# Properties

 Label 448.6.a.l.1.1 Level $448$ Weight $6$ Character 448.1 Self dual yes Analytic conductor $71.852$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 448.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: $$71.8519512762$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 448.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+10.0000 q^{3} -84.0000 q^{5} -49.0000 q^{7} -143.000 q^{9} +O(q^{10})$$ $$q+10.0000 q^{3} -84.0000 q^{5} -49.0000 q^{7} -143.000 q^{9} -336.000 q^{11} -584.000 q^{13} -840.000 q^{15} -1458.00 q^{17} +470.000 q^{19} -490.000 q^{21} +4200.00 q^{23} +3931.00 q^{25} -3860.00 q^{27} -4866.00 q^{29} +7372.00 q^{31} -3360.00 q^{33} +4116.00 q^{35} -14330.0 q^{37} -5840.00 q^{39} +6222.00 q^{41} +3704.00 q^{43} +12012.0 q^{45} +1812.00 q^{47} +2401.00 q^{49} -14580.0 q^{51} +37242.0 q^{53} +28224.0 q^{55} +4700.00 q^{57} +34302.0 q^{59} -24476.0 q^{61} +7007.00 q^{63} +49056.0 q^{65} -17452.0 q^{67} +42000.0 q^{69} -28224.0 q^{71} +3602.00 q^{73} +39310.0 q^{75} +16464.0 q^{77} -42872.0 q^{79} -3851.00 q^{81} -35202.0 q^{83} +122472. q^{85} -48660.0 q^{87} +26730.0 q^{89} +28616.0 q^{91} +73720.0 q^{93} -39480.0 q^{95} -16978.0 q^{97} +48048.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 10.0000 0.641500 0.320750 0.947164i $$-0.396065\pi$$
0.320750 + 0.947164i $$0.396065\pi$$
$$4$$ 0 0
$$5$$ −84.0000 −1.50264 −0.751319 0.659939i $$-0.770582\pi$$
−0.751319 + 0.659939i $$0.770582\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ −143.000 −0.588477
$$10$$ 0 0
$$11$$ −336.000 −0.837255 −0.418627 0.908158i $$-0.637489\pi$$
−0.418627 + 0.908158i $$0.637489\pi$$
$$12$$ 0 0
$$13$$ −584.000 −0.958417 −0.479208 0.877701i $$-0.659076\pi$$
−0.479208 + 0.877701i $$0.659076\pi$$
$$14$$ 0 0
$$15$$ −840.000 −0.963943
$$16$$ 0 0
$$17$$ −1458.00 −1.22359 −0.611794 0.791017i $$-0.709552\pi$$
−0.611794 + 0.791017i $$0.709552\pi$$
$$18$$ 0 0
$$19$$ 470.000 0.298685 0.149343 0.988786i $$-0.452284\pi$$
0.149343 + 0.988786i $$0.452284\pi$$
$$20$$ 0 0
$$21$$ −490.000 −0.242464
$$22$$ 0 0
$$23$$ 4200.00 1.65550 0.827751 0.561096i $$-0.189620\pi$$
0.827751 + 0.561096i $$0.189620\pi$$
$$24$$ 0 0
$$25$$ 3931.00 1.25792
$$26$$ 0 0
$$27$$ −3860.00 −1.01901
$$28$$ 0 0
$$29$$ −4866.00 −1.07443 −0.537214 0.843446i $$-0.680523\pi$$
−0.537214 + 0.843446i $$0.680523\pi$$
$$30$$ 0 0
$$31$$ 7372.00 1.37778 0.688892 0.724864i $$-0.258097\pi$$
0.688892 + 0.724864i $$0.258097\pi$$
$$32$$ 0 0
$$33$$ −3360.00 −0.537099
$$34$$ 0 0
$$35$$ 4116.00 0.567944
$$36$$ 0 0
$$37$$ −14330.0 −1.72085 −0.860423 0.509581i $$-0.829800\pi$$
−0.860423 + 0.509581i $$0.829800\pi$$
$$38$$ 0 0
$$39$$ −5840.00 −0.614825
$$40$$ 0 0
$$41$$ 6222.00 0.578057 0.289028 0.957321i $$-0.406668\pi$$
0.289028 + 0.957321i $$0.406668\pi$$
$$42$$ 0 0
$$43$$ 3704.00 0.305492 0.152746 0.988265i $$-0.451188\pi$$
0.152746 + 0.988265i $$0.451188\pi$$
$$44$$ 0 0
$$45$$ 12012.0 0.884268
$$46$$ 0 0
$$47$$ 1812.00 0.119650 0.0598251 0.998209i $$-0.480946\pi$$
0.0598251 + 0.998209i $$0.480946\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −14580.0 −0.784932
$$52$$ 0 0
$$53$$ 37242.0 1.82114 0.910570 0.413355i $$-0.135643\pi$$
0.910570 + 0.413355i $$0.135643\pi$$
$$54$$ 0 0
$$55$$ 28224.0 1.25809
$$56$$ 0 0
$$57$$ 4700.00 0.191607
$$58$$ 0 0
$$59$$ 34302.0 1.28289 0.641445 0.767169i $$-0.278335\pi$$
0.641445 + 0.767169i $$0.278335\pi$$
$$60$$ 0 0
$$61$$ −24476.0 −0.842201 −0.421101 0.907014i $$-0.638356\pi$$
−0.421101 + 0.907014i $$0.638356\pi$$
$$62$$ 0 0
$$63$$ 7007.00 0.222424
$$64$$ 0 0
$$65$$ 49056.0 1.44015
$$66$$ 0 0
$$67$$ −17452.0 −0.474961 −0.237481 0.971392i $$-0.576322\pi$$
−0.237481 + 0.971392i $$0.576322\pi$$
$$68$$ 0 0
$$69$$ 42000.0 1.06201
$$70$$ 0 0
$$71$$ −28224.0 −0.664466 −0.332233 0.943197i $$-0.607802\pi$$
−0.332233 + 0.943197i $$0.607802\pi$$
$$72$$ 0 0
$$73$$ 3602.00 0.0791109 0.0395555 0.999217i $$-0.487406\pi$$
0.0395555 + 0.999217i $$0.487406\pi$$
$$74$$ 0 0
$$75$$ 39310.0 0.806956
$$76$$ 0 0
$$77$$ 16464.0 0.316453
$$78$$ 0 0
$$79$$ −42872.0 −0.772869 −0.386435 0.922317i $$-0.626294\pi$$
−0.386435 + 0.922317i $$0.626294\pi$$
$$80$$ 0 0
$$81$$ −3851.00 −0.0652170
$$82$$ 0 0
$$83$$ −35202.0 −0.560883 −0.280441 0.959871i $$-0.590481\pi$$
−0.280441 + 0.959871i $$0.590481\pi$$
