# Properties

 Label 9025.2.a.bl.1.4 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 5$$ x^4 - 5*x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1805) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$1.90211$$ 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)

 $$f(q)$$ $$=$$ $$q+1.90211 q^{2} +1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} -0.726543 q^{8} +0.618034 q^{9} +O(q^{10})$$ $$q+1.90211 q^{2} +1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} -0.726543 q^{8} +0.618034 q^{9} -5.85410 q^{11} +3.07768 q^{12} -3.07768 q^{13} +8.05748 q^{14} -4.61803 q^{16} -5.23607 q^{17} +1.17557 q^{18} +8.05748 q^{21} -11.1352 q^{22} -4.09017 q^{23} -1.38197 q^{24} -5.85410 q^{26} -4.53077 q^{27} +6.85410 q^{28} -2.80017 q^{29} +1.90211 q^{31} -7.33094 q^{32} -11.1352 q^{33} -9.95959 q^{34} +1.00000 q^{36} +2.80017 q^{37} -5.85410 q^{39} -6.88191 q^{41} +15.3262 q^{42} -0.381966 q^{43} -9.47214 q^{44} -7.77997 q^{46} +1.47214 q^{47} -8.78402 q^{48} +10.9443 q^{49} -9.95959 q^{51} -4.97980 q^{52} -11.1352 q^{53} -8.61803 q^{54} -3.07768 q^{56} -5.32624 q^{58} +14.0413 q^{59} +3.94427 q^{61} +3.61803 q^{62} +2.61803 q^{63} -4.70820 q^{64} -21.1803 q^{66} -5.98385 q^{67} -8.47214 q^{68} -7.77997 q^{69} +0.171513 q^{71} -0.449028 q^{72} +1.00000 q^{73} +5.32624 q^{74} -24.7984 q^{77} -11.1352 q^{78} -5.25731 q^{79} -10.4721 q^{81} -13.0902 q^{82} +8.76393 q^{83} +13.0373 q^{84} -0.726543 q^{86} -5.32624 q^{87} +4.25325 q^{88} +7.77997 q^{89} -13.0373 q^{91} -6.61803 q^{92} +3.61803 q^{93} +2.80017 q^{94} -13.9443 q^{96} +2.62866 q^{97} +20.8172 q^{98} -3.61803 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 $$4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{58} - 20 q^{61} + 10 q^{62} + 6 q^{63} + 8 q^{64} - 40 q^{66} - 16 q^{68} + 4 q^{73} - 10 q^{74} - 50 q^{77} - 24 q^{81} - 30 q^{82} + 44 q^{83} + 10 q^{87} - 22 q^{92} + 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.90211 1.34500 0.672499 0.740098i $$-0.265221\pi$$
0.672499 + 0.740098i $$0.265221\pi$$
$$3$$ 1.90211 1.09819 0.549093 0.835761i $$-0.314973\pi$$
0.549093 + 0.835761i $$0.314973\pi$$
$$4$$ 1.61803 0.809017
$$5$$ 0 0
$$6$$ 3.61803 1.47706
$$7$$ 4.23607 1.60108 0.800542 0.599277i $$-0.204545\pi$$
0.800542 + 0.599277i $$0.204545\pi$$
$$8$$ −0.726543 −0.256872
$$9$$ 0.618034 0.206011
$$10$$ 0 0
$$11$$ −5.85410 −1.76508 −0.882539 0.470239i $$-0.844168\pi$$
−0.882539 + 0.470239i $$0.844168\pi$$
$$12$$ 3.07768 0.888451
$$13$$ −3.07768 −0.853596 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$14$$ 8.05748 2.15345
$$15$$ 0 0
$$16$$ −4.61803 −1.15451
$$17$$ −5.23607 −1.26993 −0.634967 0.772540i $$-0.718986\pi$$
−0.634967 + 0.772540i $$0.718986\pi$$
$$18$$ 1.17557 0.277085
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 8.05748 1.75829
$$22$$ −11.1352 −2.37402
$$23$$ −4.09017 −0.852859 −0.426430 0.904521i $$-0.640229\pi$$
−0.426430 + 0.904521i $$0.640229\pi$$
$$24$$ −1.38197 −0.282093
$$25$$ 0 0
$$26$$ −5.85410 −1.14808
$$27$$ −4.53077 −0.871947
$$28$$ 6.85410 1.29530
$$29$$ −2.80017 −0.519978 −0.259989 0.965612i $$-0.583719\pi$$
−0.259989 + 0.965612i $$0.583719\pi$$
$$30$$ 0 0
$$31$$ 1.90211 0.341630 0.170815 0.985303i $$-0.445360\pi$$
0.170815 + 0.985303i $$0.445360\pi$$
$$32$$ −7.33094 −1.29594
$$33$$ −11.1352 −1.93838
$$34$$ −9.95959 −1.70806
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.80017 0.460345 0.230172 0.973150i $$-0.426071\pi$$
0.230172 + 0.973150i $$0.426071\pi$$
$$38$$ 0 0
$$39$$ −5.85410 −0.937407
$$40$$ 0 0
$$41$$ −6.88191 −1.07477 −0.537387 0.843336i $$-0.680588\pi$$
−0.537387 + 0.843336i $$0.680588\pi$$
$$42$$ 15.3262 2.36489
$$43$$ −0.381966 −0.0582493 −0.0291246 0.999576i $$-0.509272\pi$$
−0.0291246 + 0.999576i $$0.509272\pi$$
$$44$$ −9.47214 −1.42798
$$45$$ 0 0
$$46$$ −7.77997 −1.14709
$$47$$ 1.47214 0.214733 0.107367 0.994220i $$-0.465758\pi$$
0.107367 + 0.994220i $$0.465758\pi$$
$$48$$ −8.78402 −1.26786
$$49$$ 10.9443 1.56347
$$50$$ 0 0
$$51$$ −9.95959 −1.39462
$$52$$ −4.97980 −0.690574
$$53$$ −11.1352 −1.52953 −0.764766 0.644308i $$-0.777146\pi$$
−0.764766 + 0.644308i $$0.777146\pi$$
$$54$$ −8.61803 −1.17277
$$55$$ 0 0
$$56$$ −3.07768 −0.411273
$$57$$ 0 0
$$58$$ −5.32624 −0.699369
$$59$$ 14.0413 1.82803 0.914013 0.405685i $$-0.132967\pi$$
0.914013 + 0.405685i $$0.132967\pi$$
$$60$$ 0 0
$$61$$ 3.94427 0.505012 0.252506 0.967595i $$-0.418745\pi$$
0.252506 + 0.967595i $$0.418745\pi$$
$$62$$ 3.61803 0.459491
$$63$$ 2.61803 0.329841
$$64$$ −4.70820 −0.588525
$$65$$ 0 0
$$66$$ −21.1803 −2.60712
$$67$$ −5.98385 −0.731044 −0.365522 0.930803i $$-0.619110\pi$$
−0.365522 + 0.930803i $$0.619110\pi$$
$$68$$ −8.47214 −1.02740
