↑

Properties

Label 1845.4.a.s.1.13
Level $1845$
Weight $4$
Character 1845.1
Self dual yes
Analytic conductor $108.859$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1845,4,Mod(1,1845)] 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("1845.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1845 = 3^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1845.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-5,0,91,-70,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(108.858523961\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 89 x^{12} + 433 x^{11} + 3100 x^{10} - 14427 x^{9} - 53983 x^{8} + 233727 x^{7} + \cdots - 2084736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 615)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-4.13962\) of defining polynomial
Character \(\chi\) \(=\) 1845.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.13962 q^{2} +9.13646 q^{4} -5.00000 q^{5} -35.0560 q^{7} +4.70451 q^{8} -20.6981 q^{10} -38.9460 q^{11} -85.2711 q^{13} -145.119 q^{14} -53.6168 q^{16} +110.536 q^{17} +135.221 q^{19} -45.6823 q^{20} -161.222 q^{22} +25.7225 q^{23} +25.0000 q^{25} -352.990 q^{26} -320.288 q^{28} -10.3593 q^{29} +61.0273 q^{31} -259.589 q^{32} +457.576 q^{34} +175.280 q^{35} -18.2373 q^{37} +559.765 q^{38} -23.5225 q^{40} +41.0000 q^{41} +383.956 q^{43} -355.829 q^{44} +106.481 q^{46} -347.875 q^{47} +885.925 q^{49} +103.491 q^{50} -779.076 q^{52} +277.934 q^{53} +194.730 q^{55} -164.921 q^{56} -42.8834 q^{58} +496.493 q^{59} -754.609 q^{61} +252.630 q^{62} -645.667 q^{64} +426.356 q^{65} +450.058 q^{67} +1009.90 q^{68} +725.593 q^{70} -139.508 q^{71} -566.451 q^{73} -75.4957 q^{74} +1235.44 q^{76} +1365.29 q^{77} +76.8373 q^{79} +268.084 q^{80} +169.724 q^{82} -949.698 q^{83} -552.678 q^{85} +1589.43 q^{86} -183.222 q^{88} +107.891 q^{89} +2989.27 q^{91} +235.013 q^{92} -1440.07 q^{94} -676.107 q^{95} +1620.46 q^{97} +3667.39 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 5 q^{2} + 91 q^{4} - 70 q^{5} + 40 q^{7} - 81 q^{8} + 25 q^{10} - 33 q^{11} + 91 q^{13} - 84 q^{14} + 331 q^{16} - 174 q^{17} + 323 q^{19} - 455 q^{20} + 110 q^{22} - 520 q^{23} + 350 q^{25} - 220 q^{26}+ \cdots + 9192 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.13962 1.46358 0.731788 − 0.681532i \(-0.238686\pi\)
0.731788 + 0.681532i \(0.238686\pi\)
\(3\) 0 0
\(4\) 9.13646 1.14206
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −35.0560 −1.89285 −0.946424 − 0.322927i \(-0.895333\pi\)
−0.946424 + 0.322927i \(0.895333\pi\)
\(8\) 4.70451 0.207912
\(9\) 0 0
\(10\) −20.6981 −0.654531
\(11\) −38.9460 −1.06751 −0.533757 − 0.845638i \(-0.679220\pi\)
−0.533757 + 0.845638i \(0.679220\pi\)
\(12\) 0 0
\(13\) −85.2711 −1.81923 −0.909614 − 0.415455i \(-0.863622\pi\)
−0.909614 + 0.415455i \(0.863622\pi\)
\(14\) −145.119 −2.77033
\(15\) 0 0
\(16\) −53.6168 −0.837762
\(17\) 110.536 1.57699 0.788495 − 0.615041i \(-0.210860\pi\)
0.788495 + 0.615041i \(0.210860\pi\)
\(18\) 0 0
\(19\) 135.221 1.63273 0.816366 − 0.577534i \(-0.195985\pi\)
0.816366 + 0.577534i \(0.195985\pi\)
