LMFDB - Embedded newform 847.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.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:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.40927$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40927 q^{2} -2.16338 q^{3} -0.0139645 q^{4} -1.83139 q^{5} +3.04878 q^{6} +1.00000 q^{7} +2.83822 q^{8} +1.68022 q^{9} +O(q^{10})$	\(q-1.40927 q^{2} -2.16338 q^{3} -0.0139645 q^{4} -1.83139 q^{5} +3.04878 q^{6} +1.00000 q^{7} +2.83822 q^{8} +1.68022 q^{9} +2.58091 q^{10} +0.0302106 q^{12} -4.64706 q^{13} -1.40927 q^{14} +3.96199 q^{15} -3.97188 q^{16} -5.47021 q^{17} -2.36788 q^{18} -5.80118 q^{19} +0.0255744 q^{20} -2.16338 q^{21} -0.719682 q^{23} -6.14014 q^{24} -1.64602 q^{25} +6.54895 q^{26} +2.85519 q^{27} -0.0139645 q^{28} -1.17247 q^{29} -5.58350 q^{30} -1.30787 q^{31} -0.0789938 q^{32} +7.70900 q^{34} -1.83139 q^{35} -0.0234634 q^{36} +2.09474 q^{37} +8.17541 q^{38} +10.0534 q^{39} -5.19787 q^{40} -0.916645 q^{41} +3.04878 q^{42} +8.02379 q^{43} -3.07713 q^{45} +1.01423 q^{46} +5.97584 q^{47} +8.59268 q^{48} +1.00000 q^{49} +2.31969 q^{50} +11.8342 q^{51} +0.0648939 q^{52} +10.1449 q^{53} -4.02373 q^{54} +2.83822 q^{56} +12.5502 q^{57} +1.65233 q^{58} -7.68081 q^{59} -0.0553272 q^{60} -6.27612 q^{61} +1.84313 q^{62} +1.68022 q^{63} +8.05508 q^{64} +8.51056 q^{65} -15.4673 q^{67} +0.0763889 q^{68} +1.55695 q^{69} +2.58091 q^{70} +13.9019 q^{71} +4.76882 q^{72} +6.01462 q^{73} -2.95205 q^{74} +3.56098 q^{75} +0.0810106 q^{76} -14.1679 q^{78} -15.6409 q^{79} +7.27404 q^{80} -11.2175 q^{81} +1.29180 q^{82} -4.37573 q^{83} +0.0302106 q^{84} +10.0181 q^{85} -11.3077 q^{86} +2.53651 q^{87} +15.3437 q^{89} +4.33650 q^{90} -4.64706 q^{91} +0.0100500 q^{92} +2.82941 q^{93} -8.42155 q^{94} +10.6242 q^{95} +0.170894 q^{96} +2.41124 q^{97} -1.40927 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9}+O(q^{10}) $ 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9	$ 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + 6 q^{10} + 9 q^{12} - 6 q^{13} + q^{14} + 11 q^{15} + q^{16} - 5 q^{17} + 8 q^{18} - 13 q^{19} + 23 q^{20} + 4 q^{21} + 16 q^{23} - 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} + 7 q^{28} + 9 q^{29} - 36 q^{30} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} + 13 q^{39} + 5 q^{40} - 10 q^{41} - q^{42} - 4 q^{43} + 35 q^{45} + 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} + 6 q^{50} + 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} + 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} + 14 q^{63} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} - 35 q^{72} - 25 q^{73} + 33 q^{74} - 13 q^{75} + 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} - 25 q^{83} + 9 q^{84} + 3 q^{85} + 4 q^{86} - 36 q^{87} + 37 q^{89} - 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} - 42 q^{94} + 21 q^{95} - 6 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) $ 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9 + 6 * q^10 + 9 * q^12 - 6 * q^13 + q^14 + 11 * q^15 + q^16 - 5 * q^17 + 8 * q^18 - 13 * q^19 + 23 * q^20 + 4 * q^21 + 16 * q^23 - 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 + 7 * q^28 + 9 * q^29 - 36 * q^30 + 9 * q^31 + 16 * q^32 - 12 * q^34 + 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 + 13 * q^39 + 5 * q^40 - 10 * q^41 - q^42 - 4 * q^43 + 35 * q^45 + 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 + 6 * q^50 + 13 * q^51 - 41 * q^52 + 37 * q^53 + 30 * q^54 + 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 + 19 * q^61 - 18 * q^62 + 14 * q^63 - 4 * q^64 - 4 * q^65 - 19 * q^67 + 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 - 35 * q^72 - 25 * q^73 + 33 * q^74 - 13 * q^75 + 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 - 25 * q^83 + 9 * q^84 + 3 * q^85 + 4 * q^86 - 36 * q^87 + 37 * q^89 - 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 - 42 * q^94 + 21 * q^95 - 6 * q^96 + 15 * q^97 + q^98

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 $

$2$	−1.40927	−0.996503	−0.498251	−	0.867033i	$-0.666024\pi$
$2$	−1.40927	−0.996503	−0.498251	+	0.867033i	$0.666024\pi$
$3$	−2.16338	−1.24903	−0.624514	−	0.781013i	$-0.714703\pi$
$3$	−2.16338	−1.24903	−0.624514	+	0.781013i	$0.714703\pi$
$4$	−0.0139645	−0.00698226
$5$	−1.83139	−0.819021	−0.409510	−	0.912305i	$-0.634300\pi$
$5$	−1.83139	−0.819021	−0.409510	+	0.912305i	$0.634300\pi$
$6$	3.04878	1.24466
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.83822	1.00346
$9$	1.68022	0.560073
$10$	2.58091	0.816157
$11$	0	0
$11$	0	0
$12$	0.0302106	0.00872104
$13$	−4.64706	−1.28886	−0.644431	−	0.764663i	$-0.722906\pi$
$13$	−4.64706	−1.28886	−0.644431	+	0.764663i	$0.722906\pi$
$14$	−1.40927	−0.376643
$15$	3.96199	1.02298
$16$	−3.97188	−0.992969
$17$	−5.47021	−1.32672	−0.663361	−	0.748300i	$-0.730871\pi$
$17$	−5.47021	−1.32672	−0.663361	+	0.748300i	$0.730871\pi$
$18$	−2.36788	−0.558114
$19$	−5.80118	−1.33088	−0.665441	−	0.746451i	$-0.731756\pi$
$19$	−5.80118	−1.33088	−0.665441	+	0.746451i	$0.731756\pi$
$20$	0.0255744	0.00571861
$21$	−2.16338	−0.472088
$22$	0	0
$23$	−0.719682	−0.150064	−0.0750321	−	0.997181i	$-0.523906\pi$
$23$	−0.719682	−0.150064	−0.0750321	+	0.997181i	$0.523906\pi$
$24$	−6.14014	−1.25335
$25$	−1.64602	−0.329205
$26$	6.54895	1.28435
$27$	2.85519	0.549482
$28$	−0.0139645	−0.00263905
$29$	−1.17247	−0.217723	−0.108861	−	0.994057i	$-0.534720\pi$
$29$	−1.17247	−0.217723	−0.108861	+	0.994057i	$0.534720\pi$
$30$	−5.58350	−1.01940
$31$	−1.30787	−0.234900	−0.117450	−	0.993079i	$-0.537472\pi$
$31$	−1.30787	−0.234900	−0.117450	+	0.993079i	$0.537472\pi$
$32$	−0.0789938	−0.0139643
$33$	0	0
$34$	7.70900	1.32208
$35$	−1.83139	−0.309561
$36$	−0.0234634	−0.00391057
$37$	2.09474	0.344373	0.172186	−	0.985064i	$-0.444917\pi$
$37$	2.09474	0.344373	0.172186	+	0.985064i	$0.444917\pi$
$38$	8.17541	1.32623
$39$	10.0534	1.60983
$40$	−5.19787	−0.821855
$41$	−0.916645	−0.143156	−0.0715780	−	0.997435i	$-0.522803\pi$
$41$	−0.916645	−0.143156	−0.0715780	+	0.997435i	$0.522803\pi$
$42$	3.04878	0.470437
$43$	8.02379	1.22362	0.611808	−	0.791006i	$-0.290442\pi$
