LMFDB - Embedded newform 215.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [215,2,Mod(1,215)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(215, 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("215.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 215 = 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	215.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:	$1.71678364346$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1933097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 5x - 4
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			1.5
Root			$2.50989$ of defining polynomial
Character	$\chi$	$=$	215.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.50989 q^{2} -3.26173 q^{3} +4.29955 q^{4} +1.00000 q^{5} -8.18658 q^{6} +3.13519 q^{7} +5.77162 q^{8} +7.63887 q^{9} +O(q^{10})$	\(q+2.50989 q^{2} -3.26173 q^{3} +4.29955 q^{4} +1.00000 q^{5} -8.18658 q^{6} +3.13519 q^{7} +5.77162 q^{8} +7.63887 q^{9} +2.50989 q^{10} -0.248163 q^{11} -14.0240 q^{12} -2.67669 q^{13} +7.86899 q^{14} -3.26173 q^{15} +5.88703 q^{16} -0.902631 q^{17} +19.1727 q^{18} -6.19474 q^{19} +4.29955 q^{20} -10.2261 q^{21} -0.622862 q^{22} -7.69647 q^{23} -18.8254 q^{24} +1.00000 q^{25} -6.71819 q^{26} -15.1307 q^{27} +13.4799 q^{28} +2.32872 q^{29} -8.18658 q^{30} +7.73623 q^{31} +3.23256 q^{32} +0.809440 q^{33} -2.26550 q^{34} +3.13519 q^{35} +32.8437 q^{36} +3.67425 q^{37} -15.5481 q^{38} +8.73063 q^{39} +5.77162 q^{40} +9.45452 q^{41} -25.6665 q^{42} -1.00000 q^{43} -1.06699 q^{44} +7.63887 q^{45} -19.3173 q^{46} -9.61888 q^{47} -19.2019 q^{48} +2.82943 q^{49} +2.50989 q^{50} +2.94414 q^{51} -11.5086 q^{52} -7.09737 q^{53} -37.9764 q^{54} -0.248163 q^{55} +18.0951 q^{56} +20.2055 q^{57} +5.84482 q^{58} -3.26173 q^{59} -14.0240 q^{60} -3.35338 q^{61} +19.4171 q^{62} +23.9493 q^{63} -3.66068 q^{64} -2.67669 q^{65} +2.03161 q^{66} -0.613476 q^{67} -3.88091 q^{68} +25.1038 q^{69} +7.86899 q^{70} -8.46512 q^{71} +44.0886 q^{72} -1.33737 q^{73} +9.22196 q^{74} -3.26173 q^{75} -26.6346 q^{76} -0.778039 q^{77} +21.9129 q^{78} +6.65447 q^{79} +5.88703 q^{80} +26.4357 q^{81} +23.7298 q^{82} +10.7162 q^{83} -43.9678 q^{84} -0.902631 q^{85} -2.50989 q^{86} -7.59564 q^{87} -1.43230 q^{88} -1.34850 q^{89} +19.1727 q^{90} -8.39193 q^{91} -33.0913 q^{92} -25.2335 q^{93} -24.1423 q^{94} -6.19474 q^{95} -10.5437 q^{96} +4.49827 q^{97} +7.10155 q^{98} -1.89568 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} - 12 q^{6} + 5 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) $ 5 * q + 2 * q^2 - q^3 + 8 * q^4 + 5 * q^5 - 12 * q^6 + 5 * q^7 + 3 * q^8 + 18 * q^9	\( 5 q + 2 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} - 12 q^{6} + 5 q^{7} + 3 q^{8} + 18 q^{9} + 2 q^{10} - 6 q^{11} + 5 q^{13} + q^{14} - q^{15} + 14 q^{16} - 17 q^{17} - 5 q^{18} - 6 q^{19} + 8 q^{20} + 20 q^{21} - 8 q^{22} + q^{23} - 45 q^{24} + 5 q^{25} + 22 q^{26} - 22 q^{27} + 26 q^{28} + 6 q^{29} - 12 q^{30} + 6 q^{31} - 7 q^{32} - 20 q^{33} + 5 q^{35} + 36 q^{36} + 5 q^{37} - 16 q^{38} - 14 q^{39} + 3 q^{40} + 2 q^{41} - 58 q^{42} - 5 q^{43} - 15 q^{44} + 18 q^{45} - 14 q^{46} - 3 q^{48} + 18 q^{49} + 2 q^{50} - 10 q^{51} - 38 q^{52} - 23 q^{53} - 56 q^{54} - 6 q^{55} - 19 q^{56} + 28 q^{57} + 12 q^{58} - q^{59} + 20 q^{61} - 3 q^{62} + 26 q^{63} - 25 q^{64} + 5 q^{65} + 13 q^{66} + 21 q^{67} - 48 q^{68} + 10 q^{69} + q^{70} + 4 q^{71} + 20 q^{72} + 5 q^{73} + 24 q^{74} - q^{75} + 32 q^{76} - 26 q^{77} + 88 q^{78} + 41 q^{79} + 14 q^{80} + 41 q^{81} + 38 q^{82} - 7 q^{83} - 33 q^{84} - 17 q^{85} - 2 q^{86} - 40 q^{87} + 12 q^{88} + 20 q^{89} - 5 q^{90} - 42 q^{91} - 52 q^{92} - 36 q^{93} - 42 q^{94} - 6 q^{95} + 9 q^{96} + 37 q^{97} - 26 q^{98} + 12 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 - q^3 + 8 * q^4 + 5 * q^5 - 12 * q^6 + 5 * q^7 + 3 * q^8 + 18 * q^9 + 2 * q^10 - 6 * q^11 + 5 * q^13 + q^14 - q^15 + 14 * q^16 - 17 * q^17 - 5 * q^18 - 6 * q^19 + 8 * q^20 + 20 * q^21 - 8 * q^22 + q^23 - 45 * q^24 + 5 * q^25 + 22 * q^26 - 22 * q^27 + 26 * q^28 + 6 * q^29 - 12 * q^30 + 6 * q^31 - 7 * q^32 - 20 * q^33 + 5 * q^35 + 36 * q^36 + 5 * q^37 - 16 * q^38 - 14 * q^39 + 3 * q^40 + 2 * q^41 - 58 * q^42 - 5 * q^43 - 15 * q^44 + 18 * q^45 - 14 * q^46 - 3 * q^48 + 18 * q^49 + 2 * q^50 - 10 * q^51 - 38 * q^52 - 23 * q^53 - 56 * q^54 - 6 * q^55 - 19 * q^56 + 28 * q^57 + 12 * q^58 - q^59 + 20 * q^61 - 3 * q^62 + 26 * q^63 - 25 * q^64 + 5 * q^65 + 13 * q^66 + 21 * q^67 - 48 * q^68 + 10 * q^69 + q^70 + 4 * q^71 + 20 * q^72 + 5 * q^73 + 24 * q^74 - q^75 + 32 * q^76 - 26 * q^77 + 88 * q^78 + 41 * q^79 + 14 * q^80 + 41 * q^81 + 38 * q^82 - 7 * q^83 - 33 * q^84 - 17 * q^85 - 2 * q^86 - 40 * q^87 + 12 * q^88 + 20 * q^89 - 5 * q^90 - 42 * q^91 - 52 * q^92 - 36 * q^93 - 42 * q^94 - 6 * q^95 + 9 * q^96 + 37 * q^97 - 26 * q^98 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.50989	1.77476	0.887380	−	0.461038i	$-0.152523\pi$
$2$	2.50989	1.77476	0.887380	+	0.461038i	$0.152523\pi$
$3$	−3.26173	−1.88316	−0.941580	−	0.336791i	$-0.890659\pi$
$3$	−3.26173	−1.88316	−0.941580	+	0.336791i	$0.890659\pi$
$4$	4.29955	2.14977
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−8.18658	−3.34216
$7$	3.13519	1.18499	0.592495	−	0.805574i	$-0.298143\pi$
$7$	3.13519	1.18499	0.592495	+	0.805574i	$0.298143\pi$
$8$	5.77162	2.04058
$9$	7.63887	2.54629
$10$	2.50989	0.793697
$11$	−0.248163	−0.0748240	−0.0374120	−	0.999300i	$-0.511911\pi$
$11$	−0.248163	−0.0748240	−0.0374120	+	0.999300i	$0.511911\pi$
$12$	−14.0240	−4.04837
$13$	−2.67669	−0.742380	−0.371190	−	0.928557i	$-0.621050\pi$
$13$	−2.67669	−0.742380	−0.371190	+	0.928557i	$0.621050\pi$
$14$	7.86899	2.10308
$15$	−3.26173	−0.842174
$16$	5.88703	1.47176
$17$	−0.902631	−0.218920	−0.109460	−	0.993991i	$-0.534912\pi$
$17$	−0.902631	−0.218920	−0.109460	+	0.993991i	$0.534912\pi$
$18$	19.1727	4.51905
$19$	−6.19474	−1.42117	−0.710585	−	0.703611i	$-0.751570\pi$
$19$	−6.19474	−1.42117	−0.710585	+	0.703611i	$0.751570\pi$
$20$	4.29955	0.961409
$21$	−10.2261	−2.23153
$22$	−0.622862	−0.132795
$23$	−7.69647	−1.60482	−0.802412	−	0.596770i	$-0.796450\pi$
$23$	−7.69647	−1.60482	−0.802412	+	0.596770i	$0.796450\pi$
$24$	−18.8254	−3.84273
