LMFDB - Embedded newform 671.2.a.b.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [671,2,Mod(1,671)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(671, 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("671.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 671 = 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	671.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:	$5.35796197563$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.2661761.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 3x^{3} + 9x^{2} - x - 1 $ x^6 - x^5 - 6x^4 + 3x^3 + 9*x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.303283$ of defining polynomial
Character	$\chi$	$=$	671.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.78997 q^{2} +0.303283 q^{3} +1.20400 q^{4} -3.29725 q^{5} +0.542868 q^{6} -1.30328 q^{7} -1.42482 q^{8} -2.90802 q^{9} +O(q^{10})$	\(q+1.78997 q^{2} +0.303283 q^{3} +1.20400 q^{4} -3.29725 q^{5} +0.542868 q^{6} -1.30328 q^{7} -1.42482 q^{8} -2.90802 q^{9} -5.90199 q^{10} -1.00000 q^{11} +0.365152 q^{12} -0.0355860 q^{13} -2.33284 q^{14} -1.00000 q^{15} -4.95838 q^{16} +1.92678 q^{17} -5.20527 q^{18} -2.26587 q^{19} -3.96989 q^{20} -0.395263 q^{21} -1.78997 q^{22} +2.13610 q^{23} -0.432123 q^{24} +5.87188 q^{25} -0.0636980 q^{26} -1.79180 q^{27} -1.56915 q^{28} +0.728102 q^{29} -1.78997 q^{30} +2.17389 q^{31} -6.02573 q^{32} -0.303283 q^{33} +3.44889 q^{34} +4.29725 q^{35} -3.50125 q^{36} -3.77408 q^{37} -4.05584 q^{38} -0.0107926 q^{39} +4.69799 q^{40} -1.24599 q^{41} -0.707510 q^{42} +1.71768 q^{43} -1.20400 q^{44} +9.58848 q^{45} +3.82356 q^{46} -4.29920 q^{47} -1.50379 q^{48} -5.30145 q^{49} +10.5105 q^{50} +0.584360 q^{51} -0.0428455 q^{52} +6.21822 q^{53} -3.20727 q^{54} +3.29725 q^{55} +1.85694 q^{56} -0.687199 q^{57} +1.30328 q^{58} +4.22425 q^{59} -1.20400 q^{60} -1.00000 q^{61} +3.89120 q^{62} +3.78997 q^{63} -0.869115 q^{64} +0.117336 q^{65} -0.542868 q^{66} -4.07553 q^{67} +2.31984 q^{68} +0.647842 q^{69} +7.69196 q^{70} -4.12244 q^{71} +4.14340 q^{72} -11.1663 q^{73} -6.75549 q^{74} +1.78084 q^{75} -2.72810 q^{76} +1.30328 q^{77} -0.0193185 q^{78} -9.50447 q^{79} +16.3490 q^{80} +8.18064 q^{81} -2.23028 q^{82} -10.8805 q^{83} -0.475896 q^{84} -6.35309 q^{85} +3.07460 q^{86} +0.220821 q^{87} +1.42482 q^{88} +0.802966 q^{89} +17.1631 q^{90} +0.0463786 q^{91} +2.57186 q^{92} +0.659303 q^{93} -7.69544 q^{94} +7.47114 q^{95} -1.82750 q^{96} +1.99413 q^{97} -9.48945 q^{98} +2.90802 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} - 6 q^{8} - 5 q^{9}+O(q^{10}) $ 6 * q - q^3 + 2 * q^4 - q^5 - q^6 - 5 * q^7 - 6 * q^8 - 5 * q^9	\( 6 q - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} - 6 q^{8} - 5 q^{9} - 7 q^{10} - 6 q^{11} - 6 q^{12} - 4 q^{13} + q^{14} - 6 q^{15} - 10 q^{16} - 5 q^{17} - 3 q^{19} - 6 q^{20} - 12 q^{21} - 3 q^{23} + 10 q^{24} - 11 q^{25} + q^{26} - 4 q^{27} + 4 q^{28} - q^{29} - 10 q^{31} + 3 q^{32} + q^{33} - 19 q^{34} + 7 q^{35} + 3 q^{36} - 19 q^{37} - 3 q^{38} + 8 q^{39} + 5 q^{40} - 7 q^{41} + q^{42} - 2 q^{43} - 2 q^{44} + 4 q^{45} - 7 q^{46} + 5 q^{47} + q^{48} - 25 q^{49} + 17 q^{50} - q^{51} + 2 q^{52} - 9 q^{53} - 11 q^{54} + q^{55} - 4 q^{56} + 11 q^{57} + 5 q^{58} - 5 q^{59} - 2 q^{60} - 6 q^{61} + 3 q^{62} + 12 q^{63} - 6 q^{64} - 11 q^{65} + q^{66} - 14 q^{67} + 18 q^{68} - 7 q^{69} + 7 q^{70} - 14 q^{71} - q^{72} - 14 q^{73} + 6 q^{74} + 6 q^{75} - 11 q^{76} + 5 q^{77} - 26 q^{78} + 5 q^{79} + 26 q^{80} - 14 q^{81} + q^{82} + 17 q^{83} - 3 q^{84} + 2 q^{85} - 7 q^{86} + 4 q^{87} + 6 q^{88} - 25 q^{89} + 22 q^{90} - 4 q^{91} + 35 q^{92} - q^{93} + 30 q^{94} + 3 q^{95} + 8 q^{96} - 24 q^{97} - 2 q^{98} + 5 q^{99}+O(q^{100}) \) 6 * q - q^3 + 2 * q^4 - q^5 - q^6 - 5 * q^7 - 6 * q^8 - 5 * q^9 - 7 * q^10 - 6 * q^11 - 6 * q^12 - 4 * q^13 + q^14 - 6 * q^15 - 10 * q^16 - 5 * q^17 - 3 * q^19 - 6 * q^20 - 12 * q^21 - 3 * q^23 + 10 * q^24 - 11 * q^25 + q^26 - 4 * q^27 + 4 * q^28 - q^29 - 10 * q^31 + 3 * q^32 + q^33 - 19 * q^34 + 7 * q^35 + 3 * q^36 - 19 * q^37 - 3 * q^38 + 8 * q^39 + 5 * q^40 - 7 * q^41 + q^42 - 2 * q^43 - 2 * q^44 + 4 * q^45 - 7 * q^46 + 5 * q^47 + q^48 - 25 * q^49 + 17 * q^50 - q^51 + 2 * q^52 - 9 * q^53 - 11 * q^54 + q^55 - 4 * q^56 + 11 * q^57 + 5 * q^58 - 5 * q^59 - 2 * q^60 - 6 * q^61 + 3 * q^62 + 12 * q^63 - 6 * q^64 - 11 * q^65 + q^66 - 14 * q^67 + 18 * q^68 - 7 * q^69 + 7 * q^70 - 14 * q^71 - q^72 - 14 * q^73 + 6 * q^74 + 6 * q^75 - 11 * q^76 + 5 * q^77 - 26 * q^78 + 5 * q^79 + 26 * q^80 - 14 * q^81 + q^82 + 17 * q^83 - 3 * q^84 + 2 * q^85 - 7 * q^86 + 4 * q^87 + 6 * q^88 - 25 * q^89 + 22 * q^90 - 4 * q^91 + 35 * q^92 - q^93 + 30 * q^94 + 3 * q^95 + 8 * q^96 - 24 * q^97 - 2 * q^98 + 5 * 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 $

$2$	1.78997	1.26570	0.632851	−	0.774274i	$-0.281885\pi$
$2$	1.78997	1.26570	0.632851	+	0.774274i	$0.281885\pi$
$3$	0.303283	0.175100	0.0875502	−	0.996160i	$-0.472096\pi$
$3$	0.303283	0.175100	0.0875502	+	0.996160i	$0.472096\pi$
$4$	1.20400	0.601999
$5$	−3.29725	−1.47458	−0.737288	−	0.675578i	$-0.763894\pi$
$5$	−3.29725	−1.47458	−0.737288	+	0.675578i	$0.763894\pi$
$6$	0.542868	0.221625
$7$	−1.30328	−0.492595	−0.246297	−	0.969194i	$-0.579214\pi$
$7$	−1.30328	−0.492595	−0.246297	+	0.969194i	$0.579214\pi$
$8$	−1.42482	−0.503750
$9$	−2.90802	−0.969340
$10$	−5.90199	−1.86637
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.365152	0.105410
$13$	−0.0355860	−0.00986978	−0.00493489	−	0.999988i	$-0.501571\pi$
$13$	−0.0355860	−0.00986978	−0.00493489	+	0.999988i	$0.501571\pi$
$14$	−2.33284	−0.623478
$15$	−1.00000	−0.258199
$16$	−4.95838	−1.23960
$17$	1.92678	0.467314	0.233657	−	0.972319i	$-0.424931\pi$
$17$	1.92678	0.467314	0.233657	+	0.972319i	$0.424931\pi$
$18$	−5.20527	−1.22689
$19$	−2.26587	−0.519826	−0.259913	−	0.965632i	$-0.583694\pi$
$19$	−2.26587	−0.519826	−0.259913	+	0.965632i	$0.583694\pi$
$20$	−3.96989	−0.887694
$21$	−0.395263	−0.0862535
$22$	−1.78997	−0.381623
$23$	2.13610	0.445408	0.222704	−	0.974886i	$-0.428512\pi$
$23$	2.13610	0.445408	0.222704	+	0.974886i	$0.428512\pi$
$24$	−0.432123	−0.0882068
$25$	5.87188	1.17438
$26$	−0.0636980	−0.0124922
$27$	−1.79180	−0.344832
$28$	−1.56915	−0.296542
