# Properties

 Label 3465.2.a.bh.1.3 Level $3465$ Weight $2$ Character 3465.1 Self dual yes Analytic conductor $27.668$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3465.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.6681643004$$ 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 $$\chi$$ $$=$$ 3465.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +O(q^{10})$$ $$q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +2.67513 q^{10} +1.00000 q^{11} +5.83146 q^{13} -2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} -1.35026 q^{19} +5.15633 q^{20} +2.67513 q^{22} +3.19394 q^{23} +1.00000 q^{25} +15.5999 q^{26} -5.15633 q^{28} +3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -14.5623 q^{34} -1.00000 q^{35} -8.54420 q^{37} -3.61213 q^{38} +8.44358 q^{40} +5.02539 q^{41} +5.89446 q^{43} +5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +1.00000 q^{49} +2.67513 q^{50} +30.0689 q^{52} +0.231548 q^{53} +1.00000 q^{55} -8.44358 q^{56} +9.66291 q^{58} -13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +18.1187 q^{64} +5.83146 q^{65} -10.8568 q^{67} -28.0689 q^{68} -2.67513 q^{70} +15.5369 q^{71} -11.3684 q^{73} -22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} +1.96968 q^{79} +12.2750 q^{80} +13.4436 q^{82} -10.6253 q^{83} -5.44358 q^{85} +15.7685 q^{86} +8.44358 q^{88} -7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} +31.6507 q^{94} -1.35026 q^{95} -0.836381 q^{97} +2.67513 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + O(q^{10})$$ $$3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.67513 1.89160 0.945802 0.324745i $$-0.105279\pi$$
0.945802 + 0.324745i $$0.105279\pi$$
$$3$$ 0 0
$$4$$ 5.15633 2.57816
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 8.44358 2.98526
$$9$$ 0 0
$$10$$ 2.67513 0.845951
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 5.83146 1.61735 0.808677 0.588252i $$-0.200184\pi$$
0.808677 + 0.588252i $$0.200184\pi$$
$$14$$ −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 $$0.729499\pi$$
$$18$$ 0 0
$$19$$ −1.35026 −0.309771 −0.154886 0.987932i $$-0.549501\pi$$
−0.154886 + 0.987932i $$0.549501\pi$$
$$20$$ 5.15633 1.15299
$$21$$ 0 0
$$22$$ 2.67513 0.570340
$$23$$ 3.19394 0.665982 0.332991 0.942930i $$-0.391942\pi$$
0.332991 + 0.942930i $$0.391942\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 15.5999 3.05939
$$27$$ 0 0
$$28$$ −5.15633 −0.974454
$$29$$ 3.61213 0.670755 0.335378 0.942084i $$-0.391136\pi$$
0.335378 + 0.942084i $$0.391136\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −14.5623 −2.49741
$$35$$ −1.00000 −0.169031
$$36$$ 0 0
$$37$$ −8.54420 −1.40466 −0.702329 0.711853i $$-0.747856\pi$$
−0.702329 + 0.711853i $$0.747856\pi$$
$$38$$ −3.61213 −0.585964
$$39$$ 0 0
$$40$$ 8.44358 1.33505
$$41$$ 5.02539 0.784834 0.392417 0.919787i $$-0.371639\pi$$
0.392417 + 0.919787i $$0.371639\pi$$
$$42$$ 0 0
$$43$$ 5.89446 0.898897 0.449448 0.893306i $$-0.351621\pi$$
0.449448 + 0.893306i $$0.351621\pi$$
$$44$$ 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 $$0.168647\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 2.67513 0.378321
$$51$$ 0 0
$$52$$ 30.0689 4.16980
$$53$$ 0.231548 0.0318056 0.0159028 0.999874i $$-0.494938\pi$$
0.0159028 + 0.999874i $$0.494938\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ −8.44358 −1.12832
$$57$$ 0 0
$$58$$ 9.66291 1.26880
$$59$$ −13.5999 −1.77056 −0.885279 0.465061i $$-0.846032\pi$$
−0.885279 + 0.465061i $$0.846032\pi$$
$$60$$ 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
$$63$$ 0 0
$$64$$ 18.1187 2.26484
$$65$$ 5.83146 0.723303
$$66$$ 0 0
$$67$$ −10.8568 −1.32638 −0.663188 0.748453i $$-0.730797\pi$$
−0.663188 + 0.748453i $$0.730797\pi$$
$$68$$ −28.0689 −3.40385
$$69$$ 0 0
$$70$$ −2.67513 −0.319739
$$71$$ 15.5369 1.84389 0.921946 0.387319i $$-0.126599\pi$$
0.921946 + 0.387319i $$0.126599\pi$$
$$72$$ 0 0
$$73$$ −11.3684 −1.33057 −0.665283 0.746591i $$-0.731689\pi$$
