LMFDB - Embedded newform 2009.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, 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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 10x^{5} - x^{4} + 30x^{3} + 7x^{2} - 25x - 11 $ x^7 - 10x^5 - x^4 + 30x^3 + 7x^2 - 25x - 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.907606$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17625 q^{2} -1.90761 q^{3} +2.73607 q^{4} -2.13420 q^{5} +4.15143 q^{6} -1.60187 q^{8} +0.638961 q^{9} +O(q^{10})$	\(q-2.17625 q^{2} -1.90761 q^{3} +2.73607 q^{4} -2.13420 q^{5} +4.15143 q^{6} -1.60187 q^{8} +0.638961 q^{9} +4.64455 q^{10} +4.92459 q^{11} -5.21934 q^{12} -0.239109 q^{13} +4.07121 q^{15} -1.98606 q^{16} -4.37833 q^{17} -1.39054 q^{18} -6.65962 q^{19} -5.83931 q^{20} -10.7171 q^{22} +4.38959 q^{23} +3.05574 q^{24} -0.445201 q^{25} +0.520362 q^{26} +4.50393 q^{27} -7.48428 q^{29} -8.85997 q^{30} +4.59534 q^{31} +7.52592 q^{32} -9.39417 q^{33} +9.52834 q^{34} +1.74824 q^{36} +4.09370 q^{37} +14.4930 q^{38} +0.456127 q^{39} +3.41871 q^{40} +1.00000 q^{41} +6.01695 q^{43} +13.4740 q^{44} -1.36367 q^{45} -9.55285 q^{46} +5.72810 q^{47} +3.78863 q^{48} +0.968869 q^{50} +8.35212 q^{51} -0.654220 q^{52} +7.81993 q^{53} -9.80169 q^{54} -10.5100 q^{55} +12.7039 q^{57} +16.2877 q^{58} +3.04167 q^{59} +11.1391 q^{60} -2.61543 q^{61} -10.0006 q^{62} -12.4062 q^{64} +0.510307 q^{65} +20.4441 q^{66} -6.95926 q^{67} -11.9794 q^{68} -8.37361 q^{69} +4.66286 q^{71} -1.02353 q^{72} +4.91552 q^{73} -8.90893 q^{74} +0.849268 q^{75} -18.2212 q^{76} -0.992646 q^{78} -5.90139 q^{79} +4.23865 q^{80} -10.5086 q^{81} -2.17625 q^{82} +15.5637 q^{83} +9.34421 q^{85} -13.0944 q^{86} +14.2770 q^{87} -7.88856 q^{88} -5.24975 q^{89} +2.96769 q^{90} +12.0102 q^{92} -8.76610 q^{93} -12.4658 q^{94} +14.2129 q^{95} -14.3565 q^{96} +0.950227 q^{97} +3.14662 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) $ 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9	$ 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) $ 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9 - 8 * q^10 - 12 * q^12 - q^13 - 4 * q^15 + 5 * q^16 + 11 * q^17 + 16 * q^18 - 9 * q^19 - 12 * q^20 + 3 * q^23 + 23 * q^24 + 31 * q^25 - 10 * q^26 - 22 * q^27 - 14 * q^29 + 6 * q^30 - 34 * q^31 - 20 * q^32 - 12 * q^33 - 15 * q^34 + q^36 + 7 * q^37 + 39 * q^38 - 22 * q^39 - 50 * q^40 + 7 * q^41 - 3 * q^43 + 26 * q^44 + 4 * q^45 - 8 * q^46 - 17 * q^47 - 17 * q^48 + 11 * q^50 + 8 * q^51 + 25 * q^52 + 24 * q^53 - 68 * q^54 - 48 * q^55 + 22 * q^57 - 38 * q^58 - 4 * q^59 - 6 * q^60 - 16 * q^61 - 24 * q^62 + 8 * q^64 - 6 * q^65 + 12 * q^66 - 24 * q^67 + 10 * q^68 - 35 * q^69 - 12 * q^71 - 11 * q^72 - 14 * q^73 - 6 * q^74 + 19 * q^75 - 42 * q^76 - 29 * q^78 - 8 * q^79 + 92 * q^80 + 15 * q^81 - q^82 - 14 * q^83 + 16 * q^85 + 35 * q^86 + 20 * q^87 + 22 * q^88 + 19 * q^89 - 24 * q^90 - 10 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^95 + 25 * q^96 - 23 * q^97 + 24 * 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 $

$2$	−2.17625	−1.53884	−0.769421	−	0.638742i	$-0.779455\pi$
$2$	−2.17625	−1.53884	−0.769421	+	0.638742i	$0.779455\pi$
$3$	−1.90761	−1.10136	−0.550678	−	0.834717i	$-0.685631\pi$
$3$	−1.90761	−1.10136	−0.550678	+	0.834717i	$0.685631\pi$
$4$	2.73607	1.36803
$5$	−2.13420	−0.954442	−0.477221	−	0.878783i	$-0.658356\pi$
$5$	−2.13420	−0.954442	−0.477221	+	0.878783i	$0.658356\pi$
$6$	4.15143	1.69481
$7$	0	0
$7$	0	0
$8$	−1.60187	−0.566347
$9$	0.638961	0.212987
$10$	4.64455	1.46874
$11$	4.92459	1.48482	0.742410	−	0.669946i	$-0.233683\pi$
$11$	4.92459	1.48482	0.742410	+	0.669946i	$0.233683\pi$
$12$	−5.21934	−1.50669
$13$	−0.239109	−0.0663170	−0.0331585	−	0.999450i	$-0.510557\pi$
$13$	−0.239109	−0.0663170	−0.0331585	+	0.999450i	$0.510557\pi$
$14$	0	0
$15$	4.07121	1.05118
$16$	−1.98606	−0.496516
$17$	−4.37833	−1.06190	−0.530950	−	0.847403i	$-0.678165\pi$
$17$	−4.37833	−1.06190	−0.530950	+	0.847403i	$0.678165\pi$
$18$	−1.39054	−0.327753
$19$	−6.65962	−1.52782	−0.763911	−	0.645322i	$-0.776723\pi$
$19$	−6.65962	−1.52782	−0.763911	+	0.645322i	$0.776723\pi$
$20$	−5.83931	−1.30571
$21$	0	0
$22$	−10.7171	−2.28490
$23$	4.38959	0.915293	0.457647	−	0.889134i	$-0.348693\pi$
$23$	4.38959	0.915293	0.457647	+	0.889134i	$0.348693\pi$
$24$	3.05574	0.623750
$25$	−0.445201	−0.0890402
$26$	0.520362	0.102051
$27$	4.50393	0.866782
$28$	0	0
$29$	−7.48428	−1.38980	−0.694898	−	0.719109i	$-0.744550\pi$
$29$	−7.48428	−1.38980	−0.694898	+	0.719109i	$0.744550\pi$
$30$	−8.85997	−1.61760
$31$	4.59534	0.825348	0.412674	−	0.910879i	$-0.364595\pi$
$31$	4.59534	0.825348	0.412674	+	0.910879i	$0.364595\pi$
$32$	7.52592	1.33041
$33$	−9.39417	−1.63532
$34$	9.52834	1.63410
$35$	0	0
$36$	1.74824	0.291374
$37$	4.09370	0.673001	0.336500	−	0.941683i	$-0.390757\pi$
$37$	4.09370	0.673001	0.336500	+	0.941683i	$0.390757\pi$
$38$	14.4930	2.35108
$39$	0.456127	0.0730387
$40$	3.41871	0.540546
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	6.01695	0.917576	0.458788	−	0.888546i	$-0.348284\pi$
$43$	6.01695	0.917576	0.458788	+	0.888546i	$0.348284\pi$
$44$	13.4740	2.03128
$45$	−1.36367	−0.203284
$46$	−9.55285	−1.40849
$47$	5.72810	0.835529	0.417765	−	0.908555i	$-0.362814\pi$
$47$	5.72810	0.835529	0.417765	+	0.908555i	$0.362814\pi$
$48$	3.78863	0.546841
$49$	0	0
$50$	0.968869	0.137019
$51$	8.35212	1.16953
$52$	−0.654220	−0.0907240
$53$	7.81993	1.07415	0.537075	−	0.843535i	$-0.319529\pi$
$53$	7.81993	1.07415	0.537075	+	0.843535i	$0.319529\pi$
$54$	−9.80169	−1.33384
