# Properties

 Label 1925.2.a.v.1.3 Level $1925$ Weight $2$ Character 1925.1 Self dual yes Analytic conductor $15.371$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1925.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.3712023891$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 385) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 1925.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.67513 q^{2} +2.48119 q^{3} +5.15633 q^{4} +6.63752 q^{6} +1.00000 q^{7} +8.44358 q^{8} +3.15633 q^{9} +O(q^{10})$$ $$q+2.67513 q^{2} +2.48119 q^{3} +5.15633 q^{4} +6.63752 q^{6} +1.00000 q^{7} +8.44358 q^{8} +3.15633 q^{9} -1.00000 q^{11} +12.7938 q^{12} -5.83146 q^{13} +2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} +8.44358 q^{18} -1.35026 q^{19} +2.48119 q^{21} -2.67513 q^{22} +3.19394 q^{23} +20.9502 q^{24} -15.5999 q^{26} +0.387873 q^{27} +5.15633 q^{28} -3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -2.48119 q^{33} -14.5623 q^{34} +16.2750 q^{36} +8.54420 q^{37} -3.61213 q^{38} -14.4690 q^{39} -5.02539 q^{41} +6.63752 q^{42} -5.89446 q^{43} -5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +30.4568 q^{48} +1.00000 q^{49} -13.5066 q^{51} -30.0689 q^{52} +0.231548 q^{53} +1.03761 q^{54} +8.44358 q^{56} -3.35026 q^{57} -9.66291 q^{58} +13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +3.15633 q^{63} +18.1187 q^{64} -6.63752 q^{66} +10.8568 q^{67} -28.0689 q^{68} +7.92478 q^{69} -15.5369 q^{71} +26.6507 q^{72} +11.3684 q^{73} +22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} -38.7064 q^{78} +1.96968 q^{79} -8.50659 q^{81} -13.4436 q^{82} -10.6253 q^{83} +12.7938 q^{84} -15.7685 q^{86} -8.96239 q^{87} -8.44358 q^{88} +7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} -13.1187 q^{93} +31.6507 q^{94} +39.5755 q^{96} +0.836381 q^{97} +2.67513 q^{98} -3.15633 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} + O(q^{10})$$ $$3 q + 3 q^{2} + 2 q^{3} + 5 q^{4} + 4 q^{6} + 3 q^{7} + 9 q^{8} - q^{9} - 3 q^{11} + 12 q^{12} - 2 q^{13} + 3 q^{14} + 5 q^{16} + 9 q^{18} + 6 q^{19} + 2 q^{21} - 3 q^{22} + 10 q^{23} + 26 q^{24} - 20 q^{26} + 2 q^{27} + 5 q^{28} - 10 q^{29} - 10 q^{31} + 11 q^{32} - 2 q^{33} - 6 q^{34} + 17 q^{36} + 16 q^{37} - 10 q^{38} - 12 q^{39} + 4 q^{42} + 2 q^{43} - 5 q^{44} + 16 q^{46} + 20 q^{47} + 34 q^{48} + 3 q^{49} - 20 q^{51} - 32 q^{52} + 12 q^{53} + 14 q^{54} + 9 q^{56} + 2 q^{58} + 14 q^{59} + 10 q^{61} - 6 q^{62} - q^{63} + 33 q^{64} - 4 q^{66} + 2 q^{67} - 26 q^{68} + 2 q^{69} - 24 q^{71} + 23 q^{72} - 4 q^{73} + 38 q^{74} - 10 q^{76} - 3 q^{77} - 42 q^{78} + 8 q^{79} - 5 q^{81} - 24 q^{82} + 10 q^{83} + 12 q^{84} - 36 q^{86} - 16 q^{87} - 9 q^{88} + 20 q^{89} - 2 q^{91} + 18 q^{92} - 18 q^{93} + 38 q^{94} + 40 q^{96} + 3 q^{98} + q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.67513 1.89160 0.945802 0.324745i $$-0.105279\pi$$
0.945802 + 0.324745i $$0.105279\pi$$
$$3$$ 2.48119 1.43252 0.716259 0.697834i $$-0.245853\pi$$
0.716259 + 0.697834i $$0.245853\pi$$
$$4$$ 5.15633 2.57816
$$5$$ 0 0
$$6$$ 6.63752 2.70976
$$7$$ 1.00000 0.377964
$$8$$ 8.44358 2.98526
$$9$$ 3.15633 1.05211
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 12.7938 3.69326
$$13$$ −5.83146 −1.61735 −0.808677 0.588252i $$-0.799816\pi$$
−0.808677 + 0.588252i $$0.799816\pi$$
$$14$$ 2.67513 0.714959
$$15$$ 0 0
$$16$$ 12.2750 3.06876
$$17$$ −5.44358 −1.32026 −0.660131 0.751150i $$-0.729499\pi$$
−0.660131 + 0.751150i $$0.729499\pi$$
$$18$$ 8.44358 1.99017
$$19$$ −1.35026 −0.309771 −0.154886 0.987932i $$-0.549501\pi$$
−0.154886 + 0.987932i $$0.549501\pi$$
$$20$$ 0 0
$$21$$ 2.48119 0.541441
$$22$$ −2.67513 −0.570340
$$23$$ 3.19394 0.665982 0.332991 0.942930i $$-0.391942\pi$$
0.332991 + 0.942930i $$0.391942\pi$$
$$24$$ 20.9502 4.27644
$$25$$ 0 0
$$26$$ −15.5999 −3.05939
$$27$$ 0.387873 0.0746462
$$28$$ 5.15633 0.974454
$$29$$ −3.61213 −0.670755 −0.335378 0.942084i $$-0.608864\pi$$
−0.335378 + 0.942084i $$0.608864\pi$$
$$30$$ 0 0
$$31$$ −5.28726 −0.949620 −0.474810 0.880088i $$-0.657483\pi$$
−0.474810 + 0.880088i $$0.657483\pi$$
$$32$$ 15.9502 2.81962
$$33$$ −2.48119 −0.431920
$$34$$ −14.5623 −2.49741
$$35$$ 0 0
$$36$$ 16.2750 2.71251
$$37$$ 8.54420 1.40466 0.702329 0.711853i $$-0.252144\pi$$
0.702329 + 0.711853i $$0.252144\pi$$
$$38$$ −3.61213 −0.585964
$$39$$ −14.4690 −2.31689
$$40$$ 0 0
$$41$$ −5.02539 −0.784834 −0.392417 0.919787i $$-0.628361\pi$$
−0.392417 + 0.919787i $$0.628361\pi$$
$$42$$ 6.63752 1.02419
$$43$$ −5.89446 −0.898897 −0.449448 0.893306i $$-0.648379\pi$$
−0.449448 + 0.893306i $$0.648379\pi$$
$$44$$ −5.15633 −0.777345
$$45$$ 0 0
$$46$$ 8.54420 1.25977
$$47$$ 11.8315 1.72580 0.862898 0.505379i $$-0.168647\pi$$
0.862898 + 0.505379i $$0.168647\pi$$
$$48$$ 30.4568 4.39605
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −13.5066 −1.89130
$$52$$ −30.0689 −4.16980