$29$	0.728102	0.135205	0.0676026	−	0.997712i	$-0.478465\pi$
$29$	0.728102	0.135205	0.0676026	+	0.997712i	$0.478465\pi$
$30$	−1.78997	−0.326803
$31$	2.17389	0.390442	0.195221	−	0.980759i	$-0.437458\pi$
$31$	2.17389	0.390442	0.195221	+	0.980759i	$0.437458\pi$
$32$	−6.02573	−1.06521
$33$	−0.303283	−0.0527947
$34$	3.44889	0.591479
$35$	4.29725	0.726368
$36$	−3.50125	−0.583542
$37$	−3.77408	−0.620454	−0.310227	−	0.950662i	$-0.600405\pi$
$37$	−3.77408	−0.620454	−0.310227	+	0.950662i	$0.600405\pi$
$38$	−4.05584	−0.657944
$39$	−0.0107926	−0.00172820
$40$	4.69799	0.742818
$41$	−1.24599	−0.194591	−0.0972953	−	0.995256i	$-0.531019\pi$
$41$	−1.24599	−0.194591	−0.0972953	+	0.995256i	$0.531019\pi$
$42$	−0.707510	−0.109171
$43$	1.71768	0.261944	0.130972	−	0.991386i	$-0.458190\pi$
$43$	1.71768	0.261944	0.130972	+	0.991386i	$0.458190\pi$
$44$	−1.20400	−0.181510
$45$	9.58848	1.42937
$46$	3.82356	0.563753
$47$	−4.29920	−0.627102	−0.313551	−	0.949571i	$-0.601519\pi$
$47$	−4.29920	−0.627102	−0.313551	+	0.949571i	$0.601519\pi$
$48$	−1.50379	−0.217054
$49$	−5.30145	−0.757351
$50$	10.5105	1.48641
$51$	0.584360	0.0818268
$52$	−0.0428455	−0.00594160
$53$	6.21822	0.854139	0.427069	−	0.904219i	$-0.359546\pi$
$53$	6.21822	0.854139	0.427069	+	0.904219i	$0.359546\pi$
$54$	−3.20727	−0.436454
$55$	3.29725	0.444602
$56$	1.85694	0.248144
$57$	−0.687199	−0.0910217
$58$	1.30328	0.171129
$59$	4.22425	0.549951	0.274975	−	0.961451i	$-0.411330\pi$
$59$	4.22425	0.549951	0.274975	+	0.961451i	$0.411330\pi$
$60$	−1.20400	−0.155436
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	3.89120	0.494183
$63$	3.78997	0.477492
$64$	−0.869115	−0.108639
$65$	0.117336	0.0145538
$66$	−0.542868	−0.0668224
$67$	−4.07553	−0.497905	−0.248953	−	0.968516i	$-0.580086\pi$
$67$	−4.07553	−0.497905	−0.248953	+	0.968516i	$0.580086\pi$
$68$	2.31984	0.281323
$69$	0.647842	0.0779910
$70$	7.69196	0.919365
$71$	−4.12244	−0.489243	−0.244622	−	0.969619i	$-0.578664\pi$
$71$	−4.12244	−0.489243	−0.244622	+	0.969619i	$0.578664\pi$
$72$	4.14340	0.488305
$73$	−11.1663	−1.30692	−0.653459	−	0.756962i	$-0.726683\pi$
$73$	−11.1663	−1.30692	−0.653459	+	0.756962i	$0.726683\pi$
$74$	−6.75549	−0.785310
$75$	1.78084	0.205634
$76$	−2.72810	−0.312935
$77$	1.30328	0.148523
$78$	−0.0193185	−0.00218739
$79$	−9.50447	−1.06934	−0.534668	−	0.845062i	$-0.679563\pi$
$79$	−9.50447	−1.06934	−0.534668	+	0.845062i	$0.679563\pi$
$80$	16.3490	1.82788
$81$	8.18064	0.908960
$82$	−2.23028	−0.246293
$83$	−10.8805	−1.19429	−0.597143	−	0.802135i	$-0.703697\pi$
$83$	−10.8805	−1.19429	−0.597143	+	0.802135i	$0.703697\pi$
$84$	−0.475896	−0.0519246
$85$	−6.35309	−0.689090
$86$	3.07460	0.331543
$87$	0.220821	0.0236745
$88$	1.42482	0.151886
$89$	0.802966	0.0851142	0.0425571	−	0.999094i	$-0.486450\pi$
$89$	0.802966	0.0851142	0.0425571	+	0.999094i	$0.486450\pi$
$90$	17.1631	1.80915
$91$	0.0463786	0.00486180
$92$	2.57186	0.268135
$93$	0.659303	0.0683665
$94$	−7.69544	−0.793724
$95$	7.47114	0.766523
$96$	−1.82750	−0.186518
$97$	1.99413	0.202473	0.101236	−	0.994862i	$-0.467720\pi$
$97$	1.99413	0.202473	0.101236	+	0.994862i	$0.467720\pi$
$98$	−9.48945	−0.958580
$99$	2.90802	0.292267
$100$	7.06974	0.706974
$101$	19.3485	1.92525	0.962624	−	0.270843i	$-0.0873024\pi$
$101$	19.3485	1.92525	0.962624	+	0.270843i	$0.0873024\pi$
$102$	1.04599	0.103568
$103$	−3.20307	−0.315608	−0.157804	−	0.987470i	$-0.550441\pi$
$103$	−3.20307	−0.315608	−0.157804	+	0.987470i	$0.550441\pi$
$104$	0.0507036	0.00497190
$105$	1.30328	0.127187
$106$	11.1304	1.08108
$107$	0.185731	0.0179553	0.00897764	−	0.999960i	$-0.497142\pi$
$107$	0.185731	0.0179553	0.00897764	+	0.999960i	$0.497142\pi$
$108$	−2.15733	−0.207589
$109$	−3.86099	−0.369815	−0.184908	−	0.982756i	$-0.559199\pi$
$109$	−3.86099	−0.369815	−0.184908	+	0.982756i	$0.559199\pi$
$110$	5.90199	0.562733
$111$	−1.14461	−0.108642
$112$	6.46218	0.610618
$113$	0.0433213	0.00407532	0.00203766	−	0.999998i	$-0.499351\pi$
$113$	0.0433213	0.00407532	0.00203766	+	0.999998i	$0.499351\pi$
$114$	−1.23007	−0.115206
$115$	−7.04326	−0.656788
$116$	0.876634	0.0813935
$117$	0.103485	0.00956717
$118$	7.56129	0.696073
$119$	−2.51114	−0.230196
$120$	1.42482	0.130068
$121$	1.00000	0.0909091
$122$	−1.78997	−0.162056
$123$	−0.377886	−0.0340729
$124$	2.61736	0.235046
$125$	−2.87481	−0.257130
$126$	6.78394	0.604362
$127$	−21.4766	−1.90574	−0.952871	−	0.303376i	$-0.901886\pi$
$127$	−21.4766	−1.90574	−0.952871	+	0.303376i	$0.901886\pi$
$128$	10.4958	0.927703
$129$	0.520944	0.0458665
$130$	0.210028	0.0184207
$131$	4.90475	0.428530	0.214265	−	0.976776i	$-0.431264\pi$
$131$	4.90475	0.428530	0.214265	+	0.976776i	$0.431264\pi$
$132$	−0.365152	−0.0317824
$133$	2.95307	0.256063
$134$	−7.29509	−0.630199
$135$	5.90802	0.508481
$136$	−2.74532	−0.235409
$137$	−18.9506	−1.61906	−0.809532	−	0.587076i	$-0.800279\pi$
$137$	−18.9506	−1.61906	−0.809532	+	0.587076i	$0.800279\pi$
$138$	1.15962	0.0987134
$139$	−10.5681	−0.896371	−0.448185	−	0.893941i	$-0.647930\pi$
$139$	−10.5681	−0.896371	−0.448185	+	0.893941i	$0.647930\pi$
$140$	5.17389	0.437273
$141$	−1.30387	−0.109806
$142$	−7.37905	−0.619236
$143$	0.0355860	0.00297585
$144$	14.4191	1.20159
$145$	−2.40074	−0.199370
$146$	−19.9874	−1.65417
$147$	−1.60784	−0.132612
$148$	−4.54398	−0.373513
$149$	−6.20363	−0.508221	−0.254111	−	0.967175i	$-0.581783\pi$
$149$	−6.20363	−0.508221	−0.254111	+	0.967175i	$0.581783\pi$
$150$	3.18765	0.260271
$151$	4.96260	0.403851	0.201925	−	0.979401i	$-0.435280\pi$
$151$	4.96260	0.403851	0.201925	+	0.979401i	$0.435280\pi$
$152$	3.22845	0.261862
$153$	−5.60312	−0.452986
$154$	2.33284	0.187986
$155$	−7.16786	−0.575736
$156$	−0.0129943	−0.00104038
$157$	6.31802	0.504233	0.252116	−	0.967697i	$-0.418873\pi$
$157$	6.31802	0.504233	0.252116	+	0.967697i	$0.418873\pi$
$158$	−17.0127	−1.35346
$159$	1.88588	0.149560
$160$	19.8684	1.57073
$161$	−2.78394	−0.219405
$162$	14.6431	1.15047
$163$	6.06276	0.474872	0.237436	−	0.971403i	$-0.423693\pi$
$163$	6.06276	0.474872	0.237436	+	0.971403i	$0.423693\pi$
$164$	−1.50017	−0.117143
$165$	1.00000	0.0778499
$166$	−19.4757	−1.51161
$167$	10.0529	0.777917	0.388959	−	0.921255i	$-0.372835\pi$
$167$	10.0529	0.777917	0.388959	+	0.921255i	$0.372835\pi$
$168$	0.563179	0.0434502
$169$	−12.9987	−0.999903