$25$	1.00000	0.200000
$26$	−6.71819	−1.31755
$27$	−15.1307	−2.91191
$28$	13.4799	2.54746
$29$	2.32872	0.432432	0.216216	−	0.976346i	$-0.430628\pi$
$29$	2.32872	0.432432	0.216216	+	0.976346i	$0.430628\pi$
$30$	−8.18658	−1.49466
$31$	7.73623	1.38947	0.694734	−	0.719266i	$-0.255522\pi$
$31$	7.73623	1.38947	0.694734	+	0.719266i	$0.255522\pi$
$32$	3.23256	0.571441
$33$	0.809440	0.140905
$34$	−2.26550	−0.388531
$35$	3.13519	0.529944
$36$	32.8437	5.47395
$37$	3.67425	0.604043	0.302021	−	0.953301i	$-0.402339\pi$
$37$	3.67425	0.604043	0.302021	+	0.953301i	$0.402339\pi$
$38$	−15.5481	−2.52224
$39$	8.73063	1.39802
$40$	5.77162	0.912573
$41$	9.45452	1.47655	0.738274	−	0.674501i	$-0.235641\pi$
$41$	9.45452	1.47655	0.738274	+	0.674501i	$0.235641\pi$
$42$	−25.6665	−3.96043
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−1.06699	−0.160855
$45$	7.63887	1.13873
$46$	−19.3173	−2.84818
$47$	−9.61888	−1.40306	−0.701529	−	0.712641i	$-0.747499\pi$
$47$	−9.61888	−1.40306	−0.701529	+	0.712641i	$0.747499\pi$
$48$	−19.2019	−2.77155
$49$	2.82943	0.404204
$50$	2.50989	0.354952
$51$	2.94414	0.412262
$52$	−11.5086	−1.59595
$53$	−7.09737	−0.974899	−0.487449	−	0.873151i	$-0.662073\pi$
$53$	−7.09737	−0.974899	−0.487449	+	0.873151i	$0.662073\pi$
$54$	−37.9764	−5.16794
$55$	−0.248163	−0.0334623
$56$	18.0951	2.41806
$57$	20.2055	2.67629
$58$	5.84482	0.767463
$59$	−3.26173	−0.424641	−0.212320	−	0.977200i	$-0.568102\pi$
$59$	−3.26173	−0.424641	−0.212320	+	0.977200i	$0.568102\pi$
$60$	−14.0240	−1.81049
$61$	−3.35338	−0.429356	−0.214678	−	0.976685i	$-0.568870\pi$
$61$	−3.35338	−0.429356	−0.214678	+	0.976685i	$0.568870\pi$
$62$	19.4171	2.46597
$63$	23.9493	3.01733
$64$	−3.66068	−0.457586
$65$	−2.67669	−0.332002
$66$	2.03161	0.250073
$67$	−0.613476	−0.0749480	−0.0374740	−	0.999298i	$-0.511931\pi$
$67$	−0.613476	−0.0749480	−0.0374740	+	0.999298i	$0.511931\pi$
$68$	−3.88091	−0.470629
$69$	25.1038	3.02214
$70$	7.86899	0.940524
$71$	−8.46512	−1.00463	−0.502313	−	0.864686i	$-0.667517\pi$
$71$	−8.46512	−1.00463	−0.502313	+	0.864686i	$0.667517\pi$
$72$	44.0886	5.19589
$73$	−1.33737	−0.156528	−0.0782638	−	0.996933i	$-0.524938\pi$
$73$	−1.33737	−0.156528	−0.0782638	+	0.996933i	$0.524938\pi$
$74$	9.22196	1.07203
$75$	−3.26173	−0.376632
$76$	−26.6346	−3.05520
$77$	−0.778039	−0.0886657
$78$	21.9129	2.48115
$79$	6.65447	0.748686	0.374343	−	0.927290i	$-0.377868\pi$
$79$	6.65447	0.748686	0.374343	+	0.927290i	$0.377868\pi$
$80$	5.88703	0.658190
$81$	26.4357	2.93730
$82$	23.7298	2.62052
$83$	10.7162	1.17626	0.588131	−	0.808766i	$-0.299864\pi$
$83$	10.7162	1.17626	0.588131	+	0.808766i	$0.299864\pi$
$84$	−43.9678	−4.79728
$85$	−0.902631	−0.0979041
$86$	−2.50989	−0.270648
$87$	−7.59564	−0.814338
$88$	−1.43230	−0.152684
$89$	−1.34850	−0.142940	−0.0714702	−	0.997443i	$-0.522769\pi$
$89$	−1.34850	−0.142940	−0.0714702	+	0.997443i	$0.522769\pi$
$90$	19.1727	2.02098
$91$	−8.39193	−0.879713
$92$	−33.0913	−3.45001
$93$	−25.2335	−2.61659
$94$	−24.1423	−2.49009
$95$	−6.19474	−0.635567
$96$	−10.5437	−1.07612
$97$	4.49827	0.456730	0.228365	−	0.973576i	$-0.426662\pi$
$97$	4.49827	0.456730	0.228365	+	0.973576i	$0.426662\pi$
$98$	7.10155	0.717365
$99$	−1.89568	−0.190523
$100$	4.29955	0.429955
$101$	6.23256	0.620163	0.310081	−	0.950710i	$-0.399644\pi$
$101$	6.23256	0.620163	0.310081	+	0.950710i	$0.399644\pi$
$102$	7.38946	0.731665
$103$	7.29211	0.718513	0.359256	−	0.933239i	$-0.383030\pi$
$103$	7.29211	0.718513	0.359256	+	0.933239i	$0.383030\pi$
$104$	−15.4488	−1.51488
$105$	−10.2261	−0.997969
$106$	−17.8136	−1.73021
$107$	5.69841	0.550886	0.275443	−	0.961317i	$-0.411175\pi$
$107$	5.69841	0.550886	0.275443	+	0.961317i	$0.411175\pi$
$108$	−65.0553	−6.25995
$109$	13.2648	1.27053	0.635267	−	0.772292i	$-0.280890\pi$
$109$	13.2648	1.27053	0.635267	+	0.772292i	$0.280890\pi$
$110$	−0.622862	−0.0593876
$111$	−11.9844	−1.13751
$112$	18.4570	1.74402
$113$	8.71451	0.819792	0.409896	−	0.912132i	$-0.365565\pi$
$113$	8.71451	0.819792	0.409896	+	0.912132i	$0.365565\pi$
$114$	50.7137	4.74977
$115$	−7.69647	−0.717699
$116$	10.0124	0.929631
$117$	−20.4469	−1.89031
$118$	−8.18658	−0.753636
$119$	−2.82992	−0.259418
$120$	−18.8254	−1.71852
$121$	−10.9384	−0.994401
$122$	−8.41660	−0.762004
$123$	−30.8381	−2.78058
$124$	33.2623	2.98704
$125$	1.00000	0.0894427
$126$	60.1101	5.35504
$127$	18.3875	1.63163	0.815815	−	0.578313i	$-0.196289\pi$
$127$	18.3875	1.63163	0.815815	+	0.578313i	$0.196289\pi$
$128$	−15.6530	−1.38355
$129$	3.26173	0.287179
$130$	−6.71819	−0.589224
$131$	6.10277	0.533202	0.266601	−	0.963807i	$-0.414099\pi$
$131$	6.10277	0.533202	0.266601	+	0.963807i	$0.414099\pi$
$132$	3.48023	0.302915
$133$	−19.4217	−1.68407
$134$	−1.53976	−0.133015
$135$	−15.1307	−1.30224
$136$	−5.20964	−0.446723
$137$	−9.57167	−0.817763	−0.408882	−	0.912587i	$-0.634081\pi$
$137$	−9.57167	−0.817763	−0.408882	+	0.912587i	$0.634081\pi$
$138$	63.0077	5.36358
$139$	−7.86277	−0.666911	−0.333456	−	0.942766i	$-0.608215\pi$
$139$	−7.86277	−0.666911	−0.333456	+	0.942766i	$0.608215\pi$
$140$	13.4799	1.13926
$141$	31.3742	2.64218
$142$	−21.2465	−1.78297
$143$	0.664255	0.0555478
$144$	44.9702	3.74752
$145$	2.32872	0.193389
$146$	−3.35666	−0.277799
$147$	−9.22881	−0.761180
$148$	15.7976	1.29856
$149$	−1.09931	−0.0900592	−0.0450296	−	0.998986i	$-0.514338\pi$
$149$	−1.09931	−0.0900592	−0.0450296	+	0.998986i	$0.514338\pi$
$150$	−8.18658	−0.668431
$151$	7.87683	0.641007	0.320504	−	0.947247i	$-0.396148\pi$
$151$	7.87683	0.641007	0.320504	+	0.947247i	$0.396148\pi$
$152$	−35.7537	−2.90000
$153$	−6.89508	−0.557434
$154$	−1.95279	−0.157360
$155$	7.73623	0.621389
$156$	37.5378	3.00543
$157$	3.04822	0.243274	0.121637	−	0.992575i	$-0.461186\pi$
$157$	3.04822	0.243274	0.121637	+	0.992575i	$0.461186\pi$
$158$	16.7020	1.32874
$159$	23.1497	1.83589
$160$	3.23256	0.255556
$161$	−24.1299	−1.90170
$162$	66.3506	5.21300
$163$	7.04394	0.551724	0.275862	−	0.961197i	$-0.411037\pi$
$163$	7.04394	0.551724	0.275862	+	0.961197i	$0.411037\pi$
$164$	40.6502	3.17425
$165$	0.809440	0.0630148
$166$	26.8966	2.08758
$167$	1.89121	0.146346	0.0731730	−	0.997319i	$-0.476687\pi$
$167$	1.89121	0.146346	0.0731730	+	0.997319i	$0.476687\pi$
$168$	−59.0214	−4.55360
$169$	−5.83534	−0.448873