$$69$$ −7.77997 −0.936598
$$70$$ 0 0
$$71$$ 0.171513 0.0203549 0.0101774 0.999948i $$-0.496760\pi$$
0.0101774 + 0.999948i $$0.496760\pi$$
$$72$$ −0.449028 −0.0529185
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 5.32624 0.619163
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −24.7984 −2.82604
$$78$$ −11.1352 −1.26081
$$79$$ −5.25731 −0.591494 −0.295747 0.955266i $$-0.595568\pi$$
−0.295747 + 0.955266i $$0.595568\pi$$
$$80$$ 0 0
$$81$$ −10.4721 −1.16357
$$82$$ −13.0902 −1.44557
$$83$$ 8.76393 0.961967 0.480983 0.876730i $$-0.340280\pi$$
0.480983 + 0.876730i $$0.340280\pi$$
$$84$$ 13.0373 1.42248
$$85$$ 0 0
$$86$$ −0.726543 −0.0783451
$$87$$ −5.32624 −0.571033
$$88$$ 4.25325 0.453398
$$89$$ 7.77997 0.824675 0.412337 0.911031i $$-0.364713\pi$$
0.412337 + 0.911031i $$0.364713\pi$$
$$90$$ 0 0
$$91$$ −13.0373 −1.36668
$$92$$ −6.61803 −0.689978
$$93$$ 3.61803 0.375173
$$94$$ 2.80017 0.288815
$$95$$ 0 0
$$96$$ −13.9443 −1.42318
$$97$$ 2.62866 0.266900 0.133450 0.991056i $$-0.457395\pi$$
0.133450 + 0.991056i $$0.457395\pi$$
$$98$$ 20.8172 2.10286
$$99$$ −3.61803 −0.363626
$$100$$ 0 0
$$101$$ 3.29180 0.327546 0.163773 0.986498i $$-0.447634\pi$$
0.163773 + 0.986498i $$0.447634\pi$$
$$102$$ −18.9443 −1.87576
$$103$$ 14.9394 1.47202 0.736011 0.676970i $$-0.236707\pi$$
0.736011 + 0.676970i $$0.236707\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ −21.1803 −2.05722
$$107$$ 8.33499 0.805774 0.402887 0.915250i $$-0.368007\pi$$
0.402887 + 0.915250i $$0.368007\pi$$
$$108$$ −7.33094 −0.705420
$$109$$ −2.17963 −0.208770 −0.104385 0.994537i $$-0.533287\pi$$
−0.104385 + 0.994537i $$0.533287\pi$$
$$110$$ 0 0
$$111$$ 5.32624 0.505544
$$112$$ −19.5623 −1.84846
$$113$$ −18.4661 −1.73714 −0.868572 0.495562i $$-0.834962\pi$$
−0.868572 + 0.495562i $$0.834962\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −4.53077 −0.420671
$$117$$ −1.90211 −0.175850
$$118$$ 26.7082 2.45869
$$119$$ −22.1803 −2.03327
$$120$$ 0 0
$$121$$ 23.2705 2.11550
$$122$$ 7.50245 0.679240
$$123$$ −13.0902 −1.18030
$$124$$ 3.07768 0.276384
$$125$$ 0 0
$$126$$ 4.97980 0.443636
$$127$$ −9.23305 −0.819301 −0.409650 0.912243i $$-0.634349\pi$$
−0.409650 + 0.912243i $$0.634349\pi$$
$$128$$ 5.70634 0.504374
$$129$$ −0.726543 −0.0639685
$$130$$ 0 0
$$131$$ −15.6180 −1.36455 −0.682277 0.731094i $$-0.739010\pi$$
−0.682277 + 0.731094i $$0.739010\pi$$
$$132$$ −18.0171 −1.56818
$$133$$ 0 0
$$134$$ −11.3820 −0.983252
$$135$$ 0 0
$$136$$ 3.80423 0.326210
$$137$$ −9.76393 −0.834189 −0.417095 0.908863i $$-0.636952\pi$$
−0.417095 + 0.908863i $$0.636952\pi$$
$$138$$ −14.7984 −1.25972
$$139$$ −22.0902 −1.87366 −0.936832 0.349780i $$-0.886256\pi$$
−0.936832 + 0.349780i $$0.886256\pi$$
$$140$$ 0 0
$$141$$ 2.80017 0.235817
$$142$$ 0.326238 0.0273773
$$143$$ 18.0171 1.50666
$$144$$ −2.85410 −0.237842
$$145$$ 0 0
$$146$$ 1.90211 0.157420
$$147$$ 20.8172 1.71698
$$148$$ 4.53077 0.372427
$$149$$ 7.09017 0.580849 0.290425 0.956898i $$-0.406203\pi$$
0.290425 + 0.956898i $$0.406203\pi$$
$$150$$ 0 0
$$151$$ 0.620541 0.0504989 0.0252495 0.999681i $$-0.491962\pi$$
0.0252495 + 0.999681i $$0.491962\pi$$
$$152$$ 0 0
$$153$$ −3.23607 −0.261621
$$154$$ −47.1693 −3.80101
$$155$$ 0 0
$$156$$ −9.47214 −0.758378
$$157$$ 16.2705 1.29853 0.649264 0.760563i $$-0.275077\pi$$
0.649264 + 0.760563i $$0.275077\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ −21.1803 −1.67971
$$160$$ 0 0
$$161$$ −17.3262 −1.36550
$$162$$ −19.9192 −1.56500
$$163$$ 25.1803 1.97228 0.986138 0.165926i $$-0.0530613\pi$$
0.986138 + 0.165926i $$0.0530613\pi$$
$$164$$ −11.1352 −0.869510
$$165$$ 0 0
$$166$$ 16.6700 1.29384
$$167$$ −0.171513 −0.0132721 −0.00663605 0.999978i $$-0.502112\pi$$
−0.00663605 + 0.999978i $$0.502112\pi$$
$$168$$ −5.85410 −0.451654
$$169$$ −3.52786 −0.271374
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −0.618034 −0.0471246
$$173$$ 13.2088 1.00425 0.502123 0.864796i $$-0.332553\pi$$
0.502123 + 0.864796i $$0.332553\pi$$
$$174$$ −10.1311 −0.768037
$$175$$ 0 0
$$176$$ 27.0344 2.03780
$$177$$ 26.7082 2.00751
$$178$$ 14.7984 1.10919
$$179$$ 10.1311 0.757234 0.378617 0.925553i $$-0.376400\pi$$
0.378617 + 0.925553i $$0.376400\pi$$
$$180$$ 0 0
$$181$$ −6.71040 −0.498780 −0.249390 0.968403i $$-0.580230\pi$$
−0.249390 + 0.968403i $$0.580230\pi$$
$$182$$ −24.7984 −1.83818
$$183$$ 7.50245 0.554597
$$184$$ 2.97168 0.219075
$$185$$ 0 0
$$186$$ 6.88191 0.504606
$$187$$ 30.6525 2.24153
$$188$$ 2.38197 0.173723
$$189$$ −19.1926 −1.39606
$$190$$ 0 0
$$191$$ 13.0000 0.940647 0.470323 0.882494i $$-0.344137\pi$$
0.470323 + 0.882494i $$0.344137\pi$$
$$192$$ −8.95554 −0.646310
$$193$$ −1.45309 −0.104595 −0.0522977 0.998632i $$-0.516654\pi$$
−0.0522977 + 0.998632i $$0.516654\pi$$
