# Properties

 Label 95.2.a.b.1.2 Level $95$ Weight $2$ Character 95.1 Self dual yes Analytic conductor $0.759$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,2,Mod(1,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("95.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 95.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: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.552409$$ of defining polynomial Character $$\chi$$ $$=$$ 95.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.14243 q^{2} -2.87834 q^{3} +2.59002 q^{4} -1.00000 q^{5} +6.16666 q^{6} +3.10482 q^{7} -1.26409 q^{8} +5.28487 q^{9} +O(q^{10})$$ $$q-2.14243 q^{2} -2.87834 q^{3} +2.59002 q^{4} -1.00000 q^{5} +6.16666 q^{6} +3.10482 q^{7} -1.26409 q^{8} +5.28487 q^{9} +2.14243 q^{10} -1.10482 q^{11} -7.45498 q^{12} +1.77353 q^{13} -6.65187 q^{14} +2.87834 q^{15} -2.47182 q^{16} +7.75669 q^{17} -11.3225 q^{18} +1.00000 q^{19} -2.59002 q^{20} -8.93674 q^{21} +2.36700 q^{22} -6.65187 q^{23} +3.63849 q^{24} +1.00000 q^{25} -3.79966 q^{26} -6.57664 q^{27} +8.04156 q^{28} +7.75669 q^{29} -6.16666 q^{30} +6.57664 q^{31} +7.82389 q^{32} +3.18005 q^{33} -16.6182 q^{34} -3.10482 q^{35} +13.6879 q^{36} -1.40652 q^{37} -2.14243 q^{38} -5.10482 q^{39} +1.26409 q^{40} +2.81995 q^{41} +19.1464 q^{42} +3.10482 q^{43} -2.86151 q^{44} -5.28487 q^{45} +14.2512 q^{46} +1.46492 q^{47} +7.11475 q^{48} +2.63990 q^{49} -2.14243 q^{50} -22.3264 q^{51} +4.59348 q^{52} -2.59348 q^{53} +14.0900 q^{54} +1.10482 q^{55} -3.92477 q^{56} -2.87834 q^{57} -16.6182 q^{58} -5.38969 q^{59} +7.45498 q^{60} +4.07523 q^{61} -14.0900 q^{62} +16.4086 q^{63} -11.8185 q^{64} -1.77353 q^{65} -6.81305 q^{66} -15.8151 q^{67} +20.0900 q^{68} +19.1464 q^{69} +6.65187 q^{70} +7.14638 q^{71} -6.68055 q^{72} +0.243310 q^{73} +3.01339 q^{74} -2.87834 q^{75} +2.59002 q^{76} -3.43026 q^{77} +10.9367 q^{78} -9.38969 q^{79} +2.47182 q^{80} +3.07523 q^{81} -6.04156 q^{82} +8.86151 q^{83} -23.1464 q^{84} -7.75669 q^{85} -6.65187 q^{86} -22.3264 q^{87} +1.39659 q^{88} +0.813048 q^{89} +11.3225 q^{90} +5.50648 q^{91} -17.2285 q^{92} -18.9298 q^{93} -3.13849 q^{94} -1.00000 q^{95} -22.5199 q^{96} +3.19689 q^{97} -5.65581 q^{98} -5.83882 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{5} + 4 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + 2 * q^3 + 8 * q^4 - 4 * q^5 + 4 * q^7 - 12 * q^8 + 8 * q^9 $$4 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 4 q^{5} + 4 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{10} + 4 q^{11} + 6 q^{12} + 2 q^{13} - 8 q^{14} - 2 q^{15} + 4 q^{16} + 4 q^{17} - 34 q^{18} + 4 q^{19} - 8 q^{20} - 4 q^{21} + 4 q^{22} - 8 q^{23} - 24 q^{24} + 4 q^{25} + 4 q^{26} - 4 q^{27} - 8 q^{28} + 4 q^{29} + 4 q^{31} - 6 q^{32} + 8 q^{33} - 4 q^{34} - 4 q^{35} + 40 q^{36} - 6 q^{37} - 2 q^{38} - 12 q^{39} + 12 q^{40} + 16 q^{41} + 28 q^{42} + 4 q^{43} + 24 q^{44} - 8 q^{45} - 12 q^{47} + 38 q^{48} + 20 q^{49} - 2 q^{50} - 36 q^{51} + 18 q^{52} - 10 q^{53} - 20 q^{54} - 4 q^{55} - 12 q^{56} + 2 q^{57} - 4 q^{58} - 6 q^{60} + 20 q^{61} + 20 q^{62} + 20 q^{63} - 4 q^{64} - 2 q^{65} - 28 q^{66} - 18 q^{67} + 4 q^{68} + 28 q^{69} + 8 q^{70} - 20 q^{71} - 52 q^{72} + 28 q^{73} + 32 q^{74} + 2 q^{75} + 8 q^{76} - 40 q^{77} + 12 q^{78} - 16 q^{79} - 4 q^{80} + 16 q^{81} + 16 q^{82} - 44 q^{84} - 4 q^{85} - 8 q^{86} - 36 q^{87} - 12 q^{88} + 4 q^{89} + 34 q^{90} - 36 q^{91} - 28 q^{92} - 40 q^{93} - 48 q^{94} - 4 q^{95} - 52 q^{96} + 30 q^{97} + 38 q^{98} - 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + 2 * q^3 + 8 * q^4 - 4 * q^5 + 4 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^10 + 4 * q^11 + 6 * q^12 + 2 * q^13 - 8 * q^14 - 2 * q^15 + 4 * q^16 + 4 * q^17 - 34 * q^18 + 4 * q^19 - 8 * q^20 - 4 * q^21 + 4 * q^22 - 8 * q^23 - 24 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^27 - 8 * q^28 + 4 * q^29 + 4 * q^31 - 6 * q^32 + 8 * q^33 - 4 * q^34 - 4 * q^35 + 40 * q^36 - 6 * q^37 - 2 * q^38 - 12 * q^39 + 12 * q^40 + 16 * q^41 + 28 * q^42 + 4 * q^43 + 24 * q^44 - 8 * q^45 - 12 * q^47 + 38 * q^48 + 20 * q^49 - 2 * q^50 - 36 * q^51 + 18 * q^52 - 10 * q^53 - 20 * q^54 - 4 * q^55 - 12 * q^56 + 2 * q^57 - 4 * q^58 - 6 * q^60 + 20 * q^61 + 20 * q^62 + 20 * q^63 - 4 * q^64 - 2 * q^65 - 28 * q^66 - 18 * q^67 + 4 * q^68 + 28 * q^69 + 8 * q^70 - 20 * q^71 - 52 * q^72 + 28 * q^73 + 32 * q^74 + 2 * q^75 + 8 * q^76 - 40 * q^77 + 12 * q^78 - 16 * q^79 - 4 * q^80 + 16 * q^81 + 16 * q^82 - 44 * q^84 - 4 * q^85 - 8 * q^86 - 36 * q^87 - 12 * q^88 + 4 * q^89 + 34 * q^90 - 36 * q^91 - 28 * q^92 - 40 * q^93 - 48 * q^94 - 4 * q^95 - 52 * q^96 + 30 * q^97 + 38 * q^98 - 4 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.14243 −1.51493 −0.757465 0.652876i $$-0.773562\pi$$
−0.757465 + 0.652876i $$0.773562\pi$$
$$3$$ −2.87834 −1.66181 −0.830907 0.556412i $$-0.812178\pi$$
−0.830907 + 0.556412i $$0.812178\pi$$
$$4$$ 2.59002 1.29501
$$5$$ −1.00000 −0.447214
$$6$$ 6.16666 2.51753
$$7$$ 3.10482 1.17351 0.586756 0.809764i $$-0.300405\pi$$
0.586756 + 0.809764i $$0.300405\pi$$
$$8$$ −1.26409 −0.446923
$$9$$ 5.28487 1.76162
$$10$$ 2.14243 0.677497
$$11$$ −1.10482 −0.333115 −0.166558 0.986032i $$-0.553265\pi$$
−0.166558 + 0.986032i $$0.553265\pi$$
$$12$$ −7.45498 −2.15207
$$13$$ 1.77353 0.491888 0.245944 0.969284i $$-0.420902\pi$$
0.245944 + 0.969284i $$0.420902\pi$$
$$14$$ −6.65187 −1.77779
$$15$$ 2.87834 0.743185
$$16$$ −2.47182 −0.617955
$$17$$ 7.75669 1.88127 0.940637 0.339415i $$-0.110229\pi$$
