# Properties

 Label 7.5.d.a Level $7$ Weight $5$ Character orbit 7.d Analytic conductor $0.724$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,5,Mod(3,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("7.3");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 7.d (of order $$6$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$0.723589741587$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 22x^{2} + 484$$ x^4 + 22*x^2 + 484 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} + \beta_1 - 2) q^{2} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{3} + ( - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_1) q^{4} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 5) q^{5} + (3 \beta_{3} - 48 \beta_{2} + 6 \beta_1 - 24) q^{6} + ( - 7 \beta_{3} + 42 \beta_{2} + 21) q^{7} + (2 \beta_{3} + 76) q^{8} + ( - 12 \beta_{2} - 6 \beta_1 - 12) q^{9}+O(q^{10})$$ q + (-2*b2 + b1 - 2) * q^2 + (b3 + b2 - b1 + 2) * q^3 + (-4*b3 + 10*b2 - 4*b1) * q^4 + (4*b3 + 5*b2 + 2*b1 - 5) * q^5 + (3*b3 - 48*b2 + 6*b1 - 24) * q^6 + (-7*b3 + 42*b2 + 21) * q^7 + (2*b3 + 76) * q^8 + (-12*b2 - 6*b1 - 12) * q^9 $$q + ( - 2 \beta_{2} + \beta_1 - 2) q^{2} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{3} + ( - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_1) q^{4} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 5) q^{5} + (3 \beta_{3} - 48 \beta_{2} + 6 \beta_1 - 24) q^{6} + ( - 7 \beta_{3} + 42 \beta_{2} + 21) q^{7} + (2 \beta_{3} + 76) q^{8} + ( - 12 \beta_{2} - 6 \beta_1 - 12) q^{9} + (\beta_{3} - 34 \beta_{2} - \beta_1 - 68) q^{10} + (17 \beta_{3} + 29 \beta_{2} + 17 \beta_1) q^{11} + ( - 28 \beta_{3} + 98 \beta_{2} - 14 \beta_1 - 98) q^{12} + ( - 14 \beta_{3} - 140 \beta_{2} - 28 \beta_1 - 70) q^{13} + (42 \beta_{3} + 112 \beta_{2} + 7 \beta_1 + 196) q^{14} + ( - 9 \beta_{3} + 117) q^{15} + ( - 36 \beta_{2} + 16 \beta_1 - 36) q^{16} + ( - 34 \beta_{3} - 41 \beta_{2} + 34 \beta_1 - 82) q^{17} - 108 \beta_{2} q^{18} + (74 \beta_{3} - 107 \beta_{2} + 37 \beta_1 + 107) q^{19} + (252 \beta_{2} + 126) q^{20} + ( - 70 \beta_{3} - 91 \beta_{2} - 56 \beta_1 - 308) q^{21} + ( - 5 \beta_{3} - 316) q^{22} + (145 \beta_{2} - 41 \beta_1 + 145) q^{23} + (78 \beta_{3} + 120 \beta_{2} - 78 \beta_1 + 240) q^{24} + ( - 60 \beta_{3} + 286 \beta_{2} - 60 \beta_1) q^{25} + ( - 84 \beta_{3} - 168 \beta_{2} - 42 \beta_1 + 168) q^{26} + (87 \beta_{3} + 78 \beta_{2} + 174 \beta_1 + 39) q^{27} + ( - 14 \beta_{3} - 826 \beta_{2} + 154 \beta_1 - 420) q^{28} + (70 \beta_{3} - 544) q^{29} + ( - 36 \beta_{2} + 99 \beta_1 - 36) q^{30} + (29 \beta_{3} + 603 \beta_{2} - 29 \beta_1 + 1206) q^{31} + ( - 36 \beta_{3} - 