# Properties

 Label 2175.2.a.q.1.1 Level $2175$ Weight $2$ Character 2175.1 Self dual yes Analytic conductor $17.367$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.a (trivial)

## 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: yes Analytic conductor: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -5.47214 q^{11} +1.61803 q^{12} +6.23607 q^{13} -1.85410 q^{14} +1.85410 q^{16} -3.47214 q^{17} -0.618034 q^{18} +7.70820 q^{19} -3.00000 q^{21} +3.38197 q^{22} -2.23607 q^{24} -3.85410 q^{26} -1.00000 q^{27} -4.85410 q^{28} +1.00000 q^{29} -8.00000 q^{31} -5.61803 q^{32} +5.47214 q^{33} +2.14590 q^{34} -1.61803 q^{36} +8.00000 q^{37} -4.76393 q^{38} -6.23607 q^{39} -4.47214 q^{41} +1.85410 q^{42} -3.23607 q^{43} +8.85410 q^{44} -6.70820 q^{47} -1.85410 q^{48} +2.00000 q^{49} +3.47214 q^{51} -10.0902 q^{52} +6.76393 q^{53} +0.618034 q^{54} +6.70820 q^{56} -7.70820 q^{57} -0.618034 q^{58} +5.23607 q^{59} -5.70820 q^{61} +4.94427 q^{62} +3.00000 q^{63} -0.236068 q^{64} -3.38197 q^{66} +11.4721 q^{67} +5.61803 q^{68} +7.23607 q^{71} +2.23607 q^{72} +8.00000 q^{73} -4.94427 q^{74} -12.4721 q^{76} -16.4164 q^{77} +3.85410 q^{78} -6.18034 q^{79} +1.00000 q^{81} +2.76393 q^{82} +3.70820 q^{83} +4.85410 q^{84} +2.00000 q^{86} -1.00000 q^{87} -12.2361 q^{88} +11.1803 q^{89} +18.7082 q^{91} +8.00000 q^{93} +4.14590 q^{94} +5.61803 q^{96} -2.76393 q^{97} -1.23607 q^{98} -5.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 $$2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} - 2 q^{11} + q^{12} + 8 q^{13} + 3 q^{14} - 3 q^{16} + 2 q^{17} + q^{18} + 2 q^{19} - 6 q^{21} + 9 q^{22} - q^{26} - 2 q^{27} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32} + 2 q^{33} + 11 q^{34} - q^{36} + 16 q^{37} - 14 q^{38} - 8 q^{39} - 3 q^{42} - 2 q^{43} + 11 q^{44} + 3 q^{48} + 4 q^{49} - 2 q^{51} - 9 q^{52} + 18 q^{53} - q^{54} - 2 q^{57} + q^{58} + 6 q^{59} + 2 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} - 9 q^{66} + 14 q^{67} + 9 q^{68} + 10 q^{71} + 16 q^{73} + 8 q^{74} - 16 q^{76} - 6 q^{77} + q^{78} + 10 q^{79} + 2 q^{81} + 10 q^{82} - 6 q^{83} + 3 q^{84} + 4 q^{86} - 2 q^{87} - 20 q^{88} + 24 q^{91} + 16 q^{93} + 15 q^{94} + 9 q^{96} - 10 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 - 2 * q^11 + q^12 + 8 * q^13 + 3 * q^14 - 3 * q^16 + 2 * q^17 + q^18 + 2 * q^19 - 6 * q^21 + 9 * q^22 - q^26 - 2 * q^27 - 3 * q^28 + 2 * q^29 - 16 * q^31 - 9 * q^32 + 2 * q^33 + 11 * q^34 - q^36 + 16 * q^37 - 14 * q^38 - 8 * q^39 - 3 * q^42 - 2 * q^43 + 11 * q^44 + 3 * q^48 + 4 * q^49 - 2 * q^51 - 9 * q^52 + 18 * q^53 - q^54 - 2 * q^57 + q^58 + 6 * q^59 + 2 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 - 9 * q^66 + 14 * q^67 + 9 * q^68 + 10 * q^71 + 16 * q^73 + 8 * q^74 - 16 * q^76 - 6 * q^77 + q^78 + 10 * q^79 + 2 * q^81 + 10 * q^82 - 6 * q^83 + 3 * q^84 + 4 * q^86 - 2 * q^87 - 20 * q^88 + 24 * q^91 + 16 * q^93 + 15 * q^94 + 9 * q^96 - 10 * q^97 + 2 * q^98 - 2 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.618034 −0.437016 −0.218508 0.975835i $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ 0.618034 0.252311
$$7$$ 3.00000 1.13389 0.566947 0.823754i $$-0.308125\pi$$
0.566947 + 0.823754i $$0.308125\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.47214 −1.64991 −0.824956 0.565198i $$-0.808800\pi$$
−0.824956 + 0.565198i $$0.808800\pi$$
$$12$$ 1.61803 0.467086
$$13$$ 6.23607 1.72957 0.864787 0.502139i $$-0.167453\pi$$
0.864787 + 0.502139i $$0.167453\pi$$
$$14$$ −1.85410 −0.495530
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ −3.47214 −0.842117 −0.421058 0.907034i $$-0.638341\pi$$
−0.421058 + 0.907034i $$0.638341\pi$$
$$18$$ −0.618034 −0.145672
$$19$$ 7.70820 1.76838 0.884192 0.467124i $$-0.154710\pi$$
0.884192 + 0.467124i $$0.154710\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 3.38197 0.721038
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −2.23607 −0.456435
$$25$$ 0 0
$$26$$ −3.85410 −0.755852
$$27$$ −1.00000 −0.192450
$$28$$ −4.85410 −0.917339
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −5.61803 −0.993137
$$33$$ 5.47214 0.952577
$$34$$ 2.14590 0.368018
$$35$$ 0 0
$$36$$ −1.61803 −0.269672
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ −4.76393 −0.772812
$$39$$ −6.23607 −0.998570
$$40$$ 0 0
$$41$$ −4.47214 −0.698430 −0.349215 0.937043i $$-0.613552\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 1.85410 0.286094
$$43$$ −3.23607 −0.493496 −0.246748 0.969080i $$-0.579362\pi$$
−0.246748 + 0.969080i $$0.579362\pi$$
$$44$$ 8.85410 1.33481
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −6.70820 −0.978492 −0.489246 0.872146i $$-0.662728\pi$$
−0.489246 + 0.872146i $$0.662728\pi$$
$$48$$ −1.85410 −0.267617
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ 3.47214 0.486196
