LMFDB - Embedded newform 845.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.51414$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51414 q^{2} +1.51414 q^{3} +4.32088 q^{4} +1.00000 q^{5} -3.80675 q^{6} -3.32088 q^{7} -5.83502 q^{8} -0.707389 q^{9} +O(q^{10})$	\(q-2.51414 q^{2} +1.51414 q^{3} +4.32088 q^{4} +1.00000 q^{5} -3.80675 q^{6} -3.32088 q^{7} -5.83502 q^{8} -0.707389 q^{9} -2.51414 q^{10} +2.83502 q^{11} +6.54241 q^{12} +8.34916 q^{14} +1.51414 q^{15} +6.02827 q^{16} -6.64177 q^{17} +1.77847 q^{18} -2.19325 q^{19} +4.32088 q^{20} -5.02827 q^{21} -7.12763 q^{22} -0.485863 q^{23} -8.83502 q^{24} +1.00000 q^{25} -5.61350 q^{27} -14.3492 q^{28} -3.32088 q^{29} -3.80675 q^{30} -3.80675 q^{31} -3.48586 q^{32} +4.29261 q^{33} +16.6983 q^{34} -3.32088 q^{35} -3.05655 q^{36} -9.32088 q^{37} +5.51414 q^{38} -5.83502 q^{40} +1.61350 q^{41} +12.6418 q^{42} -0.872368 q^{43} +12.2498 q^{44} -0.707389 q^{45} +1.22153 q^{46} +3.32088 q^{47} +9.12763 q^{48} +4.02827 q^{49} -2.51414 q^{50} -10.0565 q^{51} +11.6700 q^{53} +14.1131 q^{54} +2.83502 q^{55} +19.3774 q^{56} -3.32088 q^{57} +8.34916 q^{58} +8.83502 q^{59} +6.54241 q^{60} -3.70739 q^{61} +9.57068 q^{62} +2.34916 q^{63} -3.29261 q^{64} -10.7922 q^{66} -4.29261 q^{67} -28.6983 q^{68} -0.735663 q^{69} +8.34916 q^{70} -2.19325 q^{71} +4.12763 q^{72} -12.7357 q^{73} +23.4340 q^{74} +1.51414 q^{75} -9.47679 q^{76} -9.41478 q^{77} +0.585221 q^{79} +6.02827 q^{80} -6.37743 q^{81} -4.05655 q^{82} -7.70739 q^{83} -21.7266 q^{84} -6.64177 q^{85} +2.19325 q^{86} -5.02827 q^{87} -16.5424 q^{88} -3.41478 q^{89} +1.77847 q^{90} -2.09936 q^{92} -5.76394 q^{93} -8.34916 q^{94} -2.19325 q^{95} -5.27807 q^{96} -0.641769 q^{97} -10.1276 q^{98} -2.00546 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 2 q^{3} + 5 q^{4} + 3 q^{5} - 10 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - q^2 - 2 * q^3 + 5 * q^4 + 3 * q^5 - 10 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - q^{2} - 2 q^{3} + 5 q^{4} + 3 q^{5} - 10 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{15} + 5 q^{16} - 4 q^{17} + 17 q^{18} - 8 q^{19} + 5 q^{20} - 2 q^{21} - 12 q^{22} - 8 q^{23} - 12 q^{24} + 3 q^{25} - 14 q^{27} - 22 q^{28} - 2 q^{29} - 10 q^{30} - 10 q^{31} - 17 q^{32} + 18 q^{33} + 8 q^{34} - 2 q^{35} + 17 q^{36} - 20 q^{37} + 10 q^{38} - 3 q^{40} + 2 q^{41} + 22 q^{42} - 12 q^{43} + 12 q^{44} + 3 q^{45} - 8 q^{46} + 2 q^{47} + 18 q^{48} - q^{49} - q^{50} - 4 q^{51} + 6 q^{53} - 10 q^{54} - 6 q^{55} + 24 q^{56} - 2 q^{57} + 4 q^{58} + 12 q^{59} - 6 q^{61} - 4 q^{62} - 14 q^{63} - 15 q^{64} + 12 q^{66} - 18 q^{67} - 44 q^{68} + 16 q^{69} + 4 q^{70} - 8 q^{71} + 3 q^{72} - 20 q^{73} + 10 q^{74} - 2 q^{75} + 2 q^{76} - 18 q^{77} + 12 q^{79} + 5 q^{80} + 15 q^{81} + 14 q^{82} - 18 q^{83} - 10 q^{84} - 4 q^{85} + 8 q^{86} - 2 q^{87} - 30 q^{88} + 17 q^{90} - 10 q^{92} + 14 q^{93} - 4 q^{94} - 8 q^{95} + 22 q^{96} + 14 q^{97} - 21 q^{98} - 12 q^{99}+O(q^{100}) $ 3 * q - q^2 - 2 * q^3 + 5 * q^4 + 3 * q^5 - 10 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 - q^10 - 6 * q^11 + 4 * q^14 - 2 * q^15 + 5 * q^16 - 4 * q^17 + 17 * q^18 - 8 * q^19 + 5 * q^20 - 2 * q^21 - 12 * q^22 - 8 * q^23 - 12 * q^24 + 3 * q^25 - 14 * q^27 - 22 * q^28 - 2 * q^29 - 10 * q^30 - 10 * q^31 - 17 * q^32 + 18 * q^33 + 8 * q^34 - 2 * q^35 + 17 * q^36 - 20 * q^37 + 10 * q^38 - 3 * q^40 + 2 * q^41 + 22 * q^42 - 12 * q^43 + 12 * q^44 + 3 * q^45 - 8 * q^46 + 2 * q^47 + 18 * q^48 - q^49 - q^50 - 4 * q^51 + 6 * q^53 - 10 * q^54 - 6 * q^55 + 24 * q^56 - 2 * q^57 + 4 * q^58 + 12 * q^59 - 6 * q^61 - 4 * q^62 - 14 * q^63 - 15 * q^64 + 12 * q^66 - 18 * q^67 - 44 * q^68 + 16 * q^69 + 4 * q^70 - 8 * q^71 + 3 * q^72 - 20 * q^73 + 10 * q^74 - 2 * q^75 + 2 * q^76 - 18 * q^77 + 12 * q^79 + 5 * q^80 + 15 * q^81 + 14 * q^82 - 18 * q^83 - 10 * q^84 - 4 * q^85 + 8 * q^86 - 2 * q^87 - 30 * q^88 + 17 * q^90 - 10 * q^92 + 14 * q^93 - 4 * q^94 - 8 * q^95 + 22 * q^96 + 14 * q^97 - 21 * q^98 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.51414	−1.77776	−0.888882	−	0.458137i	$-0.848517\pi$
$2$	−2.51414	−1.77776	−0.888882	+	0.458137i	$0.848517\pi$
$3$	1.51414	0.874187	0.437094	−	0.899416i	$-0.356008\pi$
$3$	1.51414	0.874187	0.437094	+	0.899416i	$0.356008\pi$
$4$	4.32088	2.16044
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−3.80675	−1.55410
$7$	−3.32088	−1.25518	−0.627588	−	0.778545i	$-0.715958\pi$
$7$	−3.32088	−1.25518	−0.627588	+	0.778545i	$0.715958\pi$
$8$	−5.83502	−2.06299
$9$	−0.707389	−0.235796
$10$	−2.51414	−0.795040
$11$	2.83502	0.854791	0.427396	−	0.904065i	$-0.359431\pi$
$11$	2.83502	0.854791	0.427396	+	0.904065i	$0.359431\pi$
$12$	6.54241	1.88863
$13$	0	0
$13$	0	0
$14$	8.34916	2.23141
$15$	1.51414	0.390948
$16$	6.02827	1.50707
$17$	−6.64177	−1.61087	−0.805433	−	0.592687i	$-0.798067\pi$
$17$	−6.64177	−1.61087	−0.805433	+	0.592687i	$0.798067\pi$
$18$	1.77847	0.419190
$19$	−2.19325	−0.503167	−0.251583	−	0.967836i	$-0.580951\pi$
$19$	−2.19325	−0.503167	−0.251583	+	0.967836i	$0.580951\pi$
$20$	4.32088	0.966179
$21$	−5.02827	−1.09726
$22$	−7.12763	−1.51962
$23$	−0.485863	−0.101309	−0.0506547	−	0.998716i	$-0.516131\pi$
$23$	−0.485863	−0.101309	−0.0506547	+	0.998716i	$0.516131\pi$
$24$	−8.83502	−1.80344
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.61350	−1.08032
$28$	−14.3492	−2.71174
$29$	−3.32088	−0.616673	−0.308336	−	0.951277i	$-0.599772\pi$
$29$	−3.32088	−0.616673	−0.308336	+	0.951277i	$0.599772\pi$
$30$	−3.80675	−0.695014
$31$	−3.80675	−0.683712	−0.341856	−	0.939752i	$-0.611056\pi$
$31$	−3.80675	−0.683712	−0.341856	+	0.939752i	$0.611056\pi$
$32$	−3.48586	−0.616219
$33$	4.29261	0.747248
$34$	16.6983	2.86374
$35$	−3.32088	−0.561332
$36$	−3.05655	−0.509425
$37$	−9.32088	−1.53234	−0.766172	−	0.642636i	$-0.777841\pi$
$37$	−9.32088	−1.53234	−0.766172	+	0.642636i	$0.777841\pi$
$38$	5.51414	0.894511
$39$	0	0
$40$	−5.83502	−0.922598
$41$	1.61350	0.251986	0.125993	−	0.992031i	$-0.459788\pi$
$41$	1.61350	0.251986	0.125993	+	0.992031i	$0.459788\pi$
$42$	12.6418	1.95067
$43$	−0.872368	−0.133035	−0.0665174	−	0.997785i	$-0.521189\pi$
