Properties

 Label 9025.2.a.br.1.3 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.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: $$72.0649878242$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.41289040.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5$$ x^6 - 3*x^5 - 8*x^4 + 21*x^3 + 18*x^2 - 25*x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 475) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-2.24415$$ of defining polynomial Character $$\chi$$ $$=$$ 9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.311108 q^{2} -1.02983 q^{3} -1.90321 q^{4} +0.320390 q^{6} +3.28038 q^{7} +1.21432 q^{8} -1.93944 q^{9} +O(q^{10})$$ $$q-0.311108 q^{2} -1.02983 q^{3} -1.90321 q^{4} +0.320390 q^{6} +3.28038 q^{7} +1.21432 q^{8} -1.93944 q^{9} -5.16792 q^{11} +1.95999 q^{12} -3.53471 q^{13} -1.02055 q^{14} +3.42864 q^{16} +1.00928 q^{17} +0.603375 q^{18} -3.37825 q^{21} +1.60778 q^{22} +7.66351 q^{23} -1.25055 q^{24} +1.09968 q^{26} +5.08681 q^{27} -6.24326 q^{28} -4.02606 q^{29} -4.60077 q^{31} -3.49532 q^{32} +5.32210 q^{33} -0.313995 q^{34} +3.69117 q^{36} +6.48831 q^{37} +3.64017 q^{39} -6.80553 q^{41} +1.05100 q^{42} -6.31022 q^{43} +9.83565 q^{44} -2.38418 q^{46} +3.84157 q^{47} -3.53093 q^{48} +3.76091 q^{49} -1.03939 q^{51} +6.72730 q^{52} +7.11892 q^{53} -1.58255 q^{54} +3.98343 q^{56} +1.25254 q^{58} +13.4730 q^{59} +6.13700 q^{61} +1.43134 q^{62} -6.36211 q^{63} -5.76986 q^{64} -1.65575 q^{66} +11.1979 q^{67} -1.92088 q^{68} -7.89215 q^{69} -0.455404 q^{71} -2.35510 q^{72} -4.13382 q^{73} -2.01856 q^{74} -16.9528 q^{77} -1.13248 q^{78} +2.88828 q^{79} +0.579752 q^{81} +2.11725 q^{82} +5.50061 q^{83} +6.42953 q^{84} +1.96316 q^{86} +4.14617 q^{87} -6.27551 q^{88} -7.12866 q^{89} -11.5952 q^{91} -14.5853 q^{92} +4.73803 q^{93} -1.19514 q^{94} +3.59960 q^{96} +10.8250 q^{97} -1.17005 q^{98} +10.0229 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - 2 q^{7} - 6 q^{8} + 7 q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 - q^6 - 2 * q^7 - 6 * q^8 + 7 * q^9 $$6 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{6} - 2 q^{7} - 6 q^{8} + 7 q^{9} - q^{11} - 7 q^{12} - 5 q^{13} - 6 q^{14} - 6 q^{16} + 3 q^{17} - 7 q^{18} + 3 q^{21} - 9 q^{22} + 6 q^{23} + 11 q^{24} + 19 q^{26} - 18 q^{27} + 4 q^{28} + 3 q^{29} - 3 q^{31} + 6 q^{32} + 18 q^{33} - q^{34} + 13 q^{36} + 6 q^{37} + 8 q^{39} + 11 q^{41} + 11 q^{42} - 13 q^{43} + 21 q^{44} - 12 q^{46} + 6 q^{47} + 19 q^{48} + 4 q^{49} - 17 q^{51} + q^{52} - 18 q^{53} + 18 q^{54} + 4 q^{56} - 5 q^{58} + 4 q^{59} + 25 q^{61} + 21 q^{62} - 43 q^{63} - 22 q^{64} + 34 q^{66} - 6 q^{67} + q^{68} + 13 q^{69} + 18 q^{71} - 13 q^{72} - q^{73} - 6 q^{74} + 11 q^{77} - 72 q^{78} + 3 q^{79} + 2 q^{81} - 31 q^{82} + 23 q^{83} + 37 q^{84} + 9 q^{86} - 11 q^{87} - 11 q^{88} + 12 q^{89} - 11 q^{91} - 28 q^{92} + 13 q^{93} + 8 q^{94} - 13 q^{96} - 3 q^{97} + 22 q^{98} - 20 q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 - 3 * q^3 + 2 * q^4 - q^6 - 2 * q^7 - 6 * q^8 + 7 * q^9 - q^11 - 7 * q^12 - 5 * q^13 - 6 * q^14 - 6 * q^16 + 3 * q^17 - 7 * q^18 + 3 * q^21 - 9 * q^22 + 6 * q^23 + 11 * q^24 + 19 * q^26 - 18 * q^27 + 4 * q^28 + 3 * q^29 - 3 * q^31 + 6 * q^32 + 18 * q^33 - q^34 + 13 * q^36 + 6 * q^37 + 8 * q^39 + 11 * q^41 + 11 * q^42 - 13 * q^43 + 21 * q^44 - 12 * q^46 + 6 * q^47 + 19 * q^48 + 4 * q^49 - 17 * q^51 + q^52 - 18 * q^53 + 18 * q^54 + 4 * q^56 - 5 * q^58 + 4 * q^59 + 25 * q^61 + 21 * q^62 - 43 * q^63 - 22 * q^64 + 34 * q^66 - 6 * q^67 + q^68 + 13 * q^69 + 18 * q^71 - 13 * q^72 - q^73 - 6 * q^74 + 11 * q^77 - 72 * q^78 + 3 * q^79 + 2 * q^81 - 31 * q^82 + 23 * q^83 + 37 * q^84 + 9 * q^86 - 11 * q^87 - 11 * q^88 + 12 * q^89 - 11 * q^91 - 28 * q^92 + 13 * q^93 + 8 * q^94 - 13 * q^96 - 3 * q^97 + 22 * q^98 - 20 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.311108 −0.219986 −0.109993 0.993932i $$-0.535083\pi$$
−0.109993 + 0.993932i $$0.535083\pi$$
$$3$$ −1.02983 −0.594575 −0.297288 0.954788i $$-0.596082\pi$$
−0.297288 + 0.954788i $$0.596082\pi$$
$$4$$ −1.90321 −0.951606
$$5$$ 0 0
$$6$$ 0.320390 0.130799
$$7$$ 3.28038 1.23987 0.619934 0.784654i $$-0.287159\pi$$
0.619934 + 0.784654i $$0.287159\pi$$
$$8$$ 1.21432 0.429327
$$9$$ −1.93944 −0.646480
$$10$$ 0 0
$$11$$ −5.16792 −1.55819 −0.779093 0.626908i $$-0.784320\pi$$
−0.779093 + 0.626908i $$0.784320\pi$$
$$12$$ 1.95999 0.565801
$$13$$ −3.53471 −0.980352 −0.490176 0.871623i $$-0.663067\pi$$
−0.490176 + 0.871623i $$0.663067\pi$$
$$14$$ −1.02055 −0.272754
$$15$$ 0 0
$$16$$ 3.42864 0.857160
$$17$$ 1.00928 0.244787 0.122393 0.992482i $$-0.460943\pi$$
0.122393 + 0.992482i $$0.460943\pi$$
$$18$$ 0.603375 0.142217
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −3.37825 −0.737195
$$22$$ 1.60778 0.342780
$$23$$ 7.66351 1.59795 0.798976 0.601362i $$-0.205375\pi$$
0.798976 + 0.601362i $$0.205375\pi$$
$$24$$ −1.25055 −0.255267
$$25$$ 0 0
$$26$$ 1.09968 0.215664
$$27$$ 5.08681 0.978956
$$28$$ −6.24326 −1.17987
$$29$$ −4.02606 −0.747620 −0.373810 0.927505i $$-0.621949\pi$$
