# Properties

 Label 8400.2.a.dl.1.3 Level $8400$ Weight $2$ Character 8400.1 Self dual yes Analytic conductor $67.074$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.0743376979$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 8400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +5.05086 q^{11} +3.37778 q^{13} +7.18421 q^{17} -8.23506 q^{19} +1.00000 q^{21} +6.23506 q^{23} +1.00000 q^{27} -2.00000 q^{29} +4.62222 q^{31} +5.05086 q^{33} -4.85728 q^{37} +3.37778 q^{39} -3.37778 q^{41} -1.24443 q^{43} +1.00000 q^{49} +7.18421 q^{51} +4.62222 q^{53} -8.23506 q^{57} +11.6128 q^{59} +0.488863 q^{61} +1.00000 q^{63} -3.61285 q^{67} +6.23506 q^{69} +10.2953 q^{71} +16.2351 q^{73} +5.05086 q^{77} -1.24443 q^{79} +1.00000 q^{81} -11.6128 q^{83} -2.00000 q^{87} +6.99063 q^{89} +3.37778 q^{91} +4.62222 q^{93} -8.23506 q^{97} +5.05086 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 10 q^{13} + 8 q^{17} + 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} + 14 q^{31} + 2 q^{33} + 12 q^{37} + 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} + 8 q^{51} + 14 q^{53} + 2 q^{57} + 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} + 18 q^{71} + 22 q^{73} + 2 q^{77} - 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} + 10 q^{91} + 14 q^{93} + 2 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 10 * q^13 + 8 * q^17 + 2 * q^19 + 3 * q^21 - 8 * q^23 + 3 * q^27 - 6 * q^29 + 14 * q^31 + 2 * q^33 + 12 * q^37 + 10 * q^39 - 10 * q^41 - 4 * q^43 + 3 * q^49 + 8 * q^51 + 14 * q^53 + 2 * q^57 + 8 * q^59 + 2 * q^61 + 3 * q^63 + 16 * q^67 - 8 * q^69 + 18 * q^71 + 22 * q^73 + 2 * q^77 - 4 * q^79 + 3 * q^81 - 8 * q^83 - 6 * q^87 - 6 * q^89 + 10 * q^91 + 14 * q^93 + 2 * q^97 + 2 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.05086 1.52289 0.761445 0.648229i $$-0.224490\pi$$
0.761445 + 0.648229i $$0.224490\pi$$
$$12$$ 0 0
$$13$$ 3.37778 0.936829 0.468414 0.883509i $$-0.344825\pi$$
0.468414 + 0.883509i $$0.344825\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.18421 1.74243 0.871213 0.490905i $$-0.163334\pi$$
0.871213 + 0.490905i $$0.163334\pi$$
$$18$$ 0 0
$$19$$ −8.23506 −1.88925 −0.944627 0.328147i $$-0.893576\pi$$
−0.944627 + 0.328147i $$0.893576\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ 6.23506 1.30010 0.650050 0.759891i $$-0.274748\pi$$
0.650050 + 0.759891i $$0.274748\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 4.62222 0.830174 0.415087 0.909782i $$-0.363751\pi$$
0.415087 + 0.909782i $$0.363751\pi$$
$$32$$ 0 0
$$33$$ 5.05086 0.879241
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.85728 −0.798532 −0.399266 0.916835i $$-0.630735\pi$$
−0.399266 + 0.916835i $$0.630735\pi$$
$$38$$ 0 0
$$39$$ 3.37778 0.540878
$$40$$ 0 0
$$41$$ −3.37778 −0.527521 −0.263761 0.964588i $$-0.584963\pi$$
−0.263761 + 0.964588i $$0.584963\pi$$
$$42$$ 0 0
$$43$$ −1.24443 −0.189774 −0.0948870 0.995488i $$-0.530249\pi$$
−0.0948870 + 0.995488i $$0.530249\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 7.18421 1.00599
$$52$$ 0 0
$$53$$ 4.62222 0.634910 0.317455 0.948273i $$-0.397172\pi$$
0.317455 + 0.948273i $$0.397172\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −8.23506 −1.09076
$$58$$ 0 0
$$59$$ 11.6128 1.51186 0.755932 0.654650i $$-0.227184\pi$$
0.755932 + 0.654650i $$0.227184\pi$$
$$60$$ 0 0
$$61$$ 0.488863 0.0625924 0.0312962 0.999510i $$-0.490036\pi$$
0.0312962 + 0.999510i $$0.490036\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −3.61285 −0.441380 −0.220690 0.975344i $$-0.570831\pi$$
−0.220690 + 0.975344i $$0.570831\pi$$
$$68$$ 0 0
$$69$$ 6.23506 0.750613
$$70$$ 0 0
$$71$$ 10.2953 1.22183 0.610913 0.791698i $$-0.290803\pi$$
0.610913 + 0.791698i $$0.290803\pi$$
$$72$$ 0 0
$$73$$ 16.2351 1.90017 0.950085 0.311991i $$-0.100996\pi$$
0.950085 + 0.311991i $$0.100996\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.05086 0.575598
$$78$$ 0 0
$$79$$ −1.24443 −0.140009 −0.0700047 0.997547i $$-0.522301\pi$$
