# Properties

 Label 725.2.b.d.349.1 Level $725$ Weight $2$ Character 725.349 Analytic conductor $5.789$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 725.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.78915414654$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ 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: $$2^{3}$$ Twist minimal: no (minimal twist has level 145) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.1 Root $$0.403032 - 0.403032i$$ of defining polynomial Character $$\chi$$ $$=$$ 725.349 Dual form 725.2.b.d.349.6

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.67513i q^{2} -0.806063i q^{3} -5.15633 q^{4} -2.15633 q^{6} +4.15633i q^{7} +8.44358i q^{8} +2.35026 q^{9} +O(q^{10})$$ $$q-2.67513i q^{2} -0.806063i q^{3} -5.15633 q^{4} -2.15633 q^{6} +4.15633i q^{7} +8.44358i q^{8} +2.35026 q^{9} +2.80606 q^{11} +4.15633i q^{12} +1.35026i q^{13} +11.1187 q^{14} +12.2750 q^{16} +7.11871i q^{17} -6.28726i q^{18} -3.76845 q^{19} +3.35026 q^{21} -7.50659i q^{22} +4.80606i q^{23} +6.80606 q^{24} +3.61213 q^{26} -4.31265i q^{27} -21.4314i q^{28} -1.00000 q^{29} +0.231548 q^{31} -15.9502i q^{32} -2.26187i q^{33} +19.0435 q^{34} -12.1187 q^{36} +5.50659i q^{37} +10.0811i q^{38} +1.08840 q^{39} -6.96239 q^{41} -8.96239i q^{42} -3.19394i q^{43} -14.4690 q^{44} +12.8568 q^{46} -6.41819i q^{47} -9.89446i q^{48} -10.2750 q^{49} +5.73813 q^{51} -6.96239i q^{52} +6.96239i q^{53} -11.5369 q^{54} -35.0943 q^{56} +3.03761i q^{57} +2.67513i q^{58} +2.57452 q^{59} +5.35026 q^{61} -0.619421i q^{62} +9.76845i q^{63} -18.1187 q^{64} -6.05079 q^{66} +3.19394i q^{67} -36.7064i q^{68} +3.87399 q^{69} +11.3503 q^{71} +19.8446i q^{72} -11.2447i q^{73} +14.7308 q^{74} +19.4314 q^{76} +11.6629i q^{77} -2.91160i q^{78} +4.73084 q^{79} +3.57452 q^{81} +18.6253i q^{82} +2.54420i q^{83} -17.2750 q^{84} -8.54420 q^{86} +0.806063i q^{87} +23.6932i q^{88} -14.3127 q^{89} -5.61213 q^{91} -24.7816i q^{92} -0.186642i q^{93} -17.1695 q^{94} -12.8568 q^{96} +1.53102i q^{97} +27.4871i q^{98} +6.59498 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 10 q^{4} + 8 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 10 * q^4 + 8 * q^6 - 6 * q^9 $$6 q - 10 q^{4} + 8 q^{6} - 6 q^{9} + 16 q^{11} + 24 q^{14} + 10 q^{16} + 40 q^{24} + 20 q^{26} - 6 q^{29} + 24 q^{31} + 28 q^{34} - 30 q^{36} - 32 q^{39} - 20 q^{41} - 24 q^{44} + 16 q^{46} + 2 q^{49} + 16 q^{51} - 24 q^{54} - 64 q^{56} - 8 q^{59} + 12 q^{61} - 66 q^{64} + 24 q^{66} + 40 q^{69} + 48 q^{71} + 44 q^{74} + 32 q^{76} - 16 q^{79} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 44 q^{89} - 32 q^{91} - 16 q^{96} - 40 q^{99}+O(q^{100})$$ 6 * q - 10 * q^4 + 8 * q^6 - 6 * q^9 + 16 * q^11 + 24 * q^14 + 10 * q^16 + 40 * q^24 + 20 * q^26 - 6 * q^29 + 24 * q^31 + 28 * q^34 - 30 * q^36 - 32 * q^39 - 20 * q^41 - 24 * q^44 + 16 * q^46 + 2 * q^49 + 16 * q^51 - 24 * q^54 - 64 * q^56 - 8 * q^59 + 12 * q^61 - 66 * q^64 + 24 * q^66 + 40 * q^69 + 48 * q^71 + 44 * q^74 + 32 * q^76 - 16 * q^79 - 2 * q^81 - 40 * q^84 - 32 * q^86 - 44 * q^89 - 32 * q^91 - 16 * q^96 - 40 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$$.

 $$n$$ $$176$$ $$552$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$-0.605279\pi$$
0.324745 0.945802i $$-0.394721\pi$$
$$3$$ − 0.806063i − 0.465381i −0.972551 0.232690i $$-0.925247\pi$$
0.972551 0.232690i $$-0.0747529\pi$$
$$4$$ −5.15633 −2.57816
$$5$$ 0 0
$$6$$ −2.15633 −0.880316
$$7$$ 4.15633i 1.57094i 0.618898 + 0.785472i $$0.287580\pi$$
−0.618898 + 0.785472i $$0.712420\pi$$
$$8$$ 8.44358i 2.98526i
$$9$$ 2.35026 0.783421
$$10$$ 0 0
$$11$$ 2.80606 0.846060 0.423030 0.906116i $$-0.360966\pi$$
0.423030 + 0.906116i $$0.360966\pi$$
$$12$$ 4.15633i 1.19983i
$$13$$ 1.35026i 0.374495i 0.982313 + 0.187248i $$0.0599567\pi$$
−0.982313 + 0.187248i $$0.940043\pi$$
$$14$$ 11.1187 2.97160
$$15$$ 0 0
$$16$$ 12.2750 3.06876
$$17$$ 7.11871i 1.72654i 0.504741 + 0.863271i $$0.331588\pi$$
−0.504741 + 0.863271i $$0.668412\pi$$
$$18$$ − 6.28726i − 1.48192i
$$19$$ −3.76845 −0.864542 −0.432271 0.901744i $$-0.642288\pi$$
−0.432271 + 0.901744i $$0.642288\pi$$
$$20$$ 0 0
$$21$$ 3.35026 0.731087
$$22$$ − 7.50659i − 1.60041i
$$23$$ 4.80606i 1.00213i 0.865409 + 0.501067i $$0.167059\pi$$
−0.865409 + 0.501067i $$0.832941\pi$$
$$24$$ 6.80606 1.38928
$$25$$ 0 0
$$26$$ 3.61213 0.708396
$$27$$ − 4.31265i − 0.829970i
$$28$$ − 21.4314i − 4.05015i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 0.231548 0.0415872 0.0207936 0.999784i $$-0.493381\pi$$
0.0207936 + 0.999784i $$0.493381\pi$$
$$32$$ − 15.9502i − 2.81962i
$$33$$ − 2.26187i − 0.393740i
$$34$$ 19.0435 3.26593
$$35$$ 0 0
$$36$$ −12.1187 −2.01979
$$37$$ 5.50659i 0.905277i 0.891694 + 0.452639i $$0.149517\pi$$
−0.891694 + 0.452639i $$0.850483\pi$$
$$38$$ 10.0811i 1.63537i
$$39$$ 1.08840 0.174283
$$40$$ 0 0
$$41$$ −6.96239 −1.08734 −0.543671 0.839298i $$-0.682966\pi$$
−0.543671 + 0.839298i $$0.682966\pi$$
$$42$$ − 8.96239i − 1.38293i
$$43$$ − 3.19394i − 0.487071i −0.969892 0.243535i $$-0.921693\pi$$
0.969892 0.243535i $$-0.0783072\pi$$