−0.373810 + 0.927505i $$0.621949\pi$$
$$30$$ 0 0
$$31$$ −4.60077 −0.826323 −0.413162 0.910658i $$-0.635576\pi$$
−0.413162 + 0.910658i $$0.635576\pi$$
$$32$$ −3.49532 −0.617890
$$33$$ 5.32210 0.926459
$$34$$ −0.313995 −0.0538498
$$35$$ 0 0
$$36$$ 3.69117 0.615194
$$37$$ 6.48831 1.06667 0.533336 0.845904i $$-0.320938\pi$$
0.533336 + 0.845904i $$0.320938\pi$$
$$38$$ 0 0
$$39$$ 3.64017 0.582893
$$40$$ 0 0
$$41$$ −6.80553 −1.06285 −0.531423 0.847107i $$-0.678342\pi$$
−0.531423 + 0.847107i $$0.678342\pi$$
$$42$$ 1.05100 0.162173
$$43$$ −6.31022 −0.962299 −0.481150 0.876639i $$-0.659781\pi$$
−0.481150 + 0.876639i $$0.659781\pi$$
$$44$$ 9.83565 1.48278
$$45$$ 0 0
$$46$$ −2.38418 −0.351528
$$47$$ 3.84157 0.560351 0.280175 0.959949i $$-0.409607\pi$$
0.280175 + 0.959949i $$0.409607\pi$$
$$48$$ −3.53093 −0.509646
$$49$$ 3.76091 0.537273
$$50$$ 0 0
$$51$$ −1.03939 −0.145544
$$52$$ 6.72730 0.932909
$$53$$ 7.11892 0.977858 0.488929 0.872323i $$-0.337388\pi$$
0.488929 + 0.872323i $$0.337388\pi$$
$$54$$ −1.58255 −0.215357
$$55$$ 0 0
$$56$$ 3.98343 0.532309
$$57$$ 0 0
$$58$$ 1.25254 0.164466
$$59$$ 13.4730 1.75403 0.877016 0.480460i $$-0.159530\pi$$
0.877016 + 0.480460i $$0.159530\pi$$
$$60$$ 0 0
$$61$$ 6.13700 0.785763 0.392881 0.919589i $$-0.371478\pi$$
0.392881 + 0.919589i $$0.371478\pi$$
$$62$$ 1.43134 0.181780
$$63$$ −6.36211 −0.801550
$$64$$ −5.76986 −0.721232
$$65$$ 0 0
$$66$$ −1.65575 −0.203808
$$67$$ 11.1979 1.36805 0.684023 0.729460i $$-0.260229\pi$$
0.684023 + 0.729460i $$0.260229\pi$$
$$68$$ −1.92088 −0.232941
$$69$$ −7.89215 −0.950103
$$70$$ 0 0
$$71$$ −0.455404 −0.0540465 −0.0270233 0.999635i $$-0.508603\pi$$
−0.0270233 + 0.999635i $$0.508603\pi$$
$$72$$ −2.35510 −0.277551
$$73$$ −4.13382 −0.483827 −0.241914 0.970298i $$-0.577775\pi$$
−0.241914 + 0.970298i $$0.577775\pi$$
$$74$$ −2.01856 −0.234653
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −16.9528 −1.93195
$$78$$ −1.13248 −0.128229
$$79$$ 2.88828 0.324957 0.162478 0.986712i $$-0.448051\pi$$
0.162478 + 0.986712i $$0.448051\pi$$
$$80$$ 0 0
$$81$$ 0.579752 0.0644169
$$82$$ 2.11725 0.233812
$$83$$ 5.50061 0.603770 0.301885 0.953344i $$-0.402384\pi$$
0.301885 + 0.953344i $$0.402384\pi$$
$$84$$ 6.42953 0.701519
$$85$$ 0 0
$$86$$ 1.96316 0.211693
$$87$$ 4.14617 0.444516
$$88$$ −6.27551 −0.668971
$$89$$ −7.12866 −0.755636 −0.377818 0.925880i $$-0.623326\pi$$
−0.377818 + 0.925880i $$0.623326\pi$$
$$90$$ 0 0
$$91$$ −11.5952 −1.21551
$$92$$ −14.5853 −1.52062
$$93$$ 4.73803 0.491311
$$94$$ −1.19514 −0.123270
$$95$$ 0 0
$$96$$ 3.59960 0.367382
$$97$$ 10.8250 1.09912 0.549558 0.835456i $$-0.314796\pi$$
0.549558 + 0.835456i $$0.314796\pi$$
$$98$$ −1.17005 −0.118193
$$99$$ 10.0229 1.00734
$$100$$ 0 0
$$101$$ −6.18022 −0.614955 −0.307478 0.951555i $$-0.599485\pi$$
−0.307478 + 0.951555i $$0.599485\pi$$
$$102$$ 0.323363 0.0320177
$$103$$ −18.3217 −1.80529 −0.902644 0.430387i $$-0.858377\pi$$
−0.902644 + 0.430387i $$0.858377\pi$$
$$104$$ −4.29227 −0.420891
$$105$$ 0 0
$$106$$ −2.21475 −0.215116
$$107$$ −8.73445 −0.844391 −0.422196 0.906505i $$-0.638740\pi$$
−0.422196 + 0.906505i $$0.638740\pi$$
$$108$$ −9.68127 −0.931581
$$109$$ −6.09679 −0.583966 −0.291983 0.956423i $$-0.594315\pi$$
−0.291983 + 0.956423i $$0.594315\pi$$
$$110$$ 0 0
$$111$$ −6.68188 −0.634216
$$112$$ 11.2473 1.06277
$$113$$ −10.3443 −0.973111 −0.486556 0.873650i $$-0.661747\pi$$
−0.486556 + 0.873650i $$0.661747\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 7.66244 0.711440
$$117$$ 6.85536 0.633778
$$118$$ −4.19155 −0.385863
$$119$$ 3.31083 0.303503
$$120$$ 0 0
$$121$$ 15.7074 1.42794
$$122$$ −1.90927 −0.172857
$$123$$ 7.00857 0.631942
$$124$$ 8.75625 0.786334
$$125$$ 0 0
$$126$$ 1.97930 0.176330
$$127$$ −3.36713 −0.298784 −0.149392 0.988778i $$-0.547732\pi$$
−0.149392 + 0.988778i $$0.547732\pi$$
$$128$$ 8.78568 0.776552
$$129$$ 6.49848 0.572159
$$130$$ 0 0
$$131$$ 2.42178 0.211592 0.105796 0.994388i $$-0.466261\pi$$
0.105796 + 0.994388i $$0.466261\pi$$
$$132$$ −10.1291 −0.881624
$$133$$ 0 0
$$134$$ −3.48377 −0.300952
$$135$$ 0 0
$$136$$ 1.22559 0.105094
$$137$$ 11.5191 0.984144 0.492072 0.870555i $$-0.336240\pi$$
0.492072 + 0.870555i $$0.336240\pi$$
$$138$$ 2.45531 0.209010
$$139$$ 7.07553 0.600138 0.300069 0.953917i $$-0.402990\pi$$
0.300069 + 0.953917i $$0.402990\pi$$
$$140$$ 0 0
$$141$$ −3.95618 −0.333171
$$142$$ 0.141680 0.0118895
$$143$$ 18.2671 1.52757
$$144$$ −6.64964 −0.554137
$$145$$ 0 0
$$146$$ 1.28606 0.106435
$$147$$ −3.87312 −0.319449
$$148$$ −12.3486 −1.01505
$$149$$ 12.7495 1.04448 0.522240 0.852798i $$-0.325096\pi$$
0.522240 + 0.852798i $$0.325096\pi$$
$$150$$ 0 0
$$151$$ −19.6361 −1.59797 −0.798983 0.601353i $$-0.794628\pi$$
−0.798983 + 0.601353i $$0.794628\pi$$
$$152$$ 0 0
$$153$$ −1.95744 −0.158250
$$154$$ 5.27413 0.425002
$$155$$ 0 0
$$156$$ −6.92801 −0.554685
$$157$$ −3.08489 −0.246201 −0.123101 0.992394i $$-0.539284\pi$$
−0.123101 + 0.992394i $$0.539284\pi$$
$$158$$ −0.898567 −0.0714861
$$159$$ −7.33131 −0.581410
$$160$$ 0 0
$$161$$ 25.1393 1.98125
