# Properties

 Label 6525.2.a.bd.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ 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] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} -2.82843 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} -2.82843 q^{7} -3.00000 q^{8} -0.828427 q^{11} -5.65685 q^{13} -2.82843 q^{14} -1.00000 q^{16} -3.65685 q^{17} +0.828427 q^{19} -0.828427 q^{22} -8.48528 q^{23} -5.65685 q^{26} +2.82843 q^{28} +1.00000 q^{29} +4.82843 q^{31} +5.00000 q^{32} -3.65685 q^{34} -6.00000 q^{37} +0.828427 q^{38} +3.65685 q^{41} -9.65685 q^{43} +0.828427 q^{44} -8.48528 q^{46} +5.65685 q^{47} +1.00000 q^{49} +5.65685 q^{52} +4.00000 q^{53} +8.48528 q^{56} +1.00000 q^{58} +4.00000 q^{59} +11.6569 q^{61} +4.82843 q^{62} +7.00000 q^{64} -6.82843 q^{67} +3.65685 q^{68} +1.65685 q^{71} +3.65685 q^{73} -6.00000 q^{74} -0.828427 q^{76} +2.34315 q^{77} -8.82843 q^{79} +3.65685 q^{82} +12.4853 q^{83} -9.65685 q^{86} +2.48528 q^{88} +0.343146 q^{89} +16.0000 q^{91} +8.48528 q^{92} +5.65685 q^{94} -11.6569 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 4 q^{11} - 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{22} + 2 q^{29} + 4 q^{31} + 10 q^{32} + 4 q^{34} - 12 q^{37} - 4 q^{38} - 4 q^{41} - 8 q^{43} - 4 q^{44} + 2 q^{49} + 8 q^{53} + 2 q^{58} + 8 q^{59} + 12 q^{61} + 4 q^{62} + 14 q^{64} - 8 q^{67} - 4 q^{68} - 8 q^{71} - 4 q^{73} - 12 q^{74} + 4 q^{76} + 16 q^{77} - 12 q^{79} - 4 q^{82} + 8 q^{83} - 8 q^{86} - 12 q^{88} + 12 q^{89} + 32 q^{91} - 12 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 4 * q^11 - 2 * q^16 + 4 * q^17 - 4 * q^19 + 4 * q^22 + 2 * q^29 + 4 * q^31 + 10 * q^32 + 4 * q^34 - 12 * q^37 - 4 * q^38 - 4 * q^41 - 8 * q^43 - 4 * q^44 + 2 * q^49 + 8 * q^53 + 2 * q^58 + 8 * q^59 + 12 * q^61 + 4 * q^62 + 14 * q^64 - 8 * q^67 - 4 * q^68 - 8 * q^71 - 4 * q^73 - 12 * q^74 + 4 * q^76 + 16 * q^77 - 12 * q^79 - 4 * q^82 + 8 * q^83 - 8 * q^86 - 12 * q^88 + 12 * q^89 + 32 * q^91 - 12 * q^97 + 2 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −0.828427 −0.249780 −0.124890 0.992171i $$-0.539858\pi$$
−0.124890 + 0.992171i $$0.539858\pi$$
$$12$$ 0 0
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ −2.82843 −0.755929
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −3.65685 −0.886917 −0.443459 0.896295i $$-0.646249\pi$$
−0.443459 + 0.896295i $$0.646249\pi$$
$$18$$ 0 0
$$19$$ 0.828427 0.190054 0.0950271 0.995475i $$-0.469706\pi$$
0.0950271 + 0.995475i $$0.469706\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.828427 −0.176621
$$23$$ −8.48528 −1.76930 −0.884652 0.466252i $$-0.845604\pi$$
−0.884652 + 0.466252i $$0.845604\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −5.65685 −1.10940
$$27$$ 0 0
$$28$$ 2.82843 0.534522
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 4.82843 0.867211 0.433606 0.901103i $$-0.357241\pi$$
0.433606 + 0.901103i $$0.357241\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −3.65685 −0.627145
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0.828427 0.134389
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.65685 0.571105 0.285552 0.958363i $$-0.407823\pi$$
0.285552 + 0.958363i $$0.407823\pi$$
$$42$$ 0 0
$$43$$ −9.65685 −1.47266 −0.736328 0.676625i $$-0.763442\pi$$
−0.736328 + 0.676625i $$0.763442\pi$$
$$44$$ 0.828427 0.124890
$$45$$ 0 0
$$46$$ −8.48528 −1.25109
$$47$$ 5.65685 0.825137 0.412568 0.910927i $$-0.364632\pi$$
0.412568 + 0.910927i $$0.364632\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 5.65685 0.784465
$$53$$ 4.00000 0.549442 0.274721 0.961524i $$-0.411414\pi$$
0.274721 + 0.961524i $$0.411414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 8.48528 1.13389
$$57$$ 0 0
$$58$$ 1.00000 0.131306
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 11.6569 1.49251 0.746254 0.665662i $$-0.231851\pi$$
0.746254 + 0.665662i $$0.231851\pi$$
$$62$$ 4.82843 0.613211
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −6.82843 −0.834225 −0.417113 0.908855i $$-0.636958\pi$$
−0.417113 + 0.908855i $$0.636958\pi$$
$$68$$ 3.65685 0.443459
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 1.65685 0.196632 0.0983162 0.995155i $$-0.468654\pi$$
0.0983162 + 0.995155i $$0.468654\pi$$
$$72$$ 0 0
$$73$$ 3.65685 0.428002 0.214001 0.976833i $$-0.431350\pi$$
0.214001 + 0.976833i $$0.431350\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ −0.828427 −0.0950271
