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 $2$

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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: \( x^{2} + 14 \) Copy content Toggle raw display
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.

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

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:

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 \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 504 \) Copy content Toggle raw display
\( T_{7} - 1750 \) Copy content Toggle raw display

Hecke characteristic polynomials

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