↑

Properties

Label 435.4.a.i.1.5
Level $435$
Weight $4$
Character 435.1
Self dual yes
Analytic conductor $25.666$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [435,4,Mod(1,435)] 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("435.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(435, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 435.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.6658308525\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 37x^{5} + 55x^{4} + 336x^{3} - 227x^{2} - 824x - 166 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.40909\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.40909 q^{2} -3.00000 q^{3} -6.01448 q^{4} +5.00000 q^{5} -4.22726 q^{6} +22.4156 q^{7} -19.7476 q^{8} +9.00000 q^{9} +7.04543 q^{10} +11.9568 q^{11} +18.0434 q^{12} -24.3982 q^{13} +31.5855 q^{14} -15.0000 q^{15} +20.2898 q^{16} -57.0575 q^{17} +12.6818 q^{18} -101.555 q^{19} -30.0724 q^{20} -67.2467 q^{21} +16.8481 q^{22} +133.400 q^{23} +59.2428 q^{24} +25.0000 q^{25} -34.3792 q^{26} -27.0000 q^{27} -134.818 q^{28} +29.0000 q^{29} -21.1363 q^{30} +292.953 q^{31} +186.571 q^{32} -35.8703 q^{33} -80.3989 q^{34} +112.078 q^{35} -54.1303 q^{36} +393.581 q^{37} -143.100 q^{38} +73.1947 q^{39} -98.7380 q^{40} +237.918 q^{41} -94.7564 q^{42} -82.3986 q^{43} -71.9137 q^{44} +45.0000 q^{45} +187.972 q^{46} +490.781 q^{47} -60.8693 q^{48} +159.458 q^{49} +35.2271 q^{50} +171.173 q^{51} +146.743 q^{52} -416.624 q^{53} -38.0453 q^{54} +59.7838 q^{55} -442.654 q^{56} +304.666 q^{57} +40.8635 q^{58} +320.424 q^{59} +90.2172 q^{60} +612.103 q^{61} +412.796 q^{62} +201.740 q^{63} +100.576 q^{64} -121.991 q^{65} -50.5443 q^{66} +569.634 q^{67} +343.171 q^{68} -400.199 q^{69} +157.927 q^{70} -689.224 q^{71} -177.728 q^{72} +125.224 q^{73} +554.589 q^{74} -75.0000 q^{75} +610.801 q^{76} +268.018 q^{77} +103.138 q^{78} -356.958 q^{79} +101.449 q^{80} +81.0000 q^{81} +335.247 q^{82} +947.383 q^{83} +404.454 q^{84} -285.288 q^{85} -116.107 q^{86} -87.0000 q^{87} -236.117 q^{88} -1212.05 q^{89} +63.4089 q^{90} -546.901 q^{91} -802.330 q^{92} -878.858 q^{93} +691.553 q^{94} -507.776 q^{95} -559.712 q^{96} -597.340 q^{97} +224.690 q^{98} +107.611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{2} - 21 q^{3} + 22 q^{4} + 35 q^{5} + 6 q^{6} - 50 q^{7} - 33 q^{8} + 63 q^{9} - 10 q^{10} + 76 q^{11} - 66 q^{12} + 30 q^{13} + 89 q^{14} - 105 q^{15} + 138 q^{16} - 140 q^{17} - 18 q^{18} + 90 q^{19}+ \cdots + 684 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.40909 0.498187 0.249094 − 0.968479i \(-0.419867\pi\)
0.249094 + 0.968479i \(0.419867\pi\)
\(3\) −3.00000 −0.577350
\(4\) −6.01448 −0.751810
\(5\) 5.00000 0.447214
\(6\) −4.22726 −0.287628
\(7\) 22.4156 1.21033 0.605164 − 0.796101i \(-0.293108\pi\)
0.605164 + 0.796101i \(0.293108\pi\)
\(8\) −19.7476 −0.872729
\(9\) 9.00000 0.333333
\(10\) 7.04543 0.222796
\(11\) 11.9568 0.327736 0.163868 − 0.986482i \(-0.447603\pi\)
0.163868 + 0.986482i \(0.447603\pi\)
\(12\) 18.0434 0.434058
\(13\) −24.3982 −0.520527 −0.260264 − 0.965538i \(-0.583810\pi\)
−0.260264 + 0.965538i \(0.583810\pi\)
\(14\) 31.5855 0.602969
\(15\) −15.0000 −0.258199
\(16\) 20.2898 0.317027
\(17\) −57.0575 −0.814028 −0.407014 − 0.913422i \(-0.633430\pi\)
−0.407014 + 0.913422i \(0.633430\pi\)
\(18\) 12.6818 0.166062
\(19\) −101.555 −1.22623 −0.613115 − 0.789994i \(-0.710084\pi\)
−0.613115 + 0.789994i \(0.710084\pi\)
\(20\) −30.0724 −0.336220
