LMFDB - Embedded newform 4235.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4235,2,Mod(1,4235)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4235, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4235.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4235 = 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4235.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:	$33.8166452560$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.173513.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 3x^{2} + 3x - 1 $ x^5 - 2x^4 - 5x^3 + 3x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.52979$ of defining polynomial
Character	$\chi$	$=$	4235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +O(q^{10})$	\(q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +1.65369 q^{10} -0.838578 q^{12} +5.60081 q^{13} -1.65369 q^{14} +1.14142 q^{15} -4.92960 q^{16} -4.57876 q^{17} +2.80656 q^{18} +1.88149 q^{19} -0.734678 q^{20} -1.14142 q^{21} +2.08099 q^{23} -2.38836 q^{24} +1.00000 q^{25} -9.26199 q^{26} +5.36144 q^{27} +0.734678 q^{28} -5.51585 q^{29} -1.88755 q^{30} -5.46081 q^{31} +3.96713 q^{32} +7.57184 q^{34} -1.00000 q^{35} -1.24686 q^{36} -9.34694 q^{37} -3.11139 q^{38} -6.39289 q^{39} -2.09245 q^{40} +11.0136 q^{41} +1.88755 q^{42} -8.43344 q^{43} +1.69716 q^{45} -3.44131 q^{46} +1.60936 q^{47} +5.62676 q^{48} +1.00000 q^{49} -1.65369 q^{50} +5.22630 q^{51} +4.11479 q^{52} +6.21331 q^{53} -8.86613 q^{54} +2.09245 q^{56} -2.14757 q^{57} +9.12149 q^{58} +13.5862 q^{59} +0.838578 q^{60} -11.1145 q^{61} +9.03046 q^{62} -1.69716 q^{63} +3.29883 q^{64} -5.60081 q^{65} -5.60924 q^{67} -3.36392 q^{68} -2.37529 q^{69} +1.65369 q^{70} +0.696594 q^{71} -3.55121 q^{72} +13.8873 q^{73} +15.4569 q^{74} -1.14142 q^{75} +1.38229 q^{76} +10.5718 q^{78} -8.47388 q^{79} +4.92960 q^{80} -1.02820 q^{81} -18.2131 q^{82} -9.50230 q^{83} -0.838578 q^{84} +4.57876 q^{85} +13.9463 q^{86} +6.29592 q^{87} +1.49863 q^{89} -2.80656 q^{90} +5.60081 q^{91} +1.52886 q^{92} +6.23309 q^{93} -2.66137 q^{94} -1.88149 q^{95} -4.52817 q^{96} +5.93697 q^{97} -1.65369 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 5 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 5 * q^5 - 5 * q^6 + 5 * q^7 + 6 * q^8 + 3 * q^9	$ 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 11 q^{12} - 12 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 7 q^{18} - 9 q^{19} - 4 q^{20} - 2 q^{21} + 17 q^{23} - 6 q^{24} + 5 q^{25} - 11 q^{26} - 11 q^{27} + 4 q^{28} - 3 q^{29} + 5 q^{30} + 2 q^{31} + 5 q^{32} + 16 q^{34} - 5 q^{35} - 15 q^{36} + 4 q^{37} - 11 q^{38} - 2 q^{39} - 6 q^{40} + 15 q^{41} - 5 q^{42} - 4 q^{43} - 3 q^{45} + 10 q^{46} - 2 q^{47} - 10 q^{48} + 5 q^{49} - 2 q^{50} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{58} - 6 q^{59} - 11 q^{60} - 20 q^{61} - 21 q^{62} + 3 q^{63} - 26 q^{64} + 12 q^{65} + 3 q^{67} - 5 q^{68} + 2 q^{70} - 6 q^{71} - 34 q^{72} - 11 q^{73} + 15 q^{74} - 2 q^{75} - 47 q^{76} + 31 q^{78} - 19 q^{79} - 2 q^{80} + 33 q^{81} - 8 q^{83} + 11 q^{84} + 14 q^{85} + 27 q^{86} + 30 q^{87} + q^{89} + 7 q^{90} - 12 q^{91} + 44 q^{92} + 3 q^{93} - 28 q^{94} + 9 q^{95} + 4 q^{96} - 7 q^{97} - 2 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 5 * q^5 - 5 * q^6 + 5 * q^7 + 6 * q^8 + 3 * q^9 + 2 * q^10 + 11 * q^12 - 12 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 14 * q^17 - 7 * q^18 - 9 * q^19 - 4 * q^20 - 2 * q^21 + 17 * q^23 - 6 * q^24 + 5 * q^25 - 11 * q^26 - 11 * q^27 + 4 * q^28 - 3 * q^29 + 5 * q^30 + 2 * q^31 + 5 * q^32 + 16 * q^34 - 5 * q^35 - 15 * q^36 + 4 * q^37 - 11 * q^38 - 2 * q^39 - 6 * q^40 + 15 * q^41 - 5 * q^42 - 4 * q^43 - 3 * q^45 + 10 * q^46 - 2 * q^47 - 10 * q^48 + 5 * q^49 - 2 * q^50 + 18 * q^51 - 4 * q^52 + 6 * q^53 - 4 * q^54 + 6 * q^56 - 32 * q^58 - 6 * q^59 - 11 * q^60 - 20 * q^61 - 21 * q^62 + 3 * q^63 - 26 * q^64 + 12 * q^65 + 3 * q^67 - 5 * q^68 + 2 * q^70 - 6 * q^71 - 34 * q^72 - 11 * q^73 + 15 * q^74 - 2 * q^75 - 47 * q^76 + 31 * q^78 - 19 * q^79 - 2 * q^80 + 33 * q^81 - 8 * q^83 + 11 * q^84 + 14 * q^85 + 27 * q^86 + 30 * q^87 + q^89 + 7 * q^90 - 12 * q^91 + 44 * q^92 + 3 * q^93 - 28 * q^94 + 9 * q^95 + 4 * q^96 - 7 * q^97 - 2 * q^98

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 $

$2$	−1.65369	−1.16933	−0.584666	−	0.811274i	$-0.698775\pi$
$2$	−1.65369	−1.16933	−0.584666	+	0.811274i	$0.698775\pi$
$3$	−1.14142	−0.659000	−0.329500	−	0.944156i	$-0.606880\pi$
$3$	−1.14142	−0.659000	−0.329500	+	0.944156i	$0.606880\pi$
$4$	0.734678	0.367339
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.88755	0.770591
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.09245	0.739791
$9$	−1.69716	−0.565718
$10$	1.65369	0.522941
$11$	0	0
$11$	0	0
$12$	−0.838578	−0.242076
$13$	5.60081	1.55339	0.776693	−	0.629879i	$-0.216896\pi$
$13$	5.60081	1.55339	0.776693	+	0.629879i	$0.216896\pi$
$14$	−1.65369	−0.441966
$15$	1.14142	0.294714
$16$	−4.92960	−1.23240
$17$	−4.57876	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$17$	−4.57876	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$18$	2.80656	0.661513
$19$	1.88149	0.431642	0.215821	−	0.976433i	$-0.430757\pi$
$19$	1.88149	0.431642	0.215821	+	0.976433i	$0.430757\pi$
$20$	−0.734678	−0.164279
$21$	−1.14142	−0.249079
$22$	0	0
$23$	2.08099	0.433917	0.216958	−	0.976181i	$-0.430386\pi$
$23$	2.08099	0.433917	0.216958	+	0.976181i	$0.430386\pi$
$24$	−2.38836	−0.487523
$25$	1.00000	0.200000
$26$	−9.26199	−1.81642
$27$	5.36144	1.03181
$28$	0.734678	0.138841
$29$	−5.51585	−1.02427	−0.512134	−	0.858905i	$-0.671145\pi$
$29$	−5.51585	−1.02427	−0.512134	+	0.858905i	$0.671145\pi$
$30$	−1.88755	−0.344619
$31$	−5.46081	−0.980790	−0.490395	−	0.871500i	$-0.663148\pi$
$31$	−5.46081	−0.980790	−0.490395	+	0.871500i	$0.663148\pi$
$32$	3.96713	0.701296
$33$	0	0
$34$	7.57184	1.29856
$35$	−1.00000	−0.169031
$36$	−1.24686	−0.207810
$37$	−9.34694	−1.53663	−0.768314	−	0.640073i	$-0.778904\pi$