$$44$$ −14.4690 −2.18128
$$45$$ 0 0
$$46$$ 12.8568 1.89564
$$47$$ − 6.41819i − 0.936189i −0.883678 0.468095i $$-0.844941\pi$$
0.883678 0.468095i $$-0.155059\pi$$
$$48$$ − 9.89446i − 1.42814i
$$49$$ −10.2750 −1.46786
$$50$$ 0 0
$$51$$ 5.73813 0.803500
$$52$$ − 6.96239i − 0.965510i
$$53$$ 6.96239i 0.956358i 0.878263 + 0.478179i $$0.158703\pi$$
−0.878263 + 0.478179i $$0.841297\pi$$
$$54$$ −11.5369 −1.56997
$$55$$ 0 0
$$56$$ −35.0943 −4.68967
$$57$$ 3.03761i 0.402341i
$$58$$ 2.67513i 0.351262i
$$59$$ 2.57452 0.335173 0.167587 0.985857i $$-0.446403\pi$$
0.167587 + 0.985857i $$0.446403\pi$$
$$60$$ 0 0
$$61$$ 5.35026 0.685031 0.342515 0.939512i $$-0.388721\pi$$
0.342515 + 0.939512i $$0.388721\pi$$
$$62$$ − 0.619421i − 0.0786666i
$$63$$ 9.76845i 1.23071i
$$64$$ −18.1187 −2.26484
$$65$$ 0 0
$$66$$ −6.05079 −0.744800
$$67$$ 3.19394i 0.390201i 0.980783 + 0.195101i $$0.0625034\pi$$
−0.980783 + 0.195101i $$0.937497\pi$$
$$68$$ − 36.7064i − 4.45131i
$$69$$ 3.87399 0.466374
$$70$$ 0 0
$$71$$ 11.3503 1.34703 0.673514 0.739174i $$-0.264784\pi$$
0.673514 + 0.739174i $$0.264784\pi$$
$$72$$ 19.8446i 2.33871i
$$73$$ − 11.2447i − 1.31610i −0.752976 0.658048i $$-0.771383\pi$$
0.752976 0.658048i $$-0.228617\pi$$
$$74$$ 14.7308 1.71243
$$75$$ 0 0
$$76$$ 19.4314 2.22893
$$77$$ 11.6629i 1.32911i
$$78$$ − 2.91160i − 0.329674i
$$79$$ 4.73084 0.532261 0.266131 0.963937i $$-0.414255\pi$$
0.266131 + 0.963937i $$0.414255\pi$$
$$80$$ 0 0
$$81$$ 3.57452 0.397168
$$82$$ 18.6253i 2.05682i
$$83$$ 2.54420i 0.279262i 0.990204 + 0.139631i $$0.0445916\pi$$
−0.990204 + 0.139631i $$0.955408\pi$$
$$84$$ −17.2750 −1.88486
$$85$$ 0 0
$$86$$ −8.54420 −0.921345
$$87$$ 0.806063i 0.0864191i
$$88$$ 23.6932i 2.52571i
$$89$$ −14.3127 −1.51714 −0.758569 0.651593i $$-0.774101\pi$$
−0.758569 + 0.651593i $$0.774101\pi$$
$$90$$ 0 0
$$91$$ −5.61213 −0.588311
$$92$$ − 24.7816i − 2.58366i
$$93$$ − 0.186642i − 0.0193539i
$$94$$ −17.1695 −1.77090
$$95$$ 0 0
$$96$$ −12.8568 −1.31220
$$97$$ 1.53102i 0.155452i 0.996975 + 0.0777260i $$0.0247659\pi$$
−0.996975 + 0.0777260i $$0.975234\pi$$
$$98$$ 27.4871i 2.77661i
$$99$$ 6.59498 0.662821
$$100$$ 0 0
$$101$$ 2.83638 0.282230 0.141115 0.989993i $$-0.454931\pi$$
0.141115 + 0.989993i $$0.454931\pi$$
$$102$$ − 15.3503i − 1.51990i
$$103$$ − 9.89446i − 0.974930i −0.873142 0.487465i $$-0.837922\pi$$
0.873142 0.487465i $$-0.162078\pi$$
$$104$$ −11.4010 −1.11796
$$105$$ 0 0
$$106$$ 18.6253 1.80905
$$107$$ 11.6932i 1.13043i 0.824945 + 0.565214i $$0.191206\pi$$
−0.824945 + 0.565214i $$0.808794\pi$$
$$108$$ 22.2374i 2.13980i
$$109$$ 14.4993 1.38878 0.694390 0.719599i $$-0.255674\pi$$
0.694390 + 0.719599i $$0.255674\pi$$
$$110$$ 0 0
$$111$$ 4.43866 0.421299
$$112$$ 51.0191i 4.82085i
$$113$$ − 16.3938i − 1.54219i −0.636717 0.771097i $$-0.719708\pi$$
0.636717 0.771097i $$-0.280292\pi$$
$$114$$ 8.12601 0.761070
$$115$$ 0 0
$$116$$ 5.15633 0.478753
$$117$$ 3.17347i 0.293387i
$$118$$ − 6.88717i − 0.634015i
$$119$$ −29.5877 −2.71230
$$120$$ 0 0
$$121$$ −3.12601 −0.284183
$$122$$ − 14.3127i − 1.29581i
$$123$$ 5.61213i 0.506028i
$$124$$ −1.19394 −0.107219
$$125$$ 0 0
$$126$$ 26.1319 2.32801
$$127$$ − 15.3561i − 1.36264i −0.731987 0.681319i $$-0.761407\pi$$
0.731987 0.681319i $$-0.238593\pi$$
$$128$$ 16.5696i 1.46456i
$$129$$ −2.57452 −0.226673
$$130$$ 0 0
$$131$$ 1.38058 0.120622 0.0603109 0.998180i $$-0.480791\pi$$
0.0603109 + 0.998180i $$0.480791\pi$$
$$132$$ 11.6629i 1.01513i
$$133$$ − 15.6629i − 1.35815i
$$134$$ 8.54420 0.738106
$$135$$ 0 0
$$136$$ −60.1075 −5.15417
$$137$$ 6.49341i 0.554770i 0.960759 + 0.277385i $$0.0894677\pi$$
−0.960759 + 0.277385i $$0.910532\pi$$
$$138$$ − 10.3634i − 0.882194i
$$139$$ 19.0132 1.61268 0.806338 0.591455i $$-0.201446\pi$$
0.806338 + 0.591455i $$0.201446\pi$$
$$140$$ 0 0
$$141$$ −5.17347 −0.435685
$$142$$ − 30.3634i − 2.54804i
$$143$$ 3.78892i 0.316845i
$$144$$ 28.8496 2.40413
$$145$$ 0 0
$$146$$ −30.0811 −2.48953
$$147$$ 8.28233i 0.683115i
$$148$$ − 28.3938i − 2.33395i
$$149$$ −6.62530 −0.542766 −0.271383 0.962471i $$-0.587481\pi$$
−0.271383 + 0.962471i $$0.587481\pi$$
$$150$$ 0 0
$$151$$ −9.27504 −0.754792 −0.377396 0.926052i $$-0.623180\pi$$
−0.377396 + 0.926052i $$0.623180\pi$$
$$152$$ − 31.8192i − 2.58088i
$$153$$ 16.7308i 1.35261i
$$154$$ 31.1998 2.51415
$$155$$ 0 0
$$156$$ −5.61213 −0.449330
$$157$$ 5.00729i 0.399626i 0.979834 + 0.199813i $$0.0640334\pi$$
−0.979834 + 0.199813i $$0.935967\pi$$
$$158$$ − 12.6556i − 1.00683i
$$159$$ 5.61213 0.445071
$$160$$ 0 0
$$161$$ −19.9756 −1.57429
$$162$$ − 9.56230i − 0.751285i
$$163$$ 7.50659i 0.587961i 0.955811 + 0.293981i $$0.0949801\pi$$
−0.955811 + 0.293981i $$0.905020\pi$$
$$164$$ 35.9003 2.80335
$$165$$ 0 0
$$166$$ 6.80606 0.528253
$$167$$ − 21.8945i − 1.69424i −0.531398 0.847122i $$-0.678333\pi$$
0.531398 0.847122i $$-0.321667\pi$$
$$168$$ 28.2882i 2.18248i
$$169$$ 11.1768 0.859753
$$170$$ 0 0
$$171$$ −8.85685 −0.677300
$$172$$ 16.4690i 1.25575i
$$173$$ 7.02302i 0.533951i 0.963703 + 0.266975i $$0.0860242\pi$$
−0.963703 + 0.266975i $$0.913976\pi$$
$$174$$ 2.15633 0.163471
$$175$$ 0 0
