LMFDB - Embedded newform 1170.2.e.e.469.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1170,2,Mod(469,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.469"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$9.34249703649$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.e.469.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} -2.23607i q^{5} +3.23607i q^{7} -1.00000i q^{8} +2.23607 q^{10} -4.47214 q^{11} -1.00000i q^{13} -3.23607 q^{14} +1.00000 q^{16} -7.23607i q^{17} +2.76393 q^{19} +2.23607i q^{20} -4.47214i q^{22} +2.76393i q^{23} -5.00000 q^{25} +1.00000 q^{26} -3.23607i q^{28} -3.70820 q^{29} -4.00000 q^{31} +1.00000i q^{32} +7.23607 q^{34} +7.23607 q^{35} -10.9443i q^{37} +2.76393i q^{38} -2.23607 q^{40} -3.52786 q^{41} +2.47214i q^{43} +4.47214 q^{44} -2.76393 q^{46} -12.9443i q^{47} -3.47214 q^{49} -5.00000i q^{50} +1.00000i q^{52} -0.472136i q^{53} +10.0000i q^{55} +3.23607 q^{56} -3.70820i q^{58} -8.47214 q^{59} -10.9443 q^{61} -4.00000i q^{62} -1.00000 q^{64} -2.23607 q^{65} +7.23607i q^{68} +7.23607i q^{70} +2.47214 q^{71} -13.2361i q^{73} +10.9443 q^{74} -2.76393 q^{76} -14.4721i q^{77} +4.00000 q^{79} -2.23607i q^{80} -3.52786i q^{82} -4.94427i q^{83} -16.1803 q^{85} -2.47214 q^{86} +4.47214i q^{88} +0.472136 q^{89} +3.23607 q^{91} -2.76393i q^{92} +12.9443 q^{94} -6.18034i q^{95} -3.70820i q^{97} -3.47214i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 20 q^{25} + 4 q^{26} + 12 q^{29} - 16 q^{31} + 20 q^{34} + 20 q^{35} - 32 q^{41} - 20 q^{46} + 4 q^{49} + 4 q^{56} - 16 q^{59} - 8 q^{61} - 4 q^{64} - 8 q^{71}+ \cdots + 16 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 20 * q^25 + 4 * q^26 + 12 * q^29 - 16 * q^31 + 20 * q^34 + 20 * q^35 - 32 * q^41 - 20 * q^46 + 4 * q^49 + 4 * q^56 - 16 * q^59 - 8 * q^61 - 4 * q^64 - 8 * q^71 + 8 * q^74 - 20 * q^76 + 16 * q^79 - 20 * q^85 + 8 * q^86 - 16 * q^89 + 4 * q^91 + 16 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	3.23607i	1.22312i	0.791199	+	0.611559i	$0.209457\pi$
$7$	3.23607i	1.22312i	−0.791199	+	0.611559i	$0.790543\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	2.23607	0.707107
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−3.23607	−0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	− 7.23607i	− 1.75500i	−0.479573	−	0.877502i	$-0.659208\pi$
$17$	− 7.23607i	− 1.75500i	0.479573	−	0.877502i	$-0.340792\pi$
$18$	0	0
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	2.23607i	0.500000i
$21$	0	0
$22$	− 4.47214i	− 0.953463i
$23$	2.76393i	0.576320i	0.957582	+	0.288160i	$0.0930434\pi$
$23$	2.76393i	0.576320i	−0.957582	+	0.288160i	$0.906957\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	1.00000	0.196116
$27$	0	0
$28$	− 3.23607i	− 0.611559i
$29$	−3.70820	−0.688596	−0.344298	−	0.938860i	$-0.611883\pi$
$29$	−3.70820	−0.688596	−0.344298	+	0.938860i	$0.611883\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	7.23607	1.24098
$35$	7.23607	1.22312
$36$	0	0
$37$	− 10.9443i	− 1.79923i	−0.436687	−	0.899614i	$-0.643848\pi$
$37$	− 10.9443i	− 1.79923i	0.436687	−	0.899614i	$-0.356152\pi$
$38$	2.76393i	0.448369i
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−3.52786	−0.550960	−0.275480	−	0.961307i	$-0.588837\pi$
$41$	−3.52786	−0.550960	−0.275480	+	0.961307i	$0.588837\pi$
$42$	0	0
$43$	2.47214i	0.376997i	0.982073	+	0.188499i	$0.0603621\pi$
$43$	2.47214i	0.376997i	−0.982073	+	0.188499i	$0.939638\pi$
$44$	4.47214	0.674200
$45$	0	0
$46$	−2.76393	−0.407520
$47$	− 12.9443i	− 1.88812i	−0.329779	−	0.944058i	$-0.606974\pi$
$47$	− 12.9443i	− 1.88812i	0.329779	−	0.944058i	$-0.393026\pi$
$48$	0	0
$49$	−3.47214	−0.496019
$50$	− 5.00000i	− 0.707107i
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 0.472136i	− 0.0648529i	−0.999474	−	0.0324264i	$-0.989677\pi$
$53$	− 0.472136i	− 0.0648529i	0.999474	−	0.0324264i	$-0.0103235\pi$
$54$	0	0
$55$	10.0000i	1.34840i
$56$	3.23607	0.432438
$57$	0	0
$58$	− 3.70820i	− 0.486911i
$59$	−8.47214	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$59$	−8.47214	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$60$	0	0
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	−2.23607	−0.277350
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	7.23607i	0.877502i
$69$	0	0
$70$	7.23607i	0.864876i
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	− 13.2361i	− 1.54916i	−0.632473	−	0.774582i	$-0.717960\pi$
$73$	− 13.2361i	− 1.54916i	0.632473	−	0.774582i	$-0.282040\pi$
$74$	10.9443	1.27225
$75$	0	0
$76$	−2.76393	−0.317045
$77$	− 14.4721i	− 1.64925i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	− 2.23607i	− 0.250000i
$81$	0	0
$82$	− 3.52786i	− 0.389587i
$83$	− 4.94427i	− 0.542704i	−0.962480	−	0.271352i	$-0.912529\pi$
$83$	− 4.94427i	− 0.542704i	0.962480	−	0.271352i	$-0.0874708\pi$
$84$	0	0
$85$	−16.1803	−1.75500
$86$	−2.47214	−0.266577
$87$	0	0
$88$	4.47214i	0.476731i
$89$	0.472136	0.0500463	0.0250232	−	0.999687i	$-0.492034\pi$
