# Properties

 Label 6525.2.a.s.1.2 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} -1.61803 q^{4} +3.00000 q^{7} -2.23607 q^{8} +O(q^{10})$$ $$q+0.618034 q^{2} -1.61803 q^{4} +3.00000 q^{7} -2.23607 q^{8} +5.47214 q^{11} +6.23607 q^{13} +1.85410 q^{14} +1.85410 q^{16} +3.47214 q^{17} +7.70820 q^{19} +3.38197 q^{22} +3.85410 q^{26} -4.85410 q^{28} -1.00000 q^{29} -8.00000 q^{31} +5.61803 q^{32} +2.14590 q^{34} +8.00000 q^{37} +4.76393 q^{38} +4.47214 q^{41} -3.23607 q^{43} -8.85410 q^{44} +6.70820 q^{47} +2.00000 q^{49} -10.0902 q^{52} -6.76393 q^{53} -6.70820 q^{56} -0.618034 q^{58} -5.23607 q^{59} -5.70820 q^{61} -4.94427 q^{62} -0.236068 q^{64} +11.4721 q^{67} -5.61803 q^{68} -7.23607 q^{71} +8.00000 q^{73} +4.94427 q^{74} -12.4721 q^{76} +16.4164 q^{77} -6.18034 q^{79} +2.76393 q^{82} -3.70820 q^{83} -2.00000 q^{86} -12.2361 q^{88} -11.1803 q^{89} +18.7082 q^{91} +4.14590 q^{94} -2.76393 q^{97} +1.23607 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 6 q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 + 6 * q^7 $$2 q - q^{2} - q^{4} + 6 q^{7} + 2 q^{11} + 8 q^{13} - 3 q^{14} - 3 q^{16} - 2 q^{17} + 2 q^{19} + 9 q^{22} + q^{26} - 3 q^{28} - 2 q^{29} - 16 q^{31} + 9 q^{32} + 11 q^{34} + 16 q^{37} + 14 q^{38} - 2 q^{43} - 11 q^{44} + 4 q^{49} - 9 q^{52} - 18 q^{53} + q^{58} - 6 q^{59} + 2 q^{61} + 8 q^{62} + 4 q^{64} + 14 q^{67} - 9 q^{68} - 10 q^{71} + 16 q^{73} - 8 q^{74} - 16 q^{76} + 6 q^{77} + 10 q^{79} + 10 q^{82} + 6 q^{83} - 4 q^{86} - 20 q^{88} + 24 q^{91} + 15 q^{94} - 10 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 6 * q^7 + 2 * q^11 + 8 * q^13 - 3 * q^14 - 3 * q^16 - 2 * q^17 + 2 * q^19 + 9 * q^22 + q^26 - 3 * q^28 - 2 * q^29 - 16 * q^31 + 9 * q^32 + 11 * q^34 + 16 * q^37 + 14 * q^38 - 2 * q^43 - 11 * q^44 + 4 * q^49 - 9 * q^52 - 18 * q^53 + q^58 - 6 * q^59 + 2 * q^61 + 8 * q^62 + 4 * q^64 + 14 * q^67 - 9 * q^68 - 10 * q^71 + 16 * q^73 - 8 * q^74 - 16 * q^76 + 6 * q^77 + 10 * q^79 + 10 * q^82 + 6 * q^83 - 4 * q^86 - 20 * q^88 + 24 * q^91 + 15 * q^94 - 10 * q^97 - 2 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.437016 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ 0 0
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.47214 1.64991 0.824956 0.565198i $$-0.191200\pi$$
0.824956 + 0.565198i $$0.191200\pi$$
$$12$$ 0 0
$$13$$ 6.23607 1.72957 0.864787 0.502139i $$-0.167453\pi$$
0.864787 + 0.502139i $$0.167453\pi$$
$$14$$ 1.85410 0.495530
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 3.47214 0.842117 0.421058 0.907034i $$-0.361659\pi$$
0.421058 + 0.907034i $$0.361659\pi$$
$$18$$ 0 0
$$19$$ 7.70820 1.76838 0.884192 0.467124i $$-0.154710\pi$$
0.884192 + 0.467124i $$0.154710\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.38197 0.721038
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 3.85410 0.755852
$$27$$ 0 0
$$28$$ −4.85410 −0.917339
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.61803 0.993137
$$33$$ 0 0
$$34$$ 2.14590 0.368018
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 4.76393 0.772812
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.47214 0.698430 0.349215 0.937043i $$-0.386448\pi$$
0.349215 + 0.937043i $$0.386448\pi$$
$$42$$ 0 0
$$43$$ −3.23607 −0.493496 −0.246748 0.969080i $$-0.579362\pi$$
−0.246748 + 0.969080i $$0.579362\pi$$
$$44$$ −8.85410 −1.33481
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.70820 0.978492 0.489246 0.872146i $$-0.337272\pi$$
0.489246 + 0.872146i $$0.337272\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −10.0902 −1.39925
$$53$$ −6.76393 −0.929098 −0.464549 0.885548i $$-0.653783\pi$$
−0.464549 + 0.885548i $$0.653783\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.70820 −0.896421
$$57$$ 0 0
$$58$$ −0.618034 −0.0811518
$$59$$ −5.23607 −0.681678 −0.340839 0.940122i $$-0.610711\pi$$
−0.340839 + 0.940122i $$0.610711\pi$$
$$60$$ 0 0
$$61$$ −5.70820 −0.730861 −0.365430 0.930839i $$-0.619078\pi$$
−0.365430 + 0.930839i $$0.619078\pi$$
$$62$$ −4.94427 −0.627923
$$63$$ 0 0
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.4721 1.40154 0.700772 0.713385i $$-0.252839\pi$$
0.700772 + 0.713385i $$0.252839\pi$$
$$68$$ −5.61803 −0.681287
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −7.23607 −0.858763 −0.429382 0.903123i $$-0.641268\pi$$
−0.429382 + 0.903123i $$0.641268\pi$$
$$72$$ 0 0
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ 4.94427 0.574760
