LMFDB - Embedded newform 4016.2.a.k.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.09599$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16074 q^{3} +4.05107 q^{5} +4.08218 q^{7} -1.65268 q^{9} +O(q^{10})$	$q+1.16074 q^{3} +4.05107 q^{5} +4.08218 q^{7} -1.65268 q^{9} -3.43883 q^{11} +3.30116 q^{13} +4.70223 q^{15} +3.47033 q^{17} -0.926379 q^{19} +4.73835 q^{21} -4.34706 q^{23} +11.4111 q^{25} -5.40055 q^{27} +4.51229 q^{29} -5.56786 q^{31} -3.99158 q^{33} +16.5372 q^{35} -8.98564 q^{37} +3.83178 q^{39} +3.94751 q^{41} +7.51193 q^{43} -6.69513 q^{45} +5.92466 q^{47} +9.66422 q^{49} +4.02815 q^{51} -1.64638 q^{53} -13.9309 q^{55} -1.07528 q^{57} +10.8485 q^{59} +13.3511 q^{61} -6.74656 q^{63} +13.3732 q^{65} +5.72636 q^{67} -5.04581 q^{69} -0.225693 q^{71} -15.9413 q^{73} +13.2454 q^{75} -14.0379 q^{77} -6.61569 q^{79} -1.31059 q^{81} -13.0813 q^{83} +14.0586 q^{85} +5.23759 q^{87} +8.08761 q^{89} +13.4759 q^{91} -6.46284 q^{93} -3.75282 q^{95} +13.1388 q^{97} +5.68329 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.16074	0.670153	0.335077	−	0.942191i	$-0.391238\pi$
$3$	1.16074	0.670153	0.335077	+	0.942191i	$0.391238\pi$
$4$	0	0
$5$	4.05107	1.81169	0.905846	−	0.423607i	$-0.139236\pi$
$5$	4.05107	1.81169	0.905846	+	0.423607i	$0.139236\pi$
$6$	0	0
$7$	4.08218	1.54292	0.771460	−	0.636278i	$-0.219527\pi$
$7$	4.08218	1.54292	0.771460	+	0.636278i	$0.219527\pi$
$8$	0	0
$9$	−1.65268	−0.550895
$10$	0	0
$11$	−3.43883	−1.03685	−0.518423	−	0.855125i	$-0.673481\pi$
$11$	−3.43883	−1.03685	−0.518423	+	0.855125i	$0.673481\pi$
$12$	0	0
$13$	3.30116	0.915576	0.457788	−	0.889061i	$-0.348642\pi$
$13$	3.30116	0.915576	0.457788	+	0.889061i	$0.348642\pi$
$14$	0	0
$15$	4.70223	1.21411
$16$	0	0
$17$	3.47033	0.841680	0.420840	−	0.907135i	$-0.361735\pi$
$17$	3.47033	0.841680	0.420840	+	0.907135i	$0.361735\pi$
$18$	0	0
$19$	−0.926379	−0.212526	−0.106263	−	0.994338i	$-0.533889\pi$
$19$	−0.926379	−0.212526	−0.106263	+	0.994338i	$0.533889\pi$
$20$	0	0
$21$	4.73835	1.03399
$22$	0	0
$23$	−4.34706	−0.906425	−0.453213	−	0.891402i	$-0.649722\pi$
$23$	−4.34706	−0.906425	−0.453213	+	0.891402i	$0.649722\pi$
$24$	0	0
$25$	11.4111	2.28223
$26$	0	0
$27$	−5.40055	−1.03934
$28$	0	0
$29$	4.51229	0.837911	0.418955	−	0.908007i	$-0.362396\pi$
$29$	4.51229	0.837911	0.418955	+	0.908007i	$0.362396\pi$
$30$	0	0
$31$	−5.56786	−1.00002	−0.500009	−	0.866020i	$-0.666670\pi$
$31$	−5.56786	−1.00002	−0.500009	+	0.866020i	$0.666670\pi$
$32$	0	0
$33$	−3.99158	−0.694845
$34$	0	0
$35$	16.5372	2.79530
$36$	0	0
$37$	−8.98564	−1.47723	−0.738615	−	0.674127i	$-0.764520\pi$
$37$	−8.98564	−1.47723	−0.738615	+	0.674127i	$0.764520\pi$
$38$	0	0
$39$	3.83178	0.613577
$40$	0	0
$41$	3.94751	0.616497	0.308249	−	0.951306i	$-0.400257\pi$
$41$	3.94751	0.616497	0.308249	+	0.951306i	$0.400257\pi$
$42$	0	0
$43$	7.51193	1.14556	0.572779	−	0.819710i	$-0.305865\pi$
$43$	7.51193	1.14556	0.572779	+	0.819710i	$0.305865\pi$
$44$	0	0
$45$	−6.69513	−0.998052
$46$	0	0
$47$	5.92466	0.864201	0.432100	−	0.901826i	$-0.357773\pi$
$47$	5.92466	0.864201	0.432100	+	0.901826i	$0.357773\pi$
$48$	0	0
$49$	9.66422	1.38060
$50$	0	0
$51$	4.02815	0.564054
$52$	0	0
$53$	−1.64638	−0.226148	−0.113074	−	0.993587i	$-0.536070\pi$
$53$	−1.64638	−0.226148	−0.113074	+	0.993587i	$0.536070\pi$
$54$	0	0
$55$	−13.9309	−1.87844
$56$	0	0
$57$	−1.07528	−0.142425
$58$	0	0
$59$	10.8485	1.41236	0.706179	−	0.708034i	$-0.250417\pi$
$59$	10.8485	1.41236	0.706179	+	0.708034i	$0.250417\pi$
$60$	0	0
$61$	13.3511	1.70944	0.854720	−	0.519090i	$-0.173729\pi$
$61$	13.3511	1.70944	0.854720	+	0.519090i	$0.173729\pi$
$62$	0	0
$63$	−6.74656	−0.849986
$64$	0	0
$65$	13.3732	1.65874
$66$	0	0
$67$	5.72636	0.699586	0.349793	−	0.936827i	$-0.386252\pi$
$67$	5.72636	0.699586	0.349793	+	0.936827i	$0.386252\pi$
$68$	0	0
$69$	−5.04581	−0.607444
$70$	0	0
$71$	−0.225693	−0.0267849	−0.0133924	−	0.999910i	$-0.504263\pi$
$71$	−0.225693	−0.0267849	−0.0133924	+	0.999910i	$0.504263\pi$
$72$	0	0
$73$	−15.9413	−1.86579	−0.932896	−	0.360145i	$-0.882727\pi$
$73$	−15.9413	−1.86579	−0.932896	+	0.360145i	$0.882727\pi$
$74$	0	0
$75$	13.2454	1.52944
$76$	0	0
$77$	−14.0379	−1.59977
$78$	0	0
$79$	−6.61569	−0.744323	−0.372162	−	0.928168i	$-0.621383\pi$
$79$	−6.61569	−0.744323	−0.372162	+	0.928168i	$0.621383\pi$
$80$	0	0
$81$	−1.31059	−0.145621
$82$	0	0
$83$	−13.0813	−1.43586	−0.717930	−	0.696116i	$-0.754910\pi$
$83$	−13.0813	−1.43586	−0.717930	+	0.696116i	$0.754910\pi$