$89$	0.472136	0.0500463	0.0250232	+	0.999687i	$0.492034\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	− 2.76393i	− 0.288160i
$93$	0	0
$94$	12.9443	1.33510
$95$	− 6.18034i	− 0.634089i
$96$	0	0
$97$	− 3.70820i	− 0.376511i	−0.982120	−	0.188256i	$-0.939717\pi$
$97$	− 3.70820i	− 0.376511i	0.982120	−	0.188256i	$-0.0602833\pi$
$98$	− 3.47214i	− 0.350739i
$99$	0	0
$100$	5.00000	0.500000
$101$	9.23607	0.919023	0.459512	−	0.888172i	$-0.348024\pi$
$101$	9.23607	0.919023	0.459512	+	0.888172i	$0.348024\pi$
$102$	0	0
$103$	− 8.47214i	− 0.834784i	−0.908726	−	0.417392i	$-0.862944\pi$
$103$	− 8.47214i	− 0.834784i	0.908726	−	0.417392i	$-0.137056\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0.472136	0.0458579
$107$	1.52786i	0.147704i	0.997269	+	0.0738521i	$0.0235293\pi$
$107$	1.52786i	0.147704i	−0.997269	+	0.0738521i	$0.976471\pi$
$108$	0	0
$109$	2.29180	0.219514	0.109757	−	0.993958i	$-0.464993\pi$
$109$	2.29180	0.219514	0.109757	+	0.993958i	$0.464993\pi$
$110$	−10.0000	−0.953463
$111$	0	0
$112$	3.23607i	0.305780i
$113$	18.6525i	1.75468i	0.479872	+	0.877339i	$0.340683\pi$
$113$	18.6525i	1.75468i	−0.479872	+	0.877339i	$0.659317\pi$
$114$	0	0
$115$	6.18034	0.576320
$116$	3.70820	0.344298
$117$	0	0
$118$	− 8.47214i	− 0.779923i
$119$	23.4164	2.14658
$120$	0	0
$121$	9.00000	0.818182
$122$	− 10.9443i	− 0.990848i
$123$	0	0
$124$	4.00000	0.359211
$125$	11.1803i	1.00000i
$126$	0	0
$127$	21.4164i	1.90040i	0.311642	+	0.950199i	$0.399121\pi$
$127$	21.4164i	1.90040i	−0.311642	+	0.950199i	$0.600879\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	− 2.23607i	− 0.196116i
$131$	−9.70820	−0.848210	−0.424105	−	0.905613i	$-0.639411\pi$
$131$	−9.70820	−0.848210	−0.424105	+	0.905613i	$0.639411\pi$
$132$	0	0
$133$	8.94427i	0.775567i
$134$	0	0
$135$	0	0
$136$	−7.23607	−0.620488
$137$	19.8885i	1.69919i	0.527433	+	0.849596i	$0.323154\pi$
$137$	19.8885i	1.69919i	−0.527433	+	0.849596i	$0.676846\pi$
$138$	0	0
$139$	6.47214	0.548959	0.274480	−	0.961593i	$-0.411494\pi$
$139$	6.47214	0.548959	0.274480	+	0.961593i	$0.411494\pi$
$140$	−7.23607	−0.611559
$141$	0	0
$142$	2.47214i	0.207457i
$143$	4.47214i	0.373979i
$144$	0	0
$145$	8.29180i	0.688596i
$146$	13.2361	1.09542
$147$	0	0
$148$	10.9443i	0.899614i
$149$	13.5279	1.10825	0.554123	−	0.832435i	$-0.313054\pi$
$149$	13.5279	1.10825	0.554123	+	0.832435i	$0.313054\pi$
$150$	0	0
$151$	−19.4164	−1.58008	−0.790042	−	0.613052i	$-0.789942\pi$
$151$	−19.4164	−1.58008	−0.790042	+	0.613052i	$0.789942\pi$
$152$	− 2.76393i	− 0.224184i
$153$	0	0
$154$	14.4721	1.16620
$155$	8.94427i	0.718421i
$156$	0	0
$157$	− 18.9443i	− 1.51192i	−0.654619	−	0.755959i	$-0.727171\pi$
$157$	− 18.9443i	− 1.51192i	0.654619	−	0.755959i	$-0.272829\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	2.23607	0.176777
$161$	−8.94427	−0.704907
$162$	0	0
$163$	− 12.9443i	− 1.01387i	−0.861983	−	0.506937i	$-0.830778\pi$
$163$	− 12.9443i	− 1.01387i	0.861983	−	0.506937i	$-0.169222\pi$
$164$	3.52786	0.275480
$165$	0	0
$166$	4.94427	0.383750
$167$	19.4164i	1.50249i	0.660025	+	0.751243i	$0.270546\pi$
$167$	19.4164i	1.50249i	−0.660025	+	0.751243i	$0.729454\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	− 16.1803i	− 1.24098i
$171$	0	0
$172$	− 2.47214i	− 0.188499i
$173$	− 22.9443i	− 1.74442i	−0.489131	−	0.872210i	$-0.662686\pi$
$173$	− 22.9443i	− 1.74442i	0.489131	−	0.872210i	$-0.337314\pi$
$174$	0	0
$175$	− 16.1803i	− 1.22312i
$176$	−4.47214	−0.337100
$177$	0	0
$178$	0.472136i	0.0353881i
$179$	8.18034	0.611427	0.305714	−	0.952124i	$-0.401105\pi$
$179$	8.18034	0.611427	0.305714	+	0.952124i	$0.401105\pi$
$180$	0	0
$181$	17.4164	1.29455	0.647276	−	0.762256i	$-0.275908\pi$
$181$	17.4164	1.29455	0.647276	+	0.762256i	$0.275908\pi$
$182$	3.23607i	0.239873i
$183$	0	0
$184$	2.76393	0.203760
$185$	−24.4721	−1.79923
$186$	0	0
$187$	32.3607i	2.36645i
$188$	12.9443i	0.944058i
$189$	0	0
$190$	6.18034	0.448369
$191$	−22.4721	−1.62603	−0.813013	−	0.582245i	$-0.802174\pi$
$191$	−22.4721	−1.62603	−0.813013	+	0.582245i	$0.802174\pi$
$192$	0	0
$193$	8.29180i	0.596857i	0.954432	+	0.298428i	$0.0964624\pi$
$193$	8.29180i	0.596857i	−0.954432	+	0.298428i	$0.903538\pi$
$194$	3.70820	0.266234
$195$	0	0
$196$	3.47214	0.248010
$197$	14.9443i	1.06474i	0.846513	+	0.532368i	$0.178698\pi$
$197$	14.9443i	1.06474i	−0.846513	+	0.532368i	$0.821302\pi$
$198$	0	0
$199$	0.944272	0.0669377	0.0334688	−	0.999440i	$-0.489345\pi$
$199$	0.944272	0.0669377	0.0334688	+	0.999440i	$0.489345\pi$
$200$	5.00000i	0.353553i
$201$	0	0
$202$	9.23607i	0.649847i
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	7.88854i	0.550960i
$206$	8.47214	0.590282
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	−12.3607	−0.855006
$210$	0	0
$211$	13.8885	0.956127	0.478063	−	0.878325i	$-0.341339\pi$
$211$	13.8885	0.956127	0.478063	+	0.878325i	$0.341339\pi$
$212$	0.472136i	0.0324264i
$213$	0	0
$214$	−1.52786	−0.104443
$215$	5.52786	0.376997
$216$	0	0
$217$	− 12.9443i	− 0.878714i
