# Properties

 Label 338.6.a.e.1.1 Level $338$ Weight $6$ Character 338.1 Self dual yes Analytic conductor $54.210$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,6,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$54.2097310968$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +4.00000 q^{3} +16.0000 q^{4} +68.0000 q^{5} +16.0000 q^{6} -82.0000 q^{7} +64.0000 q^{8} -227.000 q^{9} +O(q^{10})$$ $$q+4.00000 q^{2} +4.00000 q^{3} +16.0000 q^{4} +68.0000 q^{5} +16.0000 q^{6} -82.0000 q^{7} +64.0000 q^{8} -227.000 q^{9} +272.000 q^{10} -390.000 q^{11} +64.0000 q^{12} -328.000 q^{14} +272.000 q^{15} +256.000 q^{16} -1738.00 q^{17} -908.000 q^{18} -1074.00 q^{19} +1088.00 q^{20} -328.000 q^{21} -1560.00 q^{22} -2104.00 q^{23} +256.000 q^{24} +1499.00 q^{25} -1880.00 q^{27} -1312.00 q^{28} -1690.00 q^{29} +1088.00 q^{30} -1430.00 q^{31} +1024.00 q^{32} -1560.00 q^{33} -6952.00 q^{34} -5576.00 q^{35} -3632.00 q^{36} -8852.00 q^{37} -4296.00 q^{38} +4352.00 q^{40} +6760.00 q^{41} -1312.00 q^{42} +16916.0 q^{43} -6240.00 q^{44} -15436.0 q^{45} -8416.00 q^{46} +25158.0 q^{47} +1024.00 q^{48} -10083.0 q^{49} +5996.00 q^{50} -6952.00 q^{51} +38214.0 q^{53} -7520.00 q^{54} -26520.0 q^{55} -5248.00 q^{56} -4296.00 q^{57} -6760.00 q^{58} -21286.0 q^{59} +4352.00 q^{60} -5458.00 q^{61} -5720.00 q^{62} +18614.0 q^{63} +4096.00 q^{64} -6240.00 q^{66} +44542.0 q^{67} -27808.0 q^{68} -8416.00 q^{69} -22304.0 q^{70} -17790.0 q^{71} -14528.0 q^{72} -31064.0 q^{73} -35408.0 q^{74} +5996.00 q^{75} -17184.0 q^{76} +31980.0 q^{77} -45360.0 q^{79} +17408.0 q^{80} +47641.0 q^{81} +27040.0 q^{82} -124546. q^{83} -5248.00 q^{84} -118184. q^{85} +67664.0 q^{86} -6760.00 q^{87} -24960.0 q^{88} +18744.0 q^{89} -61744.0 q^{90} -33664.0 q^{92} -5720.00 q^{93} +100632. q^{94} -73032.0 q^{95} +4096.00 q^{96} -121488. q^{97} -40332.0 q^{98} +88530.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ 4.00000 0.256600 0.128300 0.991735i $$-0.459048\pi$$
0.128300 + 0.991735i $$0.459048\pi$$
$$4$$ 16.0000 0.500000
$$5$$ 68.0000 1.21642 0.608210 0.793776i $$-0.291888\pi$$
0.608210 + 0.793776i $$0.291888\pi$$
$$6$$ 16.0000 0.181444
$$7$$ −82.0000 −0.632512 −0.316256 0.948674i $$-0.602426\pi$$
−0.316256 + 0.948674i $$0.602426\pi$$
$$8$$ 64.0000 0.353553
$$9$$ −227.000 −0.934156
$$10$$ 272.000 0.860140
$$11$$ −390.000 −0.971813 −0.485907 0.874011i $$-0.661511\pi$$
−0.485907 + 0.874011i $$0.661511\pi$$
$$12$$ 64.0000 0.128300
$$13$$ 0 0
$$14$$ −328.000 −0.447254
$$15$$ 272.000 0.312134
$$16$$ 256.000 0.250000
$$17$$ −1738.00 −1.45857 −0.729285 0.684210i $$-0.760147\pi$$
−0.729285 + 0.684210i $$0.760147\pi$$
$$18$$ −908.000 −0.660548
$$19$$ −1074.00 −0.682528 −0.341264 0.939968i $$-0.610855\pi$$
−0.341264 + 0.939968i $$0.610855\pi$$
$$20$$ 1088.00 0.608210
$$21$$ −328.000 −0.162303
$$22$$ −1560.00 −0.687176
$$23$$ −2104.00 −0.829328 −0.414664 0.909975i $$-0.636101\pi$$
−0.414664 + 0.909975i $$0.636101\pi$$
$$24$$ 256.000 0.0907218
$$25$$ 1499.00 0.479680
$$26$$ 0 0
$$27$$ −1880.00 −0.496305
$$28$$ −1312.00 −0.316256
$$29$$ −1690.00 −0.373157 −0.186579 0.982440i $$-0.559740\pi$$
−0.186579 + 0.982440i $$0.559740\pi$$
$$30$$ 1088.00 0.220712
$$31$$ −1430.00 −0.267259 −0.133629 0.991031i $$-0.542663\pi$$
−0.133629 + 0.991031i $$0.542663\pi$$
$$32$$ 1024.00 0.176777
$$33$$ −1560.00 −0.249367
$$34$$ −6952.00 −1.03137
$$35$$ −5576.00 −0.769401
$$36$$ −3632.00 −0.467078
$$37$$ −8852.00 −1.06301 −0.531505 0.847055i $$-0.678373\pi$$
−0.531505 + 0.847055i $$0.678373\pi$$
$$38$$ −4296.00 −0.482620
$$39$$ 0 0
$$40$$ 4352.00 0.430070
$$41$$ 6760.00 0.628040 0.314020 0.949416i $$-0.398324\pi$$
0.314020 + 0.949416i $$0.398324\pi$$
$$42$$ −1312.00 −0.114765
$$43$$ 16916.0 1.39517 0.697584 0.716503i $$-0.254258\pi$$
0.697584 + 0.716503i $$0.254258\pi$$
$$44$$ −6240.00 −0.485907
$$45$$ −15436.0 −1.13633
$$46$$ −8416.00 −0.586423
$$47$$ 25158.0 1.66124 0.830618 0.556842i $$-0.187987\pi$$
0.830618 + 0.556842i $$0.187987\pi$$
$$48$$ 1024.00 0.0641500
$$49$$ −10083.0 −0.599929
$$50$$ 5996.00 0.339185
$$51$$ −6952.00 −0.374269
$$52$$ 0 0
$$53$$ 38214.0 1.86867 0.934335 0.356395i $$-0.115994\pi$$
0.934335 + 0.356395i $$0.115994\pi$$
$$54$$ −7520.00 −0.350940
$$55$$ −26520.0 −1.18213
$$56$$ −5248.00 −0.223627
$$57$$ −4296.00 −0.175137
$$58$$ −6760.00 −0.263862
$$59$$ −21286.0 −0.796093 −0.398047 0.917365i $$-0.630312\pi$$
−0.398047 + 0.917365i $$0.630312\pi$$
$$60$$ 4352.00 0.156067
$$61$$ −5458.00 −0.187806 −0.0939029 0.995581i $$-0.529934\pi$$
−0.0939029 + 0.995581i $$0.529934\pi$$
$$62$$ −5720.00 −0.188980
$$63$$ 18614.0 0.590865
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ −6240.00 −0.176329
$$67$$ 44542.0 1.21222 0.606112 0.795379i $$-0.292728\pi$$
0.606112 + 0.795379i $$0.292728\pi$$
$$68$$ −27808.0 −0.729285
$$69$$ −8416.00 −0.212806
$$70$$ −22304.0 −0.544049
$$71$$ −17790.0 −0.418823 −0.209411 0.977828i $$-0.567155\pi$$
−0.209411 + 0.977828i $$0.567155\pi$$
$$72$$ −14528.0 −0.330274
$$73$$ −31064.0 −0.682260 −0.341130 0.940016i $$-0.610810\pi$$
−0.341130 + 0.940016i $$0.610810\pi$$
$$74$$ −35408.0 −0.751661
$$75$$ 5996.00 0.123086
