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Properties

Label 425.4.a.e.1.1
Level $425$
Weight $4$
Character 425.1
Self dual yes
Analytic conductor $25.076$
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] = [425,4,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,2] 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: \(25.0758117524\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 425.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.267949 q^{2} -0.732051 q^{3} -7.92820 q^{4} -0.196152 q^{6} +14.5885 q^{7} -4.26795 q^{8} -26.4641 q^{9} +3.51666 q^{11} +5.80385 q^{12} +88.4974 q^{13} +3.90897 q^{14} +62.2820 q^{16} +17.0000 q^{17} -7.09103 q^{18} -79.6077 q^{19} -10.6795 q^{21} +0.942286 q^{22} -153.937 q^{23} +3.12436 q^{24} +23.7128 q^{26} +39.1384 q^{27} -115.660 q^{28} -28.3257 q^{29} -209.219 q^{31} +50.8320 q^{32} -2.57437 q^{33} +4.55514 q^{34} +209.813 q^{36} -359.597 q^{37} -21.3308 q^{38} -64.7846 q^{39} +417.951 q^{41} -2.86156 q^{42} -243.520 q^{43} -27.8808 q^{44} -41.2473 q^{46} +160.144 q^{47} -45.5936 q^{48} -130.177 q^{49} -12.4449 q^{51} -701.626 q^{52} -28.8513 q^{53} +10.4871 q^{54} -62.2628 q^{56} +58.2769 q^{57} -7.58984 q^{58} -832.315 q^{59} -502.190 q^{61} -56.0601 q^{62} -386.070 q^{63} -484.636 q^{64} -0.689801 q^{66} +555.472 q^{67} -134.779 q^{68} +112.690 q^{69} -961.114 q^{71} +112.947 q^{72} -512.823 q^{73} -96.3538 q^{74} +631.146 q^{76} +51.3027 q^{77} -17.3590 q^{78} +277.906 q^{79} +685.879 q^{81} +111.990 q^{82} +288.228 q^{83} +84.6692 q^{84} -65.2511 q^{86} +20.7358 q^{87} -15.0089 q^{88} -1387.18 q^{89} +1291.04 q^{91} +1220.44 q^{92} +153.159 q^{93} +42.9103 q^{94} -37.2116 q^{96} +249.174 q^{97} -34.8808 q^{98} -93.0653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 10 q^{6} - 2 q^{7} - 12 q^{8} - 46 q^{9} - 38 q^{11} + 22 q^{12} + 80 q^{13} - 58 q^{14} - 14 q^{16} + 34 q^{17} - 80 q^{18} - 180 q^{19} - 56 q^{21} - 154 q^{22} + 42 q^{23}+ \cdots + 718 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.267949 0.0947343 0.0473672 − 0.998878i \(-0.484917\pi\)
0.0473672 + 0.998878i \(0.484917\pi\)
\(3\) −0.732051 −0.140883 −0.0704416 − 0.997516i \(-0.522441\pi\)
−0.0704416 + 0.997516i \(0.522441\pi\)
\(4\) −7.92820 −0.991025
\(5\) 0 0
\(6\) −0.196152 −0.0133465
\(7\) 14.5885 0.787703 0.393851 − 0.919174i \(-0.371142\pi\)
0.393851 + 0.919174i \(0.371142\pi\)
\(8\) −4.26795 −0.188618
\(9\) −26.4641 −0.980152
\(10\) 0 0
\(11\) 3.51666 0.0963921 0.0481960 − 0.998838i \(-0.484653\pi\)
0.0481960 + 0.998838i \(0.484653\pi\)
\(12\) 5.80385 0.139619
\(13\) 88.4974 1.88806 0.944030 − 0.329861i \(-0.107002\pi\)
0.944030 + 0.329861i \(0.107002\pi\)
\(14\) 3.90897 0.0746225
\(15\) 0 0
\(16\) 62.2820 0.973157
\(17\) 17.0000 0.242536
\(18\) −7.09103 −0.0928540
\(19\) −79.6077 −0.961224 −0.480612 − 0.876933i \(-0.659585\pi\)
−0.480612 + 0.876933i \(0.659585\pi\)
\(20\) 0 0
\(21\) −10.6795 −0.110974
\(22\) 0.942286 0.00913164
\(23\) −153.937 −1.39557 −0.697785 − 0.716307i \(-0.745831\pi\)
−0.697785 + 0.716307i \(0.745831\pi\)
\(24\) 3.12436 0.0265732
\(25\) 0 0
\(26\) 23.7128 0.178864
\(27\) 39.1384 0.278970
\(28\) −115.660 −0.780633
\(29\) −28.3257 −0.181377 −0.0906887 − 0.995879i \(-0.528907\pi\)
