# Properties

 Label 1925.2.b.n.1849.6 Level $1925$ Weight $2$ Character 1925.1849 Analytic conductor $15.371$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1925 = 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1925.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.3712023891$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 385) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.6 Root $$0.403032 + 0.403032i$$ of defining polynomial Character $$\chi$$ $$=$$ 1925.1849 Dual form 1925.2.b.n.1849.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.67513i q^{2} -2.48119i q^{3} -5.15633 q^{4} +6.63752 q^{6} +1.00000i q^{7} -8.44358i q^{8} -3.15633 q^{9} +O(q^{10})$$ $$q+2.67513i q^{2} -2.48119i q^{3} -5.15633 q^{4} +6.63752 q^{6} +1.00000i q^{7} -8.44358i q^{8} -3.15633 q^{9} -1.00000 q^{11} +12.7938i q^{12} +5.83146i q^{13} -2.67513 q^{14} +12.2750 q^{16} -5.44358i q^{17} -8.44358i q^{18} +1.35026 q^{19} +2.48119 q^{21} -2.67513i q^{22} -3.19394i q^{23} -20.9502 q^{24} -15.5999 q^{26} +0.387873i q^{27} -5.15633i q^{28} +3.61213 q^{29} -5.28726 q^{31} +15.9502i q^{32} +2.48119i q^{33} +14.5623 q^{34} +16.2750 q^{36} +8.54420i q^{37} +3.61213i q^{38} +14.4690 q^{39} -5.02539 q^{41} +6.63752i q^{42} +5.89446i q^{43} +5.15633 q^{44} +8.54420 q^{46} +11.8315i q^{47} -30.4568i q^{48} -1.00000 q^{49} -13.5066 q^{51} -30.0689i q^{52} -0.231548i q^{53} -1.03761 q^{54} +8.44358 q^{56} -3.35026i q^{57} +9.66291i q^{58} -13.5999 q^{59} -1.41327 q^{61} -14.1441i q^{62} -3.15633i q^{63} -18.1187 q^{64} -6.63752 q^{66} +10.8568i q^{67} +28.0689i q^{68} -7.92478 q^{69} -15.5369 q^{71} +26.6507i q^{72} -11.3684i q^{73} -22.8568 q^{74} -6.96239 q^{76} -1.00000i q^{77} +38.7064i q^{78} -1.96968 q^{79} -8.50659 q^{81} -13.4436i q^{82} +10.6253i q^{83} -12.7938 q^{84} -15.7685 q^{86} -8.96239i q^{87} +8.44358i q^{88} -7.22425 q^{89} -5.83146 q^{91} +16.4690i q^{92} +13.1187i q^{93} -31.6507 q^{94} +39.5755 q^{96} +0.836381i q^{97} -2.67513i q^{98} +3.15633 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} + 8 q^{6} + 2 q^{9} + O(q^{10})$$ $$6 q - 10 q^{4} + 8 q^{6} + 2 q^{9} - 6 q^{11} - 6 q^{14} + 10 q^{16} - 12 q^{19} + 4 q^{21} - 52 q^{24} - 40 q^{26} + 20 q^{29} - 20 q^{31} + 12 q^{34} + 34 q^{36} + 24 q^{39} + 10 q^{44} + 32 q^{46} - 6 q^{49} - 40 q^{51} - 28 q^{54} + 18 q^{56} - 28 q^{59} + 20 q^{61} - 66 q^{64} - 8 q^{66} - 4 q^{69} - 48 q^{71} - 76 q^{74} - 20 q^{76} - 16 q^{79} - 10 q^{81} - 24 q^{84} - 72 q^{86} - 40 q^{89} - 4 q^{91} - 76 q^{94} + 80 q^{96} - 2 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$$.

 $$n$$ $$276$$ $$1002$$ $$1751$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.67513i 1.89160i 0.324745 + 0.945802i $$0.394721\pi$$
−0.324745 + 0.945802i $$0.605279\pi$$
$$3$$ − 2.48119i − 1.43252i −0.697834 0.716259i $$-0.745853\pi$$
0.697834 0.716259i $$-0.254147\pi$$
$$4$$ −5.15633 −2.57816
$$5$$ 0 0
$$6$$ 6.63752 2.70976
$$7$$ 1.00000i 0.377964i
$$8$$ − 8.44358i − 2.98526i
$$9$$ −3.15633 −1.05211
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 12.7938i 3.69326i
$$13$$ 5.83146i 1.61735i 0.588252 + 0.808677i $$0.299816\pi$$
−0.588252 + 0.808677i $$0.700184\pi$$
$$14$$ −2.67513 −0.714959
$$15$$ 0 0
$$16$$ 12.2750 3.06876
$$17$$ − 5.44358i − 1.32026i −0.751150 0.660131i $$-0.770501\pi$$
0.751150 0.660131i $$-0.229499\pi$$
$$18$$ − 8.44358i − 1.99017i
$$19$$ 1.35026 0.309771 0.154886 0.987932i $$-0.450499\pi$$
0.154886 + 0.987932i $$0.450499\pi$$
$$20$$ 0 0
$$21$$ 2.48119 0.541441
$$22$$ − 2.67513i − 0.570340i
$$23$$ − 3.19394i − 0.665982i −0.942930 0.332991i $$-0.891942\pi$$
0.942930 0.332991i $$-0.108058\pi$$
$$24$$ −20.9502 −4.27644
$$25$$ 0 0
$$26$$ −15.5999 −3.05939
$$27$$ 0.387873i 0.0746462i
$$28$$ − 5.15633i − 0.974454i
$$29$$ 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 $$0.657483\pi$$
$$32$$ 15.9502i 2.81962i
$$33$$ 2.48119i 0.431920i
$$34$$ 14.5623 2.49741
$$35$$ 0 0
$$36$$ 16.2750 2.71251
$$37$$ 8.54420i 1.40466i 0.711853 + 0.702329i $$0.247856\pi$$
−0.711853 + 0.702329i $$0.752144\pi$$
$$38$$ 3.61213i 0.585964i
$$39$$ 14.4690 2.31689
$$40$$ 0 0
$$41$$ −5.02539 −0.784834 −0.392417 0.919787i $$-0.628361\pi$$
−0.392417 + 0.919787i $$0.628361\pi$$
$$42$$ 6.63752i 1.02419i
$$43$$ 5.89446i 0.898897i 0.893306 + 0.449448i $$0.148379\pi$$
−0.893306 + 0.449448i $$0.851621\pi$$
$$44$$ 5.15633 0.777345
$$45$$ 0 0
$$46$$ 8.54420 1.25977
$$47$$ 11.8315i 1.72580i 0.505379 + 0.862898i $$0.331353\pi$$
−0.505379 + 0.862898i $$0.668647\pi$$
$$48$$ − 30.4568i − 4.39605i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −13.5066 −1.89130
$$52$$ − 30.0689i − 4.16980i
$$53$$ − 0.231548i − 0.0318056i −0.999874 0.0159028i $$-0.994938\pi$$
