Properties

Label 3525.1.bd.a
Level $3525$
Weight $1$
Character orbit 3525.bd
Analytic conductor $1.759$
Analytic rank $0$
Dimension $44$
Projective image $D_{23}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3525.bd (of order \(46\), degree \(22\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.75920416953\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(2\) over \(\Q(\zeta_{46})\)
Coefficient field: \(\Q(\zeta_{92})\)
Defining polynomial: \(x^{44} - x^{42} + x^{40} - x^{38} + x^{36} - x^{34} + x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
Projective image: \(D_{23}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{23} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( \zeta_{92}^{3} - \zeta_{92}^{25} ) q^{2} + \zeta_{92}^{13} q^{3} + ( -\zeta_{92}^{4} + \zeta_{92}^{6} - \zeta_{92}^{28} ) q^{4} + ( \zeta_{92}^{16} - \zeta_{92}^{38} ) q^{6} + ( -\zeta_{92}^{7} + \zeta_{92}^{9} + \zeta_{92}^{29} - \zeta_{92}^{31} ) q^{8} + \zeta_{92}^{26} q^{9} +O(q^{10})\) \( q + ( \zeta_{92}^{3} - \zeta_{92}^{25} ) q^{2} + \zeta_{92}^{13} q^{3} + ( -\zeta_{92}^{4} + \zeta_{92}^{6} - \zeta_{92}^{28} ) q^{4} + ( \zeta_{92}^{16} - \zeta_{92}^{38} ) q^{6} + ( -\zeta_{92}^{7} + \zeta_{92}^{9} + \zeta_{92}^{29} - \zeta_{92}^{31} ) q^{8} + \zeta_{92}^{26} q^{9} + ( -\zeta_{92}^{17} + \zeta_{92}^{19} - \zeta_{92}^{41} ) q^{12} + ( \zeta_{92}^{8} - \zeta_{92}^{10} + \zeta_{92}^{12} + \zeta_{92}^{32} - \zeta_{92}^{34} ) q^{16} + ( \zeta_{92}^{31} - \zeta_{92}^{45} ) q^{17} + ( \zeta_{92}^{5} + \zeta_{92}^{29} ) q^{18} + ( \zeta_{92}^{30} - \zeta_{92}^{40} ) q^{19} + ( -\zeta_{92}^{23} + \zeta_{92}^{41} ) q^{23} + ( -\zeta_{92}^{20} + \zeta_{92}^{22} + \zeta_{92}^{42} - \zeta_{92}^{44} ) q^{24} + \zeta_{92}^{39} q^{27} + ( -\zeta_{92}^{2} - \zeta_{92}^{18} ) q^{31} + ( \zeta_{92}^{11} - \zeta_{92}^{13} + \zeta_{92}^{15} - \zeta_{92}^{33} + \zeta_{92}^{35} - \zeta_{92}^{37} ) q^{32} + ( \zeta_{92}^{2} + \zeta_{92}^{10} - \zeta_{92}^{24} + \zeta_{92}^{34} ) q^{34} + ( \zeta_{92}^{8} - \zeta_{92}^{30} + \zeta_{92}^{32} ) q^{36} + ( \zeta_{92}^{9} - \zeta_{92}^{19} + \zeta_{92}^{33} - \zeta_{92}^{43} ) q^{38} + ( -\zeta_{92}^{2} + \zeta_{92}^{20} - \zeta_{92}^{26} + \zeta_{92}^{44} ) q^{46} -\zeta_{92}^{21} q^{47} + ( \zeta_{92} + \zeta_{92}^{21} - \zeta_{92}^{23} + \zeta_{92}^{25} + \zeta_{92}^{45} ) q^{48} + \zeta_{92}^{14} q^{49} + ( \zeta_{92}^{12} + \zeta_{92}^{44} ) q^{51} + ( \zeta_{92} - \zeta_{92}^{7} ) q^{53} + ( \zeta_{92}^{18} + \zeta_{92}^{42} ) q^{54} + ( \zeta_{92}^{7} + \zeta_{92}^{43} ) q^{57} + ( \zeta_{92}^{4} - \zeta_{92}^{38} ) q^{61} + ( -\zeta_{92}^{5} - \zeta_{92}^{21} + \zeta_{92}^{27} + \zeta_{92}^{43} ) q^{62} + ( -\zeta_{92}^{12} + \zeta_{92}^{14} - \zeta_{92}^{16} + \zeta_{92}^{18} - \zeta_{92}^{36} + \zeta_{92}^{38} - \zeta_{92}^{40} ) q^{64} + ( -\zeta_{92}^{3} + \zeta_{92}^{5} + \zeta_{92}^{13} - \zeta_{92}^{27} - \zeta_{92}^{35} + \zeta_{92}^{37} ) q^{68} + ( -\zeta_{92}^{8} - \zeta_{92}^{36} ) q^{69} + ( -\zeta_{92}^{9} + \zeta_{92}^{11} - \zeta_{92}^{33} + \zeta_{92}^{35} ) q^{72} + ( 