LMFDB - Embedded newform 2678.2.a.r.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2678,2,Mod(1,2678)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2678, 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("2678.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2678 = 2 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2678.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:	$21.3839376613$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} - 3x^{7} + 80x^{6} + 24x^{5} - 137x^{4} - 52x^{3} + 48x^{2} + x - 2 $ x^10 - 16x^8 - 3x^7 + 80x^6 + 24x^5 - 137x^4 - 52x^3 + 48*x^2 + x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.06311$ of defining polynomial
Character	$\chi$	$=$	2678.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.06311 q^{3} +1.00000 q^{4} +3.73268 q^{5} -2.06311 q^{6} +4.82617 q^{7} -1.00000 q^{8} +1.25642 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.06311 q^{3} +1.00000 q^{4} +3.73268 q^{5} -2.06311 q^{6} +4.82617 q^{7} -1.00000 q^{8} +1.25642 q^{9} -3.73268 q^{10} +2.07657 q^{11} +2.06311 q^{12} -1.00000 q^{13} -4.82617 q^{14} +7.70093 q^{15} +1.00000 q^{16} +4.11526 q^{17} -1.25642 q^{18} -1.79201 q^{19} +3.73268 q^{20} +9.95692 q^{21} -2.07657 q^{22} -7.82459 q^{23} -2.06311 q^{24} +8.93292 q^{25} +1.00000 q^{26} -3.59720 q^{27} +4.82617 q^{28} +1.67502 q^{29} -7.70093 q^{30} +2.25409 q^{31} -1.00000 q^{32} +4.28419 q^{33} -4.11526 q^{34} +18.0146 q^{35} +1.25642 q^{36} -11.1853 q^{37} +1.79201 q^{38} -2.06311 q^{39} -3.73268 q^{40} -10.7959 q^{41} -9.95692 q^{42} -4.01442 q^{43} +2.07657 q^{44} +4.68981 q^{45} +7.82459 q^{46} +4.89486 q^{47} +2.06311 q^{48} +16.2920 q^{49} -8.93292 q^{50} +8.49024 q^{51} -1.00000 q^{52} +13.1192 q^{53} +3.59720 q^{54} +7.75117 q^{55} -4.82617 q^{56} -3.69711 q^{57} -1.67502 q^{58} -11.4396 q^{59} +7.70093 q^{60} -0.0980223 q^{61} -2.25409 q^{62} +6.06369 q^{63} +1.00000 q^{64} -3.73268 q^{65} -4.28419 q^{66} +1.67257 q^{67} +4.11526 q^{68} -16.1430 q^{69} -18.0146 q^{70} -10.5976 q^{71} -1.25642 q^{72} -3.87019 q^{73} +11.1853 q^{74} +18.4296 q^{75} -1.79201 q^{76} +10.0219 q^{77} +2.06311 q^{78} +4.29534 q^{79} +3.73268 q^{80} -11.1907 q^{81} +10.7959 q^{82} +0.687469 q^{83} +9.95692 q^{84} +15.3610 q^{85} +4.01442 q^{86} +3.45574 q^{87} -2.07657 q^{88} +5.70132 q^{89} -4.68981 q^{90} -4.82617 q^{91} -7.82459 q^{92} +4.65043 q^{93} -4.89486 q^{94} -6.68901 q^{95} -2.06311 q^{96} +5.85522 q^{97} -16.2920 q^{98} +2.60904 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{2} + 10 q^{4} + 9 q^{5} + 9 q^{7} - 10 q^{8} + 2 q^{9}+O(q^{10}) $ 10 * q - 10 * q^2 + 10 * q^4 + 9 * q^5 + 9 * q^7 - 10 * q^8 + 2 * q^9	$ 10 q - 10 q^{2} + 10 q^{4} + 9 q^{5} + 9 q^{7} - 10 q^{8} + 2 q^{9} - 9 q^{10} + 9 q^{11} - 10 q^{13} - 9 q^{14} + 9 q^{15} + 10 q^{16} + q^{17} - 2 q^{18} + 8 q^{19} + 9 q^{20} + 4 q^{21} - 9 q^{22} + 9 q^{23} + 9 q^{25} + 10 q^{26} + 9 q^{27} + 9 q^{28} + 3 q^{29} - 9 q^{30} - 3 q^{31} - 10 q^{32} - q^{34} + 12 q^{35} + 2 q^{36} + 11 q^{37} - 8 q^{38} - 9 q^{40} + 6 q^{41} - 4 q^{42} + 23 q^{43} + 9 q^{44} + 26 q^{45} - 9 q^{46} + 28 q^{47} + 33 q^{49} - 9 q^{50} - 16 q^{51} - 10 q^{52} + 19 q^{53} - 9 q^{54} + 14 q^{55} - 9 q^{56} - 10 q^{57} - 3 q^{58} - 20 q^{59} + 9 q^{60} + 25 q^{61} + 3 q^{62} + 31 q^{63} + 10 q^{64} - 9 q^{65} + 19 q^{67} + q^{68} - 24 q^{69} - 12 q^{70} - 5 q^{71} - 2 q^{72} + 29 q^{73} - 11 q^{74} + 27 q^{75} + 8 q^{76} + 31 q^{77} - 3 q^{79} + 9 q^{80} - 6 q^{81} - 6 q^{82} + 18 q^{83} + 4 q^{84} + 11 q^{85} - 23 q^{86} + 25 q^{87} - 9 q^{88} - 10 q^{89} - 26 q^{90} - 9 q^{91} + 9 q^{92} - 26 q^{93} - 28 q^{94} - 5 q^{95} + 26 q^{97} - 33 q^{98} + 57 q^{99}+O(q^{100}) $ 10 * q - 10 * q^2 + 10 * q^4 + 9 * q^5 + 9 * q^7 - 10 * q^8 + 2 * q^9 - 9 * q^10 + 9 * q^11 - 10 * q^13 - 9 * q^14 + 9 * q^15 + 10 * q^16 + q^17 - 2 * q^18 + 8 * q^19 + 9 * q^20 + 4 * q^21 - 9 * q^22 + 9 * q^23 + 9 * q^25 + 10 * q^26 + 9 * q^27 + 9 * q^28 + 3 * q^29 - 9 * q^30 - 3 * q^31 - 10 * q^32 - q^34 + 12 * q^35 + 2 * q^36 + 11 * q^37 - 8 * q^38 - 9 * q^40 + 6 * q^41 - 4 * q^42 + 23 * q^43 + 9 * q^44 + 26 * q^45 - 9 * q^46 + 28 * q^47 + 33 * q^49 - 9 * q^50 - 16 * q^51 - 10 * q^52 + 19 * q^53 - 9 * q^54 + 14 * q^55 - 9 * q^56 - 10 * q^57 - 3 * q^58 - 20 * q^59 + 9 * q^60 + 25 * q^61 + 3 * q^62 + 31 * q^63 + 10 * q^64 - 9 * q^65 + 19 * q^67 + q^68 - 24 * q^69 - 12 * q^70 - 5 * q^71 - 2 * q^72 + 29 * q^73 - 11 * q^74 + 27 * q^75 + 8 * q^76 + 31 * q^77 - 3 * q^79 + 9 * q^80 - 6 * q^81 - 6 * q^82 + 18 * q^83 + 4 * q^84 + 11 * q^85 - 23 * q^86 + 25 * q^87 - 9 * q^88 - 10 * q^89 - 26 * q^90 - 9 * q^91 + 9 * q^92 - 26 * q^93 - 28 * q^94 - 5 * q^95 + 26 * q^97 - 33 * q^98 + 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.06311	1.19114	0.595568	−	0.803305i	$-0.296927\pi$
$3$	2.06311	1.19114	0.595568	+	0.803305i	$0.296927\pi$
$4$	1.00000	0.500000
$5$	3.73268	1.66931	0.834653	−	0.550776i	$-0.185668\pi$
$5$	3.73268	1.66931	0.834653	+	0.550776i	$0.185668\pi$
$6$	−2.06311	−0.842261
$7$	4.82617	1.82412	0.912061	−	0.410054i	$-0.134490\pi$
$7$	4.82617	1.82412	0.912061	+	0.410054i	$0.134490\pi$
$8$	−1.00000	−0.353553
$9$	1.25642	0.418806
$10$	−3.73268	−1.18038
$11$	2.07657	0.626109	0.313055	−	0.949735i	$-0.398648\pi$
$11$	2.07657	0.626109	0.313055	+	0.949735i	$0.398648\pi$
$12$	2.06311	0.595568
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−4.82617	−1.28985
$15$	7.70093	1.98837
$16$	1.00000	0.250000
$17$	4.11526	0.998098	0.499049	−	0.866574i	$-0.333683\pi$
$17$	4.11526	0.998098	0.499049	+	0.866574i	$0.333683\pi$
$18$	−1.25642	−0.296141
$19$	−1.79201	−0.411116	−0.205558	−	0.978645i	$-0.565901\pi$
$19$	−1.79201	−0.411116	−0.205558	+	0.978645i	$0.565901\pi$
$20$	3.73268	0.834653
$21$	9.95692	2.17278
$22$	−2.07657	−0.442726
$23$	−7.82459	−1.63154	−0.815769	−	0.578377i	$-0.803686\pi$
$23$	−7.82459	−1.63154	−0.815769	+	0.578377i	$0.803686\pi$
$24$	−2.06311	−0.421130
$25$	8.93292	1.78658
$26$	1.00000	0.196116
$27$	−3.59720	−0.692281
$28$	4.82617	0.912061
$29$	1.67502	0.311042	0.155521	−	0.987833i	$-0.450294\pi$
$29$	1.67502	0.311042	0.155521	+	0.987833i	$0.450294\pi$
