LMFDB - Embedded newform 693.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	\(q+2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +4.47214 q^{10} +1.00000 q^{11} -1.23607 q^{13} +2.23607 q^{14} -1.00000 q^{16} -1.23607 q^{17} -2.47214 q^{19} +6.00000 q^{20} +2.23607 q^{22} +6.47214 q^{23} -1.00000 q^{25} -2.76393 q^{26} +3.00000 q^{28} +0.472136 q^{29} -7.23607 q^{31} -6.70820 q^{32} -2.76393 q^{34} +2.00000 q^{35} +0.472136 q^{37} -5.52786 q^{38} +4.47214 q^{40} +6.76393 q^{41} +8.00000 q^{43} +3.00000 q^{44} +14.4721 q^{46} -7.23607 q^{47} +1.00000 q^{49} -2.23607 q^{50} -3.70820 q^{52} -8.47214 q^{53} +2.00000 q^{55} +2.23607 q^{56} +1.05573 q^{58} -3.23607 q^{59} -2.76393 q^{61} -16.1803 q^{62} -13.0000 q^{64} -2.47214 q^{65} +5.52786 q^{67} -3.70820 q^{68} +4.47214 q^{70} +1.52786 q^{71} -5.23607 q^{73} +1.05573 q^{74} -7.41641 q^{76} +1.00000 q^{77} +8.94427 q^{79} -2.00000 q^{80} +15.1246 q^{82} -15.4164 q^{83} -2.47214 q^{85} +17.8885 q^{86} +2.23607 q^{88} -2.00000 q^{89} -1.23607 q^{91} +19.4164 q^{92} -16.1803 q^{94} -4.94427 q^{95} -9.41641 q^{97} +2.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 4 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 6 * q^4 + 4 * q^5 + 2 * q^7	$ 2 q + 6 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 4 q^{19} + 12 q^{20} + 4 q^{23} - 2 q^{25} - 10 q^{26} + 6 q^{28} - 8 q^{29} - 10 q^{31} - 10 q^{34} + 4 q^{35} - 8 q^{37} - 20 q^{38} + 18 q^{41} + 16 q^{43} + 6 q^{44} + 20 q^{46} - 10 q^{47} + 2 q^{49} + 6 q^{52} - 8 q^{53} + 4 q^{55} + 20 q^{58} - 2 q^{59} - 10 q^{61} - 10 q^{62} - 26 q^{64} + 4 q^{65} + 20 q^{67} + 6 q^{68} + 12 q^{71} - 6 q^{73} + 20 q^{74} + 12 q^{76} + 2 q^{77} - 4 q^{80} - 10 q^{82} - 4 q^{83} + 4 q^{85} - 4 q^{89} + 2 q^{91} + 12 q^{92} - 10 q^{94} + 8 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 2 * q^16 + 2 * q^17 + 4 * q^19 + 12 * q^20 + 4 * q^23 - 2 * q^25 - 10 * q^26 + 6 * q^28 - 8 * q^29 - 10 * q^31 - 10 * q^34 + 4 * q^35 - 8 * q^37 - 20 * q^38 + 18 * q^41 + 16 * q^43 + 6 * q^44 + 20 * q^46 - 10 * q^47 + 2 * q^49 + 6 * q^52 - 8 * q^53 + 4 * q^55 + 20 * q^58 - 2 * q^59 - 10 * q^61 - 10 * q^62 - 26 * q^64 + 4 * q^65 + 20 * q^67 + 6 * q^68 + 12 * q^71 - 6 * q^73 + 20 * q^74 + 12 * q^76 + 2 * q^77 - 4 * q^80 - 10 * q^82 - 4 * q^83 + 4 * q^85 - 4 * q^89 + 2 * q^91 + 12 * q^92 - 10 * q^94 + 8 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	4.47214	1.41421
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	2.23607	0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−1.23607	−0.299791	−0.149895	−	0.988702i	$-0.547894\pi$
$17$	−1.23607	−0.299791	−0.149895	+	0.988702i	$0.547894\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	6.00000	1.34164
$21$	0	0
$22$	2.23607	0.476731
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−2.76393	−0.542052
$27$	0	0
$28$	3.00000	0.566947
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	−7.23607	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$31$	−7.23607	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	−2.76393	−0.474010
$35$	2.00000	0.338062
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	−5.52786	−0.896738
$39$	0	0
$40$	4.47214	0.707107
$41$	6.76393	1.05635	0.528174	−	0.849136i	$-0.322877\pi$
$41$	6.76393	1.05635	0.528174	+	0.849136i	$0.322877\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	14.4721	2.13380
$47$	−7.23607	−1.05549	−0.527744	−	0.849403i	$-0.676962\pi$
$47$	−7.23607	−1.05549	−0.527744	+	0.849403i	$0.676962\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.23607	−0.316228
$51$	0	0
$52$	−3.70820	−0.514235
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	2.23607	0.298807
$57$	0	0
$58$	1.05573	0.138624
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	−16.1803	−2.05491
$63$	0	0
$64$	−13.0000	−1.62500
$65$	−2.47214	−0.306631
$66$	0	0
$67$	5.52786	0.675336	0.337668	−	0.941265i	$-0.390362\pi$
$67$	5.52786	0.675336	0.337668	+	0.941265i	$0.390362\pi$
$68$	−3.70820	−0.449686
$69$	0	0
$70$	4.47214	0.534522
$71$	1.52786	0.181324	0.0906621	−	0.995882i	$-0.471102\pi$
$71$	1.52786	0.181324	0.0906621	+	0.995882i	$0.471102\pi$
$72$	0	0
$73$	−5.23607	−0.612835	−0.306418	−	0.951897i	$-0.599130\pi$
$73$	−5.23607	−0.612835	−0.306418	+	0.951897i	$0.599130\pi$
$74$	1.05573	0.122726
$75$	0	0
$76$	−7.41641	−0.850720
$77$	1.00000	0.113961
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	15.1246	1.67023
$83$	−15.4164	−1.69217	−0.846085	−	0.533048i	$-0.821047\pi$
$83$	−15.4164	−1.69217	−0.846085	+	0.533048i	$0.821047\pi$
$84$	0	0
$85$	−2.47214	−0.268141
$86$	17.8885	1.92897
$87$	0	0
$88$	2.23607	0.238366
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	19.4164	2.02430
$93$	0	0
$94$	−16.1803	−1.66887
$95$	−4.94427	−0.507272
$96$	0	0
$97$	−9.41641	−0.956091	−0.478046	−	0.878335i	$-0.658655\pi$