0.999874 0.0159028i $$-0.00506223\pi$$
$$54$$ −1.03761 −0.141201
$$55$$ 0 0
$$56$$ 8.44358 1.12832
$$57$$ − 3.35026i − 0.443753i
$$58$$ 9.66291i 1.26880i
$$59$$ −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 $$0.528839\pi$$
$$62$$ − 14.1441i − 1.79630i
$$63$$ − 3.15633i − 0.397660i
$$64$$ −18.1187 −2.26484
$$65$$ 0 0
$$66$$ −6.63752 −0.817022
$$67$$ 10.8568i 1.32638i 0.748453 + 0.663188i $$0.230797\pi$$
−0.748453 + 0.663188i $$0.769203\pi$$
$$68$$ 28.0689i 3.40385i
$$69$$ −7.92478 −0.954031
$$70$$ 0 0
$$71$$ −15.5369 −1.84389 −0.921946 0.387319i $$-0.873401\pi$$
−0.921946 + 0.387319i $$0.873401\pi$$
$$72$$ 26.6507i 3.14081i
$$73$$ − 11.3684i − 1.33057i −0.746591 0.665283i $$-0.768311\pi$$
0.746591 0.665283i $$-0.231689\pi$$
$$74$$ −22.8568 −2.65705
$$75$$ 0 0
$$76$$ −6.96239 −0.798641
$$77$$ − 1.00000i − 0.113961i
$$78$$ 38.7064i 4.38264i
$$79$$ −1.96968 −0.221607 −0.110803 0.993842i $$-0.535342\pi$$
−0.110803 + 0.993842i $$0.535342\pi$$
$$80$$ 0 0
$$81$$ −8.50659 −0.945176
$$82$$ − 13.4436i − 1.48460i
$$83$$ 10.6253i 1.16628i 0.812372 + 0.583139i $$0.198176\pi$$
−0.812372 + 0.583139i $$0.801824\pi$$
$$84$$ −12.7938 −1.39592
$$85$$ 0 0
$$86$$ −15.7685 −1.70036
$$87$$ − 8.96239i − 0.960869i
$$88$$ 8.44358i 0.900089i
$$89$$ −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
$$92$$ 16.4690i 1.71701i
$$93$$ 13.1187i 1.36035i
$$94$$ −31.6507 −3.26452
$$95$$ 0 0
$$96$$ 39.5755 4.03915
$$97$$ 0.836381i 0.0849216i 0.999098 + 0.0424608i $$0.0135198\pi$$
−0.999098 + 0.0424608i $$0.986480\pi$$
$$98$$ − 2.67513i − 0.270229i
$$99$$ 3.15633 0.317223
$$100$$ 0 0
$$101$$ 7.41327 0.737648 0.368824 0.929499i $$-0.379761\pi$$
0.368824 + 0.929499i $$0.379761\pi$$
$$102$$ − 36.1319i − 3.57759i
$$103$$ 4.21933i 0.415743i 0.978156 + 0.207871i $$0.0666536\pi$$
−0.978156 + 0.207871i $$0.933346\pi$$
$$104$$ 49.2384 4.82822
$$105$$ 0 0
$$106$$ 0.619421 0.0601635
$$107$$ 11.5369i 1.11531i 0.830071 + 0.557657i $$0.188300\pi$$
−0.830071 + 0.557657i $$0.811700\pi$$
$$108$$ − 2.00000i − 0.192450i
$$109$$ 2.18664 0.209442 0.104721 0.994502i $$-0.466605\pi$$
0.104721 + 0.994502i $$0.466605\pi$$
$$110$$ 0 0
$$111$$ 21.1998 2.01220
$$112$$ 12.2750i 1.15988i
$$113$$ − 9.35026i − 0.879599i −0.898096 0.439799i $$-0.855050\pi$$
0.898096 0.439799i $$-0.144950\pi$$
$$114$$ 8.96239 0.839405
$$115$$ 0 0
$$116$$ −18.6253 −1.72932
$$117$$ − 18.4060i − 1.70163i
$$118$$ − 36.3815i − 3.34919i
$$119$$ 5.44358 0.499012
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 3.78067i − 0.342286i
$$123$$ 12.4690i 1.12429i
$$124$$ 27.2628 2.44827
$$125$$ 0 0
$$126$$ 8.44358 0.752214
$$127$$ 16.9624i 1.50517i 0.658496 + 0.752584i $$0.271193\pi$$
−0.658496 + 0.752584i $$0.728807\pi$$
$$128$$ − 16.5696i − 1.46456i
$$129$$ 14.6253 1.28769
$$130$$ 0 0
$$131$$ −9.92478 −0.867132 −0.433566 0.901122i $$-0.642745\pi$$
−0.433566 + 0.901122i $$0.642745\pi$$
$$132$$ − 12.7938i − 1.11356i
$$133$$ 1.35026i 0.117083i
$$134$$ −29.0435 −2.50898
$$135$$ 0 0
$$136$$ −45.9633 −3.94132
$$137$$ 10.9927i 0.939170i 0.882887 + 0.469585i $$0.155596\pi$$
−0.882887 + 0.469585i $$0.844404\pi$$
$$138$$ − 21.1998i − 1.80465i
$$139$$ −6.88717 −0.584162 −0.292081 0.956394i $$-0.594348\pi$$
−0.292081 + 0.956394i $$0.594348\pi$$
$$140$$ 0 0
$$141$$ 29.3561 2.47223
$$142$$ − 41.5633i − 3.48791i
$$143$$ − 5.83146i − 0.487651i
$$144$$ −38.7440 −3.22867
$$145$$ 0 0
$$146$$ 30.4119 2.51690
$$147$$ 2.48119i 0.204645i
$$148$$ − 44.0567i − 3.62144i
$$149$$ −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 $$0.542148\pi$$
$$152$$ − 11.4010i − 0.924747i
$$153$$ 17.1817i 1.38906i
$$154$$ 2.67513 0.215568
$$155$$ 0 0
$$156$$ −74.6067 −5.97332
$$157$$ 5.42548i 0.433001i 0.976283 + 0.216500i $$0.0694643\pi$$
−0.976283 + 0.216500i $$0.930536\pi$$
$$158$$ − 5.26916i − 0.419192i
$$159$$ −0.574515 −0.0455620
$$160$$ 0 0
$$161$$ 3.19394 0.251717
$$162$$ − 22.7562i − 1.78790i
$$163$$ 3.38058i 0.264787i 0.991197 + 0.132394i $$0.0422663\pi$$
−0.991197 + 0.132394i $$0.957734\pi$$
$$164$$ 25.9126 2.02343
$$165$$ 0 0
$$166$$ −28.4241 −2.20614
$$167$$ − 11.2750i − 0.872489i −0.899828 0.436244i $$-0.856308\pi$$
0.899828 0.436244i $$-0.143692\pi$$
$$168$$ − 20.9502i − 1.61634i
$$169$$ −21.0059 −1.61584
$$170$$ 0 0
$$171$$ −4.26187 −0.325913
$$172$$ − 30.3938i − 2.31750i
$$173$$ − 8.98049i − 0.682774i −0.939923 0.341387i $$-0.889103\pi$$
0.939923 0.341387i $$-0.110897\pi$$
$$174$$ 23.9756 1.81758
$$175$$ 0 0
$$176$$ −12.2750 −0.925266
$$177$$ 33.7440i 2.53636i
$$178$$ − 19.3258i − 1.44853i
$$179$$ 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 $$0.636652\pi$$
$$182$$ − 15.5999i − 1.15634i
$$183$$ 3.50659i 0.259214i
$$184$$ −26.9683 −1.98813
$$185$$ 0 0
$$186$$ −35.0943 −2.57324