$43$	−0.872368	−0.133035	−0.0665174	+	0.997785i	$0.521189\pi$
$44$	12.2498	1.84673
$45$	−0.707389	−0.105451
$46$	1.22153	0.180104
$47$	3.32088	0.484401	0.242200	−	0.970226i	$-0.422131\pi$
$47$	3.32088	0.484401	0.242200	+	0.970226i	$0.422131\pi$
$48$	9.12763	1.31746
$49$	4.02827	0.575468
$50$	−2.51414	−0.355553
$51$	−10.0565	−1.40820
$52$	0	0
$53$	11.6700	1.60300	0.801502	−	0.597992i	$-0.204035\pi$
$53$	11.6700	1.60300	0.801502	+	0.597992i	$0.204035\pi$
$54$	14.1131	1.92055
$55$	2.83502	0.382274
$56$	19.3774	2.58942
$57$	−3.32088	−0.439862
$58$	8.34916	1.09630
$59$	8.83502	1.15022	0.575111	−	0.818075i	$-0.304959\pi$
$59$	8.83502	1.15022	0.575111	+	0.818075i	$0.304959\pi$
$60$	6.54241	0.844622
$61$	−3.70739	−0.474683	−0.237341	−	0.971426i	$-0.576276\pi$
$61$	−3.70739	−0.474683	−0.237341	+	0.971426i	$0.576276\pi$
$62$	9.57068	1.21548
$63$	2.34916	0.295966
$64$	−3.29261	−0.411576
$65$	0	0
$66$	−10.7922	−1.32843
$67$	−4.29261	−0.524426	−0.262213	−	0.965010i	$-0.584452\pi$
$67$	−4.29261	−0.524426	−0.262213	+	0.965010i	$0.584452\pi$
$68$	−28.6983	−3.48018
$69$	−0.735663	−0.0885634
$70$	8.34916	0.997915
$71$	−2.19325	−0.260291	−0.130146	−	0.991495i	$-0.541544\pi$
$71$	−2.19325	−0.260291	−0.130146	+	0.991495i	$0.541544\pi$
$72$	4.12763	0.486446
$73$	−12.7357	−1.49060	−0.745298	−	0.666731i	$-0.767693\pi$
$73$	−12.7357	−1.49060	−0.745298	+	0.666731i	$0.767693\pi$
$74$	23.4340	2.72414
$75$	1.51414	0.174837
$76$	−9.47679	−1.08706
$77$	−9.41478	−1.07291
$78$	0	0
$79$	0.585221	0.0658425	0.0329213	−	0.999458i	$-0.489519\pi$
$79$	0.585221	0.0658425	0.0329213	+	0.999458i	$0.489519\pi$
$80$	6.02827	0.673982
$81$	−6.37743	−0.708604
$82$	−4.05655	−0.447971
$83$	−7.70739	−0.845996	−0.422998	−	0.906131i	$-0.639022\pi$
$83$	−7.70739	−0.845996	−0.422998	+	0.906131i	$0.639022\pi$
$84$	−21.7266	−2.37057
$85$	−6.64177	−0.720401
$86$	2.19325	0.236504
$87$	−5.02827	−0.539088
$88$	−16.5424	−1.76343
$89$	−3.41478	−0.361966	−0.180983	−	0.983486i	$-0.557928\pi$
$89$	−3.41478	−0.361966	−0.180983	+	0.983486i	$0.557928\pi$
$90$	1.77847	0.187468
$91$	0	0
$92$	−2.09936	−0.218873
$93$	−5.76394	−0.597692
$94$	−8.34916	−0.861150
$95$	−2.19325	−0.225023
$96$	−5.27807	−0.538691
$97$	−0.641769	−0.0651618	−0.0325809	−	0.999469i	$-0.510373\pi$
$97$	−0.641769	−0.0651618	−0.0325809	+	0.999469i	$0.510373\pi$
$98$	−10.1276	−1.02305
$99$	−2.00546	−0.201557
$100$	4.32088	0.432088
$101$	6.97173	0.693713	0.346856	−	0.937918i	$-0.387249\pi$
$101$	6.97173	0.693713	0.346856	+	0.937918i	$0.387249\pi$
$102$	25.2835	2.50344
$103$	−7.18418	−0.707878	−0.353939	−	0.935268i	$-0.615158\pi$
$103$	−7.18418	−0.707878	−0.353939	+	0.935268i	$0.615158\pi$
$104$	0	0
$105$	−5.02827	−0.490709
$106$	−29.3401	−2.84976
$107$	−9.57068	−0.925233	−0.462617	−	0.886558i	$-0.653089\pi$
$107$	−9.57068	−0.925233	−0.462617	+	0.886558i	$0.653089\pi$
$108$	−24.2553	−2.33396
$109$	10.6983	1.02471	0.512356	−	0.858773i	$-0.328773\pi$
$109$	10.6983	1.02471	0.512356	+	0.858773i	$0.328773\pi$
$110$	−7.12763	−0.679593
$111$	−14.1131	−1.33956
$112$	−20.0192	−1.89164
$113$	−3.22699	−0.303570	−0.151785	−	0.988414i	$-0.548502\pi$
$113$	−3.22699	−0.303570	−0.151785	+	0.988414i	$0.548502\pi$
$114$	8.34916	0.781970
$115$	−0.485863	−0.0453070
$116$	−14.3492	−1.33229
$117$	0	0
$118$	−22.2125	−2.04482
$119$	22.0565	2.02192
$120$	−8.83502	−0.806523
$121$	−2.96265	−0.269332
$122$	9.32088	0.843873
$123$	2.44305	0.220283
$124$	−16.4485	−1.47712
$125$	1.00000	0.0894427
$126$	−5.90611	−0.526158
$127$	5.90064	0.523597	0.261799	−	0.965123i	$-0.415684\pi$
$127$	5.90064	0.523597	0.261799	+	0.965123i	$0.415684\pi$
$128$	15.2498	1.34790
$129$	−1.32088	−0.116297
$130$	0	0
$131$	3.41478	0.298351	0.149175	−	0.988811i	$-0.452338\pi$
$131$	3.41478	0.298351	0.149175	+	0.988811i	$0.452338\pi$
$132$	18.5479	1.61439
$133$	7.28354	0.631563
$134$	10.7922	0.932305
$135$	−5.61350	−0.483133
$136$	38.7549	3.32320
$137$	15.0848	1.28878	0.644392	−	0.764696i	$-0.277111\pi$
$137$	15.0848	1.28878	0.644392	+	0.764696i	$0.277111\pi$
$138$	1.84956	0.157445
$139$	−20.6983	−1.75561	−0.877804	−	0.479020i	$-0.840992\pi$
$139$	−20.6983	−1.75561	−0.877804	+	0.479020i	$0.840992\pi$
$140$	−14.3492	−1.21273
$141$	5.02827	0.423457
$142$	5.51414	0.462736
$143$	0	0
$144$	−4.26434	−0.355361
$145$	−3.32088	−0.275784
$146$	32.0192	2.64993
$147$	6.09936	0.503067
$148$	−40.2745	−3.31054
$149$	−0.641769	−0.0525758	−0.0262879	−	0.999654i	$-0.508369\pi$
$149$	−0.641769	−0.0525758	−0.0262879	+	0.999654i	$0.508369\pi$
$150$	−3.80675	−0.310820
$151$	−16.6363	−1.35384	−0.676922	−	0.736055i	$-0.736686\pi$
$151$	−16.6363	−1.35384	−0.676922	+	0.736055i	$0.736686\pi$
$152$	12.7977	1.03803
$153$	4.69832	0.379836
$154$	23.6700	1.90739
$155$	−3.80675	−0.305765
$156$	0	0
$157$	10.2553	0.818459	0.409230	−	0.912431i	$-0.365798\pi$
$157$	10.2553	0.818459	0.409230	+	0.912431i	$0.365798\pi$
$158$	−1.47133	−0.117052
$159$	17.6700	1.40133
$160$	−3.48586	−0.275582
$161$	1.61350	0.127161
$162$	16.0337	1.25973
$163$	1.37743	0.107889	0.0539444	−	0.998544i	$-0.482821\pi$
$163$	1.37743	0.107889	0.0539444	+	0.998544i	$0.482821\pi$
$164$	6.97173	0.544400
$165$	4.29261	0.334179
$166$	19.3774	1.50398
$167$	−17.5761	−1.36008	−0.680042	−	0.733174i	$-0.738038\pi$
$167$	−17.5761	−1.36008	−0.680042	+	0.733174i	$0.738038\pi$
$168$	29.3401	2.26364
$169$	0	0
$170$	16.6983	1.28070
$171$	1.55148	0.118645
$172$	−3.76940	−0.287414
$173$	6.31181	0.479878	0.239939	−	0.970788i	$-0.422873\pi$
$173$	6.31181	0.479878	0.239939	+	0.970788i	$0.422873\pi$
$174$	12.6418	0.958370
$175$	−3.32088	−0.251035
$176$	17.0903	1.28823
$177$	13.3774	1.00551
$178$	8.58522	0.643490
$179$	−9.08482	−0.679031	−0.339516	−	0.940600i	$-0.610263\pi$
$179$	−9.08482	−0.679031	−0.339516	+	0.940600i	$0.610263\pi$
$180$	−3.05655	−0.227822
$181$	12.0192	0.893380	0.446690	−	0.894689i	$-0.352603\pi$
$181$	12.0192	0.893380	0.446690	+	0.894689i	$0.352603\pi$
$182$	0	0
$183$	−5.61350	−0.414962
$184$	2.83502	0.209001
$185$	−9.32088	−0.685285
$186$	14.4913	1.06256
$187$	−18.8296	−1.37695
$188$	14.3492	1.04652
$189$	18.6418	1.35599