$218$	2.29180i	0.155220i
$219$	0	0
$220$	− 10.0000i	− 0.674200i
$221$	−7.23607	−0.486751
$222$	0	0
$223$	− 16.7639i	− 1.12260i	−0.827614	−	0.561298i	$-0.810302\pi$
$223$	− 16.7639i	− 1.12260i	0.827614	−	0.561298i	$-0.189698\pi$
$224$	−3.23607	−0.216219
$225$	0	0
$226$	−18.6525	−1.24074
$227$	− 7.05573i	− 0.468305i	−0.972200	−	0.234153i	$-0.924768\pi$
$227$	− 7.05573i	− 0.468305i	0.972200	−	0.234153i	$-0.0752315\pi$
$228$	0	0
$229$	−9.70820	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$229$	−9.70820	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$230$	6.18034i	0.407520i
$231$	0	0
$232$	3.70820i	0.243456i
$233$	20.1803i	1.32206i	0.750360	+	0.661029i	$0.229880\pi$
$233$	20.1803i	1.32206i	−0.750360	+	0.661029i	$0.770120\pi$
$234$	0	0
$235$	−28.9443	−1.88812
$236$	8.47214	0.551489
$237$	0	0
$238$	23.4164i	1.51786i
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	3.52786	0.227250	0.113625	−	0.993524i	$-0.463754\pi$
$241$	3.52786	0.227250	0.113625	+	0.993524i	$0.463754\pi$
$242$	9.00000i	0.578542i
$243$	0	0
$244$	10.9443	0.700635
$245$	7.76393i	0.496019i
$246$	0	0
$247$	− 2.76393i	− 0.175865i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	−11.1803	−0.707107
$251$	−2.29180	−0.144657	−0.0723284	−	0.997381i	$-0.523043\pi$
$251$	−2.29180	−0.144657	−0.0723284	+	0.997381i	$0.523043\pi$
$252$	0	0
$253$	− 12.3607i	− 0.777109i
$254$	−21.4164	−1.34378
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 0.763932i	− 0.0476528i	−0.999716	−	0.0238264i	$-0.992415\pi$
$257$	− 0.763932i	− 0.0476528i	0.999716	−	0.0238264i	$-0.00758489\pi$
$258$	0	0
$259$	35.4164	2.20067
$260$	2.23607	0.138675
$261$	0	0
$262$	− 9.70820i	− 0.599775i
$263$	10.1803i	0.627747i	0.949465	+	0.313873i	$0.101627\pi$
$263$	10.1803i	0.627747i	−0.949465	+	0.313873i	$0.898373\pi$
$264$	0	0
$265$	−1.05573	−0.0648529
$266$	−8.94427	−0.548408
$267$	0	0
$268$	0	0
$269$	6.76393	0.412404	0.206202	−	0.978509i	$-0.433890\pi$
$269$	6.76393	0.412404	0.206202	+	0.978509i	$0.433890\pi$
$270$	0	0
$271$	−11.4164	−0.693497	−0.346749	−	0.937958i	$-0.612714\pi$
$271$	−11.4164	−0.693497	−0.346749	+	0.937958i	$0.612714\pi$
$272$	− 7.23607i	− 0.438751i
$273$	0	0
$274$	−19.8885	−1.20151
$275$	22.3607	1.34840
$276$	0	0
$277$	20.8328i	1.25172i	0.779934	+	0.625861i	$0.215252\pi$
$277$	20.8328i	1.25172i	−0.779934	+	0.625861i	$0.784748\pi$
$278$	6.47214i	0.388173i
$279$	0	0
$280$	− 7.23607i	− 0.432438i
$281$	−24.4721	−1.45989	−0.729943	−	0.683508i	$-0.760453\pi$
$281$	−24.4721	−1.45989	−0.729943	+	0.683508i	$0.760453\pi$
$282$	0	0
$283$	11.4164i	0.678635i	0.940672	+	0.339318i	$0.110196\pi$
$283$	11.4164i	0.678635i	−0.940672	+	0.339318i	$0.889804\pi$
$284$	−2.47214	−0.146694
$285$	0	0
$286$	−4.47214	−0.264443
$287$	− 11.4164i	− 0.673889i
$288$	0	0
$289$	−35.3607	−2.08004
$290$	−8.29180	−0.486911
$291$	0	0
$292$	13.2361i	0.774582i
$293$	2.94427i	0.172006i	0.996295	+	0.0860031i	$0.0274095\pi$
$293$	2.94427i	0.172006i	−0.996295	+	0.0860031i	$0.972591\pi$
$294$	0	0
$295$	18.9443i	1.10298i
$296$	−10.9443	−0.636123
$297$	0	0
$298$	13.5279i	0.783648i
$299$	2.76393	0.159842
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 19.4164i	− 1.11729i
$303$	0	0
$304$	2.76393	0.158522
$305$	24.4721i	1.40127i
$306$	0	0
$307$	− 24.3607i	− 1.39034i	−0.718847	−	0.695169i	$-0.755330\pi$
$307$	− 24.3607i	− 1.39034i	0.718847	−	0.695169i	$-0.244670\pi$
$308$	14.4721i	0.824626i
$309$	0	0
$310$	−8.94427	−0.508001
$311$	22.4721	1.27428	0.637139	−	0.770749i	$-0.280118\pi$
$311$	22.4721	1.27428	0.637139	+	0.770749i	$0.280118\pi$
$312$	0	0
$313$	19.4164i	1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$313$	19.4164i	1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$314$	18.9443	1.06909
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 13.4164i	− 0.753541i	−0.926307	−	0.376770i	$-0.877035\pi$
$317$	− 13.4164i	− 0.753541i	0.926307	−	0.376770i	$-0.122965\pi$
$318$	0	0
$319$	16.5836	0.928503
$320$	2.23607i	0.125000i
$321$	0	0
$322$	− 8.94427i	− 0.498445i
$323$	− 20.0000i	− 1.11283i
$324$	0	0
$325$	5.00000i	0.277350i
$326$	12.9443	0.716917
$327$	0	0
$328$	3.52786i	0.194794i
$329$	41.8885	2.30939
$330$	0	0
$331$	−2.76393	−0.151919	−0.0759597	−	0.997111i	$-0.524202\pi$
$331$	−2.76393	−0.151919	−0.0759597	+	0.997111i	$0.524202\pi$
$332$	4.94427i	0.271352i
$333$	0	0
$334$	−19.4164	−1.06242
$335$	0	0
$336$	0	0
$337$	− 6.47214i	− 0.352560i	−0.984340	−	0.176280i	$-0.943594\pi$
$337$	− 6.47214i	− 0.352560i	0.984340	−	0.176280i	$-0.0564064\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	16.1803	0.877502
$341$	17.8885	0.968719
$342$	0	0
$343$	11.4164i	0.616428i
$344$	2.47214	0.133289
$345$	0	0
$346$	22.9443	1.23349
$347$	− 1.52786i	− 0.0820200i	−0.999159	−	0.0410100i	$-0.986942\pi$
$347$	− 1.52786i	− 0.0820200i	0.999159	−	0.0410100i	$-0.0130576\pi$
$348$	0	0
$349$	15.2361	0.815568	0.407784	−	0.913078i	$-0.366302\pi$
