# Properties

 Label 3630.2.a.bs.1.1 Level $3630$ Weight $2$ Character 3630.1 Self dual yes Analytic conductor $28.986$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3630.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.9856959337$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.52625.1 Defining polynomial: $$x^{4} - x^{3} - 24 x^{2} + 19 x + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 330) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.15753$$ of defining polynomial Character $$\chi$$ $$=$$ 3630.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.10899 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.10899 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -1.33343 q^{13} -5.10899 q^{14} -1.00000 q^{15} +1.00000 q^{16} -0.775565 q^{17} +1.00000 q^{18} +0.0785371 q^{19} -1.00000 q^{20} -5.10899 q^{21} +6.64849 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.33343 q^{26} +1.00000 q^{27} -5.10899 q^{28} +2.01882 q^{29} -1.00000 q^{30} +8.49096 q^{31} +1.00000 q^{32} -0.775565 q^{34} +5.10899 q^{35} +1.00000 q^{36} -3.93310 q^{37} +0.0785371 q^{38} -1.33343 q^{39} -1.00000 q^{40} +11.8962 q^{41} -5.10899 q^{42} +6.01163 q^{43} -1.00000 q^{45} +6.64849 q^{46} +6.56950 q^{47} +1.00000 q^{48} +19.1018 q^{49} +1.00000 q^{50} -0.775565 q^{51} -1.33343 q^{52} +10.3451 q^{53} +1.00000 q^{54} -5.10899 q^{56} +0.0785371 q^{57} +2.01882 q^{58} -5.32624 q^{59} -1.00000 q^{60} +3.74585 q^{61} +8.49096 q^{62} -5.10899 q^{63} +1.00000 q^{64} +1.33343 q^{65} +0.588036 q^{67} -0.775565 q^{68} +6.64849 q^{69} +5.10899 q^{70} -2.31506 q^{71} +1.00000 q^{72} -10.9443 q^{73} -3.93310 q^{74} +1.00000 q^{75} +0.0785371 q^{76} -1.33343 q^{78} -12.3267 q^{79} -1.00000 q^{80} +1.00000 q^{81} +11.8962 q^{82} +8.00000 q^{83} -5.10899 q^{84} +0.775565 q^{85} +6.01163 q^{86} +2.01882 q^{87} -9.67821 q^{89} -1.00000 q^{90} +6.81247 q^{91} +6.64849 q^{92} +8.49096 q^{93} +6.56950 q^{94} -0.0785371 q^{95} +1.00000 q^{96} -13.4541 q^{97} +19.1018 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} + 2q^{13} + q^{14} - 4q^{15} + 4q^{16} + 11q^{17} + 4q^{18} + q^{19} - 4q^{20} + q^{21} + 4q^{24} + 4q^{25} + 2q^{26} + 4q^{27} + q^{28} + 9q^{29} - 4q^{30} + 17q^{31} + 4q^{32} + 11q^{34} - q^{35} + 4q^{36} + 8q^{37} + q^{38} + 2q^{39} - 4q^{40} - 11q^{41} + q^{42} + q^{43} - 4q^{45} + 10q^{47} + 4q^{48} + 31q^{49} + 4q^{50} + 11q^{51} + 2q^{52} + 11q^{53} + 4q^{54} + q^{56} + q^{57} + 9q^{58} + 10q^{59} - 4q^{60} - 10q^{61} + 17q^{62} + q^{63} + 4q^{64} - 2q^{65} + 9q^{67} + 11q^{68} - q^{70} + 10q^{71} + 4q^{72} - 8q^{73} + 8q^{74} + 4q^{75} + q^{76} + 2q^{78} - 7q^{79} - 4q^{80} + 4q^{81} - 11q^{82} + 32q^{83} + q^{84} - 11q^{85} + q^{86} + 9q^{87} - 23q^{89} - 4q^{90} + 48q^{91} + 17q^{93} + 10q^{94} - q^{95} + 4q^{96} - 2q^{97} + 31q^{98} + O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ −5.10899 −1.93102 −0.965509 0.260370i $$-0.916155\pi$$
−0.965509 + 0.260370i $$0.916155\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0
$$12$$ 1.00000 0.288675
$$13$$ −1.33343 −0.369826 −0.184913 0.982755i $$-0.559200\pi$$
−0.184913 + 0.982755i $$0.559200\pi$$
$$14$$ −5.10899 −1.36544
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −0.775565 −0.188102 −0.0940511 0.995567i $$-0.529982\pi$$
−0.0940511 + 0.995567i $$0.529982\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0.0785371 0.0180176 0.00900882 0.999959i $$-0.497132\pi$$
0.00900882 + 0.999959i $$0.497132\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −5.10899 −1.11487
$$22$$ 0 0
$$23$$ 6.64849 1.38631 0.693153 0.720791i $$-0.256221\pi$$
0.693153 + 0.720791i $$0.256221\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ −1.33343 −0.261507
$$27$$ 1.00000 0.192450
$$28$$ −5.10899 −0.965509
$$29$$ 2.01882 0.374886 0.187443 0.982275i $$-0.439980\pi$$
0.187443 + 0.982275i $$0.439980\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 8.49096 1.52502 0.762511 0.646976i $$-0.223967\pi$$
0.762511 + 0.646976i $$0.223967\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −0.775565 −0.133008
$$35$$ 5.10899 0.863577
$$36$$ 1.00000 0.166667
$$37$$ −3.93310 −0.646597 −0.323298 0.946297i $$-0.604792\pi$$
−0.323298 + 0.946297i $$0.604792\pi$$
$$38$$ 0.0785371 0.0127404
$$39$$ −1.33343 −0.213519
$$40$$ −1.00000 −0.158114
$$41$$ 11.8962 1.85787 0.928936 0.370239i $$-0.120724\pi$$
0.928936 + 0.370239i $$0.120724\pi$$
$$42$$ −5.10899 −0.788335
$$43$$ 6.01163 0.916765 0.458383 0.888755i $$-0.348429\pi$$
0.458383 + 0.888755i $$0.348429\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 6.64849 0.980266
$$47$$ 6.56950 0.958259 0.479130 0.877744i $$-0.340952\pi$$
0.479130 + 0.877744i $$0.340952\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 19.1018 2.72883
$$50$$ 1.00000 0.141421
$$51$$ −0.775565 −0.108601
$$52$$ −1.33343 −0.184913
$$53$$ 10.3451 1.42100 0.710502 0.703696i $$-0.248468\pi$$
0.710502 + 0.703696i $$0.248468\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −5.10899 −0.682718
$$57$$ 0.0785371 0.0104025
$$58$$ 2.01882 0.265084
$$59$$ −5.32624 −0.693417 −0.346709 0.937973i $$-0.612701\pi$$