$$162$$ −0.180365 −0.0141708
$$163$$ 22.4721 1.76015 0.880077 0.474831i $$-0.157491\pi$$
0.880077 + 0.474831i $$0.157491\pi$$
$$164$$ 12.9524 1.01141
$$165$$ 0 0
$$166$$ −1.71128 −0.132821
$$167$$ −10.5397 −0.815590 −0.407795 0.913074i $$-0.633702\pi$$
−0.407795 + 0.913074i $$0.633702\pi$$
$$168$$ −4.10228 −0.316498
$$169$$ −0.505830 −0.0389100
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 12.0097 0.915730
$$173$$ −22.9381 −1.74395 −0.871977 0.489547i $$-0.837162\pi$$
−0.871977 + 0.489547i $$0.837162\pi$$
$$174$$ −1.28991 −0.0977876
$$175$$ 0 0
$$176$$ −17.7189 −1.33561
$$177$$ −13.8749 −1.04290
$$178$$ 2.21778 0.166230
$$179$$ 23.2705 1.73932 0.869661 0.493649i $$-0.164337\pi$$
0.869661 + 0.493649i $$0.164337\pi$$
$$180$$ 0 0
$$181$$ −15.9669 −1.18681 −0.593406 0.804904i $$-0.702217\pi$$
−0.593406 + 0.804904i $$0.702217\pi$$
$$182$$ 3.60736 0.267395
$$183$$ −6.32010 −0.467195
$$184$$ 9.30595 0.686044
$$185$$ 0 0
$$186$$ −1.47404 −0.108082
$$187$$ −5.21589 −0.381423
$$188$$ −7.31133 −0.533233
$$189$$ 16.6867 1.21378
$$190$$ 0 0
$$191$$ −2.44600 −0.176986 −0.0884930 0.996077i $$-0.528205\pi$$
−0.0884930 + 0.996077i $$0.528205\pi$$
$$192$$ 5.94200 0.428827
$$193$$ 19.0456 1.37093 0.685466 0.728104i $$-0.259598\pi$$
0.685466 + 0.728104i $$0.259598\pi$$
$$194$$ −3.36775 −0.241791
$$195$$ 0 0
$$196$$ −7.15782 −0.511273
$$197$$ 6.18524 0.440680 0.220340 0.975423i $$-0.429283\pi$$
0.220340 + 0.975423i $$0.429283\pi$$
$$198$$ −3.11819 −0.221600
$$199$$ 7.21050 0.511138 0.255569 0.966791i $$-0.417737\pi$$
0.255569 + 0.966791i $$0.417737\pi$$
$$200$$ 0 0
$$201$$ −11.5320 −0.813407
$$202$$ 1.92272 0.135282
$$203$$ −13.2070 −0.926950
$$204$$ 1.97819 0.138501
$$205$$ 0 0
$$206$$ 5.70002 0.397139
$$207$$ −14.8629 −1.03304
$$208$$ −12.1192 −0.840318
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −19.6467 −1.35253 −0.676266 0.736658i $$-0.736403\pi$$
−0.676266 + 0.736658i $$0.736403\pi$$
$$212$$ −13.5488 −0.930536
$$213$$ 0.468991 0.0321347
$$214$$ 2.71736 0.185755
$$215$$ 0 0
$$216$$ 6.17701 0.420292
$$217$$ −15.0923 −1.02453
$$218$$ 1.89676 0.128465
$$219$$ 4.25715 0.287672
$$220$$ 0 0
$$221$$ −3.56752 −0.239977
$$222$$ 2.07879 0.139519
$$223$$ 11.3338 0.758967 0.379484 0.925198i $$-0.376102\pi$$
0.379484 + 0.925198i $$0.376102\pi$$
$$224$$ −11.4660 −0.766103
$$225$$ 0 0
$$226$$ 3.21820 0.214071
$$227$$ 11.1369 0.739180 0.369590 0.929195i $$-0.379498\pi$$
0.369590 + 0.929195i $$0.379498\pi$$
$$228$$ 0 0
$$229$$ 3.51221 0.232093 0.116047 0.993244i $$-0.462978\pi$$
0.116047 + 0.993244i $$0.462978\pi$$
$$230$$ 0 0
$$231$$ 17.4585 1.14869
$$232$$ −4.88892 −0.320973
$$233$$ −6.18998 −0.405519 −0.202759 0.979229i $$-0.564991\pi$$
−0.202759 + 0.979229i $$0.564991\pi$$
$$234$$ −2.13276 −0.139423
$$235$$ 0 0
$$236$$ −25.6419 −1.66915
$$237$$ −2.97445 −0.193211
$$238$$ −1.03003 −0.0667666
$$239$$ 12.1221 0.784116 0.392058 0.919940i $$-0.371763\pi$$
0.392058 + 0.919940i $$0.371763\pi$$
$$240$$ 0 0
$$241$$ −5.10204 −0.328651 −0.164326 0.986406i $$-0.552545\pi$$
−0.164326 + 0.986406i $$0.552545\pi$$
$$242$$ −4.88669 −0.314128
$$243$$ −15.8575 −1.01726
$$244$$ −11.6800 −0.747737
$$245$$ 0 0
$$246$$ −2.18042 −0.139019
$$247$$ 0 0
$$248$$ −5.58681 −0.354763
$$249$$ −5.66472 −0.358987
$$250$$ 0 0
$$251$$ 25.4244 1.60477 0.802385 0.596806i $$-0.203564\pi$$
0.802385 + 0.596806i $$0.203564\pi$$
$$252$$ 12.1084 0.762760
$$253$$ −39.6044 −2.48991
$$254$$ 1.04754 0.0657285
$$255$$ 0 0
$$256$$ 8.80642 0.550401
$$257$$ −18.4499 −1.15087 −0.575436 0.817847i $$-0.695167\pi$$
−0.575436 + 0.817847i $$0.695167\pi$$
$$258$$ −2.02173 −0.125867
$$259$$ 21.2841 1.32253
$$260$$ 0 0
$$261$$ 7.80830 0.483322
$$262$$ −0.753434 −0.0465473
$$263$$ −12.9506 −0.798568 −0.399284 0.916827i $$-0.630741\pi$$
−0.399284 + 0.916827i $$0.630741\pi$$
$$264$$ 6.46273 0.397754
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 7.34134 0.449283
$$268$$ −21.3121 −1.30184
$$269$$ 0.156494 0.00954163 0.00477081 0.999989i $$-0.498481\pi$$
0.00477081 + 0.999989i $$0.498481\pi$$
$$270$$ 0 0
$$271$$ 27.8191 1.68989 0.844944 0.534855i $$-0.179634\pi$$
0.844944 + 0.534855i $$0.179634\pi$$
$$272$$ 3.46046 0.209821
$$273$$ 11.9411 0.722711
$$274$$ −3.58368 −0.216498
$$275$$ 0 0
$$276$$ 15.0204 0.904124
$$277$$ 0.524062 0.0314878 0.0157439 0.999876i $$-0.494988\pi$$
0.0157439 + 0.999876i $$0.494988\pi$$
$$278$$ −2.20125 −0.132022
$$279$$ 8.92293 0.534202
$$280$$ 0 0
$$281$$ −4.88826 −0.291609 −0.145804 0.989313i $$-0.546577\pi$$
−0.145804 + 0.989313i $$0.546577\pi$$
$$282$$ 1.23080 0.0732931
$$283$$ 17.8786 1.06277 0.531387 0.847129i $$-0.321671\pi$$
0.531387 + 0.847129i $$0.321671\pi$$
$$284$$ 0.866730 0.0514310
$$285$$ 0 0
$$286$$ −5.68304 −0.336045
$$287$$ −22.3248 −1.31779
$$288$$ 6.77896 0.399454
$$289$$ −15.9814 −0.940079
$$290$$ 0 0
$$291$$ −11.1480 −0.653507
$$292$$ 7.86754 0.460413
$$293$$ −7.25399 −0.423783 −0.211891 0.977293i $$-0.567962\pi$$
−0.211891 + 0.977293i $$0.567962\pi$$
$$294$$ 1.20496 0.0702745
$$295$$ 0 0
$$296$$ 7.87888 0.457951
