LMFDB - Embedded newform 4334.2.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, 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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 23 x^{13} + 19 x^{12} + 194 x^{11} - 124 x^{10} - 761 x^{9} + 353 x^{8} + 1417 x^{7} - 465 x^{6} - 1128 x^{5} + 288 x^{4} + 316 x^{3} - 79 x^{2} - 20 x + 4 $ x^15 - x^14 - 23x^13 + 19x^12 + 194x^11 - 124x^10 - 761x^9 + 353x^8 + 1417x^7 - 465x^6 - 1128x^5 + 288x^4 + 316x^3 - 79x^2 - 20*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.03566$ of defining polynomial
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.03566 q^{3} +1.00000 q^{4} +2.52120 q^{5} +1.03566 q^{6} -1.73580 q^{7} -1.00000 q^{8} -1.92741 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.03566 q^{3} +1.00000 q^{4} +2.52120 q^{5} +1.03566 q^{6} -1.73580 q^{7} -1.00000 q^{8} -1.92741 q^{9} -2.52120 q^{10} +1.00000 q^{11} -1.03566 q^{12} -0.661596 q^{13} +1.73580 q^{14} -2.61111 q^{15} +1.00000 q^{16} -2.09650 q^{17} +1.92741 q^{18} +0.592039 q^{19} +2.52120 q^{20} +1.79770 q^{21} -1.00000 q^{22} -1.91698 q^{23} +1.03566 q^{24} +1.35646 q^{25} +0.661596 q^{26} +5.10312 q^{27} -1.73580 q^{28} +5.77853 q^{29} +2.61111 q^{30} +4.37227 q^{31} -1.00000 q^{32} -1.03566 q^{33} +2.09650 q^{34} -4.37630 q^{35} -1.92741 q^{36} -7.16641 q^{37} -0.592039 q^{38} +0.685188 q^{39} -2.52120 q^{40} -1.33596 q^{41} -1.79770 q^{42} +9.79286 q^{43} +1.00000 q^{44} -4.85938 q^{45} +1.91698 q^{46} +1.79226 q^{47} -1.03566 q^{48} -3.98701 q^{49} -1.35646 q^{50} +2.17126 q^{51} -0.661596 q^{52} +9.08836 q^{53} -5.10312 q^{54} +2.52120 q^{55} +1.73580 q^{56} -0.613152 q^{57} -5.77853 q^{58} -9.26718 q^{59} -2.61111 q^{60} -2.08729 q^{61} -4.37227 q^{62} +3.34559 q^{63} +1.00000 q^{64} -1.66802 q^{65} +1.03566 q^{66} +0.726579 q^{67} -2.09650 q^{68} +1.98534 q^{69} +4.37630 q^{70} -7.98201 q^{71} +1.92741 q^{72} +11.4544 q^{73} +7.16641 q^{74} -1.40483 q^{75} +0.592039 q^{76} -1.73580 q^{77} -0.685188 q^{78} -14.6692 q^{79} +2.52120 q^{80} +0.497122 q^{81} +1.33596 q^{82} -3.52418 q^{83} +1.79770 q^{84} -5.28569 q^{85} -9.79286 q^{86} -5.98460 q^{87} -1.00000 q^{88} -13.6381 q^{89} +4.85938 q^{90} +1.14840 q^{91} -1.91698 q^{92} -4.52819 q^{93} -1.79226 q^{94} +1.49265 q^{95} +1.03566 q^{96} -3.68679 q^{97} +3.98701 q^{98} -1.92741 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) $ 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9 + 7 * q^10 + 15 * q^11 - q^12 - q^13 - q^14 - 6 * q^15 + 15 * q^16 - 6 * q^17 - 2 * q^18 - 14 * q^19 - 7 * q^20 - 3 * q^21 - 15 * q^22 + 2 * q^23 + q^24 - 10 * q^25 + q^26 - 7 * q^27 + q^28 + 8 * q^29 + 6 * q^30 - 33 * q^31 - 15 * q^32 - q^33 + 6 * q^34 - 8 * q^35 + 2 * q^36 - 9 * q^37 + 14 * q^38 - 9 * q^39 + 7 * q^40 - 10 * q^41 + 3 * q^42 - 6 * q^43 + 15 * q^44 - 20 * q^45 - 2 * q^46 - q^47 - q^48 - 30 * q^49 + 10 * q^50 + 12 * q^51 - q^52 + 6 * q^53 + 7 * q^54 - 7 * q^55 - q^56 - 24 * q^57 - 8 * q^58 - 15 * q^59 - 6 * q^60 - 25 * q^61 + 33 * q^62 + 12 * q^63 + 15 * q^64 + 31 * q^65 + q^66 - 13 * q^67 - 6 * q^68 - 43 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 4 * q^73 + 9 * q^74 - 5 * q^75 - 14 * q^76 + q^77 + 9 * q^78 - 20 * q^79 - 7 * q^80 + 11 * q^81 + 10 * q^82 + q^83 - 3 * q^84 - q^85 + 6 * q^86 + 22 * q^87 - 15 * q^88 - 41 * q^89 + 20 * q^90 - 31 * q^91 + 2 * q^92 + 14 * q^93 + q^94 + 41 * q^95 + q^96 - 57 * q^97 + 30 * q^98 + 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.03566	−0.597939	−0.298969	−	0.954263i	$-0.596643\pi$
$3$	−1.03566	−0.597939	−0.298969	+	0.954263i	$0.596643\pi$
$4$	1.00000	0.500000
$5$	2.52120	1.12752	0.563758	−	0.825940i	$-0.309355\pi$
$5$	2.52120	1.12752	0.563758	+	0.825940i	$0.309355\pi$
$6$	1.03566	0.422807
$7$	−1.73580	−0.656070	−0.328035	−	0.944666i	$-0.606386\pi$
$7$	−1.73580	−0.656070	−0.328035	+	0.944666i	$0.606386\pi$
$8$	−1.00000	−0.353553
$9$	−1.92741	−0.642469
$10$	−2.52120	−0.797274
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.03566	−0.298969
$13$	−0.661596	−0.183494	−0.0917468	−	0.995782i	$-0.529245\pi$
$13$	−0.661596	−0.183494	−0.0917468	+	0.995782i	$0.529245\pi$
$14$	1.73580	0.463911
$15$	−2.61111	−0.674185
$16$	1.00000	0.250000
$17$	−2.09650	−0.508475	−0.254237	−	0.967142i	$-0.581824\pi$
$17$	−2.09650	−0.508475	−0.254237	+	0.967142i	$0.581824\pi$
$18$	1.92741	0.454294
$19$	0.592039	0.135823	0.0679116	−	0.997691i	$-0.478366\pi$
$19$	0.592039	0.135823	0.0679116	+	0.997691i	$0.478366\pi$
$20$	2.52120	0.563758
$21$	1.79770	0.392290
$22$	−1.00000	−0.213201
$23$	−1.91698	−0.399718	−0.199859	−	0.979825i	$-0.564048\pi$
$23$	−1.91698	−0.399718	−0.199859	+	0.979825i	$0.564048\pi$
$24$	1.03566	0.211403
$25$	1.35646	0.271291
$26$	0.661596	0.129750
$27$	5.10312	0.982096
$28$	−1.73580	−0.328035
$29$	5.77853	1.07305	0.536523	−	0.843885i	$-0.319737\pi$
$29$	5.77853	1.07305	0.536523	+	0.843885i	$0.319737\pi$
$30$	2.61111	0.476721
$31$	4.37227	0.785283	0.392642	−	0.919692i	$-0.371561\pi$
$31$	4.37227	0.785283	0.392642	+	0.919692i	$0.371561\pi$
$32$	−1.00000	−0.176777
$33$	−1.03566	−0.180285
$34$	2.09650	0.359546
$35$	−4.37630	−0.739729
$36$	−1.92741	−0.321235
$37$	−7.16641	−1.17815	−0.589075	−	0.808078i	$-0.700508\pi$
$37$	−7.16641	−1.17815	−0.589075	+	0.808078i	$0.700508\pi$
$38$	−0.592039	−0.0960415
$39$	0.685188	0.109718
$40$	−2.52120	−0.398637
$41$	−1.33596	−0.208642	−0.104321	−	0.994544i	$-0.533267\pi$
$41$	−1.33596	−0.208642	−0.104321	+	0.994544i	$0.533267\pi$
$42$	−1.79770	−0.277391
$43$	9.79286	1.49340	0.746698	−	0.665163i	$-0.231638\pi$
$43$	9.79286	1.49340	0.746698	+	0.665163i	$0.231638\pi$
$44$	1.00000	0.150756
$45$	−4.85938	−0.724394
$46$	1.91698	0.282644
$47$	1.79226	0.261427	0.130714	−	0.991420i	$-0.458273\pi$
$47$	1.79226	0.261427	0.130714	+	0.991420i	$0.458273\pi$
$48$	−1.03566	−0.149485
$49$	−3.98701	−0.569572
$50$	−1.35646	−0.191832
$51$	2.17126	0.304037
$52$	−0.661596	−0.0917468
$53$	9.08836	1.24838	0.624191	−	0.781272i	$-0.285429\pi$
$53$	9.08836	1.24838	0.624191	+	0.781272i	$0.285429\pi$
