# Properties

 Label 8470.2.a.cs.1.2 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.4400.1 Defining polynomial: $$x^{4} - 7 x^{2} + 11$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.54336$$ of defining polynomial Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -2.54336 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.54336 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.46869 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -2.54336 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.54336 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.46869 q^{9} +1.00000 q^{10} -2.54336 q^{12} -6.42254 q^{13} +1.00000 q^{14} -2.54336 q^{15} +1.00000 q^{16} -3.95385 q^{17} +3.46869 q^{18} +2.46869 q^{19} +1.00000 q^{20} -2.54336 q^{21} +4.99442 q^{23} -2.54336 q^{24} +1.00000 q^{25} -6.42254 q^{26} -1.19205 q^{27} +1.00000 q^{28} -9.04402 q^{29} -2.54336 q^{30} +7.88910 q^{31} +1.00000 q^{32} -3.95385 q^{34} +1.00000 q^{35} +3.46869 q^{36} +11.0440 q^{37} +2.46869 q^{38} +16.3348 q^{39} +1.00000 q^{40} -6.95385 q^{41} -2.54336 q^{42} +3.02295 q^{43} +3.46869 q^{45} +4.99442 q^{46} -8.37984 q^{47} -2.54336 q^{48} +1.00000 q^{49} +1.00000 q^{50} +10.0561 q^{51} -6.42254 q^{52} +10.0616 q^{53} -1.19205 q^{54} +1.00000 q^{56} -6.27877 q^{57} -9.04402 q^{58} -4.09575 q^{59} -2.54336 q^{60} -3.59696 q^{61} +7.88910 q^{62} +3.46869 q^{63} +1.00000 q^{64} -6.42254 q^{65} -1.28222 q^{67} -3.95385 q^{68} -12.7026 q^{69} +1.00000 q^{70} -5.90025 q^{71} +3.46869 q^{72} -10.2521 q^{73} +11.0440 q^{74} -2.54336 q^{75} +2.46869 q^{76} +16.3348 q^{78} -6.21099 q^{79} +1.00000 q^{80} -7.37426 q^{81} -6.95385 q^{82} -13.3383 q^{83} -2.54336 q^{84} -3.95385 q^{85} +3.02295 q^{86} +23.0022 q^{87} +14.7432 q^{89} +3.46869 q^{90} -6.42254 q^{91} +4.99442 q^{92} -20.0648 q^{93} -8.37984 q^{94} +2.46869 q^{95} -2.54336 q^{96} -11.9603 q^{97} +1.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} + 4q^{7} + 4q^{8} + 6q^{9} + O(q^{10})$$ $$4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} + 4q^{7} + 4q^{8} + 6q^{9} + 4q^{10} - 4q^{12} - 14q^{13} + 4q^{14} - 4q^{15} + 4q^{16} - 12q^{17} + 6q^{18} + 2q^{19} + 4q^{20} - 4q^{21} - 4q^{24} + 4q^{25} - 14q^{26} - 22q^{27} + 4q^{28} - 10q^{29} - 4q^{30} - 18q^{31} + 4q^{32} - 12q^{34} + 4q^{35} + 6q^{36} + 18q^{37} + 2q^{38} + 30q^{39} + 4q^{40} - 24q^{41} - 4q^{42} - 10q^{43} + 6q^{45} - 8q^{47} - 4q^{48} + 4q^{49} + 4q^{50} - 14q^{52} + 20q^{53} - 22q^{54} + 4q^{56} - 30q^{57} - 10q^{58} - 14q^{59} - 4q^{60} - 14q^{61} - 18q^{62} + 6q^{63} + 4q^{64} - 14q^{65} - 12q^{68} - 4q^{69} + 4q^{70} - 14q^{71} + 6q^{72} - 30q^{73} + 18q^{74} - 4q^{75} + 2q^{76} + 30q^{78} - 8q^{79} + 4q^{80} + 16q^{81} - 24q^{82} - 8q^{83} - 4q^{84} - 12q^{85} - 10q^{86} - 2q^{87} - 4q^{89} + 6q^{90} - 14q^{91} - 2q^{93} - 8q^{94} + 2q^{95} - 4q^{96} - 10q^{97} + 4q^{98} + O(q^{100})$$

## 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
$$3$$ −2.54336 −1.46841 −0.734205 0.678927i $$-0.762445\pi$$
−0.734205 + 0.678927i $$0.762445\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −2.54336 −1.03832
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 3.46869 1.15623
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ −2.54336 −0.734205
$$13$$ −6.42254 −1.78129 −0.890646 0.454697i $$-0.849747\pi$$
−0.890646 + 0.454697i $$0.849747\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −2.54336 −0.656693
$$16$$ 1.00000 0.250000
$$17$$ −3.95385 −0.958950 −0.479475 0.877556i $$-0.659173\pi$$
−0.479475 + 0.877556i $$0.659173\pi$$
$$18$$ 3.46869 0.817578
$$19$$ 2.46869 0.566356 0.283178 0.959067i $$-0.408611\pi$$
0.283178 + 0.959067i $$0.408611\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −2.54336 −0.555007
$$22$$ 0 0
$$23$$ 4.99442 1.04141 0.520705 0.853737i $$-0.325669\pi$$
0.520705 + 0.853737i $$0.325669\pi$$
$$24$$ −2.54336 −0.519162
$$25$$ 1.00000 0.200000
$$26$$ −6.42254 −1.25956
$$27$$ −1.19205 −0.229410
$$28$$ 1.00000 0.188982
$$29$$ −9.04402 −1.67943 −0.839716 0.543026i $$-0.817279\pi$$
−0.839716 + 0.543026i $$0.817279\pi$$
$$30$$ −2.54336 −0.464352
$$31$$ 7.88910 1.41692 0.708462 0.705749i $$-0.249389\pi$$
0.708462 + 0.705749i $$0.249389\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −3.95385 −0.678080
$$35$$ 1.00000 0.169031
$$36$$ 3.46869 0.578115
$$37$$ 11.0440 1.81563 0.907813 0.419376i $$-0.137751\pi$$
0.907813 + 0.419376i $$0.137751\pi$$
$$38$$ 2.46869 0.400474
$$39$$ 16.3348 2.61567
$$40$$ 1.00000 0.158114
$$41$$ −6.95385 −1.08601 −0.543004 0.839730i $$-0.682713\pi$$
−0.543004 + 0.839730i $$0.682713\pi$$
$$42$$ −2.54336 −0.392449
$$43$$ 3.02295 0.460995 0.230497 0.973073i $$-0.425965\pi$$
0.230497 + 0.973073i $$0.425965\pi$$
$$44$$ 0 0
$$45$$ 3.46869 0.517082
$$46$$ 4.99442 0.736388
$$47$$ −8.37984 −1.22232 −0.611162 0.791505i $$-0.709298\pi$$
−0.611162 + 0.791505i $$0.709298\pi$$
$$48$$ −2.54336 −0.367103
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ 10.0561 1.40813
$$52$$ −6.42254 −0.890646
$$53$$ 10.0616 1.38207 0.691037 0.722820i $$-0.257154\pi$$
