LMFDB - Embedded newform 8048.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11496 q^{3} +0.544032 q^{5} +4.06617 q^{7} +6.70298 q^{9} +O(q^{10})$	$q-3.11496 q^{3} +0.544032 q^{5} +4.06617 q^{7} +6.70298 q^{9} +2.60847 q^{11} -2.55433 q^{13} -1.69464 q^{15} -7.12126 q^{17} +0.832421 q^{19} -12.6660 q^{21} -0.370152 q^{23} -4.70403 q^{25} -11.5347 q^{27} +0.116099 q^{29} -4.33684 q^{31} -8.12528 q^{33} +2.21213 q^{35} +10.0662 q^{37} +7.95665 q^{39} -7.49744 q^{41} +11.6950 q^{43} +3.64663 q^{45} -4.33828 q^{47} +9.53378 q^{49} +22.1824 q^{51} -0.218817 q^{53} +1.41909 q^{55} -2.59296 q^{57} +8.73530 q^{59} -2.76890 q^{61} +27.2555 q^{63} -1.38964 q^{65} +6.35707 q^{67} +1.15301 q^{69} +12.8476 q^{71} -0.0547269 q^{73} +14.6529 q^{75} +10.6065 q^{77} -7.48916 q^{79} +15.8211 q^{81} +1.01360 q^{83} -3.87419 q^{85} -0.361643 q^{87} -2.49359 q^{89} -10.3864 q^{91} +13.5091 q^{93} +0.452863 q^{95} +0.239287 q^{97} +17.4845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	−3.11496	−1.79842	−0.899212	−	0.437513i	$-0.855859\pi$
$3$	−3.11496	−1.79842	−0.899212	+	0.437513i	$0.855859\pi$
$4$	0	0
$5$	0.544032	0.243298	0.121649	−	0.992573i	$-0.461182\pi$
$5$	0.544032	0.243298	0.121649	+	0.992573i	$0.461182\pi$
$6$	0	0
$7$	4.06617	1.53687	0.768435	−	0.639928i	$-0.221036\pi$
$7$	4.06617	1.53687	0.768435	+	0.639928i	$0.221036\pi$
$8$	0	0
$9$	6.70298	2.23433
$10$	0	0
$11$	2.60847	0.786483	0.393241	−	0.919435i	$-0.371354\pi$
$11$	2.60847	0.786483	0.393241	+	0.919435i	$0.371354\pi$
$12$	0	0
$13$	−2.55433	−0.708444	−0.354222	−	0.935161i	$-0.615254\pi$
$13$	−2.55433	−0.708444	−0.354222	+	0.935161i	$0.615254\pi$
$14$	0	0
$15$	−1.69464	−0.437553
$16$	0	0
$17$	−7.12126	−1.72716	−0.863579	−	0.504213i	$-0.831783\pi$
$17$	−7.12126	−1.72716	−0.863579	+	0.504213i	$0.831783\pi$
$18$	0	0
$19$	0.832421	0.190970	0.0954852	−	0.995431i	$-0.469560\pi$
$19$	0.832421	0.190970	0.0954852	+	0.995431i	$0.469560\pi$
$20$	0	0
$21$	−12.6660	−2.76394
$22$	0	0
$23$	−0.370152	−0.0771820	−0.0385910	−	0.999255i	$-0.512287\pi$
$23$	−0.370152	−0.0771820	−0.0385910	+	0.999255i	$0.512287\pi$
$24$	0	0
$25$	−4.70403	−0.940806
$26$	0	0
$27$	−11.5347	−2.21985
$28$	0	0
$29$	0.116099	0.0215590	0.0107795	−	0.999942i	$-0.496569\pi$
$29$	0.116099	0.0215590	0.0107795	+	0.999942i	$0.496569\pi$
$30$	0	0
$31$	−4.33684	−0.778920	−0.389460	−	0.921043i	$-0.627338\pi$
$31$	−4.33684	−0.778920	−0.389460	+	0.921043i	$0.627338\pi$
$32$	0	0
$33$	−8.12528	−1.41443
$34$	0	0
$35$	2.21213	0.373918
$36$	0	0
$37$	10.0662	1.65487	0.827437	−	0.561559i	$-0.189798\pi$
$37$	10.0662	1.65487	0.827437	+	0.561559i	$0.189798\pi$
$38$	0	0
$39$	7.95665	1.27408
$40$	0	0
$41$	−7.49744	−1.17090	−0.585452	−	0.810707i	$-0.699083\pi$
$41$	−7.49744	−1.17090	−0.585452	+	0.810707i	$0.699083\pi$
$42$	0	0
$43$	11.6950	1.78347	0.891733	−	0.452562i	$-0.149490\pi$
$43$	11.6950	1.78347	0.891733	+	0.452562i	$0.149490\pi$
$44$	0	0
$45$	3.64663	0.543608
$46$	0	0
$47$	−4.33828	−0.632803	−0.316401	−	0.948625i	$-0.602475\pi$
$47$	−4.33828	−0.632803	−0.316401	+	0.948625i	$0.602475\pi$
$48$	0	0
$49$	9.53378	1.36197
$50$	0	0
$51$	22.1824	3.10616
$52$	0	0
$53$	−0.218817	−0.0300569	−0.0150284	−	0.999887i	$-0.504784\pi$
$53$	−0.218817	−0.0300569	−0.0150284	+	0.999887i	$0.504784\pi$
$54$	0	0
$55$	1.41909	0.191350
$56$	0	0
$57$	−2.59296	−0.343446
$58$	0	0
$59$	8.73530	1.13724	0.568619	−	0.822601i	$-0.307478\pi$
$59$	8.73530	1.13724	0.568619	+	0.822601i	$0.307478\pi$
$60$	0	0
$61$	−2.76890	−0.354521	−0.177260	−	0.984164i	$-0.556723\pi$
$61$	−2.76890	−0.354521	−0.177260	+	0.984164i	$0.556723\pi$
$62$	0	0
$63$	27.2555	3.43387
$64$	0	0
$65$	−1.38964	−0.172363
$66$	0	0
$67$	6.35707	0.776639	0.388320	−	0.921525i	$-0.373056\pi$
$67$	6.35707	0.776639	0.388320	+	0.921525i	$0.373056\pi$
$68$	0	0
$69$	1.15301	0.138806
$70$	0	0
$71$	12.8476	1.52473	0.762367	−	0.647145i	$-0.224037\pi$
$71$	12.8476	1.52473	0.762367	+	0.647145i	$0.224037\pi$
$72$	0	0
$73$	−0.0547269	−0.00640529	−0.00320265	−	0.999995i	$-0.501019\pi$
$73$	−0.0547269	−0.00640529	−0.00320265	+	0.999995i	$0.501019\pi$
$74$	0	0
$75$	14.6529	1.69197
$76$	0	0
$77$	10.6065	1.20872
$78$	0	0
$79$	−7.48916	−0.842596	−0.421298	−	0.906922i	$-0.638425\pi$
$79$	−7.48916	−0.842596	−0.421298	+	0.906922i	$0.638425\pi$
$80$	0	0
$81$	15.8211	1.75789
$82$	0	0
$83$	1.01360	0.111257	0.0556286	−	0.998452i	$-0.482284\pi$
$83$	1.01360	0.111257	0.0556286	+	0.998452i	$0.482284\pi$
