# Properties

 Label 9025.2.a.m.1.2 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 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 $$-0.618034$$ 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)

 $$f(q)$$ $$=$$ $$q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +0.854102 q^{6} -0.236068 q^{7} +1.47214 q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-0.381966 q^{2} -2.23607 q^{3} -1.85410 q^{4} +0.854102 q^{6} -0.236068 q^{7} +1.47214 q^{8} +2.00000 q^{9} -3.47214 q^{11} +4.14590 q^{12} +2.00000 q^{13} +0.0901699 q^{14} +3.14590 q^{16} -1.85410 q^{17} -0.763932 q^{18} +0.527864 q^{21} +1.32624 q^{22} -3.23607 q^{23} -3.29180 q^{24} -0.763932 q^{26} +2.23607 q^{27} +0.437694 q^{28} +6.00000 q^{29} +4.85410 q^{31} -4.14590 q^{32} +7.76393 q^{33} +0.708204 q^{34} -3.70820 q^{36} -10.8541 q^{37} -4.47214 q^{39} -8.61803 q^{41} -0.201626 q^{42} +9.56231 q^{43} +6.43769 q^{44} +1.23607 q^{46} +5.38197 q^{47} -7.03444 q^{48} -6.94427 q^{49} +4.14590 q^{51} -3.70820 q^{52} +8.61803 q^{53} -0.854102 q^{54} -0.347524 q^{56} -2.29180 q^{58} -7.23607 q^{59} +14.5623 q^{61} -1.85410 q^{62} -0.472136 q^{63} -4.70820 q^{64} -2.96556 q^{66} -4.70820 q^{67} +3.43769 q^{68} +7.23607 q^{69} -13.4721 q^{71} +2.94427 q^{72} -2.70820 q^{73} +4.14590 q^{74} +0.819660 q^{77} +1.70820 q^{78} +1.00000 q^{79} -11.0000 q^{81} +3.29180 q^{82} +10.0902 q^{83} -0.978714 q^{84} -3.65248 q^{86} -13.4164 q^{87} -5.11146 q^{88} -9.70820 q^{89} -0.472136 q^{91} +6.00000 q^{92} -10.8541 q^{93} -2.05573 q^{94} +9.27051 q^{96} -9.61803 q^{97} +2.65248 q^{98} -6.94427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 $$2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} - 11 q^{14} + 13 q^{16} + 3 q^{17} - 6 q^{18} + 10 q^{21} - 13 q^{22} - 2 q^{23} - 20 q^{24} - 6 q^{26} + 21 q^{28} + 12 q^{29} + 3 q^{31} - 15 q^{32} + 20 q^{33} - 12 q^{34} + 6 q^{36} - 15 q^{37} - 15 q^{41} - 25 q^{42} - q^{43} + 33 q^{44} - 2 q^{46} + 13 q^{47} + 15 q^{48} + 4 q^{49} + 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} - 32 q^{56} - 18 q^{58} - 10 q^{59} + 9 q^{61} + 3 q^{62} + 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} + 27 q^{68} + 10 q^{69} - 18 q^{71} - 12 q^{72} + 8 q^{73} + 15 q^{74} + 24 q^{77} - 10 q^{78} + 2 q^{79} - 22 q^{81} + 20 q^{82} + 9 q^{83} + 45 q^{84} + 24 q^{86} - 46 q^{88} - 6 q^{89} + 8 q^{91} + 12 q^{92} - 15 q^{93} - 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 + 2 * q^11 + 15 * q^12 + 4 * q^13 - 11 * q^14 + 13 * q^16 + 3 * q^17 - 6 * q^18 + 10 * q^21 - 13 * q^22 - 2 * q^23 - 20 * q^24 - 6 * q^26 + 21 * q^28 + 12 * q^29 + 3 * q^31 - 15 * q^32 + 20 * q^33 - 12 * q^34 + 6 * q^36 - 15 * q^37 - 15 * q^41 - 25 * q^42 - q^43 + 33 * q^44 - 2 * q^46 + 13 * q^47 + 15 * q^48 + 4 * q^49 + 15 * q^51 + 6 * q^52 + 15 * q^53 + 5 * q^54 - 32 * q^56 - 18 * q^58 - 10 * q^59 + 9 * q^61 + 3 * q^62 + 8 * q^63 + 4 * q^64 - 35 * q^66 + 4 * q^67 + 27 * q^68 + 10 * q^69 - 18 * q^71 - 12 * q^72 + 8 * q^73 + 15 * q^74 + 24 * q^77 - 10 * q^78 + 2 * q^79 - 22 * q^81 + 20 * q^82 + 9 * q^83 + 45 * q^84 + 24 * q^86 - 46 * q^88 - 6 * q^89 + 8 * q^91 + 12 * q^92 - 15 * q^93 - 22 * q^94 - 15 * q^96 - 17 * q^97 - 26 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.381966 −0.270091 −0.135045 0.990839i $$-0.543118\pi$$
−0.135045 + 0.990839i $$0.543118\pi$$
$$3$$ −2.23607 −1.29099 −0.645497 0.763763i $$-0.723350\pi$$
−0.645497 + 0.763763i $$0.723350\pi$$
$$4$$ −1.85410 −0.927051
$$5$$ 0 0
$$6$$ 0.854102 0.348686
$$7$$ −0.236068 −0.0892253 −0.0446127 0.999004i $$-0.514205\pi$$
−0.0446127 + 0.999004i $$0.514205\pi$$
$$8$$ 1.47214 0.520479
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −3.47214 −1.04689 −0.523444 0.852060i $$-0.675353\pi$$
−0.523444 + 0.852060i $$0.675353\pi$$
$$12$$ 4.14590 1.19682
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0.0901699 0.0240989
$$15$$ 0 0
$$16$$ 3.14590 0.786475
$$17$$ −1.85410 −0.449686 −0.224843 0.974395i $$-0.572187\pi$$
−0.224843 + 0.974395i $$0.572187\pi$$
$$18$$ −0.763932 −0.180061
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 0.527864 0.115189
$$22$$ 1.32624 0.282755
$$23$$ −3.23607 −0.674767 −0.337383 0.941367i $$-0.609542\pi$$
−0.337383 + 0.941367i $$0.609542\pi$$
$$24$$ −3.29180 −0.671935
$$25$$ 0 0
$$26$$ −0.763932 −0.149819
$$27$$ 2.23607 0.430331
$$28$$ 0.437694 0.0827164
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.85410 0.871822 0.435911 0.899990i $$-0.356426\pi$$
0.435911 + 0.899990i $$0.356426\pi$$
$$32$$ −4.14590 −0.732898
$$33$$ 7.76393 1.35153
$$34$$ 0.708204 0.121456
$$35$$ 0 0
$$36$$ −3.70820 −0.618034
$$37$$ −10.8541 −1.78440 −0.892202 0.451637i $$-0.850840\pi$$
−0.892202 + 0.451637i $$0.850840\pi$$
$$38$$ 0 0
$$39$$ −4.47214 −0.716115
$$40$$ 0 0
$$41$$ −8.61803 −1.34591 −0.672955 0.739683i $$-0.734975\pi$$
−0.672955 + 0.739683i $$0.734975\pi$$
