# Properties

 Label 9408.2.a.dt.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 147) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} -2.00000 q^{11} -2.58579 q^{13} -3.41421 q^{15} -2.24264 q^{17} -2.82843 q^{19} +7.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +6.82843 q^{29} +1.17157 q^{31} -2.00000 q^{33} +4.00000 q^{37} -2.58579 q^{39} +6.24264 q^{41} +5.65685 q^{43} -3.41421 q^{45} +2.82843 q^{47} -2.24264 q^{51} +2.00000 q^{53} +6.82843 q^{55} -2.82843 q^{57} -1.17157 q^{59} -12.2426 q^{61} +8.82843 q^{65} -5.65685 q^{67} +7.65685 q^{69} -9.31371 q^{71} +13.8995 q^{73} +6.65685 q^{75} -13.6569 q^{79} +1.00000 q^{81} +7.31371 q^{83} +7.65685 q^{85} +6.82843 q^{87} -14.2426 q^{89} +1.17157 q^{93} +9.65685 q^{95} +2.58579 q^{97} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 $$2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} - 8 q^{13} - 4 q^{15} + 4 q^{17} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} + 8 q^{31} - 4 q^{33} + 8 q^{37} - 8 q^{39} + 4 q^{41} - 4 q^{45} + 4 q^{51} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 16 q^{61} + 12 q^{65} + 4 q^{69} + 4 q^{71} + 8 q^{73} + 2 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 8 q^{87} - 20 q^{89} + 8 q^{93} + 8 q^{95} + 8 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 - 4 * q^11 - 8 * q^13 - 4 * q^15 + 4 * q^17 + 4 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 + 8 * q^31 - 4 * q^33 + 8 * q^37 - 8 * q^39 + 4 * q^41 - 4 * q^45 + 4 * q^51 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 16 * q^61 + 12 * q^65 + 4 * q^69 + 4 * q^71 + 8 * q^73 + 2 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 8 * q^87 - 20 * q^89 + 8 * q^93 + 8 * q^95 + 8 * q^97 - 4 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −3.41421 −1.52688 −0.763441 0.645877i $$-0.776492\pi$$
−0.763441 + 0.645877i $$0.776492\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −2.58579 −0.717168 −0.358584 0.933497i $$-0.616740\pi$$
−0.358584 + 0.933497i $$0.616740\pi$$
$$14$$ 0 0
$$15$$ −3.41421 −0.881546
$$16$$ 0 0
$$17$$ −2.24264 −0.543920 −0.271960 0.962309i $$-0.587672\pi$$
−0.271960 + 0.962309i $$0.587672\pi$$
$$18$$ 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
$$23$$ 7.65685 1.59656 0.798282 0.602284i $$-0.205742\pi$$
0.798282 + 0.602284i $$0.205742\pi$$
$$24$$ 0 0
$$25$$ 6.65685 1.33137
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 6.82843 1.26801 0.634004 0.773330i $$-0.281410\pi$$
0.634004 + 0.773330i $$0.281410\pi$$
$$30$$ 0 0
$$31$$ 1.17157 0.210421 0.105210 0.994450i $$-0.466448\pi$$
0.105210 + 0.994450i $$0.466448\pi$$
$$32$$ 0 0
$$33$$ −2.00000 −0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 0 0
$$39$$ −2.58579 −0.414057
$$40$$ 0 0
$$41$$ 6.24264 0.974937 0.487468 0.873141i $$-0.337920\pi$$
0.487468 + 0.873141i $$0.337920\pi$$
$$42$$ 0 0
$$43$$ 5.65685 0.862662 0.431331 0.902194i $$-0.358044\pi$$
0.431331 + 0.902194i $$0.358044\pi$$
$$44$$ 0 0
$$45$$ −3.41421 −0.508961
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.24264 −0.314033
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 6.82843 0.920745
$$56$$ 0 0
$$57$$ −2.82843 −0.374634
$$58$$ 0 0
$$59$$ −1.17157 −0.152526 −0.0762629 0.997088i $$-0.524299\pi$$
−0.0762629 + 0.997088i $$0.524299\pi$$
$$60$$ 0 0
$$61$$ −12.2426 −1.56751 −0.783755 0.621070i $$-0.786698\pi$$
−0.783755 + 0.621070i $$0.786698\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 8.82843 1.09503
$$66$$ 0 0
$$67$$ −5.65685 −0.691095 −0.345547 0.938401i $$-0.612307\pi$$
−0.345547 + 0.938401i $$0.612307\pi$$
$$68$$ 0 0
$$69$$ 7.65685 0.921777
$$70$$ 0 0
$$71$$ −9.31371 −1.10533 −0.552667 0.833402i $$-0.686390\pi$$
−0.552667 + 0.833402i $$0.686390\pi$$
$$72$$ 0 0
$$73$$ 13.8995 1.62681 0.813406 0.581696i $$-0.197611\pi$$
0.813406 + 0.581696i $$0.197611\pi$$
$$74$$ 0 0
$$75$$ 6.65685 0.768667
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.6569 −1.53652 −0.768258 0.640140i $$-0.778876\pi$$
−0.768258 + 0.640140i $$0.778876\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 7.31371 0.802784 0.401392 0.915906i $$-0.368527\pi$$
