# Properties

 Label 2925.2.a.o.1.1 Level $2925$ Weight $2$ Character 2925.1 Self dual yes Analytic conductor $23.356$ 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] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.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: $$23.3562425912$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 975) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2925.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{7} -3.00000 q^{8} +1.00000 q^{11} -1.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} -5.00000 q^{17} -8.00000 q^{19} +1.00000 q^{22} -1.00000 q^{26} -3.00000 q^{28} -1.00000 q^{29} +3.00000 q^{31} +5.00000 q^{32} -5.00000 q^{34} +8.00000 q^{37} -8.00000 q^{38} +2.00000 q^{41} -8.00000 q^{43} -1.00000 q^{44} -11.0000 q^{47} +2.00000 q^{49} +1.00000 q^{52} -11.0000 q^{53} -9.00000 q^{56} -1.00000 q^{58} -5.00000 q^{59} +1.00000 q^{61} +3.00000 q^{62} +7.00000 q^{64} -3.00000 q^{67} +5.00000 q^{68} -16.0000 q^{71} +4.00000 q^{73} +8.00000 q^{74} +8.00000 q^{76} +3.00000 q^{77} +12.0000 q^{79} +2.00000 q^{82} -3.00000 q^{83} -8.00000 q^{86} -3.00000 q^{88} -3.00000 q^{91} -11.0000 q^{94} +2.00000 q^{97} +2.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 3.00000 0.801784
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −3.00000 −0.566947
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −5.00000 −0.857493
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ −8.00000 −1.29777
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −11.0000 −1.60451 −0.802257 0.596978i $$-0.796368\pi$$
−0.802257 + 0.596978i $$0.796368\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −11.0000 −1.51097 −0.755483 0.655168i $$-0.772598\pi$$
−0.755483 + 0.655168i $$0.772598\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −9.00000 −1.20268
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ −5.00000 −0.650945 −0.325472 0.945552i $$-0.605523\pi$$
−0.325472 + 0.945552i $$0.605523\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 5.00000 0.606339
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 3.00000 0.341882
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.00000 0.220863
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −11.0000 −1.13456
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 2.00000 0.202031
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 3.00000 0.294174
$$105$$ 0 0
$$106$$ −11.0000 −1.06841
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −3.00000 −0.283473
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ 0 0
$$118$$ −5.00000 −0.460287
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 1.00000 0.0905357
$$123$$ 0 0
$$124$$ −3.00000 −0.269408
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ −24.0000 −2.08106
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ 15.0000 1.28624
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ −1.00000 −0.0836242
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ −8.00000 −0.657596
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −17.0000 −1.38344 −0.691720 0.722166i $$-0.743147\pi$$
−0.691720 + 0.722166i $$0.743147\pi$$
$$152$$ 24.0000 1.94666
$$153$$ 0 0
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −3.00000 −0.232845
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ −21.0000 −1.59660 −0.798300 0.602260i $$-0.794267\pi$$
−0.798300 + 0.602260i $$0.794267\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −26.0000 −1.94333 −0.971666 0.236360i $$-0.924046\pi$$
−0.971666 + 0.236360i $$0.924046\pi$$
$$180$$ 0 0
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ −3.00000 −0.222375
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.00000 −0.365636
$$188$$ 11.0000 0.802257
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −9.00000 −0.633238
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 11.0000 0.755483
$$213$$ 0 0
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 9.00000 0.610960
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.00000 0.336336
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 15.0000 1.00223
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 23.0000 1.52656 0.763282 0.646066i $$-0.223587\pi$$
0.763282 + 0.646066i $$0.223587\pi$$
$$228$$ 0 0
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 5.00000 0.325472
$$237$$ 0 0
$$238$$ −15.0000 −0.972306
$$239$$ 19.0000 1.22901 0.614504 0.788914i $$-0.289356\pi$$
0.614504 + 0.788914i $$0.289356\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ 0 0
$$244$$ −1.00000 −0.0640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000 0.509028
