LMFDB - Embedded newform 8752.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8752,2,Mod(1,8752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8752, 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("8752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8752 = 2^{4} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8752.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:	$69.8850718490$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 8 x^{18} - 6 x^{17} + 187 x^{16} - 201 x^{15} - 1757 x^{14} + 3000 x^{13} + 8703 x^{12} + \cdots - 26 $ x^19 - 8x^18 - 6x^17 + 187x^16 - 201x^15 - 1757x^14 + 3000x^13 + 8703x^12 - 17349x^11 - 25437x^10 + 50901x^9 + 47173x^8 - 77021x^7 - 55807x^6 + 51357x^5 + 33800x^4 - 6455x^3 - 2126x^2 + 502x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 2188)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.26589$ of defining polynomial
Character	$\chi$	$=$	8752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26589 q^{3} +3.32291 q^{5} +2.61413 q^{7} -1.39752 q^{9} +O(q^{10})$	$q-1.26589 q^{3} +3.32291 q^{5} +2.61413 q^{7} -1.39752 q^{9} +5.93209 q^{11} -5.46673 q^{13} -4.20645 q^{15} +1.40358 q^{17} +0.119745 q^{19} -3.30920 q^{21} +2.06717 q^{23} +6.04174 q^{25} +5.56678 q^{27} +7.76994 q^{29} -3.19575 q^{31} -7.50938 q^{33} +8.68651 q^{35} -10.7109 q^{37} +6.92029 q^{39} -3.99884 q^{41} +4.99498 q^{43} -4.64383 q^{45} +9.89052 q^{47} -0.166344 q^{49} -1.77678 q^{51} +1.77162 q^{53} +19.7118 q^{55} -0.151584 q^{57} -11.8303 q^{59} -0.508285 q^{61} -3.65329 q^{63} -18.1655 q^{65} +0.992391 q^{67} -2.61681 q^{69} -4.90277 q^{71} +10.7080 q^{73} -7.64819 q^{75} +15.5072 q^{77} +11.8378 q^{79} -2.85439 q^{81} +12.8799 q^{83} +4.66396 q^{85} -9.83591 q^{87} +8.27868 q^{89} -14.2907 q^{91} +4.04547 q^{93} +0.397901 q^{95} +2.09006 q^{97} -8.29019 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9	$ 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9} + 10 q^{11} - 4 q^{13} + 11 q^{15} - 32 q^{17} + 14 q^{19} - 8 q^{21} + 48 q^{23} + 14 q^{25} + 47 q^{27} - 10 q^{29} - 7 q^{31} - 16 q^{33} + 25 q^{35} + 8 q^{37} + 11 q^{39} - 9 q^{41} + 4 q^{43} + 8 q^{45} + 53 q^{47} + 6 q^{49} + q^{51} + 4 q^{53} + 31 q^{55} - 22 q^{57} + 14 q^{59} + 14 q^{61} + 45 q^{63} - 36 q^{65} + 29 q^{67} + 18 q^{69} + 41 q^{71} - 34 q^{73} + 61 q^{75} + 13 q^{77} + 12 q^{79} + 11 q^{81} + 87 q^{83} - 25 q^{85} + 39 q^{87} - 46 q^{89} + 35 q^{91} - 24 q^{93} + 44 q^{95} - 46 q^{97} + 51 q^{99}+O(q^{100}) $ 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9 + 10 * q^11 - 4 * q^13 + 11 * q^15 - 32 * q^17 + 14 * q^19 - 8 * q^21 + 48 * q^23 + 14 * q^25 + 47 * q^27 - 10 * q^29 - 7 * q^31 - 16 * q^33 + 25 * q^35 + 8 * q^37 + 11 * q^39 - 9 * q^41 + 4 * q^43 + 8 * q^45 + 53 * q^47 + 6 * q^49 + q^51 + 4 * q^53 + 31 * q^55 - 22 * q^57 + 14 * q^59 + 14 * q^61 + 45 * q^63 - 36 * q^65 + 29 * q^67 + 18 * q^69 + 41 * q^71 - 34 * q^73 + 61 * q^75 + 13 * q^77 + 12 * q^79 + 11 * q^81 + 87 * q^83 - 25 * q^85 + 39 * q^87 - 46 * q^89 + 35 * q^91 - 24 * q^93 + 44 * q^95 - 46 * q^97 + 51 * 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
$3$	−1.26589	−0.730863	−0.365432	−	0.930838i	$-0.619079\pi$
$3$	−1.26589	−0.730863	−0.365432	+	0.930838i	$0.619079\pi$
$4$	0	0
$5$	3.32291	1.48605	0.743025	−	0.669263i	$-0.233390\pi$
$5$	3.32291	1.48605	0.743025	+	0.669263i	$0.233390\pi$
$6$	0	0
$7$	2.61413	0.988047	0.494023	−	0.869449i	$-0.335526\pi$
$7$	2.61413	0.988047	0.494023	+	0.869449i	$0.335526\pi$
$8$	0	0
$9$	−1.39752	−0.465839
$10$	0	0
$11$	5.93209	1.78859	0.894296	−	0.447477i	$-0.147677\pi$
$11$	5.93209	1.78859	0.894296	+	0.447477i	$0.147677\pi$
$12$	0	0
$13$	−5.46673	−1.51620	−0.758099	−	0.652139i	$-0.773872\pi$
$13$	−5.46673	−1.51620	−0.758099	+	0.652139i	$0.773872\pi$
$14$	0	0
$15$	−4.20645	−1.08610
$16$	0	0
$17$	1.40358	0.340417	0.170209	−	0.985408i	$-0.445556\pi$
$17$	1.40358	0.340417	0.170209	+	0.985408i	$0.445556\pi$
$18$	0	0
$19$	0.119745	0.0274713	0.0137356	−	0.999906i	$-0.495628\pi$
$19$	0.119745	0.0274713	0.0137356	+	0.999906i	$0.495628\pi$
$20$	0	0
$21$	−3.30920	−0.722127
$22$	0	0
$23$	2.06717	0.431035	0.215517	−	0.976500i	$-0.430856\pi$
$23$	2.06717	0.431035	0.215517	+	0.976500i	$0.430856\pi$
$24$	0	0
$25$	6.04174	1.20835
$26$	0	0
$27$	5.56678	1.07133
$28$	0	0
$29$	7.76994	1.44284	0.721421	−	0.692496i	$-0.243489\pi$
$29$	7.76994	1.44284	0.721421	+	0.692496i	$0.243489\pi$
$30$	0	0
$31$	−3.19575	−0.573973	−0.286986	−	0.957935i	$-0.592653\pi$
$31$	−3.19575	−0.573973	−0.286986	+	0.957935i	$0.592653\pi$
$32$	0	0
$33$	−7.50938	−1.30722
$34$	0	0
$35$	8.68651	1.46829
$36$	0	0
$37$	−10.7109	−1.76087	−0.880433	−	0.474170i	$-0.842748\pi$
$37$	−10.7109	−1.76087	−0.880433	+	0.474170i	$0.842748\pi$
$38$	0	0
$39$	6.92029	1.10813
$40$	0	0
$41$	−3.99884	−0.624513	−0.312257	−	0.949998i	$-0.601085\pi$