$37$	−9.34694	−1.53663	−0.768314	+	0.640073i	$0.778904\pi$
$38$	−3.11139	−0.504734
$39$	−6.39289	−1.02368
$40$	−2.09245	−0.330845
$41$	11.0136	1.72004	0.860020	−	0.510261i	$-0.170451\pi$
$41$	11.0136	1.72004	0.860020	+	0.510261i	$0.170451\pi$
$42$	1.88755	0.291256
$43$	−8.43344	−1.28609	−0.643044	−	0.765829i	$-0.722329\pi$
$43$	−8.43344	−1.28609	−0.643044	+	0.765829i	$0.722329\pi$
$44$	0	0
$45$	1.69716	0.252997
$46$	−3.44131	−0.507393
$47$	1.60936	0.234749	0.117375	−	0.993088i	$-0.462552\pi$
$47$	1.60936	0.234749	0.117375	+	0.993088i	$0.462552\pi$
$48$	5.62676	0.812153
$49$	1.00000	0.142857
$50$	−1.65369	−0.233867
$51$	5.22630	0.731829
$52$	4.11479	0.570619
$53$	6.21331	0.853463	0.426732	−	0.904378i	$-0.359665\pi$
$53$	6.21331	0.853463	0.426732	+	0.904378i	$0.359665\pi$
$54$	−8.86613	−1.20653
$55$	0	0
$56$	2.09245	0.279615
$57$	−2.14757	−0.284453
$58$	9.12149	1.19771
$59$	13.5862	1.76877	0.884386	−	0.466756i	$-0.154577\pi$
$59$	13.5862	1.76877	0.884386	+	0.466756i	$0.154577\pi$
$60$	0.838578	0.108260
$61$	−11.1145	−1.42307	−0.711533	−	0.702653i	$-0.751999\pi$
$61$	−11.1145	−1.42307	−0.711533	+	0.702653i	$0.751999\pi$
$62$	9.03046	1.14687
$63$	−1.69716	−0.213821
$64$	3.29883	0.412353
$65$	−5.60081	−0.694695
$66$	0	0
$67$	−5.60924	−0.685277	−0.342639	−	0.939467i	$-0.611321\pi$
$67$	−5.60924	−0.685277	−0.342639	+	0.939467i	$0.611321\pi$
$68$	−3.36392	−0.407935
$69$	−2.37529	−0.285951
$70$	1.65369	0.197653
$71$	0.696594	0.0826705	0.0413353	−	0.999145i	$-0.486839\pi$
$71$	0.696594	0.0826705	0.0413353	+	0.999145i	$0.486839\pi$
$72$	−3.55121	−0.418514
$73$	13.8873	1.62539	0.812694	−	0.582690i	$-0.198000\pi$
$73$	13.8873	1.62539	0.812694	+	0.582690i	$0.198000\pi$
$74$	15.4569	1.79683
$75$	−1.14142	−0.131800
$76$	1.38229	0.158559
$77$	0	0
$78$	10.5718	1.19702
$79$	−8.47388	−0.953386	−0.476693	−	0.879070i	$-0.658165\pi$
$79$	−8.47388	−0.953386	−0.476693	+	0.879070i	$0.658165\pi$
$80$	4.92960	0.551146
$81$	−1.02820	−0.114244
$82$	−18.2131	−2.01130
$83$	−9.50230	−1.04301	−0.521506	−	0.853247i	$-0.674630\pi$
$83$	−9.50230	−1.04301	−0.521506	+	0.853247i	$0.674630\pi$
$84$	−0.838578	−0.0914963
$85$	4.57876	0.496637
$86$	13.9463	1.50386
$87$	6.29592	0.674993
$88$	0	0
$89$	1.49863	0.158854	0.0794272	−	0.996841i	$-0.474691\pi$
$89$	1.49863	0.158854	0.0794272	+	0.996841i	$0.474691\pi$
$90$	−2.80656	−0.295838
$91$	5.60081	0.587125
$92$	1.52886	0.159395
$93$	6.23309	0.646341
$94$	−2.66137	−0.274500
$95$	−1.88149	−0.193036
$96$	−4.52817	−0.462154
$97$	5.93697	0.602808	0.301404	−	0.953497i	$-0.402545\pi$
$97$	5.93697	0.602808	0.301404	+	0.953497i	$0.402545\pi$
$98$	−1.65369	−0.167048
$99$	0	0
$100$	0.734678	0.0734678
$101$	−3.15826	−0.314259	−0.157129	−	0.987578i	$-0.550224\pi$
$101$	−3.15826	−0.314259	−0.157129	+	0.987578i	$0.550224\pi$
$102$	−8.64266	−0.855751
$103$	1.24686	0.122857	0.0614285	−	0.998111i	$-0.480434\pi$
$103$	1.24686	0.122857	0.0614285	+	0.998111i	$0.480434\pi$
$104$	11.7194	1.14918
$105$	1.14142	0.111391
$106$	−10.2749	−0.997983
$107$	10.2684	0.992687	0.496343	−	0.868126i	$-0.334676\pi$
$107$	10.2684	0.992687	0.496343	+	0.868126i	$0.334676\pi$
$108$	3.93893	0.379024
$109$	−5.67907	−0.543956	−0.271978	−	0.962303i	$-0.587678\pi$
$109$	−5.67907	−0.543956	−0.271978	+	0.962303i	$0.587678\pi$
$110$	0	0
$111$	10.6688	1.01264
$112$	−4.92960	−0.465804
$113$	17.9797	1.69139	0.845694	−	0.533669i	$-0.179187\pi$
$113$	17.9797	1.69139	0.845694	+	0.533669i	$0.179187\pi$
$114$	3.55141	0.332620
$115$	−2.08099	−0.194053
$116$	−4.05238	−0.376254
$117$	−9.50545	−0.878779
$118$	−22.4673	−2.06828
$119$	−4.57876	−0.419734
$120$	2.38836	0.218027
$121$	0	0
$122$	18.3799	1.66404
$123$	−12.5712	−1.13351
$124$	−4.01194	−0.360282
$125$	−1.00000	−0.0894427
$126$	2.80656	0.250028
$127$	−12.0909	−1.07289	−0.536447	−	0.843934i	$-0.680234\pi$
$127$	−12.0909	−1.07289	−0.536447	+	0.843934i	$0.680234\pi$
$128$	−13.3895	−1.18347
$129$	9.62612	0.847533
$130$	9.26199	0.812330
$131$	−1.04441	−0.0912502	−0.0456251	−	0.998959i	$-0.514528\pi$
$131$	−1.04441	−0.0912502	−0.0456251	+	0.998959i	$0.514528\pi$
$132$	0	0
$133$	1.88149	0.163146
$134$	9.27592	0.801317
$135$	−5.36144	−0.461439
$136$	−9.58081	−0.821548
$137$	−1.43820	−0.122874	−0.0614368	−	0.998111i	$-0.519568\pi$
$137$	−1.43820	−0.122874	−0.0614368	+	0.998111i	$0.519568\pi$
$138$	3.92798	0.334372
$139$	21.6489	1.83623	0.918117	−	0.396310i	$-0.129709\pi$
$139$	21.6489	1.83623	0.918117	+	0.396310i	$0.129709\pi$
$140$	−0.734678	−0.0620916
$141$	−1.83696	−0.154700
$142$	−1.15195	−0.0966694
$143$	0	0
$144$	8.36630	0.697192
$145$	5.51585	0.458067
$146$	−22.9653	−1.90062
$147$	−1.14142	−0.0941429
$148$	−6.86699	−0.564463
$149$	12.6540	1.03666	0.518329	−	0.855181i	$-0.326554\pi$
$149$	12.6540	1.03666	0.518329	+	0.855181i	$0.326554\pi$
$150$	1.88755	0.154118
$151$	−2.48546	−0.202264	−0.101132	−	0.994873i	$-0.532246\pi$
$151$	−2.48546	−0.202264	−0.101132	+	0.994873i	$0.532246\pi$
$152$	3.93691	0.319325
$153$	7.77087	0.628238
$154$	0	0
$155$	5.46081	0.438623
$156$	−4.69672	−0.376038
$157$	−14.2773	−1.13945	−0.569727	−	0.821834i	$-0.692951\pi$
$157$	−14.2773	−1.13945	−0.569727	+	0.821834i	$0.692951\pi$
$158$	14.0131	1.11483
$159$	−7.09201	−0.562433
$160$	−3.96713	−0.313629
$161$	2.08099	0.164005
$162$	1.70032	0.133589
$163$	−9.75758	−0.764272	−0.382136	−	0.924106i	$-0.624811\pi$
$163$	−9.75758	−0.764272	−0.382136	+	0.924106i	$0.624811\pi$
$164$	8.09147	0.631837
$165$	0	0
$166$	15.7138	1.21963
$167$	−11.3890	−0.881308	−0.440654	−	0.897677i	$-0.645253\pi$
$167$	−11.3890	−0.881308	−0.440654	+	0.897677i	$0.645253\pi$
$168$	−2.38836	−0.184266
$169$	18.3691	1.41301
$170$	−7.57184	−0.580733
$171$	−3.19317	−0.244188
$172$	−6.19586	−0.472430
$173$	−10.3544	−0.787227	−0.393613	−	0.919276i	$-0.628775\pi$
$173$	−10.3544	−0.787227	−0.393613	+	0.919276i	$0.628775\pi$
$174$	−10.4115	−0.789292
$175$	1.00000	0.0755929
$176$	0	0
