LMFDB - Embedded newform 4864.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.31955$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31955 q^{3} -0.319551 q^{5} +1.93923 q^{7} -1.25879 q^{9} +O(q^{10})$	$q+1.31955 q^{3} -0.319551 q^{5} +1.93923 q^{7} -1.25879 q^{9} -3.25879 q^{11} +4.25879 q^{13} -0.421664 q^{15} +3.00000 q^{17} -1.00000 q^{19} +2.55892 q^{21} -5.61968 q^{23} -4.89789 q^{25} -5.61968 q^{27} -6.55892 q^{29} -6.93923 q^{31} -4.30013 q^{33} -0.619684 q^{35} -7.57834 q^{37} +5.61968 q^{39} -4.30013 q^{41} +7.13726 q^{43} +0.402246 q^{45} -6.31955 q^{47} -3.23937 q^{49} +3.95865 q^{51} -5.31955 q^{53} +1.04135 q^{55} -1.31955 q^{57} -1.74121 q^{59} +2.19802 q^{61} -2.44108 q^{63} -1.36090 q^{65} +8.68045 q^{67} -7.41546 q^{69} +9.15667 q^{71} +15.9136 q^{73} -6.46301 q^{75} -6.31955 q^{77} -6.93923 q^{79} -3.63910 q^{81} +15.1178 q^{83} -0.958652 q^{85} -8.65483 q^{87} -5.69987 q^{89} +8.25879 q^{91} -9.15667 q^{93} +0.319551 q^{95} +17.1178 q^{97} +4.10211 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 2 q^{5} - 3 q^{7} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 + 2 * q^5 - 3 * q^7 + 8 * q^9	$ 3 q + q^{3} + 2 q^{5} - 3 q^{7} + 8 q^{9} + 2 q^{11} + q^{13} - 16 q^{15} + 9 q^{17} - 3 q^{19} - 7 q^{21} - 11 q^{23} + 3 q^{25} - 11 q^{27} - 5 q^{29} - 12 q^{31} - 10 q^{33} + 4 q^{35} - 8 q^{37} + 11 q^{39} - 10 q^{41} - 8 q^{43} + 16 q^{45} - 16 q^{47} + 2 q^{49} + 3 q^{51} - 13 q^{53} + 12 q^{55} - q^{57} - 17 q^{59} - 14 q^{61} - 22 q^{63} - 10 q^{65} + 29 q^{67} + 19 q^{69} - 2 q^{71} - 17 q^{73} - 43 q^{75} - 16 q^{77} - 12 q^{79} - 5 q^{81} + 16 q^{83} + 6 q^{85} + 27 q^{87} - 20 q^{89} + 13 q^{91} + 2 q^{93} - 2 q^{95} + 22 q^{97} + 30 q^{99}+O(q^{100}) $ 3 * q + q^3 + 2 * q^5 - 3 * q^7 + 8 * q^9 + 2 * q^11 + q^13 - 16 * q^15 + 9 * q^17 - 3 * q^19 - 7 * q^21 - 11 * q^23 + 3 * q^25 - 11 * q^27 - 5 * q^29 - 12 * q^31 - 10 * q^33 + 4 * q^35 - 8 * q^37 + 11 * q^39 - 10 * q^41 - 8 * q^43 + 16 * q^45 - 16 * q^47 + 2 * q^49 + 3 * q^51 - 13 * q^53 + 12 * q^55 - q^57 - 17 * q^59 - 14 * q^61 - 22 * q^63 - 10 * q^65 + 29 * q^67 + 19 * q^69 - 2 * q^71 - 17 * q^73 - 43 * q^75 - 16 * q^77 - 12 * q^79 - 5 * q^81 + 16 * q^83 + 6 * q^85 + 27 * q^87 - 20 * q^89 + 13 * q^91 + 2 * q^93 - 2 * q^95 + 22 * q^97 + 30 * 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
$3$	1.31955	0.761843	0.380922	−	0.924607i	$-0.375607\pi$
$3$	1.31955	0.761843	0.380922	+	0.924607i	$0.375607\pi$
$4$	0	0
$5$	−0.319551	−0.142907	−0.0714537	−	0.997444i	$-0.522764\pi$
$5$	−0.319551	−0.142907	−0.0714537	+	0.997444i	$0.522764\pi$
$6$	0	0
$7$	1.93923	0.732962	0.366481	−	0.930426i	$-0.380562\pi$
$7$	1.93923	0.732962	0.366481	+	0.930426i	$0.380562\pi$
$8$	0	0
$9$	−1.25879	−0.419595
$10$	0	0
$11$	−3.25879	−0.982561	−0.491280	−	0.871001i	$-0.663471\pi$
$11$	−3.25879	−0.982561	−0.491280	+	0.871001i	$0.663471\pi$
$12$	0	0
$13$	4.25879	1.18117	0.590587	−	0.806974i	$-0.298896\pi$
$13$	4.25879	1.18117	0.590587	+	0.806974i	$0.298896\pi$
$14$	0	0
$15$	−0.421664	−0.108873
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.55892	0.558402
$22$	0	0
$23$	−5.61968	−1.17179	−0.585893	−	0.810389i	$-0.699256\pi$
$23$	−5.61968	−1.17179	−0.585893	+	0.810389i	$0.699256\pi$
$24$	0	0
$25$	−4.89789	−0.979577
$26$	0	0
$27$	−5.61968	−1.08151
$28$	0	0
$29$	−6.55892	−1.21796	−0.608980	−	0.793185i	$-0.708421\pi$
$29$	−6.55892	−1.21796	−0.608980	+	0.793185i	$0.708421\pi$
$30$	0	0
$31$	−6.93923	−1.24632	−0.623162	−	0.782093i	$-0.714152\pi$
$31$	−6.93923	−1.24632	−0.623162	+	0.782093i	$0.714152\pi$
$32$	0	0
$33$	−4.30013	−0.748557
$34$	0	0
$35$	−0.619684	−0.104746
$36$	0	0
$37$	−7.57834	−1.24587	−0.622935	−	0.782273i	$-0.714060\pi$
$37$	−7.57834	−1.24587	−0.622935	+	0.782273i	$0.714060\pi$
$38$	0	0
$39$	5.61968	0.899870
$40$	0	0
$41$	−4.30013	−0.671568	−0.335784	−	0.941939i	$-0.609001\pi$
$41$	−4.30013	−0.671568	−0.335784	+	0.941939i	$0.609001\pi$
$42$	0	0
$43$	7.13726	1.08842	0.544211	−	0.838949i	$-0.316829\pi$
$43$	7.13726	1.08842	0.544211	+	0.838949i	$0.316829\pi$
$44$	0	0
$45$	0.402246	0.0599633
$46$	0	0
$47$	−6.31955	−0.921801	−0.460901	−	0.887452i	$-0.652474\pi$
$47$	−6.31955	−0.921801	−0.460901	+	0.887452i	$0.652474\pi$
$48$	0	0
$49$	−3.23937	−0.462767
$50$	0	0
$51$	3.95865	0.554322
$52$	0	0
$53$	−5.31955	−0.730696	−0.365348	−	0.930871i	$-0.619050\pi$
$53$	−5.31955	−0.730696	−0.365348	+	0.930871i	$0.619050\pi$
$54$	0	0
$55$	1.04135	0.140415
$56$	0	0
$57$	−1.31955	−0.174779
$58$	0	0
$59$	−1.74121	−0.226687	−0.113343	−	0.993556i	$-0.536156\pi$
$59$	−1.74121	−0.226687	−0.113343	+	0.993556i	$0.536156\pi$
$60$	0	0
$61$	2.19802	0.281428	0.140714	−	0.990050i	$-0.455060\pi$
$61$	2.19802	0.281428	0.140714	+	0.990050i	$0.455060\pi$
$62$	0	0
$63$	−2.44108	−0.307547
