# Properties

 Label 448.2.a.j.1.1 Level $448$ Weight $2$ Character 448.1 Self dual yes Analytic conductor $3.577$ 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] = [448,2,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 448.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$3.57729801055$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 224) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 448.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +O(q^{10})$$ $$q-1.23607 q^{3} -3.23607 q^{5} +1.00000 q^{7} -1.47214 q^{9} +6.47214 q^{11} -0.763932 q^{13} +4.00000 q^{15} +4.47214 q^{17} +1.23607 q^{19} -1.23607 q^{21} +4.00000 q^{23} +5.47214 q^{25} +5.52786 q^{27} +4.47214 q^{29} -2.47214 q^{31} -8.00000 q^{33} -3.23607 q^{35} +4.47214 q^{37} +0.944272 q^{39} -8.47214 q^{41} -6.47214 q^{43} +4.76393 q^{45} +10.4721 q^{47} +1.00000 q^{49} -5.52786 q^{51} +10.0000 q^{53} -20.9443 q^{55} -1.52786 q^{57} +9.23607 q^{59} -11.2361 q^{61} -1.47214 q^{63} +2.47214 q^{65} -4.00000 q^{67} -4.94427 q^{69} -4.94427 q^{71} -2.94427 q^{73} -6.76393 q^{75} +6.47214 q^{77} +12.9443 q^{79} -2.41641 q^{81} +9.23607 q^{83} -14.4721 q^{85} -5.52786 q^{87} -6.00000 q^{89} -0.763932 q^{91} +3.05573 q^{93} -4.00000 q^{95} +12.4721 q^{97} -9.52786 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + 4 q^{11} - 6 q^{13} + 8 q^{15} - 2 q^{19} + 2 q^{21} + 8 q^{23} + 2 q^{25} + 20 q^{27} + 4 q^{31} - 16 q^{33} - 2 q^{35} - 16 q^{39} - 8 q^{41} - 4 q^{43} + 14 q^{45} + 12 q^{47} + 2 q^{49} - 20 q^{51} + 20 q^{53} - 24 q^{55} - 12 q^{57} + 14 q^{59} - 18 q^{61} + 6 q^{63} - 4 q^{65} - 8 q^{67} + 8 q^{69} + 8 q^{71} + 12 q^{73} - 18 q^{75} + 4 q^{77} + 8 q^{79} + 22 q^{81} + 14 q^{83} - 20 q^{85} - 20 q^{87} - 12 q^{89} - 6 q^{91} + 24 q^{93} - 8 q^{95} + 16 q^{97} - 28 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 + 4 * q^11 - 6 * q^13 + 8 * q^15 - 2 * q^19 + 2 * q^21 + 8 * q^23 + 2 * q^25 + 20 * q^27 + 4 * q^31 - 16 * q^33 - 2 * q^35 - 16 * q^39 - 8 * q^41 - 4 * q^43 + 14 * q^45 + 12 * q^47 + 2 * q^49 - 20 * q^51 + 20 * q^53 - 24 * q^55 - 12 * q^57 + 14 * q^59 - 18 * q^61 + 6 * q^63 - 4 * q^65 - 8 * q^67 + 8 * q^69 + 8 * q^71 + 12 * q^73 - 18 * q^75 + 4 * q^77 + 8 * q^79 + 22 * q^81 + 14 * q^83 - 20 * q^85 - 20 * q^87 - 12 * q^89 - 6 * q^91 + 24 * q^93 - 8 * q^95 + 16 * q^97 - 28 * 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 0
$$3$$ −1.23607 −0.713644 −0.356822 0.934172i $$-0.616140\pi$$
−0.356822 + 0.934172i $$0.616140\pi$$
$$4$$ 0 0
$$5$$ −3.23607 −1.44721 −0.723607 0.690212i $$-0.757517\pi$$
−0.723607 + 0.690212i $$0.757517\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ −1.47214 −0.490712
$$10$$ 0 0
$$11$$ 6.47214 1.95142 0.975711 0.219061i $$-0.0702993\pi$$
0.975711 + 0.219061i $$0.0702993\pi$$
$$12$$ 0 0
$$13$$ −0.763932 −0.211877 −0.105938 0.994373i $$-0.533785\pi$$
−0.105938 + 0.994373i $$0.533785\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 0 0
$$17$$ 4.47214 1.08465 0.542326 0.840168i $$-0.317544\pi$$
0.542326 + 0.840168i $$0.317544\pi$$
$$18$$ 0 0
$$19$$ 1.23607 0.283573 0.141787 0.989897i $$-0.454715\pi$$
0.141787 + 0.989897i $$0.454715\pi$$
$$20$$ 0 0
$$21$$ −1.23607 −0.269732
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 5.47214 1.09443
$$26$$ 0 0
$$27$$ 5.52786 1.06384
$$28$$ 0 0
$$29$$ 4.47214 0.830455 0.415227 0.909718i $$-0.363702\pi$$
0.415227 + 0.909718i $$0.363702\pi$$
$$30$$ 0 0
$$31$$ −2.47214 −0.444009 −0.222004 0.975046i $$-0.571260\pi$$
−0.222004 + 0.975046i $$0.571260\pi$$
$$32$$ 0 0
$$33$$ −8.00000 −1.39262
$$34$$ 0 0
$$35$$ −3.23607 −0.546995
$$36$$ 0 0
$$37$$ 4.47214 0.735215 0.367607 0.929981i $$-0.380177\pi$$
0.367607 + 0.929981i $$0.380177\pi$$
$$38$$ 0 0
$$39$$ 0.944272 0.151205
$$40$$ 0 0
$$41$$ −8.47214 −1.32313 −0.661563 0.749890i $$-0.730106\pi$$
−0.661563 + 0.749890i $$0.730106\pi$$
$$42$$ 0 0
$$43$$ −6.47214 −0.986991 −0.493496 0.869748i $$-0.664281\pi$$
−0.493496 + 0.869748i $$0.664281\pi$$
$$44$$ 0 0
$$45$$ 4.76393 0.710165
$$46$$ 0 0
$$47$$ 10.4721 1.52752 0.763759 0.645501i $$-0.223352\pi$$
0.763759 + 0.645501i $$0.223352\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −5.52786 −0.774056
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ −20.9443 −2.82413
$$56$$ 0 0
$$57$$ −1.52786 −0.202371