−0.346709 + 0.937973i $$0.612701\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 3.74585 0.479607 0.239803 0.970821i $$-0.422917\pi$$
0.239803 + 0.970821i $$0.422917\pi$$
$$62$$ 8.49096 1.07835
$$63$$ −5.10899 −0.643673
$$64$$ 1.00000 0.125000
$$65$$ 1.33343 0.165391
$$66$$ 0 0
$$67$$ 0.588036 0.0718400 0.0359200 0.999355i $$-0.488564\pi$$
0.0359200 + 0.999355i $$0.488564\pi$$
$$68$$ −0.775565 −0.0940511
$$69$$ 6.64849 0.800384
$$70$$ 5.10899 0.610641
$$71$$ −2.31506 −0.274747 −0.137374 0.990519i $$-0.543866\pi$$
−0.137374 + 0.990519i $$0.543866\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −10.9443 −1.28093 −0.640465 0.767987i $$-0.721258\pi$$
−0.640465 + 0.767987i $$0.721258\pi$$
$$74$$ −3.93310 −0.457213
$$75$$ 1.00000 0.115470
$$76$$ 0.0785371 0.00900882
$$77$$ 0 0
$$78$$ −1.33343 −0.150981
$$79$$ −12.3267 −1.38686 −0.693431 0.720523i $$-0.743902\pi$$
−0.693431 + 0.720523i $$0.743902\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 11.8962 1.31371
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ −5.10899 −0.557437
$$85$$ 0.775565 0.0841218
$$86$$ 6.01163 0.648251
$$87$$ 2.01882 0.216440
$$88$$ 0 0
$$89$$ −9.67821 −1.02589 −0.512944 0.858422i $$-0.671445\pi$$
−0.512944 + 0.858422i $$0.671445\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 6.81247 0.714141
$$92$$ 6.64849 0.693153
$$93$$ 8.49096 0.880471
$$94$$ 6.56950 0.677592
$$95$$ −0.0785371 −0.00805773
$$96$$ 1.00000 0.102062
$$97$$ −13.4541 −1.36605 −0.683026 0.730394i $$-0.739336\pi$$
−0.683026 + 0.730394i $$0.739336\pi$$
$$98$$ 19.1018 1.92957
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 10.6408 1.05880 0.529402 0.848371i $$-0.322416\pi$$
0.529402 + 0.848371i $$0.322416\pi$$
$$102$$ −0.775565 −0.0767924
$$103$$ 16.7087 1.64635 0.823177 0.567785i $$-0.192200\pi$$
0.823177 + 0.567785i $$0.192200\pi$$
$$104$$ −1.33343 −0.130753
$$105$$ 5.10899 0.498587
$$106$$ 10.3451 1.00480
$$107$$ −17.5511 −1.69673 −0.848366 0.529410i $$-0.822413\pi$$
−0.848366 + 0.529410i $$0.822413\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 8.94427 0.856706 0.428353 0.903612i $$-0.359094\pi$$
0.428353 + 0.903612i $$0.359094\pi$$
$$110$$ 0 0
$$111$$ −3.93310 −0.373313
$$112$$ −5.10899 −0.482754
$$113$$ −5.72703 −0.538753 −0.269377 0.963035i $$-0.586818\pi$$
−0.269377 + 0.963035i $$0.586818\pi$$
$$114$$ 0.0785371 0.00735567
$$115$$ −6.64849 −0.619975
$$116$$ 2.01882 0.187443
$$117$$ −1.33343 −0.123275
$$118$$ −5.32624 −0.490320
$$119$$ 3.96236 0.363229
$$120$$ −1.00000 −0.0912871
$$121$$ 0 0
$$122$$ 3.74585 0.339133
$$123$$ 11.8962 1.07264
$$124$$ 8.49096 0.762511
$$125$$ −1.00000 −0.0894427
$$126$$ −5.10899 −0.455145
$$127$$ 6.96281 0.617850 0.308925 0.951086i $$-0.400031\pi$$
0.308925 + 0.951086i $$0.400031\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 6.01163 0.529295
$$130$$ 1.33343 0.116949
$$131$$ 2.19916 0.192142 0.0960709 0.995374i $$-0.469372\pi$$
0.0960709 + 0.995374i $$0.469372\pi$$
$$132$$ 0 0
$$133$$ −0.401245 −0.0347924
$$134$$ 0.588036 0.0507985
$$135$$ −1.00000 −0.0860663
$$136$$ −0.775565 −0.0665041
$$137$$ −2.17590 −0.185899 −0.0929497 0.995671i $$-0.529630\pi$$
−0.0929497 + 0.995671i $$0.529630\pi$$
$$138$$ 6.64849 0.565957
$$139$$ −10.3074 −0.874264 −0.437132 0.899397i $$-0.644006\pi$$
−0.437132 + 0.899397i $$0.644006\pi$$
$$140$$ 5.10899 0.431789
$$141$$ 6.56950 0.553251
$$142$$ −2.31506 −0.194276
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −2.01882 −0.167654
$$146$$ −10.9443 −0.905754
$$147$$ 19.1018 1.57549
$$148$$ −3.93310 −0.323298
$$149$$ 15.2477 1.24914 0.624570 0.780969i $$-0.285274\pi$$
0.624570 + 0.780969i $$0.285274\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −6.56921 −0.534595 −0.267297 0.963614i $$-0.586131\pi$$
−0.267297 + 0.963614i $$0.586131\pi$$
$$152$$ 0.0785371 0.00637020
$$153$$ −0.775565 −0.0627007
$$154$$ 0 0
$$155$$ −8.49096 −0.682010
$$156$$ −1.33343 −0.106760
$$157$$ −9.30815 −0.742872 −0.371436 0.928459i $$-0.621134\pi$$
−0.371436 + 0.928459i $$0.621134\pi$$
$$158$$ −12.3267 −0.980659
$$159$$ 10.3451 0.820417
$$160$$ −1.00000 −0.0790569
$$161$$ −33.9671 −2.67698
$$162$$ 1.00000 0.0785674
$$163$$ −18.7018 −1.46483 −0.732417 0.680856i $$-0.761608\pi$$
−0.732417 + 0.680856i $$0.761608\pi$$
$$164$$ 11.8962 0.928936
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 2.37478 0.183766 0.0918829 0.995770i $$-0.470711\pi$$
0.0918829 + 0.995770i $$0.470711\pi$$
$$168$$ −5.10899 −0.394167
$$169$$ −11.2220 −0.863229
$$170$$ 0.775565 0.0594831
$$171$$ 0.0785371 0.00600588
$$172$$ 6.01163 0.458383
$$173$$ −2.91427 −0.221568 −0.110784 0.993845i $$-0.535336\pi$$
−0.110784 + 0.993845i $$0.535336\pi$$
$$174$$ 2.01882 0.153047
$$175$$ −5.10899 −0.386204
$$176$$ 0 0
$$177$$ −5.32624 −0.400345
$$178$$ −9.67821 −0.725412
$$179$$ 1.56231 0.116772 0.0583861 0.998294i $$-0.481405\pi$$
0.0583861 + 0.998294i $$0.481405\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 18.4721 1.37302 0.686512 0.727119i $$-0.259141\pi$$
0.686512 + 0.727119i $$0.259141\pi$$
$$182$$ 6.81247 0.504974
$$183$$ 3.74585 0.276901
$$184$$ 6.64849 0.490133