$43$	8.02379	1.22362	0.611808	+	0.791006i	$0.290442\pi$
$44$	0	0
$45$	−3.07713	−0.458711
$46$	1.01423	0.149539
$47$	5.97584	0.871665	0.435833	−	0.900028i	$-0.356454\pi$
$47$	5.97584	0.871665	0.435833	+	0.900028i	$0.356454\pi$
$48$	8.59268	1.24025
$49$	1.00000	0.142857
$50$	2.31969	0.328053
$51$	11.8342	1.65711
$52$	0.0648939	0.00899916
$53$	10.1449	1.39351	0.696757	−	0.717307i	$-0.254626\pi$
$53$	10.1449	1.39351	0.696757	+	0.717307i	$0.254626\pi$
$54$	−4.02373	−0.547560
$55$	0	0
$56$	2.83822	0.379272
$57$	12.5502	1.66231
$58$	1.65233	0.216962
$59$	−7.68081	−0.999956	−0.499978	−	0.866038i	$-0.666659\pi$
$59$	−7.68081	−0.999956	−0.499978	+	0.866038i	$0.666659\pi$
$60$	−0.0553272	−0.00714271
$61$	−6.27612	−0.803575	−0.401788	−	0.915733i	$-0.631611\pi$
$61$	−6.27612	−0.803575	−0.401788	+	0.915733i	$0.631611\pi$
$62$	1.84313	0.234078
$63$	1.68022	0.211688
$64$	8.05508	1.00688
$65$	8.51056	1.05560
$66$	0	0
$67$	−15.4673	−1.88963	−0.944814	−	0.327608i	$-0.893758\pi$
$67$	−15.4673	−1.88963	−0.944814	+	0.327608i	$0.893758\pi$
$68$	0.0763889	0.00926351
$69$	1.55695	0.187434
$70$	2.58091	0.308478
$71$	13.9019	1.64985	0.824927	−	0.565240i	$-0.191216\pi$
$71$	13.9019	1.64985	0.824927	+	0.565240i	$0.191216\pi$
$72$	4.76882	0.562011
$73$	6.01462	0.703959	0.351979	−	0.936008i	$-0.385509\pi$
$73$	6.01462	0.703959	0.351979	+	0.936008i	$0.385509\pi$
$74$	−2.95205	−0.343169
$75$	3.56098	0.411186
$76$	0.0810106	0.00929255
$77$	0	0
$78$	−14.1679	−1.60420
$79$	−15.6409	−1.75974	−0.879872	−	0.475211i	$-0.842372\pi$
$79$	−15.6409	−1.75974	−0.879872	+	0.475211i	$0.842372\pi$
$80$	7.27404	0.813262
$81$	−11.2175	−1.24639
$82$	1.29180	0.142655
$83$	−4.37573	−0.480299	−0.240149	−	0.970736i	$-0.577196\pi$
$83$	−4.37573	−0.480299	−0.240149	+	0.970736i	$0.577196\pi$
$84$	0.0302106	0.00329624
$85$	10.0181	1.08661
$86$	−11.3077	−1.21934
$87$	2.53651	0.271942
$88$	0	0
$89$	15.3437	1.62643	0.813215	−	0.581963i	$-0.197715\pi$
$89$	15.3437	1.62643	0.813215	+	0.581963i	$0.197715\pi$
$90$	4.33650	0.457107
$91$	−4.64706	−0.487144
$92$	0.0100500	0.00104779
$93$	2.82941	0.293396
$94$	−8.42155	−0.868617
$95$	10.6242	1.09002
$96$	0.170894	0.0174418
$97$	2.41124	0.244824	0.122412	−	0.992479i	$-0.460937\pi$
$97$	2.41124	0.244824	0.122412	+	0.992479i	$0.460937\pi$
$98$	−1.40927	−0.142358
$99$	0	0
$100$	0.0229859	0.00229859
$101$	11.8959	1.18368	0.591842	−	0.806054i	$-0.298401\pi$
$101$	11.8959	1.18368	0.591842	+	0.806054i	$0.298401\pi$
$102$	−16.6775	−1.65132
$103$	0.396314	0.0390500	0.0195250	−	0.999809i	$-0.493785\pi$
$103$	0.396314	0.0390500	0.0195250	+	0.999809i	$0.493785\pi$
$104$	−13.1893	−1.29332
$105$	3.96199	0.386650
$106$	−14.2969	−1.38864
$107$	−3.26935	−0.316060	−0.158030	−	0.987434i	$-0.550514\pi$
$107$	−3.26935	−0.316060	−0.158030	+	0.987434i	$0.550514\pi$
$108$	−0.0398714	−0.00383662
$109$	−2.84638	−0.272634	−0.136317	−	0.990665i	$-0.543527\pi$
$109$	−2.84638	−0.272634	−0.136317	+	0.990665i	$0.543527\pi$
$110$	0	0
$111$	−4.53172	−0.430132
$112$	−3.97188	−0.375307
$113$	14.5445	1.36823	0.684117	−	0.729372i	$-0.260188\pi$
$113$	14.5445	1.36823	0.684117	+	0.729372i	$0.260188\pi$
$114$	−17.6865	−1.65650
$115$	1.31802	0.122906
$116$	0.0163730	0.00152020
$117$	−7.80807	−0.721856
$118$	10.8243	0.996459
$119$	−5.47021	−0.501454
$120$	11.2450	1.02652
$121$	0	0
$122$	8.84474	0.800765
$123$	1.98305	0.178806
$124$	0.0182637	0.00164013
$125$	12.1714	1.08865
$126$	−2.36788	−0.210947
$127$	5.03287	0.446595	0.223298	−	0.974750i	$-0.428318\pi$
$127$	5.03287	0.446595	0.223298	+	0.974750i	$0.428318\pi$
$128$	−11.1938	−0.989399
$129$	−17.3585	−1.52833
$130$	−11.9937	−1.05191
$131$	−0.180053	−0.0157313	−0.00786565	−	0.999969i	$-0.502504\pi$
$131$	−0.180053	−0.0157313	−0.00786565	+	0.999969i	$0.502504\pi$
$132$	0	0
$133$	−5.80118	−0.503026
$134$	21.7975	1.88302
$135$	−5.22896	−0.450037
$136$	−15.5256	−1.33131
$137$	8.32395	0.711163	0.355582	−	0.934645i	$-0.384283\pi$
$137$	8.32395	0.711163	0.355582	+	0.934645i	$0.384283\pi$
$138$	−2.19416	−0.186779
$139$	−6.96119	−0.590441	−0.295220	−	0.955429i	$-0.595393\pi$
$139$	−6.96119	−0.590441	−0.295220	+	0.955429i	$0.595393\pi$
$140$	0.0255744	0.00216143
$141$	−12.9280	−1.08873
$142$	−19.5915	−1.64408
$143$	0	0
$144$	−6.67362	−0.556135
$145$	2.14725	0.178320
$146$	−8.47622	−0.701497
$147$	−2.16338	−0.178433
$148$	−0.0292520	−0.00240450
$149$	−3.21431	−0.263327	−0.131663	−	0.991294i	$-0.542032\pi$
$149$	−3.21431	−0.263327	−0.131663	+	0.991294i	$0.542032\pi$
$150$	−5.01837	−0.409748
$151$	22.2670	1.81206	0.906032	−	0.423210i	$-0.139097\pi$
$151$	22.2670	1.81206	0.906032	+	0.423210i	$0.139097\pi$
$152$	−16.4650	−1.33549
$153$	−9.19115	−0.743061
$154$	0	0
$155$	2.39521	0.192388
$156$	−0.140390	−0.0112402
$157$	13.2548	1.05785	0.528923	−	0.848670i	$-0.322596\pi$
$157$	13.2548	1.05785	0.528923	+	0.848670i	$0.322596\pi$
$158$	22.0423	1.75359
$159$	−21.9474	−1.74054
$160$	0.144668	0.0114370
$161$	−0.719682	−0.0567189
$162$	15.8085	1.24203
$163$	13.7183	1.07450	0.537252	−	0.843422i	$-0.319462\pi$
$163$	13.7183	1.07450	0.537252	+	0.843422i	$0.319462\pi$
$164$	0.0128005	0.000999552 0
$165$	0	0
$166$	6.16657	0.478619
$167$	9.30860	0.720321	0.360160	−	0.932890i	$-0.382722\pi$
$167$	9.30860	0.720321	0.360160	+	0.932890i	$0.382722\pi$
$168$	−6.14014	−0.473722
$169$	8.59513	0.661164
$170$	−14.1181	−1.08281
$171$	−9.74724	−0.745390
$172$	−0.112048	−0.00854361
$173$	10.5057	0.798732	0.399366	−	0.916792i	$-0.369230\pi$
$173$	10.5057	0.798732	0.399366	+	0.916792i	$0.369230\pi$
$174$	−3.57462	−0.270991
$175$	−1.64602	−0.124428
$176$	0	0
$177$	16.6165	1.24897
$178$	−21.6234	−1.62074
$179$	−8.32331	−0.622113	−0.311057	−	0.950391i	$-0.600683\pi$
$179$	−8.32331	−0.622113	−0.311057	+	0.950391i	$0.600683\pi$
$180$	0.0429706	0.00320284
$181$	−14.8030	−1.10030	−0.550148	−	0.835067i	$-0.685429\pi$
$181$	−14.8030	−1.10030	−0.550148	+	0.835067i	$0.685429\pi$
$182$	6.54895	0.485440
$183$	13.5776	1.00369
$184$	−2.04261	−0.150583
$185$	−3.83628	−0.282049
$186$	−3.98740	−0.292370
$187$	0	0
$188$	−0.0834496	−0.00608619
$189$	2.85519	0.207685
$190$	−14.9723	−1.08621