$177$	−15.5076	−1.16562
$178$	−2.47826	−0.185754
$179$	18.8195	1.40664	0.703318	−	0.710875i	$-0.251701\pi$
$179$	18.8195	1.40664	0.703318	+	0.710875i	$0.251701\pi$
$180$	1.24686	0.0929356
$181$	−7.75369	−0.576327	−0.288164	−	0.957581i	$-0.593045\pi$
$181$	−7.75369	−0.576327	−0.288164	+	0.957581i	$0.593045\pi$
$182$	−9.26199	−0.686544
$183$	12.6863	0.937801
$184$	4.35436	0.321008
$185$	9.34694	0.687201
$186$	−10.3076	−0.755788
$187$	0	0
$188$	1.18236	0.0862325
$189$	5.36144	0.389987
$190$	3.11139	0.225724
$191$	15.9410	1.15345	0.576723	−	0.816939i	$-0.304331\pi$
$191$	15.9410	1.15345	0.576723	+	0.816939i	$0.304331\pi$
$192$	−3.76535	−0.271741
$193$	−8.52230	−0.613448	−0.306724	−	0.951798i	$-0.599233\pi$
$193$	−8.52230	−0.613448	−0.306724	+	0.951798i	$0.599233\pi$
$194$	−9.81789	−0.704883
$195$	6.39289	0.457804
$196$	0.734678	0.0524770
$197$	19.0556	1.35765	0.678826	−	0.734299i	$-0.262489\pi$
$197$	19.0556	1.35765	0.678826	+	0.734299i	$0.262489\pi$
$198$	0	0
$199$	−12.7118	−0.901118	−0.450559	−	0.892747i	$-0.648775\pi$
$199$	−12.7118	−0.901118	−0.450559	+	0.892747i	$0.648775\pi$
$200$	2.09245	0.147958
$201$	6.40251	0.451598
$202$	5.22277	0.367473
$203$	−5.51585	−0.387137
$204$	3.83965	0.268829
$205$	−11.0136	−0.769225
$206$	−2.06192	−0.143661
$207$	−3.53177	−0.245475
$208$	−27.6098	−1.91439
$209$	0	0
$210$	−1.88755	−0.130254
$211$	28.4506	1.95862	0.979311	−	0.202363i	$-0.0648620\pi$
$211$	28.4506	1.95862	0.979311	+	0.202363i	$0.0648620\pi$
$212$	4.56478	0.313510
$213$	−0.795108	−0.0544799
$214$	−16.9808	−1.16078
$215$	8.43344	0.575156
$216$	11.2185	0.763323
$217$	−5.46081	−0.370704
$218$	9.39140	0.636066
$219$	−15.8513	−1.07113
$220$	0	0
$221$	−25.6448	−1.72506
$222$	−17.6429	−1.18411
$223$	−23.9632	−1.60470	−0.802348	−	0.596856i	$-0.796416\pi$
$223$	−23.9632	−1.60470	−0.802348	+	0.596856i	$0.796416\pi$
$224$	3.96713	0.265065
$225$	−1.69716	−0.113144
$226$	−29.7328	−1.97779
$227$	−20.6007	−1.36732	−0.683658	−	0.729803i	$-0.739612\pi$
$227$	−20.6007	−1.36732	−0.683658	+	0.729803i	$0.739612\pi$
$228$	−1.57777	−0.104490
$229$	0.903360	0.0596957	0.0298479	−	0.999554i	$-0.490498\pi$
$229$	0.903360	0.0596957	0.0298479	+	0.999554i	$0.490498\pi$
$230$	3.44131	0.226913
$231$	0	0
$232$	−11.5416	−0.757745
$233$	−23.2459	−1.52289	−0.761444	−	0.648231i	$-0.775509\pi$
$233$	−23.2459	−1.52289	−0.761444	+	0.648231i	$0.775509\pi$
$234$	15.7190	1.02759
$235$	−1.60936	−0.104983
$236$	9.98148	0.649739
$237$	9.67228	0.628282
$238$	7.57184	0.490809
$239$	−8.21214	−0.531199	−0.265600	−	0.964083i	$-0.585570\pi$
$239$	−8.21214	−0.531199	−0.265600	+	0.964083i	$0.585570\pi$
$240$	−5.62676	−0.363206
$241$	8.33040	0.536608	0.268304	−	0.963334i	$-0.413537\pi$
$241$	8.33040	0.536608	0.268304	+	0.963334i	$0.413537\pi$
$242$	0	0
$243$	−14.9107	−0.956522
$244$	−8.16557	−0.522747
$245$	−1.00000	−0.0638877
$246$	20.7888	1.32545
$247$	10.5378	0.670507
$248$	−11.4264	−0.725580
$249$	10.8461	0.687346
$250$	1.65369	0.104588
$251$	−9.77541	−0.617018	−0.308509	−	0.951221i	$-0.599830\pi$
$251$	−9.77541	−0.617018	−0.308509	+	0.951221i	$0.599830\pi$
$252$	−1.24686	−0.0785450
$253$	0	0
$254$	19.9946	1.25457
$255$	−5.22630	−0.327284
$256$	15.5443	0.971521
$257$	4.10056	0.255786	0.127893	−	0.991788i	$-0.459179\pi$
$257$	4.10056	0.255786	0.127893	+	0.991788i	$0.459179\pi$
$258$	−15.9186	−0.991047
$259$	−9.34694	−0.580791
$260$	−4.11479	−0.255189
$261$	9.36126	0.579448
$262$	1.72712	0.106702
$263$	−25.6453	−1.58136	−0.790678	−	0.612232i	$-0.790272\pi$
$263$	−25.6453	−1.58136	−0.790678	+	0.612232i	$0.790272\pi$
$264$	0	0
$265$	−6.21331	−0.381680
$266$	−3.11139	−0.190771
$267$	−1.71057	−0.104685
$268$	−4.12098	−0.251729
$269$	13.7027	0.835470	0.417735	−	0.908569i	$-0.362824\pi$
$269$	13.7027	0.835470	0.417735	+	0.908569i	$0.362824\pi$
$270$	8.86613	0.539576
$271$	8.60468	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$271$	8.60468	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$272$	22.5715	1.36860
$273$	−6.39289	−0.386915
$274$	2.37833	0.143680
$275$	0	0
$276$	−1.74507	−0.105041
$277$	−28.8920	−1.73595	−0.867975	−	0.496607i	$-0.834579\pi$
$277$	−28.8920	−1.73595	−0.867975	+	0.496607i	$0.834579\pi$
$278$	−35.8005	−2.14717
$279$	9.26784	0.554851
$280$	−2.09245	−0.125048
$281$	−7.98397	−0.476284	−0.238142	−	0.971230i	$-0.576538\pi$
$281$	−7.98397	−0.476284	−0.238142	+	0.971230i	$0.576538\pi$
$282$	3.03775	0.180895
$283$	7.39498	0.439586	0.219793	−	0.975547i	$-0.429462\pi$
$283$	7.39498	0.439586	0.219793	+	0.975547i	$0.429462\pi$
$284$	0.511772	0.0303681
$285$	2.14757	0.127211
$286$	0	0
$287$	11.0136	0.650114
$288$	−6.73283	−0.396736
$289$	3.96507	0.233239
$290$	−9.12149	−0.535632
$291$	−6.77659	−0.397251
$292$	10.2027	0.597068
$293$	−13.7881	−0.805508	−0.402754	−	0.915308i	$-0.631947\pi$
$293$	−13.7881	−0.805508	−0.402754	+	0.915308i	$0.631947\pi$
$294$	1.88755	0.110084
$295$	−13.5862	−0.791019
$296$	−19.5580	−1.13678
$297$	0	0
$298$	−20.9258	−1.21220
$299$	11.6552	0.674040
$300$	−0.838578	−0.0484153
$301$	−8.43344	−0.486096
$302$	4.11017	0.236514
$303$	3.60491	0.207097
$304$	−9.27498	−0.531957
$305$	11.1145	0.636414
$306$	−12.8506	−0.734619
$307$	5.89759	0.336593	0.168297	−	0.985736i	$-0.446173\pi$
$307$	5.89759	0.336593	0.168297	+	0.985736i	$0.446173\pi$
$308$	0	0
$309$	−1.42320	−0.0809628
$310$	−9.03046	−0.512896
$311$	−26.0053	−1.47462	−0.737311	−	0.675553i	$-0.763905\pi$
$311$	−26.0053	−1.47462	−0.737311	+	0.675553i	$0.763905\pi$
$312$	−13.3768	−0.757311
$313$	−12.1886	−0.688938	−0.344469	−	0.938798i	$-0.611941\pi$
$313$	−12.1886	−0.688938	−0.344469	+	0.938798i	$0.611941\pi$
$314$	23.6102	1.33240
$315$	1.69716	0.0956239
$316$	−6.22557	−0.350216
$317$	13.5894	0.763258	0.381629	−	0.924316i	$-0.375363\pi$
$317$	13.5894	0.763258	0.381629	+	0.924316i	$0.375363\pi$
$318$	11.7280	0.657671
$319$	0	0
$320$	−3.29883	−0.184410
$321$	−11.7206	−0.654181
$322$	−3.44131	−0.191777