$$84$$ 0 0
$$85$$ 122472. 1.83861
$$86$$ 0 0
$$87$$ −48660.0 −0.689246
$$88$$ 0 0
$$89$$ 26730.0 0.357704 0.178852 0.983876i $$-0.442762\pi$$
0.178852 + 0.983876i $$0.442762\pi$$
$$90$$ 0 0
$$91$$ 28616.0 0.362248
$$92$$ 0 0
$$93$$ 73720.0 0.883849
$$94$$ 0 0
$$95$$ −39480.0 −0.448816
$$96$$ 0 0
$$97$$ −16978.0 −0.183213 −0.0916067 0.995795i $$-0.529200\pi$$
−0.0916067 + 0.995795i $$0.529200\pi$$
$$98$$ 0 0
$$99$$ 48048.0 0.492705
$$100$$ 0 0
$$101$$ −99204.0 −0.967667 −0.483833 0.875160i $$-0.660756\pi$$
−0.483833 + 0.875160i $$0.660756\pi$$
$$102$$ 0 0
$$103$$ 131644. 1.22267 0.611333 0.791373i $$-0.290634\pi$$
0.611333 + 0.791373i $$0.290634\pi$$
$$104$$ 0 0
$$105$$ 41160.0 0.364336
$$106$$ 0 0
$$107$$ 48852.0 0.412499 0.206250 0.978499i $$-0.433874\pi$$
0.206250 + 0.978499i $$0.433874\pi$$
$$108$$ 0 0
$$109$$ 56374.0 0.454478 0.227239 0.973839i $$-0.427030\pi$$
0.227239 + 0.973839i $$0.427030\pi$$
$$110$$ 0 0
$$111$$ −143300. −1.10392
$$112$$ 0 0
$$113$$ 8742.00 0.0644043 0.0322021 0.999481i $$-0.489748\pi$$
0.0322021 + 0.999481i $$0.489748\pi$$
$$114$$ 0 0
$$115$$ −352800. −2.48762
$$116$$ 0 0
$$117$$ 83512.0 0.564007
$$118$$ 0 0
$$119$$ 71442.0 0.462473
$$120$$ 0 0
$$121$$ −48155.0 −0.299005
$$122$$ 0 0
$$123$$ 62220.0 0.370823
$$124$$ 0 0
$$125$$ −67704.0 −0.387560
$$126$$ 0 0
$$127$$ −315992. −1.73847 −0.869234 0.494401i $$-0.835388\pi$$
−0.869234 + 0.494401i $$0.835388\pi$$
$$128$$ 0 0
$$129$$ 37040.0 0.195973
$$130$$ 0 0
$$131$$ −24666.0 −0.125580 −0.0627900 0.998027i $$-0.520000\pi$$
−0.0627900 + 0.998027i $$0.520000\pi$$
$$132$$ 0 0
$$133$$ −23030.0 −0.112892
$$134$$ 0 0
$$135$$ 324240. 1.53120
$$136$$ 0 0
$$137$$ 303234. 1.38031 0.690155 0.723662i $$-0.257542\pi$$
0.690155 + 0.723662i $$0.257542\pi$$
$$138$$ 0 0
$$139$$ 250586. 1.10007 0.550034 0.835142i $$-0.314615\pi$$
0.550034 + 0.835142i $$0.314615\pi$$
$$140$$ 0 0
$$141$$ 18120.0 0.0767557
$$142$$ 0 0
$$143$$ 196224. 0.802439
$$144$$ 0 0
$$145$$ 408744. 1.61448
$$146$$ 0 0
$$147$$ 24010.0 0.0916429
$$148$$ 0 0
$$149$$ 60594.0 0.223596 0.111798 0.993731i $$-0.464339\pi$$
0.111798 + 0.993731i $$0.464339\pi$$
$$150$$ 0 0
$$151$$ −124448. −0.444166 −0.222083 0.975028i $$-0.571286\pi$$
−0.222083 + 0.975028i $$0.571286\pi$$
$$152$$ 0 0
$$153$$ 208494. 0.720054
$$154$$ 0 0
$$155$$ −619248. −2.07031
$$156$$ 0 0
$$157$$ −76040.0 −0.246203 −0.123101 0.992394i $$-0.539284\pi$$
−0.123101 + 0.992394i $$0.539284\pi$$
$$158$$ 0 0
$$159$$ 372420. 1.16826
$$160$$ 0 0
$$161$$ −205800. −0.625721
$$162$$ 0 0
$$163$$ 124256. 0.366310 0.183155 0.983084i $$-0.441369\pi$$
0.183155 + 0.983084i $$0.441369\pi$$
$$164$$ 0 0
$$165$$ 282240. 0.807065
$$166$$ 0 0
$$167$$ 72420.0 0.200940 0.100470 0.994940i $$-0.467965\pi$$
0.100470 + 0.994940i $$0.467965\pi$$
$$168$$ 0 0
$$169$$ −30237.0 −0.0814370
$$170$$ 0 0
$$171$$ −67210.0 −0.175770
$$172$$ 0 0
$$173$$ 441552. 1.12167 0.560837 0.827926i $$-0.310479\pi$$
0.560837 + 0.827926i $$0.310479\pi$$
$$174$$ 0 0
$$175$$ −192619. −0.475449
$$176$$ 0 0
$$177$$ 343020. 0.822974
$$178$$ 0 0
$$179$$ −10692.0 −0.0249417 −0.0124709 0.999922i $$-0.503970\pi$$
−0.0124709 + 0.999922i $$0.503970\pi$$
$$180$$ 0 0
$$181$$ 546064. 1.23893 0.619465 0.785024i $$-0.287349\pi$$
0.619465 + 0.785024i $$0.287349\pi$$
$$182$$ 0 0
$$183$$ −244760. −0.540272
$$184$$ 0 0
$$185$$ 1.20372e6 2.58581
$$186$$ 0 0
$$187$$ 489888. 1.02445
$$188$$ 0 0
$$189$$ 189140. 0.385149
$$190$$ 0 0
$$191$$ 575976. 1.14241 0.571204 0.820808i $$-0.306477\pi$$
0.571204 + 0.820808i $$0.306477\pi$$
$$192$$ 0 0
$$193$$ −413938. −0.799912 −0.399956 0.916534i $$-0.630975\pi$$
−0.399956 + 0.916534i $$0.630975\pi$$
$$194$$ 0 0
$$195$$ 490560. 0.923859
$$196$$ 0 0
$$197$$ 494946. 0.908641 0.454320 0.890838i $$-0.349882\pi$$
0.454320 + 0.890838i $$0.349882\pi$$
$$198$$ 0 0
$$199$$ −520364. −0.931482 −0.465741 0.884921i $$-0.654212\pi$$
−0.465741 + 0.884921i $$0.654212\pi$$
$$200$$ 0 0
$$201$$ −174520. −0.304688
$$202$$ 0 0
$$203$$ 238434. 0.406095
$$204$$ 0 0
$$205$$ −522648. −0.868610
$$206$$ 0 0
$$207$$ −600600. −0.974225
$$208$$ 0 0
$$209$$ −157920. −0.250076
$$210$$ 0 0
$$211$$ 183284. 0.283412 0.141706 0.989909i $$-0.454741\pi$$
0.141706 + 0.989909i $$0.454741\pi$$
$$212$$ 0 0
$$213$$ −282240. −0.426255
$$214$$ 0 0
