# Properties

 Label 4900.2.e.p.2549.3 Level $4900$ Weight $2$ Character 4900.2549 Analytic conductor $39.127$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, 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("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.e (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: $$39.1266969904$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{41}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 196) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2549.3 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 4900.2549 Dual form 4900.2.e.p.2549.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.82843i q^{3} -5.00000 q^{9} +O(q^{10})$$ $$q+2.82843i q^{3} -5.00000 q^{9} +4.00000 q^{11} -4.24264i q^{13} +1.41421i q^{17} -2.82843 q^{19} +4.00000i q^{23} -5.65685i q^{27} -8.00000 q^{29} +11.3137i q^{33} -8.00000i q^{37} +12.0000 q^{39} -7.07107 q^{41} +4.00000i q^{43} +5.65685i q^{47} -4.00000 q^{51} -10.0000i q^{53} -8.00000i q^{57} -14.1421 q^{59} -7.07107 q^{61} -11.3137 q^{69} +7.07107i q^{73} -8.00000 q^{79} +1.00000 q^{81} +14.1421i q^{83} -22.6274i q^{87} -7.07107 q^{89} -1.41421i q^{97} -20.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{9}+O(q^{10})$$ 4 * q - 20 * q^9 $$4 q - 20 q^{9} + 16 q^{11} - 32 q^{29} + 48 q^{39} - 16 q^{51} - 32 q^{79} + 4 q^{81} - 80 q^{99}+O(q^{100})$$ 4 * q - 20 * q^9 + 16 * q^11 - 32 * q^29 + 48 * q^39 - 16 * q^51 - 32 * q^79 + 4 * q^81 - 80 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$2451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.82843i 1.63299i 0.577350 + 0.816497i $$0.304087\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −5.00000 −1.66667
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ − 4.24264i − 1.17670i −0.808608 0.588348i $$-0.799778\pi$$
0.808608 0.588348i $$-0.200222\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.41421i 0.342997i 0.985184 + 0.171499i $$0.0548609\pi$$
−0.985184 + 0.171499i $$0.945139\pi$$
$$18$$ 0 0
$$19$$ −2.82843 −0.648886 −0.324443 0.945905i $$-0.605177\pi$$
−0.324443 + 0.945905i $$0.605177\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.65685i − 1.08866i
$$28$$ 0 0
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 11.3137i 1.96946i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 0 0
$$39$$ 12.0000 1.92154
$$40$$ 0 0
$$41$$ −7.07107 −1.10432 −0.552158 0.833740i $$-0.686195\pi$$
−0.552158 + 0.833740i $$0.686195\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.65685i 0.825137i 0.910927 + 0.412568i $$0.135368\pi$$
−0.910927 + 0.412568i $$0.864632\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ − 10.0000i − 1.37361i −0.726844 0.686803i $$-0.759014\pi$$
0.726844 0.686803i $$-0.240986\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 8.00000i − 1.05963i
$$58$$ 0 0
$$59$$ −14.1421 −1.84115 −0.920575 0.390567i $$-0.872279\pi$$
−0.920575 + 0.390567i $$0.872279\pi$$
$$60$$ 0 0
$$61$$ −7.07107 −0.905357 −0.452679 0.891674i $$-0.649532\pi$$
−0.452679 + 0.891674i $$0.649532\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ −11.3137 −1.36201
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 7.07107i 0.827606i 0.910366 + 0.413803i $$0.135800\pi$$
−0.910366 + 0.413803i $$0.864200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 14.1421i 1.55230i 0.630548 + 0.776151i $$0.282830\pi$$
−0.630548 + 0.776151i $$0.717170\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 22.6274i − 2.42591i
$$88$$ 0 0
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 1.41421i − 0.143592i −0.997419 0.0717958i $$-0.977127\pi$$
0.997419 0.0717958i $$-0.0228730\pi$$
$$98$$ 0 0
$$99$$ −20.0000 −2.01008
$$100$$ 0 0
$$101$$ −12.7279 −1.26648 −0.633238 0.773957i $$-0.718274\pi$$