$$176$$ 34.4445 2.59635
$$177$$ − 2.07522i − 0.155983i
$$178$$ 38.2882i 2.86982i
$$179$$ −4.77575 −0.356956 −0.178478 0.983944i $$-0.557117\pi$$
−0.178478 + 0.983944i $$0.557117\pi$$
$$180$$ 0 0
$$181$$ 1.87399 0.139293 0.0696464 0.997572i $$-0.477813\pi$$
0.0696464 + 0.997572i $$0.477813\pi$$
$$182$$ 15.0132i 1.11285i
$$183$$ − 4.31265i − 0.318800i
$$184$$ −40.5804 −2.99163
$$185$$ 0 0
$$186$$ −0.499293 −0.0366099
$$187$$ 19.9756i 1.46076i
$$188$$ 33.0943i 2.41365i
$$189$$ 17.9248 1.30384
$$190$$ 0 0
$$191$$ 19.1187 1.38338 0.691691 0.722194i $$-0.256866\pi$$
0.691691 + 0.722194i $$0.256866\pi$$
$$192$$ 14.6048i 1.05401i
$$193$$ 19.8945i 1.43203i 0.698083 + 0.716017i $$0.254037\pi$$
−0.698083 + 0.716017i $$0.745963\pi$$
$$194$$ 4.09569 0.294053
$$195$$ 0 0
$$196$$ 52.9814 3.78439
$$197$$ 13.5369i 0.964464i 0.876043 + 0.482232i $$0.160174\pi$$
−0.876043 + 0.482232i $$0.839826\pi$$
$$198$$ − 17.6424i − 1.25379i
$$199$$ −2.57452 −0.182503 −0.0912513 0.995828i $$-0.529087\pi$$
−0.0912513 + 0.995828i $$0.529087\pi$$
$$200$$ 0 0
$$201$$ 2.57452 0.181592
$$202$$ − 7.58769i − 0.533868i
$$203$$ − 4.15633i − 0.291717i
$$204$$ −29.5877 −2.07155
$$205$$ 0 0
$$206$$ −26.4690 −1.84418
$$207$$ 11.2955i 0.785092i
$$208$$ 16.5745i 1.14924i
$$209$$ −10.5745 −0.731455
$$210$$ 0 0
$$211$$ 11.8945 0.818848 0.409424 0.912344i $$-0.365730\pi$$
0.409424 + 0.912344i $$0.365730\pi$$
$$212$$ − 35.9003i − 2.46565i
$$213$$ − 9.14903i − 0.626881i
$$214$$ 31.2809 2.13832
$$215$$ 0 0
$$216$$ 36.4142 2.47767
$$217$$ 0.962389i 0.0653312i
$$218$$ − 38.7875i − 2.62702i
$$219$$ −9.06396 −0.612486
$$220$$ 0 0
$$221$$ −9.61213 −0.646582
$$222$$ − 11.8740i − 0.796930i
$$223$$ − 1.11871i − 0.0749146i −0.999298 0.0374573i $$-0.988074\pi$$
0.999298 0.0374573i $$-0.0119258\pi$$
$$224$$ 66.2941 4.42946
$$225$$ 0 0
$$226$$ −43.8554 −2.91722
$$227$$ − 0.0303172i − 0.00201222i −0.999999 0.00100611i $$-0.999680\pi$$
0.999999 0.00100611i $$-0.000320255\pi$$
$$228$$ − 15.6629i − 1.03730i
$$229$$ 5.84955 0.386549 0.193275 0.981145i $$-0.438089\pi$$
0.193275 + 0.981145i $$0.438089\pi$$
$$230$$ 0 0
$$231$$ 9.40105 0.618543
$$232$$ − 8.44358i − 0.554348i
$$233$$ 26.1016i 1.70997i 0.518652 + 0.854985i $$0.326434\pi$$
−0.518652 + 0.854985i $$0.673566\pi$$
$$234$$ 8.48944 0.554972
$$235$$ 0 0
$$236$$ −13.2750 −0.864131
$$237$$ − 3.81336i − 0.247704i
$$238$$ 79.1509i 5.13059i
$$239$$ −1.42548 −0.0922069 −0.0461035 0.998937i $$-0.514680\pi$$
−0.0461035 + 0.998937i $$0.514680\pi$$
$$240$$ 0 0
$$241$$ −0.0752228 −0.00484553 −0.00242276 0.999997i $$-0.500771\pi$$
−0.00242276 + 0.999997i $$0.500771\pi$$
$$242$$ 8.36248i 0.537561i
$$243$$ − 15.8192i − 1.01480i
$$244$$ −27.5877 −1.76612
$$245$$ 0 0
$$246$$ 15.0132 0.957205
$$247$$ − 5.08840i − 0.323767i
$$248$$ 1.95509i 0.124149i
$$249$$ 2.05079 0.129963
$$250$$ 0 0
$$251$$ 16.9829 1.07195 0.535974 0.844234i $$-0.319944\pi$$
0.535974 + 0.844234i $$0.319944\pi$$
$$252$$ − 50.3693i − 3.17297i
$$253$$ 13.4861i 0.847865i
$$254$$ −41.0797 −2.57757
$$255$$ 0 0
$$256$$ 8.08840 0.505525
$$257$$ − 25.1998i − 1.57192i −0.618276 0.785961i $$-0.712169\pi$$
0.618276 0.785961i $$-0.287831\pi$$
$$258$$ 6.88717i 0.428776i
$$259$$ −22.8872 −1.42214
$$260$$ 0 0
$$261$$ −2.35026 −0.145478
$$262$$ − 3.69323i − 0.228168i
$$263$$ − 16.1319i − 0.994735i −0.867540 0.497367i $$-0.834300\pi$$
0.867540 0.497367i $$-0.165700\pi$$
$$264$$ 19.0982 1.17542
$$265$$ 0 0
$$266$$ −41.9003 −2.56907
$$267$$ 11.5369i 0.706047i
$$268$$ − 16.4690i − 1.00600i
$$269$$ −5.28963 −0.322514 −0.161257 0.986912i $$-0.551555\pi$$
−0.161257 + 0.986912i $$0.551555\pi$$
$$270$$ 0 0
$$271$$ 1.13330 0.0688432 0.0344216 0.999407i $$-0.489041\pi$$
0.0344216 + 0.999407i $$0.489041\pi$$
$$272$$ 87.3825i 5.29834i
$$273$$ 4.52373i 0.273789i
$$274$$ 17.3707 1.04940
$$275$$ 0 0
$$276$$ −19.9756 −1.20239
$$277$$ 16.3634i 0.983184i 0.870826 + 0.491592i $$0.163585\pi$$
−0.870826 + 0.491592i $$0.836415\pi$$
$$278$$ − 50.8627i − 3.05054i
$$279$$ 0.544198 0.0325803
$$280$$ 0 0
$$281$$ −24.8265 −1.48103 −0.740513 0.672042i $$-0.765418\pi$$
−0.740513 + 0.672042i $$0.765418\pi$$
$$282$$ 13.8397i 0.824142i
$$283$$ − 4.18076i − 0.248521i −0.992250 0.124260i $$-0.960344\pi$$
0.992250 0.124260i $$-0.0396558\pi$$
$$284$$ −58.5256 −3.47286
$$285$$ 0 0
$$286$$ 10.1359 0.599346
$$287$$ − 28.9380i − 1.70815i
$$288$$ − 37.4871i − 2.20895i
$$289$$ −33.6761 −1.98095
$$290$$ 0 0
$$291$$ 1.23410 0.0723444
$$292$$ 57.9814i 3.39311i
$$293$$ 23.6180i 1.37978i 0.723915 + 0.689889i $$0.242341\pi$$
−0.723915 + 0.689889i $$0.757659\pi$$
$$294$$ 22.1563 1.29218
$$295$$ 0 0
$$296$$ −46.4953 −2.70249
$$297$$ − 12.1016i − 0.702204i
$$298$$ 17.7235i 1.02670i
$$299$$ −6.48944 −0.375294
$$300$$ 0 0
$$301$$ 13.2750 0.765161
$$302$$ 24.8119i 1.42777i
$$303$$ − 2.28630i − 0.131345i
$$304$$ −46.2579 −2.65307
$$305$$ 0 0
$$306$$ 44.7572 2.55860
$$307$$ − 32.5052i − 1.85517i −0.373614 0.927584i $$-0.621882\pi$$
0.373614 0.927584i $$-0.378118\pi$$
$$308$$ − 60.1378i − 3.42667i
$$309$$ −7.97556 −0.453714
$$310$$ 0 0
$$311$$ −9.31994 −0.528486 −0.264243 0.964456i $$-0.585122\pi$$