$84$	0	0
$85$	14.0586	1.52486
$86$	0	0
$87$	5.23759	0.561529
$88$	0	0
$89$	8.08761	0.857285	0.428643	−	0.903474i	$-0.358992\pi$
$89$	8.08761	0.857285	0.428643	+	0.903474i	$0.358992\pi$
$90$	0	0
$91$	13.4759	1.41266
$92$	0	0
$93$	−6.46284	−0.670165
$94$	0	0
$95$	−3.75282	−0.385032
$96$	0	0
$97$	13.1388	1.33404	0.667020	−	0.745040i	$-0.267570\pi$
$97$	13.1388	1.33404	0.667020	+	0.745040i	$0.267570\pi$
$98$	0	0
$99$	5.68329	0.571192
$100$	0	0
$101$	−14.5864	−1.45140	−0.725702	−	0.688009i	$-0.758485\pi$
$101$	−14.5864	−1.45140	−0.725702	+	0.688009i	$0.758485\pi$
$102$	0	0
$103$	6.75547	0.665636	0.332818	−	0.942991i	$-0.392001\pi$
$103$	6.75547	0.665636	0.332818	+	0.942991i	$0.392001\pi$
$104$	0	0
$105$	19.1954	1.87328
$106$	0	0
$107$	1.59124	0.153831	0.0769156	−	0.997038i	$-0.475493\pi$
$107$	1.59124	0.153831	0.0769156	+	0.997038i	$0.475493\pi$
$108$	0	0
$109$	0.179880	0.0172294	0.00861469	−	0.999963i	$-0.497258\pi$
$109$	0.179880	0.0172294	0.00861469	+	0.999963i	$0.497258\pi$
$110$	0	0
$111$	−10.4300	−0.989971
$112$	0	0
$113$	−1.59051	−0.149623	−0.0748113	−	0.997198i	$-0.523835\pi$
$113$	−1.59051	−0.149623	−0.0748113	+	0.997198i	$0.523835\pi$
$114$	0	0
$115$	−17.6102	−1.64216
$116$	0	0
$117$	−5.45577	−0.504386
$118$	0	0
$119$	14.1665	1.29864
$120$	0	0
$121$	0.825527	0.0750479
$122$	0	0
$123$	4.58203	0.413148
$124$	0	0
$125$	25.9720	2.32301
$126$	0	0
$127$	10.9388	0.970664	0.485332	−	0.874330i	$-0.338699\pi$
$127$	10.9388	0.970664	0.485332	+	0.874330i	$0.338699\pi$
$128$	0	0
$129$	8.71939	0.767700
$130$	0	0
$131$	−6.19307	−0.541091	−0.270546	−	0.962707i	$-0.587204\pi$
$131$	−6.19307	−0.541091	−0.270546	+	0.962707i	$0.587204\pi$
$132$	0	0
$133$	−3.78165	−0.327911
$134$	0	0
$135$	−21.8780	−1.88296
$136$	0	0
$137$	−8.63041	−0.737346	−0.368673	−	0.929559i	$-0.620188\pi$
$137$	−8.63041	−0.737346	−0.368673	+	0.929559i	$0.620188\pi$
$138$	0	0
$139$	−11.7476	−0.996422	−0.498211	−	0.867056i	$-0.666009\pi$
$139$	−11.7476	−0.996422	−0.498211	+	0.867056i	$0.666009\pi$
$140$	0	0
$141$	6.87699	0.579147
$142$	0	0
$143$	−11.3521	−0.949311
$144$	0	0
$145$	18.2796	1.51804
$146$	0	0
$147$	11.2176	0.925216
$148$	0	0
$149$	−6.03119	−0.494094	−0.247047	−	0.969004i	$-0.579460\pi$
$149$	−6.03119	−0.494094	−0.247047	+	0.969004i	$0.579460\pi$
$150$	0	0
$151$	12.0074	0.977147	0.488574	−	0.872523i	$-0.337517\pi$
$151$	12.0074	0.977147	0.488574	+	0.872523i	$0.337517\pi$
$152$	0	0
$153$	−5.73537	−0.463677
$154$	0	0
$155$	−22.5558	−1.81172
$156$	0	0
$157$	3.00940	0.240176	0.120088	−	0.992763i	$-0.461682\pi$
$157$	3.00940	0.240176	0.120088	+	0.992763i	$0.461682\pi$
$158$	0	0
$159$	−1.91102	−0.151554
$160$	0	0
$161$	−17.7455	−1.39854
$162$	0	0
$163$	10.6635	0.835233	0.417617	−	0.908623i	$-0.362866\pi$
$163$	10.6635	0.835233	0.417617	+	0.908623i	$0.362866\pi$
$164$	0	0
$165$	−16.1702	−1.25885
$166$	0	0
$167$	−25.4807	−1.97175	−0.985877	−	0.167473i	$-0.946439\pi$
$167$	−25.4807	−1.97175	−0.985877	+	0.167473i	$0.946439\pi$
$168$	0	0
$169$	−2.10236	−0.161720
$170$	0	0
$171$	1.53101	0.117079
$172$	0	0
$173$	7.24408	0.550757	0.275378	−	0.961336i	$-0.411197\pi$
$173$	7.24408	0.550757	0.275378	+	0.961336i	$0.411197\pi$
$174$	0	0
$175$	46.5824	3.52130
$176$	0	0
$177$	12.5923	0.946496
$178$	0	0
$179$	−5.10289	−0.381408	−0.190704	−	0.981648i	$-0.561077\pi$
$179$	−5.10289	−0.381408	−0.190704	+	0.981648i	$0.561077\pi$
$180$	0	0
$181$	−22.2093	−1.65081	−0.825403	−	0.564543i	$-0.809052\pi$
$181$	−22.2093	−1.65081	−0.825403	+	0.564543i	$0.809052\pi$
$182$	0	0
$183$	15.4972	1.14559
$184$	0	0
$185$	−36.4014	−2.67629
$186$	0	0
$187$	−11.9339	−0.872692
$188$	0	0
$189$	−22.0461	−1.60361
$190$	0	0
$191$	4.46817	0.323305	0.161653	−	0.986848i	$-0.448318\pi$
$191$	4.46817	0.323305	0.161653	+	0.986848i	$0.448318\pi$
$192$	0	0
$193$	−10.1067	−0.727499	−0.363750	−	0.931497i	$-0.618504\pi$
$193$	−10.1067	−0.727499	−0.363750	+	0.931497i	$0.618504\pi$
$194$	0	0
$195$	15.5228	1.11161
$196$	0	0
$197$	−18.5467	−1.32140	−0.660698	−	0.750651i	$-0.729740\pi$
$197$	−18.5467	−1.32140	−0.660698	+	0.750651i	$0.729740\pi$
$198$	0	0
$199$	−0.698802	−0.0495367	−0.0247684	−	0.999693i	$-0.507885\pi$
$199$	−0.698802	−0.0495367	−0.0247684	+	0.999693i	$0.507885\pi$
$200$	0	0
$201$	6.64681	0.468830
$202$	0	0
$203$	18.4200	1.29283
$204$	0	0
$205$	15.9916	1.11690
$206$	0	0
$207$	7.18432	0.499345
$208$	0	0
$209$	3.18566	0.220356
$210$	0	0