$$42$$ −0.201626 −0.0311116
$$43$$ 9.56231 1.45824 0.729119 0.684387i $$-0.239930\pi$$
0.729119 + 0.684387i $$0.239930\pi$$
$$44$$ 6.43769 0.970519
$$45$$ 0 0
$$46$$ 1.23607 0.182248
$$47$$ 5.38197 0.785040 0.392520 0.919743i $$-0.371603\pi$$
0.392520 + 0.919743i $$0.371603\pi$$
$$48$$ −7.03444 −1.01533
$$49$$ −6.94427 −0.992039
$$50$$ 0 0
$$51$$ 4.14590 0.580542
$$52$$ −3.70820 −0.514235
$$53$$ 8.61803 1.18378 0.591889 0.806019i $$-0.298382\pi$$
0.591889 + 0.806019i $$0.298382\pi$$
$$54$$ −0.854102 −0.116229
$$55$$ 0 0
$$56$$ −0.347524 −0.0464399
$$57$$ 0 0
$$58$$ −2.29180 −0.300928
$$59$$ −7.23607 −0.942056 −0.471028 0.882118i $$-0.656117\pi$$
−0.471028 + 0.882118i $$0.656117\pi$$
$$60$$ 0 0
$$61$$ 14.5623 1.86451 0.932256 0.361799i $$-0.117837\pi$$
0.932256 + 0.361799i $$0.117837\pi$$
$$62$$ −1.85410 −0.235471
$$63$$ −0.472136 −0.0594835
$$64$$ −4.70820 −0.588525
$$65$$ 0 0
$$66$$ −2.96556 −0.365035
$$67$$ −4.70820 −0.575199 −0.287599 0.957751i $$-0.592857\pi$$
−0.287599 + 0.957751i $$0.592857\pi$$
$$68$$ 3.43769 0.416882
$$69$$ 7.23607 0.871120
$$70$$ 0 0
$$71$$ −13.4721 −1.59885 −0.799424 0.600767i $$-0.794862\pi$$
−0.799424 + 0.600767i $$0.794862\pi$$
$$72$$ 2.94427 0.346986
$$73$$ −2.70820 −0.316971 −0.158486 0.987361i $$-0.550661\pi$$
−0.158486 + 0.987361i $$0.550661\pi$$
$$74$$ 4.14590 0.481951
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0.819660 0.0934089
$$78$$ 1.70820 0.193416
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 3.29180 0.363518
$$83$$ 10.0902 1.10754 0.553770 0.832670i $$-0.313189\pi$$
0.553770 + 0.832670i $$0.313189\pi$$
$$84$$ −0.978714 −0.106786
$$85$$ 0 0
$$86$$ −3.65248 −0.393857
$$87$$ −13.4164 −1.43839
$$88$$ −5.11146 −0.544883
$$89$$ −9.70820 −1.02907 −0.514534 0.857470i $$-0.672035\pi$$
−0.514534 + 0.857470i $$0.672035\pi$$
$$90$$ 0 0
$$91$$ −0.472136 −0.0494933
$$92$$ 6.00000 0.625543
$$93$$ −10.8541 −1.12552
$$94$$ −2.05573 −0.212032
$$95$$ 0 0
$$96$$ 9.27051 0.946167
$$97$$ −9.61803 −0.976563 −0.488282 0.872686i $$-0.662376\pi$$
−0.488282 + 0.872686i $$0.662376\pi$$
$$98$$ 2.65248 0.267941
$$99$$ −6.94427 −0.697926
$$100$$ 0 0
$$101$$ 16.0344 1.59549 0.797743 0.602997i $$-0.206027\pi$$
0.797743 + 0.602997i $$0.206027\pi$$
$$102$$ −1.58359 −0.156799
$$103$$ −3.23607 −0.318859 −0.159430 0.987209i $$-0.550966\pi$$
−0.159430 + 0.987209i $$0.550966\pi$$
$$104$$ 2.94427 0.288710
$$105$$ 0 0
$$106$$ −3.29180 −0.319727
$$107$$ 13.1459 1.27086 0.635431 0.772158i $$-0.280822\pi$$
0.635431 + 0.772158i $$0.280822\pi$$
$$108$$ −4.14590 −0.398939
$$109$$ −9.94427 −0.952489 −0.476244 0.879313i $$-0.658002\pi$$
−0.476244 + 0.879313i $$0.658002\pi$$
$$110$$ 0 0
$$111$$ 24.2705 2.30365
$$112$$ −0.742646 −0.0701734
$$113$$ −6.32624 −0.595122 −0.297561 0.954703i $$-0.596173\pi$$
−0.297561 + 0.954703i $$0.596173\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −11.1246 −1.03289
$$117$$ 4.00000 0.369800
$$118$$ 2.76393 0.254441
$$119$$ 0.437694 0.0401234
$$120$$ 0 0
$$121$$ 1.05573 0.0959753
$$122$$ −5.56231 −0.503588
$$123$$ 19.2705 1.73756
$$124$$ −9.00000 −0.808224
$$125$$ 0 0
$$126$$ 0.180340 0.0160660
$$127$$ 5.85410 0.519468 0.259734 0.965680i $$-0.416365\pi$$
0.259734 + 0.965680i $$0.416365\pi$$
$$128$$ 10.0902 0.891853
$$129$$ −21.3820 −1.88258
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ −14.3951 −1.25293
$$133$$ 0 0
$$134$$ 1.79837 0.155356
$$135$$ 0 0
$$136$$ −2.72949 −0.234052
$$137$$ 4.76393 0.407010 0.203505 0.979074i $$-0.434767\pi$$
0.203505 + 0.979074i $$0.434767\pi$$
$$138$$ −2.76393 −0.235282
$$139$$ −14.0000 −1.18746 −0.593732 0.804663i $$-0.702346\pi$$
−0.593732 + 0.804663i $$0.702346\pi$$
$$140$$ 0 0
$$141$$ −12.0344 −1.01348
$$142$$ 5.14590 0.431834
$$143$$ −6.94427 −0.580709
$$144$$ 6.29180 0.524316
$$145$$ 0 0
$$146$$ 1.03444 0.0856110
$$147$$ 15.5279 1.28072
$$148$$ 20.1246 1.65423
$$149$$ −0.527864 −0.0432443 −0.0216222 0.999766i $$-0.506883\pi$$
−0.0216222 + 0.999766i $$0.506883\pi$$
$$150$$ 0 0
$$151$$ −18.7984 −1.52979 −0.764895 0.644155i $$-0.777209\pi$$
−0.764895 + 0.644155i $$0.777209\pi$$
$$152$$ 0 0
$$153$$ −3.70820 −0.299791
$$154$$ −0.313082 −0.0252289
$$155$$ 0 0
$$156$$ 8.29180 0.663875
$$157$$ −24.1803 −1.92980 −0.964901 0.262615i $$-0.915415\pi$$
−0.964901 + 0.262615i $$0.915415\pi$$
$$158$$ −0.381966 −0.0303876
$$159$$ −19.2705 −1.52825
$$160$$ 0 0
$$161$$ 0.763932 0.0602063
$$162$$ 4.20163 0.330111
$$163$$ 7.23607 0.566773 0.283386 0.959006i $$-0.408542\pi$$
0.283386 + 0.959006i $$0.408542\pi$$
$$164$$ 15.9787 1.24773
$$165$$ 0 0
$$166$$ −3.85410 −0.299136
$$167$$ 0.763932 0.0591148 0.0295574 0.999563i $$-0.490590\pi$$
0.0295574 + 0.999563i $$0.490590\pi$$
$$168$$ 0.777088 0.0599536
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −17.7295 −1.35186
$$173$$ 11.6525 0.885921 0.442961 0.896541i $$-0.353928\pi$$
0.442961 + 0.896541i $$0.353928\pi$$
