LMFDB - Embedded newform 8041.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75592 q^{2} +2.83072 q^{3} +5.59510 q^{4} -1.76173 q^{5} -7.80125 q^{6} +2.47832 q^{7} -9.90779 q^{8} +5.01300 q^{9} +O(q^{10})$	\(q-2.75592 q^{2} +2.83072 q^{3} +5.59510 q^{4} -1.76173 q^{5} -7.80125 q^{6} +2.47832 q^{7} -9.90779 q^{8} +5.01300 q^{9} +4.85519 q^{10} -1.00000 q^{11} +15.8382 q^{12} -0.921197 q^{13} -6.83004 q^{14} -4.98697 q^{15} +16.1149 q^{16} +1.00000 q^{17} -13.8154 q^{18} +1.06613 q^{19} -9.85705 q^{20} +7.01543 q^{21} +2.75592 q^{22} -6.87843 q^{23} -28.0462 q^{24} -1.89631 q^{25} +2.53874 q^{26} +5.69825 q^{27} +13.8664 q^{28} -1.83349 q^{29} +13.7437 q^{30} +3.24055 q^{31} -24.5958 q^{32} -2.83072 q^{33} -2.75592 q^{34} -4.36613 q^{35} +28.0482 q^{36} +4.84851 q^{37} -2.93817 q^{38} -2.60765 q^{39} +17.4549 q^{40} -9.01731 q^{41} -19.3340 q^{42} +1.00000 q^{43} -5.59510 q^{44} -8.83155 q^{45} +18.9564 q^{46} -10.9088 q^{47} +45.6168 q^{48} -0.857947 q^{49} +5.22607 q^{50} +2.83072 q^{51} -5.15418 q^{52} +12.6434 q^{53} -15.7039 q^{54} +1.76173 q^{55} -24.5547 q^{56} +3.01792 q^{57} +5.05295 q^{58} +9.80491 q^{59} -27.9026 q^{60} -13.8630 q^{61} -8.93069 q^{62} +12.4238 q^{63} +35.5542 q^{64} +1.62290 q^{65} +7.80125 q^{66} -8.65927 q^{67} +5.59510 q^{68} -19.4709 q^{69} +12.0327 q^{70} +2.10228 q^{71} -49.6678 q^{72} -14.5623 q^{73} -13.3621 q^{74} -5.36792 q^{75} +5.96510 q^{76} -2.47832 q^{77} +7.18648 q^{78} +9.90070 q^{79} -28.3901 q^{80} +1.09117 q^{81} +24.8510 q^{82} -12.1513 q^{83} +39.2520 q^{84} -1.76173 q^{85} -2.75592 q^{86} -5.19010 q^{87} +9.90779 q^{88} +1.88208 q^{89} +24.3391 q^{90} -2.28302 q^{91} -38.4854 q^{92} +9.17309 q^{93} +30.0638 q^{94} -1.87823 q^{95} -69.6239 q^{96} +4.30806 q^{97} +2.36443 q^{98} -5.01300 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) $ 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * 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 $

$2$	−2.75592	−1.94873	−0.974365	−	0.224974i	$-0.927770\pi$
$2$	−2.75592	−1.94873	−0.974365	+	0.224974i	$0.927770\pi$
$3$	2.83072	1.63432	0.817160	−	0.576411i	$-0.195547\pi$
$3$	2.83072	1.63432	0.817160	+	0.576411i	$0.195547\pi$
$4$	5.59510	2.79755
$5$	−1.76173	−0.787870	−0.393935	−	0.919138i	$-0.628886\pi$
$5$	−1.76173	−0.787870	−0.393935	+	0.919138i	$0.628886\pi$
$6$	−7.80125	−3.18485
$7$	2.47832	0.936716	0.468358	−	0.883539i	$-0.344846\pi$
$7$	2.47832	0.936716	0.468358	+	0.883539i	$0.344846\pi$
$8$	−9.90779	−3.50293
$9$	5.01300	1.67100
$10$	4.85519	1.53535
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	15.8382	4.57209
$13$	−0.921197	−0.255494	−0.127747	−	0.991807i	$-0.540775\pi$
$13$	−0.921197	−0.255494	−0.127747	+	0.991807i	$0.540775\pi$
$14$	−6.83004	−1.82541
$15$	−4.98697	−1.28763
$16$	16.1149	4.02873
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−13.8154	−3.25633
$19$	1.06613	0.244587	0.122294	−	0.992494i	$-0.460975\pi$
$19$	1.06613	0.244587	0.122294	+	0.992494i	$0.460975\pi$
$20$	−9.85705	−2.20410
$21$	7.01543	1.53089
$22$	2.75592	0.587564
$23$	−6.87843	−1.43425	−0.717125	−	0.696944i	$-0.754543\pi$
$23$	−6.87843	−1.43425	−0.717125	+	0.696944i	$0.754543\pi$
$24$	−28.0462	−5.72491
$25$	−1.89631	−0.379261
$26$	2.53874	0.497889
$27$	5.69825	1.09663
$28$	13.8664	2.62051
$29$	−1.83349	−0.340470	−0.170235	−	0.985403i	$-0.554453\pi$
$29$	−1.83349	−0.340470	−0.170235	+	0.985403i	$0.554453\pi$
$30$	13.7437	2.50924
$31$	3.24055	0.582019	0.291010	−	0.956720i	$-0.406009\pi$
$31$	3.24055	0.582019	0.291010	+	0.956720i	$0.406009\pi$
$32$	−24.5958	−4.34796
$33$	−2.83072	−0.492766
$34$	−2.75592	−0.472636
$35$	−4.36613	−0.738010
$36$	28.0482	4.67470
$37$	4.84851	0.797090	0.398545	−	0.917149i	$-0.369515\pi$
$37$	4.84851	0.797090	0.398545	+	0.917149i	$0.369515\pi$
$38$	−2.93817	−0.476634
$39$	−2.60765	−0.417559
$40$	17.4549	2.75986
$41$	−9.01731	−1.40827	−0.704134	−	0.710067i	$-0.748664\pi$
$41$	−9.01731	−1.40827	−0.704134	+	0.710067i	$0.748664\pi$
$42$	−19.3340	−2.98330
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−5.59510	−0.843492
$45$	−8.83155	−1.31653
$46$	18.9564	2.79497
$47$	−10.9088	−1.59121	−0.795607	−	0.605814i	$-0.792848\pi$
$47$	−10.9088	−1.59121	−0.795607	+	0.605814i	$0.792848\pi$
$48$	45.6168	6.58422
$49$	−0.857947	−0.122564
$50$	5.22607	0.739077
$51$	2.83072	0.396381
$52$	−5.15418	−0.714757
$53$	12.6434	1.73671	0.868354	−	0.495945i	$-0.165178\pi$
$53$	12.6434	1.73671	0.868354	+	0.495945i	$0.165178\pi$
$54$	−15.7039	−2.13703
$55$	1.76173	0.237552
$56$	−24.5547	−3.28125
$57$	3.01792	0.399734
$58$	5.05295	0.663484
$59$	9.80491	1.27649	0.638245	−	0.769833i	$-0.279661\pi$
$59$	9.80491	1.27649	0.638245	+	0.769833i	$0.279661\pi$
$60$	−27.9026	−3.60221
$61$	−13.8630	−1.77497	−0.887485	−	0.460837i	$-0.847549\pi$
$61$	−13.8630	−1.77497	−0.887485	+	0.460837i	$0.847549\pi$
$62$	−8.93069	−1.13420
$63$	12.4238	1.56525
$64$	35.5542	4.44428
$65$	1.62290	0.201296
$66$	7.80125	0.960267
$67$	−8.65927	−1.05790	−0.528949	−	0.848654i	$-0.677414\pi$
$67$	−8.65927	−1.05790	−0.528949	+	0.848654i	$0.677414\pi$
$68$	5.59510	0.678505
$69$	−19.4709	−2.34402
$70$	12.0327	1.43818
$71$	2.10228	0.249494	0.124747	−	0.992189i	$-0.460188\pi$
$71$	2.10228	0.249494	0.124747	+	0.992189i	$0.460188\pi$
$72$	−49.6678	−5.85340
$73$	−14.5623	−1.70439	−0.852194	−	0.523226i	$-0.824728\pi$
$73$	−14.5623	−1.70439	−0.852194	+	0.523226i	$0.824728\pi$
$74$	−13.3621	−1.55331
$75$	−5.36792	−0.619834
$76$	5.96510	0.684244
$77$	−2.47832	−0.282430
$78$	7.18648	0.813709
$79$	9.90070	1.11392	0.556958	−	0.830541i	$-0.311968\pi$
$79$	9.90070	1.11392	0.556958	+	0.830541i	$0.311968\pi$
