LMFDB - Embedded newform 1911.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,2,Mod(1,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, 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("1911.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1911.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:	$15.2594118263$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.375116.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 7x^{2} + 2x - 2 $ x^5 - x^4 - 6x^3 + 7x^2 + 2*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.17362$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32155 q^{2} +1.00000 q^{3} -0.253495 q^{4} -2.02568 q^{5} +1.32155 q^{6} -2.97812 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.32155 q^{2} +1.00000 q^{3} -0.253495 q^{4} -2.02568 q^{5} +1.32155 q^{6} -2.97812 q^{8} +1.00000 q^{9} -2.67705 q^{10} -0.0680592 q^{11} -0.253495 q^{12} +1.00000 q^{13} -2.02568 q^{15} -3.42875 q^{16} +6.04617 q^{17} +1.32155 q^{18} +5.34724 q^{19} +0.513501 q^{20} -0.0899440 q^{22} +5.55316 q^{23} -2.97812 q^{24} -0.896603 q^{25} +1.32155 q^{26} +1.00000 q^{27} -2.38258 q^{29} -2.67705 q^{30} +3.17526 q^{31} +1.42495 q^{32} -0.0680592 q^{33} +7.99035 q^{34} -0.253495 q^{36} +5.29448 q^{37} +7.06666 q^{38} +1.00000 q^{39} +6.03272 q^{40} +2.36912 q^{41} -0.103923 q^{43} +0.0172527 q^{44} -2.02568 q^{45} +7.33881 q^{46} +4.06155 q^{47} -3.42875 q^{48} -1.18491 q^{50} +6.04617 q^{51} -0.253495 q^{52} -7.95968 q^{53} +1.32155 q^{54} +0.137867 q^{55} +5.34724 q^{57} -3.14870 q^{58} +6.88298 q^{59} +0.513501 q^{60} +8.04617 q^{61} +4.19627 q^{62} +8.74065 q^{64} -2.02568 q^{65} -0.0899440 q^{66} -4.66740 q^{67} -1.53267 q^{68} +5.55316 q^{69} +0.522494 q^{71} -2.97812 q^{72} +6.87611 q^{73} +6.99694 q^{74} -0.896603 q^{75} -1.35550 q^{76} +1.32155 q^{78} +11.7336 q^{79} +6.94557 q^{80} +1.00000 q^{81} +3.13092 q^{82} -12.8208 q^{83} -12.2476 q^{85} -0.137340 q^{86} -2.38258 q^{87} +0.202688 q^{88} +4.97951 q^{89} -2.67705 q^{90} -1.40770 q^{92} +3.17526 q^{93} +5.36756 q^{94} -10.8318 q^{95} +1.42495 q^{96} +5.09702 q^{97} -0.0680592 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 6 * q^4 + 3 * q^5 + 3 * q^8 + 5 * q^9	$ 5 q + 5 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{8} + 5 q^{9} + 2 q^{10} - q^{11} + 6 q^{12} + 5 q^{13} + 3 q^{15} + 13 q^{17} + 7 q^{19} + 13 q^{20} - 19 q^{22} - 4 q^{23} + 3 q^{24} + 16 q^{25} + 5 q^{27} - 12 q^{29} + 2 q^{30} + 6 q^{31} + 21 q^{32} - q^{33} + 7 q^{34} + 6 q^{36} + 11 q^{37} + 14 q^{38} + 5 q^{39} + 11 q^{40} + 10 q^{41} + 10 q^{43} - 29 q^{44} + 3 q^{45} + q^{46} - 4 q^{47} - 29 q^{50} + 13 q^{51} + 6 q^{52} - 9 q^{53} - 12 q^{55} + 7 q^{57} + 34 q^{58} + 7 q^{59} + 13 q^{60} + 23 q^{61} - 24 q^{62} - 13 q^{64} + 3 q^{65} - 19 q^{66} + 25 q^{67} + 20 q^{68} - 4 q^{69} - 27 q^{71} + 3 q^{72} + 18 q^{73} + 15 q^{74} + 16 q^{75} + 2 q^{76} + 8 q^{79} + 41 q^{80} + 5 q^{81} + 26 q^{82} + 12 q^{83} + 10 q^{85} - 19 q^{86} - 12 q^{87} - 36 q^{88} + 29 q^{89} + 2 q^{90} - 50 q^{92} + 6 q^{93} - 2 q^{94} - 33 q^{95} + 21 q^{96} + 13 q^{97} - q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 6 * q^4 + 3 * q^5 + 3 * q^8 + 5 * q^9 + 2 * q^10 - q^11 + 6 * q^12 + 5 * q^13 + 3 * q^15 + 13 * q^17 + 7 * q^19 + 13 * q^20 - 19 * q^22 - 4 * q^23 + 3 * q^24 + 16 * q^25 + 5 * q^27 - 12 * q^29 + 2 * q^30 + 6 * q^31 + 21 * q^32 - q^33 + 7 * q^34 + 6 * q^36 + 11 * q^37 + 14 * q^38 + 5 * q^39 + 11 * q^40 + 10 * q^41 + 10 * q^43 - 29 * q^44 + 3 * q^45 + q^46 - 4 * q^47 - 29 * q^50 + 13 * q^51 + 6 * q^52 - 9 * q^53 - 12 * q^55 + 7 * q^57 + 34 * q^58 + 7 * q^59 + 13 * q^60 + 23 * q^61 - 24 * q^62 - 13 * q^64 + 3 * q^65 - 19 * q^66 + 25 * q^67 + 20 * q^68 - 4 * q^69 - 27 * q^71 + 3 * q^72 + 18 * q^73 + 15 * q^74 + 16 * q^75 + 2 * q^76 + 8 * q^79 + 41 * q^80 + 5 * q^81 + 26 * q^82 + 12 * q^83 + 10 * q^85 - 19 * q^86 - 12 * q^87 - 36 * q^88 + 29 * q^89 + 2 * q^90 - 50 * q^92 + 6 * q^93 - 2 * q^94 - 33 * q^95 + 21 * q^96 + 13 * q^97 - 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 $

$2$	1.32155	0.934480	0.467240	−	0.884131i	$-0.345248\pi$
$2$	1.32155	0.934480	0.467240	+	0.884131i	$0.345248\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.253495	−0.126747
$5$	−2.02568	−0.905914	−0.452957	−	0.891532i	$-0.649631\pi$
$5$	−2.02568	−0.905914	−0.452957	+	0.891532i	$0.649631\pi$
$6$	1.32155	0.539522
$7$	0	0
$7$	0	0
$8$	−2.97812	−1.05292
$9$	1.00000	0.333333
$10$	−2.67705	−0.846558
$11$	−0.0680592	−0.0205206	−0.0102603	−	0.999947i	$-0.503266\pi$
$11$	−0.0680592	−0.0205206	−0.0102603	+	0.999947i	$0.503266\pi$
$12$	−0.253495	−0.0731777
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.02568	−0.523029
$16$	−3.42875	−0.857188
$17$	6.04617	1.46641	0.733206	−	0.680006i	$-0.238023\pi$
$17$	6.04617	1.46641	0.733206	+	0.680006i	$0.238023\pi$
$18$	1.32155	0.311493
$19$	5.34724	1.22674	0.613370	−	0.789795i	$-0.289813\pi$
$19$	5.34724	1.22674	0.613370	+	0.789795i	$0.289813\pi$
$20$	0.513501	0.114822
$21$	0	0
$22$	−0.0899440	−0.0191761
$23$	5.55316	1.15791	0.578957	−	0.815358i	$-0.303460\pi$
$23$	5.55316	1.15791	0.578957	+	0.815358i	$0.303460\pi$
$24$	−2.97812	−0.607905
$25$	−0.896603	−0.179321
$26$	1.32155	0.259178
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.38258	−0.442433	−0.221217	−	0.975225i	$-0.571003\pi$
$29$	−2.38258	−0.442433	−0.221217	+	0.975225i	$0.571003\pi$
$30$	−2.67705	−0.488760
$31$	3.17526	0.570293	0.285146	−	0.958484i	$-0.407958\pi$
$31$	3.17526	0.570293	0.285146	+	0.958484i	$0.407958\pi$
$32$	1.42495	0.251898
$33$	−0.0680592	−0.0118476
$34$	7.99035	1.37033
$35$	0	0
$36$	−0.253495	−0.0422491
$37$	5.29448	0.870406	0.435203	−	0.900332i	$-0.356677\pi$
$37$	5.29448	0.870406	0.435203	+	0.900332i	$0.356677\pi$
$38$	7.06666	1.14636
$39$	1.00000	0.160128
$40$	6.03272	0.953857