$323$	−8.61488	−0.479345
$324$	−0.755394	−0.0419663
$325$	5.60081	0.310677
$326$	16.1360	0.893689
$327$	6.48222	0.358468
$328$	23.0454	1.27247
$329$	1.60936	0.0887268
$330$	0	0
$331$	18.3402	1.00807	0.504034	−	0.863684i	$-0.331849\pi$
$331$	18.3402	1.00807	0.504034	+	0.863684i	$0.331849\pi$
$332$	−6.98113	−0.383139
$333$	15.8632	0.869299
$334$	18.8338	1.03054
$335$	5.60924	0.306465
$336$	5.62676	0.306965
$337$	−25.2973	−1.37803	−0.689015	−	0.724747i	$-0.741957\pi$
$337$	−25.2973	−1.37803	−0.689015	+	0.724747i	$0.741957\pi$
$338$	−30.3767	−1.65228
$339$	−20.5224	−1.11462
$340$	3.36392	0.182434
$341$	0	0
$342$	5.28051	0.285537
$343$	1.00000	0.0539949
$344$	−17.6465	−0.951437
$345$	2.37529	0.127881
$346$	17.1228	0.920530
$347$	−14.4944	−0.778098	−0.389049	−	0.921217i	$-0.627196\pi$
$347$	−14.4944	−0.778098	−0.389049	+	0.921217i	$0.627196\pi$
$348$	4.62547	0.247951
$349$	−2.97717	−0.159364	−0.0796822	−	0.996820i	$-0.525391\pi$
$349$	−2.97717	−0.159364	−0.0796822	+	0.996820i	$0.525391\pi$
$350$	−1.65369	−0.0883932
$351$	30.0284	1.60280
$352$	0	0
$353$	14.0782	0.749304	0.374652	−	0.927165i	$-0.377762\pi$
$353$	14.0782	0.749304	0.374652	+	0.927165i	$0.377762\pi$
$354$	25.6447	1.36300
$355$	−0.696594	−0.0369714
$356$	1.10101	0.0583534
$357$	5.22630	0.276605
$358$	−31.1216	−1.64483
$359$	−15.3313	−0.809154	−0.404577	−	0.914504i	$-0.632581\pi$
$359$	−15.3313	−0.809154	−0.404577	+	0.914504i	$0.632581\pi$
$360$	3.55121	0.187165
$361$	−15.4600	−0.813685
$362$	12.8222	0.673918
$363$	0	0
$364$	4.11479	0.215674
$365$	−13.8873	−0.726896
$366$	−20.9792	−1.09660
$367$	13.7180	0.716072	0.358036	−	0.933708i	$-0.383447\pi$
$367$	13.7180	0.716072	0.358036	+	0.933708i	$0.383447\pi$
$368$	−10.2585	−0.534759
$369$	−18.6918	−0.973058
$370$	−15.4569	−0.803567
$371$	6.21331	0.322579
$372$	4.57931	0.237426
$373$	−12.6631	−0.655670	−0.327835	−	0.944735i	$-0.606319\pi$
$373$	−12.6631	−0.655670	−0.327835	+	0.944735i	$0.606319\pi$
$374$	0	0
$375$	1.14142	0.0589428
$376$	3.36750	0.173665
$377$	−30.8933	−1.59108
$378$	−8.86613	−0.456025
$379$	12.0373	0.618314	0.309157	−	0.951011i	$-0.399953\pi$
$379$	12.0373	0.618314	0.309157	+	0.951011i	$0.399953\pi$
$380$	−1.38229	−0.0709098
$381$	13.8008	0.707038
$382$	−26.3613	−1.34876
$383$	−9.96182	−0.509025	−0.254513	−	0.967069i	$-0.581915\pi$
$383$	−9.96182	−0.509025	−0.254513	+	0.967069i	$0.581915\pi$
$384$	15.2830	0.779910
$385$	0	0
$386$	14.0932	0.717325
$387$	14.3129	0.727564
$388$	4.36176	0.221435
$389$	22.1252	1.12179	0.560895	−	0.827887i	$-0.310457\pi$
$389$	22.1252	1.12179	0.560895	+	0.827887i	$0.310457\pi$
$390$	−10.5718	−0.535326
$391$	−9.52837	−0.481870
$392$	2.09245	0.105684
$393$	1.19211	0.0601339
$394$	−31.5119	−1.58755
$395$	8.47388	0.426367
$396$	0	0
$397$	−26.4968	−1.32984	−0.664918	−	0.746916i	$-0.731533\pi$
$397$	−26.4968	−1.32984	−0.664918	+	0.746916i	$0.731533\pi$
$398$	21.0214	1.05371
$399$	−2.14757	−0.107513
$400$	−4.92960	−0.246480
$401$	−2.98212	−0.148920	−0.0744600	−	0.997224i	$-0.523723\pi$
$401$	−2.98212	−0.148920	−0.0744600	+	0.997224i	$0.523723\pi$
$402$	−10.5877	−0.528068
$403$	−30.5850	−1.52355
$404$	−2.32031	−0.115439
$405$	1.02820	0.0510915
$406$	9.12149	0.452692
$407$	0	0
$408$	10.9358	0.541400
$409$	−1.71895	−0.0849966	−0.0424983	−	0.999097i	$-0.513532\pi$
$409$	−1.71895	−0.0849966	−0.0424983	+	0.999097i	$0.513532\pi$
$410$	18.2131	0.899480
$411$	1.64159	0.0809738
$412$	0.916042	0.0451302
$413$	13.5862	0.668533
$414$	5.84043	0.287042
$415$	9.50230	0.466449
$416$	22.2191	1.08938
$417$	−24.7105	−1.21008
$418$	0	0
$419$	−24.6056	−1.20206	−0.601031	−	0.799226i	$-0.705243\pi$
$419$	−24.6056	−1.20206	−0.601031	+	0.799226i	$0.705243\pi$
$420$	0.838578	0.0409184
$421$	−29.8732	−1.45593	−0.727965	−	0.685615i	$-0.759533\pi$
$421$	−29.8732	−1.45593	−0.727965	+	0.685615i	$0.759533\pi$
$422$	−47.0484	−2.29028
$423$	−2.73133	−0.132802
$424$	13.0010	0.631385
$425$	−4.57876	−0.222103
$426$	1.31486	0.0637052
$427$	−11.1145	−0.537868
$428$	7.54399	0.364652
$429$	0	0
$430$	−13.9463	−0.672549
$431$	15.5900	0.750943	0.375472	−	0.926834i	$-0.377481\pi$
$431$	15.5900	0.750943	0.375472	+	0.926834i	$0.377481\pi$
$432$	−26.4298	−1.27160
$433$	10.6888	0.513671	0.256835	−	0.966455i	$-0.417320\pi$
$433$	10.6888	0.513671	0.256835	+	0.966455i	$0.417320\pi$
$434$	9.03046	0.433476
$435$	−6.29592	−0.301866
$436$	−4.17229	−0.199816
$437$	3.91536	0.187297
$438$	26.2131	1.25251
$439$	−18.1537	−0.866427	−0.433214	−	0.901291i	$-0.642620\pi$
$439$	−18.1537	−0.866427	−0.433214	+	0.901291i	$0.642620\pi$
$440$	0	0
$441$	−1.69716	−0.0808169
$442$	42.4084	2.01716
$443$	−9.93668	−0.472106	−0.236053	−	0.971740i	$-0.575854\pi$
$443$	−9.93668	−0.472106	−0.236053	+	0.971740i	$0.575854\pi$
$444$	7.83814	0.371982
$445$	−1.49863	−0.0710418
$446$	39.6277	1.87642
$447$	−14.4436	−0.683158
$448$	3.29883	0.155855
$449$	−18.0920	−0.853813	−0.426906	−	0.904296i	$-0.640397\pi$
$449$	−18.0920	−0.853813	−0.426906	+	0.904296i	$0.640397\pi$
$450$	2.80656	0.132303
$451$	0	0
$452$	13.2093	0.621312
$453$	2.83696	0.133292
$454$	34.0671	1.59885
$455$	−5.60081	−0.262570
$456$	−4.49367	−0.210436
$457$	15.9901	0.747987	0.373993	−	0.927431i	$-0.377988\pi$
$457$	15.9901	0.747987	0.373993	+	0.927431i	$0.377988\pi$
$458$	−1.49387	−0.0698041
$459$	−24.5487	−1.14584
$460$	−1.52886	−0.0712834
$461$	−27.9612	−1.30228	−0.651140	−	0.758957i	$-0.725709\pi$
$461$	−27.9612	−1.30228	−0.651140	+	0.758957i	$0.725709\pi$
$462$	0	0
$463$	−30.8389	−1.43320	−0.716602	−	0.697482i	$-0.754304\pi$
$463$	−30.8389	−1.43320	−0.716602	+	0.697482i	$0.754304\pi$
$464$	27.1910	1.26231
$465$	−6.23309	−0.289053
$466$	38.4414	1.78076
$467$	−8.41735	−0.389509	−0.194754	−	0.980852i	$-0.562391\pi$
$467$	−8.41735	−0.389509	−0.194754	+	0.980852i	$0.562391\pi$
$468$	−6.98344	−0.322810
$469$	−5.60924	−0.259011
$470$	2.66137	0.122760
$471$	16.2964	0.750900
$472$	28.4284	1.30852
$473$	0	0
$474$	−15.9949	−0.734671