$211$	−15.4389	−1.06286	−0.531429	−	0.847103i	$-0.678345\pi$
$211$	−15.4389	−1.06286	−0.531429	+	0.847103i	$0.678345\pi$
$212$	0	0
$213$	−0.261971	−0.0179500
$214$	0	0
$215$	30.4313	2.07540
$216$	0	0
$217$	−22.7290	−1.54295
$218$	0	0
$219$	−18.5037	−1.25037
$220$	0	0
$221$	11.4561	0.770622
$222$	0	0
$223$	−8.95349	−0.599570	−0.299785	−	0.954007i	$-0.596915\pi$
$223$	−8.95349	−0.599570	−0.299785	+	0.954007i	$0.596915\pi$
$224$	0	0
$225$	−18.8590	−1.25727
$226$	0	0
$227$	−8.82293	−0.585599	−0.292799	−	0.956174i	$-0.594587\pi$
$227$	−8.82293	−0.585599	−0.292799	+	0.956174i	$0.594587\pi$
$228$	0	0
$229$	−5.21374	−0.344534	−0.172267	−	0.985050i	$-0.555109\pi$
$229$	−5.21374	−0.344534	−0.172267	+	0.985050i	$0.555109\pi$
$230$	0	0
$231$	−16.2944	−1.07209
$232$	0	0
$233$	4.64734	0.304458	0.152229	−	0.988345i	$-0.451355\pi$
$233$	4.64734	0.304458	0.152229	+	0.988345i	$0.451355\pi$
$234$	0	0
$235$	24.0012	1.56567
$236$	0	0
$237$	−7.67909	−0.498811
$238$	0	0
$239$	−15.7121	−1.01633	−0.508165	−	0.861259i	$-0.669676\pi$
$239$	−15.7121	−1.01633	−0.508165	+	0.861259i	$0.669676\pi$
$240$	0	0
$241$	3.13692	0.202067	0.101033	−	0.994883i	$-0.467785\pi$
$241$	3.13692	0.202067	0.101033	+	0.994883i	$0.467785\pi$
$242$	0	0
$243$	14.6804	0.941749
$244$	0	0
$245$	39.1504	2.50123
$246$	0	0
$247$	−3.05812	−0.194584
$248$	0	0
$249$	−15.1840	−0.962246
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	14.9488	0.939822
$254$	0	0
$255$	16.3183	1.02189
$256$	0	0
$257$	−22.5920	−1.40925	−0.704626	−	0.709578i	$-0.748885\pi$
$257$	−22.5920	−1.40925	−0.704626	+	0.709578i	$0.748885\pi$
$258$	0	0
$259$	−36.6810	−2.27925
$260$	0	0
$261$	−7.45738	−0.461600
$262$	0	0
$263$	7.43775	0.458631	0.229316	−	0.973352i	$-0.426351\pi$
$263$	7.43775	0.458631	0.229316	+	0.973352i	$0.426351\pi$
$264$	0	0
$265$	−6.66960	−0.409710
$266$	0	0
$267$	9.38761	0.574513
$268$	0	0
$269$	3.44877	0.210275	0.105138	−	0.994458i	$-0.466472\pi$
$269$	3.44877	0.210275	0.105138	+	0.994458i	$0.466472\pi$
$270$	0	0
$271$	−24.6826	−1.49936	−0.749680	−	0.661800i	$-0.769793\pi$
$271$	−24.6826	−1.49936	−0.749680	+	0.661800i	$0.769793\pi$
$272$	0	0
$273$	15.6420	0.946700
$274$	0	0
$275$	−39.2410	−2.36632
$276$	0	0
$277$	29.0079	1.74291	0.871456	−	0.490473i	$-0.163176\pi$
$277$	29.0079	1.74291	0.871456	+	0.490473i	$0.163176\pi$
$278$	0	0
$279$	9.20191	0.550904
$280$	0	0
$281$	−8.46837	−0.505181	−0.252590	−	0.967573i	$-0.581283\pi$
$281$	−8.46837	−0.505181	−0.252590	+	0.967573i	$0.581283\pi$
$282$	0	0
$283$	3.02133	0.179599	0.0897997	−	0.995960i	$-0.471377\pi$
$283$	3.02133	0.179599	0.0897997	+	0.995960i	$0.471377\pi$
$284$	0	0
$285$	−4.35605	−0.258030
$286$	0	0
$287$	16.1145	0.951206
$288$	0	0
$289$	−4.95678	−0.291575
$290$	0	0
$291$	15.2507	0.894011
$292$	0	0
$293$	−17.7444	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$293$	−17.7444	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$294$	0	0
$295$	43.9481	2.55876
$296$	0	0
$297$	18.5716	1.07763
$298$	0	0
$299$	−14.3503	−0.829901
$300$	0	0
$301$	30.6651	1.76751
$302$	0	0
$303$	−16.9310	−0.972663
$304$	0	0
$305$	54.0864	3.09698
$306$	0	0
$307$	19.8629	1.13364	0.566819	−	0.823843i	$-0.308174\pi$
$307$	19.8629	1.13364	0.566819	+	0.823843i	$0.308174\pi$
$308$	0	0
$309$	7.84134	0.446078
$310$	0	0
$311$	12.1928	0.691393	0.345696	−	0.938346i	$-0.387643\pi$
$311$	12.1928	0.691393	0.345696	+	0.938346i	$0.387643\pi$
$312$	0	0
$313$	5.65841	0.319832	0.159916	−	0.987131i	$-0.448878\pi$
$313$	5.65841	0.319832	0.159916	+	0.987131i	$0.448878\pi$
$314$	0	0
$315$	−27.3308	−1.53991
$316$	0	0
$317$	−31.1746	−1.75094	−0.875471	−	0.483271i	$-0.839449\pi$
$317$	−31.1746	−1.75094	−0.875471	+	0.483271i	$0.839449\pi$
$318$	0	0
$319$	−15.5170	−0.868784
$320$	0	0
$321$	1.84702	0.103090
$322$	0	0
$323$	−3.21485	−0.178879
$324$	0	0
$325$	37.6700	2.08956
$326$	0	0
$327$	0.208794	0.0115463
$328$	0	0
$329$	24.1856	1.33339
$330$	0	0
$331$	−11.6232	−0.638868	−0.319434	−	0.947609i	$-0.603493\pi$
$331$	−11.6232	−0.638868	−0.319434	+	0.947609i	$0.603493\pi$
$332$	0	0
$333$	14.8504	0.813798
$334$	0	0
$335$	23.1979	1.26744
$336$	0	0
$337$	20.6209	1.12329	0.561647	−	0.827377i	$-0.310168\pi$
$337$	20.6209	1.12329	0.561647	+	0.827377i	$0.310168\pi$
$338$	0	0
$339$	−1.84617	−0.100270
$340$	0	0
$341$	19.1469	1.03686
$342$	0	0
$343$	10.8758	0.587240
$344$	0	0
$345$	−20.4409	−1.10050
$346$	0	0
$347$	16.4071	0.880781	0.440391	−	0.897806i	$-0.354840\pi$
$347$	16.4071	0.880781	0.440391	+	0.897806i	$0.354840\pi$