\(20\) −45.6823 −0.510744
\(21\) 0 0
\(22\) −161.222 −1.56239
\(23\) 25.7225 0.233196 0.116598 − 0.993179i \(-0.462801\pi\)
0.116598 + 0.993179i \(0.462801\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −352.990 −2.66258
\(27\) 0 0
\(28\) −320.288 −2.16174
\(29\) −10.3593 −0.0663333 −0.0331667 − 0.999450i \(-0.510559\pi\)
−0.0331667 + 0.999450i \(0.510559\pi\)
\(30\) 0 0
\(31\) 61.0273 0.353575 0.176788 − 0.984249i \(-0.443429\pi\)
0.176788 + 0.984249i \(0.443429\pi\)
\(32\) −259.589 −1.43404
\(33\) 0 0
\(34\) 457.576 2.30805
\(35\) 175.280 0.846507
\(36\) 0 0
\(37\) −18.2373 −0.0810324 −0.0405162 − 0.999179i \(-0.512900\pi\)
−0.0405162 + 0.999179i \(0.512900\pi\)
\(38\) 559.765 2.38963
\(39\) 0 0
\(40\) −23.5225 −0.0929810
\(41\) 41.0000 0.156174
\(42\) 0 0
\(43\) 383.956 1.36169 0.680845 − 0.732427i \(-0.261613\pi\)
0.680845 + 0.732427i \(0.261613\pi\)
\(44\) −355.829 −1.21916
\(45\) 0 0
\(46\) 106.481 0.341301
\(47\) −347.875 −1.07963 −0.539817 − 0.841782i \(-0.681507\pi\)
−0.539817 + 0.841782i \(0.681507\pi\)
\(48\) 0 0
\(49\) 885.925 2.58287
\(50\) 103.491 0.292715
\(51\) 0 0
\(52\) −779.076 −2.07766
\(53\) 277.934 0.720323 0.360161 − 0.932890i \(-0.382722\pi\)
0.360161 + 0.932890i \(0.382722\pi\)
\(54\) 0 0
\(55\) 194.730 0.477407
\(56\) −164.921 −0.393545
\(57\) 0 0
\(58\) −42.8834 −0.0970839
\(59\) 496.493 1.09556 0.547779 − 0.836623i \(-0.315474\pi\)
0.547779 + 0.836623i \(0.315474\pi\)
\(60\) 0 0
\(61\) −754.609 −1.58390 −0.791949 − 0.610588i \(-0.790933\pi\)
−0.791949 + 0.610588i \(0.790933\pi\)
\(62\) 252.630 0.517484
\(63\) 0 0
\(64\) −645.667 −1.26107
\(65\) 426.356 0.813583
\(66\) 0 0
\(67\) 450.058 0.820647 0.410324 − 0.911940i \(-0.365416\pi\)
0.410324 + 0.911940i \(0.365416\pi\)
\(68\) 1009.90 1.80101
\(69\) 0 0
\(70\) 725.593 1.23893
\(71\) −139.508 −0.233192 −0.116596 − 0.993179i \(-0.537198\pi\)
−0.116596 + 0.993179i \(0.537198\pi\)
\(72\) 0 0
\(73\) −566.451 −0.908193 −0.454097 − 0.890952i \(-0.650038\pi\)
−0.454097 + 0.890952i \(0.650038\pi\)
\(74\) −75.4957 −0.118597
\(75\) 0 0
\(76\) 1235.44 1.86467
\(77\) 1365.29 2.02064
\(78\) 0 0
\(79\) 76.8373 0.109429 0.0547143 − 0.998502i \(-0.482575\pi\)
0.0547143 + 0.998502i \(0.482575\pi\)
\(80\) 268.084 0.374659
\(81\) 0 0
\(82\) 169.724 0.228572
\(83\) −949.698 −1.25594 −0.627969 − 0.778238i \(-0.716114\pi\)
−0.627969 + 0.778238i \(0.716114\pi\)
\(84\) 0 0
\(85\) −552.678 −0.705252
\(86\) 1589.43 1.99294
\(87\) 0 0
\(88\) −183.222 −0.221949
\(89\) 107.891 0.128499 0.0642497 − 0.997934i \(-0.479535\pi\)
0.0642497 + 0.997934i \(0.479535\pi\)
\(90\) 0 0
\(91\) 2989.27 3.44352
\(92\) 235.013 0.266323
\(93\) 0 0
\(94\) −1440.07 −1.58013
\(95\) −676.107 −0.730180
\(96\) 0 0
\(97\) 1620.46 1.69622 0.848109 − 0.529822i \(-0.177741\pi\)
0.848109 + 0.529822i \(0.177741\pi\)
\(98\) 3667.39 3.78023
\(99\) 0 0
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 1845.4.a.s.1.13 14
3.2 odd 2 615.4.a.k.1.2 ✓ 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.4.a.k.1.2 ✓ 14 3.2 odd 2
1845.4.a.s.1.13 14 1.1 even 1 trivial