$$174$$ 5.12461 0.388496
$$175$$ 0 0
$$176$$ −10.9230 −0.823351
$$177$$ 16.1803 1.21619
$$178$$ 3.70820 0.277942
$$179$$ −11.1803 −0.835658 −0.417829 0.908526i $$-0.637209\pi$$
−0.417829 + 0.908526i $$0.637209\pi$$
$$180$$ 0 0
$$181$$ −5.29180 −0.393336 −0.196668 0.980470i $$-0.563012\pi$$
−0.196668 + 0.980470i $$0.563012\pi$$
$$182$$ 0.180340 0.0133677
$$183$$ −32.5623 −2.40707
$$184$$ −4.76393 −0.351202
$$185$$ 0 0
$$186$$ 4.14590 0.303992
$$187$$ 6.43769 0.470771
$$188$$ −9.97871 −0.727772
$$189$$ −0.527864 −0.0383965
$$190$$ 0 0
$$191$$ −0.0557281 −0.00403234 −0.00201617 0.999998i $$-0.500642\pi$$
−0.00201617 + 0.999998i $$0.500642\pi$$
$$192$$ 10.5279 0.759783
$$193$$ 10.2705 0.739287 0.369644 0.929174i $$-0.379480\pi$$
0.369644 + 0.929174i $$0.379480\pi$$
$$194$$ 3.67376 0.263761
$$195$$ 0 0
$$196$$ 12.8754 0.919671
$$197$$ −1.47214 −0.104885 −0.0524427 0.998624i $$-0.516701\pi$$
−0.0524427 + 0.998624i $$0.516701\pi$$
$$198$$ 2.65248 0.188503
$$199$$ −7.41641 −0.525735 −0.262868 0.964832i $$-0.584668\pi$$
−0.262868 + 0.964832i $$0.584668\pi$$
$$200$$ 0 0
$$201$$ 10.5279 0.742578
$$202$$ −6.12461 −0.430926
$$203$$ −1.41641 −0.0994123
$$204$$ −7.68692 −0.538192
$$205$$ 0 0
$$206$$ 1.23607 0.0861209
$$207$$ −6.47214 −0.449845
$$208$$ 6.29180 0.436258
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −7.76393 −0.534491 −0.267246 0.963628i $$-0.586113\pi$$
−0.267246 + 0.963628i $$0.586113\pi$$
$$212$$ −15.9787 −1.09742
$$213$$ 30.1246 2.06410
$$214$$ −5.02129 −0.343248
$$215$$ 0 0
$$216$$ 3.29180 0.223978
$$217$$ −1.14590 −0.0777886
$$218$$ 3.79837 0.257258
$$219$$ 6.05573 0.409208
$$220$$ 0 0
$$221$$ −3.70820 −0.249441
$$222$$ −9.27051 −0.622196
$$223$$ 16.8885 1.13094 0.565470 0.824769i $$-0.308695\pi$$
0.565470 + 0.824769i $$0.308695\pi$$
$$224$$ 0.978714 0.0653931
$$225$$ 0 0
$$226$$ 2.41641 0.160737
$$227$$ −5.52786 −0.366897 −0.183449 0.983029i $$-0.558726\pi$$
−0.183449 + 0.983029i $$0.558726\pi$$
$$228$$ 0 0
$$229$$ −10.6525 −0.703935 −0.351968 0.936012i $$-0.614487\pi$$
−0.351968 + 0.936012i $$0.614487\pi$$
$$230$$ 0 0
$$231$$ −1.83282 −0.120590
$$232$$ 8.83282 0.579903
$$233$$ 25.4721 1.66874 0.834368 0.551208i $$-0.185833\pi$$
0.834368 + 0.551208i $$0.185833\pi$$
$$234$$ −1.52786 −0.0998796
$$235$$ 0 0
$$236$$ 13.4164 0.873334
$$237$$ −2.23607 −0.145248
$$238$$ −0.167184 −0.0108369
$$239$$ −12.7082 −0.822025 −0.411013 0.911630i $$-0.634825\pi$$
−0.411013 + 0.911630i $$0.634825\pi$$
$$240$$ 0 0
$$241$$ −13.0000 −0.837404 −0.418702 0.908124i $$-0.637515\pi$$
−0.418702 + 0.908124i $$0.637515\pi$$
$$242$$ −0.403252 −0.0259220
$$243$$ 17.8885 1.14755
$$244$$ −27.0000 −1.72850
$$245$$ 0 0
$$246$$ −7.36068 −0.469300
$$247$$ 0 0
$$248$$ 7.14590 0.453765
$$249$$ −22.5623 −1.42983
$$250$$ 0 0
$$251$$ 17.0902 1.07872 0.539361 0.842075i $$-0.318666\pi$$
0.539361 + 0.842075i $$0.318666\pi$$
$$252$$ 0.875388 0.0551443
$$253$$ 11.2361 0.706406
$$254$$ −2.23607 −0.140303
$$255$$ 0 0
$$256$$ 5.56231 0.347644
$$257$$ −28.7984 −1.79639 −0.898197 0.439594i $$-0.855122\pi$$
−0.898197 + 0.439594i $$0.855122\pi$$
$$258$$ 8.16718 0.508467
$$259$$ 2.56231 0.159214
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 4.58359 0.283175
$$263$$ 20.3820 1.25681 0.628403 0.777888i $$-0.283709\pi$$
0.628403 + 0.777888i $$0.283709\pi$$
$$264$$ 11.4296 0.703441
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 21.7082 1.32852
$$268$$ 8.72949 0.533238
$$269$$ −13.1803 −0.803620 −0.401810 0.915723i $$-0.631619\pi$$
−0.401810 + 0.915723i $$0.631619\pi$$
$$270$$ 0 0
$$271$$ 4.32624 0.262800 0.131400 0.991329i $$-0.458053\pi$$
0.131400 + 0.991329i $$0.458053\pi$$
$$272$$ −5.83282 −0.353666
$$273$$ 1.05573 0.0638956
$$274$$ −1.81966 −0.109930
$$275$$ 0 0
$$276$$ −13.4164 −0.807573
$$277$$ −12.0000 −0.721010 −0.360505 0.932757i $$-0.617396\pi$$
−0.360505 + 0.932757i $$0.617396\pi$$
$$278$$ 5.34752 0.320723
$$279$$ 9.70820 0.581215
$$280$$ 0 0
$$281$$ 2.38197 0.142096 0.0710481 0.997473i $$-0.477366\pi$$
0.0710481 + 0.997473i $$0.477366\pi$$
$$282$$ 4.59675 0.273732
$$283$$ −28.2361 −1.67846 −0.839230 0.543777i $$-0.816994\pi$$
−0.839230 + 0.543777i $$0.816994\pi$$
$$284$$ 24.9787 1.48221
$$285$$ 0 0
$$286$$ 2.65248 0.156844
$$287$$ 2.03444 0.120089
$$288$$ −8.29180 −0.488599
$$289$$ −13.5623 −0.797783
$$290$$ 0 0
$$291$$ 21.5066 1.26074
$$292$$ 5.02129 0.293849
$$293$$ 4.52786 0.264521 0.132260 0.991215i $$-0.457777\pi$$
0.132260 + 0.991215i $$0.457777\pi$$
$$294$$ −5.93112 −0.345910
$$295$$ 0 0
$$296$$ −15.9787 −0.928744
$$297$$ −7.76393 −0.450509
$$298$$ 0.201626 0.0116799
$$299$$ −6.47214 −0.374293
$$300$$ 0 0
$$301$$ −2.25735 −0.130112
$$302$$ 7.18034 0.413182
$$303$$ −35.8541 −2.05976
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 1.41641 0.0809706
$$307$$ 12.4377 0.709857 0.354928 0.934894i $$-0.384505\pi$$
0.354928 + 0.934894i $$0.384505\pi$$