$55$	−10.5100	−1.41717
$56$	0	0
$57$	12.7039	1.68268
$58$	16.2877	2.13868
$59$	3.04167	0.395992	0.197996	−	0.980203i	$-0.436557\pi$
$59$	3.04167	0.395992	0.197996	+	0.980203i	$0.436557\pi$
$60$	11.1391	1.43805
$61$	−2.61543	−0.334871	−0.167436	−	0.985883i	$-0.553549\pi$
$61$	−2.61543	−0.334871	−0.167436	+	0.985883i	$0.553549\pi$
$62$	−10.0006	−1.27008
$63$	0	0
$64$	−12.4062	−1.55077
$65$	0.510307	0.0632958
$66$	20.4441	2.51649
$67$	−6.95926	−0.850208	−0.425104	−	0.905144i	$-0.639763\pi$
$67$	−6.95926	−0.850208	−0.425104	+	0.905144i	$0.639763\pi$
$68$	−11.9794	−1.45272
$69$	−8.37361	−1.00806
$70$	0	0
$71$	4.66286	0.553379	0.276690	−	0.960959i	$-0.410763\pi$
$71$	4.66286	0.553379	0.276690	+	0.960959i	$0.410763\pi$
$72$	−1.02353	−0.120625
$73$	4.91552	0.575318	0.287659	−	0.957733i	$-0.407123\pi$
$73$	4.91552	0.575318	0.287659	+	0.957733i	$0.407123\pi$
$74$	−8.90893	−1.03564
$75$	0.849268	0.0980650
$76$	−18.2212	−2.09011
$77$	0	0
$78$	−0.992646	−0.112395
$79$	−5.90139	−0.663959	−0.331979	−	0.943287i	$-0.607716\pi$
$79$	−5.90139	−0.663959	−0.331979	+	0.943287i	$0.607716\pi$
$80$	4.23865	0.473896
$81$	−10.5086	−1.16762
$82$	−2.17625	−0.240327
$83$	15.5637	1.70834	0.854170	−	0.519994i	$-0.174066\pi$
$83$	15.5637	1.70834	0.854170	+	0.519994i	$0.174066\pi$
$84$	0	0
$85$	9.34421	1.01352
$86$	−13.0944	−1.41200
$87$	14.2770	1.53066
$88$	−7.88856	−0.840923
$89$	−5.24975	−0.556473	−0.278236	−	0.960513i	$-0.589750\pi$
$89$	−5.24975	−0.556473	−0.278236	+	0.960513i	$0.589750\pi$
$90$	2.96769	0.312822
$91$	0	0
$92$	12.0102	1.25215
$93$	−8.76610	−0.909003
$94$	−12.4658	−1.28575
$95$	14.2129	1.45822
$96$	−14.3565	−1.46525
$97$	0.950227	0.0964809	0.0482404	−	0.998836i	$-0.484639\pi$
$97$	0.950227	0.0964809	0.0482404	+	0.998836i	$0.484639\pi$
$98$	0	0
$99$	3.14662	0.316247
$100$	−1.21810	−0.121810
$101$	−12.4673	−1.24055	−0.620273	−	0.784386i	$-0.712978\pi$
$101$	−12.4673	−1.24055	−0.620273	+	0.784386i	$0.712978\pi$
$102$	−18.1763	−1.79972
$103$	−11.7848	−1.16119	−0.580597	−	0.814191i	$-0.697181\pi$
$103$	−11.7848	−1.16119	−0.580597	+	0.814191i	$0.697181\pi$
$104$	0.383023	0.0375585
$105$	0	0
$106$	−17.0181	−1.65295
$107$	1.05166	0.101668	0.0508341	−	0.998707i	$-0.483812\pi$
$107$	1.05166	0.101668	0.0508341	+	0.998707i	$0.483812\pi$
$108$	12.3231	1.18579
$109$	9.79953	0.938625	0.469313	−	0.883032i	$-0.344502\pi$
$109$	9.79953	0.938625	0.469313	+	0.883032i	$0.344502\pi$
$110$	22.8725	2.18081
$111$	−7.80917	−0.741214
$112$	0	0
$113$	6.57739	0.618749	0.309374	−	0.950940i	$-0.399880\pi$
$113$	6.57739	0.618749	0.309374	+	0.950940i	$0.399880\pi$
$114$	−27.6469	−2.58937
$115$	−9.36826	−0.873594
$116$	−20.4775	−1.90129
$117$	−0.152782	−0.0141247
$118$	−6.61944	−0.609369
$119$	0	0
$120$	−6.52155	−0.595334
$121$	13.2516	1.20469
$122$	5.69183	0.515314
$123$	−1.90761	−0.172003
$124$	12.5732	1.12910
$125$	11.6211	1.03943
$126$	0	0
$127$	−12.8781	−1.14274	−0.571371	−	0.820692i	$-0.693588\pi$
$127$	−12.8781	−1.14274	−0.571371	+	0.820692i	$0.693588\pi$
$128$	11.9471	1.05598
$129$	−11.4780	−1.01058
$130$	−1.11056	−0.0974022
$131$	16.8118	1.46886	0.734428	−	0.678686i	$-0.237450\pi$
$131$	16.8118	1.46886	0.734428	+	0.678686i	$0.237450\pi$
$132$	−25.7031	−2.23717
$133$	0	0
$134$	15.1451	1.30834
$135$	−9.61228	−0.827293
$136$	7.01352	0.601404
$137$	−18.7370	−1.60081	−0.800406	−	0.599458i	$-0.795383\pi$
$137$	−18.7370	−1.60081	−0.800406	+	0.599458i	$0.795383\pi$
$138$	18.2231	1.55125
$139$	2.08549	0.176889	0.0884445	−	0.996081i	$-0.471810\pi$
$139$	2.08549	0.176889	0.0884445	+	0.996081i	$0.471810\pi$
$140$	0	0
$141$	−10.9270	−0.920216
$142$	−10.1475	−0.851563
$143$	−1.17752	−0.0984688
$144$	−1.26902	−0.105751
$145$	15.9729	1.32648
$146$	−10.6974	−0.885324
$147$	0	0
$148$	11.2007	0.920688
$149$	−3.81434	−0.312483	−0.156241	−	0.987719i	$-0.549938\pi$
$149$	−3.81434	−0.312483	−0.156241	+	0.987719i	$0.549938\pi$
$150$	−1.84822	−0.150907
$151$	−5.72342	−0.465765	−0.232883	−	0.972505i	$-0.574816\pi$
$151$	−5.72342	−0.465765	−0.232883	+	0.972505i	$0.574816\pi$
$152$	10.6679	0.865278
$153$	−2.79758	−0.226171
$154$	0	0
$155$	−9.80737	−0.787747
$156$	1.24799	0.0999195
$157$	−18.0841	−1.44327	−0.721633	−	0.692276i	$-0.756608\pi$
$157$	−18.0841	−1.44327	−0.721633	+	0.692276i	$0.756608\pi$
$158$	12.8429	1.02173
$159$	−14.9173	−1.18302
$160$	−16.0618	−1.26980
$161$	0	0
$162$	22.8694	1.79679
$163$	3.53244	0.276682	0.138341	−	0.990385i	$-0.455823\pi$
$163$	3.53244	0.276682	0.138341	+	0.990385i	$0.455823\pi$
$164$	2.73607	0.213651
$165$	20.0490	1.56081
$166$	−33.8706	−2.62887
$167$	22.0487	1.70618	0.853090	−	0.521764i	$-0.174726\pi$
$167$	22.0487	1.70618	0.853090	+	0.521764i	$0.174726\pi$
$168$	0	0
$169$	−12.9428	−0.995602
$170$	−20.3354	−1.55965
$171$	−4.25524	−0.325406
$172$	16.4628	1.25528
$173$	17.1151	1.30124	0.650619	−	0.759404i	$-0.274509\pi$
$173$	17.1151	1.30124	0.650619	+	0.759404i	$0.274509\pi$
$174$	−31.0704	−2.35544
$175$	0	0
$176$	−9.78054	−0.737236
$177$	−5.80231	−0.436129
$178$	11.4248	0.856324
$179$	−14.1730	−1.05934	−0.529671	−	0.848203i	$-0.677685\pi$
$179$	−14.1730	−1.05934	−0.529671	+	0.848203i	$0.677685\pi$
$180$	−3.73109	−0.278099
$181$	−19.4752	−1.44758	−0.723789	−	0.690022i	$-0.757601\pi$
$181$	−19.4752	−1.44758	−0.723789	+	0.690022i	$0.757601\pi$
$182$	0	0
$183$	4.98921	0.368813
$184$	−7.03156	−0.518374
$185$	−8.73677	−0.642340
$186$	19.0772	1.39881
$187$	−21.5615	−1.57673
$188$	15.6725	1.14303
$189$	0	0
$190$	−30.9309	−2.24397
$191$	26.1416	1.89154	0.945768	−	0.324842i	$-0.105311\pi$
