Properties

 Label 3850.2.c.d Level $3850$ Weight $2$ Character orbit 3850.c Analytic conductor $30.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} + i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + 2*i * q^3 - q^4 - 2 * q^6 + i * q^7 - i * q^8 - q^9 $$q + i q^{2} + 2 i q^{3} - q^{4} - 2 q^{6} + i q^{7} - i q^{8} - q^{9} + q^{11} - 2 i q^{12} - 4 i q^{13} - q^{14} + q^{16} - i q^{18} - 4 q^{19} - 2 q^{21} + i q^{22} + 4 i q^{23} + 2 q^{24} + 4 q^{26} + 4 i q^{27} - i q^{28} - 2 q^{29} - 10 q^{31} + i q^{32} + 2 i q^{33} + q^{36} + 6 i q^{37} - 4 i q^{38} + 8 q^{39} - 2 i q^{42} - 4 i q^{43} - q^{44} - 4 q^{46} - 10 i q^{47} + 2 i q^{48} - q^{49} + 4 i q^{52} - 14 i q^{53} - 4 q^{54} + q^{56} - 8 i q^{57} - 2 i q^{58} - 10 q^{59} - 8 q^{61} - 10 i q^{62} - i q^{63} - q^{64} - 2 q^{66} - 8 i q^{67} - 8 q^{69} - 4 q^{71} + i q^{72} + 4 i q^{73} - 6 q^{74} + 4 q^{76} + i q^{77} + 8 i q^{78} - 16 q^{79} - 11 q^{81} + 4 i q^{83} + 2 q^{84} + 4 q^{86} - 4 i q^{87} - i q^{88} - 10 q^{89} + 4 q^{91} - 4 i q^{92} - 20 i q^{93} + 10 q^{94} - 2 q^{96} - 6 i q^{97} - i q^{98} - q^{99} +O(q^{100})$$ q + i * q^2 + 2*i * q^3 - q^4 - 2 * q^6 + i * q^7 - i * q^8 - q^9 + q^11 - 2*i * q^12 - 4*i * q^13 - q^14 + q^16 - i * q^18 - 4 * q^19 - 2 * q^21 + i * q^22 + 4*i * q^23 + 2 * q^24 + 4 * q^26 + 4*i * q^27 - i * q^28 - 2 * q^29 - 10 * q^31 + i * q^32 + 2*i * q^33 + q^36 + 6*i * q^37 - 4*i * q^38 + 8 * q^39 - 2*i * q^42 - 4*i * q^43 - q^44 - 4 * q^46 - 10*i * q^47 + 2*i * q^48 - q^49 + 4*i * q^52 - 14*i * q^53 - 4 * q^54 + q^56 - 8*i * q^57 - 2*i * q^58 - 10 * q^59 - 8 * q^61 - 10*i * q^62 - i * q^63 - q^64 - 2 * q^66 - 8*i * q^67 - 8 * q^69 - 4 * q^71 + i * q^72 + 4*i * q^73 - 6 * q^74 + 4 * q^76 + i * q^77 + 8*i * q^78 - 16 * q^79 - 11 * q^81 + 4*i * q^83 + 2 * q^84 + 4 * q^86 - 4*i * q^87 - i * q^88 - 10 * q^89 + 4 * q^91 - 4*i * q^92 - 20*i * q^93 + 10 * q^94 - 2 * q^96 - 6*i * q^97 - i * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 4 q^{24} + 8 q^{26} - 4 q^{29} - 20 q^{31} + 2 q^{36} + 16 q^{39} - 2 q^{44} - 8 q^{46} - 2 q^{49} - 8 q^{54} + 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} - 8 q^{71} - 12 q^{74} + 8 q^{76} - 32 q^{79} - 22 q^{81} + 4 q^{84} + 8 q^{86} - 20 q^{89} + 8 q^{91} + 20 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 2 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 4 * q^24 + 8 * q^26 - 4 * q^29 - 20 * q^31 + 2 * q^36 + 16 * q^39 - 2 * q^44 - 8 * q^46 - 2 * q^49 - 8 * q^54 + 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 - 4 * q^66 - 16 * q^69 - 8 * q^71 - 12 * q^74 + 8 * q^76 - 32 * q^79 - 22 * q^81 + 4 * q^84 + 8 * q^86 - 20 * q^89 + 8 * q^91 + 20 * q^94 - 4 * q^96 - 2 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 −2.00000 1.00000i 1.00000i −1.00000 0
1849.2 1.00000i 2.00000i −1.00000 0 −2.00000 1.00000i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3850.2.c.d 2
5.b even 2 1 inner 3850.2.c.d 2
5.c odd 4 1 154.2.a.b 1
5.c odd 4 1 3850.2.a.o 1
15.e even 4 1 1386.2.a.f 1
20.e even 4 1 1232.2.a.c 1
35.f even 4 1 1078.2.a.b 1
35.k even 12 2 1078.2.e.l 2
35.l odd 12 2 1078.2.e.h 2
40.i odd 4 1 4928.2.a.d 1
40.k even 4 1 4928.2.a.bf 1
55.e even 4 1 1694.2.a.i 1
105.k odd 4 1 9702.2.a.bz 1
140.j odd 4 1 8624.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.b 1 5.c odd 4 1
1078.2.a.b 1 35.f even 4 1
1078.2.e.h 2 35.l odd 12 2
1078.2.e.l 2 35.k even 12 2
1232.2.a.c 1 20.e even 4 1
1386.2.a.f 1 15.e even 4 1
1694.2.a.i 1 55.e even 4 1
3850.2.a.o 1 5.c odd 4 1
3850.2.c.d 2 1.a even 1 1 trivial
3850.2.c.d 2 5.b even 2 1 inner
4928.2.a.d 1 40.i odd 4 1
4928.2.a.bf 1 40.k even 4 1
8624.2.a.z 1 140.j odd 4 1
9702.2.a.bz 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3850, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}$$ T17 $$T_{19} + 4$$ T19 + 4 $$T_{37}^{2} + 36$$ T37^2 + 36

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 2)^{2}$$
$31$ $$(T + 10)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} + 196$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 36$$