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Properties

Label 3775.1.d.d.301.3
Level $3775$
Weight $1$
Character 3775.301
Self dual yes
Analytic conductor $1.884$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -151
Inner twists $2$

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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] = [3775,1,Mod(301,3775)] 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("3775.301"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3775 = 5^{2} \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3775.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,0,2,0,0,0,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.88397042269\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 151)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.3442951.1
Artin image: $D_{14}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

Embedding invariants

Embedding label 301.3
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 3775.301

$q$-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)
 
\(f(q)\) \(=\) \(q+1.80194 q^{2} +2.24698 q^{4} +2.24698 q^{8} +1.00000 q^{9} +1.24698 q^{11} +1.80194 q^{16} -1.24698 q^{17} +1.80194 q^{18} -1.80194 q^{19} +2.24698 q^{22} -1.80194 q^{29} -0.445042 q^{31} +1.00000 q^{32} -2.24698 q^{34} +2.24698 q^{36} -1.24698 q^{37} -3.24698 q^{38} +0.445042 q^{43} +2.80194 q^{44} +1.80194 q^{47} +1.00000 q^{49} -3.24698 q^{58} -0.445042 q^{59} -0.801938 q^{62} -2.80194 q^{68} +2.24698 q^{72} -2.24698 q^{74} -4.04892 q^{76} +1.00000 q^{81} +0.801938 q^{86} +2.80194 q^{88} +3.24698 q^{94} +1.80194 q^{97} +1.80194 q^{98} +1.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{4} + 2 q^{8} + 3 q^{9} - q^{11} + q^{16} + q^{17} + q^{18} - q^{19} + 2 q^{22} - q^{29} - q^{31} + 3 q^{32} - 2 q^{34} + 2 q^{36} + q^{37} - 5 q^{38} + q^{43} + 4 q^{44} + q^{47}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3775\mathbb{Z}\right)^\times\).

\(n\) \(152\) \(3026\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.80194 1.80194 0.900969 − 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 2.24698 2.24698
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.24698 2.24698
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 1.24698 1.24698 0.623490 − 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.80194 1.80194
\(17\) −1.24698 −1.24698 −0.623490 − 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(18\) 1.80194 1.80194
\(19\) −1.80194 −1.80194 −0.900969 − 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.24698 2.24698
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.80194 −1.80194 −0.900969 − 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(30\) 0 0
\(31\) −0.445042 −0.445042 −0.222521 − 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) −2.24698 −2.24698
\(35\) 0 0
\(36\) 2.24698 2.24698
\(37\) −1.24698 −1.24698 −0.623490 − 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(38\) −3.24698 −3.24698
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0.445042 0.445042 0.222521 − 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(44\) 2.80194 2.80194
\(45\) 0 0
\(46\) 0 0
\(47\) 1.80194 1.80194 0.900969 − 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −3.24698 −3.24698
\(59\) −0.445042 −0.445042 −0.222521 − 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −0.801938 −0.801938
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −2.80194 −2.80194
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.24698 2.24698
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −2.24698 −2.24698
\(75\) 0 0
\(76\) −4.04892 −4.04892
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.801938 0.801938
\(87\) 0 0
\(88\) 2.80194 2.80194
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 3.24698 3.24698
\(95\) 0 0
\(96\) 0 0
\(97\) 1.80194 1.80194 0.900969 − 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(98\) 1.80194 1.80194
\(99\) 1.24698 1.24698
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 3775.1.d.d.301.3 3
5.2 odd 4 3775.1.c.c.3774.6 6
5.3 odd 4 3775.1.c.c.3774.1 6
5.4 even 2 151.1.b.a.150.1 ✓ 3
15.14 odd 2 1359.1.d.b.1207.3 3
20.19 odd 2 2416.1.e.a.2113.2 3
151.150 odd 2 CM 3775.1.d.d.301.3 3
755.452 even 4 3775.1.c.c.3774.6 6
755.603 even 4 3775.1.c.c.3774.1 6
755.754 odd 2 151.1.b.a.150.1 ✓ 3
2265.2264 even 2 1359.1.d.b.1207.3 3
3020.3019 even 2 2416.1.e.a.2113.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
151.1.b.a.150.1 ✓ 3 5.4 even 2
151.1.b.a.150.1 ✓ 3 755.754 odd 2
1359.1.d.b.1207.3 3 15.14 odd 2
1359.1.d.b.1207.3 3 2265.2264 even 2
2416.1.e.a.2113.2 3 20.19 odd 2
2416.1.e.a.2113.2 3 3020.3019 even 2
3775.1.c.c.3774.1 6 5.3 odd 4
3775.1.c.c.3774.1 6 755.603 even 4
3775.1.c.c.3774.6 6 5.2 odd 4
3775.1.c.c.3774.6 6 755.452 even 4
3775.1.d.d.301.3 3 1.1 even 1 trivial
3775.1.d.d.301.3 3 151.150 odd 2 CM