−0.0700047 + 0.997547i $$0.522301\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −11.6128 −1.27468 −0.637338 0.770585i $$-0.719964\pi$$
−0.637338 + 0.770585i $$0.719964\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ 6.99063 0.741006 0.370503 0.928831i $$-0.379185\pi$$
0.370503 + 0.928831i $$0.379185\pi$$
$$90$$ 0 0
$$91$$ 3.37778 0.354088
$$92$$ 0 0
$$93$$ 4.62222 0.479301
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −8.23506 −0.836144 −0.418072 0.908414i $$-0.637294\pi$$
−0.418072 + 0.908414i $$0.637294\pi$$
$$98$$ 0 0
$$99$$ 5.05086 0.507630
$$100$$ 0 0
$$101$$ −9.47949 −0.943245 −0.471622 0.881801i $$-0.656331\pi$$
−0.471622 + 0.881801i $$0.656331\pi$$
$$102$$ 0 0
$$103$$ −16.8573 −1.66100 −0.830499 0.557021i $$-0.811944\pi$$
−0.830499 + 0.557021i $$0.811944\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −15.4795 −1.49646 −0.748230 0.663440i $$-0.769096\pi$$
−0.748230 + 0.663440i $$0.769096\pi$$
$$108$$ 0 0
$$109$$ −1.61285 −0.154483 −0.0772414 0.997012i $$-0.524611\pi$$
−0.0772414 + 0.997012i $$0.524611\pi$$
$$110$$ 0 0
$$111$$ −4.85728 −0.461033
$$112$$ 0 0
$$113$$ 1.86665 0.175599 0.0877997 0.996138i $$-0.472016\pi$$
0.0877997 + 0.996138i $$0.472016\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.37778 0.312276
$$118$$ 0 0
$$119$$ 7.18421 0.658575
$$120$$ 0 0
$$121$$ 14.5111 1.31919
$$122$$ 0 0
$$123$$ −3.37778 −0.304565
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.8573 −1.14090 −0.570450 0.821333i $$-0.693231\pi$$
−0.570450 + 0.821333i $$0.693231\pi$$
$$128$$ 0 0
$$129$$ −1.24443 −0.109566
$$130$$ 0 0
$$131$$ −21.7146 −1.89721 −0.948605 0.316463i $$-0.897505\pi$$
−0.948605 + 0.316463i $$0.897505\pi$$
$$132$$ 0 0
$$133$$ −8.23506 −0.714071
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 9.47949 0.809888 0.404944 0.914342i $$-0.367291\pi$$
0.404944 + 0.914342i $$0.367291\pi$$
$$138$$ 0 0
$$139$$ −10.1334 −0.859500 −0.429750 0.902948i $$-0.641398\pi$$
−0.429750 + 0.902948i $$0.641398\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 17.0607 1.42669
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 7.24443 0.593487 0.296743 0.954957i $$-0.404099\pi$$
0.296743 + 0.954957i $$0.404099\pi$$
$$150$$ 0 0
$$151$$ 8.85728 0.720795 0.360398 0.932799i $$-0.382641\pi$$
0.360398 + 0.932799i $$0.382641\pi$$
$$152$$ 0 0
$$153$$ 7.18421 0.580809
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.4795 1.07578 0.537890 0.843015i $$-0.319221\pi$$
0.537890 + 0.843015i $$0.319221\pi$$
$$158$$ 0 0
$$159$$ 4.62222 0.366566
$$160$$ 0 0
$$161$$ 6.23506 0.491392
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.10171 −0.162635 −0.0813176 0.996688i $$-0.525913\pi$$
−0.0813176 + 0.996688i $$0.525913\pi$$
$$168$$ 0 0
$$169$$ −1.59057 −0.122352
$$170$$ 0 0
$$171$$ −8.23506 −0.629751
$$172$$ 0 0
$$173$$ 11.1842 0.850320 0.425160 0.905118i $$-0.360218\pi$$
0.425160 + 0.905118i $$0.360218\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 11.6128 0.872875
$$178$$ 0 0
$$179$$ −20.6637 −1.54448 −0.772239 0.635332i $$-0.780863\pi$$
−0.772239 + 0.635332i $$0.780863\pi$$
$$180$$ 0 0
$$181$$ 24.9590 1.85519 0.927594 0.373591i $$-0.121874\pi$$
0.927594 + 0.373591i $$0.121874\pi$$
$$182$$ 0 0
$$183$$ 0.488863 0.0361378
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 36.2864 2.65352
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −9.52098 −0.688914 −0.344457 0.938802i $$-0.611937\pi$$
−0.344457 + 0.938802i $$0.611937\pi$$
$$192$$ 0 0
$$193$$ −22.9590 −1.65262 −0.826312 0.563212i $$-0.809565\pi$$
−0.826312 + 0.563212i $$0.809565\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −19.8479 −1.41411 −0.707053 0.707161i $$-0.749976\pi$$
−0.707053 + 0.707161i $$0.749976\pi$$
$$198$$ 0 0
$$199$$ 8.23506 0.583768 0.291884 0.956454i $$-0.405718\pi$$
0.291884 + 0.956454i $$0.405718\pi$$
$$200$$ 0 0
$$201$$ −3.61285 −0.254831
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.23506 0.433367
$$208$$ 0 0
$$209$$ −41.5941 −2.87712
$$210$$ 0 0
$$211$$ 11.6128 0.799461 0.399731 0.916633i $$-0.369104\pi$$
0.399731 + 0.916633i $$0.369104\pi$$