$349$	15.2361	0.815568	0.407784	+	0.913078i	$0.366302\pi$
$350$	16.1803	0.864876
$351$	0	0
$352$	− 4.47214i	− 0.238366i
$353$	− 26.9443i	− 1.43410i	−0.697022	−	0.717049i	$-0.745492\pi$
$353$	− 26.9443i	− 1.43410i	0.697022	−	0.717049i	$-0.254508\pi$
$354$	0	0
$355$	− 5.52786i	− 0.293389i
$356$	−0.472136	−0.0250232
$357$	0	0
$358$	8.18034i	0.432344i
$359$	6.47214	0.341586	0.170793	−	0.985307i	$-0.445367\pi$
$359$	6.47214	0.341586	0.170793	+	0.985307i	$0.445367\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	17.4164i	0.915386i
$363$	0	0
$364$	−3.23607	−0.169616
$365$	−29.5967	−1.54916
$366$	0	0
$367$	0.472136i	0.0246453i	0.999924	+	0.0123226i	$0.00392252\pi$
$367$	0.472136i	0.0246453i	−0.999924	+	0.0123226i	$0.996077\pi$
$368$	2.76393i	0.144080i
$369$	0	0
$370$	− 24.4721i	− 1.27225i
$371$	1.52786	0.0793227
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	−32.3607	−1.67333
$375$	0	0
$376$	−12.9443	−0.667550
$377$	3.70820i	0.190982i
$378$	0	0
$379$	35.1246	1.80423	0.902115	−	0.431496i	$-0.142014\pi$
$379$	35.1246	1.80423	0.902115	+	0.431496i	$0.142014\pi$
$380$	6.18034i	0.317045i
$381$	0	0
$382$	− 22.4721i	− 1.14977i
$383$	5.52786i	0.282461i	0.989977	+	0.141230i	$0.0451058\pi$
$383$	5.52786i	0.282461i	−0.989977	+	0.141230i	$0.954894\pi$
$384$	0	0
$385$	−32.3607	−1.64925
$386$	−8.29180	−0.422041
$387$	0	0
$388$	3.70820i	0.188256i
$389$	12.6525	0.641506	0.320753	−	0.947163i	$-0.396064\pi$
$389$	12.6525	0.641506	0.320753	+	0.947163i	$0.396064\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	3.47214i	0.175369i
$393$	0	0
$394$	−14.9443	−0.752882
$395$	− 8.94427i	− 0.450035i
$396$	0	0
$397$	17.4164i	0.874104i	0.899436	+	0.437052i	$0.143978\pi$
$397$	17.4164i	0.874104i	−0.899436	+	0.437052i	$0.856022\pi$
$398$	0.944272i	0.0473321i
$399$	0	0
$400$	−5.00000	−0.250000
$401$	24.4721	1.22208	0.611040	−	0.791600i	$-0.290751\pi$
$401$	24.4721	1.22208	0.611040	+	0.791600i	$0.290751\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−9.23607	−0.459512
$405$	0	0
$406$	12.0000	0.595550
$407$	48.9443i	2.42608i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	−7.88854	−0.389587
$411$	0	0
$412$	8.47214i	0.417392i
$413$	− 27.4164i	− 1.34907i
$414$	0	0
$415$	−11.0557	−0.542704
$416$	1.00000	0.0490290
$417$	0	0
$418$	− 12.3607i	− 0.604581i
$419$	27.5967	1.34819	0.674095	−	0.738645i	$-0.264534\pi$
$419$	27.5967	1.34819	0.674095	+	0.738645i	$0.264534\pi$
$420$	0	0
$421$	−13.7082	−0.668097	−0.334048	−	0.942556i	$-0.608415\pi$
$421$	−13.7082	−0.668097	−0.334048	+	0.942556i	$0.608415\pi$
$422$	13.8885i	0.676084i
$423$	0	0
$424$	−0.472136	−0.0229289
$425$	36.1803i	1.75500i
$426$	0	0
$427$	− 35.4164i	− 1.71392i
$428$	− 1.52786i	− 0.0738521i
$429$	0	0
$430$	5.52786i	0.266577i
$431$	14.8328	0.714472	0.357236	−	0.934014i	$-0.383719\pi$
$431$	14.8328	0.714472	0.357236	+	0.934014i	$0.383719\pi$
$432$	0	0
$433$	9.52786i	0.457880i	0.973441	+	0.228940i	$0.0735259\pi$
$433$	9.52786i	0.457880i	−0.973441	+	0.228940i	$0.926474\pi$
$434$	12.9443	0.621345
$435$	0	0
$436$	−2.29180	−0.109757
$437$	7.63932i	0.365438i
$438$	0	0
$439$	4.94427	0.235977	0.117989	−	0.993015i	$-0.462355\pi$
$439$	4.94427	0.235977	0.117989	+	0.993015i	$0.462355\pi$
$440$	10.0000	0.476731
$441$	0	0
$442$	− 7.23607i	− 0.344185i
$443$	− 22.4721i	− 1.06768i	−0.845584	−	0.533842i	$-0.820748\pi$
$443$	− 22.4721i	− 1.06768i	0.845584	−	0.533842i	$-0.179252\pi$
$444$	0	0
$445$	− 1.05573i	− 0.0500463i
$446$	16.7639	0.793795
$447$	0	0
$448$	− 3.23607i	− 0.152890i
$449$	0.472136	0.0222815	0.0111407	−	0.999938i	$-0.496454\pi$
$449$	0.472136	0.0222815	0.0111407	+	0.999938i	$0.496454\pi$
$450$	0	0
$451$	15.7771	0.742914
$452$	− 18.6525i	− 0.877339i
$453$	0	0
$454$	7.05573	0.331142
$455$	− 7.23607i	− 0.339232i
$456$	0	0
$457$	− 3.70820i	− 0.173462i	−0.996232	−	0.0867312i	$-0.972358\pi$
$457$	− 3.70820i	− 0.173462i	0.996232	−	0.0867312i	$-0.0276421\pi$
$458$	− 9.70820i	− 0.453635i
$459$	0	0
$460$	−6.18034	−0.288160
$461$	−41.8885	−1.95094	−0.975472	−	0.220124i	$-0.929354\pi$
$461$	−41.8885	−1.95094	−0.975472	+	0.220124i	$0.929354\pi$
$462$	0	0
$463$	22.2918i	1.03599i	0.855384	+	0.517994i	$0.173321\pi$
$463$	22.2918i	1.03599i	−0.855384	+	0.517994i	$0.826679\pi$
$464$	−3.70820	−0.172149
$465$	0	0
$466$	−20.1803	−0.934836
$467$	− 6.47214i	− 0.299495i	−0.988724	−	0.149747i	$-0.952154\pi$
$467$	− 6.47214i	− 0.299495i	0.988724	−	0.149747i	$-0.0478460\pi$
$468$	0	0
$469$	0	0
$470$	− 28.9443i	− 1.33510i
$471$	0	0
$472$	8.47214i	0.389962i
$473$	− 11.0557i	− 0.508343i
$474$	0	0
$475$	−13.8197	−0.634089
$476$	−23.4164	−1.07329
$477$	0	0
$478$	− 21.8885i	− 1.00116i
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−10.9443	−0.499016
$482$	3.52786i	0.160690i
$483$	0	0
$484$	−9.00000	−0.409091
$485$	−8.29180	−0.376511
$486$	0	0
$487$	− 23.5967i	− 1.06927i	−0.845083	−	0.534635i	$-0.820449\pi$
$487$	− 23.5967i	− 1.06927i	0.845083	−	0.534635i	$-0.179551\pi$
$488$	10.9443i	0.495424i
