LMFDB - Embedded newform 2005.2.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27896 q^{2} +1.59149 q^{3} -0.364250 q^{4} +1.00000 q^{5} -2.03546 q^{6} -4.69998 q^{7} +3.02379 q^{8} -0.467169 q^{9} +O(q^{10})$	\(q-1.27896 q^{2} +1.59149 q^{3} -0.364250 q^{4} +1.00000 q^{5} -2.03546 q^{6} -4.69998 q^{7} +3.02379 q^{8} -0.467169 q^{9} -1.27896 q^{10} -1.89034 q^{11} -0.579700 q^{12} -1.30484 q^{13} +6.01111 q^{14} +1.59149 q^{15} -3.13882 q^{16} +3.12337 q^{17} +0.597492 q^{18} +3.65896 q^{19} -0.364250 q^{20} -7.47996 q^{21} +2.41768 q^{22} +0.240346 q^{23} +4.81233 q^{24} +1.00000 q^{25} +1.66884 q^{26} -5.51795 q^{27} +1.71197 q^{28} -2.22982 q^{29} -2.03546 q^{30} +0.715968 q^{31} -2.03314 q^{32} -3.00846 q^{33} -3.99468 q^{34} -4.69998 q^{35} +0.170166 q^{36} +2.81362 q^{37} -4.67968 q^{38} -2.07663 q^{39} +3.02379 q^{40} +1.42885 q^{41} +9.56661 q^{42} +12.1535 q^{43} +0.688558 q^{44} -0.467169 q^{45} -0.307394 q^{46} +2.13584 q^{47} -4.99539 q^{48} +15.0898 q^{49} -1.27896 q^{50} +4.97080 q^{51} +0.475287 q^{52} -4.88454 q^{53} +7.05727 q^{54} -1.89034 q^{55} -14.2118 q^{56} +5.82319 q^{57} +2.85186 q^{58} +12.9252 q^{59} -0.579700 q^{60} -1.98176 q^{61} -0.915698 q^{62} +2.19569 q^{63} +8.87796 q^{64} -1.30484 q^{65} +3.84771 q^{66} +5.47249 q^{67} -1.13769 q^{68} +0.382508 q^{69} +6.01111 q^{70} -2.76469 q^{71} -1.41262 q^{72} +10.4174 q^{73} -3.59851 q^{74} +1.59149 q^{75} -1.33278 q^{76} +8.88458 q^{77} +2.65594 q^{78} -0.645939 q^{79} -3.13882 q^{80} -7.38025 q^{81} -1.82745 q^{82} -1.64415 q^{83} +2.72458 q^{84} +3.12337 q^{85} -15.5439 q^{86} -3.54873 q^{87} -5.71600 q^{88} +8.86433 q^{89} +0.597492 q^{90} +6.13271 q^{91} -0.0875461 q^{92} +1.13945 q^{93} -2.73166 q^{94} +3.65896 q^{95} -3.23572 q^{96} +2.07128 q^{97} -19.2994 q^{98} +0.883109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9}+O(q^{10}) $ 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9	\( 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9} + 11 q^{10} + 38 q^{11} + 24 q^{12} + 17 q^{13} + 14 q^{14} + 13 q^{15} + 47 q^{16} + 22 q^{17} + 18 q^{18} + 6 q^{19} + 43 q^{20} + 4 q^{21} + 18 q^{22} + 45 q^{23} - 19 q^{24} + 37 q^{25} - q^{26} + 43 q^{27} + 46 q^{28} + 17 q^{29} - 2 q^{30} - 13 q^{31} + 50 q^{32} + 12 q^{33} - 30 q^{34} + 34 q^{35} + 43 q^{36} + 9 q^{37} + q^{38} - 7 q^{39} + 27 q^{40} + 12 q^{41} - 38 q^{42} + 53 q^{43} + 36 q^{44} + 50 q^{45} - 15 q^{46} + 39 q^{47} - 6 q^{48} + 37 q^{49} + 11 q^{50} + 38 q^{51} + 17 q^{52} + 19 q^{53} - 62 q^{54} + 38 q^{55} + 18 q^{56} + 6 q^{57} - 13 q^{58} + 33 q^{59} + 24 q^{60} - 11 q^{61} + q^{62} + 98 q^{63} + 15 q^{64} + 17 q^{65} - 70 q^{66} + 49 q^{67} + 32 q^{68} - 36 q^{69} + 14 q^{70} + 19 q^{71} + 5 q^{72} + 39 q^{73} + 21 q^{74} + 13 q^{75} - 33 q^{76} + 34 q^{77} - 22 q^{78} - 9 q^{79} + 47 q^{80} + 49 q^{81} + 14 q^{82} + 114 q^{83} - 50 q^{84} + 22 q^{85} + 9 q^{86} + 56 q^{87} + 14 q^{88} + 2 q^{89} + 18 q^{90} - 29 q^{91} + 47 q^{92} - 7 q^{93} - 52 q^{94} + 6 q^{95} - 89 q^{96} + 4 q^{97} + 14 q^{98} + 57 q^{99}+O(q^{100}) \) 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9 + 11 * q^10 + 38 * q^11 + 24 * q^12 + 17 * q^13 + 14 * q^14 + 13 * q^15 + 47 * q^16 + 22 * q^17 + 18 * q^18 + 6 * q^19 + 43 * q^20 + 4 * q^21 + 18 * q^22 + 45 * q^23 - 19 * q^24 + 37 * q^25 - q^26 + 43 * q^27 + 46 * q^28 + 17 * q^29 - 2 * q^30 - 13 * q^31 + 50 * q^32 + 12 * q^33 - 30 * q^34 + 34 * q^35 + 43 * q^36 + 9 * q^37 + q^38 - 7 * q^39 + 27 * q^40 + 12 * q^41 - 38 * q^42 + 53 * q^43 + 36 * q^44 + 50 * q^45 - 15 * q^46 + 39 * q^47 - 6 * q^48 + 37 * q^49 + 11 * q^50 + 38 * q^51 + 17 * q^52 + 19 * q^53 - 62 * q^54 + 38 * q^55 + 18 * q^56 + 6 * q^57 - 13 * q^58 + 33 * q^59 + 24 * q^60 - 11 * q^61 + q^62 + 98 * q^63 + 15 * q^64 + 17 * q^65 - 70 * q^66 + 49 * q^67 + 32 * q^68 - 36 * q^69 + 14 * q^70 + 19 * q^71 + 5 * q^72 + 39 * q^73 + 21 * q^74 + 13 * q^75 - 33 * q^76 + 34 * q^77 - 22 * q^78 - 9 * q^79 + 47 * q^80 + 49 * q^81 + 14 * q^82 + 114 * q^83 - 50 * q^84 + 22 * q^85 + 9 * q^86 + 56 * q^87 + 14 * q^88 + 2 * q^89 + 18 * q^90 - 29 * q^91 + 47 * q^92 - 7 * q^93 - 52 * q^94 + 6 * q^95 - 89 * q^96 + 4 * q^97 + 14 * q^98 + 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.27896	−0.904364	−0.452182	−	0.891926i	$-0.649354\pi$
$2$	−1.27896	−0.904364	−0.452182	+	0.891926i	$0.649354\pi$
$3$	1.59149	0.918846	0.459423	−	0.888218i	$-0.348056\pi$
$3$	1.59149	0.918846	0.459423	+	0.888218i	$0.348056\pi$
$4$	−0.364250	−0.182125
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.03546	−0.830971
$7$	−4.69998	−1.77643	−0.888213	−	0.459431i	$-0.848053\pi$
$7$	−4.69998	−1.77643	−0.888213	+	0.459431i	$0.848053\pi$
$8$	3.02379	1.06907
$9$	−0.467169	−0.155723
$10$	−1.27896	−0.404444
$11$	−1.89034	−0.569960	−0.284980	−	0.958534i	$-0.591987\pi$
$11$	−1.89034	−0.569960	−0.284980	+	0.958534i	$0.591987\pi$
$12$	−0.579700	−0.167345
$13$	−1.30484	−0.361897	−0.180948	−	0.983493i	$-0.557917\pi$
$13$	−1.30484	−0.361897	−0.180948	+	0.983493i	$0.557917\pi$
$14$	6.01111	1.60654
$15$	1.59149	0.410920
$16$	−3.13882	−0.784705
$17$	3.12337	0.757529	0.378764	−	0.925493i	$-0.376349\pi$
$17$	3.12337	0.757529	0.378764	+	0.925493i	$0.376349\pi$
$18$	0.597492	0.140830
$19$	3.65896	0.839423	0.419712	−	0.907658i	$-0.362131\pi$
$19$	3.65896	0.839423	0.419712	+	0.907658i	$0.362131\pi$
$20$	−0.364250	−0.0814488
$21$	−7.47996	−1.63226
$22$	2.41768	0.515451
$23$	0.240346	0.0501156	0.0250578	−	0.999686i	$-0.492023\pi$
$23$	0.240346	0.0501156	0.0250578	+	0.999686i	$0.492023\pi$
$24$	4.81233	0.982312
$25$	1.00000	0.200000
$26$	1.66884	0.327286
$27$	−5.51795	−1.06193
$28$	1.71197	0.323532
$29$	−2.22982	−0.414067	−0.207034	−	0.978334i	$-0.566381\pi$
$29$	−2.22982	−0.414067	−0.207034	+	0.978334i	$0.566381\pi$
$30$	−2.03546	−0.371622
$31$	0.715968	0.128592	0.0642958	−	0.997931i	$-0.479520\pi$
$31$	0.715968	0.128592	0.0642958	+	0.997931i	$0.479520\pi$
$32$	−2.03314	−0.359412
$33$	−3.00846	−0.523705
$34$	−3.99468	−0.685082
$35$	−4.69998	−0.794442
$36$	0.170166	0.0283611
$37$	2.81362	0.462556	0.231278	−	0.972888i	$-0.425709\pi$
$37$	2.81362	0.462556	0.231278	+	0.972888i	$0.425709\pi$