$$212$$ 0 0
$$213$$ 10.2953 0.705421
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.62222 0.313776
$$218$$ 0 0
$$219$$ 16.2351 1.09706
$$220$$ 0 0
$$221$$ 24.2667 1.63236
$$222$$ 0 0
$$223$$ −2.48886 −0.166667 −0.0833333 0.996522i $$-0.526557\pi$$
−0.0833333 + 0.996522i $$0.526557\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0.857279 0.0568996 0.0284498 0.999595i $$-0.490943\pi$$
0.0284498 + 0.999595i $$0.490943\pi$$
$$228$$ 0 0
$$229$$ −14.4701 −0.956213 −0.478106 0.878302i $$-0.658677\pi$$
−0.478106 + 0.878302i $$0.658677\pi$$
$$230$$ 0 0
$$231$$ 5.05086 0.332322
$$232$$ 0 0
$$233$$ −3.96836 −0.259976 −0.129988 0.991516i $$-0.541494\pi$$
−0.129988 + 0.991516i $$0.541494\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −1.24443 −0.0808345
$$238$$ 0 0
$$239$$ −1.90813 −0.123427 −0.0617135 0.998094i $$-0.519656\pi$$
−0.0617135 + 0.998094i $$0.519656\pi$$
$$240$$ 0 0
$$241$$ 0.755569 0.0486705 0.0243352 0.999704i $$-0.492253\pi$$
0.0243352 + 0.999704i $$0.492253\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −27.8163 −1.76991
$$248$$ 0 0
$$249$$ −11.6128 −0.735934
$$250$$ 0 0
$$251$$ 9.12399 0.575901 0.287950 0.957645i $$-0.407026\pi$$
0.287950 + 0.957645i $$0.407026\pi$$
$$252$$ 0 0
$$253$$ 31.4924 1.97991
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.28592 −0.0802134 −0.0401067 0.999195i $$-0.512770\pi$$
−0.0401067 + 0.999195i $$0.512770\pi$$
$$258$$ 0 0
$$259$$ −4.85728 −0.301817
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −7.00937 −0.432216 −0.216108 0.976369i $$-0.569336\pi$$
−0.216108 + 0.976369i $$0.569336\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.99063 0.427820
$$268$$ 0 0
$$269$$ 23.1941 1.41417 0.707083 0.707130i $$-0.250011\pi$$
0.707083 + 0.707130i $$0.250011\pi$$
$$270$$ 0 0
$$271$$ −7.11108 −0.431967 −0.215984 0.976397i $$-0.569296\pi$$
−0.215984 + 0.976397i $$0.569296\pi$$
$$272$$ 0 0
$$273$$ 3.37778 0.204433
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.34614 0.441387 0.220693 0.975343i $$-0.429168\pi$$
0.220693 + 0.975343i $$0.429168\pi$$
$$278$$ 0 0
$$279$$ 4.62222 0.276725
$$280$$ 0 0
$$281$$ −19.9813 −1.19198 −0.595991 0.802991i $$-0.703241\pi$$
−0.595991 + 0.802991i $$0.703241\pi$$
$$282$$ 0 0
$$283$$ −4.85728 −0.288735 −0.144368 0.989524i $$-0.546115\pi$$
−0.144368 + 0.989524i $$0.546115\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3.37778 −0.199384
$$288$$ 0 0
$$289$$ 34.6128 2.03605
$$290$$ 0 0
$$291$$ −8.23506 −0.482748
$$292$$ 0 0
$$293$$ −11.1842 −0.653388 −0.326694 0.945130i $$-0.605935\pi$$
−0.326694 + 0.945130i $$0.605935\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.05086 0.293080
$$298$$ 0 0
$$299$$ 21.0607 1.21797
$$300$$ 0 0
$$301$$ −1.24443 −0.0717278
$$302$$ 0 0
$$303$$ −9.47949 −0.544583
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 23.3461 1.33243 0.666217 0.745758i $$-0.267912\pi$$
0.666217 + 0.745758i $$0.267912\pi$$
$$308$$ 0 0
$$309$$ −16.8573 −0.958977
$$310$$ 0 0
$$311$$ 32.8573 1.86317 0.931583 0.363530i $$-0.118428\pi$$
0.931583 + 0.363530i $$0.118428\pi$$
$$312$$ 0 0
$$313$$ 28.8256 1.62932 0.814661 0.579938i $$-0.196923\pi$$
0.814661 + 0.579938i $$0.196923\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 25.3590 1.42431 0.712153 0.702024i $$-0.247720\pi$$
0.712153 + 0.702024i $$0.247720\pi$$
$$318$$ 0 0
$$319$$ −10.1017 −0.565587
$$320$$ 0 0
$$321$$ −15.4795 −0.863981
$$322$$ 0 0
$$323$$ −59.1624 −3.29188
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −1.61285 −0.0891907
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1.89829 −0.104339 −0.0521697 0.998638i $$-0.516614\pi$$
−0.0521697 + 0.998638i $$0.516614\pi$$
$$332$$ 0 0
$$333$$ −4.85728 −0.266177
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.2257 1.26518 0.632592 0.774485i $$-0.281991\pi$$
0.632592 + 0.774485i $$0.281991\pi$$
$$338$$ 0 0
$$339$$ 1.86665 0.101382
$$340$$ 0 0
$$341$$ 23.3461 1.26426
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7.47949 −0.401520 −0.200760 0.979640i $$-0.564341\pi$$