$97$	−9.41641	−0.956091	−0.478046	+	0.878335i	$0.658655\pi$
$98$	2.23607	0.225877
$99$	0	0
$100$	−3.00000	−0.300000
$101$	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
$103$	−5.70820	−0.562446	−0.281223	−	0.959642i	$-0.590740\pi$
$103$	−5.70820	−0.562446	−0.281223	+	0.959642i	$0.590740\pi$
$104$	−2.76393	−0.271026
$105$	0	0
$106$	−18.9443	−1.84003
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	4.47214	0.428353	0.214176	−	0.976795i	$-0.431293\pi$
$109$	4.47214	0.428353	0.214176	+	0.976795i	$0.431293\pi$
$110$	4.47214	0.426401
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	12.9443	1.20706
$116$	1.41641	0.131510
$117$	0	0
$118$	−7.23607	−0.666134
$119$	−1.23607	−0.113310
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.18034	−0.559542
$123$	0	0
$124$	−21.7082	−1.94945
$125$	−12.0000	−1.07331
$126$	0	0
$127$	20.9443	1.85850	0.929252	−	0.369447i	$-0.120453\pi$
$127$	20.9443	1.85850	0.929252	+	0.369447i	$0.120453\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	−5.52786	−0.484826
$131$	13.8885	1.21345	0.606724	−	0.794913i	$-0.292483\pi$
$131$	13.8885	1.21345	0.606724	+	0.794913i	$0.292483\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	12.3607	1.06780
$135$	0	0
$136$	−2.76393	−0.237005
$137$	−7.52786	−0.643149	−0.321574	−	0.946884i	$-0.604212\pi$
$137$	−7.52786	−0.643149	−0.321574	+	0.946884i	$0.604212\pi$
$138$	0	0
$139$	−10.4721	−0.888235	−0.444117	−	0.895969i	$-0.646483\pi$
$139$	−10.4721	−0.888235	−0.444117	+	0.895969i	$0.646483\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	3.41641	0.286699
$143$	−1.23607	−0.103365
$144$	0	0
$145$	0.944272	0.0784175
$146$	−11.7082	−0.968978
$147$	0	0
$148$	1.41641	0.116428
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−8.94427	−0.727875	−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427	−0.727875	−0.363937	+	0.931423i	$0.618568\pi$
$152$	−5.52786	−0.448369
$153$	0	0
$154$	2.23607	0.180187
$155$	−14.4721	−1.16243
$156$	0	0
$157$	−6.94427	−0.554213	−0.277107	−	0.960839i	$-0.589376\pi$
$157$	−6.94427	−0.554213	−0.277107	+	0.960839i	$0.589376\pi$
$158$	20.0000	1.59111
$159$	0	0
$160$	−13.4164	−1.06066
$161$	6.47214	0.510076
$162$	0	0
$163$	23.4164	1.83411	0.917057	−	0.398755i	$-0.130558\pi$
$163$	23.4164	1.83411	0.917057	+	0.398755i	$0.130558\pi$
$164$	20.2918	1.58452
$165$	0	0
$166$	−34.4721	−2.67556
$167$	12.9443	1.00166	0.500829	−	0.865546i	$-0.333029\pi$
$167$	12.9443	1.00166	0.500829	+	0.865546i	$0.333029\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	−5.52786	−0.423968
$171$	0	0
$172$	24.0000	1.82998
$173$	17.2361	1.31043	0.655217	−	0.755441i	$-0.272577\pi$
$173$	17.2361	1.31043	0.655217	+	0.755441i	$0.272577\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−4.47214	−0.335201
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	1.41641	0.105281	0.0526404	−	0.998614i	$-0.483236\pi$
$181$	1.41641	0.105281	0.0526404	+	0.998614i	$0.483236\pi$
$182$	−2.76393	−0.204876
$183$	0	0
$184$	14.4721	1.06690
$185$	0.944272	0.0694243
$186$	0	0
$187$	−1.23607	−0.0903902
$188$	−21.7082	−1.58323
$189$	0	0
$190$	−11.0557	−0.802067
$191$	20.9443	1.51547	0.757737	−	0.652560i	$-0.226305\pi$
$191$	20.9443	1.51547	0.757737	+	0.652560i	$0.226305\pi$
$192$	0	0
$193$	−23.8885	−1.71954	−0.859768	−	0.510686i	$-0.829392\pi$
$193$	−23.8885	−1.71954	−0.859768	+	0.510686i	$0.829392\pi$
$194$	−21.0557	−1.51171
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	20.1803	1.43055	0.715273	−	0.698845i	$-0.246302\pi$
$199$	20.1803	1.43055	0.715273	+	0.698845i	$0.246302\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	20.6525	1.45310
$203$	0.472136	0.0331374
$204$	0	0
$205$	13.5279	0.944827
$206$	−12.7639	−0.889305
$207$	0	0
$208$	1.23607	0.0857059
$209$	−2.47214	−0.171001
$210$	0	0
$211$	21.8885	1.50687	0.753435	−	0.657523i	$-0.228396\pi$
$211$	21.8885	1.50687	0.753435	+	0.657523i	$0.228396\pi$
$212$	−25.4164	−1.74561
$213$	0	0
$214$	8.94427	0.611418
$215$	16.0000	1.09119
$216$	0	0
$217$	−7.23607	−0.491216
$218$	10.0000	0.677285
$219$	0	0
$220$	6.00000	0.404520
$221$	1.52786	0.102775
$222$	0	0
$223$	12.1803	0.815656	0.407828	−	0.913059i	$-0.366286\pi$
$223$	12.1803	0.815656	0.407828	+	0.913059i	$0.366286\pi$
$224$	−6.70820	−0.448211
$225$	0	0
$226$	−4.47214	−0.297482
$227$	29.8885	1.98377	0.991886	−	0.127129i	$-0.0405763\pi$
$227$	29.8885	1.98377	0.991886	+	0.127129i	$0.0405763\pi$
$228$	0	0
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	28.9443	1.90853
$231$	0	0
$232$	1.05573	0.0693119
$233$	17.4164	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$233$	17.4164	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$234$	0	0
$235$	−14.4721	−0.944058
$236$	−9.70820	−0.631950
$237$	0	0
$238$	−2.76393	−0.179159
$239$	−25.8885	−1.67459	−0.837295	−	0.546751i	$-0.815864\pi$
$239$	−25.8885	−1.67459	−0.837295	+	0.546751i	$0.815864\pi$
$240$	0	0
