# Properties

 Label 1425.2.c.b.799.1 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (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: $$11.3786822880$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.b.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -5.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -5.00000i q^{7} -1.00000 q^{9} +1.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} -10.0000 q^{14} -4.00000 q^{16} -1.00000i q^{17} +2.00000i q^{18} +1.00000 q^{19} +5.00000 q^{21} -2.00000i q^{22} +4.00000i q^{23} -4.00000 q^{26} -1.00000i q^{27} +10.0000i q^{28} +2.00000 q^{29} -6.00000 q^{31} +8.00000i q^{32} +1.00000i q^{33} -2.00000 q^{34} +2.00000 q^{36} -2.00000i q^{38} +2.00000 q^{39} -10.0000i q^{42} +1.00000i q^{43} -2.00000 q^{44} +8.00000 q^{46} -9.00000i q^{47} -4.00000i q^{48} -18.0000 q^{49} +1.00000 q^{51} +4.00000i q^{52} -10.0000i q^{53} -2.00000 q^{54} +1.00000i q^{57} -4.00000i q^{58} +8.00000 q^{59} -1.00000 q^{61} +12.0000i q^{62} +5.00000i q^{63} +8.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +11.0000i q^{73} -2.00000 q^{76} -5.00000i q^{77} -4.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -10.0000 q^{84} +2.00000 q^{86} +2.00000i q^{87} +6.00000 q^{89} -10.0000 q^{91} -8.00000i q^{92} -6.00000i q^{93} -18.0000 q^{94} -8.00000 q^{96} -10.0000i q^{97} +36.0000i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{11} - 20 q^{14} - 8 q^{16} + 2 q^{19} + 10 q^{21} - 8 q^{26} + 4 q^{29} - 12 q^{31} - 4 q^{34} + 4 q^{36} + 4 q^{39} - 4 q^{44} + 16 q^{46} - 36 q^{49} + 2 q^{51} - 4 q^{54} + 16 q^{59} - 2 q^{61} + 16 q^{64} + 4 q^{66} - 8 q^{69} - 24 q^{71} - 4 q^{76} - 32 q^{79} + 2 q^{81} - 20 q^{84} + 4 q^{86} + 12 q^{89} - 20 q^{91} - 36 q^{94} - 16 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^11 - 20 * q^14 - 8 * q^16 + 2 * q^19 + 10 * q^21 - 8 * q^26 + 4 * q^29 - 12 * q^31 - 4 * q^34 + 4 * q^36 + 4 * q^39 - 4 * q^44 + 16 * q^46 - 36 * q^49 + 2 * q^51 - 4 * q^54 + 16 * q^59 - 2 * q^61 + 16 * q^64 + 4 * q^66 - 8 * q^69 - 24 * q^71 - 4 * q^76 - 32 * q^79 + 2 * q^81 - 20 * q^84 + 4 * q^86 + 12 * q^89 - 20 * q^91 - 36 * q^94 - 16 * q^96 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 1.41421i −0.707107 0.707107i $$-0.750000\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ − 5.00000i − 1.88982i −0.327327 0.944911i $$-0.606148\pi$$
0.327327 0.944911i $$-0.393852\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −10.0000 −2.67261
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ − 1.00000i − 0.242536i −0.992620 0.121268i $$-0.961304\pi$$
0.992620 0.121268i $$-0.0386960\pi$$
$$18$$ 2.00000i 0.471405i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 5.00000 1.09109
$$22$$ − 2.00000i − 0.426401i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ − 1.00000i − 0.192450i
$$28$$ 10.0000i 1.88982i
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 8.00000i 1.41421i
$$33$$ 1.00000i 0.174078i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ − 2.00000i − 0.324443i
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ − 10.0000i − 1.54303i
$$43$$ 1.00000i 0.152499i 0.997089 + 0.0762493i $$0.0242945\pi$$
−0.997089 + 0.0762493i $$0.975706\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ − 9.00000i − 1.31278i −0.754420 0.656392i $$-0.772082\pi$$
0.754420 0.656392i $$-0.227918\pi$$
$$48$$ − 4.00000i − 0.577350i
$$49$$ −18.0000 −2.57143
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ 4.00000i 0.554700i
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000i 0.132453i
$$58$$ − 4.00000i − 0.525226i
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 12.0000i 1.52400i
$$63$$ 5.00000i 0.629941i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000i 1.28745i 0.765256 + 0.643726i $$0.222612\pi$$
−0.765256 + 0.643726i $$0.777388\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ − 5.00000i − 0.569803i
$$78$$ − 4.00000i − 0.452911i
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ −10.0000 −1.09109