0.940637 + 0.339415i $$0.110229\pi$$
$$18$$ −11.3225 −2.66874
$$19$$ 1.00000 0.229416
$$20$$ −2.59002 −0.579147
$$21$$ −8.93674 −1.95016
$$22$$ 2.36700 0.504647
$$23$$ −6.65187 −1.38701 −0.693505 0.720451i $$-0.743935\pi$$
−0.693505 + 0.720451i $$0.743935\pi$$
$$24$$ 3.63849 0.742703
$$25$$ 1.00000 0.200000
$$26$$ −3.79966 −0.745175
$$27$$ −6.57664 −1.26567
$$28$$ 8.04156 1.51971
$$29$$ 7.75669 1.44038 0.720191 0.693776i $$-0.244054\pi$$
0.720191 + 0.693776i $$0.244054\pi$$
$$30$$ −6.16666 −1.12587
$$31$$ 6.57664 1.18120 0.590600 0.806965i $$-0.298891\pi$$
0.590600 + 0.806965i $$0.298891\pi$$
$$32$$ 7.82389 1.38308
$$33$$ 3.18005 0.553576
$$34$$ −16.6182 −2.85000
$$35$$ −3.10482 −0.524810
$$36$$ 13.6879 2.28132
$$37$$ −1.40652 −0.231231 −0.115616 0.993294i $$-0.536884\pi$$
−0.115616 + 0.993294i $$0.536884\pi$$
$$38$$ −2.14243 −0.347549
$$39$$ −5.10482 −0.817425
$$40$$ 1.26409 0.199870
$$41$$ 2.81995 0.440402 0.220201 0.975454i $$-0.429329\pi$$
0.220201 + 0.975454i $$0.429329\pi$$
$$42$$ 19.1464 2.95435
$$43$$ 3.10482 0.473480 0.236740 0.971573i $$-0.423921\pi$$
0.236740 + 0.971573i $$0.423921\pi$$
$$44$$ −2.86151 −0.431389
$$45$$ −5.28487 −0.787822
$$46$$ 14.2512 2.10122
$$47$$ 1.46492 0.213680 0.106840 0.994276i $$-0.465927\pi$$
0.106840 + 0.994276i $$0.465927\pi$$
$$48$$ 7.11475 1.02693
$$49$$ 2.63990 0.377129
$$50$$ −2.14243 −0.302986
$$51$$ −22.3264 −3.12633
$$52$$ 4.59348 0.637001
$$53$$ −2.59348 −0.356241 −0.178121 0.984009i $$-0.557002\pi$$
−0.178121 + 0.984009i $$0.557002\pi$$
$$54$$ 14.0900 1.91741
$$55$$ 1.10482 0.148974
$$56$$ −3.92477 −0.524469
$$57$$ −2.87834 −0.381246
$$58$$ −16.6182 −2.18208
$$59$$ −5.38969 −0.701678 −0.350839 0.936436i $$-0.614103\pi$$
−0.350839 + 0.936436i $$0.614103\pi$$
$$60$$ 7.45498 0.962434
$$61$$ 4.07523 0.521780 0.260890 0.965369i $$-0.415984\pi$$
0.260890 + 0.965369i $$0.415984\pi$$
$$62$$ −14.0900 −1.78943
$$63$$ 16.4086 2.06728
$$64$$ −11.8185 −1.47732
$$65$$ −1.77353 −0.219979
$$66$$ −6.81305 −0.838628
$$67$$ −15.8151 −1.93212 −0.966060 0.258318i $$-0.916832\pi$$
−0.966060 + 0.258318i $$0.916832\pi$$
$$68$$ 20.0900 2.43627
$$69$$ 19.1464 2.30495
$$70$$ 6.65187 0.795051
$$71$$ 7.14638 0.848119 0.424059 0.905634i $$-0.360605\pi$$
0.424059 + 0.905634i $$0.360605\pi$$
$$72$$ −6.68055 −0.787310
$$73$$ 0.243310 0.0284773 0.0142387 0.999899i $$-0.495468\pi$$
0.0142387 + 0.999899i $$0.495468\pi$$
$$74$$ 3.01339 0.350299
$$75$$ −2.87834 −0.332363
$$76$$ 2.59002 0.297096
$$77$$ −3.43026 −0.390915
$$78$$ 10.9367 1.23834
$$79$$ −9.38969 −1.05642 −0.528211 0.849113i $$-0.677137\pi$$
−0.528211 + 0.849113i $$0.677137\pi$$
$$80$$ 2.47182 0.276358
$$81$$ 3.07523 0.341692
$$82$$ −6.04156 −0.667178
$$83$$ 8.86151 0.972677 0.486338 0.873771i $$-0.338332\pi$$
0.486338 + 0.873771i $$0.338332\pi$$
$$84$$ −23.1464 −2.52548
$$85$$ −7.75669 −0.841331
$$86$$ −6.65187 −0.717290
$$87$$ −22.3264 −2.39364
$$88$$ 1.39659 0.148877
$$89$$ 0.813048 0.0861829 0.0430914 0.999071i $$-0.486279\pi$$
0.0430914 + 0.999071i $$0.486279\pi$$
$$90$$ 11.3225 1.19349
$$91$$ 5.50648 0.577236
$$92$$ −17.2285 −1.79620
$$93$$ −18.9298 −1.96293
$$94$$ −3.13849 −0.323711
$$95$$ −1.00000 −0.102598
$$96$$ −22.5199 −2.29842
$$97$$ 3.19689 0.324595 0.162297 0.986742i $$-0.448110\pi$$
0.162297 + 0.986742i $$0.448110\pi$$
$$98$$ −5.65581 −0.571323
$$99$$ −5.83882 −0.586824
$$100$$ 2.59002 0.259002
$$101$$ −3.01887 −0.300389 −0.150195 0.988656i $$-0.547990\pi$$
−0.150195 + 0.988656i $$0.547990\pi$$
$$102$$ 47.8329 4.73616
$$103$$ 6.05839 0.596951 0.298476 0.954417i $$-0.403522\pi$$
0.298476 + 0.954417i $$0.403522\pi$$
$$104$$ −2.24190 −0.219836
$$105$$ 8.93674 0.872136
$$106$$ 5.55635 0.539681
$$107$$ 20.3848 1.97068 0.985338 0.170616i $$-0.0545759\pi$$
0.985338 + 0.170616i $$0.0545759\pi$$
$$108$$ −17.0337 −1.63906
$$109$$ 4.72710 0.452774 0.226387 0.974037i $$-0.427309\pi$$
0.226387 + 0.974037i $$0.427309\pi$$
$$110$$ −2.36700 −0.225685
$$111$$ 4.04846 0.384263
$$112$$ −7.67456 −0.725177
$$113$$ −7.98316 −0.750993 −0.375496 0.926824i $$-0.622528\pi$$
−0.375496 + 0.926824i $$0.622528\pi$$
$$114$$ 6.16666 0.577561
$$115$$ 6.65187 0.620290
$$116$$ 20.0900 1.86531
$$117$$ 9.37285 0.866520
$$118$$ 11.5471 1.06299
$$119$$ 24.0831 2.20770
$$120$$ −3.63849 −0.332147
$$121$$ −9.77938 −0.889034
$$122$$ −8.73091 −0.790460
$$123$$ −8.11679 −0.731866
$$124$$ 17.0337 1.52967
$$125$$ −1.00000 −0.0894427
$$126$$ −35.1543 −3.13179
$$127$$ 4.63503 0.411293 0.205646 0.978626i $$-0.434070\pi$$
0.205646 + 0.978626i $$0.434070\pi$$
$$128$$ 9.67265 0.854950
$$129$$ −8.93674 −0.786836
$$130$$ 3.79966 0.333252
$$131$$ 15.5134 1.35541 0.677705 0.735334i $$-0.262975\pi$$
0.677705 + 0.735334i $$0.262975\pi$$
$$132$$ 8.23641 0.716887
$$133$$ 3.10482 0.269222
$$134$$ 33.8828 2.92703
$$135$$ 6.57664 0.566027
$$136$$ −9.80515 −0.840785
$$137$$ −0.813048 −0.0694634 −0.0347317 0.999397i $$-0.511058\pi$$
−0.0347317 + 0.999397i $$0.511058\pi$$
$$138$$ −41.0199 −3.49184
$$139$$ −12.7687 −1.08302 −0.541512 0.840693i $$-0.682148\pi$$
−0.541512 + 0.840693i $$0.682148\pi$$
$$140$$ −8.04156 −0.679636
$$141$$ −4.21654 −0.355097
$$142$$ −15.3106 −1.28484
$$143$$ −1.95942 −0.163855
$$144$$ −13.0632 −1.08860
$$145$$ −7.75669 −0.644158
$$146$$ −0.521277 −0.0431412
$$147$$ −7.59854 −0.626717
$$148$$ −3.64293 −0.299447
$$149$$ −13.7982 −1.13040 −0.565198 0.824955i $$-0.691200\pi$$
−0.565198 + 0.824955i $$0.691200\pi$$
$$150$$ 6.16666 0.503506