$$215$$ −311136. −0.459044
$$216$$ 0 0
$$217$$ −361228. −0.520753
$$218$$ 0 0
$$219$$ 36020.0 0.0507497
$$220$$ 0 0
$$221$$ 851472. 1.17271
$$222$$ 0 0
$$223$$ 1.27746e6 1.72023 0.860115 0.510100i $$-0.170392\pi$$
0.860115 + 0.510100i $$0.170392\pi$$
$$224$$ 0 0
$$225$$ −562133. −0.740257
$$226$$ 0 0
$$227$$ −1.28764e6 −1.65856 −0.829279 0.558835i $$-0.811248\pi$$
−0.829279 + 0.558835i $$0.811248\pi$$
$$228$$ 0 0
$$229$$ −350936. −0.442221 −0.221110 0.975249i $$-0.570968\pi$$
−0.221110 + 0.975249i $$0.570968\pi$$
$$230$$ 0 0
$$231$$ 164640. 0.203004
$$232$$ 0 0
$$233$$ 836154. 1.00901 0.504506 0.863408i $$-0.331675\pi$$
0.504506 + 0.863408i $$0.331675\pi$$
$$234$$ 0 0
$$235$$ −152208. −0.179791
$$236$$ 0 0
$$237$$ −428720. −0.495796
$$238$$ 0 0
$$239$$ −774336. −0.876869 −0.438434 0.898763i $$-0.644467\pi$$
−0.438434 + 0.898763i $$0.644467\pi$$
$$240$$ 0 0
$$241$$ −1.15285e6 −1.27859 −0.639293 0.768963i $$-0.720773\pi$$
−0.639293 + 0.768963i $$0.720773\pi$$
$$242$$ 0 0
$$243$$ 899470. 0.977172
$$244$$ 0 0
$$245$$ −201684. −0.214663
$$246$$ 0 0
$$247$$ −274480. −0.286265
$$248$$ 0 0
$$249$$ −352020. −0.359806
$$250$$ 0 0
$$251$$ 1.35801e6 1.36056 0.680282 0.732951i $$-0.261858\pi$$
0.680282 + 0.732951i $$0.261858\pi$$
$$252$$ 0 0
$$253$$ −1.41120e6 −1.38608
$$254$$ 0 0
$$255$$ 1.22472e6 1.17947
$$256$$ 0 0
$$257$$ −317742. −0.300083 −0.150042 0.988680i $$-0.547941\pi$$
−0.150042 + 0.988680i $$0.547941\pi$$
$$258$$ 0 0
$$259$$ 702170. 0.650418
$$260$$ 0 0
$$261$$ 695838. 0.632276
$$262$$ 0 0
$$263$$ −1.05101e6 −0.936951 −0.468475 0.883477i $$-0.655196\pi$$
−0.468475 + 0.883477i $$0.655196\pi$$
$$264$$ 0 0
$$265$$ −3.12833e6 −2.73651
$$266$$ 0 0
$$267$$ 267300. 0.229467
$$268$$ 0 0
$$269$$ −1.18958e6 −1.00234 −0.501169 0.865349i $$-0.667097\pi$$
−0.501169 + 0.865349i $$0.667097\pi$$
$$270$$ 0 0
$$271$$ 1.43008e6 1.18287 0.591435 0.806353i $$-0.298562\pi$$
0.591435 + 0.806353i $$0.298562\pi$$
$$272$$ 0 0
$$273$$ 286160. 0.232382
$$274$$ 0 0
$$275$$ −1.32082e6 −1.05320
$$276$$ 0 0
$$277$$ −63302.0 −0.0495699 −0.0247849 0.999693i $$-0.507890\pi$$
−0.0247849 + 0.999693i $$0.507890\pi$$
$$278$$ 0 0
$$279$$ −1.05420e6 −0.810795
$$280$$ 0 0
$$281$$ −496614. −0.375192 −0.187596 0.982246i $$-0.560070\pi$$
−0.187596 + 0.982246i $$0.560070\pi$$
$$282$$ 0 0
$$283$$ −1.15842e6 −0.859803 −0.429902 0.902876i $$-0.641452\pi$$
−0.429902 + 0.902876i $$0.641452\pi$$
$$284$$ 0 0
$$285$$ −394800. −0.287915
$$286$$ 0 0
$$287$$ −304878. −0.218485
$$288$$ 0 0
$$289$$ 705907. 0.497168
$$290$$ 0 0
$$291$$ −169780. −0.117531
$$292$$ 0 0
$$293$$ −1.43886e6 −0.979151 −0.489575 0.871961i $$-0.662848\pi$$
−0.489575 + 0.871961i $$0.662848\pi$$
$$294$$ 0 0
$$295$$ −2.88137e6 −1.92772
$$296$$ 0 0
$$297$$ 1.29696e6 0.853170
$$298$$ 0 0
$$299$$ −2.45280e6 −1.58666
$$300$$ 0 0
$$301$$ −181496. −0.115465
$$302$$ 0 0
$$303$$ −992040. −0.620758
$$304$$ 0 0
$$305$$ 2.05598e6 1.26552
$$306$$ 0 0
$$307$$ −989098. −0.598954 −0.299477 0.954104i $$-0.596812\pi$$
−0.299477 + 0.954104i $$0.596812\pi$$
$$308$$ 0 0
$$309$$ 1.31644e6 0.784341
$$310$$ 0 0
$$311$$ 2.22050e6 1.30182 0.650909 0.759155i $$-0.274388\pi$$
0.650909 + 0.759155i $$0.274388\pi$$
$$312$$ 0 0
$$313$$ 2.33008e6 1.34434 0.672171 0.740396i $$-0.265362\pi$$
0.672171 + 0.740396i $$0.265362\pi$$
$$314$$ 0 0
$$315$$ −588588. −0.334222
$$316$$ 0 0
$$317$$ −427542. −0.238963 −0.119481 0.992836i $$-0.538123\pi$$
−0.119481 + 0.992836i $$0.538123\pi$$
$$318$$ 0 0
$$319$$ 1.63498e6 0.899569
$$320$$ 0 0
$$321$$ 488520. 0.264618
$$322$$ 0 0
$$323$$ −685260. −0.365468
$$324$$ 0 0
$$325$$ −2.29570e6 −1.20561
$$326$$ 0 0
$$327$$ 563740. 0.291548
$$328$$ 0 0
$$329$$ −88788.0 −0.0452235
$$330$$ 0 0
$$331$$ −396616. −0.198976 −0.0994879 0.995039i $$-0.531720\pi$$
−0.0994879 + 0.995039i $$0.531720\pi$$
$$332$$ 0 0
$$333$$ 2.04919e6 1.01268
$$334$$ 0 0
$$335$$ 1.46597e6 0.713695
$$336$$ 0 0
$$337$$ −3.21819e6 −1.54361 −0.771805 0.635860i $$-0.780646\pi$$
−0.771805 + 0.635860i $$0.780646\pi$$
$$338$$ 0 0
$$339$$ 87420.0 0.0413154
$$340$$ 0 0
$$341$$ −2.47699e6 −1.15356
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ −3.52800e6 −1.59581
$$346$$ 0 0
$$347$$ 2.78018e6 1.23951 0.619755 0.784796i $$-0.287232\pi$$
0.619755 + 0.784796i $$0.287232\pi$$
$$348$$ 0 0