−0.665283 + 0.746591i $$0.731689\pi$$
$$74$$ −22.8568 −2.65705
$$75$$ 0 0
$$76$$ −6.96239 −0.798641
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ 1.96968 0.221607 0.110803 0.993842i $$-0.464658\pi$$
0.110803 + 0.993842i $$0.464658\pi$$
$$80$$ 12.2750 1.37239
$$81$$ 0 0
$$82$$ 13.4436 1.48460
$$83$$ −10.6253 −1.16628 −0.583139 0.812372i $$-0.698176\pi$$
−0.583139 + 0.812372i $$0.698176\pi$$
$$84$$ 0 0
$$85$$ −5.44358 −0.590439
$$86$$ 15.7685 1.70036
$$87$$ 0 0
$$88$$ 8.44358 0.900089
$$89$$ −7.22425 −0.765769 −0.382885 0.923796i $$-0.625069\pi$$
−0.382885 + 0.923796i $$0.625069\pi$$
$$90$$ 0 0
$$91$$ −5.83146 −0.611303
$$92$$ 16.4690 1.71701
$$93$$ 0 0
$$94$$ 31.6507 3.26452
$$95$$ −1.35026 −0.138534
$$96$$ 0 0
$$97$$ −0.836381 −0.0849216 −0.0424608 0.999098i $$-0.513520\pi$$
−0.0424608 + 0.999098i $$0.513520\pi$$
$$98$$ 2.67513 0.270229
$$99$$ 0 0
$$100$$ 5.15633 0.515633
$$101$$ −7.41327 −0.737648 −0.368824 0.929499i $$-0.620239\pi$$
−0.368824 + 0.929499i $$0.620239\pi$$
$$102$$ 0 0
$$103$$ 4.21933 0.415743 0.207871 0.978156i $$-0.433346\pi$$
0.207871 + 0.978156i $$0.433346\pi$$
$$104$$ 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 $$0.311700\pi$$
$$108$$ 0 0
$$109$$ −2.18664 −0.209442 −0.104721 0.994502i $$-0.533395\pi$$
−0.104721 + 0.994502i $$0.533395\pi$$
$$110$$ 2.67513 0.255064
$$111$$ 0 0
$$112$$ −12.2750 −1.15988
$$113$$ 9.35026 0.879599 0.439799 0.898096i $$-0.355050\pi$$
0.439799 + 0.898096i $$0.355050\pi$$
$$114$$ 0 0
$$115$$ 3.19394 0.297836
$$116$$ 18.6253 1.72932
$$117$$ 0 0
$$118$$ −36.3815 −3.34919
$$119$$ 5.44358 0.499012
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −3.78067 −0.342286
$$123$$ 0 0
$$124$$ −27.2628 −2.44827
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.9624 −1.50517 −0.752584 0.658496i $$-0.771193\pi$$
−0.752584 + 0.658496i $$0.771193\pi$$
$$128$$ 16.5696 1.46456
$$129$$ 0 0
$$130$$ 15.5999 1.36820
$$131$$ 9.92478 0.867132 0.433566 0.901122i $$-0.357255\pi$$
0.433566 + 0.901122i $$0.357255\pi$$
$$132$$ 0 0
$$133$$ 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 $$0.344404\pi$$
$$138$$ 0 0
$$139$$ 6.88717 0.584162 0.292081 0.956394i $$-0.405652\pi$$
0.292081 + 0.956394i $$0.405652\pi$$
$$140$$ −5.15633 −0.435789
$$141$$ 0 0
$$142$$ 41.5633 3.48791
$$143$$ 5.83146 0.487651
$$144$$ 0 0
$$145$$ 3.61213 0.299971
$$146$$ −30.4119 −2.51690
$$147$$ 0 0
$$148$$ −44.0567 −3.62144
$$149$$ −22.8119 −1.86883 −0.934414 0.356190i $$-0.884076\pi$$
−0.934414 + 0.356190i $$0.884076\pi$$
$$150$$ 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
$$153$$ 0 0
$$154$$ −2.67513 −0.215568
$$155$$ −5.28726 −0.424683
$$156$$ 0 0
$$157$$ −5.42548 −0.433001 −0.216500 0.976283i $$-0.569464\pi$$
−0.216500 + 0.976283i $$0.569464\pi$$
$$158$$ 5.26916 0.419192
$$159$$ 0 0
$$160$$ 15.9502 1.26097
$$161$$ −3.19394 −0.251717
$$162$$ 0 0
$$163$$ 3.38058 0.264787 0.132394 0.991197i $$-0.457734\pi$$
0.132394 + 0.991197i $$0.457734\pi$$
$$164$$ 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 $$0.643692\pi$$
$$168$$ 0 0
$$169$$ 21.0059 1.61584
$$170$$ −14.5623 −1.11688
$$171$$ 0 0
$$172$$ 30.3938 2.31750
$$173$$ 8.98049 0.682774 0.341387 0.939923i $$-0.389103\pi$$
0.341387 + 0.939923i $$0.389103\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 12.2750 0.925266
$$177$$ 0 0
$$178$$ −19.3258 −1.44853
$$179$$ 26.2374 1.96108 0.980539 0.196326i $$-0.0629010\pi$$
0.980539 + 0.196326i $$0.0629010\pi$$
$$180$$ 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
$$183$$ 0 0
$$184$$ 26.9683 1.98813
$$185$$ −8.54420 −0.628182
$$186$$ 0 0
$$187$$ −5.44358 −0.398074
$$188$$ 61.0068 4.44938
$$189$$ 0 0
$$190$$ −3.61213 −0.262051
$$191$$ −11.1998 −0.810390 −0.405195 0.914230i $$-0.632796\pi$$
−0.405195 + 0.914230i $$0.632796\pi$$
$$192$$ 0 0
$$193$$ 0.604833 0.0435368 0.0217684 0.999763i $$-0.493070\pi$$
0.0217684 + 0.999763i $$0.493070\pi$$
$$194$$ −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 $$0.683558\pi$$
