# Properties

 Label 2175.2.c.j.349.2 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.c (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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.2 Root $$-0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.j.349.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} -5.47214 q^{11} +1.61803i q^{12} -6.23607i q^{13} +1.85410 q^{14} +1.85410 q^{16} -3.47214i q^{17} +0.618034i q^{18} -7.70820 q^{19} -3.00000 q^{21} +3.38197i q^{22} +2.23607 q^{24} -3.85410 q^{26} -1.00000i q^{27} +4.85410i q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.61803i q^{32} -5.47214i q^{33} -2.14590 q^{34} -1.61803 q^{36} +8.00000i q^{37} +4.76393i q^{38} +6.23607 q^{39} -4.47214 q^{41} +1.85410i q^{42} +3.23607i q^{43} -8.85410 q^{44} -6.70820i q^{47} +1.85410i q^{48} -2.00000 q^{49} +3.47214 q^{51} -10.0902i q^{52} -6.76393i q^{53} -0.618034 q^{54} +6.70820 q^{56} -7.70820i q^{57} +0.618034i q^{58} -5.23607 q^{59} -5.70820 q^{61} +4.94427i q^{62} -3.00000i q^{63} +0.236068 q^{64} -3.38197 q^{66} +11.4721i q^{67} -5.61803i q^{68} +7.23607 q^{71} +2.23607i q^{72} -8.00000i q^{73} +4.94427 q^{74} -12.4721 q^{76} -16.4164i q^{77} -3.85410i q^{78} +6.18034 q^{79} +1.00000 q^{81} +2.76393i q^{82} -3.70820i q^{83} -4.85410 q^{84} +2.00000 q^{86} -1.00000i q^{87} +12.2361i q^{88} -11.1803 q^{89} +18.7082 q^{91} -8.00000i q^{93} -4.14590 q^{94} +5.61803 q^{96} -2.76393i q^{97} +1.23607i q^{98} +5.47214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 $$4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 4 q^{11} - 6 q^{14} - 6 q^{16} - 4 q^{19} - 12 q^{21} - 2 q^{26} - 4 q^{29} - 32 q^{31} - 22 q^{34} - 2 q^{36} + 16 q^{39} - 22 q^{44} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 12 q^{59} + 4 q^{61} - 8 q^{64} - 18 q^{66} + 20 q^{71} - 16 q^{74} - 32 q^{76} - 20 q^{79} + 4 q^{81} - 6 q^{84} + 8 q^{86} + 48 q^{91} - 30 q^{94} + 18 q^{96} + 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 4 * q^11 - 6 * q^14 - 6 * q^16 - 4 * q^19 - 12 * q^21 - 2 * q^26 - 4 * q^29 - 32 * q^31 - 22 * q^34 - 2 * q^36 + 16 * q^39 - 22 * q^44 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 12 * q^59 + 4 * q^61 - 8 * q^64 - 18 * q^66 + 20 * q^71 - 16 * q^74 - 32 * q^76 - 20 * q^79 + 4 * q^81 - 6 * q^84 + 8 * q^86 + 48 * q^91 - 30 * q^94 + 18 * q^96 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.618034i − 0.437016i −0.975835 0.218508i $$-0.929881\pi$$
0.975835 0.218508i $$-0.0701190\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.61803 0.809017
$$5$$ 0 0
$$6$$ 0.618034 0.252311
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ − 2.23607i − 0.790569i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −5.47214 −1.64991 −0.824956 0.565198i $$-0.808800\pi$$
−0.824956 + 0.565198i $$0.808800\pi$$
$$12$$ 1.61803i 0.467086i
$$13$$ − 6.23607i − 1.72957i −0.502139 0.864787i $$-0.667453\pi$$
0.502139 0.864787i $$-0.332547\pi$$
$$14$$ 1.85410 0.495530
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ − 3.47214i − 0.842117i −0.907034 0.421058i $$-0.861659\pi$$
0.907034 0.421058i $$-0.138341\pi$$
$$18$$ 0.618034i 0.145672i
$$19$$ −7.70820 −1.76838 −0.884192 0.467124i $$-0.845290\pi$$
−0.884192 + 0.467124i $$0.845290\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 3.38197i 0.721038i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −3.85410 −0.755852
$$27$$ − 1.00000i − 0.192450i
$$28$$ 4.85410i 0.917339i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 5.61803i − 0.993137i
$$33$$ − 5.47214i − 0.952577i
$$34$$ −2.14590 −0.368018
$$35$$ 0 0
$$36$$ −1.61803 −0.269672
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ 4.76393i 0.772812i
$$39$$ 6.23607 0.998570
$$40$$ 0 0
$$41$$ −4.47214 −0.698430 −0.349215 0.937043i $$-0.613552\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 1.85410i 0.286094i
$$43$$ 3.23607i 0.493496i 0.969080 + 0.246748i $$0.0793619\pi$$
−0.969080 + 0.246748i $$0.920638\pi$$
$$44$$ −8.85410 −1.33481
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 6.70820i − 0.978492i −0.872146 0.489246i $$-0.837272\pi$$
0.872146 0.489246i $$-0.162728\pi$$
$$48$$ 1.85410i 0.267617i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 3.47214 0.486196
$$52$$ − 10.0902i − 1.39925i
$$53$$ − 6.76393i − 0.929098i −0.885548 0.464549i $$-0.846217\pi$$
0.885548 0.464549i $$-0.153783\pi$$
$$54$$ −0.618034 −0.0841038
$$55$$ 0 0
$$56$$ 6.70820 0.896421
$$57$$ − 7.70820i − 1.02098i
$$58$$ 0.618034i 0.0811518i
$$59$$ −5.23607 −0.681678 −0.340839 0.940122i $$-0.610711\pi$$
−0.340839 + 0.940122i $$0.610711\pi$$
$$60$$ 0 0
$$61$$ −5.70820 −0.730861 −0.365430 0.930839i $$-0.619078\pi$$
−0.365430 + 0.930839i $$0.619078\pi$$
$$62$$ 4.94427i 0.627923i
$$63$$ − 3.00000i − 0.377964i
$$64$$ 0.236068 0.0295085
$$65$$ 0 0
$$66$$ −3.38197 −0.416291
$$67$$ 11.4721i 1.40154i 0.713385 + 0.700772i $$0.247161\pi$$