$$58$$ 0 0
$$59$$ 9.23607 1.20243 0.601217 0.799086i $$-0.294683\pi$$
0.601217 + 0.799086i $$0.294683\pi$$
$$60$$ 0 0
$$61$$ −11.2361 −1.43863 −0.719316 0.694683i $$-0.755544\pi$$
−0.719316 + 0.694683i $$0.755544\pi$$
$$62$$ 0 0
$$63$$ −1.47214 −0.185472
$$64$$ 0 0
$$65$$ 2.47214 0.306631
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −4.94427 −0.595220
$$70$$ 0 0
$$71$$ −4.94427 −0.586777 −0.293389 0.955993i $$-0.594783\pi$$
−0.293389 + 0.955993i $$0.594783\pi$$
$$72$$ 0 0
$$73$$ −2.94427 −0.344601 −0.172300 0.985044i $$-0.555120\pi$$
−0.172300 + 0.985044i $$0.555120\pi$$
$$74$$ 0 0
$$75$$ −6.76393 −0.781032
$$76$$ 0 0
$$77$$ 6.47214 0.737568
$$78$$ 0 0
$$79$$ 12.9443 1.45634 0.728172 0.685394i $$-0.240370\pi$$
0.728172 + 0.685394i $$0.240370\pi$$
$$80$$ 0 0
$$81$$ −2.41641 −0.268490
$$82$$ 0 0
$$83$$ 9.23607 1.01379 0.506895 0.862008i $$-0.330793\pi$$
0.506895 + 0.862008i $$0.330793\pi$$
$$84$$ 0 0
$$85$$ −14.4721 −1.56972
$$86$$ 0 0
$$87$$ −5.52786 −0.592649
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −0.763932 −0.0800818
$$92$$ 0 0
$$93$$ 3.05573 0.316864
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 12.4721 1.26635 0.633177 0.774007i $$-0.281751\pi$$
0.633177 + 0.774007i $$0.281751\pi$$
$$98$$ 0 0
$$99$$ −9.52786 −0.957586
$$100$$ 0 0
$$101$$ 1.70820 0.169973 0.0849863 0.996382i $$-0.472915\pi$$
0.0849863 + 0.996382i $$0.472915\pi$$
$$102$$ 0 0
$$103$$ −5.52786 −0.544677 −0.272338 0.962202i $$-0.587797\pi$$
−0.272338 + 0.962202i $$0.587797\pi$$
$$104$$ 0 0
$$105$$ 4.00000 0.390360
$$106$$ 0 0
$$107$$ −8.94427 −0.864675 −0.432338 0.901712i $$-0.642311\pi$$
−0.432338 + 0.901712i $$0.642311\pi$$
$$108$$ 0 0
$$109$$ −8.47214 −0.811483 −0.405742 0.913988i $$-0.632987\pi$$
−0.405742 + 0.913988i $$0.632987\pi$$
$$110$$ 0 0
$$111$$ −5.52786 −0.524682
$$112$$ 0 0
$$113$$ −12.4721 −1.17328 −0.586640 0.809848i $$-0.699550\pi$$
−0.586640 + 0.809848i $$0.699550\pi$$
$$114$$ 0 0
$$115$$ −12.9443 −1.20706
$$116$$ 0 0
$$117$$ 1.12461 0.103970
$$118$$ 0 0
$$119$$ 4.47214 0.409960
$$120$$ 0 0
$$121$$ 30.8885 2.80805
$$122$$ 0 0
$$123$$ 10.4721 0.944241
$$124$$ 0 0
$$125$$ −1.52786 −0.136656
$$126$$ 0 0
$$127$$ 8.94427 0.793676 0.396838 0.917889i $$-0.370108\pi$$
0.396838 + 0.917889i $$0.370108\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −11.7082 −1.02295 −0.511475 0.859298i $$-0.670901\pi$$
−0.511475 + 0.859298i $$0.670901\pi$$
$$132$$ 0 0
$$133$$ 1.23607 0.107181
$$134$$ 0 0
$$135$$ −17.8885 −1.53960
$$136$$ 0 0
$$137$$ 14.9443 1.27678 0.638388 0.769715i $$-0.279602\pi$$
0.638388 + 0.769715i $$0.279602\pi$$
$$138$$ 0 0
$$139$$ −1.23607 −0.104842 −0.0524210 0.998625i $$-0.516694\pi$$
−0.0524210 + 0.998625i $$0.516694\pi$$
$$140$$ 0 0
$$141$$ −12.9443 −1.09010
$$142$$ 0 0
$$143$$ −4.94427 −0.413461
$$144$$ 0 0
$$145$$ −14.4721 −1.20185
$$146$$ 0 0
$$147$$ −1.23607 −0.101949
$$148$$ 0 0
$$149$$ −2.94427 −0.241204 −0.120602 0.992701i $$-0.538483\pi$$
−0.120602 + 0.992701i $$0.538483\pi$$
$$150$$ 0 0
$$151$$ −8.94427 −0.727875 −0.363937 0.931423i $$-0.618568\pi$$
−0.363937 + 0.931423i $$0.618568\pi$$
$$152$$ 0 0
$$153$$ −6.58359 −0.532252
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ −0.763932 −0.0609684 −0.0304842 0.999535i $$-0.509705\pi$$
−0.0304842 + 0.999535i $$0.509705\pi$$
$$158$$ 0 0
$$159$$ −12.3607 −0.980266
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ 3.41641 0.267594 0.133797 0.991009i $$-0.457283\pi$$
0.133797 + 0.991009i $$0.457283\pi$$
$$164$$ 0 0
$$165$$ 25.8885 2.01542
$$166$$ 0 0
$$167$$ 23.4164 1.81202 0.906008 0.423261i $$-0.139114\pi$$
0.906008 + 0.423261i $$0.139114\pi$$
$$168$$ 0 0
$$169$$ −12.4164 −0.955108
$$170$$ 0 0
$$171$$ −1.81966 −0.139153
$$172$$ 0 0
$$173$$ −5.70820 −0.433987 −0.216993 0.976173i $$-0.569625\pi$$
−0.216993 + 0.976173i $$0.569625\pi$$
$$174$$ 0 0
$$175$$ 5.47214 0.413655
$$176$$ 0 0
$$177$$ −11.4164 −0.858110
$$178$$ 0 0
$$179$$ 7.05573 0.527370 0.263685 0.964609i $$-0.415062\pi$$
0.263685 + 0.964609i $$0.415062\pi$$
$$180$$ 0 0
$$181$$ 12.1803 0.905358 0.452679 0.891674i $$-0.350468\pi$$
0.452679 + 0.891674i $$0.350468\pi$$
$$182$$ 0 0
$$183$$ 13.8885 1.02667
$$184$$ 0 0
$$185$$ −14.4721 −1.06401
$$186$$ 0 0
$$187$$ 28.9443 2.11661
$$188$$ 0 0
$$189$$ 5.52786 0.402093
$$190$$ 0 0
