# Properties

 Label 475.2.a.i.1.3 Level $475$ Weight $2$ Character 475.1 Self dual yes Analytic conductor $3.793$ 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] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.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: $$3.79289409601$$ 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: no (minimal twist has level 95) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.552409$$ of defining polynomial Character $$\chi$$ $$=$$ 475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.14243 q^{2} +2.87834 q^{3} +2.59002 q^{4} +6.16666 q^{6} -3.10482 q^{7} +1.26409 q^{8} +5.28487 q^{9} +O(q^{10})$$ $$q+2.14243 q^{2} +2.87834 q^{3} +2.59002 q^{4} +6.16666 q^{6} -3.10482 q^{7} +1.26409 q^{8} +5.28487 q^{9} -1.10482 q^{11} +7.45498 q^{12} -1.77353 q^{13} -6.65187 q^{14} -2.47182 q^{16} -7.75669 q^{17} +11.3225 q^{18} +1.00000 q^{19} -8.93674 q^{21} -2.36700 q^{22} +6.65187 q^{23} +3.63849 q^{24} -3.79966 q^{26} +6.57664 q^{27} -8.04156 q^{28} +7.75669 q^{29} +6.57664 q^{31} -7.82389 q^{32} -3.18005 q^{33} -16.6182 q^{34} +13.6879 q^{36} +1.40652 q^{37} +2.14243 q^{38} -5.10482 q^{39} +2.81995 q^{41} -19.1464 q^{42} -3.10482 q^{43} -2.86151 q^{44} +14.2512 q^{46} -1.46492 q^{47} -7.11475 q^{48} +2.63990 q^{49} -22.3264 q^{51} -4.59348 q^{52} +2.59348 q^{53} +14.0900 q^{54} -3.92477 q^{56} +2.87834 q^{57} +16.6182 q^{58} -5.38969 q^{59} +4.07523 q^{61} +14.0900 q^{62} -16.4086 q^{63} -11.8185 q^{64} -6.81305 q^{66} +15.8151 q^{67} -20.0900 q^{68} +19.1464 q^{69} +7.14638 q^{71} +6.68055 q^{72} -0.243310 q^{73} +3.01339 q^{74} +2.59002 q^{76} +3.43026 q^{77} -10.9367 q^{78} -9.38969 q^{79} +3.07523 q^{81} +6.04156 q^{82} -8.86151 q^{83} -23.1464 q^{84} -6.65187 q^{86} +22.3264 q^{87} -1.39659 q^{88} +0.813048 q^{89} +5.50648 q^{91} +17.2285 q^{92} +18.9298 q^{93} -3.13849 q^{94} -22.5199 q^{96} -3.19689 q^{97} +5.65581 q^{98} -5.83882 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^7 + 12 * q^8 + 8 * q^9 $$4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} + 4 q^{27} + 8 q^{28} + 4 q^{29} + 4 q^{31} + 6 q^{32} - 8 q^{33} - 4 q^{34} + 40 q^{36} + 6 q^{37} + 2 q^{38} - 12 q^{39} + 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} - 38 q^{48} + 20 q^{49} - 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 12 q^{56} - 2 q^{57} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} + 18 q^{67} - 4 q^{68} + 28 q^{69} - 20 q^{71} + 52 q^{72} - 28 q^{73} + 32 q^{74} + 8 q^{76} + 40 q^{77} - 12 q^{78} - 16 q^{79} + 16 q^{81} - 16 q^{82} - 44 q^{84} - 8 q^{86} + 36 q^{87} + 12 q^{88} + 4 q^{89} - 36 q^{91} + 28 q^{92} + 40 q^{93} - 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^7 + 12 * q^8 + 8 * q^9 + 4 * q^11 - 6 * q^12 - 2 * q^13 - 8 * q^14 + 4 * q^16 - 4 * q^17 + 34 * q^18 + 4 * q^19 - 4 * q^21 - 4 * q^22 + 8 * q^23 - 24 * q^24 + 4 * q^26 + 4 * q^27 + 8 * q^28 + 4 * q^29 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 4 * q^34 + 40 * q^36 + 6 * q^37 + 2 * q^38 - 12 * q^39 + 16 * q^41 - 28 * q^42 - 4 * q^43 + 24 * q^44 + 12 * q^47 - 38 * q^48 + 20 * q^49 - 36 * q^51 - 18 * q^52 + 10 * q^53 - 20 * q^54 - 12 * q^56 - 2 * q^57 + 4 * q^58 + 20 * q^61 - 20 * q^62 - 20 * q^63 - 4 * q^64 - 28 * q^66 + 18 * q^67 - 4 * q^68 + 28 * q^69 - 20 * q^71 + 52 * q^72 - 28 * q^73 + 32 * q^74 + 8 * q^76 + 40 * q^77 - 12 * q^78 - 16 * q^79 + 16 * q^81 - 16 * q^82 - 44 * q^84 - 8 * q^86 + 36 * q^87 + 12 * q^88 + 4 * q^89 - 36 * q^91 + 28 * q^92 + 40 * q^93 - 48 * q^94 - 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 $$-0.226438\pi$$
0.757465 + 0.652876i $$0.226438\pi$$
$$3$$ 2.87834 1.66181 0.830907 0.556412i $$-0.187822\pi$$
0.830907 + 0.556412i $$0.187822\pi$$
$$4$$ 2.59002 1.29501
$$5$$ 0 0
$$6$$ 6.16666 2.51753
$$7$$ −3.10482 −1.17351 −0.586756 0.809764i $$-0.699595\pi$$
−0.586756 + 0.809764i $$0.699595\pi$$
$$8$$ 1.26409 0.446923
$$9$$ 5.28487 1.76162
$$10$$ 0 0
$$11$$ −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 $$-0.579098\pi$$
−0.245944 + 0.969284i $$0.579098\pi$$