0.691037 + 0.722820i $$0.257154\pi$$
$$54$$ −1.19205 −0.162217
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −6.27877 −0.831644
$$58$$ −9.04402 −1.18754
$$59$$ −4.09575 −0.533221 −0.266610 0.963804i $$-0.585904\pi$$
−0.266610 + 0.963804i $$0.585904\pi$$
$$60$$ −2.54336 −0.328347
$$61$$ −3.59696 −0.460544 −0.230272 0.973126i $$-0.573962\pi$$
−0.230272 + 0.973126i $$0.573962\pi$$
$$62$$ 7.88910 1.00192
$$63$$ 3.46869 0.437014
$$64$$ 1.00000 0.125000
$$65$$ −6.42254 −0.796618
$$66$$ 0 0
$$67$$ −1.28222 −0.156648 −0.0783239 0.996928i $$-0.524957\pi$$
−0.0783239 + 0.996928i $$0.524957\pi$$
$$68$$ −3.95385 −0.479475
$$69$$ −12.7026 −1.52922
$$70$$ 1.00000 0.119523
$$71$$ −5.90025 −0.700231 −0.350116 0.936707i $$-0.613858\pi$$
−0.350116 + 0.936707i $$0.613858\pi$$
$$72$$ 3.46869 0.408789
$$73$$ −10.2521 −1.19992 −0.599960 0.800030i $$-0.704817\pi$$
−0.599960 + 0.800030i $$0.704817\pi$$
$$74$$ 11.0440 1.28384
$$75$$ −2.54336 −0.293682
$$76$$ 2.46869 0.283178
$$77$$ 0 0
$$78$$ 16.3348 1.84956
$$79$$ −6.21099 −0.698791 −0.349396 0.936975i $$-0.613613\pi$$
−0.349396 + 0.936975i $$0.613613\pi$$
$$80$$ 1.00000 0.111803
$$81$$ −7.37426 −0.819362
$$82$$ −6.95385 −0.767924
$$83$$ −13.3383 −1.46407 −0.732034 0.681268i $$-0.761429\pi$$
−0.732034 + 0.681268i $$0.761429\pi$$
$$84$$ −2.54336 −0.277504
$$85$$ −3.95385 −0.428855
$$86$$ 3.02295 0.325973
$$87$$ 23.0022 2.46610
$$88$$ 0 0
$$89$$ 14.7432 1.56278 0.781388 0.624045i $$-0.214512\pi$$
0.781388 + 0.624045i $$0.214512\pi$$
$$90$$ 3.46869 0.365632
$$91$$ −6.42254 −0.673265
$$92$$ 4.99442 0.520705
$$93$$ −20.0648 −2.08063
$$94$$ −8.37984 −0.864314
$$95$$ 2.46869 0.253282
$$96$$ −2.54336 −0.259581
$$97$$ −11.9603 −1.21439 −0.607194 0.794554i $$-0.707705\pi$$
−0.607194 + 0.794554i $$0.707705\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 2.26114 0.224992 0.112496 0.993652i $$-0.464115\pi$$
0.112496 + 0.993652i $$0.464115\pi$$
$$102$$ 10.0561 0.995699
$$103$$ 13.4359 1.32388 0.661940 0.749557i $$-0.269733\pi$$
0.661940 + 0.749557i $$0.269733\pi$$
$$104$$ −6.42254 −0.629782
$$105$$ −2.54336 −0.248207
$$106$$ 10.0616 0.977274
$$107$$ −13.3879 −1.29426 −0.647128 0.762381i $$-0.724030\pi$$
−0.647128 + 0.762381i $$0.724030\pi$$
$$108$$ −1.19205 −0.114705
$$109$$ 8.67624 0.831033 0.415516 0.909586i $$-0.363601\pi$$
0.415516 + 0.909586i $$0.363601\pi$$
$$110$$ 0 0
$$111$$ −28.0889 −2.66608
$$112$$ 1.00000 0.0944911
$$113$$ −9.16671 −0.862332 −0.431166 0.902273i $$-0.641898\pi$$
−0.431166 + 0.902273i $$0.641898\pi$$
$$114$$ −6.27877 −0.588061
$$115$$ 4.99442 0.465732
$$116$$ −9.04402 −0.839716
$$117$$ −22.2778 −2.05958
$$118$$ −4.09575 −0.377044
$$119$$ −3.95385 −0.362449
$$120$$ −2.54336 −0.232176
$$121$$ 0 0
$$122$$ −3.59696 −0.325653
$$123$$ 17.6862 1.59471
$$124$$ 7.88910 0.708462
$$125$$ 1.00000 0.0894427
$$126$$ 3.46869 0.309015
$$127$$ −15.2797 −1.35585 −0.677926 0.735130i $$-0.737121\pi$$
−0.677926 + 0.735130i $$0.737121\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −7.68845 −0.676930
$$130$$ −6.42254 −0.563294
$$131$$ 5.65461 0.494045 0.247023 0.969010i $$-0.420548\pi$$
0.247023 + 0.969010i $$0.420548\pi$$
$$132$$ 0 0
$$133$$ 2.46869 0.214063
$$134$$ −1.28222 −0.110767
$$135$$ −1.19205 −0.102595
$$136$$ −3.95385 −0.339040
$$137$$ −3.95288 −0.337717 −0.168859 0.985640i $$-0.554008\pi$$
−0.168859 + 0.985640i $$0.554008\pi$$
$$138$$ −12.7026 −1.08132
$$139$$ 21.0254 1.78335 0.891676 0.452673i $$-0.149530\pi$$
0.891676 + 0.452673i $$0.149530\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 21.3130 1.79487
$$142$$ −5.90025 −0.495138
$$143$$ 0 0
$$144$$ 3.46869 0.289057
$$145$$ −9.04402 −0.751065
$$146$$ −10.2521 −0.848472
$$147$$ −2.54336 −0.209773
$$148$$ 11.0440 0.907813
$$149$$ 8.55328 0.700712 0.350356 0.936617i $$-0.386061\pi$$
0.350356 + 0.936617i $$0.386061\pi$$
$$150$$ −2.54336 −0.207665
$$151$$ 2.21155 0.179973 0.0899866 0.995943i $$-0.471318\pi$$
0.0899866 + 0.995943i $$0.471318\pi$$
$$152$$ 2.46869 0.200237
$$153$$ −13.7147 −1.10877
$$154$$ 0 0
$$155$$ 7.88910 0.633668
$$156$$ 16.3348 1.30783
$$157$$ −13.3863 −1.06834 −0.534172 0.845376i $$-0.679377\pi$$
−0.534172 + 0.845376i $$0.679377\pi$$
$$158$$ −6.21099 −0.494120
$$159$$ −25.5904 −2.02945
$$160$$ 1.00000 0.0790569
$$161$$ 4.99442 0.393616
$$162$$ −7.37426 −0.579377
$$163$$ 2.05304 0.160807 0.0804033 0.996762i $$-0.474379\pi$$
0.0804033 + 0.996762i $$0.474379\pi$$
$$164$$ −6.95385 −0.543004
$$165$$ 0 0
$$166$$ −13.3383 −1.03525
$$167$$ −21.2017 −1.64064 −0.820319 0.571906i $$-0.806204\pi$$
−0.820319 + 0.571906i $$0.806204\pi$$
$$168$$ −2.54336 −0.196225
$$169$$ 28.2490 2.17300
$$170$$ −3.95385 −0.303246
$$171$$ 8.56312 0.654838
$$172$$ 3.02295 0.230497
$$173$$ 5.91230 0.449504 0.224752 0.974416i $$-0.427843\pi$$
0.224752 + 0.974416i $$0.427843\pi$$
$$174$$ 23.0022 1.74379
$$175$$ 1.00000 0.0755929
$$176$$ 0 0
$$177$$ 10.4170 0.782987
$$178$$ 14.7432 1.10505
$$179$$ −17.8879 −1.33701 −0.668504 0.743709i $$-0.733065\pi$$
−0.668504 + 0.743709i $$0.733065\pi$$
$$180$$ 3.46869 0.258541