$$76$$ −17184.0 −0.341264
$$77$$ 31980.0 0.614684
$$78$$ 0 0
$$79$$ −45360.0 −0.817721 −0.408861 0.912597i $$-0.634074\pi$$
−0.408861 + 0.912597i $$0.634074\pi$$
$$80$$ 17408.0 0.304105
$$81$$ 47641.0 0.806805
$$82$$ 27040.0 0.444091
$$83$$ −124546. −1.98442 −0.992212 0.124559i $$-0.960248\pi$$
−0.992212 + 0.124559i $$0.960248\pi$$
$$84$$ −5248.00 −0.0811513
$$85$$ −118184. −1.77424
$$86$$ 67664.0 0.986533
$$87$$ −6760.00 −0.0957522
$$88$$ −24960.0 −0.343588
$$89$$ 18744.0 0.250834 0.125417 0.992104i $$-0.459973\pi$$
0.125417 + 0.992104i $$0.459973\pi$$
$$90$$ −61744.0 −0.803505
$$91$$ 0 0
$$92$$ −33664.0 −0.414664
$$93$$ −5720.00 −0.0685786
$$94$$ 100632. 1.17467
$$95$$ −73032.0 −0.830241
$$96$$ 4096.00 0.0453609
$$97$$ −121488. −1.31100 −0.655502 0.755193i $$-0.727543\pi$$
−0.655502 + 0.755193i $$0.727543\pi$$
$$98$$ −40332.0 −0.424214
$$99$$ 88530.0 0.907826
$$100$$ 23984.0 0.239840
$$101$$ 14218.0 0.138687 0.0693434 0.997593i $$-0.477910\pi$$
0.0693434 + 0.997593i $$0.477910\pi$$
$$102$$ −27808.0 −0.264648
$$103$$ 62776.0 0.583043 0.291521 0.956564i $$-0.405839\pi$$
0.291521 + 0.956564i $$0.405839\pi$$
$$104$$ 0 0
$$105$$ −22304.0 −0.197428
$$106$$ 152856. 1.32135
$$107$$ −79252.0 −0.669192 −0.334596 0.942362i $$-0.608600\pi$$
−0.334596 + 0.942362i $$0.608600\pi$$
$$108$$ −30080.0 −0.248152
$$109$$ −218084. −1.75816 −0.879078 0.476677i $$-0.841841\pi$$
−0.879078 + 0.476677i $$0.841841\pi$$
$$110$$ −106080. −0.835895
$$111$$ −35408.0 −0.272768
$$112$$ −20992.0 −0.158128
$$113$$ 44234.0 0.325882 0.162941 0.986636i $$-0.447902\pi$$
0.162941 + 0.986636i $$0.447902\pi$$
$$114$$ −17184.0 −0.123840
$$115$$ −143072. −1.00881
$$116$$ −27040.0 −0.186579
$$117$$ 0 0
$$118$$ −85144.0 −0.562923
$$119$$ 142516. 0.922563
$$120$$ 17408.0 0.110356
$$121$$ −8951.00 −0.0555787
$$122$$ −21832.0 −0.132799
$$123$$ 27040.0 0.161155
$$124$$ −22880.0 −0.133629
$$125$$ −110568. −0.632928
$$126$$ 74456.0 0.417805
$$127$$ 310432. 1.70788 0.853940 0.520372i $$-0.174207\pi$$
0.853940 + 0.520372i $$0.174207\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 67664.0 0.358000
$$130$$ 0 0
$$131$$ 310372. 1.58017 0.790086 0.612996i $$-0.210036\pi$$
0.790086 + 0.612996i $$0.210036\pi$$
$$132$$ −24960.0 −0.124684
$$133$$ 88068.0 0.431707
$$134$$ 178168. 0.857171
$$135$$ −127840. −0.603716
$$136$$ −111232. −0.515683
$$137$$ −281032. −1.27925 −0.639623 0.768688i $$-0.720910\pi$$
−0.639623 + 0.768688i $$0.720910\pi$$
$$138$$ −33664.0 −0.150476
$$139$$ 363820. 1.59716 0.798582 0.601886i $$-0.205584\pi$$
0.798582 + 0.601886i $$0.205584\pi$$
$$140$$ −89216.0 −0.384700
$$141$$ 100632. 0.426273
$$142$$ −71160.0 −0.296152
$$143$$ 0 0
$$144$$ −58112.0 −0.233539
$$145$$ −114920. −0.453916
$$146$$ −124256. −0.482431
$$147$$ −40332.0 −0.153942
$$148$$ −141632. −0.531505
$$149$$ −274204. −1.01183 −0.505916 0.862583i $$-0.668845\pi$$
−0.505916 + 0.862583i $$0.668845\pi$$
$$150$$ 23984.0 0.0870349
$$151$$ 344030. 1.22787 0.613937 0.789355i $$-0.289585\pi$$
0.613937 + 0.789355i $$0.289585\pi$$
$$152$$ −68736.0 −0.241310
$$153$$ 394526. 1.36253
$$154$$ 127920. 0.434647
$$155$$ −97240.0 −0.325099
$$156$$ 0 0
$$157$$ 20518.0 0.0664333 0.0332167 0.999448i $$-0.489425\pi$$
0.0332167 + 0.999448i $$0.489425\pi$$
$$158$$ −181440. −0.578216
$$159$$ 152856. 0.479501
$$160$$ 69632.0 0.215035
$$161$$ 172528. 0.524560
$$162$$ 190564. 0.570497
$$163$$ 36626.0 0.107974 0.0539872 0.998542i $$-0.482807\pi$$
0.0539872 + 0.998542i $$0.482807\pi$$
$$164$$ 108160. 0.314020
$$165$$ −106080. −0.303336
$$166$$ −498184. −1.40320
$$167$$ −269442. −0.747608 −0.373804 0.927508i $$-0.621947\pi$$
−0.373804 + 0.927508i $$0.621947\pi$$
$$168$$ −20992.0 −0.0573827
$$169$$ 0 0
$$170$$ −472736. −1.25457
$$171$$ 243798. 0.637588
$$172$$ 270656. 0.697584
$$173$$ −282654. −0.718026 −0.359013 0.933333i $$-0.616887\pi$$
−0.359013 + 0.933333i $$0.616887\pi$$
$$174$$ −27040.0 −0.0677070
$$175$$ −122918. −0.303403
$$176$$ −99840.0 −0.242953
$$177$$ −85144.0 −0.204278
$$178$$ 74976.0 0.177367
$$179$$ −333780. −0.778624 −0.389312 0.921106i $$-0.627287\pi$$
−0.389312 + 0.921106i $$0.627287\pi$$
$$180$$ −246976. −0.568164
$$181$$ 459938. 1.04352 0.521762 0.853091i $$-0.325275\pi$$
0.521762 + 0.853091i $$0.325275\pi$$
$$182$$ 0 0
$$183$$ −21832.0 −0.0481910
$$184$$ −134656. −0.293212
$$185$$ −601936. −1.29307
$$186$$ −22880.0 −0.0484924
$$187$$ 677820. 1.41746
$$188$$ 402528. 0.830618
$$189$$ 154160. 0.313919
$$190$$ −292128. −0.587069
$$191$$ −917088. −1.81898 −0.909489 0.415727i $$-0.863527\pi$$
−0.909489 + 0.415727i $$0.863527\pi$$
$$192$$ 16384.0 0.0320750
$$193$$ 639056. 1.23494 0.617470 0.786595i $$-0.288158\pi$$
0.617470 + 0.786595i $$0.288158\pi$$
$$194$$ −485952. −0.927020
$$195$$ 0 0
$$196$$ −161328. −0.299964
$$197$$ 358292. 0.657766 0.328883 0.944371i $$-0.393328\pi$$
0.328883 + 0.944371i $$0.393328\pi$$
$$198$$ 354120. 0.641930
$$199$$ −370440. −0.663109 −0.331555 0.943436i $$-0.607573\pi$$
−0.331555 + 0.943436i $$0.607573\pi$$
$$200$$ 95936.0 0.169592
$$201$$ 178168. 0.311057
$$202$$ 56872.0 0.0980664
$$203$$ 138580. 0.236026
$$204$$ −111232. −0.187135
