LMFDB - Embedded newform 1071.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1071,2,Mod(1,1071)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1071, 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("1071.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1071 = 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1071.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:	$8.55197805648$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1071.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48119 q^{2} +4.15633 q^{4} -3.48119 q^{5} +1.00000 q^{7} -5.35026 q^{8} +O(q^{10})$	\(q-2.48119 q^{2} +4.15633 q^{4} -3.48119 q^{5} +1.00000 q^{7} -5.35026 q^{8} +8.63752 q^{10} +1.96239 q^{11} -2.67513 q^{13} -2.48119 q^{14} +4.96239 q^{16} +1.00000 q^{17} -1.86907 q^{19} -14.4690 q^{20} -4.86907 q^{22} +8.11871 q^{23} +7.11871 q^{25} +6.63752 q^{26} +4.15633 q^{28} +2.54420 q^{29} -6.24965 q^{31} -1.61213 q^{32} -2.48119 q^{34} -3.48119 q^{35} +1.70545 q^{37} +4.63752 q^{38} +18.6253 q^{40} -12.0254 q^{41} -1.57452 q^{43} +8.15633 q^{44} -20.1441 q^{46} -1.36248 q^{47} +1.00000 q^{49} -17.6629 q^{50} -11.1187 q^{52} +7.18172 q^{53} -6.83146 q^{55} -5.35026 q^{56} -6.31265 q^{58} -4.71274 q^{59} +13.9878 q^{61} +15.5066 q^{62} -5.92478 q^{64} +9.31265 q^{65} +7.89446 q^{67} +4.15633 q^{68} +8.63752 q^{70} -12.0811 q^{71} -0.836381 q^{73} -4.23155 q^{74} -7.76845 q^{76} +1.96239 q^{77} -12.2193 q^{79} -17.2750 q^{80} +29.8373 q^{82} -1.22425 q^{83} -3.48119 q^{85} +3.90668 q^{86} -10.4993 q^{88} -13.7381 q^{89} -2.67513 q^{91} +33.7440 q^{92} +3.38058 q^{94} +6.50659 q^{95} +4.88717 q^{97} -2.48119 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 2 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 2 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 3 q - 2 q^{2} + 2 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 10 q^{10} - 5 q^{11} - 3 q^{13} - 2 q^{14} + 4 q^{16} + 3 q^{17} - q^{19} - 12 q^{20} - 10 q^{22} + 3 q^{23} + 4 q^{26} + 2 q^{28} - 2 q^{29} - 2 q^{31} - 4 q^{32} - 2 q^{34} - 5 q^{35} - 2 q^{37} - 2 q^{38} + 14 q^{40} - 21 q^{41} + 7 q^{43} + 14 q^{44} - 24 q^{46} - 20 q^{47} + 3 q^{49} - 22 q^{50} - 12 q^{52} - 4 q^{53} - 5 q^{55} - 6 q^{56} + 2 q^{58} - 20 q^{59} + 16 q^{61} + 26 q^{62} + 4 q^{64} + 7 q^{65} + 4 q^{67} + 2 q^{68} + 10 q^{70} - 4 q^{71} - 24 q^{74} - 12 q^{76} - 5 q^{77} - 22 q^{79} - 20 q^{80} + 20 q^{82} - 2 q^{83} - 5 q^{85} + 18 q^{86} + 2 q^{88} - 32 q^{89} - 3 q^{91} + 38 q^{92} - 2 q^{94} - q^{95} - 18 q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 2 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 10 * q^10 - 5 * q^11 - 3 * q^13 - 2 * q^14 + 4 * q^16 + 3 * q^17 - q^19 - 12 * q^20 - 10 * q^22 + 3 * q^23 + 4 * q^26 + 2 * q^28 - 2 * q^29 - 2 * q^31 - 4 * q^32 - 2 * q^34 - 5 * q^35 - 2 * q^37 - 2 * q^38 + 14 * q^40 - 21 * q^41 + 7 * q^43 + 14 * q^44 - 24 * q^46 - 20 * q^47 + 3 * q^49 - 22 * q^50 - 12 * q^52 - 4 * q^53 - 5 * q^55 - 6 * q^56 + 2 * q^58 - 20 * q^59 + 16 * q^61 + 26 * q^62 + 4 * q^64 + 7 * q^65 + 4 * q^67 + 2 * q^68 + 10 * q^70 - 4 * q^71 - 24 * q^74 - 12 * q^76 - 5 * q^77 - 22 * q^79 - 20 * q^80 + 20 * q^82 - 2 * q^83 - 5 * q^85 + 18 * q^86 + 2 * q^88 - 32 * q^89 - 3 * q^91 + 38 * q^92 - 2 * q^94 - q^95 - 18 * q^97 - 2 * q^98

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 $

$2$	−2.48119	−1.75447	−0.877235	−	0.480062i	$-0.840614\pi$
$2$	−2.48119	−1.75447	−0.877235	+	0.480062i	$0.840614\pi$
$3$	0	0
$3$	0	0
$4$	4.15633	2.07816
$5$	−3.48119	−1.55684	−0.778419	−	0.627745i	$-0.783978\pi$
$5$	−3.48119	−1.55684	−0.778419	+	0.627745i	$0.783978\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−5.35026	−1.89160
$9$	0	0
$10$	8.63752	2.73142
$11$	1.96239	0.591682	0.295841	−	0.955237i	$-0.404400\pi$
$11$	1.96239	0.591682	0.295841	+	0.955237i	$0.404400\pi$
$12$	0	0
$13$	−2.67513	−0.741948	−0.370974	−	0.928643i	$-0.620976\pi$
$13$	−2.67513	−0.741948	−0.370974	+	0.928643i	$0.620976\pi$
$14$	−2.48119	−0.663127
$15$	0	0
$16$	4.96239	1.24060
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.86907	−0.428793	−0.214397	−	0.976747i	$-0.568779\pi$
$19$	−1.86907	−0.428793	−0.214397	+	0.976747i	$0.568779\pi$
$20$	−14.4690	−3.23536
$21$	0	0
$22$	−4.86907	−1.03809
$23$	8.11871	1.69287	0.846434	−	0.532493i	$-0.178745\pi$
$23$	8.11871	1.69287	0.846434	+	0.532493i	$0.178745\pi$
$24$	0	0
$25$	7.11871	1.42374
$26$	6.63752	1.30172
$27$	0	0
$28$	4.15633	0.785472
$29$	2.54420	0.472446	0.236223	−	0.971699i	$-0.424090\pi$
$29$	2.54420	0.472446	0.236223	+	0.971699i	$0.424090\pi$
$30$	0	0
$31$	−6.24965	−1.12247	−0.561235	−	0.827657i	$-0.689674\pi$
$31$	−6.24965	−1.12247	−0.561235	+	0.827657i	$0.689674\pi$
$32$	−1.61213	−0.284986
$33$	0	0
$34$	−2.48119	−0.425521
$35$	−3.48119	−0.588429
$36$	0	0
$37$	1.70545	0.280374	0.140187	−	0.990125i	$-0.455230\pi$
$37$	1.70545	0.280374	0.140187	+	0.990125i	$0.455230\pi$
$38$	4.63752	0.752305
$39$	0	0
$40$	18.6253	2.94492
$41$	−12.0254	−1.87805	−0.939025	−	0.343848i	$-0.888270\pi$
$41$	−12.0254	−1.87805	−0.939025	+	0.343848i	$0.888270\pi$
$42$	0	0
$43$	−1.57452	−0.240111	−0.120056	−	0.992767i	$-0.538307\pi$
$43$	−1.57452	−0.240111	−0.120056	+	0.992767i	$0.538307\pi$
$44$	8.15633	1.22961
$45$	0	0