$$77$$ 2.34315 0.267026
$$78$$ 0 0
$$79$$ −8.82843 −0.993276 −0.496638 0.867958i $$-0.665432\pi$$
−0.496638 + 0.867958i $$0.665432\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 3.65685 0.403832
$$83$$ 12.4853 1.37044 0.685219 0.728337i $$-0.259707\pi$$
0.685219 + 0.728337i $$0.259707\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −9.65685 −1.04133
$$87$$ 0 0
$$88$$ 2.48528 0.264932
$$89$$ 0.343146 0.0363734 0.0181867 0.999835i $$-0.494211\pi$$
0.0181867 + 0.999835i $$0.494211\pi$$
$$90$$ 0 0
$$91$$ 16.0000 1.67726
$$92$$ 8.48528 0.884652
$$93$$ 0 0
$$94$$ 5.65685 0.583460
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −11.6569 −1.18357 −0.591787 0.806094i $$-0.701577\pi$$
−0.591787 + 0.806094i $$0.701577\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −17.3137 −1.72278 −0.861389 0.507946i $$-0.830405\pi$$
−0.861389 + 0.507946i $$0.830405\pi$$
$$102$$ 0 0
$$103$$ −12.4853 −1.23021 −0.615106 0.788445i $$-0.710887\pi$$
−0.615106 + 0.788445i $$0.710887\pi$$
$$104$$ 16.9706 1.66410
$$105$$ 0 0
$$106$$ 4.00000 0.388514
$$107$$ 14.8284 1.43352 0.716759 0.697321i $$-0.245625\pi$$
0.716759 + 0.697321i $$0.245625\pi$$
$$108$$ 0 0
$$109$$ 9.31371 0.892091 0.446046 0.895010i $$-0.352832\pi$$
0.446046 + 0.895010i $$0.352832\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.82843 0.267261
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 10.3431 0.948155
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 11.6569 1.05536
$$123$$ 0 0
$$124$$ −4.82843 −0.433606
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 13.6569 1.21185 0.605925 0.795522i $$-0.292803\pi$$
0.605925 + 0.795522i $$0.292803\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 4.82843 0.421862 0.210931 0.977501i $$-0.432351\pi$$
0.210931 + 0.977501i $$0.432351\pi$$
$$132$$ 0 0
$$133$$ −2.34315 −0.203177
$$134$$ −6.82843 −0.589886
$$135$$ 0 0
$$136$$ 10.9706 0.940718
$$137$$ −3.65685 −0.312426 −0.156213 0.987723i $$-0.549929\pi$$
−0.156213 + 0.987723i $$0.549929\pi$$
$$138$$ 0 0
$$139$$ −15.3137 −1.29889 −0.649446 0.760408i $$-0.724999\pi$$
−0.649446 + 0.760408i $$0.724999\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1.65685 0.139040
$$143$$ 4.68629 0.391887
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.65685 0.302643
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ 3.31371 0.269666 0.134833 0.990868i $$-0.456950\pi$$
0.134833 + 0.990868i $$0.456950\pi$$
$$152$$ −2.48528 −0.201583
$$153$$ 0 0
$$154$$ 2.34315 0.188816
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.34315 −0.346621 −0.173310 0.984867i $$-0.555446\pi$$
−0.173310 + 0.984867i $$0.555446\pi$$
$$158$$ −8.82843 −0.702352
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 24.0000 1.89146
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −3.65685 −0.285552
$$165$$ 0 0
$$166$$ 12.4853 0.969046
$$167$$ −0.485281 −0.0375522 −0.0187761 0.999824i $$-0.505977\pi$$
−0.0187761 + 0.999824i $$0.505977\pi$$
$$168$$ 0 0
$$169$$ 19.0000 1.46154
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 9.65685 0.736328
$$173$$ 16.0000 1.21646 0.608229 0.793762i $$-0.291880\pi$$
0.608229 + 0.793762i $$0.291880\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.828427 0.0624450
$$177$$ 0 0
$$178$$ 0.343146 0.0257199
$$179$$ 7.31371 0.546652 0.273326 0.961921i $$-0.411876\pi$$
0.273326 + 0.961921i $$0.411876\pi$$
$$180$$ 0 0
$$181$$ −17.3137 −1.28692 −0.643459 0.765481i $$-0.722501\pi$$
−0.643459 + 0.765481i $$0.722501\pi$$
$$182$$ 16.0000 1.18600
$$183$$ 0 0
$$184$$ 25.4558 1.87663
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.02944 0.221534
$$188$$ −5.65685 −0.412568
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.1421 1.16800 0.584002 0.811752i $$-0.301486\pi$$
0.584002 + 0.811752i $$0.301486\pi$$
$$192$$ 0 0
$$193$$ 17.3137 1.24627 0.623134 0.782115i $$-0.285859\pi$$
0.623134 + 0.782115i $$0.285859\pi$$
$$194$$ −11.6569 −0.836913
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ −16.0000 −1.13995 −0.569976 0.821661i $$-0.693048\pi$$
−0.569976 + 0.821661i $$0.693048\pi$$
$$198$$ 0 0
$$199$$ −12.9706 −0.919459 −0.459729 0.888059i $$-0.652054\pi$$
−0.459729 + 0.888059i $$0.652054\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −17.3137 −1.21819
$$203$$ −2.82843 −0.198517
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −12.4853 −0.869891
$$207$$ 0 0
$$208$$ 5.65685 0.392232
$$209$$ −0.686292 −0.0474718