$$53$$ 0.231548 0.0318056 0.0159028 0.999874i $$-0.494938\pi$$
0.0159028 + 0.999874i $$0.494938\pi$$
$$54$$ 1.03761 0.141201
$$55$$ 0 0
$$56$$ 8.44358 1.12832
$$57$$ −3.35026 −0.443753
$$58$$ −9.66291 −1.26880
$$59$$ 13.5999 1.77056 0.885279 0.465061i $$-0.153968\pi$$
0.885279 + 0.465061i $$0.153968\pi$$
$$60$$ 0 0
$$61$$ −1.41327 −0.180950 −0.0904751 0.995899i $$-0.528839\pi$$
−0.0904751 + 0.995899i $$0.528839\pi$$
$$62$$ −14.1441 −1.79630
$$63$$ 3.15633 0.397660
$$64$$ 18.1187 2.26484
$$65$$ 0 0
$$66$$ −6.63752 −0.817022
$$67$$ 10.8568 1.32638 0.663188 0.748453i $$-0.269203\pi$$
0.663188 + 0.748453i $$0.269203\pi$$
$$68$$ −28.0689 −3.40385
$$69$$ 7.92478 0.954031
$$70$$ 0 0
$$71$$ −15.5369 −1.84389 −0.921946 0.387319i $$-0.873401\pi$$
−0.921946 + 0.387319i $$0.873401\pi$$
$$72$$ 26.6507 3.14081
$$73$$ 11.3684 1.33057 0.665283 0.746591i $$-0.268311\pi$$
0.665283 + 0.746591i $$0.268311\pi$$
$$74$$ 22.8568 2.65705
$$75$$ 0 0
$$76$$ −6.96239 −0.798641
$$77$$ −1.00000 −0.113961
$$78$$ −38.7064 −4.38264
$$79$$ 1.96968 0.221607 0.110803 0.993842i $$-0.464658\pi$$
0.110803 + 0.993842i $$0.464658\pi$$
$$80$$ 0 0
$$81$$ −8.50659 −0.945176
$$82$$ −13.4436 −1.48460
$$83$$ −10.6253 −1.16628 −0.583139 0.812372i $$-0.698176\pi$$
−0.583139 + 0.812372i $$0.698176\pi$$
$$84$$ 12.7938 1.39592
$$85$$ 0 0
$$86$$ −15.7685 −1.70036
$$87$$ −8.96239 −0.960869
$$88$$ −8.44358 −0.900089
$$89$$ 7.22425 0.765769 0.382885 0.923796i $$-0.374931\pi$$
0.382885 + 0.923796i $$0.374931\pi$$
$$90$$ 0 0
$$91$$ −5.83146 −0.611303
$$92$$ 16.4690 1.71701
$$93$$ −13.1187 −1.36035
$$94$$ 31.6507 3.26452
$$95$$ 0 0
$$96$$ 39.5755 4.03915
$$97$$ 0.836381 0.0849216 0.0424608 0.999098i $$-0.486480\pi$$
0.0424608 + 0.999098i $$0.486480\pi$$
$$98$$ 2.67513 0.270229
$$99$$ −3.15633 −0.317223
$$100$$ 0 0
$$101$$ 7.41327 0.737648 0.368824 0.929499i $$-0.379761\pi$$
0.368824 + 0.929499i $$0.379761\pi$$
$$102$$ −36.1319 −3.57759
$$103$$ −4.21933 −0.415743 −0.207871 0.978156i $$-0.566654\pi$$
−0.207871 + 0.978156i $$0.566654\pi$$
$$104$$ −49.2384 −4.82822
$$105$$ 0 0
$$106$$ 0.619421 0.0601635
$$107$$ 11.5369 1.11531 0.557657 0.830071i $$-0.311700\pi$$
0.557657 + 0.830071i $$0.311700\pi$$
$$108$$ 2.00000 0.192450
$$109$$ −2.18664 −0.209442 −0.104721 0.994502i $$-0.533395\pi$$
−0.104721 + 0.994502i $$0.533395\pi$$
$$110$$ 0 0
$$111$$ 21.1998 2.01220
$$112$$ 12.2750 1.15988
$$113$$ 9.35026 0.879599 0.439799 0.898096i $$-0.355050\pi$$
0.439799 + 0.898096i $$0.355050\pi$$
$$114$$ −8.96239 −0.839405
$$115$$ 0 0
$$116$$ −18.6253 −1.72932
$$117$$ −18.4060 −1.70163
$$118$$ 36.3815 3.34919
$$119$$ −5.44358 −0.499012
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −3.78067 −0.342286
$$123$$ −12.4690 −1.12429
$$124$$ −27.2628 −2.44827
$$125$$ 0 0
$$126$$ 8.44358 0.752214
$$127$$ 16.9624 1.50517 0.752584 0.658496i $$-0.228807\pi$$
0.752584 + 0.658496i $$0.228807\pi$$
$$128$$ 16.5696 1.46456
$$129$$ −14.6253 −1.28769
$$130$$ 0 0
$$131$$ −9.92478 −0.867132 −0.433566 0.901122i $$-0.642745\pi$$
−0.433566 + 0.901122i $$0.642745\pi$$
$$132$$ −12.7938 −1.11356
$$133$$ −1.35026 −0.117083
$$134$$ 29.0435 2.50898
$$135$$ 0 0
$$136$$ −45.9633 −3.94132
$$137$$ 10.9927 0.939170 0.469585 0.882887i $$-0.344404\pi$$
0.469585 + 0.882887i $$0.344404\pi$$
$$138$$ 21.1998 1.80465
$$139$$ 6.88717 0.584162 0.292081 0.956394i $$-0.405652\pi$$
0.292081 + 0.956394i $$0.405652\pi$$
$$140$$ 0 0
$$141$$ 29.3561 2.47223
$$142$$ −41.5633 −3.48791
$$143$$ 5.83146 0.487651
$$144$$ 38.7440 3.22867
$$145$$ 0 0
$$146$$ 30.4119 2.51690
$$147$$ 2.48119 0.204645
$$148$$ 44.0567 3.62144
$$149$$ 22.8119 1.86883 0.934414 0.356190i $$-0.115924\pi$$
0.934414 + 0.356190i $$0.115924\pi$$
$$150$$ 0 0
$$151$$ −3.24472 −0.264052 −0.132026 0.991246i $$-0.542148\pi$$
−0.132026 + 0.991246i $$0.542148\pi$$
$$152$$ −11.4010 −0.924747
$$153$$ −17.1817 −1.38906
$$154$$ −2.67513 −0.215568
$$155$$ 0 0
$$156$$ −74.6067 −5.97332
$$157$$ 5.42548 0.433001 0.216500 0.976283i $$-0.430536\pi$$
0.216500 + 0.976283i $$0.430536\pi$$
$$158$$ 5.26916 0.419192
$$159$$ 0.574515 0.0455620
$$160$$ 0 0
$$161$$ 3.19394 0.251717
$$162$$ −22.7562 −1.78790
$$163$$ −3.38058 −0.264787 −0.132394 0.991197i $$-0.542266\pi$$
−0.132394 + 0.991197i $$0.542266\pi$$
$$164$$ −25.9126 −2.02343
$$165$$ 0 0
$$166$$ −28.4241 −2.20614
$$167$$ −11.2750 −0.872489 −0.436244 0.899828i $$-0.643692\pi$$
−0.436244 + 0.899828i $$0.643692\pi$$
$$168$$ 20.9502 1.61634
$$169$$ 21.0059 1.61584
$$170$$ 0 0
$$171$$ −4.26187 −0.325913
$$172$$ −30.3938 −2.31750
$$173$$ 8.98049 0.682774 0.341387 0.939923i $$-0.389103\pi$$
0.341387 + 0.939923i $$0.389103\pi$$
$$174$$ −23.9756 −1.81758
$$175$$ 0 0
$$176$$ −12.2750 −0.925266
$$177$$ 33.7440 2.53636
$$178$$ 19.3258 1.44853
$$179$$ −26.2374 −1.96108 −0.980539 0.196326i $$-0.937099\pi$$
−0.980539 + 0.196326i $$0.937099\pi$$