$30$	−7.70093	−1.40599
$31$	2.25409	0.404846	0.202423	−	0.979298i	$-0.435118\pi$
$31$	2.25409	0.404846	0.202423	+	0.979298i	$0.435118\pi$
$32$	−1.00000	−0.176777
$33$	4.28419	0.745781
$34$	−4.11526	−0.705762
$35$	18.0146	3.04502
$36$	1.25642	0.209403
$37$	−11.1853	−1.83885	−0.919426	−	0.393263i	$-0.871346\pi$
$37$	−11.1853	−1.83885	−0.919426	+	0.393263i	$0.871346\pi$
$38$	1.79201	0.290703
$39$	−2.06311	−0.330362
$40$	−3.73268	−0.590189
$41$	−10.7959	−1.68604	−0.843018	−	0.537886i	$-0.819223\pi$
$41$	−10.7959	−1.68604	−0.843018	+	0.537886i	$0.819223\pi$
$42$	−9.95692	−1.53639
$43$	−4.01442	−0.612193	−0.306097	−	0.952000i	$-0.599023\pi$
$43$	−4.01442	−0.612193	−0.306097	+	0.952000i	$0.599023\pi$
$44$	2.07657	0.313055
$45$	4.68981	0.699116
$46$	7.82459	1.15367
$47$	4.89486	0.713989	0.356994	−	0.934106i	$-0.383801\pi$
$47$	4.89486	0.713989	0.356994	+	0.934106i	$0.383801\pi$
$48$	2.06311	0.297784
$49$	16.2920	2.32742
$50$	−8.93292	−1.26331
$51$	8.49024	1.18887
$52$	−1.00000	−0.138675
$53$	13.1192	1.80206	0.901030	−	0.433756i	$-0.142812\pi$
$53$	13.1192	1.80206	0.901030	+	0.433756i	$0.142812\pi$
$54$	3.59720	0.489517
$55$	7.75117	1.04517
$56$	−4.82617	−0.644925
$57$	−3.69711	−0.489695
$58$	−1.67502	−0.219940
$59$	−11.4396	−1.48931	−0.744653	−	0.667451i	$-0.767385\pi$
$59$	−11.4396	−1.48931	−0.744653	+	0.667451i	$0.767385\pi$
$60$	7.70093	0.994186
$61$	−0.0980223	−0.0125505	−0.00627524	−	0.999980i	$-0.501997\pi$
$61$	−0.0980223	−0.0125505	−0.00627524	+	0.999980i	$0.501997\pi$
$62$	−2.25409	−0.286270
$63$	6.06369	0.763953
$64$	1.00000	0.125000
$65$	−3.73268	−0.462982
$66$	−4.28419	−0.527347
$67$	1.67257	0.204336	0.102168	−	0.994767i	$-0.467422\pi$
$67$	1.67257	0.204336	0.102168	+	0.994767i	$0.467422\pi$
$68$	4.11526	0.499049
$69$	−16.1430	−1.94339
$70$	−18.0146	−2.15315
$71$	−10.5976	−1.25771	−0.628854	−	0.777524i	$-0.716476\pi$
$71$	−10.5976	−1.25771	−0.628854	+	0.777524i	$0.716476\pi$
$72$	−1.25642	−0.148070
$73$	−3.87019	−0.452971	−0.226486	−	0.974014i	$-0.572724\pi$
$73$	−3.87019	−0.452971	−0.226486	+	0.974014i	$0.572724\pi$
$74$	11.1853	1.30026
$75$	18.4296	2.12807
$76$	−1.79201	−0.205558
$77$	10.0219	1.14210
$78$	2.06311	0.233601
$79$	4.29534	0.483263	0.241632	−	0.970368i	$-0.422317\pi$
$79$	4.29534	0.483263	0.241632	+	0.970368i	$0.422317\pi$
$80$	3.73268	0.417327
$81$	−11.1907	−1.24341
$82$	10.7959	1.19221
$83$	0.687469	0.0754595	0.0377298	−	0.999288i	$-0.487987\pi$
$83$	0.687469	0.0754595	0.0377298	+	0.999288i	$0.487987\pi$
$84$	9.95692	1.08639
$85$	15.3610	1.66613
$86$	4.01442	0.432886
$87$	3.45574	0.370494
$88$	−2.07657	−0.221363
$89$	5.70132	0.604338	0.302169	−	0.953254i	$-0.402289\pi$
$89$	5.70132	0.604338	0.302169	+	0.953254i	$0.402289\pi$
$90$	−4.68981	−0.494349
$91$	−4.82617	−0.505921
$92$	−7.82459	−0.815769
$93$	4.65043	0.482227
$94$	−4.89486	−0.504866
$95$	−6.68901	−0.686278
$96$	−2.06311	−0.210565
$97$	5.85522	0.594508	0.297254	−	0.954798i	$-0.403929\pi$
$97$	5.85522	0.594508	0.297254	+	0.954798i	$0.403929\pi$
$98$	−16.2920	−1.64574
$99$	2.60904	0.262218
$100$	8.93292	0.893292
$101$	−9.99949	−0.994986	−0.497493	−	0.867468i	$-0.665746\pi$
$101$	−9.99949	−0.994986	−0.497493	+	0.867468i	$0.665746\pi$
$102$	−8.49024	−0.840659
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	37.1660	3.62703
$106$	−13.1192	−1.27425
$107$	−2.95877	−0.286035	−0.143018	−	0.989720i	$-0.545681\pi$
$107$	−2.95877	−0.286035	−0.143018	+	0.989720i	$0.545681\pi$
$108$	−3.59720	−0.346141
$109$	−13.9763	−1.33868	−0.669341	−	0.742955i	$-0.733424\pi$
$109$	−13.9763	−1.33868	−0.669341	+	0.742955i	$0.733424\pi$
$110$	−7.75117	−0.739045
$111$	−23.0765	−2.19032
$112$	4.82617	0.456031
$113$	−13.3990	−1.26047	−0.630234	−	0.776405i	$-0.717041\pi$
$113$	−13.3990	−1.26047	−0.630234	+	0.776405i	$0.717041\pi$
$114$	3.69711	0.346266
$115$	−29.2067	−2.72354
$116$	1.67502	0.155521
$117$	−1.25642	−0.116156
$118$	11.4396	1.05310
$119$	19.8610	1.82065
$120$	−7.70093	−0.702996
$121$	−6.68786	−0.607987
$122$	0.0980223	0.00887453
$123$	−22.2731	−2.00830
$124$	2.25409	0.202423
$125$	14.6803	1.31305
$126$	−6.06369	−0.540197
$127$	−12.4951	−1.10876	−0.554381	−	0.832263i	$-0.687045\pi$
$127$	−12.4951	−1.10876	−0.554381	+	0.832263i	$0.687045\pi$
$128$	−1.00000	−0.0883883
$129$	−8.28218	−0.729205
$130$	3.73268	0.327378
$131$	−8.73304	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$131$	−8.73304	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$132$	4.28419	0.372891
$133$	−8.64856	−0.749925
$134$	−1.67257	−0.144488
$135$	−13.4272	−1.15563
$136$	−4.11526	−0.352881
$137$	−5.14623	−0.439672	−0.219836	−	0.975537i	$-0.570552\pi$
$137$	−5.14623	−0.439672	−0.219836	+	0.975537i	$0.570552\pi$
$138$	16.1430	1.37418
$139$	5.18949	0.440167	0.220084	−	0.975481i	$-0.429367\pi$
$139$	5.18949	0.440167	0.220084	+	0.975481i	$0.429367\pi$
$140$	18.0146	1.52251
$141$	10.0986	0.850458
$142$	10.5976	0.889333
$143$	−2.07657	−0.173651
$144$	1.25642	0.104702
$145$	6.25230	0.519225
$146$	3.87019	0.320299
$147$	33.6121	2.77228
$148$	−11.1853	−0.919426
$149$	17.7571	1.45472	0.727360	−	0.686257i	$-0.240747\pi$
$149$	17.7571	1.45472	0.727360	+	0.686257i	$0.240747\pi$
$150$	−18.4296	−1.50477
$151$	−0.566707	−0.0461180	−0.0230590	−	0.999734i	$-0.507341\pi$
$151$	−0.566707	−0.0461180	−0.0230590	+	0.999734i	$0.507341\pi$
$152$	1.79201	0.145351
$153$	5.17049	0.418009
$154$	−10.0219	−0.807586
$155$	8.41380	0.675812
$156$	−2.06311	−0.165181
$157$	−9.68021	−0.772565	−0.386282	−	0.922381i	$-0.626241\pi$
$157$	−9.68021	−0.772565	−0.386282	+	0.922381i	$0.626241\pi$
$158$	−4.29534	−0.341719
$159$	27.0663	2.14650
$160$	−3.73268	−0.295094
$161$	−37.7628	−2.97613
$162$	11.1907	0.879222
$163$	22.5165	1.76363	0.881815	−	0.471596i	$-0.156322\pi$
$163$	22.5165	1.76363	0.881815	+	0.471596i	$0.156322\pi$
$164$	−10.7959	−0.843018
$165$	15.9915	1.24494
$166$	−0.687469	−0.0533579
$167$	17.2812	1.33726	0.668630	−	0.743595i	$-0.266881\pi$
$167$	17.2812	1.33726	0.668630	+	0.743595i	$0.266881\pi$
$168$	−9.95692	−0.768193
$169$	1.00000	0.0769231
$170$	−15.3610	−1.17813
$171$	−2.25151	−0.172178
$172$	−4.01442	−0.306097