$$194$$ 5.00000 0.358979
$$195$$ 0 0
$$196$$ 17.7082 1.26487
$$197$$ −14.8885 −1.06076 −0.530382 0.847759i $$-0.677952\pi$$
−0.530382 + 0.847759i $$0.677952\pi$$
$$198$$ −6.88191 −0.489076
$$199$$ 13.4164 0.951064 0.475532 0.879698i $$-0.342256\pi$$
0.475532 + 0.879698i $$0.342256\pi$$
$$200$$ 0 0
$$201$$ −11.3820 −0.802822
$$202$$ 6.26137 0.440548
$$203$$ −11.8617 −0.832529
$$204$$ −16.1150 −1.12827
$$205$$ 0 0
$$206$$ 28.4164 1.97986
$$207$$ −2.52786 −0.175699
$$208$$ 14.2128 0.985484
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1.73060 −0.119139 −0.0595697 0.998224i $$-0.518973\pi$$
−0.0595697 + 0.998224i $$0.518973\pi$$
$$212$$ −18.0171 −1.23742
$$213$$ 0.326238 0.0223535
$$214$$ 15.8541 1.08376
$$215$$ 0 0
$$216$$ 3.29180 0.223978
$$217$$ 8.05748 0.546977
$$218$$ −4.14590 −0.280796
$$219$$ 1.90211 0.128533
$$220$$ 0 0
$$221$$ 16.1150 1.08401
$$222$$ 10.1311 0.679955
$$223$$ 14.8334 0.993317 0.496659 0.867946i $$-0.334560\pi$$
0.496659 + 0.867946i $$0.334560\pi$$
$$224$$ −31.0543 −2.07491
$$225$$ 0 0
$$226$$ −35.1246 −2.33645
$$227$$ −17.3965 −1.15465 −0.577324 0.816515i $$-0.695903\pi$$
−0.577324 + 0.816515i $$0.695903\pi$$
$$228$$ 0 0
$$229$$ −18.1459 −1.19911 −0.599557 0.800332i $$-0.704657\pi$$
−0.599557 + 0.800332i $$0.704657\pi$$
$$230$$ 0 0
$$231$$ −47.1693 −3.10351
$$232$$ 2.03444 0.133568
$$233$$ −1.76393 −0.115559 −0.0577795 0.998329i $$-0.518402\pi$$
−0.0577795 + 0.998329i $$0.518402\pi$$
$$234$$ −3.61803 −0.236518
$$235$$ 0 0
$$236$$ 22.7194 1.47890
$$237$$ −10.0000 −0.649570
$$238$$ −42.1895 −2.73474
$$239$$ −23.2148 −1.50164 −0.750820 0.660507i $$-0.770341\pi$$
−0.750820 + 0.660507i $$0.770341\pi$$
$$240$$ 0 0
$$241$$ −6.32688 −0.407550 −0.203775 0.979018i $$-0.565321\pi$$
−0.203775 + 0.979018i $$0.565321\pi$$
$$242$$ 44.2631 2.84534
$$243$$ −6.32688 −0.405870
$$244$$ 6.38197 0.408564
$$245$$ 0 0
$$246$$ −24.8990 −1.58750
$$247$$ 0 0
$$248$$ −1.38197 −0.0877549
$$249$$ 16.6700 1.05642
$$250$$ 0 0
$$251$$ −3.00000 −0.189358 −0.0946792 0.995508i $$-0.530183\pi$$
−0.0946792 + 0.995508i $$0.530183\pi$$
$$252$$ 4.23607 0.266847
$$253$$ 23.9443 1.50536
$$254$$ −17.5623 −1.10196
$$255$$ 0 0
$$256$$ 20.2705 1.26691
$$257$$ 3.24920 0.202679 0.101340 0.994852i $$-0.467687\pi$$
0.101340 + 0.994852i $$0.467687\pi$$
$$258$$ −1.38197 −0.0860374
$$259$$ 11.8617 0.737051
$$260$$ 0 0
$$261$$ −1.73060 −0.107121
$$262$$ −29.7073 −1.83532
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ 8.09017 0.497916
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 14.7984 0.905646
$$268$$ −9.68208 −0.591427
$$269$$ 19.6417 1.19757 0.598787 0.800908i $$-0.295650\pi$$
0.598787 + 0.800908i $$0.295650\pi$$
$$270$$ 0 0
$$271$$ 8.61803 0.523508 0.261754 0.965135i $$-0.415699\pi$$
0.261754 + 0.965135i $$0.415699\pi$$
$$272$$ 24.1803 1.46615
$$273$$ −24.7984 −1.50087
$$274$$ −18.5721 −1.12198
$$275$$ 0 0
$$276$$ −12.5882 −0.757724
$$277$$ −24.8328 −1.49206 −0.746030 0.665913i $$-0.768042\pi$$
−0.746030 + 0.665913i $$0.768042\pi$$
$$278$$ −42.0180 −2.52007
$$279$$ 1.17557 0.0703796
$$280$$ 0 0
$$281$$ −25.0705 −1.49558 −0.747790 0.663935i $$-0.768885\pi$$
−0.747790 + 0.663935i $$0.768885\pi$$
$$282$$ 5.32624 0.317173
$$283$$ 28.2705 1.68051 0.840254 0.542193i $$-0.182406\pi$$
0.840254 + 0.542193i $$0.182406\pi$$
$$284$$ 0.277515 0.0164675
$$285$$ 0 0
$$286$$ 34.2705 2.02646
$$287$$ −29.1522 −1.72080
$$288$$ −4.53077 −0.266978
$$289$$ 10.4164 0.612730
$$290$$ 0 0
$$291$$ 5.00000 0.293105
$$292$$ 1.61803 0.0946883
$$293$$ −13.8293 −0.807918 −0.403959 0.914777i $$-0.632366\pi$$
−0.403959 + 0.914777i $$0.632366\pi$$
$$294$$ 39.5967 2.30933
$$295$$ 0 0
$$296$$ −2.03444 −0.118250
$$297$$ 26.5236 1.53905
$$298$$ 13.4863 0.781241
$$299$$ 12.5882 0.727997
$$300$$ 0 0
$$301$$ −1.61803 −0.0932619
$$302$$ 1.18034 0.0679209
$$303$$ 6.26137 0.359706
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −6.15537 −0.351879
$$307$$ −18.3601 −1.04787 −0.523933 0.851759i $$-0.675536\pi$$
−0.523933 + 0.851759i $$0.675536\pi$$
$$308$$ −40.1246 −2.28631
$$309$$ 28.4164 1.61655
$$310$$ 0 0
$$311$$ −2.76393 −0.156728 −0.0783641 0.996925i $$-0.524970\pi$$
−0.0783641 + 0.996925i $$0.524970\pi$$
$$312$$ 4.25325 0.240793
$$313$$ −3.27051 −0.184860 −0.0924301 0.995719i $$-0.529463\pi$$
−0.0924301 + 0.995719i $$0.529463\pi$$
$$314$$ 30.9483 1.74652
$$315$$ 0 0
$$316$$ −8.50651 −0.478528
$$317$$ 16.4580 0.924373 0.462186 0.886783i $$-0.347065\pi$$
0.462186 + 0.886783i $$0.347065\pi$$
$$318$$ −40.2874 −2.25921
$$319$$ 16.3925 0.917802
$$320$$ 0 0
$$321$$ 15.8541 0.884890
$$322$$ −32.9565 −1.83659
$$323$$ 0 0
$$324$$ −16.9443 −0.941348
$$325$$ 0 0
$$326$$ 47.8959 2.65271
$$327$$ −4.14590 −0.229269