$$180$$ 0 0
$$181$$ −11.1998 −0.832476 −0.416238 0.909256i $$-0.636652\pi$$
−0.416238 + 0.909256i $$0.636652\pi$$
$$182$$ −15.5999 −1.15634
$$183$$ −3.50659 −0.259214
$$184$$ 26.9683 1.98813
$$185$$ 0 0
$$186$$ −35.0943 −2.57324
$$187$$ 5.44358 0.398074
$$188$$ 61.0068 4.44938
$$189$$ 0.387873 0.0282136
$$190$$ 0 0
$$191$$ 11.1998 0.810390 0.405195 0.914230i $$-0.367204\pi$$
0.405195 + 0.914230i $$0.367204\pi$$
$$192$$ 44.9560 3.24442
$$193$$ −0.604833 −0.0435368 −0.0217684 0.999763i $$-0.506930\pi$$
−0.0217684 + 0.999763i $$0.506930\pi$$
$$194$$ 2.23743 0.160638
$$195$$ 0 0
$$196$$ 5.15633 0.368309
$$197$$ −15.3054 −1.09046 −0.545231 0.838286i $$-0.683558\pi$$
−0.545231 + 0.838286i $$0.683558\pi$$
$$198$$ −8.44358 −0.600059
$$199$$ −12.5623 −0.890518 −0.445259 0.895402i $$-0.646888\pi$$
−0.445259 + 0.895402i $$0.646888\pi$$
$$200$$ 0 0
$$201$$ 26.9380 1.90006
$$202$$ 19.8315 1.39534
$$203$$ −3.61213 −0.253522
$$204$$ −69.6444 −4.87608
$$205$$ 0 0
$$206$$ −11.2873 −0.786421
$$207$$ 10.0811 0.700685
$$208$$ −71.5814 −4.96327
$$209$$ 1.35026 0.0933996
$$210$$ 0 0
$$211$$ −4.43866 −0.305570 −0.152785 0.988259i $$-0.548824\pi$$
−0.152785 + 0.988259i $$0.548824\pi$$
$$212$$ 1.19394 0.0819999
$$213$$ −38.5501 −2.64141
$$214$$ 30.8627 2.10973
$$215$$ 0 0
$$216$$ 3.27504 0.222838
$$217$$ −5.28726 −0.358922
$$218$$ −5.84955 −0.396182
$$219$$ 28.2071 1.90606
$$220$$ 0 0
$$221$$ 31.7440 2.13533
$$222$$ 56.7123 3.80628
$$223$$ 7.78067 0.521032 0.260516 0.965469i $$-0.416107\pi$$
0.260516 + 0.965469i $$0.416107\pi$$
$$224$$ 15.9502 1.06572
$$225$$ 0 0
$$226$$ 25.0132 1.66385
$$227$$ 10.4485 0.693492 0.346746 0.937959i $$-0.387287\pi$$
0.346746 + 0.937959i $$0.387287\pi$$
$$228$$ −17.2750 −1.14407
$$229$$ −29.4518 −1.94623 −0.973116 0.230316i $$-0.926024\pi$$
−0.973116 + 0.230316i $$0.926024\pi$$
$$230$$ 0 0
$$231$$ −2.48119 −0.163251
$$232$$ −30.4993 −2.00238
$$233$$ 8.73084 0.571976 0.285988 0.958233i $$-0.407678\pi$$
0.285988 + 0.958233i $$0.407678\pi$$
$$234$$ −49.2384 −3.21881
$$235$$ 0 0
$$236$$ 70.1255 4.56478
$$237$$ 4.88717 0.317456
$$238$$ −14.5623 −0.943933
$$239$$ −21.2144 −1.37225 −0.686123 0.727486i $$-0.740689\pi$$
−0.686123 + 0.727486i $$0.740689\pi$$
$$240$$ 0 0
$$241$$ −9.33804 −0.601516 −0.300758 0.953700i $$-0.597240\pi$$
−0.300758 + 0.953700i $$0.597240\pi$$
$$242$$ 2.67513 0.171964
$$243$$ −22.2701 −1.42863
$$244$$ −7.28726 −0.466519
$$245$$ 0 0
$$246$$ −33.3561 −2.12671
$$247$$ 7.87399 0.501010
$$248$$ −44.6434 −2.83486
$$249$$ −26.3634 −1.67071
$$250$$ 0 0
$$251$$ 1.87636 0.118435 0.0592174 0.998245i $$-0.481139\pi$$
0.0592174 + 0.998245i $$0.481139\pi$$
$$252$$ 16.2750 1.02523
$$253$$ −3.19394 −0.200801
$$254$$ 45.3766 2.84718
$$255$$ 0 0
$$256$$ 8.08840 0.505525
$$257$$ −27.1392 −1.69290 −0.846448 0.532472i $$-0.821263\pi$$
−0.846448 + 0.532472i $$0.821263\pi$$
$$258$$ −39.1246 −2.43579
$$259$$ 8.54420 0.530911
$$260$$ 0 0
$$261$$ −11.4010 −0.705707
$$262$$ −26.5501 −1.64027
$$263$$ −12.8119 −0.790018 −0.395009 0.918677i $$-0.629259\pi$$
−0.395009 + 0.918677i $$0.629259\pi$$
$$264$$ −20.9502 −1.28939
$$265$$ 0 0
$$266$$ −3.61213 −0.221474
$$267$$ 17.9248 1.09698
$$268$$ 55.9814 3.41961
$$269$$ −6.26187 −0.381793 −0.190896 0.981610i $$-0.561139\pi$$
−0.190896 + 0.981610i $$0.561139\pi$$
$$270$$ 0 0
$$271$$ −5.73813 −0.348567 −0.174283 0.984696i $$-0.555761\pi$$
−0.174283 + 0.984696i $$0.555761\pi$$
$$272$$ −66.8202 −4.05157
$$273$$ −14.4690 −0.875702
$$274$$ 29.4069 1.77654
$$275$$ 0 0
$$276$$ 40.8627 2.45965
$$277$$ 8.35756 0.502157 0.251078 0.967967i $$-0.419215\pi$$
0.251078 + 0.967967i $$0.419215\pi$$
$$278$$ 18.4241 1.10500
$$279$$ −16.6883 −0.999103
$$280$$ 0 0
$$281$$ −8.44851 −0.503996 −0.251998 0.967728i $$-0.581088\pi$$
−0.251998 + 0.967728i $$0.581088\pi$$
$$282$$ 78.5315 4.67648
$$283$$ −0.836381 −0.0497177 −0.0248588 0.999691i $$-0.507914\pi$$
−0.0248588 + 0.999691i $$0.507914\pi$$
$$284$$ −80.1133 −4.75385
$$285$$ 0 0
$$286$$ 15.5999 0.922442
$$287$$ −5.02539 −0.296640
$$288$$ 50.3439 2.96654
$$289$$ 12.6326 0.743094
$$290$$ 0 0
$$291$$ 2.07522 0.121652
$$292$$ 58.6190 3.43042
$$293$$ 2.71862 0.158824 0.0794118 0.996842i $$-0.474696\pi$$
0.0794118 + 0.996842i $$0.474696\pi$$
$$294$$ 6.63752 0.387108
$$295$$ 0 0
$$296$$ 72.1436 4.19326
$$297$$ −0.387873 −0.0225067
$$298$$ 61.0249 3.53508
$$299$$ −18.6253 −1.07713
$$300$$ 0 0
$$301$$ −5.89446 −0.339751
$$302$$ −8.68006 −0.499481
$$303$$ 18.3938 1.05669
$$304$$ −16.5745 −0.950614
$$305$$ 0 0
$$306$$ −45.9633 −2.62755
$$307$$ 8.36344 0.477326 0.238663 0.971102i $$-0.423291\pi$$
0.238663 + 0.971102i $$0.423291\pi$$
$$308$$ −5.15633 −0.293809
$$309$$ −10.4690 −0.595559
$$310$$ 0 0
$$311$$ 4.43629 0.251559 0.125779 0.992058i $$-0.459857\pi$$
0.125779 + 0.992058i $$0.459857\pi$$
$$312$$ −122.170 −6.91651
$$313$$ −29.7889 −1.68377 −0.841885 0.539658i $$-0.818554\pi$$