$$181$$ 11.0710 0.822899 0.411449 0.911433i $$-0.365023\pi$$
0.411449 + 0.911433i $$0.365023\pi$$
$$182$$ −6.42254 −0.476070
$$183$$ 9.14837 0.676267
$$184$$ 4.99442 0.368194
$$185$$ 11.0440 0.811973
$$186$$ −20.0648 −1.47123
$$187$$ 0 0
$$188$$ −8.37984 −0.611162
$$189$$ −1.19205 −0.0867087
$$190$$ 2.46869 0.179098
$$191$$ −9.55976 −0.691720 −0.345860 0.938286i $$-0.612413\pi$$
−0.345860 + 0.938286i $$0.612413\pi$$
$$192$$ −2.54336 −0.183551
$$193$$ 16.6846 1.20098 0.600491 0.799631i $$-0.294972\pi$$
0.600491 + 0.799631i $$0.294972\pi$$
$$194$$ −11.9603 −0.858701
$$195$$ 16.3348 1.16976
$$196$$ 1.00000 0.0714286
$$197$$ −0.533440 −0.0380060 −0.0190030 0.999819i $$-0.506049\pi$$
−0.0190030 + 0.999819i $$0.506049\pi$$
$$198$$ 0 0
$$199$$ −2.81443 −0.199510 −0.0997548 0.995012i $$-0.531806\pi$$
−0.0997548 + 0.995012i $$0.531806\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 3.26114 0.230023
$$202$$ 2.26114 0.159094
$$203$$ −9.04402 −0.634766
$$204$$ 10.0561 0.704066
$$205$$ −6.95385 −0.485678
$$206$$ 13.4359 0.936124
$$207$$ 17.3241 1.20411
$$208$$ −6.42254 −0.445323
$$209$$ 0 0
$$210$$ −2.54336 −0.175509
$$211$$ 3.20239 0.220461 0.110231 0.993906i $$-0.464841\pi$$
0.110231 + 0.993906i $$0.464841\pi$$
$$212$$ 10.0616 0.691037
$$213$$ 15.0065 1.02823
$$214$$ −13.3879 −0.915177
$$215$$ 3.02295 0.206163
$$216$$ −1.19205 −0.0811086
$$217$$ 7.88910 0.535547
$$218$$ 8.67624 0.587629
$$219$$ 26.0749 1.76198
$$220$$ 0 0
$$221$$ 25.3938 1.70817
$$222$$ −28.0889 −1.88521
$$223$$ −0.768535 −0.0514649 −0.0257325 0.999669i $$-0.508192\pi$$
−0.0257325 + 0.999669i $$0.508192\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 3.46869 0.231246
$$226$$ −9.16671 −0.609761
$$227$$ −6.58738 −0.437220 −0.218610 0.975812i $$-0.570152\pi$$
−0.218610 + 0.975812i $$0.570152\pi$$
$$228$$ −6.27877 −0.415822
$$229$$ −23.2829 −1.53857 −0.769287 0.638903i $$-0.779389\pi$$
−0.769287 + 0.638903i $$0.779389\pi$$
$$230$$ 4.99442 0.329323
$$231$$ 0 0
$$232$$ −9.04402 −0.593769
$$233$$ 1.72239 0.112837 0.0564186 0.998407i $$-0.482032\pi$$
0.0564186 + 0.998407i $$0.482032\pi$$
$$234$$ −22.2778 −1.45635
$$235$$ −8.37984 −0.546640
$$236$$ −4.09575 −0.266610
$$237$$ 15.7968 1.02611
$$238$$ −3.95385 −0.256290
$$239$$ −11.6437 −0.753169 −0.376585 0.926382i $$-0.622902\pi$$
−0.376585 + 0.926382i $$0.622902\pi$$
$$240$$ −2.54336 −0.164173
$$241$$ 9.58236 0.617254 0.308627 0.951183i $$-0.400130\pi$$
0.308627 + 0.951183i $$0.400130\pi$$
$$242$$ 0 0
$$243$$ 22.3316 1.43257
$$244$$ −3.59696 −0.230272
$$245$$ 1.00000 0.0638877
$$246$$ 17.6862 1.12763
$$247$$ −15.8553 −1.00885
$$248$$ 7.88910 0.500958
$$249$$ 33.9241 2.14985
$$250$$ 1.00000 0.0632456
$$251$$ −8.45409 −0.533618 −0.266809 0.963749i $$-0.585969\pi$$
−0.266809 + 0.963749i $$0.585969\pi$$
$$252$$ 3.46869 0.218507
$$253$$ 0 0
$$254$$ −15.2797 −0.958732
$$255$$ 10.0561 0.629736
$$256$$ 1.00000 0.0625000
$$257$$ −20.3754 −1.27098 −0.635492 0.772108i $$-0.719203\pi$$
−0.635492 + 0.772108i $$0.719203\pi$$
$$258$$ −7.68845 −0.478662
$$259$$ 11.0440 0.686242
$$260$$ −6.42254 −0.398309
$$261$$ −31.3709 −1.94181
$$262$$ 5.65461 0.349343
$$263$$ −16.1920 −0.998444 −0.499222 0.866474i $$-0.666381\pi$$
−0.499222 + 0.866474i $$0.666381\pi$$
$$264$$ 0 0
$$265$$ 10.0616 0.618082
$$266$$ 2.46869 0.151365
$$267$$ −37.4973 −2.29480
$$268$$ −1.28222 −0.0783239
$$269$$ −26.8144 −1.63490 −0.817452 0.575996i $$-0.804614\pi$$
−0.817452 + 0.575996i $$0.804614\pi$$
$$270$$ −1.19205 −0.0725457
$$271$$ 22.4312 1.36260 0.681300 0.732004i $$-0.261415\pi$$
0.681300 + 0.732004i $$0.261415\pi$$
$$272$$ −3.95385 −0.239737
$$273$$ 16.3348 0.988630
$$274$$ −3.95288 −0.238802
$$275$$ 0 0
$$276$$ −12.7026 −0.764608
$$277$$ −22.1150 −1.32876 −0.664380 0.747395i $$-0.731305\pi$$
−0.664380 + 0.747395i $$0.731305\pi$$
$$278$$ 21.0254 1.26102
$$279$$ 27.3648 1.63829
$$280$$ 1.00000 0.0597614
$$281$$ 12.5703 0.749882 0.374941 0.927049i $$-0.377663\pi$$
0.374941 + 0.927049i $$0.377663\pi$$
$$282$$ 21.3130 1.26917
$$283$$ −5.10477 −0.303447 −0.151723 0.988423i $$-0.548482\pi$$
−0.151723 + 0.988423i $$0.548482\pi$$
$$284$$ −5.90025 −0.350116
$$285$$ −6.27877 −0.371922
$$286$$ 0 0
$$287$$ −6.95385 −0.410473
$$288$$ 3.46869 0.204395
$$289$$ −1.36707 −0.0804158
$$290$$ −9.04402 −0.531083
$$291$$ 30.4194 1.78322
$$292$$ −10.2521 −0.599960
$$293$$ −17.2556 −1.00808 −0.504041 0.863680i $$-0.668154\pi$$
−0.504041 + 0.863680i $$0.668154\pi$$
$$294$$ −2.54336 −0.148332
$$295$$ −4.09575 −0.238464
$$296$$ 11.0440 0.641921
$$297$$ 0 0
$$298$$ 8.55328 0.495478
$$299$$ −32.0769 −1.85505
$$300$$ −2.54336 −0.146841
$$301$$ 3.02295 0.174240
$$302$$ 2.21155 0.127260
$$303$$ −5.75091 −0.330381
$$304$$ 2.46869 0.141589
$$305$$ −3.59696 −0.205961
$$306$$ −13.7147 −0.784016
$$307$$ −24.8048 −1.41569 −0.707844 0.706369i $$-0.750332\pi$$
−0.707844 + 0.706369i $$0.750332\pi$$
$$308$$ 0 0
$$309$$ −34.1724 −1.94400
$$310$$ 7.88910 0.448071
$$311$$ −15.0718 −0.854645 −0.427322 0.904099i $$-0.640543\pi$$
−0.427322 + 0.904099i $$0.640543\pi$$
$$312$$ 16.3348 0.924778