$$52$$ −10.0902 −1.39925
$$53$$ 6.76393 0.929098 0.464549 0.885548i $$-0.346217\pi$$
0.464549 + 0.885548i $$0.346217\pi$$
$$54$$ 0.618034 0.0841038
$$55$$ 0 0
$$56$$ 6.70820 0.896421
$$57$$ −7.70820 −1.02098
$$58$$ −0.618034 −0.0811518
$$59$$ 5.23607 0.681678 0.340839 0.940122i $$-0.389289\pi$$
0.340839 + 0.940122i $$0.389289\pi$$
$$60$$ 0 0
$$61$$ −5.70820 −0.730861 −0.365430 0.930839i $$-0.619078\pi$$
−0.365430 + 0.930839i $$0.619078\pi$$
$$62$$ 4.94427 0.627923
$$63$$ 3.00000 0.377964
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ −3.38197 −0.416291
$$67$$ 11.4721 1.40154 0.700772 0.713385i $$-0.252839\pi$$
0.700772 + 0.713385i $$0.252839\pi$$
$$68$$ 5.61803 0.681287
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.23607 0.858763 0.429382 0.903123i $$-0.358732\pi$$
0.429382 + 0.903123i $$0.358732\pi$$
$$72$$ 2.23607 0.263523
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ −4.94427 −0.574760
$$75$$ 0 0
$$76$$ −12.4721 −1.43065
$$77$$ −16.4164 −1.87082
$$78$$ 3.85410 0.436391
$$79$$ −6.18034 −0.695343 −0.347671 0.937616i $$-0.613027\pi$$
−0.347671 + 0.937616i $$0.613027\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.76393 0.305225
$$83$$ 3.70820 0.407028 0.203514 0.979072i $$-0.434764\pi$$
0.203514 + 0.979072i $$0.434764\pi$$
$$84$$ 4.85410 0.529626
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ −1.00000 −0.107211
$$88$$ −12.2361 −1.30437
$$89$$ 11.1803 1.18511 0.592557 0.805529i $$-0.298119\pi$$
0.592557 + 0.805529i $$0.298119\pi$$
$$90$$ 0 0
$$91$$ 18.7082 1.96115
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 4.14590 0.427617
$$95$$ 0 0
$$96$$ 5.61803 0.573388
$$97$$ −2.76393 −0.280635 −0.140317 0.990107i $$-0.544812\pi$$
−0.140317 + 0.990107i $$0.544812\pi$$
$$98$$ −1.23607 −0.124862
$$99$$ −5.47214 −0.549970
$$100$$ 0 0
$$101$$ 4.23607 0.421505 0.210752 0.977540i $$-0.432409\pi$$
0.210752 + 0.977540i $$0.432409\pi$$
$$102$$ −2.14590 −0.212476
$$103$$ −7.41641 −0.730760 −0.365380 0.930858i $$-0.619061\pi$$
−0.365380 + 0.930858i $$0.619061\pi$$
$$104$$ 13.9443 1.36735
$$105$$ 0 0
$$106$$ −4.18034 −0.406031
$$107$$ −7.52786 −0.727746 −0.363873 0.931449i $$-0.618546\pi$$
−0.363873 + 0.931449i $$0.618546\pi$$
$$108$$ 1.61803 0.155695
$$109$$ −16.4164 −1.57241 −0.786203 0.617968i $$-0.787956\pi$$
−0.786203 + 0.617968i $$0.787956\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 5.56231 0.525589
$$113$$ 7.94427 0.747334 0.373667 0.927563i $$-0.378100\pi$$
0.373667 + 0.927563i $$0.378100\pi$$
$$114$$ 4.76393 0.446183
$$115$$ 0 0
$$116$$ −1.61803 −0.150231
$$117$$ 6.23607 0.576525
$$118$$ −3.23607 −0.297904
$$119$$ −10.4164 −0.954871
$$120$$ 0 0
$$121$$ 18.9443 1.72221
$$122$$ 3.52786 0.319398
$$123$$ 4.47214 0.403239
$$124$$ 12.9443 1.16243
$$125$$ 0 0
$$126$$ −1.85410 −0.165177
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 11.3820 1.00603
$$129$$ 3.23607 0.284920
$$130$$ 0 0
$$131$$ 3.47214 0.303362 0.151681 0.988430i $$-0.451531\pi$$
0.151681 + 0.988430i $$0.451531\pi$$
$$132$$ −8.85410 −0.770651
$$133$$ 23.1246 2.00516
$$134$$ −7.09017 −0.612497
$$135$$ 0 0
$$136$$ −7.76393 −0.665752
$$137$$ 10.9443 0.935032 0.467516 0.883985i $$-0.345149\pi$$
0.467516 + 0.883985i $$0.345149\pi$$
$$138$$ 0 0
$$139$$ −0.708204 −0.0600691 −0.0300345 0.999549i $$-0.509562\pi$$
−0.0300345 + 0.999549i $$0.509562\pi$$
$$140$$ 0 0
$$141$$ 6.70820 0.564933
$$142$$ −4.47214 −0.375293
$$143$$ −34.1246 −2.85364
$$144$$ 1.85410 0.154508
$$145$$ 0 0
$$146$$ −4.94427 −0.409191
$$147$$ −2.00000 −0.164957
$$148$$ −12.9443 −1.06401
$$149$$ 20.1803 1.65324 0.826619 0.562762i $$-0.190261\pi$$
0.826619 + 0.562762i $$0.190261\pi$$
$$150$$ 0 0
$$151$$ 2.47214 0.201180 0.100590 0.994928i $$-0.467927\pi$$
0.100590 + 0.994928i $$0.467927\pi$$
$$152$$ 17.2361 1.39803
$$153$$ −3.47214 −0.280706
$$154$$ 10.1459 0.817580
$$155$$ 0 0
$$156$$ 10.0902 0.807860
$$157$$ 11.7082 0.934416 0.467208 0.884147i $$-0.345260\pi$$
0.467208 + 0.884147i $$0.345260\pi$$
$$158$$ 3.81966 0.303876
$$159$$ −6.76393 −0.536415
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −0.618034 −0.0485573
$$163$$ 15.1246 1.18465 0.592326 0.805699i $$-0.298210\pi$$
0.592326 + 0.805699i $$0.298210\pi$$
$$164$$ 7.23607 0.565042
$$165$$ 0 0
$$166$$ −2.29180 −0.177878
$$167$$ 17.8885 1.38426 0.692129 0.721774i $$-0.256673\pi$$
0.692129 + 0.721774i $$0.256673\pi$$
$$168$$ −6.70820 −0.517549
$$169$$ 25.8885 1.99143
$$170$$ 0 0
$$171$$ 7.70820 0.589461
$$172$$ 5.23607 0.399246
$$173$$ 19.4164 1.47620 0.738101 0.674690i $$-0.235723\pi$$
0.738101 + 0.674690i $$0.235723\pi$$
$$174$$ 0.618034 0.0468530
$$175$$ 0 0
$$176$$ −10.1459 −0.764776
$$177$$ −5.23607 −0.393567
$$178$$ −6.90983 −0.517914
$$179$$ 4.18034 0.312453 0.156227 0.987721i $$-0.450067\pi$$
0.156227 + 0.987721i $$0.450067\pi$$
$$180$$ 0 0