$$198$$ 0 0
$$199$$ −12.5623 −0.890518 −0.445259 0.895402i $$-0.646888\pi$$
−0.445259 + 0.895402i $$0.646888\pi$$
$$200$$ 8.44358 0.597051
$$201$$ 0 0
$$202$$ −19.8315 −1.39534
$$203$$ −3.61213 −0.253522
$$204$$ 0 0
$$205$$ 5.02539 0.350989
$$206$$ 11.2873 0.786421
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 30.8627 2.10973
$$215$$ 5.89446 0.401999
$$216$$ 0 0
$$217$$ 5.28726 0.358922
$$218$$ −5.84955 −0.396182
$$219$$ 0 0
$$220$$ 5.15633 0.347639
$$221$$ −31.7440 −2.13533
$$222$$ 0 0
$$223$$ −7.78067 −0.521032 −0.260516 0.965469i $$-0.583893\pi$$
−0.260516 + 0.965469i $$0.583893\pi$$
$$224$$ −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 $$0.387287\pi$$
$$228$$ 0 0
$$229$$ −29.4518 −1.94623 −0.973116 0.230316i $$-0.926024\pi$$
−0.973116 + 0.230316i $$0.926024\pi$$
$$230$$ 8.54420 0.563388
$$231$$ 0 0
$$232$$ 30.4993 2.00238
$$233$$ 8.73084 0.571976 0.285988 0.958233i $$-0.407678\pi$$
0.285988 + 0.958233i $$0.407678\pi$$
$$234$$ 0 0
$$235$$ 11.8315 0.771799
$$236$$ −70.1255 −4.56478
$$237$$ 0 0
$$238$$ 14.5623 0.943933
$$239$$ 21.2144 1.37225 0.686123 0.727486i $$-0.259311\pi$$
0.686123 + 0.727486i $$0.259311\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ −7.28726 −0.466519
$$245$$ 1.00000 0.0638877
$$246$$ 0 0
$$247$$ −7.87399 −0.501010
$$248$$ −44.6434 −2.83486
$$249$$ 0 0
$$250$$ 2.67513 0.169190
$$251$$ −1.87636 −0.118435 −0.0592174 0.998245i $$-0.518861\pi$$
−0.0592174 + 0.998245i $$0.518861\pi$$
$$252$$ 0 0
$$253$$ 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 $$0.821263\pi$$
$$258$$ 0 0
$$259$$ 8.54420 0.530911
$$260$$ 30.0689 1.86479
$$261$$ 0 0
$$262$$ 26.5501 1.64027
$$263$$ −12.8119 −0.790018 −0.395009 0.918677i $$-0.629259\pi$$
−0.395009 + 0.918677i $$0.629259\pi$$
$$264$$ 0 0
$$265$$ 0.231548 0.0142239
$$266$$ 3.61213 0.221474
$$267$$ 0 0
$$268$$ −55.9814 −3.41961
$$269$$ 6.26187 0.381793 0.190896 0.981610i $$-0.438861\pi$$
0.190896 + 0.981610i $$0.438861\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ 29.4069 1.77654
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −8.35756 −0.502157 −0.251078 0.967967i $$-0.580785\pi$$
−0.251078 + 0.967967i $$0.580785\pi$$
$$278$$ 18.4241 1.10500
$$279$$ 0 0
$$280$$ −8.44358 −0.504601
$$281$$ 8.44851 0.503996 0.251998 0.967728i $$-0.418912\pi$$
0.251998 + 0.967728i $$0.418912\pi$$
$$282$$ 0 0
$$283$$ 0.836381 0.0497177 0.0248588 0.999691i $$-0.492086\pi$$
0.0248588 + 0.999691i $$0.492086\pi$$
$$284$$ 80.1133 4.75385
$$285$$ 0 0
$$286$$ 15.5999 0.922442
$$287$$ −5.02539 −0.296640
$$288$$ 0 0
$$289$$ 12.6326 0.743094
$$290$$ 9.66291 0.567426
$$291$$ 0 0
$$292$$ −58.6190 −3.43042
$$293$$ 2.71862 0.158824 0.0794118 0.996842i $$-0.474696\pi$$
0.0794118 + 0.996842i $$0.474696\pi$$
$$294$$ 0 0
$$295$$ −13.5999 −0.791817
$$296$$ −72.1436 −4.19326
$$297$$ 0 0
$$298$$ −61.0249 −3.53508
$$299$$ 18.6253 1.07713
$$300$$ 0 0
$$301$$ −5.89446 −0.339751
$$302$$ −8.68006 −0.499481
$$303$$ 0 0
$$304$$ −16.5745 −0.950614
$$305$$ −1.41327 −0.0809234
$$306$$ 0 0
$$307$$ −8.36344 −0.477326 −0.238663 0.971102i $$-0.576709\pi$$
−0.238663 + 0.971102i $$0.576709\pi$$
$$308$$ −5.15633 −0.293809
$$309$$ 0 0
$$310$$ −14.1441 −0.803331
$$311$$ −4.43629 −0.251559 −0.125779 0.992058i $$-0.540143\pi$$
−0.125779 + 0.992058i $$0.540143\pi$$
$$312$$ 0 0
$$313$$ 29.7889 1.68377 0.841885 0.539658i $$-0.181446\pi$$
0.841885 + 0.539658i $$0.181446\pi$$
$$314$$ −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 $$0.642370\pi$$
$$318$$ 0 0
$$319$$ 3.61213 0.202240
$$320$$ 18.1187 1.01287
$$321$$ 0 0
$$322$$ −8.54420 −0.476150
$$323$$ 7.35026 0.408980
$$324$$ 0 0
$$325$$ 5.83146 0.323471
$$326$$ 9.04349 0.500873
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ −30.1622 −1.65040
$$335$$ −10.8568 −0.593173
$$336$$ 0 0
$$337$$ −15.8700 −0.864495 −0.432248 0.901755i $$-0.642279\pi$$
−0.432248 + 0.901755i $$0.642279\pi$$
$$338$$ 56.1935 3.05652
$$339$$ 0 0