$$328$$ 5.00000 0.276079
$$329$$ 6.23607 0.343806
$$330$$ 0 0
$$331$$ −19.9192 −1.09486 −0.547429 0.836852i $$-0.684393\pi$$
−0.547429 + 0.836852i $$0.684393\pi$$
$$332$$ 14.1803 0.778247
$$333$$ 1.73060 0.0948363
$$334$$ −0.326238 −0.0178509
$$335$$ 0 0
$$336$$ −37.2097 −2.02996
$$337$$ −28.4257 −1.54845 −0.774223 0.632913i $$-0.781859\pi$$
−0.774223 + 0.632913i $$0.781859\pi$$
$$338$$ −6.71040 −0.364997
$$339$$ −35.1246 −1.90771
$$340$$ 0 0
$$341$$ −11.1352 −0.603003
$$342$$ 0 0
$$343$$ 16.7082 0.902158
$$344$$ 0.277515 0.0149626
$$345$$ 0 0
$$346$$ 25.1246 1.35071
$$347$$ −16.4721 −0.884271 −0.442135 0.896948i $$-0.645779\pi$$
−0.442135 + 0.896948i $$0.645779\pi$$
$$348$$ −8.61803 −0.461975
$$349$$ −2.56231 −0.137157 −0.0685785 0.997646i $$-0.521846\pi$$
−0.0685785 + 0.997646i $$0.521846\pi$$
$$350$$ 0 0
$$351$$ 13.9443 0.744290
$$352$$ 42.9161 2.28743
$$353$$ −3.67376 −0.195535 −0.0977673 0.995209i $$-0.531170\pi$$
−0.0977673 + 0.995209i $$0.531170\pi$$
$$354$$ 50.8020 2.70010
$$355$$ 0 0
$$356$$ 12.5882 0.667176
$$357$$ −42.1895 −2.23291
$$358$$ 19.2705 1.01848
$$359$$ 7.38197 0.389605 0.194803 0.980842i $$-0.437593\pi$$
0.194803 + 0.980842i $$0.437593\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ −12.7639 −0.670857
$$363$$ 44.2631 2.32321
$$364$$ −21.0948 −1.10567
$$365$$ 0 0
$$366$$ 14.2705 0.745931
$$367$$ −31.0689 −1.62178 −0.810891 0.585197i $$-0.801017\pi$$
−0.810891 + 0.585197i $$0.801017\pi$$
$$368$$ 18.8885 0.984633
$$369$$ −4.25325 −0.221416
$$370$$ 0 0
$$371$$ −47.1693 −2.44891
$$372$$ 5.85410 0.303521
$$373$$ −5.42882 −0.281094 −0.140547 0.990074i $$-0.544886\pi$$
−0.140547 + 0.990074i $$0.544886\pi$$
$$374$$ 58.3045 3.01485
$$375$$ 0 0
$$376$$ −1.06957 −0.0551588
$$377$$ 8.61803 0.443851
$$378$$ −36.5066 −1.87770
$$379$$ −12.8658 −0.660870 −0.330435 0.943829i $$-0.607195\pi$$
−0.330435 + 0.943829i $$0.607195\pi$$
$$380$$ 0 0
$$381$$ −17.5623 −0.899744
$$382$$ 24.7275 1.26517
$$383$$ 4.25325 0.217331 0.108666 0.994078i $$-0.465342\pi$$
0.108666 + 0.994078i $$0.465342\pi$$
$$384$$ 10.8541 0.553896
$$385$$ 0 0
$$386$$ −2.76393 −0.140680
$$387$$ −0.236068 −0.0120000
$$388$$ 4.25325 0.215926
$$389$$ −20.8541 −1.05734 −0.528672 0.848826i $$-0.677310\pi$$
−0.528672 + 0.848826i $$0.677310\pi$$
$$390$$ 0 0
$$391$$ 21.4164 1.08307
$$392$$ −7.95148 −0.401610
$$393$$ −29.7073 −1.49853
$$394$$ −28.3197 −1.42673
$$395$$ 0 0
$$396$$ −5.85410 −0.294180
$$397$$ 23.8328 1.19613 0.598067 0.801446i $$-0.295936\pi$$
0.598067 + 0.801446i $$0.295936\pi$$
$$398$$ 25.5195 1.27918
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 9.06154 0.452512 0.226256 0.974068i $$-0.427351\pi$$
0.226256 + 0.974068i $$0.427351\pi$$
$$402$$ −21.6498 −1.07979
$$403$$ −5.85410 −0.291614
$$404$$ 5.32624 0.264990
$$405$$ 0 0
$$406$$ −22.5623 −1.11975
$$407$$ −16.3925 −0.812545
$$408$$ 7.23607 0.358239
$$409$$ 21.7153 1.07375 0.536876 0.843661i $$-0.319604\pi$$
0.536876 + 0.843661i $$0.319604\pi$$
$$410$$ 0 0
$$411$$ −18.5721 −0.916094
$$412$$ 24.1724 1.19089
$$413$$ 59.4800 2.92682
$$414$$ −4.80828 −0.236314
$$415$$ 0 0
$$416$$ 22.5623 1.10621
$$417$$ −42.0180 −2.05763
$$418$$ 0 0
$$419$$ 26.1803 1.27899 0.639497 0.768794i $$-0.279143\pi$$
0.639497 + 0.768794i $$0.279143\pi$$
$$420$$ 0 0
$$421$$ −11.5842 −0.564579 −0.282289 0.959329i $$-0.591094\pi$$
−0.282289 + 0.959329i $$0.591094\pi$$
$$422$$ −3.29180 −0.160242
$$423$$ 0.909830 0.0442375
$$424$$ 8.09017 0.392893
$$425$$ 0 0
$$426$$ 0.620541 0.0300653
$$427$$ 16.7082 0.808567
$$428$$ 13.4863 0.651885
$$429$$ 34.2705 1.65460
$$430$$ 0 0
$$431$$ 6.88191 0.331490 0.165745 0.986169i $$-0.446997\pi$$
0.165745 + 0.986169i $$0.446997\pi$$
$$432$$ 20.9232 1.00667
$$433$$ 6.04937 0.290714 0.145357 0.989379i $$-0.453567\pi$$
0.145357 + 0.989379i $$0.453567\pi$$
$$434$$ 15.3262 0.735683
$$435$$ 0 0
$$436$$ −3.52671 −0.168899
$$437$$ 0 0
$$438$$ 3.61803 0.172876
$$439$$ 5.64083 0.269222 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$440$$ 0 0
$$441$$ 6.76393 0.322092
$$442$$ 30.6525 1.45799
$$443$$ −34.7771 −1.65231 −0.826155 0.563443i $$-0.809476\pi$$
−0.826155 + 0.563443i $$0.809476\pi$$
$$444$$ 8.61803 0.408994
$$445$$ 0 0
$$446$$ 28.2148 1.33601
$$447$$ 13.4863 0.637880
$$448$$ −19.9443 −0.942278
$$449$$ 5.64083 0.266207 0.133104 0.991102i $$-0.457506\pi$$
0.133104 + 0.991102i $$0.457506\pi$$
$$450$$ 0 0
$$451$$ 40.2874 1.89706
$$452$$ −29.8788 −1.40538
$$453$$ 1.18034 0.0554572
$$454$$ −33.0902 −1.55300
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5.23607 −0.244933 −0.122466 0.992473i $$-0.539080\pi$$
−0.122466 + 0.992473i $$0.539080\pi$$
$$458$$ −34.5155 −1.61281
$$459$$ 23.7234 1.10731
$$460$$ 0 0