$$313$$ −14.6737 −0.829406 −0.414703 0.909957i $$-0.636115\pi$$
−0.414703 + 0.909957i $$0.636115\pi$$
$$314$$ −13.3863 −0.755433
$$315$$ 3.46869 0.195439
$$316$$ −6.21099 −0.349396
$$317$$ −14.2689 −0.801423 −0.400712 0.916204i $$-0.631237\pi$$
−0.400712 + 0.916204i $$0.631237\pi$$
$$318$$ −25.5904 −1.43504
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 34.0502 1.90050
$$322$$ 4.99442 0.278328
$$323$$ −9.76083 −0.543107
$$324$$ −7.37426 −0.409681
$$325$$ −6.42254 −0.356258
$$326$$ 2.05304 0.113707
$$327$$ −22.0668 −1.22030
$$328$$ −6.95385 −0.383962
$$329$$ −8.37984 −0.461995
$$330$$ 0 0
$$331$$ −12.9209 −0.710197 −0.355099 0.934829i $$-0.615553\pi$$
−0.355099 + 0.934829i $$0.615553\pi$$
$$332$$ −13.3383 −0.732034
$$333$$ 38.3083 2.09928
$$334$$ −21.2017 −1.16011
$$335$$ −1.28222 −0.0700550
$$336$$ −2.54336 −0.138752
$$337$$ −20.6003 −1.12217 −0.561086 0.827758i $$-0.689616\pi$$
−0.561086 + 0.827758i $$0.689616\pi$$
$$338$$ 28.2490 1.53654
$$339$$ 23.3143 1.26626
$$340$$ −3.95385 −0.214428
$$341$$ 0 0
$$342$$ 8.56312 0.463040
$$343$$ 1.00000 0.0539949
$$344$$ 3.02295 0.162986
$$345$$ −12.7026 −0.683887
$$346$$ 5.91230 0.317847
$$347$$ 3.81140 0.204607 0.102303 0.994753i $$-0.467379\pi$$
0.102303 + 0.994753i $$0.467379\pi$$
$$348$$ 23.0022 1.23305
$$349$$ 19.7782 1.05870 0.529351 0.848403i $$-0.322435\pi$$
0.529351 + 0.848403i $$0.322435\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 7.65598 0.408646
$$352$$ 0 0
$$353$$ −2.85223 −0.151809 −0.0759044 0.997115i $$-0.524184\pi$$
−0.0759044 + 0.997115i $$0.524184\pi$$
$$354$$ 10.4170 0.553655
$$355$$ −5.90025 −0.313153
$$356$$ 14.7432 0.781388
$$357$$ 10.0561 0.532224
$$358$$ −17.8879 −0.945407
$$359$$ −16.9374 −0.893921 −0.446960 0.894554i $$-0.647494\pi$$
−0.446960 + 0.894554i $$0.647494\pi$$
$$360$$ 3.46869 0.182816
$$361$$ −12.9056 −0.679241
$$362$$ 11.0710 0.581877
$$363$$ 0 0
$$364$$ −6.42254 −0.336633
$$365$$ −10.2521 −0.536621
$$366$$ 9.14837 0.478193
$$367$$ −20.0055 −1.04428 −0.522139 0.852860i $$-0.674866\pi$$
−0.522139 + 0.852860i $$0.674866\pi$$
$$368$$ 4.99442 0.260352
$$369$$ −24.1207 −1.25568
$$370$$ 11.0440 0.574151
$$371$$ 10.0616 0.522375
$$372$$ −20.0648 −1.04031
$$373$$ 23.1627 1.19932 0.599660 0.800255i $$-0.295303\pi$$
0.599660 + 0.800255i $$0.295303\pi$$
$$374$$ 0 0
$$375$$ −2.54336 −0.131339
$$376$$ −8.37984 −0.432157
$$377$$ 58.0856 2.99156
$$378$$ −1.19205 −0.0613123
$$379$$ 32.6068 1.67490 0.837450 0.546514i $$-0.184046\pi$$
0.837450 + 0.546514i $$0.184046\pi$$
$$380$$ 2.46869 0.126641
$$381$$ 38.8617 1.99095
$$382$$ −9.55976 −0.489120
$$383$$ 5.16926 0.264137 0.132068 0.991241i $$-0.457838\pi$$
0.132068 + 0.991241i $$0.457838\pi$$
$$384$$ −2.54336 −0.129790
$$385$$ 0 0
$$386$$ 16.6846 0.849223
$$387$$ 10.4857 0.533016
$$388$$ −11.9603 −0.607194
$$389$$ −0.868284 −0.0440237 −0.0220119 0.999758i $$-0.507007\pi$$
−0.0220119 + 0.999758i $$0.507007\pi$$
$$390$$ 16.3348 0.827147
$$391$$ −19.7472 −0.998659
$$392$$ 1.00000 0.0505076
$$393$$ −14.3817 −0.725461
$$394$$ −0.533440 −0.0268743
$$395$$ −6.21099 −0.312509
$$396$$ 0 0
$$397$$ −16.7823 −0.842280 −0.421140 0.906996i $$-0.638370\pi$$
−0.421140 + 0.906996i $$0.638370\pi$$
$$398$$ −2.81443 −0.141075
$$399$$ −6.27877 −0.314332
$$400$$ 1.00000 0.0500000
$$401$$ −1.66393 −0.0830925 −0.0415463 0.999137i $$-0.513228\pi$$
−0.0415463 + 0.999137i $$0.513228\pi$$
$$402$$ 3.26114 0.162651
$$403$$ −50.6681 −2.52396
$$404$$ 2.26114 0.112496
$$405$$ −7.37426 −0.366430
$$406$$ −9.04402 −0.448847
$$407$$ 0 0
$$408$$ 10.0561 0.497850
$$409$$ 20.4530 1.01133 0.505667 0.862729i $$-0.331246\pi$$
0.505667 + 0.862729i $$0.331246\pi$$
$$410$$ −6.95385 −0.343426
$$411$$ 10.0536 0.495907
$$412$$ 13.4359 0.661940
$$413$$ −4.09575 −0.201538
$$414$$ 17.3241 0.851433
$$415$$ −13.3383 −0.654751
$$416$$ −6.42254 −0.314891
$$417$$ −53.4753 −2.61869
$$418$$ 0 0
$$419$$ 18.0870 0.883609 0.441804 0.897111i $$-0.354339\pi$$
0.441804 + 0.897111i $$0.354339\pi$$
$$420$$ −2.54336 −0.124103
$$421$$ −14.6364 −0.713333 −0.356667 0.934232i $$-0.616087\pi$$
−0.356667 + 0.934232i $$0.616087\pi$$
$$422$$ 3.20239 0.155890
$$423$$ −29.0671 −1.41329
$$424$$ 10.0616 0.488637
$$425$$ −3.95385 −0.191790
$$426$$ 15.0065 0.727066
$$427$$ −3.59696 −0.174069
$$428$$ −13.3879 −0.647128
$$429$$ 0 0
$$430$$ 3.02295 0.145779
$$431$$ 0.629485 0.0303212 0.0151606 0.999885i $$-0.495174\pi$$
0.0151606 + 0.999885i $$0.495174\pi$$
$$432$$ −1.19205 −0.0573524
$$433$$ −15.1788 −0.729445 −0.364722 0.931116i $$-0.618836\pi$$
−0.364722 + 0.931116i $$0.618836\pi$$
$$434$$ 7.88910 0.378689
$$435$$ 23.0022 1.10287
$$436$$ 8.67624 0.415516
$$437$$ 12.3297 0.589809
$$438$$ 26.0749 1.24590
$$439$$ −12.5690 −0.599888 −0.299944 0.953957i $$-0.596968\pi$$
−0.299944 + 0.953957i $$0.596968\pi$$
$$440$$ 0 0
$$441$$ 3.46869 0.165176
$$442$$ 25.3938 1.20786
$$443$$ −2.85871 −0.135821 −0.0679106 0.997691i $$-0.521633\pi$$
−0.0679106 + 0.997691i $$0.521633\pi$$
$$444$$ −28.0889 −1.33304
$$445$$ 14.7432 0.698895
$$446$$ −0.768535 −0.0363912
$$447$$ −21.7541 −1.02893
$$448$$ 1.00000 0.0472456