$80$	−28.3901	−3.17411
$81$	1.09117	0.121241
$82$	24.8510	2.74433
$83$	−12.1513	−1.33378	−0.666889	−	0.745157i	$-0.732374\pi$
$83$	−12.1513	−1.33378	−0.666889	+	0.745157i	$0.732374\pi$
$84$	39.2520	4.28274
$85$	−1.76173	−0.191087
$86$	−2.75592	−0.297179
$87$	−5.19010	−0.556437
$88$	9.90779	1.05617
$89$	1.88208	0.199500	0.0997502	−	0.995013i	$-0.468196\pi$
$89$	1.88208	0.199500	0.0997502	+	0.995013i	$0.468196\pi$
$90$	24.3391	2.56556
$91$	−2.28302	−0.239325
$92$	−38.4854	−4.01239
$93$	9.17309	0.951205
$94$	30.0638	3.10084
$95$	−1.87823	−0.192703
$96$	−69.6239	−7.10596
$97$	4.30806	0.437417	0.218709	−	0.975790i	$-0.429816\pi$
$97$	4.30806	0.437417	0.218709	+	0.975790i	$0.429816\pi$
$98$	2.36443	0.238844
$99$	−5.01300	−0.503825
$100$	−10.6100	−1.06100
$101$	−10.4147	−1.03630	−0.518149	−	0.855290i	$-0.673379\pi$
$101$	−10.4147	−1.03630	−0.518149	+	0.855290i	$0.673379\pi$
$102$	−7.80125	−0.772439
$103$	−0.613022	−0.0604028	−0.0302014	−	0.999544i	$-0.509615\pi$
$103$	−0.613022	−0.0604028	−0.0302014	+	0.999544i	$0.509615\pi$
$104$	9.12703	0.894979
$105$	−12.3593	−1.20614
$106$	−34.8443	−3.38437
$107$	−3.93219	−0.380139	−0.190070	−	0.981771i	$-0.560871\pi$
$107$	−3.93219	−0.380139	−0.190070	+	0.981771i	$0.560871\pi$
$108$	31.8822	3.06787
$109$	−18.5754	−1.77920	−0.889600	−	0.456740i	$-0.849017\pi$
$109$	−18.5754	−1.77920	−0.889600	+	0.456740i	$0.849017\pi$
$110$	−4.85519	−0.462924
$111$	13.7248	1.30270
$112$	39.9378	3.77377
$113$	7.98301	0.750978	0.375489	−	0.926827i	$-0.377475\pi$
$113$	7.98301	0.750978	0.375489	+	0.926827i	$0.377475\pi$
$114$	−8.31715	−0.778973
$115$	12.1179	1.13000
$116$	−10.2585	−0.952481
$117$	−4.61796	−0.426930
$118$	−27.0216	−2.48754
$119$	2.47832	0.227187
$120$	49.4099	4.51049
$121$	1.00000	0.0909091
$122$	38.2052	3.45894
$123$	−25.5255	−2.30156
$124$	18.1312	1.62823
$125$	12.1494	1.08668
$126$	−34.2390	−3.05025
$127$	18.7012	1.65946	0.829730	−	0.558165i	$-0.188494\pi$
$127$	18.7012	1.65946	0.829730	+	0.558165i	$0.188494\pi$
$128$	−48.7930	−4.31273
$129$	2.83072	0.249231
$130$	−4.47258	−0.392271
$131$	−16.4833	−1.44015	−0.720075	−	0.693897i	$-0.755892\pi$
$131$	−16.4833	−1.44015	−0.720075	+	0.693897i	$0.755892\pi$
$132$	−15.8382	−1.37854
$133$	2.64221	0.229109
$134$	23.8643	2.06156
$135$	−10.0388	−0.864000
$136$	−9.90779	−0.849586
$137$	−9.01685	−0.770362	−0.385181	−	0.922841i	$-0.625861\pi$
$137$	−9.01685	−0.770362	−0.385181	+	0.922841i	$0.625861\pi$
$138$	53.6603	4.56787
$139$	1.87254	0.158827	0.0794133	−	0.996842i	$-0.474695\pi$
$139$	1.87254	0.158827	0.0794133	+	0.996842i	$0.474695\pi$
$140$	−24.4289	−2.06462
$141$	−30.8798	−2.60055
$142$	−5.79371	−0.486197
$143$	0.921197	0.0770343
$144$	80.7840	6.73200
$145$	3.23011	0.268246
$146$	40.1325	3.32139
$147$	−2.42861	−0.200308
$148$	27.1279	2.22990
$149$	−17.1622	−1.40598	−0.702991	−	0.711199i	$-0.748153\pi$
$149$	−17.1622	−1.40598	−0.702991	+	0.711199i	$0.748153\pi$
$150$	14.7936	1.20789
$151$	−4.39570	−0.357717	−0.178858	−	0.983875i	$-0.557240\pi$
$151$	−4.39570	−0.357717	−0.178858	+	0.983875i	$0.557240\pi$
$152$	−10.5630	−0.856773
$153$	5.01300	0.405277
$154$	6.83004	0.550381
$155$	−5.70897	−0.458555
$156$	−14.5901	−1.16814
$157$	19.2593	1.53706	0.768528	−	0.639816i	$-0.220989\pi$
$157$	19.2593	1.53706	0.768528	+	0.639816i	$0.220989\pi$
$158$	−27.2855	−2.17072
$159$	35.7900	2.83834
$160$	43.3311	3.42563
$161$	−17.0469	−1.34349
$162$	−3.00717	−0.236266
$163$	−7.82558	−0.612947	−0.306473	−	0.951879i	$-0.599149\pi$
$163$	−7.82558	−0.612947	−0.306473	+	0.951879i	$0.599149\pi$
$164$	−50.4527	−3.93970
$165$	4.98697	0.388235
$166$	33.4880	2.59917
$167$	19.7179	1.52581	0.762907	−	0.646508i	$-0.223771\pi$
$167$	19.7179	1.52581	0.762907	+	0.646508i	$0.223771\pi$
$168$	−69.5075	−5.36262
$169$	−12.1514	−0.934723
$170$	4.85519	0.372376
$171$	5.34451	0.408705
$172$	5.59510	0.426622
$173$	0.737860	0.0560985	0.0280492	−	0.999607i	$-0.491070\pi$
$173$	0.737860	0.0560985	0.0280492	+	0.999607i	$0.491070\pi$
$174$	14.3035	1.08435
$175$	−4.69965	−0.355260
$176$	−16.1149	−1.21471
$177$	27.7550	2.08619
$178$	−5.18687	−0.388772
$179$	5.29334	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$179$	5.29334	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$180$	−49.4134	−3.68306
$181$	−2.09291	−0.155564	−0.0777822	−	0.996970i	$-0.524784\pi$
$181$	−2.09291	−0.155564	−0.0777822	+	0.996970i	$0.524784\pi$
$182$	6.29181	0.466380
$183$	−39.2422	−2.90087
$184$	68.1500	5.02409
$185$	−8.54176	−0.628003
$186$	−25.2803	−1.85364
$187$	−1.00000	−0.0731272
$188$	−61.0358	−4.45149
$189$	14.1221	1.02723
$190$	5.17627	0.375526
$191$	16.0745	1.16311	0.581554	−	0.813508i	$-0.302445\pi$
$191$	16.0745	1.16311	0.581554	+	0.813508i	$0.302445\pi$
$192$	100.644	7.26337
$193$	19.4948	1.40326	0.701631	−	0.712540i	$-0.252455\pi$
$193$	19.4948	1.40326	0.701631	+	0.712540i	$0.252455\pi$
$194$	−11.8727	−0.852408
$195$	4.59398	0.328982
$196$	−4.80029	−0.342878
$197$	14.7496	1.05087	0.525434	−	0.850835i	$-0.323903\pi$
$197$	14.7496	1.05087	0.525434	+	0.850835i	$0.323903\pi$
$198$	13.8154	0.981820
$199$	−22.6232	−1.60372	−0.801858	−	0.597515i	$-0.796155\pi$
$199$	−22.6232	−1.60372	−0.801858	+	0.597515i	$0.796155\pi$
$200$	18.7882	1.32853
$201$	−24.5120	−1.72894
$202$	28.7020	2.01947
$203$	−4.54396	−0.318924
$204$	15.8382	1.10889
$205$	15.8861	1.10953
$206$	1.68944	0.117709
$207$	−34.4815	−2.39663
$208$	−14.8450	−1.02931
$209$	−1.06613	−0.0737458
$210$	34.0612	2.35045
$211$	−4.91662	−0.338474	−0.169237	−	0.985575i	$-0.554130\pi$
$211$	−4.91662	−0.338474	−0.169237	+	0.985575i	$0.554130\pi$
$212$	70.7412	4.85852
$213$	5.95097	0.407754