$38$	−4.67968	−0.759144
$39$	−2.07663	−0.332527
$40$	3.02379	0.478103
$41$	1.42885	0.223149	0.111574	−	0.993756i	$-0.464411\pi$
$41$	1.42885	0.223149	0.111574	+	0.993756i	$0.464411\pi$
$42$	9.56661	1.47616
$43$	12.1535	1.85339	0.926695	−	0.375814i	$-0.122637\pi$
$43$	12.1535	1.85339	0.926695	+	0.375814i	$0.122637\pi$
$44$	0.688558	0.103804
$45$	−0.467169	−0.0696414
$46$	−0.307394	−0.0453228
$47$	2.13584	0.311544	0.155772	−	0.987793i	$-0.450214\pi$
$47$	2.13584	0.311544	0.155772	+	0.987793i	$0.450214\pi$
$48$	−4.99539	−0.721023
$49$	15.0898	2.15569
$50$	−1.27896	−0.180873
$51$	4.97080	0.696052
$52$	0.475287	0.0659105
$53$	−4.88454	−0.670943	−0.335472	−	0.942050i	$-0.608896\pi$
$53$	−4.88454	−0.670943	−0.335472	+	0.942050i	$0.608896\pi$
$54$	7.05727	0.960372
$55$	−1.89034	−0.254894
$56$	−14.2118	−1.89913
$57$	5.82319	0.771300
$58$	2.85186	0.374468
$59$	12.9252	1.68272	0.841361	−	0.540473i	$-0.181755\pi$
$59$	12.9252	1.68272	0.841361	+	0.540473i	$0.181755\pi$
$60$	−0.579700	−0.0748389
$61$	−1.98176	−0.253738	−0.126869	−	0.991919i	$-0.540493\pi$
$61$	−1.98176	−0.253738	−0.126869	+	0.991919i	$0.540493\pi$
$62$	−0.915698	−0.116294
$63$	2.19569	0.276630
$64$	8.87796	1.10974
$65$	−1.30484	−0.161845
$66$	3.84771	0.473620
$67$	5.47249	0.668571	0.334285	−	0.942472i	$-0.391505\pi$
$67$	5.47249	0.668571	0.334285	+	0.942472i	$0.391505\pi$
$68$	−1.13769	−0.137965
$69$	0.382508	0.0460485
$70$	6.01111	0.718465
$71$	−2.76469	−0.328109	−0.164054	−	0.986451i	$-0.552457\pi$
$71$	−2.76469	−0.328109	−0.164054	+	0.986451i	$0.552457\pi$
$72$	−1.41262	−0.166479
$73$	10.4174	1.21926	0.609631	−	0.792685i	$-0.291318\pi$
$73$	10.4174	1.21926	0.609631	+	0.792685i	$0.291318\pi$
$74$	−3.59851	−0.418319
$75$	1.59149	0.183769
$76$	−1.33278	−0.152880
$77$	8.88458	1.01249
$78$	2.65594	0.300726
$79$	−0.645939	−0.0726738	−0.0363369	−	0.999340i	$-0.511569\pi$
$79$	−0.645939	−0.0726738	−0.0363369	+	0.999340i	$0.511569\pi$
$80$	−3.13882	−0.350931
$81$	−7.38025	−0.820027
$82$	−1.82745	−0.201808
$83$	−1.64415	−0.180469	−0.0902346	−	0.995921i	$-0.528762\pi$
$83$	−1.64415	−0.180469	−0.0902346	+	0.995921i	$0.528762\pi$
$84$	2.72458	0.297276
$85$	3.12337	0.338777
$86$	−15.5439	−1.67614
$87$	−3.54873	−0.380464
$88$	−5.71600	−0.609328
$89$	8.86433	0.939617	0.469809	−	0.882768i	$-0.344323\pi$
$89$	8.86433	0.939617	0.469809	+	0.882768i	$0.344323\pi$
$90$	0.597492	0.0629812
$91$	6.13271	0.642883
$92$	−0.0875461	−0.00912731
$93$	1.13945	0.118156
$94$	−2.73166	−0.281749
$95$	3.65896	0.375401
$96$	−3.23572	−0.330244
$97$	2.07128	0.210307	0.105153	−	0.994456i	$-0.466467\pi$
$97$	2.07128	0.210307	0.105153	+	0.994456i	$0.466467\pi$
$98$	−19.2994	−1.94953
$99$	0.883109	0.0887558
$100$	−0.364250	−0.0364250
$101$	−1.96166	−0.195192	−0.0975960	−	0.995226i	$-0.531115\pi$
$101$	−1.96166	−0.195192	−0.0975960	+	0.995226i	$0.531115\pi$
$102$	−6.35748	−0.629484
$103$	2.50880	0.247200	0.123600	−	0.992332i	$-0.460556\pi$
$103$	2.50880	0.247200	0.123600	+	0.992332i	$0.460556\pi$
$104$	−3.94555	−0.386893
$105$	−7.47996	−0.729970
$106$	6.24715	0.606777
$107$	7.61420	0.736093	0.368046	−	0.929807i	$-0.380027\pi$
$107$	7.61420	0.736093	0.368046	+	0.929807i	$0.380027\pi$
$108$	2.00992	0.193404
$109$	−7.65845	−0.733546	−0.366773	−	0.930310i	$-0.619537\pi$
$109$	−7.65845	−0.733546	−0.366773	+	0.930310i	$0.619537\pi$
$110$	2.41768	0.230517
$111$	4.47783	0.425017
$112$	14.7524	1.39397
$113$	0.0280040	0.00263439	0.00131720	−	0.999999i	$-0.499581\pi$
$113$	0.0280040	0.00263439	0.00131720	+	0.999999i	$0.499581\pi$
$114$	−7.44765	−0.697536
$115$	0.240346	0.0224124
$116$	0.812213	0.0754121
$117$	0.609579	0.0563556
$118$	−16.5309	−1.52179
$119$	−14.6798	−1.34569
$120$	4.81233	0.439303
$121$	−7.42660	−0.675146
$122$	2.53460	0.229472
$123$	2.27400	0.205039
$124$	−0.260792	−0.0234198
$125$	1.00000	0.0894427
$126$	−2.80820	−0.250175
$127$	12.2700	1.08879	0.544394	−	0.838830i	$-0.316760\pi$
$127$	12.2700	1.08879	0.544394	+	0.838830i	$0.316760\pi$
$128$	−7.28831	−0.644201
$129$	19.3421	1.70298
$130$	1.66884	0.146367
$131$	5.02846	0.439338	0.219669	−	0.975574i	$-0.429502\pi$
$131$	5.02846	0.439338	0.219669	+	0.975574i	$0.429502\pi$
$132$	1.09583	0.0953798
$133$	−17.1971	−1.49117
$134$	−6.99912	−0.604631
$135$	−5.51795	−0.474910
$136$	9.44442	0.809853
$137$	8.54445	0.730002	0.365001	−	0.931007i	$-0.381069\pi$
$137$	8.54445	0.730002	0.365001	+	0.931007i	$0.381069\pi$
$138$	−0.489214	−0.0416446
$139$	12.1806	1.03315	0.516575	−	0.856242i	$-0.327207\pi$
$139$	12.1806	1.03315	0.516575	+	0.856242i	$0.327207\pi$
$140$	1.71197	0.144688
$141$	3.39916	0.286261
$142$	3.53594	0.296730
$143$	2.46659	0.206267
$144$	1.46636	0.122197
$145$	−2.22982	−0.185177
$146$	−13.3235	−1.10266
$147$	24.0153	1.98075
$148$	−1.02486	−0.0842430
$149$	3.93149	0.322080	0.161040	−	0.986948i	$-0.448515\pi$
$149$	3.93149	0.322080	0.161040	+	0.986948i	$0.448515\pi$
$150$	−2.03546	−0.166194
$151$	−1.06830	−0.0869371	−0.0434686	−	0.999055i	$-0.513841\pi$
$151$	−1.06830	−0.0869371	−0.0434686	+	0.999055i	$0.513841\pi$
$152$	11.0639	0.897404
$153$	−1.45914	−0.117965
$154$	−11.3631	−0.915662
$155$	0.715968	0.0575079
$156$	0.756413	0.0605615
$157$	−3.33629	−0.266265	−0.133132	−	0.991098i	$-0.542504\pi$
$157$	−3.33629	−0.266265	−0.133132	+	0.991098i	$0.542504\pi$
$158$	0.826132	0.0657236
$159$	−7.77368	−0.616493
$160$	−2.03314	−0.160734
$161$	−1.12962	−0.0890267
$162$	9.43907	0.741604
$163$	1.05713	0.0828008	0.0414004	−	0.999143i	$-0.486818\pi$
$163$	1.05713	0.0828008	0.0414004	+	0.999143i	$0.486818\pi$
$164$	−0.520459	−0.0406410
$165$	−3.00846	−0.234208
$166$	2.10281	0.163210
$167$	−9.80225	−0.758521	−0.379260	−	0.925290i	$-0.623822\pi$
$167$	−9.80225	−0.758521	−0.379260	+	0.925290i	$0.623822\pi$
$168$	−22.6179	−1.74501
$169$	−11.2974	−0.869031
$170$	−3.99468	−0.306378
$171$	−1.70935	−0.130717
$172$	−4.42691	−0.337549
$173$	−6.89852	−0.524485	−0.262242	−	0.965002i	$-0.584462\pi$
$173$	−6.89852	−0.524485	−0.262242	+	0.965002i	$0.584462\pi$
$174$	4.53870	0.344078
$175$	−4.69998	−0.355285
$176$	5.93345	0.447250
$177$	20.5703	1.54616
$178$	−11.3372	−0.849756
$179$	−3.80637	−0.284501	−0.142251	−	0.989831i	$-0.545434\pi$