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −10.0000 −1.04828
$$92$$ − 8.00000i − 0.834058i
$$93$$ − 6.00000i − 0.622171i
$$94$$ −18.0000 −1.85656
$$95$$ 0 0
$$96$$ −8.00000 −0.816497
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 36.0000i 3.63655i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 2.00000i 0.197066i 0.995134 + 0.0985329i $$0.0314150\pi$$
−0.995134 + 0.0985329i $$0.968585\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −20.0000 −1.94257
$$107$$ 6.00000i 0.580042i 0.957020 + 0.290021i $$0.0936623\pi$$
−0.957020 + 0.290021i $$0.906338\pi$$
$$108$$ 2.00000i 0.192450i
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 20.0000i 1.88982i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 2.00000i 0.184900i
$$118$$ − 16.0000i − 1.47292i
$$119$$ −5.00000 −0.458349
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 12.0000 1.07763
$$125$$ 0 0
$$126$$ 10.0000 0.890871
$$127$$ − 2.00000i − 0.177471i −0.996055 0.0887357i $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ 0 0
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ 7.00000 0.611593 0.305796 0.952097i $$-0.401077\pi$$
0.305796 + 0.952097i $$0.401077\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ − 5.00000i − 0.433555i
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 9.00000i − 0.768922i −0.923141 0.384461i $$-0.874387\pi$$
0.923141 0.384461i $$-0.125613\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 13.0000 1.10265 0.551323 0.834292i $$-0.314123\pi$$
0.551323 + 0.834292i $$0.314123\pi$$
$$140$$ 0 0
$$141$$ 9.00000 0.757937
$$142$$ 24.0000i 2.01404i
$$143$$ − 2.00000i − 0.167248i
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ 22.0000 1.82073
$$147$$ − 18.0000i − 1.48461i
$$148$$ 0 0
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 1.00000i 0.0808452i
$$154$$ −10.0000 −0.805823
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 32.0000i 2.54578i
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 20.0000 1.57622
$$162$$ − 2.00000i − 0.157135i
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ 10.0000i 0.773823i 0.922117 + 0.386912i $$0.126458\pi$$
−0.922117 + 0.386912i $$0.873542\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ − 2.00000i − 0.152499i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 8.00000i 0.601317i
$$178$$ − 12.0000i − 0.899438i
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ 20.0000i 1.48250i
$$183$$ − 1.00000i − 0.0739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −12.0000 −0.879883
$$187$$ − 1.00000i − 0.0731272i
$$188$$ 18.0000i 1.31278i
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 9.00000 0.651217 0.325609 0.945505i $$-0.394431\pi$$
0.325609 + 0.945505i $$0.394431\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ − 4.00000i − 0.287926i −0.989583 0.143963i $$-0.954015\pi$$
0.989583 0.143963i $$-0.0459847\pi$$
$$194$$ −20.0000 −1.43592
$$195$$ 0 0
$$196$$ 36.0000 2.57143
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ 21.0000 1.48865 0.744325 0.667817i $$-0.232771\pi$$
0.744325 + 0.667817i $$0.232771\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ − 4.00000i − 0.281439i
$$203$$ − 10.0000i − 0.701862i
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ − 4.00000i − 0.278019i
$$208$$ 8.00000i 0.554700i
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 20.0000i 1.37361i
$$213$$ − 12.0000i − 0.822226i
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 30.0000i 2.03653i
$$218$$ 8.00000i 0.541828i
$$219$$ −11.0000 −0.743311
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ − 12.0000i − 0.803579i −0.915732 0.401790i $$-0.868388\pi$$
0.915732 0.401790i $$-0.131612\pi$$
$$224$$ 40.0000 2.67261
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ − 2.00000i − 0.132453i
$$229$$ −25.0000 −1.65205 −0.826023 0.563636i $$-0.809402\pi$$
−0.826023 + 0.563636i $$0.809402\pi$$
$$230$$ 0 0
$$231$$ 5.00000 0.328976
$$232$$ 0 0
$$233$$ − 9.00000i − 0.589610i −0.955557 0.294805i $$-0.904745\pi$$