$214$	10.8368	0.740789
$215$	−1.76173	−0.120149
$216$	−56.4571	−3.84142
$217$	8.03110	0.545187
$218$	51.1923	3.46718
$219$	−41.2218	−2.78551
$220$	9.85705	0.664562
$221$	−0.921197	−0.0619664
$222$	−37.8244	−2.53861
$223$	−7.17471	−0.480454	−0.240227	−	0.970717i	$-0.577222\pi$
$223$	−7.17471	−0.480454	−0.240227	+	0.970717i	$0.577222\pi$
$224$	−60.9561	−4.07280
$225$	−9.50618	−0.633745
$226$	−22.0005	−1.46345
$227$	−4.96024	−0.329223	−0.164611	−	0.986358i	$-0.552637\pi$
$227$	−4.96024	−0.329223	−0.164611	+	0.986358i	$0.552637\pi$
$228$	16.8856	1.11827
$229$	0.879426	0.0581141	0.0290570	−	0.999578i	$-0.490750\pi$
$229$	0.879426	0.0581141	0.0290570	+	0.999578i	$0.490750\pi$
$230$	−33.3961	−2.20207
$231$	−7.01543	−0.461581
$232$	18.1658	1.19264
$233$	−9.96700	−0.652960	−0.326480	−	0.945204i	$-0.605863\pi$
$233$	−9.96700	−0.652960	−0.326480	+	0.945204i	$0.605863\pi$
$234$	12.7267	0.831972
$235$	19.2184	1.25367
$236$	54.8594	3.57104
$237$	28.0262	1.82049
$238$	−6.83004	−0.442726
$239$	2.35828	0.152544	0.0762721	−	0.997087i	$-0.475698\pi$
$239$	2.35828	0.152544	0.0762721	+	0.997087i	$0.475698\pi$
$240$	−80.3646	−5.18751
$241$	−7.25887	−0.467585	−0.233793	−	0.972286i	$-0.575114\pi$
$241$	−7.25887	−0.467585	−0.233793	+	0.972286i	$0.575114\pi$
$242$	−2.75592	−0.177157
$243$	−14.0059	−0.898482
$244$	−77.5646	−4.96556
$245$	1.51147	0.0965643
$246$	70.3463	4.48512
$247$	−0.982116	−0.0624905
$248$	−32.1067	−2.03878
$249$	−34.3970	−2.17982
$250$	−33.4829	−2.11764
$251$	16.3341	1.03100	0.515500	−	0.856890i	$-0.327606\pi$
$251$	16.3341	1.03100	0.515500	+	0.856890i	$0.327606\pi$
$252$	69.5123	4.37887
$253$	6.87843	0.432443
$254$	−51.5389	−3.23384
$255$	−4.98697	−0.312296
$256$	63.3612	3.96008
$257$	3.55239	0.221592	0.110796	−	0.993843i	$-0.464660\pi$
$257$	3.55239	0.221592	0.110796	+	0.993843i	$0.464660\pi$
$258$	−7.80125	−0.485685
$259$	12.0161	0.746646
$260$	9.08028	0.563135
$261$	−9.19127	−0.568926
$262$	45.4266	2.80646
$263$	−28.6084	−1.76407	−0.882035	−	0.471185i	$-0.843827\pi$
$263$	−28.6084	−1.76407	−0.882035	+	0.471185i	$0.843827\pi$
$264$	28.0462	1.72613
$265$	−22.2743	−1.36830
$266$	−7.28172	−0.446471
$267$	5.32766	0.326047
$268$	−48.4494	−2.95952
$269$	−3.41441	−0.208180	−0.104090	−	0.994568i	$-0.533193\pi$
$269$	−3.41441	−0.208180	−0.104090	+	0.994568i	$0.533193\pi$
$270$	27.6661	1.68370
$271$	−20.1976	−1.22692	−0.613459	−	0.789727i	$-0.710222\pi$
$271$	−20.1976	−1.22692	−0.613459	+	0.789727i	$0.710222\pi$
$272$	16.1149	0.977109
$273$	−6.46259	−0.391134
$274$	24.8497	1.50123
$275$	1.89631	0.114352
$276$	−108.942	−6.55752
$277$	−1.64326	−0.0987338	−0.0493669	−	0.998781i	$-0.515720\pi$
$277$	−1.64326	−0.0987338	−0.0493669	+	0.998781i	$0.515720\pi$
$278$	−5.16056	−0.309510
$279$	16.2449	0.972554
$280$	43.2587	2.58520
$281$	4.58914	0.273765	0.136883	−	0.990587i	$-0.456292\pi$
$281$	4.58914	0.273765	0.136883	+	0.990587i	$0.456292\pi$
$282$	85.1023	5.06777
$283$	14.4492	0.858918	0.429459	−	0.903086i	$-0.358704\pi$
$283$	14.4492	0.858918	0.429459	+	0.903086i	$0.358704\pi$
$284$	11.7624	0.697973
$285$	−5.31676	−0.314938
$286$	−2.53874	−0.150119
$287$	−22.3478	−1.31915
$288$	−123.299	−7.26544
$289$	1.00000	0.0588235
$290$	−8.90193	−0.522739
$291$	12.1949	0.714880
$292$	−81.4774	−4.76811
$293$	−29.4566	−1.72087	−0.860436	−	0.509559i	$-0.829809\pi$
$293$	−29.4566	−1.72087	−0.860436	+	0.509559i	$0.829809\pi$
$294$	6.69306	0.390347
$295$	−17.2736	−1.00571
$296$	−48.0380	−2.79215
$297$	−5.69825	−0.330646
$298$	47.2976	2.73988
$299$	6.33638	0.366442
$300$	−30.0340	−1.73401
$301$	2.47832	0.142848
$302$	12.1142	0.697093
$303$	−29.4811	−1.69364
$304$	17.1806	0.985374
$305$	24.4228	1.39845
$306$	−13.8154	−0.789775
$307$	−19.3455	−1.10411	−0.552053	−	0.833809i	$-0.686155\pi$
$307$	−19.3455	−1.10411	−0.552053	+	0.833809i	$0.686155\pi$
$308$	−13.8664	−0.790112
$309$	−1.73530	−0.0987175
$310$	15.7335	0.893601
$311$	10.9241	0.619451	0.309725	−	0.950826i	$-0.399763\pi$
$311$	10.9241	0.619451	0.309725	+	0.950826i	$0.399763\pi$
$312$	25.8361	1.46268
$313$	−34.9660	−1.97639	−0.988197	−	0.153189i	$-0.951046\pi$
$313$	−34.9660	−1.97639	−0.988197	+	0.153189i	$0.951046\pi$
$314$	−53.0770	−2.99531
$315$	−21.8874	−1.23321
$316$	55.3954	3.11623
$317$	−27.5945	−1.54986	−0.774930	−	0.632048i	$-0.782215\pi$
$317$	−27.5945	−1.54986	−0.774930	+	0.632048i	$0.782215\pi$
$318$	−98.6345	−5.53115
$319$	1.83349	0.102656
$320$	−62.6370	−3.50151
$321$	−11.1309	−0.621269
$322$	46.9799	2.61809
$323$	1.06613	0.0593211
$324$	6.10519	0.339177
$325$	1.74687	0.0968989
$326$	21.5667	1.19447
$327$	−52.5818	−2.90778
$328$	89.3417	4.93307
$329$	−27.0355	−1.49051
$330$	−13.7437	−0.756566
$331$	−15.8221	−0.869659	−0.434830	−	0.900513i	$-0.643191\pi$
$331$	−15.8221	−0.869659	−0.434830	+	0.900513i	$0.643191\pi$
$332$	−67.9876	−3.73131
$333$	24.3056	1.33194
$334$	−54.3409	−2.97340
$335$	15.2553	0.833486
$336$	113.053	6.16755
$337$	26.1285	1.42331	0.711654	−	0.702530i	$-0.247946\pi$
$337$	26.1285	1.42331	0.711654	+	0.702530i	$0.247946\pi$
$338$	33.4883	1.82152
$339$	22.5977	1.22734
$340$	−9.85705	−0.534574
$341$	−3.24055	−0.175485
$342$	−14.7290	−0.796456
$343$	−19.4745	−1.05152
$344$	−9.90779	−0.534193
$345$	34.3025	1.84679
$346$	−2.03348	−0.109321
$347$	29.3008	1.57295	0.786474	−	0.617624i	$-0.211905\pi$
$347$	29.3008	1.57295	0.786474	+	0.617624i	$0.211905\pi$
$348$	−29.0391	−1.55666
$349$	25.1926	1.34853	0.674263	−	0.738491i	$-0.264461\pi$
$349$	25.1926	1.34853	0.674263	+	0.738491i	$0.264461\pi$
$350$	12.9518	0.692305
$351$	−5.24921	−0.280182
$352$	24.5958	1.31096
$353$	16.6865	0.888131	0.444066	−	0.895994i	$-0.353536\pi$