$84$	0	0
$85$	−3.87419	−0.420215
$86$	0	0
$87$	−0.361643	−0.0387722
$88$	0	0
$89$	−2.49359	−0.264320	−0.132160	−	0.991228i	$-0.542191\pi$
$89$	−2.49359	−0.264320	−0.132160	+	0.991228i	$0.542191\pi$
$90$	0	0
$91$	−10.3864	−1.08879
$92$	0	0
$93$	13.5091	1.40083
$94$	0	0
$95$	0.452863	0.0464628
$96$	0	0
$97$	0.239287	0.0242959	0.0121480	−	0.999926i	$-0.496133\pi$
$97$	0.239287	0.0242959	0.0121480	+	0.999926i	$0.496133\pi$
$98$	0	0
$99$	17.4845	1.75726
$100$	0	0
$101$	−4.28642	−0.426515	−0.213257	−	0.976996i	$-0.568407\pi$
$101$	−4.28642	−0.426515	−0.213257	+	0.976996i	$0.568407\pi$
$102$	0	0
$103$	5.43439	0.535467	0.267733	−	0.963493i	$-0.413725\pi$
$103$	5.43439	0.535467	0.267733	+	0.963493i	$0.413725\pi$
$104$	0	0
$105$	−6.89069	−0.672463
$106$	0	0
$107$	−15.3026	−1.47936	−0.739680	−	0.672958i	$-0.765023\pi$
$107$	−15.3026	−1.47936	−0.739680	+	0.672958i	$0.765023\pi$
$108$	0	0
$109$	8.49708	0.813873	0.406936	−	0.913456i	$-0.366597\pi$
$109$	8.49708	0.813873	0.406936	+	0.913456i	$0.366597\pi$
$110$	0	0
$111$	−31.3558	−2.97616
$112$	0	0
$113$	4.72334	0.444335	0.222167	−	0.975009i	$-0.428687\pi$
$113$	4.72334	0.444335	0.222167	+	0.975009i	$0.428687\pi$
$114$	0	0
$115$	−0.201374	−0.0187782
$116$	0	0
$117$	−17.1217	−1.58290
$118$	0	0
$119$	−28.9563	−2.65442
$120$	0	0
$121$	−4.19589	−0.381445
$122$	0	0
$123$	23.3542	2.10578
$124$	0	0
$125$	−5.27930	−0.472195
$126$	0	0
$127$	18.5536	1.64637	0.823183	−	0.567775i	$-0.192196\pi$
$127$	18.5536	1.64637	0.823183	+	0.567775i	$0.192196\pi$
$128$	0	0
$129$	−36.4294	−3.20743
$130$	0	0
$131$	3.36136	0.293683	0.146842	−	0.989160i	$-0.453089\pi$
$131$	3.36136	0.293683	0.146842	+	0.989160i	$0.453089\pi$
$132$	0	0
$133$	3.38477	0.293497
$134$	0	0
$135$	−6.27522	−0.540085
$136$	0	0
$137$	12.6541	1.08111	0.540557	−	0.841307i	$-0.318213\pi$
$137$	12.6541	1.08111	0.540557	+	0.841307i	$0.318213\pi$
$138$	0	0
$139$	−1.90938	−0.161951	−0.0809757	−	0.996716i	$-0.525804\pi$
$139$	−1.90938	−0.161951	−0.0809757	+	0.996716i	$0.525804\pi$
$140$	0	0
$141$	13.5136	1.13805
$142$	0	0
$143$	−6.66289	−0.557179
$144$	0	0
$145$	0.0631614	0.00524527
$146$	0	0
$147$	−29.6973	−2.44940
$148$	0	0
$149$	−8.93385	−0.731889	−0.365945	−	0.930637i	$-0.619254\pi$
$149$	−8.93385	−0.731889	−0.365945	+	0.930637i	$0.619254\pi$
$150$	0	0
$151$	−0.476372	−0.0387666	−0.0193833	−	0.999812i	$-0.506170\pi$
$151$	−0.476372	−0.0387666	−0.0193833	+	0.999812i	$0.506170\pi$
$152$	0	0
$153$	−47.7337	−3.85904
$154$	0	0
$155$	−2.35938	−0.189510
$156$	0	0
$157$	2.99854	0.239310	0.119655	−	0.992816i	$-0.461821\pi$
$157$	2.99854	0.239310	0.119655	+	0.992816i	$0.461821\pi$
$158$	0	0
$159$	0.681608	0.0540550
$160$	0	0
$161$	−1.50510	−0.118619
$162$	0	0
$163$	19.0849	1.49485	0.747424	−	0.664347i	$-0.231290\pi$
$163$	19.0849	1.49485	0.747424	+	0.664347i	$0.231290\pi$
$164$	0	0
$165$	−4.42041	−0.344128
$166$	0	0
$167$	−0.857327	−0.0663420	−0.0331710	−	0.999450i	$-0.510561\pi$
$167$	−0.857327	−0.0663420	−0.0331710	+	0.999450i	$0.510561\pi$
$168$	0	0
$169$	−6.47539	−0.498107
$170$	0	0
$171$	5.57970	0.426691
$172$	0	0
$173$	9.35691	0.711392	0.355696	−	0.934602i	$-0.384244\pi$
$173$	9.35691	0.711392	0.355696	+	0.934602i	$0.384244\pi$
$174$	0	0
$175$	−19.1274	−1.44590
$176$	0	0
$177$	−27.2101	−2.04524
$178$	0	0
$179$	−11.5381	−0.862400	−0.431200	−	0.902256i	$-0.641910\pi$
$179$	−11.5381	−0.862400	−0.431200	+	0.902256i	$0.641910\pi$
$180$	0	0
$181$	24.5774	1.82682	0.913410	−	0.407040i	$-0.133439\pi$
$181$	24.5774	1.82682	0.913410	+	0.407040i	$0.133439\pi$
$182$	0	0
$183$	8.62501	0.637579
$184$	0	0
$185$	5.47633	0.402628
$186$	0	0
$187$	−18.5756	−1.35838
$188$	0	0
$189$	−46.9019	−3.41161
$190$	0	0
$191$	−3.99705	−0.289216	−0.144608	−	0.989489i	$-0.546192\pi$
$191$	−3.99705	−0.289216	−0.144608	+	0.989489i	$0.546192\pi$
$192$	0	0
$193$	−2.52415	−0.181692	−0.0908460	−	0.995865i	$-0.528957\pi$
$193$	−2.52415	−0.181692	−0.0908460	+	0.995865i	$0.528957\pi$
$194$	0	0
$195$	4.32867	0.309982
$196$	0	0
$197$	12.5250	0.892370	0.446185	−	0.894941i	$-0.352782\pi$
$197$	12.5250	0.892370	0.446185	+	0.894941i	$0.352782\pi$
$198$	0	0
$199$	−19.3111	−1.36893	−0.684464	−	0.729047i	$-0.739964\pi$
$199$	−19.3111	−1.36893	−0.684464	+	0.729047i	$0.739964\pi$
$200$	0	0
$201$	−19.8020	−1.39673
$202$	0	0
$203$	0.472078	0.0331334
$204$	0	0
$205$	−4.07884	−0.284879
$206$	0	0
$207$	−2.48112	−0.172450
$208$	0	0