792 \beta_{2} - 36 \beta_1) q^{32} + ( - 24 \beta_{3} - 345 \beta_{2} - 12 \beta_1 + 345) q^{33} + ( - 109 \beta_{3} + 1660 \beta_{2} - 218 \beta_1 + 830) q^{34} + (70 \beta_{3} - 7 \beta_{2} - 91 \beta_1 - 623) q^{35} + ( - 12 \beta_{3} - 408) q^{36} + ( - 135 \beta_{2} - 104 \beta_1 - 135) q^{37} + ( - 181 \beta_{3} - 1028 \beta_{2} + 181 \beta_1 - 2056) q^{38} + (168 \beta_{3} + 714 \beta_{2} + 168 \beta_1) q^{39} + (284 \beta_{3} + 292 \beta_{2} + 142 \beta_1 - 292) q^{40} + ( - 42 \beta_{3} - 1596 \beta_{2} - 84 \beta_1 - 798) q^{41} + (21 \beta_{3} + 924 \beta_{2} - 336 \beta_1 + 1974) q^{42} + ( - 350 \beta_{3} + 618) q^{43} + (1206 \beta_{2} - 54 \beta_1 + 1206) q^{44} + ( - 54 \beta_{3} + 324 \beta_{2} + 54 \beta_1 + 648) q^{45} + (227 \beta_{3} - 1192 \beta_{2} + 227 \beta_1) q^{46} + ( - 374 \beta_{3} + 257 \beta_{2} - 187 \beta_1 - 257) q^{47} + (52 \beta_{3} - 776 \beta_{2} + 104 \beta_1 - 388) q^{48} + (294 \beta_{3} + 588 \beta_1 - 245) q^{49} + (406 \beta_{3} + 1892) q^{50} + ( - 2367 \beta_{2} + 225 \beta_1 - 2367) q^{51} + (140 \beta_{3} - 532 \beta_{2} - 140 \beta_1 - 1064) q^{52} + ( - 340 \beta_{3} + 2255 \beta_{2} - 340 \beta_1) q^{53} + ( - 270 \beta_{3} + 1836 \beta_{2} - 135 \beta_1 - 1836) q^{54} + ( - 143 \beta_{3} - 1786 \beta_{2} - 286 \beta_1 - 893) q^{55} + ( - 574 \beta_{3} + 3192 \beta_{2} - 84 \beta_1 + 1288) q^{56} + (432 \beta_{3} + 2763) q^{57} + ( - 452 \beta_{2} - 404 \beta_1 - 452) q^{58} + (449 \beta_{3} + 421 \beta_{2} - 449 \beta_1 + 842) q^{59} + ( - 378 \beta_{3} + 378 \beta_{2} - 378 \beta_1) q^{60} + (480 \beta_{3} + 47 \beta_{2} + 240 \beta_1 - 47) q^{61} + (661 \beta_{3} - 3688 \beta_{2} + 1322 \beta_1 - 1844) q^{62} + ( - 252 \beta_{3} - 1176 \beta_{2} - 210 \beta_1 - 672) q^{63} + ( - 976 \beta_{3} - 1368) q^{64} + (2898 \beta_{2} + 630 \beta_1 + 2898) q^{65} + ( - 321 \beta_{3} - 426 \beta_{2} + 321 \beta_1 - 852) q^{66} + (45 \beta_{3} + 659 \beta_{2} + 45 \beta_1) q^{67} + (1008 \beta_{3} - 3402 \beta_{2} + 504 \beta_1 + 3402) q^{68} + ( - 186 \beta_{3} + 2094 \beta_{2} - 372 \beta_1 + 1047) q^{69} + (175 \beta_{3} - 2296 \beta_{2} - 301 \beta_1 - 308) q^{70} + (238 \beta_{3} - 2602) q^{71} + ( - 648 \beta_{2} - 432 \beta_1 - 648) q^{72} + ( - 272 \beta_{3} + 869 \beta_{2} + 272 \beta_1 + 1738) q^{73} + (73 \beta_{3} - 2018 \beta_{2} + 73 \beta_1) q^{74} + ( - 692 \beta_{3} + 1606 \beta_{2} - 346 \beta_1 - 1606) q^{75} + ( - 798 \beta_{3} + 8652 \beta_{2} - 1596 \beta_1 + 4326) q^{76} + (560 \beta_{3} + 2009 \beta_{2} - 154 \beta_1 - 1218) q^{77} + (378 \beta_{3} - 2268) q^{78} + ( - 4055 \beta_{2} + 351 \beta_1 - 4055) q^{79} + (8 \beta_{3} - 524 \beta_{2} - 8 \beta_1 - 1048) q^{80} + (630 \beta_{3} - 4653 \beta_{2} + 630 \beta_1) q^{81} + ( - 1428 \beta_{3} + 672 \beta_{2} - 714 \beta_1 - 672) q^{82} + ( - 84 \beta_{3} + 3864 \beta_{2} - 168 \beta_1 + 1932) q^{83} + (1372 \beta_{3} - 8330 \beta_{2} + 1568 \beta_1 - 4018) q^{84} + (264 \beta_{3} - 3873) q^{85} + (6464 \beta_{2} - 82 \beta_1 + 6464) q^{86} + ( - 474 \beta_{3} + 996 \beta_{2} + 474 \beta_1 + 1992) q^{87} + (1234 \beta_{3} + 1456 \beta_{2} + 1234 \beta_1) q^{88} + ( - 752 \beta_{3} - 5665 \beta_{2} - 376 \beta_1 + 5665) q^{89} + (216 \beta_{3} + 1080 \beta_{2} + 432 \beta_1 + 540) q^{90} + ( - 1372 \beta_{3} - 4312 \beta_{2} - 980 \beta_1 + 2254) q^{91} + ( - 990 \beta_{3} - 5058) q^{92} + (3723 \beta_{2} - 1896 \beta_1 + 3723) q^{93} + (631 \beta_{3} + 4628 \beta_{2} - 631 \beta_1 + 9256) q^{94} + (87 \beta_{3} - 3279 \beta_{2} + 87 \beta_1) q^{95} + (1512 \beta_{3} + 756 \beta_1) q^{96} + (1274 \beta_{3} - 1372 \beta_{2} + 2548 \beta_1 - 686) q^{97} + ( - 1176 \beta_{3} + 6958 \beta_{2} - 833 \beta_1 - 5978) q^{98} + ( - 378 \beta_{3} + 2592) q^{99}+O(q^{100})$$ q + (-2*b2 + b1 - 2) * q^2 + (b3 + b2 - b1 + 2) * q^3 + (-4*b3 + 10*b2 - 4*b1) * q^4 + (4*b3 + 5*b2 + 2*b1 - 5) * q^5 + (3*b3 - 48*b2 + 6*b1 - 24) * q^6 + (-7*b3 + 42*b2 + 21) * q^7 + (2*b3 + 76) * q^8 + (-12*b2 - 6*b1 - 12) * q^9 + (b3 - 34*b2 - b1 - 68) * q^10 + (17*b3 + 29*b2 + 17*b1) * q^11 + (-28*b3 + 98*b2 - 14*b1 - 98) * q^12 + (-14*b3 - 140*b2 - 28*b1 - 70) * q^13 + (42*b3 + 112*b2 + 7*b1 + 196) * q^14 + (-9*b3 + 117) * q^15 + (-36*b2 + 16*b1 - 36) * q^16 + (-34*b3 - 41*b2 + 34*b1 - 82) * q^17 - 108*b2 * q^18 + (74*b3 - 107*b2 + 37*b1 + 107) * q^19 + (252*b2 + 126) * q^20 + (-70*b3 - 91*b2 - 56*b1 - 308) * q^21 + (-5*b3 - 316) * q^22 + (145*b2 - 41*b1 + 145) * q^23 + (78*b3 + 120*b2 - 78*b1 + 240) * q^24 + (-60*b3 + 286*b2 - 60*b1) * q^25 + (-84*b3 - 168*b2 - 42*b1 + 168) * q^26 + (87*b3 + 78*b2 + 174*b1 + 39) * q^27 + (-14*b3 - 826*b2 + 154*b1 - 420) * q^28 + (70*b3 - 544) * q^29 + (-36*b2 + 99*b1 - 36) * q^30 + (29*b3 + 603*b2 - 29*b1 + 1206) * q^31 + (-36*b3 - 792*b2 - 36*b1) * q^32 + (-24*b3 - 345*b2 - 12*b1 + 345) * q^33 + (-109*b3 + 1660*b2 - 218*b1 + 830) * q^34 + (70*b3 - 7*b2 - 91*b1 - 623) * q^35 + (-12*b3 - 408) * q^36 + (-135*b2 - 104*b1 - 135) * q^37 + (-181*b3 - 1028*b2 + 181*b1 - 2056) * q^38 + (168*b3 + 714*b2 + 168*b1) * q^39 + (284*b3 + 292*b2 + 142*b1 - 292) * q^40 + (-42*b3 - 1596*b2 - 84*b1 - 798) * q^41 + (21*b3 + 924*b2 - 336*b1 + 1974) * q^42 + (-350*b3 + 618) * q^43 + (1206*b2 - 54*b1 + 1206) * q^44 + (-54*b3 + 324*b2 + 54*b1 + 648) * q^45 + (227*b3 - 1192*b2 + 