$489$	0	0
$490$	−7.76393	−0.350739
$491$	16.1803	0.730209	0.365104	−	0.930967i	$-0.381033\pi$
$491$	16.1803	0.730209	0.365104	+	0.930967i	$0.381033\pi$
$492$	0	0
$493$	26.8328i	1.20849i
$494$	2.76393	0.124355
$495$	0	0
$496$	−4.00000	−0.179605
$497$	8.00000i	0.358849i
$498$	0	0
$499$	39.1246	1.75146	0.875729	−	0.482803i	$-0.160381\pi$
$499$	39.1246	1.75146	0.875729	+	0.482803i	$0.160381\pi$
$500$	− 11.1803i	− 0.500000i
$501$	0	0
$502$	− 2.29180i	− 0.102288i
$503$	− 30.7639i	− 1.37170i	−0.727745	−	0.685848i	$-0.759431\pi$
$503$	− 30.7639i	− 1.37170i	0.727745	−	0.685848i	$-0.240569\pi$
$504$	0	0
$505$	− 20.6525i	− 0.919023i
$506$	12.3607	0.549499
$507$	0	0
$508$	− 21.4164i	− 0.950199i
$509$	−15.0557	−0.667333	−0.333667	−	0.942691i	$-0.608286\pi$
$509$	−15.0557	−0.667333	−0.333667	+	0.942691i	$0.608286\pi$
$510$	0	0
$511$	42.8328	1.89481
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	0.763932	0.0336956
$515$	−18.9443	−0.834784
$516$	0	0
$517$	57.8885i	2.54594i
$518$	35.4164i	1.55611i
$519$	0	0
$520$	2.23607i	0.0980581i
$521$	−20.8328	−0.912702	−0.456351	−	0.889800i	$-0.650844\pi$
$521$	−20.8328	−0.912702	−0.456351	+	0.889800i	$0.650844\pi$
$522$	0	0
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	9.70820	0.424105
$525$	0	0
$526$	−10.1803	−0.443884
$527$	28.9443i	1.26083i
$528$	0	0
$529$	15.3607	0.667856
$530$	− 1.05573i	− 0.0458579i
$531$	0	0
$532$	− 8.94427i	− 0.387783i
$533$	3.52786i	0.152809i
$534$	0	0
$535$	3.41641	0.147704
$536$	0	0
$537$	0	0
$538$	6.76393i	0.291614i
$539$	15.5279	0.668832
$540$	0	0
$541$	−4.76393	−0.204817	−0.102409	−	0.994742i	$-0.532655\pi$
$541$	−4.76393	−0.204817	−0.102409	+	0.994742i	$0.532655\pi$
$542$	− 11.4164i	− 0.490377i
$543$	0	0
$544$	7.23607	0.310244
$545$	− 5.12461i	− 0.219514i
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	− 19.8885i	− 0.849596i
$549$	0	0
$550$	22.3607i	0.953463i
$551$	−10.2492	−0.436632
$552$	0	0
$553$	12.9443i	0.550446i
$554$	−20.8328	−0.885102
$555$	0	0
$556$	−6.47214	−0.274480
$557$	− 19.3050i	− 0.817977i	−0.912540	−	0.408989i	$-0.865882\pi$
$557$	− 19.3050i	− 0.817977i	0.912540	−	0.408989i	$-0.134118\pi$
$558$	0	0
$559$	2.47214	0.104560
$560$	7.23607	0.305780
$561$	0	0
$562$	− 24.4721i	− 1.03229i
$563$	− 19.4164i	− 0.818304i	−0.912466	−	0.409152i	$-0.865825\pi$
$563$	− 19.4164i	− 0.818304i	0.912466	−	0.409152i	$-0.134175\pi$
$564$	0	0
$565$	41.7082	1.75468
$566$	−11.4164	−0.479867
$567$	0	0
$568$	− 2.47214i	− 0.103729i
$569$	34.9443	1.46494	0.732470	−	0.680799i	$-0.238367\pi$
$569$	34.9443	1.46494	0.732470	+	0.680799i	$0.238367\pi$
$570$	0	0
$571$	−8.58359	−0.359212	−0.179606	−	0.983739i	$-0.557482\pi$
$571$	−8.58359	−0.359212	−0.179606	+	0.983739i	$0.557482\pi$
$572$	− 4.47214i	− 0.186989i
$573$	0	0
$574$	11.4164	0.476512
$575$	− 13.8197i	− 0.576320i
$576$	0	0
$577$	− 43.1246i	− 1.79530i	−0.440708	−	0.897651i	$-0.645273\pi$
$577$	− 43.1246i	− 1.79530i	0.440708	−	0.897651i	$-0.354727\pi$
$578$	− 35.3607i	− 1.47081i
$579$	0	0
$580$	− 8.29180i	− 0.344298i
$581$	16.0000	0.663792
$582$	0	0
$583$	2.11146i	0.0874476i
$584$	−13.2361	−0.547712
$585$	0	0
$586$	−2.94427	−0.121627
$587$	2.11146i	0.0871491i	0.999050	+	0.0435746i	$0.0138746\pi$
$587$	2.11146i	0.0871491i	−0.999050	+	0.0435746i	$0.986125\pi$
$588$	0	0
$589$	−11.0557	−0.455543
$590$	−18.9443	−0.779923
$591$	0	0
$592$	− 10.9443i	− 0.449807i
$593$	3.52786i	0.144872i	0.997373	+	0.0724360i	$0.0230773\pi$
$593$	3.52786i	0.144872i	−0.997373	+	0.0724360i	$0.976923\pi$
$594$	0	0
$595$	− 52.3607i	− 2.14658i
$596$	−13.5279	−0.554123
$597$	0	0
$598$	2.76393i	0.113026i
$599$	−6.11146	−0.249707	−0.124854	−	0.992175i	$-0.539846\pi$
$599$	−6.11146	−0.249707	−0.124854	+	0.992175i	$0.539846\pi$
$600$	0	0
$601$	13.4164	0.547267	0.273633	−	0.961834i	$-0.411775\pi$
$601$	13.4164	0.547267	0.273633	+	0.961834i	$0.411775\pi$
$602$	− 8.00000i	− 0.326056i
$603$	0	0
$604$	19.4164	0.790042
$605$	− 20.1246i	− 0.818182i
$606$	0	0
$607$	1.41641i	0.0574902i	0.999587	+	0.0287451i	$0.00915111\pi$
$607$	1.41641i	0.0574902i	−0.999587	+	0.0287451i	$0.990849\pi$
$608$	2.76393i	0.112092i
$609$	0	0
$610$	−24.4721	−0.990848
$611$	−12.9443	−0.523669
$612$	0	0
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	24.3607	0.983117
$615$	0	0
$616$	−14.4721	−0.583099
$617$	− 35.8885i	− 1.44482i	−0.691466	−	0.722409i	$-0.743035\pi$
$617$	− 35.8885i	− 1.44482i	0.691466	−	0.722409i	$-0.256965\pi$
$618$	0	0
$619$	25.2361	1.01432	0.507162	−	0.861851i	$-0.330695\pi$
$619$	25.2361	1.01432	0.507162	+	0.861851i	$0.330695\pi$
$620$	− 8.94427i	− 0.359211i
$621$	0	0
$622$	22.4721i	0.901051i
$623$	1.52786i	0.0612126i
$624$	0	0
$625$	25.0000	1.00000
$626$	−19.4164	−0.776036
$627$	0	0
$628$	18.9443i	0.755959i
$629$	−79.1935	−3.15765
$630$	0	0
$631$	−12.5836	−0.500945	−0.250472	−	0.968124i	$-0.580586\pi$
$631$	−12.5836	−0.500945	−0.250472	+	0.968124i	$0.580586\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	13.4164	0.532834