$$14$$ −6.65187 −1.77779
$$15$$ 0 0
$$16$$ −2.47182 −0.617955
$$17$$ −7.75669 −1.88127 −0.940637 0.339415i $$-0.889771\pi$$
−0.940637 + 0.339415i $$0.889771\pi$$
$$18$$ 11.3225 2.66874
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −8.93674 −1.95016
$$22$$ −2.36700 −0.504647
$$23$$ 6.65187 1.38701 0.693505 0.720451i $$-0.256065\pi$$
0.693505 + 0.720451i $$0.256065\pi$$
$$24$$ 3.63849 0.742703
$$25$$ 0 0
$$26$$ −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 $$0.244054\pi$$
$$30$$ 0 0
$$31$$ 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
$$35$$ 0 0
$$36$$ 13.6879 2.28132
$$37$$ 1.40652 0.231231 0.115616 0.993294i $$-0.463116\pi$$
0.115616 + 0.993294i $$0.463116\pi$$
$$38$$ 2.14243 0.347549
$$39$$ −5.10482 −0.817425
$$40$$ 0 0
$$41$$ 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 $$-0.576079\pi$$
−0.236740 + 0.971573i $$0.576079\pi$$
$$44$$ −2.86151 −0.431389
$$45$$ 0 0
$$46$$ 14.2512 2.10122
$$47$$ −1.46492 −0.213680 −0.106840 0.994276i $$-0.534073\pi$$
−0.106840 + 0.994276i $$0.534073\pi$$
$$48$$ −7.11475 −1.02693
$$49$$ 2.63990 0.377129
$$50$$ 0 0
$$51$$ −22.3264 −3.12633
$$52$$ −4.59348 −0.637001
$$53$$ 2.59348 0.356241 0.178121 0.984009i $$-0.442998\pi$$
0.178121 + 0.984009i $$0.442998\pi$$
$$54$$ 14.0900 1.91741
$$55$$ 0 0
$$56$$ −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 $$0.614103\pi$$
$$60$$ 0 0
$$61$$ 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
$$65$$ 0 0
$$66$$ −6.81305 −0.838628
$$67$$ 15.8151 1.93212 0.966060 0.258318i $$-0.0831681\pi$$
0.966060 + 0.258318i $$0.0831681\pi$$
$$68$$ −20.0900 −2.43627
$$69$$ 19.1464 2.30495
$$70$$ 0 0
$$71$$ 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 $$-0.504532\pi$$
−0.0142387 + 0.999899i $$0.504532\pi$$
$$74$$ 3.01339 0.350299
$$75$$ 0 0
$$76$$ 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 $$0.677137\pi$$
$$80$$ 0 0
$$81$$ 3.07523 0.341692
$$82$$ 6.04156 0.667178
$$83$$ −8.86151 −0.972677 −0.486338 0.873771i $$-0.661668\pi$$
−0.486338 + 0.873771i $$0.661668\pi$$
$$84$$ −23.1464 −2.52548
$$85$$ 0 0
$$86$$ −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 $$0.486279\pi$$
$$90$$ 0 0
$$91$$ 5.50648 0.577236
$$92$$ 17.2285 1.79620
$$93$$ 18.9298 1.96293
$$94$$ −3.13849 −0.323711
$$95$$ 0 0
$$96$$ −22.5199 −2.29842
$$97$$ −3.19689 −0.324595 −0.162297 0.986742i $$-0.551890\pi$$
−0.162297 + 0.986742i $$0.551890\pi$$
$$98$$ 5.65581 0.571323
$$99$$ −5.83882 −0.586824
$$100$$ 0 0
$$101$$ −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 $$-0.596478\pi$$
−0.298476 + 0.954417i $$0.596478\pi$$
$$104$$ −2.24190 −0.219836
$$105$$ 0 0
$$106$$ 5.55635 0.539681
$$107$$ −20.3848 −1.97068 −0.985338 0.170616i $$-0.945424\pi$$
−0.985338 + 0.170616i $$0.945424\pi$$
$$108$$ 17.0337 1.63906
$$109$$ 4.72710 0.452774 0.226387 0.974037i $$-0.427309\pi$$
0.226387 + 0.974037i $$0.427309\pi$$
$$110$$ 0 0
$$111$$ 4.04846 0.384263
$$112$$ 7.67456 0.725177
$$113$$ 7.98316 0.750993 0.375496 0.926824i $$-0.377472\pi$$
0.375496 + 0.926824i $$0.377472\pi$$
$$114$$ 6.16666 0.577561
$$115$$ 0 0
$$116$$ 20.0900 1.86531
$$117$$ −9.37285 −0.866520
$$118$$ −11.5471 −1.06299
$$119$$ 24.0831 2.20770
$$120$$ 0 0
$$121$$ −9.77938 −0.889034
$$122$$ 8.73091 0.790460
$$123$$ 8.11679 0.731866
$$124$$ 17.0337 1.52967
$$125$$ 0 0
$$126$$ −35.1543 −3.13179
$$127$$ −4.63503 −0.411293 −0.205646 0.978626i $$-0.565930\pi$$
−0.205646 + 0.978626i $$0.565930\pi$$
$$128$$ −9.67265 −0.854950
$$129$$ −8.93674 −0.786836
$$130$$ 0 0
$$131$$ 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
$$135$$ 0 0
$$136$$ −9.80515 −0.840785
$$137$$ 0.813048 0.0694634 0.0347317 0.999397i $$-0.488942\pi$$
0.0347317 + 0.999397i $$0.488942\pi$$
$$138$$ 41.0199 3.49184
$$139$$ −12.7687 −1.08302 −0.541512 0.840693i $$-0.682148\pi$$
−0.541512 + 0.840693i $$0.682148\pi$$
$$140$$ 0 0
$$141$$ −4.21654 −0.355097
$$142$$ 15.3106 1.28484
$$143$$ 1.95942 0.163855
$$144$$ −13.0632 −1.08860
$$145$$ 0 0
$$146$$ −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 $$0.691200\pi$$
$$150$$ 0 0