$$151$$ 5.02269 0.408740 0.204370 0.978894i $$-0.434485\pi$$
0.204370 + 0.978894i $$0.434485\pi$$
$$152$$ −1.26409 −0.102531
$$153$$ 40.9931 3.31409
$$154$$ 7.34911 0.592208
$$155$$ −6.57664 −0.528248
$$156$$ −13.2216 −1.05858
$$157$$ 7.18695 0.573581 0.286791 0.957993i $$-0.407412\pi$$
0.286791 + 0.957993i $$0.407412\pi$$
$$158$$ 20.1168 1.60041
$$159$$ 7.46492 0.592007
$$160$$ −7.82389 −0.618533
$$161$$ −20.6529 −1.62767
$$162$$ −6.58848 −0.517640
$$163$$ 7.25528 0.568277 0.284139 0.958783i $$-0.408292\pi$$
0.284139 + 0.958783i $$0.408292\pi$$
$$164$$ 7.30374 0.570326
$$165$$ −3.18005 −0.247567
$$166$$ −18.9852 −1.47354
$$167$$ −7.33129 −0.567312 −0.283656 0.958926i $$-0.591547\pi$$
−0.283656 + 0.958926i $$0.591547\pi$$
$$168$$ 11.2968 0.871570
$$169$$ −9.85461 −0.758047
$$170$$ 16.6182 1.27456
$$171$$ 5.28487 0.404144
$$172$$ 8.04156 0.613163
$$173$$ −5.77353 −0.438953 −0.219477 0.975618i $$-0.570435\pi$$
−0.219477 + 0.975618i $$0.570435\pi$$
$$174$$ 47.8329 3.62620
$$175$$ 3.10482 0.234702
$$176$$ 2.73091 0.205850
$$177$$ 15.5134 1.16606
$$178$$ −1.74190 −0.130561
$$179$$ −4.48379 −0.335134 −0.167567 0.985861i $$-0.553591\pi$$
−0.167567 + 0.985861i $$0.553591\pi$$
$$180$$ −13.6879 −1.02024
$$181$$ −2.73400 −0.203217 −0.101608 0.994824i $$-0.532399\pi$$
−0.101608 + 0.994824i $$0.532399\pi$$
$$182$$ −11.7973 −0.874471
$$183$$ −11.7299 −0.867101
$$184$$ 8.40856 0.619887
$$185$$ 1.40652 0.103410
$$186$$ 40.5559 2.97371
$$187$$ −8.56974 −0.626681
$$188$$ 3.79418 0.276719
$$189$$ −20.4193 −1.48528
$$190$$ 2.14243 0.155429
$$191$$ −17.7230 −1.28239 −0.641196 0.767377i $$-0.721562\pi$$
−0.641196 + 0.767377i $$0.721562\pi$$
$$192$$ 34.0178 2.45502
$$193$$ −13.5371 −0.974423 −0.487212 0.873284i $$-0.661986\pi$$
−0.487212 + 0.873284i $$0.661986\pi$$
$$194$$ −6.84912 −0.491738
$$195$$ 5.10482 0.365564
$$196$$ 6.83741 0.488386
$$197$$ −21.5134 −1.53276 −0.766382 0.642385i $$-0.777945\pi$$
−0.766382 + 0.642385i $$0.777945\pi$$
$$198$$ 12.5093 0.888997
$$199$$ 7.09410 0.502888 0.251444 0.967872i $$-0.419095\pi$$
0.251444 + 0.967872i $$0.419095\pi$$
$$200$$ −1.26409 −0.0893846
$$201$$ 45.5213 3.21082
$$202$$ 6.46774 0.455068
$$203$$ 24.0831 1.69030
$$204$$ −57.8260 −4.04863
$$205$$ −2.81995 −0.196954
$$206$$ −12.9797 −0.904339
$$207$$ −35.1543 −2.44339
$$208$$ −4.38384 −0.303964
$$209$$ −1.10482 −0.0764219
$$210$$ −19.1464 −1.32123
$$211$$ 11.0604 0.761431 0.380716 0.924692i $$-0.375678\pi$$
0.380716 + 0.924692i $$0.375678\pi$$
$$212$$ −6.71717 −0.461337
$$213$$ −20.5697 −1.40942
$$214$$ −43.6731 −2.98543
$$215$$ −3.10482 −0.211747
$$216$$ 8.31346 0.565659
$$217$$ 20.4193 1.38615
$$218$$ −10.1275 −0.685921
$$219$$ −0.700331 −0.0473240
$$220$$ 2.86151 0.192923
$$221$$ 13.7567 0.925375
$$222$$ −8.67356 −0.582131
$$223$$ −12.6350 −0.846104 −0.423052 0.906105i $$-0.639041\pi$$
−0.423052 + 0.906105i $$0.639041\pi$$
$$224$$ 24.2918 1.62306
$$225$$ 5.28487 0.352325
$$226$$ 17.1034 1.13770
$$227$$ −6.62813 −0.439925 −0.219962 0.975508i $$-0.570593\pi$$
−0.219962 + 0.975508i $$0.570593\pi$$
$$228$$ −7.45498 −0.493718
$$229$$ −0.0752308 −0.00497139 −0.00248570 0.999997i $$-0.500791\pi$$
−0.00248570 + 0.999997i $$0.500791\pi$$
$$230$$ −14.2512 −0.939696
$$231$$ 9.87348 0.649627
$$232$$ −9.80515 −0.643740
$$233$$ 9.51338 0.623242 0.311621 0.950206i $$-0.399128\pi$$
0.311621 + 0.950206i $$0.399128\pi$$
$$234$$ −20.0807 −1.31272
$$235$$ −1.46492 −0.0955607
$$236$$ −13.9594 −0.908681
$$237$$ 27.0268 1.75558
$$238$$ −51.5965 −3.34450
$$239$$ −2.92984 −0.189515 −0.0947577 0.995500i $$-0.530208\pi$$
−0.0947577 + 0.995500i $$0.530208\pi$$
$$240$$ −7.11475 −0.459255
$$241$$ −9.42743 −0.607274 −0.303637 0.952788i $$-0.598201\pi$$
−0.303637 + 0.952788i $$0.598201\pi$$
$$242$$ 20.9517 1.34682
$$243$$ 10.8783 0.697846
$$244$$ 10.5549 0.675711
$$245$$ −2.63990 −0.168657
$$246$$ 17.3897 1.10873
$$247$$ 1.77353 0.112847
$$248$$ −8.31346 −0.527905
$$249$$ −25.5065 −1.61641
$$250$$ 2.14243 0.135499
$$251$$ −14.9298 −0.942363 −0.471181 0.882036i $$-0.656172\pi$$
−0.471181 + 0.882036i $$0.656172\pi$$
$$252$$ 42.4986 2.67716
$$253$$ 7.34911 0.462035
$$254$$ −9.93026 −0.623080
$$255$$ 22.3264 1.39814
$$256$$ 2.91405 0.182128
$$257$$ −15.1295 −0.943755 −0.471877 0.881664i $$-0.656424\pi$$
−0.471877 + 0.881664i $$0.656424\pi$$
$$258$$ 19.1464 1.19200
$$259$$ −4.36700 −0.271352
$$260$$ −4.59348 −0.284875
$$261$$ 40.9931 2.53741
$$262$$ −33.2364 −2.05335
$$263$$ −15.2216 −0.938605 −0.469302 0.883038i $$-0.655495\pi$$
−0.469302 + 0.883038i $$0.655495\pi$$
$$264$$ −4.01987 −0.247406
$$265$$ 2.59348 0.159316
$$266$$ −6.65187 −0.407852
$$267$$ −2.34023 −0.143220
$$268$$ −40.9615 −2.50212
$$269$$ 11.5203 0.702404 0.351202 0.936300i $$-0.385773\pi$$
0.351202 + 0.936300i $$0.385773\pi$$
$$270$$ −14.0900 −0.857491
$$271$$ 2.16118 0.131282 0.0656411 0.997843i $$-0.479091\pi$$
0.0656411 + 0.997843i $$0.479091\pi$$
$$272$$ −19.1731 −1.16254
$$273$$ −15.8495 −0.959258
$$274$$ 1.74190 0.105232
$$275$$ −1.10482 −0.0666231
$$276$$ 49.5896 2.98494
$$277$$ 23.4205 1.40720 0.703602 0.710595i $$-0.251574\pi$$
0.703602 + 0.710595i $$0.251574\pi$$
$$278$$ 27.3560 1.64070
$$279$$ 34.7567 2.08083
$$280$$ 3.92477 0.234550
$$281$$ 25.6824 1.53209 0.766043 0.642789i $$-0.222223\pi$$
0.766043 + 0.642789i $$0.222223\pi$$
$$282$$ 9.03366 0.537947
$$283$$ 7.38587 0.439045 0.219522 0.975607i $$-0.429550\pi$$
0.219522 + 0.975607i $$0.429550\pi$$
$$284$$ 18.5093 1.09832
$$285$$ 2.87834 0.170498
$$286$$ 4.19794 0.248229