−0.0906887 + 0.995879i \(0.528907\pi\)
\(30\) 0 0
\(31\) −209.219 −1.21216 −0.606079 − 0.795405i \(-0.707258\pi\)
−0.606079 + 0.795405i \(0.707258\pi\)
\(32\) 50.8320 0.280810
\(33\) −2.57437 −0.0135800
\(34\) 4.55514 0.0229765
\(35\) 0 0
\(36\) 209.813 0.971355
\(37\) −359.597 −1.59777 −0.798884 − 0.601485i \(-0.794576\pi\)
−0.798884 + 0.601485i \(0.794576\pi\)
\(38\) −21.3308 −0.0910609
\(39\) −64.7846 −0.265996
\(40\) 0 0
\(41\) 417.951 1.59202 0.796012 − 0.605280i \(-0.206939\pi\)
0.796012 + 0.605280i \(0.206939\pi\)
\(42\) −2.86156 −0.0105131
\(43\) −243.520 −0.863640 −0.431820 − 0.901960i \(-0.642128\pi\)
−0.431820 + 0.901960i \(0.642128\pi\)
\(44\) −27.8808 −0.0955270
\(45\) 0 0
\(46\) −41.2473 −0.132208
\(47\) 160.144 0.497007 0.248504 − 0.968631i \(-0.420061\pi\)
0.248504 + 0.968631i \(0.420061\pi\)
\(48\) −45.5936 −0.137101
\(49\) −130.177 −0.379525
\(50\) 0 0
\(51\) −12.4449 −0.0341692
\(52\) −701.626 −1.87111
\(53\) −28.8513 −0.0747740 −0.0373870 − 0.999301i \(-0.511903\pi\)
−0.0373870 + 0.999301i \(0.511903\pi\)
\(54\) 10.4871 0.0264281
\(55\) 0 0
\(56\) −62.2628 −0.148575
\(57\) 58.2769 0.135420
\(58\) −7.58984 −0.0171827
\(59\) −832.315 −1.83658 −0.918290 − 0.395908i \(-0.870430\pi\)
−0.918290 + 0.395908i \(0.870430\pi\)
\(60\) 0 0
\(61\) −502.190 −1.05408 −0.527039 − 0.849841i \(-0.676698\pi\)
−0.527039 + 0.849841i \(0.676698\pi\)
\(62\) −56.0601 −0.114833
\(63\) −386.070 −0.772068
\(64\) −484.636 −0.946554
\(65\) 0 0
\(66\) −0.689801 −0.00128650
\(67\) 555.472 1.01286 0.506430 − 0.862281i \(-0.330965\pi\)
0.506430 + 0.862281i \(0.330965\pi\)
\(68\) −134.779 −0.240359
\(69\) 112.690 0.196612
\(70\) 0 0
\(71\) −961.114 −1.60652 −0.803262 − 0.595625i \(-0.796904\pi\)
−0.803262 + 0.595625i \(0.796904\pi\)
\(72\) 112.947 0.184875
\(73\) −512.823 −0.822211 −0.411105 − 0.911588i \(-0.634857\pi\)
−0.411105 + 0.911588i \(0.634857\pi\)
\(74\) −96.3538 −0.151364
\(75\) 0 0
\(76\) 631.146 0.952597
\(77\) 51.3027 0.0759283
\(78\) −17.3590 −0.0251989
\(79\) 277.906 0.395783 0.197892 − 0.980224i \(-0.436591\pi\)
0.197892 + 0.980224i \(0.436591\pi\)
\(80\) 0 0
\(81\) 685.879 0.940850
\(82\) 111.990 0.150819
\(83\) 288.228 0.381170 0.190585 − 0.981671i \(-0.438961\pi\)
0.190585 + 0.981671i \(0.438961\pi\)
\(84\) 84.6692 0.109978
\(85\) 0 0
\(86\) −65.2511 −0.0818164
\(87\) 20.7358 0.0255530
\(88\) −15.0089 −0.0181813
\(89\) −1387.18 −1.65215 −0.826073 − 0.563563i \(-0.809430\pi\)
−0.826073 + 0.563563i \(0.809430\pi\)
\(90\) 0 0
\(91\) 1291.04 1.48723
\(92\) 1220.44 1.38305
\(93\) 153.159 0.170773
\(94\) 42.9103 0.0470837
\(95\) 0 0
\(96\) −37.2116 −0.0395614
\(97\) 249.174 0.260823 0.130411 − 0.991460i \(-0.458370\pi\)
0.130411 + 0.991460i \(0.458370\pi\)
\(98\) −34.8808 −0.0359540
\(99\) −93.0653 −0.0944789
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 425.4.a.e.1.1 2
5.2 odd 4 425.4.b.e.324.3 4
5.3 odd 4 425.4.b.e.324.2 4
5.4 even 2 85.4.a.d.1.2 ✓ 2
15.14 odd 2 765.4.a.i.1.1 2
20.19 odd 2 1360.4.a.m.1.1 2
85.84 even 2 1445.4.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.d.1.2 ✓ 2 5.4 even 2
425.4.a.e.1.1 2 1.1 even 1 trivial
425.4.b.e.324.2 4 5.3 odd 4
425.4.b.e.324.3 4 5.2 odd 4
765.4.a.i.1.1 2 15.14 odd 2
1360.4.a.m.1.1 2 20.19 odd 2
1445.4.a.i.1.2 2 85.84 even 2