$$151$$ 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
$$155$$ 0 0
$$156$$ −13.2216 −1.05858
$$157$$ −7.18695 −0.573581 −0.286791 0.957993i $$-0.592588\pi$$
−0.286791 + 0.957993i $$0.592588\pi$$
$$158$$ −20.1168 −1.60041
$$159$$ 7.46492 0.592007
$$160$$ 0 0
$$161$$ −20.6529 −1.62767
$$162$$ 6.58848 0.517640
$$163$$ −7.25528 −0.568277 −0.284139 0.958783i $$-0.591708\pi$$
−0.284139 + 0.958783i $$0.591708\pi$$
$$164$$ 7.30374 0.570326
$$165$$ 0 0
$$166$$ −18.9852 −1.47354
$$167$$ 7.33129 0.567312 0.283656 0.958926i $$-0.408453\pi$$
0.283656 + 0.958926i $$0.408453\pi$$
$$168$$ −11.2968 −0.871570
$$169$$ −9.85461 −0.758047
$$170$$ 0 0
$$171$$ 5.28487 0.404144
$$172$$ −8.04156 −0.613163
$$173$$ 5.77353 0.438953 0.219477 0.975618i $$-0.429565\pi$$
0.219477 + 0.975618i $$0.429565\pi$$
$$174$$ 47.8329 3.62620
$$175$$ 0 0
$$176$$ 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 $$0.553591\pi$$
$$180$$ 0 0
$$181$$ −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
$$185$$ 0 0
$$186$$ 40.5559 2.97371
$$187$$ 8.56974 0.626681
$$188$$ −3.79418 −0.276719
$$189$$ −20.4193 −1.48528
$$190$$ 0 0
$$191$$ −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 $$-0.338014\pi$$
0.487212 + 0.873284i $$0.338014\pi$$
$$194$$ −6.84912 −0.491738
$$195$$ 0 0
$$196$$ 6.83741 0.488386
$$197$$ 21.5134 1.53276 0.766382 0.642385i $$-0.222055\pi$$
0.766382 + 0.642385i $$0.222055\pi$$
$$198$$ −12.5093 −0.888997
$$199$$ 7.09410 0.502888 0.251444 0.967872i $$-0.419095\pi$$
0.251444 + 0.967872i $$0.419095\pi$$
$$200$$ 0 0
$$201$$ 45.5213 3.21082
$$202$$ −6.46774 −0.455068
$$203$$ −24.0831 −1.69030
$$204$$ −57.8260 −4.04863
$$205$$ 0 0
$$206$$ −12.9797 −0.904339
$$207$$ 35.1543 2.44339
$$208$$ 4.38384 0.303964
$$209$$ −1.10482 −0.0764219
$$210$$ 0 0
$$211$$ 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
$$215$$ 0 0
$$216$$ 8.31346 0.565659
$$217$$ −20.4193 −1.38615
$$218$$ 10.1275 0.685921
$$219$$ −0.700331 −0.0473240
$$220$$ 0 0
$$221$$ 13.7567 0.925375
$$222$$ 8.67356 0.582131
$$223$$ 12.6350 0.846104 0.423052 0.906105i $$-0.360959\pi$$
0.423052 + 0.906105i $$0.360959\pi$$
$$224$$ 24.2918 1.62306
$$225$$ 0 0
$$226$$ 17.1034 1.13770
$$227$$ 6.62813 0.439925 0.219962 0.975508i $$-0.429407\pi$$
0.219962 + 0.975508i $$0.429407\pi$$
$$228$$ 7.45498 0.493718
$$229$$ −0.0752308 −0.00497139 −0.00248570 0.999997i $$-0.500791\pi$$
−0.00248570 + 0.999997i $$0.500791\pi$$
$$230$$ 0 0
$$231$$ 9.87348 0.649627
$$232$$ 9.80515 0.643740
$$233$$ −9.51338 −0.623242 −0.311621 0.950206i $$-0.600872\pi$$
−0.311621 + 0.950206i $$0.600872\pi$$
$$234$$ −20.0807 −1.31272
$$235$$ 0 0
$$236$$ −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 $$0.530208\pi$$
$$240$$ 0 0
$$241$$ −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
$$245$$ 0 0
$$246$$ 17.3897 1.10873
$$247$$ −1.77353 −0.112847
$$248$$ 8.31346 0.527905
$$249$$ −25.5065 −1.61641
$$250$$ 0 0
$$251$$ −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
$$255$$ 0 0
$$256$$ 2.91405 0.182128
$$257$$ 15.1295 0.943755 0.471877 0.881664i $$-0.343576\pi$$
0.471877 + 0.881664i $$0.343576\pi$$
$$258$$ −19.1464 −1.19200
$$259$$ −4.36700 −0.271352
$$260$$ 0 0
$$261$$ 40.9931 2.53741
$$262$$ 33.2364 2.05335
$$263$$ 15.2216 0.938605 0.469302 0.883038i $$-0.344505\pi$$
0.469302 + 0.883038i $$0.344505\pi$$
$$264$$ −4.01987 −0.247406
$$265$$ 0 0
$$266$$ −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 $$0.385773\pi$$
$$270$$ 0 0
$$271$$ 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
$$275$$ 0 0
$$276$$ 49.5896 2.98494
$$277$$ −23.4205 −1.40720 −0.703602 0.710595i $$-0.748426\pi$$
−0.703602 + 0.710595i $$0.748426\pi$$
$$278$$ −27.3560 −1.64070
$$279$$ 34.7567 2.08083
$$280$$ 0 0
$$281$$ 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 $$-0.570450\pi$$
−0.219522 + 0.975607i $$0.570450\pi$$
$$284$$ 18.5093 1.09832
$$285$$ 0 0
$$286$$ 4.19794 0.248229
$$287$$ −8.75543 −0.516817
$$288$$ −41.3482 −2.43647
$$289$$ 43.1662 2.53919
$$290$$ 0 0
$$291$$ −9.20174 −0.539416
$$292$$ −0.630180 −0.0368785
$$293$$ 24.1522 1.41099 0.705494 0.708716i $$-0.250725\pi$$
