Properties

Label 345.4.a.g.1.1
Level $345$
Weight $4$
Character 345.1
Self dual yes
Analytic conductor $20.356$
Analytic rank $1$
Dimension $4$
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 = [4,0,12] 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: \(4\)
Coefficient field: 4.4.2081448.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} - 12x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.39895\) 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)
 
\(f(q)\) \(=\) \(q-4.39895 q^{2} +3.00000 q^{3} +11.3508 q^{4} -5.00000 q^{5} -13.1968 q^{6} -11.2695 q^{7} -14.7398 q^{8} +9.00000 q^{9} +21.9947 q^{10} +16.2312 q^{11} +34.0523 q^{12} -2.68754 q^{13} +49.5741 q^{14} -15.0000 q^{15} -25.9664 q^{16} +92.2855 q^{17} -39.5905 q^{18} -125.601 q^{19} -56.7538 q^{20} -33.8086 q^{21} -71.4003 q^{22} -23.0000 q^{23} -44.2194 q^{24} +25.0000 q^{25} +11.8223 q^{26} +27.0000 q^{27} -127.918 q^{28} -6.60481 q^{29} +65.9842 q^{30} -9.81675 q^{31} +232.143 q^{32} +48.6937 q^{33} -405.959 q^{34} +56.3476 q^{35} +102.157 q^{36} +180.078 q^{37} +552.514 q^{38} -8.06261 q^{39} +73.6990 q^{40} +114.341 q^{41} +148.722 q^{42} -310.267 q^{43} +184.237 q^{44} -45.0000 q^{45} +101.176 q^{46} -24.4063 q^{47} -77.8992 q^{48} -215.998 q^{49} -109.974 q^{50} +276.856 q^{51} -30.5056 q^{52} -24.9445 q^{53} -118.772 q^{54} -81.1561 q^{55} +166.111 q^{56} -376.804 q^{57} +29.0542 q^{58} -866.879 q^{59} -170.261 q^{60} +61.6912 q^{61} +43.1834 q^{62} -101.426 q^{63} -813.455 q^{64} +13.4377 q^{65} -214.201 q^{66} +98.7470 q^{67} +1047.51 q^{68} -69.0000 q^{69} -247.870 q^{70} -529.013 q^{71} -132.658 q^{72} -741.165 q^{73} -792.152 q^{74} +75.0000 q^{75} -1425.67 q^{76} -182.918 q^{77} +35.4670 q^{78} -1151.70 q^{79} +129.832 q^{80} +81.0000 q^{81} -502.981 q^{82} -917.445 q^{83} -383.753 q^{84} -461.427 q^{85} +1364.85 q^{86} -19.8144 q^{87} -239.245 q^{88} -290.163 q^{89} +197.953 q^{90} +30.2873 q^{91} -261.067 q^{92} -29.4502 q^{93} +107.362 q^{94} +628.006 q^{95} +696.430 q^{96} -717.714 q^{97} +950.164 q^{98} +146.081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 6 q^{4} - 20 q^{5} - 16 q^{7} - 36 q^{8} + 36 q^{9} - 20 q^{11} + 18 q^{12} - 106 q^{13} - 14 q^{14} - 60 q^{15} - 118 q^{16} + 44 q^{17} - 152 q^{19} - 30 q^{20} - 48 q^{21} - 128 q^{22}+ \cdots - 180 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.39895 −1.55526 −0.777632 − 0.628720i \(-0.783579\pi\)
−0.777632 + 0.628720i \(0.783579\pi\)
\(3\) 3.00000 0.577350
\(4\) 11.3508 1.41884
\(5\) −5.00000 −0.447214
\(6\) −13.1968 −0.897932
\(7\) −11.2695 −0.608497 −0.304248 − 0.952593i \(-0.598405\pi\)
−0.304248 + 0.952593i \(0.598405\pi\)
\(8\) −14.7398 −0.651414
\(9\) 9.00000 0.333333
\(10\) 21.9947 0.695535
\(11\) 16.2312 0.444900 0.222450 − 0.974944i \(-0.428595\pi\)
0.222450 + 0.974944i \(0.428595\pi\)
\(12\) 34.0523 0.819170
\(13\) −2.68754 −0.0573376 −0.0286688 − 0.999589i \(-0.509127\pi\)
−0.0286688 + 0.999589i \(0.509127\pi\)
\(14\) 49.5741 0.946373
\(15\) −15.0000 −0.258199
\(16\) −25.9664 −0.405725
\(17\) 92.2855 1.31662 0.658309 − 0.752748i \(-0.271272\pi\)
0.658309 + 0.752748i \(0.271272\pi\)
\(18\) −39.5905 −0.518421
\(19\) −125.601 −1.51657 −0.758287 − 0.651921i \(-0.773963\pi\)
−0.758287 + 0.651921i \(0.773963\pi\)
\(20\) −56.7538 −0.634527
\(21\) −33.8086 −0.351316
\(22\) −71.4003 −0.691937
\(23\) −23.0000 −0.208514
\(24\) −44.2194 −0.376094
