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Properties

Label 425.4.a.e.1.2
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 1.2
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)
 
\(f(q)\) \(=\) \(q+3.73205 q^{2} +2.73205 q^{3} +5.92820 q^{4} +10.1962 q^{6} -16.5885 q^{7} -7.73205 q^{8} -19.5359 q^{9} -41.5167 q^{11} +16.1962 q^{12} -8.49742 q^{13} -61.9090 q^{14} -76.2820 q^{16} +17.0000 q^{17} -72.9090 q^{18} -100.392 q^{19} -45.3205 q^{21} -154.942 q^{22} +195.937 q^{23} -21.1244 q^{24} -31.7128 q^{26} -127.138 q^{27} -98.3397 q^{28} -187.674 q^{29} +279.219 q^{31} -222.832 q^{32} -113.426 q^{33} +63.4449 q^{34} -115.813 q^{36} +7.59739 q^{37} -374.669 q^{38} -23.2154 q^{39} -73.9512 q^{41} -169.138 q^{42} +331.520 q^{43} -246.119 q^{44} +731.247 q^{46} +187.856 q^{47} -208.406 q^{48} -67.8231 q^{49} +46.4449 q^{51} -50.3744 q^{52} +192.851 q^{53} -474.487 q^{54} +128.263 q^{56} -274.277 q^{57} -700.410 q^{58} -603.685 q^{59} +426.190 q^{61} +1042.06 q^{62} +324.070 q^{63} -221.364 q^{64} -423.310 q^{66} -511.472 q^{67} +100.779 q^{68} +535.310 q^{69} -548.886 q^{71} +151.053 q^{72} -575.177 q^{73} +28.3538 q^{74} -595.146 q^{76} +688.697 q^{77} -86.6410 q^{78} -1235.91 q^{79} +180.121 q^{81} -275.990 q^{82} -536.228 q^{83} -268.669 q^{84} +1237.25 q^{86} -512.736 q^{87} +321.009 q^{88} +1183.18 q^{89} +140.959 q^{91} +1161.56 q^{92} +762.841 q^{93} +701.090 q^{94} -608.788 q^{96} -1261.17 q^{97} -253.119 q^{98} +811.065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{3} - 2 q^{4} + 10 q^{6} - 2 q^{7} - 12 q^{8} - 46 q^{9} - 38 q^{11} + 22 q^{12} + 80 q^{13} - 58 q^{14} - 14 q^{16} + 34 q^{17} - 80 q^{18} - 180 q^{19} - 56 q^{21} - 154 q^{22} + 42 q^{23}+ \cdots + 718 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.73205 1.31948 0.659740 − 0.751494i \(-0.270667\pi\)
0.659740 + 0.751494i \(0.270667\pi\)
\(3\) 2.73205 0.525783 0.262892 − 0.964825i \(-0.415324\pi\)
0.262892 + 0.964825i \(0.415324\pi\)
\(4\) 5.92820 0.741025
\(5\) 0 0
\(6\) 10.1962 0.693760
\(7\) −16.5885 −0.895693 −0.447846 − 0.894111i \(-0.647809\pi\)
−0.447846 + 0.894111i \(0.647809\pi\)
\(8\) −7.73205 −0.341712
\(9\) −19.5359 −0.723552
\(10\) 0 0
\(11\) −41.5167 −1.13798 −0.568988 − 0.822346i \(-0.692665\pi\)
−0.568988 + 0.822346i \(0.692665\pi\)
\(12\) 16.1962 0.389619
\(13\) −8.49742 −0.181289 −0.0906447 − 0.995883i \(-0.528893\pi\)
−0.0906447 + 0.995883i \(0.528893\pi\)
\(14\) −61.9090 −1.18185
\(15\) 0 0
\(16\) −76.2820 −1.19191
\(17\) 17.0000 0.242536
\(18\) −72.9090 −0.954712
\(19\) −100.392 −1.21219 −0.606094 − 0.795393i \(-0.707265\pi\)
−0.606094 + 0.795393i \(0.707265\pi\)
\(20\) 0 0
\(21\) −45.3205 −0.470940
\(22\) −154.942 −1.50154
\(23\) 195.937 1.77634 0.888168 − 0.459519i \(-0.151978\pi\)
0.888168 + 0.459519i \(0.151978\pi\)
\(24\) −21.1244 −0.179666
\(25\) 0 0
\(26\) −31.7128 −0.239207
\(27\) −127.138 −0.906215
\(28\) −98.3397 −0.663731
\(29\) −187.674 −1.20173 −0.600866 − 0.799349i \(-0.705178\pi\)
−0.600866 + 0.799349i \(0.705178\pi\)
\(30\) 0 0
\(31\) 279.219 1.61772 0.808859 − 0.588003i \(-0.200086\pi\)