$170$	−11.3719	−0.872182
$171$	6.58919	0.503888
$172$	2.06809	0.157690
$173$	3.27677	0.249128	0.124564	−	0.992212i	$-0.460247\pi$
$173$	3.27677	0.249128	0.124564	+	0.992212i	$0.460247\pi$
$174$	0.395263	0.0299648
$175$	−7.65272	−0.578491
$176$	4.95838	0.373752
$177$	1.28114	0.0962966
$178$	1.43729	0.107729
$179$	10.9545	0.818779	0.409389	−	0.912360i	$-0.365742\pi$
$179$	10.9545	0.818779	0.409389	+	0.912360i	$0.365742\pi$
$180$	11.5445	0.860477
$181$	15.2286	1.13193	0.565967	−	0.824428i	$-0.308503\pi$
$181$	15.2286	1.13193	0.565967	+	0.824428i	$0.308503\pi$
$182$	0.0830164	0.00615359
$183$	−0.303283	−0.0224193
$184$	−3.04356	−0.224374
$185$	12.4441	0.914908
$186$	1.18013	0.0865316
$187$	−1.92678	−0.140900
$188$	−5.17623	−0.377515
$189$	2.33522	0.169862
$190$	13.3731	0.970189
$191$	−13.1208	−0.949385	−0.474692	−	0.880152i	$-0.657441\pi$
$191$	−13.1208	−0.949385	−0.474692	+	0.880152i	$0.657441\pi$
$192$	−0.263588	−0.0190228
$193$	−18.7875	−1.35235	−0.676176	−	0.736740i	$-0.736364\pi$
$193$	−18.7875	−1.35235	−0.676176	+	0.736740i	$0.736364\pi$
$194$	3.56943	0.256270
$195$	0.0355860	0.00254837
$196$	−6.38294	−0.455925
$197$	−5.09288	−0.362853	−0.181426	−	0.983405i	$-0.558071\pi$
$197$	−5.09288	−0.362853	−0.181426	+	0.983405i	$0.558071\pi$
$198$	5.20527	0.369923
$199$	16.4052	1.16293	0.581466	−	0.813571i	$-0.302479\pi$
$199$	16.4052	1.16293	0.581466	+	0.813571i	$0.302479\pi$
$200$	−8.36637	−0.591592
$201$	−1.23604	−0.0871834
$202$	34.6333	2.43679
$203$	−0.948923	−0.0666014
$204$	0.703569	0.0492597
$205$	4.10834	0.286939
$206$	−5.73341	−0.399465
$207$	−6.21182	−0.431751
$208$	0.176449	0.0122345
$209$	2.26587	0.156733
$210$	2.33284	0.160981
$211$	−13.6511	−0.939784	−0.469892	−	0.882724i	$-0.655707\pi$
$211$	−13.6511	−0.939784	−0.469892	+	0.882724i	$0.655707\pi$
$212$	7.48673	0.514191
$213$	−1.25026	−0.0856667
$214$	0.332453	0.0227260
$215$	−5.66364	−0.386257
$216$	2.55299	0.173709
$217$	−2.83319	−0.192329
$218$	−6.91106	−0.468076
$219$	−3.38655	−0.228842
$220$	3.96989	0.267650
$221$	−0.0685665	−0.00461228
$222$	−2.04882	−0.137508
$223$	17.7804	1.19067	0.595333	−	0.803479i	$-0.297020\pi$
$223$	17.7804	1.19067	0.595333	+	0.803479i	$0.297020\pi$
$224$	7.85323	0.524716
$225$	−17.0755	−1.13837
$226$	0.0775439	0.00515814
$227$	12.9238	0.857781	0.428890	−	0.903357i	$-0.358905\pi$
$227$	12.9238	0.857781	0.428890	+	0.903357i	$0.358905\pi$
$228$	−0.827386	−0.0547950
$229$	−12.9805	−0.857779	−0.428889	−	0.903357i	$-0.641095\pi$
$229$	−12.9805	−0.857779	−0.428889	+	0.903357i	$0.641095\pi$
$230$	−12.6072	−0.831297
$231$	0.395263	0.0260064
$232$	−1.03741	−0.0681096
$233$	27.0571	1.77257	0.886284	−	0.463143i	$-0.153278\pi$
$233$	27.0571	1.77257	0.886284	+	0.463143i	$0.153278\pi$
$234$	0.185235	0.0121092
$235$	14.1755	0.924710
$236$	5.08599	0.331070
$237$	−2.88254	−0.187241
$238$	−4.49488	−0.291360
$239$	−1.56932	−0.101511	−0.0507554	−	0.998711i	$-0.516163\pi$
$239$	−1.56932	−0.101511	−0.0507554	+	0.998711i	$0.516163\pi$
$240$	4.95838	0.320062
$241$	−14.3442	−0.923992	−0.461996	−	0.886882i	$-0.652867\pi$
$241$	−14.3442	−0.923992	−0.461996	+	0.886882i	$0.652867\pi$
$242$	1.78997	0.115064
$243$	7.85645	0.503991
$244$	−1.20400	−0.0770781
$245$	17.4802	1.11677
$246$	−0.676406	−0.0431261
$247$	0.0806332	0.00513057
$248$	−3.09740	−0.196685
$249$	−3.29985	−0.209120
$250$	−5.14582	−0.325450
$251$	−16.4022	−1.03530	−0.517648	−	0.855594i	$-0.673192\pi$
$251$	−16.4022	−1.03530	−0.517648	+	0.855594i	$0.673192\pi$
$252$	4.56312	0.287450
$253$	−2.13610	−0.134295
$254$	−38.4425	−2.41210
$255$	−1.92678	−0.120660
$256$	20.5254	1.28283
$257$	−6.92303	−0.431847	−0.215923	−	0.976410i	$-0.569276\pi$
$257$	−6.92303	−0.431847	−0.215923	+	0.976410i	$0.569276\pi$
$258$	0.932474	0.0580533
$259$	4.91869	0.305633
$260$	0.141273	0.00876135
$261$	−2.11734	−0.131060
$262$	8.77936	0.542391
$263$	10.5250	0.649000	0.324500	−	0.945886i	$-0.394804\pi$
$263$	10.5250	0.649000	0.324500	+	0.945886i	$0.394804\pi$
$264$	0.432123	0.0265953
$265$	−20.5031	−1.25949
$266$	5.28591	0.324100
$267$	0.243526	0.0149035
$268$	−4.90693	−0.299739
$269$	−14.9611	−0.912195	−0.456097	−	0.889930i	$-0.650753\pi$
$269$	−14.9611	−0.912195	−0.456097	+	0.889930i	$0.650753\pi$
$270$	10.5752	0.643585
$271$	10.7753	0.654556	0.327278	−	0.944928i	$-0.393869\pi$
$271$	10.7753	0.654556	0.327278	+	0.944928i	$0.393869\pi$
$272$	−9.55373	−0.579280
$273$	0.0140658	0.000851303 0
$274$	−33.9211	−2.04925
$275$	−5.87188	−0.354088
$276$	0.780001	0.0469506
$277$	−1.22770	−0.0737652	−0.0368826	−	0.999320i	$-0.511743\pi$
$277$	−1.22770	−0.0737652	−0.0368826	+	0.999320i	$0.511743\pi$
$278$	−18.9165	−1.13454
$279$	−6.32171	−0.378471
$280$	−6.12281	−0.365908
$281$	−17.8839	−1.06686	−0.533431	−	0.845844i	$-0.679098\pi$
$281$	−17.8839	−1.06686	−0.533431	+	0.845844i	$0.679098\pi$
$282$	−2.33389	−0.138981
$283$	1.98520	0.118008	0.0590041	−	0.998258i	$-0.481208\pi$
$283$	1.98520	0.118008	0.0590041	+	0.998258i	$0.481208\pi$
$284$	−4.96341	−0.294524
$285$	2.26587	0.134218
$286$	0.0636980	0.00376654
$287$	1.62387	0.0958542
$288$	17.5229	1.03255
$289$	−13.2875	−0.781618
$290$	−4.29725	−0.252343
$291$	0.604785	0.0354531
$292$	−13.4442	−0.786764
$293$	6.56630	0.383607	0.191804	−	0.981433i	$-0.438566\pi$
$293$	6.56630	0.383607	0.191804	+	0.981433i	$0.438566\pi$
$294$	−2.87799	−0.167848
$295$	−13.9284	−0.810945
$296$	5.37738	0.312554
$297$	1.79180	0.103971
$298$	−11.1043	−0.643256
$299$	−0.0760153	−0.00439608
$300$	2.14413	0.123791
$301$	−2.23863	−0.129032
$302$	8.88292	0.511155
$303$	5.86806	0.337111
$304$	11.2350	0.644374
$305$	3.29725	0.188800
$306$	−10.0294	−0.573345
$307$	30.2223	1.72488	0.862439	−	0.506160i	$-0.168936\pi$
$307$	30.2223	1.72488	0.862439	+	0.506160i	$0.168936\pi$
$308$	1.56915	0.0894107
$309$	−0.971436	−0.0552631
$310$	−12.8303	−0.728710
$311$	33.0774	1.87565	0.937823	−	0.347115i	$-0.112839\pi$
$311$	33.0774	1.87565	0.937823	+	0.347115i	$0.112839\pi$
$312$	0.0153775	0.000870582 0
$313$	0.442885	0.0250333	0.0125167	−	0.999922i	$-0.496016\pi$
$313$	0.442885	0.0250333	0.0125167	+	0.999922i	$0.496016\pi$
$314$	11.3091	0.638208
$315$	−12.4965	−0.704098
$316$	−11.4434	−0.643740
$317$	−26.4432	−1.48520	−0.742600	−	0.669735i	$-0.766408\pi$