$$449$$ 28.4736 1.34375 0.671877 0.740663i $$-0.265488\pi$$
0.671877 + 0.740663i $$0.265488\pi$$
$$450$$ 3.46869 0.163516
$$451$$ 0 0
$$452$$ −9.16671 −0.431166
$$453$$ −5.62477 −0.264275
$$454$$ −6.58738 −0.309161
$$455$$ −6.42254 −0.301093
$$456$$ −6.27877 −0.294030
$$457$$ −7.63383 −0.357096 −0.178548 0.983931i $$-0.557140\pi$$
−0.178548 + 0.983931i $$0.557140\pi$$
$$458$$ −23.2829 −1.08794
$$459$$ 4.71318 0.219992
$$460$$ 4.99442 0.232866
$$461$$ 4.72213 0.219931 0.109966 0.993935i $$-0.464926\pi$$
0.109966 + 0.993935i $$0.464926\pi$$
$$462$$ 0 0
$$463$$ −19.4328 −0.903117 −0.451558 0.892242i $$-0.649132\pi$$
−0.451558 + 0.892242i $$0.649132\pi$$
$$464$$ −9.04402 −0.419858
$$465$$ −20.0648 −0.930485
$$466$$ 1.72239 0.0797880
$$467$$ −13.5440 −0.626740 −0.313370 0.949631i $$-0.601458\pi$$
−0.313370 + 0.949631i $$0.601458\pi$$
$$468$$ −22.2778 −1.02979
$$469$$ −1.28222 −0.0592073
$$470$$ −8.37984 −0.386533
$$471$$ 34.0462 1.56877
$$472$$ −4.09575 −0.188522
$$473$$ 0 0
$$474$$ 15.7968 0.725571
$$475$$ 2.46869 0.113271
$$476$$ −3.95385 −0.181224
$$477$$ 34.9007 1.59799
$$478$$ −11.6437 −0.532571
$$479$$ −4.33024 −0.197854 −0.0989269 0.995095i $$-0.531541\pi$$
−0.0989269 + 0.995095i $$0.531541\pi$$
$$480$$ −2.54336 −0.116088
$$481$$ −70.9307 −3.23416
$$482$$ 9.58236 0.436465
$$483$$ −12.7026 −0.577990
$$484$$ 0 0
$$485$$ −11.9603 −0.543090
$$486$$ 22.3316 1.01298
$$487$$ −36.5760 −1.65742 −0.828708 0.559682i $$-0.810923\pi$$
−0.828708 + 0.559682i $$0.810923\pi$$
$$488$$ −3.59696 −0.162827
$$489$$ −5.22163 −0.236130
$$490$$ 1.00000 0.0451754
$$491$$ 31.7227 1.43163 0.715813 0.698292i $$-0.246056\pi$$
0.715813 + 0.698292i $$0.246056\pi$$
$$492$$ 17.6862 0.797354
$$493$$ 35.7587 1.61049
$$494$$ −15.8553 −0.713362
$$495$$ 0 0
$$496$$ 7.88910 0.354231
$$497$$ −5.90025 −0.264662
$$498$$ 33.9241 1.52018
$$499$$ 36.7983 1.64732 0.823659 0.567085i $$-0.191929\pi$$
0.823659 + 0.567085i $$0.191929\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 53.9236 2.40913
$$502$$ −8.45409 −0.377325
$$503$$ −9.16098 −0.408468 −0.204234 0.978922i $$-0.565470\pi$$
−0.204234 + 0.978922i $$0.565470\pi$$
$$504$$ 3.46869 0.154508
$$505$$ 2.26114 0.100620
$$506$$ 0 0
$$507$$ −71.8475 −3.19086
$$508$$ −15.2797 −0.677926
$$509$$ 14.9999 0.664860 0.332430 0.943128i $$-0.392131\pi$$
0.332430 + 0.943128i $$0.392131\pi$$
$$510$$ 10.0561 0.445290
$$511$$ −10.2521 −0.453527
$$512$$ 1.00000 0.0441942
$$513$$ −2.94280 −0.129928
$$514$$ −20.3754 −0.898721
$$515$$ 13.4359 0.592057
$$516$$ −7.68845 −0.338465
$$517$$ 0 0
$$518$$ 11.0440 0.485246
$$519$$ −15.0371 −0.660057
$$520$$ −6.42254 −0.281647
$$521$$ −16.8142 −0.736642 −0.368321 0.929699i $$-0.620067\pi$$
−0.368321 + 0.929699i $$0.620067\pi$$
$$522$$ −31.3709 −1.37307
$$523$$ −1.20886 −0.0528598 −0.0264299 0.999651i $$-0.508414\pi$$
−0.0264299 + 0.999651i $$0.508414\pi$$
$$524$$ 5.65461 0.247023
$$525$$ −2.54336 −0.111001
$$526$$ −16.1920 −0.706007
$$527$$ −31.1923 −1.35876
$$528$$ 0 0
$$529$$ 1.94427 0.0845336
$$530$$ 10.0616 0.437050
$$531$$ −14.2069 −0.616526
$$532$$ 2.46869 0.107031
$$533$$ 44.6614 1.93450
$$534$$ −37.4973 −1.62267
$$535$$ −13.3879 −0.578809
$$536$$ −1.28222 −0.0553834
$$537$$ 45.4955 1.96328
$$538$$ −26.8144 −1.15605
$$539$$ 0 0
$$540$$ −1.19205 −0.0512976
$$541$$ 4.66793 0.200690 0.100345 0.994953i $$-0.468005\pi$$
0.100345 + 0.994953i $$0.468005\pi$$
$$542$$ 22.4312 0.963504
$$543$$ −28.1575 −1.20835
$$544$$ −3.95385 −0.169520
$$545$$ 8.67624 0.371649
$$546$$ 16.3348 0.699067
$$547$$ −22.1545 −0.947260 −0.473630 0.880724i $$-0.657057\pi$$
−0.473630 + 0.880724i $$0.657057\pi$$
$$548$$ −3.95288 −0.168859
$$549$$ −12.4767 −0.532494
$$550$$ 0 0
$$551$$ −22.3269 −0.951157
$$552$$ −12.7026 −0.540660
$$553$$ −6.21099 −0.264118
$$554$$ −22.1150 −0.939576
$$555$$ −28.0889 −1.19231
$$556$$ 21.0254 0.891676
$$557$$ 37.8026 1.60175 0.800875 0.598832i $$-0.204368\pi$$
0.800875 + 0.598832i $$0.204368\pi$$
$$558$$ 27.3648 1.15845
$$559$$ −19.4150 −0.821167
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ 12.5703 0.530247
$$563$$ 9.89847 0.417171 0.208585 0.978004i $$-0.433114\pi$$
0.208585 + 0.978004i $$0.433114\pi$$
$$564$$ 21.3130 0.897437
$$565$$ −9.16671 −0.385647
$$566$$ −5.10477 −0.214569
$$567$$ −7.37426 −0.309690
$$568$$ −5.90025 −0.247569
$$569$$ −4.95198 −0.207598 −0.103799 0.994598i $$-0.533100\pi$$
−0.103799 + 0.994598i $$0.533100\pi$$
$$570$$ −6.27877 −0.262989
$$571$$ −8.53293 −0.357092 −0.178546 0.983932i $$-0.557139\pi$$
−0.178546 + 0.983932i $$0.557139\pi$$
$$572$$ 0 0
$$573$$ 24.3139 1.01573
$$574$$ −6.95385 −0.290248
$$575$$ 4.99442 0.208282
$$576$$ 3.46869 0.144529
$$577$$ −9.82861 −0.409170 −0.204585 0.978849i $$-0.565585\pi$$
−0.204585 + 0.978849i $$0.565585\pi$$
$$578$$ −1.36707 −0.0568626
$$579$$ −42.4349 −1.76354
$$580$$ −9.04402 −0.375532
$$581$$ −13.3383 −0.553365
$$582$$ 30.4194 1.26093
$$583$$ 0 0
$$584$$ −10.2521 −0.424236
$$585$$ −22.2778 −0.921074
$$586$$ −17.2556 −0.712821
$$587$$ 20.0651 0.828175 0.414088 0.910237i $$-0.364101\pi$$
0.414088 + 0.910237i $$0.364101\pi$$