$64$	0	0
$65$	−1.36090	−0.168799
$66$	0	0
$67$	8.68045	1.06049	0.530243	−	0.847846i	$-0.322101\pi$
$67$	8.68045	1.06049	0.530243	+	0.847846i	$0.322101\pi$
$68$	0	0
$69$	−7.41546	−0.892716
$70$	0	0
$71$	9.15667	1.08670	0.543349	−	0.839507i	$-0.317156\pi$
$71$	9.15667	1.08670	0.543349	+	0.839507i	$0.317156\pi$
$72$	0	0
$73$	15.9136	1.86255	0.931274	−	0.364320i	$-0.118699\pi$
$73$	15.9136	1.86255	0.931274	+	0.364320i	$0.118699\pi$
$74$	0	0
$75$	−6.46301	−0.746284
$76$	0	0
$77$	−6.31955	−0.720180
$78$	0	0
$79$	−6.93923	−0.780725	−0.390362	−	0.920661i	$-0.627650\pi$
$79$	−6.93923	−0.780725	−0.390362	+	0.920661i	$0.627650\pi$
$80$	0	0
$81$	−3.63910	−0.404345
$82$	0	0
$83$	15.1178	1.65940	0.829699	−	0.558211i	$-0.188512\pi$
$83$	15.1178	1.65940	0.829699	+	0.558211i	$0.188512\pi$
$84$	0	0
$85$	−0.958652	−0.103980
$86$	0	0
$87$	−8.65483	−0.927895
$88$	0	0
$89$	−5.69987	−0.604185	−0.302092	−	0.953279i	$-0.597685\pi$
$89$	−5.69987	−0.604185	−0.302092	+	0.953279i	$0.597685\pi$
$90$	0	0
$91$	8.25879	0.865756
$92$	0	0
$93$	−9.15667	−0.949503
$94$	0	0
$95$	0.319551	0.0327852
$96$	0	0
$97$	17.1178	1.73805	0.869027	−	0.494765i	$-0.164746\pi$
$97$	17.1178	1.73805	0.869027	+	0.494765i	$0.164746\pi$
$98$	0	0
$99$	4.10211	0.412278
$100$	0	0
$101$	7.36090	0.732437	0.366218	−	0.930529i	$-0.380652\pi$
$101$	7.36090	0.732437	0.366218	+	0.930529i	$0.380652\pi$
$102$	0	0
$103$	−15.1178	−1.48960	−0.744802	−	0.667285i	$-0.767456\pi$
$103$	−15.1178	−1.48960	−0.744802	+	0.667285i	$0.767456\pi$
$104$	0	0
$105$	−0.817705	−0.0797998
$106$	0	0
$107$	−12.1373	−1.17335	−0.586676	−	0.809821i	$-0.699564\pi$
$107$	−12.1373	−1.17335	−0.586676	+	0.809821i	$0.699564\pi$
$108$	0	0
$109$	9.07649	0.869370	0.434685	−	0.900583i	$-0.356860\pi$
$109$	9.07649	0.869370	0.434685	+	0.900583i	$0.356860\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	−4.81770	−0.453212	−0.226606	−	0.973987i	$-0.572763\pi$
$113$	−4.81770	−0.453212	−0.226606	+	0.973987i	$0.572763\pi$
$114$	0	0
$115$	1.79577	0.167457
$116$	0	0
$117$	−5.36090	−0.495615
$118$	0	0
$119$	5.81770	0.533308
$120$	0	0
$121$	−0.380316	−0.0345742
$122$	0	0
$123$	−5.67424	−0.511629
$124$	0	0
$125$	3.16288	0.282896
$126$	0	0
$127$	−18.7350	−1.66246	−0.831232	−	0.555926i	$-0.812364\pi$
$127$	−18.7350	−1.66246	−0.831232	+	0.555926i	$0.812364\pi$
$128$	0	0
$129$	9.41797	0.829206
$130$	0	0
$131$	−12.4155	−1.08474	−0.542372	−	0.840139i	$-0.682474\pi$
$131$	−12.4155	−1.08474	−0.542372	+	0.840139i	$0.682474\pi$
$132$	0	0
$133$	−1.93923	−0.168153
$134$	0	0
$135$	1.79577	0.154556
$136$	0	0
$137$	12.2782	1.04900	0.524499	−	0.851411i	$-0.324253\pi$
$137$	12.2782	1.04900	0.524499	+	0.851411i	$0.324253\pi$
$138$	0	0
$139$	−13.2588	−1.12460	−0.562298	−	0.826935i	$-0.690083\pi$
$139$	−13.2588	−1.12460	−0.562298	+	0.826935i	$0.690083\pi$
$140$	0	0
$141$	−8.33897	−0.702268
$142$	0	0
$143$	−13.8785	−1.16058
$144$	0	0
$145$	2.09591	0.174056
$146$	0	0
$147$	−4.27451	−0.352556
$148$	0	0
$149$	−7.55892	−0.619251	−0.309625	−	0.950859i	$-0.600204\pi$
$149$	−7.55892	−0.619251	−0.309625	+	0.950859i	$0.600204\pi$
$150$	0	0
$151$	0.396041	0.0322294	0.0161147	−	0.999870i	$-0.494870\pi$
$151$	0.396041	0.0322294	0.0161147	+	0.999870i	$0.494870\pi$
$152$	0	0
$153$	−3.77636	−0.305300
$154$	0	0
$155$	2.21744	0.178109
$156$	0	0
$157$	−11.0351	−0.880700	−0.440350	−	0.897826i	$-0.645146\pi$
$157$	−11.0351	−0.880700	−0.440350	+	0.897826i	$0.645146\pi$
$158$	0	0
$159$	−7.01942	−0.556676
$160$	0	0
$161$	−10.8979	−0.858874
$162$	0	0
$163$	3.03514	0.237731	0.118865	−	0.992910i	$-0.462074\pi$
$163$	3.03514	0.237731	0.118865	+	0.992910i	$0.462074\pi$
$164$	0	0
$165$	1.37411	0.106974
$166$	0	0
$167$	−9.97438	−0.771841	−0.385920	−	0.922532i	$-0.626116\pi$
$167$	−9.97438	−0.771841	−0.385920	+	0.922532i	$0.626116\pi$
$168$	0	0
$169$	5.13726	0.395173
$170$	0	0
$171$	1.25879	0.0962617
$172$	0	0
$173$	−1.75694	−0.133578	−0.0667888	−	0.997767i	$-0.521275\pi$
$173$	−1.75694	−0.133578	−0.0667888	+	0.997767i	$0.521275\pi$
$174$	0	0
$175$	−9.49815	−0.717993
$176$	0	0
$177$	−2.29762	−0.172700
$178$	0	0
$179$	4.30013	0.321407	0.160704	−	0.987003i	$-0.448624\pi$
$179$	4.30013	0.321407	0.160704	+	0.987003i	$0.448624\pi$
$180$	0	0
$181$	14.4349	1.07294	0.536468	−	0.843921i	$-0.319758\pi$
$181$	14.4349	1.07294	0.536468	+	0.843921i	$0.319758\pi$
$182$	0	0
$183$	2.90040	0.214404
$184$	0	0
$185$	2.42166	0.178044
$186$	0	0
$187$	−9.77636	−0.714918
$188$	0	0
$189$	−10.8979	−0.792705
$190$	0	0
$191$	−23.7350	−1.71741	−0.858703	−	0.512474i	$-0.828729\pi$
$191$	−23.7350	−1.71741	−0.858703	+	0.512474i	$0.828729\pi$
$192$	0	0
$193$	−20.6962	−1.48974	−0.744872	−	0.667208i	$-0.767489\pi$
$193$	−20.6962	−1.48974	−0.744872	+	0.667208i	$0.767489\pi$
$194$	0	0
$195$	−1.79577	−0.128598