$$75$$ 0 0
$$76$$ −12.4721 −1.43065
$$77$$ 16.4164 1.87082
$$78$$ 0 0
$$79$$ −6.18034 −0.695343 −0.347671 0.937616i $$-0.613027\pi$$
−0.347671 + 0.937616i $$0.613027\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.76393 0.305225
$$83$$ −3.70820 −0.407028 −0.203514 0.979072i $$-0.565236\pi$$
−0.203514 + 0.979072i $$0.565236\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.00000 −0.215666
$$87$$ 0 0
$$88$$ −12.2361 −1.30437
$$89$$ −11.1803 −1.18511 −0.592557 0.805529i $$-0.701881\pi$$
−0.592557 + 0.805529i $$0.701881\pi$$
$$90$$ 0 0
$$91$$ 18.7082 1.96115
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 4.14590 0.427617
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.76393 −0.280635 −0.140317 0.990107i $$-0.544812\pi$$
−0.140317 + 0.990107i $$0.544812\pi$$
$$98$$ 1.23607 0.124862
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.23607 −0.421505 −0.210752 0.977540i $$-0.567591\pi$$
−0.210752 + 0.977540i $$0.567591\pi$$
$$102$$ 0 0
$$103$$ −7.41641 −0.730760 −0.365380 0.930858i $$-0.619061\pi$$
−0.365380 + 0.930858i $$0.619061\pi$$
$$104$$ −13.9443 −1.36735
$$105$$ 0 0
$$106$$ −4.18034 −0.406031
$$107$$ 7.52786 0.727746 0.363873 0.931449i $$-0.381454\pi$$
0.363873 + 0.931449i $$0.381454\pi$$
$$108$$ 0 0
$$109$$ −16.4164 −1.57241 −0.786203 0.617968i $$-0.787956\pi$$
−0.786203 + 0.617968i $$0.787956\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 5.56231 0.525589
$$113$$ −7.94427 −0.747334 −0.373667 0.927563i $$-0.621900\pi$$
−0.373667 + 0.927563i $$0.621900\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.61803 0.150231
$$117$$ 0 0
$$118$$ −3.23607 −0.297904
$$119$$ 10.4164 0.954871
$$120$$ 0 0
$$121$$ 18.9443 1.72221
$$122$$ −3.52786 −0.319398
$$123$$ 0 0
$$124$$ 12.9443 1.16243
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ −11.3820 −1.00603
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.47214 −0.303362 −0.151681 0.988430i $$-0.548469\pi$$
−0.151681 + 0.988430i $$0.548469\pi$$
$$132$$ 0 0
$$133$$ 23.1246 2.00516
$$134$$ 7.09017 0.612497
$$135$$ 0 0
$$136$$ −7.76393 −0.665752
$$137$$ −10.9443 −0.935032 −0.467516 0.883985i $$-0.654851\pi$$
−0.467516 + 0.883985i $$0.654851\pi$$
$$138$$ 0 0
$$139$$ −0.708204 −0.0600691 −0.0300345 0.999549i $$-0.509562\pi$$
−0.0300345 + 0.999549i $$0.509562\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.47214 −0.375293
$$143$$ 34.1246 2.85364
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 4.94427 0.409191
$$147$$ 0 0
$$148$$ −12.9443 −1.06401
$$149$$ −20.1803 −1.65324 −0.826619 0.562762i $$-0.809739\pi$$
−0.826619 + 0.562762i $$0.809739\pi$$
$$150$$ 0 0
$$151$$ 2.47214 0.201180 0.100590 0.994928i $$-0.467927\pi$$
0.100590 + 0.994928i $$0.467927\pi$$
$$152$$ −17.2361 −1.39803
$$153$$ 0 0
$$154$$ 10.1459 0.817580
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.7082 0.934416 0.467208 0.884147i $$-0.345260\pi$$
0.467208 + 0.884147i $$0.345260\pi$$
$$158$$ −3.81966 −0.303876
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 15.1246 1.18465 0.592326 0.805699i $$-0.298210\pi$$
0.592326 + 0.805699i $$0.298210\pi$$
$$164$$ −7.23607 −0.565042
$$165$$ 0 0
$$166$$ −2.29180 −0.177878
$$167$$ −17.8885 −1.38426 −0.692129 0.721774i $$-0.743327\pi$$
−0.692129 + 0.721774i $$0.743327\pi$$
$$168$$ 0 0
$$169$$ 25.8885 1.99143
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 5.23607 0.399246
$$173$$ −19.4164 −1.47620 −0.738101 0.674690i $$-0.764277\pi$$
−0.738101 + 0.674690i $$0.764277\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 10.1459 0.764776
$$177$$ 0 0
$$178$$ −6.90983 −0.517914
$$179$$ −4.18034 −0.312453 −0.156227 0.987721i $$-0.549933\pi$$
−0.156227 + 0.987721i $$0.549933\pi$$
$$180$$ 0 0
$$181$$ 8.41641 0.625587 0.312793 0.949821i $$-0.398735\pi$$
0.312793 + 0.949821i $$0.398735\pi$$
$$182$$ 11.5623 0.857055
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 19.0000 1.38942
$$188$$ −10.8541 −0.791617
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ −1.70820 −0.122642
$$195$$ 0 0
$$196$$ −3.23607 −0.231148
$$197$$ 14.9443 1.06474 0.532368 0.846513i $$-0.321302\pi$$
0.532368 + 0.846513i $$0.321302\pi$$
$$198$$ 0 0
$$199$$ 20.7082 1.46797 0.733983 0.679168i $$-0.237659\pi$$
0.733983 + 0.679168i $$0.237659\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −2.61803 −0.184204
$$203$$ −3.00000 −0.210559
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.58359 −0.319354
$$207$$ 0 0