$41$	2.36912	0.369995	0.184997	−	0.982739i	$-0.440772\pi$
$41$	2.36912	0.369995	0.184997	+	0.982739i	$0.440772\pi$
$42$	0	0
$43$	−0.103923	−0.0158482	−0.00792408	−	0.999969i	$-0.502522\pi$
$43$	−0.103923	−0.0158482	−0.00792408	+	0.999969i	$0.502522\pi$
$44$	0.0172527	0.00260094
$45$	−2.02568	−0.301971
$46$	7.33881	1.08205
$47$	4.06155	0.592438	0.296219	−	0.955120i	$-0.404274\pi$
$47$	4.06155	0.592438	0.296219	+	0.955120i	$0.404274\pi$
$48$	−3.42875	−0.494898
$49$	0	0
$50$	−1.18491	−0.167571
$51$	6.04617	0.846634
$52$	−0.253495	−0.0351534
$53$	−7.95968	−1.09335	−0.546673	−	0.837346i	$-0.684106\pi$
$53$	−7.95968	−1.09335	−0.546673	+	0.837346i	$0.684106\pi$
$54$	1.32155	0.179841
$55$	0.137867	0.0185899
$56$	0	0
$57$	5.34724	0.708259
$58$	−3.14870	−0.413445
$59$	6.88298	0.896087	0.448044	−	0.894012i	$-0.352121\pi$
$59$	6.88298	0.896087	0.448044	+	0.894012i	$0.352121\pi$
$60$	0.513501	0.0662926
$61$	8.04617	1.03021	0.515104	−	0.857128i	$-0.327753\pi$
$61$	8.04617	1.03021	0.515104	+	0.857128i	$0.327753\pi$
$62$	4.19627	0.532927
$63$	0	0
$64$	8.74065	1.09258
$65$	−2.02568	−0.251255
$66$	−0.0899440	−0.0110713
$67$	−4.66740	−0.570213	−0.285107	−	0.958496i	$-0.592029\pi$
$67$	−4.66740	−0.570213	−0.285107	+	0.958496i	$0.592029\pi$
$68$	−1.53267	−0.185864
$69$	5.55316	0.668522
$70$	0	0
$71$	0.522494	0.0620087	0.0310043	−	0.999519i	$-0.490129\pi$
$71$	0.522494	0.0620087	0.0310043	+	0.999519i	$0.490129\pi$
$72$	−2.97812	−0.350974
$73$	6.87611	0.804788	0.402394	−	0.915467i	$-0.368178\pi$
$73$	6.87611	0.804788	0.402394	+	0.915467i	$0.368178\pi$
$74$	6.99694	0.813377
$75$	−0.896603	−0.103531
$76$	−1.35550	−0.155486
$77$	0	0
$78$	1.32155	0.149637
$79$	11.7336	1.32013	0.660067	−	0.751206i	$-0.270528\pi$
$79$	11.7336	1.32013	0.660067	+	0.751206i	$0.270528\pi$
$80$	6.94557	0.776538
$81$	1.00000	0.111111
$82$	3.13092	0.345753
$83$	−12.8208	−1.40726	−0.703631	−	0.710565i	$-0.748439\pi$
$83$	−12.8208	−1.40726	−0.703631	+	0.710565i	$0.748439\pi$
$84$	0	0
$85$	−12.2476	−1.32844
$86$	−0.137340	−0.0148098
$87$	−2.38258	−0.255439
$88$	0.202688	0.0216066
$89$	4.97951	0.527827	0.263913	−	0.964546i	$-0.414987\pi$
$89$	4.97951	0.527827	0.263913	+	0.964546i	$0.414987\pi$
$90$	−2.67705	−0.282186
$91$	0	0
$92$	−1.40770	−0.146763
$93$	3.17526	0.329259
$94$	5.36756	0.553621
$95$	−10.8318	−1.11132
$96$	1.42495	0.145433
$97$	5.09702	0.517524	0.258762	−	0.965941i	$-0.416686\pi$
$97$	5.09702	0.517524	0.258762	+	0.965941i	$0.416686\pi$
$98$	0	0
$99$	−0.0680592	−0.00684021
$100$	0.227284	0.0227284
$101$	−19.1526	−1.90576	−0.952879	−	0.303350i	$-0.901895\pi$
$101$	−19.1526	−1.90576	−0.952879	+	0.303350i	$0.901895\pi$
$102$	7.99035	0.791162
$103$	16.5999	1.63563	0.817817	−	0.575479i	$-0.195184\pi$
$103$	16.5999	1.63563	0.817817	+	0.575479i	$0.195184\pi$
$104$	−2.97812	−0.292028
$105$	0	0
$106$	−10.5191	−1.02171
$107$	−12.9073	−1.24779	−0.623896	−	0.781507i	$-0.714451\pi$
$107$	−12.9073	−1.24779	−0.623896	+	0.781507i	$0.714451\pi$
$108$	−0.253495	−0.0243926
$109$	−20.3143	−1.94576	−0.972879	−	0.231315i	$-0.925697\pi$
$109$	−20.3143	−1.94576	−0.972879	+	0.231315i	$0.925697\pi$
$110$	0.182198	0.0173719
$111$	5.29448	0.502529
$112$	0	0
$113$	−8.38961	−0.789228	−0.394614	−	0.918847i	$-0.629122\pi$
$113$	−8.38961	−0.789228	−0.394614	+	0.918847i	$0.629122\pi$
$114$	7.06666	0.661854
$115$	−11.2490	−1.04897
$116$	0.603971	0.0560773
$117$	1.00000	0.0924500
$118$	9.09623	0.837376
$119$	0	0
$120$	6.03272	0.550710
$121$	−10.9954	−0.999579
$122$	10.6335	0.962708
$123$	2.36912	0.213617
$124$	−0.804911	−0.0722831
$125$	11.9447	1.06836
$126$	0	0
$127$	−10.0551	−0.892245	−0.446122	−	0.894972i	$-0.647195\pi$
$127$	−10.0551	−0.892245	−0.446122	+	0.894972i	$0.647195\pi$
$128$	8.70134	0.769097
$129$	−0.103923	−0.00914994
$130$	−2.67705	−0.234793
$131$	7.23314	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$131$	7.23314	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$132$	0.0172527	0.00150165
$133$	0	0
$134$	−6.16822	−0.532853
$135$	−2.02568	−0.174343
$136$	−18.0062	−1.54402
$137$	20.6391	1.76332	0.881660	−	0.471885i	$-0.156426\pi$
$137$	20.6391	1.76332	0.881660	+	0.471885i	$0.156426\pi$
$138$	7.33881	0.624721
$139$	−15.2921	−1.29706	−0.648528	−	0.761191i	$-0.724615\pi$
$139$	−15.2921	−1.29706	−0.648528	+	0.761191i	$0.724615\pi$
$140$	0	0
$141$	4.06155	0.342044
$142$	0.690504	0.0579458
$143$	−0.0680592	−0.00569140
$144$	−3.42875	−0.285729
$145$	4.82635	0.400806
$146$	9.08715	0.752058
$147$	0	0
$148$	−1.34212	−0.110322
$149$	8.68409	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$149$	8.68409	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$150$	−1.18491	−0.0967474
$151$	13.5314	1.10117	0.550585	−	0.834779i	$-0.314405\pi$
$151$	13.5314	1.10117	0.550585	+	0.834779i	$0.314405\pi$
$152$	−15.9247	−1.29166
$153$	6.04617	0.488804
$154$	0	0
$155$	−6.43207	−0.516636
$156$	−0.253495	−0.0202958
$157$	20.9079	1.66863	0.834317	−	0.551285i	$-0.185862\pi$
$157$	20.9079	1.66863	0.834317	+	0.551285i	$0.185862\pi$
$158$	15.5066	1.23364
$159$	−7.95968	−0.631243
$160$	−2.88650	−0.228198
$161$	0	0
$162$	1.32155	0.103831
$163$	22.8665	1.79104	0.895520	−	0.445022i	$-0.146804\pi$
$163$	22.8665	1.79104	0.895520	+	0.445022i	$0.146804\pi$
$164$	−0.600560	−0.0468959
$165$	0.137867	0.0107329
$166$	−16.9433	−1.31506
$167$	−12.7013	−0.982859	−0.491430	−	0.870917i	$-0.663526\pi$
$167$	−12.7013	−0.982859	−0.491430	+	0.870917i	$0.663526\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−16.1859	−1.24140
$171$	5.34724	0.408914
$172$	0.0263440	0.00200871
$173$	−14.5930	−1.10948	−0.554741	−	0.832023i	$-0.687183\pi$
$173$	−14.5930	−1.10948	−0.554741	+	0.832023i	$0.687183\pi$
$174$	−3.14870	−0.238703
$175$	0	0
$176$	0.233358	0.0175900
$177$	6.88298	0.517356
$178$	6.58069	0.493244
$179$	6.28687	0.469903	0.234951	−	0.972007i	$-0.424507\pi$
$179$	6.28687	0.469903	0.234951	+	0.972007i	$0.424507\pi$