$41$	−3.99884	−0.624513	−0.312257	+	0.949998i	$0.601085\pi$
$42$	0	0
$43$	4.99498	0.761727	0.380863	−	0.924631i	$-0.375627\pi$
$43$	4.99498	0.761727	0.380863	+	0.924631i	$0.375627\pi$
$44$	0	0
$45$	−4.64383	−0.692261
$46$	0	0
$47$	9.89052	1.44268	0.721341	−	0.692581i	$-0.243526\pi$
$47$	9.89052	1.44268	0.721341	+	0.692581i	$0.243526\pi$
$48$	0	0
$49$	−0.166344	−0.0237635
$50$	0	0
$51$	−1.77678	−0.248798
$52$	0	0
$53$	1.77162	0.243351	0.121675	−	0.992570i	$-0.461173\pi$
$53$	1.77162	0.243351	0.121675	+	0.992570i	$0.461173\pi$
$54$	0	0
$55$	19.7118	2.65794
$56$	0	0
$57$	−0.151584	−0.0200778
$58$	0	0
$59$	−11.8303	−1.54018	−0.770088	−	0.637937i	$-0.779788\pi$
$59$	−11.8303	−1.54018	−0.770088	+	0.637937i	$0.779788\pi$
$60$	0	0
$61$	−0.508285	−0.0650792	−0.0325396	−	0.999470i	$-0.510360\pi$
$61$	−0.508285	−0.0650792	−0.0325396	+	0.999470i	$0.510360\pi$
$62$	0	0
$63$	−3.65329	−0.460271
$64$	0	0
$65$	−18.1655	−2.25315
$66$	0	0
$67$	0.992391	0.121240	0.0606200	−	0.998161i	$-0.480692\pi$
$67$	0.992391	0.121240	0.0606200	+	0.998161i	$0.480692\pi$
$68$	0	0
$69$	−2.61681	−0.315027
$70$	0	0
$71$	−4.90277	−0.581852	−0.290926	−	0.956746i	$-0.593963\pi$
$71$	−4.90277	−0.581852	−0.290926	+	0.956746i	$0.593963\pi$
$72$	0	0
$73$	10.7080	1.25328	0.626641	−	0.779308i	$-0.284429\pi$
$73$	10.7080	1.25328	0.626641	+	0.779308i	$0.284429\pi$
$74$	0	0
$75$	−7.64819	−0.883137
$76$	0	0
$77$	15.5072	1.76721
$78$	0	0
$79$	11.8378	1.33185	0.665927	−	0.746017i	$-0.268036\pi$
$79$	11.8378	1.33185	0.665927	+	0.746017i	$0.268036\pi$
$80$	0	0
$81$	−2.85439	−0.317155
$82$	0	0
$83$	12.8799	1.41376	0.706878	−	0.707335i	$-0.250103\pi$
$83$	12.8799	1.41376	0.706878	+	0.707335i	$0.250103\pi$
$84$	0	0
$85$	4.66396	0.505877
$86$	0	0
$87$	−9.83591	−1.05452
$88$	0	0
$89$	8.27868	0.877538	0.438769	−	0.898600i	$-0.355415\pi$
$89$	8.27868	0.877538	0.438769	+	0.898600i	$0.355415\pi$
$90$	0	0
$91$	−14.2907	−1.49807
$92$	0	0
$93$	4.04547	0.419496
$94$	0	0
$95$	0.397901	0.0408238
$96$	0	0
$97$	2.09006	0.212214	0.106107	−	0.994355i	$-0.466161\pi$
$97$	2.09006	0.212214	0.106107	+	0.994355i	$0.466161\pi$
$98$	0	0
$99$	−8.29019	−0.833196
$100$	0	0
$101$	9.14269	0.909732	0.454866	−	0.890560i	$-0.349687\pi$
$101$	9.14269	0.909732	0.454866	+	0.890560i	$0.349687\pi$
$102$	0	0
$103$	12.4985	1.23151	0.615757	−	0.787936i	$-0.288850\pi$
$103$	12.4985	1.23151	0.615757	+	0.787936i	$0.288850\pi$
$104$	0	0
$105$	−10.9962	−1.07312
$106$	0	0
$107$	−9.84910	−0.952149	−0.476074	−	0.879405i	$-0.657941\pi$
$107$	−9.84910	−0.952149	−0.476074	+	0.879405i	$0.657941\pi$
$108$	0	0
$109$	14.9596	1.43287	0.716437	−	0.697652i	$-0.245772\pi$
$109$	14.9596	1.43287	0.716437	+	0.697652i	$0.245772\pi$
$110$	0	0
$111$	13.5589	1.28695
$112$	0	0
$113$	−18.2990	−1.72143	−0.860713	−	0.509091i	$-0.829982\pi$
$113$	−18.2990	−1.72143	−0.860713	+	0.509091i	$0.829982\pi$
$114$	0	0
$115$	6.86902	0.640539
$116$	0	0
$117$	7.63985	0.706304
$118$	0	0
$119$	3.66912	0.336348
$120$	0	0
$121$	24.1896	2.19906
$122$	0	0
$123$	5.06209	0.456434
$124$	0	0
$125$	3.46160	0.309615
$126$	0	0
$127$	−6.49515	−0.576351	−0.288176	−	0.957578i	$-0.593049\pi$
$127$	−6.49515	−0.576351	−0.288176	+	0.957578i	$0.593049\pi$
$128$	0	0
$129$	−6.32310	−0.556718
$130$	0	0
$131$	−5.37809	−0.469886	−0.234943	−	0.972009i	$-0.575490\pi$
$131$	−5.37809	−0.469886	−0.234943	+	0.972009i	$0.575490\pi$
$132$	0	0
$133$	0.313028	0.0271429
$134$	0	0
$135$	18.4979	1.59205
$136$	0	0
$137$	−6.23667	−0.532834	−0.266417	−	0.963858i	$-0.585840\pi$
$137$	−6.23667	−0.532834	−0.266417	+	0.963858i	$0.585840\pi$
$138$	0	0
$139$	15.1867	1.28812	0.644061	−	0.764974i	$-0.277248\pi$
$139$	15.1867	1.28812	0.644061	+	0.764974i	$0.277248\pi$
$140$	0	0
$141$	−12.5203	−1.05440
$142$	0	0
$143$	−32.4291	−2.71186
$144$	0	0
$145$	25.8188	2.14414
$146$	0	0
$147$	0.210574	0.0173679
$148$	0	0
$149$	12.7671	1.04592	0.522962	−	0.852356i	$-0.324827\pi$
$149$	12.7671	1.04592	0.522962	+	0.852356i	$0.324827\pi$
$150$	0	0
$151$	21.0028	1.70919	0.854593	−	0.519298i	$-0.173806\pi$
$151$	21.0028	1.70919	0.854593	+	0.519298i	$0.173806\pi$
$152$	0	0
$153$	−1.96152	−0.158580
$154$	0	0
$155$	−10.6192	−0.852953
$156$	0	0
$157$	−2.32027	−0.185178	−0.0925889	−	0.995704i	$-0.529514\pi$
$157$	−2.32027	−0.185178	−0.0925889	+	0.995704i	$0.529514\pi$
$158$	0	0
$159$	−2.24268	−0.177856
$160$	0	0
$161$	5.40384	0.425882
$162$	0	0
$163$	12.8724	1.00824	0.504121	−	0.863633i	$-0.331816\pi$
$163$	12.8724	1.00824	0.504121	+	0.863633i	$0.331816\pi$
$164$	0	0
$165$	−24.9530	−1.94259
$166$	0	0
$167$	−12.2502	−0.947946	−0.473973	−	0.880539i	$-0.657181\pi$
$167$	−12.2502	−0.947946	−0.473973	+	0.880539i	$0.657181\pi$
$168$	0	0
$169$	16.8851	1.29886
$170$	0	0
$171$	−0.167345	−0.0127972
$172$	0	0
$173$	3.08085	0.234232	0.117116	−	0.993118i	$-0.462635\pi$
$173$	3.08085	0.234232	0.117116	+	0.993118i	$0.462635\pi$