$54$	−5.10312	−0.694447
$55$	2.52120	0.339959
$56$	1.73580	0.231956
$57$	−0.613152	−0.0812139
$58$	−5.77853	−0.758759
$59$	−9.26718	−1.20648	−0.603242	−	0.797558i	$-0.706125\pi$
$59$	−9.26718	−1.20648	−0.603242	+	0.797558i	$0.706125\pi$
$60$	−2.61111	−0.337093
$61$	−2.08729	−0.267250	−0.133625	−	0.991032i	$-0.542662\pi$
$61$	−2.08729	−0.267250	−0.133625	+	0.991032i	$0.542662\pi$
$62$	−4.37227	−0.555279
$63$	3.34559	0.421505
$64$	1.00000	0.125000
$65$	−1.66802	−0.206892
$66$	1.03566	0.127481
$67$	0.726579	0.0887657	0.0443829	−	0.999015i	$-0.485868\pi$
$67$	0.726579	0.0887657	0.0443829	+	0.999015i	$0.485868\pi$
$68$	−2.09650	−0.254237
$69$	1.98534	0.239007
$70$	4.37630	0.523067
$71$	−7.98201	−0.947290	−0.473645	−	0.880716i	$-0.657062\pi$
$71$	−7.98201	−0.947290	−0.473645	+	0.880716i	$0.657062\pi$
$72$	1.92741	0.227147
$73$	11.4544	1.34063	0.670316	−	0.742076i	$-0.266159\pi$
$73$	11.4544	1.34063	0.670316	+	0.742076i	$0.266159\pi$
$74$	7.16641	0.833078
$75$	−1.40483	−0.162216
$76$	0.592039	0.0679116
$77$	−1.73580	−0.197813
$78$	−0.685188	−0.0775823
$79$	−14.6692	−1.65041	−0.825205	−	0.564833i	$-0.808940\pi$
$79$	−14.6692	−1.65041	−0.825205	+	0.564833i	$0.808940\pi$
$80$	2.52120	0.281879
$81$	0.497122	0.0552358
$82$	1.33596	0.147532
$83$	−3.52418	−0.386829	−0.193414	−	0.981117i	$-0.561956\pi$
$83$	−3.52418	−0.386829	−0.193414	+	0.981117i	$0.561956\pi$
$84$	1.79770	0.196145
$85$	−5.28569	−0.573313
$86$	−9.79286	−1.05599
$87$	−5.98460	−0.641616
$88$	−1.00000	−0.106600
$89$	−13.6381	−1.44563	−0.722816	−	0.691040i	$-0.757153\pi$
$89$	−13.6381	−1.44563	−0.722816	+	0.691040i	$0.757153\pi$
$90$	4.85938	0.512224
$91$	1.14840	0.120385
$92$	−1.91698	−0.199859
$93$	−4.52819	−0.469551
$94$	−1.79226	−0.184857
$95$	1.49265	0.153143
$96$	1.03566	0.105702
$97$	−3.68679	−0.374337	−0.187168	−	0.982328i	$-0.559931\pi$
$97$	−3.68679	−0.374337	−0.187168	+	0.982328i	$0.559931\pi$
$98$	3.98701	0.402748
$99$	−1.92741	−0.193712
$100$	1.35646	0.135646
$101$	−11.3290	−1.12728	−0.563641	−	0.826020i	$-0.690600\pi$
$101$	−11.3290	−1.12728	−0.563641	+	0.826020i	$0.690600\pi$
$102$	−2.17126	−0.214987
$103$	−3.18047	−0.313381	−0.156690	−	0.987648i	$-0.550082\pi$
$103$	−3.18047	−0.313381	−0.156690	+	0.987648i	$0.550082\pi$
$104$	0.661596	0.0648748
$105$	4.53236	0.442313
$106$	−9.08836	−0.882740
$107$	−7.42503	−0.717805	−0.358902	−	0.933375i	$-0.616849\pi$
$107$	−7.42503	−0.717805	−0.358902	+	0.933375i	$0.616849\pi$
$108$	5.10312	0.491048
$109$	−9.03529	−0.865423	−0.432712	−	0.901532i	$-0.642443\pi$
$109$	−9.03529	−0.865423	−0.432712	+	0.901532i	$0.642443\pi$
$110$	−2.52120	−0.240387
$111$	7.42196	0.704462
$112$	−1.73580	−0.164017
$113$	0.179700	0.0169048	0.00845239	−	0.999964i	$-0.497309\pi$
$113$	0.179700	0.0169048	0.00845239	+	0.999964i	$0.497309\pi$
$114$	0.613152	0.0574269
$115$	−4.83310	−0.450689
$116$	5.77853	0.536523
$117$	1.27516	0.117889
$118$	9.26718	0.853113
$119$	3.63909	0.333595
$120$	2.61111	0.238360
$121$	1.00000	0.0909091
$122$	2.08729	0.188974
$123$	1.38360	0.124755
$124$	4.37227	0.392642
$125$	−9.18611	−0.821630
$126$	−3.34559	−0.298049
$127$	18.8269	1.67062	0.835309	−	0.549781i	$-0.185289\pi$
$127$	18.8269	1.67062	0.835309	+	0.549781i	$0.185289\pi$
$128$	−1.00000	−0.0883883
$129$	−10.1421	−0.892960
$130$	1.66802	0.146295
$131$	−6.83617	−0.597279	−0.298639	−	0.954366i	$-0.596533\pi$
$131$	−6.83617	−0.597279	−0.298639	+	0.954366i	$0.596533\pi$
$132$	−1.03566	−0.0901427
$133$	−1.02766	−0.0891095
$134$	−0.726579	−0.0627668
$135$	12.8660	1.10733
$136$	2.09650	0.179773
$137$	6.70217	0.572605	0.286302	−	0.958139i	$-0.407574\pi$
$137$	6.70217	0.572605	0.286302	+	0.958139i	$0.407574\pi$
$138$	−1.98534	−0.169004
$139$	−4.47060	−0.379191	−0.189596	−	0.981862i	$-0.560718\pi$
$139$	−4.47060	−0.379191	−0.189596	+	0.981862i	$0.560718\pi$
$140$	−4.37630	−0.369864
$141$	−1.85617	−0.156318
$142$	7.98201	0.669835
$143$	−0.661596	−0.0553254
$144$	−1.92741	−0.160617
$145$	14.5688	1.20988
$146$	−11.4544	−0.947970
$147$	4.12918	0.340569
$148$	−7.16641	−0.589075
$149$	−3.72372	−0.305059	−0.152530	−	0.988299i	$-0.548742\pi$
$149$	−3.72372	−0.305059	−0.152530	+	0.988299i	$0.548742\pi$
$150$	1.40483	0.114704
$151$	22.8119	1.85641	0.928204	−	0.372072i	$-0.121353\pi$
$151$	22.8119	1.85641	0.928204	+	0.372072i	$0.121353\pi$
$152$	−0.592039	−0.0480207
$153$	4.04080	0.326680
$154$	1.73580	0.139875
$155$	11.0234	0.885419
$156$	0.685188	0.0548590
$157$	−11.3580	−0.906468	−0.453234	−	0.891392i	$-0.649730\pi$
$157$	−11.3580	−0.906468	−0.453234	+	0.891392i	$0.649730\pi$
$158$	14.6692	1.16702
$159$	−9.41246	−0.746456
$160$	−2.52120	−0.199318
$161$	3.32749	0.262243
$162$	−0.497122	−0.0390576
$163$	−16.5868	−1.29918	−0.649590	−	0.760285i	$-0.725059\pi$
$163$	−16.5868	−1.29918	−0.649590	+	0.760285i	$0.725059\pi$
$164$	−1.33596	−0.104321
$165$	−2.61111	−0.203275
$166$	3.52418	0.273529
$167$	−11.2683	−0.871970	−0.435985	−	0.899954i	$-0.643600\pi$
$167$	−11.2683	−0.871970	−0.435985	+	0.899954i	$0.643600\pi$
$168$	−1.79770	−0.138695
$169$	−12.5623	−0.966330
$170$	5.28569	0.405394
$171$	−1.14110	−0.0872622
$172$	9.79286	0.746698
$173$	22.3955	1.70270	0.851348	−	0.524601i	$-0.175785\pi$
$173$	22.3955	1.70270	0.851348	+	0.524601i	$0.175785\pi$
$174$	5.98460	0.453691
$175$	−2.35453	−0.177986
$176$	1.00000	0.0753778
$177$	9.59765	0.721404
$178$	13.6381	1.02222
$179$	10.1374	0.757708	0.378854	−	0.925456i	$-0.376318\pi$
$179$	10.1374	0.757708	0.378854	+	0.925456i	$0.376318\pi$
$180$	−4.85938	−0.362197
$181$	−7.64184	−0.568013	−0.284007	−	0.958822i	$-0.591664\pi$
$181$	−7.64184	−0.568013	−0.284007	+	0.958822i	$0.591664\pi$
$182$	−1.14840	−0.0851248
$183$	2.16172	0.159799
$184$	1.91698	0.141322
$185$	−18.0680	−1.32838
$186$	4.52819	0.332023
$187$	−2.09650	−0.153311
$188$	1.79226	0.130714
$189$	−8.85799	−0.644324
$190$	−1.49265	−0.108288
$191$	0.618048	0.0447204	0.0223602	−	0.999750i	$-0.492882\pi$
$191$	0.618048	0.0447204	0.0223602	+	0.999750i	$0.492882\pi$