$$287$$ 8.75543 0.516817
$$288$$ 41.3482 2.43647
$$289$$ 43.1662 2.53919
$$290$$ 16.6182 0.975854
$$291$$ −9.20174 −0.539416
$$292$$ 0.630180 0.0368785
$$293$$ −24.1522 −1.41099 −0.705494 0.708716i $$-0.749275\pi$$
−0.705494 + 0.708716i $$0.749275\pi$$
$$294$$ 16.2794 0.949433
$$295$$ 5.38969 0.313800
$$296$$ 1.77797 0.103343
$$297$$ 7.26600 0.421616
$$298$$ 29.5618 1.71247
$$299$$ −11.7973 −0.682253
$$300$$ −7.45498 −0.430414
$$301$$ 9.63990 0.555635
$$302$$ −10.7608 −0.619213
$$303$$ 8.68936 0.499190
$$304$$ −2.47182 −0.141769
$$305$$ −4.07523 −0.233347
$$306$$ −87.8250 −5.02062
$$307$$ 4.38482 0.250255 0.125127 0.992141i $$-0.460066\pi$$
0.125127 + 0.992141i $$0.460066\pi$$
$$308$$ −8.88447 −0.506239
$$309$$ −17.4381 −0.992022
$$310$$ 14.0900 0.800259
$$311$$ −15.7123 −0.890963 −0.445481 0.895291i $$-0.646967\pi$$
−0.445481 + 0.895291i $$0.646967\pi$$
$$312$$ 6.45295 0.365326
$$313$$ 24.7794 1.40061 0.700307 0.713842i $$-0.253047\pi$$
0.700307 + 0.713842i $$0.253047\pi$$
$$314$$ −15.3976 −0.868935
$$315$$ −16.4086 −0.924518
$$316$$ −24.3195 −1.36808
$$317$$ −27.3798 −1.53780 −0.768900 0.639369i $$-0.779196\pi$$
−0.768900 + 0.639369i $$0.779196\pi$$
$$318$$ −15.9931 −0.896848
$$319$$ −8.56974 −0.479813
$$320$$ 11.8185 0.660676
$$321$$ −58.6745 −3.27489
$$322$$ 44.2474 2.46581
$$323$$ 7.75669 0.431594
$$324$$ 7.96492 0.442496
$$325$$ 1.77353 0.0983775
$$326$$ −15.5440 −0.860900
$$327$$ −13.6062 −0.752426
$$328$$ −3.56467 −0.196826
$$329$$ 4.54831 0.250756
$$330$$ 6.81305 0.375046
$$331$$ 4.70033 0.258354 0.129177 0.991622i $$-0.458767\pi$$
0.129177 + 0.991622i $$0.458767\pi$$
$$332$$ 22.9515 1.25963
$$333$$ −7.43329 −0.407342
$$334$$ 15.7068 0.859439
$$335$$ 15.8151 0.864070
$$336$$ 22.0900 1.20511
$$337$$ −20.6360 −1.12412 −0.562058 0.827098i $$-0.689990\pi$$
−0.562058 + 0.827098i $$0.689990\pi$$
$$338$$ 21.1128 1.14839
$$339$$ 22.9783 1.24801
$$340$$ −20.0900 −1.08953
$$341$$ −7.26600 −0.393476
$$342$$ −11.3225 −0.612250
$$343$$ −13.5373 −0.730947
$$344$$ −3.92477 −0.211609
$$345$$ −19.1464 −1.03081
$$346$$ 12.3694 0.664983
$$347$$ 30.0148 1.61128 0.805639 0.592407i $$-0.201822\pi$$
0.805639 + 0.592407i $$0.201822\pi$$
$$348$$ −57.8260 −3.09980
$$349$$ −14.7340 −0.788693 −0.394347 0.918962i $$-0.629029\pi$$
−0.394347 + 0.918962i $$0.629029\pi$$
$$350$$ −6.65187 −0.355557
$$351$$ −11.6638 −0.622570
$$352$$ −8.64399 −0.460726
$$353$$ −29.6302 −1.57705 −0.788527 0.615000i $$-0.789156\pi$$
−0.788527 + 0.615000i $$0.789156\pi$$
$$354$$ −33.2364 −1.76649
$$355$$ −7.14638 −0.379290
$$356$$ 2.10581 0.111608
$$357$$ −69.3195 −3.66878
$$358$$ 9.60623 0.507705
$$359$$ 21.7577 1.14833 0.574163 0.818741i $$-0.305328\pi$$
0.574163 + 0.818741i $$0.305328\pi$$
$$360$$ 6.68055 0.352096
$$361$$ 1.00000 0.0526316
$$362$$ 5.85742 0.307859
$$363$$ 28.1484 1.47741
$$364$$ 14.2619 0.747527
$$365$$ −0.243310 −0.0127355
$$366$$ 25.1306 1.31360
$$367$$ 30.0347 1.56780 0.783898 0.620889i $$-0.213228\pi$$
0.783898 + 0.620889i $$0.213228\pi$$
$$368$$ 16.4422 0.857111
$$369$$ 14.9031 0.775823
$$370$$ −3.01339 −0.156658
$$371$$ −8.05227 −0.418053
$$372$$ −49.0287 −2.54202
$$373$$ −37.3055 −1.93161 −0.965803 0.259277i $$-0.916516\pi$$
−0.965803 + 0.259277i $$0.916516\pi$$
$$374$$ 18.3601 0.949378
$$375$$ 2.87834 0.148637
$$376$$ −1.85179 −0.0954987
$$377$$ 13.7567 0.708506
$$378$$ 43.7470 2.25010
$$379$$ −14.7794 −0.759166 −0.379583 0.925158i $$-0.623932\pi$$
−0.379583 + 0.925158i $$0.623932\pi$$
$$380$$ −2.59002 −0.132865
$$381$$ −13.3412 −0.683492
$$382$$ 37.9704 1.94273
$$383$$ −29.0020 −1.48193 −0.740967 0.671541i $$-0.765633\pi$$
−0.740967 + 0.671541i $$0.765633\pi$$
$$384$$ −27.8412 −1.42077
$$385$$ 3.43026 0.174822
$$386$$ 29.0024 1.47618
$$387$$ 16.4086 0.834094
$$388$$ 8.28001 0.420354
$$389$$ 21.1987 1.07481 0.537407 0.843323i $$-0.319404\pi$$
0.537407 + 0.843323i $$0.319404\pi$$
$$390$$ −10.9367 −0.553803
$$391$$ −51.5965 −2.60935
$$392$$ −3.33707 −0.168548
$$393$$ −44.6529 −2.25244
$$394$$ 46.0910 2.32203
$$395$$ 9.38969 0.472446
$$396$$ −15.1227 −0.759944
$$397$$ 10.3856 0.521238 0.260619 0.965442i $$-0.416073\pi$$
0.260619 + 0.965442i $$0.416073\pi$$
$$398$$ −15.1987 −0.761840
$$399$$ −8.93674 −0.447397
$$400$$ −2.47182 −0.123591
$$401$$ 25.6638 1.28159 0.640796 0.767712i $$-0.278605\pi$$
0.640796 + 0.767712i $$0.278605\pi$$
$$402$$ −97.5263 −4.86417
$$403$$ 11.6638 0.581017
$$404$$ −7.81896 −0.389008
$$405$$ −3.07523 −0.152809
$$406$$ −51.5965 −2.56069
$$407$$ 1.55395 0.0770266
$$408$$ 28.2226 1.39723
$$409$$ 5.71611 0.282644 0.141322 0.989964i $$-0.454865\pi$$
0.141322 + 0.989964i $$0.454865\pi$$
$$410$$ 6.04156 0.298371
$$411$$ 2.34023 0.115435
$$412$$ 15.6914 0.773059
$$413$$ −16.7340 −0.823427
$$414$$ 75.3157 3.70156
$$415$$ −8.86151 −0.434994
$$416$$ 13.8759 0.680321
$$417$$ 36.7526 1.79978
$$418$$ 2.36700 0.115774
$$419$$ 15.4303 0.753818 0.376909 0.926250i $$-0.376987\pi$$
0.376909 + 0.926250i $$0.376987\pi$$
$$420$$ 23.1464 1.12943
$$421$$ −1.18005 −0.0575121 −0.0287561 0.999586i $$-0.509155\pi$$
−0.0287561 + 0.999586i $$0.509155\pi$$
$$422$$ −23.6962 −1.15352
$$423$$ 7.74190 0.376424
$$424$$ 3.27839 0.159213
$$425$$ 7.75669 0.376255
$$426$$ 44.0693 2.13517
$$427$$ 12.6529 0.612315
$$428$$ 52.7972 2.55205
$$429$$ 5.63990 0.272297
$$430$$ 6.65187 0.320782
$$431$$ −14.0255 −0.675585 −0.337792 0.941221i $$-0.609680\pi$$
−0.337792 + 0.941221i $$0.609680\pi$$
$$432$$ 16.2563 0.782130
$$433$$ 25.5894 1.22975 0.614874 0.788625i $$-0.289207\pi$$