$170$	−2.26550	−0.173756
$171$	−47.3208	−3.61871
$172$	−4.29955	−0.327838
$173$	7.44534	0.566059	0.283029	−	0.959111i	$-0.408661\pi$
$173$	7.44534	0.566059	0.283029	+	0.959111i	$0.408661\pi$
$174$	−19.0642	−1.44525
$175$	3.13519	0.236998
$176$	−1.46094	−0.110123
$177$	10.6389	0.799666
$178$	−3.38458	−0.253685
$179$	−10.1784	−0.760771	−0.380385	−	0.924828i	$-0.624209\pi$
$179$	−10.1784	−0.760771	−0.380385	+	0.924828i	$0.624209\pi$
$180$	32.8437	2.44802
$181$	13.0309	0.968580	0.484290	−	0.874908i	$-0.339078\pi$
$181$	13.0309	0.968580	0.484290	+	0.874908i	$0.339078\pi$
$182$	−21.0628	−1.56128
$183$	10.9378	0.808545
$184$	−44.4211	−3.27477
$185$	3.67425	0.270136
$186$	−63.3333	−4.64382
$187$	0.224000	0.0163805
$188$	−41.3569	−3.01626
$189$	−47.4377	−3.45058
$190$	−15.5481	−1.12798
$191$	−13.0247	−0.942431	−0.471216	−	0.882018i	$-0.656185\pi$
$191$	−13.0247	−0.942431	−0.471216	+	0.882018i	$0.656185\pi$
$192$	11.9402	0.861706
$193$	11.6569	0.839083	0.419541	−	0.907736i	$-0.362191\pi$
$193$	11.6569	0.839083	0.419541	+	0.907736i	$0.362191\pi$
$194$	11.2902	0.810587
$195$	8.73063	0.625213
$196$	12.1653	0.868947
$197$	−15.3026	−1.09026	−0.545132	−	0.838350i	$-0.683521\pi$
$197$	−15.3026	−1.09026	−0.545132	+	0.838350i	$0.683521\pi$
$198$	−4.75796	−0.338133
$199$	0.718193	0.0509113	0.0254557	−	0.999676i	$-0.491896\pi$
$199$	0.718193	0.0509113	0.0254557	+	0.999676i	$0.491896\pi$
$200$	5.77162	0.408115
$201$	2.00099	0.141139
$202$	15.6430	1.10064
$203$	7.30097	0.512428
$204$	12.6585	0.886269
$205$	9.45452	0.660332
$206$	18.3024	1.27519
$207$	−58.7923	−4.08635
$208$	−15.7577	−1.09260
$209$	1.53730	0.106338
$210$	−25.6665	−1.77116
$211$	20.2812	1.39621	0.698107	−	0.715993i	$-0.254026\pi$
$211$	20.2812	1.39621	0.698107	+	0.715993i	$0.254026\pi$
$212$	−30.5155	−2.09581
$213$	27.6109	1.89187
$214$	14.3024	0.977691
$215$	−1.00000	−0.0681994
$216$	−87.3287	−5.94197
$217$	24.2546	1.64651
$218$	33.2931	2.25490
$219$	4.36214	0.294766
$220$	−1.06699	−0.0719364
$221$	2.41606	0.162522
$222$	−30.0795	−2.01881
$223$	22.4003	1.50003	0.750017	−	0.661419i	$-0.230045\pi$
$223$	22.4003	1.50003	0.750017	+	0.661419i	$0.230045\pi$
$224$	10.1347	0.677153
$225$	7.63887	0.509258
$226$	21.8725	1.45493
$227$	−23.0343	−1.52884	−0.764419	−	0.644720i	$-0.776974\pi$
$227$	−23.0343	−1.52884	−0.764419	+	0.644720i	$0.776974\pi$
$228$	86.8748	5.75342
$229$	12.6648	0.836911	0.418456	−	0.908237i	$-0.362572\pi$
$229$	12.6648	0.836911	0.418456	+	0.908237i	$0.362572\pi$
$230$	−19.3173	−1.27374
$231$	2.53775	0.166972
$232$	13.4405	0.882410
$233$	−30.2611	−1.98247	−0.991236	−	0.132105i	$-0.957826\pi$
$233$	−30.2611	−1.98247	−0.991236	+	0.132105i	$0.957826\pi$
$234$	−51.3194	−3.35485
$235$	−9.61888	−0.627467
$236$	−14.0240	−0.912882
$237$	−21.7051	−1.40990
$238$	−7.10279	−0.460406
$239$	18.5033	1.19688	0.598438	−	0.801169i	$-0.295788\pi$
$239$	18.5033	1.19688	0.598438	+	0.801169i	$0.295788\pi$
$240$	−19.2019	−1.23948
$241$	−16.0840	−1.03606	−0.518031	−	0.855362i	$-0.673335\pi$
$241$	−16.0840	−1.03606	−0.518031	+	0.855362i	$0.673335\pi$
$242$	−27.4542	−1.76482
$243$	−40.8338	−2.61949
$244$	−14.4180	−0.923018
$245$	2.82943	0.180765
$246$	−77.4002	−4.93486
$247$	16.5814	1.05505
$248$	44.6506	2.83532
$249$	−34.9535	−2.21509
$250$	2.50989	0.158739
$251$	−12.4459	−0.785576	−0.392788	−	0.919629i	$-0.628489\pi$
$251$	−12.4459	−0.785576	−0.392788	+	0.919629i	$0.628489\pi$
$252$	102.971	6.48658
$253$	1.90998	0.120079
$254$	46.1507	2.89575
$255$	2.94414	0.184369
$256$	−31.9660	−1.99788
$257$	4.68647	0.292334	0.146167	−	0.989260i	$-0.453306\pi$
$257$	4.68647	0.292334	0.146167	+	0.989260i	$0.453306\pi$
$258$	8.18658	0.509674
$259$	11.5195	0.715785
$260$	−11.5086	−0.713730
$261$	17.7888	1.10110
$262$	15.3173	0.946305
$263$	17.3959	1.07268	0.536339	−	0.844003i	$-0.319807\pi$
$263$	17.3959	1.07268	0.536339	+	0.844003i	$0.319807\pi$
$264$	4.67178	0.287528
$265$	−7.09737	−0.435988
$266$	−48.7463	−2.98883
$267$	4.39843	0.269180
$268$	−2.63767	−0.161121
$269$	−20.0133	−1.22023	−0.610115	−	0.792313i	$-0.708877\pi$
$269$	−20.0133	−1.22023	−0.610115	+	0.792313i	$0.708877\pi$
$270$	−37.9764	−2.31117
$271$	14.0575	0.853933	0.426966	−	0.904267i	$-0.359582\pi$
$271$	14.0575	0.853933	0.426966	+	0.904267i	$0.359582\pi$
$272$	−5.31381	−0.322197
$273$	27.3722	1.65664
$274$	−24.0238	−1.45133
$275$	−0.248163	−0.0149648
$276$	107.935	6.49692
$277$	−10.8297	−0.650695	−0.325347	−	0.945595i	$-0.605481\pi$
$277$	−10.8297	−0.650695	−0.325347	+	0.945595i	$0.605481\pi$
$278$	−19.7347	−1.18361
$279$	59.0961	3.53799
$280$	18.0951	1.08139
$281$	−7.76915	−0.463469	−0.231734	−	0.972779i	$-0.574440\pi$
$281$	−7.76915	−0.463469	−0.231734	+	0.972779i	$0.574440\pi$
$282$	78.7457	4.68924
$283$	7.32924	0.435678	0.217839	−	0.975985i	$-0.430099\pi$
$283$	7.32924	0.435678	0.217839	+	0.975985i	$0.430099\pi$
$284$	−36.3962	−2.15972
$285$	20.2055	1.19687
$286$	1.66721	0.0985840
$287$	29.6417	1.74970
$288$	24.6931	1.45505
$289$	−16.1853	−0.952074
$290$	5.84482	0.343220
$291$	−14.6721	−0.860095
$292$	−5.75010	−0.336499
$293$	−2.25307	−0.131626	−0.0658129	−	0.997832i	$-0.520964\pi$
$293$	−2.25307	−0.131626	−0.0658129	+	0.997832i	$0.520964\pi$
$294$	−23.1633	−1.35091
$295$	−3.26173	−0.189905
$296$	21.2064	1.23259
$297$	3.75488	0.217880
$298$	−2.75915	−0.159834
$299$	20.6010	1.19139
$300$	−14.0240	−0.809674
$301$	−3.13519	−0.180709
$302$	19.7700	1.13763
$303$	−20.3289	−1.16787
$304$	−36.4686	−2.09162
$305$	−3.35338	−0.192014
$306$	−17.3059	−0.989312
$307$	17.5142	0.999586	0.499793	−	0.866145i	$-0.333409\pi$
$307$	17.5142	0.999586	0.499793	+	0.866145i	$0.333409\pi$
$308$	−3.34522	−0.190611
$309$	−23.7849	−1.35307
$310$	19.4171	1.10282
$311$	−11.8628	−0.672676	−0.336338	−	0.941741i	$-0.609188\pi$
$311$	−11.8628	−0.672676	−0.336338	+	0.941741i	$0.609188\pi$
$312$	50.3898	2.85276
$313$	−28.1905	−1.59342	−0.796709	−	0.604363i	$-0.793428\pi$
$313$	−28.1905	−1.59342	−0.796709	+	0.604363i	$0.793428\pi$
$314$	7.65069	0.431753
$315$	23.9493	1.34939
$316$	28.6112	1.60951
$317$	−6.60104	−0.370752	−0.185376	−	0.982668i	$-0.559350\pi$
$317$	−6.60104	−0.370752	−0.185376	+	0.982668i	$0.559350\pi$
$318$	58.1032	3.25826
$319$	−0.577901	−0.0323563
$320$	−3.66068	−0.204638