$$205$$ 459680. 0.763961
$$206$$ 251104. 0.412274
$$207$$ 477608. 0.774722
$$208$$ 0 0
$$209$$ 418860. 0.663290
$$210$$ −89216.0 −0.139603
$$211$$ −177228. −0.274048 −0.137024 0.990568i $$-0.543754\pi$$
−0.137024 + 0.990568i $$0.543754\pi$$
$$212$$ 611424. 0.934335
$$213$$ −71160.0 −0.107470
$$214$$ −317008. −0.473190
$$215$$ 1.15029e6 1.69711
$$216$$ −120320. −0.175470
$$217$$ 117260. 0.169044
$$218$$ −872336. −1.24320
$$219$$ −124256. −0.175068
$$220$$ −424320. −0.591067
$$221$$ 0 0
$$222$$ −141632. −0.192876
$$223$$ −1.11297e6 −1.49872 −0.749359 0.662164i $$-0.769638\pi$$
−0.749359 + 0.662164i $$0.769638\pi$$
$$224$$ −83968.0 −0.111813
$$225$$ −340273. −0.448096
$$226$$ 176936. 0.230433
$$227$$ 1.39158e6 1.79244 0.896219 0.443612i $$-0.146303\pi$$
0.896219 + 0.443612i $$0.146303\pi$$
$$228$$ −68736.0 −0.0875683
$$229$$ −909796. −1.14645 −0.573225 0.819398i $$-0.694308\pi$$
−0.573225 + 0.819398i $$0.694308\pi$$
$$230$$ −572288. −0.713337
$$231$$ 127920. 0.157728
$$232$$ −108160. −0.131931
$$233$$ −266154. −0.321176 −0.160588 0.987022i $$-0.551339\pi$$
−0.160588 + 0.987022i $$0.551339\pi$$
$$234$$ 0 0
$$235$$ 1.71074e6 2.02076
$$236$$ −340576. −0.398047
$$237$$ −181440. −0.209827
$$238$$ 570064. 0.652351
$$239$$ −254614. −0.288328 −0.144164 0.989554i $$-0.546049\pi$$
−0.144164 + 0.989554i $$0.546049\pi$$
$$240$$ 69632.0 0.0780334
$$241$$ 313600. 0.347803 0.173902 0.984763i $$-0.444363\pi$$
0.173902 + 0.984763i $$0.444363\pi$$
$$242$$ −35804.0 −0.0393001
$$243$$ 647404. 0.703331
$$244$$ −87328.0 −0.0939029
$$245$$ −685644. −0.729766
$$246$$ 108160. 0.113954
$$247$$ 0 0
$$248$$ −91520.0 −0.0944902
$$249$$ −498184. −0.509204
$$250$$ −442272. −0.447548
$$251$$ 1.07127e6 1.07328 0.536641 0.843811i $$-0.319693\pi$$
0.536641 + 0.843811i $$0.319693\pi$$
$$252$$ 297824. 0.295433
$$253$$ 820560. 0.805952
$$254$$ 1.24173e6 1.20765
$$255$$ −472736. −0.455269
$$256$$ 65536.0 0.0625000
$$257$$ 188382. 0.177913 0.0889563 0.996036i $$-0.471647\pi$$
0.0889563 + 0.996036i $$0.471647\pi$$
$$258$$ 270656. 0.253144
$$259$$ 725864. 0.672366
$$260$$ 0 0
$$261$$ 383630. 0.348587
$$262$$ 1.24149e6 1.11735
$$263$$ −1.48678e6 −1.32543 −0.662714 0.748873i $$-0.730596\pi$$
−0.662714 + 0.748873i $$0.730596\pi$$
$$264$$ −99840.0 −0.0881647
$$265$$ 2.59855e6 2.27309
$$266$$ 352272. 0.305263
$$267$$ 74976.0 0.0643642
$$268$$ 712672. 0.606112
$$269$$ 743990. 0.626883 0.313441 0.949608i $$-0.398518\pi$$
0.313441 + 0.949608i $$0.398518\pi$$
$$270$$ −511360. −0.426891
$$271$$ 455590. 0.376835 0.188417 0.982089i $$-0.439664\pi$$
0.188417 + 0.982089i $$0.439664\pi$$
$$272$$ −444928. −0.364643
$$273$$ 0 0
$$274$$ −1.12413e6 −0.904564
$$275$$ −584610. −0.466159
$$276$$ −134656. −0.106403
$$277$$ −460198. −0.360367 −0.180184 0.983633i $$-0.557669\pi$$
−0.180184 + 0.983633i $$0.557669\pi$$
$$278$$ 1.45528e6 1.12937
$$279$$ 324610. 0.249661
$$280$$ −356864. −0.272024
$$281$$ 49240.0 0.0372008 0.0186004 0.999827i $$-0.494079\pi$$
0.0186004 + 0.999827i $$0.494079\pi$$
$$282$$ 402528. 0.301421
$$283$$ 544196. 0.403914 0.201957 0.979394i $$-0.435270\pi$$
0.201957 + 0.979394i $$0.435270\pi$$
$$284$$ −284640. −0.209411
$$285$$ −292128. −0.213040
$$286$$ 0 0
$$287$$ −554320. −0.397243
$$288$$ −232448. −0.165137
$$289$$ 1.60079e6 1.12743
$$290$$ −459680. −0.320967
$$291$$ −485952. −0.336404
$$292$$ −497024. −0.341130
$$293$$ 1.02504e6 0.697542 0.348771 0.937208i $$-0.386599\pi$$
0.348771 + 0.937208i $$0.386599\pi$$
$$294$$ −161328. −0.108853
$$295$$ −1.44745e6 −0.968385
$$296$$ −566528. −0.375831
$$297$$ 733200. 0.482316
$$298$$ −1.09682e6 −0.715473
$$299$$ 0 0
$$300$$ 95936.0 0.0615430
$$301$$ −1.38711e6 −0.882461
$$302$$ 1.37612e6 0.868238
$$303$$ 56872.0 0.0355870
$$304$$ −274944. −0.170632
$$305$$ −371144. −0.228451
$$306$$ 1.57810e6 0.963456
$$307$$ −1.57766e6 −0.955362 −0.477681 0.878533i $$-0.658523\pi$$
−0.477681 + 0.878533i $$0.658523\pi$$
$$308$$ 511680. 0.307342
$$309$$ 251104. 0.149609
$$310$$ −388960. −0.229880
$$311$$ 330088. 0.193521 0.0967606 0.995308i $$-0.469152\pi$$
0.0967606 + 0.995308i $$0.469152\pi$$
$$312$$ 0 0
$$313$$ −1.78677e6 −1.03088 −0.515438 0.856927i $$-0.672371\pi$$
−0.515438 + 0.856927i $$0.672371\pi$$
$$314$$ 82072.0 0.0469754
$$315$$ 1.26575e6 0.718741
$$316$$ −725760. −0.408861
$$317$$ −182148. −0.101807 −0.0509033 0.998704i $$-0.516210\pi$$
−0.0509033 + 0.998704i $$0.516210\pi$$
$$318$$ 611424. 0.339059
$$319$$ 659100. 0.362639
$$320$$ 278528. 0.152053
$$321$$ −317008. −0.171715
$$322$$ 690112. 0.370920
$$323$$ 1.86661e6 0.995515
$$324$$ 762256. 0.403402
$$325$$ 0 0
$$326$$ 146504. 0.0763494
$$327$$ −872336. −0.451143
$$328$$ 432640. 0.222046
$$329$$ −2.06296e6 −1.05075
$$330$$ −424320. −0.214491
$$331$$ 216230. 0.108479 0.0542395 0.998528i $$-0.482727\pi$$
0.0542395 + 0.998528i $$0.482727\pi$$
$$332$$ −1.99274e6 −0.992212
$$333$$ 2.00940e6 0.993017
$$334$$ −1.07777e6 −0.528639
$$335$$ 3.02886e6 1.47457
$$336$$ −83968.0 −0.0405757
$$337$$ 2.05314e6 0.984791 0.492396 0.870371i $$-0.336121\pi$$
0.492396 + 0.870371i $$0.336121\pi$$
$$338$$ 0 0
$$339$$ 176936. 0.0836213
$$340$$ −1.89094e6 −0.887118
$$341$$ 557700. 0.259726
