Properties

Label 175.10.a.c
Level $175$
Weight $10$
Character orbit 175.a
Self dual yes
Analytic conductor $90.131$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,10,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(90.1312713287\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 12) q^{2} + (54 \beta + 87) q^{3} + (24 \beta - 360) q^{4} + (735 \beta + 1476) q^{6} - 2401 q^{7} + ( - 584 \beta - 10272) q^{8} + (9396 \beta + 11214) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 12) q^{2} + (54 \beta + 87) q^{3} + (24 \beta - 360) q^{4} + (735 \beta + 1476) q^{6} - 2401 q^{7} + ( - 584 \beta - 10272) q^{8} + (9396 \beta + 11214) q^{9} + ( - 9124 \beta + 9283) q^{11} + ( - 17352 \beta - 20952) q^{12} + ( - 18334 \beta + 25545) q^{13} + ( - 2401 \beta - 28812) q^{14} + ( - 29568 \beta + 56384) q^{16} + ( - 15186 \beta + 186955) q^{17} + (123966 \beta + 209736) q^{18} + ( - 31138 \beta - 71638) q^{19} + ( - 129654 \beta - 208887) q^{21} + ( - 100205 \beta + 38404) q^{22} + ( - 821442 \beta + 249454) q^{23} + ( - 605496 \beta - 1145952) q^{24} + ( - 194463 \beta + 159868) q^{26} + (360126 \beta + 3322269) q^{27} + ( - 57624 \beta + 864360) q^{28} + (296772 \beta - 5788777) q^{29} + (2191070 \beta - 1976880) q^{31} + (576 \beta + 5699328) q^{32} + ( - 292506 \beta - 3133947) q^{33} + (4723 \beta + 2121972) q^{34} + ( - 3113424 \beta - 2233008) q^{36} + (3997070 \beta + 1602706) q^{37} + ( - 445294 \beta - 1108760) q^{38} + ( - 215628 \beta - 5697873) q^{39} + ( - 10352502 \beta + 529496) q^{41} + ( - 1764735 \beta - 3543876) q^{42} + ( - 9725678 \beta - 7974090) q^{43} + (3507432 \beta - 5093688) q^{44} + ( - 9607850 \beta - 3578088) q^{46} + ( - 9479722 \beta - 32750645) q^{47} + (472320 \beta - 7867968) q^{48} + 5764801 q^{49} + (8774388 \beta + 9704733) q^{51} + (7213320 \beta - 12716328) q^{52} + ( - 5256968 \beta + 12557344) q^{53} + (7643781 \beta + 42748236) q^{54} + (1402184 \beta + 24663072) q^{56} + ( - 6577458 \beta - 19684122) q^{57} + ( - 2227513 \beta - 67091148) q^{58} + ( - 39092800 \beta - 58079604) q^{59} + ( - 15794106 \beta - 22344272) q^{61} + (24315960 \beta - 6194000) q^{62} + ( - 22559796 \beta - 26924814) q^{63} + (20845056 \beta + 39527936) q^{64} + ( - 6644019 \beta - 39947412) q^{66} + (76121736 \beta - 59046248) q^{67} + (9953880 \beta - 70219512) q^{68} + ( - 57994938 \beta - 333160446) q^{69} + (76576000 \beta - 147082912) q^{71} + ( - 103064688 \beta - 159088320) q^{72} + (17548324 \beta + 28709666) q^{73} + (49567546 \beta + 51209032) q^{74} + (9490368 \beta + 19811184) q^{76} + (21906724 \beta - 22288483) q^{77} + ( - 8285409 \beta - 70099500) q^{78} + ( - 28471816 \beta - 346426427) q^{79} + (25792020 \beta + 223886673) q^{81} + ( - 123700528 \beta - 76466064) q^{82} + (92242900 \beta + 270339964) q^{83} + (41662152 \beta + 50305752) q^{84} + ( - 124682226 \beta - 173494504) q^{86} + ( - 286774794 \beta - 375418095) q^{87} + (88300456 \beta - 52727648) q^{88} + (97352322 \beta - 389521852) q^{89} + (44019934 \beta - 61333545) q^{91} + (301706016 \beta - 247520304) q^{92} + (83871570 \beta + 774553680) q^{93} + ( - 146507309 \beta - 468845516) q^{94} + (307813824 \beta + 496090368) q^{96} + (45681470 \beta + 1336519703) q^{97} + (5764801 \beta + 69177612) q^{98} + ( - 15093468 \beta - 581733270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9} + 18566 q^{11} - 41904 q^{12} + 51090 q^{13} - 57624 q^{14} + 112768 q^{16} + 373910 q^{17} + 419472 q^{18} - 143276 q^{19} - 417774 q^{21} + 76808 q^{22} + 498908 q^{23} - 2291904 q^{24} + 319736 q^{26} + 6644538 q^{27} + 1728720 q^{28} - 11577554 q^{29} - 3953760 q^{31} + 11398656 q^{32} - 6267894 q^{33} + 4243944 q^{34} - 4466016 q^{36} + 3205412 q^{37} - 2217520 q^{38} - 11395746 q^{39} + 1058992 q^{41} - 7087752 q^{42} - 15948180 q^{43} - 10187376 q^{44} - 7156176 q^{46} - 65501290 q^{47} - 15735936 q^{48} + 11529602 q^{49} + 19409466 q^{51} - 25432656 q^{52} + 25114688 q^{53} + 85496472 q^{54} + 49326144 q^{56} - 39368244 q^{57} - 134182296 q^{58} - 116159208 q^{59} - 44688544 q^{61} - 12388000 q^{62} - 53849628 q^{63} + 79055872 q^{64} - 79894824 q^{66} - 118092496 q^{67} - 140439024 q^{68} - 666320892 q^{69} - 294165824 q^{71} - 318176640 q^{72} + 57419332 q^{73} + 102418064 q^{74} + 39622368 q^{76} - 44576966 q^{77} - 140199000 q^{78} - 692852854 q^{79} + 447773346 q^{81} - 152932128 q^{82} + 540679928 q^{83} + 100611504 q^{84} - 346989008 q^{86} - 750836190 q^{87} - 105455296 q^{88} - 779043704 q^{89} - 122667090 q^{91} - 495040608 q^{92} + 1549107360 q^{93} - 937691032 q^{94} + 992180736 q^{96} + 2673039406 q^{97} + 138355224 q^{98} - 1163466540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
9.17157 −65.7351 −427.882 0 −602.894 −2401.00 −8620.20 −15361.9 0
1.2 14.8284 239.735 −292.118 0 3554.89 −2401.00 −11923.8 37789.9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.10.a.c 2
5.b even 2 1 35.10.a.b 2
5.c odd 4 2 175.10.b.c 4
15.d odd 2 1 315.10.a.b 2
35.c odd 2 1 245.10.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.b 2 5.b even 2 1
175.10.a.c 2 1.a even 1 1 trivial
175.10.b.c 4 5.c odd 4 2
245.10.a.c 2 35.c odd 2 1
315.10.a.b 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 24T_{2} + 136 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$3$ \( T^{2} - 174T - 15759 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18566 T - 579804919 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 2036537423 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 33107255257 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 2624597308 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 5335908376796 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 32805350195857 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 34498247424800 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 125243882156764 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 857114015266016 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 693124389149372 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 353683714337753 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 63398812089856 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 88\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 42\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 25\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 75\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 17\!\cdots\!09 \) Copy content Toggle raw display
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