$353$	16.6865	0.888131	0.444066	+	0.895994i	$0.353536\pi$
$354$	−76.4906	−4.06543
$355$	−3.70365	−0.196569
$356$	10.5304	0.558112
$357$	7.01543	0.371296
$358$	−14.5880	−0.771001
$359$	−16.4783	−0.869691	−0.434846	−	0.900505i	$-0.643197\pi$
$359$	−16.4783	−0.869691	−0.434846	+	0.900505i	$0.643197\pi$
$360$	87.5012	4.61172
$361$	−17.8634	−0.940177
$362$	5.76788	0.303153
$363$	2.83072	0.148574
$364$	−12.7737	−0.669524
$365$	25.6548	1.34284
$366$	108.148	5.65301
$367$	−19.1797	−1.00117	−0.500587	−	0.865686i	$-0.666882\pi$
$367$	−19.1797	−1.00117	−0.500587	+	0.865686i	$0.666882\pi$
$368$	−110.845	−5.77820
$369$	−45.2038	−2.35321
$370$	23.5404	1.22381
$371$	31.3344	1.62680
$372$	51.3243	2.66104
$373$	3.58225	0.185482	0.0927410	−	0.995690i	$-0.470437\pi$
$373$	3.58225	0.185482	0.0927410	+	0.995690i	$0.470437\pi$
$374$	2.75592	0.142505
$375$	34.3917	1.77598
$376$	108.082	5.57392
$377$	1.68900	0.0869881
$378$	−38.9193	−2.00179
$379$	7.29998	0.374975	0.187487	−	0.982267i	$-0.439966\pi$
$379$	7.29998	0.374975	0.187487	+	0.982267i	$0.439966\pi$
$380$	−10.5089	−0.539095
$381$	52.9379	2.71209
$382$	−44.3000	−2.26658
$383$	7.91922	0.404653	0.202327	−	0.979318i	$-0.435150\pi$
$383$	7.91922	0.404653	0.202327	+	0.979318i	$0.435150\pi$
$384$	−138.120	−7.04838
$385$	4.36613	0.222518
$386$	−53.7260	−2.73458
$387$	5.01300	0.254825
$388$	24.1040	1.22370
$389$	5.96354	0.302364	0.151182	−	0.988506i	$-0.451692\pi$
$389$	5.96354	0.302364	0.151182	+	0.988506i	$0.451692\pi$
$390$	−12.6606	−0.641097
$391$	−6.87843	−0.347857
$392$	8.50036	0.429333
$393$	−46.6596	−2.35366
$394$	−40.6488	−2.04786
$395$	−17.4424	−0.877621
$396$	−28.0482	−1.40948
$397$	−21.4910	−1.07860	−0.539302	−	0.842112i	$-0.681312\pi$
$397$	−21.4910	−1.07860	−0.539302	+	0.842112i	$0.681312\pi$
$398$	62.3477	3.12521
$399$	7.47937	0.374437
$400$	−30.5588	−1.52794
$401$	9.86409	0.492589	0.246295	−	0.969195i	$-0.420787\pi$
$401$	9.86409	0.492589	0.246295	+	0.969195i	$0.420787\pi$
$402$	67.5531	3.36924
$403$	−2.98518	−0.148702
$404$	−58.2711	−2.89910
$405$	−1.92234	−0.0955220
$406$	12.5228	0.621496
$407$	−4.84851	−0.240332
$408$	−28.0462	−1.38850
$409$	6.52919	0.322848	0.161424	−	0.986885i	$-0.448391\pi$
$409$	6.52919	0.322848	0.161424	+	0.986885i	$0.448391\pi$
$410$	−43.7807	−2.16218
$411$	−25.5242	−1.25902
$412$	−3.42991	−0.168980
$413$	24.2997	1.19571
$414$	95.0284	4.67039
$415$	21.4073	1.05084
$416$	22.6576	1.11088
$417$	5.30064	0.259573
$418$	2.93817	0.143711
$419$	14.8248	0.724239	0.362120	−	0.932132i	$-0.382053\pi$
$419$	14.8248	0.724239	0.362120	+	0.932132i	$0.382053\pi$
$420$	−69.1515	−3.37425
$421$	−11.5694	−0.563856	−0.281928	−	0.959436i	$-0.590974\pi$
$421$	−11.5694	−0.563856	−0.281928	+	0.959436i	$0.590974\pi$
$422$	13.5498	0.659595
$423$	−54.6858	−2.65892
$424$	−125.268	−6.08357
$425$	−1.89631	−0.0919843
$426$	−16.4004	−0.794602
$427$	−34.3568	−1.66264
$428$	−22.0010	−1.06346
$429$	2.60765	0.125899
$430$	4.85519	0.234138
$431$	−27.3899	−1.31932	−0.659662	−	0.751563i	$-0.729300\pi$
$431$	−27.3899	−1.31932	−0.659662	+	0.751563i	$0.729300\pi$
$432$	91.8267	4.41801
$433$	36.4841	1.75332	0.876658	−	0.481114i	$-0.159768\pi$
$433$	36.4841	1.75332	0.876658	+	0.481114i	$0.159768\pi$
$434$	−22.1331	−1.06242
$435$	9.14355	0.438400
$436$	−103.931	−4.97740
$437$	−7.33330	−0.350799
$438$	113.604	5.42821
$439$	−18.0335	−0.860692	−0.430346	−	0.902664i	$-0.641608\pi$
$439$	−18.0335	−0.860692	−0.430346	+	0.902664i	$0.641608\pi$
$440$	−17.4549	−0.832128
$441$	−4.30089	−0.204804
$442$	2.53874	0.120756
$443$	−7.71254	−0.366434	−0.183217	−	0.983073i	$-0.558651\pi$
$443$	−7.71254	−0.366434	−0.183217	+	0.983073i	$0.558651\pi$
$444$	76.7915	3.64436
$445$	−3.31572	−0.157180
$446$	19.7729	0.936276
$447$	−48.5815	−2.29782
$448$	88.1146	4.16302
$449$	11.6899	0.551681	0.275841	−	0.961203i	$-0.411044\pi$
$449$	11.6899	0.551681	0.275841	+	0.961203i	$0.411044\pi$
$450$	26.1983	1.23500
$451$	9.01731	0.424609
$452$	44.6657	2.10090
$453$	−12.4430	−0.584623
$454$	13.6700	0.641566
$455$	4.02206	0.188557
$456$	−29.9010	−1.40024
$457$	4.54533	0.212621	0.106311	−	0.994333i	$-0.466096\pi$
$457$	4.54533	0.212621	0.106311	+	0.994333i	$0.466096\pi$
$458$	−2.42363	−0.113249
$459$	5.69825	0.265971
$460$	67.8010	3.16124
$461$	−4.53893	−0.211399	−0.105700	−	0.994398i	$-0.533708\pi$
$461$	−4.53893	−0.211399	−0.105700	+	0.994398i	$0.533708\pi$
$462$	19.3340	0.899498
$463$	−5.08023	−0.236098	−0.118049	−	0.993008i	$-0.537664\pi$
$463$	−5.08023	−0.236098	−0.118049	+	0.993008i	$0.537664\pi$
$464$	−29.5465	−1.37166
$465$	−16.1605	−0.749426
$466$	27.4683	1.27244
$467$	21.1311	0.977832	0.488916	−	0.872331i	$-0.337393\pi$
$467$	21.1311	0.977832	0.488916	+	0.872331i	$0.337393\pi$
$468$	−25.8379	−1.19436
$469$	−21.4604	−0.990950
$470$	−52.9643	−2.44306
$471$	54.5177	2.51204
$472$	−97.1451	−4.47146
$473$	−1.00000	−0.0459800
$474$	−77.2378	−3.54765
$475$	−2.02171	−0.0927624
$476$	13.8664	0.635566
$477$	63.3815	2.90204
$478$	−6.49922	−0.297267
$479$	38.4202	1.75546	0.877732	−	0.479152i	$-0.159056\pi$
$479$	38.4202	1.75546	0.877732	+	0.479152i	$0.159056\pi$
$480$	122.659	5.59857
$481$	−4.46643	−0.203652
$482$	20.0049	0.911197
$483$	−48.2551	−2.19568
$484$	5.59510	0.254323
$485$	−7.58965	−0.344628
$486$	38.5993	1.75090
$487$	14.5613	0.659835	0.329917	−	0.944010i	$-0.392979\pi$
$487$	14.5613	0.659835	0.329917	+	0.944010i	$0.392979\pi$
$488$	137.351	6.21760
$489$	−22.1521	−1.00175
$490$	−4.16549	−0.188178
$491$	10.5346	0.475420	0.237710	−	0.971336i	$-0.423603\pi$
$491$	10.5346	0.475420	0.237710	+	0.971336i	$0.423603\pi$
$492$	−142.818	−6.43872
$493$	−1.83349	−0.0825761
$494$	2.70663	0.121777