$$185$$ 3.93310 0.289167
$$186$$ 8.49096 0.622587
$$187$$ 0 0
$$188$$ 6.56950 0.479130
$$189$$ −5.10899 −0.371625
$$190$$ −0.0785371 −0.00569768
$$191$$ 20.7278 1.49981 0.749904 0.661546i $$-0.230100\pi$$
0.749904 + 0.661546i $$0.230100\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −1.37079 −0.0986716 −0.0493358 0.998782i $$-0.515710\pi$$
−0.0493358 + 0.998782i $$0.515710\pi$$
$$194$$ −13.4541 −0.965945
$$195$$ 1.33343 0.0954887
$$196$$ 19.1018 1.36441
$$197$$ 5.01135 0.357044 0.178522 0.983936i $$-0.442868\pi$$
0.178522 + 0.983936i $$0.442868\pi$$
$$198$$ 0 0
$$199$$ 23.3086 1.65230 0.826152 0.563448i $$-0.190525\pi$$
0.826152 + 0.563448i $$0.190525\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0.588036 0.0414768
$$202$$ 10.6408 0.748687
$$203$$ −10.3141 −0.723911
$$204$$ −0.775565 −0.0543004
$$205$$ −11.8962 −0.830866
$$206$$ 16.7087 1.16415
$$207$$ 6.64849 0.462102
$$208$$ −1.33343 −0.0924566
$$209$$ 0 0
$$210$$ 5.10899 0.352554
$$211$$ 20.0842 1.38265 0.691326 0.722543i $$-0.257027\pi$$
0.691326 + 0.722543i $$0.257027\pi$$
$$212$$ 10.3451 0.710502
$$213$$ −2.31506 −0.158625
$$214$$ −17.5511 −1.19977
$$215$$ −6.01163 −0.409990
$$216$$ 1.00000 0.0680414
$$217$$ −43.3802 −2.94484
$$218$$ 8.94427 0.605783
$$219$$ −10.9443 −0.739545
$$220$$ 0 0
$$221$$ 1.03416 0.0695651
$$222$$ −3.93310 −0.263972
$$223$$ 9.10899 0.609983 0.304992 0.952355i $$-0.401346\pi$$
0.304992 + 0.952355i $$0.401346\pi$$
$$224$$ −5.10899 −0.341359
$$225$$ 1.00000 0.0666667
$$226$$ −5.72703 −0.380956
$$227$$ −20.6901 −1.37325 −0.686626 0.727011i $$-0.740909\pi$$
−0.686626 + 0.727011i $$0.740909\pi$$
$$228$$ 0.0785371 0.00520124
$$229$$ 10.9819 0.725705 0.362853 0.931846i $$-0.381803\pi$$
0.362853 + 0.931846i $$0.381803\pi$$
$$230$$ −6.64849 −0.438388
$$231$$ 0 0
$$232$$ 2.01882 0.132542
$$233$$ 2.96309 0.194119 0.0970594 0.995279i $$-0.469056\pi$$
0.0970594 + 0.995279i $$0.469056\pi$$
$$234$$ −1.33343 −0.0871689
$$235$$ −6.56950 −0.428547
$$236$$ −5.32624 −0.346709
$$237$$ −12.3267 −0.800705
$$238$$ 3.96236 0.256841
$$239$$ −8.56978 −0.554333 −0.277166 0.960822i $$-0.589395\pi$$
−0.277166 + 0.960822i $$0.589395\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −7.97000 −0.513393 −0.256696 0.966492i $$-0.582634\pi$$
−0.256696 + 0.966492i $$0.582634\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 3.74585 0.239803
$$245$$ −19.1018 −1.22037
$$246$$ 11.8962 0.758473
$$247$$ −0.104723 −0.00666339
$$248$$ 8.49096 0.539176
$$249$$ 8.00000 0.506979
$$250$$ −1.00000 −0.0632456
$$251$$ 20.2596 1.27878 0.639388 0.768884i $$-0.279188\pi$$
0.639388 + 0.768884i $$0.279188\pi$$
$$252$$ −5.10899 −0.321836
$$253$$ 0 0
$$254$$ 6.96281 0.436886
$$255$$ 0.775565 0.0485678
$$256$$ 1.00000 0.0625000
$$257$$ −19.9638 −1.24531 −0.622655 0.782497i $$-0.713946\pi$$
−0.622655 + 0.782497i $$0.713946\pi$$
$$258$$ 6.01163 0.374268
$$259$$ 20.0942 1.24859
$$260$$ 1.33343 0.0826957
$$261$$ 2.01882 0.124962
$$262$$ 2.19916 0.135865
$$263$$ 0.696114 0.0429242 0.0214621 0.999770i $$-0.493168\pi$$
0.0214621 + 0.999770i $$0.493168\pi$$
$$264$$ 0 0
$$265$$ −10.3451 −0.635492
$$266$$ −0.401245 −0.0246019
$$267$$ −9.67821 −0.592297
$$268$$ 0.588036 0.0359200
$$269$$ 5.92175 0.361055 0.180528 0.983570i $$-0.442219\pi$$
0.180528 + 0.983570i $$0.442219\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 13.9146 0.845249 0.422625 0.906305i $$-0.361109\pi$$
0.422625 + 0.906305i $$0.361109\pi$$
$$272$$ −0.775565 −0.0470255
$$273$$ 6.81247 0.412309
$$274$$ −2.17590 −0.131451
$$275$$ 0 0
$$276$$ 6.64849 0.400192
$$277$$ 10.1025 0.607003 0.303501 0.952831i $$-0.401844\pi$$
0.303501 + 0.952831i $$0.401844\pi$$
$$278$$ −10.3074 −0.618198
$$279$$ 8.49096 0.508340
$$280$$ 5.10899 0.305321
$$281$$ −28.9081 −1.72451 −0.862256 0.506472i $$-0.830949\pi$$
−0.862256 + 0.506472i $$0.830949\pi$$
$$282$$ 6.56950 0.391208
$$283$$ 13.1211 0.779967 0.389984 0.920822i $$-0.372481\pi$$
0.389984 + 0.920822i $$0.372481\pi$$
$$284$$ −2.31506 −0.137374
$$285$$ −0.0785371 −0.00465213
$$286$$ 0 0
$$287$$ −60.7775 −3.58759
$$288$$ 1.00000 0.0589256
$$289$$ −16.3985 −0.964618
$$290$$ −2.01882 −0.118549
$$291$$ −13.4541 −0.788691
$$292$$ −10.9443 −0.640465
$$293$$ −22.6601 −1.32382 −0.661909 0.749584i $$-0.730254\pi$$
−0.661909 + 0.749584i $$0.730254\pi$$
$$294$$ 19.1018 1.11404
$$295$$ 5.32624 0.310106
$$296$$ −3.93310 −0.228607
$$297$$ 0 0
$$298$$ 15.2477 0.883276
$$299$$ −8.86528 −0.512692
$$300$$ 1.00000 0.0577350
$$301$$ −30.7134 −1.77029
$$302$$ −6.56921 −0.378016
$$303$$ 10.6408 0.611300
$$304$$ 0.0785371 0.00450441
$$305$$ −3.74585 −0.214487
$$306$$ −0.775565 −0.0443361
$$307$$ 29.3158 1.67314 0.836571 0.547859i $$-0.184557\pi$$
0.836571 + 0.547859i $$0.184557\pi$$
$$308$$ 0 0
$$309$$ 16.7087 0.950522
$$310$$ −8.49096 −0.482254
$$311$$ 11.3570 0.643995 0.321998 0.946741i $$-0.395646\pi$$
0.321998 + 0.946741i $$0.395646\pi$$
$$312$$ −1.33343 −0.0754905
$$313$$ −3.17664 −0.179554 −0.0897770 0.995962i $$-0.528615\pi$$
−0.0897770 + 0.995962i $$0.528615\pi$$
$$314$$ −9.30815 −0.525290
$$315$$ 5.10899 0.287859
$$316$$ −12.3267 −0.693431