$173$	−20.9838	−1.59537	−0.797686	−	0.603073i	$-0.793943\pi$
$173$	−20.9838	−1.59537	−0.797686	+	0.603073i	$0.793943\pi$
$174$	−3.45574	−0.261979
$175$	43.1118	3.25895
$176$	2.07657	0.156527
$177$	−23.6011	−1.77397
$178$	−5.70132	−0.427332
$179$	5.61170	0.419438	0.209719	−	0.977762i	$-0.432745\pi$
$179$	5.61170	0.419438	0.209719	+	0.977762i	$0.432745\pi$
$180$	4.68981	0.349558
$181$	4.42165	0.328659	0.164329	−	0.986406i	$-0.447454\pi$
$181$	4.42165	0.328659	0.164329	+	0.986406i	$0.447454\pi$
$182$	4.82617	0.357740
$183$	−0.202231	−0.0149493
$184$	7.82459	0.576836
$185$	−41.7512	−3.06961
$186$	−4.65043	−0.340986
$187$	8.54563	0.624918
$188$	4.89486	0.356994
$189$	−17.3607	−1.26281
$190$	6.68901	0.485272
$191$	−13.8174	−0.999795	−0.499897	−	0.866085i	$-0.666629\pi$
$191$	−13.8174	−0.999795	−0.499897	+	0.866085i	$0.666629\pi$
$192$	2.06311	0.148892
$193$	15.9315	1.14677	0.573387	−	0.819285i	$-0.305629\pi$
$193$	15.9315	1.14677	0.573387	+	0.819285i	$0.305629\pi$
$194$	−5.85522	−0.420380
$195$	−7.70093	−0.551475
$196$	16.2920	1.16371
$197$	2.48945	0.177366	0.0886832	−	0.996060i	$-0.471734\pi$
$197$	2.48945	0.177366	0.0886832	+	0.996060i	$0.471734\pi$
$198$	−2.60904	−0.185416
$199$	2.43175	0.172382	0.0861911	−	0.996279i	$-0.472530\pi$
$199$	2.43175	0.172382	0.0861911	+	0.996279i	$0.472530\pi$
$200$	−8.93292	−0.631653
$201$	3.45069	0.243393
$202$	9.99949	0.703561
$203$	8.08391	0.567380
$204$	8.49024	0.594435
$205$	−40.2977	−2.81451
$206$	−1.00000	−0.0696733
$207$	−9.83095	−0.683298
$208$	−1.00000	−0.0693375
$209$	−3.72123	−0.257403
$210$	−37.1660	−2.56470
$211$	21.4043	1.47353	0.736766	−	0.676148i	$-0.236352\pi$
$211$	21.4043	1.47353	0.736766	+	0.676148i	$0.236352\pi$
$212$	13.1192	0.901030
$213$	−21.8641	−1.49810
$214$	2.95877	0.202257
$215$	−14.9846	−1.02194
$216$	3.59720	0.244758
$217$	10.8786	0.738489
$218$	13.9763	0.946592
$219$	−7.98462	−0.539551
$220$	7.75117	0.522584
$221$	−4.11526	−0.276823
$222$	23.0765	1.54879
$223$	−13.6216	−0.912167	−0.456084	−	0.889937i	$-0.650748\pi$
$223$	−13.6216	−0.912167	−0.456084	+	0.889937i	$0.650748\pi$
$224$	−4.82617	−0.322462
$225$	11.2235	0.748232
$226$	13.3990	0.891285
$227$	−11.4079	−0.757170	−0.378585	−	0.925566i	$-0.623589\pi$
$227$	−11.4079	−0.757170	−0.378585	+	0.925566i	$0.623589\pi$
$228$	−3.69711	−0.244847
$229$	18.4383	1.21844	0.609218	−	0.793003i	$-0.291483\pi$
$229$	18.4383	1.21844	0.609218	+	0.793003i	$0.291483\pi$
$230$	29.2067	1.92583
$231$	20.6762	1.36040
$232$	−1.67502	−0.109970
$233$	4.37546	0.286646	0.143323	−	0.989676i	$-0.454221\pi$
$233$	4.37546	0.286646	0.143323	+	0.989676i	$0.454221\pi$
$234$	1.25642	0.0821346
$235$	18.2710	1.19187
$236$	−11.4396	−0.744653
$237$	8.86175	0.575633
$238$	−19.8610	−1.28740
$239$	−7.67219	−0.496273	−0.248136	−	0.968725i	$-0.579818\pi$
$239$	−7.67219	−0.496273	−0.248136	+	0.968725i	$0.579818\pi$
$240$	7.70093	0.497093
$241$	11.6249	0.748828	0.374414	−	0.927262i	$-0.377844\pi$
$241$	11.6249	0.748828	0.374414	+	0.927262i	$0.377844\pi$
$242$	6.68786	0.429912
$243$	−12.2960	−0.788787
$244$	−0.0980223	−0.00627524
$245$	60.8127	3.88518
$246$	22.2731	1.42008
$247$	1.79201	0.114023
$248$	−2.25409	−0.143135
$249$	1.41832	0.0898826
$250$	−14.6803	−0.928467
$251$	8.42574	0.531828	0.265914	−	0.963997i	$-0.414326\pi$
$251$	8.42574	0.531828	0.265914	+	0.963997i	$0.414326\pi$
$252$	6.06369	0.381977
$253$	−16.2483	−1.02152
$254$	12.4951	0.784013
$255$	31.6914	1.98459
$256$	1.00000	0.0625000
$257$	10.0346	0.625943	0.312972	−	0.949762i	$-0.398676\pi$
$257$	10.0346	0.625943	0.312972	+	0.949762i	$0.398676\pi$
$258$	8.28218	0.515626
$259$	−53.9822	−3.35429
$260$	−3.73268	−0.231491
$261$	2.10452	0.130266
$262$	8.73304	0.539529
$263$	−3.51573	−0.216789	−0.108395	−	0.994108i	$-0.534571\pi$
$263$	−3.51573	−0.216789	−0.108395	+	0.994108i	$0.534571\pi$
$264$	−4.28419	−0.263674
$265$	48.9698	3.00819
$266$	8.64856	0.530277
$267$	11.7624	0.719849
$268$	1.67257	0.102168
$269$	28.3811	1.73043	0.865215	−	0.501402i	$-0.167182\pi$
$269$	28.3811	1.73043	0.865215	+	0.501402i	$0.167182\pi$
$270$	13.4272	0.817154
$271$	20.5274	1.24695	0.623476	−	0.781843i	$-0.285720\pi$
$271$	20.5274	1.24695	0.623476	+	0.781843i	$0.285720\pi$
$272$	4.11526	0.249524
$273$	−9.95692	−0.602620
$274$	5.14623	0.310895
$275$	18.5498	1.11860
$276$	−16.1430	−0.971693
$277$	−9.06421	−0.544616	−0.272308	−	0.962210i	$-0.587787\pi$
$277$	−9.06421	−0.544616	−0.272308	+	0.962210i	$0.587787\pi$
$278$	−5.18949	−0.311245
$279$	2.83208	0.169552
$280$	−18.0146	−1.07658
$281$	8.41196	0.501816	0.250908	−	0.968011i	$-0.419271\pi$
$281$	8.41196	0.501816	0.250908	+	0.968011i	$0.419271\pi$
$282$	−10.0986	−0.601365
$283$	21.3040	1.26639	0.633195	−	0.773992i	$-0.281743\pi$
$283$	21.3040	1.26639	0.633195	+	0.773992i	$0.281743\pi$
$284$	−10.5976	−0.628854
$285$	−13.8002	−0.817450
$286$	2.07657	0.122790
$287$	−52.1029	−3.07554
$288$	−1.25642	−0.0740351
$289$	−0.0646111	−0.00380065
$290$	−6.25230	−0.367148
$291$	12.0800	0.708140
$292$	−3.87019	−0.226486
$293$	1.99787	0.116717	0.0583583	−	0.998296i	$-0.481413\pi$
$293$	1.99787	0.116717	0.0583583	+	0.998296i	$0.481413\pi$
$294$	−33.6121	−1.96030
$295$	−42.7003	−2.48611
$296$	11.1853	0.650132
$297$	−7.46983	−0.433444
$298$	−17.7571	−1.02864
$299$	7.82459	0.452507
$300$	18.4296	1.06403
$301$	−19.3743	−1.11672
$302$	0.566707	0.0326103
$303$	−20.6300	−1.18516
$304$	−1.79201	−0.102779
$305$	−0.365886	−0.0209506
$306$	−5.17049	−0.295577
$307$	−10.8863	−0.621313	−0.310656	−	0.950522i	$-0.600549\pi$
$307$	−10.8863	−0.621313	−0.310656	+	0.950522i	$0.600549\pi$
$308$	10.0219	0.571050
$309$	2.06311	0.117366
$310$	−8.41380	−0.477872
$311$	4.91045	0.278446	0.139223	−	0.990261i	$-0.455540\pi$
$311$	4.91045	0.278446	0.139223	+	0.990261i	$0.455540\pi$
$312$	2.06311	0.116801
$313$	18.1198	1.02419	0.512095	−	0.858929i	$-0.328870\pi$
$313$	18.1198	1.02419	0.512095	+	0.858929i	$0.328870\pi$
$314$	9.68021	0.546286
$315$	22.6338	1.27527
$316$	4.29534	0.241632
$317$	−6.66471	−0.374328	−0.187164	−	0.982329i	$-0.559930\pi$
$317$	−6.66471	−0.374328	−0.187164	+	0.982329i	$0.559930\pi$
$318$	−27.0663	−1.51780
$319$	3.47828	0.194747