$180$	0.513501	0.0382741
$181$	−15.1782	−1.12819	−0.564093	−	0.825711i	$-0.690774\pi$
$181$	−15.1782	−1.12819	−0.564093	+	0.825711i	$0.690774\pi$
$182$	0	0
$183$	8.04617	0.594790
$184$	−16.5380	−1.21919
$185$	−10.7249	−0.788513
$186$	4.19627	0.307686
$187$	−0.411498	−0.0300917
$188$	−1.02958	−0.0750900
$189$	0	0
$190$	−14.3148	−1.03851
$191$	−20.9458	−1.51559	−0.757794	−	0.652494i	$-0.773723\pi$
$191$	−20.9458	−1.51559	−0.757794	+	0.652494i	$0.773723\pi$
$192$	8.74065	0.630802
$193$	3.22834	0.232381	0.116190	−	0.993227i	$-0.462932\pi$
$193$	3.22834	0.232381	0.116190	+	0.993227i	$0.462932\pi$
$194$	6.73598	0.483615
$195$	−2.02568	−0.145062
$196$	0	0
$197$	−11.2173	−0.799199	−0.399599	−	0.916690i	$-0.630851\pi$
$197$	−11.2173	−0.799199	−0.399599	+	0.916690i	$0.630851\pi$
$198$	−0.0899440	−0.00639204
$199$	−15.3611	−1.08892	−0.544461	−	0.838786i	$-0.683266\pi$
$199$	−15.3611	−1.08892	−0.544461	+	0.838786i	$0.683266\pi$
$200$	2.67019	0.188811
$201$	−4.66740	−0.329213
$202$	−25.3112	−1.78089
$203$	0	0
$204$	−1.53267	−0.107309
$205$	−4.79910	−0.335183
$206$	21.9376	1.52847
$207$	5.55316	0.385972
$208$	−3.42875	−0.237741
$209$	−0.363929	−0.0251735
$210$	0	0
$211$	17.9977	1.23902	0.619508	−	0.784991i	$-0.287332\pi$
$211$	17.9977	1.23902	0.619508	+	0.784991i	$0.287332\pi$
$212$	2.01774	0.138579
$213$	0.522494	0.0358007
$214$	−17.0577	−1.16604
$215$	0.210516	0.0143571
$216$	−2.97812	−0.202635
$217$	0	0
$218$	−26.8465	−1.81827
$219$	6.87611	0.464645
$220$	−0.0349485	−0.00235622
$221$	6.04617	0.406710
$222$	6.99694	0.469604
$223$	−2.97864	−0.199465	−0.0997323	−	0.995014i	$-0.531799\pi$
$223$	−2.97864	−0.199465	−0.0997323	+	0.995014i	$0.531799\pi$
$224$	0	0
$225$	−0.896603	−0.0597735
$226$	−11.0873	−0.737518
$227$	11.8511	0.786587	0.393293	−	0.919413i	$-0.371336\pi$
$227$	11.8511	0.786587	0.393293	+	0.919413i	$0.371336\pi$
$228$	−1.35550	−0.0897700
$229$	−6.44618	−0.425975	−0.212988	−	0.977055i	$-0.568319\pi$
$229$	−6.44618	−0.425975	−0.212988	+	0.977055i	$0.568319\pi$
$230$	−14.8661	−0.980242
$231$	0	0
$232$	7.09559	0.465848
$233$	7.15529	0.468759	0.234379	−	0.972145i	$-0.424694\pi$
$233$	7.15529	0.468759	0.234379	+	0.972145i	$0.424694\pi$
$234$	1.32155	0.0863927
$235$	−8.22741	−0.536697
$236$	−1.74480	−0.113577
$237$	11.7336	0.762180
$238$	0	0
$239$	0.280704	0.0181573	0.00907863	−	0.999959i	$-0.497110\pi$
$239$	0.280704	0.0181573	0.00907863	+	0.999959i	$0.497110\pi$
$240$	6.94557	0.448334
$241$	−7.31950	−0.471490	−0.235745	−	0.971815i	$-0.575753\pi$
$241$	−7.31950	−0.471490	−0.235745	+	0.971815i	$0.575753\pi$
$242$	−14.5310	−0.934086
$243$	1.00000	0.0641500
$244$	−2.03966	−0.130576
$245$	0	0
$246$	3.13092	0.199620
$247$	5.34724	0.340237
$248$	−9.45628	−0.600474
$249$	−12.8208	−0.812483
$250$	15.7855	0.998363
$251$	−0.159014	−0.0100369	−0.00501843	−	0.999987i	$-0.501597\pi$
$251$	−0.159014	−0.0100369	−0.00501843	+	0.999987i	$0.501597\pi$
$252$	0	0
$253$	−0.377944	−0.0237611
$254$	−13.2883	−0.833785
$255$	−12.2476	−0.766977
$256$	−5.98201	−0.373876
$257$	2.93243	0.182920	0.0914598	−	0.995809i	$-0.470847\pi$
$257$	2.93243	0.182920	0.0914598	+	0.995809i	$0.470847\pi$
$258$	−0.137340	−0.00855043
$259$	0	0
$260$	0.513501	0.0318459
$261$	−2.38258	−0.147478
$262$	9.55898	0.590556
$263$	−5.65775	−0.348872	−0.174436	−	0.984669i	$-0.555810\pi$
$263$	−5.65775	−0.348872	−0.174436	+	0.984669i	$0.555810\pi$
$264$	0.202688	0.0124746
$265$	16.1238	0.990477
$266$	0	0
$267$	4.97951	0.304741
$268$	1.18316	0.0722731
$269$	28.9030	1.76225	0.881123	−	0.472887i	$-0.156788\pi$
$269$	28.9030	1.76225	0.881123	+	0.472887i	$0.156788\pi$
$270$	−2.67705	−0.162920
$271$	14.3474	0.871541	0.435770	−	0.900058i	$-0.356476\pi$
$271$	14.3474	0.871541	0.435770	+	0.900058i	$0.356476\pi$
$272$	−20.7308	−1.25699
$273$	0	0
$274$	27.2757	1.64779
$275$	0.0610221	0.00367977
$276$	−1.40770	−0.0847335
$277$	15.5811	0.936178	0.468089	−	0.883681i	$-0.344943\pi$
$277$	15.5811	0.936178	0.468089	+	0.883681i	$0.344943\pi$
$278$	−20.2093	−1.21207
$279$	3.17526	0.190098
$280$	0	0
$281$	−11.6351	−0.694093	−0.347047	−	0.937848i	$-0.612815\pi$
$281$	−11.6351	−0.694093	−0.347047	+	0.937848i	$0.612815\pi$
$282$	5.36756	0.319633
$283$	−29.9742	−1.78178	−0.890890	−	0.454219i	$-0.849918\pi$
$283$	−29.9742	−1.78178	−0.890890	+	0.454219i	$0.849918\pi$
$284$	−0.132450	−0.00785944
$285$	−10.8318	−0.641621
$286$	−0.0899440	−0.00531850
$287$	0	0
$288$	1.42495	0.0839660
$289$	19.5562	1.15037
$290$	6.37828	0.374545
$291$	5.09702	0.298792
$292$	−1.74306	−0.102005
$293$	−3.83767	−0.224199	−0.112100	−	0.993697i	$-0.535758\pi$
$293$	−3.83767	−0.224199	−0.112100	+	0.993697i	$0.535758\pi$
$294$	0	0
$295$	−13.9427	−0.811778
$296$	−15.7676	−0.916471
$297$	−0.0680592	−0.00394920
$298$	11.4765	0.664815
$299$	5.55316	0.321148
$300$	0.227284	0.0131223
$301$	0	0
$302$	17.8825	1.02902
$303$	−19.1526	−1.10029
$304$	−18.3343	−1.05155
$305$	−16.2990	−0.933279
$306$	7.99035	0.456778
$307$	−0.566409	−0.0323266	−0.0161633	−	0.999869i	$-0.505145\pi$
$307$	−0.566409	−0.0323266	−0.0161633	+	0.999869i	$0.505145\pi$
$308$	0	0
$309$	16.5999	0.944333
$310$	−8.50032	−0.482786
$311$	−20.3044	−1.15136	−0.575680	−	0.817675i	$-0.695262\pi$
$311$	−20.3044	−1.15136	−0.575680	+	0.817675i	$0.695262\pi$
$312$	−2.97812	−0.168603
$313$	−12.8487	−0.726252	−0.363126	−	0.931740i	$-0.618291\pi$
$313$	−12.8487	−0.726252	−0.363126	+	0.931740i	$0.618291\pi$
$314$	27.6310	1.55930
$315$	0	0
$316$	−2.97441	−0.167324
$317$	−33.2621	−1.86819	−0.934094	−	0.357027i	$-0.883790\pi$
$317$	−33.2621	−1.86819	−0.934094	+	0.357027i	$0.883790\pi$
$318$	−10.5191	−0.589884
$319$	0.162156	0.00907901
$320$	−17.7058	−0.989784
$321$	−12.9073	−0.720413
$322$	0	0
$323$	32.3303	1.79891
$324$	−0.253495	−0.0140830
$325$	−0.896603	−0.0497346
$326$	30.2193	1.67369
$327$	−20.3143	−1.12338
$328$	−7.05552	−0.389576
