Properties

 Label 75.9.c.c Level $75$ Weight $9$ Character orbit 75.c Analytic conductor $30.553$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-14})$$ Defining polynomial: $$x^{2} + 14$$ x^2 + 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - 3 \beta - 45) q^{3} - 248 q^{4} + ( - 45 \beta + 1512) q^{6} + 1750 q^{7} + 8 \beta q^{8} + (270 \beta - 2511) q^{9}+O(q^{10})$$ q + b * q^2 + (-3*b - 45) * q^3 - 248 * q^4 + (-45*b + 1512) * q^6 + 1750 * q^7 + 8*b * q^8 + (270*b - 2511) * q^9 $$q + \beta q^{2} + ( - 3 \beta - 45) q^{3} - 248 q^{4} + ( - 45 \beta + 1512) q^{6} + 1750 q^{7} + 8 \beta q^{8} + (270 \beta - 2511) q^{9} - 310 \beta q^{11} + (744 \beta + 11160) q^{12} - 25730 q^{13} + 1750 \beta q^{14} - 67520 q^{16} + 3336 \beta q^{17} + ( - 2511 \beta - 136080) q^{18} + 18938 q^{19} + ( - 5250 \beta - 78750) q^{21} + 156240 q^{22} - 20956 \beta q^{23} + ( - 360 \beta + 12096) q^{24} - 25730 \beta q^{26} + ( - 4617 \beta + 521235) q^{27} - 434000 q^{28} - 20530 \beta q^{29} - 351478 q^{31} - 65472 \beta q^{32} + (13950 \beta - 468720) q^{33} - 1681344 q^{34} + ( - 66960 \beta + 622728) q^{36} - 1335170 q^{37} + 18938 \beta q^{38} + (77190 \beta + 1157850) q^{39} - 83540 \beta q^{41} + ( - 78750 \beta + 2646000) q^{42} + 3526150 q^{43} + 76880 \beta q^{44} + 10561824 q^{46} - 181784 \beta q^{47} + (202560 \beta + 3038400) q^{48} - 2702301 q^{49} + ( - 150120 \beta + 5044032) q^{51} + 6381040 q^{52} - 294066 \beta q^{53} + (521235 \beta + 2326968) q^{54} + 14000 \beta q^{56} + ( - 56814 \beta - 852210) q^{57} + 10347120 q^{58} - 610910 \beta q^{59} + 753602 q^{61} - 351478 \beta q^{62} + (472500 \beta - 4394250) q^{63} + 15712768 q^{64} + ( - 468720 \beta - 7030800) q^{66} - 2268890 q^{67} - 827328 \beta q^{68} + (943020 \beta - 31685472) q^{69} - 758220 \beta q^{71} + ( - 20088 \beta - 1088640) q^{72} - 27672770 q^{73} - 1335170 \beta q^{74} - 4696624 q^{76} - 542500 \beta q^{77} + (1157850 \beta - 38903760) q^{78} - 22980982 q^{79} + ( - 1355940 \beta - 30436479) q^{81} + 42104160 q^{82} - 2066606 \beta q^{83} + (1302000 \beta + 19530000) q^{84} + 3526150 \beta q^{86} + (923850 \beta - 31041360) q^{87} + 1249920 q^{88} - 3234540 \beta q^{89} - 45027500 q^{91} + 5197088 \beta q^{92} + (1054434 \beta + 15816510) q^{93} + 91619136 q^{94} + (2946240 \beta - 98993664) q^{96} - 147271010 q^{97} - 2702301 \beta q^{98} + (778410 \beta + 42184800) q^{99} +O(q^{100})$$ q + b * q^2 + (-3*b - 45) * q^3 - 248 * q^4 + (-45*b + 1512) * q^6 + 1750 * q^7 + 8*b * q^8 + (270*b - 2511) * q^9 - 310*b * q^11 + (744*b + 11160) * q^12 - 25730 * q^13 + 1750*b * q^14 - 67520 * q^16 + 3336*b * q^17 + (-2511*b - 136080) * q^18 + 