$$187$$ 5.44358i 0.398074i
$$188$$ − 61.0068i − 4.44938i
$$189$$ −0.387873 −0.0282136
$$190$$ 0 0
$$191$$ 11.1998 0.810390 0.405195 0.914230i $$-0.367204\pi$$
0.405195 + 0.914230i $$0.367204\pi$$
$$192$$ 44.9560i 3.24442i
$$193$$ 0.604833i 0.0435368i 0.999763 + 0.0217684i $$0.00692965\pi$$
−0.999763 + 0.0217684i $$0.993070\pi$$
$$194$$ −2.23743 −0.160638
$$195$$ 0 0
$$196$$ 5.15633 0.368309
$$197$$ − 15.3054i − 1.09046i −0.838286 0.545231i $$-0.816442\pi$$
0.838286 0.545231i $$-0.183558\pi$$
$$198$$ 8.44358i 0.600059i
$$199$$ 12.5623 0.890518 0.445259 0.895402i $$-0.353112\pi$$
0.445259 + 0.895402i $$0.353112\pi$$
$$200$$ 0 0
$$201$$ 26.9380 1.90006
$$202$$ 19.8315i 1.39534i
$$203$$ 3.61213i 0.253522i
$$204$$ 69.6444 4.87608
$$205$$ 0 0
$$206$$ −11.2873 −0.786421
$$207$$ 10.0811i 0.700685i
$$208$$ 71.5814i 4.96327i
$$209$$ −1.35026 −0.0933996
$$210$$ 0 0
$$211$$ −4.43866 −0.305570 −0.152785 0.988259i $$-0.548824\pi$$
−0.152785 + 0.988259i $$0.548824\pi$$
$$212$$ 1.19394i 0.0819999i
$$213$$ 38.5501i 2.64141i
$$214$$ −30.8627 −2.10973
$$215$$ 0 0
$$216$$ 3.27504 0.222838
$$217$$ − 5.28726i − 0.358922i
$$218$$ 5.84955i 0.396182i
$$219$$ −28.2071 −1.90606
$$220$$ 0 0
$$221$$ 31.7440 2.13533
$$222$$ 56.7123i 3.80628i
$$223$$ − 7.78067i − 0.521032i −0.965469 0.260516i $$-0.916107\pi$$
0.965469 0.260516i $$-0.0838927\pi$$
$$224$$ −15.9502 −1.06572
$$225$$ 0 0
$$226$$ 25.0132 1.66385
$$227$$ 10.4485i 0.693492i 0.937959 + 0.346746i $$0.112713\pi$$
−0.937959 + 0.346746i $$0.887287\pi$$
$$228$$ 17.2750i 1.14407i
$$229$$ 29.4518 1.94623 0.973116 0.230316i $$-0.0739759\pi$$
0.973116 + 0.230316i $$0.0739759\pi$$
$$230$$ 0 0
$$231$$ −2.48119 −0.163251
$$232$$ − 30.4993i − 2.00238i
$$233$$ − 8.73084i − 0.571976i −0.958233 0.285988i $$-0.907678\pi$$
0.958233 0.285988i $$-0.0923218\pi$$
$$234$$ 49.2384 3.21881
$$235$$ 0 0
$$236$$ 70.1255 4.56478
$$237$$ 4.88717i 0.317456i
$$238$$ 14.5623i 0.943933i
$$239$$ 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 $$0.597240\pi$$
$$242$$ 2.67513i 0.171964i
$$243$$ 22.2701i 1.42863i
$$244$$ 7.28726 0.466519
$$245$$ 0 0
$$246$$ −33.3561 −2.12671
$$247$$ 7.87399i 0.501010i
$$248$$ 44.6434i 2.83486i
$$249$$ 26.3634 1.67071
$$250$$ 0 0
$$251$$ 1.87636 0.118435 0.0592174 0.998245i $$-0.481139\pi$$
0.0592174 + 0.998245i $$0.481139\pi$$
$$252$$ 16.2750i 1.02523i
$$253$$ 3.19394i 0.200801i
$$254$$ −45.3766 −2.84718
$$255$$ 0 0
$$256$$ 8.08840 0.505525
$$257$$ − 27.1392i − 1.69290i −0.532472 0.846448i $$-0.678737\pi$$
0.532472 0.846448i $$-0.321263\pi$$
$$258$$ 39.1246i 2.43579i
$$259$$ −8.54420 −0.530911
$$260$$ 0 0
$$261$$ −11.4010 −0.705707
$$262$$ − 26.5501i − 1.64027i
$$263$$ 12.8119i 0.790018i 0.918677 + 0.395009i $$0.129259\pi$$
−0.918677 + 0.395009i $$0.870741\pi$$
$$264$$ 20.9502 1.28939
$$265$$ 0 0
$$266$$ −3.61213 −0.221474
$$267$$ 17.9248i 1.09698i
$$268$$ − 55.9814i − 3.41961i
$$269$$ 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 $$0.555761\pi$$
$$272$$ − 66.8202i − 4.05157i
$$273$$ 14.4690i 0.875702i
$$274$$ −29.4069 −1.77654
$$275$$ 0 0
$$276$$ 40.8627 2.45965
$$277$$ 8.35756i 0.502157i 0.967967 + 0.251078i $$0.0807852\pi$$
−0.967967 + 0.251078i $$0.919215\pi$$
$$278$$ − 18.4241i − 1.10500i
$$279$$ 16.6883 0.999103
$$280$$ 0 0
$$281$$ −8.44851 −0.503996 −0.251998 0.967728i $$-0.581088\pi$$
−0.251998 + 0.967728i $$0.581088\pi$$
$$282$$ 78.5315i 4.67648i
$$283$$ 0.836381i 0.0497177i 0.999691 + 0.0248588i $$0.00791363\pi$$
−0.999691 + 0.0248588i $$0.992086\pi$$
$$284$$ 80.1133 4.75385
$$285$$ 0 0
$$286$$ 15.5999 0.922442
$$287$$ − 5.02539i − 0.296640i
$$288$$ − 50.3439i − 2.96654i
$$289$$ −12.6326 −0.743094
$$290$$ 0 0
$$291$$ 2.07522 0.121652
$$292$$ 58.6190i 3.43042i
$$293$$ − 2.71862i − 0.158824i −0.996842 0.0794118i $$-0.974696\pi$$
0.996842 0.0794118i $$-0.0253042\pi$$
$$294$$ −6.63752 −0.387108
$$295$$ 0 0
$$296$$ 72.1436 4.19326
$$297$$ − 0.387873i − 0.0225067i
$$298$$ − 61.0249i − 3.53508i
$$299$$ 18.6253 1.07713
$$300$$ 0 0
$$301$$ −5.89446 −0.339751
$$302$$ − 8.68006i − 0.499481i
$$303$$ − 18.3938i − 1.05669i
$$304$$ 16.5745 0.950614
$$305$$ 0 0
$$306$$ −45.9633 −2.62755
$$307$$ 8.36344i 0.477326i 0.971102 + 0.238663i $$0.0767092\pi$$
−0.971102 + 0.238663i $$0.923291\pi$$
$$308$$ 5.15633i 0.293809i
$$309$$ 10.4690 0.595559
$$310$$ 0 0
$$311$$ 4.43629 0.251559 0.125779 0.992058i $$-0.459857\pi$$
0.125779 + 0.992058i $$0.459857\pi$$
$$312$$ − 122.170i − 6.91651i
$$313$$ 29.7889i 1.68377i 0.539658 + 0.841885i $$0.318554\pi$$
−0.539658 + 0.841885i $$0.681446\pi$$
$$314$$ −14.5139 −0.819066
$$315$$ 0 0
$$316$$ 10.1563 0.571338
$$317$$ − 15.4010i − 0.865009i −0.901632 0.432504i $$-0.857630\pi$$
0.901632 0.432504i $$-0.142370\pi$$
$$318$$ − 1.53690i − 0.0861853i