$$210$$ 0 0
$$211$$ 1.51472 0.104278 0.0521388 0.998640i $$-0.483396\pi$$
0.0521388 + 0.998640i $$0.483396\pi$$
$$212$$ −4.00000 −0.274721
$$213$$ 0 0
$$214$$ 14.8284 1.01365
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −13.6569 −0.927088
$$218$$ 9.31371 0.630804
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.6863 1.39151
$$222$$ 0 0
$$223$$ −17.1716 −1.14989 −0.574947 0.818191i $$-0.694977\pi$$
−0.574947 + 0.818191i $$0.694977\pi$$
$$224$$ −14.1421 −0.944911
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ −13.1716 −0.874228 −0.437114 0.899406i $$-0.643999\pi$$
−0.437114 + 0.899406i $$0.643999\pi$$
$$228$$ 0 0
$$229$$ −4.34315 −0.287003 −0.143502 0.989650i $$-0.545836\pi$$
−0.143502 + 0.989650i $$0.545836\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −3.00000 −0.196960
$$233$$ 24.9706 1.63588 0.817938 0.575306i $$-0.195117\pi$$
0.817938 + 0.575306i $$0.195117\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 10.3431 0.670447
$$239$$ −19.3137 −1.24930 −0.624650 0.780905i $$-0.714758\pi$$
−0.624650 + 0.780905i $$0.714758\pi$$
$$240$$ 0 0
$$241$$ −17.3137 −1.11527 −0.557637 0.830085i $$-0.688292\pi$$
−0.557637 + 0.830085i $$0.688292\pi$$
$$242$$ −10.3137 −0.662990
$$243$$ 0 0
$$244$$ −11.6569 −0.746254
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.68629 −0.298182
$$248$$ −14.4853 −0.919816
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.14214 −0.261449 −0.130725 0.991419i $$-0.541730\pi$$
−0.130725 + 0.991419i $$0.541730\pi$$
$$252$$ 0 0
$$253$$ 7.02944 0.441937
$$254$$ 13.6569 0.856907
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 20.9706 1.30811 0.654054 0.756448i $$-0.273067\pi$$
0.654054 + 0.756448i $$0.273067\pi$$
$$258$$ 0 0
$$259$$ 16.9706 1.05450
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4.82843 0.298301
$$263$$ 27.3137 1.68424 0.842118 0.539294i $$-0.181309\pi$$
0.842118 + 0.539294i $$0.181309\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.34315 −0.143667
$$267$$ 0 0
$$268$$ 6.82843 0.417113
$$269$$ −22.0000 −1.34136 −0.670682 0.741745i $$-0.733998\pi$$
−0.670682 + 0.741745i $$0.733998\pi$$
$$270$$ 0 0
$$271$$ 26.4853 1.60887 0.804433 0.594043i $$-0.202469\pi$$
0.804433 + 0.594043i $$0.202469\pi$$
$$272$$ 3.65685 0.221729
$$273$$ 0 0
$$274$$ −3.65685 −0.220919
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −9.65685 −0.580224 −0.290112 0.956993i $$-0.593693\pi$$
−0.290112 + 0.956993i $$0.593693\pi$$
$$278$$ −15.3137 −0.918455
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 29.3137 1.74871 0.874355 0.485288i $$-0.161285\pi$$
0.874355 + 0.485288i $$0.161285\pi$$
$$282$$ 0 0
$$283$$ 26.1421 1.55399 0.776994 0.629508i $$-0.216743\pi$$
0.776994 + 0.629508i $$0.216743\pi$$
$$284$$ −1.65685 −0.0983162
$$285$$ 0 0
$$286$$ 4.68629 0.277106
$$287$$ −10.3431 −0.610537
$$288$$ 0 0
$$289$$ −3.62742 −0.213377
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.65685 −0.214001
$$293$$ −16.6274 −0.971384 −0.485692 0.874130i $$-0.661432\pi$$
−0.485692 + 0.874130i $$0.661432\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ 0 0
$$298$$ −2.00000 −0.115857
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ 27.3137 1.57434
$$302$$ 3.31371 0.190682
$$303$$ 0 0
$$304$$ −0.828427 −0.0475136
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 17.6569 1.00773 0.503865 0.863782i $$-0.331911\pi$$
0.503865 + 0.863782i $$0.331911\pi$$
$$308$$ −2.34315 −0.133513
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5.51472 0.312711 0.156356 0.987701i $$-0.450025\pi$$
0.156356 + 0.987701i $$0.450025\pi$$
$$312$$ 0 0
$$313$$ −28.0000 −1.58265 −0.791327 0.611393i $$-0.790609\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$314$$ −4.34315 −0.245098
$$315$$ 0 0
$$316$$ 8.82843 0.496638
$$317$$ −11.6569 −0.654714 −0.327357 0.944901i $$-0.606158\pi$$
−0.327357 + 0.944901i $$0.606158\pi$$
$$318$$ 0 0
$$319$$ −0.828427 −0.0463830
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 24.0000 1.33747
$$323$$ −3.02944 −0.168562
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ −10.9706 −0.605748
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 8.82843 0.485254 0.242627 0.970120i $$-0.421991\pi$$
0.242627 + 0.970120i $$0.421991\pi$$
$$332$$ −12.4853 −0.685219
$$333$$ 0 0
$$334$$ −0.485281 −0.0265534
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −16.3431 −0.890268 −0.445134 0.895464i $$-0.646844\pi$$