−0.633238 + 0.773957i $$0.718274\pi$$
$$102$$ 0 0
$$103$$ 11.3137i 1.11477i 0.830253 + 0.557386i $$0.188196\pi$$
−0.830253 + 0.557386i $$0.811804\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 0 0
$$109$$ 8.00000 0.766261 0.383131 0.923694i $$-0.374846\pi$$
0.383131 + 0.923694i $$0.374846\pi$$
$$110$$ 0 0
$$111$$ 22.6274 2.14770
$$112$$ 0 0
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 21.2132i 1.96116i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ − 20.0000i − 1.80334i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 20.0000i − 1.77471i −0.461084 0.887357i $$-0.652539\pi$$
0.461084 0.887357i $$-0.347461\pi$$
$$128$$ 0 0
$$129$$ −11.3137 −0.996116
$$130$$ 0 0
$$131$$ −8.48528 −0.741362 −0.370681 0.928760i $$-0.620876\pi$$
−0.370681 + 0.928760i $$0.620876\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ −2.82843 −0.239904 −0.119952 0.992780i $$-0.538274\pi$$
−0.119952 + 0.992780i $$0.538274\pi$$
$$140$$ 0 0
$$141$$ −16.0000 −1.34744
$$142$$ 0 0
$$143$$ − 16.9706i − 1.41915i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ − 7.07107i − 0.571662i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.07107i 0.564333i 0.959366 + 0.282166i $$0.0910530\pi$$
−0.959366 + 0.282166i $$0.908947\pi$$
$$158$$ 0 0
$$159$$ 28.2843 2.24309
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 5.65685i − 0.437741i −0.975754 0.218870i $$-0.929763\pi$$
0.975754 0.218870i $$-0.0702371\pi$$
$$168$$ 0 0
$$169$$ −5.00000 −0.384615
$$170$$ 0 0
$$171$$ 14.1421 1.08148
$$172$$ 0 0
$$173$$ 4.24264i 0.322562i 0.986909 + 0.161281i $$0.0515625\pi$$
−0.986909 + 0.161281i $$0.948437\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 40.0000i − 3.00658i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −21.2132 −1.57676 −0.788382 0.615185i $$-0.789081\pi$$
−0.788382 + 0.615185i $$0.789081\pi$$
$$182$$ 0 0
$$183$$ − 20.0000i − 1.47844i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5.65685i 0.413670i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 10.0000i 0.719816i 0.932988 + 0.359908i $$0.117192\pi$$
−0.932988 + 0.359908i $$0.882808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 10.0000i − 0.712470i −0.934396 0.356235i $$-0.884060\pi$$
0.934396 0.356235i $$-0.115940\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 20.0000i − 1.39010i
$$208$$ 0 0
$$209$$ −11.3137 −0.782586
$$210$$ 0 0
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −20.0000 −1.35147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ − 16.9706i − 1.13643i −0.822879 0.568216i $$-0.807634\pi$$
0.822879 0.568216i $$-0.192366\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 8.48528i 0.563188i 0.959534 + 0.281594i $$0.0908631\pi$$
−0.959534 + 0.281594i $$0.909137\pi$$
$$228$$ 0 0
$$229$$ 21.2132 1.40181 0.700904 0.713256i $$-0.252780\pi$$
0.700904 + 0.713256i $$0.252780\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 22.6274i − 1.46981i
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 12.7279 0.819878 0.409939 0.912113i $$-0.365550\pi$$
0.409939 + 0.912113i $$0.365550\pi$$
$$242$$ 0 0
$$243$$ − 14.1421i − 0.907218i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 12.0000i 0.763542i
$$248$$ 0 0
$$249$$ −40.0000 −2.53490
$$250$$ 0 0
$$251$$ 19.7990 1.24970 0.624851 0.780744i $$-0.285160\pi$$
0.624851 + 0.780744i $$0.285160\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 21.2132i 1.32324i 0.749838 + 0.661622i $$0.230131\pi$$
−0.749838 + 0.661622i $$0.769869\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 40.0000 2.47594
$$262$$ 0 0
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 20.0000i − 1.22398i
$$268$$ 0 0
$$269$$ 18.3848 1.12094 0.560470 0.828175i $$-0.310621\pi$$
