# Properties

 Label 2550.2.d.v.2449.3 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.d (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: $$20.3618525154$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.3 Root $$-2.37228i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.v.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.37228i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.37228 q^{11} +1.00000i q^{12} -2.00000i q^{13} +2.37228 q^{14} +1.00000 q^{16} -1.00000i q^{17} -1.00000i q^{18} +2.37228 q^{19} -2.37228 q^{21} +4.37228i q^{22} +1.37228i q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000i q^{27} +2.37228i q^{28} -8.74456 q^{29} +9.11684 q^{31} +1.00000i q^{32} -4.37228i q^{33} +1.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} +2.37228i q^{38} -2.00000 q^{39} -1.37228 q^{41} -2.37228i q^{42} -3.62772i q^{43} -4.37228 q^{44} -1.37228 q^{46} -1.62772i q^{47} -1.00000i q^{48} +1.37228 q^{49} -1.00000 q^{51} +2.00000i q^{52} -5.74456i q^{53} -1.00000 q^{54} -2.37228 q^{56} -2.37228i q^{57} -8.74456i q^{58} -10.1168 q^{59} -8.11684 q^{61} +9.11684i q^{62} +2.37228i q^{63} -1.00000 q^{64} +4.37228 q^{66} +0.372281i q^{67} +1.00000i q^{68} +1.37228 q^{69} -1.37228 q^{71} +1.00000i q^{72} -8.00000i q^{73} +1.00000 q^{74} -2.37228 q^{76} -10.3723i q^{77} -2.00000i q^{78} +11.1168 q^{79} +1.00000 q^{81} -1.37228i q^{82} -1.37228i q^{83} +2.37228 q^{84} +3.62772 q^{86} +8.74456i q^{87} -4.37228i q^{88} +2.74456 q^{89} -4.74456 q^{91} -1.37228i q^{92} -9.11684i q^{93} +1.62772 q^{94} +1.00000 q^{96} -12.7446i q^{97} +1.37228i q^{98} -4.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 2 q^{14} + 4 q^{16} - 2 q^{19} + 2 q^{21} - 4 q^{24} + 8 q^{26} - 12 q^{29} + 2 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{39} + 6 q^{41} - 6 q^{44} + 6 q^{46} - 6 q^{49} - 4 q^{51} - 4 q^{54} + 2 q^{56} - 6 q^{59} + 2 q^{61} - 4 q^{64} + 6 q^{66} - 6 q^{69} + 6 q^{71} + 4 q^{74} + 2 q^{76} + 10 q^{79} + 4 q^{81} - 2 q^{84} + 26 q^{86} - 12 q^{89} + 4 q^{91} + 18 q^{94} + 4 q^{96} - 6 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 6 * q^11 - 2 * q^14 + 4 * q^16 - 2 * q^19 + 2 * q^21 - 4 * q^24 + 8 * q^26 - 12 * q^29 + 2 * q^31 + 4 * q^34 + 4 * q^36 - 8 * q^39 + 6 * q^41 - 6 * q^44 + 6 * q^46 - 6 * q^49 - 4 * q^51 - 4 * q^54 + 2 * q^56 - 6 * q^59 + 2 * q^61 - 4 * q^64 + 6 * q^66 - 6 * q^69 + 6 * q^71 + 4 * q^74 + 2 * q^76 + 10 * q^79 + 4 * q^81 - 2 * q^84 + 26 * q^86 - 12 * q^89 + 4 * q^91 + 18 * q^94 + 4 * q^96 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 2.37228i − 0.896638i −0.893874 0.448319i $$-0.852023\pi$$
0.893874 0.448319i $$-0.147977\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.37228 1.31829 0.659146 0.752015i $$-0.270918\pi$$
0.659146 + 0.752015i $$0.270918\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 2.37228 0.634019
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ − 1.00000i − 0.235702i
$$19$$ 2.37228 0.544239 0.272119 0.962264i $$-0.412275\pi$$
0.272119 + 0.962264i $$0.412275\pi$$
$$20$$ 0 0
$$21$$ −2.37228 −0.517674
$$22$$ 4.37228i 0.932174i
$$23$$ 1.37228i 0.286140i 0.989713 + 0.143070i $$0.0456975\pi$$
−0.989713 + 0.143070i $$0.954303\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 2.37228i 0.448319i
$$29$$ −8.74456 −1.62382 −0.811912 0.583779i $$-0.801573\pi$$
−0.811912 + 0.583779i $$0.801573\pi$$
$$30$$ 0 0
$$31$$ 9.11684 1.63743 0.818717 0.574198i $$-0.194686\pi$$
0.818717 + 0.574198i $$0.194686\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 4.37228i − 0.761116i
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 1.00000i − 0.164399i −0.996616 0.0821995i $$-0.973806\pi$$
0.996616 0.0821995i $$-0.0261945\pi$$
$$38$$ 2.37228i 0.384835i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −1.37228 −0.214314 −0.107157 0.994242i $$-0.534175\pi$$
−0.107157 + 0.994242i $$0.534175\pi$$
$$42$$ − 2.37228i − 0.366051i
$$43$$ − 3.62772i − 0.553222i −0.960982 0.276611i $$-0.910789\pi$$
0.960982 0.276611i $$-0.0892113\pi$$
$$44$$ −4.37228 −0.659146
$$45$$ 0 0
$$46$$ −1.37228 −0.202332
$$47$$ − 1.62772i − 0.237427i −0.992929 0.118714i $$-0.962123\pi$$
0.992929 0.118714i $$-0.0378770\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 1.37228 0.196040
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000i 0.277350i
$$53$$ − 5.74456i − 0.789076i −0.918880 0.394538i $$-0.870905\pi$$
0.918880 0.394538i $$-0.129095\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.37228 −0.317009
$$57$$ − 2.37228i − 0.314216i
$$58$$ − 8.74456i − 1.14822i
$$59$$ −10.1168 −1.31710 −0.658550 0.752537i $$-0.728830\pi$$
−0.658550 + 0.752537i $$0.728830\pi$$
$$60$$ 0 0
$$61$$ −8.11684 −1.03926 −0.519628 0.854393i $$-0.673929\pi$$
−0.519628 + 0.854393i $$0.673929\pi$$
$$62$$ 9.11684i 1.15784i
$$63$$ 2.37228i 0.298879i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.37228 0.538191