$196$	0	0
$197$	15.7569	1.12264	0.561318	−	0.827600i	$-0.310295\pi$
$197$	15.7569	1.12264	0.561318	+	0.827600i	$0.310295\pi$
$198$	0	0
$199$	−9.85654	−0.698712	−0.349356	−	0.936990i	$-0.613600\pi$
$199$	−9.85654	−0.698712	−0.349356	+	0.936990i	$0.613600\pi$
$200$	0	0
$201$	11.4543	0.807924
$202$	0	0
$203$	−12.7193	−0.892719
$204$	0	0
$205$	1.37411	0.0959721
$206$	0	0
$207$	7.07398	0.491675
$208$	0	0
$209$	3.25879	0.225415
$210$	0	0
$211$	12.1373	0.835563	0.417782	−	0.908548i	$-0.362808\pi$
$211$	12.1373	0.835563	0.417782	+	0.908548i	$0.362808\pi$
$212$	0	0
$213$	12.0827	0.827893
$214$	0	0
$215$	−2.28072	−0.155544
$216$	0	0
$217$	−13.4568	−0.913508
$218$	0	0
$219$	20.9988	1.41897
$220$	0	0
$221$	12.7764	0.859431
$222$	0	0
$223$	18.5746	1.24385	0.621925	−	0.783077i	$-0.286351\pi$
$223$	18.5746	1.24385	0.621925	+	0.783077i	$0.286351\pi$
$224$	0	0
$225$	6.16539	0.411026
$226$	0	0
$227$	−21.2939	−1.41333	−0.706664	−	0.707549i	$-0.749801\pi$
$227$	−21.2939	−1.41333	−0.706664	+	0.707549i	$0.749801\pi$
$228$	0	0
$229$	11.4762	0.758370	0.379185	−	0.925321i	$-0.376204\pi$
$229$	11.4762	0.758370	0.379185	+	0.925321i	$0.376204\pi$
$230$	0	0
$231$	−8.33897	−0.548664
$232$	0	0
$233$	−17.6548	−1.15661	−0.578303	−	0.815822i	$-0.696285\pi$
$233$	−17.6548	−1.15661	−0.578303	+	0.815822i	$0.696285\pi$
$234$	0	0
$235$	2.01942	0.131732
$236$	0	0
$237$	−9.15667	−0.594790
$238$	0	0
$239$	−11.5432	−0.746667	−0.373334	−	0.927697i	$-0.621785\pi$
$239$	−11.5432	−0.746667	−0.373334	+	0.927697i	$0.621785\pi$
$240$	0	0
$241$	9.57834	0.616995	0.308497	−	0.951225i	$-0.400174\pi$
$241$	9.57834	0.616995	0.308497	+	0.951225i	$0.400174\pi$
$242$	0	0
$243$	12.0571	0.773462
$244$	0	0
$245$	1.03514	0.0661328
$246$	0	0
$247$	−4.25879	−0.270980
$248$	0	0
$249$	19.9488	1.26420
$250$	0	0
$251$	0.223643	0.0141162	0.00705811	−	0.999975i	$-0.497753\pi$
$251$	0.223643	0.0141162	0.00705811	+	0.999975i	$0.497753\pi$
$252$	0	0
$253$	18.3133	1.15135
$254$	0	0
$255$	−1.26499	−0.0792168
$256$	0	0
$257$	−0.0826952	−0.00515838	−0.00257919	−	0.999997i	$-0.500821\pi$
$257$	−0.0826952	−0.00515838	−0.00257919	+	0.999997i	$0.500821\pi$
$258$	0	0
$259$	−14.6962	−0.913176
$260$	0	0
$261$	8.25627	0.511050
$262$	0	0
$263$	−1.04135	−0.0642122	−0.0321061	−	0.999484i	$-0.510221\pi$
$263$	−1.04135	−0.0642122	−0.0321061	+	0.999484i	$0.510221\pi$
$264$	0	0
$265$	1.69987	0.104422
$266$	0	0
$267$	−7.52126	−0.460294
$268$	0	0
$269$	13.8528	0.844623	0.422312	−	0.906451i	$-0.361219\pi$
$269$	13.8528	0.844623	0.422312	+	0.906451i	$0.361219\pi$
$270$	0	0
$271$	12.2976	0.747027	0.373514	−	0.927625i	$-0.378153\pi$
$271$	12.2976	0.747027	0.373514	+	0.927625i	$0.378153\pi$
$272$	0	0
$273$	10.8979	0.659570
$274$	0	0
$275$	15.9612	0.962494
$276$	0	0
$277$	11.6804	0.701810	0.350905	−	0.936411i	$-0.385874\pi$
$277$	11.6804	0.701810	0.350905	+	0.936411i	$0.385874\pi$
$278$	0	0
$279$	8.73501	0.522951
$280$	0	0
$281$	1.23937	0.0739345	0.0369673	−	0.999316i	$-0.488230\pi$
$281$	1.23937	0.0739345	0.0369673	+	0.999316i	$0.488230\pi$
$282$	0	0
$283$	14.6585	0.871359	0.435679	−	0.900102i	$-0.356508\pi$
$283$	14.6585	0.871359	0.435679	+	0.900102i	$0.356508\pi$
$284$	0	0
$285$	0.421664	0.0249772
$286$	0	0
$287$	−8.33897	−0.492234
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	22.5879	1.32412
$292$	0	0
$293$	−20.0157	−1.16933	−0.584666	−	0.811274i	$-0.698774\pi$
$293$	−20.0157	−1.16933	−0.584666	+	0.811274i	$0.698774\pi$
$294$	0	0
$295$	0.556406	0.0323952
$296$	0	0
$297$	18.3133	1.06265
$298$	0	0
$299$	−23.9330	−1.38408
$300$	0	0
$301$	13.8408	0.797771
$302$	0	0
$303$	9.71308	0.558002
$304$	0	0
$305$	−0.702379	−0.0402181
$306$	0	0
$307$	−19.4180	−1.10824	−0.554121	−	0.832436i	$-0.686946\pi$
$307$	−19.4180	−1.10824	−0.554121	+	0.832436i	$0.686946\pi$
$308$	0	0
$309$	−19.9488	−1.13485
$310$	0	0
$311$	−29.8565	−1.69301	−0.846505	−	0.532382i	$-0.821297\pi$
$311$	−29.8565	−1.69301	−0.846505	+	0.532382i	$0.821297\pi$
$312$	0	0
$313$	−17.4543	−0.986575	−0.493288	−	0.869866i	$-0.664205\pi$
$313$	−17.4543	−0.986575	−0.493288	+	0.869866i	$0.664205\pi$
$314$	0	0
$315$	0.780049	0.0439508
$316$	0	0
$317$	11.5370	0.647982	0.323991	−	0.946060i	$-0.394975\pi$
$317$	11.5370	0.647982	0.323991	+	0.946060i	$0.394975\pi$
$318$	0	0
$319$	21.3741	1.19672
$320$	0	0
$321$	−16.0157	−0.893911
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−20.8591	−1.15705
$326$	0	0
$327$	11.9769	0.662324
$328$	0	0
$329$	−12.2551	−0.675645
$330$	0	0
$331$	11.3196	0.622179	0.311089	−	0.950381i	$-0.399306\pi$
$331$	11.3196	0.622179	0.311089	+	0.950381i	$0.399306\pi$
$332$	0	0
$333$	9.53950	0.522761
$334$	0	0
$335$	−2.77384	−0.151551
$336$	0	0
$337$	4.51757	0.246088	0.123044	−	0.992401i	$-0.460734\pi$
$337$	4.51757	0.246088	0.123044	+	0.992401i	$0.460734\pi$