$$317$$ 4.73101 0.265720 0.132860 0.991135i $$-0.457584\pi$$
0.132860 + 0.991135i $$0.457584\pi$$
$$318$$ 10.3451 0.580122
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ −17.5511 −0.979609
$$322$$ −33.9671 −1.89291
$$323$$ −0.0609106 −0.00338916
$$324$$ 1.00000 0.0555556
$$325$$ −1.33343 −0.0739652
$$326$$ −18.7018 −1.03579
$$327$$ 8.94427 0.494619
$$328$$ 11.8962 0.656857
$$329$$ −33.5635 −1.85042
$$330$$ 0 0
$$331$$ 30.0424 1.65128 0.825639 0.564199i $$-0.190815\pi$$
0.825639 + 0.564199i $$0.190815\pi$$
$$332$$ 8.00000 0.439057
$$333$$ −3.93310 −0.215532
$$334$$ 2.37478 0.129942
$$335$$ −0.588036 −0.0321278
$$336$$ −5.10899 −0.278718
$$337$$ 25.3570 1.38128 0.690641 0.723198i $$-0.257328\pi$$
0.690641 + 0.723198i $$0.257328\pi$$
$$338$$ −11.2220 −0.610395
$$339$$ −5.72703 −0.311049
$$340$$ 0.775565 0.0420609
$$341$$ 0 0
$$342$$ 0.0785371 0.00424680
$$343$$ −61.8280 −3.33840
$$344$$ 6.01163 0.324126
$$345$$ −6.64849 −0.357943
$$346$$ −2.91427 −0.156672
$$347$$ 3.17664 0.170531 0.0852654 0.996358i $$-0.472826\pi$$
0.0852654 + 0.996358i $$0.472826\pi$$
$$348$$ 2.01882 0.108220
$$349$$ 6.95865 0.372488 0.186244 0.982504i $$-0.440368\pi$$
0.186244 + 0.982504i $$0.440368\pi$$
$$350$$ −5.10899 −0.273087
$$351$$ −1.33343 −0.0711731
$$352$$ 0 0
$$353$$ 8.01163 0.426416 0.213208 0.977007i $$-0.431609\pi$$
0.213208 + 0.977007i $$0.431609\pi$$
$$354$$ −5.32624 −0.283086
$$355$$ 2.31506 0.122871
$$356$$ −9.67821 −0.512944
$$357$$ 3.96236 0.209710
$$358$$ 1.56231 0.0825704
$$359$$ 0.180340 0.00951798 0.00475899 0.999989i $$-0.498485\pi$$
0.00475899 + 0.999989i $$0.498485\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −18.9938 −0.999675
$$362$$ 18.4721 0.970874
$$363$$ 0 0
$$364$$ 6.81247 0.357070
$$365$$ 10.9443 0.572849
$$366$$ 3.74585 0.195799
$$367$$ 30.8705 1.61142 0.805712 0.592307i $$-0.201783\pi$$
0.805712 + 0.592307i $$0.201783\pi$$
$$368$$ 6.64849 0.346576
$$369$$ 11.8962 0.619291
$$370$$ 3.93310 0.204472
$$371$$ −52.8528 −2.74398
$$372$$ 8.49096 0.440236
$$373$$ −16.1018 −0.833720 −0.416860 0.908971i $$-0.636869\pi$$
−0.416860 + 0.908971i $$0.636869\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 6.56950 0.338796
$$377$$ −2.69195 −0.138643
$$378$$ −5.10899 −0.262778
$$379$$ 32.5487 1.67191 0.835956 0.548796i $$-0.184914\pi$$
0.835956 + 0.548796i $$0.184914\pi$$
$$380$$ −0.0785371 −0.00402887
$$381$$ 6.96281 0.356716
$$382$$ 20.7278 1.06052
$$383$$ 2.10825 0.107727 0.0538634 0.998548i $$-0.482846\pi$$
0.0538634 + 0.998548i $$0.482846\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −1.37079 −0.0697714
$$387$$ 6.01163 0.305588
$$388$$ −13.4541 −0.683026
$$389$$ −12.6032 −0.639008 −0.319504 0.947585i $$-0.603516\pi$$
−0.319504 + 0.947585i $$0.603516\pi$$
$$390$$ 1.33343 0.0675207
$$391$$ −5.15633 −0.260767
$$392$$ 19.1018 0.964787
$$393$$ 2.19916 0.110933
$$394$$ 5.01135 0.252468
$$395$$ 12.3267 0.620223
$$396$$ 0 0
$$397$$ −30.6341 −1.53748 −0.768741 0.639560i $$-0.779116\pi$$
−0.768741 + 0.639560i $$0.779116\pi$$
$$398$$ 23.3086 1.16836
$$399$$ −0.401245 −0.0200874
$$400$$ 1.00000 0.0500000
$$401$$ −19.9671 −0.997108 −0.498554 0.866859i $$-0.666136\pi$$
−0.498554 + 0.866859i $$0.666136\pi$$
$$402$$ 0.588036 0.0293285
$$403$$ −11.3221 −0.563993
$$404$$ 10.6408 0.529402
$$405$$ −1.00000 −0.0496904
$$406$$ −10.3141 −0.511883
$$407$$ 0 0
$$408$$ −0.775565 −0.0383962
$$409$$ 7.13962 0.353032 0.176516 0.984298i $$-0.443517\pi$$
0.176516 + 0.984298i $$0.443517\pi$$
$$410$$ −11.8962 −0.587511
$$411$$ −2.17590 −0.107329
$$412$$ 16.7087 0.823177
$$413$$ 27.2117 1.33900
$$414$$ 6.64849 0.326755
$$415$$ −8.00000 −0.392705
$$416$$ −1.33343 −0.0653767
$$417$$ −10.3074 −0.504756
$$418$$ 0 0
$$419$$ −22.3691 −1.09280 −0.546400 0.837524i $$-0.684002\pi$$
−0.546400 + 0.837524i $$0.684002\pi$$
$$420$$ 5.10899 0.249293
$$421$$ −6.06091 −0.295391 −0.147695 0.989033i $$-0.547186\pi$$
−0.147695 + 0.989033i $$0.547186\pi$$
$$422$$ 20.0842 0.977682
$$423$$ 6.56950 0.319420
$$424$$ 10.3451 0.502401
$$425$$ −0.775565 −0.0376204
$$426$$ −2.31506 −0.112165
$$427$$ −19.1375 −0.926129
$$428$$ −17.5511 −0.848366
$$429$$ 0 0
$$430$$ −6.01163 −0.289907
$$431$$ −37.1623 −1.79004 −0.895021 0.446023i $$-0.852840\pi$$
−0.895021 + 0.446023i $$0.852840\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 30.6540 1.47314 0.736568 0.676364i $$-0.236445\pi$$
0.736568 + 0.676364i $$0.236445\pi$$
$$434$$ −43.3802 −2.08232
$$435$$ −2.01882 −0.0967951
$$436$$ 8.94427 0.428353
$$437$$ 0.522153 0.0249780
$$438$$ −10.9443 −0.522938
$$439$$ −7.06017 −0.336964 −0.168482 0.985705i $$-0.553886\pi$$
−0.168482 + 0.985705i $$0.553886\pi$$
$$440$$ 0 0
$$441$$ 19.1018 0.909610
$$442$$ 1.03416 0.0491900
$$443$$ 14.5925 0.693310 0.346655 0.937993i $$-0.387318\pi$$
0.346655 + 0.937993i $$0.387318\pi$$
$$444$$ −3.93310 −0.186656
$$445$$ 9.67821 0.458791
$$446$$ 9.10899 0.431323
$$447$$ 15.2477 0.721192
$$448$$ −5.10899 −0.241377
$$449$$ 12.7687 0.602590 0.301295 0.953531i $$-0.402581\pi$$
0.301295 + 0.953531i $$0.402581\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −5.72703 −0.269377
$$453$$ −6.56921 −0.308649