1 + \zeta_{92}^{12} - \zeta_{92}^{22} - \zeta_{92}^{34} + \zeta_{92}^{36} + \zeta_{92}^{44} ) q^{76} + ( -\zeta_{92}^{16} - \zeta_{92}^{32} ) q^{79} -\zeta_{92}^{6} q^{81} + ( -\zeta_{92}^{27} - \zeta_{92}^{35} ) q^{83} + ( -\zeta_{92} - \zeta_{92}^{5} + \zeta_{92}^{23} + \zeta_{92}^{27} - \zeta_{92}^{29} - \zeta_{92}^{45} ) q^{92} + ( -\zeta_{92}^{15} - \zeta_{92}^{31} ) q^{93} + ( -1 - \zeta_{92}^{24} ) q^{94} + ( 1 - \zeta_{92}^{2} + \zeta_{92}^{4} + \zeta_{92}^{24} - \zeta_{92}^{26} + \zeta_{92}^{28} ) q^{96} + ( \zeta_{92}^{17} - \zeta_{92}^{39} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44q + 6q^{4} - 4q^{6} + 2q^{9} + O(q^{10}) \) \( 44q + 6q^{4} - 4q^{6} + 2q^{9} - 10q^{16} + 4q^{19} + 8q^{24} - 4q^{31} + 8q^{34} - 6q^{36} - 8q^{46} + 2q^{49} - 4q^{51} + 4q^{54} - 4q^{61} + 14q^{64} + 4q^{69} + 34q^{76} + 4q^{79} - 2q^{81} - 42q^{94} + 34q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times\).

\(n\) \(1552\) \(2026\) \(2351\)
\(\chi(n)\) \(1\) \(-\zeta_{92}^{22}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−0.942261 0.334880i
0.942261 + 0.334880i
0.398401 + 0.917211i
−0.398401 0.917211i
0.269797 + 0.962917i
−0.269797 0.962917i
0.887885 0.460065i
−0.887885 + 0.460065i
−0.997669 0.0682424i
0.997669 + 0.0682424i
−0.631088 0.775711i
0.631088 + 0.775711i
0.519584 + 0.854419i
−0.519584 0.854419i
0.136167 + 0.990686i
−0.136167 0.990686i
−0.979084 + 0.203456i
0.979084 0.203456i
0.269797 0.962917i
−0.269797 + 0.962917i
−1.15067 0.0787081i 0.269797 + 0.962917i 0.327165 + 0.0449678i 0 −0.234658 1.12924i 0 0.756317 + 0.157164i −0.854419 + 0.519584i 0
101.2 1.15067 + 0.0787081i −0.269797 0.962917i 0.327165 + 0.0449678i 0 −0.234658 1.12924i 0 −0.756317 0.157164i −0.854419 + 0.519584i 0
251.1 −0.211425 + 0.347674i −0.816970 + 0.576680i 0.383889 + 0.740871i 0 −0.0277687 0.405963i 0 −0.744708 0.0509395i 0.334880 0.942261i 0
251.2 0.211425 0.347674i 0.816970 0.576680i 0.383889 + 0.740871i 0 −0.0277687 0.405963i 0 0.744708 + 0.0509395i 0.334880 0.942261i 0
401.1 −1.25042 1.53697i −0.398401 0.917211i −0.595279 + 2.86464i 0 −0.911560 + 1.75923i 0 3.38799 1.75551i −0.682553 + 0.730836i 0
401.2 1.25042 + 1.53697i 0.398401 + 0.917211i −0.595279 + 2.86464i 0 −0.911560 + 1.75923i 0 −3.38799 + 1.75551i −0.682553 + 0.730836i 0
476.1 −0.680803 1.56737i 0.997669 + 0.0682424i −1.31059 + 1.40330i 0 −0.572255 1.61017i 0 1.48157 + 0.526549i 0.990686 + 0.136167i 0
476.2 0.680803 + 1.56737i −0.997669 0.0682424i −1.31059 + 1.40330i 0 −0.572255 1.61017i 0 −1.48157 0.526549i 0.990686 + 0.136167i 0
551.1 −1.11525 + 0.787230i −0.631088 0.775711i 0.289174 0.813657i 0 1.31448 + 0.368301i 0 −0.0502675 0.179407i −0.203456 + 0.979084i 0
551.2 1.11525 0.787230i 0.631088 + 0.775711i 0.289174 0.813657i 0 1.31448 + 0.368301i 0 0.0502675 + 0.179407i −0.203456 + 0.979084i 0