$46$	−20.1441	−2.97009
$47$	−1.36248	−0.198738	−0.0993691	−	0.995051i	$-0.531682\pi$
$47$	−1.36248	−0.198738	−0.0993691	+	0.995051i	$0.531682\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−17.6629	−2.49791
$51$	0	0
$52$	−11.1187	−1.54189
$53$	7.18172	0.986485	0.493242	−	0.869892i	$-0.335812\pi$
$53$	7.18172	0.986485	0.493242	+	0.869892i	$0.335812\pi$
$54$	0	0
$55$	−6.83146	−0.921153
$56$	−5.35026	−0.714959
$57$	0	0
$58$	−6.31265	−0.828892
$59$	−4.71274	−0.613547	−0.306773	−	0.951783i	$-0.599249\pi$
$59$	−4.71274	−0.613547	−0.306773	+	0.951783i	$0.599249\pi$
$60$	0	0
$61$	13.9878	1.79095	0.895476	−	0.445110i	$-0.146835\pi$
$61$	13.9878	1.79095	0.895476	+	0.445110i	$0.146835\pi$
$62$	15.5066	1.96934
$63$	0	0
$64$	−5.92478	−0.740597
$65$	9.31265	1.15509
$66$	0	0
$67$	7.89446	0.964462	0.482231	−	0.876044i	$-0.339827\pi$
$67$	7.89446	0.964462	0.482231	+	0.876044i	$0.339827\pi$
$68$	4.15633	0.504028
$69$	0	0
$70$	8.63752	1.03238
$71$	−12.0811	−1.43376	−0.716882	−	0.697195i	$-0.754431\pi$
$71$	−12.0811	−1.43376	−0.716882	+	0.697195i	$0.754431\pi$
$72$	0	0
$73$	−0.836381	−0.0978909	−0.0489455	−	0.998801i	$-0.515586\pi$
$73$	−0.836381	−0.0978909	−0.0489455	+	0.998801i	$0.515586\pi$
$74$	−4.23155	−0.491907
$75$	0	0
$76$	−7.76845	−0.891103
$77$	1.96239	0.223635
$78$	0	0
$79$	−12.2193	−1.37478	−0.687391	−	0.726288i	$-0.741244\pi$
$79$	−12.2193	−1.37478	−0.687391	+	0.726288i	$0.741244\pi$
$80$	−17.2750	−1.93141
$81$	0	0
$82$	29.8373	3.29498
$83$	−1.22425	−0.134379	−0.0671897	−	0.997740i	$-0.521403\pi$
$83$	−1.22425	−0.134379	−0.0671897	+	0.997740i	$0.521403\pi$
$84$	0	0
$85$	−3.48119	−0.377589
$86$	3.90668	0.421268
$87$	0	0
$88$	−10.4993	−1.11923
$89$	−13.7381	−1.45624	−0.728120	−	0.685450i	$-0.759606\pi$
$89$	−13.7381	−1.45624	−0.728120	+	0.685450i	$0.759606\pi$
$90$	0	0
$91$	−2.67513	−0.280430
$92$	33.7440	3.51806
$93$	0	0
$94$	3.38058	0.348680
$95$	6.50659	0.667562
$96$	0	0
$97$	4.88717	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$97$	4.88717	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$98$	−2.48119	−0.250638
$99$	0	0
$100$	29.5877	2.95877
$101$	−2.71274	−0.269928	−0.134964	−	0.990851i	$-0.543092\pi$
$101$	−2.71274	−0.269928	−0.134964	+	0.990851i	$0.543092\pi$
$102$	0	0
$103$	−5.63752	−0.555481	−0.277741	−	0.960656i	$-0.589586\pi$
$103$	−5.63752	−0.555481	−0.277741	+	0.960656i	$0.589586\pi$
$104$	14.3127	1.40347
$105$	0	0
$106$	−17.8192	−1.73076
$107$	−16.1695	−1.56316	−0.781582	−	0.623802i	$-0.785587\pi$
$107$	−16.1695	−1.56316	−0.781582	+	0.623802i	$0.785587\pi$
$108$	0	0
$109$	−18.4568	−1.76784	−0.883918	−	0.467641i	$-0.845104\pi$
$109$	−18.4568	−1.76784	−0.883918	+	0.467641i	$0.845104\pi$
$110$	16.9502	1.61614
$111$	0	0
$112$	4.96239	0.468902
$113$	−15.0508	−1.41586	−0.707929	−	0.706283i	$-0.750371\pi$
$113$	−15.0508	−1.41586	−0.707929	+	0.706283i	$0.750371\pi$
$114$	0	0
$115$	−28.2628	−2.63552
$116$	10.5745	0.981819
$117$	0	0
$118$	11.6932	1.07645
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.14903	−0.649912
$122$	−34.7064	−3.14217
$123$	0	0
$124$	−25.9756	−2.33267
$125$	−7.37565	−0.659699
$126$	0	0
$127$	−16.3054	−1.44687	−0.723433	−	0.690394i	$-0.757437\pi$
$127$	−16.3054	−1.44687	−0.723433	+	0.690394i	$0.757437\pi$
$128$	17.9248	1.58434
$129$	0	0
$130$	−23.1065	−2.02657
$131$	−14.2424	−1.24436	−0.622180	−	0.782874i	$-0.713753\pi$
$131$	−14.2424	−1.24436	−0.622180	+	0.782874i	$0.713753\pi$
$132$	0	0
$133$	−1.86907	−0.162069
$134$	−19.5877	−1.69212
$135$	0	0
$136$	−5.35026	−0.458781
$137$	7.18172	0.613576	0.306788	−	0.951778i	$-0.400746\pi$
$137$	7.18172	0.613576	0.306788	+	0.951778i	$0.400746\pi$
$138$	0	0
$139$	8.77575	0.744349	0.372175	−	0.928163i	$-0.378612\pi$
$139$	8.77575	0.744349	0.372175	+	0.928163i	$0.378612\pi$
$140$	−14.4690	−1.22285
$141$	0	0
$142$	29.9756	2.51549
$143$	−5.24965	−0.438997
$144$	0	0
$145$	−8.85685	−0.735521
$146$	2.07522	0.171747
$147$	0	0
$148$	7.08840	0.582663
$149$	−0.574515	−0.0470661	−0.0235331	−	0.999723i	$-0.507492\pi$
$149$	−0.574515	−0.0470661	−0.0235331	+	0.999723i	$0.507492\pi$
$150$	0	0
$151$	16.8568	1.37179	0.685895	−	0.727700i	$-0.259411\pi$
$151$	16.8568	1.37179	0.685895	+	0.727700i	$0.259411\pi$
$152$	10.0000	0.811107
$153$	0	0
$154$	−4.86907	−0.392361
$155$	21.7562	1.74750
$156$	0	0
$157$	−21.7186	−1.73333	−0.866667	−	0.498886i	$-0.833742\pi$
$157$	−21.7186	−1.73333	−0.866667	+	0.498886i	$0.833742\pi$
$158$	30.3185	2.41201
$159$	0	0
$160$	5.61213	0.443678
$161$	8.11871	0.639844
$162$	0	0
$163$	0.312650	0.0244887	0.0122443	−	0.999925i	$-0.496102\pi$
$163$	0.312650	0.0244887	0.0122443	+	0.999925i	$0.496102\pi$
$164$	−49.9814	−3.90289
$165$	0	0
$166$	3.03761	0.235764
$167$	−18.8519	−1.45881	−0.729403	−	0.684084i	$-0.760202\pi$
$167$	−18.8519	−1.45881	−0.729403	+	0.684084i	$0.760202\pi$
$168$	0	0
$169$	−5.84367	−0.449513
$170$	8.63752	0.662468
$171$	0	0
$172$	−6.54420	−0.498990
$173$	10.1065	0.768383	0.384191	−	0.923254i	$-0.374480\pi$
$173$	10.1065	0.768383	0.384191	+	0.923254i	$0.374480\pi$
$174$	0	0
$175$	7.11871	0.538124
$176$	9.73813	0.734040
$177$	0	0
$178$	34.0870	2.55493
$179$	10.9624	0.819367	0.409684	−	0.912228i	$-0.365639\pi$
$179$	10.9624	0.819367	0.409684	+	0.912228i	$0.365639\pi$
$180$	0	0