0.560470 + 0.828175i $$0.310621\pi$$
$$270$$ 0 0
$$271$$ 28.2843 1.71815 0.859074 0.511852i $$-0.171040\pi$$
0.859074 + 0.511852i $$0.171040\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 16.0000 0.954480 0.477240 0.878773i $$-0.341637\pi$$
0.477240 + 0.878773i $$0.341637\pi$$
$$282$$ 0 0
$$283$$ 2.82843i 0.168133i 0.996460 + 0.0840663i $$0.0267907\pi$$
−0.996460 + 0.0840663i $$0.973209\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 15.0000 0.882353
$$290$$ 0 0
$$291$$ 4.00000 0.234484
$$292$$ 0 0
$$293$$ − 32.5269i − 1.90024i −0.311881 0.950121i $$-0.600959\pi$$
0.311881 0.950121i $$-0.399041\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 22.6274i − 1.31298i
$$298$$ 0 0
$$299$$ 16.9706 0.981433
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ − 36.0000i − 2.06815i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 19.7990i − 1.12999i −0.825095 0.564994i $$-0.808878\pi$$
0.825095 0.564994i $$-0.191122\pi$$
$$308$$ 0 0
$$309$$ −32.0000 −1.82042
$$310$$ 0 0
$$311$$ −22.6274 −1.28308 −0.641542 0.767088i $$-0.721705\pi$$
−0.641542 + 0.767088i $$0.721705\pi$$
$$312$$ 0 0
$$313$$ − 4.24264i − 0.239808i −0.992785 0.119904i $$-0.961741\pi$$
0.992785 0.119904i $$-0.0382587\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 0 0
$$319$$ −32.0000 −1.79166
$$320$$ 0 0
$$321$$ −22.6274 −1.26294
$$322$$ 0 0
$$323$$ − 4.00000i − 0.222566i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 22.6274i 1.25130i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 40.0000i 2.19199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 0 0
$$339$$ 16.9706 0.921714
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ 4.24264 0.227103 0.113552 0.993532i $$-0.463777\pi$$
0.113552 + 0.993532i $$0.463777\pi$$
$$350$$ 0 0
$$351$$ −24.0000 −1.28103
$$352$$ 0 0
$$353$$ − 9.89949i − 0.526897i −0.964673 0.263448i $$-0.915140\pi$$
0.964673 0.263448i $$-0.0848599\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ 14.1421i 0.742270i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 5.65685i 0.295285i 0.989041 + 0.147643i $$0.0471686\pi$$
−0.989041 + 0.147643i $$0.952831\pi$$
$$368$$ 0 0
$$369$$ 35.3553 1.84053
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 33.9411i 1.74806i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 56.5685 2.89809
$$382$$ 0 0
$$383$$ 5.65685i 0.289052i 0.989501 + 0.144526i $$0.0461657\pi$$
−0.989501 + 0.144526i $$0.953834\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 20.0000i − 1.01666i
$$388$$ 0 0
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ −5.65685 −0.286079
$$392$$ 0 0
$$393$$ − 24.0000i − 1.21064i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 15.5563i 0.780751i 0.920656 + 0.390375i $$0.127655\pi$$
−0.920656 + 0.390375i $$0.872345\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 32.0000i − 1.58618i
$$408$$ 0 0
$$409$$ −38.1838 −1.88807 −0.944033 0.329851i $$-0.893001\pi$$
−0.944033 + 0.329851i $$0.893001\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 8.00000i − 0.391762i
$$418$$ 0 0
$$419$$ −14.1421 −0.690889 −0.345444 0.938439i $$-0.612272\pi$$
−0.345444 + 0.938439i $$0.612272\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ − 28.2843i − 1.37523i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 48.0000 2.31746
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 0 0
$$433$$ 21.2132i 1.01944i 0.860340 + 0.509721i $$0.170251\pi$$
−0.860340 + 0.509721i $$0.829749\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 11.3137i − 0.541208i
$$438$$ 0 0
$$439$$ 16.9706 0.809961 0.404980 0.914325i $$-0.367278\pi$$
0.404980 + 0.914325i $$0.367278\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 16.0000i 0.760183i 0.924949 + 0.380091i $$0.124107\pi$$