$$342$$ 975192. 0.450843
$$343$$ 2.20498e6 1.01197
$$344$$ 1.08262e6 0.493266
$$345$$ −572288. −0.258861
$$346$$ −1.13062e6 −0.507721
$$347$$ 4.28819e6 1.91183 0.955917 0.293637i $$-0.0948658\pi$$
0.955917 + 0.293637i $$0.0948658\pi$$
$$348$$ −108160. −0.0478761
$$349$$ −3.55152e6 −1.56081 −0.780405 0.625274i $$-0.784987\pi$$
−0.780405 + 0.625274i $$0.784987\pi$$
$$350$$ −491672. −0.214539
$$351$$ 0 0
$$352$$ −399360. −0.171794
$$353$$ 2.08678e6 0.891335 0.445667 0.895199i $$-0.352966\pi$$
0.445667 + 0.895199i $$0.352966\pi$$
$$354$$ −340576. −0.144446
$$355$$ −1.20972e6 −0.509465
$$356$$ 299904. 0.125417
$$357$$ 570064. 0.236730
$$358$$ −1.33512e6 −0.550570
$$359$$ 500654. 0.205023 0.102511 0.994732i $$-0.467312\pi$$
0.102511 + 0.994732i $$0.467312\pi$$
$$360$$ −987904. −0.401752
$$361$$ −1.32262e6 −0.534156
$$362$$ 1.83975e6 0.737884
$$363$$ −35804.0 −0.0142615
$$364$$ 0 0
$$365$$ −2.11235e6 −0.829916
$$366$$ −87328.0 −0.0340762
$$367$$ −1.28027e6 −0.496178 −0.248089 0.968737i $$-0.579802\pi$$
−0.248089 + 0.968737i $$0.579802\pi$$
$$368$$ −538624. −0.207332
$$369$$ −1.53452e6 −0.586687
$$370$$ −2.40774e6 −0.914336
$$371$$ −3.13355e6 −1.18196
$$372$$ −91520.0 −0.0342893
$$373$$ −405666. −0.150972 −0.0754860 0.997147i $$-0.524051\pi$$
−0.0754860 + 0.997147i $$0.524051\pi$$
$$374$$ 2.71128e6 1.00229
$$375$$ −442272. −0.162409
$$376$$ 1.61011e6 0.587336
$$377$$ 0 0
$$378$$ 616640. 0.221974
$$379$$ 4.66217e6 1.66721 0.833604 0.552363i $$-0.186274\pi$$
0.833604 + 0.552363i $$0.186274\pi$$
$$380$$ −1.16851e6 −0.415121
$$381$$ 1.24173e6 0.438242
$$382$$ −3.66835e6 −1.28621
$$383$$ −4.35473e6 −1.51692 −0.758462 0.651717i $$-0.774049\pi$$
−0.758462 + 0.651717i $$0.774049\pi$$
$$384$$ 65536.0 0.0226805
$$385$$ 2.17464e6 0.747714
$$386$$ 2.55622e6 0.873234
$$387$$ −3.83993e6 −1.30331
$$388$$ −1.94381e6 −0.655502
$$389$$ −786990. −0.263691 −0.131845 0.991270i $$-0.542090\pi$$
−0.131845 + 0.991270i $$0.542090\pi$$
$$390$$ 0 0
$$391$$ 3.65675e6 1.20963
$$392$$ −645312. −0.212107
$$393$$ 1.24149e6 0.405472
$$394$$ 1.43317e6 0.465111
$$395$$ −3.08448e6 −0.994693
$$396$$ 1.41648e6 0.453913
$$397$$ 3.97023e6 1.26427 0.632134 0.774859i $$-0.282179\pi$$
0.632134 + 0.774859i $$0.282179\pi$$
$$398$$ −1.48176e6 −0.468889
$$399$$ 352272. 0.110776
$$400$$ 383744. 0.119920
$$401$$ 344640. 0.107030 0.0535149 0.998567i $$-0.482958\pi$$
0.0535149 + 0.998567i $$0.482958\pi$$
$$402$$ 712672. 0.219950
$$403$$ 0 0
$$404$$ 227488. 0.0693434
$$405$$ 3.23959e6 0.981414
$$406$$ 554320. 0.166896
$$407$$ 3.45228e6 1.03305
$$408$$ −444928. −0.132324
$$409$$ −2.55466e6 −0.755137 −0.377568 0.925982i $$-0.623240\pi$$
−0.377568 + 0.925982i $$0.623240\pi$$
$$410$$ 1.83872e6 0.540202
$$411$$ −1.12413e6 −0.328255
$$412$$ 1.00442e6 0.291521
$$413$$ 1.74545e6 0.503539
$$414$$ 1.91043e6 0.547811
$$415$$ −8.46913e6 −2.41390
$$416$$ 0 0
$$417$$ 1.45528e6 0.409833
$$418$$ 1.67544e6 0.469017
$$419$$ −2.51894e6 −0.700943 −0.350472 0.936573i $$-0.613979\pi$$
−0.350472 + 0.936573i $$0.613979\pi$$
$$420$$ −356864. −0.0987142
$$421$$ −4.83670e6 −1.32998 −0.664988 0.746854i $$-0.731563\pi$$
−0.664988 + 0.746854i $$0.731563\pi$$
$$422$$ −708912. −0.193781
$$423$$ −5.71087e6 −1.55185
$$424$$ 2.44570e6 0.660675
$$425$$ −2.60526e6 −0.699647
$$426$$ −284640. −0.0759927
$$427$$ 447556. 0.118789
$$428$$ −1.26803e6 −0.334596
$$429$$ 0 0
$$430$$ 4.60115e6 1.20004
$$431$$ −219110. −0.0568158 −0.0284079 0.999596i $$-0.509044\pi$$
−0.0284079 + 0.999596i $$0.509044\pi$$
$$432$$ −481280. −0.124076
$$433$$ 3.03477e6 0.777867 0.388934 0.921266i $$-0.372844\pi$$
0.388934 + 0.921266i $$0.372844\pi$$
$$434$$ 469040. 0.119532
$$435$$ −459680. −0.116475
$$436$$ −3.48934e6 −0.879078
$$437$$ 2.25970e6 0.566039
$$438$$ −497024. −0.123792
$$439$$ −4.16940e6 −1.03255 −0.516276 0.856422i $$-0.672682\pi$$
−0.516276 + 0.856422i $$0.672682\pi$$
$$440$$ −1.69728e6 −0.417948
$$441$$ 2.28884e6 0.560427
$$442$$ 0 0
$$443$$ −6.30548e6 −1.52654 −0.763271 0.646079i $$-0.776408\pi$$
−0.763271 + 0.646079i $$0.776408\pi$$
$$444$$ −566528. −0.136384
$$445$$ 1.27459e6 0.305120
$$446$$ −4.45186e6 −1.05975
$$447$$ −1.09682e6 −0.259636
$$448$$ −335872. −0.0790640
$$449$$ −7.41586e6 −1.73598 −0.867991 0.496579i $$-0.834589\pi$$
−0.867991 + 0.496579i $$0.834589\pi$$
$$450$$ −1.36109e6 −0.316852
$$451$$ −2.63640e6 −0.610337
$$452$$ 707744. 0.162941
$$453$$ 1.37612e6 0.315073
$$454$$ 5.56633e6 1.26745
$$455$$ 0 0
$$456$$ −274944. −0.0619202
$$457$$ −4.71529e6 −1.05613 −0.528065 0.849204i $$-0.677082\pi$$
−0.528065 + 0.849204i $$0.677082\pi$$
$$458$$ −3.63918e6 −0.810663
$$459$$ 3.26744e6 0.723896
$$460$$ −2.28915e6 −0.504406
$$461$$ 3.34566e6 0.733212 0.366606 0.930376i $$-0.380520\pi$$
0.366606 + 0.930376i $$0.380520\pi$$
$$462$$ 511680. 0.111530
$$463$$ −1.65791e6 −0.359426 −0.179713 0.983719i $$-0.557517\pi$$
−0.179713 + 0.983719i $$0.557517\pi$$
$$464$$ −432640. −0.0932893
$$465$$ −388960. −0.0834205
$$466$$ −1.06462e6 −0.227106
$$467$$ −823668. −0.174767 −0.0873836 0.996175i $$-0.527851\pi$$
−0.0873836 + 0.996175i $$0.527851\pi$$
$$468$$ 0 0
$$469$$ −3.65244e6 −0.766746