$$208$$ 11.5623 0.801702
$$209$$ 42.1803 2.91768
$$210$$ 0 0
$$211$$ −27.8885 −1.91993 −0.959963 0.280126i $$-0.909624\pi$$
−0.959963 + 0.280126i $$0.909624\pi$$
$$212$$ 10.9443 0.751656
$$213$$ 0 0
$$214$$ 4.65248 0.318037
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −24.0000 −1.62923
$$218$$ −10.1459 −0.687167
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 21.6525 1.45650
$$222$$ 0 0
$$223$$ 13.0000 0.870544 0.435272 0.900299i $$-0.356652\pi$$
0.435272 + 0.900299i $$0.356652\pi$$
$$224$$ 16.8541 1.12611
$$225$$ 0 0
$$226$$ −4.90983 −0.326597
$$227$$ −5.81966 −0.386264 −0.193132 0.981173i $$-0.561865\pi$$
−0.193132 + 0.981173i $$0.561865\pi$$
$$228$$ 0 0
$$229$$ 9.41641 0.622254 0.311127 0.950368i $$-0.399294\pi$$
0.311127 + 0.950368i $$0.399294\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.23607 0.146805
$$233$$ 1.41641 0.0927920 0.0463960 0.998923i $$-0.485226\pi$$
0.0463960 + 0.998923i $$0.485226\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 8.47214 0.551489
$$237$$ 0 0
$$238$$ 6.43769 0.417294
$$239$$ −11.8885 −0.769006 −0.384503 0.923124i $$-0.625627\pi$$
−0.384503 + 0.923124i $$0.625627\pi$$
$$240$$ 0 0
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ 11.7082 0.752632
$$243$$ 0 0
$$244$$ 9.23607 0.591279
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 48.0689 3.05855
$$248$$ 17.8885 1.13592
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 24.8885 1.57095 0.785475 0.618893i $$-0.212418\pi$$
0.785475 + 0.618893i $$0.212418\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 3.70820 0.232673
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ −9.70820 −0.605581 −0.302791 0.953057i $$-0.597918\pi$$
−0.302791 + 0.953057i $$0.597918\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −2.14590 −0.132574
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 14.2918 0.876286
$$267$$ 0 0
$$268$$ −18.5623 −1.13387
$$269$$ 30.2361 1.84353 0.921763 0.387754i $$-0.126749\pi$$
0.921763 + 0.387754i $$0.126749\pi$$
$$270$$ 0 0
$$271$$ 14.3607 0.872349 0.436175 0.899862i $$-0.356333\pi$$
0.436175 + 0.899862i $$0.356333\pi$$
$$272$$ 6.43769 0.390343
$$273$$ 0 0
$$274$$ −6.76393 −0.408624
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.70820 −0.523225 −0.261613 0.965173i $$-0.584254\pi$$
−0.261613 + 0.965173i $$0.584254\pi$$
$$278$$ −0.437694 −0.0262511
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 23.1246 1.37950 0.689749 0.724048i $$-0.257721\pi$$
0.689749 + 0.724048i $$0.257721\pi$$
$$282$$ 0 0
$$283$$ −18.4721 −1.09805 −0.549027 0.835804i $$-0.685002\pi$$
−0.549027 + 0.835804i $$0.685002\pi$$
$$284$$ 11.7082 0.694754
$$285$$ 0 0
$$286$$ 21.0902 1.24709
$$287$$ 13.4164 0.791946
$$288$$ 0 0
$$289$$ −4.94427 −0.290840
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −12.9443 −0.757506
$$293$$ −17.9443 −1.04832 −0.524158 0.851621i $$-0.675620\pi$$
−0.524158 + 0.851621i $$0.675620\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −17.8885 −1.03975
$$297$$ 0 0
$$298$$ −12.4721 −0.722491
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −9.70820 −0.559572
$$302$$ 1.52786 0.0879187
$$303$$ 0 0
$$304$$ 14.2918 0.819691
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −24.9443 −1.42364 −0.711822 0.702360i $$-0.752130\pi$$
−0.711822 + 0.702360i $$0.752130\pi$$
$$308$$ −26.5623 −1.51353
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −23.4721 −1.33098 −0.665491 0.746406i $$-0.731778\pi$$
−0.665491 + 0.746406i $$0.731778\pi$$
$$312$$ 0 0
$$313$$ −1.18034 −0.0667168 −0.0333584 0.999443i $$-0.510620\pi$$
−0.0333584 + 0.999443i $$0.510620\pi$$
$$314$$ 7.23607 0.408355
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ −28.4164 −1.59602 −0.798012 0.602641i $$-0.794115\pi$$
−0.798012 + 0.602641i $$0.794115\pi$$
$$318$$ 0 0
$$319$$ −5.47214 −0.306381
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 26.7639 1.48919
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9.34752 0.517711
$$327$$ 0 0
$$328$$ −10.0000 −0.552158
$$329$$ 20.1246 1.10951
$$330$$ 0 0
$$331$$ −15.8885 −0.873313 −0.436657 0.899628i $$-0.643838\pi$$
−0.436657 + 0.899628i $$0.643838\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 0 0
$$334$$ −11.0557 −0.604943
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 20.4721 1.11519 0.557594 0.830114i $$-0.311725\pi$$
0.557594 + 0.830114i $$0.311725\pi$$
$$338$$ 16.0000 0.870285