$348$	0	0
$349$	−24.5646	−1.31491	−0.657456	−	0.753493i	$-0.728367\pi$
$349$	−24.5646	−1.31491	−0.657456	+	0.753493i	$0.728367\pi$
$350$	0	0
$351$	−17.8281	−0.951593
$352$	0	0
$353$	17.6572	0.939798	0.469899	−	0.882720i	$-0.344290\pi$
$353$	17.6572	0.939798	0.469899	+	0.882720i	$0.344290\pi$
$354$	0	0
$355$	−0.914298	−0.0485259
$356$	0	0
$357$	16.4437	0.870291
$358$	0	0
$359$	18.0143	0.950758	0.475379	−	0.879781i	$-0.342311\pi$
$359$	18.0143	0.950758	0.475379	+	0.879781i	$0.342311\pi$
$360$	0	0
$361$	−18.1418	−0.954833
$362$	0	0
$363$	0.958221	0.0502936
$364$	0	0
$365$	−64.5794	−3.38024
$366$	0	0
$367$	31.2259	1.62998	0.814990	−	0.579476i	$-0.196743\pi$
$367$	31.2259	1.62998	0.814990	+	0.579476i	$0.196743\pi$
$368$	0	0
$369$	−6.52398	−0.339625
$370$	0	0
$371$	−6.72082	−0.348928
$372$	0	0
$373$	29.6683	1.53617	0.768083	−	0.640350i	$-0.221211\pi$
$373$	29.6683	1.53617	0.768083	+	0.640350i	$0.221211\pi$
$374$	0	0
$375$	30.1467	1.55677
$376$	0	0
$377$	14.8958	0.767171
$378$	0	0
$379$	14.6089	0.750407	0.375204	−	0.926942i	$-0.377573\pi$
$379$	14.6089	0.750407	0.375204	+	0.926942i	$0.377573\pi$
$380$	0	0
$381$	12.6971	0.650494
$382$	0	0
$383$	−5.23024	−0.267253	−0.133626	−	0.991032i	$-0.542662\pi$
$383$	−5.23024	−0.267253	−0.133626	+	0.991032i	$0.542662\pi$
$384$	0	0
$385$	−56.8686	−2.89829
$386$	0	0
$387$	−12.4148	−0.631082
$388$	0	0
$389$	12.2360	0.620388	0.310194	−	0.950673i	$-0.399606\pi$
$389$	12.2360	0.620388	0.310194	+	0.950673i	$0.399606\pi$
$390$	0	0
$391$	−15.0858	−0.762920
$392$	0	0
$393$	−7.18854	−0.362614
$394$	0	0
$395$	−26.8006	−1.34848
$396$	0	0
$397$	−7.37026	−0.369903	−0.184951	−	0.982748i	$-0.559213\pi$
$397$	−7.37026	−0.369903	−0.184951	+	0.982748i	$0.559213\pi$
$398$	0	0
$399$	−4.38951	−0.219750
$400$	0	0
$401$	−5.00535	−0.249955	−0.124978	−	0.992160i	$-0.539886\pi$
$401$	−5.00535	−0.249955	−0.124978	+	0.992160i	$0.539886\pi$
$402$	0	0
$403$	−18.3804	−0.915592
$404$	0	0
$405$	−5.30927	−0.263820
$406$	0	0
$407$	30.9001	1.53166
$408$	0	0
$409$	0.116239	0.00574765	0.00287382	−	0.999996i	$-0.499085\pi$
$409$	0.116239	0.00574765	0.00287382	+	0.999996i	$0.499085\pi$
$410$	0	0
$411$	−10.0177	−0.494135
$412$	0	0
$413$	44.2857	2.17916
$414$	0	0
$415$	−52.9932	−2.60134
$416$	0	0
$417$	−13.6359	−0.667755
$418$	0	0
$419$	28.4764	1.39116	0.695582	−	0.718447i	$-0.255147\pi$
$419$	28.4764	1.39116	0.695582	+	0.718447i	$0.255147\pi$
$420$	0	0
$421$	29.6456	1.44484	0.722418	−	0.691457i	$-0.243031\pi$
$421$	29.6456	1.44484	0.722418	+	0.691457i	$0.243031\pi$
$422$	0	0
$423$	−9.79159	−0.476084
$424$	0	0
$425$	39.6005	1.92091
$426$	0	0
$427$	54.5018	2.63753
$428$	0	0
$429$	−13.1768	−0.636184
$430$	0	0
$431$	5.16208	0.248649	0.124324	−	0.992242i	$-0.460324\pi$
$431$	5.16208	0.248649	0.124324	+	0.992242i	$0.460324\pi$
$432$	0	0
$433$	−3.30665	−0.158907	−0.0794536	−	0.996839i	$-0.525318\pi$
$433$	−3.30665	−0.158907	−0.0794536	+	0.996839i	$0.525318\pi$
$434$	0	0
$435$	21.2178	1.01732
$436$	0	0
$437$	4.02703	0.192639
$438$	0	0
$439$	7.84246	0.374300	0.187150	−	0.982331i	$-0.440075\pi$
$439$	7.84246	0.374300	0.187150	+	0.982331i	$0.440075\pi$
$440$	0	0
$441$	−15.9719	−0.760567
$442$	0	0
$443$	40.8864	1.94257	0.971286	−	0.237914i	$-0.0764635\pi$
$443$	40.8864	1.94257	0.971286	+	0.237914i	$0.0764635\pi$
$444$	0	0
$445$	32.7635	1.55314
$446$	0	0
$447$	−7.00064	−0.331119
$448$	0	0
$449$	8.93257	0.421554	0.210777	−	0.977534i	$-0.432401\pi$
$449$	8.93257	0.421554	0.210777	+	0.977534i	$0.432401\pi$
$450$	0	0
$451$	−13.5748	−0.639212
$452$	0	0
$453$	13.9375	0.654839
$454$	0	0
$455$	54.5919	2.55931
$456$	0	0
$457$	5.79253	0.270963	0.135481	−	0.990780i	$-0.456742\pi$
$457$	5.79253	0.270963	0.135481	+	0.990780i	$0.456742\pi$
$458$	0	0
$459$	−18.7417	−0.874789
$460$	0	0
$461$	−25.9384	−1.20807	−0.604036	−	0.796957i	$-0.706442\pi$
$461$	−25.9384	−1.20807	−0.604036	+	0.796957i	$0.706442\pi$
$462$	0	0
$463$	−31.9374	−1.48426	−0.742128	−	0.670258i	$-0.766184\pi$
$463$	−31.9374	−1.48426	−0.742128	+	0.670258i	$0.766184\pi$
$464$	0	0
$465$	−26.1814	−1.21413
$466$	0	0
$467$	9.12045	0.422044	0.211022	−	0.977481i	$-0.432321\pi$
$467$	9.12045	0.422044	0.211022	+	0.977481i	$0.432321\pi$
$468$	0	0
$469$	23.3761	1.07941
$470$	0	0
$471$	3.49313	0.160955
$472$	0	0
$473$	−25.8322	−1.18777
$474$	0	0
$475$	−10.5710	−0.485033
$476$	0	0
$477$	2.72095	0.124584
$478$	0	0
$479$	25.2680	1.15452	0.577262	−	0.816559i	$-0.304121\pi$