−0.264243 + 0.964456i $$0.585122\pi$$
$$312$$ 9.18997i 0.520279i
$$313$$ 9.60228i 0.542753i 0.962473 + 0.271376i $$0.0874788\pi$$
−0.962473 + 0.271376i $$0.912521\pi$$
$$314$$ 13.3952 0.755933
$$315$$ 0 0
$$316$$ −24.3938 −1.37226
$$317$$ − 19.3707i − 1.08797i −0.839095 0.543984i $$-0.816915\pi$$
0.839095 0.543984i $$-0.183085\pi$$
$$318$$ − 15.0132i − 0.841897i
$$319$$ −2.80606 −0.157109
$$320$$ 0 0
$$321$$ 9.42548 0.526079
$$322$$ 53.4372i 2.97794i
$$323$$ − 26.8265i − 1.49267i
$$324$$ −18.4314 −1.02396
$$325$$ 0 0
$$326$$ 20.0811 1.11219
$$327$$ − 11.6873i − 0.646312i
$$328$$ − 58.7875i − 3.24600i
$$329$$ 26.6761 1.47070
$$330$$ 0 0
$$331$$ −29.5428 −1.62382 −0.811909 0.583784i $$-0.801572\pi$$
−0.811909 + 0.583784i $$0.801572\pi$$
$$332$$ − 13.1187i − 0.719983i
$$333$$ 12.9419i 0.709213i
$$334$$ −58.5705 −3.20484
$$335$$ 0 0
$$336$$ 41.1246 2.24353
$$337$$ − 12.5442i − 0.683326i −0.939823 0.341663i $$-0.889010\pi$$
0.939823 0.341663i $$-0.110990\pi$$
$$338$$ − 29.8994i − 1.62631i
$$339$$ −13.2144 −0.717708
$$340$$ 0 0
$$341$$ 0.649738 0.0351853
$$342$$ 23.6932i 1.28118i
$$343$$ − 13.6121i − 0.734986i
$$344$$ 26.9683 1.45403
$$345$$ 0 0
$$346$$ 18.7875 1.01002
$$347$$ 25.0943i 1.34713i 0.739127 + 0.673566i $$0.235238\pi$$
−0.739127 + 0.673566i $$0.764762\pi$$
$$348$$ − 4.15633i − 0.222802i
$$349$$ 17.0738 0.913940 0.456970 0.889482i $$-0.348935\pi$$
0.456970 + 0.889482i $$0.348935\pi$$
$$350$$ 0 0
$$351$$ 5.82321 0.310820
$$352$$ − 44.7572i − 2.38557i
$$353$$ 5.66291i 0.301406i 0.988579 + 0.150703i $$0.0481538\pi$$
−0.988579 + 0.150703i $$0.951846\pi$$
$$354$$ −5.55149 −0.295058
$$355$$ 0 0
$$356$$ 73.8007 3.91143
$$357$$ 23.8496i 1.26225i
$$358$$ 12.7757i 0.675219i
$$359$$ −0.755278 −0.0398621 −0.0199310 0.999801i $$-0.506345\pi$$
−0.0199310 + 0.999801i $$0.506345\pi$$
$$360$$ 0 0
$$361$$ −4.79877 −0.252567
$$362$$ − 5.01317i − 0.263487i
$$363$$ 2.51976i 0.132253i
$$364$$ 28.9380 1.51676
$$365$$ 0 0
$$366$$ −11.5369 −0.603044
$$367$$ − 11.4460i − 0.597474i −0.954335 0.298737i $$-0.903435\pi$$
0.954335 0.298737i $$-0.0965653\pi$$
$$368$$ 58.9946i 3.07531i
$$369$$ −16.3634 −0.851846
$$370$$ 0 0
$$371$$ −28.9380 −1.50238
$$372$$ 0.962389i 0.0498975i
$$373$$ 3.86414i 0.200078i 0.994984 + 0.100039i $$0.0318967\pi$$
−0.994984 + 0.100039i $$0.968103\pi$$
$$374$$ 53.4372 2.76317
$$375$$ 0 0
$$376$$ 54.1925 2.79477
$$377$$ − 1.35026i − 0.0695420i
$$378$$ − 47.9511i − 2.46634i
$$379$$ −12.1055 −0.621820 −0.310910 0.950439i $$-0.600634\pi$$
−0.310910 + 0.950439i $$0.600634\pi$$
$$380$$ 0 0
$$381$$ −12.3780 −0.634145
$$382$$ − 51.1451i − 2.61681i
$$383$$ − 10.0205i − 0.512022i −0.966674 0.256011i $$-0.917592\pi$$
0.966674 0.256011i $$-0.0824083\pi$$
$$384$$ 13.3561 0.681578
$$385$$ 0 0
$$386$$ 53.2203 2.70884
$$387$$ − 7.50659i − 0.381581i
$$388$$ − 7.89446i − 0.400780i
$$389$$ 25.6629 1.30116 0.650581 0.759437i $$-0.274526\pi$$
0.650581 + 0.759437i $$0.274526\pi$$
$$390$$ 0 0
$$391$$ −34.2130 −1.73023
$$392$$ − 86.7581i − 4.38195i
$$393$$ − 1.11283i − 0.0561351i
$$394$$ 36.2130 1.82438
$$395$$ 0 0
$$396$$ −34.0059 −1.70886
$$397$$ − 27.7137i − 1.39091i −0.718569 0.695455i $$-0.755203\pi$$
0.718569 0.695455i $$-0.244797\pi$$
$$398$$ 6.88717i 0.345222i
$$399$$ −12.6253 −0.632056
$$400$$ 0 0
$$401$$ 7.42548 0.370811 0.185406 0.982662i $$-0.440640\pi$$
0.185406 + 0.982662i $$0.440640\pi$$
$$402$$ − 6.88717i − 0.343501i
$$403$$ 0.312650i 0.0155742i
$$404$$ −14.6253 −0.727636
$$405$$ 0 0
$$406$$ −11.1187 −0.551812
$$407$$ 15.4518i 0.765919i
$$408$$ 48.4504i 2.39865i
$$409$$ 33.1998 1.64163 0.820813 0.571198i $$-0.193521\pi$$
0.820813 + 0.571198i $$0.193521\pi$$
$$410$$ 0 0
$$411$$ 5.23410 0.258179
$$412$$ 51.0191i 2.51353i
$$413$$ 10.7005i 0.526538i
$$414$$ 30.2170 1.48508
$$415$$ 0 0
$$416$$ 21.5369 1.05593
$$417$$ − 15.3258i − 0.750509i
$$418$$ 28.2882i 1.38362i
$$419$$ −16.5599 −0.809005 −0.404503 0.914537i $$-0.632555\pi$$
−0.404503 + 0.914537i $$0.632555\pi$$
$$420$$ 0 0
$$421$$ −8.82653 −0.430179 −0.215089 0.976594i $$-0.569004\pi$$
−0.215089 + 0.976594i $$0.569004\pi$$
$$422$$ − 31.8192i − 1.54894i
$$423$$ − 15.0844i − 0.733430i
$$424$$ −58.7875 −2.85497
$$425$$ 0 0
$$426$$ −24.4749 −1.18581
$$427$$ 22.2374i 1.07614i
$$428$$ − 60.2941i − 2.91442i
$$429$$ 3.05411 0.147454
$$430$$ 0 0
$$431$$ 4.25202 0.204812 0.102406 0.994743i $$-0.467346\pi$$
0.102406 + 0.994743i $$0.467346\pi$$
$$432$$ − 52.9380i − 2.54698i
$$433$$ − 1.81924i − 0.0874270i −0.999044 0.0437135i $$-0.986081\pi$$
0.999044 0.0437135i $$-0.0139189\pi$$
$$434$$ 2.57452 0.123581
$$435$$ 0 0
$$436$$ −74.7631 −3.58050
$$437$$ − 18.1114i − 0.866387i
$$438$$ 24.2473i 1.15858i
$$439$$ −14.1114 −0.673501 −0.336751 0.941594i $$-0.609328\pi$$
−0.336751 + 0.941594i $$0.609328\pi$$
$$440$$ 0 0
$$441$$ −24.1490 −1.14995
$$442$$ 25.7137i 1.22308i
$$443$$ − 17.2809i − 0.821041i −0.911851 0.410521i $$-0.865347\pi$$
0.911851 0.410521i $$-0.134653\pi$$
$$444$$ −22.8872 −1.08618
$$445$$ 0 0
$$446$$ −2.99271 −0.141709
$$447$$ 5.34041i 0.252593i
$$448$$ − 75.3073i − 3.55793i