$$454$$ −20.6901 −0.971035
$$455$$ −6.81247 −0.319374
$$456$$ 0.0785371 0.00367783
$$457$$ 17.3426 0.811252 0.405626 0.914039i $$-0.367053\pi$$
0.405626 + 0.914039i $$0.367053\pi$$
$$458$$ 10.9819 0.513151
$$459$$ −0.775565 −0.0362003
$$460$$ −6.64849 −0.309987
$$461$$ 2.99355 0.139424 0.0697118 0.997567i $$-0.477792\pi$$
0.0697118 + 0.997567i $$0.477792\pi$$
$$462$$ 0 0
$$463$$ −16.2589 −0.755614 −0.377807 0.925884i $$-0.623322\pi$$
−0.377807 + 0.925884i $$0.623322\pi$$
$$464$$ 2.01882 0.0937215
$$465$$ −8.49096 −0.393759
$$466$$ 2.96309 0.137263
$$467$$ −18.5707 −0.859349 −0.429675 0.902984i $$-0.641372\pi$$
−0.429675 + 0.902984i $$0.641372\pi$$
$$468$$ −1.33343 −0.0616377
$$469$$ −3.00427 −0.138724
$$470$$ −6.56950 −0.303028
$$471$$ −9.30815 −0.428897
$$472$$ −5.32624 −0.245160
$$473$$ 0 0
$$474$$ −12.3267 −0.566184
$$475$$ 0.0785371 0.00360353
$$476$$ 3.96236 0.181614
$$477$$ 10.3451 0.473668
$$478$$ −8.56978 −0.391973
$$479$$ 30.5554 1.39611 0.698056 0.716043i $$-0.254049\pi$$
0.698056 + 0.716043i $$0.254049\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 5.24450 0.239129
$$482$$ −7.97000 −0.363024
$$483$$ −33.9671 −1.54556
$$484$$ 0 0
$$485$$ 13.4541 0.610917
$$486$$ 1.00000 0.0453609
$$487$$ −0.508303 −0.0230334 −0.0115167 0.999934i $$-0.503666\pi$$
−0.0115167 + 0.999934i $$0.503666\pi$$
$$488$$ 3.74585 0.169567
$$489$$ −18.7018 −0.845723
$$490$$ −19.1018 −0.862931
$$491$$ −39.3943 −1.77784 −0.888921 0.458061i $$-0.848544\pi$$
−0.888921 + 0.458061i $$0.848544\pi$$
$$492$$ 11.8962 0.536322
$$493$$ −1.56573 −0.0705168
$$494$$ −0.104723 −0.00471173
$$495$$ 0 0
$$496$$ 8.49096 0.381255
$$497$$ 11.8276 0.530542
$$498$$ 8.00000 0.358489
$$499$$ 21.4608 0.960717 0.480358 0.877072i $$-0.340507\pi$$
0.480358 + 0.877072i $$0.340507\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 2.37478 0.106097
$$502$$ 20.2596 0.904231
$$503$$ 23.5514 1.05011 0.525053 0.851070i $$-0.324046\pi$$
0.525053 + 0.851070i $$0.324046\pi$$
$$504$$ −5.10899 −0.227573
$$505$$ −10.6408 −0.473511
$$506$$ 0 0
$$507$$ −11.2220 −0.498385
$$508$$ 6.96281 0.308925
$$509$$ 38.7178 1.71614 0.858069 0.513535i $$-0.171664\pi$$
0.858069 + 0.513535i $$0.171664\pi$$
$$510$$ 0.775565 0.0343426
$$511$$ 55.9142 2.47350
$$512$$ 1.00000 0.0441942
$$513$$ 0.0785371 0.00346750
$$514$$ −19.9638 −0.880567
$$515$$ −16.7087 −0.736272
$$516$$ 6.01163 0.264647
$$517$$ 0 0
$$518$$ 20.0942 0.882887
$$519$$ −2.91427 −0.127922
$$520$$ 1.33343 0.0584747
$$521$$ 2.53921 0.111245 0.0556225 0.998452i $$-0.482286\pi$$
0.0556225 + 0.998452i $$0.482286\pi$$
$$522$$ 2.01882 0.0883615
$$523$$ −37.3202 −1.63190 −0.815950 0.578122i $$-0.803786\pi$$
−0.815950 + 0.578122i $$0.803786\pi$$
$$524$$ 2.19916 0.0960709
$$525$$ −5.10899 −0.222975
$$526$$ 0.696114 0.0303520
$$527$$ −6.58529 −0.286860
$$528$$ 0 0
$$529$$ 21.2024 0.921844
$$530$$ −10.3451 −0.449361
$$531$$ −5.32624 −0.231139
$$532$$ −0.401245 −0.0173962
$$533$$ −15.8627 −0.687090
$$534$$ −9.67821 −0.418817
$$535$$ 17.5511 0.758802
$$536$$ 0.588036 0.0253993
$$537$$ 1.56231 0.0674185
$$538$$ 5.92175 0.255305
$$539$$ 0 0
$$540$$ −1.00000 −0.0430331
$$541$$ 9.23515 0.397050 0.198525 0.980096i $$-0.436385\pi$$
0.198525 + 0.980096i $$0.436385\pi$$
$$542$$ 13.9146 0.597681
$$543$$ 18.4721 0.792715
$$544$$ −0.775565 −0.0332521
$$545$$ −8.94427 −0.383131
$$546$$ 6.81247 0.291547
$$547$$ 25.3758 1.08499 0.542495 0.840059i $$-0.317480\pi$$
0.542495 + 0.840059i $$0.317480\pi$$
$$548$$ −2.17590 −0.0929497
$$549$$ 3.74585 0.159869
$$550$$ 0 0
$$551$$ 0.158552 0.00675456
$$552$$ 6.64849 0.282978
$$553$$ 62.9770 2.67805
$$554$$ 10.1025 0.429216
$$555$$ 3.93310 0.166951
$$556$$ −10.3074 −0.437132
$$557$$ 10.6107 0.449589 0.224794 0.974406i $$-0.427829\pi$$
0.224794 + 0.974406i $$0.427829\pi$$
$$558$$ 8.49096 0.359451
$$559$$ −8.01608 −0.339044
$$560$$ 5.10899 0.215894
$$561$$ 0 0
$$562$$ −28.9081 −1.21941
$$563$$ 4.64450 0.195742 0.0978712 0.995199i $$-0.468797\pi$$
0.0978712 + 0.995199i $$0.468797\pi$$
$$564$$ 6.56950 0.276626
$$565$$ 5.72703 0.240938
$$566$$ 13.1211 0.551520
$$567$$ −5.10899 −0.214558
$$568$$ −2.31506 −0.0971378
$$569$$ 35.1105 1.47191 0.735954 0.677032i $$-0.236734\pi$$
0.735954 + 0.677032i $$0.236734\pi$$
$$570$$ −0.0785371 −0.00328956
$$571$$ −17.7758 −0.743896 −0.371948 0.928254i $$-0.621310\pi$$
−0.371948 + 0.928254i $$0.621310\pi$$
$$572$$ 0 0
$$573$$ 20.7278 0.865915
$$574$$ −60.7775 −2.53681
$$575$$ 6.64849 0.277261
$$576$$ 1.00000 0.0416667
$$577$$ 3.63439 0.151302 0.0756509 0.997134i $$-0.475897\pi$$
0.0756509 + 0.997134i $$0.475897\pi$$
$$578$$ −16.3985 −0.682088
$$579$$ −1.37079 −0.0569681
$$580$$ −2.01882 −0.0838270
$$581$$ −40.8719 −1.69565
$$582$$ −13.4541 −0.557688
$$583$$ 0 0
$$584$$ −10.9443 −0.452877
$$585$$ 1.33343 0.0551304
$$586$$ −22.6601 −0.936081
$$587$$ 7.39314 0.305148 0.152574 0.988292i $$-0.451244\pi$$
0.152574 + 0.988292i $$0.451244\pi$$
$$588$$ 19.1018 0.787745
$$589$$ 0.666855 0.0274773
$$590$$ 5.32624 0.219278
$$591$$ 5.01135 0.206139
$$592$$ −3.93310 −0.161649
$$593$$ 35.9315 1.47553 0.737766 0.675057i $$-0.235881\pi$$