$$67$$ 0.372281i 0.0454814i 0.999741 + 0.0227407i $$0.00723921\pi$$
−0.999741 + 0.0227407i $$0.992761\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 1.37228 0.165203
$$70$$ 0 0
$$71$$ −1.37228 −0.162860 −0.0814299 0.996679i $$-0.525949\pi$$
−0.0814299 + 0.996679i $$0.525949\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −2.37228 −0.272119
$$77$$ − 10.3723i − 1.18203i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 11.1168 1.25074 0.625371 0.780327i $$-0.284948\pi$$
0.625371 + 0.780327i $$0.284948\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 1.37228i − 0.151543i
$$83$$ − 1.37228i − 0.150627i −0.997160 0.0753137i $$-0.976004\pi$$
0.997160 0.0753137i $$-0.0239958\pi$$
$$84$$ 2.37228 0.258837
$$85$$ 0 0
$$86$$ 3.62772 0.391187
$$87$$ 8.74456i 0.937516i
$$88$$ − 4.37228i − 0.466087i
$$89$$ 2.74456 0.290923 0.145462 0.989364i $$-0.453533\pi$$
0.145462 + 0.989364i $$0.453533\pi$$
$$90$$ 0 0
$$91$$ −4.74456 −0.497365
$$92$$ − 1.37228i − 0.143070i
$$93$$ − 9.11684i − 0.945373i
$$94$$ 1.62772 0.167886
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 12.7446i − 1.29401i −0.762484 0.647007i $$-0.776020\pi$$
0.762484 0.647007i $$-0.223980\pi$$
$$98$$ 1.37228i 0.138621i
$$99$$ −4.37228 −0.439431
$$100$$ 0 0
$$101$$ 13.1168 1.30517 0.652587 0.757713i $$-0.273684\pi$$
0.652587 + 0.757713i $$0.273684\pi$$
$$102$$ − 1.00000i − 0.0990148i
$$103$$ − 15.3723i − 1.51468i −0.653023 0.757338i $$-0.726500\pi$$
0.653023 0.757338i $$-0.273500\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 5.74456 0.557961
$$107$$ − 10.3723i − 1.00273i −0.865237 0.501363i $$-0.832832\pi$$
0.865237 0.501363i $$-0.167168\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −9.11684 −0.873235 −0.436618 0.899647i $$-0.643824\pi$$
−0.436618 + 0.899647i $$0.643824\pi$$
$$110$$ 0 0
$$111$$ −1.00000 −0.0949158
$$112$$ − 2.37228i − 0.224160i
$$113$$ 11.7446i 1.10484i 0.833567 + 0.552418i $$0.186295\pi$$
−0.833567 + 0.552418i $$0.813705\pi$$
$$114$$ 2.37228 0.222185
$$115$$ 0 0
$$116$$ 8.74456 0.811912
$$117$$ 2.00000i 0.184900i
$$118$$ − 10.1168i − 0.931331i
$$119$$ −2.37228 −0.217467
$$120$$ 0 0
$$121$$ 8.11684 0.737895
$$122$$ − 8.11684i − 0.734865i
$$123$$ 1.37228i 0.123734i
$$124$$ −9.11684 −0.818717
$$125$$ 0 0
$$126$$ −2.37228 −0.211340
$$127$$ 1.48913i 0.132139i 0.997815 + 0.0660693i $$0.0210458\pi$$
−0.997815 + 0.0660693i $$0.978954\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −3.62772 −0.319403
$$130$$ 0 0
$$131$$ 17.4891 1.52803 0.764016 0.645197i $$-0.223225\pi$$
0.764016 + 0.645197i $$0.223225\pi$$
$$132$$ 4.37228i 0.380558i
$$133$$ − 5.62772i − 0.487985i
$$134$$ −0.372281 −0.0321602
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 5.48913i 0.468968i 0.972120 + 0.234484i $$0.0753400\pi$$
−0.972120 + 0.234484i $$0.924660\pi$$
$$138$$ 1.37228i 0.116816i
$$139$$ −18.1168 −1.53665 −0.768325 0.640060i $$-0.778910\pi$$
−0.768325 + 0.640060i $$0.778910\pi$$
$$140$$ 0 0
$$141$$ −1.62772 −0.137079
$$142$$ − 1.37228i − 0.115159i
$$143$$ − 8.74456i − 0.731257i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 8.00000 0.662085
$$147$$ − 1.37228i − 0.113184i
$$148$$ 1.00000i 0.0821995i
$$149$$ 6.86141 0.562108 0.281054 0.959692i $$-0.409316\pi$$
0.281054 + 0.959692i $$0.409316\pi$$
$$150$$ 0 0
$$151$$ −14.1168 −1.14881 −0.574406 0.818570i $$-0.694767\pi$$
−0.574406 + 0.818570i $$0.694767\pi$$
$$152$$ − 2.37228i − 0.192417i
$$153$$ 1.00000i 0.0808452i
$$154$$ 10.3723 0.835822
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 18.7446i − 1.49598i −0.663711 0.747989i $$-0.731019\pi$$
0.663711 0.747989i $$-0.268981\pi$$
$$158$$ 11.1168i 0.884409i
$$159$$ −5.74456 −0.455573
$$160$$ 0 0
$$161$$ 3.25544 0.256564
$$162$$ 1.00000i 0.0785674i
$$163$$ 2.11684i 0.165804i 0.996558 + 0.0829020i $$0.0264188\pi$$
−0.996558 + 0.0829020i $$0.973581\pi$$
$$164$$ 1.37228 0.107157
$$165$$ 0 0
$$166$$ 1.37228 0.106510
$$167$$ − 20.7446i − 1.60526i −0.596476 0.802631i $$-0.703433\pi$$
0.596476 0.802631i $$-0.296567\pi$$
$$168$$ 2.37228i 0.183025i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ −2.37228 −0.181413
$$172$$ 3.62772i 0.276611i
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ −8.74456 −0.662924
$$175$$ 0 0
$$176$$ 4.37228 0.329573
$$177$$ 10.1168i 0.760429i
$$178$$ 2.74456i 0.205714i
$$179$$ 10.1168 0.756168 0.378084 0.925771i $$-0.376583\pi$$
0.378084 + 0.925771i $$0.376583\pi$$
$$180$$ 0 0
$$181$$ −0.489125 −0.0363564 −0.0181782 0.999835i $$-0.505787\pi$$
−0.0181782 + 0.999835i $$0.505787\pi$$
$$182$$ − 4.74456i − 0.351690i
$$183$$ 8.11684i 0.600014i
$$184$$ 1.37228 0.101166
$$185$$ 0 0
$$186$$ 9.11684 0.668479
$$187$$ − 4.37228i − 0.319733i
$$188$$ 1.62772i 0.118714i
$$189$$ 2.37228 0.172558
$$190$$ 0 0
$$191$$ −27.3505 −1.97902 −0.989508 0.144481i $$-0.953849\pi$$
−0.989508 + 0.144481i $$0.953849\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 8.00000i − 0.575853i −0.957653 0.287926i $$-0.907034\pi$$