$$449$$ −9.35026 −0.441266 −0.220633 0.975357i $$-0.570812\pi$$
−0.220633 + 0.975357i $$0.570812\pi$$
$$450$$ 0 0
$$451$$ −19.5369 −0.919957
$$452$$ 84.5315i 3.97603i
$$453$$ 7.47627i 0.351266i
$$454$$ −0.0811024 −0.00380632
$$455$$ 0 0
$$456$$ −25.6483 −1.20109
$$457$$ 17.6629i 0.826236i 0.910677 + 0.413118i $$0.135560\pi$$
−0.910677 + 0.413118i $$0.864440\pi$$
$$458$$ − 15.6483i − 0.731198i
$$459$$ 30.7005 1.43298
$$460$$ 0 0
$$461$$ 15.5633 0.724853 0.362426 0.932012i $$-0.381948\pi$$
0.362426 + 0.932012i $$0.381948\pi$$
$$462$$ − 25.1490i − 1.17004i
$$463$$ 2.98286i 0.138625i 0.997595 + 0.0693126i $$0.0220806\pi$$
−0.997595 + 0.0693126i $$0.977919\pi$$
$$464$$ −12.2750 −0.569854
$$465$$ 0 0
$$466$$ 69.8251 3.23459
$$467$$ − 34.5804i − 1.60019i −0.599873 0.800095i $$-0.704782\pi$$
0.599873 0.800095i $$-0.295218\pi$$
$$468$$ − 16.3634i − 0.756400i
$$469$$ −13.2750 −0.612984
$$470$$ 0 0
$$471$$ 4.03620 0.185978
$$472$$ 21.7381i 1.00058i
$$473$$ − 8.96239i − 0.412091i
$$474$$ −10.2012 −0.468558
$$475$$ 0 0
$$476$$ 152.564 6.99275
$$477$$ 16.3634i 0.749230i
$$478$$ 3.81336i 0.174419i
$$479$$ 34.1925 1.56230 0.781148 0.624346i $$-0.214634\pi$$
0.781148 + 0.624346i $$0.214634\pi$$
$$480$$ 0 0
$$481$$ −7.43533 −0.339022
$$482$$ 0.201231i 0.00916581i
$$483$$ 16.1016i 0.732647i
$$484$$ 16.1187 0.732669
$$485$$ 0 0
$$486$$ −42.3185 −1.91961
$$487$$ − 38.4953i − 1.74439i −0.489159 0.872195i $$-0.662696\pi$$
0.489159 0.872195i $$-0.337304\pi$$
$$488$$ 45.1754i 2.04499i
$$489$$ 6.05079 0.273626
$$490$$ 0 0
$$491$$ −27.4676 −1.23959 −0.619797 0.784762i $$-0.712785\pi$$
−0.619797 + 0.784762i $$0.712785\pi$$
$$492$$ − 28.9380i − 1.30462i
$$493$$ − 7.11871i − 0.320611i
$$494$$ −13.6121 −0.612439
$$495$$ 0 0
$$496$$ 2.84226 0.127621
$$497$$ 47.1754i 2.11610i
$$498$$ − 5.48612i − 0.245839i
$$499$$ 32.1016 1.43706 0.718532 0.695494i $$-0.244814\pi$$
0.718532 + 0.695494i $$0.244814\pi$$
$$500$$ 0 0
$$501$$ −17.6483 −0.788469
$$502$$ − 45.4314i − 2.02770i
$$503$$ 9.74401i 0.434464i 0.976120 + 0.217232i $$0.0697028\pi$$
−0.976120 + 0.217232i $$0.930297\pi$$
$$504$$ −82.4807 −3.67398
$$505$$ 0 0
$$506$$ 36.0771 1.60382
$$507$$ − 9.00920i − 0.400113i
$$508$$ 79.1813i 3.51310i
$$509$$ 8.57452 0.380059 0.190029 0.981778i $$-0.439142\pi$$
0.190029 + 0.981778i $$0.439142\pi$$
$$510$$ 0 0
$$511$$ 46.7367 2.06751
$$512$$ 11.5017i 0.508306i
$$513$$ 16.2520i 0.717544i
$$514$$ −67.4128 −2.97345
$$515$$ 0 0
$$516$$ 13.2750 0.584401
$$517$$ − 18.0098i − 0.792072i
$$518$$ 61.2262i 2.69012i
$$519$$ 5.66100 0.248490
$$520$$ 0 0
$$521$$ −37.8251 −1.65715 −0.828574 0.559879i $$-0.810848\pi$$
−0.828574 + 0.559879i $$0.810848\pi$$
$$522$$ 6.28726i 0.275186i
$$523$$ − 13.5818i − 0.593891i −0.954894 0.296946i $$-0.904032\pi$$
0.954894 0.296946i $$-0.0959680\pi$$
$$524$$ −7.11871 −0.310982
$$525$$ 0 0
$$526$$ −43.1549 −1.88164
$$527$$ 1.64832i 0.0718021i
$$528$$ − 27.7645i − 1.20829i
$$529$$ −0.0982457 −0.00427155
$$530$$ 0 0
$$531$$ 6.05079 0.262582
$$532$$ 80.7631i 3.50152i
$$533$$ − 9.40105i − 0.407205i
$$534$$ 30.8627 1.33556
$$535$$ 0 0
$$536$$ −26.9683 −1.16485
$$537$$ 3.84955i 0.166121i
$$538$$ 14.1504i 0.610069i
$$539$$ −28.8324 −1.24190
$$540$$ 0 0
$$541$$ 42.3127 1.81916 0.909581 0.415526i $$-0.136402\pi$$
0.909581 + 0.415526i $$0.136402\pi$$
$$542$$ − 3.03173i − 0.130224i
$$543$$ − 1.51056i − 0.0648242i
$$544$$ 113.545 4.86819
$$545$$ 0 0
$$546$$ 12.1016 0.517899
$$547$$ 36.4690i 1.55930i 0.626215 + 0.779650i $$0.284603\pi$$
−0.626215 + 0.779650i $$0.715397\pi$$
$$548$$ − 33.4821i − 1.43029i
$$549$$ 12.5745 0.536667
$$550$$ 0 0
$$551$$ 3.76845 0.160541
$$552$$ 32.7104i 1.39225i
$$553$$ 19.6629i 0.836152i
$$554$$ 43.7743 1.85979
$$555$$ 0 0
$$556$$ −98.0381 −4.15774
$$557$$ − 19.5223i − 0.827187i −0.910462 0.413594i $$-0.864273\pi$$
0.910462 0.413594i $$-0.135727\pi$$
$$558$$ − 1.45580i − 0.0616290i
$$559$$ 4.31265 0.182406
$$560$$ 0 0
$$561$$ 16.1016 0.679809
$$562$$ 66.4142i 2.80151i
$$563$$ − 2.94192i − 0.123987i −0.998077 0.0619936i $$-0.980254\pi$$
0.998077 0.0619936i $$-0.0197458\pi$$
$$564$$ 26.6761 1.12327
$$565$$ 0 0
$$566$$ −11.1841 −0.470102
$$567$$ 14.8568i 0.623929i
$$568$$ 95.8369i 4.02123i
$$569$$ −2.49929 −0.104776 −0.0523879 0.998627i $$-0.516683\pi$$
−0.0523879 + 0.998627i $$0.516683\pi$$
$$570$$ 0 0
$$571$$ 43.8007 1.83300 0.916501 0.400033i $$-0.131001\pi$$
0.916501 + 0.400033i $$0.131001\pi$$
$$572$$ − 19.5369i − 0.816879i
$$573$$ − 15.4109i − 0.643799i
$$574$$ −77.4128 −3.23115
$$575$$ 0 0
$$576$$ −42.5837 −1.77432
$$577$$ 23.9062i 0.995229i 0.867398 + 0.497614i $$0.165791\pi$$
−0.867398 + 0.497614i $$0.834209\pi$$
$$578$$ 90.0879i 3.74716i
$$579$$ 16.0362 0.666442
$$580$$ 0 0
$$581$$ −10.5745 −0.438705
$$582$$ − 3.30139i − 0.136847i
$$583$$ 19.5369i 0.809136i
$$584$$ 94.9457 3.92888
$$585$$ 0 0
$$586$$ 63.1813 2.60999
$$587$$ 3.71767i 0.153445i 0.997053 + 0.0767223i $$0.0244455\pi$$
−0.997053 + 0.0767223i $$0.975555\pi$$
$$588$$ − 42.7064i − 1.76118i
$$589$$ −0.872577 −0.0359539
$$590$$ 0 0
$$591$$ 10.9116 0.448843
$$592$$ 67.5936i 2.77808i
