# Properties

 Label 2352.2.a.be.1.1 Level $2352$ Weight $2$ Character 2352.1 Self dual yes Analytic conductor $18.781$ 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] = [2352,2,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.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: $$18.7808145554$$ 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 $$\chi$$ $$=$$ 2352.1

## $q$-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 $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 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 $$-0.412328\pi$$
0.271960 + 0.962309i $$0.412328\pi$$
$$18$$ 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 $$-0.718590\pi$$
−0.634004 + 0.773330i $$0.718590\pi$$
$$30$$ 0 0
$$31$$ −1.17157 −0.210421 −0.105210 0.994450i $$-0.533552\pi$$
−0.105210 + 0.994450i $$0.533552\pi$$
$$32$$ 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 $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −2.58579 −0.414057
$$40$$ 0 0
$$41$$ −6.24264 −0.974937 −0.487468 0.873141i $$-0.662080\pi$$
−0.487468 + 0.873141i $$0.662080\pi$$
$$42$$ 0 0
$$43$$ −5.65685 −0.862662 −0.431331 0.902194i $$-0.641956\pi$$
−0.431331 + 0.902194i $$0.641956\pi$$
$$44$$ 0 0
$$45$$ −3.41421 −0.508961
$$46$$ 0 0
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 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 $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 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 $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ 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 $$-0.802389\pi$$
−0.813406 + 0.581696i $$0.802389\pi$$
$$74$$ 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 $$-0.227706\pi$$
0.754858 + 0.655888i $$0.227706\pi$$
$$90$$ 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 $$-0.541907\pi$$
−0.131273 + 0.991346i $$0.541907\pi$$
$$98$$ 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 $$-0.429076\pi$$
0.220974 + 0.975280i $$0.429076\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0.343146 0.0331732 0.0165866 0.999862i $$-0.494720\pi$$
0.0165866 + 0.999862i $$0.494720\pi$$
$$108$$ 0 0
$$109$$ −5.65685 −0.541828 −0.270914 0.962604i $$-0.587326\pi$$
−0.270914 + 0.962604i $$0.587326\pi$$
$$110$$ 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 $$-0.249058\pi$$
0.709197 + 0.705010i $$0.249058\pi$$
$$150$$ 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 $$-0.353886\pi$$
0.443079 + 0.896483i $$0.353886\pi$$
$$164$$ 0 0
$$165$$ −6.82843 −0.531592
$$166$$ 0 0
$$167$$ −19.7990 −1.53209 −0.766046 0.642786i $$-0.777779\pi$$
−0.766046 + 0.642786i $$0.777779\pi$$
$$168$$ 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 $$-0.237365\pi$$
0.734611 + 0.678488i $$0.237365\pi$$
$$180$$ 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 $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ 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 $$-0.647317\pi$$
−0.446465 + 0.894801i $$0.647317\pi$$
$$212$$ 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 $$-0.184844\pi$$
0.836076 + 0.548613i $$0.184844\pi$$
$$224$$ 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 $$-0.324760\pi$$
0.523140 + 0.852247i $$0.324760\pi$$
$$242$$ 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 $$-0.757698\pi$$
−0.724000 + 0.689800i $$0.757698\pi$$
$$258$$ 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 $$-0.599675\pi$$
−0.308045 + 0.951372i $$0.599675\pi$$
$$272$$ 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 $$-0.590269\pi$$
−0.279803 + 0.960057i $$0.590269\pi$$
$$278$$ 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 $$-0.444284\pi$$
0.174144 + 0.984720i $$0.444284\pi$$
$$312$$ 0 0
$$313$$ −1.89949 −0.107366 −0.0536829 0.998558i $$-0.517096\pi$$