$179$	−3.80637	−0.284501	−0.142251	+	0.989831i	$0.545434\pi$
$180$	0.170166	0.0126835
$181$	−4.23389	−0.314703	−0.157351	−	0.987543i	$-0.550296\pi$
$181$	−4.23389	−0.314703	−0.157351	+	0.987543i	$0.550296\pi$
$182$	−7.84352	−0.581400
$183$	−3.15394	−0.233146
$184$	0.726756	0.0535772
$185$	2.81362	0.206861
$186$	−1.45732	−0.106856
$187$	−5.90424	−0.431761
$188$	−0.777979	−0.0567399
$189$	25.9343	1.88644
$190$	−4.67968	−0.339500
$191$	6.56467	0.475003	0.237501	−	0.971387i	$-0.423672\pi$
$191$	6.56467	0.475003	0.237501	+	0.971387i	$0.423672\pi$
$192$	14.1292	1.01968
$193$	−5.86249	−0.421991	−0.210996	−	0.977487i	$-0.567671\pi$
$193$	−5.86249	−0.421991	−0.210996	+	0.977487i	$0.567671\pi$
$194$	−2.64910	−0.190194
$195$	−2.07663	−0.148711
$196$	−5.49648	−0.392606
$197$	13.5963	0.968694	0.484347	−	0.874876i	$-0.339057\pi$
$197$	13.5963	0.968694	0.484347	+	0.874876i	$0.339057\pi$
$198$	−1.12946	−0.0802676
$199$	−2.91751	−0.206817	−0.103409	−	0.994639i	$-0.532975\pi$
$199$	−2.91751	−0.206817	−0.103409	+	0.994639i	$0.532975\pi$
$200$	3.02379	0.213814
$201$	8.70939	0.614313
$202$	2.50889	0.176525
$203$	10.4801	0.735560
$204$	−1.81062	−0.126769
$205$	1.42885	0.0997952
$206$	−3.20867	−0.223558
$207$	−0.112282	−0.00780415
$208$	4.09565	0.283982
$209$	−6.91669	−0.478437
$210$	9.56661	0.660159
$211$	0.851179	0.0585976	0.0292988	−	0.999571i	$-0.490673\pi$
$211$	0.851179	0.0585976	0.0292988	+	0.999571i	$0.490673\pi$
$212$	1.77920	0.122196
$213$	−4.39997	−0.301481
$214$	−9.73829	−0.665696
$215$	12.1535	0.828861
$216$	−16.6851	−1.13528
$217$	−3.36504	−0.228434
$218$	9.79488	0.663393
$219$	16.5791	1.12031
$220$	0.688558	0.0464226
$221$	−4.07549	−0.274147
$222$	−5.72699	−0.384370
$223$	13.4631	0.901555	0.450777	−	0.892636i	$-0.351147\pi$
$223$	13.4631	0.901555	0.450777	+	0.892636i	$0.351147\pi$
$224$	9.55574	0.638470
$225$	−0.467169	−0.0311446
$226$	−0.0358161	−0.00238245
$227$	4.02584	0.267204	0.133602	−	0.991035i	$-0.457346\pi$
$227$	4.02584	0.267204	0.133602	+	0.991035i	$0.457346\pi$
$228$	−2.12110	−0.140473
$229$	3.20846	0.212021	0.106010	−	0.994365i	$-0.466192\pi$
$229$	3.20846	0.212021	0.106010	+	0.994365i	$0.466192\pi$
$230$	−0.307394	−0.0202690
$231$	14.1397	0.930324
$232$	−6.74251	−0.442668
$233$	23.8981	1.56562	0.782808	−	0.622263i	$-0.213787\pi$
$233$	23.8981	1.56562	0.782808	+	0.622263i	$0.213787\pi$
$234$	−0.779630	−0.0509660
$235$	2.13584	0.139327
$236$	−4.70802	−0.306466
$237$	−1.02800	−0.0667760
$238$	18.7749	1.21700
$239$	23.5738	1.52486	0.762431	−	0.647069i	$-0.224006\pi$
$239$	23.5738	1.52486	0.762431	+	0.647069i	$0.224006\pi$
$240$	−4.99539	−0.322451
$241$	−20.8905	−1.34568	−0.672839	−	0.739789i	$-0.734925\pi$
$241$	−20.8905	−1.34568	−0.672839	+	0.739789i	$0.734925\pi$
$242$	9.49836	0.610578
$243$	4.80830	0.308452
$244$	0.721856	0.0462121
$245$	15.0898	0.964055
$246$	−2.90836	−0.185430
$247$	−4.77435	−0.303784
$248$	2.16494	0.137474
$249$	−2.61665	−0.165823
$250$	−1.27896	−0.0808888
$251$	23.9213	1.50990	0.754951	−	0.655782i	$-0.227661\pi$
$251$	23.9213	1.50990	0.754951	+	0.655782i	$0.227661\pi$
$252$	−0.799779	−0.0503814
$253$	−0.454336	−0.0285639
$254$	−15.6929	−0.984660
$255$	4.97080	0.311284
$256$	−8.43443	−0.527152
$257$	−24.6514	−1.53771	−0.768855	−	0.639423i	$-0.779173\pi$
$257$	−24.6514	−1.53771	−0.768855	+	0.639423i	$0.779173\pi$
$258$	−24.7379	−1.54011
$259$	−13.2240	−0.821696
$260$	0.475287	0.0294761
$261$	1.04170	0.0644798
$262$	−6.43122	−0.397322
$263$	5.00368	0.308540	0.154270	−	0.988029i	$-0.450697\pi$
$263$	5.00368	0.308540	0.154270	+	0.988029i	$0.450697\pi$
$264$	−9.09694	−0.559878
$265$	−4.88454	−0.300055
$266$	21.9944	1.34856
$267$	14.1075	0.863363
$268$	−1.99335	−0.121764
$269$	−20.8682	−1.27235	−0.636177	−	0.771543i	$-0.719485\pi$
$269$	−20.8682	−1.27235	−0.636177	+	0.771543i	$0.719485\pi$
$270$	7.05727	0.429492
$271$	16.1412	0.980507	0.490253	−	0.871580i	$-0.336904\pi$
$271$	16.1412	0.980507	0.490253	+	0.871580i	$0.336904\pi$
$272$	−9.80370	−0.594437
$273$	9.76013	0.590710
$274$	−10.9281	−0.660188
$275$	−1.89034	−0.113992
$276$	−0.139329	−0.00838659
$277$	32.8275	1.97241	0.986205	−	0.165528i	$-0.0529327\pi$
$277$	32.8275	1.97241	0.986205	+	0.165528i	$0.0529327\pi$
$278$	−15.5786	−0.934343
$279$	−0.334478	−0.0200247
$280$	−14.2118	−0.849316
$281$	−21.4938	−1.28222	−0.641108	−	0.767451i	$-0.721525\pi$
$281$	−21.4938	−1.28222	−0.641108	+	0.767451i	$0.721525\pi$
$282$	−4.34740	−0.258884
$283$	18.3042	1.08807	0.544036	−	0.839062i	$-0.316895\pi$
$283$	18.3042	1.08807	0.544036	+	0.839062i	$0.316895\pi$
$284$	1.00704	0.0597568
$285$	5.82319	0.344936
$286$	−3.15468	−0.186540
$287$	−6.71557	−0.396408
$288$	0.949821	0.0559687
$289$	−7.24456	−0.426150
$290$	2.85186	0.167467
$291$	3.29642	0.193240
$292$	−3.79454	−0.222058
$293$	26.2643	1.53438	0.767188	−	0.641423i	$-0.221656\pi$
$293$	26.2643	1.53438	0.767188	+	0.641423i	$0.221656\pi$
$294$	−30.7147	−1.79132
$295$	12.9252	0.752536
$296$	8.50779	0.494505
$297$	10.4308	0.605258
$298$	−5.02824	−0.291278
$299$	−0.313612	−0.0181367
$300$	−0.579700	−0.0334690
$301$	−57.1212	−3.29241
$302$	1.36632	0.0786228
$303$	−3.12195	−0.179351
$304$	−11.4848	−0.658700
$305$	−1.98176	−0.113475
$306$	1.86619	0.106683
$307$	−25.0844	−1.43164	−0.715820	−	0.698285i	$-0.753947\pi$
$307$	−25.0844	−1.43164	−0.715820	+	0.698285i	$0.753947\pi$
$308$	−3.23621	−0.184400
$309$	3.99273	0.227138
$310$	−0.915698	−0.0520081
$311$	19.8222	1.12401	0.562006	−	0.827133i	$-0.310030\pi$
$311$	19.8222	1.12401	0.562006	+	0.827133i	$0.310030\pi$
$312$	−6.27930	−0.355495
$313$	21.4929	1.21485	0.607424	−	0.794377i	$-0.292203\pi$
$313$	21.4929	1.21485	0.607424	+	0.794377i	$0.292203\pi$
$314$	4.26699	0.240800
$315$	2.19569	0.123713
$316$	0.235283	0.0132357
$317$	10.4437	0.586575	0.293288	−	0.956024i	$-0.405251\pi$
$317$	10.4437	0.586575	0.293288	+	0.956024i	$0.405251\pi$
$318$	9.94226	0.557535
$319$	4.21512	0.236002
$320$	8.87796	0.496293
$321$	12.1179	0.676356
$322$	1.44475	0.0805126
$323$	11.4283	0.635887
$324$	2.68826	0.149348
$325$	−1.30484	−0.0723793
$326$	−1.35203	−0.0748821
$327$	−12.1883	−0.674015
$328$	4.32054	0.238562