−0.445134 + 0.895464i $$0.646844\pi$$
$$338$$ 19.0000 1.03346
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 28.9706 1.56199
$$345$$ 0 0
$$346$$ 16.0000 0.860165
$$347$$ −22.8284 −1.22549 −0.612747 0.790279i $$-0.709936\pi$$
−0.612747 + 0.790279i $$0.709936\pi$$
$$348$$ 0 0
$$349$$ 13.3137 0.712666 0.356333 0.934359i $$-0.384027\pi$$
0.356333 + 0.934359i $$0.384027\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −4.14214 −0.220777
$$353$$ 24.9706 1.32905 0.664524 0.747266i $$-0.268634\pi$$
0.664524 + 0.747266i $$0.268634\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −0.343146 −0.0181867
$$357$$ 0 0
$$358$$ 7.31371 0.386542
$$359$$ −20.1421 −1.06306 −0.531531 0.847039i $$-0.678383\pi$$
−0.531531 + 0.847039i $$0.678383\pi$$
$$360$$ 0 0
$$361$$ −18.3137 −0.963879
$$362$$ −17.3137 −0.909988
$$363$$ 0 0
$$364$$ −16.0000 −0.838628
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3.31371 −0.172974 −0.0864871 0.996253i $$-0.527564\pi$$
−0.0864871 + 0.996253i $$0.527564\pi$$
$$368$$ 8.48528 0.442326
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −11.3137 −0.587378
$$372$$ 0 0
$$373$$ 29.6569 1.53557 0.767787 0.640705i $$-0.221358\pi$$
0.767787 + 0.640705i $$0.221358\pi$$
$$374$$ 3.02944 0.156648
$$375$$ 0 0
$$376$$ −16.9706 −0.875190
$$377$$ −5.65685 −0.291343
$$378$$ 0 0
$$379$$ −11.1716 −0.573845 −0.286923 0.957954i $$-0.592632\pi$$
−0.286923 + 0.957954i $$0.592632\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 16.1421 0.825904
$$383$$ −11.5147 −0.588375 −0.294187 0.955748i $$-0.595049\pi$$
−0.294187 + 0.955748i $$0.595049\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 17.3137 0.881245
$$387$$ 0 0
$$388$$ 11.6569 0.591787
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 31.0294 1.56923
$$392$$ −3.00000 −0.151523
$$393$$ 0 0
$$394$$ −16.0000 −0.806068
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 24.9706 1.25324 0.626618 0.779326i $$-0.284439\pi$$
0.626618 + 0.779326i $$0.284439\pi$$
$$398$$ −12.9706 −0.650156
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ 0 0
$$403$$ −27.3137 −1.36059
$$404$$ 17.3137 0.861389
$$405$$ 0 0
$$406$$ −2.82843 −0.140372
$$407$$ 4.97056 0.246382
$$408$$ 0 0
$$409$$ −28.6274 −1.41553 −0.707767 0.706446i $$-0.750297\pi$$
−0.707767 + 0.706446i $$0.750297\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 12.4853 0.615106
$$413$$ −11.3137 −0.556711
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −28.2843 −1.38675
$$417$$ 0 0
$$418$$ −0.686292 −0.0335676
$$419$$ 8.97056 0.438241 0.219120 0.975698i $$-0.429681\pi$$
0.219120 + 0.975698i $$0.429681\pi$$
$$420$$ 0 0
$$421$$ 30.9706 1.50941 0.754706 0.656063i $$-0.227779\pi$$
0.754706 + 0.656063i $$0.227779\pi$$
$$422$$ 1.51472 0.0737353
$$423$$ 0 0
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −32.9706 −1.59556
$$428$$ −14.8284 −0.716759
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.9706 0.624770 0.312385 0.949956i $$-0.398872\pi$$
0.312385 + 0.949956i $$0.398872\pi$$
$$432$$ 0 0
$$433$$ −31.9411 −1.53499 −0.767496 0.641053i $$-0.778498\pi$$
−0.767496 + 0.641053i $$0.778498\pi$$
$$434$$ −13.6569 −0.655550
$$435$$ 0 0
$$436$$ −9.31371 −0.446046
$$437$$ −7.02944 −0.336264
$$438$$ 0 0
$$439$$ 14.6274 0.698129 0.349064 0.937099i $$-0.386499\pi$$
0.349064 + 0.937099i $$0.386499\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 20.6863 0.983947
$$443$$ −12.9706 −0.616250 −0.308125 0.951346i $$-0.599702\pi$$
−0.308125 + 0.951346i $$0.599702\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −17.1716 −0.813098
$$447$$ 0 0
$$448$$ −19.7990 −0.935414
$$449$$ 7.65685 0.361349 0.180675 0.983543i $$-0.442172\pi$$
0.180675 + 0.983543i $$0.442172\pi$$
$$450$$ 0 0
$$451$$ −3.02944 −0.142651
$$452$$ −14.0000 −0.658505
$$453$$ 0 0
$$454$$ −13.1716 −0.618173
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.6274 0.684242 0.342121 0.939656i $$-0.388855\pi$$
0.342121 + 0.939656i $$0.388855\pi$$
$$458$$ −4.34315 −0.202942
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ −30.8284 −1.43272 −0.716359 0.697732i $$-0.754193\pi$$
−0.716359 + 0.697732i $$0.754193\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 24.9706 1.15674
$$467$$ −24.2843 −1.12374 −0.561871 0.827225i $$-0.689918\pi$$
−0.561871 + 0.827225i $$0.689918\pi$$
$$468$$ 0 0
$$469$$ 19.3137 0.891824
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 8.00000 0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −10.3431 −0.474077
