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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,4,Mod(24,575)] 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("575.24"); 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([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{109})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 55x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.3
Root \(5.72015i\) of defining polynomial
Character \(\chi\) \(=\) 575.24
Dual form 575.4.b.f.24.2

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{2} -6.72015i q^{3} -1.00000 q^{4} +20.1605 q^{6} -26.6008i q^{7} +21.0000i q^{8} -18.1605 q^{9} -39.6008 q^{11} +6.72015i q^{12} -23.1605i q^{13} +79.8023 q^{14} -71.0000 q^{16} +2.95893i q^{17} -54.4814i q^{18} -32.3620 q^{19} -178.761 q^{21} -118.802i q^{22} -23.0000i q^{23} +141.123 q^{24} +69.4814 q^{26} -59.4031i q^{27} +26.6008i q^{28} +162.798 q^{29} -241.243 q^{31} -45.0000i q^{32} +266.123i q^{33} -8.87678 q^{34} +18.1605 q^{36} +180.164i q^{37} -97.0860i q^{38} -155.642 q^{39} -353.922 q^{41} -536.284i q^{42} +365.761i q^{43} +39.6008 q^{44} +69.0000 q^{46} +195.291i q^{47} +477.131i q^{48} -364.601 q^{49} +19.8844 q^{51} +23.1605i q^{52} -461.687i q^{53} +178.209 q^{54} +558.616 q^{56} +217.478i q^{57} +488.395i q^{58} +290.888 q^{59} -301.049 q^{61} -723.728i q^{62} +483.082i q^{63} -433.000 q^{64} -798.370 q^{66} +366.732i q^{67} -2.95893i q^{68} -154.564 q^{69} -8.14513 q^{71} -381.370i q^{72} +360.650i q^{73} -540.493 q^{74} +32.3620 q^{76} +1053.41i q^{77} -466.926i q^{78} -1243.87 q^{79} -889.530 q^{81} -1061.77i q^{82} -1481.70i q^{83} +178.761 q^{84} -1097.28 q^{86} -1094.03i q^{87} -831.616i q^{88} -829.628 q^{89} -616.086 q^{91} +23.0000i q^{92} +1621.19i q^{93} -585.874 q^{94} -302.407 q^{96} +390.191i q^{97} -1093.80i q^{98} +719.168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 18 q^{6} - 10 q^{9} - 54 q^{11} + 6 q^{14} - 284 q^{16} + 142 q^{19} - 548 q^{21} + 126 q^{24} + 90 q^{26} + 860 q^{29} - 610 q^{31} - 474 q^{34} + 10 q^{36} - 372 q^{39} - 1186 q^{41} + 54 q^{44}+ \cdots + 1770 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000i 1.06066i 0.847791 + 0.530330i \(0.177932\pi\)
−0.847791 + 0.530330i \(0.822068\pi\)
\(3\) − 6.72015i − 1.29329i −0.762789 − 0.646647i \(-0.776171\pi\)
0.762789 − 0.646647i \(-0.223829\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 20.1605 1.37175
\(7\) − 26.6008i − 1.43631i −0.695885 − 0.718153i \(-0.744988\pi\)
0.695885 − 0.718153i \(-0.255012\pi\)
\(8\) 21.0000i 0.928078i
\(9\) −18.1605 −0.672610
\(10\) 0 0
\(11\) −39.6008 −1.08546 −0.542731 − 0.839907i \(-0.682610\pi\)
−0.542731 + 0.839907i \(0.682610\pi\)
\(12\) 6.72015i 0.161662i
\(13\) − 23.1605i − 0.494120i −0.969000 − 0.247060i \(-0.920536\pi\)
0.969000 − 0.247060i \(-0.0794644\pi\)
\(14\) 79.8023 1.52343
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 2.95893i 0.0422144i 0.999777 + 0.0211072i \(0.00671913\pi\)
−0.999777 + 0.0211072i \(0.993281\pi\)
\(18\) − 54.4814i − 0.713410i
\(19\) −32.3620 −0.390755 −0.195378 − 0.980728i \(-0.562593\pi\)
−0.195378 + 0.980728i \(0.562593\pi\)
\(20\) 0 0
\(21\) −178.761 −1.85757
\(22\) − 118.802i − 1.15131i
\(23\) − 23.0000i − 0.208514i
\(24\) 141.123 1.20028
\(25\) 0 0
\(26\) 69.4814 0.524093
\(27\) − 59.4031i − 0.423412i