0.957653 0.287926i $$-0.0929658\pi$$
$$194$$ 12.7446 0.915006
$$195$$ 0 0
$$196$$ −1.37228 −0.0980201
$$197$$ − 9.25544i − 0.659423i −0.944082 0.329711i $$-0.893049\pi$$
0.944082 0.329711i $$-0.106951\pi$$
$$198$$ − 4.37228i − 0.310725i
$$199$$ −21.1168 −1.49693 −0.748467 0.663172i $$-0.769210\pi$$
−0.748467 + 0.663172i $$0.769210\pi$$
$$200$$ 0 0
$$201$$ 0.372281 0.0262587
$$202$$ 13.1168i 0.922898i
$$203$$ 20.7446i 1.45598i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 15.3723 1.07104
$$207$$ − 1.37228i − 0.0953801i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 10.3723 0.717466
$$210$$ 0 0
$$211$$ 10.2337 0.704516 0.352258 0.935903i $$-0.385414\pi$$
0.352258 + 0.935903i $$0.385414\pi$$
$$212$$ 5.74456i 0.394538i
$$213$$ 1.37228i 0.0940272i
$$214$$ 10.3723 0.709035
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 21.6277i − 1.46819i
$$218$$ − 9.11684i − 0.617471i
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ − 1.00000i − 0.0671156i
$$223$$ 8.11684i 0.543544i 0.962362 + 0.271772i $$0.0876097\pi$$
−0.962362 + 0.271772i $$0.912390\pi$$
$$224$$ 2.37228 0.158505
$$225$$ 0 0
$$226$$ −11.7446 −0.781237
$$227$$ − 4.37228i − 0.290199i −0.989417 0.145099i $$-0.953650\pi$$
0.989417 0.145099i $$-0.0463501\pi$$
$$228$$ 2.37228i 0.157108i
$$229$$ 12.2337 0.808425 0.404212 0.914665i $$-0.367546\pi$$
0.404212 + 0.914665i $$0.367546\pi$$
$$230$$ 0 0
$$231$$ −10.3723 −0.682446
$$232$$ 8.74456i 0.574109i
$$233$$ 10.6277i 0.696245i 0.937449 + 0.348122i $$0.113181\pi$$
−0.937449 + 0.348122i $$0.886819\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 10.1168 0.658550
$$237$$ − 11.1168i − 0.722117i
$$238$$ − 2.37228i − 0.153772i
$$239$$ 13.1168 0.848458 0.424229 0.905555i $$-0.360545\pi$$
0.424229 + 0.905555i $$0.360545\pi$$
$$240$$ 0 0
$$241$$ 28.2337 1.81869 0.909346 0.416041i $$-0.136583\pi$$
0.909346 + 0.416041i $$0.136583\pi$$
$$242$$ 8.11684i 0.521770i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 8.11684 0.519628
$$245$$ 0 0
$$246$$ −1.37228 −0.0874935
$$247$$ − 4.74456i − 0.301889i
$$248$$ − 9.11684i − 0.578920i
$$249$$ −1.37228 −0.0869648
$$250$$ 0 0
$$251$$ 8.74456 0.551952 0.275976 0.961165i $$-0.410999\pi$$
0.275976 + 0.961165i $$0.410999\pi$$
$$252$$ − 2.37228i − 0.149440i
$$253$$ 6.00000i 0.377217i
$$254$$ −1.48913 −0.0934360
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 14.2337i − 0.887873i −0.896058 0.443937i $$-0.853582\pi$$
0.896058 0.443937i $$-0.146418\pi$$
$$258$$ − 3.62772i − 0.225852i
$$259$$ −2.37228 −0.147406
$$260$$ 0 0
$$261$$ 8.74456 0.541275
$$262$$ 17.4891i 1.08048i
$$263$$ 25.6277i 1.58027i 0.612931 + 0.790136i $$0.289990\pi$$
−0.612931 + 0.790136i $$0.710010\pi$$
$$264$$ −4.37228 −0.269095
$$265$$ 0 0
$$266$$ 5.62772 0.345058
$$267$$ − 2.74456i − 0.167965i
$$268$$ − 0.372281i − 0.0227407i
$$269$$ −11.4891 −0.700504 −0.350252 0.936655i $$-0.613904\pi$$
−0.350252 + 0.936655i $$0.613904\pi$$
$$270$$ 0 0
$$271$$ −2.62772 −0.159623 −0.0798113 0.996810i $$-0.525432\pi$$
−0.0798113 + 0.996810i $$0.525432\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 4.74456i 0.287154i
$$274$$ −5.48913 −0.331610
$$275$$ 0 0
$$276$$ −1.37228 −0.0826016
$$277$$ − 13.0000i − 0.781094i −0.920583 0.390547i $$-0.872286\pi$$
0.920583 0.390547i $$-0.127714\pi$$
$$278$$ − 18.1168i − 1.08658i
$$279$$ −9.11684 −0.545811
$$280$$ 0 0
$$281$$ 0.510875 0.0304762 0.0152381 0.999884i $$-0.495149\pi$$
0.0152381 + 0.999884i $$0.495149\pi$$
$$282$$ − 1.62772i − 0.0969292i
$$283$$ − 12.1168i − 0.720272i −0.932900 0.360136i $$-0.882730\pi$$
0.932900 0.360136i $$-0.117270\pi$$
$$284$$ 1.37228 0.0814299
$$285$$ 0 0
$$286$$ 8.74456 0.517077
$$287$$ 3.25544i 0.192162i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −12.7446 −0.747099
$$292$$ 8.00000i 0.468165i
$$293$$ 28.1168i 1.64260i 0.570494 + 0.821302i $$0.306752\pi$$
−0.570494 + 0.821302i $$0.693248\pi$$
$$294$$ 1.37228 0.0800331
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ 4.37228i 0.253705i
$$298$$ 6.86141i 0.397471i
$$299$$ 2.74456 0.158722
$$300$$ 0 0
$$301$$ −8.60597 −0.496040
$$302$$ − 14.1168i − 0.812333i
$$303$$ − 13.1168i − 0.753543i
$$304$$ 2.37228 0.136060
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ 16.2337i 0.926506i 0.886226 + 0.463253i $$0.153318\pi$$
−0.886226 + 0.463253i $$0.846682\pi$$
$$308$$ 10.3723i 0.591016i
$$309$$ −15.3723 −0.874499
$$310$$ 0 0
$$311$$ −28.1168 −1.59436 −0.797180 0.603742i $$-0.793676\pi$$
−0.797180 + 0.603742i $$0.793676\pi$$
$$312$$ 2.00000i 0.113228i
$$313$$ − 4.74456i − 0.268179i −0.990969 0.134089i $$-0.957189\pi$$
0.990969 0.134089i $$-0.0428109\pi$$
$$314$$ 18.7446 1.05782
$$315$$ 0 0
$$316$$ −11.1168 −0.625371
$$317$$ 14.7446i 0.828137i 0.910246 + 0.414069i $$0.135893\pi$$
−0.910246 + 0.414069i $$0.864107\pi$$
$$318$$ − 5.74456i − 0.322139i
$$319$$ −38.2337 −2.14068
$$320$$ 0 0
$$321$$ −10.3723 −0.578924
$$322$$ 3.25544i 0.181418i