$$181$$ 8.41641 0.625587 0.312793 0.949821i $$-0.398735\pi$$
0.312793 + 0.949821i $$0.398735\pi$$
$$182$$ −11.5623 −0.857055
$$183$$ 5.70820 0.421963
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.94427 −0.362532
$$187$$ 19.0000 1.38942
$$188$$ 10.8541 0.791617
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0.236068 0.0170367
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 1.70820 0.122642
$$195$$ 0 0
$$196$$ −3.23607 −0.231148
$$197$$ −14.9443 −1.06474 −0.532368 0.846513i $$-0.678698\pi$$
−0.532368 + 0.846513i $$0.678698\pi$$
$$198$$ 3.38197 0.240346
$$199$$ 20.7082 1.46797 0.733983 0.679168i $$-0.237659\pi$$
0.733983 + 0.679168i $$0.237659\pi$$
$$200$$ 0 0
$$201$$ −11.4721 −0.809182
$$202$$ −2.61803 −0.184204
$$203$$ 3.00000 0.210559
$$204$$ −5.61803 −0.393341
$$205$$ 0 0
$$206$$ 4.58359 0.319354
$$207$$ 0 0
$$208$$ 11.5623 0.801702
$$209$$ −42.1803 −2.91768
$$210$$ 0 0
$$211$$ −27.8885 −1.91993 −0.959963 0.280126i $$-0.909624\pi$$
−0.959963 + 0.280126i $$0.909624\pi$$
$$212$$ −10.9443 −0.751656
$$213$$ −7.23607 −0.495807
$$214$$ 4.65248 0.318037
$$215$$ 0 0
$$216$$ −2.23607 −0.152145
$$217$$ −24.0000 −1.62923
$$218$$ 10.1459 0.687167
$$219$$ −8.00000 −0.540590
$$220$$ 0 0
$$221$$ −21.6525 −1.45650
$$222$$ 4.94427 0.331838
$$223$$ 13.0000 0.870544 0.435272 0.900299i $$-0.356652\pi$$
0.435272 + 0.900299i $$0.356652\pi$$
$$224$$ −16.8541 −1.12611
$$225$$ 0 0
$$226$$ −4.90983 −0.326597
$$227$$ 5.81966 0.386264 0.193132 0.981173i $$-0.438135\pi$$
0.193132 + 0.981173i $$0.438135\pi$$
$$228$$ 12.4721 0.825987
$$229$$ 9.41641 0.622254 0.311127 0.950368i $$-0.399294\pi$$
0.311127 + 0.950368i $$0.399294\pi$$
$$230$$ 0 0
$$231$$ 16.4164 1.08012
$$232$$ 2.23607 0.146805
$$233$$ −1.41641 −0.0927920 −0.0463960 0.998923i $$-0.514774\pi$$
−0.0463960 + 0.998923i $$0.514774\pi$$
$$234$$ −3.85410 −0.251951
$$235$$ 0 0
$$236$$ −8.47214 −0.551489
$$237$$ 6.18034 0.401456
$$238$$ 6.43769 0.417294
$$239$$ 11.8885 0.769006 0.384503 0.923124i $$-0.374373\pi$$
0.384503 + 0.923124i $$0.374373\pi$$
$$240$$ 0 0
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ −11.7082 −0.752632
$$243$$ −1.00000 −0.0641500
$$244$$ 9.23607 0.591279
$$245$$ 0 0
$$246$$ −2.76393 −0.176222
$$247$$ 48.0689 3.05855
$$248$$ −17.8885 −1.13592
$$249$$ −3.70820 −0.234998
$$250$$ 0 0
$$251$$ −24.8885 −1.57095 −0.785475 0.618893i $$-0.787582\pi$$
−0.785475 + 0.618893i $$0.787582\pi$$
$$252$$ −4.85410 −0.305780
$$253$$ 0 0
$$254$$ −3.70820 −0.232673
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ 9.70820 0.605581 0.302791 0.953057i $$-0.402082\pi$$
0.302791 + 0.953057i $$0.402082\pi$$
$$258$$ −2.00000 −0.124515
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ −2.14590 −0.132574
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 12.2361 0.753078
$$265$$ 0 0
$$266$$ −14.2918 −0.876286
$$267$$ −11.1803 −0.684226
$$268$$ −18.5623 −1.13387
$$269$$ −30.2361 −1.84353 −0.921763 0.387754i $$-0.873251\pi$$
−0.921763 + 0.387754i $$0.873251\pi$$
$$270$$ 0 0
$$271$$ 14.3607 0.872349 0.436175 0.899862i $$-0.356333\pi$$
0.436175 + 0.899862i $$0.356333\pi$$
$$272$$ −6.43769 −0.390343
$$273$$ −18.7082 −1.13227
$$274$$ −6.76393 −0.408624
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.70820 −0.523225 −0.261613 0.965173i $$-0.584254\pi$$
−0.261613 + 0.965173i $$0.584254\pi$$
$$278$$ 0.437694 0.0262511
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −23.1246 −1.37950 −0.689749 0.724048i $$-0.742279\pi$$
−0.689749 + 0.724048i $$0.742279\pi$$
$$282$$ −4.14590 −0.246885
$$283$$ −18.4721 −1.09805 −0.549027 0.835804i $$-0.685002\pi$$
−0.549027 + 0.835804i $$0.685002\pi$$
$$284$$ −11.7082 −0.694754
$$285$$ 0 0
$$286$$ 21.0902 1.24709
$$287$$ −13.4164 −0.791946
$$288$$ −5.61803 −0.331046
$$289$$ −4.94427 −0.290840
$$290$$ 0 0
$$291$$ 2.76393 0.162025
$$292$$ −12.9443 −0.757506
$$293$$ 17.9443 1.04832 0.524158 0.851621i $$-0.324380\pi$$
0.524158 + 0.851621i $$0.324380\pi$$
$$294$$ 1.23607 0.0720889
$$295$$ 0 0
$$296$$ 17.8885 1.03975
$$297$$ 5.47214 0.317526
$$298$$ −12.4721 −0.722491
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −9.70820 −0.559572
$$302$$ −1.52786 −0.0879187
$$303$$ −4.23607 −0.243356
$$304$$ 14.2918 0.819691
$$305$$ 0 0
$$306$$ 2.14590 0.122673
$$307$$ −24.9443 −1.42364 −0.711822 0.702360i $$-0.752130\pi$$
−0.711822 + 0.702360i $$0.752130\pi$$
$$308$$ 26.5623 1.51353
$$309$$ 7.41641 0.421905
$$310$$ 0 0
$$311$$ 23.4721 1.33098 0.665491 0.746406i $$-0.268222\pi$$
0.665491 + 0.746406i $$0.268222\pi$$
$$312$$ −13.9443 −0.789439
$$313$$ −1.18034 −0.0667168 −0.0333584 0.999443i $$-0.510620\pi$$
−0.0333584 + 0.999443i $$0.510620\pi$$
$$314$$ −7.23607 −0.408355
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 28.4164 1.59602 0.798012 0.602641i $$-0.205885\pi$$
