# Properties

 Label 1925.2.b.d.1849.2 Level $1925$ Weight $2$ Character 1925.1849 Analytic conductor $15.371$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1849,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1925.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1925.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$15.3712023891$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1925.1849 Dual form 1925.2.b.d.1849.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} -1.00000i q^{18} -2.00000 q^{21} +1.00000i q^{22} +4.00000i q^{23} +6.00000 q^{24} +4.00000 q^{26} -4.00000i q^{27} -1.00000i q^{28} +6.00000 q^{29} +10.0000 q^{31} +5.00000i q^{32} -2.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} -8.00000 q^{39} +4.00000 q^{41} -2.00000i q^{42} -12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} -10.0000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +8.00000 q^{51} -4.00000i q^{52} +6.00000i q^{53} +4.00000 q^{54} +3.00000 q^{56} +6.00000i q^{58} -2.00000 q^{59} +10.0000i q^{62} +1.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} +4.00000i q^{68} +8.00000 q^{69} -12.0000 q^{71} -3.00000i q^{72} +8.00000i q^{73} +6.00000 q^{74} -1.00000i q^{77} -8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} +4.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} -12.0000i q^{87} +3.00000i q^{88} +6.00000 q^{89} -4.00000 q^{91} +4.00000i q^{92} -20.0000i q^{93} +10.0000 q^{94} +10.0000 q^{96} -10.0000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 $$2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{14} - 2 q^{16} - 4 q^{21} + 12 q^{24} + 8 q^{26} + 12 q^{29} + 20 q^{31} - 8 q^{34} - 2 q^{36} - 16 q^{39} + 8 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49} + 16 q^{51} + 8 q^{54} + 6 q^{56} - 4 q^{59} - 14 q^{64} + 4 q^{66} + 16 q^{69} - 24 q^{71} + 12 q^{74} - 16 q^{79} - 22 q^{81} - 4 q^{84} + 24 q^{86} + 12 q^{89} - 8 q^{91} + 20 q^{94} + 20 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^14 - 2 * q^16 - 4 * q^21 + 12 * q^24 + 8 * q^26 + 12 * q^29 + 20 * q^31 - 8 * q^34 - 2 * q^36 - 16 * q^39 + 8 * q^41 + 2 * q^44 - 8 * q^46 - 2 * q^49 + 16 * q^51 + 8 * q^54 + 6 * q^56 - 4 * q^59 - 14 * q^64 + 4 * q^66 + 16 * q^69 - 24 * q^71 + 12 * q^74 - 16 * q^79 - 22 * q^81 - 4 * q^84 + 24 * q^86 + 12 * q^89 - 8 * q^91 + 20 * q^94 + 20 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$$.

 $$n$$ $$276$$ $$1002$$ $$1751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ − 2.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ − 1.00000i − 0.377964i
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 4.00000i − 1.10940i −0.832050 0.554700i $$-0.812833\pi$$
0.832050 0.554700i $$-0.187167\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 1.00000i 0.213201i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 6.00000 1.22474
$$25$$ 0 0
$$26$$ 4.00000 0.784465
$$27$$ − 4.00000i − 0.769800i
$$28$$ − 1.00000i − 0.188982i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ − 2.00000i − 0.348155i
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 12.0000i − 1.82998i −0.403473 0.914991i $$-0.632197\pi$$
0.403473 0.914991i $$-0.367803\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ − 10.0000i − 1.45865i −0.684167 0.729325i $$-0.739834\pi$$
0.684167 0.729325i $$-0.260166\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 8.00000 1.12022
$$52$$ − 4.00000i − 0.554700i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ −2.00000 −0.260378 −0.130189 0.991489i $$-0.541558\pi$$
−0.130189 + 0.991489i $$0.541558\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 10.0000i 1.27000i
$$63$$ 1.00000i 0.125988i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 8.00000i 0.936329i 0.883641 + 0.468165i $$0.155085\pi$$
−0.883641 + 0.468165i $$0.844915\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1.00000i − 0.113961i
$$78$$ − 8.00000i − 0.905822i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 4.00000i 0.441726i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ − 12.0000i − 1.28654i