$479$	25.2680	1.15452	0.577262	+	0.816559i	$0.304121\pi$
$480$	0	0
$481$	−29.6630	−1.35252
$482$	0	0
$483$	−20.5979	−0.937237
$484$	0	0
$485$	53.2260	2.41687
$486$	0	0
$487$	0.546027	0.0247428	0.0123714	−	0.999923i	$-0.496062\pi$
$487$	0.546027	0.0247428	0.0123714	+	0.999923i	$0.496062\pi$
$488$	0	0
$489$	12.3776	0.559734
$490$	0	0
$491$	−16.1230	−0.727621	−0.363811	−	0.931473i	$-0.618525\pi$
$491$	−16.1230	−0.727621	−0.363811	+	0.931473i	$0.618525\pi$
$492$	0	0
$493$	15.6591	0.705252
$494$	0	0
$495$	23.0234	1.03482
$496$	0	0
$497$	−0.921321	−0.0413269
$498$	0	0
$499$	−3.89091	−0.174181	−0.0870905	−	0.996200i	$-0.527757\pi$
$499$	−3.89091	−0.174181	−0.0870905	+	0.996200i	$0.527757\pi$
$500$	0	0
$501$	−29.5764	−1.32138
$502$	0	0
$503$	15.5163	0.691837	0.345918	−	0.938265i	$-0.387567\pi$
$503$	15.5163	0.691837	0.345918	+	0.938265i	$0.387567\pi$
$504$	0	0
$505$	−59.0906	−2.62950
$506$	0	0
$507$	−2.44029	−0.108377
$508$	0	0
$509$	−38.3301	−1.69895	−0.849476	−	0.527627i	$-0.823082\pi$
$509$	−38.3301	−1.69895	−0.849476	+	0.527627i	$0.823082\pi$
$510$	0	0
$511$	−65.0755	−2.87877
$512$	0	0
$513$	5.00296	0.220886
$514$	0	0
$515$	27.3669	1.20593
$516$	0	0
$517$	−20.3739	−0.896042
$518$	0	0
$519$	8.40849	0.369092
$520$	0	0
$521$	2.09370	0.0917265	0.0458632	−	0.998948i	$-0.485396\pi$
$521$	2.09370	0.0917265	0.0458632	+	0.998948i	$0.485396\pi$
$522$	0	0
$523$	−6.52431	−0.285288	−0.142644	−	0.989774i	$-0.545560\pi$
$523$	−6.52431	−0.285288	−0.142644	+	0.989774i	$0.545560\pi$
$524$	0	0
$525$	54.0700	2.35981
$526$	0	0
$527$	−19.3223	−0.841694
$528$	0	0
$529$	−4.10305	−0.178394
$530$	0	0
$531$	−17.9292	−0.778060
$532$	0	0
$533$	13.0313	0.564450
$534$	0	0
$535$	6.44623	0.278695
$536$	0	0
$537$	−5.92312	−0.255602
$538$	0	0
$539$	−33.2336	−1.43147
$540$	0	0
$541$	26.7677	1.15083	0.575417	−	0.817860i	$-0.304840\pi$
$541$	26.7677	1.15083	0.575417	+	0.817860i	$0.304840\pi$
$542$	0	0
$543$	−25.7793	−1.10629
$544$	0	0
$545$	0.728706	0.0312144
$546$	0	0
$547$	2.53136	0.108233	0.0541166	−	0.998535i	$-0.482766\pi$
$547$	2.53136	0.108233	0.0541166	+	0.998535i	$0.482766\pi$
$548$	0	0
$549$	−22.0652	−0.941721
$550$	0	0
$551$	−4.18009	−0.178078
$552$	0	0
$553$	−27.0065	−1.14843
$554$	0	0
$555$	−42.2526	−1.79352
$556$	0	0
$557$	14.3378	0.607512	0.303756	−	0.952750i	$-0.401759\pi$
$557$	14.3378	0.607512	0.303756	+	0.952750i	$0.401759\pi$
$558$	0	0
$559$	24.7981	1.04885
$560$	0	0
$561$	−13.8521	−0.584837
$562$	0	0
$563$	19.9184	0.839462	0.419731	−	0.907649i	$-0.362124\pi$
$563$	19.9184	0.839462	0.419731	+	0.907649i	$0.362124\pi$
$564$	0	0
$565$	−6.44327	−0.271070
$566$	0	0
$567$	−5.35005	−0.224681
$568$	0	0
$569$	−11.7787	−0.493791	−0.246895	−	0.969042i	$-0.579410\pi$
$569$	−11.7787	−0.493791	−0.246895	+	0.969042i	$0.579410\pi$
$570$	0	0
$571$	5.75953	0.241029	0.120514	−	0.992712i	$-0.461546\pi$
$571$	5.75953	0.241029	0.120514	+	0.992712i	$0.461546\pi$
$572$	0	0
$573$	5.18638	0.216664
$574$	0	0
$575$	−49.6050	−2.06867
$576$	0	0
$577$	−25.0291	−1.04197	−0.520987	−	0.853565i	$-0.674436\pi$
$577$	−25.0291	−1.04197	−0.520987	+	0.853565i	$0.674436\pi$
$578$	0	0
$579$	−11.7313	−0.487536
$580$	0	0
$581$	−53.4003	−2.21542
$582$	0	0
$583$	5.66161	0.234480
$584$	0	0
$585$	−22.1017	−0.913792
$586$	0	0
$587$	−43.7992	−1.80779	−0.903894	−	0.427757i	$-0.859304\pi$
$587$	−43.7992	−1.80779	−0.903894	+	0.427757i	$0.859304\pi$
$588$	0	0
$589$	5.15795	0.212530
$590$	0	0
$591$	−21.5279	−0.885538
$592$	0	0
$593$	−23.3228	−0.957753	−0.478877	−	0.877882i	$-0.658956\pi$
$593$	−23.3228	−0.957753	−0.478877	+	0.877882i	$0.658956\pi$
$594$	0	0
$595$	57.3896	2.35275
$596$	0	0
$597$	−0.811127	−0.0331972
$598$	0	0
$599$	−28.4638	−1.16300	−0.581499	−	0.813547i	$-0.697534\pi$
$599$	−28.4638	−1.16300	−0.581499	+	0.813547i	$0.697534\pi$
$600$	0	0
$601$	25.0784	1.02297	0.511484	−	0.859293i	$-0.329096\pi$
$601$	25.0784	1.02297	0.511484	+	0.859293i	$0.329096\pi$
$602$	0	0
$603$	−9.46387	−0.385398
$604$	0	0
$605$	3.34426	0.135964
$606$	0	0
$607$	−31.6166	−1.28328	−0.641639	−	0.767007i	$-0.721745\pi$
$607$	−31.6166	−1.28328	−0.641639	+	0.767007i	$0.721745\pi$
$608$	0	0
$609$	21.3808	0.866394
$610$	0	0
$611$	19.5582	0.791242
$612$	0	0
$613$	−32.6768	−1.31980	−0.659902	−	0.751351i	$-0.729402\pi$
$613$	−32.6768	−1.31980	−0.659902	+	0.751351i	$0.729402\pi$
$614$	0	0
$615$	18.5621	0.748496
$616$	0	0
$617$	−8.72830	−0.351388	−0.175694	−	0.984445i	$-0.556217\pi$