$$340$$ −28.0689 −1.52225
$$341$$ −5.28726 −0.286321
$$342$$ 0 0
$$343$$ −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 $$0.558352\pi$$
$$348$$ 0 0
$$349$$ 26.7489 1.43184 0.715919 0.698183i $$-0.246008\pi$$
0.715919 + 0.698183i $$0.246008\pi$$
$$350$$ −2.67513 −0.142992
$$351$$ 0 0
$$352$$ 15.9502 0.850147
$$353$$ 16.8627 0.897512 0.448756 0.893654i $$-0.351867\pi$$
0.448756 + 0.893654i $$0.351867\pi$$
$$354$$ 0 0
$$355$$ 15.5369 0.824613
$$356$$ −37.2506 −1.97428
$$357$$ 0 0
$$358$$ 70.1886 3.70958
$$359$$ 3.79289 0.200181 0.100091 0.994978i $$-0.468087\pi$$
0.100091 + 0.994978i $$0.468087\pi$$
$$360$$ 0 0
$$361$$ −17.1768 −0.904042
$$362$$ −29.9610 −1.57471
$$363$$ 0 0
$$364$$ −30.0689 −1.57604
$$365$$ −11.3684 −0.595047
$$366$$ 0 0
$$367$$ −6.36977 −0.332500 −0.166250 0.986084i $$-0.553166\pi$$
−0.166250 + 0.986084i $$0.553166\pi$$
$$368$$ 39.2057 2.04374
$$369$$ 0 0
$$370$$ −22.8568 −1.18827
$$371$$ −0.231548 −0.0120214
$$372$$ 0 0
$$373$$ −21.3317 −1.10451 −0.552257 0.833674i $$-0.686233\pi$$
−0.552257 + 0.833674i $$0.686233\pi$$
$$374$$ −14.5623 −0.752998
$$375$$ 0 0
$$376$$ 99.8999 5.15194
$$377$$ 21.0640 1.08485
$$378$$ 0 0
$$379$$ 24.7875 1.27325 0.636624 0.771174i $$-0.280330\pi$$
0.636624 + 0.771174i $$0.280330\pi$$
$$380$$ −6.96239 −0.357163
$$381$$ 0 0
$$382$$ −29.9610 −1.53294
$$383$$ 5.45817 0.278900 0.139450 0.990229i $$-0.455467\pi$$
0.139450 + 0.990229i $$0.455467\pi$$
$$384$$ 0 0
$$385$$ −1.00000 −0.0509647
$$386$$ 1.61801 0.0823544
$$387$$ 0 0
$$388$$ −4.31265 −0.218942
$$389$$ 13.7235 0.695811 0.347906 0.937530i $$-0.386893\pi$$
0.347906 + 0.937530i $$0.386893\pi$$
$$390$$ 0 0
$$391$$ −17.3865 −0.879271
$$392$$ 8.44358 0.426465
$$393$$ 0 0
$$394$$ −40.9438 −2.06272
$$395$$ 1.96968 0.0991055
$$396$$ 0 0
$$397$$ 2.11142 0.105969 0.0529846 0.998595i $$-0.483127\pi$$
0.0529846 + 0.998595i $$0.483127\pi$$
$$398$$ −33.6058 −1.68451
$$399$$ 0 0
$$400$$ 12.2750 0.613752
$$401$$ −19.1490 −0.956257 −0.478128 0.878290i $$-0.658685\pi$$
−0.478128 + 0.878290i $$0.658685\pi$$
$$402$$ 0 0
$$403$$ −30.8324 −1.53587
$$404$$ −38.2252 −1.90178
$$405$$ 0 0
$$406$$ −9.66291 −0.479562
$$407$$ −8.54420 −0.423520
$$408$$ 0 0
$$409$$ −18.6883 −0.924077 −0.462039 0.886860i $$-0.652882\pi$$
−0.462039 + 0.886860i $$0.652882\pi$$
$$410$$ 13.4436 0.663931
$$411$$ 0 0
$$412$$ 21.7562 1.07185
$$413$$ 13.5999 0.669208
$$414$$ 0 0
$$415$$ −10.6253 −0.521575
$$416$$ 93.0127 4.56032
$$417$$ 0 0
$$418$$ −3.61213 −0.176675
$$419$$ 0.773377 0.0377819 0.0188910 0.999822i $$-0.493986\pi$$
0.0188910 + 0.999822i $$0.493986\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ 1.95509 0.0949478
$$425$$ −5.44358 −0.264053
$$426$$ 0 0
$$427$$ 1.41327 0.0683927
$$428$$ 59.4880 2.87546
$$429$$ 0 0
$$430$$ 15.7685 0.760422
$$431$$ 24.7308 1.19124 0.595621 0.803265i $$-0.296906\pi$$
0.595621 + 0.803265i $$0.296906\pi$$
$$432$$ 0 0
$$433$$ −18.5599 −0.891933 −0.445967 0.895050i $$-0.647140\pi$$
−0.445967 + 0.895050i $$0.647140\pi$$
$$434$$ 14.1441 0.678939
$$435$$ 0 0
$$436$$ −11.2750 −0.539976
$$437$$ −4.31265 −0.206302
$$438$$ 0 0
$$439$$ −1.42548 −0.0680347 −0.0340173 0.999421i $$-0.510830\pi$$
−0.0340173 + 0.999421i $$0.510830\pi$$
$$440$$ 8.44358 0.402532
$$441$$ 0 0
$$442$$ −84.9194 −4.03920
$$443$$ −40.1925 −1.90960 −0.954802 0.297242i $$-0.903933\pi$$
−0.954802 + 0.297242i $$0.903933\pi$$
$$444$$ 0 0
$$445$$ −7.22425 −0.342462
$$446$$ −20.8143 −0.985586
$$447$$ 0 0
$$448$$ −18.1187 −0.856029
$$449$$ −12.6556 −0.597256 −0.298628 0.954370i $$-0.596529\pi$$
−0.298628 + 0.954370i $$0.596529\pi$$
$$450$$ 0 0
$$451$$ 5.02539 0.236636
$$452$$ 48.2130 2.26775
$$453$$ 0 0
$$454$$ 27.9511 1.31181
$$455$$ −5.83146 −0.273383
$$456$$ 0 0
$$457$$ −0.544198 −0.0254565 −0.0127283 0.999919i $$-0.504052\pi$$
−0.0127283 + 0.999919i $$0.504052\pi$$
$$458$$ −78.7875 −3.68150
$$459$$ 0 0
$$460$$ 16.4690 0.767870
$$461$$ 11.5755 0.539123 0.269562 0.962983i $$-0.413121\pi$$