\(21\) −67.2467 −0.698783
\(22\) 16.8481 0.163274
\(23\) 133.400 1.20938 0.604691 − 0.796460i \(-0.293297\pi\)
0.604691 + 0.796460i \(0.293297\pi\)
\(24\) 59.2428 0.503870
\(25\) 25.0000 0.200000
\(26\) −34.3792 −0.259320
\(27\) −27.0000 −0.192450
\(28\) −134.818 −0.909936
\(29\) 29.0000 0.185695
\(30\) −21.1363 −0.128631
\(31\) 292.953 1.69729 0.848643 − 0.528966i \(-0.177420\pi\)
0.848643 + 0.528966i \(0.177420\pi\)
\(32\) 186.571 1.03067
\(33\) −35.8703 −0.189219
\(34\) −80.3989 −0.405538
\(35\) 112.078 0.541275
\(36\) −54.1303 −0.250603
\(37\) 393.581 1.74876 0.874382 − 0.485238i \(-0.161267\pi\)
0.874382 + 0.485238i \(0.161267\pi\)
\(38\) −143.100 −0.610892
\(39\) 73.1947 0.300527
\(40\) −98.7380 −0.390296
\(41\) 237.918 0.906257 0.453128 − 0.891445i \(-0.350308\pi\)
0.453128 + 0.891445i \(0.350308\pi\)
\(42\) −94.7564 −0.348125
\(43\) −82.3986 −0.292225 −0.146112 − 0.989268i \(-0.546676\pi\)
−0.146112 + 0.989268i \(0.546676\pi\)
\(44\) −71.9137 −0.246395
\(45\) 45.0000 0.149071
\(46\) 187.972 0.602498
\(47\) 490.781 1.52314 0.761572 − 0.648080i \(-0.224428\pi\)
0.761572 + 0.648080i \(0.224428\pi\)
\(48\) −60.8693 −0.183036
\(49\) 159.458 0.464893
\(50\) 35.2271 0.0996374
\(51\) 171.173 0.469979
\(52\) 146.743 0.391338
\(53\) −416.624 −1.07977 −0.539885 − 0.841739i \(-0.681532\pi\)
−0.539885 + 0.841739i \(0.681532\pi\)
\(54\) −38.0453 −0.0958761
\(55\) 59.7838 0.146568
\(56\) −442.654 −1.05629
\(57\) 304.666 0.707964
\(58\) 40.8635 0.0925110
\(59\) 320.424 0.707046 0.353523 − 0.935426i \(-0.384984\pi\)
0.353523 + 0.935426i \(0.384984\pi\)
\(60\) 90.2172 0.194116
\(61\) 612.103 1.28478 0.642392 − 0.766376i \(-0.277942\pi\)
0.642392 + 0.766376i \(0.277942\pi\)
\(62\) 412.796 0.845566
\(63\) 201.740 0.403442
\(64\) 100.576 0.196438
\(65\) −121.991 −0.232787
\(66\) −50.5443 −0.0942663
\(67\) 569.634 1.03868 0.519342 − 0.854567i \(-0.326177\pi\)
0.519342 + 0.854567i \(0.326177\pi\)
\(68\) 343.171 0.611994
\(69\) −400.199 −0.698237
\(70\) 157.927 0.269656
\(71\) −689.224 −1.15205 −0.576027 − 0.817431i \(-0.695398\pi\)
−0.576027 + 0.817431i \(0.695398\pi\)
\(72\) −177.728 −0.290910
\(73\) 125.224 0.200772 0.100386 − 0.994949i \(-0.467992\pi\)
0.100386 + 0.994949i \(0.467992\pi\)
\(74\) 554.589 0.871212
\(75\) −75.0000 −0.115470
\(76\) 610.801 0.921891
\(77\) 268.018 0.396668
\(78\) 103.138 0.149718
\(79\) −356.958 −0.508366 −0.254183 − 0.967156i \(-0.581807\pi\)
−0.254183 + 0.967156i \(0.581807\pi\)
\(80\) 101.449 0.141779
\(81\) 81.0000 0.111111
\(82\) 335.247 0.451485
\(83\) 947.383 1.25288 0.626439 − 0.779471i \(-0.284512\pi\)
0.626439 + 0.779471i \(0.284512\pi\)
\(84\) 404.454 0.525352
\(85\) −285.288 −0.364045
\(86\) −116.107 −0.145583
\(87\) −87.0000 −0.107211
\(88\) −236.117 −0.286025
\(89\) −1212.05 −1.44356 −0.721779 − 0.692123i \(-0.756675\pi\)
−0.721779 + 0.692123i \(0.756675\pi\)
\(90\) 63.4089 0.0742653
\(91\) −546.901 −0.630009
\(92\) −802.330 −0.909224
\(93\) −878.858 −0.979929
\(94\) 691.553 0.758811
\(95\) −507.776 −0.548387
\(96\) −559.712 −0.595056
\(97\) −597.340 −0.625265 −0.312633 − 0.949874i \(-0.601211\pi\)
−0.312633 + 0.949874i \(0.601211\pi\)
\(98\) 224.690 0.231603
\(99\) 107.611 0.109245
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 435.4.a.i.1.5 ✓ 7
3.2 odd 2 1305.4.a.n.1.3 7
5.4 even 2 2175.4.a.n.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.i.1.5 ✓ 7 1.1 even 1 trivial
1305.4.a.n.1.3 7 3.2 odd 2
2175.4.a.n.1.3 7 5.4 even 2