$190$	5.51414	0.400038
$191$	13.2835	0.961163	0.480582	−	0.876950i	$-0.340426\pi$
$191$	13.2835	0.961163	0.480582	+	0.876950i	$0.340426\pi$
$192$	−4.98546	−0.359795
$193$	−16.3684	−1.17822	−0.589110	−	0.808053i	$-0.700522\pi$
$193$	−16.3684	−1.17822	−0.589110	+	0.808053i	$0.700522\pi$
$194$	1.61350	0.115842
$195$	0	0
$196$	17.4057	1.24326
$197$	−14.8970	−1.06137	−0.530685	−	0.847569i	$-0.678065\pi$
$197$	−14.8970	−1.06137	−0.530685	+	0.847569i	$0.678065\pi$
$198$	5.04201	0.358320
$199$	−16.3118	−1.15631	−0.578157	−	0.815926i	$-0.696228\pi$
$199$	−16.3118	−1.15631	−0.578157	+	0.815926i	$0.696228\pi$
$200$	−5.83502	−0.412598
$201$	−6.49960	−0.458446
$202$	−17.5279	−1.23326
$203$	11.0283	0.774033
$204$	−43.4532	−3.04233
$205$	1.61350	0.112691
$206$	18.0620	1.25844
$207$	0.343694	0.0238884
$208$	0	0
$209$	−6.21792	−0.430102
$210$	12.6418	0.872365
$211$	11.8013	0.812434	0.406217	−	0.913777i	$-0.366848\pi$
$211$	11.8013	0.812434	0.406217	+	0.913777i	$0.366848\pi$
$212$	50.4249	3.46320
$213$	−3.32088	−0.227543
$214$	24.0620	1.64485
$215$	−0.872368	−0.0594950
$216$	32.7549	2.22869
$217$	12.6418	0.858179
$218$	−26.8970	−1.82170
$219$	−19.2835	−1.30306
$220$	12.2498	0.825881
$221$	0	0
$222$	35.4823	2.38141
$223$	20.0192	1.34058	0.670292	−	0.742097i	$-0.266169\pi$
$223$	20.0192	1.34058	0.670292	+	0.742097i	$0.266169\pi$
$224$	11.5761	0.773464
$225$	−0.707389	−0.0471593
$226$	8.11310	0.539675
$227$	22.4623	1.49087	0.745436	−	0.666577i	$-0.232241\pi$
$227$	22.4623	1.49087	0.745436	+	0.666577i	$0.232241\pi$
$228$	−14.3492	−0.950296
$229$	10.3684	0.685160	0.342580	−	0.939489i	$-0.388699\pi$
$229$	10.3684	0.685160	0.342580	+	0.939489i	$0.388699\pi$
$230$	1.22153	0.0805451
$231$	−14.2553	−0.937928
$232$	19.3774	1.27219
$233$	11.3582	0.744102	0.372051	−	0.928212i	$-0.378655\pi$
$233$	11.3582	0.744102	0.372051	+	0.928212i	$0.378655\pi$
$234$	0	0
$235$	3.32088	0.216631
$236$	38.1751	2.48499
$237$	0.886105	0.0575587
$238$	−55.4532	−3.59450
$239$	−28.1186	−1.81884	−0.909419	−	0.415881i	$-0.863473\pi$
$239$	−28.1186	−1.81884	−0.909419	+	0.415881i	$0.863473\pi$
$240$	9.12763	0.589186
$241$	15.0848	0.971699	0.485849	−	0.874043i	$-0.338510\pi$
$241$	15.0848	0.971699	0.485849	+	0.874043i	$0.338510\pi$
$242$	7.44852	0.478809
$243$	7.18418	0.460865
$244$	−16.0192	−1.02552
$245$	4.02827	0.257357
$246$	−6.14217	−0.391610
$247$	0	0
$248$	22.2125	1.41049
$249$	−11.6700	−0.739559
$250$	−2.51414	−0.159008
$251$	1.15951	0.0731879	0.0365940	−	0.999330i	$-0.488349\pi$
$251$	1.15951	0.0731879	0.0365940	+	0.999330i	$0.488349\pi$
$252$	10.1504	0.639418
$253$	−1.37743	−0.0865984
$254$	−14.8350	−0.930832
$255$	−10.0565	−0.629765
$256$	−31.7549	−1.98468
$257$	−0.829557	−0.0517464	−0.0258732	−	0.999665i	$-0.508237\pi$
$257$	−0.829557	−0.0517464	−0.0258732	+	0.999665i	$0.508237\pi$
$258$	3.32088	0.206749
$259$	30.9536	1.92336
$260$	0	0
$261$	2.34916	0.145409
$262$	−8.58522	−0.530397
$263$	−25.6272	−1.58024	−0.790121	−	0.612950i	$-0.789983\pi$
$263$	−25.6272	−1.58024	−0.790121	+	0.612950i	$0.789983\pi$
$264$	−25.0475	−1.54157
$265$	11.6700	0.716885
$266$	−18.3118	−1.12277
$267$	−5.17044	−0.316426
$268$	−18.5479	−1.13299
$269$	5.02827	0.306579	0.153290	−	0.988181i	$-0.451013\pi$
$269$	5.02827	0.306579	0.153290	+	0.988181i	$0.451013\pi$
$270$	14.1131	0.858896
$271$	18.8916	1.14758	0.573791	−	0.819002i	$-0.305472\pi$
$271$	18.8916	1.14758	0.573791	+	0.819002i	$0.305472\pi$
$272$	−40.0384	−2.42768
$273$	0	0
$274$	−37.9253	−2.29115
$275$	2.83502	0.170958
$276$	−3.17872	−0.191336
$277$	23.7831	1.42899	0.714495	−	0.699640i	$-0.246656\pi$
$277$	23.7831	1.42899	0.714495	+	0.699640i	$0.246656\pi$
$278$	52.0384	3.12106
$279$	2.69285	0.161217
$280$	19.3774	1.15802
$281$	−31.5953	−1.88482	−0.942410	−	0.334459i	$-0.891446\pi$
$281$	−31.5953	−1.88482	−0.942410	+	0.334459i	$0.891446\pi$
$282$	−12.6418	−0.752806
$283$	6.35462	0.377743	0.188872	−	0.982002i	$-0.439517\pi$
$283$	6.35462	0.377743	0.188872	+	0.982002i	$0.439517\pi$
$284$	−9.47679	−0.562344
$285$	−3.32088	−0.196712
$286$	0	0
$287$	−5.35823	−0.316286
$288$	2.46586	0.145302
$289$	27.1131	1.59489
$290$	8.34916	0.490279
$291$	−0.971726	−0.0569636
$292$	−55.0293	−3.22035
$293$	8.03735	0.469547	0.234773	−	0.972050i	$-0.424565\pi$
$293$	8.03735	0.469547	0.234773	+	0.972050i	$0.424565\pi$
$294$	−15.3346	−0.894333
$295$	8.83502	0.514395
$296$	54.3876	3.16121
$297$	−15.9144	−0.923446
$298$	1.61350	0.0934673
$299$	0	0
$300$	6.54241	0.377726
$301$	2.89703	0.166982
$302$	41.8259	2.40681
$303$	10.5561	0.606435
$304$	−13.2215	−0.758307
$305$	−3.70739	−0.212284
$306$	−11.8122	−0.675259
$307$	15.8205	0.902923	0.451461	−	0.892291i	$-0.350903\pi$
$307$	15.8205	0.902923	0.451461	+	0.892291i	$0.350903\pi$
$308$	−40.6802	−2.31797
$309$	−10.8778	−0.618818
$310$	9.57068	0.543578
$311$	24.3118	1.37860	0.689298	−	0.724478i	$-0.257919\pi$
$311$	24.3118	1.37860	0.689298	+	0.724478i	$0.257919\pi$
$312$	0	0
$313$	−11.2270	−0.634587	−0.317294	−	0.948327i	$-0.602774\pi$
$313$	−11.2270	−0.634587	−0.317294	+	0.948327i	$0.602774\pi$
$314$	−25.7831	−1.45503
$315$	2.34916	0.132360
$316$	2.52867	0.142249
$317$	16.1504	0.907099	0.453550	−	0.891231i	$-0.350157\pi$
$317$	16.1504	0.907099	0.453550	+	0.891231i	$0.350157\pi$
$318$	−44.4249	−2.49123
$319$	−9.41478	−0.527126
$320$	−3.29261	−0.184063
$321$	−14.4913	−0.808827
$322$	−4.05655	−0.226063
$323$	14.5671	0.810534
$324$	−27.5561	−1.53090
$325$	0	0
$326$	−3.46305	−0.191801
$327$	16.1987	0.895791
$328$	−9.41478	−0.519844
$329$	−11.0283	−0.608008
$330$	−10.7922	−0.594092
$331$	−30.0620	−1.65236	−0.826179	−	0.563408i	$-0.809490\pi$
$331$	−30.0620	−1.65236	−0.826179	+	0.563408i	$0.809490\pi$
$332$	−33.3027	−1.82773
$333$	6.59349	0.361321
$334$	44.1888	2.41791
$335$	−4.29261	−0.234530
$336$	−30.3118	−1.65364
$337$	−17.4713	−0.951724	−0.475862	−	0.879520i	$-0.657864\pi$
$337$	−17.4713	−0.951724	−0.475862	+	0.879520i	$0.657864\pi$
$338$	0	0
$339$	−4.88611	−0.265377
$340$	−28.6983	−1.55638
$341$	−10.7922	−0.584431
$342$	−3.90064	−0.210923
$343$	9.86876	0.532863
$344$	5.09029	0.274450
$345$	−0.735663	−0.0396068