0.401392 + 0.915906i $$0.368527\pi$$
$$84$$ 0 0
$$85$$ 7.65685 0.830502
$$86$$ 0 0
$$87$$ 6.82843 0.732084
$$88$$ 0 0
$$89$$ −14.2426 −1.50972 −0.754858 0.655888i $$-0.772294\pi$$
−0.754858 + 0.655888i $$0.772294\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 1.17157 0.121486
$$94$$ 0 0
$$95$$ 9.65685 0.990772
$$96$$ 0 0
$$97$$ 2.58579 0.262547 0.131273 0.991346i $$-0.458093\pi$$
0.131273 + 0.991346i $$0.458093\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ −2.92893 −0.291440 −0.145720 0.989326i $$-0.546550\pi$$
−0.145720 + 0.989326i $$0.546550\pi$$
$$102$$ 0 0
$$103$$ −4.48528 −0.441948 −0.220974 0.975280i $$-0.570924\pi$$
−0.220974 + 0.975280i $$0.570924\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −0.343146 −0.0331732 −0.0165866 0.999862i $$-0.505280\pi$$
−0.0165866 + 0.999862i $$0.505280\pi$$
$$108$$ 0 0
$$109$$ 5.65685 0.541828 0.270914 0.962604i $$-0.412674\pi$$
0.270914 + 0.962604i $$0.412674\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ −5.31371 −0.499872 −0.249936 0.968262i $$-0.580410\pi$$
−0.249936 + 0.968262i $$0.580410\pi$$
$$114$$ 0 0
$$115$$ −26.1421 −2.43777
$$116$$ 0 0
$$117$$ −2.58579 −0.239056
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 6.24264 0.562880
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ 1.65685 0.147022 0.0735110 0.997294i $$-0.476580\pi$$
0.0735110 + 0.997294i $$0.476580\pi$$
$$128$$ 0 0
$$129$$ 5.65685 0.498058
$$130$$ 0 0
$$131$$ −15.3137 −1.33796 −0.668982 0.743278i $$-0.733270\pi$$
−0.668982 + 0.743278i $$0.733270\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −3.41421 −0.293849
$$136$$ 0 0
$$137$$ 14.1421 1.20824 0.604122 0.796892i $$-0.293524\pi$$
0.604122 + 0.796892i $$0.293524\pi$$
$$138$$ 0 0
$$139$$ −17.6569 −1.49763 −0.748817 0.662776i $$-0.769378\pi$$
−0.748817 + 0.662776i $$0.769378\pi$$
$$140$$ 0 0
$$141$$ 2.82843 0.238197
$$142$$ 0 0
$$143$$ 5.17157 0.432469
$$144$$ 0 0
$$145$$ −23.3137 −1.93610
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.3137 −1.41839 −0.709197 0.705010i $$-0.750942\pi$$
−0.709197 + 0.705010i $$0.750942\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ −2.24264 −0.181307
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −11.7574 −0.938339 −0.469170 0.883108i $$-0.655447\pi$$
−0.469170 + 0.883108i $$0.655447\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −11.3137 −0.886158 −0.443079 0.896483i $$-0.646114\pi$$
−0.443079 + 0.896483i $$0.646114\pi$$
$$164$$ 0 0
$$165$$ 6.82843 0.531592
$$166$$ 0 0
$$167$$ 19.7990 1.53209 0.766046 0.642786i $$-0.222221\pi$$
0.766046 + 0.642786i $$0.222221\pi$$
$$168$$ 0 0
$$169$$ −6.31371 −0.485670
$$170$$ 0 0
$$171$$ −2.82843 −0.216295
$$172$$ 0 0
$$173$$ −21.0711 −1.60200 −0.801002 0.598662i $$-0.795699\pi$$
−0.801002 + 0.598662i $$0.795699\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.17157 −0.0880608
$$178$$ 0 0
$$179$$ −19.6569 −1.46922 −0.734611 0.678488i $$-0.762635\pi$$
−0.734611 + 0.678488i $$0.762635\pi$$
$$180$$ 0 0
$$181$$ 2.58579 0.192200 0.0961000 0.995372i $$-0.469363\pi$$
0.0961000 + 0.995372i $$0.469363\pi$$
$$182$$ 0 0
$$183$$ −12.2426 −0.905002
$$184$$ 0 0
$$185$$ −13.6569 −1.00407
$$186$$ 0 0
$$187$$ 4.48528 0.327996
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 5.31371 0.382489 0.191245 0.981542i $$-0.438748\pi$$
0.191245 + 0.981542i $$0.438748\pi$$
$$194$$ 0 0
$$195$$ 8.82843 0.632217
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −21.6569 −1.53521 −0.767607 0.640921i $$-0.778553\pi$$
−0.767607 + 0.640921i $$0.778553\pi$$
$$200$$ 0 0
$$201$$ −5.65685 −0.399004
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −21.3137 −1.48861
$$206$$ 0 0
$$207$$ 7.65685 0.532188
$$208$$ 0 0
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ 12.9706 0.892930 0.446465 0.894801i $$-0.352683\pi$$
0.446465 + 0.894801i $$0.352683\pi$$
$$212$$ 0 0
$$213$$ −9.31371 −0.638165
$$214$$ 0 0
$$215$$ −19.3137 −1.31718
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 13.8995 0.939241
$$220$$ 0 0
$$221$$ 5.79899 0.390082
$$222$$ 0 0
$$223$$ −24.9706 −1.67215 −0.836076 0.548613i $$-0.815156\pi$$
