Properties

Label 195.4.a.f.1.3
Level $195$
Weight $4$
Character 195.1
Self dual yes
Analytic conductor $11.505$
Analytic rank $1$
Dimension $3$
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] = [195,4,Mod(1,195)] 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("195.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(195, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 195.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-2,-9] 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: \(11.5053724511\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.85577\) of defining polynomial
Character \(\chi\) \(=\) 195.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.15544 q^{2} -3.00000 q^{3} +1.95678 q^{4} +5.00000 q^{5} -9.46631 q^{6} -19.9360 q^{7} -19.0690 q^{8} +9.00000 q^{9} +15.7772 q^{10} -1.50953 q^{11} -5.87034 q^{12} -13.0000 q^{13} -62.9067 q^{14} -15.0000 q^{15} -75.8253 q^{16} -116.862 q^{17} +28.3989 q^{18} -34.6342 q^{19} +9.78390 q^{20} +59.8079 q^{21} -4.76323 q^{22} -58.3298 q^{23} +57.2070 q^{24} +25.0000 q^{25} -41.0207 q^{26} -27.0000 q^{27} -39.0103 q^{28} +31.9905 q^{29} -47.3315 q^{30} +303.785 q^{31} -86.7098 q^{32} +4.52859 q^{33} -368.750 q^{34} -99.6799 q^{35} +17.6110 q^{36} -389.815 q^{37} -109.286 q^{38} +39.0000 q^{39} -95.3450 q^{40} -215.507 q^{41} +188.720 q^{42} +256.886 q^{43} -2.95382 q^{44} +45.0000 q^{45} -184.056 q^{46} +396.849 q^{47} +227.476 q^{48} +54.4432 q^{49} +78.8859 q^{50} +350.586 q^{51} -25.4381 q^{52} +53.3907 q^{53} -85.1968 q^{54} -7.54765 q^{55} +380.159 q^{56} +103.903 q^{57} +100.944 q^{58} -668.048 q^{59} -29.3517 q^{60} +396.782 q^{61} +958.574 q^{62} -179.424 q^{63} +332.995 q^{64} -65.0000 q^{65} +14.2897 q^{66} +70.7923 q^{67} -228.673 q^{68} +174.989 q^{69} -314.534 q^{70} +1019.60 q^{71} -171.621 q^{72} -25.1060 q^{73} -1230.04 q^{74} -75.0000 q^{75} -67.7715 q^{76} +30.0940 q^{77} +123.062 q^{78} -577.460 q^{79} -379.126 q^{80} +81.0000 q^{81} -680.020 q^{82} +756.471 q^{83} +117.031 q^{84} -584.309 q^{85} +810.587 q^{86} -95.9715 q^{87} +28.7852 q^{88} -823.421 q^{89} +141.995 q^{90} +259.168 q^{91} -114.139 q^{92} -911.354 q^{93} +1252.23 q^{94} -173.171 q^{95} +260.129 q^{96} -1119.23 q^{97} +171.792 q^{98} -13.5858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 9 q^{3} + 10 q^{4} + 15 q^{5} + 6 q^{6} - 30 q^{7} - 54 q^{8} + 27 q^{9} - 10 q^{10} + 34 q^{11} - 30 q^{12} - 39 q^{13} + 64 q^{14} - 45 q^{15} + 50 q^{16} - 174 q^{17} - 18 q^{18} - 86 q^{19}+ \cdots + 306 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.15544 1.11562 0.557808 − 0.829970i \(-0.311643\pi\)
0.557808 + 0.829970i \(0.311643\pi\)
\(3\) −3.00000 −0.577350
\(4\) 1.95678 0.244598
\(5\) 5.00000 0.447214
\(6\) −9.46631 −0.644101
\(7\) −19.9360 −1.07644 −0.538221 − 0.842804i \(-0.680903\pi\)
−0.538221 + 0.842804i \(0.680903\pi\)
\(8\) −19.0690 −0.842739
\(9\) 9.00000 0.333333
\(10\) 15.7772 0.498918
\(11\) −1.50953 −0.0413764 −0.0206882 − 0.999786i \(-0.506586\pi\)
−0.0206882 + 0.999786i \(0.506586\pi\)
\(12\) −5.87034 −0.141218
\(13\) −13.0000 −0.277350
\(14\) −62.9067 −1.20089
\(15\) −15.0000 −0.258199
\(16\) −75.8253 −1.18477
\(17\) −116.862 −1.66725 −0.833623 − 0.552334i \(-0.813737\pi\)
−0.833623 + 0.552334i \(0.813737\pi\)
\(18\) 28.3989 0.371872
\(19\) −34.6342 −0.418191 −0.209096 − 0.977895i \(-0.567052\pi\)
−0.209096 + 0.977895i \(0.567052\pi\)
\(20\) 9.78390 0.109387
\(21\) 59.8079 0.621484
\(22\) −4.76323 −0.0461601