$174$	0	0
$175$	15.7939	1.19390
$176$	0	0
$177$	14.9759	1.12566
$178$	0	0
$179$	−22.6376	−1.69202	−0.846008	−	0.533170i	$-0.821001\pi$
$179$	−22.6376	−1.69202	−0.846008	+	0.533170i	$0.821001\pi$
$180$	0	0
$181$	−25.8014	−1.91780	−0.958900	−	0.283743i	$-0.908424\pi$
$181$	−25.8014	−1.91780	−0.958900	+	0.283743i	$0.908424\pi$
$182$	0	0
$183$	0.643434	0.0475640
$184$	0	0
$185$	−35.5915	−2.61674
$186$	0	0
$187$	8.32613	0.608867
$188$	0	0
$189$	14.5523	1.05852
$190$	0	0
$191$	−3.31251	−0.239685	−0.119842	−	0.992793i	$-0.538239\pi$
$191$	−3.31251	−0.239685	−0.119842	+	0.992793i	$0.538239\pi$
$192$	0	0
$193$	11.7387	0.844972	0.422486	−	0.906369i	$-0.361158\pi$
$193$	11.7387	0.844972	0.422486	+	0.906369i	$0.361158\pi$
$194$	0	0
$195$	22.9955	1.64674
$196$	0	0
$197$	−0.743061	−0.0529409	−0.0264705	−	0.999650i	$-0.508427\pi$
$197$	−0.743061	−0.0529409	−0.0264705	+	0.999650i	$0.508427\pi$
$198$	0	0
$199$	−2.58604	−0.183319	−0.0916597	−	0.995790i	$-0.529217\pi$
$199$	−2.58604	−0.183319	−0.0916597	+	0.995790i	$0.529217\pi$
$200$	0	0
$201$	−1.25626	−0.0886098
$202$	0	0
$203$	20.3116	1.42560
$204$	0	0
$205$	−13.2878	−0.928058
$206$	0	0
$207$	−2.88891	−0.200793
$208$	0	0
$209$	0.710335	0.0491349
$210$	0	0
$211$	−1.06045	−0.0730041	−0.0365020	−	0.999334i	$-0.511622\pi$
$211$	−1.06045	−0.0730041	−0.0365020	+	0.999334i	$0.511622\pi$
$212$	0	0
$213$	6.20638	0.425254
$214$	0	0
$215$	16.5979	1.13196
$216$	0	0
$217$	−8.35408	−0.567112
$218$	0	0
$219$	−13.5552	−0.915978
$220$	0	0
$221$	−7.67297	−0.516140
$222$	0	0
$223$	0.215093	0.0144037	0.00720183	−	0.999974i	$-0.497708\pi$
$223$	0.215093	0.0144037	0.00720183	+	0.999974i	$0.497708\pi$
$224$	0	0
$225$	−8.44343	−0.562896
$226$	0	0
$227$	14.3453	0.952129	0.476065	−	0.879410i	$-0.342063\pi$
$227$	14.3453	0.952129	0.476065	+	0.879410i	$0.342063\pi$
$228$	0	0
$229$	−28.2381	−1.86602	−0.933012	−	0.359846i	$-0.882829\pi$
$229$	−28.2381	−1.86602	−0.933012	+	0.359846i	$0.882829\pi$
$230$	0	0
$231$	−19.6305	−1.29159
$232$	0	0
$233$	16.5571	1.08469	0.542347	−	0.840154i	$-0.317536\pi$
$233$	16.5571	1.08469	0.542347	+	0.840154i	$0.317536\pi$
$234$	0	0
$235$	32.8653	2.14390
$236$	0	0
$237$	−14.9853	−0.973402
$238$	0	0
$239$	12.5848	0.814045	0.407023	−	0.913418i	$-0.366567\pi$
$239$	12.5848	0.814045	0.407023	+	0.913418i	$0.366567\pi$
$240$	0	0
$241$	4.25094	0.273827	0.136913	−	0.990583i	$-0.456282\pi$
$241$	4.25094	0.273827	0.136913	+	0.990583i	$0.456282\pi$
$242$	0	0
$243$	−13.0870	−0.839531
$244$	0	0
$245$	−0.552748	−0.0353137
$246$	0	0
$247$	−0.654611	−0.0416519
$248$	0	0
$249$	−16.3046	−1.03326
$250$	0	0
$251$	4.42141	0.279077	0.139538	−	0.990217i	$-0.455438\pi$
$251$	4.42141	0.279077	0.139538	+	0.990217i	$0.455438\pi$
$252$	0	0
$253$	12.2626	0.770945
$254$	0	0
$255$	−5.90407	−0.369727
$256$	0	0
$257$	−15.8452	−0.988394	−0.494197	−	0.869350i	$-0.664538\pi$
$257$	−15.8452	−0.988394	−0.494197	+	0.869350i	$0.664538\pi$
$258$	0	0
$259$	−27.9997	−1.73982
$260$	0	0
$261$	−10.8586	−0.672133
$262$	0	0
$263$	−19.5315	−1.20436	−0.602181	−	0.798360i	$-0.705701\pi$
$263$	−19.5315	−1.20436	−0.602181	+	0.798360i	$0.705701\pi$
$264$	0	0
$265$	5.88693	0.361631
$266$	0	0
$267$	−10.4799	−0.641360
$268$	0	0
$269$	−0.980166	−0.0597618	−0.0298809	−	0.999553i	$-0.509513\pi$
$269$	−0.980166	−0.0597618	−0.0298809	+	0.999553i	$0.509513\pi$
$270$	0	0
$271$	−28.2041	−1.71328	−0.856639	−	0.515916i	$-0.827452\pi$
$271$	−28.2041	−1.71328	−0.856639	+	0.515916i	$0.827452\pi$
$272$	0	0
$273$	18.0905	1.09489
$274$	0	0
$275$	35.8401	2.16124
$276$	0	0
$277$	25.5477	1.53501	0.767506	−	0.641042i	$-0.221497\pi$
$277$	25.5477	1.53501	0.767506	+	0.641042i	$0.221497\pi$
$278$	0	0
$279$	4.46611	0.267379
$280$	0	0
$281$	−9.76916	−0.582779	−0.291390	−	0.956604i	$-0.594118\pi$
$281$	−9.76916	−0.582779	−0.291390	+	0.956604i	$0.594118\pi$
$282$	0	0
$283$	1.72788	0.102712	0.0513558	−	0.998680i	$-0.483646\pi$
$283$	1.72788	0.102712	0.0513558	+	0.998680i	$0.483646\pi$
$284$	0	0
$285$	−0.503699	−0.0298366
$286$	0	0
$287$	−10.4535	−0.617048
$288$	0	0
$289$	−15.0300	−0.884116
$290$	0	0
$291$	−2.64579	−0.155099
$292$	0	0
$293$	19.1416	1.11827	0.559133	−	0.829078i	$-0.311134\pi$
$293$	19.1416	1.11827	0.559133	+	0.829078i	$0.311134\pi$
$294$	0	0
$295$	−39.3111	−2.28878
$296$	0	0
$297$	33.0226	1.91617
$298$	0	0
$299$	−11.3007	−0.653534
$300$	0	0
$301$	13.0575	0.752621
$302$	0	0
$303$	−11.5737	−0.664889
$304$	0	0
$305$	−1.68899	−0.0967111
$306$	0	0
$307$	13.9621	0.796861	0.398431	−	0.917198i	$-0.369555\pi$
$307$	13.9621	0.796861	0.398431	+	0.917198i	$0.369555\pi$
$308$	0	0
$309$	−15.8217	−0.900068
$310$	0	0
$311$	7.82007	0.443436	0.221718	−	0.975111i	$-0.428834\pi$
$311$	7.82007	0.443436	0.221718	+	0.975111i	$0.428834\pi$
$312$	0	0
$313$	−3.74808	−0.211854	−0.105927	−	0.994374i	$-0.533781\pi$
$313$	−3.74808	−0.211854	−0.105927	+	0.994374i	$0.533781\pi$
