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Properties

Label 1225.4.a.z.1.3
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $4$
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] = [1225,4,Mod(1,1225)] 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("1225.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,-3,36,0,-1] 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: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 32x^{2} - 35x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.50478\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} +2.15229 q^{6} -7.94792 q^{8} -8.82008 q^{9} +54.8800 q^{11} -33.0239 q^{12} +16.0073 q^{13} +57.9496 q^{16} -0.422056 q^{17} -4.45223 q^{18} -127.501 q^{19} +27.7025 q^{22} -51.1101 q^{23} -33.8883 q^{24} +8.08024 q^{26} -152.729 q^{27} +41.4750 q^{29} -192.354 q^{31} +92.8353 q^{32} +233.997 q^{33} -0.213047 q^{34} +68.3132 q^{36} +189.232 q^{37} -64.3605 q^{38} +68.2519 q^{39} +76.3187 q^{41} -294.499 q^{43} -425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +247.085 q^{48} -1.79956 q^{51} -123.980 q^{52} +661.316 q^{53} -77.0953 q^{54} -543.639 q^{57} +20.9359 q^{58} -410.312 q^{59} -46.0495 q^{61} -97.0974 q^{62} -416.735 q^{64} +118.118 q^{66} -10.4074 q^{67} +3.26890 q^{68} -217.923 q^{69} -491.117 q^{71} +70.1012 q^{72} -814.540 q^{73} +95.5215 q^{74} +987.522 q^{76} +34.4525 q^{78} -858.725 q^{79} -413.064 q^{81} +38.5244 q^{82} -1055.80 q^{83} -148.658 q^{86} +176.841 q^{87} -436.182 q^{88} -341.567 q^{89} +395.858 q^{92} -820.159 q^{93} +272.733 q^{94} +395.831 q^{96} -1417.21 q^{97} -484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 3 q^{3} + 36 q^{4} - q^{6} - 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 160 q^{16} - 53 q^{17} - 433 q^{18} + 29 q^{19} + 152 q^{22} - 295 q^{23} + 21 q^{24} - 700 q^{26}+ \cdots + 2383 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.504784 0.178468 0.0892340 − 0.996011i \(-0.471558\pi\)
0.0892340 + 0.996011i \(0.471558\pi\)
\(3\) 4.26379 0.820567 0.410284 − 0.911958i \(-0.365430\pi\)
0.410284 + 0.911958i \(0.365430\pi\)
\(4\) −7.74519 −0.968149
\(5\) 0 0
\(6\) 2.15229 0.146445
\(7\) 0 0
\(8\) −7.94792 −0.351252
\(9\) −8.82008 −0.326670
\(10\) 0 0
\(11\) 54.8800 1.50427 0.752134 − 0.659010i \(-0.229025\pi\)
0.752134 + 0.659010i \(0.229025\pi\)
\(12\) −33.0239 −0.794431
\(13\) 16.0073 0.341510 0.170755 − 0.985313i \(-0.445379\pi\)
0.170755 + 0.985313i \(0.445379\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 57.9496 0.905462
\(17\) −0.422056 −0.00602139 −0.00301069 − 0.999995i \(-0.500958\pi\)
−0.00301069 + 0.999995i \(0.500958\pi\)
\(18\) −4.45223 −0.0583000
\(19\) −127.501 −1.53952 −0.769758 − 0.638336i \(-0.779623\pi\)
−0.769758 + 0.638336i \(0.779623\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 27.7025 0.268464
\(23\) −51.1101 −0.463357 −0.231678 − 0.972792i \(-0.574422\pi\)
−0.231678 + 0.972792i \(0.574422\pi\)
\(24\) −33.8883 −0.288226
\(25\) 0 0
\(26\) 8.08024 0.0609487
\(27\) −152.729 −1.08862
\(28\) 0 0
\(29\) 41.4750 0.265576 0.132788 − 0.991144i \(-0.457607\pi\)
0.132788 + 0.991144i \(0.457607\pi\)
\(30\) 0 0