0.798012 + 0.602641i $$0.205885\pi$$
$$318$$ 4.18034 0.234422
$$319$$ −5.47214 −0.306381
$$320$$ 0 0
$$321$$ 7.52786 0.420164
$$322$$ 0 0
$$323$$ −26.7639 −1.48919
$$324$$ −1.61803 −0.0898908
$$325$$ 0 0
$$326$$ −9.34752 −0.517711
$$327$$ 16.4164 0.907829
$$328$$ −10.0000 −0.552158
$$329$$ −20.1246 −1.10951
$$330$$ 0 0
$$331$$ −15.8885 −0.873313 −0.436657 0.899628i $$-0.643838\pi$$
−0.436657 + 0.899628i $$0.643838\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 8.00000 0.438397
$$334$$ −11.0557 −0.604943
$$335$$ 0 0
$$336$$ −5.56231 −0.303449
$$337$$ 20.4721 1.11519 0.557594 0.830114i $$-0.311725\pi$$
0.557594 + 0.830114i $$0.311725\pi$$
$$338$$ −16.0000 −0.870285
$$339$$ −7.94427 −0.431474
$$340$$ 0 0
$$341$$ 43.7771 2.37066
$$342$$ −4.76393 −0.257604
$$343$$ −15.0000 −0.809924
$$344$$ −7.23607 −0.390143
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ −17.5279 −0.940945 −0.470473 0.882415i $$-0.655917\pi$$
−0.470473 + 0.882415i $$0.655917\pi$$
$$348$$ 1.61803 0.0867357
$$349$$ −18.9443 −1.01406 −0.507032 0.861927i $$-0.669257\pi$$
−0.507032 + 0.861927i $$0.669257\pi$$
$$350$$ 0 0
$$351$$ −6.23607 −0.332857
$$352$$ 30.7426 1.63859
$$353$$ −3.52786 −0.187769 −0.0938846 0.995583i $$-0.529928\pi$$
−0.0938846 + 0.995583i $$0.529928\pi$$
$$354$$ 3.23607 0.171995
$$355$$ 0 0
$$356$$ −18.0902 −0.958777
$$357$$ 10.4164 0.551295
$$358$$ −2.58359 −0.136547
$$359$$ −4.58359 −0.241913 −0.120956 0.992658i $$-0.538596\pi$$
−0.120956 + 0.992658i $$0.538596\pi$$
$$360$$ 0 0
$$361$$ 40.4164 2.12718
$$362$$ −5.20163 −0.273391
$$363$$ −18.9443 −0.994316
$$364$$ −30.2705 −1.58661
$$365$$ 0 0
$$366$$ −3.52786 −0.184404
$$367$$ −15.4164 −0.804730 −0.402365 0.915479i $$-0.631812\pi$$
−0.402365 + 0.915479i $$0.631812\pi$$
$$368$$ 0 0
$$369$$ −4.47214 −0.232810
$$370$$ 0 0
$$371$$ 20.2918 1.05350
$$372$$ −12.9443 −0.671129
$$373$$ −15.8885 −0.822678 −0.411339 0.911483i $$-0.634939\pi$$
−0.411339 + 0.911483i $$0.634939\pi$$
$$374$$ −11.7426 −0.607198
$$375$$ 0 0
$$376$$ −15.0000 −0.773566
$$377$$ 6.23607 0.321174
$$378$$ 1.85410 0.0953647
$$379$$ 6.94427 0.356703 0.178352 0.983967i $$-0.442924\pi$$
0.178352 + 0.983967i $$0.442924\pi$$
$$380$$ 0 0
$$381$$ −6.00000 −0.307389
$$382$$ −7.41641 −0.379456
$$383$$ 20.0689 1.02547 0.512736 0.858546i $$-0.328632\pi$$
0.512736 + 0.858546i $$0.328632\pi$$
$$384$$ −11.3820 −0.580834
$$385$$ 0 0
$$386$$ −3.70820 −0.188743
$$387$$ −3.23607 −0.164499
$$388$$ 4.47214 0.227038
$$389$$ 22.2361 1.12741 0.563707 0.825975i $$-0.309375\pi$$
0.563707 + 0.825975i $$0.309375\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 4.47214 0.225877
$$393$$ −3.47214 −0.175146
$$394$$ 9.23607 0.465306
$$395$$ 0 0
$$396$$ 8.85410 0.444935
$$397$$ 6.58359 0.330421 0.165211 0.986258i $$-0.447170\pi$$
0.165211 + 0.986258i $$0.447170\pi$$
$$398$$ −12.7984 −0.641525
$$399$$ −23.1246 −1.15768
$$400$$ 0 0
$$401$$ −36.6525 −1.83034 −0.915169 0.403071i $$-0.867943\pi$$
−0.915169 + 0.403071i $$0.867943\pi$$
$$402$$ 7.09017 0.353626
$$403$$ −49.8885 −2.48513
$$404$$ −6.85410 −0.341004
$$405$$ 0 0
$$406$$ −1.85410 −0.0920175
$$407$$ −43.7771 −2.16995
$$408$$ 7.76393 0.384372
$$409$$ 8.65248 0.427837 0.213919 0.976851i $$-0.431377\pi$$
0.213919 + 0.976851i $$0.431377\pi$$
$$410$$ 0 0
$$411$$ −10.9443 −0.539841
$$412$$ 12.0000 0.591198
$$413$$ 15.7082 0.772950
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −35.0344 −1.71770
$$417$$ 0.708204 0.0346809
$$418$$ 26.0689 1.27507
$$419$$ 34.3607 1.67863 0.839315 0.543646i $$-0.182957\pi$$
0.839315 + 0.543646i $$0.182957\pi$$
$$420$$ 0 0
$$421$$ −1.81966 −0.0886848 −0.0443424 0.999016i $$-0.514119\pi$$
−0.0443424 + 0.999016i $$0.514119\pi$$
$$422$$ 17.2361 0.839039
$$423$$ −6.70820 −0.326164
$$424$$ 15.1246 0.734516
$$425$$ 0 0
$$426$$ 4.47214 0.216676
$$427$$ −17.1246 −0.828718
$$428$$ 12.1803 0.588759
$$429$$ 34.1246 1.64755
$$430$$ 0 0
$$431$$ 34.4721 1.66046 0.830232 0.557418i $$-0.188208\pi$$
0.830232 + 0.557418i $$0.188208\pi$$
$$432$$ −1.85410 −0.0892055
$$433$$ 4.65248 0.223584 0.111792 0.993732i $$-0.464341\pi$$
0.111792 + 0.993732i $$0.464341\pi$$
$$434$$ 14.8328 0.711998
$$435$$ 0 0
$$436$$ 26.5623 1.27210
$$437$$ 0 0
$$438$$ 4.94427 0.236246
$$439$$ −10.1246 −0.483221 −0.241611 0.970373i $$-0.577676\pi$$
−0.241611 + 0.970373i $$0.577676\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 13.3820 0.636515
$$443$$ −23.7639 −1.12906 −0.564529 0.825413i $$-0.690942\pi$$
−0.564529 + 0.825413i $$0.690942\pi$$
$$444$$ 12.9443 0.614308
$$445$$ 0 0
$$446$$ −8.03444 −0.380442
$$447$$ −20.1803 −0.954497
$$448$$ −0.708204 −0.0334595
$$449$$ 1.76393 0.0832451 0.0416225 0.999133i $$-0.486747\pi$$
0.0416225 + 0.999133i $$0.486747\pi$$
$$450$$ 0 0
$$451$$ 24.4721 1.15235
$$452$$ −12.8541 −0.604606
$$453$$ −2.47214 −0.116151