0.955557 0.294805i $$-0.0952546\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −16.0000 −1.04151
$$237$$ − 16.0000i − 1.03931i
$$238$$ 10.0000i 0.648204i
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ 0 0
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 20.0000i 1.28565i
$$243$$ 1.00000i 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 2.00000i − 0.127257i
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 7.00000 0.441836 0.220918 0.975292i $$-0.429095\pi$$
0.220918 + 0.975292i $$0.429095\pi$$
$$252$$ − 10.0000i − 0.629941i
$$253$$ 4.00000i 0.251478i
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 8.00000i − 0.499026i −0.968371 0.249513i $$-0.919729\pi$$
0.968371 0.249513i $$-0.0802706\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ − 14.0000i − 0.864923i
$$263$$ − 23.0000i − 1.41824i −0.705087 0.709120i $$-0.749092\pi$$
0.705087 0.709120i $$-0.250908\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −10.0000 −0.613139
$$267$$ 6.00000i 0.367194i
$$268$$ − 16.0000i − 0.977356i
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 4.00000i 0.242536i
$$273$$ − 10.0000i − 0.605228i
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 11.0000i − 0.660926i −0.943819 0.330463i $$-0.892795\pi$$
0.943819 0.330463i $$-0.107205\pi$$
$$278$$ − 26.0000i − 1.55938i
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ − 18.0000i − 1.07188i
$$283$$ 13.0000i 0.772770i 0.922338 + 0.386385i $$0.126276\pi$$
−0.922338 + 0.386385i $$0.873724\pi$$
$$284$$ 24.0000 1.42414
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 0 0
$$288$$ − 8.00000i − 0.471405i
$$289$$ 16.0000 0.941176
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ − 22.0000i − 1.28745i
$$293$$ 28.0000i 1.63578i 0.575376 + 0.817889i $$0.304856\pi$$
−0.575376 + 0.817889i $$0.695144\pi$$
$$294$$ −36.0000 −2.09956
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 1.00000i − 0.0580259i
$$298$$ − 42.0000i − 2.43299i
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 5.00000 0.288195
$$302$$ 0 0
$$303$$ 2.00000i 0.114897i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 10.0000i 0.569803i
$$309$$ −2.00000 −0.113776
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 0 0
$$313$$ 2.00000i 0.113047i 0.998401 + 0.0565233i $$0.0180015\pi$$
−0.998401 + 0.0565233i $$0.981998\pi$$
$$314$$ −36.0000 −2.03160
$$315$$ 0 0
$$316$$ 32.0000 1.80014
$$317$$ − 4.00000i − 0.224662i −0.993671 0.112331i $$-0.964168\pi$$
0.993671 0.112331i $$-0.0358318\pi$$
$$318$$ − 20.0000i − 1.12154i
$$319$$ 2.00000 0.111979
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ − 40.0000i − 2.22911i
$$323$$ − 1.00000i − 0.0556415i
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 4.00000i − 0.221201i
$$328$$ 0 0
$$329$$ −45.0000 −2.48093
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 24.0000i 1.31717i
$$333$$ 0 0
$$334$$ 20.0000 1.09435
$$335$$ 0 0
$$336$$ −20.0000 −1.09109
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ − 18.0000i − 0.979071i
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 2.00000i 0.108148i
$$343$$ 55.0000i 2.96972i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 25.0000i − 1.34207i −0.741426 0.671035i $$-0.765850\pi$$
0.741426 0.671035i $$-0.234150\pi$$
$$348$$ − 4.00000i − 0.214423i
$$349$$ −9.00000 −0.481759 −0.240879 0.970555i $$-0.577436\pi$$
−0.240879 + 0.970555i $$0.577436\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 8.00000i 0.426401i
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 16.0000 0.850390
$$355$$ 0 0
$$356$$ −12.0000 −0.635999
$$357$$ − 5.00000i − 0.264628i
$$358$$ − 36.0000i − 1.90266i
$$359$$ −37.0000 −1.95279 −0.976393 0.216003i $$-0.930698\pi$$
−0.976393 + 0.216003i $$0.930698\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 28.0000i 1.47165i
$$363$$ − 10.0000i − 0.524864i
$$364$$ 20.0000 1.04828
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ − 16.0000i − 0.834058i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −50.0000 −2.59587