−0.841885 + 0.539658i $$0.818554\pi$$
$$314$$ 14.5139 0.819066
$$315$$ 0 0
$$316$$ 10.1563 0.571338
$$317$$ −15.4010 −0.865009 −0.432504 0.901632i $$-0.642370\pi$$
−0.432504 + 0.901632i $$0.642370\pi$$
$$318$$ 1.53690 0.0861853
$$319$$ 3.61213 0.202240
$$320$$ 0 0
$$321$$ 28.6253 1.59771
$$322$$ 8.54420 0.476150
$$323$$ 7.35026 0.408980
$$324$$ −43.8627 −2.43682
$$325$$ 0 0
$$326$$ −9.04349 −0.500873
$$327$$ −5.42548 −0.300030
$$328$$ −42.4323 −2.34293
$$329$$ 11.8315 0.652289
$$330$$ 0 0
$$331$$ 6.26187 0.344183 0.172092 0.985081i $$-0.444947\pi$$
0.172092 + 0.985081i $$0.444947\pi$$
$$332$$ −54.7875 −3.00685
$$333$$ 26.9683 1.47785
$$334$$ −30.1622 −1.65040
$$335$$ 0 0
$$336$$ 30.4568 1.66155
$$337$$ 15.8700 0.864495 0.432248 0.901755i $$-0.357721\pi$$
0.432248 + 0.901755i $$0.357721\pi$$
$$338$$ 56.1935 3.05652
$$339$$ 23.1998 1.26004
$$340$$ 0 0
$$341$$ 5.28726 0.286321
$$342$$ −11.4010 −0.616498
$$343$$ 1.00000 0.0539949
$$344$$ −49.7704 −2.68344
$$345$$ 0 0
$$346$$ 24.0240 1.29154
$$347$$ −6.79147 −0.364585 −0.182293 0.983244i $$-0.558352\pi$$
−0.182293 + 0.983244i $$0.558352\pi$$
$$348$$ −46.2130 −2.47728
$$349$$ 26.7489 1.43184 0.715919 0.698183i $$-0.246008\pi$$
0.715919 + 0.698183i $$0.246008\pi$$
$$350$$ 0 0
$$351$$ −2.26187 −0.120729
$$352$$ −15.9502 −0.850147
$$353$$ 16.8627 0.897512 0.448756 0.893654i $$-0.351867\pi$$
0.448756 + 0.893654i $$0.351867\pi$$
$$354$$ 90.2697 4.79778
$$355$$ 0 0
$$356$$ 37.2506 1.97428
$$357$$ −13.5066 −0.714844
$$358$$ −70.1886 −3.70958
$$359$$ −3.79289 −0.200181 −0.100091 0.994978i $$-0.531913\pi$$
−0.100091 + 0.994978i $$0.531913\pi$$
$$360$$ 0 0
$$361$$ −17.1768 −0.904042
$$362$$ −29.9610 −1.57471
$$363$$ 2.48119 0.130229
$$364$$ −30.0689 −1.57604
$$365$$ 0 0
$$366$$ −9.38058 −0.490331
$$367$$ 6.36977 0.332500 0.166250 0.986084i $$-0.446834\pi$$
0.166250 + 0.986084i $$0.446834\pi$$
$$368$$ 39.2057 2.04374
$$369$$ −15.8618 −0.825731
$$370$$ 0 0
$$371$$ 0.231548 0.0120214
$$372$$ −67.6444 −3.50720
$$373$$ 21.3317 1.10451 0.552257 0.833674i $$-0.313767\pi$$
0.552257 + 0.833674i $$0.313767\pi$$
$$374$$ 14.5623 0.752998
$$375$$ 0 0
$$376$$ 99.8999 5.15194
$$377$$ 21.0640 1.08485
$$378$$ 1.03761 0.0533690
$$379$$ 24.7875 1.27325 0.636624 0.771174i $$-0.280330\pi$$
0.636624 + 0.771174i $$0.280330\pi$$
$$380$$ 0 0
$$381$$ 42.0870 2.15618
$$382$$ 29.9610 1.53294
$$383$$ 5.45817 0.278900 0.139450 0.990229i $$-0.455467\pi$$
0.139450 + 0.990229i $$0.455467\pi$$
$$384$$ 41.1124 2.09801
$$385$$ 0 0
$$386$$ −1.61801 −0.0823544
$$387$$ −18.6048 −0.945737
$$388$$ 4.31265 0.218942
$$389$$ −13.7235 −0.695811 −0.347906 0.937530i $$-0.613107\pi$$
−0.347906 + 0.937530i $$0.613107\pi$$
$$390$$ 0 0
$$391$$ −17.3865 −0.879271
$$392$$ 8.44358 0.426465
$$393$$ −24.6253 −1.24218
$$394$$ −40.9438 −2.06272
$$395$$ 0 0
$$396$$ −16.2750 −0.817851
$$397$$ −2.11142 −0.105969 −0.0529846 0.998595i $$-0.516873\pi$$
−0.0529846 + 0.998595i $$0.516873\pi$$
$$398$$ −33.6058 −1.68451
$$399$$ −3.35026 −0.167723
$$400$$ 0 0
$$401$$ 19.1490 0.956257 0.478128 0.878290i $$-0.341315\pi$$
0.478128 + 0.878290i $$0.341315\pi$$
$$402$$ 72.0625 3.59415
$$403$$ 30.8324 1.53587
$$404$$ 38.2252 1.90178
$$405$$ 0 0
$$406$$ −9.66291 −0.479562
$$407$$ −8.54420 −0.423520
$$408$$ −114.044 −5.64602
$$409$$ −18.6883 −0.924077 −0.462039 0.886860i $$-0.652882\pi$$
−0.462039 + 0.886860i $$0.652882\pi$$
$$410$$ 0 0
$$411$$ 27.2750 1.34538
$$412$$ −21.7562 −1.07185
$$413$$ 13.5999 0.669208
$$414$$ 26.9683 1.32542
$$415$$ 0 0
$$416$$ −93.0127 −4.56032
$$417$$ 17.0884 0.836822
$$418$$ 3.61213 0.176675
$$419$$ −0.773377 −0.0377819 −0.0188910 0.999822i $$-0.506014\pi$$
−0.0188910 + 0.999822i $$0.506014\pi$$
$$420$$ 0 0
$$421$$ −10.5198 −0.512702 −0.256351 0.966584i $$-0.582520\pi$$
−0.256351 + 0.966584i $$0.582520\pi$$
$$422$$ −11.8740 −0.578017
$$423$$ 37.3439 1.81572
$$424$$ 1.95509 0.0949478
$$425$$ 0 0
$$426$$ −103.127 −4.99650
$$427$$ −1.41327 −0.0683927
$$428$$ 59.4880 2.87546
$$429$$ 14.4690 0.698569
$$430$$ 0 0
$$431$$ −24.7308 −1.19124 −0.595621 0.803265i $$-0.703094\pi$$
−0.595621 + 0.803265i $$0.703094\pi$$
$$432$$ 4.76116 0.229071
$$433$$ 18.5599 0.891933 0.445967 0.895050i $$-0.352860\pi$$
0.445967 + 0.895050i $$0.352860\pi$$
$$434$$ −14.1441 −0.678939
$$435$$ 0 0
$$436$$ −11.2750 −0.539976
$$437$$ −4.31265 −0.206302
$$438$$ 75.4577 3.60551
$$439$$ −1.42548 −0.0680347 −0.0340173 0.999421i $$-0.510830\pi$$
−0.0340173 + 0.999421i $$0.510830\pi$$
$$440$$ 0 0
$$441$$ 3.15633 0.150301
$$442$$ 84.9194 4.03920
$$443$$ −40.1925 −1.90960 −0.954802 0.297242i $$-0.903933\pi$$
−0.954802 + 0.297242i $$0.903933\pi$$
$$444$$ 109.313 5.18777
$$445$$ 0 0
$$446$$ 20.8143 0.985586
$$447$$ 56.6009 2.67713
$$448$$ 18.1187 0.856029
$$449$$ 12.6556 0.597256 0.298628 0.954370i $$-0.403471\pi$$
0.298628 + 0.954370i $$0.403471\pi$$