$191$	−9.60676	−0.695121	−0.347560	−	0.937658i	$-0.612990\pi$
$191$	−9.60676	−0.695121	−0.347560	+	0.937658i	$0.612990\pi$
$192$	−17.4262	−1.25763
$193$	−1.48781	−0.107095	−0.0535474	−	0.998565i	$-0.517053\pi$
$193$	−1.48781	−0.107095	−0.0535474	+	0.998565i	$0.517053\pi$
$194$	−3.39808	−0.243968
$195$	−18.4116	−1.31848
$196$	−0.0139645	−0.000997465 0
$197$	−14.0434	−1.00055	−0.500274	−	0.865867i	$-0.666767\pi$
$197$	−14.0434	−1.00055	−0.500274	+	0.865867i	$0.666767\pi$
$198$	0	0
$199$	−4.28729	−0.303918	−0.151959	−	0.988387i	$-0.548558\pi$
$199$	−4.28729	−0.303918	−0.151959	+	0.988387i	$0.548558\pi$
$200$	−4.67177	−0.330344
$201$	33.4616	2.36020
$202$	−16.7645	−1.17954
$203$	−1.17247	−0.0822915
$204$	−0.165258	−0.0115704
$205$	1.67873	0.117248
$206$	−0.558512	−0.0389134
$207$	−1.20922	−0.0840468
$208$	18.4575	1.27980
$209$	0	0
$210$	−5.58350	−0.385298
$211$	−1.45527	−0.100185	−0.0500925	−	0.998745i	$-0.515952\pi$
$211$	−1.45527	−0.100185	−0.0500925	+	0.998745i	$0.515952\pi$
$212$	−0.141669	−0.00972987
$213$	−30.0751	−2.06071
$214$	4.60739	0.314955
$215$	−14.6947	−1.00217
$216$	8.10365	0.551383
$217$	−1.30787	−0.0887837
$218$	4.01132	0.271681
$219$	−13.0119	−0.879264
$220$	0	0
$221$	25.4204	1.70996
$222$	6.38640	0.428627
$223$	−4.85642	−0.325210	−0.162605	−	0.986691i	$-0.551990\pi$
$223$	−4.85642	−0.325210	−0.162605	+	0.986691i	$0.551990\pi$
$224$	−0.0789938	−0.00527799
$225$	−2.76568	−0.184379
$226$	−20.4971	−1.36345
$227$	0.397144	0.0263594	0.0131797	−	0.999913i	$-0.495805\pi$
$227$	0.397144	0.0263594	0.0131797	+	0.999913i	$0.495805\pi$
$228$	−0.175257	−0.0116067
$229$	2.18963	0.144695	0.0723475	−	0.997379i	$-0.476951\pi$
$229$	2.18963	0.144695	0.0723475	+	0.997379i	$0.476951\pi$
$230$	−1.85744	−0.122476
$231$	0	0
$232$	−3.32773	−0.218476
$233$	−1.26016	−0.0825555	−0.0412778	−	0.999148i	$-0.513143\pi$
$233$	−1.26016	−0.0825555	−0.0412778	+	0.999148i	$0.513143\pi$
$234$	11.0037	0.719332
$235$	−10.9441	−0.713912
$236$	0.107259	0.00698195
$237$	33.8373	2.19797
$238$	7.70900	0.499700
$239$	−11.1617	−0.721987	−0.360994	−	0.932568i	$-0.617562\pi$
$239$	−11.1617	−0.721987	−0.360994	+	0.932568i	$0.617562\pi$
$240$	−15.7365	−1.01579
$241$	−21.4843	−1.38392	−0.691962	−	0.721934i	$-0.743254\pi$
$241$	−21.4843	−1.38392	−0.691962	+	0.721934i	$0.743254\pi$
$242$	0	0
$243$	15.7022	1.00730
$244$	0.0876430	0.00561077
$245$	−1.83139	−0.117003
$246$	−2.79465	−0.178181
$247$	26.9584	1.71532
$248$	−3.71200	−0.235712
$249$	9.46637	0.599907
$250$	−17.1528	−1.08484
$251$	−0.423820	−0.0267513	−0.0133756	−	0.999911i	$-0.504258\pi$
$251$	−0.423820	−0.0267513	−0.0133756	+	0.999911i	$0.504258\pi$
$252$	−0.0234634	−0.00147806
$253$	0	0
$254$	−7.09267	−0.445033
$255$	−21.6729	−1.35721
$256$	−0.335132	−0.0209457
$257$	17.4401	1.08788	0.543941	−	0.839124i	$-0.316932\pi$
$257$	17.4401	1.08788	0.543941	+	0.839124i	$0.316932\pi$
$258$	24.4628	1.52299
$259$	2.09474	0.130161
$260$	−0.118846	−0.00737050
$261$	−1.97001	−0.121941
$262$	0.253743	0.0156763
$263$	1.51519	0.0934307	0.0467153	−	0.998908i	$-0.485125\pi$
$263$	1.51519	0.0934307	0.0467153	+	0.998908i	$0.485125\pi$
$264$	0	0
$265$	−18.5793	−1.14132
$266$	8.17541	0.501267
$267$	−33.1943	−2.03146
$268$	0.215993	0.0131939
$269$	−2.03103	−0.123834	−0.0619170	−	0.998081i	$-0.519721\pi$
$269$	−2.03103	−0.123834	−0.0619170	+	0.998081i	$0.519721\pi$
$270$	7.36900	0.448463
$271$	7.60444	0.461937	0.230968	−	0.972961i	$-0.425811\pi$
$271$	7.60444	0.461937	0.230968	+	0.972961i	$0.425811\pi$
$272$	21.7270	1.31739
$273$	10.0534	0.608457
$274$	−11.7307	−0.708676
$275$	0	0
$276$	−0.0217420	−0.00130872
$277$	14.4268	0.866823	0.433411	−	0.901196i	$-0.357310\pi$
$277$	14.4268	0.866823	0.433411	+	0.901196i	$0.357310\pi$
$278$	9.81019	0.588376
$279$	−2.19750	−0.131561
$280$	−5.19787	−0.310632
$281$	17.7496	1.05886	0.529428	−	0.848355i	$-0.322407\pi$
$281$	17.7496	1.05886	0.529428	+	0.848355i	$0.322407\pi$
$282$	18.2190	1.08493
$283$	−31.1361	−1.85085	−0.925426	−	0.378929i	$-0.876292\pi$
$283$	−31.1361	−1.85085	−0.925426	+	0.378929i	$0.876292\pi$
$284$	−0.194133	−0.0115197
$285$	−22.9842	−1.36147
$286$	0	0
$287$	−0.916645	−0.0541079
$288$	−0.132727	−0.00782100
$289$	12.9232	0.760190
$290$	−3.02605	−0.177696
$291$	−5.21643	−0.305793
$292$	−0.0839913	−0.00491522
$293$	−24.0303	−1.40386	−0.701932	−	0.712244i	$-0.747679\pi$
$293$	−24.0303	−1.40386	−0.701932	+	0.712244i	$0.747679\pi$
$294$	3.04878	0.177809
$295$	14.0665	0.818985
$296$	5.94532	0.345565
$297$	0	0
$298$	4.52983	0.262406
$299$	3.34441	0.193412
$300$	−0.0497273	−0.00287101
$301$	8.02379	0.462484
$302$	−31.3802	−1.80573
$303$	−25.7353	−1.47845
$304$	23.0416	1.32152
$305$	11.4940	0.658145
$306$	12.9528	0.740462
$307$	−5.46298	−0.311789	−0.155894	−	0.987774i	$-0.549826\pi$
$307$	−5.46298	−0.311789	−0.155894	+	0.987774i	$0.549826\pi$
$308$	0	0
$309$	−0.857378	−0.0487745
$310$	−3.37549	−0.191715
$311$	13.8884	0.787541	0.393770	−	0.919209i	$-0.371170\pi$
$311$	13.8884	0.787541	0.393770	+	0.919209i	$0.371170\pi$
$312$	28.5336	1.61540
$313$	27.4486	1.55148	0.775742	−	0.631050i	$-0.217376\pi$
$313$	27.4486	1.55148	0.775742	+	0.631050i	$0.217376\pi$
$314$	−18.6795	−1.05415
$315$	−3.07713	−0.173377
$316$	0.218418	0.0122870
$317$	7.82570	0.439535	0.219768	−	0.975552i	$-0.429470\pi$
$317$	7.82570	0.439535	0.219768	+	0.975552i	$0.429470\pi$
$318$	30.9297	1.73445
$319$	0	0
$320$	−14.7520	−0.824659
$321$	7.07286	0.394768
$322$	1.01423	0.0565206
$323$	31.7337	1.76571
$324$	0.156647	0.00870262
$325$	7.64917	0.424299
$326$	−19.3328	−1.07075
$327$	6.15782	0.340528
$328$	−2.60164	−0.143651
$329$	5.97584	0.329458
$330$	0	0
$331$	−28.1462	−1.54705	−0.773527	−	0.633764i	$-0.781509\pi$
$331$	−28.1462	−1.54705	−0.773527	+	0.633764i	$0.781509\pi$
$332$	0.0611049	0.00335357
$333$	3.51962	0.192874
$334$	−13.1183	−0.717802
$335$	28.3265	1.54764
$336$	8.59268	0.468769
$337$	−24.9789	−1.36069	−0.680345	−	0.732892i	$-0.738170\pi$
$337$	−24.9789	−1.36069	−0.680345	+	0.732892i	$0.738170\pi$
$338$	−12.1128	−0.658852
$339$	−31.4654	−1.70896
$340$	−0.139898	−0.00758701
$341$	0	0
$342$	13.7365	0.742783
$343$	1.00000	0.0539949
$344$	22.7732	1.22785