$$372$$ 12.0000i 0.622171i
$$373$$ − 16.0000i − 0.828449i −0.910175 0.414224i $$-0.864053\pi$$
0.910175 0.414224i $$-0.135947\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4.00000i − 0.206010i
$$378$$ 10.0000i 0.514344i
$$379$$ −34.0000 −1.74646 −0.873231 0.487306i $$-0.837980\pi$$
−0.873231 + 0.487306i $$0.837980\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ − 18.0000i − 0.920960i
$$383$$ 34.0000i 1.73732i 0.495410 + 0.868659i $$0.335018\pi$$
−0.495410 + 0.868659i $$0.664982\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ − 1.00000i − 0.0508329i
$$388$$ 20.0000i 1.01535i
$$389$$ 27.0000 1.36895 0.684477 0.729034i $$-0.260031\pi$$
0.684477 + 0.729034i $$0.260031\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 0 0
$$393$$ 7.00000i 0.353103i
$$394$$ −4.00000 −0.201517
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ 25.0000i 1.25471i 0.778732 + 0.627357i $$0.215863\pi$$
−0.778732 + 0.627357i $$0.784137\pi$$
$$398$$ − 42.0000i − 2.10527i
$$399$$ 5.00000 0.250313
$$400$$ 0 0
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 16.0000i 0.798007i
$$403$$ 12.0000i 0.597763i
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 9.00000 0.443937
$$412$$ − 4.00000i − 0.197066i
$$413$$ − 40.0000i − 1.96827i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 16.0000 0.784465
$$417$$ 13.0000i 0.636613i
$$418$$ − 2.00000i − 0.0978232i
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 26.0000 1.26716 0.633581 0.773676i $$-0.281584\pi$$
0.633581 + 0.773676i $$0.281584\pi$$
$$422$$ − 24.0000i − 1.16830i
$$423$$ 9.00000i 0.437595i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ 5.00000i 0.241967i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 2.00000 0.0965609
$$430$$ 0 0
$$431$$ −34.0000 −1.63772 −0.818861 0.573992i $$-0.805394\pi$$
−0.818861 + 0.573992i $$0.805394\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 6.00000i − 0.288342i −0.989553 0.144171i $$-0.953949\pi$$
0.989553 0.144171i $$-0.0460515\pi$$
$$434$$ 60.0000 2.88009
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ 4.00000i 0.191346i
$$438$$ 22.0000i 1.05120i
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ 0 0
$$441$$ 18.0000 0.857143
$$442$$ 4.00000i 0.190261i
$$443$$ 5.00000i 0.237557i 0.992921 + 0.118779i $$0.0378979\pi$$
−0.992921 + 0.118779i $$0.962102\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ 21.0000i 0.993266i
$$448$$ − 40.0000i − 1.88982i
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 4.00000i 0.188144i
$$453$$ 0 0
$$454$$ 36.0000 1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 29.0000i − 1.35656i −0.734802 0.678281i $$-0.762725\pi$$
0.734802 0.678281i $$-0.237275\pi$$
$$458$$ 50.0000i 2.33635i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 27.0000 1.25752 0.628758 0.777601i $$-0.283564\pi$$
0.628758 + 0.777601i $$0.283564\pi$$
$$462$$ − 10.0000i − 0.465242i
$$463$$ − 17.0000i − 0.790057i −0.918669 0.395029i $$-0.870735\pi$$
0.918669 0.395029i $$-0.129265\pi$$
$$464$$ −8.00000 −0.371391
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ − 5.00000i − 0.231372i −0.993286 0.115686i $$-0.963093\pi$$
0.993286 0.115686i $$-0.0369067\pi$$
$$468$$ − 4.00000i − 0.184900i
$$469$$ 40.0000 1.84703
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 1.00000i 0.0459800i
$$474$$ −32.0000 −1.46981
$$475$$ 0 0
$$476$$ 10.0000 0.458349
$$477$$ 10.0000i 0.457869i
$$478$$ − 6.00000i − 0.274434i
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 40.0000i − 1.82195i
$$483$$ 20.0000i 0.910032i
$$484$$ 20.0000 0.909091
$$485$$ 0 0
$$486$$ 2.00000 0.0907218
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ − 2.00000i − 0.0900755i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 24.0000 1.07763
$$497$$ 60.0000i 2.69137i
$$498$$ − 24.0000i − 1.07547i
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ −10.0000 −0.446767
$$502$$ − 14.0000i − 0.624851i
$$503$$ − 16.0000i − 0.713405i −0.934218 0.356702i $$-0.883901\pi$$