$$349$$ 338800. 0.148895 0.0744475 0.997225i $$-0.476281\pi$$
0.0744475 + 0.997225i $$0.476281\pi$$
$$350$$ 0 0
$$351$$ 2.25424e6 0.976635
$$352$$ 0 0
$$353$$ −362046. −0.154642 −0.0773209 0.997006i $$-0.524637\pi$$
−0.0773209 + 0.997006i $$0.524637\pi$$
$$354$$ 0 0
$$355$$ 2.37082e6 0.998451
$$356$$ 0 0
$$357$$ 714420. 0.296676
$$358$$ 0 0
$$359$$ −876528. −0.358946 −0.179473 0.983763i $$-0.557439\pi$$
−0.179473 + 0.983763i $$0.557439\pi$$
$$360$$ 0 0
$$361$$ −2.25520e6 −0.910787
$$362$$ 0 0
$$363$$ −481550. −0.191812
$$364$$ 0 0
$$365$$ −302568. −0.118875
$$366$$ 0 0
$$367$$ −2.98062e6 −1.15516 −0.577578 0.816335i $$-0.696002\pi$$
−0.577578 + 0.816335i $$0.696002\pi$$
$$368$$ 0 0
$$369$$ −889746. −0.340173
$$370$$ 0 0
$$371$$ −1.82486e6 −0.688326
$$372$$ 0 0
$$373$$ −3.91441e6 −1.45678 −0.728391 0.685162i $$-0.759732\pi$$
−0.728391 + 0.685162i $$0.759732\pi$$
$$374$$ 0 0
$$375$$ −677040. −0.248620
$$376$$ 0 0
$$377$$ 2.84174e6 1.02975
$$378$$ 0 0
$$379$$ 3.60661e6 1.28974 0.644868 0.764294i $$-0.276912\pi$$
0.644868 + 0.764294i $$0.276912\pi$$
$$380$$ 0 0
$$381$$ −3.15992e6 −1.11523
$$382$$ 0 0
$$383$$ 2.66644e6 0.928826 0.464413 0.885619i $$-0.346265\pi$$
0.464413 + 0.885619i $$0.346265\pi$$
$$384$$ 0 0
$$385$$ −1.38298e6 −0.475513
$$386$$ 0 0
$$387$$ −529672. −0.179775
$$388$$ 0 0
$$389$$ 213366. 0.0714910 0.0357455 0.999361i $$-0.488619\pi$$
0.0357455 + 0.999361i $$0.488619\pi$$
$$390$$ 0 0
$$391$$ −6.12360e6 −2.02565
$$392$$ 0 0
$$393$$ −246660. −0.0805596
$$394$$ 0 0
$$395$$ 3.60125e6 1.16134
$$396$$ 0 0
$$397$$ 4.09408e6 1.30371 0.651854 0.758345i $$-0.273992\pi$$
0.651854 + 0.758345i $$0.273992\pi$$
$$398$$ 0 0
$$399$$ −230300. −0.0724205
$$400$$ 0 0
$$401$$ 942366. 0.292657 0.146328 0.989236i $$-0.453254\pi$$
0.146328 + 0.989236i $$0.453254\pi$$
$$402$$ 0 0
$$403$$ −4.30525e6 −1.32049
$$404$$ 0 0
$$405$$ 323484. 0.0979976
$$406$$ 0 0
$$407$$ 4.81488e6 1.44079
$$408$$ 0 0
$$409$$ −4.84561e6 −1.43232 −0.716160 0.697936i $$-0.754102\pi$$
−0.716160 + 0.697936i $$0.754102\pi$$
$$410$$ 0 0
$$411$$ 3.03234e6 0.885469
$$412$$ 0 0
$$413$$ −1.68080e6 −0.484887
$$414$$ 0 0
$$415$$ 2.95697e6 0.842804
$$416$$ 0 0
$$417$$ 2.50586e6 0.705694
$$418$$ 0 0
$$419$$ −1.73485e6 −0.482754 −0.241377 0.970431i $$-0.577599\pi$$
−0.241377 + 0.970431i $$0.577599\pi$$
$$420$$ 0 0
$$421$$ 1.65145e6 0.454109 0.227055 0.973882i $$-0.427090\pi$$
0.227055 + 0.973882i $$0.427090\pi$$
$$422$$ 0 0
$$423$$ −259116. −0.0704115
$$424$$ 0 0
$$425$$ −5.73140e6 −1.53918
$$426$$ 0 0
$$427$$ 1.19932e6 0.318322
$$428$$ 0 0
$$429$$ 1.96224e6 0.514765
$$430$$ 0 0
$$431$$ −4.14360e6 −1.07445 −0.537223 0.843440i $$-0.680527\pi$$
−0.537223 + 0.843440i $$0.680527\pi$$
$$432$$ 0 0
$$433$$ −3.03966e6 −0.779121 −0.389561 0.921001i $$-0.627373\pi$$
−0.389561 + 0.921001i $$0.627373\pi$$
$$434$$ 0 0
$$435$$ 4.08744e6 1.03569
$$436$$ 0 0
$$437$$ 1.97400e6 0.494474
$$438$$ 0 0
$$439$$ −2.54271e6 −0.629703 −0.314852 0.949141i $$-0.601955\pi$$
−0.314852 + 0.949141i $$0.601955\pi$$
$$440$$ 0 0
$$441$$ −343343. −0.0840682
$$442$$ 0 0
$$443$$ −2.43210e6 −0.588806 −0.294403 0.955681i $$-0.595121\pi$$
−0.294403 + 0.955681i $$0.595121\pi$$
$$444$$ 0 0
$$445$$ −2.24532e6 −0.537500
$$446$$ 0 0
$$447$$ 605940. 0.143437
$$448$$ 0 0
$$449$$ 1.82853e6 0.428042 0.214021 0.976829i $$-0.431344\pi$$
0.214021 + 0.976829i $$0.431344\pi$$
$$450$$ 0 0
$$451$$ −2.09059e6 −0.483981
$$452$$ 0 0
$$453$$ −1.24448e6 −0.284933
$$454$$ 0 0
$$455$$ −2.40374e6 −0.544327
$$456$$ 0 0
$$457$$ 1.58063e6 0.354030 0.177015 0.984208i $$-0.443356\pi$$
0.177015 + 0.984208i $$0.443356\pi$$
$$458$$ 0 0
$$459$$ 5.62788e6 1.24685
$$460$$ 0 0
$$461$$ −5.09604e6 −1.11681 −0.558407 0.829567i $$-0.688587\pi$$
−0.558407 + 0.829567i $$0.688587\pi$$
$$462$$ 0 0
$$463$$ 7.02338e6 1.52263 0.761313 0.648384i $$-0.224555\pi$$
0.761313 + 0.648384i $$0.224555\pi$$
$$464$$ 0 0
$$465$$ −6.19248e6 −1.32810
$$466$$ 0 0
$$467$$ −4.24845e6 −0.901443 −0.450722 0.892665i $$-0.648833\pi$$
−0.450722 + 0.892665i $$0.648833\pi$$
$$468$$ 0 0
$$469$$ 855148. 0.179518
$$470$$ 0 0
$$471$$ −760400. −0.157939
$$472$$ 0 0
$$473$$ −1.24454e6 −0.255775
$$474$$ 0 0
$$475$$ 1.84757e6 0.375722
$$476$$ 0 0
$$477$$ −5.32561e6 −1.07170
$$478$$ 0 0
$$479$$ −559284. −0.111377 −0.0556883 0.998448i $$-0.517735\pi$$