$191$	26.1416	1.89154	0.945768	+	0.324842i	$0.105311\pi$
$192$	23.6661	1.70795
$193$	17.5030	1.25989	0.629947	−	0.776638i	$-0.283077\pi$
$193$	17.5030	1.25989	0.629947	+	0.776638i	$0.283077\pi$
$194$	−2.06793	−0.148469
$195$	−0.973464	−0.0697112
$196$	0	0
$197$	0.888388	0.0632950	0.0316475	−	0.999499i	$-0.489925\pi$
$197$	0.888388	0.0632950	0.0316475	+	0.999499i	$0.489925\pi$
$198$	−6.84783	−0.486654
$199$	−23.1635	−1.64202	−0.821010	−	0.570914i	$-0.806589\pi$
$199$	−23.1635	−1.64202	−0.821010	+	0.570914i	$0.806589\pi$
$200$	0.713155	0.0504277
$201$	13.2755	0.936383
$202$	27.1320	1.90900
$203$	0	0
$204$	22.8520	1.59996
$205$	−2.13420	−0.149059
$206$	25.6467	1.78689
$207$	2.80478	0.194946
$208$	0.474887	0.0329275
$209$	−32.7959	−2.26854
$210$	0	0
$211$	−10.1208	−0.696743	−0.348372	−	0.937357i	$-0.613265\pi$
$211$	−10.1208	−0.696743	−0.348372	+	0.937357i	$0.613265\pi$
$212$	21.3959	1.46947
$213$	−8.89489	−0.609468
$214$	−2.28868	−0.156451
$215$	−12.8414	−0.875773
$216$	−7.21472	−0.490900
$217$	0	0
$218$	−21.3262	−1.44440
$219$	−9.37688	−0.633631
$220$	−28.7562	−1.93874
$221$	1.04690	0.0704221
$222$	16.9947	1.14061
$223$	−25.1767	−1.68596	−0.842978	−	0.537947i	$-0.819200\pi$
$223$	−25.1767	−1.68596	−0.842978	+	0.537947i	$0.819200\pi$
$224$	0	0
$225$	−0.284466	−0.0189644
$226$	−14.3141	−0.952157
$227$	−22.6244	−1.50163	−0.750817	−	0.660511i	$-0.770340\pi$
$227$	−22.6244	−1.50163	−0.750817	+	0.660511i	$0.770340\pi$
$228$	34.7588	2.30196
$229$	−19.6654	−1.29953	−0.649763	−	0.760137i	$-0.725132\pi$
$229$	−19.6654	−1.29953	−0.649763	+	0.760137i	$0.725132\pi$
$230$	20.3877	1.34432
$231$	0	0
$232$	11.9888	0.787107
$233$	−14.9513	−0.979490	−0.489745	−	0.871866i	$-0.662910\pi$
$233$	−14.9513	−0.979490	−0.489745	+	0.871866i	$0.662910\pi$
$234$	0.332491	0.0217356
$235$	−12.2249	−0.797464
$236$	8.32223	0.541731
$237$	11.2575	0.731256
$238$	0	0
$239$	−8.31838	−0.538071	−0.269036	−	0.963130i	$-0.586705\pi$
$239$	−8.31838	−0.538071	−0.269036	+	0.963130i	$0.586705\pi$
$240$	−8.08568	−0.521928
$241$	−10.8581	−0.699431	−0.349716	−	0.936856i	$-0.613722\pi$
$241$	−10.8581	−0.699431	−0.349716	+	0.936856i	$0.613722\pi$
$242$	−28.8387	−1.85383
$243$	6.53449	0.419188
$244$	−7.15599	−0.458115
$245$	0	0
$246$	4.15143	0.264686
$247$	1.59238	0.101321
$248$	−7.36115	−0.467434
$249$	−29.6894	−1.88149
$250$	−25.2905	−1.59951
$251$	−15.5522	−0.981647	−0.490824	−	0.871259i	$-0.663304\pi$
$251$	−15.5522	−0.981647	−0.490824	+	0.871259i	$0.663304\pi$
$252$	0	0
$253$	21.6169	1.35904
$254$	28.0259	1.75850
$255$	−17.8251	−1.11625
$256$	−1.18754	−0.0742212
$257$	6.82687	0.425849	0.212924	−	0.977069i	$-0.431701\pi$
$257$	6.82687	0.425849	0.212924	+	0.977069i	$0.431701\pi$
$258$	24.9789	1.55512
$259$	0	0
$260$	1.39623	0.0865908
$261$	−4.78216	−0.296008
$262$	−36.5868	−2.26034
$263$	−10.6509	−0.656760	−0.328380	−	0.944546i	$-0.606503\pi$
$263$	−10.6509	−0.656760	−0.328380	+	0.944546i	$0.606503\pi$
$264$	15.0483	0.926157
$265$	−16.6893	−1.02521
$266$	0	0
$267$	10.0145	0.612875
$268$	−19.0410	−1.16311
$269$	−2.05209	−0.125118	−0.0625590	−	0.998041i	$-0.519926\pi$
$269$	−2.05209	−0.125118	−0.0625590	+	0.998041i	$0.519926\pi$
$270$	20.9187	1.27307
$271$	−0.788473	−0.0478963	−0.0239482	−	0.999713i	$-0.507624\pi$
$271$	−0.788473	−0.0478963	−0.0239482	+	0.999713i	$0.507624\pi$
$272$	8.69563	0.527250
$273$	0	0
$274$	40.7765	2.46340
$275$	−2.19243	−0.132209
$276$	−22.9108	−1.37907
$277$	19.1241	1.14905	0.574527	−	0.818485i	$-0.305186\pi$
$277$	19.1241	1.14905	0.574527	+	0.818485i	$0.305186\pi$
$278$	−4.53855	−0.272204
$279$	2.93624	0.175788
$280$	0	0
$281$	4.91418	0.293155	0.146578	−	0.989199i	$-0.453174\pi$
$281$	4.91418	0.293155	0.146578	+	0.989199i	$0.453174\pi$
$282$	23.7798	1.41607
$283$	−25.4044	−1.51013	−0.755067	−	0.655647i	$-0.772396\pi$
$283$	−25.4044	−1.51013	−0.755067	+	0.655647i	$0.772396\pi$
$284$	12.7579	0.757042
$285$	−27.1127	−1.60602
$286$	2.56257	0.151528
$287$	0	0
$288$	4.80877	0.283359
$289$	2.16974	0.127632
$290$	−34.7611	−2.04124
$291$	−1.81266	−0.106260
$292$	13.4492	0.787055
$293$	23.5302	1.37465	0.687323	−	0.726352i	$-0.258786\pi$
$293$	23.5302	1.37465	0.687323	+	0.726352i	$0.258786\pi$
$294$	0	0
$295$	−6.49153	−0.377951
$296$	−6.55759	−0.381152
$297$	22.1800	1.28701
$298$	8.30096	0.480862
$299$	−1.04959	−0.0606995
$300$	2.32366	0.134156
$301$	0	0
$302$	12.4556	0.716739
$303$	23.7828	1.36628
$304$	13.2264	0.758588
$305$	5.58184	0.319615
$306$	6.08824	0.348041
$307$	12.4338	0.709637	0.354818	−	0.934935i	$-0.384543\pi$
$307$	12.4338	0.709637	0.354818	+	0.934935i	$0.384543\pi$
$308$	0	0
$309$	22.4808	1.27889
$310$	21.3433	1.21222
$311$	−13.8997	−0.788181	−0.394090	−	0.919072i	$-0.628940\pi$
$311$	−13.8997	−0.788181	−0.394090	+	0.919072i	$0.628940\pi$
$312$	−0.730656	−0.0413653
$313$	24.4460	1.38177	0.690885	−	0.722965i	$-0.257221\pi$
$313$	24.4460	1.38177	0.690885	+	0.722965i	$0.257221\pi$
$314$	39.3555	2.22096
$315$	0	0
$316$	−16.1466	−0.908319
$317$	−6.21542	−0.349093	−0.174546	−	0.984649i	$-0.555846\pi$
$317$	−6.21542	−0.349093	−0.174546	+	0.984649i	$0.555846\pi$
$318$	32.4639	1.82048
$319$	−36.8570	−2.06359
$320$	26.4772	1.48012
$321$	−2.00616	−0.111973
$322$	0	0
$323$	29.1580	1.62239
$324$	−28.7523	−1.59735
$325$	0.106452	0.00590488
$326$	−7.68749	−0.425771
$327$	−18.6937	−1.03376
$328$	−1.60187	−0.0884486
$329$	0	0
$330$	−43.6317	−2.40185
$331$	6.87529	0.377900	0.188950	−	0.981987i	$-0.439492\pi$
$331$	6.87529	0.377900	0.188950	+	0.981987i	$0.439492\pi$