−0.924949 + 0.380091i $$0.875893\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 28.2843i − 1.33780i
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −28.2843 −1.33185
$$452$$ 0 0
$$453$$ − 11.3137i − 0.531564i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 30.0000i 1.40334i 0.712502 + 0.701670i $$0.247562\pi$$
−0.712502 + 0.701670i $$0.752438\pi$$
$$458$$ 0 0
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ 7.07107 0.329332 0.164666 0.986349i $$-0.447345\pi$$
0.164666 + 0.986349i $$0.447345\pi$$
$$462$$ 0 0
$$463$$ − 40.0000i − 1.85896i −0.368875 0.929479i $$-0.620257\pi$$
0.368875 0.929479i $$-0.379743\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 19.7990i − 0.916188i −0.888904 0.458094i $$-0.848532\pi$$
0.888904 0.458094i $$-0.151468\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 50.0000i 2.28934i
$$478$$ 0 0
$$479$$ 11.3137 0.516937 0.258468 0.966020i $$-0.416782\pi$$
0.258468 + 0.966020i $$0.416782\pi$$
$$480$$ 0 0
$$481$$ −33.9411 −1.54758
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 12.0000i − 0.543772i −0.962329 0.271886i $$-0.912353\pi$$
0.962329 0.271886i $$-0.0876473\pi$$
$$488$$ 0 0
$$489$$ 11.3137 0.511624
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ 0 0
$$493$$ − 11.3137i − 0.509544i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ 0 0
$$503$$ 39.5980i 1.76559i 0.469762 + 0.882793i $$0.344340\pi$$
−0.469762 + 0.882793i $$0.655660\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 14.1421i − 0.628074i
$$508$$ 0 0
$$509$$ 18.3848 0.814891 0.407445 0.913230i $$-0.366420\pi$$
0.407445 + 0.913230i $$0.366420\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 16.0000i 0.706417i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 22.6274i 0.995153i
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ −41.0122 −1.79678 −0.898388 0.439202i $$-0.855261\pi$$
−0.898388 + 0.439202i $$0.855261\pi$$
$$522$$ 0 0
$$523$$ − 42.4264i − 1.85518i −0.373603 0.927589i $$-0.621878\pi$$
0.373603 0.927589i $$-0.378122\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 70.7107 3.06858
$$532$$ 0 0
$$533$$ 30.0000i 1.29944i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ 0 0
$$543$$ − 60.0000i − 2.57485i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 0 0
$$549$$ 35.3553 1.50893
$$550$$ 0 0
$$551$$ 22.6274 0.963960
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 0 0
$$559$$ 16.9706 0.717778
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ − 14.1421i − 0.596020i −0.954563 0.298010i $$-0.903677\pi$$
0.954563 0.298010i $$-0.0963229\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −40.0000 −1.67689 −0.838444 0.544988i $$-0.816534\pi$$
−0.838444 + 0.544988i $$0.816534\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ − 45.2548i − 1.89055i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12.7279i 0.529870i 0.964266 + 0.264935i $$0.0853506\pi$$
−0.964266 + 0.264935i $$0.914649\pi$$
$$578$$ 0 0
$$579$$ −28.2843 −1.17545
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 40.0000i − 1.65663i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 25.4558i 1.05068i 0.850894 + 0.525338i $$0.176061\pi$$
−0.850894 + 0.525338i $$0.823939\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 28.2843 1.16346
$$592$$ 0 0
$$593$$ 9.89949i 0.406524i 0.979124 + 0.203262i $$0.0651542\pi$$
−0.979124 + 0.203262i $$0.934846\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ −29.6985 −1.21143 −0.605713 0.795683i $$-0.707112\pi$$
−0.605713 + 0.795683i $$0.707112\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 33.9411i − 1.37763i −0.724938 0.688814i $$-0.758132\pi$$
0.724938 0.688814i $$-0.241868\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ − 24.0000i − 0.969351i −0.874694 0.484675i $$-0.838938\pi$$