0.705494 + 0.708716i $$0.250725\pi$$
$$294$$ 16.2794 0.949433
$$295$$ 0 0
$$296$$ 1.77797 0.103343
$$297$$ −7.26600 −0.421616
$$298$$ −29.5618 −1.71247
$$299$$ −11.7973 −0.682253
$$300$$ 0 0
$$301$$ 9.63990 0.555635
$$302$$ 10.7608 0.619213
$$303$$ −8.68936 −0.499190
$$304$$ −2.47182 −0.141769
$$305$$ 0 0
$$306$$ −87.8250 −5.02062
$$307$$ −4.38482 −0.250255 −0.125127 0.992141i $$-0.539934\pi$$
−0.125127 + 0.992141i $$0.539934\pi$$
$$308$$ 8.88447 0.506239
$$309$$ −17.4381 −0.992022
$$310$$ 0 0
$$311$$ −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 $$-0.746953\pi$$
−0.700307 + 0.713842i $$0.746953\pi$$
$$314$$ −15.3976 −0.868935
$$315$$ 0 0
$$316$$ −24.3195 −1.36808
$$317$$ 27.3798 1.53780 0.768900 0.639369i $$-0.220804\pi$$
0.768900 + 0.639369i $$0.220804\pi$$
$$318$$ 15.9931 0.896848
$$319$$ −8.56974 −0.479813
$$320$$ 0 0
$$321$$ −58.6745 −3.27489
$$322$$ −44.2474 −2.46581
$$323$$ −7.75669 −0.431594
$$324$$ 7.96492 0.442496
$$325$$ 0 0
$$326$$ −15.5440 −0.860900
$$327$$ 13.6062 0.752426
$$328$$ 3.56467 0.196826
$$329$$ 4.54831 0.250756
$$330$$ 0 0
$$331$$ 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
$$335$$ 0 0
$$336$$ 22.0900 1.20511
$$337$$ 20.6360 1.12412 0.562058 0.827098i $$-0.310010\pi$$
0.562058 + 0.827098i $$0.310010\pi$$
$$338$$ −21.1128 −1.14839
$$339$$ 22.9783 1.24801
$$340$$ 0 0
$$341$$ −7.26600 −0.393476
$$342$$ 11.3225 0.612250
$$343$$ 13.5373 0.730947
$$344$$ −3.92477 −0.211609
$$345$$ 0 0
$$346$$ 12.3694 0.664983
$$347$$ −30.0148 −1.61128 −0.805639 0.592407i $$-0.798178\pi$$
−0.805639 + 0.592407i $$0.798178\pi$$
$$348$$ 57.8260 3.09980
$$349$$ −14.7340 −0.788693 −0.394347 0.918962i $$-0.629029\pi$$
−0.394347 + 0.918962i $$0.629029\pi$$
$$350$$ 0 0
$$351$$ −11.6638 −0.622570
$$352$$ 8.64399 0.460726
$$353$$ 29.6302 1.57705 0.788527 0.615000i $$-0.210844\pi$$
0.788527 + 0.615000i $$0.210844\pi$$
$$354$$ −33.2364 −1.76649
$$355$$ 0 0
$$356$$ 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 $$0.305328\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −5.85742 −0.307859
$$363$$ −28.1484 −1.47741
$$364$$ 14.2619 0.747527
$$365$$ 0 0
$$366$$ 25.1306 1.31360
$$367$$ −30.0347 −1.56780 −0.783898 0.620889i $$-0.786772\pi$$
−0.783898 + 0.620889i $$0.786772\pi$$
$$368$$ −16.4422 −0.857111
$$369$$ 14.9031 0.775823
$$370$$ 0 0
$$371$$ −8.05227 −0.418053
$$372$$ 49.0287 2.54202
$$373$$ 37.3055 1.93161 0.965803 0.259277i $$-0.0834843\pi$$
0.965803 + 0.259277i $$0.0834843\pi$$
$$374$$ 18.3601 0.949378
$$375$$ 0 0
$$376$$ −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 $$0.623932\pi$$
$$380$$ 0 0
$$381$$ −13.3412 −0.683492
$$382$$ −37.9704 −1.94273
$$383$$ 29.0020 1.48193 0.740967 0.671541i $$-0.234367\pi$$
0.740967 + 0.671541i $$0.234367\pi$$
$$384$$ −27.8412 −1.42077
$$385$$ 0 0
$$386$$ 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 $$0.319404\pi$$
$$390$$ 0 0
$$391$$ −51.5965 −2.60935
$$392$$ 3.33707 0.168548
$$393$$ 44.6529 2.25244
$$394$$ 46.0910 2.32203
$$395$$ 0 0
$$396$$ −15.1227 −0.759944
$$397$$ −10.3856 −0.521238 −0.260619 0.965442i $$-0.583927\pi$$
−0.260619 + 0.965442i $$0.583927\pi$$
$$398$$ 15.1987 0.761840
$$399$$ −8.93674 −0.447397
$$400$$ 0 0
$$401$$ 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
$$405$$ 0 0
$$406$$ −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 $$0.454865\pi$$
$$410$$ 0 0
$$411$$ 2.34023 0.115435
$$412$$ −15.6914 −0.773059
$$413$$ 16.7340 0.823427
$$414$$ 75.3157 3.70156
$$415$$ 0 0
$$416$$ 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 $$0.376987\pi$$
$$420$$ 0 0
$$421$$ −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
$$425$$ 0 0
$$426$$ 44.0693 2.13517
$$427$$ −12.6529 −0.612315
$$428$$ −52.7972 −2.55205
$$429$$ 5.63990 0.272297
$$430$$ 0 0
$$431$$ −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 $$-0.710793\pi$$
−0.614874 + 0.788625i $$0.710793\pi$$
$$434$$ −43.7470 −2.09992
$$435$$ 0 0
$$436$$ 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 $$0.675696\pi$$
$$440$$ 0 0
$$441$$ 13.9515 0.664358
$$442$$ 29.4728 1.40188