$338$	0	0
$339$	−6.35721	−0.345276
$340$	0	0
$341$	22.6135	1.22459
$342$	0	0
$343$	−19.8565	−1.07215
$344$	0	0
$345$	2.36962	0.127576
$346$	0	0
$347$	17.6937	0.949846	0.474923	−	0.880027i	$-0.342476\pi$
$347$	17.6937	0.949846	0.474923	+	0.880027i	$0.342476\pi$
$348$	0	0
$349$	11.7193	0.627319	0.313659	−	0.949536i	$-0.398445\pi$
$349$	11.7193	0.627319	0.313659	+	0.949536i	$0.398445\pi$
$350$	0	0
$351$	−23.9330	−1.27745
$352$	0	0
$353$	−9.37662	−0.499067	−0.249534	−	0.968366i	$-0.580277\pi$
$353$	−9.37662	−0.499067	−0.249534	+	0.968366i	$0.580277\pi$
$354$	0	0
$355$	−2.92602	−0.155297
$356$	0	0
$357$	7.67676	0.406297
$358$	0	0
$359$	−28.4568	−1.50189	−0.750946	−	0.660363i	$-0.770402\pi$
$359$	−28.4568	−1.50189	−0.750946	+	0.660363i	$0.770402\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.501846	−0.0263401
$364$	0	0
$365$	−5.08521	−0.266172
$366$	0	0
$367$	8.60027	0.448930	0.224465	−	0.974482i	$-0.427937\pi$
$367$	8.60027	0.448930	0.224465	+	0.974482i	$0.427937\pi$
$368$	0	0
$369$	5.41295	0.281787
$370$	0	0
$371$	−10.3159	−0.535573
$372$	0	0
$373$	−12.0157	−0.622151	−0.311075	−	0.950385i	$-0.600689\pi$
$373$	−12.0157	−0.622151	−0.311075	+	0.950385i	$0.600689\pi$
$374$	0	0
$375$	4.17358	0.215523
$376$	0	0
$377$	−27.9330	−1.43862
$378$	0	0
$379$	23.5370	1.20901	0.604507	−	0.796600i	$-0.293370\pi$
$379$	23.5370	1.20901	0.604507	+	0.796600i	$0.293370\pi$
$380$	0	0
$381$	−24.7218	−1.26654
$382$	0	0
$383$	−12.0571	−0.616088	−0.308044	−	0.951372i	$-0.599674\pi$
$383$	−12.0571	−0.616088	−0.308044	+	0.951372i	$0.599674\pi$
$384$	0	0
$385$	2.01942	0.102919
$386$	0	0
$387$	−8.98427	−0.456696
$388$	0	0
$389$	−17.3986	−0.882142	−0.441071	−	0.897472i	$-0.645401\pi$
$389$	−17.3986	−0.882142	−0.441071	+	0.897472i	$0.645401\pi$
$390$	0	0
$391$	−16.8591	−0.852599
$392$	0	0
$393$	−16.3828	−0.826404
$394$	0	0
$395$	2.21744	0.111571
$396$	0	0
$397$	14.8371	0.744654	0.372327	−	0.928102i	$-0.378560\pi$
$397$	14.8371	0.744654	0.372327	+	0.928102i	$0.378560\pi$
$398$	0	0
$399$	−2.55892	−0.128106
$400$	0	0
$401$	−2.81770	−0.140709	−0.0703547	−	0.997522i	$-0.522413\pi$
$401$	−2.81770	−0.140709	−0.0703547	+	0.997522i	$0.522413\pi$
$402$	0	0
$403$	−29.5527	−1.47213
$404$	0	0
$405$	1.16288	0.0577839
$406$	0	0
$407$	24.6962	1.22414
$408$	0	0
$409$	3.33528	0.164919	0.0824594	−	0.996594i	$-0.473723\pi$
$409$	3.33528	0.164919	0.0824594	+	0.996594i	$0.473723\pi$
$410$	0	0
$411$	16.2017	0.799172
$412$	0	0
$413$	−3.37662	−0.166153
$414$	0	0
$415$	−4.83092	−0.237140
$416$	0	0
$417$	−17.4956	−0.856765
$418$	0	0
$419$	20.2357	0.988577	0.494289	−	0.869298i	$-0.335429\pi$
$419$	20.2357	0.988577	0.494289	+	0.869298i	$0.335429\pi$
$420$	0	0
$421$	5.83712	0.284484	0.142242	−	0.989832i	$-0.454569\pi$
$421$	5.83712	0.284484	0.142242	+	0.989832i	$0.454569\pi$
$422$	0	0
$423$	7.95496	0.386783
$424$	0	0
$425$	−14.6937	−0.712747
$426$	0	0
$427$	4.26248	0.206276
$428$	0	0
$429$	−18.3133	−0.884177
$430$	0	0
$431$	−13.1955	−0.635605	−0.317803	−	0.948157i	$-0.602945\pi$
$431$	−13.1955	−0.635605	−0.317803	+	0.948157i	$0.602945\pi$
$432$	0	0
$433$	16.9781	0.815914	0.407957	−	0.913001i	$-0.366241\pi$
$433$	16.9781	0.815914	0.407957	+	0.913001i	$0.366241\pi$
$434$	0	0
$435$	2.76566	0.132603
$436$	0	0
$437$	5.61968	0.268826
$438$	0	0
$439$	6.38283	0.304636	0.152318	−	0.988332i	$-0.451326\pi$
$439$	6.38283	0.304636	0.152318	+	0.988332i	$0.451326\pi$
$440$	0	0
$441$	4.07767	0.194175
$442$	0	0
$443$	18.3766	0.873100	0.436550	−	0.899680i	$-0.356200\pi$
$443$	18.3766	0.873100	0.436550	+	0.899680i	$0.356200\pi$
$444$	0	0
$445$	1.82140	0.0863425
$446$	0	0
$447$	−9.97438	−0.471772
$448$	0	0
$449$	31.7569	1.49870	0.749351	−	0.662173i	$-0.230365\pi$
$449$	31.7569	1.49870	0.749351	+	0.662173i	$0.230365\pi$
$450$	0	0
$451$	14.0132	0.659856
$452$	0	0
$453$	0.522596	0.0245537
$454$	0	0
$455$	−2.63910	−0.123723
$456$	0	0
$457$	30.9173	1.44625	0.723125	−	0.690717i	$-0.242705\pi$
$457$	30.9173	1.44625	0.723125	+	0.690717i	$0.242705\pi$
$458$	0	0
$459$	−16.8591	−0.786913
$460$	0	0
$461$	24.2369	1.12882	0.564411	−	0.825494i	$-0.309103\pi$
$461$	24.2369	1.12882	0.564411	+	0.825494i	$0.309103\pi$
$462$	0	0
$463$	20.5941	0.957087	0.478544	−	0.878064i	$-0.341165\pi$
$463$	20.5941	0.957087	0.478544	+	0.878064i	$0.341165\pi$
$464$	0	0
$465$	2.92602	0.135691
$466$	0	0
$467$	−1.85905	−0.0860267	−0.0430133	−	0.999074i	$-0.513696\pi$
$467$	−1.85905	−0.0860267	−0.0430133	+	0.999074i	$0.513696\pi$
$468$	0	0
$469$	16.8334	0.777296
$470$	0	0
$471$	−14.5614	−0.670955
$472$	0	0
$473$	−23.2588	−1.06944
$474$	0	0
$475$	4.89789	0.224730
$476$	0	0
$477$	6.69617	0.306597
$478$	0	0
$479$	6.91361	0.315891	0.157946	−	0.987448i	$-0.449513\pi$
$479$	6.91361	0.315891	0.157946	+	0.987448i	$0.449513\pi$
$480$	0	0