−0.200760 + 0.979640i $$0.564341\pi$$
$$348$$ 0 0
$$349$$ 29.2257 1.56442 0.782208 0.623018i $$-0.214094\pi$$
0.782208 + 0.623018i $$0.214094\pi$$
$$350$$ 0 0
$$351$$ 3.37778 0.180293
$$352$$ 0 0
$$353$$ −30.4099 −1.61856 −0.809278 0.587426i $$-0.800141\pi$$
−0.809278 + 0.587426i $$0.800141\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 7.18421 0.380229
$$358$$ 0 0
$$359$$ 16.1936 0.854664 0.427332 0.904095i $$-0.359454\pi$$
0.427332 + 0.904095i $$0.359454\pi$$
$$360$$ 0 0
$$361$$ 48.8163 2.56928
$$362$$ 0 0
$$363$$ 14.5111 0.761637
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 5.51114 0.287679 0.143840 0.989601i $$-0.454055\pi$$
0.143840 + 0.989601i $$0.454055\pi$$
$$368$$ 0 0
$$369$$ −3.37778 −0.175840
$$370$$ 0 0
$$371$$ 4.62222 0.239973
$$372$$ 0 0
$$373$$ 1.63158 0.0844802 0.0422401 0.999107i $$-0.486551\pi$$
0.0422401 + 0.999107i $$0.486551\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.75557 −0.347929
$$378$$ 0 0
$$379$$ −20.8573 −1.07137 −0.535683 0.844419i $$-0.679946\pi$$
−0.535683 + 0.844419i $$0.679946\pi$$
$$380$$ 0 0
$$381$$ −12.8573 −0.658698
$$382$$ 0 0
$$383$$ −20.0830 −1.02619 −0.513096 0.858331i $$-0.671502\pi$$
−0.513096 + 0.858331i $$0.671502\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.24443 −0.0632580
$$388$$ 0 0
$$389$$ 33.2257 1.68461 0.842305 0.539001i $$-0.181198\pi$$
0.842305 + 0.539001i $$0.181198\pi$$
$$390$$ 0 0
$$391$$ 44.7940 2.26533
$$392$$ 0 0
$$393$$ −21.7146 −1.09535
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.7239 1.14048 0.570241 0.821478i $$-0.306850\pi$$
0.570241 + 0.821478i $$0.306850\pi$$
$$398$$ 0 0
$$399$$ −8.23506 −0.412269
$$400$$ 0 0
$$401$$ 12.7556 0.636983 0.318491 0.947926i $$-0.396824\pi$$
0.318491 + 0.947926i $$0.396824\pi$$
$$402$$ 0 0
$$403$$ 15.6128 0.777731
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.5334 −1.21608
$$408$$ 0 0
$$409$$ 27.4479 1.35721 0.678604 0.734504i $$-0.262585\pi$$
0.678604 + 0.734504i $$0.262585\pi$$
$$410$$ 0 0
$$411$$ 9.47949 0.467589
$$412$$ 0 0
$$413$$ 11.6128 0.571431
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −10.1334 −0.496232
$$418$$ 0 0
$$419$$ −2.36842 −0.115705 −0.0578524 0.998325i $$-0.518425\pi$$
−0.0578524 + 0.998325i $$0.518425\pi$$
$$420$$ 0 0
$$421$$ 39.3274 1.91670 0.958350 0.285596i $$-0.0921914\pi$$
0.958350 + 0.285596i $$0.0921914\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.488863 0.0236577
$$428$$ 0 0
$$429$$ 17.0607 0.823698
$$430$$ 0 0
$$431$$ −8.19358 −0.394671 −0.197335 0.980336i $$-0.563229\pi$$
−0.197335 + 0.980336i $$0.563229\pi$$
$$432$$ 0 0
$$433$$ −14.6035 −0.701798 −0.350899 0.936413i $$-0.614124\pi$$
−0.350899 + 0.936413i $$0.614124\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −51.3461 −2.45622
$$438$$ 0 0
$$439$$ −5.27607 −0.251813 −0.125907 0.992042i $$-0.540184\pi$$
−0.125907 + 0.992042i $$0.540184\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −18.5018 −0.879046 −0.439523 0.898231i $$-0.644852\pi$$
−0.439523 + 0.898231i $$0.644852\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 7.24443 0.342650
$$448$$ 0 0
$$449$$ 11.7146 0.552844 0.276422 0.961036i $$-0.410851\pi$$
0.276422 + 0.961036i $$0.410851\pi$$
$$450$$ 0 0
$$451$$ −17.0607 −0.803357
$$452$$ 0 0
$$453$$ 8.85728 0.416151
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.9590 −0.886864 −0.443432 0.896308i $$-0.646239\pi$$
−0.443432 + 0.896308i $$0.646239\pi$$
$$458$$ 0 0
$$459$$ 7.18421 0.335330
$$460$$ 0 0
$$461$$ −15.1111 −0.703793 −0.351897 0.936039i $$-0.614463\pi$$
−0.351897 + 0.936039i $$0.614463\pi$$
$$462$$ 0 0
$$463$$ 22.5718 1.04900 0.524501 0.851410i $$-0.324252\pi$$
0.524501 + 0.851410i $$0.324252\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.3684 1.03509 0.517543 0.855657i $$-0.326847\pi$$
0.517543 + 0.855657i $$0.326847\pi$$
$$468$$ 0 0
$$469$$ −3.61285 −0.166826
$$470$$ 0 0
$$471$$ 13.4795 0.621102
$$472$$ 0 0
$$473$$ −6.28544 −0.289005
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 4.62222 0.211637