$$470$$ 6.84298e6 1.42890
$$471$$ 82072.0 0.0170468
$$472$$ −1.36230e6 −0.281462
$$473$$ −6.59724e6 −1.35584
$$474$$ −725760. −0.148370
$$475$$ −1.60993e6 −0.327395
$$476$$ 2.28026e6 0.461282
$$477$$ −8.67458e6 −1.74563
$$478$$ −1.01846e6 −0.203879
$$479$$ −3.59011e6 −0.714938 −0.357469 0.933925i $$-0.616360\pi$$
−0.357469 + 0.933925i $$0.616360\pi$$
$$480$$ 278528. 0.0551780
$$481$$ 0 0
$$482$$ 1.25440e6 0.245934
$$483$$ 690112. 0.134602
$$484$$ −143216. −0.0277893
$$485$$ −8.26118e6 −1.59473
$$486$$ 2.58962e6 0.497330
$$487$$ −9.67688e6 −1.84890 −0.924449 0.381306i $$-0.875474\pi$$
−0.924449 + 0.381306i $$0.875474\pi$$
$$488$$ −349312. −0.0663994
$$489$$ 146504. 0.0277062
$$490$$ −2.74258e6 −0.516022
$$491$$ −3.45633e6 −0.647011 −0.323506 0.946226i $$-0.604861\pi$$
−0.323506 + 0.946226i $$0.604861\pi$$
$$492$$ 432640. 0.0805775
$$493$$ 2.93722e6 0.544276
$$494$$ 0 0
$$495$$ 6.02004e6 1.10430
$$496$$ −366080. −0.0668147
$$497$$ 1.45878e6 0.264910
$$498$$ −1.99274e6 −0.360061
$$499$$ 2.09109e6 0.375942 0.187971 0.982175i $$-0.439809\pi$$
0.187971 + 0.982175i $$0.439809\pi$$
$$500$$ −1.76909e6 −0.316464
$$501$$ −1.07777e6 −0.191836
$$502$$ 4.28507e6 0.758925
$$503$$ 5.58626e6 0.984468 0.492234 0.870463i $$-0.336180\pi$$
0.492234 + 0.870463i $$0.336180\pi$$
$$504$$ 1.19130e6 0.208902
$$505$$ 966824. 0.168702
$$506$$ 3.28224e6 0.569894
$$507$$ 0 0
$$508$$ 4.96691e6 0.853940
$$509$$ 4.15504e6 0.710854 0.355427 0.934704i $$-0.384335\pi$$
0.355427 + 0.934704i $$0.384335\pi$$
$$510$$ −1.89094e6 −0.321924
$$511$$ 2.54725e6 0.431538
$$512$$ 262144. 0.0441942
$$513$$ 2.01912e6 0.338742
$$514$$ 753528. 0.125803
$$515$$ 4.26877e6 0.709226
$$516$$ 1.08262e6 0.179000
$$517$$ −9.81162e6 −1.61441
$$518$$ 2.90346e6 0.475435
$$519$$ −1.13062e6 −0.184245
$$520$$ 0 0
$$521$$ −9.84416e6 −1.58886 −0.794428 0.607359i $$-0.792229\pi$$
−0.794428 + 0.607359i $$0.792229\pi$$
$$522$$ 1.53452e6 0.246488
$$523$$ 481324. 0.0769455 0.0384728 0.999260i $$-0.487751\pi$$
0.0384728 + 0.999260i $$0.487751\pi$$
$$524$$ 4.96595e6 0.790086
$$525$$ −491672. −0.0778533
$$526$$ −5.94710e6 −0.937219
$$527$$ 2.48534e6 0.389816
$$528$$ −399360. −0.0623419
$$529$$ −2.00953e6 −0.312216
$$530$$ 1.03942e7 1.60732
$$531$$ 4.83192e6 0.743676
$$532$$ 1.40909e6 0.215853
$$533$$ 0 0
$$534$$ 299904. 0.0455123
$$535$$ −5.38914e6 −0.814019
$$536$$ 2.85069e6 0.428586
$$537$$ −1.33512e6 −0.199795
$$538$$ 2.97596e6 0.443273
$$539$$ 3.93237e6 0.583019
$$540$$ −2.04544e6 −0.301858
$$541$$ −263980. −0.0387773 −0.0193887 0.999812i $$-0.506172\pi$$
−0.0193887 + 0.999812i $$0.506172\pi$$
$$542$$ 1.82236e6 0.266462
$$543$$ 1.83975e6 0.267769
$$544$$ −1.77971e6 −0.257841
$$545$$ −1.48297e7 −2.13866
$$546$$ 0 0
$$547$$ 2.80023e6 0.400152 0.200076 0.979780i $$-0.435881\pi$$
0.200076 + 0.979780i $$0.435881\pi$$
$$548$$ −4.49651e6 −0.639623
$$549$$ 1.23897e6 0.175440
$$550$$ −2.33844e6 −0.329625
$$551$$ 1.81506e6 0.254690
$$552$$ −538624. −0.0752381
$$553$$ 3.71952e6 0.517219
$$554$$ −1.84079e6 −0.254818
$$555$$ −2.40774e6 −0.331801
$$556$$ 5.82112e6 0.798582
$$557$$ 2.70983e6 0.370087 0.185043 0.982730i $$-0.440757\pi$$
0.185043 + 0.982730i $$0.440757\pi$$
$$558$$ 1.29844e6 0.176537
$$559$$ 0 0
$$560$$ −1.42746e6 −0.192350
$$561$$ 2.71128e6 0.363720
$$562$$ 196960. 0.0263049
$$563$$ 1.14870e7 1.52733 0.763667 0.645610i $$-0.223397\pi$$
0.763667 + 0.645610i $$0.223397\pi$$
$$564$$ 1.61011e6 0.213137
$$565$$ 3.00791e6 0.396409
$$566$$ 2.17678e6 0.285611
$$567$$ −3.90656e6 −0.510314
$$568$$ −1.13856e6 −0.148076
$$569$$ −7.85065e6 −1.01654 −0.508271 0.861197i $$-0.669715\pi$$
−0.508271 + 0.861197i $$0.669715\pi$$
$$570$$ −1.16851e6 −0.150642
$$571$$ 6.34071e6 0.813856 0.406928 0.913460i $$-0.366600\pi$$
0.406928 + 0.913460i $$0.366600\pi$$
$$572$$ 0 0
$$573$$ −3.66835e6 −0.466750
$$574$$ −2.21728e6 −0.280893
$$575$$ −3.15390e6 −0.397812
$$576$$ −929792. −0.116770
$$577$$ 7.20867e6 0.901396 0.450698 0.892676i $$-0.351175\pi$$
0.450698 + 0.892676i $$0.351175\pi$$
$$578$$ 6.40315e6 0.797212
$$579$$ 2.55622e6 0.316886
$$580$$ −1.83872e6 −0.226958
$$581$$ 1.02128e7 1.25517
$$582$$ −1.94381e6 −0.237873
$$583$$ −1.49035e7 −1.81600
$$584$$ −1.98810e6 −0.241216
$$585$$ 0 0
$$586$$ 4.10014e6 0.493236
$$587$$ 2.48138e6 0.297234 0.148617 0.988895i $$-0.452518\pi$$
0.148617 + 0.988895i $$0.452518\pi$$
$$588$$ −645312. −0.0769709
$$589$$ 1.53582e6 0.182411
$$590$$ −5.78979e6 −0.684751
$$591$$ 1.43317e6 0.168783
$$592$$ −2.26611e6 −0.265752
$$593$$ −1.38811e7 −1.62102 −0.810508 0.585728i $$-0.800809\pi$$
−0.810508 + 0.585728i $$0.800809\pi$$
$$594$$ 2.93280e6 0.341049
$$595$$ 9.69109e6 1.12223
$$596$$ −4.38726e6 −0.505916
$$597$$ −1.48176e6 −0.170154
$$598$$ 0 0
$$599$$ 3.85356e6 0.438829 0.219414 0.975632i $$-0.429585\pi$$
0.219414 + 0.975632i $$0.429585\pi$$
$$600$$ 383744. 0.0435175
$$601$$ 1.32728e6 0.149892 0.0749458 0.997188i $$-0.476122\pi$$
0.0749458 + 0.997188i $$0.476122\pi$$
$$602$$ −5.54845e6 −0.623994
$$603$$ −1.01110e7 −1.13241
$$604$$ 5.50448e6 0.613937
$$605$$ −608668. −0.0676071
$$606$$ 227488. 0.0251638
$$607$$ 9.73197e6 1.07208 0.536042 0.844191i $$-0.319919\pi$$