$329$	0	0
$330$	0.182198	0.0100297
$331$	29.6302	1.62862	0.814312	−	0.580427i	$-0.197114\pi$
$331$	29.6302	1.62862	0.814312	+	0.580427i	$0.197114\pi$
$332$	3.25000	0.178367
$333$	5.29448	0.290135
$334$	−16.7855	−0.918462
$335$	9.45467	0.516564
$336$	0	0
$337$	23.6462	1.28809	0.644044	−	0.764988i	$-0.277255\pi$
$337$	23.6462	1.28809	0.644044	+	0.764988i	$0.277255\pi$
$338$	1.32155	0.0718831
$339$	−8.38961	−0.455661
$340$	3.10471	0.168377
$341$	−0.216105	−0.0117028
$342$	7.06666	0.382121
$343$	0	0
$344$	0.309496	0.0166869
$345$	−11.2490	−0.605624
$346$	−19.2854	−1.03679
$347$	−2.00651	−0.107715	−0.0538576	−	0.998549i	$-0.517152\pi$
$347$	−2.00651	−0.107715	−0.0538576	+	0.998549i	$0.517152\pi$
$348$	0.603971	0.0323762
$349$	26.7671	1.43281	0.716406	−	0.697684i	$-0.245786\pi$
$349$	26.7671	1.43281	0.716406	+	0.697684i	$0.245786\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−0.0969811	−0.00516911
$353$	13.7846	0.733678	0.366839	−	0.930284i	$-0.380440\pi$
$353$	13.7846	0.733678	0.366839	+	0.930284i	$0.380440\pi$
$354$	9.09623	0.483459
$355$	−1.05841	−0.0561745
$356$	−1.26228	−0.0669007
$357$	0	0
$358$	8.30844	0.439115
$359$	−0.397741	−0.0209920	−0.0104960	−	0.999945i	$-0.503341\pi$
$359$	−0.397741	−0.0209920	−0.0104960	+	0.999945i	$0.503341\pi$
$360$	6.03272	0.317952
$361$	9.59296	0.504893
$362$	−20.0588	−1.05427
$363$	−10.9954	−0.577107
$364$	0	0
$365$	−13.9288	−0.729068
$366$	10.6335	0.555820
$367$	−16.7443	−0.874044	−0.437022	−	0.899451i	$-0.643967\pi$
$367$	−16.7443	−0.874044	−0.437022	+	0.899451i	$0.643967\pi$
$368$	−19.0404	−0.992550
$369$	2.36912	0.123332
$370$	−14.1736	−0.736849
$371$	0	0
$372$	−0.804911	−0.0417327
$373$	10.4258	0.539829	0.269915	−	0.962884i	$-0.413005\pi$
$373$	10.4258	0.539829	0.269915	+	0.962884i	$0.413005\pi$
$374$	−0.543817	−0.0281201
$375$	11.9447	0.616819
$376$	−12.0958	−0.623791
$377$	−2.38258	−0.122709
$378$	0	0
$379$	25.6729	1.31873	0.659365	−	0.751823i	$-0.270825\pi$
$379$	25.6729	1.31873	0.659365	+	0.751823i	$0.270825\pi$
$380$	2.74581	0.140857
$381$	−10.0551	−0.515138
$382$	−27.6811	−1.41629
$383$	−6.80495	−0.347717	−0.173858	−	0.984771i	$-0.555623\pi$
$383$	−6.80495	−0.347717	−0.173858	+	0.984771i	$0.555623\pi$
$384$	8.70134	0.444038
$385$	0	0
$386$	4.26642	0.217155
$387$	−0.103923	−0.00528272
$388$	−1.29207	−0.0655948
$389$	−36.7570	−1.86365	−0.931827	−	0.362903i	$-0.881786\pi$
$389$	−36.7570	−1.86365	−0.931827	+	0.362903i	$0.881786\pi$
$390$	−2.67705	−0.135558
$391$	33.5754	1.69798
$392$	0	0
$393$	7.23314	0.364863
$394$	−14.8243	−0.746835
$395$	−23.7686	−1.19593
$396$	0.0172527	0.000866979 0
$397$	−22.6998	−1.13927	−0.569635	−	0.821898i	$-0.692916\pi$
$397$	−22.6998	−1.13927	−0.569635	+	0.821898i	$0.692916\pi$
$398$	−20.3006	−1.01758
$399$	0	0
$400$	3.07423	0.153711
$401$	17.4714	0.872482	0.436241	−	0.899830i	$-0.356309\pi$
$401$	17.4714	0.872482	0.436241	+	0.899830i	$0.356309\pi$
$402$	−6.16822	−0.307643
$403$	3.17526	0.158171
$404$	4.85509	0.241550
$405$	−2.02568	−0.100657
$406$	0	0
$407$	−0.360338	−0.0178613
$408$	−18.0062	−0.891440
$409$	30.4054	1.50345	0.751726	−	0.659476i	$-0.229222\pi$
$409$	30.4054	1.50345	0.751726	+	0.659476i	$0.229222\pi$
$410$	−6.34226	−0.313222
$411$	20.6391	1.01805
$412$	−4.20798	−0.207312
$413$	0	0
$414$	7.33881	0.360683
$415$	25.9708	1.27486
$416$	1.42495	0.0698640
$417$	−15.2921	−0.748856
$418$	−0.480952	−0.0235241
$419$	26.1442	1.27723	0.638615	−	0.769526i	$-0.279508\pi$
$419$	26.1442	1.27723	0.638615	+	0.769526i	$0.279508\pi$
$420$	0	0
$421$	−22.0532	−1.07481	−0.537404	−	0.843325i	$-0.680595\pi$
$421$	−22.0532	−1.07481	−0.537404	+	0.843325i	$0.680595\pi$
$422$	23.7850	1.15783
$423$	4.06155	0.197479
$424$	23.7048	1.15121
$425$	−5.42102	−0.262958
$426$	0.690504	0.0334550
$427$	0	0
$428$	3.27193	0.158154
$429$	−0.0680592	−0.00328593
$430$	0.278208	0.0134164
$431$	−23.9824	−1.15519	−0.577597	−	0.816322i	$-0.696009\pi$
$431$	−23.9824	−1.15519	−0.577597	+	0.816322i	$0.696009\pi$
$432$	−3.42875	−0.164966
$433$	21.7131	1.04347	0.521733	−	0.853109i	$-0.325286\pi$
$433$	21.7131	1.04347	0.521733	+	0.853109i	$0.325286\pi$
$434$	0	0
$435$	4.82635	0.231406
$436$	5.14957	0.246620
$437$	29.6941	1.42046
$438$	9.08715	0.434201
$439$	−9.12988	−0.435746	−0.217873	−	0.975977i	$-0.569912\pi$
$439$	−9.12988	−0.435746	−0.217873	+	0.975977i	$0.569912\pi$
$440$	−0.410582	−0.0195737
$441$	0	0
$442$	7.99035	0.380062
$443$	4.47401	0.212567	0.106283	−	0.994336i	$-0.466105\pi$
$443$	4.47401	0.212567	0.106283	+	0.994336i	$0.466105\pi$
$444$	−1.34212	−0.0636943
$445$	−10.0869	−0.478166
$446$	−3.93644	−0.186396
$447$	8.68409	0.410743
$448$	0	0
$449$	−18.9055	−0.892207	−0.446103	−	0.894981i	$-0.647189\pi$
$449$	−18.9055	−0.892207	−0.446103	+	0.894981i	$0.647189\pi$
$450$	−1.18491	−0.0558572
$451$	−0.161241	−0.00759253
$452$	2.12672	0.100033
$453$	13.5314	0.635761
$454$	15.6619	0.735049
$455$	0	0
$456$	−15.9247	−0.745742
$457$	−25.9288	−1.21290	−0.606449	−	0.795122i	$-0.707407\pi$
$457$	−25.9288	−1.21290	−0.606449	+	0.795122i	$0.707407\pi$
$458$	−8.51897	−0.398065
$459$	6.04617	0.282211
$460$	2.85155	0.132954
$461$	−30.8552	−1.43707	−0.718535	−	0.695491i	$-0.755187\pi$
$461$	−30.8552	−1.43707	−0.718535	+	0.695491i	$0.755187\pi$
$462$	0	0
$463$	33.5654	1.55992	0.779958	−	0.625832i	$-0.215240\pi$
$463$	33.5654	1.55992	0.779958	+	0.625832i	$0.215240\pi$
$464$	8.16926	0.379248
$465$	−6.43207	−0.298280
$466$	9.45611	0.438046
$467$	0.00156731	7.25266e−5 0	3.62633e−5	−	1.00000i	$-0.499988\pi$
$467$	0.00156731	7.25266e−5 0	3.62633e−5		1.00000i	$0.499988\pi$
$468$	−0.253495	−0.0117178
$469$	0	0
$470$	−10.8730	−0.501533
$471$	20.9079	0.963386
$472$	−20.4983	−0.943511
$473$	0.00707294	0.000325214 0
$474$	15.5066	0.712242
$475$	−4.79435	−0.219980
$476$	0	0
$477$	−7.95968	−0.364449
$478$	0.370966	0.0169676
$479$	11.4560	0.523437	0.261718	−	0.965144i	$-0.415711\pi$