$$443$$ 19.3721 0.920395 0.460197 0.887817i $$-0.347779\pi$$
0.460197 + 0.887817i $$0.347779\pi$$
$$444$$ 10.4856 0.497625
$$445$$ 0 0
$$446$$ 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 $$0.649531\pi$$
$$450$$ 0 0
$$451$$ −3.11553 −0.146705
$$452$$ 20.6766 0.972545
$$453$$ 14.4570 0.679250
$$454$$ 14.2003 0.666455
$$455$$ 0 0
$$456$$ 3.63849 0.170388
$$457$$ −8.36010 −0.391069 −0.195534 0.980697i $$-0.562644\pi$$
−0.195534 + 0.980697i $$0.562644\pi$$
$$458$$ −0.161177 −0.00753131
$$459$$ −51.0130 −2.38108
$$460$$ 0 0
$$461$$ −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 $$-0.771053\pi$$
−0.752294 + 0.658827i $$0.771053\pi$$
$$464$$ −19.1731 −0.890091
$$465$$ 0 0
$$466$$ −20.3818 −0.944168
$$467$$ −15.2216 −0.704372 −0.352186 0.935930i $$-0.614562\pi$$
−0.352186 + 0.935930i $$0.614562\pi$$
$$468$$ −24.2759 −1.12215
$$469$$ −49.1030 −2.26736
$$470$$ 0 0
$$471$$ −20.6865 −0.953185
$$472$$ −6.81305 −0.313596
$$473$$ 3.43026 0.157724
$$474$$ −57.9031 −2.65958
$$475$$ 0 0
$$476$$ 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 $$0.651441\pi$$
$$480$$ 0 0
$$481$$ −2.49451 −0.113740
$$482$$ −20.1977 −0.919978
$$483$$ −59.4460 −2.70489
$$484$$ −25.3288 −1.15131
$$485$$ 0 0
$$486$$ −23.3061 −1.05719
$$487$$ 31.6508 1.43424 0.717118 0.696952i $$-0.245461\pi$$
0.717118 + 0.696952i $$0.245461\pi$$
$$488$$ 5.15146 0.233195
$$489$$ −20.8832 −0.944371
$$490$$ 0 0
$$491$$ 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
$$495$$ 0 0
$$496$$ −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 $$0.764783\pi$$
$$500$$ 0 0
$$501$$ 21.1020 0.942767
$$502$$ −31.9862 −1.42761
$$503$$ 3.54396 0.158017 0.0790087 0.996874i $$-0.474825\pi$$
0.0790087 + 0.996874i $$0.474825\pi$$
$$504$$ −20.7419 −0.923917
$$505$$ 0 0
$$506$$ −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 $$0.514938\pi$$
$$510$$ 0 0
$$511$$ 0.755435 0.0334185
$$512$$ 25.5885 1.13086
$$513$$ 6.57664 0.290366
$$514$$ 32.4140 1.42972
$$515$$ 0 0
$$516$$ −23.1464 −1.01896
$$517$$ 1.61847 0.0711802
$$518$$ −9.35601 −0.411080
$$519$$ 16.6182 0.729458
$$520$$ 0 0
$$521$$ −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 $$-0.798017\pi$$
−0.805340 + 0.592814i $$0.798017\pi$$
$$524$$ 40.1800 1.75527
$$525$$ 0 0
$$526$$ 32.6113 1.42192
$$527$$ −51.0130 −2.22216
$$528$$ 7.86051 0.342085
$$529$$ 21.2474 0.923799
$$530$$ 0 0
$$531$$ −28.4838 −1.23609
$$532$$ −8.04156 −0.348646
$$533$$ −5.00125 −0.216628
$$534$$ 5.01379 0.216968
$$535$$ 0 0
$$536$$ 19.9917 0.863509
$$537$$ −12.9059 −0.556931
$$538$$ 24.6814 1.06409
$$539$$ −2.91661 −0.125627
$$540$$ 0 0
$$541$$ −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
$$545$$ 0 0
$$546$$ 33.9566 1.45321
$$547$$ −17.6240 −0.753550 −0.376775 0.926305i $$-0.622967\pi$$
−0.376775 + 0.926305i $$0.622967\pi$$
$$548$$ 2.10581 0.0899559
$$549$$ 21.5371 0.919179
$$550$$ 0 0
$$551$$ 7.75669 0.330446
$$552$$ 24.2027 1.03014
$$553$$ 29.1533 1.23972
$$554$$ −50.1769 −2.13181
$$555$$ 0 0
$$556$$ −33.0711 −1.40253
$$557$$ 34.6865 1.46972 0.734858 0.678221i $$-0.237249\pi$$
0.734858 + 0.678221i $$0.237249\pi$$
$$558$$ 74.4639 3.15231
$$559$$ 5.50648 0.232899
$$560$$ 0 0
$$561$$ 24.6667 1.04143
$$562$$ 55.0229 2.32100
$$563$$ −13.5073 −0.569263 −0.284632 0.958637i $$-0.591871\pi$$
−0.284632 + 0.958637i $$0.591871\pi$$
$$564$$ −10.9209 −0.459855
$$565$$ 0 0
$$566$$ −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 $$0.262727\pi$$
$$570$$ 0 0
$$571$$ −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
$$575$$ 0 0
$$576$$ −62.4594 −2.60248
$$577$$ −14.9059 −0.620541 −0.310270 0.950648i $$-0.600420\pi$$
−0.310270 + 0.950648i $$0.600420\pi$$
$$578$$ 92.4808 3.84669
$$579$$ 38.9645 1.61931
$$580$$ 0 0
$$581$$ 27.5134 1.14145
$$582$$ −19.7141 −0.817177
$$583$$ −2.86532 −0.118669
$$584$$ −0.307566 −0.0127272
$$585$$ 0 0
$$586$$ 51.7446 2.13755
$$587$$ −7.37207 −0.304278 −0.152139 0.988359i $$-0.548616\pi$$
−0.152139 + 0.988359i $$0.548616\pi$$
$$588$$ 19.6804 0.811607
$$589$$ 6.57664 0.270986
$$590$$ 0 0
$$591$$ 61.9229 2.54717