$209$	2.17134	0.150195
$210$	0	0
$211$	15.9644	1.09903	0.549517	−	0.835483i	$-0.314812\pi$
$211$	15.9644	1.09903	0.549517	+	0.835483i	$0.314812\pi$
$212$	0	0
$213$	−40.0199	−2.74212
$214$	0	0
$215$	6.36243	0.433914
$216$	0	0
$217$	−17.6344	−1.19710
$218$	0	0
$219$	0.170472	0.0115194
$220$	0	0
$221$	18.1901	1.22360
$222$	0	0
$223$	3.28709	0.220120	0.110060	−	0.993925i	$-0.464896\pi$
$223$	3.28709	0.220120	0.110060	+	0.993925i	$0.464896\pi$
$224$	0	0
$225$	−31.5310	−2.10207
$226$	0	0
$227$	3.49893	0.232232	0.116116	−	0.993236i	$-0.462955\pi$
$227$	3.49893	0.232232	0.116116	+	0.993236i	$0.462955\pi$
$228$	0	0
$229$	19.3346	1.27767	0.638834	−	0.769345i	$-0.279417\pi$
$229$	19.3346	1.27767	0.638834	+	0.769345i	$0.279417\pi$
$230$	0	0
$231$	−33.0388	−2.17379
$232$	0	0
$233$	3.60251	0.236009	0.118004	−	0.993013i	$-0.462350\pi$
$233$	3.60251	0.236009	0.118004	+	0.993013i	$0.462350\pi$
$234$	0	0
$235$	−2.36016	−0.153960
$236$	0	0
$237$	23.3284	1.51534
$238$	0	0
$239$	12.8720	0.832619	0.416310	−	0.909223i	$-0.363323\pi$
$239$	12.8720	0.832619	0.416310	+	0.909223i	$0.363323\pi$
$240$	0	0
$241$	−17.0737	−1.09982	−0.549908	−	0.835226i	$-0.685337\pi$
$241$	−17.0737	−1.09982	−0.549908	+	0.835226i	$0.685337\pi$
$242$	0	0
$243$	−14.6780	−0.941594
$244$	0	0
$245$	5.18667	0.331364
$246$	0	0
$247$	−2.12628	−0.135292
$248$	0	0
$249$	−3.15733	−0.200088
$250$	0	0
$251$	27.4865	1.73493	0.867466	−	0.497496i	$-0.165747\pi$
$251$	27.4865	1.73493	0.867466	+	0.497496i	$0.165747\pi$
$252$	0	0
$253$	−0.965529	−0.0607023
$254$	0	0
$255$	12.0679	0.755724
$256$	0	0
$257$	3.74116	0.233367	0.116683	−	0.993169i	$-0.462774\pi$
$257$	3.74116	0.233367	0.116683	+	0.993169i	$0.462774\pi$
$258$	0	0
$259$	40.9309	2.54332
$260$	0	0
$261$	0.778209	0.0481699
$262$	0	0
$263$	−7.02615	−0.433251	−0.216625	−	0.976255i	$-0.569505\pi$
$263$	−7.02615	−0.433251	−0.216625	+	0.976255i	$0.569505\pi$
$264$	0	0
$265$	−0.119044	−0.00731279
$266$	0	0
$267$	7.76745	0.475360
$268$	0	0
$269$	−6.55451	−0.399635	−0.199818	−	0.979833i	$-0.564035\pi$
$269$	−6.55451	−0.399635	−0.199818	+	0.979833i	$0.564035\pi$
$270$	0	0
$271$	−8.14696	−0.494893	−0.247446	−	0.968902i	$-0.579591\pi$
$271$	−8.14696	−0.494893	−0.247446	+	0.968902i	$0.579591\pi$
$272$	0	0
$273$	32.3531	1.95810
$274$	0	0
$275$	−12.2703	−0.739928
$276$	0	0
$277$	4.85761	0.291866	0.145933	−	0.989295i	$-0.453382\pi$
$277$	4.85761	0.291866	0.145933	+	0.989295i	$0.453382\pi$
$278$	0	0
$279$	−29.0698	−1.74036
$280$	0	0
$281$	21.4279	1.27828	0.639141	−	0.769089i	$-0.279290\pi$
$281$	21.4279	1.27828	0.639141	+	0.769089i	$0.279290\pi$
$282$	0	0
$283$	−23.7790	−1.41351	−0.706757	−	0.707457i	$-0.749842\pi$
$283$	−23.7790	−1.41351	−0.706757	+	0.707457i	$0.749842\pi$
$284$	0	0
$285$	−1.41065	−0.0835597
$286$	0	0
$287$	−30.4859	−1.79953
$288$	0	0
$289$	33.7123	1.98308
$290$	0	0
$291$	−0.745369	−0.0436943
$292$	0	0
$293$	−9.93680	−0.580514	−0.290257	−	0.956949i	$-0.593741\pi$
$293$	−9.93680	−0.580514	−0.290257	+	0.956949i	$0.593741\pi$
$294$	0	0
$295$	4.75228	0.276688
$296$	0	0
$297$	−30.0878	−1.74587
$298$	0	0
$299$	0.945491	0.0546791
$300$	0	0
$301$	47.5538	2.74095
$302$	0	0
$303$	13.3520	0.767054
$304$	0	0
$305$	−1.50637	−0.0862543
$306$	0	0
$307$	15.8282	0.903361	0.451681	−	0.892180i	$-0.350825\pi$
$307$	15.8282	0.903361	0.451681	+	0.892180i	$0.350825\pi$
$308$	0	0
$309$	−16.9279	−0.962996
$310$	0	0
$311$	1.74402	0.0988942	0.0494471	−	0.998777i	$-0.484254\pi$
$311$	1.74402	0.0988942	0.0494471	+	0.998777i	$0.484254\pi$
$312$	0	0
$313$	28.9429	1.63595	0.817975	−	0.575254i	$-0.195097\pi$
$313$	28.9429	1.63595	0.817975	+	0.575254i	$0.195097\pi$
$314$	0	0
$315$	14.8279	0.835455
$316$	0	0
$317$	−0.504520	−0.0283366	−0.0141683	−	0.999900i	$-0.504510\pi$
$317$	−0.504520	−0.0283366	−0.0141683	+	0.999900i	$0.504510\pi$
$318$	0	0
$319$	0.302840	0.0169558
$320$	0	0
$321$	47.6671	2.66052
$322$	0	0
$323$	−5.92788	−0.329836
$324$	0	0
$325$	12.0157	0.666509
$326$	0	0
$327$	−26.4681	−1.46369
$328$	0	0
$329$	−17.6402	−0.972535
$330$	0	0
$331$	−8.07805	−0.444010	−0.222005	−	0.975046i	$-0.571260\pi$
$331$	−8.07805	−0.444010	−0.222005	+	0.975046i	$0.571260\pi$
$332$	0	0
$333$	67.4736	3.69753
$334$	0	0
$335$	3.45844	0.188955
$336$	0	0
$337$	34.0352	1.85402	0.927008	−	0.375041i	$-0.122372\pi$
$337$	34.0352	1.85402	0.927008	+	0.375041i	$0.122372\pi$
$338$	0	0
$339$	−14.7130	−0.799102
$340$	0	0
$341$	−11.3125	−0.612607
$342$	0	0