−0.0536829 + 0.998558i $$0.517096\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ 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 $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 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 $$-0.852245\pi$$
−0.894187 + 0.447694i $$0.852245\pi$$
$$348$$ 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 $$-0.371803\pi$$
0.391944 + 0.919989i $$0.371803\pi$$
$$354$$ 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 $$-0.527564\pi$$
−0.0864871 + 0.996253i $$0.527564\pi$$
$$368$$ 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 $$-0.589227\pi$$
−0.276658 + 0.960969i $$0.589227\pi$$
$$374$$ 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 $$-0.571615\pi$$
−0.223092 + 0.974797i $$0.571615\pi$$
$$380$$ 0 0
$$381$$ 1.65685 0.0848832
$$382$$ 0 0
$$383$$ −18.3431 −0.937291 −0.468645 0.883386i $$-0.655258\pi$$
−0.468645 + 0.883386i $$0.655258\pi$$
$$384$$ 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 $$-0.652122\pi$$
−0.459921 + 0.887960i $$0.652122\pi$$
$$390$$ 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 $$-0.322405\pi$$
0.529432 + 0.848352i $$0.322405\pi$$
$$410$$ 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 $$-0.367207\pi$$
0.405185 + 0.914235i $$0.367207\pi$$
$$422$$ 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 $$-0.338310\pi$$
0.486400 + 0.873736i $$0.338310\pi$$
$$434$$ 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 $$-0.402098\pi$$
0.302742 + 0.953073i $$0.402098\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −34.9706 −1.66150 −0.830751 0.556645i $$-0.812089\pi$$
−0.830751 + 0.556645i $$0.812089\pi$$
$$444$$ 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 $$-0.486486\pi$$
0.0424440 + 0.999099i $$0.486486\pi$$
$$480$$ 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 $$-0.536202\pi$$
−0.113488 + 0.993539i $$0.536202\pi$$
$$492$$ 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 $$-0.523631\pi$$
−0.0741710 + 0.997246i $$0.523631\pi$$
$$500$$ 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 $$-0.217366\pi$$
0.775762 + 0.631025i $$0.217366\pi$$
$$522$$ 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 $$-0.378608\pi$$
0.372187 + 0.928158i $$0.378608\pi$$
$$542$$ 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 $$-0.209886\pi$$
0.790374 + 0.612625i $$0.209886\pi$$
$$548$$ 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 $$-0.314312\pi$$
0.550828 + 0.834619i $$0.314312\pi$$
$$558$$ 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 $$-0.654851\pi$$
−0.467516 + 0.883985i $$0.654851\pi$$
$$572$$ 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 $$-0.250665\pi$$
0.705627 + 0.708583i $$0.250665\pi$$
$$578$$ 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 $$-0.454560\pi$$
0.142269 + 0.989828i $$0.454560\pi$$
$$594$$ 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 $$-0.400547\pi$$
0.307381 + 0.951587i $$0.400547\pi$$
$$602$$ 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 $$-0.621418\pi$$
−0.372263 + 0.928127i $$0.621418\pi$$
$$608$$ 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 $$-0.469830\pi$$
0.0946388 + 0.995512i $$0.469830\pi$$
$$614$$ 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 $$-0.323179\pi$$
0.527367 + 0.849638i $$0.323179\pi$$
$$648$$ 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 $$-0.246932\pi$$
0.713890 + 0.700258i $$0.246932\pi$$
$$654$$ 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 $$-0.558065\pi$$
−0.181405 + 0.983408i $$0.558065\pi$$
$$660$$ 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 $$-0.377274\pi$$
0.376074 + 0.926590i $$0.377274\pi$$
$$684$$ 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 $$-0.605122\pi$$
−0.324281 + 0.945961i $$0.605122\pi$$
$$702$$ 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 $$-0.738603\pi$$
−0.681342 + 0.731965i $$0.738603\pi$$
$$710$$ 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 $$-0.214164\pi$$
0.782070 + 0.623191i $$0.214164\pi$$