$$450$$ 0 0
$$451$$ 5.02539 0.236636
$$452$$ 48.2130 2.26775
$$453$$ −8.05079 −0.378259
$$454$$ 27.9511 1.31181
$$455$$ 0 0
$$456$$ −28.2882 −1.32472
$$457$$ 0.544198 0.0254565 0.0127283 0.999919i $$-0.495948\pi$$
0.0127283 + 0.999919i $$0.495948\pi$$
$$458$$ −78.7875 −3.68150
$$459$$ −2.11142 −0.0985526
$$460$$ 0 0
$$461$$ −11.5755 −0.539123 −0.269562 0.962983i $$-0.586879\pi$$
−0.269562 + 0.962983i $$0.586879\pi$$
$$462$$ −6.63752 −0.308805
$$463$$ −23.7948 −1.10584 −0.552919 0.833235i $$-0.686486\pi$$
−0.552919 + 0.833235i $$0.686486\pi$$
$$464$$ −44.3390 −2.05839
$$465$$ 0 0
$$466$$ 23.3561 1.08195
$$467$$ 2.66784 0.123453 0.0617264 0.998093i $$-0.480339\pi$$
0.0617264 + 0.998093i $$0.480339\pi$$
$$468$$ −94.9072 −4.38709
$$469$$ 10.8568 0.501323
$$470$$ 0 0
$$471$$ 13.4617 0.620282
$$472$$ 114.832 5.28557
$$473$$ 5.89446 0.271028
$$474$$ 13.0738 0.600500
$$475$$ 0 0
$$476$$ −28.0689 −1.28654
$$477$$ 0.730841 0.0334629
$$478$$ −56.7513 −2.59574
$$479$$ −10.7104 −0.489369 −0.244685 0.969603i $$-0.578684\pi$$
−0.244685 + 0.969603i $$0.578684\pi$$
$$480$$ 0 0
$$481$$ −49.8251 −2.27183
$$482$$ −24.9805 −1.13783
$$483$$ 7.92478 0.360590
$$484$$ 5.15633 0.234378
$$485$$ 0 0
$$486$$ −59.5755 −2.70240
$$487$$ −17.4314 −0.789891 −0.394945 0.918705i $$-0.629236\pi$$
−0.394945 + 0.918705i $$0.629236\pi$$
$$488$$ −11.9330 −0.540183
$$489$$ −8.38787 −0.379313
$$490$$ 0 0
$$491$$ −28.3693 −1.28029 −0.640145 0.768254i $$-0.721126\pi$$
−0.640145 + 0.768254i $$0.721126\pi$$
$$492$$ −64.2941 −2.89860
$$493$$ 19.6629 0.885573
$$494$$ 21.0640 0.947712
$$495$$ 0 0
$$496$$ −64.9013 −2.91415
$$497$$ −15.5369 −0.696925
$$498$$ −70.5256 −3.16033
$$499$$ 27.4763 1.23001 0.615003 0.788524i $$-0.289155\pi$$
0.615003 + 0.788524i $$0.289155\pi$$
$$500$$ 0 0
$$501$$ −27.9756 −1.24986
$$502$$ 5.01951 0.224032
$$503$$ −20.2981 −0.905046 −0.452523 0.891753i $$-0.649476\pi$$
−0.452523 + 0.891753i $$0.649476\pi$$
$$504$$ 26.6507 1.18712
$$505$$ 0 0
$$506$$ −8.54420 −0.379836
$$507$$ 52.1197 2.31472
$$508$$ 87.4636 3.88057
$$509$$ −24.2619 −1.07539 −0.537694 0.843140i $$-0.680704\pi$$
−0.537694 + 0.843140i $$0.680704\pi$$
$$510$$ 0 0
$$511$$ 11.3684 0.502907
$$512$$ −11.5017 −0.508306
$$513$$ −0.523730 −0.0231233
$$514$$ −72.6009 −3.20229
$$515$$ 0 0
$$516$$ −75.4128 −3.31986
$$517$$ −11.8315 −0.520347
$$518$$ 22.8568 1.00427
$$519$$ 22.2823 0.978086
$$520$$ 0 0
$$521$$ 2.20123 0.0964377 0.0482188 0.998837i $$-0.484646\pi$$
0.0482188 + 0.998837i $$0.484646\pi$$
$$522$$ −30.4993 −1.33492
$$523$$ 22.1378 0.968017 0.484008 0.875063i $$-0.339180\pi$$
0.484008 + 0.875063i $$0.339180\pi$$
$$524$$ −51.1754 −2.23561
$$525$$ 0 0
$$526$$ −34.2736 −1.49440
$$527$$ 28.7816 1.25375
$$528$$ −30.4568 −1.32546
$$529$$ −12.7988 −0.556468
$$530$$ 0 0
$$531$$ 42.9257 1.86282
$$532$$ −6.96239 −0.301858
$$533$$ 29.3054 1.26936
$$534$$ 47.9511 2.07505
$$535$$ 0 0
$$536$$ 91.6707 3.95957
$$537$$ −65.1002 −2.80928
$$538$$ −16.7513 −0.722200
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ 23.0640 0.991597 0.495799 0.868438i $$-0.334875\pi$$
0.495799 + 0.868438i $$0.334875\pi$$
$$542$$ −15.3503 −0.659350
$$543$$ −27.7889 −1.19254
$$544$$ −86.8261 −3.72264
$$545$$ 0 0
$$546$$ −38.7064 −1.65648
$$547$$ −21.3766 −0.913998 −0.456999 0.889467i $$-0.651076\pi$$
−0.456999 + 0.889467i $$0.651076\pi$$
$$548$$ 56.6820 2.42133
$$549$$ −4.46073 −0.190379
$$550$$ 0 0
$$551$$ 4.87732 0.207781
$$552$$ 66.9135 2.84803
$$553$$ 1.96968 0.0837594
$$554$$ 22.3576 0.949882
$$555$$ 0 0
$$556$$ 35.5125 1.50606
$$557$$ 9.19394 0.389560 0.194780 0.980847i $$-0.437601\pi$$
0.194780 + 0.980847i $$0.437601\pi$$
$$558$$ −44.6434 −1.88991
$$559$$ 34.3733 1.45384
$$560$$ 0 0
$$561$$ 13.5066 0.570249
$$562$$ −22.6009 −0.953360
$$563$$ 9.79877 0.412969 0.206484 0.978450i $$-0.433798\pi$$
0.206484 + 0.978450i $$0.433798\pi$$
$$564$$ 151.370 6.37382
$$565$$ 0 0
$$566$$ −2.23743 −0.0940461
$$567$$ −8.50659 −0.357243
$$568$$ −131.187 −5.50449
$$569$$ −33.5125 −1.40492 −0.702458 0.711725i $$-0.747914\pi$$
−0.702458 + 0.711725i $$0.747914\pi$$
$$570$$ 0 0
$$571$$ 43.1392 1.80532 0.902659 0.430356i $$-0.141612\pi$$
0.902659 + 0.430356i $$0.141612\pi$$
$$572$$ 30.0689 1.25724
$$573$$ 27.7889 1.16090
$$574$$ −13.4436 −0.561124
$$575$$ 0 0
$$576$$ 57.1886 2.38286
$$577$$ 14.8510 0.618254 0.309127 0.951021i $$-0.399963\pi$$
0.309127 + 0.951021i $$0.399963\pi$$
$$578$$ 33.7938 1.40564
$$579$$ −1.50071 −0.0623673
$$580$$ 0 0
$$581$$ −10.6253 −0.440812
$$582$$ 5.55149 0.230117
$$583$$ −0.231548 −0.00958974
$$584$$ 95.9897 3.97208
$$585$$ 0 0
$$586$$ 7.27267 0.300431
$$587$$ −14.7938 −0.610607 −0.305304 0.952255i $$-0.598758\pi$$
−0.305304 + 0.952255i $$0.598758\pi$$
$$588$$ 12.7938 0.527609
$$589$$ 7.13918 0.294165
$$590$$ 0 0
$$591$$ −37.9756 −1.56211
$$592$$ 104.880 4.31056