$321$	−18.5867	−1.03741
$322$	−60.5634	−3.37507
$323$	5.59156	0.311123
$324$	113.661	6.31453
$325$	−2.67669	−0.148476
$326$	17.6795	0.979178
$327$	−43.2661	−2.39262
$328$	54.5679	3.01301
$329$	−30.1570	−1.66261
$330$	2.03161	0.111836
$331$	−9.68209	−0.532176	−0.266088	−	0.963949i	$-0.585731\pi$
$331$	−9.68209	−0.532176	−0.266088	+	0.963949i	$0.585731\pi$
$332$	46.0750	2.52870
$333$	28.0671	1.53807
$334$	4.74672	0.259729
$335$	−0.613476	−0.0335178
$336$	−60.2016	−3.28427
$337$	29.2544	1.59359	0.796794	−	0.604251i	$-0.206527\pi$
$337$	29.2544	1.59359	0.796794	+	0.604251i	$0.206527\pi$
$338$	−14.6461	−0.796641
$339$	−28.4244	−1.54380
$340$	−3.88091	−0.210472
$341$	−1.91985	−0.103966
$342$	−118.770	−6.42234
$343$	−13.0756	−0.706013
$344$	−5.77162	−0.311185
$345$	25.1038	1.35154
$346$	18.6870	1.00462
$347$	8.34899	0.448197	0.224099	−	0.974566i	$-0.428056\pi$
$347$	8.34899	0.448197	0.224099	+	0.974566i	$0.428056\pi$
$348$	−32.6578	−1.75064
$349$	−22.2663	−1.19189	−0.595944	−	0.803026i	$-0.703222\pi$
$349$	−22.2663	−1.19189	−0.595944	+	0.803026i	$0.703222\pi$
$350$	7.86899	0.420615
$351$	40.5002	2.16174
$352$	−0.802202	−0.0427575
$353$	18.3351	0.975880	0.487940	−	0.872877i	$-0.337748\pi$
$353$	18.3351	0.975880	0.487940	+	0.872877i	$0.337748\pi$
$354$	26.7024	1.41922
$355$	−8.46512	−0.449282
$356$	−5.79793	−0.307290
$357$	9.23043	0.488526
$358$	−25.5467	−1.35019
$359$	−28.8857	−1.52453	−0.762265	−	0.647265i	$-0.775913\pi$
$359$	−28.8857	−1.52453	−0.762265	+	0.647265i	$0.775913\pi$
$360$	44.0886	2.32367
$361$	19.3748	1.01973
$362$	32.7061	1.71900
$363$	35.6781	1.87262
$364$	−36.0815	−1.89119
$365$	−1.33737	−0.0700013
$366$	27.4527	1.43497
$367$	22.7231	1.18613	0.593067	−	0.805153i	$-0.297917\pi$
$367$	22.7231	1.18613	0.593067	+	0.805153i	$0.297917\pi$
$368$	−45.3093	−2.36191
$369$	72.2218	3.75972
$370$	9.22196	0.479427
$371$	−22.2516	−1.15525
$372$	−108.493	−5.62508
$373$	−30.1880	−1.56308	−0.781539	−	0.623857i	$-0.785565\pi$
$373$	−30.1880	−1.56308	−0.781539	+	0.623857i	$0.785565\pi$
$374$	0.562214	0.0290714
$375$	−3.26173	−0.168435
$376$	−55.5165	−2.86304
$377$	−6.23325	−0.321029
$378$	−119.063	−6.12396
$379$	14.7592	0.758128	0.379064	−	0.925370i	$-0.376246\pi$
$379$	14.7592	0.758128	0.379064	+	0.925370i	$0.376246\pi$
$380$	−26.6346	−1.36633
$381$	−59.9751	−3.07262
$382$	−32.6905	−1.67259
$383$	−14.0955	−0.720248	−0.360124	−	0.932904i	$-0.617266\pi$
$383$	−14.0955	−0.720248	−0.360124	+	0.932904i	$0.617266\pi$
$384$	51.0559	2.60544
$385$	−0.778039	−0.0396525
$386$	29.2576	1.48917
$387$	−7.63887	−0.388305
$388$	19.3405	0.981867
$389$	30.9837	1.57094	0.785468	−	0.618902i	$-0.212422\pi$
$389$	30.9837	1.57094	0.785468	+	0.618902i	$0.212422\pi$
$390$	21.9129	1.10960
$391$	6.94707	0.351328
$392$	16.3304	0.824808
$393$	−19.9056	−1.00410
$394$	−38.4078	−1.93496
$395$	6.65447	0.334823
$396$	−8.15059	−0.409582
$397$	31.0726	1.55949	0.779744	−	0.626099i	$-0.215349\pi$
$397$	31.0726	1.55949	0.779744	+	0.626099i	$0.215349\pi$
$398$	1.80259	0.0903554
$399$	63.3483	3.17138
$400$	5.88703	0.294351
$401$	−4.92956	−0.246170	−0.123085	−	0.992396i	$-0.539279\pi$
$401$	−4.92956	−0.246170	−0.123085	+	0.992396i	$0.539279\pi$
$402$	5.02227	0.250488
$403$	−20.7075	−1.03151
$404$	26.7972	1.33321
$405$	26.4357	1.31360
$406$	18.3246	0.909437
$407$	−0.911813	−0.0451969
$408$	16.9924	0.841251
$409$	−7.61888	−0.376729	−0.188365	−	0.982099i	$-0.560319\pi$
$409$	−7.61888	−0.376729	−0.188365	+	0.982099i	$0.560319\pi$
$410$	23.7298	1.17193
$411$	31.2202	1.53998
$412$	31.3528	1.54464
$413$	−10.2261	−0.503195
$414$	−147.562	−7.25229
$415$	10.7162	0.526040
$416$	−8.65256	−0.424226
$417$	25.6462	1.25590
$418$	3.85847	0.188724
$419$	21.8373	1.06682	0.533410	−	0.845857i	$-0.320910\pi$
$419$	21.8373	1.06682	0.533410	+	0.845857i	$0.320910\pi$
$420$	−43.9678	−2.14541
$421$	−13.5012	−0.658009	−0.329005	−	0.944328i	$-0.606713\pi$
$421$	−13.5012	−0.658009	−0.329005	+	0.944328i	$0.606713\pi$
$422$	50.9036	2.47795
$423$	−73.4773	−3.57259
$424$	−40.9633	−1.98935
$425$	−0.902631	−0.0437840
$426$	69.3004	3.35761
$427$	−10.5135	−0.508783
$428$	24.5006	1.18428
$429$	−2.16662	−0.104605
$430$	−2.50989	−0.121038
$431$	−7.29975	−0.351617	−0.175808	−	0.984424i	$-0.556254\pi$
$431$	−7.29975	−0.351617	−0.175808	+	0.984424i	$0.556254\pi$
$432$	−89.0749	−4.28562
$433$	10.1310	0.486865	0.243433	−	0.969918i	$-0.421726\pi$
$433$	10.1310	0.486865	0.243433	+	0.969918i	$0.421726\pi$
$434$	60.8763	2.92216
$435$	−7.59564	−0.364183
$436$	57.0326	2.73136
$437$	47.6776	2.28073
$438$	10.9485	0.523140
$439$	−17.6650	−0.843103	−0.421552	−	0.906804i	$-0.638514\pi$
$439$	−17.6650	−0.843103	−0.421552	+	0.906804i	$0.638514\pi$
$440$	−1.43230	−0.0682823
$441$	21.6136	1.02922
$442$	6.06405	0.288437
$443$	36.7934	1.74811	0.874054	−	0.485828i	$-0.161482\pi$
$443$	36.7934	1.74811	0.874054	+	0.485828i	$0.161482\pi$
$444$	−51.5275	−2.44539
$445$	−1.34850	−0.0639249
$446$	56.2223	2.66220
$447$	3.58566	0.169596
$448$	−11.4769	−0.542235
$449$	0.788959	0.0372333	0.0186166	−	0.999827i	$-0.494074\pi$
$449$	0.788959	0.0372333	0.0186166	+	0.999827i	$0.494074\pi$
$450$	19.1727	0.903810
$451$	−2.34626	−0.110481
$452$	37.4685	1.76237
$453$	−25.6921	−1.20712
$454$	−57.8135	−2.71332
$455$	−8.39193	−0.393420
$456$	116.619	5.46117
$457$	23.7442	1.11070	0.555352	−	0.831615i	$-0.312583\pi$
$457$	23.7442	1.11070	0.555352	+	0.831615i	$0.312583\pi$
$458$	31.7872	1.48532
$459$	13.6575	0.637475
$460$	−33.0913	−1.54289
$461$	10.5205	0.489988	0.244994	−	0.969525i	$-0.421214\pi$
$461$	10.5205	0.489988	0.244994	+	0.969525i	$0.421214\pi$
$462$	6.36947	0.296335
$463$	−33.7411	−1.56808	−0.784042	−	0.620708i	$-0.786845\pi$
$463$	−33.7411	−1.56808	−0.784042	+	0.620708i	$0.786845\pi$
$464$	13.7092	0.636435
$465$	−25.2335	−1.17017
$466$	−75.9521	−3.51841
$467$	19.2665	0.891548	0.445774	−	0.895146i	$-0.352929\pi$
$467$	19.2665	0.891548	0.445774	+	0.895146i	$0.352929\pi$
$468$	−87.9123	−4.06375
$469$	−1.92336	−0.0888127
$470$	−24.1423	−1.11360
$471$	−9.94245	−0.458124
$472$	−18.8254	−0.866511
$473$	0.248163	0.0114105
$474$	−54.4773	−2.50223
$475$	−6.19474	−0.284234
$476$	−12.1674	−0.557691
$477$	−54.2158	−2.48237
$478$	46.4412	2.12417