$$88$$ 3.00000i 0.319801i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 4.00000i 0.417029i
$$93$$ − 20.0000i − 2.07390i
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ 10.0000 1.02062
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 8.00000i 0.792118i
$$103$$ − 14.0000i − 1.37946i −0.724066 0.689730i $$-0.757729\pi$$
0.724066 0.689730i $$-0.242271\pi$$
$$104$$ 12.0000 1.17670
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ 1.00000i 0.0944911i
$$113$$ − 18.0000i − 1.69330i −0.532152 0.846649i $$-0.678617\pi$$
0.532152 0.846649i $$-0.321383\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ 4.00000i 0.369800i
$$118$$ − 2.00000i − 0.184115i
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 8.00000i − 0.721336i
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ −24.0000 −2.11308
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −12.0000 −1.02899
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −20.0000 −1.68430
$$142$$ − 12.0000i − 1.00702i
$$143$$ − 4.00000i − 0.334497i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −8.00000 −0.662085
$$147$$ 2.00000i 0.164957i
$$148$$ − 6.00000i − 0.493197i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ − 4.00000i − 0.323381i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ −8.00000 −0.640513
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ − 11.0000i − 0.864242i
$$163$$ 8.00000i 0.626608i 0.949653 + 0.313304i $$0.101436\pi$$
−0.949653 + 0.313304i $$0.898564\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ − 6.00000i − 0.462910i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 12.0000i − 0.914991i
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ 12.0000 0.909718
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 4.00000i 0.300658i
$$178$$ 6.00000i 0.449719i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ − 4.00000i − 0.296500i
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 0 0
$$186$$ 20.0000 1.46647
$$187$$ 4.00000i 0.292509i
$$188$$ − 10.0000i − 0.729325i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 14.0000i 1.01036i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ 22.0000i 1.56744i 0.621117 + 0.783718i $$0.286679\pi$$
−0.621117 + 0.783718i $$0.713321\pi$$
$$198$$ − 1.00000i − 0.0710669i
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ − 4.00000i − 0.281439i
$$203$$ − 6.00000i − 0.421117i
$$204$$ 8.00000 0.560112
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ − 4.00000i − 0.278019i
$$208$$ 4.00000i 0.277350i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 24.0000i 1.64445i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 12.0000 0.816497
$$217$$ − 10.0000i − 0.678844i
$$218$$ 14.0000i 0.948200i
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ 16.0000 1.07628
$$222$$ − 12.0000i − 0.805387i
$$223$$ − 22.0000i − 1.47323i −0.676313 0.736614i $$-0.736423\pi$$
0.676313 0.736614i $$-0.263577\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 18.0000i 1.18176i
$$233$$ 18.0000i 1.17922i 0.807688 + 0.589610i $$0.200718\pi$$
−0.807688 + 0.589610i $$0.799282\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −2.00000 −0.130189
$$237$$ 16.0000i 1.03931i
$$238$$ 4.00000i 0.259281i
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ 10.0000i 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ 0 0
$$248$$ 30.0000i 1.90500i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ 1.00000i 0.0629941i
$$253$$ 4.00000i 0.251478i
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ − 24.0000i − 1.49417i
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 12.0000i − 0.734388i
$$268$$ 8.00000i 0.488678i
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ − 4.00000i − 0.242536i
$$273$$ 8.00000i 0.484182i
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ − 20.0000i − 1.19098i
$$283$$ − 4.00000i − 0.237775i −0.992908 0.118888i $$-0.962067\pi$$