$617$	−8.72830	−0.351388	−0.175694	+	0.984445i	$0.556217\pi$
$618$	0	0
$619$	16.5505	0.665222	0.332611	−	0.943064i	$-0.392070\pi$
$619$	16.5505	0.665222	0.332611	+	0.943064i	$0.392070\pi$
$620$	0	0
$621$	23.4765	0.942081
$622$	0	0
$623$	33.0151	1.32272
$624$	0	0
$625$	48.1585	1.92634
$626$	0	0
$627$	3.69772	0.147673
$628$	0	0
$629$	−31.1832	−1.24335
$630$	0	0
$631$	−12.1413	−0.483338	−0.241669	−	0.970359i	$-0.577695\pi$
$631$	−12.1413	−0.483338	−0.241669	+	0.970359i	$0.577695\pi$
$632$	0	0
$633$	−17.9206	−0.712278
$634$	0	0
$635$	44.3139	1.75854
$636$	0	0
$637$	31.9031	1.26405
$638$	0	0
$639$	0.373000	0.0147556
$640$	0	0
$641$	−35.7536	−1.41218	−0.706092	−	0.708120i	$-0.749544\pi$
$641$	−35.7536	−1.41218	−0.706092	+	0.708120i	$0.749544\pi$
$642$	0	0
$643$	39.1387	1.54348	0.771741	−	0.635938i	$-0.219386\pi$
$643$	39.1387	1.54348	0.771741	+	0.635938i	$0.219386\pi$
$644$	0	0
$645$	35.3229	1.39084
$646$	0	0
$647$	−3.99748	−0.157157	−0.0785785	−	0.996908i	$-0.525038\pi$
$647$	−3.99748	−0.157157	−0.0785785	+	0.996908i	$0.525038\pi$
$648$	0	0
$649$	−37.3062	−1.46440
$650$	0	0
$651$	−26.3825	−1.03401
$652$	0	0
$653$	−24.3079	−0.951243	−0.475621	−	0.879650i	$-0.657777\pi$
$653$	−24.3079	−0.951243	−0.475621	+	0.879650i	$0.657777\pi$
$654$	0	0
$655$	−25.0885	−0.980291
$656$	0	0
$657$	26.3460	1.02785
$658$	0	0
$659$	−37.2796	−1.45221	−0.726104	−	0.687585i	$-0.758671\pi$
$659$	−37.2796	−1.45221	−0.726104	+	0.687585i	$0.758671\pi$
$660$	0	0
$661$	−40.0665	−1.55841	−0.779203	−	0.626772i	$-0.784376\pi$
$661$	−40.0665	−1.55841	−0.779203	+	0.626772i	$0.784376\pi$
$662$	0	0
$663$	13.2976	0.516435
$664$	0	0
$665$	−15.3197	−0.594073
$666$	0	0
$667$	−19.6152	−0.759503
$668$	0	0
$669$	−10.3927	−0.401804
$670$	0	0
$671$	−45.9123	−1.77242
$672$	0	0
$673$	−15.4069	−0.593891	−0.296946	−	0.954894i	$-0.595968\pi$
$673$	−15.4069	−0.593891	−0.296946	+	0.954894i	$0.595968\pi$
$674$	0	0
$675$	−61.6265	−2.37201
$676$	0	0
$677$	39.6370	1.52338	0.761688	−	0.647944i	$-0.224371\pi$
$677$	39.6370	1.52338	0.761688	+	0.647944i	$0.224371\pi$
$678$	0	0
$679$	53.6348	2.05832
$680$	0	0
$681$	−10.2411	−0.392441
$682$	0	0
$683$	−33.1968	−1.27024	−0.635120	−	0.772413i	$-0.719049\pi$
$683$	−33.1968	−1.27024	−0.635120	+	0.772413i	$0.719049\pi$
$684$	0	0
$685$	−34.9624	−1.33584
$686$	0	0
$687$	−6.05180	−0.230891
$688$	0	0
$689$	−5.43496	−0.207055
$690$	0	0
$691$	20.9557	0.797194	0.398597	−	0.917126i	$-0.369497\pi$
$691$	20.9557	0.797194	0.398597	+	0.917126i	$0.369497\pi$
$692$	0	0
$693$	23.2002	0.881304
$694$	0	0
$695$	−47.5905	−1.80521
$696$	0	0
$697$	13.6992	0.518893
$698$	0	0
$699$	5.39435	0.204033
$700$	0	0
$701$	7.42397	0.280400	0.140200	−	0.990123i	$-0.455226\pi$
$701$	7.42397	0.280400	0.140200	+	0.990123i	$0.455226\pi$
$702$	0	0
$703$	8.32411	0.313950
$704$	0	0
$705$	27.8591	1.04924
$706$	0	0
$707$	−59.5445	−2.23940
$708$	0	0
$709$	4.81340	0.180771	0.0903855	−	0.995907i	$-0.471190\pi$
$709$	4.81340	0.180771	0.0903855	+	0.995907i	$0.471190\pi$
$710$	0	0
$711$	10.9336	0.410044
$712$	0	0
$713$	24.2038	0.906441
$714$	0	0
$715$	−45.9882	−1.71986
$716$	0	0
$717$	−18.2376	−0.681098
$718$	0	0
$719$	−20.3515	−0.758983	−0.379492	−	0.925195i	$-0.623901\pi$
$719$	−20.3515	−0.758983	−0.379492	+	0.925195i	$0.623901\pi$
$720$	0	0
$721$	27.5771	1.02702
$722$	0	0
$723$	3.64114	0.135416
$724$	0	0
$725$	51.4904	1.91230
$726$	0	0
$727$	−6.53128	−0.242232	−0.121116	−	0.992638i	$-0.538647\pi$
$727$	−6.53128	−0.242232	−0.121116	+	0.992638i	$0.538647\pi$
$728$	0	0
$729$	20.9719	0.776737
$730$	0	0
$731$	26.0689	0.964194
$732$	0	0
$733$	25.4795	0.941106	0.470553	−	0.882372i	$-0.344054\pi$
$733$	25.4795	0.941106	0.470553	+	0.882372i	$0.344054\pi$
$734$	0	0
$735$	45.4434	1.67621
$736$	0	0
$737$	−19.6920	−0.725363
$738$	0	0
$739$	2.59427	0.0954317	0.0477159	−	0.998861i	$-0.484806\pi$
$739$	2.59427	0.0954317	0.0477159	+	0.998861i	$0.484806\pi$
$740$	0	0
$741$	−3.54968	−0.130401
$742$	0	0
$743$	−46.8293	−1.71800	−0.859000	−	0.511976i	$-0.828914\pi$
$743$	−46.8293	−1.71800	−0.859000	+	0.511976i	$0.828914\pi$
$744$	0	0
$745$	−24.4327	−0.895146
$746$	0	0
$747$	21.6192	0.791007
$748$	0	0
$749$	6.49574	0.237349
$750$	0	0
$751$	−2.02981	−0.0740687	−0.0370343	−	0.999314i	$-0.511791\pi$
$751$	−2.02981	−0.0740687	−0.0370343	+	0.999314i	$0.511791\pi$
$752$	0	0
$753$	−1.16074	−0.0422997
$754$	0	0
$755$	48.6427	1.77029
$756$	0	0
$757$	12.7633	0.463889	0.231945	−	0.972729i	$-0.425491\pi$