$345$	−2.85137	−0.153513
$346$	−14.8053	−0.795939
$347$	20.6492	1.10851	0.554254	−	0.832347i	$-0.313003\pi$
$347$	20.6492	1.10851	0.554254	+	0.832347i	$0.313003\pi$
$348$	−0.0354211	−0.00189877
$349$	−5.99721	−0.321023	−0.160512	−	0.987034i	$-0.551314\pi$
$349$	−5.99721	−0.321023	−0.160512	+	0.987034i	$0.551314\pi$
$350$	2.31969	0.123993
$351$	−13.2682	−0.708206
$352$	0	0
$353$	24.0382	1.27942	0.639712	−	0.768615i	$-0.279054\pi$
$353$	24.0382	1.27942	0.639712	+	0.768615i	$0.279054\pi$
$354$	−23.4171	−1.24461
$355$	−25.4598	−1.35126
$356$	−0.214267	−0.0113562
$357$	11.8342	0.626330
$358$	11.7298	0.619938
$359$	−10.9501	−0.577925	−0.288963	−	0.957340i	$-0.593310\pi$
$359$	−10.9501	−0.577925	−0.288963	+	0.957340i	$0.593310\pi$
$360$	−8.73355	−0.460299
$361$	14.6536	0.771244
$362$	20.8613	1.09645
$363$	0	0
$364$	0.0648939	0.00340136
$365$	−11.0151	−0.576557
$366$	−19.1345	−1.00018
$367$	10.6178	0.554243	0.277121	−	0.960835i	$-0.410620\pi$
$367$	10.6178	0.554243	0.277121	+	0.960835i	$0.410620\pi$
$368$	2.85849	0.149009
$369$	−1.54016	−0.0801777
$370$	5.40634	0.281062
$371$	10.1449	0.526699
$372$	−0.0395113	−0.00204857
$373$	36.6036	1.89526	0.947631	−	0.319367i	$-0.103470\pi$
$373$	36.6036	1.89526	0.947631	+	0.319367i	$0.103470\pi$
$374$	0	0
$375$	−26.3315	−1.35975
$376$	16.9607	0.874682
$377$	5.44855	0.280615
$378$	−4.02373	−0.206958
$379$	−12.6578	−0.650186	−0.325093	−	0.945682i	$-0.605396\pi$
$379$	−12.6578	−0.650186	−0.325093	+	0.945682i	$0.605396\pi$
$380$	−0.148362	−0.00761080
$381$	−10.8880	−0.557810
$382$	13.5385	0.692690
$383$	−15.4679	−0.790372	−0.395186	−	0.918601i	$-0.629320\pi$
$383$	−15.4679	−0.790372	−0.395186	+	0.918601i	$0.629320\pi$
$384$	24.2164	1.23579
$385$	0	0
$386$	2.09672	0.106720
$387$	13.4817	0.685314
$388$	−0.0336718	−0.00170943
$389$	−12.7130	−0.644572	−0.322286	−	0.946642i	$-0.604451\pi$
$389$	−12.7130	−0.644572	−0.322286	+	0.946642i	$0.604451\pi$
$390$	25.9468	1.31387
$391$	3.93682	0.199093
$392$	2.83822	0.143352
$393$	0.389523	0.0196488
$394$	19.7909	0.997049
$395$	28.6446	1.44127
$396$	0	0
$397$	−18.9574	−0.951445	−0.475722	−	0.879596i	$-0.657813\pi$
$397$	−18.9574	−0.951445	−0.475722	+	0.879596i	$0.657813\pi$
$398$	6.04193	0.302855
$399$	12.5502	0.628294
$400$	6.53780	0.326890
$401$	−8.68208	−0.433563	−0.216781	−	0.976220i	$-0.569556\pi$
$401$	−8.68208	−0.433563	−0.216781	+	0.976220i	$0.569556\pi$
$402$	−47.1564	−2.35194
$403$	6.07772	0.302753
$404$	−0.166120	−0.00826478
$405$	20.5436	1.02082
$406$	1.65233	0.0820037
$407$	0	0
$408$	33.5879	1.66285
$409$	5.71406	0.282542	0.141271	−	0.989971i	$-0.454881\pi$
$409$	5.71406	0.282542	0.141271	+	0.989971i	$0.454881\pi$
$410$	−2.36578	−0.116838
$411$	−18.0079	−0.888263
$412$	−0.00553433	−0.000272657 0
$413$	−7.68081	−0.377948
$414$	1.70412	0.0837529
$415$	8.01365	0.393375
$416$	0.367089	0.0179980
$417$	15.0597	0.737477
$418$	0	0
$419$	−27.1909	−1.32836	−0.664181	−	0.747571i	$-0.731220\pi$
$419$	−27.1909	−1.32836	−0.664181	+	0.747571i	$0.731220\pi$
$420$	−0.0553272	−0.00269969
$421$	−23.9651	−1.16799	−0.583993	−	0.811759i	$-0.698510\pi$
$421$	−23.9651	−1.16799	−0.583993	+	0.811759i	$0.698510\pi$
$422$	2.05087	0.0998347
$423$	10.0407	0.488196
$424$	28.7935	1.39834
$425$	9.00410	0.436763
$426$	42.3839	2.05351
$427$	−6.27612	−0.303723
$428$	0.0456549	0.00220681
$429$	0	0
$430$	20.7087	0.998663
$431$	16.4732	0.793484	0.396742	−	0.917930i	$-0.370141\pi$
$431$	16.4732	0.793484	0.396742	+	0.917930i	$0.370141\pi$
$432$	−11.3405	−0.545618
$433$	20.0909	0.965509	0.482754	−	0.875756i	$-0.339636\pi$
$433$	20.0909	0.965509	0.482754	+	0.875756i	$0.339636\pi$
$434$	1.84313	0.0884732
$435$	−4.64533	−0.222726
$436$	0.0397484	0.00190360
$437$	4.17500	0.199718
$438$	18.3373	0.876189
$439$	26.7682	1.27758	0.638788	−	0.769383i	$-0.279436\pi$
$439$	26.7682	1.27758	0.638788	+	0.769383i	$0.279436\pi$
$440$	0	0
$441$	1.68022	0.0800104
$442$	−35.8241	−1.70398
$443$	26.2153	1.24553	0.622764	−	0.782410i	$-0.286010\pi$
$443$	26.2153	1.24553	0.622764	+	0.782410i	$0.286010\pi$
$444$	0.0632832	0.00300329
$445$	−28.1003	−1.33208
$446$	6.84400	0.324073
$447$	6.95378	0.328903
$448$	8.05508	0.380567
$449$	9.74740	0.460008	0.230004	−	0.973190i	$-0.426126\pi$
$449$	9.74740	0.460008	0.230004	+	0.973190i	$0.426126\pi$
$450$	3.89758	0.183734
$451$	0	0
$452$	−0.203107	−0.00955336
$453$	−48.1720	−2.26332
$454$	−0.559682	−0.0262672
$455$	8.51056	0.398981
$456$	35.6200	1.66806
$457$	11.8853	0.555971	0.277986	−	0.960585i	$-0.410333\pi$
$457$	11.8853	0.555971	0.277986	+	0.960585i	$0.410333\pi$
$458$	−3.08578	−0.144189
$459$	−15.6185	−0.729009
$460$	−0.0184055	−0.000858159 0
$461$	9.14737	0.426035	0.213018	−	0.977048i	$-0.431671\pi$
$461$	9.14737	0.426035	0.213018	+	0.977048i	$0.431671\pi$
$462$	0	0
$463$	38.9342	1.80943	0.904713	−	0.426021i	$-0.140085\pi$
$463$	38.9342	1.80943	0.904713	+	0.426021i	$0.140085\pi$
$464$	4.65692	0.216192
$465$	−5.18174	−0.240298
$466$	1.77590	0.0822668
$467$	−20.7834	−0.961742	−0.480871	−	0.876791i	$-0.659680\pi$
$467$	−20.7834	−0.961742	−0.480871	+	0.876791i	$0.659680\pi$
$468$	0.109036	0.00504019
$469$	−15.4673	−0.714212
$470$	15.4231	0.711415
$471$	−28.6751	−1.32128
$472$	−21.7998	−1.00342
$473$	0	0
$474$	−47.6858	−2.19028
$475$	9.54887	0.438132
$476$	0.0763889	0.00350128
$477$	17.0457	0.780469
$478$	15.7298	0.719462
$479$	−24.5363	−1.12109	−0.560546	−	0.828123i	$-0.689409\pi$
$479$	−24.5363	−1.12109	−0.560546	+	0.828123i	$0.689409\pi$
$480$	−0.312972	−0.0142852
$481$	−9.73437	−0.443849
$482$	30.2771	1.37908
$483$	1.55695	0.0708436
$484$	0	0
$485$	−4.41591	−0.200516
$486$	−22.1286	−1.00377
$487$	−12.3713	−0.560598	−0.280299	−	0.959913i	$-0.590434\pi$
$487$	−12.3713	−0.560598	−0.280299	+	0.959913i	$0.590434\pi$
$488$	−17.8130	−0.806356
$489$	−29.6780	−1.34209
$490$	2.58091	0.116594
$491$	16.5251	0.745766	0.372883	−	0.927878i	$-0.378369\pi$
$491$	16.5251	0.745766	0.372883	+	0.927878i	$0.378369\pi$
$492$	−0.0276924	−0.00124847
$493$	6.41368	0.288858
$494$	−37.9916	−1.70932
$495$	0	0
$496$	5.19468	0.233248
$497$	13.9019	0.623586
$498$	−13.3407	−0.597809