0.992908 0.118888i $$-0.0379328\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ − 4.00000i − 0.236113i
$$288$$ − 5.00000i − 0.294628i
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −20.0000 −1.17242
$$292$$ 8.00000i 0.468165i
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ − 4.00000i − 0.232104i
$$298$$ 10.0000i 0.579284i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ − 16.0000i − 0.920697i
$$303$$ 8.00000i 0.459588i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ 20.0000i 1.14146i 0.821138 + 0.570730i $$0.193340\pi$$
−0.821138 + 0.570730i $$0.806660\pi$$
$$308$$ − 1.00000i − 0.0569803i
$$309$$ −28.0000 −1.59286
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ − 24.0000i − 1.35873i
$$313$$ − 2.00000i − 0.113047i −0.998401 0.0565233i $$-0.981998\pi$$
0.998401 0.0565233i $$-0.0180015\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 12.0000i 0.672927i
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 24.0000 1.33955
$$322$$ 4.00000i 0.222911i
$$323$$ 0 0
$$324$$ −11.0000 −0.611111
$$325$$ 0 0
$$326$$ −8.00000 −0.443079
$$327$$ − 28.0000i − 1.54840i
$$328$$ 12.0000i 0.662589i
$$329$$ −10.0000 −0.551318
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 6.00000i 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ −36.0000 −1.95525
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 36.0000 1.94099
$$345$$ 0 0
$$346$$ 12.0000 0.645124
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ − 12.0000i − 0.643268i
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 5.00000i 0.266501i
$$353$$ 30.0000i 1.59674i 0.602168 + 0.798369i $$0.294304\pi$$
−0.602168 + 0.798369i $$0.705696\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ − 8.00000i − 0.423405i
$$358$$ − 12.0000i − 0.634220i
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 10.0000i 0.525588i
$$363$$ − 2.00000i − 0.104973i
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 22.0000i 1.14839i 0.818718 + 0.574195i $$0.194685\pi$$
−0.818718 + 0.574195i $$0.805315\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −4.00000 −0.208232
$$370$$ 0 0
$$371$$ 6.00000 0.311504
$$372$$ − 20.0000i − 1.03695i
$$373$$ 26.0000i 1.34623i 0.739538 + 0.673114i $$0.235044\pi$$
−0.739538 + 0.673114i $$0.764956\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ 30.0000 1.54713
$$377$$ − 24.0000i − 1.23606i
$$378$$ − 4.00000i − 0.205738i
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 8.00000i 0.409316i
$$383$$ − 2.00000i − 0.102195i −0.998694 0.0510976i $$-0.983728\pi$$
0.998694 0.0510976i $$-0.0162720\pi$$
$$384$$ 6.00000 0.306186
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 12.0000i 0.609994i
$$388$$ − 10.0000i − 0.507673i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 3.00000i − 0.151523i
$$393$$ − 24.0000i − 1.21064i
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ 22.0000i 1.10415i 0.833795 + 0.552074i $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ 18.0000i 0.902258i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 16.0000i 0.798007i
$$403$$ − 40.0000i − 1.99254i
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ − 6.00000i − 0.297409i
$$408$$ 24.0000i 1.18818i
$$409$$ −24.0000 −1.18672 −0.593362 0.804936i $$-0.702200\pi$$
−0.593362 + 0.804936i $$0.702200\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ − 14.0000i − 0.689730i
$$413$$ 2.00000i 0.0984136i
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 20.0000 0.980581
$$417$$ − 16.0000i − 0.783523i
$$418$$ 0 0
$$419$$ −2.00000 −0.0977064 −0.0488532 0.998806i $$-0.515557\pi$$
−0.0488532 + 0.998806i $$0.515557\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 10.0000i 0.486217i
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 26.0000i 1.24948i 0.780833 + 0.624740i $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ 10.0000 0.480015
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 16.0000i 0.764510i
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 16.0000i 0.761042i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −12.0000 −0.569495