$346$	−15.8688	−0.853110
$347$	33.2407	1.78446	0.892228	−	0.451585i	$-0.149141\pi$
$347$	33.2407	1.78446	0.892228	+	0.451585i	$0.149141\pi$
$348$	−21.7266	−1.16467
$349$	20.8970	1.11859	0.559296	−	0.828968i	$-0.311071\pi$
$349$	20.8970	1.11859	0.559296	+	0.828968i	$0.311071\pi$
$350$	8.34916	0.446281
$351$	0	0
$352$	−9.88250	−0.526739
$353$	−12.5479	−0.667856	−0.333928	−	0.942599i	$-0.608374\pi$
$353$	−12.5479	−0.667856	−0.333928	+	0.942599i	$0.608374\pi$
$354$	−33.6327	−1.78756
$355$	−2.19325	−0.116406
$356$	−14.7549	−0.782006
$357$	33.3966	1.76754
$358$	22.8405	1.20716
$359$	2.97719	0.157130	0.0785650	−	0.996909i	$-0.474966\pi$
$359$	2.97719	0.157130	0.0785650	+	0.996909i	$0.474966\pi$
$360$	4.12763	0.217545
$361$	−14.1896	−0.746823
$362$	−30.2179	−1.58822
$363$	−4.48586	−0.235447
$364$	0	0
$365$	−12.7357	−0.666615
$366$	14.1131	0.737703
$367$	−30.5990	−1.59725	−0.798626	−	0.601827i	$-0.794440\pi$
$367$	−30.5990	−1.59725	−0.798626	+	0.601827i	$0.794440\pi$
$368$	−2.92892	−0.152680
$369$	−1.14137	−0.0594173
$370$	23.4340	1.21827
$371$	−38.7549	−2.01205
$372$	−24.9053	−1.29128
$373$	18.2553	0.945222	0.472611	−	0.881271i	$-0.343312\pi$
$373$	18.2553	0.945222	0.472611	+	0.881271i	$0.343312\pi$
$374$	47.3401	2.44790
$375$	1.51414	0.0781897
$376$	−19.3774	−0.999315
$377$	0	0
$378$	−46.8680	−2.41063
$379$	−15.9945	−0.821584	−0.410792	−	0.911729i	$-0.634748\pi$
$379$	−15.9945	−0.821584	−0.410792	+	0.911729i	$0.634748\pi$
$380$	−9.47679	−0.486149
$381$	8.93438	0.457722
$382$	−33.3966	−1.70872
$383$	2.16137	0.110441	0.0552204	−	0.998474i	$-0.482414\pi$
$383$	2.16137	0.110441	0.0552204	+	0.998474i	$0.482414\pi$
$384$	23.0903	1.17832
$385$	−9.41478	−0.479822
$386$	41.1523	2.09460
$387$	0.617104	0.0313691
$388$	−2.77301	−0.140778
$389$	−0.453981	−0.0230177	−0.0115089	−	0.999934i	$-0.503663\pi$
$389$	−0.453981	−0.0230177	−0.0115089	+	0.999934i	$0.503663\pi$
$390$	0	0
$391$	3.22699	0.163196
$392$	−23.5051	−1.18719
$393$	5.17044	0.260814
$394$	37.4532	1.88686
$395$	0.585221	0.0294457
$396$	−8.66538	−0.435452
$397$	29.2462	1.46782	0.733912	−	0.679244i	$-0.237692\pi$
$397$	29.2462	1.46782	0.733912	+	0.679244i	$0.237692\pi$
$398$	41.0101	2.05565
$399$	11.0283	0.552104
$400$	6.02827	0.301414
$401$	−15.2270	−0.760400	−0.380200	−	0.924904i	$-0.624145\pi$
$401$	−15.2270	−0.760400	−0.380200	+	0.924904i	$0.624145\pi$
$402$	16.3409	0.815009
$403$	0	0
$404$	30.1240	1.49873
$405$	−6.37743	−0.316897
$406$	−27.7266	−1.37605
$407$	−26.4249	−1.30983
$408$	58.6802	2.90510
$409$	23.6700	1.17041	0.585204	−	0.810886i	$-0.301014\pi$
$409$	23.6700	1.17041	0.585204	+	0.810886i	$0.301014\pi$
$410$	−4.05655	−0.200339
$411$	22.8405	1.12664
$412$	−31.0420	−1.52933
$413$	−29.3401	−1.44373
$414$	−0.864095	−0.0424679
$415$	−7.70739	−0.378341
$416$	0	0
$417$	−31.3401	−1.53473
$418$	15.6327	0.764620
$419$	−5.67004	−0.277000	−0.138500	−	0.990362i	$-0.544228\pi$
$419$	−5.67004	−0.277000	−0.138500	+	0.990362i	$0.544228\pi$
$420$	−21.7266	−1.06015
$421$	26.3009	1.28183	0.640913	−	0.767613i	$-0.278556\pi$
$421$	26.3009	1.28183	0.640913	+	0.767613i	$0.278556\pi$
$422$	−29.6700	−1.44432
$423$	−2.34916	−0.114220
$424$	−68.0950	−3.30698
$425$	−6.64177	−0.322173
$426$	8.34916	0.404518
$427$	12.3118	0.595810
$428$	−41.3538	−1.99891
$429$	0	0
$430$	2.19325	0.105768
$431$	−5.09029	−0.245190	−0.122595	−	0.992457i	$-0.539122\pi$
$431$	−5.09029	−0.245190	−0.122595	+	0.992457i	$0.539122\pi$
$432$	−33.8397	−1.62811
$433$	24.0565	1.15608	0.578042	−	0.816007i	$-0.303817\pi$
$433$	24.0565	1.15608	0.578042	+	0.816007i	$0.303817\pi$
$434$	−31.7831	−1.52564
$435$	−5.02827	−0.241087
$436$	46.2262	2.21383
$437$	1.06562	0.0509755
$438$	48.4815	2.31653
$439$	−7.21606	−0.344404	−0.172202	−	0.985062i	$-0.555088\pi$
$439$	−7.21606	−0.344404	−0.172202	+	0.985062i	$0.555088\pi$
$440$	−16.5424	−0.788628
$441$	−2.84956	−0.135693
$442$	0	0
$443$	27.3829	1.30100	0.650500	−	0.759506i	$-0.274559\pi$
$443$	27.3829	1.30100	0.650500	+	0.759506i	$0.274559\pi$
$444$	−60.9811	−2.89403
$445$	−3.41478	−0.161876
$446$	−50.3310	−2.38324
$447$	−0.971726	−0.0459611
$448$	10.9344	0.516601
$449$	−1.42571	−0.0672833	−0.0336416	−	0.999434i	$-0.510710\pi$
$449$	−1.42571	−0.0672833	−0.0336416	+	0.999434i	$0.510710\pi$
$450$	1.77847	0.0838381
$451$	4.57429	0.215395
$452$	−13.9435	−0.655845
$453$	−25.1896	−1.18351
$454$	−56.4732	−2.65042
$455$	0	0
$456$	19.3774	0.907431
$457$	35.3219	1.65229	0.826145	−	0.563457i	$-0.190529\pi$
$457$	35.3219	1.65229	0.826145	+	0.563457i	$0.190529\pi$
$458$	−26.0675	−1.21805
$459$	37.2835	1.74025
$460$	−2.09936	−0.0978831
$461$	−29.1979	−1.35988	−0.679941	−	0.733267i	$-0.737995\pi$
$461$	−29.1979	−1.35988	−0.679941	+	0.733267i	$0.737995\pi$
$462$	35.8397	1.66741
$463$	3.32088	0.154335	0.0771673	−	0.997018i	$-0.475412\pi$
$463$	3.32088	0.154335	0.0771673	+	0.997018i	$0.475412\pi$
$464$	−20.0192	−0.929368
$465$	−5.76394	−0.267296
$466$	−28.5561	−1.32284
$467$	21.5525	0.997333	0.498666	−	0.866794i	$-0.333823\pi$
$467$	21.5525	0.997333	0.498666	+	0.866794i	$0.333823\pi$
$468$	0	0
$469$	14.2553	0.658247
$470$	−8.34916	−0.385118
$471$	15.5279	0.715487
$472$	−51.5525	−2.37290
$473$	−2.47318	−0.113717
$474$	−2.22779	−0.102326
$475$	−2.19325	−0.100633
$476$	95.3038	4.36824
$477$	−8.25526	−0.377983
$478$	70.6939	3.23346
$479$	−5.42024	−0.247657	−0.123829	−	0.992304i	$-0.539517\pi$
$479$	−5.42024	−0.247657	−0.123829	+	0.992304i	$0.539517\pi$
$480$	−5.27807	−0.240910
$481$	0	0
$482$	−37.9253	−1.72745
$483$	2.44305	0.111163
$484$	−12.8013	−0.581877
$485$	−0.641769	−0.0291412
$486$	−18.0620	−0.819309
$487$	−17.4521	−0.790831	−0.395416	−	0.918502i	$-0.629399\pi$
$487$	−17.4521	−0.790831	−0.395416	+	0.918502i	$0.629399\pi$
$488$	21.6327	0.979266
$489$	2.08562	0.0943150
$490$	−10.1276	−0.457520
$491$	24.6236	1.11125	0.555624	−	0.831433i	$-0.312479\pi$
$491$	24.6236	1.11125	0.555624	+	0.831433i	$0.312479\pi$
$492$	10.5561	0.475908
$493$	22.0565	0.993377
$494$	0	0
$495$	−2.00546	−0.0901389
$496$	−22.9481	−1.03040
$497$	7.28354	0.326711
$498$	29.3401	1.31476