$320$	3.73268	0.208663
$321$	−6.10426	−0.340707
$322$	37.7628	2.10444
$323$	−7.37460	−0.410334
$324$	−11.1907	−0.621704
$325$	−8.93292	−0.495509
$326$	−22.5165	−1.24707
$327$	−28.8345	−1.59455
$328$	10.7959	0.596104
$329$	23.6235	1.30240
$330$	−15.9915	−0.880304
$331$	28.4398	1.56319	0.781597	−	0.623784i	$-0.214405\pi$
$331$	28.4398	1.56319	0.781597	+	0.623784i	$0.214405\pi$
$332$	0.687469	0.0377298
$333$	−14.0534	−0.770122
$334$	−17.2812	−0.945586
$335$	6.24316	0.341100
$336$	9.95692	0.543195
$337$	31.5311	1.71761	0.858805	−	0.512303i	$-0.171207\pi$
$337$	31.5311	1.71761	0.858805	+	0.512303i	$0.171207\pi$
$338$	−1.00000	−0.0543928
$339$	−27.6435	−1.50139
$340$	15.3610	0.833066
$341$	4.68077	0.253478
$342$	2.25151	0.121748
$343$	44.8446	2.42138
$344$	4.01442	0.216443
$345$	−60.2566	−3.24411
$346$	20.9838	1.12810
$347$	8.47263	0.454835	0.227417	−	0.973797i	$-0.426972\pi$
$347$	8.47263	0.454835	0.227417	+	0.973797i	$0.426972\pi$
$348$	3.45574	0.185247
$349$	2.04891	0.109675	0.0548377	−	0.998495i	$-0.482536\pi$
$349$	2.04891	0.109675	0.0548377	+	0.998495i	$0.482536\pi$
$350$	−43.1118	−2.30442
$351$	3.59720	0.192004
$352$	−2.07657	−0.110681
$353$	−31.7850	−1.69174	−0.845872	−	0.533386i	$-0.820919\pi$
$353$	−31.7850	−1.69174	−0.845872	+	0.533386i	$0.820919\pi$
$354$	23.6011	1.25438
$355$	−39.5576	−2.09950
$356$	5.70132	0.302169
$357$	40.9754	2.16865
$358$	−5.61170	−0.296588
$359$	18.1747	0.959224	0.479612	−	0.877481i	$-0.340777\pi$
$359$	18.1747	0.959224	0.479612	+	0.877481i	$0.340777\pi$
$360$	−4.68981	−0.247175
$361$	−15.7887	−0.830984
$362$	−4.42165	−0.232397
$363$	−13.7978	−0.724196
$364$	−4.82617	−0.252960
$365$	−14.4462	−0.756148
$366$	0.202231	0.0105708
$367$	−30.1842	−1.57560	−0.787802	−	0.615929i	$-0.788781\pi$
$367$	−30.1842	−1.57560	−0.787802	+	0.615929i	$0.788781\pi$
$368$	−7.82459	−0.407885
$369$	−13.5642	−0.706122
$370$	41.7512	2.17054
$371$	63.3155	3.28718
$372$	4.65043	0.241114
$373$	20.3058	1.05139	0.525696	−	0.850673i	$-0.323805\pi$
$373$	20.3058	1.05139	0.525696	+	0.850673i	$0.323805\pi$
$374$	−8.54563	−0.441884
$375$	30.2872	1.56402
$376$	−4.89486	−0.252433
$377$	−1.67502	−0.0862677
$378$	17.3607	0.892939
$379$	7.04774	0.362018	0.181009	−	0.983481i	$-0.442064\pi$
$379$	7.04774	0.362018	0.181009	+	0.983481i	$0.442064\pi$
$380$	−6.68901	−0.343139
$381$	−25.7788	−1.32069
$382$	13.8174	0.706962
$383$	1.49704	0.0764951	0.0382476	−	0.999268i	$-0.487822\pi$
$383$	1.49704	0.0764951	0.0382476	+	0.999268i	$0.487822\pi$
$384$	−2.06311	−0.105283
$385$	37.4085	1.90651
$386$	−15.9315	−0.810891
$387$	−5.04379	−0.256390
$388$	5.85522	0.297254
$389$	25.7794	1.30707	0.653534	−	0.756897i	$-0.273286\pi$
$389$	25.7794	1.30707	0.653534	+	0.756897i	$0.273286\pi$
$390$	7.70093	0.389952
$391$	−32.2002	−1.62844
$392$	−16.2920	−0.822868
$393$	−18.0172	−0.908848
$394$	−2.48945	−0.125417
$395$	16.0331	0.806715
$396$	2.60904	0.131109
$397$	38.8389	1.94927	0.974634	−	0.223806i	$-0.0718480\pi$
$397$	38.8389	1.94927	0.974634	+	0.223806i	$0.0718480\pi$
$398$	−2.43175	−0.121893
$399$	−17.8429	−0.893263
$400$	8.93292	0.446646
$401$	3.86100	0.192809	0.0964045	−	0.995342i	$-0.469266\pi$
$401$	3.86100	0.192809	0.0964045	+	0.995342i	$0.469266\pi$
$402$	−3.45069	−0.172105
$403$	−2.25409	−0.112284
$404$	−9.99949	−0.497493
$405$	−41.7712	−2.07563
$406$	−8.08391	−0.401198
$407$	−23.2270	−1.15132
$408$	−8.49024	−0.420329
$409$	26.7751	1.32394	0.661971	−	0.749530i	$-0.269720\pi$
$409$	26.7751	1.32394	0.661971	+	0.749530i	$0.269720\pi$
$410$	40.2977	1.99016
$411$	−10.6172	−0.523709
$412$	1.00000	0.0492665
$413$	−55.2094	−2.71668
$414$	9.83095	0.483165
$415$	2.56610	0.125965
$416$	1.00000	0.0490290
$417$	10.7065	0.524299
$418$	3.72123	0.182011
$419$	−15.5075	−0.757592	−0.378796	−	0.925480i	$-0.623662\pi$
$419$	−15.5075	−0.757592	−0.378796	+	0.925480i	$0.623662\pi$
$420$	37.1660	1.81352
$421$	−0.984257	−0.0479697	−0.0239849	−	0.999712i	$-0.507635\pi$
$421$	−0.984257	−0.0479697	−0.0239849	+	0.999712i	$0.507635\pi$
$422$	−21.4043	−1.04194
$423$	6.14999	0.299023
$424$	−13.1192	−0.637125
$425$	36.7613	1.78319
$426$	21.8641	1.05932
$427$	−0.473073	−0.0228936
$428$	−2.95877	−0.143018
$429$	−4.28419	−0.206843
$430$	14.9846	0.722619
$431$	−2.81027	−0.135366	−0.0676829	−	0.997707i	$-0.521561\pi$
$431$	−2.81027	−0.135366	−0.0676829	+	0.997707i	$0.521561\pi$
$432$	−3.59720	−0.173070
$433$	−0.0600336	−0.00288503	−0.00144251	−	0.999999i	$-0.500459\pi$
$433$	−0.0600336	−0.00288503	−0.00144251	+	0.999999i	$0.500459\pi$
$434$	−10.8786	−0.522191
$435$	12.8992	0.618468
$436$	−13.9763	−0.669341
$437$	14.0217	0.670751
$438$	7.98462	0.381520
$439$	−5.56695	−0.265696	−0.132848	−	0.991136i	$-0.542412\pi$
$439$	−5.56695	−0.265696	−0.132848	+	0.991136i	$0.542412\pi$
$440$	−7.75117	−0.369523
$441$	20.4695	0.974739
$442$	4.11526	0.195743
$443$	−34.2099	−1.62536	−0.812680	−	0.582710i	$-0.801992\pi$
$443$	−34.2099	−1.62536	−0.812680	+	0.582710i	$0.801992\pi$
$444$	−23.0765	−1.09516
$445$	21.2812	1.00883
$446$	13.6216	0.645000
$447$	36.6349	1.73277
$448$	4.82617	0.228015
$449$	14.3445	0.676959	0.338479	−	0.940974i	$-0.390087\pi$
$449$	14.3445	0.676959	0.338479	+	0.940974i	$0.390087\pi$
$450$	−11.2235	−0.529080
$451$	−22.4184	−1.05564
$452$	−13.3990	−0.630234
$453$	−1.16918	−0.0549328
$454$	11.4079	0.535400
$455$	−18.0146	−0.844536
$456$	3.69711	0.173133
$457$	−40.0081	−1.87150	−0.935751	−	0.352662i	$-0.885276\pi$
$457$	−40.0081	−1.87150	−0.935751	+	0.352662i	$0.885276\pi$
$458$	−18.4383	−0.861564
$459$	−14.8034	−0.690965
$460$	−29.2067	−1.36177
$461$	29.4475	1.37151	0.685753	−	0.727835i	$-0.259473\pi$
$461$	29.4475	1.37151	0.685753	+	0.727835i	$0.259473\pi$
$462$	−20.6762	−0.961945
$463$	20.3908	0.947640	0.473820	−	0.880622i	$-0.342875\pi$
$463$	20.3908	0.947640	0.473820	+	0.880622i	$0.342875\pi$
$464$	1.67502	0.0777606
$465$	17.3586	0.804985
$466$	−4.37546	−0.202689
$467$	30.7647	1.42362	0.711811	−	0.702372i	$-0.247875\pi$
$467$	30.7647	1.42362	0.711811	+	0.702372i	$0.247875\pi$
$468$	−1.25642	−0.0580779
$469$	8.07209	0.372735
$470$	−18.2710	−0.842777
$471$	−19.9713	−0.920230
$472$	11.4396	0.526549