$$319$$ −3.61213 −0.202240
$$320$$ 0 0
$$321$$ 28.6253 1.59771
$$322$$ 8.54420i 0.476150i
$$323$$ − 7.35026i − 0.408980i
$$324$$ 43.8627 2.43682
$$325$$ 0 0
$$326$$ −9.04349 −0.500873
$$327$$ − 5.42548i − 0.300030i
$$328$$ 42.4323i 2.34293i
$$329$$ −11.8315 −0.652289
$$330$$ 0 0
$$331$$ 6.26187 0.344183 0.172092 0.985081i $$-0.444947\pi$$
0.172092 + 0.985081i $$0.444947\pi$$
$$332$$ − 54.7875i − 3.00685i
$$333$$ − 26.9683i − 1.47785i
$$334$$ 30.1622 1.65040
$$335$$ 0 0
$$336$$ 30.4568 1.66155
$$337$$ 15.8700i 0.864495i 0.901755 + 0.432248i $$0.142279\pi$$
−0.901755 + 0.432248i $$0.857721\pi$$
$$338$$ − 56.1935i − 3.05652i
$$339$$ −23.1998 −1.26004
$$340$$ 0 0
$$341$$ 5.28726 0.286321
$$342$$ − 11.4010i − 0.616498i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 49.7704 2.68344
$$345$$ 0 0
$$346$$ 24.0240 1.29154
$$347$$ − 6.79147i − 0.364585i −0.983244 0.182293i $$-0.941648\pi$$
0.983244 0.182293i $$-0.0583519\pi$$
$$348$$ 46.2130i 2.47728i
$$349$$ −26.7489 −1.43184 −0.715919 0.698183i $$-0.753992\pi$$
−0.715919 + 0.698183i $$0.753992\pi$$
$$350$$ 0 0
$$351$$ −2.26187 −0.120729
$$352$$ − 15.9502i − 0.850147i
$$353$$ − 16.8627i − 0.897512i −0.893654 0.448756i $$-0.851867\pi$$
0.893654 0.448756i $$-0.148133\pi$$
$$354$$ −90.2697 −4.79778
$$355$$ 0 0
$$356$$ 37.2506 1.97428
$$357$$ − 13.5066i − 0.714844i
$$358$$ 70.1886i 3.70958i
$$359$$ 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
$$362$$ − 29.9610i − 1.57471i
$$363$$ − 2.48119i − 0.130229i
$$364$$ 30.0689 1.57604
$$365$$ 0 0
$$366$$ −9.38058 −0.490331
$$367$$ 6.36977i 0.332500i 0.986084 + 0.166250i $$0.0531658\pi$$
−0.986084 + 0.166250i $$0.946834\pi$$
$$368$$ − 39.2057i − 2.04374i
$$369$$ 15.8618 0.825731
$$370$$ 0 0
$$371$$ 0.231548 0.0120214
$$372$$ − 67.6444i − 3.50720i
$$373$$ − 21.3317i − 1.10451i −0.833674 0.552257i $$-0.813767\pi$$
0.833674 0.552257i $$-0.186233\pi$$
$$374$$ −14.5623 −0.752998
$$375$$ 0 0
$$376$$ 99.8999 5.15194
$$377$$ 21.0640i 1.08485i
$$378$$ − 1.03761i − 0.0533690i
$$379$$ −24.7875 −1.27325 −0.636624 0.771174i $$-0.719670\pi$$
−0.636624 + 0.771174i $$0.719670\pi$$
$$380$$ 0 0
$$381$$ 42.0870 2.15618
$$382$$ 29.9610i 1.53294i
$$383$$ − 5.45817i − 0.278900i −0.990229 0.139450i $$-0.955467\pi$$
0.990229 0.139450i $$-0.0445334\pi$$
$$384$$ −41.1124 −2.09801
$$385$$ 0 0
$$386$$ −1.61801 −0.0823544
$$387$$ − 18.6048i − 0.945737i
$$388$$ − 4.31265i − 0.218942i
$$389$$ 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
$$392$$ 8.44358i 0.426465i
$$393$$ 24.6253i 1.24218i
$$394$$ 40.9438 2.06272
$$395$$ 0 0
$$396$$ −16.2750 −0.817851
$$397$$ − 2.11142i − 0.105969i −0.998595 0.0529846i $$-0.983127\pi$$
0.998595 0.0529846i $$-0.0168734\pi$$
$$398$$ 33.6058i 1.68451i
$$399$$ 3.35026 0.167723
$$400$$ 0 0
$$401$$ 19.1490 0.956257 0.478128 0.878290i $$-0.341315\pi$$
0.478128 + 0.878290i $$0.341315\pi$$
$$402$$ 72.0625i 3.59415i
$$403$$ − 30.8324i − 1.53587i
$$404$$ −38.2252 −1.90178
$$405$$ 0 0
$$406$$ −9.66291 −0.479562
$$407$$ − 8.54420i − 0.423520i
$$408$$ 114.044i 5.64602i
$$409$$ 18.6883 0.924077 0.462039 0.886860i $$-0.347118\pi$$
0.462039 + 0.886860i $$0.347118\pi$$
$$410$$ 0 0
$$411$$ 27.2750 1.34538
$$412$$ − 21.7562i − 1.07185i
$$413$$ − 13.5999i − 0.669208i
$$414$$ −26.9683 −1.32542
$$415$$ 0 0
$$416$$ −93.0127 −4.56032
$$417$$ 17.0884i 0.836822i
$$418$$ − 3.61213i − 0.176675i
$$419$$ 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 $$0.582520\pi$$
$$422$$ − 11.8740i − 0.578017i
$$423$$ − 37.3439i − 1.81572i
$$424$$ −1.95509 −0.0949478
$$425$$ 0 0
$$426$$ −103.127 −4.99650
$$427$$ − 1.41327i − 0.0683927i
$$428$$ − 59.4880i − 2.87546i
$$429$$ −14.4690 −0.698569
$$430$$ 0 0
$$431$$ −24.7308 −1.19124 −0.595621 0.803265i $$-0.703094\pi$$
−0.595621 + 0.803265i $$0.703094\pi$$
$$432$$ 4.76116i 0.229071i
$$433$$ − 18.5599i − 0.891933i −0.895050 0.445967i $$-0.852860\pi$$
0.895050 0.445967i $$-0.147140\pi$$
$$434$$ 14.1441 0.678939
$$435$$ 0 0
$$436$$ −11.2750 −0.539976
$$437$$ − 4.31265i − 0.206302i
$$438$$ − 75.4577i − 3.60551i
$$439$$ 1.42548 0.0680347 0.0340173 0.999421i $$-0.489170\pi$$
0.0340173 + 0.999421i $$0.489170\pi$$
$$440$$ 0 0
$$441$$ 3.15633 0.150301
$$442$$ 84.9194i 4.03920i
$$443$$ 40.1925i 1.90960i 0.297242 + 0.954802i $$0.403933\pi$$
−0.297242 + 0.954802i $$0.596067\pi$$
$$444$$ −109.313 −5.18777
$$445$$ 0 0
$$446$$ 20.8143 0.985586
$$447$$ 56.6009i 2.67713i
$$448$$ − 18.1187i − 0.856029i
$$449$$ −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
$$452$$ 48.2130i 2.26775i
$$453$$ 8.05079i 0.378259i
$$454$$ −27.9511 −1.31181
$$455$$ 0 0
$$456$$ −28.2882 −1.32472
$$457$$ 0.544198i 0.0254565i 0.999919 + 0.0127283i $$0.00405164\pi$$
−0.999919 + 0.0127283i $$0.995948\pi$$