$181$	22.7513	1.69109	0.845546	−	0.533903i	$-0.179275\pi$
$181$	22.7513	1.69109	0.845546	+	0.533903i	$0.179275\pi$
$182$	6.63752	0.492006
$183$	0	0
$184$	−43.4372	−3.20224
$185$	−5.93700	−0.436497
$186$	0	0
$187$	1.96239	0.143504
$188$	−5.66291	−0.413010
$189$	0	0
$190$	−16.1441	−1.17122
$191$	−7.40105	−0.535521	−0.267760	−	0.963486i	$-0.586284\pi$
$191$	−7.40105	−0.535521	−0.267760	+	0.963486i	$0.586284\pi$
$192$	0	0
$193$	−7.44358	−0.535801	−0.267900	−	0.963447i	$-0.586330\pi$
$193$	−7.44358	−0.535801	−0.267900	+	0.963447i	$0.586330\pi$
$194$	−12.1260	−0.870597
$195$	0	0
$196$	4.15633	0.296880
$197$	0.112834	0.00803910	0.00401955	−	0.999992i	$-0.498721\pi$
$197$	0.112834	0.00803910	0.00401955	+	0.999992i	$0.498721\pi$
$198$	0	0
$199$	−8.89938	−0.630861	−0.315430	−	0.948949i	$-0.602149\pi$
$199$	−8.89938	−0.630861	−0.315430	+	0.948949i	$0.602149\pi$
$200$	−38.0870	−2.69316
$201$	0	0
$202$	6.73084	0.473580
$203$	2.54420	0.178568
$204$	0	0
$205$	41.8627	2.92382
$206$	13.9878	0.974575
$207$	0	0
$208$	−13.2750	−0.920458
$209$	−3.66784	−0.253710
$210$	0	0
$211$	−24.6434	−1.69652	−0.848261	−	0.529579i	$-0.822350\pi$
$211$	−24.6434	−1.69652	−0.848261	+	0.529579i	$0.822350\pi$
$212$	29.8496	2.05008
$213$	0	0
$214$	40.1197	2.74252
$215$	5.48119	0.373814
$216$	0	0
$217$	−6.24965	−0.424254
$218$	45.7948	3.10162
$219$	0	0
$220$	−28.3938	−1.91431
$221$	−2.67513	−0.179949
$222$	0	0
$223$	−17.1890	−1.15106	−0.575531	−	0.817780i	$-0.695204\pi$
$223$	−17.1890	−1.15106	−0.575531	+	0.817780i	$0.695204\pi$
$224$	−1.61213	−0.107715
$225$	0	0
$226$	37.3439	2.48408
$227$	22.1368	1.46927	0.734636	−	0.678462i	$-0.237353\pi$
$227$	22.1368	1.46927	0.734636	+	0.678462i	$0.237353\pi$
$228$	0	0
$229$	−0.357556	−0.0236280	−0.0118140	−	0.999930i	$-0.503761\pi$
$229$	−0.357556	−0.0236280	−0.0118140	+	0.999930i	$0.503761\pi$
$230$	70.1255	4.62394
$231$	0	0
$232$	−13.6121	−0.893680
$233$	−11.2677	−0.738175	−0.369087	−	0.929395i	$-0.620330\pi$
$233$	−11.2677	−0.738175	−0.369087	+	0.929395i	$0.620330\pi$
$234$	0	0
$235$	4.74306	0.309403
$236$	−19.5877	−1.27505
$237$	0	0
$238$	−2.48119	−0.160832
$239$	23.6810	1.53180	0.765899	−	0.642961i	$-0.222294\pi$
$239$	23.6810	1.53180	0.765899	+	0.642961i	$0.222294\pi$
$240$	0	0
$241$	−2.70052	−0.173956	−0.0869780	−	0.996210i	$-0.527721\pi$
$241$	−2.70052	−0.173956	−0.0869780	+	0.996210i	$0.527721\pi$
$242$	17.7381	1.14025
$243$	0	0
$244$	58.1378	3.72189
$245$	−3.48119	−0.222405
$246$	0	0
$247$	5.00000	0.318142
$248$	33.4372	2.12327
$249$	0	0
$250$	18.3004	1.15742
$251$	18.6375	1.17639	0.588195	−	0.808719i	$-0.299839\pi$
$251$	18.6375	1.17639	0.588195	+	0.808719i	$0.299839\pi$
$252$	0	0
$253$	15.9321	1.00164
$254$	40.4568	2.53848
$255$	0	0
$256$	−32.6253	−2.03908
$257$	−12.7127	−0.792999	−0.396500	−	0.918035i	$-0.629775\pi$
$257$	−12.7127	−0.792999	−0.396500	+	0.918035i	$0.629775\pi$
$258$	0	0
$259$	1.70545	0.105971
$260$	38.7064	2.40047
$261$	0	0
$262$	35.3380	2.18319
$263$	6.11142	0.376846	0.188423	−	0.982088i	$-0.439662\pi$
$263$	6.11142	0.376846	0.188423	+	0.982088i	$0.439662\pi$
$264$	0	0
$265$	−25.0010	−1.53580
$266$	4.63752	0.284345
$267$	0	0
$268$	32.8119	2.00431
$269$	17.0992	1.04256	0.521278	−	0.853387i	$-0.325455\pi$
$269$	17.0992	1.04256	0.521278	+	0.853387i	$0.325455\pi$
$270$	0	0
$271$	−14.2062	−0.862962	−0.431481	−	0.902122i	$-0.642009\pi$
$271$	−14.2062	−0.862962	−0.431481	+	0.902122i	$0.642009\pi$
$272$	4.96239	0.300889
$273$	0	0
$274$	−17.8192	−1.07650
$275$	13.9697	0.842404
$276$	0	0
$277$	28.0362	1.68453	0.842266	−	0.539062i	$-0.181221\pi$
$277$	28.0362	1.68453	0.842266	+	0.539062i	$0.181221\pi$
$278$	−21.7743	−1.30594
$279$	0	0
$280$	18.6253	1.11307
$281$	−31.0640	−1.85312	−0.926560	−	0.376146i	$-0.877249\pi$
$281$	−31.0640	−1.85312	−0.926560	+	0.376146i	$0.877249\pi$
$282$	0	0
$283$	7.59991	0.451768	0.225884	−	0.974154i	$-0.427473\pi$
$283$	7.59991	0.451768	0.225884	+	0.974154i	$0.427473\pi$
$284$	−50.2130	−2.97959
$285$	0	0
$286$	13.0254	0.770208
$287$	−12.0254	−0.709836
$288$	0	0
$289$	1.00000	0.0588235
$290$	21.9756	1.29045
$291$	0	0
$292$	−3.47627	−0.203433
$293$	11.2144	0.655153	0.327576	−	0.944825i	$-0.393768\pi$
$293$	11.2144	0.655153	0.327576	+	0.944825i	$0.393768\pi$
$294$	0	0
$295$	16.4060	0.955193
$296$	−9.12459	−0.530356
$297$	0	0
$298$	1.42548	0.0825761
$299$	−21.7186	−1.25602
$300$	0	0
$301$	−1.57452	−0.0907536
$302$	−41.8251	−2.40677
$303$	0	0
$304$	−9.27504	−0.531960
$305$	−48.6942	−2.78822
$306$	0	0
$307$	−0.775746	−0.0442742	−0.0221371	−	0.999755i	$-0.507047\pi$
$307$	−0.775746	−0.0442742	−0.0221371	+	0.999755i	$0.507047\pi$
$308$	8.15633	0.464750
$309$	0	0
$310$	−53.9814	−3.06594
$311$	−13.7988	−0.782456	−0.391228	−	0.920294i	$-0.627950\pi$
$311$	−13.7988	−0.782456	−0.391228	+	0.920294i	$0.627950\pi$
$312$	0	0
$313$	17.8618	1.00961	0.504804	−	0.863234i	$-0.331565\pi$
$313$	17.8618	1.00961	0.504804	+	0.863234i	$0.331565\pi$
$314$	53.8881	3.04108
$315$	0	0
$316$	−50.7875	−2.85702
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	4.99271	0.279538
$320$	20.6253	1.15299
$321$	0	0
$322$	−20.1441	−1.12259
$323$	−1.86907	−0.103998
$324$	0	0
$325$	−19.0435	−1.05634
$326$	−0.775746	−0.0429646
$327$	0	0
$328$	64.3390	3.55253
$329$	−1.36248	−0.0751160
$330$	0	0