$$297$$ −26.2882 −1.52540
$$298$$ −3.96647 −0.229772
$$299$$ −27.0883 −1.56656
$$300$$ 0 0
$$301$$ −20.6999 −1.19312
$$302$$ 6.10896 0.351531
$$303$$ 6.36461 0.365637
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0.608976 0.0348128
$$307$$ −14.3803 −0.820727 −0.410364 0.911922i $$-0.634598\pi$$
−0.410364 + 0.911922i $$0.634598\pi$$
$$308$$ 32.2647 1.83845
$$309$$ 18.8683 1.07338
$$310$$ 0 0
$$311$$ 8.52590 0.483459 0.241730 0.970344i $$-0.422285\pi$$
0.241730 + 0.970344i $$0.422285\pi$$
$$312$$ 4.42033 0.250252
$$313$$ 21.3619 1.20744 0.603722 0.797195i $$-0.293684\pi$$
0.603722 + 0.797195i $$0.293684\pi$$
$$314$$ 0.959733 0.0541609
$$315$$ 0 0
$$316$$ −5.49701 −0.309231
$$317$$ −28.0928 −1.57785 −0.788924 0.614490i $$-0.789362\pi$$
−0.788924 + 0.614490i $$0.789362\pi$$
$$318$$ 2.28083 0.127902
$$319$$ 20.8063 1.16493
$$320$$ 0 0
$$321$$ 8.99504 0.502054
$$322$$ −7.82102 −0.435848
$$323$$ 0 0
$$324$$ −1.10339 −0.0612995
$$325$$ 0 0
$$326$$ −6.99126 −0.387210
$$327$$ 6.27868 0.347212
$$328$$ −8.26409 −0.456308
$$329$$ 12.6018 0.694761
$$330$$ 0 0
$$331$$ 15.1725 0.833957 0.416979 0.908916i $$-0.363089\pi$$
0.416979 + 0.908916i $$0.363089\pi$$
$$332$$ −10.4688 −0.574552
$$333$$ −12.5837 −0.689582
$$334$$ 3.27900 0.179419
$$335$$ 0 0
$$336$$ −11.5828 −0.631894
$$337$$ 17.5008 0.953330 0.476665 0.879085i $$-0.341845\pi$$
0.476665 + 0.879085i $$0.341845\pi$$
$$338$$ 0.157368 0.00855967
$$339$$ 10.6529 0.578588
$$340$$ 0 0
$$341$$ 23.7764 1.28757
$$342$$ 0 0
$$343$$ −10.6254 −0.573720
$$344$$ −7.66262 −0.413141
$$345$$ 0 0
$$346$$ 7.13623 0.383646
$$347$$ −17.3687 −0.932401 −0.466201 0.884679i $$-0.654377\pi$$
−0.466201 + 0.884679i $$0.654377\pi$$
$$348$$ −7.89105 −0.423004
$$349$$ −34.2563 −1.83370 −0.916850 0.399232i $$-0.869277\pi$$
−0.916850 + 0.399232i $$0.869277\pi$$
$$350$$ 0 0
$$351$$ −17.9804 −0.959722
$$352$$ 18.0635 0.962788
$$353$$ −34.9287 −1.85907 −0.929534 0.368736i $$-0.879791\pi$$
−0.929534 + 0.368736i $$0.879791\pi$$
$$354$$ 4.31660 0.229425
$$355$$ 0 0
$$356$$ 13.5673 0.719068
$$357$$ −3.40961 −0.180456
$$358$$ −7.23965 −0.382627
$$359$$ −0.976546 −0.0515401 −0.0257701 0.999668i $$-0.508204\pi$$
−0.0257701 + 0.999668i $$0.508204\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 4.96743 0.261082
$$363$$ −16.1760 −0.849020
$$364$$ 22.0681 1.15668
$$365$$ 0 0
$$366$$ 1.96623 0.102777
$$367$$ −9.86635 −0.515019 −0.257510 0.966276i $$-0.582902\pi$$
−0.257510 + 0.966276i $$0.582902\pi$$
$$368$$ 26.2754 1.36970
$$369$$ 13.1989 0.687109
$$370$$ 0 0
$$371$$ 23.3528 1.21242
$$372$$ −9.01748 −0.467535
$$373$$ 12.9521 0.670634 0.335317 0.942105i $$-0.391157\pi$$
0.335317 + 0.942105i $$0.391157\pi$$
$$374$$ 1.62270 0.0839080
$$375$$ 0 0
$$376$$ 4.66490 0.240574
$$377$$ 14.2309 0.732931
$$378$$ −5.19136 −0.267015
$$379$$ 5.21597 0.267926 0.133963 0.990986i $$-0.457230\pi$$
0.133963 + 0.990986i $$0.457230\pi$$
$$380$$ 0 0
$$381$$ 3.46758 0.177650
$$382$$ 0.760969 0.0389345
$$383$$ −33.5314 −1.71337 −0.856686 0.515838i $$-0.827481\pi$$
−0.856686 + 0.515838i $$0.827481\pi$$
$$384$$ −9.04780 −0.461718
$$385$$ 0 0
$$386$$ −5.92524 −0.301587
$$387$$ 12.2383 0.622107
$$388$$ −20.6023 −1.04593
$$389$$ 27.7406 1.40651 0.703253 0.710940i $$-0.251730\pi$$
0.703253 + 0.710940i $$0.251730\pi$$
$$390$$ 0 0
$$391$$ 7.73464 0.391158
$$392$$ 4.56695 0.230666
$$393$$ −2.49403 −0.125807
$$394$$ −1.92428 −0.0969437
$$395$$ 0 0
$$396$$ −19.0757 −0.958588
$$397$$ 5.49942 0.276008 0.138004 0.990432i $$-0.455931\pi$$
0.138004 + 0.990432i $$0.455931\pi$$
$$398$$ −2.24324 −0.112444
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −8.51309 −0.425124 −0.212562 0.977148i $$-0.568181\pi$$
−0.212562 + 0.977148i $$0.568181\pi$$
$$402$$ 3.58770 0.178938
$$403$$ 16.2624 0.810088
$$404$$ 11.7623 0.585195
$$405$$ 0 0
$$406$$ 4.10880 0.203917
$$407$$ −33.5311 −1.66207
$$408$$ −1.26216 −0.0624860
$$409$$ −21.4706 −1.06165 −0.530826 0.847481i $$-0.678118\pi$$
−0.530826 + 0.847481i $$0.678118\pi$$
$$410$$ 0 0
$$411$$ −11.8628 −0.585148
$$412$$ 34.8700 1.71792
$$413$$ 44.1965 2.17477
$$414$$ 4.62397 0.227256
$$415$$ 0 0
$$416$$ 12.3549 0.605750
$$417$$ −7.28662 −0.356827
$$418$$ 0 0
$$419$$ −5.09378 −0.248848 −0.124424 0.992229i $$-0.539708\pi$$
−0.124424 + 0.992229i $$0.539708\pi$$
$$420$$ 0 0
$$421$$ 5.65823 0.275765 0.137883 0.990449i $$-0.455970\pi$$
0.137883 + 0.990449i $$0.455970\pi$$
$$422$$ 6.11223 0.297539
$$423$$ −7.45050 −0.362256
$$424$$ 8.64464 0.419821
$$425$$ 0 0
$$426$$ −0.145907 −0.00706920
$$427$$ 20.1317 0.974242
$$428$$ 16.6235 0.803528
$$429$$ −18.8121 −0.908256
$$430$$ 0 0
$$431$$ −38.3668 −1.84807 −0.924033 0.382313i $$-0.875128\pi$$
−0.924033 + 0.382313i $$0.875128\pi$$
$$432$$ 17.4408 0.839122
$$433$$ −6.96250 −0.334596 −0.167298 0.985906i $$-0.553504\pi$$
−0.167298 + 0.985906i $$0.553504\pi$$
$$434$$ 4.69533 0.225383
$$435$$ 0 0
$$436$$ 11.6035 0.555706
$$437$$ 0 0
$$438$$ −1.32443 −0.0632839
$$439$$ 9.52692 0.454695 0.227347 0.973814i $$-0.426995\pi$$
0.227347 + 0.973814i $$0.426995\pi$$