−0.713385 + 0.700772i $$0.752839\pi$$
$$68$$ − 5.61803i − 0.681287i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 7.23607 0.858763 0.429382 0.903123i $$-0.358732\pi$$
0.429382 + 0.903123i $$0.358732\pi$$
$$72$$ 2.23607i 0.263523i
$$73$$ − 8.00000i − 0.936329i −0.883641 0.468165i $$-0.844915\pi$$
0.883641 0.468165i $$-0.155085\pi$$
$$74$$ 4.94427 0.574760
$$75$$ 0 0
$$76$$ −12.4721 −1.43065
$$77$$ − 16.4164i − 1.87082i
$$78$$ − 3.85410i − 0.436391i
$$79$$ 6.18034 0.695343 0.347671 0.937616i $$-0.386973\pi$$
0.347671 + 0.937616i $$0.386973\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.76393i 0.305225i
$$83$$ − 3.70820i − 0.407028i −0.979072 0.203514i $$-0.934764\pi$$
0.979072 0.203514i $$-0.0652363\pi$$
$$84$$ −4.85410 −0.529626
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ − 1.00000i − 0.107211i
$$88$$ 12.2361i 1.30437i
$$89$$ −11.1803 −1.18511 −0.592557 0.805529i $$-0.701881\pi$$
−0.592557 + 0.805529i $$0.701881\pi$$
$$90$$ 0 0
$$91$$ 18.7082 1.96115
$$92$$ 0 0
$$93$$ − 8.00000i − 0.829561i
$$94$$ −4.14590 −0.427617
$$95$$ 0 0
$$96$$ 5.61803 0.573388
$$97$$ − 2.76393i − 0.280635i −0.990107 0.140317i $$-0.955188\pi$$
0.990107 0.140317i $$-0.0448123\pi$$
$$98$$ 1.23607i 0.124862i
$$99$$ 5.47214 0.549970
$$100$$ 0 0
$$101$$ 4.23607 0.421505 0.210752 0.977540i $$-0.432409\pi$$
0.210752 + 0.977540i $$0.432409\pi$$
$$102$$ − 2.14590i − 0.212476i
$$103$$ 7.41641i 0.730760i 0.930858 + 0.365380i $$0.119061\pi$$
−0.930858 + 0.365380i $$0.880939\pi$$
$$104$$ −13.9443 −1.36735
$$105$$ 0 0
$$106$$ −4.18034 −0.406031
$$107$$ − 7.52786i − 0.727746i −0.931449 0.363873i $$-0.881454\pi$$
0.931449 0.363873i $$-0.118546\pi$$
$$108$$ − 1.61803i − 0.155695i
$$109$$ 16.4164 1.57241 0.786203 0.617968i $$-0.212044\pi$$
0.786203 + 0.617968i $$0.212044\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 5.56231i 0.525589i
$$113$$ − 7.94427i − 0.747334i −0.927563 0.373667i $$-0.878100\pi$$
0.927563 0.373667i $$-0.121900\pi$$
$$114$$ −4.76393 −0.446183
$$115$$ 0 0
$$116$$ −1.61803 −0.150231
$$117$$ 6.23607i 0.576525i
$$118$$ 3.23607i 0.297904i
$$119$$ 10.4164 0.954871
$$120$$ 0 0
$$121$$ 18.9443 1.72221
$$122$$ 3.52786i 0.319398i
$$123$$ − 4.47214i − 0.403239i
$$124$$ −12.9443 −1.16243
$$125$$ 0 0
$$126$$ −1.85410 −0.165177
$$127$$ 6.00000i 0.532414i 0.963916 + 0.266207i $$0.0857705\pi$$
−0.963916 + 0.266207i $$0.914230\pi$$
$$128$$ − 11.3820i − 1.00603i
$$129$$ −3.23607 −0.284920
$$130$$ 0 0
$$131$$ 3.47214 0.303362 0.151681 0.988430i $$-0.451531\pi$$
0.151681 + 0.988430i $$0.451531\pi$$
$$132$$ − 8.85410i − 0.770651i
$$133$$ − 23.1246i − 2.00516i
$$134$$ 7.09017 0.612497
$$135$$ 0 0
$$136$$ −7.76393 −0.665752
$$137$$ 10.9443i 0.935032i 0.883985 + 0.467516i $$0.154851\pi$$
−0.883985 + 0.467516i $$0.845149\pi$$
$$138$$ 0 0
$$139$$ 0.708204 0.0600691 0.0300345 0.999549i $$-0.490438\pi$$
0.0300345 + 0.999549i $$0.490438\pi$$
$$140$$ 0 0
$$141$$ 6.70820 0.564933
$$142$$ − 4.47214i − 0.375293i
$$143$$ 34.1246i 2.85364i
$$144$$ −1.85410 −0.154508
$$145$$ 0 0
$$146$$ −4.94427 −0.409191
$$147$$ − 2.00000i − 0.164957i
$$148$$ 12.9443i 1.06401i
$$149$$ −20.1803 −1.65324 −0.826619 0.562762i $$-0.809739\pi$$
−0.826619 + 0.562762i $$0.809739\pi$$
$$150$$ 0 0
$$151$$ 2.47214 0.201180 0.100590 0.994928i $$-0.467927\pi$$
0.100590 + 0.994928i $$0.467927\pi$$
$$152$$ 17.2361i 1.39803i
$$153$$ 3.47214i 0.280706i
$$154$$ −10.1459 −0.817580
$$155$$ 0 0
$$156$$ 10.0902 0.807860
$$157$$ 11.7082i 0.934416i 0.884147 + 0.467208i $$0.154740\pi$$
−0.884147 + 0.467208i $$0.845260\pi$$
$$158$$ − 3.81966i − 0.303876i
$$159$$ 6.76393 0.536415
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 0.618034i − 0.0485573i
$$163$$ − 15.1246i − 1.18465i −0.805699 0.592326i $$-0.798210\pi$$
0.805699 0.592326i $$-0.201790\pi$$
$$164$$ −7.23607 −0.565042
$$165$$ 0 0
$$166$$ −2.29180 −0.177878
$$167$$ 17.8885i 1.38426i 0.721774 + 0.692129i $$0.243327\pi$$
−0.721774 + 0.692129i $$0.756673\pi$$
$$168$$ 6.70820i 0.517549i
$$169$$ −25.8885 −1.99143
$$170$$ 0 0
$$171$$ 7.70820 0.589461
$$172$$ 5.23607i 0.399246i
$$173$$ − 19.4164i − 1.47620i −0.674690 0.738101i $$-0.735723\pi$$
0.674690 0.738101i $$-0.264277\pi$$
$$174$$ −0.618034 −0.0468530
$$175$$ 0 0
$$176$$ −10.1459 −0.764776
$$177$$ − 5.23607i − 0.393567i
$$178$$ 6.90983i 0.517914i
$$179$$ −4.18034 −0.312453 −0.156227 0.987721i $$-0.549933\pi$$
−0.156227 + 0.987721i $$0.549933\pi$$
$$180$$ 0 0
$$181$$ 8.41641 0.625587 0.312793 0.949821i $$-0.398735\pi$$
0.312793 + 0.949821i $$0.398735\pi$$
$$182$$ − 11.5623i − 0.857055i
$$183$$ − 5.70820i − 0.421963i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.94427 −0.362532
$$187$$ 19.0000i 1.38942i
$$188$$ − 10.8541i − 0.791617i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0.236068i 0.0170367i
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ −1.70820 −0.122642
$$195$$ 0 0