$331$	−13.4182	−0.737530	−0.368765	−	0.929523i	$-0.620219\pi$
$331$	−13.4182	−0.737530	−0.368765	+	0.929523i	$0.620219\pi$
$332$	−5.08840	−0.279262
$333$	0	0
$334$	46.7753	2.55943
$335$	−27.4821	−1.50151
$336$	0	0
$337$	28.3815	1.54604	0.773020	−	0.634381i	$-0.218745\pi$
$337$	28.3815	1.54604	0.773020	+	0.634381i	$0.218745\pi$
$338$	14.4993	0.788658
$339$	0	0
$340$	−14.4690	−0.784690
$341$	−12.2642	−0.664146
$342$	0	0
$343$	1.00000	0.0539949
$344$	8.42407	0.454195
$345$	0	0
$346$	−25.0762	−1.34810
$347$	−6.65562	−0.357292	−0.178646	−	0.983913i	$-0.557172\pi$
$347$	−6.65562	−0.357292	−0.178646	+	0.983913i	$0.557172\pi$
$348$	0	0
$349$	26.7054	1.42951	0.714755	−	0.699375i	$-0.246538\pi$
$349$	26.7054	1.42951	0.714755	+	0.699375i	$0.246538\pi$
$350$	−17.6629	−0.944122
$351$	0	0
$352$	−3.16362	−0.168621
$353$	−5.16125	−0.274706	−0.137353	−	0.990522i	$-0.543859\pi$
$353$	−5.16125	−0.274706	−0.137353	+	0.990522i	$0.543859\pi$
$354$	0	0
$355$	42.0567	2.23214
$356$	−57.1002	−3.02630
$357$	0	0
$358$	−27.1998	−1.43755
$359$	11.5940	0.611909	0.305955	−	0.952046i	$-0.401024\pi$
$359$	11.5940	0.611909	0.305955	+	0.952046i	$0.401024\pi$
$360$	0	0
$361$	−15.5066	−0.816136
$362$	−56.4504	−2.96697
$363$	0	0
$364$	−11.1187	−0.582779
$365$	2.91160	0.152400
$366$	0	0
$367$	23.8496	1.24494	0.622468	−	0.782645i	$-0.286130\pi$
$367$	23.8496	1.24494	0.622468	+	0.782645i	$0.286130\pi$
$368$	40.2882	2.10017
$369$	0	0
$370$	14.7308	0.765820
$371$	7.18172	0.372856
$372$	0	0
$373$	3.35614	0.173774	0.0868872	−	0.996218i	$-0.472308\pi$
$373$	3.35614	0.173774	0.0868872	+	0.996218i	$0.472308\pi$
$374$	−4.86907	−0.251773
$375$	0	0
$376$	7.28963	0.375934
$377$	−6.80606	−0.350530
$378$	0	0
$379$	16.7005	0.857848	0.428924	−	0.903341i	$-0.358893\pi$
$379$	16.7005	0.857848	0.428924	+	0.903341i	$0.358893\pi$
$380$	27.0435	1.38730
$381$	0	0
$382$	18.3634	0.939555
$383$	24.1500	1.23401	0.617003	−	0.786961i	$-0.288347\pi$
$383$	24.1500	1.23401	0.617003	+	0.786961i	$0.288347\pi$
$384$	0	0
$385$	−6.83146	−0.348163
$386$	18.4690	0.940046
$387$	0	0
$388$	20.3127	1.03122
$389$	−36.4993	−1.85059	−0.925294	−	0.379251i	$-0.876182\pi$
$389$	−36.4993	−1.85059	−0.925294	+	0.379251i	$0.876182\pi$
$390$	0	0
$391$	8.11871	0.410581
$392$	−5.35026	−0.270229
$393$	0	0
$394$	−0.279964	−0.0141044
$395$	42.5379	2.14031
$396$	0	0
$397$	2.45088	0.123006	0.0615030	−	0.998107i	$-0.480411\pi$
$397$	2.45088	0.123006	0.0615030	+	0.998107i	$0.480411\pi$
$398$	22.0811	1.10683
$399$	0	0
$400$	35.3258	1.76629
$401$	−19.2520	−0.961400	−0.480700	−	0.876885i	$-0.659617\pi$
$401$	−19.2520	−0.961400	−0.480700	+	0.876885i	$0.659617\pi$
$402$	0	0
$403$	16.7186	0.832814
$404$	−11.2750	−0.560954
$405$	0	0
$406$	−6.31265	−0.313292
$407$	3.34675	0.165892
$408$	0	0
$409$	−4.04983	−0.200251	−0.100126	−	0.994975i	$-0.531924\pi$
$409$	−4.04983	−0.200251	−0.100126	+	0.994975i	$0.531924\pi$
$410$	−103.870	−5.12975
$411$	0	0
$412$	−23.4314	−1.15438
$413$	−4.71274	−0.231899
$414$	0	0
$415$	4.26187	0.209207
$416$	4.31265	0.211445
$417$	0	0
$418$	9.10062	0.445126
$419$	31.7948	1.55328	0.776639	−	0.629946i	$-0.216923\pi$
$419$	31.7948	1.55328	0.776639	+	0.629946i	$0.216923\pi$
$420$	0	0
$421$	−8.89446	−0.433489	−0.216745	−	0.976228i	$-0.569544\pi$
$421$	−8.89446	−0.433489	−0.216745	+	0.976228i	$0.569544\pi$
$422$	61.1451	2.97650
$423$	0	0
$424$	−38.4241	−1.86604
$425$	7.11871	0.345308
$426$	0	0
$427$	13.9878	0.676916
$428$	−67.2057	−3.24851
$429$	0	0
$430$	−13.5999	−0.655846
$431$	16.1866	0.779683	0.389842	−	0.920882i	$-0.372530\pi$
$431$	16.1866	0.779683	0.389842	+	0.920882i	$0.372530\pi$
$432$	0	0
$433$	−22.0557	−1.05993	−0.529965	−	0.848020i	$-0.677795\pi$
$433$	−22.0557	−1.05993	−0.529965	+	0.848020i	$0.677795\pi$
$434$	15.5066	0.744340
$435$	0	0
$436$	−76.7123	−3.67385
$437$	−15.1744	−0.725891
$438$	0	0
$439$	14.3004	0.682522	0.341261	−	0.939969i	$-0.389146\pi$
$439$	14.3004	0.682522	0.341261	+	0.939969i	$0.389146\pi$
$440$	36.5501	1.74246
$441$	0	0
$442$	6.63752	0.315715
$443$	−32.9706	−1.56648	−0.783241	−	0.621718i	$-0.786435\pi$
$443$	−32.9706	−1.56648	−0.783241	+	0.621718i	$0.786435\pi$
$444$	0	0
$445$	47.8251	2.26713
$446$	42.6493	2.01950
$447$	0	0
$448$	−5.92478	−0.279919
$449$	1.43136	0.0675502	0.0337751	−	0.999429i	$-0.489247\pi$
$449$	1.43136	0.0675502	0.0337751	+	0.999429i	$0.489247\pi$
$450$	0	0
$451$	−23.5985	−1.11121
$452$	−62.5560	−2.94238
$453$	0	0
$454$	−54.9257	−2.57779
$455$	9.31265	0.436584
$456$	0	0
$457$	−40.4577	−1.89253	−0.946266	−	0.323389i	$-0.895178\pi$
$457$	−40.4577	−1.89253	−0.946266	+	0.323389i	$0.895178\pi$
$458$	0.887166	0.0414545
$459$	0	0
$460$	−117.469	−5.47704
$461$	−29.9779	−1.39621	−0.698106	−	0.715995i	$-0.745973\pi$
$461$	−29.9779	−1.39621	−0.698106	+	0.715995i	$0.745973\pi$
$462$	0	0
$463$	−20.7308	−0.963444	−0.481722	−	0.876324i	$-0.659988\pi$
$463$	−20.7308	−0.963444	−0.481722	+	0.876324i	$0.659988\pi$
$464$	12.6253	0.586115
$465$	0	0
$466$	27.9575	1.29510
$467$	−18.2882	−0.846278	−0.423139	−	0.906065i	$-0.639072\pi$
$467$	−18.2882	−0.846278	−0.423139	+	0.906065i	$0.639072\pi$
$468$	0	0
$469$	7.89446	0.364532
$470$	−11.7685	−0.542838
$471$	0	0
$472$	25.2144	1.16059
$473$	−3.08981	−0.142070
$474$	0	0
$475$	−13.3054	−0.610492
$476$	4.15633	0.190505
$477$	0	0
$478$	−58.7572	−2.68749