$$593$$ − 25.5125i − 1.04767i −0.851819 0.523836i $$-0.824501\pi$$
0.851819 0.523836i $$-0.175499\pi$$
$$594$$ −32.3733 −1.32829
$$595$$ 0 0
$$596$$ 34.1622 1.39934
$$597$$ 2.07522i 0.0849332i
$$598$$ 17.3601i 0.709908i
$$599$$ 15.6834 0.640806 0.320403 0.947281i $$-0.396182\pi$$
0.320403 + 0.947281i $$0.396182\pi$$
$$600$$ 0 0
$$601$$ 13.1392 0.535958 0.267979 0.963425i $$-0.413644\pi$$
0.267979 + 0.963425i $$0.413644\pi$$
$$602$$ − 35.5125i − 1.44738i
$$603$$ 7.50659i 0.305692i
$$604$$ 47.8251 1.94598
$$605$$ 0 0
$$606$$ −6.11616 −0.248452
$$607$$ − 18.1465i − 0.736543i −0.929718 0.368271i $$-0.879950\pi$$
0.929718 0.368271i $$-0.120050\pi$$
$$608$$ 60.1075i 2.43768i
$$609$$ −3.35026 −0.135759
$$610$$ 0 0
$$611$$ 8.66624 0.350598
$$612$$ − 86.2697i − 3.48724i
$$613$$ − 32.5501i − 1.31469i −0.753592 0.657343i $$-0.771680\pi$$
0.753592 0.657343i $$-0.228320\pi$$
$$614$$ −86.9556 −3.50924
$$615$$ 0 0
$$616$$ −98.4768 −3.96774
$$617$$ − 20.2433i − 0.814965i −0.913213 0.407482i $$-0.866407\pi$$
0.913213 0.407482i $$-0.133593\pi$$
$$618$$ 21.3357i 0.858247i
$$619$$ 16.2071 0.651419 0.325709 0.945470i $$-0.394397\pi$$
0.325709 + 0.945470i $$0.394397\pi$$
$$620$$ 0 0
$$621$$ 20.7269 0.831741
$$622$$ 24.9321i 0.999685i
$$623$$ − 59.4880i − 2.38334i
$$624$$ 13.3601 0.534832
$$625$$ 0 0
$$626$$ 25.6873 1.02667
$$627$$ 8.52373i 0.340405i
$$628$$ − 25.8192i − 1.03030i
$$629$$ −39.1998 −1.56300
$$630$$ 0 0
$$631$$ 2.13586 0.0850271 0.0425136 0.999096i $$-0.486463\pi$$
0.0425136 + 0.999096i $$0.486463\pi$$
$$632$$ 39.9452i 1.58894i
$$633$$ − 9.58769i − 0.381076i
$$634$$ −51.8192 −2.05800
$$635$$ 0 0
$$636$$ −28.9380 −1.14746
$$637$$ − 13.8740i − 0.549708i
$$638$$ 7.50659i 0.297189i
$$639$$ 26.6761 1.05529
$$640$$ 0 0
$$641$$ −14.0362 −0.554396 −0.277198 0.960813i $$-0.589406\pi$$
−0.277198 + 0.960813i $$0.589406\pi$$
$$642$$ − 25.2144i − 0.995133i
$$643$$ − 43.8799i − 1.73045i −0.501381 0.865227i $$-0.667174\pi$$
0.501381 0.865227i $$-0.332826\pi$$
$$644$$ 103.000 4.05879
$$645$$ 0 0
$$646$$ −71.7645 −2.82354
$$647$$ 22.5560i 0.886766i 0.896332 + 0.443383i $$0.146222\pi$$
−0.896332 + 0.443383i $$0.853778\pi$$
$$648$$ 30.1817i 1.18565i
$$649$$ 7.22425 0.283577
$$650$$ 0 0
$$651$$ 0.775746 0.0304039
$$652$$ − 38.7064i − 1.51586i
$$653$$ 27.3054i 1.06854i 0.845314 + 0.534271i $$0.179414\pi$$
−0.845314 + 0.534271i $$0.820586\pi$$
$$654$$ −31.2652 −1.22257
$$655$$ 0 0
$$656$$ −85.4636 −3.33679
$$657$$ − 26.4280i − 1.03106i
$$658$$ − 71.3620i − 2.78198i
$$659$$ −14.0665 −0.547954 −0.273977 0.961736i $$-0.588339\pi$$
−0.273977 + 0.961736i $$0.588339\pi$$
$$660$$ 0 0
$$661$$ 38.9741 1.51592 0.757959 0.652302i $$-0.226197\pi$$
0.757959 + 0.652302i $$0.226197\pi$$
$$662$$ 79.0308i 3.07162i
$$663$$ 7.74798i 0.300907i
$$664$$ −21.4821 −0.833669
$$665$$ 0 0
$$666$$ 34.6213 1.34155
$$667$$ − 4.80606i − 0.186092i
$$668$$ 112.895i 4.36804i
$$669$$ −0.901754 −0.0348638
$$670$$ 0 0
$$671$$ 15.0132 0.579577
$$672$$ − 53.4372i − 2.06139i
$$673$$ 20.3390i 0.784011i 0.919963 + 0.392005i $$0.128219\pi$$
−0.919963 + 0.392005i $$0.871781\pi$$
$$674$$ −33.5574 −1.29258
$$675$$ 0 0
$$676$$ −57.6312 −2.21658
$$677$$ 19.1841i 0.737304i 0.929567 + 0.368652i $$0.120181\pi$$
−0.929567 + 0.368652i $$0.879819\pi$$
$$678$$ 35.3503i 1.35762i
$$679$$ −6.36344 −0.244206
$$680$$ 0 0
$$681$$ −0.0244376 −0.000936449 0
$$682$$ − 1.73813i − 0.0665566i
$$683$$ − 24.1319i − 0.923381i −0.887041 0.461691i $$-0.847243\pi$$
0.887041 0.461691i $$-0.152757\pi$$
$$684$$ 45.6688 1.74619
$$685$$ 0 0
$$686$$ −36.4142 −1.39030
$$687$$ − 4.71511i − 0.179893i
$$688$$ − 39.2057i − 1.49470i
$$689$$ −9.40105 −0.358151
$$690$$ 0 0
$$691$$ −4.28821 −0.163131 −0.0815657 0.996668i $$-0.525992\pi$$
−0.0815657 + 0.996668i $$0.525992\pi$$
$$692$$ − 36.2130i − 1.37661i
$$693$$ 27.4109i 1.04125i
$$694$$ 67.1305 2.54824
$$695$$ 0 0
$$696$$ −6.80606 −0.257983
$$697$$ − 49.5633i − 1.87734i
$$698$$ − 45.6747i − 1.72881i
$$699$$ 21.0395 0.795788
$$700$$ 0 0
$$701$$ −14.1260 −0.533532 −0.266766 0.963761i $$-0.585955\pi$$
−0.266766 + 0.963761i $$0.585955\pi$$
$$702$$ − 15.5778i − 0.587948i
$$703$$ − 20.7513i − 0.782650i
$$704$$ −50.8423 −1.91619
$$705$$ 0 0
$$706$$ 15.1490 0.570141
$$707$$ 11.7889i 0.443368i
$$708$$ 10.7005i 0.402150i
$$709$$ −6.75131 −0.253551 −0.126775 0.991931i $$-0.540463\pi$$
−0.126775 + 0.991931i $$0.540463\pi$$
$$710$$ 0 0
$$711$$ 11.1187 0.416984
$$712$$ − 120.850i − 4.52905i
$$713$$ 1.11283i 0.0416760i
$$714$$ 63.8007 2.38768
$$715$$ 0 0
$$716$$ 24.6253 0.920291
$$717$$ 1.14903i 0.0429113i
$$718$$ 2.02047i 0.0754032i
$$719$$ 43.8251 1.63440 0.817201 0.576353i $$-0.195525\pi$$
0.817201 + 0.576353i $$0.195525\pi$$
$$720$$ 0 0
$$721$$ 41.1246 1.53156
$$722$$ 12.8373i 0.477756i
$$723$$ 0.0606343i 0.00225502i
$$724$$ −9.66291 −0.359119
$$725$$ 0 0
$$726$$ 6.74069 0.250170
$$727$$ 14.8813i 0.551916i 0.961170 + 0.275958i $$0.0889951\pi$$
−0.961170 + 0.275958i $$0.911005\pi$$
$$728$$ − 47.3865i − 1.75626i
$$729$$ −2.02776 −0.0751023
$$730$$ 0 0
$$731$$ 22.7367 0.840948
$$732$$ 22.2374i 0.821919i