$$458$$ 78.7875i 3.68150i
$$459$$ 2.11142 0.0985526
$$460$$ 0 0
$$461$$ −11.5755 −0.539123 −0.269562 0.962983i $$-0.586879\pi$$
−0.269562 + 0.962983i $$0.586879\pi$$
$$462$$ − 6.63752i − 0.308805i
$$463$$ 23.7948i 1.10584i 0.833235 + 0.552919i $$0.186486\pi$$
−0.833235 + 0.552919i $$0.813514\pi$$
$$464$$ 44.3390 2.05839
$$465$$ 0 0
$$466$$ 23.3561 1.08195
$$467$$ 2.66784i 0.123453i 0.998093 + 0.0617264i $$0.0196606\pi$$
−0.998093 + 0.0617264i $$0.980339\pi$$
$$468$$ 94.9072i 4.38709i
$$469$$ −10.8568 −0.501323
$$470$$ 0 0
$$471$$ 13.4617 0.620282
$$472$$ 114.832i 5.28557i
$$473$$ − 5.89446i − 0.271028i
$$474$$ −13.0738 −0.600500
$$475$$ 0 0
$$476$$ −28.0689 −1.28654
$$477$$ 0.730841i 0.0334629i
$$478$$ 56.7513i 2.59574i
$$479$$ 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
$$482$$ − 24.9805i − 1.13783i
$$483$$ − 7.92478i − 0.360590i
$$484$$ −5.15633 −0.234378
$$485$$ 0 0
$$486$$ −59.5755 −2.70240
$$487$$ − 17.4314i − 0.789891i −0.918705 0.394945i $$-0.870764\pi$$
0.918705 0.394945i $$-0.129236\pi$$
$$488$$ 11.9330i 0.540183i
$$489$$ 8.38787 0.379313
$$490$$ 0 0
$$491$$ −28.3693 −1.28029 −0.640145 0.768254i $$-0.721126\pi$$
−0.640145 + 0.768254i $$0.721126\pi$$
$$492$$ − 64.2941i − 2.89860i
$$493$$ − 19.6629i − 0.885573i
$$494$$ −21.0640 −0.947712
$$495$$ 0 0
$$496$$ −64.9013 −2.91415
$$497$$ − 15.5369i − 0.696925i
$$498$$ 70.5256i 3.16033i
$$499$$ −27.4763 −1.23001 −0.615003 0.788524i $$-0.710845\pi$$
−0.615003 + 0.788524i $$0.710845\pi$$
$$500$$ 0 0
$$501$$ −27.9756 −1.24986
$$502$$ 5.01951i 0.224032i
$$503$$ 20.2981i 0.905046i 0.891753 + 0.452523i $$0.149476\pi$$
−0.891753 + 0.452523i $$0.850524\pi$$
$$504$$ −26.6507 −1.18712
$$505$$ 0 0
$$506$$ −8.54420 −0.379836
$$507$$ 52.1197i 2.31472i
$$508$$ − 87.4636i − 3.88057i
$$509$$ 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
$$512$$ − 11.5017i − 0.508306i
$$513$$ 0.523730i 0.0231233i
$$514$$ 72.6009 3.20229
$$515$$ 0 0
$$516$$ −75.4128 −3.31986
$$517$$ − 11.8315i − 0.520347i
$$518$$ − 22.8568i − 1.00427i
$$519$$ −22.2823 −0.978086
$$520$$ 0 0
$$521$$ 2.20123 0.0964377 0.0482188 0.998837i $$-0.484646\pi$$
0.0482188 + 0.998837i $$0.484646\pi$$
$$522$$ − 30.4993i − 1.33492i
$$523$$ − 22.1378i − 0.968017i −0.875063 0.484008i $$-0.839180\pi$$
0.875063 0.484008i $$-0.160820\pi$$
$$524$$ 51.1754 2.23561
$$525$$ 0 0
$$526$$ −34.2736 −1.49440
$$527$$ 28.7816i 1.25375i
$$528$$ 30.4568i 1.32546i
$$529$$ 12.7988 0.556468
$$530$$ 0 0
$$531$$ 42.9257 1.86282
$$532$$ − 6.96239i − 0.301858i
$$533$$ − 29.3054i − 1.26936i
$$534$$ −47.9511 −2.07505
$$535$$ 0 0
$$536$$ 91.6707 3.95957
$$537$$ − 65.1002i − 2.80928i
$$538$$ 16.7513i 0.722200i
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 23.0640 0.991597 0.495799 0.868438i $$-0.334875\pi$$
0.495799 + 0.868438i $$0.334875\pi$$
$$542$$ − 15.3503i − 0.659350i
$$543$$ 27.7889i 1.19254i
$$544$$ 86.8261 3.72264
$$545$$ 0 0
$$546$$ −38.7064 −1.65648
$$547$$ − 21.3766i − 0.913998i −0.889467 0.456999i $$-0.848924\pi$$
0.889467 0.456999i $$-0.151076\pi$$
$$548$$ − 56.6820i − 2.42133i
$$549$$ 4.46073 0.190379
$$550$$ 0 0
$$551$$ 4.87732 0.207781
$$552$$ 66.9135i 2.84803i
$$553$$ − 1.96968i − 0.0837594i
$$554$$ −22.3576 −0.949882
$$555$$ 0 0
$$556$$ 35.5125 1.50606
$$557$$ 9.19394i 0.389560i 0.980847 + 0.194780i $$0.0623992\pi$$
−0.980847 + 0.194780i $$0.937601\pi$$
$$558$$ 44.6434i 1.88991i
$$559$$ −34.3733 −1.45384
$$560$$ 0 0
$$561$$ 13.5066 0.570249
$$562$$ − 22.6009i − 0.953360i
$$563$$ − 9.79877i − 0.412969i −0.978450 0.206484i $$-0.933798\pi$$
0.978450 0.206484i $$-0.0662023\pi$$
$$564$$ −151.370 −6.37382
$$565$$ 0 0
$$566$$ −2.23743 −0.0940461
$$567$$ − 8.50659i − 0.357243i
$$568$$ 131.187i 5.50449i
$$569$$ 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 $$0.141612\pi$$
$$572$$ 30.0689i 1.25724i
$$573$$ − 27.7889i − 1.16090i
$$574$$ 13.4436 0.561124
$$575$$ 0 0
$$576$$ 57.1886 2.38286
$$577$$ 14.8510i 0.618254i 0.951021 + 0.309127i $$0.100037\pi$$
−0.951021 + 0.309127i $$0.899963\pi$$
$$578$$ − 33.7938i − 1.40564i
$$579$$ 1.50071 0.0623673
$$580$$ 0 0
$$581$$ −10.6253 −0.440812
$$582$$ 5.55149i 0.230117i
$$583$$ 0.231548i 0.00958974i
$$584$$ −95.9897 −3.97208
$$585$$ 0 0
$$586$$ 7.27267 0.300431
$$587$$ − 14.7938i − 0.610607i −0.952255 0.305304i $$-0.901242\pi$$
0.952255 0.305304i $$-0.0987580\pi$$
$$588$$ − 12.7938i − 0.527609i
$$589$$ −7.13918 −0.294165
$$590$$ 0 0
$$591$$ −37.9756 −1.56211
$$592$$ 104.880i 4.31056i
$$593$$ − 27.4191i − 1.12597i −0.826467 0.562985i $$-0.809653\pi$$
0.826467 0.562985i $$-0.190347\pi$$
$$594$$ 1.03761 0.0425737
$$595$$ 0 0
$$596$$ 117.626 4.81814
$$597$$ − 31.1695i − 1.27568i
$$598$$ 49.8251i 2.03750i