−0.0556883 + 0.998448i $$0.517735\pi$$
$$480$$ 0 0
$$481$$ 8.36872e6 1.64929
$$482$$ 0 0
$$483$$ −2.05800e6 −0.401400
$$484$$ 0 0
$$485$$ 1.42615e6 0.275303
$$486$$ 0 0
$$487$$ 1.32057e6 0.252312 0.126156 0.992010i $$-0.459736\pi$$
0.126156 + 0.992010i $$0.459736\pi$$
$$488$$ 0 0
$$489$$ 1.24256e6 0.234988
$$490$$ 0 0
$$491$$ 6.27193e6 1.17408 0.587040 0.809558i $$-0.300293\pi$$
0.587040 + 0.809558i $$0.300293\pi$$
$$492$$ 0 0
$$493$$ 7.09463e6 1.31466
$$494$$ 0 0
$$495$$ −4.03603e6 −0.740358
$$496$$ 0 0
$$497$$ 1.38298e6 0.251144
$$498$$ 0 0
$$499$$ −3.93785e6 −0.707959 −0.353979 0.935253i $$-0.615172\pi$$
−0.353979 + 0.935253i $$0.615172\pi$$
$$500$$ 0 0
$$501$$ 724200. 0.128903
$$502$$ 0 0
$$503$$ 7.59830e6 1.33905 0.669525 0.742790i $$-0.266498\pi$$
0.669525 + 0.742790i $$0.266498\pi$$
$$504$$ 0 0
$$505$$ 8.33314e6 1.45405
$$506$$ 0 0
$$507$$ −302370. −0.0522419
$$508$$ 0 0
$$509$$ 7.82664e6 1.33900 0.669501 0.742812i $$-0.266508\pi$$
0.669501 + 0.742812i $$0.266508\pi$$
$$510$$ 0 0
$$511$$ −176498. −0.0299011
$$512$$ 0 0
$$513$$ −1.81420e6 −0.304363
$$514$$ 0 0
$$515$$ −1.10581e7 −1.83722
$$516$$ 0 0
$$517$$ −608832. −0.100178
$$518$$ 0 0
$$519$$ 4.41552e6 0.719554
$$520$$ 0 0
$$521$$ 8.94454e6 1.44366 0.721828 0.692072i $$-0.243302\pi$$
0.721828 + 0.692072i $$0.243302\pi$$
$$522$$ 0 0
$$523$$ 4.07481e6 0.651407 0.325704 0.945472i $$-0.394399\pi$$
0.325704 + 0.945472i $$0.394399\pi$$
$$524$$ 0 0
$$525$$ −1.92619e6 −0.305001
$$526$$ 0 0
$$527$$ −1.07484e7 −1.68584
$$528$$ 0 0
$$529$$ 1.12037e7 1.74069
$$530$$ 0 0
$$531$$ −4.90519e6 −0.754952
$$532$$ 0 0
$$533$$ −3.63365e6 −0.554019
$$534$$ 0 0
$$535$$ −4.10357e6 −0.619837
$$536$$ 0 0
$$537$$ −106920. −0.0160001
$$538$$ 0 0
$$539$$ −806736. −0.119608
$$540$$ 0 0
$$541$$ 1.18676e7 1.74329 0.871644 0.490140i $$-0.163054\pi$$
0.871644 + 0.490140i $$0.163054\pi$$
$$542$$ 0 0
$$543$$ 5.46064e6 0.794775
$$544$$ 0 0
$$545$$ −4.73542e6 −0.682915
$$546$$ 0 0
$$547$$ −5.37801e6 −0.768516 −0.384258 0.923226i $$-0.625543\pi$$
−0.384258 + 0.923226i $$0.625543\pi$$
$$548$$ 0 0
$$549$$ 3.50007e6 0.495616
$$550$$ 0 0
$$551$$ −2.28702e6 −0.320916
$$552$$ 0 0
$$553$$ 2.10073e6 0.292117
$$554$$ 0 0
$$555$$ 1.20372e7 1.65880
$$556$$ 0 0
$$557$$ 5.64878e6 0.771466 0.385733 0.922611i $$-0.373949\pi$$
0.385733 + 0.922611i $$0.373949\pi$$
$$558$$ 0 0
$$559$$ −2.16314e6 −0.292789
$$560$$ 0 0
$$561$$ 4.89888e6 0.657188
$$562$$ 0 0
$$563$$ 4.56407e6 0.606850 0.303425 0.952855i $$-0.401870\pi$$
0.303425 + 0.952855i $$0.401870\pi$$
$$564$$ 0 0
$$565$$ −734328. −0.0967763
$$566$$ 0 0
$$567$$ 188699. 0.0246497
$$568$$ 0 0
$$569$$ 8.00165e6 1.03609 0.518047 0.855352i $$-0.326659\pi$$
0.518047 + 0.855352i $$0.326659\pi$$
$$570$$ 0 0
$$571$$ −1.37164e7 −1.76055 −0.880275 0.474464i $$-0.842642\pi$$
−0.880275 + 0.474464i $$0.842642\pi$$
$$572$$ 0 0
$$573$$ 5.75976e6 0.732855
$$574$$ 0 0
$$575$$ 1.65102e7 2.08249
$$576$$ 0 0
$$577$$ 6.09797e6 0.762510 0.381255 0.924470i $$-0.375492\pi$$
0.381255 + 0.924470i $$0.375492\pi$$
$$578$$ 0 0
$$579$$ −4.13938e6 −0.513144
$$580$$ 0 0
$$581$$ 1.72490e6 0.211994
$$582$$ 0 0
$$583$$ −1.25133e7 −1.52476
$$584$$ 0 0
$$585$$ −7.01501e6 −0.847498
$$586$$ 0 0
$$587$$ −8.08462e6 −0.968422 −0.484211 0.874951i $$-0.660893\pi$$
−0.484211 + 0.874951i $$0.660893\pi$$
$$588$$ 0 0
$$589$$ 3.46484e6 0.411524
$$590$$ 0 0
$$591$$ 4.94946e6 0.582893
$$592$$ 0 0
$$593$$ 1.41575e6 0.165330 0.0826649 0.996577i $$-0.473657\pi$$
0.0826649 + 0.996577i $$0.473657\pi$$
$$594$$ 0 0
$$595$$ −6.00113e6 −0.694929
$$596$$ 0 0
$$597$$ −5.20364e6 −0.597546
$$598$$ 0 0
$$599$$ −8.75460e6 −0.996941 −0.498470 0.866907i $$-0.666105\pi$$
−0.498470 + 0.866907i $$0.666105\pi$$
$$600$$ 0 0
$$601$$ 8.70276e6 0.982813 0.491407 0.870930i $$-0.336483\pi$$
0.491407 + 0.870930i $$0.336483\pi$$
$$602$$ 0 0
$$603$$ 2.49564e6 0.279504
$$604$$ 0 0
$$605$$ 4.04502e6 0.449296
$$606$$ 0 0
$$607$$ 1.69578e7 1.86809 0.934045 0.357157i $$-0.116254\pi$$
0.934045 + 0.357157i $$0.116254\pi$$
$$608$$ 0 0
$$609$$ 2.38434e6 0.260510
$$610$$ 0 0
$$611$$ −1.05821e6 −0.114675
$$612$$ 0 0
$$613$$ −1.76743e7 −1.89973 −0.949866 0.312658i $$-0.898780\pi$$
−0.949866 + 0.312658i $$0.898780\pi$$
$$614$$ 0 0
$$615$$ −5.22648e6 −0.557213
$$616$$ 0 0