$332$	42.5834	2.33707
$333$	2.61572	0.143340
$334$	−47.9835	−2.62554
$335$	14.8524	0.811475
$336$	0	0
$337$	−30.7360	−1.67430	−0.837149	−	0.546975i	$-0.815779\pi$
$337$	−30.7360	−1.67430	−0.837149	+	0.546975i	$0.815779\pi$
$338$	28.1668	1.53207
$339$	−12.5471	−0.681463
$340$	25.5664	1.38653
$341$	22.6302	1.22549
$342$	9.26047	0.500749
$343$	0	0
$344$	−9.63838	−0.519667
$345$	17.8709	0.962139
$346$	−37.2468	−2.00240
$347$	−37.1323	−1.99337	−0.996684	−	0.0813678i	$-0.974071\pi$
$347$	−37.1323	−1.99337	−0.996684	+	0.0813678i	$0.974071\pi$
$348$	39.0630	2.09400
$349$	1.97185	0.105551	0.0527754	−	0.998606i	$-0.483193\pi$
$349$	1.97185	0.105551	0.0527754	+	0.998606i	$0.483193\pi$
$350$	0	0
$351$	−1.07693	−0.0574824
$352$	37.0620	1.97541
$353$	−15.1347	−0.805538	−0.402769	−	0.915302i	$-0.631952\pi$
$353$	−15.1347	−0.805538	−0.402769	+	0.915302i	$0.631952\pi$
$354$	12.6273	0.671133
$355$	−9.95145	−0.528168
$356$	−14.3637	−0.761274
$357$	0	0
$358$	30.8441	1.63016
$359$	31.5328	1.66424	0.832120	−	0.554596i	$-0.187127\pi$
$359$	31.5328	1.66424	0.832120	+	0.554596i	$0.187127\pi$
$360$	2.18442	0.115129
$361$	25.3505	1.33424
$362$	42.3828	2.22759
$363$	−25.2788	−1.32679
$364$	0	0
$365$	−10.4907	−0.549108
$366$	−10.8578	−0.567544
$367$	14.5883	0.761504	0.380752	−	0.924677i	$-0.375665\pi$
$367$	14.5883	0.761504	0.380752	+	0.924677i	$0.375665\pi$
$368$	−8.71801	−0.454457
$369$	0.638961	0.0332630
$370$	19.0134	0.988460
$371$	0	0
$372$	−23.9847	−1.24355
$373$	1.64417	0.0851318	0.0425659	−	0.999094i	$-0.486447\pi$
$373$	1.64417	0.0851318	0.0425659	+	0.999094i	$0.486447\pi$
$374$	46.9231	2.42634
$375$	−22.1685	−1.14478
$376$	−9.17568	−0.473200
$377$	1.78956	0.0921671
$378$	0	0
$379$	11.0808	0.569181	0.284591	−	0.958649i	$-0.408142\pi$
$379$	11.0808	0.569181	0.284591	+	0.958649i	$0.408142\pi$
$380$	38.8876	1.99489
$381$	24.5662	1.25857
$382$	−56.8906	−2.91078
$383$	−34.7806	−1.77720	−0.888602	−	0.458679i	$-0.848323\pi$
$383$	−34.7806	−1.77720	−0.888602	+	0.458679i	$0.848323\pi$
$384$	−22.7903	−1.16301
$385$	0	0
$386$	−38.0909	−1.93878
$387$	3.84459	0.195432
$388$	2.59989	0.131989
$389$	32.2133	1.63328	0.816641	−	0.577146i	$-0.195834\pi$
$389$	32.2133	1.63328	0.816641	+	0.577146i	$0.195834\pi$
$390$	2.11850	0.107275
$391$	−19.2191	−0.971950
$392$	0	0
$393$	−32.0703	−1.61774
$394$	−1.93336	−0.0974010
$395$	12.5947	0.633710
$396$	8.60937	0.432637
$397$	21.9800	1.10314	0.551571	−	0.834128i	$-0.314028\pi$
$397$	21.9800	1.10314	0.551571	+	0.834128i	$0.314028\pi$
$398$	50.4097	2.52681
$399$	0	0
$400$	0.884197	0.0442099
$401$	−31.9672	−1.59637	−0.798183	−	0.602416i	$-0.794205\pi$
$401$	−31.9672	−1.59637	−0.798183	+	0.602416i	$0.794205\pi$
$402$	−28.8909	−1.44095
$403$	−1.09879	−0.0547346
$404$	−34.1115	−1.69711
$405$	22.4275	1.11443
$406$	0	0
$407$	20.1598	0.999285
$408$	−13.3790	−0.662361
$409$	33.6351	1.66315	0.831575	−	0.555412i	$-0.187439\pi$
$409$	33.6351	1.66315	0.831575	+	0.555412i	$0.187439\pi$
$410$	4.64455	0.229378
$411$	35.7429	1.76307
$412$	−32.2441	−1.58855
$413$	0	0
$414$	−6.10390	−0.299990
$415$	−33.2161	−1.63051
$416$	−1.79952	−0.0882286
$417$	−3.97830	−0.194818
$418$	71.3721	3.49092
$419$	32.9582	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$419$	32.9582	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$420$	0	0
$421$	−22.5858	−1.10076	−0.550382	−	0.834913i	$-0.685518\pi$
$421$	−22.5858	−1.10076	−0.550382	+	0.834913i	$0.685518\pi$
$422$	22.0254	1.07218
$423$	3.66003	0.177957
$424$	−12.5265	−0.608342
$425$	1.94923	0.0945518
$426$	19.3575	0.937875
$427$	0	0
$428$	2.87742	0.139086
$429$	2.24624	0.108449
$430$	27.9460	1.34768
$431$	−3.11663	−0.150123	−0.0750614	−	0.997179i	$-0.523915\pi$
$431$	−3.11663	−0.150123	−0.0750614	+	0.997179i	$0.523915\pi$
$432$	−8.94509	−0.430371
$433$	−29.4001	−1.41288	−0.706440	−	0.707773i	$-0.749700\pi$
$433$	−29.4001	−1.41288	−0.706440	+	0.707773i	$0.749700\pi$
$434$	0	0
$435$	−30.4700	−1.46093
$436$	26.8122	1.28407
$437$	−29.2330	−1.39840
$438$	20.4064	0.975057
$439$	20.0996	0.959301	0.479650	−	0.877460i	$-0.340763\pi$
$439$	20.0996	0.959301	0.479650	+	0.877460i	$0.340763\pi$
$440$	16.8357	0.802613
$441$	0	0
$442$	−2.27832	−0.108368
$443$	−40.5696	−1.92752	−0.963761	−	0.266769i	$-0.914044\pi$
$443$	−40.5696	−1.92752	−0.963761	+	0.266769i	$0.914044\pi$
$444$	−21.3664	−1.01401
$445$	11.2040	0.531121
$446$	54.7908	2.59442
$447$	7.27626	0.344155
$448$	0	0
$449$	−3.14475	−0.148410	−0.0742050	−	0.997243i	$-0.523642\pi$
$449$	−3.14475	−0.148410	−0.0742050	+	0.997243i	$0.523642\pi$
$450$	0.619069	0.0291832
$451$	4.92459	0.231890
$452$	17.9962	0.846470
$453$	10.9180	0.512974
$454$	49.2364	2.31078
$455$	0	0
$456$	−20.3501	−0.952979
$457$	−30.2554	−1.41529	−0.707644	−	0.706569i	$-0.750242\pi$
$457$	−30.2554	−1.41529	−0.707644	+	0.706569i	$0.750242\pi$
$458$	42.7969	1.99977
$459$	−19.7197	−0.920436
$460$	−25.6322	−1.19511
$461$	13.1445	0.612201	0.306100	−	0.951999i	$-0.400976\pi$
$461$	13.1445	0.612201	0.306100	+	0.951999i	$0.400976\pi$
$462$	0	0
$463$	−1.39124	−0.0646564	−0.0323282	−	0.999477i	$-0.510292\pi$
$463$	−1.39124	−0.0646564	−0.0323282	+	0.999477i	$0.510292\pi$
$464$	14.8642	0.690055
$465$	18.7086	0.867591
$466$	32.5377	1.50728
$467$	−13.3724	−0.618799	−0.309400	−	0.950932i	$-0.600128\pi$
$467$	−13.3724	−0.618799	−0.309400	+	0.950932i	$0.600128\pi$
$468$	−0.418021	−0.0193230
$469$	0	0
$470$	26.6044	1.22717
$471$	34.4973	1.58955
$472$	−4.87237	−0.224269
$473$	29.6310	1.36243
$474$	−24.4992	−1.12529
$475$	2.96487	0.136038