0.269562 + 0.962983i $$0.413121\pi$$
$$462$$ 0 0
$$463$$ 23.7948 1.10584 0.552919 0.833235i $$-0.313514\pi$$
0.552919 + 0.833235i $$0.313514\pi$$
$$464$$ 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 $$0.480339\pi$$
$$468$$ 0 0
$$469$$ 10.8568 0.501323
$$470$$ 31.6507 1.45994
$$471$$ 0 0
$$472$$ −114.832 −5.28557
$$473$$ 5.89446 0.271028
$$474$$ 0 0
$$475$$ −1.35026 −0.0619543
$$476$$ 28.0689 1.28654
$$477$$ 0 0
$$478$$ 56.7513 2.59574
$$479$$ 10.7104 0.489369 0.244685 0.969603i $$-0.421316\pi$$
0.244685 + 0.969603i $$0.421316\pi$$
$$480$$ 0 0
$$481$$ −49.8251 −2.27183
$$482$$ −24.9805 −1.13783
$$483$$ 0 0
$$484$$ 5.15633 0.234378
$$485$$ −0.836381 −0.0379781
$$486$$ 0 0
$$487$$ 17.4314 0.789891 0.394945 0.918705i $$-0.370764\pi$$
0.394945 + 0.918705i $$0.370764\pi$$
$$488$$ −11.9330 −0.540183
$$489$$ 0 0
$$490$$ 2.67513 0.120850
$$491$$ 28.3693 1.28029 0.640145 0.768254i $$-0.278874\pi$$
0.640145 + 0.768254i $$0.278874\pi$$
$$492$$ 0 0
$$493$$ −19.6629 −0.885573
$$494$$ −21.0640 −0.947712
$$495$$ 0 0
$$496$$ −64.9013 −2.91415
$$497$$ −15.5369 −0.696925
$$498$$ 0 0
$$499$$ 27.4763 1.23001 0.615003 0.788524i $$-0.289155\pi$$
0.615003 + 0.788524i $$0.289155\pi$$
$$500$$ 5.15633 0.230598
$$501$$ 0 0
$$502$$ −5.01951 −0.224032
$$503$$ −20.2981 −0.905046 −0.452523 0.891753i $$-0.649476\pi$$
−0.452523 + 0.891753i $$0.649476\pi$$
$$504$$ 0 0
$$505$$ −7.41327 −0.329886
$$506$$ 8.54420 0.379836
$$507$$ 0 0
$$508$$ −87.4636 −3.88057
$$509$$ 24.2619 1.07539 0.537694 0.843140i $$-0.319296\pi$$
0.537694 + 0.843140i $$0.319296\pi$$
$$510$$ 0 0
$$511$$ 11.3684 0.502907
$$512$$ −11.5017 −0.508306
$$513$$ 0 0
$$514$$ −72.6009 −3.20229
$$515$$ 4.21933 0.185926
$$516$$ 0 0
$$517$$ 11.8315 0.520347
$$518$$ 22.8568 1.00427
$$519$$ 0 0
$$520$$ 49.2384 2.15925
$$521$$ −2.20123 −0.0964377 −0.0482188 0.998837i $$-0.515354\pi$$
−0.0482188 + 0.998837i $$0.515354\pi$$
$$522$$ 0 0
$$523$$ −22.1378 −0.968017 −0.484008 0.875063i $$-0.660820\pi$$
−0.484008 + 0.875063i $$0.660820\pi$$
$$524$$ 51.1754 2.23561
$$525$$ 0 0
$$526$$ −34.2736 −1.49440
$$527$$ 28.7816 1.25375
$$528$$ 0 0
$$529$$ −12.7988 −0.556468
$$530$$ 0.619421 0.0269059
$$531$$ 0 0
$$532$$ 6.96239 0.301858
$$533$$ 29.3054 1.26936
$$534$$ 0 0
$$535$$ 11.5369 0.498784
$$536$$ −91.6707 −3.95957
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ −86.8261 −3.72264
$$545$$ −2.18664 −0.0936655
$$546$$ 0 0
$$547$$ 21.3766 0.913998 0.456999 0.889467i $$-0.348924\pi$$
0.456999 + 0.889467i $$0.348924\pi$$
$$548$$ 56.6820 2.42133
$$549$$ 0 0
$$550$$ 2.67513 0.114068
$$551$$ −4.87732 −0.207781
$$552$$ 0 0
$$553$$ −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 $$0.437601\pi$$
$$558$$ 0 0
$$559$$ 34.3733 1.45384
$$560$$ −12.2750 −0.518715
$$561$$ 0 0
$$562$$ 22.6009 0.953360
$$563$$ 9.79877 0.412969 0.206484 0.978450i $$-0.433798\pi$$
0.206484 + 0.978450i $$0.433798\pi$$
$$564$$ 0 0
$$565$$ 9.35026 0.393368
$$566$$ 2.23743 0.0940461
$$567$$ 0 0
$$568$$ 131.187 5.50449
$$569$$ 33.5125 1.40492 0.702458 0.711725i $$-0.252086\pi$$
0.702458 + 0.711725i $$0.252086\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ −13.4436 −0.561124
$$575$$ 3.19394 0.133196
$$576$$ 0 0
$$577$$ −14.8510 −0.618254 −0.309127 0.951021i $$-0.600037\pi$$
−0.309127 + 0.951021i $$0.600037\pi$$
$$578$$ 33.7938 1.40564
$$579$$ 0 0
$$580$$ 18.6253 0.773374
$$581$$ 10.6253 0.440812
$$582$$ 0 0
$$583$$ 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 $$0.598758\pi$$
$$588$$ 0 0
$$589$$ 7.13918 0.294165
$$590$$ −36.3815 −1.49780
$$591$$ 0 0
$$592$$ −104.880 −4.31056
$$593$$ 27.4191 1.12597 0.562985 0.826467i $$-0.309653\pi$$
0.562985 + 0.826467i $$0.309653\pi$$
$$594$$ 0 0
$$595$$ 5.44358 0.223165
$$596$$ −117.626 −4.81814
$$597$$ 0 0
$$598$$ 49.8251 2.03750
$$599$$ −11.3258 −0.462761 −0.231380 0.972863i $$-0.574324\pi$$
−0.231380 + 0.972863i $$0.574324\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ −16.7308 −0.680768