$$248$$ −9.00000 −0.571501
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 30.0000 1.89358 0.946792 0.321847i $$-0.104304\pi$$
0.946792 + 0.321847i $$0.104304\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.00000 −0.370681
$$263$$ 14.0000 0.863277 0.431638 0.902047i $$-0.357936\pi$$
0.431638 + 0.902047i $$0.357936\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −24.0000 −1.47153
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ −17.0000 −1.03651 −0.518254 0.855227i $$-0.673418\pi$$
−0.518254 + 0.855227i $$0.673418\pi$$
$$270$$ 0 0
$$271$$ −5.00000 −0.303728 −0.151864 0.988401i $$-0.548528\pi$$
−0.151864 + 0.988401i $$0.548528\pi$$
$$272$$ 5.00000 0.303170
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ −10.0000 −0.599760
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 18.0000 1.06999 0.534994 0.844856i $$-0.320314\pi$$
0.534994 + 0.844856i $$0.320314\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ −1.00000 −0.0591312
$$287$$ 6.00000 0.354169
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.00000 −0.234082
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −24.0000 −1.39497
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −24.0000 −1.38334
$$302$$ −17.0000 −0.978240
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 29.0000 1.63918 0.819588 0.572953i $$-0.194202\pi$$
0.819588 + 0.572953i $$0.194202\pi$$
$$314$$ 11.0000 0.620766
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −16.0000 −0.898650 −0.449325 0.893368i $$-0.648335\pi$$
−0.449325 + 0.893368i $$0.648335\pi$$
$$318$$ 0 0
$$319$$ −1.00000 −0.0559893
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 40.0000 2.22566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ −33.0000 −1.81935
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 3.00000 0.164646
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −5.00000 −0.272367 −0.136184 0.990684i $$-0.543484\pi$$
−0.136184 + 0.990684i $$0.543484\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.00000 0.162459
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ −21.0000 −1.12897
$$347$$ −2.00000 −0.107366 −0.0536828 0.998558i $$-0.517096\pi$$
−0.0536828 + 0.998558i $$0.517096\pi$$
$$348$$ 0 0
$$349$$ 16.0000 0.856460 0.428230 0.903670i $$-0.359137\pi$$
0.428230 + 0.903670i $$0.359137\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −26.0000 −1.37414
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 5.00000 0.262794
$$363$$ 0 0
$$364$$ 3.00000 0.157243
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 12.0000 0.626395 0.313197 0.949688i $$-0.398600\pi$$
0.313197 + 0.949688i $$0.398600\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −33.0000 −1.71327
$$372$$ 0 0
$$373$$ 31.0000 1.60512 0.802560 0.596572i $$-0.203471\pi$$
0.802560 + 0.596572i $$0.203471\pi$$
$$374$$ −5.00000 −0.258544
$$375$$ 0 0
$$376$$ 33.0000 1.70185
$$377$$ 1.00000 0.0515026
$$378$$ 0 0
$$379$$ −35.0000 −1.79783 −0.898915 0.438124i $$-0.855643\pi$$
−0.898915 + 0.438124i $$0.855643\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −10.0000 −0.511645
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −6.00000 −0.303046
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ −3.00000 −0.149441
$$404$$ 9.00000 0.447767
$$405$$ 0 0
$$406$$ −3.00000 −0.148888
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ −15.0000 −0.738102
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ 0 0
$$418$$ −8.00000 −0.391293
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 28.0000 1.36464 0.682318 0.731055i $$-0.260972\pi$$
0.682318 + 0.731055i $$0.260972\pi$$
$$422$$ −16.0000 −0.778868
$$423$$ 0 0
$$424$$ 33.0000 1.60262
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3.00000 0.145180
$$428$$ −18.0000 −0.870063
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 0 0
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ 9.00000 0.432014
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 4.00000 0.190910 0.0954548 0.995434i $$-0.469569\pi$$
0.0954548 + 0.995434i $$0.469569\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 5.00000 0.237826
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ 21.0000 0.992157
$$449$$ 28.0000 1.32140 0.660701 0.750649i $$-0.270259\pi$$
0.660701 + 0.750649i $$0.270259\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 2.00000 0.0940721
$$453$$ 0 0
$$454$$ 23.0000 1.07944
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 34.0000 1.59045 0.795226 0.606313i $$-0.207352\pi$$
0.795226 + 0.606313i $$0.207352\pi$$
$$458$$ −18.0000 −0.841085
