 + 0.0868888i
\(70\) 0 0
\(71\) 12.9317 + 9.39541i 1.53471 + 1.11503i 0.953548 + 0.301240i \(0.0974006\pi\)
0.581159 + 0.813790i \(0.302599\pi\)
\(72\) 0 0
\(73\) 9.80610 − 7.12455i 1.14772 − 0.833866i 0.159542 − 0.987191i \(-0.448998\pi\)
0.988176 + 0.153325i \(0.0489983\pi\)
\(74\) 0 0
\(75\) −0.309017 + 0.951057i −0.0356822 + 0.109819i
\(76\) 0 0
\(77\) −4.96829 + 15.2908i −0.566189 + 1.74255i
\(78\) 0 0
\(79\) 8.77049 + 6.37213i 0.986757 + 0.716921i 0.959208 − 0.282700i \(-0.0912300\pi\)
0.0275483 + 0.999620i \(0.491230\pi\)
\(80\) 0 0
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 0 0
\(83\) 0.606376 + 1.86623i 0.0665585 + 0.204846i 0.978804 − 0.204797i \(-0.0656534\pi\)
−0.912246 + 0.409643i \(0.865653\pi\)
\(84\) 0 0
\(85\) −2.48359 − 1.80443i −0.269383 − 0.195718i
\(86\) 0 0
\(87\) 6.41549 0.687813
\(88\) 0 0
\(89\) −7.27230 + 5.28364i −0.770862 + 0.560064i −0.902223 − 0.431270i \(-0.858066\pi\)
0.131360 + 0.991335i \(0.458066\pi\)
\(90\) 0 0
\(91\) 1.20355 + 3.70416i 0.126167 + 0.388301i
\(92\) 0 0
\(93\) −3.46769 + 4.35604i −0.359583 + 0.451701i
\(94\) 0 0
\(95\) 1.73199 + 5.33053i 0.177699 + 0.546901i
\(96\) 0 0
\(97\) 10.0874 − 7.32895i 1.02422 − 0.744142i 0.0570791 − 0.998370i \(-0.481821\pi\)
0.967144 + 0.254228i \(0.0818213\pi\)
\(98\) 0 0
\(99\) −4.74795 −0.477187
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 1860.2.z.b.841.1 yes 16
31.8 even 5 inner 1860.2.z.b.721.1 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.z.b.721.1 ✓ 16 31.8 even 5 inner
1860.2.z.b.841.1 yes 16 1.1 even 1 trivial