$499$	−32.3629	−1.44876	−0.724381	−	0.689400i	$-0.757874\pi$
$499$	−32.3629	−1.44876	−0.724381	+	0.689400i	$0.757874\pi$
$500$	4.32088	0.193236
$501$	−26.6127	−1.18897
$502$	−2.91518	−0.130111
$503$	−26.7411	−1.19233	−0.596164	−	0.802863i	$-0.703309\pi$
$503$	−26.7411	−1.19233	−0.596164	+	0.802863i	$0.703309\pi$
$504$	−13.7074	−0.610576
$505$	6.97173	0.310238
$506$	3.46305	0.153951
$507$	0	0
$508$	25.4960	1.13120
$509$	−9.74474	−0.431928	−0.215964	−	0.976401i	$-0.569289\pi$
$509$	−9.74474	−0.431928	−0.215964	+	0.976401i	$0.569289\pi$
$510$	25.2835	1.11957
$511$	42.2937	1.87096
$512$	49.3365	2.18038
$513$	12.3118	0.543580
$514$	2.08562	0.0919928
$515$	−7.18418	−0.316573
$516$	−5.70739	−0.251254
$517$	9.41478	0.414061
$518$	−77.8215	−3.41928
$519$	9.55695	0.419503
$520$	0	0
$521$	−11.4340	−0.500932	−0.250466	−	0.968125i	$-0.580584\pi$
$521$	−11.4340	−0.500932	−0.250466	+	0.968125i	$0.580584\pi$
$522$	−5.90611	−0.258503
$523$	33.6272	1.47042	0.735208	−	0.677841i	$-0.237084\pi$
$523$	33.6272	1.47042	0.735208	+	0.677841i	$0.237084\pi$
$524$	14.7549	0.644569
$525$	−5.02827	−0.219452
$526$	64.4304	2.80930
$527$	25.2835	1.10137
$528$	25.8770	1.12615
$529$	−22.7639	−0.989736
$530$	−29.3401	−1.27445
$531$	−6.24980	−0.271218
$532$	31.4713	1.36446
$533$	0	0
$534$	12.9992	0.562530
$535$	−9.57068	−0.413777
$536$	25.0475	1.08189
$537$	−13.7557	−0.593601
$538$	−12.6418	−0.545025
$539$	11.4202	0.491905
$540$	−24.2553	−1.04378
$541$	−26.4431	−1.13688	−0.568438	−	0.822726i	$-0.692452\pi$
$541$	−26.4431	−1.13688	−0.568438	+	0.822726i	$0.692452\pi$
$542$	−47.4960	−2.04013
$543$	18.1987	0.780982
$544$	23.1523	0.992647
$545$	10.6983	0.458266
$546$	0	0
$547$	−21.9681	−0.939289	−0.469644	−	0.882856i	$-0.655618\pi$
$547$	−21.9681	−0.939289	−0.469644	+	0.882856i	$0.655618\pi$
$548$	65.1798	2.78434
$549$	2.62257	0.111928
$550$	−7.12763	−0.303923
$551$	7.28354	0.310289
$552$	4.29261	0.182706
$553$	−1.94345	−0.0826440
$554$	−59.7941	−2.54041
$555$	−14.1131	−0.599068
$556$	−89.4350	−3.79289
$557$	39.3027	1.66531	0.832655	−	0.553792i	$-0.186820\pi$
$557$	39.3027	1.66531	0.832655	+	0.553792i	$0.186820\pi$
$558$	−6.77020	−0.286605
$559$	0	0
$560$	−20.0192	−0.845966
$561$	−28.5105	−1.20372
$562$	79.4350	3.35076
$563$	−23.0420	−0.971105	−0.485552	−	0.874208i	$-0.661382\pi$
$563$	−23.0420	−0.971105	−0.485552	+	0.874208i	$0.661382\pi$
$564$	21.7266	0.914854
$565$	−3.22699	−0.135761
$566$	−15.9764	−0.671538
$567$	21.1787	0.889422
$568$	12.7977	0.536979
$569$	−3.63270	−0.152291	−0.0761453	−	0.997097i	$-0.524261\pi$
$569$	−3.63270	−0.152291	−0.0761453	+	0.997097i	$0.524261\pi$
$570$	8.34916	0.349708
$571$	−43.3785	−1.81533	−0.907667	−	0.419692i	$-0.862138\pi$
$571$	−43.3785	−1.81533	−0.907667	+	0.419692i	$0.862138\pi$
$572$	0	0
$573$	20.1131	0.840237
$574$	13.4713	0.562282
$575$	−0.485863	−0.0202619
$576$	2.32916	0.0970482
$577$	−8.84876	−0.368379	−0.184189	−	0.982891i	$-0.558966\pi$
$577$	−8.84876	−0.368379	−0.184189	+	0.982891i	$0.558966\pi$
$578$	−68.1660	−2.83533
$579$	−24.7839	−1.02999
$580$	−14.3492	−0.595816
$581$	25.5953	1.06187
$582$	2.44305	0.101268
$583$	33.0848	1.37023
$584$	74.3129	3.07509
$585$	0	0
$586$	−20.2070	−0.834743
$587$	15.3209	0.632361	0.316180	−	0.948699i	$-0.397600\pi$
$587$	15.3209	0.632361	0.316180	+	0.948699i	$0.397600\pi$
$588$	26.3546	1.08685
$589$	8.34916	0.344021
$590$	−22.2125	−0.914472
$591$	−22.5561	−0.927836
$592$	−56.1888	−2.30935
$593$	4.52867	0.185970	0.0929852	−	0.995667i	$-0.470359\pi$
$593$	4.52867	0.185970	0.0929852	+	0.995667i	$0.470359\pi$
$594$	40.0109	1.64167
$595$	22.0565	0.904230
$596$	−2.77301	−0.113587
$597$	−24.6983	−1.01083
$598$	0	0
$599$	23.0283	0.940910	0.470455	−	0.882424i	$-0.344090\pi$
$599$	23.0283	0.940910	0.470455	+	0.882424i	$0.344090\pi$
$600$	−8.83502	−0.360688
$601$	−12.0565	−0.491797	−0.245898	−	0.969296i	$-0.579083\pi$
$601$	−12.0565	−0.491797	−0.245898	+	0.969296i	$0.579083\pi$
$602$	−7.28354	−0.296855
$603$	3.03655	0.123658
$604$	−71.8836	−2.92490
$605$	−2.96265	−0.120449
$606$	−26.5396	−1.07810
$607$	−8.99639	−0.365152	−0.182576	−	0.983192i	$-0.558444\pi$
$607$	−8.99639	−0.365152	−0.182576	+	0.983192i	$0.558444\pi$
$608$	7.64538	0.310061
$609$	16.6983	0.676650
$610$	9.32088	0.377392
$611$	0	0
$612$	20.3009	0.820615
$613$	14.1131	0.570023	0.285011	−	0.958524i	$-0.408003\pi$
$613$	14.1131	0.570023	0.285011	+	0.958524i	$0.408003\pi$
$614$	−39.7749	−1.60518
$615$	2.44305	0.0985134
$616$	54.9354	2.21341
$617$	2.46120	0.0990841	0.0495420	−	0.998772i	$-0.484224\pi$
$617$	2.46120	0.0990841	0.0495420	+	0.998772i	$0.484224\pi$
$618$	27.3484	1.10011
$619$	−5.73205	−0.230391	−0.115195	−	0.993343i	$-0.536749\pi$
$619$	−5.73205	−0.230391	−0.115195	+	0.993343i	$0.536749\pi$
$620$	−16.4485	−0.660588
$621$	2.72739	0.109446
$622$	−61.1232	−2.45082
$623$	11.3401	0.454331
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.2262	1.12815
$627$	−9.41478	−0.375990
$628$	44.3118	1.76823
$629$	61.9072	2.46840
$630$	−5.90611	−0.235305
$631$	5.27807	0.210117	0.105058	−	0.994466i	$-0.466497\pi$
$631$	5.27807	0.210117	0.105058	+	0.994466i	$0.466497\pi$
$632$	−3.41478	−0.135833
$633$	17.8688	0.710219
$634$	−40.6044	−1.61261
$635$	5.90064	0.234160
$636$	76.3502	3.02748
$637$	0	0
$638$	23.6700	0.937106
$639$	1.55148	0.0613757
$640$	15.2498	0.602801
$641$	−42.9992	−1.69837	−0.849183	−	0.528098i	$-0.822905\pi$
$641$	−42.9992	−1.69837	−0.849183	+	0.528098i	$0.822905\pi$
$642$	36.4332	1.43790
$643$	−43.3593	−1.70992	−0.854962	−	0.518691i	$-0.826419\pi$
$643$	−43.3593	−1.70992	−0.854962	+	0.518691i	$0.826419\pi$
$644$	6.97173	0.274724
$645$	−1.32088	−0.0520098
$646$	−36.6236	−1.44094
$647$	−19.6454	−0.772339	−0.386170	−	0.922428i	$-0.626202\pi$
$647$	−19.6454	−0.772339	−0.386170	+	0.922428i	$0.626202\pi$
$648$	37.2125	1.46184
$649$	25.0475	0.983199
$650$	0	0
$651$	19.1414	0.750209
$652$	5.95173	0.233088
$653$	9.93252	0.388690	0.194345	−	0.980933i	$-0.437742\pi$
$653$	9.93252	0.388690	0.194345	+	0.980933i	$0.437742\pi$
$654$	−40.7258	−1.59250
$655$	3.41478	0.133426
$656$	9.72659	0.379760
$657$	9.00907	0.351477
$658$	27.7266	1.08090
