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Properties

Label 115.4.a.c.1.1
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 1.1
Root \(5.72015\) 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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{2} -6.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} +20.1605 q^{6} +26.6008 q^{7} +21.0000 q^{8} +18.1605 q^{9} -15.0000 q^{10} -39.6008 q^{11} -6.72015 q^{12} -23.1605 q^{13} -79.8023 q^{14} -33.6008 q^{15} -71.0000 q^{16} -2.95893 q^{17} -54.4814 q^{18} +32.3620 q^{19} +5.00000 q^{20} -178.761 q^{21} +118.802 q^{22} -23.0000 q^{23} -141.123 q^{24} +25.0000 q^{25} +69.4814 q^{26} +59.4031 q^{27} +26.6008 q^{28} -162.798 q^{29} +100.802 q^{30} -241.243 q^{31} +45.0000 q^{32} +266.123 q^{33} +8.87678 q^{34} +133.004 q^{35} +18.1605 q^{36} -180.164 q^{37} -97.0860 q^{38} +155.642 q^{39} +105.000 q^{40} -353.922 q^{41} +536.284 q^{42} +365.761 q^{43} -39.6008 q^{44} +90.8023 q^{45} +69.0000 q^{46} -195.291 q^{47} +477.131 q^{48} +364.601 q^{49} -75.0000 q^{50} +19.8844 q^{51} -23.1605 q^{52} -461.687 q^{53} -178.209 q^{54} -198.004 q^{55} +558.616 q^{56} -217.478 q^{57} +488.395 q^{58} -290.888 q^{59} -33.6008 q^{60} -301.049 q^{61} +723.728 q^{62} +483.082 q^{63} +433.000 q^{64} -115.802 q^{65} -798.370 q^{66} -366.732 q^{67} -2.95893 q^{68} +154.564 q^{69} -399.011 q^{70} -8.14513 q^{71} +381.370 q^{72} +360.650 q^{73} +540.493 q^{74} -168.004 q^{75} +32.3620 q^{76} -1053.41 q^{77} -466.926 q^{78} +1243.87 q^{79} -355.000 q^{80} -889.530 q^{81} +1061.77 q^{82} -1481.70 q^{83} -178.761 q^{84} -14.7946 q^{85} -1097.28 q^{86} +1094.03 q^{87} -831.616 q^{88} +829.628 q^{89} -272.407 q^{90} -616.086 q^{91} -23.0000 q^{92} +1621.19 q^{93} +585.874 q^{94} +161.810 q^{95} -302.407 q^{96} -390.191 q^{97} -1093.80 q^{98} -719.168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9} - 30 q^{10} - 27 q^{11} - 3 q^{12} - 15 q^{13} - 3 q^{14} - 15 q^{15} - 142 q^{16} - 79 q^{17} - 15 q^{18} - 71 q^{19}+ \cdots - 885 q^{99}+O(q^{100}) \) 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 \(0.677932\pi\)
\(3\) −6.72015 −1.29329 −0.646647 − 0.762789i \(-0.723829\pi\)
−0.646647 + 0.762789i \(0.723829\pi\)
\(4\) 1.00000 0.125000
\(5\) 5.00000 0.447214
\(6\) 20.1605 1.37175
\(7\) 26.6008 1.43631 0.718153 − 0.695885i \(-0.244988\pi\)
0.718153 + 0.695885i \(0.244988\pi\)
\(8\) 21.0000 0.928078
\(9\) 18.1605 0.672610
\(10\) −15.0000 −0.474342
\(11\) −39.6008 −1.08546 −0.542731 − 0.839907i \(-0.682610\pi\)
−0.542731 + 0.839907i \(0.682610\pi\)
\(12\) −6.72015 −0.161662
\(13\) −23.1605 −0.494120 −0.247060 − 0.969000i \(-0.579464\pi\)
−0.247060 + 0.969000i \(0.579464\pi\)
\(14\) −79.8023 −1.52343
\(15\) −33.6008 −0.578379
\(16\) −71.0000 −1.10938
\(17\) −2.95893 −0.0422144 −0.0211072 − 0.999777i \(-0.506719\pi\)
−0.0211072 + 0.999777i \(0.506719\pi\)
\(18\) −54.4814 −0.713410
\(19\) 32.3620 0.390755 0.195378 − 0.980728i \(-0.437407\pi\)
0.195378 + 0.980728i \(0.437407\pi\)
\(20\) 5.00000 0.0559017
\(21\) −178.761 −1.85757
\(22\) 118.802 1.15131
\(23\) −23.0000 −0.208514
\(24\) −141.123 −1.20028
\(25\) 25.0000 0.200000
\(26\) 69.4814 0.524093
\(27\) 59.4031 0.423412
\(28\) 26.6008 0.179538
\(29\) −162.798 −1.04245 −0.521223 − 0.853421i \(-0.674524\pi\)
−0.521223 + 0.853421i \(0.674524\pi\)