$$617$$ −9.70636e6 −1.02646 −0.513232 0.858250i $$-0.671552\pi$$
−0.513232 + 0.858250i $$0.671552\pi$$
$$618$$ 0 0
$$619$$ 1.48739e7 1.56027 0.780133 0.625613i $$-0.215151\pi$$
0.780133 + 0.625613i $$0.215151\pi$$
$$620$$ 0 0
$$621$$ −1.62120e7 −1.68697
$$622$$ 0 0
$$623$$ −1.30977e6 −0.135199
$$624$$ 0 0
$$625$$ −6.59724e6 −0.675557
$$626$$ 0 0
$$627$$ −1.57920e6 −0.160424
$$628$$ 0 0
$$629$$ 2.08931e7 2.10561
$$630$$ 0 0
$$631$$ −1.26353e7 −1.26331 −0.631656 0.775248i $$-0.717625\pi$$
−0.631656 + 0.775248i $$0.717625\pi$$
$$632$$ 0 0
$$633$$ 1.83284e6 0.181809
$$634$$ 0 0
$$635$$ 2.65433e7 2.61229
$$636$$ 0 0
$$637$$ −1.40218e6 −0.136917
$$638$$ 0 0
$$639$$ 4.03603e6 0.391023
$$640$$ 0 0
$$641$$ 6.23398e6 0.599267 0.299634 0.954054i $$-0.403136\pi$$
0.299634 + 0.954054i $$0.403136\pi$$
$$642$$ 0 0
$$643$$ 1.06874e7 1.01940 0.509701 0.860352i $$-0.329756\pi$$
0.509701 + 0.860352i $$0.329756\pi$$
$$644$$ 0 0
$$645$$ −3.11136e6 −0.294477
$$646$$ 0 0
$$647$$ −1.83258e7 −1.72109 −0.860544 0.509376i $$-0.829876\pi$$
−0.860544 + 0.509376i $$0.829876\pi$$
$$648$$ 0 0
$$649$$ −1.15255e7 −1.07411
$$650$$ 0 0
$$651$$ −3.61228e6 −0.334063
$$652$$ 0 0
$$653$$ 7.28857e6 0.668897 0.334448 0.942414i $$-0.391450\pi$$
0.334448 + 0.942414i $$0.391450\pi$$
$$654$$ 0 0
$$655$$ 2.07194e6 0.188701
$$656$$ 0 0
$$657$$ −515086. −0.0465550
$$658$$ 0 0
$$659$$ 4.54337e6 0.407534 0.203767 0.979019i $$-0.434681\pi$$
0.203767 + 0.979019i $$0.434681\pi$$
$$660$$ 0 0
$$661$$ 2.10021e7 1.86964 0.934821 0.355120i $$-0.115560\pi$$
0.934821 + 0.355120i $$0.115560\pi$$
$$662$$ 0 0
$$663$$ 8.51472e6 0.752292
$$664$$ 0 0
$$665$$ 1.93452e6 0.169636
$$666$$ 0 0
$$667$$ −2.04372e7 −1.77872
$$668$$ 0 0
$$669$$ 1.27746e7 1.10353
$$670$$ 0 0
$$671$$ 8.22394e6 0.705137
$$672$$ 0 0
$$673$$ 3.46923e6 0.295253 0.147627 0.989043i $$-0.452837\pi$$
0.147627 + 0.989043i $$0.452837\pi$$
$$674$$ 0 0
$$675$$ −1.51737e7 −1.28183
$$676$$ 0 0
$$677$$ 1.80916e7 1.51707 0.758536 0.651631i $$-0.225915\pi$$
0.758536 + 0.651631i $$0.225915\pi$$
$$678$$ 0 0
$$679$$ 831922. 0.0692481
$$680$$ 0 0
$$681$$ −1.28764e7 −1.06397
$$682$$ 0 0
$$683$$ 4.67752e6 0.383675 0.191838 0.981427i $$-0.438555\pi$$
0.191838 + 0.981427i $$0.438555\pi$$
$$684$$ 0 0
$$685$$ −2.54717e7 −2.07411
$$686$$ 0 0
$$687$$ −3.50936e6 −0.283685
$$688$$ 0 0
$$689$$ −2.17493e7 −1.74541
$$690$$ 0 0
$$691$$ 1.68960e7 1.34614 0.673069 0.739579i $$-0.264976\pi$$
0.673069 + 0.739579i $$0.264976\pi$$
$$692$$ 0 0
$$693$$ −2.35435e6 −0.186225
$$694$$ 0 0
$$695$$ −2.10492e7 −1.65300
$$696$$ 0 0
$$697$$ −9.07168e6 −0.707303
$$698$$ 0 0
$$699$$ 8.36154e6 0.647282
$$700$$ 0 0
$$701$$ −2.40964e6 −0.185207 −0.0926035 0.995703i $$-0.529519\pi$$
−0.0926035 + 0.995703i $$0.529519\pi$$
$$702$$ 0 0
$$703$$ −6.73510e6 −0.513991
$$704$$ 0 0
$$705$$ −1.52208e6 −0.115336
$$706$$ 0 0
$$707$$ 4.86100e6 0.365744
$$708$$ 0 0
$$709$$ 5.77010e6 0.431090 0.215545 0.976494i $$-0.430847\pi$$
0.215545 + 0.976494i $$0.430847\pi$$
$$710$$ 0 0
$$711$$ 6.13070e6 0.454816
$$712$$ 0 0
$$713$$ 3.09624e7 2.28092
$$714$$ 0 0
$$715$$ −1.64828e7 −1.20578
$$716$$ 0 0
$$717$$ −7.74336e6 −0.562512
$$718$$ 0 0
$$719$$ 1.43716e7 1.03677 0.518385 0.855147i $$-0.326533\pi$$
0.518385 + 0.855147i $$0.326533\pi$$
$$720$$ 0 0
$$721$$ −6.45056e6 −0.462124
$$722$$ 0 0
$$723$$ −1.15285e7 −0.820214
$$724$$ 0 0
$$725$$ −1.91282e7 −1.35154
$$726$$ 0 0
$$727$$ 1.40705e7 0.987353 0.493676 0.869646i $$-0.335653\pi$$
0.493676 + 0.869646i $$0.335653\pi$$
$$728$$ 0 0
$$729$$ 9.93049e6 0.692073
$$730$$ 0 0
$$731$$ −5.40043e6 −0.373796
$$732$$ 0 0
$$733$$ 3.75000e6 0.257793 0.128897 0.991658i $$-0.458856\pi$$
0.128897 + 0.991658i $$0.458856\pi$$
$$734$$ 0 0
$$735$$ −2.01684e6 −0.137706
$$736$$ 0 0
$$737$$ 5.86387e6 0.397664
$$738$$ 0 0
$$739$$ 2.61318e7 1.76019 0.880093 0.474802i $$-0.157480\pi$$
0.880093 + 0.474802i $$0.157480\pi$$
$$740$$ 0 0
$$741$$ −2.74480e6 −0.183639
$$742$$ 0 0
$$743$$ 159072. 0.0105711 0.00528557 0.999986i $$-0.498318\pi$$
0.00528557 + 0.999986i $$0.498318\pi$$
$$744$$ 0 0
$$745$$ −5.08990e6 −0.335984
$$746$$ 0 0
$$747$$ 5.03389e6 0.330067
$$748$$ 0 0
$$749$$ −2.39375e6 −0.155910
$$750$$ 0 0
$$751$$ 2.65311e7 1.71654 0.858272 0.513196i $$-0.171539\pi$$
0.858272 + 0.513196i $$0.171539\pi$$
$$752$$ 0 0
$$753$$ 1.35801e7 0.872802