$$477$$ 0 0
$$478$$ −19.3137 −0.883388
$$479$$ 1.51472 0.0692093 0.0346046 0.999401i $$-0.488983\pi$$
0.0346046 + 0.999401i $$0.488983\pi$$
$$480$$ 0 0
$$481$$ 33.9411 1.54758
$$482$$ −17.3137 −0.788618
$$483$$ 0 0
$$484$$ 10.3137 0.468805
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 21.1716 0.959376 0.479688 0.877439i $$-0.340750\pi$$
0.479688 + 0.877439i $$0.340750\pi$$
$$488$$ −34.9706 −1.58304
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −38.4853 −1.73682 −0.868408 0.495850i $$-0.834857\pi$$
−0.868408 + 0.495850i $$0.834857\pi$$
$$492$$ 0 0
$$493$$ −3.65685 −0.164696
$$494$$ −4.68629 −0.210846
$$495$$ 0 0
$$496$$ −4.82843 −0.216803
$$497$$ −4.68629 −0.210209
$$498$$ 0 0
$$499$$ 29.6569 1.32762 0.663812 0.747900i $$-0.268938\pi$$
0.663812 + 0.747900i $$0.268938\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4.14214 −0.184873
$$503$$ −36.2843 −1.61784 −0.808918 0.587922i $$-0.799946\pi$$
−0.808918 + 0.587922i $$0.799946\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 7.02944 0.312497
$$507$$ 0 0
$$508$$ −13.6569 −0.605925
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −10.3431 −0.457554
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 20.9706 0.924972
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.68629 −0.206103
$$518$$ 16.9706 0.745644
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 16.6274 0.728460 0.364230 0.931309i $$-0.381332\pi$$
0.364230 + 0.931309i $$0.381332\pi$$
$$522$$ 0 0
$$523$$ 31.7990 1.39047 0.695236 0.718781i $$-0.255300\pi$$
0.695236 + 0.718781i $$0.255300\pi$$
$$524$$ −4.82843 −0.210931
$$525$$ 0 0
$$526$$ 27.3137 1.19093
$$527$$ −17.6569 −0.769145
$$528$$ 0 0
$$529$$ 49.0000 2.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.34315 0.101588
$$533$$ −20.6863 −0.896023
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 20.4853 0.884829
$$537$$ 0 0
$$538$$ −22.0000 −0.948487
$$539$$ −0.828427 −0.0356829
$$540$$ 0 0
$$541$$ 12.3431 0.530673 0.265337 0.964156i $$-0.414517\pi$$
0.265337 + 0.964156i $$0.414517\pi$$
$$542$$ 26.4853 1.13764
$$543$$ 0 0
$$544$$ −18.2843 −0.783932
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 14.1421 0.604674 0.302337 0.953201i $$-0.402233\pi$$
0.302337 + 0.953201i $$0.402233\pi$$
$$548$$ 3.65685 0.156213
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.828427 0.0352922
$$552$$ 0 0
$$553$$ 24.9706 1.06186
$$554$$ −9.65685 −0.410280
$$555$$ 0 0
$$556$$ 15.3137 0.649446
$$557$$ 15.3137 0.648863 0.324431 0.945909i $$-0.394827\pi$$
0.324431 + 0.945909i $$0.394827\pi$$
$$558$$ 0 0
$$559$$ 54.6274 2.31049
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 29.3137 1.23652
$$563$$ 10.6274 0.447892 0.223946 0.974602i $$-0.428106\pi$$
0.223946 + 0.974602i $$0.428106\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 26.1421 1.09884
$$567$$ 0 0
$$568$$ −4.97056 −0.208560
$$569$$ −30.9706 −1.29835 −0.649177 0.760638i $$-0.724886\pi$$
−0.649177 + 0.760638i $$0.724886\pi$$
$$570$$ 0 0
$$571$$ −39.3137 −1.64523 −0.822614 0.568601i $$-0.807485\pi$$
−0.822614 + 0.568601i $$0.807485\pi$$
$$572$$ −4.68629 −0.195944
$$573$$ 0 0
$$574$$ −10.3431 −0.431715
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −5.31371 −0.221213 −0.110606 0.993864i $$-0.535279\pi$$
−0.110606 + 0.993864i $$0.535279\pi$$
$$578$$ −3.62742 −0.150881
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −35.3137 −1.46506
$$582$$ 0 0
$$583$$ −3.31371 −0.137240
$$584$$ −10.9706 −0.453965
$$585$$ 0 0
$$586$$ −16.6274 −0.686872
$$587$$ −19.7990 −0.817192 −0.408596 0.912715i $$-0.633981\pi$$
−0.408596 + 0.912715i $$0.633981\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ −18.3431 −0.753263 −0.376631 0.926363i $$-0.622918\pi$$
−0.376631 + 0.926363i $$0.622918\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 0 0
$$598$$ 48.0000 1.96287
$$599$$ 21.1127 0.862641 0.431321 0.902199i $$-0.358048\pi$$
0.431321 + 0.902199i $$0.358048\pi$$
$$600$$ 0 0
$$601$$ −21.3137 −0.869404 −0.434702 0.900574i $$-0.643146\pi$$
−0.434702 + 0.900574i $$0.643146\pi$$
$$602$$ 27.3137 1.11322
$$603$$ 0 0
$$604$$ −3.31371 −0.134833
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −24.9706 −1.01352 −0.506762 0.862086i $$-0.669158\pi$$
−0.506762 + 0.862086i $$0.669158\pi$$
$$608$$ 4.14214 0.167986
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −32.0000 −1.29458