$317$	−26.4432	−1.48520	−0.742600	+	0.669735i	$0.766408\pi$
$318$	3.37567	0.189298
$319$	−0.728102	−0.0407659
$320$	2.86569	0.160197
$321$	0.0563289	0.00314397
$322$	−4.98318	−0.277702
$323$	−4.36584	−0.242922
$324$	9.84948	0.547193
$325$	−0.208957	−0.0115908
$326$	10.8522	0.601046
$327$	−1.17097	−0.0647548
$328$	1.77531	0.0980249
$329$	5.60307	0.308907
$330$	1.78997	0.0985347
$331$	20.7875	1.14258	0.571291	−	0.820748i	$-0.306443\pi$
$331$	20.7875	1.14258	0.571291	+	0.820748i	$0.306443\pi$
$332$	−13.1001	−0.718959
$333$	10.9751	0.601431
$334$	17.9944	0.984611
$335$	13.4381	0.734200
$336$	1.95987	0.106919
$337$	1.69750	0.0924689	0.0462345	−	0.998931i	$-0.485278\pi$
$337$	1.69750	0.0924689	0.0462345	+	0.998931i	$0.485278\pi$
$338$	−23.2674	−1.26558
$339$	0.0131386	0.000713591 0
$340$	−7.64912	−0.414832
$341$	−2.17389	−0.117723
$342$	11.7945	0.637771
$343$	16.0323	0.865661
$344$	−2.44739	−0.131954
$345$	−2.13610	−0.115004
$346$	5.86533	0.315322
$347$	−27.3674	−1.46916	−0.734579	−	0.678523i	$-0.762620\pi$
$347$	−27.3674	−1.46916	−0.734579	+	0.678523i	$0.762620\pi$
$348$	0.265868	0.0142520
$349$	−9.21577	−0.493309	−0.246654	−	0.969103i	$-0.579331\pi$
$349$	−9.21577	−0.493309	−0.246654	+	0.969103i	$0.579331\pi$
$350$	−13.6981	−0.732197
$351$	0.0637630	0.00340342
$352$	6.02573	0.321172
$353$	−10.1051	−0.537838	−0.268919	−	0.963163i	$-0.586666\pi$
$353$	−10.1051	−0.537838	−0.268919	+	0.963163i	$0.586666\pi$
$354$	2.29321	0.121883
$355$	13.5927	0.721427
$356$	0.966770	0.0512387
$357$	−0.761587	−0.0403074
$358$	19.6083	1.03633
$359$	1.13204	0.0597469	0.0298735	−	0.999554i	$-0.490490\pi$
$359$	1.13204	0.0597469	0.0298735	+	0.999554i	$0.490490\pi$
$360$	−13.6619	−0.720043
$361$	−13.8658	−0.729781
$362$	27.2588	1.43269
$363$	0.303283	0.0159182
$364$	0.0558398	0.00292680
$365$	36.8182	1.92715
$366$	−0.542868	−0.0283761
$367$	11.2697	0.588274	0.294137	−	0.955763i	$-0.404968\pi$
$367$	11.2697	0.588274	0.294137	+	0.955763i	$0.404968\pi$
$368$	−10.5916	−0.552126
$369$	3.62336	0.188624
$370$	22.2746	1.15800
$371$	−8.10410	−0.420744
$372$	0.793800	0.0411566
$373$	−11.4459	−0.592648	−0.296324	−	0.955087i	$-0.595761\pi$
$373$	−11.4459	−0.592648	−0.296324	+	0.955087i	$0.595761\pi$
$374$	−3.44889	−0.178338
$375$	−0.871879	−0.0450236
$376$	6.12558	0.315903
$377$	−0.0259103	−0.00133445
$378$	4.17998	0.214995
$379$	−6.60741	−0.339400	−0.169700	−	0.985496i	$-0.554280\pi$
$379$	−6.60741	−0.339400	−0.169700	+	0.985496i	$0.554280\pi$
$380$	8.99524	0.461446
$381$	−6.51349	−0.333696
$382$	−23.4858	−1.20164
$383$	−24.2362	−1.23841	−0.619207	−	0.785228i	$-0.712546\pi$
$383$	−24.2362	−1.23841	−0.619207	+	0.785228i	$0.712546\pi$
$384$	3.18318	0.162441
$385$	−4.29725	−0.219008
$386$	−33.6291	−1.71167
$387$	−4.99506	−0.253913
$388$	2.40093	0.121889
$389$	−1.30910	−0.0663741	−0.0331870	−	0.999449i	$-0.510566\pi$
$389$	−1.30910	−0.0663741	−0.0331870	+	0.999449i	$0.510566\pi$
$390$	0.0636980	0.00322547
$391$	4.11580	0.208145
$392$	7.55362	0.381515
$393$	1.48752	0.0750357
$394$	−9.11612	−0.459263
$395$	31.3386	1.57682
$396$	3.50125	0.175945
$397$	−16.2549	−0.815809	−0.407905	−	0.913025i	$-0.633740\pi$
$397$	−16.2549	−0.815809	−0.407905	+	0.913025i	$0.633740\pi$
$398$	29.3648	1.47192
$399$	0.895614	0.0448368
$400$	−29.1150	−1.45575
$401$	17.0169	0.849783	0.424891	−	0.905244i	$-0.360312\pi$
$401$	17.0169	0.849783	0.424891	+	0.905244i	$0.360312\pi$
$402$	−2.21247	−0.110348
$403$	−0.0773600	−0.00385358
$404$	23.2956	1.15900
$405$	−26.9736	−1.34033
$406$	−1.69855	−0.0842974
$407$	3.77408	0.187074
$408$	−0.832608	−0.0412202
$409$	−7.55140	−0.373393	−0.186696	−	0.982418i	$-0.559778\pi$
$409$	−7.55140	−0.373393	−0.186696	+	0.982418i	$0.559778\pi$
$410$	7.35380	0.363179
$411$	−5.74740	−0.283499
$412$	−3.85649	−0.189996
$413$	−5.50540	−0.270903
$414$	−11.1190	−0.546468
$415$	35.8756	1.76106
$416$	0.214432	0.0105134
$417$	−3.20511	−0.156955
$418$	4.05584	0.198378
$419$	12.4435	0.607906	0.303953	−	0.952687i	$-0.401693\pi$
$419$	12.4435	0.607906	0.303953	+	0.952687i	$0.401693\pi$
$420$	1.56915	0.0765667
$421$	−35.4065	−1.72561	−0.862803	−	0.505540i	$-0.831293\pi$
$421$	−35.4065	−1.72561	−0.862803	+	0.505540i	$0.831293\pi$
$422$	−24.4352	−1.18949
$423$	12.5021	0.607875
$424$	−8.85985	−0.430272
$425$	11.3138	0.548802
$426$	−2.23794	−0.108428
$427$	1.30328	0.0630703
$428$	0.223620	0.0108091
$429$	0.0107926	0.000521073 0
$430$	−10.1377	−0.488886
$431$	4.01538	0.193414	0.0967069	−	0.995313i	$-0.469169\pi$
$431$	4.01538	0.193414	0.0967069	+	0.995313i	$0.469169\pi$
$432$	8.88444	0.427453
$433$	−27.4451	−1.31893	−0.659463	−	0.751737i	$-0.729216\pi$
$433$	−27.4451	−1.31893	−0.659463	+	0.751737i	$0.729216\pi$
$434$	−5.07133	−0.243432
$435$	−0.728102	−0.0349098
$436$	−4.64862	−0.222629
$437$	−4.84012	−0.231534
$438$	−6.06183	−0.289645
$439$	−23.7738	−1.13466	−0.567331	−	0.823490i	$-0.692024\pi$
$439$	−23.7738	−1.13466	−0.567331	+	0.823490i	$0.692024\pi$
$440$	−4.69799	−0.223968
$441$	15.4167	0.734130
$442$	−0.122732	−0.00583777
$443$	−25.4055	−1.20705	−0.603526	−	0.797344i	$-0.706238\pi$
$443$	−25.4055	−1.20705	−0.603526	+	0.797344i	$0.706238\pi$
$444$	−1.37811	−0.0654023
$445$	−2.64758	−0.125507
$446$	31.8265	1.50703
$447$	−1.88145	−0.0889898
$448$	1.13270	0.0535152
$449$	−11.0871	−0.523232	−0.261616	−	0.965172i	$-0.584255\pi$
$449$	−11.0871	−0.523232	−0.261616	+	0.965172i	$0.584255\pi$
$450$	−30.5647	−1.44084
$451$	1.24599	0.0586713
$452$	0.0521588	0.00245334
$453$	1.50507	0.0707144
$454$	23.1332	1.08569
$455$	−0.152922	−0.00716910
$456$	0.979134	0.0458522
$457$	14.4756	0.677141	0.338570	−	0.940941i	$-0.390057\pi$
$457$	14.4756	0.677141	0.338570	+	0.940941i	$0.390057\pi$
$458$	−23.2348	−1.08569
$459$	−3.45241	−0.161145
$460$	−8.48008	−0.395386
$461$	40.6049	1.89116	0.945579	−	0.325393i	$-0.105497\pi$
$461$	40.6049	1.89116	0.945579	+	0.325393i	$0.105497\pi$
$462$	0.707510	0.0329163
$463$	−17.5851	−0.817250	−0.408625	−	0.912702i	$-0.633992\pi$
$463$	−17.5851	−0.817250	−0.408625	+	0.912702i	$0.633992\pi$
$464$	−3.61021	−0.167600
$465$	−2.17389	−0.100812
$466$	48.4314	2.24354
$467$	11.2972	0.522770	0.261385	−	0.965235i	$-0.415821\pi$
$467$	11.2972	0.522770	0.261385	+	0.965235i	$0.415821\pi$
$468$	0.124596	0.00575943
