Newspace parameters
Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 605.m (of order \(20\), degree \(8\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.83094932229\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 55) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
112.1 | −0.665529 | − | 1.30617i | −0.130227 | + | 0.822224i | −0.0875924 | + | 0.120561i | −1.11862 | − | 1.93615i | 1.16064 | − | 0.377114i | 4.16343 | − | 0.659422i | −2.68004 | − | 0.424477i | 2.19408 | + | 0.712899i | −1.78448 | + | 2.74968i |
112.2 | 0.261423 | + | 0.513072i | 0.120415 | − | 0.760272i | 0.980670 | − | 1.34978i | −1.76986 | + | 1.36660i | 0.421554 | − | 0.136971i | −1.17850 | + | 0.186656i | 2.08639 | + | 0.330452i | 2.28966 | + | 0.743954i | −1.16385 | − | 0.550801i |
112.3 | 0.474334 | + | 0.930933i | −0.440550 | + | 2.78152i | 0.533928 | − | 0.734888i | 2.23541 | + | 0.0540419i | −2.79838 | + | 0.909249i | 0.543058 | − | 0.0860119i | 3.00128 | + | 0.475357i | −4.68963 | − | 1.52375i | 1.01002 | + | 2.10665i |
112.4 | 1.15100 | + | 2.25897i | 0.313634 | − | 1.98021i | −2.60258 | + | 3.58214i | 1.91314 | + | 1.15755i | 4.83422 | − | 1.57073i | 1.78576 | − | 0.282837i | −6.07934 | − | 0.962874i | −0.969677 | − | 0.315067i | −0.412838 | + | 5.65406i |
118.1 | −1.50135 | − | 0.237790i | −0.361933 | + | 0.710333i | 0.295389 | + | 0.0959778i | −1.89470 | − | 1.18749i | 0.712297 | − | 0.980393i | −0.170602 | + | 0.0869260i | 2.28811 | + | 1.16585i | 1.38978 | + | 1.91287i | 2.56223 | + | 2.23337i |
118.2 | −0.482327 | − | 0.0763931i | 0.517260 | − | 1.01518i | −1.67531 | − | 0.544341i | 2.05331 | + | 0.885381i | −0.327041 | + | 0.450133i | 2.59854 | − | 1.32402i | 1.63669 | + | 0.833936i | 1.00032 | + | 1.37683i | −0.922733 | − | 0.583903i |
118.3 | 1.22307 | + | 0.193716i | 1.15501 | − | 2.26684i | −0.443732 | − | 0.144177i | −2.23029 | + | 0.160637i | 1.85178 | − | 2.54876i | −3.09022 | + | 1.57455i | −2.72149 | − | 1.38667i | −2.04114 | − | 2.80938i | −2.75893 | − | 0.235572i |
118.4 | 2.40264 | + | 0.380541i | −0.271495 | + | 0.532840i | 3.72576 | + | 1.21057i | 1.85044 | − | 1.25533i | −0.855073 | + | 1.17691i | −2.30033 | + | 1.17208i | 4.15610 | + | 2.11764i | 1.55315 | + | 2.13772i | 4.92366 | − | 2.31195i |
233.1 | −1.30617 | + | 0.665529i | −0.822224 | − | 0.130227i | 0.0875924 | − | 0.120561i | −0.233059 | − | 2.22389i | 1.16064 | − | 0.377114i | −0.659422 | − | 4.16343i | 0.424477 | − | 2.68004i | −2.19408 | − | 0.712899i | 1.78448 | + | 2.74968i |
233.2 | 0.513072 | − | 0.261423i | 0.760272 | + | 0.120415i | −0.980670 | + | 1.34978i | 2.23511 | + | 0.0653109i | 0.421554 | − | 0.136971i | 0.186656 | + | 1.17850i | −0.330452 | + | 2.08639i | −2.28966 | − | 0.743954i | 1.16385 | − | 0.550801i |