$$592$$ −3.47668 −0.142890
$$593$$ 17.6853 0.726247 0.363124 0.931741i $$-0.381710\pi$$
0.363124 + 0.931741i $$0.381710\pi$$
$$594$$ −15.5669 −0.638718
$$595$$ 0 0
$$596$$ −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 $$0.465424\pi$$
$$600$$ 0 0
$$601$$ 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
$$605$$ 0 0
$$606$$ −18.6164 −0.756239
$$607$$ 7.24535 0.294080 0.147040 0.989131i $$-0.453025\pi$$
0.147040 + 0.989131i $$0.453025\pi$$
$$608$$ −7.82389 −0.317301
$$609$$ −69.3195 −2.80897
$$610$$ 0 0
$$611$$ 2.59807 0.105107
$$612$$ −106.173 −4.29179
$$613$$ −20.8962 −0.843988 −0.421994 0.906599i $$-0.638670\pi$$
−0.421994 + 0.906599i $$0.638670\pi$$
$$614$$ −9.39419 −0.379119
$$615$$ 0 0
$$616$$ 4.33616 0.174709
$$617$$ 22.0494 0.887677 0.443839 0.896107i $$-0.353616\pi$$
0.443839 + 0.896107i $$0.353616\pi$$
$$618$$ −37.3601 −1.50284
$$619$$ −0.0484607 −0.00194780 −0.000973900 1.00000i $$-0.500310\pi$$
−0.000973900 1.00000i $$0.500310\pi$$
$$620$$ 0 0
$$621$$ 43.7470 1.75550
$$622$$ −33.6626 −1.34975
$$623$$ −2.52437 −0.101137
$$624$$ 12.6182 0.505132
$$625$$ 0 0
$$626$$ −53.0882 −2.12183
$$627$$ −3.18005 −0.126999
$$628$$ −18.6144 −0.742795
$$629$$ −10.9100 −0.435009
$$630$$ 0 0
$$631$$ −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
$$635$$ 0 0
$$636$$ 19.3343 0.766656
$$637$$ −4.68193 −0.185505
$$638$$ −18.3601 −0.726883
$$639$$ 37.7677 1.49407
$$640$$ 0 0
$$641$$ −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 $$-0.655527\pi$$
−0.469392 + 0.882990i $$0.655527\pi$$
$$644$$ −53.4914 −2.10786
$$645$$ 0 0
$$646$$ −16.6182 −0.653834
$$647$$ 48.4580 1.90508 0.952540 0.304412i $$-0.0984601\pi$$
0.952540 + 0.304412i $$0.0984601\pi$$
$$648$$ 3.88737 0.152710
$$649$$ 5.95463 0.233740
$$650$$ 0 0
$$651$$ −58.7737 −2.30352
$$652$$ −18.7914 −0.735926
$$653$$ −22.2351 −0.870128 −0.435064 0.900399i $$-0.643274\pi$$
−0.435064 + 0.900399i $$0.643274\pi$$
$$654$$ 29.1504 1.13987
$$655$$ 0 0
$$656$$ −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 $$0.566874\pi$$
$$660$$ 0 0
$$661$$ −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
$$665$$ 0 0
$$666$$ 15.9253 0.617095
$$667$$ 51.5965 1.99782
$$668$$ 18.9882 0.734677
$$669$$ 36.3680 1.40607
$$670$$ 0 0
$$671$$ −4.50239 −0.173813
$$672$$ 69.9201 2.69723
$$673$$ 43.2772 1.66821 0.834106 0.551604i $$-0.185984\pi$$
0.834106 + 0.551604i $$0.185984\pi$$
$$674$$ 44.2113 1.70296
$$675$$ 0 0
$$676$$ −25.5237 −0.981680
$$677$$ −11.5330 −0.443251 −0.221625 0.975132i $$-0.571136\pi$$
−0.221625 + 0.975132i $$0.571136\pi$$
$$678$$ 49.2295 1.89065
$$679$$ 9.92575 0.380915
$$680$$ 0 0
$$681$$ 19.0780 0.731072
$$682$$ −15.5669 −0.596088
$$683$$ −0.916090 −0.0350532 −0.0175266 0.999846i $$-0.505579\pi$$
−0.0175266 + 0.999846i $$0.505579\pi$$
$$684$$ 13.6879 0.523372
$$685$$ 0 0
$$686$$ 29.0028 1.10733
$$687$$ −0.216540 −0.00826153
$$688$$ 7.67456 0.292590
$$689$$ −4.59960 −0.175231
$$690$$ 0 0
$$691$$ 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
$$695$$ 0 0
$$696$$ 28.2226 1.06978
$$697$$ −21.8735 −0.828517
$$698$$ −31.5666 −1.19481
$$699$$ −27.3828 −1.03571
$$700$$ 0 0
$$701$$ 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
$$705$$ 0 0
$$706$$ 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 $$0.437234\pi$$
$$710$$ 0 0
$$711$$ −49.6233 −1.86102
$$712$$ 1.02777 0.0385171
$$713$$ 43.7470 1.63834
$$714$$ 148.513 5.55794
$$715$$ 0 0
$$716$$ −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 $$0.453936\pi$$
$$720$$ 0 0
$$721$$ 18.8102 0.700529
$$722$$ 2.14243 0.0797331
$$723$$ −27.1354 −1.00918
$$724$$ −7.08114 −0.263168
$$725$$ 0 0
$$726$$ −60.3061 −2.23817
$$727$$ 39.5241 1.46587 0.732934 0.680300i $$-0.238151\pi$$
0.732934 + 0.680300i $$0.238151\pi$$
$$728$$ 6.96068 0.257980
$$729$$ −40.5373 −1.50138
$$730$$ 0 0
$$731$$ 24.0831 0.890746
$$732$$ 30.3808 1.12291
$$733$$ −50.7498 −1.87449 −0.937243 0.348677i $$-0.886631\pi$$
−0.937243 + 0.348677i $$0.886631\pi$$
$$734$$ −64.3473 −2.37510
$$735$$ 0 0
$$736$$ −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 $$0.497545\pi$$