$$461$$ −8.00000 −0.372597 −0.186299 0.982493i $$-0.559649\pi$$
−0.186299 + 0.982493i $$0.559649\pi$$
$$462$$ −89.7214 −4.17422
$$463$$ −0.145898 −0.00678046 −0.00339023 0.999994i $$-0.501079\pi$$
−0.00339023 + 0.999994i $$0.501079\pi$$
$$464$$ 12.9313 0.600319
$$465$$ 0 0
$$466$$ −3.35520 −0.155427
$$467$$ 9.18034 0.424815 0.212408 0.977181i $$-0.431870\pi$$
0.212408 + 0.977181i $$0.431870\pi$$
$$468$$ −3.07768 −0.142266
$$469$$ −25.3480 −1.17046
$$470$$ 0 0
$$471$$ 30.9483 1.42602
$$472$$ −10.2016 −0.469568
$$473$$ 2.23607 0.102815
$$474$$ −19.0211 −0.873669
$$475$$ 0 0
$$476$$ −35.8885 −1.64495
$$477$$ −6.88191 −0.315101
$$478$$ −44.1571 −2.01970
$$479$$ −11.4377 −0.522602 −0.261301 0.965257i $$-0.584151\pi$$
−0.261301 + 0.965257i $$0.584151\pi$$
$$480$$ 0 0
$$481$$ −8.61803 −0.392949
$$482$$ −12.0344 −0.548154
$$483$$ −32.9565 −1.49957
$$484$$ 37.6525 1.71148
$$485$$ 0 0
$$486$$ −12.0344 −0.545893
$$487$$ 21.0292 0.952926 0.476463 0.879195i $$-0.341919\pi$$
0.476463 + 0.879195i $$0.341919\pi$$
$$488$$ −2.86568 −0.129723
$$489$$ 47.8959 2.16593
$$490$$ 0 0
$$491$$ 37.2705 1.68199 0.840997 0.541039i $$-0.181969\pi$$
0.840997 + 0.541039i $$0.181969\pi$$
$$492$$ −21.1803 −0.954883
$$493$$ 14.6619 0.660338
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.78402 −0.394414
$$497$$ 0.726543 0.0325899
$$498$$ 31.7082 1.42088
$$499$$ −29.5279 −1.32185 −0.660924 0.750453i $$-0.729836\pi$$
−0.660924 + 0.750453i $$0.729836\pi$$
$$500$$ 0 0
$$501$$ −0.326238 −0.0145752
$$502$$ −5.70634 −0.254686
$$503$$ −31.6525 −1.41131 −0.705657 0.708554i $$-0.749348\pi$$
−0.705657 + 0.708554i $$0.749348\pi$$
$$504$$ −1.90211 −0.0847268
$$505$$ 0 0
$$506$$ 45.5447 2.02471
$$507$$ −6.71040 −0.298019
$$508$$ −14.9394 −0.662828
$$509$$ 30.1563 1.33665 0.668327 0.743868i $$-0.267011\pi$$
0.668327 + 0.743868i $$0.267011\pi$$
$$510$$ 0 0
$$511$$ 4.23607 0.187393
$$512$$ 27.1441 1.19961
$$513$$ 0 0
$$514$$ 6.18034 0.272603
$$515$$ 0 0
$$516$$ −1.17557 −0.0517516
$$517$$ −8.61803 −0.379021
$$518$$ 22.5623 0.991331
$$519$$ 25.1246 1.10285
$$520$$ 0 0
$$521$$ −32.6789 −1.43169 −0.715845 0.698259i $$-0.753958\pi$$
−0.715845 + 0.698259i $$0.753958\pi$$
$$522$$ −3.29180 −0.144078
$$523$$ 6.53888 0.285925 0.142963 0.989728i $$-0.454337\pi$$
0.142963 + 0.989728i $$0.454337\pi$$
$$524$$ −25.2705 −1.10395
$$525$$ 0 0
$$526$$ 11.4127 0.497616
$$527$$ −9.95959 −0.433847
$$528$$ 51.4226 2.23788
$$529$$ −6.27051 −0.272631
$$530$$ 0 0
$$531$$ 8.67802 0.376594
$$532$$ 0 0
$$533$$ 21.1803 0.917422
$$534$$ 28.1482 1.21809
$$535$$ 0 0
$$536$$ 4.34752 0.187784
$$537$$ 19.2705 0.831584
$$538$$ 37.3607 1.61073
$$539$$ −64.0689 −2.75964
$$540$$ 0 0
$$541$$ −24.5967 −1.05750 −0.528748 0.848779i $$-0.677338\pi$$
−0.528748 + 0.848779i $$0.677338\pi$$
$$542$$ 16.3925 0.704117
$$543$$ −12.7639 −0.547753
$$544$$ 38.3853 1.64576
$$545$$ 0 0
$$546$$ −47.1693 −2.01866
$$547$$ 21.2663 0.909280 0.454640 0.890675i $$-0.349768\pi$$
0.454640 + 0.890675i $$0.349768\pi$$
$$548$$ −15.7984 −0.674873
$$549$$ 2.43769 0.104038
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 5.65248 0.240585
$$553$$ −22.2703 −0.947031
$$554$$ −47.2348 −2.00682
$$555$$ 0 0
$$556$$ −35.7426 −1.51583
$$557$$ 31.0689 1.31643 0.658215 0.752830i $$-0.271312\pi$$
0.658215 + 0.752830i $$0.271312\pi$$
$$558$$ 2.23607 0.0946603
$$559$$ 1.17557 0.0497213
$$560$$ 0 0
$$561$$ 58.3045 2.46162
$$562$$ −47.6869 −2.01155
$$563$$ 4.35926 0.183721 0.0918604 0.995772i $$-0.470719\pi$$
0.0918604 + 0.995772i $$0.470719\pi$$
$$564$$ 4.53077 0.190780
$$565$$ 0 0
$$566$$ 53.7737 2.26028
$$567$$ −44.3607 −1.86297
$$568$$ −0.124612 −0.00522859
$$569$$ −25.7970 −1.08147 −0.540734 0.841194i $$-0.681853\pi$$
−0.540734 + 0.841194i $$0.681853\pi$$
$$570$$ 0 0
$$571$$ −25.8541 −1.08196 −0.540980 0.841035i $$-0.681947\pi$$
−0.540980 + 0.841035i $$0.681947\pi$$
$$572$$ 29.1522 1.21892
$$573$$ 24.7275 1.03300
$$574$$ −55.4508 −2.31447
$$575$$ 0 0
$$576$$ −2.90983 −0.121243
$$577$$ −8.52786 −0.355020 −0.177510 0.984119i $$-0.556804\pi$$
−0.177510 + 0.984119i $$0.556804\pi$$
$$578$$ 19.8132 0.824120
$$579$$ −2.76393 −0.114865
$$580$$ 0 0
$$581$$ 37.1246 1.54019
$$582$$ 9.51057 0.394226
$$583$$ 65.1864 2.69974
$$584$$ −0.726543 −0.0300645
$$585$$ 0 0
$$586$$ −26.3050 −1.08665
$$587$$ −36.4164 −1.50307 −0.751533 0.659695i $$-0.770685\pi$$
−0.751533 + 0.659695i $$0.770685\pi$$
$$588$$ 33.6830 1.38906
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −28.3197 −1.16492
$$592$$ −12.9313 −0.531472
$$593$$ −20.7082 −0.850384 −0.425192 0.905103i $$-0.639793\pi$$
−0.425192 + 0.905103i $$0.639793\pi$$
$$594$$ 50.4508 2.07002
$$595$$ 0 0
$$596$$ 11.4721 0.469917
$$597$$ 25.5195 1.04444