$329$	−10.0384	−0.553435
$330$	3.84771	0.211809
$331$	−26.9789	−1.48289	−0.741447	−	0.671011i	$-0.765860\pi$
$331$	−26.9789	−1.48289	−0.741447	+	0.671011i	$0.765860\pi$
$332$	0.598883	0.0328680
$333$	−1.31443	−0.0720305
$334$	12.5367	0.685979
$335$	5.47249	0.298994
$336$	23.4783	1.28084
$337$	−15.1950	−0.827726	−0.413863	−	0.910339i	$-0.635821\pi$
$337$	−15.1950	−0.827726	−0.413863	+	0.910339i	$0.635821\pi$
$338$	14.4490	0.785921
$339$	0.0445680	0.00242060
$340$	−1.13769	−0.0616998
$341$	−1.35343	−0.0732921
$342$	2.18620	0.118216
$343$	−38.0222	−2.05300
$344$	36.7496	1.98141
$345$	0.382508	0.0205935
$346$	8.82296	0.474326
$347$	2.14907	0.115368	0.0576840	−	0.998335i	$-0.481628\pi$
$347$	2.14907	0.115368	0.0576840	+	0.998335i	$0.481628\pi$
$348$	1.29263	0.0692920
$349$	10.9360	0.585392	0.292696	−	0.956206i	$-0.405448\pi$
$349$	10.9360	0.585392	0.292696	+	0.956206i	$0.405448\pi$
$350$	6.01111	0.321307
$351$	7.20003	0.384309
$352$	3.84334	0.204851
$353$	−4.05902	−0.216040	−0.108020	−	0.994149i	$-0.534451\pi$
$353$	−4.05902	−0.216040	−0.108020	+	0.994149i	$0.534451\pi$
$354$	−26.3087	−1.39829
$355$	−2.76469	−0.146735
$356$	−3.22883	−0.171128
$357$	−23.3627	−1.23649
$358$	4.86821	0.257293
$359$	−20.6454	−1.08962	−0.544812	−	0.838558i	$-0.683399\pi$
$359$	−20.6454	−1.08962	−0.544812	+	0.838558i	$0.683399\pi$
$360$	−1.41262	−0.0744517
$361$	−5.61201	−0.295369
$362$	5.41500	0.284606
$363$	−11.8193	−0.620355
$364$	−2.23384	−0.117085
$365$	10.4174	0.545271
$366$	4.03378	0.210849
$367$	29.4934	1.53954	0.769771	−	0.638320i	$-0.220370\pi$
$367$	29.4934	1.53954	0.769771	+	0.638320i	$0.220370\pi$
$368$	−0.754403	−0.0393260
$369$	−0.667514	−0.0347494
$370$	−3.59851	−0.187078
$371$	22.9573	1.19188
$372$	−0.415046	−0.0215192
$373$	20.3978	1.05616	0.528080	−	0.849195i	$-0.322912\pi$
$373$	20.3978	1.05616	0.528080	+	0.849195i	$0.322912\pi$
$374$	7.55131	0.390469
$375$	1.59149	0.0821840
$376$	6.45832	0.333063
$377$	2.90955	0.149850
$378$	−33.1690	−1.70603
$379$	−22.4373	−1.15253	−0.576263	−	0.817264i	$-0.695490\pi$
$379$	−22.4373	−1.15253	−0.576263	+	0.817264i	$0.695490\pi$
$380$	−1.33278	−0.0683700
$381$	19.5276	1.00043
$382$	−8.39598	−0.429576
$383$	−0.214813	−0.0109764	−0.00548822	−	0.999985i	$-0.501747\pi$
$383$	−0.214813	−0.0109764	−0.00548822	+	0.999985i	$0.501747\pi$
$384$	−11.5992	−0.591922
$385$	8.88458	0.452800
$386$	7.49792	0.381634
$387$	−5.67773	−0.288615
$388$	−0.754465	−0.0383022
$389$	−33.3122	−1.68899	−0.844497	−	0.535560i	$-0.820100\pi$
$389$	−33.3122	−1.68899	−0.844497	+	0.535560i	$0.820100\pi$
$390$	2.65594	0.134489
$391$	0.750690	0.0379640
$392$	45.6286	2.30459
$393$	8.00273	0.403684
$394$	−17.3891	−0.876052
$395$	−0.645939	−0.0325007
$396$	−0.321673	−0.0161647
$397$	8.30984	0.417059	0.208529	−	0.978016i	$-0.433132\pi$
$397$	8.30984	0.417059	0.208529	+	0.978016i	$0.433132\pi$
$398$	3.73140	0.187038
$399$	−27.3689	−1.37016
$400$	−3.13882	−0.156941
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	−11.1390	−0.555563
$403$	−0.934222	−0.0465369
$404$	0.714533	0.0355494
$405$	−7.38025	−0.366727
$406$	−13.4037	−0.665214
$407$	−5.31870	−0.263638
$408$	15.0307	0.744129
$409$	16.5390	0.817801	0.408900	−	0.912579i	$-0.365912\pi$
$409$	16.5390	0.817801	0.408900	+	0.912579i	$0.365912\pi$
$410$	−1.82745	−0.0902512
$411$	13.5984	0.670759
$412$	−0.913832	−0.0450213
$413$	−60.7484	−2.98923
$414$	0.143605	0.00705780
$415$	−1.64415	−0.0807083
$416$	2.65292	0.130070
$417$	19.3853	0.949304
$418$	8.84620	0.432682
$419$	15.0760	0.736512	0.368256	−	0.929724i	$-0.379955\pi$
$419$	15.0760	0.736512	0.368256	+	0.929724i	$0.379955\pi$
$420$	2.72458	0.132946
$421$	−28.0404	−1.36661	−0.683303	−	0.730135i	$-0.739457\pi$
$421$	−28.0404	−1.36661	−0.683303	+	0.730135i	$0.739457\pi$
$422$	−1.08863	−0.0529936
$423$	−0.997796	−0.0485145
$424$	−14.7698	−0.717287
$425$	3.12337	0.151506
$426$	5.62741	0.272649
$427$	9.31423	0.450747
$428$	−2.77348	−0.134061
$429$	3.92554	0.189527
$430$	−15.5439	−0.749593
$431$	−18.2463	−0.878892	−0.439446	−	0.898269i	$-0.644825\pi$
$431$	−18.2463	−0.878892	−0.439446	+	0.898269i	$0.644825\pi$
$432$	17.3199	0.833303
$433$	−9.66498	−0.464469	−0.232235	−	0.972660i	$-0.574604\pi$
$433$	−9.66498	−0.464469	−0.232235	+	0.972660i	$0.574604\pi$
$434$	4.30376	0.206587
$435$	−3.54873	−0.170149
$436$	2.78959	0.133597
$437$	0.879417	0.0420682
$438$	−21.2041	−1.01317
$439$	−37.6843	−1.79857	−0.899286	−	0.437360i	$-0.855913\pi$
$439$	−37.6843	−1.79857	−0.899286	+	0.437360i	$0.855913\pi$
$440$	−5.71600	−0.272500
$441$	−7.04951	−0.335691
$442$	5.21241	0.247929
$443$	5.92101	0.281316	0.140658	−	0.990058i	$-0.455078\pi$
$443$	5.92101	0.281316	0.140658	+	0.990058i	$0.455078\pi$
$444$	−1.63105	−0.0774063
$445$	8.86433	0.420210
$446$	−17.2188	−0.815334
$447$	6.25692	0.295942
$448$	−41.7263	−1.97138
$449$	21.5690	1.01790	0.508951	−	0.860795i	$-0.330033\pi$
$449$	21.5690	1.01790	0.508951	+	0.860795i	$0.330033\pi$
$450$	0.597492	0.0281661
$451$	−2.70102	−0.127186
$452$	−0.0102005	−0.000479789 0
$453$	−1.70019	−0.0798818
$454$	−5.14890	−0.241650
$455$	6.13271	0.287506
$456$	17.6081	0.824575
$457$	−29.3519	−1.37302	−0.686511	−	0.727119i	$-0.740859\pi$
$457$	−29.3519	−1.37302	−0.686511	+	0.727119i	$0.740859\pi$
$458$	−4.10351	−0.191744
$459$	−17.2346	−0.804443
$460$	−0.0875461	−0.00408186
$461$	−0.328538	−0.0153015	−0.00765077	−	0.999971i	$-0.502435\pi$
$461$	−0.328538	−0.0153015	−0.00765077	+	0.999971i	$0.502435\pi$
$462$	−18.0842	−0.841351
$463$	5.52563	0.256798	0.128399	−	0.991723i	$-0.459016\pi$
$463$	5.52563	0.256798	0.128399	+	0.991723i	$0.459016\pi$
$464$	6.99901	0.324921
$465$	1.13945	0.0528409
$466$	−30.5648	−1.41589
$467$	−3.95742	−0.183127	−0.0915637	−	0.995799i	$-0.529187\pi$
$467$	−3.95742	−0.183127	−0.0915637	+	0.995799i	$0.529187\pi$
$468$	−0.222039	−0.0102638
$469$	−25.7206	−1.18767
$470$	−2.73166	−0.126002
$471$	−5.30966	−0.244656
$472$	39.0832	1.79895
$473$	−22.9743	−1.05636
$474$	1.31478	0.0603898
$475$	3.65896	0.167885
$476$	5.34712	0.245085
$477$	2.28191	0.104481
$478$	−30.1500	−1.37903
$479$	−33.8572	−1.54697	−0.773487	−	0.633812i	$-0.781489\pi$