776.1 −0.0911989 0.663521i −0.519584 + 0.854419i 0.530974 0.148772i 0 0.614311 + 0.266833i 0 −0.413970 0.953056i −0.460065 0.887885i 0
776.2 0.0911989 + 0.663521i 0.519584 0.854419i 0.530974 0.148772i 0 0.614311 + 0.266833i 0 0.413970 + 0.953056i −0.460065 0.887885i 0
1001.1 −1.88555 0.391823i 0.730836 + 0.682553i 2.48458 + 1.07920i 0 −1.11059 1.57335i 0 −2.68860 1.89782i 0.0682424 + 0.997669i 0
1001.2 1.88555 + 0.391823i −0.730836 0.682553i 2.48458 + 1.07920i 0 −1.11059 1.57335i 0 2.68860 + 1.89782i 0.0682424 + 0.997669i 0
1076.1 −0.128604 + 0.0457060i 0.979084 0.203456i −0.761261 + 0.619332i 0 −0.116615 + 0.0709153i 0 0.140510 0.231058i 0.917211 0.398401i 0
1076.2 0.128604 0.0457060i −0.979084 + 0.203456i −0.761261 + 0.619332i 0 −0.116615 + 0.0709153i 0 −0.140510 + 0.231058i 0.917211 0.398401i 0
1226.1 −0.418569 + 1.49389i 0.887885 + 0.460065i −1.20209 0.731009i 0 −1.05893 + 1.13384i 0 0.461371 0.430891i 0.576680 + 0.816970i 0
1226.2 0.418569 1.49389i −0.887885 0.460065i −1.20209 0.731009i 0 −1.05893 + 1.13384i 0 −0.461371 + 0.430891i 0.576680 + 0.816970i 0
1301.1 −1.25042 + 1.53697i −0.398401 + 0.917211i −0.595279 2.86464i 0 −0.911560 1.75923i 0 3.38799 + 1.75551i −0.682553 0.730836i 0
1301.2 1.25042 1.53697i 0.398401 0.917211i −0.595279 2.86464i 0 −0.911560 1.75923i 0 −3.38799 1.75551i −0.682553 0.730836i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3401.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
47.c even 23 1 inner
141.h odd 46 1 inner
235.i even 46 1 inner
705.p odd 46 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.1.bd.a 44
3.b odd 2 1 inner 3525.1.bd.a 44
5.b even 2 1 inner 3525.1.bd.a 44
5.c odd 4 1 705.1.p.a 22
5.c odd 4 1 705.1.p.b yes 22
15.d odd 2 1 CM 3525.1.bd.a 44
15.e even 4 1 705.1.p.a 22
15.e even 4 1 705.1.p.b yes 22
47.c even 23 1 inner 3525.1.bd.a 44
141.h odd 46 1 inner 3525.1.bd.a 44
235.i even 46 1 inner 3525.1.bd.a 44
235.k odd 92 1 705.1.p.a 22
235.k odd 92 1 705.1.p.b yes 22
705.p odd 46 1 inner 3525.1.bd.a 44
705.w even 92 1 705.1.p.a 22
705.w even 92 1 705.1.p.b yes 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
705.1.p.a 22 5.c odd 4 1
705.1.p.a 22 15.e even 4 1
705.1.p.a 22 235.k odd 92 1
705.1.p.a 22 705.w even 92 1
705.1.p.b yes 22 5.c odd 4 1
705.1.p.b yes 22 15.e even 4 1
705.1.p.b yes 22 235.k odd 92 1
705.1.p.b yes 22 705.w even 92 1
3525.1.bd.a 44 1.a even 1 1 trivial
3525.1.bd.a 44 3.b odd 2 1 inner
3525.1.bd.a 44 5.b even 2 1 inner
3525.1.bd.a 44 15.d odd 2 1 CM
3525.1.bd.a 44 47.c even 23 1 inner
3525.1.bd.a 44 141.h odd 46 1 inner
3525.1.bd.a 44 235.i even 46 1 inner
3525.1.bd.a 44 705.p odd 46 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3525, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 75 T^{2} + 2244 T^{4} + 19633 T^{6} + 141757 T^{8} + 292144 T^{10} + 55534 T^{12} - 17644 T^{14} + 189032 T^{16} - 720146 T^{18} + 374444 T^{20} - 94669 T^{22} + 23788 T^{24} + 9854 T^{26} + 19145 T^{28} - 4608 T^{30} + 1152 T^{32} - 288 T^{34} + 210 T^{36} - 64 T^{38} + 16 T^{40} - 4 T^{42} + T^{44} \)