−0.836076 + 0.548613i $$0.815156\pi$$
$$224$$ 0 0
$$225$$ 6.65685 0.443790
$$226$$ 0 0
$$227$$ 23.7990 1.57959 0.789797 0.613368i $$-0.210186\pi$$
0.789797 + 0.613368i $$0.210186\pi$$
$$228$$ 0 0
$$229$$ 0.242641 0.0160341 0.00801707 0.999968i $$-0.497448\pi$$
0.00801707 + 0.999968i $$0.497448\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.14214 −0.402385 −0.201192 0.979552i $$-0.564482\pi$$
−0.201192 + 0.979552i $$0.564482\pi$$
$$234$$ 0 0
$$235$$ −9.65685 −0.629944
$$236$$ 0 0
$$237$$ −13.6569 −0.887108
$$238$$ 0 0
$$239$$ 15.6569 1.01276 0.506379 0.862311i $$-0.330984\pi$$
0.506379 + 0.862311i $$0.330984\pi$$
$$240$$ 0 0
$$241$$ −16.2426 −1.04628 −0.523140 0.852247i $$-0.675240\pi$$
−0.523140 + 0.852247i $$0.675240\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.31371 0.465360
$$248$$ 0 0
$$249$$ 7.31371 0.463487
$$250$$ 0 0
$$251$$ −12.4853 −0.788064 −0.394032 0.919097i $$-0.628920\pi$$
−0.394032 + 0.919097i $$0.628920\pi$$
$$252$$ 0 0
$$253$$ −15.3137 −0.962765
$$254$$ 0 0
$$255$$ 7.65685 0.479491
$$256$$ 0 0
$$257$$ 23.2132 1.44800 0.724000 0.689800i $$-0.242302\pi$$
0.724000 + 0.689800i $$0.242302\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.82843 0.422669
$$262$$ 0 0
$$263$$ −5.31371 −0.327657 −0.163829 0.986489i $$-0.552384\pi$$
−0.163829 + 0.986489i $$0.552384\pi$$
$$264$$ 0 0
$$265$$ −6.82843 −0.419467
$$266$$ 0 0
$$267$$ −14.2426 −0.871635
$$268$$ 0 0
$$269$$ 14.7279 0.897977 0.448989 0.893537i $$-0.351784\pi$$
0.448989 + 0.893537i $$0.351784\pi$$
$$270$$ 0 0
$$271$$ 10.1421 0.616091 0.308045 0.951372i $$-0.400325\pi$$
0.308045 + 0.951372i $$0.400325\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −13.3137 −0.802847
$$276$$ 0 0
$$277$$ 9.31371 0.559607 0.279803 0.960057i $$-0.409731\pi$$
0.279803 + 0.960057i $$0.409731\pi$$
$$278$$ 0 0
$$279$$ 1.17157 0.0701402
$$280$$ 0 0
$$281$$ 0.485281 0.0289495 0.0144747 0.999895i $$-0.495392\pi$$
0.0144747 + 0.999895i $$0.495392\pi$$
$$282$$ 0 0
$$283$$ −8.48528 −0.504398 −0.252199 0.967675i $$-0.581154\pi$$
−0.252199 + 0.967675i $$0.581154\pi$$
$$284$$ 0 0
$$285$$ 9.65685 0.572023
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −11.9706 −0.704151
$$290$$ 0 0
$$291$$ 2.58579 0.151581
$$292$$ 0 0
$$293$$ −16.5858 −0.968952 −0.484476 0.874805i $$-0.660990\pi$$
−0.484476 + 0.874805i $$0.660990\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 0 0
$$297$$ −2.00000 −0.116052
$$298$$ 0 0
$$299$$ −19.7990 −1.14501
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −2.92893 −0.168263
$$304$$ 0 0
$$305$$ 41.7990 2.39340
$$306$$ 0 0
$$307$$ 30.1421 1.72030 0.860151 0.510039i $$-0.170369\pi$$
0.860151 + 0.510039i $$0.170369\pi$$
$$308$$ 0 0
$$309$$ −4.48528 −0.255159
$$310$$ 0 0
$$311$$ −6.14214 −0.348289 −0.174144 0.984720i $$-0.555716\pi$$
−0.174144 + 0.984720i $$0.555716\pi$$
$$312$$ 0 0
$$313$$ 1.89949 0.107366 0.0536829 0.998558i $$-0.482904\pi$$
0.0536829 + 0.998558i $$0.482904\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10.0000 −0.561656 −0.280828 0.959758i $$-0.590609\pi$$
−0.280828 + 0.959758i $$0.590609\pi$$
$$318$$ 0 0
$$319$$ −13.6569 −0.764637
$$320$$ 0 0
$$321$$ −0.343146 −0.0191525
$$322$$ 0 0
$$323$$ 6.34315 0.352942
$$324$$ 0 0
$$325$$ −17.2132 −0.954817
$$326$$ 0 0
$$327$$ 5.65685 0.312825
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 4.00000 0.219199
$$334$$ 0 0
$$335$$ 19.3137 1.05522
$$336$$ 0 0
$$337$$ −29.6569 −1.61551 −0.807756 0.589517i $$-0.799318\pi$$
−0.807756 + 0.589517i $$0.799318\pi$$
$$338$$ 0 0
$$339$$ −5.31371 −0.288601
$$340$$ 0 0
$$341$$ −2.34315 −0.126888
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −26.1421 −1.40745
$$346$$ 0 0
$$347$$ 33.3137 1.78837 0.894187 0.447694i $$-0.147755\pi$$
0.894187 + 0.447694i $$0.147755\pi$$
$$348$$ 0 0
$$349$$ 9.89949 0.529908 0.264954 0.964261i $$-0.414643\pi$$
0.264954 + 0.964261i $$0.414643\pi$$
$$350$$ 0 0
$$351$$ −2.58579 −0.138019
$$352$$ 0 0
$$353$$ −14.7279 −0.783888 −0.391944 0.919989i $$-0.628197\pi$$
−0.391944 + 0.919989i $$0.628197\pi$$
$$354$$ 0 0
$$355$$ 31.7990 1.68772