$343$	10.3028	0.556298
$344$	0	0
$345$	0.627273	0.0337712
$346$	0	0
$347$	−26.3874	−1.41655	−0.708275	−	0.705936i	$-0.750526\pi$
$347$	−26.3874	−1.41655	−0.708275	+	0.705936i	$0.750526\pi$
$348$	0	0
$349$	−36.3265	−1.94452	−0.972258	−	0.233912i	$-0.924847\pi$
$349$	−36.3265	−1.94452	−0.972258	+	0.233912i	$0.924847\pi$
$350$	0	0
$351$	29.4633	1.57264
$352$	0	0
$353$	15.6978	0.835511	0.417755	−	0.908560i	$-0.362817\pi$
$353$	15.6978	0.835511	0.417755	+	0.908560i	$0.362817\pi$
$354$	0	0
$355$	6.98952	0.370965
$356$	0	0
$357$	90.1977	4.77377
$358$	0	0
$359$	−22.9202	−1.20968	−0.604841	−	0.796346i	$-0.706763\pi$
$359$	−22.9202	−1.20968	−0.604841	+	0.796346i	$0.706763\pi$
$360$	0	0
$361$	−18.3071	−0.963530
$362$	0	0
$363$	13.0700	0.685999
$364$	0	0
$365$	−0.0297731	−0.00155840
$366$	0	0
$367$	20.1516	1.05190	0.525952	−	0.850514i	$-0.323709\pi$
$367$	20.1516	1.05190	0.525952	+	0.850514i	$0.323709\pi$
$368$	0	0
$369$	−50.2552	−2.61618
$370$	0	0
$371$	−0.889750	−0.0461935
$372$	0	0
$373$	5.84927	0.302864	0.151432	−	0.988468i	$-0.451612\pi$
$373$	5.84927	0.302864	0.151432	+	0.988468i	$0.451612\pi$
$374$	0	0
$375$	16.4448	0.849206
$376$	0	0
$377$	−0.296555	−0.0152734
$378$	0	0
$379$	−9.02864	−0.463770	−0.231885	−	0.972743i	$-0.574489\pi$
$379$	−9.02864	−0.463770	−0.231885	+	0.972743i	$0.574489\pi$
$380$	0	0
$381$	−57.7938	−2.96087
$382$	0	0
$383$	28.8573	1.47454	0.737270	−	0.675598i	$-0.236114\pi$
$383$	28.8573	1.47454	0.737270	+	0.675598i	$0.236114\pi$
$384$	0	0
$385$	5.77026	0.294080
$386$	0	0
$387$	78.3912	3.98485
$388$	0	0
$389$	−33.4134	−1.69413	−0.847063	−	0.531492i	$-0.821631\pi$
$389$	−33.4134	−1.69413	−0.847063	+	0.531492i	$0.821631\pi$
$390$	0	0
$391$	2.63595	0.133306
$392$	0	0
$393$	−10.4705	−0.528167
$394$	0	0
$395$	−4.07434	−0.205002
$396$	0	0
$397$	−28.3461	−1.42265	−0.711324	−	0.702864i	$-0.751904\pi$
$397$	−28.3461	−1.42265	−0.711324	+	0.702864i	$0.751904\pi$
$398$	0	0
$399$	−10.5434	−0.527831
$400$	0	0
$401$	38.4895	1.92207	0.961036	−	0.276422i	$-0.0891486\pi$
$401$	38.4895	1.92207	0.961036	+	0.276422i	$0.0891486\pi$
$402$	0	0
$403$	11.0777	0.551822
$404$	0	0
$405$	8.60715	0.427693
$406$	0	0
$407$	26.2574	1.30153
$408$	0	0
$409$	1.43137	0.0707767	0.0353884	−	0.999374i	$-0.488733\pi$
$409$	1.43137	0.0707767	0.0353884	+	0.999374i	$0.488733\pi$
$410$	0	0
$411$	−39.4171	−1.94430
$412$	0	0
$413$	35.5192	1.74779
$414$	0	0
$415$	0.551431	0.0270687
$416$	0	0
$417$	5.94764	0.291257
$418$	0	0
$419$	25.6655	1.25384	0.626921	−	0.779082i	$-0.284315\pi$
$419$	25.6655	1.25384	0.626921	+	0.779082i	$0.284315\pi$
$420$	0	0
$421$	−12.4672	−0.607616	−0.303808	−	0.952733i	$-0.598258\pi$
$421$	−12.4672	−0.607616	−0.303808	+	0.952733i	$0.598258\pi$
$422$	0	0
$423$	−29.0794	−1.41389
$424$	0	0
$425$	33.4986	1.62492
$426$	0	0
$427$	−11.2588	−0.544852
$428$	0	0
$429$	20.7547	1.00204
$430$	0	0
$431$	6.84481	0.329703	0.164852	−	0.986318i	$-0.447285\pi$
$431$	6.84481	0.329703	0.164852	+	0.986318i	$0.447285\pi$
$432$	0	0
$433$	30.0106	1.44222	0.721108	−	0.692822i	$-0.243633\pi$
$433$	30.0106	1.44222	0.721108	+	0.692822i	$0.243633\pi$
$434$	0	0
$435$	−0.196745	−0.00943322
$436$	0	0
$437$	−0.308122	−0.0147395
$438$	0	0
$439$	−31.3539	−1.49644	−0.748219	−	0.663451i	$-0.769091\pi$
$439$	−31.3539	−1.49644	−0.748219	+	0.663451i	$0.769091\pi$
$440$	0	0
$441$	63.9048	3.04308
$442$	0	0
$443$	12.4312	0.590626	0.295313	−	0.955401i	$-0.404576\pi$
$443$	12.4312	0.590626	0.295313	+	0.955401i	$0.404576\pi$
$444$	0	0
$445$	−1.35659	−0.0643087
$446$	0	0
$447$	27.8286	1.31625
$448$	0	0
$449$	0.488482	0.0230529	0.0115265	−	0.999934i	$-0.496331\pi$
$449$	0.488482	0.0230529	0.0115265	+	0.999934i	$0.496331\pi$
$450$	0	0
$451$	−19.5568	−0.920895
$452$	0	0
$453$	1.48388	0.0697187
$454$	0	0
$455$	−5.65051	−0.264900
$456$	0	0
$457$	−23.7703	−1.11193	−0.555963	−	0.831207i	$-0.687650\pi$
$457$	−23.7703	−1.11193	−0.555963	+	0.831207i	$0.687650\pi$
$458$	0	0
$459$	82.1413	3.83403
$460$	0	0
$461$	−8.58507	−0.399847	−0.199923	−	0.979812i	$-0.564069\pi$
$461$	−8.58507	−0.399847	−0.199923	+	0.979812i	$0.564069\pi$
$462$	0	0
$463$	21.2154	0.985963	0.492981	−	0.870040i	$-0.335907\pi$
$463$	21.2154	0.985963	0.492981	+	0.870040i	$0.335907\pi$
$464$	0	0
$465$	7.34938	0.340819
$466$	0	0
$467$	−8.69231	−0.402232	−0.201116	−	0.979567i	$-0.564457\pi$
$467$	−8.69231	−0.402232	−0.201116	+	0.979567i	$0.564457\pi$
$468$	0	0
$469$	25.8489	1.19359