227*b1) * q^46 + (-374*b3 + 257*b2 - 187*b1 - 257) * q^47 + (52*b3 - 776*b2 + 104*b1 - 388) * q^48 + (294*b3 + 588*b1 - 245) * q^49 + (406*b3 + 1892) * q^50 + (-2367*b2 + 225*b1 - 2367) * q^51 + (140*b3 - 532*b2 - 140*b1 - 1064) * q^52 + (-340*b3 + 2255*b2 - 340*b1) * q^53 + (-270*b3 + 1836*b2 - 135*b1 - 1836) * q^54 + (-143*b3 - 1786*b2 - 286*b1 - 893) * q^55 + (-574*b3 + 3192*b2 - 84*b1 + 1288) * q^56 + (432*b3 + 2763) * q^57 + (-452*b2 - 404*b1 - 452) * q^58 + (449*b3 + 421*b2 - 449*b1 + 842) * q^59 + (-378*b3 + 378*b2 - 378*b1) * q^60 + (480*b3 + 47*b2 + 240*b1 - 47) * q^61 + (661*b3 - 3688*b2 + 1322*b1 - 1844) * q^62 + (-252*b3 - 1176*b2 - 210*b1 - 672) * q^63 + (-976*b3 - 1368) * q^64 + (2898*b2 + 630*b1 + 2898) * q^65 + (-321*b3 - 426*b2 + 321*b1 - 852) * q^66 + (45*b3 + 659*b2 + 45*b1) * q^67 + (1008*b3 - 3402*b2 + 504*b1 + 3402) * q^68 + (-186*b3 + 2094*b2 - 372*b1 + 1047) * q^69 + (175*b3 - 2296*b2 - 301*b1 - 308) * q^70 + (238*b3 - 2602) * q^71 + (-648*b2 - 432*b1 - 648) * q^72 + (-272*b3 + 869*b2 + 272*b1 + 1738) * q^73 + (73*b3 - 2018*b2 + 73*b1) * q^74 + (-692*b3 + 1606*b2 - 346*b1 - 1606) * q^75 + (-798*b3 + 8652*b2 - 1596*b1 + 4326) * q^76 + (560*b3 + 2009*b2 - 154*b1 - 1218) * q^77 + (378*b3 - 2268) * q^78 + (-4055*b2 + 351*b1 - 4055) * q^79 + (8*b3 - 524*b2 - 8*b1 - 1048) * q^80 + (630*b3 - 4653*b2 + 630*b1) * q^81 + (-1428*b3 + 672*b2 - 714*b1 - 672) * q^82 + (-84*b3 + 3864*b2 - 168*b1 + 1932) * q^83 + (1372*b3 - 8330*b2 + 1568*b1 - 4018) * q^84 + (264*b3 - 3873) * q^85 + (6464*b2 - 82*b1 + 6464) * q^86 + (-474*b3 + 996*b2 + 474*b1 + 1992) * q^87 + (1234*b3 + 1456*b2 + 1234*b1) * q^88 + (-752*b3 - 5665*b2 - 376*b1 + 5665) * q^89 + (216*b3 + 1080*b2 + 432*b1 + 540) * q^90 + (-1372*b3 - 4312*b2 - 980*b1 + 2254) * q^91 + (-990*b3 - 5058) * q^92 + (3723*b2 - 1896*b1 + 3723) * q^93 + (631*b3 + 4628*b2 - 631*b1 + 9256) * q^94 + (87*b3 - 3279*b2 + 87*b1) * q^95 + (1512*b3 + 756*b1) * q^96 + (1274*b3 - 1372*b2 + 2548*b1 - 686) * q^97 + (-1176*b3 + 6958*b2 - 833*b1 - 5978) * q^98 + (-378*b3 + 2592) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 6 q^{3} - 20 q^{4} - 30 q^{5} + 304 q^{8} - 24 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 6 * q^3 - 20 * q^4 - 30 * q^5 + 304 * q^8 - 24 * q^9 $$4 q - 4 q^{2} + 6 q^{3} - 20 q^{4} - 30 q^{5} + 304 q^{8} - 24 q^{9} - 204 q^{10} - 58 q^{11} - 588 q^{12} + 560 q^{14} + 468 q^{15} - 72 q^{16} - 246 q^{17} + 216 q^{18} + 642 q^{19} - 1050 q^{21} - 1264 q^{22} + 290 q^{23} + 720 q^{24} - 572 q^{25} + 1008 q^{26} - 28 q^{28} - 2176 q^{29} - 72 q^{30} + 3618 q^{31} + 1584 q^{32} + 2070 q^{33} - 2478 q^{35} - 1632 q^{36} - 270 q^{37} - 