$314$	0	0
$315$	−12.1395	−0.683986
$316$	0	0
$317$	3.36673	0.189094	0.0945471	−	0.995520i	$-0.469860\pi$
$317$	3.36673	0.189094	0.0945471	+	0.995520i	$0.469860\pi$
$318$	0	0
$319$	46.0920	2.58066
$320$	0	0
$321$	12.4679	0.695890
$322$	0	0
$323$	0.168071	0.00935170
$324$	0	0
$325$	−33.0285	−1.83209
$326$	0	0
$327$	−18.9373	−1.04724
$328$	0	0
$329$	25.8551	1.42544
$330$	0	0
$331$	−11.7177	−0.644065	−0.322033	−	0.946729i	$-0.604366\pi$
$331$	−11.7177	−0.644065	−0.322033	+	0.946729i	$0.604366\pi$
$332$	0	0
$333$	14.9687	0.820281
$334$	0	0
$335$	3.29763	0.180169
$336$	0	0
$337$	−17.4564	−0.950912	−0.475456	−	0.879739i	$-0.657717\pi$
$337$	−17.4564	−0.950912	−0.475456	+	0.879739i	$0.657717\pi$
$338$	0	0
$339$	23.1646	1.25813
$340$	0	0
$341$	−18.9574	−1.02660
$342$	0	0
$343$	−18.7337	−1.01153
$344$	0	0
$345$	−8.69544	−0.468147
$346$	0	0
$347$	18.6545	1.00143	0.500713	−	0.865614i	$-0.333071\pi$
$347$	18.6545	1.00143	0.500713	+	0.865614i	$0.333071\pi$
$348$	0	0
$349$	25.5985	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$349$	25.5985	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$350$	0	0
$351$	−30.4321	−1.62434
$352$	0	0
$353$	−17.7612	−0.945331	−0.472666	−	0.881242i	$-0.656708\pi$
$353$	−17.7612	−0.945331	−0.472666	+	0.881242i	$0.656708\pi$
$354$	0	0
$355$	−16.2915	−0.864661
$356$	0	0
$357$	−4.64471	−0.245824
$358$	0	0
$359$	−22.8587	−1.20644	−0.603219	−	0.797576i	$-0.706115\pi$
$359$	−22.8587	−1.20644	−0.603219	+	0.797576i	$0.706115\pi$
$360$	0	0
$361$	−18.9857	−0.999245
$362$	0	0
$363$	−30.6215	−1.60721
$364$	0	0
$365$	35.5819	1.86244
$366$	0	0
$367$	−12.3210	−0.643150	−0.321575	−	0.946884i	$-0.604212\pi$
$367$	−12.3210	−0.643150	−0.321575	+	0.946884i	$0.604212\pi$
$368$	0	0
$369$	5.58844	0.290923
$370$	0	0
$371$	4.63124	0.240442
$372$	0	0
$373$	−24.5453	−1.27091	−0.635455	−	0.772138i	$-0.719187\pi$
$373$	−24.5453	−1.27091	−0.635455	+	0.772138i	$0.719187\pi$
$374$	0	0
$375$	−4.38201	−0.226286
$376$	0	0
$377$	−42.4762	−2.18764
$378$	0	0
$379$	26.3897	1.35555	0.677774	−	0.735270i	$-0.262945\pi$
$379$	26.3897	1.35555	0.677774	+	0.735270i	$0.262945\pi$
$380$	0	0
$381$	8.22216	0.421234
$382$	0	0
$383$	35.6256	1.82038	0.910192	−	0.414186i	$-0.135934\pi$
$383$	35.6256	1.82038	0.910192	+	0.414186i	$0.135934\pi$
$384$	0	0
$385$	51.5291	2.62617
$386$	0	0
$387$	−6.98057	−0.354842
$388$	0	0
$389$	−37.5289	−1.90279	−0.951396	−	0.307970i	$-0.900350\pi$
$389$	−37.5289	−1.90279	−0.951396	+	0.307970i	$0.900350\pi$
$390$	0	0
$391$	2.90143	0.146732
$392$	0	0
$393$	6.80809	0.343423
$394$	0	0
$395$	39.3359	1.97920
$396$	0	0
$397$	16.0798	0.807024	0.403512	−	0.914974i	$-0.367789\pi$
$397$	16.0798	0.807024	0.403512	+	0.914974i	$0.367789\pi$
$398$	0	0
$399$	−0.396259	−0.0198378
$400$	0	0
$401$	−8.69022	−0.433969	−0.216985	−	0.976175i	$-0.569622\pi$
$401$	−8.69022	−0.433969	−0.216985	+	0.976175i	$0.569622\pi$
$402$	0	0
$403$	17.4703	0.870257
$404$	0	0
$405$	−9.48489	−0.471308
$406$	0	0
$407$	−63.5382	−3.14947
$408$	0	0
$409$	10.8177	0.534902	0.267451	−	0.963572i	$-0.413819\pi$
$409$	10.8177	0.534902	0.267451	+	0.963572i	$0.413819\pi$
$410$	0	0
$411$	7.89495	0.389429
$412$	0	0
$413$	−30.9259	−1.52177
$414$	0	0
$415$	42.7989	2.10091
$416$	0	0
$417$	−19.2248	−0.941441
$418$	0	0
$419$	−2.97662	−0.145417	−0.0727086	−	0.997353i	$-0.523164\pi$
$419$	−2.97662	−0.145417	−0.0727086	+	0.997353i	$0.523164\pi$
$420$	0	0
$421$	−24.9939	−1.21813	−0.609064	−	0.793121i	$-0.708455\pi$
$421$	−24.9939	−1.21813	−0.609064	+	0.793121i	$0.708455\pi$
$422$	0	0
$423$	−13.8222	−0.672057
$424$	0	0
$425$	8.48004	0.411342
$426$	0	0
$427$	−1.32872	−0.0643013
$428$	0	0
$429$	41.0518	1.98200
$430$	0	0
$431$	−2.30166	−0.110867	−0.0554335	−	0.998462i	$-0.517654\pi$
$431$	−2.30166	−0.110867	−0.0554335	+	0.998462i	$0.517654\pi$
$432$	0	0
$433$	26.9697	1.29608	0.648040	−	0.761606i	$-0.275589\pi$
$433$	26.9697	1.29608	0.648040	+	0.761606i	$0.275589\pi$
$434$	0	0
$435$	−32.6839	−1.56707
$436$	0	0
$437$	0.247532	0.0118411
$438$	0	0
$439$	16.7454	0.799212	0.399606	−	0.916687i	$-0.369147\pi$
$439$	16.7454	0.799212	0.399606	+	0.916687i	$0.369147\pi$
$440$	0	0
$441$	0.232469	0.0110700
$442$	0	0
$443$	11.1215	0.528400	0.264200	−	0.964468i	$-0.414892\pi$
$443$	11.1215	0.528400	0.264200	+	0.964468i	$0.414892\pi$
$444$	0	0
$445$	27.5093	1.30407
$446$	0	0
$447$	−16.1618	−0.764427
$448$	0	0
$449$	27.8215	1.31298	0.656489	−	0.754335i	$-0.272041\pi$
$449$	27.8215	1.31298	0.656489	+	0.754335i	$0.272041\pi$
$450$	0	0
$451$	−23.7214	−1.11700
$452$	0	0
$453$	−26.5873	−1.24918
$454$	0	0
$455$	−47.4868	−2.22622
$456$	0	0
$457$	7.96988	0.372815	0.186408	−	0.982472i	$-0.440316\pi$
$457$	7.96988	0.372815	0.186408	+	0.982472i	$0.440316\pi$
$458$	0	0
$459$	7.81340	0.364698
$460$	0	0
$461$	27.6472	1.28766	0.643828	−	0.765170i	$-0.277345\pi$
$461$	27.6472	1.28766	0.643828	+	0.765170i	$0.277345\pi$