$$754$$ 0 0
$$755$$ 1.04536e7 0.667421
$$756$$ 0 0
$$757$$ 1.52032e7 0.964260 0.482130 0.876100i $$-0.339863\pi$$
0.482130 + 0.876100i $$0.339863\pi$$
$$758$$ 0 0
$$759$$ −1.41120e7 −0.889169
$$760$$ 0 0
$$761$$ 4.71380e6 0.295059 0.147530 0.989058i $$-0.452868\pi$$
0.147530 + 0.989058i $$0.452868\pi$$
$$762$$ 0 0
$$763$$ −2.76233e6 −0.171776
$$764$$ 0 0
$$765$$ −1.75135e7 −1.08198
$$766$$ 0 0
$$767$$ −2.00324e7 −1.22954
$$768$$ 0 0
$$769$$ −1.58977e6 −0.0969434 −0.0484717 0.998825i $$-0.515435\pi$$
−0.0484717 + 0.998825i $$0.515435\pi$$
$$770$$ 0 0
$$771$$ −3.17742e6 −0.192504
$$772$$ 0 0
$$773$$ 9.69095e6 0.583334 0.291667 0.956520i $$-0.405790\pi$$
0.291667 + 0.956520i $$0.405790\pi$$
$$774$$ 0 0
$$775$$ 2.89793e7 1.73314
$$776$$ 0 0
$$777$$ 7.02170e6 0.417244
$$778$$ 0 0
$$779$$ 2.92434e6 0.172657
$$780$$ 0 0
$$781$$ 9.48326e6 0.556327
$$782$$ 0 0
$$783$$ 1.87828e7 1.09485
$$784$$ 0 0
$$785$$ 6.38736e6 0.369954
$$786$$ 0 0
$$787$$ −1.57170e6 −0.0904549 −0.0452275 0.998977i $$-0.514401\pi$$
−0.0452275 + 0.998977i $$0.514401\pi$$
$$788$$ 0 0
$$789$$ −1.05101e7 −0.601054
$$790$$ 0 0
$$791$$ −428358. −0.0243425
$$792$$ 0 0
$$793$$ 1.42940e7 0.807180
$$794$$ 0 0
$$795$$ −3.12833e7 −1.75547
$$796$$ 0 0
$$797$$ 2.25298e6 0.125635 0.0628175 0.998025i $$-0.479991\pi$$
0.0628175 + 0.998025i $$0.479991\pi$$
$$798$$ 0 0
$$799$$ −2.64190e6 −0.146403
$$800$$ 0 0
$$801$$ −3.82239e6 −0.210501
$$802$$ 0 0
$$803$$ −1.21027e6 −0.0662360
$$804$$ 0 0
$$805$$ 1.72872e7 0.940232
$$806$$ 0 0
$$807$$ −1.18958e7 −0.643000
$$808$$ 0 0
$$809$$ −2.37938e7 −1.27818 −0.639090 0.769132i $$-0.720689\pi$$
−0.639090 + 0.769132i $$0.720689\pi$$
$$810$$ 0 0
$$811$$ 5.32300e6 0.284187 0.142093 0.989853i $$-0.454617\pi$$
0.142093 + 0.989853i $$0.454617\pi$$
$$812$$ 0 0
$$813$$ 1.43008e7 0.758812
$$814$$ 0 0
$$815$$ −1.04375e7 −0.550431
$$816$$ 0 0
$$817$$ 1.74088e6 0.0912460
$$818$$ 0 0
$$819$$ −4.09209e6 −0.213174
$$820$$ 0 0
$$821$$ −1.48802e7 −0.770464 −0.385232 0.922820i $$-0.625879\pi$$
−0.385232 + 0.922820i $$0.625879\pi$$
$$822$$ 0 0
$$823$$ −2.00601e7 −1.03236 −0.516182 0.856479i $$-0.672647\pi$$
−0.516182 + 0.856479i $$0.672647\pi$$
$$824$$ 0 0
$$825$$ −1.32082e7 −0.675628
$$826$$ 0 0
$$827$$ 1.21539e7 0.617949 0.308975 0.951070i $$-0.400014\pi$$
0.308975 + 0.951070i $$0.400014\pi$$
$$828$$ 0 0
$$829$$ −3.21197e7 −1.62325 −0.811625 0.584179i $$-0.801417\pi$$
−0.811625 + 0.584179i $$0.801417\pi$$
$$830$$ 0 0
$$831$$ −633020. −0.0317991
$$832$$ 0 0
$$833$$ −3.50066e6 −0.174798
$$834$$ 0 0
$$835$$ −6.08328e6 −0.301941
$$836$$ 0 0
$$837$$ −2.84559e7 −1.40397
$$838$$ 0 0
$$839$$ 1.01320e6 0.0496922 0.0248461 0.999691i $$-0.492090\pi$$
0.0248461 + 0.999691i $$0.492090\pi$$
$$840$$ 0 0
$$841$$ 3.16681e6 0.154394
$$842$$ 0 0
$$843$$ −4.96614e6 −0.240686
$$844$$ 0 0
$$845$$ 2.53991e6 0.122370
$$846$$ 0 0
$$847$$ 2.35960e6 0.113013
$$848$$ 0 0
$$849$$ −1.15842e7 −0.551564
$$850$$ 0 0
$$851$$ −6.01860e7 −2.84886
$$852$$ 0 0
$$853$$ −234824. −0.0110502 −0.00552510 0.999985i $$-0.501759\pi$$
−0.00552510 + 0.999985i $$0.501759\pi$$
$$854$$ 0 0
$$855$$ 5.64564e6 0.264118
$$856$$ 0 0
$$857$$ 2.83802e7 1.31997 0.659985 0.751279i $$-0.270563\pi$$
0.659985 + 0.751279i $$0.270563\pi$$
$$858$$ 0 0
$$859$$ 4.00081e7 1.84997 0.924986 0.380001i $$-0.124076\pi$$
0.924986 + 0.380001i $$0.124076\pi$$
$$860$$ 0 0
$$861$$ −3.04878e6 −0.140158
$$862$$ 0 0
$$863$$ 2.08030e7 0.950823 0.475411 0.879764i $$-0.342299\pi$$
0.475411 + 0.879764i $$0.342299\pi$$
$$864$$ 0 0
$$865$$ −3.70904e7 −1.68547
$$866$$ 0 0
$$867$$ 7.05907e6 0.318933
$$868$$ 0 0
$$869$$ 1.44050e7 0.647088
$$870$$ 0 0
$$871$$ 1.01920e7 0.455211
$$872$$ 0 0
$$873$$ 2.42785e6 0.107817
$$874$$ 0 0
$$875$$ 3.31750e6 0.146484
$$876$$ 0 0
$$877$$ −3.03559e7 −1.33273 −0.666367 0.745624i $$-0.732152\pi$$
−0.666367 + 0.745624i $$0.732152\pi$$
$$878$$ 0 0
$$879$$ −1.43886e7 −0.628125
$$880$$ 0 0
$$881$$ −2.58936e7 −1.12396 −0.561981 0.827150i $$-0.689961\pi$$
−0.561981 + 0.827150i $$0.689961\pi$$
$$882$$ 0 0
$$883$$ −1.88813e7 −0.814950 −0.407475 0.913216i $$-0.633591\pi$$
−0.407475 + 0.913216i $$0.633591\pi$$
$$884$$ 0 0
$$885$$ −2.88137e7 −1.23663
$$886$$ 0 0
$$887$$ 2.34431e7 1.00048 0.500238 0.865888i $$-0.333246\pi$$
0.500238 + 0.865888i $$0.333246\pi$$
$$888$$ 0 0