0.536042 + 0.844191i $$0.319919\pi$$
$$608$$ −1.09978e6 −0.120655
$$609$$ 554320. 0.0605644
$$610$$ −1.48458e6 −0.161539
$$611$$ 0 0
$$612$$ 6.31242e6 0.681267
$$613$$ −1.40465e7 −1.50979 −0.754894 0.655846i $$-0.772312\pi$$
−0.754894 + 0.655846i $$0.772312\pi$$
$$614$$ −6.31065e6 −0.675543
$$615$$ 1.83872e6 0.196032
$$616$$ 2.04672e6 0.217323
$$617$$ −3.72561e6 −0.393989 −0.196995 0.980405i $$-0.563118\pi$$
−0.196995 + 0.980405i $$0.563118\pi$$
$$618$$ 1.00442e6 0.105789
$$619$$ 8.96911e6 0.940855 0.470428 0.882439i $$-0.344100\pi$$
0.470428 + 0.882439i $$0.344100\pi$$
$$620$$ −1.55584e6 −0.162550
$$621$$ 3.95552e6 0.411599
$$622$$ 1.32035e6 0.136840
$$623$$ −1.53701e6 −0.158656
$$624$$ 0 0
$$625$$ −1.22030e7 −1.24959
$$626$$ −7.14706e6 −0.728940
$$627$$ 1.67544e6 0.170200
$$628$$ 328288. 0.0332167
$$629$$ 1.53848e7 1.55047
$$630$$ 5.06301e6 0.508226
$$631$$ 1.72189e7 1.72160 0.860800 0.508943i $$-0.169964\pi$$
0.860800 + 0.508943i $$0.169964\pi$$
$$632$$ −2.90304e6 −0.289108
$$633$$ −708912. −0.0703207
$$634$$ −728592. −0.0719882
$$635$$ 2.11094e7 2.07750
$$636$$ 2.44570e6 0.239751
$$637$$ 0 0
$$638$$ 2.63640e6 0.256425
$$639$$ 4.03833e6 0.391246
$$640$$ 1.11411e6 0.107517
$$641$$ −8.51692e6 −0.818724 −0.409362 0.912372i $$-0.634249\pi$$
−0.409362 + 0.912372i $$0.634249\pi$$
$$642$$ −1.26803e6 −0.121421
$$643$$ 8.14145e6 0.776559 0.388280 0.921542i $$-0.373069\pi$$
0.388280 + 0.921542i $$0.373069\pi$$
$$644$$ 2.76045e6 0.262280
$$645$$ 4.60115e6 0.435479
$$646$$ 7.46645e6 0.703935
$$647$$ 2.39391e6 0.224826 0.112413 0.993662i $$-0.464142\pi$$
0.112413 + 0.993662i $$0.464142\pi$$
$$648$$ 3.04902e6 0.285248
$$649$$ 8.30154e6 0.773654
$$650$$ 0 0
$$651$$ 469040. 0.0433768
$$652$$ 586016. 0.0539872
$$653$$ 1.17900e7 1.08201 0.541003 0.841020i $$-0.318045\pi$$
0.541003 + 0.841020i $$0.318045\pi$$
$$654$$ −3.48934e6 −0.319006
$$655$$ 2.11053e7 1.92215
$$656$$ 1.73056e6 0.157010
$$657$$ 7.05153e6 0.637338
$$658$$ −8.25182e6 −0.742994
$$659$$ 4.84562e6 0.434646 0.217323 0.976100i $$-0.430267\pi$$
0.217323 + 0.976100i $$0.430267\pi$$
$$660$$ −1.69728e6 −0.151668
$$661$$ −1.14461e7 −1.01895 −0.509476 0.860485i $$-0.670161\pi$$
−0.509476 + 0.860485i $$0.670161\pi$$
$$662$$ 864920. 0.0767063
$$663$$ 0 0
$$664$$ −7.97094e6 −0.701600
$$665$$ 5.98862e6 0.525137
$$666$$ 8.03762e6 0.702169
$$667$$ 3.55576e6 0.309470
$$668$$ −4.31107e6 −0.373804
$$669$$ −4.45186e6 −0.384571
$$670$$ 1.21154e7 1.04268
$$671$$ 2.12862e6 0.182512
$$672$$ −335872. −0.0286913
$$673$$ 5.34001e6 0.454469 0.227234 0.973840i $$-0.427032\pi$$
0.227234 + 0.973840i $$0.427032\pi$$
$$674$$ 8.21257e6 0.696353
$$675$$ −2.81812e6 −0.238067
$$676$$ 0 0
$$677$$ −7.06132e6 −0.592126 −0.296063 0.955168i $$-0.595674\pi$$
−0.296063 + 0.955168i $$0.595674\pi$$
$$678$$ 707744. 0.0591292
$$679$$ 9.96202e6 0.829226
$$680$$ −7.56378e6 −0.627287
$$681$$ 5.56633e6 0.459940
$$682$$ 2.23080e6 0.183654
$$683$$ 3.50035e6 0.287117 0.143559 0.989642i $$-0.454145\pi$$
0.143559 + 0.989642i $$0.454145\pi$$
$$684$$ 3.90077e6 0.318794
$$685$$ −1.91102e7 −1.55610
$$686$$ 8.81992e6 0.715574
$$687$$ −3.63918e6 −0.294179
$$688$$ 4.33050e6 0.348792
$$689$$ 0 0
$$690$$ −2.28915e6 −0.183042
$$691$$ 302510. 0.0241015 0.0120508 0.999927i $$-0.496164\pi$$
0.0120508 + 0.999927i $$0.496164\pi$$
$$692$$ −4.52246e6 −0.359013
$$693$$ −7.25946e6 −0.574211
$$694$$ 1.71528e7 1.35187
$$695$$ 2.47398e7 1.94282
$$696$$ −432640. −0.0338535
$$697$$ −1.17489e7 −0.916040
$$698$$ −1.42061e7 −1.10366
$$699$$ −1.06462e6 −0.0824138
$$700$$ −1.96669e6 −0.151702
$$701$$ 1.03212e7 0.793294 0.396647 0.917971i $$-0.370174\pi$$
0.396647 + 0.917971i $$0.370174\pi$$
$$702$$ 0 0
$$703$$ 9.50705e6 0.725533
$$704$$ −1.59744e6 −0.121477
$$705$$ 6.84298e6 0.518528
$$706$$ 8.34714e6 0.630269
$$707$$ −1.16588e6 −0.0877211
$$708$$ −1.36230e6 −0.102139
$$709$$ −5.27524e6 −0.394118 −0.197059 0.980392i $$-0.563139\pi$$
−0.197059 + 0.980392i $$0.563139\pi$$
$$710$$ −4.83888e6 −0.360246
$$711$$ 1.02967e7 0.763880
$$712$$ 1.19962e6 0.0886834
$$713$$ 3.00872e6 0.221645
$$714$$ 2.28026e6 0.167393
$$715$$ 0 0
$$716$$ −5.34048e6 −0.389312
$$717$$ −1.01846e6 −0.0739851
$$718$$ 2.00262e6 0.144973
$$719$$ 5.02216e6 0.362300 0.181150 0.983455i $$-0.442018\pi$$
0.181150 + 0.983455i $$0.442018\pi$$
$$720$$ −3.95162e6 −0.284082
$$721$$ −5.14763e6 −0.368782
$$722$$ −5.29049e6 −0.377705
$$723$$ 1.25440e6 0.0892463
$$724$$ 7.35901e6 0.521762
$$725$$ −2.53331e6 −0.178996
$$726$$ −143216. −0.0100844
$$727$$ −8.80441e6 −0.617823 −0.308912 0.951091i $$-0.599965\pi$$
−0.308912 + 0.951091i $$0.599965\pi$$
$$728$$ 0 0
$$729$$ −8.98715e6 −0.626330
$$730$$ −8.44941e6 −0.586839
$$731$$ −2.94000e7 −2.03495
$$732$$ −349312. −0.0240955
$$733$$ 3.05052e6 0.209708 0.104854 0.994488i $$-0.466563\pi$$
0.104854 + 0.994488i $$0.466563\pi$$
$$734$$ −5.12109e6 −0.350850
$$735$$ −2.74258e6 −0.187258
$$736$$ −2.15450e6 −0.146606
$$737$$ −1.73714e7 −1.17806
$$738$$ −6.13808e6 −0.414851
$$739$$ −7.62605e6 −0.513675 −0.256837 0.966455i $$-0.582680\pi$$
−0.256837 + 0.966455i $$0.582680\pi$$
$$740$$ −9.63098e6 −0.646533
$$741$$ 0 0