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ 17.7235 0.719377 0.359688 0.933073i $$-0.382883\pi$$
0.359688 + 0.933073i $$0.382883\pi$$
$$608$$ −21.5369 −0.873437
$$609$$ 0 0
$$610$$ −3.78067 −0.153075
$$611$$ 68.9946 2.79122
$$612$$ 0 0
$$613$$ 22.2941 0.900450 0.450225 0.892915i $$-0.351344\pi$$
0.450225 + 0.892915i $$0.351344\pi$$
$$614$$ −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 $$0.285964\pi$$
$$618$$ 0 0
$$619$$ 32.4119 1.30274 0.651371 0.758759i $$-0.274194\pi$$
0.651371 + 0.758759i $$0.274194\pi$$
$$620$$ −27.2628 −1.09490
$$621$$ 0 0
$$622$$ −11.8677 −0.475850
$$623$$ 7.22425 0.289434
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 79.6893 3.18502
$$627$$ 0 0
$$628$$ −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
$$633$$ 0 0
$$634$$ −41.1998 −1.63625
$$635$$ −16.9624 −0.673132
$$636$$ 0 0
$$637$$ 5.83146 0.231051
$$638$$ 9.66291 0.382558
$$639$$ 0 0
$$640$$ 16.5696 0.654971
$$641$$ −19.4460 −0.768069 −0.384034 0.923319i $$-0.625466\pi$$
−0.384034 + 0.923319i $$0.625466\pi$$
$$642$$ 0 0
$$643$$ 5.29314 0.208741 0.104370 0.994538i $$-0.466717\pi$$
0.104370 + 0.994538i $$0.466717\pi$$
$$644$$ −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 $$0.742344\pi$$
$$648$$ 0 0
$$649$$ −13.5999 −0.533843
$$650$$ 15.5999 0.611879
$$651$$ 0 0
$$652$$ 17.4314 0.682665
$$653$$ −27.7988 −1.08785 −0.543925 0.839134i $$-0.683062\pi$$
−0.543925 + 0.839134i $$0.683062\pi$$
$$654$$ 0 0
$$655$$ 9.92478 0.387793
$$656$$ 61.6869 2.40847
$$657$$ 0 0
$$658$$ −31.6507 −1.23387
$$659$$ −19.6180 −0.764209 −0.382105 0.924119i $$-0.624801\pi$$
−0.382105 + 0.924119i $$0.624801\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ −89.7156 −3.48164
$$665$$ 1.35026 0.0523609
$$666$$ 0 0
$$667$$ 11.5369 0.446711
$$668$$ −58.1378 −2.24942
$$669$$ 0 0
$$670$$ −29.0435 −1.12205
$$671$$ −1.41327 −0.0545585
$$672$$ 0 0
$$673$$ −21.0679 −0.812109 −0.406054 0.913849i $$-0.633096\pi$$
−0.406054 + 0.913849i $$0.633096\pi$$
$$674$$ −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 $$0.730847\pi$$
$$678$$ 0 0
$$679$$ 0.836381 0.0320973
$$680$$ −45.9633 −1.76261
$$681$$ 0 0
$$682$$ −14.1441 −0.541606
$$683$$ −33.7802 −1.29256 −0.646282 0.763099i $$-0.723677\pi$$
−0.646282 + 0.763099i $$0.723677\pi$$
$$684$$ 0 0
$$685$$ 10.9927 0.420010
$$686$$ −2.67513 −0.102137
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ −18.1681 −0.689651
$$695$$ 6.88717 0.261245
$$696$$ 0 0
$$697$$ −27.3561 −1.03619
$$698$$ 71.5569 2.70847
$$699$$ 0 0
$$700$$ −5.15633 −0.194891
$$701$$ −40.5256 −1.53063 −0.765316 0.643655i $$-0.777417\pi$$
−0.765316 + 0.643655i $$0.777417\pi$$
$$702$$ 0 0
$$703$$ 11.5369 0.435123
$$704$$ 18.1187 0.682875
$$705$$ 0 0
$$706$$ 45.1100 1.69774
$$707$$ 7.41327 0.278805
$$708$$ 0 0
$$709$$ −0.850969 −0.0319588 −0.0159794 0.999872i $$-0.505087\pi$$
−0.0159794 + 0.999872i $$0.505087\pi$$
$$710$$ 41.5633 1.55984
$$711$$ 0 0
$$712$$ −60.9986 −2.28602
$$713$$ −16.8872 −0.632429
$$714$$ 0 0
$$715$$ 5.83146 0.218084
$$716$$ 135.289 5.05598
$$717$$ 0 0
$$718$$ 10.1465 0.378663
$$719$$ −22.5769 −0.841976 −0.420988 0.907066i $$-0.638317\pi$$
−0.420988 + 0.907066i $$0.638317\pi$$
$$720$$ 0 0
$$721$$ −4.21933 −0.157136
$$722$$ −45.9502 −1.71009
$$723$$ 0 0
$$724$$ −57.7499 −2.14626
$$725$$ 3.61213 0.134151
$$726$$ 0 0
$$727$$ 12.5174 0.464244 0.232122 0.972687i $$-0.425433\pi$$
0.232122 + 0.972687i $$0.425433\pi$$
$$728$$ −49.2384 −1.82490
$$729$$ 0 0
$$730$$ −30.4119 −1.12559
$$731$$ −32.0870 −1.18678
$$732$$ 0 0
$$733$$ 16.6678 0.615641 0.307820 0.951445i $$-0.400400\pi$$
0.307820 + 0.951445i $$0.400400\pi$$
$$734$$ −17.0400 −0.628957
$$735$$ 0 0
$$736$$ 50.9438 1.87781
$$737$$ −10.8568 −0.399917
$$738$$ 0 0
$$739$$ 42.7005 1.57076 0.785382 0.619011i $$-0.212467\pi$$
0.785382 + 0.619011i $$0.212467\pi$$
$$740$$ −44.0567 −1.61956
$$741$$ 0 0
$$742$$ −0.619421 −0.0227397
$$743$$ 19.6873 0.722259 0.361129 0.932516i $$-0.382391\pi$$