$479$	11.4560	0.523437	0.261718	+	0.965144i	$0.415711\pi$
$480$	−2.88650	−0.131750
$481$	5.29448	0.241407
$482$	−9.67311	−0.440598
$483$	0	0
$484$	2.78727	0.126694
$485$	−10.3249	−0.468832
$486$	1.32155	0.0599469
$487$	15.5551	0.704868	0.352434	−	0.935837i	$-0.385354\pi$
$487$	15.5551	0.704868	0.352434	+	0.935837i	$0.385354\pi$
$488$	−23.9624	−1.08473
$489$	22.8665	1.03406
$490$	0	0
$491$	27.2263	1.22871	0.614354	−	0.789031i	$-0.289417\pi$
$491$	27.2263	1.22871	0.614354	+	0.789031i	$0.289417\pi$
$492$	−0.600560	−0.0270754
$493$	−14.4055	−0.648790
$494$	7.06666	0.317944
$495$	0.137867	0.00619664
$496$	−10.8872	−0.488848
$497$	0	0
$498$	−16.9433	−0.759249
$499$	−29.0216	−1.29918	−0.649592	−	0.760283i	$-0.725060\pi$
$499$	−29.0216	−1.29918	−0.649592	+	0.760283i	$0.725060\pi$
$500$	−3.02791	−0.135412
$501$	−12.7013	−0.567454
$502$	−0.210145	−0.00937924
$503$	−1.79018	−0.0798202	−0.0399101	−	0.999203i	$-0.512707\pi$
$503$	−1.79018	−0.0798202	−0.0399101	+	0.999203i	$0.512707\pi$
$504$	0	0
$505$	38.7972	1.72645
$506$	−0.499474	−0.0222043
$507$	1.00000	0.0444116
$508$	2.54891	0.113090
$509$	20.6027	0.913197	0.456598	−	0.889673i	$-0.349068\pi$
$509$	20.6027	0.913197	0.456598	+	0.889673i	$0.349068\pi$
$510$	−16.1859	−0.716725
$511$	0	0
$512$	−25.3082	−1.11848
$513$	5.34724	0.236086
$514$	3.87536	0.170935
$515$	−33.6261	−1.48174
$516$	0.0263440	0.00115973
$517$	−0.276426	−0.0121572
$518$	0	0
$519$	−14.5930	−0.640560
$520$	6.03272	0.264552
$521$	−36.5356	−1.60065	−0.800327	−	0.599564i	$-0.795341\pi$
$521$	−36.5356	−1.60065	−0.800327	+	0.599564i	$0.795341\pi$
$522$	−3.14870	−0.137815
$523$	10.9667	0.479541	0.239770	−	0.970830i	$-0.422928\pi$
$523$	10.9667	0.479541	0.239770	+	0.970830i	$0.422928\pi$
$524$	−1.83356	−0.0800995
$525$	0	0
$526$	−7.47702	−0.326014
$527$	19.1982	0.836285
$528$	0.233358	0.0101556
$529$	7.83763	0.340767
$530$	21.3085	0.925581
$531$	6.88298	0.298696
$532$	0	0
$533$	2.36912	0.102618
$534$	6.58069	0.284774
$535$	26.1460	1.13039
$536$	13.9000	0.600391
$537$	6.28687	0.271299
$538$	38.1969	1.64678
$539$	0	0
$540$	0.513501	0.0220975
$541$	−6.32308	−0.271850	−0.135925	−	0.990719i	$-0.543401\pi$
$541$	−6.32308	−0.271850	−0.135925	+	0.990719i	$0.543401\pi$
$542$	18.9608	0.814437
$543$	−15.1782	−0.651358
$544$	8.61550	0.369387
$545$	41.1504	1.76269
$546$	0	0
$547$	−9.45925	−0.404448	−0.202224	−	0.979339i	$-0.564817\pi$
$547$	−9.45925	−0.404448	−0.202224	+	0.979339i	$0.564817\pi$
$548$	−5.23191	−0.223496
$549$	8.04617	0.343402
$550$	0.0806440	0.00343867
$551$	−12.7402	−0.542751
$552$	−16.5380	−0.703902
$553$	0	0
$554$	20.5913	0.874840
$555$	−10.7249	−0.455248
$556$	3.87646	0.164399
$557$	−18.9196	−0.801650	−0.400825	−	0.916155i	$-0.631277\pi$
$557$	−18.9196	−0.801650	−0.400825	+	0.916155i	$0.631277\pi$
$558$	4.19627	0.177642
$559$	−0.103923	−0.00439549
$560$	0	0
$561$	−0.411498	−0.0173735
$562$	−15.3764	−0.648616
$563$	16.0560	0.676681	0.338341	−	0.941024i	$-0.390134\pi$
$563$	16.0560	0.676681	0.338341	+	0.941024i	$0.390134\pi$
$564$	−1.02958	−0.0433532
$565$	16.9947	0.714973
$566$	−39.6125	−1.66504
$567$	0	0
$568$	−1.55605	−0.0652903
$569$	−19.8836	−0.833562	−0.416781	−	0.909007i	$-0.636842\pi$
$569$	−19.8836	−0.833562	−0.416781	+	0.909007i	$0.636842\pi$
$570$	−14.3148	−0.599582
$571$	−2.94453	−0.123225	−0.0616123	−	0.998100i	$-0.519624\pi$
$571$	−2.94453	−0.123225	−0.0616123	+	0.998100i	$0.519624\pi$
$572$	0.0172527	0.000721370 0
$573$	−20.9458	−0.875025
$574$	0	0
$575$	−4.97898	−0.207638
$576$	8.74065	0.364194
$577$	28.0259	1.16673	0.583367	−	0.812209i	$-0.301735\pi$
$577$	28.0259	1.16673	0.583367	+	0.812209i	$0.301735\pi$
$578$	25.8446	1.07499
$579$	3.22834	0.134165
$580$	−1.22345	−0.0508012
$581$	0	0
$582$	6.73598	0.279215
$583$	0.541730	0.0224361
$584$	−20.4779	−0.847380
$585$	−2.02568	−0.0837517
$586$	−5.07169	−0.209510
$587$	−20.3286	−0.839051	−0.419525	−	0.907744i	$-0.637803\pi$
$587$	−20.3286	−0.839051	−0.419525	+	0.907744i	$0.637803\pi$
$588$	0	0
$589$	16.9789	0.699601
$590$	−18.4261	−0.758590
$591$	−11.2173	−0.461418
$592$	−18.1534	−0.746102
$593$	−2.97922	−0.122342	−0.0611710	−	0.998127i	$-0.519483\pi$
$593$	−2.97922	−0.122342	−0.0611710	+	0.998127i	$0.519483\pi$
$594$	−0.0899440	−0.00369045
$595$	0	0
$596$	−2.20137	−0.0901717
$597$	−15.3611	−0.628690
$598$	7.33881	0.300106
$599$	−43.7897	−1.78920	−0.894600	−	0.446868i	$-0.852539\pi$
$599$	−43.7897	−1.78920	−0.894600	+	0.446868i	$0.852539\pi$
$600$	2.67019	0.109010
$601$	−30.8592	−1.25877	−0.629387	−	0.777092i	$-0.716694\pi$
$601$	−30.8592	−1.25877	−0.629387	+	0.777092i	$0.716694\pi$
$602$	0	0
$603$	−4.66740	−0.190071
$604$	−3.43014	−0.139571
$605$	22.2731	0.905532
$606$	−25.3112	−1.02820
$607$	−8.96895	−0.364039	−0.182019	−	0.983295i	$-0.558263\pi$
$607$	−8.96895	−0.364039	−0.182019	+	0.983295i	$0.558263\pi$
$608$	7.61955	0.309014
$609$	0	0
$610$	−21.5400	−0.872130
$611$	4.06155	0.164313
$612$	−1.53267	−0.0619547
$613$	21.7341	0.877832	0.438916	−	0.898528i	$-0.355363\pi$
$613$	21.7341	0.877832	0.438916	+	0.898528i	$0.355363\pi$
$614$	−0.748540	−0.0302086
$615$	−4.79910	−0.193518
$616$	0	0
$617$	16.2684	0.654941	0.327471	−	0.944861i	$-0.393804\pi$
$617$	16.2684	0.654941	0.327471	+	0.944861i	$0.393804\pi$
$618$	21.9376	0.882460
$619$	42.8926	1.72400	0.862000	−	0.506908i	$-0.169212\pi$
$619$	42.8926	1.72400	0.862000	+	0.506908i	$0.169212\pi$
$620$	1.63050	0.0654823
$621$	5.55316	0.222841
$622$	−26.8334	−1.07592
$623$	0	0
$624$	−3.42875	−0.137260
$625$	−19.7131	−0.788524
$626$	−16.9803	−0.678668
$627$	−0.363929	−0.0145339
$628$	−5.30005	−0.211495
$629$	32.0113	1.27637
$630$	0	0
$631$	−29.3809	−1.16963	−0.584817	−	0.811165i	$-0.698834\pi$
$631$	−29.3809	−1.16963	−0.584817	+	0.811165i	$0.698834\pi$
$632$	−34.9441	−1.39000
$633$	17.9977	0.715346
$634$	−43.9577	−1.74578
$635$	20.3684	0.808297