$$588$$ −2.54336 −0.104886
$$589$$ 19.4757 0.802484
$$590$$ −4.09575 −0.168619
$$591$$ 1.35673 0.0558085
$$592$$ 11.0440 0.453906
$$593$$ −3.85286 −0.158218 −0.0791090 0.996866i $$-0.525208\pi$$
−0.0791090 + 0.996866i $$0.525208\pi$$
$$594$$ 0 0
$$595$$ −3.95385 −0.162092
$$596$$ 8.55328 0.350356
$$597$$ 7.15811 0.292962
$$598$$ −32.0769 −1.31172
$$599$$ 7.38576 0.301774 0.150887 0.988551i $$-0.451787\pi$$
0.150887 + 0.988551i $$0.451787\pi$$
$$600$$ −2.54336 −0.103832
$$601$$ −25.7196 −1.04912 −0.524562 0.851372i $$-0.675771\pi$$
−0.524562 + 0.851372i $$0.675771\pi$$
$$602$$ 3.02295 0.123206
$$603$$ −4.44762 −0.181121
$$604$$ 2.21155 0.0899866
$$605$$ 0 0
$$606$$ −5.75091 −0.233615
$$607$$ −3.02418 −0.122748 −0.0613738 0.998115i $$-0.519548\pi$$
−0.0613738 + 0.998115i $$0.519548\pi$$
$$608$$ 2.46869 0.100119
$$609$$ 23.0022 0.932097
$$610$$ −3.59696 −0.145637
$$611$$ 53.8198 2.17732
$$612$$ −13.7147 −0.554383
$$613$$ 20.7030 0.836185 0.418093 0.908404i $$-0.362699\pi$$
0.418093 + 0.908404i $$0.362699\pi$$
$$614$$ −24.8048 −1.00104
$$615$$ 17.6862 0.713175
$$616$$ 0 0
$$617$$ 16.2405 0.653817 0.326909 0.945056i $$-0.393993\pi$$
0.326909 + 0.945056i $$0.393993\pi$$
$$618$$ −34.1724 −1.37461
$$619$$ −35.9490 −1.44491 −0.722457 0.691416i $$-0.756987\pi$$
−0.722457 + 0.691416i $$0.756987\pi$$
$$620$$ 7.88910 0.316834
$$621$$ −5.95359 −0.238909
$$622$$ −15.0718 −0.604325
$$623$$ 14.7432 0.590674
$$624$$ 16.3348 0.653917
$$625$$ 1.00000 0.0400000
$$626$$ −14.6737 −0.586479
$$627$$ 0 0
$$628$$ −13.3863 −0.534172
$$629$$ −43.6664 −1.74109
$$630$$ 3.46869 0.138196
$$631$$ −41.4726 −1.65100 −0.825499 0.564403i $$-0.809106\pi$$
−0.825499 + 0.564403i $$0.809106\pi$$
$$632$$ −6.21099 −0.247060
$$633$$ −8.14483 −0.323728
$$634$$ −14.2689 −0.566692
$$635$$ −15.2797 −0.606355
$$636$$ −25.5904 −1.01473
$$637$$ −6.42254 −0.254470
$$638$$ 0 0
$$639$$ −20.4661 −0.809628
$$640$$ 1.00000 0.0395285
$$641$$ 3.17884 0.125557 0.0627783 0.998027i $$-0.480004\pi$$
0.0627783 + 0.998027i $$0.480004\pi$$
$$642$$ 34.0502 1.34386
$$643$$ 20.0750 0.791680 0.395840 0.918320i $$-0.370454\pi$$
0.395840 + 0.918320i $$0.370454\pi$$
$$644$$ 4.99442 0.196808
$$645$$ −7.68845 −0.302732
$$646$$ −9.76083 −0.384035
$$647$$ −38.1738 −1.50077 −0.750383 0.661003i $$-0.770131\pi$$
−0.750383 + 0.661003i $$0.770131\pi$$
$$648$$ −7.37426 −0.289688
$$649$$ 0 0
$$650$$ −6.42254 −0.251913
$$651$$ −20.0648 −0.786403
$$652$$ 2.05304 0.0804033
$$653$$ 44.2896 1.73319 0.866594 0.499014i $$-0.166304\pi$$
0.866594 + 0.499014i $$0.166304\pi$$
$$654$$ −22.0668 −0.862880
$$655$$ 5.65461 0.220944
$$656$$ −6.95385 −0.271502
$$657$$ −35.5614 −1.38738
$$658$$ −8.37984 −0.326680
$$659$$ −30.2781 −1.17947 −0.589734 0.807598i $$-0.700767\pi$$
−0.589734 + 0.807598i $$0.700767\pi$$
$$660$$ 0 0
$$661$$ 7.16382 0.278640 0.139320 0.990247i $$-0.455508\pi$$
0.139320 + 0.990247i $$0.455508\pi$$
$$662$$ −12.9209 −0.502185
$$663$$ −64.5855 −2.50829
$$664$$ −13.3383 −0.517626
$$665$$ 2.46869 0.0957317
$$666$$ 38.3083 1.48442
$$667$$ −45.1697 −1.74898
$$668$$ −21.2017 −0.820319
$$669$$ 1.95466 0.0755717
$$670$$ −1.28222 −0.0495364
$$671$$ 0 0
$$672$$ −2.54336 −0.0981123
$$673$$ −22.0439 −0.849730 −0.424865 0.905257i $$-0.639678\pi$$
−0.424865 + 0.905257i $$0.639678\pi$$
$$674$$ −20.6003 −0.793495
$$675$$ −1.19205 −0.0458819
$$676$$ 28.2490 1.08650
$$677$$ −34.8050 −1.33767 −0.668833 0.743413i $$-0.733206\pi$$
−0.668833 + 0.743413i $$0.733206\pi$$
$$678$$ 23.3143 0.895379
$$679$$ −11.9603 −0.458995
$$680$$ −3.95385 −0.151623
$$681$$ 16.7541 0.642018
$$682$$ 0 0
$$683$$ 35.0185 1.33995 0.669973 0.742385i $$-0.266306\pi$$
0.669973 + 0.742385i $$0.266306\pi$$
$$684$$ 8.56312 0.327419
$$685$$ −3.95288 −0.151032
$$686$$ 1.00000 0.0381802
$$687$$ 59.2167 2.25926
$$688$$ 3.02295 0.115249
$$689$$ −64.6213 −2.46188
$$690$$ −12.7026 −0.483581
$$691$$ −19.5535 −0.743849 −0.371925 0.928263i $$-0.621302\pi$$
−0.371925 + 0.928263i $$0.621302\pi$$
$$692$$ 5.91230 0.224752
$$693$$ 0 0
$$694$$ 3.81140 0.144679
$$695$$ 21.0254 0.797540
$$696$$ 23.0022 0.871897
$$697$$ 27.4945 1.04143
$$698$$ 19.7782 0.748616
$$699$$ −4.38065 −0.165691
$$700$$ 1.00000 0.0377964
$$701$$ 13.9458 0.526726 0.263363 0.964697i $$-0.415168\pi$$
0.263363 + 0.964697i $$0.415168\pi$$
$$702$$ 7.65598 0.288956
$$703$$ 27.2643 1.02829
$$704$$ 0 0
$$705$$ 21.3130 0.802692
$$706$$ −2.85223 −0.107345
$$707$$ 2.26114 0.0850391
$$708$$ 10.4170 0.391493
$$709$$ −0.549195 −0.0206255 −0.0103127 0.999947i $$-0.503283\pi$$
−0.0103127 + 0.999947i $$0.503283\pi$$
$$710$$ −5.90025 −0.221433
$$711$$ −21.5440 −0.807963
$$712$$ 14.7432 0.552525
$$713$$ 39.4015 1.47560
$$714$$ 10.0561 0.376339
$$715$$ 0 0
$$716$$ −17.8879 −0.668504
$$717$$ 29.6142 1.10596
$$718$$ −16.9374 −0.632097
$$719$$ −25.8914 −0.965586 −0.482793 0.875735i $$-0.660378\pi$$
−0.482793 + 0.875735i $$0.660378\pi$$
$$720$$ 3.46869 0.129270
$$721$$ 13.4359 0.500379
$$722$$ −12.9056 −0.480296
$$723$$ −24.3714 −0.906383
$$724$$ 11.0710 0.411449
$$725$$ −9.04402 −0.335886
$$726$$ 0 0