\(31\) −192.354 −1.11445 −0.557224 − 0.830362i \(-0.688134\pi\)
−0.557224 + 0.830362i \(0.688134\pi\)
\(32\) 92.8353 0.512848
\(33\) 233.997 1.23435
\(34\) −0.213047 −0.00107462
\(35\) 0 0
\(36\) 68.3132 0.316265
\(37\) 189.232 0.840801 0.420400 − 0.907339i \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(38\) −64.3605 −0.274754
\(39\) 68.2519 0.280232
\(40\) 0 0
\(41\) 76.3187 0.290707 0.145353 − 0.989380i \(-0.453568\pi\)
0.145353 + 0.989380i \(0.453568\pi\)
\(42\) 0 0
\(43\) −294.499 −1.04443 −0.522216 − 0.852813i \(-0.674895\pi\)
−0.522216 + 0.852813i \(0.674895\pi\)
\(44\) −425.056 −1.45636
\(45\) 0 0
\(46\) −25.7996 −0.0826943
\(47\) 540.297 1.67682 0.838408 − 0.545043i \(-0.183487\pi\)
0.838408 + 0.545043i \(0.183487\pi\)
\(48\) 247.085 0.742992
\(49\) 0 0
\(50\) 0 0
\(51\) −1.79956 −0.00494095
\(52\) −123.980 −0.330633
\(53\) 661.316 1.71394 0.856969 − 0.515368i \(-0.172345\pi\)
0.856969 + 0.515368i \(0.172345\pi\)
\(54\) −77.0953 −0.194284
\(55\) 0 0
\(56\) 0 0
\(57\) −543.639 −1.26328
\(58\) 20.9359 0.0473969
\(59\) −410.312 −0.905390 −0.452695 − 0.891665i \(-0.649537\pi\)
−0.452695 + 0.891665i \(0.649537\pi\)
\(60\) 0 0
\(61\) −46.0495 −0.0966563 −0.0483281 − 0.998832i \(-0.515389\pi\)
−0.0483281 + 0.998832i \(0.515389\pi\)
\(62\) −97.0974 −0.198893
\(63\) 0 0
\(64\) −416.735 −0.813935
\(65\) 0 0
\(66\) 118.118 0.220292
\(67\) −10.4074 −0.0189771 −0.00948854 − 0.999955i \(-0.503020\pi\)
−0.00948854 + 0.999955i \(0.503020\pi\)
\(68\) 3.26890 0.00582960
\(69\) −217.923 −0.380215
\(70\) 0 0
\(71\) −491.117 −0.820913 −0.410456 − 0.911880i \(-0.634631\pi\)
−0.410456 + 0.911880i \(0.634631\pi\)
\(72\) 70.1012 0.114743
\(73\) −814.540 −1.30595 −0.652977 − 0.757378i \(-0.726480\pi\)
−0.652977 + 0.757378i \(0.726480\pi\)
\(74\) 95.5215 0.150056
\(75\) 0 0
\(76\) 987.522 1.49048
\(77\) 0 0
\(78\) 34.4525 0.0500125
\(79\) −858.725 −1.22296 −0.611482 − 0.791258i \(-0.709426\pi\)
−0.611482 + 0.791258i \(0.709426\pi\)
\(80\) 0 0
\(81\) −413.064 −0.566617
\(82\) 38.5244 0.0518818
\(83\) −1055.80 −1.39626 −0.698129 − 0.715972i \(-0.745984\pi\)
−0.698129 + 0.715972i \(0.745984\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −148.658 −0.186398
\(87\) 176.841 0.217923
\(88\) −436.182 −0.528376
\(89\) −341.567 −0.406809 −0.203405 − 0.979095i \(-0.565201\pi\)
−0.203405 + 0.979095i \(0.565201\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 395.858 0.448598
\(93\) −820.159 −0.914479
\(94\) 272.733 0.299258
\(95\) 0 0
\(96\) 395.831 0.420826
\(97\) −1417.21 −1.48346 −0.741731 − 0.670697i \(-0.765995\pi\)
−0.741731 + 0.670697i \(0.765995\pi\)
\(98\) 0 0
\(99\) −484.046 −0.491398
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 1225.4.a.z.1.3 4
5.4 even 2 1225.4.a.bd.1.2 4
7.6 odd 2 175.4.a.g.1.3 ✓ 4
21.20 even 2 1575.4.a.bl.1.2 4
35.13 even 4 175.4.b.f.99.4 8
35.27 even 4 175.4.b.f.99.5 8
35.34 odd 2 175.4.a.h.1.2 yes 4
105.104 even 2 1575.4.a.bg.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.3 ✓ 4 7.6 odd 2
175.4.a.h.1.2 yes 4 35.34 odd 2
175.4.b.f.99.4 8 35.13 even 4
175.4.b.f.99.5 8 35.27 even 4
1225.4.a.z.1.3 4 1.1 even 1 trivial
1225.4.a.bd.1.2 4 5.4 even 2
1575.4.a.bg.1.3 4 105.104 even 2
1575.4.a.bl.1.2 4 21.20 even 2