$469$	5.31157	0.245265
$470$	25.3738	1.17041
$471$	1.91615	0.0882913
$472$	−6.01880	−0.277038
$473$	−1.71768	−0.0789791
$474$	−5.15967	−0.236991
$475$	−13.3049	−0.610471
$476$	−3.02341	−0.138578
$477$	−18.0827	−0.827951
$478$	−2.80904	−0.128482
$479$	−17.3760	−0.793931	−0.396966	−	0.917834i	$-0.629937\pi$
$479$	−17.3760	−0.793931	−0.396966	+	0.917834i	$0.629937\pi$
$480$	6.02573	0.275036
$481$	0.134304	0.00612375
$482$	−25.6757	−1.16950
$483$	−0.844322	−0.0384180
$484$	1.20400	0.0547272
$485$	−6.57514	−0.298562
$486$	14.0628	0.637902
$487$	24.9379	1.13005	0.565023	−	0.825075i	$-0.308867\pi$
$487$	24.9379	1.13005	0.565023	+	0.825075i	$0.308867\pi$
$488$	1.42482	0.0644986
$489$	1.83873	0.0831503
$490$	31.2891	1.41350
$491$	−17.5904	−0.793844	−0.396922	−	0.917852i	$-0.629922\pi$
$491$	−17.5904	−0.793844	−0.396922	+	0.917852i	$0.629922\pi$
$492$	−0.454975	−0.0205118
$493$	1.40290	0.0631832
$494$	0.144331	0.00649377
$495$	−9.58848	−0.430970
$496$	−10.7790	−0.483990
$497$	5.37270	0.240999
$498$	−5.90665	−0.264683
$499$	−30.8929	−1.38296	−0.691478	−	0.722397i	$-0.743040\pi$
$499$	−30.8929	−1.38296	−0.691478	+	0.722397i	$0.743040\pi$
$500$	−3.46126	−0.154792
$501$	3.04887	0.136214
$502$	−29.3594	−1.31038
$503$	29.0557	1.29553	0.647766	−	0.761840i	$-0.275704\pi$
$503$	29.0557	1.29553	0.647766	+	0.761840i	$0.275704\pi$
$504$	−5.40003	−0.240536
$505$	−63.7969	−2.83892
$506$	−3.82356	−0.169978
$507$	−3.94229	−0.175083
$508$	−25.8578	−1.14726
$509$	−3.20982	−0.142273	−0.0711364	−	0.997467i	$-0.522663\pi$
$509$	−3.20982	−0.142273	−0.0711364	+	0.997467i	$0.522663\pi$
$510$	−3.44889	−0.152719
$511$	14.5529	0.643781
$512$	15.7483	0.695982
$513$	4.05998	0.179253
$514$	−12.3920	−0.546589
$515$	10.5613	0.465388
$516$	0.627215	0.0276116
$517$	4.29920	0.189078
$518$	8.80432	0.386839
$519$	0.993788	0.0436225
$520$	−0.167183	−0.00733145
$521$	6.46365	0.283178	0.141589	−	0.989926i	$-0.454779\pi$
$521$	6.46365	0.283178	0.141589	+	0.989926i	$0.454779\pi$
$522$	−3.78997	−0.165883
$523$	11.4436	0.500392	0.250196	−	0.968195i	$-0.419505\pi$
$523$	11.4436	0.500392	0.250196	+	0.968195i	$0.419505\pi$
$524$	5.90531	0.257975
$525$	−2.32094	−0.101294
$526$	18.8395	0.821440
$527$	4.18861	0.182459
$528$	1.50379	0.0654442
$529$	−18.4371	−0.801612
$530$	−36.6999	−1.59414
$531$	−12.2842	−0.533089
$532$	3.55549	0.154150
$533$	0.0443397	0.00192057
$534$	0.435904	0.0188634
$535$	−0.612401	−0.0264764
$536$	5.80690	0.250820
$537$	3.32231	0.143368
$538$	−26.7800	−1.15457
$539$	5.30145	0.228350
$540$	7.11325	0.306105
$541$	36.9979	1.59067	0.795333	−	0.606173i	$-0.207296\pi$
$541$	36.9979	1.59067	0.795333	+	0.606173i	$0.207296\pi$
$542$	19.2876	0.828472
$543$	4.61858	0.198202
$544$	−11.6103	−0.497786
$545$	12.7307	0.545321
$546$	0.0251775	0.00107750
$547$	7.41098	0.316871	0.158435	−	0.987369i	$-0.449355\pi$
$547$	7.41098	0.316871	0.158435	+	0.987369i	$0.449355\pi$
$548$	−22.8166	−0.974675
$549$	2.90802	0.124111
$550$	−10.5105	−0.448169
$551$	−1.64978	−0.0702832
$552$	−0.923058	−0.0392880
$553$	12.3870	0.526749
$554$	−2.19754	−0.0933647
$555$	3.77408	0.160201
$556$	−12.7239	−0.539615
$557$	8.68648	0.368058	0.184029	−	0.982921i	$-0.441086\pi$
$557$	8.68648	0.368058	0.184029	+	0.982921i	$0.441086\pi$
$558$	−11.3157	−0.479031
$559$	−0.0611255	−0.00258533
$560$	−21.3074	−0.900403
$561$	−0.584360	−0.0246717
$562$	−32.0116	−1.35033
$563$	25.4557	1.07283	0.536414	−	0.843955i	$-0.319779\pi$
$563$	25.4557	1.07283	0.536414	+	0.843955i	$0.319779\pi$
$564$	−1.56986	−0.0661030
$565$	−0.142841	−0.00600938
$566$	3.55346	0.149363
$567$	−10.6617	−0.447749
$568$	5.87373	0.246456
$569$	−1.09261	−0.0458046	−0.0229023	−	0.999738i	$-0.507291\pi$
$569$	−1.09261	−0.0458046	−0.0229023	+	0.999738i	$0.507291\pi$
$570$	4.05584	0.169880
$571$	−11.2254	−0.469767	−0.234883	−	0.972024i	$-0.575471\pi$
$571$	−11.2254	−0.469767	−0.234883	+	0.972024i	$0.575471\pi$
$572$	0.0428455	0.00179146
$573$	−3.97930	−0.166238
$574$	2.90669	0.121323
$575$	12.5429	0.523076
$576$	2.52740	0.105308
$577$	−42.1606	−1.75517	−0.877585	−	0.479422i	$-0.840846\pi$
$577$	−42.1606	−1.75517	−0.877585	+	0.479422i	$0.840846\pi$
$578$	−23.7843	−0.989295
$579$	−5.69792	−0.236797
$580$	−2.89049	−0.120021
$581$	14.1803	0.588298
$582$	1.08255	0.0448730
$583$	−6.21822	−0.257532
$584$	15.9100	0.658360
$585$	−0.341216	−0.0141075
$586$	11.7535	0.485532
$587$	4.60083	0.189897	0.0949483	−	0.995482i	$-0.469731\pi$
$587$	4.60083	0.189897	0.0949483	+	0.995482i	$0.469731\pi$
$588$	−1.93584	−0.0798326
$589$	−4.92574	−0.202962
$590$	−24.9315	−1.02641
$591$	−1.54458	−0.0635357
$592$	18.7133	0.769113
$593$	26.3663	1.08274	0.541368	−	0.840786i	$-0.317907\pi$
$593$	26.3663	1.08274	0.541368	+	0.840786i	$0.317907\pi$
$594$	3.20727	0.131596
$595$	8.27988	0.339442
$596$	−7.46917	−0.305949
$597$	4.97541	0.203630
$598$	−0.136065	−0.00556412
$599$	35.7105	1.45909	0.729546	−	0.683932i	$-0.239731\pi$
$599$	35.7105	1.45909	0.729546	+	0.683932i	$0.239731\pi$
$600$	−2.53738	−0.103588
$601$	31.0559	1.26680	0.633399	−	0.773825i	$-0.281659\pi$
$601$	31.0559	1.26680	0.633399	+	0.773825i	$0.281659\pi$
$602$	−4.00708	−0.163316
$603$	11.8517	0.482640
$604$	5.97497	0.243118
$605$	−3.29725	−0.134052
$606$	10.5037	0.426682
$607$	−0.337080	−0.0136816	−0.00684082	−	0.999977i	$-0.502178\pi$
$607$	−0.337080	−0.0136816	−0.00684082	+	0.999977i	$0.502178\pi$
$608$	13.6535	0.553723
$609$	−0.287792	−0.0116619
$610$	5.90199	0.238965
$611$	0.152991	0.00618936
$612$	−6.74615	−0.272697
$613$	14.8519	0.599864	0.299932	−	0.953961i	$-0.403036\pi$
$613$	14.8519	0.599864	0.299932	+	0.953961i	$0.403036\pi$
$614$	54.0971	2.18318
$615$	1.24599	0.0502431
$616$	−1.85694	−0.0748184
$617$	−37.9886	−1.52936	−0.764682	−	0.644408i	$-0.777104\pi$
$617$	−37.9886	−1.52936	−0.764682	+	0.644408i	$0.777104\pi$
$618$	−1.73884	−0.0699465
$619$	−20.3441	−0.817699	−0.408850	−	0.912602i	$-0.634070\pi$
$619$	−20.3441	−0.817699	−0.408850	+	0.912602i	$0.634070\pi$
$620$	−8.63009	−0.346593
$621$	−3.82746	−0.153591
$622$	59.2075	2.37401
$623$	−1.04649	−0.0419268
$624$	0.0535140	0.00214227
$625$	−19.8804	−0.795217
$626$	0.792752	0.0316847
$627$	0.687199	0.0274441
$628$	7.60688	0.303548
$629$	−7.27183	−0.289947
$630$	−22.3684	−0.891177