$479$	−9.69667	−0.443052	−0.221526	−	0.975154i	$-0.571104\pi$
$479$	−9.69667	−0.443052	−0.221526	+	0.975154i	$0.571104\pi$
$480$	−10.5437	−0.481253
$481$	−9.83482	−0.448429
$482$	−40.3691	−1.83876
$483$	78.7052	3.58121
$484$	−47.0303	−2.13774
$485$	4.49827	0.204256
$486$	−102.488	−4.64897
$487$	−0.683509	−0.0309728	−0.0154864	−	0.999880i	$-0.504930\pi$
$487$	−0.683509	−0.0309728	−0.0154864	+	0.999880i	$0.504930\pi$
$488$	−19.3544	−0.876133
$489$	−22.9754	−1.03898
$490$	7.10155	0.320815
$491$	−29.1038	−1.31344	−0.656718	−	0.754137i	$-0.728056\pi$
$491$	−29.1038	−1.31344	−0.656718	+	0.754137i	$0.728056\pi$
$492$	−132.590	−5.97761
$493$	−2.10197	−0.0946680
$494$	41.6174	1.87246
$495$	−1.89568	−0.0852047
$496$	45.5434	2.04496
$497$	−26.5398	−1.19047
$498$	−87.7294	−3.93125
$499$	30.7637	1.37717	0.688586	−	0.725155i	$-0.258232\pi$
$499$	30.7637	1.37717	0.688586	+	0.725155i	$0.258232\pi$
$500$	4.29955	0.192282
$501$	−6.16860	−0.275593
$502$	−31.2378	−1.39421
$503$	4.02345	0.179397	0.0896983	−	0.995969i	$-0.471410\pi$
$503$	4.02345	0.179397	0.0896983	+	0.995969i	$0.471410\pi$
$504$	138.226	6.15709
$505$	6.23256	0.277345
$506$	4.79384	0.213112
$507$	19.0333	0.845298
$508$	79.0581	3.50764
$509$	12.8168	0.568094	0.284047	−	0.958810i	$-0.408323\pi$
$509$	12.8168	0.568094	0.284047	+	0.958810i	$0.408323\pi$
$510$	7.38946	0.327211
$511$	−4.19292	−0.185484
$512$	−48.9252	−2.16221
$513$	93.7308	4.13832
$514$	11.7625	0.518823
$515$	7.29211	0.321329
$516$	14.0240	0.617370
$517$	2.38705	0.104982
$518$	28.9126	1.27035
$519$	−24.2847	−1.06598
$520$	−15.4488	−0.677476
$521$	−10.8754	−0.476460	−0.238230	−	0.971209i	$-0.576567\pi$
$521$	−10.8754	−0.476460	−0.238230	+	0.971209i	$0.576567\pi$
$522$	44.6478	1.95418
$523$	−35.9454	−1.57178	−0.785892	−	0.618364i	$-0.787796\pi$
$523$	−35.9454	−1.57178	−0.785892	+	0.618364i	$0.787796\pi$
$524$	26.2392	1.14626
$525$	−10.2261	−0.446305
$526$	43.6618	1.90375
$527$	−6.98296	−0.304183
$528$	4.76520	0.207379
$529$	36.2356	1.57546
$530$	−17.8136	−0.773774
$531$	−24.9159	−1.08126
$532$	−83.5045	−3.62038
$533$	−25.3068	−1.09616
$534$	11.0396	0.477729
$535$	5.69841	0.246364
$536$	−3.54075	−0.152937
$537$	33.1992	1.43265
$538$	−50.2311	−2.16562
$539$	−0.702159	−0.0302441
$540$	−65.0553	−2.79953
$541$	26.1499	1.12427	0.562135	−	0.827045i	$-0.309980\pi$
$541$	26.1499	1.12427	0.562135	+	0.827045i	$0.309980\pi$
$542$	35.2828	1.51553
$543$	−42.5033	−1.82399
$544$	−2.91781	−0.125100
$545$	13.2648	0.568200
$546$	68.7012	2.94014
$547$	0.231141	0.00988287	0.00494143	−	0.999988i	$-0.498427\pi$
$547$	0.231141	0.00988287	0.00494143	+	0.999988i	$0.498427\pi$
$548$	−41.1539	−1.75801
$549$	−25.6160	−1.09326
$550$	−0.622862	−0.0265589
$551$	−14.4258	−0.614559
$552$	144.889	6.16690
$553$	20.8630	0.887186
$554$	−27.1814	−1.15483
$555$	−11.9844	−0.508709
$556$	−33.8064	−1.43371
$557$	−3.96726	−0.168098	−0.0840491	−	0.996462i	$-0.526785\pi$
$557$	−3.96726	−0.168098	−0.0840491	+	0.996462i	$0.526785\pi$
$558$	148.325	6.27908
$559$	2.67669	0.113212
$560$	18.4570	0.779949
$561$	−0.730626	−0.0308470
$562$	−19.4997	−0.822546
$563$	12.1492	0.512026	0.256013	−	0.966673i	$-0.417591\pi$
$563$	12.1492	0.512026	0.256013	+	0.966673i	$0.417591\pi$
$564$	134.895	5.68009
$565$	8.71451	0.366622
$566$	18.3956	0.773225
$567$	82.8809	3.48067
$568$	−48.8574	−2.05001
$569$	−42.4665	−1.78029	−0.890143	−	0.455681i	$-0.849396\pi$
$569$	−42.4665	−1.78029	−0.890143	+	0.455681i	$0.849396\pi$
$570$	50.7137	2.12416
$571$	−23.4374	−0.980824	−0.490412	−	0.871491i	$-0.663154\pi$
$571$	−23.4374	−0.980824	−0.490412	+	0.871491i	$0.663154\pi$
$572$	2.85600	0.119415
$573$	42.4829	1.77475
$574$	74.3975	3.10529
$575$	−7.69647	−0.320965
$576$	−27.9635	−1.16514
$577$	−41.0241	−1.70785	−0.853927	−	0.520393i	$-0.825786\pi$
$577$	−41.0241	−1.70785	−0.853927	+	0.520393i	$0.825786\pi$
$578$	−40.6232	−1.68970
$579$	−38.0217	−1.58013
$580$	10.0124	0.415744
$581$	33.5975	1.39386
$582$	−36.8254	−1.52646
$583$	1.76130	0.0729458
$584$	−7.71880	−0.319406
$585$	−20.4469	−0.845374
$586$	−5.65496	−0.233604
$587$	−29.4970	−1.21747	−0.608735	−	0.793374i	$-0.708323\pi$
$587$	−29.4970	−1.21747	−0.608735	+	0.793374i	$0.708323\pi$
$588$	−39.6797	−1.63637
$589$	−47.9239	−1.97467
$590$	−8.18658	−0.337036
$591$	49.9129	2.05314
$592$	21.6304	0.889004
$593$	17.7334	0.728223	0.364111	−	0.931355i	$-0.381373\pi$
$593$	17.7334	0.728223	0.364111	+	0.931355i	$0.381373\pi$
$594$	9.42435	0.386686
$595$	−2.82992	−0.116015
$596$	−4.72655	−0.193607
$597$	−2.34255	−0.0958742
$598$	51.7064	2.11443
$599$	−16.9852	−0.693997	−0.346998	−	0.937866i	$-0.612799\pi$
$599$	−16.9852	−0.693997	−0.346998	+	0.937866i	$0.612799\pi$
$600$	−18.8254	−0.768546
$601$	5.70922	0.232884	0.116442	−	0.993197i	$-0.462851\pi$
$601$	5.70922	0.232884	0.116442	+	0.993197i	$0.462851\pi$
$602$	−7.86899	−0.320716
$603$	−4.68626	−0.190839
$604$	33.8668	1.37802
$605$	−10.9384	−0.444710
$606$	−51.0233	−2.07268
$607$	4.37105	0.177415	0.0887077	−	0.996058i	$-0.471726\pi$
$607$	4.37105	0.177415	0.0887077	+	0.996058i	$0.471726\pi$
$608$	−20.0249	−0.812116
$609$	−23.8138	−0.964983
$610$	−8.41660	−0.340778
$611$	25.7467	1.04160
$612$	−29.6457	−1.19836
$613$	35.0848	1.41706	0.708532	−	0.705679i	$-0.249358\pi$
$613$	35.0848	1.41706	0.708532	+	0.705679i	$0.249358\pi$
$614$	43.9586	1.77403
$615$	−30.8381	−1.24351
$616$	−4.49054	−0.180929
$617$	23.0537	0.928108	0.464054	−	0.885807i	$-0.346394\pi$
$617$	23.0537	0.928108	0.464054	+	0.885807i	$0.346394\pi$
$618$	−59.6974	−2.40138
$619$	−29.3047	−1.17785	−0.588927	−	0.808186i	$-0.700449\pi$
$619$	−29.3047	−1.17785	−0.588927	+	0.808186i	$0.700449\pi$
$620$	33.2623	1.33585
$621$	116.453	4.67310
$622$	−29.7743	−1.19384
$623$	−4.22780	−0.169383
$624$	51.3974	2.05754
$625$	1.00000	0.0400000
$626$	−70.7550	−2.82794
$627$	−5.01427	−0.200251
$628$	13.1060	0.522985
$629$	−3.31649	−0.132237
$630$	60.1101	2.39484
$631$	−31.7883	−1.26547	−0.632737	−	0.774367i	$-0.718069\pi$
$631$	−31.7883	−1.26547	−0.632737	+	0.774367i	$0.718069\pi$
$632$	38.4070	1.52775
$633$	−66.1517	−2.62929
$634$	−16.5679	−0.657995
$635$	18.3875	0.729687
$636$	99.5332	3.94675
$637$	−7.57349	−0.300073
$638$	−1.45047	−0.0574246
$639$	−64.6639	−2.55807
$640$	−15.6530	−0.618741