$481$	−32.2745	−1.47159
$482$	0	0
$483$	−14.3803	−0.654327
$484$	0	0
$485$	−5.47002	−0.248381
$486$	0	0
$487$	−39.5527	−1.79230	−0.896152	−	0.443747i	$-0.853649\pi$
$487$	−39.5527	−1.79230	−0.896152	+	0.443747i	$0.853649\pi$
$488$	0	0
$489$	4.00502	0.181113
$490$	0	0
$491$	−11.7958	−0.532336	−0.266168	−	0.963927i	$-0.585758\pi$
$491$	−11.7958	−0.532336	−0.266168	+	0.963927i	$0.585758\pi$
$492$	0	0
$493$	−19.6768	−0.886197
$494$	0	0
$495$	−1.31083	−0.0589176
$496$	0	0
$497$	17.7569	0.796508
$498$	0	0
$499$	1.01573	0.0454701	0.0227351	−	0.999742i	$-0.492763\pi$
$499$	1.01573	0.0454701	0.0227351	+	0.999742i	$0.492763\pi$
$500$	0	0
$501$	−13.1617	−0.588021
$502$	0	0
$503$	−43.0897	−1.92127	−0.960637	−	0.277805i	$-0.910393\pi$
$503$	−43.0897	−1.92127	−0.960637	+	0.277805i	$0.910393\pi$
$504$	0	0
$505$	−2.35218	−0.104671
$506$	0	0
$507$	6.77887	0.301060
$508$	0	0
$509$	35.1881	1.55969	0.779843	−	0.625975i	$-0.215299\pi$
$509$	35.1881	1.55969	0.779843	+	0.625975i	$0.215299\pi$
$510$	0	0
$511$	30.8602	1.36518
$512$	0	0
$513$	5.61968	0.248115
$514$	0	0
$515$	4.83092	0.212876
$516$	0	0
$517$	20.5941	0.905726
$518$	0	0
$519$	−2.31837	−0.101765
$520$	0	0
$521$	14.2174	0.622877	0.311439	−	0.950266i	$-0.399189\pi$
$521$	14.2174	0.622877	0.311439	+	0.950266i	$0.399189\pi$
$522$	0	0
$523$	−4.38032	−0.191538	−0.0957689	−	0.995404i	$-0.530531\pi$
$523$	−4.38032	−0.191538	−0.0957689	+	0.995404i	$0.530531\pi$
$524$	0	0
$525$	−12.5333	−0.546998
$526$	0	0
$527$	−20.8177	−0.906833
$528$	0	0
$529$	8.58085	0.373080
$530$	0	0
$531$	2.19182	0.0951167
$532$	0	0
$533$	−18.3133	−0.793239
$534$	0	0
$535$	3.87847	0.167681
$536$	0	0
$537$	5.67424	0.244862
$538$	0	0
$539$	10.5564	0.454697
$540$	0	0
$541$	14.9198	0.641453	0.320727	−	0.947172i	$-0.396073\pi$
$541$	14.9198	0.641453	0.320727	+	0.947172i	$0.396073\pi$
$542$	0	0
$543$	19.0476	0.817409
$544$	0	0
$545$	−2.90040	−0.124239
$546$	0	0
$547$	−46.0703	−1.96982	−0.984912	−	0.173058i	$-0.944635\pi$
$547$	−46.0703	−1.96982	−0.984912	+	0.173058i	$0.944635\pi$
$548$	0	0
$549$	−2.76684	−0.118086
$550$	0	0
$551$	6.55892	0.279419
$552$	0	0
$553$	−13.4568	−0.572242
$554$	0	0
$555$	3.19551	0.135642
$556$	0	0
$557$	−44.2295	−1.87406	−0.937031	−	0.349245i	$-0.886438\pi$
$557$	−44.2295	−1.87406	−0.937031	+	0.349245i	$0.886438\pi$
$558$	0	0
$559$	30.3960	1.28562
$560$	0	0
$561$	−12.9004	−0.544655
$562$	0	0
$563$	−2.61348	−0.110145	−0.0550725	−	0.998482i	$-0.517539\pi$
$563$	−2.61348	−0.110145	−0.0550725	+	0.998482i	$0.517539\pi$
$564$	0	0
$565$	1.53950	0.0647673
$566$	0	0
$567$	−7.05707	−0.296369
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	33.4312	1.39905	0.699526	−	0.714607i	$-0.253395\pi$
$571$	33.4312	1.39905	0.699526	+	0.714607i	$0.253395\pi$
$572$	0	0
$573$	−31.3196	−1.30839
$574$	0	0
$575$	27.5246	1.14785
$576$	0	0
$577$	12.6742	0.527636	0.263818	−	0.964573i	$-0.415018\pi$
$577$	12.6742	0.527636	0.263818	+	0.964573i	$0.415018\pi$
$578$	0	0
$579$	−27.3097	−1.13495
$580$	0	0
$581$	29.3170	1.21628
$582$	0	0
$583$	17.3353	0.717954
$584$	0	0
$585$	1.71308	0.0708271
$586$	0	0
$587$	21.5721	0.890377	0.445189	−	0.895437i	$-0.353137\pi$
$587$	21.5721	0.890377	0.445189	+	0.895437i	$0.353137\pi$
$588$	0	0
$589$	6.93923	0.285926
$590$	0	0
$591$	20.7921	0.855272
$592$	0	0
$593$	−4.76063	−0.195496	−0.0977479	−	0.995211i	$-0.531164\pi$
$593$	−4.76063	−0.195496	−0.0977479	+	0.995211i	$0.531164\pi$
$594$	0	0
$595$	−1.85905	−0.0762137
$596$	0	0
$597$	−13.0062	−0.532309
$598$	0	0
$599$	40.9268	1.67222	0.836112	−	0.548558i	$-0.184823\pi$
$599$	40.9268	1.67222	0.836112	+	0.548558i	$0.184823\pi$
$600$	0	0
$601$	−25.5139	−1.04073	−0.520366	−	0.853943i	$-0.674205\pi$
$601$	−25.5139	−1.04073	−0.520366	+	0.853943i	$0.674205\pi$
$602$	0	0
$603$	−10.9268	−0.444975
$604$	0	0
$605$	0.121530	0.00494091
$606$	0	0
$607$	33.1955	1.34736	0.673682	−	0.739021i	$-0.264712\pi$
$607$	33.1955	1.34736	0.673682	+	0.739021i	$0.264712\pi$
$608$	0	0
$609$	−16.7837	−0.680112
$610$	0	0
$611$	−26.9136	−1.08881
$612$	0	0
$613$	39.3159	1.58795	0.793976	−	0.607949i	$-0.208007\pi$
$613$	39.3159	1.58795	0.793976	+	0.607949i	$0.208007\pi$
$614$	0	0
$615$	1.81321	0.0731157
$616$	0	0
$617$	2.45932	0.0990084	0.0495042	−	0.998774i	$-0.484236\pi$
$617$	2.45932	0.0990084	0.0495042	+	0.998774i	$0.484236\pi$
$618$	0	0
$619$	−45.5139	−1.82936	−0.914679	−	0.404182i	$-0.867556\pi$
$619$	−45.5139	−1.82936	−0.914679	+	0.404182i	$0.867556\pi$
$620$	0	0
$621$	31.5808	1.26730
$622$	0	0
$623$	−11.0534	−0.442844
$624$	0	0
$625$	23.4787	0.939149
$626$	0	0
$627$	4.30013	0.171731
$628$	0	0
$629$	−22.7350	−0.906504
$630$	0	0
$631$	−24.6329	−0.980620	−0.490310	−	0.871548i	$-0.663116\pi$
$631$	−24.6329	−0.980620	−0.490310	+	0.871548i	$0.663116\pi$
$632$	0	0
$633$	16.0157	0.636568