$$191$$ 4.94427 0.357755 0.178877 0.983871i $$-0.442753\pi$$
0.178877 + 0.983871i $$0.442753\pi$$
$$192$$ 0 0
$$193$$ 0.472136 0.0339851 0.0169925 0.999856i $$-0.494591\pi$$
0.0169925 + 0.999856i $$0.494591\pi$$
$$194$$ 0 0
$$195$$ −3.05573 −0.218825
$$196$$ 0 0
$$197$$ −10.9443 −0.779747 −0.389874 0.920868i $$-0.627481\pi$$
−0.389874 + 0.920868i $$0.627481\pi$$
$$198$$ 0 0
$$199$$ 15.4164 1.09284 0.546420 0.837511i $$-0.315990\pi$$
0.546420 + 0.837511i $$0.315990\pi$$
$$200$$ 0 0
$$201$$ 4.94427 0.348742
$$202$$ 0 0
$$203$$ 4.47214 0.313882
$$204$$ 0 0
$$205$$ 27.4164 1.91484
$$206$$ 0 0
$$207$$ −5.88854 −0.409282
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ 6.11146 0.418750
$$214$$ 0 0
$$215$$ 20.9443 1.42839
$$216$$ 0 0
$$217$$ −2.47214 −0.167820
$$218$$ 0 0
$$219$$ 3.63932 0.245922
$$220$$ 0 0
$$221$$ −3.41641 −0.229812
$$222$$ 0 0
$$223$$ 12.9443 0.866813 0.433406 0.901199i $$-0.357312\pi$$
0.433406 + 0.901199i $$0.357312\pi$$
$$224$$ 0 0
$$225$$ −8.05573 −0.537049
$$226$$ 0 0
$$227$$ 17.2361 1.14400 0.571999 0.820254i $$-0.306168\pi$$
0.571999 + 0.820254i $$0.306168\pi$$
$$228$$ 0 0
$$229$$ −23.5967 −1.55932 −0.779658 0.626205i $$-0.784607\pi$$
−0.779658 + 0.626205i $$0.784607\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ −15.8885 −1.04089 −0.520447 0.853894i $$-0.674235\pi$$
−0.520447 + 0.853894i $$0.674235\pi$$
$$234$$ 0 0
$$235$$ −33.8885 −2.21064
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ −13.8885 −0.898375 −0.449188 0.893437i $$-0.648287\pi$$
−0.449188 + 0.893437i $$0.648287\pi$$
$$240$$ 0 0
$$241$$ 12.4721 0.803401 0.401700 0.915771i $$-0.368419\pi$$
0.401700 + 0.915771i $$0.368419\pi$$
$$242$$ 0 0
$$243$$ −13.5967 −0.872232
$$244$$ 0 0
$$245$$ −3.23607 −0.206745
$$246$$ 0 0
$$247$$ −0.944272 −0.0600826
$$248$$ 0 0
$$249$$ −11.4164 −0.723485
$$250$$ 0 0
$$251$$ −4.29180 −0.270896 −0.135448 0.990784i $$-0.543247\pi$$
−0.135448 + 0.990784i $$0.543247\pi$$
$$252$$ 0 0
$$253$$ 25.8885 1.62760
$$254$$ 0 0
$$255$$ 17.8885 1.12022
$$256$$ 0 0
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 0 0
$$259$$ 4.47214 0.277885
$$260$$ 0 0
$$261$$ −6.58359 −0.407514
$$262$$ 0 0
$$263$$ 11.0557 0.681725 0.340863 0.940113i $$-0.389281\pi$$
0.340863 + 0.940113i $$0.389281\pi$$
$$264$$ 0 0
$$265$$ −32.3607 −1.98790
$$266$$ 0 0
$$267$$ 7.41641 0.453877
$$268$$ 0 0
$$269$$ 4.18034 0.254880 0.127440 0.991846i $$-0.459324\pi$$
0.127440 + 0.991846i $$0.459324\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0.944272 0.0571499
$$274$$ 0 0
$$275$$ 35.4164 2.13569
$$276$$ 0 0
$$277$$ −7.88854 −0.473977 −0.236988 0.971512i $$-0.576160\pi$$
−0.236988 + 0.971512i $$0.576160\pi$$
$$278$$ 0 0
$$279$$ 3.63932 0.217880
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 6.18034 0.367383 0.183692 0.982984i $$-0.441195\pi$$
0.183692 + 0.982984i $$0.441195\pi$$
$$284$$ 0 0
$$285$$ 4.94427 0.292873
$$286$$ 0 0
$$287$$ −8.47214 −0.500094
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ −15.4164 −0.903726
$$292$$ 0 0
$$293$$ 12.7639 0.745677 0.372838 0.927896i $$-0.378385\pi$$
0.372838 + 0.927896i $$0.378385\pi$$
$$294$$ 0 0
$$295$$ −29.8885 −1.74018
$$296$$ 0 0
$$297$$ 35.7771 2.07600
$$298$$ 0 0
$$299$$ −3.05573 −0.176717
$$300$$ 0 0
$$301$$ −6.47214 −0.373048
$$302$$ 0 0
$$303$$ −2.11146 −0.121300
$$304$$ 0 0
$$305$$ 36.3607 2.08201
$$306$$ 0 0
$$307$$ −1.81966 −0.103853 −0.0519267 0.998651i $$-0.516536\pi$$
−0.0519267 + 0.998651i $$0.516536\pi$$
$$308$$ 0 0
$$309$$ 6.83282 0.388705
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −8.47214 −0.478873 −0.239437 0.970912i $$-0.576963\pi$$
−0.239437 + 0.970912i $$0.576963\pi$$
$$314$$ 0 0
$$315$$ 4.76393 0.268417
$$316$$ 0 0
$$317$$ −9.05573 −0.508620 −0.254310 0.967123i $$-0.581848\pi$$
−0.254310 + 0.967123i $$0.581848\pi$$
$$318$$ 0 0
$$319$$ 28.9443 1.62057
$$320$$ 0 0
$$321$$ 11.0557 0.617071
$$322$$ 0 0
$$323$$ 5.52786 0.307579
$$324$$ 0 0
$$325$$ −4.18034 −0.231884
$$326$$ 0 0
$$327$$ 10.4721 0.579110
$$328$$ 0 0
$$329$$ 10.4721 0.577348
$$330$$ 0 0
$$331$$ −22.4721 −1.23518 −0.617590 0.786500i $$-0.711891\pi$$
−0.617590 + 0.786500i $$0.711891\pi$$
$$332$$ 0 0
$$333$$ −6.58359 −0.360779
$$334$$ 0 0
$$335$$ 12.9443 0.707221
$$336$$ 0 0
$$337$$ 10.3607 0.564382 0.282191 0.959358i $$-0.408939\pi$$