$$308$$ −1.51973 −0.0865948
$$309$$ 7.23607 0.411646
$$310$$ 0 0
$$311$$ 11.7639 0.667071 0.333536 0.942737i $$-0.391758\pi$$
0.333536 + 0.942737i $$0.391758\pi$$
$$312$$ −6.58359 −0.372723
$$313$$ 2.05573 0.116197 0.0580983 0.998311i $$-0.481496\pi$$
0.0580983 + 0.998311i $$0.481496\pi$$
$$314$$ 9.23607 0.521221
$$315$$ 0 0
$$316$$ −1.85410 −0.104301
$$317$$ 6.94427 0.390029 0.195015 0.980800i $$-0.437525\pi$$
0.195015 + 0.980800i $$0.437525\pi$$
$$318$$ 7.36068 0.412766
$$319$$ −20.8328 −1.16641
$$320$$ 0 0
$$321$$ −29.3951 −1.64068
$$322$$ −0.291796 −0.0162612
$$323$$ 0 0
$$324$$ 20.3951 1.13306
$$325$$ 0 0
$$326$$ −2.76393 −0.153080
$$327$$ 22.2361 1.22966
$$328$$ −12.6869 −0.700518
$$329$$ −1.27051 −0.0700455
$$330$$ 0 0
$$331$$ 6.56231 0.360697 0.180348 0.983603i $$-0.442277\pi$$
0.180348 + 0.983603i $$0.442277\pi$$
$$332$$ −18.7082 −1.02675
$$333$$ −21.7082 −1.18960
$$334$$ −0.291796 −0.0159664
$$335$$ 0 0
$$336$$ 1.66061 0.0905935
$$337$$ −8.70820 −0.474366 −0.237183 0.971465i $$-0.576224\pi$$
−0.237183 + 0.971465i $$0.576224\pi$$
$$338$$ 3.43769 0.186986
$$339$$ 14.1459 0.768300
$$340$$ 0 0
$$341$$ −16.8541 −0.912701
$$342$$ 0 0
$$343$$ 3.29180 0.177740
$$344$$ 14.0770 0.758982
$$345$$ 0 0
$$346$$ −4.45085 −0.239279
$$347$$ 24.5279 1.31672 0.658362 0.752701i $$-0.271249\pi$$
0.658362 + 0.752701i $$0.271249\pi$$
$$348$$ 24.8754 1.33346
$$349$$ 31.0689 1.66308 0.831540 0.555465i $$-0.187460\pi$$
0.831540 + 0.555465i $$0.187460\pi$$
$$350$$ 0 0
$$351$$ 4.47214 0.238705
$$352$$ 14.3951 0.767263
$$353$$ −25.4721 −1.35574 −0.677872 0.735179i $$-0.737098\pi$$
−0.677872 + 0.735179i $$0.737098\pi$$
$$354$$ −6.18034 −0.328481
$$355$$ 0 0
$$356$$ 18.0000 0.953998
$$357$$ −0.978714 −0.0517990
$$358$$ 4.27051 0.225703
$$359$$ 26.5967 1.40372 0.701861 0.712314i $$-0.252353\pi$$
0.701861 + 0.712314i $$0.252353\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 2.02129 0.106236
$$363$$ −2.36068 −0.123904
$$364$$ 0.875388 0.0458828
$$365$$ 0 0
$$366$$ 12.4377 0.650129
$$367$$ 3.52786 0.184153 0.0920765 0.995752i $$-0.470650\pi$$
0.0920765 + 0.995752i $$0.470650\pi$$
$$368$$ −10.1803 −0.530687
$$369$$ −17.2361 −0.897274
$$370$$ 0 0
$$371$$ −2.03444 −0.105623
$$372$$ 20.1246 1.04341
$$373$$ −19.8541 −1.02801 −0.514003 0.857788i $$-0.671838\pi$$
−0.514003 + 0.857788i $$0.671838\pi$$
$$374$$ −2.45898 −0.127151
$$375$$ 0 0
$$376$$ 7.92299 0.408597
$$377$$ 12.0000 0.618031
$$378$$ 0.201626 0.0103705
$$379$$ 30.5410 1.56879 0.784393 0.620264i $$-0.212974\pi$$
0.784393 + 0.620264i $$0.212974\pi$$
$$380$$ 0 0
$$381$$ −13.0902 −0.670630
$$382$$ 0.0212862 0.00108910
$$383$$ −36.0689 −1.84303 −0.921517 0.388338i $$-0.873049\pi$$
−0.921517 + 0.388338i $$0.873049\pi$$
$$384$$ −22.5623 −1.15138
$$385$$ 0 0
$$386$$ −3.92299 −0.199675
$$387$$ 19.1246 0.972159
$$388$$ 17.8328 0.905324
$$389$$ 4.47214 0.226746 0.113373 0.993552i $$-0.463834\pi$$
0.113373 + 0.993552i $$0.463834\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ −10.2229 −0.516335
$$393$$ 26.8328 1.35354
$$394$$ 0.562306 0.0283286
$$395$$ 0 0
$$396$$ 12.8754 0.647013
$$397$$ −20.0344 −1.00550 −0.502750 0.864432i $$-0.667678\pi$$
−0.502750 + 0.864432i $$0.667678\pi$$
$$398$$ 2.83282 0.141996
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 29.8885 1.49256 0.746281 0.665631i $$-0.231837\pi$$
0.746281 + 0.665631i $$0.231837\pi$$
$$402$$ −4.02129 −0.200564
$$403$$ 9.70820 0.483600
$$404$$ −29.7295 −1.47910
$$405$$ 0 0
$$406$$ 0.541020 0.0268504
$$407$$ 37.6869 1.86807
$$408$$ 6.10333 0.302160
$$409$$ 36.5623 1.80789 0.903945 0.427649i $$-0.140658\pi$$
0.903945 + 0.427649i $$0.140658\pi$$
$$410$$ 0 0
$$411$$ −10.6525 −0.525448
$$412$$ 6.00000 0.295599
$$413$$ 1.70820 0.0840552
$$414$$ 2.47214 0.121499
$$415$$ 0 0
$$416$$ −8.29180 −0.406539
$$417$$ 31.3050 1.53301
$$418$$ 0 0
$$419$$ −18.4721 −0.902423 −0.451211 0.892417i $$-0.649008\pi$$
−0.451211 + 0.892417i $$0.649008\pi$$
$$420$$ 0 0
$$421$$ −15.0000 −0.731055 −0.365528 0.930800i $$-0.619111\pi$$
−0.365528 + 0.930800i $$0.619111\pi$$
$$422$$ 2.96556 0.144361
$$423$$ 10.7639 0.523360
$$424$$ 12.6869 0.616131
$$425$$ 0 0
$$426$$ −11.5066 −0.557496
$$427$$ −3.43769 −0.166362
$$428$$ −24.3738 −1.17815
$$429$$ 15.5279 0.749692
$$430$$ 0 0
$$431$$ 20.6525 0.994795 0.497397 0.867523i $$-0.334289\pi$$
0.497397 + 0.867523i $$0.334289\pi$$
$$432$$ 7.03444 0.338445
$$433$$ 11.3607 0.545959 0.272980 0.962020i $$-0.411991\pi$$
0.272980 + 0.962020i $$0.411991\pi$$
$$434$$ 0.437694 0.0210100
$$435$$ 0 0
$$436$$ 18.4377 0.883005
$$437$$ 0 0
$$438$$ −2.31308 −0.110523
$$439$$ −1.50658 −0.0719050 −0.0359525 0.999353i $$-0.511447\pi$$
−0.0359525 + 0.999353i $$0.511447\pi$$
$$440$$ 0 0
$$441$$ −13.8885 −0.661359
$$442$$ 1.41641 0.0673717
$$443$$ 28.3262 1.34582 0.672910 0.739724i $$-0.265044\pi$$
0.672910 + 0.739724i $$0.265044\pi$$