$659$	10.1806	0.396579	0.198289	−	0.980144i	$-0.436461\pi$
$659$	10.1806	0.396579	0.198289	+	0.980144i	$0.436461\pi$
$660$	18.5479	0.721975
$661$	32.9427	1.28132	0.640660	−	0.767824i	$-0.278661\pi$
$661$	32.9427	1.28132	0.640660	+	0.767824i	$0.278661\pi$
$662$	75.5800	2.93750
$663$	0	0
$664$	44.9728	1.74528
$665$	7.28354	0.282443
$666$	−16.5769	−0.642344
$667$	1.61350	0.0624748
$668$	−75.9445	−2.93838
$669$	30.3118	1.17192
$670$	10.7922	0.416939
$671$	−10.5105	−0.405754
$672$	17.5279	0.676152
$673$	−11.2270	−0.432769	−0.216384	−	0.976308i	$-0.569426\pi$
$673$	−11.2270	−0.432769	−0.216384	+	0.976308i	$0.569426\pi$
$674$	43.9253	1.69194
$675$	−5.61350	−0.216064
$676$	0	0
$677$	−15.7447	−0.605119	−0.302560	−	0.953130i	$-0.597841\pi$
$677$	−15.7447	−0.605119	−0.302560	+	0.953130i	$0.597841\pi$
$678$	12.2843	0.471777
$679$	2.13124	0.0817895
$680$	38.7549	1.48618
$681$	34.0109	1.30330
$682$	27.1331	1.03898
$683$	−50.3492	−1.92656	−0.963279	−	0.268504i	$-0.913471\pi$
$683$	−50.3492	−1.92656	−0.963279	+	0.268504i	$0.913471\pi$
$684$	6.70378	0.256325
$685$	15.0848	0.576361
$686$	−24.8114	−0.947304
$687$	15.6991	0.598959
$688$	−5.25887	−0.200493
$689$	0	0
$690$	1.84956	0.0704115
$691$	−8.19325	−0.311686	−0.155843	−	0.987782i	$-0.549809\pi$
$691$	−8.19325	−0.311686	−0.155843	+	0.987782i	$0.549809\pi$
$692$	27.2726	1.03675
$693$	6.65991	0.252989
$694$	−83.5717	−3.17234
$695$	−20.6983	−0.785132
$696$	29.3401	1.11213
$697$	−10.7165	−0.405915
$698$	−52.5380	−1.98859
$699$	17.1979	0.650485
$700$	−14.3492	−0.542347
$701$	−8.25526	−0.311797	−0.155899	−	0.987773i	$-0.549827\pi$
$701$	−8.25526	−0.311797	−0.155899	+	0.987773i	$0.549827\pi$
$702$	0	0
$703$	20.4431	0.771024
$704$	−9.33462	−0.351812
$705$	5.02827	0.189376
$706$	31.5471	1.18729
$707$	−23.1523	−0.870732
$708$	57.8023	2.17234
$709$	−37.7831	−1.41898	−0.709488	−	0.704718i	$-0.751074\pi$
$709$	−37.7831	−1.41898	−0.709488	+	0.704718i	$0.751074\pi$
$710$	5.51414	0.206942
$711$	−0.413979	−0.0155254
$712$	19.9253	0.746732
$713$	1.84956	0.0692665
$714$	−83.9637	−3.14226
$715$	0	0
$716$	−39.2545	−1.46701
$717$	−42.5753	−1.59001
$718$	−7.48506	−0.279340
$719$	−22.2443	−0.829574	−0.414787	−	0.909919i	$-0.636144\pi$
$719$	−22.2443	−0.829574	−0.414787	+	0.909919i	$0.636144\pi$
$720$	−4.26434	−0.158922
$721$	23.8578	0.888512
$722$	35.6747	1.32768
$723$	22.8405	0.849447
$724$	51.9336	1.93010
$725$	−3.32088	−0.123335
$726$	11.2781	0.418569
$727$	−30.0812	−1.11565	−0.557825	−	0.829958i	$-0.688364\pi$
$727$	−30.0812	−1.11565	−0.557825	+	0.829958i	$0.688364\pi$
$728$	0	0
$729$	30.0101	1.11149
$730$	32.0192	1.18508
$731$	5.79407	0.214301
$732$	−24.2553	−0.896500
$733$	35.0101	1.29313	0.646564	−	0.762859i	$-0.276205\pi$
$733$	35.0101	1.29313	0.646564	+	0.762859i	$0.276205\pi$
$734$	76.9300	2.83954
$735$	6.09936	0.224978
$736$	1.69365	0.0624288
$737$	−12.1696	−0.448275
$738$	2.86956	0.105630
$739$	−42.6856	−1.57022	−0.785108	−	0.619359i	$-0.787393\pi$
$739$	−42.6856	−1.57022	−0.785108	+	0.619359i	$0.787393\pi$
$740$	−40.2745	−1.48052
$741$	0	0
$742$	97.4350	3.57695
$743$	−13.6892	−0.502210	−0.251105	−	0.967960i	$-0.580794\pi$
$743$	−13.6892	−0.502210	−0.251105	+	0.967960i	$0.580794\pi$
$744$	33.6327	1.23303
$745$	−0.641769	−0.0235126
$746$	−45.8962	−1.68038
$747$	5.45213	0.199483
$748$	−81.3603	−2.97483
$749$	31.7831	1.16133
$750$	−3.80675	−0.139003
$751$	17.7831	0.648916	0.324458	−	0.945900i	$-0.394818\pi$
$751$	17.7831	0.648916	0.324458	+	0.945900i	$0.394818\pi$
$752$	20.0192	0.730025
$753$	1.75566	0.0639799
$754$	0	0
$755$	−16.6363	−0.605457
$756$	80.5489	2.92954
$757$	−26.9717	−0.980304	−0.490152	−	0.871637i	$-0.663059\pi$
$757$	−26.9717	−0.980304	−0.490152	+	0.871637i	$0.663059\pi$
$758$	40.2125	1.46058
$759$	−2.08562	−0.0757032
$760$	12.7977	0.464220
$761$	35.1523	1.27427	0.637135	−	0.770752i	$-0.280119\pi$
$761$	35.1523	1.27427	0.637135	+	0.770752i	$0.280119\pi$
$762$	−22.4623	−0.813722
$763$	−35.5279	−1.28620
$764$	57.3966	2.07654
$765$	4.69832	0.169868
$766$	−5.43398	−0.196338
$767$	0	0
$768$	−48.0812	−1.73498
$769$	−23.1523	−0.834893	−0.417447	−	0.908701i	$-0.637075\pi$
$769$	−23.1523	−0.834893	−0.417447	+	0.908701i	$0.637075\pi$
$770$	23.6700	0.853009
$771$	−1.25606	−0.0452360
$772$	−70.7258	−2.54548
$773$	41.6218	1.49703	0.748515	−	0.663117i	$-0.230767\pi$
$773$	41.6218	1.49703	0.748515	+	0.663117i	$0.230767\pi$
$774$	−1.55148	−0.0557669
$775$	−3.80675	−0.136742
$776$	3.74474	0.134428
$777$	46.8680	1.68138
$778$	1.14137	0.0409201
$779$	−3.53880	−0.126791
$780$	0	0
$781$	−6.21792	−0.222495
$782$	−8.11310	−0.290124
$783$	18.6418	0.666202
$784$	24.2835	0.867269
$785$	10.2553	0.366026
$786$	−12.9992	−0.463666
$787$	−43.2353	−1.54117	−0.770585	−	0.637337i	$-0.780036\pi$
$787$	−43.2353	−1.54117	−0.770585	+	0.637337i	$0.780036\pi$
$788$	−64.3684	−2.29303
$789$	−38.8031	−1.38143
$790$	−1.47133	−0.0523474
$791$	10.7165	0.381034
$792$	11.7019	0.415810
$793$	0	0
$794$	−73.5289	−2.60944
$795$	17.6700	0.626692
$796$	−70.4815	−2.49815
$797$	−14.4431	−0.511599	−0.255800	−	0.966730i	$-0.582339\pi$
$797$	−14.4431	−0.511599	−0.255800	+	0.966730i	$0.582339\pi$
$798$	−27.7266	−0.981511
$799$	−22.0565	−0.780305
$800$	−3.48586	−0.123244
$801$	2.41558	0.0853503
$802$	38.2827	1.35181
$803$	−36.1059	−1.27415
$804$	−28.0840	−0.990447
$805$	1.61350	0.0568682
$806$	0	0
$807$	7.61350	0.268008
$808$	−40.6802	−1.43112
$809$	17.8880	0.628907	0.314454	−	0.949273i	$-0.398179\pi$
$809$	17.8880	0.628907	0.314454	+	0.949273i	$0.398179\pi$
$810$	16.0337	0.563368
$811$	8.69285	0.305247	0.152624	−	0.988284i	$-0.451228\pi$
$811$	8.69285	0.305247	0.152624	+	0.988284i	$0.451228\pi$
$812$	47.6519	1.67225
$813$	28.6044	1.00320
$814$	66.4358	2.32857
$815$	1.37743	0.0482493
$816$	−60.6236	−2.12225
$817$	1.91332	0.0669387
$818$	−59.5097	−2.08071
$819$	0	0
$820$	6.97173	0.243463
$821$	9.41478	0.328578	0.164289	−	0.986412i	$-0.447467\pi$
$821$	9.41478	0.328578	0.164289	+	0.986412i	$0.447467\pi$
$822$	−57.4241	−2.00290
$823$	−30.3365	−1.05746	−0.528732	−	0.848789i	$-0.677332\pi$
$823$	−30.3365	−1.05746	−0.528732	+	0.848789i	$0.677332\pi$