0.874694 0.484675i $$-0.161062\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 8.00000i − 0.322068i −0.986949 0.161034i $$-0.948517\pi$$
0.986949 0.161034i $$-0.0514829\pi$$
$$618$$ 0 0
$$619$$ 31.1127 1.25052 0.625262 0.780415i $$-0.284992\pi$$
0.625262 + 0.780415i $$0.284992\pi$$
$$620$$ 0 0
$$621$$ 22.6274 0.908007
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 32.0000i − 1.27796i
$$628$$ 0 0
$$629$$ 11.3137 0.451107
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 0 0
$$633$$ 67.8823i 2.69808i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 8.00000 0.315981 0.157991 0.987441i $$-0.449498\pi$$
0.157991 + 0.987441i $$0.449498\pi$$
$$642$$ 0 0
$$643$$ − 2.82843i − 0.111542i −0.998444 0.0557711i $$-0.982238\pi$$
0.998444 0.0557711i $$-0.0177617\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ −56.5685 −2.22051
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 24.0000i 0.939193i 0.882881 + 0.469596i $$0.155601\pi$$
−0.882881 + 0.469596i $$0.844399\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 35.3553i − 1.37934i
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −21.2132 −0.825098 −0.412549 0.910935i $$-0.635361\pi$$
−0.412549 + 0.910935i $$0.635361\pi$$
$$662$$ 0 0
$$663$$ 16.9706i 0.659082i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 32.0000i − 1.23904i
$$668$$ 0 0
$$669$$ 48.0000 1.85579
$$670$$ 0 0
$$671$$ −28.2843 −1.09190
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 12.7279i 0.489174i 0.969627 + 0.244587i $$0.0786523\pi$$
−0.969627 + 0.244587i $$0.921348\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 60.0000i 2.28914i
$$688$$ 0 0
$$689$$ −42.4264 −1.61632
$$690$$ 0 0
$$691$$ −42.4264 −1.61398 −0.806988 0.590567i $$-0.798904\pi$$
−0.806988 + 0.590567i $$0.798904\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 10.0000i − 0.378777i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 0 0
$$703$$ 22.6274i 0.853409i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 8.00000 0.300446 0.150223 0.988652i $$-0.452001\pi$$
0.150223 + 0.988652i $$0.452001\pi$$
$$710$$ 0 0
$$711$$ 40.0000 1.50012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 33.9411i 1.26755i
$$718$$ 0 0
$$719$$ −39.5980 −1.47676 −0.738378 0.674387i $$-0.764408\pi$$
−0.738378 + 0.674387i $$0.764408\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 36.0000i 1.33885i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.2843i 1.04901i 0.851409 + 0.524503i $$0.175749\pi$$
−0.851409 + 0.524503i $$0.824251\pi$$
$$728$$ 0 0
$$729$$ 43.0000 1.59259
$$730$$ 0 0
$$731$$ −5.65685 −0.209226
$$732$$ 0 0
$$733$$ 38.1838i 1.41035i 0.709034 + 0.705175i $$0.249131\pi$$
−0.709034 + 0.705175i $$0.750869\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ −33.9411 −1.24686
$$742$$ 0 0
$$743$$ 20.0000i 0.733729i 0.930274 + 0.366864i $$0.119569\pi$$
−0.930274 + 0.366864i $$0.880431\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 70.7107i − 2.58717i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 56.0000i 2.04075i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 40.0000i − 1.45382i −0.686730 0.726912i $$-0.740955\pi$$
0.686730 0.726912i $$-0.259045\pi$$
$$758$$ 0 0
$$759$$ −45.2548 −1.64265
$$760$$ 0 0
$$761$$ 41.0122 1.48669 0.743345 0.668908i $$-0.233238\pi$$
0.743345 + 0.668908i $$0.233238\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 60.0000i 2.16647i
$$768$$ 0 0
$$769$$ −46.6690 −1.68293 −0.841464 0.540312i $$-0.818306\pi$$
−0.841464 + 0.540312i $$0.818306\pi$$
$$770$$ 0 0
$$771$$ −60.0000 −2.16085
$$772$$ 0 0
$$773$$ − 24.0416i − 0.864717i −0.901702 0.432359i $$-0.857681\pi$$
