Properties

Label 345.4.a.i.1.6
Level $345$
Weight $4$
Character 345.1
Self dual yes
Analytic conductor $20.356$
Analytic rank $1$
Dimension $6$
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] = [345,4,Mod(1,345)] 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("345.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(345, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 345.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-4,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.3556589520\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 34x^{4} + 38x^{3} + 293x^{2} - 12x - 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.00313\) of defining polynomial
Character \(\chi\) \(=\) 345.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00313 q^{2} -3.00000 q^{3} +8.02508 q^{4} +5.00000 q^{5} -12.0094 q^{6} -23.1506 q^{7} +0.100409 q^{8} +9.00000 q^{9} +20.0157 q^{10} +7.66754 q^{11} -24.0752 q^{12} -50.0262 q^{13} -92.6749 q^{14} -15.0000 q^{15} -63.7987 q^{16} -1.23884 q^{17} +36.0282 q^{18} -77.2627 q^{19} +40.1254 q^{20} +69.4517 q^{21} +30.6942 q^{22} -23.0000 q^{23} -0.301228 q^{24} +25.0000 q^{25} -200.261 q^{26} -27.0000 q^{27} -185.785 q^{28} +102.938 q^{29} -60.0470 q^{30} -192.252 q^{31} -256.198 q^{32} -23.0026 q^{33} -4.95923 q^{34} -115.753 q^{35} +72.2257 q^{36} -432.028 q^{37} -309.293 q^{38} +150.079 q^{39} +0.502046 q^{40} +430.822 q^{41} +278.025 q^{42} +276.135 q^{43} +61.5327 q^{44} +45.0000 q^{45} -92.0721 q^{46} -462.407 q^{47} +191.396 q^{48} +192.949 q^{49} +100.078 q^{50} +3.71651 q^{51} -401.464 q^{52} +331.215 q^{53} -108.085 q^{54} +38.3377 q^{55} -2.32453 q^{56} +231.788 q^{57} +412.076 q^{58} +245.539 q^{59} -120.376 q^{60} -267.456 q^{61} -769.609 q^{62} -208.355 q^{63} -515.206 q^{64} -250.131 q^{65} -92.0826 q^{66} -98.3575 q^{67} -9.94177 q^{68} +69.0000 q^{69} -463.374 q^{70} +413.288 q^{71} +0.903684 q^{72} -265.319 q^{73} -1729.47 q^{74} -75.0000 q^{75} -620.040 q^{76} -177.508 q^{77} +600.784 q^{78} -151.251 q^{79} -318.994 q^{80} +81.0000 q^{81} +1724.64 q^{82} -85.3964 q^{83} +557.356 q^{84} -6.19418 q^{85} +1105.40 q^{86} -308.815 q^{87} +0.769892 q^{88} +234.816 q^{89} +180.141 q^{90} +1158.13 q^{91} -184.577 q^{92} +576.755 q^{93} -1851.08 q^{94} -386.314 q^{95} +768.594 q^{96} +917.819 q^{97} +772.401 q^{98} +69.0079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} - 18 q^{3} + 26 q^{4} + 30 q^{5} + 12 q^{6} - 40 q^{7} - 54 q^{8} + 54 q^{9} - 20 q^{10} - 48 q^{11} - 78 q^{12} - 56 q^{13} - 50 q^{14} - 90 q^{15} + 42 q^{16} - 84 q^{17} - 36 q^{18} + 80 q^{19}+ \cdots - 432 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00313 1.41532 0.707661 − 0.706552i \(-0.249750\pi\)
0.707661 + 0.706552i \(0.249750\pi\)
\(3\) −3.00000 −0.577350
\(4\) 8.02508 1.00314
\(5\) 5.00000 0.447214
\(6\) −12.0094 −0.817136
\(7\) −23.1506 −1.25001 −0.625007 − 0.780619i \(-0.714904\pi\)
−0.625007 + 0.780619i \(0.714904\pi\)
\(8\) 0.100409 0.00443751
\(9\) 9.00000 0.333333
\(10\) 20.0157 0.632951
\(11\) 7.66754 0.210168 0.105084 − 0.994463i \(-0.466489\pi\)
0.105084 + 0.994463i \(0.466489\pi\)
\(12\) −24.0752 −0.579160
\(13\) −50.0262 −1.06729 −0.533645 − 0.845709i \(-0.679178\pi\)
−0.533645 + 0.845709i \(0.679178\pi\)
\(14\) −92.6749 −1.76917
\(15\) −15.0000 −0.258199
\(16\) −63.7987 −0.996855
\(17\) −1.23884 −0.0176742 −0.00883712 − 0.999961i \(-0.502813\pi\)
−0.00883712 + 0.999961i \(0.502813\pi\)
\(18\) 36.0282 0.471774
\(19\) −77.2627 −0.932910 −0.466455 − 0.884545i \(-0.654469\pi\)