$635$	47.8885	1.90040
$636$	0	0
$637$	3.47214i	0.137571i
$638$	16.5836i	0.656551i
$639$	0	0
$640$	−2.23607	−0.0883883
$641$	23.3050	0.920490	0.460245	−	0.887792i	$-0.347762\pi$
$641$	23.3050	0.920490	0.460245	+	0.887792i	$0.347762\pi$
$642$	0	0
$643$	− 19.0557i	− 0.751485i	−0.926724	−	0.375742i	$-0.877388\pi$
$643$	− 19.0557i	− 0.751485i	0.926724	−	0.375742i	$-0.122612\pi$
$644$	8.94427	0.352454
$645$	0	0
$646$	20.0000	0.786889
$647$	3.70820i	0.145785i	0.997340	+	0.0728923i	$0.0232229\pi$
$647$	3.70820i	0.145785i	−0.997340	+	0.0728923i	$0.976777\pi$
$648$	0	0
$649$	37.8885	1.48726
$650$	−5.00000	−0.196116
$651$	0	0
$652$	12.9443i	0.506937i
$653$	− 30.3607i	− 1.18811i	−0.804426	−	0.594053i	$-0.797527\pi$
$653$	− 30.3607i	− 1.18811i	0.804426	−	0.594053i	$-0.202473\pi$
$654$	0	0
$655$	21.7082i	0.848210i
$656$	−3.52786	−0.137740
$657$	0	0
$658$	41.8885i	1.63299i
$659$	−10.0689	−0.392228	−0.196114	−	0.980581i	$-0.562832\pi$
$659$	−10.0689	−0.392228	−0.196114	+	0.980581i	$0.562832\pi$
$660$	0	0
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	− 2.76393i	− 0.107423i
$663$	0	0
$664$	−4.94427	−0.191875
$665$	20.0000	0.775567
$666$	0	0
$667$	− 10.2492i	− 0.396852i
$668$	− 19.4164i	− 0.751243i
$669$	0	0
$670$	0	0
$671$	48.9443	1.88947
$672$	0	0
$673$	− 44.0000i	− 1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$	− 44.0000i	− 1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	6.47214	0.249297
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 27.5279i	− 1.05798i	−0.848628	−	0.528991i	$-0.822571\pi$
$677$	− 27.5279i	− 1.05798i	0.848628	−	0.528991i	$-0.177429\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	16.1803i	0.620488i
$681$	0	0
$682$	17.8885i	0.684988i
$683$	− 41.8885i	− 1.60282i	−0.598115	−	0.801410i	$-0.704083\pi$
$683$	− 41.8885i	− 1.60282i	0.598115	−	0.801410i	$-0.295917\pi$
$684$	0	0
$685$	44.4721	1.69919
$686$	−11.4164	−0.435880
$687$	0	0
$688$	2.47214i	0.0942493i
$689$	−0.472136	−0.0179869
$690$	0	0
$691$	0.291796	0.0111004	0.00555022	−	0.999985i	$-0.498233\pi$
$691$	0.291796	0.0111004	0.00555022	+	0.999985i	$0.498233\pi$
$692$	22.9443i	0.872210i
$693$	0	0
$694$	1.52786	0.0579969
$695$	− 14.4721i	− 0.548959i
$696$	0	0
$697$	25.5279i	0.966937i
$698$	15.2361i	0.576694i
$699$	0	0
$700$	16.1803i	0.611559i
$701$	24.0689	0.909069	0.454535	−	0.890729i	$-0.349806\pi$
$701$	24.0689	0.909069	0.454535	+	0.890729i	$0.349806\pi$
$702$	0	0
$703$	− 30.2492i	− 1.14087i
$704$	4.47214	0.168550
$705$	0	0
$706$	26.9443	1.01406
$707$	29.8885i	1.12407i
$708$	0	0
$709$	−32.5410	−1.22210	−0.611052	−	0.791591i	$-0.709253\pi$
$709$	−32.5410	−1.22210	−0.611052	+	0.791591i	$0.709253\pi$
$710$	5.52786	0.207457
$711$	0	0
$712$	− 0.472136i	− 0.0176940i
$713$	− 11.0557i	− 0.414040i
$714$	0	0
$715$	10.0000	0.373979
$716$	−8.18034	−0.305714
$717$	0	0
$718$	6.47214i	0.241538i
$719$	20.9443	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$719$	20.9443	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$720$	0	0
$721$	27.4164	1.02104
$722$	− 11.3607i	− 0.422801i
$723$	0	0
$724$	−17.4164	−0.647276
$725$	18.5410	0.688596
$726$	0	0
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	− 3.23607i	− 0.119937i
$729$	0	0
$730$	− 29.5967i	− 1.09542i
$731$	17.8885	0.661632
$732$	0	0
$733$	− 33.4164i	− 1.23426i	−0.786860	−	0.617132i	$-0.788295\pi$
$733$	− 33.4164i	− 1.23426i	0.786860	−	0.617132i	$-0.211705\pi$
$734$	−0.472136	−0.0174269
$735$	0	0
$736$	−2.76393	−0.101880
$737$	0	0
$738$	0	0
$739$	0.291796	0.0107339	0.00536695	−	0.999986i	$-0.498292\pi$
$739$	0.291796	0.0107339	0.00536695	+	0.999986i	$0.498292\pi$
$740$	24.4721	0.899614
$741$	0	0
$742$	1.52786i	0.0560897i
$743$	28.3607i	1.04045i	0.854029	+	0.520226i	$0.174152\pi$
$743$	28.3607i	1.04045i	−0.854029	+	0.520226i	$0.825848\pi$
$744$	0	0
$745$	− 30.2492i	− 1.10825i
$746$	2.00000	0.0732252
$747$	0	0
$748$	− 32.3607i	− 1.18322i
$749$	−4.94427	−0.180660
$750$	0	0
$751$	−46.8328	−1.70895	−0.854477	−	0.519489i	$-0.826122\pi$
$751$	−46.8328	−1.70895	−0.854477	+	0.519489i	$0.826122\pi$
$752$	− 12.9443i	− 0.472029i
$753$	0	0
$754$	−3.70820	−0.135045
$755$	43.4164i	1.58008i
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	35.1246i	1.27578i
$759$	0	0
$760$	−6.18034	−0.224184
$761$	−7.52786	−0.272885	−0.136442	−	0.990648i	$-0.543567\pi$
$761$	−7.52786	−0.272885	−0.136442	+	0.990648i	$0.543567\pi$
$762$	0	0
$763$	7.41641i	0.268492i
$764$	22.4721	0.813013
$765$	0	0
$766$	−5.52786	−0.199730
$767$	8.47214i	0.305911i
$768$	0	0
$769$	4.83282	0.174276	0.0871379	−	0.996196i	$-0.472228\pi$
$769$	4.83282	0.174276	0.0871379	+	0.996196i	$0.472228\pi$
$770$	− 32.3607i	− 1.16620i
$771$	0	0
$772$	− 8.29180i	− 0.298428i
$773$	− 2.94427i	− 0.105898i	−0.998597	−	0.0529491i	$-0.983138\pi$
$773$	− 2.94427i	− 0.105898i	0.998597	−	0.0529491i	$-0.0168621\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	−3.70820	−0.133117
$777$	0	0
