# Properties

 Label 26.8.a.d Level $26$ Weight $8$ Character orbit 26.a Self dual yes Analytic conductor $8.122$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,8,Mod(1,26)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 26.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: $$8.12201066259$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{105})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 26$$ x^2 - x - 26 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$\beta = \sqrt{105}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 q^{2} + ( - 7 \beta - 6) q^{3} + 64 q^{4} + ( - 36 \beta - 73) q^{5} + (56 \beta + 48) q^{6} + ( - 27 \beta - 890) q^{7} - 512 q^{8} + (84 \beta + 2994) q^{9}+O(q^{10})$$ q - 8 * q^2 + (-7*b - 6) * q^3 + 64 * q^4 + (-36*b - 73) * q^5 + (56*b + 48) * q^6 + (-27*b - 890) * q^7 - 512 * q^8 + (84*b + 2994) * q^9 $$q - 8 q^{2} + ( - 7 \beta - 6) q^{3} + 64 q^{4} + ( - 36 \beta - 73) q^{5} + (56 \beta + 48) q^{6} + ( - 27 \beta - 890) q^{7} - 512 q^{8} + (84 \beta + 2994) q^{9} + (288 \beta + 584) q^{10} + ( - 90 \beta + 5452) q^{11} + ( - 448 \beta - 384) q^{12} - 2197 q^{13} + (216 \beta + 7120) q^{14} + (727 \beta + 26898) q^{15} + 4096 q^{16} + ( - 2304 \beta - 7059) q^{17} + ( - 672 \beta - 23952) q^{18} + ( - 1386 \beta + 27204) q^{19} + ( - 2304 \beta - 4672) q^{20} + (6392 \beta + 25185) q^{21} + (720 \beta - 43616) q^{22} + (3564 \beta - 78232) q^{23} + (3584 \beta + 3072) q^{24} + (5256 \beta + 63284) q^{25} + 17576 q^{26} + ( - 6153 \beta - 66582) q^{27} + ( - 1728 \beta - 56960) q^{28} + ( - 14544 \beta - 72934) q^{29} + ( - 5816 \beta - 215184) q^{30} + ( - 2808 \beta + 22900) q^{31} - 32768 q^{32} + ( - 37624 \beta + 33438) q^{33} + (18432 \beta + 56472) q^{34} + (34011 \beta + 167030) q^{35} + (5376 \beta + 191616) q^{36} + (14544 \beta + 174279) q^{37} + (11088 \beta - 217632) q^{38} + (15379 \beta + 13182) q^{39} + (18432 \beta + 37376) q^{40} + (9864 \beta - 78120) q^{41} + ( - 51136 \beta - 201480) q^{42} + (36891 \beta + 368626) q^{43} + ( - 5760 \beta + 348928) q^{44} + ( - 113916 \beta - 536082) q^{45} + ( - 28512 \beta + 625856) q^{46} + (44037 \beta - 307598) q^{47} + ( - 28672 \beta - 24576) q^{48} + (48060 \beta + 45102) q^{49} + ( - 42048 \beta - 506272) q^{50} + (63237 \beta + 1735794) q^{51} - 140608 q^{52} + (76104 \beta - 768012) q^{53} + (49224 \beta + 532656) q^{54} + ( - 189702 \beta - 57796) q^{55} + (13824 \beta + 455680) q^{56} + ( - 182112 \beta + 855486) q^{57} + (116352 \beta + 583472) q^{58} + (202734 \beta - 881236) q^{59} + (46528 \beta + 1721472) q^{60} + ( - 90864 \beta - 2230136) q^{61} + (22464 \beta - 183200) q^{62} + ( - 155598 \beta - 2902800) q^{63} + 262144 q^{64} + (79092 \beta + 160381) q^{65} + (300992 \beta - 267504) q^{66} + ( - 3114 \beta + 963340) q^{67} + ( - 147456 \beta - 451776) q^{68} + (526240 \beta - 2150148) q^{69} + ( - 272088 \beta - 1336240) q^{70} + ( - 