$$440$$ 0 0
$$441$$ −7.29407 −0.347337
$$442$$ 1.10988 0.0527917
$$443$$ −34.7038 −1.64883 −0.824414 0.565987i $$-0.808495\pi$$
−0.824414 + 0.565987i $$0.808495\pi$$
$$444$$ 12.7170 0.603524
$$445$$ 0 0
$$446$$ −3.52603 −0.166963
$$447$$ −13.1299 −0.621023
$$448$$ −18.9273 −0.894233
$$449$$ −8.84228 −0.417293 −0.208646 0.977991i $$-0.566906\pi$$
−0.208646 + 0.977991i $$0.566906\pi$$
$$450$$ 0 0
$$451$$ 35.1704 1.65611
$$452$$ 19.6874 0.926019
$$453$$ 20.2220 0.950111
$$454$$ −3.46477 −0.162610
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.664998 −0.0311073 −0.0155536 0.999879i $$-0.504951\pi$$
−0.0155536 + 0.999879i $$0.504951\pi$$
$$458$$ −1.09268 −0.0510574
$$459$$ 5.13402 0.239636
$$460$$ 0 0
$$461$$ −39.5392 −1.84152 −0.920762 0.390124i $$-0.872432\pi$$
−0.920762 + 0.390124i $$0.872432\pi$$
$$462$$ −5.43149 −0.252696
$$463$$ 15.5782 0.723981 0.361991 0.932182i $$-0.382097\pi$$
0.361991 + 0.932182i $$0.382097\pi$$
$$464$$ −13.8039 −0.640830
$$465$$ 0 0
$$466$$ 1.92575 0.0892086
$$467$$ −41.0358 −1.89891 −0.949456 0.313901i $$-0.898364\pi$$
−0.949456 + 0.313901i $$0.898364\pi$$
$$468$$ −13.0472 −0.603107
$$469$$ 36.7335 1.69620
$$470$$ 0 0
$$471$$ 3.17693 0.146385
$$472$$ 16.3605 0.753053
$$473$$ 32.6107 1.49944
$$474$$ 0.925375 0.0425039
$$475$$ 0 0
$$476$$ −6.30121 −0.288816
$$477$$ −13.8067 −0.632166
$$478$$ −3.77129 −0.172495
$$479$$ −23.5587 −1.07643 −0.538213 0.842809i $$-0.680900\pi$$
−0.538213 + 0.842809i $$0.680900\pi$$
$$480$$ 0 0
$$481$$ −22.9343 −1.04571
$$482$$ 1.58728 0.0722988
$$483$$ −25.8893 −1.17800
$$484$$ −29.8945 −1.35884
$$485$$ 0 0
$$486$$ 4.93338 0.223783
$$487$$ 23.9166 1.08377 0.541883 0.840454i $$-0.317712\pi$$
0.541883 + 0.840454i $$0.317712\pi$$
$$488$$ 7.45228 0.337349
$$489$$ −23.1426 −1.04654
$$490$$ 0 0
$$491$$ 16.3475 0.737751 0.368875 0.929479i $$-0.379743\pi$$
0.368875 + 0.929479i $$0.379743\pi$$
$$492$$ −13.3388 −0.601360
$$493$$ −4.06343 −0.183007
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −15.7744 −0.708291
$$497$$ −1.49390 −0.0670105
$$498$$ 1.76234 0.0789723
$$499$$ −32.0708 −1.43569 −0.717844 0.696204i $$-0.754871\pi$$
−0.717844 + 0.696204i $$0.754871\pi$$
$$500$$ 0 0
$$501$$ 10.8542 0.484930
$$502$$ −7.90972 −0.353028
$$503$$ −14.7254 −0.656572 −0.328286 0.944578i $$-0.606471\pi$$
−0.328286 + 0.944578i $$0.606471\pi$$
$$504$$ −7.72563 −0.344127
$$505$$ 0 0
$$506$$ 12.3212 0.547746
$$507$$ 0.520921 0.0231349
$$508$$ 6.40836 0.284325
$$509$$ −22.0663 −0.978073 −0.489037 0.872263i $$-0.662652\pi$$
−0.489037 + 0.872263i $$0.662652\pi$$
$$510$$ 0 0
$$511$$ −13.5605 −0.599882
$$512$$ −20.3111 −0.897633
$$513$$ 0 0
$$514$$ 5.73990 0.253176
$$515$$ 0 0
$$516$$ −12.3680 −0.544470
$$517$$ −19.8529 −0.873131
$$518$$ −6.62166 −0.290939
$$519$$ 23.6225 1.03691
$$520$$ 0 0
$$521$$ −14.7781 −0.647442 −0.323721 0.946153i $$-0.604934\pi$$
−0.323721 + 0.946153i $$0.604934\pi$$
$$522$$ −2.42922 −0.106324
$$523$$ 12.7100 0.555768 0.277884 0.960615i $$-0.410367\pi$$
0.277884 + 0.960615i $$0.410367\pi$$
$$524$$ −4.60916 −0.201352
$$525$$ 0 0
$$526$$ 4.02903 0.175674
$$527$$ −4.64348 −0.202273
$$528$$ 18.2476 0.794124
$$529$$ 35.7294 1.55345
$$530$$ 0 0
$$531$$ −26.1301 −1.13395
$$532$$ 0 0
$$533$$ 24.0556 1.04196
$$534$$ −2.28395 −0.0988361
$$535$$ 0 0
$$536$$ 13.5979 0.587339
$$537$$ −23.9648 −1.03416
$$538$$ −0.0486866 −0.00209903
$$539$$ −19.4361 −0.837172
$$540$$ 0 0
$$541$$ −30.2345 −1.29988 −0.649941 0.759985i $$-0.725206\pi$$
−0.649941 + 0.759985i $$0.725206\pi$$
$$542$$ −8.65473 −0.371752
$$543$$ 16.4433 0.705649
$$544$$ −3.52776 −0.151251
$$545$$ 0 0
$$546$$ −3.71498 −0.158987
$$547$$ 7.80744 0.333822 0.166911 0.985972i $$-0.446621\pi$$
0.166911 + 0.985972i $$0.446621\pi$$
$$548$$ −21.9233 −0.936517
$$549$$ −11.9024 −0.507980
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −9.58359 −0.407905
$$553$$ 9.47467 0.402904
$$554$$ −0.163040 −0.00692689
$$555$$ 0 0
$$556$$ −13.4662 −0.571095
$$557$$ −2.41495 −0.102325 −0.0511623 0.998690i $$-0.516293\pi$$
−0.0511623 + 0.998690i $$0.516293\pi$$
$$558$$ −2.77599 −0.117517
$$559$$ 22.3048 0.943392
$$560$$ 0 0
$$561$$ 5.37150 0.226785
$$562$$ 1.52078 0.0641500
$$563$$ −25.3392 −1.06792 −0.533960 0.845510i $$-0.679297\pi$$
−0.533960 + 0.845510i $$0.679297\pi$$
$$564$$ 7.52946 0.317047
$$565$$ 0 0
$$566$$ −5.56218 −0.233796
$$567$$ 1.90181 0.0798685
$$568$$ −0.553006 −0.0232036
$$569$$ −45.1046 −1.89088 −0.945441 0.325793i $$-0.894369\pi$$
−0.945441 + 0.325793i $$0.894369\pi$$
$$570$$ 0 0
$$571$$ −9.73299 −0.407313 −0.203656 0.979042i $$-0.565283\pi$$
−0.203656 + 0.979042i $$0.565283\pi$$
$$572$$ −34.7661 −1.45365
$$573$$ 2.51897 0.105232
$$574$$ 6.94541 0.289896
$$575$$ 0 0
$$576$$ 11.1903 0.466262
$$577$$ 8.83145 0.367658 0.183829 0.982958i $$-0.441151\pi$$
0.183829 + 0.982958i $$0.441151\pi$$
$$578$$ 4.97192 0.206805
$$579$$ −19.6138 −0.815123
$$580$$ 0 0
$$581$$ 18.0441 0.748596
$$582$$ 3.46823 0.143763
$$583$$ −36.7900 −1.52369
$$584$$ −5.01978 −0.207720
$$585$$ 0 0
$$586$$ 2.25677 0.0932265