$$742$$ −1.25342e7 −0.835770
$$743$$ −2.18236e7 −1.45029 −0.725146 0.688595i $$-0.758228\pi$$
−0.725146 + 0.688595i $$0.758228\pi$$
$$744$$ −366080. −0.0242462
$$745$$ −1.86459e7 −1.23081
$$746$$ −1.62266e6 −0.106753
$$747$$ 2.82719e7 1.85376
$$748$$ 1.08451e7 0.708729
$$749$$ 6.49866e6 0.423272
$$750$$ −1.76909e6 −0.114841
$$751$$ −1.69030e7 −1.09361 −0.546807 0.837259i $$-0.684157\pi$$
−0.546807 + 0.837259i $$0.684157\pi$$
$$752$$ 6.44045e6 0.415309
$$753$$ 4.28507e6 0.275404
$$754$$ 0 0
$$755$$ 2.33940e7 1.49361
$$756$$ 2.46656e6 0.156959
$$757$$ −8.90252e6 −0.564642 −0.282321 0.959320i $$-0.591104\pi$$
−0.282321 + 0.959320i $$0.591104\pi$$
$$758$$ 1.86487e7 1.17889
$$759$$ 3.28224e6 0.206807
$$760$$ −4.67405e6 −0.293535
$$761$$ 6.98052e6 0.436944 0.218472 0.975843i $$-0.429893\pi$$
0.218472 + 0.975843i $$0.429893\pi$$
$$762$$ 4.96691e6 0.309884
$$763$$ 1.78829e7 1.11206
$$764$$ −1.46734e7 −0.909489
$$765$$ 2.68278e7 1.65741
$$766$$ −1.74189e7 −1.07263
$$767$$ 0 0
$$768$$ 262144. 0.0160375
$$769$$ −2.67789e7 −1.63296 −0.816481 0.577372i $$-0.804078\pi$$
−0.816481 + 0.577372i $$0.804078\pi$$
$$770$$ 8.69856e6 0.528714
$$771$$ 753528. 0.0456524
$$772$$ 1.02249e7 0.617470
$$773$$ 710244. 0.0427522 0.0213761 0.999772i $$-0.493195\pi$$
0.0213761 + 0.999772i $$0.493195\pi$$
$$774$$ −1.53597e7 −0.921576
$$775$$ −2.14357e6 −0.128199
$$776$$ −7.77523e6 −0.463510
$$777$$ 2.90346e6 0.172529
$$778$$ −3.14796e6 −0.186458
$$779$$ −7.26024e6 −0.428654
$$780$$ 0 0
$$781$$ 6.93810e6 0.407017
$$782$$ 1.46270e7 0.855340
$$783$$ 3.17720e6 0.185200
$$784$$ −2.58125e6 −0.149982
$$785$$ 1.39522e6 0.0808109
$$786$$ 4.96595e6 0.286712
$$787$$ 5.18538e6 0.298431 0.149215 0.988805i $$-0.452325\pi$$
0.149215 + 0.988805i $$0.452325\pi$$
$$788$$ 5.73267e6 0.328883
$$789$$ −5.94710e6 −0.340105
$$790$$ −1.23379e7 −0.703354
$$791$$ −3.62719e6 −0.206124
$$792$$ 5.66592e6 0.320965
$$793$$ 0 0
$$794$$ 1.58809e7 0.893973
$$795$$ 1.03942e7 0.583275
$$796$$ −5.92704e6 −0.331555
$$797$$ 2.93628e6 0.163739 0.0818695 0.996643i $$-0.473911\pi$$
0.0818695 + 0.996643i $$0.473911\pi$$
$$798$$ 1.40909e6 0.0783305
$$799$$ −4.37246e7 −2.42303
$$800$$ 1.53498e6 0.0847962
$$801$$ −4.25489e6 −0.234319
$$802$$ 1.37856e6 0.0756815
$$803$$ 1.21150e7 0.663030
$$804$$ 2.85069e6 0.155528
$$805$$ 1.17319e7 0.638085
$$806$$ 0 0
$$807$$ 2.97596e6 0.160858
$$808$$ 909952. 0.0490332
$$809$$ −1.25821e7 −0.675900 −0.337950 0.941164i $$-0.609733\pi$$
−0.337950 + 0.941164i $$0.609733\pi$$
$$810$$ 1.29584e7 0.693964
$$811$$ −2.08048e7 −1.11074 −0.555369 0.831604i $$-0.687423\pi$$
−0.555369 + 0.831604i $$0.687423\pi$$
$$812$$ 2.21728e6 0.118013
$$813$$ 1.82236e6 0.0966958
$$814$$ 1.38091e7 0.730474
$$815$$ 2.49057e6 0.131342
$$816$$ −1.77971e6 −0.0935674
$$817$$ −1.81678e7 −0.952241
$$818$$ −1.02187e7 −0.533962
$$819$$ 0 0
$$820$$ 7.35488e6 0.381980
$$821$$ 2.11600e7 1.09562 0.547808 0.836604i $$-0.315463\pi$$
0.547808 + 0.836604i $$0.315463\pi$$
$$822$$ −4.49651e6 −0.232111
$$823$$ 2.20857e7 1.13661 0.568306 0.822817i $$-0.307599\pi$$
0.568306 + 0.822817i $$0.307599\pi$$
$$824$$ 4.01766e6 0.206137
$$825$$ −2.33844e6 −0.119617
$$826$$ 6.98181e6 0.356056
$$827$$ −1.69119e7 −0.859864 −0.429932 0.902861i $$-0.641462\pi$$
−0.429932 + 0.902861i $$0.641462\pi$$
$$828$$ 7.64173e6 0.387361
$$829$$ 2.34520e7 1.18521 0.592604 0.805494i $$-0.298100\pi$$
0.592604 + 0.805494i $$0.298100\pi$$
$$830$$ −3.38765e7 −1.70688
$$831$$ −1.84079e6 −0.0924703
$$832$$ 0 0
$$833$$ 1.75243e7 0.875038
$$834$$ 5.82112e6 0.289795
$$835$$ −1.83221e7 −0.909406
$$836$$ 6.70176e6 0.331645
$$837$$ 2.68840e6 0.132642
$$838$$ −1.00758e7 −0.495642
$$839$$ 725134. 0.0355642 0.0177821 0.999842i $$-0.494339\pi$$
0.0177821 + 0.999842i $$0.494339\pi$$
$$840$$ −1.42746e6 −0.0698015
$$841$$ −1.76550e7 −0.860754
$$842$$ −1.93468e7 −0.940435
$$843$$ 196960. 0.00954573
$$844$$ −2.83565e6 −0.137024
$$845$$ 0 0
$$846$$ −2.28435e7 −1.09733
$$847$$ 733982. 0.0351542
$$848$$ 9.78278e6 0.467168
$$849$$ 2.17678e6 0.103644
$$850$$ −1.04210e7 −0.494725
$$851$$ 1.86246e7 0.881583
$$852$$ −1.13856e6 −0.0537350
$$853$$ −1.03218e7 −0.485719 −0.242859 0.970062i $$-0.578085\pi$$
−0.242859 + 0.970062i $$0.578085\pi$$
$$854$$ 1.79022e6 0.0839968
$$855$$ 1.65783e7 0.775575
$$856$$ −5.07213e6 −0.236595
$$857$$ 3.71067e7 1.72584 0.862919 0.505343i $$-0.168634\pi$$
0.862919 + 0.505343i $$0.168634\pi$$
$$858$$ 0 0
$$859$$ 3.47061e7 1.60481 0.802405 0.596780i $$-0.203554\pi$$
0.802405 + 0.596780i $$0.203554\pi$$
$$860$$ 1.84046e7 0.848556
$$861$$ −2.21728e6 −0.101932
$$862$$ −876440. −0.0401748
$$863$$ −1.92294e7 −0.878897 −0.439448 0.898268i $$-0.644826\pi$$
−0.439448 + 0.898268i $$0.644826\pi$$
$$864$$ −1.92512e6 −0.0877351
$$865$$ −1.92205e7 −0.873421
$$866$$ 1.21391e7 0.550035
$$867$$ 6.40315e6 0.289298
$$868$$ 1.87616e6 0.0845222
$$869$$ 1.76904e7 0.794673
$$870$$ −1.83872e6 −0.0823602
$$871$$ 0 0
$$872$$ −1.39574e7 −0.621602
$$873$$ 2.75778e7 1.22468
$$874$$ 9.03878e6 0.400250
$$875$$ 9.06658e6 0.400335
$$876$$ −1.98810e6 −0.0875341
$$877$$ −3.84616e7 −1.68861 −0.844303 0.535866i $$-0.819985\pi$$
−0.844303 + 0.535866i $$0.819985\pi$$