$241$	27.1246	1.74725	0.873625	−	0.486600i	$-0.161763\pi$
$241$	27.1246	1.74725	0.873625	+	0.486600i	$0.161763\pi$
$242$	2.23607	0.143740
$243$	0	0
$244$	−8.29180	−0.530828
$245$	2.00000	0.127775
$246$	0	0
$247$	3.05573	0.194431
$248$	−16.1803	−1.02745
$249$	0	0
$250$	−26.8328	−1.69706
$251$	−17.7082	−1.11773	−0.558866	−	0.829258i	$-0.688763\pi$
$251$	−17.7082	−1.11773	−0.558866	+	0.829258i	$0.688763\pi$
$252$	0	0
$253$	6.47214	0.406900
$254$	46.8328	2.93855
$255$	0	0
$256$	−9.00000	−0.562500
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0.472136	0.0293371
$260$	−7.41641	−0.459946
$261$	0	0
$262$	31.0557	1.91863
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−16.9443	−1.04088
$266$	−5.52786	−0.338935
$267$	0	0
$268$	16.5836	1.01300
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	1.23607	0.0749476
$273$	0	0
$274$	−16.8328	−1.01691
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	−23.4164	−1.40442
$279$	0	0
$280$	4.47214	0.267261
$281$	12.4721	0.744025	0.372013	−	0.928228i	$-0.378668\pi$
$281$	12.4721	0.744025	0.372013	+	0.928228i	$0.378668\pi$
$282$	0	0
$283$	−5.88854	−0.350038	−0.175019	−	0.984565i	$-0.555999\pi$
$283$	−5.88854	−0.350038	−0.175019	+	0.984565i	$0.555999\pi$
$284$	4.58359	0.271986
$285$	0	0
$286$	−2.76393	−0.163435
$287$	6.76393	0.399262
$288$	0	0
$289$	−15.4721	−0.910126
$290$	2.11146	0.123989
$291$	0	0
$292$	−15.7082	−0.919253
$293$	−15.1246	−0.883589	−0.441795	−	0.897116i	$-0.645658\pi$
$293$	−15.1246	−0.883589	−0.441795	+	0.897116i	$0.645658\pi$
$294$	0	0
$295$	−6.47214	−0.376822
$296$	1.05573	0.0613629
$297$	0	0
$298$	−31.3050	−1.81345
$299$	−8.00000	−0.462652
$300$	0	0
$301$	8.00000	0.461112
$302$	−20.0000	−1.15087
$303$	0	0
$304$	2.47214	0.141787
$305$	−5.52786	−0.316525
$306$	0	0
$307$	8.94427	0.510477	0.255238	−	0.966878i	$-0.417846\pi$
$307$	8.94427	0.510477	0.255238	+	0.966878i	$0.417846\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	−32.3607	−1.83796
$311$	21.7082	1.23096	0.615480	−	0.788153i	$-0.288962\pi$
$311$	21.7082	1.23096	0.615480	+	0.788153i	$0.288962\pi$
$312$	0	0
$313$	−2.94427	−0.166420	−0.0832100	−	0.996532i	$-0.526517\pi$
$313$	−2.94427	−0.166420	−0.0832100	+	0.996532i	$0.526517\pi$
$314$	−15.5279	−0.876288
$315$	0	0
$316$	26.8328	1.50946
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	0.472136	0.0264345
$320$	−26.0000	−1.45344
$321$	0	0
$322$	14.4721	0.806501
$323$	3.05573	0.170025
$324$	0	0
$325$	1.23607	0.0685647
$326$	52.3607	2.89999
$327$	0	0
$328$	15.1246	0.835117
$329$	−7.23607	−0.398937
$330$	0	0
$331$	21.8885	1.20310	0.601552	−	0.798834i	$-0.294549\pi$
$331$	21.8885	1.20310	0.601552	+	0.798834i	$0.294549\pi$
$332$	−46.2492	−2.53826
$333$	0	0
$334$	28.9443	1.58376
$335$	11.0557	0.604039
$336$	0	0
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	−25.6525	−1.39531
$339$	0	0
$340$	−7.41641	−0.402211
$341$	−7.23607	−0.391855
$342$	0	0
$343$	1.00000	0.0539949
$344$	17.8885	0.964486
$345$	0	0
$346$	38.5410	2.07198
$347$	−3.05573	−0.164040	−0.0820200	−	0.996631i	$-0.526137\pi$
$347$	−3.05573	−0.164040	−0.0820200	+	0.996631i	$0.526137\pi$
$348$	0	0
$349$	−2.76393	−0.147950	−0.0739749	−	0.997260i	$-0.523568\pi$
$349$	−2.76393	−0.147950	−0.0739749	+	0.997260i	$0.523568\pi$
$350$	−2.23607	−0.119523
$351$	0	0
$352$	−6.70820	−0.357548
$353$	15.8885	0.845662	0.422831	−	0.906209i	$-0.361036\pi$
$353$	15.8885	0.845662	0.422831	+	0.906209i	$0.361036\pi$
$354$	0	0
$355$	3.05573	0.162181
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−20.0000	−1.05703
$359$	7.05573	0.372387	0.186194	−	0.982513i	$-0.440385\pi$
$359$	7.05573	0.372387	0.186194	+	0.982513i	$0.440385\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	3.16718	0.166464
$363$	0	0
$364$	−3.70820	−0.194363
$365$	−10.4721	−0.548137
$366$	0	0
$367$	17.1246	0.893897	0.446949	−	0.894560i	$-0.352511\pi$
$367$	17.1246	0.893897	0.446949	+	0.894560i	$0.352511\pi$
$368$	−6.47214	−0.337383
$369$	0	0
$370$	2.11146	0.109769
$371$	−8.47214	−0.439851
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−2.76393	−0.142920
$375$	0	0
$376$	−16.1803	−0.834437
$377$	−0.583592	−0.0300565
$378$	0	0
$379$	−25.3050	−1.29983	−0.649914	−	0.760008i	$-0.725195\pi$
$379$	−25.3050	−1.29983	−0.649914	+	0.760008i	$0.725195\pi$
$380$	−14.8328	−0.760907
$381$	0	0
$382$	46.8328	2.39618
$383$	−26.6525	−1.36188	−0.680939	−	0.732340i	$-0.738428\pi$
$383$	−26.6525	−1.36188	−0.680939	+	0.732340i	$0.738428\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	−53.4164	−2.71882
$387$	0	0
$388$	−28.2492	−1.43414
$389$	19.8885	1.00839	0.504195	−	0.863590i	$-0.331789\pi$
$389$	19.8885	1.00839	0.504195	+	0.863590i	$0.331789\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	2.23607	0.112938
$393$	0	0
$394$	4.47214	0.225303
$395$	17.8885	0.900070
$396$	0	0
$397$	−0.111456	−0.00559383	−0.00279691	−	0.999996i	$-0.500890\pi$