0.737766 + 0.675057i $$0.235881\pi$$
$$594$$ 0 0
$$595$$ −3.96236 −0.162441
$$596$$ 15.2477 0.624570
$$597$$ 23.3086 0.953958
$$598$$ −8.86528 −0.362528
$$599$$ −20.5554 −0.839871 −0.419935 0.907554i $$-0.637947\pi$$
−0.419935 + 0.907554i $$0.637947\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 5.12136 0.208905 0.104452 0.994530i $$-0.466691\pi$$
0.104452 + 0.994530i $$0.466691\pi$$
$$602$$ −30.7134 −1.25178
$$603$$ 0.588036 0.0239467
$$604$$ −6.56921 −0.267297
$$605$$ 0 0
$$606$$ 10.6408 0.432255
$$607$$ 37.3946 1.51780 0.758900 0.651207i $$-0.225737\pi$$
0.758900 + 0.651207i $$0.225737\pi$$
$$608$$ 0.0785371 0.00318510
$$609$$ −10.3141 −0.417950
$$610$$ −3.74585 −0.151665
$$611$$ −8.75995 −0.354389
$$612$$ −0.775565 −0.0313504
$$613$$ 8.39685 0.339145 0.169573 0.985518i $$-0.445761\pi$$
0.169573 + 0.985518i $$0.445761\pi$$
$$614$$ 29.3158 1.18309
$$615$$ −11.8962 −0.479701
$$616$$ 0 0
$$617$$ −25.9029 −1.04281 −0.521406 0.853309i $$-0.674592\pi$$
−0.521406 + 0.853309i $$0.674592\pi$$
$$618$$ 16.7087 0.672121
$$619$$ 25.1441 1.01063 0.505313 0.862936i $$-0.331377\pi$$
0.505313 + 0.862936i $$0.331377\pi$$
$$620$$ −8.49096 −0.341005
$$621$$ 6.64849 0.266795
$$622$$ 11.3570 0.455373
$$623$$ 49.4459 1.98101
$$624$$ −1.33343 −0.0533798
$$625$$ 1.00000 0.0400000
$$626$$ −3.17664 −0.126964
$$627$$ 0 0
$$628$$ −9.30815 −0.371436
$$629$$ 3.05037 0.121626
$$630$$ 5.10899 0.203547
$$631$$ −42.0436 −1.67373 −0.836864 0.547411i $$-0.815613\pi$$
−0.836864 + 0.547411i $$0.815613\pi$$
$$632$$ −12.3267 −0.490330
$$633$$ 20.0842 0.798274
$$634$$ 4.73101 0.187893
$$635$$ −6.96281 −0.276311
$$636$$ 10.3451 0.410208
$$637$$ −25.4709 −1.00919
$$638$$ 0 0
$$639$$ −2.31506 −0.0915824
$$640$$ −1.00000 −0.0395285
$$641$$ −2.11562 −0.0835619 −0.0417809 0.999127i $$-0.513303\pi$$
−0.0417809 + 0.999127i $$0.513303\pi$$
$$642$$ −17.5511 −0.692688
$$643$$ −39.4827 −1.55704 −0.778522 0.627617i $$-0.784030\pi$$
−0.778522 + 0.627617i $$0.784030\pi$$
$$644$$ −33.9671 −1.33849
$$645$$ −6.01163 −0.236708
$$646$$ −0.0609106 −0.00239649
$$647$$ −16.1454 −0.634743 −0.317371 0.948301i $$-0.602800\pi$$
−0.317371 + 0.948301i $$0.602800\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ −1.33343 −0.0523013
$$651$$ −43.3802 −1.70021
$$652$$ −18.7018 −0.732417
$$653$$ −1.29134 −0.0505340 −0.0252670 0.999681i $$-0.508044\pi$$
−0.0252670 + 0.999681i $$0.508044\pi$$
$$654$$ 8.94427 0.349749
$$655$$ −2.19916 −0.0859284
$$656$$ 11.8962 0.464468
$$657$$ −10.9443 −0.426977
$$658$$ −33.5635 −1.30844
$$659$$ 23.7015 0.923278 0.461639 0.887068i $$-0.347262\pi$$
0.461639 + 0.887068i $$0.347262\pi$$
$$660$$ 0 0
$$661$$ −17.0796 −0.664318 −0.332159 0.943223i $$-0.607777\pi$$
−0.332159 + 0.943223i $$0.607777\pi$$
$$662$$ 30.0424 1.16763
$$663$$ 1.03416 0.0401634
$$664$$ 8.00000 0.310460
$$665$$ 0.401245 0.0155596
$$666$$ −3.93310 −0.152404
$$667$$ 13.4221 0.519707
$$668$$ 2.37478 0.0918829
$$669$$ 9.10899 0.352174
$$670$$ −0.588036 −0.0227178
$$671$$ 0 0
$$672$$ −5.10899 −0.197084
$$673$$ 8.51069 0.328063 0.164032 0.986455i $$-0.447550\pi$$
0.164032 + 0.986455i $$0.447550\pi$$
$$674$$ 25.3570 0.976714
$$675$$ 1.00000 0.0384900
$$676$$ −11.2220 −0.431614
$$677$$ −22.7125 −0.872911 −0.436456 0.899726i $$-0.643766\pi$$
−0.436456 + 0.899726i $$0.643766\pi$$
$$678$$ −5.72703 −0.219945
$$679$$ 68.7367 2.63787
$$680$$ 0.775565 0.0297416
$$681$$ −20.6901 −0.792847
$$682$$ 0 0
$$683$$ −11.0419 −0.422507 −0.211254 0.977431i $$-0.567755\pi$$
−0.211254 + 0.977431i $$0.567755\pi$$
$$684$$ 0.0785371 0.00300294
$$685$$ 2.17590 0.0831367
$$686$$ −61.8280 −2.36060
$$687$$ 10.9819 0.418986
$$688$$ 6.01163 0.229191
$$689$$ −13.7944 −0.525524
$$690$$ −6.64849 −0.253104
$$691$$ 9.49934 0.361372 0.180686 0.983541i $$-0.442168\pi$$
0.180686 + 0.983541i $$0.442168\pi$$
$$692$$ −2.91427 −0.110784
$$693$$ 0 0
$$694$$ 3.17664 0.120583
$$695$$ 10.3074 0.390983
$$696$$ 2.01882 0.0765233
$$697$$ −9.22627 −0.349470
$$698$$ 6.95865 0.263389
$$699$$ 2.96309 0.112075
$$700$$ −5.10899 −0.193102
$$701$$ −25.7924 −0.974165 −0.487082 0.873356i $$-0.661939\pi$$
−0.487082 + 0.873356i $$0.661939\pi$$
$$702$$ −1.33343 −0.0503270
$$703$$ −0.308894 −0.0116501
$$704$$ 0 0
$$705$$ −6.56950 −0.247422
$$706$$ 8.01163 0.301522
$$707$$ −54.3640 −2.04457
$$708$$ −5.32624 −0.200172
$$709$$ −13.7924 −0.517984 −0.258992 0.965880i $$-0.583390\pi$$
−0.258992 + 0.965880i $$0.583390\pi$$
$$710$$ 2.31506 0.0868827
$$711$$ −12.3267 −0.462287
$$712$$ −9.67821 −0.362706
$$713$$ 56.4520 2.11415
$$714$$ 3.96236 0.148287
$$715$$ 0 0
$$716$$ 1.56231 0.0583861
$$717$$ −8.56978 −0.320044
$$718$$ 0.180340 0.00673022
$$719$$ −19.4035 −0.723629 −0.361814 0.932250i $$-0.617843\pi$$
−0.361814 + 0.932250i $$0.617843\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −85.3644 −3.17914
$$722$$ −18.9938 −0.706877
$$723$$ −7.97000 −0.296408
$$724$$ 18.4721 0.686512
$$725$$ 2.01882 0.0749772
$$726$$ 0 0
$$727$$ −1.37752 −0.0510895 −0.0255447 0.999674i $$-0.508132\pi$$
−0.0255447 + 0.999674i $$0.508132\pi$$
$$728$$ 6.81247 0.252487
$$729$$ 1.00000 0.0370370
$$730$$ 10.9443 0.405066
$$731$$ −4.66241 −0.172446