$$593$$ 27.4191 1.12597 0.562985 0.826467i $$-0.309653\pi$$
0.562985 + 0.826467i $$0.309653\pi$$
$$594$$ −1.03761 −0.0425737
$$595$$ 0 0
$$596$$ 117.626 4.81814
$$597$$ −31.1695 −1.27568
$$598$$ −49.8251 −2.03750
$$599$$ 11.3258 0.462761 0.231380 0.972863i $$-0.425676\pi$$
0.231380 + 0.972863i $$0.425676\pi$$
$$600$$ 0 0
$$601$$ 15.5393 0.633860 0.316930 0.948449i $$-0.397348\pi$$
0.316930 + 0.948449i $$0.397348\pi$$
$$602$$ −15.7685 −0.642674
$$603$$ 34.2677 1.39549
$$604$$ −16.7308 −0.680768
$$605$$ 0 0
$$606$$ 49.2057 1.99884
$$607$$ −17.7235 −0.719377 −0.359688 0.933073i $$-0.617117\pi$$
−0.359688 + 0.933073i $$0.617117\pi$$
$$608$$ −21.5369 −0.873437
$$609$$ −8.96239 −0.363174
$$610$$ 0 0
$$611$$ −68.9946 −2.79122
$$612$$ −88.5945 −3.58122
$$613$$ −22.2941 −0.900450 −0.450225 0.892915i $$-0.648656\pi$$
−0.450225 + 0.892915i $$0.648656\pi$$
$$614$$ 22.3733 0.902912
$$615$$ 0 0
$$616$$ −8.44358 −0.340202
$$617$$ 30.9438 1.24575 0.622876 0.782321i $$-0.285964\pi$$
0.622876 + 0.782321i $$0.285964\pi$$
$$618$$ −28.0059 −1.12656
$$619$$ 32.4119 1.30274 0.651371 0.758759i $$-0.274194\pi$$
0.651371 + 0.758759i $$0.274194\pi$$
$$620$$ 0 0
$$621$$ 1.23884 0.0497130
$$622$$ 11.8677 0.475850
$$623$$ 7.22425 0.289434
$$624$$ −177.607 −7.10998
$$625$$ 0 0
$$626$$ −79.6893 −3.18502
$$627$$ 3.35026 0.133797
$$628$$ 27.9756 1.11635
$$629$$ −46.5111 −1.85452
$$630$$ 0 0
$$631$$ −27.3258 −1.08782 −0.543912 0.839142i $$-0.683057\pi$$
−0.543912 + 0.839142i $$0.683057\pi$$
$$632$$ 16.6312 0.661553
$$633$$ −11.0132 −0.437734
$$634$$ −41.1998 −1.63625
$$635$$ 0 0
$$636$$ 2.96239 0.117466
$$637$$ −5.83146 −0.231051
$$638$$ 9.66291 0.382558
$$639$$ −49.0395 −1.93997
$$640$$ 0 0
$$641$$ 19.4460 0.768069 0.384034 0.923319i $$-0.374534\pi$$
0.384034 + 0.923319i $$0.374534\pi$$
$$642$$ 76.5764 3.02223
$$643$$ −5.29314 −0.208741 −0.104370 0.994538i $$-0.533283\pi$$
−0.104370 + 0.994538i $$0.533283\pi$$
$$644$$ 16.4690 0.648969
$$645$$ 0 0
$$646$$ 19.6629 0.773627
$$647$$ −35.0966 −1.37979 −0.689896 0.723909i $$-0.742344\pi$$
−0.689896 + 0.723909i $$0.742344\pi$$
$$648$$ −71.8261 −2.82159
$$649$$ −13.5999 −0.533843
$$650$$ 0 0
$$651$$ −13.1187 −0.514163
$$652$$ −17.4314 −0.682665
$$653$$ −27.7988 −1.08785 −0.543925 0.839134i $$-0.683062\pi$$
−0.543925 + 0.839134i $$0.683062\pi$$
$$654$$ −14.5139 −0.567538
$$655$$ 0 0
$$656$$ −61.6869 −2.40847
$$657$$ 35.8822 1.39990
$$658$$ 31.6507 1.23387
$$659$$ 19.6180 0.764209 0.382105 0.924119i $$-0.375199\pi$$
0.382105 + 0.924119i $$0.375199\pi$$
$$660$$ 0 0
$$661$$ 21.5633 0.838713 0.419357 0.907822i $$-0.362256\pi$$
0.419357 + 0.907822i $$0.362256\pi$$
$$662$$ 16.7513 0.651058
$$663$$ 78.7631 3.05890
$$664$$ −89.7156 −3.48164
$$665$$ 0 0
$$666$$ 72.1436 2.79551
$$667$$ −11.5369 −0.446711
$$668$$ −58.1378 −2.24942
$$669$$ 19.3054 0.746388
$$670$$ 0 0
$$671$$ 1.41327 0.0545585
$$672$$ 39.5755 1.52666
$$673$$ 21.0679 0.812109 0.406054 0.913849i $$-0.366904\pi$$
0.406054 + 0.913849i $$0.366904\pi$$
$$674$$ 42.4544 1.63528
$$675$$ 0 0
$$676$$ 108.313 4.16589
$$677$$ −34.5174 −1.32661 −0.663306 0.748349i $$-0.730847\pi$$
−0.663306 + 0.748349i $$0.730847\pi$$
$$678$$ 62.0625 2.38350
$$679$$ 0.836381 0.0320973
$$680$$ 0 0
$$681$$ 25.9248 0.993440
$$682$$ 14.1441 0.541606
$$683$$ −33.7802 −1.29256 −0.646282 0.763099i $$-0.723677\pi$$
−0.646282 + 0.763099i $$0.723677\pi$$
$$684$$ −21.9756 −0.840257
$$685$$ 0 0
$$686$$ 2.67513 0.102137
$$687$$ −73.0757 −2.78801
$$688$$ −72.3547 −2.75850
$$689$$ −1.35026 −0.0514409
$$690$$ 0 0
$$691$$ −13.8618 −0.527327 −0.263663 0.964615i $$-0.584931\pi$$
−0.263663 + 0.964615i $$0.584931\pi$$
$$692$$ 46.3063 1.76030
$$693$$ −3.15633 −0.119899
$$694$$ −18.1681 −0.689651
$$695$$ 0 0
$$696$$ −75.6747 −2.86844
$$697$$ 27.3561 1.03619
$$698$$ 71.5569 2.70847
$$699$$ 21.6629 0.819367
$$700$$ 0 0
$$701$$ 40.5256 1.53063 0.765316 0.643655i $$-0.222583\pi$$
0.765316 + 0.643655i $$0.222583\pi$$
$$702$$ −6.05079 −0.228372
$$703$$ −11.5369 −0.435123
$$704$$ −18.1187 −0.682875
$$705$$ 0 0
$$706$$ 45.1100 1.69774
$$707$$ 7.41327 0.278805
$$708$$ 173.995 6.53914
$$709$$ −0.850969 −0.0319588 −0.0159794 0.999872i $$-0.505087\pi$$
−0.0159794 + 0.999872i $$0.505087\pi$$
$$710$$ 0 0
$$711$$ 6.21696 0.233154
$$712$$ 60.9986 2.28602
$$713$$ −16.8872 −0.632429
$$714$$ −36.1319 −1.35220
$$715$$ 0 0
$$716$$ −135.289 −5.05598
$$717$$ −52.6371 −1.96577
$$718$$ −10.1465 −0.378663
$$719$$ 22.5769 0.841976 0.420988 0.907066i $$-0.361683\pi$$
0.420988 + 0.907066i $$0.361683\pi$$
$$720$$ 0 0
$$721$$ −4.21933 −0.157136
$$722$$ −45.9502 −1.71009
$$723$$ −23.1695 −0.861683
$$724$$ −57.7499 −2.14626
$$725$$ 0 0
$$726$$ 6.63752 0.246341
$$727$$ −12.5174 −0.464244 −0.232122 0.972687i $$-0.574567\pi$$
−0.232122 + 0.972687i $$0.574567\pi$$
$$728$$ −49.2384 −1.82490
$$729$$ −29.7367 −1.10136
$$730$$ 0 0
$$731$$ 32.0870 1.18678