0.614874 + 0.788625i $$0.289207\pi$$
$$434$$ −43.7470 −2.09992
$$435$$ 22.3264 1.07047
$$436$$ 12.2433 0.586348
$$437$$ −6.65187 −0.318202
$$438$$ 1.50041 0.0716925
$$439$$ −21.9732 −1.04873 −0.524363 0.851495i $$-0.675696\pi$$
−0.524363 + 0.851495i $$0.675696\pi$$
$$440$$ −1.39659 −0.0665798
$$441$$ 13.9515 0.664358
$$442$$ −29.4728 −1.40188
$$443$$ −19.3721 −0.920395 −0.460197 0.887817i $$-0.652221\pi$$
−0.460197 + 0.887817i $$0.652221\pi$$
$$444$$ 10.4856 0.497625
$$445$$ −0.813048 −0.0385422
$$446$$ 27.0697 1.28179
$$447$$ 39.7161 1.87851
$$448$$ −36.6944 −1.73365
$$449$$ −19.1841 −0.905355 −0.452677 0.891674i $$-0.649531\pi$$
−0.452677 + 0.891674i $$0.649531\pi$$
$$450$$ −11.3225 −0.533747
$$451$$ −3.11553 −0.146705
$$452$$ −20.6766 −0.972545
$$453$$ −14.4570 −0.679250
$$454$$ 14.2003 0.666455
$$455$$ −5.50648 −0.258148
$$456$$ 3.63849 0.170388
$$457$$ 8.36010 0.391069 0.195534 0.980697i $$-0.437356\pi$$
0.195534 + 0.980697i $$0.437356\pi$$
$$458$$ 0.161177 0.00753131
$$459$$ −51.0130 −2.38108
$$460$$ 17.2285 0.803283
$$461$$ −25.0268 −1.16561 −0.582806 0.812611i $$-0.698045\pi$$
−0.582806 + 0.812611i $$0.698045\pi$$
$$462$$ −21.1533 −0.984140
$$463$$ 32.3749 1.50459 0.752294 0.658827i $$-0.228947\pi$$
0.752294 + 0.658827i $$0.228947\pi$$
$$464$$ −19.1731 −0.890091
$$465$$ 18.9298 0.877850
$$466$$ −20.3818 −0.944168
$$467$$ 15.2216 0.704372 0.352186 0.935930i $$-0.385438\pi$$
0.352186 + 0.935930i $$0.385438\pi$$
$$468$$ 24.2759 1.12215
$$469$$ −49.1030 −2.26736
$$470$$ 3.13849 0.144768
$$471$$ −20.6865 −0.953185
$$472$$ 6.81305 0.313596
$$473$$ −3.43026 −0.157724
$$474$$ −57.9031 −2.65958
$$475$$ 1.00000 0.0458831
$$476$$ 62.3759 2.85899
$$477$$ −13.7062 −0.627563
$$478$$ 6.27698 0.287103
$$479$$ −20.0485 −0.916038 −0.458019 0.888943i $$-0.651441\pi$$
−0.458019 + 0.888943i $$0.651441\pi$$
$$480$$ 22.5199 1.02789
$$481$$ −2.49451 −0.113740
$$482$$ 20.1977 0.919978
$$483$$ 59.4460 2.70489
$$484$$ −25.3288 −1.15131
$$485$$ −3.19689 −0.145163
$$486$$ −23.3061 −1.05719
$$487$$ −31.6508 −1.43424 −0.717118 0.696952i $$-0.754539\pi$$
−0.717118 + 0.696952i $$0.754539\pi$$
$$488$$ −5.15146 −0.233195
$$489$$ −20.8832 −0.944371
$$490$$ 5.65581 0.255504
$$491$$ 24.8033 1.11936 0.559679 0.828710i $$-0.310924\pi$$
0.559679 + 0.828710i $$0.310924\pi$$
$$492$$ −21.0227 −0.947776
$$493$$ 60.1662 2.70975
$$494$$ −3.79966 −0.170955
$$495$$ 5.83882 0.262436
$$496$$ −16.2563 −0.729928
$$497$$ 22.1882 0.995277
$$498$$ 54.6460 2.44874
$$499$$ −33.0237 −1.47834 −0.739171 0.673518i $$-0.764783\pi$$
−0.739171 + 0.673518i $$0.764783\pi$$
$$500$$ −2.59002 −0.115829
$$501$$ 21.1020 0.942767
$$502$$ 31.9862 1.42761
$$503$$ −3.54396 −0.158017 −0.0790087 0.996874i $$-0.525175\pi$$
−0.0790087 + 0.996874i $$0.525175\pi$$
$$504$$ −20.7419 −0.923917
$$505$$ 3.01887 0.134338
$$506$$ −15.7450 −0.699950
$$507$$ 28.3650 1.25973
$$508$$ 12.0049 0.532629
$$509$$ −2.11679 −0.0938250 −0.0469125 0.998899i $$-0.514938\pi$$
−0.0469125 + 0.998899i $$0.514938\pi$$
$$510$$ −47.8329 −2.11808
$$511$$ 0.755435 0.0334185
$$512$$ −25.5885 −1.13086
$$513$$ −6.57664 −0.290366
$$514$$ 32.4140 1.42972
$$515$$ −6.05839 −0.266965
$$516$$ −23.1464 −1.01896
$$517$$ −1.61847 −0.0711802
$$518$$ 9.35601 0.411080
$$519$$ 16.6182 0.729458
$$520$$ 2.24190 0.0983136
$$521$$ −8.60748 −0.377101 −0.188550 0.982064i $$-0.560379\pi$$
−0.188550 + 0.982064i $$0.560379\pi$$
$$522$$ −87.8250 −3.84400
$$523$$ 36.8349 1.61068 0.805340 0.592814i $$-0.201983\pi$$
0.805340 + 0.592814i $$0.201983\pi$$
$$524$$ 40.1800 1.75527
$$525$$ −8.93674 −0.390031
$$526$$ 32.6113 1.42192
$$527$$ 51.0130 2.22216
$$528$$ −7.86051 −0.342085
$$529$$ 21.2474 0.923799
$$530$$ −5.55635 −0.241353
$$531$$ −28.4838 −1.23609
$$532$$ 8.04156 0.348646
$$533$$ 5.00125 0.216628
$$534$$ 5.01379 0.216968
$$535$$ −20.3848 −0.881313
$$536$$ 19.9917 0.863509
$$537$$ 12.9059 0.556931
$$538$$ −24.6814 −1.06409
$$539$$ −2.91661 −0.125627
$$540$$ 17.0337 0.733012
$$541$$ −32.0079 −1.37613 −0.688063 0.725651i $$-0.741539\pi$$
−0.688063 + 0.725651i $$0.741539\pi$$
$$542$$ −4.63018 −0.198883
$$543$$ 7.86941 0.337709
$$544$$ 60.6875 2.60196
$$545$$ −4.72710 −0.202487
$$546$$ 33.9566 1.45321
$$547$$ 17.6240 0.753550 0.376775 0.926305i $$-0.377033\pi$$
0.376775 + 0.926305i $$0.377033\pi$$
$$548$$ −2.10581 −0.0899559
$$549$$ 21.5371 0.919179
$$550$$ 2.36700 0.100929
$$551$$ 7.75669 0.330446
$$552$$ −24.2027 −1.03014
$$553$$ −29.1533 −1.23972
$$554$$ −50.1769 −2.13181
$$555$$ −4.04846 −0.171848
$$556$$ −33.0711 −1.40253
$$557$$ −34.6865 −1.46972 −0.734858 0.678221i $$-0.762751\pi$$
−0.734858 + 0.678221i $$0.762751\pi$$
$$558$$ −74.4639 −3.15231
$$559$$ 5.50648 0.232899
$$560$$ 7.67456 0.324309
$$561$$ 24.6667 1.04143
$$562$$ −55.0229 −2.32100
$$563$$ 13.5073 0.569263 0.284632 0.958637i $$-0.408129\pi$$
0.284632 + 0.958637i $$0.408129\pi$$
$$564$$ −10.9209 −0.459855
$$565$$ 7.98316 0.335854
$$566$$ −15.8238 −0.665122
$$567$$ 9.54803 0.400980
$$568$$ −9.03366 −0.379044
$$569$$ 32.3588 1.35655 0.678276 0.734807i $$-0.262727\pi$$
0.678276 + 0.734807i $$0.262727\pi$$
$$570$$ −6.16666 −0.258293
$$571$$ −8.93293 −0.373831 −0.186916 0.982376i $$-0.559849\pi$$
−0.186916 + 0.982376i $$0.559849\pi$$
$$572$$ −5.07496 −0.212195
$$573$$ 51.0130 2.13110
$$574$$ −18.7579 −0.782941
$$575$$ −6.65187 −0.277402
$$576$$ −62.4594 −2.60248
$$577$$ 14.9059 0.620541 0.310270 0.950648i $$-0.399580\pi$$
0.310270 + 0.950648i $$0.399580\pi$$
$$578$$ −92.4808 −3.84669