$$445$$ 0 0
$$446$$ 22.0000 1.04173
$$447$$ − 20.0000i − 0.945968i
$$448$$ 7.00000i 0.330719i
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ − 18.0000i − 0.846649i
$$453$$ 32.0000i 1.50349i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ − 18.0000i − 0.841085i
$$459$$ 16.0000 0.746816
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ − 2.00000i − 0.0930484i
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 30.0000i 1.38823i 0.719862 + 0.694117i $$0.244205\pi$$
−0.719862 + 0.694117i $$0.755795\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 28.0000 1.29017
$$472$$ − 6.00000i − 0.276172i
$$473$$ − 12.0000i − 0.551761i
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 0 0
$$481$$ −24.0000 −1.09431
$$482$$ − 20.0000i − 0.910975i
$$483$$ − 8.00000i − 0.364013i
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −10.0000 −0.453609
$$487$$ − 28.0000i − 1.26880i −0.773004 0.634401i $$-0.781247\pi$$
0.773004 0.634401i $$-0.218753\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ − 8.00000i − 0.360668i
$$493$$ 24.0000i 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 2.00000i − 0.0892644i
$$503$$ − 4.00000i − 0.178351i −0.996016 0.0891756i $$-0.971577\pi$$
0.996016 0.0891756i $$-0.0284232\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −4.00000 −0.177822
$$507$$ 6.00000i 0.266469i
$$508$$ 8.00000i 0.354943i
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ −24.0000 −1.05654
$$517$$ − 10.0000i − 0.439799i
$$518$$ − 6.00000i − 0.263625i
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 40.0000i 1.74243i
$$528$$ 2.00000i 0.0870388i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 2.00000 0.0867926
$$532$$ 0 0
$$533$$ − 16.0000i − 0.693037i
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ 24.0000i 1.03568i
$$538$$ − 10.0000i − 0.431131i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −26.0000 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$542$$ − 4.00000i − 0.171815i
$$543$$ − 20.0000i − 0.858282i
$$544$$ −20.0000 −0.857493
$$545$$ 0 0
$$546$$ −8.00000 −0.342368
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ − 10.0000i − 0.427179i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 24.0000i 1.02151i
$$553$$ 8.00000i 0.340195i
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 22.0000i 0.932170i 0.884740 + 0.466085i $$0.154336\pi$$
−0.884740 + 0.466085i $$0.845664\pi$$
$$558$$ − 10.0000i − 0.423334i
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 6.00000i 0.253095i
$$563$$ 32.0000i 1.34864i 0.738440 + 0.674320i $$0.235563\pi$$
−0.738440 + 0.674320i $$0.764437\pi$$
$$564$$ −20.0000 −0.842152
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 11.0000i 0.461957i
$$568$$ − 36.0000i − 1.51053i
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ − 16.0000i − 0.668410i
$$574$$ 4.00000 0.166957
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ − 18.0000i − 0.749350i −0.927156 0.374675i $$-0.877754\pi$$
0.927156 0.374675i $$-0.122246\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 28.0000 1.16364
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 20.0000i − 0.829027i
$$583$$ 6.00000i 0.248495i
$$584$$ −24.0000 −0.993127
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ − 2.00000i − 0.0825488i −0.999148 0.0412744i $$-0.986858\pi$$
0.999148 0.0412744i $$-0.0131418\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 44.0000 1.80992
$$592$$ 6.00000i 0.246598i
$$593$$ − 32.0000i − 1.31408i −0.753855 0.657041i $$-0.771808\pi$$
0.753855 0.657041i $$-0.228192\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ − 36.0000i − 1.47338i
$$598$$ 16.0000i 0.654289i
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ − 12.0000i − 0.489083i
$$603$$ − 8.00000i − 0.325785i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −8.00000 −0.324978
$$607$$ − 40.0000i − 1.62355i −0.583970 0.811775i $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ 0 0