0.282191 + 0.959358i $$0.408939\pi$$
$$338$$ 0 0
$$339$$ 15.4164 0.837304
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ −6.47214 −0.347442 −0.173721 0.984795i $$-0.555579\pi$$
−0.173721 + 0.984795i $$0.555579\pi$$
$$348$$ 0 0
$$349$$ −26.6525 −1.42667 −0.713337 0.700821i $$-0.752817\pi$$
−0.713337 + 0.700821i $$0.752817\pi$$
$$350$$ 0 0
$$351$$ −4.22291 −0.225402
$$352$$ 0 0
$$353$$ −15.8885 −0.845662 −0.422831 0.906209i $$-0.638964\pi$$
−0.422831 + 0.906209i $$0.638964\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 0 0
$$357$$ −5.52786 −0.292566
$$358$$ 0 0
$$359$$ −16.9443 −0.894284 −0.447142 0.894463i $$-0.647558\pi$$
−0.447142 + 0.894463i $$0.647558\pi$$
$$360$$ 0 0
$$361$$ −17.4721 −0.919586
$$362$$ 0 0
$$363$$ −38.1803 −2.00395
$$364$$ 0 0
$$365$$ 9.52786 0.498711
$$366$$ 0 0
$$367$$ −22.8328 −1.19186 −0.595932 0.803035i $$-0.703217\pi$$
−0.595932 + 0.803035i $$0.703217\pi$$
$$368$$ 0 0
$$369$$ 12.4721 0.649273
$$370$$ 0 0
$$371$$ 10.0000 0.519174
$$372$$ 0 0
$$373$$ −2.94427 −0.152449 −0.0762243 0.997091i $$-0.524287\pi$$
−0.0762243 + 0.997091i $$0.524287\pi$$
$$374$$ 0 0
$$375$$ 1.88854 0.0975240
$$376$$ 0 0
$$377$$ −3.41641 −0.175954
$$378$$ 0 0
$$379$$ 4.58359 0.235443 0.117722 0.993047i $$-0.462441\pi$$
0.117722 + 0.993047i $$0.462441\pi$$
$$380$$ 0 0
$$381$$ −11.0557 −0.566402
$$382$$ 0 0
$$383$$ −15.4164 −0.787742 −0.393871 0.919166i $$-0.628864\pi$$
−0.393871 + 0.919166i $$0.628864\pi$$
$$384$$ 0 0
$$385$$ −20.9443 −1.06742
$$386$$ 0 0
$$387$$ 9.52786 0.484329
$$388$$ 0 0
$$389$$ 4.47214 0.226746 0.113373 0.993552i $$-0.463834\pi$$
0.113373 + 0.993552i $$0.463834\pi$$
$$390$$ 0 0
$$391$$ 17.8885 0.904663
$$392$$ 0 0
$$393$$ 14.4721 0.730023
$$394$$ 0 0
$$395$$ −41.8885 −2.10764
$$396$$ 0 0
$$397$$ 15.2361 0.764676 0.382338 0.924022i $$-0.375119\pi$$
0.382338 + 0.924022i $$0.375119\pi$$
$$398$$ 0 0
$$399$$ −1.52786 −0.0764889
$$400$$ 0 0
$$401$$ −23.5279 −1.17493 −0.587463 0.809251i $$-0.699873\pi$$
−0.587463 + 0.809251i $$0.699873\pi$$
$$402$$ 0 0
$$403$$ 1.88854 0.0940751
$$404$$ 0 0
$$405$$ 7.81966 0.388562
$$406$$ 0 0
$$407$$ 28.9443 1.43471
$$408$$ 0 0
$$409$$ −21.4164 −1.05897 −0.529487 0.848318i $$-0.677615\pi$$
−0.529487 + 0.848318i $$0.677615\pi$$
$$410$$ 0 0
$$411$$ −18.4721 −0.911163
$$412$$ 0 0
$$413$$ 9.23607 0.454477
$$414$$ 0 0
$$415$$ −29.8885 −1.46717
$$416$$ 0 0
$$417$$ 1.52786 0.0748198
$$418$$ 0 0
$$419$$ −22.1803 −1.08358 −0.541790 0.840514i $$-0.682253\pi$$
−0.541790 + 0.840514i $$0.682253\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 0 0
$$423$$ −15.4164 −0.749571
$$424$$ 0 0
$$425$$ 24.4721 1.18707
$$426$$ 0 0
$$427$$ −11.2361 −0.543751
$$428$$ 0 0
$$429$$ 6.11146 0.295064
$$430$$ 0 0
$$431$$ −28.0000 −1.34871 −0.674356 0.738406i $$-0.735579\pi$$
−0.674356 + 0.738406i $$0.735579\pi$$
$$432$$ 0 0
$$433$$ 9.41641 0.452524 0.226262 0.974067i $$-0.427349\pi$$
0.226262 + 0.974067i $$0.427349\pi$$
$$434$$ 0 0
$$435$$ 17.8885 0.857690
$$436$$ 0 0
$$437$$ 4.94427 0.236517
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ −1.47214 −0.0701017
$$442$$ 0 0
$$443$$ 13.8885 0.659865 0.329932 0.944005i $$-0.392974\pi$$
0.329932 + 0.944005i $$0.392974\pi$$
$$444$$ 0 0
$$445$$ 19.4164 0.920426
$$446$$ 0 0
$$447$$ 3.63932 0.172134
$$448$$ 0 0
$$449$$ −7.88854 −0.372283 −0.186142 0.982523i $$-0.559598\pi$$
−0.186142 + 0.982523i $$0.559598\pi$$
$$450$$ 0 0
$$451$$ −54.8328 −2.58198
$$452$$ 0 0
$$453$$ 11.0557 0.519443
$$454$$ 0 0
$$455$$ 2.47214 0.115896
$$456$$ 0 0
$$457$$ −7.52786 −0.352139 −0.176069 0.984378i $$-0.556338\pi$$
−0.176069 + 0.984378i $$0.556338\pi$$
$$458$$ 0 0
$$459$$ 24.7214 1.15389
$$460$$ 0 0
$$461$$ −21.7082 −1.01105 −0.505526 0.862811i $$-0.668702\pi$$
−0.505526 + 0.862811i $$0.668702\pi$$
$$462$$ 0 0
$$463$$ −35.7771 −1.66270 −0.831351 0.555748i $$-0.812432\pi$$
−0.831351 + 0.555748i $$0.812432\pi$$
$$464$$ 0 0
$$465$$ −9.88854 −0.458570
$$466$$ 0 0
$$467$$ 32.0689 1.48397 0.741985 0.670416i $$-0.233884\pi$$
0.741985 + 0.670416i $$0.233884\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0.944272 0.0435098
$$472$$ 0 0
$$473$$ −41.8885 −1.92604
$$474$$ 0 0
$$475$$ 6.76393 0.310350
$$476$$ 0 0
$$477$$ −14.7214 −0.674045
$$478$$ 0 0
$$479$$ −8.58359 −0.392194 −0.196097 0.980584i $$-0.562827\pi$$
