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Properties

Label 2175.4.a.n.1.3
Level $2175$
Weight $4$
Character 2175.1
Self dual yes
Analytic conductor $128.329$
Analytic rank $0$
Dimension $7$
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] = [2175,4,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,2,21] 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: \(128.329154262\)
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: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.40909\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.40909 q^{2} +3.00000 q^{3} -6.01448 q^{4} -4.22726 q^{6} -22.4156 q^{7} +19.7476 q^{8} +9.00000 q^{9} +11.9568 q^{11} -18.0434 q^{12} +24.3982 q^{13} +31.5855 q^{14} +20.2898 q^{16} +57.0575 q^{17} -12.6818 q^{18} -101.555 q^{19} -67.2467 q^{21} -16.8481 q^{22} -133.400 q^{23} +59.2428 q^{24} -34.3792 q^{26} +27.0000 q^{27} +134.818 q^{28} +29.0000 q^{29} +292.953 q^{31} -186.571 q^{32} +35.8703 q^{33} -80.3989 q^{34} -54.1303 q^{36} -393.581 q^{37} +143.100 q^{38} +73.1947 q^{39} +237.918 q^{41} +94.7564 q^{42} +82.3986 q^{43} -71.9137 q^{44} +187.972 q^{46} -490.781 q^{47} +60.8693 q^{48} +159.458 q^{49} +171.173 q^{51} -146.743 q^{52} +416.624 q^{53} -38.0453 q^{54} -442.654 q^{56} -304.666 q^{57} -40.8635 q^{58} +320.424 q^{59} +612.103 q^{61} -412.796 q^{62} -201.740 q^{63} +100.576 q^{64} -50.5443 q^{66} -569.634 q^{67} -343.171 q^{68} -400.199 q^{69} -689.224 q^{71} +177.728 q^{72} -125.224 q^{73} +554.589 q^{74} +610.801 q^{76} -268.018 q^{77} -103.138 q^{78} -356.958 q^{79} +81.0000 q^{81} -335.247 q^{82} -947.383 q^{83} +404.454 q^{84} -116.107 q^{86} +87.0000 q^{87} +236.117 q^{88} -1212.05 q^{89} -546.901 q^{91} +802.330 q^{92} +878.858 q^{93} +691.553 q^{94} -559.712 q^{96} +597.340 q^{97} -224.690 q^{98} +107.611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 21 q^{3} + 22 q^{4} + 6 q^{6} + 50 q^{7} + 33 q^{8} + 63 q^{9} + 76 q^{11} + 66 q^{12} - 30 q^{13} + 89 q^{14} + 138 q^{16} + 140 q^{17} + 18 q^{18} + 90 q^{19} + 150 q^{21} - 61 q^{22}+ \cdots + 684 q^{99}+O(q^{100}) \) 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 \(-0.580133\pi\)
−0.249094 + 0.968479i \(0.580133\pi\)
\(3\) 3.00000 0.577350
\(4\) −6.01448 −0.751810
\(5\) 0 0
\(6\) −4.22726 −0.287628
\(7\) −22.4156 −1.21033 −0.605164 − 0.796101i \(-0.706892\pi\)
−0.605164 + 0.796101i \(0.706892\pi\)
\(8\) 19.7476 0.872729
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 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 \(-0.416190\pi\)
0.260264 + 0.965538i \(0.416190\pi\)
\(14\) 31.5855 0.602969
\(15\) 0 0
\(16\) 20.2898 0.317027
\(17\) 57.0575 0.814028 0.407014 − 0.913422i \(-0.366570\pi\)
0.407014 + 0.913422i \(0.366570\pi\)
\(18\) −12.6818 −0.166062
\(19\) −101.555 −1.22623 −0.613115 − 0.789994i \(-0.710084\pi\)
−0.613115 + 0.789994i \(0.710084\pi\)
\(20\) 0 0
\(21\) −67.2467 −0.698783
\(22\) −16.8481 −0.163274
\(23\) −133.400 −1.20938 −0.604691 − 0.796460i \(-0.706703\pi\)
−0.604691 + 0.796460i \(0.706703\pi\)
\(24\) 59.2428 0.503870
\(25\) 0 0
\(26\) −34.3792 −0.259320
\(27\) 27.0000 0.192450
\(28\) 134.818 0.909936
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) −54.1303 −0.250603
\(37\) −393.581 −1.74876 −0.874382 − 0.485238i \(-0.838733\pi\)
−0.874382 + 0.485238i \(0.838733\pi\)
\(38\) 143.100 0.610892
\(39\) 73.1947 0.300527
\(40\) 0 0
\(41\) 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 \(-0.453324\pi\)
0.146112 + 0.989268i \(0.453324\pi\)
\(44\) −71.9137 −0.246395
\(45\) 0 0
\(46\) 187.972 0.602498
\(47\) −490.781 −1.52314 −0.761572 − 0.648080i \(-0.775572\pi\)
−0.761572 + 0.648080i \(0.775572\pi\)
\(48\) 60.8693 0.183036
\(49\) 159.458 0.464893
\(50\) 0 0
\(51\) 171.173 0.469979
\(52\) −146.743 −0.391338
\(53\) 416.624 1.07977 0.539885 − 0.841739i \(-0.318468\pi\)
0.539885 + 0.841739i \(0.318468\pi\)
\(54\) −38.0453 −0.0958761
\(55\) 0 0
\(56\) −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 \(0.384984\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) −50.5443 −0.0942663
\(67\) −569.634 −1.03868 −0.519342 − 0.854567i \(-0.673823\pi\)
−0.519342 + 0.854567i \(0.673823\pi\)
\(68\) −343.171 −0.611994
\(69\) −400.199 −0.698237
\(70\) 0 0
\(71\) −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 \(-0.532008\pi\)
−0.100386 + 0.994949i \(0.532008\pi\)
\(74\) 554.589 0.871212
\(75\) 0 0
\(76\) 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 \(0.581807\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −335.247 −0.451485
\(83\) −947.383 −1.25288 −0.626439 − 0.779471i \(-0.715488\pi\)
−0.626439 + 0.779471i \(0.715488\pi\)
\(84\) 404.454 0.525352
\(85\) 0 0
\(86\) −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 \(0.756675\pi\)
\(90\) 0 0
\(91\) −546.901 −0.630009
\(92\) 802.330 0.909224
\(93\) 878.858 0.979929
\(94\) 691.553 0.758811
\(95\) 0 0
\(96\) −559.712 −0.595056
\(97\) 597.340 0.625265 0.312633 − 0.949874i \(-0.398789\pi\)
0.312633 + 0.949874i \(0.398789\pi\)
\(98\) −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 2175.4.a.n.1.3 7
5.4 even 2 435.4.a.i.1.5 ✓ 7
15.14 odd 2 1305.4.a.n.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.4.a.i.1.5 ✓ 7 5.4 even 2
1305.4.a.n.1.3 7 15.14 odd 2
2175.4.a.n.1.3 7 1.1 even 1 trivial