$479$	−15.8994	−0.726461	−0.363231	−	0.931699i	$-0.618326\pi$
$479$	−15.8994	−0.726461	−0.363231	+	0.931699i	$0.618326\pi$
$480$	0	0
$481$	−4.56230	−0.208023
$482$	6.70052	0.305200
$483$	0	0
$484$	−29.7137	−1.35062
$485$	−17.0132	−0.772528
$486$	0	0
$487$	39.6991	1.79894	0.899469	−	0.436984i	$-0.143953\pi$
$487$	39.6991	1.79894	0.899469	+	0.436984i	$0.143953\pi$
$488$	−74.8383	−3.38777
$489$	0	0
$490$	8.63752	0.390203
$491$	−16.9380	−0.764399	−0.382200	−	0.924080i	$-0.624833\pi$
$491$	−16.9380	−0.764399	−0.382200	+	0.924080i	$0.624833\pi$
$492$	0	0
$493$	2.54420	0.114585
$494$	−12.4060	−0.558171
$495$	0	0
$496$	−31.0132	−1.39253
$497$	−12.0811	−0.541912
$498$	0	0
$499$	14.1685	0.634271	0.317136	−	0.948380i	$-0.397279\pi$
$499$	14.1685	0.634271	0.317136	+	0.948380i	$0.397279\pi$
$500$	−30.6556	−1.37096
$501$	0	0
$502$	−46.2433	−2.06394
$503$	20.4337	0.911095	0.455548	−	0.890211i	$-0.349443\pi$
$503$	20.4337	0.911095	0.455548	+	0.890211i	$0.349443\pi$
$504$	0	0
$505$	9.44358	0.420234
$506$	−39.5306	−1.75735
$507$	0	0
$508$	−67.7704	−3.00682
$509$	−4.78560	−0.212118	−0.106059	−	0.994360i	$-0.533823\pi$
$509$	−4.78560	−0.212118	−0.106059	+	0.994360i	$0.533823\pi$
$510$	0	0
$511$	−0.836381	−0.0369993
$512$	45.1002	1.99316
$513$	0	0
$514$	31.5428	1.39129
$515$	19.6253	0.864794
$516$	0	0
$517$	−2.67372	−0.117590
$518$	−4.23155	−0.185924
$519$	0	0
$520$	−49.8251	−2.18498
$521$	31.8651	1.39604	0.698018	−	0.716081i	$-0.254066\pi$
$521$	31.8651	1.39604	0.698018	+	0.716081i	$0.254066\pi$
$522$	0	0
$523$	−41.2506	−1.80376	−0.901881	−	0.431984i	$-0.857814\pi$
$523$	−41.2506	−1.80376	−0.901881	+	0.431984i	$0.857814\pi$
$524$	−59.1958	−2.58598
$525$	0	0
$526$	−15.1636	−0.661165
$527$	−6.24965	−0.272239
$528$	0	0
$529$	42.9135	1.86580
$530$	62.0322	2.69451
$531$	0	0
$532$	−7.76845	−0.336805
$533$	32.1695	1.39342
$534$	0	0
$535$	56.2892	2.43359
$536$	−42.2374	−1.82438
$537$	0	0
$538$	−42.4264	−1.82913
$539$	1.96239	0.0845261
$540$	0	0
$541$	−14.0508	−0.604090	−0.302045	−	0.953294i	$-0.597669\pi$
$541$	−14.0508	−0.604090	−0.302045	+	0.953294i	$0.597669\pi$
$542$	35.2482	1.51404
$543$	0	0
$544$	−1.61213	−0.0691194
$545$	64.2516	2.75223
$546$	0	0
$547$	−32.8202	−1.40329	−0.701645	−	0.712527i	$-0.747551\pi$
$547$	−32.8202	−1.40329	−0.701645	+	0.712527i	$0.747551\pi$
$548$	29.8496	1.27511
$549$	0	0
$550$	−34.6615	−1.47797
$551$	−4.75528	−0.202582
$552$	0	0
$553$	−12.2193	−0.519619
$554$	−69.5633	−2.95546
$555$	0	0
$556$	36.4749	1.54688
$557$	−10.5320	−0.446254	−0.223127	−	0.974789i	$-0.571627\pi$
$557$	−10.5320	−0.446254	−0.223127	+	0.974789i	$0.571627\pi$
$558$	0	0
$559$	4.21203	0.178150
$560$	−17.2750	−0.730004
$561$	0	0
$562$	77.0757	3.25124
$563$	−12.7659	−0.538018	−0.269009	−	0.963138i	$-0.586696\pi$
$563$	−12.7659	−0.538018	−0.269009	+	0.963138i	$0.586696\pi$
$564$	0	0
$565$	52.3947	2.20426
$566$	−18.8568	−0.792612
$567$	0	0
$568$	64.6371	2.71211
$569$	−17.8134	−0.746775	−0.373387	−	0.927676i	$-0.621804\pi$
$569$	−17.8134	−0.746775	−0.373387	+	0.927676i	$0.621804\pi$
$570$	0	0
$571$	−2.23743	−0.0936334	−0.0468167	−	0.998903i	$-0.514908\pi$
$571$	−2.23743	−0.0936334	−0.0468167	+	0.998903i	$0.514908\pi$
$572$	−21.8192	−0.912308
$573$	0	0
$574$	29.8373	1.24539
$575$	57.7948	2.41021
$576$	0	0
$577$	−31.7127	−1.32022	−0.660109	−	0.751170i	$-0.729490\pi$
$577$	−31.7127	−1.32022	−0.660109	+	0.751170i	$0.729490\pi$
$578$	−2.48119	−0.103204
$579$	0	0
$580$	−36.8119	−1.52853
$581$	−1.22425	−0.0507906
$582$	0	0
$583$	14.0933	0.583686
$584$	4.47486	0.185171
$585$	0	0
$586$	−27.8251	−1.14944
$587$	3.10062	0.127976	0.0639880	−	0.997951i	$-0.479618\pi$
$587$	3.10062	0.127976	0.0639880	+	0.997951i	$0.479618\pi$
$588$	0	0
$589$	11.6810	0.481308
$590$	−40.7064	−1.67586
$591$	0	0
$592$	8.46310	0.347831
$593$	−13.0860	−0.537379	−0.268689	−	0.963227i	$-0.586591\pi$
$593$	−13.0860	−0.537379	−0.268689	+	0.963227i	$0.586591\pi$
$594$	0	0
$595$	−3.48119	−0.142715
$596$	−2.38787	−0.0978111
$597$	0	0
$598$	53.8881	2.20365
$599$	4.85097	0.198205	0.0991026	−	0.995077i	$-0.468403\pi$
$599$	4.85097	0.198205	0.0991026	+	0.995077i	$0.468403\pi$
$600$	0	0
$601$	−38.3366	−1.56378	−0.781892	−	0.623414i	$-0.785745\pi$
$601$	−38.3366	−1.56378	−0.781892	+	0.623414i	$0.785745\pi$
$602$	3.90668	0.159224
$603$	0	0
$604$	70.0625	2.85080
$605$	24.8872	1.01181
$606$	0	0
$607$	−7.22662	−0.293320	−0.146660	−	0.989187i	$-0.546852\pi$
$607$	−7.22662	−0.293320	−0.146660	+	0.989187i	$0.546852\pi$
$608$	3.01317	0.122200
$609$	0	0
$610$	120.820	4.89185
$611$	3.64481	0.147453
$612$	0	0
$613$	6.13918	0.247959	0.123980	−	0.992285i	$-0.460434\pi$
$613$	6.13918	0.247959	0.123980	+	0.992285i	$0.460434\pi$
$614$	1.92478	0.0776777
$615$	0	0
$616$	−10.4993	−0.423029
$617$	35.3561	1.42338	0.711692	−	0.702491i	$-0.247929\pi$
$617$	35.3561	1.42338	0.711692	+	0.702491i	$0.247929\pi$
$618$	0	0
$619$	−15.5901	−0.626617	−0.313309	−	0.949651i	$-0.601437\pi$
$619$	−15.5901	−0.626617	−0.313309	+	0.949651i	$0.601437\pi$
$620$	90.4260	3.63159
$621$	0	0
$622$	34.2374	1.37280
$623$	−13.7381	−0.550407
$624$	0	0
$625$	−9.91748	−0.396699
$626$	−44.3185	−1.77132
$627$	0	0
$628$	−90.2697	−3.60215
$629$	1.70545	0.0680007
$630$	0	0
$631$	−25.7513	−1.02514	−0.512572	−	0.858644i	$-0.671307\pi$