0.934218 0.356702i $$-0.116099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ 9.00000i 0.399704i
$$508$$ 4.00000i 0.177471i
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 55.0000 2.43306
$$512$$ − 32.0000i − 1.41421i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −16.0000 −0.705730
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ − 9.00000i − 0.395820i
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 34.0000i 1.48672i 0.668894 + 0.743358i $$0.266768\pi$$
−0.668894 + 0.743358i $$0.733232\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ −46.0000 −2.00570
$$527$$ 6.00000i 0.261364i
$$528$$ − 4.00000i − 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 10.0000i 0.433555i
$$533$$ 0 0
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 18.0000i 0.776757i
$$538$$ − 28.0000i − 1.20717i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 3.00000 0.128980 0.0644900 0.997918i $$-0.479458\pi$$
0.0644900 + 0.997918i $$0.479458\pi$$
$$542$$ − 24.0000i − 1.03089i
$$543$$ − 14.0000i − 0.600798i
$$544$$ 8.00000 0.342997
$$545$$ 0 0
$$546$$ −20.0000 −0.855921
$$547$$ − 26.0000i − 1.11168i −0.831289 0.555840i $$-0.812397\pi$$
0.831289 0.555840i $$-0.187603\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ 2.00000 0.0852029
$$552$$ 0 0
$$553$$ 80.0000i 3.40195i
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −26.0000 −1.10265
$$557$$ − 41.0000i − 1.73723i −0.495491 0.868613i $$-0.665012\pi$$
0.495491 0.868613i $$-0.334988\pi$$
$$558$$ − 12.0000i − 0.508001i
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 1.00000 0.0422200
$$562$$ − 20.0000i − 0.843649i
$$563$$ 12.0000i 0.505740i 0.967500 + 0.252870i $$0.0813744\pi$$
−0.967500 + 0.252870i $$0.918626\pi$$
$$564$$ −18.0000 −0.757937
$$565$$ 0 0
$$566$$ 26.0000 1.09286
$$567$$ − 5.00000i − 0.209980i
$$568$$ 0 0
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 9.00000i 0.375980i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 27.0000i 1.12402i 0.827129 + 0.562012i $$0.189973\pi$$
−0.827129 + 0.562012i $$0.810027\pi$$
$$578$$ − 32.0000i − 1.33102i
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −60.0000 −2.48922
$$582$$ − 20.0000i − 0.829027i
$$583$$ − 10.0000i − 0.414158i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 56.0000 2.31334
$$587$$ 7.00000i 0.288921i 0.989511 + 0.144460i $$0.0461446\pi$$
−0.989511 + 0.144460i $$0.953855\pi$$
$$588$$ 36.0000i 1.48461i
$$589$$ −6.00000 −0.247226
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ −42.0000 −1.72039
$$597$$ 21.0000i 0.859473i
$$598$$ − 16.0000i − 0.654289i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ − 10.0000i − 0.407570i
$$603$$ − 8.00000i − 0.325785i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 4.00000 0.162489
$$607$$ 26.0000i 1.05531i 0.849460 + 0.527654i $$0.176928\pi$$
−0.849460 + 0.527654i $$0.823072\pi$$
$$608$$ 8.00000i 0.324443i
$$609$$ 10.0000 0.405220
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 33.0000i − 1.33286i −0.745569 0.666429i $$-0.767822\pi$$
0.745569 0.666429i $$-0.232178\pi$$
$$614$$ −24.0000 −0.968561
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 27.0000i 1.08698i 0.839416 + 0.543490i $$0.182897\pi$$
−0.839416 + 0.543490i $$0.817103\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 42.0000i 1.68405i
$$623$$ − 30.0000i − 1.20192i
$$624$$ −8.00000 −0.320256
$$625$$ 0 0
$$626$$ 4.00000 0.159872
$$627$$ 1.00000i 0.0399362i
$$628$$ 36.0000i 1.43656i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 15.0000 0.597141 0.298570 0.954388i $$-0.403490\pi$$
0.298570 + 0.954388i $$0.403490\pi$$
$$632$$ 0 0
$$633$$ 12.0000i 0.476957i
$$634$$ −8.00000 −0.317721
$$635$$ 0 0
$$636$$ −20.0000 −0.793052
$$637$$ 36.0000i 1.42637i
$$638$$ − 4.00000i − 0.158362i
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 1.00000i 0.0394362i 0.999806 + 0.0197181i $$0.00627687\pi$$
−0.999806 + 0.0197181i $$0.993723\pi$$
$$644$$ −40.0000 −1.57622