$757$	12.7633	0.463889	0.231945	+	0.972729i	$0.425491\pi$
$758$	0	0
$759$	17.3517	0.629825
$760$	0	0
$761$	−30.1458	−1.09278	−0.546392	−	0.837529i	$-0.683999\pi$
$761$	−30.1458	−1.09278	−0.546392	+	0.837529i	$0.683999\pi$
$762$	0	0
$763$	0.734304	0.0265836
$764$	0	0
$765$	−23.2344	−0.840040
$766$	0	0
$767$	35.8127	1.29312
$768$	0	0
$769$	−39.8518	−1.43709	−0.718546	−	0.695480i	$-0.755192\pi$
$769$	−39.8518	−1.43709	−0.718546	+	0.695480i	$0.755192\pi$
$770$	0	0
$771$	−26.2235	−0.944415
$772$	0	0
$773$	45.8906	1.65057	0.825285	−	0.564716i	$-0.191014\pi$
$773$	45.8906	1.65057	0.825285	+	0.564716i	$0.191014\pi$
$774$	0	0
$775$	−63.5357	−2.28227
$776$	0	0
$777$	−42.5771	−1.52745
$778$	0	0
$779$	−3.65689	−0.131022
$780$	0	0
$781$	0.776120	0.0277718
$782$	0	0
$783$	−24.3688	−0.870872
$784$	0	0
$785$	12.1913	0.435126
$786$	0	0
$787$	−1.79616	−0.0640262	−0.0320131	−	0.999487i	$-0.510192\pi$
$787$	−1.79616	−0.0640262	−0.0320131	+	0.999487i	$0.510192\pi$
$788$	0	0
$789$	8.63329	0.307353
$790$	0	0
$791$	−6.49276	−0.230856
$792$	0	0
$793$	44.0742	1.56512
$794$	0	0
$795$	−7.74166	−0.274568
$796$	0	0
$797$	−3.16469	−0.112099	−0.0560495	−	0.998428i	$-0.517850\pi$
$797$	−3.16469	−0.112099	−0.0560495	+	0.998428i	$0.517850\pi$
$798$	0	0
$799$	20.5606	0.727380
$800$	0	0
$801$	−13.3663	−0.472274
$802$	0	0
$803$	54.8195	1.93454
$804$	0	0
$805$	−71.8882	−2.53373
$806$	0	0
$807$	4.00313	0.140917
$808$	0	0
$809$	1.79181	0.0629965	0.0314983	−	0.999504i	$-0.489972\pi$
$809$	1.79181	0.0629965	0.0314983	+	0.999504i	$0.489972\pi$
$810$	0	0
$811$	15.4104	0.541131	0.270565	−	0.962702i	$-0.412789\pi$
$811$	15.4104	0.541131	0.270565	+	0.962702i	$0.412789\pi$
$812$	0	0
$813$	−28.6501	−1.00480
$814$	0	0
$815$	43.1987	1.51319
$816$	0	0
$817$	−6.95889	−0.243461
$818$	0	0
$819$	−22.2715	−0.778228
$820$	0	0
$821$	0.00349149	0.000121854 0	6.09270e−5	−	1.00000i	$-0.499981\pi$
$821$	0.00349149	0.000121854 0	6.09270e−5		1.00000i	$0.499981\pi$
$822$	0	0
$823$	11.6467	0.405978	0.202989	−	0.979181i	$-0.434934\pi$
$823$	11.6467	0.405978	0.202989	+	0.979181i	$0.434934\pi$
$824$	0	0
$825$	−45.5485	−1.58580
$826$	0	0
$827$	−37.2500	−1.29531	−0.647655	−	0.761934i	$-0.724250\pi$
$827$	−37.2500	−1.29531	−0.647655	+	0.761934i	$0.724250\pi$
$828$	0	0
$829$	48.3248	1.67839	0.839194	−	0.543831i	$-0.183027\pi$
$829$	48.3248	1.67839	0.839194	+	0.543831i	$0.183027\pi$
$830$	0	0
$831$	33.6706	1.16802
$832$	0	0
$833$	33.5381	1.16203
$834$	0	0
$835$	−103.224	−3.57221
$836$	0	0
$837$	30.0695	1.03935
$838$	0	0
$839$	2.58539	0.0892576	0.0446288	−	0.999004i	$-0.485789\pi$
$839$	2.58539	0.0892576	0.0446288	+	0.999004i	$0.485789\pi$
$840$	0	0
$841$	−8.63927	−0.297906
$842$	0	0
$843$	−9.82957	−0.338549
$844$	0	0
$845$	−8.51679	−0.292987
$846$	0	0
$847$	3.36995	0.115793
$848$	0	0
$849$	3.50698	0.120359
$850$	0	0
$851$	39.0611	1.33900
$852$	0	0
$853$	−14.8072	−0.506987	−0.253494	−	0.967337i	$-0.581580\pi$
$853$	−14.8072	−0.506987	−0.253494	+	0.967337i	$0.581580\pi$
$854$	0	0
$855$	6.20223	0.212112
$856$	0	0
$857$	5.04528	0.172344	0.0861718	−	0.996280i	$-0.472537\pi$
$857$	5.04528	0.172344	0.0861718	+	0.996280i	$0.472537\pi$
$858$	0	0
$859$	50.0869	1.70894	0.854472	−	0.519497i	$-0.173881\pi$
$859$	50.0869	1.70894	0.854472	+	0.519497i	$0.173881\pi$
$860$	0	0
$861$	18.7047	0.637454
$862$	0	0
$863$	17.8667	0.608191	0.304095	−	0.952642i	$-0.401646\pi$
$863$	17.8667	0.608191	0.304095	+	0.952642i	$0.401646\pi$
$864$	0	0
$865$	29.3462	0.997802
$866$	0	0
$867$	−5.75353	−0.195400
$868$	0	0
$869$	22.7502	0.771748
$870$	0	0
$871$	18.9036	0.640525
$872$	0	0
$873$	−21.7142	−0.734915
$874$	0	0
$875$	106.022	3.58421
$876$	0	0
$877$	52.4459	1.77097	0.885486	−	0.464666i	$-0.153825\pi$
$877$	52.4459	1.77097	0.885486	+	0.464666i	$0.153825\pi$
$878$	0	0
$879$	−20.5967	−0.694709
$880$	0	0
$881$	18.6321	0.627731	0.313866	−	0.949467i	$-0.398376\pi$
$881$	18.6321	0.627731	0.313866	+	0.949467i	$0.398376\pi$
$882$	0	0
$883$	21.8329	0.734735	0.367368	−	0.930076i	$-0.380259\pi$
$883$	21.8329	0.734735	0.367368	+	0.930076i	$0.380259\pi$
$884$	0	0
$885$	51.0123	1.71476
$886$	0	0
$887$	8.38890	0.281672	0.140836	−	0.990033i	$-0.455021\pi$
$887$	8.38890	0.281672	0.140836	+	0.990033i	$0.455021\pi$
$888$	0	0
$889$	44.6543	1.49766
$890$	0	0
$891$	4.50687	0.150986
$892$	0	0
$893$	−5.48848	−0.183665
$894$	0	0
$895$	−20.6721	−0.690993
$896$	0	0
$897$	−16.6570	−0.556161
$898$	0	0
$899$	−25.1238	−0.837925
$900$	0	0
