# Properties

 Label 260.2.p.d Level $260$ Weight $2$ Character orbit 260.p Analytic conductor $2.076$ Analytic rank $0$ Dimension $64$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64 q+O(q^{10})$$ 64 * q $$\operatorname{Tr}(f)(q) =$$ $$64 q - 24 q^{12} - 4 q^{16} - 40 q^{17} - 36 q^{22} + 44 q^{26} + 24 q^{30} + 28 q^{36} + 16 q^{38} - 44 q^{40} + 8 q^{42} - 44 q^{48} + 56 q^{52} - 48 q^{53} - 64 q^{56} + 80 q^{61} + 20 q^{62} - 72 q^{65} - 24 q^{66} - 76 q^{68} - 112 q^{77} - 20 q^{78} + 80 q^{81} + 52 q^{82} - 152 q^{88} - 64 q^{90} + 56 q^{92}+O(q^{100})$$ 64 * q - 24 * q^12 - 4 * q^16 - 40 * q^17 - 36 * q^22 + 44 * q^26 + 24 * q^30 + 28 * q^36 + 16 * q^38 - 44 * q^40 + 8 * q^42 - 44 * q^48 + 56 * q^52 - 48 * q^53 - 64 * q^56 + 80 * q^61 + 20 * q^62 - 72 * q^65 - 24 * q^66 - 76 * q^68 - 112 * q^77 - 20 * q^78 + 80 * q^81 + 52 * q^82 - 152 * q^88 - 64 * q^90 + 56 * q^92

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
103.1 −1.41384 0.0324895i −2.32949 + 2.32949i 1.99789 + 0.0918700i 2.23599 + 0.0191440i 3.36921 3.21784i 1.97121 1.97121i −2.82171 0.194800i 7.85303i −3.16071 0.0997126i
103.2 −1.36804 0.358430i 1.80624 1.80624i 1.74306 + 0.980692i 1.12997 1.92955i −3.11841 + 1.82359i −0.163803 + 0.163803i −2.03306 1.96639i 3.52499i −2.23745 + 2.23468i
103.3 −1.35124 + 0.417327i 0.561614 0.561614i 1.65168 1.12781i −2.09189 + 0.789927i −0.524497 + 0.993250i 1.48711 1.48711i −1.76114 + 2.21323i 2.36918i 2.49698 1.94038i
103.4 −1.31170 + 0.528615i −0.834468 + 0.834468i 1.44113 1.38677i −0.462357 2.18774i 0.653463 1.53569i −1.21162 + 1.21162i −1.15727 + 2.58084i 1.60733i 1.76295 + 2.62526i
103.5 −1.27985 + 0.601658i 1.55255 1.55255i 1.27602 1.54006i 1.95736 + 1.08109i −1.05293 + 2.92114i 3.02953 3.02953i −0.706517 + 2.73876i 1.82084i −3.15556 0.205972i
103.6 −1.23237 0.693735i −0.828304 + 0.828304i 1.03746 + 1.70987i −1.31937 + 1.80534i 1.59540 0.446152i 2.94084 2.94084i −0.0923349 2.82692i 1.62783i 2.87838 1.30955i
103.7 −1.22720 0.702829i −1.63423 + 1.63423i 1.01206 + 1.72503i −1.85457 1.24923i 3.15411 0.856948i −1.80181 + 1.80181i −0.0296052 2.82827i 2.34139i 1.39793 + 2.83651i
103.8 −1.10269 + 0.885485i −1.59124 + 1.59124i 0.431834 1.95282i −0.349775 + 2.20854i 0.345619 3.16365i −1.15539 + 1.15539i 1.25302 + 2.53573i 2.06407i −1.56994 2.74505i
103.9 −0.885485 + 1.10269i 1.59124 1.59124i −0.431834 1.95282i 0.349775 2.20854i 0.345619 + 3.16365i −1.15539 + 1.15539i 2.53573 + 1.25302i 2.06407i 2.12561 + 2.34132i
103.10 −0.702829 1.22720i 1.63423 1.63423i −1.01206 + 1.72503i −1.85457 1.24923i −3.15411 0.856948i 1.80181 1.80181i 2.82827 + 0.0296052i 2.34139i −0.229622 + 3.15393i
103.11 −0.693735 1.23237i 0.828304 0.828304i −1.03746 + 1.70987i −1.31937 + 1.80534i −1.59540 0.446152i −2.94084 + 2.94084i 2.82692 + 0.0923349i 1.62783i 3.14014 + 0.373519i
103.12 −0.601658 + 1.27985i −1.55255 + 1.55255i −1.27602 1.54006i −1.95736 1.08109i −1.05293 2.92114i 3.02953 3.02953i 2.73876 0.706517i 1.82084i 2.56129 1.85467i
103.13 −0.528615 + 1.31170i 0.834468 0.834468i −1.44113 1.38677i 0.462357 + 2.18774i 0.653463 + 1.53569i −1.21162 + 1.21162i 2.58084 1.15727i 1.60733i −3.11408 0.549999i
103.14 −0.417327 + 1.35124i −0.561614 + 0.561614i −1.65168 1.12781i 2.09189 0.789927i −0.524497 0.993250i 1.48711 1.48711i 2.21323 1.76114i 2.36918i 0.194375 + 3.15630i
103.15 −0.358430 1.36804i −1.80624 + 1.80624i −1.74306 + 0.980692i 1.12997 1.92955i 3.11841 + 1.82359i 0.163803 0.163803i 1.96639 + 2.03306i 3.52499i −3.04471 0.854237i
103.16 −0.0324895 1.41384i 2.32949 2.32949i −1.99789 + 0.0918700i 2.23599 + 0.0191440i −3.36921 3.21784i −1.97121 + 1.97121i 0.194800 + 2.82171i 7.85303i −0.0455796 3.16195i
103.17 0.0324895 + 1.41384i 2.32949 2.32949i −1.99789 + 0.0918700i −2.23599 0.0191440i 3.36921 + 3.21784i 1.97121 1.97121i −0.194800 2.82171i 7.85303i −0.0455796 3.16195i
103.18 0.358430 + 1.36804i −1.80624 + 1.80624i −1.74306 + 0.980692i −1.12997 + 1.92955i −3.11841 1.82359i −0.163803 + 0.163803i −1.96639 2.03306i 3.52499i −3.04471 0.854237i
103.19 0.417327 1.35124i −0.561614 + 0.561614i −1.65168 1.12781i −2.09189 + 0.789927i 0.524497 + 0.993250i −1.48711 + 1.48711i −2.21323 + 1.76114i 2.36918i 0.194375 + 3.15630i
103.20 0.528615 1.31170i 0.834468 0.834468i −1.44113 1.38677i −0.462357 2.18774i −0.653463 1.53569i 1.21162 1.21162i −2.58084 + 1.15727i 1.60733i −3.11408 0.549999i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 207.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
20.e even 4 1 inner
52.b odd 2 1 inner
65.h odd 4 1 inner
260.p even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.p.d 64
4.b odd 2 1 inner 260.2.p.d 64
5.c odd 4 1 inner 260.2.p.d 64
13.b even 2 1 inner 260.2.p.d 64
20.e even 4 1 inner 260.2.p.d 64
52.b odd 2 1 inner 260.2.p.d 64
65.h odd 4 1 inner 260.2.p.d 64
260.p even 4 1 inner 260.2.p.d 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.p.d 64 1.a even 1 1 trivial
260.2.p.d 64 4.b odd 2 1 inner
260.2.p.d 64 5.c odd 4 1 inner
260.2.p.d 64 13.b even 2 1 inner
260.2.p.d 64 20.e even 4 1 inner
260.2.p.d 64 52.b odd 2 1 inner
260.2.p.d 64 65.h odd 4 1 inner
260.2.p.d 64 260.p even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(260, [\chi])$$:

 $$T_{3}^{32} + 242 T_{3}^{28} + 20429 T_{3}^{24} + 807668 T_{3}^{20} + 15868452 T_{3}^{16} + 142690872 T_{3}^{12} + 426686656 T_{3}^{8} + 459588976 T_{3}^{4} + 123921424$$ T3^32 + 242*T3^28 + 20429*T3^24 + 807668*T3^20 + 15868452*T3^16 + 142690872*T3^12 + 426686656*T3^8 + 459588976*T3^4 + 123921424 $$T_{7}^{32} + 774 T_{7}^{28} + 195049 T_{7}^{24} + 18186004 T_{7}^{20} + 742170832 T_{7}^{16} + 13687235424 T_{7}^{12} + 109263546432 T_{7}^{8} + 308856934656 T_{7}^{4} + \cdots + 888516864$$ T7^32 + 774*T7^28 + 195049*T7^24 + 18186004*T7^20 + 742170832*T7^16 + 13687235424*T7^12 + 109263546432*T7^8 + 308856934656*T7^4 + 888516864 $$T_{37}^{32} + 8258 T_{37}^{28} + 11169457 T_{37}^{24} + 5348990192 T_{37}^{20} + 1048121669152 T_{37}^{16} + 93888672291936 T_{37}^{12} + \cdots + 98\!\cdots\!56$$ T37^32 + 8258*T37^28 + 11169457*T37^24 + 5348990192*T37^20 + 1048121669152*T37^16 + 93888672291936*T37^12 + 3843927144027648*T37^8 + 58594009261632768*T37^4 + 9888469522534656