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −43.7771 −2.37066
$$342$$ 0 0
$$343$$ −15.0000 −0.809924
$$344$$ 7.23607 0.390143
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 17.5279 0.940945 0.470473 0.882415i $$-0.344083\pi$$
0.470473 + 0.882415i $$0.344083\pi$$
$$348$$ 0 0
$$349$$ −18.9443 −1.01406 −0.507032 0.861927i $$-0.669257\pi$$
−0.507032 + 0.861927i $$0.669257\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 30.7426 1.63859
$$353$$ 3.52786 0.187769 0.0938846 0.995583i $$-0.470072\pi$$
0.0938846 + 0.995583i $$0.470072\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 18.0902 0.958777
$$357$$ 0 0
$$358$$ −2.58359 −0.136547
$$359$$ 4.58359 0.241913 0.120956 0.992658i $$-0.461404\pi$$
0.120956 + 0.992658i $$0.461404\pi$$
$$360$$ 0 0
$$361$$ 40.4164 2.12718
$$362$$ 5.20163 0.273391
$$363$$ 0 0
$$364$$ −30.2705 −1.58661
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −15.4164 −0.804730 −0.402365 0.915479i $$-0.631812\pi$$
−0.402365 + 0.915479i $$0.631812\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −20.2918 −1.05350
$$372$$ 0 0
$$373$$ −15.8885 −0.822678 −0.411339 0.911483i $$-0.634939\pi$$
−0.411339 + 0.911483i $$0.634939\pi$$
$$374$$ 11.7426 0.607198
$$375$$ 0 0
$$376$$ −15.0000 −0.773566
$$377$$ −6.23607 −0.321174
$$378$$ 0 0
$$379$$ 6.94427 0.356703 0.178352 0.983967i $$-0.442924\pi$$
0.178352 + 0.983967i $$0.442924\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −7.41641 −0.379456
$$383$$ −20.0689 −1.02547 −0.512736 0.858546i $$-0.671368\pi$$
−0.512736 + 0.858546i $$0.671368\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 3.70820 0.188743
$$387$$ 0 0
$$388$$ 4.47214 0.227038
$$389$$ −22.2361 −1.12741 −0.563707 0.825975i $$-0.690625\pi$$
−0.563707 + 0.825975i $$0.690625\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −4.47214 −0.225877
$$393$$ 0 0
$$394$$ 9.23607 0.465306
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.58359 0.330421 0.165211 0.986258i $$-0.447170\pi$$
0.165211 + 0.986258i $$0.447170\pi$$
$$398$$ 12.7984 0.641525
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 36.6525 1.83034 0.915169 0.403071i $$-0.132057\pi$$
0.915169 + 0.403071i $$0.132057\pi$$
$$402$$ 0 0
$$403$$ −49.8885 −2.48513
$$404$$ 6.85410 0.341004
$$405$$ 0 0
$$406$$ −1.85410 −0.0920175
$$407$$ 43.7771 2.16995
$$408$$ 0 0
$$409$$ 8.65248 0.427837 0.213919 0.976851i $$-0.431377\pi$$
0.213919 + 0.976851i $$0.431377\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 12.0000 0.591198
$$413$$ −15.7082 −0.772950
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 35.0344 1.71770
$$417$$ 0 0
$$418$$ 26.0689 1.27507
$$419$$ −34.3607 −1.67863 −0.839315 0.543646i $$-0.817043\pi$$
−0.839315 + 0.543646i $$0.817043\pi$$
$$420$$ 0 0
$$421$$ −1.81966 −0.0886848 −0.0443424 0.999016i $$-0.514119\pi$$
−0.0443424 + 0.999016i $$0.514119\pi$$
$$422$$ −17.2361 −0.839039
$$423$$ 0 0
$$424$$ 15.1246 0.734516
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −17.1246 −0.828718
$$428$$ −12.1803 −0.588759
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −34.4721 −1.66046 −0.830232 0.557418i $$-0.811792\pi$$
−0.830232 + 0.557418i $$0.811792\pi$$
$$432$$ 0 0
$$433$$ 4.65248 0.223584 0.111792 0.993732i $$-0.464341\pi$$
0.111792 + 0.993732i $$0.464341\pi$$
$$434$$ −14.8328 −0.711998
$$435$$ 0 0
$$436$$ 26.5623 1.27210
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −10.1246 −0.483221 −0.241611 0.970373i $$-0.577676\pi$$
−0.241611 + 0.970373i $$0.577676\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 13.3820 0.636515
$$443$$ 23.7639 1.12906 0.564529 0.825413i $$-0.309058\pi$$
0.564529 + 0.825413i $$0.309058\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.03444 0.380442
$$447$$ 0 0
$$448$$ −0.708204 −0.0334595
$$449$$ −1.76393 −0.0832451 −0.0416225 0.999133i $$-0.513253\pi$$
−0.0416225 + 0.999133i $$0.513253\pi$$
$$450$$ 0 0
$$451$$ 24.4721 1.15235
$$452$$ 12.8541 0.604606
$$453$$ 0 0
$$454$$ −3.59675 −0.168804
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.12461 −0.286497 −0.143249 0.989687i $$-0.545755\pi$$
−0.143249 + 0.989687i $$0.545755\pi$$
$$458$$ 5.81966 0.271935
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2.36068 −0.109948 −0.0549739 0.998488i $$-0.517508\pi$$
−0.0549739 + 0.998488i $$0.517508\pi$$
$$462$$ 0 0
$$463$$ 2.52786 0.117480 0.0587399 0.998273i $$-0.481292\pi$$
0.0587399 + 0.998273i $$0.481292\pi$$
$$464$$ −1.85410 −0.0860745