$499$	−13.7410	−0.615129	−0.307565	−	0.951527i	$-0.599514\pi$
$499$	−13.7410	−0.615129	−0.307565	+	0.951527i	$0.599514\pi$
$500$	−0.169968	−0.00760121
$501$	−20.1380	−0.899701
$502$	0.597276	0.0266577
$503$	−22.5968	−1.00754	−0.503770	−	0.863838i	$-0.668054\pi$
$503$	−22.5968	−1.00754	−0.503770	+	0.863838i	$0.668054\pi$
$504$	4.76882	0.212420
$505$	−21.7859	−0.969461
$506$	0	0
$507$	−18.5946	−0.825813
$508$	−0.0702816	−0.00311824
$509$	−21.4636	−0.951358	−0.475679	−	0.879619i	$-0.657798\pi$
$509$	−21.4636	−0.951358	−0.475679	+	0.879619i	$0.657798\pi$
$510$	30.5429	1.35246
$511$	6.01462	0.266071
$512$	22.8598	1.01027
$513$	−16.5635	−0.731295
$514$	−24.5777	−1.08408
$515$	−0.725804	−0.0319827
$516$	0.242403	0.0106712
$517$	0	0
$518$	−2.95205	−0.129706
$519$	−22.7278	−0.997639
$520$	24.1548	1.05926
$521$	34.7116	1.52074	0.760371	−	0.649489i	$-0.225017\pi$
$521$	34.7116	1.52074	0.760371	+	0.649489i	$0.225017\pi$
$522$	2.77627	0.121514
$523$	19.7281	0.862651	0.431326	−	0.902196i	$-0.358046\pi$
$523$	19.7281	0.862651	0.431326	+	0.902196i	$0.358046\pi$
$524$	0.00251435	0.000109840 0
$525$	3.56098	0.155414
$526$	−2.13531	−0.0931039
$527$	7.15430	0.311646
$528$	0	0
$529$	−22.4821	−0.977481
$530$	26.1832	1.13733
$531$	−12.9054	−0.560048
$532$	0.0810106	0.00351226
$533$	4.25970	0.184508
$534$	46.7797	2.02435
$535$	5.98745	0.258860
$536$	−43.8994	−1.89617
$537$	18.0065	0.777037
$538$	2.86226	0.123401
$539$	0	0
$540$	0.0730199	0.00314227
$541$	−5.54942	−0.238588	−0.119294	−	0.992859i	$-0.538063\pi$
$541$	−5.54942	−0.238588	−0.119294	+	0.992859i	$0.538063\pi$
$542$	−10.7167	−0.460321
$543$	32.0245	1.37430
$544$	0.432113	0.0185267
$545$	5.21283	0.223293
$546$	−14.1679	−0.606329
$547$	−8.44671	−0.361155	−0.180578	−	0.983561i	$-0.557797\pi$
$547$	−8.44671	−0.361155	−0.180578	+	0.983561i	$0.557797\pi$
$548$	−0.116240	−0.00496553
$549$	−10.5453	−0.450061
$550$	0	0
$551$	6.80173	0.289763
$552$	4.41895	0.188083
$553$	−15.6409	−0.665120
$554$	−20.3312	−0.863791
$555$	8.29933	0.352287
$556$	0.0972097	0.00412261
$557$	−12.1835	−0.516231	−0.258115	−	0.966114i	$-0.583102\pi$
$557$	−12.1835	−0.516231	−0.258115	+	0.966114i	$0.583102\pi$
$558$	3.09686	0.131101
$559$	−37.2870	−1.57707
$560$	7.27404	0.307384
$561$	0	0
$562$	−25.0140	−1.05515
$563$	−27.3535	−1.15281	−0.576407	−	0.817163i	$-0.695546\pi$
$563$	−27.3535	−1.15281	−0.576407	+	0.817163i	$0.695546\pi$
$564$	0.180533	0.00760183
$565$	−26.6366	−1.12061
$566$	43.8792	1.84438
$567$	−11.2175	−0.471092
$568$	39.4566	1.65556
$569$	7.13524	0.299125	0.149562	−	0.988752i	$-0.452214\pi$
$569$	7.13524	0.299125	0.149562	+	0.988752i	$0.452214\pi$
$570$	32.3909	1.35670
$571$	32.4839	1.35941	0.679705	−	0.733486i	$-0.262108\pi$
$571$	32.4839	1.35941	0.679705	+	0.733486i	$0.262108\pi$
$572$	0	0
$573$	20.7831	0.868226
$574$	1.29180	0.0539186
$575$	1.18461	0.0494018
$576$	13.5343	0.563928
$577$	34.7819	1.44799	0.723995	−	0.689806i	$-0.242304\pi$
$577$	34.7819	1.44799	0.723995	+	0.689806i	$0.242304\pi$
$578$	−18.2123	−0.757532
$579$	3.21870	0.133764
$580$	−0.0299853	−0.00124507
$581$	−4.37573	−0.181536
$582$	7.35135	0.304723
$583$	0	0
$584$	17.0708	0.706395
$585$	14.2996	0.591215
$586$	33.8651	1.39896
$587$	−14.4749	−0.597445	−0.298722	−	0.954340i	$-0.596560\pi$
$587$	−14.4749	−0.597445	−0.298722	+	0.954340i	$0.596560\pi$
$588$	0.0302106	0.00124586
$589$	7.58716	0.312623
$590$	−19.8235	−0.816120
$591$	30.3812	1.24971
$592$	−8.32004	−0.341952
$593$	−15.0291	−0.617169	−0.308585	−	0.951197i	$-0.599855\pi$
$593$	−15.0291	−0.617169	−0.308585	+	0.951197i	$0.599855\pi$
$594$	0	0
$595$	10.0181	0.410701
$596$	0.0448863	0.00183862
$597$	9.27503	0.379602
$598$	−4.71316	−0.192736
$599$	1.76045	0.0719302	0.0359651	−	0.999353i	$-0.488549\pi$
$599$	1.76045	0.0719302	0.0359651	+	0.999353i	$0.488549\pi$
$600$	10.1068	0.412609
$601$	23.4365	0.955993	0.477997	−	0.878362i	$-0.341363\pi$
$601$	23.4365	0.955993	0.477997	+	0.878362i	$0.341363\pi$
$602$	−11.3077	−0.460866
$603$	−25.9884	−1.05833
$604$	−0.310948	−0.0126523
$605$	0	0
$606$	36.2679	1.47328
$607$	−25.2785	−1.02602	−0.513012	−	0.858382i	$-0.671470\pi$
$607$	−25.2785	−1.02602	−0.513012	+	0.858382i	$0.671470\pi$
$608$	0.458257	0.0185848
$609$	2.53651	0.102784
$610$	−16.1981	−0.655843
$611$	−27.7700	−1.12346
$612$	0.128350	0.00518824
$613$	1.16094	0.0468900	0.0234450	−	0.999725i	$-0.492537\pi$
$613$	1.16094	0.0468900	0.0234450	+	0.999725i	$0.492537\pi$
$614$	7.69880	0.310698
$615$	−3.63174	−0.146446
$616$	0	0
$617$	12.9711	0.522197	0.261098	−	0.965312i	$-0.415915\pi$
$617$	12.9711	0.522197	0.261098	+	0.965312i	$0.415915\pi$
$618$	1.20827	0.0486039
$619$	45.7920	1.84053	0.920267	−	0.391291i	$-0.127971\pi$
$619$	45.7920	1.84053	0.920267	+	0.391291i	$0.127971\pi$
$620$	−0.0334479	−0.00134330
$621$	−2.05483	−0.0824575
$622$	−19.5725	−0.784787
$623$	15.3437	0.614733
$624$	−39.9307	−1.59851
$625$	−14.0605	−0.562419
$626$	−38.6824	−1.54606
$627$	0	0
$628$	−0.185097	−0.00738615
$629$	−11.4587	−0.456887
$630$	4.33650	0.172770
$631$	−12.6207	−0.502421	−0.251211	−	0.967932i	$-0.580829\pi$
$631$	−12.6207	−0.502421	−0.251211	+	0.967932i	$0.580829\pi$
$632$	−44.3924	−1.76583
$633$	3.14831	0.125134
$634$	−11.0285	−0.437998
$635$	−9.21713	−0.365771
$636$	0.306484	0.0121529
$637$	−4.64706	−0.184123
$638$	0	0
$639$	23.3582	0.924038
$640$	20.5001	0.810338
$641$	27.9567	1.10422	0.552112	−	0.833770i	$-0.313822\pi$
$641$	27.9567	1.10422	0.552112	+	0.833770i	$0.313822\pi$
$642$	−9.96755	−0.393388
$643$	−49.6981	−1.95990	−0.979950	−	0.199243i	$-0.936152\pi$
$643$	−49.6981	−1.95990	−0.979950	+	0.199243i	$0.936152\pi$
$644$	0.0100500	0.000396026 0
$645$	31.7902	1.25174
$646$	−44.7212	−1.75953
$647$	−11.0067	−0.432718	−0.216359	−	0.976314i	$-0.569418\pi$
$647$	−11.0067	−0.432718	−0.216359	+	0.976314i	$0.569418\pi$
$648$	−31.8377	−1.25070
$649$	0	0
$650$	−10.7797	−0.422816
$651$	2.82941	0.110893
$652$	−0.191570	−0.00750246
$653$	28.9364	1.13237	0.566185	−	0.824278i	$-0.308419\pi$
$653$	28.9364	1.13237	0.566185	+	0.824278i	$0.308419\pi$
$654$	−8.67801	−0.339337
$655$	0.329747	0.0128843
$656$	3.64080	0.142149
$657$	10.1059	0.394268
$658$	−8.42155	−0.328306