$192$	−1.03566	−0.0747424
$193$	−16.5394	−1.19053	−0.595265	−	0.803529i	$-0.702953\pi$
$193$	−16.5394	−1.19053	−0.595265	+	0.803529i	$0.702953\pi$
$194$	3.68679	0.264696
$195$	1.72750	0.123709
$196$	−3.98701	−0.284786
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	1.92741	0.136975
$199$	−3.36737	−0.238706	−0.119353	−	0.992852i	$-0.538082\pi$
$199$	−3.36737	−0.238706	−0.119353	+	0.992852i	$0.538082\pi$
$200$	−1.35646	−0.0959159
$201$	−0.752489	−0.0530765
$202$	11.3290	0.797109
$203$	−10.0304	−0.703994
$204$	2.17126	0.152018
$205$	−3.36822	−0.235247
$206$	3.18047	0.221594
$207$	3.69481	0.256807
$208$	−0.661596	−0.0458734
$209$	0.592039	0.0409522
$210$	−4.53236	−0.312762
$211$	15.0329	1.03491	0.517453	−	0.855712i	$-0.326880\pi$
$211$	15.0329	1.03491	0.517453	+	0.855712i	$0.326880\pi$
$212$	9.08836	0.624191
$213$	8.26665	0.566421
$214$	7.42503	0.507565
$215$	24.6898	1.68383
$216$	−5.10312	−0.347223
$217$	−7.58938	−0.515201
$218$	9.03529	0.611947
$219$	−11.8628	−0.801616
$220$	2.52120	0.169979
$221$	1.38703	0.0933019
$222$	−7.42196	−0.498130
$223$	3.89916	0.261107	0.130553	−	0.991441i	$-0.458325\pi$
$223$	3.89916	0.261107	0.130553	+	0.991441i	$0.458325\pi$
$224$	1.73580	0.115978
$225$	−2.61444	−0.174296
$226$	−0.179700	−0.0119535
$227$	15.1464	1.00530	0.502652	−	0.864489i	$-0.332358\pi$
$227$	15.1464	1.00530	0.502652	+	0.864489i	$0.332358\pi$
$228$	−0.613152	−0.0406070
$229$	−24.1376	−1.59506	−0.797529	−	0.603281i	$-0.793860\pi$
$229$	−24.1376	−1.59506	−0.797529	+	0.603281i	$0.793860\pi$
$230$	4.83310	0.318685
$231$	1.79770	0.118280
$232$	−5.77853	−0.379379
$233$	−8.42912	−0.552210	−0.276105	−	0.961127i	$-0.589044\pi$
$233$	−8.42912	−0.552210	−0.276105	+	0.961127i	$0.589044\pi$
$234$	−1.27516	−0.0833601
$235$	4.51864	0.294764
$236$	−9.26718	−0.603242
$237$	15.1923	0.986844
$238$	−3.63909	−0.235887
$239$	17.0649	1.10384	0.551918	−	0.833899i	$-0.313896\pi$
$239$	17.0649	1.10384	0.551918	+	0.833899i	$0.313896\pi$
$240$	−2.61111	−0.168546
$241$	−2.91512	−0.187779	−0.0938896	−	0.995583i	$-0.529930\pi$
$241$	−2.91512	−0.187779	−0.0938896	+	0.995583i	$0.529930\pi$
$242$	−1.00000	−0.0642824
$243$	−15.8242	−1.01512
$244$	−2.08729	−0.133625
$245$	−10.0520	−0.642202
$246$	−1.38360	−0.0882151
$247$	−0.391691	−0.0249227
$248$	−4.37227	−0.277640
$249$	3.64985	0.231300
$250$	9.18611	0.580980
$251$	−26.6776	−1.68387	−0.841936	−	0.539577i	$-0.818584\pi$
$251$	−26.6776	−1.68387	−0.841936	+	0.539577i	$0.818584\pi$
$252$	3.34559	0.210752
$253$	−1.91698	−0.120520
$254$	−18.8269	−1.18130
$255$	5.47418	0.342806
$256$	1.00000	0.0625000
$257$	4.66062	0.290722	0.145361	−	0.989379i	$-0.453566\pi$
$257$	4.66062	0.290722	0.145361	+	0.989379i	$0.453566\pi$
$258$	10.1421	0.631418
$259$	12.4394	0.772949
$260$	−1.66802	−0.103446
$261$	−11.1376	−0.689400
$262$	6.83617	0.422340
$263$	9.70084	0.598179	0.299090	−	0.954225i	$-0.403317\pi$
$263$	9.70084	0.598179	0.299090	+	0.954225i	$0.403317\pi$
$264$	1.03566	0.0637405
$265$	22.9136	1.40757
$266$	1.02766	0.0630099
$267$	14.1244	0.864400
$268$	0.726579	0.0443829
$269$	−5.70499	−0.347839	−0.173920	−	0.984760i	$-0.555643\pi$
$269$	−5.70499	−0.347839	−0.173920	+	0.984760i	$0.555643\pi$
$270$	−12.8660	−0.783000
$271$	−15.8705	−0.964065	−0.482032	−	0.876153i	$-0.660101\pi$
$271$	−15.8705	−0.964065	−0.482032	+	0.876153i	$0.660101\pi$
$272$	−2.09650	−0.127119
$273$	−1.18935	−0.0719826
$274$	−6.70217	−0.404893
$275$	1.35646	0.0817974
$276$	1.98534	0.119504
$277$	−29.8963	−1.79630	−0.898148	−	0.439693i	$-0.855087\pi$
$277$	−29.8963	−1.79630	−0.898148	+	0.439693i	$0.855087\pi$
$278$	4.47060	0.268129
$279$	−8.42715	−0.504520
$280$	4.37630	0.261534
$281$	−29.3852	−1.75297	−0.876486	−	0.481428i	$-0.840118\pi$
$281$	−29.3852	−1.75297	−0.876486	+	0.481428i	$0.840118\pi$
$282$	1.85617	0.110533
$283$	21.0954	1.25399	0.626996	−	0.779023i	$-0.284284\pi$
$283$	21.0954	1.25399	0.626996	+	0.779023i	$0.284284\pi$
$284$	−7.98201	−0.473645
$285$	−1.54588	−0.0915700
$286$	0.661596	0.0391210
$287$	2.31896	0.136884
$288$	1.92741	0.113574
$289$	−12.6047	−0.741453
$290$	−14.5688	−0.855512
$291$	3.81826	0.223830
$292$	11.4544	0.670316
$293$	−22.8019	−1.33210	−0.666051	−	0.745906i	$-0.732017\pi$
$293$	−22.8019	−1.33210	−0.666051	+	0.745906i	$0.732017\pi$
$294$	−4.12918	−0.240819
$295$	−23.3644	−1.36033
$296$	7.16641	0.416539
$297$	5.10312	0.296113
$298$	3.72372	0.215710
$299$	1.26827	0.0733458
$300$	−1.40483	−0.0811078
$301$	−16.9984	−0.979773
$302$	−22.8119	−1.31268
$303$	11.7330	0.674046
$304$	0.592039	0.0339558
$305$	−5.26248	−0.301329
$306$	−4.04080	−0.230997
$307$	21.6573	1.23605	0.618023	−	0.786160i	$-0.287934\pi$
$307$	21.6573	1.23605	0.618023	+	0.786160i	$0.287934\pi$
$308$	−1.73580	−0.0989063
$309$	3.29388	0.187383
$310$	−11.0234	−0.626086
$311$	5.96867	0.338452	0.169226	−	0.985577i	$-0.445873\pi$
$311$	5.96867	0.338452	0.169226	+	0.985577i	$0.445873\pi$
$312$	−0.685188	−0.0387911
$313$	23.2868	1.31625	0.658124	−	0.752909i	$-0.271350\pi$
$313$	23.2868	1.31625	0.658124	+	0.752909i	$0.271350\pi$
$314$	11.3580	0.640970
$315$	8.43491	0.475253
$316$	−14.6692	−0.825205
$317$	−17.5188	−0.983954	−0.491977	−	0.870608i	$-0.663726\pi$
$317$	−17.5188	−0.983954	−0.491977	+	0.870608i	$0.663726\pi$
$318$	9.41246	0.527824
$319$	5.77853	0.323536
$320$	2.52120	0.140939
$321$	7.68981	0.429203
$322$	−3.32749	−0.185434
$323$	−1.24121	−0.0690627
$324$	0.497122	0.0276179
$325$	−0.897425	−0.0497802
$326$	16.5868	0.918658
$327$	9.35749	0.517470
$328$	1.33596	0.0737660
$329$	−3.11100	−0.171515
$330$	2.61111	0.143737
$331$	−19.7977	−1.08818	−0.544089	−	0.839027i	$-0.683125\pi$
$331$	−19.7977	−1.08818	−0.544089	+	0.839027i	$0.683125\pi$
$332$	−3.52418	−0.193414
$333$	13.8126	0.756925
$334$	11.2683	0.616576
$335$	1.83185	0.100085
$336$	1.79770	0.0980724
$337$	5.87540	0.320053	0.160027	−	0.987113i	$-0.448842\pi$
$337$	5.87540	0.320053	0.160027	+	0.987113i	$0.448842\pi$
$338$	12.5623	0.683299
$339$	−0.186109	−0.0101080
$340$	−5.28569	−0.286657
$341$	4.37227	0.236772