$$196$$ −3.23607 −0.231148
$$197$$ − 14.9443i − 1.06474i −0.846513 0.532368i $$-0.821302\pi$$
0.846513 0.532368i $$-0.178698\pi$$
$$198$$ − 3.38197i − 0.240346i
$$199$$ −20.7082 −1.46797 −0.733983 0.679168i $$-0.762341\pi$$
−0.733983 + 0.679168i $$0.762341\pi$$
$$200$$ 0 0
$$201$$ −11.4721 −0.809182
$$202$$ − 2.61803i − 0.184204i
$$203$$ − 3.00000i − 0.210559i
$$204$$ 5.61803 0.393341
$$205$$ 0 0
$$206$$ 4.58359 0.319354
$$207$$ 0 0
$$208$$ − 11.5623i − 0.801702i
$$209$$ 42.1803 2.91768
$$210$$ 0 0
$$211$$ −27.8885 −1.91993 −0.959963 0.280126i $$-0.909624\pi$$
−0.959963 + 0.280126i $$0.909624\pi$$
$$212$$ − 10.9443i − 0.751656i
$$213$$ 7.23607i 0.495807i
$$214$$ −4.65248 −0.318037
$$215$$ 0 0
$$216$$ −2.23607 −0.152145
$$217$$ − 24.0000i − 1.62923i
$$218$$ − 10.1459i − 0.687167i
$$219$$ 8.00000 0.540590
$$220$$ 0 0
$$221$$ −21.6525 −1.45650
$$222$$ 4.94427i 0.331838i
$$223$$ − 13.0000i − 0.870544i −0.900299 0.435272i $$-0.856652\pi$$
0.900299 0.435272i $$-0.143348\pi$$
$$224$$ 16.8541 1.12611
$$225$$ 0 0
$$226$$ −4.90983 −0.326597
$$227$$ 5.81966i 0.386264i 0.981173 + 0.193132i $$0.0618646\pi$$
−0.981173 + 0.193132i $$0.938135\pi$$
$$228$$ − 12.4721i − 0.825987i
$$229$$ −9.41641 −0.622254 −0.311127 0.950368i $$-0.600706\pi$$
−0.311127 + 0.950368i $$0.600706\pi$$
$$230$$ 0 0
$$231$$ 16.4164 1.08012
$$232$$ 2.23607i 0.146805i
$$233$$ 1.41641i 0.0927920i 0.998923 + 0.0463960i $$0.0147736\pi$$
−0.998923 + 0.0463960i $$0.985226\pi$$
$$234$$ 3.85410 0.251951
$$235$$ 0 0
$$236$$ −8.47214 −0.551489
$$237$$ 6.18034i 0.401456i
$$238$$ − 6.43769i − 0.417294i
$$239$$ −11.8885 −0.769006 −0.384503 0.923124i $$-0.625627\pi$$
−0.384503 + 0.923124i $$0.625627\pi$$
$$240$$ 0 0
$$241$$ 7.00000 0.450910 0.225455 0.974254i $$-0.427613\pi$$
0.225455 + 0.974254i $$0.427613\pi$$
$$242$$ − 11.7082i − 0.752632i
$$243$$ 1.00000i 0.0641500i
$$244$$ −9.23607 −0.591279
$$245$$ 0 0
$$246$$ −2.76393 −0.176222
$$247$$ 48.0689i 3.05855i
$$248$$ 17.8885i 1.13592i
$$249$$ 3.70820 0.234998
$$250$$ 0 0
$$251$$ −24.8885 −1.57095 −0.785475 0.618893i $$-0.787582\pi$$
−0.785475 + 0.618893i $$0.787582\pi$$
$$252$$ − 4.85410i − 0.305780i
$$253$$ 0 0
$$254$$ 3.70820 0.232673
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ 9.70820i 0.605581i 0.953057 + 0.302791i $$0.0979183\pi$$
−0.953057 + 0.302791i $$0.902082\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ − 2.14590i − 0.132574i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ −12.2361 −0.753078
$$265$$ 0 0
$$266$$ −14.2918 −0.876286
$$267$$ − 11.1803i − 0.684226i
$$268$$ 18.5623i 1.13387i
$$269$$ 30.2361 1.84353 0.921763 0.387754i $$-0.126749\pi$$
0.921763 + 0.387754i $$0.126749\pi$$
$$270$$ 0 0
$$271$$ 14.3607 0.872349 0.436175 0.899862i $$-0.356333\pi$$
0.436175 + 0.899862i $$0.356333\pi$$
$$272$$ − 6.43769i − 0.390343i
$$273$$ 18.7082i 1.13227i
$$274$$ 6.76393 0.408624
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 8.70820i − 0.523225i −0.965173 0.261613i $$-0.915746\pi$$
0.965173 0.261613i $$-0.0842543\pi$$
$$278$$ − 0.437694i − 0.0262511i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −23.1246 −1.37950 −0.689749 0.724048i $$-0.742279\pi$$
−0.689749 + 0.724048i $$0.742279\pi$$
$$282$$ − 4.14590i − 0.246885i
$$283$$ 18.4721i 1.09805i 0.835804 + 0.549027i $$0.185002\pi$$
−0.835804 + 0.549027i $$0.814998\pi$$
$$284$$ 11.7082 0.694754
$$285$$ 0 0
$$286$$ 21.0902 1.24709
$$287$$ − 13.4164i − 0.791946i
$$288$$ 5.61803i 0.331046i
$$289$$ 4.94427 0.290840
$$290$$ 0 0
$$291$$ 2.76393 0.162025
$$292$$ − 12.9443i − 0.757506i
$$293$$ − 17.9443i − 1.04832i −0.851621 0.524158i $$-0.824380\pi$$
0.851621 0.524158i $$-0.175620\pi$$
$$294$$ −1.23607 −0.0720889
$$295$$ 0 0
$$296$$ 17.8885 1.03975
$$297$$ 5.47214i 0.317526i
$$298$$ 12.4721i 0.722491i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −9.70820 −0.559572
$$302$$ − 1.52786i − 0.0879187i
$$303$$ 4.23607i 0.243356i
$$304$$ −14.2918 −0.819691
$$305$$ 0 0
$$306$$ 2.14590 0.122673
$$307$$ − 24.9443i − 1.42364i −0.702360 0.711822i $$-0.747870\pi$$
0.702360 0.711822i $$-0.252130\pi$$
$$308$$ − 26.5623i − 1.51353i
$$309$$ −7.41641 −0.421905
$$310$$ 0 0
$$311$$ 23.4721 1.33098 0.665491 0.746406i $$-0.268222\pi$$
0.665491 + 0.746406i $$0.268222\pi$$
$$312$$ − 13.9443i − 0.789439i
$$313$$ 1.18034i 0.0667168i 0.999443 + 0.0333584i $$0.0106203\pi$$
−0.999443 + 0.0333584i $$0.989380\pi$$
$$314$$ 7.23607 0.408355
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 28.4164i 1.59602i 0.602641 + 0.798012i $$0.294115\pi$$
−0.602641 + 0.798012i $$0.705885\pi$$
$$318$$ − 4.18034i − 0.234422i
$$319$$ 5.47214 0.306381
$$320$$ 0 0
$$321$$ 7.52786 0.420164
$$322$$ 0 0
$$323$$ 26.7639i 1.48919i
$$324$$ 1.61803 0.0898908
$$325$$ 0 0
$$326$$ −9.34752 −0.517711
$$327$$ 16.4164i 0.907829i
$$328$$ 10.0000i 0.552158i
$$329$$ 20.1246 1.10951
$$330$$ 0 0