$659$	10.8405	0.422288	0.211144	−	0.977455i	$-0.432281\pi$
$659$	10.8405	0.422288	0.211144	+	0.977455i	$0.432281\pi$
$660$	0	0
$661$	20.3444	0.791305	0.395652	−	0.918400i	$-0.370519\pi$
$661$	20.3444	0.791305	0.395652	+	0.918400i	$0.370519\pi$
$662$	39.6655	1.54164
$663$	−54.9940	−2.13579
$664$	−12.4193	−0.481961
$665$	10.6242	0.411989
$666$	−4.96008	−0.192199
$667$	0.843809	0.0326724
$668$	−0.129990	−0.00502946
$669$	10.5063	0.406197
$670$	−39.9197	−1.54223
$671$	0	0
$672$	0.170894	0.00659237
$673$	−12.0788	−0.465604	−0.232802	−	0.972524i	$-0.574789\pi$
$673$	−12.0788	−0.465604	−0.232802	+	0.972524i	$0.574789\pi$
$674$	35.2020	1.35593
$675$	−4.69971	−0.180892
$676$	−0.120027	−0.00461642
$677$	3.39630	0.130530	0.0652651	−	0.997868i	$-0.479211\pi$
$677$	3.39630	0.130530	0.0652651	+	0.997868i	$0.479211\pi$
$678$	44.3431	1.70299
$679$	2.41124	0.0925349
$680$	28.4335	1.09037
$681$	−0.859173	−0.0329236
$682$	0	0
$683$	−4.75643	−0.182000	−0.0909999	−	0.995851i	$-0.529006\pi$
$683$	−4.75643	−0.182000	−0.0909999	+	0.995851i	$0.529006\pi$
$684$	0.136115	0.00520451
$685$	−15.2444	−0.582458
$686$	−1.40927	−0.0538061
$687$	−4.73701	−0.180728
$688$	−31.8695	−1.21501
$689$	−47.1441	−1.79605
$690$	4.01835	0.152976
$691$	6.63388	0.252365	0.126182	−	0.992007i	$-0.459728\pi$
$691$	6.63388	0.252365	0.126182	+	0.992007i	$0.459728\pi$
$692$	−0.146707	−0.00557695
$693$	0	0
$694$	−29.1003	−1.10463
$695$	12.7486	0.483583
$696$	7.19916	0.272883
$697$	5.01425	0.189928
$698$	8.45168	0.319901
$699$	2.72620	0.103114
$700$	0.0229859	0.000868786 0
$701$	3.03003	0.114443	0.0572213	−	0.998362i	$-0.481776\pi$
$701$	3.03003	0.114443	0.0572213	+	0.998362i	$0.481776\pi$
$702$	18.6985	0.705729
$703$	−12.1519	−0.458319
$704$	0	0
$705$	23.6762	0.891697
$706$	−33.8762	−1.27495
$707$	11.8959	0.447390
$708$	−0.232041	−0.00872065
$709$	−13.6570	−0.512901	−0.256451	−	0.966557i	$-0.582553\pi$
$709$	−13.6570	−0.512901	−0.256451	+	0.966557i	$0.582553\pi$
$710$	35.8796	1.34654
$711$	−26.2802	−0.985584
$712$	43.5488	1.63206
$713$	0.941248	0.0352500
$714$	−16.6775	−0.624140
$715$	0	0
$716$	0.116231	0.00434376
$717$	24.1469	0.901783
$718$	15.4317	0.575904
$719$	−1.90029	−0.0708689	−0.0354344	−	0.999372i	$-0.511281\pi$
$719$	−1.90029	−0.0708689	−0.0354344	+	0.999372i	$0.511281\pi$
$720$	12.2220	0.455486
$721$	0.396314	0.0147595
$722$	−20.6509	−0.768547
$723$	46.4787	1.72856
$724$	0.206716	0.00768255
$725$	1.92992	0.0716754
$726$	0	0
$727$	−13.8211	−0.512595	−0.256298	−	0.966598i	$-0.582503\pi$
$727$	−13.8211	−0.512595	−0.256298	+	0.966598i	$0.582503\pi$
$728$	−13.1893	−0.488830
$729$	−0.317279	−0.0117511
$730$	15.5232	0.574540
$731$	−43.8918	−1.62340
$732$	−0.189605	−0.00700801
$733$	48.3744	1.78675	0.893374	−	0.449314i	$-0.148331\pi$
$733$	48.3744	1.78675	0.893374	+	0.449314i	$0.148331\pi$
$734$	−14.9633	−0.552304
$735$	3.96199	0.146140
$736$	0.0568504	0.00209553
$737$	0	0
$738$	2.17050	0.0798973
$739$	23.2081	0.853724	0.426862	−	0.904317i	$-0.359619\pi$
$739$	23.2081	0.853724	0.426862	+	0.904317i	$0.359619\pi$
$740$	0.0535717	0.00196934
$741$	−58.3213	−2.14249
$742$	−14.2969	−0.524857
$743$	−44.8311	−1.64469	−0.822347	−	0.568986i	$-0.807336\pi$
$743$	−44.8311	−1.64469	−0.822347	+	0.568986i	$0.807336\pi$
$744$	8.03048	0.294412
$745$	5.88665	0.215670
$746$	−51.5843	−1.88863
$747$	−7.35218	−0.269002
$748$	0	0
$749$	−3.26935	−0.119460
$750$	37.1081	1.35500
$751$	−41.1856	−1.50288	−0.751442	−	0.659799i	$-0.770641\pi$
$751$	−41.1856	−1.50288	−0.751442	+	0.659799i	$0.770641\pi$
$752$	−23.7353	−0.865537
$753$	0.916884	0.0334131
$754$	−7.67847	−0.279633
$755$	−40.7795	−1.48412
$756$	−0.0398714	−0.00145011
$757$	−21.8888	−0.795561	−0.397781	−	0.917481i	$-0.630219\pi$
$757$	−21.8888	−0.795561	−0.397781	+	0.917481i	$0.630219\pi$
$758$	17.8382	0.647912
$759$	0	0
$760$	30.1537	1.09379
$761$	−35.7154	−1.29468	−0.647340	−	0.762201i	$-0.724119\pi$
$761$	−35.7154	−1.29468	−0.647340	+	0.762201i	$0.724119\pi$
$762$	15.3441	0.555859
$763$	−2.84638	−0.103046
$764$	0.134154	0.00485351
$765$	16.8325	0.608582
$766$	21.7984	0.787608
$767$	35.6931	1.28880
$768$	0.725018	0.0261618
$769$	5.30246	0.191212	0.0956058	−	0.995419i	$-0.469521\pi$
$769$	5.30246	0.191212	0.0956058	+	0.995419i	$0.469521\pi$
$770$	0	0
$771$	−37.7295	−1.35880
$772$	0.0207765	0.000747763 0
$773$	−49.8912	−1.79446	−0.897231	−	0.441561i	$-0.854425\pi$
$773$	−49.8912	−1.79446	−0.897231	+	0.441561i	$0.854425\pi$
$774$	−18.9993	−0.682917
$775$	2.15278	0.0773300
$776$	6.84362	0.245672
$777$	−4.53172	−0.162574
$778$	17.9160	0.642318
$779$	5.31762	0.190524
$780$	0.257109	0.00920597
$781$	0	0
$782$	−5.54803	−0.198397
$783$	−3.34764	−0.119635
$784$	−3.97188	−0.141853
$785$	−24.2746	−0.866398
$786$	−0.548942	−0.0195801
$787$	35.8570	1.27816	0.639081	−	0.769139i	$-0.279315\pi$
$787$	35.8570	1.27816	0.639081	+	0.769139i	$0.279315\pi$
$788$	0.196109	0.00698609
$789$	−3.27794	−0.116698
$790$	−40.3679	−1.43623
$791$	14.5445	0.517144
$792$	0	0
$793$	29.1655	1.03570
$794$	26.7161	0.948117
$795$	40.1941	1.42554
$796$	0.0598699	0.00212203
$797$	14.7413	0.522163	0.261081	−	0.965317i	$-0.415921\pi$
$797$	14.7413	0.522163	0.261081	+	0.965317i	$0.415921\pi$
$798$	−17.6865	−0.626096
$799$	−32.6891	−1.15646
$800$	0.130026	0.00459710
$801$	25.7808	0.910919
$802$	12.2354	0.432046
$803$	0	0
$804$	−0.467275	−0.0164795
$805$	1.31802	0.0464540
$806$	−8.56514	−0.301694
$807$	4.39389	0.154672
$808$	33.7630	1.18778
$809$	−26.9758	−0.948420	−0.474210	−	0.880412i	$-0.657266\pi$
$809$	−26.9758	−0.948420	−0.474210	+	0.880412i	$0.657266\pi$
$810$	−28.9515	−1.01725
$811$	1.46440	0.0514219	0.0257109	−	0.999669i	$-0.491815\pi$
$811$	1.46440	0.0514219	0.0257109	+	0.999669i	$0.491815\pi$
$812$	0.0163730	0.000574581 0
$813$	−16.4513	−0.576972
$814$	0	0
$815$	−25.1236	−0.880041
$816$	−47.0038	−1.64546
$817$	−46.5474	−1.62849
$818$	−8.05264	−0.281554
$819$	−7.80807	−0.272836
$820$	−0.0234427	−0.000818654 0
$821$	30.6287	1.06895	0.534474	−	0.845185i	$-0.320510\pi$
$821$	30.6287	1.06895	0.534474	+	0.845185i	$0.320510\pi$
$822$	25.3779	0.885157
$823$	−24.5338	−0.855195	−0.427598	−	0.903969i	$-0.640640\pi$