$342$	1.14110	0.0617037
$343$	19.0712	1.02975
$344$	−9.79286	−0.527996
$345$	5.00545	0.269484
$346$	−22.3955	−1.20399
$347$	−15.4392	−0.828818	−0.414409	−	0.910091i	$-0.636012\pi$
$347$	−15.4392	−0.828818	−0.414409	+	0.910091i	$0.636012\pi$
$348$	−5.98460	−0.320808
$349$	−14.7457	−0.789319	−0.394659	−	0.918828i	$-0.629137\pi$
$349$	−14.7457	−0.789319	−0.394659	+	0.918828i	$0.629137\pi$
$350$	2.35453	0.125855
$351$	−3.37620	−0.180208
$352$	−1.00000	−0.0533002
$353$	24.4420	1.30092	0.650458	−	0.759542i	$-0.274577\pi$
$353$	24.4420	1.30092	0.650458	+	0.759542i	$0.274577\pi$
$354$	−9.59765	−0.510109
$355$	−20.1242	−1.06808
$356$	−13.6381	−0.722816
$357$	−3.76886	−0.199469
$358$	−10.1374	−0.535780
$359$	5.59776	0.295438	0.147719	−	0.989029i	$-0.452807\pi$
$359$	5.59776	0.295438	0.147719	+	0.989029i	$0.452807\pi$
$360$	4.85938	0.256112
$361$	−18.6495	−0.981552
$362$	7.64184	0.401646
$363$	−1.03566	−0.0543581
$364$	1.14840	0.0601923
$365$	28.8787	1.51158
$366$	−2.16172	−0.112995
$367$	8.51249	0.444348	0.222174	−	0.975007i	$-0.428685\pi$
$367$	8.51249	0.444348	0.222174	+	0.975007i	$0.428685\pi$
$368$	−1.91698	−0.0999296
$369$	2.57494	0.134046
$370$	18.0680	0.939308
$371$	−15.7756	−0.819026
$372$	−4.52819	−0.234776
$373$	−37.2003	−1.92616	−0.963080	−	0.269217i	$-0.913235\pi$
$373$	−37.2003	−1.92616	−0.963080	+	0.269217i	$0.913235\pi$
$374$	2.09650	0.108407
$375$	9.51369	0.491285
$376$	−1.79226	−0.0924286
$377$	−3.82305	−0.196897
$378$	8.85799	0.455606
$379$	12.1776	0.625519	0.312760	−	0.949832i	$-0.398747\pi$
$379$	12.1776	0.625519	0.312760	+	0.949832i	$0.398747\pi$
$380$	1.49265	0.0765714
$381$	−19.4983	−0.998927
$382$	−0.618048	−0.0316221
$383$	−25.9801	−1.32752	−0.663761	−	0.747944i	$-0.731041\pi$
$383$	−25.9801	−1.32752	−0.663761	+	0.747944i	$0.731041\pi$
$384$	1.03566	0.0528508
$385$	−4.37630	−0.223037
$386$	16.5394	0.841832
$387$	−18.8748	−0.959461
$388$	−3.68679	−0.187168
$389$	24.9045	1.26271	0.631354	−	0.775495i	$-0.282500\pi$
$389$	24.9045	1.26271	0.631354	+	0.775495i	$0.282500\pi$
$390$	−1.72750	−0.0874752
$391$	4.01895	0.203247
$392$	3.98701	0.201374
$393$	7.07995	0.357136
$394$	−1.00000	−0.0503793
$395$	−36.9839	−1.86086
$396$	−1.92741	−0.0968559
$397$	14.9156	0.748593	0.374297	−	0.927309i	$-0.377884\pi$
$397$	14.9156	0.748593	0.374297	+	0.927309i	$0.377884\pi$
$398$	3.36737	0.168791
$399$	1.06431	0.0532820
$400$	1.35646	0.0678228
$401$	−4.56349	−0.227890	−0.113945	−	0.993487i	$-0.536349\pi$
$401$	−4.56349	−0.227890	−0.113945	+	0.993487i	$0.536349\pi$
$402$	0.752489	0.0375307
$403$	−2.89268	−0.144094
$404$	−11.3290	−0.563641
$405$	1.25334	0.0622792
$406$	10.0304	0.497799
$407$	−7.16641	−0.355226
$408$	−2.17126	−0.107493
$409$	1.19069	0.0588758	0.0294379	−	0.999567i	$-0.490628\pi$
$409$	1.19069	0.0588758	0.0294379	+	0.999567i	$0.490628\pi$
$410$	3.36822	0.166345
$411$	−6.94117	−0.342383
$412$	−3.18047	−0.156690
$413$	16.0859	0.791538
$414$	−3.69481	−0.181590
$415$	−8.88516	−0.436155
$416$	0.661596	0.0324374
$417$	4.63002	0.226733
$418$	−0.592039	−0.0289576
$419$	−12.1040	−0.591319	−0.295659	−	0.955293i	$-0.595539\pi$
$419$	−12.1040	−0.591319	−0.295659	+	0.955293i	$0.595539\pi$
$420$	4.53236	0.221156
$421$	−30.6982	−1.49614	−0.748069	−	0.663621i	$-0.769019\pi$
$421$	−30.6982	−1.49614	−0.748069	+	0.663621i	$0.769019\pi$
$422$	−15.0329	−0.731789
$423$	−3.45441	−0.167959
$424$	−9.08836	−0.441370
$425$	−2.84380	−0.137945
$426$	−8.26665	−0.400520
$427$	3.62311	0.175335
$428$	−7.42503	−0.358902
$429$	0.685188	0.0330812
$430$	−24.6898	−1.19065
$431$	−9.16476	−0.441451	−0.220725	−	0.975336i	$-0.570842\pi$
$431$	−9.16476	−0.441451	−0.220725	+	0.975336i	$0.570842\pi$
$432$	5.10312	0.245524
$433$	−20.5838	−0.989193	−0.494596	−	0.869123i	$-0.664684\pi$
$433$	−20.5838	−0.989193	−0.494596	+	0.869123i	$0.664684\pi$
$434$	7.58938	0.364302
$435$	−15.0884	−0.723432
$436$	−9.03529	−0.432712
$437$	−1.13493	−0.0542910
$438$	11.8628	0.566828
$439$	−19.8541	−0.947586	−0.473793	−	0.880636i	$-0.657115\pi$
$439$	−19.8541	−0.947586	−0.473793	+	0.880636i	$0.657115\pi$
$440$	−2.52120	−0.120194
$441$	7.68459	0.365933
$442$	−1.38703	−0.0659744
$443$	1.65311	0.0785416	0.0392708	−	0.999229i	$-0.487497\pi$
$443$	1.65311	0.0785416	0.0392708	+	0.999229i	$0.487497\pi$
$444$	7.42196	0.352231
$445$	−34.3843	−1.62997
$446$	−3.89916	−0.184630
$447$	3.85651	0.182407
$448$	−1.73580	−0.0820087
$449$	−34.3912	−1.62302	−0.811511	−	0.584338i	$-0.801354\pi$
$449$	−34.3912	−1.62302	−0.811511	+	0.584338i	$0.801354\pi$
$450$	2.61444	0.123246
$451$	−1.33596	−0.0629079
$452$	0.179700	0.00845239
$453$	−23.6254	−1.11002
$454$	−15.1464	−0.710858
$455$	2.89534	0.135736
$456$	0.613152	0.0287135
$457$	−9.36018	−0.437851	−0.218925	−	0.975742i	$-0.570255\pi$
$457$	−9.36018	−0.437851	−0.218925	+	0.975742i	$0.570255\pi$
$458$	24.1376	1.12788
$459$	−10.6987	−0.499371
$460$	−4.83310	−0.225344
$461$	−31.5116	−1.46764	−0.733821	−	0.679343i	$-0.762265\pi$
$461$	−31.5116	−1.46764	−0.733821	+	0.679343i	$0.762265\pi$
$462$	−1.79770	−0.0836364
$463$	−31.6852	−1.47254	−0.736268	−	0.676690i	$-0.763414\pi$
$463$	−31.6852	−1.47254	−0.736268	+	0.676690i	$0.763414\pi$
$464$	5.77853	0.268262
$465$	−11.4165	−0.529426
$466$	8.42912	0.390472
$467$	8.22554	0.380632	0.190316	−	0.981723i	$-0.439049\pi$
$467$	8.22554	0.380632	0.190316	+	0.981723i	$0.439049\pi$
$468$	1.27516	0.0589445
$469$	−1.26119	−0.0582365
$470$	−4.51864	−0.208429
$471$	11.7630	0.542012
$472$	9.26718	0.426556
$473$	9.79286	0.450276
$474$	−15.1923	−0.697804
$475$	0.803075	0.0368476
$476$	3.63909	0.166798
$477$	−17.5170	−0.802047
$478$	−17.0649	−0.780529
$479$	−28.3633	−1.29595	−0.647976	−	0.761661i	$-0.724384\pi$
$479$	−28.3633	−1.29595	−0.647976	+	0.761661i	$0.724384\pi$
$480$	2.61111	0.119180
$481$	4.74126	0.216183
$482$	2.91512	0.132780
$483$	−3.44615	−0.156805
$484$	1.00000	0.0454545
$485$	−9.29514	−0.422070
$486$	15.8242	0.717801
$487$	3.84730	0.174338	0.0871689	−	0.996194i	$-0.472218\pi$
$487$	3.84730	0.174338	0.0871689	+	0.996194i	$0.472218\pi$