−0.196097 + 0.980584i $$0.562827\pi$$
$$480$$ 0 0
$$481$$ −3.41641 −0.155775
$$482$$ 0 0
$$483$$ −4.94427 −0.224972
$$484$$ 0 0
$$485$$ −40.3607 −1.83268
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 0 0
$$489$$ −4.22291 −0.190967
$$490$$ 0 0
$$491$$ −37.8885 −1.70989 −0.854943 0.518722i $$-0.826408\pi$$
−0.854943 + 0.518722i $$0.826408\pi$$
$$492$$ 0 0
$$493$$ 20.0000 0.900755
$$494$$ 0 0
$$495$$ 30.8328 1.38583
$$496$$ 0 0
$$497$$ −4.94427 −0.221781
$$498$$ 0 0
$$499$$ −21.8885 −0.979866 −0.489933 0.871760i $$-0.662979\pi$$
−0.489933 + 0.871760i $$0.662979\pi$$
$$500$$ 0 0
$$501$$ −28.9443 −1.29313
$$502$$ 0 0
$$503$$ −4.94427 −0.220454 −0.110227 0.993906i $$-0.535158\pi$$
−0.110227 + 0.993906i $$0.535158\pi$$
$$504$$ 0 0
$$505$$ −5.52786 −0.245987
$$506$$ 0 0
$$507$$ 15.3475 0.681607
$$508$$ 0 0
$$509$$ 41.1246 1.82282 0.911408 0.411503i $$-0.134996\pi$$
0.911408 + 0.411503i $$0.134996\pi$$
$$510$$ 0 0
$$511$$ −2.94427 −0.130247
$$512$$ 0 0
$$513$$ 6.83282 0.301676
$$514$$ 0 0
$$515$$ 17.8885 0.788263
$$516$$ 0 0
$$517$$ 67.7771 2.98083
$$518$$ 0 0
$$519$$ 7.05573 0.309712
$$520$$ 0 0
$$521$$ −6.58359 −0.288432 −0.144216 0.989546i $$-0.546066\pi$$
−0.144216 + 0.989546i $$0.546066\pi$$
$$522$$ 0 0
$$523$$ 4.29180 0.187667 0.0938336 0.995588i $$-0.470088\pi$$
0.0938336 + 0.995588i $$0.470088\pi$$
$$524$$ 0 0
$$525$$ −6.76393 −0.295202
$$526$$ 0 0
$$527$$ −11.0557 −0.481595
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −13.5967 −0.590049
$$532$$ 0 0
$$533$$ 6.47214 0.280339
$$534$$ 0 0
$$535$$ 28.9443 1.25137
$$536$$ 0 0
$$537$$ −8.72136 −0.376354
$$538$$ 0 0
$$539$$ 6.47214 0.278775
$$540$$ 0 0
$$541$$ 5.05573 0.217363 0.108681 0.994077i $$-0.465337\pi$$
0.108681 + 0.994077i $$0.465337\pi$$
$$542$$ 0 0
$$543$$ −15.0557 −0.646103
$$544$$ 0 0
$$545$$ 27.4164 1.17439
$$546$$ 0 0
$$547$$ 4.58359 0.195980 0.0979901 0.995187i $$-0.468759\pi$$
0.0979901 + 0.995187i $$0.468759\pi$$
$$548$$ 0 0
$$549$$ 16.5410 0.705954
$$550$$ 0 0
$$551$$ 5.52786 0.235495
$$552$$ 0 0
$$553$$ 12.9443 0.550446
$$554$$ 0 0
$$555$$ 17.8885 0.759326
$$556$$ 0 0
$$557$$ −9.05573 −0.383704 −0.191852 0.981424i $$-0.561449\pi$$
−0.191852 + 0.981424i $$0.561449\pi$$
$$558$$ 0 0
$$559$$ 4.94427 0.209120
$$560$$ 0 0
$$561$$ −35.7771 −1.51051
$$562$$ 0 0
$$563$$ 17.8197 0.751009 0.375505 0.926821i $$-0.377469\pi$$
0.375505 + 0.926821i $$0.377469\pi$$
$$564$$ 0 0
$$565$$ 40.3607 1.69799
$$566$$ 0 0
$$567$$ −2.41641 −0.101480
$$568$$ 0 0
$$569$$ 26.3607 1.10510 0.552549 0.833481i $$-0.313655\pi$$
0.552549 + 0.833481i $$0.313655\pi$$
$$570$$ 0 0
$$571$$ 14.4721 0.605640 0.302820 0.953048i $$-0.402072\pi$$
0.302820 + 0.953048i $$0.402072\pi$$
$$572$$ 0 0
$$573$$ −6.11146 −0.255310
$$574$$ 0 0
$$575$$ 21.8885 0.912815
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ −0.583592 −0.0242533
$$580$$ 0 0
$$581$$ 9.23607 0.383177
$$582$$ 0 0
$$583$$ 64.7214 2.68048
$$584$$ 0 0
$$585$$ −3.63932 −0.150467
$$586$$ 0 0
$$587$$ 3.70820 0.153054 0.0765270 0.997068i $$-0.475617\pi$$
0.0765270 + 0.997068i $$0.475617\pi$$
$$588$$ 0 0
$$589$$ −3.05573 −0.125909
$$590$$ 0 0
$$591$$ 13.5279 0.556462
$$592$$ 0 0
$$593$$ 32.8328 1.34828 0.674141 0.738603i $$-0.264514\pi$$
0.674141 + 0.738603i $$0.264514\pi$$
$$594$$ 0 0
$$595$$ −14.4721 −0.593300
$$596$$ 0 0
$$597$$ −19.0557 −0.779899
$$598$$ 0 0
$$599$$ 17.8885 0.730906 0.365453 0.930830i $$-0.380914\pi$$
0.365453 + 0.930830i $$0.380914\pi$$
$$600$$ 0 0
$$601$$ 29.7771 1.21463 0.607316 0.794460i $$-0.292246\pi$$
0.607316 + 0.794460i $$0.292246\pi$$
$$602$$ 0 0
$$603$$ 5.88854 0.239800
$$604$$ 0 0
$$605$$ −99.9574 −4.06385
$$606$$ 0 0
$$607$$ 9.88854 0.401364 0.200682 0.979656i $$-0.435684\pi$$
0.200682 + 0.979656i $$0.435684\pi$$
$$608$$ 0 0
$$609$$ −5.52786 −0.224000
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −29.4164 −1.18812 −0.594059 0.804422i $$-0.702475\pi$$
−0.594059 + 0.804422i $$0.702475\pi$$
$$614$$ 0 0
$$615$$ −33.8885 −1.36652
$$616$$ 0 0
$$617$$ 34.3607 1.38331 0.691654 0.722229i $$-0.256882\pi$$
0.691654 + 0.722229i $$0.256882\pi$$
$$618$$ 0 0
$$619$$ 48.0689 1.93205 0.966026 0.258446i $$-0.0832103\pi$$
0.966026 + 0.258446i $$0.0832103\pi$$
$$620$$ 0 0
$$621$$ 22.1115 0.887302
$$622$$ 0 0
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ −22.4164 −0.896656
