Properties

Label 95.4.a.c.1.1
Level $95$
Weight $4$
Character 95.1
Self dual yes
Analytic conductor $5.605$
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] = [95,4,Mod(1,95)] 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("95.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,3,7] 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 95.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +7.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +21.0000 q^{6} +11.0000 q^{7} -21.0000 q^{8} +22.0000 q^{9} +15.0000 q^{10} -36.0000 q^{11} +7.00000 q^{12} +65.0000 q^{13} +33.0000 q^{14} +35.0000 q^{15} -71.0000 q^{16} -87.0000 q^{17} +66.0000 q^{18} +19.0000 q^{19} +5.00000 q^{20} +77.0000 q^{21} -108.000 q^{22} -129.000 q^{23} -147.000 q^{24} +25.0000 q^{25} +195.000 q^{26} -35.0000 q^{27} +11.0000 q^{28} +231.000 q^{29} +105.000 q^{30} +110.000 q^{31} -45.0000 q^{32} -252.000 q^{33} -261.000 q^{34} +55.0000 q^{35} +22.0000 q^{36} -142.000 q^{37} +57.0000 q^{38} +455.000 q^{39} -105.000 q^{40} -330.000 q^{41} +231.000 q^{42} +74.0000 q^{43} -36.0000 q^{44} +110.000 q^{45} -387.000 q^{46} -336.000 q^{47} -497.000 q^{48} -222.000 q^{49} +75.0000 q^{50} -609.000 q^{51} +65.0000 q^{52} +501.000 q^{53} -105.000 q^{54} -180.000 q^{55} -231.000 q^{56} +133.000 q^{57} +693.000 q^{58} +633.000 q^{59} +35.0000 q^{60} -88.0000 q^{61} +330.000 q^{62} +242.000 q^{63} +433.000 q^{64} +325.000 q^{65} -756.000 q^{66} +119.000 q^{67} -87.0000 q^{68} -903.000 q^{69} +165.000 q^{70} -204.000 q^{71} -462.000 q^{72} +407.000 q^{73} -426.000 q^{74} +175.000 q^{75} +19.0000 q^{76} -396.000 q^{77} +1365.00 q^{78} +1262.00 q^{79} -355.000 q^{80} -839.000 q^{81} -990.000 q^{82} +270.000 q^{83} +77.0000 q^{84} -435.000 q^{85} +222.000 q^{86} +1617.00 q^{87} +756.000 q^{88} -30.0000 q^{89} +330.000 q^{90} +715.000 q^{91} -129.000 q^{92} +770.000 q^{93} -1008.00 q^{94} +95.0000 q^{95} -315.000 q^{96} +1406.00 q^{97} -666.000 q^{98} -792.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 − 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 7.00000 1.34715 0.673575 − 0.739119i \(-0.264758\pi\)
0.673575 + 0.739119i \(0.264758\pi\)
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 21.0000 1.42887
\(7\) 11.0000 0.593944 0.296972 − 0.954886i \(-0.404023\pi\)
0.296972 + 0.954886i \(0.404023\pi\)
\(8\) −21.0000 −0.928078
\(9\) 22.0000 0.814815
\(10\) 15.0000 0.474342
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 7.00000 0.168394
\(13\) 65.0000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 33.0000 0.629973
\(15\) 35.0000 0.602464
\(16\) −71.0000 −1.10938
\(17\) −87.0000 −1.24121 −0.620606 − 0.784123i \(-0.713113\pi\)
−0.620606 + 0.784123i \(0.713113\pi\)
\(18\) 66.0000 0.864242
\(19\) 19.0000 0.229416
\(20\) 5.00000 0.0559017
\(21\) 77.0000 0.800132
\(22\) −108.000 −1.04662
\(23\) −129.000 −1.16949 −0.584747 − 0.811216i \(-0.698806\pi\)
−0.584747 + 0.811216i \(0.698806\pi\)
\(24\) −147.000 −1.25026
\(25\) 25.0000 0.200000
\(26\) 195.000 1.47087
\(27\) −35.0000 −0.249472
\(28\) 11.0000 0.0742430
\(29\) 231.000 1.47916 0.739580 − 0.673069i \(-0.235024\pi\)
0.739580 + 0.673069i \(0.235024\pi\)
\(30\) 105.000 0.639010
\(31\) 110.000 0.637309 0.318655 − 0.947871i \(-0.396769\pi\)