0.361129 + 0.932516i $$0.382391\pi$$
$$744$$ 0 0
$$745$$ −22.8119 −0.835765
$$746$$ −57.0651 −2.08930
$$747$$ 0 0
$$748$$ −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
$$753$$ 0 0
$$754$$ 56.3488 2.05210
$$755$$ −3.24472 −0.118088
$$756$$ 0 0
$$757$$ −40.5863 −1.47513 −0.737567 0.675274i $$-0.764025\pi$$
−0.737567 + 0.675274i $$0.764025\pi$$
$$758$$ 66.3098 2.40848
$$759$$ 0 0
$$760$$ −11.4010 −0.413559
$$761$$ −21.8472 −0.791960 −0.395980 0.918259i $$-0.629595\pi$$
−0.395980 + 0.918259i $$0.629595\pi$$
$$762$$ 0 0
$$763$$ 2.18664 0.0791618
$$764$$ −57.7499 −2.08932
$$765$$ 0 0
$$766$$ 14.6013 0.527567
$$767$$ −79.3073 −2.86362
$$768$$ 0 0
$$769$$ 45.2892 1.63317 0.816585 0.577226i $$-0.195865\pi$$
0.816585 + 0.577226i $$0.195865\pi$$
$$770$$ −2.67513 −0.0964050
$$771$$ 0 0
$$772$$ 3.11871 0.112245
$$773$$ 33.8153 1.21625 0.608125 0.793841i $$-0.291922\pi$$
0.608125 + 0.793841i $$0.291922\pi$$
$$774$$ 0 0
$$775$$ −5.28726 −0.189924
$$776$$ −7.06205 −0.253513
$$777$$ 0 0
$$778$$ 36.7123 1.31620
$$779$$ −6.78560 −0.243119
$$780$$ 0 0
$$781$$ 15.5369 0.555954
$$782$$ −46.5111 −1.66323
$$783$$ 0 0
$$784$$ 12.2750 0.438394
$$785$$ −5.42548 −0.193644
$$786$$ 0 0
$$787$$ 1.27504 0.0454502 0.0227251 0.999742i $$-0.492766\pi$$
0.0227251 + 0.999742i $$0.492766\pi$$
$$788$$ −78.9194 −2.81139
$$789$$ 0 0
$$790$$ 5.26916 0.187468
$$791$$ −9.35026 −0.332457
$$792$$ 0 0
$$793$$ −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 $$0.228525\pi$$
$$798$$ 0 0
$$799$$ −64.4055 −2.27850
$$800$$ 15.9502 0.563924
$$801$$ 0 0
$$802$$ −51.2262 −1.80886
$$803$$ −11.3684 −0.401181
$$804$$ 0 0
$$805$$ −3.19394 −0.112571
$$806$$ −82.4807 −2.90526
$$807$$ 0 0
$$808$$ −62.5945 −2.20207
$$809$$ 14.7151 0.517356 0.258678 0.965964i $$-0.416713\pi$$
0.258678 + 0.965964i $$0.416713\pi$$
$$810$$ 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
$$813$$ 0 0
$$814$$ −22.8568 −0.801132
$$815$$ 3.38058 0.118417
$$816$$ 0 0
$$817$$ −7.95906 −0.278452
$$818$$ −49.9937 −1.74799
$$819$$ 0 0
$$820$$ 25.9126 0.904906
$$821$$ −2.64974 −0.0924765 −0.0462383 0.998930i $$-0.514723\pi$$
−0.0462383 + 0.998930i $$0.514723\pi$$
$$822$$ 0 0
$$823$$ 5.76845 0.201076 0.100538 0.994933i $$-0.467944\pi$$
0.100538 + 0.994933i $$0.467944\pi$$
$$824$$ 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 $$0.424628\pi$$
$$828$$ 0 0
$$829$$ −4.70052 −0.163256 −0.0816280 0.996663i $$-0.526012\pi$$
−0.0816280 + 0.996663i $$0.526012\pi$$
$$830$$ −28.4241 −0.986614
$$831$$ 0 0
$$832$$ 105.658 3.66305
$$833$$ −5.44358 −0.188609
$$834$$ 0 0
$$835$$ −11.2750 −0.390189
$$836$$ −6.96239 −0.240799
$$837$$ 0 0
$$838$$ 2.06888 0.0714684
$$839$$ 38.8045 1.33968 0.669839 0.742506i $$-0.266363\pi$$
0.669839 + 0.742506i $$0.266363\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ −28.1417 −0.969828
$$843$$ 0 0
$$844$$ −22.8872 −0.787809
$$845$$ 21.0059 0.722624
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 2.84226 0.0976036
$$849$$ 0 0
$$850$$ −14.5623 −0.499483
$$851$$ −27.2896 −0.935476
$$852$$ 0 0
$$853$$ 20.6824 0.708153 0.354076 0.935217i $$-0.384795\pi$$
0.354076 + 0.935217i $$0.384795\pi$$
$$854$$ 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 $$0.648565\pi$$
$$858$$ 0 0
$$859$$ −8.51151 −0.290409 −0.145205 0.989402i $$-0.546384\pi$$
−0.145205 + 0.989402i $$0.546384\pi$$
$$860$$ 30.3938 1.03642
$$861$$ 0 0
$$862$$ 66.1582 2.25336
$$863$$ −7.56722 −0.257591 −0.128796 0.991671i $$-0.541111\pi$$
−0.128796 + 0.991671i $$0.541111\pi$$
$$864$$ 0 0
$$865$$ 8.98049 0.305346
$$866$$ −49.6502 −1.68718
$$867$$ 0 0
$$868$$ 27.2628 0.925360
$$869$$ 1.96968 0.0668169
$$870$$ 0 0
$$871$$ −63.3112 −2.14522
$$872$$ −18.4631 −0.625239
$$873$$ 0 0
$$874$$ −11.5369 −0.390242
$$875$$ −1.00000 −0.0338062
$$876$$ 0 0
$$877$$ 17.2955 0.584028 0.292014 0.956414i $$-0.405675\pi$$
0.292014 + 0.956414i $$0.405675\pi$$
$$878$$ −3.81336 −0.128695
$$879$$ 0 0
$$880$$ 12.2750 0.413791