$634$	0	0
$635$	5.98679	0.237578
$636$	0	0
$637$	−13.7958	−0.546608
$638$	0	0
$639$	−11.5263	−0.455973
$640$	0	0
$641$	30.2357	1.19424	0.597119	−	0.802153i	$-0.296312\pi$
$641$	30.2357	1.19424	0.597119	+	0.802153i	$0.296312\pi$
$642$	0	0
$643$	−23.4945	−0.926531	−0.463266	−	0.886220i	$-0.653322\pi$
$643$	−23.4945	−0.926531	−0.463266	+	0.886220i	$0.653322\pi$
$644$	0	0
$645$	−3.00952	−0.118500
$646$	0	0
$647$	−31.0959	−1.22251	−0.611253	−	0.791435i	$-0.709334\pi$
$647$	−31.0959	−1.22251	−0.611253	+	0.791435i	$0.709334\pi$
$648$	0	0
$649$	5.67424	0.222734
$650$	0	0
$651$	−17.7569	−0.695949
$652$	0	0
$653$	21.9550	0.859164	0.429582	−	0.903028i	$-0.358661\pi$
$653$	21.9550	0.859164	0.429582	+	0.903028i	$0.358661\pi$
$654$	0	0
$655$	3.96737	0.155018
$656$	0	0
$657$	−20.0318	−0.781516
$658$	0	0
$659$	32.2976	1.25814	0.629068	−	0.777350i	$-0.283437\pi$
$659$	32.2976	1.25814	0.629068	+	0.777350i	$0.283437\pi$
$660$	0	0
$661$	−19.5808	−0.761607	−0.380803	−	0.924656i	$-0.624353\pi$
$661$	−19.5808	−0.761607	−0.380803	+	0.924656i	$0.624353\pi$
$662$	0	0
$663$	16.8591	0.654751
$664$	0	0
$665$	0.619684	0.0240303
$666$	0	0
$667$	36.8591	1.42719
$668$	0	0
$669$	24.5102	0.947619
$670$	0	0
$671$	−7.16288	−0.276520
$672$	0	0
$673$	40.8309	1.57392	0.786958	−	0.617006i	$-0.211655\pi$
$673$	40.8309	1.57392	0.786958	+	0.617006i	$0.211655\pi$
$674$	0	0
$675$	27.5246	1.05942
$676$	0	0
$677$	4.43739	0.170543	0.0852714	−	0.996358i	$-0.472824\pi$
$677$	4.43739	0.170543	0.0852714	+	0.996358i	$0.472824\pi$
$678$	0	0
$679$	33.1955	1.27393
$680$	0	0
$681$	−28.0984	−1.07673
$682$	0	0
$683$	13.8785	0.531045	0.265522	−	0.964105i	$-0.414456\pi$
$683$	13.8785	0.531045	0.265522	+	0.964105i	$0.414456\pi$
$684$	0	0
$685$	−3.92351	−0.149910
$686$	0	0
$687$	15.1435	0.577759
$688$	0	0
$689$	−22.6548	−0.863080
$690$	0	0
$691$	−36.4543	−1.38679	−0.693393	−	0.720559i	$-0.743885\pi$
$691$	−36.4543	−1.38679	−0.693393	+	0.720559i	$0.743885\pi$
$692$	0	0
$693$	7.95496	0.302184
$694$	0	0
$695$	4.23686	0.160713
$696$	0	0
$697$	−12.9004	−0.488637
$698$	0	0
$699$	−23.2964	−0.881152
$700$	0	0
$701$	3.11784	0.117759	0.0588796	−	0.998265i	$-0.481247\pi$
$701$	3.11784	0.117759	0.0588796	+	0.998265i	$0.481247\pi$
$702$	0	0
$703$	7.57834	0.285822
$704$	0	0
$705$	2.66472	0.100359
$706$	0	0
$707$	14.2745	0.536848
$708$	0	0
$709$	−24.3133	−0.913107	−0.456553	−	0.889696i	$-0.650916\pi$
$709$	−24.3133	−0.913107	−0.456553	+	0.889696i	$0.650916\pi$
$710$	0	0
$711$	8.73501	0.327588
$712$	0	0
$713$	38.9963	1.46042
$714$	0	0
$715$	4.43488	0.165855
$716$	0	0
$717$	−15.2318	−0.568843
$718$	0	0
$719$	−18.0608	−0.673553	−0.336776	−	0.941585i	$-0.609337\pi$
$719$	−18.0608	−0.673553	−0.336776	+	0.941585i	$0.609337\pi$
$720$	0	0
$721$	−29.3170	−1.09182
$722$	0	0
$723$	12.6391	0.470053
$724$	0	0
$725$	32.1248	1.19309
$726$	0	0
$727$	−27.4531	−1.01818	−0.509090	−	0.860713i	$-0.670018\pi$
$727$	−27.4531	−1.01818	−0.509090	+	0.860713i	$0.670018\pi$
$728$	0	0
$729$	26.8272	0.993601
$730$	0	0
$731$	21.4118	0.791943
$732$	0	0
$733$	−22.6003	−0.834760	−0.417380	−	0.908732i	$-0.637052\pi$
$733$	−22.6003	−0.834760	−0.417380	+	0.908732i	$0.637052\pi$
$734$	0	0
$735$	1.36592	0.0503828
$736$	0	0
$737$	−28.2877	−1.04199
$738$	0	0
$739$	6.13356	0.225627	0.112813	−	0.993616i	$-0.464014\pi$
$739$	6.13356	0.225627	0.112813	+	0.993616i	$0.464014\pi$
$740$	0	0
$741$	−5.61968	−0.206444
$742$	0	0
$743$	0.817705	0.0299987	0.0149993	−	0.999888i	$-0.495225\pi$
$743$	0.817705	0.0299987	0.0149993	+	0.999888i	$0.495225\pi$
$744$	0	0
$745$	2.41546	0.0884956
$746$	0	0
$747$	−19.0301	−0.696276
$748$	0	0
$749$	−23.5370	−0.860023
$750$	0	0
$751$	28.6003	1.04364	0.521819	−	0.853056i	$-0.325254\pi$
$751$	28.6003	1.04364	0.521819	+	0.853056i	$0.325254\pi$
$752$	0	0
$753$	0.295108	0.0107543
$754$	0	0
$755$	−0.126555	−0.00460582
$756$	0	0
$757$	−22.9901	−0.835589	−0.417795	−	0.908541i	$-0.637197\pi$
$757$	−22.9901	−0.835589	−0.417795	+	0.908541i	$0.637197\pi$
$758$	0	0
$759$	24.1654	0.877148
$760$	0	0
$761$	46.6669	1.69167	0.845836	−	0.533443i	$-0.179102\pi$
$761$	46.6669	1.69167	0.845836	+	0.533443i	$0.179102\pi$
$762$	0	0
$763$	17.6014	0.637215
$764$	0	0
$765$	1.20674	0.0436297
$766$	0	0
$767$	−7.41546	−0.267757
$768$	0	0
$769$	−0.674244	−0.0243139	−0.0121569	−	0.999926i	$-0.503870\pi$
$769$	−0.674244	−0.0243139	−0.0121569	+	0.999926i	$0.503870\pi$
$770$	0	0
$771$	−0.109120	−0.00392988
$772$	0	0
$773$	−51.5684	−1.85479	−0.927394	−	0.374086i	$-0.877956\pi$
$773$	−51.5684	−1.85479	−0.927394	+	0.374086i	$0.877956\pi$
$774$	0	0
$775$	33.9876	1.22087
$776$	0	0
$777$	−19.3923	−0.695697
$778$	0	0
$779$	4.30013	0.154068
$780$	0	0
$781$	−29.8396	−1.06775
$782$	0	0
$783$	36.8591	1.31724
$784$	0	0
$785$	3.52629	0.125859
$786$	0	0