18938 * q^19 + (-5250*b - 78750) * q^21 + 156240 * q^22 - 20956*b * q^23 + (-360*b + 12096) * q^24 - 25730*b * q^26 + (-4617*b + 521235) * q^27 - 434000 * q^28 - 20530*b * q^29 - 351478 * q^31 - 65472*b * q^32 + (13950*b - 468720) * q^33 - 1681344 * q^34 + (-66960*b + 622728) * q^36 - 1335170 * q^37 + 18938*b * q^38 + (77190*b + 1157850) * q^39 - 83540*b * q^41 + (-78750*b + 2646000) * q^42 + 3526150 * q^43 + 76880*b * q^44 + 10561824 * q^46 - 181784*b * q^47 + (202560*b + 3038400) * q^48 - 2702301 * q^49 + (-150120*b + 5044032) * q^51 + 6381040 * q^52 - 294066*b * q^53 + (521235*b + 2326968) * q^54 + 14000*b * q^56 + (-56814*b - 852210) * q^57 + 10347120 * q^58 - 610910*b * q^59 + 753602 * q^61 - 351478*b * q^62 + (472500*b - 4394250) * q^63 + 15712768 * q^64 + (-468720*b - 7030800) * q^66 - 2268890 * q^67 - 827328*b * q^68 + (943020*b - 31685472) * q^69 - 758220*b * q^71 + (-20088*b - 1088640) * q^72 - 27672770 * q^73 - 1335170*b * q^74 - 4696624 * q^76 - 542500*b * q^77 + (1157850*b - 38903760) * q^78 - 22980982 * q^79 + (-1355940*b - 30436479) * q^81 + 42104160 * q^82 - 2066606*b * q^83 + (1302000*b + 19530000) * q^84 + 3526150*b * q^86 + (923850*b - 31041360) * q^87 + 1249920 * q^88 - 3234540*b * q^89 - 45027500 * q^91 + 5197088*b * q^92 + (1054434*b + 15816510) * q^93 + 91619136 * q^94 + (2946240*b - 98993664) * q^96 - 147271010 * q^97 - 2702301*b * q^98 + (778410*b + 42184800) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 90 q^{3} - 496 q^{4} + 3024 q^{6} + 3500 q^{7} - 5022 q^{9}+O(q^{10})$$ 2 * q - 90 * q^3 - 496 * q^4 + 3024 * q^6 + 3500 * q^7 - 5022 * q^9 $$2 q - 90 q^{3} - 496 q^{4} + 3024 q^{6} + 3500 q^{7} - 5022 q^{9} + 22320 q^{12} - 51460 q^{13} - 135040 q^{16} - 272160 q^{18} + 37876 q^{19} - 157500 q^{21} + 312480 q^{22} + 24192 q^{24} + 1042470 q^{27} - 868000 q^{28} - 702956 q^{31} - 937440 q^{33} - 3362688 q^{34} + 1245456 q^{36} - 2670340 q^{37} + 2315700 q^{39} + 5292000 q^{42} + 7052300 q^{43} + 21123648 q^{46} + 6076800 q^{48} - 5404602 q^{49} + 10088064 q^{51} + 12762080 q^{52} + 4653936 q^{54} - 1704420 q^{57} + 20694240 q^{58} + 1507204 q^{61} - 8788500 q^{63} + 31425536 q^{64} - 14061600 q^{66} - 4537780 q^{67} - 63370944 q^{69} - 2177280 q^{72} - 55345540 q^{73} - 9393248 q^{76} - 77807520 q^{78} - 45961964 q^{79} - 60872958 q^{81} + 84208320 q^{82} + 39060000 q^{84} - 62082720 q^{87} + 2499840 q^{88} - 90055000 q^{91} + 31633020 q^{93} + 183238272 q^{94} - 197987328 q^{96} - 294542020 q^{97} + 84369600 q^{99}+O(q^{100})$$ 2 * q - 90 * q^3 - 496 * q^4 + 3024 * q^6 + 3500 * q^7 - 5022 * q^9 + 22320 * q^12 - 51460 * q^13 - 135040 * q^16 - 272160 * q^18 + 37876 * q^19 - 157500 * q^21 + 312480 * q^22 + 24192 * q^24 + 1042470 * q^27 - 868000 * q^28 - 702956 * q^31 - 937440 * q^33 - 3362688 * q^34 + 1245456 * q^36 - 2670340 * q^37 + 2315700 * q^39 + 5292000 * q^42 + 7052300 * q^43 + 21123648 * q^46 + 6076800 * q^48 - 5404602 * q^49 + 10088064 * q^51 + 12762080 * q^52 + 4653936 * q^54 - 1704420 * q^57 + 20694240 * q^58 + 1507204 * q^61 - 8788500 * q^63 + 31425536 * q^64 - 14061600 * q^66 - 4537780 * q^67 - 63370944 * q^69 - 2177280 * q^72 - 55345540 * q^73 - 9393248 * q^76 - 77807520 * q^78 - 45961964 * q^79 - 60872958 * q^81 + 84208320 * q^82 + 39060000 * q^84 - 62082720 * q^87 + 2499840 * q^88 - 90055000 * q^91 + 31633020 * q^93 + 183238272 * q^94 - 197987328 * q^96 - 294542020 * q^97 + 84369600 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 − 3.74166i 3.74166i
22.4499i −45.0000 + 67.3498i −248.000 0 1512.00 + 1010.25i 1750.00 179.600i −2511.00 6061.48i 0
26.2 22.4499i −45.0000 67.3498i −248.000 0 1512.00 1010.25i 1750.00 179.600i −2511.00 + 6061.48i 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
3.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.c.c 2
3.b odd 2 1 inner 75.9.c.c 2
5.b even 2 1 3.9.b.a 2
5.c odd 4 2 75.9.d.b 4
15.d odd 2 1 3.9.b.a 2
15.e even 4 2 75.9.d.b 4
20.d odd 2 1 48.9.e.b 2
40.e odd 2 1 192.9.e.f 2
40.f even 2 1 192.9.e.e 2
45.h odd 6 2 81.9.d.d 4
45.j even 6 2 81.9.d.d 4
60.h even 2 1 48.9.e.b 2
120.i odd 2 1 192.9.e.e 2
120.m even 2 1 192.9.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 5.b even 2 1
3.9.b.a 2 15.d odd 2 1
48.9.e.b 2 20.d odd 2 1
48.9.e.b 2 60.h even 2 1
75.9.c.c 2 1.a even 1 1 trivial
75.9.c.c 2 3.b odd 2 1 inner
75.9.d.b 4 5.c odd 4 2
75.9.d.b 4 15.e even 4 2
81.9.d.d 4 45.h odd 6 2
81.9.d.d 4 45.j even 6 2
192.9.e.e 2 40.f even 2 1
192.9.e.e 2 120.i odd 2 1
192.9.e.f 2 40.e odd 2 1
192.9.e.f 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{2} + 504$$ T2^2 + 504 $$T_{7} - 1750$$ T7 - 1750

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 504$$
$3$ $$T^{2} + 90T + 6561$$
$5$ $$T^{2}$$
$7$ $$(T - 1750)^{2}$$
$11$ $$T^{2} + 48434400$$
$13$ $$(T + 25730)^{2}$$
$17$ $$T^{2} + 5608963584$$
$19$ $$(T - 18938)^{2}$$
$23$ $$T^{2} + 221333583744$$
$29$ $$T^{2} + 212426373600$$
$31$ $$(T + 351478)^{2}$$
$37$ $$(T + 1335170)^{2}$$
$41$ $$T^{2} + 3517381526400$$
$43$ $$(T - 3526150)^{2}$$
$47$ $$T^{2} + 16654893018624$$
$53$ $$T^{2} + 43583305427424$$
$59$ $$T^{2} + \cdots + 188098358162400$$
$61$ $$(T - 753602)^{2}$$
$67$ $$(T + 2268890)^{2}$$
$71$ $$T^{2} + \cdots + 289748374473600$$
$73$ $$(T + 27672770)^{2}$$
$79$ $$(T + 22980982)^{2}$$
$83$ $$T^{2} + 21\!\cdots\!44$$
$89$ $$T^{2} + 52\!\cdots\!00$$
$97$ $$(T + 147271010)^{2}$$