$$612$$ 0 0
$$613$$ 4.97056 0.200759 0.100380 0.994949i $$-0.467994\pi$$
0.100380 + 0.994949i $$0.467994\pi$$
$$614$$ 17.6569 0.712573
$$615$$ 0 0
$$616$$ −7.02944 −0.283224
$$617$$ 21.3137 0.858058 0.429029 0.903291i $$-0.358856\pi$$
0.429029 + 0.903291i $$0.358856\pi$$
$$618$$ 0 0
$$619$$ 15.8579 0.637381 0.318691 0.947859i $$-0.396757\pi$$
0.318691 + 0.947859i $$0.396757\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 5.51472 0.221120
$$623$$ −0.970563 −0.0388848
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −28.0000 −1.11911
$$627$$ 0 0
$$628$$ 4.34315 0.173310
$$629$$ 21.9411 0.874850
$$630$$ 0 0
$$631$$ −4.97056 −0.197875 −0.0989375 0.995094i $$-0.531544\pi$$
−0.0989375 + 0.995094i $$0.531544\pi$$
$$632$$ 26.4853 1.05353
$$633$$ 0 0
$$634$$ −11.6569 −0.462953
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −5.65685 −0.224133
$$638$$ −0.828427 −0.0327977
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 26.2843 1.03817 0.519083 0.854724i $$-0.326273\pi$$
0.519083 + 0.854724i $$0.326273\pi$$
$$642$$ 0 0
$$643$$ 21.4558 0.846136 0.423068 0.906098i $$-0.360953\pi$$
0.423068 + 0.906098i $$0.360953\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ −3.02944 −0.119192
$$647$$ 6.82843 0.268453 0.134227 0.990951i $$-0.457145\pi$$
0.134227 + 0.990951i $$0.457145\pi$$
$$648$$ 0 0
$$649$$ −3.31371 −0.130074
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ −9.31371 −0.364474 −0.182237 0.983255i $$-0.558334\pi$$
−0.182237 + 0.983255i $$0.558334\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −3.65685 −0.142776
$$657$$ 0 0
$$658$$ −16.0000 −0.623745
$$659$$ 19.4558 0.757892 0.378946 0.925419i $$-0.376287\pi$$
0.378946 + 0.925419i $$0.376287\pi$$
$$660$$ 0 0
$$661$$ −31.9411 −1.24237 −0.621183 0.783666i $$-0.713348\pi$$
−0.621183 + 0.783666i $$0.713348\pi$$
$$662$$ 8.82843 0.343127
$$663$$ 0 0
$$664$$ −37.4558 −1.45357
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.48528 −0.328551
$$668$$ 0.485281 0.0187761
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −9.65685 −0.372799
$$672$$ 0 0
$$673$$ 4.00000 0.154189 0.0770943 0.997024i $$-0.475436\pi$$
0.0770943 + 0.997024i $$0.475436\pi$$
$$674$$ −16.3431 −0.629514
$$675$$ 0 0
$$676$$ −19.0000 −0.730769
$$677$$ −8.62742 −0.331579 −0.165789 0.986161i $$-0.553017\pi$$
−0.165789 + 0.986161i $$0.553017\pi$$
$$678$$ 0 0
$$679$$ 32.9706 1.26529
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −4.00000 −0.153168
$$683$$ −22.1421 −0.847245 −0.423623 0.905839i $$-0.639242\pi$$
−0.423623 + 0.905839i $$0.639242\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 16.9706 0.647939
$$687$$ 0 0
$$688$$ 9.65685 0.368164
$$689$$ −22.6274 −0.862036
$$690$$ 0 0
$$691$$ −28.2843 −1.07598 −0.537992 0.842950i $$-0.680817\pi$$
−0.537992 + 0.842950i $$0.680817\pi$$
$$692$$ −16.0000 −0.608229
$$693$$ 0 0
$$694$$ −22.8284 −0.866555
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −13.3726 −0.506523
$$698$$ 13.3137 0.503931
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −21.3137 −0.805008 −0.402504 0.915418i $$-0.631860\pi$$
−0.402504 + 0.915418i $$0.631860\pi$$
$$702$$ 0 0
$$703$$ −4.97056 −0.187468
$$704$$ −5.79899 −0.218558
$$705$$ 0 0
$$706$$ 24.9706 0.939780
$$707$$ 48.9706 1.84173
$$708$$ 0 0
$$709$$ −8.62742 −0.324009 −0.162005 0.986790i $$-0.551796\pi$$
−0.162005 + 0.986790i $$0.551796\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −1.02944 −0.0385798
$$713$$ −40.9706 −1.53436
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −7.31371 −0.273326
$$717$$ 0 0
$$718$$ −20.1421 −0.751698
$$719$$ 4.68629 0.174769 0.0873846 0.996175i $$-0.472149\pi$$
0.0873846 + 0.996175i $$0.472149\pi$$
$$720$$ 0 0
$$721$$ 35.3137 1.31515
$$722$$ −18.3137 −0.681566
$$723$$ 0 0
$$724$$ 17.3137 0.643459
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 46.6274 1.72932 0.864658 0.502362i $$-0.167535\pi$$
0.864658 + 0.502362i $$0.167535\pi$$
$$728$$ −48.0000 −1.77900
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 35.3137 1.30612
$$732$$ 0 0
$$733$$ −29.3137 −1.08273 −0.541363 0.840789i $$-0.682092\pi$$
−0.541363 + 0.840789i $$0.682092\pi$$
$$734$$ −3.31371 −0.122311
$$735$$ 0 0
$$736$$ −42.4264 −1.56386
$$737$$ 5.65685 0.208373
$$738$$ 0 0
$$739$$ −33.1127 −1.21807 −0.609035 0.793143i $$-0.708443\pi$$
−0.609035 + 0.793143i $$0.708443\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −11.3137 −0.415339