$$454$$ −3.59675 −0.168804
$$455$$ 0 0
$$456$$ −17.2361 −0.807153
$$457$$ −6.12461 −0.286497 −0.143249 0.989687i $$-0.545755\pi$$
−0.143249 + 0.989687i $$0.545755\pi$$
$$458$$ −5.81966 −0.271935
$$459$$ 3.47214 0.162065
$$460$$ 0 0
$$461$$ 2.36068 0.109948 0.0549739 0.998488i $$-0.482492\pi$$
0.0549739 + 0.998488i $$0.482492\pi$$
$$462$$ −10.1459 −0.472030
$$463$$ 2.52786 0.117480 0.0587399 0.998273i $$-0.481292\pi$$
0.0587399 + 0.998273i $$0.481292\pi$$
$$464$$ 1.85410 0.0860745
$$465$$ 0 0
$$466$$ 0.875388 0.0405516
$$467$$ 12.9443 0.598989 0.299495 0.954098i $$-0.403182\pi$$
0.299495 + 0.954098i $$0.403182\pi$$
$$468$$ −10.0902 −0.466418
$$469$$ 34.4164 1.58920
$$470$$ 0 0
$$471$$ −11.7082 −0.539486
$$472$$ 11.7082 0.538914
$$473$$ 17.7082 0.814224
$$474$$ −3.81966 −0.175443
$$475$$ 0 0
$$476$$ 16.8541 0.772506
$$477$$ 6.76393 0.309699
$$478$$ −7.34752 −0.336068
$$479$$ −6.47214 −0.295719 −0.147860 0.989008i $$-0.547238\pi$$
−0.147860 + 0.989008i $$0.547238\pi$$
$$480$$ 0 0
$$481$$ 49.8885 2.27472
$$482$$ −4.32624 −0.197055
$$483$$ 0 0
$$484$$ −30.6525 −1.39329
$$485$$ 0 0
$$486$$ 0.618034 0.0280346
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ −12.7639 −0.577796
$$489$$ −15.1246 −0.683959
$$490$$ 0 0
$$491$$ 25.8885 1.16833 0.584167 0.811634i $$-0.301421\pi$$
0.584167 + 0.811634i $$0.301421\pi$$
$$492$$ −7.23607 −0.326227
$$493$$ −3.47214 −0.156377
$$494$$ −29.7082 −1.33664
$$495$$ 0 0
$$496$$ −14.8328 −0.666013
$$497$$ 21.7082 0.973746
$$498$$ 2.29180 0.102698
$$499$$ −23.7639 −1.06382 −0.531910 0.846801i $$-0.678525\pi$$
−0.531910 + 0.846801i $$0.678525\pi$$
$$500$$ 0 0
$$501$$ −17.8885 −0.799201
$$502$$ 15.3820 0.686531
$$503$$ 30.5967 1.36424 0.682121 0.731240i $$-0.261058\pi$$
0.682121 + 0.731240i $$0.261058\pi$$
$$504$$ 6.70820 0.298807
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −25.8885 −1.14975
$$508$$ −9.70820 −0.430732
$$509$$ 6.18034 0.273939 0.136969 0.990575i $$-0.456264\pi$$
0.136969 + 0.990575i $$0.456264\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ −18.7082 −0.826794
$$513$$ −7.70820 −0.340326
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ −5.23607 −0.230505
$$517$$ 36.7082 1.61442
$$518$$ −14.8328 −0.651717
$$519$$ −19.4164 −0.852286
$$520$$ 0 0
$$521$$ −20.7639 −0.909684 −0.454842 0.890572i $$-0.650304\pi$$
−0.454842 + 0.890572i $$0.650304\pi$$
$$522$$ −0.618034 −0.0270506
$$523$$ 29.8328 1.30450 0.652249 0.758005i $$-0.273826\pi$$
0.652249 + 0.758005i $$0.273826\pi$$
$$524$$ −5.61803 −0.245425
$$525$$ 0 0
$$526$$ 14.8328 0.646741
$$527$$ 27.7771 1.20999
$$528$$ 10.1459 0.441544
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 5.23607 0.227226
$$532$$ −37.4164 −1.62221
$$533$$ −27.8885 −1.20799
$$534$$ 6.90983 0.299018
$$535$$ 0 0
$$536$$ 25.6525 1.10802
$$537$$ −4.18034 −0.180395
$$538$$ 18.6869 0.805650
$$539$$ −10.9443 −0.471403
$$540$$ 0 0
$$541$$ 38.3607 1.64925 0.824627 0.565677i $$-0.191385\pi$$
0.824627 + 0.565677i $$0.191385\pi$$
$$542$$ −8.87539 −0.381231
$$543$$ −8.41641 −0.361183
$$544$$ 19.5066 0.836338
$$545$$ 0 0
$$546$$ 11.5623 0.494821
$$547$$ −27.4721 −1.17462 −0.587312 0.809361i $$-0.699814\pi$$
−0.587312 + 0.809361i $$0.699814\pi$$
$$548$$ −17.7082 −0.756457
$$549$$ −5.70820 −0.243620
$$550$$ 0 0
$$551$$ 7.70820 0.328381
$$552$$ 0 0
$$553$$ −18.5410 −0.788444
$$554$$ 5.38197 0.228658
$$555$$ 0 0
$$556$$ 1.14590 0.0485969
$$557$$ −2.76393 −0.117112 −0.0585558 0.998284i $$-0.518650\pi$$
−0.0585558 + 0.998284i $$0.518650\pi$$
$$558$$ 4.94427 0.209308
$$559$$ −20.1803 −0.853537
$$560$$ 0 0
$$561$$ −19.0000 −0.802181
$$562$$ 14.2918 0.602863
$$563$$ −44.1246 −1.85963 −0.929815 0.368026i $$-0.880034\pi$$
−0.929815 + 0.368026i $$0.880034\pi$$
$$564$$ −10.8541 −0.457040
$$565$$ 0 0
$$566$$ 11.4164 0.479867
$$567$$ 3.00000 0.125988
$$568$$ 16.1803 0.678912
$$569$$ −5.18034 −0.217171 −0.108586 0.994087i $$-0.534632\pi$$
−0.108586 + 0.994087i $$0.534632\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 55.2148 2.30865
$$573$$ −12.0000 −0.501307
$$574$$ 8.29180 0.346093
$$575$$ 0 0
$$576$$ −0.236068 −0.00983617
$$577$$ 29.3050 1.21998 0.609991 0.792409i $$-0.291173\pi$$
0.609991 + 0.792409i $$0.291173\pi$$
$$578$$ 3.05573 0.127102
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 11.1246 0.461527
$$582$$ −1.70820 −0.0708073
$$583$$ −37.0132 −1.53293
$$584$$ 17.8885 0.740233
$$585$$ 0 0
$$586$$ −11.0902 −0.458131
$$587$$ 1.81966 0.0751054 0.0375527 0.999295i $$-0.488044\pi$$
0.0375527 + 0.999295i $$0.488044\pi$$
$$588$$ 3.23607 0.133453
$$589$$ −61.6656 −2.54089
$$590$$ 0 0
$$591$$ 14.9443 0.614725
$$592$$ 14.8328 0.609625
$$593$$ 24.6525 1.01236 0.506178 0.862429i $$-0.331058\pi$$
0.506178 + 0.862429i $$0.331058\pi$$
$$594$$ −3.38197 −0.138764
$$595$$ 0 0
$$596$$ −32.6525 −1.33750