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0.343146 0.0181105 0.00905527 0.999959i $$-0.497118\pi$$
0.00905527 + 0.999959i $$0.497118\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ −7.00000 −0.367405
$$364$$ 0 0
$$365$$ −47.4558 −2.48395
$$366$$ 0 0
$$367$$ 3.31371 0.172974 0.0864871 0.996253i $$-0.472436\pi$$
0.0864871 + 0.996253i $$0.472436\pi$$
$$368$$ 0 0
$$369$$ 6.24264 0.324979
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.6863 0.553315 0.276658 0.960969i $$-0.410773\pi$$
0.276658 + 0.960969i $$0.410773\pi$$
$$374$$ 0 0
$$375$$ −5.65685 −0.292119
$$376$$ 0 0
$$377$$ −17.6569 −0.909374
$$378$$ 0 0
$$379$$ 8.68629 0.446185 0.223092 0.974797i $$-0.428385\pi$$
0.223092 + 0.974797i $$0.428385\pi$$
$$380$$ 0 0
$$381$$ 1.65685 0.0848832
$$382$$ 0 0
$$383$$ 18.3431 0.937291 0.468645 0.883386i $$-0.344742\pi$$
0.468645 + 0.883386i $$0.344742\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 5.65685 0.287554
$$388$$ 0 0
$$389$$ 18.1421 0.919843 0.459921 0.887960i $$-0.347878\pi$$
0.459921 + 0.887960i $$0.347878\pi$$
$$390$$ 0 0
$$391$$ −17.1716 −0.868404
$$392$$ 0 0
$$393$$ −15.3137 −0.772474
$$394$$ 0 0
$$395$$ 46.6274 2.34608
$$396$$ 0 0
$$397$$ 2.38478 0.119688 0.0598442 0.998208i $$-0.480940\pi$$
0.0598442 + 0.998208i $$0.480940\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6.14214 −0.306724 −0.153362 0.988170i $$-0.549010\pi$$
−0.153362 + 0.988170i $$0.549010\pi$$
$$402$$ 0 0
$$403$$ −3.02944 −0.150907
$$404$$ 0 0
$$405$$ −3.41421 −0.169654
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ −21.4142 −1.05886 −0.529432 0.848352i $$-0.677595\pi$$
−0.529432 + 0.848352i $$0.677595\pi$$
$$410$$ 0 0
$$411$$ 14.1421 0.697580
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −24.9706 −1.22576
$$416$$ 0 0
$$417$$ −17.6569 −0.864660
$$418$$ 0 0
$$419$$ 33.1716 1.62054 0.810269 0.586059i $$-0.199321\pi$$
0.810269 + 0.586059i $$0.199321\pi$$
$$420$$ 0 0
$$421$$ −16.6274 −0.810371 −0.405185 0.914235i $$-0.632793\pi$$
−0.405185 + 0.914235i $$0.632793\pi$$
$$422$$ 0 0
$$423$$ 2.82843 0.137523
$$424$$ 0 0
$$425$$ −14.9289 −0.724160
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 5.17157 0.249686
$$430$$ 0 0
$$431$$ 26.9706 1.29913 0.649563 0.760308i $$-0.274952\pi$$
0.649563 + 0.760308i $$0.274952\pi$$
$$432$$ 0 0
$$433$$ −20.2426 −0.972799 −0.486400 0.873736i $$-0.661690\pi$$
−0.486400 + 0.873736i $$0.661690\pi$$
$$434$$ 0 0
$$435$$ −23.3137 −1.11781
$$436$$ 0 0
$$437$$ −21.6569 −1.03599
$$438$$ 0 0
$$439$$ −12.6863 −0.605484 −0.302742 0.953073i $$-0.597902\pi$$
−0.302742 + 0.953073i $$0.597902\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 34.9706 1.66150 0.830751 0.556645i $$-0.187911\pi$$
0.830751 + 0.556645i $$0.187911\pi$$
$$444$$ 0 0
$$445$$ 48.6274 2.30516
$$446$$ 0 0
$$447$$ −17.3137 −0.818910
$$448$$ 0 0
$$449$$ −5.31371 −0.250769 −0.125385 0.992108i $$-0.540017\pi$$
−0.125385 + 0.992108i $$0.540017\pi$$
$$450$$ 0 0
$$451$$ −12.4853 −0.587909
$$452$$ 0 0
$$453$$ −12.0000 −0.563809
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ 0 0
$$459$$ −2.24264 −0.104678
$$460$$ 0 0
$$461$$ 16.5858 0.772477 0.386239 0.922399i $$-0.373774\pi$$
0.386239 + 0.922399i $$0.373774\pi$$
$$462$$ 0 0
$$463$$ 26.6274 1.23748 0.618741 0.785595i $$-0.287643\pi$$
0.618741 + 0.785595i $$0.287643\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ 0.201010 0.00930164 0.00465082 0.999989i $$-0.498520\pi$$
0.00465082 + 0.999989i $$0.498520\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −11.7574 −0.541751
$$472$$ 0 0
$$473$$ −11.3137 −0.520205
$$474$$ 0 0
$$475$$ −18.8284 −0.863907
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ 0 0
$$479$$ −1.85786 −0.0848880 −0.0424440 0.999099i $$-0.513514\pi$$
−0.0424440 + 0.999099i $$0.513514\pi$$
$$480$$ 0 0
$$481$$ −10.3431 −0.471607
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −8.82843 −0.400878
$$486$$ 0 0
$$487$$ −26.6274 −1.20660 −0.603302 0.797513i $$-0.706149\pi$$
−0.603302 + 0.797513i $$0.706149\pi$$
$$488$$ 0 0
$$489$$ −11.3137 −0.511624
$$490$$ 0 0
$$491$$ 5.02944 0.226975 0.113488 0.993539i $$-0.463798\pi$$
0.113488 + 0.993539i $$0.463798\pi$$