$$626$$ 0 0
$$627$$ −9.88854 −0.394910
$$628$$ 0 0
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −44.9443 −1.78920 −0.894602 0.446865i $$-0.852541\pi$$
−0.894602 + 0.446865i $$0.852541\pi$$
$$632$$ 0 0
$$633$$ −14.8328 −0.589551
$$634$$ 0 0
$$635$$ −28.9443 −1.14862
$$636$$ 0 0
$$637$$ −0.763932 −0.0302681
$$638$$ 0 0
$$639$$ 7.27864 0.287939
$$640$$ 0 0
$$641$$ 14.5836 0.576017 0.288009 0.957628i $$-0.407007\pi$$
0.288009 + 0.957628i $$0.407007\pi$$
$$642$$ 0 0
$$643$$ 43.7082 1.72368 0.861842 0.507177i $$-0.169311\pi$$
0.861842 + 0.507177i $$0.169311\pi$$
$$644$$ 0 0
$$645$$ −25.8885 −1.01936
$$646$$ 0 0
$$647$$ −20.3607 −0.800461 −0.400230 0.916415i $$-0.631070\pi$$
−0.400230 + 0.916415i $$0.631070\pi$$
$$648$$ 0 0
$$649$$ 59.7771 2.34646
$$650$$ 0 0
$$651$$ 3.05573 0.119763
$$652$$ 0 0
$$653$$ 9.41641 0.368493 0.184246 0.982880i $$-0.441016\pi$$
0.184246 + 0.982880i $$0.441016\pi$$
$$654$$ 0 0
$$655$$ 37.8885 1.48043
$$656$$ 0 0
$$657$$ 4.33437 0.169100
$$658$$ 0 0
$$659$$ −37.3050 −1.45319 −0.726597 0.687064i $$-0.758899\pi$$
−0.726597 + 0.687064i $$0.758899\pi$$
$$660$$ 0 0
$$661$$ 22.6525 0.881079 0.440540 0.897733i $$-0.354787\pi$$
0.440540 + 0.897733i $$0.354787\pi$$
$$662$$ 0 0
$$663$$ 4.22291 0.164004
$$664$$ 0 0
$$665$$ −4.00000 −0.155113
$$666$$ 0 0
$$667$$ 17.8885 0.692647
$$668$$ 0 0
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −72.7214 −2.80738
$$672$$ 0 0
$$673$$ −2.94427 −0.113493 −0.0567467 0.998389i $$-0.518073\pi$$
−0.0567467 + 0.998389i $$0.518073\pi$$
$$674$$ 0 0
$$675$$ 30.2492 1.16429
$$676$$ 0 0
$$677$$ −19.8197 −0.761731 −0.380866 0.924630i $$-0.624374\pi$$
−0.380866 + 0.924630i $$0.624374\pi$$
$$678$$ 0 0
$$679$$ 12.4721 0.478637
$$680$$ 0 0
$$681$$ −21.3050 −0.816408
$$682$$ 0 0
$$683$$ −23.7771 −0.909805 −0.454902 0.890541i $$-0.650326\pi$$
−0.454902 + 0.890541i $$0.650326\pi$$
$$684$$ 0 0
$$685$$ −48.3607 −1.84777
$$686$$ 0 0
$$687$$ 29.1672 1.11280
$$688$$ 0 0
$$689$$ −7.63932 −0.291035
$$690$$ 0 0
$$691$$ 14.1803 0.539446 0.269723 0.962938i $$-0.413068\pi$$
0.269723 + 0.962938i $$0.413068\pi$$
$$692$$ 0 0
$$693$$ −9.52786 −0.361934
$$694$$ 0 0
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ −37.8885 −1.43513
$$698$$ 0 0
$$699$$ 19.6393 0.742827
$$700$$ 0 0
$$701$$ 41.4164 1.56428 0.782138 0.623105i $$-0.214129\pi$$
0.782138 + 0.623105i $$0.214129\pi$$
$$702$$ 0 0
$$703$$ 5.52786 0.208487
$$704$$ 0 0
$$705$$ 41.8885 1.57761
$$706$$ 0 0
$$707$$ 1.70820 0.0642436
$$708$$ 0 0
$$709$$ −1.63932 −0.0615660 −0.0307830 0.999526i $$-0.509800\pi$$
−0.0307830 + 0.999526i $$0.509800\pi$$
$$710$$ 0 0
$$711$$ −19.0557 −0.714646
$$712$$ 0 0
$$713$$ −9.88854 −0.370329
$$714$$ 0 0
$$715$$ 16.0000 0.598366
$$716$$ 0 0
$$717$$ 17.1672 0.641120
$$718$$ 0 0
$$719$$ 51.1935 1.90920 0.954598 0.297898i $$-0.0962856\pi$$
0.954598 + 0.297898i $$0.0962856\pi$$
$$720$$ 0 0
$$721$$ −5.52786 −0.205868
$$722$$ 0 0
$$723$$ −15.4164 −0.573342
$$724$$ 0 0
$$725$$ 24.4721 0.908872
$$726$$ 0 0
$$727$$ 12.3607 0.458432 0.229216 0.973376i $$-0.426384\pi$$
0.229216 + 0.973376i $$0.426384\pi$$
$$728$$ 0 0
$$729$$ 24.0557 0.890953
$$730$$ 0 0
$$731$$ −28.9443 −1.07054
$$732$$ 0 0
$$733$$ 4.76393 0.175960 0.0879799 0.996122i $$-0.471959\pi$$
0.0879799 + 0.996122i $$0.471959\pi$$
$$734$$ 0 0
$$735$$ 4.00000 0.147542
$$736$$ 0 0
$$737$$ −25.8885 −0.953617
$$738$$ 0 0
$$739$$ 3.41641 0.125675 0.0628373 0.998024i $$-0.479985\pi$$
0.0628373 + 0.998024i $$0.479985\pi$$
$$740$$ 0 0
$$741$$ 1.16718 0.0428776
$$742$$ 0 0
$$743$$ 24.9443 0.915117 0.457558 0.889180i $$-0.348724\pi$$
0.457558 + 0.889180i $$0.348724\pi$$
$$744$$ 0 0
$$745$$ 9.52786 0.349074
$$746$$ 0 0
$$747$$ −13.5967 −0.497479
$$748$$ 0 0
$$749$$ −8.94427 −0.326817
$$750$$ 0 0
$$751$$ 36.0000 1.31366 0.656829 0.754039i $$-0.271897\pi$$
0.656829 + 0.754039i $$0.271897\pi$$
$$752$$ 0 0
$$753$$ 5.30495 0.193323
$$754$$ 0 0
$$755$$ 28.9443 1.05339
$$756$$ 0 0
$$757$$ −39.3050 −1.42856 −0.714281 0.699859i $$-0.753246\pi$$
−0.714281 + 0.699859i $$0.753246\pi$$
$$758$$ 0 0
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ −3.52786 −0.127885 −0.0639425 0.997954i $$-0.520367\pi$$
−0.0639425 + 0.997954i $$0.520367\pi$$
$$762$$ 0 0
$$763$$ −8.47214 −0.306712
$$764$$ 0 0
$$765$$ 21.3050 0.770282