$$720$$ 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 $$-0.425628\pi$$
0.231527 + 0.972829i $$0.425628\pi$$
$$728$$ 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 $$-0.527470\pi$$
−0.0861940 + 0.996278i $$0.527470\pi$$
$$740$$ 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 $$-0.341098\pi$$
0.478729 + 0.877963i $$0.341098\pi$$
$$758$$ 0 0
$$759$$ 15.3137 0.555852
$$760$$ 0 0
$$761$$ 18.5269 0.671600 0.335800 0.941933i $$-0.390993\pi$$
0.335800 + 0.941933i $$0.390993\pi$$
$$762$$ 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 $$-0.679868\pi$$
−0.535477 + 0.844550i $$0.679868\pi$$
$$770$$ 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 $$-0.559707\pi$$
−0.186477 + 0.982459i $$0.559707\pi$$
$$822$$ 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 $$-0.810860\pi$$
−0.828596 + 0.559848i $$0.810860\pi$$
$$828$$ 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 $$-0.159279\pi$$
0.877396 + 0.479767i $$0.159279\pi$$
$$840$$ 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 $$-0.584801\pi$$
−0.263270 + 0.964722i $$0.584801\pi$$
$$858$$ 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 $$-0.561179\pi$$
−0.191018 + 0.981586i $$0.561179\pi$$
$$878$$ 0 0
$$879$$ −16.5858 −0.559425
$$880$$ 0 0
$$881$$ −21.7574 −0.733024 −0.366512 0.930413i $$-0.619448\pi$$
−0.366512 + 0.930413i $$0.619448\pi$$
$$882$$ 0 0
$$883$$ 4.68629 0.157706 0.0788531 0.996886i $$-0.474874\pi$$
0.0788531 + 0.996886i $$0.474874\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ 0 0
$$887$$ −2.82843 −0.0949693 −0.0474846 0.998872i $$-0.515121\pi$$
−0.0474846 + 0.998872i $$0.515121\pi$$
$$888$$ 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 $$-0.414418\pi$$
0.265636 + 0.964073i $$0.414418\pi$$
$$908$$ 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 $$-0.277577\pi$$
0.643272 + 0.765638i $$0.277577\pi$$
$$930$$ 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 $$-0.666519\pi$$
−0.499597 + 0.866258i $$0.666519\pi$$
$$938$$ 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 $$-0.666147\pi$$
−0.498585 + 0.866841i $$0.666147\pi$$
$$948$$ 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 $$-0.822965\pi$$
−0.849283 + 0.527938i $$0.822965\pi$$
$$984$$ 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 2352.2.a.be.1.1 2
3.2 odd 2 7056.2.a.cv.1.2 2
4.3 odd 2 147.2.a.d.1.2 2
7.2 even 3 2352.2.q.bb.1537.2 4
7.3 odd 6 2352.2.q.bd.961.1 4
7.4 even 3 2352.2.q.bb.961.2 4
7.5 odd 6 2352.2.q.bd.1537.1 4
7.6 odd 2 2352.2.a.bc.1.2 2
8.3 odd 2 9408.2.a.ef.1.2 2
8.5 even 2 9408.2.a.dq.1.2 2
12.11 even 2 441.2.a.j.1.1 2
20.19 odd 2 3675.2.a.bf.1.1 2
21.20 even 2 7056.2.a.cf.1.1 2
28.3 even 6 147.2.e.d.79.1 4
28.11 odd 6 147.2.e.e.79.1 4
28.19 even 6 147.2.e.d.67.1 4
28.23 odd 6 147.2.e.e.67.1 4
28.27 even 2 147.2.a.e.1.2 yes 2
56.13 odd 2 9408.2.a.dt.1.1 2
56.27 even 2 9408.2.a.di.1.1 2
84.11 even 6 441.2.e.f.226.2 4
84.23 even 6 441.2.e.f.361.2 4
84.47 odd 6 441.2.e.g.361.2 4
84.59 odd 6 441.2.e.g.226.2 4
84.83 odd 2 441.2.a.i.1.1 2
140.139 even 2 3675.2.a.bd.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.2 2 4.3 odd 2
147.2.a.e.1.2 yes 2 28.27 even 2
147.2.e.d.67.1 4 28.19 even 6
147.2.e.d.79.1 4 28.3 even 6
147.2.e.e.67.1 4 28.23 odd 6
147.2.e.e.79.1 4 28.11 odd 6
441.2.a.i.1.1 2 84.83 odd 2
441.2.a.j.1.1 2 12.11 even 2
441.2.e.f.226.2 4 84.11 even 6
441.2.e.f.361.2 4 84.23 even 6
441.2.e.g.226.2 4 84.59 odd 6
441.2.e.g.361.2 4 84.47 odd 6
2352.2.a.bc.1.2 2 7.6 odd 2
2352.2.a.be.1.1 2 1.1 even 1 trivial
2352.2.q.bb.961.2 4 7.4 even 3
2352.2.q.bb.1537.2 4 7.2 even 3
2352.2.q.bd.961.1 4 7.3 odd 6
2352.2.q.bd.1537.1 4 7.5 odd 6
3675.2.a.bd.1.1 2 140.139 even 2
3675.2.a.bf.1.1 2 20.19 odd 2
7056.2.a.cf.1.1 2 21.20 even 2
7056.2.a.cv.1.2 2 3.2 odd 2
9408.2.a.di.1.1 2 56.27 even 2
9408.2.a.dq.1.2 2 8.5 even 2
9408.2.a.dt.1.1 2 56.13 odd 2
9408.2.a.ef.1.2 2 8.3 odd 2