$479$	−33.8572	−1.54697	−0.773487	+	0.633812i	$0.781489\pi$
$480$	−3.23572	−0.147690
$481$	−3.67131	−0.167397
$482$	26.7183	1.21698
$483$	−1.79778	−0.0818018
$484$	2.70514	0.122961
$485$	2.07128	0.0940521
$486$	−6.14964	−0.278953
$487$	12.1862	0.552212	0.276106	−	0.961127i	$-0.410956\pi$
$487$	12.1862	0.552212	0.276106	+	0.961127i	$0.410956\pi$
$488$	−5.99242	−0.271264
$489$	1.68241	0.0760812
$490$	−19.2994	−0.871857
$491$	41.8553	1.88891	0.944453	−	0.328648i	$-0.106593\pi$
$491$	41.8553	1.88891	0.944453	+	0.328648i	$0.106593\pi$
$492$	−0.828304	−0.0373428
$493$	−6.96456	−0.313668
$494$	6.10622	0.274732
$495$	0.883109	0.0396928
$496$	−2.24730	−0.100907
$497$	12.9940	0.582861
$498$	3.34660	0.149965
$499$	−3.07664	−0.137729	−0.0688647	−	0.997626i	$-0.521938\pi$
$499$	−3.07664	−0.137729	−0.0688647	+	0.997626i	$0.521938\pi$
$500$	−0.364250	−0.0162898
$501$	−15.6002	−0.696963
$502$	−30.5945	−1.36550
$503$	24.3922	1.08759	0.543797	−	0.839217i	$-0.316986\pi$
$503$	24.3922	1.08759	0.543797	+	0.839217i	$0.316986\pi$
$504$	6.63930	0.295738
$505$	−1.96166	−0.0872925
$506$	0.581080	0.0258322
$507$	−17.9797	−0.798505
$508$	−4.46935	−0.198296
$509$	−33.5382	−1.48655	−0.743277	−	0.668984i	$-0.766729\pi$
$509$	−33.5382	−1.48655	−0.743277	+	0.668984i	$0.766729\pi$
$510$	−6.35748	−0.281514
$511$	−48.9615	−2.16593
$512$	25.3640	1.12094
$513$	−20.1900	−0.891409
$514$	31.5282	1.39065
$515$	2.50880	0.110551
$516$	−7.04537	−0.310155
$517$	−4.03746	−0.177567
$518$	16.9130	0.743113
$519$	−10.9789	−0.481921
$520$	−3.94555	−0.173024
$521$	−37.5997	−1.64727	−0.823636	−	0.567118i	$-0.808058\pi$
$521$	−37.5997	−1.64727	−0.823636	+	0.567118i	$0.808058\pi$
$522$	−1.33230	−0.0583132
$523$	18.5395	0.810678	0.405339	−	0.914167i	$-0.367154\pi$
$523$	18.5395	0.810678	0.405339	+	0.914167i	$0.367154\pi$
$524$	−1.83162	−0.0800146
$525$	−7.47996	−0.326452
$526$	−6.39953	−0.279033
$527$	2.23623	0.0974119
$528$	9.44301	0.410954
$529$	−22.9422	−0.997488
$530$	6.24715	0.271359
$531$	−6.03827	−0.262039
$532$	6.26403	0.271580
$533$	−1.86442	−0.0807568
$534$	−18.0429	−0.780795
$535$	7.61420	0.329191
$536$	16.5477	0.714750
$537$	−6.05778	−0.261413
$538$	26.6896	1.15067
$539$	−28.5250	−1.22866
$540$	2.00992	0.0864930
$541$	−42.9011	−1.84446	−0.922231	−	0.386640i	$-0.873636\pi$
$541$	−42.9011	−1.84446	−0.922231	+	0.386640i	$0.873636\pi$
$542$	−20.6440	−0.886735
$543$	−6.73819	−0.289163
$544$	−6.35026	−0.272265
$545$	−7.65845	−0.328052
$546$	−12.4829	−0.534217
$547$	−3.43202	−0.146742	−0.0733712	−	0.997305i	$-0.523376\pi$
$547$	−3.43202	−0.146742	−0.0733712	+	0.997305i	$0.523376\pi$
$548$	−3.11232	−0.132952
$549$	0.925815	0.0395128
$550$	2.41768	0.103090
$551$	−8.15883	−0.347578
$552$	1.15662	0.0492292
$553$	3.03590	0.129100
$554$	−41.9851	−1.78378
$555$	4.47783	0.190073
$556$	−4.43680	−0.188162
$557$	−16.3381	−0.692268	−0.346134	−	0.938185i	$-0.612506\pi$
$557$	−16.3381	−0.692268	−0.346134	+	0.938185i	$0.612506\pi$
$558$	0.427785	0.0181096
$559$	−15.8583	−0.670736
$560$	14.7524	0.623403
$561$	−9.39652	−0.396722
$562$	27.4899	1.15959
$563$	20.9462	0.882775	0.441388	−	0.897317i	$-0.354486\pi$
$563$	20.9462	0.882775	0.441388	+	0.897317i	$0.354486\pi$
$564$	−1.23814	−0.0521352
$565$	0.0280040	0.00117814
$566$	−23.4104	−0.984014
$567$	34.6870	1.45672
$568$	−8.35986	−0.350772
$569$	28.2992	1.18637	0.593183	−	0.805068i	$-0.297871\pi$
$569$	28.2992	1.18637	0.593183	+	0.805068i	$0.297871\pi$
$570$	−7.44765	−0.311948
$571$	−14.6035	−0.611136	−0.305568	−	0.952170i	$-0.598846\pi$
$571$	−14.6035	−0.611136	−0.305568	+	0.952170i	$0.598846\pi$
$572$	−0.898456	−0.0375663
$573$	10.4476	0.436454
$574$	8.58897	0.358497
$575$	0.240346	0.0100231
$576$	−4.14751	−0.172813
$577$	−10.0194	−0.417115	−0.208557	−	0.978010i	$-0.566877\pi$
$577$	−10.0194	−0.417115	−0.208557	+	0.978010i	$0.566877\pi$
$578$	9.26553	0.385395
$579$	−9.33008	−0.387745
$580$	0.812213	0.0337253
$581$	7.72749	0.320590
$582$	−4.21600	−0.174759
$583$	9.23346	0.382411
$584$	31.5000	1.30348
$585$	0.609579	0.0252030
$586$	−33.5911	−1.38763
$587$	−21.0098	−0.867166	−0.433583	−	0.901114i	$-0.642751\pi$
$587$	−21.0098	−0.867166	−0.433583	+	0.901114i	$0.642751\pi$
$588$	−8.74758	−0.360744
$589$	2.61970	0.107943
$590$	−16.5309	−0.680567
$591$	21.6383	0.890080
$592$	−8.83144	−0.362970
$593$	−11.3229	−0.464974	−0.232487	−	0.972599i	$-0.574686\pi$
$593$	−11.3229	−0.464974	−0.232487	+	0.972599i	$0.574686\pi$
$594$	−13.3407	−0.547374
$595$	−14.6798	−0.601813
$596$	−1.43205	−0.0586589
$597$	−4.64319	−0.190033
$598$	0.401099	0.0164022
$599$	22.5851	0.922803	0.461402	−	0.887191i	$-0.347347\pi$
$599$	22.5851	0.922803	0.461402	+	0.887191i	$0.347347\pi$
$600$	4.81233	0.196462
$601$	21.8415	0.890935	0.445467	−	0.895298i	$-0.353038\pi$
$601$	21.8415	0.890935	0.445467	+	0.895298i	$0.353038\pi$
$602$	73.0560	2.97754
$603$	−2.55658	−0.104112
$604$	0.389129	0.0158334
$605$	−7.42660	−0.301934
$606$	3.99286	0.162199
$607$	−7.93373	−0.322020	−0.161010	−	0.986953i	$-0.551475\pi$
$607$	−7.93373	−0.322020	−0.161010	+	0.986953i	$0.551475\pi$
$608$	−7.43919	−0.301699
$609$	16.6790	0.675866
$610$	2.53460	0.102623
$611$	−2.78692	−0.112747
$612$	0.531493	0.0214843
$613$	31.5368	1.27376	0.636879	−	0.770964i	$-0.280225\pi$
$613$	31.5368	1.27376	0.636879	+	0.770964i	$0.280225\pi$
$614$	32.0820	1.29472
$615$	2.27400	0.0916964
$616$	26.8651	1.08243
$617$	22.3910	0.901428	0.450714	−	0.892668i	$-0.351169\pi$
$617$	22.3910	0.901428	0.450714	+	0.892668i	$0.351169\pi$
$618$	−5.10655	−0.205416
$619$	3.26853	0.131373	0.0656866	−	0.997840i	$-0.479076\pi$
$619$	3.26853	0.131373	0.0656866	+	0.997840i	$0.479076\pi$
$620$	−0.260792	−0.0104736
$621$	−1.32622	−0.0532193
$622$	−25.3519	−1.01652
$623$	−41.6622	−1.66916
$624$	6.51817	0.260936
$625$	1.00000	0.0400000
$626$	−27.4886	−1.09867
$627$	−11.0078	−0.439610
$628$	1.21524	0.0484935
$629$	8.78797	0.350399
$630$	−2.80820	−0.111882
$631$	10.5635	0.420525	0.210262	−	0.977645i	$-0.432568\pi$
$631$	10.5635	0.420525	0.210262	+	0.977645i	$0.432568\pi$
$632$	−1.95318	−0.0776935
$633$	1.35464	0.0538421
$634$	−13.3571	−0.530478
$635$	12.2700	0.486920
$636$	2.83157	0.112279
$637$	−19.6898	−0.780138
