Properties

Label 4.33.b.a
Level 4
Weight 33
Character orbit 4.b
Self dual Yes
Analytic conductor 25.947
Analytic rank 0
Dimension 1
CM disc. -4
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 33 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(25.9466620569\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 65536q^{2} \) \(\mathstrut +\mathstrut 4294967296q^{4} \) \(\mathstrut -\mathstrut 196496109694q^{5} \) \(\mathstrut +\mathstrut 281474976710656q^{8} \) \(\mathstrut +\mathstrut 1853020188851841q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 65536q^{2} \) \(\mathstrut +\mathstrut 4294967296q^{4} \) \(\mathstrut -\mathstrut 196496109694q^{5} \) \(\mathstrut +\mathstrut 281474976710656q^{8} \) \(\mathstrut +\mathstrut 1853020188851841q^{9} \) \(\mathstrut -\mathstrut 12877569044905984q^{10} \) \(\mathstrut +\mathstrut 1330087744899070082q^{13} \) \(\mathstrut +\mathstrut 18446744073709551616q^{16} \) \(\mathstrut +\mathstrut 1427124567881986562q^{17} \) \(\mathstrut +\mathstrut 121439531096594251776q^{18} \) \(\mathstrut -\mathstrut 843944364926958567424q^{20} \) \(\mathstrut +\mathstrut 15327656759489517883011q^{25} \) \(\mathstrut +\mathstrut 87168630449705456893952q^{26} \) \(\mathstrut +\mathstrut 462878764200641031680642q^{29} \) \(\mathstrut +\mathstrut 1208925819614629174706176q^{32} \) \(\mathstrut +\mathstrut 93528035680713871327232q^{34} \) \(\mathstrut +\mathstrut 7958661109946400884391936q^{36} \) \(\mathstrut +\mathstrut 13364603395730595798238082q^{37} \) \(\mathstrut -\mathstrut 55308737899853156674699264q^{40} \) \(\mathstrut -\mathstrut 117479780930606773712920318q^{41} \) \(\mathstrut -\mathstrut 364111258293827945049846654q^{45} \) \(\mathstrut +\mathstrut 1104427674243920646305299201q^{49} \) \(\mathstrut +\mathstrut 1004513313389905043981008896q^{50} \) \(\mathstrut +\mathstrut 5712683365151896823002038272q^{52} \) \(\mathstrut -\mathstrut 6730923570418671225390264958q^{53} \) \(\mathstrut +\mathstrut 30335222690653210652222554112q^{58} \) \(\mathstrut -\mathstrut 71388302423745245269146679678q^{61} \) \(\mathstrut +\mathstrut 79228162514264337593543950336q^{64} \) \(\mathstrut -\mathstrut 261357067424332763791265574908q^{65} \) \(\mathstrut +\mathstrut 6129453346371264271301476352q^{68} \) \(\mathstrut +\mathstrut 521578814501447328359509917696q^{72} \) \(\mathstrut +\mathstrut 606523506722898114294021899522q^{73} \) \(\mathstrut +\mathstrut 875862648142600326233330941952q^{74} \) \(\mathstrut -\mathstrut 3624713447004776475833090965504q^{80} \) \(\mathstrut +\mathstrut 3433683820292512484657849089281q^{81} \) \(\mathstrut -\mathstrut 7699154923068245522049945960448q^{82} \) \(\mathstrut -\mathstrut 280424425637541180733385932028q^{85} \) \(\mathstrut +\mathstrut 17408498582555430732193248126722q^{89} \) \(\mathstrut -\mathstrut 23862395423544308206786750316544q^{90} \) \(\mathstrut +\mathstrut 84097663609016849910038850800642q^{97} \) \(\mathstrut +\mathstrut 72379772059249583476264088436736q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
65536.0 0 4.29497e9 −1.96496e11 0 0 2.81475e14 1.85302e15 −1.28776e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{33}^{\mathrm{new}}(4, [\chi])\).