$$598$$ 23.9443 0.979154
$$599$$ −30.0503 −1.22782 −0.613911 0.789375i $$-0.710405\pi$$
−0.613911 + 0.789375i $$0.710405\pi$$
$$600$$ 0 0
$$601$$ 10.5146 0.428900 0.214450 0.976735i $$-0.431204\pi$$
0.214450 + 0.976735i $$0.431204\pi$$
$$602$$ −3.07768 −0.125437
$$603$$ −3.69822 −0.150603
$$604$$ 1.00406 0.0408545
$$605$$ 0 0
$$606$$ 11.9098 0.483804
$$607$$ 16.2210 0.658389 0.329194 0.944262i $$-0.393223\pi$$
0.329194 + 0.944262i $$0.393223\pi$$
$$608$$ 0 0
$$609$$ −22.5623 −0.914271
$$610$$ 0 0
$$611$$ −4.53077 −0.183295
$$612$$ −5.23607 −0.211656
$$613$$ −26.5279 −1.07145 −0.535725 0.844392i $$-0.679962\pi$$
−0.535725 + 0.844392i $$0.679962\pi$$
$$614$$ −34.9230 −1.40938
$$615$$ 0 0
$$616$$ 18.0171 0.725929
$$617$$ −21.6180 −0.870309 −0.435155 0.900356i $$-0.643306\pi$$
−0.435155 + 0.900356i $$0.643306\pi$$
$$618$$ 54.0512 2.17426
$$619$$ 39.5410 1.58929 0.794644 0.607076i $$-0.207658\pi$$
0.794644 + 0.607076i $$0.207658\pi$$
$$620$$ 0 0
$$621$$ 18.5316 0.743648
$$622$$ −5.25731 −0.210799
$$623$$ 32.9565 1.32037
$$624$$ 27.0344 1.08224
$$625$$ 0 0
$$626$$ −6.22088 −0.248636
$$627$$ 0 0
$$628$$ 26.3262 1.05053
$$629$$ −14.6619 −0.584607
$$630$$ 0 0
$$631$$ −41.1803 −1.63936 −0.819682 0.572819i $$-0.805850\pi$$
−0.819682 + 0.572819i $$0.805850\pi$$
$$632$$ 3.81966 0.151938
$$633$$ −3.29180 −0.130837
$$634$$ 31.3050 1.24328
$$635$$ 0 0
$$636$$ −34.2705 −1.35891
$$637$$ −33.6830 −1.33457
$$638$$ 31.1803 1.23444
$$639$$ 0.106001 0.00419334
$$640$$ 0 0
$$641$$ −36.6952 −1.44937 −0.724686 0.689079i $$-0.758015\pi$$
−0.724686 + 0.689079i $$0.758015\pi$$
$$642$$ 30.1563 1.19017
$$643$$ 38.4721 1.51719 0.758596 0.651561i $$-0.225885\pi$$
0.758596 + 0.651561i $$0.225885\pi$$
$$644$$ −28.0344 −1.10471
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −8.52786 −0.335265 −0.167632 0.985850i $$-0.553612\pi$$
−0.167632 + 0.985850i $$0.553612\pi$$
$$648$$ 7.60845 0.298888
$$649$$ −82.1994 −3.22661
$$650$$ 0 0
$$651$$ 15.3262 0.600683
$$652$$ 40.7426 1.59561
$$653$$ 13.7984 0.539972 0.269986 0.962864i $$-0.412981\pi$$
0.269986 + 0.962864i $$0.412981\pi$$
$$654$$ −7.88597 −0.308366
$$655$$ 0 0
$$656$$ 31.7809 1.24084
$$657$$ 0.618034 0.0241118
$$658$$ 11.8617 0.462417
$$659$$ 7.26543 0.283021 0.141510 0.989937i $$-0.454804\pi$$
0.141510 + 0.989937i $$0.454804\pi$$
$$660$$ 0 0
$$661$$ −29.3238 −1.14056 −0.570281 0.821450i $$-0.693166\pi$$
−0.570281 + 0.821450i $$0.693166\pi$$
$$662$$ −37.8885 −1.47258
$$663$$ 30.6525 1.19044
$$664$$ −6.36737 −0.247102
$$665$$ 0 0
$$666$$ 3.29180 0.127555
$$667$$ 11.4532 0.443468
$$668$$ −0.277515 −0.0107374
$$669$$ 28.2148 1.09085
$$670$$ 0 0
$$671$$ −23.0902 −0.891386
$$672$$ −59.0689 −2.27863
$$673$$ −24.2380 −0.934304 −0.467152 0.884177i $$-0.654720\pi$$
−0.467152 + 0.884177i $$0.654720\pi$$
$$674$$ −54.0689 −2.08266
$$675$$ 0 0
$$676$$ −5.70820 −0.219546
$$677$$ −33.5770 −1.29047 −0.645235 0.763985i $$-0.723240\pi$$
−0.645235 + 0.763985i $$0.723240\pi$$
$$678$$ −66.8110 −2.56586
$$679$$ 11.1352 0.427328
$$680$$ 0 0
$$681$$ −33.0902 −1.26802
$$682$$ −21.1803 −0.811037
$$683$$ −28.0422 −1.07300 −0.536502 0.843899i $$-0.680255\pi$$
−0.536502 + 0.843899i $$0.680255\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 31.7809 1.21340
$$687$$ −34.5155 −1.31685
$$688$$ 1.76393 0.0672493
$$689$$ 34.2705 1.30560
$$690$$ 0 0
$$691$$ −12.2361 −0.465482 −0.232741 0.972539i $$-0.574769\pi$$
−0.232741 + 0.972539i $$0.574769\pi$$
$$692$$ 21.3723 0.812452
$$693$$ −15.3262 −0.582196
$$694$$ −31.3319 −1.18934
$$695$$ 0 0
$$696$$ 3.86974 0.146682
$$697$$ 36.0341 1.36489
$$698$$ −4.87380 −0.184476
$$699$$ −3.35520 −0.126905
$$700$$ 0 0
$$701$$ −31.9098 −1.20522 −0.602609 0.798037i $$-0.705872\pi$$
−0.602609 + 0.798037i $$0.705872\pi$$
$$702$$ 26.5236 1.00107
$$703$$ 0 0
$$704$$ 27.5623 1.03879
$$705$$ 0 0
$$706$$ −6.98791 −0.262993
$$707$$ 13.9443 0.524428
$$708$$ 43.2148 1.62411
$$709$$ 36.1803 1.35878 0.679391 0.733777i $$-0.262244\pi$$
0.679391 + 0.733777i $$0.262244\pi$$
$$710$$ 0 0
$$711$$ −3.24920 −0.121854
$$712$$ −5.65248 −0.211835
$$713$$ −7.77997 −0.291362
$$714$$ −80.2492 −3.00325
$$715$$ 0 0
$$716$$ 16.3925 0.612616
$$717$$ −44.1571 −1.64908
$$718$$ 14.0413 0.524018
$$719$$ −30.9098 −1.15274 −0.576371 0.817188i $$-0.695532\pi$$
−0.576371 + 0.817188i $$0.695532\pi$$
$$720$$ 0 0
$$721$$ 63.2843 2.35683
$$722$$ 0 0
$$723$$ −12.0344 −0.447566
$$724$$ −10.8576 −0.403521
$$725$$ 0 0
$$726$$ 84.1935 3.12471
$$727$$ −34.3607 −1.27437 −0.637184 0.770712i $$-0.719901\pi$$
−0.637184 + 0.770712i $$0.719901\pi$$
$$728$$ 9.47214 0.351061
$$729$$ 19.3820 0.717851
$$730$$ 0 0
$$731$$ 2.00000 0.0739727
$$732$$ 12.1392 0.448679
$$733$$ 11.0000 0.406294 0.203147 0.979148i $$-0.434883\pi$$