6168 q^{38} - 1428 q^{39} - 1752 q^{40} + 6048 q^{42} + 2472 q^{43} + 2412 q^{44} + 1944 q^{45} + 2384 q^{46} - 1542 q^{47} - 980 q^{49} + 7568 q^{50} - 4734 q^{51} - 3192 q^{52} - 4510 q^{53} - 11016 q^{54} - 1232 q^{56} + 11052 q^{57} - 904 q^{58} + 2526 q^{59} - 756 q^{60} - 282 q^{61} - 336 q^{63} - 5472 q^{64} + 5796 q^{65} - 2556 q^{66} - 1318 q^{67} + 20412 q^{68} + 3360 q^{70} - 10408 q^{71} - 1296 q^{72} + 5214 q^{73} + 4036 q^{74} - 9636 q^{75} - 8890 q^{77} - 9072 q^{78} - 8110 q^{79} - 3144 q^{80} + 9306 q^{81} - 4032 q^{82} + 588 q^{84} - 15492 q^{85} + 12928 q^{86} + 5976 q^{87} - 2912 q^{88} + 33990 q^{89} + 17640 q^{91} - 20232 q^{92} + 7446 q^{93} + 27768 q^{94} + 6558 q^{95} - 37828 q^{98} + 10368 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 6 * q^3 - 20 * q^4 - 30 * q^5 + 304 * q^8 - 24 * q^9 - 204 * q^10 - 58 * q^11 - 588 * q^12 + 560 * q^14 + 468 * q^15 - 72 * q^16 - 246 * q^17 + 216 * q^18 + 642 * q^19 - 1050 * q^21 - 1264 * q^22 + 290 * q^23 + 720 * q^24 - 572 * q^25 + 1008 * q^26 - 28 * q^28 - 2176 * q^29 - 72 * q^30 + 3618 * q^31 + 1584 * q^32 + 2070 * q^33 - 2478 * q^35 - 1632 * q^36 - 270 * q^37 - 6168 * q^38 - 1428 * q^39 - 1752 * q^40 + 6048 * q^42 + 2472 * q^43 + 2412 * q^44 + 1944 * q^45 + 2384 * q^46 - 1542 * q^47 - 980 * q^49 + 7568 * q^50 - 4734 * q^51 - 3192 * q^52 - 4510 * q^53 - 11016 * q^54 - 1232 * q^56 + 11052 * q^57 - 904 * q^58 + 2526 * q^59 - 756 * q^60 - 282 * q^61 - 336 * q^63 - 5472 * q^64 + 5796 * q^65 - 2556 * q^66 - 1318 * q^67 + 20412 * q^68 + 3360 * q^70 - 10408 * q^71 - 1296 * q^72 + 5214 * q^73 + 4036 * q^74 - 9636 * q^75 - 8890 * q^77 - 9072 * q^78 - 8110 * q^79 - 3144 * q^80 + 9306 * q^81 - 4032 * q^82 + 588 * q^84 - 15492 * q^85 + 12928 * q^86 + 5976 * q^87 - 2912 * q^88 + 33990 * q^89 + 17640 * q^91 - 20232 * q^92 + 7446 * q^93 + 27768 * q^94 + 6558 * q^95 - 37828 * q^98 + 10368 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 22x^{2} + 484$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 22$$ (v^2) / 22 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 22$$ (v^3) / 22
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$22\beta_{2}$$ 22*b2 $$\nu^{3}$$ $$=$$ $$22\beta_{3}$$ 22*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −2.34521 − 4.06202i 2.34521 + 4.06202i −2.34521 + 4.06202i 2.34521 − 4.06202i
−3.34521 5.79407i 8.53562 + 4.92804i −14.3808 + 24.9083i 6.57125 3.79391i 65.9413i −32.8329 + 36.3731i 85.3808 8.07125 + 13.9798i −43.9644 25.3828i
3.2 1.34521 + 2.32997i −5.53562 3.19599i 4.38083 7.58782i −21.5712 + 12.4542i 17.1971i 32.8329 + 36.3731i 66.6192 −20.0712 34.7644i −58.0356 33.5069i