$475$	1.88149	0.0863285
$476$	−3.36392	−0.154185
$477$	−10.5449	−0.482820
$478$	13.5803	0.621149
$479$	−29.6865	−1.35641	−0.678205	−	0.734873i	$-0.737242\pi$
$479$	−29.6865	−1.35641	−0.678205	+	0.734873i	$0.737242\pi$
$480$	4.52817	0.206682
$481$	−52.3505	−2.38698
$482$	−13.7759	−0.627474
$483$	−2.37529	−0.108079
$484$	0	0
$485$	−5.93697	−0.269584
$486$	24.6576	1.11849
$487$	37.8313	1.71430	0.857150	−	0.515067i	$-0.172233\pi$
$487$	37.8313	1.71430	0.857150	+	0.515067i	$0.172233\pi$
$488$	−23.2565	−1.05277
$489$	11.1375	0.503656
$490$	1.65369	0.0747059
$491$	−18.6985	−0.843850	−0.421925	−	0.906631i	$-0.638645\pi$
$491$	−18.6985	−0.843850	−0.421925	+	0.906631i	$0.638645\pi$
$492$	−9.23578	−0.416381
$493$	25.2558	1.13746
$494$	−17.4263	−0.784046
$495$	0	0
$496$	26.9196	1.20873
$497$	0.696594	0.0312465
$498$	−17.9361	−0.803736
$499$	22.8758	1.02406	0.512030	−	0.858967i	$-0.328894\pi$
$499$	22.8758	1.02406	0.512030	+	0.858967i	$0.328894\pi$
$500$	−0.734678	−0.0328558
$501$	12.9997	0.580782
$502$	16.1655	0.721499
$503$	11.6087	0.517607	0.258804	−	0.965930i	$-0.416672\pi$
$503$	11.6087	0.517607	0.258804	+	0.965930i	$0.416672\pi$
$504$	−3.55121	−0.158183
$505$	3.15826	0.140541
$506$	0	0
$507$	−20.9669	−0.931173
$508$	−8.88292	−0.394116
$509$	−13.8562	−0.614163	−0.307082	−	0.951683i	$-0.599353\pi$
$509$	−13.8562	−0.614163	−0.307082	+	0.951683i	$0.599353\pi$
$510$	8.64266	0.382704
$511$	13.8873	0.614339
$512$	1.07349	0.0474422
$513$	10.0875	0.445373
$514$	−6.78105	−0.299099
$515$	−1.24686	−0.0549433
$516$	7.07210	0.311332
$517$	0	0
$518$	15.4569	0.679138
$519$	11.8187	0.518783
$520$	−11.7194	−0.513930
$521$	−25.2028	−1.10415	−0.552077	−	0.833793i	$-0.686165\pi$
$521$	−25.2028	−1.10415	−0.552077	+	0.833793i	$0.686165\pi$
$522$	−15.4806	−0.677567
$523$	28.9229	1.26471	0.632355	−	0.774679i	$-0.282089\pi$
$523$	28.9229	1.26471	0.632355	+	0.774679i	$0.282089\pi$
$524$	−0.767303	−0.0335198
$525$	−1.14142	−0.0498157
$526$	42.4093	1.84913
$527$	25.0038	1.08918
$528$	0	0
$529$	−18.6695	−0.811716
$530$	10.2749	0.446311
$531$	−23.0579	−1.00063
$532$	1.38229	0.0599297
$533$	61.6852	2.67188
$534$	2.82874	0.122412
$535$	−10.2684	−0.443943
$536$	−11.7370	−0.506962
$537$	−21.4810	−0.926974
$538$	−22.6600	−0.976943
$539$	0	0
$540$	−3.93893	−0.169505
$541$	−13.9092	−0.598003	−0.299001	−	0.954253i	$-0.596653\pi$
$541$	−13.9092	−0.598003	−0.299001	+	0.954253i	$0.596653\pi$
$542$	−14.2294	−0.611207
$543$	8.85023	0.379800
$544$	−18.1645	−0.778798
$545$	5.67907	0.243265
$546$	10.5718	0.452433
$547$	−12.4603	−0.532766	−0.266383	−	0.963867i	$-0.585829\pi$
$547$	−12.4603	−0.532766	−0.266383	+	0.963867i	$0.585829\pi$
$548$	−1.05661	−0.0451363
$549$	18.8630	0.805054
$550$	0	0
$551$	−10.3780	−0.442118
$552$	−4.97017	−0.211544
$553$	−8.47388	−0.360346
$554$	47.7783	2.02990
$555$	−10.6688	−0.452866
$556$	15.9050	0.674520
$557$	−37.5067	−1.58921	−0.794605	−	0.607127i	$-0.792322\pi$
$557$	−37.5067	−1.58921	−0.794605	+	0.607127i	$0.792322\pi$
$558$	−15.3261	−0.648806
$559$	−47.2341	−1.99779
$560$	4.92960	0.208314
$561$	0	0
$562$	13.2030	0.556934
$563$	24.5527	1.03477	0.517387	−	0.855752i	$-0.326905\pi$
$563$	24.5527	1.03477	0.517387	+	0.855752i	$0.326905\pi$
$564$	−1.34957	−0.0568272
$565$	−17.9797	−0.756411
$566$	−12.2290	−0.514022
$567$	−1.02820	−0.0431802
$568$	1.45759	0.0611590
$569$	39.6609	1.66267	0.831336	−	0.555771i	$-0.187577\pi$
$569$	39.6609	1.66267	0.831336	+	0.555771i	$0.187577\pi$
$570$	−3.55141	−0.148752
$571$	−31.3014	−1.30992	−0.654961	−	0.755663i	$-0.727315\pi$
$571$	−31.3014	−1.30992	−0.654961	+	0.755663i	$0.727315\pi$
$572$	0	0
$573$	−18.1954	−0.760122
$574$	−18.2131	−0.760199
$575$	2.08099	0.0867834
$576$	−5.59862	−0.233276
$577$	−33.8941	−1.41103	−0.705516	−	0.708694i	$-0.749285\pi$
$577$	−33.8941	−1.41103	−0.705516	+	0.708694i	$0.749285\pi$
$578$	−6.55698	−0.272734
$579$	9.72754	0.404263
$580$	4.05238	0.168266
$581$	−9.50230	−0.394222
$582$	11.2064	0.464518
$583$	0	0
$584$	29.0585	1.20245
$585$	9.50545	0.393002
$586$	22.8011	0.941907
$587$	−32.3331	−1.33453	−0.667264	−	0.744821i	$-0.732535\pi$
$587$	−32.3331	−1.33453	−0.667264	+	0.744821i	$0.732535\pi$
$588$	−0.838578	−0.0345824
$589$	−10.2744	−0.423351
$590$	22.4673	0.924964
$591$	−21.7504	−0.894693
$592$	46.0767	1.89374
$593$	−25.6741	−1.05431	−0.527153	−	0.849770i	$-0.676741\pi$
$593$	−25.6741	−1.05431	−0.527153	+	0.849770i	$0.676741\pi$
$594$	0	0
$595$	4.57876	0.187711
$596$	9.29663	0.380805
$597$	14.5096	0.593837
$598$	−19.2741	−0.788177
$599$	−21.7518	−0.888756	−0.444378	−	0.895839i	$-0.646575\pi$
$599$	−21.7518	−0.888756	−0.444378	+	0.895839i	$0.646575\pi$
$600$	−2.38836	−0.0975046
$601$	15.6850	0.639806	0.319903	−	0.947450i	$-0.396350\pi$
$601$	15.6850	0.639806	0.319903	+	0.947450i	$0.396350\pi$
$602$	13.9463	0.568407
$603$	9.51975	0.387674
$604$	−1.82601	−0.0742994
$605$	0	0
$606$	−5.96139	−0.242165
$607$	−22.9458	−0.931343	−0.465671	−	0.884958i	$-0.654187\pi$
$607$	−22.9458	−0.931343	−0.465671	+	0.884958i	$0.654187\pi$
$608$	7.46409	0.302709
$609$	6.29592	0.255123
$610$	−18.3799	−0.744180
$611$	9.01372	0.364656
$612$	5.70909	0.230776
$613$	−40.0317	−1.61687	−0.808433	−	0.588588i	$-0.799684\pi$
$613$	−40.0317	−1.61687	−0.808433	+	0.588588i	$0.799684\pi$
$614$	−9.75276	−0.393589
$615$	12.5712	0.506920
$616$	0	0
$617$	14.7371	0.593293	0.296647	−	0.954987i	$-0.404132\pi$
$617$	14.7371	0.593293	0.296647	+	0.954987i	$0.404132\pi$
$618$	2.35352	0.0946725
$619$	25.4079	1.02123	0.510616	−	0.859809i	$-0.329417\pi$
$619$	25.4079	1.02123	0.510616	+	0.859809i	$0.329417\pi$
$620$	4.01194	0.161123
$621$	11.1571	0.447719
$622$	43.0045	1.72432
$623$	1.49863	0.0600413
$624$	31.5144	1.26159
$625$	1.00000	0.0400000
$626$	20.1560	0.805597
$627$	0	0
$628$	−10.4892	−0.418566
$629$	42.7974	1.70645
$630$	−2.80656	−0.111816
$631$	−26.1912	−1.04265	−0.521327	−	0.853357i	$-0.674563\pi$