$824$	41.9198	1.46035
$825$	4.29261	0.149450
$826$	73.7650	2.56661
$827$	18.2361	0.634130	0.317065	−	0.948404i	$-0.397303\pi$
$827$	18.2361	0.634130	0.317065	+	0.948404i	$0.397303\pi$
$828$	1.48506	0.0516095
$829$	29.8023	1.03508	0.517539	−	0.855660i	$-0.326848\pi$
$829$	29.8023	1.03508	0.517539	+	0.855660i	$0.326848\pi$
$830$	19.3774	0.672600
$831$	36.0109	1.24921
$832$	0	0
$833$	−26.7549	−0.927001
$834$	78.7933	2.72839
$835$	−17.5761	−0.608248
$836$	−26.8669	−0.929211
$837$	21.3692	0.738626
$838$	14.2553	0.492440
$839$	46.1004	1.59156	0.795782	−	0.605584i	$-0.207060\pi$
$839$	46.1004	1.59156	0.795782	+	0.605584i	$0.207060\pi$
$840$	29.3401	1.01233
$841$	−17.9717	−0.619715
$842$	−66.1240	−2.27878
$843$	−47.8397	−1.64769
$844$	50.9920	1.75522
$845$	0	0
$846$	5.90611	0.203056
$847$	9.83863	0.338059
$848$	70.3502	2.41584
$849$	9.62177	0.330218
$850$	16.6983	0.572748
$851$	4.52867	0.155241
$852$	−14.3492	−0.491594
$853$	19.3774	0.663471	0.331735	−	0.943373i	$-0.392366\pi$
$853$	19.3774	0.663471	0.331735	+	0.943373i	$0.392366\pi$
$854$	−30.9536	−1.05921
$855$	1.55148	0.0530596
$856$	55.8452	1.90875
$857$	−43.4713	−1.48495	−0.742476	−	0.669873i	$-0.766349\pi$
$857$	−43.4713	−1.48495	−0.742476	+	0.669873i	$0.766349\pi$
$858$	0	0
$859$	−27.7375	−0.946392	−0.473196	−	0.880957i	$-0.656900\pi$
$859$	−27.7375	−0.946392	−0.473196	+	0.880957i	$0.656900\pi$
$860$	−3.76940	−0.128536
$861$	−8.11310	−0.276494
$862$	12.7977	0.435891
$863$	−3.69646	−0.125829	−0.0629145	−	0.998019i	$-0.520040\pi$
$863$	−3.69646	−0.125829	−0.0629145	+	0.998019i	$0.520040\pi$
$864$	19.5679	0.665713
$865$	6.31181	0.214608
$866$	−60.4815	−2.05524
$867$	41.0529	1.39423
$868$	54.6236	1.85405
$869$	1.65911	0.0562816
$870$	12.6418	0.428596
$871$	0	0
$872$	−62.4249	−2.11397
$873$	0.453981	0.0153649
$874$	−2.67912	−0.0906224
$875$	−3.32088	−0.112266
$876$	−83.3219	−2.81519
$877$	20.5671	0.694501	0.347250	−	0.937772i	$-0.387115\pi$
$877$	20.5671	0.694501	0.347250	+	0.937772i	$0.387115\pi$
$878$	18.1422	0.612269
$879$	12.1696	0.410472
$880$	17.0903	0.576113
$881$	−14.5369	−0.489762	−0.244881	−	0.969553i	$-0.578749\pi$
$881$	−14.5369	−0.489762	−0.244881	+	0.969553i	$0.578749\pi$
$882$	7.16418	0.241230
$883$	−36.2871	−1.22116	−0.610580	−	0.791955i	$-0.709064\pi$
$883$	−36.2871	−1.22116	−0.610580	+	0.791955i	$0.709064\pi$
$884$	0	0
$885$	13.3774	0.449677
$886$	−68.8444	−2.31287
$887$	−45.2407	−1.51903	−0.759517	−	0.650487i	$-0.774565\pi$
$887$	−45.2407	−1.51903	−0.759517	+	0.650487i	$0.774565\pi$
$888$	82.3502	2.76349
$889$	−19.5953	−0.657207
$890$	8.58522	0.287777
$891$	−18.0802	−0.605708
$892$	86.5007	2.89626
$893$	−7.28354	−0.243734
$894$	2.44305	0.0817079
$895$	−9.08482	−0.303672
$896$	−50.6428	−1.69186
$897$	0	0
$898$	3.58442	0.119614
$899$	12.6418	0.421627
$900$	−3.05655	−0.101885
$901$	−77.5097	−2.58222
$902$	−11.5004	−0.382921
$903$	4.38650	0.145974
$904$	18.8296	0.626262
$905$	12.0192	0.399532
$906$	63.3302	2.10401
$907$	−11.2973	−0.375120	−0.187560	−	0.982253i	$-0.560058\pi$
$907$	−11.2973	−0.375120	−0.187560	+	0.982253i	$0.560058\pi$
$908$	97.0568	3.22094
$909$	−4.93172	−0.163575
$910$	0	0
$911$	−41.1979	−1.36495	−0.682474	−	0.730910i	$-0.739096\pi$
$911$	−41.1979	−1.36495	−0.682474	+	0.730910i	$0.739096\pi$
$912$	−20.0192	−0.662902
$913$	−21.8506	−0.723150
$914$	−88.8042	−2.93738
$915$	−5.61350	−0.185576
$916$	44.8005	1.48025
$917$	−11.3401	−0.374483
$918$	−93.7359	−3.09375
$919$	1.74474	0.0575535	0.0287768	−	0.999586i	$-0.490839\pi$
$919$	1.74474	0.0575535	0.0287768	+	0.999586i	$0.490839\pi$
$920$	2.83502	0.0934679
$921$	23.9544	0.789324
$922$	73.4076	2.41755
$923$	0	0
$924$	−61.5953	−2.02634
$925$	−9.32088	−0.306469
$926$	−8.34916	−0.274370
$927$	5.08201	0.166915
$928$	11.5761	0.380006
$929$	−15.5569	−0.510407	−0.255203	−	0.966887i	$-0.582142\pi$
$929$	−15.5569	−0.510407	−0.255203	+	0.966887i	$0.582142\pi$
$930$	14.4913	0.475189
$931$	−8.83502	−0.289556
$932$	49.0776	1.60759
$933$	36.8114	1.20515
$934$	−54.1860	−1.77302
$935$	−18.8296	−0.615792
$936$	0	0
$937$	26.0384	0.850638	0.425319	−	0.905044i	$-0.360162\pi$
$937$	26.0384	0.850638	0.425319	+	0.905044i	$0.360162\pi$
$938$	−35.8397	−1.17021
$939$	−16.9992	−0.554748
$940$	14.3492	0.468018
$941$	−50.8789	−1.65860	−0.829302	−	0.558800i	$-0.811262\pi$
$941$	−50.8789	−1.65860	−0.829302	+	0.558800i	$0.811262\pi$
$942$	−39.0392	−1.27197
$943$	−0.783938	−0.0255285
$944$	53.2599	1.73346
$945$	18.6418	0.606417
$946$	6.21792	0.202162
$947$	−37.8770	−1.23084	−0.615419	−	0.788200i	$-0.711013\pi$
$947$	−37.8770	−1.23084	−0.615419	+	0.788200i	$0.711013\pi$
$948$	3.82876	0.124352
$949$	0	0
$950$	5.51414	0.178902
$951$	24.4540	0.792975
$952$	−128.700	−4.17120
$953$	−32.0950	−1.03966	−0.519829	−	0.854271i	$-0.674004\pi$
$953$	−32.0950	−1.03966	−0.519829	+	0.854271i	$0.674004\pi$
$954$	20.7549	0.671964
$955$	13.2835	0.429845
$956$	−121.497	−3.92950
$957$	−14.2553	−0.460807
$958$	13.6272	0.440276
$959$	−50.0950	−1.61765
$960$	−4.98546	−0.160905
$961$	−16.5087	−0.532538
$962$	0	0
$963$	6.77020	0.218167
$964$	65.1798	2.09930
$965$	−16.3684	−0.526916
$966$	−6.14217	−0.197621
$967$	24.4057	0.784835	0.392417	−	0.919787i	$-0.371639\pi$
$967$	24.4057	0.784835	0.392417	+	0.919787i	$0.371639\pi$
$968$	17.2871	0.555630
$969$	22.0565	0.708558
$970$	1.61350	0.0518062
$971$	−36.9354	−1.18531	−0.592657	−	0.805455i	$-0.701921\pi$
$971$	−36.9354	−1.18531	−0.592657	+	0.805455i	$0.701921\pi$
$972$	31.0420	0.995673
$973$	68.7367	2.20360
$974$	43.8770	1.40591
$975$	0	0
$976$	−22.3492	−0.715379
$977$	25.1715	0.805308	0.402654	−	0.915352i	$-0.368088\pi$
$977$	25.1715	0.805308	0.402654	+	0.915352i	$0.368088\pi$
$978$	−5.24354	−0.167670
$979$	−9.68097	−0.309405
$980$	17.4057	0.556005
$981$	−7.56788	−0.241624
$982$	−61.9072	−1.97554
$983$	−13.1896	−0.420684	−0.210342	−	0.977628i	$-0.567458\pi$
$983$	−13.1896	−0.420684	−0.210342	+	0.977628i	$0.567458\pi$
$984$	−14.2553	−0.454441
$985$	−14.8970	−0.474659
$986$	−55.4532	−1.76599
$987$	−16.6983	−0.531513
$988$	0	0
$989$	0.423851	0.0134777
$990$	5.04201	0.160246