$$766$$ 0 0
$$767$$ −7.05573 −0.254768
$$768$$ 0 0
$$769$$ −18.3607 −0.662103 −0.331052 0.943613i $$-0.607403\pi$$
−0.331052 + 0.943613i $$0.607403\pi$$
$$770$$ 0 0
$$771$$ 17.3050 0.623223
$$772$$ 0 0
$$773$$ −40.1803 −1.44519 −0.722593 0.691274i $$-0.757050\pi$$
−0.722593 + 0.691274i $$0.757050\pi$$
$$774$$ 0 0
$$775$$ −13.5279 −0.485935
$$776$$ 0 0
$$777$$ −5.52786 −0.198311
$$778$$ 0 0
$$779$$ −10.4721 −0.375203
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ 24.7214 0.883469
$$784$$ 0 0
$$785$$ 2.47214 0.0882343
$$786$$ 0 0
$$787$$ 12.2918 0.438155 0.219078 0.975707i $$-0.429695\pi$$
0.219078 + 0.975707i $$0.429695\pi$$
$$788$$ 0 0
$$789$$ −13.6656 −0.486509
$$790$$ 0 0
$$791$$ −12.4721 −0.443458
$$792$$ 0 0
$$793$$ 8.58359 0.304812
$$794$$ 0 0
$$795$$ 40.0000 1.41865
$$796$$ 0 0
$$797$$ 52.1803 1.84832 0.924161 0.382003i $$-0.124765\pi$$
0.924161 + 0.382003i $$0.124765\pi$$
$$798$$ 0 0
$$799$$ 46.8328 1.65683
$$800$$ 0 0
$$801$$ 8.83282 0.312092
$$802$$ 0 0
$$803$$ −19.0557 −0.672462
$$804$$ 0 0
$$805$$ −12.9443 −0.456226
$$806$$ 0 0
$$807$$ −5.16718 −0.181894
$$808$$ 0 0
$$809$$ −17.4164 −0.612328 −0.306164 0.951979i $$-0.599046\pi$$
−0.306164 + 0.951979i $$0.599046\pi$$
$$810$$ 0 0
$$811$$ 53.0132 1.86154 0.930772 0.365601i $$-0.119136\pi$$
0.930772 + 0.365601i $$0.119136\pi$$
$$812$$ 0 0
$$813$$ 29.6656 1.04042
$$814$$ 0 0
$$815$$ −11.0557 −0.387265
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ 1.12461 0.0392971
$$820$$ 0 0
$$821$$ −4.11146 −0.143491 −0.0717454 0.997423i $$-0.522857\pi$$
−0.0717454 + 0.997423i $$0.522857\pi$$
$$822$$ 0 0
$$823$$ −19.7771 −0.689386 −0.344693 0.938715i $$-0.612017\pi$$
−0.344693 + 0.938715i $$0.612017\pi$$
$$824$$ 0 0
$$825$$ −43.7771 −1.52412
$$826$$ 0 0
$$827$$ −15.0557 −0.523539 −0.261769 0.965130i $$-0.584306\pi$$
−0.261769 + 0.965130i $$0.584306\pi$$
$$828$$ 0 0
$$829$$ −11.2361 −0.390245 −0.195122 0.980779i $$-0.562510\pi$$
−0.195122 + 0.980779i $$0.562510\pi$$
$$830$$ 0 0
$$831$$ 9.75078 0.338251
$$832$$ 0 0
$$833$$ 4.47214 0.154950
$$834$$ 0 0
$$835$$ −75.7771 −2.62237
$$836$$ 0 0
$$837$$ −13.6656 −0.472353
$$838$$ 0 0
$$839$$ −21.5279 −0.743224 −0.371612 0.928388i $$-0.621195\pi$$
−0.371612 + 0.928388i $$0.621195\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0 0
$$843$$ −32.1378 −1.10688
$$844$$ 0 0
$$845$$ 40.1803 1.38225
$$846$$ 0 0
$$847$$ 30.8885 1.06134
$$848$$ 0 0
$$849$$ −7.63932 −0.262181
$$850$$ 0 0
$$851$$ 17.8885 0.613211
$$852$$ 0 0
$$853$$ −13.7082 −0.469360 −0.234680 0.972073i $$-0.575404\pi$$
−0.234680 + 0.972073i $$0.575404\pi$$
$$854$$ 0 0
$$855$$ 5.88854 0.201384
$$856$$ 0 0
$$857$$ −27.5279 −0.940334 −0.470167 0.882577i $$-0.655806\pi$$
−0.470167 + 0.882577i $$0.655806\pi$$
$$858$$ 0 0
$$859$$ 33.8197 1.15391 0.576956 0.816775i $$-0.304240\pi$$
0.576956 + 0.816775i $$0.304240\pi$$
$$860$$ 0 0
$$861$$ 10.4721 0.356889
$$862$$ 0 0
$$863$$ −20.9443 −0.712951 −0.356476 0.934305i $$-0.616022\pi$$
−0.356476 + 0.934305i $$0.616022\pi$$
$$864$$ 0 0
$$865$$ 18.4721 0.628071
$$866$$ 0 0
$$867$$ −3.70820 −0.125937
$$868$$ 0 0
$$869$$ 83.7771 2.84194
$$870$$ 0 0
$$871$$ 3.05573 0.103539
$$872$$ 0 0
$$873$$ −18.3607 −0.621415
$$874$$ 0 0
$$875$$ −1.52786 −0.0516512
$$876$$ 0 0
$$877$$ −13.4164 −0.453040 −0.226520 0.974007i $$-0.572735\pi$$
−0.226520 + 0.974007i $$0.572735\pi$$
$$878$$ 0 0
$$879$$ −15.7771 −0.532148
$$880$$ 0 0
$$881$$ 24.8328 0.836639 0.418319 0.908300i $$-0.362619\pi$$
0.418319 + 0.908300i $$0.362619\pi$$
$$882$$ 0 0
$$883$$ −23.0557 −0.775887 −0.387944 0.921683i $$-0.626814\pi$$
−0.387944 + 0.921683i $$0.626814\pi$$
$$884$$ 0 0
$$885$$ 36.9443 1.24187
$$886$$ 0 0
$$887$$ −33.3050 −1.11827 −0.559135 0.829076i $$-0.688867\pi$$
−0.559135 + 0.829076i $$0.688867\pi$$
$$888$$ 0 0
$$889$$ 8.94427 0.299981
$$890$$ 0 0
$$891$$ −15.6393 −0.523937
$$892$$ 0 0
$$893$$ 12.9443 0.433164
$$894$$ 0 0
$$895$$ −22.8328 −0.763217
$$896$$ 0 0
$$897$$ 3.77709 0.126113
$$898$$ 0 0
$$899$$ −11.0557 −0.368729
$$900$$ 0 0
$$901$$ 44.7214 1.48988
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ −39.4164 −1.31025
$$906$$ 0 0
$$907$$ −0.944272 −0.0313540 −0.0156770 0.999877i $$-0.504990\pi$$
−0.0156770 + 0.999877i $$0.504990\pi$$
$$908$$ 0 0
$$909$$ −2.51471 −0.0834076
$$910$$ 0 0