$$444$$ −45.0000 −2.13561
$$445$$ 0 0
$$446$$ −6.45085 −0.305457
$$447$$ 1.18034 0.0558282
$$448$$ 1.11146 0.0525114
$$449$$ 0.201626 0.00951533 0.00475766 0.999989i $$-0.498486\pi$$
0.00475766 + 0.999989i $$0.498486\pi$$
$$450$$ 0 0
$$451$$ 29.9230 1.40902
$$452$$ 11.7295 0.551709
$$453$$ 42.0344 1.97495
$$454$$ 2.11146 0.0990955
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.5623 0.681196 0.340598 0.940209i $$-0.389371\pi$$
0.340598 + 0.940209i $$0.389371\pi$$
$$458$$ 4.06888 0.190126
$$459$$ −4.14590 −0.193514
$$460$$ 0 0
$$461$$ 21.0344 0.979672 0.489836 0.871815i $$-0.337057\pi$$
0.489836 + 0.871815i $$0.337057\pi$$
$$462$$ 0.700073 0.0325704
$$463$$ 40.8673 1.89926 0.949631 0.313370i $$-0.101458\pi$$
0.949631 + 0.313370i $$0.101458\pi$$
$$464$$ 18.8754 0.876268
$$465$$ 0 0
$$466$$ −9.72949 −0.450710
$$467$$ 36.2705 1.67840 0.839200 0.543824i $$-0.183024\pi$$
0.839200 + 0.543824i $$0.183024\pi$$
$$468$$ −7.41641 −0.342824
$$469$$ 1.11146 0.0513223
$$470$$ 0 0
$$471$$ 54.0689 2.49136
$$472$$ −10.6525 −0.490320
$$473$$ −33.2016 −1.52661
$$474$$ 0.854102 0.0392302
$$475$$ 0 0
$$476$$ −0.811529 −0.0371964
$$477$$ 17.2361 0.789185
$$478$$ 4.85410 0.222021
$$479$$ 14.7426 0.673609 0.336804 0.941575i $$-0.390654\pi$$
0.336804 + 0.941575i $$0.390654\pi$$
$$480$$ 0 0
$$481$$ −21.7082 −0.989809
$$482$$ 4.96556 0.226175
$$483$$ −1.70820 −0.0777260
$$484$$ −1.95743 −0.0889740
$$485$$ 0 0
$$486$$ −6.83282 −0.309943
$$487$$ −30.1246 −1.36508 −0.682538 0.730850i $$-0.739124\pi$$
−0.682538 + 0.730850i $$0.739124\pi$$
$$488$$ 21.4377 0.970439
$$489$$ −16.1803 −0.731700
$$490$$ 0 0
$$491$$ −32.2361 −1.45479 −0.727397 0.686217i $$-0.759270\pi$$
−0.727397 + 0.686217i $$0.759270\pi$$
$$492$$ −35.7295 −1.61081
$$493$$ −11.1246 −0.501027
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 15.2705 0.685666
$$497$$ 3.18034 0.142658
$$498$$ 8.61803 0.386183
$$499$$ 38.5967 1.72783 0.863914 0.503640i $$-0.168006\pi$$
0.863914 + 0.503640i $$0.168006\pi$$
$$500$$ 0 0
$$501$$ −1.70820 −0.0763169
$$502$$ −6.52786 −0.291353
$$503$$ −23.7426 −1.05863 −0.529316 0.848425i $$-0.677551\pi$$
−0.529316 + 0.848425i $$0.677551\pi$$
$$504$$ −0.695048 −0.0309599
$$505$$ 0 0
$$506$$ −4.29180 −0.190794
$$507$$ 20.1246 0.893765
$$508$$ −10.8541 −0.481573
$$509$$ −16.0344 −0.710714 −0.355357 0.934731i $$-0.615641\pi$$
−0.355357 + 0.934731i $$0.615641\pi$$
$$510$$ 0 0
$$511$$ 0.639320 0.0282819
$$512$$ −22.3050 −0.985749
$$513$$ 0 0
$$514$$ 11.0000 0.485189
$$515$$ 0 0
$$516$$ 39.6443 1.74524
$$517$$ −18.6869 −0.821850
$$518$$ −0.978714 −0.0430022
$$519$$ −26.0557 −1.14372
$$520$$ 0 0
$$521$$ 15.7639 0.690630 0.345315 0.938487i $$-0.387772\pi$$
0.345315 + 0.938487i $$0.387772\pi$$
$$522$$ −4.58359 −0.200618
$$523$$ −4.61803 −0.201933 −0.100966 0.994890i $$-0.532193\pi$$
−0.100966 + 0.994890i $$0.532193\pi$$
$$524$$ 22.2492 0.971962
$$525$$ 0 0
$$526$$ −7.78522 −0.339452
$$527$$ −9.00000 −0.392046
$$528$$ 24.4245 1.06294
$$529$$ −12.5279 −0.544690
$$530$$ 0 0
$$531$$ −14.4721 −0.628037
$$532$$ 0 0
$$533$$ −17.2361 −0.746577
$$534$$ −8.29180 −0.358821
$$535$$ 0 0
$$536$$ −6.93112 −0.299379
$$537$$ 25.0000 1.07883
$$538$$ 5.03444 0.217050
$$539$$ 24.1115 1.03855
$$540$$ 0 0
$$541$$ 38.6525 1.66180 0.830900 0.556422i $$-0.187826\pi$$
0.830900 + 0.556422i $$0.187826\pi$$
$$542$$ −1.65248 −0.0709799
$$543$$ 11.8328 0.507795
$$544$$ 7.68692 0.329574
$$545$$ 0 0
$$546$$ −0.403252 −0.0172576
$$547$$ 20.4164 0.872943 0.436471 0.899718i $$-0.356228\pi$$
0.436471 + 0.899718i $$0.356228\pi$$
$$548$$ −8.83282 −0.377319
$$549$$ 29.1246 1.24301
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 10.6525 0.453399
$$553$$ −0.236068 −0.0100386
$$554$$ 4.58359 0.194738
$$555$$ 0 0
$$556$$ 25.9574 1.10084
$$557$$ −4.49342 −0.190392 −0.0951962 0.995459i $$-0.530348\pi$$
−0.0951962 + 0.995459i $$0.530348\pi$$
$$558$$ −3.70820 −0.156981
$$559$$ 19.1246 0.808885
$$560$$ 0 0
$$561$$ −14.3951 −0.607763
$$562$$ −0.909830 −0.0383789
$$563$$ 32.5967 1.37379 0.686895 0.726757i $$-0.258973\pi$$
0.686895 + 0.726757i $$0.258973\pi$$
$$564$$ 22.3131 0.939550
$$565$$ 0 0
$$566$$ 10.7852 0.453337
$$567$$ 2.59675 0.109053
$$568$$ −19.8328 −0.832166
$$569$$ 11.2918 0.473377 0.236688 0.971586i $$-0.423938\pi$$
0.236688 + 0.971586i $$0.423938\pi$$
$$570$$ 0 0
$$571$$ −36.2492 −1.51698 −0.758491 0.651683i $$-0.774063\pi$$
−0.758491 + 0.651683i $$0.774063\pi$$
$$572$$ 12.8754 0.538347
$$573$$ 0.124612 0.00520573
$$574$$ −0.777088 −0.0324350
$$575$$ 0 0
$$576$$ −9.41641 −0.392350
$$577$$ 15.3050 0.637153 0.318577 0.947897i $$-0.396795\pi$$
0.318577 + 0.947897i $$0.396795\pi$$
$$578$$ 5.18034 0.215474
$$579$$ −22.9656 −0.954416
$$580$$ 0 0
$$581$$ −2.38197 −0.0988206
$$582$$ −8.21478 −0.340514
$$583$$ −29.9230 −1.23928
$$584$$ −3.98684 −0.164977
$$585$$ 0 0
$$586$$ −1.72949 −0.0714446