$631$	−26.1912	−1.04265	−0.521327	+	0.853357i	$0.674563\pi$
$632$	−17.7311	−0.705307
$633$	−32.4742	−1.29073
$634$	−22.4726	−0.892503
$635$	12.0909	0.479813
$636$	−5.21034	−0.206603
$637$	5.60081	0.221912
$638$	0	0
$639$	−1.18223	−0.0467683
$640$	13.3895	0.529266
$641$	8.22192	0.324746	0.162373	−	0.986729i	$-0.448085\pi$
$641$	8.22192	0.324746	0.162373	+	0.986729i	$0.448085\pi$
$642$	19.3822	0.764955
$643$	31.8519	1.25612	0.628059	−	0.778166i	$-0.283850\pi$
$643$	31.8519	1.25612	0.628059	+	0.778166i	$0.283850\pi$
$644$	1.52886	0.0602455
$645$	−9.62612	−0.379028
$646$	14.2463	0.560513
$647$	−22.5883	−0.888036	−0.444018	−	0.896018i	$-0.646447\pi$
$647$	−22.5883	−0.888036	−0.444018	+	0.896018i	$0.646447\pi$
$648$	−2.15145	−0.0845168
$649$	0	0
$650$	−9.26199	−0.363285
$651$	6.23309	0.244294
$652$	−7.16867	−0.280747
$653$	−0.763209	−0.0298667	−0.0149333	−	0.999888i	$-0.504754\pi$
$653$	−0.763209	−0.0298667	−0.0149333	+	0.999888i	$0.504754\pi$
$654$	−10.7196	−0.419168
$655$	1.04441	0.0408083
$656$	−54.2928	−2.11978
$657$	−23.5690	−0.919512
$658$	−2.66137	−0.103751
$659$	42.2863	1.64724	0.823619	−	0.567143i	$-0.191951\pi$
$659$	42.2863	1.64724	0.823619	+	0.567143i	$0.191951\pi$
$660$	0	0
$661$	−17.2730	−0.671843	−0.335921	−	0.941890i	$-0.609048\pi$
$661$	−17.2730	−0.671843	−0.335921	+	0.941890i	$0.609048\pi$
$662$	−30.3289	−1.17877
$663$	29.2715	1.13681
$664$	−19.8830	−0.771612
$665$	−1.88149	−0.0729609
$666$	−26.2328	−1.01650
$667$	−11.4784	−0.444447
$668$	−8.36725	−0.323739
$669$	27.3522	1.05750
$670$	−9.27592	−0.358360
$671$	0	0
$672$	−4.52817	−0.174678
$673$	39.3343	1.51623	0.758113	−	0.652124i	$-0.226122\pi$
$673$	39.3343	1.51623	0.758113	+	0.652124i	$0.226122\pi$
$674$	41.8338	1.61138
$675$	5.36144	0.206362
$676$	13.4954	0.519053
$677$	−4.86816	−0.187098	−0.0935492	−	0.995615i	$-0.529821\pi$
$677$	−4.86816	−0.187098	−0.0935492	+	0.995615i	$0.529821\pi$
$678$	33.9376	1.30337
$679$	5.93697	0.227840
$680$	9.58081	0.367407
$681$	23.5141	0.901062
$682$	0	0
$683$	28.9705	1.10853	0.554263	−	0.832341i	$-0.313000\pi$
$683$	28.9705	1.10853	0.554263	+	0.832341i	$0.313000\pi$
$684$	−2.34595	−0.0896998
$685$	1.43820	0.0549507
$686$	−1.65369	−0.0631380
$687$	−1.03112	−0.0393395
$688$	41.5735	1.58498
$689$	34.7996	1.32576
$690$	−3.92798	−0.149536
$691$	−15.4179	−0.586526	−0.293263	−	0.956032i	$-0.594741\pi$
$691$	−15.4179	−0.586526	−0.293263	+	0.956032i	$0.594741\pi$
$692$	−7.60711	−0.289179
$693$	0	0
$694$	23.9691	0.909855
$695$	−21.6489	−0.821189
$696$	13.1739	0.499354
$697$	−50.4288	−1.91013
$698$	4.92331	0.186350
$699$	26.5334	1.00358
$700$	0.734678	0.0277682
$701$	−6.78227	−0.256163	−0.128081	−	0.991764i	$-0.540882\pi$
$701$	−6.78227	−0.256163	−0.128081	+	0.991764i	$0.540882\pi$
$702$	−49.6576	−1.87420
$703$	−17.5861	−0.663274
$704$	0	0
$705$	1.83696	0.0691838
$706$	−23.2808	−0.876186
$707$	−3.15826	−0.118779
$708$	−11.3931	−0.428178
$709$	−21.1656	−0.794891	−0.397446	−	0.917626i	$-0.630103\pi$
$709$	−21.1656	−0.794891	−0.397446	+	0.917626i	$0.630103\pi$
$710$	1.15195	0.0432319
$711$	14.3815	0.539348
$712$	3.13580	0.117519
$713$	−11.3639	−0.425581
$714$	−8.64266	−0.323443
$715$	0	0
$716$	13.8263	0.516712
$717$	9.37352	0.350061
$718$	25.3531	0.946170
$719$	32.6186	1.21647	0.608234	−	0.793758i	$-0.291878\pi$
$719$	32.6186	1.21647	0.608234	+	0.793758i	$0.291878\pi$
$720$	−8.36630	−0.311794
$721$	1.24686	0.0464356
$722$	25.5660	0.951468
$723$	−9.50851	−0.353625
$724$	−5.69646	−0.211707
$725$	−5.51585	−0.204854
$726$	0	0
$727$	22.4245	0.831678	0.415839	−	0.909438i	$-0.363488\pi$
$727$	22.4245	0.831678	0.415839	+	0.909438i	$0.363488\pi$
$728$	11.7194	0.434350
$729$	20.1040	0.744593
$730$	22.9653	0.849983
$731$	38.6147	1.42822
$732$	9.32037	0.344491
$733$	−28.7607	−1.06230	−0.531150	−	0.847278i	$-0.678240\pi$
$733$	−28.7607	−1.06230	−0.531150	+	0.847278i	$0.678240\pi$
$734$	−22.6852	−0.837326
$735$	1.14142	0.0421020
$736$	8.25556	0.304304
$737$	0	0
$738$	30.9104	1.13783
$739$	2.12241	0.0780742	0.0390371	−	0.999238i	$-0.487571\pi$
$739$	2.12241	0.0780742	0.0390371	+	0.999238i	$0.487571\pi$
$740$	6.86699	0.252436
$741$	−12.0281	−0.441865
$742$	−10.2749	−0.377202
$743$	38.6935	1.41953	0.709763	−	0.704440i	$-0.248802\pi$
$743$	38.6935	1.41953	0.709763	+	0.704440i	$0.248802\pi$
$744$	13.0424	0.478158
$745$	−12.6540	−0.463607
$746$	20.9408	0.766696
$747$	16.1269	0.590052
$748$	0	0
$749$	10.2684	0.375200
$750$	−1.88755	−0.0689237
$751$	2.01833	0.0736498	0.0368249	−	0.999322i	$-0.488276\pi$
$751$	2.01833	0.0736498	0.0368249	+	0.999322i	$0.488276\pi$
$752$	−7.93350	−0.289305
$753$	11.1579	0.406615
$754$	51.0878	1.86051
$755$	2.48546	0.0904551
$756$	3.93893	0.143257
$757$	39.9006	1.45021	0.725107	−	0.688637i	$-0.241790\pi$
$757$	39.9006	1.45021	0.725107	+	0.688637i	$0.241790\pi$
$758$	−19.9059	−0.723015
$759$	0	0
$760$	−3.93691	−0.142807
$761$	−28.9628	−1.04990	−0.524950	−	0.851133i	$-0.675916\pi$
$761$	−28.9628	−1.04990	−0.524950	+	0.851133i	$0.675916\pi$
$762$	−22.8222	−0.826762
$763$	−5.67907	−0.205596
$764$	11.7115	0.423706
$765$	−7.77087	−0.280956
$766$	16.4737	0.595220
$767$	76.0937	2.74759
$768$	−17.7427	−0.640233
$769$	−5.26499	−0.189860	−0.0949302	−	0.995484i	$-0.530263\pi$
$769$	−5.26499	−0.189860	−0.0949302	+	0.995484i	$0.530263\pi$
$770$	0	0
$771$	−4.68047	−0.168563
$772$	−6.26114	−0.225344
$773$	−0.286828	−0.0103165	−0.00515825	−	0.999987i	$-0.501642\pi$
$773$	−0.286828	−0.0103165	−0.00515825	+	0.999987i	$0.501642\pi$
$774$	−23.6690	−0.850764
$775$	−5.46081	−0.196158
$776$	12.4228	0.445952
$777$	10.6688	0.382741
$778$	−36.5881	−1.31175
$779$	20.7220	0.742442
$780$	4.69672	0.168169
$781$	0	0
$782$	15.7569	0.563467
$783$	−29.5729	−1.05685
$784$	−4.92960	−0.176057
$785$	14.2773	0.509579
$786$	−1.97137	−0.0703166
$787$	−27.4622	−0.978923	−0.489462	−	0.872025i	$-0.662807\pi$
$787$	−27.4622	−0.978923	−0.489462	+	0.872025i	$0.662807\pi$
$788$	13.9997	0.498718
$789$	29.2721	1.04211
