# Properties

 Label 770.2.w.b Level $770$ Weight $2$ Character orbit 770.w Analytic conductor $6.148$ Analytic rank $0$ Dimension $96$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.w (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$24$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 24 q^{2} - 24 q^{4} + 3 q^{7} + 24 q^{8} - 18 q^{9}+O(q^{10})$$ 96 * q + 24 * q^2 - 24 * q^4 + 3 * q^7 + 24 * q^8 - 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$96 q + 24 q^{2} - 24 q^{4} + 3 q^{7} + 24 q^{8} - 18 q^{9} + 4 q^{11} - 3 q^{14} - 6 q^{15} - 24 q^{16} + 28 q^{18} - 4 q^{22} + 38 q^{25} - 2 q^{28} + 6 q^{30} - 96 q^{32} + 46 q^{35} - 28 q^{36} + 10 q^{39} - 24 q^{43} - 16 q^{44} + 37 q^{49} - 8 q^{50} - 30 q^{51} + 40 q^{53} + 2 q^{56} - 20 q^{57} + 14 q^{60} + 63 q^{63} - 24 q^{64} - 40 q^{65} + 39 q^{70} - 20 q^{71} + 18 q^{72} - 16 q^{77} - 126 q^{81} + 60 q^{85} - 36 q^{86} - 14 q^{88} - 31 q^{91} - 10 q^{92} + 20 q^{93} - 44 q^{95} + 18 q^{98} + 100 q^{99}+O(q^{100})$$ 96 * q + 24 * q^2 - 24 * q^4 + 3 * q^7 + 24 * q^8 - 18 * q^9 + 4 * q^11 - 3 * q^14 - 6 * q^15 - 24 * q^16 + 28 * q^18 - 4 * q^22 + 38 * q^25 - 2 * q^28 + 6 * q^30 - 96 * q^32 + 46 * q^35 - 28 * q^36 + 10 * q^39 - 24 * q^43 - 16 * q^44 + 37 * q^49 - 8 * q^50 - 30 * q^51 + 40 * q^53 + 2 * q^56 - 20 * q^57 + 14 * q^60 + 63 * q^63 - 24 * q^64 - 40 * q^65 + 39 * q^70 - 20 * q^71 + 18 * q^72 - 16 * q^77 - 126 * q^81 + 60 * q^85 - 36 * q^86 - 14 * q^88 - 31 * q^91 - 10 * q^92 + 20 * q^93 - 44 * q^95 + 18 * q^98 + 100 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 0.809017 + 0.587785i −0.968385 + 2.98038i 0.309017 + 0.951057i −2.19353 0.434105i −2.53526 + 1.84198i −2.64551 0.0356561i −0.309017 + 0.951057i −5.51786 4.00896i −1.51944 1.64052i
139.2 0.809017 + 0.587785i −0.934488 + 2.87606i 0.309017 + 0.951057i 2.04566 + 0.902928i −2.44652 + 1.77750i 2.63792 0.203429i −0.309017 + 0.951057i −4.97140 3.61193i 1.12425 + 1.93289i
139.3 0.809017 + 0.587785i −0.822398 + 2.53108i 0.309017 + 0.951057i −2.15934 0.580738i −2.15307 + 1.56429i 2.22188 + 1.43641i −0.309017 + 0.951057i −3.30298 2.39976i −1.40559 1.73905i
139.4 0.809017 + 0.587785i −0.732958 + 2.25581i 0.309017 + 0.951057i 0.464104 + 2.18737i −1.91891 + 1.39417i −2.63063 0.282464i −0.309017 + 0.951057i −2.12441 1.54347i −0.910238 + 2.04242i
139.5 0.809017 + 0.587785i −0.704096 + 2.16698i 0.309017 + 0.951057i 1.29948 1.81971i −1.84335 + 1.33927i −0.555990 2.58667i −0.309017 + 0.951057i −1.77302 1.28817i 2.12090 0.708357i
139.6 0.809017 + 0.587785i −0.497687 + 1.53172i 0.309017 + 0.951057i 2.22362 0.235570i −1.30296 + 0.946658i −1.37179 + 2.26234i −0.309017 + 0.951057i 0.328565 + 0.238717i 1.93741 + 1.11643i
139.7 0.809017 + 0.587785i −0.463552 + 1.42667i 0.309017 + 0.951057i −0.665551 + 2.13472i −1.21359 + 0.881728i 0.576225 + 2.58224i −0.309017 + 0.951057i 0.606556 + 0.440689i −1.79320 + 1.33583i
139.8 0.809017 + 0.587785i −0.407445 + 1.25399i 0.309017 + 0.951057i −1.72447 1.42345i −1.06670 + 0.775006i 2.48344 0.912422i −0.309017 + 0.951057i 1.02058 + 0.741496i −0.558447 2.16521i