0.808859 + 0.588003i \(0.200086\pi\)
\(32\) −222.832 −1.23098
\(33\) −113.426 −0.598329
\(34\) 63.4449 0.320021
\(35\) 0 0
\(36\) −115.813 −0.536170
\(37\) 7.59739 0.0337568 0.0168784 − 0.999858i \(-0.494627\pi\)
0.0168784 + 0.999858i \(0.494627\pi\)
\(38\) −374.669 −1.59946
\(39\) −23.2154 −0.0953189
\(40\) 0 0
\(41\) −73.9512 −0.281689 −0.140844 − 0.990032i \(-0.544982\pi\)
−0.140844 + 0.990032i \(0.544982\pi\)
\(42\) −169.138 −0.621396
\(43\) 331.520 1.17573 0.587865 − 0.808959i \(-0.299969\pi\)
0.587865 + 0.808959i \(0.299969\pi\)
\(44\) −246.119 −0.843270
\(45\) 0 0
\(46\) 731.247 2.34384
\(47\) 187.856 0.583014 0.291507 − 0.956569i \(-0.405843\pi\)
0.291507 + 0.956569i \(0.405843\pi\)
\(48\) −208.406 −0.626685
\(49\) −67.8231 −0.197735
\(50\) 0 0
\(51\) 46.4449 0.127521
\(52\) −50.3744 −0.134340
\(53\) 192.851 0.499814 0.249907 − 0.968270i \(-0.419600\pi\)
0.249907 + 0.968270i \(0.419600\pi\)
\(54\) −474.487 −1.19573
\(55\) 0 0
\(56\) 128.263 0.306069
\(57\) −274.277 −0.637348
\(58\) −700.410 −1.58566
\(59\) −603.685 −1.33209 −0.666043 − 0.745914i \(-0.732013\pi\)
−0.666043 + 0.745914i \(0.732013\pi\)
\(60\) 0 0
\(61\) 426.190 0.894557 0.447279 − 0.894395i \(-0.352393\pi\)
0.447279 + 0.894395i \(0.352393\pi\)
\(62\) 1042.06 2.13454
\(63\) 324.070 0.648080
\(64\) −221.364 −0.432352
\(65\) 0 0
\(66\) −423.310 −0.789483
\(67\) −511.472 −0.932630 −0.466315 − 0.884619i \(-0.654419\pi\)
−0.466315 + 0.884619i \(0.654419\pi\)
\(68\) 100.779 0.179725
\(69\) 535.310 0.933968
\(70\) 0 0
\(71\) −548.886 −0.917476 −0.458738 − 0.888572i \(-0.651698\pi\)
−0.458738 + 0.888572i \(0.651698\pi\)
\(72\) 151.053 0.247246
\(73\) −575.177 −0.922183 −0.461092 − 0.887353i \(-0.652542\pi\)
−0.461092 + 0.887353i \(0.652542\pi\)
\(74\) 28.3538 0.0445414
\(75\) 0 0
\(76\) −595.146 −0.898262
\(77\) 688.697 1.01928
\(78\) −86.6410 −0.125771
\(79\) −1235.91 −1.76013 −0.880065 − 0.474853i \(-0.842501\pi\)
−0.880065 + 0.474853i \(0.842501\pi\)
\(80\) 0 0
\(81\) 180.121 0.247079
\(82\) −275.990 −0.371682
\(83\) −536.228 −0.709141 −0.354570 − 0.935029i \(-0.615373\pi\)
−0.354570 + 0.935029i \(0.615373\pi\)
\(84\) −268.669 −0.348979
\(85\) 0 0
\(86\) 1237.25 1.55135
\(87\) −512.736 −0.631851
\(88\) 321.009 0.388860
\(89\) 1183.18 1.40918 0.704590 − 0.709615i \(-0.251131\pi\)
0.704590 + 0.709615i \(0.251131\pi\)
\(90\) 0 0
\(91\) 140.959 0.162379
\(92\) 1161.56 1.31631
\(93\) 762.841 0.850569
\(94\) 701.090 0.769275
\(95\) 0 0
\(96\) −608.788 −0.647231
\(97\) −1261.17 −1.32013 −0.660066 − 0.751207i \(-0.729472\pi\)
−0.660066 + 0.751207i \(0.729472\pi\)
\(98\) −253.119 −0.260907
\(99\) 811.065 0.823385
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.1.2 2
5.2 odd 4 425.4.b.e.324.4 4
5.3 odd 4 425.4.b.e.324.1 4
5.4 even 2 85.4.a.d.1.1 ✓ 2
15.14 odd 2 765.4.a.i.1.2 2
20.19 odd 2 1360.4.a.m.1.2 2
85.84 even 2 1445.4.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.d.1.1 ✓ 2 5.4 even 2
425.4.a.e.1.2 2 1.1 even 1 trivial
425.4.b.e.324.1 4 5.3 odd 4
425.4.b.e.324.4 4 5.2 odd 4
765.4.a.i.1.2 2 15.14 odd 2
1360.4.a.m.1.2 2 20.19 odd 2
1445.4.a.i.1.1 2 85.84 even 2