$$740$$ 0 0
$$741$$ −5.10482 −0.187530
$$742$$ −17.2515 −0.633321
$$743$$ −19.5511 −0.717259 −0.358630 0.933480i $$-0.616756\pi$$
−0.358630 + 0.933480i $$0.616756\pi$$
$$744$$ 23.9290 0.877280
$$745$$ 0 0
$$746$$ 79.9246 2.92625
$$747$$ −46.8319 −1.71349
$$748$$ 22.1958 0.811560
$$749$$ 63.2912 2.31261
$$750$$ 0 0
$$751$$ 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
$$755$$ 0 0
$$756$$ −52.8864 −1.92346
$$757$$ 48.0535 1.74653 0.873267 0.487241i $$-0.161997\pi$$
0.873267 + 0.487241i $$0.161997\pi$$
$$758$$ −31.6638 −1.15008
$$759$$ −21.1533 −0.767815
$$760$$ 0 0
$$761$$ 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
$$765$$ 0 0
$$766$$ 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 $$0.577551\pi$$
$$770$$ 0 0
$$771$$ 43.5480 1.56834
$$772$$ 35.0615 1.26189
$$773$$ −0.133625 −0.00480617 −0.00240309 0.999997i $$-0.500765\pi$$
−0.00240309 + 0.999997i $$0.500765\pi$$
$$774$$ −35.1543 −1.26359
$$775$$ 0 0
$$776$$ −4.04115 −0.145069
$$777$$ −12.5697 −0.450937
$$778$$ 45.4167 1.62827
$$779$$ 2.81995 0.101035
$$780$$ 0 0
$$781$$ −7.89545 −0.282522
$$782$$ −110.542 −3.95298
$$783$$ 51.0130 1.82305
$$784$$ −6.52536 −0.233049
$$785$$ 0 0
$$786$$ 95.6658 3.41229
$$787$$ 4.84060 0.172549 0.0862744 0.996271i $$-0.472504\pi$$
0.0862744 + 0.996271i $$0.472504\pi$$
$$788$$ 55.7202 1.98495
$$789$$ 43.8130 1.55979
$$790$$ 0 0
$$791$$ −24.7863 −0.881299
$$792$$ −7.38079 −0.262265
$$793$$ −7.22753 −0.256657
$$794$$ −22.2505 −0.789640
$$795$$ 0 0
$$796$$ 18.3739 0.651246
$$797$$ −11.6993 −0.414410 −0.207205 0.978298i $$-0.566437\pi$$
−0.207205 + 0.978298i $$0.566437\pi$$
$$798$$ −19.1464 −0.677774
$$799$$ 11.3629 0.401991
$$800$$ 0 0
$$801$$ 4.29685 0.151822
$$802$$ 54.9831 1.94152
$$803$$ 0.268814 0.00948624
$$804$$ 117.901 4.15806
$$805$$ 0 0
$$806$$ −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 $$0.698357\pi$$
$$810$$ 0 0
$$811$$ −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
$$815$$ 0 0
$$816$$ 55.1869 1.93193
$$817$$ −3.10482 −0.108624
$$818$$ 12.2464 0.428185
$$819$$ 29.1010 1.01687
$$820$$ 0 0
$$821$$ 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 $$-0.572629\pi$$
−0.226198 + 0.974081i $$0.572629\pi$$
$$824$$ −7.65835 −0.266791
$$825$$ 0 0
$$826$$ 35.8515 1.24743
$$827$$ 12.5167 0.435248 0.217624 0.976033i $$-0.430169\pi$$
0.217624 + 0.976033i $$0.430169\pi$$
$$828$$ 91.0504 3.16422
$$829$$ −19.4391 −0.675149 −0.337574 0.941299i $$-0.609606\pi$$
−0.337574 + 0.941299i $$0.609606\pi$$
$$830$$ 0 0
$$831$$ −67.4124 −2.33851
$$832$$ 20.9605 0.726674
$$833$$ −20.4769 −0.709482
$$834$$ −78.7400 −2.72654
$$835$$ 0 0
$$836$$ −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 $$0.774544\pi$$
$$840$$ 0 0
$$841$$ 31.1662 1.07470
$$842$$ −2.52818 −0.0871268
$$843$$ 73.9229 2.54604
$$844$$ 28.6468 0.986063
$$845$$ 0 0
$$846$$ −16.5865 −0.570256
$$847$$ 30.3632 1.04329
$$848$$ −6.41061 −0.220141
$$849$$ −21.2591 −0.729610
$$850$$ 0 0
$$851$$ 9.35601 0.320720
$$852$$ 53.2761 1.82521
$$853$$ −3.30374 −0.113118 −0.0565590 0.998399i $$-0.518013\pi$$
−0.0565590 + 0.998399i $$0.518013\pi$$
$$854$$ −27.1079 −0.927614
$$855$$ 0 0
$$856$$ −25.7682 −0.880740
$$857$$ −6.02374 −0.205767 −0.102883 0.994693i $$-0.532807\pi$$
−0.102883 + 0.994693i $$0.532807\pi$$
$$858$$ 12.0831 0.412511
$$859$$ −36.0158 −1.22884 −0.614421 0.788978i $$-0.710610\pi$$
−0.614421 + 0.788978i $$0.710610\pi$$
$$860$$ 0 0
$$861$$ −25.2012 −0.858853
$$862$$ −30.0487 −1.02346
$$863$$ −29.9250 −1.01866 −0.509329 0.860572i $$-0.670106\pi$$
−0.509329 + 0.860572i $$0.670106\pi$$
$$864$$ −51.4549 −1.75053
$$865$$ 0 0
$$866$$ −54.8236 −1.86298
$$867$$ 124.247 4.21966
$$868$$ −52.8864 −1.79508
$$869$$ 10.3739 0.351911
$$870$$ 0 0
$$871$$ −28.0485 −0.950386
$$872$$ 5.97548 0.202355
$$873$$ −16.8951 −0.571813
$$874$$ 14.2512 0.482054
$$875$$ 0 0
$$876$$ −1.81388 −0.0612852
$$877$$ −0.618980 −0.0209015 −0.0104507 0.999945i $$-0.503327\pi$$
−0.0104507 + 0.999945i $$0.503327\pi$$
$$878$$ −47.0762 −1.58874
$$879$$ 69.5184 2.34480
$$880$$ 0 0