$$732$$ 3.74585 0.138451
$$733$$ −21.8778 −0.808076 −0.404038 0.914742i $$-0.632394\pi$$
−0.404038 + 0.914742i $$0.632394\pi$$
$$734$$ 30.8705 1.13945
$$735$$ −19.1018 −0.704581
$$736$$ 6.64849 0.245067
$$737$$ 0 0
$$738$$ 11.8962 0.437905
$$739$$ −0.525400 −0.0193272 −0.00966358 0.999953i $$-0.503076\pi$$
−0.00966358 + 0.999953i $$0.503076\pi$$
$$740$$ 3.93310 0.144583
$$741$$ −0.104723 −0.00384711
$$742$$ −52.8528 −1.94029
$$743$$ −47.8868 −1.75680 −0.878399 0.477929i $$-0.841388\pi$$
−0.878399 + 0.477929i $$0.841388\pi$$
$$744$$ 8.49096 0.311294
$$745$$ −15.2477 −0.558633
$$746$$ −16.1018 −0.589529
$$747$$ 8.00000 0.292705
$$748$$ 0 0
$$749$$ 89.6686 3.27642
$$750$$ −1.00000 −0.0365148
$$751$$ 36.1543 1.31929 0.659645 0.751577i $$-0.270707\pi$$
0.659645 + 0.751577i $$0.270707\pi$$
$$752$$ 6.56950 0.239565
$$753$$ 20.2596 0.738301
$$754$$ −2.69195 −0.0980352
$$755$$ 6.56921 0.239078
$$756$$ −5.10899 −0.185812
$$757$$ −43.6114 −1.58508 −0.792542 0.609818i $$-0.791243\pi$$
−0.792542 + 0.609818i $$0.791243\pi$$
$$758$$ 32.5487 1.18222
$$759$$ 0 0
$$760$$ −0.0785371 −0.00284884
$$761$$ 1.10226 0.0399569 0.0199784 0.999800i $$-0.493640\pi$$
0.0199784 + 0.999800i $$0.493640\pi$$
$$762$$ 6.96281 0.252236
$$763$$ −45.6962 −1.65431
$$764$$ 20.7278 0.749904
$$765$$ 0.775565 0.0280406
$$766$$ 2.10825 0.0761743
$$767$$ 7.10215 0.256444
$$768$$ 1.00000 0.0360844
$$769$$ 26.9983 0.973583 0.486791 0.873518i $$-0.338167\pi$$
0.486791 + 0.873518i $$0.338167\pi$$
$$770$$ 0 0
$$771$$ −19.9638 −0.718980
$$772$$ −1.37079 −0.0493358
$$773$$ −23.5077 −0.845512 −0.422756 0.906244i $$-0.638937\pi$$
−0.422756 + 0.906244i $$0.638937\pi$$
$$774$$ 6.01163 0.216084
$$775$$ 8.49096 0.305004
$$776$$ −13.4541 −0.482972
$$777$$ 20.0942 0.720874
$$778$$ −12.6032 −0.451847
$$779$$ 0.934292 0.0334745
$$780$$ 1.33343 0.0477444
$$781$$ 0 0
$$782$$ −5.15633 −0.184390
$$783$$ 2.01882 0.0721468
$$784$$ 19.1018 0.682207
$$785$$ 9.30815 0.332222
$$786$$ 2.19916 0.0784415
$$787$$ 32.3634 1.15363 0.576816 0.816874i $$-0.304295\pi$$
0.576816 + 0.816874i $$0.304295\pi$$
$$788$$ 5.01135 0.178522
$$789$$ 0.696114 0.0247823
$$790$$ 12.3267 0.438564
$$791$$ 29.2593 1.04034
$$792$$ 0 0
$$793$$ −4.99482 −0.177371
$$794$$ −30.6341 −1.08716
$$795$$ −10.3451 −0.366901
$$796$$ 23.3086 0.826152
$$797$$ −40.4031 −1.43115 −0.715575 0.698536i $$-0.753835\pi$$
−0.715575 + 0.698536i $$0.753835\pi$$
$$798$$ −0.401245 −0.0142039
$$799$$ −5.09507 −0.180251
$$800$$ 1.00000 0.0353553
$$801$$ −9.67821 −0.341963
$$802$$ −19.9671 −0.705062
$$803$$ 0 0
$$804$$ 0.588036 0.0207384
$$805$$ 33.9671 1.19718
$$806$$ −11.3221 −0.398803
$$807$$ 5.92175 0.208455
$$808$$ 10.6408 0.374344
$$809$$ −12.2365 −0.430213 −0.215107 0.976591i $$-0.569010\pi$$
−0.215107 + 0.976591i $$0.569010\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 15.1684 0.532635 0.266318 0.963885i $$-0.414193\pi$$
0.266318 + 0.963885i $$0.414193\pi$$
$$812$$ −10.3141 −0.361956
$$813$$ 13.9146 0.488005
$$814$$ 0 0
$$815$$ 18.7018 0.655094
$$816$$ −0.775565 −0.0271502
$$817$$ 0.472136 0.0165179
$$818$$ 7.13962 0.249631
$$819$$ 6.81247 0.238047
$$820$$ −11.8962 −0.415433
$$821$$ 14.4730 0.505113 0.252556 0.967582i $$-0.418729\pi$$
0.252556 + 0.967582i $$0.418729\pi$$
$$822$$ −2.17590 −0.0758931
$$823$$ 3.54640 0.123620 0.0618099 0.998088i $$-0.480313\pi$$
0.0618099 + 0.998088i $$0.480313\pi$$
$$824$$ 16.7087 0.582074
$$825$$ 0 0
$$826$$ 27.2117 0.946816
$$827$$ 37.2217 1.29432 0.647162 0.762352i $$-0.275956\pi$$
0.647162 + 0.762352i $$0.275956\pi$$
$$828$$ 6.64849 0.231051
$$829$$ −40.8705 −1.41949 −0.709745 0.704459i $$-0.751190\pi$$
−0.709745 + 0.704459i $$0.751190\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 10.1025 0.350453
$$832$$ −1.33343 −0.0462283
$$833$$ −14.8147 −0.513299
$$834$$ −10.3074 −0.356917
$$835$$ −2.37478 −0.0821825
$$836$$ 0 0
$$837$$ 8.49096 0.293490
$$838$$ −22.3691 −0.772727
$$839$$ −16.4136 −0.566661 −0.283330 0.959022i $$-0.591439\pi$$
−0.283330 + 0.959022i $$0.591439\pi$$
$$840$$ 5.10899 0.176277
$$841$$ −24.9244 −0.859461
$$842$$ −6.06091 −0.208873
$$843$$ −28.9081 −0.995648
$$844$$ 20.0842 0.691326
$$845$$ 11.2220 0.386048
$$846$$ 6.56950 0.225864
$$847$$ 0 0
$$848$$ 10.3451 0.355251
$$849$$ 13.1211 0.450314
$$850$$ −0.775565 −0.0266017
$$851$$ −26.1491 −0.896381
$$852$$ −2.31506 −0.0793127
$$853$$ −18.1880 −0.622745 −0.311372 0.950288i $$-0.600789\pi$$
−0.311372 + 0.950288i $$0.600789\pi$$
$$854$$ −19.1375 −0.654872
$$855$$ −0.0785371 −0.00268591
$$856$$ −17.5511 −0.599885
$$857$$ −48.5679 −1.65905 −0.829525 0.558470i $$-0.811389\pi$$
−0.829525 + 0.558470i $$0.811389\pi$$
$$858$$ 0 0
$$859$$ −29.1564 −0.994805 −0.497402 0.867520i $$-0.665713\pi$$
−0.497402 + 0.867520i $$0.665713\pi$$
$$860$$ −6.01163 −0.204995
$$861$$ −60.7775 −2.07129
$$862$$ −37.1623 −1.26575
$$863$$ −29.7157 −1.01153 −0.505767 0.862670i $$-0.668790\pi$$
−0.505767 + 0.862670i $$0.668790\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 2.91427 0.0990883
$$866$$ 30.6540 1.04166
$$867$$ −16.3985 −0.556922
$$868$$ −43.3802 −1.47242
$$869$$ 0 0
$$870$$ −2.01882 −0.0684445