$$331$$ −15.8885 −0.873313 −0.436657 0.899628i $$-0.643838\pi$$
−0.436657 + 0.899628i $$0.643838\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ − 8.00000i − 0.438397i
$$334$$ 11.0557 0.604943
$$335$$ 0 0
$$336$$ −5.56231 −0.303449
$$337$$ 20.4721i 1.11519i 0.830114 + 0.557594i $$0.188275\pi$$
−0.830114 + 0.557594i $$0.811725\pi$$
$$338$$ 16.0000i 0.870285i
$$339$$ 7.94427 0.431474
$$340$$ 0 0
$$341$$ 43.7771 2.37066
$$342$$ − 4.76393i − 0.257604i
$$343$$ 15.0000i 0.809924i
$$344$$ 7.23607 0.390143
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 17.5279i − 0.940945i −0.882415 0.470473i $$-0.844083\pi$$
0.882415 0.470473i $$-0.155917\pi$$
$$348$$ − 1.61803i − 0.0867357i
$$349$$ 18.9443 1.01406 0.507032 0.861927i $$-0.330743\pi$$
0.507032 + 0.861927i $$0.330743\pi$$
$$350$$ 0 0
$$351$$ −6.23607 −0.332857
$$352$$ 30.7426i 1.63859i
$$353$$ 3.52786i 0.187769i 0.995583 + 0.0938846i $$0.0299285\pi$$
−0.995583 + 0.0938846i $$0.970072\pi$$
$$354$$ −3.23607 −0.171995
$$355$$ 0 0
$$356$$ −18.0902 −0.958777
$$357$$ 10.4164i 0.551295i
$$358$$ 2.58359i 0.136547i
$$359$$ 4.58359 0.241913 0.120956 0.992658i $$-0.461404\pi$$
0.120956 + 0.992658i $$0.461404\pi$$
$$360$$ 0 0
$$361$$ 40.4164 2.12718
$$362$$ − 5.20163i − 0.273391i
$$363$$ 18.9443i 0.994316i
$$364$$ 30.2705 1.58661
$$365$$ 0 0
$$366$$ −3.52786 −0.184404
$$367$$ − 15.4164i − 0.804730i −0.915479 0.402365i $$-0.868188\pi$$
0.915479 0.402365i $$-0.131812\pi$$
$$368$$ 0 0
$$369$$ 4.47214 0.232810
$$370$$ 0 0
$$371$$ 20.2918 1.05350
$$372$$ − 12.9443i − 0.671129i
$$373$$ 15.8885i 0.822678i 0.911483 + 0.411339i $$0.134939\pi$$
−0.911483 + 0.411339i $$0.865061\pi$$
$$374$$ 11.7426 0.607198
$$375$$ 0 0
$$376$$ −15.0000 −0.773566
$$377$$ 6.23607i 0.321174i
$$378$$ − 1.85410i − 0.0953647i
$$379$$ −6.94427 −0.356703 −0.178352 0.983967i $$-0.557076\pi$$
−0.178352 + 0.983967i $$0.557076\pi$$
$$380$$ 0 0
$$381$$ −6.00000 −0.307389
$$382$$ − 7.41641i − 0.379456i
$$383$$ − 20.0689i − 1.02547i −0.858546 0.512736i $$-0.828632\pi$$
0.858546 0.512736i $$-0.171368\pi$$
$$384$$ 11.3820 0.580834
$$385$$ 0 0
$$386$$ −3.70820 −0.188743
$$387$$ − 3.23607i − 0.164499i
$$388$$ − 4.47214i − 0.227038i
$$389$$ −22.2361 −1.12741 −0.563707 0.825975i $$-0.690625\pi$$
−0.563707 + 0.825975i $$0.690625\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 4.47214i 0.225877i
$$393$$ 3.47214i 0.175146i
$$394$$ −9.23607 −0.465306
$$395$$ 0 0
$$396$$ 8.85410 0.444935
$$397$$ 6.58359i 0.330421i 0.986258 + 0.165211i $$0.0528304\pi$$
−0.986258 + 0.165211i $$0.947170\pi$$
$$398$$ 12.7984i 0.641525i
$$399$$ 23.1246 1.15768
$$400$$ 0 0
$$401$$ −36.6525 −1.83034 −0.915169 0.403071i $$-0.867943\pi$$
−0.915169 + 0.403071i $$0.867943\pi$$
$$402$$ 7.09017i 0.353626i
$$403$$ 49.8885i 2.48513i
$$404$$ 6.85410 0.341004
$$405$$ 0 0
$$406$$ −1.85410 −0.0920175
$$407$$ − 43.7771i − 2.16995i
$$408$$ − 7.76393i − 0.384372i
$$409$$ −8.65248 −0.427837 −0.213919 0.976851i $$-0.568623\pi$$
−0.213919 + 0.976851i $$0.568623\pi$$
$$410$$ 0 0
$$411$$ −10.9443 −0.539841
$$412$$ 12.0000i 0.591198i
$$413$$ − 15.7082i − 0.772950i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −35.0344 −1.71770
$$417$$ 0.708204i 0.0346809i
$$418$$ − 26.0689i − 1.27507i
$$419$$ −34.3607 −1.67863 −0.839315 0.543646i $$-0.817043\pi$$
−0.839315 + 0.543646i $$0.817043\pi$$
$$420$$ 0 0
$$421$$ −1.81966 −0.0886848 −0.0443424 0.999016i $$-0.514119\pi$$
−0.0443424 + 0.999016i $$0.514119\pi$$
$$422$$ 17.2361i 0.839039i
$$423$$ 6.70820i 0.326164i
$$424$$ −15.1246 −0.734516
$$425$$ 0 0
$$426$$ 4.47214 0.216676
$$427$$ − 17.1246i − 0.828718i
$$428$$ − 12.1803i − 0.588759i
$$429$$ −34.1246 −1.64755
$$430$$ 0 0
$$431$$ 34.4721 1.66046 0.830232 0.557418i $$-0.188208\pi$$
0.830232 + 0.557418i $$0.188208\pi$$
$$432$$ − 1.85410i − 0.0892055i
$$433$$ − 4.65248i − 0.223584i −0.993732 0.111792i $$-0.964341\pi$$
0.993732 0.111792i $$-0.0356590\pi$$
$$434$$ −14.8328 −0.711998
$$435$$ 0 0
$$436$$ 26.5623 1.27210
$$437$$ 0 0
$$438$$ − 4.94427i − 0.236246i
$$439$$ 10.1246 0.483221 0.241611 0.970373i $$-0.422324\pi$$
0.241611 + 0.970373i $$0.422324\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 13.3820i 0.636515i
$$443$$ 23.7639i 1.12906i 0.825413 + 0.564529i $$0.190942\pi$$
−0.825413 + 0.564529i $$0.809058\pi$$
$$444$$ −12.9443 −0.614308
$$445$$ 0 0
$$446$$ −8.03444 −0.380442
$$447$$ − 20.1803i − 0.954497i
$$448$$ 0.708204i 0.0334595i
$$449$$ −1.76393 −0.0832451 −0.0416225 0.999133i $$-0.513253\pi$$
−0.0416225 + 0.999133i $$0.513253\pi$$
$$450$$ 0 0
$$451$$ 24.4721 1.15235
$$452$$ − 12.8541i − 0.604606i
$$453$$ 2.47214i 0.116151i
$$454$$ 3.59675 0.168804
$$455$$ 0 0
$$456$$ −17.2361 −0.807153
$$457$$ − 6.12461i − 0.286497i −0.989687 0.143249i $$-0.954245\pi$$
0.989687 0.143249i $$-0.0457549\pi$$
$$458$$ 5.81966i 0.271935i
$$459$$ −3.47214 −0.162065
$$460$$ 0 0
$$461$$ 2.36068 0.109948 0.0549739 0.998488i $$-0.482492\pi$$