$631$	−25.7513	−1.02514	−0.512572	+	0.858644i	$0.671307\pi$
$632$	65.3766	2.60054
$633$	0	0
$634$	−24.8119	−0.985408
$635$	56.7621	2.25254
$636$	0	0
$637$	−2.67513	−0.105993
$638$	−12.3879	−0.490441
$639$	0	0
$640$	−62.3996	−2.46656
$641$	2.87002	0.113359	0.0566795	−	0.998392i	$-0.481949\pi$
$641$	2.87002	0.113359	0.0566795	+	0.998392i	$0.481949\pi$
$642$	0	0
$643$	−31.0132	−1.22304	−0.611520	−	0.791229i	$-0.709442\pi$
$643$	−31.0132	−1.22304	−0.611520	+	0.791229i	$0.709442\pi$
$644$	33.7440	1.32970
$645$	0	0
$646$	4.63752	0.182461
$647$	−7.78892	−0.306214	−0.153107	−	0.988210i	$-0.548928\pi$
$647$	−7.78892	−0.306214	−0.153107	+	0.988210i	$0.548928\pi$
$648$	0	0
$649$	−9.24823	−0.363025
$650$	47.2506	1.85332
$651$	0	0
$652$	1.29948	0.0508914
$653$	24.2955	0.950757	0.475378	−	0.879781i	$-0.342311\pi$
$653$	24.2955	0.950757	0.475378	+	0.879781i	$0.342311\pi$
$654$	0	0
$655$	49.5804	1.93727
$656$	−59.6747	−2.32990
$657$	0	0
$658$	3.38058	0.131789
$659$	39.0167	1.51987	0.759937	−	0.649997i	$-0.225230\pi$
$659$	39.0167	1.51987	0.759937	+	0.649997i	$0.225230\pi$
$660$	0	0
$661$	−28.0313	−1.09029	−0.545145	−	0.838342i	$-0.683525\pi$
$661$	−28.0313	−1.09029	−0.545145	+	0.838342i	$0.683525\pi$
$662$	33.2931	1.29397
$663$	0	0
$664$	6.55008	0.254192
$665$	6.50659	0.252315
$666$	0	0
$667$	20.6556	0.799789
$668$	−78.3547	−3.03164
$669$	0	0
$670$	68.1886	2.63435
$671$	27.4495	1.05967
$672$	0	0
$673$	30.0870	1.15977	0.579884	−	0.814699i	$-0.303098\pi$
$673$	30.0870	1.15977	0.579884	+	0.814699i	$0.303098\pi$
$674$	−70.4201	−2.71248
$675$	0	0
$676$	−24.2882	−0.934162
$677$	11.9502	0.459282	0.229641	−	0.973275i	$-0.426245\pi$
$677$	11.9502	0.459282	0.229641	+	0.973275i	$0.426245\pi$
$678$	0	0
$679$	4.88717	0.187552
$680$	18.6253	0.714248
$681$	0	0
$682$	30.4299	1.16522
$683$	−0.825117	−0.0315722	−0.0157861	−	0.999875i	$-0.505025\pi$
$683$	−0.825117	−0.0315722	−0.0157861	+	0.999875i	$0.505025\pi$
$684$	0	0
$685$	−25.0010	−0.955237
$686$	−2.48119	−0.0947324
$687$	0	0
$688$	−7.81336	−0.297881
$689$	−19.2120	−0.731920
$690$	0	0
$691$	44.4528	1.69106	0.845532	−	0.533925i	$-0.179284\pi$
$691$	44.4528	1.69106	0.845532	+	0.533925i	$0.179284\pi$
$692$	42.0059	1.59682
$693$	0	0
$694$	16.5139	0.626858
$695$	−30.5501	−1.15883
$696$	0	0
$697$	−12.0254	−0.455494
$698$	−66.2614	−2.50803
$699$	0	0
$700$	29.5877	1.11831
$701$	−23.1998	−0.876245	−0.438122	−	0.898915i	$-0.644356\pi$
$701$	−23.1998	−0.876245	−0.438122	+	0.898915i	$0.644356\pi$
$702$	0	0
$703$	−3.18760	−0.120223
$704$	−11.6267	−0.438198
$705$	0	0
$706$	12.8061	0.481963
$707$	−2.71274	−0.102023
$708$	0	0
$709$	−16.2981	−0.612087	−0.306043	−	0.952018i	$-0.599005\pi$
$709$	−16.2981	−0.612087	−0.306043	+	0.952018i	$0.599005\pi$
$710$	−104.351	−3.91621
$711$	0	0
$712$	73.5026	2.75463
$713$	−50.7391	−1.90019
$714$	0	0
$715$	18.2750	0.683448
$716$	45.5633	1.70278
$717$	0	0
$718$	−28.7670	−1.07358
$719$	14.9067	0.555925	0.277963	−	0.960592i	$-0.410341\pi$
$719$	14.9067	0.555925	0.277963	+	0.960592i	$0.410341\pi$
$720$	0	0
$721$	−5.63752	−0.209952
$722$	38.4749	1.43189
$723$	0	0
$724$	94.5618	3.51436
$725$	18.1114	0.672641
$726$	0	0
$727$	45.6286	1.69227	0.846136	−	0.532967i	$-0.178923\pi$
$727$	45.6286	1.69227	0.846136	+	0.532967i	$0.178923\pi$
$728$	14.3127	0.530462
$729$	0	0
$730$	−7.22425	−0.267382
$731$	−1.57452	−0.0582356
$732$	0	0
$733$	33.9102	1.25250	0.626251	−	0.779622i	$-0.284589\pi$
$733$	33.9102	1.25250	0.626251	+	0.779622i	$0.284589\pi$
$734$	−59.1754	−2.18420
$735$	0	0
$736$	−13.0884	−0.482445
$737$	15.4920	0.570655
$738$	0	0
$739$	28.6239	1.05295	0.526473	−	0.850192i	$-0.323514\pi$
$739$	28.6239	1.05295	0.526473	+	0.850192i	$0.323514\pi$
$740$	−24.6761	−0.907111
$741$	0	0
$742$	−17.8192	−0.654165
$743$	15.1432	0.555548	0.277774	−	0.960646i	$-0.410403\pi$
$743$	15.1432	0.555548	0.277774	+	0.960646i	$0.410403\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	−8.32724	−0.304882
$747$	0	0
$748$	8.15633	0.298225
$749$	−16.1695	−0.590821
$750$	0	0
$751$	28.2193	1.02974	0.514869	−	0.857269i	$-0.327841\pi$
$751$	28.2193	1.02974	0.514869	+	0.857269i	$0.327841\pi$
$752$	−6.76116	−0.246554
$753$	0	0
$754$	16.8872	0.614994
$755$	−58.6820	−2.13566
$756$	0	0
$757$	−7.60957	−0.276575	−0.138287	−	0.990392i	$-0.544160\pi$
$757$	−7.60957	−0.276575	−0.138287	+	0.990392i	$0.544160\pi$
$758$	−41.4372	−1.50507
$759$	0	0
$760$	−34.8119	−1.26276
$761$	−23.6121	−0.855939	−0.427969	−	0.903793i	$-0.640771\pi$
$761$	−23.6121	−0.855939	−0.427969	+	0.903793i	$0.640771\pi$
$762$	0	0
$763$	−18.4568	−0.668179
$764$	−30.7612	−1.11290
$765$	0	0
$766$	−59.9208	−2.16503
$767$	12.6072	0.455220
$768$	0	0
$769$	16.0517	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$769$	16.0517	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$770$	16.9502	0.610842
$771$	0	0
$772$	−30.9380	−1.11348
$773$	22.7635	0.818747	0.409374	−	0.912367i	$-0.365747\pi$
$773$	22.7635	0.818747	0.409374	+	0.912367i	$0.365747\pi$
$774$	0	0
$775$	−44.4894	−1.59811
$776$	−26.1476	−0.938645
$777$	0	0
$778$	90.5618	3.24680
$779$	22.4763	0.805296
$780$	0	0
$781$	−23.7078	−0.848332
$782$	−20.1441	−0.720352
$783$	0	0
$784$	4.96239	0.177228
$785$	75.6067	2.69852
$786$	0	0
$787$	−11.2774	−0.401996	−0.200998	−	0.979592i	$-0.564419\pi$