139.9 0.809017 + 0.587785i −0.403178 + 1.24085i 0.309017 + 0.951057i 2.07187 0.841043i −1.05553 + 0.766890i 2.64445 + 0.0829819i −0.309017 + 0.951057i 1.04989 + 0.762787i 2.17053 + 0.537397i
139.10 0.809017 + 0.587785i −0.266549 + 0.820354i 0.309017 + 0.951057i −0.466255 2.18692i −0.697835 + 0.507006i −1.52414 + 2.16264i −0.309017 + 0.951057i 1.82512 + 1.32603i 0.908230 2.04331i
139.11 0.809017 + 0.587785i −0.208297 + 0.641072i 0.309017 + 0.951057i −1.77458 + 1.36046i −0.545328 + 0.396204i −1.13890 2.38807i −0.309017 + 0.951057i 2.05947 + 1.49629i −2.23533 + 0.0575660i
139.12 0.809017 + 0.587785i −0.0561872 + 0.172927i 0.309017 + 0.951057i −1.27672 1.83575i −0.147100 + 0.106874i −2.32414 1.26426i −0.309017 + 0.951057i 2.40030 + 1.74392i 0.0461423 2.23559i
139.13 0.809017 + 0.587785i 0.0561872 0.172927i 0.309017 + 0.951057i 1.27672 + 1.83575i 0.147100 0.106874i 1.13715 2.38891i −0.309017 + 0.951057i 2.40030 + 1.74392i −0.0461423 + 2.23559i
139.14 0.809017 + 0.587785i 0.208297 0.641072i 0.309017 + 0.951057i 1.77458 1.36046i 0.545328 0.396204i −0.482283 2.60142i −0.309017 + 0.951057i 2.05947 + 1.49629i 2.23533 0.0575660i
139.15 0.809017 + 0.587785i 0.266549 0.820354i 0.309017 + 0.951057i 0.466255 + 2.18692i 0.697835 0.507006i 2.50422 + 0.853746i −0.309017 + 0.951057i 1.82512 + 1.32603i −0.908230 + 2.04331i
139.16 0.809017 + 0.587785i 0.403178 1.24085i 0.309017 + 0.951057i −2.07187 + 0.841043i 1.05553 0.766890i −2.09063 + 1.62150i −0.309017 + 0.951057i 1.04989 + 0.762787i −2.17053 0.537397i
139.17 0.809017 + 0.587785i 0.407445 1.25399i 0.309017 + 0.951057i 1.72447 + 1.42345i 1.06670 0.775006i −2.54546 + 0.721566i −0.309017 + 0.951057i 1.02058 + 0.741496i 0.558447 + 2.16521i
139.18 0.809017 + 0.587785i 0.463552 1.42667i 0.309017 + 0.951057i 0.665551 2.13472i 1.21359 0.881728i 1.05163 + 2.42777i −0.309017 + 0.951057i 0.606556 + 0.440689i 1.79320 1.33583i
139.19 0.809017 + 0.587785i 0.497687 1.53172i 0.309017 + 0.951057i −2.22362 + 0.235570i 1.30296 0.946658i 2.43957 + 1.02395i −0.309017 + 0.951057i 0.328565 + 0.238717i −1.93741 1.11643i
139.20 0.809017 + 0.587785i 0.704096 2.16698i 0.309017 + 0.951057i −1.29948 + 1.81971i 1.84335 1.33927i −1.07060 2.41946i −0.309017 + 0.951057i −1.77302 1.28817i −2.12090 + 0.708357i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
55.h odd 10 1 inner
385.v even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.w.b yes 96
5.b even 2 1 770.2.w.a 96
7.b odd 2 1 inner 770.2.w.b yes 96
11.d odd 10 1 770.2.w.a 96
35.c odd 2 1 770.2.w.a 96
55.h odd 10 1 inner 770.2.w.b yes 96
77.l even 10 1 770.2.w.a 96
385.v even 10 1 inner 770.2.w.b yes 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.w.a 96 5.b even 2 1
770.2.w.a 96 11.d odd 10 1
770.2.w.a 96 35.c odd 2 1
770.2.w.a 96 77.l even 10 1
770.2.w.b yes 96 1.a even 1 1 trivial
770.2.w.b yes 96 7.b odd 2 1 inner
770.2.w.b yes 96 55.h odd 10 1 inner
770.2.w.b yes 96 385.v even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{37}^{48} - 86 T_{37}^{46} - 605 T_{37}^{45} + 23337 T_{37}^{44} + 52030 T_{37}^{43} - 2995717 T_{37}^{42} - 5554150 T_{37}^{41} + 344028691 T_{37}^{40} + 354526025 T_{37}^{39} - 33649229469 T_{37}^{38} + \cdots + 88\!\cdots\!56$$ acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$.