$$587$$ 43.3262 1.78827 0.894133 0.447802i $$-0.147793\pi$$
0.894133 + 0.447802i $$0.147793\pi$$
$$588$$ −28.7902 −1.18729
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 3.29180 0.135406
$$592$$ −34.1459 −1.40339
$$593$$ 36.5967 1.50285 0.751424 0.659819i $$-0.229367\pi$$
0.751424 + 0.659819i $$0.229367\pi$$
$$594$$ 2.96556 0.121678
$$595$$ 0 0
$$596$$ 0.978714 0.0400897
$$597$$ 16.5836 0.678721
$$598$$ 2.47214 0.101093
$$599$$ 17.0689 0.697416 0.348708 0.937231i $$-0.386621\pi$$
0.348708 + 0.937231i $$0.386621\pi$$
$$600$$ 0 0
$$601$$ −41.6869 −1.70044 −0.850222 0.526424i $$-0.823533\pi$$
−0.850222 + 0.526424i $$0.823533\pi$$
$$602$$ 0.862233 0.0351420
$$603$$ −9.41641 −0.383466
$$604$$ 34.8541 1.41819
$$605$$ 0 0
$$606$$ 13.6950 0.556323
$$607$$ 19.9787 0.810911 0.405455 0.914115i $$-0.367113\pi$$
0.405455 + 0.914115i $$0.367113\pi$$
$$608$$ 0 0
$$609$$ 3.16718 0.128341
$$610$$ 0 0
$$611$$ 10.7639 0.435462
$$612$$ 6.87539 0.277921
$$613$$ −30.5066 −1.23215 −0.616075 0.787688i $$-0.711278\pi$$
−0.616075 + 0.787688i $$0.711278\pi$$
$$614$$ −4.75078 −0.191726
$$615$$ 0 0
$$616$$ 1.20665 0.0486174
$$617$$ 28.5066 1.14763 0.573816 0.818984i $$-0.305463\pi$$
0.573816 + 0.818984i $$0.305463\pi$$
$$618$$ −2.76393 −0.111182
$$619$$ −1.50658 −0.0605545 −0.0302772 0.999542i $$-0.509639\pi$$
−0.0302772 + 0.999542i $$0.509639\pi$$
$$620$$ 0 0
$$621$$ −7.23607 −0.290373
$$622$$ −4.49342 −0.180170
$$623$$ 2.29180 0.0918189
$$624$$ −14.0689 −0.563206
$$625$$ 0 0
$$626$$ −0.785218 −0.0313836
$$627$$ 0 0
$$628$$ 44.8328 1.78902
$$629$$ 20.1246 0.802421
$$630$$ 0 0
$$631$$ 12.9787 0.516674 0.258337 0.966055i $$-0.416825\pi$$
0.258337 + 0.966055i $$0.416825\pi$$
$$632$$ 1.47214 0.0585584
$$633$$ 17.3607 0.690025
$$634$$ −2.65248 −0.105343
$$635$$ 0 0
$$636$$ 35.7295 1.41677
$$637$$ −13.8885 −0.550284
$$638$$ 7.95743 0.315038
$$639$$ −26.9443 −1.06590
$$640$$ 0 0
$$641$$ 30.7082 1.21290 0.606451 0.795121i $$-0.292593\pi$$
0.606451 + 0.795121i $$0.292593\pi$$
$$642$$ 11.2279 0.443131
$$643$$ 24.3050 0.958494 0.479247 0.877680i $$-0.340910\pi$$
0.479247 + 0.877680i $$0.340910\pi$$
$$644$$ −1.41641 −0.0558143
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16.5066 −0.648941 −0.324470 0.945896i $$-0.605186\pi$$
−0.324470 + 0.945896i $$0.605186\pi$$
$$648$$ −16.1935 −0.636141
$$649$$ 25.1246 0.986227
$$650$$ 0 0
$$651$$ 2.56231 0.100425
$$652$$ −13.4164 −0.525427
$$653$$ 26.1246 1.02234 0.511168 0.859481i $$-0.329213\pi$$
0.511168 + 0.859481i $$0.329213\pi$$
$$654$$ −8.49342 −0.332119
$$655$$ 0 0
$$656$$ −27.1115 −1.05852
$$657$$ −5.41641 −0.211314
$$658$$ 0.485292 0.0189186
$$659$$ −43.0689 −1.67773 −0.838863 0.544343i $$-0.816779\pi$$
−0.838863 + 0.544343i $$0.816779\pi$$
$$660$$ 0 0
$$661$$ −1.58359 −0.0615946 −0.0307973 0.999526i $$-0.509805\pi$$
−0.0307973 + 0.999526i $$0.509805\pi$$
$$662$$ −2.50658 −0.0974209
$$663$$ 8.29180 0.322027
$$664$$ 14.8541 0.576451
$$665$$ 0 0
$$666$$ 8.29180 0.321301
$$667$$ −19.4164 −0.751806
$$668$$ −1.41641 −0.0548025
$$669$$ −37.7639 −1.46004
$$670$$ 0 0
$$671$$ −50.5623 −1.95194
$$672$$ −2.18847 −0.0844221
$$673$$ −25.1803 −0.970631 −0.485315 0.874339i $$-0.661295\pi$$
−0.485315 + 0.874339i $$0.661295\pi$$
$$674$$ 3.32624 0.128122
$$675$$ 0 0
$$676$$ 16.6869 0.641805
$$677$$ −24.9098 −0.957363 −0.478681 0.877989i $$-0.658885\pi$$
−0.478681 + 0.877989i $$0.658885\pi$$
$$678$$ −5.40325 −0.207511
$$679$$ 2.27051 0.0871342
$$680$$ 0 0
$$681$$ 12.3607 0.473662
$$682$$ 6.43769 0.246512
$$683$$ −12.2016 −0.466882 −0.233441 0.972371i $$-0.574999\pi$$
−0.233441 + 0.972371i $$0.574999\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.25735 −0.0480060
$$687$$ 23.8197 0.908777
$$688$$ 30.0820 1.14687
$$689$$ 17.2361 0.656642
$$690$$ 0 0
$$691$$ 23.6525 0.899783 0.449891 0.893083i $$-0.351463\pi$$
0.449891 + 0.893083i $$0.351463\pi$$
$$692$$ −21.6049 −0.821294
$$693$$ 1.63932 0.0622726
$$694$$ −9.36881 −0.355635
$$695$$ 0 0
$$696$$ −19.7508 −0.748651
$$697$$ 15.9787 0.605237
$$698$$ −11.8673 −0.449182
$$699$$ −56.9574 −2.15433
$$700$$ 0 0
$$701$$ −31.0557 −1.17296 −0.586479 0.809964i $$-0.699486\pi$$
−0.586479 + 0.809964i $$0.699486\pi$$
$$702$$ −1.70820 −0.0644720
$$703$$ 0 0
$$704$$ 16.3475 0.616120
$$705$$ 0 0
$$706$$ 9.72949 0.366174
$$707$$ −3.78522 −0.142358
$$708$$ −30.0000 −1.12747
$$709$$ −44.8328 −1.68373 −0.841866 0.539687i $$-0.818543\pi$$
−0.841866 + 0.539687i $$0.818543\pi$$
$$710$$ 0 0
$$711$$ 2.00000 0.0750059
$$712$$ −14.2918 −0.535608
$$713$$ −15.7082 −0.588277
$$714$$ 0.373835 0.0139904
$$715$$ 0 0
$$716$$ 20.7295 0.774697
$$717$$ 28.4164 1.06123
$$718$$ −10.1591 −0.379133
$$719$$ −7.76393 −0.289546 −0.144773 0.989465i $$-0.546245\pi$$
−0.144773 + 0.989465i $$0.546245\pi$$
$$720$$ 0 0
$$721$$ 0.763932 0.0284503
$$722$$ 0 0
$$723$$ 29.0689 1.08108
$$724$$ 9.81153 0.364643
$$725$$ 0 0
$$726$$ 0.901699 0.0334652