$636$	2.01774	0.0800085
$637$	0	0
$638$	0.214298	0.00848415
$639$	0.522494	0.0206696
$640$	−17.6262	−0.696736
$641$	−34.4794	−1.36186	−0.680928	−	0.732351i	$-0.738423\pi$
$641$	−34.4794	−1.36186	−0.680928	+	0.732351i	$0.738423\pi$
$642$	−17.0577	−0.673212
$643$	−13.2947	−0.524291	−0.262146	−	0.965028i	$-0.584430\pi$
$643$	−13.2947	−0.524291	−0.262146	+	0.965028i	$0.584430\pi$
$644$	0	0
$645$	0.210516	0.00828905
$646$	42.7263	1.68104
$647$	−29.7605	−1.17001	−0.585003	−	0.811031i	$-0.698907\pi$
$647$	−29.7605	−1.17001	−0.585003	+	0.811031i	$0.698907\pi$
$648$	−2.97812	−0.116991
$649$	−0.468450	−0.0183883
$650$	−1.18491	−0.0464760
$651$	0	0
$652$	−5.79653	−0.227010
$653$	−12.6655	−0.495638	−0.247819	−	0.968806i	$-0.579714\pi$
$653$	−12.6655	−0.495638	−0.247819	+	0.968806i	$0.579714\pi$
$654$	−26.8465	−1.04978
$655$	−14.6520	−0.572503
$656$	−8.12313	−0.317155
$657$	6.87611	0.268263
$658$	0	0
$659$	−40.5989	−1.58151	−0.790755	−	0.612133i	$-0.790312\pi$
$659$	−40.5989	−1.58151	−0.790755	+	0.612133i	$0.790312\pi$
$660$	−0.0349485	−0.00136037
$661$	−3.09978	−0.120568	−0.0602838	−	0.998181i	$-0.519201\pi$
$661$	−3.09978	−0.120568	−0.0602838	+	0.998181i	$0.519201\pi$
$662$	39.1579	1.52192
$663$	6.04617	0.234814
$664$	38.1817	1.48174
$665$	0	0
$666$	6.99694	0.271126
$667$	−13.2308	−0.512300
$668$	3.21972	0.124575
$669$	−2.97864	−0.115161
$670$	12.4949	0.482719
$671$	−0.547617	−0.0211405
$672$	0	0
$673$	33.8836	1.30612	0.653058	−	0.757308i	$-0.273486\pi$
$673$	33.8836	1.30612	0.653058	+	0.757308i	$0.273486\pi$
$674$	31.2497	1.20369
$675$	−0.896603	−0.0345103
$676$	−0.253495	−0.00974980
$677$	−27.3008	−1.04925	−0.524627	−	0.851332i	$-0.675795\pi$
$677$	−27.3008	−1.04925	−0.524627	+	0.851332i	$0.675795\pi$
$678$	−11.0873	−0.425806
$679$	0	0
$680$	36.4749	1.39875
$681$	11.8511	0.454136
$682$	−0.285595	−0.0109360
$683$	−9.65538	−0.369453	−0.184726	−	0.982790i	$-0.559140\pi$
$683$	−9.65538	−0.369453	−0.184726	+	0.982790i	$0.559140\pi$
$684$	−1.35550	−0.0518287
$685$	−41.8084	−1.59742
$686$	0	0
$687$	−6.44618	−0.245937
$688$	0.356327	0.0135848
$689$	−7.95968	−0.303240
$690$	−14.8661	−0.565943
$691$	−48.9876	−1.86358	−0.931788	−	0.363003i	$-0.881751\pi$
$691$	−48.9876	−1.86358	−0.931788	+	0.363003i	$0.881751\pi$
$692$	3.69924	0.140624
$693$	0	0
$694$	−2.65171	−0.100658
$695$	30.9769	1.17502
$696$	7.09559	0.268958
$697$	14.3241	0.542565
$698$	35.3742	1.33893
$699$	7.15529	0.270638
$700$	0	0
$701$	11.7557	0.444007	0.222004	−	0.975046i	$-0.428740\pi$
$701$	11.7557	0.444007	0.222004	+	0.975046i	$0.428740\pi$
$702$	1.32155	0.0498788
$703$	28.3108	1.06776
$704$	−0.594882	−0.0224205
$705$	−8.22741	−0.309862
$706$	18.2170	0.685608
$707$	0	0
$708$	−1.74480	−0.0655736
$709$	−40.7758	−1.53137	−0.765684	−	0.643217i	$-0.777599\pi$
$709$	−40.7758	−1.53137	−0.765684	+	0.643217i	$0.777599\pi$
$710$	−1.39874	−0.0524939
$711$	11.7336	0.440045
$712$	−14.8296	−0.555761
$713$	17.6327	0.660350
$714$	0	0
$715$	0.137867	0.00515592
$716$	−1.59369	−0.0595590
$717$	0.280704	0.0104831
$718$	−0.525637	−0.0196166
$719$	14.6273	0.545507	0.272754	−	0.962084i	$-0.412066\pi$
$719$	14.6273	0.545507	0.272754	+	0.962084i	$0.412066\pi$
$720$	6.94557	0.258846
$721$	0	0
$722$	12.6776	0.471812
$723$	−7.31950	−0.272215
$724$	3.84759	0.142995
$725$	2.13622	0.0793374
$726$	−14.5310	−0.539295
$727$	49.8840	1.85009	0.925047	−	0.379852i	$-0.124025\pi$
$727$	49.8840	1.85009	0.925047	+	0.379852i	$0.124025\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−18.4077	−0.681300
$731$	−0.628338	−0.0232399
$732$	−2.03966	−0.0753881
$733$	6.67359	0.246495	0.123247	−	0.992376i	$-0.460669\pi$
$733$	6.67359	0.246495	0.123247	+	0.992376i	$0.460669\pi$
$734$	−22.1285	−0.816777
$735$	0	0
$736$	7.91299	0.291677
$737$	0.317660	0.0117011
$738$	3.13092	0.115251
$739$	22.5737	0.830385	0.415193	−	0.909734i	$-0.363714\pi$
$739$	22.5737	0.830385	0.415193	+	0.909734i	$0.363714\pi$
$740$	2.71872	0.0999420
$741$	5.34724	0.196436
$742$	0	0
$743$	−22.6974	−0.832686	−0.416343	−	0.909208i	$-0.636688\pi$
$743$	−22.6974	−0.832686	−0.416343	+	0.909208i	$0.636688\pi$
$744$	−9.45628	−0.346684
$745$	−17.5912	−0.644493
$746$	13.7783	0.504459
$747$	−12.8208	−0.469087
$748$	0.104313	0.00381405
$749$	0	0
$750$	15.7855	0.576405
$751$	31.0982	1.13479	0.567395	−	0.823446i	$-0.307951\pi$
$751$	31.0982	1.13479	0.567395	+	0.823446i	$0.307951\pi$
$752$	−13.9260	−0.507830
$753$	−0.159014	−0.00579478
$754$	−3.14870	−0.114669
$755$	−27.4104	−0.997565
$756$	0	0
$757$	−16.9262	−0.615192	−0.307596	−	0.951517i	$-0.599525\pi$
$757$	−16.9262	−0.615192	−0.307596	+	0.951517i	$0.599525\pi$
$758$	33.9282	1.23233
$759$	−0.377944	−0.0137185
$760$	32.2584	1.17014
$761$	36.4050	1.31968	0.659840	−	0.751406i	$-0.270624\pi$
$761$	36.4050	1.31968	0.659840	+	0.751406i	$0.270624\pi$
$762$	−13.2883	−0.481386
$763$	0	0
$764$	5.30966	0.192097
$765$	−12.2476	−0.442814
$766$	−8.99311	−0.324934
$767$	6.88298	0.248530
$768$	−5.98201	−0.215857
$769$	9.82130	0.354165	0.177083	−	0.984196i	$-0.443334\pi$
$769$	9.82130	0.354165	0.177083	+	0.984196i	$0.443334\pi$
$770$	0	0
$771$	2.93243	0.105609
$772$	−0.818367	−0.0294537
$773$	−9.40372	−0.338228	−0.169114	−	0.985596i	$-0.554091\pi$
$773$	−9.40372	−0.338228	−0.169114	+	0.985596i	$0.554091\pi$
$774$	−0.137340	−0.00493659
$775$	−2.84694	−0.102265
$776$	−15.1795	−0.544912
$777$	0	0
$778$	−48.5764	−1.74155
$779$	12.6683	0.453888
$780$	0.513501	0.0183863
$781$	−0.0355606	−0.00127246
$782$	44.3717	1.58673
$783$	−2.38258	−0.0851463
$784$	0	0
$785$	−42.3529	−1.51164
$786$	9.55898	0.340958
$787$	−3.16564	−0.112843	−0.0564214	−	0.998407i	$-0.517969\pi$
$787$	−3.16564	−0.112843	−0.0564214	+	0.998407i	$0.517969\pi$
$788$	2.84352	0.101296
$789$	−5.65775	−0.201421
$790$	−31.4115	−1.11757
$791$	0	0
$792$	0.202688	0.00720221
$793$	8.04617	0.285728
$794$	−29.9990	−1.06463
$795$	16.1238	0.571852