$$878$$ −1.66776e7 −0.730125
$$879$$ 4.10014e6 0.178989
$$880$$ −6.78912e6 −0.295534
$$881$$ −3.29337e7 −1.42955 −0.714777 0.699353i $$-0.753472\pi$$
−0.714777 + 0.699353i $$0.753472\pi$$
$$882$$ 9.15536e6 0.396282
$$883$$ −2.67529e7 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$884$$ 0 0
$$885$$ −5.78979e6 −0.248488
$$886$$ −2.52219e7 −1.07943
$$887$$ −1.05284e7 −0.449317 −0.224659 0.974438i $$-0.572127\pi$$
−0.224659 + 0.974438i $$0.572127\pi$$
$$888$$ −2.26611e6 −0.0964382
$$889$$ −2.54554e7 −1.08025
$$890$$ 5.09837e6 0.215753
$$891$$ −1.85800e7 −0.784063
$$892$$ −1.78075e7 −0.749359
$$893$$ −2.70197e7 −1.13384
$$894$$ −4.38726e6 −0.183590
$$895$$ −2.26970e7 −0.947134
$$896$$ −1.34349e6 −0.0559067
$$897$$ 0 0
$$898$$ −2.96634e7 −1.22753
$$899$$ 2.41670e6 0.0997295
$$900$$ −5.44437e6 −0.224048
$$901$$ −6.64159e7 −2.72559
$$902$$ −1.05456e7 −0.431574
$$903$$ −5.54845e6 −0.226439
$$904$$ 2.83098e6 0.115217
$$905$$ 3.12758e7 1.26937
$$906$$ 5.50448e6 0.222790
$$907$$ 2.53255e7 1.02221 0.511104 0.859519i $$-0.329237\pi$$
0.511104 + 0.859519i $$0.329237\pi$$
$$908$$ 2.22653e7 0.896219
$$909$$ −3.22749e6 −0.129555
$$910$$ 0 0
$$911$$ 6.02395e6 0.240484 0.120242 0.992745i $$-0.461633\pi$$
0.120242 + 0.992745i $$0.461633\pi$$
$$912$$ −1.09978e6 −0.0437842
$$913$$ 4.85729e7 1.92849
$$914$$ −1.88612e7 −0.746797
$$915$$ −1.48458e6 −0.0586205
$$916$$ −1.45567e7 −0.573225
$$917$$ −2.54505e7 −0.999478
$$918$$ 1.30698e7 0.511871
$$919$$ 9.75228e6 0.380906 0.190453 0.981696i $$-0.439004\pi$$
0.190453 + 0.981696i $$0.439004\pi$$
$$920$$ −9.15661e6 −0.356669
$$921$$ −6.31065e6 −0.245146
$$922$$ 1.33826e7 0.518459
$$923$$ 0 0
$$924$$ 2.04672e6 0.0788639
$$925$$ −1.32691e7 −0.509904
$$926$$ −6.63166e6 −0.254153
$$927$$ −1.42502e7 −0.544653
$$928$$ −1.73056e6 −0.0659655
$$929$$ 2.30543e7 0.876419 0.438210 0.898873i $$-0.355613\pi$$
0.438210 + 0.898873i $$0.355613\pi$$
$$930$$ −1.55584e6 −0.0589872
$$931$$ 1.08291e7 0.409468
$$932$$ −4.25846e6 −0.160588
$$933$$ 1.32035e6 0.0496576
$$934$$ −3.29467e6 −0.123579
$$935$$ 4.60918e7 1.72423
$$936$$ 0 0
$$937$$ −4.03783e7 −1.50245 −0.751223 0.660049i $$-0.770536\pi$$
−0.751223 + 0.660049i $$0.770536\pi$$
$$938$$ −1.46098e7 −0.542171
$$939$$ −7.14706e6 −0.264523
$$940$$ 2.73719e7 1.01038
$$941$$ −4.02522e7 −1.48189 −0.740944 0.671567i $$-0.765622\pi$$
−0.740944 + 0.671567i $$0.765622\pi$$
$$942$$ 328288. 0.0120539
$$943$$ −1.42230e7 −0.520851
$$944$$ −5.44922e6 −0.199023
$$945$$ 1.04829e7 0.381857
$$946$$ −2.63890e7 −0.958726
$$947$$ 1.06360e7 0.385393 0.192697 0.981258i $$-0.438277\pi$$
0.192697 + 0.981258i $$0.438277\pi$$
$$948$$ −2.90304e6 −0.104914
$$949$$ 0 0
$$950$$ −6.43970e6 −0.231503
$$951$$ −728592. −0.0261236
$$952$$ 9.12102e6 0.326175
$$953$$ −90234.0 −0.00321838 −0.00160919 0.999999i $$-0.500512\pi$$
−0.00160919 + 0.999999i $$0.500512\pi$$
$$954$$ −3.46983e7 −1.23435
$$955$$ −6.23620e7 −2.21264
$$956$$ −4.07382e6 −0.144164
$$957$$ 2.63640e6 0.0930532
$$958$$ −1.43604e7 −0.505538
$$959$$ 2.30446e7 0.809139
$$960$$ 1.11411e6 0.0390167
$$961$$ −2.65843e7 −0.928573
$$962$$ 0 0
$$963$$ 1.79902e7 0.625130
$$964$$ 5.01760e6 0.173902
$$965$$ 4.34558e7 1.50221
$$966$$ 2.76045e6 0.0951780
$$967$$ 1.20331e7 0.413821 0.206910 0.978360i $$-0.433659\pi$$
0.206910 + 0.978360i $$0.433659\pi$$
$$968$$ −572864. −0.0196500
$$969$$ 7.46645e6 0.255449
$$970$$ −3.30447e7 −1.12765
$$971$$ 1.84061e7 0.626489 0.313245 0.949672i $$-0.398584\pi$$
0.313245 + 0.949672i $$0.398584\pi$$
$$972$$ 1.03585e7 0.351665
$$973$$ −2.98332e7 −1.01023
$$974$$ −3.87075e7 −1.30737
$$975$$ 0 0
$$976$$ −1.39725e6 −0.0469514
$$977$$ 4.66720e7 1.56430 0.782150 0.623090i $$-0.214123\pi$$
0.782150 + 0.623090i $$0.214123\pi$$
$$978$$ 586016. 0.0195913
$$979$$ −7.31016e6 −0.243764
$$980$$ −1.09703e7 −0.364883
$$981$$ 4.95051e7 1.64239
$$982$$ −1.38253e7 −0.457506
$$983$$ 1.98925e7 0.656608 0.328304 0.944572i $$-0.393523\pi$$
0.328304 + 0.944572i $$0.393523\pi$$
$$984$$ 1.73056e6 0.0569769
$$985$$ 2.43639e7 0.800121
$$986$$ 1.17489e7 0.384861
$$987$$ −8.25182e6 −0.269623
$$988$$ 0 0
$$989$$ −3.55913e7 −1.15705
$$990$$ 2.40802e7 0.780857
$$991$$ −4.58344e7 −1.48254 −0.741271 0.671206i $$-0.765777\pi$$
−0.741271 + 0.671206i $$0.765777\pi$$
$$992$$ −1.46432e6 −0.0472451
$$993$$ 864920. 0.0278357
$$994$$ 5.83512e6 0.187320
$$995$$ −2.51899e7 −0.806620
$$996$$ −7.97094e6 −0.254602
$$997$$ 2.51716e7 0.801999 0.400999 0.916078i $$-0.368663\pi$$
0.400999 + 0.916078i $$0.368663\pi$$
$$998$$ 8.36434e6 0.265831
$$999$$ 1.66418e7 0.527577
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 338.6.a.e.1.1 1
13.5 odd 4 26.6.b.b.25.1 2
13.8 odd 4 26.6.b.b.25.2 yes 2
13.12 even 2 338.6.a.b.1.1 1
39.5 even 4 234.6.b.a.181.2 2
39.8 even 4 234.6.b.a.181.1 2
52.31 even 4 208.6.f.a.129.1 2
52.47 even 4 208.6.f.a.129.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.b.b.25.1 2 13.5 odd 4
26.6.b.b.25.2 yes 2 13.8 odd 4
208.6.f.a.129.1 2 52.31 even 4
208.6.f.a.129.2 2 52.47 even 4
234.6.b.a.181.1 2 39.8 even 4
234.6.b.a.181.2 2 39.5 even 4
338.6.a.b.1.1 1 13.12 even 2
338.6.a.e.1.1 1 1.1 even 1 trivial