$$465$$ 0 0
$$466$$ 0.875388 0.0405516
$$467$$ −12.9443 −0.598989 −0.299495 0.954098i $$-0.596818\pi$$
−0.299495 + 0.954098i $$0.596818\pi$$
$$468$$ 0 0
$$469$$ 34.4164 1.58920
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 11.7082 0.538914
$$473$$ −17.7082 −0.814224
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −16.8541 −0.772506
$$477$$ 0 0
$$478$$ −7.34752 −0.336068
$$479$$ 6.47214 0.295719 0.147860 0.989008i $$-0.452762\pi$$
0.147860 + 0.989008i $$0.452762\pi$$
$$480$$ 0 0
$$481$$ 49.8885 2.27472
$$482$$ 4.32624 0.197055
$$483$$ 0 0
$$484$$ −30.6525 −1.39329
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 12.7639 0.577796
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −25.8885 −1.16833 −0.584167 0.811634i $$-0.698579\pi$$
−0.584167 + 0.811634i $$0.698579\pi$$
$$492$$ 0 0
$$493$$ −3.47214 −0.156377
$$494$$ 29.7082 1.33664
$$495$$ 0 0
$$496$$ −14.8328 −0.666013
$$497$$ −21.7082 −0.973746
$$498$$ 0 0
$$499$$ −23.7639 −1.06382 −0.531910 0.846801i $$-0.678525\pi$$
−0.531910 + 0.846801i $$0.678525\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 15.3820 0.686531
$$503$$ −30.5967 −1.36424 −0.682121 0.731240i $$-0.738942\pi$$
−0.682121 + 0.731240i $$0.738942\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −9.70820 −0.430732
$$509$$ −6.18034 −0.273939 −0.136969 0.990575i $$-0.543736\pi$$
−0.136969 + 0.990575i $$0.543736\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ 18.7082 0.826794
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 36.7082 1.61442
$$518$$ 14.8328 0.651717
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20.7639 0.909684 0.454842 0.890572i $$-0.349696\pi$$
0.454842 + 0.890572i $$0.349696\pi$$
$$522$$ 0 0
$$523$$ 29.8328 1.30450 0.652249 0.758005i $$-0.273826\pi$$
0.652249 + 0.758005i $$0.273826\pi$$
$$524$$ 5.61803 0.245425
$$525$$ 0 0
$$526$$ 14.8328 0.646741
$$527$$ −27.7771 −1.20999
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −37.4164 −1.62221
$$533$$ 27.8885 1.20799
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −25.6525 −1.10802
$$537$$ 0 0
$$538$$ 18.6869 0.805650
$$539$$ 10.9443 0.471403
$$540$$ 0 0
$$541$$ 38.3607 1.64925 0.824627 0.565677i $$-0.191385\pi$$
0.824627 + 0.565677i $$0.191385\pi$$
$$542$$ 8.87539 0.381231
$$543$$ 0 0
$$544$$ 19.5066 0.836338
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −27.4721 −1.17462 −0.587312 0.809361i $$-0.699814\pi$$
−0.587312 + 0.809361i $$0.699814\pi$$
$$548$$ 17.7082 0.756457
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7.70820 −0.328381
$$552$$ 0 0
$$553$$ −18.5410 −0.788444
$$554$$ −5.38197 −0.228658
$$555$$ 0 0
$$556$$ 1.14590 0.0485969
$$557$$ 2.76393 0.117112 0.0585558 0.998284i $$-0.481350\pi$$
0.0585558 + 0.998284i $$0.481350\pi$$
$$558$$ 0 0
$$559$$ −20.1803 −0.853537
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 14.2918 0.602863
$$563$$ 44.1246 1.85963 0.929815 0.368026i $$-0.119966\pi$$
0.929815 + 0.368026i $$0.119966\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −11.4164 −0.479867
$$567$$ 0 0
$$568$$ 16.1803 0.678912
$$569$$ 5.18034 0.217171 0.108586 0.994087i $$-0.465368\pi$$
0.108586 + 0.994087i $$0.465368\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ −55.2148 −2.30865
$$573$$ 0 0
$$574$$ 8.29180 0.346093
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 29.3050 1.21998 0.609991 0.792409i $$-0.291173\pi$$
0.609991 + 0.792409i $$0.291173\pi$$
$$578$$ −3.05573 −0.127102
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −11.1246 −0.461527
$$582$$ 0 0
$$583$$ −37.0132 −1.53293
$$584$$ −17.8885 −0.740233
$$585$$ 0 0
$$586$$ −11.0902 −0.458131
$$587$$ −1.81966 −0.0751054 −0.0375527 0.999295i $$-0.511956\pi$$
−0.0375527 + 0.999295i $$0.511956\pi$$
$$588$$ 0 0
$$589$$ −61.6656 −2.54089
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 14.8328 0.609625
$$593$$ −24.6525 −1.01236 −0.506178 0.862429i $$-0.668942\pi$$
−0.506178 + 0.862429i $$0.668942\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 32.6525 1.33750
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 8.88854 0.363176 0.181588 0.983375i $$-0.441876\pi$$
0.181588 + 0.983375i $$0.441876\pi$$
$$600$$ 0 0
$$601$$ 8.11146 0.330873 0.165437 0.986220i $$-0.447097\pi$$
0.165437 + 0.986220i $$0.447097\pi$$
$$602$$ −6.00000 −0.244542
$$603$$ 0 0
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −40.6525 −1.65003 −0.825017 0.565109i $$-0.808834\pi$$