$796$	3.89397	0.138018
$797$	31.1127	1.10207	0.551035	−	0.834482i	$-0.314233\pi$
$797$	31.1127	1.10207	0.551035	+	0.834482i	$0.314233\pi$
$798$	0	0
$799$	24.5568	0.868758
$800$	−1.27762	−0.0451705
$801$	4.97951	0.175942
$802$	23.0895	0.815317
$803$	−0.467983	−0.0165148
$804$	1.18316	0.0417269
$805$	0	0
$806$	4.19627	0.147807
$807$	28.9030	1.01743
$808$	57.0388	2.00662
$809$	50.5941	1.77879	0.889397	−	0.457136i	$-0.151125\pi$
$809$	50.5941	1.77879	0.889397	+	0.457136i	$0.151125\pi$
$810$	−2.67705	−0.0940620
$811$	−0.221250	−0.00776913	−0.00388456	−	0.999992i	$-0.501236\pi$
$811$	−0.221250	−0.00776913	−0.00388456	+	0.999992i	$0.501236\pi$
$812$	0	0
$813$	14.3474	0.503184
$814$	−0.476206	−0.0166910
$815$	−46.3202	−1.62253
$816$	−20.7308	−0.725724
$817$	−0.555703	−0.0194416
$818$	40.1824	1.40495
$819$	0	0
$820$	1.21655	0.0424836
$821$	−1.75036	−0.0610879	−0.0305440	−	0.999533i	$-0.509724\pi$
$821$	−1.75036	−0.0610879	−0.0305440	+	0.999533i	$0.509724\pi$
$822$	27.2757	0.951350
$823$	12.8447	0.447739	0.223869	−	0.974619i	$-0.428131\pi$
$823$	12.8447	0.447739	0.223869	+	0.974619i	$0.428131\pi$
$824$	−49.4363	−1.72220
$825$	0.0610221	0.00212452
$826$	0	0
$827$	−12.4757	−0.433823	−0.216911	−	0.976191i	$-0.569598\pi$
$827$	−12.4757	−0.433823	−0.216911	+	0.976191i	$0.569598\pi$
$828$	−1.40770	−0.0489209
$829$	3.44858	0.119774	0.0598871	−	0.998205i	$-0.480926\pi$
$829$	3.44858	0.119774	0.0598871	+	0.998205i	$0.480926\pi$
$830$	34.3219	1.19133
$831$	15.5811	0.540503
$832$	8.74065	0.303028
$833$	0	0
$834$	−20.2093	−0.699791
$835$	25.7289	0.890386
$836$	0.0922541	0.00319068
$837$	3.17526	0.109753
$838$	34.5510	1.19355
$839$	−17.4568	−0.602675	−0.301338	−	0.953518i	$-0.597433\pi$
$839$	−17.4568	−0.602675	−0.301338	+	0.953518i	$0.597433\pi$
$840$	0	0
$841$	−23.3233	−0.804253
$842$	−29.1445	−1.00439
$843$	−11.6351	−0.400735
$844$	−4.56233	−0.157042
$845$	−2.02568	−0.0696857
$846$	5.36756	0.184540
$847$	0	0
$848$	27.2917	0.937202
$849$	−29.9742	−1.02871
$850$	−7.16417	−0.245729
$851$	29.4011	1.00786
$852$	−0.132450	−0.00453765
$853$	−31.5211	−1.07926	−0.539632	−	0.841901i	$-0.681436\pi$
$853$	−31.5211	−1.07926	−0.539632	+	0.841901i	$0.681436\pi$
$854$	0	0
$855$	−10.8318	−0.370440
$856$	38.4393	1.31383
$857$	−7.85300	−0.268253	−0.134127	−	0.990964i	$-0.542823\pi$
$857$	−7.85300	−0.268253	−0.134127	+	0.990964i	$0.542823\pi$
$858$	−0.0899440	−0.00307064
$859$	−5.37755	−0.183480	−0.0917398	−	0.995783i	$-0.529243\pi$
$859$	−5.37755	−0.183480	−0.0917398	+	0.995783i	$0.529243\pi$
$860$	−0.0533647	−0.00181972
$861$	0	0
$862$	−31.6941	−1.07951
$863$	−53.7344	−1.82914	−0.914570	−	0.404427i	$-0.867471\pi$
$863$	−53.7344	−1.82914	−0.914570	+	0.404427i	$0.867471\pi$
$864$	1.42495	0.0484778
$865$	29.5607	1.00509
$866$	28.6951	0.975098
$867$	19.5562	0.664164
$868$	0	0
$869$	−0.798581	−0.0270900
$870$	6.37828	0.216244
$871$	−4.66740	−0.158149
$872$	60.4983	2.04873
$873$	5.09702	0.172508
$874$	39.2423	1.32739
$875$	0	0
$876$	−1.74306	−0.0588925
$877$	−28.8873	−0.975454	−0.487727	−	0.872996i	$-0.662174\pi$
$877$	−28.8873	−0.975454	−0.487727	+	0.872996i	$0.662174\pi$
$878$	−12.0656	−0.407195
$879$	−3.83767	−0.129441
$880$	−0.472710	−0.0159351
$881$	30.9653	1.04325	0.521623	−	0.853176i	$-0.325327\pi$
$881$	30.9653	1.04325	0.521623	+	0.853176i	$0.325327\pi$
$882$	0	0
$883$	6.60585	0.222305	0.111152	−	0.993803i	$-0.464546\pi$
$883$	6.60585	0.222305	0.111152	+	0.993803i	$0.464546\pi$
$884$	−1.53267	−0.0515494
$885$	−13.9427	−0.468680
$886$	5.91265	0.198639
$887$	25.3939	0.852642	0.426321	−	0.904572i	$-0.359809\pi$
$887$	25.3939	0.852642	0.426321	+	0.904572i	$0.359809\pi$
$888$	−15.7676	−0.529125
$889$	0	0
$890$	−13.3304	−0.446836
$891$	−0.0680592	−0.00228007
$892$	0.755070	0.0252816
$893$	21.7181	0.726767
$894$	11.4765	0.383831
$895$	−12.7352	−0.425691
$896$	0	0
$897$	5.55316	0.185415
$898$	−24.9847	−0.833749
$899$	−7.56529	−0.252317
$900$	0.227284	0.00757614
$901$	−48.1256	−1.60330
$902$	−0.213088	−0.00709507
$903$	0	0
$904$	24.9852	0.830997
$905$	30.7462	1.02204
$906$	17.8825	0.594106
$907$	−2.53313	−0.0841111	−0.0420556	−	0.999115i	$-0.513391\pi$
$907$	−2.53313	−0.0841111	−0.0420556	+	0.999115i	$0.513391\pi$
$908$	−3.00420	−0.0996978
$909$	−19.1526	−0.635253
$910$	0	0
$911$	−49.8354	−1.65112	−0.825560	−	0.564314i	$-0.809141\pi$
$911$	−49.8354	−1.65112	−0.825560	+	0.564314i	$0.809141\pi$
$912$	−18.3343	−0.607111
$913$	0.872572	0.0288779
$914$	−34.2663	−1.13343
$915$	−16.2990	−0.538829
$916$	1.63407	0.0539913
$917$	0	0
$918$	7.99035	0.263721
$919$	47.1155	1.55420	0.777098	−	0.629379i	$-0.216691\pi$
$919$	47.1155	1.55420	0.777098	+	0.629379i	$0.216691\pi$
$920$	33.5007	1.10449
$921$	−0.566409	−0.0186638
$922$	−40.7768	−1.34291
$923$	0.522494	0.0171981
$924$	0	0
$925$	−4.74704	−0.156082
$926$	44.3585	1.45771
$927$	16.5999	0.545211
$928$	−3.39505	−0.111448
$929$	41.0724	1.34754	0.673771	−	0.738940i	$-0.264673\pi$
$929$	41.0724	1.34754	0.673771	+	0.738940i	$0.264673\pi$
$930$	−8.50032	−0.278737
$931$	0	0
$932$	−1.81383	−0.0594140
$933$	−20.3044	−0.664738
$934$	0.00207129	6.77746e−5 0
$935$	0.833565	0.0272605
$936$	−2.97812	−0.0973427
$937$	−15.0742	−0.492451	−0.246226	−	0.969213i	$-0.579190\pi$
$937$	−15.0742	−0.492451	−0.246226	+	0.969213i	$0.579190\pi$
$938$	0	0
$939$	−12.8487	−0.419302
$940$	2.08561	0.0680250
$941$	40.5379	1.32150	0.660749	−	0.750607i	$-0.270239\pi$
$941$	40.5379	1.32150	0.660749	+	0.750607i	$0.270239\pi$
$942$	27.6310	0.900265
$943$	13.1561	0.428423
$944$	−23.6000	−0.768115
$945$	0	0
$946$	0.00934727	0.000303906 0
$947$	−9.04941	−0.294066	−0.147033	−	0.989132i	$-0.546972\pi$
$947$	−9.04941	−0.294066	−0.147033	+	0.989132i	$0.546972\pi$
$948$	−2.97441	−0.0966044
$949$	6.87611	0.223208
$950$	−6.33599	−0.205567
$951$	−33.2621	−1.07860
$952$	0	0
$953$	−19.6398	−0.636195	−0.318098	−	0.948058i	$-0.603044\pi$