$$478$$ 0 0
$$479$$ 32.8573 1.50129 0.750644 0.660707i $$-0.229744\pi$$
0.750644 + 0.660707i $$0.229744\pi$$
$$480$$ 0 0
$$481$$ −16.4068 −0.748088
$$482$$ 0 0
$$483$$ 6.23506 0.283705
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3.40943 −0.154496 −0.0772479 0.997012i $$-0.524613\pi$$
−0.0772479 + 0.997012i $$0.524613\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 33.7877 1.52482 0.762409 0.647096i $$-0.224017\pi$$
0.762409 + 0.647096i $$0.224017\pi$$
$$492$$ 0 0
$$493$$ −14.3684 −0.647121
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10.2953 0.461807
$$498$$ 0 0
$$499$$ −13.6316 −0.610233 −0.305117 0.952315i $$-0.598695\pi$$
−0.305117 + 0.952315i $$0.598695\pi$$
$$500$$ 0 0
$$501$$ −2.10171 −0.0938975
$$502$$ 0 0
$$503$$ −7.34614 −0.327548 −0.163774 0.986498i $$-0.552367\pi$$
−0.163774 + 0.986498i $$0.552367\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.59057 −0.0706398
$$508$$ 0 0
$$509$$ −15.9684 −0.707785 −0.353892 0.935286i $$-0.615142\pi$$
−0.353892 + 0.935286i $$0.615142\pi$$
$$510$$ 0 0
$$511$$ 16.2351 0.718197
$$512$$ 0 0
$$513$$ −8.23506 −0.363587
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 11.1842 0.490932
$$520$$ 0 0
$$521$$ −1.09234 −0.0478564 −0.0239282 0.999714i $$-0.507617\pi$$
−0.0239282 + 0.999714i $$0.507617\pi$$
$$522$$ 0 0
$$523$$ −17.5111 −0.765709 −0.382854 0.923809i $$-0.625059\pi$$
−0.382854 + 0.923809i $$0.625059\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 33.2070 1.44652
$$528$$ 0 0
$$529$$ 15.8760 0.690262
$$530$$ 0 0
$$531$$ 11.6128 0.503955
$$532$$ 0 0
$$533$$ −11.4094 −0.494197
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −20.6637 −0.891705
$$538$$ 0 0
$$539$$ 5.05086 0.217556
$$540$$ 0 0
$$541$$ −15.3274 −0.658977 −0.329488 0.944160i $$-0.606876\pi$$
−0.329488 + 0.944160i $$0.606876\pi$$
$$542$$ 0 0
$$543$$ 24.9590 1.07109
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 5.32741 0.227783 0.113892 0.993493i $$-0.463668\pi$$
0.113892 + 0.993493i $$0.463668\pi$$
$$548$$ 0 0
$$549$$ 0.488863 0.0208641
$$550$$ 0 0
$$551$$ 16.4701 0.701651
$$552$$ 0 0
$$553$$ −1.24443 −0.0529186
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −0.355509 −0.0150634 −0.00753171 0.999972i $$-0.502397\pi$$
−0.00753171 + 0.999972i $$0.502397\pi$$
$$558$$ 0 0
$$559$$ −4.20342 −0.177786
$$560$$ 0 0
$$561$$ 36.2864 1.53201
$$562$$ 0 0
$$563$$ −16.4701 −0.694133 −0.347067 0.937841i $$-0.612822\pi$$
−0.347067 + 0.937841i $$0.612822\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −20.9590 −0.878647 −0.439323 0.898329i $$-0.644782\pi$$
−0.439323 + 0.898329i $$0.644782\pi$$
$$570$$ 0 0
$$571$$ −17.5111 −0.732818 −0.366409 0.930454i $$-0.619413\pi$$
−0.366409 + 0.930454i $$0.619413\pi$$
$$572$$ 0 0
$$573$$ −9.52098 −0.397745
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −0.152089 −0.00633155 −0.00316577 0.999995i $$-0.501008\pi$$
−0.00316577 + 0.999995i $$0.501008\pi$$
$$578$$ 0 0
$$579$$ −22.9590 −0.954143
$$580$$ 0 0
$$581$$ −11.6128 −0.481782
$$582$$ 0 0
$$583$$ 23.3461 0.966898
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 34.1847 1.41095 0.705476 0.708733i $$-0.250733\pi$$
0.705476 + 0.708733i $$0.250733\pi$$
$$588$$ 0 0
$$589$$ −38.0642 −1.56841
$$590$$ 0 0
$$591$$ −19.8479 −0.816434
$$592$$ 0 0
$$593$$ −19.3047 −0.792747 −0.396374 0.918089i $$-0.629731\pi$$
−0.396374 + 0.918089i $$0.629731\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 8.23506 0.337039
$$598$$ 0 0
$$599$$ 25.9081 1.05858 0.529289 0.848442i $$-0.322459\pi$$
0.529289 + 0.848442i $$0.322459\pi$$
$$600$$ 0 0
$$601$$ −22.7368 −0.927455 −0.463727 0.885978i $$-0.653488\pi$$
−0.463727 + 0.885978i $$0.653488\pi$$
$$602$$ 0 0
$$603$$ −3.61285 −0.147127
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −32.9403 −1.33700 −0.668502 0.743711i $$-0.733064\pi$$
−0.668502 + 0.743711i $$0.733064\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 13.1240 0.530073 0.265036 0.964238i $$-0.414616\pi$$