$787$	−11.2774	−0.401996	−0.200998	+	0.979592i	$0.564419\pi$
$788$	0.468976	0.0167066
$789$	0	0
$790$	−105.545	−3.75511
$791$	−15.0508	−0.535144
$792$	0	0
$793$	−37.4191	−1.32879
$794$	−6.08110	−0.215810
$795$	0	0
$796$	−36.9887	−1.31103
$797$	−36.3752	−1.28848	−0.644238	−	0.764825i	$-0.722825\pi$
$797$	−36.3752	−1.28848	−0.644238	+	0.764825i	$0.722825\pi$
$798$	0	0
$799$	−1.36248	−0.0482011
$800$	−11.4763	−0.405747
$801$	0	0
$802$	47.7680	1.68675
$803$	−1.64130	−0.0579204
$804$	0	0
$805$	−28.2628	−0.996134
$806$	−41.4821	−1.46115
$807$	0	0
$808$	14.5139	0.510597
$809$	−2.62198	−0.0921838	−0.0460919	−	0.998937i	$-0.514677\pi$
$809$	−2.62198	−0.0921838	−0.0460919	+	0.998937i	$0.514677\pi$
$810$	0	0
$811$	−36.5599	−1.28379	−0.641896	−	0.766791i	$-0.721852\pi$
$811$	−36.5599	−1.28379	−0.641896	+	0.766791i	$0.721852\pi$
$812$	10.5745	0.371093
$813$	0	0
$814$	−8.30394	−0.291053
$815$	−1.08840	−0.0381249
$816$	0	0
$817$	2.94288	0.102958
$818$	10.0484	0.351335
$819$	0	0
$820$	173.995	6.07617
$821$	−49.3039	−1.72072	−0.860360	−	0.509687i	$-0.829761\pi$
$821$	−49.3039	−1.72072	−0.860360	+	0.509687i	$0.829761\pi$
$822$	0	0
$823$	5.46802	0.190603	0.0953016	−	0.995448i	$-0.469618\pi$
$823$	5.46802	0.190603	0.0953016	+	0.995448i	$0.469618\pi$
$824$	30.1622	1.05075
$825$	0	0
$826$	11.6932	0.406859
$827$	−3.67609	−0.127830	−0.0639150	−	0.997955i	$-0.520359\pi$
$827$	−3.67609	−0.127830	−0.0639150	+	0.997955i	$0.520359\pi$
$828$	0	0
$829$	6.99859	0.243071	0.121535	−	0.992587i	$-0.461218\pi$
$829$	6.99859	0.243071	0.121535	+	0.992587i	$0.461218\pi$
$830$	−10.5745	−0.367047
$831$	0	0
$832$	15.8496	0.549484
$833$	1.00000	0.0346479
$834$	0	0
$835$	65.6272	2.27112
$836$	−15.2447	−0.527250
$837$	0	0
$838$	−78.8891	−2.72518
$839$	−14.5247	−0.501448	−0.250724	−	0.968059i	$-0.580669\pi$
$839$	−14.5247	−0.501448	−0.250724	+	0.968059i	$0.580669\pi$
$840$	0	0
$841$	−22.5271	−0.776795
$842$	22.0689	0.760544
$843$	0	0
$844$	−102.426	−3.52565
$845$	20.3430	0.699819
$846$	0	0
$847$	−7.14903	−0.245644
$848$	35.6385	1.22383
$849$	0	0
$850$	−17.6629	−0.605833
$851$	13.8460	0.474636
$852$	0	0
$853$	4.04842	0.138615	0.0693076	−	0.997595i	$-0.477921\pi$
$853$	4.04842	0.138615	0.0693076	+	0.997595i	$0.477921\pi$
$854$	−34.7064	−1.18763
$855$	0	0
$856$	86.5111	2.95689
$857$	−29.9511	−1.02311	−0.511555	−	0.859250i	$-0.670931\pi$
$857$	−29.9511	−1.02311	−0.511555	+	0.859250i	$0.670931\pi$
$858$	0	0
$859$	2.07522	0.0708057	0.0354028	−	0.999373i	$-0.488729\pi$
$859$	2.07522	0.0708057	0.0354028	+	0.999373i	$0.488729\pi$
$860$	22.7816	0.776847
$861$	0	0
$862$	−40.1622	−1.36793
$863$	54.5682	1.85752	0.928761	−	0.370679i	$-0.120875\pi$
$863$	54.5682	1.85752	0.928761	+	0.370679i	$0.120875\pi$
$864$	0	0
$865$	−35.1827	−1.19625
$866$	54.7245	1.85961
$867$	0	0
$868$	−25.9756	−0.881668
$869$	−23.9791	−0.813434
$870$	0	0
$871$	−21.1187	−0.715580
$872$	98.7485	3.34405
$873$	0	0
$874$	37.6507	1.27355
$875$	−7.37565	−0.249343
$876$	0	0
$877$	−54.8300	−1.85148	−0.925739	−	0.378162i	$-0.876556\pi$
$877$	−54.8300	−1.85148	−0.925739	+	0.378162i	$0.876556\pi$
$878$	−35.4821	−1.19746
$879$	0	0
$880$	−33.9003	−1.14278
$881$	−5.50071	−0.185324	−0.0926618	−	0.995698i	$-0.529538\pi$
$881$	−5.50071	−0.185324	−0.0926618	+	0.995698i	$0.529538\pi$
$882$	0	0
$883$	39.8080	1.33964	0.669822	−	0.742521i	$-0.266370\pi$
$883$	39.8080	1.33964	0.669822	+	0.742521i	$0.266370\pi$
$884$	−11.1187	−0.373963
$885$	0	0
$886$	81.8066	2.74835
$887$	−1.81431	−0.0609187	−0.0304593	−	0.999536i	$-0.509697\pi$
$887$	−1.81431	−0.0609187	−0.0304593	+	0.999536i	$0.509697\pi$
$888$	0	0
$889$	−16.3054	−0.546864
$890$	−118.663	−3.97761
$891$	0	0
$892$	−71.4431	−2.39209
$893$	2.54657	0.0852176
$894$	0	0
$895$	−38.1622	−1.27562
$896$	17.9248	0.598825
$897$	0	0
$898$	−3.55149	−0.118515
$899$	−15.9003	−0.530306
$900$	0	0
$901$	7.18172	0.239258
$902$	58.5524	1.94958
$903$	0	0
$904$	80.5256	2.67824
$905$	−79.2017	−2.63275
$906$	0	0
$907$	−0.378024	−0.0125521	−0.00627604	−	0.999980i	$-0.501998\pi$
$907$	−0.378024	−0.0125521	−0.00627604	+	0.999980i	$0.501998\pi$
$908$	92.0078	3.05339
$909$	0	0
$910$	−23.1065	−0.765973
$911$	−29.2130	−0.967870	−0.483935	−	0.875104i	$-0.660793\pi$
$911$	−29.2130	−0.967870	−0.483935	+	0.875104i	$0.660793\pi$
$912$	0	0
$913$	−2.40246	−0.0795099
$914$	100.383	3.32039
$915$	0	0
$916$	−1.48612	−0.0491027
$917$	−14.2424	−0.470324
$918$	0	0
$919$	24.4617	0.806916	0.403458	−	0.914998i	$-0.367808\pi$
$919$	24.4617	0.806916	0.403458	+	0.914998i	$0.367808\pi$
$920$	151.213	4.98536
$921$	0	0
$922$	74.3811	2.44961
$923$	32.3185	1.06378
$924$	0	0
$925$	12.1406	0.399180
$926$	51.4372	1.69033
$927$	0	0
$928$	−4.10157	−0.134641
$929$	−41.7245	−1.36894	−0.684468	−	0.729043i	$-0.739966\pi$
$929$	−41.7245	−1.36894	−0.684468	+	0.729043i	$0.739966\pi$
$930$	0	0
$931$	−1.86907	−0.0612562
$932$	−46.8324	−1.53405
$933$	0	0
$934$	45.3766	1.48477
$935$	−6.83146	−0.223413
$936$	0	0
$937$	−11.0289	−0.360299	−0.180149	−	0.983639i	$-0.557658\pi$
$937$	−11.0289	−0.360299	−0.180149	+	0.983639i	$0.557658\pi$
$938$	−19.5877	−0.639561
$939$	0	0
$940$	19.7137	0.642990
$941$	39.0592	1.27329	0.636647	−	0.771155i	$-0.280321\pi$
$941$	39.0592	1.27329	0.636647	+	0.771155i	$0.280321\pi$
$942$	0	0