−0.825017 + 0.565109i $$0.808834\pi$$
$$608$$ 43.3050 1.75625
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 41.8328 1.69237
$$612$$ 0 0
$$613$$ −7.29180 −0.294513 −0.147256 0.989098i $$-0.547044\pi$$
−0.147256 + 0.989098i $$0.547044\pi$$
$$614$$ −15.4164 −0.622156
$$615$$ 0 0
$$616$$ −36.7082 −1.47902
$$617$$ 43.3050 1.74339 0.871696 0.490047i $$-0.163020\pi$$
0.871696 + 0.490047i $$0.163020\pi$$
$$618$$ 0 0
$$619$$ −3.52786 −0.141797 −0.0708984 0.997484i $$-0.522587\pi$$
−0.0708984 + 0.997484i $$0.522587\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −14.5066 −0.581661
$$623$$ −33.5410 −1.34379
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −0.729490 −0.0291563
$$627$$ 0 0
$$628$$ −18.9443 −0.755959
$$629$$ 27.7771 1.10755
$$630$$ 0 0
$$631$$ −38.4853 −1.53208 −0.766038 0.642796i $$-0.777774\pi$$
−0.766038 + 0.642796i $$0.777774\pi$$
$$632$$ 13.8197 0.549717
$$633$$ 0 0
$$634$$ −17.5623 −0.697488
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 12.4721 0.494164
$$638$$ −3.38197 −0.133893
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −46.2361 −1.82621 −0.913107 0.407719i $$-0.866324\pi$$
−0.913107 + 0.407719i $$0.866324\pi$$
$$642$$ 0 0
$$643$$ 35.8328 1.41311 0.706554 0.707659i $$-0.250249\pi$$
0.706554 + 0.707659i $$0.250249\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 16.5410 0.650798
$$647$$ 11.2361 0.441735 0.220868 0.975304i $$-0.429111\pi$$
0.220868 + 0.975304i $$0.429111\pi$$
$$648$$ 0 0
$$649$$ −28.6525 −1.12471
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −24.4721 −0.958403
$$653$$ 8.88854 0.347836 0.173918 0.984760i $$-0.444357\pi$$
0.173918 + 0.984760i $$0.444357\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 8.29180 0.323740
$$657$$ 0 0
$$658$$ 12.4377 0.484872
$$659$$ 21.0000 0.818044 0.409022 0.912525i $$-0.365870\pi$$
0.409022 + 0.912525i $$0.365870\pi$$
$$660$$ 0 0
$$661$$ −37.8328 −1.47153 −0.735763 0.677239i $$-0.763176\pi$$
−0.735763 + 0.677239i $$0.763176\pi$$
$$662$$ −9.81966 −0.381652
$$663$$ 0 0
$$664$$ 8.29180 0.321784
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 28.9443 1.11989
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −31.2361 −1.20586
$$672$$ 0 0
$$673$$ 33.2918 1.28330 0.641652 0.766996i $$-0.278249\pi$$
0.641652 + 0.766996i $$0.278249\pi$$
$$674$$ 12.6525 0.487355
$$675$$ 0 0
$$676$$ −41.8885 −1.61110
$$677$$ −20.8885 −0.802812 −0.401406 0.915900i $$-0.631478\pi$$
−0.401406 + 0.915900i $$0.631478\pi$$
$$678$$ 0 0
$$679$$ −8.29180 −0.318210
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −27.0557 −1.03602
$$683$$ 7.41641 0.283781 0.141890 0.989882i $$-0.454682\pi$$
0.141890 + 0.989882i $$0.454682\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −9.27051 −0.353950
$$687$$ 0 0
$$688$$ −6.00000 −0.228748
$$689$$ −42.1803 −1.60694
$$690$$ 0 0
$$691$$ −34.7082 −1.32036 −0.660181 0.751106i $$-0.729520\pi$$
−0.660181 + 0.751106i $$0.729520\pi$$
$$692$$ 31.4164 1.19427
$$693$$ 0 0
$$694$$ 10.8328 0.411208
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 15.5279 0.588160
$$698$$ −11.7082 −0.443162
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5.12461 0.193554 0.0967770 0.995306i $$-0.469147\pi$$
0.0967770 + 0.995306i $$0.469147\pi$$
$$702$$ 0 0
$$703$$ 61.6656 2.32576
$$704$$ −1.29180 −0.0486864
$$705$$ 0 0
$$706$$ 2.18034 0.0820582
$$707$$ −12.7082 −0.477941
$$708$$ 0 0
$$709$$ 19.8885 0.746930 0.373465 0.927644i $$-0.378170\pi$$
0.373465 + 0.927644i $$0.378170\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 25.0000 0.936915
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.76393 0.252780
$$717$$ 0 0
$$718$$ 2.83282 0.105720
$$719$$ 11.2361 0.419035 0.209517 0.977805i $$-0.432811\pi$$
0.209517 + 0.977805i $$0.432811\pi$$
$$720$$ 0 0
$$721$$ −22.2492 −0.828604
$$722$$ 24.9787 0.929611
$$723$$ 0 0
$$724$$ −13.6180 −0.506110
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −2.11146 −0.0783096 −0.0391548 0.999233i $$-0.512467\pi$$
−0.0391548 + 0.999233i $$0.512467\pi$$
$$728$$ −41.8328 −1.55043
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −11.2361 −0.415581
$$732$$ 0 0
$$733$$ 14.2918 0.527880 0.263940 0.964539i $$-0.414978\pi$$
0.263940 + 0.964539i $$0.414978\pi$$
$$734$$ −9.52786 −0.351680
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 62.7771 2.31242
$$738$$ 0 0
$$739$$ 41.4853 1.52606 0.763031 0.646362i $$-0.223711\pi$$