$641$	42.7217	1.68740	0.843702	−	0.536811i	$-0.180371\pi$
$641$	42.7217	1.68740	0.843702	+	0.536811i	$0.180371\pi$
$642$	−46.6505	−1.84115
$643$	−5.88234	−0.231977	−0.115988	−	0.993251i	$-0.537004\pi$
$643$	−5.88234	−0.231977	−0.115988	+	0.993251i	$0.537004\pi$
$644$	−103.748	−4.08823
$645$	3.26173	0.128430
$646$	14.0342	0.552169
$647$	39.6468	1.55868	0.779338	−	0.626604i	$-0.215555\pi$
$647$	39.6468	1.55868	0.779338	+	0.626604i	$0.215555\pi$
$648$	152.577	5.99377
$649$	0.809440	0.0317733
$650$	−6.71819	−0.263509
$651$	−79.1118	−3.10064
$652$	30.2858	1.18608
$653$	−39.0899	−1.52971	−0.764854	−	0.644204i	$-0.777189\pi$
$653$	−39.0899	−1.52971	−0.764854	+	0.644204i	$0.777189\pi$
$654$	−108.593	−4.24633
$655$	6.10277	0.238455
$656$	55.6590	2.17312
$657$	−10.2160	−0.398564
$658$	−75.6908	−2.95074
$659$	−49.9625	−1.94626	−0.973131	−	0.230254i	$-0.926044\pi$
$659$	−49.9625	−1.94626	−0.973131	+	0.230254i	$0.926044\pi$
$660$	3.48023	0.135468
$661$	−12.3002	−0.478421	−0.239210	−	0.970968i	$-0.576889\pi$
$661$	−12.3002	−0.478421	−0.239210	+	0.970968i	$0.576889\pi$
$662$	−24.3010	−0.944485
$663$	−7.88053	−0.306055
$664$	61.8501	2.40025
$665$	−19.4217	−0.753141
$666$	70.4453	2.72970
$667$	−17.9229	−0.693977
$668$	8.13134	0.314611
$669$	−73.0636	−2.82480
$670$	−1.53976	−0.0594860
$671$	0.832184	0.0321261
$672$	−33.0566	−1.27519
$673$	14.3751	0.554118	0.277059	−	0.960853i	$-0.410640\pi$
$673$	14.3751	0.554118	0.277059	+	0.960853i	$0.410640\pi$
$674$	73.4253	2.82824
$675$	−15.1307	−0.582381
$676$	−25.0893	−0.964975
$677$	−19.8543	−0.763062	−0.381531	−	0.924356i	$-0.624603\pi$
$677$	−19.8543	−0.763062	−0.381531	+	0.924356i	$0.624603\pi$
$678$	−71.3420	−2.73987
$679$	14.1029	0.541221
$680$	−5.20964	−0.199781
$681$	75.1315	2.87905
$682$	−4.81861	−0.184514
$683$	33.8908	1.29680	0.648398	−	0.761302i	$-0.275439\pi$
$683$	33.8908	1.29680	0.648398	+	0.761302i	$0.275439\pi$
$684$	−203.458	−7.77941
$685$	−9.57167	−0.365715
$686$	−32.8182	−1.25300
$687$	−41.3090	−1.57604
$688$	−5.88703	−0.224441
$689$	18.9974	0.723745
$690$	63.0077	2.39866
$691$	14.3619	0.546354	0.273177	−	0.961964i	$-0.411926\pi$
$691$	14.3619	0.546354	0.273177	+	0.961964i	$0.411926\pi$
$692$	32.0116	1.21690
$693$	−5.94333	−0.225769
$694$	20.9551	0.795443
$695$	−7.86277	−0.298252
$696$	−43.8391	−1.66172
$697$	−8.53394	−0.323246
$698$	−55.8860	−2.11532
$699$	98.7035	3.73331
$700$	13.4799	0.509493
$701$	11.8354	0.447015	0.223508	−	0.974702i	$-0.428249\pi$
$701$	11.8354	0.447015	0.223508	+	0.974702i	$0.428249\pi$
$702$	101.651	3.83657
$703$	−22.7610	−0.858448
$704$	0.908446	0.0342384
$705$	31.3742	1.18162
$706$	46.0192	1.73195
$707$	19.5403	0.734887
$708$	45.7423	1.71910
$709$	−22.9382	−0.861462	−0.430731	−	0.902480i	$-0.641744\pi$
$709$	−22.9382	−0.861462	−0.430731	+	0.902480i	$0.641744\pi$
$710$	−21.2465	−0.797368
$711$	50.8326	1.90637
$712$	−7.78301	−0.291681
$713$	−59.5417	−2.22985
$714$	23.1674	0.867017
$715$	0.664255	0.0248417
$716$	−43.7626	−1.63549
$717$	−60.3526	−2.25391
$718$	−72.5000	−2.70567
$719$	40.1157	1.49606	0.748032	−	0.663663i	$-0.230999\pi$
$719$	40.1157	1.49606	0.748032	+	0.663663i	$0.230999\pi$
$720$	44.9702	1.67594
$721$	22.8622	0.851431
$722$	48.6286	1.80977
$723$	52.4616	1.95107
$724$	56.0270	2.08223
$725$	2.32872	0.0864864
$726$	89.5482	3.32345
$727$	10.3290	0.383082	0.191541	−	0.981485i	$-0.438652\pi$
$727$	10.3290	0.383082	0.191541	+	0.981485i	$0.438652\pi$
$728$	−48.4350	−1.79512
$729$	53.8817	1.99562
$730$	−3.35666	−0.124235
$731$	0.902631	0.0333850
$732$	47.0276	1.73819
$733$	−27.0182	−0.997938	−0.498969	−	0.866620i	$-0.666288\pi$
$733$	−27.0182	−0.997938	−0.498969	+	0.866620i	$0.666288\pi$
$734$	57.0324	2.10510
$735$	−9.22881	−0.340410
$736$	−24.8793	−0.917063
$737$	0.152242	0.00560791
$738$	181.269	6.67260
$739$	20.7027	0.761560	0.380780	−	0.924666i	$-0.375655\pi$
$739$	20.7027	0.761560	0.380780	+	0.924666i	$0.375655\pi$
$740$	15.7976	0.580732
$741$	−54.0839	−1.98682
$742$	−55.8491	−2.05029
$743$	40.3655	1.48087	0.740433	−	0.672130i	$-0.234620\pi$
$743$	40.3655	1.48087	0.740433	+	0.672130i	$0.234620\pi$
$744$	−145.638	−5.33935
$745$	−1.09931	−0.0402757
$746$	−75.7686	−2.77409
$747$	81.8600	2.99510
$748$	0.963098	0.0352143
$749$	17.8656	0.652795
$750$	−8.18658	−0.298932
$751$	9.23655	0.337046	0.168523	−	0.985698i	$-0.446100\pi$
$751$	9.23655	0.337046	0.168523	+	0.985698i	$0.446100\pi$
$752$	−56.6266	−2.06496
$753$	40.5950	1.47936
$754$	−15.6448	−0.569749
$755$	7.87683	0.286667
$756$	−203.961	−7.41798
$757$	10.6745	0.387973	0.193986	−	0.981004i	$-0.437858\pi$
$757$	10.6745	0.387973	0.193986	+	0.981004i	$0.437858\pi$
$758$	37.0439	1.34550
$759$	−6.22983	−0.226129
$760$	−35.7537	−1.29692
$761$	−21.3301	−0.773217	−0.386608	−	0.922244i	$-0.626353\pi$
$761$	−21.3301	−0.773217	−0.386608	+	0.922244i	$0.626353\pi$
$762$	−150.531	−5.45316
$763$	41.5876	1.50557
$764$	−56.0002	−2.02601
$765$	−6.89508	−0.249292
$766$	−35.3782	−1.27827
$767$	8.73063	0.315245
$768$	104.265	3.76232
$769$	25.6062	0.923385	0.461692	−	0.887040i	$-0.347242\pi$
$769$	25.6062	0.923385	0.461692	+	0.887040i	$0.347242\pi$
$770$	−1.95279	−0.0703737
$771$	−15.2860	−0.550512
$772$	50.1195	1.80384
$773$	−21.4476	−0.771416	−0.385708	−	0.922621i	$-0.626043\pi$
$773$	−21.4476	−0.771416	−0.385708	+	0.922621i	$0.626043\pi$
$774$	−19.1727	−0.689149
$775$	7.73623	0.277894
$776$	25.9623	0.931992
$777$	−37.5734	−1.34794
$778$	77.7657	2.78803
$779$	−58.5683	−2.09843
$780$	37.5378	1.34407
$781$	2.10073	0.0751700
$782$	17.4364	0.623524
$783$	−35.2351	−1.25920
$784$	16.6569	0.594890
$785$	3.04822	0.108796
$786$	−49.9608	−1.78204
$787$	−43.6900	−1.55738	−0.778691	−	0.627408i	$-0.784116\pi$
$787$	−43.6900	−1.55738	−0.778691	+	0.627408i	$0.784116\pi$
$788$	−65.7943	−2.34382
$789$	−56.7407	−2.02002
$790$	16.7020	0.594230
$791$	27.3217	0.971446
$792$	−10.9412	−0.388777
$793$	8.97594	0.318745
$794$	77.9888	2.76772
$795$	23.1497	0.821035
$796$	3.08791	0.109448
$797$	−46.8097	−1.65809	−0.829043	−	0.559185i	$-0.811114\pi$
$797$	−46.8097	−1.65809	−0.829043	+	0.559185i	$0.811114\pi$
$798$	158.997	5.62844
$799$	8.68230	0.307158
$800$	3.23256	0.114288
$801$	−10.3010	−0.363968
$802$	−12.3726	−0.436893
$803$	0.331886	0.0117120