$$579$$ 38.9645 1.61931
$$580$$ −20.0900 −0.834193
$$581$$ 27.5134 1.14145
$$582$$ 19.7141 0.817177
$$583$$ 2.86532 0.118669
$$584$$ −0.307566 −0.0127272
$$585$$ −9.37285 −0.387520
$$586$$ 51.7446 2.13755
$$587$$ 7.37207 0.304278 0.152139 0.988359i $$-0.451384\pi$$
0.152139 + 0.988359i $$0.451384\pi$$
$$588$$ −19.6804 −0.811607
$$589$$ 6.57664 0.270986
$$590$$ −11.5471 −0.475385
$$591$$ 61.9229 2.54717
$$592$$ 3.47668 0.142890
$$593$$ −17.6853 −0.726247 −0.363124 0.931741i $$-0.618290\pi$$
−0.363124 + 0.931741i $$0.618290\pi$$
$$594$$ −15.5669 −0.638718
$$595$$ −24.0831 −0.987312
$$596$$ −35.7378 −1.46388
$$597$$ −20.4193 −0.835705
$$598$$ 25.2749 1.03357
$$599$$ 5.30657 0.216821 0.108410 0.994106i $$-0.465424\pi$$
0.108410 + 0.994106i $$0.465424\pi$$
$$600$$ 3.63849 0.148541
$$601$$ 43.7875 1.78613 0.893065 0.449927i $$-0.148550\pi$$
0.893065 + 0.449927i $$0.148550\pi$$
$$602$$ −20.6529 −0.841747
$$603$$ −83.5806 −3.40367
$$604$$ 13.0089 0.529324
$$605$$ 9.77938 0.397588
$$606$$ −18.6164 −0.756239
$$607$$ −7.24535 −0.294080 −0.147040 0.989131i $$-0.546975\pi$$
−0.147040 + 0.989131i $$0.546975\pi$$
$$608$$ 7.82389 0.317301
$$609$$ −69.3195 −2.80897
$$610$$ 8.73091 0.353504
$$611$$ 2.59807 0.105107
$$612$$ 106.173 4.29179
$$613$$ 20.8962 0.843988 0.421994 0.906599i $$-0.361330\pi$$
0.421994 + 0.906599i $$0.361330\pi$$
$$614$$ −9.39419 −0.379119
$$615$$ 8.11679 0.327301
$$616$$ 4.33616 0.174709
$$617$$ −22.0494 −0.887677 −0.443839 0.896107i $$-0.646384\pi$$
−0.443839 + 0.896107i $$0.646384\pi$$
$$618$$ 37.3601 1.50284
$$619$$ −0.0484607 −0.00194780 −0.000973900 1.00000i $$-0.500310\pi$$
−0.000973900 1.00000i $$0.500310\pi$$
$$620$$ −17.0337 −0.684088
$$621$$ 43.7470 1.75550
$$622$$ 33.6626 1.34975
$$623$$ 2.52437 0.101137
$$624$$ 12.6182 0.505132
$$625$$ 1.00000 0.0400000
$$626$$ −53.0882 −2.12183
$$627$$ 3.18005 0.126999
$$628$$ 18.6144 0.742795
$$629$$ −10.9100 −0.435009
$$630$$ 35.1543 1.40058
$$631$$ −32.6182 −1.29851 −0.649255 0.760571i $$-0.724919\pi$$
−0.649255 + 0.760571i $$0.724919\pi$$
$$632$$ 11.8694 0.472140
$$633$$ −31.8357 −1.26536
$$634$$ 58.6593 2.32966
$$635$$ −4.63503 −0.183936
$$636$$ 19.3343 0.766656
$$637$$ 4.68193 0.185505
$$638$$ 18.3601 0.726883
$$639$$ 37.7677 1.49407
$$640$$ −9.67265 −0.382345
$$641$$ −13.4136 −0.529806 −0.264903 0.964275i $$-0.585340\pi$$
−0.264903 + 0.964275i $$0.585340\pi$$
$$642$$ 125.706 4.96123
$$643$$ 23.8052 0.938783 0.469392 0.882990i $$-0.344473\pi$$
0.469392 + 0.882990i $$0.344473\pi$$
$$644$$ −53.4914 −2.10786
$$645$$ 8.93674 0.351884
$$646$$ −16.6182 −0.653834
$$647$$ −48.4580 −1.90508 −0.952540 0.304412i $$-0.901540\pi$$
−0.952540 + 0.304412i $$0.901540\pi$$
$$648$$ −3.88737 −0.152710
$$649$$ 5.95463 0.233740
$$650$$ −3.79966 −0.149035
$$651$$ −58.7737 −2.30352
$$652$$ 18.7914 0.735926
$$653$$ 22.2351 0.870128 0.435064 0.900399i $$-0.356726\pi$$
0.435064 + 0.900399i $$0.356726\pi$$
$$654$$ 29.1504 1.13987
$$655$$ −15.5134 −0.606158
$$656$$ −6.97041 −0.272149
$$657$$ 1.28586 0.0501663
$$658$$ −9.74445 −0.379878
$$659$$ −10.7072 −0.417095 −0.208547 0.978012i $$-0.566874\pi$$
−0.208547 + 0.978012i $$0.566874\pi$$
$$660$$ −8.23641 −0.320602
$$661$$ −44.7980 −1.74244 −0.871220 0.490893i $$-0.836670\pi$$
−0.871220 + 0.490893i $$0.836670\pi$$
$$662$$ −10.0702 −0.391388
$$663$$ −39.5965 −1.53780
$$664$$ −11.2017 −0.434712
$$665$$ −3.10482 −0.120400
$$666$$ 15.9253 0.617095
$$667$$ −51.5965 −1.99782
$$668$$ −18.9882 −0.734677
$$669$$ 36.3680 1.40607
$$670$$ −33.8828 −1.30901
$$671$$ −4.50239 −0.173813
$$672$$ −69.9201 −2.69723
$$673$$ −43.2772 −1.66821 −0.834106 0.551604i $$-0.814016\pi$$
−0.834106 + 0.551604i $$0.814016\pi$$
$$674$$ 44.2113 1.70296
$$675$$ −6.57664 −0.253135
$$676$$ −25.5237 −0.981680
$$677$$ 11.5330 0.443251 0.221625 0.975132i $$-0.428864\pi$$
0.221625 + 0.975132i $$0.428864\pi$$
$$678$$ −49.2295 −1.89065
$$679$$ 9.92575 0.380915
$$680$$ 9.80515 0.376010
$$681$$ 19.0780 0.731072
$$682$$ 15.5669 0.596088
$$683$$ 0.916090 0.0350532 0.0175266 0.999846i $$-0.494421\pi$$
0.0175266 + 0.999846i $$0.494421\pi$$
$$684$$ 13.6879 0.523372
$$685$$ 0.813048 0.0310650
$$686$$ 29.0028 1.10733
$$687$$ 0.216540 0.00826153
$$688$$ −7.67456 −0.292590
$$689$$ −4.59960 −0.175231
$$690$$ 41.0199 1.56160
$$691$$ 14.8278 0.564077 0.282039 0.959403i $$-0.408989\pi$$
0.282039 + 0.959403i $$0.408989\pi$$
$$692$$ −14.9536 −0.568450
$$693$$ −18.1285 −0.688644
$$694$$ −64.3047 −2.44097
$$695$$ 12.7687 0.484343
$$696$$ 28.2226 1.06978
$$697$$ 21.8735 0.828517
$$698$$ 31.5666 1.19481
$$699$$ −27.3828 −1.03571
$$700$$ 8.04156 0.303942
$$701$$ 36.1722 1.36620 0.683102 0.730323i $$-0.260631\pi$$
0.683102 + 0.730323i $$0.260631\pi$$
$$702$$ 24.9890 0.943150
$$703$$ −1.40652 −0.0530481
$$704$$ 13.0573 0.492117
$$705$$ 4.21654 0.158804
$$706$$ 63.4807 2.38913
$$707$$ −9.37305 −0.352510
$$708$$ 40.1800 1.51006
$$709$$ 10.4331 0.391823 0.195911 0.980622i $$-0.437234\pi$$
0.195911 + 0.980622i $$0.437234\pi$$
$$710$$ 15.3106 0.574598
$$711$$ −49.6233 −1.86102
$$712$$ −1.02777 −0.0385171
$$713$$ −43.7470 −1.63834
$$714$$ 148.513 5.55794
$$715$$ 1.95942 0.0732783
$$716$$ −11.6131 −0.434003
$$717$$ 8.43308 0.314939
$$718$$ −46.6144 −1.73963
$$719$$ 7.73373 0.288420 0.144210 0.989547i $$-0.453936\pi$$
0.144210 + 0.989547i $$0.453936\pi$$
$$720$$ 13.0632 0.486839
$$721$$ 18.8102 0.700529
$$722$$ −2.14243 −0.0797331
$$723$$ 27.1354 1.00918
$$724$$ −7.08114 −0.263168
$$725$$ 7.75669 0.288076