$787$	40.2902	1.43619	0.718096	−	0.695944i	$-0.245014\pi$
$787$	40.2902	1.43619	0.718096	+	0.695944i	$0.245014\pi$
$788$	0	0
$789$	−1.37411	−0.0489196
$790$	0	0
$791$	−9.34266	−0.332187
$792$	0	0
$793$	9.36090	0.332415
$794$	0	0
$795$	2.24306	0.0795532
$796$	0	0
$797$	16.0728	0.569328	0.284664	−	0.958627i	$-0.408118\pi$
$797$	16.0728	0.569328	0.284664	+	0.958627i	$0.408118\pi$
$798$	0	0
$799$	−18.9587	−0.670709
$800$	0	0
$801$	7.17491	0.253513
$802$	0	0
$803$	−51.8591	−1.83007
$804$	0	0
$805$	3.48243	0.122739
$806$	0	0
$807$	18.2795	0.643470
$808$	0	0
$809$	43.7094	1.53674	0.768370	−	0.640006i	$-0.221068\pi$
$809$	43.7094	1.53674	0.768370	+	0.640006i	$0.221068\pi$
$810$	0	0
$811$	6.30265	0.221316	0.110658	−	0.993859i	$-0.464704\pi$
$811$	6.30265	0.221316	0.110658	+	0.993859i	$0.464704\pi$
$812$	0	0
$813$	16.2273	0.569117
$814$	0	0
$815$	−0.969882	−0.0339735
$816$	0	0
$817$	−7.13726	−0.249701
$818$	0	0
$819$	−10.3960	−0.363267
$820$	0	0
$821$	49.1505	1.71536	0.857682	−	0.514181i	$-0.171904\pi$
$821$	49.1505	1.71536	0.857682	+	0.514181i	$0.171904\pi$
$822$	0	0
$823$	44.9231	1.56592	0.782961	−	0.622071i	$-0.213708\pi$
$823$	44.9231	1.56592	0.782961	+	0.622071i	$0.213708\pi$
$824$	0	0
$825$	21.0616	0.733270
$826$	0	0
$827$	−44.3547	−1.54236	−0.771182	−	0.636615i	$-0.780334\pi$
$827$	−44.3547	−1.54236	−0.771182	+	0.636615i	$0.780334\pi$
$828$	0	0
$829$	−33.4857	−1.16301	−0.581504	−	0.813544i	$-0.697536\pi$
$829$	−33.4857	−1.16301	−0.581504	+	0.813544i	$0.697536\pi$
$830$	0	0
$831$	15.4129	0.534669
$832$	0	0
$833$	−9.71810	−0.336712
$834$	0	0
$835$	3.18732	0.110302
$836$	0	0
$837$	38.9963	1.34791
$838$	0	0
$839$	−16.9136	−0.583923	−0.291961	−	0.956430i	$-0.594308\pi$
$839$	−16.9136	−0.583923	−0.291961	+	0.956430i	$0.594308\pi$
$840$	0	0
$841$	14.0194	0.483428
$842$	0	0
$843$	1.63541	0.0563265
$844$	0	0
$845$	−1.64161	−0.0564732
$846$	0	0
$847$	−0.737522	−0.0253416
$848$	0	0
$849$	19.3427	0.663838
$850$	0	0
$851$	42.5879	1.45989
$852$	0	0
$853$	−44.1530	−1.51177	−0.755885	−	0.654705i	$-0.772793\pi$
$853$	−44.1530	−1.51177	−0.755885	+	0.654705i	$0.772793\pi$
$854$	0	0
$855$	−0.402246	−0.0137565
$856$	0	0
$857$	−46.3887	−1.58461	−0.792303	−	0.610128i	$-0.791118\pi$
$857$	−46.3887	−1.58461	−0.792303	+	0.610128i	$0.791118\pi$
$858$	0	0
$859$	34.3378	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$859$	34.3378	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$860$	0	0
$861$	−11.0037	−0.375005
$862$	0	0
$863$	24.0132	0.817419	0.408710	−	0.912664i	$-0.365979\pi$
$863$	24.0132	0.817419	0.408710	+	0.912664i	$0.365979\pi$
$864$	0	0
$865$	0.561431	0.0190892
$866$	0	0
$867$	−10.5564	−0.358514
$868$	0	0
$869$	22.6135	0.767110
$870$	0	0
$871$	36.9682	1.25262
$872$	0	0
$873$	−21.5477	−0.729279
$874$	0	0
$875$	6.13356	0.207352
$876$	0	0
$877$	−44.7325	−1.51051	−0.755255	−	0.655432i	$-0.772487\pi$
$877$	−44.7325	−1.51051	−0.755255	+	0.655432i	$0.772487\pi$
$878$	0	0
$879$	−26.4118	−0.890847
$880$	0	0
$881$	−24.6073	−0.829040	−0.414520	−	0.910040i	$-0.636051\pi$
$881$	−24.6073	−0.829040	−0.414520	+	0.910040i	$0.636051\pi$
$882$	0	0
$883$	−27.9806	−0.941622	−0.470811	−	0.882234i	$-0.656039\pi$
$883$	−27.9806	−0.941622	−0.470811	+	0.882234i	$0.656039\pi$
$884$	0	0
$885$	0.734207	0.0246801
$886$	0	0
$887$	−46.3060	−1.55480	−0.777401	−	0.629005i	$-0.783462\pi$
$887$	−46.3060	−1.55480	−0.777401	+	0.629005i	$0.783462\pi$
$888$	0	0
$889$	−36.3316	−1.21852
$890$	0	0
$891$	11.8591	0.397293
$892$	0	0
$893$	6.31955	0.211476
$894$	0	0
$895$	−1.37411	−0.0459315
$896$	0	0
$897$	−31.5808	−1.05445
$898$	0	0
$899$	45.5139	1.51797
$900$	0	0
$901$	−15.9587	−0.531660
$902$	0	0
$903$	18.2637	0.607776
$904$	0	0
$905$	−4.61268	−0.153331
$906$	0	0
$907$	−6.99380	−0.232225	−0.116113	−	0.993236i	$-0.537043\pi$
$907$	−6.99380	−0.232225	−0.116113	+	0.993236i	$0.537043\pi$
$908$	0	0
$909$	−9.26579	−0.307327
$910$	0	0
$911$	2.19182	0.0726181	0.0363090	−	0.999341i	$-0.488440\pi$
$911$	2.19182	0.0726181	0.0363090	+	0.999341i	$0.488440\pi$
$912$	0	0
$913$	−49.2658	−1.63046
$914$	0	0
$915$	−0.926825	−0.0306399
$916$	0	0
$917$	−24.0765	−0.795076
$918$	0	0
$919$	−29.3716	−0.968880	−0.484440	−	0.874825i	$-0.660977\pi$
$919$	−29.3716	−0.968880	−0.484440	+	0.874825i	$0.660977\pi$
$920$	0	0
$921$	−25.6230	−0.844307
$922$	0	0
$923$	38.9963	1.28358
$924$	0	0
$925$	37.1178	1.22043
$926$	0	0
$927$	19.0301	0.625031
$928$	0	0
$929$	21.7288	0.712899	0.356449	−	0.934315i	$-0.383987\pi$
$929$	21.7288	0.712899	0.356449	+	0.934315i	$0.383987\pi$
$930$	0	0
$931$	3.23937	0.106166
$932$	0	0
$933$	−39.3972	−1.28981
$934$	0	0
$935$	3.12404	0.102167
$936$	0	0
$937$	−0.917305	−0.0299670	−0.0149835	−	0.999888i	$-0.504770\pi$
$937$	−0.917305	−0.0299670	−0.0149835	+	0.999888i	$0.504770\pi$
$938$	0	0
$939$	−23.0318	−0.751615