$495$	8.83155	0.396949
$496$	52.2211	2.34480
$497$	5.21011	0.233705
$498$	94.7953	4.24788
$499$	−2.83366	−0.126852	−0.0634260	−	0.997987i	$-0.520203\pi$
$499$	−2.83366	−0.126852	−0.0634260	+	0.997987i	$0.520203\pi$
$500$	67.9772	3.04003
$501$	55.8158	2.49367
$502$	−45.0155	−2.00914
$503$	−31.6956	−1.41324	−0.706619	−	0.707594i	$-0.749781\pi$
$503$	−31.6956	−1.41324	−0.706619	+	0.707594i	$0.749781\pi$
$504$	−123.092	−5.48297
$505$	18.3478	0.816469
$506$	−18.9564	−0.842714
$507$	−34.3973	−1.52764
$508$	104.635	4.64242
$509$	−13.4644	−0.596800	−0.298400	−	0.954441i	$-0.596453\pi$
$509$	−13.4644	−0.596800	−0.298400	+	0.954441i	$0.596453\pi$
$510$	13.7437	0.608581
$511$	−36.0900	−1.59653
$512$	−77.0324	−3.40438
$513$	6.07508	0.268221
$514$	−9.79009	−0.431822
$515$	1.07998	0.0475896
$516$	15.8382	0.697237
$517$	10.9088	0.479769
$518$	−33.1155	−1.45501
$519$	2.08868	0.0916829
$520$	−16.0794	−0.705127
$521$	22.9484	1.00539	0.502694	−	0.864464i	$-0.332342\pi$
$521$	22.9484	1.00539	0.502694	+	0.864464i	$0.332342\pi$
$522$	25.3304	1.10868
$523$	3.71790	0.162573	0.0812863	−	0.996691i	$-0.474097\pi$
$523$	3.71790	0.162573	0.0812863	+	0.996691i	$0.474097\pi$
$524$	−92.2254	−4.02889
$525$	−13.3034	−0.580608
$526$	78.8425	3.43769
$527$	3.24055	0.141160
$528$	−45.6168	−1.98522
$529$	24.3127	1.05708
$530$	61.3862	2.66645
$531$	49.1520	2.13302
$532$	14.7834	0.640942
$533$	8.30672	0.359804
$534$	−14.6826	−0.635378
$535$	6.92746	0.299500
$536$	85.7943	3.70575
$537$	14.9840	0.646607
$538$	9.40984	0.405687
$539$	0.857947	0.0369544
$540$	−56.1679	−2.41708
$541$	−30.7807	−1.32337	−0.661683	−	0.749783i	$-0.730158\pi$
$541$	−30.7807	−1.32337	−0.661683	+	0.749783i	$0.730158\pi$
$542$	55.6630	2.39093
$543$	−5.92444	−0.254242
$544$	−24.5958	−1.05454
$545$	32.7248	1.40178
$546$	17.8104	0.762214
$547$	15.8214	0.676475	0.338237	−	0.941061i	$-0.390169\pi$
$547$	15.8214	0.676475	0.338237	+	0.941061i	$0.390169\pi$
$548$	−50.4502	−2.15512
$549$	−69.4950	−2.96597
$550$	−5.22607	−0.222840
$551$	−1.95474	−0.0832746
$552$	192.914	8.21096
$553$	24.5371	1.04342
$554$	4.52869	0.192406
$555$	−24.1794	−1.02636
$556$	10.4770	0.444325
$557$	−36.5420	−1.54833	−0.774167	−	0.632982i	$-0.781831\pi$
$557$	−36.5420	−1.54833	−0.774167	+	0.632982i	$0.781831\pi$
$558$	−44.7695	−1.89525
$559$	−0.921197	−0.0389625
$560$	−70.3597	−2.97324
$561$	−2.83072	−0.119513
$562$	−12.6473	−0.533495
$563$	−0.692623	−0.0291906	−0.0145953	−	0.999893i	$-0.504646\pi$
$563$	−0.692623	−0.0291906	−0.0145953	+	0.999893i	$0.504646\pi$
$564$	−172.776	−7.27516
$565$	−14.0639	−0.591673
$566$	−39.8209	−1.67380
$567$	2.70426	0.113568
$568$	−20.8289	−0.873963
$569$	7.22464	0.302873	0.151436	−	0.988467i	$-0.451610\pi$
$569$	7.22464	0.302873	0.151436	+	0.988467i	$0.451610\pi$
$570$	14.6526	0.613729
$571$	25.9170	1.08459	0.542297	−	0.840187i	$-0.317555\pi$
$571$	25.9170	1.08459	0.542297	+	0.840187i	$0.317555\pi$
$572$	5.15418	0.215507
$573$	45.5024	1.90089
$574$	61.5886	2.57066
$575$	13.0436	0.543956
$576$	178.233	7.42639
$577$	25.9861	1.08181	0.540907	−	0.841082i	$-0.318081\pi$
$577$	25.9861	1.08181	0.540907	+	0.841082i	$0.318081\pi$
$578$	−2.75592	−0.114631
$579$	55.1843	2.29338
$580$	18.0728	0.750431
$581$	−30.1148	−1.24937
$582$	−33.6083	−1.39311
$583$	−12.6434	−0.523637
$584$	144.280	5.97036
$585$	8.13560	0.336366
$586$	81.1800	3.35351
$587$	10.7970	0.445638	0.222819	−	0.974860i	$-0.428474\pi$
$587$	10.7970	0.445638	0.222819	+	0.974860i	$0.428474\pi$
$588$	−13.5883	−0.560372
$589$	3.45485	0.142354
$590$	47.6047	1.95985
$591$	41.7521	1.71745
$592$	78.1332	3.21126
$593$	−7.78692	−0.319771	−0.159885	−	0.987136i	$-0.551112\pi$
$593$	−7.78692	−0.319771	−0.159885	+	0.987136i	$0.551112\pi$
$594$	15.7039	0.644339
$595$	−4.36613	−0.178994
$596$	−96.0241	−3.93330
$597$	−64.0400	−2.62098
$598$	−17.4626	−0.714097
$599$	−9.17818	−0.375010	−0.187505	−	0.982264i	$-0.560040\pi$
$599$	−9.17818	−0.375010	−0.187505	+	0.982264i	$0.560040\pi$
$600$	53.1842	2.17124
$601$	39.3717	1.60601	0.803003	−	0.595976i	$-0.203234\pi$
$601$	39.3717	1.60601	0.803003	+	0.595976i	$0.203234\pi$
$602$	−6.83004	−0.278372
$603$	−43.4089	−1.76775
$604$	−24.5943	−1.00073
$605$	−1.76173	−0.0716245
$606$	81.2475	3.30045
$607$	30.4587	1.23628	0.618140	−	0.786068i	$-0.287886\pi$
$607$	30.4587	1.23628	0.618140	+	0.786068i	$0.287886\pi$
$608$	−26.2223	−1.06346
$609$	−12.8627	−0.521223
$610$	−67.3073	−2.72519
$611$	10.0492	0.406545
$612$	28.0482	1.13378
$613$	−9.14982	−0.369558	−0.184779	−	0.982780i	$-0.559157\pi$
$613$	−9.14982	−0.369558	−0.184779	+	0.982780i	$0.559157\pi$
$614$	53.3146	2.15160
$615$	44.9691	1.81333
$616$	24.5547	0.989335
$617$	−16.7057	−0.672545	−0.336273	−	0.941765i	$-0.609166\pi$
$617$	−16.7057	−0.672545	−0.336273	+	0.941765i	$0.609166\pi$
$618$	4.78234	0.192374
$619$	28.6779	1.15266	0.576332	−	0.817216i	$-0.304484\pi$
$619$	28.6779	1.15266	0.576332	+	0.817216i	$0.304484\pi$
$620$	−31.9422	−1.28283
$621$	−39.1950	−1.57284
$622$	−30.1060	−1.20714
$623$	4.66440	0.186875
$624$	−42.0221	−1.68223
$625$	−11.9225	−0.476900
$626$	96.3634	3.85146
$627$	−3.01792	−0.120524
$628$	107.757	4.29999
$629$	4.84851	0.193323
$630$	60.3199	2.40320
$631$	−18.6284	−0.741587	−0.370793	−	0.928715i	$-0.620914\pi$
$631$	−18.6284	−0.741587	−0.370793	+	0.928715i	$0.620914\pi$
$632$	−98.0941	−3.90197
$633$	−13.9176	−0.553175
$634$	76.0481	3.02026
$635$	−32.9464	−1.30744
$636$	200.249	7.94038
$637$	0.790338	0.0313143
$638$	−5.05295	−0.200048
$639$	10.5387	0.416905
$640$	85.9601	3.39787
$641$	14.6407	0.578275	0.289137	−	0.957288i	$-0.406632\pi$
$641$	14.6407	0.578275	0.289137	+	0.957288i	$0.406632\pi$