$488$	2.08729	0.0944871
$489$	17.1783	0.776830
$490$	10.0520	0.454105
$491$	26.6876	1.20439	0.602197	−	0.798347i	$-0.294292\pi$
$491$	26.6876	1.20439	0.602197	+	0.798347i	$0.294292\pi$
$492$	1.38360	0.0623775
$493$	−12.1147	−0.545618
$494$	0.391691	0.0176230
$495$	−4.85938	−0.218413
$496$	4.37227	0.196321
$497$	13.8551	0.621488
$498$	−3.64985	−0.163554
$499$	10.1351	0.453708	0.226854	−	0.973929i	$-0.427156\pi$
$499$	10.1351	0.453708	0.226854	+	0.973929i	$0.427156\pi$
$500$	−9.18611	−0.410815
$501$	11.6702	0.521385
$502$	26.6776	1.19068
$503$	16.8789	0.752595	0.376297	−	0.926499i	$-0.377197\pi$
$503$	16.8789	0.752595	0.376297	+	0.926499i	$0.377197\pi$
$504$	−3.34559	−0.149024
$505$	−28.5628	−1.27103
$506$	1.91698	0.0852203
$507$	13.0103	0.577806
$508$	18.8269	0.835309
$509$	20.4750	0.907536	0.453768	−	0.891120i	$-0.350079\pi$
$509$	20.4750	0.907536	0.453768	+	0.891120i	$0.350079\pi$
$510$	−5.47418	−0.242401
$511$	−19.8825	−0.879548
$512$	−1.00000	−0.0441942
$513$	3.02125	0.133391
$514$	−4.66062	−0.205571
$515$	−8.01860	−0.353342
$516$	−10.1421	−0.446480
$517$	1.79226	0.0788234
$518$	−12.4394	−0.546557
$519$	−23.1941	−1.01811
$520$	1.66802	0.0731473
$521$	17.7597	0.778068	0.389034	−	0.921223i	$-0.372809\pi$
$521$	17.7597	0.778068	0.389034	+	0.921223i	$0.372809\pi$
$522$	11.1376	0.487479
$523$	−8.12089	−0.355102	−0.177551	−	0.984112i	$-0.556817\pi$
$523$	−8.12089	−0.355102	−0.177551	+	0.984112i	$0.556817\pi$
$524$	−6.83617	−0.298639
$525$	2.43850	0.106425
$526$	−9.70084	−0.422977
$527$	−9.16645	−0.399297
$528$	−1.03566	−0.0450713
$529$	−19.3252	−0.840225
$530$	−22.9136	−0.995303
$531$	17.8616	0.775129
$532$	−1.02766	−0.0445547
$533$	0.883865	0.0382844
$534$	−14.1244	−0.611223
$535$	−18.7200	−0.809336
$536$	−0.726579	−0.0313834
$537$	−10.4989	−0.453063
$538$	5.70499	0.245960
$539$	−3.98701	−0.171733
$540$	12.8660	0.553664
$541$	40.4394	1.73863	0.869313	−	0.494262i	$-0.164562\pi$
$541$	40.4394	1.73863	0.869313	+	0.494262i	$0.164562\pi$
$542$	15.8705	0.681697
$543$	7.91435	0.339637
$544$	2.09650	0.0898865
$545$	−22.7798	−0.975778
$546$	1.18935	0.0508994
$547$	−1.92266	−0.0822072	−0.0411036	−	0.999155i	$-0.513087\pi$
$547$	−1.92266	−0.0822072	−0.0411036	+	0.999155i	$0.513087\pi$
$548$	6.70217	0.286302
$549$	4.02306	0.171700
$550$	−1.35646	−0.0578395
$551$	3.42112	0.145745
$552$	−1.98534	−0.0845018
$553$	25.4627	1.08278
$554$	29.8963	1.27017
$555$	18.7123	0.794291
$556$	−4.47060	−0.189596
$557$	−8.43674	−0.357476	−0.178738	−	0.983897i	$-0.557201\pi$
$557$	−8.43674	−0.357476	−0.178738	+	0.983897i	$0.557201\pi$
$558$	8.42715	0.356750
$559$	−6.47891	−0.274029
$560$	−4.37630	−0.184932
$561$	2.17126	0.0916706
$562$	29.3852	1.23954
$563$	8.87788	0.374158	0.187079	−	0.982345i	$-0.440098\pi$
$563$	8.87788	0.374158	0.187079	+	0.982345i	$0.440098\pi$
$564$	−1.85617	−0.0781588
$565$	0.453061	0.0190604
$566$	−21.0954	−0.886706
$567$	−0.862903	−0.0362385
$568$	7.98201	0.334918
$569$	10.6005	0.444397	0.222199	−	0.975001i	$-0.428677\pi$
$569$	10.6005	0.444397	0.222199	+	0.975001i	$0.428677\pi$
$570$	1.54588	0.0647497
$571$	14.0366	0.587411	0.293706	−	0.955896i	$-0.405111\pi$
$571$	14.0366	0.587411	0.293706	+	0.955896i	$0.405111\pi$
$572$	−0.661596	−0.0276627
$573$	−0.640088	−0.0267401
$574$	−2.31896	−0.0967913
$575$	−2.60030	−0.108440
$576$	−1.92741	−0.0803086
$577$	−33.8714	−1.41009	−0.705043	−	0.709164i	$-0.749072\pi$
$577$	−33.8714	−1.41009	−0.705043	+	0.709164i	$0.749072\pi$
$578$	12.6047	0.524287
$579$	17.1292	0.711864
$580$	14.5688	0.604939
$581$	6.11726	0.253787
$582$	−3.81826	−0.158272
$583$	9.08836	0.376401
$584$	−11.4544	−0.473985
$585$	3.21495	0.132922
$586$	22.8019	0.941939
$587$	−39.3259	−1.62315	−0.811577	−	0.584245i	$-0.801391\pi$
$587$	−39.3259	−1.62315	−0.811577	+	0.584245i	$0.801391\pi$
$588$	4.12918	0.170285
$589$	2.58856	0.106660
$590$	23.3644	0.961898
$591$	−1.03566	−0.0426014
$592$	−7.16641	−0.294537
$593$	23.3274	0.957943	0.478972	−	0.877830i	$-0.341010\pi$
$593$	23.3274	0.957943	0.478972	+	0.877830i	$0.341010\pi$
$594$	−5.10312	−0.209384
$595$	9.17489	0.376134
$596$	−3.72372	−0.152530
$597$	3.48745	0.142732
$598$	−1.26827	−0.0518633
$599$	11.5160	0.470530	0.235265	−	0.971931i	$-0.424404\pi$
$599$	11.5160	0.470530	0.235265	+	0.971931i	$0.424404\pi$
$600$	1.40483	0.0573519
$601$	15.7163	0.641082	0.320541	−	0.947235i	$-0.396135\pi$
$601$	15.7163	0.641082	0.320541	+	0.947235i	$0.396135\pi$
$602$	16.9984	0.692804
$603$	−1.40041	−0.0570292
$604$	22.8119	0.928204
$605$	2.52120	0.102501
$606$	−11.7330	−0.476622
$607$	2.39409	0.0971733	0.0485866	−	0.998819i	$-0.484528\pi$
$607$	2.39409	0.0971733	0.0485866	+	0.998819i	$0.484528\pi$
$608$	−0.592039	−0.0240104
$609$	10.3881	0.420945
$610$	5.26248	0.213071
$611$	−1.18575	−0.0479703
$612$	4.04080	0.163340
$613$	−20.8827	−0.843445	−0.421722	−	0.906725i	$-0.638574\pi$
$613$	−20.8827	−0.843445	−0.421722	+	0.906725i	$0.638574\pi$
$614$	−21.6573	−0.874016
$615$	3.48833	0.140663
$616$	1.73580	0.0699373
$617$	−32.4607	−1.30682	−0.653410	−	0.757004i	$-0.726662\pi$
$617$	−32.4607	−1.30682	−0.653410	+	0.757004i	$0.726662\pi$
$618$	−3.29388	−0.132499
$619$	10.5885	0.425590	0.212795	−	0.977097i	$-0.431743\pi$
$619$	10.5885	0.425590	0.212795	+	0.977097i	$0.431743\pi$
$620$	11.0234	0.442710
$621$	−9.78259	−0.392562
$622$	−5.96867	−0.239322
$623$	23.6729	0.948436
$624$	0.685188	0.0274295
$625$	−29.9423	−1.19769
$626$	−23.2868	−0.930728
$627$	−0.613152	−0.0244869
$628$	−11.3580	−0.453234
$629$	15.0243	0.599060
$630$	−8.43491	−0.336055
$631$	44.3252	1.76456	0.882279	−	0.470727i	$-0.156008\pi$
$631$	44.3252	1.76456	0.882279	+	0.470727i	$0.156008\pi$
$632$	14.6692	0.583508
$633$	−15.5690	−0.618810
$634$	17.5188	0.695760
$635$	47.4664	1.88365
$636$	−9.41246	−0.373228
$637$	2.63779	0.104513
$638$	−5.77853	−0.228774
$639$	15.3846	0.608604
$640$	−2.52120	−0.0996592
$641$	−27.7088	−1.09443	−0.547215	−	0.836992i	$-0.684312\pi$
$641$	−27.7088	−1.09443	−0.547215	+	0.836992i	$0.684312\pi$