$$727$$ −41.6185 −1.54355 −0.771773 0.635898i $$-0.780630\pi$$
−0.771773 + 0.635898i $$0.780630\pi$$
$$728$$ −6.42254 −0.238035
$$729$$ −34.6744 −1.28424
$$730$$ −10.2521 −0.379448
$$731$$ −11.9523 −0.442071
$$732$$ 9.14837 0.338134
$$733$$ 2.16850 0.0800954 0.0400477 0.999198i $$-0.487249\pi$$
0.0400477 + 0.999198i $$0.487249\pi$$
$$734$$ −20.0055 −0.738417
$$735$$ −2.54336 −0.0938133
$$736$$ 4.99442 0.184097
$$737$$ 0 0
$$738$$ −24.1207 −0.887897
$$739$$ −3.06748 −0.112839 −0.0564196 0.998407i $$-0.517968\pi$$
−0.0564196 + 0.998407i $$0.517968\pi$$
$$740$$ 11.0440 0.405986
$$741$$ 40.3257 1.48140
$$742$$ 10.0616 0.369375
$$743$$ −38.9833 −1.43016 −0.715079 0.699044i $$-0.753609\pi$$
−0.715079 + 0.699044i $$0.753609\pi$$
$$744$$ −20.0648 −0.735613
$$745$$ 8.55328 0.313368
$$746$$ 23.1627 0.848047
$$747$$ −46.2664 −1.69280
$$748$$ 0 0
$$749$$ −13.3879 −0.489183
$$750$$ −2.54336 −0.0928704
$$751$$ 24.2379 0.884455 0.442228 0.896903i $$-0.354188\pi$$
0.442228 + 0.896903i $$0.354188\pi$$
$$752$$ −8.37984 −0.305581
$$753$$ 21.5018 0.783570
$$754$$ 58.0856 2.11535
$$755$$ 2.21155 0.0804865
$$756$$ −1.19205 −0.0433544
$$757$$ −24.2156 −0.880132 −0.440066 0.897965i $$-0.645045\pi$$
−0.440066 + 0.897965i $$0.645045\pi$$
$$758$$ 32.6068 1.18433
$$759$$ 0 0
$$760$$ 2.46869 0.0895488
$$761$$ 2.07860 0.0753493 0.0376746 0.999290i $$-0.488005\pi$$
0.0376746 + 0.999290i $$0.488005\pi$$
$$762$$ 38.8617 1.40781
$$763$$ 8.67624 0.314101
$$764$$ −9.55976 −0.345860
$$765$$ −13.7147 −0.495855
$$766$$ 5.16926 0.186773
$$767$$ 26.3051 0.949822
$$768$$ −2.54336 −0.0917757
$$769$$ −14.7508 −0.531929 −0.265964 0.963983i $$-0.585690\pi$$
−0.265964 + 0.963983i $$0.585690\pi$$
$$770$$ 0 0
$$771$$ 51.8221 1.86633
$$772$$ 16.6846 0.600491
$$773$$ 49.5573 1.78245 0.891225 0.453561i $$-0.149847\pi$$
0.891225 + 0.453561i $$0.149847\pi$$
$$774$$ 10.4857 0.376899
$$775$$ 7.88910 0.283385
$$776$$ −11.9603 −0.429351
$$777$$ −28.0889 −1.00769
$$778$$ −0.868284 −0.0311295
$$779$$ −17.1669 −0.615068
$$780$$ 16.3348 0.584881
$$781$$ 0 0
$$782$$ −19.7472 −0.706159
$$783$$ 10.7809 0.385278
$$784$$ 1.00000 0.0357143
$$785$$ −13.3863 −0.477778
$$786$$ −14.3817 −0.512979
$$787$$ 31.0744 1.10768 0.553842 0.832622i $$-0.313161\pi$$
0.553842 + 0.832622i $$0.313161\pi$$
$$788$$ −0.533440 −0.0190030
$$789$$ 41.1822 1.46613
$$790$$ −6.21099 −0.220977
$$791$$ −9.16671 −0.325931
$$792$$ 0 0
$$793$$ 23.1016 0.820363
$$794$$ −16.7823 −0.595582
$$795$$ −25.5904 −0.907598
$$796$$ −2.81443 −0.0997548
$$797$$ 18.3839 0.651191 0.325595 0.945509i $$-0.394435\pi$$
0.325595 + 0.945509i $$0.394435\pi$$
$$798$$ −6.27877 −0.222266
$$799$$ 33.1326 1.17215
$$800$$ 1.00000 0.0353553
$$801$$ 51.1396 1.80693
$$802$$ −1.66393 −0.0587553
$$803$$ 0 0
$$804$$ 3.26114 0.115012
$$805$$ 4.99442 0.176030
$$806$$ −50.6681 −1.78471
$$807$$ 68.1988 2.40071
$$808$$ 2.26114 0.0795468
$$809$$ −41.4458 −1.45716 −0.728578 0.684963i $$-0.759818\pi$$
−0.728578 + 0.684963i $$0.759818\pi$$
$$810$$ −7.37426 −0.259105
$$811$$ 7.70150 0.270436 0.135218 0.990816i $$-0.456827\pi$$
0.135218 + 0.990816i $$0.456827\pi$$
$$812$$ −9.04402 −0.317383
$$813$$ −57.0507 −2.00086
$$814$$ 0 0
$$815$$ 2.05304 0.0719149
$$816$$ 10.0561 0.352033
$$817$$ 7.46272 0.261087
$$818$$ 20.4530 0.715122
$$819$$ −22.2778 −0.778449
$$820$$ −6.95385 −0.242839
$$821$$ −48.2959 −1.68554 −0.842769 0.538275i $$-0.819076\pi$$
−0.842769 + 0.538275i $$0.819076\pi$$
$$822$$ 10.0536 0.350660
$$823$$ 36.7609 1.28140 0.640702 0.767789i $$-0.278643\pi$$
0.640702 + 0.767789i $$0.278643\pi$$
$$824$$ 13.4359 0.468062
$$825$$ 0 0
$$826$$ −4.09575 −0.142509
$$827$$ −29.9018 −1.03979 −0.519893 0.854231i $$-0.674028\pi$$
−0.519893 + 0.854231i $$0.674028\pi$$
$$828$$ 17.3241 0.602054
$$829$$ 18.8334 0.654110 0.327055 0.945005i $$-0.393944\pi$$
0.327055 + 0.945005i $$0.393944\pi$$
$$830$$ −13.3383 −0.462979
$$831$$ 56.2464 1.95117
$$832$$ −6.42254 −0.222662
$$833$$ −3.95385 −0.136993
$$834$$ −53.4753 −1.85170
$$835$$ −21.2017 −0.733716
$$836$$ 0 0
$$837$$ −9.40419 −0.325056
$$838$$ 18.0870 0.624806
$$839$$ 7.76812 0.268185 0.134093 0.990969i $$-0.457188\pi$$
0.134093 + 0.990969i $$0.457188\pi$$
$$840$$ −2.54336 −0.0877543
$$841$$ 52.7943 1.82049
$$842$$ −14.6364 −0.504403
$$843$$ −31.9708 −1.10113
$$844$$ 3.20239 0.110231
$$845$$ 28.2490 0.971796
$$846$$ −29.0671 −0.999346
$$847$$ 0 0
$$848$$ 10.0616 0.345518
$$849$$ 12.9833 0.445585
$$850$$ −3.95385 −0.135616
$$851$$ 55.1585 1.89081
$$852$$ 15.0065 0.514113
$$853$$ −24.1273 −0.826104 −0.413052 0.910707i $$-0.635537\pi$$
−0.413052 + 0.910707i $$0.635537\pi$$
$$854$$ −3.59696 −0.123085
$$855$$ 8.56312 0.292853
$$856$$ −13.3879 −0.457589
$$857$$ 22.2559 0.760246 0.380123 0.924936i $$-0.375882\pi$$
0.380123 + 0.924936i $$0.375882\pi$$
$$858$$ 0 0
$$859$$ 5.09623 0.173881 0.0869406 0.996214i $$-0.472291\pi$$
0.0869406 + 0.996214i $$0.472291\pi$$
$$860$$ 3.02295 0.103082
$$861$$ 17.6862 0.602743
$$862$$ 0.629485 0.0214403
$$863$$ 14.7292 0.501389 0.250694 0.968066i $$-0.419341\pi$$
0.250694 + 0.968066i $$0.419341\pi$$