$$871$$ −0.784103 −0.0265683
$$872$$ 8.94427 0.302891
$$873$$ −13.4541 −0.455351
$$874$$ 0.522153 0.0176621
$$875$$ 5.10899 0.172715
$$876$$ −10.9443 −0.369773
$$877$$ −26.7194 −0.902249 −0.451125 0.892461i $$-0.648977\pi$$
−0.451125 + 0.892461i $$0.648977\pi$$
$$878$$ −7.06017 −0.238269
$$879$$ −22.6601 −0.764307
$$880$$ 0 0
$$881$$ 2.72012 0.0916431 0.0458216 0.998950i $$-0.485409\pi$$
0.0458216 + 0.998950i $$0.485409\pi$$
$$882$$ 19.1018 0.643191
$$883$$ 34.7250 1.16859 0.584295 0.811541i $$-0.301371\pi$$
0.584295 + 0.811541i $$0.301371\pi$$
$$884$$ 1.03416 0.0347825
$$885$$ 5.32624 0.179040
$$886$$ 14.5925 0.490244
$$887$$ 37.2657 1.25126 0.625630 0.780120i $$-0.284842\pi$$
0.625630 + 0.780120i $$0.284842\pi$$
$$888$$ −3.93310 −0.131986
$$889$$ −35.5730 −1.19308
$$890$$ 9.67821 0.324414
$$891$$ 0 0
$$892$$ 9.10899 0.304992
$$893$$ 0.515949 0.0172656
$$894$$ 15.2477 0.509959
$$895$$ −1.56231 −0.0522221
$$896$$ −5.10899 −0.170679
$$897$$ −8.86528 −0.296003
$$898$$ 12.7687 0.426096
$$899$$ 17.1417 0.571709
$$900$$ 1.00000 0.0333333
$$901$$ −8.02327 −0.267294
$$902$$ 0 0
$$903$$ −30.7134 −1.02208
$$904$$ −5.72703 −0.190478
$$905$$ −18.4721 −0.614035
$$906$$ −6.56921 −0.218247
$$907$$ −39.7936 −1.32133 −0.660663 0.750682i $$-0.729725\pi$$
−0.660663 + 0.750682i $$0.729725\pi$$
$$908$$ −20.6901 −0.686626
$$909$$ 10.6408 0.352934
$$910$$ −6.81247 −0.225831
$$911$$ −12.8234 −0.424857 −0.212429 0.977177i $$-0.568137\pi$$
−0.212429 + 0.977177i $$0.568137\pi$$
$$912$$ 0.0785371 0.00260062
$$913$$ 0 0
$$914$$ 17.3426 0.573642
$$915$$ −3.74585 −0.123834
$$916$$ 10.9819 0.362853
$$917$$ −11.2355 −0.371029
$$918$$ −0.775565 −0.0255975
$$919$$ 37.1139 1.22427 0.612137 0.790752i $$-0.290310\pi$$
0.612137 + 0.790752i $$0.290310\pi$$
$$920$$ −6.64849 −0.219194
$$921$$ 29.3158 0.965988
$$922$$ 2.99355 0.0985873
$$923$$ 3.08697 0.101609
$$924$$ 0 0
$$925$$ −3.93310 −0.129319
$$926$$ −16.2589 −0.534300
$$927$$ 16.7087 0.548784
$$928$$ 2.01882 0.0662711
$$929$$ 9.97000 0.327105 0.163553 0.986535i $$-0.447705\pi$$
0.163553 + 0.986535i $$0.447705\pi$$
$$930$$ −8.49096 −0.278429
$$931$$ 1.50020 0.0491670
$$932$$ 2.96309 0.0970594
$$933$$ 11.3570 0.371811
$$934$$ −18.5707 −0.607652
$$935$$ 0 0
$$936$$ −1.33343 −0.0435844
$$937$$ −11.9767 −0.391263 −0.195631 0.980677i $$-0.562676\pi$$
−0.195631 + 0.980677i $$0.562676\pi$$
$$938$$ −3.00427 −0.0980929
$$939$$ −3.17664 −0.103666
$$940$$ −6.56950 −0.214273
$$941$$ −5.50996 −0.179619 −0.0898097 0.995959i $$-0.528626\pi$$
−0.0898097 + 0.995959i $$0.528626\pi$$
$$942$$ −9.30815 −0.303276
$$943$$ 79.0917 2.57558
$$944$$ −5.32624 −0.173354
$$945$$ 5.10899 0.166196
$$946$$ 0 0
$$947$$ −27.3946 −0.890206 −0.445103 0.895479i $$-0.646833\pi$$
−0.445103 + 0.895479i $$0.646833\pi$$
$$948$$ −12.3267 −0.400352
$$949$$ 14.5934 0.473722
$$950$$ 0.0785371 0.00254808
$$951$$ 4.73101 0.153414
$$952$$ 3.96236 0.128421
$$953$$ −7.98985 −0.258816 −0.129408 0.991591i $$-0.541308\pi$$
−0.129408 + 0.991591i $$0.541308\pi$$
$$954$$ 10.3451 0.334934
$$955$$ −20.7278 −0.670735
$$956$$ −8.56978 −0.277166
$$957$$ 0 0
$$958$$ 30.5554 0.987200
$$959$$ 11.1166 0.358975
$$960$$ −1.00000 −0.0322749
$$961$$ 41.0964 1.32569
$$962$$ 5.24450 0.169089
$$963$$ −17.5511 −0.565577
$$964$$ −7.97000 −0.256696
$$965$$ 1.37079 0.0441273
$$966$$ −33.9671 −1.09287
$$967$$ 19.0966 0.614106 0.307053 0.951692i $$-0.400657\pi$$
0.307053 + 0.951692i $$0.400657\pi$$
$$968$$ 0 0
$$969$$ −0.0609106 −0.00195673
$$970$$ 13.4541 0.431984
$$971$$ 0.149777 0.00480656 0.00240328 0.999997i $$-0.499235\pi$$
0.00240328 + 0.999997i $$0.499235\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 52.6605 1.68822
$$974$$ −0.508303 −0.0162871
$$975$$ −1.33343 −0.0427039
$$976$$ 3.74585 0.119902
$$977$$ −19.1384 −0.612292 −0.306146 0.951984i $$-0.599040\pi$$
−0.306146 + 0.951984i $$0.599040\pi$$
$$978$$ −18.7018 −0.598016
$$979$$ 0 0
$$980$$ −19.1018 −0.610185
$$981$$ 8.94427 0.285569
$$982$$ −39.3943 −1.25712
$$983$$ 8.58575 0.273843 0.136921 0.990582i $$-0.456279\pi$$
0.136921 + 0.990582i $$0.456279\pi$$
$$984$$ 11.8962 0.379237
$$985$$ −5.01135 −0.159675
$$986$$ −1.56573 −0.0498629
$$987$$ −33.5635 −1.06834
$$988$$ −0.104723 −0.00333170
$$989$$ 39.9683 1.27092
$$990$$ 0 0
$$991$$ −11.3086 −0.359230 −0.179615 0.983737i $$-0.557485\pi$$
−0.179615 + 0.983737i $$0.557485\pi$$
$$992$$ 8.49096 0.269588
$$993$$ 30.0424 0.953366
$$994$$ 11.8276 0.375150
$$995$$ −23.3086 −0.738933
$$996$$ 8.00000 0.253490
$$997$$ 37.0321 1.17282 0.586409 0.810015i $$-0.300541\pi$$
0.586409 + 0.810015i $$0.300541\pi$$
$$998$$ 21.4608 0.679329
$$999$$ −3.93310 −0.124438
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 3630.2.a.bs.1.1 4
11.2 odd 10 330.2.m.f.301.2 yes 8
11.6 odd 10 330.2.m.f.91.2 8
11.10 odd 2 3630.2.a.bq.1.4 4
33.2 even 10 990.2.n.i.631.2 8
33.17 even 10 990.2.n.i.91.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
330.2.m.f.91.2 8 11.6 odd 10
330.2.m.f.301.2 yes 8 11.2 odd 10
990.2.n.i.91.2 8 33.17 even 10
990.2.n.i.631.2 8 33.2 even 10
3630.2.a.bq.1.4 4 11.10 odd 2
3630.2.a.bs.1.1 4 1.1 even 1 trivial