$$323$$ − 2.37228i − 0.131997i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −2.11684 −0.117241
$$327$$ 9.11684i 0.504163i
$$328$$ 1.37228i 0.0757716i
$$329$$ −3.86141 −0.212886
$$330$$ 0 0
$$331$$ 14.6060 0.802817 0.401408 0.915899i $$-0.368521\pi$$
0.401408 + 0.915899i $$0.368521\pi$$
$$332$$ 1.37228i 0.0753137i
$$333$$ 1.00000i 0.0547997i
$$334$$ 20.7446 1.13509
$$335$$ 0 0
$$336$$ −2.37228 −0.129419
$$337$$ − 24.2337i − 1.32009i −0.751225 0.660047i $$-0.770537\pi$$
0.751225 0.660047i $$-0.229463\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 11.7446 0.637877
$$340$$ 0 0
$$341$$ 39.8614 2.15862
$$342$$ − 2.37228i − 0.128278i
$$343$$ − 19.8614i − 1.07242i
$$344$$ −3.62772 −0.195593
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 22.3723i 1.20101i 0.799622 + 0.600503i $$0.205033\pi$$
−0.799622 + 0.600503i $$0.794967\pi$$
$$348$$ − 8.74456i − 0.468758i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 4.37228i 0.233043i
$$353$$ 15.2554i 0.811965i 0.913881 + 0.405983i $$0.133071\pi$$
−0.913881 + 0.405983i $$0.866929\pi$$
$$354$$ −10.1168 −0.537704
$$355$$ 0 0
$$356$$ −2.74456 −0.145462
$$357$$ 2.37228i 0.125554i
$$358$$ 10.1168i 0.534692i
$$359$$ 13.6277 0.719243 0.359622 0.933098i $$-0.382906\pi$$
0.359622 + 0.933098i $$0.382906\pi$$
$$360$$ 0 0
$$361$$ −13.3723 −0.703804
$$362$$ − 0.489125i − 0.0257079i
$$363$$ − 8.11684i − 0.426024i
$$364$$ 4.74456 0.248683
$$365$$ 0 0
$$366$$ −8.11684 −0.424274
$$367$$ 14.6060i 0.762425i 0.924487 + 0.381213i $$0.124493\pi$$
−0.924487 + 0.381213i $$0.875507\pi$$
$$368$$ 1.37228i 0.0715351i
$$369$$ 1.37228 0.0714381
$$370$$ 0 0
$$371$$ −13.6277 −0.707516
$$372$$ 9.11684i 0.472686i
$$373$$ 1.76631i 0.0914562i 0.998954 + 0.0457281i $$0.0145608\pi$$
−0.998954 + 0.0457281i $$0.985439\pi$$
$$374$$ 4.37228 0.226085
$$375$$ 0 0
$$376$$ −1.62772 −0.0839432
$$377$$ 17.4891i 0.900736i
$$378$$ 2.37228i 0.122017i
$$379$$ 19.6060 1.00709 0.503545 0.863969i $$-0.332029\pi$$
0.503545 + 0.863969i $$0.332029\pi$$
$$380$$ 0 0
$$381$$ 1.48913 0.0762902
$$382$$ − 27.3505i − 1.39937i
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ 3.62772i 0.184407i
$$388$$ 12.7446i 0.647007i
$$389$$ −17.7446 −0.899685 −0.449843 0.893108i $$-0.648520\pi$$
−0.449843 + 0.893108i $$0.648520\pi$$
$$390$$ 0 0
$$391$$ 1.37228 0.0693992
$$392$$ − 1.37228i − 0.0693107i
$$393$$ − 17.4891i − 0.882210i
$$394$$ 9.25544 0.466282
$$395$$ 0 0
$$396$$ 4.37228 0.219715
$$397$$ − 1.00000i − 0.0501886i −0.999685 0.0250943i $$-0.992011\pi$$
0.999685 0.0250943i $$-0.00798860\pi$$
$$398$$ − 21.1168i − 1.05849i
$$399$$ −5.62772 −0.281738
$$400$$ 0 0
$$401$$ 4.11684 0.205585 0.102793 0.994703i $$-0.467222\pi$$
0.102793 + 0.994703i $$0.467222\pi$$
$$402$$ 0.372281i 0.0185677i
$$403$$ − 18.2337i − 0.908285i
$$404$$ −13.1168 −0.652587
$$405$$ 0 0
$$406$$ −20.7446 −1.02954
$$407$$ − 4.37228i − 0.216726i
$$408$$ 1.00000i 0.0495074i
$$409$$ −14.8614 −0.734849 −0.367425 0.930053i $$-0.619760\pi$$
−0.367425 + 0.930053i $$0.619760\pi$$
$$410$$ 0 0
$$411$$ 5.48913 0.270759
$$412$$ 15.3723i 0.757338i
$$413$$ 24.0000i 1.18096i
$$414$$ 1.37228 0.0674439
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 18.1168i 0.887186i
$$418$$ 10.3723i 0.507325i
$$419$$ −10.9783 −0.536323 −0.268161 0.963374i $$-0.586416\pi$$
−0.268161 + 0.963374i $$0.586416\pi$$
$$420$$ 0 0
$$421$$ −18.2337 −0.888656 −0.444328 0.895864i $$-0.646557\pi$$
−0.444328 + 0.895864i $$0.646557\pi$$
$$422$$ 10.2337i 0.498168i
$$423$$ 1.62772i 0.0791424i
$$424$$ −5.74456 −0.278981
$$425$$ 0 0
$$426$$ −1.37228 −0.0664872
$$427$$ 19.2554i 0.931836i
$$428$$ 10.3723i 0.501363i
$$429$$ −8.74456 −0.422191
$$430$$ 0 0
$$431$$ 12.0000 0.578020 0.289010 0.957326i $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 40.0951i 1.92685i 0.267983 + 0.963424i $$0.413643\pi$$
−0.267983 + 0.963424i $$0.586357\pi$$
$$434$$ 21.6277 1.03816
$$435$$ 0 0
$$436$$ 9.11684 0.436618
$$437$$ 3.25544i 0.155729i
$$438$$ − 8.00000i − 0.382255i
$$439$$ −2.51087 −0.119838 −0.0599188 0.998203i $$-0.519084\pi$$
−0.0599188 + 0.998203i $$0.519084\pi$$
$$440$$ 0 0
$$441$$ −1.37228 −0.0653467
$$442$$ − 2.00000i − 0.0951303i
$$443$$ 7.37228i 0.350268i 0.984545 + 0.175134i $$0.0560358\pi$$
−0.984545 + 0.175134i $$0.943964\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 0 0
$$446$$ −8.11684 −0.384344
$$447$$ − 6.86141i − 0.324533i
$$448$$ 2.37228i 0.112080i
$$449$$ 33.8614 1.59802 0.799009 0.601319i $$-0.205358\pi$$
0.799009 + 0.601319i $$0.205358\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ − 11.7446i − 0.552418i
$$453$$ 14.1168i 0.663267i
$$454$$ 4.37228 0.205201
$$455$$ 0 0
$$456$$ −2.37228 −0.111092
$$457$$ 23.0000i 1.07589i 0.842978 + 0.537947i $$0.180800\pi$$
−0.842978 + 0.537947i $$0.819200\pi$$
$$458$$ 12.2337i 0.571643i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 34.7228 1.61720 0.808601 0.588357i $$-0.200225\pi$$