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ −40.0000 −1.61823
$$612$$ − 4.00000i − 0.161690i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ − 28.0000i − 1.12633i
$$619$$ −14.0000 −0.562708 −0.281354 0.959604i $$-0.590783\pi$$
−0.281354 + 0.959604i $$0.590783\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ − 18.0000i − 0.721734i
$$623$$ − 6.00000i − 0.240385i
$$624$$ 8.00000 0.320256
$$625$$ 0 0
$$626$$ 2.00000 0.0799361
$$627$$ 0 0
$$628$$ 14.0000i 0.558661i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ − 24.0000i − 0.954669i
$$633$$ 24.0000i 0.953914i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 4.00000i 0.158486i
$$638$$ 6.00000i 0.237542i
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 24.0000i 0.947204i
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 22.0000i − 0.864909i −0.901656 0.432455i $$-0.857648\pi$$
0.901656 0.432455i $$-0.142352\pi$$
$$648$$ − 33.0000i − 1.29636i
$$649$$ −2.00000 −0.0785069
$$650$$ 0 0
$$651$$ −20.0000 −0.783862
$$652$$ 8.00000i 0.313304i
$$653$$ 26.0000i 1.01746i 0.860927 + 0.508729i $$0.169885\pi$$
−0.860927 + 0.508729i $$0.830115\pi$$
$$654$$ 28.0000 1.09489
$$655$$ 0 0
$$656$$ −4.00000 −0.156174
$$657$$ − 8.00000i − 0.312110i
$$658$$ − 10.0000i − 0.389841i
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ − 32.0000i − 1.24278i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 24.0000i 0.929284i
$$668$$ 0 0
$$669$$ −44.0000 −1.70114
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 10.0000i − 0.385758i
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −3.00000 −0.115385
$$677$$ − 12.0000i − 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ − 36.0000i − 1.38257i
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ 10.0000i 0.382920i
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 36.0000i 1.37349i
$$688$$ 12.0000i 0.457496i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ −46.0000 −1.74992 −0.874961 0.484193i $$-0.839113\pi$$
−0.874961 + 0.484193i $$0.839113\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ 1.00000i 0.0379869i
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 36.0000 1.36458
$$697$$ 16.0000i 0.606043i
$$698$$ 0 0
$$699$$ 36.0000 1.36165
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ − 16.0000i − 0.603881i
$$703$$ 0 0
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 4.00000i 0.150435i
$$708$$ 4.00000i 0.150329i
$$709$$ 34.0000 1.27690 0.638448 0.769665i $$-0.279577\pi$$
0.638448 + 0.769665i $$0.279577\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 18.0000i 0.674579i
$$713$$ 40.0000i 1.49801i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ − 16.0000i − 0.597115i
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ − 19.0000i − 0.707107i
$$723$$ 40.0000i 1.48762i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ 2.00000 0.0742270
$$727$$ 18.0000i 0.667583i 0.942647 + 0.333792i $$0.108328\pi$$
−0.942647 + 0.333792i $$0.891672\pi$$
$$728$$ − 12.0000i − 0.444750i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ 8.00000i 0.294684i
$$738$$ − 4.00000i − 0.147242i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 6.00000i 0.220267i
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 60.0000 2.19971
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 4.00000i 0.146254i
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 10.0000i 0.364662i
$$753$$ 4.00000i 0.145768i
$$754$$ 24.0000 0.874028
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ 8.00000i 0.290573i
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ − 14.0000i − 0.506834i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 2.00000 0.0722629
$$767$$ 8.00000i 0.288863i
$$768$$ 34.0000i 1.22687i
$$769$$ −32.0000 −1.15395 −0.576975 0.816762i $$-0.695767\pi$$
−0.576975 + 0.816762i $$0.695767\pi$$
$$770$$ 0 0
$$771$$ −28.0000 −1.00840
$$772$$ 14.0000i 0.503871i
$$773$$ 30.0000i 1.07903i 0.841978 + 0.539513i $$0.181391\pi$$
−0.841978 + 0.539513i $$0.818609\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ 30.0000 1.07694
