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Properties

Label 115.4.a.c.1.2
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $1$
Dimension $2$
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] = [115,4,Mod(1,115)] 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("115.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-6] 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: \(6.78521965066\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.72015\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} +3.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} -11.1605 q^{6} -25.6008 q^{7} +21.0000 q^{8} -13.1605 q^{9} -15.0000 q^{10} +12.6008 q^{11} +3.72015 q^{12} +8.16046 q^{13} +76.8023 q^{14} +18.6008 q^{15} -71.0000 q^{16} -76.0411 q^{17} +39.4814 q^{18} -103.362 q^{19} +5.00000 q^{20} -95.2388 q^{21} -37.8023 q^{22} -23.0000 q^{23} +78.1232 q^{24} +25.0000 q^{25} -24.4814 q^{26} -149.403 q^{27} -25.6008 q^{28} -267.202 q^{29} -55.8023 q^{30} -63.7574 q^{31} +45.0000 q^{32} +46.8768 q^{33} +228.123 q^{34} -128.004 q^{35} -13.1605 q^{36} +112.164 q^{37} +310.086 q^{38} +30.3582 q^{39} +105.000 q^{40} -239.078 q^{41} +285.716 q^{42} +282.239 q^{43} +12.6008 q^{44} -65.8023 q^{45} +69.0000 q^{46} +577.291 q^{47} -264.131 q^{48} +312.399 q^{49} -75.0000 q^{50} -282.884 q^{51} +8.16046 q^{52} -2.31326 q^{53} +448.209 q^{54} +63.0038 q^{55} -537.616 q^{56} -384.522 q^{57} +801.605 q^{58} +272.888 q^{59} +18.6008 q^{60} +294.049 q^{61} +191.272 q^{62} +336.918 q^{63} +433.000 q^{64} +40.8023 q^{65} -140.630 q^{66} +426.732 q^{67} -76.0411 q^{68} -85.5635 q^{69} +384.011 q^{70} -1020.85 q^{71} -276.370 q^{72} -286.650 q^{73} -336.493 q^{74} +93.0038 q^{75} -103.362 q^{76} -322.589 q^{77} -91.0745 q^{78} -551.866 q^{79} -355.000 q^{80} -200.470 q^{81} +717.235 q^{82} +21.7021 q^{83} -95.2388 q^{84} -380.205 q^{85} -846.716 q^{86} -994.031 q^{87} +264.616 q^{88} -1049.63 q^{89} +197.407 q^{90} -208.914 q^{91} -23.0000 q^{92} -237.187 q^{93} -1731.87 q^{94} -516.810 q^{95} +167.407 q^{96} +1729.19 q^{97} -937.198 q^{98} -165.832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9} - 30 q^{10} - 27 q^{11} - 3 q^{12} - 15 q^{13} - 3 q^{14} - 15 q^{15} - 142 q^{16} - 79 q^{17} - 15 q^{18} - 71 q^{19}+ \cdots - 885 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 − 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 3.72015 0.715944 0.357972 − 0.933732i \(-0.383468\pi\)
0.357972 + 0.933732i \(0.383468\pi\)
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) −11.1605 −0.759373
\(7\) −25.6008 −1.38231 −0.691156 − 0.722706i \(-0.742898\pi\)
−0.691156 + 0.722706i \(0.742898\pi\)
\(8\) 21.0000 0.928078
\(9\) −13.1605 −0.487424
\(10\) −15.0000 −0.474342
\(11\) 12.6008 0.345389 0.172694 − 0.984975i \(-0.444753\pi\)
0.172694 + 0.984975i \(0.444753\pi\)
\(12\) 3.72015 0.0894930
\(13\) 8.16046 0.174100 0.0870502 − 0.996204i \(-0.472256\pi\)
0.0870502 + 0.996204i \(0.472256\pi\)
\(14\) 76.8023 1.46616
\(15\) 18.6008 0.320180
\(16\) −71.0000 −1.10938
\(17\) −76.0411 −1.08486 −0.542431 − 0.840100i \(-0.682496\pi\)
−0.542431 + 0.840100i \(0.682496\pi\)
\(18\) 39.4814 0.516992
\(19\) −103.362 −1.24805 −0.624023 − 0.781406i \(-0.714503\pi\)
−0.624023 + 0.781406i \(0.714503\pi\)
\(20\) 5.00000 0.0559017
\(21\) −95.2388 −0.989657
\(22\) −37.8023 −0.366340
\(23\) −23.0000 −0.208514
\(24\) 78.1232 0.664451
\(25\) 25.0000 0.200000
\(26\) −24.4814 −0.184661
\(27\) −149.403 −1.06491