$943$	−97.6307	−3.17929
$944$	−23.3865	−0.761164
$945$	0	0
$946$	7.66642	0.249257
$947$	25.4821	0.828059	0.414029	−	0.910264i	$-0.364121\pi$
$947$	25.4821	0.828059	0.414029	+	0.910264i	$0.364121\pi$
$948$	0	0
$949$	2.23743	0.0726300
$950$	33.0132	1.07109
$951$	0	0
$952$	−5.35026	−0.173403
$953$	−7.54515	−0.244411	−0.122206	−	0.992505i	$-0.538997\pi$
$953$	−7.54515	−0.244411	−0.122206	+	0.992505i	$0.538997\pi$
$954$	0	0
$955$	25.7645	0.833719
$956$	98.4260	3.18332
$957$	0	0
$958$	39.4495	1.27455
$959$	7.18172	0.231910
$960$	0	0
$961$	8.05808	0.259938
$962$	11.3199	0.364970
$963$	0	0
$964$	−11.2243	−0.361509
$965$	25.9126	0.834155
$966$	0	0
$967$	−23.4953	−0.755559	−0.377779	−	0.925896i	$-0.623312\pi$
$967$	−23.4953	−0.755559	−0.377779	+	0.925896i	$0.623312\pi$
$968$	38.2492	1.22938
$969$	0	0
$970$	42.2130	1.35538
$971$	38.6155	1.23923	0.619614	−	0.784906i	$-0.287289\pi$
$971$	38.6155	1.23923	0.619614	+	0.784906i	$0.287289\pi$
$972$	0	0
$973$	8.77575	0.281338
$974$	−98.5012	−3.15618
$975$	0	0
$976$	69.4128	2.22185
$977$	38.4847	1.23123	0.615617	−	0.788045i	$-0.288907\pi$
$977$	38.4847	1.23123	0.615617	+	0.788045i	$0.288907\pi$
$978$	0	0
$979$	−26.9596	−0.861631
$980$	−14.4690	−0.462194
$981$	0	0
$982$	42.0263	1.34111
$983$	−11.1500	−0.355629	−0.177815	−	0.984064i	$-0.556903\pi$
$983$	−11.1500	−0.355629	−0.177815	+	0.984064i	$0.556903\pi$
$984$	0	0
$985$	−0.392798	−0.0125156
$986$	−6.31265	−0.201036
$987$	0	0
$988$	20.7816	0.661152
$989$	−12.7830	−0.406477
$990$	0	0
$991$	5.41564	0.172033	0.0860167	−	0.996294i	$-0.472586\pi$
$991$	5.41564	0.172033	0.0860167	+	0.996294i	$0.472586\pi$
$992$	10.0752	0.319889
$993$	0	0
$994$	29.9756	0.950767
$995$	30.9805	0.982147
$996$	0	0
$997$	−4.75131	−0.150475	−0.0752377	−	0.997166i	$-0.523972\pi$
$997$	−4.75131	−0.150475	−0.0752377	+	0.997166i	$0.523972\pi$
$998$	−35.1549	−1.11281
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1071.2.a.g.1.1	✓	3
3.2	odd	2		1071.2.a.i.1.3	yes	3
7.6	odd	2		7497.2.a.z.1.1		3
21.20	even	2		7497.2.a.bd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1071.2.a.g.1.1	✓	3	1.1	even	1	trivial
1071.2.a.i.1.3	yes	3	3.2	odd	2
7497.2.a.z.1.1		3	7.6	odd	2
7497.2.a.bd.1.3		3	21.20	even	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:	\( N \)	\(=\)	\( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1071.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48119 q^{2} +4.15633 q^{4} -3.48119 q^{5} +1.00000 q^{7} -5.35026 q^{8} +O(q^{10})\)	\(q-2.48119 q^{2} +4.15633 q^{4} -3.48119 q^{5} +1.00000 q^{7} -5.35026 q^{8} +8.63752 q^{10} +1.96239 q^{11} -2.67513 q^{13} -2.48119 q^{14} +4.96239 q^{16} +1.00000 q^{17} -1.86907 q^{19} -14.4690 q^{20} -4.86907 q^{22} +8.11871 q^{23} +7.11871 q^{25} +6.63752 q^{26} +4.15633 q^{28} +2.54420 q^{29} -6.24965 q^{31} -1.61213 q^{32} -2.48119 q^{34} -3.48119 q^{35} +1.70545 q^{37} +4.63752 q^{38} +18.6253 q^{40} -12.0254 q^{41} -1.57452 q^{43} +8.15633 q^{44} -20.1441 q^{46} -1.36248 q^{47} +1.00000 q^{49} -17.6629 q^{50} -11.1187 q^{52} +7.18172 q^{53} -6.83146 q^{55} -5.35026 q^{56} -6.31265 q^{58} -4.71274 q^{59} +13.9878 q^{61} +15.5066 q^{62} -5.92478 q^{64} +9.31265 q^{65} +7.89446 q^{67} +4.15633 q^{68} +8.63752 q^{70} -12.0811 q^{71} -0.836381 q^{73} -4.23155 q^{74} -7.76845 q^{76} +1.96239 q^{77} -12.2193 q^{79} -17.2750 q^{80} +29.8373 q^{82} -1.22425 q^{83} -3.48119 q^{85} +3.90668 q^{86} -10.4993 q^{88} -13.7381 q^{89} -2.67513 q^{91} +33.7440 q^{92} +3.38058 q^{94} +6.50659 q^{95} +4.88717 q^{97} -2.48119 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 2 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 2 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 3 q - 2 q^{2} + 2 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 10 q^{10} - 5 q^{11} - 3 q^{13} - 2 q^{14} + 4 q^{16} + 3 q^{17} - q^{19} - 12 q^{20} - 10 q^{22} + 3 q^{23} + 4 q^{26} + 2 q^{28} - 2 q^{29} - 2 q^{31} - 4 q^{32} - 2 q^{34} - 5 q^{35} - 2 q^{37} - 2 q^{38} + 14 q^{40} - 21 q^{41} + 7 q^{43} + 14 q^{44} - 24 q^{46} - 20 q^{47} + 3 q^{49} - 22 q^{50} - 12 q^{52} - 4 q^{53} - 5 q^{55} - 6 q^{56} + 2 q^{58} - 20 q^{59} + 16 q^{61} + 26 q^{62} + 4 q^{64} + 7 q^{65} + 4 q^{67} + 2 q^{68} + 10 q^{70} - 4 q^{71} - 24 q^{74} - 12 q^{76} - 5 q^{77} - 22 q^{79} - 20 q^{80} + 20 q^{82} - 2 q^{83} - 5 q^{85} + 18 q^{86} + 2 q^{88} - 32 q^{89} - 3 q^{91} + 38 q^{92} - 2 q^{94} - q^{95} - 18 q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 2 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 10 * q^10 - 5 * q^11 - 3 * q^13 - 2 * q^14 + 4 * q^16 + 3 * q^17 - q^19 - 12 * q^20 - 10 * q^22 + 3 * q^23 + 4 * q^26 + 2 * q^28 - 2 * q^29 - 2 * q^31 - 4 * q^32 - 2 * q^34 - 5 * q^35 - 2 * q^37 - 2 * q^38 + 14 * q^40 - 21 * q^41 + 7 * q^43 + 14 * q^44 - 24 * q^46 - 20 * q^47 + 3 * q^49 - 22 * q^50 - 12 * q^52 - 4 * q^53 - 5 * q^55 - 6 * q^56 + 2 * q^58 - 20 * q^59 + 16 * q^61 + 26 * q^62 + 4 * q^64 + 7 * q^65 + 4 * q^67 + 2 * q^68 + 10 * q^70 - 4 * q^71 - 24 * q^74 - 12 * q^76 - 5 * q^77 - 22 * q^79 - 20 * q^80 + 20 * q^82 - 2 * q^83 - 5 * q^85 + 18 * q^86 + 2 * q^88 - 32 * q^89 - 3 * q^91 + 38 * q^92 - 2 * q^94 - q^95 - 18 * q^97 - 2 * q^98

Introduction

L-functions

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