$462$	0	0
$463$	8.29073	0.385303	0.192651	−	0.981267i	$-0.438291\pi$
$463$	8.29073	0.385303	0.192651	+	0.981267i	$0.438291\pi$
$464$	0	0
$465$	13.4427	0.623392
$466$	0	0
$467$	−7.54679	−0.349224	−0.174612	−	0.984637i	$-0.555867\pi$
$467$	−7.54679	−0.349224	−0.174612	+	0.984637i	$0.555867\pi$
$468$	0	0
$469$	2.59424	0.119791
$470$	0	0
$471$	2.93721	0.135340
$472$	0	0
$473$	29.6306	1.36242
$474$	0	0
$475$	0.723466	0.0331949
$476$	0	0
$477$	−2.47587	−0.113362
$478$	0	0
$479$	25.1968	1.15127	0.575635	−	0.817707i	$-0.304755\pi$
$479$	25.1968	1.15127	0.575635	+	0.817707i	$0.304755\pi$
$480$	0	0
$481$	58.5538	2.66982
$482$	0	0
$483$	−6.84068	−0.311262
$484$	0	0
$485$	6.94509	0.315360
$486$	0	0
$487$	−2.40382	−0.108928	−0.0544638	−	0.998516i	$-0.517345\pi$
$487$	−2.40382	−0.108928	−0.0544638	+	0.998516i	$0.517345\pi$
$488$	0	0
$489$	−16.2950	−0.736888
$490$	0	0
$491$	39.0662	1.76303	0.881516	−	0.472155i	$-0.156524\pi$
$491$	39.0662	1.76303	0.881516	+	0.472155i	$0.156524\pi$
$492$	0	0
$493$	10.9057	0.491168
$494$	0	0
$495$	−27.5476	−1.23817
$496$	0	0
$497$	−12.8165	−0.574897
$498$	0	0
$499$	18.5159	0.828884	0.414442	−	0.910076i	$-0.363977\pi$
$499$	18.5159	0.828884	0.414442	+	0.910076i	$0.363977\pi$
$500$	0	0
$501$	15.5074	0.692819
$502$	0	0
$503$	38.0165	1.69507	0.847535	−	0.530740i	$-0.178086\pi$
$503$	38.0165	1.69507	0.847535	+	0.530740i	$0.178086\pi$
$504$	0	0
$505$	30.3803	1.35191
$506$	0	0
$507$	−21.3748	−0.949286
$508$	0	0
$509$	−7.28596	−0.322945	−0.161472	−	0.986877i	$-0.551624\pi$
$509$	−7.28596	−0.322945	−0.161472	+	0.986877i	$0.551624\pi$
$510$	0	0
$511$	27.9922	1.23830
$512$	0	0
$513$	0.666592	0.0294308
$514$	0	0
$515$	41.5314	1.83009
$516$	0	0
$517$	58.6714	2.58037
$518$	0	0
$519$	−3.90002	−0.171192
$520$	0	0
$521$	0.224342	0.00982859	0.00491429	−	0.999988i	$-0.498436\pi$
$521$	0.224342	0.00982859	0.00491429	+	0.999988i	$0.498436\pi$
$522$	0	0
$523$	−23.8785	−1.04413	−0.522066	−	0.852905i	$-0.674839\pi$
$523$	−23.8785	−1.04413	−0.522066	+	0.852905i	$0.674839\pi$
$524$	0	0
$525$	−19.9933	−0.872580
$526$	0	0
$527$	−4.48547	−0.195390
$528$	0	0
$529$	−18.7268	−0.814209
$530$	0	0
$531$	16.5331	0.717474
$532$	0	0
$533$	21.8606	0.946886
$534$	0	0
$535$	−32.7277	−1.41494
$536$	0	0
$537$	28.6568	1.23663
$538$	0	0
$539$	−0.986769	−0.0425032
$540$	0	0
$541$	28.1582	1.21062	0.605309	−	0.795991i	$-0.293050\pi$
$541$	28.1582	1.21062	0.605309	+	0.795991i	$0.293050\pi$
$542$	0	0
$543$	32.6618	1.40165
$544$	0	0
$545$	49.7096	2.12932
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	0	0
$549$	0.710337	0.0303165
$550$	0	0
$551$	0.930409	0.0396368
$552$	0	0
$553$	30.9454	1.31593
$554$	0	0
$555$	45.0550	1.91248
$556$	0	0
$557$	−4.36660	−0.185019	−0.0925093	−	0.995712i	$-0.529489\pi$
$557$	−4.36660	−0.185019	−0.0925093	+	0.995712i	$0.529489\pi$
$558$	0	0
$559$	−27.3062	−1.15493
$560$	0	0
$561$	−10.5400	−0.444998
$562$	0	0
$563$	43.7186	1.84252	0.921260	−	0.388947i	$-0.127161\pi$
$563$	43.7186	1.84252	0.921260	+	0.388947i	$0.127161\pi$
$564$	0	0
$565$	−60.8060	−2.55813
$566$	0	0
$567$	−7.46174	−0.313364
$568$	0	0
$569$	−25.8834	−1.08509	−0.542543	−	0.840028i	$-0.682539\pi$
$569$	−25.8834	−1.08509	−0.542543	+	0.840028i	$0.682539\pi$
$570$	0	0
$571$	4.06880	0.170274	0.0851369	−	0.996369i	$-0.472867\pi$
$571$	4.06880	0.170274	0.0851369	+	0.996369i	$0.472867\pi$
$572$	0	0
$573$	4.19328	0.175177
$574$	0	0
$575$	12.4893	0.520840
$576$	0	0
$577$	25.4150	1.05804	0.529020	−	0.848610i	$-0.322560\pi$
$577$	25.4150	1.05804	0.529020	+	0.848610i	$0.322560\pi$
$578$	0	0
$579$	−14.8600	−0.617559
$580$	0	0
$581$	33.6698	1.39686
$582$	0	0
$583$	10.5094	0.435255
$584$	0	0
$585$	25.3865	1.04960
$586$	0	0
$587$	−20.9066	−0.862910	−0.431455	−	0.902135i	$-0.642000\pi$
$587$	−20.9066	−0.862910	−0.431455	+	0.902135i	$0.642000\pi$
$588$	0	0
$589$	−0.382673	−0.0157678
$590$	0	0
$591$	0.940635	0.0386926
$592$	0	0
$593$	34.1652	1.40300	0.701498	−	0.712671i	$-0.252515\pi$
$593$	34.1652	1.40300	0.701498	+	0.712671i	$0.252515\pi$
$594$	0	0
$595$	12.1922	0.499830
$596$	0	0
$597$	3.27365	0.133981
$598$	0	0
$599$	33.0622	1.35088	0.675441	−	0.737414i	$-0.263953\pi$
$599$	33.0622	1.35088	0.675441	+	0.737414i	$0.263953\pi$
$600$	0	0
$601$	12.1961	0.497489	0.248744	−	0.968569i	$-0.419982\pi$
$601$	12.1961	0.497489	0.248744	+	0.968569i	$0.419982\pi$
$602$	0	0
$603$	−1.38688	−0.0564783
$604$	0	0
$605$	80.3800	3.26791
$606$	0	0
$607$	6.23889	0.253229	0.126614	−	0.991952i	$-0.459589\pi$
$607$	6.23889	0.253229	0.126614	+	0.991952i	$0.459589\pi$
$608$	0	0
$609$	−25.7123	−1.04192
$610$	0	0
$611$	−54.0688	−2.18739
$612$	0	0
$613$	−13.9095	−0.561799	−0.280899	−	0.959737i	$-0.590633\pi$
$613$	−13.9095	−0.561799	−0.280899	+	0.959737i	$0.590633\pi$
$614$	0	0
$615$	16.8209	0.678284
$616$	0	0
$617$	−21.9438	−0.883424	−0.441712	−	0.897157i	$-0.645629\pi$