0.265036 + 0.964238i $$0.414616\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.2538 −0.734870 −0.367435 0.930049i $$-0.619764\pi$$
−0.367435 + 0.930049i $$0.619764\pi$$
$$618$$ 0 0
$$619$$ −7.64449 −0.307258 −0.153629 0.988129i $$-0.549096\pi$$
−0.153629 + 0.988129i $$0.549096\pi$$
$$620$$ 0 0
$$621$$ 6.23506 0.250204
$$622$$ 0 0
$$623$$ 6.99063 0.280074
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −41.5941 −1.66111
$$628$$ 0 0
$$629$$ −34.8957 −1.39138
$$630$$ 0 0
$$631$$ −28.6735 −1.14148 −0.570738 0.821132i $$-0.693343\pi$$
−0.570738 + 0.821132i $$0.693343\pi$$
$$632$$ 0 0
$$633$$ 11.6128 0.461569
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 3.37778 0.133833
$$638$$ 0 0
$$639$$ 10.2953 0.407275
$$640$$ 0 0
$$641$$ 15.4479 0.610153 0.305077 0.952328i $$-0.401318\pi$$
0.305077 + 0.952328i $$0.401318\pi$$
$$642$$ 0 0
$$643$$ 25.8350 1.01883 0.509417 0.860520i $$-0.329861\pi$$
0.509417 + 0.860520i $$0.329861\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −23.6128 −0.928317 −0.464158 0.885752i $$-0.653643\pi$$
−0.464158 + 0.885752i $$0.653643\pi$$
$$648$$ 0 0
$$649$$ 58.6548 2.30240
$$650$$ 0 0
$$651$$ 4.62222 0.181159
$$652$$ 0 0
$$653$$ −36.7052 −1.43639 −0.718193 0.695844i $$-0.755030\pi$$
−0.718193 + 0.695844i $$0.755030\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 16.2351 0.633390
$$658$$ 0 0
$$659$$ −37.9911 −1.47992 −0.739962 0.672649i $$-0.765156\pi$$
−0.739962 + 0.672649i $$0.765156\pi$$
$$660$$ 0 0
$$661$$ −49.2257 −1.91466 −0.957329 0.289001i $$-0.906677\pi$$
−0.957329 + 0.289001i $$0.906677\pi$$
$$662$$ 0 0
$$663$$ 24.2667 0.942441
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −12.4701 −0.482845
$$668$$ 0 0
$$669$$ −2.48886 −0.0962250
$$670$$ 0 0
$$671$$ 2.46917 0.0953214
$$672$$ 0 0
$$673$$ 11.2257 0.432719 0.216359 0.976314i $$-0.430582\pi$$
0.216359 + 0.976314i $$0.430582\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −9.55262 −0.367137 −0.183569 0.983007i $$-0.558765\pi$$
−0.183569 + 0.983007i $$0.558765\pi$$
$$678$$ 0 0
$$679$$ −8.23506 −0.316033
$$680$$ 0 0
$$681$$ 0.857279 0.0328510
$$682$$ 0 0
$$683$$ −17.9684 −0.687540 −0.343770 0.939054i $$-0.611704\pi$$
−0.343770 + 0.939054i $$0.611704\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −14.4701 −0.552070
$$688$$ 0 0
$$689$$ 15.6128 0.594802
$$690$$ 0 0
$$691$$ −0.355509 −0.0135242 −0.00676211 0.999977i $$-0.502152\pi$$
−0.00676211 + 0.999977i $$0.502152\pi$$
$$692$$ 0 0
$$693$$ 5.05086 0.191866
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −24.2667 −0.919167
$$698$$ 0 0
$$699$$ −3.96836 −0.150097
$$700$$ 0 0
$$701$$ 13.2257 0.499528 0.249764 0.968307i $$-0.419647\pi$$
0.249764 + 0.968307i $$0.419647\pi$$
$$702$$ 0 0
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −9.47949 −0.356513
$$708$$ 0 0
$$709$$ −10.9777 −0.412277 −0.206139 0.978523i $$-0.566090\pi$$
−0.206139 + 0.978523i $$0.566090\pi$$
$$710$$ 0 0
$$711$$ −1.24443 −0.0466698
$$712$$ 0 0
$$713$$ 28.8198 1.07931
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −1.90813 −0.0712606
$$718$$ 0 0
$$719$$ 21.9813 0.819763 0.409881 0.912139i $$-0.365570\pi$$
0.409881 + 0.912139i $$0.365570\pi$$
$$720$$ 0 0
$$721$$ −16.8573 −0.627798
$$722$$ 0 0
$$723$$ 0.755569 0.0280999
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −13.0607 −0.484395 −0.242197 0.970227i $$-0.577868\pi$$
−0.242197 + 0.970227i $$0.577868\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.94025 −0.330667
$$732$$ 0 0
$$733$$ −13.5625 −0.500941 −0.250471 0.968124i $$-0.580585\pi$$
−0.250471 + 0.968124i $$0.580585\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −18.2480 −0.672173
$$738$$ 0 0
$$739$$ −10.2854 −0.378356 −0.189178 0.981943i $$-0.560582\pi$$
−0.189178 + 0.981943i $$0.560582\pi$$
$$740$$ 0 0
$$741$$ −27.8163 −1.02186
$$742$$ 0 0
$$743$$ 21.4608 0.787319 0.393659 0.919256i $$-0.371209\pi$$
0.393659 + 0.919256i $$0.371209\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −11.6128 −0.424892
$$748$$ 0 0
$$749$$ −15.4795 −0.565608
$$750$$ 0 0