$397$	−0.111456	−0.00559383	−0.00279691	+	0.999996i	$0.500890\pi$
$398$	45.1246	2.26189
$399$	0	0
$400$	1.00000	0.0500000
$401$	−5.05573	−0.252471	−0.126236	−	0.992000i	$-0.540290\pi$
$401$	−5.05573	−0.252471	−0.126236	+	0.992000i	$0.540290\pi$
$402$	0	0
$403$	8.94427	0.445546
$404$	27.7082	1.37853
$405$	0	0
$406$	1.05573	0.0523949
$407$	0.472136	0.0234029
$408$	0	0
$409$	−31.1246	−1.53901	−0.769507	−	0.638639i	$-0.779498\pi$
$409$	−31.1246	−1.53901	−0.769507	+	0.638639i	$0.779498\pi$
$410$	30.2492	1.49390
$411$	0	0
$412$	−17.1246	−0.843669
$413$	−3.23607	−0.159236
$414$	0	0
$415$	−30.8328	−1.51352
$416$	8.29180	0.406539
$417$	0	0
$418$	−5.52786	−0.270377
$419$	6.65248	0.324995	0.162497	−	0.986709i	$-0.448045\pi$
$419$	6.65248	0.324995	0.162497	+	0.986709i	$0.448045\pi$
$420$	0	0
$421$	−22.3607	−1.08979	−0.544896	−	0.838503i	$-0.683431\pi$
$421$	−22.3607	−1.08979	−0.544896	+	0.838503i	$0.683431\pi$
$422$	48.9443	2.38257
$423$	0	0
$424$	−18.9443	−0.920015
$425$	1.23607	0.0599581
$426$	0	0
$427$	−2.76393	−0.133756
$428$	12.0000	0.580042
$429$	0	0
$430$	35.7771	1.72532
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	0.472136	0.0226894	0.0113447	−	0.999936i	$-0.496389\pi$
$433$	0.472136	0.0226894	0.0113447	+	0.999936i	$0.496389\pi$
$434$	−16.1803	−0.776681
$435$	0	0
$436$	13.4164	0.642529
$437$	−16.0000	−0.765384
$438$	0	0
$439$	1.52786	0.0729210	0.0364605	−	0.999335i	$-0.488392\pi$
$439$	1.52786	0.0729210	0.0364605	+	0.999335i	$0.488392\pi$
$440$	4.47214	0.213201
$441$	0	0
$442$	3.41641	0.162502
$443$	7.05573	0.335228	0.167614	−	0.985853i	$-0.446394\pi$
$443$	7.05573	0.335228	0.167614	+	0.985853i	$0.446394\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	27.2361	1.28967
$447$	0	0
$448$	−13.0000	−0.614192
$449$	19.5279	0.921577	0.460788	−	0.887510i	$-0.347567\pi$
$449$	19.5279	0.921577	0.460788	+	0.887510i	$0.347567\pi$
$450$	0	0
$451$	6.76393	0.318501
$452$	−6.00000	−0.282216
$453$	0	0
$454$	66.8328	3.13662
$455$	−2.47214	−0.115896
$456$	0	0
$457$	24.8328	1.16163	0.580815	−	0.814036i	$-0.302734\pi$
$457$	24.8328	1.16163	0.580815	+	0.814036i	$0.302734\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	38.8328	1.81059
$461$	−10.1803	−0.474146	−0.237073	−	0.971492i	$-0.576188\pi$
$461$	−10.1803	−0.474146	−0.237073	+	0.971492i	$0.576188\pi$
$462$	0	0
$463$	−14.4721	−0.672577	−0.336289	−	0.941759i	$-0.609172\pi$
$463$	−14.4721	−0.672577	−0.336289	+	0.941759i	$0.609172\pi$
$464$	−0.472136	−0.0219184
$465$	0	0
$466$	38.9443	1.80406
$467$	34.0689	1.57652	0.788260	−	0.615342i	$-0.210982\pi$
$467$	34.0689	1.57652	0.788260	+	0.615342i	$0.210982\pi$
$468$	0	0
$469$	5.52786	0.255253
$470$	−32.3607	−1.49269
$471$	0	0
$472$	−7.23607	−0.333067
$473$	8.00000	0.367840
$474$	0	0
$475$	2.47214	0.113429
$476$	−3.70820	−0.169965
$477$	0	0
$478$	−57.8885	−2.64776
$479$	−22.4721	−1.02678	−0.513389	−	0.858156i	$-0.671610\pi$
$479$	−22.4721	−1.02678	−0.513389	+	0.858156i	$0.671610\pi$
$480$	0	0
$481$	−0.583592	−0.0266095
$482$	60.6525	2.76264
$483$	0	0
$484$	3.00000	0.136364
$485$	−18.8328	−0.855154
$486$	0	0
$487$	8.36068	0.378859	0.189429	−	0.981894i	$-0.439336\pi$
$487$	8.36068	0.378859	0.189429	+	0.981894i	$0.439336\pi$
$488$	−6.18034	−0.279771
$489$	0	0
$490$	4.47214	0.202031
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−0.583592	−0.0262837
$494$	6.83282	0.307423
$495$	0	0
$496$	7.23607	0.324909
$497$	1.52786	0.0685341
$498$	0	0
$499$	10.4721	0.468797	0.234399	−	0.972141i	$-0.424688\pi$
$499$	10.4721	0.468797	0.234399	+	0.972141i	$0.424688\pi$
$500$	−36.0000	−1.60997
$501$	0	0
$502$	−39.5967	−1.76729
$503$	3.41641	0.152330	0.0761650	−	0.997095i	$-0.475732\pi$
$503$	3.41641	0.152330	0.0761650	+	0.997095i	$0.475732\pi$
$504$	0	0
$505$	18.4721	0.821999
$506$	14.4721	0.643365
$507$	0	0
$508$	62.8328	2.78776
$509$	−31.5279	−1.39745	−0.698724	−	0.715391i	$-0.746248\pi$
$509$	−31.5279	−1.39745	−0.698724	+	0.715391i	$0.746248\pi$
$510$	0	0
$511$	−5.23607	−0.231630
$512$	11.1803	0.494106
$513$	0	0
$514$	13.4164	0.591772
$515$	−11.4164	−0.503067
$516$	0	0
$517$	−7.23607	−0.318242
$518$	1.05573	0.0463860
$519$	0	0
$520$	−5.52786	−0.242413
$521$	14.3607	0.629153	0.314576	−	0.949232i	$-0.398138\pi$
$521$	14.3607	0.629153	0.314576	+	0.949232i	$0.398138\pi$
$522$	0	0
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	41.6656	1.82017
$525$	0	0
$526$	0	0
$527$	8.94427	0.389619
$528$	0	0
$529$	18.8885	0.821241
$530$	−37.8885	−1.64577
$531$	0	0
$532$	−7.41641	−0.321542
$533$	−8.36068	−0.362141
$534$	0	0
$535$	8.00000	0.345870
$536$	12.3607	0.533900
$537$	0	0
$538$	30.0000	1.29339
$539$	1.00000	0.0430730
$540$	0	0
$541$	−32.8328	−1.41159	−0.705797	−	0.708415i	$-0.749411\pi$
$541$	−32.8328	−1.41159	−0.705797	+	0.708415i	$0.749411\pi$
$542$	−3.41641	−0.146747
$543$	0	0
$544$	8.29180	0.355508
$545$	8.94427	0.383131