$$726$$ −60.3061 −2.23817
$$727$$ −39.5241 −1.46587 −0.732934 0.680300i $$-0.761849\pi$$
−0.732934 + 0.680300i $$0.761849\pi$$
$$728$$ −6.96068 −0.257980
$$729$$ −40.5373 −1.50138
$$730$$ 0.521277 0.0192933
$$731$$ 24.0831 0.890746
$$732$$ −30.3808 −1.12291
$$733$$ 50.7498 1.87449 0.937243 0.348677i $$-0.113369\pi$$
0.937243 + 0.348677i $$0.113369\pi$$
$$734$$ −64.3473 −2.37510
$$735$$ 7.59854 0.280277
$$736$$ −52.0435 −1.91835
$$737$$ 17.4728 0.643619
$$738$$ −31.9288 −1.17532
$$739$$ 0.419276 0.0154233 0.00771165 0.999970i $$-0.497545\pi$$
0.00771165 + 0.999970i $$0.497545\pi$$
$$740$$ 3.64293 0.133917
$$741$$ −5.10482 −0.187530
$$742$$ 17.2515 0.633321
$$743$$ 19.5511 0.717259 0.358630 0.933480i $$-0.383244\pi$$
0.358630 + 0.933480i $$0.383244\pi$$
$$744$$ 23.9290 0.877280
$$745$$ 13.7982 0.505529
$$746$$ 79.9246 2.92625
$$747$$ 46.8319 1.71349
$$748$$ −22.1958 −0.811560
$$749$$ 63.2912 2.31261
$$750$$ −6.16666 −0.225175
$$751$$ 8.76768 0.319937 0.159969 0.987122i $$-0.448861\pi$$
0.159969 + 0.987122i $$0.448861\pi$$
$$752$$ −3.62102 −0.132045
$$753$$ 42.9732 1.56603
$$754$$ −29.4728 −1.07334
$$755$$ −5.02269 −0.182794
$$756$$ −52.8864 −1.92346
$$757$$ −48.0535 −1.74653 −0.873267 0.487241i $$-0.838003\pi$$
−0.873267 + 0.487241i $$0.838003\pi$$
$$758$$ 31.6638 1.15008
$$759$$ −21.1533 −0.767815
$$760$$ 1.26409 0.0458533
$$761$$ 29.3947 1.06556 0.532779 0.846254i $$-0.321148\pi$$
0.532779 + 0.846254i $$0.321148\pi$$
$$762$$ 28.5827 1.03544
$$763$$ 14.6768 0.531336
$$764$$ −45.9031 −1.66071
$$765$$ −40.9931 −1.48211
$$766$$ 62.1350 2.24503
$$767$$ −9.55875 −0.345146
$$768$$ −8.38765 −0.302663
$$769$$ −13.3790 −0.482458 −0.241229 0.970468i $$-0.577551\pi$$
−0.241229 + 0.970468i $$0.577551\pi$$
$$770$$ −7.34911 −0.264844
$$771$$ 43.5480 1.56834
$$772$$ −35.0615 −1.26189
$$773$$ 0.133625 0.00480617 0.00240309 0.999997i $$-0.499235\pi$$
0.00240309 + 0.999997i $$0.499235\pi$$
$$774$$ −35.1543 −1.26359
$$775$$ 6.57664 0.236240
$$776$$ −4.04115 −0.145069
$$777$$ 12.5697 0.450937
$$778$$ −45.4167 −1.62827
$$779$$ 2.81995 0.101035
$$780$$ 13.2216 0.473410
$$781$$ −7.89545 −0.282522
$$782$$ 110.542 3.95298
$$783$$ −51.0130 −1.82305
$$784$$ −6.52536 −0.233049
$$785$$ −7.18695 −0.256513
$$786$$ 95.6658 3.41229
$$787$$ −4.84060 −0.172549 −0.0862744 0.996271i $$-0.527496\pi$$
−0.0862744 + 0.996271i $$0.527496\pi$$
$$788$$ −55.7202 −1.98495
$$789$$ 43.8130 1.55979
$$790$$ −20.1168 −0.715723
$$791$$ −24.7863 −0.881299
$$792$$ 7.38079 0.262265
$$793$$ 7.22753 0.256657
$$794$$ −22.2505 −0.789640
$$795$$ −7.46492 −0.264753
$$796$$ 18.3739 0.651246
$$797$$ 11.6993 0.414410 0.207205 0.978298i $$-0.433563\pi$$
0.207205 + 0.978298i $$0.433563\pi$$
$$798$$ 19.1464 0.677774
$$799$$ 11.3629 0.401991
$$800$$ 7.82389 0.276616
$$801$$ 4.29685 0.151822
$$802$$ −54.9831 −1.94152
$$803$$ −0.268814 −0.00948624
$$804$$ 117.901 4.15806
$$805$$ 20.6529 0.727917
$$806$$ −24.9890 −0.880200
$$807$$ −33.1593 −1.16726
$$808$$ 3.81613 0.134251
$$809$$ −33.1987 −1.16720 −0.583601 0.812040i $$-0.698357\pi$$
−0.583601 + 0.812040i $$0.698357\pi$$
$$810$$ 6.58848 0.231496
$$811$$ −26.5766 −0.933232 −0.466616 0.884460i $$-0.654527\pi$$
−0.466616 + 0.884460i $$0.654527\pi$$
$$812$$ 62.3759 2.18896
$$813$$ −6.22061 −0.218166
$$814$$ −3.32924 −0.116690
$$815$$ −7.25528 −0.254141
$$816$$ 55.1869 1.93193
$$817$$ 3.10482 0.108624
$$818$$ −12.2464 −0.428185
$$819$$ 29.1010 1.01687
$$820$$ −7.30374 −0.255058
$$821$$ 25.5807 0.892773 0.446387 0.894840i $$-0.352711\pi$$
0.446387 + 0.894840i $$0.352711\pi$$
$$822$$ −5.01379 −0.174876
$$823$$ 12.9783 0.452395 0.226198 0.974081i $$-0.427371\pi$$
0.226198 + 0.974081i $$0.427371\pi$$
$$824$$ −7.65835 −0.266791
$$825$$ 3.18005 0.110715
$$826$$ 35.8515 1.24743
$$827$$ −12.5167 −0.435248 −0.217624 0.976033i $$-0.569831\pi$$
−0.217624 + 0.976033i $$0.569831\pi$$
$$828$$ −91.0504 −3.16422
$$829$$ −19.4391 −0.675149 −0.337574 0.941299i $$-0.609606\pi$$
−0.337574 + 0.941299i $$0.609606\pi$$
$$830$$ 18.9852 0.658986
$$831$$ −67.4124 −2.33851
$$832$$ −20.9605 −0.726674
$$833$$ 20.4769 0.709482
$$834$$ −78.7400 −2.72654
$$835$$ 7.33129 0.253710
$$836$$ −2.86151 −0.0989673
$$837$$ −43.2522 −1.49501
$$838$$ −33.0583 −1.14198
$$839$$ −43.9972 −1.51895 −0.759475 0.650536i $$-0.774544\pi$$
−0.759475 + 0.650536i $$0.774544\pi$$
$$840$$ −11.2968 −0.389778
$$841$$ 31.1662 1.07470
$$842$$ 2.52818 0.0871268
$$843$$ −73.9229 −2.54604
$$844$$ 28.6468 0.986063
$$845$$ 9.85461 0.339009
$$846$$ −16.5865 −0.570256
$$847$$ −30.3632 −1.04329
$$848$$ 6.41061 0.220141
$$849$$ −21.2591 −0.729610
$$850$$ −16.6182 −0.569999
$$851$$ 9.35601 0.320720
$$852$$ −53.2761 −1.82521
$$853$$ 3.30374 0.113118 0.0565590 0.998399i $$-0.481987\pi$$
0.0565590 + 0.998399i $$0.481987\pi$$
$$854$$ −27.1079 −0.927614
$$855$$ −5.28487 −0.180739
$$856$$ −25.7682 −0.880740
$$857$$ 6.02374 0.205767 0.102883 0.994693i $$-0.467193\pi$$
0.102883 + 0.994693i $$0.467193\pi$$
$$858$$ −12.0831 −0.412511
$$859$$ −36.0158 −1.22884 −0.614421 0.788978i $$-0.710610\pi$$
−0.614421 + 0.788978i $$0.710610\pi$$
$$860$$ −8.04156 −0.274215
$$861$$ −25.2012 −0.858853
$$862$$ 30.0487 1.02346
$$863$$ 29.9250 1.01866 0.509329 0.860572i $$-0.329894\pi$$
0.509329 + 0.860572i $$0.329894\pi$$
$$864$$ −51.4549 −1.75053
$$865$$ 5.77353 0.196306
$$866$$ −54.8236 −1.86298
$$867$$ −124.247 −4.21966
$$868$$ 52.8864 1.79508
$$869$$ 10.3739 0.351911
$$870$$ −47.8329 −1.62169
$$871$$ −28.0485 −0.950386
$$872$$ −5.97548 −0.202355