$804$	8.60336	0.303417
$805$	−24.1299	−0.850467
$806$	−51.9735	−1.83069
$807$	65.2778	2.29789
$808$	35.9720	1.26549
$809$	2.62610	0.0923288	0.0461644	−	0.998934i	$-0.485300\pi$
$809$	2.62610	0.0923288	0.0461644	+	0.998934i	$0.485300\pi$
$810$	66.3506	2.33132
$811$	36.1962	1.27102	0.635510	−	0.772093i	$-0.280790\pi$
$811$	36.1962	1.27102	0.635510	+	0.772093i	$0.280790\pi$
$812$	31.3909	1.10160
$813$	−45.8518	−1.60809
$814$	−2.28855	−0.0802136
$815$	7.04394	0.246739
$816$	17.3322	0.606749
$817$	6.19474	0.216726
$818$	−19.1226	−0.668604
$819$	−64.1048	−2.24000
$820$	40.6502	1.41957
$821$	−31.4660	−1.09817	−0.549085	−	0.835767i	$-0.685024\pi$
$821$	−31.4660	−1.09817	−0.549085	+	0.835767i	$0.685024\pi$
$822$	78.3592	2.73309
$823$	−43.9257	−1.53115	−0.765577	−	0.643344i	$-0.777546\pi$
$823$	−43.9257	−1.53115	−0.765577	+	0.643344i	$0.777546\pi$
$824$	42.0873	1.46618
$825$	0.809440	0.0281811
$826$	−25.6665	−0.893051
$827$	28.7775	1.00069	0.500346	−	0.865825i	$-0.333206\pi$
$827$	28.7775	1.00069	0.500346	+	0.865825i	$0.333206\pi$
$828$	−252.780	−8.78472
$829$	31.2180	1.08425	0.542123	−	0.840299i	$-0.317621\pi$
$829$	31.2180	1.08425	0.542123	+	0.840299i	$0.317621\pi$
$830$	26.8966	0.933595
$831$	35.3236	1.22536
$832$	9.79851	0.339702
$833$	−2.55393	−0.0884883
$834$	64.3692	2.22892
$835$	1.89121	0.0654479
$836$	6.60972	0.228602
$837$	−117.055	−4.04600
$838$	54.8092	1.89335
$839$	−16.8160	−0.580554	−0.290277	−	0.956943i	$-0.593747\pi$
$839$	−16.8160	−0.580554	−0.290277	+	0.956943i	$0.593747\pi$
$840$	−59.0214	−2.03643
$841$	−23.5771	−0.813003
$842$	−33.8866	−1.16781
$843$	25.3408	0.872785
$844$	87.2000	3.00155
$845$	−5.83534	−0.200742
$846$	−184.420	−6.34049
$847$	−34.2940	−1.17836
$848$	−41.7824	−1.43481
$849$	−23.9060	−0.820452
$850$	−2.26550	−0.0777062
$851$	−28.2787	−0.969383
$852$	118.715	4.06709
$853$	−20.6282	−0.706295	−0.353148	−	0.935568i	$-0.614889\pi$
$853$	−20.6282	−0.706295	−0.353148	+	0.935568i	$0.614889\pi$
$854$	−26.3877	−0.902967
$855$	−47.3208	−1.61834
$856$	32.8891	1.12412
$857$	−32.8803	−1.12317	−0.561585	−	0.827419i	$-0.689808\pi$
$857$	−32.8803	−1.12317	−0.561585	+	0.827419i	$0.689808\pi$
$858$	−5.43797	−0.185649
$859$	−43.1433	−1.47203	−0.736016	−	0.676964i	$-0.763295\pi$
$859$	−43.1433	−1.47203	−0.736016	+	0.676964i	$0.763295\pi$
$860$	−4.29955	−0.146613
$861$	−96.6833	−3.29496
$862$	−18.3216	−0.624036
$863$	−45.6650	−1.55445	−0.777227	−	0.629221i	$-0.783374\pi$
$863$	−45.6650	−1.55445	−0.777227	+	0.629221i	$0.783374\pi$
$864$	−48.9109	−1.66398
$865$	7.44534	0.253149
$866$	25.4277	0.864070
$867$	52.7919	1.79291
$868$	104.284	3.53962
$869$	−1.65139	−0.0560197
$870$	−19.0642	−0.646338
$871$	1.64208	0.0556399
$872$	76.5592	2.59262
$873$	34.3617	1.16297
$874$	119.666	4.04775
$875$	3.13519	0.105989
$876$	18.7553	0.633681
$877$	15.9707	0.539294	0.269647	−	0.962959i	$-0.413093\pi$
$877$	15.9707	0.539294	0.269647	+	0.962959i	$0.413093\pi$
$878$	−44.3372	−1.49631
$879$	7.34891	0.247872
$880$	−1.46094	−0.0492484
$881$	13.7670	0.463823	0.231912	−	0.972737i	$-0.425502\pi$
$881$	13.7670	0.463823	0.231912	+	0.972737i	$0.425502\pi$
$882$	54.2478	1.82662
$883$	12.0482	0.405455	0.202728	−	0.979235i	$-0.435019\pi$
$883$	12.0482	0.405455	0.202728	+	0.979235i	$0.435019\pi$
$884$	10.3880	0.349385
$885$	10.6389	0.357622
$886$	92.3475	3.10247
$887$	−14.1405	−0.474791	−0.237395	−	0.971413i	$-0.576294\pi$
$887$	−14.1405	−0.474791	−0.237395	+	0.971413i	$0.576294\pi$
$888$	−69.1694	−2.32117
$889$	57.6484	1.93347
$890$	−3.38458	−0.113451
$891$	−6.56036	−0.219780
$892$	96.3111	3.22474
$893$	59.5864	1.99398
$894$	8.99961	0.300992
$895$	−10.1784	−0.340227
$896$	−49.0753	−1.63949
$897$	−67.1950	−2.24358
$898$	1.98020	0.0660802
$899$	18.0155	0.600850
$900$	32.8437	1.09479
$901$	6.40630	0.213425
$902$	−5.88886	−0.196078
$903$	10.2261	0.340305
$904$	50.2968	1.67285
$905$	13.0309	0.433162
$906$	−64.4843	−2.14235
$907$	2.93772	0.0975455	0.0487727	−	0.998810i	$-0.484469\pi$
$907$	2.93772	0.0975455	0.0487727	+	0.998810i	$0.484469\pi$
$908$	−99.0370	−3.28666
$909$	47.6097	1.57911
$910$	−21.0628	−0.698226
$911$	1.04732	0.0346992	0.0173496	−	0.999849i	$-0.494477\pi$
$911$	1.04732	0.0346992	0.0173496	+	0.999849i	$0.494477\pi$
$912$	118.951	3.93885
$913$	−2.65938	−0.0880125
$914$	59.5952	1.97124
$915$	10.9378	0.361592
$916$	54.4528	1.79917
$917$	19.1334	0.631839
$918$	34.2787	1.13137
$919$	15.7325	0.518966	0.259483	−	0.965748i	$-0.416448\pi$
$919$	15.7325	0.518966	0.259483	+	0.965748i	$0.416448\pi$
$920$	−44.4211	−1.46452
$921$	−57.1264	−1.88238
$922$	26.4053	0.869611
$923$	22.6585	0.745813
$924$	10.9112	0.358952
$925$	3.67425	0.120809
$926$	−84.6866	−2.78297
$927$	55.7034	1.82954
$928$	7.52772	0.247109
$929$	−26.5084	−0.869711	−0.434855	−	0.900500i	$-0.643201\pi$
$929$	−26.5084	−0.869711	−0.434855	+	0.900500i	$0.643201\pi$
$930$	−63.3333	−2.07678
$931$	−17.5275	−0.574442
$932$	−130.109	−4.26187
$933$	38.6931	1.26676
$934$	48.3568	1.58228
$935$	0.224000	0.00732557
$936$	−118.011	−3.85732
$937$	25.7747	0.842021	0.421011	−	0.907056i	$-0.361676\pi$
$937$	25.7747	0.842021	0.421011	+	0.907056i	$0.361676\pi$
$938$	−4.82743	−0.157621
$939$	91.9496	3.00066
$940$	−41.3569	−1.34891
$941$	−33.5036	−1.09219	−0.546093	−	0.837725i	$-0.683885\pi$
$941$	−33.5036	−1.09219	−0.546093	+	0.837725i	$0.683885\pi$
$942$	−24.9545	−0.813060
$943$	−72.7664	−2.36960
$944$	−19.2019	−0.624968
$945$	−47.4377	−1.54315
$946$	0.622862	0.0202510
$947$	−12.5066	−0.406411	−0.203205	−	0.979136i	$-0.565136\pi$
$947$	−12.5066	−0.406411	−0.203205	+	0.979136i	$0.565136\pi$
$948$	−93.3220	−3.03096
$949$	3.57973	0.116203
$950$	−15.5481	−0.504447
$951$	21.5308	0.698184
$952$	−16.3332	−0.529363
$953$	13.3246	0.431628	0.215814	−	0.976435i	$-0.430760\pi$
$953$	13.3246	0.431628	0.215814	+	0.976435i	$0.430760\pi$
$954$	−136.076	−4.40562
$955$	−13.0247	−0.421468
$956$	79.5557	2.57302
$957$	1.88496	0.0609320
$958$	−24.3376	−0.786312
$959$	−30.0090	−0.969042
$960$	11.9402	0.385367
$961$	28.8493	0.930623
$962$	−24.6843	−0.795854
$963$	43.5294	1.40272
$964$	−69.1540	−2.22730
$965$	11.6569	0.375249
$966$	197.541	6.35579
$967$	−35.1437	−1.13014	−0.565072	−	0.825042i	$-0.691152\pi$