$$743$$ 16.9706 0.622590 0.311295 0.950313i $$-0.399237\pi$$
0.311295 + 0.950313i $$0.399237\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 29.6569 1.08581
$$747$$ 0 0
$$748$$ −3.02944 −0.110767
$$749$$ −41.9411 −1.53250
$$750$$ 0 0
$$751$$ 11.1716 0.407656 0.203828 0.979007i $$-0.434662\pi$$
0.203828 + 0.979007i $$0.434662\pi$$
$$752$$ −5.65685 −0.206284
$$753$$ 0 0
$$754$$ −5.65685 −0.206010
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18.2843 −0.664553 −0.332277 0.943182i $$-0.607817\pi$$
−0.332277 + 0.943182i $$0.607817\pi$$
$$758$$ −11.1716 −0.405770
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 0 0
$$763$$ −26.3431 −0.953686
$$764$$ −16.1421 −0.584002
$$765$$ 0 0
$$766$$ −11.5147 −0.416044
$$767$$ −22.6274 −0.817029
$$768$$ 0 0
$$769$$ 4.62742 0.166869 0.0834345 0.996513i $$-0.473411\pi$$
0.0834345 + 0.996513i $$0.473411\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −17.3137 −0.623134
$$773$$ 38.2843 1.37699 0.688495 0.725241i $$-0.258272\pi$$
0.688495 + 0.725241i $$0.258272\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 34.9706 1.25537
$$777$$ 0 0
$$778$$ 26.0000 0.932145
$$779$$ 3.02944 0.108541
$$780$$ 0 0
$$781$$ −1.37258 −0.0491149
$$782$$ 31.0294 1.10961
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 35.7990 1.27610 0.638048 0.769997i $$-0.279742\pi$$
0.638048 + 0.769997i $$0.279742\pi$$
$$788$$ 16.0000 0.569976
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −39.5980 −1.40794
$$792$$ 0 0
$$793$$ −65.9411 −2.34164
$$794$$ 24.9706 0.886172
$$795$$ 0 0
$$796$$ 12.9706 0.459729
$$797$$ 13.3137 0.471596 0.235798 0.971802i $$-0.424230\pi$$
0.235798 + 0.971802i $$0.424230\pi$$
$$798$$ 0 0
$$799$$ −20.6863 −0.731828
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 10.0000 0.353112
$$803$$ −3.02944 −0.106907
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −27.3137 −0.962084
$$807$$ 0 0
$$808$$ 51.9411 1.82728
$$809$$ −20.3431 −0.715227 −0.357613 0.933870i $$-0.616409\pi$$
−0.357613 + 0.933870i $$0.616409\pi$$
$$810$$ 0 0
$$811$$ 39.5980 1.39047 0.695237 0.718781i $$-0.255300\pi$$
0.695237 + 0.718781i $$0.255300\pi$$
$$812$$ 2.82843 0.0992583
$$813$$ 0 0
$$814$$ 4.97056 0.174218
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ −28.6274 −1.00093
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −43.2548 −1.50960 −0.754802 0.655953i $$-0.772267\pi$$
−0.754802 + 0.655953i $$0.772267\pi$$
$$822$$ 0 0
$$823$$ 8.00000 0.278862 0.139431 0.990232i $$-0.455473\pi$$
0.139431 + 0.990232i $$0.455473\pi$$
$$824$$ 37.4558 1.30484
$$825$$ 0 0
$$826$$ −11.3137 −0.393654
$$827$$ −9.65685 −0.335802 −0.167901 0.985804i $$-0.553699\pi$$
−0.167901 + 0.985804i $$0.553699\pi$$
$$828$$ 0 0
$$829$$ 22.2843 0.773965 0.386982 0.922087i $$-0.373517\pi$$
0.386982 + 0.922087i $$0.373517\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −39.5980 −1.37281
$$833$$ −3.65685 −0.126702
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0.686292 0.0237359
$$837$$ 0 0
$$838$$ 8.97056 0.309883
$$839$$ 14.2010 0.490273 0.245137 0.969489i $$-0.421167\pi$$
0.245137 + 0.969489i $$0.421167\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 30.9706 1.06732
$$843$$ 0 0
$$844$$ −1.51472 −0.0521388
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 29.1716 1.00235
$$848$$ −4.00000 −0.137361
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 50.9117 1.74523
$$852$$ 0 0
$$853$$ −50.0000 −1.71197 −0.855984 0.517003i $$-0.827048\pi$$
−0.855984 + 0.517003i $$0.827048\pi$$
$$854$$ −32.9706 −1.12823
$$855$$ 0 0
$$856$$ −44.4853 −1.52048
$$857$$ 6.34315 0.216678 0.108339 0.994114i $$-0.465447\pi$$
0.108339 + 0.994114i $$0.465447\pi$$
$$858$$ 0 0
$$859$$ 22.4853 0.767188 0.383594 0.923502i $$-0.374686\pi$$
0.383594 + 0.923502i $$0.374686\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 12.9706 0.441779
$$863$$ 1.17157 0.0398808 0.0199404 0.999801i $$-0.493652\pi$$
0.0199404 + 0.999801i $$0.493652\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −31.9411 −1.08540
$$867$$ 0 0
$$868$$ 13.6569 0.463544
$$869$$ 7.31371 0.248101
$$870$$ 0 0
$$871$$ 38.6274 1.30884
$$872$$ −27.9411 −0.946206
$$873$$ 0 0
$$874$$ −7.02944 −0.237774
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −43.5980 −1.47220 −0.736100 0.676873i $$-0.763335\pi$$
−0.736100 + 0.676873i $$0.763335\pi$$
$$878$$ 14.6274 0.493651