0.0549739 + 0.998488i $$0.482492\pi$$
$$462$$ − 10.1459i − 0.472030i
$$463$$ − 2.52786i − 0.117480i −0.998273 0.0587399i $$-0.981292\pi$$
0.998273 0.0587399i $$-0.0187083\pi$$
$$464$$ −1.85410 −0.0860745
$$465$$ 0 0
$$466$$ 0.875388 0.0405516
$$467$$ 12.9443i 0.598989i 0.954098 + 0.299495i $$0.0968181\pi$$
−0.954098 + 0.299495i $$0.903182\pi$$
$$468$$ 10.0902i 0.466418i
$$469$$ −34.4164 −1.58920
$$470$$ 0 0
$$471$$ −11.7082 −0.539486
$$472$$ 11.7082i 0.538914i
$$473$$ − 17.7082i − 0.814224i
$$474$$ 3.81966 0.175443
$$475$$ 0 0
$$476$$ 16.8541 0.772506
$$477$$ 6.76393i 0.309699i
$$478$$ 7.34752i 0.336068i
$$479$$ 6.47214 0.295719 0.147860 0.989008i $$-0.452762\pi$$
0.147860 + 0.989008i $$0.452762\pi$$
$$480$$ 0 0
$$481$$ 49.8885 2.27472
$$482$$ − 4.32624i − 0.197055i
$$483$$ 0 0
$$484$$ 30.6525 1.39329
$$485$$ 0 0
$$486$$ 0.618034 0.0280346
$$487$$ 12.0000i 0.543772i 0.962329 + 0.271886i $$0.0876473\pi$$
−0.962329 + 0.271886i $$0.912353\pi$$
$$488$$ 12.7639i 0.577796i
$$489$$ 15.1246 0.683959
$$490$$ 0 0
$$491$$ 25.8885 1.16833 0.584167 0.811634i $$-0.301421\pi$$
0.584167 + 0.811634i $$0.301421\pi$$
$$492$$ − 7.23607i − 0.326227i
$$493$$ 3.47214i 0.156377i
$$494$$ 29.7082 1.33664
$$495$$ 0 0
$$496$$ −14.8328 −0.666013
$$497$$ 21.7082i 0.973746i
$$498$$ − 2.29180i − 0.102698i
$$499$$ 23.7639 1.06382 0.531910 0.846801i $$-0.321475\pi$$
0.531910 + 0.846801i $$0.321475\pi$$
$$500$$ 0 0
$$501$$ −17.8885 −0.799201
$$502$$ 15.3820i 0.686531i
$$503$$ − 30.5967i − 1.36424i −0.731240 0.682121i $$-0.761058\pi$$
0.731240 0.682121i $$-0.238942\pi$$
$$504$$ −6.70820 −0.298807
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 25.8885i − 1.14975i
$$508$$ 9.70820i 0.430732i
$$509$$ −6.18034 −0.273939 −0.136969 0.990575i $$-0.543736\pi$$
−0.136969 + 0.990575i $$0.543736\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ − 18.7082i − 0.826794i
$$513$$ 7.70820i 0.340326i
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ −5.23607 −0.230505
$$517$$ 36.7082i 1.61442i
$$518$$ 14.8328i 0.651717i
$$519$$ 19.4164 0.852286
$$520$$ 0 0
$$521$$ −20.7639 −0.909684 −0.454842 0.890572i $$-0.650304\pi$$
−0.454842 + 0.890572i $$0.650304\pi$$
$$522$$ − 0.618034i − 0.0270506i
$$523$$ − 29.8328i − 1.30450i −0.758005 0.652249i $$-0.773826\pi$$
0.758005 0.652249i $$-0.226174\pi$$
$$524$$ 5.61803 0.245425
$$525$$ 0 0
$$526$$ 14.8328 0.646741
$$527$$ 27.7771i 1.20999i
$$528$$ − 10.1459i − 0.441544i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 5.23607 0.227226
$$532$$ − 37.4164i − 1.62221i
$$533$$ 27.8885i 1.20799i
$$534$$ −6.90983 −0.299018
$$535$$ 0 0
$$536$$ 25.6525 1.10802
$$537$$ − 4.18034i − 0.180395i
$$538$$ − 18.6869i − 0.805650i
$$539$$ 10.9443 0.471403
$$540$$ 0 0
$$541$$ 38.3607 1.64925 0.824627 0.565677i $$-0.191385\pi$$
0.824627 + 0.565677i $$0.191385\pi$$
$$542$$ − 8.87539i − 0.381231i
$$543$$ 8.41641i 0.361183i
$$544$$ −19.5066 −0.836338
$$545$$ 0 0
$$546$$ 11.5623 0.494821
$$547$$ − 27.4721i − 1.17462i −0.809361 0.587312i $$-0.800186\pi$$
0.809361 0.587312i $$-0.199814\pi$$
$$548$$ 17.7082i 0.756457i
$$549$$ 5.70820 0.243620
$$550$$ 0 0
$$551$$ 7.70820 0.328381
$$552$$ 0 0
$$553$$ 18.5410i 0.788444i
$$554$$ −5.38197 −0.228658
$$555$$ 0 0
$$556$$ 1.14590 0.0485969
$$557$$ − 2.76393i − 0.117112i −0.998284 0.0585558i $$-0.981350\pi$$
0.998284 0.0585558i $$-0.0186496\pi$$
$$558$$ − 4.94427i − 0.209308i
$$559$$ 20.1803 0.853537
$$560$$ 0 0
$$561$$ −19.0000 −0.802181
$$562$$ 14.2918i 0.602863i
$$563$$ 44.1246i 1.85963i 0.368026 + 0.929815i $$0.380034\pi$$
−0.368026 + 0.929815i $$0.619966\pi$$
$$564$$ 10.8541 0.457040
$$565$$ 0 0
$$566$$ 11.4164 0.479867
$$567$$ 3.00000i 0.125988i
$$568$$ − 16.1803i − 0.678912i
$$569$$ 5.18034 0.217171 0.108586 0.994087i $$-0.465368\pi$$
0.108586 + 0.994087i $$0.465368\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 55.2148i 2.30865i
$$573$$ 12.0000i 0.501307i
$$574$$ −8.29180 −0.346093
$$575$$ 0 0
$$576$$ −0.236068 −0.00983617
$$577$$ 29.3050i 1.21998i 0.792409 + 0.609991i $$0.208827\pi$$
−0.792409 + 0.609991i $$0.791173\pi$$
$$578$$ − 3.05573i − 0.127102i
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ 11.1246 0.461527
$$582$$ − 1.70820i − 0.0708073i
$$583$$ 37.0132i 1.53293i
$$584$$ −17.8885 −0.740233
$$585$$ 0 0
$$586$$ −11.0902 −0.458131
$$587$$ 1.81966i 0.0751054i 0.999295 + 0.0375527i $$0.0119562\pi$$
−0.999295 + 0.0375527i $$0.988044\pi$$
$$588$$ − 3.23607i − 0.133453i
$$589$$ 61.6656 2.54089
$$590$$ 0 0
$$591$$ 14.9443 0.614725
$$592$$ 14.8328i 0.609625i
$$593$$ − 24.6525i − 1.01236i −0.862429 0.506178i $$-0.831058\pi$$
0.862429 0.506178i $$-0.168942\pi$$
$$594$$ 3.38197 0.138764
$$595$$ 0 0
$$596$$ −32.6525 −1.33750
$$597$$ − 20.7082i − 0.847530i
$$598$$ 0 0
$$599$$ 8.88854 0.363176 0.181588 0.983375i $$-0.441876\pi$$
0.181588 + 0.983375i $$0.441876\pi$$
$$600$$ 0 0
$$601$$ 8.11146 0.330873 0.165437 0.986220i $$-0.447097\pi$$