$$727$$ 19.6869 0.730147 0.365074 0.930979i $$-0.381044\pi$$
0.365074 + 0.930979i $$0.381044\pi$$
$$728$$ −0.695048 −0.0257602
$$729$$ −7.00000 −0.259259
$$730$$ 0 0
$$731$$ −17.7295 −0.655749
$$732$$ 60.3738 2.23148
$$733$$ −19.1591 −0.707656 −0.353828 0.935311i $$-0.615120\pi$$
−0.353828 + 0.935311i $$0.615120\pi$$
$$734$$ −1.34752 −0.0497380
$$735$$ 0 0
$$736$$ 13.4164 0.494535
$$737$$ 16.3475 0.602169
$$738$$ 6.58359 0.242345
$$739$$ −18.5623 −0.682825 −0.341413 0.939913i $$-0.610905\pi$$
−0.341413 + 0.939913i $$0.610905\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.777088 0.0285278
$$743$$ −13.2361 −0.485584 −0.242792 0.970078i $$-0.578063\pi$$
−0.242792 + 0.970078i $$0.578063\pi$$
$$744$$ −15.9787 −0.585808
$$745$$ 0 0
$$746$$ 7.58359 0.277655
$$747$$ 20.1803 0.738360
$$748$$ −11.9361 −0.436429
$$749$$ −3.10333 −0.113393
$$750$$ 0 0
$$751$$ 47.5623 1.73557 0.867787 0.496937i $$-0.165542\pi$$
0.867787 + 0.496937i $$0.165542\pi$$
$$752$$ 16.9311 0.617414
$$753$$ −38.2148 −1.39262
$$754$$ −4.58359 −0.166925
$$755$$ 0 0
$$756$$ 0.978714 0.0355955
$$757$$ 28.8885 1.04997 0.524986 0.851111i $$-0.324071\pi$$
0.524986 + 0.851111i $$0.324071\pi$$
$$758$$ −11.6656 −0.423715
$$759$$ −25.1246 −0.911966
$$760$$ 0 0
$$761$$ 29.6180 1.07365 0.536826 0.843693i $$-0.319623\pi$$
0.536826 + 0.843693i $$0.319623\pi$$
$$762$$ 5.00000 0.181131
$$763$$ 2.34752 0.0849861
$$764$$ 0.103326 0.00373819
$$765$$ 0 0
$$766$$ 13.7771 0.497786
$$767$$ −14.4721 −0.522559
$$768$$ −12.4377 −0.448807
$$769$$ −18.2705 −0.658851 −0.329426 0.944181i $$-0.606855\pi$$
−0.329426 + 0.944181i $$0.606855\pi$$
$$770$$ 0 0
$$771$$ 64.3951 2.31913
$$772$$ −19.0426 −0.685357
$$773$$ −18.9656 −0.682144 −0.341072 0.940037i $$-0.610790\pi$$
−0.341072 + 0.940037i $$0.610790\pi$$
$$774$$ −7.30495 −0.262571
$$775$$ 0 0
$$776$$ −14.1591 −0.508280
$$777$$ −5.72949 −0.205544
$$778$$ −1.70820 −0.0612421
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 46.7771 1.67382
$$782$$ −2.29180 −0.0819545
$$783$$ 13.4164 0.479463
$$784$$ −21.8460 −0.780213
$$785$$ 0 0
$$786$$ −10.2492 −0.365578
$$787$$ −5.59675 −0.199503 −0.0997513 0.995012i $$-0.531805\pi$$
−0.0997513 + 0.995012i $$0.531805\pi$$
$$788$$ 2.72949 0.0972341
$$789$$ −45.5755 −1.62253
$$790$$ 0 0
$$791$$ 1.49342 0.0531000
$$792$$ −10.2229 −0.363255
$$793$$ 29.1246 1.03425
$$794$$ 7.65248 0.271576
$$795$$ 0 0
$$796$$ 13.7508 0.487383
$$797$$ 54.8673 1.94350 0.971749 0.236017i $$-0.0758420\pi$$
0.971749 + 0.236017i $$0.0758420\pi$$
$$798$$ 0 0
$$799$$ −9.97871 −0.353022
$$800$$ 0 0
$$801$$ −19.4164 −0.686045
$$802$$ −11.4164 −0.403127
$$803$$ 9.40325 0.331834
$$804$$ −19.5197 −0.688408
$$805$$ 0 0
$$806$$ −3.70820 −0.130616
$$807$$ 29.4721 1.03747
$$808$$ 23.6049 0.830417
$$809$$ 2.23607 0.0786160 0.0393080 0.999227i $$-0.487485\pi$$
0.0393080 + 0.999227i $$0.487485\pi$$
$$810$$ 0 0
$$811$$ −7.38197 −0.259216 −0.129608 0.991565i $$-0.541372\pi$$
−0.129608 + 0.991565i $$0.541372\pi$$
$$812$$ 2.62616 0.0921603
$$813$$ −9.67376 −0.339274
$$814$$ −14.3951 −0.504549
$$815$$ 0 0
$$816$$ 13.0426 0.456581
$$817$$ 0 0
$$818$$ −13.9656 −0.488294
$$819$$ −0.944272 −0.0329955
$$820$$ 0 0
$$821$$ −36.9230 −1.28862 −0.644311 0.764764i $$-0.722856\pi$$
−0.644311 + 0.764764i $$0.722856\pi$$
$$822$$ 4.06888 0.141919
$$823$$ −5.74265 −0.200176 −0.100088 0.994979i $$-0.531912\pi$$
−0.100088 + 0.994979i $$0.531912\pi$$
$$824$$ −4.76393 −0.165959
$$825$$ 0 0
$$826$$ −0.652476 −0.0227025
$$827$$ −49.2492 −1.71256 −0.856282 0.516509i $$-0.827231\pi$$
−0.856282 + 0.516509i $$0.827231\pi$$
$$828$$ 12.0000 0.417029
$$829$$ 1.00000 0.0347314 0.0173657 0.999849i $$-0.494472\pi$$
0.0173657 + 0.999849i $$0.494472\pi$$
$$830$$ 0 0
$$831$$ 26.8328 0.930820
$$832$$ −9.41641 −0.326455
$$833$$ 12.8754 0.446106
$$834$$ −11.9574 −0.414052
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 10.8541 0.375173
$$838$$ 7.05573 0.243736
$$839$$ 49.7639 1.71804 0.859021 0.511941i $$-0.171073\pi$$
0.859021 + 0.511941i $$0.171073\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 5.72949 0.197451
$$843$$ −5.32624 −0.183445
$$844$$ 14.3951 0.495501
$$845$$ 0 0
$$846$$ −4.11146 −0.141355
$$847$$ −0.249224 −0.00856342
$$848$$ 27.1115 0.931011
$$849$$ 63.1378 2.16688
$$850$$ 0 0
$$851$$ 35.1246 1.20406
$$852$$ −55.8541 −1.91353
$$853$$ 40.1459 1.37457 0.687285 0.726388i $$-0.258802\pi$$
0.687285 + 0.726388i $$0.258802\pi$$
$$854$$ 1.31308 0.0449328
$$855$$ 0 0
$$856$$ 19.3525 0.661457
$$857$$ 47.1803 1.61165 0.805825 0.592154i $$-0.201722\pi$$
0.805825 + 0.592154i $$0.201722\pi$$
$$858$$ −5.93112 −0.202485
$$859$$ 23.0902 0.787826 0.393913 0.919148i $$-0.371121\pi$$
0.393913 + 0.919148i $$0.371121\pi$$
$$860$$ 0 0
$$861$$ −4.54915 −0.155035
$$862$$ −7.88854 −0.268685
$$863$$ 32.9230 1.12071 0.560356 0.828252i $$-0.310664\pi$$
0.560356 + 0.828252i $$0.310664\pi$$