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −10.9706 −0.369608 −0.184804 0.982775i $$-0.559165\pi$$
−0.184804 + 0.982775i $$0.559165\pi$$
$$882$$ 0 0
$$883$$ 14.1421 0.475921 0.237960 0.971275i $$-0.423521\pi$$
0.237960 + 0.971275i $$0.423521\pi$$
$$884$$ −20.6863 −0.695755
$$885$$ 0 0
$$886$$ −12.9706 −0.435755
$$887$$ −40.9706 −1.37566 −0.687828 0.725873i $$-0.741436\pi$$
−0.687828 + 0.725873i $$0.741436\pi$$
$$888$$ 0 0
$$889$$ −38.6274 −1.29552
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 17.1716 0.574947
$$893$$ 4.68629 0.156821
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8.48528 0.283473
$$897$$ 0 0
$$898$$ 7.65685 0.255513
$$899$$ 4.82843 0.161037
$$900$$ 0 0
$$901$$ −14.6274 −0.487310
$$902$$ −3.02944 −0.100869
$$903$$ 0 0
$$904$$ −42.0000 −1.39690
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −9.65685 −0.320651 −0.160325 0.987064i $$-0.551254\pi$$
−0.160325 + 0.987064i $$0.551254\pi$$
$$908$$ 13.1716 0.437114
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −20.1421 −0.667339 −0.333669 0.942690i $$-0.608287\pi$$
−0.333669 + 0.942690i $$0.608287\pi$$
$$912$$ 0 0
$$913$$ −10.3431 −0.342308
$$914$$ 14.6274 0.483832
$$915$$ 0 0
$$916$$ 4.34315 0.143502
$$917$$ −13.6569 −0.450989
$$918$$ 0 0
$$919$$ −12.9706 −0.427859 −0.213930 0.976849i $$-0.568626\pi$$
−0.213930 + 0.976849i $$0.568626\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −6.00000 −0.197599
$$923$$ −9.37258 −0.308502
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −30.8284 −1.01308
$$927$$ 0 0
$$928$$ 5.00000 0.164133
$$929$$ −1.31371 −0.0431014 −0.0215507 0.999768i $$-0.506860\pi$$
−0.0215507 + 0.999768i $$0.506860\pi$$
$$930$$ 0 0
$$931$$ 0.828427 0.0271506
$$932$$ −24.9706 −0.817938
$$933$$ 0 0
$$934$$ −24.2843 −0.794606
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 46.6274 1.52325 0.761626 0.648017i $$-0.224402\pi$$
0.761626 + 0.648017i $$0.224402\pi$$
$$938$$ 19.3137 0.630615
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 11.9411 0.389270 0.194635 0.980876i $$-0.437648\pi$$
0.194635 + 0.980876i $$0.437648\pi$$
$$942$$ 0 0
$$943$$ −31.0294 −1.01046
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 36.9706 1.20138 0.600691 0.799481i $$-0.294892\pi$$
0.600691 + 0.799481i $$0.294892\pi$$
$$948$$ 0 0
$$949$$ −20.6863 −0.671505
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −31.0294 −1.00567
$$953$$ 23.0294 0.745997 0.372998 0.927832i $$-0.378330\pi$$
0.372998 + 0.927832i $$0.378330\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 19.3137 0.624650
$$957$$ 0 0
$$958$$ 1.51472 0.0489383
$$959$$ 10.3431 0.333998
$$960$$ 0 0
$$961$$ −7.68629 −0.247945
$$962$$ 33.9411 1.09431
$$963$$ 0 0
$$964$$ 17.3137 0.557637
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −24.0000 −0.771788 −0.385894 0.922543i $$-0.626107\pi$$
−0.385894 + 0.922543i $$0.626107\pi$$
$$968$$ 30.9411 0.994485
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 21.1127 0.677539 0.338769 0.940869i $$-0.389989\pi$$
0.338769 + 0.940869i $$0.389989\pi$$
$$972$$ 0 0
$$973$$ 43.3137 1.38857
$$974$$ 21.1716 0.678381
$$975$$ 0 0
$$976$$ −11.6569 −0.373127
$$977$$ 52.9706 1.69468 0.847339 0.531052i $$-0.178203\pi$$
0.847339 + 0.531052i $$0.178203\pi$$
$$978$$ 0 0
$$979$$ −0.284271 −0.00908535
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −38.4853 −1.22811
$$983$$ −29.6569 −0.945907 −0.472953 0.881087i $$-0.656812\pi$$
−0.472953 + 0.881087i $$0.656812\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −3.65685 −0.116458
$$987$$ 0 0
$$988$$ 4.68629 0.149091
$$989$$ 81.9411 2.60558
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 24.1421 0.766514
$$993$$ 0 0
$$994$$ −4.68629 −0.148640
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 58.9706 1.86762 0.933808 0.357774i $$-0.116464\pi$$
0.933808 + 0.357774i $$0.116464\pi$$
$$998$$ 29.6569 0.938771
$$999$$ 0 0
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 6525.2.a.bd.1.1 2
3.2 odd 2 2175.2.a.k.1.1 2
5.2 odd 4 1305.2.c.g.784.3 4
5.3 odd 4 1305.2.c.g.784.2 4
5.4 even 2 6525.2.a.n.1.2 2
15.2 even 4 435.2.c.c.349.1 4
15.8 even 4 435.2.c.c.349.4 yes 4
15.14 odd 2 2175.2.a.s.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.c.349.1 4 15.2 even 4
435.2.c.c.349.4 yes 4 15.8 even 4
1305.2.c.g.784.2 4 5.3 odd 4
1305.2.c.g.784.3 4 5.2 odd 4
2175.2.a.k.1.1 2 3.2 odd 2
2175.2.a.s.1.2 2 15.14 odd 2
6525.2.a.n.1.2 2 5.4 even 2
6525.2.a.bd.1.1 2 1.1 even 1 trivial