$953$	−19.6398	−0.636195	−0.318098	+	0.948058i	$0.603044\pi$
$954$	−10.5191	−0.340570
$955$	42.4297	1.37299
$956$	−0.0711571	−0.00230139
$957$	0.162156	0.00524177
$958$	15.1397	0.489141
$959$	0	0
$960$	−17.7058	−0.571452
$961$	−20.9178	−0.674766
$962$	6.99694	0.225590
$963$	−12.9073	−0.415931
$964$	1.85546	0.0597602
$965$	−6.53959	−0.210517
$966$	0	0
$967$	−51.6182	−1.65993	−0.829964	−	0.557816i	$-0.811639\pi$
$967$	−51.6182	−1.65993	−0.829964	+	0.557816i	$0.811639\pi$
$968$	32.7455	1.05248
$969$	32.3303	1.03860
$970$	−13.6450	−0.438114
$971$	15.0659	0.483487	0.241743	−	0.970340i	$-0.422281\pi$
$971$	15.0659	0.483487	0.241743	+	0.970340i	$0.422281\pi$
$972$	−0.253495	−0.00813085
$973$	0	0
$974$	20.5569	0.658685
$975$	−0.896603	−0.0287143
$976$	−27.5883	−0.883081
$977$	−9.68172	−0.309746	−0.154873	−	0.987934i	$-0.549497\pi$
$977$	−9.68172	−0.309746	−0.154873	+	0.987934i	$0.549497\pi$
$978$	30.2193	0.966305
$979$	−0.338902	−0.0108313
$980$	0	0
$981$	−20.3143	−0.648586
$982$	35.9811	1.14820
$983$	2.87074	0.0915625	0.0457812	−	0.998951i	$-0.485422\pi$
$983$	2.87074	0.0915625	0.0457812	+	0.998951i	$0.485422\pi$
$984$	−7.05552	−0.224922
$985$	22.7227	0.724005
$986$	−19.0376	−0.606281
$987$	0	0
$988$	−1.35550	−0.0431241
$989$	−0.577103	−0.0183508
$990$	0.182198	0.00579064
$991$	−41.1053	−1.30575	−0.652876	−	0.757465i	$-0.726438\pi$
$991$	−41.1053	−1.30575	−0.652876	+	0.757465i	$0.726438\pi$
$992$	4.52458	0.143656
$993$	29.6302	0.940287
$994$	0	0
$995$	31.1168	0.986470
$996$	3.25000	0.102980
$997$	−15.8129	−0.500799	−0.250399	−	0.968143i	$-0.580562\pi$
$997$	−15.8129	−0.500799	−0.250399	+	0.968143i	$0.580562\pi$
$998$	−38.3536	−1.21406
$999$	5.29448	0.167510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.2.a.u.1.4		5
3.2	odd	2		5733.2.a.bp.1.2		5
7.3	odd	6		273.2.i.e.79.2	✓	10
7.5	odd	6		273.2.i.e.235.2	yes	10
7.6	odd	2		1911.2.a.t.1.4		5
21.5	even	6		819.2.j.g.235.4		10
21.17	even	6		819.2.j.g.352.4		10
21.20	even	2		5733.2.a.bq.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.i.e.79.2	✓	10	7.3	odd	6
273.2.i.e.235.2	yes	10	7.5	odd	6
819.2.j.g.235.4		10	21.5	even	6
819.2.j.g.352.4		10	21.17	even	6
1911.2.a.t.1.4		5	7.6	odd	2
1911.2.a.u.1.4		5	1.1	even	1	trivial
5733.2.a.bp.1.2		5	3.2	odd	2
5733.2.a.bq.1.2		5	21.20	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:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q+1.32155 q^{2} +1.00000 q^{3} -0.253495 q^{4} -2.02568 q^{5} +1.32155 q^{6} -2.97812 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.32155 q^{2} +1.00000 q^{3} -0.253495 q^{4} -2.02568 q^{5} +1.32155 q^{6} -2.97812 q^{8} +1.00000 q^{9} -2.67705 q^{10} -0.0680592 q^{11} -0.253495 q^{12} +1.00000 q^{13} -2.02568 q^{15} -3.42875 q^{16} +6.04617 q^{17} +1.32155 q^{18} +5.34724 q^{19} +0.513501 q^{20} -0.0899440 q^{22} +5.55316 q^{23} -2.97812 q^{24} -0.896603 q^{25} +1.32155 q^{26} +1.00000 q^{27} -2.38258 q^{29} -2.67705 q^{30} +3.17526 q^{31} +1.42495 q^{32} -0.0680592 q^{33} +7.99035 q^{34} -0.253495 q^{36} +5.29448 q^{37} +7.06666 q^{38} +1.00000 q^{39} +6.03272 q^{40} +2.36912 q^{41} -0.103923 q^{43} +0.0172527 q^{44} -2.02568 q^{45} +7.33881 q^{46} +4.06155 q^{47} -3.42875 q^{48} -1.18491 q^{50} +6.04617 q^{51} -0.253495 q^{52} -7.95968 q^{53} +1.32155 q^{54} +0.137867 q^{55} +5.34724 q^{57} -3.14870 q^{58} +6.88298 q^{59} +0.513501 q^{60} +8.04617 q^{61} +4.19627 q^{62} +8.74065 q^{64} -2.02568 q^{65} -0.0899440 q^{66} -4.66740 q^{67} -1.53267 q^{68} +5.55316 q^{69} +0.522494 q^{71} -2.97812 q^{72} +6.87611 q^{73} +6.99694 q^{74} -0.896603 q^{75} -1.35550 q^{76} +1.32155 q^{78} +11.7336 q^{79} +6.94557 q^{80} +1.00000 q^{81} +3.13092 q^{82} -12.8208 q^{83} -12.2476 q^{85} -0.137340 q^{86} -2.38258 q^{87} +0.202688 q^{88} +4.97951 q^{89} -2.67705 q^{90} -1.40770 q^{92} +3.17526 q^{93} +5.36756 q^{94} -10.8318 q^{95} +1.42495 q^{96} +5.09702 q^{97} -0.0680592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 6 * q^4 + 3 * q^5 + 3 * q^8 + 5 * q^9	\( 5 q + 5 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{8} + 5 q^{9} + 2 q^{10} - q^{11} + 6 q^{12} + 5 q^{13} + 3 q^{15} + 13 q^{17} + 7 q^{19} + 13 q^{20} - 19 q^{22} - 4 q^{23} + 3 q^{24} + 16 q^{25} + 5 q^{27} - 12 q^{29} + 2 q^{30} + 6 q^{31} + 21 q^{32} - q^{33} + 7 q^{34} + 6 q^{36} + 11 q^{37} + 14 q^{38} + 5 q^{39} + 11 q^{40} + 10 q^{41} + 10 q^{43} - 29 q^{44} + 3 q^{45} + q^{46} - 4 q^{47} - 29 q^{50} + 13 q^{51} + 6 q^{52} - 9 q^{53} - 12 q^{55} + 7 q^{57} + 34 q^{58} + 7 q^{59} + 13 q^{60} + 23 q^{61} - 24 q^{62} - 13 q^{64} + 3 q^{65} - 19 q^{66} + 25 q^{67} + 20 q^{68} - 4 q^{69} - 27 q^{71} + 3 q^{72} + 18 q^{73} + 15 q^{74} + 16 q^{75} + 2 q^{76} + 8 q^{79} + 41 q^{80} + 5 q^{81} + 26 q^{82} + 12 q^{83} + 10 q^{85} - 19 q^{86} - 12 q^{87} - 36 q^{88} + 29 q^{89} + 2 q^{90} - 50 q^{92} + 6 q^{93} - 2 q^{94} - 33 q^{95} + 21 q^{96} + 13 q^{97} - q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 6 * q^4 + 3 * q^5 + 3 * q^8 + 5 * q^9 + 2 * q^10 - q^11 + 6 * q^12 + 5 * q^13 + 3 * q^15 + 13 * q^17 + 7 * q^19 + 13 * q^20 - 19 * q^22 - 4 * q^23 + 3 * q^24 + 16 * q^25 + 5 * q^27 - 12 * q^29 + 2 * q^30 + 6 * q^31 + 21 * q^32 - q^33 + 7 * q^34 + 6 * q^36 + 11 * q^37 + 14 * q^38 + 5 * q^39 + 11 * q^40 + 10 * q^41 + 10 * q^43 - 29 * q^44 + 3 * q^45 + q^46 - 4 * q^47 - 29 * q^50 + 13 * q^51 + 6 * q^52 - 9 * q^53 - 12 * q^55 + 7 * q^57 + 34 * q^58 + 7 * q^59 + 13 * q^60 + 23 * q^61 - 24 * q^62 - 13 * q^64 + 3 * q^65 - 19 * q^66 + 25 * q^67 + 20 * q^68 - 4 * q^69 - 27 * q^71 + 3 * q^72 + 18 * q^73 + 15 * q^74 + 16 * q^75 + 2 * q^76 + 8 * q^79 + 41 * q^80 + 5 * q^81 + 26 * q^82 + 12 * q^83 + 10 * q^85 - 19 * q^86 - 12 * q^87 - 36 * q^88 + 29 * q^89 + 2 * q^90 - 50 * q^92 + 6 * q^93 - 2 * q^94 - 33 * q^95 + 21 * q^96 + 13 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more