$638$	−5.39099	−0.213432
$639$	1.29158	0.0510941
$640$	−7.28831	−0.288096
$641$	16.6932	0.659344	0.329672	−	0.944096i	$-0.393062\pi$
$641$	16.6932	0.659344	0.329672	+	0.944096i	$0.393062\pi$
$642$	−15.4984	−0.611672
$643$	−10.3107	−0.406614	−0.203307	−	0.979115i	$-0.565169\pi$
$643$	−10.3107	−0.406614	−0.203307	+	0.979115i	$0.565169\pi$
$644$	0.411465	0.0162140
$645$	19.3421	0.761595
$646$	−14.6164	−0.575074
$647$	23.9154	0.940211	0.470106	−	0.882610i	$-0.344216\pi$
$647$	23.9154	0.940211	0.470106	+	0.882610i	$0.344216\pi$
$648$	−22.3163	−0.876668
$649$	−24.4331	−0.959084
$650$	1.66884	0.0654573
$651$	−5.35542	−0.209895
$652$	−0.385060	−0.0150801
$653$	10.5992	0.414781	0.207390	−	0.978258i	$-0.433503\pi$
$653$	10.5992	0.414781	0.207390	+	0.978258i	$0.433503\pi$
$654$	15.5884	0.609556
$655$	5.02846	0.196478
$656$	−4.48490	−0.175106
$657$	−4.86668	−0.189867
$658$	12.8387	0.500507
$659$	−22.3492	−0.870602	−0.435301	−	0.900285i	$-0.643358\pi$
$659$	−22.3492	−0.870602	−0.435301	+	0.900285i	$0.643358\pi$
$660$	1.09583	0.0426552
$661$	28.1748	1.09587	0.547936	−	0.836520i	$-0.315414\pi$
$661$	28.1748	1.09587	0.547936	+	0.836520i	$0.315414\pi$
$662$	34.5050	1.34108
$663$	−6.48609	−0.251899
$664$	−4.97158	−0.192935
$665$	−17.1971	−0.666873
$666$	1.68111	0.0651418
$667$	−0.535929	−0.0207512
$668$	3.57047	0.138146
$669$	21.4263	0.828390
$670$	−6.99912	−0.270399
$671$	3.74620	0.144620
$672$	15.2078	0.586655
$673$	−3.84262	−0.148122	−0.0740611	−	0.997254i	$-0.523596\pi$
$673$	−3.84262	−0.148122	−0.0740611	+	0.997254i	$0.523596\pi$
$674$	19.4339	0.748566
$675$	−5.51795	−0.212386
$676$	4.11508	0.158272
$677$	−14.1698	−0.544589	−0.272294	−	0.962214i	$-0.587782\pi$
$677$	−14.1698	−0.544589	−0.272294	+	0.962214i	$0.587782\pi$
$678$	−0.0570008	−0.00218910
$679$	−9.73500	−0.373595
$680$	9.44442	0.362177
$681$	6.40707	0.245519
$682$	1.73098	0.0662827
$683$	−50.1864	−1.92033	−0.960165	−	0.279433i	$-0.909854\pi$
$683$	−50.1864	−1.92033	−0.960165	+	0.279433i	$0.909854\pi$
$684$	0.622632	0.0238069
$685$	8.54445	0.326467
$686$	48.6290	1.85666
$687$	5.10622	0.194815
$688$	−38.1476	−1.45436
$689$	6.37353	0.242812
$690$	−0.489214	−0.0186240
$691$	20.8678	0.793848	0.396924	−	0.917851i	$-0.370078\pi$
$691$	20.8678	0.793848	0.396924	+	0.917851i	$0.370078\pi$
$692$	2.51279	0.0955219
$693$	−4.15060	−0.157668
$694$	−2.74858	−0.104335
$695$	12.1806	0.462038
$696$	−10.7306	−0.406743
$697$	4.46283	0.169042
$698$	−13.9868	−0.529408
$699$	38.0335	1.43856
$700$	1.71197	0.0647064
$701$	32.9343	1.24391	0.621956	−	0.783052i	$-0.286338\pi$
$701$	32.9343	1.24391	0.621956	+	0.783052i	$0.286338\pi$
$702$	−9.20858	−0.347556
$703$	10.2949	0.388280
$704$	−16.7824	−0.632510
$705$	3.39916	0.128020
$706$	5.19134	0.195379
$707$	9.21975	0.346744
$708$	−7.49276	−0.281595
$709$	−10.5568	−0.396468	−0.198234	−	0.980155i	$-0.563521\pi$
$709$	−10.5568	−0.396468	−0.198234	+	0.980155i	$0.563521\pi$
$710$	3.53594	0.132702
$711$	0.301762	0.0113170
$712$	26.8039	1.00452
$713$	0.172080	0.00644445
$714$	29.8801	1.11823
$715$	2.46659	0.0922452
$716$	1.38647	0.0518148
$717$	37.5174	1.40111
$718$	26.4048	0.985417
$719$	32.6940	1.21928	0.609641	−	0.792678i	$-0.291314\pi$
$719$	32.6940	1.21928	0.609641	+	0.792678i	$0.291314\pi$
$720$	1.46636	0.0546480
$721$	−11.7913	−0.439132
$722$	7.17756	0.267121
$723$	−33.2470	−1.23647
$724$	1.54220	0.0573153
$725$	−2.22982	−0.0828135
$726$	15.1165	0.561027
$727$	−17.7587	−0.658634	−0.329317	−	0.944220i	$-0.606818\pi$
$727$	−17.7587	−0.658634	−0.329317	+	0.944220i	$0.606818\pi$
$728$	18.5440	0.687288
$729$	29.7931	1.10345
$730$	−13.3235	−0.493123
$731$	37.9599	1.40400
$732$	1.14882	0.0424617
$733$	−2.57234	−0.0950115	−0.0475057	−	0.998871i	$-0.515127\pi$
$733$	−2.57234	−0.0950115	−0.0475057	+	0.998871i	$0.515127\pi$
$734$	−37.7210	−1.39231
$735$	24.0153	0.885818
$736$	−0.488658	−0.0180122
$737$	−10.3449	−0.381058
$738$	0.853727	0.0314261
$739$	0.664789	0.0244547	0.0122273	−	0.999925i	$-0.496108\pi$
$739$	0.664789	0.0244547	0.0122273	+	0.999925i	$0.496108\pi$
$740$	−1.02486	−0.0376746
$741$	−7.59831	−0.279131
$742$	−29.3615	−1.07790
$743$	−5.43400	−0.199354	−0.0996770	−	0.995020i	$-0.531781\pi$
$743$	−5.43400	−0.199354	−0.0996770	+	0.995020i	$0.531781\pi$
$744$	3.44547	0.126317
$745$	3.93149	0.144039
$746$	−26.0881	−0.955153
$747$	0.768097	0.0281032
$748$	2.15062	0.0786345
$749$	−35.7866	−1.30762
$750$	−2.03546	−0.0743243
$751$	−48.4047	−1.76631	−0.883156	−	0.469079i	$-0.844586\pi$
$751$	−48.4047	−1.76631	−0.883156	+	0.469079i	$0.844586\pi$
$752$	−6.70401	−0.244470
$753$	38.0705	1.38737
$754$	−3.72121	−0.135519
$755$	−1.06830	−0.0388795
$756$	−9.44657	−0.343569
$757$	3.25934	0.118463	0.0592314	−	0.998244i	$-0.481135\pi$
$757$	3.25934	0.118463	0.0592314	+	0.998244i	$0.481135\pi$
$758$	28.6965	1.04230
$759$	−0.723070	−0.0262458
$760$	11.0639	0.401331
$761$	46.6845	1.69231	0.846157	−	0.532934i	$-0.178911\pi$
$761$	46.6845	1.69231	0.846157	+	0.532934i	$0.178911\pi$
$762$	−24.9751	−0.904751
$763$	35.9946	1.30309
$764$	−2.39118	−0.0865100
$765$	−1.45914	−0.0527554
$766$	0.274738	0.00992670
$767$	−16.8653	−0.608972
$768$	−13.4233	−0.484371
$769$	−5.80547	−0.209350	−0.104675	−	0.994506i	$-0.533380\pi$
$769$	−5.80547	−0.209350	−0.104675	+	0.994506i	$0.533380\pi$
$770$	−11.3631	−0.409496
$771$	−39.2324	−1.41292
$772$	2.13541	0.0768553
$773$	−38.2494	−1.37574	−0.687868	−	0.725836i	$-0.741453\pi$
$773$	−38.2494	−1.37574	−0.687868	+	0.725836i	$0.741453\pi$
$774$	7.26162	0.261013
$775$	0.715968	0.0257183
$776$	6.26313	0.224833
$777$	−21.0457	−0.755012
$778$	42.6051	1.52747
$779$	5.22810	0.187316
$780$	0.756413	0.0270839
$781$	5.22622	0.187009
$782$	−0.960105	−0.0343333
$783$	12.3040	0.439711
$784$	−47.3643	−1.69158
$785$	−3.33629	−0.119077
$786$	−10.2352	−0.365078
$787$	−0.877412	−0.0312764	−0.0156382	−	0.999878i	$-0.504978\pi$
$787$	−0.877412	−0.0312764	−0.0156382	+	0.999878i	$0.504978\pi$
$788$	−4.95244	−0.176424
$789$	7.96330	0.283501
$790$	0.826132	0.0293925
$791$	−0.131618	−0.00467981
$792$	2.67034	0.0948863
$793$	2.58587	0.0918269
$794$	−10.6280	−0.377173
$795$	−7.77368	−0.275704
$796$	1.06271	0.0376666