$476$	0	0
$477$	4.99663	0.228780
$478$	18.1029	0.828007
$479$	−29.3559	−1.34131	−0.670653	−	0.741772i	$-0.733986\pi$
$479$	−29.3559	−1.34131	−0.670653	+	0.741772i	$0.733986\pi$
$480$	30.6396	1.39850
$481$	−0.978843	−0.0446314
$482$	23.6299	1.07631
$483$	0	0
$484$	36.2572	1.64806
$485$	−2.02797	−0.0920854
$486$	−14.2207	−0.645064
$487$	11.4732	0.519901	0.259950	−	0.965622i	$-0.416294\pi$
$487$	11.4732	0.519901	0.259950	+	0.965622i	$0.416294\pi$
$488$	4.18958	0.189653
$489$	−6.73851	−0.304726
$490$	0	0
$491$	−23.3769	−1.05498	−0.527491	−	0.849560i	$-0.676867\pi$
$491$	−23.3769	−1.05498	−0.527491	+	0.849560i	$0.676867\pi$
$492$	−5.21934	−0.235306
$493$	32.7686	1.47582
$494$	−3.46542	−0.155916
$495$	−6.71551	−0.301840
$496$	−9.12664	−0.409798
$497$	0	0
$498$	64.6117	2.89532
$499$	18.8341	0.843130	0.421565	−	0.906798i	$-0.361481\pi$
$499$	18.8341	0.843130	0.421565	+	0.906798i	$0.361481\pi$
$500$	31.7962	1.42197
$501$	−42.0602	−1.87911
$502$	33.8455	1.51060
$503$	−16.6573	−0.742712	−0.371356	−	0.928491i	$-0.621107\pi$
$503$	−16.6573	−0.742712	−0.371356	+	0.928491i	$0.621107\pi$
$504$	0	0
$505$	26.6078	1.18403
$506$	−47.0439	−2.09136
$507$	24.6898	1.09651
$508$	−35.2352	−1.56331
$509$	−33.9430	−1.50450	−0.752248	−	0.658880i	$-0.771030\pi$
$509$	−33.9430	−1.50450	−0.752248	+	0.658880i	$0.771030\pi$
$510$	38.7918	1.71773
$511$	0	0
$512$	−21.3098	−0.941768
$513$	−29.9945	−1.32429
$514$	−14.8570	−0.655314
$515$	25.1512	1.10829
$516$	−31.4045	−1.38251
$517$	28.2085	1.24061
$518$	0	0
$519$	−32.6489	−1.43313
$520$	−0.817446	−0.0358474
$521$	6.31381	0.276613	0.138307	−	0.990389i	$-0.455834\pi$
$521$	6.31381	0.276613	0.138307	+	0.990389i	$0.455834\pi$
$522$	10.4072	0.455510
$523$	−9.25239	−0.404579	−0.202289	−	0.979326i	$-0.564838\pi$
$523$	−9.25239	−0.404579	−0.202289	+	0.979326i	$0.564838\pi$
$524$	45.9983	2.00945
$525$	0	0
$526$	23.1789	1.01065
$527$	−20.1199	−0.876437
$528$	18.6574	0.811960
$529$	−3.73149	−0.162239
$530$	36.3200	1.57764
$531$	1.94351	0.0843411
$532$	0	0
$533$	−0.239109	−0.0103570
$534$	−21.7940	−0.943118
$535$	−2.24446	−0.0970363
$536$	11.1478	0.481513
$537$	27.0366	1.16671
$538$	4.46586	0.192537
$539$	0	0
$540$	−26.2999	−1.13177
$541$	−20.9677	−0.901473	−0.450737	−	0.892657i	$-0.648839\pi$
$541$	−20.9677	−0.901473	−0.450737	+	0.892657i	$0.648839\pi$
$542$	1.71592	0.0737049
$543$	37.1509	1.59430
$544$	−32.9509	−1.41276
$545$	−20.9141	−0.895863
$546$	0	0
$547$	46.1308	1.97241	0.986205	−	0.165527i	$-0.0529326\pi$
$547$	46.1308	1.97241	0.986205	+	0.165527i	$0.0529326\pi$
$548$	−51.2658	−2.18997
$549$	−1.67116	−0.0713232
$550$	4.77128	0.203448
$551$	49.8424	2.12336
$552$	13.4135	0.570914
$553$	0	0
$554$	−41.6188	−1.76821
$555$	16.6663	0.707446
$556$	5.70605	0.241990
$557$	1.15248	0.0488323	0.0244162	−	0.999702i	$-0.492227\pi$
$557$	1.15248	0.0488323	0.0244162	+	0.999702i	$0.492227\pi$
$558$	−6.39001	−0.270511
$559$	−1.43871	−0.0608509
$560$	0	0
$561$	41.1308	1.73654
$562$	−10.6945	−0.451120
$563$	−8.90809	−0.375431	−0.187716	−	0.982223i	$-0.560108\pi$
$563$	−8.90809	−0.375431	−0.187716	+	0.982223i	$0.560108\pi$
$564$	−29.8969	−1.25889
$565$	−14.0375	−0.590560
$566$	55.2864	2.32386
$567$	0	0
$568$	−7.46930	−0.313405
$569$	−41.0681	−1.72166	−0.860832	−	0.508889i	$-0.830056\pi$
$569$	−41.0681	−1.72166	−0.860832	+	0.508889i	$0.830056\pi$
$570$	59.0040	2.47141
$571$	−3.59358	−0.150387	−0.0751933	−	0.997169i	$-0.523957\pi$
$571$	−3.59358	−0.150387	−0.0751933	+	0.997169i	$0.523957\pi$
$572$	−3.22176	−0.134709
$573$	−49.8678	−2.08326
$574$	0	0
$575$	−1.95425	−0.0814979
$576$	−7.92705	−0.330294
$577$	13.3905	0.557453	0.278727	−	0.960371i	$-0.410088\pi$
$577$	13.3905	0.557453	0.278727	+	0.960371i	$0.410088\pi$
$578$	−4.72190	−0.196405
$579$	−33.3888	−1.38759
$580$	43.7030	1.81467
$581$	0	0
$582$	3.94480	0.163517
$583$	38.5099	1.59492
$584$	−7.87403	−0.325830
$585$	0.326066	0.0134812
$586$	−51.2075	−2.11536
$587$	−28.7637	−1.18721	−0.593603	−	0.804758i	$-0.702295\pi$
$587$	−28.7637	−1.18721	−0.593603	+	0.804758i	$0.702295\pi$
$588$	0	0
$589$	−30.6032	−1.26098
$590$	14.1272	0.581608
$591$	−1.69469	−0.0697104
$592$	−8.13036	−0.334156
$593$	20.7120	0.850541	0.425271	−	0.905066i	$-0.360179\pi$
$593$	20.7120	0.850541	0.425271	+	0.905066i	$0.360179\pi$
$594$	−48.2693	−1.98051
$595$	0	0
$596$	−10.4363	−0.427488
$597$	44.1869	1.80845
$598$	2.28418	0.0934070
$599$	−21.6546	−0.884782	−0.442391	−	0.896822i	$-0.645870\pi$
$599$	−21.6546	−0.884782	−0.442391	+	0.896822i	$0.645870\pi$
$600$	−1.36042	−0.0555388
$601$	26.4258	1.07793	0.538966	−	0.842328i	$-0.318815\pi$
$601$	26.4258	1.07793	0.538966	+	0.842328i	$0.318815\pi$
$602$	0	0
$603$	−4.44669	−0.181083
$604$	−15.6597	−0.637183
$605$	−28.2815	−1.14981
$606$	−51.7573	−2.10250
$607$	17.2018	0.698199	0.349100	−	0.937086i	$-0.386488\pi$
$607$	17.2018	0.698199	0.349100	+	0.937086i	$0.386488\pi$
$608$	−50.1197	−2.03262
$609$	0	0
$610$	−12.1475	−0.491837
$611$	−1.36964	−0.0554098
$612$	−7.65437	−0.309410
$613$	13.2085	0.533485	0.266743	−	0.963768i	$-0.414053\pi$
$613$	13.2085	0.533485	0.266743	+	0.963768i	$0.414053\pi$
$614$	−27.0592	−1.09202
$615$	4.07121	0.164167
$616$	0	0
$617$	0.0723031	0.00291081	0.00145541	−	0.999999i	$-0.499537\pi$
$617$	0.0723031	0.00291081	0.00145541	+	0.999999i	$0.499537\pi$
$618$	−48.9239	−1.96801
$619$	7.84318	0.315244	0.157622	−	0.987500i	$-0.449617\pi$
$619$	7.84318	0.315244	0.157622	+	0.987500i	$0.449617\pi$
$620$	−26.8336	−1.07767
$621$	19.7704	0.793360
$622$	30.2493	1.21289
$623$	0	0