0.808601 + 0.588357i $$0.200225\pi$$
$$462$$ − 10.3723i − 0.482562i
$$463$$ 19.6060i 0.911167i 0.890193 + 0.455583i $$0.150569\pi$$
−0.890193 + 0.455583i $$0.849431\pi$$
$$464$$ −8.74456 −0.405956
$$465$$ 0 0
$$466$$ −10.6277 −0.492320
$$467$$ 16.1168i 0.745799i 0.927872 + 0.372899i $$0.121636\pi$$
−0.927872 + 0.372899i $$0.878364\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 0.883156 0.0407804
$$470$$ 0 0
$$471$$ −18.7446 −0.863704
$$472$$ 10.1168i 0.465665i
$$473$$ − 15.8614i − 0.729308i
$$474$$ 11.1168 0.510614
$$475$$ 0 0
$$476$$ 2.37228 0.108733
$$477$$ 5.74456i 0.263025i
$$478$$ 13.1168i 0.599950i
$$479$$ 31.7228 1.44945 0.724726 0.689037i $$-0.241966\pi$$
0.724726 + 0.689037i $$0.241966\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 28.2337i 1.28601i
$$483$$ − 3.25544i − 0.148128i
$$484$$ −8.11684 −0.368947
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 12.2337i − 0.554361i −0.960818 0.277181i $$-0.910600\pi$$
0.960818 0.277181i $$-0.0894001\pi$$
$$488$$ 8.11684i 0.367432i
$$489$$ 2.11684 0.0957270
$$490$$ 0 0
$$491$$ −31.3723 −1.41581 −0.707906 0.706307i $$-0.750360\pi$$
−0.707906 + 0.706307i $$0.750360\pi$$
$$492$$ − 1.37228i − 0.0618672i
$$493$$ 8.74456i 0.393835i
$$494$$ 4.74456 0.213468
$$495$$ 0 0
$$496$$ 9.11684 0.409358
$$497$$ 3.25544i 0.146026i
$$498$$ − 1.37228i − 0.0614934i
$$499$$ −18.6277 −0.833891 −0.416946 0.908931i $$-0.636899\pi$$
−0.416946 + 0.908931i $$0.636899\pi$$
$$500$$ 0 0
$$501$$ −20.7446 −0.926799
$$502$$ 8.74456i 0.390289i
$$503$$ 9.60597i 0.428309i 0.976800 + 0.214154i $$0.0686996\pi$$
−0.976800 + 0.214154i $$0.931300\pi$$
$$504$$ 2.37228 0.105670
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ − 9.00000i − 0.399704i
$$508$$ − 1.48913i − 0.0660693i
$$509$$ 19.6277 0.869983 0.434992 0.900434i $$-0.356751\pi$$
0.434992 + 0.900434i $$0.356751\pi$$
$$510$$ 0 0
$$511$$ −18.9783 −0.839548
$$512$$ 1.00000i 0.0441942i
$$513$$ 2.37228i 0.104739i
$$514$$ 14.2337 0.627821
$$515$$ 0 0
$$516$$ 3.62772 0.159701
$$517$$ − 7.11684i − 0.312998i
$$518$$ − 2.37228i − 0.104232i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ −27.3505 −1.19825 −0.599124 0.800656i $$-0.704484\pi$$
−0.599124 + 0.800656i $$0.704484\pi$$
$$522$$ 8.74456i 0.382739i
$$523$$ − 16.2337i − 0.709850i −0.934895 0.354925i $$-0.884506\pi$$
0.934895 0.354925i $$-0.115494\pi$$
$$524$$ −17.4891 −0.764016
$$525$$ 0 0
$$526$$ −25.6277 −1.11742
$$527$$ − 9.11684i − 0.397136i
$$528$$ − 4.37228i − 0.190279i
$$529$$ 21.1168 0.918124
$$530$$ 0 0
$$531$$ 10.1168 0.439034
$$532$$ 5.62772i 0.243993i
$$533$$ 2.74456i 0.118880i
$$534$$ 2.74456 0.118769
$$535$$ 0 0
$$536$$ 0.372281 0.0160801
$$537$$ − 10.1168i − 0.436574i
$$538$$ − 11.4891i − 0.495331i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 5.00000 0.214967 0.107483 0.994207i $$-0.465721\pi$$
0.107483 + 0.994207i $$0.465721\pi$$
$$542$$ − 2.62772i − 0.112870i
$$543$$ 0.489125i 0.0209904i
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ −4.74456 −0.203049
$$547$$ 30.6277i 1.30955i 0.755825 + 0.654773i $$0.227236\pi$$
−0.755825 + 0.654773i $$0.772764\pi$$
$$548$$ − 5.48913i − 0.234484i
$$549$$ 8.11684 0.346418
$$550$$ 0 0
$$551$$ −20.7446 −0.883748
$$552$$ − 1.37228i − 0.0584082i
$$553$$ − 26.3723i − 1.12146i
$$554$$ 13.0000 0.552317
$$555$$ 0 0
$$556$$ 18.1168 0.768325
$$557$$ − 24.2554i − 1.02774i −0.857869 0.513868i $$-0.828212\pi$$
0.857869 0.513868i $$-0.171788\pi$$
$$558$$ − 9.11684i − 0.385947i
$$559$$ −7.25544 −0.306872
$$560$$ 0 0
$$561$$ −4.37228 −0.184598
$$562$$ 0.510875i 0.0215499i
$$563$$ − 18.3505i − 0.773383i −0.922209 0.386691i $$-0.873618\pi$$
0.922209 0.386691i $$-0.126382\pi$$
$$564$$ 1.62772 0.0685393
$$565$$ 0 0
$$566$$ 12.1168 0.509309
$$567$$ − 2.37228i − 0.0996265i
$$568$$ 1.37228i 0.0575796i
$$569$$ 9.25544 0.388008 0.194004 0.981001i $$-0.437853\pi$$
0.194004 + 0.981001i $$0.437853\pi$$
$$570$$ 0 0
$$571$$ −32.6277 −1.36543 −0.682714 0.730686i $$-0.739200\pi$$
−0.682714 + 0.730686i $$0.739200\pi$$
$$572$$ 8.74456i 0.365629i
$$573$$ 27.3505i 1.14258i
$$574$$ −3.25544 −0.135879
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 7.00000i − 0.291414i −0.989328 0.145707i $$-0.953454\pi$$
0.989328 0.145707i $$-0.0465456\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ −3.25544 −0.135058
$$582$$ − 12.7446i − 0.528279i
$$583$$ − 25.1168i − 1.04023i
$$584$$ −8.00000 −0.331042
$$585$$ 0 0
$$586$$ −28.1168 −1.16150
$$587$$ 30.8614i 1.27379i 0.770952 + 0.636893i $$0.219781\pi$$
−0.770952 + 0.636893i $$0.780219\pi$$
$$588$$ 1.37228i 0.0565919i
$$589$$ 21.6277 0.891155
$$590$$ 0 0
$$591$$ −9.25544 −0.380718
$$592$$ − 1.00000i − 0.0410997i
$$593$$ 40.4674i 1.66180i 0.556425 + 0.830898i $$0.312173\pi$$
−0.556425 + 0.830898i $$0.687827\pi$$
$$594$$ −4.37228 −0.179397
$$595$$ 0 0
$$596$$ −6.86141 −0.281054
$$597$$ 21.1168i 0.864255i
$$598$$ 2.74456i 0.112234i
$$599$$ 1.62772 0.0665068 0.0332534 0.999447i $$-0.489413\pi$$
