Properties

Label 105.10.a.d.1.4
Level $105$
Weight $10$
Character 105.1
Self dual yes
Analytic conductor $54.079$
Analytic rank $1$
Dimension $4$
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] = [105,10,Mod(1,105)] 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("105.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.0787627972\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1253x^{2} - 1039x + 42996 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-33.9278\) of defining polynomial
Character \(\chi\) \(=\) 105.1

$q$-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)
 
\(f(q)\) \(=\) \(q+30.9278 q^{2} +81.0000 q^{3} +444.529 q^{4} -625.000 q^{5} +2505.15 q^{6} +2401.00 q^{7} -2086.72 q^{8} +6561.00 q^{9} -19329.9 q^{10} -51657.7 q^{11} +36006.9 q^{12} -129921. q^{13} +74257.7 q^{14} -50625.0 q^{15} -292137. q^{16} -259513. q^{17} +202917. q^{18} +90316.4 q^{19} -277831. q^{20} +194481. q^{21} -1.59766e6 q^{22} +751692. q^{23} -169024. q^{24} +390625. q^{25} -4.01818e6 q^{26} +531441. q^{27} +1.06731e6 q^{28} -2.20510e6 q^{29} -1.56572e6 q^{30} +3.10032e6 q^{31} -7.96675e6 q^{32} -4.18427e6 q^{33} -8.02616e6 q^{34} -1.50062e6 q^{35} +2.91656e6 q^{36} -1.42524e7 q^{37} +2.79329e6 q^{38} -1.05236e7 q^{39} +1.30420e6 q^{40} -1.45691e7 q^{41} +6.01487e6 q^{42} +1.89091e7 q^{43} -2.29633e7 q^{44} -4.10062e6 q^{45} +2.32482e7 q^{46} -1.43175e7 q^{47} -2.36631e7 q^{48} +5.76480e6 q^{49} +1.20812e7 q^{50} -2.10205e7 q^{51} -5.77538e7 q^{52} -3.44220e7 q^{53} +1.64363e7 q^{54} +3.22860e7 q^{55} -5.01022e6 q^{56} +7.31563e6 q^{57} -6.81989e7 q^{58} -3.61330e7 q^{59} -2.25043e7 q^{60} -9.33007e7 q^{61} +9.58860e7 q^{62} +1.57530e7 q^{63} -9.68200e7 q^{64} +8.12008e7 q^{65} -1.29410e8 q^{66} +2.05762e8 q^{67} -1.15361e8 q^{68} +6.08870e7 q^{69} -4.64110e7 q^{70} -1.92397e8 q^{71} -1.36910e7 q^{72} +8.11669e7 q^{73} -4.40795e8 q^{74} +3.16406e7 q^{75} +4.01483e7 q^{76} -1.24030e8 q^{77} -3.25473e8 q^{78} +1.07122e8 q^{79} +1.82585e8 q^{80} +4.30467e7 q^{81} -4.50592e8 q^{82} +3.54922e8 q^{83} +8.64525e7 q^{84} +1.62195e8 q^{85} +5.84818e8 q^{86} -1.78613e8 q^{87} +1.07795e8 q^{88} +4.42433e8 q^{89} -1.26823e8 q^{90} -3.11941e8 q^{91} +3.34149e8 q^{92} +2.51126e8 q^{93} -4.42810e8 q^{94} -5.64478e7 q^{95} -6.45306e8 q^{96} -6.40463e8 q^{97} +1.78293e8 q^{98} -3.38926e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 13 q^{2} + 324 q^{3} + 501 q^{4} - 2500 q^{5} - 1053 q^{6} + 9604 q^{7} - 16263 q^{8} + 26244 q^{9} + 8125 q^{10} - 87062 q^{11} + 40581 q^{12} + 39494 q^{13} - 31213 q^{14} - 202500 q^{15} + 328849 q^{16}+ \cdots - 571213782 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 30.9278 1.36683 0.683414 − 0.730031i \(-0.260494\pi\)
0.683414 + 0.730031i \(0.260494\pi\)
\(3\) 81.0000 0.577350
\(4\) 444.529 0.868221
\(5\) −625.000 −0.447214
\(6\) 2505.15 0.789139
\(7\) 2401.00 0.377964
\(8\) −2086.72 −0.180119
\(9\) 6561.00 0.333333
\(10\) −19329.9 −0.611264
\(11\) −51657.7 −1.06382 −0.531909 − 0.846801i \(-0.678525\pi\)
−0.531909 + 0.846801i \(0.678525\pi\)
\(12\) 36006.9 0.501268
\(13\) −129921. −1.26164 −0.630820 − 0.775930i \(-0.717281\pi\)
−0.630820 + 0.775930i \(0.717281\pi\)
\(14\) 74257.7 0.516613
\(15\) −50625.0 −0.258199
\(16\) −292137. −1.11441
\(17\) −259513. −0.753595 −0.376798 − 0.926296i \(-0.622975\pi\)
−0.376798 + 0.926296i \(0.622975\pi\)
\(18\) 202917. 0.455610
\(19\) 90316.4 0.158992 0.0794961 − 0.996835i \(-0.474669\pi\)
0.0794961 + 0.996835i \(0.474669\pi\)
\(20\) −277831. −0.388280
\(21\) 194481. 0.218218
\(22\) −1.59766e6 −1.45406