$473$	−8.33622	−0.383300
$474$	−8.86175	−0.407034
$475$	−16.0079	−0.734492
$476$	19.8610	0.910326
$477$	16.4832	0.754714
$478$	7.67219	0.350918
$479$	3.05825	0.139735	0.0698675	−	0.997556i	$-0.477742\pi$
$479$	3.05825	0.139735	0.0698675	+	0.997556i	$0.477742\pi$
$480$	−7.70093	−0.351498
$481$	11.1853	0.510006
$482$	−11.6249	−0.529502
$483$	−77.9088	−3.54497
$484$	−6.68786	−0.303994
$485$	21.8557	0.992416
$486$	12.2960	0.557756
$487$	−14.8476	−0.672810	−0.336405	−	0.941717i	$-0.609211\pi$
$487$	−14.8476	−0.672810	−0.336405	+	0.941717i	$0.609211\pi$
$488$	0.0980223	0.00443726
$489$	46.4540	2.10072
$490$	−60.8127	−2.74724
$491$	−5.40321	−0.243844	−0.121922	−	0.992540i	$-0.538906\pi$
$491$	−5.40321	−0.243844	−0.121922	+	0.992540i	$0.538906\pi$
$492$	−22.2731	−1.00415
$493$	6.89313	0.310451
$494$	−1.79201	−0.0806264
$495$	9.73871	0.437723
$496$	2.25409	0.101212
$497$	−51.1460	−2.29421
$498$	−1.41832	−0.0635566
$499$	9.35930	0.418980	0.209490	−	0.977811i	$-0.432820\pi$
$499$	9.35930	0.418980	0.209490	+	0.977811i	$0.432820\pi$
$500$	14.6803	0.656525
$501$	35.6530	1.59286
$502$	−8.42574	−0.376059
$503$	−4.72499	−0.210677	−0.105338	−	0.994436i	$-0.533593\pi$
$503$	−4.72499	−0.210677	−0.105338	+	0.994436i	$0.533593\pi$
$504$	−6.06369	−0.270098
$505$	−37.3249	−1.66094
$506$	16.2483	0.722325
$507$	2.06311	0.0916259
$508$	−12.4951	−0.554381
$509$	−34.7091	−1.53845	−0.769226	−	0.638976i	$-0.779358\pi$
$509$	−34.7091	−1.53845	−0.769226	+	0.638976i	$0.779358\pi$
$510$	−31.6914	−1.40332
$511$	−18.6782	−0.826275
$512$	−1.00000	−0.0441942
$513$	6.44622	0.284608
$514$	−10.0346	−0.442609
$515$	3.73268	0.164482
$516$	−8.28218	−0.364603
$517$	10.1645	0.447035
$518$	53.9822	2.37184
$519$	−43.2920	−1.90031
$520$	3.73268	0.163689
$521$	14.2636	0.624901	0.312450	−	0.949934i	$-0.398850\pi$
$521$	14.2636	0.624901	0.312450	+	0.949934i	$0.398850\pi$
$522$	−2.10452	−0.0921123
$523$	3.35008	0.146489	0.0732444	−	0.997314i	$-0.476665\pi$
$523$	3.35008	0.146489	0.0732444	+	0.997314i	$0.476665\pi$
$524$	−8.73304	−0.381504
$525$	88.9444	3.88185
$526$	3.51573	0.153293
$527$	9.27617	0.404076
$528$	4.28419	0.186445
$529$	38.2241	1.66192
$530$	−48.9698	−2.12711
$531$	−14.3729	−0.623731
$532$	−8.64856	−0.374963
$533$	10.7959	0.467622
$534$	−11.7624	−0.509010
$535$	−11.0441	−0.477480
$536$	−1.67257	−0.0722438
$537$	11.5776	0.499608
$538$	−28.3811	−1.22360
$539$	33.8314	1.45722
$540$	−13.4272	−0.577815
$541$	26.8839	1.15583	0.577914	−	0.816098i	$-0.303867\pi$
$541$	26.8839	1.15583	0.577914	+	0.816098i	$0.303867\pi$
$542$	−20.5274	−0.881728
$543$	9.12235	0.391477
$544$	−4.11526	−0.176440
$545$	−52.1689	−2.23467
$546$	9.95692	0.426117
$547$	−8.98413	−0.384134	−0.192067	−	0.981382i	$-0.561519\pi$
$547$	−8.98413	−0.384134	−0.192067	+	0.981382i	$0.561519\pi$
$548$	−5.14623	−0.219836
$549$	−0.123157	−0.00525621
$550$	−18.5498	−0.790967
$551$	−3.00165	−0.127874
$552$	16.1430	0.687090
$553$	20.7301	0.881532
$554$	9.06421	0.385101
$555$	−86.1372	−3.65632
$556$	5.18949	0.220084
$557$	−14.2471	−0.603670	−0.301835	−	0.953360i	$-0.597599\pi$
$557$	−14.2471	−0.603670	−0.301835	+	0.953360i	$0.597599\pi$
$558$	−2.83208	−0.119891
$559$	4.01442	0.169792
$560$	18.0146	0.761255
$561$	17.6306	0.744363
$562$	−8.41196	−0.354837
$563$	26.5928	1.12075	0.560377	−	0.828238i	$-0.310656\pi$
$563$	26.5928	1.12075	0.560377	+	0.828238i	$0.310656\pi$
$564$	10.0986	0.425229
$565$	−50.0140	−2.10411
$566$	−21.3040	−0.895473
$567$	−54.0081	−2.26813
$568$	10.5976	0.444667
$569$	37.9664	1.59163	0.795817	−	0.605537i	$-0.207042\pi$
$569$	37.9664	1.59163	0.795817	+	0.605537i	$0.207042\pi$
$570$	13.8002	0.578025
$571$	45.6023	1.90840	0.954199	−	0.299172i	$-0.0967106\pi$
$571$	45.6023	1.90840	0.954199	+	0.299172i	$0.0967106\pi$
$572$	−2.07657	−0.0868257
$573$	−28.5069	−1.19089
$574$	52.1029	2.17473
$575$	−69.8964	−2.91488
$576$	1.25642	0.0523508
$577$	32.6712	1.36012	0.680060	−	0.733156i	$-0.261954\pi$
$577$	32.6712	1.36012	0.680060	+	0.733156i	$0.261954\pi$
$578$	0.0646111	0.00268747
$579$	32.8684	1.36596
$580$	6.25230	0.259613
$581$	3.31784	0.137647
$582$	−12.0800	−0.500730
$583$	27.2429	1.12829
$584$	3.87019	0.160150
$585$	−4.68981	−0.193900
$586$	−1.99787	−0.0825311
$587$	−5.30857	−0.219108	−0.109554	−	0.993981i	$-0.534942\pi$
$587$	−5.30857	−0.219108	−0.109554	+	0.993981i	$0.534942\pi$
$588$	33.6121	1.38614
$589$	−4.03935	−0.166439
$590$	42.7003	1.75794
$591$	5.13602	0.211267
$592$	−11.1853	−0.459713
$593$	11.6750	0.479434	0.239717	−	0.970843i	$-0.422945\pi$
$593$	11.6750	0.479434	0.239717	+	0.970843i	$0.422945\pi$
$594$	7.46983	0.306491
$595$	74.1347	3.03923
$596$	17.7571	0.727360
$597$	5.01697	0.205331
$598$	−7.82459	−0.319971
$599$	−42.9216	−1.75373	−0.876864	−	0.480738i	$-0.840369\pi$
$599$	−42.9216	−1.75373	−0.876864	+	0.480738i	$0.840369\pi$
$600$	−18.4296	−0.752385
$601$	3.40711	0.138979	0.0694895	−	0.997583i	$-0.477863\pi$
$601$	3.40711	0.138979	0.0694895	+	0.997583i	$0.477863\pi$
$602$	19.3743	0.789637
$603$	2.10144	0.0855773
$604$	−0.566707	−0.0230590
$605$	−24.9637	−1.01492
$606$	20.6300	0.838038
$607$	31.3195	1.27122	0.635610	−	0.772010i	$-0.280749\pi$
$607$	31.3195	1.27122	0.635610	+	0.772010i	$0.280749\pi$
$608$	1.79201	0.0726756
$609$	16.6780	0.675827
$610$	0.365886	0.0148143
$611$	−4.89486	−0.198025
$612$	5.17049	0.209005
$613$	19.6358	0.793082	0.396541	−	0.918017i	$-0.370210\pi$
$613$	19.6358	0.793082	0.396541	+	0.918017i	$0.370210\pi$
$614$	10.8863	0.439335
$615$	−83.1384	−3.35247
$616$	−10.0219	−0.403793
$617$	−38.9679	−1.56879	−0.784394	−	0.620263i	$-0.787026\pi$
$617$	−38.9679	−1.56879	−0.784394	+	0.620263i	$0.787026\pi$
$618$	−2.06311	−0.0829904
$619$	4.62397	0.185853	0.0929265	−	0.995673i	$-0.470378\pi$
$619$	4.62397	0.185853	0.0929265	+	0.995673i	$0.470378\pi$
$620$	8.41380	0.337906
$621$	28.1466	1.12948
$622$	−4.91045	−0.196891
$623$	27.5155	1.10239
$624$	−2.06311	−0.0825905
$625$	10.1325	0.405299
$626$	−18.1198	−0.724211
$627$	−7.67731	−0.306602
$628$	−9.68021	−0.386282
$629$	−46.0305	−1.83535
$630$	−22.6338	−0.901754
$631$	−31.6009	−1.25801	−0.629006	−	0.777400i	$-0.716538\pi$