$$733$$ 7.17935i 0.265175i 0.991171 + 0.132588i $$0.0423286\pi$$
−0.991171 + 0.132588i $$0.957671\pi$$
$$734$$ −30.6194 −1.13018
$$735$$ 0 0
$$736$$ 76.6575 2.82563
$$737$$ 8.96239i 0.330134i
$$738$$ 43.7743i 1.61136i
$$739$$ −16.1709 −0.594857 −0.297428 0.954744i $$-0.596129\pi$$
−0.297428 + 0.954744i $$0.596129\pi$$
$$740$$ 0 0
$$741$$ −4.10157 −0.150675
$$742$$ 77.4128i 2.84191i
$$743$$ 27.8192i 1.02059i 0.860000 + 0.510294i $$0.170464\pi$$
−0.860000 + 0.510294i $$0.829536\pi$$
$$744$$ 1.57593 0.0577764
$$745$$ 0 0
$$746$$ 10.3371 0.378468
$$747$$ 5.97953i 0.218780i
$$748$$ − 103.000i − 3.76607i
$$749$$ −48.6009 −1.77584
$$750$$ 0 0
$$751$$ −17.6326 −0.643423 −0.321711 0.946838i $$-0.604258\pi$$
−0.321711 + 0.946838i $$0.604258\pi$$
$$752$$ − 78.7835i − 2.87294i
$$753$$ − 13.6893i − 0.498864i
$$754$$ −3.61213 −0.131546
$$755$$ 0 0
$$756$$ −92.4260 −3.36150
$$757$$ − 1.53102i − 0.0556460i −0.999613 0.0278230i $$-0.991143\pi$$
0.999613 0.0278230i $$-0.00885748\pi$$
$$758$$ 32.3839i 1.17624i
$$759$$ 10.8707 0.394580
$$760$$ 0 0
$$761$$ −34.4749 −1.24971 −0.624856 0.780740i $$-0.714842\pi$$
−0.624856 + 0.780740i $$0.714842\pi$$
$$762$$ 33.1128i 1.19955i
$$763$$ 60.2638i 2.18170i
$$764$$ −98.5823 −3.56658
$$765$$ 0 0
$$766$$ −26.8061 −0.968542
$$767$$ 3.47627i 0.125521i
$$768$$ − 6.51976i − 0.235262i
$$769$$ 19.3404 0.697433 0.348717 0.937228i $$-0.386618\pi$$
0.348717 + 0.937228i $$0.386618\pi$$
$$770$$ 0 0
$$771$$ −20.3127 −0.731542
$$772$$ − 102.582i − 3.69202i
$$773$$ 36.2677i 1.30446i 0.758021 + 0.652230i $$0.226166\pi$$
−0.758021 + 0.652230i $$0.773834\pi$$
$$774$$ −20.0811 −0.721800
$$775$$ 0 0
$$776$$ −12.9273 −0.464064
$$777$$ 18.4485i 0.661837i
$$778$$ − 68.6516i − 2.46128i
$$779$$ 26.2374 0.940053
$$780$$ 0 0
$$781$$ 31.8496 1.13967
$$782$$ 91.5242i 3.27290i
$$783$$ 4.31265i 0.154122i
$$784$$ −126.127 −4.50452
$$785$$ 0 0
$$786$$ −2.97698 −0.106185
$$787$$ − 32.0059i − 1.14089i −0.821337 0.570443i $$-0.806771\pi$$
0.821337 0.570443i $$-0.193229\pi$$
$$788$$ − 69.8007i − 2.48655i
$$789$$ −13.0033 −0.462931
$$790$$ 0 0
$$791$$ 68.1378 2.42270
$$792$$ 55.6853i 1.97869i
$$793$$ 7.22425i 0.256541i
$$794$$ −74.1378 −2.63105
$$795$$ 0 0
$$796$$ 13.2750 0.470521
$$797$$ 41.6932i 1.47685i 0.674336 + 0.738425i $$0.264430\pi$$
−0.674336 + 0.738425i $$0.735570\pi$$
$$798$$ 33.7743i 1.19560i
$$799$$ 45.6893 1.61637
$$800$$ 0 0
$$801$$ −33.6385 −1.18856
$$802$$ − 19.8641i − 0.701427i
$$803$$ − 31.5534i − 1.11350i
$$804$$ −13.2750 −0.468175
$$805$$ 0 0
$$806$$ 0.836381 0.0294603
$$807$$ 4.26378i 0.150092i
$$808$$ 23.9492i 0.842530i
$$809$$ −30.6371 −1.07714 −0.538571 0.842580i $$-0.681036\pi$$
−0.538571 + 0.842580i $$0.681036\pi$$
$$810$$ 0 0
$$811$$ 25.4617 0.894081 0.447040 0.894514i $$-0.352478\pi$$
0.447040 + 0.894514i $$0.352478\pi$$
$$812$$ 21.4314i 0.752093i
$$813$$ − 0.913513i − 0.0320383i
$$814$$ 41.3357 1.44881
$$815$$ 0 0
$$816$$ 70.4358 2.46575
$$817$$ 12.0362i 0.421093i
$$818$$ − 88.8139i − 3.10530i
$$819$$ −13.1900 −0.460895
$$820$$ 0 0
$$821$$ −32.7005 −1.14126 −0.570628 0.821209i $$-0.693300\pi$$
−0.570628 + 0.821209i $$0.693300\pi$$
$$822$$ − 14.0019i − 0.488373i
$$823$$ − 31.1041i − 1.08422i −0.840307 0.542111i $$-0.817625\pi$$
0.840307 0.542111i $$-0.182375\pi$$
$$824$$ 83.5447 2.91042
$$825$$ 0 0
$$826$$ 28.6253 0.996002
$$827$$ 1.58181i 0.0550049i 0.999622 + 0.0275025i $$0.00875541\pi$$
−0.999622 + 0.0275025i $$0.991245\pi$$
$$828$$ − 58.2433i − 2.02409i
$$829$$ 0.111420 0.00386976 0.00193488 0.999998i $$-0.499384\pi$$
0.00193488 + 0.999998i $$0.499384\pi$$
$$830$$ 0 0
$$831$$ 13.1900 0.457555
$$832$$ − 24.4650i − 0.848171i
$$833$$ − 73.1451i − 2.53433i
$$834$$ −40.9986 −1.41966
$$835$$ 0 0
$$836$$ 54.5256 1.88581
$$837$$ − 0.998585i − 0.0345162i
$$838$$ 44.3000i 1.53032i
$$839$$ −28.9829 −1.00060 −0.500300 0.865852i $$-0.666777\pi$$
−0.500300 + 0.865852i $$0.666777\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 23.6121i 0.813728i
$$843$$ 20.0118i 0.689242i
$$844$$ −61.3317 −2.11112
$$845$$ 0 0
$$846$$ −40.3528 −1.38736
$$847$$ − 12.9927i − 0.446435i
$$848$$ 85.4636i 2.93483i
$$849$$ −3.36996 −0.115657
$$850$$ 0 0
$$851$$ −26.4650 −0.907209
$$852$$ 47.1754i 1.61620i
$$853$$ − 7.77319i − 0.266149i −0.991106 0.133075i $$-0.957515\pi$$
0.991106 0.133075i $$-0.0424850\pi$$
$$854$$ 59.4880 2.03564
$$855$$ 0 0
$$856$$ −98.7328 −3.37462
$$857$$ 13.8740i 0.473927i 0.971519 + 0.236963i $$0.0761521\pi$$
−0.971519 + 0.236963i $$0.923848\pi$$
$$858$$ − 8.17014i − 0.278924i
$$859$$ 15.2809 0.521378 0.260689 0.965423i $$-0.416050\pi$$
0.260689 + 0.965423i $$0.416050\pi$$
$$860$$ 0 0
$$861$$ −23.3258 −0.794942
$$862$$ − 11.3747i − 0.387424i
$$863$$ 31.2301i 1.06309i 0.847031 + 0.531543i $$0.178388\pi$$
−0.847031 + 0.531543i $$0.821612\pi$$
$$864$$ −68.7875 −2.34020
$$865$$ 0 0
$$866$$ −4.86670 −0.165377
$$867$$ 27.1451i 0.921895i
$$868$$ − 4.96239i − 0.168434i
$$869$$ 13.2750 0.450325
$$870$$ 0 0
$$871$$ −4.31265 −0.146129
$$872$$ 122.426i 4.14587i
$$873$$ 3.59831i 0.121784i
$$874$$ −48.4504 −1.63886
$$875$$ 0 0
$$876$$ 46.7367 1.57909