$631$	47.7247	1.89989	0.949945	−	0.312419i	$-0.101139\pi$
$631$	47.7247	1.89989	0.949945	+	0.312419i	$0.101139\pi$
$632$	13.5422	0.538678
$633$	−4.14016	−0.164556
$634$	−47.3327	−1.87982
$635$	70.8139	2.81016
$636$	2.27060	0.0900350
$637$	0.188658	0.00747489
$638$	−1.30328	−0.0515975
$639$	11.9881	0.474243
$640$	−34.6072	−1.36797
$641$	−12.7823	−0.504871	−0.252435	−	0.967614i	$-0.581232\pi$
$641$	−12.7823	−0.504871	−0.252435	+	0.967614i	$0.581232\pi$
$642$	0.100827	0.00397933
$643$	8.65673	0.341388	0.170694	−	0.985324i	$-0.445399\pi$
$643$	8.65673	0.341388	0.170694	+	0.985324i	$0.445399\pi$
$644$	−3.35186	−0.132082
$645$	−1.71768	−0.0676337
$646$	−7.81473	−0.307466
$647$	34.6968	1.36407	0.682036	−	0.731319i	$-0.261095\pi$
$647$	34.6968	1.36407	0.682036	+	0.731319i	$0.261095\pi$
$648$	−11.6559	−0.457888
$649$	−4.22425	−0.165816
$650$	−0.374027	−0.0146705
$651$	−0.859258	−0.0336770
$652$	7.29956	0.285873
$653$	23.7469	0.929288	0.464644	−	0.885498i	$-0.346182\pi$
$653$	23.7469	0.929288	0.464644	+	0.885498i	$0.346182\pi$
$654$	−2.09600	−0.0819603
$655$	−16.1722	−0.631900
$656$	6.17808	0.241214
$657$	32.4719	1.26685
$658$	10.0293	0.390984
$659$	27.0082	1.05209	0.526046	−	0.850456i	$-0.323674\pi$
$659$	27.0082	1.05209	0.526046	+	0.850456i	$0.323674\pi$
$660$	1.20400	0.0468656
$661$	−13.1903	−0.513042	−0.256521	−	0.966539i	$-0.582576\pi$
$661$	−13.1903	−0.513042	−0.256521	+	0.966539i	$0.582576\pi$
$662$	37.2089	1.44617
$663$	−0.0207950	−0.000807613 0
$664$	15.5027	0.601621
$665$	−9.73701	−0.377585
$666$	19.6451	0.761232
$667$	1.55530	0.0602214
$668$	12.1037	0.468306
$669$	5.39250	0.208486
$670$	24.0537	0.929277
$671$	1.00000	0.0386046
$672$	2.38175	0.0918779
$673$	−15.9967	−0.616629	−0.308314	−	0.951284i	$-0.599765\pi$
$673$	−15.9967	−0.616629	−0.308314	+	0.951284i	$0.599765\pi$
$674$	3.03848	0.117038
$675$	−10.5212	−0.404963
$676$	−15.6505	−0.601941
$677$	43.4258	1.66899	0.834494	−	0.551017i	$-0.185760\pi$
$677$	43.4258	1.66899	0.834494	+	0.551017i	$0.185760\pi$
$678$	0.0235177	0.000903193 0
$679$	−2.59891	−0.0997371
$680$	9.05201	0.347129
$681$	3.91956	0.150198
$682$	−3.89120	−0.149002
$683$	27.9259	1.06855	0.534277	−	0.845310i	$-0.320584\pi$
$683$	27.9259	1.06855	0.534277	+	0.845310i	$0.320584\pi$
$684$	7.93338	0.303340
$685$	62.4851	2.38743
$686$	28.6973	1.09567
$687$	−3.93678	−0.150197
$688$	−8.51693	−0.324705
$689$	−0.221282	−0.00843016
$690$	−3.82356	−0.145560
$691$	37.2569	1.41732	0.708660	−	0.705550i	$-0.249300\pi$
$691$	37.2569	1.41732	0.708660	+	0.705550i	$0.249300\pi$
$692$	3.94523	0.149975
$693$	−3.78997	−0.143969
$694$	−48.9869	−1.85952
$695$	34.8455	1.32177
$696$	−0.314630	−0.0119260
$697$	−2.40075	−0.0909348
$698$	−16.4960	−0.624382
$699$	8.20594	0.310377
$700$	−9.21386	−0.348251
$701$	21.9274	0.828187	0.414093	−	0.910234i	$-0.364099\pi$
$701$	21.9274	0.828187	0.414093	+	0.910234i	$0.364099\pi$
$702$	0.114134	0.00430771
$703$	8.55156	0.322528
$704$	0.869115	0.0327560
$705$	4.29920	0.161917
$706$	−18.0878	−0.680742
$707$	−25.2166	−0.948366
$708$	1.54249	0.0579705
$709$	14.8274	0.556856	0.278428	−	0.960457i	$-0.410187\pi$
$709$	14.8274	0.556856	0.278428	+	0.960457i	$0.410187\pi$
$710$	24.3306	0.913111
$711$	27.6392	1.03655
$712$	−1.14408	−0.0428763
$713$	4.64364	0.173906
$714$	−1.36322	−0.0510172
$715$	−0.117336	−0.00438812
$716$	13.1892	0.492904
$717$	−0.475947	−0.0177746
$718$	2.02633	0.0756218
$719$	16.4056	0.611824	0.305912	−	0.952060i	$-0.401039\pi$
$719$	16.4056	0.611824	0.305912	+	0.952060i	$0.401039\pi$
$720$	−47.5434	−1.77184
$721$	4.17451	0.155467
$722$	−24.8195	−0.923685
$723$	−4.35035	−0.161791
$724$	18.3352	0.681424
$725$	4.27533	0.158782
$726$	0.542868	0.0201477
$727$	−26.8201	−0.994704	−0.497352	−	0.867549i	$-0.665694\pi$
$727$	−26.8201	−0.994704	−0.497352	+	0.867549i	$0.665694\pi$
$728$	−0.0660812	−0.00244913
$729$	−22.1592	−0.820711
$730$	65.9035	2.43920
$731$	3.30960	0.122410
$732$	−0.365152	−0.0134964
$733$	−32.9387	−1.21662	−0.608309	−	0.793701i	$-0.708152\pi$
$733$	−32.9387	−1.21662	−0.608309	+	0.793701i	$0.708152\pi$
$734$	20.1725	0.744579
$735$	5.30145	0.195547
$736$	−12.8716	−0.474452
$737$	4.07553	0.150124
$738$	6.48570	0.238742
$739$	20.3955	0.750261	0.375131	−	0.926972i	$-0.377598\pi$
$739$	20.3955	0.750261	0.375131	+	0.926972i	$0.377598\pi$
$740$	14.9827	0.550774
$741$	0.0244547	0.000898364 0
$742$	−14.5061	−0.532536
$743$	19.1928	0.704115	0.352058	−	0.935978i	$-0.385482\pi$
$743$	19.1928	0.704115	0.352058	+	0.935978i	$0.385482\pi$
$744$	−0.939387	−0.0344396
$745$	20.4550	0.749411
$746$	−20.4879	−0.750116
$747$	31.6406	1.15767
$748$	−2.31984	−0.0848219
$749$	−0.242060	−0.00884467
$750$	−1.56064	−0.0569865
$751$	−35.0052	−1.27736	−0.638678	−	0.769474i	$-0.720519\pi$
$751$	−35.0052	−1.27736	−0.638678	+	0.769474i	$0.720519\pi$
$752$	21.3171	0.777353
$753$	−4.97450	−0.181281
$754$	−0.0463786	−0.00168901
$755$	−16.3630	−0.595509
$756$	2.81161	0.102257
$757$	13.5266	0.491634	0.245817	−	0.969316i	$-0.420944\pi$
$757$	13.5266	0.491634	0.245817	+	0.969316i	$0.420944\pi$
$758$	−11.8271	−0.429579
$759$	−0.647842	−0.0235152
$760$	−10.6450	−0.386136
$761$	−8.73887	−0.316784	−0.158392	−	0.987376i	$-0.550631\pi$
$761$	−8.73887	−0.316784	−0.158392	+	0.987376i	$0.550631\pi$
$762$	−11.6590	−0.422360
$763$	5.03196	0.182169
$764$	−15.7974	−0.571529
$765$	18.4749	0.667962
$766$	−43.3822	−1.56746
$767$	−0.150324	−0.00542790
$768$	6.22499	0.224625
$769$	−34.7249	−1.25221	−0.626107	−	0.779737i	$-0.715353\pi$
$769$	−34.7249	−1.25221	−0.626107	+	0.779737i	$0.715353\pi$
$770$	−7.69196	−0.277199
$771$	−2.09964	−0.0756165
$772$	−22.6201	−0.814115
$773$	−17.1154	−0.615599	−0.307799	−	0.951451i	$-0.599593\pi$
$773$	−17.1154	−0.615599	−0.307799	+	0.951451i	$0.599593\pi$
$774$	−8.94101	−0.321378
$775$	12.7648	0.458525
$776$	−2.84127	−0.101996
$777$	1.49175	0.0535164
$778$	−2.34325	−0.0840098
$779$	2.82324	0.101153
$780$	0.0428455	0.00153412
$781$	4.12244	0.147512
$782$	7.36717	0.263449
$783$	−1.30461	−0.0466231
$784$	26.2866	0.938809
$785$	−20.8321	−0.743530
$786$	2.66263	0.0949728
$787$	−47.4678	−1.69204	−0.846022	−	0.533148i	$-0.821009\pi$
$787$	−47.4678	−1.69204	−0.846022	+	0.533148i	$0.821009\pi$
$788$	−6.13183	−0.218437