$642$	−7.68981	−0.303493
$643$	−11.9777	−0.472355	−0.236178	−	0.971710i	$-0.575895\pi$
$643$	−11.9777	−0.472355	−0.236178	+	0.971710i	$0.575895\pi$
$644$	3.32749	0.131122
$645$	−25.5702	−1.00683
$646$	1.24121	0.0488347
$647$	7.51854	0.295584	0.147792	−	0.989018i	$-0.452783\pi$
$647$	7.51854	0.295584	0.147792	+	0.989018i	$0.452783\pi$
$648$	−0.497122	−0.0195288
$649$	−9.26718	−0.363769
$650$	0.897425	0.0351999
$651$	7.86002	0.308059
$652$	−16.5868	−0.649590
$653$	−7.55273	−0.295561	−0.147781	−	0.989020i	$-0.547213\pi$
$653$	−7.55273	−0.295561	−0.147781	+	0.989020i	$0.547213\pi$
$654$	−9.35749	−0.365907
$655$	−17.2354	−0.673441
$656$	−1.33596	−0.0521605
$657$	−22.0772	−0.861314
$658$	3.11100	0.121279
$659$	32.6551	1.27206	0.636031	−	0.771663i	$-0.280575\pi$
$659$	32.6551	1.27206	0.636031	+	0.771663i	$0.280575\pi$
$660$	−2.61111	−0.101637
$661$	−41.2651	−1.60503	−0.802513	−	0.596634i	$-0.796504\pi$
$661$	−41.2651	−1.60503	−0.802513	+	0.596634i	$0.796504\pi$
$662$	19.7977	0.769458
$663$	−1.43649	−0.0557888
$664$	3.52418	0.136765
$665$	−2.59094	−0.100472
$666$	−13.8126	−0.535227
$667$	−11.0773	−0.428917
$668$	−11.2683	−0.435985
$669$	−4.03820	−0.156126
$670$	−1.83185	−0.0707706
$671$	−2.08729	−0.0805789
$672$	−1.79770	−0.0693477
$673$	26.2045	1.01011	0.505055	−	0.863087i	$-0.331472\pi$
$673$	26.2045	1.01011	0.505055	+	0.863087i	$0.331472\pi$
$674$	−5.87540	−0.226312
$675$	6.92216	0.266434
$676$	−12.5623	−0.483165
$677$	1.09914	0.0422433	0.0211216	−	0.999777i	$-0.493276\pi$
$677$	1.09914	0.0422433	0.0211216	+	0.999777i	$0.493276\pi$
$678$	0.186109	0.00714746
$679$	6.39952	0.245591
$680$	5.28569	0.202697
$681$	−15.6866	−0.601111
$682$	−4.37227	−0.167423
$683$	46.5909	1.78275	0.891376	−	0.453265i	$-0.149741\pi$
$683$	46.5909	1.78275	0.891376	+	0.453265i	$0.149741\pi$
$684$	−1.14110	−0.0436311
$685$	16.8975	0.645621
$686$	−19.0712	−0.728143
$687$	24.9984	0.953747
$688$	9.79286	0.373349
$689$	−6.01282	−0.229070
$690$	−5.00545	−0.190554
$691$	−14.3112	−0.544425	−0.272212	−	0.962237i	$-0.587755\pi$
$691$	−14.3112	−0.544425	−0.272212	+	0.962237i	$0.587755\pi$
$692$	22.3955	0.851348
$693$	3.34559	0.127088
$694$	15.4392	0.586063
$695$	−11.2713	−0.427544
$696$	5.98460	0.226846
$697$	2.80083	0.106089
$698$	14.7457	0.558133
$699$	8.72971	0.330188
$700$	−2.35453	−0.0889930
$701$	45.7544	1.72812	0.864059	−	0.503390i	$-0.167914\pi$
$701$	45.7544	1.72812	0.864059	+	0.503390i	$0.167914\pi$
$702$	3.37620	0.127427
$703$	−4.24280	−0.160020
$704$	1.00000	0.0376889
$705$	−4.67978	−0.176251
$706$	−24.4420	−0.919887
$707$	19.6649	0.739576
$708$	9.59765	0.360702
$709$	0.740212	0.0277993	0.0138996	−	0.999903i	$-0.495575\pi$
$709$	0.740212	0.0277993	0.0138996	+	0.999903i	$0.495575\pi$
$710$	20.1242	0.755249
$711$	28.2735	1.06034
$712$	13.6381	0.511108
$713$	−8.38157	−0.313892
$714$	3.76886	0.141046
$715$	−1.66802	−0.0623802
$716$	10.1374	0.378854
$717$	−17.6734	−0.660026
$718$	−5.59776	−0.208906
$719$	44.6697	1.66590	0.832950	−	0.553348i	$-0.186650\pi$
$719$	44.6697	1.66590	0.832950	+	0.553348i	$0.186650\pi$
$720$	−4.85938	−0.181098
$721$	5.52065	0.205600
$722$	18.6495	0.694062
$723$	3.01907	0.112280
$724$	−7.64184	−0.284007
$725$	7.83833	0.291108
$726$	1.03566	0.0384370
$727$	38.8867	1.44223	0.721113	−	0.692817i	$-0.243631\pi$
$727$	38.8867	1.44223	0.721113	+	0.692817i	$0.243631\pi$
$728$	−1.14840	−0.0425624
$729$	14.8971	0.551746
$730$	−28.8787	−1.06885
$731$	−20.5307	−0.759355
$732$	2.16172	0.0798996
$733$	24.5229	0.905775	0.452888	−	0.891568i	$-0.350394\pi$
$733$	24.5229	0.905775	0.452888	+	0.891568i	$0.350394\pi$
$734$	−8.51249	−0.314202
$735$	10.4105	0.383997
$736$	1.91698	0.0706609
$737$	0.726579	0.0267639
$738$	−2.57494	−0.0947848
$739$	−37.8714	−1.39312	−0.696560	−	0.717498i	$-0.745287\pi$
$739$	−37.8714	−1.39312	−0.696560	+	0.717498i	$0.745287\pi$
$740$	−18.0680	−0.664191
$741$	0.405658	0.0149022
$742$	15.7756	0.579139
$743$	9.06047	0.332396	0.166198	−	0.986092i	$-0.446851\pi$
$743$	9.06047	0.332396	0.166198	+	0.986092i	$0.446851\pi$
$744$	4.52819	0.166011
$745$	−9.38826	−0.343959
$746$	37.2003	1.36200
$747$	6.79252	0.248525
$748$	−2.09650	−0.0766555
$749$	12.8884	0.470930
$750$	−9.51369	−0.347391
$751$	4.38806	0.160123	0.0800614	−	0.996790i	$-0.474488\pi$
$751$	4.38806	0.160123	0.0800614	+	0.996790i	$0.474488\pi$
$752$	1.79226	0.0653569
$753$	27.6289	1.00685
$754$	3.82305	0.139227
$755$	57.5134	2.09313
$756$	−8.85799	−0.322162
$757$	−27.9012	−1.01409	−0.507043	−	0.861921i	$-0.669261\pi$
$757$	−27.9012	−1.01409	−0.507043	+	0.861921i	$0.669261\pi$
$758$	−12.1776	−0.442309
$759$	1.98534	0.0720634
$760$	−1.49265	−0.0541441
$761$	23.6515	0.857367	0.428684	−	0.903455i	$-0.358978\pi$
$761$	23.6515	0.857367	0.428684	+	0.903455i	$0.358978\pi$
$762$	19.4983	0.706348
$763$	15.6834	0.567778
$764$	0.618048	0.0223602
$765$	10.1877	0.368336
$766$	25.9801	0.938700
$767$	6.13112	0.221382
$768$	−1.03566	−0.0373712
$769$	−21.6221	−0.779712	−0.389856	−	0.920876i	$-0.627475\pi$
$769$	−21.6221	−0.779712	−0.389856	+	0.920876i	$0.627475\pi$
$770$	4.37630	0.157711
$771$	−4.82682	−0.173834
$772$	−16.5394	−0.595265
$773$	−16.1644	−0.581392	−0.290696	−	0.956816i	$-0.593887\pi$
$773$	−16.1644	−0.581392	−0.290696	+	0.956816i	$0.593887\pi$
$774$	18.8748	0.678442
$775$	5.93080	0.213040
$776$	3.68679	0.132348
$777$	−12.8830	−0.462176
$778$	−24.9045	−0.892869
$779$	−0.790941	−0.0283384
$780$	1.72750	0.0618543
$781$	−7.98201	−0.285619
$782$	−4.01895	−0.143717
$783$	29.4886	1.05384
$784$	−3.98701	−0.142393
$785$	−28.6358	−1.02206
$786$	−7.07995	−0.252533
$787$	4.06955	0.145064	0.0725318	−	0.997366i	$-0.476892\pi$
$787$	4.06955	0.145064	0.0725318	+	0.997366i	$0.476892\pi$
$788$	1.00000	0.0356235
$789$	−10.0468	−0.357675
$790$	36.9839	1.31583
$791$	−0.311924	−0.0110907
$792$	1.92741	0.0684874
$793$	1.38094	0.0490387
$794$	−14.9156	−0.529335
$795$	−23.7307	−0.841641
$796$	−3.36737	−0.119353
$797$	−4.88414	−0.173005	−0.0865026	−	0.996252i	$-0.527569\pi$