$$873$$ 16.8951 0.571813
$$874$$ 14.2512 0.482054
$$875$$ −3.10482 −0.104962
$$876$$ −1.81388 −0.0612852
$$877$$ 0.618980 0.0209015 0.0104507 0.999945i $$-0.496673\pi$$
0.0104507 + 0.999945i $$0.496673\pi$$
$$878$$ 47.0762 1.58874
$$879$$ 69.5184 2.34480
$$880$$ −2.73091 −0.0920591
$$881$$ −21.2423 −0.715672 −0.357836 0.933784i $$-0.616485\pi$$
−0.357836 + 0.933784i $$0.616485\pi$$
$$882$$ −29.8902 −1.00646
$$883$$ −48.8971 −1.64552 −0.822760 0.568389i $$-0.807567\pi$$
−0.822760 + 0.568389i $$0.807567\pi$$
$$884$$ 35.6302 1.19837
$$885$$ −15.5134 −0.521477
$$886$$ 41.5034 1.39433
$$887$$ 58.2797 1.95684 0.978421 0.206622i $$-0.0662469\pi$$
0.978421 + 0.206622i $$0.0662469\pi$$
$$888$$ −5.11762 −0.171736
$$889$$ 14.3909 0.482657
$$890$$ 1.74190 0.0583887
$$891$$ −3.39757 −0.113823
$$892$$ −32.7251 −1.09572
$$893$$ 1.46492 0.0490216
$$894$$ −85.0892 −2.84581
$$895$$ 4.48379 0.149877
$$896$$ 30.0318 1.00329
$$897$$ 33.9566 1.13378
$$898$$ 41.1007 1.37155
$$899$$ 51.0130 1.70138
$$900$$ 13.6879 0.456265
$$901$$ −20.1168 −0.670187
$$902$$ 6.67483 0.222247
$$903$$ −27.7470 −0.923361
$$904$$ 10.0914 0.335636
$$905$$ 2.73400 0.0908814
$$906$$ 30.9732 1.02902
$$907$$ −36.5154 −1.21247 −0.606237 0.795284i $$-0.707322\pi$$
−0.606237 + 0.795284i $$0.707322\pi$$
$$908$$ −17.1670 −0.569708
$$909$$ −15.9543 −0.529172
$$910$$ 11.7973 0.391076
$$911$$ 6.62735 0.219574 0.109787 0.993955i $$-0.464983\pi$$
0.109787 + 0.993955i $$0.464983\pi$$
$$912$$ 7.11475 0.235593
$$913$$ −9.79036 −0.324014
$$914$$ −17.9110 −0.592442
$$915$$ 11.7299 0.387779
$$916$$ −0.194850 −0.00643802
$$917$$ 48.1662 1.59059
$$918$$ 109.292 3.60717
$$919$$ 7.91688 0.261154 0.130577 0.991438i $$-0.458317\pi$$
0.130577 + 0.991438i $$0.458317\pi$$
$$920$$ −8.40856 −0.277222
$$921$$ −12.6210 −0.415877
$$922$$ 53.6182 1.76582
$$923$$ 12.6743 0.417179
$$924$$ 25.5726 0.841275
$$925$$ −1.40652 −0.0462462
$$926$$ −69.3611 −2.27935
$$927$$ 32.0178 1.05160
$$928$$ 60.6875 1.99217
$$929$$ 35.8597 1.17652 0.588259 0.808673i $$-0.299814\pi$$
0.588259 + 0.808673i $$0.299814\pi$$
$$930$$ −40.5559 −1.32988
$$931$$ 2.63990 0.0865192
$$932$$ 24.6399 0.807106
$$933$$ 45.2254 1.48061
$$934$$ −32.6113 −1.06707
$$935$$ 8.56974 0.280260
$$936$$ −11.8481 −0.387268
$$937$$ −11.4420 −0.373793 −0.186896 0.982380i $$-0.559843\pi$$
−0.186896 + 0.982380i $$0.559843\pi$$
$$938$$ 105.200 3.43490
$$939$$ −71.3236 −2.32756
$$940$$ −3.79418 −0.123752
$$941$$ −0.360100 −0.0117389 −0.00586945 0.999983i $$-0.501868\pi$$
−0.00586945 + 0.999983i $$0.501868\pi$$
$$942$$ 44.3195 1.44401
$$943$$ −18.7579 −0.610843
$$944$$ 13.3223 0.433605
$$945$$ 20.4193 0.664239
$$946$$ 7.34911 0.238940
$$947$$ 2.63807 0.0857256 0.0428628 0.999081i $$-0.486352\pi$$
0.0428628 + 0.999081i $$0.486352\pi$$
$$948$$ 70.0000 2.27349
$$949$$ 0.431517 0.0140076
$$950$$ −2.14243 −0.0695097
$$951$$ 78.8084 2.55554
$$952$$ −30.4432 −0.986670
$$953$$ 10.4430 0.338282 0.169141 0.985592i $$-0.445901\pi$$
0.169141 + 0.985592i $$0.445901\pi$$
$$954$$ 29.3646 0.950714
$$955$$ 17.7230 0.573503
$$956$$ −7.58835 −0.245425
$$957$$ 24.6667 0.797360
$$958$$ 42.9525 1.38773
$$959$$ −2.52437 −0.0815160
$$960$$ −34.0178 −1.09792
$$961$$ 12.2522 0.395232
$$962$$ 5.34432 0.172308
$$963$$ 107.731 3.47159
$$964$$ −24.4173 −0.786428
$$965$$ 13.5371 0.435775
$$966$$ −127.359 −4.09772
$$967$$ 33.2732 1.06999 0.534996 0.844854i $$-0.320313\pi$$
0.534996 + 0.844854i $$0.320313\pi$$
$$968$$ 12.3620 0.397330
$$969$$ −22.3264 −0.717228
$$970$$ 6.84912 0.219912
$$971$$ 23.5657 0.756258 0.378129 0.925753i $$-0.376568\pi$$
0.378129 + 0.925753i $$0.376568\pi$$
$$972$$ 28.1752 0.903719
$$973$$ −39.6444 −1.27094
$$974$$ 67.8098 2.17277
$$975$$ −5.10482 −0.163485
$$976$$ −10.0732 −0.322437
$$977$$ 0.402439 0.0128752 0.00643759 0.999979i $$-0.497951\pi$$
0.00643759 + 0.999979i $$0.497951\pi$$
$$978$$ 44.7409 1.43066
$$979$$ −0.898271 −0.0287089
$$980$$ −6.83741 −0.218413
$$981$$ 24.9821 0.797617
$$982$$ −53.1395 −1.69575
$$983$$ 11.4927 0.366561 0.183281 0.983061i $$-0.441328\pi$$
0.183281 + 0.983061i $$0.441328\pi$$
$$984$$ 10.2603 0.327088
$$985$$ 21.5134 0.685473
$$986$$ −128.902 −4.10508
$$987$$ −13.0916 −0.416710
$$988$$ 4.59348 0.146138
$$989$$ −20.6529 −0.656723
$$990$$ −12.5093 −0.397571
$$991$$ 6.05761 0.192426 0.0962132 0.995361i $$-0.469327\pi$$
0.0962132 + 0.995361i $$0.469327\pi$$
$$992$$ 51.4549 1.63370
$$993$$ −13.5292 −0.429335
$$994$$ −47.5368 −1.50777
$$995$$ −7.09410 −0.224898
$$996$$ −66.0624 −2.09327
$$997$$ −48.1086 −1.52362 −0.761808 0.647803i $$-0.775688\pi$$
−0.761808 + 0.647803i $$0.775688\pi$$
$$998$$ 70.7510 2.23959
$$999$$ 9.25020 0.292663
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 95.2.a.b.1.2 4
3.2 odd 2 855.2.a.m.1.3 4
4.3 odd 2 1520.2.a.t.1.4 4
5.2 odd 4 475.2.b.e.324.2 8
5.3 odd 4 475.2.b.e.324.7 8
5.4 even 2 475.2.a.i.1.3 4
7.6 odd 2 4655.2.a.y.1.2 4
8.3 odd 2 6080.2.a.ch.1.1 4
8.5 even 2 6080.2.a.cc.1.4 4
15.14 odd 2 4275.2.a.bo.1.2 4
19.18 odd 2 1805.2.a.p.1.3 4
20.19 odd 2 7600.2.a.cf.1.1 4
95.94 odd 2 9025.2.a.bf.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.2 4 1.1 even 1 trivial
475.2.a.i.1.3 4 5.4 even 2
475.2.b.e.324.2 8 5.2 odd 4
475.2.b.e.324.7 8 5.3 odd 4
855.2.a.m.1.3 4 3.2 odd 2
1520.2.a.t.1.4 4 4.3 odd 2
1805.2.a.p.1.3 4 19.18 odd 2
4275.2.a.bo.1.2 4 15.14 odd 2
4655.2.a.y.1.2 4 7.6 odd 2
6080.2.a.cc.1.4 4 8.5 even 2
6080.2.a.ch.1.1 4 8.3 odd 2
7600.2.a.cf.1.1 4 20.19 odd 2
9025.2.a.bf.1.2 4 95.94 odd 2