\(23\) 751692. 0.560099 0.280049 − 0.959986i \(-0.409649\pi\)
0.280049 + 0.959986i \(0.409649\pi\)
\(24\) −169024. −0.103992
\(25\) 390625. 0.200000
\(26\) −4.01818e6 −1.72444
\(27\) 531441. 0.192450
\(28\) 1.06731e6 0.328157
\(29\) −2.20510e6 −0.578945 −0.289472 − 0.957186i \(-0.593480\pi\)
−0.289472 + 0.957186i \(0.593480\pi\)
\(30\) −1.56572e6 −0.352914
\(31\) 3.10032e6 0.602946 0.301473 − 0.953475i \(-0.402522\pi\)
0.301473 + 0.953475i \(0.402522\pi\)
\(32\) −7.96675e6 −1.34309
\(33\) −4.18427e6 −0.614196
\(34\) −8.02616e6 −1.03004
\(35\) −1.50062e6 −0.169031
\(36\) 2.91656e6 0.289407
\(37\) −1.42524e7 −1.25020 −0.625101 − 0.780544i \(-0.714942\pi\)
−0.625101 + 0.780544i \(0.714942\pi\)
\(38\) 2.79329e6 0.217315
\(39\) −1.05236e7 −0.728408
\(40\) 1.30420e6 0.0805517
\(41\) −1.45691e7 −0.805205 −0.402603 − 0.915375i \(-0.631894\pi\)
−0.402603 + 0.915375i \(0.631894\pi\)
\(42\) 6.01487e6 0.298267
\(43\) 1.89091e7 0.843458 0.421729 − 0.906722i \(-0.361423\pi\)
0.421729 + 0.906722i \(0.361423\pi\)
\(44\) −2.29633e7 −0.923630
\(45\) −4.10062e6 −0.149071
\(46\) 2.32482e7 0.765559
\(47\) −1.43175e7 −0.427985 −0.213992 − 0.976835i \(-0.568647\pi\)
−0.213992 + 0.976835i \(0.568647\pi\)
\(48\) −2.36631e7 −0.643407
\(49\) 5.76480e6 0.142857
\(50\) 1.20812e7 0.273366
\(51\) −2.10205e7 −0.435089
\(52\) −5.77538e7 −1.09538
\(53\) −3.44220e7 −0.599232 −0.299616 − 0.954060i \(-0.596858\pi\)
−0.299616 + 0.954060i \(0.596858\pi\)
\(54\) 1.64363e7 0.263046
\(55\) 3.22860e7 0.475754
\(56\) −5.01022e6 −0.0680786
\(57\) 7.31563e6 0.0917941
\(58\) −6.81989e7 −0.791319
\(59\) −3.61330e7 −0.388213 −0.194107 − 0.980980i \(-0.562181\pi\)
−0.194107 + 0.980980i \(0.562181\pi\)
\(60\) −2.25043e7 −0.224174
\(61\) −9.33007e7 −0.862781 −0.431390 − 0.902165i \(-0.641977\pi\)
−0.431390 + 0.902165i \(0.641977\pi\)
\(62\) 9.58860e7 0.824124
\(63\) 1.57530e7 0.125988
\(64\) −9.68200e7 −0.721365
\(65\) 8.12008e7 0.564222
\(66\) −1.29410e8 −0.839501
\(67\) 2.05762e8 1.24746 0.623732 − 0.781638i \(-0.285616\pi\)
0.623732 + 0.781638i \(0.285616\pi\)
\(68\) −1.15361e8 −0.654288
\(69\) 6.08870e7 0.323373
\(70\) −4.64110e7 −0.231036
\(71\) −1.92397e8 −0.898535 −0.449268 − 0.893397i \(-0.648315\pi\)
−0.449268 + 0.893397i \(0.648315\pi\)
\(72\) −1.36910e7 −0.0600397
\(73\) 8.11669e7 0.334523 0.167262 − 0.985913i \(-0.446508\pi\)
0.167262 + 0.985913i \(0.446508\pi\)
\(74\) −4.40795e8 −1.70881
\(75\) 3.16406e7 0.115470
\(76\) 4.01483e7 0.138040
\(77\) −1.24030e8 −0.402086
\(78\) −3.25473e8 −0.995609
\(79\) 1.07122e8 0.309427 0.154713 − 0.987959i \(-0.450555\pi\)
0.154713 + 0.987959i \(0.450555\pi\)
\(80\) 1.82585e8 0.498381
\(81\) 4.30467e7 0.111111
\(82\) −4.50592e8 −1.10058
\(83\) 3.54922e8 0.820883 0.410441 − 0.911887i \(-0.365375\pi\)
0.410441 + 0.911887i \(0.365375\pi\)
\(84\) 8.64525e7 0.189461
\(85\) 1.62195e8 0.337018
\(86\) 5.84818e8 1.15286
\(87\) −1.78613e8 −0.334254
\(88\) 1.07795e8 0.191614
\(89\) 4.42433e8 0.747468 0.373734 − 0.927536i \(-0.378077\pi\)
0.373734 + 0.927536i \(0.378077\pi\)
\(90\) −1.26823e8 −0.203755
\(91\) −3.11941e8 −0.476855
\(92\) 3.34149e8 0.486290
\(93\) 2.51126e8 0.348111
\(94\) −4.42810e8 −0.584982
\(95\) −5.64478e7 −0.0711034
\(96\) −6.45306e8 −0.775435
\(97\) −6.40463e8 −0.734549 −0.367275 − 0.930113i \(-0.619709\pi\)
−0.367275 + 0.930113i \(0.619709\pi\)
\(98\) 1.78293e8 0.195261
\(99\) −3.38926e8 −0.354606
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 105.10.a.d.1.4 ✓ 4
3.2 odd 2 315.10.a.e.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.10.a.d.1.4 ✓ 4 1.1 even 1 trivial
315.10.a.e.1.1 4 3.2 odd 2