$470$	0	0
$471$	−9.34034	−0.430380
$472$	0	0
$473$	30.5060	1.40267
$474$	0	0
$475$	−3.91573	−0.179666
$476$	0	0
$477$	−1.46673	−0.0671569
$478$	0	0
$479$	15.2454	0.696581	0.348291	−	0.937387i	$-0.386762\pi$
$479$	15.2454	0.696581	0.348291	+	0.937387i	$0.386762\pi$
$480$	0	0
$481$	−25.7124	−1.17239
$482$	0	0
$483$	4.68834	0.213327
$484$	0	0
$485$	0.130180	0.00591115
$486$	0	0
$487$	14.7583	0.668761	0.334380	−	0.942438i	$-0.391473\pi$
$487$	14.7583	0.668761	0.334380	+	0.942438i	$0.391473\pi$
$488$	0	0
$489$	−59.4489	−2.68837
$490$	0	0
$491$	−4.88193	−0.220318	−0.110159	−	0.993914i	$-0.535136\pi$
$491$	−4.88193	−0.220318	−0.110159	+	0.993914i	$0.535136\pi$
$492$	0	0
$493$	−0.826770	−0.0372358
$494$	0	0
$495$	9.51213	0.427539
$496$	0	0
$497$	52.2408	2.34332
$498$	0	0
$499$	39.8728	1.78495	0.892475	−	0.451096i	$-0.148967\pi$
$499$	39.8728	1.78495	0.892475	+	0.451096i	$0.148967\pi$
$500$	0	0
$501$	2.67054	0.119311
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−2.33195	−0.103770
$506$	0	0
$507$	20.1706	0.895807
$508$	0	0
$509$	−38.3023	−1.69772	−0.848860	−	0.528618i	$-0.822710\pi$
$509$	−38.3023	−1.69772	−0.848860	+	0.528618i	$0.822710\pi$
$510$	0	0
$511$	−0.222529	−0.00984410
$512$	0	0
$513$	−9.60168	−0.423925
$514$	0	0
$515$	2.95648	0.130278
$516$	0	0
$517$	−11.3163	−0.497688
$518$	0	0
$519$	−29.1464	−1.27939
$520$	0	0
$521$	15.1754	0.664847	0.332424	−	0.943130i	$-0.392134\pi$
$521$	15.1754	0.664847	0.332424	+	0.943130i	$0.392134\pi$
$522$	0	0
$523$	−38.9182	−1.70177	−0.850887	−	0.525349i	$-0.823935\pi$
$523$	−38.9182	−1.70177	−0.850887	+	0.525349i	$0.823935\pi$
$524$	0	0
$525$	59.5811	2.60033
$526$	0	0
$527$	30.8838	1.34532
$528$	0	0
$529$	−22.8630	−0.994043
$530$	0	0
$531$	58.5526	2.54096
$532$	0	0
$533$	19.1510	0.829520
$534$	0	0
$535$	−8.32511	−0.359926
$536$	0	0
$537$	35.9408	1.55096
$538$	0	0
$539$	24.8686	1.07116
$540$	0	0
$541$	35.1147	1.50970	0.754850	−	0.655898i	$-0.227710\pi$
$541$	35.1147	1.50970	0.754850	+	0.655898i	$0.227710\pi$
$542$	0	0
$543$	−76.5575	−3.28540
$544$	0	0
$545$	4.62268	0.198014
$546$	0	0
$547$	28.1530	1.20373	0.601867	−	0.798596i	$-0.294424\pi$
$547$	28.1530	1.20373	0.601867	+	0.798596i	$0.294424\pi$
$548$	0	0
$549$	−18.5599	−0.792116
$550$	0	0
$551$	0.0966430	0.00411713
$552$	0	0
$553$	−30.4522	−1.29496
$554$	0	0
$555$	−17.0586	−0.724096
$556$	0	0
$557$	−14.7916	−0.626740	−0.313370	−	0.949631i	$-0.601458\pi$
$557$	−14.7916	−0.626740	−0.313370	+	0.949631i	$0.601458\pi$
$558$	0	0
$559$	−29.8728	−1.26349
$560$	0	0
$561$	57.8622	2.44294
$562$	0	0
$563$	38.3498	1.61625	0.808125	−	0.589011i	$-0.200482\pi$
$563$	38.3498	1.61625	0.808125	+	0.589011i	$0.200482\pi$
$564$	0	0
$565$	2.56965	0.108106
$566$	0	0
$567$	64.3312	2.70165
$568$	0	0
$569$	−31.2247	−1.30901	−0.654504	−	0.756059i	$-0.727122\pi$
$569$	−31.2247	−1.30901	−0.654504	+	0.756059i	$0.727122\pi$
$570$	0	0
$571$	−8.85600	−0.370612	−0.185306	−	0.982681i	$-0.559328\pi$
$571$	−8.85600	−0.370612	−0.185306	+	0.982681i	$0.559328\pi$
$572$	0	0
$573$	12.4506	0.520133
$574$	0	0
$575$	1.74121	0.0726133
$576$	0	0
$577$	22.0244	0.916889	0.458445	−	0.888723i	$-0.348407\pi$
$577$	22.0244	0.916889	0.458445	+	0.888723i	$0.348407\pi$
$578$	0	0
$579$	7.86262	0.326759
$580$	0	0
$581$	4.12148	0.170988
$582$	0	0
$583$	−0.570778	−0.0236392
$584$	0	0
$585$	−9.31472	−0.385116
$586$	0	0
$587$	22.2179	0.917030	0.458515	−	0.888687i	$-0.348382\pi$
$587$	22.2179	0.917030	0.458515	+	0.888687i	$0.348382\pi$
$588$	0	0
$589$	−3.61008	−0.148751
$590$	0	0
$591$	−39.0149	−1.60486
$592$	0	0
$593$	38.3818	1.57615	0.788076	−	0.615578i	$-0.211078\pi$
$593$	38.3818	1.57615	0.788076	+	0.615578i	$0.211078\pi$
$594$	0	0
$595$	−15.7531	−0.645815
$596$	0	0
$597$	60.1534	2.46191
$598$	0	0
$599$	−15.8958	−0.649486	−0.324743	−	0.945802i	$-0.605278\pi$
$599$	−15.8958	−0.649486	−0.324743	+	0.945802i	$0.605278\pi$
$600$	0	0
$601$	33.6843	1.37401	0.687006	−	0.726652i	$-0.258925\pi$
$601$	33.6843	1.37401	0.687006	+	0.726652i	$0.258925\pi$
$602$	0	0
$603$	42.6113	1.73527
$604$	0	0
$605$	−2.28270	−0.0928049
$606$	0	0
$607$	2.28606	0.0927883	0.0463941	−	0.998923i	$-0.485227\pi$
$607$	2.28606	0.0927883	0.0463941	+	0.998923i	$0.485227\pi$
$608$	0	0
$609$	−1.47050	−0.0595879
$610$	0	0
$611$	11.0814	0.448305
$612$	0	0
$613$	15.7993	0.638127	0.319063	−	0.947733i	$-0.396632\pi$
$613$	15.7993	0.638127	0.319063	+	0.947733i	$0.396632\pi$
$614$	0	0