$642$	30.6760	1.21069
$643$	44.5045	1.75509	0.877543	−	0.479499i	$-0.159182\pi$
$643$	44.5045	1.75509	0.877543	+	0.479499i	$0.159182\pi$
$644$	−95.3791	−3.75846
$645$	−4.98697	−0.196362
$646$	−2.93817	−0.115601
$647$	12.2137	0.480172	0.240086	−	0.970752i	$-0.422824\pi$
$647$	12.2137	0.480172	0.240086	+	0.970752i	$0.422824\pi$
$648$	−10.8111	−0.424699
$649$	−9.80491	−0.384876
$650$	−4.81423	−0.188830
$651$	22.7338	0.891009
$652$	−43.7849	−1.71475
$653$	20.9830	0.821127	0.410563	−	0.911832i	$-0.365332\pi$
$653$	20.9830	0.821127	0.410563	+	0.911832i	$0.365332\pi$
$654$	144.911	5.66648
$655$	29.0391	1.13465
$656$	−145.313	−5.67352
$657$	−73.0008	−2.84803
$658$	74.5076	2.90461
$659$	0.330303	0.0128668	0.00643338	−	0.999979i	$-0.497952\pi$
$659$	0.330303	0.0128668	0.00643338	+	0.999979i	$0.497952\pi$
$660$	27.9026	1.08611
$661$	−20.7386	−0.806637	−0.403319	−	0.915060i	$-0.632143\pi$
$661$	−20.7386	−0.806637	−0.403319	+	0.915060i	$0.632143\pi$
$662$	43.6043	1.69473
$663$	−2.60765	−0.101273
$664$	120.393	4.67214
$665$	−4.65486	−0.180508
$666$	−66.9842	−2.59559
$667$	12.6115	0.488320
$668$	110.323	4.26854
$669$	−20.3096	−0.785216
$670$	−42.0424	−1.62424
$671$	13.8630	0.535174
$672$	−172.550	−6.65626
$673$	2.11347	0.0814682	0.0407341	−	0.999170i	$-0.487030\pi$
$673$	2.11347	0.0814682	0.0407341	+	0.999170i	$0.487030\pi$
$674$	−72.0080	−2.77364
$675$	−10.8056	−0.415908
$676$	−67.9882	−2.61493
$677$	23.5772	0.906144	0.453072	−	0.891474i	$-0.350328\pi$
$677$	23.5772	0.906144	0.453072	+	0.891474i	$0.350328\pi$
$678$	−62.2774	−2.39175
$679$	10.6767	0.409736
$680$	17.4549	0.669364
$681$	−14.0411	−0.538055
$682$	8.93069	0.341974
$683$	39.3453	1.50551	0.752754	−	0.658302i	$-0.228725\pi$
$683$	39.3453	1.50551	0.752754	+	0.658302i	$0.228725\pi$
$684$	29.9031	1.14337
$685$	15.8853	0.606945
$686$	53.6701	2.04913
$687$	2.48941	0.0949770
$688$	16.1149	0.614375
$689$	−11.6471	−0.443718
$690$	−94.5350	−3.59889
$691$	−39.4293	−1.49996	−0.749981	−	0.661460i	$-0.769937\pi$
$691$	−39.4293	−1.49996	−0.749981	+	0.661460i	$0.769937\pi$
$692$	4.12840	0.156938
$693$	−12.4238	−0.471941
$694$	−80.7506	−3.06525
$695$	−3.29891	−0.125135
$696$	51.4224	1.94916
$697$	−9.01731	−0.341555
$698$	−69.4287	−2.62791
$699$	−28.2138	−1.06715
$700$	−26.2950	−0.993856
$701$	−32.9071	−1.24288	−0.621441	−	0.783461i	$-0.713453\pi$
$701$	−32.9071	−1.24288	−0.621441	+	0.783461i	$0.713453\pi$
$702$	14.4664	0.545999
$703$	5.16914	0.194958
$704$	−35.5542	−1.34000
$705$	54.4019	2.04890
$706$	−45.9866	−1.73073
$707$	−25.8109	−0.970717
$708$	155.292	5.83623
$709$	18.2810	0.686557	0.343279	−	0.939234i	$-0.388463\pi$
$709$	18.2810	0.686557	0.343279	+	0.939234i	$0.388463\pi$
$710$	10.2070	0.383060
$711$	49.6322	1.86135
$712$	−18.6473	−0.698837
$713$	−22.2899	−0.834762
$714$	−19.3340	−0.723556
$715$	−1.62290	−0.0606930
$716$	29.6167	1.10683
$717$	6.67563	0.249306
$718$	45.4129	1.69479
$719$	−11.2974	−0.421322	−0.210661	−	0.977559i	$-0.567562\pi$
$719$	−11.2974	−0.421322	−0.210661	+	0.977559i	$0.567562\pi$
$720$	−142.320	−5.30394
$721$	−1.51926	−0.0565803
$722$	49.2300	1.83215
$723$	−20.5479	−0.764183
$724$	−11.7100	−0.435199
$725$	3.47685	0.129127
$726$	−7.80125	−0.289532
$727$	−36.4839	−1.35311	−0.676556	−	0.736391i	$-0.736529\pi$
$727$	−36.4839	−1.35311	−0.676556	+	0.736391i	$0.736529\pi$
$728$	22.6197	0.838340
$729$	−42.9205	−1.58965
$730$	−70.7027	−2.61682
$731$	1.00000	0.0369863
$732$	−219.564	−8.11532
$733$	22.7033	0.838567	0.419283	−	0.907855i	$-0.362281\pi$
$733$	22.7033	0.838567	0.419283	+	0.907855i	$0.362281\pi$
$734$	52.8578	1.95102
$735$	4.27856	0.157817
$736$	169.180	6.23607
$737$	8.65927	0.318968
$738$	124.578	4.58578
$739$	−24.4395	−0.899020	−0.449510	−	0.893275i	$-0.648401\pi$
$739$	−24.4395	−0.899020	−0.449510	+	0.893275i	$0.648401\pi$
$740$	−47.7920	−1.75687
$741$	−2.78010	−0.102129
$742$	−86.3551	−3.17020
$743$	−11.1211	−0.407992	−0.203996	−	0.978972i	$-0.565393\pi$
$743$	−11.1211	−0.407992	−0.203996	+	0.978972i	$0.565393\pi$
$744$	−90.8851	−3.33201
$745$	30.2352	1.10773
$746$	−9.87240	−0.361454
$747$	−60.9144	−2.22874
$748$	−5.59510	−0.204577
$749$	−9.74521	−0.356082
$750$	−94.7808	−3.46090
$751$	−20.5367	−0.749397	−0.374698	−	0.927147i	$-0.622254\pi$
$751$	−20.5367	−0.749397	−0.374698	+	0.927147i	$0.622254\pi$
$752$	−175.794	−6.41056
$753$	46.2373	1.68498
$754$	−4.65476	−0.169516
$755$	7.74403	0.281834
$756$	79.0143	2.87372
$757$	−1.18717	−0.0431485	−0.0215742	−	0.999767i	$-0.506868\pi$
$757$	−1.18717	−0.0431485	−0.0215742	+	0.999767i	$0.506868\pi$
$758$	−20.1182	−0.730724
$759$	19.4709	0.706750
$760$	18.6092	0.675025
$761$	12.3912	0.449180	0.224590	−	0.974453i	$-0.427896\pi$
$761$	12.3912	0.449180	0.224590	+	0.974453i	$0.427896\pi$
$762$	−145.892	−5.28513
$763$	−46.0357	−1.66660
$764$	89.9382	3.25385
$765$	−8.83155	−0.319306
$766$	−21.8247	−0.788560
$767$	−9.03225	−0.326136
$768$	179.358	6.47203
$769$	24.6363	0.888408	0.444204	−	0.895926i	$-0.353487\pi$
$769$	24.6363	0.888408	0.444204	+	0.895926i	$0.353487\pi$
$770$	−12.0327	−0.433628
$771$	10.0558	0.362152
$772$	109.075	3.92569
$773$	−33.1192	−1.19122	−0.595608	−	0.803275i	$-0.703089\pi$
$773$	−33.1192	−1.19122	−0.595608	+	0.803275i	$0.703089\pi$
$774$	−13.8154	−0.496585
$775$	−6.14507	−0.220737
$776$	−42.6834	−1.53224
$777$	34.0144	1.22026
$778$	−16.4350	−0.589225
$779$	−9.61363	−0.344444
$780$	25.7038	0.920343
$781$	−2.10228	−0.0752254
$782$	18.9564	0.677879
$783$	−10.4477	−0.373369
$784$	−13.8257	−0.493776
$785$	−33.9296	−1.21100
$786$	128.590	4.58666
$787$	40.5431	1.44520	0.722602	−	0.691264i	$-0.242946\pi$