$624$	−0.905896	−0.0362649
$625$	−22.5758	−0.903032
$626$	−53.2006	−2.12633
$627$	62.5616	2.49847
$628$	−49.4793	−1.97444
$629$	−17.9236	−0.714660
$630$	0	0
$631$	−42.8842	−1.70719	−0.853596	−	0.520936i	$-0.825583\pi$
$631$	−42.8842	−1.70719	−0.853596	+	0.520936i	$0.825583\pi$
$632$	9.45328	0.376031
$633$	19.3065	0.767363
$634$	13.5263	0.537198
$635$	27.4843	1.09068
$636$	−40.8149	−1.61842
$637$	0	0
$638$	80.2100	3.17555
$639$	2.97938	0.117863
$640$	−25.4974	−1.00787
$641$	−30.5054	−1.20489	−0.602446	−	0.798160i	$-0.705807\pi$
$641$	−30.5054	−1.20489	−0.602446	+	0.798160i	$0.705807\pi$
$642$	4.36591	0.172309
$643$	30.2314	1.19221	0.596106	−	0.802906i	$-0.296714\pi$
$643$	30.2314	1.19221	0.596106	+	0.802906i	$0.296714\pi$
$644$	0	0
$645$	24.4962	0.964539
$646$	−63.4551	−2.49661
$647$	−4.52447	−0.177875	−0.0889376	−	0.996037i	$-0.528347\pi$
$647$	−4.52447	−0.177875	−0.0889376	+	0.996037i	$0.528347\pi$
$648$	16.8334	0.661280
$649$	14.9790	0.587977
$650$	−0.231666	−0.00908668
$651$	0	0
$652$	9.66501	0.378511
$653$	−26.6485	−1.04284	−0.521418	−	0.853301i	$-0.674597\pi$
$653$	−26.6485	−1.04284	−0.521418	+	0.853301i	$0.674597\pi$
$654$	40.6821	1.59080
$655$	−35.8798	−1.40194
$656$	−1.98606	−0.0775427
$657$	3.14083	0.122535
$658$	0	0
$659$	9.16573	0.357046	0.178523	−	0.983936i	$-0.442868\pi$
$659$	9.16573	0.357046	0.178523	+	0.983936i	$0.442868\pi$
$660$	54.8555	2.13525
$661$	2.92291	0.113688	0.0568441	−	0.998383i	$-0.481896\pi$
$661$	2.92291	0.113688	0.0568441	+	0.998383i	$0.481896\pi$
$662$	−14.9624	−0.581529
$663$	−1.99707	−0.0775598
$664$	−24.9311	−0.967514
$665$	0	0
$666$	−5.69246	−0.220578
$667$	−32.8529	−1.27207
$668$	60.3268	2.33411
$669$	48.0272	1.85684
$670$	−32.3226	−1.24873
$671$	−12.8799	−0.497223
$672$	0	0
$673$	−17.4742	−0.673580	−0.336790	−	0.941580i	$-0.609341\pi$
$673$	−17.4742	−0.673580	−0.336790	+	0.941580i	$0.609341\pi$
$674$	66.8893	2.57648
$675$	−2.00515	−0.0771784
$676$	−35.4125	−1.36202
$677$	17.7771	0.683230	0.341615	−	0.939840i	$-0.389026\pi$
$677$	17.7771	0.683230	0.341615	+	0.939840i	$0.389026\pi$
$678$	27.3056	1.04866
$679$	0	0
$680$	−14.9682	−0.574005
$681$	43.1584	1.65383
$682$	−49.2489	−1.88584
$683$	−3.14513	−0.120345	−0.0601725	−	0.998188i	$-0.519165\pi$
$683$	−3.14513	−0.120345	−0.0601725	+	0.998188i	$0.519165\pi$
$684$	−11.6426	−0.445167
$685$	39.9885	1.52788
$686$	0	0
$687$	37.5138	1.43124
$688$	−11.9500	−0.455591
$689$	−1.86982	−0.0712344
$690$	−38.8917	−1.48058
$691$	−30.4847	−1.15969	−0.579846	−	0.814726i	$-0.696887\pi$
$691$	−30.4847	−1.15969	−0.579846	+	0.814726i	$0.696887\pi$
$692$	46.8282	1.78014
$693$	0	0
$694$	80.8093	3.06748
$695$	−4.45085	−0.168830
$696$	−22.8700	−0.866885
$697$	−4.37833	−0.165841
$698$	−4.29125	−0.162426
$699$	28.5211	1.07877
$700$	0	0
$701$	18.7300	0.707422	0.353711	−	0.935355i	$-0.384920\pi$
$701$	18.7300	0.707422	0.353711	+	0.935355i	$0.384920\pi$
$702$	2.34368	0.0884564
$703$	−27.2625	−1.02823
$704$	−61.0952	−2.30261
$705$	23.3203	0.878293
$706$	32.9369	1.23960
$707$	0	0
$708$	−15.8755	−0.596639
$709$	42.9153	1.61172	0.805859	−	0.592108i	$-0.201704\pi$
$709$	42.9153	1.61172	0.805859	+	0.592108i	$0.201704\pi$
$710$	21.6569	0.812768
$711$	−3.77076	−0.141415
$712$	8.40943	0.315157
$713$	20.1717	0.755435
$714$	0	0
$715$	2.51305	0.0939828
$716$	−38.7784	−1.44922
$717$	15.8682	0.592608
$718$	−68.6234	−2.56100
$719$	−12.7065	−0.473874	−0.236937	−	0.971525i	$-0.576143\pi$
$719$	−12.7065	−0.473874	−0.236937	+	0.971525i	$0.576143\pi$
$720$	2.70833	0.100934
$721$	0	0
$722$	−55.1691	−2.05318
$723$	20.7130	0.770324
$724$	−53.2854	−1.98034
$725$	3.33201	0.123748
$726$	55.0130	2.04172
$727$	15.7827	0.585347	0.292673	−	0.956212i	$-0.405455\pi$
$727$	15.7827	0.585347	0.292673	+	0.956212i	$0.405455\pi$
$728$	0	0
$729$	19.0606	0.705948
$730$	22.8304	0.844990
$731$	−26.3442	−0.974374
$732$	13.6508	0.504549
$733$	−0.248845	−0.00919132	−0.00459566	−	0.999989i	$-0.501463\pi$
$733$	−0.248845	−0.00919132	−0.00459566	+	0.999989i	$0.501463\pi$
$734$	−31.7478	−1.17183
$735$	0	0
$736$	33.0357	1.21771
$737$	−34.2715	−1.26241
$738$	−1.39054	−0.0511865
$739$	33.3554	1.22700	0.613499	−	0.789695i	$-0.289761\pi$
$739$	33.3554	1.22700	0.613499	+	0.789695i	$0.289761\pi$
$740$	−23.9044	−0.878744
$741$	−3.03763	−0.111590
$742$	0	0
$743$	−47.4140	−1.73945	−0.869725	−	0.493537i	$-0.835704\pi$
$743$	−47.4140	−1.73945	−0.869725	+	0.493537i	$0.835704\pi$
$744$	14.0422	0.514811
$745$	8.14056	0.298247
$746$	−3.57812	−0.131004
$747$	9.94461	0.363854
$748$	−58.9936	−2.15702
$749$	0	0
$750$	48.2443	1.76163
$751$	9.81101	0.358009	0.179004	−	0.983848i	$-0.442712\pi$
$751$	9.81101	0.358009	0.179004	+	0.983848i	$0.442712\pi$
$752$	−11.3764	−0.414853
$753$	29.6675	1.08114
$754$	−3.89453	−0.141831
$755$	12.2149	0.444546
$756$	0	0
$757$	−13.8555	−0.503587	−0.251794	−	0.967781i	$-0.581020\pi$
$757$	−13.8555	−0.503587	−0.251794	+	0.967781i	$0.581020\pi$
$758$	−24.1145	−0.875880
$759$	−41.2366	−1.49679
$760$	−22.7673	−0.825857
$761$	−4.65478	−0.168736	−0.0843678	−	0.996435i	$-0.526887\pi$
$761$	−4.65478	−0.168736	−0.0843678	+	0.996435i	$0.526887\pi$
$762$	−53.4623	−1.93674
$763$	0	0
$764$	71.5251	2.58769
$765$	5.97059	0.215867
$766$	75.6912	2.73484
$767$	−0.727293	−0.0262610
$768$	2.26536	0.0817440
$769$	−30.8078	−1.11096	−0.555479	−	0.831531i	$-0.687465\pi$
$769$	−30.8078	−1.11096	−0.555479	+	0.831531i	$0.687465\pi$
$770$	0	0
$771$	−13.0230	−0.469011
$772$	47.8894	1.72358
$773$	50.9759	1.83348	0.916738	−	0.399488i	$-0.130812\pi$