5.1 −3.34521 + 5.79407i 8.53562 4.92804i −14.3808 24.9083i 6.57125 + 3.79391i 65.9413i −32.8329 36.3731i 85.3808 8.07125 13.9798i −43.9644 + 25.3828i
5.2 1.34521 2.32997i −5.53562 + 3.19599i 4.38083 + 7.58782i −21.5712 12.4542i 17.1971i 32.8329 36.3731i 66.6192 −20.0712 + 34.7644i −58.0356 + 33.5069i
 $$n$$: 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
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.5.d.a 4
3.b odd 2 1 63.5.m.d 4
4.b odd 2 1 112.5.s.a 4
5.b even 2 1 175.5.i.a 4
5.c odd 4 2 175.5.j.a 8
7.b odd 2 1 49.5.d.b 4
7.c even 3 1 49.5.b.a 4
7.c even 3 1 49.5.d.b 4
7.d odd 6 1 inner 7.5.d.a 4
7.d odd 6 1 49.5.b.a 4
21.g even 6 1 63.5.m.d 4
21.g even 6 1 441.5.d.d 4
21.h odd 6 1 441.5.d.d 4
28.f even 6 1 112.5.s.a 4
28.f even 6 1 784.5.c.c 4
28.g odd 6 1 784.5.c.c 4
35.i odd 6 1 175.5.i.a 4
35.k even 12 2 175.5.j.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.5.d.a 4 1.a even 1 1 trivial
7.5.d.a 4 7.d odd 6 1 inner
49.5.b.a 4 7.c even 3 1
49.5.b.a 4 7.d odd 6 1
49.5.d.b 4 7.b odd 2 1
49.5.d.b 4 7.c even 3 1
63.5.m.d 4 3.b odd 2 1
63.5.m.d 4 21.g even 6 1
112.5.s.a 4 4.b odd 2 1
112.5.s.a 4 28.f even 6 1
175.5.i.a 4 5.b even 2 1
175.5.i.a 4 35.i odd 6 1
175.5.j.a 8 5.c odd 4 2
175.5.j.a 8 35.k even 12 2
441.5.d.d 4 21.g even 6 1
441.5.d.d 4 21.h odd 6 1
784.5.c.c 4 28.f even 6 1
784.5.c.c 4 28.g odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} + 34 T^{2} - 72 T + 324$$
$3$ $$T^{4} - 6 T^{3} - 51 T^{2} + \cdots + 3969$$
$5$ $$T^{4} + 30 T^{3} + 111 T^{2} + \cdots + 35721$$
$7$ $$T^{4} + 490 T^{2} + 5764801$$
$11$ $$T^{4} + 58 T^{3} + 8881 T^{2} + \cdots + 30437289$$
$13$ $$T^{4} + 55272 T^{2} + \cdots + 3111696$$
$17$ $$T^{4} + 246 T^{3} + \cdots + 5076990009$$
$19$ $$T^{4} - 642 T^{3} + \cdots + 3136784049$$
$23$ $$T^{4} - 290 T^{3} + \cdots + 254625849$$
$29$ $$(T^{2} + 1088 T + 188136)^{2}$$
$31$ $$T^{4} - 3618 T^{3} + \cdots + 1071889573041$$
$37$ $$T^{4} + 270 T^{3} + \cdots + 48279954529$$
$41$ $$T^{4} + 4053672 T^{2} + \cdots + 3218392944144$$
$43$ $$(T^{2} - 1236 T - 2313076)^{2}$$
$47$ $$T^{4} + 1542 T^{3} + \cdots + 4451285577249$$
$53$ $$T^{4} + 4510 T^{3} + \cdots + 6460874330625$$
$59$ $$T^{4} + \cdots + 163173619767249$$
$61$ $$T^{4} + 282 T^{3} + \cdots + 14401820070729$$
$67$ $$T^{4} + 1318 T^{3} + \cdots + 151890252361$$
$71$ $$(T^{2} + 5204 T + 5524236)^{2}$$
$73$ $$T^{4} - 5214 T^{3} + \cdots + 6851102086521$$
$79$ $$T^{4} + \cdots + 188584385155609$$
$83$ $$T^{4} + \cdots + 115179601694976$$
$89$ $$T^{4} - 33990 T^{3} + \cdots + 75\!\cdots\!81$$
$97$ $$T^{4} + 217069608 T^{2} + \cdots + 11\!\cdots\!84$$
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