$$864$$ −9.27051 −0.315389
$$865$$ 0 0
$$866$$ −4.33939 −0.147459
$$867$$ 30.3262 1.02993
$$868$$ 2.12461 0.0721140
$$869$$ −3.47214 −0.117784
$$870$$ 0 0
$$871$$ −9.41641 −0.319063
$$872$$ −14.6393 −0.495750
$$873$$ −19.2361 −0.651042
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −11.2279 −0.379357
$$877$$ 15.9656 0.539119 0.269559 0.962984i $$-0.413122\pi$$
0.269559 + 0.962984i $$0.413122\pi$$
$$878$$ 0.575462 0.0194209
$$879$$ −10.1246 −0.341495
$$880$$ 0 0
$$881$$ −1.47214 −0.0495975 −0.0247988 0.999692i $$-0.507894\pi$$
−0.0247988 + 0.999692i $$0.507894\pi$$
$$882$$ 5.30495 0.178627
$$883$$ 49.7771 1.67513 0.837566 0.546336i $$-0.183978\pi$$
0.837566 + 0.546336i $$0.183978\pi$$
$$884$$ 6.87539 0.231244
$$885$$ 0 0
$$886$$ −10.8197 −0.363494
$$887$$ −19.0557 −0.639829 −0.319914 0.947446i $$-0.603654\pi$$
−0.319914 + 0.947446i $$0.603654\pi$$
$$888$$ 35.7295 1.19900
$$889$$ −1.38197 −0.0463497
$$890$$ 0 0
$$891$$ 38.1935 1.27953
$$892$$ −31.3131 −1.04844
$$893$$ 0 0
$$894$$ −0.450850 −0.0150787
$$895$$ 0 0
$$896$$ −2.38197 −0.0795759
$$897$$ 14.4721 0.483211
$$898$$ −0.0770143 −0.00257000
$$899$$ 29.1246 0.971360
$$900$$ 0 0
$$901$$ −15.9787 −0.532328
$$902$$ −11.4296 −0.380563
$$903$$ 5.04760 0.167974
$$904$$ −9.31308 −0.309749
$$905$$ 0 0
$$906$$ −16.0557 −0.533416
$$907$$ 54.9574 1.82483 0.912416 0.409265i $$-0.134215\pi$$
0.912416 + 0.409265i $$0.134215\pi$$
$$908$$ 10.2492 0.340132
$$909$$ 32.0689 1.06366
$$910$$ 0 0
$$911$$ 16.1459 0.534937 0.267469 0.963567i $$-0.413813\pi$$
0.267469 + 0.963567i $$0.413813\pi$$
$$912$$ 0 0
$$913$$ −35.0344 −1.15947
$$914$$ −5.56231 −0.183985
$$915$$ 0 0
$$916$$ 19.7508 0.652584
$$917$$ 2.83282 0.0935478
$$918$$ 1.58359 0.0522663
$$919$$ 7.50658 0.247619 0.123810 0.992306i $$-0.460489\pi$$
0.123810 + 0.992306i $$0.460489\pi$$
$$920$$ 0 0
$$921$$ −27.8115 −0.916421
$$922$$ −8.03444 −0.264600
$$923$$ −26.9443 −0.886882
$$924$$ 3.39823 0.111793
$$925$$ 0 0
$$926$$ −15.6099 −0.512973
$$927$$ −6.47214 −0.212573
$$928$$ −24.8754 −0.816575
$$929$$ 16.5279 0.542262 0.271131 0.962543i $$-0.412602\pi$$
0.271131 + 0.962543i $$0.412602\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −47.2279 −1.54700
$$933$$ −26.3050 −0.861185
$$934$$ −13.8541 −0.453320
$$935$$ 0 0
$$936$$ 5.88854 0.192473
$$937$$ −26.8328 −0.876590 −0.438295 0.898831i $$-0.644417\pi$$
−0.438295 + 0.898831i $$0.644417\pi$$
$$938$$ −0.424538 −0.0138617
$$939$$ −4.59675 −0.150009
$$940$$ 0 0
$$941$$ 17.2574 0.562574 0.281287 0.959624i $$-0.409239\pi$$
0.281287 + 0.959624i $$0.409239\pi$$
$$942$$ −20.6525 −0.672894
$$943$$ 27.8885 0.908176
$$944$$ −22.7639 −0.740903
$$945$$ 0 0
$$946$$ 12.6819 0.412324
$$947$$ 22.6869 0.737226 0.368613 0.929583i $$-0.379833\pi$$
0.368613 + 0.929583i $$0.379833\pi$$
$$948$$ 4.14590 0.134653
$$949$$ −5.41641 −0.175824
$$950$$ 0 0
$$951$$ −15.5279 −0.503525
$$952$$ 0.644345 0.0208833
$$953$$ 13.4164 0.434600 0.217300 0.976105i $$-0.430275\pi$$
0.217300 + 0.976105i $$0.430275\pi$$
$$954$$ −6.58359 −0.213152
$$955$$ 0 0
$$956$$ 23.5623 0.762059
$$957$$ 46.5836 1.50583
$$958$$ −5.63119 −0.181935
$$959$$ −1.12461 −0.0363156
$$960$$ 0 0
$$961$$ −7.43769 −0.239926
$$962$$ 8.29180 0.267338
$$963$$ 26.2918 0.847241
$$964$$ 24.1033 0.776316
$$965$$ 0 0
$$966$$ 0.652476 0.0209931
$$967$$ −5.09017 −0.163689 −0.0818444 0.996645i $$-0.526081\pi$$
−0.0818444 + 0.996645i $$0.526081\pi$$
$$968$$ 1.55418 0.0499531
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −43.7984 −1.40556 −0.702778 0.711409i $$-0.748057\pi$$
−0.702778 + 0.711409i $$0.748057\pi$$
$$972$$ −33.1672 −1.06384
$$973$$ 3.30495 0.105952
$$974$$ 11.5066 0.368695
$$975$$ 0 0
$$976$$ 45.8115 1.46639
$$977$$ −19.2148 −0.614735 −0.307368 0.951591i $$-0.599448\pi$$
−0.307368 + 0.951591i $$0.599448\pi$$
$$978$$ 6.18034 0.197625
$$979$$ 33.7082 1.07732
$$980$$ 0 0
$$981$$ −19.8885 −0.634992
$$982$$ 12.3131 0.392926
$$983$$ −33.9787 −1.08375 −0.541876 0.840458i $$-0.682286\pi$$
−0.541876 + 0.840458i $$0.682286\pi$$
$$984$$ 28.3688 0.904365
$$985$$ 0 0
$$986$$ 4.24922 0.135323
$$987$$ 2.84095 0.0904283
$$988$$ 0 0
$$989$$ −30.9443 −0.983971
$$990$$ 0 0
$$991$$ 11.4164 0.362654 0.181327 0.983423i $$-0.441961\pi$$
0.181327 + 0.983423i $$0.441961\pi$$
$$992$$ −20.1246 −0.638957
$$993$$ −14.6738 −0.465658
$$994$$ −1.21478 −0.0385305
$$995$$ 0 0
$$996$$ 41.8328 1.32552
$$997$$ 35.7426 1.13198 0.565990 0.824412i $$-0.308494\pi$$
0.565990 + 0.824412i $$0.308494\pi$$
$$998$$ −14.7426 −0.466670
$$999$$ −24.2705 −0.767885
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.m.1.2 yes 2
5.4 even 2 9025.2.a.t.1.1 yes 2
19.18 odd 2 9025.2.a.u.1.1 yes 2
95.94 odd 2 9025.2.a.l.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
9025.2.a.l.1.2 2 95.94 odd 2
9025.2.a.m.1.2 yes 2 1.1 even 1 trivial
9025.2.a.t.1.1 yes 2 5.4 even 2
9025.2.a.u.1.1 yes 2 19.18 odd 2