233.3 | 0.930933 | − | 0.474334i | −2.78152 | − | 0.440550i | −0.533928 | + | 0.734888i | −1.77672 | + | 1.35766i | −2.79838 | + | 0.909249i | −0.0860119 | − | 0.543058i | −0.475357 | + | 3.00128i | 4.68963 | + | 1.52375i | −1.01002 | + | 2.10665i |
233.4 | 2.25897 | − | 1.15100i | 1.98021 | + | 0.313634i | 2.60258 | − | 3.58214i | −0.867371 | + | 2.06099i | 4.83422 | − | 1.57073i | −0.282837 | − | 1.78576i | 0.962874 | − | 6.07934i | 0.969677 | + | 0.315067i | 0.412838 | + | 5.65406i |
282.1 | −1.50135 | + | 0.237790i | −0.361933 | − | 0.710333i | 0.295389 | − | 0.0959778i | −1.89470 | + | 1.18749i | 0.712297 | + | 0.980393i | −0.170602 | − | 0.0869260i | 2.28811 | − | 1.16585i | 1.38978 | − | 1.91287i | 2.56223 | − | 2.23337i |
282.2 | −0.482327 | + | 0.0763931i | 0.517260 | + | 1.01518i | −1.67531 | + | 0.544341i | 2.05331 | − | 0.885381i | −0.327041 | − | 0.450133i | 2.59854 | + | 1.32402i | 1.63669 | − | 0.833936i | 1.00032 | − | 1.37683i | −0.922733 | + | 0.583903i |
282.3 | 1.22307 | − | 0.193716i | 1.15501 | + | 2.26684i | −0.443732 | + | 0.144177i | −2.23029 | − | 0.160637i | 1.85178 | + | 2.54876i | −3.09022 | − | 1.57455i | −2.72149 | + | 1.38667i | −2.04114 | + | 2.80938i | −2.75893 | + | 0.235572i |
282.4 | 2.40264 | − | 0.380541i | −0.271495 | − | 0.532840i | 3.72576 | − | 1.21057i | 1.85044 | + | 1.25533i | −0.855073 | − | 1.17691i | −2.30033 | − | 1.17208i | 4.15610 | − | 2.11764i | 1.55315 | − | 2.13772i | 4.92366 | + | 2.31195i |
403.1 | −0.380541 | − | 2.40264i | 0.532840 | − | 0.271495i | −3.72576 | + | 1.21057i | −0.622076 | − | 2.14779i | −0.855073 | − | 1.17691i | −1.17208 | + | 2.30033i | 2.11764 | + | 4.15610i | −1.55315 | + | 2.13772i | −4.92366 | + | 2.31195i |
403.2 | −0.193716 | − | 1.22307i | −2.26684 | + | 1.15501i | 0.443732 | − | 0.144177i | −0.536423 | + | 2.17077i | 1.85178 | + | 2.54876i | −1.57455 | + | 3.09022i | −1.38667 | − | 2.72149i | 2.04114 | − | 2.80938i | 2.75893 | + | 0.235572i |
403.3 | 0.0763931 | + | 0.482327i | −1.01518 | + | 0.517260i | 1.67531 | − | 0.544341i | 1.47656 | − | 1.67922i | −0.327041 | − | 0.450133i | 1.32402 | − | 2.59854i | 0.833936 | + | 1.63669i | −1.00032 | + | 1.37683i | 0.922733 | + | 0.583903i |
403.4 | 0.237790 | + | 1.50135i | 0.710333 | − | 0.361933i | −0.295389 | + | 0.0959778i | −1.71486 | + | 1.43501i | 0.712297 | + | 0.980393i | −0.0869260 | + | 0.170602i | 1.16585 | + | 2.28811i | −1.38978 | + | 1.91287i | −2.56223 | − | 2.23337i |
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.d | odd | 10 | 1 | inner |
55.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 605.2.m.e | 32 | |
5.c | odd | 4 | 1 | inner | 605.2.m.e | 32 | |
11.b | odd | 2 | 1 | 55.2.l.a | ✓ | 32 | |
11.c | even | 5 | 1 | 55.2.l.a | ✓ | 32 | |
11.c | even | 5 | 1 | 605.2.e.b | 32 | ||
11.c | even | 5 | 1 | 605.2.m.c | 32 | ||
11.c | even | 5 | 1 | 605.2.m.d | 32 | ||
11.d | odd | 10 | 1 | 605.2.e.b | 32 | ||