0.0332534 + 0.999447i $$0.489413\pi$$
$$600$$ 0 0
$$601$$ 36.9783 1.50837 0.754187 0.656660i $$-0.228031\pi$$
0.754187 + 0.656660i $$0.228031\pi$$
$$602$$ − 8.60597i − 0.350753i
$$603$$ − 0.372281i − 0.0151605i
$$604$$ 14.1168 0.574406
$$605$$ 0 0
$$606$$ 13.1168 0.532835
$$607$$ 40.2337i 1.63304i 0.577321 + 0.816518i $$0.304098\pi$$
−0.577321 + 0.816518i $$0.695902\pi$$
$$608$$ 2.37228i 0.0962087i
$$609$$ 20.7446 0.840612
$$610$$ 0 0
$$611$$ −3.25544 −0.131701
$$612$$ − 1.00000i − 0.0404226i
$$613$$ 3.48913i 0.140924i 0.997514 + 0.0704622i $$0.0224474\pi$$
−0.997514 + 0.0704622i $$0.977553\pi$$
$$614$$ −16.2337 −0.655138
$$615$$ 0 0
$$616$$ −10.3723 −0.417911
$$617$$ 44.8397i 1.80518i 0.430505 + 0.902588i $$0.358336\pi$$
−0.430505 + 0.902588i $$0.641664\pi$$
$$618$$ − 15.3723i − 0.618364i
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ −1.37228 −0.0550678
$$622$$ − 28.1168i − 1.12738i
$$623$$ − 6.51087i − 0.260853i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ 4.74456 0.189631
$$627$$ − 10.3723i − 0.414229i
$$628$$ 18.7446i 0.747989i
$$629$$ −1.00000 −0.0398726
$$630$$ 0 0
$$631$$ −34.8614 −1.38781 −0.693905 0.720066i $$-0.744111\pi$$
−0.693905 + 0.720066i $$0.744111\pi$$
$$632$$ − 11.1168i − 0.442204i
$$633$$ − 10.2337i − 0.406753i
$$634$$ −14.7446 −0.585581
$$635$$ 0 0
$$636$$ 5.74456 0.227787
$$637$$ − 2.74456i − 0.108744i
$$638$$ − 38.2337i − 1.51369i
$$639$$ 1.37228 0.0542866
$$640$$ 0 0
$$641$$ 40.9783 1.61854 0.809272 0.587434i $$-0.199862\pi$$
0.809272 + 0.587434i $$0.199862\pi$$
$$642$$ − 10.3723i − 0.409361i
$$643$$ − 24.1168i − 0.951075i −0.879695 0.475538i $$-0.842254\pi$$
0.879695 0.475538i $$-0.157746\pi$$
$$644$$ −3.25544 −0.128282
$$645$$ 0 0
$$646$$ 2.37228 0.0933362
$$647$$ − 25.7228i − 1.01127i −0.862748 0.505634i $$-0.831259\pi$$
0.862748 0.505634i $$-0.168741\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −44.2337 −1.73632
$$650$$ 0 0
$$651$$ −21.6277 −0.847657
$$652$$ − 2.11684i − 0.0829020i
$$653$$ 22.9783i 0.899208i 0.893228 + 0.449604i $$0.148435\pi$$
−0.893228 + 0.449604i $$0.851565\pi$$
$$654$$ −9.11684 −0.356497
$$655$$ 0 0
$$656$$ −1.37228 −0.0535786
$$657$$ 8.00000i 0.312110i
$$658$$ − 3.86141i − 0.150533i
$$659$$ −6.51087 −0.253628 −0.126814 0.991927i $$-0.540475\pi$$
−0.126814 + 0.991927i $$0.540475\pi$$
$$660$$ 0 0
$$661$$ −13.2554 −0.515577 −0.257788 0.966201i $$-0.582994\pi$$
−0.257788 + 0.966201i $$0.582994\pi$$
$$662$$ 14.6060i 0.567677i
$$663$$ 2.00000i 0.0776736i
$$664$$ −1.37228 −0.0532548
$$665$$ 0 0
$$666$$ −1.00000 −0.0387492
$$667$$ − 12.0000i − 0.464642i
$$668$$ 20.7446i 0.802631i
$$669$$ 8.11684 0.313815
$$670$$ 0 0
$$671$$ −35.4891 −1.37004
$$672$$ − 2.37228i − 0.0915127i
$$673$$ − 39.7228i − 1.53120i −0.643316 0.765601i $$-0.722442\pi$$
0.643316 0.765601i $$-0.277558\pi$$
$$674$$ 24.2337 0.933447
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 24.0000i − 0.922395i −0.887298 0.461197i $$-0.847420\pi$$
0.887298 0.461197i $$-0.152580\pi$$
$$678$$ 11.7446i 0.451047i
$$679$$ −30.2337 −1.16026
$$680$$ 0 0
$$681$$ −4.37228 −0.167546
$$682$$ 39.8614i 1.52637i
$$683$$ − 9.25544i − 0.354149i −0.984197 0.177075i $$-0.943337\pi$$
0.984197 0.177075i $$-0.0566634\pi$$
$$684$$ 2.37228 0.0907064
$$685$$ 0 0
$$686$$ 19.8614 0.758312
$$687$$ − 12.2337i − 0.466744i
$$688$$ − 3.62772i − 0.138305i
$$689$$ −11.4891 −0.437701
$$690$$ 0 0
$$691$$ 39.3723 1.49779 0.748896 0.662687i $$-0.230584\pi$$
0.748896 + 0.662687i $$0.230584\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ 10.3723i 0.394010i
$$694$$ −22.3723 −0.849240
$$695$$ 0 0
$$696$$ 8.74456 0.331462
$$697$$ 1.37228i 0.0519789i
$$698$$ 10.0000i 0.378506i
$$699$$ 10.6277 0.401977
$$700$$ 0 0
$$701$$ 9.60597 0.362812 0.181406 0.983408i $$-0.441935\pi$$
0.181406 + 0.983408i $$0.441935\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ − 2.37228i − 0.0894723i
$$704$$ −4.37228 −0.164787
$$705$$ 0 0
$$706$$ −15.2554 −0.574146
$$707$$ − 31.1168i − 1.17027i
$$708$$ − 10.1168i − 0.380214i
$$709$$ 8.37228 0.314428 0.157214 0.987565i $$-0.449749\pi$$
0.157214 + 0.987565i $$0.449749\pi$$
$$710$$ 0 0
$$711$$ −11.1168 −0.416914
$$712$$ − 2.74456i − 0.102857i
$$713$$ 12.5109i 0.468536i
$$714$$ −2.37228 −0.0887804
$$715$$ 0 0
$$716$$ −10.1168 −0.378084
$$717$$ − 13.1168i − 0.489858i
$$718$$ 13.6277i 0.508582i
$$719$$ 27.2554 1.01646 0.508228 0.861222i $$-0.330301\pi$$
0.508228 + 0.861222i $$0.330301\pi$$
$$720$$ 0 0
$$721$$ −36.4674 −1.35812
$$722$$ − 13.3723i − 0.497665i
$$723$$ − 28.2337i − 1.05002i
$$724$$ 0.489125 0.0181782
$$725$$ 0 0
$$726$$ 8.11684 0.301244
$$727$$ − 31.2554i − 1.15920i −0.814901 0.579600i $$-0.803209\pi$$
0.814901 0.579600i $$-0.196791\pi$$
$$728$$ 4.74456i 0.175845i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −3.62772 −0.134176
$$732$$ − 8.11684i − 0.300007i
$$733$$ 1.25544i 0.0463706i 0.999731 + 0.0231853i $$0.00738078\pi$$