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−22.5836	−0.964723
$549$	0	0
$550$	−2.23607	−0.0953463
$551$	−1.16718	−0.0497237
$552$	0	0
$553$	8.94427	0.380349
$554$	35.5279	1.50943
$555$	0	0
$556$	−31.4164	−1.33235
$557$	21.0557	0.892160	0.446080	−	0.894993i	$-0.352820\pi$
$557$	21.0557	0.892160	0.446080	+	0.894993i	$0.352820\pi$
$558$	0	0
$559$	−9.88854	−0.418241
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	27.8885	1.17641
$563$	−39.4164	−1.66120	−0.830602	−	0.556867i	$-0.812003\pi$
$563$	−39.4164	−1.66120	−0.830602	+	0.556867i	$0.812003\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	−13.1672	−0.553458
$567$	0	0
$568$	3.41641	0.143349
$569$	−16.4721	−0.690548	−0.345274	−	0.938502i	$-0.612214\pi$
$569$	−16.4721	−0.690548	−0.345274	+	0.938502i	$0.612214\pi$
$570$	0	0
$571$	−32.9443	−1.37867	−0.689337	−	0.724440i	$-0.742098\pi$
$571$	−32.9443	−1.37867	−0.689337	+	0.724440i	$0.742098\pi$
$572$	−3.70820	−0.155048
$573$	0	0
$574$	15.1246	0.631289
$575$	−6.47214	−0.269907
$576$	0	0
$577$	−28.4721	−1.18531	−0.592655	−	0.805456i	$-0.701920\pi$
$577$	−28.4721	−1.18531	−0.592655	+	0.805456i	$0.701920\pi$
$578$	−34.5967	−1.43903
$579$	0	0
$580$	2.83282	0.117626
$581$	−15.4164	−0.639580
$582$	0	0
$583$	−8.47214	−0.350880
$584$	−11.7082	−0.484489
$585$	0	0
$586$	−33.8197	−1.39708
$587$	13.1246	0.541711	0.270855	−	0.962620i	$-0.412693\pi$
$587$	13.1246	0.541711	0.270855	+	0.962620i	$0.412693\pi$
$588$	0	0
$589$	17.8885	0.737085
$590$	−14.4721	−0.595808
$591$	0	0
$592$	−0.472136	−0.0194047
$593$	32.2918	1.32607	0.663033	−	0.748591i	$-0.269269\pi$
$593$	32.2918	1.32607	0.663033	+	0.748591i	$0.269269\pi$
$594$	0	0
$595$	−2.47214	−0.101348
$596$	−42.0000	−1.72039
$597$	0	0
$598$	−17.8885	−0.731517
$599$	−3.41641	−0.139591	−0.0697953	−	0.997561i	$-0.522235\pi$
$599$	−3.41641	−0.139591	−0.0697953	+	0.997561i	$0.522235\pi$
$600$	0	0
$601$	3.12461	0.127456	0.0637278	−	0.997967i	$-0.479701\pi$
$601$	3.12461	0.127456	0.0637278	+	0.997967i	$0.479701\pi$
$602$	17.8885	0.729083
$603$	0	0
$604$	−26.8328	−1.09181
$605$	2.00000	0.0813116
$606$	0	0
$607$	−4.94427	−0.200682	−0.100341	−	0.994953i	$-0.531993\pi$
$607$	−4.94427	−0.200682	−0.100341	+	0.994953i	$0.531993\pi$
$608$	16.5836	0.672553
$609$	0	0
$610$	−12.3607	−0.500469
$611$	8.94427	0.361847
$612$	0	0
$613$	−47.3050	−1.91063	−0.955315	−	0.295591i	$-0.904483\pi$
$613$	−47.3050	−1.91063	−0.955315	+	0.295591i	$0.904483\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	2.23607	0.0900937
$617$	−33.4164	−1.34529	−0.672647	−	0.739964i	$-0.734843\pi$
$617$	−33.4164	−1.34529	−0.672647	+	0.739964i	$0.734843\pi$
$618$	0	0
$619$	−29.1246	−1.17062	−0.585308	−	0.810811i	$-0.699027\pi$
$619$	−29.1246	−1.17062	−0.585308	+	0.810811i	$0.699027\pi$
$620$	−43.4164	−1.74364
$621$	0	0
$622$	48.5410	1.94632
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−6.58359	−0.263133
$627$	0	0
$628$	−20.8328	−0.831320
$629$	−0.583592	−0.0232693
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	20.0000	0.795557
$633$	0	0
$634$	−31.3050	−1.24328
$635$	41.8885	1.66230
$636$	0	0
$637$	−1.23607	−0.0489748
$638$	1.05573	0.0417967
$639$	0	0
$640$	−31.3050	−1.23744
$641$	24.4721	0.966591	0.483296	−	0.875457i	$-0.339440\pi$
$641$	24.4721	0.966591	0.483296	+	0.875457i	$0.339440\pi$
$642$	0	0
$643$	−29.1246	−1.14856	−0.574281	−	0.818658i	$-0.694718\pi$
$643$	−29.1246	−1.14856	−0.574281	+	0.818658i	$0.694718\pi$
$644$	19.4164	0.765114
$645$	0	0
$646$	6.83282	0.268834
$647$	22.0689	0.867617	0.433809	−	0.901005i	$-0.357169\pi$
$647$	22.0689	0.867617	0.433809	+	0.901005i	$0.357169\pi$
$648$	0	0
$649$	−3.23607	−0.127027
$650$	2.76393	0.108410
$651$	0	0
$652$	70.2492	2.75117
$653$	−42.9443	−1.68054	−0.840270	−	0.542169i	$-0.817603\pi$
$653$	−42.9443	−1.68054	−0.840270	+	0.542169i	$0.817603\pi$
$654$	0	0
$655$	27.7771	1.08534
$656$	−6.76393	−0.264087
$657$	0	0
$658$	−16.1803	−0.630775
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	12.8328	0.499139	0.249569	−	0.968357i	$-0.419711\pi$
$661$	12.8328	0.499139	0.249569	+	0.968357i	$0.419711\pi$
$662$	48.9443	1.90227
$663$	0	0
$664$	−34.4721	−1.33778
$665$	−4.94427	−0.191731
$666$	0	0
$667$	3.05573	0.118318
$668$	38.8328	1.50249
$669$	0	0
$670$	24.7214	0.955069
$671$	−2.76393	−0.106700
$672$	0	0
$673$	5.41641	0.208787	0.104394	−	0.994536i	$-0.466710\pi$
$673$	5.41641	0.208787	0.104394	+	0.994536i	$0.466710\pi$
$674$	−45.7771	−1.76327
$675$	0	0
$676$	−34.4164	−1.32371
$677$	−3.70820	−0.142518	−0.0712589	−	0.997458i	$-0.522702\pi$
$677$	−3.70820	−0.142518	−0.0712589	+	0.997458i	$0.522702\pi$
$678$	0	0
$679$	−9.41641	−0.361369
$680$	−5.52786	−0.211984
$681$	0	0
$682$	−16.1803	−0.619577
$683$	−29.8885	−1.14365	−0.571827	−	0.820374i	$-0.693765\pi$
$683$	−29.8885	−1.14365	−0.571827	+	0.820374i	$0.693765\pi$
$684$	0	0
$685$	−15.0557	−0.575250
$686$	2.23607	0.0853735