$$751$$ −46.0642 −1.68091 −0.840454 0.541883i $$-0.817712\pi$$
−0.840454 + 0.541883i $$0.817712\pi$$
$$752$$ 0 0
$$753$$ 9.12399 0.332497
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −27.1052 −0.985157 −0.492579 0.870268i $$-0.663946\pi$$
−0.492579 + 0.870268i $$0.663946\pi$$
$$758$$ 0 0
$$759$$ 31.4924 1.14310
$$760$$ 0 0
$$761$$ 39.4608 1.43045 0.715226 0.698894i $$-0.246324\pi$$
0.715226 + 0.698894i $$0.246324\pi$$
$$762$$ 0 0
$$763$$ −1.61285 −0.0583890
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 39.2257 1.41636
$$768$$ 0 0
$$769$$ −7.51114 −0.270859 −0.135429 0.990787i $$-0.543241\pi$$
−0.135429 + 0.990787i $$0.543241\pi$$
$$770$$ 0 0
$$771$$ −1.28592 −0.0463112
$$772$$ 0 0
$$773$$ −1.46965 −0.0528596 −0.0264298 0.999651i $$-0.508414\pi$$
−0.0264298 + 0.999651i $$0.508414\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −4.85728 −0.174254
$$778$$ 0 0
$$779$$ 27.8163 0.996621
$$780$$ 0 0
$$781$$ 52.0000 1.86071
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −16.2034 −0.577590 −0.288795 0.957391i $$-0.593255\pi$$
−0.288795 + 0.957391i $$0.593255\pi$$
$$788$$ 0 0
$$789$$ −7.00937 −0.249540
$$790$$ 0 0
$$791$$ 1.86665 0.0663703
$$792$$ 0 0
$$793$$ 1.65127 0.0586384
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0.161933 0.00573597 0.00286799 0.999996i $$-0.499087\pi$$
0.00286799 + 0.999996i $$0.499087\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 6.99063 0.247002
$$802$$ 0 0
$$803$$ 82.0010 2.89375
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 23.1941 0.816469
$$808$$ 0 0
$$809$$ −42.9403 −1.50970 −0.754849 0.655898i $$-0.772290\pi$$
−0.754849 + 0.655898i $$0.772290\pi$$
$$810$$ 0 0
$$811$$ 41.2958 1.45009 0.725045 0.688701i $$-0.241819\pi$$
0.725045 + 0.688701i $$0.241819\pi$$
$$812$$ 0 0
$$813$$ −7.11108 −0.249396
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 10.2480 0.358531
$$818$$ 0 0
$$819$$ 3.37778 0.118029
$$820$$ 0 0
$$821$$ −43.2070 −1.50793 −0.753967 0.656913i $$-0.771862\pi$$
−0.753967 + 0.656913i $$0.771862\pi$$
$$822$$ 0 0
$$823$$ 11.1427 0.388411 0.194205 0.980961i $$-0.437787\pi$$
0.194205 + 0.980961i $$0.437787\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −36.9906 −1.28629 −0.643145 0.765744i $$-0.722371\pi$$
−0.643145 + 0.765744i $$0.722371\pi$$
$$828$$ 0 0
$$829$$ −22.6735 −0.787485 −0.393742 0.919221i $$-0.628820\pi$$
−0.393742 + 0.919221i $$0.628820\pi$$
$$830$$ 0 0
$$831$$ 7.34614 0.254835
$$832$$ 0 0
$$833$$ 7.18421 0.248918
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.62222 0.159767
$$838$$ 0 0
$$839$$ 21.8983 0.756013 0.378006 0.925803i $$-0.376610\pi$$
0.378006 + 0.925803i $$0.376610\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −19.9813 −0.688191
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14.5111 0.498609
$$848$$ 0 0
$$849$$ −4.85728 −0.166701
$$850$$ 0 0
$$851$$ −30.2854 −1.03817
$$852$$ 0 0
$$853$$ 52.4010 1.79418 0.897088 0.441851i $$-0.145678\pi$$
0.897088 + 0.441851i $$0.145678\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −24.5116 −0.837301 −0.418650 0.908147i $$-0.637497\pi$$
−0.418650 + 0.908147i $$0.637497\pi$$
$$858$$ 0 0
$$859$$ −57.4795 −1.96118 −0.980588 0.196082i $$-0.937178\pi$$
−0.980588 + 0.196082i $$0.937178\pi$$
$$860$$ 0 0
$$861$$ −3.37778 −0.115115
$$862$$ 0 0
$$863$$ −21.1941 −0.721454 −0.360727 0.932671i $$-0.617471\pi$$
−0.360727 + 0.932671i $$0.617471\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 34.6128 1.17551
$$868$$ 0 0
$$869$$ −6.28544 −0.213219
$$870$$ 0 0
$$871$$ −12.2034 −0.413497
$$872$$ 0 0
$$873$$ −8.23506 −0.278715
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.38715 0.283214 0.141607 0.989923i $$-0.454773\pi$$
0.141607 + 0.989923i $$0.454773\pi$$
$$878$$ 0 0
$$879$$ −11.1842 −0.377234
$$880$$ 0 0
$$881$$ 29.1753 0.982941 0.491471 0.870894i $$-0.336459\pi$$
0.491471 + 0.870894i $$0.336459\pi$$
$$882$$ 0 0
$$883$$ 19.8796 0.669000 0.334500 0.942396i $$-0.391433\pi$$
0.334500 + 0.942396i $$0.391433\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −14.6923 −0.493319 −0.246659 0.969102i $$-0.579333\pi$$