$$492$$ 0 0
$$493$$ −15.3137 −0.689695
$$494$$ 0 0
$$495$$ 6.82843 0.306915
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 3.31371 0.148342 0.0741710 0.997246i $$-0.476369\pi$$
0.0741710 + 0.997246i $$0.476369\pi$$
$$500$$ 0 0
$$501$$ 19.7990 0.884554
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 10.0000 0.444994
$$506$$ 0 0
$$507$$ −6.31371 −0.280402
$$508$$ 0 0
$$509$$ −5.55635 −0.246281 −0.123140 0.992389i $$-0.539297\pi$$
−0.123140 + 0.992389i $$0.539297\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2.82843 −0.124878
$$514$$ 0 0
$$515$$ 15.3137 0.674803
$$516$$ 0 0
$$517$$ −5.65685 −0.248788
$$518$$ 0 0
$$519$$ −21.0711 −0.924917
$$520$$ 0 0
$$521$$ −35.4142 −1.55152 −0.775762 0.631025i $$-0.782634\pi$$
−0.775762 + 0.631025i $$0.782634\pi$$
$$522$$ 0 0
$$523$$ −25.6569 −1.12190 −0.560948 0.827851i $$-0.689563\pi$$
−0.560948 + 0.827851i $$0.689563\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.62742 −0.114452
$$528$$ 0 0
$$529$$ 35.6274 1.54902
$$530$$ 0 0
$$531$$ −1.17157 −0.0508419
$$532$$ 0 0
$$533$$ −16.1421 −0.699194
$$534$$ 0 0
$$535$$ 1.17157 0.0506515
$$536$$ 0 0
$$537$$ −19.6569 −0.848256
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.3137 −0.744374 −0.372187 0.928158i $$-0.621392\pi$$
−0.372187 + 0.928158i $$0.621392\pi$$
$$542$$ 0 0
$$543$$ 2.58579 0.110967
$$544$$ 0 0
$$545$$ −19.3137 −0.827308
$$546$$ 0 0
$$547$$ −36.9706 −1.58075 −0.790374 0.612625i $$-0.790114\pi$$
−0.790374 + 0.612625i $$0.790114\pi$$
$$548$$ 0 0
$$549$$ −12.2426 −0.522503
$$550$$ 0 0
$$551$$ −19.3137 −0.822792
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −13.6569 −0.579701
$$556$$ 0 0
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 0 0
$$559$$ −14.6274 −0.618674
$$560$$ 0 0
$$561$$ 4.48528 0.189369
$$562$$ 0 0
$$563$$ −1.17157 −0.0493759 −0.0246880 0.999695i $$-0.507859\pi$$
−0.0246880 + 0.999695i $$0.507859\pi$$
$$564$$ 0 0
$$565$$ 18.1421 0.763245
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −16.4853 −0.691099 −0.345549 0.938401i $$-0.612307\pi$$
−0.345549 + 0.938401i $$0.612307\pi$$
$$570$$ 0 0
$$571$$ 22.3431 0.935032 0.467516 0.883985i $$-0.345149\pi$$
0.467516 + 0.883985i $$0.345149\pi$$
$$572$$ 0 0
$$573$$ 18.0000 0.751961
$$574$$ 0 0
$$575$$ 50.9706 2.12562
$$576$$ 0 0
$$577$$ −33.8995 −1.41125 −0.705627 0.708583i $$-0.749335\pi$$
−0.705627 + 0.708583i $$0.749335\pi$$
$$578$$ 0 0
$$579$$ 5.31371 0.220830
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −4.00000 −0.165663
$$584$$ 0 0
$$585$$ 8.82843 0.365011
$$586$$ 0 0
$$587$$ −22.8284 −0.942230 −0.471115 0.882072i $$-0.656148\pi$$
−0.471115 + 0.882072i $$0.656148\pi$$
$$588$$ 0 0
$$589$$ −3.31371 −0.136539
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ −6.92893 −0.284537 −0.142269 0.989828i $$-0.545440\pi$$
−0.142269 + 0.989828i $$0.545440\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −21.6569 −0.886356
$$598$$ 0 0
$$599$$ 2.00000 0.0817178 0.0408589 0.999165i $$-0.486991\pi$$
0.0408589 + 0.999165i $$0.486991\pi$$
$$600$$ 0 0
$$601$$ −15.0711 −0.614762 −0.307381 0.951587i $$-0.599453\pi$$
−0.307381 + 0.951587i $$0.599453\pi$$
$$602$$ 0 0
$$603$$ −5.65685 −0.230365
$$604$$ 0 0
$$605$$ 23.8995 0.971653
$$606$$ 0 0
$$607$$ 18.3431 0.744525 0.372263 0.928127i $$-0.378582\pi$$
0.372263 + 0.928127i $$0.378582\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −7.31371 −0.295881
$$612$$ 0 0
$$613$$ −4.68629 −0.189278 −0.0946388 0.995512i $$-0.530170\pi$$
−0.0946388 + 0.995512i $$0.530170\pi$$
$$614$$ 0 0
$$615$$ −21.3137 −0.859452
$$616$$ 0 0
$$617$$ −24.4853 −0.985740 −0.492870 0.870103i $$-0.664052\pi$$
−0.492870 + 0.870103i $$0.664052\pi$$
$$618$$ 0 0
$$619$$ 28.9706 1.16443 0.582213 0.813037i $$-0.302187\pi$$
0.582213 + 0.813037i $$0.302187\pi$$
$$620$$ 0 0
$$621$$ 7.65685 0.307259
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −13.9706 −0.558823
$$626$$ 0 0
$$627$$ 5.65685 0.225913
$$628$$ 0 0
$$629$$ −8.97056 −0.357680
$$630$$ 0 0
$$631$$ −23.3137 −0.928104 −0.464052 0.885808i $$-0.653605\pi$$
−0.464052 + 0.885808i $$0.653605\pi$$
$$632$$ 0 0
$$633$$ 12.9706 0.515534