$687$	0	0
$688$	−8.00000	−0.304997
$689$	10.4721	0.398957
$690$	0	0
$691$	−48.5410	−1.84659	−0.923294	−	0.384095i	$-0.874514\pi$
$691$	−48.5410	−1.84659	−0.923294	+	0.384095i	$0.874514\pi$
$692$	51.7082	1.96565
$693$	0	0
$694$	−6.83282	−0.259370
$695$	−20.9443	−0.794462
$696$	0	0
$697$	−8.36068	−0.316683
$698$	−6.18034	−0.233929
$699$	0	0
$700$	−3.00000	−0.113389
$701$	24.4721	0.924300	0.462150	−	0.886802i	$-0.347078\pi$
$701$	24.4721	0.924300	0.462150	+	0.886802i	$0.347078\pi$
$702$	0	0
$703$	−1.16718	−0.0440212
$704$	−13.0000	−0.489956
$705$	0	0
$706$	35.5279	1.33711
$707$	9.23607	0.347358
$708$	0	0
$709$	2.94427	0.110574	0.0552872	−	0.998470i	$-0.482393\pi$
$709$	2.94427	0.110574	0.0552872	+	0.998470i	$0.482393\pi$
$710$	6.83282	0.256431
$711$	0	0
$712$	−4.47214	−0.167600
$713$	−46.8328	−1.75390
$714$	0	0
$715$	−2.47214	−0.0924526
$716$	−26.8328	−1.00279
$717$	0	0
$718$	15.7771	0.588796
$719$	−33.4853	−1.24879	−0.624395	−	0.781108i	$-0.714655\pi$
$719$	−33.4853	−1.24879	−0.624395	+	0.781108i	$0.714655\pi$
$720$	0	0
$721$	−5.70820	−0.212585
$722$	−28.8197	−1.07256
$723$	0	0
$724$	4.24922	0.157921
$725$	−0.472136	−0.0175347
$726$	0	0
$727$	−51.0132	−1.89197	−0.945987	−	0.324206i	$-0.894903\pi$
$727$	−51.0132	−1.89197	−0.945987	+	0.324206i	$0.894903\pi$
$728$	−2.76393	−0.102438
$729$	0	0
$730$	−23.4164	−0.866680
$731$	−9.88854	−0.365741
$732$	0	0
$733$	13.2361	0.488885	0.244443	−	0.969664i	$-0.421395\pi$
$733$	13.2361	0.488885	0.244443	+	0.969664i	$0.421395\pi$
$734$	38.2918	1.41338
$735$	0	0
$736$	−43.4164	−1.60035
$737$	5.52786	0.203621
$738$	0	0
$739$	7.05573	0.259549	0.129775	−	0.991544i	$-0.458575\pi$
$739$	7.05573	0.259549	0.129775	+	0.991544i	$0.458575\pi$
$740$	2.83282	0.104136
$741$	0	0
$742$	−18.9443	−0.695466
$743$	33.8885	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$743$	33.8885	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$744$	0	0
$745$	−28.0000	−1.02584
$746$	13.4164	0.491210
$747$	0	0
$748$	−3.70820	−0.135585
$749$	4.00000	0.146157
$750$	0	0
$751$	38.4721	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$751$	38.4721	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$752$	7.23607	0.263872
$753$	0	0
$754$	−1.30495	−0.0475235
$755$	−17.8885	−0.651031
$756$	0	0
$757$	−19.8885	−0.722861	−0.361431	−	0.932399i	$-0.617712\pi$
$757$	−19.8885	−0.722861	−0.361431	+	0.932399i	$0.617712\pi$
$758$	−56.5836	−2.05521
$759$	0	0
$760$	−11.0557	−0.401033
$761$	−17.5967	−0.637882	−0.318941	−	0.947775i	$-0.603327\pi$
$761$	−17.5967	−0.637882	−0.318941	+	0.947775i	$0.603327\pi$
$762$	0	0
$763$	4.47214	0.161902
$764$	62.8328	2.27321
$765$	0	0
$766$	−59.5967	−2.15332
$767$	4.00000	0.144432
$768$	0	0
$769$	31.7082	1.14343	0.571714	−	0.820453i	$-0.306279\pi$
$769$	31.7082	1.14343	0.571714	+	0.820453i	$0.306279\pi$
$770$	4.47214	0.161165
$771$	0	0
$772$	−71.6656	−2.57930
$773$	−6.36068	−0.228778	−0.114389	−	0.993436i	$-0.536491\pi$
$773$	−6.36068	−0.228778	−0.114389	+	0.993436i	$0.536491\pi$
$774$	0	0
$775$	7.23607	0.259927
$776$	−21.0557	−0.755857
$777$	0	0
$778$	44.4721	1.59440
$779$	−16.7214	−0.599105
$780$	0	0
$781$	1.52786	0.0546713
$782$	−17.8885	−0.639693
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	−13.8885	−0.495703
$786$	0	0
$787$	−16.5836	−0.591141	−0.295571	−	0.955321i	$-0.595510\pi$
$787$	−16.5836	−0.591141	−0.295571	+	0.955321i	$0.595510\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	40.0000	1.42314
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	3.41641	0.121320
$794$	−0.249224	−0.00884461
$795$	0	0
$796$	60.5410	2.14582
$797$	−2.94427	−0.104291	−0.0521457	−	0.998639i	$-0.516606\pi$
$797$	−2.94427	−0.104291	−0.0521457	+	0.998639i	$0.516606\pi$
$798$	0	0
$799$	8.94427	0.316426
$800$	6.70820	0.237171
$801$	0	0
$802$	−11.3050	−0.399192
$803$	−5.23607	−0.184777
$804$	0	0
$805$	12.9443	0.456226
$806$	20.0000	0.704470
$807$	0	0
$808$	20.6525	0.726552
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0	0
$811$	18.8328	0.661310	0.330655	−	0.943752i	$-0.392730\pi$
$811$	18.8328	0.661310	0.330655	+	0.943752i	$0.392730\pi$
$812$	1.41641	0.0497062
$813$	0	0
$814$	1.05573	0.0370033
$815$	46.8328	1.64048
$816$	0	0
$817$	−19.7771	−0.691913
$818$	−69.5967	−2.43339
$819$	0	0
$820$	40.5836	1.41724
$821$	8.83282	0.308267	0.154134	−	0.988050i	$-0.450741\pi$
$821$	8.83282	0.308267	0.154134	+	0.988050i	$0.450741\pi$
$822$	0	0
$823$	−49.8885	−1.73901	−0.869503	−	0.493928i	$-0.835561\pi$
$823$	−49.8885	−1.73901	−0.869503	+	0.493928i	$0.835561\pi$
$824$	−12.7639	−0.444653
$825$	0	0
$826$	−7.23607	−0.251775
$827$	−4.94427	−0.171929	−0.0859646	−	0.996298i	$-0.527397\pi$
$827$	−4.94427	−0.171929	−0.0859646	+	0.996298i	$0.527397\pi$
$828$	0	0
$829$	−16.8328	−0.584628	−0.292314	−	0.956322i	$-0.594425\pi$
$829$	−16.8328	−0.584628	−0.292314	+	0.956322i	$0.594425\pi$
$830$	−68.9443	−2.39309