$$587$$ 26.9932 1.11413 0.557065 0.830469i $$-0.311927\pi$$
0.557065 + 0.830469i $$0.311927\pi$$
$$588$$ 7.37137 0.303990
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −6.36978 −0.262018
$$592$$ 22.2461 0.914308
$$593$$ 21.2339 0.871974 0.435987 0.899953i $$-0.356399\pi$$
0.435987 + 0.899953i $$0.356399\pi$$
$$594$$ 8.17847 0.335567
$$595$$ 0 0
$$596$$ −24.2650 −0.993934
$$597$$ −7.42562 −0.303910
$$598$$ 8.42738 0.344621
$$599$$ −30.3294 −1.23922 −0.619612 0.784908i $$-0.712710\pi$$
−0.619612 + 0.784908i $$0.712710\pi$$
$$600$$ 0 0
$$601$$ −10.7285 −0.437624 −0.218812 0.975767i $$-0.570218\pi$$
−0.218812 + 0.975767i $$0.570218\pi$$
$$602$$ 6.43991 0.262471
$$603$$ −21.7177 −0.884415
$$604$$ 37.3717 1.52063
$$605$$ 0 0
$$606$$ −1.98008 −0.0804352
$$607$$ −8.81498 −0.357789 −0.178894 0.983868i $$-0.557252\pi$$
−0.178894 + 0.983868i $$0.557252\pi$$
$$608$$ 0 0
$$609$$ 13.6010 0.551142
$$610$$ 0 0
$$611$$ −13.5788 −0.549341
$$612$$ 3.72543 0.150591
$$613$$ −24.1307 −0.974631 −0.487315 0.873226i $$-0.662024\pi$$
−0.487315 + 0.873226i $$0.662024\pi$$
$$614$$ 4.47383 0.180549
$$615$$ 0 0
$$616$$ −20.5861 −0.829436
$$617$$ 15.4650 0.622595 0.311298 0.950312i $$-0.399236\pi$$
0.311298 + 0.950312i $$0.399236\pi$$
$$618$$ −5.87008 −0.236129
$$619$$ 0.0390990 0.00157152 0.000785761 1.00000i $$-0.499750\pi$$
0.000785761 1.00000i $$0.499750\pi$$
$$620$$ 0 0
$$621$$ 38.9828 1.56433
$$622$$ −2.65247 −0.106354
$$623$$ −23.3847 −0.936889
$$624$$ 12.4808 0.499633
$$625$$ 0 0
$$626$$ −6.64584 −0.265621
$$627$$ 0 0
$$628$$ 5.87120 0.234286
$$629$$ 6.54853 0.261107
$$630$$ 0 0
$$631$$ 6.73516 0.268122 0.134061 0.990973i $$-0.457198\pi$$
0.134061 + 0.990973i $$0.457198\pi$$
$$632$$ 3.50730 0.139513
$$633$$ 20.2328 0.804182
$$634$$ 8.73989 0.347105
$$635$$ 0 0
$$636$$ 13.9530 0.553274
$$637$$ −13.2937 −0.526717
$$638$$ −6.47301 −0.256269
$$639$$ 0.883229 0.0349400
$$640$$ 0 0
$$641$$ 27.5788 1.08930 0.544649 0.838664i $$-0.316663\pi$$
0.544649 + 0.838664i $$0.316663\pi$$
$$642$$ −2.79843 −0.110445
$$643$$ −11.9081 −0.469611 −0.234806 0.972042i $$-0.575445\pi$$
−0.234806 + 0.972042i $$0.575445\pi$$
$$644$$ −47.8453 −1.88537
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 11.6648 0.458590 0.229295 0.973357i $$-0.426358\pi$$
0.229295 + 0.973357i $$0.426358\pi$$
$$648$$ 0.704005 0.0276559
$$649$$ −69.6273 −2.73311
$$650$$ 0 0
$$651$$ 15.5426 0.609161
$$652$$ −42.7692 −1.67497
$$653$$ 40.7527 1.59477 0.797387 0.603468i $$-0.206215\pi$$
0.797387 + 0.603468i $$0.206215\pi$$
$$654$$ −1.95335 −0.0763819
$$655$$ 0 0
$$656$$ −23.3337 −0.911029
$$657$$ 8.01730 0.312785
$$658$$ −3.92053 −0.152838
$$659$$ −23.1446 −0.901585 −0.450793 0.892629i $$-0.648859\pi$$
−0.450793 + 0.892629i $$0.648859\pi$$
$$660$$ 0 0
$$661$$ 4.88970 0.190187 0.0950936 0.995468i $$-0.469685\pi$$
0.0950936 + 0.995468i $$0.469685\pi$$
$$662$$ −4.72029 −0.183459
$$663$$ 3.67395 0.142685
$$664$$ 6.67950 0.259215
$$665$$ 0 0
$$666$$ 3.91488 0.151699
$$667$$ −30.8537 −1.19466
$$668$$ 20.0594 0.776120
$$669$$ −11.6719 −0.451263
$$670$$ 0 0
$$671$$ −31.7155 −1.22436
$$672$$ 11.8081 0.455506
$$673$$ −29.5149 −1.13771 −0.568857 0.822436i $$-0.692614\pi$$
−0.568857 + 0.822436i $$0.692614\pi$$
$$674$$ −5.44464 −0.209720
$$675$$ 0 0
$$676$$ 0.962701 0.0370270
$$677$$ 29.4248 1.13089 0.565443 0.824788i $$-0.308705\pi$$
0.565443 + 0.824788i $$0.308705\pi$$
$$678$$ −3.31421 −0.127282
$$679$$ 35.5103 1.36276
$$680$$ 0 0
$$681$$ −11.4691 −0.439498
$$682$$ −7.39703 −0.283247
$$683$$ −29.9433 −1.14575 −0.572875 0.819643i $$-0.694172\pi$$
−0.572875 + 0.819643i $$0.694172\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 3.30566 0.126211
$$687$$ −3.61699 −0.137997
$$688$$ −21.6355 −0.824844
$$689$$ −25.1633 −0.958645
$$690$$ 0 0
$$691$$ −24.2225 −0.921467 −0.460733 0.887539i $$-0.652414\pi$$
−0.460733 + 0.887539i $$0.652414\pi$$
$$692$$ 43.6561 1.65956
$$693$$ 32.8789 1.24896
$$694$$ 5.40354 0.205116
$$695$$ 0 0
$$696$$ 5.03478 0.190843
$$697$$ −6.86870 −0.260171
$$698$$ 10.6574 0.403389
$$699$$ 6.37465 0.241111
$$700$$ 0 0
$$701$$ −14.8539 −0.561024 −0.280512 0.959851i $$-0.590504\pi$$
−0.280512 + 0.959851i $$0.590504\pi$$
$$702$$ 5.59384 0.211126
$$703$$ 0 0
$$704$$ 29.8182 1.12381
$$705$$ 0 0
$$706$$ 10.8666 0.408970
$$707$$ −20.2735 −0.762463
$$708$$ 26.4070 0.992434
$$709$$ −29.8009 −1.11920 −0.559598 0.828764i $$-0.689044\pi$$
−0.559598 + 0.828764i $$0.689044\pi$$
$$710$$ 0 0
$$711$$ −5.60165 −0.210078
$$712$$ −8.65647 −0.324415
$$713$$ −35.2581 −1.32043
$$714$$ 1.06076 0.0396978
$$715$$ 0 0
$$716$$ −44.2888 −1.65515
$$717$$ −12.4838 −0.466216
$$718$$ 0.303811 0.0113381
$$719$$ −25.8925 −0.965628 −0.482814 0.875723i $$-0.660385\pi$$
−0.482814 + 0.875723i $$0.660385\pi$$
$$720$$ 0 0
$$721$$ −60.1021 −2.23832
$$722$$ 0 0
$$723$$ 5.25426 0.195408
$$724$$ 30.3884 1.12938
$$725$$ 0 0
$$726$$ 5.03248 0.186773
$$727$$ −9.34885 −0.346729 −0.173365 0.984858i $$-0.555464\pi$$
−0.173365 + 0.984858i $$0.555464\pi$$
$$728$$ −14.0803 −0.521850
$$729$$ 14.5913 0.540419