$773$	50.9759	1.83348	0.916738	+	0.399488i	$0.130812\pi$
$774$	−8.36680	−0.300739
$775$	−2.04585	−0.0734891
$776$	−1.52214	−0.0546417
$777$	0	0
$778$	−70.1043	−2.51336
$779$	−6.65962	−0.238606
$780$	−2.66347	−0.0953674
$781$	22.9626	0.821668
$782$	41.8255	1.49568
$783$	−33.7087	−1.20465
$784$	0	0
$785$	38.5950	1.37751
$786$	69.7931	2.48944
$787$	1.62952	0.0580862	0.0290431	−	0.999578i	$-0.490754\pi$
$787$	1.62952	0.0580862	0.0290431	+	0.999578i	$0.490754\pi$
$788$	2.43069	0.0865898
$789$	20.3176	0.723327
$790$	−27.4093	−0.975180
$791$	0	0
$792$	−5.04048	−0.179106
$793$	0.625373	0.0222077
$794$	−47.8339	−1.69756
$795$	31.8365	1.12913
$796$	−63.3771	−2.24634
$797$	−15.9224	−0.564002	−0.282001	−	0.959414i	$-0.590998\pi$
$797$	−15.9224	−0.564002	−0.282001	+	0.959414i	$0.590998\pi$
$798$	0	0
$799$	−25.0795	−0.887248
$800$	−3.35054	−0.118460
$801$	−3.35439	−0.118521
$802$	69.5686	2.45655
$803$	24.2069	0.854243
$804$	36.3227	1.28100
$805$	0	0
$806$	2.39124	0.0842280
$807$	3.91458	0.137800
$808$	19.9711	0.702580
$809$	49.7454	1.74895	0.874477	−	0.485066i	$-0.161205\pi$
$809$	49.7454	1.74895	0.874477	+	0.485066i	$0.161205\pi$
$810$	−48.8078	−1.71493
$811$	−40.2744	−1.41422	−0.707112	−	0.707101i	$-0.750002\pi$
$811$	−40.2744	−1.41422	−0.707112	+	0.707101i	$0.750002\pi$
$812$	0	0
$813$	1.50410	0.0527510
$814$	−43.8728	−1.53774
$815$	−7.53893	−0.264077
$816$	−16.5878	−0.580691
$817$	−40.0706	−1.40189
$818$	−73.1985	−2.55933
$819$	0	0
$820$	−5.83931	−0.203918
$821$	−19.4744	−0.679663	−0.339831	−	0.940486i	$-0.610370\pi$
$821$	−19.4744	−0.679663	−0.339831	+	0.940486i	$0.610370\pi$
$822$	−77.7854	−2.71308
$823$	−11.1807	−0.389735	−0.194867	−	0.980830i	$-0.562428\pi$
$823$	−11.1807	−0.389735	−0.194867	+	0.980830i	$0.562428\pi$
$824$	18.8778	0.657639
$825$	4.18229	0.145609
$826$	0	0
$827$	−1.14084	−0.0396708	−0.0198354	−	0.999803i	$-0.506314\pi$
$827$	−1.14084	−0.0396708	−0.0198354	+	0.999803i	$0.506314\pi$
$828$	7.67407	0.266692
$829$	−9.06071	−0.314691	−0.157346	−	0.987544i	$-0.550294\pi$
$829$	−9.06071	−0.314691	−0.157346	+	0.987544i	$0.550294\pi$
$830$	72.2865	2.50910
$831$	−36.4812	−1.26552
$832$	2.96643	0.102842
$833$	0	0
$834$	8.65777	0.299794
$835$	−47.0563	−1.62845
$836$	−89.7318	−3.10344
$837$	20.6971	0.715397
$838$	−71.7253	−2.47771
$839$	20.1326	0.695054	0.347527	−	0.937670i	$-0.387022\pi$
$839$	20.1326	0.695054	0.347527	+	0.937670i	$0.387022\pi$
$840$	0	0
$841$	27.0144	0.931530
$842$	49.1524	1.69390
$843$	−9.37431	−0.322869
$844$	−27.6912	−0.953169
$845$	27.6225	0.950245
$846$	−7.96515	−0.273847
$847$	0	0
$848$	−15.5309	−0.533332
$849$	48.4616	1.66320
$850$	−4.24202	−0.145500
$851$	17.9697	0.615993
$852$	−24.3370	−0.833773
$853$	12.7584	0.436840	0.218420	−	0.975855i	$-0.429910\pi$
$853$	12.7584	0.436840	0.218420	+	0.975855i	$0.429910\pi$
$854$	0	0
$855$	9.08152	0.310581
$856$	−1.68463	−0.0575795
$857$	53.2622	1.81940	0.909702	−	0.415262i	$-0.136310\pi$
$857$	53.2622	1.81940	0.909702	+	0.415262i	$0.136310\pi$
$858$	−4.88837	−0.166886
$859$	−11.8487	−0.404273	−0.202136	−	0.979357i	$-0.564788\pi$
$859$	−11.8487	−0.404273	−0.202136	+	0.979357i	$0.564788\pi$
$860$	−35.1348	−1.19809
$861$	0	0
$862$	6.78257	0.231015
$863$	−34.5191	−1.17504	−0.587522	−	0.809208i	$-0.699897\pi$
$863$	−34.5191	−1.17504	−0.587522	+	0.809208i	$0.699897\pi$
$864$	33.8962	1.15317
$865$	−36.5271	−1.24196
$866$	63.9821	2.17420
$867$	−4.13901	−0.140568
$868$	0	0
$869$	−29.0619	−0.985859
$870$	66.3105	2.24814
$871$	1.66402	0.0563833
$872$	−15.6976	−0.531588
$873$	0.607158	0.0205492
$874$	63.6184	2.15192
$875$	0	0
$876$	−25.6558	−0.866829
$877$	−5.62471	−0.189933	−0.0949665	−	0.995480i	$-0.530274\pi$
$877$	−5.62471	−0.189933	−0.0949665	+	0.995480i	$0.530274\pi$
$878$	−43.7417	−1.47621
$879$	−44.8863	−1.51398
$880$	20.8736	0.703649
$881$	−31.4624	−1.06000	−0.529998	−	0.847999i	$-0.677807\pi$
$881$	−31.4624	−1.06000	−0.529998	+	0.847999i	$0.677807\pi$
$882$	0	0
$883$	−14.3636	−0.483374	−0.241687	−	0.970354i	$-0.577701\pi$
$883$	−14.3636	−0.483374	−0.241687	+	0.970354i	$0.577701\pi$
$884$	2.86439	0.0963398
$885$	12.3833	0.416259
$886$	88.2897	2.96615
$887$	42.1649	1.41576	0.707879	−	0.706333i	$-0.249652\pi$
$887$	42.1649	1.41576	0.707879	+	0.706333i	$0.249652\pi$
$888$	12.5093	0.419785
$889$	0	0
$890$	−24.3827	−0.817311
$891$	−51.7506	−1.73371
$892$	−68.8852	−2.30645
$893$	−38.1470	−1.27654
$894$	−15.8350	−0.529601
$895$	30.2481	1.01108
$896$	0	0
$897$	2.00221	0.0668518
$898$	6.84377	0.228379
$899$	−34.3928	−1.14706
$900$	−0.778319	−0.0259440
$901$	−34.2382	−1.14064
$902$	−10.7171	−0.356842
$903$	0	0
$904$	−10.5361	−0.350427
$905$	41.5638	1.38163
$906$	−23.7604	−0.789386
$907$	−39.2596	−1.30359	−0.651796	−	0.758394i	$-0.725984\pi$
$907$	−39.2596	−1.30359	−0.651796	+	0.758394i	$0.725984\pi$
$908$	−61.9019	−2.05429
$909$	−7.96614	−0.264220
$910$	0	0
$911$	−30.6058	−1.01402	−0.507008	−	0.861942i	$-0.669248\pi$
$911$	−30.6058	−1.01402	−0.507008	+	0.861942i	$0.669248\pi$
$912$	−25.2308	−0.835476
$913$	76.6449	2.53658
$914$	65.8434	2.17790
$915$	−10.6479	−0.352010
$916$	−53.8059	−1.77780
$917$	0	0
$918$	42.9150	1.41641
$919$	−32.8355	−1.08314	−0.541572	−	0.840654i	$-0.682171\pi$
$919$	−32.8355	−1.08314	−0.541572	+	0.840654i	$0.682171\pi$
$920$	15.0067	0.494758
$921$	−23.7189	−0.781563
$922$	−28.6057	−0.942080
$923$	−1.11493	−0.0366985
$924$	0	0
$925$	−1.82252	−0.0599241
$926$	3.02769	0.0994959
$927$	−7.53004	−0.247319
$928$	−56.3260	−1.84899
$929$	−23.2519	−0.762871	−0.381435	−	0.924395i	$-0.624570\pi$