0.318655 + 0.947871i \(0.396769\pi\)
\(32\) −45.0000 −0.248592
\(33\) −252.000 −1.32932
\(34\) −261.000 −1.31650
\(35\) 55.0000 0.265620
\(36\) 22.0000 0.101852
\(37\) −142.000 −0.630937 −0.315468 − 0.948936i \(-0.602162\pi\)
−0.315468 + 0.948936i \(0.602162\pi\)
\(38\) 57.0000 0.243332
\(39\) 455.000 1.86816
\(40\) −105.000 −0.415049
\(41\) −330.000 −1.25701 −0.628504 − 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(42\) 231.000 0.848668
\(43\) 74.0000 0.262439 0.131220 − 0.991353i \(-0.458111\pi\)
0.131220 + 0.991353i \(0.458111\pi\)
\(44\) −36.0000 −0.123346
\(45\) 110.000 0.364396
\(46\) −387.000 −1.24044
\(47\) −336.000 −1.04278 −0.521390 − 0.853319i \(-0.674586\pi\)
−0.521390 + 0.853319i \(0.674586\pi\)
\(48\) −497.000 −1.49450
\(49\) −222.000 −0.647230
\(50\) 75.0000 0.212132
\(51\) −609.000 −1.67210
\(52\) 65.0000 0.173344
\(53\) 501.000 1.29845 0.649223 − 0.760598i \(-0.275094\pi\)
0.649223 + 0.760598i \(0.275094\pi\)
\(54\) −105.000 −0.264605
\(55\) −180.000 −0.441294
\(56\) −231.000 −0.551226
\(57\) 133.000 0.309058
\(58\) 693.000 1.56889
\(59\) 633.000 1.39677 0.698386 − 0.715721i \(-0.253902\pi\)
0.698386 + 0.715721i \(0.253902\pi\)
\(60\) 35.0000 0.0753080
\(61\) −88.0000 −0.184709 −0.0923545 − 0.995726i \(-0.529439\pi\)
−0.0923545 + 0.995726i \(0.529439\pi\)
\(62\) 330.000 0.675968
\(63\) 242.000 0.483955
\(64\) 433.000 0.845703
\(65\) 325.000 0.620174
\(66\) −756.000 −1.40996
\(67\) 119.000 0.216988 0.108494 − 0.994097i \(-0.465397\pi\)
0.108494 + 0.994097i \(0.465397\pi\)
\(68\) −87.0000 −0.155151
\(69\) −903.000 −1.57548
\(70\) 165.000 0.281732
\(71\) −204.000 −0.340991 −0.170495 − 0.985358i \(-0.554537\pi\)
−0.170495 + 0.985358i \(0.554537\pi\)
\(72\) −462.000 −0.756211
\(73\) 407.000 0.652544 0.326272 − 0.945276i \(-0.394207\pi\)
0.326272 + 0.945276i \(0.394207\pi\)
\(74\) −426.000 −0.669209
\(75\) 175.000 0.269430
\(76\) 19.0000 0.0286770
\(77\) −396.000 −0.586083
\(78\) 1365.00 1.98148
\(79\) 1262.00 1.79729 0.898646 − 0.438674i \(-0.144552\pi\)
0.898646 + 0.438674i \(0.144552\pi\)
\(80\) −355.000 −0.496128
\(81\) −839.000 −1.15089
\(82\) −990.000 −1.33326
\(83\) 270.000 0.357064 0.178532 − 0.983934i \(-0.442865\pi\)
0.178532 + 0.983934i \(0.442865\pi\)
\(84\) 77.0000 0.100017
\(85\) −435.000 −0.555087
\(86\) 222.000 0.278359
\(87\) 1617.00 1.99265
\(88\) 756.000 0.915794
\(89\) −30.0000 −0.0357303 −0.0178651 − 0.999840i \(-0.505687\pi\)
−0.0178651 + 0.999840i \(0.505687\pi\)
\(90\) 330.000 0.386501
\(91\) 715.000 0.823652
\(92\) −129.000 −0.146187
\(93\) 770.000 0.858551
\(94\) −1008.00 −1.10603
\(95\) 95.0000 0.102598
\(96\) −315.000 −0.334891
\(97\) 1406.00 1.47173 0.735864 − 0.677129i \(-0.236776\pi\)
0.735864 + 0.677129i \(0.236776\pi\)
\(98\) −666.000 −0.686491
\(99\) −792.000 −0.804030
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 95.4.a.c.1.1 ✓ 1
3.2 odd 2 855.4.a.c.1.1 1
4.3 odd 2 1520.4.a.a.1.1 1
5.2 odd 4 475.4.b.d.324.2 2
5.3 odd 4 475.4.b.d.324.1 2
5.4 even 2 475.4.a.b.1.1 1
19.18 odd 2 1805.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.a.c.1.1 ✓ 1 1.1 even 1 trivial
475.4.a.b.1.1 1 5.4 even 2
475.4.b.d.324.1 2 5.3 odd 4
475.4.b.d.324.2 2 5.2 odd 4
855.4.a.c.1.1 1 3.2 odd 2
1520.4.a.a.1.1 1 4.3 odd 2
1805.4.a.c.1.1 1 19.18 odd 2