$631$	−31.6009	−1.25801	−0.629006	+	0.777400i	$0.716538\pi$
$632$	−4.29534	−0.170859
$633$	44.1594	1.75518
$634$	6.66471	0.264690
$635$	−46.6403	−1.85086
$636$	27.0663	1.07325
$637$	−16.2920	−0.645511
$638$	−3.47828	−0.137707
$639$	−13.3151	−0.526735
$640$	−3.73268	−0.147547
$641$	−25.5901	−1.01075	−0.505374	−	0.862901i	$-0.668645\pi$
$641$	−25.5901	−1.01075	−0.505374	+	0.862901i	$0.668645\pi$
$642$	6.10426	0.240916
$643$	−30.6406	−1.20835	−0.604174	−	0.796853i	$-0.706497\pi$
$643$	−30.6406	−1.20835	−0.604174	+	0.796853i	$0.706497\pi$
$644$	−37.7628	−1.48806
$645$	−30.9148	−1.21727
$646$	7.37460	0.290150
$647$	−45.3568	−1.78316	−0.891579	−	0.452865i	$-0.850402\pi$
$647$	−45.3568	−1.78316	−0.891579	+	0.452865i	$0.850402\pi$
$648$	11.1907	0.439611
$649$	−23.7551	−0.932468
$650$	8.93292	0.350378
$651$	22.4438	0.879641
$652$	22.5165	0.881815
$653$	22.6630	0.886872	0.443436	−	0.896306i	$-0.353759\pi$
$653$	22.6630	0.886872	0.443436	+	0.896306i	$0.353759\pi$
$654$	28.8345	1.12752
$655$	−32.5977	−1.27370
$656$	−10.7959	−0.421509
$657$	−4.86258	−0.189707
$658$	−23.6235	−0.920938
$659$	−25.6367	−0.998664	−0.499332	−	0.866411i	$-0.666421\pi$
$659$	−25.6367	−0.998664	−0.499332	+	0.866411i	$0.666421\pi$
$660$	15.9915	0.622469
$661$	−47.4867	−1.84702	−0.923508	−	0.383578i	$-0.874692\pi$
$661$	−47.4867	−1.84702	−0.923508	+	0.383578i	$0.874692\pi$
$662$	−28.4398	−1.10535
$663$	−8.49024	−0.329733
$664$	−0.687469	−0.0266790
$665$	−32.2823	−1.25185
$666$	14.0534	0.544559
$667$	−13.1063	−0.507478
$668$	17.2812	0.668630
$669$	−28.1028	−1.08652
$670$	−6.24316	−0.241194
$671$	−0.203550	−0.00785797
$672$	−9.95692	−0.384097
$673$	2.10260	0.0810494	0.0405247	−	0.999179i	$-0.487097\pi$
$673$	2.10260	0.0810494	0.0405247	+	0.999179i	$0.487097\pi$
$674$	−31.5311	−1.21453
$675$	−32.1335	−1.23682
$676$	1.00000	0.0384615
$677$	−14.7822	−0.568126	−0.284063	−	0.958806i	$-0.591683\pi$
$677$	−14.7822	−0.568126	−0.284063	+	0.958806i	$0.591683\pi$
$678$	27.6435	1.06164
$679$	28.2583	1.08445
$680$	−15.3610	−0.589066
$681$	−23.5358	−0.901893
$682$	−4.68077	−0.179236
$683$	−28.0613	−1.07374	−0.536869	−	0.843666i	$-0.680393\pi$
$683$	−28.0613	−1.07374	−0.536869	+	0.843666i	$0.680393\pi$
$684$	−2.25151	−0.0860888
$685$	−19.2092	−0.733947
$686$	−44.8446	−1.71218
$687$	38.0402	1.45132
$688$	−4.01442	−0.153048
$689$	−13.1192	−0.499802
$690$	60.2566	2.29393
$691$	−43.7847	−1.66565	−0.832824	−	0.553537i	$-0.813278\pi$
$691$	−43.7847	−1.66565	−0.832824	+	0.553537i	$0.813278\pi$
$692$	−20.9838	−0.797686
$693$	12.5917	0.478318
$694$	−8.47263	−0.321617
$695$	19.3707	0.734774
$696$	−3.45574	−0.130989
$697$	−44.4279	−1.68283
$698$	−2.04891	−0.0775522
$699$	9.02704	0.341434
$700$	43.1118	1.62947
$701$	−41.5429	−1.56905	−0.784527	−	0.620095i	$-0.787094\pi$
$701$	−41.5429	−1.56905	−0.784527	+	0.620095i	$0.787094\pi$
$702$	−3.59720	−0.135768
$703$	20.0442	0.755981
$704$	2.07657	0.0782636
$705$	37.6950	1.41968
$706$	31.7850	1.19624
$707$	−48.2593	−1.81498
$708$	−23.6011	−0.886984
$709$	51.8506	1.94729	0.973645	−	0.228067i	$-0.0732406\pi$
$709$	51.8506	1.94729	0.973645	+	0.228067i	$0.0732406\pi$
$710$	39.5576	1.48457
$711$	5.39674	0.202394
$712$	−5.70132	−0.213666
$713$	−17.6373	−0.660522
$714$	−40.9754	−1.53346
$715$	−7.75117	−0.289877
$716$	5.61170	0.209719
$717$	−15.8286	−0.591128
$718$	−18.1747	−0.678274
$719$	−40.2193	−1.49993	−0.749964	−	0.661478i	$-0.769929\pi$
$719$	−40.2193	−1.49993	−0.749964	+	0.661478i	$0.769929\pi$
$720$	4.68981	0.174779
$721$	4.82617	0.179736
$722$	15.7887	0.587594
$723$	23.9835	0.891957
$724$	4.42165	0.164329
$725$	14.9628	0.555704
$726$	13.7978	0.512084
$727$	19.7429	0.732224	0.366112	−	0.930571i	$-0.380689\pi$
$727$	19.7429	0.732224	0.366112	+	0.930571i	$0.380689\pi$
$728$	4.82617	0.178870
$729$	8.20408	0.303855
$730$	14.4462	0.534677
$731$	−16.5204	−0.611029
$732$	−0.202231	−0.00747466
$733$	5.30689	0.196014	0.0980072	−	0.995186i	$-0.468753\pi$
$733$	5.30689	0.196014	0.0980072	+	0.995186i	$0.468753\pi$
$734$	30.1842	1.11412
$735$	125.463	4.62778
$736$	7.82459	0.288418
$737$	3.47320	0.127937
$738$	13.5642	0.499304
$739$	−3.00980	−0.110717	−0.0553587	−	0.998467i	$-0.517630\pi$
$739$	−3.00980	−0.110717	−0.0553587	+	0.998467i	$0.517630\pi$
$740$	−41.7512	−1.53480
$741$	3.69711	0.135817
$742$	−63.3155	−2.32439
$743$	34.3445	1.25998	0.629989	−	0.776604i	$-0.283059\pi$
$743$	34.3445	1.25998	0.629989	+	0.776604i	$0.283059\pi$
$744$	−4.65043	−0.170493
$745$	66.2817	2.42837
$746$	−20.3058	−0.743446
$747$	0.863748	0.0316029
$748$	8.54563	0.312459
$749$	−14.2795	−0.521763
$750$	−30.2872	−1.10593
$751$	−7.01137	−0.255848	−0.127924	−	0.991784i	$-0.540831\pi$
$751$	−7.01137	−0.255848	−0.127924	+	0.991784i	$0.540831\pi$
$752$	4.89486	0.178497
$753$	17.3832	0.633480
$754$	1.67502	0.0610005
$755$	−2.11534	−0.0769850
$756$	−17.3607	−0.631403
$757$	25.8093	0.938055	0.469027	−	0.883184i	$-0.344605\pi$
$757$	25.8093	0.938055	0.469027	+	0.883184i	$0.344605\pi$
$758$	−7.04774	−0.255986
$759$	−33.5220	−1.21677
$760$	6.68901	0.242636
$761$	32.0275	1.16100	0.580499	−	0.814261i	$-0.302858\pi$
$761$	32.0275	1.16100	0.580499	+	0.814261i	$0.302858\pi$
$762$	25.7788	0.933867
$763$	−67.4519	−2.44192
$764$	−13.8174	−0.499897
$765$	19.2998	0.697786
$766$	−1.49704	−0.0540902
$767$	11.4396	0.413059
$768$	2.06311	0.0744460
$769$	15.3384	0.553117	0.276558	−	0.960997i	$-0.410806\pi$
$769$	15.3384	0.553117	0.276558	+	0.960997i	$0.410806\pi$
$770$	−37.4085	−1.34811
$771$	20.7025	0.745584
$772$	15.9315	0.573387
$773$	23.6355	0.850111	0.425056	−	0.905167i	$-0.360255\pi$
$773$	23.6355	0.850111	0.425056	+	0.905167i	$0.360255\pi$
$774$	5.04379	0.181295
$775$	20.1356	0.723292
$776$	−5.85522	−0.210190
$777$	−111.371	−3.99542
$778$	−25.7794	−0.924237
$779$	19.3464	0.693155
$780$	−7.70093	−0.275738
$781$	−22.0067	−0.787462
$782$	32.2002	1.15148
$783$	−6.02536	−0.215329
$784$	16.2920	0.581856
$785$	−36.1331	−1.28965
$786$	18.0172	0.642652
$787$	11.2602	0.401381	0.200691	−	0.979655i	$-0.435681\pi$
$787$	11.2602	0.401381	0.200691	+	0.979655i	$0.435681\pi$
$788$	2.48945	0.0886832
$789$	−7.25334	−0.258226
$790$	−16.0331	−0.570434