$615$	12.7054	0.512333
$616$	0	0
$617$	−19.6329	−0.790390	−0.395195	−	0.918597i	$-0.629323\pi$
$617$	−19.6329	−0.790390	−0.395195	+	0.918597i	$0.629323\pi$
$618$	0	0
$619$	33.1487	1.33236	0.666180	−	0.745791i	$-0.267928\pi$
$619$	33.1487	1.33236	0.666180	+	0.745791i	$0.267928\pi$
$620$	0	0
$621$	4.26957	0.171332
$622$	0	0
$623$	−10.1394	−0.406226
$624$	0	0
$625$	20.6480	0.825922
$626$	0	0
$627$	−6.76365	−0.270114
$628$	0	0
$629$	−71.6840	−2.85823
$630$	0	0
$631$	18.4604	0.734898	0.367449	−	0.930044i	$-0.380231\pi$
$631$	18.4604	0.734898	0.367449	+	0.930044i	$0.380231\pi$
$632$	0	0
$633$	−49.7285	−1.97653
$634$	0	0
$635$	10.0938	0.400558
$636$	0	0
$637$	−24.3524	−0.964878
$638$	0	0
$639$	86.1175	3.40676
$640$	0	0
$641$	−27.9324	−1.10326	−0.551632	−	0.834087i	$-0.685995\pi$
$641$	−27.9324	−1.10326	−0.551632	+	0.834087i	$0.685995\pi$
$642$	0	0
$643$	−15.9331	−0.628340	−0.314170	−	0.949367i	$-0.601726\pi$
$643$	−15.9331	−0.628340	−0.314170	+	0.949367i	$0.601726\pi$
$644$	0	0
$645$	−19.8187	−0.780362
$646$	0	0
$647$	28.7045	1.12849	0.564246	−	0.825607i	$-0.309167\pi$
$647$	28.7045	1.12849	0.564246	+	0.825607i	$0.309167\pi$
$648$	0	0
$649$	22.7857	0.894419
$650$	0	0
$651$	54.9304	2.15289
$652$	0	0
$653$	−10.9511	−0.428551	−0.214276	−	0.976773i	$-0.568739\pi$
$653$	−10.9511	−0.428551	−0.214276	+	0.976773i	$0.568739\pi$
$654$	0	0
$655$	1.82869	0.0714527
$656$	0	0
$657$	−0.366833	−0.0143115
$658$	0	0
$659$	45.7335	1.78152	0.890762	−	0.454470i	$-0.150171\pi$
$659$	45.7335	1.78152	0.890762	+	0.454470i	$0.150171\pi$
$660$	0	0
$661$	1.40358	0.0545928	0.0272964	−	0.999627i	$-0.491310\pi$
$661$	1.40358	0.0545928	0.0272964	+	0.999627i	$0.491310\pi$
$662$	0	0
$663$	−56.6613	−2.20054
$664$	0	0
$665$	1.84142	0.0714072
$666$	0	0
$667$	−0.0429742	−0.00166397
$668$	0	0
$669$	−10.2392	−0.395869
$670$	0	0
$671$	−7.22258	−0.278825
$672$	0	0
$673$	−3.36295	−0.129632	−0.0648160	−	0.997897i	$-0.520646\pi$
$673$	−3.36295	−0.129632	−0.0648160	+	0.997897i	$0.520646\pi$
$674$	0	0
$675$	54.2594	2.08844
$676$	0	0
$677$	10.0645	0.386811	0.193405	−	0.981119i	$-0.438047\pi$
$677$	10.0645	0.386811	0.193405	+	0.981119i	$0.438047\pi$
$678$	0	0
$679$	0.972982	0.0373396
$680$	0	0
$681$	−10.8990	−0.417652
$682$	0	0
$683$	−49.5438	−1.89574	−0.947871	−	0.318653i	$-0.896769\pi$
$683$	−49.5438	−1.89574	−0.947871	+	0.318653i	$0.896769\pi$
$684$	0	0
$685$	6.88424	0.263033
$686$	0	0
$687$	−60.2266	−2.29779
$688$	0	0
$689$	0.558932	0.0212936
$690$	0	0
$691$	44.3212	1.68606	0.843029	−	0.537868i	$-0.180770\pi$
$691$	44.3212	1.68606	0.843029	+	0.537868i	$0.180770\pi$
$692$	0	0
$693$	71.0951	2.70068
$694$	0	0
$695$	−1.03876	−0.0394025
$696$	0	0
$697$	53.3912	2.02234
$698$	0	0
$699$	−11.2217	−0.424443
$700$	0	0
$701$	−33.6302	−1.27020	−0.635098	−	0.772431i	$-0.719040\pi$
$701$	−33.6302	−1.27020	−0.635098	+	0.772431i	$0.719040\pi$
$702$	0	0
$703$	8.37931	0.316032
$704$	0	0
$705$	7.35180	0.276885
$706$	0	0
$707$	−17.4293	−0.655498
$708$	0	0
$709$	−35.8225	−1.34534	−0.672670	−	0.739942i	$-0.734853\pi$
$709$	−35.8225	−1.34534	−0.672670	+	0.739942i	$0.734853\pi$
$710$	0	0
$711$	−50.1997	−1.88264
$712$	0	0
$713$	1.60529	0.0601186
$714$	0	0
$715$	−3.62482	−0.135561
$716$	0	0
$717$	−40.0957	−1.49740
$718$	0	0
$719$	−49.4640	−1.84470	−0.922348	−	0.386359i	$-0.873733\pi$
$719$	−49.4640	−1.84470	−0.922348	+	0.386359i	$0.873733\pi$
$720$	0	0
$721$	22.0972	0.822942
$722$	0	0
$723$	53.1840	1.97793
$724$	0	0
$725$	−0.546132	−0.0202828
$726$	0	0
$727$	11.3440	0.420726	0.210363	−	0.977623i	$-0.432535\pi$
$727$	11.3440	0.420726	0.210363	+	0.977623i	$0.432535\pi$
$728$	0	0
$729$	−1.74176	−0.0645095
$730$	0	0
$731$	−83.2829	−3.08033
$732$	0	0
$733$	14.1414	0.522325	0.261162	−	0.965295i	$-0.415894\pi$
$733$	14.1414	0.522325	0.261162	+	0.965295i	$0.415894\pi$
$734$	0	0
$735$	−16.1563	−0.595934
$736$	0	0
$737$	16.5822	0.610813
$738$	0	0
$739$	16.5404	0.608447	0.304223	−	0.952601i	$-0.401603\pi$
$739$	16.5404	0.608447	0.304223	+	0.952601i	$0.401603\pi$
$740$	0	0
$741$	6.62328	0.243312
$742$	0	0
$743$	28.8396	1.05802	0.529012	−	0.848614i	$-0.322563\pi$
$743$	28.8396	1.05802	0.529012	+	0.848614i	$0.322563\pi$
$744$	0	0
$745$	−4.86029	−0.178067
$746$	0	0
$747$	6.79415	0.248585
$748$	0	0
$749$	−62.2232	−2.27358
$750$	0	0
$751$	4.44249	0.162109	0.0810544	−	0.996710i	$-0.474171\pi$
$751$	4.44249	0.162109	0.0810544	+	0.996710i	$0.474171\pi$
$752$	0	0
$753$	−85.6194	−3.12014