$617$	−21.9438	−0.883424	−0.441712	+	0.897157i	$0.645629\pi$
$618$	0	0
$619$	−1.20832	−0.0485664	−0.0242832	−	0.999705i	$-0.507730\pi$
$619$	−1.20832	−0.0485664	−0.0242832	+	0.999705i	$0.507730\pi$
$620$	0	0
$621$	11.5075	0.461779
$622$	0	0
$623$	21.6415	0.867049
$624$	0	0
$625$	−18.7061	−0.748244
$626$	0	0
$627$	−0.899208	−0.0359109
$628$	0	0
$629$	−15.0336	−0.599429
$630$	0	0
$631$	−45.3580	−1.80567	−0.902836	−	0.429986i	$-0.858519\pi$
$631$	−45.3580	−1.80567	−0.902836	+	0.429986i	$0.858519\pi$
$632$	0	0
$633$	1.34241	0.0533560
$634$	0	0
$635$	−21.5828	−0.856487
$636$	0	0
$637$	0.909360	0.0360301
$638$	0	0
$639$	6.85171	0.271049
$640$	0	0
$641$	−4.63284	−0.182986	−0.0914931	−	0.995806i	$-0.529164\pi$
$641$	−4.63284	−0.182986	−0.0914931	+	0.995806i	$0.529164\pi$
$642$	0	0
$643$	35.3186	1.39283	0.696414	−	0.717640i	$-0.254778\pi$
$643$	35.3186	1.39283	0.696414	+	0.717640i	$0.254778\pi$
$644$	0	0
$645$	−21.0111	−0.827311
$646$	0	0
$647$	−0.338377	−0.0133030	−0.00665149	−	0.999978i	$-0.502117\pi$
$647$	−0.338377	−0.0133030	−0.00665149	+	0.999978i	$0.502117\pi$
$648$	0	0
$649$	−70.1785	−2.75475
$650$	0	0
$651$	10.5754	0.414481
$652$	0	0
$653$	−32.6860	−1.27910	−0.639550	−	0.768749i	$-0.720879\pi$
$653$	−32.6860	−1.27910	−0.639550	+	0.768749i	$0.720879\pi$
$654$	0	0
$655$	−17.8709	−0.698275
$656$	0	0
$657$	−14.9647	−0.583828
$658$	0	0
$659$	−33.2902	−1.29680	−0.648402	−	0.761298i	$-0.724562\pi$
$659$	−33.2902	−1.29680	−0.648402	+	0.761298i	$0.724562\pi$
$660$	0	0
$661$	48.8032	1.89823	0.949113	−	0.314937i	$-0.101983\pi$
$661$	48.8032	1.89823	0.949113	+	0.314937i	$0.101983\pi$
$662$	0	0
$663$	9.71315	0.377227
$664$	0	0
$665$	1.04016	0.0403358
$666$	0	0
$667$	16.0618	0.621915
$668$	0	0
$669$	−0.272284	−0.0105271
$670$	0	0
$671$	−3.01519	−0.116400
$672$	0	0
$673$	−27.5442	−1.06175	−0.530875	−	0.847450i	$-0.678137\pi$
$673$	−27.5442	−1.06175	−0.530875	+	0.847450i	$0.678137\pi$
$674$	0	0
$675$	33.6330	1.29454
$676$	0	0
$677$	−29.5315	−1.13499	−0.567493	−	0.823378i	$-0.692087\pi$
$677$	−29.5315	−1.13499	−0.567493	+	0.823378i	$0.692087\pi$
$678$	0	0
$679$	5.46368	0.209677
$680$	0	0
$681$	−18.1596	−0.695876
$682$	0	0
$683$	−5.68705	−0.217609	−0.108804	−	0.994063i	$-0.534702\pi$
$683$	−5.68705	−0.217609	−0.108804	+	0.994063i	$0.534702\pi$
$684$	0	0
$685$	−20.7239	−0.791819
$686$	0	0
$687$	35.7463	1.36381
$688$	0	0
$689$	−9.68496	−0.368968
$690$	0	0
$691$	−29.2584	−1.11304	−0.556521	−	0.830834i	$-0.687864\pi$
$691$	−29.2584	−1.11304	−0.556521	+	0.830834i	$0.687864\pi$
$692$	0	0
$693$	−21.6716	−0.823237
$694$	0	0
$695$	50.4642	1.91422
$696$	0	0
$697$	−5.61267	−0.212595
$698$	0	0
$699$	−20.9596	−0.792763
$700$	0	0
$701$	−24.7324	−0.934130	−0.467065	−	0.884223i	$-0.654689\pi$
$701$	−24.7324	−0.934130	−0.467065	+	0.884223i	$0.654689\pi$
$702$	0	0
$703$	−1.28258	−0.0483733
$704$	0	0
$705$	−41.6040	−1.56690
$706$	0	0
$707$	23.9001	0.898857
$708$	0	0
$709$	18.3632	0.689646	0.344823	−	0.938668i	$-0.387939\pi$
$709$	18.3632	0.689646	0.344823	+	0.938668i	$0.387939\pi$
$710$	0	0
$711$	−16.5435	−0.620429
$712$	0	0
$713$	−6.60615	−0.247402
$714$	0	0
$715$	−107.759	−4.02996
$716$	0	0
$717$	−15.9310	−0.594956
$718$	0	0
$719$	−6.49411	−0.242189	−0.121095	−	0.992641i	$-0.538640\pi$
$719$	−6.49411	−0.242189	−0.121095	+	0.992641i	$0.538640\pi$
$720$	0	0
$721$	32.6726	1.21679
$722$	0	0
$723$	−5.38123	−0.200130
$724$	0	0
$725$	46.9440	1.74346
$726$	0	0
$727$	−27.4174	−1.01685	−0.508427	−	0.861105i	$-0.669773\pi$
$727$	−27.4174	−1.01685	−0.508427	+	0.861105i	$0.669773\pi$
$728$	0	0
$729$	25.1299	0.930737
$730$	0	0
$731$	7.01083	0.259305
$732$	0	0
$733$	49.5386	1.82975	0.914874	−	0.403739i	$-0.132289\pi$
$733$	49.5386	1.82975	0.914874	+	0.403739i	$0.132289\pi$
$734$	0	0
$735$	0.699719	0.0258095
$736$	0	0
$737$	5.88695	0.216849
$738$	0	0
$739$	27.2696	1.00313	0.501564	−	0.865121i	$-0.332758\pi$
$739$	27.2696	1.00313	0.501564	+	0.865121i	$0.332758\pi$
$740$	0	0
$741$	0.828667	0.0304419
$742$	0	0
$743$	19.6732	0.721741	0.360870	−	0.932616i	$-0.382480\pi$
$743$	19.6732	0.721741	0.360870	+	0.932616i	$0.382480\pi$
$744$	0	0
$745$	42.4240	1.55430
$746$	0	0
$747$	−17.9999	−0.658583
$748$	0	0
$749$	−25.7468	−0.940767
$750$	0	0
$751$	−37.1918	−1.35715	−0.678573	−	0.734533i	$-0.737401\pi$
$751$	−37.1918	−1.35715	−0.678573	+	0.734533i	$0.737401\pi$
$752$	0	0
$753$	−5.59703	−0.203967
$754$	0	0
$755$	69.7906	2.53994
$756$	0	0
$757$	−32.2236	−1.17119	−0.585593	−	0.810605i	$-0.699138\pi$
$757$	−32.2236	−1.17119	−0.585593	+	0.810605i	$0.699138\pi$
$758$	0	0
$759$	−15.5232	−0.563455
$760$	0	0
$761$	−37.3177	−1.35277	−0.676384	−	0.736549i	$-0.736454\pi$
$761$	−37.3177	−1.35277	−0.676384	+	0.736549i	$0.736454\pi$
$762$	0	0
$763$	39.1064	1.41575
$764$	0	0
$765$	−6.51796	−0.235657
$766$	0	0
$767$	64.6732	2.33521
$768$	0	0
$769$	−24.0679	−0.867911	−0.433956	−	0.900934i	$-0.642883\pi$