$$864$$ −1.19205 −0.0405543
$$865$$ 5.91230 0.201024
$$866$$ −15.1788 −0.515795
$$867$$ 3.47695 0.118083
$$868$$ 7.88910 0.267774
$$869$$ 0 0
$$870$$ 23.0022 0.779848
$$871$$ 8.23510 0.279036
$$872$$ 8.67624 0.293814
$$873$$ −41.4867 −1.40411
$$874$$ 12.3297 0.417058
$$875$$ 1.00000 0.0338062
$$876$$ 26.0749 0.880988
$$877$$ 50.2137 1.69560 0.847798 0.530319i $$-0.177928\pi$$
0.847798 + 0.530319i $$0.177928\pi$$
$$878$$ −12.5690 −0.424185
$$879$$ 43.8872 1.48028
$$880$$ 0 0
$$881$$ 11.6181 0.391423 0.195711 0.980662i $$-0.437298\pi$$
0.195711 + 0.980662i $$0.437298\pi$$
$$882$$ 3.46869 0.116797
$$883$$ 2.75951 0.0928650 0.0464325 0.998921i $$-0.485215\pi$$
0.0464325 + 0.998921i $$0.485215\pi$$
$$884$$ 25.3938 0.854085
$$885$$ 10.4170 0.350162
$$886$$ −2.85871 −0.0960401
$$887$$ 34.8529 1.17024 0.585122 0.810945i $$-0.301046\pi$$
0.585122 + 0.810945i $$0.301046\pi$$
$$888$$ −28.0889 −0.942603
$$889$$ −15.2797 −0.512464
$$890$$ 14.7432 0.494193
$$891$$ 0 0
$$892$$ −0.768535 −0.0257325
$$893$$ −20.6872 −0.692271
$$894$$ −21.7541 −0.727566
$$895$$ −17.8879 −0.597928
$$896$$ 1.00000 0.0334077
$$897$$ 81.5831 2.72398
$$898$$ 28.4736 0.950178
$$899$$ −71.3492 −2.37963
$$900$$ 3.46869 0.115623
$$901$$ −39.7822 −1.32534
$$902$$ 0 0
$$903$$ −7.68845 −0.255855
$$904$$ −9.16671 −0.304880
$$905$$ 11.0710 0.368011
$$906$$ −5.62477 −0.186870
$$907$$ −5.25497 −0.174488 −0.0872442 0.996187i $$-0.527806\pi$$
−0.0872442 + 0.996187i $$0.527806\pi$$
$$908$$ −6.58738 −0.218610
$$909$$ 7.84321 0.260143
$$910$$ −6.42254 −0.212905
$$911$$ −11.1799 −0.370407 −0.185204 0.982700i $$-0.559294\pi$$
−0.185204 + 0.982700i $$0.559294\pi$$
$$912$$ −6.27877 −0.207911
$$913$$ 0 0
$$914$$ −7.63383 −0.252505
$$915$$ 9.14837 0.302436
$$916$$ −23.2829 −0.769287
$$917$$ 5.65461 0.186732
$$918$$ 4.71318 0.155558
$$919$$ 24.3862 0.804428 0.402214 0.915546i $$-0.368241\pi$$
0.402214 + 0.915546i $$0.368241\pi$$
$$920$$ 4.99442 0.164661
$$921$$ 63.0877 2.07881
$$922$$ 4.72213 0.155515
$$923$$ 37.8946 1.24732
$$924$$ 0 0
$$925$$ 11.0440 0.363125
$$926$$ −19.4328 −0.638600
$$927$$ 46.6050 1.53071
$$928$$ −9.04402 −0.296885
$$929$$ 19.9250 0.653718 0.326859 0.945073i $$-0.394010\pi$$
0.326859 + 0.945073i $$0.394010\pi$$
$$930$$ −20.0648 −0.657952
$$931$$ 2.46869 0.0809080
$$932$$ 1.72239 0.0564186
$$933$$ 38.3331 1.25497
$$934$$ −13.5440 −0.443172
$$935$$ 0 0
$$936$$ −22.2778 −0.728173
$$937$$ 47.9680 1.56705 0.783523 0.621363i $$-0.213421\pi$$
0.783523 + 0.621363i $$0.213421\pi$$
$$938$$ −1.28222 −0.0418659
$$939$$ 37.3205 1.21791
$$940$$ −8.37984 −0.273320
$$941$$ 43.8482 1.42941 0.714704 0.699427i $$-0.246561\pi$$
0.714704 + 0.699427i $$0.246561\pi$$
$$942$$ 34.0462 1.10929
$$943$$ −34.7305 −1.13098
$$944$$ −4.09575 −0.133305
$$945$$ −1.19205 −0.0387773
$$946$$ 0 0
$$947$$ −27.0791 −0.879951 −0.439976 0.898010i $$-0.645013\pi$$
−0.439976 + 0.898010i $$0.645013\pi$$
$$948$$ 15.7968 0.513056
$$949$$ 65.8447 2.13741
$$950$$ 2.46869 0.0800949
$$951$$ 36.2911 1.17682
$$952$$ −3.95385 −0.128145
$$953$$ 30.8977 1.00088 0.500438 0.865772i $$-0.333172\pi$$
0.500438 + 0.865772i $$0.333172\pi$$
$$954$$ 34.9007 1.12995
$$955$$ −9.55976 −0.309347
$$956$$ −11.6437 −0.376585
$$957$$ 0 0
$$958$$ −4.33024 −0.139904
$$959$$ −3.95288 −0.127645
$$960$$ −2.54336 −0.0820867
$$961$$ 31.2379 1.00767
$$962$$ −70.9307 −2.28690
$$963$$ −46.4384 −1.49646
$$964$$ 9.58236 0.308627
$$965$$ 16.6846 0.537096
$$966$$ −12.7026 −0.408700
$$967$$ −33.0146 −1.06168 −0.530839 0.847473i $$-0.678123\pi$$
−0.530839 + 0.847473i $$0.678123\pi$$
$$968$$ 0 0
$$969$$ 24.8253 0.797504
$$970$$ −11.9603 −0.384023
$$971$$ 17.7082 0.568283 0.284142 0.958782i $$-0.408291\pi$$
0.284142 + 0.958782i $$0.408291\pi$$
$$972$$ 22.3316 0.716285
$$973$$ 21.0254 0.674044
$$974$$ −36.5760 −1.17197
$$975$$ 16.3348 0.523134
$$976$$ −3.59696 −0.115136
$$977$$ 14.4633 0.462722 0.231361 0.972868i $$-0.425682\pi$$
0.231361 + 0.972868i $$0.425682\pi$$
$$978$$ −5.22163 −0.166969
$$979$$ 0 0
$$980$$ 1.00000 0.0319438
$$981$$ 30.0952 0.960865
$$982$$ 31.7227 1.01231
$$983$$ −12.3858 −0.395047 −0.197523 0.980298i $$-0.563290\pi$$
−0.197523 + 0.980298i $$0.563290\pi$$
$$984$$ 17.6862 0.563814
$$985$$ −0.533440 −0.0169968
$$986$$ 35.7587 1.13879
$$987$$ 21.3130 0.678399
$$988$$ −15.8553 −0.504423
$$989$$ 15.0979 0.480085
$$990$$ 0 0
$$991$$ −48.6112 −1.54419 −0.772093 0.635509i $$-0.780790\pi$$
−0.772093 + 0.635509i $$0.780790\pi$$
$$992$$ 7.88910 0.250479
$$993$$ 32.8625 1.04286
$$994$$ −5.90025 −0.187145
$$995$$ −2.81443 −0.0892234
$$996$$ 33.9241 1.07493
$$997$$ 35.2749 1.11717 0.558583 0.829449i $$-0.311345\pi$$
0.558583 + 0.829449i $$0.311345\pi$$
$$998$$ 36.7983 1.16483
$$999$$ −13.1650 −0.416522
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 8470.2.a.cs.1.2 4
11.2 odd 10 770.2.n.f.631.1 yes 8
11.6 odd 10 770.2.n.f.421.1 8
11.10 odd 2 8470.2.a.co.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.421.1 8 11.6 odd 10
770.2.n.f.631.1 yes 8 11.2 odd 10
8470.2.a.co.1.2 4 11.10 odd 2
8470.2.a.cs.1.2 4 1.1 even 1 trivial