\(30\) 100.802 0.613463
\(31\) −241.243 −1.39769 −0.698846 − 0.715272i \(-0.746303\pi\)
−0.698846 + 0.715272i \(0.746303\pi\)
\(32\) 45.0000 0.248592
\(33\) 266.123 1.40382
\(34\) 8.87678 0.0447752
\(35\) 133.004 0.642336
\(36\) 18.1605 0.0840762
\(37\) −180.164 −0.800509 −0.400254 − 0.916404i \(-0.631078\pi\)
−0.400254 + 0.916404i \(0.631078\pi\)
\(38\) −97.0860 −0.414459
\(39\) 155.642 0.639042
\(40\) 105.000 0.415049
\(41\) −353.922 −1.34813 −0.674064 − 0.738673i \(-0.735453\pi\)
−0.674064 + 0.738673i \(0.735453\pi\)
\(42\) 536.284 1.97025
\(43\) 365.761 1.29716 0.648582 − 0.761145i \(-0.275362\pi\)
0.648582 + 0.761145i \(0.275362\pi\)
\(44\) −39.6008 −0.135683
\(45\) 90.8023 0.300800
\(46\) 69.0000 0.221163
\(47\) −195.291 −0.606089 −0.303044 − 0.952976i \(-0.598003\pi\)
−0.303044 + 0.952976i \(0.598003\pi\)
\(48\) 477.131 1.43475
\(49\) 364.601 1.06298
\(50\) −75.0000 −0.212132
\(51\) 19.8844 0.0545957
\(52\) −23.1605 −0.0617650
\(53\) −461.687 −1.19656 −0.598279 − 0.801288i \(-0.704148\pi\)
−0.598279 + 0.801288i \(0.704148\pi\)
\(54\) −178.209 −0.449096
\(55\) −198.004 −0.485433
\(56\) 558.616 1.33300
\(57\) −217.478 −0.505361
\(58\) 488.395 1.10568
\(59\) −290.888 −0.641872 −0.320936 − 0.947101i \(-0.603997\pi\)
−0.320936 + 0.947101i \(0.603997\pi\)
\(60\) −33.6008 −0.0722973
\(61\) −301.049 −0.631891 −0.315945 − 0.948777i \(-0.602322\pi\)
−0.315945 + 0.948777i \(0.602322\pi\)
\(62\) 723.728 1.48248
\(63\) 483.082 0.966073
\(64\) 433.000 0.845703
\(65\) −115.802 −0.220977
\(66\) −798.370 −1.48898
\(67\) −366.732 −0.668707 −0.334354 − 0.942448i \(-0.608518\pi\)
−0.334354 + 0.942448i \(0.608518\pi\)
\(68\) −2.95893 −0.00527680
\(69\) 154.564 0.269670
\(70\) −399.011 −0.681300
\(71\) −8.14513 −0.0136148 −0.00680739 − 0.999977i \(-0.502167\pi\)
−0.00680739 + 0.999977i \(0.502167\pi\)
\(72\) 381.370 0.624234
\(73\) 360.650 0.578231 0.289115 − 0.957294i \(-0.406639\pi\)
0.289115 + 0.957294i \(0.406639\pi\)
\(74\) 540.493 0.849068
\(75\) −168.004 −0.258659
\(76\) 32.3620 0.0488444
\(77\) −1053.41 −1.55906
\(78\) −466.926 −0.677806
\(79\) 1243.87 1.77147 0.885734 − 0.464194i \(-0.153656\pi\)
0.885734 + 0.464194i \(0.153656\pi\)
\(80\) −355.000 −0.496128
\(81\) −889.530 −1.22021
\(82\) 1061.77 1.42991
\(83\) −1481.70 −1.95949 −0.979747 − 0.200241i \(-0.935827\pi\)
−0.979747 + 0.200241i \(0.935827\pi\)
\(84\) −178.761 −0.232196
\(85\) −14.7946 −0.0188789
\(86\) −1097.28 −1.37585
\(87\) 1094.03 1.34819
\(88\) −831.616 −1.00739
\(89\) 829.628 0.988094 0.494047 − 0.869435i \(-0.335517\pi\)
0.494047 + 0.869435i \(0.335517\pi\)
\(90\) −272.407 −0.319047
\(91\) −616.086 −0.709707
\(92\) −23.0000 −0.0260643
\(93\) 1621.19 1.80763
\(94\) 585.874 0.642854
\(95\) 161.810 0.174751
\(96\) −302.407 −0.321503
\(97\) −390.191 −0.408432 −0.204216 − 0.978926i \(-0.565464\pi\)
−0.204216 + 0.978926i \(0.565464\pi\)
\(98\) −1093.80 −1.12746
\(99\) −719.168 −0.730092
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.1.1 ✓ 2
3.2 odd 2 1035.4.a.g.1.2 2
4.3 odd 2 1840.4.a.h.1.2 2
5.2 odd 4 575.4.b.f.24.2 4
5.3 odd 4 575.4.b.f.24.3 4
5.4 even 2 575.4.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.1 ✓ 2 1.1 even 1 trivial
575.4.a.h.1.2 2 5.4 even 2
575.4.b.f.24.2 4 5.2 odd 4
575.4.b.f.24.3 4 5.3 odd 4
1035.4.a.g.1.2 2 3.2 odd 2
1840.4.a.h.1.2 2 4.3 odd 2