$823$	−24.5338	−0.855195	−0.427598	+	0.903969i	$0.640640\pi$
$824$	1.12482	0.0391851
$825$	0	0
$826$	10.8243	0.376626
$827$	−6.56686	−0.228352	−0.114176	−	0.993461i	$-0.536423\pi$
$827$	−6.56686	−0.228352	−0.114176	+	0.993461i	$0.536423\pi$
$828$	0.0168862	0.000586837 0
$829$	−19.8442	−0.689219	−0.344610	−	0.938746i	$-0.611989\pi$
$829$	−19.8442	−0.689219	−0.344610	+	0.938746i	$0.611989\pi$
$830$	−11.2934	−0.391999
$831$	−31.2107	−1.08269
$832$	−37.4324	−1.29773
$833$	−5.47021	−0.189532
$834$	−21.2232	−0.734898
$835$	−17.0476	−0.589958
$836$	0	0
$837$	−3.73421	−0.129073
$838$	38.3193	1.32372
$839$	−7.96051	−0.274827	−0.137414	−	0.990514i	$-0.543879\pi$
$839$	−7.96051	−0.274827	−0.137414	+	0.990514i	$0.543879\pi$
$840$	11.2450	0.387988
$841$	−27.6253	−0.952597
$842$	33.7732	1.16390
$843$	−38.3993	−1.32254
$844$	0.0203222	0.000699518 0
$845$	−15.7410	−0.541507
$846$	−14.1500	−0.486489
$847$	0	0
$848$	−40.2944	−1.38372
$849$	67.3593	2.31177
$850$	−12.6892	−0.435236
$851$	−1.50755	−0.0516780
$852$	0.419985	0.0143884
$853$	34.0732	1.16664	0.583322	−	0.812241i	$-0.301753\pi$
$853$	34.0732	1.16664	0.583322	+	0.812241i	$0.301753\pi$
$854$	8.84474	0.302661
$855$	17.8510	0.610490
$856$	−9.27913	−0.317154
$857$	24.8539	0.848992	0.424496	−	0.905430i	$-0.360451\pi$
$857$	24.8539	0.848992	0.424496	+	0.905430i	$0.360451\pi$
$858$	0	0
$859$	2.05654	0.0701683	0.0350841	−	0.999384i	$-0.488830\pi$
$859$	2.05654	0.0701683	0.0350841	+	0.999384i	$0.488830\pi$
$860$	0.205204	0.00699739
$861$	1.98305	0.0675823
$862$	−23.2151	−0.790709
$863$	−0.259476	−0.00883265	−0.00441633	−	0.999990i	$-0.501406\pi$
$863$	−0.259476	−0.00883265	−0.00441633	+	0.999990i	$0.501406\pi$
$864$	−0.225542	−0.00767311
$865$	−19.2400	−0.654178
$866$	−28.3135	−0.962132
$867$	−27.9579	−0.949500
$868$	0.0182637	0.000619910 0
$869$	0	0
$870$	6.54651	0.221947
$871$	71.8773	2.43547
$872$	−8.07865	−0.273578
$873$	4.05141	0.137119
$874$	−5.88370	−0.199019
$875$	12.1714	0.411470
$876$	0.181705	0.00613925
$877$	18.5930	0.627840	0.313920	−	0.949449i	$-0.398358\pi$
$877$	18.5930	0.627840	0.313920	+	0.949449i	$0.398358\pi$
$878$	−37.7235	−1.27311
$879$	51.9867	1.75347
$880$	0	0
$881$	6.45292	0.217404	0.108702	−	0.994074i	$-0.465330\pi$
$881$	6.45292	0.217404	0.108702	+	0.994074i	$0.465330\pi$
$882$	−2.36788	−0.0797306
$883$	−0.278487	−0.00937185	−0.00468592	−	0.999989i	$-0.501492\pi$
$883$	−0.278487	−0.00937185	−0.00468592	+	0.999989i	$0.501492\pi$
$884$	−0.354983	−0.0119394
$885$	−30.4313	−1.02294
$886$	−36.9444	−1.24117
$887$	30.5570	1.02600	0.513001	−	0.858388i	$-0.328534\pi$
$887$	30.5570	1.02600	0.513001	+	0.858388i	$0.328534\pi$
$888$	−12.8620	−0.431620
$889$	5.03287	0.168797
$890$	39.6008	1.32742
$891$	0	0
$892$	0.0678176	0.00227070
$893$	−34.6669	−1.16008
$894$	−9.79974	−0.327752
$895$	15.2432	0.509524
$896$	−11.1938	−0.373958
$897$	−7.23522	−0.241577
$898$	−13.7367	−0.458400
$899$	1.53344	0.0511430
$900$	0.0386214	0.00128738
$901$	−55.4949	−1.84880
$902$	0	0
$903$	−17.3585	−0.577655
$904$	41.2805	1.37297
$905$	27.1099	0.901165
$906$	67.8873	2.25540
$907$	31.9347	1.06037	0.530187	−	0.847881i	$-0.322122\pi$
$907$	31.9347	1.06037	0.530187	+	0.847881i	$0.322122\pi$
$908$	−0.00554592	−0.000184048 0
$909$	19.9877	0.662949
$910$	−11.9937	−0.397586
$911$	2.80042	0.0927821	0.0463911	−	0.998923i	$-0.485228\pi$
$911$	2.80042	0.0927821	0.0463911	+	0.998923i	$0.485228\pi$
$912$	−49.8477	−1.65062
$913$	0	0
$914$	−16.7496	−0.554027
$915$	−24.8659	−0.822042
$916$	−0.0305772	−0.00101030
$917$	−0.180053	−0.00594587
$918$	22.0107	0.726460
$919$	−15.8339	−0.522311	−0.261156	−	0.965297i	$-0.584104\pi$
$919$	−15.8339	−0.522311	−0.261156	+	0.965297i	$0.584104\pi$
$920$	3.74081	0.123331
$921$	11.8185	0.389433
$922$	−12.8911	−0.424545
$923$	−64.6030	−2.12643
$924$	0	0
$925$	−3.44799	−0.113369
$926$	−54.8687	−1.80310
$927$	0.665894	0.0218708
$928$	0.0926181	0.00304034
$929$	−27.4834	−0.901699	−0.450850	−	0.892600i	$-0.648879\pi$
$929$	−27.4834	−0.901699	−0.450850	+	0.892600i	$0.648879\pi$
$930$	7.30247	0.239457
$931$	−5.80118	−0.190126
$932$	0.0175975	0.000576424 0
$933$	−30.0460	−0.983661
$934$	29.2894	0.958379
$935$	0	0
$936$	−22.1610	−0.724354
$937$	−12.3086	−0.402105	−0.201053	−	0.979580i	$-0.564436\pi$
$937$	−12.3086	−0.402105	−0.201053	+	0.979580i	$0.564436\pi$
$938$	21.7975	0.711714
$939$	−59.3817	−1.93785
$940$	0.152829	0.00498472
$941$	1.46014	0.0475991	0.0237996	−	0.999717i	$-0.492424\pi$
$941$	1.46014	0.0475991	0.0237996	+	0.999717i	$0.492424\pi$
$942$	40.4109	1.31666
$943$	0.659693	0.0214826
$944$	30.5072	0.992925
$945$	−5.22896	−0.170098
$946$	0	0
$947$	11.0714	0.359771	0.179885	−	0.983688i	$-0.442427\pi$
$947$	11.0714	0.359771	0.179885	+	0.983688i	$0.442427\pi$
$948$	−0.472522	−0.0153468
$949$	−27.9503	−0.907305
$950$	−13.4569	−0.436600
$951$	−16.9300	−0.548992
$952$	−15.5256	−0.503189
$953$	−14.8234	−0.480176	−0.240088	−	0.970751i	$-0.577176\pi$
$953$	−14.8234	−0.480176	−0.240088	+	0.970751i	$0.577176\pi$
$954$	−24.0220	−0.777739
$955$	17.5937	0.569318
$956$	0.155867	0.00504110
$957$	0	0
$958$	34.5782	1.11717
$959$	8.32395	0.268794
$960$	31.9141	1.03002
$961$	−29.2895	−0.944822
$962$	13.7183	0.442297
$963$	−5.49323	−0.177017
$964$	0.300018	0.00966292
$965$	2.72475	0.0877129
$966$	−2.19416	−0.0705958
$967$	16.5193	0.531224	0.265612	−	0.964080i	$-0.414426\pi$
$967$	16.5193	0.531224	0.265612	+	0.964080i	$0.414426\pi$
$968$	0	0
$969$	−68.6520	−2.20542
$970$	6.22320	0.199815
$971$	40.7993	1.30931	0.654656	−	0.755927i	$-0.272813\pi$
$971$	40.7993	1.30931	0.654656	+	0.755927i	$0.272813\pi$
$972$	−0.219274	−0.00703320
$973$	−6.96119	−0.223166
$974$	17.4345	0.558637
$975$	−16.5481	−0.529962
$976$	24.9280	0.797925
$977$	49.1145	1.57131	0.785656	−	0.618664i	$-0.212326\pi$
$977$	49.1145	1.57131	0.785656	+	0.618664i	$0.212326\pi$
$978$	41.8243	1.33739
$979$	0	0
$980$	0.0255744	0.000816945 0
$981$	−4.78255	−0.152695
$982$	−23.2883	−0.743158
$983$	1.10679	0.0353011	0.0176505	−	0.999844i	$-0.494381\pi$
$983$	1.10679	0.0353011	0.0176505	+	0.999844i	$0.494381\pi$
$984$	5.62833	0.179425
