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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(1,575)] 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("575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,6,3] 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: \(33.9260982533\)
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: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.72015\) of defining polynomial
Character \(\chi\) \(=\) 575.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} +6.72015 q^{3} +1.00000 q^{4} +20.1605 q^{6} -26.6008 q^{7} -21.0000 q^{8} +18.1605 q^{9} -39.6008 q^{11} +6.72015 q^{12} +23.1605 q^{13} -79.8023 q^{14} -71.0000 q^{16} +2.95893 q^{17} +54.4814 q^{18} +32.3620 q^{19} -178.761 q^{21} -118.802 q^{22} +23.0000 q^{23} -141.123 q^{24} +69.4814 q^{26} -59.4031 q^{27} -26.6008 q^{28} -162.798 q^{29} -241.243 q^{31} -45.0000 q^{32} -266.123 q^{33} +8.87678 q^{34} +18.1605 q^{36} +180.164 q^{37} +97.0860 q^{38} +155.642 q^{39} -353.922 q^{41} -536.284 q^{42} -365.761 q^{43} -39.6008 q^{44} +69.0000 q^{46} +195.291 q^{47} -477.131 q^{48} +364.601 q^{49} +19.8844 q^{51} +23.1605 q^{52} +461.687 q^{53} -178.209 q^{54} +558.616 q^{56} +217.478 q^{57} -488.395 q^{58} -290.888 q^{59} -301.049 q^{61} -723.728 q^{62} -483.082 q^{63} +433.000 q^{64} -798.370 q^{66} +366.732 q^{67} +2.95893 q^{68} +154.564 q^{69} -8.14513 q^{71} -381.370 q^{72} -360.650 q^{73} +540.493 q^{74} +32.3620 q^{76} +1053.41 q^{77} +466.926 q^{78} +1243.87 q^{79} -889.530 q^{81} -1061.77 q^{82} +1481.70 q^{83} -178.761 q^{84} -1097.28 q^{86} -1094.03 q^{87} +831.616 q^{88} +829.628 q^{89} -616.086 q^{91} +23.0000 q^{92} -1621.19 q^{93} +585.874 q^{94} -302.407 q^{96} +390.191 q^{97} +1093.80 q^{98} -719.168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 3 q^{3} + 2 q^{4} + 9 q^{6} - q^{7} - 42 q^{8} + 5 q^{9} - 27 q^{11} + 3 q^{12} + 15 q^{13} - 3 q^{14} - 142 q^{16} + 79 q^{17} + 15 q^{18} - 71 q^{19} - 274 q^{21} - 81 q^{22} + 46 q^{23}+ \cdots - 885 q^{99}+O(q^{100}) \) 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 \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 6.72015 1.29329 0.646647 − 0.762789i \(-0.276171\pi\)
0.646647 + 0.762789i \(0.276171\pi\)
\(4\) 1.00000 0.125000
\(5\) 0 0
\(6\) 20.1605 1.37175
\(7\) −26.6008 −1.43631 −0.718153 − 0.695885i \(-0.755012\pi\)
−0.718153 + 0.695885i \(0.755012\pi\)
\(8\) −21.0000 −0.928078
\(9\) 18.1605 0.672610
\(10\) 0 0
\(11\) −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 \(-0.420536\pi\)
0.247060 + 0.969000i \(0.420536\pi\)
\(14\) −79.8023 −1.52343
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 2.95893 0.0422144 0.0211072 − 0.999777i \(-0.493281\pi\)
0.0211072 + 0.999777i \(0.493281\pi\)
\(18\) 54.4814 0.713410
\(19\) 32.3620 0.390755 0.195378 − 0.980728i \(-0.437407\pi\)
0.195378 + 0.980728i \(0.437407\pi\)
\(20\) 0 0
\(21\) −178.761 −1.85757
\(22\) −118.802 −1.15131
\(23\) 23.0000 0.208514
\(24\) −141.123 −1.20028
\(25\) 0 0
\(26\) 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 \(0.674524\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 18.1605 0.0840762
\(37\) 180.164 0.800509 0.400254 − 0.916404i \(-0.368922\pi\)
0.400254 + 0.916404i \(0.368922\pi\)
\(38\) 97.0860 0.414459
\(39\) 155.642 0.639042
\(40\) 0 0
\(41\) −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 \(-0.724638\pi\)
−0.648582 + 0.761145i \(0.724638\pi\)
\(44\) −39.6008 −0.135683
\(45\) 0 0
\(46\) 69.0000 0.221163
\(47\) 195.291 0.606089 0.303044 − 0.952976i \(-0.401997\pi\)
0.303044 + 0.952976i \(0.401997\pi\)
\(48\) −477.131 −1.43475
\(49\) 364.601 1.06298
\(50\) 0 0
\(51\) 19.8844 0.0545957
\(52\) 23.1605 0.0617650
\(53\) 461.687 1.19656 0.598279 − 0.801288i \(-0.295852\pi\)
0.598279 + 0.801288i \(0.295852\pi\)
\(54\) −178.209 −0.449096
\(55\) 0 0
\(56\) 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 \(0.603997\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) −798.370 −1.48898
\(67\) 366.732 0.668707 0.334354 − 0.942448i \(-0.391482\pi\)
0.334354 + 0.942448i \(0.391482\pi\)
\(68\) 2.95893 0.00527680
\(69\) 154.564 0.269670
\(70\) 0 0
\(71\) −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 \(-0.593361\pi\)
−0.289115 + 0.957294i \(0.593361\pi\)
\(74\) 540.493 0.849068
\(75\) 0 0
\(76\) 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 \(0.153656\pi\)
\(80\) 0 0
\(81\) −889.530 −1.22021
\(82\) −1061.77 −1.42991
\(83\) 1481.70 1.95949 0.979747 − 0.200241i \(-0.0641727\pi\)
0.979747 + 0.200241i \(0.0641727\pi\)
\(84\) −178.761 −0.232196
\(85\) 0 0
\(86\) −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 \(0.335517\pi\)
\(90\) 0 0
\(91\) −616.086 −0.709707
\(92\) 23.0000 0.0260643
\(93\) −1621.19 −1.80763
\(94\) 585.874 0.642854
\(95\) 0 0
\(96\) −302.407 −0.321503
\(97\) 390.191 0.408432 0.204216 − 0.978926i \(-0.434536\pi\)
0.204216 + 0.978926i \(0.434536\pi\)
\(98\) 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 575.4.a.h.1.2 2
5.2 odd 4 575.4.b.f.24.3 4
5.3 odd 4 575.4.b.f.24.2 4
5.4 even 2 115.4.a.c.1.1 ✓ 2
15.14 odd 2 1035.4.a.g.1.2 2
20.19 odd 2 1840.4.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.1 ✓ 2 5.4 even 2
575.4.a.h.1.2 2 1.1 even 1 trivial
575.4.b.f.24.2 4 5.3 odd 4
575.4.b.f.24.3 4 5.2 odd 4
1035.4.a.g.1.2 2 15.14 odd 2
1840.4.a.h.1.2 2 20.19 odd 2