$929$	−23.2519	−0.762871	−0.381435	+	0.924395i	$0.624570\pi$
$930$	−40.7146	−1.33508
$931$	0	0
$932$	−40.9077	−1.33998
$933$	26.5152	0.868068
$934$	29.1016	0.952234
$935$	46.0164	1.50490
$936$	0.244737	0.00799946
$937$	−41.6403	−1.36033	−0.680164	−	0.733060i	$-0.738092\pi$
$937$	−41.6403	−1.36033	−0.680164	+	0.733060i	$0.738092\pi$
$938$	0	0
$939$	−46.6333	−1.52182
$940$	−33.4482	−1.09096
$941$	5.97037	0.194628	0.0973142	−	0.995254i	$-0.468975\pi$
$941$	5.97037	0.194628	0.0973142	+	0.995254i	$0.468975\pi$
$942$	−75.0747	−2.44607
$943$	4.38959	0.142945
$944$	−6.04095	−0.196616
$945$	0	0
$946$	−64.4845	−2.09657
$947$	40.2690	1.30857	0.654283	−	0.756250i	$-0.272971\pi$
$947$	40.2690	1.30857	0.654283	+	0.756250i	$0.272971\pi$
$948$	30.8014	1.00038
$949$	−1.17535	−0.0381534
$950$	−6.45230	−0.209340
$951$	11.8566	0.384476
$952$	0	0
$953$	7.50566	0.243132	0.121566	−	0.992583i	$-0.461208\pi$
$953$	7.50566	0.243132	0.121566	+	0.992583i	$0.461208\pi$
$954$	−10.8739	−0.352056
$955$	−55.7912	−1.80536
$956$	−22.7597	−0.736100
$957$	70.3086	2.27275
$958$	63.8858	2.06406
$959$	0	0
$960$	−50.5081	−1.63014
$961$	−9.88282	−0.318801
$962$	2.13021	0.0686807
$963$	0.671972	0.0216540
$964$	−29.7085	−0.956846
$965$	−37.3549	−1.20250
$966$	0	0
$967$	−12.2423	−0.393687	−0.196844	−	0.980435i	$-0.563069\pi$
$967$	−12.2423	−0.393687	−0.196844	+	0.980435i	$0.563069\pi$
$968$	−21.2273	−0.682272
$969$	−55.6220	−1.78683
$970$	4.41337	0.141705
$971$	42.0361	1.34900	0.674501	−	0.738274i	$-0.264359\pi$
$971$	42.0361	1.34900	0.674501	+	0.738274i	$0.264359\pi$
$972$	17.8788	0.573464
$973$	0	0
$974$	−24.9686	−0.800045
$975$	−0.203068	−0.00650338
$976$	5.19440	0.166269
$977$	−24.4650	−0.782704	−0.391352	−	0.920241i	$-0.627993\pi$
$977$	−24.4650	−0.782704	−0.391352	+	0.920241i	$0.627993\pi$
$978$	14.6647	0.468925
$979$	−25.8529	−0.826261
$980$	0	0
$981$	6.26152	0.199915
$982$	50.8739	1.62345
$983$	−18.4075	−0.587107	−0.293553	−	0.955943i	$-0.594838\pi$
$983$	−18.4075	−0.587107	−0.293553	+	0.955943i	$0.594838\pi$
$984$	3.05574	0.0974134
$985$	−1.89600	−0.0604114
$986$	−71.3127	−2.27106
$987$	0	0
$988$	4.35686	0.138610
$989$	26.4119	0.839851
$990$	14.6146	0.464484
$991$	−6.23837	−0.198168	−0.0990841	−	0.995079i	$-0.531591\pi$
$991$	−6.23837	−0.198168	−0.0990841	+	0.995079i	$0.531591\pi$
$992$	34.5842	1.09805
$993$	−13.1154	−0.416203
$994$	0	0
$995$	49.4356	1.56721
$996$	−81.2324	−2.57395
$997$	−6.92770	−0.219402	−0.109701	−	0.993965i	$-0.534989\pi$
$997$	−6.92770	−0.219402	−0.109701	+	0.993965i	$0.534989\pi$
$998$	−40.9877	−1.29744
$999$	18.4378	0.583345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.p.1.2	✓	7
7.6	odd	2		2009.2.a.q.1.2	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.p.1.2	✓	7	1.1	even	1	trivial
2009.2.a.q.1.2	yes	7	7.6	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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17625 q^{2} -1.90761 q^{3} +2.73607 q^{4} -2.13420 q^{5} +4.15143 q^{6} -1.60187 q^{8} +0.638961 q^{9} +O(q^{10})\)	\(q-2.17625 q^{2} -1.90761 q^{3} +2.73607 q^{4} -2.13420 q^{5} +4.15143 q^{6} -1.60187 q^{8} +0.638961 q^{9} +4.64455 q^{10} +4.92459 q^{11} -5.21934 q^{12} -0.239109 q^{13} +4.07121 q^{15} -1.98606 q^{16} -4.37833 q^{17} -1.39054 q^{18} -6.65962 q^{19} -5.83931 q^{20} -10.7171 q^{22} +4.38959 q^{23} +3.05574 q^{24} -0.445201 q^{25} +0.520362 q^{26} +4.50393 q^{27} -7.48428 q^{29} -8.85997 q^{30} +4.59534 q^{31} +7.52592 q^{32} -9.39417 q^{33} +9.52834 q^{34} +1.74824 q^{36} +4.09370 q^{37} +14.4930 q^{38} +0.456127 q^{39} +3.41871 q^{40} +1.00000 q^{41} +6.01695 q^{43} +13.4740 q^{44} -1.36367 q^{45} -9.55285 q^{46} +5.72810 q^{47} +3.78863 q^{48} +0.968869 q^{50} +8.35212 q^{51} -0.654220 q^{52} +7.81993 q^{53} -9.80169 q^{54} -10.5100 q^{55} +12.7039 q^{57} +16.2877 q^{58} +3.04167 q^{59} +11.1391 q^{60} -2.61543 q^{61} -10.0006 q^{62} -12.4062 q^{64} +0.510307 q^{65} +20.4441 q^{66} -6.95926 q^{67} -11.9794 q^{68} -8.37361 q^{69} +4.66286 q^{71} -1.02353 q^{72} +4.91552 q^{73} -8.90893 q^{74} +0.849268 q^{75} -18.2212 q^{76} -0.992646 q^{78} -5.90139 q^{79} +4.23865 q^{80} -10.5086 q^{81} -2.17625 q^{82} +15.5637 q^{83} +9.34421 q^{85} -13.0944 q^{86} +14.2770 q^{87} -7.88856 q^{88} -5.24975 q^{89} +2.96769 q^{90} +12.0102 q^{92} -8.76610 q^{93} -12.4658 q^{94} +14.2129 q^{95} -14.3565 q^{96} +0.950227 q^{97} +3.14662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9	\( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) \) 7 * q - q^2 - 7 * q^3 + 9 * q^4 - 4 * q^5 + 4 * q^6 - 12 * q^8 + 6 * q^9 - 8 * q^10 - 12 * q^12 - q^13 - 4 * q^15 + 5 * q^16 + 11 * q^17 + 16 * q^18 - 9 * q^19 - 12 * q^20 + 3 * q^23 + 23 * q^24 + 31 * q^25 - 10 * q^26 - 22 * q^27 - 14 * q^29 + 6 * q^30 - 34 * q^31 - 20 * q^32 - 12 * q^33 - 15 * q^34 + q^36 + 7 * q^37 + 39 * q^38 - 22 * q^39 - 50 * q^40 + 7 * q^41 - 3 * q^43 + 26 * q^44 + 4 * q^45 - 8 * q^46 - 17 * q^47 - 17 * q^48 + 11 * q^50 + 8 * q^51 + 25 * q^52 + 24 * q^53 - 68 * q^54 - 48 * q^55 + 22 * q^57 - 38 * q^58 - 4 * q^59 - 6 * q^60 - 16 * q^61 - 24 * q^62 + 8 * q^64 - 6 * q^65 + 12 * q^66 - 24 * q^67 + 10 * q^68 - 35 * q^69 - 12 * q^71 - 11 * q^72 - 14 * q^73 - 6 * q^74 + 19 * q^75 - 42 * q^76 - 29 * q^78 - 8 * q^79 + 92 * q^80 + 15 * q^81 - q^82 - 14 * q^83 + 16 * q^85 + 35 * q^86 + 20 * q^87 + 22 * q^88 + 19 * q^89 - 24 * q^90 - 10 * q^92 + 2 * q^93 + 20 * q^94 - 8 * q^95 + 25 * q^96 - 23 * q^97 + 24 * q^99

Introduction

L-functions

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