0.901702 0.432359i $$-0.142319\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 20.0000 0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 45.2548i 1.61728i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 48.0833i 1.71398i 0.515330 + 0.856992i $$0.327669\pi$$
−0.515330 + 0.856992i $$0.672331\pi$$
$$788$$ 0 0
$$789$$ −67.8823 −2.41667
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 30.0000i 1.06533i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 29.6985i − 1.05197i −0.850493 0.525987i $$-0.823696\pi$$
0.850493 0.525987i $$-0.176304\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 0 0
$$801$$ 35.3553 1.24922
$$802$$ 0 0
$$803$$ 28.2843i 0.998130i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 52.0000i 1.83049i
$$808$$ 0 0
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ 8.48528 0.297959 0.148979 0.988840i $$-0.452401\pi$$
0.148979 + 0.988840i $$0.452401\pi$$
$$812$$ 0 0
$$813$$ 80.0000i 2.80572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 11.3137i − 0.395817i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ − 40.0000i − 1.39431i −0.716919 0.697156i $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 48.0000i 1.66912i 0.550914 + 0.834562i $$0.314279\pi$$
−0.550914 + 0.834562i $$0.685721\pi$$
$$828$$ 0 0
$$829$$ 7.07107 0.245588 0.122794 0.992432i $$-0.460815\pi$$
0.122794 + 0.992432i $$0.460815\pi$$
$$830$$ 0 0
$$831$$ −62.2254 −2.15858
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 11.3137 0.390593 0.195296 0.980744i $$-0.437433\pi$$
0.195296 + 0.980744i $$0.437433\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ 45.2548i 1.55866i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ 0 0
$$853$$ − 12.7279i − 0.435796i −0.975972 0.217898i $$-0.930080\pi$$
0.975972 0.217898i $$-0.0699200\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7.07107i 0.241543i 0.992680 + 0.120772i $$0.0385368\pi$$
−0.992680 + 0.120772i $$0.961463\pi$$
$$858$$ 0 0
$$859$$ −14.1421 −0.482523 −0.241262 0.970460i $$-0.577561\pi$$
−0.241262 + 0.970460i $$0.577561\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 40.0000i 1.36162i 0.732462 + 0.680808i $$0.238371\pi$$
−0.732462 + 0.680808i $$0.761629\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 42.4264i 1.44088i
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 7.07107i 0.239319i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.00000i 0.270141i 0.990836 + 0.135070i $$0.0431261\pi$$
−0.990836 + 0.135070i $$0.956874\pi$$
$$878$$ 0 0
$$879$$ 92.0000 3.10308
$$880$$ 0 0
$$881$$ −21.2132 −0.714691 −0.357345 0.933972i $$-0.616318\pi$$
−0.357345 + 0.933972i $$0.616318\pi$$
$$882$$ 0 0
$$883$$ 40.0000i 1.34611i 0.739594 + 0.673054i $$0.235018\pi$$
−0.739594 + 0.673054i $$0.764982\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 28.2843i 0.949693i 0.880069 + 0.474846i $$0.157496\pi$$
−0.880069 + 0.474846i $$0.842504\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 0 0
$$893$$ − 16.0000i − 0.535420i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 48.0000i 1.60267i
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 14.1421 0.471143
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 8.00000i − 0.265636i −0.991140 0.132818i $$-0.957597\pi$$
0.991140 0.132818i $$-0.0424025\pi$$
$$908$$ 0 0
$$909$$ 63.6396 2.11079
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 0 0
$$913$$ 56.5685i 1.87215i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 56.0000 1.84526
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 56.5685i − 1.85795i
$$928$$ 0 0
$$929$$ −35.3553 −1.15997 −0.579986 0.814627i $$-0.696942\pi$$
−0.579986 + 0.814627i $$0.696942\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ − 64.0000i − 2.09527i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 15.5563i 0.508204i 0.967177 + 0.254102i $$0.0817799\pi$$