0.763031 + 0.646362i $$0.223711\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −12.5410 −0.460395
$$743$$ −10.8197 −0.396935 −0.198467 0.980107i $$-0.563596\pi$$
−0.198467 + 0.980107i $$0.563596\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −9.81966 −0.359523
$$747$$ 0 0
$$748$$ −30.7426 −1.12406
$$749$$ 22.5836 0.825186
$$750$$ 0 0
$$751$$ 43.4164 1.58429 0.792144 0.610335i $$-0.208965\pi$$
0.792144 + 0.610335i $$0.208965\pi$$
$$752$$ 12.4377 0.453556
$$753$$ 0 0
$$754$$ −3.85410 −0.140358
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 34.8328 1.26602 0.633010 0.774144i $$-0.281819\pi$$
0.633010 + 0.774144i $$0.281819\pi$$
$$758$$ 4.29180 0.155885
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33.0132 −1.19673 −0.598363 0.801225i $$-0.704182\pi$$
−0.598363 + 0.801225i $$0.704182\pi$$
$$762$$ 0 0
$$763$$ −49.2492 −1.78294
$$764$$ 19.4164 0.702461
$$765$$ 0 0
$$766$$ −12.4033 −0.448148
$$767$$ −32.6525 −1.17901
$$768$$ 0 0
$$769$$ 34.6525 1.24960 0.624800 0.780785i $$-0.285180\pi$$
0.624800 + 0.780785i $$0.285180\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −9.70820 −0.349406
$$773$$ −52.2492 −1.87927 −0.939637 0.342173i $$-0.888837\pi$$
−0.939637 + 0.342173i $$0.888837\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.18034 0.221861
$$777$$ 0 0
$$778$$ −13.7426 −0.492698
$$779$$ 34.4721 1.23509
$$780$$ 0 0
$$781$$ −39.5967 −1.41688
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 3.70820 0.132436
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −1.88854 −0.0673193 −0.0336597 0.999433i $$-0.510716\pi$$
−0.0336597 + 0.999433i $$0.510716\pi$$
$$788$$ −24.1803 −0.861389
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −23.8328 −0.847397
$$792$$ 0 0
$$793$$ −35.5967 −1.26408
$$794$$ 4.06888 0.144399
$$795$$ 0 0
$$796$$ −33.5066 −1.18761
$$797$$ 6.94427 0.245979 0.122989 0.992408i $$-0.460752\pi$$
0.122989 + 0.992408i $$0.460752\pi$$
$$798$$ 0 0
$$799$$ 23.2918 0.824005
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 22.6525 0.799887
$$803$$ 43.7771 1.54486
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −30.8328 −1.08604
$$807$$ 0 0
$$808$$ 9.47214 0.333229
$$809$$ −44.2361 −1.55526 −0.777629 0.628724i $$-0.783578\pi$$
−0.777629 + 0.628724i $$0.783578\pi$$
$$810$$ 0 0
$$811$$ −22.7082 −0.797393 −0.398696 0.917083i $$-0.630537\pi$$
−0.398696 + 0.917083i $$0.630537\pi$$
$$812$$ 4.85410 0.170346
$$813$$ 0 0
$$814$$ 27.0557 0.948303
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.9443 −0.872690
$$818$$ 5.34752 0.186972
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.5836 0.369370 0.184685 0.982798i $$-0.440874\pi$$
0.184685 + 0.982798i $$0.440874\pi$$
$$822$$ 0 0
$$823$$ 28.7639 1.00265 0.501324 0.865260i $$-0.332847\pi$$
0.501324 + 0.865260i $$0.332847\pi$$
$$824$$ 16.5836 0.577717
$$825$$ 0 0
$$826$$ −9.70820 −0.337792
$$827$$ −7.41641 −0.257894 −0.128947 0.991652i $$-0.541160\pi$$
−0.128947 + 0.991652i $$0.541160\pi$$
$$828$$ 0 0
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.47214 −0.0510371
$$833$$ 6.94427 0.240605
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −68.2492 −2.36045
$$837$$ 0 0
$$838$$ −21.2361 −0.733588
$$839$$ −14.8885 −0.514010 −0.257005 0.966410i $$-0.582736\pi$$
−0.257005 + 0.966410i $$0.582736\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −1.12461 −0.0387567
$$843$$ 0 0
$$844$$ 45.1246 1.55325
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 56.8328 1.95280
$$848$$ −12.5410 −0.430660
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 28.7639 0.984858 0.492429 0.870353i $$-0.336109\pi$$
0.492429 + 0.870353i $$0.336109\pi$$
$$854$$ −10.5836 −0.362163
$$855$$ 0 0
$$856$$ −16.8328 −0.575334
$$857$$ −23.8885 −0.816017 −0.408009 0.912978i $$-0.633777\pi$$
−0.408009 + 0.912978i $$0.633777\pi$$
$$858$$ 0 0
$$859$$ −15.7082 −0.535957 −0.267979 0.963425i $$-0.586356\pi$$
−0.267979 + 0.963425i $$0.586356\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −21.3050 −0.725650
$$863$$ 15.5967 0.530919 0.265460 0.964122i $$-0.414476\pi$$
0.265460 + 0.964122i $$0.414476\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 2.87539 0.0977097
$$867$$ 0 0
$$868$$ 38.8328 1.31807
$$869$$ −33.8197 −1.14725
$$870$$ 0 0
$$871$$ 71.5410 2.42407
$$872$$ 36.7082 1.24310
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −48.8328 −1.64897 −0.824484 0.565886i $$-0.808534\pi$$
−0.824484 + 0.565886i $$0.808534\pi$$
$$878$$ −6.25735 −0.211175