$787$	40.5431	1.44520	0.722602	+	0.691264i	$0.242946\pi$
$788$	82.5255	2.93985
$789$	−80.9825	−2.88305
$790$	48.0698	1.71025
$791$	19.7844	0.703453
$792$	49.6678	1.76487
$793$	12.7705	0.453494
$794$	59.2276	2.10191
$795$	−63.0524	−2.23624
$796$	−126.579	−4.48647
$797$	−29.8682	−1.05799	−0.528993	−	0.848626i	$-0.677430\pi$
$797$	−29.8682	−1.05799	−0.528993	+	0.848626i	$0.677430\pi$
$798$	−20.6125	−0.729676
$799$	−10.9088	−0.385926
$800$	46.6411	1.64901
$801$	9.43488	0.333365
$802$	−27.1846	−0.959923
$803$	14.5623	0.513892
$804$	−137.147	−4.83680
$805$	30.0321	1.05849
$806$	8.22692	0.289781
$807$	−9.66525	−0.340233
$808$	103.186	3.63009
$809$	7.52998	0.264740	0.132370	−	0.991200i	$-0.457741\pi$
$809$	7.52998	0.264740	0.132370	+	0.991200i	$0.457741\pi$
$810$	5.29782	0.186147
$811$	0.326093	0.0114507	0.00572533	−	0.999984i	$-0.498178\pi$
$811$	0.326093	0.0114507	0.00572533	+	0.999984i	$0.498178\pi$
$812$	−25.4239	−0.892204
$813$	−57.1739	−2.00518
$814$	13.3621	0.468341
$815$	13.7866	0.482922
$816$	45.6168	1.59691
$817$	1.06613	0.0372992
$818$	−17.9939	−0.629143
$819$	−11.4448	−0.399912
$820$	88.8841	3.10397
$821$	15.3574	0.535978	0.267989	−	0.963422i	$-0.413641\pi$
$821$	15.3574	0.535978	0.267989	+	0.963422i	$0.413641\pi$
$822$	70.3427	2.45348
$823$	−26.6683	−0.929598	−0.464799	−	0.885416i	$-0.653873\pi$
$823$	−26.6683	−0.929598	−0.464799	+	0.885416i	$0.653873\pi$
$824$	6.07369	0.211587
$825$	5.36792	0.186887
$826$	−66.9680	−2.33011
$827$	3.81560	0.132681	0.0663406	−	0.997797i	$-0.478868\pi$
$827$	3.81560	0.132681	0.0663406	+	0.997797i	$0.478868\pi$
$828$	−192.928	−6.70470
$829$	−23.2155	−0.806307	−0.403154	−	0.915132i	$-0.632086\pi$
$829$	−23.2155	−0.806307	−0.403154	+	0.915132i	$0.632086\pi$
$830$	−58.9968	−2.04781
$831$	−4.65161	−0.161363
$832$	−32.7524	−1.13549
$833$	−0.857947	−0.0297261
$834$	−14.6081	−0.505838
$835$	−34.7376	−1.20214
$836$	−5.96510	−0.206307
$837$	18.4654	0.638259
$838$	−40.8560	−1.41135
$839$	−55.2486	−1.90739	−0.953697	−	0.300769i	$-0.902757\pi$
$839$	−55.2486	−1.90739	−0.953697	+	0.300769i	$0.902757\pi$
$840$	122.453	4.22504
$841$	−25.6383	−0.884080
$842$	31.8842	1.09880
$843$	12.9906	0.447420
$844$	−27.5090	−0.946898
$845$	21.4075	0.736440
$846$	150.710	5.18151
$847$	2.47832	0.0851560
$848$	203.748	6.99672
$849$	40.9018	1.40375
$850$	5.22607	0.179253
$851$	−33.3501	−1.14323
$852$	33.2962	1.14071
$853$	−31.5328	−1.07966	−0.539831	−	0.841774i	$-0.681512\pi$
$853$	−31.5328	−1.07966	−0.539831	+	0.841774i	$0.681512\pi$
$854$	94.6846	3.24004
$855$	−9.41559	−0.322006
$856$	38.9593	1.33160
$857$	32.5709	1.11260	0.556300	−	0.830981i	$-0.312220\pi$
$857$	32.5709	1.11260	0.556300	+	0.830981i	$0.312220\pi$
$858$	−7.18648	−0.245343
$859$	−20.9748	−0.715652	−0.357826	−	0.933788i	$-0.616482\pi$
$859$	−20.9748	−0.715652	−0.357826	+	0.933788i	$0.616482\pi$
$860$	−9.85705	−0.336123
$861$	−63.2603	−2.15591
$862$	75.4843	2.57100
$863$	−18.0096	−0.613055	−0.306527	−	0.951862i	$-0.599167\pi$
$863$	−18.0096	−0.613055	−0.306527	+	0.951862i	$0.599167\pi$
$864$	−140.153	−4.76810
$865$	−1.29991	−0.0441983
$866$	−100.547	−3.41674
$867$	2.83072	0.0961364
$868$	44.9348	1.52519
$869$	−9.90070	−0.335858
$870$	−25.1989	−0.854323
$871$	7.97689	0.270287
$872$	184.041	6.23242
$873$	21.5963	0.730925
$874$	20.2100	0.683613
$875$	30.1101	1.01791
$876$	−230.640	−7.79261
$877$	−29.9665	−1.01190	−0.505949	−	0.862563i	$-0.668858\pi$
$877$	−29.9665	−1.01190	−0.505949	+	0.862563i	$0.668858\pi$
$878$	49.6989	1.67726
$879$	−83.3835	−2.81245
$880$	28.3901	0.957030
$881$	−24.2676	−0.817597	−0.408799	−	0.912625i	$-0.634052\pi$
$881$	−24.2676	−0.817597	−0.408799	+	0.912625i	$0.634052\pi$
$882$	11.8529	0.399108
$883$	−18.3773	−0.618447	−0.309223	−	0.950989i	$-0.600069\pi$
$883$	−18.3773	−0.618447	−0.309223	+	0.950989i	$0.600069\pi$
$884$	−5.15418	−0.173354
$885$	−48.8968	−1.64365
$886$	21.2551	0.714080
$887$	47.2937	1.58797	0.793984	−	0.607939i	$-0.208004\pi$
$887$	47.2937	1.58797	0.793984	+	0.607939i	$0.208004\pi$
$888$	−135.982	−4.56327
$889$	46.3474	1.55444
$890$	9.13786	0.306302
$891$	−1.09117	−0.0365555
$892$	−40.1432	−1.34409
$893$	−11.6302	−0.389190
$894$	133.887	4.47784
$895$	−9.32544	−0.311715
$896$	−120.925	−4.03981
$897$	17.9366	0.598884
$898$	−32.2165	−1.07508
$899$	−5.94150	−0.198160
$900$	−53.1880	−1.77293
$901$	12.6434	0.421214
$902$	−24.8510	−0.827447
$903$	7.01543	0.233459
$904$	−79.0940	−2.63063
$905$	3.68714	0.122565
$906$	34.2919	1.13927
$907$	−43.0248	−1.42862	−0.714308	−	0.699831i	$-0.753259\pi$
$907$	−43.0248	−1.42862	−0.714308	+	0.699831i	$0.753259\pi$
$908$	−27.7530	−0.921017
$909$	−52.2088	−1.73166
$910$	−11.0845	−0.367447
$911$	−45.2369	−1.49877	−0.749383	−	0.662137i	$-0.769650\pi$
$911$	−45.2369	−1.49877	−0.749383	+	0.662137i	$0.769650\pi$
$912$	48.6335	1.61042
$913$	12.1513	0.402149
$914$	−12.5266	−0.414342
$915$	69.1342	2.28551
$916$	4.92047	0.162577
$917$	−40.8507	−1.34901
$918$	−15.7039	−0.518306
$919$	−39.9984	−1.31943	−0.659713	−	0.751517i	$-0.729322\pi$
$919$	−39.9984	−1.31943	−0.659713	+	0.751517i	$0.729322\pi$
$920$	−120.062	−3.95833
$921$	−54.7617	−1.80446
$922$	12.5089	0.411960
$923$	−1.93661	−0.0637443
$924$	−39.2520	−1.29130
$925$	−9.19425	−0.302305
$926$	14.0007	0.460091
$927$	−3.07308	−0.100933
$928$	45.0961	1.48035
$929$	−9.65282	−0.316699	−0.158349	−	0.987383i	$-0.550617\pi$
$929$	−9.65282	−0.316699	−0.158349	+	0.987383i	$0.550617\pi$
$930$	44.5371	1.46043
$931$	−0.914683	−0.0299775
$932$	−55.7663	−1.82669
$933$	30.9232	1.01238
$934$	−58.2357	−1.90553
$935$	1.76173	0.0576147
$936$	45.7538	1.49551
$937$	42.0166	1.37262	0.686312	−	0.727308i	$-0.259229\pi$