\(25\) 25.0000 0.200000
\(26\) 11.8223 0.0891751
\(27\) 27.0000 0.192450
\(28\) −127.918 −0.863363
\(29\) −6.60481 −0.0422925 −0.0211462 − 0.999776i \(-0.506732\pi\)
−0.0211462 + 0.999776i \(0.506732\pi\)
\(30\) 65.9842 0.401567
\(31\) −9.81675 −0.0568755 −0.0284377 − 0.999596i \(-0.509053\pi\)
−0.0284377 + 0.999596i \(0.509053\pi\)
\(32\) 232.143 1.28242
\(33\) 48.6937 0.256863
\(34\) −405.959 −2.04769
\(35\) 56.3476 0.272128
\(36\) 102.157 0.472948
\(37\) 180.078 0.800123 0.400062 − 0.916488i \(-0.368989\pi\)
0.400062 + 0.916488i \(0.368989\pi\)
\(38\) 552.514 2.35867
\(39\) −8.06261 −0.0331039
\(40\) 73.6990 0.291321
\(41\) 114.341 0.435539 0.217769 − 0.976000i \(-0.430122\pi\)
0.217769 + 0.976000i \(0.430122\pi\)
\(42\) 148.722 0.546389
\(43\) −310.267 −1.10035 −0.550177 − 0.835048i \(-0.685440\pi\)
−0.550177 + 0.835048i \(0.685440\pi\)
\(44\) 184.237 0.631244
\(45\) −45.0000 −0.149071
\(46\) 101.176 0.324295
\(47\) −24.4063 −0.0757452 −0.0378726 − 0.999283i \(-0.512058\pi\)
−0.0378726 + 0.999283i \(0.512058\pi\)
\(48\) −77.8992 −0.234245
\(49\) −215.998 −0.629731
\(50\) −109.974 −0.311053
\(51\) 276.856 0.760150
\(52\) −30.5056 −0.0813532
\(53\) −24.9445 −0.0646488 −0.0323244 − 0.999477i \(-0.510291\pi\)
−0.0323244 + 0.999477i \(0.510291\pi\)
\(54\) −118.772 −0.299311
\(55\) −81.1561 −0.198965
\(56\) 166.111 0.396383
\(57\) −376.804 −0.875594
\(58\) 29.0542 0.0657760
\(59\) −866.879 −1.91285 −0.956424 − 0.291982i \(-0.905685\pi\)
−0.956424 + 0.291982i \(0.905685\pi\)
\(60\) −170.261 −0.366344
\(61\) 61.6912 0.129488 0.0647439 − 0.997902i \(-0.479377\pi\)
0.0647439 + 0.997902i \(0.479377\pi\)
\(62\) 43.1834 0.0884563
\(63\) −101.426 −0.202832
\(64\) −813.455 −1.58878
\(65\) 13.4377 0.0256422
\(66\) −214.201 −0.399490
\(67\) 98.7470 0.180058 0.0900288 − 0.995939i \(-0.471304\pi\)
0.0900288 + 0.995939i \(0.471304\pi\)
\(68\) 1047.51 1.86808
\(69\) −69.0000 −0.120386
\(70\) −247.870 −0.423231
\(71\) −529.013 −0.884258 −0.442129 − 0.896951i \(-0.645777\pi\)
−0.442129 + 0.896951i \(0.645777\pi\)
\(72\) −132.658 −0.217138
\(73\) −741.165 −1.18831 −0.594156 − 0.804350i \(-0.702514\pi\)
−0.594156 + 0.804350i \(0.702514\pi\)
\(74\) −792.152 −1.24440
\(75\) 75.0000 0.115470
\(76\) −1425.67 −2.15178
\(77\) −182.918 −0.270720
\(78\) 35.4670 0.0514853
\(79\) −1151.70 −1.64021 −0.820103 − 0.572216i \(-0.806084\pi\)
−0.820103 + 0.572216i \(0.806084\pi\)
\(80\) 129.832 0.181446
\(81\) 81.0000 0.111111
\(82\) −502.981 −0.677378
\(83\) −917.445 −1.21328 −0.606642 − 0.794975i \(-0.707484\pi\)
−0.606642 + 0.794975i \(0.707484\pi\)
\(84\) −383.753 −0.498463
\(85\) −461.427 −0.588810
\(86\) 1364.85 1.71134
\(87\) −19.8144 −0.0244176
\(88\) −239.245 −0.289814
\(89\) −290.163 −0.345587 −0.172793 − 0.984958i \(-0.555279\pi\)
−0.172793 + 0.984958i \(0.555279\pi\)
\(90\) 197.953 0.231845
\(91\) 30.2873 0.0348898
\(92\) −261.067 −0.295850
\(93\) −29.4502 −0.0328371
\(94\) 107.362 0.117804
\(95\) 628.006 0.678233
\(96\) 696.430 0.740407
\(97\) −717.714 −0.751266 −0.375633 − 0.926768i \(-0.622575\pi\)
−0.375633 + 0.926768i \(0.622575\pi\)
\(98\) 950.164 0.979398
\(99\) 146.081 0.148300
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.g.1.1 ✓ 4
3.2 odd 2 1035.4.a.i.1.4 4
5.4 even 2 1725.4.a.q.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
345.4.a.g.1.1 ✓ 4 1.1 even 1 trivial
1035.4.a.i.1.4 4 3.2 odd 2
1725.4.a.q.1.4 4 5.4 even 2