$797$	−4.88414	−0.173005	−0.0865026	+	0.996252i	$0.527569\pi$
$798$	−1.06431	−0.0376761
$799$	−3.75746	−0.132929
$800$	−1.35646	−0.0479580
$801$	26.2861	0.928774
$802$	4.56349	0.161142
$803$	11.4544	0.404216
$804$	−0.752489	−0.0265382
$805$	8.38928	0.295683
$806$	2.89268	0.101890
$807$	5.90843	0.207987
$808$	11.3290	0.398554
$809$	24.8920	0.875155	0.437577	−	0.899181i	$-0.355837\pi$
$809$	24.8920	0.875155	0.437577	+	0.899181i	$0.355837\pi$
$810$	−1.25334	−0.0440380
$811$	−48.8768	−1.71630	−0.858148	−	0.513402i	$-0.828385\pi$
$811$	−48.8768	−1.71630	−0.858148	+	0.513402i	$0.828385\pi$
$812$	−10.0304	−0.351997
$813$	16.4365	0.576452
$814$	7.16641	0.251182
$815$	−41.8187	−1.46484
$816$	2.17126	0.0760092
$817$	5.79776	0.202838
$818$	−1.19069	−0.0416315
$819$	−2.21343	−0.0773434
$820$	−3.36822	−0.117623
$821$	44.9222	1.56779	0.783897	−	0.620891i	$-0.213229\pi$
$821$	44.9222	1.56779	0.783897	+	0.620891i	$0.213229\pi$
$822$	6.94117	0.242101
$823$	9.49265	0.330893	0.165446	−	0.986219i	$-0.447094\pi$
$823$	9.49265	0.330893	0.165446	+	0.986219i	$0.447094\pi$
$824$	3.18047	0.110797
$825$	−1.40483	−0.0489098
$826$	−16.0859	−0.559702
$827$	−34.7253	−1.20752	−0.603759	−	0.797167i	$-0.706331\pi$
$827$	−34.7253	−1.20752	−0.603759	+	0.797167i	$0.706331\pi$
$828$	3.69481	0.128403
$829$	−20.4794	−0.711278	−0.355639	−	0.934623i	$-0.615737\pi$
$829$	−20.4794	−0.711278	−0.355639	+	0.934623i	$0.615737\pi$
$830$	8.88516	0.308408
$831$	30.9625	1.07408
$832$	−0.661596	−0.0229367
$833$	8.35874	0.289613
$834$	−4.63002	−0.160324
$835$	−28.4097	−0.983160
$836$	0.592039	0.0204761
$837$	22.3122	0.771224
$838$	12.1040	0.418125
$839$	32.1767	1.11086	0.555431	−	0.831563i	$-0.312553\pi$
$839$	32.1767	1.11086	0.555431	+	0.831563i	$0.312553\pi$
$840$	−4.53236	−0.156381
$841$	4.39146	0.151430
$842$	30.6982	1.05793
$843$	30.4330	1.04817
$844$	15.0329	0.517453
$845$	−31.6721	−1.08955
$846$	3.45441	0.118765
$847$	−1.73580	−0.0596427
$848$	9.08836	0.312096
$849$	−21.8477	−0.749810
$850$	2.84380	0.0975417
$851$	13.7379	0.470928
$852$	8.26665	0.283211
$853$	−26.5986	−0.910718	−0.455359	−	0.890308i	$-0.650489\pi$
$853$	−26.5986	−0.910718	−0.455359	+	0.890308i	$0.650489\pi$
$854$	−3.62311	−0.123980
$855$	−2.87695	−0.0983895
$856$	7.42503	0.253782
$857$	−9.38910	−0.320725	−0.160363	−	0.987058i	$-0.551266\pi$
$857$	−9.38910	−0.320725	−0.160363	+	0.987058i	$0.551266\pi$
$858$	−0.685188	−0.0233919
$859$	42.6884	1.45651	0.728254	−	0.685307i	$-0.240332\pi$
$859$	42.6884	1.45651	0.728254	+	0.685307i	$0.240332\pi$
$860$	24.6898	0.841914
$861$	−2.40165	−0.0818480
$862$	9.16476	0.312153
$863$	18.8089	0.640263	0.320131	−	0.947373i	$-0.396273\pi$
$863$	18.8089	0.640263	0.320131	+	0.947373i	$0.396273\pi$
$864$	−5.10312	−0.173612
$865$	56.4635	1.91982
$866$	20.5838	0.699465
$867$	13.0542	0.443344
$868$	−7.58938	−0.257600
$869$	−14.6692	−0.497617
$870$	15.0884	0.511544
$871$	−0.480701	−0.0162879
$872$	9.03529	0.305973
$873$	7.10594	0.240500
$874$	1.13493	0.0383895
$875$	15.9452	0.539047
$876$	−11.8628	−0.400808
$877$	32.3311	1.09174	0.545872	−	0.837868i	$-0.316198\pi$
$877$	32.3311	1.09174	0.545872	+	0.837868i	$0.316198\pi$
$878$	19.8541	0.670044
$879$	23.6151	0.796516
$880$	2.52120	0.0849897
$881$	34.1548	1.15070	0.575351	−	0.817906i	$-0.304865\pi$
$881$	34.1548	1.15070	0.575351	+	0.817906i	$0.304865\pi$
$882$	−7.68459	−0.258753
$883$	−25.3413	−0.852802	−0.426401	−	0.904534i	$-0.640219\pi$
$883$	−25.3413	−0.852802	−0.426401	+	0.904534i	$0.640219\pi$
$884$	1.38703	0.0466509
$885$	24.1976	0.813394
$886$	−1.65311	−0.0555373
$887$	11.0158	0.369874	0.184937	−	0.982750i	$-0.440792\pi$
$887$	11.0158	0.369874	0.184937	+	0.982750i	$0.440792\pi$
$888$	−7.42196	−0.249065
$889$	−32.6797	−1.09604
$890$	34.3843	1.15256
$891$	0.497122	0.0166542
$892$	3.89916	0.130553
$893$	1.06109	0.0355079
$894$	−3.85651	−0.128981
$895$	25.5585	0.854328
$896$	1.73580	0.0579889
$897$	−1.31349	−0.0438563
$898$	34.3912	1.14765
$899$	25.2653	0.842646
$900$	−2.61444	−0.0871481
$901$	−19.0537	−0.634771
$902$	1.33596	0.0444826
$903$	17.6046	0.585844
$904$	−0.179700	−0.00597675
$905$	−19.2666	−0.640444
$906$	23.6254	0.784901
$907$	−9.23996	−0.306808	−0.153404	−	0.988164i	$-0.549024\pi$
$907$	−9.23996	−0.306808	−0.153404	+	0.988164i	$0.549024\pi$
$908$	15.1464	0.502652
$909$	21.8357	0.724244
$910$	−2.89534	−0.0959795
$911$	−46.6779	−1.54651	−0.773254	−	0.634097i	$-0.781372\pi$
$911$	−46.6779	−1.54651	−0.773254	+	0.634097i	$0.781372\pi$
$912$	−0.613152	−0.0203035
$913$	−3.52418	−0.116633
$914$	9.36018	0.309607
$915$	5.45014	0.180176
$916$	−24.1376	−0.797529
$917$	11.8662	0.391857
$918$	10.6987	0.353109
$919$	7.79819	0.257239	0.128619	−	0.991694i	$-0.458945\pi$
$919$	7.79819	0.257239	0.128619	+	0.991694i	$0.458945\pi$
$920$	4.83310	0.159343
$921$	−22.4296	−0.739080
$922$	31.5116	1.03778
$923$	5.28086	0.173822
$924$	1.79770	0.0591399
$925$	−9.72092	−0.319622
$926$	31.6852	1.04124
$927$	6.13006	0.201338
$928$	−5.77853	−0.189690
$929$	12.0048	0.393866	0.196933	−	0.980417i	$-0.436902\pi$
$929$	12.0048	0.393866	0.196933	+	0.980417i	$0.436902\pi$
$930$	11.4165	0.374361
$931$	−2.36046	−0.0773611
$932$	−8.42912	−0.276105
$933$	−6.18151	−0.202374
$934$	−8.22554	−0.269148
$935$	−5.28569	−0.172861
$936$	−1.27516	−0.0416800
$937$	−7.27247	−0.237581	−0.118791	−	0.992919i	$-0.537902\pi$
$937$	−7.27247	−0.237581	−0.118791	+	0.992919i	$0.537902\pi$
$938$	1.26119	0.0411794
$939$	−24.1172	−0.787036
$940$	4.51864	0.147382
$941$	56.0480	1.82711	0.913556	−	0.406714i	$-0.133325\pi$
$941$	56.0480	1.82711	0.913556	+	0.406714i	$0.133325\pi$
$942$	−11.7630	−0.383261
$943$	2.56101	0.0833980
$944$	−9.26718	−0.301621
$945$	−22.3328	−0.726485
$946$	−9.79286	−0.318393
$947$	20.6617	0.671415	0.335707	−	0.941966i	$-0.391025\pi$
$947$	20.6617	0.671415	0.335707	+	0.941966i	$0.391025\pi$
$948$	15.1923	0.493422
$949$	−7.57815	−0.245997
$950$	−0.803075	−0.0260552
$951$	18.1435	0.588344
$952$	−3.63909	−0.117944
$953$	51.5227	1.66898	0.834492	−	0.551020i	$-0.185761\pi$