0.165437 + 0.986220i $$0.447097\pi$$
$$602$$ 6.00000i 0.244542i
$$603$$ − 11.4721i − 0.467181i
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 2.61803 0.106350
$$607$$ − 40.6525i − 1.65003i −0.565109 0.825017i $$-0.691166\pi$$
0.565109 0.825017i $$-0.308834\pi$$
$$608$$ 43.3050i 1.75625i
$$609$$ 3.00000 0.121566
$$610$$ 0 0
$$611$$ −41.8328 −1.69237
$$612$$ 5.61803i 0.227096i
$$613$$ 7.29180i 0.294513i 0.989098 + 0.147256i $$0.0470443\pi$$
−0.989098 + 0.147256i $$0.952956\pi$$
$$614$$ −15.4164 −0.622156
$$615$$ 0 0
$$616$$ −36.7082 −1.47902
$$617$$ − 43.3050i − 1.74339i −0.490047 0.871696i $$-0.663020\pi$$
0.490047 0.871696i $$-0.336980\pi$$
$$618$$ 4.58359i 0.184379i
$$619$$ 3.52786 0.141797 0.0708984 0.997484i $$-0.477413\pi$$
0.0708984 + 0.997484i $$0.477413\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 14.5066i − 0.581661i
$$623$$ − 33.5410i − 1.34379i
$$624$$ 11.5623 0.462863
$$625$$ 0 0
$$626$$ 0.729490 0.0291563
$$627$$ 42.1803i 1.68452i
$$628$$ 18.9443i 0.755959i
$$629$$ 27.7771 1.10755
$$630$$ 0 0
$$631$$ −38.4853 −1.53208 −0.766038 0.642796i $$-0.777774\pi$$
−0.766038 + 0.642796i $$0.777774\pi$$
$$632$$ − 13.8197i − 0.549717i
$$633$$ − 27.8885i − 1.10847i
$$634$$ 17.5623 0.697488
$$635$$ 0 0
$$636$$ 10.9443 0.433969
$$637$$ 12.4721i 0.494164i
$$638$$ − 3.38197i − 0.133893i
$$639$$ −7.23607 −0.286254
$$640$$ 0 0
$$641$$ 46.2361 1.82621 0.913107 0.407719i $$-0.133676\pi$$
0.913107 + 0.407719i $$0.133676\pi$$
$$642$$ − 4.65248i − 0.183619i
$$643$$ − 35.8328i − 1.41311i −0.707659 0.706554i $$-0.750249\pi$$
0.707659 0.706554i $$-0.249751\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 16.5410 0.650798
$$647$$ − 11.2361i − 0.441735i −0.975304 0.220868i $$-0.929111\pi$$
0.975304 0.220868i $$-0.0708889\pi$$
$$648$$ − 2.23607i − 0.0878410i
$$649$$ 28.6525 1.12471
$$650$$ 0 0
$$651$$ 24.0000 0.940634
$$652$$ − 24.4721i − 0.958403i
$$653$$ 8.88854i 0.347836i 0.984760 + 0.173918i $$0.0556427\pi$$
−0.984760 + 0.173918i $$0.944357\pi$$
$$654$$ 10.1459 0.396736
$$655$$ 0 0
$$656$$ −8.29180 −0.323740
$$657$$ 8.00000i 0.312110i
$$658$$ − 12.4377i − 0.484872i
$$659$$ 21.0000 0.818044 0.409022 0.912525i $$-0.365870\pi$$
0.409022 + 0.912525i $$0.365870\pi$$
$$660$$ 0 0
$$661$$ −37.8328 −1.47153 −0.735763 0.677239i $$-0.763176\pi$$
−0.735763 + 0.677239i $$0.763176\pi$$
$$662$$ 9.81966i 0.381652i
$$663$$ − 21.6525i − 0.840912i
$$664$$ −8.29180 −0.321784
$$665$$ 0 0
$$666$$ −4.94427 −0.191587
$$667$$ 0 0
$$668$$ 28.9443i 1.11989i
$$669$$ 13.0000 0.502609
$$670$$ 0 0
$$671$$ 31.2361 1.20586
$$672$$ 16.8541i 0.650161i
$$673$$ − 33.2918i − 1.28330i −0.766996 0.641652i $$-0.778249\pi$$
0.766996 0.641652i $$-0.221751\pi$$
$$674$$ 12.6525 0.487355
$$675$$ 0 0
$$676$$ −41.8885 −1.61110
$$677$$ 20.8885i 0.802812i 0.915900 + 0.401406i $$0.131478\pi$$
−0.915900 + 0.401406i $$0.868522\pi$$
$$678$$ − 4.90983i − 0.188561i
$$679$$ 8.29180 0.318210
$$680$$ 0 0
$$681$$ −5.81966 −0.223010
$$682$$ − 27.0557i − 1.03602i
$$683$$ 7.41641i 0.283781i 0.989882 + 0.141890i $$0.0453181\pi$$
−0.989882 + 0.141890i $$0.954682\pi$$
$$684$$ 12.4721 0.476884
$$685$$ 0 0
$$686$$ 9.27051 0.353950
$$687$$ − 9.41641i − 0.359258i
$$688$$ 6.00000i 0.228748i
$$689$$ −42.1803 −1.60694
$$690$$ 0 0
$$691$$ −34.7082 −1.32036 −0.660181 0.751106i $$-0.729520\pi$$
−0.660181 + 0.751106i $$0.729520\pi$$
$$692$$ − 31.4164i − 1.19427i
$$693$$ 16.4164i 0.623608i
$$694$$ −10.8328 −0.411208
$$695$$ 0 0
$$696$$ −2.23607 −0.0847579
$$697$$ 15.5279i 0.588160i
$$698$$ − 11.7082i − 0.443162i
$$699$$ −1.41641 −0.0535735
$$700$$ 0 0
$$701$$ −5.12461 −0.193554 −0.0967770 0.995306i $$-0.530853\pi$$
−0.0967770 + 0.995306i $$0.530853\pi$$
$$702$$ 3.85410i 0.145464i
$$703$$ − 61.6656i − 2.32576i
$$704$$ −1.29180 −0.0486864
$$705$$ 0 0
$$706$$ 2.18034 0.0820582
$$707$$ 12.7082i 0.477941i
$$708$$ − 8.47214i − 0.318402i
$$709$$ −19.8885 −0.746930 −0.373465 0.927644i $$-0.621830\pi$$
−0.373465 + 0.927644i $$0.621830\pi$$
$$710$$ 0 0
$$711$$ −6.18034 −0.231781
$$712$$ 25.0000i 0.936915i
$$713$$ 0 0
$$714$$ 6.43769 0.240925
$$715$$ 0 0
$$716$$ −6.76393 −0.252780
$$717$$ − 11.8885i − 0.443986i
$$718$$ − 2.83282i − 0.105720i
$$719$$ 11.2361 0.419035 0.209517 0.977805i $$-0.432811\pi$$
0.209517 + 0.977805i $$0.432811\pi$$
$$720$$ 0 0
$$721$$ −22.2492 −0.828604
$$722$$ − 24.9787i − 0.929611i
$$723$$ 7.00000i 0.260333i
$$724$$ 13.6180 0.506110
$$725$$ 0 0
$$726$$ 11.7082 0.434532
$$727$$ − 2.11146i − 0.0783096i −0.999233 0.0391548i $$-0.987533\pi$$
0.999233 0.0391548i $$-0.0124665\pi$$
$$728$$ − 41.8328i − 1.55043i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 11.2361 0.415581
$$732$$ − 9.23607i − 0.341375i
$$733$$ − 14.2918i − 0.527880i −0.964539 0.263940i $$-0.914978\pi$$
0.964539 0.263940i $$-0.0850220\pi$$
$$734$$ −9.52786 −0.351680
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 62.7771i − 2.31242i