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 0 0
$$463$$ 9.00000 0.418265 0.209133 0.977887i $$-0.432936\pi$$
0.209133 + 0.977887i $$0.432936\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 32.0000 1.48078 0.740392 0.672176i $$-0.234640\pi$$
0.740392 + 0.672176i $$0.234640\pi$$
$$468$$ 0 0
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 15.0000 0.690431
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 15.0000 0.687524
$$477$$ 0 0
$$478$$ 19.0000 0.869040
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 10.0000 0.454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −37.0000 −1.67663 −0.838315 0.545186i $$-0.816459\pi$$
−0.838315 + 0.545186i $$0.816459\pi$$
$$488$$ −3.00000 −0.135804
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 26.0000 1.17336 0.586682 0.809818i $$-0.300434\pi$$
0.586682 + 0.809818i $$0.300434\pi$$
$$492$$ 0 0
$$493$$ 5.00000 0.225189
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −3.00000 −0.134704
$$497$$ −48.0000 −2.15309
$$498$$ 0 0
$$499$$ −41.0000 −1.83541 −0.917706 0.397260i $$-0.869961\pi$$
−0.917706 + 0.397260i $$0.869961\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 30.0000 1.33897
$$503$$ 26.0000 1.15928 0.579641 0.814872i $$-0.303193\pi$$
0.579641 + 0.814872i $$0.303193\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −3.00000 −0.132324
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −11.0000 −0.483779
$$518$$ 24.0000 1.05450
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ −26.0000 −1.13690 −0.568450 0.822718i $$-0.692457\pi$$
−0.568450 + 0.822718i $$0.692457\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 14.0000 0.610429
$$527$$ −15.0000 −0.653410
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 24.0000 1.04053
$$533$$ −2.00000 −0.0866296
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 9.00000 0.388741
$$537$$ 0 0
$$538$$ −17.0000 −0.732922
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ 12.0000 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$542$$ −5.00000 −0.214768
$$543$$ 0 0
$$544$$ −25.0000 −1.07187
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 22.0000 0.940652 0.470326 0.882493i $$-0.344136\pi$$
0.470326 + 0.882493i $$0.344136\pi$$
$$548$$ 12.0000 0.512615
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 36.0000 1.53088
$$554$$ −6.00000 −0.254916
$$555$$ 0 0
$$556$$ 10.0000 0.424094
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 18.0000 0.756596
$$567$$ 0 0
$$568$$ 48.0000 2.01404
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 1.00000 0.0418121
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −9.00000 −0.373383
$$582$$ 0 0
$$583$$ −11.0000 −0.455573
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ −45.0000 −1.85735 −0.928674 0.370896i $$-0.879051\pi$$
−0.928674 + 0.370896i $$0.879051\pi$$
$$588$$ 0 0
$$589$$ −24.0000 −0.988903
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −8.00000 −0.328798
$$593$$ 26.0000 1.06769 0.533846 0.845582i $$-0.320746\pi$$
0.533846 + 0.845582i $$0.320746\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −46.0000 −1.87951 −0.939755 0.341850i $$-0.888947\pi$$
−0.939755 + 0.341850i $$0.888947\pi$$
$$600$$ 0 0
$$601$$ 5.00000 0.203954 0.101977 0.994787i $$-0.467483\pi$$
0.101977 + 0.994787i $$0.467483\pi$$
$$602$$ −24.0000 −0.978167
$$603$$ 0 0
$$604$$ 17.0000 0.691720
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −38.0000 −1.54237 −0.771186 0.636610i $$-0.780336\pi$$
−0.771186 + 0.636610i $$0.780336\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 11.0000 0.445012
$$612$$ 0 0
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ −9.00000 −0.362620
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −6.00000 −0.240578
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 29.0000 1.15907
$$627$$ 0 0
$$628$$ −11.0000 −0.438948
$$629$$ −40.0000 −1.59490
$$630$$ 0 0
$$631$$ −44.0000 −1.75161 −0.875806 0.482663i $$-0.839670\pi$$
−0.875806 + 0.482663i $$0.839670\pi$$
$$632$$ −36.0000 −1.43200
$$633$$ 0 0
$$634$$ −16.0000 −0.635441
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ −1.00000 −0.0395904
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 15.0000 0.592464 0.296232 0.955116i $$-0.404270\pi$$
0.296232 + 0.955116i $$0.404270\pi$$
$$642$$ 0 0
$$643$$ 12.0000 0.473234 0.236617 0.971603i $$-0.423961\pi$$
0.236617 + 0.971603i $$0.423961\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 40.0000 1.57378
$$647$$ 16.0000 0.629025 0.314512 0.949253i $$-0.398159\pi$$
0.314512 + 0.949253i $$0.398159\pi$$
$$648$$ 0 0
$$649$$ −5.00000 −0.196267