$$911$$ −34.8328 −1.15406 −0.577031 0.816722i $$-0.695789\pi$$
−0.577031 + 0.816722i $$0.695789\pi$$
$$912$$ 0 0
$$913$$ 59.7771 1.97833
$$914$$ 0 0
$$915$$ −44.9443 −1.48581
$$916$$ 0 0
$$917$$ −11.7082 −0.386639
$$918$$ 0 0
$$919$$ −35.7771 −1.18018 −0.590089 0.807338i $$-0.700907\pi$$
−0.590089 + 0.807338i $$0.700907\pi$$
$$920$$ 0 0
$$921$$ 2.24922 0.0741144
$$922$$ 0 0
$$923$$ 3.77709 0.124324
$$924$$ 0 0
$$925$$ 24.4721 0.804639
$$926$$ 0 0
$$927$$ 8.13777 0.267279
$$928$$ 0 0
$$929$$ −47.3050 −1.55203 −0.776013 0.630717i $$-0.782761\pi$$
−0.776013 + 0.630717i $$0.782761\pi$$
$$930$$ 0 0
$$931$$ 1.23607 0.0405105
$$932$$ 0 0
$$933$$ −9.88854 −0.323736
$$934$$ 0 0
$$935$$ −93.6656 −3.06319
$$936$$ 0 0
$$937$$ −9.05573 −0.295838 −0.147919 0.988999i $$-0.547257\pi$$
−0.147919 + 0.988999i $$0.547257\pi$$
$$938$$ 0 0
$$939$$ 10.4721 0.341745
$$940$$ 0 0
$$941$$ 35.5967 1.16042 0.580210 0.814467i $$-0.302970\pi$$
0.580210 + 0.814467i $$0.302970\pi$$
$$942$$ 0 0
$$943$$ −33.8885 −1.10356
$$944$$ 0 0
$$945$$ −17.8885 −0.581914
$$946$$ 0 0
$$947$$ 4.58359 0.148947 0.0744734 0.997223i $$-0.476272\pi$$
0.0744734 + 0.997223i $$0.476272\pi$$
$$948$$ 0 0
$$949$$ 2.24922 0.0730129
$$950$$ 0 0
$$951$$ 11.1935 0.362974
$$952$$ 0 0
$$953$$ 51.8885 1.68083 0.840417 0.541940i $$-0.182310\pi$$
0.840417 + 0.541940i $$0.182310\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ −35.7771 −1.15651
$$958$$ 0 0
$$959$$ 14.9443 0.482576
$$960$$ 0 0
$$961$$ −24.8885 −0.802856
$$962$$ 0 0
$$963$$ 13.1672 0.424307
$$964$$ 0 0
$$965$$ −1.52786 −0.0491837
$$966$$ 0 0
$$967$$ 29.8885 0.961151 0.480575 0.876953i $$-0.340428\pi$$
0.480575 + 0.876953i $$0.340428\pi$$
$$968$$ 0 0
$$969$$ −6.83282 −0.219502
$$970$$ 0 0
$$971$$ −22.7639 −0.730529 −0.365265 0.930904i $$-0.619022\pi$$
−0.365265 + 0.930904i $$0.619022\pi$$
$$972$$ 0 0
$$973$$ −1.23607 −0.0396265
$$974$$ 0 0
$$975$$ 5.16718 0.165482
$$976$$ 0 0
$$977$$ −12.8328 −0.410558 −0.205279 0.978703i $$-0.565810\pi$$
−0.205279 + 0.978703i $$0.565810\pi$$
$$978$$ 0 0
$$979$$ −38.8328 −1.24110
$$980$$ 0 0
$$981$$ 12.4721 0.398205
$$982$$ 0 0
$$983$$ −5.52786 −0.176311 −0.0881557 0.996107i $$-0.528097\pi$$
−0.0881557 + 0.996107i $$0.528097\pi$$
$$984$$ 0 0
$$985$$ 35.4164 1.12846
$$986$$ 0 0
$$987$$ −12.9443 −0.412021
$$988$$ 0 0
$$989$$ −25.8885 −0.823208
$$990$$ 0 0
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 0 0
$$993$$ 27.7771 0.881479
$$994$$ 0 0
$$995$$ −49.8885 −1.58157
$$996$$ 0 0
$$997$$ −18.0689 −0.572247 −0.286124 0.958193i $$-0.592367\pi$$
−0.286124 + 0.958193i $$0.592367\pi$$
$$998$$ 0 0
$$999$$ 24.7214 0.782149
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 448.2.a.j.1.1 2
3.2 odd 2 4032.2.a.bw.1.2 2
4.3 odd 2 448.2.a.i.1.2 2
7.6 odd 2 3136.2.a.bf.1.2 2
8.3 odd 2 224.2.a.d.1.1 yes 2
8.5 even 2 224.2.a.c.1.2 2
12.11 even 2 4032.2.a.bv.1.2 2
16.3 odd 4 1792.2.b.m.897.3 4
16.5 even 4 1792.2.b.k.897.3 4
16.11 odd 4 1792.2.b.m.897.2 4
16.13 even 4 1792.2.b.k.897.2 4
24.5 odd 2 2016.2.a.r.1.1 2
24.11 even 2 2016.2.a.o.1.1 2
28.27 even 2 3136.2.a.by.1.1 2
40.19 odd 2 5600.2.a.z.1.2 2
40.29 even 2 5600.2.a.bk.1.1 2
56.3 even 6 1568.2.i.w.961.1 4
56.5 odd 6 1568.2.i.n.1537.2 4
56.11 odd 6 1568.2.i.m.961.2 4
56.13 odd 2 1568.2.a.v.1.1 2
56.19 even 6 1568.2.i.w.1537.1 4
56.27 even 2 1568.2.a.k.1.2 2
56.37 even 6 1568.2.i.v.1537.1 4
56.45 odd 6 1568.2.i.n.961.2 4
56.51 odd 6 1568.2.i.m.1537.2 4
56.53 even 6 1568.2.i.v.961.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.2 2 8.5 even 2
224.2.a.d.1.1 yes 2 8.3 odd 2
448.2.a.i.1.2 2 4.3 odd 2
448.2.a.j.1.1 2 1.1 even 1 trivial
1568.2.a.k.1.2 2 56.27 even 2
1568.2.a.v.1.1 2 56.13 odd 2
1568.2.i.m.961.2 4 56.11 odd 6
1568.2.i.m.1537.2 4 56.51 odd 6
1568.2.i.n.961.2 4 56.45 odd 6
1568.2.i.n.1537.2 4 56.5 odd 6
1568.2.i.v.961.1 4 56.53 even 6
1568.2.i.v.1537.1 4 56.37 even 6
1568.2.i.w.961.1 4 56.3 even 6
1568.2.i.w.1537.1 4 56.19 even 6
1792.2.b.k.897.2 4 16.13 even 4
1792.2.b.k.897.3 4 16.5 even 4
1792.2.b.m.897.2 4 16.11 odd 4
1792.2.b.m.897.3 4 16.3 odd 4
2016.2.a.o.1.1 2 24.11 even 2
2016.2.a.r.1.1 2 24.5 odd 2
3136.2.a.bf.1.2 2 7.6 odd 2
3136.2.a.by.1.1 2 28.27 even 2
4032.2.a.bv.1.2 2 12.11 even 2
4032.2.a.bw.1.2 2 3.2 odd 2
5600.2.a.z.1.2 2 40.19 odd 2
5600.2.a.bk.1.1 2 40.29 even 2