$$738$$ − 2.76393i − 0.101742i
$$739$$ −41.4853 −1.52606 −0.763031 0.646362i $$-0.776289\pi$$
−0.763031 + 0.646362i $$0.776289\pi$$
$$740$$ 0 0
$$741$$ −48.0689 −1.76585
$$742$$ − 12.5410i − 0.460395i
$$743$$ − 10.8197i − 0.396935i −0.980107 0.198467i $$-0.936404\pi$$
0.980107 0.198467i $$-0.0635964\pi$$
$$744$$ −17.8885 −0.655826
$$745$$ 0 0
$$746$$ 9.81966 0.359523
$$747$$ 3.70820i 0.135676i
$$748$$ 30.7426i 1.12406i
$$749$$ 22.5836 0.825186
$$750$$ 0 0
$$751$$ 43.4164 1.58429 0.792144 0.610335i $$-0.208965\pi$$
0.792144 + 0.610335i $$0.208965\pi$$
$$752$$ − 12.4377i − 0.453556i
$$753$$ − 24.8885i − 0.906989i
$$754$$ 3.85410 0.140358
$$755$$ 0 0
$$756$$ 4.85410 0.176542
$$757$$ 34.8328i 1.26602i 0.774144 + 0.633010i $$0.218181\pi$$
−0.774144 + 0.633010i $$0.781819\pi$$
$$758$$ 4.29180i 0.155885i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 33.0132 1.19673 0.598363 0.801225i $$-0.295818\pi$$
0.598363 + 0.801225i $$0.295818\pi$$
$$762$$ 3.70820i 0.134334i
$$763$$ 49.2492i 1.78294i
$$764$$ 19.4164 0.702461
$$765$$ 0 0
$$766$$ −12.4033 −0.448148
$$767$$ 32.6525i 1.17901i
$$768$$ − 6.56231i − 0.236797i
$$769$$ −34.6525 −1.24960 −0.624800 0.780785i $$-0.714820\pi$$
−0.624800 + 0.780785i $$0.714820\pi$$
$$770$$ 0 0
$$771$$ −9.70820 −0.349632
$$772$$ − 9.70820i − 0.349406i
$$773$$ − 52.2492i − 1.87927i −0.342173 0.939637i $$-0.611163\pi$$
0.342173 0.939637i $$-0.388837\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ −6.18034 −0.221861
$$777$$ − 24.0000i − 0.860995i
$$778$$ 13.7426i 0.492698i
$$779$$ 34.4721 1.23509
$$780$$ 0 0
$$781$$ −39.5967 −1.41688
$$782$$ 0 0
$$783$$ 1.00000i 0.0357371i
$$784$$ −3.70820 −0.132436
$$785$$ 0 0
$$786$$ 2.14590 0.0765416
$$787$$ − 1.88854i − 0.0673193i −0.999433 0.0336597i $$-0.989284\pi$$
0.999433 0.0336597i $$-0.0107162\pi$$
$$788$$ − 24.1803i − 0.861389i
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 23.8328 0.847397
$$792$$ − 12.2361i − 0.434790i
$$793$$ 35.5967i 1.26408i
$$794$$ 4.06888 0.144399
$$795$$ 0 0
$$796$$ −33.5066 −1.18761
$$797$$ − 6.94427i − 0.245979i −0.992408 0.122989i $$-0.960752\pi$$
0.992408 0.122989i $$-0.0392481\pi$$
$$798$$ − 14.2918i − 0.505924i
$$799$$ −23.2918 −0.824005
$$800$$ 0 0
$$801$$ 11.1803 0.395038
$$802$$ 22.6525i 0.799887i
$$803$$ 43.7771i 1.54486i
$$804$$ −18.5623 −0.654642
$$805$$ 0 0
$$806$$ 30.8328 1.08604
$$807$$ 30.2361i 1.06436i
$$808$$ − 9.47214i − 0.333229i
$$809$$ −44.2361 −1.55526 −0.777629 0.628724i $$-0.783578\pi$$
−0.777629 + 0.628724i $$0.783578\pi$$
$$810$$ 0 0
$$811$$ −22.7082 −0.797393 −0.398696 0.917083i $$-0.630537\pi$$
−0.398696 + 0.917083i $$0.630537\pi$$
$$812$$ − 4.85410i − 0.170346i
$$813$$ 14.3607i 0.503651i
$$814$$ −27.0557 −0.948303
$$815$$ 0 0
$$816$$ 6.43769 0.225364
$$817$$ − 24.9443i − 0.872690i
$$818$$ 5.34752i 0.186972i
$$819$$ −18.7082 −0.653718
$$820$$ 0 0
$$821$$ −10.5836 −0.369370 −0.184685 0.982798i $$-0.559126\pi$$
−0.184685 + 0.982798i $$0.559126\pi$$
$$822$$ 6.76393i 0.235919i
$$823$$ − 28.7639i − 1.00265i −0.865260 0.501324i $$-0.832847\pi$$
0.865260 0.501324i $$-0.167153\pi$$
$$824$$ 16.5836 0.577717
$$825$$ 0 0
$$826$$ −9.70820 −0.337792
$$827$$ 7.41641i 0.257894i 0.991652 + 0.128947i $$0.0411597\pi$$
−0.991652 + 0.128947i $$0.958840\pi$$
$$828$$ 0 0
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ 8.70820 0.302084
$$832$$ − 1.47214i − 0.0510371i
$$833$$ 6.94427i 0.240605i
$$834$$ 0.437694 0.0151561
$$835$$ 0 0
$$836$$ 68.2492 2.36045
$$837$$ 8.00000i 0.276520i
$$838$$ 21.2361i 0.733588i
$$839$$ −14.8885 −0.514010 −0.257005 0.966410i $$-0.582736\pi$$
−0.257005 + 0.966410i $$0.582736\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 1.12461i 0.0387567i
$$843$$ − 23.1246i − 0.796454i
$$844$$ −45.1246 −1.55325
$$845$$ 0 0
$$846$$ 4.14590 0.142539
$$847$$ 56.8328i 1.95280i
$$848$$ − 12.5410i − 0.430660i
$$849$$ −18.4721 −0.633962
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 11.7082i 0.401116i
$$853$$ − 28.7639i − 0.984858i −0.870353 0.492429i $$-0.836109\pi$$
0.870353 0.492429i $$-0.163891\pi$$
$$854$$ −10.5836 −0.362163
$$855$$ 0 0
$$856$$ −16.8328 −0.575334
$$857$$ 23.8885i 0.816017i 0.912978 + 0.408009i $$0.133777\pi$$
−0.912978 + 0.408009i $$0.866223\pi$$
$$858$$ 21.0902i 0.720007i
$$859$$ 15.7082 0.535957 0.267979 0.963425i $$-0.413644\pi$$
0.267979 + 0.963425i $$0.413644\pi$$
$$860$$ 0 0
$$861$$ 13.4164 0.457230
$$862$$ − 21.3050i − 0.725650i
$$863$$ 15.5967i 0.530919i 0.964122 + 0.265460i $$0.0855237\pi$$
−0.964122 + 0.265460i $$0.914476\pi$$
$$864$$ −5.61803 −0.191129
$$865$$ 0 0
$$866$$ −2.87539 −0.0977097
$$867$$ 4.94427i 0.167916i
$$868$$ − 38.8328i − 1.31807i
$$869$$ −33.8197 −1.14725
$$870$$ 0 0
$$871$$ 71.5410 2.42407
$$872$$ − 36.7082i − 1.24310i
$$873$$ 2.76393i 0.0935449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 12.9443 0.437346