$3$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} \)
$5$ \( T^{44} \)
$7$ \( T^{44} \)
$11$ \( T^{44} \)
$13$ \( T^{44} \)
$17$ \( 1 - 6 T^{2} + 2911 T^{4} + 3510 T^{6} + 112593 T^{8} - 83239 T^{10} + 565812 T^{12} + 25688 T^{14} + 551144 T^{16} - 358908 T^{18} + 235202 T^{20} + 95932 T^{22} - 14714 T^{24} - 59353 T^{26} + 14936 T^{28} + 11952 T^{30} - 2988 T^{32} - 1668 T^{34} + 417 T^{36} + 120 T^{38} - 30 T^{40} - 4 T^{42} + T^{44} \)
$19$ \( ( 1 - 12 T + 98 T^{2} - 348 T^{3} + 519 T^{4} - 340 T^{5} + 170 T^{6} - 85 T^{7} + 54 T^{8} + 341 T^{9} + 554 T^{10} - 208 T^{11} + 104 T^{12} - 52 T^{13} + 26 T^{14} - 13 T^{15} + 41 T^{16} - 32 T^{17} + 16 T^{18} - 8 T^{19} + 4 T^{20} - 2 T^{21} + T^{22} )^{2} \)
$23$ \( 1 - 121 T^{2} + 4521 T^{4} - 51161 T^{6} + 260161 T^{8} - 628661 T^{10} + 858441 T^{12} - 561916 T^{14} + 385636 T^{16} + 91064 T^{18} + 264136 T^{20} + 271974 T^{22} + 270026 T^{24} + 218510 T^{26} + 149670 T^{28} + 85161 T^{30} + 39999 T^{32} + 15237 T^{34} + 4603 T^{36} + 1063 T^{38} + 177 T^{40} + 19 T^{42} + T^{44} \)
$29$ \( T^{44} \)
$31$ \( ( 1 + 12 T + 98 T^{2} + 348 T^{3} + 519 T^{4} + 340 T^{5} + 170 T^{6} + 85 T^{7} + 54 T^{8} - 341 T^{9} + 554 T^{10} + 208 T^{11} + 104 T^{12} + 52 T^{13} + 26 T^{14} + 13 T^{15} + 41 T^{16} + 32 T^{17} + 16 T^{18} + 8 T^{19} + 4 T^{20} + 2 T^{21} + T^{22} )^{2} \)
$37$ \( T^{44} \)
$41$ \( T^{44} \)
$43$ \( T^{44} \)
$47$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} - T^{26} + T^{28} - T^{30} + T^{32} - T^{34} + T^{36} - T^{38} + T^{40} - T^{42} + T^{44} \)
$53$ \( 1 - 29 T^{2} + 2635 T^{4} + 19817 T^{6} + 13256 T^{8} - 310663 T^{10} + 178423 T^{12} + 177971 T^{14} + 556319 T^{16} - 140316 T^{18} + 419662 T^{20} - 114173 T^{22} + 32367 T^{24} + 36120 T^{26} - 9030 T^{28} - 882 T^{30} + 232 T^{32} - 58 T^{34} + 256 T^{36} - 64 T^{38} + 16 T^{40} - 4 T^{42} + T^{44} \)
$59$ \( T^{44} \)
$61$ \( ( 1 + 12 T + 52 T^{2} + 26 T^{3} + 174 T^{4} - 396 T^{5} - 198 T^{6} + 39 T^{7} + 1158 T^{8} + 579 T^{9} + 301 T^{10} - 896 T^{11} - 448 T^{12} - 224 T^{13} + 348 T^{14} + 174 T^{15} + 87 T^{16} - 60 T^{17} - 30 T^{18} - 15 T^{19} + 4 T^{20} + 2 T^{21} + T^{22} )^{2} \)
$67$ \( T^{44} \)
$71$ \( T^{44} \)
$73$ \( T^{44} \)
$79$ \( ( 1 - 12 T + 121 T^{2} - 716 T^{3} + 2704 T^{4} - 6826 T^{5} + 11785 T^{6} - 14115 T^{7} + 11991 T^{8} - 7640 T^{9} + 4073 T^{10} - 2048 T^{11} + 1024 T^{12} - 512 T^{13} + 256 T^{14} - 128 T^{15} + 64 T^{16} - 32 T^{17} + 16 T^{18} - 8 T^{19} + 4 T^{20} - 2 T^{21} + T^{22} )^{2} \)
$83$ \( 1 + 109 T^{2} + 3486 T^{4} + 18575 T^{6} + 63442 T^{8} - 108332 T^{10} + 486163 T^{12} - 1331059 T^{14} + 2576386 T^{16} - 3480859 T^{18} + 3219636 T^{20} - 2000978 T^{22} + 838264 T^{24} - 254140 T^{26} + 65490 T^{28} - 16384 T^{30} + 4096 T^{32} - 1024 T^{34} + 256 T^{36} - 64 T^{38} + 16 T^{40} - 4 T^{42} + T^{44} \)
$89$ \( T^{44} \)
$97$ \( T^{44} \)
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