11.d | odd | 10 | 1 | 605.2.m.c | 32 | ||
11.d | odd | 10 | 1 | 605.2.m.d | 32 | ||
11.d | odd | 10 | 1 | inner | 605.2.m.e | 32 | |
33.d | even | 2 | 1 | 495.2.bj.a | 32 | ||
33.h | odd | 10 | 1 | 495.2.bj.a | 32 | ||
44.c | even | 2 | 1 | 880.2.cm.a | 32 | ||
44.h | odd | 10 | 1 | 880.2.cm.a | 32 | ||
55.d | odd | 2 | 1 | 275.2.bm.b | 32 | ||
55.e | even | 4 | 1 | 55.2.l.a | ✓ | 32 | |
55.e | even | 4 | 1 | 275.2.bm.b | 32 | ||
55.j | even | 10 | 1 | 275.2.bm.b | 32 | ||
55.k | odd | 20 | 1 | 55.2.l.a | ✓ | 32 | |
55.k | odd | 20 | 1 | 275.2.bm.b | 32 | ||
55.k | odd | 20 | 1 | 605.2.e.b | 32 | ||
55.k | odd | 20 | 1 | 605.2.m.c | 32 | ||
55.k | odd | 20 | 1 | 605.2.m.d | 32 | ||
55.l | even | 20 | 1 | 605.2.e.b | 32 | ||
55.l | even | 20 | 1 | 605.2.m.c | 32 | ||
55.l | even | 20 | 1 | 605.2.m.d | 32 | ||
55.l | even | 20 | 1 | inner | 605.2.m.e | 32 | |
165.l | odd | 4 | 1 | 495.2.bj.a | 32 | ||
165.v | even | 20 | 1 | 495.2.bj.a | 32 | ||
220.i | odd | 4 | 1 | 880.2.cm.a | 32 | ||
220.v | even | 20 | 1 | 880.2.cm.a | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
55.2.l.a | ✓ | 32 | 11.b | odd | 2 | 1 | |
55.2.l.a | ✓ | 32 | 11.c | even | 5 | 1 | |
55.2.l.a | ✓ | 32 | 55.e | even | 4 | 1 | |
55.2.l.a | ✓ | 32 | 55.k | odd | 20 | 1 | |
275.2.bm.b | 32 | 55.d | odd | 2 | 1 | ||
275.2.bm.b | 32 | 55.e | even | 4 | 1 | ||
275.2.bm.b | 32 | 55.j | even | 10 | 1 | ||
275.2.bm.b | 32 | 55.k | odd | 20 | 1 | ||
495.2.bj.a | 32 | 33.d | even | 2 | 1 | ||
495.2.bj.a | 32 | 33.h | odd | 10 | 1 | ||
495.2.bj.a | 32 | 165.l | odd | 4 | 1 | ||
495.2.bj.a | 32 | 165.v | even | 20 | 1 | ||
605.2.e.b | 32 | 11.c | even | 5 | 1 | ||
605.2.e.b | 32 | 11.d | odd | 10 | 1 | ||
605.2.e.b | 32 | 55.k | odd | 20 | 1 | ||
605.2.e.b | 32 | 55.l | even | 20 | 1 | ||
605.2.m.c | 32 | 11.c | even | 5 | 1 | ||
605.2.m.c | 32 | 11.d | odd | 10 | 1 | ||
605.2.m.c | 32 | 55.k | odd | 20 | 1 | ||
605.2.m.c | 32 | 55.l | even | 20 | 1 | ||
605.2.m.d | 32 | 11.c | even | 5 | 1 | ||
605.2.m.d | 32 | 11.d | odd | 10 | 1 | ||
605.2.m.d | 32 | 55.k | odd | 20 | 1 | ||
605.2.m.d | 32 | 55.l | even | 20 | 1 | ||
605.2.m.e | 32 | 1.a | even | 1 | 1 | trivial | |
605.2.m.e | 32 | 5.c | odd | 4 | 1 | inner | |
605.2.m.e | 32 | 11.d | odd | 10 | 1 | inner | |
605.2.m.e | 32 | 55.l | even | 20 | 1 | inner | |
880.2.cm.a | 32 | 44.c | even | 2 | 1 | ||
880.2.cm.a | 32 | 44.h | odd | 10 | 1 | ||
880.2.cm.a | 32 | 220.i | odd | 4 | 1 | ||
880.2.cm.a | 32 | 220.v | even | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{32} - 10 T_{2}^{31} + 50 T_{2}^{30} - 170 T_{2}^{29} + 430 T_{2}^{28} - 800 T_{2}^{27} + 950 T_{2}^{26} - 60 T_{2}^{25} - 2445 T_{2}^{24} + 4890 T_{2}^{23} - 4950 T_{2}^{22} + 1500 T_{2}^{21} + 9160 T_{2}^{20} - 19450 T_{2}^{19} + \cdots + 625 \)
acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).