$754$	0	0
$755$	−0.259161	−0.00943184
$756$	0	0
$757$	−8.21775	−0.298679	−0.149340	−	0.988786i	$-0.547715\pi$
$757$	−8.21775	−0.298679	−0.149340	+	0.988786i	$0.547715\pi$
$758$	0	0
$759$	3.00759	0.109168
$760$	0	0
$761$	−48.5835	−1.76115	−0.880576	−	0.473905i	$-0.842844\pi$
$761$	−48.5835	−1.76115	−0.880576	+	0.473905i	$0.842844\pi$
$762$	0	0
$763$	34.5506	1.25082
$764$	0	0
$765$	−25.9686	−0.938898
$766$	0	0
$767$	−22.3128	−0.805670
$768$	0	0
$769$	9.96818	0.359462	0.179731	−	0.983716i	$-0.442477\pi$
$769$	9.96818	0.359462	0.179731	+	0.983716i	$0.442477\pi$
$770$	0	0
$771$	−11.6536	−0.419693
$772$	0	0
$773$	35.3040	1.26980	0.634899	−	0.772595i	$-0.281042\pi$
$773$	35.3040	1.26980	0.634899	+	0.772595i	$0.281042\pi$
$774$	0	0
$775$	20.4006	0.732813
$776$	0	0
$777$	−127.498	−4.57398
$778$	0	0
$779$	−6.24102	−0.223608
$780$	0	0
$781$	33.5127	1.19918
$782$	0	0
$783$	−1.33916	−0.0478577
$784$	0	0
$785$	1.63130	0.0582236
$786$	0	0
$787$	−27.9639	−0.996804	−0.498402	−	0.866946i	$-0.666080\pi$
$787$	−27.9639	−0.996804	−0.498402	+	0.866946i	$0.666080\pi$
$788$	0	0
$789$	21.8862	0.779169
$790$	0	0
$791$	19.2059	0.682884
$792$	0	0
$793$	7.07268	0.251158
$794$	0	0
$795$	0.370816	0.0131515
$796$	0	0
$797$	33.0070	1.16917	0.584583	−	0.811334i	$-0.301258\pi$
$797$	33.0070	1.16917	0.584583	+	0.811334i	$0.301258\pi$
$798$	0	0
$799$	30.8940	1.09295
$800$	0	0
$801$	−16.7145	−0.590579
$802$	0	0
$803$	−0.142753	−0.00503765
$804$	0	0
$805$	−0.818823	−0.0288597
$806$	0	0
$807$	20.4170	0.718714
$808$	0	0
$809$	−26.6876	−0.938288	−0.469144	−	0.883122i	$-0.655437\pi$
$809$	−26.6876	−0.938288	−0.469144	+	0.883122i	$0.655437\pi$
$810$	0	0
$811$	−19.8883	−0.698371	−0.349186	−	0.937054i	$-0.613542\pi$
$811$	−19.8883	−0.698371	−0.349186	+	0.937054i	$0.613542\pi$
$812$	0	0
$813$	25.3775	0.890027
$814$	0	0
$815$	10.3828	0.363694
$816$	0	0
$817$	9.73513	0.340589
$818$	0	0
$819$	−69.6196	−2.43271
$820$	0	0
$821$	41.9862	1.46533	0.732664	−	0.680590i	$-0.238277\pi$
$821$	41.9862	1.46533	0.732664	+	0.680590i	$0.238277\pi$
$822$	0	0
$823$	20.3429	0.709108	0.354554	−	0.935036i	$-0.384633\pi$
$823$	20.3429	0.709108	0.354554	+	0.935036i	$0.384633\pi$
$824$	0	0
$825$	38.2216	1.33070
$826$	0	0
$827$	21.0590	0.732292	0.366146	−	0.930558i	$-0.380677\pi$
$827$	21.0590	0.732292	0.366146	+	0.930558i	$0.380677\pi$
$828$	0	0
$829$	44.7041	1.55264	0.776319	−	0.630341i	$-0.217085\pi$
$829$	44.7041	1.55264	0.776319	+	0.630341i	$0.217085\pi$
$830$	0	0
$831$	−15.1313	−0.524898
$832$	0	0
$833$	−67.8925	−2.35233
$834$	0	0
$835$	−0.466413	−0.0161409
$836$	0	0
$837$	50.0240	1.72908
$838$	0	0
$839$	52.0847	1.79816	0.899081	−	0.437781i	$-0.144236\pi$
$839$	52.0847	1.79816	0.899081	+	0.437781i	$0.144236\pi$
$840$	0	0
$841$	−28.9865	−0.999535
$842$	0	0
$843$	−66.7471	−2.29889
$844$	0	0
$845$	−3.52281	−0.121188
$846$	0	0
$847$	−17.0612	−0.586231
$848$	0	0
$849$	74.0706	2.54210
$850$	0	0
$851$	−3.72602	−0.127726
$852$	0	0
$853$	−20.4155	−0.699012	−0.349506	−	0.936934i	$-0.613651\pi$
$853$	−20.4155	−0.699012	−0.349506	+	0.936934i	$0.613651\pi$
$854$	0	0
$855$	3.03553	0.103813
$856$	0	0
$857$	−29.9634	−1.02353	−0.511765	−	0.859126i	$-0.671008\pi$
$857$	−29.9634	−1.02353	−0.511765	+	0.859126i	$0.671008\pi$
$858$	0	0
$859$	37.1094	1.26616	0.633079	−	0.774087i	$-0.281791\pi$
$859$	37.1094	1.26616	0.633079	+	0.774087i	$0.281791\pi$
$860$	0	0
$861$	94.9624	3.23631
$862$	0	0
$863$	20.3550	0.692893	0.346446	−	0.938070i	$-0.387388\pi$
$863$	20.3550	0.692893	0.346446	+	0.938070i	$0.387388\pi$
$864$	0	0
$865$	5.09045	0.173081
$866$	0	0
$867$	−105.013	−3.56641
$868$	0	0
$869$	−19.5352	−0.662687
$870$	0	0
$871$	−16.2381	−0.550206
$872$	0	0
$873$	1.60394	0.0542850
$874$	0	0
$875$	−21.4665	−0.725702
$876$	0	0
$877$	7.27733	0.245738	0.122869	−	0.992423i	$-0.460790\pi$
$877$	7.27733	0.245738	0.122869	+	0.992423i	$0.460790\pi$
$878$	0	0
$879$	30.9527	1.04401
$880$	0	0
$881$	26.9712	0.908684	0.454342	−	0.890827i	$-0.349874\pi$
$881$	26.9712	0.908684	0.454342	+	0.890827i	$0.349874\pi$
$882$	0	0
$883$	31.7951	1.06999	0.534995	−	0.844855i	$-0.320314\pi$
$883$	31.7951	1.06999	0.534995	+	0.844855i	$0.320314\pi$
$884$	0	0
$885$	−14.8032	−0.497603
$886$	0	0
$887$	−56.9627	−1.91262	−0.956310	−	0.292354i	$-0.905562\pi$
$887$	−56.9627	−1.91262	−0.956310	+	0.292354i	$0.905562\pi$
$888$	0	0
$889$	75.4422	2.53025
$890$	0	0
$891$	41.2687	1.38255
$892$	0	0
$893$	−3.61127	−0.120847