−0.246659 + 0.969102i $$0.579333\pi$$
$$888$$ 0 0
$$889$$ −12.8573 −0.431219
$$890$$ 0 0
$$891$$ 5.05086 0.169210
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 21.0607 0.703196
$$898$$ 0 0
$$899$$ −9.24443 −0.308319
$$900$$ 0 0
$$901$$ 33.2070 1.10628
$$902$$ 0 0
$$903$$ −1.24443 −0.0414121
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 19.8796 0.660090 0.330045 0.943965i $$-0.392936\pi$$
0.330045 + 0.943965i $$0.392936\pi$$
$$908$$ 0 0
$$909$$ −9.47949 −0.314415
$$910$$ 0 0
$$911$$ 53.3372 1.76714 0.883571 0.468297i $$-0.155132\pi$$
0.883571 + 0.468297i $$0.155132\pi$$
$$912$$ 0 0
$$913$$ −58.6548 −1.94119
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −21.7146 −0.717078
$$918$$ 0 0
$$919$$ 21.1240 0.696816 0.348408 0.937343i $$-0.386722\pi$$
0.348408 + 0.937343i $$0.386722\pi$$
$$920$$ 0 0
$$921$$ 23.3461 0.769282
$$922$$ 0 0
$$923$$ 34.7753 1.14464
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −16.8573 −0.553666
$$928$$ 0 0
$$929$$ 2.72393 0.0893691 0.0446846 0.999001i $$-0.485772\pi$$
0.0446846 + 0.999001i $$0.485772\pi$$
$$930$$ 0 0
$$931$$ −8.23506 −0.269893
$$932$$ 0 0
$$933$$ 32.8573 1.07570
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −15.3145 −0.500303 −0.250151 0.968207i $$-0.580480\pi$$
−0.250151 + 0.968207i $$0.580480\pi$$
$$938$$ 0 0
$$939$$ 28.8256 0.940689
$$940$$ 0 0
$$941$$ −30.6035 −0.997645 −0.498822 0.866704i $$-0.666234\pi$$
−0.498822 + 0.866704i $$0.666234\pi$$
$$942$$ 0 0
$$943$$ −21.0607 −0.685831
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.74266 0.0891245 0.0445623 0.999007i $$-0.485811\pi$$
0.0445623 + 0.999007i $$0.485811\pi$$
$$948$$ 0 0
$$949$$ 54.8385 1.78013
$$950$$ 0 0
$$951$$ 25.3590 0.822323
$$952$$ 0 0
$$953$$ −47.8479 −1.54995 −0.774973 0.631994i $$-0.782237\pi$$
−0.774973 + 0.631994i $$0.782237\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −10.1017 −0.326542
$$958$$ 0 0
$$959$$ 9.47949 0.306109
$$960$$ 0 0
$$961$$ −9.63512 −0.310810
$$962$$ 0 0
$$963$$ −15.4795 −0.498820
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 38.7181 1.24509 0.622545 0.782584i $$-0.286099\pi$$
0.622545 + 0.782584i $$0.286099\pi$$
$$968$$ 0 0
$$969$$ −59.1624 −1.90057
$$970$$ 0 0
$$971$$ −13.7778 −0.442152 −0.221076 0.975257i $$-0.570957\pi$$
−0.221076 + 0.975257i $$0.570957\pi$$
$$972$$ 0 0
$$973$$ −10.1334 −0.324860
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 4.03164 0.128984 0.0644918 0.997918i $$-0.479457\pi$$
0.0644918 + 0.997918i $$0.479457\pi$$
$$978$$ 0 0
$$979$$ 35.3087 1.12847
$$980$$ 0 0
$$981$$ −1.61285 −0.0514943
$$982$$ 0 0
$$983$$ −47.4924 −1.51477 −0.757386 0.652967i $$-0.773524\pi$$
−0.757386 + 0.652967i $$0.773524\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −7.75911 −0.246725
$$990$$ 0 0
$$991$$ −1.16146 −0.0368949 −0.0184474 0.999830i $$-0.505872\pi$$
−0.0184474 + 0.999830i $$0.505872\pi$$
$$992$$ 0 0
$$993$$ −1.89829 −0.0602404
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −18.4572 −0.584546 −0.292273 0.956335i $$-0.594412\pi$$
−0.292273 + 0.956335i $$0.594412\pi$$
$$998$$ 0 0
$$999$$ −4.85728 −0.153678
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 8400.2.a.dl.1.3 3
4.3 odd 2 4200.2.a.bn.1.1 3
5.2 odd 4 1680.2.t.j.1009.2 6
5.3 odd 4 1680.2.t.j.1009.5 6
5.4 even 2 8400.2.a.di.1.3 3
15.2 even 4 5040.2.t.z.1009.4 6
15.8 even 4 5040.2.t.z.1009.3 6
20.3 even 4 840.2.t.d.169.2 6
20.7 even 4 840.2.t.d.169.5 yes 6
20.19 odd 2 4200.2.a.bp.1.1 3
60.23 odd 4 2520.2.t.k.1009.3 6
60.47 odd 4 2520.2.t.k.1009.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.d.169.2 6 20.3 even 4
840.2.t.d.169.5 yes 6 20.7 even 4
1680.2.t.j.1009.2 6 5.2 odd 4
1680.2.t.j.1009.5 6 5.3 odd 4
2520.2.t.k.1009.3 6 60.23 odd 4
2520.2.t.k.1009.4 6 60.47 odd 4
4200.2.a.bn.1.1 3 4.3 odd 2
4200.2.a.bp.1.1 3 20.19 odd 2
5040.2.t.z.1009.3 6 15.8 even 4
5040.2.t.z.1009.4 6 15.2 even 4
8400.2.a.di.1.3 3 5.4 even 2
8400.2.a.dl.1.3 3 1.1 even 1 trivial