$831$	0	0
$832$	16.0689	0.557088
$833$	−1.23607	−0.0428272
$834$	0	0
$835$	25.8885	0.895910
$836$	−7.41641	−0.256502
$837$	0	0
$838$	14.8754	0.513862
$839$	14.0689	0.485712	0.242856	−	0.970062i	$-0.421916\pi$
$839$	14.0689	0.485712	0.242856	+	0.970062i	$0.421916\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	−50.0000	−1.72311
$843$	0	0
$844$	65.6656	2.26030
$845$	−22.9443	−0.789307
$846$	0	0
$847$	1.00000	0.0343604
$848$	8.47214	0.290934
$849$	0	0
$850$	2.76393	0.0948021
$851$	3.05573	0.104749
$852$	0	0
$853$	0.652476	0.0223403	0.0111702	−	0.999938i	$-0.496444\pi$
$853$	0.652476	0.0223403	0.0111702	+	0.999938i	$0.496444\pi$
$854$	−6.18034	−0.211487
$855$	0	0
$856$	8.94427	0.305709
$857$	−10.7639	−0.367689	−0.183844	−	0.982955i	$-0.558854\pi$
$857$	−10.7639	−0.367689	−0.183844	+	0.982955i	$0.558854\pi$
$858$	0	0
$859$	40.5410	1.38324	0.691621	−	0.722261i	$-0.256897\pi$
$859$	40.5410	1.38324	0.691621	+	0.722261i	$0.256897\pi$
$860$	48.0000	1.63679
$861$	0	0
$862$	−26.8328	−0.913929
$863$	20.9443	0.712951	0.356476	−	0.934305i	$-0.383978\pi$
$863$	20.9443	0.712951	0.356476	+	0.934305i	$0.383978\pi$
$864$	0	0
$865$	34.4721	1.17209
$866$	1.05573	0.0358751
$867$	0	0
$868$	−21.7082	−0.736824
$869$	8.94427	0.303414
$870$	0	0
$871$	−6.83282	−0.231521
$872$	10.0000	0.338643
$873$	0	0
$874$	−35.7771	−1.21018
$875$	−12.0000	−0.405674
$876$	0	0
$877$	41.4164	1.39853	0.699266	−	0.714861i	$-0.253510\pi$
$877$	41.4164	1.39853	0.699266	+	0.714861i	$0.253510\pi$
$878$	3.41641	0.115298
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	−29.4164	−0.991064	−0.495532	−	0.868590i	$-0.665027\pi$
$881$	−29.4164	−0.991064	−0.495532	+	0.868590i	$0.665027\pi$
$882$	0	0
$883$	8.94427	0.300999	0.150499	−	0.988610i	$-0.451912\pi$
$883$	8.94427	0.300999	0.150499	+	0.988610i	$0.451912\pi$
$884$	4.58359	0.154163
$885$	0	0
$886$	15.7771	0.530042
$887$	−40.3607	−1.35518	−0.677589	−	0.735440i	$-0.736975\pi$
$887$	−40.3607	−1.35518	−0.677589	+	0.735440i	$0.736975\pi$
$888$	0	0
$889$	20.9443	0.702448
$890$	−8.94427	−0.299813
$891$	0	0
$892$	36.5410	1.22348
$893$	17.8885	0.598617
$894$	0	0
$895$	−17.8885	−0.597948
$896$	−15.6525	−0.522913
$897$	0	0
$898$	43.6656	1.45714
$899$	−3.41641	−0.113944
$900$	0	0
$901$	10.4721	0.348877
$902$	15.1246	0.503594
$903$	0	0
$904$	−4.47214	−0.148741
$905$	2.83282	0.0941660
$906$	0	0
$907$	13.5279	0.449185	0.224593	−	0.974453i	$-0.427895\pi$
$907$	13.5279	0.449185	0.224593	+	0.974453i	$0.427895\pi$
$908$	89.6656	2.97566
$909$	0	0
$910$	−5.52786	−0.183247
$911$	−33.5279	−1.11083	−0.555414	−	0.831574i	$-0.687440\pi$
$911$	−33.5279	−1.11083	−0.555414	+	0.831574i	$0.687440\pi$
$912$	0	0
$913$	−15.4164	−0.510209
$914$	55.5279	1.83670
$915$	0	0
$916$	13.4164	0.443291
$917$	13.8885	0.458640
$918$	0	0
$919$	6.11146	0.201598	0.100799	−	0.994907i	$-0.467860\pi$
$919$	6.11146	0.201598	0.100799	+	0.994907i	$0.467860\pi$
$920$	28.9443	0.954264
$921$	0	0
$922$	−22.7639	−0.749690
$923$	−1.88854	−0.0621622
$924$	0	0
$925$	−0.472136	−0.0155237
$926$	−32.3607	−1.06344
$927$	0	0
$928$	−3.16718	−0.103968
$929$	−28.2492	−0.926827	−0.463413	−	0.886142i	$-0.653376\pi$
$929$	−28.2492	−0.926827	−0.463413	+	0.886142i	$0.653376\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	52.2492	1.71148
$933$	0	0
$934$	76.1803	2.49270
$935$	−2.47214	−0.0808475
$936$	0	0
$937$	20.6525	0.674687	0.337343	−	0.941382i	$-0.390472\pi$
$937$	20.6525	0.674687	0.337343	+	0.941382i	$0.390472\pi$
$938$	12.3607	0.403591
$939$	0	0
$940$	−43.4164	−1.41609
$941$	41.5967	1.35602	0.678008	−	0.735055i	$-0.262844\pi$
$941$	41.5967	1.35602	0.678008	+	0.735055i	$0.262844\pi$
$942$	0	0
$943$	43.7771	1.42558
$944$	3.23607	0.105325
$945$	0	0
$946$	17.8885	0.581607
$947$	58.8328	1.91181	0.955905	−	0.293677i	$-0.0948789\pi$
$947$	58.8328	1.91181	0.955905	+	0.293677i	$0.0948789\pi$
$948$	0	0
$949$	6.47214	0.210094
$950$	5.52786	0.179348
$951$	0	0
$952$	−2.76393	−0.0895796
$953$	−5.05573	−0.163771	−0.0818855	−	0.996642i	$-0.526094\pi$
$953$	−5.05573	−0.163771	−0.0818855	+	0.996642i	$0.526094\pi$
$954$	0	0
$955$	41.8885	1.35548
$956$	−77.6656	−2.51189
$957$	0	0
$958$	−50.2492	−1.62348
$959$	−7.52786	−0.243087
$960$	0	0
$961$	21.3607	0.689054
$962$	−1.30495	−0.0420733
$963$	0	0
$964$	81.3738	2.62087
$965$	−47.7771	−1.53800
$966$	0	0
$967$	21.8885	0.703888	0.351944	−	0.936021i	$-0.385521\pi$
$967$	21.8885	0.703888	0.351944	+	0.936021i	$0.385521\pi$
$968$	2.23607	0.0718699
$969$	0	0
$970$	−42.1115	−1.35212
$971$	29.1246	0.934653	0.467327	−	0.884085i	$-0.345217\pi$
$971$	29.1246	0.934653	0.467327	+	0.884085i	$0.345217\pi$
$972$	0	0
$973$	−10.4721	−0.335721
$974$	18.6950	0.599028
$975$	0	0
$976$	2.76393	0.0884713
$977$	−5.05573	−0.161747	−0.0808735	−	0.996724i	$-0.525771\pi$
$977$	−5.05573	−0.161747	−0.0808735	+	0.996724i	$0.525771\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	6.00000	0.191663