$$881$$ −20.4504 −0.688992 −0.344496 0.938788i $$-0.611950\pi$$
−0.344496 + 0.938788i $$0.611950\pi$$
$$882$$ 0 0
$$883$$ −49.6589 −1.67116 −0.835578 0.549371i $$-0.814867\pi$$
−0.835578 + 0.549371i $$0.814867\pi$$
$$884$$ −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 $$0.209613\pi$$
$$888$$ 0 0
$$889$$ 16.9624 0.568900
$$890$$ −19.3258 −0.647803
$$891$$ 0 0
$$892$$ −40.1197 −1.34331
$$893$$ −15.9756 −0.534602
$$894$$ 0 0
$$895$$ 26.2374 0.877020
$$896$$ −16.5696 −0.553551
$$897$$ 0 0
$$898$$ −33.8554 −1.12977
$$899$$ −19.0982 −0.636962
$$900$$ 0 0
$$901$$ −1.26045 −0.0419917
$$902$$ 13.4436 0.447622
$$903$$ 0 0
$$904$$ 78.9497 2.62583
$$905$$ −11.1998 −0.372294
$$906$$ 0 0
$$907$$ 14.4591 0.480107 0.240054 0.970760i $$-0.422835\pi$$
0.240054 + 0.970760i $$0.422835\pi$$
$$908$$ 53.8759 1.78793
$$909$$ 0 0
$$910$$ −15.5999 −0.517132
$$911$$ 31.5369 1.04486 0.522432 0.852681i $$-0.325025\pi$$
0.522432 + 0.852681i $$0.325025\pi$$
$$912$$ 0 0
$$913$$ −10.6253 −0.351646
$$914$$ −1.45580 −0.0481536
$$915$$ 0 0
$$916$$ −151.863 −5.01770
$$917$$ −9.92478 −0.327745
$$918$$ 0 0
$$919$$ 5.26328 0.173620 0.0868098 0.996225i $$-0.472333\pi$$
0.0868098 + 0.996225i $$0.472333\pi$$
$$920$$ 26.9683 0.889117
$$921$$ 0 0
$$922$$ 30.9659 1.01981
$$923$$ 90.6028 2.98223
$$924$$ 0 0
$$925$$ −8.54420 −0.280932
$$926$$ 63.6542 2.09181
$$927$$ 0 0
$$928$$ 57.6140 1.89127
$$929$$ −26.0508 −0.854699 −0.427349 0.904087i $$-0.640553\pi$$
−0.427349 + 0.904087i $$0.640553\pi$$
$$930$$ 0 0
$$931$$ −1.35026 −0.0442530
$$932$$ 45.0191 1.47465
$$933$$ 0 0
$$934$$ 7.13681 0.233524
$$935$$ −5.44358 −0.178024
$$936$$ 0 0
$$937$$ −29.3439 −0.958624 −0.479312 0.877645i $$-0.659114\pi$$
−0.479312 + 0.877645i $$0.659114\pi$$
$$938$$ 29.0435 0.948304
$$939$$ 0 0
$$940$$ 61.0068 1.98982
$$941$$ 28.6375 0.933556 0.466778 0.884374i $$-0.345415\pi$$
0.466778 + 0.884374i $$0.345415\pi$$
$$942$$ 0 0
$$943$$ 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 $$0.171599\pi$$
$$948$$ 0 0
$$949$$ −66.2941 −2.15200
$$950$$ −3.61213 −0.117193
$$951$$ 0 0
$$952$$ 45.9633 1.48968
$$953$$ −37.1939 −1.20483 −0.602415 0.798183i $$-0.705795\pi$$
−0.602415 + 0.798183i $$0.705795\pi$$
$$954$$ 0 0
$$955$$ −11.1998 −0.362418
$$956$$ 109.388 3.53787
$$957$$ 0 0
$$958$$ 28.6516 0.925693
$$959$$ −10.9927 −0.354973
$$960$$ 0 0
$$961$$ −3.04491 −0.0982228
$$962$$ −133.289 −4.29740
$$963$$ 0 0
$$964$$ −48.1500 −1.55081
$$965$$ 0.604833 0.0194703
$$966$$ 0 0
$$967$$ 4.07125 0.130923 0.0654613 0.997855i $$-0.479148\pi$$
0.0654613 + 0.997855i $$0.479148\pi$$
$$968$$ 8.44358 0.271387
$$969$$ 0 0
$$970$$ −2.23743 −0.0718395
$$971$$ −0.773377 −0.0248188 −0.0124094 0.999923i $$-0.503950\pi$$
−0.0124094 + 0.999923i $$0.503950\pi$$
$$972$$ 0 0
$$973$$ −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 $$0.292834\pi$$
$$978$$ 0 0
$$979$$ −7.22425 −0.230888
$$980$$ 5.15633 0.164713
$$981$$ 0 0
$$982$$ 75.8916 2.42180
$$983$$ 15.5794 0.496907 0.248453 0.968644i $$-0.420078\pi$$
0.248453 + 0.968644i $$0.420078\pi$$
$$984$$ 0 0
$$985$$ −15.3054 −0.487669
$$986$$ −52.6009 −1.67515
$$987$$ 0 0
$$988$$ −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
$$993$$ 0 0
$$994$$ −41.5633 −1.31831
$$995$$ −12.5623 −0.398252
$$996$$ 0 0
$$997$$ 50.4060 1.59637 0.798187 0.602410i $$-0.205793\pi$$
0.798187 + 0.602410i $$0.205793\pi$$
$$998$$ 73.5026 2.32668
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3465.2.a.bh.1.3 3
3.2 odd 2 385.2.a.f.1.1 3
12.11 even 2 6160.2.a.bn.1.3 3
15.2 even 4 1925.2.b.n.1849.1 6
15.8 even 4 1925.2.b.n.1849.6 6
15.14 odd 2 1925.2.a.v.1.3 3
21.20 even 2 2695.2.a.g.1.1 3
33.32 even 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 3.2 odd 2
1925.2.a.v.1.3 3 15.14 odd 2
1925.2.b.n.1849.1 6 15.2 even 4
1925.2.b.n.1849.6 6 15.8 even 4
2695.2.a.g.1.1 3 21.20 even 2
3465.2.a.bh.1.3 3 1.1 even 1 trivial
4235.2.a.q.1.3 3 33.32 even 2
6160.2.a.bn.1.3 3 12.11 even 2