$937$	42.0166	1.37262	0.686312	+	0.727308i	$0.259229\pi$
$938$	59.1432	1.93109
$939$	−98.9790	−3.23006
$940$	107.529	3.50720
$941$	−52.7524	−1.71968	−0.859840	−	0.510564i	$-0.829436\pi$
$941$	−52.7524	−1.71968	−0.859840	+	0.510564i	$0.829436\pi$
$942$	−150.246	−4.89529
$943$	62.0249	2.01981
$944$	158.005	5.14263
$945$	−24.8793	−0.809323
$946$	2.75592	0.0896027
$947$	−21.2503	−0.690542	−0.345271	−	0.938503i	$-0.612213\pi$
$947$	−21.2503	−0.690542	−0.345271	+	0.938503i	$0.612213\pi$
$948$	156.809	5.09292
$949$	13.4147	0.435461
$950$	5.57167	0.180769
$951$	−78.1123	−2.53296
$952$	−24.5547	−0.795821
$953$	11.8297	0.383202	0.191601	−	0.981473i	$-0.438632\pi$
$953$	11.8297	0.383202	0.191601	+	0.981473i	$0.438632\pi$
$954$	−174.674	−5.65529
$955$	−28.3189	−0.916378
$956$	13.1948	0.426750
$957$	5.19010	0.167772
$958$	−105.883	−3.42093
$959$	−22.3466	−0.721610
$960$	−177.308	−5.72259
$961$	−20.4989	−0.661254
$962$	12.3091	0.396862
$963$	−19.7121	−0.635213
$964$	−40.6141	−1.30809
$965$	−34.3445	−1.10559
$966$	132.987	4.27879
$967$	−11.4167	−0.367136	−0.183568	−	0.983007i	$-0.558765\pi$
$967$	−11.4167	−0.367136	−0.183568	+	0.983007i	$0.558765\pi$
$968$	−9.90779	−0.318449
$969$	3.01792	0.0969496
$970$	20.9165	0.671587
$971$	51.9125	1.66595	0.832975	−	0.553310i	$-0.186636\pi$
$971$	51.9125	1.66595	0.832975	+	0.553310i	$0.186636\pi$
$972$	−78.3646	−2.51355
$973$	4.64074	0.148775
$974$	−40.1297	−1.28584
$975$	4.94491	0.158364
$976$	−223.400	−7.15087
$977$	−43.5467	−1.39318	−0.696592	−	0.717468i	$-0.745301\pi$
$977$	−43.5467	−1.39318	−0.696592	+	0.717468i	$0.745301\pi$
$978$	61.0493	1.95214
$979$	−1.88208	−0.0601516
$980$	8.45682	0.270143
$981$	−93.1185	−2.97304
$982$	−29.0325	−0.926465
$983$	50.1447	1.59937	0.799684	−	0.600422i	$-0.205001\pi$
$983$	50.1447	1.59937	0.799684	+	0.600422i	$0.205001\pi$
$984$	252.902	8.06221
$985$	−25.9849	−0.827947
$986$	5.05295	0.160919
$987$	−76.5300	−2.43598
$988$	−5.49503	−0.174820
$989$	−6.87843	−0.218721
$990$	−24.3391	−0.773546
$991$	−24.4273	−0.775960	−0.387980	−	0.921668i	$-0.626827\pi$
$991$	−24.4273	−0.775960	−0.387980	+	0.921668i	$0.626827\pi$
$992$	−79.7038	−2.53060
$993$	−44.7879	−1.42130
$994$	−14.3586	−0.455429
$995$	39.8560	1.26352
$996$	−192.454	−6.09815
$997$	−55.0435	−1.74325	−0.871623	−	0.490176i	$-0.836932\pi$
$997$	−55.0435	−1.74325	−0.871623	+	0.490176i	$0.836932\pi$
$998$	7.80934	0.247200
$999$	27.6280	0.874111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.g.1.3	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.g.1.3	✓	69	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75592 q^{2} +2.83072 q^{3} +5.59510 q^{4} -1.76173 q^{5} -7.80125 q^{6} +2.47832 q^{7} -9.90779 q^{8} +5.01300 q^{9} +O(q^{10})\)	\(q-2.75592 q^{2} +2.83072 q^{3} +5.59510 q^{4} -1.76173 q^{5} -7.80125 q^{6} +2.47832 q^{7} -9.90779 q^{8} +5.01300 q^{9} +4.85519 q^{10} -1.00000 q^{11} +15.8382 q^{12} -0.921197 q^{13} -6.83004 q^{14} -4.98697 q^{15} +16.1149 q^{16} +1.00000 q^{17} -13.8154 q^{18} +1.06613 q^{19} -9.85705 q^{20} +7.01543 q^{21} +2.75592 q^{22} -6.87843 q^{23} -28.0462 q^{24} -1.89631 q^{25} +2.53874 q^{26} +5.69825 q^{27} +13.8664 q^{28} -1.83349 q^{29} +13.7437 q^{30} +3.24055 q^{31} -24.5958 q^{32} -2.83072 q^{33} -2.75592 q^{34} -4.36613 q^{35} +28.0482 q^{36} +4.84851 q^{37} -2.93817 q^{38} -2.60765 q^{39} +17.4549 q^{40} -9.01731 q^{41} -19.3340 q^{42} +1.00000 q^{43} -5.59510 q^{44} -8.83155 q^{45} +18.9564 q^{46} -10.9088 q^{47} +45.6168 q^{48} -0.857947 q^{49} +5.22607 q^{50} +2.83072 q^{51} -5.15418 q^{52} +12.6434 q^{53} -15.7039 q^{54} +1.76173 q^{55} -24.5547 q^{56} +3.01792 q^{57} +5.05295 q^{58} +9.80491 q^{59} -27.9026 q^{60} -13.8630 q^{61} -8.93069 q^{62} +12.4238 q^{63} +35.5542 q^{64} +1.62290 q^{65} +7.80125 q^{66} -8.65927 q^{67} +5.59510 q^{68} -19.4709 q^{69} +12.0327 q^{70} +2.10228 q^{71} -49.6678 q^{72} -14.5623 q^{73} -13.3621 q^{74} -5.36792 q^{75} +5.96510 q^{76} -2.47832 q^{77} +7.18648 q^{78} +9.90070 q^{79} -28.3901 q^{80} +1.09117 q^{81} +24.8510 q^{82} -12.1513 q^{83} +39.2520 q^{84} -1.76173 q^{85} -2.75592 q^{86} -5.19010 q^{87} +9.90779 q^{88} +1.88208 q^{89} +24.3391 q^{90} -2.28302 q^{91} -38.4854 q^{92} +9.17309 q^{93} +30.0638 q^{94} -1.87823 q^{95} -69.6239 q^{96} +4.30806 q^{97} +2.36443 q^{98} -5.01300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9	\( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) 69 * q - 11 * q^2 - 3 * q^3 + 65 * q^4 - 6 * q^5 - 10 * q^6 - 11 * q^7 - 33 * q^8 + 56 * q^9 - q^10 - 69 * q^11 - 3 * q^12 - 28 * q^13 - 15 * q^14 - 45 * q^15 + 53 * q^16 + 69 * q^17 - 17 * q^18 - 32 * q^19 - 21 * q^20 - 38 * q^21 + 11 * q^22 - 41 * q^23 - 11 * q^24 + 67 * q^25 - 6 * q^26 - 3 * q^27 - 21 * q^28 - 22 * q^29 - 22 * q^30 - 27 * q^31 - 87 * q^32 + 3 * q^33 - 11 * q^34 - 44 * q^35 + 59 * q^36 + 24 * q^37 - 22 * q^38 - 59 * q^39 + q^40 - 43 * q^41 - 15 * q^42 + 69 * q^43 - 65 * q^44 - 12 * q^45 - 21 * q^46 - 99 * q^47 + 2 * q^48 + 64 * q^49 - 78 * q^50 - 3 * q^51 - 57 * q^52 - 50 * q^53 + 20 * q^54 + 6 * q^55 - 59 * q^56 - 15 * q^57 + 22 * q^58 - 82 * q^59 - 86 * q^60 - 24 * q^61 + 15 * q^62 - 63 * q^63 + 63 * q^64 - 23 * q^65 + 10 * q^66 - 54 * q^67 + 65 * q^68 + 36 * q^69 + 9 * q^70 - 128 * q^71 - 69 * q^72 + 2 * q^73 - 58 * q^74 - 31 * q^75 - 76 * q^76 + 11 * q^77 - 19 * q^78 - 43 * q^79 - 19 * q^80 + 49 * q^81 - 2 * q^82 - 62 * q^83 - 82 * q^84 - 6 * q^85 - 11 * q^86 - 62 * q^87 + 33 * q^88 - 49 * q^89 - 37 * q^90 - 2 * q^91 - 96 * q^92 - 29 * q^93 - 75 * q^94 - 133 * q^95 - 86 * q^96 + 5 * q^97 - 72 * q^98 - 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more