$985$	25.7188	0.819470
$986$	−9.03860	−0.287848
$987$	−12.9280	−0.411503
$988$	−0.376461	−0.0119768
$989$	−5.77458	−0.183621
$990$	0	0
$991$	41.7851	1.32735	0.663674	−	0.748022i	$-0.268996\pi$
$991$	41.7851	1.32735	0.663674	+	0.748022i	$0.268996\pi$
$992$	0.103313	0.00328020
$993$	60.8909	1.93231
$994$	−19.5915	−0.621405
$995$	7.85168	0.248915
$996$	−0.132193	−0.00418870
$997$	44.1945	1.39965	0.699827	−	0.714313i	$-0.253261\pi$
$997$	44.1945	1.39965	0.699827	+	0.714313i	$0.253261\pi$
$998$	19.3647	0.612978
$999$	5.98088	0.189227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.p.1.3		8
3.2	odd	2		7623.2.a.ct.1.6		8
7.6	odd	2		5929.2.a.bt.1.3		8
11.2	odd	10		847.2.f.v.323.2		16
11.3	even	5		77.2.f.b.64.2	✓	16
11.4	even	5		77.2.f.b.71.2	yes	16
11.5	even	5		847.2.f.w.729.3		16
11.6	odd	10		847.2.f.v.729.2		16
11.7	odd	10		847.2.f.x.148.3		16
11.8	odd	10		847.2.f.x.372.3		16
11.9	even	5		847.2.f.w.323.3		16
11.10	odd	2		847.2.a.o.1.6		8
33.14	odd	10		693.2.m.i.64.3		16
33.26	odd	10		693.2.m.i.379.3		16
33.32	even	2		7623.2.a.cw.1.3		8
77.3	odd	30		539.2.q.f.471.2		32
77.4	even	15		539.2.q.g.324.3		32
77.25	even	15		539.2.q.g.471.2		32
77.26	odd	30		539.2.q.f.214.2		32
77.37	even	15		539.2.q.g.214.2		32
77.47	odd	30		539.2.q.f.361.3		32
77.48	odd	10		539.2.f.e.148.2		16
77.58	even	15		539.2.q.g.361.3		32
77.59	odd	30		539.2.q.f.324.3		32
77.69	odd	10		539.2.f.e.295.2		16
77.76	even	2		5929.2.a.bs.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.2	✓	16	11.3	even	5
77.2.f.b.71.2	yes	16	11.4	even	5
539.2.f.e.148.2		16	77.48	odd	10
539.2.f.e.295.2		16	77.69	odd	10
539.2.q.f.214.2		32	77.26	odd	30
539.2.q.f.324.3		32	77.59	odd	30
539.2.q.f.361.3		32	77.47	odd	30
539.2.q.f.471.2		32	77.3	odd	30
539.2.q.g.214.2		32	77.37	even	15
539.2.q.g.324.3		32	77.4	even	15
539.2.q.g.361.3		32	77.58	even	15
539.2.q.g.471.2		32	77.25	even	15
693.2.m.i.64.3		16	33.14	odd	10
693.2.m.i.379.3		16	33.26	odd	10
847.2.a.o.1.6		8	11.10	odd	2
847.2.a.p.1.3		8	1.1	even	1	trivial
847.2.f.v.323.2		16	11.2	odd	10
847.2.f.v.729.2		16	11.6	odd	10
847.2.f.w.323.3		16	11.9	even	5
847.2.f.w.729.3		16	11.5	even	5
847.2.f.x.148.3		16	11.7	odd	10
847.2.f.x.372.3		16	11.8	odd	10
5929.2.a.bs.1.6		8	77.76	even	2
5929.2.a.bt.1.3		8	7.6	odd	2
7623.2.a.ct.1.6		8	3.2	odd	2
7623.2.a.cw.1.3		8	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40927 q^{2} -2.16338 q^{3} -0.0139645 q^{4} -1.83139 q^{5} +3.04878 q^{6} +1.00000 q^{7} +2.83822 q^{8} +1.68022 q^{9} +O(q^{10})\)	\(q-1.40927 q^{2} -2.16338 q^{3} -0.0139645 q^{4} -1.83139 q^{5} +3.04878 q^{6} +1.00000 q^{7} +2.83822 q^{8} +1.68022 q^{9} +2.58091 q^{10} +0.0302106 q^{12} -4.64706 q^{13} -1.40927 q^{14} +3.96199 q^{15} -3.97188 q^{16} -5.47021 q^{17} -2.36788 q^{18} -5.80118 q^{19} +0.0255744 q^{20} -2.16338 q^{21} -0.719682 q^{23} -6.14014 q^{24} -1.64602 q^{25} +6.54895 q^{26} +2.85519 q^{27} -0.0139645 q^{28} -1.17247 q^{29} -5.58350 q^{30} -1.30787 q^{31} -0.0789938 q^{32} +7.70900 q^{34} -1.83139 q^{35} -0.0234634 q^{36} +2.09474 q^{37} +8.17541 q^{38} +10.0534 q^{39} -5.19787 q^{40} -0.916645 q^{41} +3.04878 q^{42} +8.02379 q^{43} -3.07713 q^{45} +1.01423 q^{46} +5.97584 q^{47} +8.59268 q^{48} +1.00000 q^{49} +2.31969 q^{50} +11.8342 q^{51} +0.0648939 q^{52} +10.1449 q^{53} -4.02373 q^{54} +2.83822 q^{56} +12.5502 q^{57} +1.65233 q^{58} -7.68081 q^{59} -0.0553272 q^{60} -6.27612 q^{61} +1.84313 q^{62} +1.68022 q^{63} +8.05508 q^{64} +8.51056 q^{65} -15.4673 q^{67} +0.0763889 q^{68} +1.55695 q^{69} +2.58091 q^{70} +13.9019 q^{71} +4.76882 q^{72} +6.01462 q^{73} -2.95205 q^{74} +3.56098 q^{75} +0.0810106 q^{76} -14.1679 q^{78} -15.6409 q^{79} +7.27404 q^{80} -11.2175 q^{81} +1.29180 q^{82} -4.37573 q^{83} +0.0302106 q^{84} +10.0181 q^{85} -11.3077 q^{86} +2.53651 q^{87} +15.3437 q^{89} +4.33650 q^{90} -4.64706 q^{91} +0.0100500 q^{92} +2.82941 q^{93} -8.42155 q^{94} +10.6242 q^{95} +0.170894 q^{96} +2.41124 q^{97} -1.40927 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9}+O(q^{10}) \) 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9	\( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + 6 q^{10} + 9 q^{12} - 6 q^{13} + q^{14} + 11 q^{15} + q^{16} - 5 q^{17} + 8 q^{18} - 13 q^{19} + 23 q^{20} + 4 q^{21} + 16 q^{23} - 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} + 7 q^{28} + 9 q^{29} - 36 q^{30} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} + 13 q^{39} + 5 q^{40} - 10 q^{41} - q^{42} - 4 q^{43} + 35 q^{45} + 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} + 6 q^{50} + 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} + 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} + 14 q^{63} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} - 35 q^{72} - 25 q^{73} + 33 q^{74} - 13 q^{75} + 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} - 25 q^{83} + 9 q^{84} + 3 q^{85} + 4 q^{86} - 36 q^{87} + 37 q^{89} - 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} - 42 q^{94} + 21 q^{95} - 6 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) \) 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9 + 6 * q^10 + 9 * q^12 - 6 * q^13 + q^14 + 11 * q^15 + q^16 - 5 * q^17 + 8 * q^18 - 13 * q^19 + 23 * q^20 + 4 * q^21 + 16 * q^23 - 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 + 7 * q^28 + 9 * q^29 - 36 * q^30 + 9 * q^31 + 16 * q^32 - 12 * q^34 + 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 + 13 * q^39 + 5 * q^40 - 10 * q^41 - q^42 - 4 * q^43 + 35 * q^45 + 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 + 6 * q^50 + 13 * q^51 - 41 * q^52 + 37 * q^53 + 30 * q^54 + 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 + 19 * q^61 - 18 * q^62 + 14 * q^63 - 4 * q^64 - 4 * q^65 - 19 * q^67 + 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 - 35 * q^72 - 25 * q^73 + 33 * q^74 - 13 * q^75 + 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 - 25 * q^83 + 9 * q^84 + 3 * q^85 + 4 * q^86 - 36 * q^87 + 37 * q^89 - 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 - 42 * q^94 + 21 * q^95 - 6 * q^96 + 15 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more