$$877$$ − 48.8328i − 1.64897i −0.565886 0.824484i $$-0.691466\pi$$
0.565886 0.824484i $$-0.308534\pi$$
$$878$$ − 6.25735i − 0.211175i
$$879$$ 17.9443 0.605245
$$880$$ 0 0
$$881$$ 12.7082 0.428150 0.214075 0.976817i $$-0.431326\pi$$
0.214075 + 0.976817i $$0.431326\pi$$
$$882$$ − 1.23607i − 0.0416206i
$$883$$ − 32.0000i − 1.07689i −0.842662 0.538443i $$-0.819013\pi$$
0.842662 0.538443i $$-0.180987\pi$$
$$884$$ −35.0344 −1.17834
$$885$$ 0 0
$$886$$ 14.6869 0.493417
$$887$$ 44.5967i 1.49741i 0.662902 + 0.748706i $$0.269325\pi$$
−0.662902 + 0.748706i $$0.730675\pi$$
$$888$$ 17.8885i 0.600300i
$$889$$ −18.0000 −0.603701
$$890$$ 0 0
$$891$$ −5.47214 −0.183323
$$892$$ − 21.0344i − 0.704285i
$$893$$ 51.7082i 1.73035i
$$894$$ −12.4721 −0.417131
$$895$$ 0 0
$$896$$ 34.1459 1.14073
$$897$$ 0 0
$$898$$ 1.09017i 0.0363794i
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ −23.4853 −0.782409
$$902$$ − 15.1246i − 0.503594i
$$903$$ − 9.70820i − 0.323069i
$$904$$ −17.7639 −0.590820
$$905$$ 0 0
$$906$$ 1.52786 0.0507599
$$907$$ − 26.7639i − 0.888682i −0.895858 0.444341i $$-0.853438\pi$$
0.895858 0.444341i $$-0.146562\pi$$
$$908$$ 9.41641i 0.312494i
$$909$$ −4.23607 −0.140502
$$910$$ 0 0
$$911$$ −18.0557 −0.598213 −0.299106 0.954220i $$-0.596689\pi$$
−0.299106 + 0.954220i $$0.596689\pi$$
$$912$$ − 14.2918i − 0.473249i
$$913$$ 20.2918i 0.671560i
$$914$$ −3.78522 −0.125204
$$915$$ 0 0
$$916$$ −15.2361 −0.503414
$$917$$ 10.4164i 0.343980i
$$918$$ 2.14590i 0.0708252i
$$919$$ −20.1246 −0.663850 −0.331925 0.943306i $$-0.607698\pi$$
−0.331925 + 0.943306i $$0.607698\pi$$
$$920$$ 0 0
$$921$$ 24.9443 0.821942
$$922$$ − 1.45898i − 0.0480490i
$$923$$ − 45.1246i − 1.48529i
$$924$$ 26.5623 0.873836
$$925$$ 0 0
$$926$$ −1.56231 −0.0513406
$$927$$ − 7.41641i − 0.243587i
$$928$$ 5.61803i 0.184421i
$$929$$ 56.8328 1.86462 0.932312 0.361655i $$-0.117788\pi$$
0.932312 + 0.361655i $$0.117788\pi$$
$$930$$ 0 0
$$931$$ 15.4164 0.505252
$$932$$ 2.29180i 0.0750703i
$$933$$ 23.4721i 0.768443i
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 13.9443 0.455783
$$937$$ − 19.7639i − 0.645660i −0.946457 0.322830i $$-0.895366\pi$$
0.946457 0.322830i $$-0.104634\pi$$
$$938$$ 21.2705i 0.694507i
$$939$$ −1.18034 −0.0385189
$$940$$ 0 0
$$941$$ −46.0689 −1.50180 −0.750901 0.660414i $$-0.770381\pi$$
−0.750901 + 0.660414i $$0.770381\pi$$
$$942$$ 7.23607i 0.235764i
$$943$$ 0 0
$$944$$ −9.70820 −0.315975
$$945$$ 0 0
$$946$$ −10.9443 −0.355829
$$947$$ 12.7082i 0.412961i 0.978451 + 0.206481i $$0.0662010\pi$$
−0.978451 + 0.206481i $$0.933799\pi$$
$$948$$ 10.0000i 0.324785i
$$949$$ −49.8885 −1.61945
$$950$$ 0 0
$$951$$ −28.4164 −0.921465
$$952$$ − 23.2918i − 0.754891i
$$953$$ 19.0132i 0.615897i 0.951403 + 0.307948i $$0.0996424\pi$$
−0.951403 + 0.307948i $$0.900358\pi$$
$$954$$ 4.18034 0.135344
$$955$$ 0 0
$$956$$ −19.2361 −0.622139
$$957$$ 5.47214i 0.176889i
$$958$$ − 4.00000i − 0.129234i
$$959$$ −32.8328 −1.06023
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 30.8328i − 0.994090i
$$963$$ 7.52786i 0.242582i
$$964$$ 11.3262 0.364794
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 35.1246i 1.12953i 0.825251 + 0.564766i $$0.191033\pi$$
−0.825251 + 0.564766i $$0.808967\pi$$
$$968$$ − 42.3607i − 1.36152i
$$969$$ −26.7639 −0.859781
$$970$$ 0 0
$$971$$ −23.0557 −0.739894 −0.369947 0.929053i $$-0.620624\pi$$
−0.369947 + 0.929053i $$0.620624\pi$$
$$972$$ 1.61803i 0.0518985i
$$973$$ 2.12461i 0.0681119i
$$974$$ 7.41641 0.237637
$$975$$ 0 0
$$976$$ −10.5836 −0.338773
$$977$$ 22.4721i 0.718947i 0.933155 + 0.359474i $$0.117044\pi$$
−0.933155 + 0.359474i $$0.882956\pi$$
$$978$$ − 9.34752i − 0.298901i
$$979$$ 61.1803 1.95533
$$980$$ 0 0
$$981$$ −16.4164 −0.524136
$$982$$ − 16.0000i − 0.510581i
$$983$$ 37.5279i 1.19695i 0.801140 + 0.598476i $$0.204227\pi$$
−0.801140 + 0.598476i $$0.795773\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ 2.14590 0.0683393
$$987$$ 20.1246i 0.640573i
$$988$$ 77.7771i 2.47442i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 22.4853 0.714269 0.357134 0.934053i $$-0.383754\pi$$
0.357134 + 0.934053i $$0.383754\pi$$
$$992$$ 44.9443i 1.42698i
$$993$$ − 15.8885i − 0.504208i
$$994$$ 13.4164 0.425543
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ − 7.41641i − 0.234880i −0.993080 0.117440i $$-0.962531\pi$$
0.993080 0.117440i $$-0.0374688\pi$$
$$998$$ − 14.6869i − 0.464906i
$$999$$ 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.c.j.349.2 4
5.2 odd 4 435.2.a.e.1.2 2
5.3 odd 4 2175.2.a.q.1.1 2
5.4 even 2 inner 2175.2.c.j.349.3 4
15.2 even 4 1305.2.a.k.1.1 2
15.8 even 4 6525.2.a.s.1.2 2
20.7 even 4 6960.2.a.bu.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.e.1.2 2 5.2 odd 4
1305.2.a.k.1.1 2 15.2 even 4
2175.2.a.q.1.1 2 5.3 odd 4
2175.2.c.j.349.2 4 1.1 even 1 trivial
2175.2.c.j.349.3 4 5.4 even 2 inner
6525.2.a.s.1.2 2 15.8 even 4
6960.2.a.bu.1.2 2 20.7 even 4