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −20.0000 −0.783260
$$653$$ 35.0000 1.36966 0.684828 0.728705i $$-0.259877\pi$$
0.684828 + 0.728705i $$0.259877\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 0 0
$$658$$ −33.0000 −1.28647
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ 0 0
$$664$$ 9.00000 0.349268
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1.00000 0.0386046
$$672$$ 0 0
$$673$$ −45.0000 −1.73462 −0.867311 0.497766i $$-0.834154\pi$$
−0.867311 + 0.497766i $$0.834154\pi$$
$$674$$ −5.00000 −0.192593
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 3.00000 0.114876
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ 11.0000 0.419067
$$690$$ 0 0
$$691$$ −29.0000 −1.10321 −0.551606 0.834105i $$-0.685985\pi$$
−0.551606 + 0.834105i $$0.685985\pi$$
$$692$$ 21.0000 0.798300
$$693$$ 0 0
$$694$$ −2.00000 −0.0759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −10.0000 −0.378777
$$698$$ 16.0000 0.605609
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −31.0000 −1.17085 −0.585427 0.810725i $$-0.699073\pi$$
−0.585427 + 0.810725i $$0.699073\pi$$
$$702$$ 0 0
$$703$$ −64.0000 −2.41381
$$704$$ 7.00000 0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ −27.0000 −1.01544
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 26.0000 0.971666
$$717$$ 0 0
$$718$$ −15.0000 −0.559795
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ 45.0000 1.67473
$$723$$ 0 0
$$724$$ −5.00000 −0.185824
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.00000 0.0741759 0.0370879 0.999312i $$-0.488192\pi$$
0.0370879 + 0.999312i $$0.488192\pi$$
$$728$$ 9.00000 0.333562
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 40.0000 1.47945
$$732$$ 0 0
$$733$$ 20.0000 0.738717 0.369358 0.929287i $$-0.379577\pi$$
0.369358 + 0.929287i $$0.379577\pi$$
$$734$$ 12.0000 0.442928
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3.00000 −0.110506
$$738$$ 0 0
$$739$$ 7.00000 0.257499 0.128750 0.991677i $$-0.458904\pi$$
0.128750 + 0.991677i $$0.458904\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −33.0000 −1.21147
$$743$$ 13.0000 0.476924 0.238462 0.971152i $$-0.423357\pi$$
0.238462 + 0.971152i $$0.423357\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 31.0000 1.13499
$$747$$ 0 0
$$748$$ 5.00000 0.182818
$$749$$ 54.0000 1.97312
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 11.0000 0.401129
$$753$$ 0 0
$$754$$ 1.00000 0.0364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −23.0000 −0.835949 −0.417975 0.908459i $$-0.637260\pi$$
−0.417975 + 0.908459i $$0.637260\pi$$
$$758$$ −35.0000 −1.27126
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 24.0000 0.869999 0.435000 0.900431i $$-0.356748\pi$$
0.435000 + 0.900431i $$0.356748\pi$$
$$762$$ 0 0
$$763$$ −30.0000 −1.08607
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 5.00000 0.180540
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 10.0000 0.359908
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −16.0000 −0.573259
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −17.0000 −0.605985 −0.302992 0.952993i $$-0.597986\pi$$
−0.302992 + 0.952993i $$0.597986\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ −1.00000 −0.0355110
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ −43.0000 −1.52314 −0.761569 0.648084i $$-0.775571\pi$$
−0.761569 + 0.648084i $$0.775571\pi$$
$$798$$ 0 0
$$799$$ 55.0000 1.94576
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 4.00000 0.141157
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3.00000 −0.105670
$$807$$ 0 0
$$808$$ 27.0000 0.949857
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ −33.0000 −1.15879 −0.579393 0.815048i $$-0.696710\pi$$
−0.579393 + 0.815048i $$0.696710\pi$$
$$812$$ 3.00000 0.105279
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 64.0000 2.23908
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ −42.0000 −1.46314
$$825$$ 0 0
$$826$$ −15.0000 −0.521917
$$827$$ 1.00000 0.0347734 0.0173867 0.999849i $$-0.494465\pi$$
0.0173867 + 0.999849i $$0.494465\pi$$
$$828$$ 0 0
$$829$$ 51.0000 1.77130 0.885652 0.464350i $$-0.153712\pi$$
0.885652 + 0.464350i $$0.153712\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −7.00000 −0.242681
$$833$$ −10.0000 −0.346479
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 8.00000 0.276686
$$837$$ 0 0
$$838$$ 26.0000 0.898155
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 28.0000 0.964944
$$843$$ 0 0
$$844$$ 16.0000 0.550743
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −30.0000 −1.03081
$$848$$ 11.0000 0.377742