$$599$$ −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 $$0.397348\pi$$
$$602$$ − 15.7685i − 0.642674i
$$603$$ − 34.2677i − 1.39549i
$$604$$ 16.7308 0.680768
$$605$$ 0 0
$$606$$ 49.2057 1.99884
$$607$$ − 17.7235i − 0.719377i −0.933073 0.359688i $$-0.882883\pi$$
0.933073 0.359688i $$-0.117117\pi$$
$$608$$ 21.5369i 0.873437i
$$609$$ 8.96239 0.363174
$$610$$ 0 0
$$611$$ −68.9946 −2.79122
$$612$$ − 88.5945i − 3.58122i
$$613$$ 22.2941i 0.900450i 0.892915 + 0.450225i $$0.148656\pi$$
−0.892915 + 0.450225i $$0.851344\pi$$
$$614$$ −22.3733 −0.902912
$$615$$ 0 0
$$616$$ −8.44358 −0.340202
$$617$$ 30.9438i 1.24575i 0.782321 + 0.622876i $$0.214036\pi$$
−0.782321 + 0.622876i $$0.785964\pi$$
$$618$$ 28.0059i 1.12656i
$$619$$ −32.4119 −1.30274 −0.651371 0.758759i $$-0.725806\pi$$
−0.651371 + 0.758759i $$0.725806\pi$$
$$620$$ 0 0
$$621$$ 1.23884 0.0497130
$$622$$ 11.8677i 0.475850i
$$623$$ − 7.22425i − 0.289434i
$$624$$ 177.607 7.10998
$$625$$ 0 0
$$626$$ −79.6893 −3.18502
$$627$$ 3.35026i 0.133797i
$$628$$ − 27.9756i − 1.11635i
$$629$$ 46.5111 1.85452
$$630$$ 0 0
$$631$$ −27.3258 −1.08782 −0.543912 0.839142i $$-0.683057\pi$$
−0.543912 + 0.839142i $$0.683057\pi$$
$$632$$ 16.6312i 0.661553i
$$633$$ 11.0132i 0.437734i
$$634$$ 41.1998 1.63625
$$635$$ 0 0
$$636$$ 2.96239 0.117466
$$637$$ − 5.83146i − 0.231051i
$$638$$ − 9.66291i − 0.382558i
$$639$$ 49.0395 1.93997
$$640$$ 0 0
$$641$$ 19.4460 0.768069 0.384034 0.923319i $$-0.374534\pi$$
0.384034 + 0.923319i $$0.374534\pi$$
$$642$$ 76.5764i 3.02223i
$$643$$ 5.29314i 0.208741i 0.994538 + 0.104370i $$0.0332828\pi$$
−0.994538 + 0.104370i $$0.966717\pi$$
$$644$$ −16.4690 −0.648969
$$645$$ 0 0
$$646$$ 19.6629 0.773627
$$647$$ − 35.0966i − 1.37979i −0.723909 0.689896i $$-0.757656\pi$$
0.723909 0.689896i $$-0.242344\pi$$
$$648$$ 71.8261i 2.82159i
$$649$$ 13.5999 0.533843
$$650$$ 0 0
$$651$$ −13.1187 −0.514163
$$652$$ − 17.4314i − 0.682665i
$$653$$ 27.7988i 1.08785i 0.839134 + 0.543925i $$0.183062\pi$$
−0.839134 + 0.543925i $$0.816938\pi$$
$$654$$ 14.5139 0.567538
$$655$$ 0 0
$$656$$ −61.6869 −2.40847
$$657$$ 35.8822i 1.39990i
$$658$$ − 31.6507i − 1.23387i
$$659$$ −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 $$0.362256\pi$$
$$662$$ 16.7513i 0.651058i
$$663$$ − 78.7631i − 3.05890i
$$664$$ 89.7156 3.48164
$$665$$ 0 0
$$666$$ 72.1436 2.79551
$$667$$ − 11.5369i − 0.446711i
$$668$$ 58.1378i 2.24942i
$$669$$ −19.3054 −0.746388
$$670$$ 0 0
$$671$$ 1.41327 0.0545585
$$672$$ 39.5755i 1.52666i
$$673$$ − 21.0679i − 0.812109i −0.913849 0.406054i $$-0.866904\pi$$
0.913849 0.406054i $$-0.133096\pi$$
$$674$$ −42.4544 −1.63528
$$675$$ 0 0
$$676$$ 108.313 4.16589
$$677$$ − 34.5174i − 1.32661i −0.748349 0.663306i $$-0.769153\pi$$
0.748349 0.663306i $$-0.230847\pi$$
$$678$$ − 62.0625i − 2.38350i
$$679$$ −0.836381 −0.0320973
$$680$$ 0 0
$$681$$ 25.9248 0.993440
$$682$$ 14.1441i 0.541606i
$$683$$ 33.7802i 1.29256i 0.763099 + 0.646282i $$0.223677\pi$$
−0.763099 + 0.646282i $$0.776323\pi$$
$$684$$ 21.9756 0.840257
$$685$$ 0 0
$$686$$ 2.67513 0.102137
$$687$$ − 73.0757i − 2.78801i
$$688$$ 72.3547i 2.75850i
$$689$$ 1.35026 0.0514409
$$690$$ 0 0
$$691$$ −13.8618 −0.527327 −0.263663 0.964615i $$-0.584931\pi$$
−0.263663 + 0.964615i $$0.584931\pi$$
$$692$$ 46.3063i 1.76030i
$$693$$ 3.15633i 0.119899i
$$694$$ 18.1681 0.689651
$$695$$ 0 0
$$696$$ −75.6747 −2.86844
$$697$$ 27.3561i 1.03619i
$$698$$ − 71.5569i − 2.70847i
$$699$$ −21.6629 −0.819367
$$700$$ 0 0
$$701$$ 40.5256 1.53063 0.765316 0.643655i $$-0.222583\pi$$
0.765316 + 0.643655i $$0.222583\pi$$
$$702$$ − 6.05079i − 0.228372i
$$703$$ 11.5369i 0.435123i
$$704$$ 18.1187 0.682875
$$705$$ 0 0
$$706$$ 45.1100 1.69774
$$707$$ 7.41327i 0.278805i
$$708$$ − 173.995i − 6.53914i
$$709$$ 0.850969 0.0319588 0.0159794 0.999872i $$-0.494913\pi$$
0.0159794 + 0.999872i $$0.494913\pi$$
$$710$$ 0 0
$$711$$ 6.21696 0.233154
$$712$$ 60.9986i 2.28602i
$$713$$ 16.8872i 0.632429i
$$714$$ 36.1319 1.35220
$$715$$ 0 0
$$716$$ −135.289 −5.05598
$$717$$ − 52.6371i − 1.96577i
$$718$$ 10.1465i 0.378663i
$$719$$ −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
$$722$$ − 45.9502i − 1.71009i
$$723$$ 23.1695i 0.861683i
$$724$$ 57.7499 2.14626
$$725$$ 0 0
$$726$$ 6.63752 0.246341
$$727$$ − 12.5174i − 0.464244i −0.972687 0.232122i $$-0.925433\pi$$
0.972687 0.232122i $$-0.0745669\pi$$
$$728$$ 49.2384i 1.82490i
$$729$$ 29.7367 1.10136
$$730$$ 0 0
$$731$$ 32.0870 1.18678
$$732$$ − 18.0811i − 0.668297i
$$733$$ 16.6678i 0.615641i 0.951445 + 0.307820i $$0.0995996\pi$$
−0.951445 + 0.307820i $$0.900400\pi$$
$$734$$ −17.0400 −0.628957
$$735$$ 0 0
$$736$$ 50.9438 1.87781
$$737$$ − 10.8568i − 0.399917i
$$738$$ 42.4323i 1.56196i
$$739$$ −42.7005 −1.57076 −0.785382 0.619011i $$-0.787533\pi$$