$769$	−24.0679	−0.867911	−0.433956	+	0.900934i	$0.642883\pi$
$770$	0	0
$771$	20.0583	0.722381
$772$	0	0
$773$	5.55647	0.199852	0.0999262	−	0.994995i	$-0.468139\pi$
$773$	5.55647	0.199852	0.0999262	+	0.994995i	$0.468139\pi$
$774$	0	0
$775$	−19.3079	−0.693559
$776$	0	0
$777$	35.4446	1.27157
$778$	0	0
$779$	−0.478839	−0.0171562
$780$	0	0
$781$	−29.0837	−1.04069
$782$	0	0
$783$	43.2536	1.54576
$784$	0	0
$785$	−7.71005	−0.275184
$786$	0	0
$787$	23.7031	0.844925	0.422463	−	0.906380i	$-0.361166\pi$
$787$	23.7031	0.844925	0.422463	+	0.906380i	$0.361166\pi$
$788$	0	0
$789$	24.7247	0.880223
$790$	0	0
$791$	−47.8359	−1.70085
$792$	0	0
$793$	2.77866	0.0986730
$794$	0	0
$795$	−7.45222	−0.264303
$796$	0	0
$797$	23.1343	0.819458	0.409729	−	0.912207i	$-0.365623\pi$
$797$	23.1343	0.819458	0.409729	+	0.912207i	$0.365623\pi$
$798$	0	0
$799$	13.8821	0.491113
$800$	0	0
$801$	−11.5696	−0.408792
$802$	0	0
$803$	63.5211	2.24161
$804$	0	0
$805$	17.9565	0.632883
$806$	0	0
$807$	1.24078	0.0436777
$808$	0	0
$809$	−4.87505	−0.171398	−0.0856988	−	0.996321i	$-0.527312\pi$
$809$	−4.87505	−0.171398	−0.0856988	+	0.996321i	$0.527312\pi$
$810$	0	0
$811$	−35.6789	−1.25285	−0.626427	−	0.779480i	$-0.715483\pi$
$811$	−35.6789	−1.25285	−0.626427	+	0.779480i	$0.715483\pi$
$812$	0	0
$813$	35.7034	1.25217
$814$	0	0
$815$	42.7738	1.49830
$816$	0	0
$817$	0.598121	0.0209256
$818$	0	0
$819$	19.9715	0.697862
$820$	0	0
$821$	2.50052	0.0872689	0.0436344	−	0.999048i	$-0.486106\pi$
$821$	2.50052	0.0872689	0.0436344	+	0.999048i	$0.486106\pi$
$822$	0	0
$823$	−30.5845	−1.06611	−0.533055	−	0.846081i	$-0.678956\pi$
$823$	−30.5845	−1.06611	−0.533055	+	0.846081i	$0.678956\pi$
$824$	0	0
$825$	−45.3697	−1.57957
$826$	0	0
$827$	−4.54627	−0.158089	−0.0790446	−	0.996871i	$-0.525187\pi$
$827$	−4.54627	−0.158089	−0.0790446	+	0.996871i	$0.525187\pi$
$828$	0	0
$829$	1.88059	0.0653157	0.0326579	−	0.999467i	$-0.489603\pi$
$829$	1.88059	0.0653157	0.0326579	+	0.999467i	$0.489603\pi$
$830$	0	0
$831$	−32.3406	−1.12188
$832$	0	0
$833$	−0.233477	−0.00808950
$834$	0	0
$835$	−40.7062	−1.40870
$836$	0	0
$837$	−17.7900	−0.614913
$838$	0	0
$839$	−18.9156	−0.653038	−0.326519	−	0.945191i	$-0.605876\pi$
$839$	−18.9156	−0.653038	−0.326519	+	0.945191i	$0.605876\pi$
$840$	0	0
$841$	31.3720	1.08179
$842$	0	0
$843$	12.3667	0.425932
$844$	0	0
$845$	56.1078	1.93017
$846$	0	0
$847$	63.2348	2.17277
$848$	0	0
$849$	−2.18730	−0.0750681
$850$	0	0
$851$	−22.1413	−0.758994
$852$	0	0
$853$	16.3250	0.558957	0.279479	−	0.960152i	$-0.409838\pi$
$853$	16.3250	0.558957	0.279479	+	0.960152i	$0.409838\pi$
$854$	0	0
$855$	−0.556073	−0.0190173
$856$	0	0
$857$	24.0237	0.820633	0.410316	−	0.911943i	$-0.365418\pi$
$857$	24.0237	0.820633	0.410316	+	0.911943i	$0.365418\pi$
$858$	0	0
$859$	−38.2650	−1.30559	−0.652793	−	0.757537i	$-0.726403\pi$
$859$	−38.2650	−1.30559	−0.652793	+	0.757537i	$0.726403\pi$
$860$	0	0
$861$	13.2330	0.450978
$862$	0	0
$863$	20.4587	0.696421	0.348211	−	0.937416i	$-0.386789\pi$
$863$	20.4587	0.696421	0.348211	+	0.937416i	$0.386789\pi$
$864$	0	0
$865$	10.2374	0.348081
$866$	0	0
$867$	19.0263	0.646168
$868$	0	0
$869$	70.2227	2.38214
$870$	0	0
$871$	−5.42514	−0.183824
$872$	0	0
$873$	−2.92090	−0.0988574
$874$	0	0
$875$	9.04906	0.305914
$876$	0	0
$877$	18.7881	0.634430	0.317215	−	0.948354i	$-0.397252\pi$
$877$	18.7881	0.634430	0.317215	+	0.948354i	$0.397252\pi$
$878$	0	0
$879$	−24.2312	−0.817300
$880$	0	0
$881$	35.4503	1.19435	0.597175	−	0.802111i	$-0.296290\pi$
$881$	35.4503	1.19435	0.597175	+	0.802111i	$0.296290\pi$
$882$	0	0
$883$	10.6796	0.359396	0.179698	−	0.983722i	$-0.442488\pi$
$883$	10.6796	0.359396	0.179698	+	0.983722i	$0.442488\pi$
$884$	0	0
$885$	49.7636	1.67279
$886$	0	0
$887$	−16.2170	−0.544514	−0.272257	−	0.962225i	$-0.587770\pi$
$887$	−16.2170	−0.544514	−0.272257	+	0.962225i	$0.587770\pi$
$888$	0	0
$889$	−16.9791	−0.569462
$890$	0	0
$891$	−16.9325	−0.567260
$892$	0	0
$893$	1.18434	0.0396323
$894$	0	0
$895$	−75.2228	−2.51442
$896$	0	0
$897$	14.3054	0.477644
$898$	0	0
$899$	−24.8308	−0.828153
$900$	0	0
$901$	2.48660	0.0828407
$902$	0	0
$903$	−16.5294	−0.550063
$904$	0	0
$905$	−85.7357	−2.84995
$906$	0	0
$907$	37.2392	1.23651	0.618253	−	0.785979i	$-0.287841\pi$
$907$	37.2392	1.23651	0.618253	+	0.785979i	$0.287841\pi$
$908$	0	0
$909$	−12.7771	−0.423789
$910$	0	0
$911$	−16.9638	−0.562035	−0.281017	−	0.959703i	$-0.590672\pi$
$911$	−16.9638	−0.562035	−0.281017	+	0.959703i	$0.590672\pi$
$912$	0	0
$913$	76.4049	2.52863
$914$	0	0
$915$	2.13807	0.0706826
$916$	0	0
$917$	−14.0590	−0.464270
$918$	0	0
$919$	−8.94711	−0.295138	−0.147569	−	0.989052i	$-0.547145\pi$
$919$	−8.94711	−0.295138	−0.147569	+	0.989052i	$0.547145\pi$
$920$	0	0
$921$	−17.6746	−0.582396
$922$	0	0
$923$	26.8021	0.882202
$924$	0	0
$925$	−64.7127	−2.12774
$926$	0	0
$927$	−17.4669	−0.573687
$928$	0	0