$$730$$ 0 0
$$731$$ −6.36879 −0.235558
$$732$$ 12.0285 0.444586
$$733$$ −7.20434 −0.266098 −0.133049 0.991109i $$-0.542477\pi$$
−0.133049 + 0.991109i $$0.542477\pi$$
$$734$$ 3.06950 0.113297
$$735$$ 0 0
$$736$$ −26.7864 −0.987360
$$737$$ −57.8701 −2.13167
$$738$$ −4.10629 −0.151155
$$739$$ 12.9740 0.477258 0.238629 0.971111i $$-0.423302\pi$$
0.238629 + 0.971111i $$0.423302\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −7.26523 −0.266715
$$743$$ −9.23100 −0.338652 −0.169326 0.985560i $$-0.554159\pi$$
−0.169326 + 0.985560i $$0.554159\pi$$
$$744$$ 5.75349 0.210933
$$745$$ 0 0
$$746$$ −4.02950 −0.147530
$$747$$ −10.6681 −0.390326
$$748$$ 9.92694 0.362965
$$749$$ −28.6523 −1.04693
$$750$$ 0 0
$$751$$ 14.9226 0.544534 0.272267 0.962222i $$-0.412227\pi$$
0.272267 + 0.962222i $$0.412227\pi$$
$$752$$ 13.1714 0.480310
$$753$$ −26.1829 −0.954157
$$754$$ −4.42736 −0.161235
$$755$$ 0 0
$$756$$ −31.7583 −1.15504
$$757$$ −24.4976 −0.890382 −0.445191 0.895436i $$-0.646864\pi$$
−0.445191 + 0.895436i $$0.646864\pi$$
$$758$$ −1.62273 −0.0589402
$$759$$ 40.7860 1.48044
$$760$$ 0 0
$$761$$ −15.8968 −0.576260 −0.288130 0.957591i $$-0.593034\pi$$
−0.288130 + 0.957591i $$0.593034\pi$$
$$762$$ −1.07879 −0.0390805
$$763$$ −19.9998 −0.724041
$$764$$ 4.65525 0.168421
$$765$$ 0 0
$$766$$ 10.4319 0.376919
$$767$$ −47.6231 −1.71957
$$768$$ −9.06916 −0.327255
$$769$$ −29.5850 −1.06686 −0.533432 0.845843i $$-0.679098\pi$$
−0.533432 + 0.845843i $$0.679098\pi$$
$$770$$ 0 0
$$771$$ 19.0003 0.684280
$$772$$ −36.2478 −1.30459
$$773$$ −14.1310 −0.508257 −0.254129 0.967170i $$-0.581789\pi$$
−0.254129 + 0.967170i $$0.581789\pi$$
$$774$$ −3.80743 −0.136855
$$775$$ 0 0
$$776$$ 13.1451 0.471880
$$777$$ −21.9191 −0.786345
$$778$$ −8.63032 −0.309412
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 2.35349 0.0842145
$$782$$ −2.40631 −0.0860494
$$783$$ −20.4798 −0.731887
$$784$$ 12.8948 0.460529
$$785$$ 0 0
$$786$$ 0.775912 0.0276759
$$787$$ −24.2465 −0.864293 −0.432146 0.901803i $$-0.642244\pi$$
−0.432146 + 0.901803i $$0.642244\pi$$
$$788$$ −11.7718 −0.419354
$$789$$ 13.3370 0.474809
$$790$$ 0 0
$$791$$ −33.9333 −1.20653
$$792$$ 12.1710 0.432477
$$793$$ −21.6925 −0.770324
$$794$$ −1.71091 −0.0607180
$$795$$ 0 0
$$796$$ −13.7231 −0.486402
$$797$$ 27.9258 0.989184 0.494592 0.869125i $$-0.335318\pi$$
0.494592 + 0.869125i $$0.335318\pi$$
$$798$$ 0 0
$$799$$ 3.87723 0.137166
$$800$$ 0 0
$$801$$ 13.8256 0.488504
$$802$$ 2.64849 0.0935214
$$803$$ 21.3633 0.753893
$$804$$ 21.9479 0.774043
$$805$$ 0 0
$$806$$ −5.05936 −0.178208
$$807$$ −0.161163 −0.00567321
$$808$$ −7.50476 −0.264017
$$809$$ 8.78914 0.309010 0.154505 0.987992i $$-0.450622\pi$$
0.154505 + 0.987992i $$0.450622\pi$$
$$810$$ 0 0
$$811$$ −14.0993 −0.495092 −0.247546 0.968876i $$-0.579624\pi$$
−0.247546 + 0.968876i $$0.579624\pi$$
$$812$$ 25.1357 0.882091
$$813$$ −28.6490 −1.00477
$$814$$ 10.4318 0.365633
$$815$$ 0 0
$$816$$ −3.56370 −0.124755
$$817$$ 0 0
$$818$$ 6.67967 0.233549
$$819$$ 22.4882 0.785801
$$820$$ 0 0
$$821$$ −30.5326 −1.06559 −0.532797 0.846243i $$-0.678859\pi$$
−0.532797 + 0.846243i $$0.678859\pi$$
$$822$$ 3.69060 0.128725
$$823$$ 14.3583 0.500498 0.250249 0.968182i $$-0.419488\pi$$
0.250249 + 0.968182i $$0.419488\pi$$
$$824$$ −22.2484 −0.775059
$$825$$ 0 0
$$826$$ −13.7499 −0.478420
$$827$$ 8.27364 0.287703 0.143851 0.989599i $$-0.454051\pi$$
0.143851 + 0.989599i $$0.454051\pi$$
$$828$$ 28.2873 0.983052
$$829$$ 13.1498 0.456710 0.228355 0.973578i $$-0.426665\pi$$
0.228355 + 0.973578i $$0.426665\pi$$
$$830$$ 0 0
$$831$$ −0.539697 −0.0187219
$$832$$ 20.3948 0.707062
$$833$$ 3.79582 0.131517
$$834$$ 2.26693 0.0784972
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −23.4032 −0.808934
$$838$$ 1.58472 0.0547431
$$839$$ −8.41534 −0.290530 −0.145265 0.989393i $$-0.546403\pi$$
−0.145265 + 0.989393i $$0.546403\pi$$
$$840$$ 0 0
$$841$$ −12.7909 −0.441064
$$842$$ −1.76032 −0.0606646
$$843$$ 5.03410 0.173384
$$844$$ 37.3918 1.28708
$$845$$ 0 0
$$846$$ 2.31791 0.0796914
$$847$$ 51.5262 1.77046
$$848$$ 24.4082 0.838181
$$849$$ −18.4120 −0.631899
$$850$$ 0 0
$$851$$ 49.7232 1.70449
$$852$$ −0.892589 −0.0305796
$$853$$ −52.0363 −1.78169 −0.890845 0.454308i $$-0.849887\pi$$
−0.890845 + 0.454308i $$0.849887\pi$$
$$854$$ −6.26314 −0.214320
$$855$$ 0 0
$$856$$ −10.6064 −0.362520
$$857$$ 21.4595 0.733043 0.366522 0.930410i $$-0.380549\pi$$
0.366522 + 0.930410i $$0.380549\pi$$
$$858$$ 5.85259 0.199804
$$859$$ 18.1479 0.619199 0.309599 0.950867i $$-0.399805\pi$$
0.309599 + 0.950867i $$0.399805\pi$$
$$860$$ 0 0
$$861$$ 22.9908 0.783525
$$862$$ 11.9362 0.406549
$$863$$ 13.7867 0.469303 0.234652 0.972080i $$-0.424605\pi$$
0.234652 + 0.972080i $$0.424605\pi$$
$$864$$ −17.7800 −0.604888
$$865$$ 0 0
$$866$$ 2.16609 0.0736066
$$867$$ 16.4581 0.558948
$$868$$ 28.7238 0.974951
$$869$$ −14.9264 −0.506343
$$870$$ 0 0
$$871$$ −39.5815 −1.34117
$$872$$ −7.40345 −0.250712
$$873$$ −20.9945 −0.710557
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −8.10226 −0.273750