\(28\) −25.6008 −0.172789
\(29\) −267.202 −1.71097 −0.855484 − 0.517829i \(-0.826740\pi\)
−0.855484 + 0.517829i \(0.826740\pi\)
\(30\) −55.8023 −0.339602
\(31\) −63.7574 −0.369392 −0.184696 − 0.982796i \(-0.559130\pi\)
−0.184696 + 0.982796i \(0.559130\pi\)
\(32\) 45.0000 0.248592
\(33\) 46.8768 0.247279
\(34\) 228.123 1.15067
\(35\) −128.004 −0.618188
\(36\) −13.1605 −0.0609281
\(37\) 112.164 0.498370 0.249185 − 0.968456i \(-0.419837\pi\)
0.249185 + 0.968456i \(0.419837\pi\)
\(38\) 310.086 1.32375
\(39\) 30.3582 0.124646
\(40\) 105.000 0.415049
\(41\) −239.078 −0.910677 −0.455339 − 0.890318i \(-0.650482\pi\)
−0.455339 + 0.890318i \(0.650482\pi\)
\(42\) 285.716 1.04969
\(43\) 282.239 1.00095 0.500477 − 0.865750i \(-0.333158\pi\)
0.500477 + 0.865750i \(0.333158\pi\)
\(44\) 12.6008 0.0431736
\(45\) −65.8023 −0.217983
\(46\) 69.0000 0.221163
\(47\) 577.291 1.79163 0.895815 − 0.444427i \(-0.146593\pi\)
0.895815 + 0.444427i \(0.146593\pi\)
\(48\) −264.131 −0.794250
\(49\) 312.399 0.910785
\(50\) −75.0000 −0.212132
\(51\) −282.884 −0.776701
\(52\) 8.16046 0.0217625
\(53\) −2.31326 −0.00599529 −0.00299764 − 0.999996i \(-0.500954\pi\)
−0.00299764 + 0.999996i \(0.500954\pi\)
\(54\) 448.209 1.12951
\(55\) 63.0038 0.154462
\(56\) −537.616 −1.28289
\(57\) −384.522 −0.893531
\(58\) 801.605 1.81476
\(59\) 272.888 0.602153 0.301077 − 0.953600i \(-0.402654\pi\)
0.301077 + 0.953600i \(0.402654\pi\)
\(60\) 18.6008 0.0400225
\(61\) 294.049 0.617198 0.308599 − 0.951192i \(-0.400140\pi\)
0.308599 + 0.951192i \(0.400140\pi\)
\(62\) 191.272 0.391800
\(63\) 336.918 0.673772
\(64\) 433.000 0.845703
\(65\) 40.8023 0.0778600
\(66\) −140.630 −0.262279
\(67\) 426.732 0.778113 0.389056 − 0.921214i \(-0.372801\pi\)
0.389056 + 0.921214i \(0.372801\pi\)
\(68\) −76.0411 −0.135608
\(69\) −85.5635 −0.149285
\(70\) 384.011 0.655688
\(71\) −1020.85 −1.70638 −0.853191 − 0.521598i \(-0.825336\pi\)
−0.853191 + 0.521598i \(0.825336\pi\)
\(72\) −276.370 −0.452368
\(73\) −286.650 −0.459586 −0.229793 − 0.973240i \(-0.573805\pi\)
−0.229793 + 0.973240i \(0.573805\pi\)
\(74\) −336.493 −0.528601
\(75\) 93.0038 0.143189
\(76\) −103.362 −0.156006
\(77\) −322.589 −0.477435
\(78\) −91.0745 −0.132207
\(79\) −551.866 −0.785947 −0.392974 − 0.919550i \(-0.628554\pi\)
−0.392974 + 0.919550i \(0.628554\pi\)
\(80\) −355.000 −0.496128
\(81\) −200.470 −0.274993
\(82\) 717.235 0.965919
\(83\) 21.7021 0.0287001 0.0143501 − 0.999897i \(-0.495432\pi\)
0.0143501 + 0.999897i \(0.495432\pi\)
\(84\) −95.2388 −0.123707
\(85\) −380.205 −0.485165
\(86\) −846.716 −1.06167
\(87\) −994.031 −1.22496
\(88\) 264.616 0.320547
\(89\) −1049.63 −1.25012 −0.625058 − 0.780578i \(-0.714925\pi\)
−0.625058 + 0.780578i \(0.714925\pi\)
\(90\) 197.407 0.231206
\(91\) −208.914 −0.240661
\(92\) −23.0000 −0.0260643
\(93\) −237.187 −0.264464
\(94\) −1731.87 −1.90031
\(95\) −516.810 −0.558143
\(96\) 167.407 0.177978
\(97\) 1729.19 1.81003 0.905014 − 0.425381i \(-0.139860\pi\)
0.905014 + 0.425381i \(0.139860\pi\)
\(98\) −937.198 −0.966033
\(99\) −165.832 −0.168351
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 115.4.a.c.1.2 ✓ 2
3.2 odd 2 1035.4.a.g.1.1 2
4.3 odd 2 1840.4.a.h.1.1 2
5.2 odd 4 575.4.b.f.24.1 4
5.3 odd 4 575.4.b.f.24.4 4
5.4 even 2 575.4.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.2 ✓ 2 1.1 even 1 trivial
575.4.a.h.1.1 2 5.4 even 2
575.4.b.f.24.1 4 5.2 odd 4
575.4.b.f.24.4 4 5.3 odd 4
1035.4.a.g.1.1 2 3.2 odd 2
1840.4.a.h.1.1 2 4.3 odd 2