$$597$$ −20.7082 −0.847530
$$598$$ 0 0
$$599$$ −8.88854 −0.363176 −0.181588 0.983375i $$-0.558124\pi$$
−0.181588 + 0.983375i $$0.558124\pi$$
$$600$$ 0 0
$$601$$ 8.11146 0.330873 0.165437 0.986220i $$-0.447097\pi$$
0.165437 + 0.986220i $$0.447097\pi$$
$$602$$ 6.00000 0.244542
$$603$$ 11.4721 0.467181
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 2.61803 0.106350
$$607$$ −40.6525 −1.65003 −0.825017 0.565109i $$-0.808834\pi$$
−0.825017 + 0.565109i $$0.808834\pi$$
$$608$$ −43.3050 −1.75625
$$609$$ −3.00000 −0.121566
$$610$$ 0 0
$$611$$ −41.8328 −1.69237
$$612$$ 5.61803 0.227096
$$613$$ −7.29180 −0.294513 −0.147256 0.989098i $$-0.547044\pi$$
−0.147256 + 0.989098i $$0.547044\pi$$
$$614$$ 15.4164 0.622156
$$615$$ 0 0
$$616$$ −36.7082 −1.47902
$$617$$ −43.3050 −1.74339 −0.871696 0.490047i $$-0.836980\pi$$
−0.871696 + 0.490047i $$0.836980\pi$$
$$618$$ −4.58359 −0.184379
$$619$$ −3.52786 −0.141797 −0.0708984 0.997484i $$-0.522587\pi$$
−0.0708984 + 0.997484i $$0.522587\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −14.5066 −0.581661
$$623$$ 33.5410 1.34379
$$624$$ −11.5623 −0.462863
$$625$$ 0 0
$$626$$ 0.729490 0.0291563
$$627$$ 42.1803 1.68452
$$628$$ −18.9443 −0.755959
$$629$$ −27.7771 −1.10755
$$630$$ 0 0
$$631$$ −38.4853 −1.53208 −0.766038 0.642796i $$-0.777774\pi$$
−0.766038 + 0.642796i $$0.777774\pi$$
$$632$$ −13.8197 −0.549717
$$633$$ 27.8885 1.10847
$$634$$ −17.5623 −0.697488
$$635$$ 0 0
$$636$$ 10.9443 0.433969
$$637$$ 12.4721 0.494164
$$638$$ 3.38197 0.133893
$$639$$ 7.23607 0.286254
$$640$$ 0 0
$$641$$ 46.2361 1.82621 0.913107 0.407719i $$-0.133676\pi$$
0.913107 + 0.407719i $$0.133676\pi$$
$$642$$ −4.65248 −0.183619
$$643$$ 35.8328 1.41311 0.706554 0.707659i $$-0.250249\pi$$
0.706554 + 0.707659i $$0.250249\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 16.5410 0.650798
$$647$$ −11.2361 −0.441735 −0.220868 0.975304i $$-0.570889\pi$$
−0.220868 + 0.975304i $$0.570889\pi$$
$$648$$ 2.23607 0.0878410
$$649$$ −28.6525 −1.12471
$$650$$ 0 0
$$651$$ 24.0000 0.940634
$$652$$ −24.4721 −0.958403
$$653$$ −8.88854 −0.347836 −0.173918 0.984760i $$-0.555643\pi$$
−0.173918 + 0.984760i $$0.555643\pi$$
$$654$$ −10.1459 −0.396736
$$655$$ 0 0
$$656$$ −8.29180 −0.323740
$$657$$ 8.00000 0.312110
$$658$$ 12.4377 0.484872
$$659$$ −21.0000 −0.818044 −0.409022 0.912525i $$-0.634130\pi$$
−0.409022 + 0.912525i $$0.634130\pi$$
$$660$$ 0 0
$$661$$ −37.8328 −1.47153 −0.735763 0.677239i $$-0.763176\pi$$
−0.735763 + 0.677239i $$0.763176\pi$$
$$662$$ 9.81966 0.381652
$$663$$ 21.6525 0.840912
$$664$$ 8.29180 0.321784
$$665$$ 0 0
$$666$$ −4.94427 −0.191587
$$667$$ 0 0
$$668$$ −28.9443 −1.11989
$$669$$ −13.0000 −0.502609
$$670$$ 0 0
$$671$$ 31.2361 1.20586
$$672$$ 16.8541 0.650161
$$673$$ 33.2918 1.28330 0.641652 0.766996i $$-0.278249\pi$$
0.641652 + 0.766996i $$0.278249\pi$$
$$674$$ −12.6525 −0.487355
$$675$$ 0 0
$$676$$ −41.8885 −1.61110
$$677$$ 20.8885 0.802812 0.401406 0.915900i $$-0.368522\pi$$
0.401406 + 0.915900i $$0.368522\pi$$
$$678$$ 4.90983 0.188561
$$679$$ −8.29180 −0.318210
$$680$$ 0 0
$$681$$ −5.81966 −0.223010
$$682$$ −27.0557 −1.03602
$$683$$ −7.41641 −0.283781 −0.141890 0.989882i $$-0.545318\pi$$
−0.141890 + 0.989882i $$0.545318\pi$$
$$684$$ −12.4721 −0.476884
$$685$$ 0 0
$$686$$ 9.27051 0.353950
$$687$$ −9.41641 −0.359258
$$688$$ −6.00000 −0.228748
$$689$$ 42.1803 1.60694
$$690$$ 0 0
$$691$$ −34.7082 −1.32036 −0.660181 0.751106i $$-0.729520\pi$$
−0.660181 + 0.751106i $$0.729520\pi$$
$$692$$ −31.4164 −1.19427
$$693$$ −16.4164 −0.623608
$$694$$ 10.8328 0.411208
$$695$$ 0 0
$$696$$ −2.23607 −0.0847579
$$697$$ 15.5279 0.588160
$$698$$ 11.7082 0.443162
$$699$$ 1.41641 0.0535735
$$700$$ 0 0
$$701$$ −5.12461 −0.193554 −0.0967770 0.995306i $$-0.530853\pi$$
−0.0967770 + 0.995306i $$0.530853\pi$$
$$702$$ 3.85410 0.145464
$$703$$ 61.6656 2.32576
$$704$$ 1.29180 0.0486864
$$705$$ 0 0
$$706$$ 2.18034 0.0820582
$$707$$ 12.7082 0.477941
$$708$$ 8.47214 0.318402
$$709$$ 19.8885 0.746930 0.373465 0.927644i $$-0.378170\pi$$
0.373465 + 0.927644i $$0.378170\pi$$
$$710$$ 0 0
$$711$$ −6.18034 −0.231781
$$712$$ 25.0000 0.936915
$$713$$ 0 0
$$714$$ −6.43769 −0.240925
$$715$$ 0 0
$$716$$ −6.76393 −0.252780
$$717$$ −11.8885 −0.443986
$$718$$ 2.83282 0.105720
$$719$$ −11.2361 −0.419035 −0.209517 0.977805i $$-0.567189\pi$$
−0.209517 + 0.977805i $$0.567189\pi$$
$$720$$ 0 0
$$721$$ −22.2492 −0.828604
$$722$$ −24.9787 −0.929611
$$723$$ −7.00000 −0.260333
$$724$$ −13.6180 −0.506110
$$725$$ 0 0
$$726$$ 11.7082 0.434532
$$727$$ −2.11146 −0.0783096 −0.0391548 0.999233i $$-0.512467\pi$$
−0.0391548 + 0.999233i $$0.512467\pi$$
$$728$$ 41.8328 1.55043
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 11.2361 0.415581
$$732$$ −9.23607 −0.341375
$$733$$ 14.2918 0.527880 0.263940 0.964539i $$-0.414978\pi$$