$967$	−35.1437	−1.13014	−0.565072	+	0.825042i	$0.691152\pi$
$968$	−63.1324	−2.02915
$969$	−18.2382	−0.585894
$970$	11.2902	0.362505
$971$	54.0062	1.73314	0.866571	−	0.499054i	$-0.166319\pi$
$971$	54.0062	1.73314	0.866571	+	0.499054i	$0.166319\pi$
$972$	−175.567	−5.63131
$973$	−24.6513	−0.790284
$974$	−1.71553	−0.0549692
$975$	8.73063	0.279604
$976$	−19.7414	−0.631907
$977$	0.447973	0.0143319	0.00716596	−	0.999974i	$-0.497719\pi$
$977$	0.447973	0.0143319	0.00716596	+	0.999974i	$0.497719\pi$
$978$	−57.6658	−1.84395
$979$	0.334647	0.0106954
$980$	12.1653	0.388605
$981$	101.328	3.23515
$982$	−73.0473	−2.33103
$983$	43.5886	1.39026	0.695130	−	0.718884i	$-0.255347\pi$
$983$	43.5886	1.39026	0.695130	+	0.718884i	$0.255347\pi$
$984$	−177.986	−5.67397
$985$	−15.3026	−0.487581
$986$	−5.27572	−0.168013
$987$	98.3640	3.13096
$988$	71.2925	2.26812
$989$	7.69647	0.244733
$990$	−4.75796	−0.151218
$991$	−47.3742	−1.50489	−0.752445	−	0.658655i	$-0.771126\pi$
$991$	−47.3742	−1.50489	−0.752445	+	0.658655i	$0.771126\pi$
$992$	25.0078	0.794000
$993$	31.5803	1.00217
$994$	−66.6119	−2.11280
$995$	0.718193	0.0227682
$996$	−150.284	−4.76194
$997$	−9.73338	−0.308259	−0.154130	−	0.988051i	$-0.549257\pi$
$997$	−9.73338	−0.308259	−0.154130	+	0.988051i	$0.549257\pi$
$998$	77.2135	2.44415
$999$	−55.5940	−1.75892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	215.2.a.c.1.5	✓	5
3.2	odd	2		1935.2.a.u.1.1		5
4.3	odd	2		3440.2.a.w.1.5		5
5.2	odd	4		1075.2.b.h.474.10		10
5.3	odd	4		1075.2.b.h.474.1		10
5.4	even	2		1075.2.a.m.1.1		5
15.14	odd	2		9675.2.a.ch.1.5		5
43.42	odd	2		9245.2.a.l.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.c.1.5	✓	5	1.1	even	1	trivial
1075.2.a.m.1.1		5	5.4	even	2
1075.2.b.h.474.1		10	5.3	odd	4
1075.2.b.h.474.10		10	5.2	odd	4
1935.2.a.u.1.1		5	3.2	odd	2
3440.2.a.w.1.5		5	4.3	odd	2
9245.2.a.l.1.1		5	43.42	odd	2
9675.2.a.ch.1.5		5	15.14	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:	\( N \)	\(=\)	\( 215 = 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	215.a (trivial)

\(f(q)\)	\(=\)	\(q+2.50989 q^{2} -3.26173 q^{3} +4.29955 q^{4} +1.00000 q^{5} -8.18658 q^{6} +3.13519 q^{7} +5.77162 q^{8} +7.63887 q^{9} +O(q^{10})\)	\(q+2.50989 q^{2} -3.26173 q^{3} +4.29955 q^{4} +1.00000 q^{5} -8.18658 q^{6} +3.13519 q^{7} +5.77162 q^{8} +7.63887 q^{9} +2.50989 q^{10} -0.248163 q^{11} -14.0240 q^{12} -2.67669 q^{13} +7.86899 q^{14} -3.26173 q^{15} +5.88703 q^{16} -0.902631 q^{17} +19.1727 q^{18} -6.19474 q^{19} +4.29955 q^{20} -10.2261 q^{21} -0.622862 q^{22} -7.69647 q^{23} -18.8254 q^{24} +1.00000 q^{25} -6.71819 q^{26} -15.1307 q^{27} +13.4799 q^{28} +2.32872 q^{29} -8.18658 q^{30} +7.73623 q^{31} +3.23256 q^{32} +0.809440 q^{33} -2.26550 q^{34} +3.13519 q^{35} +32.8437 q^{36} +3.67425 q^{37} -15.5481 q^{38} +8.73063 q^{39} +5.77162 q^{40} +9.45452 q^{41} -25.6665 q^{42} -1.00000 q^{43} -1.06699 q^{44} +7.63887 q^{45} -19.3173 q^{46} -9.61888 q^{47} -19.2019 q^{48} +2.82943 q^{49} +2.50989 q^{50} +2.94414 q^{51} -11.5086 q^{52} -7.09737 q^{53} -37.9764 q^{54} -0.248163 q^{55} +18.0951 q^{56} +20.2055 q^{57} +5.84482 q^{58} -3.26173 q^{59} -14.0240 q^{60} -3.35338 q^{61} +19.4171 q^{62} +23.9493 q^{63} -3.66068 q^{64} -2.67669 q^{65} +2.03161 q^{66} -0.613476 q^{67} -3.88091 q^{68} +25.1038 q^{69} +7.86899 q^{70} -8.46512 q^{71} +44.0886 q^{72} -1.33737 q^{73} +9.22196 q^{74} -3.26173 q^{75} -26.6346 q^{76} -0.778039 q^{77} +21.9129 q^{78} +6.65447 q^{79} +5.88703 q^{80} +26.4357 q^{81} +23.7298 q^{82} +10.7162 q^{83} -43.9678 q^{84} -0.902631 q^{85} -2.50989 q^{86} -7.59564 q^{87} -1.43230 q^{88} -1.34850 q^{89} +19.1727 q^{90} -8.39193 q^{91} -33.0913 q^{92} -25.2335 q^{93} -24.1423 q^{94} -6.19474 q^{95} -10.5437 q^{96} +4.49827 q^{97} +7.10155 q^{98} -1.89568 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} - 12 q^{6} + 5 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) 5 * q + 2 * q^2 - q^3 + 8 * q^4 + 5 * q^5 - 12 * q^6 + 5 * q^7 + 3 * q^8 + 18 * q^9	\( 5 q + 2 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} - 12 q^{6} + 5 q^{7} + 3 q^{8} + 18 q^{9} + 2 q^{10} - 6 q^{11} + 5 q^{13} + q^{14} - q^{15} + 14 q^{16} - 17 q^{17} - 5 q^{18} - 6 q^{19} + 8 q^{20} + 20 q^{21} - 8 q^{22} + q^{23} - 45 q^{24} + 5 q^{25} + 22 q^{26} - 22 q^{27} + 26 q^{28} + 6 q^{29} - 12 q^{30} + 6 q^{31} - 7 q^{32} - 20 q^{33} + 5 q^{35} + 36 q^{36} + 5 q^{37} - 16 q^{38} - 14 q^{39} + 3 q^{40} + 2 q^{41} - 58 q^{42} - 5 q^{43} - 15 q^{44} + 18 q^{45} - 14 q^{46} - 3 q^{48} + 18 q^{49} + 2 q^{50} - 10 q^{51} - 38 q^{52} - 23 q^{53} - 56 q^{54} - 6 q^{55} - 19 q^{56} + 28 q^{57} + 12 q^{58} - q^{59} + 20 q^{61} - 3 q^{62} + 26 q^{63} - 25 q^{64} + 5 q^{65} + 13 q^{66} + 21 q^{67} - 48 q^{68} + 10 q^{69} + q^{70} + 4 q^{71} + 20 q^{72} + 5 q^{73} + 24 q^{74} - q^{75} + 32 q^{76} - 26 q^{77} + 88 q^{78} + 41 q^{79} + 14 q^{80} + 41 q^{81} + 38 q^{82} - 7 q^{83} - 33 q^{84} - 17 q^{85} - 2 q^{86} - 40 q^{87} + 12 q^{88} + 20 q^{89} - 5 q^{90} - 42 q^{91} - 52 q^{92} - 36 q^{93} - 42 q^{94} - 6 q^{95} + 9 q^{96} + 37 q^{97} - 26 q^{98} + 12 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 - q^3 + 8 * q^4 + 5 * q^5 - 12 * q^6 + 5 * q^7 + 3 * q^8 + 18 * q^9 + 2 * q^10 - 6 * q^11 + 5 * q^13 + q^14 - q^15 + 14 * q^16 - 17 * q^17 - 5 * q^18 - 6 * q^19 + 8 * q^20 + 20 * q^21 - 8 * q^22 + q^23 - 45 * q^24 + 5 * q^25 + 22 * q^26 - 22 * q^27 + 26 * q^28 + 6 * q^29 - 12 * q^30 + 6 * q^31 - 7 * q^32 - 20 * q^33 + 5 * q^35 + 36 * q^36 + 5 * q^37 - 16 * q^38 - 14 * q^39 + 3 * q^40 + 2 * q^41 - 58 * q^42 - 5 * q^43 - 15 * q^44 + 18 * q^45 - 14 * q^46 - 3 * q^48 + 18 * q^49 + 2 * q^50 - 10 * q^51 - 38 * q^52 - 23 * q^53 - 56 * q^54 - 6 * q^55 - 19 * q^56 + 28 * q^57 + 12 * q^58 - q^59 + 20 * q^61 - 3 * q^62 + 26 * q^63 - 25 * q^64 + 5 * q^65 + 13 * q^66 + 21 * q^67 - 48 * q^68 + 10 * q^69 + q^70 + 4 * q^71 + 20 * q^72 + 5 * q^73 + 24 * q^74 - q^75 + 32 * q^76 - 26 * q^77 + 88 * q^78 + 41 * q^79 + 14 * q^80 + 41 * q^81 + 38 * q^82 - 7 * q^83 - 33 * q^84 - 17 * q^85 - 2 * q^86 - 40 * q^87 + 12 * q^88 + 20 * q^89 - 5 * q^90 - 42 * q^91 - 52 * q^92 - 36 * q^93 - 42 * q^94 - 6 * q^95 + 9 * q^96 + 37 * q^97 - 26 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more