$$777$$ 12.0000i 0.430498i
$$778$$ − 6.00000i − 0.215110i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ − 16.0000i − 0.572159i
$$783$$ − 24.0000i − 0.857690i
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 24.0000 0.856052
$$787$$ 16.0000i 0.570338i 0.958477 + 0.285169i $$0.0920498\pi$$
−0.958477 + 0.285169i $$0.907950\pi$$
$$788$$ 22.0000i 0.783718i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ − 3.00000i − 0.106600i
$$793$$ 0 0
$$794$$ −22.0000 −0.780751
$$795$$ 0 0
$$796$$ 18.0000 0.637993
$$797$$ 14.0000i 0.495905i 0.968772 + 0.247953i $$0.0797578\pi$$
−0.968772 + 0.247953i $$0.920242\pi$$
$$798$$ 0 0
$$799$$ 40.0000 1.41510
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 22.0000i − 0.776847i
$$803$$ 8.00000i 0.282314i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 40.0000 1.40894
$$807$$ 20.0000i 0.704033i
$$808$$ − 12.0000i − 0.422159i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ 8.00000i 0.280572i
$$814$$ 6.00000 0.210300
$$815$$ 0 0
$$816$$ −8.00000 −0.280056
$$817$$ 0 0
$$818$$ − 24.0000i − 0.839140i
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ 46.0000 1.60541 0.802706 0.596376i $$-0.203393\pi$$
0.802706 + 0.596376i $$0.203393\pi$$
$$822$$ − 20.0000i − 0.697580i
$$823$$ − 24.0000i − 0.836587i −0.908312 0.418294i $$-0.862628\pi$$
0.908312 0.418294i $$-0.137372\pi$$
$$824$$ 42.0000 1.46314
$$825$$ 0 0
$$826$$ −2.00000 −0.0695889
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 44.0000 1.52634
$$832$$ 28.0000i 0.970725i
$$833$$ − 4.00000i − 0.138592i
$$834$$ 16.0000 0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 40.0000i − 1.38260i
$$838$$ − 2.00000i − 0.0690889i
$$839$$ −34.0000 −1.17381 −0.586905 0.809656i $$-0.699654\pi$$
−0.586905 + 0.809656i $$0.699654\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 14.0000i − 0.482472i
$$843$$ − 12.0000i − 0.413302i
$$844$$ −12.0000 −0.413057
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 6.00000i − 0.206041i
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 24.0000i 0.822226i
$$853$$ − 44.0000i − 1.50653i −0.657716 0.753266i $$-0.728477\pi$$
0.657716 0.753266i $$-0.271523\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ 56.0000i 1.91292i 0.291858 + 0.956462i $$0.405727\pi$$
−0.291858 + 0.956462i $$0.594273\pi$$
$$858$$ − 8.00000i − 0.273115i
$$859$$ −6.00000 −0.204717 −0.102359 0.994748i $$-0.532639\pi$$
−0.102359 + 0.994748i $$0.532639\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 20.0000 0.680414
$$865$$ 0 0
$$866$$ −26.0000 −0.883516
$$867$$ − 2.00000i − 0.0679236i
$$868$$ − 10.0000i − 0.339422i
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 32.0000 1.08428
$$872$$ 42.0000i 1.42230i
$$873$$ 10.0000i 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 16.0000 0.540590
$$877$$ 42.0000i 1.41824i 0.705088 + 0.709120i $$0.250907\pi$$
−0.705088 + 0.709120i $$0.749093\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ 48.0000 1.61900
$$880$$ 0 0
$$881$$ −34.0000 −1.14549 −0.572745 0.819734i $$-0.694121\pi$$
−0.572745 + 0.819734i $$0.694121\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ − 28.0000i − 0.942275i −0.882060 0.471138i $$-0.843844\pi$$
0.882060 0.471138i $$-0.156156\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ − 28.0000i − 0.940148i −0.882627 0.470074i $$-0.844227\pi$$
0.882627 0.470074i $$-0.155773\pi$$
$$888$$ − 36.0000i − 1.20808i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −11.0000 −0.368514
$$892$$ − 22.0000i − 0.736614i
$$893$$ 0 0
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ − 32.0000i − 1.06845i
$$898$$ 10.0000i 0.333704i
$$899$$ 60.0000 2.00111
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 4.00000i 0.133185i
$$903$$ 24.0000i 0.798670i
$$904$$ 54.0000 1.79601
$$905$$ 0 0
$$906$$ −32.0000 −1.06313
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −18.0000 −0.594737
$$917$$ − 12.0000i − 0.396275i