\(23\) −58.3298 −0.528809 −0.264404 − 0.964412i \(-0.585175\pi\)
−0.264404 + 0.964412i \(0.585175\pi\)
\(24\) 57.2070 0.486555
\(25\) 25.0000 0.200000
\(26\) −41.0207 −0.309416
\(27\) −27.0000 −0.192450
\(28\) −39.0103 −0.263295
\(29\) 31.9905 0.204844 0.102422 − 0.994741i \(-0.467341\pi\)
0.102422 + 0.994741i \(0.467341\pi\)
\(30\) −47.3315 −0.288051
\(31\) 303.785 1.76004 0.880022 − 0.474933i \(-0.157528\pi\)
0.880022 + 0.474933i \(0.157528\pi\)
\(32\) −86.7098 −0.479009
\(33\) 4.52859 0.0238887
\(34\) −368.750 −1.86000
\(35\) −99.6799 −0.481399
\(36\) 17.6110 0.0815325
\(37\) −389.815 −1.73203 −0.866016 − 0.500016i \(-0.833328\pi\)
−0.866016 + 0.500016i \(0.833328\pi\)
\(38\) −109.286 −0.466541
\(39\) 39.0000 0.160128
\(40\) −95.3450 −0.376884
\(41\) −215.507 −0.820893 −0.410447 − 0.911885i \(-0.634627\pi\)
−0.410447 + 0.911885i \(0.634627\pi\)
\(42\) 188.720 0.693337
\(43\) 256.886 0.911040 0.455520 − 0.890226i \(-0.349453\pi\)
0.455520 + 0.890226i \(0.349453\pi\)
\(44\) −2.95382 −0.0101206
\(45\) 45.0000 0.149071
\(46\) −184.056 −0.589947
\(47\) 396.849 1.23162 0.615812 − 0.787893i \(-0.288828\pi\)
0.615812 + 0.787893i \(0.288828\pi\)
\(48\) 227.476 0.684027
\(49\) 54.4432 0.158727
\(50\) 78.8859 0.223123
\(51\) 350.586 0.962584
\(52\) −25.4381 −0.0678391
\(53\) 53.3907 0.138373 0.0691866 − 0.997604i \(-0.477960\pi\)
0.0691866 + 0.997604i \(0.477960\pi\)
\(54\) −85.1968 −0.214700
\(55\) −7.54765 −0.0185041
\(56\) 380.159 0.907159
\(57\) 103.903 0.241443
\(58\) 100.944 0.228528
\(59\) −668.048 −1.47411 −0.737055 − 0.675833i \(-0.763784\pi\)
−0.737055 + 0.675833i \(0.763784\pi\)
\(60\) −29.3517 −0.0631548
\(61\) 396.782 0.832832 0.416416 − 0.909174i \(-0.363286\pi\)
0.416416 + 0.909174i \(0.363286\pi\)
\(62\) 958.574 1.96353
\(63\) −179.424 −0.358814
\(64\) 332.995 0.650380
\(65\) −65.0000 −0.124035
\(66\) 14.2897 0.0266506
\(67\) 70.7923 0.129084 0.0645422 − 0.997915i \(-0.479441\pi\)
0.0645422 + 0.997915i \(0.479441\pi\)
\(68\) −228.673 −0.407804
\(69\) 174.989 0.305308
\(70\) −314.534 −0.537056
\(71\) 1019.60 1.70428 0.852139 − 0.523315i \(-0.175305\pi\)
0.852139 + 0.523315i \(0.175305\pi\)
\(72\) −171.621 −0.280913
\(73\) −25.1060 −0.0402526 −0.0201263 − 0.999797i \(-0.506407\pi\)
−0.0201263 + 0.999797i \(0.506407\pi\)
\(74\) −1230.04 −1.93228
\(75\) −75.0000 −0.115470
\(76\) −67.7715 −0.102289
\(77\) 30.0940 0.0445393
\(78\) 123.062 0.178641
\(79\) −577.460 −0.822397 −0.411198 − 0.911546i \(-0.634890\pi\)
−0.411198 + 0.911546i \(0.634890\pi\)
\(80\) −379.126 −0.529845
\(81\) 81.0000 0.111111
\(82\) −680.020 −0.915801
\(83\) 756.471 1.00040 0.500202 − 0.865909i \(-0.333259\pi\)
0.500202 + 0.865909i \(0.333259\pi\)
\(84\) 117.031 0.152013
\(85\) −584.309 −0.745615
\(86\) 810.587 1.01637
\(87\) −95.9715 −0.118267
\(88\) 28.7852 0.0348695
\(89\) −823.421 −0.980701 −0.490351 − 0.871525i \(-0.663131\pi\)
−0.490351 + 0.871525i \(0.663131\pi\)
\(90\) 141.995 0.166306
\(91\) 259.168 0.298551
\(92\) −114.139 −0.129345
\(93\) −911.354 −1.01616
\(94\) 1252.23 1.37402
\(95\) −173.171 −0.187021
\(96\) 260.129 0.276556
\(97\) −1119.23 −1.17155 −0.585777 − 0.810473i \(-0.699210\pi\)
−0.585777 + 0.810473i \(0.699210\pi\)
\(98\) 171.792 0.177078
\(99\) −13.5858 −0.0137921
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 195.4.a.f.1.3 ✓ 3
3.2 odd 2 585.4.a.k.1.1 3
5.4 even 2 975.4.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.4.a.f.1.3 ✓ 3 1.1 even 1 trivial
585.4.a.k.1.1 3 3.2 odd 2
975.4.a.n.1.1 3 5.4 even 2