−0.785382 + 0.619011i $$0.787533\pi$$
$$740$$ 0 0
$$741$$ 19.5369 0.717706
$$742$$ 0.619421i 0.0227397i
$$743$$ − 19.6873i − 0.722259i −0.932516 0.361129i $$-0.882391\pi$$
0.932516 0.361129i $$-0.117609\pi$$
$$744$$ 110.769 4.06099
$$745$$ 0 0
$$746$$ 57.0651 2.08930
$$747$$ − 33.5369i − 1.22705i
$$748$$ − 28.0689i − 1.02630i
$$749$$ −11.5369 −0.421549
$$750$$ 0 0
$$751$$ −5.85940 −0.213813 −0.106906 0.994269i $$-0.534094\pi$$
−0.106906 + 0.994269i $$0.534094\pi$$
$$752$$ 145.232i 5.29605i
$$753$$ − 4.65562i − 0.169660i
$$754$$ −56.3488 −2.05210
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 40.5863i 1.47513i 0.675274 + 0.737567i $$0.264025\pi$$
−0.675274 + 0.737567i $$0.735975\pi$$
$$758$$ − 66.3098i − 2.40848i
$$759$$ 7.92478 0.287651
$$760$$ 0 0
$$761$$ 21.8472 0.791960 0.395980 0.918259i $$-0.370405\pi$$
0.395980 + 0.918259i $$0.370405\pi$$
$$762$$ 112.588i 4.07864i
$$763$$ 2.18664i 0.0791618i
$$764$$ −57.7499 −2.08932
$$765$$ 0 0
$$766$$ 14.6013 0.527567
$$767$$ − 79.3073i − 2.86362i
$$768$$ − 20.0689i − 0.724173i
$$769$$ −45.2892 −1.63317 −0.816585 0.577226i $$-0.804135\pi$$
−0.816585 + 0.577226i $$0.804135\pi$$
$$770$$ 0 0
$$771$$ −67.3376 −2.42510
$$772$$ − 3.11871i − 0.112245i
$$773$$ − 33.8153i − 1.21625i −0.793841 0.608125i $$-0.791922\pi$$
0.793841 0.608125i $$-0.208078\pi$$
$$774$$ 49.7704 1.78896
$$775$$ 0 0
$$776$$ 7.06205 0.253513
$$777$$ 21.1998i 0.760539i
$$778$$ 36.7123i 1.31620i
$$779$$ −6.78560 −0.243119
$$780$$ 0 0
$$781$$ 15.5369 0.555954
$$782$$ − 46.5111i − 1.66323i
$$783$$ 1.40105i 0.0500693i
$$784$$ −12.2750 −0.438394
$$785$$ 0 0
$$786$$ −65.8759 −2.34972
$$787$$ − 1.27504i − 0.0454502i −0.999742 0.0227251i $$-0.992766\pi$$
0.999742 0.0227251i $$-0.00723425\pi$$
$$788$$ 78.9194i 2.81139i
$$789$$ 31.7889 1.13172
$$790$$ 0 0
$$791$$ 9.35026 0.332457
$$792$$ − 26.6507i − 0.946991i
$$793$$ − 8.24140i − 0.292661i
$$794$$ 5.64832 0.200452
$$795$$ 0 0
$$796$$ −64.7753 −2.29590
$$797$$ 42.5256i 1.50634i 0.657829 + 0.753168i $$0.271475\pi$$
−0.657829 + 0.753168i $$0.728525\pi$$
$$798$$ 8.96239i 0.317265i
$$799$$ 64.4055 2.27850
$$800$$ 0 0
$$801$$ 22.8021 0.805672
$$802$$ 51.2262i 1.80886i
$$803$$ 11.3684i 0.401181i
$$804$$ −138.901 −4.89865
$$805$$ 0 0
$$806$$ 82.4807 2.90526
$$807$$ − 15.5369i − 0.546925i
$$808$$ − 62.5945i − 2.20207i
$$809$$ 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 $$0.136831\pi$$
$$812$$ − 18.6253i − 0.653620i
$$813$$ 14.2374i 0.499328i
$$814$$ 22.8568 0.801132
$$815$$ 0 0
$$816$$ −165.794 −5.80395
$$817$$ 7.95906i 0.278452i
$$818$$ 49.9937i 1.74799i
$$819$$ 18.4060 0.643157
$$820$$ 0 0
$$821$$ 2.64974 0.0924765 0.0462383 0.998930i $$-0.485277\pi$$
0.0462383 + 0.998930i $$0.485277\pi$$
$$822$$ 72.9643i 2.54492i
$$823$$ 5.76845i 0.201076i 0.994933 + 0.100538i $$0.0320563\pi$$
−0.994933 + 0.100538i $$0.967944\pi$$
$$824$$ 35.6263 1.24110
$$825$$ 0 0
$$826$$ 36.3815 1.26588
$$827$$ 13.4920i 0.469163i 0.972096 + 0.234581i $$0.0753719\pi$$
−0.972096 + 0.234581i $$0.924628\pi$$
$$828$$ − 51.9814i − 1.80648i
$$829$$ 4.70052 0.163256 0.0816280 0.996663i $$-0.473988\pi$$
0.0816280 + 0.996663i $$0.473988\pi$$
$$830$$ 0 0
$$831$$ 20.7367 0.719349
$$832$$ − 105.658i − 3.66305i
$$833$$ 5.44358i 0.188609i
$$834$$ −45.7137 −1.58294
$$835$$ 0 0
$$836$$ 6.96239 0.240799
$$837$$ − 2.05079i − 0.0708855i
$$838$$ 2.06888i 0.0714684i
$$839$$ 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
$$842$$ − 28.1417i − 0.969828i
$$843$$ 20.9624i 0.721983i
$$844$$ 22.8872 0.787809
$$845$$ 0 0
$$846$$ 99.8999 3.43463
$$847$$ 1.00000i 0.0343604i
$$848$$ − 2.84226i − 0.0976036i
$$849$$ 2.07522 0.0712215
$$850$$ 0 0
$$851$$ 27.2896 0.935476
$$852$$ − 198.777i − 6.80998i
$$853$$ 20.6824i 0.708153i 0.935217 + 0.354076i $$0.115205\pi$$
−0.935217 + 0.354076i $$0.884795\pi$$
$$854$$ 3.78067 0.129372
$$855$$ 0 0
$$856$$ 97.4128 3.32950
$$857$$ − 26.3453i − 0.899940i −0.893044 0.449970i $$-0.851435\pi$$
0.893044 0.449970i $$-0.148565\pi$$
$$858$$ − 38.7064i − 1.32141i
$$859$$ 8.51151 0.290409 0.145205 0.989402i $$-0.453616\pi$$
0.145205 + 0.989402i $$0.453616\pi$$
$$860$$ 0 0
$$861$$ −12.4690 −0.424942
$$862$$ − 66.1582i − 2.25336i
$$863$$ 7.56722i 0.257591i 0.991671 + 0.128796i $$0.0411111\pi$$
−0.991671 + 0.128796i $$0.958889\pi$$
$$864$$ −6.18664 −0.210474
$$865$$ 0 0
$$866$$ 49.6502 1.68718
$$867$$ 31.3439i 1.06450i
$$868$$ 27.2628i 0.925360i
$$869$$ 1.96968 0.0668169
$$870$$ 0 0
$$871$$ −63.3112 −2.14522
$$872$$ − 18.4631i − 0.625239i
$$873$$ − 2.63989i − 0.0893467i
$$874$$ 11.5369 0.390242
$$875$$ 0 0
$$876$$ 145.445 4.91413
$$877$$ − 17.2955i − 0.584028i −0.956414 0.292014i $$-0.905675\pi$$
0.956414 0.292014i $$-0.0943254\pi$$
$$878$$ 3.81336i 0.128695i