0.203147 + 0.979148i $$0.434883\pi$$
$$734$$ −59.0965 −2.18129
$$735$$ 0 0
$$736$$ 29.9848 1.10525
$$737$$ 35.0301 1.29035
$$738$$ −8.09017 −0.297803
$$739$$ −27.7639 −1.02131 −0.510656 0.859785i $$-0.670598\pi$$
−0.510656 + 0.859785i $$0.670598\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −89.7214 −3.29377
$$743$$ 5.98385 0.219526 0.109763 0.993958i $$-0.464991\pi$$
0.109763 + 0.993958i $$0.464991\pi$$
$$744$$ −2.62866 −0.0963712
$$745$$ 0 0
$$746$$ −10.3262 −0.378070
$$747$$ 5.41641 0.198176
$$748$$ 49.5967 1.81344
$$749$$ 35.3076 1.29011
$$750$$ 0 0
$$751$$ 50.4590 1.84127 0.920637 0.390419i $$-0.127670\pi$$
0.920637 + 0.390419i $$0.127670\pi$$
$$752$$ −6.79837 −0.247911
$$753$$ −5.70634 −0.207951
$$754$$ 16.3925 0.596979
$$755$$ 0 0
$$756$$ −31.0543 −1.12944
$$757$$ −2.97871 −0.108263 −0.0541316 0.998534i $$-0.517239\pi$$
−0.0541316 + 0.998534i $$0.517239\pi$$
$$758$$ −24.4721 −0.888868
$$759$$ 45.5447 1.65317
$$760$$ 0 0
$$761$$ −18.8197 −0.682212 −0.341106 0.940025i $$-0.610802\pi$$
−0.341106 + 0.940025i $$0.610802\pi$$
$$762$$ −33.4055 −1.21015
$$763$$ −9.23305 −0.334259
$$764$$ 21.0344 0.760999
$$765$$ 0 0
$$766$$ 8.09017 0.292310
$$767$$ −43.2148 −1.56040
$$768$$ 38.5568 1.39130
$$769$$ −13.2705 −0.478547 −0.239273 0.970952i $$-0.576909\pi$$
−0.239273 + 0.970952i $$0.576909\pi$$
$$770$$ 0 0
$$771$$ 6.18034 0.222580
$$772$$ −2.35114 −0.0846194
$$773$$ 32.5074 1.16921 0.584606 0.811318i $$-0.301249\pi$$
0.584606 + 0.811318i $$0.301249\pi$$
$$774$$ −0.449028 −0.0161400
$$775$$ 0 0
$$776$$ −1.90983 −0.0685589
$$777$$ 22.5623 0.809418
$$778$$ −39.6669 −1.42213
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −1.00406 −0.0359280
$$782$$ 40.7364 1.45673
$$783$$ 12.6869 0.453393
$$784$$ −50.5410 −1.80504
$$785$$ 0 0
$$786$$ −56.5066 −2.01552
$$787$$ −20.1312 −0.717599 −0.358800 0.933415i $$-0.616814\pi$$
−0.358800 + 0.933415i $$0.616814\pi$$
$$788$$ −24.0902 −0.858177
$$789$$ 11.4127 0.406302
$$790$$ 0 0
$$791$$ −78.2237 −2.78131
$$792$$ 2.62866 0.0934052
$$793$$ −12.1392 −0.431076
$$794$$ 45.3327 1.60880
$$795$$ 0 0
$$796$$ 21.7082 0.769427
$$797$$ 23.7234 0.840326 0.420163 0.907449i $$-0.361973\pi$$
0.420163 + 0.907449i $$0.361973\pi$$
$$798$$ 0 0
$$799$$ −7.70820 −0.272697
$$800$$ 0 0
$$801$$ 4.80828 0.169892
$$802$$ 17.2361 0.608627
$$803$$ −5.85410 −0.206587
$$804$$ −18.4164 −0.649497
$$805$$ 0 0
$$806$$ −11.1352 −0.392219
$$807$$ 37.3607 1.31516
$$808$$ −2.39163 −0.0841372
$$809$$ −53.2148 −1.87093 −0.935466 0.353417i $$-0.885020\pi$$
−0.935466 + 0.353417i $$0.885020\pi$$
$$810$$ 0 0
$$811$$ −13.1433 −0.461523 −0.230761 0.973010i $$-0.574122\pi$$
−0.230761 + 0.973010i $$0.574122\pi$$
$$812$$ −19.1926 −0.673530
$$813$$ 16.3925 0.574909
$$814$$ −31.1803 −1.09287
$$815$$ 0 0
$$816$$ 45.9937 1.61010
$$817$$ 0 0
$$818$$ 41.3050 1.44419
$$819$$ −8.05748 −0.281551
$$820$$ 0 0
$$821$$ −27.4508 −0.958041 −0.479021 0.877804i $$-0.659008\pi$$
−0.479021 + 0.877804i $$0.659008\pi$$
$$822$$ −35.3262 −1.23214
$$823$$ 16.7639 0.584354 0.292177 0.956364i $$-0.405620\pi$$
0.292177 + 0.956364i $$0.405620\pi$$
$$824$$ −10.8541 −0.378121
$$825$$ 0 0
$$826$$ 113.138 3.93657
$$827$$ 9.23305 0.321065 0.160532 0.987031i $$-0.448679\pi$$
0.160532 + 0.987031i $$0.448679\pi$$
$$828$$ −4.09017 −0.142143
$$829$$ 25.9686 0.901925 0.450963 0.892543i $$-0.351081\pi$$
0.450963 + 0.892543i $$0.351081\pi$$
$$830$$ 0 0
$$831$$ −47.2348 −1.63856
$$832$$ 14.4904 0.502363
$$833$$ −57.3050 −1.98550
$$834$$ −79.9230 −2.76751
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −8.61803 −0.297883
$$838$$ 49.7980 1.72024
$$839$$ −4.08174 −0.140917 −0.0704587 0.997515i $$-0.522446\pi$$
−0.0704587 + 0.997515i $$0.522446\pi$$
$$840$$ 0 0
$$841$$ −21.1591 −0.729623
$$842$$ −22.0344 −0.759357
$$843$$ −47.6869 −1.64242
$$844$$ −2.80017 −0.0963858
$$845$$ 0 0
$$846$$ 1.73060 0.0594992
$$847$$ 98.5755 3.38709
$$848$$ 51.4226 1.76586
$$849$$ 53.7737 1.84551
$$850$$ 0 0
$$851$$ −11.4532 −0.392610
$$852$$ 0.527864 0.0180843
$$853$$ 15.4721 0.529756 0.264878 0.964282i $$-0.414668\pi$$
0.264878 + 0.964282i $$0.414668\pi$$
$$854$$ 31.7809 1.08752
$$855$$ 0 0
$$856$$ −6.05573 −0.206981
$$857$$ −6.22088 −0.212501 −0.106251 0.994339i $$-0.533885\pi$$
−0.106251 + 0.994339i $$0.533885\pi$$
$$858$$ 65.1864 2.22543
$$859$$ −22.7639 −0.776695 −0.388348 0.921513i $$-0.626954\pi$$
−0.388348 + 0.921513i $$0.626954\pi$$
$$860$$ 0 0
$$861$$ −55.4508 −1.88976
$$862$$ 13.0902 0.445853
$$863$$ −36.3772 −1.23829 −0.619147 0.785275i $$-0.712521\pi$$
−0.619147 + 0.785275i $$0.712521\pi$$
$$864$$ 33.2148 1.12999
$$865$$ 0 0
$$866$$ 11.5066 0.391009
$$867$$ 19.8132 0.672891
$$868$$ 13.0373 0.442514