$790$	−14.0131	−0.498565
$791$	17.9797	0.639284
$792$	0	0
$793$	−62.2502	−2.21057
$794$	43.8174	1.55502
$795$	7.09201	0.251528
$796$	−9.33911	−0.331016
$797$	22.6304	0.801609	0.400805	−	0.916164i	$-0.368731\pi$
$797$	22.6304	0.801609	0.400805	+	0.916164i	$0.368731\pi$
$798$	3.55141	0.125718
$799$	−7.36887	−0.260692
$800$	3.96713	0.140259
$801$	−2.54341	−0.0898668
$802$	4.93149	0.174137
$803$	0	0
$804$	4.70378	0.165890
$805$	−2.08099	−0.0733453
$806$	50.5779	1.78153
$807$	−15.6406	−0.550575
$808$	−6.60849	−0.232486
$809$	37.2584	1.30994	0.654968	−	0.755656i	$-0.272682\pi$
$809$	37.2584	1.30994	0.654968	+	0.755656i	$0.272682\pi$
$810$	−1.70032	−0.0597430
$811$	0.190250	0.00668059	0.00334030	−	0.999994i	$-0.498937\pi$
$811$	0.190250	0.00668059	0.00334030	+	0.999994i	$0.498937\pi$
$812$	−4.05238	−0.142211
$813$	−9.82158	−0.344458
$814$	0	0
$815$	9.75758	0.341793
$816$	−25.7636	−0.901906
$817$	−15.8674	−0.555130
$818$	2.84260	0.0993893
$819$	−9.50545	−0.332147
$820$	−8.09147	−0.282566
$821$	30.0724	1.04953	0.524767	−	0.851246i	$-0.324152\pi$
$821$	30.0724	1.04953	0.524767	+	0.851246i	$0.324152\pi$
$822$	−2.71468	−0.0946853
$823$	20.6546	0.719975	0.359987	−	0.932957i	$-0.382781\pi$
$823$	20.6546	0.719975	0.359987	+	0.932957i	$0.382781\pi$
$824$	2.60899	0.0908885
$825$	0	0
$826$	−22.4673	−0.781738
$827$	−25.3871	−0.882795	−0.441397	−	0.897312i	$-0.645517\pi$
$827$	−25.3871	−0.882795	−0.441397	+	0.897312i	$0.645517\pi$
$828$	−2.59471	−0.0901724
$829$	6.53298	0.226900	0.113450	−	0.993544i	$-0.463810\pi$
$829$	6.53298	0.226900	0.113450	+	0.993544i	$0.463810\pi$
$830$	−15.7138	−0.545435
$831$	32.9779	1.14399
$832$	18.4761	0.640544
$833$	−4.57876	−0.158645
$834$	40.8634	1.41498
$835$	11.3890	0.394133
$836$	0	0
$837$	−29.2778	−1.01199
$838$	40.6899	1.40561
$839$	−13.3320	−0.460271	−0.230136	−	0.973159i	$-0.573917\pi$
$839$	−13.3320	−0.460271	−0.230136	+	0.973159i	$0.573917\pi$
$840$	2.38836	0.0824064
$841$	1.42465	0.0491257
$842$	49.4008	1.70247
$843$	9.11308	0.313871
$844$	20.9020	0.719478
$845$	−18.3691	−0.631916
$846$	4.51677	0.155290
$847$	0	0
$848$	−30.6291	−1.05181
$849$	−8.44080	−0.289687
$850$	7.57184	0.259712
$851$	−19.4509	−0.666769
$852$	−0.584148	−0.0200126
$853$	−45.3221	−1.55180	−0.775899	−	0.630857i	$-0.782704\pi$
$853$	−45.3221	−1.55180	−0.775899	+	0.630857i	$0.782704\pi$
$854$	18.3799	0.628947
$855$	3.19317	0.109204
$856$	21.4861	0.734381
$857$	−26.0895	−0.891199	−0.445599	−	0.895232i	$-0.647009\pi$
$857$	−26.0895	−0.891199	−0.445599	+	0.895232i	$0.647009\pi$
$858$	0	0
$859$	40.0079	1.36505	0.682527	−	0.730861i	$-0.260881\pi$
$859$	40.0079	1.36505	0.682527	+	0.730861i	$0.260881\pi$
$860$	6.19586	0.211277
$861$	−12.5712	−0.428425
$862$	−25.7809	−0.878103
$863$	58.6608	1.99684	0.998418	−	0.0562309i	$-0.0179083\pi$
$863$	58.6608	1.99684	0.998418	+	0.0562309i	$0.0179083\pi$
$864$	21.2695	0.723603
$865$	10.3544	0.352059
$866$	−17.6759	−0.600652
$867$	−4.52582	−0.153705
$868$	−4.01194	−0.136174
$869$	0	0
$870$	10.4115	0.352982
$871$	−31.4163	−1.06450
$872$	−11.8832	−0.402414
$873$	−10.0760	−0.341020
$874$	−6.47477	−0.219012
$875$	−1.00000	−0.0338062
$876$	−11.6456	−0.393468
$877$	−33.3513	−1.12619	−0.563096	−	0.826392i	$-0.690390\pi$
$877$	−33.3513	−1.12619	−0.563096	+	0.826392i	$0.690390\pi$
$878$	30.0205	1.01314
$879$	15.7380	0.530830
$880$	0	0
$881$	21.4875	0.723933	0.361967	−	0.932191i	$-0.382105\pi$
$881$	21.4875	0.723933	0.361967	+	0.932191i	$0.382105\pi$
$882$	2.80656	0.0945019
$883$	−43.7638	−1.47277	−0.736385	−	0.676563i	$-0.763469\pi$
$883$	−43.7638	−1.47277	−0.736385	+	0.676563i	$0.763469\pi$
$884$	−18.8407	−0.633680
$885$	15.5076	0.521282
$886$	16.4321	0.552049
$887$	4.54756	0.152692	0.0763460	−	0.997081i	$-0.475675\pi$
$887$	4.54756	0.152692	0.0763460	+	0.997081i	$0.475675\pi$
$888$	22.3239	0.749141
$889$	−12.0909	−0.405516
$890$	2.47826	0.0830715
$891$	0	0
$892$	−17.6053	−0.589467
$893$	3.02799	0.101328
$894$	23.8851	0.798839
$895$	−18.8195	−0.629067
$896$	−13.3895	−0.447311
$897$	−13.3036	−0.444193
$898$	29.9184	0.998391
$899$	30.1210	1.00459
$900$	−1.24686	−0.0415621
$901$	−28.4493	−0.947782
$902$	0	0
$903$	9.62612	0.320337
$904$	37.6215	1.25127
$905$	7.75369	0.257741
$906$	−4.69144	−0.155863
$907$	−17.4034	−0.577869	−0.288935	−	0.957349i	$-0.593301\pi$
$907$	−17.4034	−0.577869	−0.288935	+	0.957349i	$0.593301\pi$
$908$	−15.1349	−0.502268
$909$	5.36006	0.177782
$910$	9.26199	0.307032
$911$	4.17760	0.138410	0.0692050	−	0.997602i	$-0.477954\pi$
$911$	4.17760	0.138410	0.0692050	+	0.997602i	$0.477954\pi$
$912$	10.5867	0.350560
$913$	0	0
$914$	−26.4427	−0.874645
$915$	−12.6863	−0.419397
$916$	0.663678	0.0219286
$917$	−1.04441	−0.0344894
$918$	40.5959	1.33987
$919$	−18.4017	−0.607018	−0.303509	−	0.952829i	$-0.598158\pi$
$919$	−18.4017	−0.607018	−0.303509	+	0.952829i	$0.598158\pi$
$920$	−4.35436	−0.143559
$921$	−6.73164	−0.221815
$922$	46.2390	1.52280
$923$	3.90149	0.128419
$924$	0	0
$925$	−9.34694	−0.307326
$926$	50.9978	1.67589
$927$	−2.11612	−0.0695025
$928$	−21.8821	−0.718315
$929$	−52.1959	−1.71249	−0.856245	−	0.516569i	$-0.827209\pi$
$929$	−52.1959	−1.71249	−0.856245	+	0.516569i	$0.827209\pi$
$930$	10.3076	0.337999
$931$	1.88149	0.0616632
$932$	−17.0782	−0.559416
$933$	29.6830	0.971777
$934$	13.9197	0.455465
$935$	0	0
$936$	−19.8896	−0.650113
$937$	12.3471	0.403363	0.201681	−	0.979451i	$-0.435359\pi$
$937$	12.3471	0.403363	0.201681	+	0.979451i	$0.435359\pi$
$938$	9.27592	0.302869
$939$	13.9123	0.454010
$940$	−1.18236	−0.0385643
$941$	−13.8509	−0.451525	−0.225763	−	0.974182i	$-0.572487\pi$
$941$	−13.8509	−0.451525	−0.225763	+	0.974182i	$0.572487\pi$
$942$	−26.9492	−0.878052
$943$	22.9193	0.746354
$944$	−66.9746	−2.17984
$945$	−5.36144	−0.174408
$946$	0	0
$947$	−21.7800	−0.707756	−0.353878	−	0.935292i	$-0.615137\pi$
$947$	−21.7800	−0.707756	−0.353878	+	0.935292i	$0.615137\pi$
$948$	7.10601	0.230792
$949$	77.7803	2.52486