$$732$$ −18.0811 −0.668297
$$733$$ −16.6678 −0.615641 −0.307820 0.951445i $$-0.599600\pi$$
−0.307820 + 0.951445i $$0.599600\pi$$
$$734$$ 17.0400 0.628957
$$735$$ 0 0
$$736$$ 50.9438 1.87781
$$737$$ −10.8568 −0.399917
$$738$$ −42.4323 −1.56196
$$739$$ 42.7005 1.57076 0.785382 0.619011i $$-0.212467\pi$$
0.785382 + 0.619011i $$0.212467\pi$$
$$740$$ 0 0
$$741$$ 19.5369 0.717706
$$742$$ 0.619421 0.0227397
$$743$$ 19.6873 0.722259 0.361129 0.932516i $$-0.382391\pi$$
0.361129 + 0.932516i $$0.382391\pi$$
$$744$$ −110.769 −4.06099
$$745$$ 0 0
$$746$$ 57.0651 2.08930
$$747$$ −33.5369 −1.22705
$$748$$ 28.0689 1.02630
$$749$$ 11.5369 0.421549
$$750$$ 0 0
$$751$$ −5.85940 −0.213813 −0.106906 0.994269i $$-0.534094\pi$$
−0.106906 + 0.994269i $$0.534094\pi$$
$$752$$ 145.232 5.29605
$$753$$ 4.65562 0.169660
$$754$$ 56.3488 2.05210
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 40.5863 1.47513 0.737567 0.675274i $$-0.235975\pi$$
0.737567 + 0.675274i $$0.235975\pi$$
$$758$$ 66.3098 2.40848
$$759$$ −7.92478 −0.287651
$$760$$ 0 0
$$761$$ 21.8472 0.791960 0.395980 0.918259i $$-0.370405\pi$$
0.395980 + 0.918259i $$0.370405\pi$$
$$762$$ 112.588 4.07864
$$763$$ −2.18664 −0.0791618
$$764$$ 57.7499 2.08932
$$765$$ 0 0
$$766$$ 14.6013 0.527567
$$767$$ −79.3073 −2.86362
$$768$$ 20.0689 0.724173
$$769$$ 45.2892 1.63317 0.816585 0.577226i $$-0.195865\pi$$
0.816585 + 0.577226i $$0.195865\pi$$
$$770$$ 0 0
$$771$$ −67.3376 −2.42510
$$772$$ −3.11871 −0.112245
$$773$$ 33.8153 1.21625 0.608125 0.793841i $$-0.291922\pi$$
0.608125 + 0.793841i $$0.291922\pi$$
$$774$$ −49.7704 −1.78896
$$775$$ 0 0
$$776$$ 7.06205 0.253513
$$777$$ 21.1998 0.760539
$$778$$ −36.7123 −1.31620
$$779$$ 6.78560 0.243119
$$780$$ 0 0
$$781$$ 15.5369 0.555954
$$782$$ −46.5111 −1.66323
$$783$$ −1.40105 −0.0500693
$$784$$ 12.2750 0.438394
$$785$$ 0 0
$$786$$ −65.8759 −2.34972
$$787$$ −1.27504 −0.0454502 −0.0227251 0.999742i $$-0.507234\pi$$
−0.0227251 + 0.999742i $$0.507234\pi$$
$$788$$ −78.9194 −2.81139
$$789$$ −31.7889 −1.13172
$$790$$ 0 0
$$791$$ 9.35026 0.332457
$$792$$ −26.6507 −0.946991
$$793$$ 8.24140 0.292661
$$794$$ −5.64832 −0.200452
$$795$$ 0 0
$$796$$ −64.7753 −2.29590
$$797$$ 42.5256 1.50634 0.753168 0.657829i $$-0.228525\pi$$
0.753168 + 0.657829i $$0.228525\pi$$
$$798$$ −8.96239 −0.317265
$$799$$ −64.4055 −2.27850
$$800$$ 0 0
$$801$$ 22.8021 0.805672
$$802$$ 51.2262 1.80886
$$803$$ −11.3684 −0.401181
$$804$$ 138.901 4.89865
$$805$$ 0 0
$$806$$ 82.4807 2.90526
$$807$$ −15.5369 −0.546925
$$808$$ 62.5945 2.20207
$$809$$ −14.7151 −0.517356 −0.258678 0.965964i $$-0.583287\pi$$
−0.258678 + 0.965964i $$0.583287\pi$$
$$810$$ 0 0
$$811$$ 51.7743 1.81804 0.909021 0.416750i $$-0.136831\pi$$
0.909021 + 0.416750i $$0.136831\pi$$
$$812$$ −18.6253 −0.653620
$$813$$ −14.2374 −0.499328
$$814$$ −22.8568 −0.801132
$$815$$ 0 0
$$816$$ −165.794 −5.80395
$$817$$ 7.95906 0.278452
$$818$$ −49.9937 −1.74799
$$819$$ −18.4060 −0.643157
$$820$$ 0 0
$$821$$ 2.64974 0.0924765 0.0462383 0.998930i $$-0.485277\pi$$
0.0462383 + 0.998930i $$0.485277\pi$$
$$822$$ 72.9643 2.54492
$$823$$ −5.76845 −0.201076 −0.100538 0.994933i $$-0.532056\pi$$
−0.100538 + 0.994933i $$0.532056\pi$$
$$824$$ −35.6263 −1.24110
$$825$$ 0 0
$$826$$ 36.3815 1.26588
$$827$$ 13.4920 0.469163 0.234581 0.972096i $$-0.424628\pi$$
0.234581 + 0.972096i $$0.424628\pi$$
$$828$$ 51.9814 1.80648
$$829$$ −4.70052 −0.163256 −0.0816280 0.996663i $$-0.526012\pi$$
−0.0816280 + 0.996663i $$0.526012\pi$$
$$830$$ 0 0
$$831$$ 20.7367 0.719349
$$832$$ −105.658 −3.66305
$$833$$ −5.44358 −0.188609
$$834$$ 45.7137 1.58294
$$835$$ 0 0
$$836$$ 6.96239 0.240799
$$837$$ −2.05079 −0.0708855
$$838$$ −2.06888 −0.0714684
$$839$$ −38.8045 −1.33968 −0.669839 0.742506i $$-0.733637\pi$$
−0.669839 + 0.742506i $$0.733637\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ −28.1417 −0.969828
$$843$$ −20.9624 −0.721983
$$844$$ −22.8872 −0.787809
$$845$$ 0 0
$$846$$ 99.8999 3.43463
$$847$$ 1.00000 0.0343604
$$848$$ 2.84226 0.0976036
$$849$$ −2.07522 −0.0712215
$$850$$ 0 0
$$851$$ 27.2896 0.935476
$$852$$ −198.777 −6.80998
$$853$$ −20.6824 −0.708153 −0.354076 0.935217i $$-0.615205\pi$$
−0.354076 + 0.935217i $$0.615205\pi$$
$$854$$ −3.78067 −0.129372
$$855$$ 0 0
$$856$$ 97.4128 3.32950
$$857$$ −26.3453 −0.899940 −0.449970 0.893044i $$-0.648565\pi$$
−0.449970 + 0.893044i $$0.648565\pi$$
$$858$$ 38.7064 1.32141
$$859$$ −8.51151 −0.290409 −0.145205 0.989402i $$-0.546384\pi$$
−0.145205 + 0.989402i $$0.546384\pi$$
$$860$$ 0 0
$$861$$ −12.4690 −0.424942
$$862$$ −66.1582 −2.25336
$$863$$ −7.56722 −0.257591 −0.128796 0.991671i $$-0.541111\pi$$
−0.128796 + 0.991671i $$0.541111\pi$$
$$864$$ 6.18664 0.210474
$$865$$ 0 0
$$866$$ 49.6502 1.68718
$$867$$ 31.3439 1.06450
$$868$$ −27.2628 −0.925360
$$869$$ −1.96968 −0.0668169
$$870$$ 0 0
$$871$$ −63.3112 −2.14522
$$872$$ −18.4631 −0.625239