$991$	23.0667	0.732737	0.366369	−	0.930470i	$-0.380601\pi$
$991$	23.0667	0.732737	0.366369	+	0.930470i	$0.380601\pi$
$992$	13.2698	0.421317
$993$	−45.5180	−1.44447
$994$	−18.3118	−0.580815
$995$	−16.3118	−0.517119
$996$	−50.4249	−1.59777
$997$	−0.630841	−0.0199789	−0.00998947	−	0.999950i	$-0.503180\pi$
$997$	−0.630841	−0.0199789	−0.00998947	+	0.999950i	$0.503180\pi$
$998$	81.3648	2.57556
$999$	52.3227	1.65542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.i.1.1		3
3.2	odd	2		7605.2.a.cc.1.3		3
5.4	even	2		4225.2.a.be.1.3		3
13.2	odd	12		845.2.m.h.316.1		12
13.3	even	3		845.2.e.k.191.3		6
13.4	even	6		845.2.e.i.146.1		6
13.5	odd	4		65.2.c.a.51.6	yes	6
13.6	odd	12		845.2.m.h.361.6		12
13.7	odd	12		845.2.m.h.361.1		12
13.8	odd	4		65.2.c.a.51.1	✓	6
13.9	even	3		845.2.e.k.146.3		6
13.10	even	6		845.2.e.i.191.1		6
13.11	odd	12		845.2.m.h.316.6		12
13.12	even	2		845.2.a.k.1.3		3
39.5	even	4		585.2.b.g.181.1		6
39.8	even	4		585.2.b.g.181.6		6
39.38	odd	2		7605.2.a.bs.1.1		3
52.31	even	4		1040.2.k.d.961.1		6
52.47	even	4		1040.2.k.d.961.2		6
65.8	even	4		325.2.d.e.324.2		6
65.18	even	4		325.2.d.f.324.6		6
65.34	odd	4		325.2.c.g.51.6		6
65.44	odd	4		325.2.c.g.51.1		6
65.47	even	4		325.2.d.f.324.5		6
65.57	even	4		325.2.d.e.324.1		6
65.64	even	2		4225.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.c.a.51.1	✓	6	13.8	odd	4
65.2.c.a.51.6	yes	6	13.5	odd	4
325.2.c.g.51.1		6	65.44	odd	4
325.2.c.g.51.6		6	65.34	odd	4
325.2.d.e.324.1		6	65.57	even	4
325.2.d.e.324.2		6	65.8	even	4
325.2.d.f.324.5		6	65.47	even	4
325.2.d.f.324.6		6	65.18	even	4
585.2.b.g.181.1		6	39.5	even	4
585.2.b.g.181.6		6	39.8	even	4
845.2.a.i.1.1		3	1.1	even	1	trivial
845.2.a.k.1.3		3	13.12	even	2
845.2.e.i.146.1		6	13.4	even	6
845.2.e.i.191.1		6	13.10	even	6
845.2.e.k.146.3		6	13.9	even	3
845.2.e.k.191.3		6	13.3	even	3
845.2.m.h.316.1		12	13.2	odd	12
845.2.m.h.316.6		12	13.11	odd	12
845.2.m.h.361.1		12	13.7	odd	12
845.2.m.h.361.6		12	13.6	odd	12
1040.2.k.d.961.1		6	52.31	even	4
1040.2.k.d.961.2		6	52.47	even	4
4225.2.a.bc.1.1		3	65.64	even	2
4225.2.a.be.1.3		3	5.4	even	2
7605.2.a.bs.1.1		3	39.38	odd	2
7605.2.a.cc.1.3		3	3.2	odd	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51414 q^{2} +1.51414 q^{3} +4.32088 q^{4} +1.00000 q^{5} -3.80675 q^{6} -3.32088 q^{7} -5.83502 q^{8} -0.707389 q^{9} +O(q^{10})\)	\(q-2.51414 q^{2} +1.51414 q^{3} +4.32088 q^{4} +1.00000 q^{5} -3.80675 q^{6} -3.32088 q^{7} -5.83502 q^{8} -0.707389 q^{9} -2.51414 q^{10} +2.83502 q^{11} +6.54241 q^{12} +8.34916 q^{14} +1.51414 q^{15} +6.02827 q^{16} -6.64177 q^{17} +1.77847 q^{18} -2.19325 q^{19} +4.32088 q^{20} -5.02827 q^{21} -7.12763 q^{22} -0.485863 q^{23} -8.83502 q^{24} +1.00000 q^{25} -5.61350 q^{27} -14.3492 q^{28} -3.32088 q^{29} -3.80675 q^{30} -3.80675 q^{31} -3.48586 q^{32} +4.29261 q^{33} +16.6983 q^{34} -3.32088 q^{35} -3.05655 q^{36} -9.32088 q^{37} +5.51414 q^{38} -5.83502 q^{40} +1.61350 q^{41} +12.6418 q^{42} -0.872368 q^{43} +12.2498 q^{44} -0.707389 q^{45} +1.22153 q^{46} +3.32088 q^{47} +9.12763 q^{48} +4.02827 q^{49} -2.51414 q^{50} -10.0565 q^{51} +11.6700 q^{53} +14.1131 q^{54} +2.83502 q^{55} +19.3774 q^{56} -3.32088 q^{57} +8.34916 q^{58} +8.83502 q^{59} +6.54241 q^{60} -3.70739 q^{61} +9.57068 q^{62} +2.34916 q^{63} -3.29261 q^{64} -10.7922 q^{66} -4.29261 q^{67} -28.6983 q^{68} -0.735663 q^{69} +8.34916 q^{70} -2.19325 q^{71} +4.12763 q^{72} -12.7357 q^{73} +23.4340 q^{74} +1.51414 q^{75} -9.47679 q^{76} -9.41478 q^{77} +0.585221 q^{79} +6.02827 q^{80} -6.37743 q^{81} -4.05655 q^{82} -7.70739 q^{83} -21.7266 q^{84} -6.64177 q^{85} +2.19325 q^{86} -5.02827 q^{87} -16.5424 q^{88} -3.41478 q^{89} +1.77847 q^{90} -2.09936 q^{92} -5.76394 q^{93} -8.34916 q^{94} -2.19325 q^{95} -5.27807 q^{96} -0.641769 q^{97} -10.1276 q^{98} -2.00546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 2 q^{3} + 5 q^{4} + 3 q^{5} - 10 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - q^2 - 2 * q^3 + 5 * q^4 + 3 * q^5 - 10 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - q^{2} - 2 q^{3} + 5 q^{4} + 3 q^{5} - 10 q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{15} + 5 q^{16} - 4 q^{17} + 17 q^{18} - 8 q^{19} + 5 q^{20} - 2 q^{21} - 12 q^{22} - 8 q^{23} - 12 q^{24} + 3 q^{25} - 14 q^{27} - 22 q^{28} - 2 q^{29} - 10 q^{30} - 10 q^{31} - 17 q^{32} + 18 q^{33} + 8 q^{34} - 2 q^{35} + 17 q^{36} - 20 q^{37} + 10 q^{38} - 3 q^{40} + 2 q^{41} + 22 q^{42} - 12 q^{43} + 12 q^{44} + 3 q^{45} - 8 q^{46} + 2 q^{47} + 18 q^{48} - q^{49} - q^{50} - 4 q^{51} + 6 q^{53} - 10 q^{54} - 6 q^{55} + 24 q^{56} - 2 q^{57} + 4 q^{58} + 12 q^{59} - 6 q^{61} - 4 q^{62} - 14 q^{63} - 15 q^{64} + 12 q^{66} - 18 q^{67} - 44 q^{68} + 16 q^{69} + 4 q^{70} - 8 q^{71} + 3 q^{72} - 20 q^{73} + 10 q^{74} - 2 q^{75} + 2 q^{76} - 18 q^{77} + 12 q^{79} + 5 q^{80} + 15 q^{81} + 14 q^{82} - 18 q^{83} - 10 q^{84} - 4 q^{85} + 8 q^{86} - 2 q^{87} - 30 q^{88} + 17 q^{90} - 10 q^{92} + 14 q^{93} - 4 q^{94} - 8 q^{95} + 22 q^{96} + 14 q^{97} - 21 q^{98} - 12 q^{99}+O(q^{100}) \) 3 * q - q^2 - 2 * q^3 + 5 * q^4 + 3 * q^5 - 10 * q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 - q^10 - 6 * q^11 + 4 * q^14 - 2 * q^15 + 5 * q^16 - 4 * q^17 + 17 * q^18 - 8 * q^19 + 5 * q^20 - 2 * q^21 - 12 * q^22 - 8 * q^23 - 12 * q^24 + 3 * q^25 - 14 * q^27 - 22 * q^28 - 2 * q^29 - 10 * q^30 - 10 * q^31 - 17 * q^32 + 18 * q^33 + 8 * q^34 - 2 * q^35 + 17 * q^36 - 20 * q^37 + 10 * q^38 - 3 * q^40 + 2 * q^41 + 22 * q^42 - 12 * q^43 + 12 * q^44 + 3 * q^45 - 8 * q^46 + 2 * q^47 + 18 * q^48 - q^49 - q^50 - 4 * q^51 + 6 * q^53 - 10 * q^54 - 6 * q^55 + 24 * q^56 - 2 * q^57 + 4 * q^58 + 12 * q^59 - 6 * q^61 - 4 * q^62 - 14 * q^63 - 15 * q^64 + 12 * q^66 - 18 * q^67 - 44 * q^68 + 16 * q^69 + 4 * q^70 - 8 * q^71 + 3 * q^72 - 20 * q^73 + 10 * q^74 - 2 * q^75 + 2 * q^76 - 18 * q^77 + 12 * q^79 + 5 * q^80 + 15 * q^81 + 14 * q^82 - 18 * q^83 - 10 * q^84 - 4 * q^85 + 8 * q^86 - 2 * q^87 - 30 * q^88 + 17 * q^90 - 10 * q^92 + 14 * q^93 - 4 * q^94 - 8 * q^95 + 22 * q^96 + 14 * q^97 - 21 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more