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −24.0000 −0.821744 −0.410872 0.911693i $$-0.634776\pi$$
−0.410872 + 0.911693i $$0.634776\pi$$
$$854$$ 3.00000 0.102658
$$855$$ 0 0
$$856$$ −54.0000 −1.84568
$$857$$ −38.0000 −1.29806 −0.649028 0.760765i $$-0.724824\pi$$
−0.649028 + 0.760765i $$0.724824\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ −7.00000 −0.238283 −0.119141 0.992877i $$-0.538014\pi$$
−0.119141 + 0.992877i $$0.538014\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 0 0
$$868$$ −9.00000 −0.305480
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 30.0000 1.01593
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −24.0000 −0.810422 −0.405211 0.914223i $$-0.632802\pi$$
−0.405211 + 0.914223i $$0.632802\pi$$
$$878$$ 4.00000 0.134993
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25.0000 0.842271 0.421136 0.906998i $$-0.361632\pi$$
0.421136 + 0.906998i $$0.361632\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ −5.00000 −0.168168
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ 88.0000 2.94481
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −9.00000 −0.300669
$$897$$ 0 0
$$898$$ 28.0000 0.934372
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ 55.0000 1.83232
$$902$$ 2.00000 0.0665927
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ −23.0000 −0.763282
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −56.0000 −1.85536 −0.927681 0.373373i $$-0.878201\pi$$
−0.927681 + 0.373373i $$0.878201\pi$$
$$912$$ 0 0
$$913$$ −3.00000 −0.0992855
$$914$$ 34.0000 1.12462
$$915$$ 0 0
$$916$$ 18.0000 0.594737
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ 22.0000 0.725713 0.362857 0.931845i $$-0.381802\pi$$
0.362857 + 0.931845i $$0.381802\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −42.0000 −1.38320
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 9.00000 0.295758
$$927$$ 0 0
$$928$$ −5.00000 −0.164133
$$929$$ 26.0000 0.853032 0.426516 0.904480i $$-0.359741\pi$$
0.426516 + 0.904480i $$0.359741\pi$$
$$930$$ 0 0
$$931$$ −16.0000 −0.524379
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 32.0000 1.04707
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −45.0000 −1.47009 −0.735043 0.678021i $$-0.762838\pi$$
−0.735043 + 0.678021i $$0.762838\pi$$
$$938$$ −9.00000 −0.293860
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 56.0000 1.82555 0.912774 0.408465i $$-0.133936\pi$$
0.912774 + 0.408465i $$0.133936\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 5.00000 0.162736
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 21.0000 0.682408 0.341204 0.939989i $$-0.389165\pi$$
0.341204 + 0.939989i $$0.389165\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 45.0000 1.45846
$$953$$ 37.0000 1.19855 0.599274 0.800544i $$-0.295456\pi$$
0.599274 + 0.800544i $$0.295456\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −19.0000 −0.614504
$$957$$ 0 0
$$958$$ 15.0000 0.484628
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −8.00000 −0.257930
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −37.0000 −1.18984 −0.594920 0.803785i $$-0.702816\pi$$
−0.594920 + 0.803785i $$0.702816\pi$$
$$968$$ 30.0000 0.964237
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −54.0000 −1.73294 −0.866471 0.499227i $$-0.833617\pi$$
−0.866471 + 0.499227i $$0.833617\pi$$
$$972$$ 0 0
$$973$$ −30.0000 −0.961756
$$974$$ −37.0000 −1.18556
$$975$$ 0 0
$$976$$ −1.00000 −0.0320092
$$977$$ 28.0000 0.895799 0.447900 0.894084i $$-0.352172\pi$$
0.447900 + 0.894084i $$0.352172\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 26.0000 0.829693
$$983$$ −21.0000 −0.669796 −0.334898 0.942254i $$-0.608702\pi$$
−0.334898 + 0.942254i $$0.608702\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 5.00000 0.159232
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 15.0000 0.476250
$$993$$ 0 0
$$994$$ −48.0000 −1.52247
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −3.00000 −0.0950110 −0.0475055 0.998871i $$-0.515127\pi$$
−0.0475055 + 0.998871i $$0.515127\pi$$
$$998$$ −41.0000 −1.29783
$$999$$ 0 0
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 2925.2.a.o.1.1 1
3.2 odd 2 975.2.a.d.1.1 1
5.2 odd 4 2925.2.c.i.2224.2 2
5.3 odd 4 2925.2.c.i.2224.1 2
5.4 even 2 2925.2.a.b.1.1 1
15.2 even 4 975.2.c.g.274.1 2
15.8 even 4 975.2.c.g.274.2 2
15.14 odd 2 975.2.a.k.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
975.2.a.d.1.1 1 3.2 odd 2
975.2.a.k.1.1 yes 1 15.14 odd 2
975.2.c.g.274.1 2 15.2 even 4
975.2.c.g.274.2 2 15.8 even 4
2925.2.a.b.1.1 1 5.4 even 2
2925.2.a.o.1.1 1 1.1 even 1 trivial
2925.2.c.i.2224.1 2 5.3 odd 4
2925.2.c.i.2224.2 2 5.2 odd 4