$778$	12.6525i	0.453613i
$779$	−9.75078	−0.349358
$780$	0	0
$781$	−11.0557	−0.395605
$782$	20.0000i	0.715199i
$783$	0	0
$784$	−3.47214	−0.124005
$785$	−42.3607	−1.51192
$786$	0	0
$787$	28.9443i	1.03175i	0.856663	+	0.515876i	$0.172533\pi$
$787$	28.9443i	1.03175i	−0.856663	+	0.515876i	$0.827467\pi$
$788$	− 14.9443i	− 0.532368i
$789$	0	0
$790$	8.94427	0.318223
$791$	−60.3607	−2.14618
$792$	0	0
$793$	10.9443i	0.388642i
$794$	−17.4164	−0.618085
$795$	0	0
$796$	−0.944272	−0.0334688
$797$	− 9.05573i	− 0.320770i	−0.987055	−	0.160385i	$-0.948726\pi$
$797$	− 9.05573i	− 0.320770i	0.987055	−	0.160385i	$-0.0512736\pi$
$798$	0	0
$799$	−93.6656	−3.31365
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	24.4721i	0.864141i
$803$	59.1935i	2.08889i
$804$	0	0
$805$	20.0000i	0.704907i
$806$	−4.00000	−0.140894
$807$	0	0
$808$	− 9.23607i	− 0.324924i
$809$	31.5279	1.10846	0.554230	−	0.832363i	$-0.313013\pi$
$809$	31.5279	1.10846	0.554230	+	0.832363i	$0.313013\pi$
$810$	0	0
$811$	4.29180	0.150705	0.0753527	−	0.997157i	$-0.475992\pi$
$811$	4.29180	0.150705	0.0753527	+	0.997157i	$0.475992\pi$
$812$	12.0000i	0.421117i
$813$	0	0
$814$	−48.9443	−1.71550
$815$	−28.9443	−1.01387
$816$	0	0
$817$	6.83282i	0.239050i
$818$	− 6.00000i	− 0.209785i
$819$	0	0
$820$	− 7.88854i	− 0.275480i
$821$	30.4721	1.06348	0.531742	−	0.846906i	$-0.321537\pi$
$821$	30.4721	1.06348	0.531742	+	0.846906i	$0.321537\pi$
$822$	0	0
$823$	− 48.2492i	− 1.68186i	−0.541142	−	0.840931i	$-0.682008\pi$
$823$	− 48.2492i	− 1.68186i	0.541142	−	0.840931i	$-0.317992\pi$
$824$	−8.47214	−0.295141
$825$	0	0
$826$	27.4164	0.953939
$827$	41.8885i	1.45661i	0.685254	+	0.728304i	$0.259691\pi$
$827$	41.8885i	1.45661i	−0.685254	+	0.728304i	$0.740309\pi$
$828$	0	0
$829$	−42.7214	−1.48377	−0.741887	−	0.670525i	$-0.766069\pi$
$829$	−42.7214	−1.48377	−0.741887	+	0.670525i	$0.766069\pi$
$830$	− 11.0557i	− 0.383750i
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	25.1246i	0.870516i
$834$	0	0
$835$	43.4164	1.50249
$836$	12.3607	0.427503
$837$	0	0
$838$	27.5967i	0.953314i
$839$	−51.7771	−1.78754	−0.893772	−	0.448522i	$-0.851951\pi$
$839$	−51.7771	−1.78754	−0.893772	+	0.448522i	$0.851951\pi$
$840$	0	0
$841$	−15.2492	−0.525835
$842$	− 13.7082i	− 0.472416i
$843$	0	0
$844$	−13.8885	−0.478063
$845$	2.23607i	0.0769231i
$846$	0	0
$847$	29.1246i	1.00073i
$848$	− 0.472136i	− 0.0162132i
$849$	0	0
$850$	−36.1803	−1.24098
$851$	30.2492	1.03693
$852$	0	0
$853$	− 41.4164i	− 1.41807i	−0.705173	−	0.709035i	$-0.749131\pi$
$853$	− 41.4164i	− 1.41807i	0.705173	−	0.709035i	$-0.250869\pi$
$854$	35.4164	1.21192
$855$	0	0
$856$	1.52786	0.0522213
$857$	0.180340i	0.00616029i	0.999995	+	0.00308015i	$0.000980443\pi$
$857$	0.180340i	0.00616029i	−0.999995	+	0.00308015i	$0.999020\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	−5.52786	−0.188499
$861$	0	0
$862$	14.8328i	0.505208i
$863$	22.4721i	0.764960i	0.923964	+	0.382480i	$0.124930\pi$
$863$	22.4721i	0.764960i	−0.923964	+	0.382480i	$0.875070\pi$
$864$	0	0
$865$	−51.3050	−1.74442
$866$	−9.52786	−0.323770
$867$	0	0
$868$	12.9443i	0.439357i
$869$	−17.8885	−0.606827
$870$	0	0
$871$	0	0
$872$	− 2.29180i	− 0.0776100i
$873$	0	0
$874$	−7.63932	−0.258404
$875$	−36.1803	−1.22312
$876$	0	0
$877$	24.4721i	0.826365i	0.910648	+	0.413183i	$0.135583\pi$
$877$	24.4721i	0.826365i	−0.910648	+	0.413183i	$0.864417\pi$
$878$	4.94427i	0.166861i
$879$	0	0
$880$	10.0000i	0.337100i
$881$	−34.3607	−1.15764	−0.578820	−	0.815455i	$-0.696487\pi$
$881$	−34.3607	−1.15764	−0.578820	+	0.815455i	$0.696487\pi$
$882$	0	0
$883$	− 25.8885i	− 0.871219i	−0.900136	−	0.435609i	$-0.856533\pi$
$883$	− 25.8885i	− 0.871219i	0.900136	−	0.435609i	$-0.143467\pi$
$884$	7.23607	0.243375
$885$	0	0
$886$	22.4721	0.754966
$887$	10.1803i	0.341822i	0.985286	+	0.170911i	$0.0546711\pi$
$887$	10.1803i	0.341822i	−0.985286	+	0.170911i	$0.945329\pi$
$888$	0	0
$889$	−69.3050	−2.32441
$890$	1.05573	0.0353881
$891$	0	0
$892$	16.7639i	0.561298i
$893$	− 35.7771i	− 1.19723i
$894$	0	0
$895$	− 18.2918i	− 0.611427i
$896$	3.23607	0.108109
$897$	0	0
$898$	0.472136i	0.0157554i
$899$	14.8328	0.494702
$900$	0	0
$901$	−3.41641	−0.113817
$902$	15.7771i	0.525320i
$903$	0	0
$904$	18.6525	0.620372
$905$	− 38.9443i	− 1.29455i
$906$	0	0
$907$	45.5279i	1.51173i	0.654729	+	0.755864i	$0.272783\pi$
$907$	45.5279i	1.51173i	−0.654729	+	0.755864i	$0.727217\pi$
$908$	7.05573i	0.234153i
$909$	0	0
$910$	7.23607	0.239873
$911$	11.0557	0.366293	0.183146	−	0.983086i	$-0.441372\pi$
$911$	11.0557	0.366293	0.183146	+	0.983086i	$0.441372\pi$
$912$	0	0
$913$	22.1115i	0.731782i
$914$	3.70820	0.122656
$915$	0	0
$916$	9.70820	0.320768
$917$	− 31.4164i	− 1.03746i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	− 6.18034i	− 0.203760i
$921$	0	0
$922$	− 41.8885i	− 1.37953i
$923$	− 2.47214i	− 0.0813713i
$924$	0	0
$925$	54.7214i	1.79923i
$926$	−22.2918	−0.732554
$927$	0	0
$928$	− 3.70820i	− 0.121728i
$929$	32.4721	1.06538	0.532688	−	0.846312i	$-0.321182\pi$