−0.466455 + 0.884545i \(0.654469\pi\)
\(20\) 40.1254 0.448616
\(21\) 69.4517 0.721696
\(22\) 30.6942 0.297456
\(23\) −23.0000 −0.208514
\(24\) −0.301228 −0.00256199
\(25\) 25.0000 0.200000
\(26\) −200.261 −1.51056
\(27\) −27.0000 −0.192450
\(28\) −185.785 −1.25393
\(29\) 102.938 0.659144 0.329572 − 0.944131i \(-0.393096\pi\)
0.329572 + 0.944131i \(0.393096\pi\)
\(30\) −60.0470 −0.365434
\(31\) −192.252 −1.11385 −0.556926 − 0.830562i \(-0.688019\pi\)
−0.556926 + 0.830562i \(0.688019\pi\)
\(32\) −256.198 −1.41531
\(33\) −23.0026 −0.121341
\(34\) −4.95923 −0.0250147
\(35\) −115.753 −0.559023
\(36\) 72.2257 0.334378
\(37\) −432.028 −1.91959 −0.959797 − 0.280697i \(-0.909435\pi\)
−0.959797 + 0.280697i \(0.909435\pi\)
\(38\) −309.293 −1.32037
\(39\) 150.079 0.616200
\(40\) 0.502046 0.00198451
\(41\) 430.822 1.64105 0.820525 − 0.571610i \(-0.193681\pi\)
0.820525 + 0.571610i \(0.193681\pi\)
\(42\) 278.025 1.02143
\(43\) 276.135 0.979306 0.489653 − 0.871917i \(-0.337124\pi\)
0.489653 + 0.871917i \(0.337124\pi\)
\(44\) 61.5327 0.210827
\(45\) 45.0000 0.149071
\(46\) −92.0721 −0.295115
\(47\) −462.407 −1.43508 −0.717542 − 0.696515i \(-0.754733\pi\)
−0.717542 + 0.696515i \(0.754733\pi\)
\(48\) 191.396 0.575534
\(49\) 192.949 0.562534
\(50\) 100.078 0.283064
\(51\) 3.71651 0.0102042
\(52\) −401.464 −1.07064
\(53\) 331.215 0.858412 0.429206 − 0.903207i \(-0.358793\pi\)
0.429206 + 0.903207i \(0.358793\pi\)
\(54\) −108.085 −0.272379
\(55\) 38.3377 0.0939901
\(56\) −2.32453 −0.00554694
\(57\) 231.788 0.538616
\(58\) 412.076 0.932900
\(59\) 245.539 0.541805 0.270902 − 0.962607i \(-0.412678\pi\)
0.270902 + 0.962607i \(0.412678\pi\)
\(60\) −120.376 −0.259008
\(61\) −267.456 −0.561380 −0.280690 − 0.959798i \(-0.590563\pi\)
−0.280690 + 0.959798i \(0.590563\pi\)
\(62\) −769.609 −1.57646
\(63\) −208.355 −0.416671
\(64\) −515.206 −1.00626
\(65\) −250.131 −0.477306
\(66\) −92.0826 −0.171736
\(67\) −98.3575 −0.179347 −0.0896737 − 0.995971i \(-0.528582\pi\)
−0.0896737 + 0.995971i \(0.528582\pi\)
\(68\) −9.94177 −0.0177297
\(69\) 69.0000 0.120386
\(70\) −463.374 −0.791197
\(71\) 413.288 0.690821 0.345411 − 0.938452i \(-0.387740\pi\)
0.345411 + 0.938452i \(0.387740\pi\)
\(72\) 0.903684 0.00147917
\(73\) −265.319 −0.425386 −0.212693 − 0.977119i \(-0.568223\pi\)
−0.212693 + 0.977119i \(0.568223\pi\)
\(74\) −1729.47 −2.71684
\(75\) −75.0000 −0.115470
\(76\) −620.040 −0.935835
\(77\) −177.508 −0.262713
\(78\) 600.784 0.872121
\(79\) −151.251 −0.215406 −0.107703 − 0.994183i \(-0.534350\pi\)
−0.107703 + 0.994183i \(0.534350\pi\)
\(80\) −318.994 −0.445807
\(81\) 81.0000 0.111111
\(82\) 1724.64 2.32261
\(83\) −85.3964 −0.112933 −0.0564667 − 0.998404i \(-0.517983\pi\)
−0.0564667 + 0.998404i \(0.517983\pi\)
\(84\) 557.356 0.723958
\(85\) −6.19418 −0.00790416
\(86\) 1105.40 1.38603
\(87\) −308.815 −0.380557
\(88\) 0.769892 0.000932623 0
\(89\) 234.816 0.279668 0.139834 − 0.990175i \(-0.455343\pi\)
0.139834 + 0.990175i \(0.455343\pi\)
\(90\) 180.141 0.210984
\(91\) 1158.13 1.33413
\(92\) −184.577 −0.209168
\(93\) 576.755 0.643083
\(94\) −1851.08 −2.03111
\(95\) −386.314 −0.417210
\(96\) 768.594 0.817128
\(97\) 917.819 0.960726 0.480363 − 0.877070i \(-0.340505\pi\)
0.480363 + 0.877070i \(0.340505\pi\)
\(98\) 772.401 0.796166
\(99\) 69.0079 0.0700561
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 345.4.a.i.1.6 ✓ 6
3.2 odd 2 1035.4.a.q.1.1 6
5.4 even 2 1725.4.a.x.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.4.a.i.1.6 ✓ 6 1.1 even 1 trivial
1035.4.a.q.1.1 6 3.2 odd 2
1725.4.a.x.1.1 6 5.4 even 2