−0.999731 + 0.0231853i $$0.992619\pi$$
$$734$$ −14.6060 −0.539116
$$735$$ 0 0
$$736$$ −1.37228 −0.0505830
$$737$$ 1.62772i 0.0599578i
$$738$$ 1.37228i 0.0505144i
$$739$$ −33.6277 −1.23702 −0.618508 0.785779i $$-0.712262\pi$$
−0.618508 + 0.785779i $$0.712262\pi$$
$$740$$ 0 0
$$741$$ −4.74456 −0.174296
$$742$$ − 13.6277i − 0.500289i
$$743$$ − 22.6277i − 0.830130i −0.909792 0.415065i $$-0.863759\pi$$
0.909792 0.415065i $$-0.136241\pi$$
$$744$$ −9.11684 −0.334240
$$745$$ 0 0
$$746$$ −1.76631 −0.0646693
$$747$$ 1.37228i 0.0502091i
$$748$$ 4.37228i 0.159866i
$$749$$ −24.6060 −0.899083
$$750$$ 0 0
$$751$$ −32.4674 −1.18475 −0.592376 0.805662i $$-0.701810\pi$$
−0.592376 + 0.805662i $$0.701810\pi$$
$$752$$ − 1.62772i − 0.0593568i
$$753$$ − 8.74456i − 0.318670i
$$754$$ −17.4891 −0.636916
$$755$$ 0 0
$$756$$ −2.37228 −0.0862790
$$757$$ 34.2337i 1.24424i 0.782920 + 0.622122i $$0.213729\pi$$
−0.782920 + 0.622122i $$0.786271\pi$$
$$758$$ 19.6060i 0.712121i
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −17.4891 −0.633980 −0.316990 0.948429i $$-0.602672\pi$$
−0.316990 + 0.948429i $$0.602672\pi$$
$$762$$ 1.48913i 0.0539453i
$$763$$ 21.6277i 0.782976i
$$764$$ 27.3505 0.989508
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 20.2337i 0.730596i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 13.0000 0.468792 0.234396 0.972141i $$-0.424689\pi$$
0.234396 + 0.972141i $$0.424689\pi$$
$$770$$ 0 0
$$771$$ −14.2337 −0.512614
$$772$$ 8.00000i 0.287926i
$$773$$ 17.1386i 0.616432i 0.951316 + 0.308216i $$0.0997320\pi$$
−0.951316 + 0.308216i $$0.900268\pi$$
$$774$$ −3.62772 −0.130396
$$775$$ 0 0
$$776$$ −12.7446 −0.457503
$$777$$ 2.37228i 0.0851051i
$$778$$ − 17.7446i − 0.636173i
$$779$$ −3.25544 −0.116638
$$780$$ 0 0
$$781$$ −6.00000 −0.214697
$$782$$ 1.37228i 0.0490727i
$$783$$ − 8.74456i − 0.312505i
$$784$$ 1.37228 0.0490100
$$785$$ 0 0
$$786$$ 17.4891 0.623816
$$787$$ 44.3505i 1.58093i 0.612510 + 0.790463i $$0.290160\pi$$
−0.612510 + 0.790463i $$0.709840\pi$$
$$788$$ 9.25544i 0.329711i
$$789$$ 25.6277 0.912371
$$790$$ 0 0
$$791$$ 27.8614 0.990638
$$792$$ 4.37228i 0.155362i
$$793$$ 16.2337i 0.576475i
$$794$$ 1.00000 0.0354887
$$795$$ 0 0
$$796$$ 21.1168 0.748467
$$797$$ 6.25544i 0.221579i 0.993844 + 0.110789i $$0.0353379\pi$$
−0.993844 + 0.110789i $$0.964662\pi$$
$$798$$ − 5.62772i − 0.199219i
$$799$$ −1.62772 −0.0575845
$$800$$ 0 0
$$801$$ −2.74456 −0.0969744
$$802$$ 4.11684i 0.145371i
$$803$$ − 34.9783i − 1.23436i
$$804$$ −0.372281 −0.0131293
$$805$$ 0 0
$$806$$ 18.2337 0.642254
$$807$$ 11.4891i 0.404436i
$$808$$ − 13.1168i − 0.461449i
$$809$$ −18.6060 −0.654151 −0.327076 0.944998i $$-0.606063\pi$$
−0.327076 + 0.944998i $$0.606063\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ − 20.7446i − 0.727991i
$$813$$ 2.62772i 0.0921581i
$$814$$ 4.37228 0.153248
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ − 8.60597i − 0.301085i
$$818$$ − 14.8614i − 0.519617i
$$819$$ 4.74456 0.165788
$$820$$ 0 0
$$821$$ −44.7446 −1.56160 −0.780798 0.624784i $$-0.785187\pi$$
−0.780798 + 0.624784i $$0.785187\pi$$
$$822$$ 5.48913i 0.191455i
$$823$$ 20.4674i 0.713448i 0.934210 + 0.356724i $$0.116106\pi$$
−0.934210 + 0.356724i $$0.883894\pi$$
$$824$$ −15.3723 −0.535519
$$825$$ 0 0
$$826$$ −24.0000 −0.835067
$$827$$ − 12.0951i − 0.420588i −0.977638 0.210294i $$-0.932558\pi$$
0.977638 0.210294i $$-0.0674421\pi$$
$$828$$ 1.37228i 0.0476901i
$$829$$ −40.7446 −1.41512 −0.707559 0.706655i $$-0.750203\pi$$
−0.707559 + 0.706655i $$0.750203\pi$$
$$830$$ 0 0
$$831$$ −13.0000 −0.450965
$$832$$ 2.00000i 0.0693375i
$$833$$ − 1.37228i − 0.0475467i
$$834$$ −18.1168 −0.627335
$$835$$ 0 0
$$836$$ −10.3723 −0.358733
$$837$$ 9.11684i 0.315124i
$$838$$ − 10.9783i − 0.379237i
$$839$$ −7.37228 −0.254519 −0.127260 0.991869i $$-0.540618\pi$$
−0.127260 + 0.991869i $$0.540618\pi$$
$$840$$ 0 0
$$841$$ 47.4674 1.63681
$$842$$ − 18.2337i − 0.628374i
$$843$$ − 0.510875i − 0.0175955i
$$844$$ −10.2337 −0.352258
$$845$$ 0 0
$$846$$ −1.62772 −0.0559621
$$847$$ − 19.2554i − 0.661625i
$$848$$ − 5.74456i − 0.197269i
$$849$$ −12.1168 −0.415849
$$850$$ 0 0
$$851$$ 1.37228 0.0470412
$$852$$ − 1.37228i − 0.0470136i
$$853$$ − 17.8614i − 0.611563i −0.952102 0.305781i $$-0.901082\pi$$
0.952102 0.305781i $$-0.0989177\pi$$
$$854$$ −19.2554 −0.658908
$$855$$ 0 0
$$856$$ −10.3723 −0.354517
$$857$$ 18.2554i 0.623594i 0.950149 + 0.311797i $$0.100931\pi$$
−0.950149 + 0.311797i $$0.899069\pi$$
$$858$$ − 8.74456i − 0.298534i
$$859$$ 13.3505 0.455514 0.227757 0.973718i $$-0.426861\pi$$
0.227757 + 0.973718i $$0.426861\pi$$
$$860$$ 0 0
$$861$$ 3.25544 0.110945
$$862$$ 12.0000i 0.408722i
$$863$$ 13.6277i 0.463893i 0.972729 + 0.231946i $$0.0745094\pi$$
−0.972729 + 0.231946i $$0.925491\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −40.0951 −1.36249
$$867$$ 1.00000i 0.0339618i
$$868$$ 21.6277i 0.734093i
$$869$$ 48.6060 1.64884
$$870$$ 0 0