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ − 39.0000i − 1.53325i −0.642096 0.766624i $$-0.721935\pi$$
0.642096 0.766624i $$-0.278065\pi$$
$$648$$ 0 0
$$649$$ 8.00000 0.314027
$$650$$ 0 0
$$651$$ −30.0000 −1.17579
$$652$$ 0 0
$$653$$ − 3.00000i − 0.117399i −0.998276 0.0586995i $$-0.981305\pi$$
0.998276 0.0586995i $$-0.0186954\pi$$
$$654$$ −8.00000 −0.312825
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 11.0000i − 0.429151i
$$658$$ 90.0000i 3.50857i
$$659$$ 14.0000 0.545363 0.272681 0.962104i $$-0.412090\pi$$
0.272681 + 0.962104i $$0.412090\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 8.00000i 0.310929i
$$663$$ − 2.00000i − 0.0776736i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.00000i 0.309761i
$$668$$ − 20.0000i − 0.773823i
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ −1.00000 −0.0386046
$$672$$ 40.0000i 1.54303i
$$673$$ 24.0000i 0.925132i 0.886585 + 0.462566i $$0.153071\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ −28.0000 −1.07852
$$675$$ 0 0
$$676$$ −18.0000 −0.692308
$$677$$ 34.0000i 1.30673i 0.757045 + 0.653363i $$0.226642\pi$$
−0.757045 + 0.653363i $$0.773358\pi$$
$$678$$ − 4.00000i − 0.153619i
$$679$$ −50.0000 −1.91882
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 12.0000i 0.459504i
$$683$$ 6.00000i 0.229584i 0.993390 + 0.114792i $$0.0366201\pi$$
−0.993390 + 0.114792i $$0.963380\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 110.000 4.19982
$$687$$ − 25.0000i − 0.953809i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −31.0000 −1.17930 −0.589648 0.807661i $$-0.700733\pi$$
−0.589648 + 0.807661i $$0.700733\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ 5.00000i 0.189934i
$$694$$ −50.0000 −1.89797
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 18.0000i 0.681310i
$$699$$ 9.00000 0.340411
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 4.00000i 0.150970i
$$703$$ 0 0
$$704$$ 8.00000 0.301511
$$705$$ 0 0
$$706$$ 4.00000 0.150542
$$707$$ − 10.0000i − 0.376089i
$$708$$ − 16.0000i − 0.601317i
$$709$$ 42.0000 1.57734 0.788672 0.614815i $$-0.210769\pi$$
0.788672 + 0.614815i $$0.210769\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ − 24.0000i − 0.898807i
$$714$$ −10.0000 −0.374241
$$715$$ 0 0
$$716$$ −36.0000 −1.34538
$$717$$ 3.00000i 0.112037i
$$718$$ 74.0000i 2.76166i
$$719$$ −33.0000 −1.23069 −0.615346 0.788257i $$-0.710984\pi$$
−0.615346 + 0.788257i $$0.710984\pi$$
$$720$$ 0 0
$$721$$ 10.0000 0.372419
$$722$$ − 2.00000i − 0.0744323i
$$723$$ 20.0000i 0.743808i
$$724$$ 28.0000 1.04061
$$725$$ 0 0
$$726$$ −20.0000 −0.742270
$$727$$ − 23.0000i − 0.853023i −0.904482 0.426511i $$-0.859742\pi$$
0.904482 0.426511i $$-0.140258\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 1.00000 0.0369863
$$732$$ 2.00000i 0.0739221i
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ −32.0000 −1.17954
$$737$$ 8.00000i 0.294684i
$$738$$ 0 0
$$739$$ 5.00000 0.183928 0.0919640 0.995762i $$-0.470686\pi$$
0.0919640 + 0.995762i $$0.470686\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 100.000i 3.67112i
$$743$$ 8.00000i 0.293492i 0.989174 + 0.146746i $$0.0468799\pi$$
−0.989174 + 0.146746i $$0.953120\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ 12.0000i 0.439057i
$$748$$ 2.00000i 0.0731272i
$$749$$ 30.0000 1.09618
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 36.0000i 1.31278i
$$753$$ 7.00000i 0.255094i
$$754$$ −8.00000 −0.291343
$$755$$ 0 0
$$756$$ 10.0000 0.363696
$$757$$ − 17.0000i − 0.617876i −0.951082 0.308938i $$-0.900027\pi$$
0.951082 0.308938i $$-0.0999735\pi$$
$$758$$ 68.0000i 2.46987i
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ 15.0000 0.543750 0.271875 0.962333i $$-0.412356\pi$$
0.271875 + 0.962333i $$0.412356\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 20.0000i 0.724049i
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 68.0000 2.45694
$$767$$ − 16.0000i − 0.577727i
$$768$$ 16.0000i 0.577350i
$$769$$ −11.0000 −0.396670 −0.198335 0.980134i $$-0.563553\pi$$