$789$	3.19205	0.113640
$790$	56.0953	1.99578
$791$	−0.0564599	−0.00200748
$792$	−4.14340	−0.147229
$793$	0.0355860	0.00126370
$794$	−29.0958	−1.03257
$795$	−6.21822	−0.220538
$796$	19.7518	0.700084
$797$	34.3099	1.21532	0.607659	−	0.794198i	$-0.292109\pi$
$797$	34.3099	1.21532	0.607659	+	0.794198i	$0.292109\pi$
$798$	1.60312	0.0567500
$799$	−8.28362	−0.293053
$800$	−35.3823	−1.25095
$801$	−2.33504	−0.0825046
$802$	30.4597	1.07557
$803$	11.1663	0.394051
$804$	−1.48819	−0.0524844
$805$	9.17936	0.323530
$806$	−0.138472	−0.00487747
$807$	−4.53744	−0.159726
$808$	−27.5681	−0.969843
$809$	−15.5735	−0.547536	−0.273768	−	0.961796i	$-0.588270\pi$
$809$	−15.5735	−0.547536	−0.273768	+	0.961796i	$0.588270\pi$
$810$	−48.2820	−1.69646
$811$	44.9854	1.57965	0.789826	−	0.613330i	$-0.210171\pi$
$811$	44.9854	1.57965	0.789826	+	0.613330i	$0.210171\pi$
$812$	−1.14250	−0.0400940
$813$	3.26798	0.114613
$814$	6.75549	0.236780
$815$	−19.9905	−0.700235
$816$	−2.89748	−0.101432
$817$	−3.89204	−0.136165
$818$	−13.5168	−0.472603
$819$	−0.134870	−0.00471274
$820$	4.94643	0.172737
$821$	39.6672	1.38439	0.692197	−	0.721708i	$-0.256643\pi$
$821$	39.6672	1.38439	0.692197	+	0.721708i	$0.256643\pi$
$822$	−10.2877	−0.358824
$823$	19.8303	0.691242	0.345621	−	0.938374i	$-0.387668\pi$
$823$	19.8303	0.691242	0.345621	+	0.938374i	$0.387668\pi$
$824$	4.56380	0.158987
$825$	−1.78084	−0.0620009
$826$	−9.85450	−0.342882
$827$	3.96396	0.137840	0.0689201	−	0.997622i	$-0.478045\pi$
$827$	3.96396	0.137840	0.0689201	+	0.997622i	$0.478045\pi$
$828$	−7.47902	−0.259914
$829$	−47.8899	−1.66329	−0.831643	−	0.555311i	$-0.812599\pi$
$829$	−47.8899	−1.66329	−0.831643	+	0.555311i	$0.812599\pi$
$830$	64.2163	2.22898
$831$	−0.372340	−0.0129163
$832$	0.0309283	0.00107225
$833$	−10.2148	−0.353920
$834$	−5.73705	−0.198658
$835$	−33.1470	−1.14710
$836$	2.72810	0.0943534
$837$	−3.89517	−0.134637
$838$	22.2736	0.769427
$839$	−33.3141	−1.15013	−0.575065	−	0.818108i	$-0.695023\pi$
$839$	−33.3141	−1.15013	−0.575065	+	0.818108i	$0.695023\pi$
$840$	−1.85694	−0.0640706
$841$	−28.4699	−0.981720
$842$	−63.3766	−2.18410
$843$	−5.42387	−0.186808
$844$	−16.4360	−0.565749
$845$	42.8601	1.47443
$846$	22.3785	0.769388
$847$	−1.30328	−0.0447813
$848$	−30.8323	−1.05879
$849$	0.602078	0.0206633
$850$	20.2515	0.694619
$851$	−8.06181	−0.276355
$852$	−1.50532	−0.0515713
$853$	26.2955	0.900340	0.450170	−	0.892943i	$-0.351363\pi$
$853$	26.2955	0.900340	0.450170	+	0.892943i	$0.351363\pi$
$854$	2.33284	0.0798281
$855$	−21.7262	−0.743021
$856$	−0.264633	−0.00904497
$857$	20.4602	0.698907	0.349453	−	0.936954i	$-0.386367\pi$
$857$	20.4602	0.698907	0.349453	+	0.936954i	$0.386367\pi$
$858$	0.0193185	0.000659522 0
$859$	−22.2969	−0.760759	−0.380379	−	0.924831i	$-0.624207\pi$
$859$	−22.2969	−0.760759	−0.380379	+	0.924831i	$0.624207\pi$
$860$	−6.81901	−0.232526
$861$	0.492493	0.0167841
$862$	7.18741	0.244804
$863$	−24.5802	−0.836721	−0.418360	−	0.908281i	$-0.637395\pi$
$863$	−24.5802	−0.836721	−0.418360	+	0.908281i	$0.637395\pi$
$864$	10.7969	0.367318
$865$	−10.8043	−0.367359
$866$	−49.1259	−1.66937
$867$	−4.02987	−0.136862
$868$	−3.41116	−0.115782
$869$	9.50447	0.322417
$870$	−1.30328	−0.0441854
$871$	0.145032	0.00491422
$872$	5.50121	0.186294
$873$	−5.79896	−0.196265
$874$	−8.66368	−0.293053
$875$	3.74668	0.126661
$876$	−4.07740	−0.137763
$877$	−8.27809	−0.279531	−0.139766	−	0.990185i	$-0.544635\pi$
$877$	−8.27809	−0.279531	−0.139766	+	0.990185i	$0.544635\pi$
$878$	−42.5544	−1.43614
$879$	1.99145	0.0671698
$880$	−16.3490	−0.551126
$881$	−32.4376	−1.09285	−0.546425	−	0.837508i	$-0.684011\pi$
$881$	−32.4376	−1.09285	−0.546425	+	0.837508i	$0.684011\pi$
$882$	27.5955	0.929189
$883$	−31.8072	−1.07040	−0.535198	−	0.844726i	$-0.679763\pi$
$883$	−31.8072	−1.07040	−0.535198	+	0.844726i	$0.679763\pi$
$884$	−0.0825540	−0.00277659
$885$	−4.22425	−0.141997
$886$	−45.4751	−1.52777
$887$	−25.6006	−0.859583	−0.429792	−	0.902928i	$-0.641413\pi$
$887$	−25.6006	−0.859583	−0.429792	+	0.902928i	$0.641413\pi$
$888$	1.63087	0.0547283
$889$	27.9901	0.938758
$890$	−4.73910	−0.158855
$891$	−8.18064	−0.274062
$892$	21.4076	0.716780
$893$	9.74141	0.325984
$894$	−3.36775	−0.112634
$895$	−36.1198	−1.20735
$896$	−13.6790	−0.456982
$897$	−0.0230541	−0.000769755 0
$898$	−19.8456	−0.662255
$899$	1.58281	0.0527898
$900$	−20.5589	−0.685298
$901$	11.9812	0.399151
$902$	2.23028	0.0742603
$903$	−0.678937	−0.0225936
$904$	−0.0617250	−0.00205294
$905$	−50.2126	−1.66912
$906$	2.69404	0.0895033
$907$	53.9552	1.79155	0.895777	−	0.444505i	$-0.146620\pi$
$907$	53.9552	1.79155	0.895777	+	0.444505i	$0.146620\pi$
$908$	15.5602	0.516383
$909$	−56.2658	−1.86622
$910$	−0.273726	−0.00907394
$911$	−23.2323	−0.769722	−0.384861	−	0.922975i	$-0.625751\pi$
$911$	−23.2323	−0.769722	−0.384861	+	0.922975i	$0.625751\pi$
$912$	3.40740	0.112830
$913$	10.8805	0.360091
$914$	25.9110	0.857058
$915$	1.00000	0.0330590
$916$	−15.6286	−0.516382
$917$	−6.39227	−0.211091
$918$	−6.17972	−0.203961
$919$	2.13813	0.0705304	0.0352652	−	0.999378i	$-0.488772\pi$
$919$	2.13813	0.0705304	0.0352652	+	0.999378i	$0.488772\pi$
$920$	10.0354	0.330857
$921$	9.16591	0.302027
$922$	72.6816	2.39364
$923$	0.146701	0.00482873
$924$	0.475896	0.0156558
$925$	−22.1609	−0.728647
$926$	−31.4769	−1.03439
$927$	9.31459	0.305931
$928$	−4.38735	−0.144022
$929$	52.5616	1.72449	0.862245	−	0.506491i	$-0.169058\pi$
$929$	52.5616	1.72449	0.862245	+	0.506491i	$0.169058\pi$
$930$	−3.89120	−0.127597
$931$	12.0124	0.393690
$932$	32.5767	1.06708
$933$	10.0318	0.328426
$934$	20.2216	0.661671
$935$	6.35309	0.207768
$936$	−0.147447	−0.00481946
$937$	25.8234	0.843615	0.421807	−	0.906686i	$-0.361396\pi$
$937$	25.8234	0.843615	0.421807	+	0.906686i	$0.361396\pi$
$938$	9.50756	0.310433
$939$	0.134319	0.00438335
$940$	17.0673	0.556675
$941$	26.8656	0.875794	0.437897	−	0.899025i	$-0.355724\pi$
$941$	26.8656	0.875794	0.437897	+	0.899025i	$0.355724\pi$
$942$	3.42985	0.111750
$943$	−2.66155	−0.0866721
$944$	−20.9455	−0.681717
$945$	−7.69982	−0.250475
$946$	−3.07460	−0.0999640
$947$	23.9326	0.777705	0.388852	−	0.921300i	$-0.372872\pi$
$947$	23.9326	0.777705	0.388852	+	0.921300i	$0.372872\pi$
$948$	−3.47058	−0.112719
$949$	0.397364	0.0128990
$950$	−23.8154	−0.772674