$894$	0	0
$895$	−6.27710	−0.209820
$896$	0	0
$897$	−2.94517	−0.0983363
$898$	0	0
$899$	−0.503502	−0.0167927
$900$	0	0
$901$	1.55825	0.0519130
$902$	0	0
$903$	−148.128	−4.92940
$904$	0	0
$905$	13.3709	0.444462
$906$	0	0
$907$	−22.6342	−0.751556	−0.375778	−	0.926710i	$-0.622624\pi$
$907$	−22.6342	−0.751556	−0.375778	+	0.926710i	$0.622624\pi$
$908$	0	0
$909$	−28.7318	−0.952974
$910$	0	0
$911$	29.2195	0.968084	0.484042	−	0.875045i	$-0.339168\pi$
$911$	29.2195	0.968084	0.484042	+	0.875045i	$0.339168\pi$
$912$	0	0
$913$	2.64395	0.0875019
$914$	0	0
$915$	4.69227	0.155122
$916$	0	0
$917$	13.6679	0.451353
$918$	0	0
$919$	−26.3423	−0.868952	−0.434476	−	0.900683i	$-0.643066\pi$
$919$	−26.3423	−0.868952	−0.434476	+	0.900683i	$0.643066\pi$
$920$	0	0
$921$	−49.3041	−1.62463
$922$	0	0
$923$	−32.8171	−1.08019
$924$	0	0
$925$	−47.3517	−1.55691
$926$	0	0
$927$	36.4267	1.19641
$928$	0	0
$929$	37.6392	1.23490	0.617451	−	0.786610i	$-0.288166\pi$
$929$	37.6392	1.23490	0.617451	+	0.786610i	$0.288166\pi$
$930$	0	0
$931$	7.93611	0.260096
$932$	0	0
$933$	−5.43255	−0.177854
$934$	0	0
$935$	−10.1057	−0.330492
$936$	0	0
$937$	21.6564	0.707483	0.353742	−	0.935343i	$-0.384909\pi$
$937$	21.6564	0.707483	0.353742	+	0.935343i	$0.384909\pi$
$938$	0	0
$939$	−90.1560	−2.94213
$940$	0	0
$941$	−36.8472	−1.20118	−0.600592	−	0.799556i	$-0.705068\pi$
$941$	−36.8472	−1.20118	−0.600592	+	0.799556i	$0.705068\pi$
$942$	0	0
$943$	2.77519	0.0903727
$944$	0	0
$945$	−25.5161	−0.830040
$946$	0	0
$947$	−25.3700	−0.824414	−0.412207	−	0.911090i	$-0.635242\pi$
$947$	−25.3700	−0.824414	−0.412207	+	0.911090i	$0.635242\pi$
$948$	0	0
$949$	0.139791	0.00453779
$950$	0	0
$951$	1.57156	0.0509613
$952$	0	0
$953$	−41.2401	−1.33590	−0.667950	−	0.744207i	$-0.732828\pi$
$953$	−41.2401	−1.33590	−0.667950	+	0.744207i	$0.732828\pi$
$954$	0	0
$955$	−2.17452	−0.0703658
$956$	0	0
$957$	−0.943335	−0.0304937
$958$	0	0
$959$	51.4539	1.66153
$960$	0	0
$961$	−12.1918	−0.393283
$962$	0	0
$963$	−102.573	−3.30538
$964$	0	0
$965$	−1.37321	−0.0442053
$966$	0	0
$967$	−42.9090	−1.37986	−0.689930	−	0.723876i	$-0.742359\pi$
$967$	−42.9090	−1.37986	−0.689930	+	0.723876i	$0.742359\pi$
$968$	0	0
$969$	18.4651	0.593185
$970$	0	0
$971$	0.734797	0.0235808	0.0117904	−	0.999930i	$-0.496247\pi$
$971$	0.734797	0.0235808	0.0117904	+	0.999930i	$0.496247\pi$
$972$	0	0
$973$	−7.76387	−0.248898
$974$	0	0
$975$	−37.4283	−1.19866
$976$	0	0
$977$	−33.0223	−1.05648	−0.528239	−	0.849096i	$-0.677147\pi$
$977$	−33.0223	−1.05648	−0.528239	+	0.849096i	$0.677147\pi$
$978$	0	0
$979$	−6.50446	−0.207883
$980$	0	0
$981$	56.9558	1.81846
$982$	0	0
$983$	4.13640	0.131931	0.0659653	−	0.997822i	$-0.478987\pi$
$983$	4.13640	0.131931	0.0659653	+	0.997822i	$0.478987\pi$
$984$	0	0
$985$	6.81400	0.217112
$986$	0	0
$987$	54.9485	1.74903
$988$	0	0
$989$	−4.32891	−0.137651
$990$	0	0
$991$	43.6651	1.38707	0.693533	−	0.720425i	$-0.256053\pi$
$991$	43.6651	1.38707	0.693533	+	0.720425i	$0.256053\pi$
$992$	0	0
$993$	25.1628	0.798518
$994$	0	0
$995$	−10.5059	−0.333058
$996$	0	0
$997$	23.0119	0.728795	0.364397	−	0.931244i	$-0.381275\pi$
$997$	23.0119	0.728795	0.364397	+	0.931244i	$0.381275\pi$
$998$	0	0
$999$	−116.110	−3.67356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.1		29
4.3	odd	2		4024.2.a.e.1.29	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.29	✓	29	4.3	odd	2
8048.2.a.w.1.1		29	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-3.11496 q^{3} +0.544032 q^{5} +4.06617 q^{7} +6.70298 q^{9} +O(q^{10})\)	\(q-3.11496 q^{3} +0.544032 q^{5} +4.06617 q^{7} +6.70298 q^{9} +2.60847 q^{11} -2.55433 q^{13} -1.69464 q^{15} -7.12126 q^{17} +0.832421 q^{19} -12.6660 q^{21} -0.370152 q^{23} -4.70403 q^{25} -11.5347 q^{27} +0.116099 q^{29} -4.33684 q^{31} -8.12528 q^{33} +2.21213 q^{35} +10.0662 q^{37} +7.95665 q^{39} -7.49744 q^{41} +11.6950 q^{43} +3.64663 q^{45} -4.33828 q^{47} +9.53378 q^{49} +22.1824 q^{51} -0.218817 q^{53} +1.41909 q^{55} -2.59296 q^{57} +8.73530 q^{59} -2.76890 q^{61} +27.2555 q^{63} -1.38964 q^{65} +6.35707 q^{67} +1.15301 q^{69} +12.8476 q^{71} -0.0547269 q^{73} +14.6529 q^{75} +10.6065 q^{77} -7.48916 q^{79} +15.8211 q^{81} +1.01360 q^{83} -3.87419 q^{85} -0.361643 q^{87} -2.49359 q^{89} -10.3864 q^{91} +13.5091 q^{93} +0.452863 q^{95} +0.239287 q^{97} +17.4845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

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