$953$	51.5227	1.66898	0.834492	+	0.551020i	$0.185761\pi$
$954$	17.5170	0.567133
$955$	1.55822	0.0504229
$956$	17.0649	0.551918
$957$	−5.98460	−0.193455
$958$	28.3633	0.916377
$959$	−11.6336	−0.375669
$960$	−2.61111	−0.0842732
$961$	−11.8832	−0.383330
$962$	−4.74126	−0.152864
$963$	14.3111	0.461167
$964$	−2.91512	−0.0938896
$965$	−41.6991	−1.34234
$966$	3.44615	0.110878
$967$	10.1949	0.327845	0.163922	−	0.986473i	$-0.447585\pi$
$967$	10.1949	0.327845	0.163922	+	0.986473i	$0.447585\pi$
$968$	−1.00000	−0.0321412
$969$	1.28547	0.0412953
$970$	9.29514	0.298449
$971$	38.2263	1.22674	0.613369	−	0.789796i	$-0.289814\pi$
$971$	38.2263	1.22674	0.613369	+	0.789796i	$0.289814\pi$
$972$	−15.8242	−0.507562
$973$	7.76005	0.248776
$974$	−3.84730	−0.123275
$975$	0.929428	0.0297655
$976$	−2.08729	−0.0668125
$977$	−0.921778	−0.0294903	−0.0147452	−	0.999891i	$-0.504694\pi$
$977$	−0.921778	−0.0294903	−0.0147452	+	0.999891i	$0.504694\pi$
$978$	−17.1783	−0.549302
$979$	−13.6381	−0.435875
$980$	−10.0520	−0.321101
$981$	17.4147	0.556008
$982$	−26.6876	−0.851636
$983$	−10.6976	−0.341199	−0.170600	−	0.985340i	$-0.554570\pi$
$983$	−10.6976	−0.341199	−0.170600	+	0.985340i	$0.554570\pi$
$984$	−1.38360	−0.0441076
$985$	2.52120	0.0803322
$986$	12.1147	0.385810
$987$	3.22193	0.102555
$988$	−0.391691	−0.0124613
$989$	−18.7727	−0.596938
$990$	4.85938	0.154441
$991$	38.0455	1.20855	0.604277	−	0.796774i	$-0.293462\pi$
$991$	38.0455	1.20855	0.604277	+	0.796774i	$0.293462\pi$
$992$	−4.37227	−0.138820
$993$	20.5037	0.650664
$994$	−13.8551	−0.439459
$995$	−8.48981	−0.269145
$996$	3.64985	0.115650
$997$	−53.6949	−1.70053	−0.850267	−	0.526352i	$-0.823559\pi$
$997$	−53.6949	−1.70053	−0.850267	+	0.526352i	$0.823559\pi$
$998$	−10.1351	−0.320820
$999$	−36.5710	−1.15706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.a.1.5	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.a.1.5	✓	15	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.03566 q^{3} +1.00000 q^{4} +2.52120 q^{5} +1.03566 q^{6} -1.73580 q^{7} -1.00000 q^{8} -1.92741 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.03566 q^{3} +1.00000 q^{4} +2.52120 q^{5} +1.03566 q^{6} -1.73580 q^{7} -1.00000 q^{8} -1.92741 q^{9} -2.52120 q^{10} +1.00000 q^{11} -1.03566 q^{12} -0.661596 q^{13} +1.73580 q^{14} -2.61111 q^{15} +1.00000 q^{16} -2.09650 q^{17} +1.92741 q^{18} +0.592039 q^{19} +2.52120 q^{20} +1.79770 q^{21} -1.00000 q^{22} -1.91698 q^{23} +1.03566 q^{24} +1.35646 q^{25} +0.661596 q^{26} +5.10312 q^{27} -1.73580 q^{28} +5.77853 q^{29} +2.61111 q^{30} +4.37227 q^{31} -1.00000 q^{32} -1.03566 q^{33} +2.09650 q^{34} -4.37630 q^{35} -1.92741 q^{36} -7.16641 q^{37} -0.592039 q^{38} +0.685188 q^{39} -2.52120 q^{40} -1.33596 q^{41} -1.79770 q^{42} +9.79286 q^{43} +1.00000 q^{44} -4.85938 q^{45} +1.91698 q^{46} +1.79226 q^{47} -1.03566 q^{48} -3.98701 q^{49} -1.35646 q^{50} +2.17126 q^{51} -0.661596 q^{52} +9.08836 q^{53} -5.10312 q^{54} +2.52120 q^{55} +1.73580 q^{56} -0.613152 q^{57} -5.77853 q^{58} -9.26718 q^{59} -2.61111 q^{60} -2.08729 q^{61} -4.37227 q^{62} +3.34559 q^{63} +1.00000 q^{64} -1.66802 q^{65} +1.03566 q^{66} +0.726579 q^{67} -2.09650 q^{68} +1.98534 q^{69} +4.37630 q^{70} -7.98201 q^{71} +1.92741 q^{72} +11.4544 q^{73} +7.16641 q^{74} -1.40483 q^{75} +0.592039 q^{76} -1.73580 q^{77} -0.685188 q^{78} -14.6692 q^{79} +2.52120 q^{80} +0.497122 q^{81} +1.33596 q^{82} -3.52418 q^{83} +1.79770 q^{84} -5.28569 q^{85} -9.79286 q^{86} -5.98460 q^{87} -1.00000 q^{88} -13.6381 q^{89} +4.85938 q^{90} +1.14840 q^{91} -1.91698 q^{92} -4.52819 q^{93} -1.79226 q^{94} +1.49265 q^{95} +1.03566 q^{96} -3.68679 q^{97} +3.98701 q^{98} -1.92741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9}+O(q^{10}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9	\( 15 q - 15 q^{2} - q^{3} + 15 q^{4} - 7 q^{5} + q^{6} + q^{7} - 15 q^{8} + 2 q^{9} + 7 q^{10} + 15 q^{11} - q^{12} - q^{13} - q^{14} - 6 q^{15} + 15 q^{16} - 6 q^{17} - 2 q^{18} - 14 q^{19} - 7 q^{20} - 3 q^{21} - 15 q^{22} + 2 q^{23} + q^{24} - 10 q^{25} + q^{26} - 7 q^{27} + q^{28} + 8 q^{29} + 6 q^{30} - 33 q^{31} - 15 q^{32} - q^{33} + 6 q^{34} - 8 q^{35} + 2 q^{36} - 9 q^{37} + 14 q^{38} - 9 q^{39} + 7 q^{40} - 10 q^{41} + 3 q^{42} - 6 q^{43} + 15 q^{44} - 20 q^{45} - 2 q^{46} - q^{47} - q^{48} - 30 q^{49} + 10 q^{50} + 12 q^{51} - q^{52} + 6 q^{53} + 7 q^{54} - 7 q^{55} - q^{56} - 24 q^{57} - 8 q^{58} - 15 q^{59} - 6 q^{60} - 25 q^{61} + 33 q^{62} + 12 q^{63} + 15 q^{64} + 31 q^{65} + q^{66} - 13 q^{67} - 6 q^{68} - 43 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 4 q^{73} + 9 q^{74} - 5 q^{75} - 14 q^{76} + q^{77} + 9 q^{78} - 20 q^{79} - 7 q^{80} + 11 q^{81} + 10 q^{82} + q^{83} - 3 q^{84} - q^{85} + 6 q^{86} + 22 q^{87} - 15 q^{88} - 41 q^{89} + 20 q^{90} - 31 q^{91} + 2 q^{92} + 14 q^{93} + q^{94} + 41 q^{95} + q^{96} - 57 q^{97} + 30 q^{98} + 2 q^{99}+O(q^{100}) \) 15 * q - 15 * q^2 - q^3 + 15 * q^4 - 7 * q^5 + q^6 + q^7 - 15 * q^8 + 2 * q^9 + 7 * q^10 + 15 * q^11 - q^12 - q^13 - q^14 - 6 * q^15 + 15 * q^16 - 6 * q^17 - 2 * q^18 - 14 * q^19 - 7 * q^20 - 3 * q^21 - 15 * q^22 + 2 * q^23 + q^24 - 10 * q^25 + q^26 - 7 * q^27 + q^28 + 8 * q^29 + 6 * q^30 - 33 * q^31 - 15 * q^32 - q^33 + 6 * q^34 - 8 * q^35 + 2 * q^36 - 9 * q^37 + 14 * q^38 - 9 * q^39 + 7 * q^40 - 10 * q^41 + 3 * q^42 - 6 * q^43 + 15 * q^44 - 20 * q^45 - 2 * q^46 - q^47 - q^48 - 30 * q^49 + 10 * q^50 + 12 * q^51 - q^52 + 6 * q^53 + 7 * q^54 - 7 * q^55 - q^56 - 24 * q^57 - 8 * q^58 - 15 * q^59 - 6 * q^60 - 25 * q^61 + 33 * q^62 + 12 * q^63 + 15 * q^64 + 31 * q^65 + q^66 - 13 * q^67 - 6 * q^68 - 43 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 4 * q^73 + 9 * q^74 - 5 * q^75 - 14 * q^76 + q^77 + 9 * q^78 - 20 * q^79 - 7 * q^80 + 11 * q^81 + 10 * q^82 + q^83 - 3 * q^84 - q^85 + 6 * q^86 + 22 * q^87 - 15 * q^88 - 41 * q^89 + 20 * q^90 - 31 * q^91 + 2 * q^92 + 14 * q^93 + q^94 + 41 * q^95 + q^96 - 57 * q^97 + 30 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more