−0.198335 + 0.980134i $$0.563553\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ 8.00000i 0.287926i
$$773$$ − 20.0000i − 0.719350i −0.933078 0.359675i $$-0.882888\pi$$
0.933078 0.359675i $$-0.117112\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ − 54.0000i − 1.93599i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −12.0000 −0.429394
$$782$$ − 8.00000i − 0.286079i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 72.0000 2.57143
$$785$$ 0 0
$$786$$ 14.0000 0.499363
$$787$$ 40.0000i 1.42585i 0.701242 + 0.712923i $$0.252629\pi$$
−0.701242 + 0.712923i $$0.747371\pi$$
$$788$$ 4.00000i 0.142494i
$$789$$ 23.0000 0.818822
$$790$$ 0 0
$$791$$ −10.0000 −0.355559
$$792$$ 0 0
$$793$$ 2.00000i 0.0710221i
$$794$$ 50.0000 1.77443
$$795$$ 0 0
$$796$$ −42.0000 −1.48865
$$797$$ − 44.0000i − 1.55856i −0.626676 0.779280i $$-0.715585\pi$$
0.626676 0.779280i $$-0.284415\pi$$
$$798$$ − 10.0000i − 0.353996i
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 72.0000i − 2.54241i
$$803$$ 11.0000i 0.388182i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 24.0000 0.845364
$$807$$ 14.0000i 0.492823i
$$808$$ 0 0
$$809$$ 55.0000 1.93370 0.966849 0.255351i $$-0.0821909\pi$$
0.966849 + 0.255351i $$0.0821909\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 20.0000i 0.701862i
$$813$$ 12.0000i 0.420858i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −4.00000 −0.140028
$$817$$ 1.00000i 0.0349856i
$$818$$ − 28.0000i − 0.978997i
$$819$$ 10.0000 0.349428
$$820$$ 0 0
$$821$$ −45.0000 −1.57051 −0.785255 0.619172i $$-0.787468\pi$$
−0.785255 + 0.619172i $$0.787468\pi$$
$$822$$ − 18.0000i − 0.627822i
$$823$$ − 43.0000i − 1.49889i −0.662069 0.749443i $$-0.730321\pi$$
0.662069 0.749443i $$-0.269679\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −80.0000 −2.78356
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ −52.0000 −1.80603 −0.903017 0.429604i $$-0.858653\pi$$
−0.903017 + 0.429604i $$0.858653\pi$$
$$830$$ 0 0
$$831$$ 11.0000 0.381586
$$832$$ − 16.0000i − 0.554700i
$$833$$ 18.0000i 0.623663i
$$834$$ 26.0000 0.900306
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 6.00000i 0.207390i
$$838$$ 56.0000i 1.93449i
$$839$$ −54.0000 −1.86429 −0.932144 0.362089i $$-0.882064\pi$$
−0.932144 + 0.362089i $$0.882064\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 52.0000i − 1.79204i
$$843$$ 10.0000i 0.344418i
$$844$$ −24.0000 −0.826114
$$845$$ 0 0
$$846$$ 18.0000 0.618853
$$847$$ 50.0000i 1.71802i
$$848$$ 40.0000i 1.37361i
$$849$$ −13.0000 −0.446159
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 24.0000i 0.822226i
$$853$$ 14.0000i 0.479351i 0.970853 + 0.239675i $$0.0770410\pi$$
−0.970853 + 0.239675i $$0.922959\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 8.00000i − 0.273275i −0.990621 0.136637i $$-0.956370\pi$$
0.990621 0.136637i $$-0.0436295\pi$$
$$858$$ − 4.00000i − 0.136558i
$$859$$ −27.0000 −0.921228 −0.460614 0.887601i $$-0.652371\pi$$
−0.460614 + 0.887601i $$0.652371\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 68.0000i 2.31609i
$$863$$ 44.0000i 1.49778i 0.662696 + 0.748889i $$0.269412\pi$$
−0.662696 + 0.748889i $$0.730588\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 0 0
$$866$$ −12.0000 −0.407777
$$867$$ 16.0000i 0.543388i
$$868$$ − 60.0000i − 2.03653i
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 0 0
$$873$$ 10.0000i 0.338449i
$$874$$ 8.00000 0.270604
$$875$$ 0 0
$$876$$ 22.0000 0.743311
$$877$$ − 6.00000i − 0.202606i −0.994856 0.101303i $$-0.967699\pi$$
0.994856 0.101303i $$-0.0323011\pi$$
$$878$$ 52.0000i 1.75491i
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −37.0000 −1.24656 −0.623281 0.781998i $$-0.714201\pi$$
−0.623281 + 0.781998i $$0.714201\pi$$
$$882$$ − 36.0000i − 1.21218i
$$883$$ − 35.0000i − 1.17784i −0.808190 0.588922i $$-0.799553\pi$$
0.808190 0.588922i $$-0.200447\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 10.0000 0.335957
$$887$$ − 12.0000i − 0.402921i −0.979497 0.201460i $$-0.935431\pi$$
0.979497 0.201460i $$-0.0645687\pi$$