$$634$$ 0 0
$$635$$ −5.65685 −0.224485
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −9.31371 −0.368445
$$640$$ 0 0
$$641$$ −10.8284 −0.427697 −0.213849 0.976867i $$-0.568600\pi$$
−0.213849 + 0.976867i $$0.568600\pi$$
$$642$$ 0 0
$$643$$ −34.4264 −1.35764 −0.678822 0.734302i $$-0.737509\pi$$
−0.678822 + 0.734302i $$0.737509\pi$$
$$644$$ 0 0
$$645$$ −19.3137 −0.760477
$$646$$ 0 0
$$647$$ −26.8284 −1.05473 −0.527367 0.849638i $$-0.676821\pi$$
−0.527367 + 0.849638i $$0.676821\pi$$
$$648$$ 0 0
$$649$$ 2.34315 0.0919765
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −36.4853 −1.42778 −0.713890 0.700258i $$-0.753068\pi$$
−0.713890 + 0.700258i $$0.753068\pi$$
$$654$$ 0 0
$$655$$ 52.2843 2.04292
$$656$$ 0 0
$$657$$ 13.8995 0.542271
$$658$$ 0 0
$$659$$ 9.31371 0.362811 0.181405 0.983408i $$-0.441935\pi$$
0.181405 + 0.983408i $$0.441935\pi$$
$$660$$ 0 0
$$661$$ 23.5563 0.916236 0.458118 0.888891i $$-0.348524\pi$$
0.458118 + 0.888891i $$0.348524\pi$$
$$662$$ 0 0
$$663$$ 5.79899 0.225214
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 52.2843 2.02446
$$668$$ 0 0
$$669$$ −24.9706 −0.965418
$$670$$ 0 0
$$671$$ 24.4853 0.945244
$$672$$ 0 0
$$673$$ 23.3137 0.898677 0.449339 0.893361i $$-0.351660\pi$$
0.449339 + 0.893361i $$0.351660\pi$$
$$674$$ 0 0
$$675$$ 6.65685 0.256222
$$676$$ 0 0
$$677$$ −31.4142 −1.20735 −0.603673 0.797232i $$-0.706297\pi$$
−0.603673 + 0.797232i $$0.706297\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 23.7990 0.911979
$$682$$ 0 0
$$683$$ −19.6569 −0.752149 −0.376074 0.926590i $$-0.622726\pi$$
−0.376074 + 0.926590i $$0.622726\pi$$
$$684$$ 0 0
$$685$$ −48.2843 −1.84485
$$686$$ 0 0
$$687$$ 0.242641 0.00925732
$$688$$ 0 0
$$689$$ −5.17157 −0.197021
$$690$$ 0 0
$$691$$ −0.686292 −0.0261078 −0.0130539 0.999915i $$-0.504155\pi$$
−0.0130539 + 0.999915i $$0.504155\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 60.2843 2.28671
$$696$$ 0 0
$$697$$ −14.0000 −0.530288
$$698$$ 0 0
$$699$$ −6.14214 −0.232317
$$700$$ 0 0
$$701$$ 17.1716 0.648561 0.324281 0.945961i $$-0.394878\pi$$
0.324281 + 0.945961i $$0.394878\pi$$
$$702$$ 0 0
$$703$$ −11.3137 −0.426705
$$704$$ 0 0
$$705$$ −9.65685 −0.363698
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 36.2843 1.36268 0.681342 0.731965i $$-0.261397\pi$$
0.681342 + 0.731965i $$0.261397\pi$$
$$710$$ 0 0
$$711$$ −13.6569 −0.512172
$$712$$ 0 0
$$713$$ 8.97056 0.335950
$$714$$ 0 0
$$715$$ −17.6569 −0.660329
$$716$$ 0 0
$$717$$ 15.6569 0.584716
$$718$$ 0 0
$$719$$ −41.9411 −1.56414 −0.782070 0.623191i $$-0.785836\pi$$
−0.782070 + 0.623191i $$0.785836\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −16.2426 −0.604070
$$724$$ 0 0
$$725$$ 45.4558 1.68819
$$726$$ 0 0
$$727$$ −12.4853 −0.463053 −0.231527 0.972829i $$-0.574372\pi$$
−0.231527 + 0.972829i $$0.574372\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −12.6863 −0.469219
$$732$$ 0 0
$$733$$ −49.6985 −1.83566 −0.917828 0.396979i $$-0.870059\pi$$
−0.917828 + 0.396979i $$0.870059\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 11.3137 0.416746
$$738$$ 0 0
$$739$$ 4.68629 0.172388 0.0861940 0.996278i $$-0.472530\pi$$
0.0861940 + 0.996278i $$0.472530\pi$$
$$740$$ 0 0
$$741$$ 7.31371 0.268676
$$742$$ 0 0
$$743$$ 50.9706 1.86993 0.934964 0.354742i $$-0.115431\pi$$
0.934964 + 0.354742i $$0.115431\pi$$
$$744$$ 0 0
$$745$$ 59.1127 2.16572
$$746$$ 0 0
$$747$$ 7.31371 0.267595
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −13.6569 −0.498346 −0.249173 0.968459i $$-0.580159\pi$$
−0.249173 + 0.968459i $$0.580159\pi$$
$$752$$ 0 0
$$753$$ −12.4853 −0.454989
$$754$$ 0 0
$$755$$ 40.9706 1.49107
$$756$$ 0 0
$$757$$ −26.3431 −0.957458 −0.478729 0.877963i $$-0.658902\pi$$
−0.478729 + 0.877963i $$0.658902\pi$$
$$758$$ 0 0
$$759$$ −15.3137 −0.555852
$$760$$ 0 0
$$761$$ −18.5269 −0.671600 −0.335800 0.941933i $$-0.609007\pi$$
−0.335800 + 0.941933i $$0.609007\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 7.65685 0.276834
$$766$$ 0 0
$$767$$ 3.02944 0.109387
$$768$$ 0 0
$$769$$ 29.6985 1.07095 0.535477 0.844550i $$-0.320132\pi$$
0.535477 + 0.844550i $$0.320132\pi$$