$940$	0	0
$941$	−19.3716	−0.631496	−0.315748	−	0.948843i	$-0.602255\pi$
$941$	−19.3716	−0.631496	−0.315748	+	0.948843i	$0.602255\pi$
$942$	0	0
$943$	24.1654	0.786933
$944$	0	0
$945$	3.48243	0.113283
$946$	0	0
$947$	46.7532	1.51928	0.759638	−	0.650346i	$-0.225376\pi$
$947$	46.7532	1.51928	0.759638	+	0.650346i	$0.225376\pi$
$948$	0	0
$949$	67.7727	2.19999
$950$	0	0
$951$	15.2236	0.493660
$952$	0	0
$953$	−13.7313	−0.444801	−0.222400	−	0.974955i	$-0.571389\pi$
$953$	−13.7313	−0.444801	−0.222400	+	0.974955i	$0.571389\pi$
$954$	0	0
$955$	7.58454	0.245430
$956$	0	0
$957$	28.2042	0.911713
$958$	0	0
$959$	23.8103	0.768875
$960$	0	0
$961$	17.1530	0.553322
$962$	0	0
$963$	15.2782	0.492333
$964$	0	0
$965$	6.61348	0.212895
$966$	0	0
$967$	8.47371	0.272496	0.136248	−	0.990675i	$-0.456496\pi$
$967$	8.47371	0.272496	0.136248	+	0.990675i	$0.456496\pi$
$968$	0	0
$969$	−3.95865	−0.127170
$970$	0	0
$971$	−59.1054	−1.89678	−0.948392	−	0.317101i	$-0.897291\pi$
$971$	−59.1054	−1.89678	−0.948392	+	0.317101i	$0.897291\pi$
$972$	0	0
$973$	−25.7119	−0.824286
$974$	0	0
$975$	−27.5246	−0.881492
$976$	0	0
$977$	−38.9268	−1.24538	−0.622690	−	0.782469i	$-0.713960\pi$
$977$	−38.9268	−1.24538	−0.622690	+	0.782469i	$0.713960\pi$
$978$	0	0
$979$	18.5746	0.593648
$980$	0	0
$981$	−11.4254	−0.364784
$982$	0	0
$983$	51.5089	1.64288	0.821439	−	0.570297i	$-0.193172\pi$
$983$	51.5089	1.64288	0.821439	+	0.570297i	$0.193172\pi$
$984$	0	0
$985$	−5.03514	−0.160433
$986$	0	0
$987$	−16.1712	−0.514736
$988$	0	0
$989$	−40.1091	−1.27540
$990$	0	0
$991$	28.9707	0.920284	0.460142	−	0.887845i	$-0.347798\pi$
$991$	28.9707	0.920284	0.460142	+	0.887845i	$0.347798\pi$
$992$	0	0
$993$	14.9367	0.474003
$994$	0	0
$995$	3.14967	0.0998511
$996$	0	0
$997$	−27.9111	−0.883953	−0.441977	−	0.897027i	$-0.645723\pi$
$997$	−27.9111	−0.883953	−0.441977	+	0.897027i	$0.645723\pi$
$998$	0	0
$999$	42.5879	1.34742

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bf.1.2		3
4.3	odd	2		4864.2.a.bd.1.2		3
8.3	odd	2		4864.2.a.be.1.2		3
8.5	even	2		4864.2.a.bc.1.2		3
16.3	odd	4		2432.2.c.f.1217.3	✓	6
16.5	even	4		2432.2.c.g.1217.3	yes	6
16.11	odd	4		2432.2.c.f.1217.4	yes	6
16.13	even	4		2432.2.c.g.1217.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.f.1217.3	✓	6	16.3	odd	4
2432.2.c.f.1217.4	yes	6	16.11	odd	4
2432.2.c.g.1217.3	yes	6	16.5	even	4
2432.2.c.g.1217.4	yes	6	16.13	even	4
4864.2.a.bc.1.2		3	8.5	even	2
4864.2.a.bd.1.2		3	4.3	odd	2
4864.2.a.be.1.2		3	8.3	odd	2
4864.2.a.bf.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.31955 q^{3} -0.319551 q^{5} +1.93923 q^{7} -1.25879 q^{9} +O(q^{10})\)	\(q+1.31955 q^{3} -0.319551 q^{5} +1.93923 q^{7} -1.25879 q^{9} -3.25879 q^{11} +4.25879 q^{13} -0.421664 q^{15} +3.00000 q^{17} -1.00000 q^{19} +2.55892 q^{21} -5.61968 q^{23} -4.89789 q^{25} -5.61968 q^{27} -6.55892 q^{29} -6.93923 q^{31} -4.30013 q^{33} -0.619684 q^{35} -7.57834 q^{37} +5.61968 q^{39} -4.30013 q^{41} +7.13726 q^{43} +0.402246 q^{45} -6.31955 q^{47} -3.23937 q^{49} +3.95865 q^{51} -5.31955 q^{53} +1.04135 q^{55} -1.31955 q^{57} -1.74121 q^{59} +2.19802 q^{61} -2.44108 q^{63} -1.36090 q^{65} +8.68045 q^{67} -7.41546 q^{69} +9.15667 q^{71} +15.9136 q^{73} -6.46301 q^{75} -6.31955 q^{77} -6.93923 q^{79} -3.63910 q^{81} +15.1178 q^{83} -0.958652 q^{85} -8.65483 q^{87} -5.69987 q^{89} +8.25879 q^{91} -9.15667 q^{93} +0.319551 q^{95} +17.1178 q^{97} +4.10211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 2 q^{5} - 3 q^{7} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 + 2 * q^5 - 3 * q^7 + 8 * q^9	\( 3 q + q^{3} + 2 q^{5} - 3 q^{7} + 8 q^{9} + 2 q^{11} + q^{13} - 16 q^{15} + 9 q^{17} - 3 q^{19} - 7 q^{21} - 11 q^{23} + 3 q^{25} - 11 q^{27} - 5 q^{29} - 12 q^{31} - 10 q^{33} + 4 q^{35} - 8 q^{37} + 11 q^{39} - 10 q^{41} - 8 q^{43} + 16 q^{45} - 16 q^{47} + 2 q^{49} + 3 q^{51} - 13 q^{53} + 12 q^{55} - q^{57} - 17 q^{59} - 14 q^{61} - 22 q^{63} - 10 q^{65} + 29 q^{67} + 19 q^{69} - 2 q^{71} - 17 q^{73} - 43 q^{75} - 16 q^{77} - 12 q^{79} - 5 q^{81} + 16 q^{83} + 6 q^{85} + 27 q^{87} - 20 q^{89} + 13 q^{91} + 2 q^{93} - 2 q^{95} + 22 q^{97} + 30 q^{99}+O(q^{100}) \) 3 * q + q^3 + 2 * q^5 - 3 * q^7 + 8 * q^9 + 2 * q^11 + q^13 - 16 * q^15 + 9 * q^17 - 3 * q^19 - 7 * q^21 - 11 * q^23 + 3 * q^25 - 11 * q^27 - 5 * q^29 - 12 * q^31 - 10 * q^33 + 4 * q^35 - 8 * q^37 + 11 * q^39 - 10 * q^41 - 8 * q^43 + 16 * q^45 - 16 * q^47 + 2 * q^49 + 3 * q^51 - 13 * q^53 + 12 * q^55 - q^57 - 17 * q^59 - 14 * q^61 - 22 * q^63 - 10 * q^65 + 29 * q^67 + 19 * q^69 - 2 * q^71 - 17 * q^73 - 43 * q^75 - 16 * q^77 - 12 * q^79 - 5 * q^81 + 16 * q^83 + 6 * q^85 + 27 * q^87 - 20 * q^89 + 13 * q^91 + 2 * q^93 - 2 * q^95 + 22 * q^97 + 30 * q^99

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