$950$	−3.11139	−0.100947
$951$	−15.5113	−0.502988
$952$	−9.58081	−0.310516
$953$	−7.56150	−0.244941	−0.122471	−	0.992472i	$-0.539082\pi$
$953$	−7.56150	−0.244941	−0.122471	+	0.992472i	$0.539082\pi$
$954$	17.4380	0.564577
$955$	−15.9410	−0.515837
$956$	−6.03328	−0.195130
$957$	0	0
$958$	49.0921	1.58609
$959$	−1.43820	−0.0464419
$960$	3.76535	0.121526
$961$	−1.17956	−0.0380503
$962$	86.5713	2.79117
$963$	−17.4271	−0.561581
$964$	6.12016	0.197117
$965$	8.52230	0.274343
$966$	3.92798	0.126381
$967$	24.0488	0.773357	0.386679	−	0.922215i	$-0.373622\pi$
$967$	24.0488	0.773357	0.386679	+	0.922215i	$0.373622\pi$
$968$	0	0
$969$	9.83321	0.315888
$970$	9.81789	0.315233
$971$	16.0049	0.513622	0.256811	−	0.966462i	$-0.417328\pi$
$971$	16.0049	0.513622	0.256811	+	0.966462i	$0.417328\pi$
$972$	−10.9546	−0.351368
$973$	21.6489	0.694031
$974$	−62.5611	−2.00459
$975$	−6.39289	−0.204736
$976$	54.7901	1.75379
$977$	16.4515	0.526330	0.263165	−	0.964751i	$-0.415234\pi$
$977$	16.4515	0.526330	0.263165	+	0.964751i	$0.415234\pi$
$978$	−18.4180	−0.588941
$979$	0	0
$980$	−0.734678	−0.0234684
$981$	9.63827	0.307726
$982$	30.9214	0.986741
$983$	30.2550	0.964984	0.482492	−	0.875900i	$-0.339732\pi$
$983$	30.2550	0.964984	0.482492	+	0.875900i	$0.339732\pi$
$984$	−26.3045	−0.838558
$985$	−19.0556	−0.607160
$986$	−41.7651	−1.33007
$987$	−1.83696	−0.0584710
$988$	7.74192	0.246303
$989$	−17.5499	−0.558055
$990$	0	0
$991$	−13.0778	−0.415428	−0.207714	−	0.978190i	$-0.566602\pi$
$991$	−13.0778	−0.415428	−0.207714	+	0.978190i	$0.566602\pi$
$992$	−21.6637	−0.687824
$993$	−20.9339	−0.664317
$994$	−1.15195	−0.0365376
$995$	12.7118	0.402992
$996$	7.96841	0.252489
$997$	−31.8380	−1.00832	−0.504160	−	0.863610i	$-0.668198\pi$
$997$	−31.8380	−1.00832	−0.504160	+	0.863610i	$0.668198\pi$
$998$	−37.8294	−1.19747
$999$	−50.1131	−1.58551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4235.2.a.y.1.2	✓	5
11.10	odd	2		4235.2.a.be.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4235.2.a.y.1.2	✓	5	1.1	even	1	trivial
4235.2.a.be.1.4	yes	5	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4235.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +O(q^{10})\)	\(q-1.65369 q^{2} -1.14142 q^{3} +0.734678 q^{4} -1.00000 q^{5} +1.88755 q^{6} +1.00000 q^{7} +2.09245 q^{8} -1.69716 q^{9} +1.65369 q^{10} -0.838578 q^{12} +5.60081 q^{13} -1.65369 q^{14} +1.14142 q^{15} -4.92960 q^{16} -4.57876 q^{17} +2.80656 q^{18} +1.88149 q^{19} -0.734678 q^{20} -1.14142 q^{21} +2.08099 q^{23} -2.38836 q^{24} +1.00000 q^{25} -9.26199 q^{26} +5.36144 q^{27} +0.734678 q^{28} -5.51585 q^{29} -1.88755 q^{30} -5.46081 q^{31} +3.96713 q^{32} +7.57184 q^{34} -1.00000 q^{35} -1.24686 q^{36} -9.34694 q^{37} -3.11139 q^{38} -6.39289 q^{39} -2.09245 q^{40} +11.0136 q^{41} +1.88755 q^{42} -8.43344 q^{43} +1.69716 q^{45} -3.44131 q^{46} +1.60936 q^{47} +5.62676 q^{48} +1.00000 q^{49} -1.65369 q^{50} +5.22630 q^{51} +4.11479 q^{52} +6.21331 q^{53} -8.86613 q^{54} +2.09245 q^{56} -2.14757 q^{57} +9.12149 q^{58} +13.5862 q^{59} +0.838578 q^{60} -11.1145 q^{61} +9.03046 q^{62} -1.69716 q^{63} +3.29883 q^{64} -5.60081 q^{65} -5.60924 q^{67} -3.36392 q^{68} -2.37529 q^{69} +1.65369 q^{70} +0.696594 q^{71} -3.55121 q^{72} +13.8873 q^{73} +15.4569 q^{74} -1.14142 q^{75} +1.38229 q^{76} +10.5718 q^{78} -8.47388 q^{79} +4.92960 q^{80} -1.02820 q^{81} -18.2131 q^{82} -9.50230 q^{83} -0.838578 q^{84} +4.57876 q^{85} +13.9463 q^{86} +6.29592 q^{87} +1.49863 q^{89} -2.80656 q^{90} +5.60081 q^{91} +1.52886 q^{92} +6.23309 q^{93} -2.66137 q^{94} -1.88149 q^{95} -4.52817 q^{96} +5.93697 q^{97} -1.65369 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 5 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 5 * q^5 - 5 * q^6 + 5 * q^7 + 6 * q^8 + 3 * q^9	\( 5 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 5 q^{5} - 5 q^{6} + 5 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{10} + 11 q^{12} - 12 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} - 14 q^{17} - 7 q^{18} - 9 q^{19} - 4 q^{20} - 2 q^{21} + 17 q^{23} - 6 q^{24} + 5 q^{25} - 11 q^{26} - 11 q^{27} + 4 q^{28} - 3 q^{29} + 5 q^{30} + 2 q^{31} + 5 q^{32} + 16 q^{34} - 5 q^{35} - 15 q^{36} + 4 q^{37} - 11 q^{38} - 2 q^{39} - 6 q^{40} + 15 q^{41} - 5 q^{42} - 4 q^{43} - 3 q^{45} + 10 q^{46} - 2 q^{47} - 10 q^{48} + 5 q^{49} - 2 q^{50} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{58} - 6 q^{59} - 11 q^{60} - 20 q^{61} - 21 q^{62} + 3 q^{63} - 26 q^{64} + 12 q^{65} + 3 q^{67} - 5 q^{68} + 2 q^{70} - 6 q^{71} - 34 q^{72} - 11 q^{73} + 15 q^{74} - 2 q^{75} - 47 q^{76} + 31 q^{78} - 19 q^{79} - 2 q^{80} + 33 q^{81} - 8 q^{83} + 11 q^{84} + 14 q^{85} + 27 q^{86} + 30 q^{87} + q^{89} + 7 q^{90} - 12 q^{91} + 44 q^{92} + 3 q^{93} - 28 q^{94} + 9 q^{95} + 4 q^{96} - 7 q^{97} - 2 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 5 * q^5 - 5 * q^6 + 5 * q^7 + 6 * q^8 + 3 * q^9 + 2 * q^10 + 11 * q^12 - 12 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 - 14 * q^17 - 7 * q^18 - 9 * q^19 - 4 * q^20 - 2 * q^21 + 17 * q^23 - 6 * q^24 + 5 * q^25 - 11 * q^26 - 11 * q^27 + 4 * q^28 - 3 * q^29 + 5 * q^30 + 2 * q^31 + 5 * q^32 + 16 * q^34 - 5 * q^35 - 15 * q^36 + 4 * q^37 - 11 * q^38 - 2 * q^39 - 6 * q^40 + 15 * q^41 - 5 * q^42 - 4 * q^43 - 3 * q^45 + 10 * q^46 - 2 * q^47 - 10 * q^48 + 5 * q^49 - 2 * q^50 + 18 * q^51 - 4 * q^52 + 6 * q^53 - 4 * q^54 + 6 * q^56 - 32 * q^58 - 6 * q^59 - 11 * q^60 - 20 * q^61 - 21 * q^62 + 3 * q^63 - 26 * q^64 + 12 * q^65 + 3 * q^67 - 5 * q^68 + 2 * q^70 - 6 * q^71 - 34 * q^72 - 11 * q^73 + 15 * q^74 - 2 * q^75 - 47 * q^76 + 31 * q^78 - 19 * q^79 - 2 * q^80 + 33 * q^81 - 8 * q^83 + 11 * q^84 + 14 * q^85 + 27 * q^86 + 30 * q^87 + q^89 + 7 * q^90 - 12 * q^91 + 44 * q^92 + 3 * q^93 - 28 * q^94 + 9 * q^95 + 4 * q^96 - 7 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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