$797$	3.10164	0.109866	0.0549329	−	0.998490i	$-0.482506\pi$
$797$	3.10164	0.109866	0.0549329	+	0.998490i	$0.482506\pi$
$798$	35.0038	1.23912
$799$	6.67101	0.236003
$800$	−2.03314	−0.0718825
$801$	−4.14114	−0.146320
$802$	1.27896	0.0451618
$803$	−19.6924	−0.694931
$804$	−3.17240	−0.111882
$805$	−1.12962	−0.0398140
$806$	1.19484	0.0420863
$807$	−33.2114	−1.16910
$808$	−5.93164	−0.208674
$809$	45.6850	1.60620	0.803100	−	0.595845i	$-0.203183\pi$
$809$	45.6850	1.60620	0.803100	+	0.595845i	$0.203183\pi$
$810$	9.43907	0.331655
$811$	−3.02692	−0.106290	−0.0531448	−	0.998587i	$-0.516924\pi$
$811$	−3.02692	−0.106290	−0.0531448	+	0.998587i	$0.516924\pi$
$812$	−3.81739	−0.133964
$813$	25.6885	0.900934
$814$	6.80243	0.238425
$815$	1.05713	0.0370296
$816$	−15.6025	−0.546196
$817$	44.4691	1.55578
$818$	−21.1528	−0.739590
$819$	−2.86501	−0.100112
$820$	−0.520459	−0.0181752
$821$	−8.78388	−0.306560	−0.153280	−	0.988183i	$-0.548984\pi$
$821$	−8.78388	−0.306560	−0.153280	+	0.988183i	$0.548984\pi$
$822$	−17.3919	−0.606611
$823$	28.7400	1.00181	0.500906	−	0.865501i	$-0.333000\pi$
$823$	28.7400	1.00181	0.500906	+	0.865501i	$0.333000\pi$
$824$	7.58609	0.264274
$825$	−3.00846	−0.104741
$826$	77.6950	2.70336
$827$	10.0222	0.348505	0.174252	−	0.984701i	$-0.444249\pi$
$827$	10.0222	0.348505	0.174252	+	0.984701i	$0.444249\pi$
$828$	0.0408988	0.00142133
$829$	−21.8640	−0.759369	−0.379685	−	0.925116i	$-0.623967\pi$
$829$	−21.8640	−0.759369	−0.379685	+	0.925116i	$0.623967\pi$
$830$	2.10281	0.0729897
$831$	52.2445	1.81234
$832$	−11.5843	−0.401613
$833$	47.1312	1.63300
$834$	−24.7932	−0.858517
$835$	−9.80225	−0.339221
$836$	2.51941	0.0871355
$837$	−3.95068	−0.136555
$838$	−19.2817	−0.666075
$839$	−29.2129	−1.00854	−0.504270	−	0.863546i	$-0.668238\pi$
$839$	−29.2129	−1.00854	−0.504270	+	0.863546i	$0.668238\pi$
$840$	−22.6179	−0.780390
$841$	−24.0279	−0.828548
$842$	35.8627	1.23591
$843$	−34.2072	−1.17816
$844$	−0.310042	−0.0106721
$845$	−11.2974	−0.388642
$846$	1.27615	0.0438748
$847$	34.9049	1.19935
$848$	15.3317	0.526493
$849$	29.1309	0.999771
$850$	−3.99468	−0.137016
$851$	0.676241	0.0231813
$852$	1.60269	0.0549073
$853$	12.9547	0.443559	0.221779	−	0.975097i	$-0.428814\pi$
$853$	12.9547	0.443559	0.221779	+	0.975097i	$0.428814\pi$
$854$	−11.9126	−0.407639
$855$	−1.70935	−0.0584586
$856$	23.0238	0.786936
$857$	19.4220	0.663443	0.331722	−	0.943377i	$-0.392371\pi$
$857$	19.4220	0.663443	0.331722	+	0.943377i	$0.392371\pi$
$858$	−5.02063	−0.171402
$859$	25.4387	0.867957	0.433979	−	0.900923i	$-0.357109\pi$
$859$	25.4387	0.867957	0.433979	+	0.900923i	$0.357109\pi$
$860$	−4.42691	−0.150956
$861$	−10.6877	−0.364237
$862$	23.3363	0.794839
$863$	55.9941	1.90606	0.953031	−	0.302872i	$-0.0979456\pi$
$863$	55.9941	1.90606	0.953031	+	0.302872i	$0.0979456\pi$
$864$	11.2188	0.381671
$865$	−6.89852	−0.234557
$866$	12.3612	0.420049
$867$	−11.5296	−0.391566
$868$	1.22572	0.0416035
$869$	1.22105	0.0414211
$870$	4.53870	0.153876
$871$	−7.14070	−0.241953
$872$	−23.1575	−0.784213
$873$	−0.967639	−0.0327496
$874$	−1.12474	−0.0380450
$875$	−4.69998	−0.158888
$876$	−6.03895	−0.204037
$877$	3.81340	0.128769	0.0643847	−	0.997925i	$-0.479492\pi$
$877$	3.81340	0.128769	0.0643847	+	0.997925i	$0.479492\pi$
$878$	48.1969	1.62657
$879$	41.7992	1.40985
$880$	5.93345	0.200016
$881$	24.4672	0.824322	0.412161	−	0.911111i	$-0.364774\pi$
$881$	24.4672	0.824322	0.412161	+	0.911111i	$0.364774\pi$
$882$	9.01607	0.303587
$883$	−41.8018	−1.40674	−0.703372	−	0.710822i	$-0.748323\pi$
$883$	−41.8018	−1.40674	−0.703372	+	0.710822i	$0.748323\pi$
$884$	1.48450	0.0499291
$885$	20.5703	0.691465
$886$	−7.57276	−0.254412
$887$	−52.8195	−1.77351	−0.886753	−	0.462244i	$-0.847044\pi$
$887$	−52.8195	−1.77351	−0.886753	+	0.462244i	$0.847044\pi$
$888$	13.5400	0.454374
$889$	−57.6688	−1.93415
$890$	−11.3372	−0.380023
$891$	13.9512	0.467383
$892$	−4.90393	−0.164196
$893$	7.81494	0.261517
$894$	−8.00237	−0.267639
$895$	−3.80637	−0.127233
$896$	34.2549	1.14438
$897$	−0.499110	−0.0166648
$898$	−27.5860	−0.920555
$899$	−1.59648	−0.0532456
$900$	0.170166	0.00567221
$901$	−15.2562	−0.508259
$902$	3.45450	0.115022
$903$	−90.9077	−3.02522
$904$	0.0846782	0.00281636
$905$	−4.23389	−0.140739
$906$	2.17448	0.0722422
$907$	−43.7025	−1.45112	−0.725559	−	0.688160i	$-0.758419\pi$
$907$	−43.7025	−1.45112	−0.725559	+	0.688160i	$0.758419\pi$
$908$	−1.46641	−0.0486646
$909$	0.916424	0.0303959
$910$	−7.84352	−0.260010
$911$	−9.86149	−0.326726	−0.163363	−	0.986566i	$-0.552234\pi$
$911$	−9.86149	−0.326726	−0.163363	+	0.986566i	$0.552234\pi$
$912$	−18.2779	−0.605243
$913$	3.10801	0.102860
$914$	37.5400	1.24171
$915$	−3.15394	−0.104266
$916$	−1.16868	−0.0386144
$917$	−23.6337	−0.780453
$918$	22.0425	0.727510
$919$	1.35881	0.0448231	0.0224115	−	0.999749i	$-0.492866\pi$
$919$	1.35881	0.0448231	0.0224115	+	0.999749i	$0.492866\pi$
$920$	0.726756	0.0239604
$921$	−39.9214	−1.31546
$922$	0.420188	0.0138382
$923$	3.60747	0.118741
$924$	−5.15039	−0.169435
$925$	2.81362	0.0925111
$926$	−7.06708	−0.232239
$927$	−1.17203	−0.0384946
$928$	4.53354	0.148821
$929$	9.33911	0.306406	0.153203	−	0.988195i	$-0.451041\pi$
$929$	9.33911	0.306406	0.153203	+	0.988195i	$0.451041\pi$
$930$	−1.45732	−0.0477874
$931$	55.2132	1.80954
$932$	−8.70489	−0.285138
$933$	31.5467	1.03279
$934$	5.06139	0.165614
$935$	−5.90424	−0.193089
$936$	1.84324	0.0602482
$937$	18.8030	0.614268	0.307134	−	0.951666i	$-0.400630\pi$
$937$	18.8030	0.614268	0.307134	+	0.951666i	$0.400630\pi$
$938$	32.8957	1.07408
$939$	34.2056	1.11626
$940$	−0.777979	−0.0253749
$941$	−3.80645	−0.124087	−0.0620433	−	0.998073i	$-0.519762\pi$
$941$	−3.80645	−0.124087	−0.0620433	+	0.998073i	$0.519762\pi$
$942$	6.79086	0.221258
$943$	0.343418	0.0111832
$944$	−40.5700	−1.32044
$945$	25.9343	0.843643
$946$	29.3833	0.955332
$947$	−1.90837	−0.0620137	−0.0310069	−	0.999519i	$-0.509871\pi$
$947$	−1.90837	−0.0620137	−0.0310069	+	0.999519i	$0.509871\pi$
$948$	0.374450	0.0121616
$949$	−13.5930	−0.441247
$950$	−4.67968	−0.151829
$951$	16.6210	0.538972
$952$	−44.3886	−1.43864
$953$	40.2831	1.30490	0.652450	−	0.757832i	$-0.273741\pi$
$953$	40.2831	1.30490	0.652450	+	0.757832i	$0.273741\pi$
