Newspace parameters
Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 775.k (of order \(5\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.18840615665\) |
Analytic rank: | \(0\) |
Dimension: | \(36\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
101.1 | −0.756851 | + | 2.32935i | −0.554304 | − | 1.70597i | −3.23501 | − | 2.35037i | 0 | 4.39333 | 2.00183 | + | 1.45441i | 3.96032 | − | 2.87734i | −0.176040 | + | 0.127900i | 0 | ||||||
101.2 | −0.679709 | + | 2.09193i | 0.127997 | + | 0.393934i | −2.29613 | − | 1.66823i | 0 | −0.911081 | −2.45457 | − | 1.78335i | 1.49152 | − | 1.08366i | 2.28825 | − | 1.66251i | 0 | ||||||
101.3 | −0.593461 | + | 1.82648i | 0.885070 | + | 2.72397i | −1.36582 | − | 0.992323i | 0 | −5.50054 | 0.299634 | + | 0.217697i | −0.484381 | + | 0.351923i | −4.20959 | + | 3.05845i | 0 | ||||||
101.4 | −0.216539 | + | 0.666437i | −0.898636 | − | 2.76572i | 1.22078 | + | 0.886952i | 0 | 2.03777 | 0.447464 | + | 0.325102i | −1.98926 | + | 1.44528i | −4.41459 | + | 3.20739i | 0 | ||||||
101.5 | −0.140176 | + | 0.431417i | 0.352463 | + | 1.08477i | 1.45156 | + | 1.05462i | 0 | −0.517395 | 1.66302 | + | 1.20825i | −1.39243 | + | 1.01166i | 1.37455 | − | 0.998672i | 0 | ||||||
101.6 | 0.237416 | − | 0.730690i | 0.201777 | + | 0.621004i | 1.14049 | + | 0.828616i | 0 | 0.501667 | −3.69321 | − | 2.68327i | 2.11936 | − | 1.53980i | 2.08212 | − | 1.51275i | 0 | ||||||
101.7 | 0.262890 | − | 0.809091i | −0.441915 | − | 1.36007i | 1.03252 | + | 0.750167i | 0 | −1.21660 | 0.894593 | + | 0.649960i | 2.25490 | − | 1.63828i | 0.772536 | − | 0.561280i | 0 | ||||||
101.8 | 0.664374 | − | 2.04473i | −0.331864 | − | 1.02137i | −2.12151 | − | 1.54137i | 0 | −2.30891 | 3.55777 | + | 2.58487i | −1.08245 | + | 0.786446i | 1.49398 | − | 1.08544i | 0 | ||||||
101.9 | 0.722056 | − | 2.22226i | −0.767639 | − | 2.36255i | −2.79904 | − | 2.03362i | 0 | −5.80447 | −4.21653 | − | 3.06349i | −2.75955 | + | 2.00493i | −2.56532 | + | 1.86381i | 0 | ||||||
126.1 | −2.10630 | − | 1.53032i | −1.44142 | + | 1.04725i | 1.47660 | + | 4.54452i | 0 | 4.63870 | −0.566161 | − | 1.74246i | 2.23532 | − | 6.87960i | 0.0539012 | − | 0.165891i | 0 | ||||||
126.2 | −1.45154 | − | 1.05460i | 0.436799 | − | 0.317353i | 0.376735 | + | 1.15947i | 0 | −0.968711 | −0.374177 | − | 1.15160i | −0.432937 | + | 1.33244i | −0.836970 | + | 2.57593i | 0 | ||||||
126.3 | −0.930073 | − | 0.675738i | 2.66102 | − | 1.93334i | −0.209619 | − | 0.645141i | 0 | −3.78137 | −0.813453 | − | 2.50355i | −0.951498 | + | 2.92841i | 2.41615 | − | 7.43615i | 0 | ||||||
126.4 | −0.750933 | − | 0.545585i | −2.30185 | + | 1.67239i | −0.351796 | − | 1.08272i | 0 | 2.64097 | 0.289318 | + | 0.890428i | −0.900201 | + | 2.77053i | 1.57457 | − | 4.84604i | 0 | ||||||
126.5 | −0.470650 | − | 0.341947i | 0.758887 | − | 0.551364i | −0.513450 | − | 1.58024i | 0 | −0.545708 | 0.739195 | + | 2.27501i | −0.658247 | + | 2.02588i | −0.655143 | + | 2.01632i | 0 | ||||||
126.6 | 0.601276 | + | 0.436853i | −1.34036 | + | 0.973831i | −0.447341 | − | 1.37678i | 0 | −1.23135 | −1.14480 | − | 3.52332i | 0.791806 | − | 2.43693i | −0.0788236 | + | 0.242594i | 0 | ||||||
126.7 | 1.19002 | + | 0.864600i | 2.40953 | − | 1.75063i | 0.0505793 | + | 0.155667i | 0 | 4.38098 | 1.26381 | + | 3.88961i | 0.834694 | − | 2.56893i | 1.81409 | − | 5.58319i | 0 | ||||||
126.8 | 1.44025 | + | 1.04641i | 1.28666 | − | 0.934816i | 0.361331 | + | 1.11206i | 0 | 2.83132 | −1.30585 | − | 4.01899i | 0.456995 | − | 1.40649i | −0.145428 | + | 0.447583i | 0 | ||||||
126.9 | 1.97795 | + | 1.43706i | −0.542210 | + | 0.393939i | 1.22909 | + | 3.78276i | 0 | −1.63858 | 0.412110 | + | 1.26835i | −1.49396 | + | 4.59795i | −0.788247 | + | 2.42597i | 0 | ||||||
326.1 | −2.10630 | + | 1.53032i | −1.44142 | − | 1.04725i | 1.47660 | − | 4.54452i | 0 | 4.63870 | −0.566161 | + | 1.74246i | 2.23532 | + | 6.87960i | 0.0539012 | + | 0.165891i | 0 | ||||||
326.2 | −1.45154 | + | 1.05460i | 0.436799 | + | 0.317353i | 0.376735 | − | 1.15947i | 0 | −0.968711 | −0.374177 | + | 1.15160i | −0.432937 | − | 1.33244i | −0.836970 | − | 2.57593i | 0 | ||||||
See all 36 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
31.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 775.2.k.f | ✓ | 36 |
5.b | even | 2 | 1 | 775.2.k.g | yes | 36 | |
5.c | odd | 4 | 2 | 775.2.bf.e | 72 | ||
31.d | even | 5 | 1 | inner | 775.2.k.f | ✓ | 36 |
155.n | even | 10 | 1 | 775.2.k.g | yes | 36 | |
155.s | odd | 20 | 2 | 775.2.bf.e | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
775.2.k.f | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
775.2.k.f | ✓ | 36 | 31.d | even | 5 | 1 | inner |
775.2.k.g | yes | 36 | 5.b | even | 2 | 1 | |
775.2.k.g | yes | 36 | 155.n | even | 10 | 1 | |
775.2.bf.e | 72 | 5.c | odd | 4 | 2 | ||
775.2.bf.e | 72 | 155.s | odd | 20 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{36} + 2 T_{2}^{35} + 16 T_{2}^{34} + 24 T_{2}^{33} + 141 T_{2}^{32} + 188 T_{2}^{31} + 1034 T_{2}^{30} + 1240 T_{2}^{29} + 6847 T_{2}^{28} + 7973 T_{2}^{27} + 34561 T_{2}^{26} + 36292 T_{2}^{25} + 128399 T_{2}^{24} + \cdots + 22201 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).