−0.967177 + 0.254102i $$0.918220\pi$$
$$938$$ 0 0
$$939$$ 12.0000 0.391605
$$940$$ 0 0
$$941$$ −7.07107 −0.230510 −0.115255 0.993336i $$-0.536769\pi$$
−0.115255 + 0.993336i $$0.536769\pi$$
$$942$$ 0 0
$$943$$ − 28.2843i − 0.921063i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 60.0000i 1.94974i 0.222779 + 0.974869i $$0.428487\pi$$
−0.222779 + 0.974869i $$0.571513\pi$$
$$948$$ 0 0
$$949$$ 30.0000 0.973841
$$950$$ 0 0
$$951$$ 5.65685 0.183436
$$952$$ 0 0
$$953$$ − 10.0000i − 0.323932i −0.986796 0.161966i $$-0.948217\pi$$
0.986796 0.161966i $$-0.0517835\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 90.5097i − 2.92576i
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 40.0000i − 1.28898i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12.0000i 0.385894i 0.981209 + 0.192947i $$0.0618045\pi$$
−0.981209 + 0.192947i $$0.938195\pi$$
$$968$$ 0 0
$$969$$ 11.3137 0.363449
$$970$$ 0 0
$$971$$ 14.1421 0.453843 0.226921 0.973913i $$-0.427134\pi$$
0.226921 + 0.973913i $$0.427134\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 48.0000i 1.53566i 0.640656 + 0.767828i $$0.278662\pi$$
−0.640656 + 0.767828i $$0.721338\pi$$
$$978$$ 0 0
$$979$$ −28.2843 −0.903969
$$980$$ 0 0
$$981$$ −40.0000 −1.27710
$$982$$ 0 0
$$983$$ − 45.2548i − 1.44341i −0.692203 0.721703i $$-0.743360\pi$$
0.692203 0.721703i $$-0.256640\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 0 0
$$993$$ − 56.5685i − 1.79515i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.41421i 0.0447886i 0.999749 + 0.0223943i $$0.00712892\pi$$
−0.999749 + 0.0223943i $$0.992871\pi$$
$$998$$ 0 0
$$999$$ −45.2548 −1.43180
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 4900.2.e.p.2549.3 4
5.2 odd 4 4900.2.a.y.1.2 2
5.3 odd 4 196.2.a.c.1.1 2
5.4 even 2 inner 4900.2.e.p.2549.1 4
7.6 odd 2 inner 4900.2.e.p.2549.2 4
15.8 even 4 1764.2.a.l.1.1 2
20.3 even 4 784.2.a.m.1.2 2
35.3 even 12 196.2.e.b.177.1 4
35.13 even 4 196.2.a.c.1.2 yes 2
35.18 odd 12 196.2.e.b.177.2 4
35.23 odd 12 196.2.e.b.165.2 4
35.27 even 4 4900.2.a.y.1.1 2
35.33 even 12 196.2.e.b.165.1 4
35.34 odd 2 inner 4900.2.e.p.2549.4 4
40.3 even 4 3136.2.a.bs.1.1 2
40.13 odd 4 3136.2.a.br.1.2 2
60.23 odd 4 7056.2.a.cr.1.1 2
105.23 even 12 1764.2.k.l.361.2 4
105.38 odd 12 1764.2.k.l.1549.1 4
105.53 even 12 1764.2.k.l.1549.2 4
105.68 odd 12 1764.2.k.l.361.1 4
105.83 odd 4 1764.2.a.l.1.2 2
140.3 odd 12 784.2.i.l.177.2 4
140.23 even 12 784.2.i.l.753.1 4
140.83 odd 4 784.2.a.m.1.1 2
140.103 odd 12 784.2.i.l.753.2 4
140.123 even 12 784.2.i.l.177.1 4
280.13 even 4 3136.2.a.br.1.1 2
280.83 odd 4 3136.2.a.bs.1.2 2
420.83 even 4 7056.2.a.cr.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
196.2.a.c.1.1 2 5.3 odd 4
196.2.a.c.1.2 yes 2 35.13 even 4
196.2.e.b.165.1 4 35.33 even 12
196.2.e.b.165.2 4 35.23 odd 12
196.2.e.b.177.1 4 35.3 even 12
196.2.e.b.177.2 4 35.18 odd 12
784.2.a.m.1.1 2 140.83 odd 4
784.2.a.m.1.2 2 20.3 even 4
784.2.i.l.177.1 4 140.123 even 12
784.2.i.l.177.2 4 140.3 odd 12
784.2.i.l.753.1 4 140.23 even 12
784.2.i.l.753.2 4 140.103 odd 12
1764.2.a.l.1.1 2 15.8 even 4
1764.2.a.l.1.2 2 105.83 odd 4
1764.2.k.l.361.1 4 105.68 odd 12
1764.2.k.l.361.2 4 105.23 even 12
1764.2.k.l.1549.1 4 105.38 odd 12
1764.2.k.l.1549.2 4 105.53 even 12
3136.2.a.br.1.1 2 280.13 even 4
3136.2.a.br.1.2 2 40.13 odd 4
3136.2.a.bs.1.1 2 40.3 even 4
3136.2.a.bs.1.2 2 280.83 odd 4
4900.2.a.y.1.1 2 35.27 even 4
4900.2.a.y.1.2 2 5.2 odd 4
4900.2.e.p.2549.1 4 5.4 even 2 inner
4900.2.e.p.2549.2 4 7.6 odd 2 inner
4900.2.e.p.2549.3 4 1.1 even 1 trivial
4900.2.e.p.2549.4 4 35.34 odd 2 inner
7056.2.a.cr.1.1 2 60.23 odd 4
7056.2.a.cr.1.2 2 420.83 even 4