Properties

Label 1805.4.a.g.1.1
Level $1805$
Weight $4$
Character 1805.1
Self dual yes
Analytic conductor $106.498$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1805,4,Mod(1,1805)] 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("1805.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,5,-7,5,5,22,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(106.498447560\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1805.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +5.00000 q^{3} -7.00000 q^{4} +5.00000 q^{5} +5.00000 q^{6} +22.0000 q^{7} -15.0000 q^{8} -2.00000 q^{9} +5.00000 q^{10} +9.00000 q^{11} -35.0000 q^{12} -54.0000 q^{13} +22.0000 q^{14} +25.0000 q^{15} +41.0000 q^{16} -54.0000 q^{17} -2.00000 q^{18} -35.0000 q^{20} +110.000 q^{21} +9.00000 q^{22} -92.0000 q^{23} -75.0000 q^{24} +25.0000 q^{25} -54.0000 q^{26} -145.000 q^{27} -154.000 q^{28} +134.000 q^{29} +25.0000 q^{30} +252.000 q^{31} +161.000 q^{32} +45.0000 q^{33} -54.0000 q^{34} +110.000 q^{35} +14.0000 q^{36} +236.000 q^{37} -270.000 q^{39} -75.0000 q^{40} +243.000 q^{41} +110.000 q^{42} +496.000 q^{43} -63.0000 q^{44} -10.0000 q^{45} -92.0000 q^{46} +502.000 q^{47} +205.000 q^{48} +141.000 q^{49} +25.0000 q^{50} -270.000 q^{51} +378.000 q^{52} -62.0000 q^{53} -145.000 q^{54} +45.0000 q^{55} -330.000 q^{56} +134.000 q^{58} -681.000 q^{59} -175.000 q^{60} -142.000 q^{61} +252.000 q^{62} -44.0000 q^{63} -167.000 q^{64} -270.000 q^{65} +45.0000 q^{66} -55.0000 q^{67} +378.000 q^{68} -460.000 q^{69} +110.000 q^{70} +974.000 q^{71} +30.0000 q^{72} +695.000 q^{73} +236.000 q^{74} +125.000 q^{75} +198.000 q^{77} -270.000 q^{78} +736.000 q^{79} +205.000 q^{80} -671.000 q^{81} +243.000 q^{82} -63.0000 q^{83} -770.000 q^{84} -270.000 q^{85} +496.000 q^{86} +670.000 q^{87} -135.000 q^{88} -726.000 q^{89} -10.0000 q^{90} -1188.00 q^{91} +644.000 q^{92} +1260.00 q^{93} +502.000 q^{94} +805.000 q^{96} +1167.00 q^{97} +141.000 q^{98} -18.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.353553 0.176777 − 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 5.00000 0.962250 0.481125 − 0.876652i \(-0.340228\pi\)
0.481125 + 0.876652i \(0.340228\pi\)
\(4\) −7.00000 −0.875000
\(5\) 5.00000 0.447214
\(6\) 5.00000 0.340207
\(7\) 22.0000 1.18789 0.593944 − 0.804506i \(-0.297570\pi\)
0.593944 + 0.804506i \(0.297570\pi\)
\(8\) −15.0000 −0.662913
\(9\) −2.00000 −0.0740741
\(10\) 5.00000 0.158114
\(11\) 9.00000 0.246691 0.123346 − 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(12\) −35.0000 −0.841969
\(13\) −54.0000 −1.15207 −0.576035 − 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 22.0000 0.419982
\(15\) 25.0000 0.430331
\(16\) 41.0000 0.640625
\(17\) −54.0000 −0.770407 −0.385204 − 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) −2.00000 −0.0261891
\(19\) 0 0
\(20\) −35.0000 −0.391312
\(21\) 110.000 1.14305
\(22\) 9.00000 0.0872185
\(23\) −92.0000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −75.0000 −0.637888
\(25\) 25.0000 0.200000
\(26\) −54.0000 −0.407318
\(27\) −145.000 −1.03353
\(28\) −154.000 −1.03940
\(29\) 134.000 0.858041 0.429020 − 0.903295i \(-0.358859\pi\)
0.429020 + 0.903295i \(0.358859\pi\)