0.263940 + 0.964539i $$0.414978\pi$$
$$734$$ 9.52786 0.351680
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −62.7771 −2.31242
$$738$$ 2.76393 0.101742
$$739$$ 41.4853 1.52606 0.763031 0.646362i $$-0.223711\pi$$
0.763031 + 0.646362i $$0.223711\pi$$
$$740$$ 0 0
$$741$$ −48.0689 −1.76585
$$742$$ −12.5410 −0.460395
$$743$$ 10.8197 0.396935 0.198467 0.980107i $$-0.436404\pi$$
0.198467 + 0.980107i $$0.436404\pi$$
$$744$$ 17.8885 0.655826
$$745$$ 0 0
$$746$$ 9.81966 0.359523
$$747$$ 3.70820 0.135676
$$748$$ −30.7426 −1.12406
$$749$$ −22.5836 −0.825186
$$750$$ 0 0
$$751$$ 43.4164 1.58429 0.792144 0.610335i $$-0.208965\pi$$
0.792144 + 0.610335i $$0.208965\pi$$
$$752$$ −12.4377 −0.453556
$$753$$ 24.8885 0.906989
$$754$$ −3.85410 −0.140358
$$755$$ 0 0
$$756$$ 4.85410 0.176542
$$757$$ 34.8328 1.26602 0.633010 0.774144i $$-0.281819\pi$$
0.633010 + 0.774144i $$0.281819\pi$$
$$758$$ −4.29180 −0.155885
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.0132 1.19673 0.598363 0.801225i $$-0.295818\pi$$
0.598363 + 0.801225i $$0.295818\pi$$
$$762$$ 3.70820 0.134334
$$763$$ −49.2492 −1.78294
$$764$$ −19.4164 −0.702461
$$765$$ 0 0
$$766$$ −12.4033 −0.448148
$$767$$ 32.6525 1.17901
$$768$$ 6.56231 0.236797
$$769$$ 34.6525 1.24960 0.624800 0.780785i $$-0.285180\pi$$
0.624800 + 0.780785i $$0.285180\pi$$
$$770$$ 0 0
$$771$$ −9.70820 −0.349632
$$772$$ −9.70820 −0.349406
$$773$$ 52.2492 1.87927 0.939637 0.342173i $$-0.111163\pi$$
0.939637 + 0.342173i $$0.111163\pi$$
$$774$$ 2.00000 0.0718885
$$775$$ 0 0
$$776$$ −6.18034 −0.221861
$$777$$ −24.0000 −0.860995
$$778$$ −13.7426 −0.492698
$$779$$ −34.4721 −1.23509
$$780$$ 0 0
$$781$$ −39.5967 −1.41688
$$782$$ 0 0
$$783$$ −1.00000 −0.0357371
$$784$$ 3.70820 0.132436
$$785$$ 0 0
$$786$$ 2.14590 0.0765416
$$787$$ −1.88854 −0.0673193 −0.0336597 0.999433i $$-0.510716\pi$$
−0.0336597 + 0.999433i $$0.510716\pi$$
$$788$$ 24.1803 0.861389
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 23.8328 0.847397
$$792$$ −12.2361 −0.434790
$$793$$ −35.5967 −1.26408
$$794$$ −4.06888 −0.144399
$$795$$ 0 0
$$796$$ −33.5066 −1.18761
$$797$$ −6.94427 −0.245979 −0.122989 0.992408i $$-0.539248\pi$$
−0.122989 + 0.992408i $$0.539248\pi$$
$$798$$ 14.2918 0.505924
$$799$$ 23.2918 0.824005
$$800$$ 0 0
$$801$$ 11.1803 0.395038
$$802$$ 22.6525 0.799887
$$803$$ −43.7771 −1.54486
$$804$$ 18.5623 0.654642
$$805$$ 0 0
$$806$$ 30.8328 1.08604
$$807$$ 30.2361 1.06436
$$808$$ 9.47214 0.333229
$$809$$ 44.2361 1.55526 0.777629 0.628724i $$-0.216422\pi$$
0.777629 + 0.628724i $$0.216422\pi$$
$$810$$ 0 0
$$811$$ −22.7082 −0.797393 −0.398696 0.917083i $$-0.630537\pi$$
−0.398696 + 0.917083i $$0.630537\pi$$
$$812$$ −4.85410 −0.170346
$$813$$ −14.3607 −0.503651
$$814$$ 27.0557 0.948303
$$815$$ 0 0
$$816$$ 6.43769 0.225364
$$817$$ −24.9443 −0.872690
$$818$$ −5.34752 −0.186972
$$819$$ 18.7082 0.653718
$$820$$ 0 0
$$821$$ −10.5836 −0.369370 −0.184685 0.982798i $$-0.559126\pi$$
−0.184685 + 0.982798i $$0.559126\pi$$
$$822$$ 6.76393 0.235919
$$823$$ 28.7639 1.00265 0.501324 0.865260i $$-0.332847\pi$$
0.501324 + 0.865260i $$0.332847\pi$$
$$824$$ −16.5836 −0.577717
$$825$$ 0 0
$$826$$ −9.70820 −0.337792
$$827$$ 7.41641 0.257894 0.128947 0.991652i $$-0.458840\pi$$
0.128947 + 0.991652i $$0.458840\pi$$
$$828$$ 0 0
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 8.70820 0.302084
$$832$$ −1.47214 −0.0510371
$$833$$ −6.94427 −0.240605
$$834$$ −0.437694 −0.0151561
$$835$$ 0 0
$$836$$ 68.2492 2.36045
$$837$$ 8.00000 0.276520
$$838$$ −21.2361 −0.733588
$$839$$ 14.8885 0.514010 0.257005 0.966410i $$-0.417264\pi$$
0.257005 + 0.966410i $$0.417264\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 1.12461 0.0387567
$$843$$ 23.1246 0.796454
$$844$$ 45.1246 1.55325
$$845$$ 0 0
$$846$$ 4.14590 0.142539
$$847$$ 56.8328 1.95280
$$848$$ 12.5410 0.430660
$$849$$ 18.4721 0.633962
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 11.7082 0.401116
$$853$$ 28.7639 0.984858 0.492429 0.870353i $$-0.336109\pi$$
0.492429 + 0.870353i $$0.336109\pi$$
$$854$$ 10.5836 0.362163
$$855$$ 0 0
$$856$$ −16.8328 −0.575334
$$857$$ 23.8885 0.816017 0.408009 0.912978i $$-0.366223\pi$$
0.408009 + 0.912978i $$0.366223\pi$$
$$858$$ −21.0902 −0.720007
$$859$$ −15.7082 −0.535957 −0.267979 0.963425i $$-0.586356\pi$$
−0.267979 + 0.963425i $$0.586356\pi$$
$$860$$ 0 0
$$861$$ 13.4164 0.457230
$$862$$ −21.3050 −0.725650
$$863$$ −15.5967 −0.530919 −0.265460 0.964122i $$-0.585524\pi$$
−0.265460 + 0.964122i $$0.585524\pi$$
$$864$$ 5.61803 0.191129
$$865$$ 0 0
$$866$$ −2.87539 −0.0977097
$$867$$ 4.94427 0.167916
$$868$$ 38.8328 1.31807
$$869$$ 33.8197 1.14725
$$870$$ 0 0
$$871$$ 71.5410 2.42407
$$872$$ −36.7082 −1.24310
$$873$$ −2.76393 −0.0935449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 12.9443 0.437346