$$881$$ −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 $$-0.192433\pi$$
0.822760 + 0.568389i $$0.192433\pi$$
$$884$$ 35.6302 1.19837
$$885$$ 0 0
$$886$$ 41.5034 1.39433
$$887$$ −58.2797 −1.95684 −0.978421 0.206622i $$-0.933753\pi$$
−0.978421 + 0.206622i $$0.933753\pi$$
$$888$$ 5.11762 0.171736
$$889$$ 14.3909 0.482657
$$890$$ 0 0
$$891$$ −3.39757 −0.113823
$$892$$ 32.7251 1.09572
$$893$$ −1.46492 −0.0490216
$$894$$ −85.0892 −2.84581
$$895$$ 0 0
$$896$$ 30.0318 1.00329
$$897$$ −33.9566 −1.13378
$$898$$ −41.1007 −1.37155
$$899$$ 51.0130 1.70138
$$900$$ 0 0
$$901$$ −20.1168 −0.670187
$$902$$ −6.67483 −0.222247
$$903$$ 27.7470 0.923361
$$904$$ 10.0914 0.335636
$$905$$ 0 0
$$906$$ 30.9732 1.02902
$$907$$ 36.5154 1.21247 0.606237 0.795284i $$-0.292678\pi$$
0.606237 + 0.795284i $$0.292678\pi$$
$$908$$ 17.1670 0.569708
$$909$$ −15.9543 −0.529172
$$910$$ 0 0
$$911$$ 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
$$915$$ 0 0
$$916$$ −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 $$0.458317\pi$$
$$920$$ 0 0
$$921$$ −12.6210 −0.415877
$$922$$ −53.6182 −1.76582
$$923$$ −12.6743 −0.417179
$$924$$ 25.5726 0.841275
$$925$$ 0 0
$$926$$ −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 $$0.299814\pi$$
$$930$$ 0 0
$$931$$ 2.63990 0.0865192
$$932$$ −24.6399 −0.807106
$$933$$ −45.2254 −1.48061
$$934$$ −32.6113 −1.06707
$$935$$ 0 0
$$936$$ −11.8481 −0.387268
$$937$$ 11.4420 0.373793 0.186896 0.982380i $$-0.440157\pi$$
0.186896 + 0.982380i $$0.440157\pi$$
$$938$$ −105.200 −3.43490
$$939$$ −71.3236 −2.32756
$$940$$ 0 0
$$941$$ −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
$$945$$ 0 0
$$946$$ 7.34911 0.238940
$$947$$ −2.63807 −0.0857256 −0.0428628 0.999081i $$-0.513648\pi$$
−0.0428628 + 0.999081i $$0.513648\pi$$
$$948$$ −70.0000 −2.27349
$$949$$ 0.431517 0.0140076
$$950$$ 0 0
$$951$$ 78.8084 2.55554
$$952$$ 30.4432 0.986670
$$953$$ −10.4430 −0.338282 −0.169141 0.985592i $$-0.554099\pi$$
−0.169141 + 0.985592i $$0.554099\pi$$
$$954$$ 29.3646 0.950714
$$955$$ 0 0
$$956$$ −7.58835 −0.245425
$$957$$ −24.6667 −0.797360
$$958$$ −42.9525 −1.38773
$$959$$ −2.52437 −0.0815160
$$960$$ 0 0
$$961$$ 12.2522 0.395232
$$962$$ −5.34432 −0.172308
$$963$$ −107.731 −3.47159
$$964$$ −24.4173 −0.786428
$$965$$ 0 0
$$966$$ −127.359 −4.09772
$$967$$ −33.2732 −1.06999 −0.534996 0.844854i $$-0.679687\pi$$
−0.534996 + 0.844854i $$0.679687\pi$$
$$968$$ −12.3620 −0.397330
$$969$$ −22.3264 −0.717228
$$970$$ 0 0
$$971$$ 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
$$975$$ 0 0
$$976$$ −10.0732 −0.322437
$$977$$ −0.402439 −0.0128752 −0.00643759 0.999979i $$-0.502049\pi$$
−0.00643759 + 0.999979i $$0.502049\pi$$
$$978$$ −44.7409 −1.43066
$$979$$ −0.898271 −0.0287089
$$980$$ 0 0
$$981$$ 24.9821 0.797617
$$982$$ 53.1395 1.69575
$$983$$ −11.4927 −0.366561 −0.183281 0.983061i $$-0.558672\pi$$
−0.183281 + 0.983061i $$0.558672\pi$$
$$984$$ 10.2603 0.327088
$$985$$ 0 0
$$986$$ −128.902 −4.10508
$$987$$ 13.0916 0.416710
$$988$$ −4.59348 −0.146138
$$989$$ −20.6529 −0.656723
$$990$$ 0 0
$$991$$ 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
$$995$$ 0 0
$$996$$ −66.0624 −2.09327
$$997$$ 48.1086 1.52362 0.761808 0.647803i $$-0.224312\pi$$
0.761808 + 0.647803i $$0.224312\pi$$
$$998$$ −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 475.2.a.i.1.3 4
3.2 odd 2 4275.2.a.bo.1.2 4
4.3 odd 2 7600.2.a.cf.1.1 4
5.2 odd 4 475.2.b.e.324.7 8
5.3 odd 4 475.2.b.e.324.2 8
5.4 even 2 95.2.a.b.1.2 4
15.14 odd 2 855.2.a.m.1.3 4
19.18 odd 2 9025.2.a.bf.1.2 4
20.19 odd 2 1520.2.a.t.1.4 4
35.34 odd 2 4655.2.a.y.1.2 4
40.19 odd 2 6080.2.a.ch.1.1 4
40.29 even 2 6080.2.a.cc.1.4 4
95.94 odd 2 1805.2.a.p.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.2 4 5.4 even 2
475.2.a.i.1.3 4 1.1 even 1 trivial
475.2.b.e.324.2 8 5.3 odd 4
475.2.b.e.324.7 8 5.2 odd 4
855.2.a.m.1.3 4 15.14 odd 2
1520.2.a.t.1.4 4 20.19 odd 2
1805.2.a.p.1.3 4 95.94 odd 2
4275.2.a.bo.1.2 4 3.2 odd 2
4655.2.a.y.1.2 4 35.34 odd 2
6080.2.a.cc.1.4 4 40.29 even 2
6080.2.a.ch.1.1 4 40.19 odd 2
7600.2.a.cf.1.1 4 4.3 odd 2
9025.2.a.bf.1.2 4 19.18 odd 2