\(30\) 25.0000 0.152145
\(31\) 252.000 1.46002 0.730009 − 0.683438i \(-0.239516\pi\)
0.730009 + 0.683438i \(0.239516\pi\)
\(32\) 161.000 0.889408
\(33\) 45.0000 0.237379
\(34\) −54.0000 −0.272380
\(35\) 110.000 0.531240
\(36\) 14.0000 0.0648148
\(37\) 236.000 1.04860 0.524299 − 0.851534i \(-0.324327\pi\)
0.524299 + 0.851534i \(0.324327\pi\)
\(38\) 0 0
\(39\) −270.000 −1.10858
\(40\) −75.0000 −0.296464
\(41\) 243.000 0.925615 0.462808 − 0.886459i \(-0.346842\pi\)
0.462808 + 0.886459i \(0.346842\pi\)
\(42\) 110.000 0.404128
\(43\) 496.000 1.75905 0.879527 − 0.475850i \(-0.157859\pi\)
0.879527 + 0.475850i \(0.157859\pi\)
\(44\) −63.0000 −0.215855
\(45\) −10.0000 −0.0331269
\(46\) −92.0000 −0.294884
\(47\) 502.000 1.55796 0.778981 − 0.627047i \(-0.215737\pi\)
0.778981 + 0.627047i \(0.215737\pi\)
\(48\) 205.000 0.616442
\(49\) 141.000 0.411079
\(50\) 25.0000 0.0707107
\(51\) −270.000 −0.741325
\(52\) 378.000 1.00806
\(53\) −62.0000 −0.160686 −0.0803430 − 0.996767i \(-0.525602\pi\)
−0.0803430 + 0.996767i \(0.525602\pi\)
\(54\) −145.000 −0.365407
\(55\) 45.0000 0.110324
\(56\) −330.000 −0.787466
\(57\) 0 0
\(58\) 134.000 0.303363
\(59\) −681.000 −1.50269 −0.751344 − 0.659910i \(-0.770594\pi\)
−0.751344 + 0.659910i \(0.770594\pi\)
\(60\) −175.000 −0.376540
\(61\) −142.000 −0.298053 −0.149027 − 0.988833i \(-0.547614\pi\)
−0.149027 + 0.988833i \(0.547614\pi\)
\(62\) 252.000 0.516194
\(63\) −44.0000 −0.0879917
\(64\) −167.000 −0.326172
\(65\) −270.000 −0.515221
\(66\) 45.0000 0.0839260
\(67\) −55.0000 −0.100288 −0.0501442 − 0.998742i \(-0.515968\pi\)
−0.0501442 + 0.998742i \(0.515968\pi\)
\(68\) 378.000 0.674106
\(69\) −460.000 −0.802572
\(70\) 110.000 0.187822
\(71\) 974.000 1.62806 0.814032 − 0.580820i \(-0.197268\pi\)
0.814032 + 0.580820i \(0.197268\pi\)
\(72\) 30.0000 0.0491046
\(73\) 695.000 1.11430 0.557148 − 0.830413i \(-0.311896\pi\)
0.557148 + 0.830413i \(0.311896\pi\)
\(74\) 236.000 0.370736
\(75\) 125.000 0.192450
\(76\) 0 0
\(77\) 198.000 0.293041
\(78\) −270.000 −0.391942
\(79\) 736.000 1.04818 0.524092 − 0.851662i \(-0.324405\pi\)
0.524092 + 0.851662i \(0.324405\pi\)
\(80\) 205.000 0.286496
\(81\) −671.000 −0.920439
\(82\) 243.000 0.327254
\(83\) −63.0000 −0.0833150 −0.0416575 − 0.999132i \(-0.513264\pi\)
−0.0416575 + 0.999132i \(0.513264\pi\)
\(84\) −770.000 −1.00017
\(85\) −270.000 −0.344537
\(86\) 496.000 0.621919
\(87\) 670.000 0.825650
\(88\) −135.000 −0.163535
\(89\) −726.000 −0.864672 −0.432336 − 0.901712i \(-0.642311\pi\)
−0.432336 + 0.901712i \(0.642311\pi\)
\(90\) −10.0000 −0.0117121
\(91\) −1188.00 −1.36853
\(92\) 644.000 0.729800
\(93\) 1260.00 1.40490
\(94\) 502.000 0.550823
\(95\) 0 0
\(96\) 805.000 0.855833
\(97\) 1167.00 1.22156 0.610778 − 0.791802i \(-0.290857\pi\)
0.610778 + 0.791802i \(0.290857\pi\)
\(98\) 141.000 0.145338
\(99\) −18.0000 −0.0182734
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 1805.4.a.g.1.1 1
19.8 odd 6 95.4.e.a.26.1 yes 2
19.12 odd 6 95.4.e.a.11.1 ✓ 2
19.18 odd 2 1805.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.e.a.11.1 ✓ 2 19.12 odd 6
95.4.e.a.26.1 yes 2 19.8 odd 6
1805.4.a.e.1.1 1 19.18 odd 2
1805.4.a.g.1.1 1 1.1 even 1 trivial