388305 \beta + 1252570) q^{71} + ( - 43008 \beta - 1532928) q^{72} + (219960 \beta + 1862702) q^{73} + ( - 116352 \beta - 1394232) q^{74} + ( - 474524 \beta - 4242864) q^{75} + ( - 88704 \beta + 1741056) q^{76} + ( - 67104 \beta - 4597130) q^{77} + ( - 123032 \beta - 105456) q^{78} + (169956 \beta + 1955696) q^{79} + ( - 147456 \beta - 299008) q^{80} + (319284 \beta - 1625931) q^{81} + ( - 78912 \beta + 624960) q^{82} + ( - 203472 \beta + 4441680) q^{83} + (409088 \beta + 1611840) q^{84} + (422316 \beta + 9224427) q^{85} + ( - 295128 \beta - 2949008) q^{86} + (597802 \beta + 11127444) q^{87} + (46080 \beta - 2791424) q^{88} + ( - 50904 \beta - 8360274) q^{89} + (911328 \beta + 4288656) q^{90} + (59319 \beta + 1955330) q^{91} + (228096 \beta - 5006848) q^{92} + ( - 143452 \beta + 1926480) q^{93} + ( - 352296 \beta + 2460784) q^{94} + ( - 878166 \beta + 3253188) q^{95} + (229376 \beta + 196608) q^{96} + ( - 900504 \beta + 5658566) q^{97} + ( - 384480 \beta - 360816) q^{98} + (188508 \beta + 15529488) q^{99}+O(q^{100})$$ q - 8 * q^2 + (-7*b - 6) * q^3 + 64 * q^4 + (-36*b - 73) * q^5 + (56*b + 48) * q^6 + (-27*b - 890) * q^7 - 512 * q^8 + (84*b + 2994) * q^9 + (288*b + 584) * q^10 + (-90*b + 5452) * q^11 + (-448*b - 384) * q^12 - 2197 * q^13 + (216*b + 7120) * q^14 + (727*b + 26898) * q^15 + 4096 * q^16 + (-2304*b - 7059) * q^17 + (-672*b - 23952) * q^18 + (-1386*b + 27204) * q^19 + (-2304*b - 4672) * q^20 + (6392*b + 25185) * q^21 + (720*b - 43616) * q^22 + (3564*b - 78232) * q^23 + (3584*b + 3072) * q^24 + (5256*b + 63284) * q^25 + 17576 * q^26 + (-6153*b - 66582) * q^27 + (-1728*b - 56960) * q^28 + (-14544*b - 72934) * q^29 + (-5816*b - 215184) * q^30 + (-2808*b + 22900) * q^31 - 32768 * q^32 + (-37624*b + 33438) * q^33 + (18432*b + 56472) * q^34 + (34011*b + 167030) * q^35 + (5376*b + 191616) * q^36 + (14544*b + 174279) * q^37 + (11088*b - 217632) * q^38 + (15379*b + 13182) * q^39 + (18432*b + 37376) * q^40 + (9864*b - 78120) * q^41 + (-51136*b - 201480) * q^42 + (36891*b + 368626) * q^43 + (-5760*b + 348928) * q^44 + (-113916*b - 536082) * q^45 + (-28512*b + 625856) * q^46 + (44037*b - 307598) * q^47 + (-28672*b - 24576) * q^48 + (48060*b + 45102) * q^49 + (-42048*b - 506272) * q^50 + (63237*b + 1735794) * q^51 - 140608 * q^52 + (76104*b - 768012) * q^53 + (49224*b + 532656) * q^54 + (-189702*b - 57796) * q^55 + (13824*b + 455680) * q^56 + (-182112*b + 855486) * q^57 + (116352*b + 583472) * q^58 + (202734*b - 881236) * q^59 + (46528*b + 1721472) * q^60 + (-90864*b - 2230136) * q^61 + (22464*b - 183200) * q^62 + (-155598*b - 2902800) * q^63 + 262144 * q^64 + (79092*b + 160381) * q^65 + (300992*b - 267504) * q^66 + (-3114*b + 963340) * q^67 + (-147456*b - 451776) * q^68 + (526240*b - 2150148) * q^69 + (-272088*b - 1336240) * q^70 + (-388305*b + 1252570) * q^71 + (-43008*b - 1532928) * q^72 + (219960*b + 1862702) * q^73 + (-116352*b - 1394232) * q^74 + (-474524*b - 4242864) * q^75 + (-88704*b + 1741056) * q^76 + (-67104*b - 4597130) * q^77 + (-123032*b - 105456) * q^78 + (169956*b + 1955696) * q^79 + (-147456*b - 299008) * q^80 + (319284*b - 1625931) * q^81 + (-78912*b + 624960) * q^82 + (-203472*b + 4441680) * q^83 + (409088*b + 1611840) * q^84 + (422316*b + 9224427) * q^85 + (-295128*b - 2949008) * q^86 + (597802*b + 11127444) * q^87 + (46080*b - 2791424) * q^88 + (-50904*b - 8360274) * q^89 + (911328*b + 4288656) * q^90 + (59319*b + 1955330) * q^91 + (228096*b - 5006848) * q^92 + (-143452*b + 1926480) * q^93 + (-352296*b + 2460784) * q^94 + (-878166*b + 3253188) * q^95 + (229376*b + 196608) * q^96 + (-900504*b + 5658566) * q^97 + (-384480*b - 360816) * q^98 + (188508*b + 15529488) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{2} - 12 q^{3} + 128 q^{4} - 146 q^{5} + 96 q^{6} - 1780 q^{7} - 1024 q^{8} + 5988 q^{9}+O(q^{10})$$ 2 * q - 16 * q^2 - 12 * q^3 + 128 * q^4 - 146 * q^5 + 96 * q^6 - 1780 * q^7 - 1024 * q^8 + 5988 * q^9 $$2 q - 16 q^{2} - 12 q^{3} + 128 q^{4} - 146 q^{5} + 96 q^{6} - 1780 q^{7} - 1024 q^{8} + 5988 q^{9} + 1168 q^{10} + 10904 q^{11} - 768 q^{12} - 4394 q^{13} + 14240 q^{14} + 53796 q^{15} + 8192 q^{16} - 14118 q^{17} - 47904 q^{18} + 54408 q^{19} - 9344 q^{20} + 50370 q^{21} - 87232 q^{22} - 156464 q^{23} + 6144 q^{24} + 126568 q^{25} + 35152 q^{26} - 133164 q^{27} - 113920 q^{28} - 145868 q^{29} - 430368 q^{30} + 45800 q^{31} - 65536 q^{32} + 66876 q^{33} + 112944 q^{34} + 334060 q^{35} + 383232 q^{36} + 348558 q^{37} - 435264 q^{38} + 26364 q^{39} + 74752 q^{40} - 156240 q^{41} - 402960 q^{42} + 737252 q^{43} + 697856 q^{44} - 1072164 q^{45} + 1251712 q^{46} - 615196 q^{47} - 49152 q^{48} + 90204 q^{49} - 1012544 q^{50} + 3471588 q^{51} - 281216 q^{52} - 1536024 q^{53} + 1065312 q^{54} - 115592 q^{55} + 911360 q^{56} + 1710972 q^{57} + 1166944 q^{58} - 1762472 q^{59} + 3442944 q^{60} - 4460272 q^{61} - 366400 q^{62} - 5805600 q^{63} + 524288 q^{64} + 320762 q^{65} - 535008 q^{66} + 1926680 q^{67} - 903552 q^{68} - 4300296 q^{69} - 2672480 q^{70} + 2505140 q^{71} - 3065856 q^{72} + 3725404 q^{73} - 2788464 q^{74} - 8485728 q^{75} + 3482112 q^{76} - 9194260 q^{77} - 210912 q^{78} + 3911392 q^{79} - 598016 q^{80} - 3251862 q^{81} + 1249920 q^{82} + 8883360 q^{83} + 3223680 q^{84} + 18448854 q^{85} - 5898016 q^{86} + 22254888 q^{87} - 5582848 q^{88} - 16720548 q^{89} + 8577312 q^{90} + 3910660 q^{91} - 10013696 q^{92} + 3852960 q^{93} + 4921568 q^{94} + 6506376 q^{95} + 393216 q^{96} + 11317132 q^{97} - 721632 q^{98} + 31058976 q^{99}+O(q^{100})$$ 2 * q - 16 * q^2 - 12 * q^3 + 128 * q^4 - 146 * q^5 + 96 * q^6 - 1780 * q^7 - 1024 * q^8 + 5988 * q^9 + 1168 * q^10 + 10904 * q^11 - 768 * q^12 - 4394 * q^13 + 14240 * q^14 + 53796 * q^15 + 8192 * q^16 - 14118 * q^17 - 47904 * q^18 + 54408 * q^19 - 9344 * q^20 + 50370 * q^21 - 87232 * q^22 - 156464 * q^23 + 6144 * q^24 + 126568 * q^25 + 35152 * q^26 - 133164 * q^27 - 113920 * q^28 - 145868 * q^29 - 430368 * q^30 + 45800 * q^31 - 65536 * q^32 + 66876 * q^33 + 112944 * q^34 + 334060 * q^35 + 383232 * q^36 + 348558 * q^37 - 435264 * q^38 + 26364 * q^39 + 74752 * q^40 - 156240 * q^41 - 402960 * q^42 + 737252 * q^43 + 697856 * q^44 - 1072164 * q^45 + 1251712 * q^46 - 615196 * q^47 - 49152 * q^48 + 90204 * q^49 - 1012544 * q^50 + 3471588 * q^51 - 281216 * q^52 - 1536024 * q^53 + 1065312 * q^54 - 115592 * q^55 + 911360 * q^56 + 1710972 * q^57 + 1166944 * q^58 - 1762472 * q^59 + 3442944 * q^60 - 4460272 * q^61 - 366400 * q^62 - 5805600 * q^63 + 524288 * q^64 + 320762 * q^65 - 535008 * q^66 + 1926680 * q^67 - 903552 * q^68 - 4300296 * q^69 - 2672480 * q^70 + 2505140 * q^71 - 3065856 * q^72 + 3725404 * q^73 - 2788464 * q^74 - 8485728 * q^75 + 3482112 * q^76 - 9194260 * q^77 - 210912 * q^78 + 3911392 * q^79 - 598016 * q^80 - 3251862 * q^81 + 1249920 * q^82 + 8883360 * q^83 + 3223680 * q^84 + 18448854 * q^85 - 5898016 * q^86 + 22254888 * q^87 - 5582848 * q^88 - 16720548 * q^89 + 8577312 * q^90 + 3910660 * q^91 - 10013696 * q^92 + 3852960 * q^93 + 4921568 * q^94 + 6506376 * q^95 + 393216 * q^96 + 11317132 * q^97 - 721632 * q^98 + 31058976 * q^99

## 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
 5.62348 −4.62348
−8.00000 −77.7287 64.0000 −441.890 621.829 −1166.67 −512.000 3854.74 3535.12
1.2 −8.00000 65.7287 64.0000 295.890 −525.829 −613.332 −512.000 2133.26 −2367.12
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$13$$ $$+1$$

## 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 26.8.a.d 2
3.b odd 2 1 234.8.a.l 2
4.b odd 2 1 208.8.a.h 2
13.b even 2 1 338.8.a.f 2
13.d odd 4 2 338.8.b.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.d 2 1.a even 1 1 trivial
208.8.a.h 2 4.b odd 2 1
234.8.a.l 2 3.b odd 2 1
338.8.a.f 2 13.b even 2 1
338.8.b.e 4 13.d odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 12T_{3} - 5109$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(26))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 8)^{2}$$
$3$ $$T^{2} + 12T - 5109$$
$5$ $$T^{2} + 146T - 130751$$
$7$ $$T^{2} + 1780 T + 715555$$
$11$ $$T^{2} - 10904 T + 28873804$$
$13$ $$(T + 2197)^{2}$$
$17$ $$T^{2} + 14118 T - 507554199$$
$19$ $$T^{2} - 54408 T + 538353036$$
$23$ $$T^{2} + \cdots + 4786525744$$
$29$ $$T^{2} + \cdots - 16891064924$$
$31$ $$T^{2} - 45800 T - 303500720$$
$37$ $$T^{2} + \cdots + 8162736561$$
$41$ $$T^{2} + \cdots - 4113607680$$
$43$ $$T^{2} + \cdots - 7014189629$$
$47$ $$T^{2} + \cdots - 109005494141$$
$53$ $$T^{2} + \cdots - 18298543536$$
$59$ $$T^{2} + \cdots - 3539035961684$$
$61$ $$T^{2} + \cdots + 4106598596416$$
$67$ $$T^{2} + \cdots + 927005771020$$
$71$ $$T^{2} + \cdots - 14263049562725$$
$73$ $$T^{2} + \cdots - 1610493427196$$
$79$ $$T^{2} + \cdots + 791817441136$$
$83$ $$T^{2} + \cdots + 15381431470080$$
$89$ $$T^{2} + \cdots + 69622103547396$$
$97$ $$T^{2} + \cdots - 53125913495324$$