$791$	−64.6657	−2.29925
$792$	−2.60904	−0.0927081
$793$	0.0980223	0.00348088
$794$	−38.8389	−1.37834
$795$	101.030	3.58317
$796$	2.43175	0.0861911
$797$	−41.4310	−1.46756	−0.733781	−	0.679386i	$-0.762246\pi$
$797$	−41.4310	−1.46756	−0.733781	+	0.679386i	$0.762246\pi$
$798$	17.8429	0.631632
$799$	20.1436	0.712631
$800$	−8.93292	−0.315826
$801$	7.16324	0.253101
$802$	−3.86100	−0.136337
$803$	−8.03671	−0.283609
$804$	3.45069	0.121696
$805$	−140.957	−4.96807
$806$	2.25409	0.0793969
$807$	58.5534	2.06118
$808$	9.99949	0.351781
$809$	−3.12243	−0.109779	−0.0548893	−	0.998492i	$-0.517481\pi$
$809$	−3.12243	−0.109779	−0.0548893	+	0.998492i	$0.517481\pi$
$810$	41.7712	1.46769
$811$	−16.4156	−0.576430	−0.288215	−	0.957566i	$-0.593062\pi$
$811$	−16.4156	−0.576430	−0.288215	+	0.957566i	$0.593062\pi$
$812$	8.08391	0.283690
$813$	42.3503	1.48529
$814$	23.2270	0.814108
$815$	84.0470	2.94404
$816$	8.49024	0.297218
$817$	7.19388	0.251682
$818$	−26.7751	−0.936168
$819$	−6.06369	−0.211883
$820$	−40.2977	−1.40726
$821$	−26.9417	−0.940272	−0.470136	−	0.882594i	$-0.655795\pi$
$821$	−26.9417	−0.940272	−0.470136	+	0.882594i	$0.655795\pi$
$822$	10.6172	0.370318
$823$	−16.5400	−0.576550	−0.288275	−	0.957548i	$-0.593082\pi$
$823$	−16.5400	−0.576550	−0.288275	+	0.957548i	$0.593082\pi$
$824$	−1.00000	−0.0348367
$825$	38.2703	1.33240
$826$	55.2094	1.92098
$827$	−28.6746	−0.997113	−0.498556	−	0.866857i	$-0.666136\pi$
$827$	−28.6746	−0.997113	−0.498556	+	0.866857i	$0.666136\pi$
$828$	−9.83095	−0.341649
$829$	−45.7871	−1.59025	−0.795126	−	0.606445i	$-0.792595\pi$
$829$	−45.7871	−1.59025	−0.795126	+	0.606445i	$0.792595\pi$
$830$	−2.56610	−0.0890707
$831$	−18.7005	−0.648712
$832$	−1.00000	−0.0346688
$833$	67.0457	2.32300
$834$	−10.7065	−0.370735
$835$	64.5053	2.23230
$836$	−3.72123	−0.128702
$837$	−8.10841	−0.280268
$838$	15.5075	0.535699
$839$	−20.2017	−0.697441	−0.348720	−	0.937227i	$-0.613384\pi$
$839$	−20.2017	−0.697441	−0.348720	+	0.937227i	$0.613384\pi$
$840$	−37.1660	−1.28235
$841$	−26.1943	−0.903253
$842$	0.984257	0.0339197
$843$	17.3548	0.597731
$844$	21.4043	0.736766
$845$	3.73268	0.128408
$846$	−6.14999	−0.211441
$847$	−32.2768	−1.10904
$848$	13.1192	0.450515
$849$	43.9524	1.50844
$850$	−36.7613	−1.26090
$851$	87.5203	3.00016
$852$	−21.8641	−0.749050
$853$	53.7237	1.83947	0.919733	−	0.392545i	$-0.128405\pi$
$853$	53.7237	1.83947	0.919733	+	0.392545i	$0.128405\pi$
$854$	0.473073	0.0161882
$855$	−8.40419	−0.287417
$856$	2.95877	0.101129
$857$	−7.57420	−0.258730	−0.129365	−	0.991597i	$-0.541294\pi$
$857$	−7.57420	−0.258730	−0.129365	+	0.991597i	$0.541294\pi$
$858$	4.28419	0.146260
$859$	−50.2517	−1.71457	−0.857283	−	0.514846i	$-0.827849\pi$
$859$	−50.2517	−1.71457	−0.857283	+	0.514846i	$0.827849\pi$
$860$	−14.9846	−0.510969
$861$	−107.494	−3.66338
$862$	2.81027	0.0957180
$863$	−47.0904	−1.60297	−0.801487	−	0.598012i	$-0.795958\pi$
$863$	−47.0904	−1.60297	−0.801487	+	0.598012i	$0.795958\pi$
$864$	3.59720	0.122379
$865$	−78.3260	−2.66317
$866$	0.0600336	0.00204002
$867$	−0.133300	−0.00452709
$868$	10.8786	0.369245
$869$	8.91957	0.302576
$870$	−12.8992	−0.437323
$871$	−1.67257	−0.0566727
$872$	13.9763	0.473296
$873$	7.35661	0.248983
$874$	−14.0217	−0.474293
$875$	70.8499	2.39516
$876$	−7.98462	−0.269775
$877$	−27.8811	−0.941478	−0.470739	−	0.882272i	$-0.656013\pi$
$877$	−27.8811	−0.941478	−0.470739	+	0.882272i	$0.656013\pi$
$878$	5.56695	0.187875
$879$	4.12181	0.139025
$880$	7.75117	0.261292
$881$	−13.3534	−0.449886	−0.224943	−	0.974372i	$-0.572220\pi$
$881$	−13.3534	−0.449886	−0.224943	+	0.974372i	$0.572220\pi$
$882$	−20.4695	−0.689244
$883$	18.1344	0.610270	0.305135	−	0.952309i	$-0.401298\pi$
$883$	18.1344	0.610270	0.305135	+	0.952309i	$0.401298\pi$
$884$	−4.11526	−0.138411
$885$	−88.0954	−2.96130
$886$	34.2099	1.14930
$887$	47.0130	1.57854	0.789270	−	0.614046i	$-0.210459\pi$
$887$	47.0130	1.57854	0.789270	+	0.614046i	$0.210459\pi$
$888$	23.0765	0.774396
$889$	−60.3036	−2.02252
$890$	−21.2812	−0.713348
$891$	−23.2382	−0.778509
$892$	−13.6216	−0.456084
$893$	−8.77165	−0.293532
$894$	−36.6349	−1.22525
$895$	20.9467	0.700171
$896$	−4.82617	−0.161231
$897$	16.1430	0.538998
$898$	−14.3445	−0.478682
$899$	3.77563	0.125924
$900$	11.2235	0.374116
$901$	53.9890	1.79863
$902$	22.4184	0.746452
$903$	−39.9713	−1.33016
$904$	13.3990	0.445643
$905$	16.5046	0.548632
$906$	1.16918	0.0388433
$907$	−27.7714	−0.922133	−0.461067	−	0.887366i	$-0.652533\pi$
$907$	−27.7714	−0.922133	−0.461067	+	0.887366i	$0.652533\pi$
$908$	−11.4079	−0.378585
$909$	−12.5635	−0.416706
$910$	18.0146	0.597177
$911$	−22.8497	−0.757046	−0.378523	−	0.925592i	$-0.623568\pi$
$911$	−22.8497	−0.757046	−0.378523	+	0.925592i	$0.623568\pi$
$912$	−3.69711	−0.122424
$913$	1.42758	0.0472459
$914$	40.0081	1.32335
$915$	−0.754863	−0.0249550
$916$	18.4383	0.609218
$917$	−42.1472	−1.39182
$918$	14.8034	0.488586
$919$	−35.8678	−1.18317	−0.591585	−	0.806243i	$-0.701498\pi$
$919$	−35.8678	−1.18317	−0.591585	+	0.806243i	$0.701498\pi$
$920$	29.2067	0.962916
$921$	−22.4596	−0.740068
$922$	−29.4475	−0.969801
$923$	10.5976	0.348825
$924$	20.6762	0.680198
$925$	−99.9174	−3.28526
$926$	−20.3908	−0.670083
$927$	1.25642	0.0412662
$928$	−1.67502	−0.0549851
$929$	13.3872	0.439221	0.219611	−	0.975588i	$-0.429521\pi$
$929$	13.3872	0.439221	0.219611	+	0.975588i	$0.429521\pi$
$930$	−17.3586	−0.569210
$931$	−29.1954	−0.956840
$932$	4.37546	0.143323
$933$	10.1308	0.331667
$934$	−30.7647	−1.00665
$935$	31.8981	1.04318
$936$	1.25642	0.0410673
$937$	47.2670	1.54415	0.772073	−	0.635534i	$-0.219220\pi$
$937$	47.2670	1.54415	0.772073	+	0.635534i	$0.219220\pi$
$938$	−8.07209	−0.263563
$939$	37.3830	1.21995
$940$	18.2710	0.595933
$941$	−40.7578	−1.32867	−0.664334	−	0.747436i	$-0.731285\pi$
$941$	−40.7578	−1.32867	−0.664334	+	0.747436i	$0.731285\pi$
$942$	19.9713	0.650701
$943$	84.4734	2.75083
$944$	−11.4396	−0.372327
$945$	−64.8020	−2.10801
$946$	8.33622	0.271034
$947$	13.8344	0.449557	0.224779	−	0.974410i	$-0.427834\pi$
$947$	13.8344	0.449557	0.224779	+	0.974410i	$0.427834\pi$
$948$	8.86175	0.287816
$949$	3.87019	0.125632
$950$	16.0079	0.519365
$951$	−13.7500	−0.445875