$$879$$ −6.74543 −0.227518
$$880$$ 0 0
$$881$$ 20.4504 0.688992 0.344496 0.938788i $$-0.388050\pi$$
0.344496 + 0.938788i $$0.388050\pi$$
$$882$$ 8.44358i 0.284310i
$$883$$ − 49.6589i − 1.67116i −0.549371 0.835578i $$-0.685133\pi$$
0.549371 0.835578i $$-0.314867\pi$$
$$884$$ −163.682 −5.50524
$$885$$ 0 0
$$886$$ −107.520 −3.61221
$$887$$ 47.1100i 1.58180i 0.611946 + 0.790900i $$0.290387\pi$$
−0.611946 + 0.790900i $$0.709613\pi$$
$$888$$ − 179.002i − 6.00693i
$$889$$ −16.9624 −0.568900
$$890$$ 0 0
$$891$$ 8.50659 0.284981
$$892$$ 40.1197i 1.34331i
$$893$$ 15.9756i 0.534602i
$$894$$ −151.415 −5.06407
$$895$$ 0 0
$$896$$ 16.5696 0.553551
$$897$$ − 46.2130i − 1.54301i
$$898$$ − 33.8554i − 1.12977i
$$899$$ −19.0982 −0.636962
$$900$$ 0 0
$$901$$ −1.26045 −0.0419917
$$902$$ 13.4436i 0.447622i
$$903$$ 14.6253i 0.486700i
$$904$$ −78.9497 −2.62583
$$905$$ 0 0
$$906$$ −21.5369 −0.715516
$$907$$ − 14.4591i − 0.480107i −0.970760 0.240054i $$-0.922835\pi$$
0.970760 0.240054i $$-0.0771651\pi$$
$$908$$ − 53.8759i − 1.78793i
$$909$$ −23.3987 −0.776085
$$910$$ 0 0
$$911$$ −31.5369 −1.04486 −0.522432 0.852681i $$-0.674975\pi$$
−0.522432 + 0.852681i $$0.674975\pi$$
$$912$$ − 41.1246i − 1.36177i
$$913$$ − 10.6253i − 0.351646i
$$914$$ −1.45580 −0.0481536
$$915$$ 0 0
$$916$$ −151.863 −5.01770
$$917$$ − 9.92478i − 0.327745i
$$918$$ 5.64832i 0.186422i
$$919$$ −5.26328 −0.173620 −0.0868098 0.996225i $$-0.527667\pi$$
−0.0868098 + 0.996225i $$0.527667\pi$$
$$920$$ 0 0
$$921$$ 20.7513 0.683779
$$922$$ − 30.9659i − 1.01981i
$$923$$ − 90.6028i − 2.98223i
$$924$$ 12.7938 0.420887
$$925$$ 0 0
$$926$$ −63.6542 −2.09181
$$927$$ − 13.3176i − 0.437407i
$$928$$ 57.6140i 1.89127i
$$929$$ −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
$$932$$ 45.0191i 1.47465i
$$933$$ − 11.0073i − 0.360363i
$$934$$ −7.13681 −0.233524
$$935$$ 0 0
$$936$$ −155.412 −5.07981
$$937$$ 29.3439i 0.958624i 0.877645 + 0.479312i $$0.159114\pi$$
−0.877645 + 0.479312i $$0.840886\pi$$
$$938$$ − 29.0435i − 0.948304i
$$939$$ 73.9121 2.41203
$$940$$ 0 0
$$941$$ −28.6375 −0.933556 −0.466778 0.884374i $$-0.654585\pi$$
−0.466778 + 0.884374i $$0.654585\pi$$
$$942$$ 36.0118i 1.17333i
$$943$$ 16.0508i 0.522685i
$$944$$ −166.939 −5.43341
$$945$$ 0 0
$$946$$ 15.7685 0.512677
$$947$$ 52.8178i 1.71635i 0.513358 + 0.858174i $$0.328401\pi$$
−0.513358 + 0.858174i $$0.671599\pi$$
$$948$$ − 25.1998i − 0.818452i
$$949$$ 66.2941 2.15200
$$950$$ 0 0
$$951$$ −38.2130 −1.23914
$$952$$ − 45.9633i − 1.48968i
$$953$$ 37.1939i 1.20483i 0.798183 + 0.602415i $$0.205795\pi$$
−0.798183 + 0.602415i $$0.794205\pi$$
$$954$$ −1.95509 −0.0632985
$$955$$ 0 0
$$956$$ −109.388 −3.53787
$$957$$ 8.96239i 0.289713i
$$958$$ 28.6516i 0.925693i
$$959$$ −10.9927 −0.354973
$$960$$ 0 0
$$961$$ −3.04491 −0.0982228
$$962$$ − 133.289i − 4.29740i
$$963$$ − 36.4142i − 1.17343i
$$964$$ 48.1500 1.55081
$$965$$ 0 0
$$966$$ 21.1998 0.682093
$$967$$ − 4.07125i − 0.130923i −0.997855 0.0654613i $$-0.979148\pi$$
0.997855 0.0654613i $$-0.0208519\pi$$
$$968$$ − 8.44358i − 0.271387i
$$969$$ −18.2374 −0.585871
$$970$$ 0 0
$$971$$ 0.773377 0.0248188 0.0124094 0.999923i $$-0.496050\pi$$
0.0124094 + 0.999923i $$0.496050\pi$$
$$972$$ − 114.832i − 3.68324i
$$973$$ − 6.88717i − 0.220792i
$$974$$ 46.6312 1.49416
$$975$$ 0 0
$$976$$ −17.3479 −0.555292
$$977$$ 37.8740i 1.21170i 0.795580 + 0.605848i $$0.207166\pi$$
−0.795580 + 0.605848i $$0.792834\pi$$
$$978$$ 22.4387i 0.717509i
$$979$$ 7.22425 0.230888
$$980$$ 0 0
$$981$$ −6.90175 −0.220356
$$982$$ − 75.8916i − 2.42180i
$$983$$ − 15.5794i − 0.496907i −0.968644 0.248453i $$-0.920078\pi$$
0.968644 0.248453i $$-0.0799223\pi$$
$$984$$ 105.283 3.35629
$$985$$ 0 0
$$986$$ 52.6009 1.67515
$$987$$ 29.3561i 0.934416i
$$988$$ − 40.6009i − 1.29169i
$$989$$ 18.8265 0.598649
$$990$$ 0 0
$$991$$ 27.0982 0.860804 0.430402 0.902637i $$-0.358372\pi$$
0.430402 + 0.902637i $$0.358372\pi$$
$$992$$ − 84.3327i − 2.67756i
$$993$$ − 15.5369i − 0.493049i
$$994$$ 41.5633 1.31831
$$995$$ 0 0
$$996$$ −135.938 −4.30737
$$997$$ − 50.4060i − 1.59637i −0.602410 0.798187i $$-0.705793\pi$$
0.602410 0.798187i $$-0.294207\pi$$
$$998$$ − 73.5026i − 2.32668i
$$999$$ −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.b.n.1849.6 6
5.2 odd 4 385.2.a.f.1.1 3
5.3 odd 4 1925.2.a.v.1.3 3
5.4 even 2 inner 1925.2.b.n.1849.1 6
15.2 even 4 3465.2.a.bh.1.3 3
20.7 even 4 6160.2.a.bn.1.3 3
35.27 even 4 2695.2.a.g.1.1 3
55.32 even 4 4235.2.a.q.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 5.2 odd 4
1925.2.a.v.1.3 3 5.3 odd 4
1925.2.b.n.1849.1 6 5.4 even 2 inner
1925.2.b.n.1849.6 6 1.1 even 1 trivial
2695.2.a.g.1.1 3 35.27 even 4
3465.2.a.bh.1.3 3 15.2 even 4
4235.2.a.q.1.3 3 55.32 even 4
6160.2.a.bn.1.3 3 20.7 even 4