$$869$$ 30.7768 1.04403
$$870$$ 0 0
$$871$$ 18.4164 0.624016
$$872$$ 1.58359 0.0536272
$$873$$ 1.62460 0.0549843
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 3.07768 0.103985
$$877$$ 11.9677 0.404121 0.202060 0.979373i $$-0.435236\pi$$
0.202060 + 0.979373i $$0.435236\pi$$
$$878$$ 10.7295 0.362103
$$879$$ −26.3050 −0.887244
$$880$$ 0 0
$$881$$ 10.3262 0.347900 0.173950 0.984755i $$-0.444347\pi$$
0.173950 + 0.984755i $$0.444347\pi$$
$$882$$ 12.8658 0.433213
$$883$$ 5.78522 0.194688 0.0973440 0.995251i $$-0.468965\pi$$
0.0973440 + 0.995251i $$0.468965\pi$$
$$884$$ 26.0746 0.876982
$$885$$ 0 0
$$886$$ −66.1500 −2.22235
$$887$$ 14.3188 0.480780 0.240390 0.970676i $$-0.422725\pi$$
0.240390 + 0.970676i $$0.422725\pi$$
$$888$$ −3.86974 −0.129860
$$889$$ −39.1118 −1.31177
$$890$$ 0 0
$$891$$ 61.3050 2.05379
$$892$$ 24.0009 0.803610
$$893$$ 0 0
$$894$$ 25.6525 0.857947
$$895$$ 0 0
$$896$$ 24.1724 0.807545
$$897$$ 23.9443 0.799476
$$898$$ 10.7295 0.358048
$$899$$ −5.32624 −0.177640
$$900$$ 0 0
$$901$$ 58.3045 1.94240
$$902$$ 76.6312 2.55154
$$903$$ −3.07768 −0.102419
$$904$$ 13.4164 0.446223
$$905$$ 0 0
$$906$$ 2.24514 0.0745898
$$907$$ 51.9371 1.72454 0.862272 0.506446i $$-0.169041\pi$$
0.862272 + 0.506446i $$0.169041\pi$$
$$908$$ −28.1482 −0.934130
$$909$$ 2.03444 0.0674782
$$910$$ 0 0
$$911$$ 34.3035 1.13653 0.568264 0.822847i $$-0.307615\pi$$
0.568264 + 0.822847i $$0.307615\pi$$
$$912$$ 0 0
$$913$$ −51.3050 −1.69795
$$914$$ −9.95959 −0.329434
$$915$$ 0 0
$$916$$ −29.3607 −0.970104
$$917$$ −66.1591 −2.18476
$$918$$ 45.1246 1.48933
$$919$$ 7.36068 0.242806 0.121403 0.992603i $$-0.461261\pi$$
0.121403 + 0.992603i $$0.461261\pi$$
$$920$$ 0 0
$$921$$ −34.9230 −1.15075
$$922$$ −15.2169 −0.501142
$$923$$ −0.527864 −0.0173749
$$924$$ −76.3215 −2.51079
$$925$$ 0 0
$$926$$ −0.277515 −0.00911969
$$927$$ 9.23305 0.303253
$$928$$ 20.5279 0.673860
$$929$$ 37.5623 1.23238 0.616190 0.787598i $$-0.288675\pi$$
0.616190 + 0.787598i $$0.288675\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −2.85410 −0.0934892
$$933$$ −5.25731 −0.172117
$$934$$ 17.4620 0.571376
$$935$$ 0 0
$$936$$ 1.38197 0.0451710
$$937$$ −23.2016 −0.757964 −0.378982 0.925404i $$-0.623726\pi$$
−0.378982 + 0.925404i $$0.623726\pi$$
$$938$$ −48.2148 −1.57427
$$939$$ −6.22088 −0.203011
$$940$$ 0 0
$$941$$ −50.7615 −1.65478 −0.827389 0.561629i $$-0.810175\pi$$
−0.827389 + 0.561629i $$0.810175\pi$$
$$942$$ 58.8673 1.91800
$$943$$ 28.1482 0.916631
$$944$$ −64.8434 −2.11047
$$945$$ 0 0
$$946$$ 4.25325 0.138285
$$947$$ −1.41641 −0.0460271 −0.0230135 0.999735i $$-0.507326\pi$$
−0.0230135 + 0.999735i $$0.507326\pi$$
$$948$$ −16.1803 −0.525513
$$949$$ −3.07768 −0.0999058
$$950$$ 0 0
$$951$$ 31.3050 1.01513
$$952$$ 16.1150 0.522289
$$953$$ −42.2300 −1.36796 −0.683982 0.729499i $$-0.739753\pi$$
−0.683982 + 0.729499i $$0.739753\pi$$
$$954$$ −13.0902 −0.423810
$$955$$ 0 0
$$956$$ −37.5623 −1.21485
$$957$$ 31.1803 1.00792
$$958$$ −21.7558 −0.702898
$$959$$ −41.3607 −1.33561
$$960$$ 0 0
$$961$$ −27.3820 −0.883289
$$962$$ −16.3925 −0.528515
$$963$$ 5.15131 0.165999
$$964$$ −10.2371 −0.329715
$$965$$ 0 0
$$966$$ −62.6869 −2.01692
$$967$$ 29.0557 0.934369 0.467185 0.884160i $$-0.345268\pi$$
0.467185 + 0.884160i $$0.345268\pi$$
$$968$$ −16.9070 −0.543412
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27.4216 −0.880002 −0.440001 0.897997i $$-0.645022\pi$$
−0.440001 + 0.897997i $$0.645022\pi$$
$$972$$ −10.2371 −0.328355
$$973$$ −93.5755 −2.99989
$$974$$ 40.0000 1.28168
$$975$$ 0 0
$$976$$ −18.2148 −0.583041
$$977$$ 36.7607 1.17608 0.588039 0.808832i $$-0.299900\pi$$
0.588039 + 0.808832i $$0.299900\pi$$
$$978$$ 91.1033 2.91316
$$979$$ −45.5447 −1.45562
$$980$$ 0 0
$$981$$ −1.34708 −0.0430091
$$982$$ 70.8927 2.26228
$$983$$ 19.6417 0.626472 0.313236 0.949675i $$-0.398587\pi$$
0.313236 + 0.949675i $$0.398587\pi$$
$$984$$ 9.51057 0.303186
$$985$$ 0 0
$$986$$ 27.8885 0.888152
$$987$$ 11.8617 0.377562
$$988$$ 0 0
$$989$$ 1.56231 0.0496784
$$990$$ 0 0
$$991$$ 57.6839 1.83239 0.916195 0.400732i $$-0.131244\pi$$
0.916195 + 0.400732i $$0.131244\pi$$
$$992$$ −13.9443 −0.442731
$$993$$ −37.8885 −1.20236
$$994$$ 1.38197 0.0438333
$$995$$ 0 0
$$996$$ 26.9726 0.854660
$$997$$ 36.9443 1.17004 0.585018 0.811020i $$-0.301087\pi$$
0.585018 + 0.811020i $$0.301087\pi$$
$$998$$ −56.1653 −1.77788
$$999$$ −12.6869 −0.401396
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.bl.1.4 4
5.4 even 2 1805.2.a.l.1.1 4
19.18 odd 2 inner 9025.2.a.bl.1.1 4
95.94 odd 2 1805.2.a.l.1.4 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.l.1.1 4 5.4 even 2
1805.2.a.l.1.4 yes 4 95.94 odd 2
9025.2.a.bl.1.1 4 19.18 odd 2 inner
9025.2.a.bl.1.4 4 1.1 even 1 trivial