$951$	−8.01978	−0.260059
$952$	3.57793	0.115961
$953$	−19.9285	−0.645547	−0.322773	−	0.946476i	$-0.604615\pi$
$953$	−19.9285	−0.645547	−0.322773	+	0.946476i	$0.604615\pi$
$954$	−32.3675	−1.04794
$955$	43.2625	1.39994
$956$	−1.88946	−0.0611094
$957$	−0.220821	−0.00713813
$958$	−31.1026	−1.00488
$959$	24.6981	0.797542
$960$	0.869115	0.0280506
$961$	−26.2742	−0.847555
$962$	0.240401	0.00775084
$963$	−0.540109	−0.0174048
$964$	−17.2704	−0.556243
$965$	61.9471	1.99415
$966$	−1.51131	−0.0486257
$967$	50.5920	1.62693	0.813464	−	0.581616i	$-0.197579\pi$
$967$	50.5920	1.62693	0.813464	+	0.581616i	$0.197579\pi$
$968$	−1.42482	−0.0457954
$969$	−1.32408	−0.0425357
$970$	−11.7693	−0.377890
$971$	−34.4456	−1.10541	−0.552706	−	0.833376i	$-0.686405\pi$
$971$	−34.4456	−1.10541	−0.552706	+	0.833376i	$0.686405\pi$
$972$	9.45915	0.303402
$973$	13.7732	0.441547
$974$	44.6382	1.43030
$975$	−0.0633730	−0.00202956
$976$	4.95838	0.158714
$977$	−38.0534	−1.21744	−0.608718	−	0.793386i	$-0.708316\pi$
$977$	−38.0534	−1.21744	−0.608718	+	0.793386i	$0.708316\pi$
$978$	3.29128	0.105243
$979$	−0.802966	−0.0256629
$980$	21.0462	0.672296
$981$	11.2278	0.358477
$982$	−31.4863	−1.00477
$983$	−36.2253	−1.15541	−0.577703	−	0.816247i	$-0.696051\pi$
$983$	−36.2253	−1.15541	−0.577703	+	0.816247i	$0.696051\pi$
$984$	0.538420	0.0171642
$985$	16.7925	0.535054
$986$	2.51114	0.0799711
$987$	1.69931	0.0540897
$988$	0.0970823	0.00308860
$989$	3.66914	0.116672
$990$	−17.1631	−0.545479
$991$	3.72773	0.118415	0.0592076	−	0.998246i	$-0.481143\pi$
$991$	3.72773	0.118415	0.0592076	+	0.998246i	$0.481143\pi$
$992$	−13.0993	−0.415902
$993$	6.30448	0.200066
$994$	9.61699	0.305032
$995$	−54.0920	−1.71483
$996$	−3.97302	−0.125890
$997$	34.1306	1.08093	0.540464	−	0.841367i	$-0.318249\pi$
$997$	34.1306	1.08093	0.540464	+	0.841367i	$0.318249\pi$
$998$	−55.2974	−1.75041
$999$	6.76239	0.213953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	671.2.a.b.1.6	✓	6
3.2	odd	2		6039.2.a.b.1.1		6
11.10	odd	2		7381.2.a.h.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.a.b.1.6	✓	6	1.1	even	1	trivial
6039.2.a.b.1.1		6	3.2	odd	2
7381.2.a.h.1.1		6	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.a (trivial)

\(f(q)\)	\(=\)	\(q+1.78997 q^{2} +0.303283 q^{3} +1.20400 q^{4} -3.29725 q^{5} +0.542868 q^{6} -1.30328 q^{7} -1.42482 q^{8} -2.90802 q^{9} +O(q^{10})\)	\(q+1.78997 q^{2} +0.303283 q^{3} +1.20400 q^{4} -3.29725 q^{5} +0.542868 q^{6} -1.30328 q^{7} -1.42482 q^{8} -2.90802 q^{9} -5.90199 q^{10} -1.00000 q^{11} +0.365152 q^{12} -0.0355860 q^{13} -2.33284 q^{14} -1.00000 q^{15} -4.95838 q^{16} +1.92678 q^{17} -5.20527 q^{18} -2.26587 q^{19} -3.96989 q^{20} -0.395263 q^{21} -1.78997 q^{22} +2.13610 q^{23} -0.432123 q^{24} +5.87188 q^{25} -0.0636980 q^{26} -1.79180 q^{27} -1.56915 q^{28} +0.728102 q^{29} -1.78997 q^{30} +2.17389 q^{31} -6.02573 q^{32} -0.303283 q^{33} +3.44889 q^{34} +4.29725 q^{35} -3.50125 q^{36} -3.77408 q^{37} -4.05584 q^{38} -0.0107926 q^{39} +4.69799 q^{40} -1.24599 q^{41} -0.707510 q^{42} +1.71768 q^{43} -1.20400 q^{44} +9.58848 q^{45} +3.82356 q^{46} -4.29920 q^{47} -1.50379 q^{48} -5.30145 q^{49} +10.5105 q^{50} +0.584360 q^{51} -0.0428455 q^{52} +6.21822 q^{53} -3.20727 q^{54} +3.29725 q^{55} +1.85694 q^{56} -0.687199 q^{57} +1.30328 q^{58} +4.22425 q^{59} -1.20400 q^{60} -1.00000 q^{61} +3.89120 q^{62} +3.78997 q^{63} -0.869115 q^{64} +0.117336 q^{65} -0.542868 q^{66} -4.07553 q^{67} +2.31984 q^{68} +0.647842 q^{69} +7.69196 q^{70} -4.12244 q^{71} +4.14340 q^{72} -11.1663 q^{73} -6.75549 q^{74} +1.78084 q^{75} -2.72810 q^{76} +1.30328 q^{77} -0.0193185 q^{78} -9.50447 q^{79} +16.3490 q^{80} +8.18064 q^{81} -2.23028 q^{82} -10.8805 q^{83} -0.475896 q^{84} -6.35309 q^{85} +3.07460 q^{86} +0.220821 q^{87} +1.42482 q^{88} +0.802966 q^{89} +17.1631 q^{90} +0.0463786 q^{91} +2.57186 q^{92} +0.659303 q^{93} -7.69544 q^{94} +7.47114 q^{95} -1.82750 q^{96} +1.99413 q^{97} -9.48945 q^{98} +2.90802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} - 6 q^{8} - 5 q^{9}+O(q^{10}) \) 6 * q - q^3 + 2 * q^4 - q^5 - q^6 - 5 * q^7 - 6 * q^8 - 5 * q^9	\( 6 q - q^{3} + 2 q^{4} - q^{5} - q^{6} - 5 q^{7} - 6 q^{8} - 5 q^{9} - 7 q^{10} - 6 q^{11} - 6 q^{12} - 4 q^{13} + q^{14} - 6 q^{15} - 10 q^{16} - 5 q^{17} - 3 q^{19} - 6 q^{20} - 12 q^{21} - 3 q^{23} + 10 q^{24} - 11 q^{25} + q^{26} - 4 q^{27} + 4 q^{28} - q^{29} - 10 q^{31} + 3 q^{32} + q^{33} - 19 q^{34} + 7 q^{35} + 3 q^{36} - 19 q^{37} - 3 q^{38} + 8 q^{39} + 5 q^{40} - 7 q^{41} + q^{42} - 2 q^{43} - 2 q^{44} + 4 q^{45} - 7 q^{46} + 5 q^{47} + q^{48} - 25 q^{49} + 17 q^{50} - q^{51} + 2 q^{52} - 9 q^{53} - 11 q^{54} + q^{55} - 4 q^{56} + 11 q^{57} + 5 q^{58} - 5 q^{59} - 2 q^{60} - 6 q^{61} + 3 q^{62} + 12 q^{63} - 6 q^{64} - 11 q^{65} + q^{66} - 14 q^{67} + 18 q^{68} - 7 q^{69} + 7 q^{70} - 14 q^{71} - q^{72} - 14 q^{73} + 6 q^{74} + 6 q^{75} - 11 q^{76} + 5 q^{77} - 26 q^{78} + 5 q^{79} + 26 q^{80} - 14 q^{81} + q^{82} + 17 q^{83} - 3 q^{84} + 2 q^{85} - 7 q^{86} + 4 q^{87} + 6 q^{88} - 25 q^{89} + 22 q^{90} - 4 q^{91} + 35 q^{92} - q^{93} + 30 q^{94} + 3 q^{95} + 8 q^{96} - 24 q^{97} - 2 q^{98} + 5 q^{99}+O(q^{100}) \) 6 * q - q^3 + 2 * q^4 - q^5 - q^6 - 5 * q^7 - 6 * q^8 - 5 * q^9 - 7 * q^10 - 6 * q^11 - 6 * q^12 - 4 * q^13 + q^14 - 6 * q^15 - 10 * q^16 - 5 * q^17 - 3 * q^19 - 6 * q^20 - 12 * q^21 - 3 * q^23 + 10 * q^24 - 11 * q^25 + q^26 - 4 * q^27 + 4 * q^28 - q^29 - 10 * q^31 + 3 * q^32 + q^33 - 19 * q^34 + 7 * q^35 + 3 * q^36 - 19 * q^37 - 3 * q^38 + 8 * q^39 + 5 * q^40 - 7 * q^41 + q^42 - 2 * q^43 - 2 * q^44 + 4 * q^45 - 7 * q^46 + 5 * q^47 + q^48 - 25 * q^49 + 17 * q^50 - q^51 + 2 * q^52 - 9 * q^53 - 11 * q^54 + q^55 - 4 * q^56 + 11 * q^57 + 5 * q^58 - 5 * q^59 - 2 * q^60 - 6 * q^61 + 3 * q^62 + 12 * q^63 - 6 * q^64 - 11 * q^65 + q^66 - 14 * q^67 + 18 * q^68 - 7 * q^69 + 7 * q^70 - 14 * q^71 - q^72 - 14 * q^73 + 6 * q^74 + 6 * q^75 - 11 * q^76 + 5 * q^77 - 26 * q^78 + 5 * q^79 + 26 * q^80 - 14 * q^81 + q^82 + 17 * q^83 - 3 * q^84 + 2 * q^85 - 7 * q^86 + 4 * q^87 + 6 * q^88 - 25 * q^89 + 22 * q^90 - 4 * q^91 + 35 * q^92 - q^93 + 30 * q^94 + 3 * q^95 + 8 * q^96 - 24 * q^97 - 2 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more