$929$	32.4721	1.06538	0.532688	+	0.846312i	$0.321182\pi$
$930$	0	0
$931$	−9.59675	−0.314521
$932$	− 20.1803i	− 0.661029i
$933$	0	0
$934$	6.47214	0.211775
$935$	72.3607	2.36645
$936$	0	0
$937$	19.7771i	0.646089i	0.946384	+	0.323045i	$0.104706\pi$
$937$	19.7771i	0.646089i	−0.946384	+	0.323045i	$0.895294\pi$
$938$	0	0
$939$	0	0
$940$	28.9443	0.944058
$941$	37.5279	1.22337	0.611687	−	0.791100i	$-0.290491\pi$
$941$	37.5279	1.22337	0.611687	+	0.791100i	$0.290491\pi$
$942$	0	0
$943$	− 9.75078i	− 0.317529i
$944$	−8.47214	−0.275745
$945$	0	0
$946$	11.0557	0.359453
$947$	− 4.94427i	− 0.160667i	−0.996768	−	0.0803336i	$-0.974401\pi$
$947$	− 4.94427i	− 0.160667i	0.996768	−	0.0803336i	$-0.0255986\pi$
$948$	0	0
$949$	−13.2361	−0.429661
$950$	− 13.8197i	− 0.448369i
$951$	0	0
$952$	− 23.4164i	− 0.758930i
$953$	29.1246i	0.943439i	0.881749	+	0.471719i	$0.156366\pi$
$953$	29.1246i	0.943439i	−0.881749	+	0.471719i	$0.843634\pi$
$954$	0	0
$955$	50.2492i	1.62603i
$956$	21.8885	0.707926
$957$	0	0
$958$	− 36.0000i	− 1.16311i
$959$	−64.3607	−2.07831
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 10.9443i	− 0.352857i
$963$	0	0
$964$	−3.52786	−0.113625
$965$	18.5410	0.596857
$966$	0	0
$967$	− 56.5410i	− 1.81824i	−0.416538	−	0.909118i	$-0.636757\pi$
$967$	− 56.5410i	− 1.81824i	0.416538	−	0.909118i	$-0.363243\pi$
$968$	− 9.00000i	− 0.289271i
$969$	0	0
$970$	− 8.29180i	− 0.266234i
$971$	−23.2361	−0.745681	−0.372840	−	0.927895i	$-0.621616\pi$
$971$	−23.2361	−0.745681	−0.372840	+	0.927895i	$0.621616\pi$
$972$	0	0
$973$	20.9443i	0.671443i
$974$	23.5967	0.756089
$975$	0	0
$976$	−10.9443	−0.350318
$977$	10.3607i	0.331468i	0.986171	+	0.165734i	$0.0529992\pi$
$977$	10.3607i	0.331468i	−0.986171	+	0.165734i	$0.947001\pi$
$978$	0	0
$979$	−2.11146	−0.0674824
$980$	− 7.76393i	− 0.248010i
$981$	0	0
$982$	16.1803i	0.516335i
$983$	− 7.41641i	− 0.236547i	−0.992981	−	0.118273i	$-0.962264\pi$
$983$	− 7.41641i	− 0.236547i	0.992981	−	0.118273i	$-0.0377359\pi$
$984$	0	0
$985$	33.4164	1.06474
$986$	−26.8328	−0.854531
$987$	0	0
$988$	2.76393i	0.0879324i
$989$	−6.83282	−0.217271
$990$	0	0
$991$	24.9443	0.792381	0.396190	−	0.918168i	$-0.370332\pi$
$991$	24.9443	0.792381	0.396190	+	0.918168i	$0.370332\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	−8.00000	−0.253745
$995$	− 2.11146i	− 0.0669377i
$996$	0	0
$997$	43.8885i	1.38996i	0.719027	+	0.694982i	$0.244588\pi$
$997$	43.8885i	1.38996i	−0.719027	+	0.694982i	$0.755412\pi$
$998$	39.1246i	1.23847i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.e.469.3		4
3.2	odd	2		390.2.e.e.79.2	✓	4
5.2	odd	4		5850.2.a.cf.1.1		2
5.3	odd	4		5850.2.a.cm.1.2		2
5.4	even	2	inner	1170.2.e.e.469.2		4
12.11	even	2		3120.2.l.k.1249.4		4
15.2	even	4		1950.2.a.bf.1.1		2
15.8	even	4		1950.2.a.be.1.2		2
15.14	odd	2		390.2.e.e.79.3	yes	4
60.59	even	2		3120.2.l.k.1249.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.2	✓	4	3.2	odd	2
390.2.e.e.79.3	yes	4	15.14	odd	2
1170.2.e.e.469.2		4	5.4	even	2	inner
1170.2.e.e.469.3		4	1.1	even	1	trivial
1950.2.a.be.1.2		2	15.8	even	4
1950.2.a.bf.1.1		2	15.2	even	4
3120.2.l.k.1249.1		4	60.59	even	2
3120.2.l.k.1249.4		4	12.11	even	2
5850.2.a.cf.1.1		2	5.2	odd	4
5850.2.a.cm.1.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.23607i q^{5} +3.23607i q^{7} -1.00000i q^{8} +2.23607 q^{10} -4.47214 q^{11} -1.00000i q^{13} -3.23607 q^{14} +1.00000 q^{16} -7.23607i q^{17} +2.76393 q^{19} +2.23607i q^{20} -4.47214i q^{22} +2.76393i q^{23} -5.00000 q^{25} +1.00000 q^{26} -3.23607i q^{28} -3.70820 q^{29} -4.00000 q^{31} +1.00000i q^{32} +7.23607 q^{34} +7.23607 q^{35} -10.9443i q^{37} +2.76393i q^{38} -2.23607 q^{40} -3.52786 q^{41} +2.47214i q^{43} +4.47214 q^{44} -2.76393 q^{46} -12.9443i q^{47} -3.47214 q^{49} -5.00000i q^{50} +1.00000i q^{52} -0.472136i q^{53} +10.0000i q^{55} +3.23607 q^{56} -3.70820i q^{58} -8.47214 q^{59} -10.9443 q^{61} -4.00000i q^{62} -1.00000 q^{64} -2.23607 q^{65} +7.23607i q^{68} +7.23607i q^{70} +2.47214 q^{71} -13.2361i q^{73} +10.9443 q^{74} -2.76393 q^{76} -14.4721i q^{77} +4.00000 q^{79} -2.23607i q^{80} -3.52786i q^{82} -4.94427i q^{83} -16.1803 q^{85} -2.47214 q^{86} +4.47214i q^{88} +0.472136 q^{89} +3.23607 q^{91} -2.76393i q^{92} +12.9443 q^{94} -6.18034i q^{95} -3.70820i q^{97} -3.47214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 20 q^{25} + 4 q^{26} + 12 q^{29} - 16 q^{31} + 20 q^{34} + 20 q^{35} - 32 q^{41} - 20 q^{46} + 4 q^{49} + 4 q^{56} - 16 q^{59} - 8 q^{61} - 4 q^{64} - 8 q^{71}+ \cdots + 16 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 20 * q^25 + 4 * q^26 + 12 * q^29 - 16 * q^31 + 20 * q^34 + 20 * q^35 - 32 * q^41 - 20 * q^46 + 4 * q^49 + 4 * q^56 - 16 * q^59 - 8 * q^61 - 4 * q^64 - 8 * q^71 + 8 * q^74 - 20 * q^76 + 16 * q^79 - 20 * q^85 + 8 * q^86 - 16 * q^89 + 4 * q^91 + 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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