$$770$$ 0 0
$$771$$ 23.2132 0.836003
$$772$$ 0 0
$$773$$ 9.55635 0.343718 0.171859 0.985122i $$-0.445023\pi$$
0.171859 + 0.985122i $$0.445023\pi$$
$$774$$ 0 0
$$775$$ 7.79899 0.280148
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −17.6569 −0.632622
$$780$$ 0 0
$$781$$ 18.6274 0.666541
$$782$$ 0 0
$$783$$ 6.82843 0.244028
$$784$$ 0 0
$$785$$ 40.1421 1.43273
$$786$$ 0 0
$$787$$ 24.6863 0.879971 0.439986 0.898005i $$-0.354984\pi$$
0.439986 + 0.898005i $$0.354984\pi$$
$$788$$ 0 0
$$789$$ −5.31371 −0.189173
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 31.6569 1.12417
$$794$$ 0 0
$$795$$ −6.82843 −0.242179
$$796$$ 0 0
$$797$$ 8.38478 0.297004 0.148502 0.988912i $$-0.452555\pi$$
0.148502 + 0.988912i $$0.452555\pi$$
$$798$$ 0 0
$$799$$ −6.34315 −0.224404
$$800$$ 0 0
$$801$$ −14.2426 −0.503239
$$802$$ 0 0
$$803$$ −27.7990 −0.981005
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 14.7279 0.518447
$$808$$ 0 0
$$809$$ 19.9411 0.701093 0.350546 0.936545i $$-0.385996\pi$$
0.350546 + 0.936545i $$0.385996\pi$$
$$810$$ 0 0
$$811$$ −17.6569 −0.620016 −0.310008 0.950734i $$-0.600332\pi$$
−0.310008 + 0.950734i $$0.600332\pi$$
$$812$$ 0 0
$$813$$ 10.1421 0.355700
$$814$$ 0 0
$$815$$ 38.6274 1.35306
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.6863 0.372954 0.186477 0.982459i $$-0.440293\pi$$
0.186477 + 0.982459i $$0.440293\pi$$
$$822$$ 0 0
$$823$$ −8.97056 −0.312694 −0.156347 0.987702i $$-0.549972\pi$$
−0.156347 + 0.987702i $$0.549972\pi$$
$$824$$ 0 0
$$825$$ −13.3137 −0.463524
$$826$$ 0 0
$$827$$ 47.6569 1.65719 0.828596 0.559848i $$-0.189140\pi$$
0.828596 + 0.559848i $$0.189140\pi$$
$$828$$ 0 0
$$829$$ 0.727922 0.0252818 0.0126409 0.999920i $$-0.495976\pi$$
0.0126409 + 0.999920i $$0.495976\pi$$
$$830$$ 0 0
$$831$$ 9.31371 0.323089
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −67.5980 −2.33932
$$836$$ 0 0
$$837$$ 1.17157 0.0404955
$$838$$ 0 0
$$839$$ −50.8284 −1.75479 −0.877396 0.479767i $$-0.840721\pi$$
−0.877396 + 0.479767i $$0.840721\pi$$
$$840$$ 0 0
$$841$$ 17.6274 0.607842
$$842$$ 0 0
$$843$$ 0.485281 0.0167140
$$844$$ 0 0
$$845$$ 21.5563 0.741561
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −8.48528 −0.291214
$$850$$ 0 0
$$851$$ 30.6274 1.04989
$$852$$ 0 0
$$853$$ 49.4975 1.69476 0.847381 0.530986i $$-0.178178\pi$$
0.847381 + 0.530986i $$0.178178\pi$$
$$854$$ 0 0
$$855$$ 9.65685 0.330257
$$856$$ 0 0
$$857$$ 15.4142 0.526540 0.263270 0.964722i $$-0.415199\pi$$
0.263270 + 0.964722i $$0.415199\pi$$
$$858$$ 0 0
$$859$$ 57.4558 1.96037 0.980184 0.198089i $$-0.0634735\pi$$
0.980184 + 0.198089i $$0.0634735\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −17.3137 −0.589365 −0.294683 0.955595i $$-0.595214\pi$$
−0.294683 + 0.955595i $$0.595214\pi$$
$$864$$ 0 0
$$865$$ 71.9411 2.44607
$$866$$ 0 0
$$867$$ −11.9706 −0.406542
$$868$$ 0 0
$$869$$ 27.3137 0.926554
$$870$$ 0 0
$$871$$ 14.6274 0.495631
$$872$$ 0 0
$$873$$ 2.58579 0.0875156
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 11.3137 0.382037 0.191018 0.981586i $$-0.438821\pi$$
0.191018 + 0.981586i $$0.438821\pi$$
$$878$$ 0 0
$$879$$ −16.5858 −0.559425
$$880$$ 0 0
$$881$$ 21.7574 0.733024 0.366512 0.930413i $$-0.380552\pi$$
0.366512 + 0.930413i $$0.380552\pi$$
$$882$$ 0 0
$$883$$ −4.68629 −0.157706 −0.0788531 0.996886i $$-0.525126\pi$$
−0.0788531 + 0.996886i $$0.525126\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ 2.82843 0.0949693 0.0474846 0.998872i $$-0.484879\pi$$
0.0474846 + 0.998872i $$0.484879\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 0 0
$$893$$ −8.00000 −0.267710
$$894$$ 0 0
$$895$$ 67.1127 2.24333
$$896$$ 0 0
$$897$$ −19.7990 −0.661069
$$898$$ 0 0
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −4.48528 −0.149426
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8.82843 −0.293467
$$906$$ 0 0
$$907$$ −16.0000 −0.531271 −0.265636 0.964073i $$-0.585582\pi$$
−0.265636 + 0.964073i $$0.585582\pi$$
$$908$$ 0 0
$$909$$ −2.92893 −0.0971465
$$910$$ 0 0
$$911$$ 1.02944 0.0341068 0.0170534 0.999855i $$-0.494571\pi$$
0.0170534 + 0.999855i $$0.494571\pi$$
$$912$$ 0 0
$$913$$ −14.6274 −0.484097