$$877$$ −48.8328 −1.64897 −0.824484 0.565886i $$-0.808534\pi$$
−0.824484 + 0.565886i $$0.808534\pi$$
$$878$$ 6.25735 0.211175
$$879$$ −17.9443 −0.605245
$$880$$ 0 0
$$881$$ 12.7082 0.428150 0.214075 0.976817i $$-0.431326\pi$$
0.214075 + 0.976817i $$0.431326\pi$$
$$882$$ −1.23607 −0.0416206
$$883$$ 32.0000 1.07689 0.538443 0.842662i $$-0.319013\pi$$
0.538443 + 0.842662i $$0.319013\pi$$
$$884$$ 35.0344 1.17834
$$885$$ 0 0
$$886$$ 14.6869 0.493417
$$887$$ 44.5967 1.49741 0.748706 0.662902i $$-0.230675\pi$$
0.748706 + 0.662902i $$0.230675\pi$$
$$888$$ −17.8885 −0.600300
$$889$$ 18.0000 0.603701
$$890$$ 0 0
$$891$$ −5.47214 −0.183323
$$892$$ −21.0344 −0.704285
$$893$$ −51.7082 −1.73035
$$894$$ 12.4721 0.417131
$$895$$ 0 0
$$896$$ 34.1459 1.14073
$$897$$ 0 0
$$898$$ −1.09017 −0.0363794
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ −23.4853 −0.782409
$$902$$ −15.1246 −0.503594
$$903$$ 9.70820 0.323069
$$904$$ 17.7639 0.590820
$$905$$ 0 0
$$906$$ 1.52786 0.0507599
$$907$$ −26.7639 −0.888682 −0.444341 0.895858i $$-0.646562\pi$$
−0.444341 + 0.895858i $$0.646562\pi$$
$$908$$ −9.41641 −0.312494
$$909$$ 4.23607 0.140502
$$910$$ 0 0
$$911$$ −18.0557 −0.598213 −0.299106 0.954220i $$-0.596689\pi$$
−0.299106 + 0.954220i $$0.596689\pi$$
$$912$$ −14.2918 −0.473249
$$913$$ −20.2918 −0.671560
$$914$$ 3.78522 0.125204
$$915$$ 0 0
$$916$$ −15.2361 −0.503414
$$917$$ 10.4164 0.343980
$$918$$ −2.14590 −0.0708252
$$919$$ 20.1246 0.663850 0.331925 0.943306i $$-0.392302\pi$$
0.331925 + 0.943306i $$0.392302\pi$$
$$920$$ 0 0
$$921$$ 24.9443 0.821942
$$922$$ −1.45898 −0.0480490
$$923$$ 45.1246 1.48529
$$924$$ −26.5623 −0.873836
$$925$$ 0 0
$$926$$ −1.56231 −0.0513406
$$927$$ −7.41641 −0.243587
$$928$$ −5.61803 −0.184421
$$929$$ −56.8328 −1.86462 −0.932312 0.361655i $$-0.882212\pi$$
−0.932312 + 0.361655i $$0.882212\pi$$
$$930$$ 0 0
$$931$$ 15.4164 0.505252
$$932$$ 2.29180 0.0750703
$$933$$ −23.4721 −0.768443
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 13.9443 0.455783
$$937$$ −19.7639 −0.645660 −0.322830 0.946457i $$-0.604634\pi$$
−0.322830 + 0.946457i $$0.604634\pi$$
$$938$$ −21.2705 −0.694507
$$939$$ 1.18034 0.0385189
$$940$$ 0 0
$$941$$ −46.0689 −1.50180 −0.750901 0.660414i $$-0.770381\pi$$
−0.750901 + 0.660414i $$0.770381\pi$$
$$942$$ 7.23607 0.235764
$$943$$ 0 0
$$944$$ 9.70820 0.315975
$$945$$ 0 0
$$946$$ −10.9443 −0.355829
$$947$$ 12.7082 0.412961 0.206481 0.978451i $$-0.433799\pi$$
0.206481 + 0.978451i $$0.433799\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 49.8885 1.61945
$$950$$ 0 0
$$951$$ −28.4164 −0.921465
$$952$$ −23.2918 −0.754891
$$953$$ −19.0132 −0.615897 −0.307948 0.951403i $$-0.599642\pi$$
−0.307948 + 0.951403i $$0.599642\pi$$
$$954$$ −4.18034 −0.135344
$$955$$ 0 0
$$956$$ −19.2361 −0.622139
$$957$$ 5.47214 0.176889
$$958$$ 4.00000 0.129234
$$959$$ 32.8328 1.06023
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −30.8328 −0.994090
$$963$$ −7.52786 −0.242582
$$964$$ −11.3262 −0.364794
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 35.1246 1.12953 0.564766 0.825251i $$-0.308967\pi$$
0.564766 + 0.825251i $$0.308967\pi$$
$$968$$ 42.3607 1.36152
$$969$$ 26.7639 0.859781
$$970$$ 0 0
$$971$$ −23.0557 −0.739894 −0.369947 0.929053i $$-0.620624\pi$$
−0.369947 + 0.929053i $$0.620624\pi$$
$$972$$ 1.61803 0.0518985
$$973$$ −2.12461 −0.0681119
$$974$$ −7.41641 −0.237637
$$975$$ 0 0
$$976$$ −10.5836 −0.338773
$$977$$ 22.4721 0.718947 0.359474 0.933155i $$-0.382956\pi$$
0.359474 + 0.933155i $$0.382956\pi$$
$$978$$ 9.34752 0.298901
$$979$$ −61.1803 −1.95533
$$980$$ 0 0
$$981$$ −16.4164 −0.524136
$$982$$ −16.0000 −0.510581
$$983$$ −37.5279 −1.19695 −0.598476 0.801140i $$-0.704227\pi$$
−0.598476 + 0.801140i $$0.704227\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ 2.14590 0.0683393
$$987$$ 20.1246 0.640573
$$988$$ −77.7771 −2.47442
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 22.4853 0.714269 0.357134 0.934053i $$-0.383754\pi$$
0.357134 + 0.934053i $$0.383754\pi$$
$$992$$ 44.9443 1.42698
$$993$$ 15.8885 0.504208
$$994$$ −13.4164 −0.425543
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ −7.41641 −0.234880 −0.117440 0.993080i $$-0.537469\pi$$
−0.117440 + 0.993080i $$0.537469\pi$$
$$998$$ 14.6869 0.464906
$$999$$ −8.00000 −0.253109
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 2175.2.a.q.1.1 2
3.2 odd 2 6525.2.a.s.1.2 2
5.2 odd 4 2175.2.c.j.349.2 4
5.3 odd 4 2175.2.c.j.349.3 4
5.4 even 2 435.2.a.e.1.2 2
15.14 odd 2 1305.2.a.k.1.1 2
20.19 odd 2 6960.2.a.bu.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.e.1.2 2 5.4 even 2
1305.2.a.k.1.1 2 15.14 odd 2
2175.2.a.q.1.1 2 1.1 even 1 trivial
2175.2.c.j.349.2 4 5.2 odd 4
2175.2.c.j.349.3 4 5.3 odd 4
6525.2.a.s.1.2 2 3.2 odd 2
6960.2.a.bu.1.2 2 20.19 odd 2