$$871$$ 0.744563 0.0252285
$$872$$ 9.11684i 0.308735i
$$873$$ 12.7446i 0.431338i
$$874$$ −3.25544 −0.110117
$$875$$ 0 0
$$876$$ 8.00000 0.270295
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ − 2.51087i − 0.0847379i
$$879$$ 28.1168 0.948358
$$880$$ 0 0
$$881$$ −34.2119 −1.15263 −0.576315 0.817228i $$-0.695510\pi$$
−0.576315 + 0.817228i $$0.695510\pi$$
$$882$$ − 1.37228i − 0.0462071i
$$883$$ 7.76631i 0.261357i 0.991425 + 0.130679i $$0.0417156\pi$$
−0.991425 + 0.130679i $$0.958284\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −7.37228 −0.247677
$$887$$ 39.6060i 1.32984i 0.746915 + 0.664919i $$0.231534\pi$$
−0.746915 + 0.664919i $$0.768466\pi$$
$$888$$ 1.00000i 0.0335578i
$$889$$ 3.53262 0.118480
$$890$$ 0 0
$$891$$ 4.37228 0.146477
$$892$$ − 8.11684i − 0.271772i
$$893$$ − 3.86141i − 0.129217i
$$894$$ 6.86141 0.229480
$$895$$ 0 0
$$896$$ −2.37228 −0.0792524
$$897$$ − 2.74456i − 0.0916383i
$$898$$ 33.8614i 1.12997i
$$899$$ −79.7228 −2.65890
$$900$$ 0 0
$$901$$ −5.74456 −0.191379
$$902$$ − 6.00000i − 0.199778i
$$903$$ 8.60597i 0.286389i
$$904$$ 11.7446 0.390618
$$905$$ 0 0
$$906$$ −14.1168 −0.469001
$$907$$ 10.3940i 0.345128i 0.984998 + 0.172564i $$0.0552052\pi$$
−0.984998 + 0.172564i $$0.944795\pi$$
$$908$$ 4.37228i 0.145099i
$$909$$ −13.1168 −0.435058
$$910$$ 0 0
$$911$$ −6.51087 −0.215715 −0.107857 0.994166i $$-0.534399\pi$$
−0.107857 + 0.994166i $$0.534399\pi$$
$$912$$ − 2.37228i − 0.0785541i
$$913$$ − 6.00000i − 0.198571i
$$914$$ −23.0000 −0.760772
$$915$$ 0 0
$$916$$ −12.2337 −0.404212
$$917$$ − 41.4891i − 1.37009i
$$918$$ 1.00000i 0.0330049i
$$919$$ 42.5842 1.40472 0.702362 0.711820i $$-0.252129\pi$$
0.702362 + 0.711820i $$0.252129\pi$$
$$920$$ 0 0
$$921$$ 16.2337 0.534918
$$922$$ 34.7228i 1.14353i
$$923$$ 2.74456i 0.0903384i
$$924$$ 10.3723 0.341223
$$925$$ 0 0
$$926$$ −19.6060 −0.644292
$$927$$ 15.3723i 0.504892i
$$928$$ − 8.74456i − 0.287054i
$$929$$ 40.7228 1.33607 0.668036 0.744129i $$-0.267135\pi$$
0.668036 + 0.744129i $$0.267135\pi$$
$$930$$ 0 0
$$931$$ 3.25544 0.106693
$$932$$ − 10.6277i − 0.348122i
$$933$$ 28.1168i 0.920504i
$$934$$ −16.1168 −0.527359
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 41.6060i 1.35921i 0.733579 + 0.679604i $$0.237848\pi$$
−0.733579 + 0.679604i $$0.762152\pi$$
$$938$$ 0.883156i 0.0288361i
$$939$$ −4.74456 −0.154833
$$940$$ 0 0
$$941$$ −22.4674 −0.732416 −0.366208 0.930533i $$-0.619344\pi$$
−0.366208 + 0.930533i $$0.619344\pi$$
$$942$$ − 18.7446i − 0.610731i
$$943$$ − 1.88316i − 0.0613240i
$$944$$ −10.1168 −0.329275
$$945$$ 0 0
$$946$$ 15.8614 0.515699
$$947$$ 34.3723i 1.11695i 0.829522 + 0.558475i $$0.188613\pi$$
−0.829522 + 0.558475i $$0.811387\pi$$
$$948$$ 11.1168i 0.361058i
$$949$$ −16.0000 −0.519382
$$950$$ 0 0
$$951$$ 14.7446 0.478125
$$952$$ 2.37228i 0.0768861i
$$953$$ − 40.9783i − 1.32742i −0.747992 0.663708i $$-0.768982\pi$$
0.747992 0.663708i $$-0.231018\pi$$
$$954$$ −5.74456 −0.185987
$$955$$ 0 0
$$956$$ −13.1168 −0.424229
$$957$$ 38.2337i 1.23592i
$$958$$ 31.7228i 1.02492i
$$959$$ 13.0217 0.420494
$$960$$ 0 0
$$961$$ 52.1168 1.68119
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 10.3723i 0.334242i
$$964$$ −28.2337 −0.909346
$$965$$ 0 0
$$966$$ 3.25544 0.104742
$$967$$ − 22.3505i − 0.718745i −0.933194 0.359372i $$-0.882991\pi$$
0.933194 0.359372i $$-0.117009\pi$$
$$968$$ − 8.11684i − 0.260885i
$$969$$ −2.37228 −0.0762087
$$970$$ 0 0
$$971$$ −6.86141 −0.220193 −0.110097 0.993921i $$-0.535116\pi$$
−0.110097 + 0.993921i $$0.535116\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 42.9783i 1.37782i
$$974$$ 12.2337 0.391993
$$975$$ 0 0
$$976$$ −8.11684 −0.259814
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ 2.11684i 0.0676892i
$$979$$ 12.0000 0.383522
$$980$$ 0 0
$$981$$ 9.11684 0.291078
$$982$$ − 31.3723i − 1.00113i
$$983$$ − 9.76631i − 0.311497i −0.987797 0.155748i $$-0.950221\pi$$
0.987797 0.155748i $$-0.0497789\pi$$
$$984$$ 1.37228 0.0437467
$$985$$ 0 0
$$986$$ −8.74456 −0.278484
$$987$$ 3.86141i 0.122910i
$$988$$ 4.74456i 0.150945i
$$989$$ 4.97825 0.158299
$$990$$ 0 0
$$991$$ 60.4674 1.92081 0.960405 0.278609i $$-0.0898732\pi$$
0.960405 + 0.278609i $$0.0898732\pi$$
$$992$$ 9.11684i 0.289460i
$$993$$ − 14.6060i − 0.463506i
$$994$$ −3.25544 −0.103256
$$995$$ 0 0
$$996$$ 1.37228 0.0434824
$$997$$ 33.1168i 1.04882i 0.851466 + 0.524410i $$0.175714\pi$$
−0.851466 + 0.524410i $$0.824286\pi$$
$$998$$ − 18.6277i − 0.589650i
$$999$$ 1.00000 0.0316386
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 2550.2.d.v.2449.3 4
5.2 odd 4 2550.2.a.bg.1.2 2
5.3 odd 4 2550.2.a.bm.1.1 yes 2
5.4 even 2 inner 2550.2.d.v.2449.2 4
15.2 even 4 7650.2.a.df.1.2 2
15.8 even 4 7650.2.a.cv.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.bg.1.2 2 5.2 odd 4
2550.2.a.bm.1.1 yes 2 5.3 odd 4
2550.2.d.v.2449.2 4 5.4 even 2 inner
2550.2.d.v.2449.3 4 1.1 even 1 trivial
7650.2.a.cv.1.1 2 15.8 even 4
7650.2.a.df.1.2 2 15.2 even 4