$$888$$ 0 0
$$889$$ −10.0000 −0.335389
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 24.0000i 0.803579i
$$893$$ − 9.00000i − 0.301174i
$$894$$ 42.0000 1.40469
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 8.00000i 0.267112i
$$898$$ − 72.0000i − 2.40267i
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −10.0000 −0.333148
$$902$$ 0 0
$$903$$ 5.00000i 0.166390i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 26.0000i − 0.863316i −0.902037 0.431658i $$-0.857929\pi$$
0.902037 0.431658i $$-0.142071\pi$$
$$908$$ − 36.0000i − 1.19470i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 12.0000i − 0.397142i
$$914$$ −58.0000 −1.91847
$$915$$ 0 0
$$916$$ 50.0000 1.65205
$$917$$ − 35.0000i − 1.15580i
$$918$$ 2.00000i 0.0660098i
$$919$$ −8.00000 −0.263896 −0.131948 0.991257i $$-0.542123\pi$$
−0.131948 + 0.991257i $$0.542123\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 54.0000i − 1.77840i
$$923$$ 24.0000i 0.789970i
$$924$$ −10.0000 −0.328976
$$925$$ 0 0
$$926$$ −34.0000 −1.11731
$$927$$ − 2.00000i − 0.0656886i
$$928$$ 16.0000i 0.525226i
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 18.0000i 0.589610i
$$933$$ − 21.0000i − 0.687509i
$$934$$ −10.0000 −0.327210
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 21.0000i 0.686040i 0.939328 + 0.343020i $$0.111450\pi$$
−0.939328 + 0.343020i $$0.888550\pi$$
$$938$$ − 80.0000i − 2.61209i
$$939$$ −2.00000 −0.0652675
$$940$$ 0 0
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ − 36.0000i − 1.17294i
$$943$$ 0 0
$$944$$ −32.0000 −1.04151
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ − 28.0000i − 0.909878i −0.890523 0.454939i $$-0.849661\pi$$
0.890523 0.454939i $$-0.150339\pi$$
$$948$$ 32.0000i 1.03931i
$$949$$ 22.0000 0.714150
$$950$$ 0 0
$$951$$ 4.00000 0.129709
$$952$$ 0 0
$$953$$ 32.0000i 1.03658i 0.855204 + 0.518291i $$0.173432\pi$$
−0.855204 + 0.518291i $$0.826568\pi$$
$$954$$ 20.0000 0.647524
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 2.00000i 0.0646508i
$$958$$ − 32.0000i − 1.03387i
$$959$$ −45.0000 −1.45313
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ − 6.00000i − 0.193347i
$$964$$ −40.0000 −1.28831
$$965$$ 0 0
$$966$$ 40.0000 1.28698
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 0 0
$$969$$ 1.00000 0.0321246
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 2.00000i − 0.0641500i
$$973$$ − 65.0000i − 2.08380i
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 44.0000i 1.40338i 0.712481 + 0.701691i $$0.247571\pi$$
−0.712481 + 0.701691i $$0.752429\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ − 45.0000i − 1.43237i
$$988$$ 4.00000i 0.127257i
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ − 48.0000i − 1.52400i
$$993$$ − 4.00000i − 0.126936i
$$994$$ 120.000 3.80617
$$995$$ 0 0
$$996$$ −24.0000 −0.760469
$$997$$ − 47.0000i − 1.48850i −0.667898 0.744252i $$-0.732806\pi$$
0.667898 0.744252i $$-0.267194\pi$$
$$998$$ 10.0000i 0.316544i
$$999$$ 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 1425.2.c.b.799.1 2
5.2 odd 4 1425.2.a.j.1.1 1
5.3 odd 4 57.2.a.a.1.1 1
5.4 even 2 inner 1425.2.c.b.799.2 2
15.2 even 4 4275.2.a.b.1.1 1
15.8 even 4 171.2.a.d.1.1 1
20.3 even 4 912.2.a.g.1.1 1
35.13 even 4 2793.2.a.b.1.1 1
40.3 even 4 3648.2.a.r.1.1 1
40.13 odd 4 3648.2.a.bh.1.1 1
55.43 even 4 6897.2.a.f.1.1 1
60.23 odd 4 2736.2.a.v.1.1 1
65.38 odd 4 9633.2.a.o.1.1 1
95.18 even 4 1083.2.a.e.1.1 1
105.83 odd 4 8379.2.a.p.1.1 1
285.113 odd 4 3249.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 5.3 odd 4
171.2.a.d.1.1 1 15.8 even 4
912.2.a.g.1.1 1 20.3 even 4
1083.2.a.e.1.1 1 95.18 even 4
1425.2.a.j.1.1 1 5.2 odd 4
1425.2.c.b.799.1 2 1.1 even 1 trivial
1425.2.c.b.799.2 2 5.4 even 2 inner
2736.2.a.v.1.1 1 60.23 odd 4
2793.2.a.b.1.1 1 35.13 even 4
3249.2.a.b.1.1 1 285.113 odd 4
3648.2.a.r.1.1 1 40.3 even 4
3648.2.a.bh.1.1 1 40.13 odd 4
4275.2.a.b.1.1 1 15.2 even 4
6897.2.a.f.1.1 1 55.43 even 4
8379.2.a.p.1.1 1 105.83 odd 4
9633.2.a.o.1.1 1 65.38 odd 4