\(28\) 26.6008i 0.179538i
\(29\) 162.798 1.04245 0.521223 − 0.853421i \(-0.325476\pi\)
0.521223 + 0.853421i \(0.325476\pi\)
\(30\) 0 0
\(31\) −241.243 −1.39769 −0.698846 − 0.715272i \(-0.746303\pi\)
−0.698846 + 0.715272i \(0.746303\pi\)
\(32\) − 45.0000i − 0.248592i
\(33\) 266.123i 1.40382i
\(34\) −8.87678 −0.0447752
\(35\) 0 0
\(36\) 18.1605 0.0840762
\(37\) 180.164i 0.800509i 0.916404 + 0.400254i \(0.131078\pi\)
−0.916404 + 0.400254i \(0.868922\pi\)
\(38\) − 97.0860i − 0.414459i
\(39\) −155.642 −0.639042
\(40\) 0 0
\(41\) −353.922 −1.34813 −0.674064 − 0.738673i \(-0.735453\pi\)
−0.674064 + 0.738673i \(0.735453\pi\)
\(42\) − 536.284i − 1.97025i
\(43\) 365.761i 1.29716i 0.761145 + 0.648582i \(0.224638\pi\)
−0.761145 + 0.648582i \(0.775362\pi\)
\(44\) 39.6008 0.135683
\(45\) 0 0
\(46\) 69.0000 0.221163
\(47\) 195.291i 0.606089i 0.952976 + 0.303044i \(0.0980030\pi\)
−0.952976 + 0.303044i \(0.901997\pi\)
\(48\) 477.131i 1.43475i
\(49\) −364.601 −1.06298
\(50\) 0 0
\(51\) 19.8844 0.0545957
\(52\) 23.1605i 0.0617650i
\(53\) − 461.687i − 1.19656i −0.801288 − 0.598279i \(-0.795852\pi\)
0.801288 − 0.598279i \(-0.204148\pi\)
\(54\) 178.209 0.449096
\(55\) 0 0
\(56\) 558.616 1.33300
\(57\) 217.478i 0.505361i
\(58\) 488.395i 1.10568i
\(59\) 290.888 0.641872 0.320936 − 0.947101i \(-0.396003\pi\)
0.320936 + 0.947101i \(0.396003\pi\)
\(60\) 0 0
\(61\) −301.049 −0.631891 −0.315945 − 0.948777i \(-0.602322\pi\)
−0.315945 + 0.948777i \(0.602322\pi\)
\(62\) − 723.728i − 1.48248i
\(63\) 483.082i 0.966073i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) −798.370 −1.48898
\(67\) 366.732i 0.668707i 0.942448 + 0.334354i \(0.108518\pi\)
−0.942448 + 0.334354i \(0.891482\pi\)
\(68\) − 2.95893i − 0.00527680i
\(69\) −154.564 −0.269670
\(70\) 0 0
\(71\) −8.14513 −0.0136148 −0.00680739 − 0.999977i \(-0.502167\pi\)
−0.00680739 + 0.999977i \(0.502167\pi\)
\(72\) − 381.370i − 0.624234i
\(73\) 360.650i 0.578231i 0.957294 + 0.289115i \(0.0933611\pi\)
−0.957294 + 0.289115i \(0.906639\pi\)
\(74\) −540.493 −0.849068
\(75\) 0 0
\(76\) 32.3620 0.0488444
\(77\) 1053.41i 1.55906i
\(78\) − 466.926i − 0.677806i
\(79\) −1243.87 −1.77147 −0.885734 − 0.464194i \(-0.846344\pi\)
−0.885734 + 0.464194i \(0.846344\pi\)
\(80\) 0 0
\(81\) −889.530 −1.22021
\(82\) − 1061.77i − 1.42991i
\(83\) − 1481.70i − 1.95949i −0.200241 − 0.979747i \(-0.564173\pi\)
0.200241 − 0.979747i \(-0.435827\pi\)
\(84\) 178.761 0.232196
\(85\) 0 0
\(86\) −1097.28 −1.37585
\(87\) − 1094.03i − 1.34819i
\(88\) − 831.616i − 1.00739i
\(89\) −829.628 −0.988094 −0.494047 − 0.869435i \(-0.664483\pi\)
−0.494047 + 0.869435i \(0.664483\pi\)
\(90\) 0 0
\(91\) −616.086 −0.709707
\(92\) 23.0000i 0.0260643i
\(93\) 1621.19i 1.80763i
\(94\) −585.874 −0.642854
\(95\) 0 0
\(96\) −302.407 −0.321503
\(97\) 390.191i 0.408432i 0.978926 + 0.204216i \(0.0654645\pi\)
−0.978926 + 0.204216i \(0.934536\pi\)
\(98\) − 1093.80i − 1.12746i
\(99\) 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.b.f.24.3 4
5.2 odd 4 115.4.a.c.1.1 ✓ 2
5.3 odd 4 575.4.a.h.1.2 2
5.4 even 2 inner 575.4.b.f.24.2 4
15.2 even 4 1035.4.a.g.1.2 2
20.7 even 4 1840.4.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.c.1.1 ✓ 2 5.2 odd 4
575.4.a.h.1.2 2 5.3 odd 4
575.4.b.f.24.2 4 5.4 even 2 inner
575.4.b.f.24.3 4 1.1 even 1 trivial
1035.4.a.g.1.2 2 15.2 even 4
1840.4.a.h.1.2 2 20.7 even 4