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −12.7082 −0.428150 −0.214075 0.976817i $$-0.568674\pi$$
−0.214075 + 0.976817i $$0.568674\pi$$
$$882$$ 0 0
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ −35.0344 −1.17834
$$885$$ 0 0
$$886$$ 14.6869 0.493417
$$887$$ −44.5967 −1.49741 −0.748706 0.662902i $$-0.769325\pi$$
−0.748706 + 0.662902i $$0.769325\pi$$
$$888$$ 0 0
$$889$$ 18.0000 0.603701
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −21.0344 −0.704285
$$893$$ 51.7082 1.73035
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −34.1459 −1.14073
$$897$$ 0 0
$$898$$ −1.09017 −0.0363794
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −23.4853 −0.782409
$$902$$ 15.1246 0.503594
$$903$$ 0 0
$$904$$ 17.7639 0.590820
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −26.7639 −0.888682 −0.444341 0.895858i $$-0.646562\pi$$
−0.444341 + 0.895858i $$0.646562\pi$$
$$908$$ 9.41641 0.312494
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.0557 0.598213 0.299106 0.954220i $$-0.403311\pi$$
0.299106 + 0.954220i $$0.403311\pi$$
$$912$$ 0 0
$$913$$ −20.2918 −0.671560
$$914$$ −3.78522 −0.125204
$$915$$ 0 0
$$916$$ −15.2361 −0.503414
$$917$$ −10.4164 −0.343980
$$918$$ 0 0
$$919$$ 20.1246 0.663850 0.331925 0.943306i $$-0.392302\pi$$
0.331925 + 0.943306i $$0.392302\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −1.45898 −0.0480490
$$923$$ −45.1246 −1.48529
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 1.56231 0.0513406
$$927$$ 0 0
$$928$$ −5.61803 −0.184421
$$929$$ 56.8328 1.86462 0.932312 0.361655i $$-0.117788\pi$$
0.932312 + 0.361655i $$0.117788\pi$$
$$930$$ 0 0
$$931$$ 15.4164 0.505252
$$932$$ −2.29180 −0.0750703
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −19.7639 −0.645660 −0.322830 0.946457i $$-0.604634\pi$$
−0.322830 + 0.946457i $$0.604634\pi$$
$$938$$ 21.2705 0.694507
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 46.0689 1.50180 0.750901 0.660414i $$-0.229619\pi$$
0.750901 + 0.660414i $$0.229619\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −9.70820 −0.315975
$$945$$ 0 0
$$946$$ −10.9443 −0.355829
$$947$$ −12.7082 −0.412961 −0.206481 0.978451i $$-0.566201\pi$$
−0.206481 + 0.978451i $$0.566201\pi$$
$$948$$ 0 0
$$949$$ 49.8885 1.61945
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −23.2918 −0.754891
$$953$$ 19.0132 0.615897 0.307948 0.951403i $$-0.400358\pi$$
0.307948 + 0.951403i $$0.400358\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 19.2361 0.622139
$$957$$ 0 0
$$958$$ 4.00000 0.129234
$$959$$ −32.8328 −1.06023
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 30.8328 0.994090
$$963$$ 0 0
$$964$$ −11.3262 −0.364794
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 35.1246 1.12953 0.564766 0.825251i $$-0.308967\pi$$
0.564766 + 0.825251i $$0.308967\pi$$
$$968$$ −42.3607 −1.36152
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 23.0557 0.739894 0.369947 0.929053i $$-0.379376\pi$$
0.369947 + 0.929053i $$0.379376\pi$$
$$972$$ 0 0
$$973$$ −2.12461 −0.0681119
$$974$$ 7.41641 0.237637
$$975$$ 0 0
$$976$$ −10.5836 −0.338773
$$977$$ −22.4721 −0.718947 −0.359474 0.933155i $$-0.617044\pi$$
−0.359474 + 0.933155i $$0.617044\pi$$
$$978$$ 0 0
$$979$$ −61.1803 −1.95533
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −16.0000 −0.510581
$$983$$ 37.5279 1.19695 0.598476 0.801140i $$-0.295773\pi$$
0.598476 + 0.801140i $$0.295773\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.14590 −0.0683393
$$987$$ 0 0
$$988$$ −77.7771 −2.47442
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 22.4853 0.714269 0.357134 0.934053i $$-0.383754\pi$$
0.357134 + 0.934053i $$0.383754\pi$$
$$992$$ −44.9443 −1.42698
$$993$$ 0 0
$$994$$ −13.4164 −0.425543
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −7.41641 −0.234880 −0.117440 0.993080i $$-0.537469\pi$$
−0.117440 + 0.993080i $$0.537469\pi$$
$$998$$ −14.6869 −0.464906
$$999$$ 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 6525.2.a.s.1.2 2
3.2 odd 2 2175.2.a.q.1.1 2
5.4 even 2 1305.2.a.k.1.1 2
15.2 even 4 2175.2.c.j.349.2 4
15.8 even 4 2175.2.c.j.349.3 4
15.14 odd 2 435.2.a.e.1.2 2
60.59 even 2 6960.2.a.bu.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.e.1.2 2 15.14 odd 2
1305.2.a.k.1.1 2 5.4 even 2
2175.2.a.q.1.1 2 3.2 odd 2
2175.2.c.j.349.2 4 15.2 even 4
2175.2.c.j.349.3 4 15.8 even 4
6525.2.a.s.1.2 2 1.1 even 1 trivial
6960.2.a.bu.1.2 2 60.59 even 2