$$877$$ 21.1937 0.715660 0.357830 0.933787i $$-0.383517\pi$$
0.357830 + 0.933787i $$0.383517\pi$$
$$878$$ −2.96390 −0.100027
$$879$$ 7.47041 0.251971
$$880$$ 0 0
$$881$$ 44.5944 1.50242 0.751212 0.660061i $$-0.229469\pi$$
0.751212 + 0.660061i $$0.229469\pi$$
$$882$$ 2.26924 0.0764093
$$883$$ −1.69355 −0.0569924 −0.0284962 0.999594i $$-0.509072\pi$$
−0.0284962 + 0.999594i $$0.509072\pi$$
$$884$$ 6.78974 0.228364
$$885$$ 0 0
$$886$$ 10.7966 0.362720
$$887$$ −48.4354 −1.62630 −0.813151 0.582053i $$-0.802250\pi$$
−0.813151 + 0.582053i $$0.802250\pi$$
$$888$$ −8.11394 −0.272286
$$889$$ −11.0455 −0.370453
$$890$$ 0 0
$$891$$ −2.99611 −0.100374
$$892$$ −21.5706 −0.722238
$$893$$ 0 0
$$894$$ 4.08481 0.136617
$$895$$ 0 0
$$896$$ 28.8204 0.962822
$$897$$ 27.8965 0.931436
$$898$$ 2.75090 0.0917988
$$899$$ 18.5230 0.617776
$$900$$ 0 0
$$901$$ 7.18499 0.239367
$$902$$ −10.9418 −0.364322
$$903$$ 21.3175 0.709402
$$904$$ −12.5613 −0.417783
$$905$$ 0 0
$$906$$ −6.29122 −0.209012
$$907$$ −27.9388 −0.927692 −0.463846 0.885916i $$-0.653531\pi$$
−0.463846 + 0.885916i $$0.653531\pi$$
$$908$$ −21.1958 −0.703408
$$909$$ 11.9862 0.397556
$$910$$ 0 0
$$911$$ 45.0862 1.49377 0.746887 0.664951i $$-0.231548\pi$$
0.746887 + 0.664951i $$0.231548\pi$$
$$912$$ 0 0
$$913$$ −28.4267 −0.940787
$$914$$ 0.206886 0.00684318
$$915$$ 0 0
$$916$$ −6.68448 −0.220861
$$917$$ 7.94436 0.262346
$$918$$ −1.59723 −0.0527166
$$919$$ −5.86849 −0.193584 −0.0967918 0.995305i $$-0.530858\pi$$
−0.0967918 + 0.995305i $$0.530858\pi$$
$$920$$ 0 0
$$921$$ 14.8093 0.487984
$$922$$ 12.3010 0.405111
$$923$$ 1.60972 0.0529846
$$924$$ −33.2273 −1.09310
$$925$$ 0 0
$$926$$ −4.84651 −0.159266
$$927$$ 35.5338 1.16708
$$928$$ 14.0723 0.461947
$$929$$ 46.6314 1.52993 0.764963 0.644075i $$-0.222757\pi$$
0.764963 + 0.644075i $$0.222757\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 11.7808 0.385894
$$933$$ −8.78026 −0.287453
$$934$$ 12.7666 0.417735
$$935$$ 0 0
$$936$$ 8.32460 0.272098
$$937$$ 51.2940 1.67570 0.837850 0.545900i $$-0.183812\pi$$
0.837850 + 0.545900i $$0.183812\pi$$
$$938$$ −11.4281 −0.373140
$$939$$ −21.9992 −0.717916
$$940$$ 0 0
$$941$$ −11.3486 −0.369954 −0.184977 0.982743i $$-0.559221\pi$$
−0.184977 + 0.982743i $$0.559221\pi$$
$$942$$ −0.988367 −0.0322027
$$943$$ −52.1543 −1.69838
$$944$$ 46.1940 1.50349
$$945$$ 0 0
$$946$$ −10.1454 −0.329857
$$947$$ 7.20935 0.234272 0.117136 0.993116i $$-0.462629\pi$$
0.117136 + 0.993116i $$0.462629\pi$$
$$948$$ 5.66101 0.183861
$$949$$ 14.6119 0.474321
$$950$$ 0 0
$$951$$ 28.9309 0.938150
$$952$$ 4.02041 0.130302
$$953$$ −20.5394 −0.665338 −0.332669 0.943044i $$-0.607949\pi$$
−0.332669 + 0.943044i $$0.607949\pi$$
$$954$$ 4.29538 0.139068
$$955$$ 0 0
$$956$$ −23.0710 −0.746170
$$957$$ −21.4271 −0.692639
$$958$$ 7.32930 0.236799
$$959$$ 37.7871 1.22021
$$960$$ 0 0
$$961$$ −9.83289 −0.317190
$$962$$ 7.13504 0.230043
$$963$$ 16.9400 0.545882
$$964$$ 9.71026 0.312746
$$965$$ 0 0
$$966$$ 8.05436 0.259145
$$967$$ 17.0669 0.548835 0.274417 0.961611i $$-0.411515\pi$$
0.274417 + 0.961611i $$0.411515\pi$$
$$968$$ 19.0738 0.613055
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −3.48171 −0.111733 −0.0558667 0.998438i $$-0.517792\pi$$
−0.0558667 + 0.998438i $$0.517792\pi$$
$$972$$ 30.1801 0.968028
$$973$$ 23.2104 0.744093
$$974$$ −7.44065 −0.238414
$$975$$ 0 0
$$976$$ 21.0416 0.673524
$$977$$ −21.9600 −0.702562 −0.351281 0.936270i $$-0.614254\pi$$
−0.351281 + 0.936270i $$0.614254\pi$$
$$978$$ 7.19984 0.230226
$$979$$ 36.8403 1.17742
$$980$$ 0 0
$$981$$ 11.8244 0.377523
$$982$$ −5.08583 −0.162295
$$983$$ 5.49704 0.175328 0.0876641 0.996150i $$-0.472060\pi$$
0.0876641 + 0.996150i $$0.472060\pi$$
$$984$$ 8.51065 0.271310
$$985$$ 0 0
$$986$$ 1.26416 0.0402592
$$987$$ −12.9778 −0.413088
$$988$$ 0 0
$$989$$ −48.3584 −1.53771
$$990$$ 0 0
$$991$$ −29.4970 −0.937003 −0.468502 0.883463i $$-0.655206\pi$$
−0.468502 + 0.883463i $$0.655206\pi$$
$$992$$ 16.0812 0.510577
$$993$$ −15.6252 −0.495850
$$994$$ 0.464764 0.0147414
$$995$$ 0 0
$$996$$ 10.7812 0.341614
$$997$$ 7.26449 0.230069 0.115034 0.993362i $$-0.463302\pi$$
0.115034 + 0.993362i $$0.463302\pi$$
$$998$$ 9.97749 0.315832
$$999$$ 33.0048 1.04422
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 9025.2.a.br.1.3 6
5.4 even 2 9025.2.a.bz.1.4 6
19.7 even 3 475.2.e.h.201.4 yes 12
19.11 even 3 475.2.e.h.26.4 yes 12
19.18 odd 2 9025.2.a.by.1.4 6
95.7 odd 12 475.2.j.d.49.7 24
95.49 even 6 475.2.e.f.26.3 12
95.64 even 6 475.2.e.f.201.3 yes 12
95.68 odd 12 475.2.j.d.349.7 24
95.83 odd 12 475.2.j.d.49.6 24
95.87 odd 12 475.2.j.d.349.6 24
95.94 odd 2 9025.2.a.bs.1.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.e.f.26.3 12 95.49 even 6
475.2.e.f.201.3 yes 12 95.64 even 6
475.2.e.h.26.4 yes 12 19.11 even 3
475.2.e.h.201.4 yes 12 19.7 even 3
475.2.j.d.49.6 24 95.83 odd 12
475.2.j.d.49.7 24 95.7 odd 12
475.2.j.d.349.6 24 95.87 odd 12
475.2.j.d.349.7 24 95.68 odd 12
9025.2.a.br.1.3 6 1.1 even 1 trivial
9025.2.a.bs.1.3 6 95.94 odd 2
9025.2.a.by.1.4 6 19.18 odd 2
9025.2.a.bz.1.4 6 5.4 even 2