$901$	−5.71349	−0.190344
$902$	0	0
$903$	35.5942	1.18450
$904$	0	0
$905$	−89.9715	−2.99075
$906$	0	0
$907$	0.786939	0.0261299	0.0130649	−	0.999915i	$-0.495841\pi$
$907$	0.786939	0.0261299	0.0130649	+	0.999915i	$0.495841\pi$
$908$	0	0
$909$	24.1067	0.799570
$910$	0	0
$911$	−20.3641	−0.674691	−0.337346	−	0.941381i	$-0.609529\pi$
$911$	−20.3641	−0.674691	−0.337346	+	0.941381i	$0.609529\pi$
$912$	0	0
$913$	44.9843	1.48876
$914$	0	0
$915$	62.7802	2.07545
$916$	0	0
$917$	−25.2812	−0.834860
$918$	0	0
$919$	17.1509	0.565755	0.282877	−	0.959156i	$-0.408711\pi$
$919$	17.1509	0.565755	0.282877	+	0.959156i	$0.408711\pi$
$920$	0	0
$921$	23.0557	0.759711
$922$	0	0
$923$	−0.745049	−0.0245236
$924$	0	0
$925$	−102.536	−3.37138
$926$	0	0
$927$	−11.1647	−0.366695
$928$	0	0
$929$	15.8349	0.519527	0.259763	−	0.965672i	$-0.416355\pi$
$929$	15.8349	0.519527	0.259763	+	0.965672i	$0.416355\pi$
$930$	0	0
$931$	−8.95273	−0.293414
$932$	0	0
$933$	14.1527	0.463339
$934$	0	0
$935$	−48.3449	−1.58105
$936$	0	0
$937$	10.6258	0.347130	0.173565	−	0.984822i	$-0.444471\pi$
$937$	10.6258	0.347130	0.173565	+	0.984822i	$0.444471\pi$
$938$	0	0
$939$	6.56794	0.214337
$940$	0	0
$941$	−10.7752	−0.351260	−0.175630	−	0.984456i	$-0.556196\pi$
$941$	−10.7752	−0.351260	−0.175630	+	0.984456i	$0.556196\pi$
$942$	0	0
$943$	−17.1601	−0.558808
$944$	0	0
$945$	−89.3100	−2.90526
$946$	0	0
$947$	−33.5270	−1.08948	−0.544740	−	0.838605i	$-0.683372\pi$
$947$	−33.5270	−1.08948	−0.544740	+	0.838605i	$0.683372\pi$
$948$	0	0
$949$	−52.6249	−1.70828
$950$	0	0
$951$	−36.1856	−1.17340
$952$	0	0
$953$	−37.7122	−1.22162	−0.610809	−	0.791778i	$-0.709156\pi$
$953$	−37.7122	−1.22162	−0.610809	+	0.791778i	$0.709156\pi$
$954$	0	0
$955$	18.1008	0.585730
$956$	0	0
$957$	−18.0112	−0.582218
$958$	0	0
$959$	−35.2309	−1.13767
$960$	0	0
$961$	0.00106578	3.43800e−5 0
$962$	0	0
$963$	−2.62982	−0.0847448
$964$	0	0
$965$	−40.9431	−1.31800
$966$	0	0
$967$	−17.6034	−0.566089	−0.283044	−	0.959107i	$-0.591344\pi$
$967$	−17.6034	−0.566089	−0.283044	+	0.959107i	$0.591344\pi$
$968$	0	0
$969$	−3.73160	−0.119876
$970$	0	0
$971$	43.1429	1.38452	0.692260	−	0.721648i	$-0.256615\pi$
$971$	43.1429	1.38452	0.692260	+	0.721648i	$0.256615\pi$
$972$	0	0
$973$	−47.9560	−1.53740
$974$	0	0
$975$	43.7251	1.40032
$976$	0	0
$977$	−19.2387	−0.615500	−0.307750	−	0.951467i	$-0.599576\pi$
$977$	−19.2387	−0.615500	−0.307750	+	0.951467i	$0.599576\pi$
$978$	0	0
$979$	−27.8119	−0.888872
$980$	0	0
$981$	−0.297285	−0.00949158
$982$	0	0
$983$	−50.8893	−1.62312	−0.811558	−	0.584272i	$-0.801380\pi$
$983$	−50.8893	−1.62312	−0.811558	+	0.584272i	$0.801380\pi$
$984$	0	0
$985$	−75.1339	−2.39396
$986$	0	0
$987$	28.0731	0.893578
$988$	0	0
$989$	−32.6548	−1.03836
$990$	0	0
$991$	−62.3178	−1.97959	−0.989796	−	0.142494i	$-0.954488\pi$
$991$	−62.3178	−1.97959	−0.989796	+	0.142494i	$0.954488\pi$
$992$	0	0
$993$	−13.4915	−0.428140
$994$	0	0
$995$	−2.83089	−0.0897454
$996$	0	0
$997$	5.03126	0.159342	0.0796708	−	0.996821i	$-0.474613\pi$
$997$	5.03126	0.159342	0.0796708	+	0.996821i	$0.474613\pi$
$998$	0	0
$999$	48.5274	1.53534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.12		17
4.3	odd	2		251.2.a.b.1.5	✓	17
12.11	even	2		2259.2.a.k.1.13		17
20.19	odd	2		6275.2.a.e.1.13		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.5	✓	17	4.3	odd	2
2259.2.a.k.1.13		17	12.11	even	2
4016.2.a.k.1.12		17	1.1	even	1	trivial
6275.2.a.e.1.13		17	20.19	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.16074 q^{3} +4.05107 q^{5} +4.08218 q^{7} -1.65268 q^{9} +O(q^{10})\)	\(q+1.16074 q^{3} +4.05107 q^{5} +4.08218 q^{7} -1.65268 q^{9} -3.43883 q^{11} +3.30116 q^{13} +4.70223 q^{15} +3.47033 q^{17} -0.926379 q^{19} +4.73835 q^{21} -4.34706 q^{23} +11.4111 q^{25} -5.40055 q^{27} +4.51229 q^{29} -5.56786 q^{31} -3.99158 q^{33} +16.5372 q^{35} -8.98564 q^{37} +3.83178 q^{39} +3.94751 q^{41} +7.51193 q^{43} -6.69513 q^{45} +5.92466 q^{47} +9.66422 q^{49} +4.02815 q^{51} -1.64638 q^{53} -13.9309 q^{55} -1.07528 q^{57} +10.8485 q^{59} +13.3511 q^{61} -6.74656 q^{63} +13.3732 q^{65} +5.72636 q^{67} -5.04581 q^{69} -0.225693 q^{71} -15.9413 q^{73} +13.2454 q^{75} -14.0379 q^{77} -6.61569 q^{79} -1.31059 q^{81} -13.0813 q^{83} +14.0586 q^{85} +5.23759 q^{87} +8.08761 q^{89} +13.4759 q^{91} -6.46284 q^{93} -3.75282 q^{95} +13.1388 q^{97} +5.68329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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