$$918$$ 16.0000i 0.528079i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 40.0000 1.31804
$$922$$ 0 0
$$923$$ 48.0000i 1.57994i
$$924$$ −2.00000 −0.0657952
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ 14.0000i 0.459820i
$$928$$ 30.0000i 0.984798i
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 18.0000i 0.589610i
$$933$$ 36.0000i 1.17859i
$$934$$ −30.0000 −0.981630
$$935$$ 0 0
$$936$$ −12.0000 −0.392232
$$937$$ − 16.0000i − 0.522697i −0.965244 0.261349i $$-0.915833\pi$$
0.965244 0.261349i $$-0.0841672\pi$$
$$938$$ 8.00000i 0.261209i
$$939$$ −4.00000 −0.130535
$$940$$ 0 0
$$941$$ −24.0000 −0.782378 −0.391189 0.920310i $$-0.627936\pi$$
−0.391189 + 0.920310i $$0.627936\pi$$
$$942$$ 28.0000i 0.912289i
$$943$$ 16.0000i 0.521032i
$$944$$ 2.00000 0.0650945
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ 16.0000i 0.519656i
$$949$$ 32.0000 1.03876
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ 12.0000i 0.388922i
$$953$$ − 34.0000i − 1.10137i −0.834714 0.550684i $$-0.814367\pi$$
0.834714 0.550684i $$-0.185633\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 12.0000i − 0.387905i
$$958$$ 4.00000i 0.129234i
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ − 24.0000i − 0.773791i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −20.0000 −0.644157
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ 40.0000i 1.28631i 0.765735 + 0.643157i $$0.222376\pi$$
−0.765735 + 0.643157i $$0.777624\pi$$
$$968$$ 3.00000i 0.0964237i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 14.0000 0.449281 0.224641 0.974442i $$-0.427879\pi$$
0.224641 + 0.974442i $$0.427879\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ − 8.00000i − 0.256468i
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 16.0000i 0.511624i
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 28.0000i 0.893516i
$$983$$ − 54.0000i − 1.72233i −0.508323 0.861166i $$-0.669735\pi$$
0.508323 0.861166i $$-0.330265\pi$$
$$984$$ 24.0000 0.765092
$$985$$ 0 0
$$986$$ −24.0000 −0.764316
$$987$$ 20.0000i 0.636607i
$$988$$ 0 0
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ 50.0000i 1.58750i
$$993$$ 40.0000i 1.26936i
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 20.0000i 0.633406i 0.948525 + 0.316703i $$0.102576\pi$$
−0.948525 + 0.316703i $$0.897424\pi$$
$$998$$ 16.0000i 0.506471i
$$999$$ −24.0000 −0.759326
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 1925.2.b.d.1849.2 2
5.2 odd 4 1925.2.a.c.1.1 1
5.3 odd 4 77.2.a.c.1.1 1
5.4 even 2 inner 1925.2.b.d.1849.1 2
15.8 even 4 693.2.a.a.1.1 1
20.3 even 4 1232.2.a.a.1.1 1
35.3 even 12 539.2.e.b.177.1 2
35.13 even 4 539.2.a.d.1.1 1
35.18 odd 12 539.2.e.a.177.1 2
35.23 odd 12 539.2.e.a.67.1 2
35.33 even 12 539.2.e.b.67.1 2
40.3 even 4 4928.2.a.bi.1.1 1
40.13 odd 4 4928.2.a.g.1.1 1
55.3 odd 20 847.2.f.e.372.1 4
55.8 even 20 847.2.f.k.372.1 4
55.13 even 20 847.2.f.k.323.1 4
55.18 even 20 847.2.f.k.148.1 4
55.28 even 20 847.2.f.k.729.1 4
55.38 odd 20 847.2.f.e.729.1 4
55.43 even 4 847.2.a.a.1.1 1
55.48 odd 20 847.2.f.e.148.1 4
55.53 odd 20 847.2.f.e.323.1 4
105.83 odd 4 4851.2.a.a.1.1 1
140.83 odd 4 8624.2.a.bc.1.1 1
165.98 odd 4 7623.2.a.n.1.1 1
385.153 odd 4 5929.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.c.1.1 1 5.3 odd 4
539.2.a.d.1.1 1 35.13 even 4
539.2.e.a.67.1 2 35.23 odd 12
539.2.e.a.177.1 2 35.18 odd 12
539.2.e.b.67.1 2 35.33 even 12
539.2.e.b.177.1 2 35.3 even 12
693.2.a.a.1.1 1 15.8 even 4
847.2.a.a.1.1 1 55.43 even 4
847.2.f.e.148.1 4 55.48 odd 20
847.2.f.e.323.1 4 55.53 odd 20
847.2.f.e.372.1 4 55.3 odd 20
847.2.f.e.729.1 4 55.38 odd 20
847.2.f.k.148.1 4 55.18 even 20
847.2.f.k.323.1 4 55.13 even 20
847.2.f.k.372.1 4 55.8 even 20
847.2.f.k.729.1 4 55.28 even 20
1232.2.a.a.1.1 1 20.3 even 4
1925.2.a.c.1.1 1 5.2 odd 4
1925.2.b.d.1849.1 2 5.4 even 2 inner
1925.2.b.d.1849.2 2 1.1 even 1 trivial
4851.2.a.a.1.1 1 105.83 odd 4
4928.2.a.g.1.1 1 40.13 odd 4
4928.2.a.bi.1.1 1 40.3 even 4
5929.2.a.b.1.1 1 385.153 odd 4
7623.2.a.n.1.1 1 165.98 odd 4
8624.2.a.bc.1.1 1 140.83 odd 4