$954$	−2.91848	−0.0944891
$955$	6.56467	0.212428
$956$	−8.58676	−0.277716
$957$	6.70832	0.216849
$958$	43.3022	1.39903
$959$	−40.1588	−1.29680
$960$	14.1292	0.456017
$961$	−30.4874	−0.983464
$962$	4.69547	0.151388
$963$	−3.55712	−0.114627
$964$	7.60939	0.245082
$965$	−5.86249	−0.188720
$966$	2.29930	0.0739786
$967$	44.3556	1.42638	0.713189	−	0.700971i	$-0.247250\pi$
$967$	44.3556	1.42638	0.713189	+	0.700971i	$0.247250\pi$
$968$	−22.4565	−0.721779
$969$	18.1880	0.584282
$970$	−2.64910	−0.0850574
$971$	−10.6323	−0.341206	−0.170603	−	0.985340i	$-0.554571\pi$
$971$	−10.6323	−0.341206	−0.170603	+	0.985340i	$0.554571\pi$
$972$	−1.75142	−0.0561769
$973$	−57.2489	−1.83531
$974$	−15.5858	−0.499400
$975$	−2.07663	−0.0665054
$976$	6.22038	0.199110
$977$	29.8772	0.955856	0.477928	−	0.878399i	$-0.341388\pi$
$977$	29.8772	0.955856	0.477928	+	0.878399i	$0.341388\pi$
$978$	−2.15174	−0.0688051
$979$	−16.7566	−0.535544
$980$	−5.49648	−0.175579
$981$	3.57779	0.114230
$982$	−53.5315	−1.70826
$983$	4.76690	0.152040	0.0760202	−	0.997106i	$-0.475779\pi$
$983$	4.76690	0.152040	0.0760202	+	0.997106i	$0.475779\pi$
$984$	6.87609	0.219202
$985$	13.5963	0.433213
$986$	8.90742	0.283670
$987$	−15.9760	−0.508521
$988$	1.73906	0.0553268
$989$	2.92104	0.0928838
$990$	−1.12946	−0.0358968
$991$	−39.1303	−1.24302	−0.621508	−	0.783408i	$-0.713480\pi$
$991$	−39.1303	−1.24302	−0.621508	+	0.783408i	$0.713480\pi$
$992$	−1.45567	−0.0462174
$993$	−42.9366	−1.36255
$994$	−16.6189	−0.527119
$995$	−2.91751	−0.0924914
$996$	0.953115	0.0302006
$997$	7.89424	0.250013	0.125007	−	0.992156i	$-0.460105\pi$
$997$	7.89424	0.250013	0.125007	+	0.992156i	$0.460105\pi$
$998$	3.93491	0.124557
$999$	−15.5254	−0.491202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.g.1.9	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.g.1.9	✓	37	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27896 q^{2} +1.59149 q^{3} -0.364250 q^{4} +1.00000 q^{5} -2.03546 q^{6} -4.69998 q^{7} +3.02379 q^{8} -0.467169 q^{9} +O(q^{10})\)	\(q-1.27896 q^{2} +1.59149 q^{3} -0.364250 q^{4} +1.00000 q^{5} -2.03546 q^{6} -4.69998 q^{7} +3.02379 q^{8} -0.467169 q^{9} -1.27896 q^{10} -1.89034 q^{11} -0.579700 q^{12} -1.30484 q^{13} +6.01111 q^{14} +1.59149 q^{15} -3.13882 q^{16} +3.12337 q^{17} +0.597492 q^{18} +3.65896 q^{19} -0.364250 q^{20} -7.47996 q^{21} +2.41768 q^{22} +0.240346 q^{23} +4.81233 q^{24} +1.00000 q^{25} +1.66884 q^{26} -5.51795 q^{27} +1.71197 q^{28} -2.22982 q^{29} -2.03546 q^{30} +0.715968 q^{31} -2.03314 q^{32} -3.00846 q^{33} -3.99468 q^{34} -4.69998 q^{35} +0.170166 q^{36} +2.81362 q^{37} -4.67968 q^{38} -2.07663 q^{39} +3.02379 q^{40} +1.42885 q^{41} +9.56661 q^{42} +12.1535 q^{43} +0.688558 q^{44} -0.467169 q^{45} -0.307394 q^{46} +2.13584 q^{47} -4.99539 q^{48} +15.0898 q^{49} -1.27896 q^{50} +4.97080 q^{51} +0.475287 q^{52} -4.88454 q^{53} +7.05727 q^{54} -1.89034 q^{55} -14.2118 q^{56} +5.82319 q^{57} +2.85186 q^{58} +12.9252 q^{59} -0.579700 q^{60} -1.98176 q^{61} -0.915698 q^{62} +2.19569 q^{63} +8.87796 q^{64} -1.30484 q^{65} +3.84771 q^{66} +5.47249 q^{67} -1.13769 q^{68} +0.382508 q^{69} +6.01111 q^{70} -2.76469 q^{71} -1.41262 q^{72} +10.4174 q^{73} -3.59851 q^{74} +1.59149 q^{75} -1.33278 q^{76} +8.88458 q^{77} +2.65594 q^{78} -0.645939 q^{79} -3.13882 q^{80} -7.38025 q^{81} -1.82745 q^{82} -1.64415 q^{83} +2.72458 q^{84} +3.12337 q^{85} -15.5439 q^{86} -3.54873 q^{87} -5.71600 q^{88} +8.86433 q^{89} +0.597492 q^{90} +6.13271 q^{91} -0.0875461 q^{92} +1.13945 q^{93} -2.73166 q^{94} +3.65896 q^{95} -3.23572 q^{96} +2.07128 q^{97} -19.2994 q^{98} +0.883109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9}+O(q^{10}) \) 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9	\( 37 q + 11 q^{2} + 13 q^{3} + 43 q^{4} + 37 q^{5} - 2 q^{6} + 34 q^{7} + 27 q^{8} + 50 q^{9} + 11 q^{10} + 38 q^{11} + 24 q^{12} + 17 q^{13} + 14 q^{14} + 13 q^{15} + 47 q^{16} + 22 q^{17} + 18 q^{18} + 6 q^{19} + 43 q^{20} + 4 q^{21} + 18 q^{22} + 45 q^{23} - 19 q^{24} + 37 q^{25} - q^{26} + 43 q^{27} + 46 q^{28} + 17 q^{29} - 2 q^{30} - 13 q^{31} + 50 q^{32} + 12 q^{33} - 30 q^{34} + 34 q^{35} + 43 q^{36} + 9 q^{37} + q^{38} - 7 q^{39} + 27 q^{40} + 12 q^{41} - 38 q^{42} + 53 q^{43} + 36 q^{44} + 50 q^{45} - 15 q^{46} + 39 q^{47} - 6 q^{48} + 37 q^{49} + 11 q^{50} + 38 q^{51} + 17 q^{52} + 19 q^{53} - 62 q^{54} + 38 q^{55} + 18 q^{56} + 6 q^{57} - 13 q^{58} + 33 q^{59} + 24 q^{60} - 11 q^{61} + q^{62} + 98 q^{63} + 15 q^{64} + 17 q^{65} - 70 q^{66} + 49 q^{67} + 32 q^{68} - 36 q^{69} + 14 q^{70} + 19 q^{71} + 5 q^{72} + 39 q^{73} + 21 q^{74} + 13 q^{75} - 33 q^{76} + 34 q^{77} - 22 q^{78} - 9 q^{79} + 47 q^{80} + 49 q^{81} + 14 q^{82} + 114 q^{83} - 50 q^{84} + 22 q^{85} + 9 q^{86} + 56 q^{87} + 14 q^{88} + 2 q^{89} + 18 q^{90} - 29 q^{91} + 47 q^{92} - 7 q^{93} - 52 q^{94} + 6 q^{95} - 89 q^{96} + 4 q^{97} + 14 q^{98} + 57 q^{99}+O(q^{100}) \) 37 * q + 11 * q^2 + 13 * q^3 + 43 * q^4 + 37 * q^5 - 2 * q^6 + 34 * q^7 + 27 * q^8 + 50 * q^9 + 11 * q^10 + 38 * q^11 + 24 * q^12 + 17 * q^13 + 14 * q^14 + 13 * q^15 + 47 * q^16 + 22 * q^17 + 18 * q^18 + 6 * q^19 + 43 * q^20 + 4 * q^21 + 18 * q^22 + 45 * q^23 - 19 * q^24 + 37 * q^25 - q^26 + 43 * q^27 + 46 * q^28 + 17 * q^29 - 2 * q^30 - 13 * q^31 + 50 * q^32 + 12 * q^33 - 30 * q^34 + 34 * q^35 + 43 * q^36 + 9 * q^37 + q^38 - 7 * q^39 + 27 * q^40 + 12 * q^41 - 38 * q^42 + 53 * q^43 + 36 * q^44 + 50 * q^45 - 15 * q^46 + 39 * q^47 - 6 * q^48 + 37 * q^49 + 11 * q^50 + 38 * q^51 + 17 * q^52 + 19 * q^53 - 62 * q^54 + 38 * q^55 + 18 * q^56 + 6 * q^57 - 13 * q^58 + 33 * q^59 + 24 * q^60 - 11 * q^61 + q^62 + 98 * q^63 + 15 * q^64 + 17 * q^65 - 70 * q^66 + 49 * q^67 + 32 * q^68 - 36 * q^69 + 14 * q^70 + 19 * q^71 + 5 * q^72 + 39 * q^73 + 21 * q^74 + 13 * q^75 - 33 * q^76 + 34 * q^77 - 22 * q^78 - 9 * q^79 + 47 * q^80 + 49 * q^81 + 14 * q^82 + 114 * q^83 - 50 * q^84 + 22 * q^85 + 9 * q^86 + 56 * q^87 + 14 * q^88 + 2 * q^89 + 18 * q^90 - 29 * q^91 + 47 * q^92 - 7 * q^93 - 52 * q^94 + 6 * q^95 - 89 * q^96 + 4 * q^97 + 14 * q^98 + 57 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more