$952$	−19.8610	−0.643698
$953$	34.3393	1.11236	0.556179	−	0.831062i	$-0.312267\pi$
$953$	34.3393	1.11236	0.556179	+	0.831062i	$0.312267\pi$
$954$	−16.4832	−0.533663
$955$	−51.5761	−1.66896
$956$	−7.67219	−0.248136
$957$	7.17608	0.231970
$958$	−3.05825	−0.0988076
$959$	−24.8366	−0.802016
$960$	7.70093	0.248546
$961$	−25.9191	−0.836100
$962$	−11.1853	−0.360629
$963$	−3.71745	−0.119793
$964$	11.6249	0.374414
$965$	59.4672	1.91432
$966$	77.9088	2.50667
$967$	14.9362	0.480316	0.240158	−	0.970734i	$-0.422801\pi$
$967$	14.9362	0.480316	0.240158	+	0.970734i	$0.422801\pi$
$968$	6.68786	0.214956
$969$	−15.2146	−0.488763
$970$	−21.8557	−0.701744
$971$	1.08750	0.0348996	0.0174498	−	0.999848i	$-0.494445\pi$
$971$	1.08750	0.0348996	0.0174498	+	0.999848i	$0.494445\pi$
$972$	−12.2960	−0.394393
$973$	25.0454	0.802919
$974$	14.8476	0.475749
$975$	−18.4296	−0.590219
$976$	−0.0980223	−0.00313762
$977$	−35.5666	−1.13788	−0.568938	−	0.822381i	$-0.692645\pi$
$977$	−35.5666	−1.13788	−0.568938	+	0.822381i	$0.692645\pi$
$978$	−46.4540	−1.48544
$979$	11.8392	0.378382
$980$	60.8127	1.94259
$981$	−17.5600	−0.560648
$982$	5.40321	0.172423
$983$	26.1630	0.834469	0.417235	−	0.908799i	$-0.362999\pi$
$983$	26.1630	0.834469	0.417235	+	0.908799i	$0.362999\pi$
$984$	22.2731	0.710041
$985$	9.29234	0.296079
$986$	−6.89313	−0.219522
$987$	48.7378	1.55134
$988$	1.79201	0.0570115
$989$	31.4112	0.998817
$990$	−9.73871	−0.309517
$991$	2.95500	0.0938685	0.0469343	−	0.998898i	$-0.485055\pi$
$991$	2.95500	0.0938685	0.0469343	+	0.998898i	$0.485055\pi$
$992$	−2.25409	−0.0715674
$993$	58.6745	1.86198
$994$	51.1460	1.62225
$995$	9.07696	0.287759
$996$	1.41832	0.0449413
$997$	48.8462	1.54697	0.773487	−	0.633813i	$-0.218511\pi$
$997$	48.8462	1.54697	0.773487	+	0.633813i	$0.218511\pi$
$998$	−9.35930	−0.296264
$999$	40.2358	1.27300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2678.2.a.r.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2678.2.a.r.1.9	✓	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2678 = 2 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2678.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.06311 q^{3} +1.00000 q^{4} +3.73268 q^{5} -2.06311 q^{6} +4.82617 q^{7} -1.00000 q^{8} +1.25642 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.06311 q^{3} +1.00000 q^{4} +3.73268 q^{5} -2.06311 q^{6} +4.82617 q^{7} -1.00000 q^{8} +1.25642 q^{9} -3.73268 q^{10} +2.07657 q^{11} +2.06311 q^{12} -1.00000 q^{13} -4.82617 q^{14} +7.70093 q^{15} +1.00000 q^{16} +4.11526 q^{17} -1.25642 q^{18} -1.79201 q^{19} +3.73268 q^{20} +9.95692 q^{21} -2.07657 q^{22} -7.82459 q^{23} -2.06311 q^{24} +8.93292 q^{25} +1.00000 q^{26} -3.59720 q^{27} +4.82617 q^{28} +1.67502 q^{29} -7.70093 q^{30} +2.25409 q^{31} -1.00000 q^{32} +4.28419 q^{33} -4.11526 q^{34} +18.0146 q^{35} +1.25642 q^{36} -11.1853 q^{37} +1.79201 q^{38} -2.06311 q^{39} -3.73268 q^{40} -10.7959 q^{41} -9.95692 q^{42} -4.01442 q^{43} +2.07657 q^{44} +4.68981 q^{45} +7.82459 q^{46} +4.89486 q^{47} +2.06311 q^{48} +16.2920 q^{49} -8.93292 q^{50} +8.49024 q^{51} -1.00000 q^{52} +13.1192 q^{53} +3.59720 q^{54} +7.75117 q^{55} -4.82617 q^{56} -3.69711 q^{57} -1.67502 q^{58} -11.4396 q^{59} +7.70093 q^{60} -0.0980223 q^{61} -2.25409 q^{62} +6.06369 q^{63} +1.00000 q^{64} -3.73268 q^{65} -4.28419 q^{66} +1.67257 q^{67} +4.11526 q^{68} -16.1430 q^{69} -18.0146 q^{70} -10.5976 q^{71} -1.25642 q^{72} -3.87019 q^{73} +11.1853 q^{74} +18.4296 q^{75} -1.79201 q^{76} +10.0219 q^{77} +2.06311 q^{78} +4.29534 q^{79} +3.73268 q^{80} -11.1907 q^{81} +10.7959 q^{82} +0.687469 q^{83} +9.95692 q^{84} +15.3610 q^{85} +4.01442 q^{86} +3.45574 q^{87} -2.07657 q^{88} +5.70132 q^{89} -4.68981 q^{90} -4.82617 q^{91} -7.82459 q^{92} +4.65043 q^{93} -4.89486 q^{94} -6.68901 q^{95} -2.06311 q^{96} +5.85522 q^{97} -16.2920 q^{98} +2.60904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} + 9 q^{5} + 9 q^{7} - 10 q^{8} + 2 q^{9}+O(q^{10}) \) 10 * q - 10 * q^2 + 10 * q^4 + 9 * q^5 + 9 * q^7 - 10 * q^8 + 2 * q^9	\( 10 q - 10 q^{2} + 10 q^{4} + 9 q^{5} + 9 q^{7} - 10 q^{8} + 2 q^{9} - 9 q^{10} + 9 q^{11} - 10 q^{13} - 9 q^{14} + 9 q^{15} + 10 q^{16} + q^{17} - 2 q^{18} + 8 q^{19} + 9 q^{20} + 4 q^{21} - 9 q^{22} + 9 q^{23} + 9 q^{25} + 10 q^{26} + 9 q^{27} + 9 q^{28} + 3 q^{29} - 9 q^{30} - 3 q^{31} - 10 q^{32} - q^{34} + 12 q^{35} + 2 q^{36} + 11 q^{37} - 8 q^{38} - 9 q^{40} + 6 q^{41} - 4 q^{42} + 23 q^{43} + 9 q^{44} + 26 q^{45} - 9 q^{46} + 28 q^{47} + 33 q^{49} - 9 q^{50} - 16 q^{51} - 10 q^{52} + 19 q^{53} - 9 q^{54} + 14 q^{55} - 9 q^{56} - 10 q^{57} - 3 q^{58} - 20 q^{59} + 9 q^{60} + 25 q^{61} + 3 q^{62} + 31 q^{63} + 10 q^{64} - 9 q^{65} + 19 q^{67} + q^{68} - 24 q^{69} - 12 q^{70} - 5 q^{71} - 2 q^{72} + 29 q^{73} - 11 q^{74} + 27 q^{75} + 8 q^{76} + 31 q^{77} - 3 q^{79} + 9 q^{80} - 6 q^{81} - 6 q^{82} + 18 q^{83} + 4 q^{84} + 11 q^{85} - 23 q^{86} + 25 q^{87} - 9 q^{88} - 10 q^{89} - 26 q^{90} - 9 q^{91} + 9 q^{92} - 26 q^{93} - 28 q^{94} - 5 q^{95} + 26 q^{97} - 33 q^{98} + 57 q^{99}+O(q^{100}) \) 10 * q - 10 * q^2 + 10 * q^4 + 9 * q^5 + 9 * q^7 - 10 * q^8 + 2 * q^9 - 9 * q^10 + 9 * q^11 - 10 * q^13 - 9 * q^14 + 9 * q^15 + 10 * q^16 + q^17 - 2 * q^18 + 8 * q^19 + 9 * q^20 + 4 * q^21 - 9 * q^22 + 9 * q^23 + 9 * q^25 + 10 * q^26 + 9 * q^27 + 9 * q^28 + 3 * q^29 - 9 * q^30 - 3 * q^31 - 10 * q^32 - q^34 + 12 * q^35 + 2 * q^36 + 11 * q^37 - 8 * q^38 - 9 * q^40 + 6 * q^41 - 4 * q^42 + 23 * q^43 + 9 * q^44 + 26 * q^45 - 9 * q^46 + 28 * q^47 + 33 * q^49 - 9 * q^50 - 16 * q^51 - 10 * q^52 + 19 * q^53 - 9 * q^54 + 14 * q^55 - 9 * q^56 - 10 * q^57 - 3 * q^58 - 20 * q^59 + 9 * q^60 + 25 * q^61 + 3 * q^62 + 31 * q^63 + 10 * q^64 - 9 * q^65 + 19 * q^67 + q^68 - 24 * q^69 - 12 * q^70 - 5 * q^71 - 2 * q^72 + 29 * q^73 - 11 * q^74 + 27 * q^75 + 8 * q^76 + 31 * q^77 - 3 * q^79 + 9 * q^80 - 6 * q^81 - 6 * q^82 + 18 * q^83 + 4 * q^84 + 11 * q^85 - 23 * q^86 + 25 * q^87 - 9 * q^88 - 10 * q^89 - 26 * q^90 - 9 * q^91 + 9 * q^92 - 26 * q^93 - 28 * q^94 - 5 * q^95 + 26 * q^97 - 33 * q^98 + 57 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more