$$873$$ 2.63989 0.0893467
$$874$$ −11.5369 −0.390242
$$875$$ 0 0
$$876$$ 145.445 4.91413
$$877$$ −17.2955 −0.584028 −0.292014 0.956414i $$-0.594325\pi$$
−0.292014 + 0.956414i $$0.594325\pi$$
$$878$$ −3.81336 −0.128695
$$879$$ 6.74543 0.227518
$$880$$ 0 0
$$881$$ 20.4504 0.688992 0.344496 0.938788i $$-0.388050\pi$$
0.344496 + 0.938788i $$0.388050\pi$$
$$882$$ 8.44358 0.284310
$$883$$ 49.6589 1.67116 0.835578 0.549371i $$-0.185133\pi$$
0.835578 + 0.549371i $$0.185133\pi$$
$$884$$ 163.682 5.50524
$$885$$ 0 0
$$886$$ −107.520 −3.61221
$$887$$ 47.1100 1.58180 0.790900 0.611946i $$-0.209613\pi$$
0.790900 + 0.611946i $$0.209613\pi$$
$$888$$ 179.002 6.00693
$$889$$ 16.9624 0.568900
$$890$$ 0 0
$$891$$ 8.50659 0.284981
$$892$$ 40.1197 1.34331
$$893$$ −15.9756 −0.534602
$$894$$ 151.415 5.06407
$$895$$ 0 0
$$896$$ 16.5696 0.553551
$$897$$ −46.2130 −1.54301
$$898$$ 33.8554 1.12977
$$899$$ 19.0982 0.636962
$$900$$ 0 0
$$901$$ −1.26045 −0.0419917
$$902$$ 13.4436 0.447622
$$903$$ −14.6253 −0.486700
$$904$$ 78.9497 2.62583
$$905$$ 0 0
$$906$$ −21.5369 −0.715516
$$907$$ −14.4591 −0.480107 −0.240054 0.970760i $$-0.577165\pi$$
−0.240054 + 0.970760i $$0.577165\pi$$
$$908$$ 53.8759 1.78793
$$909$$ 23.3987 0.776085
$$910$$ 0 0
$$911$$ −31.5369 −1.04486 −0.522432 0.852681i $$-0.674975\pi$$
−0.522432 + 0.852681i $$0.674975\pi$$
$$912$$ −41.1246 −1.36177
$$913$$ 10.6253 0.351646
$$914$$ 1.45580 0.0481536
$$915$$ 0 0
$$916$$ −151.863 −5.01770
$$917$$ −9.92478 −0.327745
$$918$$ −5.64832 −0.186422
$$919$$ 5.26328 0.173620 0.0868098 0.996225i $$-0.472333\pi$$
0.0868098 + 0.996225i $$0.472333\pi$$
$$920$$ 0 0
$$921$$ 20.7513 0.683779
$$922$$ −30.9659 −1.01981
$$923$$ 90.6028 2.98223
$$924$$ −12.7938 −0.420887
$$925$$ 0 0
$$926$$ −63.6542 −2.09181
$$927$$ −13.3176 −0.437407
$$928$$ −57.6140 −1.89127
$$929$$ 26.0508 0.854699 0.427349 0.904087i $$-0.359447\pi$$
0.427349 + 0.904087i $$0.359447\pi$$
$$930$$ 0 0
$$931$$ −1.35026 −0.0442530
$$932$$ 45.0191 1.47465
$$933$$ 11.0073 0.360363
$$934$$ 7.13681 0.233524
$$935$$ 0 0
$$936$$ −155.412 −5.07981
$$937$$ 29.3439 0.958624 0.479312 0.877645i $$-0.340886\pi$$
0.479312 + 0.877645i $$0.340886\pi$$
$$938$$ 29.0435 0.948304
$$939$$ −73.9121 −2.41203
$$940$$ 0 0
$$941$$ −28.6375 −0.933556 −0.466778 0.884374i $$-0.654585\pi$$
−0.466778 + 0.884374i $$0.654585\pi$$
$$942$$ 36.0118 1.17333
$$943$$ −16.0508 −0.522685
$$944$$ 166.939 5.43341
$$945$$ 0 0
$$946$$ 15.7685 0.512677
$$947$$ 52.8178 1.71635 0.858174 0.513358i $$-0.171599\pi$$
0.858174 + 0.513358i $$0.171599\pi$$
$$948$$ 25.1998 0.818452
$$949$$ −66.2941 −2.15200
$$950$$ 0 0
$$951$$ −38.2130 −1.23914
$$952$$ −45.9633 −1.48968
$$953$$ −37.1939 −1.20483 −0.602415 0.798183i $$-0.705795\pi$$
−0.602415 + 0.798183i $$0.705795\pi$$
$$954$$ 1.95509 0.0632985
$$955$$ 0 0
$$956$$ −109.388 −3.53787
$$957$$ 8.96239 0.289713
$$958$$ −28.6516 −0.925693
$$959$$ 10.9927 0.354973
$$960$$ 0 0
$$961$$ −3.04491 −0.0982228
$$962$$ −133.289 −4.29740
$$963$$ 36.4142 1.17343
$$964$$ −48.1500 −1.55081
$$965$$ 0 0
$$966$$ 21.1998 0.682093
$$967$$ −4.07125 −0.130923 −0.0654613 0.997855i $$-0.520852\pi$$
−0.0654613 + 0.997855i $$0.520852\pi$$
$$968$$ 8.44358 0.271387
$$969$$ 18.2374 0.585871
$$970$$ 0 0
$$971$$ 0.773377 0.0248188 0.0124094 0.999923i $$-0.496050\pi$$
0.0124094 + 0.999923i $$0.496050\pi$$
$$972$$ −114.832 −3.68324
$$973$$ 6.88717 0.220792
$$974$$ −46.6312 −1.49416
$$975$$ 0 0
$$976$$ −17.3479 −0.555292
$$977$$ 37.8740 1.21170 0.605848 0.795580i $$-0.292834\pi$$
0.605848 + 0.795580i $$0.292834\pi$$
$$978$$ −22.4387 −0.717509
$$979$$ −7.22425 −0.230888
$$980$$ 0 0
$$981$$ −6.90175 −0.220356
$$982$$ −75.8916 −2.42180
$$983$$ 15.5794 0.496907 0.248453 0.968644i $$-0.420078\pi$$
0.248453 + 0.968644i $$0.420078\pi$$
$$984$$ −105.283 −3.35629
$$985$$ 0 0
$$986$$ 52.6009 1.67515
$$987$$ 29.3561 0.934416
$$988$$ 40.6009 1.29169
$$989$$ −18.8265 −0.598649
$$990$$ 0 0
$$991$$ 27.0982 0.860804 0.430402 0.902637i $$-0.358372\pi$$
0.430402 + 0.902637i $$0.358372\pi$$
$$992$$ −84.3327 −2.67756
$$993$$ 15.5369 0.493049
$$994$$ −41.5633 −1.31831
$$995$$ 0 0
$$996$$ −135.938 −4.30737
$$997$$ −50.4060 −1.59637 −0.798187 0.602410i $$-0.794207\pi$$
−0.798187 + 0.602410i $$0.794207\pi$$
$$998$$ 73.5026 2.32668
$$999$$ 3.31406 0.104852
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 1925.2.a.v.1.3 3
5.2 odd 4 1925.2.b.n.1849.6 6
5.3 odd 4 1925.2.b.n.1849.1 6
5.4 even 2 385.2.a.f.1.1 3
15.14 odd 2 3465.2.a.bh.1.3 3
20.19 odd 2 6160.2.a.bn.1.3 3
35.34 odd 2 2695.2.a.g.1.1 3
55.54 odd 2 4235.2.a.q.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 5.4 even 2
1925.2.a.v.1.3 3 1.1 even 1 trivial
1925.2.b.n.1849.1 6 5.3 odd 4
1925.2.b.n.1849.6 6 5.2 odd 4
2695.2.a.g.1.1 3 35.34 odd 2
3465.2.a.bh.1.3 3 15.14 odd 2
4235.2.a.q.1.3 3 55.54 odd 2
6160.2.a.bn.1.3 3 20.19 odd 2