$929$	0.641919	0.0210607	0.0105303	−	0.999945i	$-0.496648\pi$
$929$	0.641919	0.0210607	0.0105303	+	0.999945i	$0.496648\pi$
$930$	0	0
$931$	−0.0199188	−0.000652814 0
$932$	0	0
$933$	−9.89937	−0.324091
$934$	0	0
$935$	27.6670	0.904808
$936$	0	0
$937$	−30.5087	−0.996675	−0.498337	−	0.866983i	$-0.666056\pi$
$937$	−30.5087	−0.996675	−0.498337	+	0.866983i	$0.666056\pi$
$938$	0	0
$939$	4.74466	0.154836
$940$	0	0
$941$	−30.2575	−0.986366	−0.493183	−	0.869926i	$-0.664167\pi$
$941$	−30.2575	−0.986366	−0.493183	+	0.869926i	$0.664167\pi$
$942$	0	0
$943$	−8.26627	−0.269187
$944$	0	0
$945$	48.3559	1.57302
$946$	0	0
$947$	9.10613	0.295910	0.147955	−	0.988994i	$-0.452731\pi$
$947$	9.10613	0.295910	0.147955	+	0.988994i	$0.452731\pi$
$948$	0	0
$949$	−58.5380	−1.90022
$950$	0	0
$951$	−4.26191	−0.138202
$952$	0	0
$953$	26.5368	0.859612	0.429806	−	0.902921i	$-0.358582\pi$
$953$	26.5368	0.859612	0.429806	+	0.902921i	$0.358582\pi$
$954$	0	0
$955$	−11.0072	−0.356184
$956$	0	0
$957$	−58.3475	−1.88611
$958$	0	0
$959$	−16.3034	−0.526465
$960$	0	0
$961$	−20.7872	−0.670555
$962$	0	0
$963$	13.7643	0.443548
$964$	0	0
$965$	39.0068	1.25567
$966$	0	0
$967$	−6.94709	−0.223403	−0.111702	−	0.993742i	$-0.535630\pi$
$967$	−6.94709	−0.223403	−0.111702	+	0.993742i	$0.535630\pi$
$968$	0	0
$969$	−0.212759	−0.00683481
$970$	0	0
$971$	59.2968	1.90292	0.951462	−	0.307767i	$-0.0995818\pi$
$971$	59.2968	1.90292	0.951462	+	0.307767i	$0.0995818\pi$
$972$	0	0
$973$	39.7001	1.27273
$974$	0	0
$975$	41.8106	1.33901
$976$	0	0
$977$	−35.6676	−1.14111	−0.570554	−	0.821260i	$-0.693272\pi$
$977$	−35.6676	−1.14111	−0.570554	+	0.821260i	$0.693272\pi$
$978$	0	0
$979$	49.1098	1.56956
$980$	0	0
$981$	−20.9064	−0.667489
$982$	0	0
$983$	35.9019	1.14509	0.572546	−	0.819873i	$-0.305956\pi$
$983$	35.9019	1.14509	0.572546	+	0.819873i	$0.305956\pi$
$984$	0	0
$985$	−2.46913	−0.0786729
$986$	0	0
$987$	−32.7297	−1.04180
$988$	0	0
$989$	10.3255	0.328330
$990$	0	0
$991$	3.33682	0.105998	0.0529988	−	0.998595i	$-0.483122\pi$
$991$	3.33682	0.105998	0.0529988	+	0.998595i	$0.483122\pi$
$992$	0	0
$993$	14.8334	0.470723
$994$	0	0
$995$	−8.59318	−0.272422
$996$	0	0
$997$	−8.97147	−0.284129	−0.142065	−	0.989857i	$-0.545374\pi$
$997$	−8.97147	−0.284129	−0.142065	+	0.989857i	$0.545374\pi$
$998$	0	0
$999$	−59.6254	−1.88647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8752.2.a.t.1.5		19
4.3	odd	2		2188.2.a.d.1.15	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2188.2.a.d.1.15	✓	19	4.3	odd	2
8752.2.a.t.1.5		19	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:	\( N \)	\(=\)	\( 8752 = 2^{4} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8752.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26589 q^{3} +3.32291 q^{5} +2.61413 q^{7} -1.39752 q^{9} +O(q^{10})\)	\(q-1.26589 q^{3} +3.32291 q^{5} +2.61413 q^{7} -1.39752 q^{9} +5.93209 q^{11} -5.46673 q^{13} -4.20645 q^{15} +1.40358 q^{17} +0.119745 q^{19} -3.30920 q^{21} +2.06717 q^{23} +6.04174 q^{25} +5.56678 q^{27} +7.76994 q^{29} -3.19575 q^{31} -7.50938 q^{33} +8.68651 q^{35} -10.7109 q^{37} +6.92029 q^{39} -3.99884 q^{41} +4.99498 q^{43} -4.64383 q^{45} +9.89052 q^{47} -0.166344 q^{49} -1.77678 q^{51} +1.77162 q^{53} +19.7118 q^{55} -0.151584 q^{57} -11.8303 q^{59} -0.508285 q^{61} -3.65329 q^{63} -18.1655 q^{65} +0.992391 q^{67} -2.61681 q^{69} -4.90277 q^{71} +10.7080 q^{73} -7.64819 q^{75} +15.5072 q^{77} +11.8378 q^{79} -2.85439 q^{81} +12.8799 q^{83} +4.66396 q^{85} -9.83591 q^{87} +8.27868 q^{89} -14.2907 q^{91} +4.04547 q^{93} +0.397901 q^{95} +2.09006 q^{97} -8.29019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9	\( 19 q + 8 q^{3} - q^{5} + 9 q^{7} + 19 q^{9} + 10 q^{11} - 4 q^{13} + 11 q^{15} - 32 q^{17} + 14 q^{19} - 8 q^{21} + 48 q^{23} + 14 q^{25} + 47 q^{27} - 10 q^{29} - 7 q^{31} - 16 q^{33} + 25 q^{35} + 8 q^{37} + 11 q^{39} - 9 q^{41} + 4 q^{43} + 8 q^{45} + 53 q^{47} + 6 q^{49} + q^{51} + 4 q^{53} + 31 q^{55} - 22 q^{57} + 14 q^{59} + 14 q^{61} + 45 q^{63} - 36 q^{65} + 29 q^{67} + 18 q^{69} + 41 q^{71} - 34 q^{73} + 61 q^{75} + 13 q^{77} + 12 q^{79} + 11 q^{81} + 87 q^{83} - 25 q^{85} + 39 q^{87} - 46 q^{89} + 35 q^{91} - 24 q^{93} + 44 q^{95} - 46 q^{97} + 51 q^{99}+O(q^{100}) \) 19 * q + 8 * q^3 - q^5 + 9 * q^7 + 19 * q^9 + 10 * q^11 - 4 * q^13 + 11 * q^15 - 32 * q^17 + 14 * q^19 - 8 * q^21 + 48 * q^23 + 14 * q^25 + 47 * q^27 - 10 * q^29 - 7 * q^31 - 16 * q^33 + 25 * q^35 + 8 * q^37 + 11 * q^39 - 9 * q^41 + 4 * q^43 + 8 * q^45 + 53 * q^47 + 6 * q^49 + q^51 + 4 * q^53 + 31 * q^55 - 22 * q^57 + 14 * q^59 + 14 * q^61 + 45 * q^63 - 36 * q^65 + 29 * q^67 + 18 * q^69 + 41 * q^71 - 34 * q^73 + 61 * q^75 + 13 * q^77 + 12 * q^79 + 11 * q^81 + 87 * q^83 - 25 * q^85 + 39 * q^87 - 46 * q^89 + 35 * q^91 - 24 * q^93 + 44 * q^95 - 46 * q^97 + 51 * q^99

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