$$889$$ 1.54836e7 0.657079
$$890$$ 0 0
$$891$$ 1.29394e6 0.0546033
$$892$$ 0 0
$$893$$ 851640. 0.0357378
$$894$$ 0 0
$$895$$ 898128. 0.0374784
$$896$$ 0 0
$$897$$ −2.45280e7 −1.01784
$$898$$ 0 0
$$899$$ −3.58722e7 −1.48033
$$900$$ 0 0
$$901$$ −5.42988e7 −2.22833
$$902$$ 0 0
$$903$$ −1.81496e6 −0.0740709
$$904$$ 0 0
$$905$$ −4.58694e7 −1.86166
$$906$$ 0 0
$$907$$ −5.60873e6 −0.226384 −0.113192 0.993573i $$-0.536108\pi$$
−0.113192 + 0.993573i $$0.536108\pi$$
$$908$$ 0 0
$$909$$ 1.41862e7 0.569450
$$910$$ 0 0
$$911$$ −2.16215e7 −0.863156 −0.431578 0.902076i $$-0.642043\pi$$
−0.431578 + 0.902076i $$0.642043\pi$$
$$912$$ 0 0
$$913$$ 1.18279e7 0.469602
$$914$$ 0 0
$$915$$ 2.05598e7 0.811834
$$916$$ 0 0
$$917$$ 1.20863e6 0.0474648
$$918$$ 0 0
$$919$$ −4.51695e7 −1.76424 −0.882119 0.471028i $$-0.843883\pi$$
−0.882119 + 0.471028i $$0.843883\pi$$
$$920$$ 0 0
$$921$$ −9.89098e6 −0.384229
$$922$$ 0 0
$$923$$ 1.64828e7 0.636835
$$924$$ 0 0
$$925$$ −5.63312e7 −2.16469
$$926$$ 0 0
$$927$$ −1.88251e7 −0.719512
$$928$$ 0 0
$$929$$ −2.28729e7 −0.869524 −0.434762 0.900545i $$-0.643168\pi$$
−0.434762 + 0.900545i $$0.643168\pi$$
$$930$$ 0 0
$$931$$ 1.12847e6 0.0426693
$$932$$ 0 0
$$933$$ 2.22050e7 0.835117
$$934$$ 0 0
$$935$$ −4.11506e7 −1.53938
$$936$$ 0 0
$$937$$ −1.79616e7 −0.668336 −0.334168 0.942514i $$-0.608455\pi$$
−0.334168 + 0.942514i $$0.608455\pi$$
$$938$$ 0 0
$$939$$ 2.33008e7 0.862395
$$940$$ 0 0
$$941$$ 1.79697e7 0.661558 0.330779 0.943708i $$-0.392689\pi$$
0.330779 + 0.943708i $$0.392689\pi$$
$$942$$ 0 0
$$943$$ 2.61324e7 0.956974
$$944$$ 0 0
$$945$$ −1.58878e7 −0.578740
$$946$$ 0 0
$$947$$ 4.32115e7 1.56576 0.782879 0.622174i $$-0.213750\pi$$
0.782879 + 0.622174i $$0.213750\pi$$
$$948$$ 0 0
$$949$$ −2.10357e6 −0.0758213
$$950$$ 0 0
$$951$$ −4.27542e6 −0.153295
$$952$$ 0 0
$$953$$ −7.50965e6 −0.267848 −0.133924 0.990992i $$-0.542758\pi$$
−0.133924 + 0.990992i $$0.542758\pi$$
$$954$$ 0 0
$$955$$ −4.83820e7 −1.71662
$$956$$ 0 0
$$957$$ 1.63498e7 0.577074
$$958$$ 0 0
$$959$$ −1.48585e7 −0.521708
$$960$$ 0 0
$$961$$ 2.57172e7 0.898288
$$962$$ 0 0
$$963$$ −6.98584e6 −0.242746
$$964$$ 0 0
$$965$$ 3.47708e7 1.20198
$$966$$ 0 0
$$967$$ 1.69305e7 0.582242 0.291121 0.956686i $$-0.405972\pi$$
0.291121 + 0.956686i $$0.405972\pi$$
$$968$$ 0 0
$$969$$ −6.85260e6 −0.234448
$$970$$ 0 0
$$971$$ 2.86144e7 0.973949 0.486974 0.873416i $$-0.338101\pi$$
0.486974 + 0.873416i $$0.338101\pi$$
$$972$$ 0 0
$$973$$ −1.22787e7 −0.415787
$$974$$ 0 0
$$975$$ −2.29570e7 −0.773400
$$976$$ 0 0
$$977$$ 3.69445e7 1.23826 0.619132 0.785287i $$-0.287485\pi$$
0.619132 + 0.785287i $$0.287485\pi$$
$$978$$ 0 0
$$979$$ −8.98128e6 −0.299489
$$980$$ 0 0
$$981$$ −8.06148e6 −0.267450
$$982$$ 0 0
$$983$$ 3.88787e7 1.28330 0.641650 0.766998i $$-0.278250\pi$$
0.641650 + 0.766998i $$0.278250\pi$$
$$984$$ 0 0
$$985$$ −4.15755e7 −1.36536
$$986$$ 0 0
$$987$$ −887880. −0.0290109
$$988$$ 0 0
$$989$$ 1.55568e7 0.505743
$$990$$ 0 0
$$991$$ −2.49212e7 −0.806092 −0.403046 0.915180i $$-0.632049\pi$$
−0.403046 + 0.915180i $$0.632049\pi$$
$$992$$ 0 0
$$993$$ −3.96616e6 −0.127643
$$994$$ 0 0
$$995$$ 4.37106e7 1.39968
$$996$$ 0 0
$$997$$ −1.01956e7 −0.324845 −0.162422 0.986721i $$-0.551931\pi$$
−0.162422 + 0.986721i $$0.551931\pi$$
$$998$$ 0 0
$$999$$ 5.53138e7 1.75356
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 448.6.a.l.1.1 1
4.3 odd 2 448.6.a.e.1.1 1
8.3 odd 2 14.6.a.a.1.1 1
8.5 even 2 112.6.a.c.1.1 1
24.5 odd 2 1008.6.a.b.1.1 1
24.11 even 2 126.6.a.f.1.1 1
40.3 even 4 350.6.c.d.99.2 2
40.19 odd 2 350.6.a.i.1.1 1
40.27 even 4 350.6.c.d.99.1 2
56.3 even 6 98.6.c.d.79.1 2
56.11 odd 6 98.6.c.c.79.1 2
56.13 odd 2 784.6.a.i.1.1 1
56.19 even 6 98.6.c.d.67.1 2
56.27 even 2 98.6.a.a.1.1 1
56.51 odd 6 98.6.c.c.67.1 2
168.83 odd 2 882.6.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.a.1.1 1 8.3 odd 2
98.6.a.a.1.1 1 56.27 even 2
98.6.c.c.67.1 2 56.51 odd 6
98.6.c.c.79.1 2 56.11 odd 6
98.6.c.d.67.1 2 56.19 even 6
98.6.c.d.79.1 2 56.3 even 6
112.6.a.c.1.1 1 8.5 even 2
126.6.a.f.1.1 1 24.11 even 2
350.6.a.i.1.1 1 40.19 odd 2
350.6.c.d.99.1 2 40.27 even 4
350.6.c.d.99.2 2 40.3 even 4
448.6.a.e.1.1 1 4.3 odd 2
448.6.a.l.1.1 1 1.1 even 1 trivial
784.6.a.i.1.1 1 56.13 odd 2
882.6.a.x.1.1 1 168.83 odd 2
1008.6.a.b.1.1 1 24.5 odd 2