$981$	0	0
$982$	0	0
$983$	−44.1803	−1.40913	−0.704567	−	0.709637i	$-0.748859\pi$
$983$	−44.1803	−1.40913	−0.704567	+	0.709637i	$0.748859\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	−1.30495	−0.0415581
$987$	0	0
$988$	9.16718	0.291647
$989$	51.7771	1.64642
$990$	0	0
$991$	−26.2492	−0.833834	−0.416917	−	0.908945i	$-0.636889\pi$
$991$	−26.2492	−0.833834	−0.416917	+	0.908945i	$0.636889\pi$
$992$	48.5410	1.54118
$993$	0	0
$994$	3.41641	0.108362
$995$	40.3607	1.27952
$996$	0	0
$997$	32.6525	1.03411	0.517057	−	0.855951i	$-0.327027\pi$
$997$	32.6525	1.03411	0.517057	+	0.855951i	$0.327027\pi$
$998$	23.4164	0.741233
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.h.1.2		2
3.2	odd	2		77.2.a.d.1.1	✓	2
7.6	odd	2		4851.2.a.y.1.2		2
11.10	odd	2		7623.2.a.bl.1.1		2
12.11	even	2		1232.2.a.m.1.1		2
15.2	even	4		1925.2.b.h.1849.1		4
15.8	even	4		1925.2.b.h.1849.4		4
15.14	odd	2		1925.2.a.r.1.2		2
21.2	odd	6		539.2.e.i.67.2		4
21.5	even	6		539.2.e.j.67.2		4
21.11	odd	6		539.2.e.i.177.2		4
21.17	even	6		539.2.e.j.177.2		4
21.20	even	2		539.2.a.f.1.1		2
24.5	odd	2		4928.2.a.bm.1.1		2
24.11	even	2		4928.2.a.bv.1.2		2
33.2	even	10		847.2.f.b.323.1		4
33.5	odd	10		847.2.f.n.729.1		4
33.8	even	10		847.2.f.m.372.1		4
33.14	odd	10		847.2.f.a.372.1		4
33.17	even	10		847.2.f.b.729.1		4
33.20	odd	10		847.2.f.n.323.1		4
33.26	odd	10		847.2.f.a.148.1		4
33.29	even	10		847.2.f.m.148.1		4
33.32	even	2		847.2.a.f.1.2		2
84.83	odd	2		8624.2.a.ce.1.2		2
231.230	odd	2		5929.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.d.1.1	✓	2	3.2	odd	2
539.2.a.f.1.1		2	21.20	even	2
539.2.e.i.67.2		4	21.2	odd	6
539.2.e.i.177.2		4	21.11	odd	6
539.2.e.j.67.2		4	21.5	even	6
539.2.e.j.177.2		4	21.17	even	6
693.2.a.h.1.2		2	1.1	even	1	trivial
847.2.a.f.1.2		2	33.32	even	2
847.2.f.a.148.1		4	33.26	odd	10
847.2.f.a.372.1		4	33.14	odd	10
847.2.f.b.323.1		4	33.2	even	10
847.2.f.b.729.1		4	33.17	even	10
847.2.f.m.148.1		4	33.29	even	10
847.2.f.m.372.1		4	33.8	even	10
847.2.f.n.323.1		4	33.20	odd	10
847.2.f.n.729.1		4	33.5	odd	10
1232.2.a.m.1.1		2	12.11	even	2
1925.2.a.r.1.2		2	15.14	odd	2
1925.2.b.h.1849.1		4	15.2	even	4
1925.2.b.h.1849.4		4	15.8	even	4
4851.2.a.y.1.2		2	7.6	odd	2
4928.2.a.bm.1.1		2	24.5	odd	2
4928.2.a.bv.1.2		2	24.11	even	2
5929.2.a.m.1.2		2	231.230	odd	2
7623.2.a.bl.1.1		2	11.10	odd	2
8624.2.a.ce.1.2		2	84.83	odd	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q+2.23607 q^{2} +3.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +4.47214 q^{10} +1.00000 q^{11} -1.23607 q^{13} +2.23607 q^{14} -1.00000 q^{16} -1.23607 q^{17} -2.47214 q^{19} +6.00000 q^{20} +2.23607 q^{22} +6.47214 q^{23} -1.00000 q^{25} -2.76393 q^{26} +3.00000 q^{28} +0.472136 q^{29} -7.23607 q^{31} -6.70820 q^{32} -2.76393 q^{34} +2.00000 q^{35} +0.472136 q^{37} -5.52786 q^{38} +4.47214 q^{40} +6.76393 q^{41} +8.00000 q^{43} +3.00000 q^{44} +14.4721 q^{46} -7.23607 q^{47} +1.00000 q^{49} -2.23607 q^{50} -3.70820 q^{52} -8.47214 q^{53} +2.00000 q^{55} +2.23607 q^{56} +1.05573 q^{58} -3.23607 q^{59} -2.76393 q^{61} -16.1803 q^{62} -13.0000 q^{64} -2.47214 q^{65} +5.52786 q^{67} -3.70820 q^{68} +4.47214 q^{70} +1.52786 q^{71} -5.23607 q^{73} +1.05573 q^{74} -7.41641 q^{76} +1.00000 q^{77} +8.94427 q^{79} -2.00000 q^{80} +15.1246 q^{82} -15.4164 q^{83} -2.47214 q^{85} +17.8885 q^{86} +2.23607 q^{88} -2.00000 q^{89} -1.23607 q^{91} +19.4164 q^{92} -16.1803 q^{94} -4.94427 q^{95} -9.41641 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 4 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 6 * q^4 + 4 * q^5 + 2 * q^7	\( 2 q + 6 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 2 q^{16} + 2 q^{17} + 4 q^{19} + 12 q^{20} + 4 q^{23} - 2 q^{25} - 10 q^{26} + 6 q^{28} - 8 q^{29} - 10 q^{31} - 10 q^{34} + 4 q^{35} - 8 q^{37} - 20 q^{38} + 18 q^{41} + 16 q^{43} + 6 q^{44} + 20 q^{46} - 10 q^{47} + 2 q^{49} + 6 q^{52} - 8 q^{53} + 4 q^{55} + 20 q^{58} - 2 q^{59} - 10 q^{61} - 10 q^{62} - 26 q^{64} + 4 q^{65} + 20 q^{67} + 6 q^{68} + 12 q^{71} - 6 q^{73} + 20 q^{74} + 12 q^{76} + 2 q^{77} - 4 q^{80} - 10 q^{82} - 4 q^{83} + 4 q^{85} - 4 q^{89} + 2 q^{91} + 12 q^{92} - 10 q^{94} + 8 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 2 * q^16 + 2 * q^17 + 4 * q^19 + 12 * q^20 + 4 * q^23 - 2 * q^25 - 10 * q^26 + 6 * q^28 - 8 * q^29 - 10 * q^31 - 10 * q^34 + 4 * q^35 - 8 * q^37 - 20 * q^38 + 18 * q^41 + 16 * q^43 + 6 * q^44 + 20 * q^46 - 10 * q^47 + 2 * q^49 + 6 * q^52 - 8 * q^53 + 4 * q^55 + 20 * q^58 - 2 * q^59 - 10 * q^61 - 10 * q^62 - 26 * q^64 + 4 * q^65 + 20 * q^67 + 6 * q^68 + 12 * q^71 - 6 * q^73 + 20 * q^74 + 12 * q^76 + 2 * q^77 - 4 * q^80 - 10 * q^82 - 4 * q^83 + 4 * q^85 - 4 * q^89 + 2 * q^91 + 12 * q^92 - 10 * q^94 + 8 * q^95 + 8 * q^97

Introduction

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