$$877$$ 4.26187i 0.143913i 0.997408 + 0.0719565i $$0.0229243\pi$$
−0.997408 + 0.0719565i $$0.977076\pi$$
$$878$$ 37.7499i 1.27400i
$$879$$ 19.0376 0.642123
$$880$$ 0 0
$$881$$ 15.2144 0.512586 0.256293 0.966599i $$-0.417499\pi$$
0.256293 + 0.966599i $$0.417499\pi$$
$$882$$ 64.6018i 2.17526i
$$883$$ − 13.7078i − 0.461305i −0.973036 0.230652i $$-0.925914\pi$$
0.973036 0.230652i $$-0.0740860\pi$$
$$884$$ 49.5633 1.66699
$$885$$ 0 0
$$886$$ −46.2287 −1.55308
$$887$$ − 12.6556i − 0.424934i −0.977168 0.212467i $$-0.931850\pi$$
0.977168 0.212467i $$-0.0681498\pi$$
$$888$$ 37.4782i 1.25769i
$$889$$ 63.8251 2.14063
$$890$$ 0 0
$$891$$ 10.0303 0.336028
$$892$$ 5.76845i 0.193142i
$$893$$ 24.1866i 0.809375i
$$894$$ 14.2863 0.477805
$$895$$ 0 0
$$896$$ −68.8686 −2.30074
$$897$$ 5.23090i 0.174655i
$$898$$ 25.0132i 0.834700i
$$899$$ −0.231548 −0.00772256
$$900$$ 0 0
$$901$$ −49.5633 −1.65119
$$902$$ 52.2638i 1.74019i
$$903$$ − 10.7005i − 0.356091i
$$904$$ 138.422 4.60385
$$905$$ 0 0
$$906$$ 20.0000 0.664455
$$907$$ 12.5540i 0.416850i 0.978038 + 0.208425i $$0.0668338\pi$$
−0.978038 + 0.208425i $$0.933166\pi$$
$$908$$ 0.156325i 0.00518783i
$$909$$ 6.66624 0.221105
$$910$$ 0 0
$$911$$ 22.8714 0.757765 0.378882 0.925445i $$-0.376309\pi$$
0.378882 + 0.925445i $$0.376309\pi$$
$$912$$ 37.2868i 1.23469i
$$913$$ 7.13918i 0.236272i
$$914$$ 47.2506 1.56291
$$915$$ 0 0
$$916$$ −30.1622 −0.996587
$$917$$ 5.73813i 0.189490i
$$918$$ − 82.1279i − 2.71063i
$$919$$ −9.67750 −0.319231 −0.159616 0.987179i $$-0.551025\pi$$
−0.159616 + 0.987179i $$0.551025\pi$$
$$920$$ 0 0
$$921$$ −26.2012 −0.863360
$$922$$ − 41.6337i − 1.37113i
$$923$$ 15.3258i 0.504456i
$$924$$ −48.4749 −1.59471
$$925$$ 0 0
$$926$$ 7.97953 0.262224
$$927$$ − 23.2546i − 0.763780i
$$928$$ 15.9502i 0.523590i
$$929$$ −51.9248 −1.70360 −0.851798 0.523870i $$-0.824488\pi$$
−0.851798 + 0.523870i $$0.824488\pi$$
$$930$$ 0 0
$$931$$ 38.7210 1.26903
$$932$$ − 134.588i − 4.40858i
$$933$$ 7.51247i 0.245947i
$$934$$ −92.5071 −3.02692
$$935$$ 0 0
$$936$$ −26.7954 −0.875837
$$937$$ − 3.58769i − 0.117205i −0.998281 0.0586024i $$-0.981336\pi$$
0.998281 0.0586024i $$-0.0186644\pi$$
$$938$$ 35.5125i 1.15952i
$$939$$ 7.74004 0.252587
$$940$$ 0 0
$$941$$ −18.6253 −0.607167 −0.303584 0.952805i $$-0.598183\pi$$
−0.303584 + 0.952805i $$0.598183\pi$$
$$942$$ − 10.7974i − 0.351797i
$$943$$ − 33.4617i − 1.08966i
$$944$$ 31.6023 1.02857
$$945$$ 0 0
$$946$$ −23.9756 −0.779513
$$947$$ 16.5950i 0.539265i 0.962963 + 0.269632i $$0.0869021\pi$$
−0.962963 + 0.269632i $$0.913098\pi$$
$$948$$ 19.6629i 0.638622i
$$949$$ 15.1833 0.492871
$$950$$ 0 0
$$951$$ −15.6140 −0.506320
$$952$$ − 249.826i − 8.09691i
$$953$$ − 12.7005i − 0.411410i −0.978614 0.205705i $$-0.934051\pi$$
0.978614 0.205705i $$-0.0659488\pi$$
$$954$$ 43.7743 1.41725
$$955$$ 0 0
$$956$$ 7.35026 0.237724
$$957$$ 2.26187i 0.0731157i
$$958$$ − 91.4695i − 2.95524i
$$959$$ −26.9887 −0.871512
$$960$$ 0 0
$$961$$ −30.9464 −0.998271
$$962$$ 19.8905i 0.641295i
$$963$$ 27.4821i 0.885600i
$$964$$ 0.387873 0.0124926
$$965$$ 0 0
$$966$$ 43.0738 1.38588
$$967$$ − 37.4314i − 1.20371i −0.798605 0.601856i $$-0.794428\pi$$
0.798605 0.601856i $$-0.205572\pi$$
$$968$$ − 26.3947i − 0.848358i
$$969$$ −21.6239 −0.694659
$$970$$ 0 0
$$971$$ 7.51644 0.241214 0.120607 0.992700i $$-0.461516\pi$$
0.120607 + 0.992700i $$0.461516\pi$$
$$972$$ 81.5691i 2.61633i
$$973$$ 79.0249i 2.53342i
$$974$$ −102.980 −3.29969
$$975$$ 0 0
$$976$$ 65.6747 2.10220
$$977$$ 2.52847i 0.0808929i 0.999182 + 0.0404465i $$0.0128780\pi$$
−0.999182 + 0.0404465i $$0.987122\pi$$
$$978$$ − 16.1866i − 0.517592i
$$979$$ −40.1622 −1.28359
$$980$$ 0 0
$$981$$ 34.0771 1.08800
$$982$$ 73.4793i 2.34482i
$$983$$ − 9.32979i − 0.297574i −0.988869 0.148787i $$-0.952463\pi$$
0.988869 0.148787i $$-0.0475369\pi$$
$$984$$ −47.3865 −1.51063
$$985$$ 0 0
$$986$$ −19.0435 −0.606468
$$987$$ − 21.5026i − 0.684436i
$$988$$ 26.2374i 0.834724i
$$989$$ 15.3503 0.488110
$$990$$ 0 0
$$991$$ 38.4241 1.22058 0.610290 0.792178i $$-0.291053\pi$$
0.610290 + 0.792178i $$0.291053\pi$$
$$992$$ − 3.69323i − 0.117260i
$$993$$ 23.8134i 0.755694i
$$994$$ 126.200 4.00283
$$995$$ 0 0
$$996$$ −10.5745 −0.335066
$$997$$ 18.8423i 0.596740i 0.954450 + 0.298370i $$0.0964430\pi$$
−0.954450 + 0.298370i $$0.903557\pi$$
$$998$$ − 85.8759i − 2.71835i
$$999$$ 23.7480 0.751353
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 725.2.b.d.349.1 6
5.2 odd 4 145.2.a.d.1.3 3
5.3 odd 4 725.2.a.d.1.1 3
5.4 even 2 inner 725.2.b.d.349.6 6
15.2 even 4 1305.2.a.o.1.1 3
15.8 even 4 6525.2.a.bh.1.3 3
20.7 even 4 2320.2.a.s.1.2 3
35.27 even 4 7105.2.a.p.1.3 3
40.27 even 4 9280.2.a.bm.1.2 3
40.37 odd 4 9280.2.a.bu.1.2 3
145.57 odd 4 4205.2.a.e.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.3 3 5.2 odd 4
725.2.a.d.1.1 3 5.3 odd 4
725.2.b.d.349.1 6 1.1 even 1 trivial
725.2.b.d.349.6 6 5.4 even 2 inner
1305.2.a.o.1.1 3 15.2 even 4
2320.2.a.s.1.2 3 20.7 even 4
4205.2.a.e.1.1 3 145.57 odd 4
6525.2.a.bh.1.3 3 15.8 even 4
7105.2.a.p.1.3 3 35.27 even 4
9280.2.a.bm.1.2 3 40.27 even 4
9280.2.a.bu.1.2 3 40.37 odd 4