$$914$$ 0 0
$$915$$ 41.7990 1.38183
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8.28427 0.273273 0.136636 0.990621i $$-0.456371\pi$$
0.136636 + 0.990621i $$0.456371\pi$$
$$920$$ 0 0
$$921$$ 30.1421 0.993217
$$922$$ 0 0
$$923$$ 24.0833 0.792710
$$924$$ 0 0
$$925$$ 26.6274 0.875504
$$926$$ 0 0
$$927$$ −4.48528 −0.147316
$$928$$ 0 0
$$929$$ −39.2132 −1.28654 −0.643272 0.765638i $$-0.722423\pi$$
−0.643272 + 0.765638i $$0.722423\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −6.14214 −0.201084
$$934$$ 0 0
$$935$$ −15.3137 −0.500812
$$936$$ 0 0
$$937$$ 30.5858 0.999194 0.499597 0.866258i $$-0.333481\pi$$
0.499597 + 0.866258i $$0.333481\pi$$
$$938$$ 0 0
$$939$$ 1.89949 0.0619877
$$940$$ 0 0
$$941$$ −35.2132 −1.14792 −0.573959 0.818884i $$-0.694593\pi$$
−0.573959 + 0.818884i $$0.694593\pi$$
$$942$$ 0 0
$$943$$ 47.7990 1.55655
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 30.6863 0.997170 0.498585 0.866841i $$-0.333853\pi$$
0.498585 + 0.866841i $$0.333853\pi$$
$$948$$ 0 0
$$949$$ −35.9411 −1.16670
$$950$$ 0 0
$$951$$ −10.0000 −0.324272
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ −61.4558 −1.98866
$$956$$ 0 0
$$957$$ −13.6569 −0.441463
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29.6274 −0.955723
$$962$$ 0 0
$$963$$ −0.343146 −0.0110577
$$964$$ 0 0
$$965$$ −18.1421 −0.584016
$$966$$ 0 0
$$967$$ −33.6569 −1.08233 −0.541166 0.840916i $$-0.682017\pi$$
−0.541166 + 0.840916i $$0.682017\pi$$
$$968$$ 0 0
$$969$$ 6.34315 0.203771
$$970$$ 0 0
$$971$$ −50.6274 −1.62471 −0.812356 0.583162i $$-0.801815\pi$$
−0.812356 + 0.583162i $$0.801815\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −17.2132 −0.551264
$$976$$ 0 0
$$977$$ −21.1716 −0.677339 −0.338669 0.940905i $$-0.609977\pi$$
−0.338669 + 0.940905i $$0.609977\pi$$
$$978$$ 0 0
$$979$$ 28.4853 0.910394
$$980$$ 0 0
$$981$$ 5.65685 0.180609
$$982$$ 0 0
$$983$$ 53.2548 1.69857 0.849283 0.527938i $$-0.177035\pi$$
0.849283 + 0.527938i $$0.177035\pi$$
$$984$$ 0 0
$$985$$ 6.82843 0.217572
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 43.3137 1.37730
$$990$$ 0 0
$$991$$ 12.9706 0.412024 0.206012 0.978550i $$-0.433951\pi$$
0.206012 + 0.978550i $$0.433951\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 73.9411 2.34409
$$996$$ 0 0
$$997$$ −26.3848 −0.835614 −0.417807 0.908536i $$-0.637201\pi$$
−0.417807 + 0.908536i $$0.637201\pi$$
$$998$$ 0 0
$$999$$ 4.00000 0.126554
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 9408.2.a.dt.1.1 2
4.3 odd 2 9408.2.a.di.1.1 2
7.6 odd 2 9408.2.a.dq.1.2 2
8.3 odd 2 147.2.a.e.1.2 yes 2
8.5 even 2 2352.2.a.bc.1.2 2
24.5 odd 2 7056.2.a.cf.1.1 2
24.11 even 2 441.2.a.i.1.1 2
28.27 even 2 9408.2.a.ef.1.2 2
40.19 odd 2 3675.2.a.bd.1.1 2
56.3 even 6 147.2.e.e.79.1 4
56.5 odd 6 2352.2.q.bb.1537.2 4
56.11 odd 6 147.2.e.d.79.1 4
56.13 odd 2 2352.2.a.be.1.1 2
56.19 even 6 147.2.e.e.67.1 4
56.27 even 2 147.2.a.d.1.2 2
56.37 even 6 2352.2.q.bd.1537.1 4
56.45 odd 6 2352.2.q.bb.961.2 4
56.51 odd 6 147.2.e.d.67.1 4
56.53 even 6 2352.2.q.bd.961.1 4
168.11 even 6 441.2.e.g.226.2 4
168.59 odd 6 441.2.e.f.226.2 4
168.83 odd 2 441.2.a.j.1.1 2
168.107 even 6 441.2.e.g.361.2 4
168.125 even 2 7056.2.a.cv.1.2 2
168.131 odd 6 441.2.e.f.361.2 4
280.139 even 2 3675.2.a.bf.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 56.27 even 2
147.2.a.e.1.2 yes 2 8.3 odd 2
147.2.e.d.67.1 4 56.51 odd 6
147.2.e.d.79.1 4 56.11 odd 6
147.2.e.e.67.1 4 56.19 even 6
147.2.e.e.79.1 4 56.3 even 6
441.2.a.i.1.1 2 24.11 even 2
441.2.a.j.1.1 2 168.83 odd 2
441.2.e.f.226.2 4 168.59 odd 6
441.2.e.f.361.2 4 168.131 odd 6
441.2.e.g.226.2 4 168.11 even 6
441.2.e.g.361.2 4 168.107 even 6
2352.2.a.bc.1.2 2 8.5 even 2
2352.2.a.be.1.1 2 56.13 odd 2
2352.2.q.bb.961.2 4 56.45 odd 6
2352.2.q.bb.1537.2 4 56.5 odd 6
2352.2.q.bd.961.1 4 56.53 even 6
2352.2.q.bd.1537.1 4 56.37 even 6
3675.2.a.bd.1.1 2 40.19 odd 2
3675.2.a.bf.1.1 2 280.139 even 2
7056.2.a.cf.1.1 2 24.5 odd 2
7056.2.a.cv.1.2 2 168.125 even 2
9408.2.a.di.1.1 2 4.3 odd 2
9408.2.a.dq.1.2 2 7.6 odd 2
9408.2.a.dt.1.1 2 1.1 even 1 trivial
9408.2.a.ef.1.2 2 28.27 even 2